LMFDB - Embedded newform 3675.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +2.23607 q^{8} +1.00000 q^{9} +2.23607 q^{11} +1.61803 q^{12} -2.47214 q^{13} +1.85410 q^{16} +6.47214 q^{17} -0.618034 q^{18} +6.47214 q^{19} -1.38197 q^{22} -0.236068 q^{23} -2.23607 q^{24} +1.52786 q^{26} -1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} -5.61803 q^{32} -2.23607 q^{33} -4.00000 q^{34} -1.61803 q^{36} +5.47214 q^{37} -4.00000 q^{38} +2.47214 q^{39} -8.94427 q^{41} -8.70820 q^{43} -3.61803 q^{44} +0.145898 q^{46} +1.52786 q^{47} -1.85410 q^{48} -6.47214 q^{51} +4.00000 q^{52} -6.00000 q^{53} +0.618034 q^{54} -6.47214 q^{57} +1.85410 q^{58} -2.47214 q^{59} +12.0000 q^{61} -2.47214 q^{62} -0.236068 q^{64} +1.38197 q^{66} +0.708204 q^{67} -10.4721 q^{68} +0.236068 q^{69} +3.76393 q^{71} +2.23607 q^{72} -5.52786 q^{73} -3.38197 q^{74} -10.4721 q^{76} -1.52786 q^{78} +10.7082 q^{79} +1.00000 q^{81} +5.52786 q^{82} -0.944272 q^{83} +5.38197 q^{86} +3.00000 q^{87} +5.00000 q^{88} +10.4721 q^{89} +0.381966 q^{92} -4.00000 q^{93} -0.944272 q^{94} +5.61803 q^{96} -4.00000 q^{97} +2.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 2 * q^9	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{9} + q^{12} + 4 q^{13} - 3 q^{16} + 4 q^{17} + q^{18} + 4 q^{19} - 5 q^{22} + 4 q^{23} + 12 q^{26} - 2 q^{27} - 6 q^{29} + 8 q^{31} - 9 q^{32} - 8 q^{34} - q^{36} + 2 q^{37} - 8 q^{38} - 4 q^{39} - 4 q^{43} - 5 q^{44} + 7 q^{46} + 12 q^{47} + 3 q^{48} - 4 q^{51} + 8 q^{52} - 12 q^{53} - q^{54} - 4 q^{57} - 3 q^{58} + 4 q^{59} + 24 q^{61} + 4 q^{62} + 4 q^{64} + 5 q^{66} - 12 q^{67} - 12 q^{68} - 4 q^{69} + 12 q^{71} - 20 q^{73} - 9 q^{74} - 12 q^{76} - 12 q^{78} + 8 q^{79} + 2 q^{81} + 20 q^{82} + 16 q^{83} + 13 q^{86} + 6 q^{87} + 10 q^{88} + 12 q^{89} + 3 q^{92} - 8 q^{93} + 16 q^{94} + 9 q^{96} - 8 q^{97}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 2 * q^9 + q^12 + 4 * q^13 - 3 * q^16 + 4 * q^17 + q^18 + 4 * q^19 - 5 * q^22 + 4 * q^23 + 12 * q^26 - 2 * q^27 - 6 * q^29 + 8 * q^31 - 9 * q^32 - 8 * q^34 - q^36 + 2 * q^37 - 8 * q^38 - 4 * q^39 - 4 * q^43 - 5 * q^44 + 7 * q^46 + 12 * q^47 + 3 * q^48 - 4 * q^51 + 8 * q^52 - 12 * q^53 - q^54 - 4 * q^57 - 3 * q^58 + 4 * q^59 + 24 * q^61 + 4 * q^62 + 4 * q^64 + 5 * q^66 - 12 * q^67 - 12 * q^68 - 4 * q^69 + 12 * q^71 - 20 * q^73 - 9 * q^74 - 12 * q^76 - 12 * q^78 + 8 * q^79 + 2 * q^81 + 20 * q^82 + 16 * q^83 + 13 * q^86 + 6 * q^87 + 10 * q^88 + 12 * q^89 + 3 * q^92 - 8 * q^93 + 16 * q^94 + 9 * q^96 - 8 * q^97

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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	0	0
$7$	0	0
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	1.61803	0.467086
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	0	0
$15$	0	0
$16$	1.85410	0.463525
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	−0.618034	−0.145672
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	0	0
$22$	−1.38197	−0.294636
$23$	−0.236068	−0.0492236	−0.0246118	−	0.999697i	$-0.507835\pi$
$23$	−0.236068	−0.0492236	−0.0246118	+	0.999697i	$0.507835\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	1.52786	0.299639
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.61803	−0.993137
$33$	−2.23607	−0.389249
$34$	−4.00000	−0.685994
$35$	0	0
$36$	−1.61803	−0.269672
$37$	5.47214	0.899614	0.449807	−	0.893126i	$-0.351493\pi$
$37$	5.47214	0.899614	0.449807	+	0.893126i	$0.351493\pi$
$38$	−4.00000	−0.648886
$39$	2.47214	0.395859
$40$	0	0
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	0	0
$43$	−8.70820	−1.32799	−0.663994	−	0.747738i	$-0.731140\pi$
$43$	−8.70820	−1.32799	−0.663994	+	0.747738i	$0.731140\pi$
$44$	−3.61803	−0.545439
$45$	0	0
$46$	0.145898	0.0215115
$47$	1.52786	0.222862	0.111431	−	0.993772i	$-0.464457\pi$
$47$	1.52786	0.222862	0.111431	+	0.993772i	$0.464457\pi$
$48$	−1.85410	−0.267617
$49$	0	0
$50$	0	0
$51$	−6.47214	−0.906280
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	0	0
$57$	−6.47214	−0.857255
$58$	1.85410	0.243456
$59$	−2.47214	−0.321845	−0.160922	−	0.986967i	$-0.551447\pi$
$59$	−2.47214	−0.321845	−0.160922	+	0.986967i	$0.551447\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	−2.47214	−0.313962
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	1.38197	0.170108
$67$	0.708204	0.0865209	0.0432604	−	0.999064i	$-0.486225\pi$
$67$	0.708204	0.0865209	0.0432604	+	0.999064i	$0.486225\pi$
$68$	−10.4721	−1.26993
$69$	0.236068	0.0284192
$70$	0	0
$71$	3.76393	0.446697	0.223348	−	0.974739i	$-0.428301\pi$
$71$	3.76393	0.446697	0.223348	+	0.974739i	$0.428301\pi$
$72$	2.23607	0.263523
$73$	−5.52786	−0.646988	−0.323494	−	0.946230i	$-0.604857\pi$
$73$	−5.52786	−0.646988	−0.323494	+	0.946230i	$0.604857\pi$
$74$	−3.38197	−0.393146
$75$	0	0
$76$	−10.4721	−1.20124
$77$	0	0
$78$	−1.52786	−0.172997
$79$	10.7082	1.20477	0.602384	−	0.798207i	$-0.294218\pi$
