LMFDB - Embedded newform 2912.2.a.t.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1025428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 2 $ x^5 - 8x^3 + 13x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.44016$ of defining polynomial
Character	$\chi$	$=$	2912.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.44016 q^{3} +1.05143 q^{5} -1.00000 q^{7} +8.83471 q^{9} +5.76901 q^{11} -1.00000 q^{13} +3.61708 q^{15} +2.81463 q^{17} -1.60628 q^{19} -3.44016 q^{21} -3.19118 q^{23} -3.89450 q^{25} +20.0723 q^{27} -0.0108018 q^{29} -1.42936 q^{31} +19.8463 q^{33} -1.05143 q^{35} -6.32386 q^{37} -3.44016 q^{39} -5.75683 q^{41} -7.02589 q^{43} +9.28905 q^{45} +7.30622 q^{47} +1.00000 q^{49} +9.68277 q^{51} +13.8034 q^{53} +6.06569 q^{55} -5.52585 q^{57} -8.08949 q^{59} -5.75683 q^{61} -8.83471 q^{63} -1.05143 q^{65} -9.71156 q^{67} -10.9782 q^{69} +3.61708 q^{71} -2.95576 q^{73} -13.3977 q^{75} -5.76901 q^{77} -11.7543 q^{79} +42.5479 q^{81} -3.88241 q^{83} +2.95937 q^{85} -0.0371598 q^{87} +12.0096 q^{89} +1.00000 q^{91} -4.91722 q^{93} -1.68888 q^{95} -10.7833 q^{97} +50.9675 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 3 q^{5} - 5 q^{7} + 6 q^{9} + 5 q^{11} - 5 q^{13} + 4 q^{17} + 7 q^{19} - 5 q^{21} - 4 q^{23} + 2 q^{25} + 23 q^{27} + 3 q^{29} + 2 q^{31} + 17 q^{33} + 3 q^{35} - q^{37} - 5 q^{39} - 7 q^{41}+ \cdots + 40 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 3 * q^5 - 5 * q^7 + 6 * q^9 + 5 * q^11 - 5 * q^13 + 4 * q^17 + 7 * q^19 - 5 * q^21 - 4 * q^23 + 2 * q^25 + 23 * q^27 + 3 * q^29 + 2 * q^31 + 17 * q^33 + 3 * q^35 - q^37 - 5 * q^39 - 7 * q^41 + 15 * q^43 - q^45 + 6 * q^47 + 5 * q^49 + 16 * q^51 + 11 * q^53 + 16 * q^55 + 18 * q^57 + 20 * q^59 - 7 * q^61 - 6 * q^63 + 3 * q^65 + 11 * q^67 + 5 * q^69 - 2 * q^73 + 19 * q^75 - 5 * q^77 + 10 * q^79 + 33 * q^81 + 9 * q^83 + 20 * q^85 + 2 * q^87 + 11 * q^89 + 5 * q^91 - 13 * q^93 - 11 * q^95 - 24 * q^97 + 40 * 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
$3$	3.44016	1.98618	0.993089	−	0.117365i	$-0.0374447\pi$
$3$	3.44016	1.98618	0.993089	+	0.117365i	$0.0374447\pi$
$4$	0	0
$5$	1.05143	0.470212	0.235106	−	0.971970i	$-0.424456\pi$
$5$	1.05143	0.470212	0.235106	+	0.971970i	$0.424456\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.83471	2.94490
$10$	0	0
$11$	5.76901	1.73942	0.869711	−	0.493561i	$-0.164305\pi$
$11$	5.76901	1.73942	0.869711	+	0.493561i	$0.164305\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.61708	0.933925
$16$	0	0
$17$	2.81463	0.682647	0.341324	−	0.939946i	$-0.389125\pi$
$17$	2.81463	0.682647	0.341324	+	0.939946i	$0.389125\pi$
$18$	0	0
$19$	−1.60628	−0.368505	−0.184252	−	0.982879i	$-0.558986\pi$
$19$	−1.60628	−0.368505	−0.184252	+	0.982879i	$0.558986\pi$
$20$	0	0
$21$	−3.44016	−0.750705
$22$	0	0
$23$	−3.19118	−0.665408	−0.332704	−	0.943031i	$-0.607961\pi$
$23$	−3.19118	−0.665408	−0.332704	+	0.943031i	$0.607961\pi$
$24$	0	0
$25$	−3.89450	−0.778900
$26$	0	0
$27$	20.0723	3.86292
$28$	0	0
$29$	−0.0108018	−0.00200584	−0.00100292	−	0.999999i	$-0.500319\pi$
$29$	−0.0108018	−0.00200584	−0.00100292	+	0.999999i	$0.500319\pi$
$30$	0	0
$31$	−1.42936	−0.256720	−0.128360	−	0.991728i	$-0.540971\pi$
$31$	−1.42936	−0.256720	−0.128360	+	0.991728i	$0.540971\pi$
$32$	0	0
$33$	19.8463	3.45480
$34$	0	0
$35$	−1.05143	−0.177724
$36$	0	0
$37$	−6.32386	−1.03964	−0.519818	−	0.854277i	$-0.674000\pi$
$37$	−6.32386	−1.03964	−0.519818	+	0.854277i	$0.674000\pi$
$38$	0	0
$39$	−3.44016	−0.550867
$40$	0	0
$41$	−5.75683	−0.899066	−0.449533	−	0.893264i	$-0.648410\pi$
$41$	−5.75683	−0.899066	−0.449533	+	0.893264i	$0.648410\pi$
$42$	0	0
$43$	−7.02589	−1.07144	−0.535719	−	0.844396i	$-0.679959\pi$
$43$	−7.02589	−1.07144	−0.535719	+	0.844396i	$0.679959\pi$
$44$	0	0
$45$	9.28905	1.38473
$46$	0	0
$47$	7.30622	1.06572	0.532861	−	0.846203i	$-0.321117\pi$
$47$	7.30622	1.06572	0.532861	+	0.846203i	$0.321117\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.68277	1.35586
$52$	0	0
$53$	13.8034	1.89604	0.948018	−	0.318215i	$-0.103083\pi$
$53$	13.8034	1.89604	0.948018	+	0.318215i	$0.103083\pi$
$54$	0	0
$55$	6.06569	0.817898
$56$	0	0
$57$	−5.52585	−0.731916
$58$	0	0
$59$	−8.08949	−1.05316	−0.526581	−	0.850125i	$-0.676526\pi$
$59$	−8.08949	−1.05316	−0.526581	+	0.850125i	$0.676526\pi$
$60$	0	0
$61$	−5.75683	−0.737087	−0.368544	−	0.929610i	$-0.620143\pi$
$61$	−5.75683	−0.737087	−0.368544	+	0.929610i	$0.620143\pi$
$62$	0	0
$63$	−8.83471	−1.11307
$64$	0	0
$65$	−1.05143	−0.130413
$66$	0	0
$67$	−9.71156	−1.18646	−0.593228	−	0.805034i	$-0.702147\pi$
$67$	−9.71156	−1.18646	−0.593228	+	0.805034i	$0.702147\pi$
$68$	0	0
$69$	−10.9782	−1.32162
$70$	0	0
