LMFDB - Embedded newform 6025.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77437 q^{2} +2.60911 q^{3} +5.69713 q^{4} -7.23863 q^{6} +1.40482 q^{7} -10.2572 q^{8} +3.80744 q^{9} +O(q^{10})$	\(q-2.77437 q^{2} +2.60911 q^{3} +5.69713 q^{4} -7.23863 q^{6} +1.40482 q^{7} -10.2572 q^{8} +3.80744 q^{9} -4.51138 q^{11} +14.8644 q^{12} +3.66906 q^{13} -3.89749 q^{14} +17.0631 q^{16} +1.12193 q^{17} -10.5633 q^{18} -7.75667 q^{19} +3.66532 q^{21} +12.5163 q^{22} -8.71847 q^{23} -26.7622 q^{24} -10.1793 q^{26} +2.10670 q^{27} +8.00345 q^{28} +5.79899 q^{29} +10.1166 q^{31} -26.8248 q^{32} -11.7707 q^{33} -3.11264 q^{34} +21.6915 q^{36} +2.30957 q^{37} +21.5199 q^{38} +9.57298 q^{39} -1.43908 q^{41} -10.1690 q^{42} -9.39680 q^{43} -25.7020 q^{44} +24.1883 q^{46} -11.9309 q^{47} +44.5194 q^{48} -5.02648 q^{49} +2.92723 q^{51} +20.9031 q^{52} -9.40902 q^{53} -5.84476 q^{54} -14.4095 q^{56} -20.2380 q^{57} -16.0885 q^{58} +3.44128 q^{59} +5.18535 q^{61} -28.0673 q^{62} +5.34877 q^{63} +40.2959 q^{64} +32.6562 q^{66} -11.1682 q^{67} +6.39177 q^{68} -22.7474 q^{69} -1.58462 q^{71} -39.0538 q^{72} -7.27458 q^{73} -6.40760 q^{74} -44.1908 q^{76} -6.33768 q^{77} -26.5590 q^{78} +8.45345 q^{79} -5.92572 q^{81} +3.99254 q^{82} +15.5007 q^{83} +20.8818 q^{84} +26.0702 q^{86} +15.1302 q^{87} +46.2743 q^{88} -4.71787 q^{89} +5.15437 q^{91} -49.6703 q^{92} +26.3953 q^{93} +33.1008 q^{94} -69.9889 q^{96} -2.12320 q^{97} +13.9453 q^{98} -17.1768 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.77437	−1.96178	−0.980888	−	0.194572i	$-0.937668\pi$
$2$	−2.77437	−1.96178	−0.980888	+	0.194572i	$0.937668\pi$
$3$	2.60911	1.50637	0.753184	−	0.657809i	$-0.228517\pi$
$3$	2.60911	1.50637	0.753184	+	0.657809i	$0.228517\pi$
$4$	5.69713	2.84857
$5$	0	0
$5$	0	0
$6$	−7.23863	−2.95516
$7$	1.40482	0.530972	0.265486	−	0.964115i	$-0.414468\pi$
$7$	1.40482	0.530972	0.265486	+	0.964115i	$0.414468\pi$
$8$	−10.2572	−3.62648
$9$	3.80744	1.26915
$10$	0	0
$11$	−4.51138	−1.36023	−0.680117	−	0.733104i	$-0.738071\pi$
$11$	−4.51138	−1.36023	−0.680117	+	0.733104i	$0.738071\pi$
$12$	14.8644	4.29099
$13$	3.66906	1.01761	0.508807	−	0.860880i	$-0.330087\pi$
$13$	3.66906	1.01761	0.508807	+	0.860880i	$0.330087\pi$
$14$	−3.89749	−1.04165
$15$	0	0
$16$	17.0631	4.26577
$17$	1.12193	0.272107	0.136054	−	0.990701i	$-0.456558\pi$
$17$	1.12193	0.272107	0.136054	+	0.990701i	$0.456558\pi$
$18$	−10.5633	−2.48978
$19$	−7.75667	−1.77950	−0.889752	−	0.456445i	$-0.849122\pi$
$19$	−7.75667	−1.77950	−0.889752	+	0.456445i	$0.849122\pi$
$20$	0	0
$21$	3.66532	0.799839
$22$	12.5163	2.66847
$23$	−8.71847	−1.81793	−0.908964	−	0.416875i	$-0.863125\pi$
$23$	−8.71847	−1.81793	−0.908964	+	0.416875i	$0.863125\pi$
$24$	−26.7622	−5.46281
$25$	0	0
$26$	−10.1793	−1.99633
$27$	2.10670	0.405434
$28$	8.00345	1.51251
$29$	5.79899	1.07684	0.538422	−	0.842675i	$-0.319021\pi$
$29$	5.79899	1.07684	0.538422	+	0.842675i	$0.319021\pi$
$30$	0	0
$31$	10.1166	1.81700	0.908499	−	0.417886i	$-0.137229\pi$
$31$	10.1166	1.81700	0.908499	+	0.417886i	$0.137229\pi$
$32$	−26.8248	−4.74201
$33$	−11.7707	−2.04901
$34$	−3.11264	−0.533814
$35$	0	0
$36$	21.6915	3.61525
$37$	2.30957	0.379691	0.189845	−	0.981814i	$-0.439201\pi$
$37$	2.30957	0.379691	0.189845	+	0.981814i	$0.439201\pi$
$38$	21.5199	3.49099
$39$	9.57298	1.53290
$40$	0	0
$41$	−1.43908	−0.224747	−0.112373	−	0.993666i	$-0.535845\pi$
$41$	−1.43908	−0.224747	−0.112373	+	0.993666i	$0.535845\pi$
$42$	−10.1690	−1.56911
$43$	−9.39680	−1.43300	−0.716499	−	0.697588i	$-0.754257\pi$
$43$	−9.39680	−1.43300	−0.716499	+	0.697588i	$0.754257\pi$
$44$	−25.7020	−3.87472
$45$	0	0
$46$	24.1883	3.56637
$47$	−11.9309	−1.74030	−0.870152	−	0.492783i	$-0.835980\pi$
$47$	−11.9309	−1.74030	−0.870152	+	0.492783i	$0.835980\pi$
$48$	44.5194	6.42582
$49$	−5.02648	−0.718069
$50$	0	0
$51$	2.92723	0.409894
$52$	20.9031	2.89874
$53$	−9.40902	−1.29243	−0.646214	−	0.763156i	$-0.723649\pi$
$53$	−9.40902	−1.29243	−0.646214	+	0.763156i	$0.723649\pi$
$54$	−5.84476	−0.795371
$55$	0	0
$56$	−14.4095	−1.92556
$57$	−20.2380	−2.68059
$58$	−16.0885	−2.11253
$59$	3.44128	0.448017	0.224008	−	0.974587i	$-0.428086\pi$
$59$	3.44128	0.448017	0.224008	+	0.974587i	$0.428086\pi$
$60$	0	0
$61$	5.18535	0.663916	0.331958	−	0.943294i	$-0.392291\pi$
$61$	5.18535	0.663916	0.331958	+	0.943294i	$0.392291\pi$
$62$	−28.0673	−3.56455
$63$	5.34877	0.673881
$64$	40.2959	5.03699
$65$	0	0
$66$	32.6562	4.01971
$67$	−11.1682	−1.36441	−0.682207	−	0.731159i	$-0.738980\pi$
$67$	−11.1682	−1.36441	−0.682207	+	0.731159i	$0.738980\pi$
$68$	6.39177	0.775116
$69$	−22.7474	−2.73847
$70$	0	0
$71$	−1.58462	−0.188060	−0.0940298	−	0.995569i	$-0.529975\pi$
$71$	−1.58462	−0.188060	−0.0940298	+	0.995569i	$0.529975\pi$
$72$	−39.0538	−4.60253
$73$	−7.27458	−0.851425	−0.425713	−	0.904858i	$-0.639977\pi$
$73$	−7.27458	−0.851425	−0.425713	+	0.904858i	$0.639977\pi$
$74$	−6.40760	−0.744868
$75$	0	0
$76$	−44.1908	−5.06903
$77$	−6.33768	−0.722246
$78$	−26.5590	−3.00721
$79$	8.45345	0.951088	0.475544	−	0.879692i	$-0.342251\pi$