$79$	10.7082	1.20477	0.602384	+	0.798207i	$0.294218\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.52786	0.610450
$83$	−0.944272	−0.103647	−0.0518237	−	0.998656i	$-0.516503\pi$
$83$	−0.944272	−0.103647	−0.0518237	+	0.998656i	$0.516503\pi$
$84$	0	0
$85$	0	0
$86$	5.38197	0.580352
$87$	3.00000	0.321634
$88$	5.00000	0.533002
$89$	10.4721	1.11004	0.555022	−	0.831836i	$-0.312710\pi$
$89$	10.4721	1.11004	0.555022	+	0.831836i	$0.312710\pi$
$90$	0	0
$91$	0	0
$92$	0.381966	0.0398227
$93$	−4.00000	−0.414781
$94$	−0.944272	−0.0973942
$95$	0	0
$96$	5.61803	0.573388
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	2.23607	0.224733
$100$	0	0
$101$	18.4721	1.83805	0.919023	−	0.394204i	$-0.128980\pi$
$101$	18.4721	1.83805	0.919023	+	0.394204i	$0.128980\pi$
$102$	4.00000	0.396059
$103$	−19.4164	−1.91316	−0.956578	−	0.291477i	$-0.905853\pi$
$103$	−19.4164	−1.91316	−0.956578	+	0.291477i	$0.905853\pi$
$104$	−5.52786	−0.542052
$105$	0	0
$106$	3.70820	0.360173
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	1.61803	0.155695
$109$	−10.4164	−0.997711	−0.498855	−	0.866685i	$-0.666246\pi$
$109$	−10.4164	−0.997711	−0.498855	+	0.866685i	$0.666246\pi$
$110$	0	0
$111$	−5.47214	−0.519392
$112$	0	0
$113$	19.4721	1.83178	0.915892	−	0.401424i	$-0.131485\pi$
$113$	19.4721	1.83178	0.915892	+	0.401424i	$0.131485\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	4.85410	0.450692
$117$	−2.47214	−0.228549
$118$	1.52786	0.140651
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	−7.41641	−0.671450
$123$	8.94427	0.806478
$124$	−6.47214	−0.581215
$125$	0	0
$126$	0	0
$127$	3.18034	0.282210	0.141105	−	0.989995i	$-0.454935\pi$
$127$	3.18034	0.282210	0.141105	+	0.989995i	$0.454935\pi$
$128$	11.3820	1.00603
$129$	8.70820	0.766715
$130$	0	0
$131$	22.4721	1.96340	0.981700	−	0.190435i	$-0.0609898\pi$
$131$	22.4721	1.96340	0.981700	+	0.190435i	$0.0609898\pi$
$132$	3.61803	0.314909
$133$	0	0
$134$	−0.437694	−0.0378110
$135$	0	0
$136$	14.4721	1.24098
$137$	−15.8885	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$137$	−15.8885	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$138$	−0.145898	−0.0124197
$139$	15.4164	1.30760	0.653801	−	0.756666i	$-0.273173\pi$
$139$	15.4164	1.30760	0.653801	+	0.756666i	$0.273173\pi$
$140$	0	0
$141$	−1.52786	−0.128669
$142$	−2.32624	−0.195214
$143$	−5.52786	−0.462263
$144$	1.85410	0.154508
$145$	0	0
$146$	3.41641	0.282744
$147$	0	0
$148$	−8.85410	−0.727803
$149$	−22.8885	−1.87510	−0.937551	−	0.347847i	$-0.886913\pi$
$149$	−22.8885	−1.87510	−0.937551	+	0.347847i	$0.886913\pi$
$150$	0	0
$151$	−6.70820	−0.545906	−0.272953	−	0.962027i	$-0.588000\pi$
$151$	−6.70820	−0.545906	−0.272953	+	0.962027i	$0.588000\pi$
$152$	14.4721	1.17385
$153$	6.47214	0.523241
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	16.9443	1.35230	0.676150	−	0.736764i	$-0.263647\pi$
$157$	16.9443	1.35230	0.676150	+	0.736764i	$0.263647\pi$
$158$	−6.61803	−0.526503
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	−0.618034	−0.0485573
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	14.4721	1.13008
$165$	0	0
$166$	0.583592	0.0452955
$167$	−13.5279	−1.04682	−0.523409	−	0.852082i	$-0.675340\pi$
$167$	−13.5279	−1.04682	−0.523409	+	0.852082i	$0.675340\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	0	0
$171$	6.47214	0.494937
$172$	14.0902	1.07437
$173$	−6.47214	−0.492067	−0.246034	−	0.969261i	$-0.579127\pi$
$173$	−6.47214	−0.492067	−0.246034	+	0.969261i	$0.579127\pi$
$174$	−1.85410	−0.140559
$175$	0	0
$176$	4.14590	0.312509
$177$	2.47214	0.185817
$178$	−6.47214	−0.485107
$179$	16.9443	1.26647	0.633237	−	0.773958i	$-0.281726\pi$
$179$	16.9443	1.26647	0.633237	+	0.773958i	$0.281726\pi$
$180$	0	0
$181$	−2.47214	−0.183752	−0.0918762	−	0.995770i	$-0.529286\pi$
$181$	−2.47214	−0.183752	−0.0918762	+	0.995770i	$0.529286\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	−0.527864	−0.0389147
$185$	0	0
$186$	2.47214	0.181266
$187$	14.4721	1.05831
$188$	−2.47214	−0.180299
$189$	0	0
$190$	0	0
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0.236068	0.0170367
$193$	−12.8885	−0.927738	−0.463869	−	0.885904i	$-0.653539\pi$
$193$	−12.8885	−0.927738	−0.463869	+	0.885904i	$0.653539\pi$
$194$	2.47214	0.177489
$195$	0	0
$196$	0	0
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	−1.38197	−0.0982120
$199$	13.8885	0.984533	0.492266	−	0.870445i	$-0.336169\pi$
$199$	13.8885	0.984533	0.492266	+	0.870445i	$0.336169\pi$
$200$	0	0
$201$	−0.708204	−0.0499529
$202$	−11.4164	−0.803256
$203$	0	0
$204$	10.4721	0.733196
$205$	0	0
$206$	12.0000	0.836080
$207$	−0.236068	−0.0164079
$208$	−4.58359	−0.317815
$209$	14.4721	1.00106
$210$	0	0
$211$	−21.8885	−1.50687	−0.753435	−	0.657523i	$-0.771604\pi$
$211$	−21.8885	−1.50687	−0.753435	+	0.657523i	$0.771604\pi$
$212$	9.70820	0.666762
$213$	−3.76393	−0.257900
$214$	5.52786	0.377877
$215$	0	0
$216$	−2.23607	−0.152145
$217$	0	0