$71$	3.61708	0.429268	0.214634	−	0.976695i	$-0.431144\pi$
$71$	3.61708	0.429268	0.214634	+	0.976695i	$0.431144\pi$
$72$	0	0
$73$	−2.95576	−0.345946	−0.172973	−	0.984927i	$-0.555337\pi$
$73$	−2.95576	−0.345946	−0.172973	+	0.984927i	$0.555337\pi$
$74$	0	0
$75$	−13.3977	−1.54703
$76$	0	0
$77$	−5.76901	−0.657440
$78$	0	0
$79$	−11.7543	−1.32246	−0.661230	−	0.750184i	$-0.729965\pi$
$79$	−11.7543	−1.32246	−0.661230	+	0.750184i	$0.729965\pi$
$80$	0	0
$81$	42.5479	4.72755
$82$	0	0
$83$	−3.88241	−0.426150	−0.213075	−	0.977036i	$-0.568348\pi$
$83$	−3.88241	−0.426150	−0.213075	+	0.977036i	$0.568348\pi$
$84$	0	0
$85$	2.95937	0.320989
$86$	0	0
$87$	−0.0371598	−0.00398395
$88$	0	0
$89$	12.0096	1.27302	0.636509	−	0.771270i	$-0.280378\pi$
$89$	12.0096	1.27302	0.636509	+	0.771270i	$0.280378\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−4.91722	−0.509892
$94$	0	0
$95$	−1.68888	−0.173276
$96$	0	0
$97$	−10.7833	−1.09488	−0.547438	−	0.836846i	$-0.684397\pi$
$97$	−10.7833	−1.09488	−0.547438	+	0.836846i	$0.684397\pi$
$98$	0	0
$99$	50.9675	5.12243
$100$	0	0
$101$	−6.49714	−0.646490	−0.323245	−	0.946315i	$-0.604774\pi$
$101$	−6.49714	−0.646490	−0.323245	+	0.946315i	$0.604774\pi$
$102$	0	0
$103$	17.5666	1.73088	0.865442	−	0.501009i	$-0.167038\pi$
$103$	17.5666	1.73088	0.865442	+	0.501009i	$0.167038\pi$
$104$	0	0
$105$	−3.61708	−0.352991
$106$	0	0
$107$	3.92213	0.379166	0.189583	−	0.981865i	$-0.439286\pi$
$107$	3.92213	0.379166	0.189583	+	0.981865i	$0.439286\pi$
$108$	0	0
$109$	−8.69148	−0.832493	−0.416247	−	0.909252i	$-0.636655\pi$
$109$	−8.69148	−0.832493	−0.416247	+	0.909252i	$0.636655\pi$
$110$	0	0
$111$	−21.7551	−2.06490
$112$	0	0
$113$	−10.2522	−0.964449	−0.482224	−	0.876048i	$-0.660171\pi$
$113$	−10.2522	−0.964449	−0.482224	+	0.876048i	$0.660171\pi$
$114$	0	0
$115$	−3.35530	−0.312883
$116$	0	0
$117$	−8.83471	−0.816769
$118$	0	0
$119$	−2.81463	−0.258016
$120$	0	0
$121$	22.2815	2.02559
$122$	0	0
$123$	−19.8044	−1.78571
$124$	0	0
$125$	−9.35192	−0.836461
$126$	0	0
$127$	21.0758	1.87017	0.935087	−	0.354418i	$-0.115321\pi$
$127$	21.0758	1.87017	0.935087	+	0.354418i	$0.115321\pi$
$128$	0	0
$129$	−24.1702	−2.12807
$130$	0	0
$131$	2.86870	0.250639	0.125320	−	0.992116i	$-0.460004\pi$
$131$	2.86870	0.250639	0.125320	+	0.992116i	$0.460004\pi$
$132$	0	0
$133$	1.60628	0.139282
$134$	0	0
$135$	21.1046	1.81639
$136$	0	0
$137$	3.98782	0.340703	0.170351	−	0.985383i	$-0.445510\pi$
$137$	3.98782	0.340703	0.170351	+	0.985383i	$0.445510\pi$
$138$	0	0
$139$	13.5921	1.15287	0.576433	−	0.817144i	$-0.304444\pi$
$139$	13.5921	1.15287	0.576433	+	0.817144i	$0.304444\pi$
$140$	0	0
$141$	25.1346	2.11671
$142$	0	0
$143$	−5.76901	−0.482429
$144$	0	0
$145$	−0.0113573	−0.000943169 0
$146$	0	0
$147$	3.44016	0.283740
$148$	0	0
$149$	18.5196	1.51718	0.758592	−	0.651566i	$-0.225888\pi$
$149$	18.5196	1.51718	0.758592	+	0.651566i	$0.225888\pi$
$150$	0	0
$151$	−4.11503	−0.334877	−0.167438	−	0.985883i	$-0.553549\pi$
$151$	−4.11503	−0.334877	−0.167438	+	0.985883i	$0.553549\pi$
$152$	0	0
$153$	24.8664	2.01033
$154$	0	0
$155$	−1.50287	−0.120713
$156$	0	0
$157$	−16.4802	−1.31527	−0.657633	−	0.753339i	$-0.728442\pi$
$157$	−16.4802	−1.31527	−0.657633	+	0.753339i	$0.728442\pi$
$158$	0	0
$159$	47.4858	3.76587
$160$	0	0
$161$	3.19118	0.251501
$162$	0	0
$163$	0.794206	0.0622070	0.0311035	−	0.999516i	$-0.490098\pi$
$163$	0.794206	0.0622070	0.0311035	+	0.999516i	$0.490098\pi$
$164$	0	0
$165$	20.8670	1.62449
$166$	0	0
$167$	18.8522	1.45883	0.729415	−	0.684072i	$-0.239793\pi$
$167$	18.8522	1.45883	0.729415	+	0.684072i	$0.239793\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−14.1910	−1.08521
$172$	0	0
$173$	−9.03725	−0.687089	−0.343545	−	0.939136i	$-0.611628\pi$
$173$	−9.03725	−0.687089	−0.343545	+	0.939136i	$0.611628\pi$
$174$	0	0
$175$	3.89450	0.294397
$176$	0	0
$177$	−27.8292	−2.09177
$178$	0	0
$179$	22.5855	1.68812	0.844060	−	0.536248i	$-0.180159\pi$
$179$	22.5855	1.68812	0.844060	+	0.536248i	$0.180159\pi$
$180$	0	0
$181$	−22.4566	−1.66918	−0.834592	−	0.550869i	$-0.814296\pi$
$181$	−22.4566	−1.66918	−0.834592	+	0.550869i	$0.814296\pi$
$182$	0	0
$183$	−19.8044	−1.46399
$184$	0	0
$185$	−6.64908	−0.488850
$186$	0	0
$187$	16.2376	1.18741
$188$	0	0
$189$	−20.0723	−1.46005
$190$	0	0
$191$	11.8711	0.858959	0.429480	−	0.903077i	$-0.358697\pi$
$191$	11.8711	0.858959	0.429480	+	0.903077i	$0.358697\pi$
$192$	0	0
$193$	−21.8508	−1.57285	−0.786426	−	0.617684i	$-0.788071\pi$
$193$	−21.8508	−1.57285	−0.786426	+	0.617684i	$0.788071\pi$
$194$	0	0
$195$	−3.61708	−0.259024
$196$	0	0
$197$	9.87660	0.703679	0.351839	−	0.936060i	$-0.385556\pi$
$197$	9.87660	0.703679	0.351839	+	0.936060i	$0.385556\pi$
