LMFDB - Embedded newform 269.4.b.a.268.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [269,4,Mod(268,269)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("269.268"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(269, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 269 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	269.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.8715137915$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			268.5
Character	$\chi$	$=$	269.268
Dual form			269.4.b.a.268.62

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.09712i q^{2} +0.677318i q^{3} -17.9807 q^{4} +15.4092 q^{5} +3.45237 q^{6} -7.81953i q^{7} +50.8727i q^{8} +26.5412 q^{9} -78.5428i q^{10} -41.5692 q^{11} -12.1786i q^{12} +66.5794 q^{13} -39.8571 q^{14} +10.4370i q^{15} +115.459 q^{16} -108.686i q^{17} -135.284i q^{18} -102.065i q^{19} -277.068 q^{20} +5.29631 q^{21} +211.883i q^{22} -25.0644 q^{23} -34.4570 q^{24} +112.445 q^{25} -339.363i q^{26} +36.2644i q^{27} +140.600i q^{28} +58.5426i q^{29} +53.1984 q^{30} -43.6955i q^{31} -181.527i q^{32} -28.1556i q^{33} -553.984 q^{34} -120.493i q^{35} -477.229 q^{36} -130.004 q^{37} -520.237 q^{38} +45.0954i q^{39} +783.909i q^{40} -144.598 q^{41} -26.9959i q^{42} -236.138 q^{43} +747.442 q^{44} +408.980 q^{45} +127.756i q^{46} +70.9175 q^{47} +78.2024i q^{48} +281.855 q^{49} -573.144i q^{50} +73.6147 q^{51} -1197.14 q^{52} +579.852 q^{53} +184.844 q^{54} -640.550 q^{55} +397.800 q^{56} +69.1303 q^{57} +298.399 q^{58} +387.518i q^{59} -187.663i q^{60} +161.665 q^{61} -222.721 q^{62} -207.540i q^{63} -1.59520 q^{64} +1025.94 q^{65} -143.512 q^{66} -558.841 q^{67} +1954.24i q^{68} -16.9766i q^{69} -614.168 q^{70} -858.333i q^{71} +1350.22i q^{72} -13.2304 q^{73} +662.645i q^{74} +76.1607i q^{75} +1835.19i q^{76} +325.052i q^{77} +229.857 q^{78} -402.737 q^{79} +1779.14 q^{80} +692.051 q^{81} +737.033i q^{82} -245.757i q^{83} -95.2311 q^{84} -1674.76i q^{85} +1203.63i q^{86} -39.6520 q^{87} -2114.74i q^{88} -221.693 q^{89} -2084.62i q^{90} -520.619i q^{91} +450.675 q^{92} +29.5957 q^{93} -361.475i q^{94} -1572.74i q^{95} +122.952 q^{96} -1434.51 q^{97} -1436.65i q^{98} -1103.30 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) $ 66 * q - 258 * q^4 - 34 * q^5 + 48 * q^6 - 564 * q^9 + 22 * q^11 + 118 * q^13 + 60 * q^14 + 1030 * q^16 + 144 * q^20 - 64 * q^21 - 80 * q^23 - 778 * q^24 + 1676 * q^25 - 1146 * q^30 + 308 * q^34 + 2030 * q^36 + 470 * q^37 + 696 * q^38 + 76 * q^41 + 426 * q^43 + 698 * q^44 + 1254 * q^45 - 1280 * q^47 - 3502 * q^49 - 640 * q^51 - 1924 * q^52 - 618 * q^53 - 3766 * q^54 + 176 * q^55 + 3164 * q^56 + 1668 * q^57 - 2170 * q^58 - 1026 * q^61 + 516 * q^62 - 2936 * q^64 - 900 * q^65 - 5296 * q^66 + 862 * q^67 - 1776 * q^70 + 3560 * q^73 + 8818 * q^78 + 2064 * q^79 - 3966 * q^80 + 7446 * q^81 - 3304 * q^84 - 776 * q^87 - 892 * q^89 - 6402 * q^92 + 7388 * q^93 + 8632 * q^96 + 4552 * q^97 - 3982 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/269\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	5.09712i		−	1.80211i	−0.433710	−	0.901053i	$-0.642796\pi$
$2$		−	5.09712i		−	1.80211i	0.433710	−	0.901053i	$-0.357204\pi$
$3$			0.677318i			0.130350i	0.997874	+	0.0651749i	$0.0207605\pi$
$3$			0.677318i			0.130350i	−0.997874	+	0.0651749i	$0.979239\pi$
$4$	−17.9807			−2.24758
$5$	15.4092			1.37824			0.689122	−	0.724645i	$-0.257996\pi$
$5$	15.4092			1.37824			0.689122	+	0.724645i	$0.257996\pi$
$6$	3.45237			0.234904
$7$		−	7.81953i		−	0.422215i	−0.977463	−	0.211107i	$-0.932293\pi$
$7$		−	7.81953i		−	0.422215i	0.977463	−	0.211107i	$-0.0677070\pi$
$8$			50.8727i			2.24828i
$9$	26.5412			0.983009
$10$		−	78.5428i		−	2.48374i
$11$	−41.5692			−1.13942			−0.569708	−	0.821847i	$-0.692944\pi$
$11$	−41.5692			−1.13942			−0.569708	+	0.821847i	$0.692944\pi$
$12$		−	12.1786i		−	0.292972i
$13$	66.5794			1.42045			0.710223	−	0.703977i	$-0.248594\pi$
$13$	66.5794			1.42045			0.710223	+	0.703977i	$0.248594\pi$
$14$	−39.8571			−0.760876
$15$			10.4370i			0.179654i
$16$	115.459			1.80405
$17$		−	108.686i		−	1.55060i	−0.631596	−	0.775298i	$-0.717600\pi$
$17$		−	108.686i		−	1.55060i	0.631596	−	0.775298i	$-0.282400\pi$
$18$		−	135.284i		−	1.77149i
$19$		−	102.065i		−	1.23238i	−0.787597	−	0.616191i	$-0.788675\pi$
$19$		−	102.065i		−	1.23238i	0.787597	−	0.616191i	$-0.211325\pi$
$20$	−277.068			−3.09772
$21$	5.29631			0.0550357
$22$			211.883i			2.05335i
$23$	−25.0644			−0.227230			−0.113615	−	0.993525i	$-0.536243\pi$
$23$	−25.0644			−0.227230			−0.113615	+	0.993525i	$0.536243\pi$
$24$	−34.4570			−0.293062
$25$	112.445			0.899557
$26$		−	339.363i		−	2.55979i
$27$			36.2644i			0.258485i
$28$			140.600i			0.948963i
$29$			58.5426i			0.374865i	0.982277	+	0.187433i	$0.0600166\pi$
$29$			58.5426i			0.374865i	−0.982277	+	0.187433i	$0.939983\pi$
$30$	53.1984			0.323755
$31$		−	43.6955i		−	0.253159i	−0.991956	−	0.126580i	$-0.959600\pi$
$31$		−	43.6955i		−	0.253159i	0.991956	−	0.126580i	$-0.0403999\pi$
$32$		−	181.527i		−	1.00281i
$33$		−	28.1556i		−	0.148523i
$34$	−553.984			−2.79434
$35$		−	120.493i		−	0.581915i
$36$	−477.229			−2.20939
$37$	−130.004			−0.577635			−0.288817	−	0.957384i	$-0.593262\pi$
$37$	−130.004			−0.577635			−0.288817	+	0.957384i	$0.593262\pi$
$38$	−520.237			−2.22088
$39$			45.0954i			0.185155i
$40$			783.909i			3.09867i
$41$	−144.598			−0.550790			−0.275395	−	0.961331i	$-0.588809\pi$
$41$	−144.598			−0.550790			−0.275395	+	0.961331i	$0.588809\pi$
$42$		−	26.9959i		−	0.0991801i
$43$	−236.138			−0.837459			−0.418729	−	0.908111i	$-0.637524\pi$
$43$	−236.138			−0.837459			−0.418729	+	0.908111i	$0.637524\pi$
$44$	747.442			2.56093
$45$	408.980			1.35483
$46$			127.756i			0.409492i
$47$	70.9175			0.220093			0.110047	−	0.993926i	$-0.464900\pi$
$47$	70.9175			0.220093			0.110047	+	0.993926i	$0.464900\pi$
$48$			78.2024i			0.235157i
$49$	281.855			0.821735
$50$		−	573.144i		−	1.62110i
$51$	73.6147			0.202120
$52$	−1197.14			−3.19257
$53$	579.852			1.50281			0.751403	−	0.659843i	$-0.229377\pi$
$53$	579.852			1.50281			0.751403	+	0.659843i	$0.229377\pi$
$54$	184.844			0.465817
$55$	−640.550			−1.57039
$56$	397.800			0.949256
$57$	69.1303			0.160641
$58$	298.399			0.675547
$59$			387.518i			0.855093i	0.903993	+	0.427547i	$0.140622\pi$
$59$			387.518i			0.855093i	−0.903993	+	0.427547i	$0.859378\pi$
$60$		−	187.663i		−	0.403787i
