LMFDB - Embedded newform 740.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [740,2,Mod(1,740)] 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("740.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	740.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$5.90892974957$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	740.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.539189 q^{3} -1.00000 q^{5} +3.80098 q^{7} -2.70928 q^{9} +2.34017 q^{11} -4.04945 q^{13} +0.539189 q^{15} +6.78765 q^{17} +5.46081 q^{19} -2.04945 q^{21} -4.34017 q^{23} +1.00000 q^{25} +3.07838 q^{27} +7.75872 q^{29} -1.61757 q^{31} -1.26180 q^{33} -3.80098 q^{35} -1.00000 q^{37} +2.18342 q^{39} -3.07838 q^{41} +11.7587 q^{43} +2.70928 q^{45} -2.14116 q^{47} +7.44748 q^{49} -3.65983 q^{51} +14.0989 q^{53} -2.34017 q^{55} -2.94441 q^{57} -7.37629 q^{59} +14.9939 q^{61} -10.2979 q^{63} +4.04945 q^{65} +14.1145 q^{67} +2.34017 q^{69} -13.2618 q^{71} +4.34017 q^{73} -0.539189 q^{75} +8.89496 q^{77} -4.72261 q^{79} +6.46800 q^{81} -2.19902 q^{83} -6.78765 q^{85} -4.18342 q^{87} -16.2557 q^{89} -15.3919 q^{91} +0.872174 q^{93} -5.46081 q^{95} +4.92162 q^{97} -6.34017 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7} - q^{9} - 4 q^{11} + 6 q^{13} + 10 q^{17} + 18 q^{19} + 12 q^{21} - 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 3 q^{37} + 2 q^{39} - 6 q^{41} + 10 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - q^9 - 4 * q^11 + 6 * q^13 + 10 * q^17 + 18 * q^19 + 12 * q^21 - 2 * q^23 + 3 * q^25 + 6 * q^27 - 2 * q^29 + 4 * q^33 - 2 * q^35 - 3 * q^37 + 2 * q^39 - 6 * q^41 + 10 * q^43 + q^45 + 14 * q^47 + 23 * q^49 - 22 * q^51 + 6 * q^53 + 4 * q^55 + 8 * q^57 + 8 * q^59 + 10 * q^61 - 4 * q^63 - 6 * q^65 + 20 * q^67 - 4 * q^69 - 32 * q^71 + 2 * q^73 + 28 * q^77 - 8 * q^79 - 13 * q^81 - 16 * q^83 - 10 * q^85 - 8 * q^87 - 6 * q^89 - 30 * q^91 + 24 * q^93 - 18 * q^95 + 18 * q^97 - 8 * 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
$3$	−0.539189	−0.311301	−0.155650	−	0.987812i	$-0.549747\pi$
$3$	−0.539189	−0.311301	−0.155650	+	0.987812i	$0.549747\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.80098	1.43664	0.718318	−	0.695714i	$-0.244912\pi$
$7$	3.80098	1.43664	0.718318	+	0.695714i	$0.244912\pi$
$8$	0	0
$9$	−2.70928	−0.903092
$10$	0	0
$11$	2.34017	0.705589	0.352794	−	0.935701i	$-0.385232\pi$
$11$	2.34017	0.705589	0.352794	+	0.935701i	$0.385232\pi$
$12$	0	0
$13$	−4.04945	−1.12311	−0.561557	−	0.827438i	$-0.689798\pi$
$13$	−4.04945	−1.12311	−0.561557	+	0.827438i	$0.689798\pi$
$14$	0	0
$15$	0.539189	0.139218
$16$	0	0
$17$	6.78765	1.64625	0.823124	−	0.567862i	$-0.192229\pi$
$17$	6.78765	1.64625	0.823124	+	0.567862i	$0.192229\pi$
$18$	0	0
$19$	5.46081	1.25280	0.626398	−	0.779503i	$-0.284528\pi$
$19$	5.46081	1.25280	0.626398	+	0.779503i	$0.284528\pi$
$20$	0	0
$21$	−2.04945	−0.447226
$22$	0	0
$23$	−4.34017	−0.904989	−0.452494	−	0.891767i	$-0.649466\pi$
$23$	−4.34017	−0.904989	−0.452494	+	0.891767i	$0.649466\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.07838	0.592434
$28$	0	0
$29$	7.75872	1.44076	0.720379	−	0.693580i	$-0.243968\pi$
$29$	7.75872	1.44076	0.720379	+	0.693580i	$0.243968\pi$
$30$	0	0
$31$	−1.61757	−0.290524	−0.145262	−	0.989393i	$-0.546402\pi$
$31$	−1.61757	−0.290524	−0.145262	+	0.989393i	$0.546402\pi$
$32$	0	0
$33$	−1.26180	−0.219650
$34$	0	0
$35$	−3.80098	−0.642484
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	2.18342	0.349627
$40$	0	0
$41$	−3.07838	−0.480762	−0.240381	−	0.970679i	$-0.577272\pi$
$41$	−3.07838	−0.480762	−0.240381	+	0.970679i	$0.577272\pi$
$42$	0	0
$43$	11.7587	1.79319	0.896594	−	0.442853i	$-0.146034\pi$
$43$	11.7587	1.79319	0.896594	+	0.442853i	$0.146034\pi$
$44$	0	0
$45$	2.70928	0.403875
$46$	0	0
$47$	−2.14116	−0.312320	−0.156160	−	0.987732i	$-0.549912\pi$
$47$	−2.14116	−0.312320	−0.156160	+	0.987732i	$0.549912\pi$
$48$	0	0
$49$	7.44748	1.06393
$50$	0	0
$51$	−3.65983	−0.512478
$52$	0	0
$53$	14.0989	1.93663	0.968316	−	0.249727i	$-0.0803410\pi$
$53$	14.0989	1.93663	0.968316	+	0.249727i	$0.0803410\pi$
$54$	0	0
$55$	−2.34017	−0.315549
$56$	0	0
$57$	−2.94441	−0.389996
$58$	0	0
$59$	−7.37629	−0.960311	−0.480156	−	0.877183i	$-0.659420\pi$
$59$	−7.37629	−0.960311	−0.480156	+	0.877183i	$0.659420\pi$
$60$	0	0
$61$	14.9939	1.91977	0.959883	−	0.280400i	$-0.0904670\pi$
$61$	14.9939	1.91977	0.959883	+	0.280400i	$0.0904670\pi$
$62$	0	0
$63$	−10.2979	−1.29742
$64$	0	0
$65$	4.04945	0.502272
$66$	0	0
$67$	14.1145	1.72436	0.862180	−	0.506602i	$-0.169099\pi$
$67$	14.1145	1.72436	0.862180	+	0.506602i	$0.169099\pi$
$68$	0	0
$69$	2.34017	0.281724
$70$	0	0
