LMFDB - Embedded newform 740.2.a.e.1.3

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			1.3
Root			$0.311108$ 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)

$f(q)$	$=$	$q+2.21432 q^{3} -1.00000 q^{5} +2.83654 q^{7} +1.90321 q^{9} -1.37778 q^{11} +4.28100 q^{13} -2.21432 q^{15} -3.33185 q^{17} +8.21432 q^{19} +6.28100 q^{21} -0.622216 q^{23} +1.00000 q^{25} -2.42864 q^{27} -5.18421 q^{29} +6.64296 q^{31} -3.05086 q^{33} -2.83654 q^{35} -1.00000 q^{37} +9.47949 q^{39} +2.42864 q^{41} -1.18421 q^{43} -1.90321 q^{45} +2.54125 q^{47} +1.04593 q^{49} -7.37778 q^{51} -2.56199 q^{53} +1.37778 q^{55} +18.1891 q^{57} +13.8272 q^{59} -14.4701 q^{61} +5.39853 q^{63} -4.28100 q^{65} -8.87802 q^{67} -1.37778 q^{69} -15.0509 q^{71} +0.622216 q^{73} +2.21432 q^{75} -3.90813 q^{77} -9.26517 q^{79} -11.0874 q^{81} -3.16346 q^{83} +3.33185 q^{85} -11.4795 q^{87} +11.4193 q^{89} +12.1432 q^{91} +14.7096 q^{93} -8.21432 q^{95} +10.4286 q^{97} -2.62222 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.21432	1.27844	0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432	1.27844	0.639219	+	0.769025i	$0.279258\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.83654	1.07211	0.536055	−	0.844183i	$-0.319914\pi$
$7$	2.83654	1.07211	0.536055	+	0.844183i	$0.319914\pi$
$8$	0	0
$9$	1.90321	0.634404
$10$	0	0
$11$	−1.37778	−0.415418	−0.207709	−	0.978191i	$-0.566601\pi$
$11$	−1.37778	−0.415418	−0.207709	+	0.978191i	$0.566601\pi$
$12$	0	0
$13$	4.28100	1.18733	0.593667	−	0.804711i	$-0.297679\pi$
$13$	4.28100	1.18733	0.593667	+	0.804711i	$0.297679\pi$
$14$	0	0
$15$	−2.21432	−0.571735
$16$	0	0
$17$	−3.33185	−0.808093	−0.404046	−	0.914739i	$-0.632397\pi$
$17$	−3.33185	−0.808093	−0.404046	+	0.914739i	$0.632397\pi$
$18$	0	0
$19$	8.21432	1.88449	0.942247	−	0.334919i	$-0.108709\pi$
$19$	8.21432	1.88449	0.942247	+	0.334919i	$0.108709\pi$
$20$	0	0
$21$	6.28100	1.37063
$22$	0	0
$23$	−0.622216	−0.129741	−0.0648705	−	0.997894i	$-0.520663\pi$
$23$	−0.622216	−0.129741	−0.0648705	+	0.997894i	$0.520663\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.42864	−0.467392
$28$	0	0
$29$	−5.18421	−0.962683	−0.481342	−	0.876533i	$-0.659850\pi$
$29$	−5.18421	−0.962683	−0.481342	+	0.876533i	$0.659850\pi$
$30$	0	0
$31$	6.64296	1.19311	0.596555	−	0.802572i	$-0.296536\pi$
$31$	6.64296	1.19311	0.596555	+	0.802572i	$0.296536\pi$
$32$	0	0
$33$	−3.05086	−0.531086
$34$	0	0
$35$	−2.83654	−0.479462
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	9.47949	1.51793
$40$	0	0
$41$	2.42864	0.379290	0.189645	−	0.981853i	$-0.439266\pi$
$41$	2.42864	0.379290	0.189645	+	0.981853i	$0.439266\pi$
$42$	0	0
$43$	−1.18421	−0.180590	−0.0902950	−	0.995915i	$-0.528781\pi$
$43$	−1.18421	−0.180590	−0.0902950	+	0.995915i	$0.528781\pi$
$44$	0	0
$45$	−1.90321	−0.283714
$46$	0	0
$47$	2.54125	0.370679	0.185340	−	0.982675i	$-0.440661\pi$
$47$	2.54125	0.370679	0.185340	+	0.982675i	$0.440661\pi$
$48$	0	0
$49$	1.04593	0.149419
$50$	0	0
$51$	−7.37778	−1.03310
$52$	0	0
$53$	−2.56199	−0.351917	−0.175958	−	0.984398i	$-0.556302\pi$
$53$	−2.56199	−0.351917	−0.175958	+	0.984398i	$0.556302\pi$
$54$	0	0
$55$	1.37778	0.185780
$56$	0	0
$57$	18.1891	2.40921
$58$	0	0
$59$	13.8272	1.80014	0.900072	−	0.435741i	$-0.143514\pi$
$59$	13.8272	1.80014	0.900072	+	0.435741i	$0.143514\pi$
$60$	0	0
$61$	−14.4701	−1.85271	−0.926355	−	0.376652i	$-0.877075\pi$
$61$	−14.4701	−1.85271	−0.926355	+	0.376652i	$0.877075\pi$
$62$	0	0
$63$	5.39853	0.680151
$64$	0	0
$65$	−4.28100	−0.530992
$66$	0	0
$67$	−8.87802	−1.08462	−0.542312	−	0.840177i	$-0.682451\pi$
$67$	−8.87802	−1.08462	−0.542312	+	0.840177i	$0.682451\pi$
$68$	0	0
$69$	−1.37778	−0.165866
$70$	0	0
$71$	−15.0509	−1.78621	−0.893104	−	0.449850i	$-0.851477\pi$
$71$	−15.0509	−1.78621	−0.893104	+	0.449850i	$0.851477\pi$
$72$	0	0
$73$	0.622216	0.0728248	0.0364124	−	0.999337i	$-0.488407\pi$
$73$	0.622216	0.0728248	0.0364124	+	0.999337i	$0.488407\pi$
$74$	0	0
$75$	2.21432	0.255688
$76$	0	0
$77$	−3.90813	−0.445373
$78$	0	0
$79$	−9.26517	−1.04241	−0.521207	−	0.853430i	$-0.674518\pi$
$79$	−9.26517	−1.04241	−0.521207	+	0.853430i	$0.674518\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	0	0
$83$	−3.16346	−0.347235	−0.173618	−	0.984813i	$-0.555546\pi$
$83$	−3.16346	−0.347235	−0.173618	+	0.984813i	$0.555546\pi$
$84$	0	0
$85$	3.33185	0.361390
$86$	0	0
$87$	−11.4795	−1.23073
$88$	0	0
$89$	11.4193	1.21044	0.605220	−	0.796058i	$-0.293085\pi$