$61$	161.665			0.339328			0.169664	−	0.985502i	$-0.445732\pi$
$61$	161.665			0.339328			0.169664	+	0.985502i	$0.445732\pi$
$62$	−222.721			−0.456220
$63$		−	207.540i		−	0.415041i
$64$	−1.59520			−0.00311562
$65$	1025.94			1.95772
$66$	−143.512			−0.267654
$67$	−558.841			−1.01900			−0.509502	−	0.860469i	$-0.670171\pi$
$67$	−558.841			−1.01900			−0.509502	+	0.860469i	$0.670171\pi$
$68$			1954.24i			3.48509i
$69$		−	16.9766i		−	0.0296194i
$70$	−614.168			−1.04867
$71$		−	858.333i		−	1.43472i	−0.696701	−	0.717362i	$-0.745350\pi$
$71$		−	858.333i		−	1.43472i	0.696701	−	0.717362i	$-0.254650\pi$
$72$			1350.22i			2.21008i
$73$	−13.2304			−0.0212124			−0.0106062	−	0.999944i	$-0.503376\pi$
$73$	−13.2304			−0.0212124			−0.0106062	+	0.999944i	$0.503376\pi$
$74$			662.645i			1.04096i
$75$			76.1607i			0.117257i
$76$			1835.19i			2.76988i
$77$			325.052i			0.481079i
$78$	229.857			0.333669
$79$	−402.737			−0.573562			−0.286781	−	0.957996i	$-0.592585\pi$
$79$	−402.737			−0.573562			−0.286781	+	0.957996i	$0.592585\pi$
$80$	1779.14			2.48642
$81$	692.051			0.949315
$82$			737.033i			0.992581i
$83$		−	245.757i		−	0.325004i	−0.986708	−	0.162502i	$-0.948044\pi$
$83$		−	245.757i		−	0.325004i	0.986708	−	0.162502i	$-0.0519564\pi$
$84$	−95.2311			−0.123697
$85$		−	1674.76i		−	2.13710i
$86$			1203.63i			1.50919i
$87$	−39.6520			−0.0488636
$88$		−	2114.74i		−	2.56172i
$89$	−221.693			−0.264039			−0.132019	−	0.991247i	$-0.542146\pi$
$89$	−221.693			−0.264039			−0.132019	+	0.991247i	$0.542146\pi$
$90$		−	2084.62i		−	2.44154i
$91$		−	520.619i		−	0.599733i
$92$	450.675			0.510718
$93$	29.5957			0.0329993
$94$		−	361.475i		−	0.396631i
$95$		−	1572.74i		−	1.69852i
$96$	122.952			0.130716
$97$	−1434.51			−1.50157			−0.750785	−	0.660547i	$-0.770325\pi$
$97$	−1434.51			−1.50157			−0.750785	+	0.660547i	$0.770325\pi$
$98$		−	1436.65i		−	1.48085i
$99$	−1103.30			−1.12006
$100$	−2021.83			−2.02183
$101$		−	1403.84i		−	1.38304i	−0.722358	−	0.691519i	$-0.756942\pi$
$101$		−	1403.84i		−	1.38304i	0.722358	−	0.691519i	$-0.243058\pi$
$102$		−	375.223i		−	0.364241i
$103$	853.549			0.816531			0.408265	−	0.912863i	$-0.366134\pi$
$103$	853.549			0.816531			0.408265	+	0.912863i	$0.366134\pi$
$104$			3387.07i			3.19355i
$105$	81.6121			0.0758526
$106$		−	2955.58i		−	2.70822i
$107$			1960.99i			1.77174i	0.463935	+	0.885869i	$0.346437\pi$
$107$			1960.99i			1.77174i	−0.463935	+	0.885869i	$0.653563\pi$
$108$		−	652.059i		−	0.580966i
$109$		−	703.837i		−	0.618490i	−0.950982	−	0.309245i	$-0.899924\pi$
$109$		−	703.837i		−	0.618490i	0.950982	−	0.309245i	$-0.100076\pi$
$110$			3264.96i			2.83002i
$111$		−	88.0539i		−	0.0752946i
$112$		−	902.835i		−	0.761696i
$113$			1091.27i			0.908479i	0.890880	+	0.454239i	$0.150089\pi$
$113$			1091.27i			0.908479i	−0.890880	+	0.454239i	$0.849911\pi$
$114$		−	352.365i		−	0.289492i
$115$	−386.223			−0.313178
$116$		−	1052.64i		−	0.842541i
$117$	1767.10			1.39631
$118$	1975.23			1.54097
$119$	−849.870			−0.654685
$120$	−530.956			−0.403912
$121$	396.999			0.298271
$122$		−	824.024i		−	0.611506i
$123$		−	97.9387i		−	0.0717954i
$124$			785.674i			0.568997i
$125$	−193.469			−0.138435
$126$	−1057.86			−0.747948
$127$	2627.26			1.83568			0.917841	−	0.396948i	$-0.129931\pi$
$127$	2627.26			1.83568			0.917841	+	0.396948i	$0.129931\pi$
$128$		−	1444.09i		−	0.997192i
$129$		−	159.941i		−	0.109163i
$130$		−	5229.33i		−	3.52802i
$131$	1293.89			0.862956			0.431478	−	0.902123i	$-0.357992\pi$
$131$	1293.89			0.862956			0.431478	+	0.902123i	$0.357992\pi$
$132$			506.256i			0.333817i
$133$	−798.098			−0.520330
$134$			2848.48i			1.83635i
$135$			558.807i			0.356255i
$136$	5529.13			3.48617
$137$		−	544.929i		−	0.339828i	−0.985459	−	0.169914i	$-0.945651\pi$
$137$		−	544.929i		−	0.339828i	0.985459	−	0.169914i	$-0.0543489\pi$
$138$	−86.5316			−0.0533773
$139$			1537.32i			0.938087i	0.883175	+	0.469044i	$0.155401\pi$
$139$			1537.32i			0.938087i	−0.883175	+	0.469044i	$0.844599\pi$
$140$			2166.54i			1.30790i
$141$			48.0337i			0.0286891i
$142$	−4375.03			−2.58552
$143$	−2767.65			−1.61848
$144$	3064.43			1.77339
$145$			902.097i			0.516656i
$146$			67.4371i			0.0382270i
$147$			190.905i			0.107113i
$148$	2337.55			1.29828
$149$	3459.00			1.90183			0.950914	−	0.309454i	$-0.100146\pi$
$149$	3459.00			1.90183			0.950914	+	0.309454i	$0.100146\pi$
$150$	388.201			0.211310
$151$	1193.82			0.643389			0.321695	−	0.946843i	$-0.395748\pi$
$151$	1193.82			0.643389			0.321695	+	0.946843i	$0.395748\pi$
$152$	5192.31			2.77074
$153$		−	2884.65i		−	1.52425i
$154$	1656.83			0.866955
$155$		−	673.314i		−	0.348915i
$156$		−	810.845i		−	0.416151i
$157$			3037.73i			1.54419i	0.635508	+	0.772094i	$0.280791\pi$
$157$			3037.73i			1.54419i	−0.635508	+	0.772094i	$0.719209\pi$
$158$			2052.80i			1.03362i
$159$			392.744i			0.195891i
$160$		−	2797.20i		−	1.38211i
$161$			195.992i			0.0959399i
$162$		−	3527.47i		−	1.71077i
$163$		−	3303.68i		−	1.58751i	−0.608236	−	0.793756i	$-0.708123\pi$
$163$		−	3303.68i		−	1.58751i	0.608236	−	0.793756i	$-0.291877\pi$
$164$	2599.96			1.23795
$165$		−	433.856i		−	0.204701i
$166$	−1252.65			−0.585692
$167$			3101.92i			1.43733i	0.695357	+	0.718665i	$0.255246\pi$
$167$			3101.92i			1.43733i	−0.695357	+	0.718665i	$0.744754\pi$
$168$			269.437i			0.123735i
$169$	2235.81			1.01767
$170$	−8536.47			−3.85128
$171$		−	2708.92i		−	1.21144i
$172$	4245.92			1.88226
$173$	1295.85			0.569488			0.284744	−	0.958604i	$-0.408091\pi$
$173$	1295.85			0.569488			0.284744	+	0.958604i	$0.408091\pi$
$174$			202.111i			0.0880574i
$175$		−	879.264i		−	0.379806i
$176$	−4799.54			−2.05556
$177$	−262.473			−0.111461
$178$			1130.00i			0.475826i
$179$			1699.81i			0.709774i	0.934909	+	0.354887i	$0.115481\pi$
$179$			1699.81i			0.709774i	−0.934909	+	0.354887i	$0.884519\pi$
$180$	−7353.74			−3.04508
$181$			2446.01i			1.00448i	0.864729	+	0.502238i	$0.167490\pi$
$181$			2446.01i			1.00448i	−0.864729	+	0.502238i	$0.832510\pi$
$182$	−2653.66			−1.08078
$183$			109.498i			0.0442314i
$184$		−	1275.09i		−	0.510876i
$185$	−2003.26			−0.796122
$186$		−	150.853i		−	0.0594682i
$187$			4517.97i			1.76677i
$188$	−1275.14			−0.494678
$189$	283.571			0.109136
$190$	−8016.45			−3.06092
$191$	4277.64			1.62052			0.810259	−	0.586072i	$-0.199326\pi$
$191$	4277.64			1.62052			0.810259	+	0.586072i	$0.199326\pi$
$192$		−	1.08046i		−	0.000406121i
$193$			3756.34i			1.40097i	0.713666	+	0.700486i	$0.247033\pi$