$71$	−13.2618	−1.57389	−0.786943	−	0.617026i	$-0.788337\pi$
$71$	−13.2618	−1.57389	−0.786943	+	0.617026i	$0.788337\pi$
$72$	0	0
$73$	4.34017	0.507979	0.253989	−	0.967207i	$-0.418257\pi$
$73$	4.34017	0.507979	0.253989	+	0.967207i	$0.418257\pi$
$74$	0	0
$75$	−0.539189	−0.0622602
$76$	0	0
$77$	8.89496	1.01367
$78$	0	0
$79$	−4.72261	−0.531335	−0.265667	−	0.964065i	$-0.585592\pi$
$79$	−4.72261	−0.531335	−0.265667	+	0.964065i	$0.585592\pi$
$80$	0	0
$81$	6.46800	0.718667
$82$	0	0
$83$	−2.19902	−0.241373	−0.120687	−	0.992691i	$-0.538510\pi$
$83$	−2.19902	−0.241373	−0.120687	+	0.992691i	$0.538510\pi$
$84$	0	0
$85$	−6.78765	−0.736224
$86$	0	0
$87$	−4.18342	−0.448509
$88$	0	0
$89$	−16.2557	−1.72310	−0.861548	−	0.507676i	$-0.830505\pi$
$89$	−16.2557	−1.72310	−0.861548	+	0.507676i	$0.830505\pi$
$90$	0	0
$91$	−15.3919	−1.61351
$92$	0	0
$93$	0.872174	0.0904402
$94$	0	0
$95$	−5.46081	−0.560267
$96$	0	0
$97$	4.92162	0.499715	0.249858	−	0.968283i	$-0.419616\pi$
$97$	4.92162	0.499715	0.249858	+	0.968283i	$0.419616\pi$
$98$	0	0
$99$	−6.34017	−0.637211
$100$	0	0
$101$	−15.6514	−1.55737	−0.778687	−	0.627412i	$-0.784114\pi$
$101$	−15.6514	−1.55737	−0.778687	+	0.627412i	$0.784114\pi$
$102$	0	0
$103$	6.09890	0.600942	0.300471	−	0.953791i	$-0.402856\pi$
$103$	6.09890	0.600942	0.300471	+	0.953791i	$0.402856\pi$
$104$	0	0
$105$	2.04945	0.200006
$106$	0	0
$107$	−19.3184	−1.86758	−0.933792	−	0.357817i	$-0.883521\pi$
$107$	−19.3184	−1.86758	−0.933792	+	0.357817i	$0.883521\pi$
$108$	0	0
$109$	−5.41855	−0.519003	−0.259502	−	0.965743i	$-0.583558\pi$
$109$	−5.41855	−0.519003	−0.259502	+	0.965743i	$0.583558\pi$
$110$	0	0
$111$	0.539189	0.0511775
$112$	0	0
$113$	−6.20620	−0.583831	−0.291915	−	0.956444i	$-0.594293\pi$
$113$	−6.20620	−0.583831	−0.291915	+	0.956444i	$0.594293\pi$
$114$	0	0
$115$	4.34017	0.404723
$116$	0	0
$117$	10.9711	1.01428
$118$	0	0
$119$	25.7998	2.36506
$120$	0	0
$121$	−5.52359	−0.502145
$122$	0	0
$123$	1.65983	0.149662
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.19902	−0.550074	−0.275037	−	0.961434i	$-0.588690\pi$
$127$	−6.19902	−0.550074	−0.275037	+	0.961434i	$0.588690\pi$
$128$	0	0
$129$	−6.34017	−0.558221
$130$	0	0
$131$	−5.64423	−0.493139	−0.246569	−	0.969125i	$-0.579303\pi$
$131$	−5.64423	−0.493139	−0.246569	+	0.969125i	$0.579303\pi$
$132$	0	0
$133$	20.7565	1.79981
$134$	0	0
$135$	−3.07838	−0.264945
$136$	0	0
$137$	11.1773	0.954939	0.477469	−	0.878648i	$-0.341554\pi$
$137$	11.1773	0.954939	0.477469	+	0.878648i	$0.341554\pi$
$138$	0	0
$139$	4.58145	0.388593	0.194297	−	0.980943i	$-0.437758\pi$
$139$	4.58145	0.388593	0.194297	+	0.980943i	$0.437758\pi$
$140$	0	0
$141$	1.15449	0.0972254
$142$	0	0
$143$	−9.47641	−0.792457
$144$	0	0
$145$	−7.75872	−0.644327
$146$	0	0
$147$	−4.01560	−0.331201
$148$	0	0
$149$	−7.65142	−0.626828	−0.313414	−	0.949617i	$-0.601473\pi$
$149$	−7.65142	−0.626828	−0.313414	+	0.949617i	$0.601473\pi$
$150$	0	0
$151$	−16.8638	−1.37235	−0.686177	−	0.727435i	$-0.740712\pi$
$151$	−16.8638	−1.37235	−0.686177	+	0.727435i	$0.740712\pi$
$152$	0	0
$153$	−18.3896	−1.48671
$154$	0	0
$155$	1.61757	0.129926
$156$	0	0
$157$	14.4969	1.15698	0.578490	−	0.815689i	$-0.303642\pi$
$157$	14.4969	1.15698	0.578490	+	0.815689i	$0.303642\pi$
$158$	0	0
$159$	−7.60197	−0.602875
$160$	0	0
$161$	−16.4969	−1.30014
$162$	0	0
$163$	−2.36683	−0.185385	−0.0926924	−	0.995695i	$-0.529547\pi$
$163$	−2.36683	−0.185385	−0.0926924	+	0.995695i	$0.529547\pi$
$164$	0	0
$165$	1.26180	0.0982306
$166$	0	0
$167$	2.68035	0.207411	0.103706	−	0.994608i	$-0.466930\pi$
$167$	2.68035	0.207411	0.103706	+	0.994608i	$0.466930\pi$
$168$	0	0
$169$	3.39803	0.261387
$170$	0	0
$171$	−14.7948	−1.13139
$172$	0	0
$173$	6.76487	0.514323	0.257162	−	0.966368i	$-0.417213\pi$
$173$	6.76487	0.514323	0.257162	+	0.966368i	$0.417213\pi$
$174$	0	0
$175$	3.80098	0.287327
$176$	0	0
$177$	3.97721	0.298946
$178$	0	0
$179$	0.382433	0.0285844	0.0142922	−	0.999898i	$-0.495450\pi$
$179$	0.382433	0.0285844	0.0142922	+	0.999898i	$0.495450\pi$
$180$	0	0
$181$	−9.70928	−0.721685	−0.360842	−	0.932627i	$-0.617511\pi$
$181$	−9.70928	−0.721685	−0.360842	+	0.932627i	$0.617511\pi$
$182$	0	0
$183$	−8.08452	−0.597625
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	15.8843	1.16157
$188$	0	0
$189$	11.7009	0.851113
$190$	0	0
$191$	0.722606	0.0522860	0.0261430	−	0.999658i	$-0.491677\pi$
$191$	0.722606	0.0522860	0.0261430	+	0.999658i	$0.491677\pi$
$192$	0	0
$193$	−15.3607	−1.10569	−0.552843	−	0.833285i	$-0.686457\pi$
$193$	−15.3607	−1.10569	−0.552843	+	0.833285i	$0.686457\pi$
$194$	0	0
$195$	−2.18342	−0.156358
$196$	0	0
$197$	−8.92162	−0.635639	−0.317820	−	0.948151i	$-0.602951\pi$
$197$	−8.92162	−0.635639	−0.317820	+	0.948151i	$0.602951\pi$