$89$	11.4193	1.21044	0.605220	+	0.796058i	$0.293085\pi$
$90$	0	0
$91$	12.1432	1.27295
$92$	0	0
$93$	14.7096	1.52532
$94$	0	0
$95$	−8.21432	−0.842771
$96$	0	0
$97$	10.4286	1.05887	0.529434	−	0.848351i	$-0.322404\pi$
$97$	10.4286	1.05887	0.529434	+	0.848351i	$0.322404\pi$
$98$	0	0
$99$	−2.62222	−0.263543
$100$	0	0
$101$	−5.39207	−0.536531	−0.268266	−	0.963345i	$-0.586451\pi$
$101$	−5.39207	−0.536531	−0.268266	+	0.963345i	$0.586451\pi$
$102$	0	0
$103$	−10.5620	−1.04070	−0.520352	−	0.853952i	$-0.674199\pi$
$103$	−10.5620	−1.04070	−0.520352	+	0.853952i	$0.674199\pi$
$104$	0	0
$105$	−6.28100	−0.612962
$106$	0	0
$107$	7.53188	0.728134	0.364067	−	0.931373i	$-0.381388\pi$
$107$	7.53188	0.728134	0.364067	+	0.931373i	$0.381388\pi$
$108$	0	0
$109$	3.80642	0.364589	0.182295	−	0.983244i	$-0.441648\pi$
$109$	3.80642	0.364589	0.182295	+	0.983244i	$0.441648\pi$
$110$	0	0
$111$	−2.21432	−0.210174
$112$	0	0
$113$	13.1383	1.23594	0.617972	−	0.786200i	$-0.287954\pi$
$113$	13.1383	1.23594	0.617972	+	0.786200i	$0.287954\pi$
$114$	0	0
$115$	0.622216	0.0580219
$116$	0	0
$117$	8.14764	0.753250
$118$	0	0
$119$	−9.45091	−0.866364
$120$	0	0
$121$	−9.10171	−0.827428
$122$	0	0
$123$	5.37778	0.484898
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.16346	−0.635655	−0.317827	−	0.948149i	$-0.602953\pi$
$127$	−7.16346	−0.635655	−0.317827	+	0.948149i	$0.602953\pi$
$128$	0	0
$129$	−2.62222	−0.230873
$130$	0	0
$131$	−15.6938	−1.37117	−0.685587	−	0.727990i	$-0.740455\pi$
$131$	−15.6938	−1.37117	−0.685587	+	0.727990i	$0.740455\pi$
$132$	0	0
$133$	23.3002	2.02038
$134$	0	0
$135$	2.42864	0.209024
$136$	0	0
$137$	−10.9906	−0.938993	−0.469497	−	0.882934i	$-0.655565\pi$
$137$	−10.9906	−0.938993	−0.469497	+	0.882934i	$0.655565\pi$
$138$	0	0
$139$	13.8064	1.17105	0.585523	−	0.810656i	$-0.300889\pi$
$139$	13.8064	1.17105	0.585523	+	0.810656i	$0.300889\pi$
$140$	0	0
$141$	5.62714	0.473890
$142$	0	0
$143$	−5.89829	−0.493240
$144$	0	0
$145$	5.18421	0.430525
$146$	0	0
$147$	2.31603	0.191023
$148$	0	0
$149$	2.60793	0.213650	0.106825	−	0.994278i	$-0.465932\pi$
$149$	2.60793	0.213650	0.106825	+	0.994278i	$0.465932\pi$
$150$	0	0
$151$	−16.7239	−1.36097	−0.680487	−	0.732760i	$-0.738232\pi$
$151$	−16.7239	−1.36097	−0.680487	+	0.732760i	$0.738232\pi$
$152$	0	0
$153$	−6.34122	−0.512657
$154$	0	0
$155$	−6.64296	−0.533575
$156$	0	0
$157$	−0.235063	−0.0187601	−0.00938005	−	0.999956i	$-0.502986\pi$
$157$	−0.235063	−0.0187601	−0.00938005	+	0.999956i	$0.502986\pi$
$158$	0	0
$159$	−5.67307	−0.449904
$160$	0	0
$161$	−1.76494	−0.139096
$162$	0	0
$163$	−16.9590	−1.32833	−0.664165	−	0.747586i	$-0.731213\pi$
$163$	−16.9590	−1.32833	−0.664165	+	0.747586i	$0.731213\pi$
$164$	0	0
$165$	3.05086	0.237509
$166$	0	0
$167$	−4.75557	−0.367997	−0.183998	−	0.982927i	$-0.558904\pi$
$167$	−4.75557	−0.367997	−0.183998	+	0.982927i	$0.558904\pi$
$168$	0	0
$169$	5.32693	0.409764
$170$	0	0
$171$	15.6336	1.19553
$172$	0	0
$173$	23.2859	1.77040	0.885198	−	0.465215i	$-0.154023\pi$
$173$	23.2859	1.77040	0.885198	+	0.465215i	$0.154023\pi$
$174$	0	0
$175$	2.83654	0.214422
$176$	0	0
$177$	30.6178	2.30137
$178$	0	0
$179$	8.64296	0.646005	0.323003	−	0.946398i	$-0.395308\pi$
$179$	8.64296	0.646005	0.323003	+	0.946398i	$0.395308\pi$
$180$	0	0
$181$	−5.09679	−0.378841	−0.189421	−	0.981896i	$-0.560661\pi$
$181$	−5.09679	−0.378841	−0.189421	+	0.981896i	$0.560661\pi$
$182$	0	0
$183$	−32.0415	−2.36857
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	4.59057	0.335696
$188$	0	0
$189$	−6.88892	−0.501095
$190$	0	0
$191$	5.26517	0.380975	0.190487	−	0.981690i	$-0.438993\pi$
$191$	5.26517	0.380975	0.190487	+	0.981690i	$0.438993\pi$
$192$	0	0
$193$	−0.488863	−0.0351891	−0.0175945	−	0.999845i	$-0.505601\pi$
$193$	−0.488863	−0.0351891	−0.0175945	+	0.999845i	$0.505601\pi$
$194$	0	0
$195$	−9.47949	−0.678841
$196$	0	0
$197$	−14.4286	−1.02800	−0.513999	−	0.857791i	$-0.671837\pi$
$197$	−14.4286	−1.02800	−0.513999	+	0.857791i	$0.671837\pi$
$198$	0	0
$199$	−14.0622	−0.996845	−0.498423	−	0.866934i	$-0.666087\pi$
$199$	−14.0622	−0.996845	−0.498423	+	0.866934i	$0.666087\pi$
$200$	0	0
$201$	−19.6588	−1.38662
$202$	0	0
$203$	−14.7052	−1.03210
$204$	0	0
$205$	−2.42864	−0.169624
$206$	0	0
$207$	−1.18421	−0.0823082
$208$	0	0
$209$	−11.3176	−0.782852
$210$	0	0
$211$	−17.7146	−1.21952	−0.609760	−	0.792586i	$-0.708734\pi$
$211$	−17.7146	−1.21952	−0.609760	+	0.792586i	$0.708734\pi$
$212$	0	0
$213$	−33.3274	−2.28356
$214$	0	0
$215$	1.18421	0.0807623
$216$	0	0
