LMFDB - Embedded newform 6664.2.a.y.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 8x^{5} + 21x^{4} + 25x^{3} - 41x^{2} - 28x + 19 $ x^7 - 3x^6 - 8x^5 + 21x^4 + 25x^3 - 41x^2 - 28x + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.55636$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.55636 q^{3} +3.12252 q^{5} -0.577758 q^{9} -5.30004 q^{11} +3.11905 q^{13} +4.85975 q^{15} -1.00000 q^{17} -5.65030 q^{19} -6.92994 q^{23} +4.75013 q^{25} -5.56826 q^{27} +3.34548 q^{29} -9.32439 q^{31} -8.24874 q^{33} +0.437652 q^{37} +4.85435 q^{39} -4.48893 q^{41} +2.07827 q^{43} -1.80406 q^{45} -10.4139 q^{47} -1.55636 q^{51} +4.15326 q^{53} -16.5495 q^{55} -8.79388 q^{57} +3.68330 q^{59} +2.38973 q^{61} +9.73930 q^{65} -1.99926 q^{67} -10.7854 q^{69} -11.0535 q^{71} +0.306192 q^{73} +7.39289 q^{75} -4.68545 q^{79} -6.93292 q^{81} +10.7572 q^{83} -3.12252 q^{85} +5.20675 q^{87} -11.0224 q^{89} -14.5121 q^{93} -17.6432 q^{95} +4.44964 q^{97} +3.06214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 2 * q^5 + 4 * q^9 - 12 * q^11 - 2 * q^13 - 6 * q^15 - 7 * q^17 + 3 * q^19 - 18 * q^23 + 15 * q^25 + 18 * q^27 + 5 * q^29 + 10 * q^31 - 21 * q^33 + 11 * q^37 - 12 * q^39 - 15 * q^43 - 13 * q^45 - 15 * q^47 - 3 * q^51 + 8 * q^53 + 9 * q^55 - 3 * q^57 - 9 * q^59 + 4 * q^61 - 38 * q^65 - 30 * q^67 + 28 * q^69 - 27 * q^71 + 5 * q^73 + 2 * q^75 + 3 * q^79 - 17 * q^81 + 12 * q^83 + 2 * q^85 - 19 * q^87 - 32 * q^89 - 9 * q^93 - 42 * q^95 + 39 * q^97 - 20 * q^99

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
$3$	1.55636	0.898562	0.449281	−	0.893390i	$-0.351680\pi$
$3$	1.55636	0.898562	0.449281	+	0.893390i	$0.351680\pi$
$4$	0	0
$5$	3.12252	1.39643	0.698217	−	0.715886i	$-0.253977\pi$
$5$	3.12252	1.39643	0.698217	+	0.715886i	$0.253977\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.577758	−0.192586
$10$	0	0
$11$	−5.30004	−1.59802	−0.799010	−	0.601317i	$-0.794643\pi$
$11$	−5.30004	−1.59802	−0.799010	+	0.601317i	$0.794643\pi$
$12$	0	0
$13$	3.11905	0.865069	0.432534	−	0.901617i	$-0.357619\pi$
$13$	3.11905	0.865069	0.432534	+	0.901617i	$0.357619\pi$
$14$	0	0
$15$	4.85975	1.25478
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.65030	−1.29627	−0.648134	−	0.761526i	$-0.724450\pi$
$19$	−5.65030	−1.29627	−0.648134	+	0.761526i	$0.724450\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.92994	−1.44499	−0.722496	−	0.691375i	$-0.757005\pi$
$23$	−6.92994	−1.44499	−0.722496	+	0.691375i	$0.757005\pi$
$24$	0	0
$25$	4.75013	0.950026
$26$	0	0
$27$	−5.56826	−1.07161
$28$	0	0
$29$	3.34548	0.621239	0.310620	−	0.950534i	$-0.399463\pi$
$29$	3.34548	0.621239	0.310620	+	0.950534i	$0.399463\pi$
$30$	0	0
$31$	−9.32439	−1.67471	−0.837355	−	0.546660i	$-0.815899\pi$
$31$	−9.32439	−1.67471	−0.837355	+	0.546660i	$0.815899\pi$
$32$	0	0
$33$	−8.24874	−1.43592
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.437652	0.0719495	0.0359748	−	0.999353i	$-0.488546\pi$
$37$	0.437652	0.0719495	0.0359748	+	0.999353i	$0.488546\pi$
$38$	0	0
$39$	4.85435	0.777318
$40$	0	0
$41$	−4.48893	−0.701053	−0.350527	−	0.936553i	$-0.613997\pi$
$41$	−4.48893	−0.701053	−0.350527	+	0.936553i	$0.613997\pi$
$42$	0	0
$43$	2.07827	0.316933	0.158466	−	0.987364i	$-0.449345\pi$
$43$	2.07827	0.316933	0.158466	+	0.987364i	$0.449345\pi$
$44$	0	0
$45$	−1.80406	−0.268934
$46$	0	0
$47$	−10.4139	−1.51903	−0.759515	−	0.650490i	$-0.774564\pi$
$47$	−10.4139	−1.51903	−0.759515	+	0.650490i	$0.774564\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.55636	−0.217933
$52$	0	0
$53$	4.15326	0.570495	0.285247	−	0.958454i	$-0.407924\pi$
$53$	4.15326	0.570495	0.285247	+	0.958454i	$0.407924\pi$
$54$	0	0
$55$	−16.5495	−2.23153
$56$	0	0
$57$	−8.79388	−1.16478
$58$	0	0
$59$	3.68330	0.479525	0.239762	−	0.970832i	$-0.422930\pi$
$59$	3.68330	0.479525	0.239762	+	0.970832i	$0.422930\pi$
$60$	0	0
$61$	2.38973	0.305973	0.152987	−	0.988228i	$-0.451111\pi$
$61$	2.38973	0.305973	0.152987	+	0.988228i	$0.451111\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.73930	1.20801
$66$	0	0
$67$	−1.99926	−0.244249	−0.122124	−	0.992515i	$-0.538971\pi$
$67$	−1.99926	−0.244249	−0.122124	+	0.992515i	$0.538971\pi$
$68$	0	0
$69$	−10.7854	−1.29842
$70$	0	0
$71$	−11.0535	−1.31181	−0.655906	−	0.754842i	$-0.727713\pi$
$71$	−11.0535	−1.31181	−0.655906	+	0.754842i	$0.727713\pi$
$72$	0	0
$73$	0.306192	0.0358371	0.0179185	−	0.999839i	$-0.494296\pi$
$73$	0.306192	0.0358371	0.0179185	+	0.999839i	$0.494296\pi$
$74$	0	0
$75$	7.39289	0.853658
