LMFDB - Embedded newform 351.2.a.f.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [351,2,Mod(1,351)] 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("351.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.80274911095$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7600.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 19 $ x^4 - 9*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.37024$ of defining polynomial
Character	$\chi$	$=$	351.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.37024 q^{2} +3.61803 q^{4} +3.83513 q^{5} -4.47214 q^{7} +3.83513 q^{8} +9.09017 q^{10} -2.92978 q^{11} -1.00000 q^{13} -10.6000 q^{14} +1.85410 q^{16} -4.74048 q^{17} +0.763932 q^{19} +13.8756 q^{20} -6.94427 q^{22} +2.92978 q^{23} +9.70820 q^{25} -2.37024 q^{26} -16.1803 q^{28} +2.92978 q^{29} +7.23607 q^{31} -3.27559 q^{32} -11.2361 q^{34} -17.1512 q^{35} +3.23607 q^{37} +1.81070 q^{38} +14.7082 q^{40} -4.74048 q^{41} +0.236068 q^{43} -10.6000 q^{44} +6.94427 q^{46} -3.83513 q^{47} +13.0000 q^{49} +23.0108 q^{50} -3.61803 q^{52} +5.85955 q^{53} -11.2361 q^{55} -17.1512 q^{56} +6.94427 q^{58} -2.02443 q^{59} +10.2361 q^{61} +17.1512 q^{62} -11.4721 q^{64} -3.83513 q^{65} -2.76393 q^{67} -17.1512 q^{68} -40.6525 q^{70} -8.57561 q^{71} -10.4721 q^{73} +7.67026 q^{74} +2.76393 q^{76} +13.1024 q^{77} -8.94427 q^{79} +7.11072 q^{80} -11.2361 q^{82} +0.905351 q^{83} -18.1803 q^{85} +0.559538 q^{86} -11.2361 q^{88} +10.3863 q^{89} +4.47214 q^{91} +10.6000 q^{92} -9.09017 q^{94} +2.92978 q^{95} +18.1803 q^{97} +30.8131 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 14 q^{10} - 4 q^{13} - 6 q^{16} + 12 q^{19} + 8 q^{22} + 12 q^{25} - 20 q^{28} + 20 q^{31} - 36 q^{34} + 4 q^{37} + 32 q^{40} - 8 q^{43} - 8 q^{46} + 52 q^{49} - 10 q^{52} - 36 q^{55} - 8 q^{58}+ \cdots + 28 q^{97}+O(q^{100}) $ 4 * q + 10 * q^4 + 14 * q^10 - 4 * q^13 - 6 * q^16 + 12 * q^19 + 8 * q^22 + 12 * q^25 - 20 * q^28 + 20 * q^31 - 36 * q^34 + 4 * q^37 + 32 * q^40 - 8 * q^43 - 8 * q^46 + 52 * q^49 - 10 * q^52 - 36 * q^55 - 8 * q^58 + 32 * q^61 - 28 * q^64 - 20 * q^67 - 100 * q^70 - 24 * q^73 + 20 * q^76 - 36 * q^82 - 28 * q^85 - 36 * q^88 - 14 * q^94 + 28 * q^97

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 $

$2$	2.37024	1.67601	0.838006	−	0.545661i	$-0.183721\pi$
$2$	2.37024	1.67601	0.838006	+	0.545661i	$0.183721\pi$
$3$	0	0
$3$	0	0
$4$	3.61803	1.80902
$5$	3.83513	1.71512	0.857561	−	0.514383i	$-0.171979\pi$
$5$	3.83513	1.71512	0.857561	+	0.514383i	$0.171979\pi$
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	3.83513	1.35592
$9$	0	0
$10$	9.09017	2.87456
$11$	−2.92978	−0.883361	−0.441680	−	0.897172i	$-0.645618\pi$
$11$	−2.92978	−0.883361	−0.441680	+	0.897172i	$0.645618\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−10.6000	−2.83298
$15$	0	0
$16$	1.85410	0.463525
$17$	−4.74048	−1.14973	−0.574867	−	0.818247i	$-0.694946\pi$
$17$	−4.74048	−1.14973	−0.574867	+	0.818247i	$0.694946\pi$
$18$	0	0
$19$	0.763932	0.175258	0.0876290	−	0.996153i	$-0.472071\pi$
$19$	0.763932	0.175258	0.0876290	+	0.996153i	$0.472071\pi$
$20$	13.8756	3.10268
$21$	0	0
$22$	−6.94427	−1.48052
$23$	2.92978	0.610901	0.305450	−	0.952208i	$-0.401193\pi$
$23$	2.92978	0.610901	0.305450	+	0.952208i	$0.401193\pi$
$24$	0	0
$25$	9.70820	1.94164
$26$	−2.37024	−0.464842
$27$	0	0
$28$	−16.1803	−3.05780
$29$	2.92978	0.544046	0.272023	−	0.962291i	$-0.412307\pi$
$29$	2.92978	0.544046	0.272023	+	0.962291i	$0.412307\pi$
$30$	0	0
$31$	7.23607	1.29964	0.649818	−	0.760090i	$-0.274845\pi$
$31$	7.23607	1.29964	0.649818	+	0.760090i	$0.274845\pi$
$32$	−3.27559	−0.579048
$33$	0	0
$34$	−11.2361	−1.92697
$35$	−17.1512	−2.89908
$36$	0	0
$37$	3.23607	0.532006	0.266003	−	0.963972i	$-0.414297\pi$
$37$	3.23607	0.532006	0.266003	+	0.963972i	$0.414297\pi$
$38$	1.81070	0.293735
$39$	0	0
$40$	14.7082	2.32557
$41$	−4.74048	−0.740338	−0.370169	−	0.928964i	$-0.620700\pi$
$41$	−4.74048	−0.740338	−0.370169	+	0.928964i	$0.620700\pi$
$42$	0	0
$43$	0.236068	0.0360000	0.0180000	−	0.999838i	$-0.494270\pi$
$43$	0.236068	0.0360000	0.0180000	+	0.999838i	$0.494270\pi$
$44$	−10.6000	−1.59801
$45$	0	0
$46$	6.94427	1.02388
$47$	−3.83513	−0.559411	−0.279705	−	0.960086i	$-0.590237\pi$
$47$	−3.83513	−0.559411	−0.279705	+	0.960086i	$0.590237\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	23.0108	3.25421
$51$	0	0
$52$	−3.61803	−0.501731
$53$	5.85955	0.804872	0.402436	−	0.915448i	$-0.368164\pi$
$53$	5.85955	0.804872	0.402436	+	0.915448i	$0.368164\pi$
$54$	0	0
$55$	−11.2361	−1.51507
$56$	−17.1512	−2.29193
$57$	0	0
$58$	6.94427	0.911828
$59$	−2.02443	−0.263558	−0.131779	−	0.991279i	$-0.542069\pi$
$59$	−2.02443	−0.263558	−0.131779	+	0.991279i	$0.542069\pi$
$60$	0	0
$61$	10.2361	1.31059	0.655297	−	0.755371i	$-0.272543\pi$
$61$	10.2361	1.31059	0.655297	+	0.755371i	$0.272543\pi$
$62$	17.1512	2.17821
$63$	0	0
$64$	−11.4721	−1.43402
$65$	−3.83513	−0.475689
$66$	0	0
$67$	−2.76393	−0.337668	−0.168834	−	0.985644i	$-0.554000\pi$