$198$	0	0
$199$	1.66641	0.118129	0.0590645	−	0.998254i	$-0.481188\pi$
$199$	1.66641	0.118129	0.0590645	+	0.998254i	$0.481188\pi$
$200$	0	0
$201$	−33.4093	−2.35651
$202$	0	0
$203$	0.0108018	0.000758135 0
$204$	0	0
$205$	−6.05289	−0.422752
$206$	0	0
$207$	−28.1932	−1.95956
$208$	0	0
$209$	−9.26662	−0.640986
$210$	0	0
$211$	−1.82834	−0.125868	−0.0629340	−	0.998018i	$-0.520046\pi$
$211$	−1.82834	−0.125868	−0.0629340	+	0.998018i	$0.520046\pi$
$212$	0	0
$213$	12.4433	0.852603
$214$	0	0
$215$	−7.38721	−0.503803
$216$	0	0
$217$	1.42936	0.0970312
$218$	0	0
$219$	−10.1683	−0.687109
$220$	0	0
$221$	−2.81463	−0.189332
$222$	0	0
$223$	−22.8878	−1.53268	−0.766339	−	0.642436i	$-0.777924\pi$
$223$	−22.8878	−1.53268	−0.766339	+	0.642436i	$0.777924\pi$
$224$	0	0
$225$	−34.4068	−2.29379
$226$	0	0
$227$	−1.84443	−0.122419	−0.0612096	−	0.998125i	$-0.519496\pi$
$227$	−1.84443	−0.122419	−0.0612096	+	0.998125i	$0.519496\pi$
$228$	0	0
$229$	2.06514	0.136468	0.0682341	−	0.997669i	$-0.478264\pi$
$229$	2.06514	0.136468	0.0682341	+	0.997669i	$0.478264\pi$
$230$	0	0
$231$	−19.8463	−1.30579
$232$	0	0
$233$	−17.8732	−1.17091	−0.585457	−	0.810703i	$-0.699085\pi$
$233$	−17.8732	−1.17091	−0.585457	+	0.810703i	$0.699085\pi$
$234$	0	0
$235$	7.68195	0.501115
$236$	0	0
$237$	−40.4366	−2.62664
$238$	0	0
$239$	−26.2742	−1.69954	−0.849769	−	0.527155i	$-0.823259\pi$
$239$	−26.2742	−1.69954	−0.849769	+	0.527155i	$0.823259\pi$
$240$	0	0
$241$	−20.1945	−1.30084	−0.650422	−	0.759573i	$-0.725408\pi$
$241$	−20.1945	−1.30084	−0.650422	+	0.759573i	$0.725408\pi$
$242$	0	0
$243$	86.1547	5.52682
$244$	0	0
$245$	1.05143	0.0671732
$246$	0	0
$247$	1.60628	0.102205
$248$	0	0
$249$	−13.3561	−0.846410
$250$	0	0
$251$	8.93528	0.563990	0.281995	−	0.959416i	$-0.409004\pi$
$251$	8.93528	0.563990	0.281995	+	0.959416i	$0.409004\pi$
$252$	0	0
$253$	−18.4100	−1.15743
$254$	0	0
$255$	10.1807	0.637542
$256$	0	0
$257$	5.31676	0.331650	0.165825	−	0.986155i	$-0.446971\pi$
$257$	5.31676	0.331650	0.165825	+	0.986155i	$0.446971\pi$
$258$	0	0
$259$	6.32386	0.392946
$260$	0	0
$261$	−0.0954303	−0.00590699
$262$	0	0
$263$	9.55913	0.589441	0.294721	−	0.955583i	$-0.404773\pi$
$263$	9.55913	0.589441	0.294721	+	0.955583i	$0.404773\pi$
$264$	0	0
$265$	14.5132	0.891540
$266$	0	0
$267$	41.3150	2.52844
$268$	0	0
$269$	5.24050	0.319519	0.159759	−	0.987156i	$-0.448928\pi$
$269$	5.24050	0.319519	0.159759	+	0.987156i	$0.448928\pi$
$270$	0	0
$271$	1.77426	0.107779	0.0538893	−	0.998547i	$-0.482838\pi$
$271$	1.77426	0.107779	0.0538893	+	0.998547i	$0.482838\pi$
$272$	0	0
$273$	3.44016	0.208208
$274$	0	0
$275$	−22.4674	−1.35484
$276$	0	0
$277$	−5.58412	−0.335517	−0.167759	−	0.985828i	$-0.553653\pi$
$277$	−5.58412	−0.335517	−0.167759	+	0.985828i	$0.553653\pi$
$278$	0	0
$279$	−12.6280	−0.756017
$280$	0	0
$281$	−8.01704	−0.478257	−0.239128	−	0.970988i	$-0.576862\pi$
$281$	−8.01704	−0.478257	−0.239128	+	0.970988i	$0.576862\pi$
$282$	0	0
$283$	15.7009	0.933321	0.466661	−	0.884436i	$-0.345457\pi$
$283$	15.7009	0.933321	0.466661	+	0.884436i	$0.345457\pi$
$284$	0	0
$285$	−5.81002	−0.344156
$286$	0	0
$287$	5.75683	0.339815
$288$	0	0
$289$	−9.07787	−0.533992
$290$	0	0
$291$	−37.0962	−2.17462
$292$	0	0
$293$	−19.2194	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$293$	−19.2194	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$294$	0	0
$295$	−8.50551	−0.495210
$296$	0	0
$297$	115.797	6.71925
$298$	0	0
$299$	3.19118	0.184551
$300$	0	0
$301$	7.02589	0.404966
$302$	0	0
$303$	−22.3512	−1.28404
$304$	0	0
$305$	−6.05289	−0.346587
$306$	0	0
$307$	10.5292	0.600932	0.300466	−	0.953793i	$-0.402858\pi$
$307$	10.5292	0.600932	0.300466	+	0.953793i	$0.402858\pi$
$308$	0	0
$309$	60.4318	3.43784
$310$	0	0
$311$	−16.9676	−0.962145	−0.481073	−	0.876681i	$-0.659753\pi$
$311$	−16.9676	−0.962145	−0.481073	+	0.876681i	$0.659753\pi$
$312$	0	0
$313$	15.4840	0.875205	0.437602	−	0.899169i	$-0.355828\pi$
$313$	15.4840	0.875205	0.437602	+	0.899169i	$0.355828\pi$
$314$	0	0
$315$	−9.28905	−0.523378
$316$	0	0
$317$	7.55857	0.424532	0.212266	−	0.977212i	$-0.431916\pi$
$317$	7.55857	0.424532	0.212266	+	0.977212i	$0.431916\pi$
$318$	0	0
$319$	−0.0623155	−0.00348900
$320$	0	0
$321$	13.4928	0.753092
$322$	0	0
$323$	−4.52107	−0.251559
$324$	0	0
$325$	3.89450	0.216028
$326$	0	0
$327$	−29.9001	−1.65348
$328$	0	0
$329$	−7.30622	−0.402805
$330$	0	0
$331$	−28.0556	−1.54207	−0.771037	−	0.636790i	$-0.780262\pi$
$331$	−28.0556	−1.54207	−0.771037	+	0.636790i	$0.780262\pi$
$332$	0	0
$333$	−55.8694	−3.06163
$334$	0	0
$335$	−10.2110	−0.557886
$336$	0	0
$337$	13.4572	0.733058	0.366529	−	0.930407i	$-0.380546\pi$