$218$	6.43769	0.436016
$219$	5.52786	0.373538
$220$	0	0
$221$	−16.0000	−1.07628
$222$	3.38197	0.226983
$223$	23.4164	1.56808	0.784039	−	0.620711i	$-0.213156\pi$
$223$	23.4164	1.56808	0.784039	+	0.620711i	$0.213156\pi$
$224$	0	0
$225$	0	0
$226$	−12.0344	−0.800519
$227$	19.4164	1.28871	0.644356	−	0.764726i	$-0.277125\pi$
$227$	19.4164	1.28871	0.644356	+	0.764726i	$0.277125\pi$
$228$	10.4721	0.693534
$229$	11.0557	0.730583	0.365292	−	0.930893i	$-0.380969\pi$
$229$	11.0557	0.730583	0.365292	+	0.930893i	$0.380969\pi$
$230$	0	0
$231$	0	0
$232$	−6.70820	−0.440415
$233$	13.4721	0.882589	0.441294	−	0.897362i	$-0.354519\pi$
$233$	13.4721	0.882589	0.441294	+	0.897362i	$0.354519\pi$
$234$	1.52786	0.0998796
$235$	0	0
$236$	4.00000	0.260378
$237$	−10.7082	−0.695573
$238$	0	0
$239$	−3.05573	−0.197659	−0.0988293	−	0.995104i	$-0.531510\pi$
$239$	−3.05573	−0.197659	−0.0988293	+	0.995104i	$0.531510\pi$
$240$	0	0
$241$	9.52786	0.613744	0.306872	−	0.951751i	$-0.400718\pi$
$241$	9.52786	0.613744	0.306872	+	0.951751i	$0.400718\pi$
$242$	3.70820	0.238372
$243$	−1.00000	−0.0641500
$244$	−19.4164	−1.24301
$245$	0	0
$246$	−5.52786	−0.352444
$247$	−16.0000	−1.01806
$248$	8.94427	0.567962
$249$	0.944272	0.0598408
$250$	0	0
$251$	−1.52786	−0.0964379	−0.0482190	−	0.998837i	$-0.515355\pi$
$251$	−1.52786	−0.0964379	−0.0482190	+	0.998837i	$0.515355\pi$
$252$	0	0
$253$	−0.527864	−0.0331865
$254$	−1.96556	−0.123330
$255$	0	0
$256$	−6.56231	−0.410144
$257$	8.94427	0.557928	0.278964	−	0.960302i	$-0.410009\pi$
$257$	8.94427	0.557928	0.278964	+	0.960302i	$0.410009\pi$
$258$	−5.38197	−0.335067
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	−13.8885	−0.858037
$263$	1.18034	0.0727829	0.0363914	−	0.999338i	$-0.488414\pi$
$263$	1.18034	0.0727829	0.0363914	+	0.999338i	$0.488414\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	0	0
$267$	−10.4721	−0.640884
$268$	−1.14590	−0.0699969
$269$	8.94427	0.545342	0.272671	−	0.962107i	$-0.412093\pi$
$269$	8.94427	0.545342	0.272671	+	0.962107i	$0.412093\pi$
$270$	0	0
$271$	23.4164	1.42245	0.711223	−	0.702967i	$-0.248142\pi$
$271$	23.4164	1.42245	0.711223	+	0.702967i	$0.248142\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	9.81966	0.593227
$275$	0	0
$276$	−0.381966	−0.0229917
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	−9.52786	−0.571443
$279$	4.00000	0.239474
$280$	0	0
$281$	25.3607	1.51289	0.756446	−	0.654057i	$-0.226934\pi$
$281$	25.3607	1.51289	0.756446	+	0.654057i	$0.226934\pi$
$282$	0.944272	0.0562306
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−6.09017	−0.361385
$285$	0	0
$286$	3.41641	0.202016
$287$	0	0
$288$	−5.61803	−0.331046
$289$	24.8885	1.46403
$290$	0	0
$291$	4.00000	0.234484
$292$	8.94427	0.523424
$293$	28.3607	1.65685	0.828424	−	0.560101i	$-0.189238\pi$
$293$	28.3607	1.65685	0.828424	+	0.560101i	$0.189238\pi$
$294$	0	0
$295$	0	0
$296$	12.2361	0.711207
$297$	−2.23607	−0.129750
$298$	14.1459	0.819450
$299$	0.583592	0.0337500
$300$	0	0
$301$	0	0
$302$	4.14590	0.238570
$303$	−18.4721	−1.06120
$304$	12.0000	0.688247
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−4.58359	−0.261599	−0.130800	−	0.991409i	$-0.541754\pi$
$307$	−4.58359	−0.261599	−0.130800	+	0.991409i	$0.541754\pi$
$308$	0	0
$309$	19.4164	1.10456
$310$	0	0
$311$	−24.3607	−1.38137	−0.690684	−	0.723157i	$-0.742690\pi$
$311$	−24.3607	−1.38137	−0.690684	+	0.723157i	$0.742690\pi$
$312$	5.52786	0.312954
$313$	19.4164	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$313$	19.4164	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$314$	−10.4721	−0.590977
$315$	0	0
$316$	−17.3262	−0.974677
$317$	9.94427	0.558526	0.279263	−	0.960215i	$-0.409910\pi$
$317$	9.94427	0.558526	0.279263	+	0.960215i	$0.409910\pi$
$318$	−3.70820	−0.207946
$319$	−6.70820	−0.375587
$320$	0	0
$321$	8.94427	0.499221
$322$	0	0
$323$	41.8885	2.33074
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	7.41641	0.410757
$327$	10.4164	0.576029
$328$	−20.0000	−1.10432
$329$	0	0
$330$	0	0
$331$	−10.8197	−0.594702	−0.297351	−	0.954768i	$-0.596103\pi$
$331$	−10.8197	−0.594702	−0.297351	+	0.954768i	$0.596103\pi$
$332$	1.52786	0.0838524
$333$	5.47214	0.299871
$334$	8.36068	0.457476
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	4.25735	0.231570
$339$	−19.4721	−1.05758
$340$	0	0
$341$	8.94427	0.484359
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−19.4721	−1.04987
$345$	0	0
$346$	4.00000	0.215041
$347$	18.7082	1.00431	0.502155	−	0.864778i	$-0.332541\pi$
$347$	18.7082	1.00431	0.502155	+	0.864778i	$0.332541\pi$
$348$	−4.85410	−0.260207
$349$	−3.41641	−0.182876	−0.0914381	−	0.995811i	$-0.529146\pi$
$349$	−3.41641	−0.182876	−0.0914381	+	0.995811i	$0.529146\pi$
$350$	0	0
$351$	2.47214	0.131953
$352$	−12.5623	−0.669573
$353$	−21.8885	−1.16501	−0.582505	−	0.812827i	$-0.697927\pi$
$353$	−21.8885	−1.16501	−0.582505	+	0.812827i	$0.697927\pi$
