LMFDB - Embedded newform 3640.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.43828$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43828 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.94523 q^{9} +O(q^{10})$	$q-2.43828 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.94523 q^{9} +1.23607 q^{11} +1.00000 q^{13} -2.43828 q^{15} +0.931341 q^{17} -7.18129 q^{19} +2.43828 q^{21} +2.65438 q^{23} +1.00000 q^{25} +0.133557 q^{27} +2.66047 q^{29} -7.04774 q^{31} -3.01388 q^{33} -1.00000 q^{35} -0.797785 q^{37} -2.43828 q^{39} -2.52691 q^{41} -3.64050 q^{43} +2.94523 q^{45} +1.05382 q^{47} +1.00000 q^{49} -2.27087 q^{51} +5.89045 q^{53} +1.23607 q^{55} +17.5100 q^{57} +12.3565 q^{59} +1.01388 q^{61} -2.94523 q^{63} +1.00000 q^{65} +0.349656 q^{67} -6.47214 q^{69} +0.249952 q^{71} +8.90433 q^{73} -2.43828 q^{75} -1.23607 q^{77} -6.75409 q^{79} -9.16132 q^{81} -4.43220 q^{83} +0.931341 q^{85} -6.48697 q^{87} -6.64830 q^{89} -1.00000 q^{91} +17.1844 q^{93} -7.18129 q^{95} -4.29488 q^{97} +3.64050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.43828	−1.40774	−0.703872	−	0.710327i	$-0.748547\pi$
$3$	−2.43828	−1.40774	−0.703872	+	0.710327i	$0.748547\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.94523	0.981742
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.43828	−0.629562
$16$	0	0
$17$	0.931341	0.225883	0.112942	−	0.993602i	$-0.463973\pi$
$17$	0.931341	0.225883	0.112942	+	0.993602i	$0.463973\pi$
$18$	0	0
$19$	−7.18129	−1.64750	−0.823751	−	0.566952i	$-0.808122\pi$
$19$	−7.18129	−1.64750	−0.823751	+	0.566952i	$0.808122\pi$
$20$	0	0
$21$	2.43828	0.532077
$22$	0	0
$23$	2.65438	0.553477	0.276738	−	0.960945i	$-0.410746\pi$
$23$	2.65438	0.553477	0.276738	+	0.960945i	$0.410746\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.133557	0.0257030
$28$	0	0
$29$	2.66047	0.494036	0.247018	−	0.969011i	$-0.420549\pi$
$29$	2.66047	0.494036	0.247018	+	0.969011i	$0.420549\pi$
$30$	0	0
$31$	−7.04774	−1.26581	−0.632905	−	0.774229i	$-0.718138\pi$
$31$	−7.04774	−1.26581	−0.632905	+	0.774229i	$0.718138\pi$
$32$	0	0
$33$	−3.01388	−0.524650
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.797785	−0.131155	−0.0655775	−	0.997847i	$-0.520889\pi$
$37$	−0.797785	−0.131155	−0.0655775	+	0.997847i	$0.520889\pi$
$38$	0	0
$39$	−2.43828	−0.390438
$40$	0	0
$41$	−2.52691	−0.394637	−0.197319	−	0.980339i	$-0.563223\pi$
$41$	−2.52691	−0.394637	−0.197319	+	0.980339i	$0.563223\pi$
$42$	0	0
$43$	−3.64050	−0.555171	−0.277585	−	0.960701i	$-0.589534\pi$
$43$	−3.64050	−0.555171	−0.277585	+	0.960701i	$0.589534\pi$
$44$	0	0
$45$	2.94523	0.439048
$46$	0	0
$47$	1.05382	0.153716	0.0768578	−	0.997042i	$-0.475511\pi$
$47$	1.05382	0.153716	0.0768578	+	0.997042i	$0.475511\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.27087	−0.317986
$52$	0	0
$53$	5.89045	0.809116	0.404558	−	0.914512i	$-0.367425\pi$
$53$	5.89045	0.809116	0.404558	+	0.914512i	$0.367425\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	0	0
$57$	17.5100	2.31926
$58$	0	0
$59$	12.3565	1.60868	0.804340	−	0.594170i	$-0.202519\pi$
$59$	12.3565	1.60868	0.804340	+	0.594170i	$0.202519\pi$
$60$	0	0
$61$	1.01388	0.129815	0.0649073	−	0.997891i	$-0.479325\pi$
$61$	1.01388	0.129815	0.0649073	+	0.997891i	$0.479325\pi$
$62$	0	0
$63$	−2.94523	−0.371063
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	0.349656	0.0427172	0.0213586	−	0.999772i	$-0.493201\pi$
$67$	0.349656	0.0427172	0.0213586	+	0.999772i	$0.493201\pi$
$68$	0	0
$69$	−6.47214	−0.779154
$70$	0	0
$71$	0.249952	0.0296638	0.0148319	−	0.999890i	$-0.495279\pi$
$71$	0.249952	0.0296638	0.0148319	+	0.999890i	$0.495279\pi$
$72$	0	0
$73$	8.90433	1.04217	0.521087	−	0.853504i	$-0.325527\pi$
$73$	8.90433	1.04217	0.521087	+	0.853504i	$0.325527\pi$
$74$	0	0
$75$	−2.43828	−0.281549
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	−6.75409	−0.759894	−0.379947	−	0.925008i	$-0.624058\pi$
$79$	−6.75409	−0.759894	−0.379947	+	0.925008i	$0.624058\pi$
$80$	0	0
$81$	−9.16132	−1.01792
$82$	0	0
$83$	−4.43220	−0.486497	−0.243248	−	0.969964i	$-0.578213\pi$
$83$	−4.43220	−0.486497	−0.243248	+	0.969964i	$0.578213\pi$
$84$	0	0
$85$	0.931341	0.101018
$86$	0	0
$87$	−6.48697	−0.695477
$88$	0	0
$89$	−6.64830	−0.704718	−0.352359	−	0.935865i	$-0.614620\pi$
$89$	−6.64830	−0.704718	−0.352359	+	0.935865i	$0.614620\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	17.1844	1.78194
$94$	0	0
$95$	−7.18129	−0.736785