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.68545	−0.527154	−0.263577	−	0.964638i	$-0.584902\pi$
$79$	−4.68545	−0.527154	−0.263577	+	0.964638i	$0.584902\pi$
$80$	0	0
$81$	−6.93292	−0.770325
$82$	0	0
$83$	10.7572	1.18075	0.590376	−	0.807129i	$-0.298980\pi$
$83$	10.7572	1.18075	0.590376	+	0.807129i	$0.298980\pi$
$84$	0	0
$85$	−3.12252	−0.338685
$86$	0	0
$87$	5.20675	0.558222
$88$	0	0
$89$	−11.0224	−1.16838	−0.584188	−	0.811618i	$-0.698587\pi$
$89$	−11.0224	−1.16838	−0.584188	+	0.811618i	$0.698587\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14.5121	−1.50483
$94$	0	0
$95$	−17.6432	−1.81015
$96$	0	0
$97$	4.44964	0.451792	0.225896	−	0.974151i	$-0.427469\pi$
$97$	4.44964	0.451792	0.225896	+	0.974151i	$0.427469\pi$
$98$	0	0
$99$	3.06214	0.307757
$100$	0	0
$101$	1.31248	0.130597	0.0652983	−	0.997866i	$-0.479200\pi$
$101$	1.31248	0.130597	0.0652983	+	0.997866i	$0.479200\pi$
$102$	0	0
$103$	−2.49874	−0.246208	−0.123104	−	0.992394i	$-0.539285\pi$
$103$	−2.49874	−0.246208	−0.123104	+	0.992394i	$0.539285\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1657	0.982757	0.491378	−	0.870946i	$-0.336493\pi$
$107$	10.1657	0.982757	0.491378	+	0.870946i	$0.336493\pi$
$108$	0	0
$109$	0.473853	0.0453869	0.0226934	−	0.999742i	$-0.492776\pi$
$109$	0.473853	0.0453869	0.0226934	+	0.999742i	$0.492776\pi$
$110$	0	0
$111$	0.681142	0.0646511
$112$	0	0
$113$	−9.75485	−0.917659	−0.458830	−	0.888524i	$-0.651731\pi$
$113$	−9.75485	−0.917659	−0.458830	+	0.888524i	$0.651731\pi$
$114$	0	0
$115$	−21.6389	−2.01784
$116$	0	0
$117$	−1.80206	−0.166600
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.0904	1.55367
$122$	0	0
$123$	−6.98637	−0.629940
$124$	0	0
$125$	−0.780222	−0.0697851
$126$	0	0
$127$	15.5883	1.38323	0.691617	−	0.722264i	$-0.256899\pi$
$127$	15.5883	1.38323	0.691617	+	0.722264i	$0.256899\pi$
$128$	0	0
$129$	3.23452	0.284784
$130$	0	0
$131$	21.9040	1.91376	0.956882	−	0.290476i	$-0.0938136\pi$
$131$	21.9040	1.91376	0.956882	+	0.290476i	$0.0938136\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−17.3870	−1.49644
$136$	0	0
$137$	17.5016	1.49526	0.747632	−	0.664113i	$-0.231191\pi$
$137$	17.5016	1.49526	0.747632	+	0.664113i	$0.231191\pi$
$138$	0	0
$139$	3.29930	0.279843	0.139921	−	0.990163i	$-0.455315\pi$
$139$	3.29930	0.279843	0.139921	+	0.990163i	$0.455315\pi$
$140$	0	0
$141$	−16.2078	−1.36494
$142$	0	0
$143$	−16.5311	−1.38240
$144$	0	0
$145$	10.4463	0.867519
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.9562	−1.14334	−0.571668	−	0.820485i	$-0.693703\pi$
$149$	−13.9562	−1.14334	−0.571668	+	0.820485i	$0.693703\pi$
$150$	0	0
$151$	10.9871	0.894119	0.447060	−	0.894504i	$-0.352471\pi$
$151$	10.9871	0.894119	0.447060	+	0.894504i	$0.352471\pi$
$152$	0	0
$153$	0.577758	0.0467090
$154$	0	0
$155$	−29.1156	−2.33862
$156$	0	0
$157$	−13.0368	−1.04045	−0.520226	−	0.854029i	$-0.674152\pi$
$157$	−13.0368	−1.04045	−0.520226	+	0.854029i	$0.674152\pi$
$158$	0	0
$159$	6.46395	0.512625
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.4847	−1.05621	−0.528103	−	0.849180i	$-0.677097\pi$
$163$	−13.4847	−1.05621	−0.528103	+	0.849180i	$0.677097\pi$
$164$	0	0
$165$	−25.7569	−2.00517
$166$	0	0
$167$	−4.87541	−0.377270	−0.188635	−	0.982047i	$-0.560406\pi$
$167$	−4.87541	−0.377270	−0.188635	+	0.982047i	$0.560406\pi$
$168$	0	0
$169$	−3.27153	−0.251656
$170$	0	0
$171$	3.26451	0.249643
$172$	0	0
$173$	−9.73683	−0.740278	−0.370139	−	0.928976i	$-0.620690\pi$
$173$	−9.73683	−0.740278	−0.370139	+	0.928976i	$0.620690\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.73252	0.430883
$178$	0	0
$179$	−21.7243	−1.62375	−0.811875	−	0.583831i	$-0.801553\pi$
$179$	−21.7243	−1.62375	−0.811875	+	0.583831i	$0.801553\pi$
$180$	0	0
$181$	20.1205	1.49555	0.747774	−	0.663953i	$-0.231123\pi$
$181$	20.1205	1.49555	0.747774	+	0.663953i	$0.231123\pi$
$182$	0	0
$183$	3.71927	0.274936
$184$	0	0
$185$	1.36658	0.100473
$186$	0	0
$187$	5.30004	0.387577
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.6906	1.35241	0.676204	−	0.736715i	$-0.263624\pi$
$191$	18.6906	1.35241	0.676204	+	0.736715i	$0.263624\pi$
$192$	0	0
$193$	8.80417	0.633738	0.316869	−	0.948469i	$-0.397368\pi$
$193$	8.80417	0.633738	0.316869	+	0.948469i	$0.397368\pi$
$194$	0	0
$195$	15.1578	1.08547
$196$	0	0
$197$	3.46584	0.246931	0.123465	−	0.992349i	$-0.460599\pi$
$197$	3.46584	0.246931	0.123465	+	0.992349i	$0.460599\pi$
$198$	0	0
$199$	−7.65289	−0.542499	−0.271250	−	0.962509i	$-0.587437\pi$
$199$	−7.65289	−0.542499	−0.271250	+	0.962509i	$0.587437\pi$
$200$	0	0
$201$	−3.11156	−0.219473