$354$	−1.52786	−0.0812051
$355$	0	0
$356$	−16.9443	−0.898045
$357$	0	0
$358$	−10.4721	−0.553470
$359$	−16.2361	−0.856907	−0.428453	−	0.903564i	$-0.640941\pi$
$359$	−16.2361	−0.856907	−0.428453	+	0.903564i	$0.640941\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	1.52786	0.0803028
$363$	6.00000	0.314918
$364$	0	0
$365$	0	0
$366$	7.41641	0.387662
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−0.437694	−0.0228164
$369$	−8.94427	−0.465620
$370$	0	0
$371$	0	0
$372$	6.47214	0.335565
$373$	−8.52786	−0.441556	−0.220778	−	0.975324i	$-0.570860\pi$
$373$	−8.52786	−0.441556	−0.220778	+	0.975324i	$0.570860\pi$
$374$	−8.94427	−0.462497
$375$	0	0
$376$	3.41641	0.176188
$377$	7.41641	0.381964
$378$	0	0
$379$	−4.70820	−0.241844	−0.120922	−	0.992662i	$-0.538585\pi$
$379$	−4.70820	−0.241844	−0.120922	+	0.992662i	$0.538585\pi$
$380$	0	0
$381$	−3.18034	−0.162934
$382$	−12.9443	−0.662287
$383$	7.41641	0.378961	0.189480	−	0.981885i	$-0.439320\pi$
$383$	7.41641	0.378961	0.189480	+	0.981885i	$0.439320\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	7.96556	0.405436
$387$	−8.70820	−0.442663
$388$	6.47214	0.328573
$389$	29.9443	1.51823	0.759117	−	0.650954i	$-0.225631\pi$
$389$	29.9443	1.51823	0.759117	+	0.650954i	$0.225631\pi$
$390$	0	0
$391$	−1.52786	−0.0772674
$392$	0	0
$393$	−22.4721	−1.13357
$394$	−12.9787	−0.653858
$395$	0	0
$396$	−3.61803	−0.181813
$397$	−25.8885	−1.29931	−0.649654	−	0.760230i	$-0.725086\pi$
$397$	−25.8885	−1.29931	−0.649654	+	0.760230i	$0.725086\pi$
$398$	−8.58359	−0.430257
$399$	0	0
$400$	0	0
$401$	1.58359	0.0790808	0.0395404	−	0.999218i	$-0.487411\pi$
$401$	1.58359	0.0790808	0.0395404	+	0.999218i	$0.487411\pi$
$402$	0.437694	0.0218302
$403$	−9.88854	−0.492583
$404$	−29.8885	−1.48701
$405$	0	0
$406$	0	0
$407$	12.2361	0.606519
$408$	−14.4721	−0.716477
$409$	−19.4164	−0.960080	−0.480040	−	0.877247i	$-0.659378\pi$
$409$	−19.4164	−0.960080	−0.480040	+	0.877247i	$0.659378\pi$
$410$	0	0
$411$	15.8885	0.783724
$412$	31.4164	1.54778
$413$	0	0
$414$	0.145898	0.00717050
$415$	0	0
$416$	13.8885	0.680942
$417$	−15.4164	−0.754945
$418$	−8.94427	−0.437479
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	−16.4164	−0.800087	−0.400043	−	0.916496i	$-0.631005\pi$
$421$	−16.4164	−0.800087	−0.400043	+	0.916496i	$0.631005\pi$
$422$	13.5279	0.658526
$423$	1.52786	0.0742873
$424$	−13.4164	−0.651558
$425$	0	0
$426$	2.32624	0.112707
$427$	0	0
$428$	14.4721	0.699537
$429$	5.52786	0.266888
$430$	0	0
$431$	−28.9443	−1.39420	−0.697098	−	0.716976i	$-0.745526\pi$
$431$	−28.9443	−1.39420	−0.697098	+	0.716976i	$0.745526\pi$
$432$	−1.85410	−0.0892055
$433$	4.94427	0.237607	0.118803	−	0.992918i	$-0.462094\pi$
$433$	4.94427	0.237607	0.118803	+	0.992918i	$0.462094\pi$
$434$	0	0
$435$	0	0
$436$	16.8541	0.807165
$437$	−1.52786	−0.0730876
$438$	−3.41641	−0.163242
$439$	0.583592	0.0278533	0.0139267	−	0.999903i	$-0.495567\pi$
$439$	0.583592	0.0278533	0.0139267	+	0.999903i	$0.495567\pi$
$440$	0	0
$441$	0	0
$442$	9.88854	0.470350
$443$	32.9443	1.56523	0.782615	−	0.622506i	$-0.213885\pi$
$443$	32.9443	1.56523	0.782615	+	0.622506i	$0.213885\pi$
$444$	8.85410	0.420197
$445$	0	0
$446$	−14.4721	−0.685275
$447$	22.8885	1.08259
$448$	0	0
$449$	−15.4721	−0.730175	−0.365088	−	0.930973i	$-0.618961\pi$
$449$	−15.4721	−0.730175	−0.365088	+	0.930973i	$0.618961\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	−31.5066	−1.48194
$453$	6.70820	0.315179
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−14.4721	−0.677720
$457$	8.05573	0.376831	0.188416	−	0.982089i	$-0.439665\pi$
$457$	8.05573	0.376831	0.188416	+	0.982089i	$0.439665\pi$
$458$	−6.83282	−0.319277
$459$	−6.47214	−0.302093
$460$	0	0
$461$	21.5279	1.00265	0.501326	−	0.865258i	$-0.332846\pi$
$461$	21.5279	1.00265	0.501326	+	0.865258i	$0.332846\pi$
$462$	0	0
$463$	−33.8885	−1.57493	−0.787467	−	0.616357i	$-0.788608\pi$
$463$	−33.8885	−1.57493	−0.787467	+	0.616357i	$0.788608\pi$
$464$	−5.56231	−0.258224
$465$	0	0
$466$	−8.32624	−0.385706
$467$	−33.8885	−1.56817	−0.784087	−	0.620650i	$-0.786869\pi$
$467$	−33.8885	−1.56817	−0.784087	+	0.620650i	$0.786869\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	0	0
$471$	−16.9443	−0.780751
$472$	−5.52786	−0.254441
$473$	−19.4721	−0.895330
$474$	6.61803	0.303976
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	1.88854	0.0863800
$479$	−21.5279	−0.983633	−0.491817	−	0.870699i	$-0.663667\pi$
$479$	−21.5279	−0.983633	−0.491817	+	0.870699i	$0.663667\pi$
$480$	0	0
$481$	−13.5279	−0.616818
$482$	−5.88854	−0.268216
$483$	0	0
$484$	9.70820	0.441282
$485$	0	0
$486$	0.618034	0.0280346
$487$	−1.29180	−0.0585369	−0.0292684	−	0.999572i	$-0.509318\pi$
$487$	−1.29180	−0.0585369	−0.0292684	+	0.999572i	$0.509318\pi$
$488$	26.8328	1.21466
$489$	12.0000	0.542659
$490$	0	0
$491$	27.1803	1.22663	0.613316	−	0.789838i	$-0.289835\pi$
$491$	27.1803	1.22663	0.613316	+	0.789838i	$0.289835\pi$