$198$	0	0
$199$	21.8732	1.55055	0.775276	−	0.631623i	$-0.217611\pi$
$199$	21.8732	1.55055	0.775276	+	0.631623i	$0.217611\pi$
$200$	0	0
$201$	−7.61038	−0.536795
$202$	0	0
$203$	29.4908	2.06985
$204$	0	0
$205$	3.07838	0.215003
$206$	0	0
$207$	11.7587	0.817288
$208$	0	0
$209$	12.7792	0.883959
$210$	0	0
$211$	4.31351	0.296954	0.148477	−	0.988916i	$-0.452563\pi$
$211$	4.31351	0.296954	0.148477	+	0.988916i	$0.452563\pi$
$212$	0	0
$213$	7.15061	0.489952
$214$	0	0
$215$	−11.7587	−0.801938
$216$	0	0
$217$	−6.14834	−0.417377
$218$	0	0
$219$	−2.34017	−0.158134
$220$	0	0
$221$	−27.4863	−1.84893
$222$	0	0
$223$	−25.9988	−1.74101	−0.870503	−	0.492162i	$-0.836207\pi$
$223$	−25.9988	−1.74101	−0.870503	+	0.492162i	$0.836207\pi$
$224$	0	0
$225$	−2.70928	−0.180618
$226$	0	0
$227$	−22.7792	−1.51191	−0.755956	−	0.654623i	$-0.772827\pi$
$227$	−22.7792	−1.51191	−0.755956	+	0.654623i	$0.772827\pi$
$228$	0	0
$229$	−25.1194	−1.65994	−0.829969	−	0.557810i	$-0.811642\pi$
$229$	−25.1194	−1.65994	−0.829969	+	0.557810i	$0.811642\pi$
$230$	0	0
$231$	−4.79606	−0.315558
$232$	0	0
$233$	7.17727	0.470199	0.235099	−	0.971971i	$-0.424458\pi$
$233$	7.17727	0.470199	0.235099	+	0.971971i	$0.424458\pi$
$234$	0	0
$235$	2.14116	0.139674
$236$	0	0
$237$	2.54638	0.165405
$238$	0	0
$239$	10.9516	0.708400	0.354200	−	0.935170i	$-0.384753\pi$
$239$	10.9516	0.708400	0.354200	+	0.935170i	$0.384753\pi$
$240$	0	0
$241$	27.8576	1.79447	0.897234	−	0.441556i	$-0.145573\pi$
$241$	27.8576	1.79447	0.897234	+	0.441556i	$0.145573\pi$
$242$	0	0
$243$	−12.7226	−0.816156
$244$	0	0
$245$	−7.44748	−0.475802
$246$	0	0
$247$	−22.1133	−1.40703
$248$	0	0
$249$	1.18568	0.0751397
$250$	0	0
$251$	23.4030	1.47718	0.738591	−	0.674154i	$-0.235492\pi$
$251$	23.4030	1.47718	0.738591	+	0.674154i	$0.235492\pi$
$252$	0	0
$253$	−10.1568	−0.638550
$254$	0	0
$255$	3.65983	0.229187
$256$	0	0
$257$	3.95055	0.246429	0.123214	−	0.992380i	$-0.460680\pi$
$257$	3.95055	0.246429	0.123214	+	0.992380i	$0.460680\pi$
$258$	0	0
$259$	−3.80098	−0.236182
$260$	0	0
$261$	−21.0205	−1.30114
$262$	0	0
$263$	−2.29791	−0.141695	−0.0708477	−	0.997487i	$-0.522570\pi$
$263$	−2.29791	−0.141695	−0.0708477	+	0.997487i	$0.522570\pi$
$264$	0	0
$265$	−14.0989	−0.866088
$266$	0	0
$267$	8.76487	0.536401
$268$	0	0
$269$	−5.61038	−0.342071	−0.171035	−	0.985265i	$-0.554711\pi$
$269$	−5.61038	−0.342071	−0.171035	+	0.985265i	$0.554711\pi$
$270$	0	0
$271$	−14.7382	−0.895282	−0.447641	−	0.894213i	$-0.647736\pi$
$271$	−14.7382	−0.895282	−0.447641	+	0.894213i	$0.647736\pi$
$272$	0	0
$273$	8.29914	0.502287
$274$	0	0
$275$	2.34017	0.141118
$276$	0	0
$277$	15.9506	0.958376	0.479188	−	0.877712i	$-0.340931\pi$
$277$	15.9506	0.958376	0.479188	+	0.877712i	$0.340931\pi$
$278$	0	0
$279$	4.38243	0.262369
$280$	0	0
$281$	28.2245	1.68373	0.841865	−	0.539688i	$-0.181458\pi$
$281$	28.2245	1.68373	0.841865	+	0.539688i	$0.181458\pi$
$282$	0	0
$283$	−23.4596	−1.39453	−0.697264	−	0.716815i	$-0.745599\pi$
$283$	−23.4596	−1.39453	−0.697264	+	0.716815i	$0.745599\pi$
$284$	0	0
$285$	2.94441	0.174412
$286$	0	0
$287$	−11.7009	−0.690680
$288$	0	0
$289$	29.0722	1.71013
$290$	0	0
$291$	−2.65368	−0.155562
$292$	0	0
$293$	1.68649	0.0985257	0.0492629	−	0.998786i	$-0.484313\pi$
$293$	1.68649	0.0985257	0.0492629	+	0.998786i	$0.484313\pi$
$294$	0	0
$295$	7.37629	0.429464
$296$	0	0
$297$	7.20394	0.418015
$298$	0	0
$299$	17.5753	1.01641
$300$	0	0
$301$	44.6947	2.57616
$302$	0	0
$303$	8.43907	0.484812
$304$	0	0
$305$	−14.9939	−0.858546
$306$	0	0
$307$	13.0628	0.745532	0.372766	−	0.927925i	$-0.378409\pi$
$307$	13.0628	0.745532	0.372766	+	0.927925i	$0.378409\pi$
$308$	0	0
$309$	−3.28846	−0.187074
$310$	0	0
$311$	−8.35577	−0.473812	−0.236906	−	0.971533i	$-0.576133\pi$
$311$	−8.35577	−0.473812	−0.236906	+	0.971533i	$0.576133\pi$
$312$	0	0
$313$	−6.59583	−0.372818	−0.186409	−	0.982472i	$-0.559685\pi$
$313$	−6.59583	−0.372818	−0.186409	+	0.982472i	$0.559685\pi$
$314$	0	0
$315$	10.2979	0.580222
$316$	0	0
$317$	20.2245	1.13592	0.567959	−	0.823057i	$-0.307733\pi$
$317$	20.2245	1.13592	0.567959	+	0.823057i	$0.307733\pi$
$318$	0	0
$319$	18.1568	1.01658
$320$	0	0
$321$	10.4163	0.581380
$322$	0	0
$323$	37.0661	2.06241
$324$	0	0
$325$	−4.04945	−0.224623
$326$	0	0
$327$	2.92162	0.161566
$328$	0	0
$329$	−8.13850	−0.448690
$330$	0	0
$331$	6.63809	0.364862	0.182431	−	0.983219i	$-0.441603\pi$
$331$	6.63809	0.364862	0.182431	+	0.983219i	$0.441603\pi$
$332$	0	0
$333$	2.70928	0.148467
$334$	0	0
$335$	−14.1145	−0.771157
$336$	0	0
$337$	10.8638	0.591787	0.295893	−	0.955221i	$-0.404383\pi$
$337$	10.8638	0.591787	0.295893	+	0.955221i	$0.404383\pi$
$338$	0	0
$339$	3.34632	0.181747