$217$	18.8430	1.27915
$218$	0	0
$219$	1.37778	0.0931020
$220$	0	0
$221$	−14.2636	−0.959476
$222$	0	0
$223$	8.28745	0.554969	0.277484	−	0.960730i	$-0.410499\pi$
$223$	8.28745	0.554969	0.277484	+	0.960730i	$0.410499\pi$
$224$	0	0
$225$	1.90321	0.126881
$226$	0	0
$227$	1.31756	0.0874496	0.0437248	−	0.999044i	$-0.486078\pi$
$227$	1.31756	0.0874496	0.0437248	+	0.999044i	$0.486078\pi$
$228$	0	0
$229$	2.69535	0.178113	0.0890567	−	0.996027i	$-0.471615\pi$
$229$	2.69535	0.178113	0.0890567	+	0.996027i	$0.471615\pi$
$230$	0	0
$231$	−8.65386	−0.569382
$232$	0	0
$233$	−14.9906	−0.982069	−0.491034	−	0.871140i	$-0.663381\pi$
$233$	−14.9906	−0.982069	−0.491034	+	0.871140i	$0.663381\pi$
$234$	0	0
$235$	−2.54125	−0.165773
$236$	0	0
$237$	−20.5161	−1.33266
$238$	0	0
$239$	−30.4909	−1.97229	−0.986145	−	0.165884i	$-0.946952\pi$
$239$	−30.4909	−1.97229	−0.986145	+	0.165884i	$0.946952\pi$
$240$	0	0
$241$	−1.74620	−0.112483	−0.0562413	−	0.998417i	$-0.517912\pi$
$241$	−1.74620	−0.112483	−0.0562413	+	0.998417i	$0.517912\pi$
$242$	0	0
$243$	−17.2652	−1.10756
$244$	0	0
$245$	−1.04593	−0.0668222
$246$	0	0
$247$	35.1655	2.23753
$248$	0	0
$249$	−7.00492	−0.443919
$250$	0	0
$251$	20.5096	1.29455	0.647277	−	0.762255i	$-0.275908\pi$
$251$	20.5096	1.29455	0.647277	+	0.762255i	$0.275908\pi$
$252$	0	0
$253$	0.857279	0.0538967
$254$	0	0
$255$	7.37778	0.462015
$256$	0	0
$257$	12.2810	0.766068	0.383034	−	0.923734i	$-0.374879\pi$
$257$	12.2810	0.766068	0.383034	+	0.923734i	$0.374879\pi$
$258$	0	0
$259$	−2.83654	−0.176254
$260$	0	0
$261$	−9.86665	−0.610730
$262$	0	0
$263$	13.3985	0.826189	0.413094	−	0.910688i	$-0.364448\pi$
$263$	13.3985	0.826189	0.413094	+	0.910688i	$0.364448\pi$
$264$	0	0
$265$	2.56199	0.157382
$266$	0	0
$267$	25.2859	1.54747
$268$	0	0
$269$	−17.6588	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$269$	−17.6588	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$270$	0	0
$271$	−12.9491	−0.786605	−0.393302	−	0.919409i	$-0.628667\pi$
$271$	−12.9491	−0.786605	−0.393302	+	0.919409i	$0.628667\pi$
$272$	0	0
$273$	26.8889	1.62739
$274$	0	0
$275$	−1.37778	−0.0830835
$276$	0	0
$277$	24.2810	1.45890	0.729452	−	0.684032i	$-0.239775\pi$
$277$	24.2810	1.45890	0.729452	+	0.684032i	$0.239775\pi$
$278$	0	0
$279$	12.6430	0.756914
$280$	0	0
$281$	13.2128	0.788209	0.394104	−	0.919066i	$-0.371055\pi$
$281$	13.2128	0.788209	0.394104	+	0.919066i	$0.371055\pi$
$282$	0	0
$283$	8.07313	0.479898	0.239949	−	0.970786i	$-0.422869\pi$
$283$	8.07313	0.479898	0.239949	+	0.970786i	$0.422869\pi$
$284$	0	0
$285$	−18.1891	−1.07743
$286$	0	0
$287$	6.88892	0.406640
$288$	0	0
$289$	−5.89877	−0.346986
$290$	0	0
$291$	23.0923	1.35370
$292$	0	0
$293$	23.7146	1.38542	0.692710	−	0.721217i	$-0.256417\pi$
$293$	23.7146	1.38542	0.692710	+	0.721217i	$0.256417\pi$
$294$	0	0
$295$	−13.8272	−0.805049
$296$	0	0
$297$	3.34614	0.194163
$298$	0	0
$299$	−2.66370	−0.154046
$300$	0	0
$301$	−3.35905	−0.193612
$302$	0	0
$303$	−11.9398	−0.685922
$304$	0	0
$305$	14.4701	0.828557
$306$	0	0
$307$	13.8874	0.792595	0.396298	−	0.918122i	$-0.370295\pi$
$307$	13.8874	0.792595	0.396298	+	0.918122i	$0.370295\pi$
$308$	0	0
$309$	−23.3876	−1.33048
$310$	0	0
$311$	1.69381	0.0960474	0.0480237	−	0.998846i	$-0.484708\pi$
$311$	1.69381	0.0960474	0.0480237	+	0.998846i	$0.484708\pi$
$312$	0	0
$313$	24.7971	1.40161	0.700806	−	0.713352i	$-0.252824\pi$
$313$	24.7971	1.40161	0.700806	+	0.713352i	$0.252824\pi$
$314$	0	0
$315$	−5.39853	−0.304173
$316$	0	0
$317$	5.21279	0.292779	0.146390	−	0.989227i	$-0.453235\pi$
$317$	5.21279	0.292779	0.146390	+	0.989227i	$0.453235\pi$
$318$	0	0
$319$	7.14272	0.399916
$320$	0	0
$321$	16.6780	0.930875
$322$	0	0
$323$	−27.3689	−1.52285
$324$	0	0
$325$	4.28100	0.237467
$326$	0	0
$327$	8.42864	0.466105
$328$	0	0
$329$	7.20834	0.397409
$330$	0	0
$331$	−12.7763	−0.702250	−0.351125	−	0.936329i	$-0.614201\pi$
$331$	−12.7763	−0.702250	−0.351125	+	0.936329i	$0.614201\pi$
$332$	0	0
$333$	−1.90321	−0.104295
$334$	0	0
$335$	8.87802	0.485058
$336$	0	0
$337$	10.7239	0.584169	0.292085	−	0.956392i	$-0.405651\pi$
$337$	10.7239	0.584169	0.292085	+	0.956392i	$0.405651\pi$
$338$	0	0
$339$	29.0923	1.58008
$340$	0	0
$341$	−9.15257	−0.495639
$342$	0	0
$343$	−16.8889	−0.911916
$344$	0	0
$345$	1.37778	0.0741774
$346$	0	0
$347$	26.7239	1.43462	0.717308	−	0.696756i	$-0.245374\pi$
$347$	26.7239	1.43462	0.717308	+	0.696756i	$0.245374\pi$
$348$	0	0
$349$	1.56691	0.0838750	0.0419375	−	0.999120i	$-0.486647\pi$