$337$	13.4572	0.733058	0.366529	+	0.930407i	$0.380546\pi$
$338$	0	0
$339$	−35.2693	−1.91557
$340$	0	0
$341$	−8.24599	−0.446545
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−11.5428	−0.621441
$346$	0	0
$347$	−14.0801	−0.755861	−0.377931	−	0.925834i	$-0.623364\pi$
$347$	−14.0801	−0.755861	−0.377931	+	0.925834i	$0.623364\pi$
$348$	0	0
$349$	19.6215	1.05031	0.525156	−	0.851006i	$-0.324007\pi$
$349$	19.6215	1.05031	0.525156	+	0.851006i	$0.324007\pi$
$350$	0	0
$351$	−20.0723	−1.07138
$352$	0	0
$353$	−9.35435	−0.497882	−0.248941	−	0.968519i	$-0.580082\pi$
$353$	−9.35435	−0.497882	−0.248941	+	0.968519i	$0.580082\pi$
$354$	0	0
$355$	3.80309	0.201847
$356$	0	0
$357$	−9.68277	−0.512467
$358$	0	0
$359$	−30.9826	−1.63520	−0.817600	−	0.575787i	$-0.804696\pi$
$359$	−30.9826	−1.63520	−0.817600	+	0.575787i	$0.804696\pi$
$360$	0	0
$361$	−16.4199	−0.864204
$362$	0	0
$363$	76.6519	4.02318
$364$	0	0
$365$	−3.10777	−0.162668
$366$	0	0
$367$	26.8021	1.39906	0.699528	−	0.714605i	$-0.253393\pi$
$367$	26.8021	1.39906	0.699528	+	0.714605i	$0.253393\pi$
$368$	0	0
$369$	−50.8599	−2.64766
$370$	0	0
$371$	−13.8034	−0.716635
$372$	0	0
$373$	27.7647	1.43760	0.718801	−	0.695216i	$-0.244691\pi$
$373$	27.7647	1.43760	0.718801	+	0.695216i	$0.244691\pi$
$374$	0	0
$375$	−32.1721	−1.66136
$376$	0	0
$377$	0.0108018	0.000556319 0
$378$	0	0
$379$	19.8055	1.01734	0.508671	−	0.860961i	$-0.330137\pi$
$379$	19.8055	1.01734	0.508671	+	0.860961i	$0.330137\pi$
$380$	0	0
$381$	72.5041	3.71450
$382$	0	0
$383$	23.0086	1.17568	0.587842	−	0.808976i	$-0.299978\pi$
$383$	23.0086	1.17568	0.587842	+	0.808976i	$0.299978\pi$
$384$	0	0
$385$	−6.06569	−0.309136
$386$	0	0
$387$	−62.0717	−3.15528
$388$	0	0
$389$	−6.21924	−0.315328	−0.157664	−	0.987493i	$-0.550396\pi$
$389$	−6.21924	−0.315328	−0.157664	+	0.987493i	$0.550396\pi$
$390$	0	0
$391$	−8.98199	−0.454239
$392$	0	0
$393$	9.86879	0.497814
$394$	0	0
$395$	−12.3588	−0.621837
$396$	0	0
$397$	−15.0764	−0.756663	−0.378332	−	0.925670i	$-0.623502\pi$
$397$	−15.0764	−0.756663	−0.378332	+	0.925670i	$0.623502\pi$
$398$	0	0
$399$	5.52585	0.276638
$400$	0	0
$401$	30.5899	1.52759	0.763793	−	0.645461i	$-0.223335\pi$
$401$	30.5899	1.52759	0.763793	+	0.645461i	$0.223335\pi$
$402$	0	0
$403$	1.42936	0.0712014
$404$	0	0
$405$	44.7360	2.22295
$406$	0	0
$407$	−36.4824	−1.80837
$408$	0	0
$409$	4.16773	0.206081	0.103040	−	0.994677i	$-0.467143\pi$
$409$	4.16773	0.206081	0.103040	+	0.994677i	$0.467143\pi$
$410$	0	0
$411$	13.7187	0.676696
$412$	0	0
$413$	8.08949	0.398058
$414$	0	0
$415$	−4.08207	−0.200381
$416$	0	0
$417$	46.7590	2.28980
$418$	0	0
$419$	10.4159	0.508850	0.254425	−	0.967093i	$-0.418114\pi$
$419$	10.4159	0.508850	0.254425	+	0.967093i	$0.418114\pi$
$420$	0	0
$421$	6.05433	0.295070	0.147535	−	0.989057i	$-0.452866\pi$
$421$	6.05433	0.295070	0.147535	+	0.989057i	$0.452866\pi$
$422$	0	0
$423$	64.5483	3.13844
$424$	0	0
$425$	−10.9616	−0.531714
$426$	0	0
$427$	5.75683	0.278593
$428$	0	0
$429$	−19.8463	−0.958190
$430$	0	0
$431$	7.38667	0.355804	0.177902	−	0.984048i	$-0.443069\pi$
$431$	7.38667	0.355804	0.177902	+	0.984048i	$0.443069\pi$
$432$	0	0
$433$	−33.1642	−1.59377	−0.796886	−	0.604130i	$-0.793521\pi$
$433$	−33.1642	−1.59377	−0.796886	+	0.604130i	$0.793521\pi$
$434$	0	0
$435$	−0.0390708	−0.00187330
$436$	0	0
$437$	5.12592	0.245206
$438$	0	0
$439$	−23.0185	−1.09861	−0.549306	−	0.835621i	$-0.685108\pi$
$439$	−23.0185	−1.09861	−0.549306	+	0.835621i	$0.685108\pi$
$440$	0	0
$441$	8.83471	0.420700
$442$	0	0
$443$	22.6940	1.07822	0.539112	−	0.842234i	$-0.318760\pi$
$443$	22.6940	1.07822	0.539112	+	0.842234i	$0.318760\pi$
$444$	0	0
$445$	12.6272	0.598588
$446$	0	0
$447$	63.7104	3.01340
$448$	0	0
$449$	−16.1045	−0.760018	−0.380009	−	0.924983i	$-0.624079\pi$
$449$	−16.1045	−0.760018	−0.380009	+	0.924983i	$0.624079\pi$
$450$	0	0
$451$	−33.2112	−1.56386
$452$	0	0
$453$	−14.1564	−0.665124
$454$	0	0
$455$	1.05143	0.0492917
$456$	0	0
$457$	−35.0112	−1.63775	−0.818876	−	0.573970i	$-0.805403\pi$
$457$	−35.0112	−1.63775	−0.818876	+	0.573970i	$0.805403\pi$
$458$	0	0
$459$	56.4961	2.63701
$460$	0	0
$461$	5.50691	0.256482	0.128241	−	0.991743i	$-0.459067\pi$
$461$	5.50691	0.256482	0.128241	+	0.991743i	$0.459067\pi$
$462$	0	0
$463$	25.5159	1.18582	0.592911	−	0.805268i	$-0.297978\pi$
$463$	25.5159	1.18582	0.592911	+	0.805268i	$0.297978\pi$
$464$	0	0
$465$	−5.17010	−0.239758
$466$	0	0
$467$	20.6965	0.957720	0.478860	−	0.877891i	$-0.341050\pi$
$467$	20.6965	0.957720	0.478860	+	0.877891i	$0.341050\pi$
$468$	0	0
$469$	9.71156	0.448438
$470$	0	0
$471$	−56.6946	−2.61235
$472$	0	0
$473$	−40.5324	−1.86368
$474$	0	0
$475$	6.25564	0.287029
$476$	0	0
$477$	121.949	5.58364