$492$	−14.4721	−0.652454
$493$	−19.4164	−0.874471
$494$	9.88854	0.444907
$495$	0	0
$496$	7.41641	0.333007
$497$	0	0
$498$	−0.583592	−0.0261514
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	13.5279	0.604380
$502$	0.944272	0.0421449
$503$	4.94427	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$504$	0	0
$505$	0	0
$506$	0.326238	0.0145030
$507$	6.88854	0.305931
$508$	−5.14590	−0.228312
$509$	18.4721	0.818763	0.409382	−	0.912363i	$-0.365745\pi$
$509$	18.4721	0.818763	0.409382	+	0.912363i	$0.365745\pi$
$510$	0	0
$511$	0	0
$512$	−18.7082	−0.826794
$513$	−6.47214	−0.285752
$514$	−5.52786	−0.243824
$515$	0	0
$516$	−14.0902	−0.620285
$517$	3.41641	0.150253
$518$	0	0
$519$	6.47214	0.284095
$520$	0	0
$521$	−3.05573	−0.133874	−0.0669369	−	0.997757i	$-0.521323\pi$
$521$	−3.05573	−0.133874	−0.0669369	+	0.997757i	$0.521323\pi$
$522$	1.85410	0.0811518
$523$	33.8885	1.48184	0.740921	−	0.671592i	$-0.234389\pi$
$523$	33.8885	1.48184	0.740921	+	0.671592i	$0.234389\pi$
$524$	−36.3607	−1.58842
$525$	0	0
$526$	−0.729490	−0.0318073
$527$	25.8885	1.12772
$528$	−4.14590	−0.180427
$529$	−22.9443	−0.997577
$530$	0	0
$531$	−2.47214	−0.107282
$532$	0	0
$533$	22.1115	0.957753
$534$	6.47214	0.280077
$535$	0	0
$536$	1.58359	0.0684008
$537$	−16.9443	−0.731199
$538$	−5.52786	−0.238323
$539$	0	0
$540$	0	0
$541$	−1.58359	−0.0680839	−0.0340420	−	0.999420i	$-0.510838\pi$
$541$	−1.58359	−0.0680839	−0.0340420	+	0.999420i	$0.510838\pi$
$542$	−14.4721	−0.621631
$543$	2.47214	0.106090
$544$	−36.3607	−1.55895
$545$	0	0
$546$	0	0
$547$	−37.1803	−1.58972	−0.794858	−	0.606795i	$-0.792455\pi$
$547$	−37.1803	−1.58972	−0.794858	+	0.606795i	$0.792455\pi$
$548$	25.7082	1.09820
$549$	12.0000	0.512148
$550$	0	0
$551$	−19.4164	−0.827167
$552$	0.527864	0.0224674
$553$	0	0
$554$	−12.2918	−0.522228
$555$	0	0
$556$	−24.9443	−1.05787
$557$	14.8885	0.630848	0.315424	−	0.948951i	$-0.397853\pi$
$557$	14.8885	0.630848	0.315424	+	0.948951i	$0.397853\pi$
$558$	−2.47214	−0.104654
$559$	21.5279	0.910532
$560$	0	0
$561$	−14.4721	−0.611014
$562$	−15.6738	−0.661158
$563$	12.9443	0.545536	0.272768	−	0.962080i	$-0.412061\pi$
$563$	12.9443	0.545536	0.272768	+	0.962080i	$0.412061\pi$
$564$	2.47214	0.104096
$565$	0	0
$566$	0	0
$567$	0	0
$568$	8.41641	0.353145
$569$	26.3050	1.10276	0.551380	−	0.834254i	$-0.314101\pi$
$569$	26.3050	1.10276	0.551380	+	0.834254i	$0.314101\pi$
$570$	0	0
$571$	16.2361	0.679458	0.339729	−	0.940523i	$-0.389665\pi$
$571$	16.2361	0.679458	0.339729	+	0.940523i	$0.389665\pi$
$572$	8.94427	0.373979
$573$	−20.9443	−0.874960
$574$	0	0
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	27.4164	1.14136	0.570680	−	0.821173i	$-0.306680\pi$
$577$	27.4164	1.14136	0.570680	+	0.821173i	$0.306680\pi$
$578$	−15.3820	−0.639805
$579$	12.8885	0.535630
$580$	0	0
$581$	0	0
$582$	−2.47214	−0.102473
$583$	−13.4164	−0.555651
$584$	−12.3607	−0.511489
$585$	0	0
$586$	−17.5279	−0.724069
$587$	−5.52786	−0.228159	−0.114080	−	0.993472i	$-0.536392\pi$
$587$	−5.52786	−0.228159	−0.114080	+	0.993472i	$0.536392\pi$
$588$	0	0
$589$	25.8885	1.06672
$590$	0	0
$591$	−21.0000	−0.863825
$592$	10.1459	0.416994
$593$	27.0557	1.11105	0.555523	−	0.831501i	$-0.312518\pi$
$593$	27.0557	1.11105	0.555523	+	0.831501i	$0.312518\pi$
$594$	1.38197	0.0567028
$595$	0	0
$596$	37.0344	1.51699
$597$	−13.8885	−0.568420
$598$	−0.360680	−0.0147493
$599$	−0.708204	−0.0289364	−0.0144682	−	0.999895i	$-0.504606\pi$
$599$	−0.708204	−0.0289364	−0.0144682	+	0.999895i	$0.504606\pi$
$600$	0	0
$601$	19.0557	0.777299	0.388650	−	0.921386i	$-0.372942\pi$
$601$	19.0557	0.777299	0.388650	+	0.921386i	$0.372942\pi$
$602$	0	0
$603$	0.708204	0.0288403
$604$	10.8541	0.441647
$605$	0	0
$606$	11.4164	0.463760
$607$	−8.58359	−0.348397	−0.174199	−	0.984711i	$-0.555733\pi$
$607$	−8.58359	−0.348397	−0.174199	+	0.984711i	$0.555733\pi$
$608$	−36.3607	−1.47462
$609$	0	0
$610$	0	0
$611$	−3.77709	−0.152805
$612$	−10.4721	−0.423311
$613$	45.2492	1.82760	0.913799	−	0.406166i	$-0.133134\pi$
$613$	45.2492	1.82760	0.913799	+	0.406166i	$0.133134\pi$
$614$	2.83282	0.114323
$615$	0	0
$616$	0	0
$617$	7.58359	0.305304	0.152652	−	0.988280i	$-0.451219\pi$
$617$	7.58359	0.305304	0.152652	+	0.988280i	$0.451219\pi$
$618$	−12.0000	−0.482711
$619$	−26.4721	−1.06400	−0.532002	−	0.846743i	$-0.678560\pi$
$619$	−26.4721	−1.06400	−0.532002	+	0.846743i	$0.678560\pi$
$620$	0	0
$621$	0.236068	0.00947308
$622$	15.0557	0.603680
$623$	0	0
$624$	4.58359	0.183491
$625$	0	0
$626$	−12.0000	−0.479616
$627$	−14.4721	−0.577961
$628$	−27.4164	−1.09403
$629$	35.4164	1.41214
$630$	0	0
$631$	31.6525	1.26007	0.630033	−	0.776569i	$-0.283042\pi$
$631$	31.6525	1.26007	0.630033	+	0.776569i	$0.283042\pi$
$632$	23.9443	0.952452
$633$	21.8885	0.869992
$634$	−6.14590	−0.244085
$635$	0	0
$636$	−9.70820	−0.384955
$637$	0	0
$638$	4.14590	0.164138
$639$	3.76393	0.148899
$640$	0	0