$193$			3756.34i			1.40097i	−0.713666	+	0.700486i	$0.752967\pi$
$194$			7311.87i			2.70599i
$195$			694.886i			0.255189i
$196$	−5067.94			−1.84692
$197$			749.442i			0.271043i	0.990774	+	0.135522i	$0.0432710\pi$
$197$			749.442i			0.271043i	−0.990774	+	0.135522i	$0.956729\pi$
$198$			5623.65i			2.01846i
$199$	224.491			0.0799687			0.0399844	−	0.999200i	$-0.487269\pi$
$199$	224.491			0.0799687			0.0399844	+	0.999200i	$0.487269\pi$
$200$			5720.36i			2.02245i
$201$		−	378.513i		−	0.132827i
$202$	−7155.52			−2.49238
$203$	457.776			0.158274
$204$	−1323.64			−0.454281
$205$	−2228.14			−0.759123
$206$		−	4350.64i		−	1.47147i
$207$	−665.240			−0.223369
$208$	7687.19			2.56255
$209$			4242.75i			1.40420i
$210$		−	415.987i		−	0.136694i
$211$	695.385			0.226883			0.113441	−	0.993545i	$-0.463813\pi$
$211$	695.385			0.226883			0.113441	+	0.993545i	$0.463813\pi$
$212$	−10426.1			−3.37768
$213$	581.364			0.187016
$214$	9995.41			3.19286
$215$	−3638.71			−1.15422
$216$	−1844.87			−0.581146
$217$	−341.678			−0.106888
$218$	−3587.55			−1.11458
$219$		−	8.96121i		−	0.00276503i
$220$	11517.5			3.52959
$221$		−	7236.22i		−	2.20254i
$222$	−448.821			−0.135689
$223$		−	3871.23i		−	1.16250i	−0.813726	−	0.581248i	$-0.802565\pi$
$223$		−	3871.23i		−	1.16250i	0.813726	−	0.581248i	$-0.197435\pi$
$224$	−1419.46			−0.423400
$225$	2984.42			0.884272
$226$	5562.34			1.63717
$227$			2281.57i			0.667107i	0.942731	+	0.333553i	$0.108248\pi$
$227$			2281.57i			0.667107i	−0.942731	+	0.333553i	$0.891752\pi$
$228$	−1243.01			−0.361054
$229$			4583.75i			1.32272i	0.750069	+	0.661359i	$0.230020\pi$
$229$			4583.75i			1.32272i	−0.750069	+	0.661359i	$0.769980\pi$
$230$			1968.63i			0.564380i
$231$	−220.163			−0.0627086
$232$	−2978.22			−0.842801
$233$	−5434.51			−1.52801			−0.764005	−	0.645210i	$-0.776770\pi$
$233$	−5434.51			−1.52801			−0.764005	+	0.645210i	$0.776770\pi$
$234$		−	9007.12i		−	2.51630i
$235$	1092.78			0.303342
$236$		−	6967.83i		−	1.92189i
$237$		−	272.781i		−	0.0747638i
$238$			4331.89i			1.17981i
$239$	1036.30			0.280471			0.140235	−	0.990118i	$-0.455214\pi$
$239$	1036.30			0.280471			0.140235	+	0.990118i	$0.455214\pi$
$240$			1205.04i			0.324104i
$241$		−	4515.54i		−	1.20694i	−0.797387	−	0.603468i	$-0.793785\pi$
$241$		−	4515.54i		−	1.20694i	0.797387	−	0.603468i	$-0.206215\pi$
$242$		−	2023.55i		−	0.537516i
$243$			1447.88i			0.382228i
$244$	−2906.84			−0.762669
$245$	4343.17			1.13255
$246$	−499.205			−0.129383
$247$		−	6795.41i		−	1.75053i
$248$	2222.91			0.569172
$249$	166.456			0.0423643
$250$			986.135i			0.249475i
$251$		−	1320.42i		−	0.332048i	−0.986122	−	0.166024i	$-0.946907\pi$
$251$		−	1320.42i		−	0.332048i	0.986122	−	0.166024i	$-0.0530929\pi$
$252$			3731.71i			0.932839i
$253$	1041.91			0.258910
$254$		−	13391.5i		−	3.30809i
$255$	1134.35			0.278571
$256$	−7373.46			−1.80016
$257$		−	1607.46i		−	0.390157i	−0.980788	−	0.195078i	$-0.937504\pi$
$257$		−	1607.46i		−	0.390157i	0.980788	−	0.195078i	$-0.0624961\pi$
$258$	−815.237			−0.196723
$259$			1016.57i			0.243886i
$260$	−18447.0			−4.40014
$261$			1553.79i			0.368496i
$262$		−	6595.09i		−	1.55514i
$263$	7501.82			1.75887			0.879434	−	0.476021i	$-0.157921\pi$
$263$	7501.82			1.75887			0.879434	+	0.476021i	$0.157921\pi$
$264$	1432.35			0.333920
$265$	8935.07			2.07123
$266$			4068.01i			0.937690i
$267$		−	150.157i		−	0.0344174i
$268$	10048.3			2.29030
$269$	−4410.02	+	129.695i	−0.999568	+	0.0293964i
$269$	−4410.02	+	129.695i	−0.999568	+	0.0293964i
$270$	2848.31			0.642010
$271$			2473.24i			0.554387i	0.960814	+	0.277193i	$0.0894043\pi$
$271$			2473.24i			0.554387i	−0.960814	+	0.277193i	$0.910596\pi$
$272$		−	12548.7i		−	2.79735i
$273$	352.625			0.0781752
$274$	−2777.57			−0.612405
$275$	−4674.23			−1.02497
$276$			305.250i			0.0665720i
$277$			7840.63i			1.70072i	0.526205	+	0.850358i	$0.323614\pi$
$277$			7840.63i			1.70072i	−0.526205	+	0.850358i	$0.676386\pi$
$278$	7835.93			1.69053
$279$		−	1159.73i		−	0.248858i
$280$	6129.80			1.30831
$281$			6850.05i			1.45423i	0.686513	+	0.727117i	$0.259140\pi$
$281$			6850.05i			1.45423i	−0.686513	+	0.727117i	$0.740860\pi$
$282$	244.834			0.0517008
$283$	3547.80			0.745212			0.372606	−	0.927990i	$-0.378464\pi$
$283$	3547.80			0.745212			0.372606	+	0.927990i	$0.378464\pi$
$284$			15433.4i			3.22466i
$285$	1065.24			0.221402
$286$			14107.1i			2.91667i
$287$			1130.69i			0.232552i
$288$		−	4817.96i		−	0.985768i
$289$	−6899.55			−1.40435
$290$	4598.10			0.931068
$291$		−	971.618i		−	0.195729i
$292$	237.892			0.0476766
$293$	822.399			0.163976			0.0819881	−	0.996633i	$-0.473873\pi$
$293$	822.399			0.163976			0.0819881	+	0.996633i	$0.473873\pi$
$294$	973.068			0.193029
$295$			5971.35i			1.17853i
$296$		−	6613.64i		−	1.29868i
$297$		−	1507.48i		−	0.294522i
$298$		−	17631.0i		−	3.42730i
$299$	−1668.77			−0.322768
$300$		−	1369.42i		−	0.263545i
$301$			1846.49i			0.353588i
$302$		−	6085.05i		−	1.15946i
$303$	950.843			0.180279
$304$		−	11784.3i		−	2.22328i
$305$	2491.13			0.467677
$306$	−14703.4			−2.74686
$307$	−10587.9			−1.96835			−0.984175	−	0.177201i	$-0.943296\pi$
$307$	−10587.9			−1.96835			−0.984175	+	0.177201i	$0.943296\pi$
$308$		−	5844.65i		−	1.08126i
$309$			578.124i			0.106435i
$310$	−3431.97			−0.628782
$311$			8778.06i			1.60051i	0.599661	+	0.800254i	$0.295302\pi$
$311$			8778.06i			1.60051i	−0.599661	+	0.800254i	$0.704698\pi$
$312$	−2294.12			−0.416279
$313$	2996.88			0.541194			0.270597	−	0.962693i	$-0.412779\pi$
$313$	2996.88			0.541194			0.270597	+	0.962693i	$0.412779\pi$
$314$	15483.7			2.78279
$315$		−	3198.03i		−	0.572028i
$316$	7241.48			1.28913
$317$			4960.67i			0.878924i	0.898261	+	0.439462i	$0.144831\pi$
$317$			4960.67i			0.878924i	−0.898261	+	0.439462i	$0.855169\pi$
$318$	2001.86			0.353016
$319$		−	2433.57i		−	0.427128i
$320$	−24.5808			−0.00429409
$321$	−1328.21			−0.230946
$322$	998.994			0.172894
$323$	−11093.0			−1.91093
$324$	−12443.5			−2.13367
$325$	7486.49			1.27777
$326$	−16839.3			−2.86086
$327$	476.722			0.0806201
$328$		−	7356.08i		−	1.23833i
$329$		−	554.541i		−	0.0929266i
$330$	−2211.42			−0.368892
$331$	−5041.48			−0.837175			−0.418587	−	0.908177i	$-0.637475\pi$
$331$	−5041.48			−0.837175			−0.418587	+	0.908177i	$0.637475\pi$
$332$			4418.88i			0.730474i
$333$	−3450.46			−0.567820
$334$	15810.9			2.59022
$335$	−8611.32			−1.40444
$336$	611.506			0.0992869
$337$		−	9988.29i		−	1.61453i	−0.590189	−	0.807265i	$-0.700947\pi$