$79$	8.45345	0.951088	0.475544	+	0.879692i	$0.342251\pi$
$80$	0	0
$81$	−5.92572	−0.658413
$82$	3.99254	0.440903
$83$	15.5007	1.70142	0.850711	−	0.525634i	$-0.176172\pi$
$83$	15.5007	1.70142	0.850711	+	0.525634i	$0.176172\pi$
$84$	20.8818	2.27840
$85$	0	0
$86$	26.0702	2.81122
$87$	15.1302	1.62212
$88$	46.2743	4.93285
$89$	−4.71787	−0.500093	−0.250047	−	0.968234i	$-0.580446\pi$
$89$	−4.71787	−0.500093	−0.250047	+	0.968234i	$0.580446\pi$
$90$	0	0
$91$	5.15437	0.540325
$92$	−49.6703	−5.17849
$93$	26.3953	2.73707
$94$	33.1008	3.41409
$95$	0	0
$96$	−69.9889	−7.14321
$97$	−2.12320	−0.215579	−0.107789	−	0.994174i	$-0.534377\pi$
$97$	−2.12320	−0.215579	−0.107789	+	0.994174i	$0.534377\pi$
$98$	13.9453	1.40869
$99$	−17.1768	−1.72634
$100$	0	0
$101$	13.1898	1.31243	0.656217	−	0.754572i	$-0.272155\pi$
$101$	13.1898	1.31243	0.656217	+	0.754572i	$0.272155\pi$
$102$	−8.12122	−0.804121
$103$	1.59393	0.157055	0.0785275	−	0.996912i	$-0.474978\pi$
$103$	1.59393	0.157055	0.0785275	+	0.996912i	$0.474978\pi$
$104$	−37.6344	−3.69036
$105$	0	0
$106$	26.1041	2.53546
$107$	1.24824	0.120672	0.0603358	−	0.998178i	$-0.480783\pi$
$107$	1.24824	0.120672	0.0603358	+	0.998178i	$0.480783\pi$
$108$	12.0021	1.15491
$109$	−9.52637	−0.912461	−0.456231	−	0.889862i	$-0.650801\pi$
$109$	−9.52637	−0.912461	−0.456231	+	0.889862i	$0.650801\pi$
$110$	0	0
$111$	6.02591	0.571954
$112$	23.9705	2.26500
$113$	4.13600	0.389083	0.194541	−	0.980894i	$-0.437678\pi$
$113$	4.13600	0.389083	0.194541	+	0.980894i	$0.437678\pi$
$114$	56.1477	5.25871
$115$	0	0
$116$	33.0376	3.06746
$117$	13.9697	1.29150
$118$	−9.54740	−0.878909
$119$	1.57611	0.144481
$120$	0	0
$121$	9.35259	0.850236
$122$	−14.3861	−1.30245
$123$	−3.75472	−0.338551
$124$	57.6357	5.17584
$125$	0	0
$126$	−14.8395	−1.32200
$127$	5.84061	0.518270	0.259135	−	0.965841i	$-0.416562\pi$
$127$	5.84061	0.518270	0.259135	+	0.965841i	$0.416562\pi$
$128$	−58.1461	−5.13944
$129$	−24.5172	−2.15862
$130$	0	0
$131$	−5.54120	−0.484137	−0.242069	−	0.970259i	$-0.577826\pi$
$131$	−5.54120	−0.484137	−0.242069	+	0.970259i	$0.577826\pi$
$132$	−67.0592	−5.83675
$133$	−10.8967	−0.944866
$134$	30.9847	2.67667
$135$	0	0
$136$	−11.5079	−0.986791
$137$	−6.33449	−0.541192	−0.270596	−	0.962693i	$-0.587221\pi$
$137$	−6.33449	−0.541192	−0.270596	+	0.962693i	$0.587221\pi$
$138$	63.1098	5.37226
$139$	−13.5007	−1.14511	−0.572557	−	0.819865i	$-0.694048\pi$
$139$	−13.5007	−1.14511	−0.572557	+	0.819865i	$0.694048\pi$
$140$	0	0
$141$	−31.1291	−2.62154
$142$	4.39632	0.368931
$143$	−16.5526	−1.38419
$144$	64.9666	5.41388
$145$	0	0
$146$	20.1824	1.67031
$147$	−13.1146	−1.08168
$148$	13.1579	1.08157
$149$	−8.92532	−0.731191	−0.365595	−	0.930774i	$-0.619135\pi$
$149$	−8.92532	−0.731191	−0.365595	+	0.930774i	$0.619135\pi$
$150$	0	0
$151$	−7.57178	−0.616183	−0.308091	−	0.951357i	$-0.599690\pi$
$151$	−7.57178	−0.616183	−0.308091	+	0.951357i	$0.599690\pi$
$152$	79.5619	6.45332
$153$	4.27167	0.345344
$154$	17.5831	1.41688
$155$	0	0
$156$	54.5385	4.36658
$157$	9.53018	0.760591	0.380296	−	0.924865i	$-0.375822\pi$
$157$	9.53018	0.760591	0.380296	+	0.924865i	$0.375822\pi$
$158$	−23.4530	−1.86582
$159$	−24.5491	−1.94687
$160$	0	0
$161$	−12.2479	−0.965268
$162$	16.4401	1.29166
$163$	−17.8173	−1.39556	−0.697780	−	0.716313i	$-0.745829\pi$
$163$	−17.8173	−1.39556	−0.697780	+	0.716313i	$0.745829\pi$
$164$	−8.19864	−0.640206
$165$	0	0
$166$	−43.0047	−3.33781
$167$	0.755169	0.0584368	0.0292184	−	0.999573i	$-0.490698\pi$
$167$	0.755169	0.0584368	0.0292184	+	0.999573i	$0.490698\pi$
$168$	−37.5960	−2.90060
$169$	0.462023	0.0355402
$170$	0	0
$171$	−29.5331	−2.25845
$172$	−53.5348	−4.08199
$173$	−6.24390	−0.474715	−0.237357	−	0.971422i	$-0.576281\pi$
$173$	−6.24390	−0.474715	−0.237357	+	0.971422i	$0.576281\pi$
$174$	−41.9767	−3.18225
$175$	0	0
$176$	−76.9781	−5.80244
$177$	8.97868	0.674879
$178$	13.0891	0.981071
$179$	−19.3747	−1.44813	−0.724067	−	0.689729i	$-0.757730\pi$
$179$	−19.3747	−1.44813	−0.724067	+	0.689729i	$0.757730\pi$
$180$	0	0
$181$	3.26289	0.242529	0.121264	−	0.992620i	$-0.461305\pi$
$181$	3.26289	0.242529	0.121264	+	0.992620i	$0.461305\pi$
$182$	−14.3001	−1.06000
$183$	13.5291	1.00010
$184$	89.4273	6.59267
$185$	0	0
$186$	−73.2305	−5.36952
$187$	−5.06145	−0.370130
$188$	−67.9721	−4.95737
$189$	2.95953	0.215274
$190$	0	0
$191$	−2.87146	−0.207772	−0.103886	−	0.994589i	$-0.533128\pi$
$191$	−2.87146	−0.207772	−0.103886	+	0.994589i	$0.533128\pi$
$192$	105.136	7.58756
$193$	−20.1635	−1.45140	−0.725699	−	0.688012i	$-0.758484\pi$
$193$	−20.1635	−1.45140	−0.725699	+	0.688012i	$0.758484\pi$
$194$	5.89055	0.422917
$195$	0	0
$196$	−28.6365	−2.04547
$197$	8.24404	0.587364	0.293682	−	0.955903i	$-0.405119\pi$
$197$	8.24404	0.587364	0.293682	+	0.955903i	$0.405119\pi$
$198$	47.6549	3.38669
$199$	1.34342	0.0952328	0.0476164	−	0.998866i	$-0.484837\pi$
$199$	1.34342	0.0952328	0.0476164	+	0.998866i	$0.484837\pi$
$200$	0	0
$201$	−29.1390	−2.05531
$202$	−36.5934	−2.57470
$203$	8.14653	0.571774
$204$	16.6768	1.16761
$205$	0	0