$67$	−2.76393	−0.337668	−0.168834	+	0.985644i	$0.554000\pi$
$68$	−17.1512	−2.07989
$69$	0	0
$70$	−40.6525	−4.85890
$71$	−8.57561	−1.01774	−0.508869	−	0.860844i	$-0.669936\pi$
$71$	−8.57561	−1.01774	−0.508869	+	0.860844i	$0.669936\pi$
$72$	0	0
$73$	−10.4721	−1.22567	−0.612835	−	0.790211i	$-0.709971\pi$
$73$	−10.4721	−1.22567	−0.612835	+	0.790211i	$0.709971\pi$
$74$	7.67026	0.891649
$75$	0	0
$76$	2.76393	0.317045
$77$	13.1024	1.49315
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	7.11072	0.795002
$81$	0	0
$82$	−11.2361	−1.24082
$83$	0.905351	0.0993752	0.0496876	−	0.998765i	$-0.484177\pi$
$83$	0.905351	0.0993752	0.0496876	+	0.998765i	$0.484177\pi$
$84$	0	0
$85$	−18.1803	−1.97193
$86$	0.559538	0.0603365
$87$	0	0
$88$	−11.2361	−1.19777
$89$	10.3863	1.10095	0.550473	−	0.834853i	$-0.314447\pi$
$89$	10.3863	1.10095	0.550473	+	0.834853i	$0.314447\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	10.6000	1.10513
$93$	0	0
$94$	−9.09017	−0.937579
$95$	2.92978	0.300589
$96$	0	0
$97$	18.1803	1.84593	0.922967	−	0.384879i	$-0.125757\pi$
$97$	18.1803	1.84593	0.922967	+	0.384879i	$0.125757\pi$
$98$	30.8131	3.11259
$99$	0	0
$100$	35.1246	3.51246
$101$	−4.74048	−0.471695	−0.235848	−	0.971790i	$-0.575787\pi$
$101$	−4.74048	−0.471695	−0.235848	+	0.971790i	$0.575787\pi$
$102$	0	0
$103$	−1.76393	−0.173805	−0.0869027	−	0.996217i	$-0.527697\pi$
$103$	−1.76393	−0.173805	−0.0869027	+	0.996217i	$0.527697\pi$
$104$	−3.83513	−0.376065
$105$	0	0
$106$	13.8885	1.34897
$107$	−7.67026	−0.741512	−0.370756	−	0.928730i	$-0.620901\pi$
$107$	−7.67026	−0.741512	−0.370756	+	0.928730i	$0.620901\pi$
$108$	0	0
$109$	8.94427	0.856706	0.428353	−	0.903612i	$-0.359094\pi$
$109$	8.94427	0.856706	0.428353	+	0.903612i	$0.359094\pi$
$110$	−26.6322	−2.53928
$111$	0	0
$112$	−8.29180	−0.783501
$113$	−7.67026	−0.721557	−0.360778	−	0.932652i	$-0.617489\pi$
$113$	−7.67026	−0.721557	−0.360778	+	0.932652i	$0.617489\pi$
$114$	0	0
$115$	11.2361	1.04777
$116$	10.6000	0.984188
$117$	0	0
$118$	−4.79837	−0.441726
$119$	21.2001	1.94341
$120$	0	0
$121$	−2.41641	−0.219673
$122$	24.2619	2.19657
$123$	0	0
$124$	26.1803	2.35106
$125$	18.0566	1.61503
$126$	0	0
$127$	−5.94427	−0.527469	−0.263734	−	0.964595i	$-0.584954\pi$
$127$	−5.94427	−0.527469	−0.263734	+	0.964595i	$0.584954\pi$
$128$	−20.6405	−1.82438
$129$	0	0
$130$	−9.09017	−0.797261
$131$	20.0810	1.75448	0.877242	−	0.480048i	$-0.159381\pi$
$131$	20.0810	1.75448	0.877242	+	0.480048i	$0.159381\pi$
$132$	0	0
$133$	−3.41641	−0.296240
$134$	−6.55118	−0.565936
$135$	0	0
$136$	−18.1803	−1.55895
$137$	−10.6000	−0.905622	−0.452811	−	0.891607i	$-0.649579\pi$
$137$	−10.6000	−0.905622	−0.452811	+	0.891607i	$0.649579\pi$
$138$	0	0
$139$	3.18034	0.269753	0.134876	−	0.990862i	$-0.456936\pi$
$139$	3.18034	0.269753	0.134876	+	0.990862i	$0.456936\pi$
$140$	−62.0537	−5.24449
$141$	0	0
$142$	−20.3262	−1.70574
$143$	2.92978	0.245000
$144$	0	0
$145$	11.2361	0.933105
$146$	−24.8215	−2.05424
$147$	0	0
$148$	11.7082	0.962408
$149$	4.95420	0.405864	0.202932	−	0.979193i	$-0.434953\pi$
$149$	4.95420	0.405864	0.202932	+	0.979193i	$0.434953\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	2.92978	0.237636
$153$	0	0
$154$	31.0557	2.50254
$155$	27.7512	2.22903
$156$	0	0
$157$	24.4164	1.94864	0.974321	−	0.225165i	$-0.0722920\pi$
$157$	24.4164	1.94864	0.974321	+	0.225165i	$0.0722920\pi$
$158$	−21.2001	−1.68659
$159$	0	0
$160$	−12.5623	−0.993137
$161$	−13.1024	−1.03261
$162$	0	0
$163$	0.944272	0.0739611	0.0369805	−	0.999316i	$-0.488226\pi$
$163$	0.944272	0.0739611	0.0369805	+	0.999316i	$0.488226\pi$
$164$	−17.1512	−1.33928
$165$	0	0
$166$	2.14590	0.166554
$167$	4.52675	0.350291	0.175145	−	0.984543i	$-0.443960\pi$
$167$	4.52675	0.350291	0.175145	+	0.984543i	$0.443960\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−43.0918	−3.30499
$171$	0	0
$172$	0.854102	0.0651247
$173$	−15.3405	−1.16632	−0.583159	−	0.812358i	$-0.698183\pi$
$173$	−15.3405	−1.16632	−0.583159	+	0.812358i	$0.698183\pi$
$174$	0	0
$175$	−43.4164	−3.28197
$176$	−5.43210	−0.409460
$177$	0	0
$178$	24.6180	1.84520
$179$	20.7726	1.55262	0.776309	−	0.630352i	$-0.217090\pi$
$179$	20.7726	1.55262	0.776309	+	0.630352i	$0.217090\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	10.6000	0.785727
$183$	0	0
$184$	11.2361	0.828334
$185$	12.4107	0.912455
$186$	0	0
$187$	13.8885	1.01563
$188$	−13.8756	−1.01198
$189$	0	0
$190$	6.94427	0.503790
$191$	−23.7024	−1.71504	−0.857522	−	0.514447i	$-0.827997\pi$
$191$	−23.7024	−1.71504	−0.857522	+	0.514447i	$0.827997\pi$
$192$	0	0
$193$	−14.1803	−1.02072	−0.510362	−	0.859960i	$-0.670488\pi$
$193$	−14.1803	−1.02072	−0.510362	+	0.859960i	$0.670488\pi$
$194$	43.0918	3.09381
$195$	0	0
$196$	47.0344	3.35960
$197$	−14.4352	−1.02846	−0.514231	−	0.857652i	$-0.671923\pi$
$197$	−14.4352	−1.02846	−0.514231	+	0.857652i	$0.671923\pi$
$198$	0	0
$199$	24.8885	1.76430	0.882151	−	0.470967i	$-0.156095\pi$
$199$	24.8885	1.76430	0.882151	+	0.470967i	$0.156095\pi$