$340$	0	0
$341$	−3.78539	−0.204990
$342$	0	0
$343$	1.70086	0.0918381
$344$	0	0
$345$	−2.34017	−0.125991
$346$	0	0
$347$	26.8638	1.44212	0.721061	−	0.692871i	$-0.243655\pi$
$347$	26.8638	1.44212	0.721061	+	0.692871i	$0.243655\pi$
$348$	0	0
$349$	−23.2846	−1.24640	−0.623198	−	0.782064i	$-0.714167\pi$
$349$	−23.2846	−1.24640	−0.623198	+	0.782064i	$0.714167\pi$
$350$	0	0
$351$	−12.4657	−0.665372
$352$	0	0
$353$	−0.156755	−0.00834325	−0.00417163	−	0.999991i	$-0.501328\pi$
$353$	−0.156755	−0.00834325	−0.00417163	+	0.999991i	$0.501328\pi$
$354$	0	0
$355$	13.2618	0.703863
$356$	0	0
$357$	−13.9109	−0.736245
$358$	0	0
$359$	2.34017	0.123510	0.0617548	−	0.998091i	$-0.480330\pi$
$359$	2.34017	0.123510	0.0617548	+	0.998091i	$0.480330\pi$
$360$	0	0
$361$	10.8205	0.569498
$362$	0	0
$363$	2.97826	0.156318
$364$	0	0
$365$	−4.34017	−0.227175
$366$	0	0
$367$	−3.77432	−0.197018	−0.0985090	−	0.995136i	$-0.531407\pi$
$367$	−3.77432	−0.197018	−0.0985090	+	0.995136i	$0.531407\pi$
$368$	0	0
$369$	8.34017	0.434172
$370$	0	0
$371$	53.5897	2.78224
$372$	0	0
$373$	24.3402	1.26029	0.630143	−	0.776479i	$-0.282996\pi$
$373$	24.3402	1.26029	0.630143	+	0.776479i	$0.282996\pi$
$374$	0	0
$375$	0.539189	0.0278436
$376$	0	0
$377$	−31.4186	−1.61814
$378$	0	0
$379$	−4.28231	−0.219968	−0.109984	−	0.993933i	$-0.535080\pi$
$379$	−4.28231	−0.219968	−0.109984	+	0.993933i	$0.535080\pi$
$380$	0	0
$381$	3.34244	0.171238
$382$	0	0
$383$	−18.9627	−0.968947	−0.484473	−	0.874806i	$-0.660989\pi$
$383$	−18.9627	−0.968947	−0.484473	+	0.874806i	$0.660989\pi$
$384$	0	0
$385$	−8.89496	−0.453329
$386$	0	0
$387$	−31.8576	−1.61941
$388$	0	0
$389$	−3.88882	−0.197171	−0.0985854	−	0.995129i	$-0.531432\pi$
$389$	−3.88882	−0.197171	−0.0985854	+	0.995129i	$0.531432\pi$
$390$	0	0
$391$	−29.4596	−1.48984
$392$	0	0
$393$	3.04331	0.153514
$394$	0	0
$395$	4.72261	0.237620
$396$	0	0
$397$	7.20847	0.361783	0.180891	−	0.983503i	$-0.442102\pi$
$397$	7.20847	0.361783	0.180891	+	0.983503i	$0.442102\pi$
$398$	0	0
$399$	−11.1917	−0.560283
$400$	0	0
$401$	24.8371	1.24031	0.620153	−	0.784481i	$-0.287071\pi$
$401$	24.8371	1.24031	0.620153	+	0.784481i	$0.287071\pi$
$402$	0	0
$403$	6.55025	0.326291
$404$	0	0
$405$	−6.46800	−0.321397
$406$	0	0
$407$	−2.34017	−0.115998
$408$	0	0
$409$	−8.07223	−0.399146	−0.199573	−	0.979883i	$-0.563956\pi$
$409$	−8.07223	−0.399146	−0.199573	+	0.979883i	$0.563956\pi$
$410$	0	0
$411$	−6.02666	−0.297273
$412$	0	0
$413$	−28.0372	−1.37962
$414$	0	0
$415$	2.19902	0.107945
$416$	0	0
$417$	−2.47027	−0.120969
$418$	0	0
$419$	−38.9048	−1.90062	−0.950312	−	0.311299i	$-0.899236\pi$
$419$	−38.9048	−1.90062	−0.950312	+	0.311299i	$0.899236\pi$
$420$	0	0
$421$	−26.2967	−1.28162	−0.640811	−	0.767699i	$-0.721402\pi$
$421$	−26.2967	−1.28162	−0.640811	+	0.767699i	$0.721402\pi$
$422$	0	0
$423$	5.80098	0.282053
$424$	0	0
$425$	6.78765	0.329250
$426$	0	0
$427$	56.9914	2.75801
$428$	0	0
$429$	5.10957	0.246693
$430$	0	0
$431$	−13.9311	−0.671036	−0.335518	−	0.942034i	$-0.608911\pi$
$431$	−13.9311	−0.671036	−0.335518	+	0.942034i	$0.608911\pi$
$432$	0	0
$433$	16.5548	0.795572	0.397786	−	0.917478i	$-0.369779\pi$
$433$	16.5548	0.795572	0.397786	+	0.917478i	$0.369779\pi$
$434$	0	0
$435$	4.18342	0.200580
$436$	0	0
$437$	−23.7009	−1.13377
$438$	0	0
$439$	−5.00946	−0.239088	−0.119544	−	0.992829i	$-0.538143\pi$
$439$	−5.00946	−0.239088	−0.119544	+	0.992829i	$0.538143\pi$
$440$	0	0
$441$	−20.1773	−0.960823
$442$	0	0
$443$	−18.3701	−0.872792	−0.436396	−	0.899755i	$-0.643745\pi$
$443$	−18.3701	−0.872792	−0.436396	+	0.899755i	$0.643745\pi$
$444$	0	0
$445$	16.2557	0.770592
$446$	0	0
$447$	4.12556	0.195132
$448$	0	0
$449$	−9.28846	−0.438349	−0.219175	−	0.975686i	$-0.570336\pi$
$449$	−9.28846	−0.438349	−0.219175	+	0.975686i	$0.570336\pi$
$450$	0	0
$451$	−7.20394	−0.339220
$452$	0	0
$453$	9.09275	0.427215
$454$	0	0
$455$	15.3919	0.721583
$456$	0	0
$457$	33.0661	1.54677	0.773383	−	0.633939i	$-0.218563\pi$
$457$	33.0661	1.54677	0.773383	+	0.633939i	$0.218563\pi$
$458$	0	0
$459$	20.8950	0.975293
$460$	0	0
$461$	−28.6537	−1.33454	−0.667268	−	0.744818i	$-0.732536\pi$
$461$	−28.6537	−1.33454	−0.667268	+	0.744818i	$0.732536\pi$
$462$	0	0
$463$	−1.40417	−0.0652575	−0.0326288	−	0.999468i	$-0.510388\pi$
$463$	−1.40417	−0.0652575	−0.0326288	+	0.999468i	$0.510388\pi$
$464$	0	0
$465$	−0.872174	−0.0404461
$466$	0	0
$467$	26.7259	1.23673	0.618364	−	0.785892i	$-0.287796\pi$
$467$	26.7259	1.23673	0.618364	+	0.785892i	$0.287796\pi$
$468$	0	0
$469$	53.6490	2.47728
$470$	0	0
$471$	−7.81658	−0.360169
$472$	0	0
$473$	27.5174	1.26525
$474$	0	0
$475$	5.46081	0.250559
$476$	0	0
$477$	−38.1978	−1.74896
$478$	0	0