$96$	0	0
$97$	−4.29488	−0.436079	−0.218040	−	0.975940i	$-0.569966\pi$
$97$	−4.29488	−0.436079	−0.218040	+	0.975940i	$0.569966\pi$
$98$	0	0
$99$	3.64050	0.365884
$100$	0	0
$101$	−5.19114	−0.516538	−0.258269	−	0.966073i	$-0.583152\pi$
$101$	−5.19114	−0.516538	−0.258269	+	0.966073i	$0.583152\pi$
$102$	0	0
$103$	−19.1394	−1.88587	−0.942933	−	0.332983i	$-0.891945\pi$
$103$	−19.1394	−1.88587	−0.942933	+	0.332983i	$0.891945\pi$
$104$	0	0
$105$	2.43828	0.237952
$106$	0	0
$107$	13.8935	1.34314	0.671569	−	0.740942i	$-0.265621\pi$
$107$	13.8935	1.34314	0.671569	+	0.740942i	$0.265621\pi$
$108$	0	0
$109$	−1.45825	−0.139675	−0.0698376	−	0.997558i	$-0.522248\pi$
$109$	−1.45825	−0.139675	−0.0698376	+	0.997558i	$0.522248\pi$
$110$	0	0
$111$	1.94523	0.184633
$112$	0	0
$113$	−3.86268	−0.363371	−0.181685	−	0.983357i	$-0.558155\pi$
$113$	−3.86268	−0.363371	−0.181685	+	0.983357i	$0.558155\pi$
$114$	0	0
$115$	2.65438	0.247522
$116$	0	0
$117$	2.94523	0.272286
$118$	0	0
$119$	−0.931341	−0.0853759
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	6.16132	0.555548
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.23607	0.109683	0.0548416	−	0.998495i	$-0.482535\pi$
$127$	1.23607	0.109683	0.0548416	+	0.998495i	$0.482535\pi$
$128$	0	0
$129$	8.87657	0.781538
$130$	0	0
$131$	−10.9165	−0.953779	−0.476890	−	0.878963i	$-0.658236\pi$
$131$	−10.9165	−0.953779	−0.476890	+	0.878963i	$0.658236\pi$
$132$	0	0
$133$	7.18129	0.622697
$134$	0	0
$135$	0.133557	0.0114947
$136$	0	0
$137$	19.8597	1.69673	0.848364	−	0.529414i	$-0.177588\pi$
$137$	19.8597	1.69673	0.848364	+	0.529414i	$0.177588\pi$
$138$	0	0
$139$	−15.9582	−1.35355	−0.676777	−	0.736188i	$-0.736624\pi$
$139$	−15.9582	−1.35355	−0.676777	+	0.736188i	$0.736624\pi$
$140$	0	0
$141$	−2.56952	−0.216392
$142$	0	0
$143$	1.23607	0.103365
$144$	0	0
$145$	2.66047	0.220940
$146$	0	0
$147$	−2.43828	−0.201106
$148$	0	0
$149$	−9.10374	−0.745808	−0.372904	−	0.927870i	$-0.621638\pi$
$149$	−9.10374	−0.745808	−0.372904	+	0.927870i	$0.621638\pi$
$150$	0	0
$151$	17.4492	1.41999	0.709997	−	0.704205i	$-0.248696\pi$
$151$	17.4492	1.41999	0.709997	+	0.704205i	$0.248696\pi$
$152$	0	0
$153$	2.74301	0.221759
$154$	0	0
$155$	−7.04774	−0.566088
$156$	0	0
$157$	12.5909	1.00486	0.502430	−	0.864618i	$-0.332440\pi$
$157$	12.5909	1.00486	0.502430	+	0.864618i	$0.332440\pi$
$158$	0	0
$159$	−14.3626	−1.13903
$160$	0	0
$161$	−2.65438	−0.209195
$162$	0	0
$163$	−5.74396	−0.449902	−0.224951	−	0.974370i	$-0.572222\pi$
$163$	−5.74396	−0.449902	−0.224951	+	0.974370i	$0.572222\pi$
$164$	0	0
$165$	−3.01388	−0.234631
$166$	0	0
$167$	2.56952	0.198835	0.0994175	−	0.995046i	$-0.468302\pi$
$167$	2.56952	0.198835	0.0994175	+	0.995046i	$0.468302\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−21.1505	−1.61742
$172$	0	0
$173$	−6.48321	−0.492910	−0.246455	−	0.969154i	$-0.579266\pi$
$173$	−6.48321	−0.492910	−0.246455	+	0.969154i	$0.579266\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−30.1286	−2.26461
$178$	0	0
$179$	−15.1444	−1.13195	−0.565974	−	0.824423i	$-0.691500\pi$
$179$	−15.1444	−1.13195	−0.565974	+	0.824423i	$0.691500\pi$
$180$	0	0
$181$	−3.41079	−0.253522	−0.126761	−	0.991933i	$-0.540458\pi$
$181$	−3.41079	−0.253522	−0.126761	+	0.991933i	$0.540458\pi$
$182$	0	0
$183$	−2.47214	−0.182746
$184$	0	0
$185$	−0.797785	−0.0586543
$186$	0	0
$187$	1.15120	0.0841842
$188$	0	0
$189$	−0.133557	−0.00971481
$190$	0	0
$191$	−8.32565	−0.602423	−0.301211	−	0.953557i	$-0.597391\pi$
$191$	−8.32565	−0.602423	−0.301211	+	0.953557i	$0.597391\pi$
$192$	0	0
$193$	−18.8199	−1.35468	−0.677342	−	0.735668i	$-0.736868\pi$
$193$	−18.8199	−1.35468	−0.677342	+	0.735668i	$0.736868\pi$
$194$	0	0
$195$	−2.43828	−0.174609
$196$	0	0
$197$	4.88764	0.348230	0.174115	−	0.984725i	$-0.444294\pi$
$197$	4.88764	0.348230	0.174115	+	0.984725i	$0.444294\pi$
$198$	0	0
$199$	−24.3151	−1.72365	−0.861827	−	0.507203i	$-0.830680\pi$
$199$	−24.3151	−1.72365	−0.861827	+	0.507203i	$0.830680\pi$
$200$	0	0
$201$	−0.852560	−0.0601349
$202$	0	0
$203$	−2.66047	−0.186728
$204$	0	0
$205$	−2.52691	−0.176487
$206$	0	0
$207$	7.81775	0.543371
$208$	0	0
$209$	−8.87657	−0.614005
$210$	0	0
$211$	3.08890	0.212649	0.106324	−	0.994331i	$-0.466092\pi$
$211$	3.08890	0.212649	0.106324	+	0.994331i	$0.466092\pi$
$212$	0	0
$213$	−0.609453	−0.0417591
$214$	0	0
$215$	−3.64050	−0.248280
$216$	0	0
$217$	7.04774	0.478432
$218$	0	0
$219$	−21.7113	−1.46711
$220$	0	0