$200$	37.2322	2.63271
$201$	0	0
$202$	−11.2361	−0.790567
$203$	−13.1024	−0.919605
$204$	0	0
$205$	−18.1803	−1.26977
$206$	−4.18094	−0.291300
$207$	0	0
$208$	−1.85410	−0.128559
$209$	−2.23815	−0.154816
$210$	0	0
$211$	16.4164	1.13015	0.565076	−	0.825039i	$-0.308847\pi$
$211$	16.4164	1.13015	0.565076	+	0.825039i	$0.308847\pi$
$212$	21.2001	1.45603
$213$	0	0
$214$	−18.1803	−1.24278
$215$	0.905351	0.0617444
$216$	0	0
$217$	−32.3607	−2.19679
$218$	21.2001	1.43585
$219$	0	0
$220$	−40.6525	−2.74079
$221$	4.74048	0.318879
$222$	0	0
$223$	−15.1246	−1.01282	−0.506409	−	0.862293i	$-0.669028\pi$
$223$	−15.1246	−1.01282	−0.506409	+	0.862293i	$0.669028\pi$
$224$	14.6489	0.978770
$225$	0	0
$226$	−18.1803	−1.20934
$227$	18.0566	1.19846	0.599228	−	0.800578i	$-0.295474\pi$
$227$	18.0566	1.19846	0.599228	+	0.800578i	$0.295474\pi$
$228$	0	0
$229$	4.76393	0.314809	0.157405	−	0.987534i	$-0.449687\pi$
$229$	4.76393	0.314809	0.157405	+	0.987534i	$0.449687\pi$
$230$	26.6322	1.75607
$231$	0	0
$232$	11.2361	0.737684
$233$	−20.0810	−1.31555	−0.657775	−	0.753215i	$-0.728502\pi$
$233$	−20.0810	−1.31555	−0.657775	+	0.753215i	$0.728502\pi$
$234$	0	0
$235$	−14.7082	−0.959457
$236$	−7.32444	−0.476781
$237$	0	0
$238$	50.2492	3.25717
$239$	−14.6489	−0.947558	−0.473779	−	0.880644i	$-0.657110\pi$
$239$	−14.6489	−0.947558	−0.473779	+	0.880644i	$0.657110\pi$
$240$	0	0
$241$	13.5279	0.871406	0.435703	−	0.900090i	$-0.356500\pi$
$241$	13.5279	0.871406	0.435703	+	0.900090i	$0.356500\pi$
$242$	−5.72746	−0.368175
$243$	0	0
$244$	37.0344	2.37089
$245$	49.8567	3.18523
$246$	0	0
$247$	−0.763932	−0.0486078
$248$	27.7512	1.76221
$249$	0	0
$250$	42.7984	2.70681
$251$	−23.0108	−1.45243	−0.726213	−	0.687469i	$-0.758722\pi$
$251$	−23.0108	−1.45243	−0.726213	+	0.687469i	$0.758722\pi$
$252$	0	0
$253$	−8.58359	−0.539646
$254$	−14.0893	−0.884044
$255$	0	0
$256$	−25.9787	−1.62367
$257$	−27.7512	−1.73108	−0.865538	−	0.500844i	$-0.833023\pi$
$257$	−27.7512	−1.73108	−0.865538	+	0.500844i	$0.833023\pi$
$258$	0	0
$259$	−14.4721	−0.899255
$260$	−13.8756	−0.860530
$261$	0	0
$262$	47.5967	2.94054
$263$	28.4429	1.75386	0.876931	−	0.480616i	$-0.159587\pi$
$263$	28.4429	1.75386	0.876931	+	0.480616i	$0.159587\pi$
$264$	0	0
$265$	22.4721	1.38045
$266$	−8.09770	−0.496502
$267$	0	0
$268$	−10.0000	−0.610847
$269$	−5.43210	−0.331201	−0.165601	−	0.986193i	$-0.552956\pi$
$269$	−5.43210	−0.331201	−0.165601	+	0.986193i	$0.552956\pi$
$270$	0	0
$271$	−1.23607	−0.0750858	−0.0375429	−	0.999295i	$-0.511953\pi$
$271$	−1.23607	−0.0750858	−0.0375429	+	0.999295i	$0.511953\pi$
$272$	−8.78933	−0.532931
$273$	0	0
$274$	−25.1246	−1.51783
$275$	−28.4429	−1.71517
$276$	0	0
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	7.53817	0.452109
$279$	0	0
$280$	−65.7771	−3.93093
$281$	−19.1756	−1.14392	−0.571961	−	0.820281i	$-0.693817\pi$
$281$	−19.1756	−1.14392	−0.571961	+	0.820281i	$0.693817\pi$
$282$	0	0
$283$	4.41641	0.262528	0.131264	−	0.991347i	$-0.458096\pi$
$283$	4.41641	0.262528	0.131264	+	0.991347i	$0.458096\pi$
$284$	−31.0268	−1.84110
$285$	0	0
$286$	6.94427	0.410623
$287$	21.2001	1.25140
$288$	0	0
$289$	5.47214	0.321890
$290$	26.6322	1.56389
$291$	0	0
$292$	−37.8885	−2.21726
$293$	1.11908	0.0653771	0.0326885	−	0.999466i	$-0.489593\pi$
$293$	1.11908	0.0653771	0.0326885	+	0.999466i	$0.489593\pi$
$294$	0	0
$295$	−7.76393	−0.452034
$296$	12.4107	0.721359
$297$	0	0
$298$	11.7426	0.680233
$299$	−2.92978	−0.169433
$300$	0	0
$301$	−1.05573	−0.0608512
$302$	0	0
$303$	0	0
$304$	1.41641	0.0812366
$305$	39.2566	2.24783
$306$	0	0
$307$	−22.9443	−1.30950	−0.654749	−	0.755846i	$-0.727226\pi$
$307$	−22.9443	−1.30950	−0.654749	+	0.755846i	$0.727226\pi$
$308$	47.4048	2.70114
$309$	0	0
$310$	65.7771	3.73589
$311$	−15.3405	−0.869881	−0.434940	−	0.900459i	$-0.643231\pi$
$311$	−15.3405	−0.869881	−0.434940	+	0.900459i	$0.643231\pi$
$312$	0	0
$313$	30.7082	1.73573	0.867865	−	0.496800i	$-0.165492\pi$
$313$	30.7082	1.73573	0.867865	+	0.496800i	$0.165492\pi$
$314$	57.8727	3.26595
$315$	0	0
$316$	−32.3607	−1.82043
$317$	−14.2214	−0.798755	−0.399378	−	0.916786i	$-0.630774\pi$
$317$	−14.2214	−0.798755	−0.399378	+	0.916786i	$0.630774\pi$
$318$	0	0
$319$	−8.58359	−0.480589
$320$	−43.9971	−2.45951
$321$	0	0
$322$	−31.0557	−1.73067
$323$	−3.62140	−0.201500
$324$	0	0
$325$	−9.70820	−0.538514
$326$	2.23815	0.123960
$327$	0	0
$328$	−18.1803	−1.00384
$329$	17.1512	0.945577
$330$	0	0
$331$	−26.9443	−1.48099	−0.740496	−	0.672061i	$-0.765409\pi$
$331$	−26.9443	−1.48099	−0.740496	+	0.672061i	$0.765409\pi$
$332$	3.27559	0.179771
$333$	0	0
$334$	10.7295	0.587092
$335$	−10.6000	−0.579142
$336$	0	0
$337$	7.65248	0.416857	0.208428	−	0.978038i	$-0.433165\pi$
$337$	7.65248	0.416857	0.208428	+	0.978038i	$0.433165\pi$
$338$	2.37024	0.128924
$339$	0	0
$340$	−65.7771	−3.56726
$341$	−21.2001	−1.14805
$342$	0	0
$343$	−26.8328	−1.44884
$344$	0.905351	0.0488132
$345$	0	0
$346$	−36.3607	−1.95476