$206$	−4.42216	−0.308107
$207$	−33.1951	−2.30722
$208$	62.6055	4.34091
$209$	34.9933	2.42054
$210$	0	0
$211$	15.6368	1.07648	0.538241	−	0.842791i	$-0.319089\pi$
$211$	15.6368	1.07648	0.538241	+	0.842791i	$0.319089\pi$
$212$	−53.6045	−3.68157
$213$	−4.13444	−0.283287
$214$	−3.46307	−0.236731
$215$	0	0
$216$	−21.6089	−1.47030
$217$	14.2120	0.964775
$218$	26.4297	1.79004
$219$	−18.9802	−1.28256
$220$	0	0
$221$	4.11642	0.276901
$222$	−16.7181	−1.12205
$223$	−27.2751	−1.82648	−0.913238	−	0.407425i	$-0.866427\pi$
$223$	−27.2751	−1.82648	−0.913238	+	0.407425i	$0.866427\pi$
$224$	−37.6841	−2.51787
$225$	0	0
$226$	−11.4748	−0.763293
$227$	−3.05952	−0.203068	−0.101534	−	0.994832i	$-0.532375\pi$
$227$	−3.05952	−0.203068	−0.101534	+	0.994832i	$0.532375\pi$
$228$	−115.299	−7.63583
$229$	7.97221	0.526818	0.263409	−	0.964684i	$-0.415153\pi$
$229$	7.97221	0.526818	0.263409	+	0.964684i	$0.415153\pi$
$230$	0	0
$231$	−16.5357	−1.08797
$232$	−59.4815	−3.90515
$233$	15.4752	1.01382	0.506908	−	0.862000i	$-0.330788\pi$
$233$	15.4752	1.01382	0.506908	+	0.862000i	$0.330788\pi$
$234$	−38.7572	−2.53364
$235$	0	0
$236$	19.6055	1.27621
$237$	22.0560	1.43269
$238$	−4.37270	−0.283440
$239$	−9.12379	−0.590169	−0.295084	−	0.955471i	$-0.595348\pi$
$239$	−9.12379	−0.590169	−0.295084	+	0.955471i	$0.595348\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−25.9476	−1.66797
$243$	−21.7809	−1.39725
$244$	29.5416	1.89121
$245$	0	0
$246$	10.4170	0.664162
$247$	−28.4597	−1.81085
$248$	−103.768	−6.58930
$249$	40.4430	2.56297
$250$	0	0
$251$	4.41082	0.278408	0.139204	−	0.990264i	$-0.455546\pi$
$251$	4.41082	0.278408	0.139204	+	0.990264i	$0.455546\pi$
$252$	30.4726	1.91960
$253$	39.3324	2.47281
$254$	−16.2040	−1.01673
$255$	0	0
$256$	80.7271	5.04545
$257$	−5.72655	−0.357212	−0.178606	−	0.983921i	$-0.557159\pi$
$257$	−5.72655	−0.357212	−0.178606	+	0.983921i	$0.557159\pi$
$258$	68.0199	4.23474
$259$	3.24453	0.201605
$260$	0	0
$261$	22.0793	1.36667
$262$	15.3734	0.949769
$263$	7.39073	0.455732	0.227866	−	0.973692i	$-0.426825\pi$
$263$	7.39073	0.455732	0.227866	+	0.973692i	$0.426825\pi$
$264$	120.735	7.43070
$265$	0	0
$266$	30.2316	1.85362
$267$	−12.3094	−0.753325
$268$	−63.6268	−3.88662
$269$	5.21758	0.318121	0.159061	−	0.987269i	$-0.449153\pi$
$269$	5.21758	0.318121	0.159061	+	0.987269i	$0.449153\pi$
$270$	0	0
$271$	17.9178	1.08843	0.544214	−	0.838946i	$-0.316828\pi$
$271$	17.9178	1.08843	0.544214	+	0.838946i	$0.316828\pi$
$272$	19.1435	1.16075
$273$	13.4483	0.813929
$274$	17.5742	1.06170
$275$	0	0
$276$	−129.595	−7.80071
$277$	13.5441	0.813783	0.406892	−	0.913476i	$-0.366613\pi$
$277$	13.5441	0.813783	0.406892	+	0.913476i	$0.366613\pi$
$278$	37.4559	2.24646
$279$	38.5184	2.30604
$280$	0	0
$281$	−10.5123	−0.627110	−0.313555	−	0.949570i	$-0.601520\pi$
$281$	−10.5123	−0.627110	−0.313555	+	0.949570i	$0.601520\pi$
$282$	86.3636	5.14288
$283$	−15.3785	−0.914159	−0.457080	−	0.889426i	$-0.651105\pi$
$283$	−15.3785	−0.914159	−0.457080	+	0.889426i	$0.651105\pi$
$284$	−9.02778	−0.535700
$285$	0	0
$286$	45.9229	2.71548
$287$	−2.02165	−0.119334
$288$	−102.134	−6.01830
$289$	−15.7413	−0.925958
$290$	0	0
$291$	−5.53967	−0.324741
$292$	−41.4443	−2.42534
$293$	29.5610	1.72697	0.863485	−	0.504375i	$-0.168277\pi$
$293$	29.5610	1.72697	0.863485	+	0.504375i	$0.168277\pi$
$294$	36.3849	2.12201
$295$	0	0
$296$	−23.6898	−1.37694
$297$	−9.50412	−0.551485
$298$	24.7622	1.43443
$299$	−31.9886	−1.84995
$300$	0	0
$301$	−13.2008	−0.760882
$302$	21.0069	1.20881
$303$	34.4136	1.97701
$304$	−132.353	−7.59095
$305$	0	0
$306$	−11.8512	−0.677488
$307$	−26.5223	−1.51371	−0.756853	−	0.653585i	$-0.773264\pi$
$307$	−26.5223	−1.51371	−0.756853	+	0.653585i	$0.773264\pi$
$308$	−36.1066	−2.05737
$309$	4.15874	0.236583
$310$	0	0
$311$	−1.84649	−0.104705	−0.0523523	−	0.998629i	$-0.516672\pi$
$311$	−1.84649	−0.104705	−0.0523523	+	0.998629i	$0.516672\pi$
$312$	−98.1922	−5.55904
$313$	15.3372	0.866909	0.433455	−	0.901175i	$-0.357294\pi$
$313$	15.3372	0.866909	0.433455	+	0.901175i	$0.357294\pi$
$314$	−26.4403	−1.49211
$315$	0	0
$316$	48.1605	2.70924
$317$	−21.2001	−1.19072	−0.595359	−	0.803460i	$-0.702990\pi$
$317$	−21.2001	−1.19072	−0.595359	+	0.803460i	$0.702990\pi$
$318$	68.1084	3.81933
$319$	−26.1615	−1.46476
$320$	0	0
$321$	3.25679	0.181776
$322$	33.9802	1.89364
$323$	−8.70243	−0.484216
$324$	−33.7596	−1.87553
$325$	0	0
$326$	49.4318	2.73778
$327$	−24.8553	−1.37450
$328$	14.7610	0.815038
$329$	−16.7608	−0.924053
$330$	0	0
$331$	2.51603	0.138294	0.0691468	−	0.997606i	$-0.477972\pi$
$331$	2.51603	0.138294	0.0691468	+	0.997606i	$0.477972\pi$
$332$	88.3095	4.84661
$333$	8.79354	0.481883
$334$	−2.09512	−0.114640
$335$	0	0
$336$	62.5417	3.41193
$337$	−23.3415	−1.27149	−0.635747	−	0.771897i	$-0.719308\pi$
$337$	−23.3415	−1.27149	−0.635747	+	0.771897i	$0.719308\pi$
$338$	−1.28182	−0.0697220
$339$	10.7913	0.586102
$340$	0	0
$341$	−45.6400	−2.47154
$342$	81.9357	4.43058
$343$	−16.8950	−0.912246
$344$	96.3850	5.19673
$345$	0	0
$346$	17.3229	0.931284