$221$	0.931341	0.0626488
$222$	0	0
$223$	−17.8885	−1.19791	−0.598953	−	0.800784i	$-0.704416\pi$
$223$	−17.8885	−1.19791	−0.598953	+	0.800784i	$0.704416\pi$
$224$	0	0
$225$	2.94523	0.196348
$226$	0	0
$227$	−2.31457	−0.153624	−0.0768118	−	0.997046i	$-0.524474\pi$
$227$	−2.31457	−0.153624	−0.0768118	+	0.997046i	$0.524474\pi$
$228$	0	0
$229$	3.50086	0.231343	0.115672	−	0.993288i	$-0.463098\pi$
$229$	3.50086	0.231343	0.115672	+	0.993288i	$0.463098\pi$
$230$	0	0
$231$	3.01388	0.198299
$232$	0	0
$233$	−23.9662	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$233$	−23.9662	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$234$	0	0
$235$	1.05382	0.0687437
$236$	0	0
$237$	16.4684	1.06974
$238$	0	0
$239$	−17.0847	−1.10512	−0.552558	−	0.833475i	$-0.686348\pi$
$239$	−17.0847	−1.10512	−0.552558	+	0.833475i	$0.686348\pi$
$240$	0	0
$241$	−4.30972	−0.277613	−0.138807	−	0.990319i	$-0.544327\pi$
$241$	−4.30972	−0.277613	−0.138807	+	0.990319i	$0.544327\pi$
$242$	0	0
$243$	21.9372	1.40727
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−7.18129	−0.456935
$248$	0	0
$249$	10.8070	0.684863
$250$	0	0
$251$	−17.8430	−1.12624	−0.563120	−	0.826375i	$-0.690399\pi$
$251$	−17.8430	−1.12624	−0.563120	+	0.826375i	$0.690399\pi$
$252$	0	0
$253$	3.28100	0.206275
$254$	0	0
$255$	−2.27087	−0.142208
$256$	0	0
$257$	8.83663	0.551214	0.275607	−	0.961270i	$-0.411121\pi$
$257$	8.83663	0.551214	0.275607	+	0.961270i	$0.411121\pi$
$258$	0	0
$259$	0.797785	0.0495719
$260$	0	0
$261$	7.83568	0.485016
$262$	0	0
$263$	−5.49101	−0.338590	−0.169295	−	0.985565i	$-0.554149\pi$
$263$	−5.49101	−0.338590	−0.169295	+	0.985565i	$0.554149\pi$
$264$	0	0
$265$	5.89045	0.361847
$266$	0	0
$267$	16.2104	0.992062
$268$	0	0
$269$	−31.5119	−1.92131	−0.960657	−	0.277739i	$-0.910415\pi$
$269$	−31.5119	−1.92131	−0.960657	+	0.277739i	$0.910415\pi$
$270$	0	0
$271$	−21.2164	−1.28880	−0.644402	−	0.764687i	$-0.722893\pi$
$271$	−21.2164	−1.28880	−0.644402	+	0.764687i	$0.722893\pi$
$272$	0	0
$273$	2.43828	0.147572
$274$	0	0
$275$	1.23607	0.0745377
$276$	0	0
$277$	−5.51379	−0.331291	−0.165646	−	0.986185i	$-0.552971\pi$
$277$	−5.51379	−0.331291	−0.165646	+	0.986185i	$0.552971\pi$
$278$	0	0
$279$	−20.7572	−1.24270
$280$	0	0
$281$	−9.10374	−0.543084	−0.271542	−	0.962427i	$-0.587534\pi$
$281$	−9.10374	−0.543084	−0.271542	+	0.962427i	$0.587534\pi$
$282$	0	0
$283$	25.3370	1.50613	0.753063	−	0.657949i	$-0.228576\pi$
$283$	25.3370	1.50613	0.753063	+	0.657949i	$0.228576\pi$
$284$	0	0
$285$	17.5100	1.03720
$286$	0	0
$287$	2.52691	0.149159
$288$	0	0
$289$	−16.1326	−0.948977
$290$	0	0
$291$	10.4721	0.613887
$292$	0	0
$293$	−8.73925	−0.510552	−0.255276	−	0.966868i	$-0.582166\pi$
$293$	−8.73925	−0.510552	−0.255276	+	0.966868i	$0.582166\pi$
$294$	0	0
$295$	12.3565	0.719423
$296$	0	0
$297$	0.165085	0.00957920
$298$	0	0
$299$	2.65438	0.153507
$300$	0	0
$301$	3.64050	0.209835
$302$	0	0
$303$	12.6575	0.727152
$304$	0	0
$305$	1.01388	0.0580548
$306$	0	0
$307$	−3.08159	−0.175876	−0.0879378	−	0.996126i	$-0.528028\pi$
$307$	−3.08159	−0.175876	−0.0879378	+	0.996126i	$0.528028\pi$
$308$	0	0
$309$	46.6674	2.65482
$310$	0	0
$311$	1.19921	0.0680012	0.0340006	−	0.999422i	$-0.489175\pi$
$311$	1.19921	0.0680012	0.0340006	+	0.999422i	$0.489175\pi$
$312$	0	0
$313$	−25.8875	−1.46325	−0.731623	−	0.681710i	$-0.761237\pi$
$313$	−25.8875	−1.46325	−0.731623	+	0.681710i	$0.761237\pi$
$314$	0	0
$315$	−2.94523	−0.165945
$316$	0	0
$317$	−8.78726	−0.493542	−0.246771	−	0.969074i	$-0.579370\pi$
$317$	−8.78726	−0.493542	−0.246771	+	0.969074i	$0.579370\pi$
$318$	0	0
$319$	3.28852	0.184122
$320$	0	0
$321$	−33.8764	−1.89079
$322$	0	0
$323$	−6.68824	−0.372143
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	3.55563	0.196627
$328$	0	0
$329$	−1.05382	−0.0580991
$330$	0	0
$331$	−24.0505	−1.32194	−0.660969	−	0.750413i	$-0.729854\pi$
$331$	−24.0505	−1.32194	−0.660969	+	0.750413i	$0.729854\pi$
$332$	0	0
$333$	−2.34966	−0.128760
$334$	0	0
$335$	0.349656	0.0191037
$336$	0	0
$337$	2.21910	0.120882	0.0604410	−	0.998172i	$-0.480749\pi$
$337$	2.21910	0.120882	0.0604410	+	0.998172i	$0.480749\pi$
$338$	0	0
$339$	9.41831	0.511533
$340$	0	0
$341$	−8.71148	−0.471753
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.47214	−0.348448
$346$	0	0
$347$	5.32593	0.285911	0.142955	−	0.989729i	$-0.454339\pi$
$347$	5.32593	0.285911	0.142955	+	0.989729i	$0.454339\pi$
$348$	0	0
$349$	19.2159	1.02860	0.514302	−	0.857609i	$-0.328051\pi$