$347$	27.7512	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$347$	27.7512	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$348$	0	0
$349$	0.180340	0.00965337	0.00482669	−	0.999988i	$-0.498464\pi$
$349$	0.180340	0.00965337	0.00482669	+	0.999988i	$0.498464\pi$
$350$	−102.907	−5.50063
$351$	0	0
$352$	9.59675	0.511508
$353$	−28.6566	−1.52524	−0.762618	−	0.646849i	$-0.776087\pi$
$353$	−28.6566	−1.52524	−0.762618	+	0.646849i	$0.776087\pi$
$354$	0	0
$355$	−32.8885	−1.74554
$356$	37.5780	1.99163
$357$	0	0
$358$	49.2361	2.60221
$359$	25.7268	1.35781	0.678905	−	0.734226i	$-0.262455\pi$
$359$	25.7268	1.35781	0.678905	+	0.734226i	$0.262455\pi$
$360$	0	0
$361$	−18.4164	−0.969285
$362$	−16.5917	−0.872039
$363$	0	0
$364$	16.1803	0.848080
$365$	−40.1620	−2.10217
$366$	0	0
$367$	7.41641	0.387133	0.193567	−	0.981087i	$-0.437994\pi$
$367$	7.41641	0.387133	0.193567	+	0.981087i	$0.437994\pi$
$368$	5.43210	0.283168
$369$	0	0
$370$	29.4164	1.52929
$371$	−26.2047	−1.36048
$372$	0	0
$373$	−8.70820	−0.450894	−0.225447	−	0.974255i	$-0.572384\pi$
$373$	−8.70820	−0.450894	−0.225447	+	0.974255i	$0.572384\pi$
$374$	32.9192	1.70221
$375$	0	0
$376$	−14.7082	−0.758518
$377$	−2.92978	−0.150891
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	10.6000	0.543770
$381$	0	0
$382$	−56.1803	−2.87444
$383$	−25.5131	−1.30366	−0.651829	−	0.758366i	$-0.725998\pi$
$383$	−25.5131	−1.30366	−0.651829	+	0.758366i	$0.725998\pi$
$384$	0	0
$385$	50.2492	2.56094
$386$	−33.6108	−1.71074
$387$	0	0
$388$	65.7771	3.33933
$389$	−11.2917	−0.572510	−0.286255	−	0.958153i	$-0.592410\pi$
$389$	−11.2917	−0.572510	−0.286255	+	0.958153i	$0.592410\pi$
$390$	0	0
$391$	−13.8885	−0.702374
$392$	49.8567	2.51814
$393$	0	0
$394$	−34.2148	−1.72372
$395$	−34.3024	−1.72594
$396$	0	0
$397$	−0.291796	−0.0146448	−0.00732241	−	0.999973i	$-0.502331\pi$
$397$	−0.291796	−0.0146448	−0.00732241	+	0.999973i	$0.502331\pi$
$398$	58.9918	2.95699
$399$	0	0
$400$	18.0000	0.900000
$401$	12.6245	0.630435	0.315218	−	0.949019i	$-0.397922\pi$
$401$	12.6245	0.630435	0.315218	+	0.949019i	$0.397922\pi$
$402$	0	0
$403$	−7.23607	−0.360454
$404$	−17.1512	−0.853305
$405$	0	0
$406$	−31.0557	−1.54127
$407$	−9.48096	−0.469954
$408$	0	0
$409$	21.7082	1.07340	0.536701	−	0.843773i	$-0.319670\pi$
$409$	21.7082	1.07340	0.536701	+	0.843773i	$0.319670\pi$
$410$	−43.0918	−2.12815
$411$	0	0
$412$	−6.38197	−0.314417
$413$	9.05351	0.445494
$414$	0	0
$415$	3.47214	0.170440
$416$	3.27559	0.160599
$417$	0	0
$418$	−5.30495	−0.259474
$419$	17.8428	0.871680	0.435840	−	0.900024i	$-0.356451\pi$
$419$	17.8428	0.871680	0.435840	+	0.900024i	$0.356451\pi$
$420$	0	0
$421$	−39.0132	−1.90138	−0.950692	−	0.310135i	$-0.899626\pi$
$421$	−39.0132	−1.90138	−0.950692	+	0.310135i	$0.899626\pi$
$422$	38.9108	1.89415
$423$	0	0
$424$	22.4721	1.09134
$425$	−46.0215	−2.23237
$426$	0	0
$427$	−45.7771	−2.21531
$428$	−27.7512	−1.34141
$429$	0	0
$430$	2.14590	0.103484
$431$	13.3161	0.641413	0.320707	−	0.947179i	$-0.396080\pi$
$431$	13.3161	0.641413	0.320707	+	0.947179i	$0.396080\pi$
$432$	0	0
$433$	24.4721	1.17606	0.588028	−	0.808841i	$-0.299905\pi$
$433$	24.4721	1.17606	0.588028	+	0.808841i	$0.299905\pi$
$434$	−76.7026	−3.68184
$435$	0	0
$436$	32.3607	1.54980
$437$	2.23815	0.107065
$438$	0	0
$439$	24.7082	1.17926	0.589629	−	0.807674i	$-0.299274\pi$
$439$	24.7082	1.17926	0.589629	+	0.807674i	$0.299274\pi$
$440$	−43.0918	−2.05432
$441$	0	0
$442$	11.2361	0.534445
$443$	2.92978	0.139198	0.0695989	−	0.997575i	$-0.477828\pi$
$443$	2.92978	0.139198	0.0695989	+	0.997575i	$0.477828\pi$
$444$	0	0
$445$	39.8328	1.88826
$446$	−35.8489	−1.69750
$447$	0	0
$448$	51.3050	2.42393
$449$	3.83513	0.180991	0.0904954	−	0.995897i	$-0.471155\pi$
$449$	3.83513	0.180991	0.0904954	+	0.995897i	$0.471155\pi$
$450$	0	0
$451$	13.8885	0.653986
$452$	−27.7512	−1.30531
$453$	0	0
$454$	42.7984	2.00863
$455$	17.1512	0.804061
$456$	0	0
$457$	−7.34752	−0.343703	−0.171851	−	0.985123i	$-0.554975\pi$
$457$	−7.34752	−0.343703	−0.171851	+	0.985123i	$0.554975\pi$
$458$	11.2917	0.527625
$459$	0	0
$460$	40.6525	1.89543
$461$	35.2078	1.63979	0.819895	−	0.572514i	$-0.194032\pi$
$461$	35.2078	1.63979	0.819895	+	0.572514i	$0.194032\pi$
$462$	0	0
$463$	−0.291796	−0.0135609	−0.00678046	−	0.999977i	$-0.502158\pi$
$463$	−0.291796	−0.0135609	−0.00678046	+	0.999977i	$0.502158\pi$
$464$	5.43210	0.252179
$465$	0	0
$466$	−47.5967	−2.20488
$467$	37.6596	1.74268	0.871340	−	0.490679i	$-0.163251\pi$
$467$	37.6596	1.74268	0.871340	+	0.490679i	$0.163251\pi$
$468$	0	0
$469$	12.3607	0.570763
$470$	−34.8620	−1.60806
$471$	0	0
$472$	−7.76393	−0.357364
$473$	−0.691626	−0.0318010
$474$	0	0
$475$	7.41641	0.340288
$476$	76.7026	3.51566
$477$	0	0
$478$	−34.7214	−1.58812
$479$	−32.7054	−1.49435	−0.747175	−	0.664627i	$-0.768590\pi$
$479$	−32.7054	−1.49435	−0.747175	+	0.664627i	$0.768590\pi$
$480$	0	0
$481$	−3.23607	−0.147552
$482$	32.0643	1.46049
$483$	0	0
$484$	−8.74265	−0.397393
$485$	69.7239	3.16600
$486$	0	0
$487$	16.4721	0.746424	0.373212	−	0.927746i	$-0.378256\pi$