$478$	0	0
$479$	−39.3007	−1.79569	−0.897847	−	0.440308i	$-0.854869\pi$
$479$	−39.3007	−1.79569	−0.897847	+	0.440308i	$0.854869\pi$
$480$	0	0
$481$	6.32386	0.288343
$482$	0	0
$483$	10.9782	0.499525
$484$	0	0
$485$	−11.3378	−0.514824
$486$	0	0
$487$	22.2391	1.00775	0.503875	−	0.863777i	$-0.331907\pi$
$487$	22.2391	1.00775	0.503875	+	0.863777i	$0.331907\pi$
$488$	0	0
$489$	2.73220	0.123554
$490$	0	0
$491$	33.8186	1.52621	0.763105	−	0.646275i	$-0.223674\pi$
$491$	33.8186	1.52621	0.763105	+	0.646275i	$0.223674\pi$
$492$	0	0
$493$	−0.0304029	−0.00136928
$494$	0	0
$495$	53.5886	2.40863
$496$	0	0
$497$	−3.61708	−0.162248
$498$	0	0
$499$	−20.8934	−0.935316	−0.467658	−	0.883909i	$-0.654902\pi$
$499$	−20.8934	−0.935316	−0.467658	+	0.883909i	$0.654902\pi$
$500$	0	0
$501$	64.8547	2.89749
$502$	0	0
$503$	−33.1978	−1.48021	−0.740107	−	0.672489i	$-0.765225\pi$
$503$	−33.1978	−1.48021	−0.740107	+	0.672489i	$0.765225\pi$
$504$	0	0
$505$	−6.83127	−0.303988
$506$	0	0
$507$	3.44016	0.152783
$508$	0	0
$509$	−12.8893	−0.571306	−0.285653	−	0.958333i	$-0.592211\pi$
$509$	−12.8893	−0.571306	−0.285653	+	0.958333i	$0.592211\pi$
$510$	0	0
$511$	2.95576	0.130755
$512$	0	0
$513$	−32.2417	−1.42351
$514$	0	0
$515$	18.4700	0.813883
$516$	0	0
$517$	42.1496	1.85374
$518$	0	0
$519$	−31.0896	−1.36468
$520$	0	0
$521$	12.0174	0.526493	0.263246	−	0.964729i	$-0.415207\pi$
$521$	12.0174	0.526493	0.263246	+	0.964729i	$0.415207\pi$
$522$	0	0
$523$	−8.60299	−0.376183	−0.188091	−	0.982152i	$-0.560230\pi$
$523$	−8.60299	−0.376183	−0.188091	+	0.982152i	$0.560230\pi$
$524$	0	0
$525$	13.3977	0.584724
$526$	0	0
$527$	−4.02311	−0.175250
$528$	0	0
$529$	−12.8163	−0.557232
$530$	0	0
$531$	−71.4683	−3.10146
$532$	0	0
$533$	5.75683	0.249356
$534$	0	0
$535$	4.12383	0.178289
$536$	0	0
$537$	77.6978	3.35291
$538$	0	0
$539$	5.76901	0.248489
$540$	0	0
$541$	−3.29905	−0.141837	−0.0709186	−	0.997482i	$-0.522593\pi$
$541$	−3.29905	−0.141837	−0.0709186	+	0.997482i	$0.522593\pi$
$542$	0	0
$543$	−77.2542	−3.31529
$544$	0	0
$545$	−9.13846	−0.391449
$546$	0	0
$547$	35.5790	1.52125	0.760624	−	0.649193i	$-0.224893\pi$
$547$	35.5790	1.52125	0.760624	+	0.649193i	$0.224893\pi$
$548$	0	0
$549$	−50.8599	−2.17065
$550$	0	0
$551$	0.0173506	0.000739160 0
$552$	0	0
$553$	11.7543	0.499843
$554$	0	0
$555$	−22.8739	−0.970943
$556$	0	0
$557$	−17.5244	−0.742533	−0.371266	−	0.928526i	$-0.621076\pi$
$557$	−17.5244	−0.742533	−0.371266	+	0.928526i	$0.621076\pi$
$558$	0	0
$559$	7.02589	0.297163
$560$	0	0
$561$	55.8600	2.35841
$562$	0	0
$563$	8.50995	0.358651	0.179326	−	0.983790i	$-0.442608\pi$
$563$	8.50995	0.358651	0.179326	+	0.983790i	$0.442608\pi$
$564$	0	0
$565$	−10.7795	−0.453496
$566$	0	0
$567$	−42.5479	−1.78684
$568$	0	0
$569$	−44.1447	−1.85064	−0.925321	−	0.379184i	$-0.876205\pi$
$569$	−44.1447	−1.85064	−0.925321	+	0.379184i	$0.876205\pi$
$570$	0	0
$571$	−15.6944	−0.656791	−0.328395	−	0.944540i	$-0.606508\pi$
$571$	−15.6944	−0.656791	−0.328395	+	0.944540i	$0.606508\pi$
$572$	0	0
$573$	40.8383	1.70605
$574$	0	0
$575$	12.4281	0.518286
$576$	0	0
$577$	−1.57236	−0.0654582	−0.0327291	−	0.999464i	$-0.510420\pi$
$577$	−1.57236	−0.0654582	−0.0327291	+	0.999464i	$0.510420\pi$
$578$	0	0
$579$	−75.1701	−3.12396
$580$	0	0
$581$	3.88241	0.161070
$582$	0	0
$583$	79.6317	3.29801
$584$	0	0
$585$	−9.28905	−0.384055
$586$	0	0
$587$	−33.2622	−1.37288	−0.686438	−	0.727188i	$-0.740827\pi$
$587$	−33.2622	−1.37288	−0.686438	+	0.727188i	$0.740827\pi$
$588$	0	0
$589$	2.29594	0.0946027
$590$	0	0
$591$	33.9771	1.39763
$592$	0	0
$593$	−28.7357	−1.18003	−0.590017	−	0.807391i	$-0.700879\pi$
$593$	−28.7357	−1.18003	−0.590017	+	0.807391i	$0.700879\pi$
$594$	0	0
$595$	−2.95937	−0.121323
$596$	0	0
$597$	5.73274	0.234625
$598$	0	0
$599$	−5.51603	−0.225379	−0.112689	−	0.993630i	$-0.535947\pi$
$599$	−5.51603	−0.225379	−0.112689	+	0.993630i	$0.535947\pi$
$600$	0	0
$601$	26.8938	1.09702	0.548510	−	0.836144i	$-0.315195\pi$
$601$	26.8938	1.09702	0.548510	+	0.836144i	$0.315195\pi$
$602$	0	0
$603$	−85.7988	−3.49400
$604$	0	0
$605$	23.4274	0.952458
$606$	0	0
$607$	14.8000	0.600712	0.300356	−	0.953827i	$-0.402895\pi$
$607$	14.8000	0.600712	0.300356	+	0.953827i	$0.402895\pi$
$608$	0	0
$609$	0.0371598	0.00150579
$610$	0	0
$611$	−7.30622	−0.295578
$612$	0	0
$613$	28.5366	1.15258	0.576292	−	0.817244i	$-0.304499\pi$
$613$	28.5366	1.15258	0.576292	+	0.817244i	$0.304499\pi$
$614$	0	0
$615$	−20.8229	−0.839661
$616$	0	0
$617$	40.5038	1.63062	0.815310	−	0.579024i	$-0.196566\pi$
$617$	40.5038	1.63062	0.815310	+	0.579024i	$0.196566\pi$
$618$	0	0
$619$	−15.8087	−0.635404	−0.317702	−	0.948191i	$-0.602911\pi$
$619$	−15.8087	−0.635404	−0.317702	+	0.948191i	$0.602911\pi$