$349$	19.2159	1.02860	0.514302	+	0.857609i	$0.328051\pi$
$350$	0	0
$351$	0.133557	0.00712872
$352$	0	0
$353$	8.60755	0.458133	0.229067	−	0.973411i	$-0.426433\pi$
$353$	8.60755	0.458133	0.229067	+	0.973411i	$0.426433\pi$
$354$	0	0
$355$	0.249952	0.0132661
$356$	0	0
$357$	2.27087	0.120187
$358$	0	0
$359$	12.3055	0.649459	0.324729	−	0.945807i	$-0.394727\pi$
$359$	12.3055	0.649459	0.324729	+	0.945807i	$0.394727\pi$
$360$	0	0
$361$	32.5710	1.71426
$362$	0	0
$363$	23.0958	1.21221
$364$	0	0
$365$	8.90433	0.466074
$366$	0	0
$367$	16.1884	0.845028	0.422514	−	0.906356i	$-0.361148\pi$
$367$	16.1884	0.845028	0.422514	+	0.906356i	$0.361148\pi$
$368$	0	0
$369$	−7.44232	−0.387432
$370$	0	0
$371$	−5.89045	−0.305817
$372$	0	0
$373$	−32.0683	−1.66043	−0.830216	−	0.557442i	$-0.811783\pi$
$373$	−32.0683	−1.66043	−0.830216	+	0.557442i	$0.811783\pi$
$374$	0	0
$375$	−2.43828	−0.125912
$376$	0	0
$377$	2.66047	0.137021
$378$	0	0
$379$	20.3101	1.04326	0.521631	−	0.853171i	$-0.325324\pi$
$379$	20.3101	1.04326	0.521631	+	0.853171i	$0.325324\pi$
$380$	0	0
$381$	−3.01388	−0.154406
$382$	0	0
$383$	−31.3967	−1.60430	−0.802149	−	0.597124i	$-0.796310\pi$
$383$	−31.3967	−1.60430	−0.802149	+	0.597124i	$0.796310\pi$
$384$	0	0
$385$	−1.23607	−0.0629959
$386$	0	0
$387$	−10.7221	−0.545034
$388$	0	0
$389$	1.58264	0.0802430	0.0401215	−	0.999195i	$-0.487226\pi$
$389$	1.58264	0.0802430	0.0401215	+	0.999195i	$0.487226\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	26.6175	1.34268
$394$	0	0
$395$	−6.75409	−0.339835
$396$	0	0
$397$	10.8210	0.543092	0.271546	−	0.962425i	$-0.412465\pi$
$397$	10.8210	0.543092	0.271546	+	0.962425i	$0.412465\pi$
$398$	0	0
$399$	−17.5100	−0.876598
$400$	0	0
$401$	−10.3470	−0.516704	−0.258352	−	0.966051i	$-0.583179\pi$
$401$	−10.3470	−0.516704	−0.258352	+	0.966051i	$0.583179\pi$
$402$	0	0
$403$	−7.04774	−0.351073
$404$	0	0
$405$	−9.16132	−0.455230
$406$	0	0
$407$	−0.986116	−0.0488800
$408$	0	0
$409$	−3.43801	−0.169998	−0.0849992	−	0.996381i	$-0.527089\pi$
$409$	−3.43801	−0.169998	−0.0849992	+	0.996381i	$0.527089\pi$
$410$	0	0
$411$	−48.4235	−2.38856
$412$	0	0
$413$	−12.3565	−0.608024
$414$	0	0
$415$	−4.43220	−0.217568
$416$	0	0
$417$	38.9105	1.90546
$418$	0	0
$419$	24.0062	1.17278	0.586389	−	0.810030i	$-0.300549\pi$
$419$	24.0062	1.17278	0.586389	+	0.810030i	$0.300549\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	3.10374	0.150909
$424$	0	0
$425$	0.931341	0.0451767
$426$	0	0
$427$	−1.01388	−0.0490653
$428$	0	0
$429$	−3.01388	−0.145512
$430$	0	0
$431$	38.5632	1.85752	0.928761	−	0.370678i	$-0.120875\pi$
$431$	38.5632	1.85752	0.928761	+	0.370678i	$0.120875\pi$
$432$	0	0
$433$	−24.2725	−1.16646	−0.583231	−	0.812306i	$-0.698212\pi$
$433$	−24.2725	−1.16646	−0.583231	+	0.812306i	$0.698212\pi$
$434$	0	0
$435$	−6.48697	−0.311027
$436$	0	0
$437$	−19.0619	−0.911854
$438$	0	0
$439$	−4.67716	−0.223229	−0.111614	−	0.993752i	$-0.535602\pi$
$439$	−4.67716	−0.223229	−0.111614	+	0.993752i	$0.535602\pi$
$440$	0	0
$441$	2.94523	0.140249
$442$	0	0
$443$	6.09294	0.289484	0.144742	−	0.989469i	$-0.453765\pi$
$443$	6.09294	0.289484	0.144742	+	0.989469i	$0.453765\pi$
$444$	0	0
$445$	−6.64830	−0.315160
$446$	0	0
$447$	22.1975	1.04991
$448$	0	0
$449$	6.10955	0.288327	0.144164	−	0.989554i	$-0.453951\pi$
$449$	6.10955	0.288327	0.144164	+	0.989554i	$0.453951\pi$
$450$	0	0
$451$	−3.12343	−0.147077
$452$	0	0
$453$	−42.5460	−1.99899
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	12.0788	0.565022	0.282511	−	0.959264i	$-0.408833\pi$
$457$	12.0788	0.565022	0.282511	+	0.959264i	$0.408833\pi$
$458$	0	0
$459$	0.124387	0.00580588
$460$	0	0
$461$	−32.1046	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$461$	−32.1046	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$462$	0	0
$463$	3.48526	0.161974	0.0809869	−	0.996715i	$-0.474193\pi$
$463$	3.48526	0.161974	0.0809869	+	0.996715i	$0.474193\pi$
$464$	0	0
$465$	17.1844	0.796906
$466$	0	0
$467$	14.0004	0.647862	0.323931	−	0.946081i	$-0.394995\pi$
$467$	14.0004	0.647862	0.323931	+	0.946081i	$0.394995\pi$
$468$	0	0
$469$	−0.349656	−0.0161456
$470$	0	0
$471$	−30.7001	−1.41458
$472$	0	0
$473$	−4.49990	−0.206906
$474$	0	0
$475$	−7.18129	−0.329500
$476$	0	0
$477$	17.3487	0.794343
$478$	0	0
$479$	−17.3351	−0.792061	−0.396030	−	0.918237i	$-0.629613\pi$
$479$	−17.3351	−0.792061	−0.396030	+	0.918237i	$0.629613\pi$
$480$	0	0
$481$	−0.797785	−0.0363759