$347$	−11.2110	−0.601839	−0.300920	−	0.953649i	$-0.597294\pi$
$347$	−11.2110	−0.601839	−0.300920	+	0.953649i	$0.597294\pi$
$348$	86.1986	4.62073
$349$	−4.23771	−0.226839	−0.113420	−	0.993547i	$-0.536180\pi$
$349$	−4.23771	−0.226839	−0.113420	+	0.993547i	$0.536180\pi$
$350$	0	0
$351$	7.72961	0.412576
$352$	121.017	6.45024
$353$	−2.36983	−0.126133	−0.0630667	−	0.998009i	$-0.520088\pi$
$353$	−2.36983	−0.126133	−0.0630667	+	0.998009i	$0.520088\pi$
$354$	−24.9102	−1.32396
$355$	0	0
$356$	−26.8783	−1.42455
$357$	4.11223	0.217642
$358$	53.7527	2.84092
$359$	19.2257	1.01469	0.507346	−	0.861742i	$-0.330627\pi$
$359$	19.2257	1.01469	0.507346	+	0.861742i	$0.330627\pi$
$360$	0	0
$361$	41.1660	2.16663
$362$	−9.05247	−0.475787
$363$	24.4019	1.28077
$364$	29.3651	1.53915
$365$	0	0
$366$	−37.5348	−1.96198
$367$	25.8754	1.35069	0.675343	−	0.737504i	$-0.263996\pi$
$367$	25.8754	1.35069	0.675343	+	0.737504i	$0.263996\pi$
$368$	−148.764	−7.75486
$369$	−5.47922	−0.285237
$370$	0	0
$371$	−13.2180	−0.686243
$372$	150.378	7.79673
$373$	20.5605	1.06458	0.532291	−	0.846561i	$-0.321331\pi$
$373$	20.5605	1.06458	0.532291	+	0.846561i	$0.321331\pi$
$374$	14.0423	0.726112
$375$	0	0
$376$	122.378	6.31117
$377$	21.2768	1.09581
$378$	−8.21083	−0.422320
$379$	4.90733	0.252072	0.126036	−	0.992026i	$-0.459774\pi$
$379$	4.90733	0.252072	0.126036	+	0.992026i	$0.459774\pi$
$380$	0	0
$381$	15.2388	0.780706
$382$	7.96650	0.407602
$383$	−1.24988	−0.0638658	−0.0319329	−	0.999490i	$-0.510166\pi$
$383$	−1.24988	−0.0638658	−0.0319329	+	0.999490i	$0.510166\pi$
$384$	−151.710	−7.74189
$385$	0	0
$386$	55.9409	2.84732
$387$	−35.7777	−1.81868
$388$	−12.0962	−0.614090
$389$	10.5629	0.535558	0.267779	−	0.963480i	$-0.413710\pi$
$389$	10.5629	0.535558	0.267779	+	0.963480i	$0.413710\pi$
$390$	0	0
$391$	−9.78150	−0.494671
$392$	51.5577	2.60406
$393$	−14.4576	−0.729289
$394$	−22.8720	−1.15228
$395$	0	0
$396$	−97.8587	−4.91758
$397$	1.70564	0.0856036	0.0428018	−	0.999084i	$-0.486372\pi$
$397$	1.70564	0.0856036	0.0428018	+	0.999084i	$0.486372\pi$
$398$	−3.72716	−0.186826
$399$	−28.4307	−1.42332
$400$	0	0
$401$	−1.51346	−0.0755785	−0.0377893	−	0.999286i	$-0.512032\pi$
$401$	−1.51346	−0.0755785	−0.0377893	+	0.999286i	$0.512032\pi$
$402$	80.8425	4.03206
$403$	37.1185	1.84901
$404$	75.1441	3.73856
$405$	0	0
$406$	−22.6015	−1.12169
$407$	−10.4193	−0.516468
$408$	−30.0252	−1.48647
$409$	−23.2510	−1.14969	−0.574844	−	0.818263i	$-0.694937\pi$
$409$	−23.2510	−1.14969	−0.574844	+	0.818263i	$0.694937\pi$
$410$	0	0
$411$	−16.5274	−0.815235
$412$	9.08085	0.447382
$413$	4.83438	0.237884
$414$	92.0954	4.52624
$415$	0	0
$416$	−98.4220	−4.82554
$417$	−35.2248	−1.72496
$418$	−97.0845	−4.74856
$419$	−21.9820	−1.07389	−0.536945	−	0.843617i	$-0.680422\pi$
$419$	−21.9820	−1.07389	−0.536945	+	0.843617i	$0.680422\pi$
$420$	0	0
$421$	−8.55661	−0.417024	−0.208512	−	0.978020i	$-0.566862\pi$
$421$	−8.55661	−0.417024	−0.208512	+	0.978020i	$0.566862\pi$
$422$	−43.3823	−2.11182
$423$	−45.4263	−2.20870
$424$	96.5104	4.68696
$425$	0	0
$426$	11.4705	0.555746
$427$	7.28448	0.352520
$428$	7.11138	0.343741
$429$	−43.1874	−2.08511
$430$	0	0
$431$	23.8701	1.14978	0.574890	−	0.818230i	$-0.305045\pi$
$431$	23.8701	1.14978	0.574890	+	0.818230i	$0.305045\pi$
$432$	35.9467	1.72949
$433$	−27.1136	−1.30300	−0.651498	−	0.758650i	$-0.725859\pi$
$433$	−27.1136	−1.30300	−0.651498	+	0.758650i	$0.725859\pi$
$434$	−39.4294	−1.89267
$435$	0	0
$436$	−54.2730	−2.59921
$437$	67.6264	3.23501
$438$	52.6580	2.51610
$439$	22.5446	1.07600	0.537998	−	0.842946i	$-0.319181\pi$
$439$	22.5446	1.07600	0.537998	+	0.842946i	$0.319181\pi$
$440$	0	0
$441$	−19.1380	−0.911335
$442$	−11.4205	−0.543217
$443$	6.72035	0.319293	0.159647	−	0.987174i	$-0.448965\pi$
$443$	6.72035	0.319293	0.159647	+	0.987174i	$0.448965\pi$
$444$	34.3304	1.62925
$445$	0	0
$446$	75.6713	3.58314
$447$	−23.2871	−1.10144
$448$	56.6085	2.67450
$449$	11.3630	0.536253	0.268126	−	0.963384i	$-0.413596\pi$
$449$	11.3630	0.536253	0.268126	+	0.963384i	$0.413596\pi$
$450$	0	0
$451$	6.49225	0.305708
$452$	23.5634	1.10833
$453$	−19.7556	−0.928198
$454$	8.48825	0.398373
$455$	0	0
$456$	207.586	9.72109
$457$	15.3462	0.717866	0.358933	−	0.933363i	$-0.383141\pi$
$457$	15.3462	0.717866	0.358933	+	0.933363i	$0.383141\pi$
$458$	−22.1179	−1.03350
$459$	2.36356	0.110322
$460$	0	0
$461$	24.0307	1.11922	0.559610	−	0.828756i	$-0.310951\pi$
$461$	24.0307	1.11922	0.559610	+	0.828756i	$0.310951\pi$
$462$	45.8761	2.13435
$463$	12.5967	0.585420	0.292710	−	0.956201i	$-0.405443\pi$
$463$	12.5967	0.585420	0.292710	+	0.956201i	$0.405443\pi$
$464$	98.9485	4.59357
$465$	0	0
$466$	−42.9340	−1.98888
$467$	−23.9912	−1.11018	−0.555090	−	0.831790i	$-0.687316\pi$
$467$	−23.9912	−1.11018	−0.555090	+	0.831790i	$0.687316\pi$
$468$	79.5875	3.67893
$469$	−15.6893	−0.724465
$470$	0	0
$471$	24.8653	1.14573
$472$	−35.2980	−1.62472
$473$	42.3926	1.94921
$474$	−61.1914	−2.81062
$475$	0	0
$476$	8.97929	0.411565
$477$	−35.8243	−1.64028
$478$	25.3128	1.15778
$479$	38.1664	1.74387	0.871933	−	0.489624i	$-0.162866\pi$