$487$	16.4721	0.746424	0.373212	+	0.927746i	$0.378256\pi$
$488$	39.2566	1.77706
$489$	0	0
$490$	118.172	5.33848
$491$	41.2811	1.86299	0.931494	−	0.363757i	$-0.118506\pi$
$491$	41.2811	1.86299	0.931494	+	0.363757i	$0.118506\pi$
$492$	0	0
$493$	−13.8885	−0.625509
$494$	−1.81070	−0.0814673
$495$	0	0
$496$	13.4164	0.602414
$497$	38.3513	1.72029
$498$	0	0
$499$	−4.36068	−0.195211	−0.0976054	−	0.995225i	$-0.531118\pi$
$499$	−4.36068	−0.195211	−0.0976054	+	0.995225i	$0.531118\pi$
$500$	65.3293	2.92161
$501$	0	0
$502$	−54.5410	−2.43428
$503$	−10.6000	−0.472632	−0.236316	−	0.971676i	$-0.575940\pi$
$503$	−10.6000	−0.472632	−0.236316	+	0.971676i	$0.575940\pi$
$504$	0	0
$505$	−18.1803	−0.809015
$506$	−20.3452	−0.904453
$507$	0	0
$508$	−21.5066	−0.954200
$509$	−12.1970	−0.540623	−0.270311	−	0.962773i	$-0.587127\pi$
$509$	−12.1970	−0.540623	−0.270311	+	0.962773i	$0.587127\pi$
$510$	0	0
$511$	46.8328	2.07176
$512$	−20.2947	−0.896908
$513$	0	0
$514$	−65.7771	−2.90130
$515$	−6.76490	−0.298097
$516$	0	0
$517$	11.2361	0.494162
$518$	−34.3024	−1.50716
$519$	0	0
$520$	−14.7082	−0.644997
$521$	−6.97863	−0.305739	−0.152870	−	0.988246i	$-0.548851\pi$
$521$	−6.97863	−0.305739	−0.152870	+	0.988246i	$0.548851\pi$
$522$	0	0
$523$	−7.65248	−0.334619	−0.167310	−	0.985904i	$-0.553508\pi$
$523$	−7.65248	−0.334619	−0.167310	+	0.985904i	$0.553508\pi$
$524$	72.6537	3.17389
$525$	0	0
$526$	67.4164	2.93950
$527$	−34.3024	−1.49424
$528$	0	0
$529$	−14.4164	−0.626800
$530$	53.2643	2.31366
$531$	0	0
$532$	−12.3607	−0.535903
$533$	4.74048	0.205333
$534$	0	0
$535$	−29.4164	−1.27178
$536$	−10.6000	−0.457852
$537$	0	0
$538$	−12.8754	−0.555097
$539$	−38.0871	−1.64053
$540$	0	0
$541$	−23.7082	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$541$	−23.7082	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$542$	−2.92978	−0.125845
$543$	0	0
$544$	15.5279	0.665752
$545$	34.3024	1.46935
$546$	0	0
$547$	−25.3050	−1.08196	−0.540981	−	0.841035i	$-0.681947\pi$
$547$	−25.3050	−1.08196	−0.540981	+	0.841035i	$0.681947\pi$
$548$	−38.3513	−1.63829
$549$	0	0
$550$	−67.4164	−2.87465
$551$	2.23815	0.0953484
$552$	0	0
$553$	40.0000	1.70097
$554$	16.5917	0.704913
$555$	0	0
$556$	11.5066	0.487988
$557$	−16.4596	−0.697415	−0.348708	−	0.937232i	$-0.613379\pi$
$557$	−16.4596	−0.697415	−0.348708	+	0.937232i	$0.613379\pi$
$558$	0	0
$559$	−0.236068	−0.00998461
$560$	−31.8001	−1.34380
$561$	0	0
$562$	−45.4508	−1.91723
$563$	2.50233	0.105461	0.0527303	−	0.998609i	$-0.483208\pi$
$563$	2.50233	0.105461	0.0527303	+	0.998609i	$0.483208\pi$
$564$	0	0
$565$	−29.4164	−1.23756
$566$	10.4679	0.440000
$567$	0	0
$568$	−32.8885	−1.37997
$569$	5.85955	0.245645	0.122823	−	0.992429i	$-0.460805\pi$
$569$	5.85955	0.245645	0.122823	+	0.992429i	$0.460805\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	10.6000	0.443210
$573$	0	0
$574$	50.2492	2.09736
$575$	28.4429	1.18615
$576$	0	0
$577$	21.4164	0.891577	0.445788	−	0.895138i	$-0.352923\pi$
$577$	21.4164	0.891577	0.445788	+	0.895138i	$0.352923\pi$
$578$	12.9703	0.539492
$579$	0	0
$580$	40.6525	1.68800
$581$	−4.04885	−0.167975
$582$	0	0
$583$	−17.1672	−0.710992
$584$	−40.1620	−1.66191
$585$	0	0
$586$	2.65248	0.109573
$587$	−8.78933	−0.362774	−0.181387	−	0.983412i	$-0.558059\pi$
$587$	−8.78933	−0.362774	−0.181387	+	0.983412i	$0.558059\pi$
$588$	0	0
$589$	5.52786	0.227772
$590$	−18.4024	−0.757614
$591$	0	0
$592$	6.00000	0.246598
$593$	14.2214	0.584004	0.292002	−	0.956418i	$-0.405679\pi$
$593$	14.2214	0.584004	0.292002	+	0.956418i	$0.405679\pi$
$594$	0	0
$595$	81.3050	3.33318
$596$	17.9245	0.734215
$597$	0	0
$598$	−6.94427	−0.283972
$599$	1.11908	0.0457242	0.0228621	−	0.999739i	$-0.492722\pi$
$599$	1.11908	0.0457242	0.0228621	+	0.999739i	$0.492722\pi$
$600$	0	0
$601$	−2.59675	−0.105924	−0.0529618	−	0.998597i	$-0.516866\pi$
$601$	−2.59675	−0.105924	−0.0529618	+	0.998597i	$0.516866\pi$
$602$	−2.50233	−0.101987
$603$	0	0
$604$	0	0
$605$	−9.26723	−0.376767
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−2.50233	−0.101483
$609$	0	0
$610$	93.0476	3.76739
$611$	3.83513	0.155153
$612$	0	0
$613$	−3.05573	−0.123420	−0.0617098	−	0.998094i	$-0.519655\pi$
$613$	−3.05573	−0.123420	−0.0617098	+	0.998094i	$0.519655\pi$
$614$	−54.3834	−2.19474
$615$	0	0
$616$	50.2492	2.02460
$617$	−30.8947	−1.24378	−0.621888	−	0.783106i	$-0.713634\pi$
$617$	−30.8947	−1.24378	−0.621888	+	0.783106i	$0.713634\pi$
$618$	0	0
$619$	−45.3050	−1.82096	−0.910480	−	0.413553i	$-0.864288\pi$
$619$	−45.3050	−1.82096	−0.910480	+	0.413553i	$0.864288\pi$
$620$	100.405	4.03236
$621$	0	0
$622$	−36.3607	−1.45793
$623$	−46.4490	−1.86094
$624$	0	0
$625$	20.7082	0.828328
$626$	72.7858	2.90911
$627$	0	0
$628$	88.3394	3.52513
$629$	−15.3405	−0.611666
$630$	0	0
$631$	19.3050	0.768518	0.384259	−	0.923225i	$-0.374457\pi$
$631$	19.3050	0.768518	0.384259	+	0.923225i	$0.374457\pi$
$632$	−34.3024	−1.36448
$633$	0	0
$634$	−33.7082	−1.33872
$635$	−22.7970	−0.904673
$636$	0	0