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−14.0168	−0.978974
$206$	0	0
$207$	4.00383	0.278285
$208$	0	0
$209$	29.9468	2.07146
$210$	0	0
$211$	23.5472	1.62106	0.810529	−	0.585699i	$-0.199180\pi$
$211$	23.5472	1.62106	0.810529	+	0.585699i	$0.199180\pi$
$212$	0	0
$213$	−17.2032	−1.17874
$214$	0	0
$215$	6.48943	0.442576
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.476544	0.0322018
$220$	0	0
$221$	−3.11905	−0.209810
$222$	0	0
$223$	13.1948	0.883590	0.441795	−	0.897116i	$-0.354342\pi$
$223$	13.1948	0.883590	0.441795	+	0.897116i	$0.354342\pi$
$224$	0	0
$225$	−2.74443	−0.182962
$226$	0	0
$227$	−15.0783	−1.00078	−0.500391	−	0.865799i	$-0.666810\pi$
$227$	−15.0783	−1.00078	−0.500391	+	0.865799i	$0.666810\pi$
$228$	0	0
$229$	−23.1560	−1.53019	−0.765097	−	0.643916i	$-0.777309\pi$
$229$	−23.1560	−1.53019	−0.765097	+	0.643916i	$0.777309\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.2044	−0.668510	−0.334255	−	0.942483i	$-0.608485\pi$
$233$	−10.2044	−0.668510	−0.334255	+	0.942483i	$0.608485\pi$
$234$	0	0
$235$	−32.5177	−2.12122
$236$	0	0
$237$	−7.29222	−0.473681
$238$	0	0
$239$	−18.6974	−1.20943	−0.604716	−	0.796441i	$-0.706713\pi$
$239$	−18.6974	−1.20943	−0.604716	+	0.796441i	$0.706713\pi$
$240$	0	0
$241$	−21.7560	−1.40143	−0.700715	−	0.713441i	$-0.747136\pi$
$241$	−21.7560	−1.40143	−0.700715	+	0.713441i	$0.747136\pi$
$242$	0	0
$243$	5.91470	0.379428
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−17.6236	−1.12136
$248$	0	0
$249$	16.7420	1.06098
$250$	0	0
$251$	27.9141	1.76192	0.880962	−	0.473188i	$-0.156897\pi$
$251$	27.9141	1.76192	0.880962	+	0.473188i	$0.156897\pi$
$252$	0	0
$253$	36.7289	2.30913
$254$	0	0
$255$	−4.85975	−0.304329
$256$	0	0
$257$	−24.8860	−1.55234	−0.776172	−	0.630522i	$-0.782841\pi$
$257$	−24.8860	−1.55234	−0.776172	+	0.630522i	$0.782841\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.93288	−0.119642
$262$	0	0
$263$	−24.4833	−1.50970	−0.754851	−	0.655896i	$-0.772291\pi$
$263$	−24.4833	−1.50970	−0.754851	+	0.655896i	$0.772291\pi$
$264$	0	0
$265$	12.9686	0.796658
$266$	0	0
$267$	−17.1548	−1.04986
$268$	0	0
$269$	28.4920	1.73719	0.868595	−	0.495523i	$-0.165024\pi$
$269$	28.4920	1.73719	0.868595	+	0.495523i	$0.165024\pi$
$270$	0	0
$271$	−13.0943	−0.795421	−0.397711	−	0.917511i	$-0.630195\pi$
$271$	−13.0943	−0.795421	−0.397711	+	0.917511i	$0.630195\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−25.1759	−1.51816
$276$	0	0
$277$	6.58218	0.395485	0.197742	−	0.980254i	$-0.436639\pi$
$277$	6.58218	0.395485	0.197742	+	0.980254i	$0.436639\pi$
$278$	0	0
$279$	5.38724	0.322526
$280$	0	0
$281$	−28.7156	−1.71303	−0.856514	−	0.516124i	$-0.827375\pi$
$281$	−28.7156	−1.71303	−0.856514	+	0.516124i	$0.827375\pi$
$282$	0	0
$283$	24.5310	1.45821	0.729107	−	0.684399i	$-0.239936\pi$
$283$	24.5310	1.45821	0.729107	+	0.684399i	$0.239936\pi$
$284$	0	0
$285$	−27.4591	−1.62653
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.92522	0.405963
$292$	0	0
$293$	3.29835	0.192692	0.0963459	−	0.995348i	$-0.469284\pi$
$293$	3.29835	0.192692	0.0963459	+	0.995348i	$0.469284\pi$
$294$	0	0
$295$	11.5012	0.669624
$296$	0	0
$297$	29.5120	1.71246
$298$	0	0
$299$	−21.6148	−1.25002
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.04268	0.117349
$304$	0	0
$305$	7.46197	0.427271
$306$	0	0
$307$	26.5890	1.51751	0.758757	−	0.651373i	$-0.225807\pi$
$307$	26.5890	1.51751	0.758757	+	0.651373i	$0.225807\pi$
$308$	0	0
$309$	−3.88893	−0.221233
$310$	0	0
$311$	15.3665	0.871354	0.435677	−	0.900103i	$-0.356509\pi$
$311$	15.3665	0.871354	0.435677	+	0.900103i	$0.356509\pi$
$312$	0	0
$313$	6.77112	0.382727	0.191363	−	0.981519i	$-0.438709\pi$
$313$	6.77112	0.382727	0.191363	+	0.981519i	$0.438709\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.6353	1.43982	0.719912	−	0.694065i	$-0.244182\pi$
$317$	25.6353	1.43982	0.719912	+	0.694065i	$0.244182\pi$
$318$	0	0
$319$	−17.7311	−0.992753
$320$	0	0
$321$	15.8215	0.883068
$322$	0	0
$323$	5.65030	0.314391
$324$	0	0
$325$	14.8159	0.821838
$326$	0	0
$327$	0.737483	0.0407829
$328$	0	0
$329$	0	0
$330$	0	0
$331$	21.9619	1.20714	0.603568	−	0.797312i	$-0.293745\pi$
$331$	21.9619	1.20714	0.603568	+	0.797312i	$0.293745\pi$
$332$	0	0
$333$	−0.252857	−0.0138565
$334$	0	0
$335$	−6.24273	−0.341077
$336$	0	0
$337$	25.4922	1.38865	0.694324	−	0.719663i	$-0.255704\pi$
$337$	25.4922	1.38865	0.694324	+	0.719663i	$0.255704\pi$
$338$	0	0
$339$	−15.1820	−0.824574
$340$	0	0
$341$	49.4196	2.67622
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−33.6778	−1.81315