$482$	0	0
$483$	6.47214	0.294492
$484$	0	0
$485$	−4.29488	−0.195020
$486$	0	0
$487$	−13.2502	−0.600425	−0.300213	−	0.953872i	$-0.597058\pi$
$487$	−13.2502	−0.600425	−0.300213	+	0.953872i	$0.597058\pi$
$488$	0	0
$489$	14.0054	0.633346
$490$	0	0
$491$	32.3213	1.45864	0.729320	−	0.684173i	$-0.239837\pi$
$491$	32.3213	1.45864	0.729320	+	0.684173i	$0.239837\pi$
$492$	0	0
$493$	2.47780	0.111595
$494$	0	0
$495$	3.64050	0.163628
$496$	0	0
$497$	−0.249952	−0.0112119
$498$	0	0
$499$	14.0783	0.630232	0.315116	−	0.949053i	$-0.397957\pi$
$499$	14.0783	0.630232	0.315116	+	0.949053i	$0.397957\pi$
$500$	0	0
$501$	−6.26521	−0.279909
$502$	0	0
$503$	−2.57451	−0.114792	−0.0573958	−	0.998352i	$-0.518280\pi$
$503$	−2.57451	−0.114792	−0.0573958	+	0.998352i	$0.518280\pi$
$504$	0	0
$505$	−5.19114	−0.231003
$506$	0	0
$507$	−2.43828	−0.108288
$508$	0	0
$509$	32.7603	1.45207	0.726036	−	0.687656i	$-0.241360\pi$
$509$	32.7603	1.45207	0.726036	+	0.687656i	$0.241360\pi$
$510$	0	0
$511$	−8.90433	−0.393905
$512$	0	0
$513$	−0.959109	−0.0423457
$514$	0	0
$515$	−19.1394	−0.843385
$516$	0	0
$517$	1.30260	0.0572881
$518$	0	0
$519$	15.8079	0.693890
$520$	0	0
$521$	−21.9878	−0.963304	−0.481652	−	0.876362i	$-0.659963\pi$
$521$	−21.9878	−0.963304	−0.481652	+	0.876362i	$0.659963\pi$
$522$	0	0
$523$	16.0187	0.700448	0.350224	−	0.936666i	$-0.386105\pi$
$523$	16.0187	0.700448	0.350224	+	0.936666i	$0.386105\pi$
$524$	0	0
$525$	2.43828	0.106415
$526$	0	0
$527$	−6.56385	−0.285926
$528$	0	0
$529$	−15.9543	−0.693663
$530$	0	0
$531$	36.3927	1.57931
$532$	0	0
$533$	−2.52691	−0.109453
$534$	0	0
$535$	13.8935	0.600670
$536$	0	0
$537$	36.9264	1.59349
$538$	0	0
$539$	1.23607	0.0532412
$540$	0	0
$541$	−10.8569	−0.466774	−0.233387	−	0.972384i	$-0.574981\pi$
$541$	−10.8569	−0.466774	−0.233387	+	0.972384i	$0.574981\pi$
$542$	0	0
$543$	8.31648	0.356894
$544$	0	0
$545$	−1.45825	−0.0624647
$546$	0	0
$547$	11.4557	0.489811	0.244906	−	0.969547i	$-0.421243\pi$
$547$	11.4557	0.489811	0.244906	+	0.969547i	$0.421243\pi$
$548$	0	0
$549$	2.98612	0.127444
$550$	0	0
$551$	−19.1056	−0.813926
$552$	0	0
$553$	6.75409	0.287213
$554$	0	0
$555$	1.94523	0.0825702
$556$	0	0
$557$	42.6760	1.80824	0.904120	−	0.427278i	$-0.140527\pi$
$557$	42.6760	1.80824	0.904120	+	0.427278i	$0.140527\pi$
$558$	0	0
$559$	−3.64050	−0.153977
$560$	0	0
$561$	−2.80695	−0.118510
$562$	0	0
$563$	−29.2471	−1.23262	−0.616309	−	0.787504i	$-0.711373\pi$
$563$	−29.2471	−1.23262	−0.616309	+	0.787504i	$0.711373\pi$
$564$	0	0
$565$	−3.86268	−0.162504
$566$	0	0
$567$	9.16132	0.384739
$568$	0	0
$569$	8.84580	0.370835	0.185418	−	0.982660i	$-0.440636\pi$
$569$	8.84580	0.370835	0.185418	+	0.982660i	$0.440636\pi$
$570$	0	0
$571$	30.3596	1.27051	0.635255	−	0.772303i	$-0.280895\pi$
$571$	30.3596	1.27051	0.635255	+	0.772303i	$0.280895\pi$
$572$	0	0
$573$	20.3003	0.848057
$574$	0	0
$575$	2.65438	0.110695
$576$	0	0
$577$	−4.17535	−0.173822	−0.0869110	−	0.996216i	$-0.527700\pi$
$577$	−4.17535	−0.173822	−0.0869110	+	0.996216i	$0.527700\pi$
$578$	0	0
$579$	45.8882	1.90705
$580$	0	0
$581$	4.43220	0.183879
$582$	0	0
$583$	7.28100	0.301548
$584$	0	0
$585$	2.94523	0.121770
$586$	0	0
$587$	−12.3828	−0.511094	−0.255547	−	0.966797i	$-0.582256\pi$
$587$	−12.3828	−0.511094	−0.255547	+	0.966797i	$0.582256\pi$
$588$	0	0
$589$	50.6119	2.08543
$590$	0	0
$591$	−11.9175	−0.490219
$592$	0	0
$593$	4.46749	0.183458	0.0917289	−	0.995784i	$-0.470761\pi$
$593$	4.46749	0.183458	0.0917289	+	0.995784i	$0.470761\pi$
$594$	0	0
$595$	−0.931341	−0.0381813
$596$	0	0
$597$	59.2872	2.42646
$598$	0	0
$599$	−7.19731	−0.294074	−0.147037	−	0.989131i	$-0.546974\pi$
$599$	−7.19731	−0.294074	−0.147037	+	0.989131i	$0.546974\pi$
$600$	0	0
$601$	−13.9126	−0.567507	−0.283753	−	0.958897i	$-0.591580\pi$
$601$	−13.9126	−0.567507	−0.283753	+	0.958897i	$0.591580\pi$
$602$	0	0
$603$	1.02981	0.0419373
$604$	0	0
$605$	−9.47214	−0.385097
$606$	0	0
$607$	1.11735	0.0453518	0.0226759	−	0.999743i	$-0.492781\pi$
$607$	1.11735	0.0453518	0.0226759	+	0.999743i	$0.492781\pi$
$608$	0	0
$609$	6.48697	0.262865
$610$	0	0
$611$	1.05382	0.0426331
$612$	0	0
$613$	13.4507	0.543270	0.271635	−	0.962400i	$-0.412436\pi$
$613$	13.4507	0.543270	0.271635	+	0.962400i	$0.412436\pi$
$614$	0	0
$615$	6.16132	0.248449
$616$	0	0
$617$	−0.741916	−0.0298684	−0.0149342	−	0.999888i	$-0.504754\pi$
$617$	−0.741916	−0.0298684	−0.0149342	+	0.999888i	$0.504754\pi$
$618$	0	0