$349$	1.56691	0.0838750	0.0419375	+	0.999120i	$0.486647\pi$
$350$	0	0
$351$	−10.3970	−0.554951
$352$	0	0
$353$	10.8573	0.577875	0.288937	−	0.957348i	$-0.406698\pi$
$353$	10.8573	0.577875	0.288937	+	0.957348i	$0.406698\pi$
$354$	0	0
$355$	15.0509	0.798816
$356$	0	0
$357$	−20.9273	−1.10759
$358$	0	0
$359$	−1.37778	−0.0727167	−0.0363583	−	0.999339i	$-0.511576\pi$
$359$	−1.37778	−0.0727167	−0.0363583	+	0.999339i	$0.511576\pi$
$360$	0	0
$361$	48.4750	2.55132
$362$	0	0
$363$	−20.1541	−1.05782
$364$	0	0
$365$	−0.622216	−0.0325683
$366$	0	0
$367$	15.5002	0.809106	0.404553	−	0.914515i	$-0.367427\pi$
$367$	15.5002	0.809106	0.404553	+	0.914515i	$0.367427\pi$
$368$	0	0
$369$	4.62222	0.240623
$370$	0	0
$371$	−7.26718	−0.377293
$372$	0	0
$373$	20.6222	1.06778	0.533889	−	0.845555i	$-0.320730\pi$
$373$	20.6222	1.06778	0.533889	+	0.845555i	$0.320730\pi$
$374$	0	0
$375$	−2.21432	−0.114347
$376$	0	0
$377$	−22.1936	−1.14303
$378$	0	0
$379$	5.08250	0.261070	0.130535	−	0.991444i	$-0.458330\pi$
$379$	5.08250	0.261070	0.130535	+	0.991444i	$0.458330\pi$
$380$	0	0
$381$	−15.8622	−0.812645
$382$	0	0
$383$	−2.16193	−0.110470	−0.0552348	−	0.998473i	$-0.517591\pi$
$383$	−2.16193	−0.110470	−0.0552348	+	0.998473i	$0.517591\pi$
$384$	0	0
$385$	3.90813	0.199177
$386$	0	0
$387$	−2.25380	−0.114567
$388$	0	0
$389$	38.3783	1.94586	0.972928	−	0.231110i	$-0.0742359\pi$
$389$	38.3783	1.94586	0.972928	+	0.231110i	$0.0742359\pi$
$390$	0	0
$391$	2.07313	0.104843
$392$	0	0
$393$	−34.7511	−1.75296
$394$	0	0
$395$	9.26517	0.466182
$396$	0	0
$397$	−27.6227	−1.38634	−0.693172	−	0.720773i	$-0.743787\pi$
$397$	−27.6227	−1.38634	−0.693172	+	0.720773i	$0.743787\pi$
$398$	0	0
$399$	51.5941	2.58294
$400$	0	0
$401$	6.38715	0.318959	0.159480	−	0.987201i	$-0.449018\pi$
$401$	6.38715	0.318959	0.159480	+	0.987201i	$0.449018\pi$
$402$	0	0
$403$	28.4385	1.41662
$404$	0	0
$405$	11.0874	0.550938
$406$	0	0
$407$	1.37778	0.0682942
$408$	0	0
$409$	26.8988	1.33006	0.665029	−	0.746817i	$-0.268419\pi$
$409$	26.8988	1.33006	0.665029	+	0.746817i	$0.268419\pi$
$410$	0	0
$411$	−24.3368	−1.20044
$412$	0	0
$413$	39.2212	1.92995
$414$	0	0
$415$	3.16346	0.155288
$416$	0	0
$417$	30.5718	1.49711
$418$	0	0
$419$	−16.4572	−0.803988	−0.401994	−	0.915642i	$-0.631683\pi$
$419$	−16.4572	−0.803988	−0.401994	+	0.915642i	$0.631683\pi$
$420$	0	0
$421$	23.6860	1.15438	0.577192	−	0.816608i	$-0.304148\pi$
$421$	23.6860	1.15438	0.577192	+	0.816608i	$0.304148\pi$
$422$	0	0
$423$	4.83654	0.235160
$424$	0	0
$425$	−3.33185	−0.161619
$426$	0	0
$427$	−41.0450	−1.98631
$428$	0	0
$429$	−13.0607	−0.630577
$430$	0	0
$431$	16.3575	0.787914	0.393957	−	0.919129i	$-0.371106\pi$
$431$	16.3575	0.787914	0.393957	+	0.919129i	$0.371106\pi$
$432$	0	0
$433$	7.46965	0.358968	0.179484	−	0.983761i	$-0.442557\pi$
$433$	7.46965	0.358968	0.179484	+	0.983761i	$0.442557\pi$
$434$	0	0
$435$	11.4795	0.550400
$436$	0	0
$437$	−5.11108	−0.244496
$438$	0	0
$439$	30.7862	1.46934	0.734672	−	0.678423i	$-0.237336\pi$
$439$	30.7862	1.46934	0.734672	+	0.678423i	$0.237336\pi$
$440$	0	0
$441$	1.99063	0.0947920
$442$	0	0
$443$	32.2973	1.53449	0.767245	−	0.641354i	$-0.221627\pi$
$443$	32.2973	1.53449	0.767245	+	0.641354i	$0.221627\pi$
$444$	0	0
$445$	−11.4193	−0.541325
$446$	0	0
$447$	5.77478	0.273138
$448$	0	0
$449$	−29.3876	−1.38689	−0.693444	−	0.720511i	$-0.743908\pi$
$449$	−29.3876	−1.38689	−0.693444	+	0.720511i	$0.743908\pi$
$450$	0	0
$451$	−3.34614	−0.157564
$452$	0	0
$453$	−37.0321	−1.73992
$454$	0	0
$455$	−12.1432	−0.569282
$456$	0	0
$457$	−31.3689	−1.46737	−0.733687	−	0.679487i	$-0.762202\pi$
$457$	−31.3689	−1.46737	−0.733687	+	0.679487i	$0.762202\pi$
$458$	0	0
$459$	8.09187	0.377696
$460$	0	0
$461$	−2.90766	−0.135423	−0.0677116	−	0.997705i	$-0.521570\pi$
$461$	−2.90766	−0.135423	−0.0677116	+	0.997705i	$0.521570\pi$
$462$	0	0
$463$	−32.7971	−1.52421	−0.762104	−	0.647454i	$-0.775834\pi$
$463$	−32.7971	−1.52421	−0.762104	+	0.647454i	$0.775834\pi$
$464$	0	0
$465$	−14.7096	−0.682143
$466$	0	0
$467$	−33.9911	−1.57292	−0.786460	−	0.617641i	$-0.788089\pi$
$467$	−33.9911	−1.57292	−0.786460	+	0.617641i	$0.788089\pi$
$468$	0	0
$469$	−25.1828	−1.16283
$470$	0	0
$471$	−0.520505	−0.0239836
$472$	0	0
$473$	1.63158	0.0750203
$474$	0	0
$475$	8.21432	0.376899
$476$	0	0
$477$	−4.87601	−0.223257
$478$	0	0
$479$	−31.2652	−1.42854	−0.714271	−	0.699869i	$-0.753242\pi$
$479$	−31.2652	−1.42854	−0.714271	+	0.699869i	$0.753242\pi$
$480$	0	0
$481$	−4.28100	−0.195197