$337$		−	9988.29i		−	1.61453i	0.590189	−	0.807265i	$-0.299053\pi$
$338$		−	11396.2i		−	1.83394i
$339$	−739.137			−0.118420
$340$			30113.3i			4.80331i
$341$			1816.39i			0.288454i
$342$	−13807.7			−2.18315
$343$		−	4886.07i		−	0.769164i
$344$		−	12013.0i		−	1.88284i
$345$		−	261.596i		−	0.0408227i
$346$		−	6605.09i		−	1.02628i
$347$	224.577			0.0347433			0.0173716	−	0.999849i	$-0.494470\pi$
$347$	224.577			0.0347433			0.0173716	+	0.999849i	$0.494470\pi$
$348$	712.969			0.109825
$349$	11624.4			1.78292			0.891458	−	0.453104i	$-0.149683\pi$
$349$	11624.4			1.78292			0.891458	+	0.453104i	$0.149683\pi$
$350$	−4481.72			−0.684451
$351$			2414.46i			0.367164i
$352$			7545.95i			1.14261i
$353$	−4505.08			−0.679267			−0.339634	−	0.940558i	$-0.610303\pi$
$353$	−4505.08			−0.679267			−0.339634	+	0.940558i	$0.610303\pi$
$354$			1337.86i			0.200865i
$355$		−	13226.3i		−	1.97740i
$356$	3986.20			0.593449
$357$		−	575.632i		−	0.0853380i
$358$	8664.13			1.27909
$359$			8129.66i			1.19517i	0.801804	+	0.597587i	$0.203874\pi$
$359$			8129.66i			1.19517i	−0.801804	+	0.597587i	$0.796126\pi$
$360$			20805.9i			3.04602i
$361$	−3558.21			−0.518765
$362$	12467.6			1.81017
$363$			268.894i			0.0388796i
$364$			9361.08i			1.34795i
$365$	−203.871			−0.0292359
$366$	558.126			0.0797097
$367$		−	11849.3i		−	1.68536i	−0.538412	−	0.842681i	$-0.680976\pi$
$367$		−	11849.3i		−	1.68536i	0.538412	−	0.842681i	$-0.319024\pi$
$368$	−2893.91			−0.409933
$369$	−3837.80			−0.541431
$370$			10210.9i			1.43470i
$371$		−	4534.17i		−	0.634507i
$372$	−532.151			−0.0741687
$373$		−	10732.7i		−	1.48986i	−0.667142	−	0.744930i	$-0.732483\pi$
$373$		−	10732.7i		−	1.48986i	0.667142	−	0.744930i	$-0.267517\pi$
$374$	23028.7			3.18391
$375$		−	131.040i		−	0.0180450i
$376$			3607.76i			0.494830i
$377$			3897.73i			0.532476i
$378$		−	1445.40i		−	0.196675i
$379$			10431.4i			1.41378i	0.707322	+	0.706891i	$0.249903\pi$
$379$			10431.4i			1.41378i	−0.707322	+	0.706891i	$0.750097\pi$
$380$			28278.9i			3.81757i
$381$			1779.49i			0.239281i
$382$		−	21803.6i		−	2.92034i
$383$			13377.3i			1.78473i	0.451318	+	0.892363i	$0.350954\pi$
$383$			13377.3i			1.78473i	−0.451318	+	0.892363i	$0.649046\pi$
$384$	978.106			0.129984
$385$			5008.80i			0.663044i
$386$	19146.5			2.52470
$387$	−6267.40			−0.823229
$388$	25793.4			3.37490
$389$	−4353.71			−0.567460			−0.283730	−	0.958904i	$-0.591572\pi$
$389$	−4353.71			−0.567460			−0.283730	+	0.958904i	$0.591572\pi$
$390$	3541.92			0.459877
$391$			2724.14i			0.352342i
$392$			14338.7i			1.84749i
$393$			876.371i			0.112486i
$394$	3820.00			0.488449
$395$	−6205.87			−0.790509
$396$	19838.0			2.51742
$397$		−	5809.96i		−	0.734492i	−0.930124	−	0.367246i	$-0.880301\pi$
$397$		−	5809.96i		−	0.734492i	0.930124	−	0.367246i	$-0.119699\pi$
$398$		−	1144.26i		−	0.144112i
$399$		−	540.566i		−	0.0678250i
$400$	12982.7			1.62284
$401$			2811.31i			0.350100i	0.984560	+	0.175050i	$0.0560087\pi$
$401$			2811.31i			0.350100i	−0.984560	+	0.175050i	$0.943991\pi$
$402$	−1929.33			−0.239369
$403$		−	2909.22i		−	0.359599i
$404$			25241.9i			3.10849i
$405$	10664.0			1.30839
$406$		−	2333.34i		−	0.285226i
$407$	5404.15			0.658167
$408$			3744.98i			0.454421i
$409$		−	9449.71i		−	1.14244i	−0.820797	−	0.571220i	$-0.806470\pi$
$409$		−	9449.71i		−	1.14244i	0.820797	−	0.571220i	$-0.193530\pi$
$410$			11357.1i			1.36802i
$411$	369.090			0.0442965
$412$	−15347.4			−1.83522
$413$	3030.21			0.361033
$414$			3390.81i			0.402534i
$415$		−	3786.93i		−	0.447935i
$416$		−	12086.0i		−	1.42443i
$417$	−1041.26			−0.122280
$418$	21625.8			2.53051
$419$	2662.28			0.310408			0.155204	−	0.987882i	$-0.450396\pi$
$419$	2662.28			0.310408			0.155204	+	0.987882i	$0.450396\pi$
$420$	−1467.44			−0.170485
$421$	−13514.6			−1.56451			−0.782256	−	0.622957i	$-0.785931\pi$
$421$	−13514.6			−1.56451			−0.782256	+	0.622957i	$0.785931\pi$
$422$		−	3544.46i		−	0.408867i
$423$	1882.24			0.216354
$424$			29498.6i			3.37872i
$425$		−	12221.1i		−	1.39485i
$426$		−	2963.28i		−	0.337022i
$427$		−	1264.14i		−	0.143270i
$428$		−	35259.9i		−	3.98213i
$429$		−	1874.58i		−	0.210969i
$430$			18546.9i			2.08003i
$431$		−	3826.16i		−	0.427609i	−0.976876	−	0.213805i	$-0.931414\pi$
$431$		−	3826.16i		−	0.427609i	0.976876	−	0.213805i	$-0.0685856\pi$
$432$			4187.06i			0.466319i
$433$	−6199.10			−0.688013			−0.344007	−	0.938967i	$-0.611784\pi$
$433$	−6199.10			−0.688013			−0.344007	+	0.938967i	$0.611784\pi$
$434$			1741.58i			0.192623i
$435$	−611.006			−0.0673460
$436$			12655.5i			1.39011i
$437$			2558.19i			0.280034i
$438$	−45.6764			−0.00498288
$439$	−5183.48			−0.563540			−0.281770	−	0.959482i	$-0.590922\pi$
$439$	−5183.48			−0.563540			−0.281770	+	0.959482i	$0.590922\pi$
$440$		−	32586.5i		−	3.53068i
$441$	7480.78			0.807772
$442$	−36883.9			−3.96920
$443$			10577.0i			1.13437i	0.823590	+	0.567186i	$0.191968\pi$
$443$			10577.0i			1.13437i	−0.823590	+	0.567186i	$0.808032\pi$
$444$			1583.27i			0.169231i
$445$	−3416.13			−0.363910
$446$	−19732.1			−2.09494
$447$			2342.84i			0.247903i
$448$			12.4737i			0.00131546i
$449$	3242.68			0.340827			0.170414	−	0.985373i	$-0.445490\pi$
$449$	3242.68			0.340827			0.170414	+	0.985373i	$0.445490\pi$
$450$		−	15212.0i		−	1.59355i
$451$	6010.82			0.627579
$452$		−	19621.8i		−	2.04188i
$453$			808.596i			0.0838657i
$454$	11629.5			1.20220
$455$		−	8022.35i		−	0.826579i
$456$			3516.84i			0.361165i
$457$	10137.3			1.03764			0.518821	−	0.854883i	$-0.326371\pi$
$457$	10137.3			1.03764			0.518821	+	0.854883i	$0.326371\pi$
$458$	23363.9			2.38368
$459$	3941.42			0.400806
$460$	6944.55			0.703894
$461$		−	5616.18i		−	0.567400i	−0.958913	−	0.283700i	$-0.908438\pi$
$461$		−	5616.18i		−	0.567400i	0.958913	−	0.283700i	$-0.0915621\pi$
$462$			1122.20i			0.113007i
$463$		−	13514.5i		−	1.35653i	−0.734817	−	0.678266i	$-0.762732\pi$
$463$		−	13514.5i		−	1.35653i	0.734817	−	0.678266i	$-0.237268\pi$
$464$			6759.27i			0.676275i
$465$	456.048			0.0454811
$466$			27700.4i			2.75364i
$467$		−	12281.6i		−	1.21697i	−0.793564	−	0.608486i	$-0.791777\pi$
$467$		−	12281.6i		−	1.21697i	0.793564	−	0.608486i	$-0.208223\pi$
$468$	−31773.6			−3.13832
$469$			4369.88i			0.430239i
$470$		−	5570.06i		−	0.546654i
$471$	−2057.51			−0.201285
$472$	−19714.1			−1.92249
$473$	9816.07			0.954215
$474$	−1390.40			−0.134732
$475$		−	11476.6i		−	1.10860i
$476$	15281.2			1.47146