$619$	0.655476	0.0263458	0.0131729	−	0.999913i	$-0.495807\pi$
$619$	0.655476	0.0263458	0.0131729	+	0.999913i	$0.495807\pi$
$620$	0	0
$621$	0.354510	0.0142260
$622$	0	0
$623$	6.64830	0.266358
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.6436	0.864361
$628$	0	0
$629$	−0.743010	−0.0296257
$630$	0	0
$631$	18.7221	0.745315	0.372657	−	0.927969i	$-0.378447\pi$
$631$	18.7221	0.745315	0.372657	+	0.927969i	$0.378447\pi$
$632$	0	0
$633$	−7.53162	−0.299355
$634$	0	0
$635$	1.23607	0.0490519
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0.736164	0.0291222
$640$	0	0
$641$	−36.1046	−1.42605	−0.713024	−	0.701140i	$-0.752675\pi$
$641$	−36.1046	−1.42605	−0.713024	+	0.701140i	$0.752675\pi$
$642$	0	0
$643$	40.2390	1.58687	0.793435	−	0.608655i	$-0.208291\pi$
$643$	40.2390	1.58687	0.793435	+	0.608655i	$0.208291\pi$
$644$	0	0
$645$	8.87657	0.349514
$646$	0	0
$647$	−46.2747	−1.81925	−0.909623	−	0.415434i	$-0.863630\pi$
$647$	−46.2747	−1.81925	−0.909623	+	0.415434i	$0.863630\pi$
$648$	0	0
$649$	15.2735	0.599536
$650$	0	0
$651$	−17.1844	−0.673509
$652$	0	0
$653$	−17.0201	−0.666046	−0.333023	−	0.942919i	$-0.608069\pi$
$653$	−17.0201	−0.666046	−0.333023	+	0.942919i	$0.608069\pi$
$654$	0	0
$655$	−10.9165	−0.426543
$656$	0	0
$657$	26.2253	1.02315
$658$	0	0
$659$	42.5053	1.65577	0.827886	−	0.560896i	$-0.189543\pi$
$659$	42.5053	1.65577	0.827886	+	0.560896i	$0.189543\pi$
$660$	0	0
$661$	−8.36525	−0.325371	−0.162685	−	0.986678i	$-0.552016\pi$
$661$	−8.36525	−0.325371	−0.162685	+	0.986678i	$0.552016\pi$
$662$	0	0
$663$	−2.27087	−0.0881934
$664$	0	0
$665$	7.18129	0.278479
$666$	0	0
$667$	7.06190	0.273438
$668$	0	0
$669$	43.6173	1.68634
$670$	0	0
$671$	1.25323	0.0483804
$672$	0	0
$673$	−17.9385	−0.691477	−0.345738	−	0.938331i	$-0.612372\pi$
$673$	−17.9385	−0.691477	−0.345738	+	0.938331i	$0.612372\pi$
$674$	0	0
$675$	0.133557	0.00514060
$676$	0	0
$677$	−30.5061	−1.17244	−0.586222	−	0.810151i	$-0.699385\pi$
$677$	−30.5061	−1.17244	−0.586222	+	0.810151i	$0.699385\pi$
$678$	0	0
$679$	4.29488	0.164822
$680$	0	0
$681$	5.64358	0.216263
$682$	0	0
$683$	9.23839	0.353497	0.176749	−	0.984256i	$-0.443442\pi$
$683$	9.23839	0.353497	0.176749	+	0.984256i	$0.443442\pi$
$684$	0	0
$685$	19.8597	0.758799
$686$	0	0
$687$	−8.53608	−0.325672
$688$	0	0
$689$	5.89045	0.224408
$690$	0	0
$691$	−20.8564	−0.793415	−0.396708	−	0.917945i	$-0.629847\pi$
$691$	−20.8564	−0.793415	−0.396708	+	0.917945i	$0.629847\pi$
$692$	0	0
$693$	−3.64050	−0.138291
$694$	0	0
$695$	−15.9582	−0.605327
$696$	0	0
$697$	−2.35342	−0.0891420
$698$	0	0
$699$	58.4365	2.21027
$700$	0	0
$701$	24.3097	0.918165	0.459083	−	0.888394i	$-0.348178\pi$
$701$	24.3097	0.918165	0.459083	+	0.888394i	$0.348178\pi$
$702$	0	0
$703$	5.72913	0.216078
$704$	0	0
$705$	−2.56952	−0.0967736
$706$	0	0
$707$	5.19114	0.195233
$708$	0	0
$709$	−31.0478	−1.16603	−0.583013	−	0.812463i	$-0.698126\pi$
$709$	−31.0478	−1.16603	−0.583013	+	0.812463i	$0.698126\pi$
$710$	0	0
$711$	−19.8923	−0.746020
$712$	0	0
$713$	−18.7074	−0.700597
$714$	0	0
$715$	1.23607	0.0462263
$716$	0	0
$717$	41.6573	1.55572
$718$	0	0
$719$	−16.8644	−0.628936	−0.314468	−	0.949268i	$-0.601826\pi$
$719$	−16.8644	−0.628936	−0.314468	+	0.949268i	$0.601826\pi$
$720$	0	0
$721$	19.1394	0.712790
$722$	0	0
$723$	10.5083	0.390808
$724$	0	0
$725$	2.66047	0.0988073
$726$	0	0
$727$	−0.348980	−0.0129429	−0.00647147	−	0.999979i	$-0.502060\pi$
$727$	−0.348980	−0.0129429	−0.00647147	+	0.999979i	$0.502060\pi$
$728$	0	0
$729$	−26.0052	−0.963156
$730$	0	0
$731$	−3.39055	−0.125404
$732$	0	0
$733$	3.97975	0.146996	0.0734978	−	0.997295i	$-0.476584\pi$
$733$	3.97975	0.146996	0.0734978	+	0.997295i	$0.476584\pi$
$734$	0	0
$735$	−2.43828	−0.0899374
$736$	0	0
$737$	0.432198	0.0159202
$738$	0	0
$739$	−23.5249	−0.865379	−0.432690	−	0.901543i	$-0.642435\pi$
$739$	−23.5249	−0.865379	−0.432690	+	0.901543i	$0.642435\pi$
$740$	0	0
$741$	17.5100	0.643247
$742$	0	0
$743$	48.4912	1.77897	0.889485	−	0.456963i	$-0.151063\pi$
$743$	48.4912	1.77897	0.889485	+	0.456963i	$0.151063\pi$
$744$	0	0
$745$	−9.10374	−0.333535
$746$	0	0
$747$	−13.0538	−0.477614
$748$	0	0
$749$	−13.8935	−0.507659
$750$	0	0
$751$	38.6192	1.40923	0.704617	−	0.709588i	$-0.251119\pi$
$751$	38.6192	1.40923	0.704617	+	0.709588i	$0.251119\pi$
$752$	0	0
$753$	43.5063	1.58546
$754$	0	0
$755$	17.4492	0.635040
$756$	0	0
$757$	4.62353	0.168045	0.0840225	−	0.996464i	$-0.473223\pi$