$482$	0	0
$483$	−3.90813	−0.177826
$484$	0	0
$485$	−10.4286	−0.473540
$486$	0	0
$487$	39.8064	1.80380	0.901901	−	0.431944i	$-0.142172\pi$
$487$	39.8064	1.80380	0.901901	+	0.431944i	$0.142172\pi$
$488$	0	0
$489$	−37.5526	−1.69819
$490$	0	0
$491$	40.6450	1.83428	0.917141	−	0.398563i	$-0.130491\pi$
$491$	40.6450	1.83428	0.917141	+	0.398563i	$0.130491\pi$
$492$	0	0
$493$	17.2730	0.777937
$494$	0	0
$495$	2.62222	0.117860
$496$	0	0
$497$	−42.6923	−1.91501
$498$	0	0
$499$	27.3254	1.22325	0.611626	−	0.791147i	$-0.290516\pi$
$499$	27.3254	1.22325	0.611626	+	0.791147i	$0.290516\pi$
$500$	0	0
$501$	−10.5303	−0.470461
$502$	0	0
$503$	22.6953	1.01194	0.505968	−	0.862552i	$-0.331135\pi$
$503$	22.6953	1.01194	0.505968	+	0.862552i	$0.331135\pi$
$504$	0	0
$505$	5.39207	0.239944
$506$	0	0
$507$	11.7955	0.523858
$508$	0	0
$509$	−2.59057	−0.114825	−0.0574126	−	0.998351i	$-0.518285\pi$
$509$	−2.59057	−0.114825	−0.0574126	+	0.998351i	$0.518285\pi$
$510$	0	0
$511$	1.76494	0.0780762
$512$	0	0
$513$	−19.9496	−0.880797
$514$	0	0
$515$	10.5620	0.465417
$516$	0	0
$517$	−3.50129	−0.153987
$518$	0	0
$519$	51.5625	2.26334
$520$	0	0
$521$	−22.4242	−0.982422	−0.491211	−	0.871041i	$-0.663446\pi$
$521$	−22.4242	−0.982422	−0.491211	+	0.871041i	$0.663446\pi$
$522$	0	0
$523$	12.1432	0.530985	0.265492	−	0.964113i	$-0.414465\pi$
$523$	12.1432	0.530985	0.265492	+	0.964113i	$0.414465\pi$
$524$	0	0
$525$	6.28100	0.274125
$526$	0	0
$527$	−22.1334	−0.964144
$528$	0	0
$529$	−22.6128	−0.983167
$530$	0	0
$531$	26.3160	1.14202
$532$	0	0
$533$	10.3970	0.450344
$534$	0	0
$535$	−7.53188	−0.325632
$536$	0	0
$537$	19.1383	0.825878
$538$	0	0
$539$	−1.44107	−0.0620713
$540$	0	0
$541$	25.5081	1.09668	0.548339	−	0.836256i	$-0.315260\pi$
$541$	25.5081	1.09668	0.548339	+	0.836256i	$0.315260\pi$
$542$	0	0
$543$	−11.2859	−0.484325
$544$	0	0
$545$	−3.80642	−0.163049
$546$	0	0
$547$	−33.6829	−1.44018	−0.720089	−	0.693882i	$-0.755899\pi$
$547$	−33.6829	−1.44018	−0.720089	+	0.693882i	$0.755899\pi$
$548$	0	0
$549$	−27.5397	−1.17537
$550$	0	0
$551$	−42.5847	−1.81417
$552$	0	0
$553$	−26.2810	−1.11758
$554$	0	0
$555$	2.21432	0.0939926
$556$	0	0
$557$	5.52543	0.234120	0.117060	−	0.993125i	$-0.462653\pi$
$557$	5.52543	0.234120	0.117060	+	0.993125i	$0.462653\pi$
$558$	0	0
$559$	−5.06959	−0.214421
$560$	0	0
$561$	10.1650	0.429166
$562$	0	0
$563$	40.6133	1.71165	0.855824	−	0.517268i	$-0.173051\pi$
$563$	40.6133	1.71165	0.855824	+	0.517268i	$0.173051\pi$
$564$	0	0
$565$	−13.1383	−0.552731
$566$	0	0
$567$	−31.4499	−1.32077
$568$	0	0
$569$	−23.1941	−0.972345	−0.486173	−	0.873863i	$-0.661607\pi$
$569$	−23.1941	−0.972345	−0.486173	+	0.873863i	$0.661607\pi$
$570$	0	0
$571$	1.21585	0.0508818	0.0254409	−	0.999676i	$-0.491901\pi$
$571$	1.21585	0.0508818	0.0254409	+	0.999676i	$0.491901\pi$
$572$	0	0
$573$	11.6588	0.487053
$574$	0	0
$575$	−0.622216	−0.0259482
$576$	0	0
$577$	13.4050	0.558057	0.279028	−	0.960283i	$-0.409988\pi$
$577$	13.4050	0.558057	0.279028	+	0.960283i	$0.409988\pi$
$578$	0	0
$579$	−1.08250	−0.0449871
$580$	0	0
$581$	−8.97328	−0.372274
$582$	0	0
$583$	3.52987	0.146192
$584$	0	0
$585$	−8.14764	−0.336864
$586$	0	0
$587$	29.0923	1.20077	0.600385	−	0.799711i	$-0.295014\pi$
$587$	29.0923	1.20077	0.600385	+	0.799711i	$0.295014\pi$
$588$	0	0
$589$	54.5674	2.24841
$590$	0	0
$591$	−31.9496	−1.31423
$592$	0	0
$593$	29.2543	1.20133	0.600665	−	0.799501i	$-0.294903\pi$
$593$	29.2543	1.20133	0.600665	+	0.799501i	$0.294903\pi$
$594$	0	0
$595$	9.45091	0.387450
$596$	0	0
$597$	−31.1383	−1.27440
$598$	0	0
$599$	−29.1941	−1.19284	−0.596418	−	0.802674i	$-0.703410\pi$
$599$	−29.1941	−1.19284	−0.596418	+	0.802674i	$0.703410\pi$
$600$	0	0
$601$	−3.02366	−0.123338	−0.0616688	−	0.998097i	$-0.519642\pi$
$601$	−3.02366	−0.123338	−0.0616688	+	0.998097i	$0.519642\pi$
$602$	0	0
$603$	−16.8968	−0.688089
$604$	0	0
$605$	9.10171	0.370037
$606$	0	0
$607$	−23.6731	−0.960860	−0.480430	−	0.877033i	$-0.659519\pi$
$607$	−23.6731	−0.960860	−0.480430	+	0.877033i	$0.659519\pi$
$608$	0	0
$609$	−32.5620	−1.31948
$610$	0	0
$611$	10.8791	0.440120
$612$	0	0
$613$	30.0830	1.21504	0.607520	−	0.794304i	$-0.292164\pi$
$613$	30.0830	1.21504	0.607520	+	0.794304i	$0.292164\pi$
$614$	0	0
$615$	−5.37778	−0.216853
$616$	0	0
$617$	−33.8163	−1.36139	−0.680696	−	0.732566i	$-0.738322\pi$
$617$	−33.8163	−1.36139	−0.680696	+	0.732566i	$0.738322\pi$
$618$	0	0
$619$	18.9590	0.762026	0.381013	−	0.924570i	$-0.375575\pi$