$346$	0	0
$347$	0.250846	0.0134661	0.00673306	−	0.999977i	$-0.497857\pi$
$347$	0.250846	0.0134661	0.00673306	+	0.999977i	$0.497857\pi$
$348$	0	0
$349$	−12.7226	−0.681024	−0.340512	−	0.940240i	$-0.610600\pi$
$349$	−12.7226	−0.681024	−0.340512	+	0.940240i	$0.610600\pi$
$350$	0	0
$351$	−17.3677	−0.927019
$352$	0	0
$353$	−28.5289	−1.51844	−0.759220	−	0.650834i	$-0.774419\pi$
$353$	−28.5289	−1.51844	−0.759220	+	0.650834i	$0.774419\pi$
$354$	0	0
$355$	−34.5149	−1.83186
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.0108200	0.000571059 0	0.000285529	−	1.00000i	$-0.499909\pi$
$359$	0.0108200	0.000571059 0	0.000285529		1.00000i	$0.499909\pi$
$360$	0	0
$361$	12.9259	0.680312
$362$	0	0
$363$	26.5987	1.39607
$364$	0	0
$365$	0.956091	0.0500441
$366$	0	0
$367$	−15.1922	−0.793025	−0.396512	−	0.918029i	$-0.629780\pi$
$367$	−15.1922	−0.793025	−0.396512	+	0.918029i	$0.629780\pi$
$368$	0	0
$369$	2.59352	0.135013
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−28.6107	−1.48141	−0.740704	−	0.671832i	$-0.765508\pi$
$373$	−28.6107	−1.48141	−0.740704	+	0.671832i	$0.765508\pi$
$374$	0	0
$375$	−1.21430	−0.0627063
$376$	0	0
$377$	10.4347	0.537415
$378$	0	0
$379$	7.05026	0.362147	0.181074	−	0.983470i	$-0.442043\pi$
$379$	7.05026	0.362147	0.181074	+	0.983470i	$0.442043\pi$
$380$	0	0
$381$	24.2609	1.24292
$382$	0	0
$383$	11.4078	0.582912	0.291456	−	0.956584i	$-0.405860\pi$
$383$	11.4078	0.582912	0.291456	+	0.956584i	$0.405860\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.20074	−0.0610369
$388$	0	0
$389$	23.1409	1.17329	0.586644	−	0.809845i	$-0.300449\pi$
$389$	23.1409	1.17329	0.586644	+	0.809845i	$0.300449\pi$
$390$	0	0
$391$	6.92994	0.350462
$392$	0	0
$393$	34.0905	1.71964
$394$	0	0
$395$	−14.6304	−0.736135
$396$	0	0
$397$	21.9523	1.10176	0.550878	−	0.834586i	$-0.314293\pi$
$397$	21.9523	1.10176	0.550878	+	0.834586i	$0.314293\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.20085	−0.109905	−0.0549526	−	0.998489i	$-0.517501\pi$
$401$	−2.20085	−0.109905	−0.0549526	+	0.998489i	$0.517501\pi$
$402$	0	0
$403$	−29.0832	−1.44874
$404$	0	0
$405$	−21.6482	−1.07571
$406$	0	0
$407$	−2.31957	−0.114977
$408$	0	0
$409$	−11.8394	−0.585419	−0.292710	−	0.956201i	$-0.594557\pi$
$409$	−11.8394	−0.585419	−0.292710	+	0.956201i	$0.594557\pi$
$410$	0	0
$411$	27.2387	1.34359
$412$	0	0
$413$	0	0
$414$	0	0
$415$	33.5894	1.64884
$416$	0	0
$417$	5.13488	0.251456
$418$	0	0
$419$	18.9750	0.926987	0.463494	−	0.886100i	$-0.346596\pi$
$419$	18.9750	0.926987	0.463494	+	0.886100i	$0.346596\pi$
$420$	0	0
$421$	5.13866	0.250443	0.125222	−	0.992129i	$-0.460036\pi$
$421$	5.13866	0.250443	0.125222	+	0.992129i	$0.460036\pi$
$422$	0	0
$423$	6.01674	0.292544
$424$	0	0
$425$	−4.75013	−0.230415
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−25.7282	−1.24217
$430$	0	0
$431$	26.4300	1.27309	0.636545	−	0.771239i	$-0.280363\pi$
$431$	26.4300	1.27309	0.636545	+	0.771239i	$0.280363\pi$
$432$	0	0
$433$	−13.1473	−0.631819	−0.315909	−	0.948789i	$-0.602310\pi$
$433$	−13.1473	−0.631819	−0.315909	+	0.948789i	$0.602310\pi$
$434$	0	0
$435$	16.2582	0.779520
$436$	0	0
$437$	39.1563	1.87310
$438$	0	0
$439$	12.4616	0.594757	0.297379	−	0.954760i	$-0.403888\pi$
$439$	12.4616	0.594757	0.297379	+	0.954760i	$0.403888\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.81102	−0.418624	−0.209312	−	0.977849i	$-0.567122\pi$
$443$	−8.81102	−0.418624	−0.209312	+	0.977849i	$0.567122\pi$
$444$	0	0
$445$	−34.4178	−1.63156
$446$	0	0
$447$	−21.7208	−1.02736
$448$	0	0
$449$	5.62799	0.265601	0.132801	−	0.991143i	$-0.457603\pi$
$449$	5.62799	0.265601	0.132801	+	0.991143i	$0.457603\pi$
$450$	0	0
$451$	23.7915	1.12030
$452$	0	0
$453$	17.0999	0.803422
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.0270	−1.68527	−0.842637	−	0.538482i	$-0.818998\pi$
$457$	−36.0270	−1.68527	−0.842637	+	0.538482i	$0.818998\pi$
$458$	0	0
$459$	5.56826	0.259904
$460$	0	0
$461$	−26.4964	−1.23406	−0.617030	−	0.786940i	$-0.711664\pi$
$461$	−26.4964	−1.23406	−0.617030	+	0.786940i	$0.711664\pi$
$462$	0	0
$463$	18.1937	0.845533	0.422766	−	0.906239i	$-0.361059\pi$
$463$	18.1937	0.845533	0.422766	+	0.906239i	$0.361059\pi$
$464$	0	0
$465$	−45.3142	−2.10140
$466$	0	0
$467$	24.4698	1.13233	0.566163	−	0.824293i	$-0.308427\pi$
$467$	24.4698	1.13233	0.566163	+	0.824293i	$0.308427\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.2899	−0.934910
$472$	0	0
$473$	−11.0149	−0.506465
$474$	0	0
$475$	−26.8397	−1.23149
$476$	0	0
$477$	−2.39958	−0.109869
$478$	0	0