$479$	−26.7226	−1.22099	−0.610494	−	0.792021i	$-0.709029\pi$
$479$	−26.7226	−1.22099	−0.610494	+	0.792021i	$0.709029\pi$
$480$	0	0
$481$	4.04945	0.184639
$482$	0	0
$483$	8.89496	0.404735
$484$	0	0
$485$	−4.92162	−0.223479
$486$	0	0
$487$	30.5814	1.38578	0.692889	−	0.721044i	$-0.256338\pi$
$487$	30.5814	1.38578	0.692889	+	0.721044i	$0.256338\pi$
$488$	0	0
$489$	1.27617	0.0577105
$490$	0	0
$491$	−23.9299	−1.07994	−0.539970	−	0.841685i	$-0.681564\pi$
$491$	−23.9299	−1.07994	−0.539970	+	0.841685i	$0.681564\pi$
$492$	0	0
$493$	52.6635	2.37185
$494$	0	0
$495$	6.34017	0.284970
$496$	0	0
$497$	−50.4079	−2.26110
$498$	0	0
$499$	43.1617	1.93218	0.966091	−	0.258202i	$-0.0831300\pi$
$499$	43.1617	1.93218	0.966091	+	0.258202i	$0.0831300\pi$
$500$	0	0
$501$	−1.44521	−0.0645673
$502$	0	0
$503$	−5.11942	−0.228263	−0.114132	−	0.993466i	$-0.536409\pi$
$503$	−5.11942	−0.228263	−0.114132	+	0.993466i	$0.536409\pi$
$504$	0	0
$505$	15.6514	0.696479
$506$	0	0
$507$	−1.83218	−0.0813700
$508$	0	0
$509$	−13.8843	−0.615410	−0.307705	−	0.951482i	$-0.599561\pi$
$509$	−13.8843	−0.615410	−0.307705	+	0.951482i	$0.599561\pi$
$510$	0	0
$511$	16.4969	0.729781
$512$	0	0
$513$	16.8104	0.742199
$514$	0	0
$515$	−6.09890	−0.268749
$516$	0	0
$517$	−5.01068	−0.220369
$518$	0	0
$519$	−3.64754	−0.160109
$520$	0	0
$521$	13.4413	0.588876	0.294438	−	0.955671i	$-0.404868\pi$
$521$	13.4413	0.588876	0.294438	+	0.955671i	$0.404868\pi$
$522$	0	0
$523$	−15.3919	−0.673040	−0.336520	−	0.941676i	$-0.609250\pi$
$523$	−15.3919	−0.673040	−0.336520	+	0.941676i	$0.609250\pi$
$524$	0	0
$525$	−2.04945	−0.0894453
$526$	0	0
$527$	−10.9795	−0.478274
$528$	0	0
$529$	−4.16290	−0.180996
$530$	0	0
$531$	19.9844	0.867249
$532$	0	0
$533$	12.4657	0.539951
$534$	0	0
$535$	19.3184	0.835209
$536$	0	0
$537$	−0.206204	−0.00889835
$538$	0	0
$539$	17.4284	0.750694
$540$	0	0
$541$	46.1666	1.98486	0.992429	−	0.122824i	$-0.0391950\pi$
$541$	46.1666	1.98486	0.992429	+	0.122824i	$0.0391950\pi$
$542$	0	0
$543$	5.23513	0.224661
$544$	0	0
$545$	5.41855	0.232105
$546$	0	0
$547$	−19.2306	−0.822241	−0.411121	−	0.911581i	$-0.634862\pi$
$547$	−19.2306	−0.822241	−0.411121	+	0.911581i	$0.634862\pi$
$548$	0	0
$549$	−40.6225	−1.73373
$550$	0	0
$551$	42.3689	1.80498
$552$	0	0
$553$	−17.9506	−0.763335
$554$	0	0
$555$	−0.539189	−0.0228873
$556$	0	0
$557$	4.63090	0.196217	0.0981087	−	0.995176i	$-0.468721\pi$
$557$	4.63090	0.196217	0.0981087	+	0.995176i	$0.468721\pi$
$558$	0	0
$559$	−47.6163	−2.01396
$560$	0	0
$561$	−8.56463	−0.361599
$562$	0	0
$563$	−16.3857	−0.690577	−0.345288	−	0.938497i	$-0.612219\pi$
$563$	−16.3857	−0.690577	−0.345288	+	0.938497i	$0.612219\pi$
$564$	0	0
$565$	6.20620	0.261097
$566$	0	0
$567$	24.5848	1.03246
$568$	0	0
$569$	6.13009	0.256987	0.128493	−	0.991710i	$-0.458986\pi$
$569$	6.13009	0.256987	0.128493	+	0.991710i	$0.458986\pi$
$570$	0	0
$571$	−19.3028	−0.807798	−0.403899	−	0.914803i	$-0.632345\pi$
$571$	−19.3028	−0.807798	−0.403899	+	0.914803i	$0.632345\pi$
$572$	0	0
$573$	−0.389621	−0.0162767
$574$	0	0
$575$	−4.34017	−0.180998
$576$	0	0
$577$	−28.2472	−1.17595	−0.587974	−	0.808880i	$-0.700074\pi$
$577$	−28.2472	−1.17595	−0.587974	+	0.808880i	$0.700074\pi$
$578$	0	0
$579$	8.28231	0.344201
$580$	0	0
$581$	−8.35842	−0.346766
$582$	0	0
$583$	32.9939	1.36647
$584$	0	0
$585$	−10.9711	−0.453598
$586$	0	0
$587$	3.34632	0.138117	0.0690586	−	0.997613i	$-0.478000\pi$
$587$	3.34632	0.138117	0.0690586	+	0.997613i	$0.478000\pi$
$588$	0	0
$589$	−8.83323	−0.363967
$590$	0	0
$591$	4.81044	0.197875
$592$	0	0
$593$	20.3090	0.833990	0.416995	−	0.908909i	$-0.363083\pi$
$593$	20.3090	0.833990	0.416995	+	0.908909i	$0.363083\pi$
$594$	0	0
$595$	−25.7998	−1.05769
$596$	0	0
$597$	−11.7938	−0.482688
$598$	0	0
$599$	0.130094	0.00531548	0.00265774	−	0.999996i	$-0.499154\pi$
$599$	0.130094	0.00531548	0.00265774	+	0.999996i	$0.499154\pi$
$600$	0	0
$601$	−39.1689	−1.59773	−0.798866	−	0.601510i	$-0.794566\pi$
$601$	−39.1689	−1.59773	−0.798866	+	0.601510i	$0.794566\pi$
$602$	0	0
$603$	−38.2401	−1.55726
$604$	0	0
$605$	5.52359	0.224566
$606$	0	0
$607$	−25.6020	−1.03915	−0.519576	−	0.854424i	$-0.673910\pi$
$607$	−25.6020	−1.03915	−0.519576	+	0.854424i	$0.673910\pi$
$608$	0	0
$609$	−15.9011	−0.644345
$610$	0	0
$611$	8.67050	0.350771
$612$	0	0
$613$	−17.8310	−0.720186	−0.360093	−	0.932916i	$-0.617255\pi$
$613$	−17.8310	−0.720186	−0.360093	+	0.932916i	$0.617255\pi$
$614$	0	0
$615$	−1.65983	−0.0669307
$616$	0	0
$617$	−8.21008	−0.330525	−0.165263	−	0.986250i	$-0.552847\pi$
$617$	−8.21008	−0.330525	−0.165263	+	0.986250i	$0.552847\pi$
$618$	0	0
$619$	4.36683	0.175518	0.0877590	−	0.996142i	$-0.472029\pi$
$619$	4.36683	0.175518	0.0877590	+	0.996142i	$0.472029\pi$