$637$	−13.0000	−0.515079
$638$	−20.3452	−0.805473
$639$	0	0
$640$	−79.1591	−3.12904
$641$	33.6108	1.32755	0.663773	−	0.747934i	$-0.268954\pi$
$641$	33.6108	1.32755	0.663773	+	0.747934i	$0.268954\pi$
$642$	0	0
$643$	17.8197	0.702739	0.351369	−	0.936237i	$-0.385716\pi$
$643$	17.8197	0.702739	0.351369	+	0.936237i	$0.385716\pi$
$644$	−47.4048	−1.86801
$645$	0	0
$646$	−8.58359	−0.337717
$647$	25.2489	0.992637	0.496319	−	0.868140i	$-0.334685\pi$
$647$	25.2489	0.992637	0.496319	+	0.868140i	$0.334685\pi$
$648$	0	0
$649$	5.93112	0.232817
$650$	−23.0108	−0.902556
$651$	0	0
$652$	3.41641	0.133797
$653$	−24.8215	−0.971339	−0.485670	−	0.874142i	$-0.661424\pi$
$653$	−24.8215	−0.971339	−0.485670	+	0.874142i	$0.661424\pi$
$654$	0	0
$655$	77.0132	3.00915
$656$	−8.78933	−0.343166
$657$	0	0
$658$	40.6525	1.58480
$659$	9.90841	0.385977	0.192988	−	0.981201i	$-0.438182\pi$
$659$	9.90841	0.385977	0.192988	+	0.981201i	$0.438182\pi$
$660$	0	0
$661$	3.70820	0.144232	0.0721162	−	0.997396i	$-0.477025\pi$
$661$	3.70820	0.144232	0.0721162	+	0.997396i	$0.477025\pi$
$662$	−63.8644	−2.48216
$663$	0	0
$664$	3.47214	0.134745
$665$	−13.1024	−0.508088
$666$	0	0
$667$	8.58359	0.332358
$668$	16.3780	0.633682
$669$	0	0
$670$	−25.1246	−0.970648
$671$	−29.9894	−1.15773
$672$	0	0
$673$	−5.94427	−0.229135	−0.114567	−	0.993415i	$-0.536548\pi$
$673$	−5.94427	−0.229135	−0.114567	+	0.993415i	$0.536548\pi$
$674$	18.1382	0.698657
$675$	0	0
$676$	3.61803	0.139155
$677$	23.7024	0.910957	0.455478	−	0.890247i	$-0.349468\pi$
$677$	23.7024	0.910957	0.455478	+	0.890247i	$0.349468\pi$
$678$	0	0
$679$	−81.3050	−3.12020
$680$	−69.7239	−2.67379
$681$	0	0
$682$	−50.2492	−1.92414
$683$	19.8673	0.760200	0.380100	−	0.924945i	$-0.375890\pi$
$683$	19.8673	0.760200	0.380100	+	0.924945i	$0.375890\pi$
$684$	0	0
$685$	−40.6525	−1.55325
$686$	−63.6002	−2.42827
$687$	0	0
$688$	0.437694	0.0166869
$689$	−5.85955	−0.223231
$690$	0	0
$691$	−11.8197	−0.449641	−0.224821	−	0.974400i	$-0.572180\pi$
$691$	−11.8197	−0.449641	−0.224821	+	0.974400i	$0.572180\pi$
$692$	−55.5025	−2.10989
$693$	0	0
$694$	65.7771	2.49686
$695$	12.1970	0.462659
$696$	0	0
$697$	22.4721	0.851193
$698$	0.427449	0.0161792
$699$	0	0
$700$	−157.082	−5.93714
$701$	−8.36188	−0.315824	−0.157912	−	0.987453i	$-0.550476\pi$
$701$	−8.36188	−0.315824	−0.157912	+	0.987453i	$0.550476\pi$
$702$	0	0
$703$	2.47214	0.0932384
$704$	33.6108	1.26675
$705$	0	0
$706$	−67.9230	−2.55632
$707$	21.2001	0.797310
$708$	0	0
$709$	−51.5967	−1.93776	−0.968878	−	0.247538i	$-0.920379\pi$
$709$	−51.5967	−1.93776	−0.968878	+	0.247538i	$0.920379\pi$
$710$	−77.9537	−2.92555
$711$	0	0
$712$	39.8328	1.49280
$713$	21.2001	0.793949
$714$	0	0
$715$	11.2361	0.420205
$716$	75.1560	2.80871
$717$	0	0
$718$	60.9787	2.27571
$719$	−8.36188	−0.311846	−0.155923	−	0.987769i	$-0.549835\pi$
$719$	−8.36188	−0.311846	−0.155923	+	0.987769i	$0.549835\pi$
$720$	0	0
$721$	7.88854	0.293785
$722$	−43.6513	−1.62453
$723$	0	0
$724$	−25.3262	−0.941242
$725$	28.4429	1.05634
$726$	0	0
$727$	−0.944272	−0.0350211	−0.0175106	−	0.999847i	$-0.505574\pi$
$727$	−0.944272	−0.0350211	−0.0175106	+	0.999847i	$0.505574\pi$
$728$	17.1512	0.635666
$729$	0	0
$730$	−95.1935	−3.52327
$731$	−1.11908	−0.0413905
$732$	0	0
$733$	−14.7639	−0.545318	−0.272659	−	0.962111i	$-0.587903\pi$
$733$	−14.7639	−0.545318	−0.272659	+	0.962111i	$0.587903\pi$
$734$	17.5787	0.648840
$735$	0	0
$736$	−9.59675	−0.353741
$737$	8.09770	0.298283
$738$	0	0
$739$	8.11146	0.298385	0.149192	−	0.988808i	$-0.452333\pi$
$739$	8.11146	0.298385	0.149192	+	0.988808i	$0.452333\pi$
$740$	44.9025	1.65065
$741$	0	0
$742$	−62.1115	−2.28018
$743$	−37.0185	−1.35808	−0.679038	−	0.734103i	$-0.737603\pi$
$743$	−37.0185	−1.35808	−0.679038	+	0.734103i	$0.737603\pi$
$744$	0	0
$745$	19.0000	0.696106
$746$	−20.6405	−0.755703
$747$	0	0
$748$	50.2492	1.83729
$749$	34.3024	1.25338
$750$	0	0
$751$	42.8328	1.56299	0.781496	−	0.623910i	$-0.214457\pi$
$751$	42.8328	1.56299	0.781496	+	0.623910i	$0.214457\pi$
$752$	−7.11072	−0.259301
$753$	0	0
$754$	−6.94427	−0.252895
$755$	0	0
$756$	0	0
$757$	1.41641	0.0514802	0.0257401	−	0.999669i	$-0.491806\pi$
$757$	1.41641	0.0514802	0.0257401	+	0.999669i	$0.491806\pi$
$758$	23.7024	0.860910
$759$	0	0
$760$	11.2361	0.407575
$761$	39.9483	1.44812	0.724062	−	0.689735i	$-0.242273\pi$
$761$	39.9483	1.44812	0.724062	+	0.689735i	$0.242273\pi$
$762$	0	0
$763$	−40.0000	−1.44810
$764$	−85.7561	−3.10255
$765$	0	0
$766$	−60.4721	−2.18495
$767$	2.02443	0.0730978
$768$	0	0
$769$	37.4164	1.34927	0.674635	−	0.738151i	$-0.264301\pi$
$769$	37.4164	1.34927	0.674635	+	0.738151i	$0.264301\pi$
$770$	119.103	4.29216
$771$	0	0
$772$	−51.3050	−1.84651
$773$	−44.6887	−1.60734	−0.803671	−	0.595074i	$-0.797123\pi$
$773$	−44.6887	−1.60734	−0.803671	+	0.595074i	$0.797123\pi$
$774$	0	0
$775$	70.2492	2.52343
$776$	69.7239	2.50294
$777$	0	0
$778$	−26.7639	−0.959533
$779$	−3.62140	−0.129750
$780$	0	0
$781$	25.1246	0.899029
$782$	−32.9192	−1.17719