$641$	−19.3607	−0.764701	−0.382350	−	0.924017i	$-0.624885\pi$
$641$	−19.3607	−0.764701	−0.382350	+	0.924017i	$0.624885\pi$
$642$	−5.52786	−0.218167
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	0	0
$645$	0	0
$646$	−25.8885	−1.01857
$647$	−33.8885	−1.33230	−0.666148	−	0.745820i	$-0.732058\pi$
$647$	−33.8885	−1.33230	−0.666148	+	0.745820i	$0.732058\pi$
$648$	2.23607	0.0878410
$649$	−5.52786	−0.216988
$650$	0	0
$651$	0	0
$652$	19.4164	0.760405
$653$	23.8885	0.934831	0.467415	−	0.884038i	$-0.345185\pi$
$653$	23.8885	0.934831	0.467415	+	0.884038i	$0.345185\pi$
$654$	−6.43769	−0.251734
$655$	0	0
$656$	−16.5836	−0.647480
$657$	−5.52786	−0.215663
$658$	0	0
$659$	−32.9443	−1.28333	−0.641663	−	0.766986i	$-0.721755\pi$
$659$	−32.9443	−1.28333	−0.641663	+	0.766986i	$0.721755\pi$
$660$	0	0
$661$	−24.3607	−0.947521	−0.473760	−	0.880654i	$-0.657104\pi$
$661$	−24.3607	−0.947521	−0.473760	+	0.880654i	$0.657104\pi$
$662$	6.68692	0.259894
$663$	16.0000	0.621389
$664$	−2.11146	−0.0819404
$665$	0	0
$666$	−3.38197	−0.131049
$667$	0.708204	0.0274218
$668$	21.8885	0.846893
$669$	−23.4164	−0.905331
$670$	0	0
$671$	26.8328	1.03587
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−1.23607	−0.0476116
$675$	0	0
$676$	11.1459	0.428688
$677$	−14.1115	−0.542347	−0.271174	−	0.962530i	$-0.587412\pi$
$677$	−14.1115	−0.542347	−0.271174	+	0.962530i	$0.587412\pi$
$678$	12.0344	0.462180
$679$	0	0
$680$	0	0
$681$	−19.4164	−0.744038
$682$	−5.52786	−0.211673
$683$	4.81966	0.184419	0.0922096	−	0.995740i	$-0.470607\pi$
$683$	4.81966	0.184419	0.0922096	+	0.995740i	$0.470607\pi$
$684$	−10.4721	−0.400412
$685$	0	0
$686$	0	0
$687$	−11.0557	−0.421802
$688$	−16.1459	−0.615557
$689$	14.8328	0.565085
$690$	0	0
$691$	−42.8328	−1.62944	−0.814719	−	0.579857i	$-0.803109\pi$
$691$	−42.8328	−1.62944	−0.814719	+	0.579857i	$0.803109\pi$
$692$	10.4721	0.398091
$693$	0	0
$694$	−11.5623	−0.438899
$695$	0	0
$696$	6.70820	0.254274
$697$	−57.8885	−2.19268
$698$	2.11146	0.0799198
$699$	−13.4721	−0.509563
$700$	0	0
$701$	−33.7771	−1.27574	−0.637871	−	0.770143i	$-0.720185\pi$
$701$	−33.7771	−1.27574	−0.637871	+	0.770143i	$0.720185\pi$
$702$	−1.52786	−0.0576655
$703$	35.4164	1.33576
$704$	−0.527864	−0.0198946
$705$	0	0
$706$	13.5279	0.509128
$707$	0	0
$708$	−4.00000	−0.150329
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	10.7082	0.401589
$712$	23.4164	0.877567
$713$	−0.944272	−0.0353633
$714$	0	0
$715$	0	0
$716$	−27.4164	−1.02460
$717$	3.05573	0.114118
$718$	10.0344	0.374482
$719$	16.9443	0.631915	0.315957	−	0.948773i	$-0.397674\pi$
$719$	16.9443	0.631915	0.315957	+	0.948773i	$0.397674\pi$
$720$	0	0
$721$	0	0
$722$	−14.1459	−0.526456
$723$	−9.52786	−0.354345
$724$	4.00000	0.148659
$725$	0	0
$726$	−3.70820	−0.137624
$727$	−45.3050	−1.68027	−0.840134	−	0.542379i	$-0.817524\pi$
$727$	−45.3050	−1.68027	−0.840134	+	0.542379i	$0.817524\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−56.3607	−2.08458
$732$	19.4164	0.717651
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	2.47214	0.0912482
$735$	0	0
$736$	1.32624	0.0488858
$737$	1.58359	0.0583324
$738$	5.52786	0.203483
$739$	−7.29180	−0.268233	−0.134117	−	0.990966i	$-0.542820\pi$
$739$	−7.29180	−0.268233	−0.134117	+	0.990966i	$0.542820\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	20.9443	0.768371	0.384185	−	0.923256i	$-0.374482\pi$
$743$	20.9443	0.768371	0.384185	+	0.923256i	$0.374482\pi$
$744$	−8.94427	−0.327913
$745$	0	0
$746$	5.27051	0.192967
$747$	−0.944272	−0.0345491
$748$	−23.4164	−0.856189
$749$	0	0
$750$	0	0
$751$	51.7771	1.88937	0.944686	−	0.327975i	$-0.106366\pi$
$751$	51.7771	1.88937	0.944686	+	0.327975i	$0.106366\pi$
$752$	2.83282	0.103302
$753$	1.52786	0.0556785
$754$	−4.58359	−0.166925
$755$	0	0
$756$	0	0
$757$	−3.47214	−0.126197	−0.0630985	−	0.998007i	$-0.520098\pi$
$757$	−3.47214	−0.126197	−0.0630985	+	0.998007i	$0.520098\pi$
$758$	2.90983	0.105690
$759$	0.527864	0.0191603
$760$	0	0
$761$	−35.0557	−1.27077	−0.635385	−	0.772196i	$-0.719158\pi$
$761$	−35.0557	−1.27077	−0.635385	+	0.772196i	$0.719158\pi$
$762$	1.96556	0.0712047
$763$	0	0
$764$	−33.8885	−1.22604
$765$	0	0
$766$	−4.58359	−0.165612
$767$	6.11146	0.220672
$768$	6.56231	0.236797
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	−8.94427	−0.322120
$772$	20.8541	0.750556
$773$	31.7771	1.14294	0.571471	−	0.820622i	$-0.306373\pi$
$773$	31.7771	1.14294	0.571471	+	0.820622i	$0.306373\pi$
$774$	5.38197	0.193451
$775$	0	0
$776$	−8.94427	−0.321081
$777$	0	0
$778$	−18.5066	−0.663493
$779$	−57.8885	−2.07407
$780$	0	0
$781$	8.41641	0.301163
$782$	0.944272	0.0337671
$783$	3.00000	0.107211
$784$	0	0
$785$	0	0
$786$	13.8885	0.495388
$787$	−42.4721	−1.51397	−0.756984	−	0.653433i	$-0.773328\pi$
$787$	−42.4721	−1.51397	−0.756984	+	0.653433i	$0.773328\pi$
$788$	−33.9787	−1.21044
$789$	−1.18034	−0.0420212
$790$	0	0
$791$	0	0
$792$	5.00000	0.177667
$793$	−29.6656	−1.05346
