LMFDB - Embedded newform 2912.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(1,2912)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.153499364.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 18x^{3} + 19x^{2} - 25x - 19 $ x^6 - 3x^5 - 7x^4 + 18x^3 + 19x^2 - 25*x - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.37463$ 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)

$f(q)$	$=$	$q-0.735766 q^{3} +0.856892 q^{5} -1.00000 q^{7} -2.45865 q^{9} -5.29340 q^{11} -1.00000 q^{13} -0.630472 q^{15} +7.13529 q^{17} -1.14311 q^{19} +0.735766 q^{21} -6.17311 q^{23} -4.26574 q^{25} +4.01629 q^{27} +1.21376 q^{29} +4.77441 q^{31} +3.89471 q^{33} -0.856892 q^{35} -4.32888 q^{37} +0.735766 q^{39} +8.47515 q^{41} +8.59628 q^{43} -2.10680 q^{45} +5.97869 q^{47} +1.00000 q^{49} -5.24990 q^{51} +10.8069 q^{53} -4.53588 q^{55} +0.841060 q^{57} +5.83536 q^{59} -2.83558 q^{61} +2.45865 q^{63} -0.856892 q^{65} -10.0391 q^{67} +4.54196 q^{69} -6.12899 q^{71} +10.0079 q^{73} +3.13858 q^{75} +5.29340 q^{77} +8.70793 q^{79} +4.42090 q^{81} -9.26469 q^{83} +6.11417 q^{85} -0.893040 q^{87} +7.67959 q^{89} +1.00000 q^{91} -3.51285 q^{93} -0.979520 q^{95} -2.46070 q^{97} +13.0146 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 3 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} - 9 q^{19} + 4 q^{21} + 11 q^{23} + 21 q^{25} - 22 q^{27} + 7 q^{29} - q^{31} + 18 q^{33} - 3 q^{35} + 16 q^{37}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 3 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 - 6 * q^13 + 2 * q^15 - 2 * q^17 - 9 * q^19 + 4 * q^21 + 11 * q^23 + 21 * q^25 - 22 * q^27 + 7 * q^29 - q^31 + 18 * q^33 - 3 * q^35 + 16 * q^37 + 4 * q^39 - q^43 + 29 * q^45 + 11 * q^47 + 6 * q^49 + 8 * q^51 + 23 * q^53 - 18 * q^55 + 10 * q^57 - 10 * q^59 + 10 * q^61 - 8 * q^63 - 3 * q^65 + 8 * q^67 + 6 * q^69 + 14 * q^71 + 11 * q^73 - 14 * q^75 + 8 * q^77 + 3 * q^79 + 30 * q^81 - 17 * q^83 + 6 * q^85 + 36 * q^87 - 5 * q^89 + 6 * q^91 - 16 * q^93 + 45 * q^95 + 27 * q^97 - 4 * 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
$3$	−0.735766	−0.424795	−0.212397	−	0.977183i	$-0.568127\pi$
$3$	−0.735766	−0.424795	−0.212397	+	0.977183i	$0.568127\pi$
$4$	0	0
$5$	0.856892	0.383214	0.191607	−	0.981472i	$-0.438630\pi$
$5$	0.856892	0.383214	0.191607	+	0.981472i	$0.438630\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.45865	−0.819550
$10$	0	0
$11$	−5.29340	−1.59602	−0.798011	−	0.602643i	$-0.794114\pi$
$11$	−5.29340	−1.59602	−0.798011	+	0.602643i	$0.794114\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.630472	−0.162787
$16$	0	0
$17$	7.13529	1.73056	0.865281	−	0.501287i	$-0.167140\pi$
$17$	7.13529	1.73056	0.865281	+	0.501287i	$0.167140\pi$