$479$	27.0700	1.23686	0.618431	−	0.785839i	$-0.287769\pi$
$479$	27.0700	1.23686	0.618431	+	0.785839i	$0.287769\pi$
$480$	0	0
$481$	1.36506	0.0622413
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.8941	0.630898
$486$	0	0
$487$	−26.9106	−1.21944	−0.609718	−	0.792618i	$-0.708717\pi$
$487$	−26.9106	−1.21944	−0.609718	+	0.792618i	$0.708717\pi$
$488$	0	0
$489$	−20.9870	−0.949067
$490$	0	0
$491$	−3.05456	−0.137850	−0.0689251	−	0.997622i	$-0.521957\pi$
$491$	−3.05456	−0.137850	−0.0689251	+	0.997622i	$0.521957\pi$
$492$	0	0
$493$	−3.34548	−0.150673
$494$	0	0
$495$	9.56159	0.429762
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15.3270	−0.686130	−0.343065	−	0.939312i	$-0.611465\pi$
$499$	−15.3270	−0.686130	−0.343065	+	0.939312i	$0.611465\pi$
$500$	0	0
$501$	−7.58786	−0.339001
$502$	0	0
$503$	0.662276	0.0295294	0.0147647	−	0.999891i	$-0.495300\pi$
$503$	0.662276	0.0295294	0.0147647	+	0.999891i	$0.495300\pi$
$504$	0	0
$505$	4.09824	0.182369
$506$	0	0
$507$	−5.09166	−0.226128
$508$	0	0
$509$	−18.6366	−0.826053	−0.413027	−	0.910719i	$-0.635528\pi$
$509$	−18.6366	−0.826053	−0.413027	+	0.910719i	$0.635528\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	31.4624	1.38910
$514$	0	0
$515$	−7.80236	−0.343813
$516$	0	0
$517$	55.1943	2.42744
$518$	0	0
$519$	−15.1540	−0.665185
$520$	0	0
$521$	−20.3120	−0.889886	−0.444943	−	0.895559i	$-0.646776\pi$
$521$	−20.3120	−0.889886	−0.444943	+	0.895559i	$0.646776\pi$
$522$	0	0
$523$	16.0813	0.703188	0.351594	−	0.936153i	$-0.385640\pi$
$523$	16.0813	0.703188	0.351594	+	0.936153i	$0.385640\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.32439	0.406177
$528$	0	0
$529$	25.0241	1.08800
$530$	0	0
$531$	−2.12806	−0.0923498
$532$	0	0
$533$	−14.0012	−0.606459
$534$	0	0
$535$	31.7426	1.37235
$536$	0	0
$537$	−33.8107	−1.45904
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21.4389	−0.921728	−0.460864	−	0.887471i	$-0.652460\pi$
$541$	−21.4389	−0.921728	−0.460864	+	0.887471i	$0.652460\pi$
$542$	0	0
$543$	31.3147	1.34384
$544$	0	0
$545$	1.47961	0.0633797
$546$	0	0
$547$	−23.3942	−1.00026	−0.500131	−	0.865950i	$-0.666715\pi$
$547$	−23.3942	−1.00026	−0.500131	+	0.865950i	$0.666715\pi$
$548$	0	0
$549$	−1.38068	−0.0589262
$550$	0	0
$551$	−18.9030	−0.805293
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.12688	0.0902810
$556$	0	0
$557$	−32.0265	−1.35701	−0.678503	−	0.734598i	$-0.737371\pi$
$557$	−32.0265	−1.35701	−0.678503	+	0.734598i	$0.737371\pi$
$558$	0	0
$559$	6.48222	0.274169
$560$	0	0
$561$	8.24874	0.348262
$562$	0	0
$563$	8.87159	0.373893	0.186947	−	0.982370i	$-0.440141\pi$
$563$	8.87159	0.373893	0.186947	+	0.982370i	$0.440141\pi$
$564$	0	0
$565$	−30.4597	−1.28145
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.3090	0.809475	0.404737	−	0.914433i	$-0.367363\pi$
$569$	19.3090	0.809475	0.404737	+	0.914433i	$0.367363\pi$
$570$	0	0
$571$	−39.6733	−1.66028	−0.830139	−	0.557557i	$-0.811739\pi$
$571$	−39.6733	−1.66028	−0.830139	+	0.557557i	$0.811739\pi$
$572$	0	0
$573$	29.0893	1.21522
$574$	0	0
$575$	−32.9181	−1.37278
$576$	0	0
$577$	−9.91280	−0.412675	−0.206338	−	0.978481i	$-0.566155\pi$
$577$	−9.91280	−0.412675	−0.206338	+	0.978481i	$0.566155\pi$
$578$	0	0
$579$	13.7024	0.569453
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−22.0124	−0.911662
$584$	0	0
$585$	−5.62696	−0.232646
$586$	0	0
$587$	−14.4801	−0.597660	−0.298830	−	0.954306i	$-0.596596\pi$
$587$	−14.4801	−0.597660	−0.298830	+	0.954306i	$0.596596\pi$
$588$	0	0
$589$	52.6856	2.17087
$590$	0	0
$591$	5.39407	0.221883
$592$	0	0
$593$	−32.4159	−1.33116	−0.665580	−	0.746327i	$-0.731816\pi$
$593$	−32.4159	−1.33116	−0.665580	+	0.746327i	$0.731816\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.9106	−0.487469
$598$	0	0
$599$	−40.4506	−1.65276	−0.826382	−	0.563109i	$-0.809605\pi$
$599$	−40.4506	−1.65276	−0.826382	+	0.563109i	$0.809605\pi$
$600$	0	0
$601$	−22.7016	−0.926019	−0.463009	−	0.886353i	$-0.653230\pi$
$601$	−22.7016	−0.926019	−0.463009	+	0.886353i	$0.653230\pi$
$602$	0	0
$603$	1.15509	0.0470389
$604$	0	0
$605$	53.3651	2.16960
$606$	0	0
$607$	18.2439	0.740498	0.370249	−	0.928933i	$-0.379272\pi$
$607$	18.2439	0.740498	0.370249	+	0.928933i	$0.379272\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.4816	−1.31407
$612$	0	0
$613$	0.399267	0.0161262	0.00806312	−	0.999967i	$-0.497433\pi$
$613$	0.399267	0.0161262	0.00806312	+	0.999967i	$0.497433\pi$
$614$	0	0
$615$	−21.8151	−0.879669
$616$	0	0
$617$	30.7131	1.23646	0.618232	−	0.785996i	$-0.287849\pi$