$619$	18.9590	0.762026	0.381013	+	0.924570i	$0.375575\pi$
$620$	0	0
$621$	1.51114	0.0606399
$622$	0	0
$623$	32.3912	1.29772
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−25.0607	−1.00083
$628$	0	0
$629$	3.33185	0.132850
$630$	0	0
$631$	−1.25533	−0.0499739	−0.0249870	−	0.999688i	$-0.507954\pi$
$631$	−1.25533	−0.0499739	−0.0249870	+	0.999688i	$0.507954\pi$
$632$	0	0
$633$	−39.2257	−1.55908
$634$	0	0
$635$	7.16346	0.284273
$636$	0	0
$637$	4.47764	0.177410
$638$	0	0
$639$	−28.6450	−1.13318
$640$	0	0
$641$	19.7891	0.781621	0.390811	−	0.920471i	$-0.372195\pi$
$641$	19.7891	0.781621	0.390811	+	0.920471i	$0.372195\pi$
$642$	0	0
$643$	2.20342	0.0868944	0.0434472	−	0.999056i	$-0.486166\pi$
$643$	2.20342	0.0868944	0.0434472	+	0.999056i	$0.486166\pi$
$644$	0	0
$645$	2.62222	0.103250
$646$	0	0
$647$	−7.64449	−0.300536	−0.150268	−	0.988645i	$-0.548014\pi$
$647$	−7.64449	−0.300536	−0.150268	+	0.988645i	$0.548014\pi$
$648$	0	0
$649$	−19.0509	−0.747811
$650$	0	0
$651$	41.7244	1.63531
$652$	0	0
$653$	−44.7753	−1.75219	−0.876096	−	0.482137i	$-0.839861\pi$
$653$	−44.7753	−1.75219	−0.876096	+	0.482137i	$0.839861\pi$
$654$	0	0
$655$	15.6938	0.613208
$656$	0	0
$657$	1.18421	0.0462004
$658$	0	0
$659$	−22.0098	−0.857382	−0.428691	−	0.903451i	$-0.641025\pi$
$659$	−22.0098	−0.857382	−0.428691	+	0.903451i	$0.641025\pi$
$660$	0	0
$661$	14.4701	0.562823	0.281411	−	0.959587i	$-0.409197\pi$
$661$	14.4701	0.562823	0.281411	+	0.959587i	$0.409197\pi$
$662$	0	0
$663$	−31.5843	−1.22663
$664$	0	0
$665$	−23.3002	−0.903543
$666$	0	0
$667$	3.22570	0.124899
$668$	0	0
$669$	18.3511	0.709493
$670$	0	0
$671$	19.9367	0.769648
$672$	0	0
$673$	−10.3872	−0.400395	−0.200198	−	0.979756i	$-0.564158\pi$
$673$	−10.3872	−0.400395	−0.200198	+	0.979756i	$0.564158\pi$
$674$	0	0
$675$	−2.42864	−0.0934784
$676$	0	0
$677$	−19.8479	−0.762817	−0.381409	−	0.924407i	$-0.624561\pi$
$677$	−19.8479	−0.762817	−0.381409	+	0.924407i	$0.624561\pi$
$678$	0	0
$679$	29.5812	1.13522
$680$	0	0
$681$	2.91750	0.111799
$682$	0	0
$683$	5.09234	0.194853	0.0974265	−	0.995243i	$-0.468939\pi$
$683$	5.09234	0.194853	0.0974265	+	0.995243i	$0.468939\pi$
$684$	0	0
$685$	10.9906	0.419930
$686$	0	0
$687$	5.96836	0.227707
$688$	0	0
$689$	−10.9679	−0.417843
$690$	0	0
$691$	41.5812	1.58182	0.790912	−	0.611930i	$-0.209607\pi$
$691$	41.5812	1.58182	0.790912	+	0.611930i	$0.209607\pi$
$692$	0	0
$693$	−7.43801	−0.282547
$694$	0	0
$695$	−13.8064	−0.523708
$696$	0	0
$697$	−8.09187	−0.306501
$698$	0	0
$699$	−33.1941	−1.25551
$700$	0	0
$701$	−20.6637	−0.780457	−0.390229	−	0.920718i	$-0.627604\pi$
$701$	−20.6637	−0.780457	−0.390229	+	0.920718i	$0.627604\pi$
$702$	0	0
$703$	−8.21432	−0.309809
$704$	0	0
$705$	−5.62714	−0.211930
$706$	0	0
$707$	−15.2948	−0.575221
$708$	0	0
$709$	14.4572	0.542952	0.271476	−	0.962445i	$-0.412488\pi$
$709$	14.4572	0.542952	0.271476	+	0.962445i	$0.412488\pi$
$710$	0	0
$711$	−17.6336	−0.661311
$712$	0	0
$713$	−4.13335	−0.154795
$714$	0	0
$715$	5.89829	0.220584
$716$	0	0
$717$	−67.5165	−2.52145
$718$	0	0
$719$	24.8988	0.928567	0.464284	−	0.885687i	$-0.346312\pi$
$719$	24.8988	0.928567	0.464284	+	0.885687i	$0.346312\pi$
$720$	0	0
$721$	−29.9595	−1.11575
$722$	0	0
$723$	−3.86665	−0.143802
$724$	0	0
$725$	−5.18421	−0.192537
$726$	0	0
$727$	−38.7239	−1.43619	−0.718095	−	0.695945i	$-0.754986\pi$
$727$	−38.7239	−1.43619	−0.718095	+	0.695945i	$0.754986\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	0	0
$731$	3.94561	0.145934
$732$	0	0
$733$	29.6958	1.09684	0.548420	−	0.836203i	$-0.315229\pi$
$733$	29.6958	1.09684	0.548420	+	0.836203i	$0.315229\pi$
$734$	0	0
$735$	−2.31603	−0.0854281
$736$	0	0
$737$	12.2320	0.450572
$738$	0	0
$739$	−7.49240	−0.275612	−0.137806	−	0.990459i	$-0.544005\pi$
$739$	−7.49240	−0.275612	−0.137806	+	0.990459i	$0.544005\pi$
$740$	0	0
$741$	77.8676	2.86054
$742$	0	0
$743$	−12.0306	−0.441359	−0.220680	−	0.975346i	$-0.570828\pi$
$743$	−12.0306	−0.441359	−0.220680	+	0.975346i	$0.570828\pi$
$744$	0	0
$745$	−2.60793	−0.0955470
$746$	0	0
$747$	−6.02074	−0.220287
$748$	0	0
$749$	21.3644	0.780640
$750$	0	0
$751$	−36.0415	−1.31517	−0.657586	−	0.753379i	$-0.728422\pi$
$751$	−36.0415	−1.31517	−0.657586	+	0.753379i	$0.728422\pi$
$752$	0	0
$753$	45.4148	1.65501
$754$	0	0
$755$	16.7239	0.608646
$756$	0	0
$757$	−41.4291	−1.50577	−0.752883	−	0.658154i	$-0.771338\pi$
$757$	−41.4291	−1.50577	−0.752883	+	0.658154i	$0.771338\pi$
$758$	0	0
$759$	1.89829	0.0689036
$760$	0	0
$761$	−16.4889	−0.597721	−0.298860	−	0.954297i	$-0.596606\pi$