$18$	0	0
$19$	−1.14311	−0.262247	−0.131123	−	0.991366i	$-0.541858\pi$
$19$	−1.14311	−0.262247	−0.131123	+	0.991366i	$0.541858\pi$
$20$	0	0
$21$	0.735766	0.160557
$22$	0	0
$23$	−6.17311	−1.28718	−0.643591	−	0.765370i	$-0.722556\pi$
$23$	−6.17311	−1.28718	−0.643591	+	0.765370i	$0.722556\pi$
$24$	0	0
$25$	−4.26574	−0.853147
$26$	0	0
$27$	4.01629	0.772935
$28$	0	0
$29$	1.21376	0.225389	0.112694	−	0.993630i	$-0.464052\pi$
$29$	1.21376	0.225389	0.112694	+	0.993630i	$0.464052\pi$
$30$	0	0
$31$	4.77441	0.857509	0.428754	−	0.903421i	$-0.358953\pi$
$31$	4.77441	0.857509	0.428754	+	0.903421i	$0.358953\pi$
$32$	0	0
$33$	3.89471	0.677981
$34$	0	0
$35$	−0.856892	−0.144841
$36$	0	0
$37$	−4.32888	−0.711663	−0.355832	−	0.934550i	$-0.615802\pi$
$37$	−4.32888	−0.711663	−0.355832	+	0.934550i	$0.615802\pi$
$38$	0	0
$39$	0.735766	0.117817
$40$	0	0
$41$	8.47515	1.32360	0.661798	−	0.749682i	$-0.269793\pi$
$41$	8.47515	1.32360	0.661798	+	0.749682i	$0.269793\pi$
$42$	0	0
$43$	8.59628	1.31092	0.655460	−	0.755230i	$-0.272475\pi$
$43$	8.59628	1.31092	0.655460	+	0.755230i	$0.272475\pi$
$44$	0	0
$45$	−2.10680	−0.314063
$46$	0	0
$47$	5.97869	0.872082	0.436041	−	0.899927i	$-0.356380\pi$
$47$	5.97869	0.872082	0.436041	+	0.899927i	$0.356380\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.24990	−0.735134
$52$	0	0
$53$	10.8069	1.48444	0.742219	−	0.670158i	$-0.233774\pi$
$53$	10.8069	1.48444	0.742219	+	0.670158i	$0.233774\pi$
$54$	0	0
$55$	−4.53588	−0.611617
$56$	0	0
$57$	0.841060	0.111401
$58$	0	0
$59$	5.83536	0.759700	0.379850	−	0.925048i	$-0.375976\pi$
$59$	5.83536	0.759700	0.379850	+	0.925048i	$0.375976\pi$
$60$	0	0
$61$	−2.83558	−0.363059	−0.181530	−	0.983385i	$-0.558105\pi$
$61$	−2.83558	−0.363059	−0.181530	+	0.983385i	$0.558105\pi$
$62$	0	0
$63$	2.45865	0.309761
$64$	0	0
$65$	−0.856892	−0.106284
$66$	0	0
$67$	−10.0391	−1.22647	−0.613235	−	0.789900i	$-0.710132\pi$
$67$	−10.0391	−1.22647	−0.613235	+	0.789900i	$0.710132\pi$
$68$	0	0
$69$	4.54196	0.546788
$70$	0	0
$71$	−6.12899	−0.727377	−0.363689	−	0.931521i	$-0.618483\pi$
$71$	−6.12899	−0.727377	−0.363689	+	0.931521i	$0.618483\pi$
$72$	0	0
$73$	10.0079	1.17133	0.585666	−	0.810553i	$-0.300833\pi$
$73$	10.0079	1.17133	0.585666	+	0.810553i	$0.300833\pi$
$74$	0	0
$75$	3.13858	0.362412
$76$	0	0
$77$	5.29340	0.603239
$78$	0	0
$79$	8.70793	0.979719	0.489860	−	0.871801i	$-0.337048\pi$
$79$	8.70793	0.979719	0.489860	+	0.871801i	$0.337048\pi$
$80$	0	0
$81$	4.42090	0.491211
$82$	0	0
$83$	−9.26469	−1.01693	−0.508466	−	0.861082i	$-0.669787\pi$
$83$	−9.26469	−1.01693	−0.508466	+	0.861082i	$0.669787\pi$
$84$	0	0
$85$	6.11417	0.663175
$86$	0	0
$87$	−0.893040	−0.0957440
$88$	0	0
$89$	7.67959	0.814035	0.407018	−	0.913420i	$-0.366569\pi$