$477$	15390.0			1.47727
$478$		−	5282.14i		−	0.505438i
$479$			13874.2i			1.32344i	0.749751	+	0.661720i	$0.230173\pi$
$479$			13874.2i			1.32344i	−0.749751	+	0.661720i	$0.769827\pi$
$480$	1894.59			0.180158
$481$	−8655.57			−0.820499
$482$	−23016.3			−2.17503
$483$	−132.749			−0.0125057
$484$	−7138.30			−0.670389
$485$	−22104.7			−2.06953
$486$	7380.01			0.688815
$487$	3698.85			0.344170			0.172085	−	0.985082i	$-0.444950\pi$
$487$	3698.85			0.344170			0.172085	+	0.985082i	$0.444950\pi$
$488$			8224.31i			0.762904i
$489$	2237.64			0.206932
$490$		−	22137.7i		−	2.04098i
$491$	15768.7			1.44935			0.724676	−	0.689089i	$-0.241989\pi$
$491$	15768.7			1.44935			0.724676	+	0.689089i	$0.241989\pi$
$492$			1761.00i			0.161366i
$493$	6362.74			0.581264
$494$	−34637.0			−3.15464
$495$	−17001.0			−1.54371
$496$		−	5045.04i		−	0.456711i
$497$	−6711.76			−0.605762
$498$		−	848.445i		−	0.0763449i
$499$			832.226i			0.0746604i	0.999303	+	0.0373302i	$0.0118853\pi$
$499$			832.226i			0.0746604i	−0.999303	+	0.0373302i	$0.988115\pi$
$500$	3478.70			0.311145
$501$	−2100.99			−0.187356
$502$	−6730.33			−0.598386
$503$		−	10011.5i		−	0.887454i	−0.896162	−	0.443727i	$-0.853656\pi$
$503$		−	10011.5i		−	0.887454i	0.896162	−	0.443727i	$-0.146344\pi$
$504$	10558.1			0.933127
$505$		−	21632.0i		−	1.90616i
$506$		−	5310.73i		−	0.466582i
$507$			1514.36i			0.132653i
$508$	−47239.9			−4.12585
$509$			932.375i			0.0811922i	0.999176	+	0.0405961i	$0.0129257\pi$
$509$			932.375i			0.0811922i	−0.999176	+	0.0405961i	$0.987074\pi$
$510$		−	5781.90i		−	0.502013i
$511$			103.456i			0.00895619i
$512$			26030.7i			2.24689i
$513$	3701.32			0.318552
$514$	−8193.40			−0.703104
$515$	13152.5			1.12538
$516$			2875.84i			0.245352i
$517$	−2947.98			−0.250778
$518$	5181.58			0.439508
$519$			877.700i			0.0742327i
$520$			52192.2i			4.40150i
$521$		−	630.658i		−	0.0530319i	−0.999648	−	0.0265159i	$-0.991559\pi$
$521$		−	630.658i		−	0.0530319i	0.999648	−	0.0265159i	$-0.00844128\pi$
$522$	7919.88			0.664068
$523$		−	8471.02i		−	0.708245i	−0.935199	−	0.354122i	$-0.884780\pi$
$523$		−	8471.02i		−	0.708245i	0.935199	−	0.354122i	$-0.115220\pi$
$524$	−23264.9			−1.93957
$525$	595.541			0.0495077
$526$		−	38237.7i		−	3.16967i
$527$	−4749.07			−0.392548
$528$		−	3250.81i		−	0.267942i
$529$	−11538.8			−0.948367
$530$		−	45543.2i		−	3.73258i
$531$			10285.2i			0.840564i
$532$	14350.3			1.16949
$533$	−9627.23			−0.782367
$534$	−765.368			−0.0620238
$535$			30217.4i			2.44189i
$536$		−	28429.8i		−	2.29100i
$537$	−1151.31			−0.0925190
$538$	661.070	+	22478.4i	0.0529754	+	1.80133i
$539$	−11716.5			−0.936298
$540$		−	10047.7i		−	0.800714i
$541$			5435.69i			0.431975i	0.976396	+	0.215988i	$0.0692970\pi$
$541$			5435.69i			0.431975i	−0.976396	+	0.215988i	$0.930703\pi$
$542$	12606.4			0.999064
$543$	−1656.72			−0.130933
$544$	−19729.4			−1.55495
$545$		−	10845.6i		−	0.852430i
$546$		−	1797.37i		−	0.140880i
$547$	10733.2			0.838974			0.419487	−	0.907761i	$-0.362210\pi$
$547$	10733.2			0.838974			0.419487	+	0.907761i	$0.362210\pi$
$548$			9798.18i			0.763791i
$549$	4290.78			0.333563
$550$			23825.1i			1.84710i
$551$	5975.14			0.461977
$552$	863.643			0.0665926
$553$			3149.21i			0.242167i
$554$	39964.7			3.06487
$555$		−	1356.84i		−	0.103774i
$556$		−	27642.1i		−	2.10843i
$557$		−	10026.5i		−	0.762725i	−0.924426	−	0.381362i	$-0.875455\pi$
$557$		−	10026.5i		−	0.762725i	0.924426	−	0.381362i	$-0.124545\pi$
$558$	−5911.30			−0.448468
$559$	−15721.9			−1.18956
$560$		−	13912.0i		−	1.04980i
$561$	−3060.10			−0.230299
$562$	34915.6			2.62068
$563$	1992.43			0.149149			0.0745747	−	0.997215i	$-0.476240\pi$
$563$	1992.43			0.149149			0.0745747	+	0.997215i	$0.476240\pi$
$564$		−	863.677i		−	0.0644812i
$565$			16815.6i			1.25211i
$566$		−	18083.6i		−	1.34295i
$567$		−	5411.51i		−	0.400815i
$568$	43665.7			3.22565
$569$		−	12209.1i		−	0.899530i	−0.893147	−	0.449765i	$-0.851508\pi$
$569$		−	12209.1i		−	0.899530i	0.893147	−	0.449765i	$-0.148492\pi$
$570$		−	5429.68i		−	0.398990i
$571$			1508.80i			0.110580i	0.998470	+	0.0552901i	$0.0176084\pi$
$571$			1508.80i			0.110580i	−0.998470	+	0.0552901i	$0.982392\pi$
$572$	49764.2			3.63767
$573$			2897.32i			0.211234i
$574$	5763.25			0.419083
$575$	−2818.36			−0.204406
$576$	−42.3386			−0.00306269
$577$			7099.70i			0.512244i	0.966645	+	0.256122i	$0.0824448\pi$
$577$			7099.70i			0.512244i	−0.966645	+	0.256122i	$0.917555\pi$
$578$			35167.9i			2.53078i
$579$	−2544.24			−0.182616
$580$		−	16220.3i		−	1.16123i
$581$	−1921.71			−0.137222
$582$	−4952.46			−0.352725
$583$	−24104.0			−1.71232
$584$		−	673.068i		−	0.0476913i
$585$	27229.6			1.92446
$586$		−	4191.87i		−	0.295502i
$587$	−6377.02			−0.448395			−0.224198	−	0.974544i	$-0.571976\pi$
$587$	−6377.02			−0.448395			−0.224198	+	0.974544i	$0.571976\pi$
$588$		−	3432.61i		−	0.240745i
$589$	−4459.77			−0.311989
$590$	30436.7			2.12383
$591$	−507.610			−0.0353305
$592$	−15010.1			−1.04208
$593$	−1396.62			−0.0967159			−0.0483579	−	0.998830i	$-0.515399\pi$
$593$	−1396.62			−0.0967159			−0.0483579	+	0.998830i	$0.515399\pi$
$594$	−7683.83			−0.530760
$595$	−13095.9			−0.902315
$596$	−62195.2			−4.27452
$597$			152.052i			0.0104239i
$598$			8505.94i			0.581661i
$599$	−15726.6			−1.07274			−0.536371	−	0.843982i	$-0.680205\pi$
$599$	−15726.6			−1.07274			−0.536371	+	0.843982i	$0.680205\pi$
$600$	−3874.50			−0.263626
$601$			14651.8i			0.994438i	0.867625	+	0.497219i	$0.165645\pi$
$601$			14651.8i			0.994438i	−0.867625	+	0.497219i	$0.834355\pi$
$602$	9411.78			0.637202
$603$	−14832.3			−1.00169
$604$	−21465.7			−1.44607
$605$	6117.45			0.411090
$606$		−	4846.56i		−	0.324881i
$607$		−	15145.2i		−	1.01272i	−0.862321	−	0.506362i	$-0.830990\pi$
$607$		−	15145.2i		−	1.01272i	0.862321	−	0.506362i	$-0.169010\pi$
$608$	−18527.5			−1.23584
$609$			310.060i			0.0206310i
$610$		−	12697.6i		−	0.842804i
$611$	4721.64			0.312630
$612$			51867.9i			3.42588i
$613$			25710.9i			1.69405i	0.531553	+	0.847025i	$0.321609\pi$
$613$			25710.9i			1.69405i	−0.531553	+	0.847025i	$0.678391\pi$
$614$			53967.8i			3.54717i
$615$		−	1509.16i		−	0.0989516i
$616$	−16536.3			−1.08160
$617$	4469.96			0.291659			0.145830	−	0.989310i	$-0.453415\pi$
$617$	4469.96			0.291659			0.145830	+	0.989310i	$0.453415\pi$
$618$	2946.77			0.191807
$619$	−24081.5			−1.56368			−0.781840	−	0.623480i	$-0.785718\pi$
$619$	−24081.5			−1.56368			−0.781840	+	0.623480i	$0.785718\pi$