$620$	0	0
$621$	−13.3607	−0.536146
$622$	0	0
$623$	−61.7875	−2.47546
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.89043	−0.275177
$628$	0	0
$629$	−6.78765	−0.270641
$630$	0	0
$631$	−13.0940	−0.521263	−0.260631	−	0.965438i	$-0.583931\pi$
$631$	−13.0940	−0.521263	−0.260631	+	0.965438i	$0.583931\pi$
$632$	0	0
$633$	−2.32580	−0.0924421
$634$	0	0
$635$	6.19902	0.246000
$636$	0	0
$637$	−30.1582	−1.19491
$638$	0	0
$639$	35.9299	1.42136
$640$	0	0
$641$	32.1171	1.26855	0.634276	−	0.773107i	$-0.281298\pi$
$641$	32.1171	1.26855	0.634276	+	0.773107i	$0.281298\pi$
$642$	0	0
$643$	−4.95282	−0.195320	−0.0976601	−	0.995220i	$-0.531136\pi$
$643$	−4.95282	−0.195320	−0.0976601	+	0.995220i	$0.531136\pi$
$644$	0	0
$645$	6.34017	0.249644
$646$	0	0
$647$	18.3812	0.722640	0.361320	−	0.932442i	$-0.382326\pi$
$647$	18.3812	0.722640	0.361320	+	0.932442i	$0.382326\pi$
$648$	0	0
$649$	−17.2618	−0.677585
$650$	0	0
$651$	3.31512	0.129930
$652$	0	0
$653$	−4.57691	−0.179109	−0.0895543	−	0.995982i	$-0.528544\pi$
$653$	−4.57691	−0.179109	−0.0895543	+	0.995982i	$0.528544\pi$
$654$	0	0
$655$	5.64423	0.220538
$656$	0	0
$657$	−11.7587	−0.458752
$658$	0	0
$659$	−5.62863	−0.219260	−0.109630	−	0.993972i	$-0.534967\pi$
$659$	−5.62863	−0.219260	−0.109630	+	0.993972i	$0.534967\pi$
$660$	0	0
$661$	−14.9939	−0.583193	−0.291597	−	0.956541i	$-0.594187\pi$
$661$	−14.9939	−0.583193	−0.291597	+	0.956541i	$0.594187\pi$
$662$	0	0
$663$	14.8203	0.575572
$664$	0	0
$665$	−20.7565	−0.804901
$666$	0	0
$667$	−33.6742	−1.30387
$668$	0	0
$669$	14.0183	0.541977
$670$	0	0
$671$	35.0882	1.35457
$672$	0	0
$673$	−28.8371	−1.11159	−0.555794	−	0.831320i	$-0.687586\pi$
$673$	−28.8371	−1.11159	−0.555794	+	0.831320i	$0.687586\pi$
$674$	0	0
$675$	3.07838	0.118487
$676$	0	0
$677$	13.3340	0.512468	0.256234	−	0.966615i	$-0.417518\pi$
$677$	13.3340	0.512468	0.256234	+	0.966615i	$0.417518\pi$
$678$	0	0
$679$	18.7070	0.717909
$680$	0	0
$681$	12.2823	0.470659
$682$	0	0
$683$	−20.6537	−0.790291	−0.395146	−	0.918618i	$-0.629306\pi$
$683$	−20.6537	−0.790291	−0.395146	+	0.918618i	$0.629306\pi$
$684$	0	0
$685$	−11.1773	−0.427062
$686$	0	0
$687$	13.5441	0.516740
$688$	0	0
$689$	−57.0928	−2.17506
$690$	0	0
$691$	30.7070	1.16815	0.584075	−	0.811700i	$-0.301457\pi$
$691$	30.7070	1.16815	0.584075	+	0.811700i	$0.301457\pi$
$692$	0	0
$693$	−24.0989	−0.915441
$694$	0	0
$695$	−4.58145	−0.173784
$696$	0	0
$697$	−20.8950	−0.791453
$698$	0	0
$699$	−3.86991	−0.146373
$700$	0	0
$701$	−0.424694	−0.0160405	−0.00802023	−	0.999968i	$-0.502553\pi$
$701$	−0.424694	−0.0160405	−0.00802023	+	0.999968i	$0.502553\pi$
$702$	0	0
$703$	−5.46081	−0.205958
$704$	0	0
$705$	−1.15449	−0.0434805
$706$	0	0
$707$	−59.4908	−2.23738
$708$	0	0
$709$	36.9048	1.38599	0.692994	−	0.720943i	$-0.256291\pi$
$709$	36.9048	1.38599	0.692994	+	0.720943i	$0.256291\pi$
$710$	0	0
$711$	12.7948	0.479844
$712$	0	0
$713$	7.02052	0.262920
$714$	0	0
$715$	9.47641	0.354398
$716$	0	0
$717$	−5.90498	−0.220525
$718$	0	0
$719$	−10.0722	−0.375631	−0.187815	−	0.982204i	$-0.560141\pi$
$719$	−10.0722	−0.375631	−0.187815	+	0.982204i	$0.560141\pi$
$720$	0	0
$721$	23.1818	0.863336
$722$	0	0
$723$	−15.0205	−0.558619
$724$	0	0
$725$	7.75872	0.288152
$726$	0	0
$727$	−38.8638	−1.44138	−0.720689	−	0.693259i	$-0.756174\pi$
$727$	−38.8638	−1.44138	−0.720689	+	0.693259i	$0.756174\pi$
$728$	0	0
$729$	−12.5441	−0.464597
$730$	0	0
$731$	79.8141	2.95203
$732$	0	0
$733$	−36.6681	−1.35437	−0.677183	−	0.735815i	$-0.736799\pi$
$733$	−36.6681	−1.35437	−0.677183	+	0.735815i	$0.736799\pi$
$734$	0	0
$735$	4.01560	0.148118
$736$	0	0
$737$	33.0304	1.21669
$738$	0	0
$739$	51.7152	1.90238	0.951188	−	0.308612i	$-0.0998645\pi$
$739$	51.7152	1.90238	0.951188	+	0.308612i	$0.0998645\pi$
$740$	0	0
$741$	11.9232	0.438011
$742$	0	0
$743$	16.3291	0.599057	0.299528	−	0.954087i	$-0.403171\pi$
$743$	16.3291	0.599057	0.299528	+	0.954087i	$0.403171\pi$
$744$	0	0
$745$	7.65142	0.280326
$746$	0	0
$747$	5.95774	0.217982
$748$	0	0
$749$	−73.4291	−2.68304
$750$	0	0
$751$	−12.0845	−0.440970	−0.220485	−	0.975390i	$-0.570764\pi$
$751$	−12.0845	−0.440970	−0.220485	+	0.975390i	$0.570764\pi$
$752$	0	0
$753$	−12.6186	−0.459848
$754$	0	0
$755$	16.8638	0.613735
$756$	0	0
$757$	2.62702	0.0954807	0.0477404	−	0.998860i	$-0.484798\pi$
$757$	2.62702	0.0954807	0.0477404	+	0.998860i	$0.484798\pi$
$758$	0	0
$759$	5.47641	0.198781
$760$	0	0
$761$	−31.3607	−1.13682	−0.568412	−	0.822744i	$-0.692442\pi$
$761$	−31.3607	−1.13682	−0.568412	+	0.822744i	$0.692442\pi$
$762$	0	0
$763$	−20.5958	−0.745619
$764$	0	0
$765$	18.3896	0.664878
$766$	0	0
$767$	29.8699	1.07854