$89$	7.67959	0.814035	0.407018	+	0.913420i	$0.366569\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−3.51285	−0.364265
$94$	0	0
$95$	−0.979520	−0.100497
$96$	0	0
$97$	−2.46070	−0.249846	−0.124923	−	0.992166i	$-0.539868\pi$
$97$	−2.46070	−0.249846	−0.124923	+	0.992166i	$0.539868\pi$
$98$	0	0
$99$	13.0146	1.30802
$100$	0	0
$101$	11.4150	1.13583	0.567917	−	0.823086i	$-0.307750\pi$
$101$	11.4150	1.13583	0.567917	+	0.823086i	$0.307750\pi$
$102$	0	0
$103$	6.90045	0.679921	0.339961	−	0.940440i	$-0.389586\pi$
$103$	6.90045	0.679921	0.339961	+	0.940440i	$0.389586\pi$
$104$	0	0
$105$	0.630472	0.0615277
$106$	0	0
$107$	−9.93632	−0.960580	−0.480290	−	0.877110i	$-0.659469\pi$
$107$	−9.93632	−0.960580	−0.480290	+	0.877110i	$0.659469\pi$
$108$	0	0
$109$	15.1873	1.45468	0.727338	−	0.686279i	$-0.240757\pi$
$109$	15.1873	1.45468	0.727338	+	0.686279i	$0.240757\pi$
$110$	0	0
$111$	3.18504	0.302311
$112$	0	0
$113$	−13.1196	−1.23419	−0.617093	−	0.786890i	$-0.711690\pi$
$113$	−13.1196	−1.23419	−0.617093	+	0.786890i	$0.711690\pi$
$114$	0	0
$115$	−5.28968	−0.493266
$116$	0	0
$117$	2.45865	0.227302
$118$	0	0
$119$	−7.13529	−0.654091
$120$	0	0
$121$	17.0201	1.54728
$122$	0	0
$123$	−6.23573	−0.562257
$124$	0	0
$125$	−7.93973	−0.710151
$126$	0	0
$127$	8.70939	0.772833	0.386417	−	0.922324i	$-0.373713\pi$
$127$	8.70939	0.772833	0.386417	+	0.922324i	$0.373713\pi$
$128$	0	0
$129$	−6.32485	−0.556872
$130$	0	0
$131$	−7.01266	−0.612699	−0.306350	−	0.951919i	$-0.599108\pi$
$131$	−7.01266	−0.612699	−0.306350	+	0.951919i	$0.599108\pi$
$132$	0	0
$133$	1.14311	0.0991200
$134$	0	0
$135$	3.44152	0.296199
$136$	0	0
$137$	−2.86804	−0.245034	−0.122517	−	0.992466i	$-0.539097\pi$
$137$	−2.86804	−0.245034	−0.122517	+	0.992466i	$0.539097\pi$
$138$	0	0
$139$	0.509801	0.0432407	0.0216204	−	0.999766i	$-0.493117\pi$
$139$	0.509801	0.0432407	0.0216204	+	0.999766i	$0.493117\pi$
$140$	0	0
$141$	−4.39892	−0.370456
$142$	0	0
$143$	5.29340	0.442657
$144$	0	0
$145$	1.04006	0.0863721
$146$	0	0
$147$	−0.735766	−0.0606849
$148$	0	0
$149$	−6.30355	−0.516407	−0.258203	−	0.966091i	$-0.583130\pi$
$149$	−6.30355	−0.516407	−0.258203	+	0.966091i	$0.583130\pi$
$150$	0	0
$151$	0.942016	0.0766602	0.0383301	−	0.999265i	$-0.487796\pi$
$151$	0.942016	0.0766602	0.0383301	+	0.999265i	$0.487796\pi$
$152$	0	0
$153$	−17.5432	−1.41828
$154$	0	0
$155$	4.09115	0.328609
$156$	0	0
$157$	−2.84255	−0.226860	−0.113430	−	0.993546i	$-0.536184\pi$
$157$	−2.84255	−0.226860	−0.113430	+	0.993546i	$0.536184\pi$
$158$	0	0
$159$	−7.95132	−0.630581
$160$	0	0
$161$	6.17311	0.486509
$162$	0	0
$163$	−3.34315	−0.261856	−0.130928	−	0.991392i	$-0.541796\pi$
$163$	−3.34315	−0.261856	−0.130928	+	0.991392i	$0.541796\pi$