$620$			12106.6i			0.784217i
$621$		−	908.946i		−	0.0587355i
$622$	44742.8			2.88428
$623$			1733.54i			0.111481i
$624$			5206.67i			0.334028i
$625$	−17036.8			−1.09035
$626$		−	15275.5i		−	0.975288i
$627$	−2873.69			−0.183037
$628$		−	54620.5i		−	3.47069i
$629$			14129.5i			0.895678i
$630$	−16300.8			−1.03085
$631$	12820.1			0.808809			0.404404	−	0.914580i	$-0.367479\pi$
$631$	12820.1			0.808809			0.404404	+	0.914580i	$0.367479\pi$
$632$		−	20488.3i		−	1.28953i
$633$			470.997i			0.0295742i
$634$	25285.1			1.58391
$635$	40484.1			2.53002
$636$		−	7061.80i		−	0.440281i
$637$	18765.7			1.16723
$638$	−12404.2			−0.769729
$639$		−	22781.2i		−	1.41035i
$640$		−	22252.3i		−	1.37437i
$641$	25892.1			1.59544			0.797718	−	0.603031i	$-0.206040\pi$
$641$	25892.1			1.59544			0.797718	+	0.603031i	$0.206040\pi$
$642$			6770.07i			0.416189i
$643$	−27862.2			−1.70883			−0.854414	−	0.519593i	$-0.826084\pi$
$643$	−27862.2			−1.70883			−0.854414	+	0.519593i	$0.826084\pi$
$644$		−	3524.06i		−	0.215633i
$645$		−	2464.56i		−	0.150453i
$646$			56542.2i			3.44369i
$647$		−	13612.2i		−	0.827124i	−0.910476	−	0.413562i	$-0.864284\pi$
$647$		−	13612.2i		−	0.827124i	0.910476	−	0.413562i	$-0.135716\pi$
$648$			35206.5i			2.13432i
$649$		−	16108.8i		−	0.974308i
$650$		−	38159.6i		−	2.30268i
$651$		−	231.425i		−	0.0139328i
$652$			59402.4i			3.56807i
$653$	−30330.7			−1.81766			−0.908830	−	0.417167i	$-0.863023\pi$
$653$	−30330.7			−1.81766			−0.908830	+	0.417167i	$0.863023\pi$
$654$		−	2429.91i		−	0.145286i
$655$	19937.8			1.18936
$656$	−16695.1			−0.993651
$657$	−351.152			−0.0208520
$658$	−2826.57			−0.167464
$659$	−16832.2			−0.994978			−0.497489	−	0.867470i	$-0.665745\pi$
$659$	−16832.2			−0.994978			−0.497489	+	0.867470i	$0.665745\pi$
$660$			7801.01i			0.460082i
$661$		−	26370.5i		−	1.55173i	−0.630900	−	0.775864i	$-0.717314\pi$
$661$		−	26370.5i		−	1.55173i	0.630900	−	0.775864i	$-0.282686\pi$
$662$			25697.1i			1.50868i
$663$	4901.22			0.287100
$664$	12502.3			0.730699
$665$	−12298.1			−0.717142
$666$			17587.4i			1.02327i
$667$		−	1467.34i		−	0.0851806i
$668$		−	55774.6i		−	3.23052i
$669$	2622.05			0.151531
$670$			43893.0i			2.53094i
$671$	−6720.27			−0.386637
$672$		−	961.425i		−	0.0551901i
$673$			22178.3i			1.27030i	0.772390	+	0.635148i	$0.219061\pi$
$673$			22178.3i			1.27030i	−0.772390	+	0.635148i	$0.780939\pi$
$674$	−50911.5			−2.90955
$675$			4077.74i			0.232522i
$676$	−40201.4			−2.28729
$677$		−	13773.6i		−	0.781926i	−0.920406	−	0.390963i	$-0.872142\pi$
$677$		−	13773.6i		−	0.781926i	0.920406	−	0.390963i	$-0.127858\pi$
$678$			3767.47i			0.213405i
$679$			11217.2i			0.633985i
$680$	85199.6			4.80479
$681$	−1545.35			−0.0869573
$682$	9258.35			0.519825
$683$			355.239i			0.0199017i	0.999950	+	0.00995085i	$0.00316751\pi$
$683$			355.239i			0.0199017i	−0.999950	+	0.00995085i	$0.996832\pi$
$684$			48708.3i			2.72282i
$685$		−	8396.93i		−	0.468365i
$686$	−24904.9			−1.38611
$687$	−3104.65			−0.172416
$688$	−27264.3			−1.51082
$689$	38606.2			2.13466
$690$	−1333.39			−0.0735669
$691$		−	8750.55i		−	0.481746i	−0.970557	−	0.240873i	$-0.922566\pi$
$691$		−	8750.55i		−	0.481746i	0.970557	−	0.240873i	$-0.0774337\pi$
$692$	−23300.2			−1.27997
$693$			8627.27i			0.472905i
$694$		−	1144.70i		−	0.0626111i
$695$			23689.0i			1.29291i
$696$		−	2017.20i		−	0.109859i
$697$			15715.7i			0.854052i
$698$		−	59250.8i		−	3.21300i
$699$		−	3680.89i		−	0.199176i
$700$			15809.8i			0.853646i
$701$			29541.3i			1.59167i	0.605515	+	0.795834i	$0.292967\pi$
$701$			29541.3i			1.59167i	−0.605515	+	0.795834i	$0.707033\pi$
$702$	12306.8			0.661668
$703$			13268.8i			0.711867i
$704$	66.3112			0.00355000
$705$			740.162i			0.0395406i
$706$			22963.0i			1.22411i
$707$	−10977.3			−0.583939
$708$	4719.43			0.250519
$709$		−	18072.1i		−	0.957282i	−0.878011	−	0.478641i	$-0.841130\pi$
$709$		−	18072.1i		−	0.957282i	0.878011	−	0.478641i	$-0.158870\pi$
$710$	−67415.8			−3.56348
$711$	−10689.1			−0.563817
$712$		−	11278.1i		−	0.593632i
$713$			1095.20i			0.0575254i
$714$	−2934.07			−0.153788
$715$	−42647.4			−2.23066
$716$		−	30563.7i		−	1.59528i
$717$			701.903i			0.0365593i
$718$	41437.9			2.15383
$719$			22330.3i			1.15825i	0.815240	+	0.579123i	$0.196605\pi$
$719$			22330.3i			1.15825i	−0.815240	+	0.579123i	$0.803395\pi$
$720$	47220.5			2.44417
$721$		−	6674.35i		−	0.344752i
$722$			18136.6i			0.934870i
$723$	3058.46			0.157324
$724$		−	43980.8i		−	2.25764i
$725$			6582.80i			0.337213i
$726$	1370.59			0.0700651
$727$	12108.4			0.617712			0.308856	−	0.951109i	$-0.400054\pi$
$727$	12108.4			0.617712			0.308856	+	0.951109i	$0.400054\pi$
$728$	26485.3			1.34837
$729$	17704.7			0.899492
$730$			1039.16i			0.0526861i
$731$			25664.8i			1.29856i
$732$		−	1968.85i		−	0.0994138i
$733$		−	24849.3i		−	1.25215i	−0.779762	−	0.626076i	$-0.784660\pi$
$733$		−	24849.3i		−	1.25215i	0.779762	−	0.626076i	$-0.215340\pi$
$734$	−60397.3			−3.03720
$735$			2941.71i			0.147628i
$736$			4549.87i			0.227868i
$737$	23230.6			1.16107
$738$			19561.8i			0.975716i
$739$			8940.00i			0.445011i	0.974931	+	0.222505i	$0.0714235\pi$
$739$			8940.00i			0.445011i	−0.974931	+	0.222505i	$0.928576\pi$
$740$	36019.9			1.78935
$741$	4602.65			0.228182
$742$	−23111.2			−1.14345
$743$	−9931.87			−0.490397			−0.245199	−	0.969473i	$-0.578853\pi$
$743$	−9931.87			−0.490397			−0.245199	+	0.969473i	$0.578853\pi$
$744$			1505.61i			0.0741915i
$745$	53300.6			2.62118
$746$	−54705.9			−2.68489
$747$		−	6522.70i		−	0.319482i
$748$		−	81236.2i		−	3.97097i
$749$	15334.0			0.748055
$750$	−667.927			−0.0325190
$751$	−18900.4			−0.918358			−0.459179	−	0.888344i	$-0.651856\pi$
$751$	−18900.4			−0.918358			−0.459179	+	0.888344i	$0.651856\pi$
$752$	8188.06			0.397058
$753$	894.343			0.0432824
$754$	19867.2			0.959577
$755$	18395.9			0.886747
$756$	−5098.79			−0.245293
$757$			2063.85i			0.0990910i	0.998772	+	0.0495455i	$0.0157773\pi$
$757$			2063.85i			0.0990910i	−0.998772	+	0.0495455i	$0.984223\pi$
$758$	53170.0			2.54779
$759$			705.702i			0.0337488i
$760$	80009.5			3.81875
$761$		−	13649.0i		−	0.650165i	−0.945686	−	0.325082i	$-0.894608\pi$
$761$		−	13649.0i		−	0.650165i	0.945686	−	0.325082i	$-0.105392\pi$
$762$	9070.28			0.431209
$763$	−5503.68			−0.261136
$764$	−76914.8			−3.64225
$765$		−	44450.3i		−	2.10079i
$766$	68186.0			3.21626
$767$			25800.7i			1.21461i
$768$		−	4994.17i		−	0.234651i