$620$	0	0
$621$	−64.0545	−2.57042
$622$	0	0
$623$	−12.0096	−0.481155
$624$	0	0
$625$	9.63965	0.385586
$626$	0	0
$627$	−31.8787	−1.27311
$628$	0	0
$629$	−17.7993	−0.709705
$630$	0	0
$631$	36.2717	1.44395	0.721977	−	0.691918i	$-0.243234\pi$
$631$	36.2717	1.44395	0.721977	+	0.691918i	$0.243234\pi$
$632$	0	0
$633$	−6.28978	−0.249996
$634$	0	0
$635$	22.1597	0.879379
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	31.9558	1.26415
$640$	0	0
$641$	−28.0970	−1.10976	−0.554882	−	0.831929i	$-0.687237\pi$
$641$	−28.0970	−1.10976	−0.554882	+	0.831929i	$0.687237\pi$
$642$	0	0
$643$	−5.54783	−0.218785	−0.109393	−	0.993999i	$-0.534891\pi$
$643$	−5.54783	−0.218785	−0.109393	+	0.993999i	$0.534891\pi$
$644$	0	0
$645$	−25.4132	−1.00064
$646$	0	0
$647$	8.32318	0.327218	0.163609	−	0.986525i	$-0.447686\pi$
$647$	8.32318	0.327218	0.163609	+	0.986525i	$0.447686\pi$
$648$	0	0
$649$	−46.6684	−1.83189
$650$	0	0
$651$	4.91722	0.192721
$652$	0	0
$653$	−14.0819	−0.551069	−0.275534	−	0.961291i	$-0.588855\pi$
$653$	−14.0819	−0.551069	−0.275534	+	0.961291i	$0.588855\pi$
$654$	0	0
$655$	3.01623	0.117854
$656$	0	0
$657$	−26.1133	−1.01878
$658$	0	0
$659$	8.64161	0.336629	0.168315	−	0.985733i	$-0.446168\pi$
$659$	8.64161	0.336629	0.168315	+	0.985733i	$0.446168\pi$
$660$	0	0
$661$	31.1208	1.21046	0.605230	−	0.796051i	$-0.293081\pi$
$661$	31.1208	1.21046	0.605230	+	0.796051i	$0.293081\pi$
$662$	0	0
$663$	−9.68277	−0.376048
$664$	0	0
$665$	1.68888	0.0654920
$666$	0	0
$667$	0.0344704	0.00133470
$668$	0	0
$669$	−78.7376	−3.04417
$670$	0	0
$671$	−33.2112	−1.28211
$672$	0	0
$673$	17.3266	0.667893	0.333946	−	0.942592i	$-0.391620\pi$
$673$	17.3266	0.667893	0.333946	+	0.942592i	$0.391620\pi$
$674$	0	0
$675$	−78.1717	−3.00883
$676$	0	0
$677$	39.3738	1.51326	0.756629	−	0.653844i	$-0.226845\pi$
$677$	39.3738	1.51326	0.756629	+	0.653844i	$0.226845\pi$
$678$	0	0
$679$	10.7833	0.413824
$680$	0	0
$681$	−6.34514	−0.243146
$682$	0	0
$683$	15.9637	0.610832	0.305416	−	0.952219i	$-0.401204\pi$
$683$	15.9637	0.610832	0.305416	+	0.952219i	$0.401204\pi$
$684$	0	0
$685$	4.19290	0.160203
$686$	0	0
$687$	7.10441	0.271050
$688$	0	0
$689$	−13.8034	−0.525866
$690$	0	0
$691$	−5.50647	−0.209476	−0.104738	−	0.994500i	$-0.533400\pi$
$691$	−5.50647	−0.209476	−0.104738	+	0.994500i	$0.533400\pi$
$692$	0	0
$693$	−50.9675	−1.93610
$694$	0	0
$695$	14.2911	0.542092
$696$	0	0
$697$	−16.2033	−0.613745
$698$	0	0
$699$	−61.4868	−2.32565
$700$	0	0
$701$	−25.3011	−0.955610	−0.477805	−	0.878466i	$-0.658568\pi$
$701$	−25.3011	−0.955610	−0.477805	+	0.878466i	$0.658568\pi$
$702$	0	0
$703$	10.1579	0.383111
$704$	0	0
$705$	26.4271	0.995304
$706$	0	0
$707$	6.49714	0.244350
$708$	0	0
$709$	37.4215	1.40539	0.702697	−	0.711489i	$-0.251979\pi$
$709$	37.4215	1.40539	0.702697	+	0.711489i	$0.251979\pi$
$710$	0	0
$711$	−103.846	−3.89451
$712$	0	0
$713$	4.56135	0.170824
$714$	0	0
$715$	−6.06569	−0.226844
$716$	0	0
$717$	−90.3876	−3.37559
$718$	0	0
$719$	19.3116	0.720203	0.360101	−	0.932913i	$-0.382742\pi$
$719$	19.3116	0.720203	0.360101	+	0.932913i	$0.382742\pi$
$720$	0	0
$721$	−17.5666	−0.654213
$722$	0	0
$723$	−69.4724	−2.58371
$724$	0	0
$725$	0.0420675	0.00156235
$726$	0	0
$727$	−5.88441	−0.218241	−0.109120	−	0.994029i	$-0.534803\pi$
$727$	−5.88441	−0.218241	−0.109120	+	0.994029i	$0.534803\pi$
$728$	0	0
$729$	168.742	6.24971
$730$	0	0
$731$	−19.7753	−0.731414
$732$	0	0
$733$	−47.4603	−1.75299	−0.876493	−	0.481414i	$-0.840123\pi$
$733$	−47.4603	−1.75299	−0.876493	+	0.481414i	$0.840123\pi$
$734$	0	0
$735$	3.61708	0.133418
$736$	0	0
$737$	−56.0261	−2.06375
$738$	0	0
$739$	6.41585	0.236011	0.118005	−	0.993013i	$-0.462350\pi$
$739$	6.41585	0.236011	0.118005	+	0.993013i	$0.462350\pi$
$740$	0	0
$741$	5.52585	0.202997
$742$	0	0
$743$	5.71193	0.209550	0.104775	−	0.994496i	$-0.466588\pi$
$743$	5.71193	0.209550	0.104775	+	0.994496i	$0.466588\pi$
$744$	0	0
$745$	19.4720	0.713399
$746$	0	0
$747$	−34.3000	−1.25497
$748$	0	0
$749$	−3.92213	−0.143311
$750$	0	0
$751$	−1.20506	−0.0439731	−0.0219866	−	0.999758i	$-0.506999\pi$
$751$	−1.20506	−0.0439731	−0.0219866	+	0.999758i	$0.506999\pi$
$752$	0	0
$753$	30.7388	1.12018
$754$	0	0
$755$	−4.32665	−0.157463
$756$	0	0
$757$	6.03447	0.219327	0.109663	−	0.993969i	$-0.465023\pi$
$757$	6.03447	0.219327	0.109663	+	0.993969i	$0.465023\pi$
$758$	0	0
$759$	−63.3333	−2.29885
$760$	0	0
$761$	−38.5211	−1.39639	−0.698194	−	0.715908i	$-0.746013\pi$
$761$	−38.5211	−1.39639	−0.698194	+	0.715908i	$0.746013\pi$
$762$	0	0
$763$	8.69148	0.314653
$764$	0	0
$765$	26.1452	0.945282
$766$	0	0
$767$	8.08949	0.292095
$768$	0	0
$769$	−47.2558	−1.70409	−0.852044	−	0.523470i	$-0.824637\pi$