$617$	30.7131	1.23646	0.618232	+	0.785996i	$0.287849\pi$
$618$	0	0
$619$	40.9692	1.64669	0.823345	−	0.567541i	$-0.192105\pi$
$619$	40.9692	1.64669	0.823345	+	0.567541i	$0.192105\pi$
$620$	0	0
$621$	38.5877	1.54847
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.1869	−1.04748
$626$	0	0
$627$	46.6079	1.86134
$628$	0	0
$629$	−0.437652	−0.0174503
$630$	0	0
$631$	−47.3665	−1.88563	−0.942815	−	0.333317i	$-0.891832\pi$
$631$	−47.3665	−1.88563	−0.942815	+	0.333317i	$0.891832\pi$
$632$	0	0
$633$	36.6478	1.45662
$634$	0	0
$635$	48.6747	1.93159
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.38627	0.252637
$640$	0	0
$641$	−44.7652	−1.76812	−0.884060	−	0.467373i	$-0.845201\pi$
$641$	−44.7652	−1.76812	−0.884060	+	0.467373i	$0.845201\pi$
$642$	0	0
$643$	−32.5353	−1.28307	−0.641534	−	0.767095i	$-0.721702\pi$
$643$	−32.5353	−1.28307	−0.641534	+	0.767095i	$0.721702\pi$
$644$	0	0
$645$	10.0999	0.397682
$646$	0	0
$647$	−48.3904	−1.90242	−0.951212	−	0.308539i	$-0.900160\pi$
$647$	−48.3904	−1.90242	−0.951212	+	0.308539i	$0.900160\pi$
$648$	0	0
$649$	−19.5216	−0.766291
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.4814	−1.07543	−0.537715	−	0.843127i	$-0.680712\pi$
$653$	−27.4814	−1.07543	−0.537715	+	0.843127i	$0.680712\pi$
$654$	0	0
$655$	68.3958	2.67244
$656$	0	0
$657$	−0.176905	−0.00690172
$658$	0	0
$659$	25.8417	1.00665	0.503325	−	0.864097i	$-0.332110\pi$
$659$	25.8417	1.00665	0.503325	+	0.864097i	$0.332110\pi$
$660$	0	0
$661$	−23.4203	−0.910943	−0.455471	−	0.890250i	$-0.650529\pi$
$661$	−23.4203	−0.910943	−0.455471	+	0.890250i	$0.650529\pi$
$662$	0	0
$663$	−4.85435	−0.188527
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.1839	−0.897686
$668$	0	0
$669$	20.5358	0.793960
$670$	0	0
$671$	−12.6656	−0.488952
$672$	0	0
$673$	−30.6382	−1.18102	−0.590508	−	0.807032i	$-0.701072\pi$
$673$	−30.6382	−1.18102	−0.590508	+	0.807032i	$0.701072\pi$
$674$	0	0
$675$	−26.4500	−1.01806
$676$	0	0
$677$	−26.2000	−1.00695	−0.503473	−	0.864011i	$-0.667945\pi$
$677$	−26.2000	−1.00695	−0.503473	+	0.864011i	$0.667945\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−23.4672	−0.899265
$682$	0	0
$683$	−37.7248	−1.44350	−0.721750	−	0.692154i	$-0.756662\pi$
$683$	−37.7248	−1.44350	−0.721750	+	0.692154i	$0.756662\pi$
$684$	0	0
$685$	54.6492	2.08804
$686$	0	0
$687$	−36.0390	−1.37497
$688$	0	0
$689$	12.9542	0.493517
$690$	0	0
$691$	−25.7920	−0.981173	−0.490587	−	0.871392i	$-0.663217\pi$
$691$	−25.7920	−0.981173	−0.490587	+	0.871392i	$0.663217\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.3021	0.390782
$696$	0	0
$697$	4.48893	0.170030
$698$	0	0
$699$	−15.8816	−0.600698
$700$	0	0
$701$	34.6224	1.30767	0.653835	−	0.756637i	$-0.273159\pi$
$701$	34.6224	1.30767	0.653835	+	0.756637i	$0.273159\pi$
$702$	0	0
$703$	−2.47287	−0.0932659
$704$	0	0
$705$	−50.6092	−1.90605
$706$	0	0
$707$	0	0
$708$	0	0
$709$	27.5781	1.03572	0.517859	−	0.855466i	$-0.326729\pi$
$709$	27.5781	1.03572	0.517859	+	0.855466i	$0.326729\pi$
$710$	0	0
$711$	2.70706	0.101523
$712$	0	0
$713$	64.6174	2.41994
$714$	0	0
$715$	−51.6186	−1.93043
$716$	0	0
$717$	−29.0997	−1.08675
$718$	0	0
$719$	−16.3062	−0.608118	−0.304059	−	0.952653i	$-0.598342\pi$
$719$	−16.3062	−0.608118	−0.304059	+	0.952653i	$0.598342\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−33.8601	−1.25927
$724$	0	0
$725$	15.8914	0.590193
$726$	0	0
$727$	−38.6398	−1.43307	−0.716536	−	0.697550i	$-0.754273\pi$
$727$	−38.6398	−1.43307	−0.716536	+	0.697550i	$0.754273\pi$
$728$	0	0
$729$	30.0041	1.11126
$730$	0	0
$731$	−2.07827	−0.0768675
$732$	0	0
$733$	6.57031	0.242680	0.121340	−	0.992611i	$-0.461281\pi$
$733$	6.57031	0.242680	0.121340	+	0.992611i	$0.461281\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.5962	0.390314
$738$	0	0
$739$	−9.55818	−0.351603	−0.175802	−	0.984426i	$-0.556252\pi$
$739$	−9.55818	−0.351603	−0.175802	+	0.984426i	$0.556252\pi$
$740$	0	0
$741$	−27.4286	−1.00761
$742$	0	0
$743$	−46.7702	−1.71583	−0.857917	−	0.513789i	$-0.828241\pi$
$743$	−46.7702	−1.71583	−0.857917	+	0.513789i	$0.828241\pi$
$744$	0	0
$745$	−43.5785	−1.59659
$746$	0	0
$747$	−6.21504	−0.227396
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−19.0754	−0.696070	−0.348035	−	0.937482i	$-0.613151\pi$
$751$	−19.0754	−0.696070	−0.348035	+	0.937482i	$0.613151\pi$
$752$	0	0
$753$	43.4443	1.58320
$754$	0	0
$755$	34.3075	1.24858
$756$	0	0
$757$	31.4442	1.14286	0.571430	−	0.820651i	$-0.306389\pi$
$757$	31.4442	1.14286	0.571430	+	0.820651i	$0.306389\pi$
$758$	0	0
$759$	57.1633	2.07490
$760$	0	0