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−2.13009	−0.0767134
$772$	0	0
$773$	−7.56093	−0.271948	−0.135974	−	0.990712i	$-0.543416\pi$
$773$	−7.56093	−0.271948	−0.135974	+	0.990712i	$0.543416\pi$
$774$	0	0
$775$	−1.61757	−0.0581047
$776$	0	0
$777$	2.04945	0.0735235
$778$	0	0
$779$	−16.8104	−0.602296
$780$	0	0
$781$	−31.0349	−1.11052
$782$	0	0
$783$	23.8843	0.853555
$784$	0	0
$785$	−14.4969	−0.517418
$786$	0	0
$787$	−23.8999	−0.851939	−0.425969	−	0.904738i	$-0.640067\pi$
$787$	−23.8999	−0.851939	−0.425969	+	0.904738i	$0.640067\pi$
$788$	0	0
$789$	1.23901	0.0441099
$790$	0	0
$791$	−23.5897	−0.838753
$792$	0	0
$793$	−60.7168	−2.15612
$794$	0	0
$795$	7.60197	0.269614
$796$	0	0
$797$	−3.99612	−0.141550	−0.0707750	−	0.997492i	$-0.522547\pi$
$797$	−3.99612	−0.141550	−0.0707750	+	0.997492i	$0.522547\pi$
$798$	0	0
$799$	−14.5334	−0.514156
$800$	0	0
$801$	44.0410	1.55611
$802$	0	0
$803$	10.1568	0.358424
$804$	0	0
$805$	16.4969	0.581440
$806$	0	0
$807$	3.02505	0.106487
$808$	0	0
$809$	−35.3607	−1.24322	−0.621608	−	0.783329i	$-0.713520\pi$
$809$	−35.3607	−1.24322	−0.621608	+	0.783329i	$0.713520\pi$
$810$	0	0
$811$	−1.84324	−0.0647251	−0.0323625	−	0.999476i	$-0.510303\pi$
$811$	−1.84324	−0.0647251	−0.0323625	+	0.999476i	$0.510303\pi$
$812$	0	0
$813$	7.94668	0.278702
$814$	0	0
$815$	2.36683	0.0829066
$816$	0	0
$817$	64.2122	2.24650
$818$	0	0
$819$	41.7009	1.45715
$820$	0	0
$821$	−11.4764	−0.400529	−0.200265	−	0.979742i	$-0.564180\pi$
$821$	−11.4764	−0.400529	−0.200265	+	0.979742i	$0.564180\pi$
$822$	0	0
$823$	−30.4235	−1.06050	−0.530248	−	0.847843i	$-0.677901\pi$
$823$	−30.4235	−1.06050	−0.530248	+	0.847843i	$0.677901\pi$
$824$	0	0
$825$	−1.26180	−0.0439301
$826$	0	0
$827$	−0.738205	−0.0256699	−0.0128349	−	0.999918i	$-0.504086\pi$
$827$	−0.738205	−0.0256699	−0.0128349	+	0.999918i	$0.504086\pi$
$828$	0	0
$829$	−34.1445	−1.18589	−0.592943	−	0.805244i	$-0.702034\pi$
$829$	−34.1445	−1.18589	−0.592943	+	0.805244i	$0.702034\pi$
$830$	0	0
$831$	−8.60036	−0.298343
$832$	0	0
$833$	50.5509	1.75149
$834$	0	0
$835$	−2.68035	−0.0927572
$836$	0	0
$837$	−4.97948	−0.172116
$838$	0	0
$839$	−26.3545	−0.909860	−0.454930	−	0.890527i	$-0.650336\pi$
$839$	−26.3545	−0.909860	−0.454930	+	0.890527i	$0.650336\pi$
$840$	0	0
$841$	31.1978	1.07579
$842$	0	0
$843$	−15.2183	−0.524147
$844$	0	0
$845$	−3.39803	−0.116896
$846$	0	0
$847$	−20.9951	−0.721399
$848$	0	0
$849$	12.6491	0.434118
$850$	0	0
$851$	4.34017	0.148779
$852$	0	0
$853$	18.7936	0.643481	0.321741	−	0.946828i	$-0.395732\pi$
$853$	18.7936	0.643481	0.321741	+	0.946828i	$0.395732\pi$
$854$	0	0
$855$	14.7948	0.505973
$856$	0	0
$857$	39.4452	1.34742	0.673711	−	0.738995i	$-0.264699\pi$
$857$	39.4452	1.34742	0.673711	+	0.738995i	$0.264699\pi$
$858$	0	0
$859$	−30.0521	−1.02536	−0.512682	−	0.858578i	$-0.671348\pi$
$859$	−30.0521	−1.02536	−0.512682	+	0.858578i	$0.671348\pi$
$860$	0	0
$861$	6.30898	0.215009
$862$	0	0
$863$	0.821503	0.0279643	0.0139821	−	0.999902i	$-0.495549\pi$
$863$	0.821503	0.0279643	0.0139821	+	0.999902i	$0.495549\pi$
$864$	0	0
$865$	−6.76487	−0.230012
$866$	0	0
$867$	−15.6754	−0.532365
$868$	0	0
$869$	−11.0517	−0.374904
$870$	0	0
$871$	−57.1559	−1.93665
$872$	0	0
$873$	−13.3340	−0.451289
$874$	0	0
$875$	−3.80098	−0.128497
$876$	0	0
$877$	−10.4436	−0.352655	−0.176328	−	0.984332i	$-0.556422\pi$
$877$	−10.4436	−0.352655	−0.176328	+	0.984332i	$0.556422\pi$
$878$	0	0
$879$	−0.909336	−0.0306711
$880$	0	0
$881$	11.9649	0.403109	0.201554	−	0.979477i	$-0.435401\pi$
$881$	11.9649	0.403109	0.201554	+	0.979477i	$0.435401\pi$
$882$	0	0
$883$	43.2905	1.45684	0.728421	−	0.685129i	$-0.240254\pi$
$883$	43.2905	1.45684	0.728421	+	0.685129i	$0.240254\pi$
$884$	0	0
$885$	−3.97721	−0.133693
$886$	0	0
$887$	8.19902	0.275296	0.137648	−	0.990481i	$-0.456046\pi$
$887$	8.19902	0.275296	0.137648	+	0.990481i	$0.456046\pi$
$888$	0	0
$889$	−23.5624	−0.790256
$890$	0	0
$891$	15.1362	0.507083
$892$	0	0
$893$	−11.6925	−0.391273
$894$	0	0
$895$	−0.382433	−0.0127833
$896$	0	0
$897$	−9.47641	−0.316408
$898$	0	0
$899$	−12.5503	−0.418574
$900$	0	0
$901$	95.6984	3.18818
$902$	0	0
$903$	−24.0989	−0.801961
$904$	0	0
$905$	9.70928	0.322747
$906$	0	0
$907$	−23.4596	−0.778963	−0.389481	−	0.921034i	$-0.627346\pi$
$907$	−23.4596	−0.778963	−0.389481	+	0.921034i	$0.627346\pi$
$908$	0	0
$909$	42.4040	1.40645
$910$	0	0
$911$	−13.5864	−0.450137	−0.225068	−	0.974343i	$-0.572261\pi$
$911$	−13.5864	−0.450137	−0.225068	+	0.974343i	$0.572261\pi$
$912$	0	0
$913$	−5.14608	−0.170310
$914$	0	0
$915$	8.08452	0.267266
$916$	0	0
$917$	−21.4536	−0.708461
$918$	0	0
$919$	27.4896	0.906797	0.453399	−	0.891308i	$-0.350211\pi$
$919$	27.4896	0.906797	0.453399	+	0.891308i	$0.350211\pi$