$479$	38.1664	1.74387	0.871933	+	0.489624i	$0.162866\pi$
$480$	0	0
$481$	8.47395	0.386379
$482$	2.77437	0.126369
$483$	−31.9560	−1.45405
$484$	53.2830	2.42195
$485$	0	0
$486$	60.4284	2.74109
$487$	−40.8403	−1.85065	−0.925326	−	0.379173i	$-0.876208\pi$
$487$	−40.8403	−1.85065	−0.925326	+	0.379173i	$0.876208\pi$
$488$	−53.1872	−2.40767
$489$	−46.4873	−2.10223
$490$	0	0
$491$	−13.4813	−0.608401	−0.304201	−	0.952608i	$-0.598389\pi$
$491$	−13.4813	−0.608401	−0.304201	+	0.952608i	$0.598389\pi$
$492$	−21.3911	−0.964386
$493$	6.50604	0.293017
$494$	78.9578	3.55248
$495$	0	0
$496$	172.621	7.75089
$497$	−2.22610	−0.0998543
$498$	−112.204	−5.02797
$499$	10.2923	0.460749	0.230374	−	0.973102i	$-0.426005\pi$
$499$	10.2923	0.460749	0.230374	+	0.973102i	$0.426005\pi$
$500$	0	0
$501$	1.97032	0.0880273
$502$	−12.2372	−0.546175
$503$	2.20458	0.0982976	0.0491488	−	0.998791i	$-0.484349\pi$
$503$	2.20458	0.0982976	0.0491488	+	0.998791i	$0.484349\pi$
$504$	−54.8635	−2.44381
$505$	0	0
$506$	−109.123	−4.85109
$507$	1.20547	0.0535367
$508$	33.2747	1.47633
$509$	−31.1063	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$509$	−31.1063	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$510$	0	0
$511$	−10.2195	−0.452083
$512$	−107.675	−4.75860
$513$	−16.3410	−0.721471
$514$	15.8876	0.700770
$515$	0	0
$516$	−139.678	−6.14898
$517$	53.8250	2.36722
$518$	−9.00152	−0.395504
$519$	−16.2910	−0.715095
$520$	0	0
$521$	36.8506	1.61446	0.807228	−	0.590240i	$-0.200967\pi$
$521$	36.8506	1.61446	0.807228	+	0.590240i	$0.200967\pi$
$522$	−61.2561	−2.68111
$523$	−11.5309	−0.504210	−0.252105	−	0.967700i	$-0.581123\pi$
$523$	−11.5309	−0.504210	−0.252105	+	0.967700i	$0.581123\pi$
$524$	−31.5690	−1.37910
$525$	0	0
$526$	−20.5046	−0.894045
$527$	11.3501	0.494419
$528$	−200.844	−8.74061
$529$	53.0118	2.30486
$530$	0	0
$531$	13.1025	0.568599
$532$	−62.0801	−2.69151
$533$	−5.28008	−0.228706
$534$	34.1509	1.47785
$535$	0	0
$536$	114.555	4.94801
$537$	−50.5507	−2.18142
$538$	−14.4755	−0.624083
$539$	22.6764	0.976741
$540$	0	0
$541$	9.23870	0.397203	0.198601	−	0.980080i	$-0.436360\pi$
$541$	9.23870	0.397203	0.198601	+	0.980080i	$0.436360\pi$
$542$	−49.7106	−2.13525
$543$	8.51323	0.365338
$544$	−30.0955	−1.29034
$545$	0	0
$546$	−37.3106	−1.59675
$547$	3.43754	0.146979	0.0734894	−	0.997296i	$-0.476587\pi$
$547$	3.43754	0.146979	0.0734894	+	0.997296i	$0.476587\pi$
$548$	−36.0885	−1.54162
$549$	19.7429	0.842606
$550$	0	0
$551$	−44.9808	−1.91625
$552$	233.325	9.93099
$553$	11.8756	0.505001
$554$	−37.5762	−1.59646
$555$	0	0
$556$	−76.9153	−3.26193
$557$	6.43272	0.272563	0.136282	−	0.990670i	$-0.456485\pi$
$557$	6.43272	0.272563	0.136282	+	0.990670i	$0.456485\pi$
$558$	−106.864	−4.52393
$559$	−34.4774	−1.45824
$560$	0	0
$561$	−13.2059	−0.557552
$562$	29.1649	1.23025
$563$	12.6443	0.532892	0.266446	−	0.963850i	$-0.414151\pi$
$563$	12.6443	0.532892	0.266446	+	0.963850i	$0.414151\pi$
$564$	−177.346	−7.46763
$565$	0	0
$566$	42.6658	1.79338
$567$	−8.32457	−0.349599
$568$	16.2538	0.681993
$569$	−31.6191	−1.32554	−0.662770	−	0.748823i	$-0.730619\pi$
$569$	−31.6191	−1.32554	−0.662770	+	0.748823i	$0.730619\pi$
$570$	0	0
$571$	17.2069	0.720088	0.360044	−	0.932935i	$-0.382762\pi$
$571$	17.2069	0.720088	0.360044	+	0.932935i	$0.382762\pi$
$572$	−94.3021	−3.94297
$573$	−7.49196	−0.312981
$574$	5.60880	0.234107
$575$	0	0
$576$	153.424	6.39268
$577$	13.3595	0.556165	0.278083	−	0.960557i	$-0.410301\pi$
$577$	13.3595	0.556165	0.278083	+	0.960557i	$0.410301\pi$
$578$	43.6721	1.81652
$579$	−52.6087	−2.18634
$580$	0	0
$581$	21.7757	0.903407
$582$	15.3691	0.637069
$583$	42.4477	1.75800
$584$	74.6170	3.08767
$585$	0	0
$586$	−82.0131	−3.38793
$587$	−25.5450	−1.05435	−0.527177	−	0.849756i	$-0.676749\pi$
$587$	−25.5450	−1.05435	−0.527177	+	0.849756i	$0.676749\pi$
$588$	−74.7158	−3.08123
$589$	−78.4713	−3.23335
$590$	0	0
$591$	21.5096	0.884786
$592$	39.4083	1.61967
$593$	8.83290	0.362724	0.181362	−	0.983416i	$-0.441949\pi$
$593$	8.83290	0.362724	0.181362	+	0.983416i	$0.441949\pi$
$594$	26.3680	1.08189
$595$	0	0
$596$	−50.8488	−2.08285
$597$	3.50514	0.143456
$598$	88.7483	3.62919
$599$	35.4129	1.44693	0.723466	−	0.690360i	$-0.242548\pi$
$599$	35.4129	1.44693	0.723466	+	0.690360i	$0.242548\pi$
$600$	0	0
$601$	−13.9495	−0.569013	−0.284506	−	0.958674i	$-0.591830\pi$
$601$	−13.9495	−0.569013	−0.284506	+	0.958674i	$0.591830\pi$
$602$	36.6239	1.49268
$603$	−42.5223	−1.73164
$604$	−43.1375	−1.75524
$605$	0	0
$606$	−95.4762	−3.87845
$607$	−12.9653	−0.526246	−0.263123	−	0.964762i	$-0.584753\pi$
$607$	−12.9653	−0.526246	−0.263123	+	0.964762i	$0.584753\pi$
$608$	208.072	8.43842
$609$	21.2552	0.861303
$610$	0	0
$611$	−43.7753	−1.77096
$612$	24.3363	0.983736
$613$	31.6071	1.27660	0.638300	−	0.769788i	$-0.279638\pi$
$613$	31.6071	1.27660	0.638300	+	0.769788i	$0.279638\pi$
$614$	73.5826	2.96955
$615$	0	0
$616$	65.0070	2.61921
$617$	−13.4667	−0.542149	−0.271074	−	0.962558i	$-0.587379\pi$
$617$	−13.4667	−0.542149	−0.271074	+	0.962558i	$0.587379\pi$