$769$	−16289.4			−0.763865			−0.381933	−	0.924190i	$-0.624741\pi$
$769$	−16289.4			−0.763865			−0.381933	+	0.924190i	$0.624741\pi$
$770$	25530.5			1.19488
$771$	1088.76			0.0508569
$772$		−	67541.5i		−	3.14880i
$773$	28653.4			1.33324			0.666618	−	0.745400i	$-0.267741\pi$
$773$	28653.4			1.33324			0.666618	+	0.745400i	$0.267741\pi$
$774$			31945.7i			1.48355i
$775$		−	4913.32i		−	0.227731i
$776$		−	72977.3i		−	3.37594i
$777$	−688.540			−0.0317905
$778$			22191.4i			1.02262i
$779$			14758.3i			0.678783i
$780$		−	12494.5i		−	0.573558i
$781$			35680.2i			1.63475i
$782$	13885.3			0.634957
$783$	−2123.01			−0.0968970
$784$	32542.7			1.48245
$785$			46809.2i			2.12827i
$786$	4466.97			0.202712
$787$	−28719.4			−1.30081			−0.650404	−	0.759588i	$-0.725400\pi$
$787$	−28719.4			−1.30081			−0.650404	+	0.759588i	$0.725400\pi$
$788$		−	13475.5i		−	0.609192i
$789$			5081.12i			0.229268i
$790$			31632.1i			1.42458i
$791$	8533.22			0.383573
$792$		−	56127.7i		−	2.51820i
$793$	10763.5			0.481998
$794$	−29614.1			−1.32363
$795$			6051.88i			0.269985i
$796$	−4036.51			−0.179736
$797$			22525.3i			1.00111i	0.865703	+	0.500557i	$0.166872\pi$
$797$			22525.3i			1.00111i	−0.865703	+	0.500557i	$0.833128\pi$
$798$	−2755.33			−0.122228
$799$		−	7707.71i		−	0.341275i
$800$		−	20411.8i		−	0.902081i
$801$	−5884.02			−0.259553
$802$	14329.6			0.630916
$803$	549.979			0.0241698
$804$			6805.92i			0.298540i
$805$			3020.08i			0.132229i
$806$	−14828.6			−0.648036
$807$	−87.8446	−	2986.99i	−0.00383182	−	0.130294i
$808$	71416.9			3.10945
$809$			4044.63i			0.175775i	0.996130	+	0.0878873i	$0.0280115\pi$
$809$			4044.63i			0.175775i	−0.996130	+	0.0878873i	$0.971988\pi$
$810$		−	54355.6i		−	2.35785i
$811$	−8135.01			−0.352230			−0.176115	−	0.984370i	$-0.556353\pi$
$811$	−8135.01			−0.352230			−0.176115	+	0.984370i	$0.556353\pi$
$812$	−8231.11			−0.355733
$813$	−1675.17			−0.0722643
$814$		−	27545.6i		−	1.18609i
$815$		−	50907.3i		−	2.18798i
$816$	8499.48			0.364634
$817$			24101.4i			1.03207i
$818$	−48166.4			−2.05880
$819$		−	13817.9i		−	0.589543i
$820$	40063.5			1.70619
$821$	2756.41			0.117174			0.0585868	−	0.998282i	$-0.481341\pi$
$821$	2756.41			0.117174			0.0585868	+	0.998282i	$0.481341\pi$
$822$		−	1881.30i		−	0.0798269i
$823$	37015.1			1.56776			0.783879	−	0.620914i	$-0.213238\pi$
$823$	37015.1			1.56776			0.783879	+	0.620914i	$0.213238\pi$
$824$			43422.3i			1.83579i
$825$		−	3165.94i		−	0.133605i
$826$		−	15445.3i		−	0.650620i
$827$	8809.10			0.370402			0.185201	−	0.982701i	$-0.440706\pi$
$827$	8809.10			0.370402			0.185201	+	0.982701i	$0.440706\pi$
$828$	11961.5			0.502040
$829$		−	9637.79i		−	0.403781i	−0.979408	−	0.201890i	$-0.935292\pi$
$829$		−	9637.79i		−	0.403781i	0.979408	−	0.201890i	$-0.0647085\pi$
$830$	−19302.5			−0.807227
$831$	−5310.60			−0.221688
$832$	−106.207			−0.00442558
$833$		−	30633.6i		−	1.27418i
$834$			5307.42i			0.220361i
$835$			47798.3i			1.98099i
$836$		−	76287.5i		−	3.15605i
$837$	1584.59			0.0654379
$838$		−	13570.0i		−	0.559389i
$839$			38850.4i			1.59865i	0.600901	+	0.799324i	$0.294809\pi$
$839$			38850.4i			1.59865i	−0.600901	+	0.799324i	$0.705191\pi$
$840$			4151.82i			0.170538i
$841$	20961.8			0.859476
$842$			68885.4i			2.81942i
$843$	−4639.66			−0.189559
$844$	−12503.5			−0.509938
$845$	34452.2			1.40259
$846$		−	9594.00i		−	0.389892i
$847$		−	3104.34i		−	0.125935i
$848$	66949.1			2.71113
$849$			2402.99i			0.0971383i
$850$	−62292.5			−2.51366
$851$	3258.47			0.131256
$852$	−10453.3			−0.420334
$853$		−	7506.03i		−	0.301291i	−0.988588	−	0.150646i	$-0.951865\pi$
$853$		−	7506.03i		−	0.301291i	0.988588	−	0.150646i	$-0.0481353\pi$
$854$	−6443.48			−0.258187
$855$		−	41742.5i		−	1.66966i
$856$	−99760.8			−3.98336
$857$			3395.38i			0.135337i	0.997708	+	0.0676687i	$0.0215561\pi$
$857$			3395.38i			0.135337i	−0.997708	+	0.0676687i	$0.978444\pi$
$858$	−9554.96			−0.380188
$859$	3604.39			0.143167			0.0715834	−	0.997435i	$-0.477195\pi$
$859$	3604.39			0.143167			0.0715834	+	0.997435i	$0.477195\pi$
$860$	65426.4			2.59421
$861$	−765.834			−0.0303131
$862$	−19502.4			−0.770596
$863$	−5377.53			−0.212113			−0.106056	−	0.994360i	$-0.533822\pi$
$863$	−5377.53			−0.212113			−0.106056	+	0.994360i	$0.533822\pi$
$864$	6582.99			0.259210
$865$	19968.0			0.784893
$866$			31597.6i			1.23987i
$867$		−	4673.19i		−	0.183056i
$868$	6143.60			0.240239
$869$	16741.4			0.653527
$870$			3114.38i			0.121365i
$871$	−37207.3			−1.44744
$872$	35806.1			1.39054
$873$	−38073.6			−1.47606
$874$	13039.4			0.504651
$875$			1512.84i			0.0584494i
$876$			161.128i			0.00621464i
$877$	18306.9			0.704880			0.352440	−	0.935834i	$-0.385352\pi$
$877$	18306.9			0.704880			0.352440	+	0.935834i	$0.385352\pi$
$878$			26420.9i			1.01556i
$879$			557.025i			0.0213743i
$880$	−73957.2			−2.83307
$881$			42005.1i			1.60634i	0.595747	+	0.803172i	$0.296856\pi$
$881$			42005.1i			1.60634i	−0.595747	+	0.803172i	$0.703144\pi$
$882$		−	38130.5i		−	1.45569i
$883$			25784.3i			0.982684i	0.870967	+	0.491342i	$0.163494\pi$
$883$			25784.3i			0.982684i	−0.870967	+	0.491342i	$0.836506\pi$
$884$			130112.i			4.95038i
$885$	−4044.50			−0.153621
$886$	53912.1			2.04426
$887$	−5342.41			−0.202233			−0.101116	−	0.994875i	$-0.532241\pi$
$887$	−5342.41			−0.202233			−0.101116	+	0.994875i	$0.532241\pi$
$888$	4479.54			0.169283
$889$		−	20543.9i		−	0.775052i
$890$			17412.4i			0.655804i
$891$	−28768.0			−1.08167
$892$			69607.3i			2.61281i
$893$		−	7238.17i		−	0.271239i
$894$	11941.8			0.446747
$895$			26192.7i			0.978242i
$896$	−11292.1			−0.421029
$897$		−	1130.29i		−	0.0420727i
$898$		−	16528.3i		−	0.614207i
$899$	2558.05			0.0949007
$900$	−53661.8			−1.98748
$901$		−	63021.5i		−	2.33025i
$902$		−	30637.9i		−	1.13096i
$903$	−1250.66			−0.0460901
$904$	−55515.8			−2.04251
$905$			37691.1i			1.38441i
$906$	4121.51			0.151135
$907$	−28927.6			−1.05901			−0.529506	−	0.848306i	$-0.677623\pi$
$907$	−28927.6			−1.05901			−0.529506	+	0.848306i	$0.677623\pi$
$908$		−	41024.2i		−	1.49938i
$909$		−	37259.5i		−	1.35954i
$910$	−40890.9			−1.48958
$911$		−	15428.8i		−	0.561120i	−0.959836	−	0.280560i	$-0.909480\pi$
$911$		−	15428.8i		−	0.561120i	0.959836	−	0.280560i	$-0.0905202\pi$
$912$	7981.71			0.289804
$913$			10215.9i			0.370315i
$914$		−	51671.0i		−	1.86994i
$915$			1687.29i			0.0609617i
$916$		−	82418.8i		−	2.97292i
$917$		−	10117.6i		−	0.364353i
$918$		−	20089.9i		−	0.722294i