$783$	0	0
$784$	24.1033	0.860833
$785$	93.6400	3.34216
$786$	0	0
$787$	1.88854	0.0673193	0.0336597	−	0.999433i	$-0.489284\pi$
$787$	1.88854	0.0673193	0.0336597	+	0.999433i	$0.489284\pi$
$788$	−52.2269	−1.86051
$789$	0	0
$790$	−81.3050	−2.89270
$791$	34.3024	1.21965
$792$	0	0
$793$	−10.2361	−0.363493
$794$	−0.691626	−0.0245449
$795$	0	0
$796$	90.0476	3.19165
$797$	36.1131	1.27919	0.639596	−	0.768711i	$-0.279102\pi$
$797$	36.1131	1.27919	0.639596	+	0.768711i	$0.279102\pi$
$798$	0	0
$799$	18.1803	0.643174
$800$	−31.8001	−1.12430
$801$	0	0
$802$	29.9230	1.05662
$803$	30.6810	1.08271
$804$	0	0
$805$	−50.2492	−1.77105
$806$	−17.1512	−0.604126
$807$	0	0
$808$	−18.1803	−0.639582
$809$	43.7834	1.53934	0.769671	−	0.638441i	$-0.220420\pi$
$809$	43.7834	1.53934	0.769671	+	0.638441i	$0.220420\pi$
$810$	0	0
$811$	33.5279	1.17732	0.588661	−	0.808380i	$-0.299655\pi$
$811$	33.5279	1.17732	0.588661	+	0.808380i	$0.299655\pi$
$812$	−47.4048	−1.66358
$813$	0	0
$814$	−22.4721	−0.787648
$815$	3.62140	0.126852
$816$	0	0
$817$	0.180340	0.00630929
$818$	51.4536	1.79903
$819$	0	0
$820$	−65.7771	−2.29704
$821$	−22.3191	−0.778943	−0.389472	−	0.921038i	$-0.627342\pi$
$821$	−22.3191	−0.778943	−0.389472	+	0.921038i	$0.627342\pi$
$822$	0	0
$823$	−7.87539	−0.274519	−0.137259	−	0.990535i	$-0.543829\pi$
$823$	−7.87539	−0.274519	−0.137259	+	0.990535i	$0.543829\pi$
$824$	−6.76490	−0.235667
$825$	0	0
$826$	21.4590	0.746653
$827$	−31.3726	−1.09093	−0.545467	−	0.838132i	$-0.683648\pi$
$827$	−31.3726	−1.09093	−0.545467	+	0.838132i	$0.683648\pi$
$828$	0	0
$829$	11.1803	0.388309	0.194155	−	0.980971i	$-0.437804\pi$
$829$	11.1803	0.388309	0.194155	+	0.980971i	$0.437804\pi$
$830$	8.22979	0.285660
$831$	0	0
$832$	11.4721	0.397725
$833$	−61.6262	−2.13522
$834$	0	0
$835$	17.3607	0.600791
$836$	−8.09770	−0.280065
$837$	0	0
$838$	42.2918	1.46095
$839$	21.6780	0.748407	0.374203	−	0.927347i	$-0.377916\pi$
$839$	21.6780	0.748407	0.374203	+	0.927347i	$0.377916\pi$
$840$	0	0
$841$	−20.4164	−0.704014
$842$	−92.4705	−3.18674
$843$	0	0
$844$	59.3951	2.04446
$845$	3.83513	0.131932
$846$	0	0
$847$	10.8065	0.371316
$848$	10.8642	0.373078
$849$	0	0
$850$	−109.082	−3.74148
$851$	9.48096	0.325003
$852$	0	0
$853$	−8.11146	−0.277731	−0.138865	−	0.990311i	$-0.544346\pi$
$853$	−8.11146	−0.277731	−0.138865	+	0.990311i	$0.544346\pi$
$854$	−108.503	−3.71288
$855$	0	0
$856$	−29.4164	−1.00543
$857$	−12.4107	−0.423943	−0.211971	−	0.977276i	$-0.567988\pi$
$857$	−12.4107	−0.423943	−0.211971	+	0.977276i	$0.567988\pi$
$858$	0	0
$859$	4.36068	0.148784	0.0743922	−	0.997229i	$-0.476298\pi$
$859$	4.36068	0.148784	0.0743922	+	0.997229i	$0.476298\pi$
$860$	3.27559	0.111697
$861$	0	0
$862$	31.5623	1.07502
$863$	−16.2459	−0.553016	−0.276508	−	0.961012i	$-0.589177\pi$
$863$	−16.2459	−0.553016	−0.276508	+	0.961012i	$0.589177\pi$
$864$	0	0
$865$	−58.8328	−2.00038
$866$	58.0048	1.97108
$867$	0	0
$868$	−117.082	−3.97402
$869$	26.2047	0.888934
$870$	0	0
$871$	2.76393	0.0936523
$872$	34.3024	1.16163
$873$	0	0
$874$	5.30495	0.179443
$875$	−80.7514	−2.72990
$876$	0	0
$877$	0.360680	0.0121793	0.00608965	−	0.999981i	$-0.498062\pi$
$877$	0.360680	0.0121793	0.00608965	+	0.999981i	$0.498062\pi$
$878$	58.5644	1.97645
$879$	0	0
$880$	−20.8328	−0.702274
$881$	−32.0643	−1.08027	−0.540136	−	0.841577i	$-0.681627\pi$
$881$	−32.0643	−1.08027	−0.540136	+	0.841577i	$0.681627\pi$
$882$	0	0
$883$	41.8885	1.40966	0.704831	−	0.709375i	$-0.251023\pi$
$883$	41.8885	1.40966	0.704831	+	0.709375i	$0.251023\pi$
$884$	17.1512	0.576858
$885$	0	0
$886$	6.94427	0.233297
$887$	23.4382	0.786978	0.393489	−	0.919329i	$-0.371268\pi$
$887$	23.4382	0.786978	0.393489	+	0.919329i	$0.371268\pi$
$888$	0	0
$889$	26.5836	0.891585
$890$	94.4133	3.16474
$891$	0	0
$892$	−54.7214	−1.83221
$893$	−2.92978	−0.0980412
$894$	0	0
$895$	79.6656	2.66293
$896$	92.3072	3.08377
$897$	0	0
$898$	9.09017	0.303343
$899$	21.2001	0.707062
$900$	0	0
$901$	−27.7771	−0.925389
$902$	32.9192	1.09609
$903$	0	0
$904$	−29.4164	−0.978375
$905$	−26.8459	−0.892388
$906$	0	0
$907$	39.8328	1.32263	0.661313	−	0.750110i	$-0.269999\pi$
$907$	39.8328	1.32263	0.661313	+	0.750110i	$0.269999\pi$
$908$	65.3293	2.16803
$909$	0	0
$910$	40.6525	1.34762
$911$	8.36188	0.277042	0.138521	−	0.990360i	$-0.455765\pi$
$911$	8.36188	0.277042	0.138521	+	0.990360i	$0.455765\pi$
$912$	0	0
$913$	−2.65248	−0.0877841
$914$	−17.4154	−0.576050
$915$	0	0
$916$	17.2361	0.569496
$917$	−89.8049	−2.96562
$918$	0	0
$919$	41.3607	1.36436	0.682181	−	0.731183i	$-0.261031\pi$
$919$	41.3607	1.36436	0.682181	+	0.731183i	$0.261031\pi$
$920$	43.0918	1.42069
$921$	0	0
$922$	83.4508	2.74831
$923$	8.57561	0.282270
$924$	0	0
$925$	31.4164	1.03297
$926$	−0.691626	−0.0227283
$927$	0	0
$928$	−9.59675	−0.315029
$929$	22.3191	0.732267	0.366134	−	0.930562i	$-0.380681\pi$
$929$	22.3191	0.732267	0.366134	+	0.930562i	$0.380681\pi$
$930$	0	0
$931$	9.93112	0.325479
$932$	−72.6537	−2.37985
$933$	0	0
$934$	89.2624	2.92075
$935$	53.2643	1.74193