$618$	−11.5379	−0.464122
$619$	−28.7042	−1.15372	−0.576860	−	0.816843i	$-0.695722\pi$
$619$	−28.7042	−1.15372	−0.576860	+	0.816843i	$0.695722\pi$
$620$	0	0
$621$	−18.3672	−0.737050
$622$	5.12283	0.205407
$623$	−6.62776	−0.265535
$624$	163.344	6.53901
$625$	0	0
$626$	−42.5511	−1.70068
$627$	91.3014	3.64623
$628$	54.2947	2.16660
$629$	2.59117	0.103317
$630$	0	0
$631$	27.6794	1.10190	0.550949	−	0.834539i	$-0.314266\pi$
$631$	27.6794	1.10190	0.550949	+	0.834539i	$0.314266\pi$
$632$	−86.7089	−3.44910
$633$	40.7981	1.62158
$634$	58.8171	2.33592
$635$	0	0
$636$	−139.860	−5.54580
$637$	−18.4425	−0.730718
$638$	72.5816	2.87353
$639$	−6.03334	−0.238675
$640$	0	0
$641$	28.9121	1.14196	0.570979	−	0.820964i	$-0.306564\pi$
$641$	28.9121	1.14196	0.570979	+	0.820964i	$0.306564\pi$
$642$	−9.03553	−0.356604
$643$	−32.5884	−1.28516	−0.642580	−	0.766218i	$-0.722136\pi$
$643$	−32.5884	−1.28516	−0.642580	+	0.766218i	$0.722136\pi$
$644$	−69.7778	−2.74963
$645$	0	0
$646$	24.1438	0.949924
$647$	25.6153	1.00704	0.503521	−	0.863983i	$-0.332038\pi$
$647$	25.6153	1.00704	0.503521	+	0.863983i	$0.332038\pi$
$648$	60.7814	2.38772
$649$	−15.5250	−0.609408
$650$	0	0
$651$	37.0807	1.45331
$652$	−101.508	−3.97534
$653$	29.2629	1.14515	0.572573	−	0.819854i	$-0.305945\pi$
$653$	29.2629	1.14515	0.572573	+	0.819854i	$0.305945\pi$
$654$	68.9579	2.69647
$655$	0	0
$656$	−24.5551	−0.958717
$657$	−27.6975	−1.08058
$658$	46.5007	1.81279
$659$	−22.1103	−0.861296	−0.430648	−	0.902520i	$-0.641715\pi$
$659$	−22.1103	−0.861296	−0.430648	+	0.902520i	$0.641715\pi$
$660$	0	0
$661$	−27.5546	−1.07175	−0.535876	−	0.844297i	$-0.680018\pi$
$661$	−27.5546	−1.07175	−0.535876	+	0.844297i	$0.680018\pi$
$662$	−6.98040	−0.271301
$663$	10.7402	0.417114
$664$	−158.994	−6.17016
$665$	0	0
$666$	−24.3965	−0.945347
$667$	−50.5583	−1.95763
$668$	4.30230	0.166461
$669$	−71.1637	−2.75135
$670$	0	0
$671$	−23.3931	−0.903080
$672$	−98.3217	−3.79284
$673$	28.6939	1.10607	0.553035	−	0.833158i	$-0.313470\pi$
$673$	28.6939	1.10607	0.553035	+	0.833158i	$0.313470\pi$
$674$	64.7581	2.49439
$675$	0	0
$676$	2.63221	0.101239
$677$	22.2281	0.854297	0.427149	−	0.904181i	$-0.359518\pi$
$677$	22.2281	0.854297	0.427149	+	0.904181i	$0.359518\pi$
$678$	−29.9390	−1.14980
$679$	−2.98272	−0.114466
$680$	0	0
$681$	−7.98262	−0.305895
$682$	126.622	4.84861
$683$	−43.8691	−1.67861	−0.839303	−	0.543664i	$-0.817037\pi$
$683$	−43.8691	−1.67861	−0.839303	+	0.543664i	$0.817037\pi$
$684$	−168.254	−6.43335
$685$	0	0
$686$	46.8731	1.78962
$687$	20.8003	0.793582
$688$	−160.338	−6.11284
$689$	−34.5223	−1.31519
$690$	0	0
$691$	−3.17108	−0.120633	−0.0603167	−	0.998179i	$-0.519211\pi$
$691$	−3.17108	−0.120633	−0.0603167	+	0.998179i	$0.519211\pi$
$692$	−35.5723	−1.35226
$693$	−24.1303	−0.916636
$694$	31.1035	1.18067
$695$	0	0
$696$	−155.194	−5.88260
$697$	−1.61454	−0.0611552
$698$	11.7570	0.445008
$699$	40.3765	1.52718
$700$	0	0
$701$	−29.2671	−1.10540	−0.552701	−	0.833379i	$-0.686403\pi$
$701$	−29.2671	−1.10540	−0.552701	+	0.833379i	$0.686403\pi$
$702$	−21.4448	−0.809382
$703$	−17.9146	−0.675661
$704$	−181.790	−6.85148
$705$	0	0
$706$	6.57479	0.247446
$707$	18.5293	0.696866
$708$	51.1527	1.92244
$709$	28.6054	1.07430	0.537149	−	0.843487i	$-0.319501\pi$
$709$	28.6054	1.07430	0.537149	+	0.843487i	$0.319501\pi$
$710$	0	0
$711$	32.1860	1.20707
$712$	48.3922	1.81358
$713$	−88.2015	−3.30317
$714$	−11.4088	−0.426965
$715$	0	0
$716$	−110.380	−4.12511
$717$	−23.8049	−0.889012
$718$	−53.3392	−1.99060
$719$	−36.9687	−1.37870	−0.689351	−	0.724428i	$-0.742104\pi$
$719$	−36.9687	−1.37870	−0.689351	+	0.724428i	$0.742104\pi$
$720$	0	0
$721$	2.23919	0.0833918
$722$	−114.210	−4.25045
$723$	−2.60911	−0.0970337
$724$	18.5891	0.690859
$725$	0	0
$726$	−67.7000	−2.51258
$727$	−19.1660	−0.710829	−0.355415	−	0.934709i	$-0.615660\pi$
$727$	−19.1660	−0.710829	−0.355415	+	0.934709i	$0.615660\pi$
$728$	−52.8695	−1.95948
$729$	−39.0516	−1.44636
$730$	0	0
$731$	−10.5425	−0.389929
$732$	77.0772	2.84886
$733$	16.7119	0.617267	0.308634	−	0.951181i	$-0.400128\pi$
$733$	16.7119	0.617267	0.308634	+	0.951181i	$0.400128\pi$
$734$	−71.7880	−2.64974
$735$	0	0
$736$	233.872	8.62062
$737$	50.3841	1.85592
$738$	15.2014	0.559570
$739$	−33.0708	−1.21653	−0.608265	−	0.793734i	$-0.708134\pi$
$739$	−33.0708	−1.21653	−0.608265	+	0.793734i	$0.708134\pi$
$740$	0	0
$741$	−74.2545	−2.72781
$742$	36.6716	1.34626
$743$	−22.4218	−0.822575	−0.411287	−	0.911506i	$-0.634921\pi$
$743$	−22.4218	−0.822575	−0.411287	+	0.911506i	$0.634921\pi$
$744$	−270.743	−9.92592
$745$	0	0
$746$	−57.0425	−2.08847
$747$	59.0180	2.15935
$748$	−28.8357	−1.05434
$749$	1.75355	0.0640733
$750$	0	0
$751$	−20.6109	−0.752101	−0.376050	−	0.926599i	$-0.622718\pi$
$751$	−20.6109	−0.752101	−0.376050	+	0.926599i	$0.622718\pi$
$752$	−203.578	−7.42374
$753$	11.5083	0.419386
$754$	−59.0299	−2.14974
$755$	0	0
$756$	16.8608	0.613223
$757$	43.4293	1.57846	0.789232	−	0.614095i	$-0.210479\pi$
$757$	43.4293	1.57846	0.789232	+	0.614095i	$0.210479\pi$
$758$	−13.6147	−0.494510
$759$	102.622	3.72496