$919$		−	37179.4i		−	1.33453i	−0.744819	−	0.667267i	$-0.767464\pi$
$919$		−	37179.4i		−	1.33453i	0.744819	−	0.667267i	$-0.232536\pi$
$920$		−	19648.2i		−	0.704111i
$921$		−	7171.37i		−	0.256574i
$922$	−28626.4			−1.02252
$923$		−	57147.2i		−	2.03795i
$924$	3958.68			0.140943
$925$	−14618.2			−0.519615
$926$	−68885.3			−2.44461
$927$	22654.3			0.802657
$928$	10627.1			0.375917
$929$		−	12319.5i		−	0.435081i	−0.976051	−	0.217541i	$-0.930197\pi$
$929$		−	12319.5i		−	0.435081i	0.976051	−	0.217541i	$-0.0698035\pi$
$930$		−	2324.53i		−	0.0819617i
$931$		−	28767.5i		−	1.01269i
$932$	97716.1			3.43433
$933$	−5945.53			−0.208626
$934$	−62601.0			−2.19311
$935$			69618.5i			2.43505i
$936$			89897.1i			3.13929i
$937$			16507.4i			0.575533i	0.957701	+	0.287766i	$0.0929126\pi$
$937$			16507.4i			0.575533i	−0.957701	+	0.287766i	$0.907087\pi$
$938$	22273.8			0.775336
$939$			2029.84i			0.0705445i
$940$	−19649.0			−0.681787
$941$			36816.2i			1.27542i	0.770276	+	0.637711i	$0.220119\pi$
$941$			36816.2i			1.27542i	−0.770276	+	0.637711i	$0.779881\pi$
$942$			10487.4i			0.362736i
$943$	3624.26			0.125156
$944$			44742.4i			1.54263i
$945$	4369.61			0.150416
$946$		−	50033.7i		−	1.71960i
$947$			1698.89i			0.0582962i	0.999575	+	0.0291481i	$0.00927944\pi$
$947$			1698.89i			0.0582962i	−0.999575	+	0.0291481i	$0.990721\pi$
$948$			4904.78i			0.168038i
$949$	−880.874			−0.0301311
$950$	−58497.8			−1.99781
$951$	−3359.95			−0.114568
$952$		−	43235.2i		−	1.47191i
$953$		−	29295.2i		−	0.995764i	−0.867245	−	0.497882i	$-0.834111\pi$
$953$		−	29295.2i		−	0.995764i	0.867245	−	0.497882i	$-0.165889\pi$
$954$		−	78444.6i		−	2.66220i
$955$	65915.1			2.23347
$956$	−18633.3			−0.630381
$957$	1648.30			0.0556761
$958$	70718.4			2.38498
$959$	−4261.09			−0.143480
$960$		−	16.6490i		−	0.000559734i
$961$	27881.7			0.935910
$962$			44118.5i			1.47863i
$963$			52047.1i			1.74164i
$964$			81192.5i			2.71269i
$965$			57882.4i			1.93088i
$966$			676.637i			0.0225367i
$967$		−	43061.2i		−	1.43201i	−0.698095	−	0.716005i	$-0.745969\pi$
$967$		−	43061.2i		−	1.43201i	0.698095	−	0.716005i	$-0.254031\pi$
$968$			20196.4i			0.670596i
$969$		−	7513.46i		−	0.249089i
$970$			112670.i			3.72951i
$971$	−28750.2			−0.950193			−0.475097	−	0.879934i	$-0.657587\pi$
$971$	−28750.2			−0.950193			−0.475097	+	0.879934i	$0.657587\pi$
$972$		−	26033.8i		−	0.859089i
$973$	12021.2			0.396075
$974$		−	18853.5i		−	0.620230i
$975$			5070.73i			0.166557i
$976$	18665.6			0.612164
$977$	−36131.7			−1.18317			−0.591584	−	0.806243i	$-0.701497\pi$
$977$	−36131.7			−1.18317			−0.591584	+	0.806243i	$0.701497\pi$
$978$		−	11405.5i		−	0.372913i
$979$	9215.62			0.300850
$980$	−78093.1			−2.54550
$981$		−	18680.7i		−	0.607981i
$982$		−	80375.1i		−	2.61189i
$983$	25454.0			0.825895			0.412948	−	0.910755i	$-0.364499\pi$
$983$	25454.0			0.825895			0.412948	+	0.910755i	$0.364499\pi$
$984$	4982.40			0.161416
$985$			11548.3i			0.373564i
$986$		−	32431.7i		−	1.04750i
$987$	375.601			0.0121130
$988$			122186.i			3.93447i
$989$	5918.66			0.190296
$990$			86656.1i			2.78193i
$991$		−	23080.3i		−	0.739829i	−0.929066	−	0.369914i	$-0.879387\pi$
$991$		−	23080.3i		−	0.739829i	0.929066	−	0.369914i	$-0.120613\pi$
$992$	−7931.93			−0.253870
$993$		−	3414.68i		−	0.109126i
$994$			34210.7i			1.09165i
$995$	3459.24			0.110216
$996$	−2992.98			−0.0952172
$997$	9046.47			0.287367			0.143683	−	0.989624i	$-0.454105\pi$
$997$	9046.47			0.287367			0.143683	+	0.989624i	$0.454105\pi$
$998$	4241.96			0.134546
$999$		−	4714.51i		−	0.149310i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	269.4.b.a.268.5	✓	66
269.268	even	2	inner	269.4.b.a.268.62	yes	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.4.b.a.268.5	✓	66	1.1	even	1	trivial
269.4.b.a.268.62	yes	66	269.268	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 269 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	269.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.09712i q^{2} +0.677318i q^{3} -17.9807 q^{4} +15.4092 q^{5} +3.45237 q^{6} -7.81953i q^{7} +50.8727i q^{8} +26.5412 q^{9} -78.5428i q^{10} -41.5692 q^{11} -12.1786i q^{12} +66.5794 q^{13} -39.8571 q^{14} +10.4370i q^{15} +115.459 q^{16} -108.686i q^{17} -135.284i q^{18} -102.065i q^{19} -277.068 q^{20} +5.29631 q^{21} +211.883i q^{22} -25.0644 q^{23} -34.4570 q^{24} +112.445 q^{25} -339.363i q^{26} +36.2644i q^{27} +140.600i q^{28} +58.5426i q^{29} +53.1984 q^{30} -43.6955i q^{31} -181.527i q^{32} -28.1556i q^{33} -553.984 q^{34} -120.493i q^{35} -477.229 q^{36} -130.004 q^{37} -520.237 q^{38} +45.0954i q^{39} +783.909i q^{40} -144.598 q^{41} -26.9959i q^{42} -236.138 q^{43} +747.442 q^{44} +408.980 q^{45} +127.756i q^{46} +70.9175 q^{47} +78.2024i q^{48} +281.855 q^{49} -573.144i q^{50} +73.6147 q^{51} -1197.14 q^{52} +579.852 q^{53} +184.844 q^{54} -640.550 q^{55} +397.800 q^{56} +69.1303 q^{57} +298.399 q^{58} +387.518i q^{59} -187.663i q^{60} +161.665 q^{61} -222.721 q^{62} -207.540i q^{63} -1.59520 q^{64} +1025.94 q^{65} -143.512 q^{66} -558.841 q^{67} +1954.24i q^{68} -16.9766i q^{69} -614.168 q^{70} -858.333i q^{71} +1350.22i q^{72} -13.2304 q^{73} +662.645i q^{74} +76.1607i q^{75} +1835.19i q^{76} +325.052i q^{77} +229.857 q^{78} -402.737 q^{79} +1779.14 q^{80} +692.051 q^{81} +737.033i q^{82} -245.757i q^{83} -95.2311 q^{84} -1674.76i q^{85} +1203.63i q^{86} -39.6520 q^{87} -2114.74i q^{88} -221.693 q^{89} -2084.62i q^{90} -520.619i q^{91} +450.675 q^{92} +29.5957 q^{93} -361.475i q^{94} -1572.74i q^{95} +122.952 q^{96} -1434.51 q^{97} -1436.65i q^{98} -1103.30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) \) 66 * q - 258 * q^4 - 34 * q^5 + 48 * q^6 - 564 * q^9 + 22 * q^11 + 118 * q^13 + 60 * q^14 + 1030 * q^16 + 144 * q^20 - 64 * q^21 - 80 * q^23 - 778 * q^24 + 1676 * q^25 - 1146 * q^30 + 308 * q^34 + 2030 * q^36 + 470 * q^37 + 696 * q^38 + 76 * q^41 + 426 * q^43 + 698 * q^44 + 1254 * q^45 - 1280 * q^47 - 3502 * q^49 - 640 * q^51 - 1924 * q^52 - 618 * q^53 - 3766 * q^54 + 176 * q^55 + 3164 * q^56 + 1668 * q^57 - 2170 * q^58 - 1026 * q^61 + 516 * q^62 - 2936 * q^64 - 900 * q^65 - 5296 * q^66 + 862 * q^67 - 1776 * q^70 + 3560 * q^73 + 8818 * q^78 + 2064 * q^79 - 3966 * q^80 + 7446 * q^81 - 3304 * q^84 - 776 * q^87 - 892 * q^89 - 6402 * q^92 + 7388 * q^93 + 8632 * q^96 + 4552 * q^97 - 3982 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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