$164$	0	0
$165$	3.33734	0.259812
$166$	0	0
$167$	−0.639968	−0.0495222	−0.0247611	−	0.999693i	$-0.507883\pi$
$167$	−0.639968	−0.0495222	−0.0247611	+	0.999693i	$0.507883\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.81050	0.214924
$172$	0	0
$173$	10.0070	0.760816	0.380408	−	0.924819i	$-0.375784\pi$
$173$	10.0070	0.760816	0.380408	+	0.924819i	$0.375784\pi$
$174$	0	0
$175$	4.26574	0.322459
$176$	0	0
$177$	−4.29346	−0.322716
$178$	0	0
$179$	−4.63890	−0.346727	−0.173364	−	0.984858i	$-0.555464\pi$
$179$	−4.63890	−0.346727	−0.173364	+	0.984858i	$0.555464\pi$
$180$	0	0
$181$	0.228760	0.0170036	0.00850179	−	0.999964i	$-0.497294\pi$
$181$	0.228760	0.0170036	0.00850179	+	0.999964i	$0.497294\pi$
$182$	0	0
$183$	2.08633	0.154226
$184$	0	0
$185$	−3.70938	−0.272719
$186$	0	0
$187$	−37.7700	−2.76201
$188$	0	0
$189$	−4.01629	−0.292142
$190$	0	0
$191$	24.4939	1.77231	0.886157	−	0.463385i	$-0.153365\pi$
$191$	24.4939	1.77231	0.886157	+	0.463385i	$0.153365\pi$
$192$	0	0
$193$	11.9131	0.857525	0.428762	−	0.903417i	$-0.358950\pi$
$193$	11.9131	0.857525	0.428762	+	0.903417i	$0.358950\pi$
$194$	0	0
$195$	0.630472	0.0451490
$196$	0	0
$197$	−6.01070	−0.428244	−0.214122	−	0.976807i	$-0.568689\pi$
$197$	−6.01070	−0.428244	−0.214122	+	0.976807i	$0.568689\pi$
$198$	0	0
$199$	12.5626	0.890536	0.445268	−	0.895397i	$-0.353108\pi$
$199$	12.5626	0.890536	0.445268	+	0.895397i	$0.353108\pi$
$200$	0	0
$201$	7.38642	0.520998
$202$	0	0
$203$	−1.21376	−0.0851890
$204$	0	0
$205$	7.26229	0.507220
$206$	0	0
$207$	15.1775	1.05491
$208$	0	0
$209$	6.05093	0.418552
$210$	0	0
$211$	−14.4316	−0.993515	−0.496758	−	0.867889i	$-0.665476\pi$
$211$	−14.4316	−0.993515	−0.496758	+	0.867889i	$0.665476\pi$
$212$	0	0
$213$	4.50950	0.308986
$214$	0	0
$215$	7.36608	0.502363
$216$	0	0
$217$	−4.77441	−0.324108
$218$	0	0
$219$	−7.36344	−0.497575
$220$	0	0
$221$	−7.13529	−0.479972
$222$	0	0
$223$	6.98634	0.467840	0.233920	−	0.972256i	$-0.424845\pi$
$223$	6.98634	0.467840	0.233920	+	0.972256i	$0.424845\pi$
$224$	0	0
$225$	10.4879	0.699196
$226$	0	0
$227$	4.82482	0.320234	0.160117	−	0.987098i	$-0.448813\pi$
$227$	4.82482	0.320234	0.160117	+	0.987098i	$0.448813\pi$
$228$	0	0
$229$	−22.8424	−1.50947	−0.754733	−	0.656032i	$-0.772233\pi$
$229$	−22.8424	−1.50947	−0.754733	+	0.656032i	$0.772233\pi$
$230$	0	0
$231$	−3.89471	−0.256253
$232$	0	0
$233$	22.9996	1.50676	0.753379	−	0.657587i	$-0.228423\pi$
$233$	22.9996	1.50676	0.753379	+	0.657587i	$0.228423\pi$
$234$	0	0
$235$	5.12309	0.334194
$236$	0	0
$237$	−6.40700	−0.416179
$238$	0	0
$239$	19.1909	1.24136	0.620678	−	0.784065i	$-0.286857\pi$
$239$	19.1909	1.24136	0.620678	+	0.784065i	$0.286857\pi$
$240$	0	0