$769$	−47.2558	−1.70409	−0.852044	+	0.523470i	$0.824637\pi$
$770$	0	0
$771$	18.2905	0.658717
$772$	0	0
$773$	0.269359	0.00968816	0.00484408	−	0.999988i	$-0.498458\pi$
$773$	0.269359	0.00968816	0.00484408	+	0.999988i	$0.498458\pi$
$774$	0	0
$775$	5.56664	0.199960
$776$	0	0
$777$	21.7551	0.780460
$778$	0	0
$779$	9.24706	0.331310
$780$	0	0
$781$	20.8670	0.746679
$782$	0	0
$783$	−0.216816	−0.00774838
$784$	0	0
$785$	−17.3278	−0.618454
$786$	0	0
$787$	−14.9545	−0.533069	−0.266534	−	0.963825i	$-0.585879\pi$
$787$	−14.9545	−0.533069	−0.266534	+	0.963825i	$0.585879\pi$
$788$	0	0
$789$	32.8850	1.17074
$790$	0	0
$791$	10.2522	0.364527
$792$	0	0
$793$	5.75683	0.204431
$794$	0	0
$795$	49.9278	1.77076
$796$	0	0
$797$	27.2080	0.963758	0.481879	−	0.876238i	$-0.339955\pi$
$797$	27.2080	0.963758	0.481879	+	0.876238i	$0.339955\pi$
$798$	0	0
$799$	20.5643	0.727512
$800$	0	0
$801$	106.101	3.74891
$802$	0	0
$803$	−17.0518	−0.601746
$804$	0	0
$805$	3.35530	0.118259
$806$	0	0
$807$	18.0282	0.634621
$808$	0	0
$809$	33.1514	1.16554	0.582770	−	0.812637i	$-0.301969\pi$
$809$	33.1514	1.16554	0.582770	+	0.812637i	$0.301969\pi$
$810$	0	0
$811$	32.5660	1.14355	0.571774	−	0.820411i	$-0.306255\pi$
$811$	32.5660	1.14355	0.571774	+	0.820411i	$0.306255\pi$
$812$	0	0
$813$	6.10374	0.214067
$814$	0	0
$815$	0.835049	0.0292505
$816$	0	0
$817$	11.2855	0.394830
$818$	0	0
$819$	8.83471	0.308710
$820$	0	0
$821$	17.5724	0.613280	0.306640	−	0.951826i	$-0.400795\pi$
$821$	17.5724	0.613280	0.306640	+	0.951826i	$0.400795\pi$
$822$	0	0
$823$	−14.0374	−0.489312	−0.244656	−	0.969610i	$-0.578675\pi$
$823$	−14.0374	−0.489312	−0.244656	+	0.969610i	$0.578675\pi$
$824$	0	0
$825$	−77.2916	−2.69095
$826$	0	0
$827$	2.20153	0.0765547	0.0382773	−	0.999267i	$-0.487813\pi$
$827$	2.20153	0.0765547	0.0382773	+	0.999267i	$0.487813\pi$
$828$	0	0
$829$	−27.5802	−0.957901	−0.478950	−	0.877842i	$-0.658983\pi$
$829$	−27.5802	−0.957901	−0.478950	+	0.877842i	$0.658983\pi$
$830$	0	0
$831$	−19.2103	−0.666396
$832$	0	0
$833$	2.81463	0.0975211
$834$	0	0
$835$	19.8217	0.685959
$836$	0	0
$837$	−28.6906	−0.991691
$838$	0	0
$839$	4.85302	0.167545	0.0837724	−	0.996485i	$-0.473303\pi$
$839$	4.85302	0.167545	0.0837724	+	0.996485i	$0.473303\pi$
$840$	0	0
$841$	−28.9999	−0.999996
$842$	0	0
$843$	−27.5799	−0.949903
$844$	0	0
$845$	1.05143	0.0361702
$846$	0	0
$847$	−22.2815	−0.765601
$848$	0	0
$849$	54.0136	1.85374
$850$	0	0
$851$	20.1806	0.691782
$852$	0	0
$853$	−44.3108	−1.51717	−0.758587	−	0.651572i	$-0.774110\pi$
$853$	−44.3108	−1.51717	−0.758587	+	0.651572i	$0.774110\pi$
$854$	0	0
$855$	−14.9208	−0.510280
$856$	0	0
$857$	33.9517	1.15977	0.579884	−	0.814699i	$-0.303098\pi$
$857$	33.9517	1.15977	0.579884	+	0.814699i	$0.303098\pi$
$858$	0	0
$859$	−5.48700	−0.187214	−0.0936071	−	0.995609i	$-0.529840\pi$
$859$	−5.48700	−0.187214	−0.0936071	+	0.995609i	$0.529840\pi$
$860$	0	0
$861$	19.8044	0.674933
$862$	0	0
$863$	−23.4104	−0.796899	−0.398449	−	0.917190i	$-0.630452\pi$
$863$	−23.4104	−0.796899	−0.398449	+	0.917190i	$0.630452\pi$
$864$	0	0
$865$	−9.50200	−0.323078
$866$	0	0
$867$	−31.2293	−1.06060
$868$	0	0
$869$	−67.8106	−2.30032
$870$	0	0
$871$	9.71156	0.329064
$872$	0	0
$873$	−95.2671	−3.22430
$874$	0	0
$875$	9.35192	0.316153
$876$	0	0
$877$	−14.0392	−0.474071	−0.237036	−	0.971501i	$-0.576176\pi$
$877$	−14.0392	−0.474071	−0.237036	+	0.971501i	$0.576176\pi$
$878$	0	0
$879$	−66.1179	−2.23010
$880$	0	0
$881$	7.62250	0.256808	0.128404	−	0.991722i	$-0.459015\pi$
$881$	7.62250	0.256808	0.128404	+	0.991722i	$0.459015\pi$
$882$	0	0
$883$	−13.4738	−0.453431	−0.226715	−	0.973961i	$-0.572799\pi$
$883$	−13.4738	−0.453431	−0.226715	+	0.973961i	$0.572799\pi$
$884$	0	0
$885$	−29.2603	−0.983575
$886$	0	0
$887$	24.8602	0.834723	0.417362	−	0.908740i	$-0.362955\pi$
$887$	24.8602	0.834723	0.417362	+	0.908740i	$0.362955\pi$
$888$	0	0
$889$	−21.0758	−0.706859
$890$	0	0
$891$	245.459	8.22320
$892$	0	0
$893$	−11.7358	−0.392723
$894$	0	0
$895$	23.7470	0.793775
$896$	0	0
$897$	10.9782	0.366551
$898$	0	0
$899$	0.0154396	0.000514939 0
$900$	0	0
$901$	38.8513	1.29432
$902$	0	0
$903$	24.1702	0.804334
$904$	0	0
$905$	−23.6114	−0.784871
$906$	0	0
$907$	37.1017	1.23194	0.615971	−	0.787769i	$-0.288764\pi$
$907$	37.1017	1.23194	0.615971	+	0.787769i	$0.288764\pi$
$908$	0	0
$909$	−57.4003	−1.90385
$910$	0	0
$911$	−35.5028	−1.17626	−0.588130	−	0.808767i	$-0.700136\pi$
$911$	−35.5028	−1.17626	−0.588130	+	0.808767i	$0.700136\pi$
$912$	0	0
$913$	−22.3977	−0.741255
$914$	0	0
$915$	−20.8229	−0.688384
$916$	0	0
$917$	−2.86870	−0.0947328
$918$	0	0
$919$	−44.9888	−1.48404	−0.742022	−	0.670376i	$-0.766133\pi$