$920$	0	0
$921$	−7.04331	−0.232085
$922$	0	0
$923$	53.7030	1.76765
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−16.5236	−0.542706
$928$	0	0
$929$	17.8622	0.586038	0.293019	−	0.956107i	$-0.405340\pi$
$929$	17.8622	0.586038	0.293019	+	0.956107i	$0.405340\pi$
$930$	0	0
$931$	40.6693	1.33288
$932$	0	0
$933$	4.50534	0.147498
$934$	0	0
$935$	−15.8843	−0.519472
$936$	0	0
$937$	−52.4678	−1.71405	−0.857025	−	0.515276i	$-0.827690\pi$
$937$	−52.4678	−1.71405	−0.857025	+	0.515276i	$0.827690\pi$
$938$	0	0
$939$	3.55640	0.116059
$940$	0	0
$941$	17.0121	0.554579	0.277289	−	0.960786i	$-0.410564\pi$
$941$	17.0121	0.554579	0.277289	+	0.960786i	$0.410564\pi$
$942$	0	0
$943$	13.3607	0.435084
$944$	0	0
$945$	−11.7009	−0.380629
$946$	0	0
$947$	−4.79153	−0.155704	−0.0778519	−	0.996965i	$-0.524806\pi$
$947$	−4.79153	−0.155704	−0.0778519	+	0.996965i	$0.524806\pi$
$948$	0	0
$949$	−17.5753	−0.570519
$950$	0	0
$951$	−10.9048	−0.353612
$952$	0	0
$953$	40.0944	1.29878	0.649392	−	0.760454i	$-0.275024\pi$
$953$	40.0944	1.29878	0.649392	+	0.760454i	$0.275024\pi$
$954$	0	0
$955$	−0.722606	−0.0233830
$956$	0	0
$957$	−9.78992	−0.316463
$958$	0	0
$959$	42.4846	1.37190
$960$	0	0
$961$	−28.3835	−0.915596
$962$	0	0
$963$	52.3390	1.68660
$964$	0	0
$965$	15.3607	0.494478
$966$	0	0
$967$	−44.3090	−1.42488	−0.712440	−	0.701733i	$-0.752410\pi$
$967$	−44.3090	−1.42488	−0.712440	+	0.701733i	$0.752410\pi$
$968$	0	0
$969$	−19.9856	−0.642031
$970$	0	0
$971$	53.8187	1.72712	0.863562	−	0.504243i	$-0.168228\pi$
$971$	53.8187	1.72712	0.863562	+	0.504243i	$0.168228\pi$
$972$	0	0
$973$	17.4140	0.558268
$974$	0	0
$975$	2.18342	0.0699253
$976$	0	0
$977$	−7.39189	−0.236487	−0.118244	−	0.992985i	$-0.537726\pi$
$977$	−7.39189	−0.236487	−0.118244	+	0.992985i	$0.537726\pi$
$978$	0	0
$979$	−38.0410	−1.21580
$980$	0	0
$981$	14.6803	0.468707
$982$	0	0
$983$	1.02174	0.0325885	0.0162942	−	0.999867i	$-0.494813\pi$
$983$	1.02174	0.0325885	0.0162942	+	0.999867i	$0.494813\pi$
$984$	0	0
$985$	8.92162	0.284267
$986$	0	0
$987$	4.38819	0.139678
$988$	0	0
$989$	−51.0349	−1.62282
$990$	0	0
$991$	47.1494	1.49775	0.748875	−	0.662711i	$-0.230594\pi$
$991$	47.1494	1.49775	0.748875	+	0.662711i	$0.230594\pi$
$992$	0	0
$993$	−3.57918	−0.113582
$994$	0	0
$995$	−21.8732	−0.693428
$996$	0	0
$997$	7.04718	0.223186	0.111593	−	0.993754i	$-0.464405\pi$
$997$	7.04718	0.223186	0.111593	+	0.993754i	$0.464405\pi$
$998$	0	0
$999$	−3.07838	−0.0973956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.2.a.e.1.2	✓	3
3.2	odd	2		6660.2.a.q.1.3		3
4.3	odd	2		2960.2.a.t.1.2		3
5.2	odd	4		3700.2.d.h.149.4		6
5.3	odd	4		3700.2.d.h.149.3		6
5.4	even	2		3700.2.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.e.1.2	✓	3	1.1	even	1	trivial
2960.2.a.t.1.2		3	4.3	odd	2
3700.2.a.i.1.2		3	5.4	even	2
3700.2.d.h.149.3		6	5.3	odd	4
3700.2.d.h.149.4		6	5.2	odd	4
6660.2.a.q.1.3		3	3.2	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:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.a (trivial)

\(f(q)\)	\(=\)	\(q-0.539189 q^{3} -1.00000 q^{5} +3.80098 q^{7} -2.70928 q^{9} +2.34017 q^{11} -4.04945 q^{13} +0.539189 q^{15} +6.78765 q^{17} +5.46081 q^{19} -2.04945 q^{21} -4.34017 q^{23} +1.00000 q^{25} +3.07838 q^{27} +7.75872 q^{29} -1.61757 q^{31} -1.26180 q^{33} -3.80098 q^{35} -1.00000 q^{37} +2.18342 q^{39} -3.07838 q^{41} +11.7587 q^{43} +2.70928 q^{45} -2.14116 q^{47} +7.44748 q^{49} -3.65983 q^{51} +14.0989 q^{53} -2.34017 q^{55} -2.94441 q^{57} -7.37629 q^{59} +14.9939 q^{61} -10.2979 q^{63} +4.04945 q^{65} +14.1145 q^{67} +2.34017 q^{69} -13.2618 q^{71} +4.34017 q^{73} -0.539189 q^{75} +8.89496 q^{77} -4.72261 q^{79} +6.46800 q^{81} -2.19902 q^{83} -6.78765 q^{85} -4.18342 q^{87} -16.2557 q^{89} -15.3919 q^{91} +0.872174 q^{93} -5.46081 q^{95} +4.92162 q^{97} -6.34017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7} - q^{9} - 4 q^{11} + 6 q^{13} + 10 q^{17} + 18 q^{19} + 12 q^{21} - 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 3 q^{37} + 2 q^{39} - 6 q^{41} + 10 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - q^9 - 4 * q^11 + 6 * q^13 + 10 * q^17 + 18 * q^19 + 12 * q^21 - 2 * q^23 + 3 * q^25 + 6 * q^27 - 2 * q^29 + 4 * q^33 - 2 * q^35 - 3 * q^37 + 2 * q^39 - 6 * q^41 + 10 * q^43 + q^45 + 14 * q^47 + 23 * q^49 - 22 * q^51 + 6 * q^53 + 4 * q^55 + 8 * q^57 + 8 * q^59 + 10 * q^61 - 4 * q^63 - 6 * q^65 + 20 * q^67 - 4 * q^69 - 32 * q^71 + 2 * q^73 + 28 * q^77 - 8 * q^79 - 13 * q^81 - 16 * q^83 - 10 * q^85 - 8 * q^87 - 6 * q^89 - 30 * q^91 + 24 * q^93 - 18 * q^95 + 18 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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