$794$	16.0000	0.567819
$795$	0	0
$796$	−22.4721	−0.796504
$797$	25.5279	0.904243	0.452122	−	0.891956i	$-0.350667\pi$
$797$	25.5279	0.904243	0.452122	+	0.891956i	$0.350667\pi$
$798$	0	0
$799$	9.88854	0.349832
$800$	0	0
$801$	10.4721	0.370015
$802$	−0.978714	−0.0345596
$803$	−12.3607	−0.436199
$804$	1.14590	0.0404127
$805$	0	0
$806$	6.11146	0.215267
$807$	−8.94427	−0.314853
$808$	41.3050	1.45310
$809$	17.4721	0.614288	0.307144	−	0.951663i	$-0.400627\pi$
$809$	17.4721	0.614288	0.307144	+	0.951663i	$0.400627\pi$
$810$	0	0
$811$	−14.8328	−0.520851	−0.260425	−	0.965494i	$-0.583863\pi$
$811$	−14.8328	−0.520851	−0.260425	+	0.965494i	$0.583863\pi$
$812$	0	0
$813$	−23.4164	−0.821249
$814$	−7.56231	−0.265059
$815$	0	0
$816$	−12.0000	−0.420084
$817$	−56.3607	−1.97181
$818$	12.0000	0.419570
$819$	0	0
$820$	0	0
$821$	25.7771	0.899627	0.449813	−	0.893123i	$-0.351491\pi$
$821$	25.7771	0.899627	0.449813	+	0.893123i	$0.351491\pi$
$822$	−9.81966	−0.342500
$823$	−49.5410	−1.72689	−0.863446	−	0.504442i	$-0.831698\pi$
$823$	−49.5410	−1.72689	−0.863446	+	0.504442i	$0.831698\pi$
$824$	−43.4164	−1.51248
$825$	0	0
$826$	0	0
$827$	9.76393	0.339525	0.169763	−	0.985485i	$-0.445700\pi$
$827$	9.76393	0.339525	0.169763	+	0.985485i	$0.445700\pi$
$828$	0.381966	0.0132742
$829$	6.47214	0.224787	0.112393	−	0.993664i	$-0.464148\pi$
$829$	6.47214	0.224787	0.112393	+	0.993664i	$0.464148\pi$
$830$	0	0
$831$	−19.8885	−0.689926
$832$	0.583592	0.0202324
$833$	0	0
$834$	9.52786	0.329923
$835$	0	0
$836$	−23.4164	−0.809873
$837$	−4.00000	−0.138260
$838$	3.63932	0.125718
$839$	43.1935	1.49121	0.745603	−	0.666391i	$-0.232162\pi$
$839$	43.1935	1.49121	0.745603	+	0.666391i	$0.232162\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	10.1459	0.349651
$843$	−25.3607	−0.873468
$844$	35.4164	1.21908
$845$	0	0
$846$	−0.944272	−0.0324647
$847$	0	0
$848$	−11.1246	−0.382021
$849$	0	0
$850$	0	0
$851$	−1.29180	−0.0442822
$852$	6.09017	0.208646
$853$	−46.8328	−1.60353	−0.801763	−	0.597643i	$-0.796104\pi$
$853$	−46.8328	−1.60353	−0.801763	+	0.597643i	$0.796104\pi$
$854$	0	0
$855$	0	0
$856$	−20.0000	−0.683586
$857$	2.11146	0.0721260	0.0360630	−	0.999350i	$-0.488518\pi$
$857$	2.11146	0.0721260	0.0360630	+	0.999350i	$0.488518\pi$
$858$	−3.41641	−0.116634
$859$	9.88854	0.337393	0.168696	−	0.985668i	$-0.446044\pi$
$859$	9.88854	0.337393	0.168696	+	0.985668i	$0.446044\pi$
$860$	0	0
$861$	0	0
$862$	17.8885	0.609286
$863$	38.1246	1.29778	0.648888	−	0.760884i	$-0.275234\pi$
$863$	38.1246	1.29778	0.648888	+	0.760884i	$0.275234\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	−3.05573	−0.103838
$867$	−24.8885	−0.845259
$868$	0	0
$869$	23.9443	0.812254
$870$	0	0
$871$	−1.75078	−0.0593228
$872$	−23.2918	−0.788760
$873$	−4.00000	−0.135379
$874$	0.944272	0.0319405
$875$	0	0
$876$	−8.94427	−0.302199
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−0.360680	−0.0121724
$879$	−28.3607	−0.956582
$880$	0	0
$881$	23.0557	0.776767	0.388384	−	0.921498i	$-0.373034\pi$
$881$	23.0557	0.776767	0.388384	+	0.921498i	$0.373034\pi$
$882$	0	0
$883$	23.5410	0.792218	0.396109	−	0.918203i	$-0.370360\pi$
$883$	23.5410	0.792218	0.396109	+	0.918203i	$0.370360\pi$
$884$	25.8885	0.870726
$885$	0	0
$886$	−20.3607	−0.684030
$887$	−26.8328	−0.900958	−0.450479	−	0.892787i	$-0.648747\pi$
$887$	−26.8328	−0.900958	−0.450479	+	0.892787i	$0.648747\pi$
$888$	−12.2361	−0.410616
$889$	0	0
$890$	0	0
$891$	2.23607	0.0749111
$892$	−37.8885	−1.26860
$893$	9.88854	0.330908
$894$	−14.1459	−0.473110
$895$	0	0
$896$	0	0
$897$	−0.583592	−0.0194856
$898$	9.56231	0.319098
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−38.8328	−1.29371
$902$	12.3607	0.411566
$903$	0	0
$904$	43.5410	1.44815
$905$	0	0
$906$	−4.14590	−0.137738
$907$	10.1115	0.335745	0.167873	−	0.985809i	$-0.446310\pi$
$907$	10.1115	0.335745	0.167873	+	0.985809i	$0.446310\pi$
$908$	−31.4164	−1.04259
$909$	18.4721	0.612682
$910$	0	0
$911$	20.0132	0.663065	0.331533	−	0.943444i	$-0.392434\pi$
$911$	20.0132	0.663065	0.331533	+	0.943444i	$0.392434\pi$
$912$	−12.0000	−0.397360
$913$	−2.11146	−0.0698790
$914$	−4.97871	−0.164681
$915$	0	0
$916$	−17.8885	−0.591054
$917$	0	0
$918$	4.00000	0.132020
$919$	−7.18034	−0.236858	−0.118429	−	0.992963i	$-0.537786\pi$
$919$	−7.18034	−0.236858	−0.118429	+	0.992963i	$0.537786\pi$
$920$	0	0
$921$	4.58359	0.151034
$922$	−13.3050	−0.438175
$923$	−9.30495	−0.306276
$924$	0	0
$925$	0	0
$926$	20.9443	0.688271
$927$	−19.4164	−0.637719
$928$	16.8541	0.553263
$929$	−31.4164	−1.03074	−0.515369	−	0.856968i	$-0.672345\pi$
$929$	−31.4164	−1.03074	−0.515369	+	0.856968i	$0.672345\pi$
$930$	0	0
$931$	0	0
$932$	−21.7984	−0.714029
$933$	24.3607	0.797533
$934$	20.9443	0.685318
$935$	0	0
$936$	−5.52786	−0.180684
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	0	0
$939$	−19.4164	−0.633631
$940$	0	0
$941$	39.7771	1.29670	0.648348	−	0.761344i	$-0.275460\pi$