$757$	4.62353	0.168045	0.0840225	+	0.996464i	$0.473223\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	18.8614	0.683725	0.341863	−	0.939750i	$-0.388942\pi$
$761$	18.8614	0.683725	0.341863	+	0.939750i	$0.388942\pi$
$762$	0	0
$763$	1.45825	0.0527923
$764$	0	0
$765$	2.74301	0.0991737
$766$	0	0
$767$	12.3565	0.446167
$768$	0	0
$769$	−6.94427	−0.250417	−0.125208	−	0.992130i	$-0.539960\pi$
$769$	−6.94427	−0.250417	−0.125208	+	0.992130i	$0.539960\pi$
$770$	0	0
$771$	−21.5462	−0.775968
$772$	0	0
$773$	27.4307	0.986612	0.493306	−	0.869856i	$-0.335788\pi$
$773$	27.4307	0.986612	0.493306	+	0.869856i	$0.335788\pi$
$774$	0	0
$775$	−7.04774	−0.253162
$776$	0	0
$777$	−1.94523	−0.0697846
$778$	0	0
$779$	18.1465	0.650165
$780$	0	0
$781$	0.308957	0.0110554
$782$	0	0
$783$	0.355323	0.0126982
$784$	0	0
$785$	12.5909	0.449387
$786$	0	0
$787$	−9.80867	−0.349641	−0.174821	−	0.984600i	$-0.555935\pi$
$787$	−9.80867	−0.349641	−0.174821	+	0.984600i	$0.555935\pi$
$788$	0	0
$789$	13.3886	0.476648
$790$	0	0
$791$	3.86268	0.137341
$792$	0	0
$793$	1.01388	0.0360041
$794$	0	0
$795$	−14.3626	−0.509388
$796$	0	0
$797$	−13.9673	−0.494748	−0.247374	−	0.968920i	$-0.579568\pi$
$797$	−13.9673	−0.494748	−0.247374	+	0.968920i	$0.579568\pi$
$798$	0	0
$799$	0.981468	0.0347218
$800$	0	0
$801$	−19.5807	−0.691851
$802$	0	0
$803$	11.0064	0.388406
$804$	0	0
$805$	−2.65438	−0.0935547
$806$	0	0
$807$	76.8349	2.70472
$808$	0	0
$809$	−6.10688	−0.214707	−0.107353	−	0.994221i	$-0.534238\pi$
$809$	−6.10688	−0.214707	−0.107353	+	0.994221i	$0.534238\pi$
$810$	0	0
$811$	26.0252	0.913870	0.456935	−	0.889500i	$-0.348947\pi$
$811$	26.0252	0.913870	0.456935	+	0.889500i	$0.348947\pi$
$812$	0	0
$813$	51.7315	1.81430
$814$	0	0
$815$	−5.74396	−0.201202
$816$	0	0
$817$	26.1435	0.914645
$818$	0	0
$819$	−2.94523	−0.102914
$820$	0	0
$821$	24.8328	0.866671	0.433336	−	0.901233i	$-0.357336\pi$
$821$	24.8328	0.866671	0.433336	+	0.901233i	$0.357336\pi$
$822$	0	0
$823$	−17.7460	−0.618585	−0.309293	−	0.950967i	$-0.600092\pi$
$823$	−17.7460	−0.618585	−0.309293	+	0.950967i	$0.600092\pi$
$824$	0	0
$825$	−3.01388	−0.104930
$826$	0	0
$827$	−4.02434	−0.139940	−0.0699700	−	0.997549i	$-0.522290\pi$
$827$	−4.02434	−0.139940	−0.0699700	+	0.997549i	$0.522290\pi$
$828$	0	0
$829$	−11.9060	−0.413514	−0.206757	−	0.978392i	$-0.566291\pi$
$829$	−11.9060	−0.413514	−0.206757	+	0.978392i	$0.566291\pi$
$830$	0	0
$831$	13.4442	0.466373
$832$	0	0
$833$	0.931341	0.0322691
$834$	0	0
$835$	2.56952	0.0889218
$836$	0	0
$837$	−0.941272	−0.0325351
$838$	0	0
$839$	19.4876	0.672786	0.336393	−	0.941722i	$-0.390793\pi$
$839$	19.4876	0.672786	0.336393	+	0.941722i	$0.390793\pi$
$840$	0	0
$841$	−21.9219	−0.755928
$842$	0	0
$843$	22.1975	0.764523
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	9.47214	0.325466
$848$	0	0
$849$	−61.7787	−2.12024
$850$	0	0
$851$	−2.11763	−0.0725913
$852$	0	0
$853$	−42.6218	−1.45934	−0.729671	−	0.683798i	$-0.760327\pi$
$853$	−42.6218	−1.45934	−0.729671	+	0.683798i	$0.760327\pi$
$854$	0	0
$855$	−21.1505	−0.723333
$856$	0	0
$857$	20.3718	0.695886	0.347943	−	0.937516i	$-0.386880\pi$
$857$	20.3718	0.695886	0.347943	+	0.937516i	$0.386880\pi$
$858$	0	0
$859$	4.16973	0.142269	0.0711347	−	0.997467i	$-0.477338\pi$
$859$	4.16973	0.142269	0.0711347	+	0.997467i	$0.477338\pi$
$860$	0	0
$861$	−6.16132	−0.209977
$862$	0	0
$863$	14.7678	0.502701	0.251350	−	0.967896i	$-0.419125\pi$
$863$	14.7678	0.502701	0.251350	+	0.967896i	$0.419125\pi$
$864$	0	0
$865$	−6.48321	−0.220436
$866$	0	0
$867$	39.3359	1.33592
$868$	0	0
$869$	−8.34851	−0.283204
$870$	0	0
$871$	0.349656	0.0118476
$872$	0	0
$873$	−12.6494	−0.428117
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−6.08119	−0.205347	−0.102674	−	0.994715i	$-0.532740\pi$
$877$	−6.08119	−0.205347	−0.102674	+	0.994715i	$0.532740\pi$
$878$	0	0
$879$	21.3088	0.718727
$880$	0	0
$881$	−26.2868	−0.885625	−0.442812	−	0.896614i	$-0.646019\pi$
$881$	−26.2868	−0.885625	−0.442812	+	0.896614i	$0.646019\pi$
$882$	0	0
$883$	31.4092	1.05700	0.528502	−	0.848932i	$-0.322754\pi$
$883$	31.4092	1.05700	0.528502	+	0.848932i	$0.322754\pi$
$884$	0	0
$885$	−30.1286	−1.01276
$886$	0	0
$887$	−18.2685	−0.613398	−0.306699	−	0.951807i	$-0.599224\pi$
$887$	−18.2685	−0.613398	−0.306699	+	0.951807i	$0.599224\pi$
$888$	0	0
$889$	−1.23607	−0.0414564
$890$	0	0
$891$	−11.3240	−0.379369
$892$	0	0
$893$	−7.56780	−0.253247
$894$	0	0
$895$	−15.1444	−0.506223
$896$	0	0
$897$	−6.47214	−0.216098