$761$	−16.4889	−0.597721	−0.298860	+	0.954297i	$0.596606\pi$
$762$	0	0
$763$	10.7971	0.390880
$764$	0	0
$765$	6.34122	0.229267
$766$	0	0
$767$	59.1941	2.13737
$768$	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
$771$	27.1941	0.979370
$772$	0	0
$773$	−27.9398	−1.00492	−0.502462	−	0.864599i	$-0.667572\pi$
$773$	−27.9398	−1.00492	−0.502462	+	0.864599i	$0.667572\pi$
$774$	0	0
$775$	6.64296	0.238622
$776$	0	0
$777$	−6.28100	−0.225329
$778$	0	0
$779$	19.9496	0.714769
$780$	0	0
$781$	20.7368	0.742022
$782$	0	0
$783$	12.5906	0.449950
$784$	0	0
$785$	0.235063	0.00838977
$786$	0	0
$787$	−6.27454	−0.223663	−0.111832	−	0.993727i	$-0.535672\pi$
$787$	−6.27454	−0.223663	−0.111832	+	0.993727i	$0.535672\pi$
$788$	0	0
$789$	29.6686	1.05623
$790$	0	0
$791$	37.2672	1.32507
$792$	0	0
$793$	−61.9466	−2.19979
$794$	0	0
$795$	5.67307	0.201203
$796$	0	0
$797$	40.9545	1.45068	0.725342	−	0.688388i	$-0.241682\pi$
$797$	40.9545	1.45068	0.725342	+	0.688388i	$0.241682\pi$
$798$	0	0
$799$	−8.46706	−0.299543
$800$	0	0
$801$	21.7333	0.767908
$802$	0	0
$803$	−0.857279	−0.0302527
$804$	0	0
$805$	1.76494	0.0622058
$806$	0	0
$807$	−39.1022	−1.37646
$808$	0	0
$809$	−20.4889	−0.720350	−0.360175	−	0.932885i	$-0.617283\pi$
$809$	−20.4889	−0.720350	−0.360175	+	0.932885i	$0.617283\pi$
$810$	0	0
$811$	−12.8573	−0.451480	−0.225740	−	0.974188i	$-0.572480\pi$
$811$	−12.8573	−0.451480	−0.225740	+	0.974188i	$0.572480\pi$
$812$	0	0
$813$	−28.6735	−1.00563
$814$	0	0
$815$	16.9590	0.594047
$816$	0	0
$817$	−9.72746	−0.340321
$818$	0	0
$819$	23.1111	0.807566
$820$	0	0
$821$	−7.89829	−0.275652	−0.137826	−	0.990456i	$-0.544012\pi$
$821$	−7.89829	−0.275652	−0.137826	+	0.990456i	$0.544012\pi$
$822$	0	0
$823$	−16.3763	−0.570840	−0.285420	−	0.958402i	$-0.592133\pi$
$823$	−16.3763	−0.570840	−0.285420	+	0.958402i	$0.592133\pi$
$824$	0	0
$825$	−3.05086	−0.106217
$826$	0	0
$827$	1.05086	0.0365418	0.0182709	−	0.999833i	$-0.494184\pi$
$827$	1.05086	0.0365418	0.0182709	+	0.999833i	$0.494184\pi$
$828$	0	0
$829$	35.7975	1.24330	0.621650	−	0.783295i	$-0.286463\pi$
$829$	35.7975	1.24330	0.621650	+	0.783295i	$0.286463\pi$
$830$	0	0
$831$	53.7659	1.86512
$832$	0	0
$833$	−3.48489	−0.120744
$834$	0	0
$835$	4.75557	0.164573
$836$	0	0
$837$	−16.1334	−0.557650
$838$	0	0
$839$	17.9813	0.620782	0.310391	−	0.950609i	$-0.399540\pi$
$839$	17.9813	0.620782	0.310391	+	0.950609i	$0.399540\pi$
$840$	0	0
$841$	−2.12399	−0.0732409
$842$	0	0
$843$	29.2573	1.00768
$844$	0	0
$845$	−5.32693	−0.183252
$846$	0	0
$847$	−25.8173	−0.887094
$848$	0	0
$849$	17.8765	0.613520
$850$	0	0
$851$	0.622216	0.0213293
$852$	0	0
$853$	−45.9210	−1.57231	−0.786153	−	0.618032i	$-0.787930\pi$
$853$	−45.9210	−1.57231	−0.786153	+	0.618032i	$0.787930\pi$
$854$	0	0
$855$	−15.6336	−0.534658
$856$	0	0
$857$	48.5303	1.65777	0.828883	−	0.559423i	$-0.188977\pi$
$857$	48.5303	1.65777	0.828883	+	0.559423i	$0.188977\pi$
$858$	0	0
$859$	−32.3861	−1.10500	−0.552500	−	0.833513i	$-0.686326\pi$
$859$	−32.3861	−1.10500	−0.552500	+	0.833513i	$0.686326\pi$
$860$	0	0
$861$	15.2543	0.519864
$862$	0	0
$863$	−11.2968	−0.384548	−0.192274	−	0.981341i	$-0.561586\pi$
$863$	−11.2968	−0.384548	−0.192274	+	0.981341i	$0.561586\pi$
$864$	0	0
$865$	−23.2859	−0.791745
$866$	0	0
$867$	−13.0618	−0.443600
$868$	0	0
$869$	12.7654	0.433037
$870$	0	0
$871$	−38.0068	−1.28781
$872$	0	0
$873$	19.8479	0.671750
$874$	0	0
$875$	−2.83654	−0.0958924
$876$	0	0
$877$	40.9086	1.38139	0.690693	−	0.723148i	$-0.257306\pi$
$877$	40.9086	1.38139	0.690693	+	0.723148i	$0.257306\pi$
$878$	0	0
$879$	52.5116	1.77117
$880$	0	0
$881$	−20.3225	−0.684682	−0.342341	−	0.939576i	$-0.611220\pi$
$881$	−20.3225	−0.684682	−0.342341	+	0.939576i	$0.611220\pi$
$882$	0	0
$883$	−36.1561	−1.21675	−0.608375	−	0.793650i	$-0.708178\pi$
$883$	−36.1561	−1.21675	−0.608375	+	0.793650i	$0.708178\pi$
$884$	0	0
$885$	−30.6178	−1.02921
$886$	0	0
$887$	9.16346	0.307679	0.153840	−	0.988096i	$-0.450836\pi$
$887$	9.16346	0.307679	0.153840	+	0.988096i	$0.450836\pi$
$888$	0	0
$889$	−20.3194	−0.681492
$890$	0	0
$891$	15.2761	0.511768
$892$	0	0
$893$	20.8746	0.698543
$894$	0	0
$895$	−8.64296	−0.288902
$896$	0	0
$897$	−5.89829	−0.196938
$898$	0	0
$899$	−34.4385	−1.14859
$900$	0	0
$901$	8.53618	0.284381
$902$	0	0
$903$	−7.43801	−0.247521
$904$	0	0
$905$	5.09679	0.169423
$906$	0	0
$907$	8.07313	0.268064	0.134032	−	0.990977i	$-0.457208\pi$
$907$	8.07313	0.268064	0.134032	+	0.990977i	$0.457208\pi$