$761$	46.6573	1.69133	0.845664	−	0.533716i	$-0.179205\pi$
$761$	46.6573	1.69133	0.845664	+	0.533716i	$0.179205\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.80406	0.0652260
$766$	0	0
$767$	11.4884	0.414822
$768$	0	0
$769$	−0.187784	−0.00677168	−0.00338584	−	0.999994i	$-0.501078\pi$
$769$	−0.187784	−0.00677168	−0.00338584	+	0.999994i	$0.501078\pi$
$770$	0	0
$771$	−38.7314	−1.39488
$772$	0	0
$773$	−11.4286	−0.411060	−0.205530	−	0.978651i	$-0.565892\pi$
$773$	−11.4286	−0.411060	−0.205530	+	0.978651i	$0.565892\pi$
$774$	0	0
$775$	−44.2921	−1.59102
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.3638	0.908753
$780$	0	0
$781$	58.5841	2.09630
$782$	0	0
$783$	−18.6285	−0.665728
$784$	0	0
$785$	−40.7077	−1.45292
$786$	0	0
$787$	−41.2970	−1.47208	−0.736040	−	0.676938i	$-0.763307\pi$
$787$	−41.2970	−1.47208	−0.736040	+	0.676938i	$0.763307\pi$
$788$	0	0
$789$	−38.1047	−1.35656
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.45368	0.264688
$794$	0	0
$795$	20.1838	0.715846
$796$	0	0
$797$	−30.7517	−1.08928	−0.544641	−	0.838670i	$-0.683334\pi$
$797$	−30.7517	−1.08928	−0.544641	+	0.838670i	$0.683334\pi$
$798$	0	0
$799$	10.4139	0.368419
$800$	0	0
$801$	6.36830	0.225013
$802$	0	0
$803$	−1.62283	−0.0572684
$804$	0	0
$805$	0	0
$806$	0	0
$807$	44.3437	1.56097
$808$	0	0
$809$	−35.7016	−1.25520	−0.627601	−	0.778535i	$-0.715963\pi$
$809$	−35.7016	−1.25520	−0.627601	+	0.778535i	$0.715963\pi$
$810$	0	0
$811$	22.3511	0.784854	0.392427	−	0.919783i	$-0.371636\pi$
$811$	22.3511	0.784854	0.392427	+	0.919783i	$0.371636\pi$
$812$	0	0
$813$	−20.3794	−0.714735
$814$	0	0
$815$	−42.1064	−1.47492
$816$	0	0
$817$	−11.7428	−0.410830
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.4483	0.923053	0.461526	−	0.887126i	$-0.347302\pi$
$821$	26.4483	0.923053	0.461526	+	0.887126i	$0.347302\pi$
$822$	0	0
$823$	8.10743	0.282607	0.141304	−	0.989966i	$-0.454871\pi$
$823$	8.10743	0.282607	0.141304	+	0.989966i	$0.454871\pi$
$824$	0	0
$825$	−39.1826	−1.36416
$826$	0	0
$827$	9.15837	0.318468	0.159234	−	0.987241i	$-0.449098\pi$
$827$	9.15837	0.318468	0.159234	+	0.987241i	$0.449098\pi$
$828$	0	0
$829$	1.01875	0.0353826	0.0176913	−	0.999843i	$-0.494368\pi$
$829$	1.01875	0.0353826	0.0176913	+	0.999843i	$0.494368\pi$
$830$	0	0
$831$	10.2442	0.355368
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15.2236	−0.526833
$836$	0	0
$837$	51.9206	1.79464
$838$	0	0
$839$	−22.6296	−0.781261	−0.390631	−	0.920548i	$-0.627743\pi$
$839$	−22.6296	−0.781261	−0.390631	+	0.920548i	$0.627743\pi$
$840$	0	0
$841$	−17.8078	−0.614062
$842$	0	0
$843$	−44.6917	−1.53926
$844$	0	0
$845$	−10.2154	−0.351421
$846$	0	0
$847$	0	0
$848$	0	0
$849$	38.1789	1.31030
$850$	0	0
$851$	−3.03290	−0.103966
$852$	0	0
$853$	10.6753	0.365516	0.182758	−	0.983158i	$-0.441497\pi$
$853$	10.6753	0.365516	0.182758	+	0.983158i	$0.441497\pi$
$854$	0	0
$855$	10.1935	0.348610
$856$	0	0
$857$	11.9265	0.407403	0.203701	−	0.979033i	$-0.434703\pi$
$857$	11.9265	0.407403	0.203701	+	0.979033i	$0.434703\pi$
$858$	0	0
$859$	6.84044	0.233393	0.116696	−	0.993168i	$-0.462770\pi$
$859$	6.84044	0.233393	0.116696	+	0.993168i	$0.462770\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.1692	−1.53758	−0.768788	−	0.639504i	$-0.779140\pi$
$863$	−45.1692	−1.53758	−0.768788	+	0.639504i	$0.779140\pi$
$864$	0	0
$865$	−30.4034	−1.03375
$866$	0	0
$867$	1.55636	0.0528566
$868$	0	0
$869$	24.8330	0.842403
$870$	0	0
$871$	−6.23580	−0.211292
$872$	0	0
$873$	−2.57081	−0.0870089
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.04830	−0.170469	−0.0852344	−	0.996361i	$-0.527164\pi$
$877$	−5.04830	−0.170469	−0.0852344	+	0.996361i	$0.527164\pi$
$878$	0	0
$879$	5.13341	0.173146
$880$	0	0
$881$	−11.7391	−0.395500	−0.197750	−	0.980253i	$-0.563363\pi$
$881$	−11.7391	−0.395500	−0.197750	+	0.980253i	$0.563363\pi$
$882$	0	0
$883$	34.8401	1.17246	0.586231	−	0.810144i	$-0.300611\pi$
$883$	34.8401	1.17246	0.586231	+	0.810144i	$0.300611\pi$
$884$	0	0
$885$	17.8999	0.601699
$886$	0	0
$887$	8.16157	0.274039	0.137019	−	0.990568i	$-0.456248\pi$
$887$	8.16157	0.274039	0.137019	+	0.990568i	$0.456248\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	36.7447	1.23099
$892$	0	0
$893$	58.8419	1.96907
$894$	0	0
$895$	−67.8345	−2.26746
$896$	0	0
$897$	−33.6404	−1.12322
$898$	0	0
$899$	−31.1945	−1.04040
$900$	0	0
$901$	−4.15326	−0.138365
$902$	0	0
$903$	0	0
$904$	0	0
$905$	62.8268	2.08843
$906$	0	0
$907$	−28.7971	−0.956191	−0.478095	−	0.878308i	$-0.658673\pi$
$907$	−28.7971	−0.956191	−0.478095	+	0.878308i	$0.658673\pi$