$241$	24.8655	1.60173	0.800863	−	0.598847i	$-0.204374\pi$
$241$	24.8655	1.60173	0.800863	+	0.598847i	$0.204374\pi$
$242$	0	0
$243$	−15.3016	−0.981599
$244$	0	0
$245$	0.856892	0.0547448
$246$	0	0
$247$	1.14311	0.0727342
$248$	0	0
$249$	6.81664	0.431987
$250$	0	0
$251$	−8.42422	−0.531732	−0.265866	−	0.964010i	$-0.585658\pi$
$251$	−8.42422	−0.531732	−0.265866	+	0.964010i	$0.585658\pi$
$252$	0	0
$253$	32.6767	2.05437
$254$	0	0
$255$	−4.49860	−0.281713
$256$	0	0
$257$	−12.0509	−0.751717	−0.375858	−	0.926677i	$-0.622652\pi$
$257$	−12.0509	−0.751717	−0.375858	+	0.926677i	$0.622652\pi$
$258$	0	0
$259$	4.32888	0.268983
$260$	0	0
$261$	−2.98420	−0.184717
$262$	0	0
$263$	26.7973	1.65239	0.826195	−	0.563384i	$-0.190501\pi$
$263$	26.7973	1.65239	0.826195	+	0.563384i	$0.190501\pi$
$264$	0	0
$265$	9.26032	0.568857
$266$	0	0
$267$	−5.65038	−0.345798
$268$	0	0
$269$	16.0222	0.976894	0.488447	−	0.872594i	$-0.337564\pi$
$269$	16.0222	0.976894	0.488447	+	0.872594i	$0.337564\pi$
$270$	0	0
$271$	−11.2249	−0.681861	−0.340931	−	0.940088i	$-0.610742\pi$
$271$	−11.2249	−0.681861	−0.340931	+	0.940088i	$0.610742\pi$
$272$	0	0
$273$	−0.735766	−0.0445306
$274$	0	0
$275$	22.5803	1.36164
$276$	0	0
$277$	0.683940	0.0410940	0.0205470	−	0.999789i	$-0.493459\pi$
$277$	0.683940	0.0410940	0.0205470	+	0.999789i	$0.493459\pi$
$278$	0	0
$279$	−11.7386	−0.702771
$280$	0	0
$281$	−3.34389	−0.199480	−0.0997399	−	0.995014i	$-0.531801\pi$
$281$	−3.34389	−0.199480	−0.0997399	+	0.995014i	$0.531801\pi$
$282$	0	0
$283$	21.6803	1.28876	0.644381	−	0.764705i	$-0.277115\pi$
$283$	21.6803	1.28876	0.644381	+	0.764705i	$0.277115\pi$
$284$	0	0
$285$	0.720697	0.0426904
$286$	0	0
$287$	−8.47515	−0.500273
$288$	0	0
$289$	33.9124	1.99485
$290$	0	0
$291$	1.81050	0.106133
$292$	0	0
$293$	21.0101	1.22743	0.613713	−	0.789529i	$-0.289675\pi$
$293$	21.0101	1.22743	0.613713	+	0.789529i	$0.289675\pi$
$294$	0	0
$295$	5.00028	0.291127
$296$	0	0
$297$	−21.2598	−1.23362
$298$	0	0
$299$	6.17311	0.357000
$300$	0	0
$301$	−8.59628	−0.495481
$302$	0	0
$303$	−8.39876	−0.482496
$304$	0	0
$305$	−2.42979	−0.139129
$306$	0	0
$307$	18.0255	1.02877	0.514385	−	0.857560i	$-0.328020\pi$
$307$	18.0255	1.02877	0.514385	+	0.857560i	$0.328020\pi$
$308$	0	0
$309$	−5.07711	−0.288827
$310$	0	0
$311$	26.5642	1.50632	0.753159	−	0.657839i	$-0.228529\pi$
$311$	26.5642	1.50632	0.753159	+	0.657839i	$0.228529\pi$
$312$	0	0
$313$	2.20952	0.124889	0.0624446	−	0.998048i	$-0.480110\pi$
$313$	2.20952	0.124889	0.0624446	+	0.998048i	$0.480110\pi$
$314$	0	0
$315$	2.10680	0.118705
$316$	0	0
$317$	−21.2159	−1.19160	−0.595801	−	0.803132i	$-0.703165\pi$
$317$	−21.2159	−1.19160	−0.595801	+	0.803132i	$0.703165\pi$
$318$	0	0
$319$	−6.42490	−0.359725
$320$	0	0
$321$	7.31081	0.408049