$760$	0	0
$761$	−34.1498	−1.23793	−0.618965	−	0.785418i	$-0.712448\pi$
$761$	−34.1498	−1.23793	−0.618965	+	0.785418i	$0.712448\pi$
$762$	−42.2780	−1.53157
$763$	−13.3828	−0.484491
$764$	−16.3591	−0.591852
$765$	0	0
$766$	3.46763	0.125290
$767$	12.6263	0.455909
$768$	210.626	7.60030
$769$	35.9716	1.29717	0.648584	−	0.761143i	$-0.275362\pi$
$769$	35.9716	1.29717	0.648584	+	0.761143i	$0.275362\pi$
$770$	0	0
$771$	−14.9412	−0.538093
$772$	−114.874	−4.13441
$773$	40.7746	1.46656	0.733281	−	0.679926i	$-0.237988\pi$
$773$	40.7746	1.46656	0.733281	+	0.679926i	$0.237988\pi$
$774$	99.2607	3.56785
$775$	0	0
$776$	21.7782	0.781791
$777$	8.46532	0.303692
$778$	−29.3053	−1.05065
$779$	11.1625	0.399937
$780$	0	0
$781$	7.14882	0.255805
$782$	27.1375	0.970435
$783$	12.2167	0.436590
$784$	−85.7672	−3.06311
$785$	0	0
$786$	40.1107	1.43070
$787$	50.9208	1.81513	0.907565	−	0.419911i	$-0.137939\pi$
$787$	50.9208	1.81513	0.907565	+	0.419911i	$0.137939\pi$
$788$	46.9674	1.67314
$789$	19.2832	0.686501
$790$	0	0
$791$	5.81034	0.206592
$792$	176.187	6.26052
$793$	19.0254	0.675610
$794$	−4.73208	−0.167935
$795$	0	0
$796$	7.65367	0.271277
$797$	−42.9083	−1.51989	−0.759945	−	0.649988i	$-0.774774\pi$
$797$	−42.9083	−1.51989	−0.759945	+	0.649988i	$0.774774\pi$
$798$	78.8774	2.79223
$799$	−13.3856	−0.473550
$800$	0	0
$801$	−17.9630	−0.634692
$802$	4.19890	0.148268
$803$	32.8184	1.15814
$804$	−166.009	−5.85469
$805$	0	0
$806$	−102.981	−3.62733
$807$	13.6132	0.479208
$808$	−135.291	−4.75951
$809$	−30.1367	−1.05955	−0.529774	−	0.848139i	$-0.677723\pi$
$809$	−30.1367	−1.05955	−0.529774	+	0.848139i	$0.677723\pi$
$810$	0	0
$811$	32.0269	1.12462	0.562309	−	0.826927i	$-0.309913\pi$
$811$	32.0269	1.12462	0.562309	+	0.826927i	$0.309913\pi$
$812$	46.4119	1.62874
$813$	46.7495	1.63957
$814$	28.9071	1.01319
$815$	0	0
$816$	49.9475	1.74851
$817$	72.8879	2.55002
$818$	64.5069	2.25543
$819$	19.6250	0.685752
$820$	0	0
$821$	−18.6121	−0.649568	−0.324784	−	0.945788i	$-0.605292\pi$
$821$	−18.6121	−0.649568	−0.324784	+	0.945788i	$0.605292\pi$
$822$	45.8531	1.59931
$823$	30.4278	1.06065	0.530323	−	0.847796i	$-0.322071\pi$
$823$	30.4278	1.06065	0.530323	+	0.847796i	$0.322071\pi$
$824$	−16.3493	−0.569556
$825$	0	0
$826$	−13.4124	−0.466676
$827$	40.2222	1.39866	0.699332	−	0.714797i	$-0.253481\pi$
$827$	40.2222	1.39866	0.699332	+	0.714797i	$0.253481\pi$
$828$	−189.117	−6.57226
$829$	−40.6524	−1.41192	−0.705958	−	0.708254i	$-0.749483\pi$
$829$	−40.6524	−1.41192	−0.705958	+	0.708254i	$0.749483\pi$
$830$	0	0
$831$	35.3379	1.22586
$832$	147.848	5.12572
$833$	−5.63935	−0.195392
$834$	97.7265	3.38399
$835$	0	0
$836$	199.362	6.89507
$837$	21.3127	0.736673
$838$	60.9862	2.10673
$839$	14.7504	0.509241	0.254621	−	0.967041i	$-0.418049\pi$
$839$	14.7504	0.509241	0.254621	+	0.967041i	$0.418049\pi$
$840$	0	0
$841$	4.62823	0.159594
$842$	23.7392	0.818107
$843$	−27.4276	−0.944658
$844$	89.0850	3.06643
$845$	0	0
$846$	126.029	4.33298
$847$	13.1387	0.451451
$848$	−160.547	−5.51320
$849$	−40.1242	−1.37706
$850$	0	0
$851$	−20.1359	−0.690250
$852$	−23.5545	−0.806962
$853$	−1.58063	−0.0541196	−0.0270598	−	0.999634i	$-0.508614\pi$
$853$	−1.58063	−0.0541196	−0.0270598	+	0.999634i	$0.508614\pi$
$854$	−20.2098	−0.691566
$855$	0	0
$856$	−12.8034	−0.437613
$857$	−46.0644	−1.57353	−0.786765	−	0.617253i	$-0.788245\pi$
$857$	−46.0644	−1.57353	−0.786765	+	0.617253i	$0.788245\pi$
$858$	119.818	4.09051
$859$	−8.26945	−0.282150	−0.141075	−	0.989999i	$-0.545056\pi$
$859$	−8.26945	−0.282150	−0.141075	+	0.989999i	$0.545056\pi$
$860$	0	0
$861$	−5.27470	−0.179761
$862$	−66.2244	−2.25561
$863$	−36.0823	−1.22825	−0.614127	−	0.789207i	$-0.710492\pi$
$863$	−36.0823	−1.22825	−0.614127	+	0.789207i	$0.710492\pi$
$864$	−56.5118	−1.92257
$865$	0	0
$866$	75.2232	2.55619
$867$	−41.0707	−1.39483
$868$	80.9678	2.74823
$869$	−38.1368	−1.29370
$870$	0	0
$871$	−40.9769	−1.38845
$872$	97.7141	3.30902
$873$	−8.08397	−0.273601
$874$	−187.621	−6.34636
$875$	0	0
$876$	−108.132	−3.65346
$877$	−6.92305	−0.233775	−0.116887	−	0.993145i	$-0.537292\pi$
$877$	−6.92305	−0.233775	−0.116887	+	0.993145i	$0.537292\pi$
$878$	−62.5472	−2.11087
$879$	77.1277	2.60145
$880$	0	0
$881$	19.8668	0.669329	0.334665	−	0.942337i	$-0.391377\pi$
$881$	19.8668	0.669329	0.334665	+	0.942337i	$0.391377\pi$
$882$	53.0960	1.78784
$883$	23.9373	0.805556	0.402778	−	0.915298i	$-0.368045\pi$
$883$	23.9373	0.805556	0.402778	+	0.915298i	$0.368045\pi$
$884$	23.4518	0.788770
$885$	0	0
$886$	−18.6447	−0.626382
$887$	12.5474	0.421301	0.210650	−	0.977561i	$-0.432442\pi$
$887$	12.5474	0.421301	0.210650	+	0.977561i	$0.432442\pi$
$888$	−61.8091	−2.07418
$889$	8.20500	0.275187
$890$	0	0
$891$	26.7332	0.895596
$892$	−155.390	−5.20284
$893$	92.5443	3.09688
$894$	64.6071	2.16078
$895$	0	0
$896$	−81.6848	−2.72890
$897$	−83.4618	−2.78671
$898$	−31.5252	−1.05201
$899$	58.6661	1.95663
$900$	0	0
$901$	−10.5562	−0.351679
$902$	−18.0119	−0.599731
$903$	−34.4423	−1.14617
$904$	−42.4239	−1.41100
$905$	0	0
$906$	54.8093	1.82092