$898$	0	0
$899$	−18.7503	−0.625357
$900$	0	0
$901$	5.48602	0.182766
$902$	0	0
$903$	−8.87657	−0.295394
$904$	0	0
$905$	−3.41079	−0.113379
$906$	0	0
$907$	28.8297	0.957276	0.478638	−	0.878012i	$-0.341131\pi$
$907$	28.8297	0.957276	0.478638	+	0.878012i	$0.341131\pi$
$908$	0	0
$909$	−15.2891	−0.507107
$910$	0	0
$911$	−19.1444	−0.634284	−0.317142	−	0.948378i	$-0.602723\pi$
$911$	−19.1444	−0.634284	−0.317142	+	0.948378i	$0.602723\pi$
$912$	0	0
$913$	−5.47850	−0.181312
$914$	0	0
$915$	−2.47214	−0.0817263
$916$	0	0
$917$	10.9165	0.360495
$918$	0	0
$919$	15.8197	0.521845	0.260923	−	0.965360i	$-0.415973\pi$
$919$	15.8197	0.521845	0.260923	+	0.965360i	$0.415973\pi$
$920$	0	0
$921$	7.51379	0.247588
$922$	0	0
$923$	0.249952	0.00822726
$924$	0	0
$925$	−0.797785	−0.0262310
$926$	0	0
$927$	−56.3700	−1.85143
$928$	0	0
$929$	32.3399	1.06104	0.530519	−	0.847673i	$-0.321997\pi$
$929$	32.3399	1.06104	0.530519	+	0.847673i	$0.321997\pi$
$930$	0	0
$931$	−7.18129	−0.235357
$932$	0	0
$933$	−2.92403	−0.0957283
$934$	0	0
$935$	1.15120	0.0376483
$936$	0	0
$937$	57.4929	1.87821	0.939106	−	0.343626i	$-0.111655\pi$
$937$	57.4929	1.87821	0.939106	+	0.343626i	$0.111655\pi$
$938$	0	0
$939$	63.1209	2.05987
$940$	0	0
$941$	13.9236	0.453897	0.226949	−	0.973907i	$-0.427125\pi$
$941$	13.9236	0.453897	0.226949	+	0.973907i	$0.427125\pi$
$942$	0	0
$943$	−6.70739	−0.218423
$944$	0	0
$945$	−0.133557	−0.00434460
$946$	0	0
$947$	−18.3836	−0.597387	−0.298693	−	0.954349i	$-0.596551\pi$
$947$	−18.3836	−0.597387	−0.298693	+	0.954349i	$0.596551\pi$
$948$	0	0
$949$	8.90433	0.289047
$950$	0	0
$951$	21.4258	0.694780
$952$	0	0
$953$	−28.8625	−0.934948	−0.467474	−	0.884007i	$-0.654836\pi$
$953$	−28.8625	−0.934948	−0.467474	+	0.884007i	$0.654836\pi$
$954$	0	0
$955$	−8.32565	−0.269412
$956$	0	0
$957$	−8.01834	−0.259196
$958$	0	0
$959$	−19.8597	−0.641303
$960$	0	0
$961$	18.6706	0.602277
$962$	0	0
$963$	40.9196	1.31862
$964$	0	0
$965$	−18.8199	−0.605834
$966$	0	0
$967$	−28.5989	−0.919680	−0.459840	−	0.888002i	$-0.652093\pi$
$967$	−28.5989	−0.919680	−0.459840	+	0.888002i	$0.652093\pi$
$968$	0	0
$969$	16.3078	0.523882
$970$	0	0
$971$	43.6432	1.40058	0.700288	−	0.713860i	$-0.253055\pi$
$971$	43.6432	1.40058	0.700288	+	0.713860i	$0.253055\pi$
$972$	0	0
$973$	15.9582	0.511595
$974$	0	0
$975$	−2.43828	−0.0780876
$976$	0	0
$977$	−31.7186	−1.01477	−0.507384	−	0.861720i	$-0.669387\pi$
$977$	−31.7186	−1.01477	−0.507384	+	0.861720i	$0.669387\pi$
$978$	0	0
$979$	−8.21775	−0.262640
$980$	0	0
$981$	−4.29488	−0.137125
$982$	0	0
$983$	53.7533	1.71446	0.857232	−	0.514930i	$-0.172182\pi$
$983$	53.7533	1.71446	0.857232	+	0.514930i	$0.172182\pi$
$984$	0	0
$985$	4.88764	0.155733
$986$	0	0
$987$	2.56952	0.0817886
$988$	0	0
$989$	−9.66327	−0.307274
$990$	0	0
$991$	4.98126	0.158235	0.0791175	−	0.996865i	$-0.474790\pi$
$991$	4.98126	0.158235	0.0791175	+	0.996865i	$0.474790\pi$
$992$	0	0
$993$	58.6420	1.86095
$994$	0	0
$995$	−24.3151	−0.770841
$996$	0	0
$997$	34.5764	1.09505	0.547524	−	0.836790i	$-0.315571\pi$
$997$	34.5764	1.09505	0.547524	+	0.836790i	$0.315571\pi$
$998$	0	0
$999$	−0.106549	−0.00337107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.s.1.1	✓	4
4.3	odd	2		7280.2.a.ca.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.s.1.1	✓	4	1.1	even	1	trivial
7280.2.a.ca.1.4		4	4.3	odd	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43828 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.94523 q^{9} +O(q^{10})\)	\(q-2.43828 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.94523 q^{9} +1.23607 q^{11} +1.00000 q^{13} -2.43828 q^{15} +0.931341 q^{17} -7.18129 q^{19} +2.43828 q^{21} +2.65438 q^{23} +1.00000 q^{25} +0.133557 q^{27} +2.66047 q^{29} -7.04774 q^{31} -3.01388 q^{33} -1.00000 q^{35} -0.797785 q^{37} -2.43828 q^{39} -2.52691 q^{41} -3.64050 q^{43} +2.94523 q^{45} +1.05382 q^{47} +1.00000 q^{49} -2.27087 q^{51} +5.89045 q^{53} +1.23607 q^{55} +17.5100 q^{57} +12.3565 q^{59} +1.01388 q^{61} -2.94523 q^{63} +1.00000 q^{65} +0.349656 q^{67} -6.47214 q^{69} +0.249952 q^{71} +8.90433 q^{73} -2.43828 q^{75} -1.23607 q^{77} -6.75409 q^{79} -9.16132 q^{81} -4.43220 q^{83} +0.931341 q^{85} -6.48697 q^{87} -6.64830 q^{89} -1.00000 q^{91} +17.1844 q^{93} -7.18129 q^{95} -4.29488 q^{97} +3.64050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * q^99

Introduction

L-functions

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