$941$	39.7771	1.29670	0.648348	+	0.761344i	$0.275460\pi$
$942$	10.4721	0.341201
$943$	2.11146	0.0687585
$944$	−4.58359	−0.149183
$945$	0	0
$946$	12.0344	0.391273
$947$	−15.0557	−0.489245	−0.244623	−	0.969618i	$-0.578664\pi$
$947$	−15.0557	−0.489245	−0.244623	+	0.969618i	$0.578664\pi$
$948$	17.3262	0.562730
$949$	13.6656	0.443605
$950$	0	0
$951$	−9.94427	−0.322465
$952$	0	0
$953$	33.3607	1.08066	0.540329	−	0.841454i	$-0.318300\pi$
$953$	33.3607	1.08066	0.540329	+	0.841454i	$0.318300\pi$
$954$	3.70820	0.120058
$955$	0	0
$956$	4.94427	0.159909
$957$	6.70820	0.216845
$958$	13.3050	0.429863
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	8.36068	0.269559
$963$	−8.94427	−0.288225
$964$	−15.4164	−0.496529
$965$	0	0
$966$	0	0
$967$	−27.7771	−0.893251	−0.446625	−	0.894721i	$-0.647374\pi$
$967$	−27.7771	−0.893251	−0.446625	+	0.894721i	$0.647374\pi$
$968$	−13.4164	−0.431220
$969$	−41.8885	−1.34565
$970$	0	0
$971$	−21.3050	−0.683708	−0.341854	−	0.939753i	$-0.611055\pi$
$971$	−21.3050	−0.683708	−0.341854	+	0.939753i	$0.611055\pi$
$972$	1.61803	0.0518985
$973$	0	0
$974$	0.798374	0.0255815
$975$	0	0
$976$	22.2492	0.712180
$977$	34.4164	1.10108	0.550539	−	0.834809i	$-0.314422\pi$
$977$	34.4164	1.10108	0.550539	+	0.834809i	$0.314422\pi$
$978$	−7.41641	−0.237151
$979$	23.4164	0.748392
$980$	0	0
$981$	−10.4164	−0.332570
$982$	−16.7984	−0.536058
$983$	40.9443	1.30592	0.652960	−	0.757393i	$-0.273527\pi$
$983$	40.9443	1.30592	0.652960	+	0.757393i	$0.273527\pi$
$984$	20.0000	0.637577
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	25.8885	0.823624
$989$	2.05573	0.0653684
$990$	0	0
$991$	9.54102	0.303080	0.151540	−	0.988451i	$-0.451577\pi$
$991$	9.54102	0.303080	0.151540	+	0.988451i	$0.451577\pi$
$992$	−22.4721	−0.713491
$993$	10.8197	0.343352
$994$	0	0
$995$	0	0
$996$	−1.52786	−0.0484122
$997$	−19.4164	−0.614924	−0.307462	−	0.951560i	$-0.599480\pi$
$997$	−19.4164	−0.614924	−0.307462	+	0.951560i	$0.599480\pi$
$998$	7.41641	0.234762
$999$	−5.47214	−0.173131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.ba.1.1	yes	2
5.4	even	2		3675.2.a.v.1.2	yes	2
7.6	odd	2		3675.2.a.bc.1.1	yes	2
35.34	odd	2		3675.2.a.u.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.2.a.u.1.2	✓	2	35.34	odd	2
3675.2.a.v.1.2	yes	2	5.4	even	2
3675.2.a.ba.1.1	yes	2	1.1	even	1	trivial
3675.2.a.bc.1.1	yes	2	7.6	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +2.23607 q^{8} +1.00000 q^{9} +2.23607 q^{11} +1.61803 q^{12} -2.47214 q^{13} +1.85410 q^{16} +6.47214 q^{17} -0.618034 q^{18} +6.47214 q^{19} -1.38197 q^{22} -0.236068 q^{23} -2.23607 q^{24} +1.52786 q^{26} -1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} -5.61803 q^{32} -2.23607 q^{33} -4.00000 q^{34} -1.61803 q^{36} +5.47214 q^{37} -4.00000 q^{38} +2.47214 q^{39} -8.94427 q^{41} -8.70820 q^{43} -3.61803 q^{44} +0.145898 q^{46} +1.52786 q^{47} -1.85410 q^{48} -6.47214 q^{51} +4.00000 q^{52} -6.00000 q^{53} +0.618034 q^{54} -6.47214 q^{57} +1.85410 q^{58} -2.47214 q^{59} +12.0000 q^{61} -2.47214 q^{62} -0.236068 q^{64} +1.38197 q^{66} +0.708204 q^{67} -10.4721 q^{68} +0.236068 q^{69} +3.76393 q^{71} +2.23607 q^{72} -5.52786 q^{73} -3.38197 q^{74} -10.4721 q^{76} -1.52786 q^{78} +10.7082 q^{79} +1.00000 q^{81} +5.52786 q^{82} -0.944272 q^{83} +5.38197 q^{86} +3.00000 q^{87} +5.00000 q^{88} +10.4721 q^{89} +0.381966 q^{92} -4.00000 q^{93} -0.944272 q^{94} +5.61803 q^{96} -4.00000 q^{97} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{9} + q^{12} + 4 q^{13} - 3 q^{16} + 4 q^{17} + q^{18} + 4 q^{19} - 5 q^{22} + 4 q^{23} + 12 q^{26} - 2 q^{27} - 6 q^{29} + 8 q^{31} - 9 q^{32} - 8 q^{34} - q^{36} + 2 q^{37} - 8 q^{38} - 4 q^{39} - 4 q^{43} - 5 q^{44} + 7 q^{46} + 12 q^{47} + 3 q^{48} - 4 q^{51} + 8 q^{52} - 12 q^{53} - q^{54} - 4 q^{57} - 3 q^{58} + 4 q^{59} + 24 q^{61} + 4 q^{62} + 4 q^{64} + 5 q^{66} - 12 q^{67} - 12 q^{68} - 4 q^{69} + 12 q^{71} - 20 q^{73} - 9 q^{74} - 12 q^{76} - 12 q^{78} + 8 q^{79} + 2 q^{81} + 20 q^{82} + 16 q^{83} + 13 q^{86} + 6 q^{87} + 10 q^{88} + 12 q^{89} + 3 q^{92} - 8 q^{93} + 16 q^{94} + 9 q^{96} - 8 q^{97}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 2 * q^9 + q^12 + 4 * q^13 - 3 * q^16 + 4 * q^17 + q^18 + 4 * q^19 - 5 * q^22 + 4 * q^23 + 12 * q^26 - 2 * q^27 - 6 * q^29 + 8 * q^31 - 9 * q^32 - 8 * q^34 - q^36 + 2 * q^37 - 8 * q^38 - 4 * q^39 - 4 * q^43 - 5 * q^44 + 7 * q^46 + 12 * q^47 + 3 * q^48 - 4 * q^51 + 8 * q^52 - 12 * q^53 - q^54 - 4 * q^57 - 3 * q^58 + 4 * q^59 + 24 * q^61 + 4 * q^62 + 4 * q^64 + 5 * q^66 - 12 * q^67 - 12 * q^68 - 4 * q^69 + 12 * q^71 - 20 * q^73 - 9 * q^74 - 12 * q^76 - 12 * q^78 + 8 * q^79 + 2 * q^81 + 20 * q^82 + 16 * q^83 + 13 * q^86 + 6 * q^87 + 10 * q^88 + 12 * q^89 + 3 * q^92 - 8 * q^93 + 16 * q^94 + 9 * q^96 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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