$907$	8.31362	0.276049	0.138025	−	0.990429i	$-0.455925\pi$
$907$	8.31362	0.276049	0.138025	+	0.990429i	$0.455925\pi$
$908$	−17.4305	−0.578452
$909$	50.2194	1.66567
$910$	0	0
$911$	−43.6232	−1.44530	−0.722650	−	0.691214i	$-0.757076\pi$
$911$	−43.6232	−1.44530	−0.722650	+	0.691214i	$0.757076\pi$
$912$	−345.322	−11.4348
$913$	−69.9296	−2.31433
$914$	−42.5761	−1.40829
$915$	0	0
$916$	45.4187	1.50068
$917$	−7.78439	−0.257063
$918$	−6.55740	−0.216426
$919$	−32.7212	−1.07937	−0.539687	−	0.841866i	$-0.681457\pi$
$919$	−32.7212	−1.07937	−0.539687	+	0.841866i	$0.681457\pi$
$920$	0	0
$921$	−69.1995	−2.28020
$922$	−66.6700	−2.19566
$923$	−5.81406	−0.191372
$924$	−94.2060	−3.09915
$925$	0	0
$926$	−34.9480	−1.14846
$927$	6.06881	0.199326
$928$	−155.557	−5.10640
$929$	44.1355	1.44804	0.724019	−	0.689780i	$-0.242293\pi$
$929$	44.1355	1.44804	0.724019	+	0.689780i	$0.242293\pi$
$930$	0	0
$931$	38.9888	1.27781
$932$	88.1644	2.88792
$933$	−4.81768	−0.157724
$934$	66.5605	2.17793
$935$	0	0
$936$	−143.291	−4.68360
$937$	−38.8796	−1.27014	−0.635071	−	0.772454i	$-0.719029\pi$
$937$	−38.8796	−1.27014	−0.635071	+	0.772454i	$0.719029\pi$
$938$	43.5280	1.42124
$939$	40.0164	1.30588
$940$	0	0
$941$	−10.5472	−0.343829	−0.171915	−	0.985112i	$-0.554995\pi$
$941$	−10.5472	−0.343829	−0.171915	+	0.985112i	$0.554995\pi$
$942$	−68.9855	−2.24767
$943$	12.5466	0.408573
$944$	58.7189	1.91114
$945$	0	0
$946$	−117.613	−3.82392
$947$	−7.38831	−0.240088	−0.120044	−	0.992769i	$-0.538304\pi$
$947$	−7.38831	−0.240088	−0.120044	+	0.992769i	$0.538304\pi$
$948$	125.656	4.08111
$949$	−26.6909	−0.866423
$950$	0	0
$951$	−55.3134	−1.79366
$952$	−16.1665	−0.523958
$953$	19.5572	0.633520	0.316760	−	0.948506i	$-0.397405\pi$
$953$	19.5572	0.633520	0.316760	+	0.948506i	$0.397405\pi$
$954$	99.3899	3.21787
$955$	0	0
$956$	−51.9794	−1.68113
$957$	−68.2580	−2.20647
$958$	−105.888	−3.42108
$959$	−8.89882	−0.287358
$960$	0	0
$961$	71.3460	2.30148
$962$	−23.5099	−0.757989
$963$	4.75259	0.153150
$964$	−5.69713	−0.183492
$965$	0	0
$966$	88.6579	2.85252
$967$	−6.68326	−0.214919	−0.107460	−	0.994209i	$-0.534272\pi$
$967$	−6.68326	−0.214919	−0.107460	+	0.994209i	$0.534272\pi$
$968$	−95.9316	−3.08336
$969$	−22.7056	−0.729408
$970$	0	0
$971$	54.7717	1.75771	0.878853	−	0.477092i	$-0.158309\pi$
$971$	54.7717	1.75771	0.878853	+	0.477092i	$0.158309\pi$
$972$	−124.089	−3.98015
$973$	−18.9660	−0.608023
$974$	113.306	3.63056
$975$	0	0
$976$	88.4779	2.83211
$977$	−46.7108	−1.49441	−0.747205	−	0.664594i	$-0.768605\pi$
$977$	−46.7108	−1.49441	−0.747205	+	0.664594i	$0.768605\pi$
$978$	128.973	4.12410
$979$	21.2841	0.680244
$980$	0	0
$981$	−36.2711	−1.15805
$982$	37.4020	1.19355
$983$	49.7532	1.58688	0.793441	−	0.608647i	$-0.208287\pi$
$983$	49.7532	1.58688	0.793441	+	0.608647i	$0.208287\pi$
$984$	38.5130	1.22775
$985$	0	0
$986$	−18.0502	−0.574835
$987$	−43.7307	−1.39196
$988$	−162.139	−5.15832
$989$	81.9257	2.60509
$990$	0	0
$991$	2.28643	0.0726309	0.0363155	−	0.999340i	$-0.488438\pi$
$991$	2.28643	0.0726309	0.0363155	+	0.999340i	$0.488438\pi$
$992$	−271.377	−8.61622
$993$	6.56459	0.208321
$994$	6.17603	0.195892
$995$	0	0
$996$	230.409	7.30079
$997$	−25.5726	−0.809893	−0.404946	−	0.914340i	$-0.632710\pi$
$997$	−25.5726	−0.809893	−0.404946	+	0.914340i	$0.632710\pi$
$998$	−28.5548	−0.903886
$999$	4.86556	0.153940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.l.1.1	✓	40
5.4	even	2		6025.2.a.o.1.40	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.1	✓	40	1.1	even	1	trivial
6025.2.a.o.1.40	yes	40	5.4	even	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77437 q^{2} +2.60911 q^{3} +5.69713 q^{4} -7.23863 q^{6} +1.40482 q^{7} -10.2572 q^{8} +3.80744 q^{9} +O(q^{10})\)	\(q-2.77437 q^{2} +2.60911 q^{3} +5.69713 q^{4} -7.23863 q^{6} +1.40482 q^{7} -10.2572 q^{8} +3.80744 q^{9} -4.51138 q^{11} +14.8644 q^{12} +3.66906 q^{13} -3.89749 q^{14} +17.0631 q^{16} +1.12193 q^{17} -10.5633 q^{18} -7.75667 q^{19} +3.66532 q^{21} +12.5163 q^{22} -8.71847 q^{23} -26.7622 q^{24} -10.1793 q^{26} +2.10670 q^{27} +8.00345 q^{28} +5.79899 q^{29} +10.1166 q^{31} -26.8248 q^{32} -11.7707 q^{33} -3.11264 q^{34} +21.6915 q^{36} +2.30957 q^{37} +21.5199 q^{38} +9.57298 q^{39} -1.43908 q^{41} -10.1690 q^{42} -9.39680 q^{43} -25.7020 q^{44} +24.1883 q^{46} -11.9309 q^{47} +44.5194 q^{48} -5.02648 q^{49} +2.92723 q^{51} +20.9031 q^{52} -9.40902 q^{53} -5.84476 q^{54} -14.4095 q^{56} -20.2380 q^{57} -16.0885 q^{58} +3.44128 q^{59} +5.18535 q^{61} -28.0673 q^{62} +5.34877 q^{63} +40.2959 q^{64} +32.6562 q^{66} -11.1682 q^{67} +6.39177 q^{68} -22.7474 q^{69} -1.58462 q^{71} -39.0538 q^{72} -7.27458 q^{73} -6.40760 q^{74} -44.1908 q^{76} -6.33768 q^{77} -26.5590 q^{78} +8.45345 q^{79} -5.92572 q^{81} +3.99254 q^{82} +15.5007 q^{83} +20.8818 q^{84} +26.0702 q^{86} +15.1302 q^{87} +46.2743 q^{88} -4.71787 q^{89} +5.15437 q^{91} -49.6703 q^{92} +26.3953 q^{93} +33.1008 q^{94} -69.9889 q^{96} -2.12320 q^{97} +13.9453 q^{98} -17.1768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more