$936$	0	0
$937$	−46.4853	−1.51861	−0.759304	−	0.650736i	$-0.774461\pi$
$937$	−46.4853	−1.51861	−0.759304	+	0.650736i	$0.774461\pi$
$938$	29.2978	0.956606
$939$	0	0
$940$	−53.2148	−1.73567
$941$	−22.3191	−0.727583	−0.363792	−	0.931480i	$-0.618518\pi$
$941$	−22.3191	−0.727583	−0.363792	+	0.931480i	$0.618518\pi$
$942$	0	0
$943$	−13.8885	−0.452273
$944$	−3.75349	−0.122166
$945$	0	0
$946$	−1.63932	−0.0532989
$947$	5.21838	0.169575	0.0847873	−	0.996399i	$-0.472979\pi$
$947$	5.21838	0.169575	0.0847873	+	0.996399i	$0.472979\pi$
$948$	0	0
$949$	10.4721	0.339940
$950$	17.5787	0.570327
$951$	0	0
$952$	81.3050	2.63511
$953$	−0.691626	−0.0224040	−0.0112020	−	0.999937i	$-0.503566\pi$
$953$	−0.691626	−0.0224040	−0.0112020	+	0.999937i	$0.503566\pi$
$954$	0	0
$955$	−90.9017	−2.94151
$956$	−53.0002	−1.71415
$957$	0	0
$958$	−77.5197	−2.50455
$959$	47.4048	1.53078
$960$	0	0
$961$	21.3607	0.689054
$962$	−7.67026	−0.247299
$963$	0	0
$964$	48.9443	1.57639
$965$	−54.3834	−1.75066
$966$	0	0
$967$	10.7639	0.346145	0.173072	−	0.984909i	$-0.444631\pi$
$967$	10.7639	0.346145	0.173072	+	0.984909i	$0.444631\pi$
$968$	−9.26723	−0.297860
$969$	0	0
$970$	165.262	5.30626
$971$	−45.3299	−1.45471	−0.727353	−	0.686264i	$-0.759250\pi$
$971$	−45.3299	−1.45471	−0.727353	+	0.686264i	$0.759250\pi$
$972$	0	0
$973$	−14.2229	−0.455966
$974$	39.0429	1.25102
$975$	0	0
$976$	18.9787	0.607494
$977$	−36.3269	−1.16220	−0.581099	−	0.813833i	$-0.697377\pi$
$977$	−36.3269	−1.16220	−0.581099	+	0.813833i	$0.697377\pi$
$978$	0	0
$979$	−30.4296	−0.972533
$980$	180.383	5.76213
$981$	0	0
$982$	97.8460	3.12239
$983$	−40.8536	−1.30303	−0.651514	−	0.758637i	$-0.725866\pi$
$983$	−40.8536	−1.30303	−0.651514	+	0.758637i	$0.725866\pi$
$984$	0	0
$985$	−55.3607	−1.76394
$986$	−32.9192	−1.04836
$987$	0	0
$988$	−2.76393	−0.0879324
$989$	0.691626	0.0219924
$990$	0	0
$991$	12.3475	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$991$	12.3475	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$992$	−23.7024	−0.752552
$993$	0	0
$994$	90.9017	2.88323
$995$	95.4507	3.02599
$996$	0	0
$997$	−15.1115	−0.478585	−0.239292	−	0.970948i	$-0.576915\pi$
$997$	−15.1115	−0.478585	−0.239292	+	0.970948i	$0.576915\pi$
$998$	−10.3359	−0.327176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	351.2.a.f.1.4	yes	4
3.2	odd	2	inner	351.2.a.f.1.1	✓	4
4.3	odd	2		5616.2.a.ch.1.4		4
5.4	even	2		8775.2.a.bo.1.1		4
9.2	odd	6		1053.2.e.p.352.4		8
9.4	even	3		1053.2.e.p.703.1		8
9.5	odd	6		1053.2.e.p.703.4		8
9.7	even	3		1053.2.e.p.352.1		8
12.11	even	2		5616.2.a.ch.1.1		4
13.12	even	2		4563.2.a.ba.1.1		4
15.14	odd	2		8775.2.a.bo.1.4		4
39.38	odd	2		4563.2.a.ba.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
351.2.a.f.1.1	✓	4	3.2	odd	2	inner
351.2.a.f.1.4	yes	4	1.1	even	1	trivial
1053.2.e.p.352.1		8	9.7	even	3
1053.2.e.p.352.4		8	9.2	odd	6
1053.2.e.p.703.1		8	9.4	even	3
1053.2.e.p.703.4		8	9.5	odd	6
4563.2.a.ba.1.1		4	13.12	even	2
4563.2.a.ba.1.4		4	39.38	odd	2
5616.2.a.ch.1.1		4	12.11	even	2
5616.2.a.ch.1.4		4	4.3	odd	2
8775.2.a.bo.1.1		4	5.4	even	2
8775.2.a.bo.1.4		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37024 q^{2} +3.61803 q^{4} +3.83513 q^{5} -4.47214 q^{7} +3.83513 q^{8} +9.09017 q^{10} -2.92978 q^{11} -1.00000 q^{13} -10.6000 q^{14} +1.85410 q^{16} -4.74048 q^{17} +0.763932 q^{19} +13.8756 q^{20} -6.94427 q^{22} +2.92978 q^{23} +9.70820 q^{25} -2.37024 q^{26} -16.1803 q^{28} +2.92978 q^{29} +7.23607 q^{31} -3.27559 q^{32} -11.2361 q^{34} -17.1512 q^{35} +3.23607 q^{37} +1.81070 q^{38} +14.7082 q^{40} -4.74048 q^{41} +0.236068 q^{43} -10.6000 q^{44} +6.94427 q^{46} -3.83513 q^{47} +13.0000 q^{49} +23.0108 q^{50} -3.61803 q^{52} +5.85955 q^{53} -11.2361 q^{55} -17.1512 q^{56} +6.94427 q^{58} -2.02443 q^{59} +10.2361 q^{61} +17.1512 q^{62} -11.4721 q^{64} -3.83513 q^{65} -2.76393 q^{67} -17.1512 q^{68} -40.6525 q^{70} -8.57561 q^{71} -10.4721 q^{73} +7.67026 q^{74} +2.76393 q^{76} +13.1024 q^{77} -8.94427 q^{79} +7.11072 q^{80} -11.2361 q^{82} +0.905351 q^{83} -18.1803 q^{85} +0.559538 q^{86} -11.2361 q^{88} +10.3863 q^{89} +4.47214 q^{91} +10.6000 q^{92} -9.09017 q^{94} +2.92978 q^{95} +18.1803 q^{97} +30.8131 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 14 q^{10} - 4 q^{13} - 6 q^{16} + 12 q^{19} + 8 q^{22} + 12 q^{25} - 20 q^{28} + 20 q^{31} - 36 q^{34} + 4 q^{37} + 32 q^{40} - 8 q^{43} - 8 q^{46} + 52 q^{49} - 10 q^{52} - 36 q^{55} - 8 q^{58}+ \cdots + 28 q^{97}+O(q^{100}) \) 4 * q + 10 * q^4 + 14 * q^10 - 4 * q^13 - 6 * q^16 + 12 * q^19 + 8 * q^22 + 12 * q^25 - 20 * q^28 + 20 * q^31 - 36 * q^34 + 4 * q^37 + 32 * q^40 - 8 * q^43 - 8 * q^46 + 52 * q^49 - 10 * q^52 - 36 * q^55 - 8 * q^58 + 32 * q^61 - 28 * q^64 - 20 * q^67 - 100 * q^70 - 24 * q^73 + 20 * q^76 - 36 * q^82 - 28 * q^85 - 36 * q^88 - 14 * q^94 + 28 * q^97

Introduction

L-functions

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Database

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