$908$	0	0
$909$	−0.758296	−0.0251511
$910$	0	0
$911$	39.7765	1.31785	0.658927	−	0.752207i	$-0.271011\pi$
$911$	39.7765	1.31785	0.658927	+	0.752207i	$0.271011\pi$
$912$	0	0
$913$	−57.0133	−1.88687
$914$	0	0
$915$	11.6135	0.383930
$916$	0	0
$917$	0	0
$918$	0	0
$919$	13.2661	0.437607	0.218804	−	0.975769i	$-0.429785\pi$
$919$	13.2661	0.437607	0.218804	+	0.975769i	$0.429785\pi$
$920$	0	0
$921$	41.3819	1.36358
$922$	0	0
$923$	−34.4765	−1.13481
$924$	0	0
$925$	2.07890	0.0683539
$926$	0	0
$927$	1.44367	0.0474162
$928$	0	0
$929$	46.1013	1.51253	0.756267	−	0.654263i	$-0.227021\pi$
$929$	46.1013	1.51253	0.756267	+	0.654263i	$0.227021\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	23.9157	0.782966
$934$	0	0
$935$	16.5495	0.541225
$936$	0	0
$937$	−38.6039	−1.26113	−0.630567	−	0.776135i	$-0.717178\pi$
$937$	−38.6039	−1.26113	−0.630567	+	0.776135i	$0.717178\pi$
$938$	0	0
$939$	10.5383	0.343904
$940$	0	0
$941$	47.5919	1.55145	0.775726	−	0.631070i	$-0.217384\pi$
$941$	47.5919	1.55145	0.775726	+	0.631070i	$0.217384\pi$
$942$	0	0
$943$	31.1080	1.01302
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.05429	0.0342599	0.0171299	−	0.999853i	$-0.494547\pi$
$947$	1.05429	0.0342599	0.0171299	+	0.999853i	$0.494547\pi$
$948$	0	0
$949$	0.955029	0.0310015
$950$	0	0
$951$	39.8977	1.29377
$952$	0	0
$953$	21.2753	0.689176	0.344588	−	0.938754i	$-0.388019\pi$
$953$	21.2753	0.689176	0.344588	+	0.938754i	$0.388019\pi$
$954$	0	0
$955$	58.3619	1.88855
$956$	0	0
$957$	−27.5960	−0.892051
$958$	0	0
$959$	0	0
$960$	0	0
$961$	55.9442	1.80465
$962$	0	0
$963$	−5.87332	−0.189265
$964$	0	0
$965$	27.4912	0.884973
$966$	0	0
$967$	19.0423	0.612359	0.306180	−	0.951974i	$-0.400949\pi$
$967$	19.0423	0.612359	0.306180	+	0.951974i	$0.400949\pi$
$968$	0	0
$969$	8.79388	0.282500
$970$	0	0
$971$	−33.8597	−1.08661	−0.543305	−	0.839536i	$-0.682827\pi$
$971$	−33.8597	−1.08661	−0.543305	+	0.839536i	$0.682827\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	23.0588	0.738473
$976$	0	0
$977$	4.27385	0.136733	0.0683663	−	0.997660i	$-0.478221\pi$
$977$	4.27385	0.136733	0.0683663	+	0.997660i	$0.478221\pi$
$978$	0	0
$979$	58.4193	1.86709
$980$	0	0
$981$	−0.273772	−0.00874088
$982$	0	0
$983$	−14.4227	−0.460014	−0.230007	−	0.973189i	$-0.573875\pi$
$983$	−14.4227	−0.460014	−0.230007	+	0.973189i	$0.573875\pi$
$984$	0	0
$985$	10.8221	0.344822
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.4023	−0.457966
$990$	0	0
$991$	59.3486	1.88527	0.942635	−	0.333826i	$-0.108340\pi$
$991$	59.3486	1.88527	0.942635	+	0.333826i	$0.108340\pi$
$992$	0	0
$993$	34.1805	1.08469
$994$	0	0
$995$	−23.8963	−0.757564
$996$	0	0
$997$	2.57483	0.0815456	0.0407728	−	0.999168i	$-0.487018\pi$
$997$	2.57483	0.0815456	0.0407728	+	0.999168i	$0.487018\pi$
$998$	0	0
$999$	−2.43696	−0.0771020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.y.1.5		7
7.3	odd	6		952.2.q.e.681.5	yes	14
7.5	odd	6		952.2.q.e.137.5	✓	14
7.6	odd	2		6664.2.a.v.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.e.137.5	✓	14	7.5	odd	6
952.2.q.e.681.5	yes	14	7.3	odd	6
6664.2.a.v.1.3		7	7.6	odd	2
6664.2.a.y.1.5		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.55636 q^{3} +3.12252 q^{5} -0.577758 q^{9} -5.30004 q^{11} +3.11905 q^{13} +4.85975 q^{15} -1.00000 q^{17} -5.65030 q^{19} -6.92994 q^{23} +4.75013 q^{25} -5.56826 q^{27} +3.34548 q^{29} -9.32439 q^{31} -8.24874 q^{33} +0.437652 q^{37} +4.85435 q^{39} -4.48893 q^{41} +2.07827 q^{43} -1.80406 q^{45} -10.4139 q^{47} -1.55636 q^{51} +4.15326 q^{53} -16.5495 q^{55} -8.79388 q^{57} +3.68330 q^{59} +2.38973 q^{61} +9.73930 q^{65} -1.99926 q^{67} -10.7854 q^{69} -11.0535 q^{71} +0.306192 q^{73} +7.39289 q^{75} -4.68545 q^{79} -6.93292 q^{81} +10.7572 q^{83} -3.12252 q^{85} +5.20675 q^{87} -11.0224 q^{89} -14.5121 q^{93} -17.6432 q^{95} +4.44964 q^{97} +3.06214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 2 * q^5 + 4 * q^9 - 12 * q^11 - 2 * q^13 - 6 * q^15 - 7 * q^17 + 3 * q^19 - 18 * q^23 + 15 * q^25 + 18 * q^27 + 5 * q^29 + 10 * q^31 - 21 * q^33 + 11 * q^37 - 12 * q^39 - 15 * q^43 - 13 * q^45 - 15 * q^47 - 3 * q^51 + 8 * q^53 + 9 * q^55 - 3 * q^57 - 9 * q^59 + 4 * q^61 - 38 * q^65 - 30 * q^67 + 28 * q^69 - 27 * q^71 + 5 * q^73 + 2 * q^75 + 3 * q^79 - 17 * q^81 + 12 * q^83 + 2 * q^85 - 19 * q^87 - 32 * q^89 - 9 * q^93 - 42 * q^95 + 39 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

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Database

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