$908$	0	0
$909$	−10.2623	−0.340378
$910$	0	0
$911$	−17.9891	−0.596005	−0.298003	−	0.954565i	$-0.596320\pi$
$911$	−17.9891	−0.596005	−0.298003	+	0.954565i	$0.596320\pi$
$912$	0	0
$913$	4.35857	0.144248
$914$	0	0
$915$	32.0415	1.05926
$916$	0	0
$917$	−44.5161	−1.47005
$918$	0	0
$919$	−50.9926	−1.68209	−0.841046	−	0.540964i	$-0.818060\pi$
$919$	−50.9926	−1.68209	−0.841046	+	0.540964i	$0.818060\pi$
$920$	0	0
$921$	30.7511	1.01328
$922$	0	0
$923$	−64.4327	−2.12083
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−20.1017	−0.660227
$928$	0	0
$929$	−42.7150	−1.40143	−0.700717	−	0.713439i	$-0.747137\pi$
$929$	−42.7150	−1.40143	−0.700717	+	0.713439i	$0.747137\pi$
$930$	0	0
$931$	8.59163	0.281579
$932$	0	0
$933$	3.75065	0.122791
$934$	0	0
$935$	−4.59057	−0.150128
$936$	0	0
$937$	49.1467	1.60555	0.802777	−	0.596279i	$-0.203355\pi$
$937$	49.1467	1.60555	0.802777	+	0.596279i	$0.203355\pi$
$938$	0	0
$939$	54.9086	1.79187
$940$	0	0
$941$	−8.11906	−0.264674	−0.132337	−	0.991205i	$-0.542248\pi$
$941$	−8.11906	−0.264674	−0.132337	+	0.991205i	$0.542248\pi$
$942$	0	0
$943$	−1.51114	−0.0492094
$944$	0	0
$945$	6.88892	0.224097
$946$	0	0
$947$	−39.6227	−1.28756	−0.643782	−	0.765209i	$-0.722636\pi$
$947$	−39.6227	−1.28756	−0.643782	+	0.765209i	$0.722636\pi$
$948$	0	0
$949$	2.66370	0.0864675
$950$	0	0
$951$	11.5428	0.374300
$952$	0	0
$953$	54.4068	1.76241	0.881205	−	0.472734i	$-0.156733\pi$
$953$	54.4068	1.76241	0.881205	+	0.472734i	$0.156733\pi$
$954$	0	0
$955$	−5.26517	−0.170377
$956$	0	0
$957$	15.8163	0.511267
$958$	0	0
$959$	−31.1753	−1.00670
$960$	0	0
$961$	13.1289	0.423513
$962$	0	0
$963$	14.3348	0.461931
$964$	0	0
$965$	0.488863	0.0157370
$966$	0	0
$967$	−53.2543	−1.71254	−0.856271	−	0.516527i	$-0.827225\pi$
$967$	−53.2543	−1.71254	−0.856271	+	0.516527i	$0.827225\pi$
$968$	0	0
$969$	−60.6035	−1.94686
$970$	0	0
$971$	−53.0232	−1.70160	−0.850798	−	0.525493i	$-0.823881\pi$
$971$	−53.0232	−1.70160	−0.850798	+	0.525493i	$0.823881\pi$
$972$	0	0
$973$	39.1624	1.25549
$974$	0	0
$975$	9.47949	0.303587
$976$	0	0
$977$	20.1432	0.644438	0.322219	−	0.946665i	$-0.395571\pi$
$977$	20.1432	0.644438	0.322219	+	0.946665i	$0.395571\pi$
$978$	0	0
$979$	−15.7333	−0.502838
$980$	0	0
$981$	7.24443	0.231297
$982$	0	0
$983$	24.1541	0.770396	0.385198	−	0.922834i	$-0.374133\pi$
$983$	24.1541	0.770396	0.385198	+	0.922834i	$0.374133\pi$
$984$	0	0
$985$	14.4286	0.459735
$986$	0	0
$987$	15.9616	0.508063
$988$	0	0
$989$	0.736833	0.0234299
$990$	0	0
$991$	−27.6149	−0.877215	−0.438607	−	0.898679i	$-0.644528\pi$
$991$	−27.6149	−0.877215	−0.438607	+	0.898679i	$0.644528\pi$
$992$	0	0
$993$	−28.2908	−0.897783
$994$	0	0
$995$	14.0622	0.445803
$996$	0	0
$997$	14.2034	0.449827	0.224913	−	0.974379i	$-0.427790\pi$
$997$	14.2034	0.449827	0.224913	+	0.974379i	$0.427790\pi$
$998$	0	0
$999$	2.42864	0.0768388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.2.a.e.1.3	✓	3
3.2	odd	2		6660.2.a.q.1.2		3
4.3	odd	2		2960.2.a.t.1.1		3
5.2	odd	4		3700.2.d.h.149.1		6
5.3	odd	4		3700.2.d.h.149.6		6
5.4	even	2		3700.2.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.e.1.3	✓	3	1.1	even	1	trivial
2960.2.a.t.1.1		3	4.3	odd	2
3700.2.a.i.1.1		3	5.4	even	2
3700.2.d.h.149.1		6	5.2	odd	4
3700.2.d.h.149.6		6	5.3	odd	4
6660.2.a.q.1.2		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)

\(f(q)\)	\(=\)	\(q+2.21432 q^{3} -1.00000 q^{5} +2.83654 q^{7} +1.90321 q^{9} -1.37778 q^{11} +4.28100 q^{13} -2.21432 q^{15} -3.33185 q^{17} +8.21432 q^{19} +6.28100 q^{21} -0.622216 q^{23} +1.00000 q^{25} -2.42864 q^{27} -5.18421 q^{29} +6.64296 q^{31} -3.05086 q^{33} -2.83654 q^{35} -1.00000 q^{37} +9.47949 q^{39} +2.42864 q^{41} -1.18421 q^{43} -1.90321 q^{45} +2.54125 q^{47} +1.04593 q^{49} -7.37778 q^{51} -2.56199 q^{53} +1.37778 q^{55} +18.1891 q^{57} +13.8272 q^{59} -14.4701 q^{61} +5.39853 q^{63} -4.28100 q^{65} -8.87802 q^{67} -1.37778 q^{69} -15.0509 q^{71} +0.622216 q^{73} +2.21432 q^{75} -3.90813 q^{77} -9.26517 q^{79} -11.0874 q^{81} -3.16346 q^{83} +3.33185 q^{85} -11.4795 q^{87} +11.4193 q^{89} +12.1432 q^{91} +14.7096 q^{93} -8.21432 q^{95} +10.4286 q^{97} -2.62222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7} - q^{9} - 4 q^{11} + 6 q^{13} + 10 q^{17} + 18 q^{19} + 12 q^{21} - 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 3 q^{37} + 2 q^{39} - 6 q^{41} + 10 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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