$919$	−44.9888	−1.48404	−0.742022	+	0.670376i	$0.766133\pi$
$920$	0	0
$921$	36.2221	1.19356
$922$	0	0
$923$	−3.61708	−0.119058
$924$	0	0
$925$	24.6283	0.809773
$926$	0	0
$927$	155.195	5.09728
$928$	0	0
$929$	44.1538	1.44864	0.724320	−	0.689464i	$-0.242154\pi$
$929$	44.1538	1.44864	0.724320	+	0.689464i	$0.242154\pi$
$930$	0	0
$931$	−1.60628	−0.0526436
$932$	0	0
$933$	−58.3713	−1.91099
$934$	0	0
$935$	17.0727	0.558336
$936$	0	0
$937$	−2.32343	−0.0759032	−0.0379516	−	0.999280i	$-0.512083\pi$
$937$	−2.32343	−0.0759032	−0.0379516	+	0.999280i	$0.512083\pi$
$938$	0	0
$939$	53.2673	1.73831
$940$	0	0
$941$	2.14275	0.0698515	0.0349257	−	0.999390i	$-0.488881\pi$
$941$	2.14275	0.0698515	0.0349257	+	0.999390i	$0.488881\pi$
$942$	0	0
$943$	18.3711	0.598246
$944$	0	0
$945$	−21.1046	−0.686532
$946$	0	0
$947$	−33.9204	−1.10226	−0.551132	−	0.834418i	$-0.685804\pi$
$947$	−33.9204	−1.10226	−0.551132	+	0.834418i	$0.685804\pi$
$948$	0	0
$949$	2.95576	0.0959480
$950$	0	0
$951$	26.0027	0.843195
$952$	0	0
$953$	4.88157	0.158129	0.0790647	−	0.996869i	$-0.474807\pi$
$953$	4.88157	0.158129	0.0790647	+	0.996869i	$0.474807\pi$
$954$	0	0
$955$	12.4815	0.403893
$956$	0	0
$957$	−0.214375	−0.00692977
$958$	0	0
$959$	−3.98782	−0.128773
$960$	0	0
$961$	−28.9569	−0.934095
$962$	0	0
$963$	34.6508	1.11661
$964$	0	0
$965$	−22.9745	−0.739575
$966$	0	0
$967$	41.2856	1.32766	0.663828	−	0.747886i	$-0.268931\pi$
$967$	41.2856	1.32766	0.663828	+	0.747886i	$0.268931\pi$
$968$	0	0
$969$	−15.5532	−0.499641
$970$	0	0
$971$	−52.3660	−1.68050	−0.840252	−	0.542196i	$-0.817593\pi$
$971$	−52.3660	−1.68050	−0.840252	+	0.542196i	$0.817593\pi$
$972$	0	0
$973$	−13.5921	−0.435743
$974$	0	0
$975$	13.3977	0.429070
$976$	0	0
$977$	47.3847	1.51597	0.757986	−	0.652271i	$-0.226183\pi$
$977$	47.3847	1.51597	0.757986	+	0.652271i	$0.226183\pi$
$978$	0	0
$979$	69.2836	2.21431
$980$	0	0
$981$	−76.7867	−2.45161
$982$	0	0
$983$	−10.3959	−0.331577	−0.165789	−	0.986161i	$-0.553017\pi$
$983$	−10.3959	−0.331577	−0.165789	+	0.986161i	$0.553017\pi$
$984$	0	0
$985$	10.3845	0.330878
$986$	0	0
$987$	−25.1346	−0.800042
$988$	0	0
$989$	22.4209	0.712943
$990$	0	0
$991$	36.9544	1.17389	0.586947	−	0.809625i	$-0.300330\pi$
$991$	36.9544	1.17389	0.586947	+	0.809625i	$0.300330\pi$
$992$	0	0
$993$	−96.5157	−3.06283
$994$	0	0
$995$	1.75211	0.0555457
$996$	0	0
$997$	25.1668	0.797039	0.398520	−	0.917160i	$-0.369524\pi$
$997$	25.1668	0.797039	0.398520	+	0.917160i	$0.369524\pi$
$998$	0	0
$999$	−126.935	−4.01603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.a.t.1.5	yes	5
4.3	odd	2		2912.2.a.q.1.1	✓	5
8.3	odd	2		5824.2.a.cl.1.5		5
8.5	even	2		5824.2.a.ci.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.a.q.1.1	✓	5	4.3	odd	2
2912.2.a.t.1.5	yes	5	1.1	even	1	trivial
5824.2.a.ci.1.1		5	8.5	even	2
5824.2.a.cl.1.5		5	8.3	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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.a (trivial)

\(f(q)\)	\(=\)	\(q+3.44016 q^{3} +1.05143 q^{5} -1.00000 q^{7} +8.83471 q^{9} +5.76901 q^{11} -1.00000 q^{13} +3.61708 q^{15} +2.81463 q^{17} -1.60628 q^{19} -3.44016 q^{21} -3.19118 q^{23} -3.89450 q^{25} +20.0723 q^{27} -0.0108018 q^{29} -1.42936 q^{31} +19.8463 q^{33} -1.05143 q^{35} -6.32386 q^{37} -3.44016 q^{39} -5.75683 q^{41} -7.02589 q^{43} +9.28905 q^{45} +7.30622 q^{47} +1.00000 q^{49} +9.68277 q^{51} +13.8034 q^{53} +6.06569 q^{55} -5.52585 q^{57} -8.08949 q^{59} -5.75683 q^{61} -8.83471 q^{63} -1.05143 q^{65} -9.71156 q^{67} -10.9782 q^{69} +3.61708 q^{71} -2.95576 q^{73} -13.3977 q^{75} -5.76901 q^{77} -11.7543 q^{79} +42.5479 q^{81} -3.88241 q^{83} +2.95937 q^{85} -0.0371598 q^{87} +12.0096 q^{89} +1.00000 q^{91} -4.91722 q^{93} -1.68888 q^{95} -10.7833 q^{97} +50.9675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 3 q^{5} - 5 q^{7} + 6 q^{9} + 5 q^{11} - 5 q^{13} + 4 q^{17} + 7 q^{19} - 5 q^{21} - 4 q^{23} + 2 q^{25} + 23 q^{27} + 3 q^{29} + 2 q^{31} + 17 q^{33} + 3 q^{35} - q^{37} - 5 q^{39} - 7 q^{41}+ \cdots + 40 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 3 * q^5 - 5 * q^7 + 6 * q^9 + 5 * q^11 - 5 * q^13 + 4 * q^17 + 7 * q^19 - 5 * q^21 - 4 * q^23 + 2 * q^25 + 23 * q^27 + 3 * q^29 + 2 * q^31 + 17 * q^33 + 3 * q^35 - q^37 - 5 * q^39 - 7 * q^41 + 15 * q^43 - q^45 + 6 * q^47 + 5 * q^49 + 16 * q^51 + 11 * q^53 + 16 * q^55 + 18 * q^57 + 20 * q^59 - 7 * q^61 - 6 * q^63 + 3 * q^65 + 11 * q^67 + 5 * q^69 - 2 * q^73 + 19 * q^75 - 5 * q^77 + 10 * q^79 + 33 * q^81 + 9 * q^83 + 20 * q^85 + 2 * q^87 + 11 * q^89 + 5 * q^91 - 13 * q^93 - 11 * q^95 - 24 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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