$322$	0	0
$323$	−8.15641	−0.453835
$324$	0	0
$325$	4.26574	0.236620
$326$	0	0
$327$	−11.1743	−0.617939
$328$	0	0
$329$	−5.97869	−0.329616
$330$	0	0
$331$	−3.72122	−0.204537	−0.102268	−	0.994757i	$-0.532610\pi$
$331$	−3.72122	−0.204537	−0.102268	+	0.994757i	$0.532610\pi$
$332$	0	0
$333$	10.6432	0.583243
$334$	0	0
$335$	−8.60242	−0.470000
$336$	0	0
$337$	−13.8460	−0.754241	−0.377120	−	0.926164i	$-0.623086\pi$
$337$	−13.8460	−0.754241	−0.377120	+	0.926164i	$0.623086\pi$
$338$	0	0
$339$	9.65293	0.524275
$340$	0	0
$341$	−25.2729	−1.36860
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.89197	0.209537
$346$	0	0
$347$	−4.34530	−0.233268	−0.116634	−	0.993175i	$-0.537210\pi$
$347$	−4.34530	−0.233268	−0.116634	+	0.993175i	$0.537210\pi$
$348$	0	0
$349$	−15.4939	−0.829370	−0.414685	−	0.909965i	$-0.636108\pi$
$349$	−15.4939	−0.829370	−0.414685	+	0.909965i	$0.636108\pi$
$350$	0	0
$351$	−4.01629	−0.214374
$352$	0	0
$353$	−32.3836	−1.72360	−0.861802	−	0.507246i	$-0.830664\pi$
$353$	−32.3836	−1.72360	−0.861802	+	0.507246i	$0.830664\pi$
$354$	0	0
$355$	−5.25188	−0.278741
$356$	0	0
$357$	5.24990	0.277854
$358$	0	0
$359$	−24.2941	−1.28219	−0.641096	−	0.767461i	$-0.721520\pi$
$359$	−24.2941	−1.28219	−0.641096	+	0.767461i	$0.721520\pi$
$360$	0	0
$361$	−17.6933	−0.931227
$362$	0	0
$363$	−12.5228	−0.657278
$364$	0	0
$365$	8.57566	0.448870
$366$	0	0
$367$	12.3079	0.642467	0.321233	−	0.947000i	$-0.395903\pi$
$367$	12.3079	0.642467	0.321233	+	0.947000i	$0.395903\pi$
$368$	0	0
$369$	−20.8374	−1.08475
$370$	0	0
$371$	−10.8069	−0.561065
$372$	0	0
$373$	14.8051	0.766579	0.383290	−	0.923628i	$-0.374791\pi$
$373$	14.8051	0.766579	0.383290	+	0.923628i	$0.374791\pi$
$374$	0	0
$375$	5.84179	0.301668
$376$	0	0
$377$	−1.21376	−0.0625116
$378$	0	0
$379$	23.0467	1.18383	0.591915	−	0.806000i	$-0.298372\pi$
$379$	23.0467	1.18383	0.591915	+	0.806000i	$0.298372\pi$
$380$	0	0
$381$	−6.40807	−0.328295
$382$	0	0
$383$	−9.68193	−0.494724	−0.247362	−	0.968923i	$-0.579564\pi$
$383$	−9.68193	−0.494724	−0.247362	+	0.968923i	$0.579564\pi$
$384$	0	0
$385$	4.53588	0.231170
$386$	0	0
$387$	−21.1352	−1.07436
$388$	0	0
$389$	−8.72433	−0.442341	−0.221171	−	0.975235i	$-0.570988\pi$
$389$	−8.72433	−0.442341	−0.221171	+	0.975235i	$0.570988\pi$
$390$	0	0
$391$	−44.0469	−2.22755
$392$	0	0
$393$	5.15968	0.260271
$394$	0	0
$395$	7.46176	0.375442
$396$	0	0
$397$	34.1651	1.71470	0.857350	−	0.514735i	$-0.172110\pi$
$397$	34.1651	1.71470	0.857350	+	0.514735i	$0.172110\pi$
$398$	0	0
$399$	−0.841060	−0.0421057
$400$	0	0
$401$	5.31326	0.265331	0.132666	−	0.991161i	$-0.457646\pi$
$401$	5.31326	0.265331	0.132666	+	0.991161i	$0.457646\pi$
$402$	0	0
$403$	−4.77441	−0.237830
$404$	0	0
$405$	3.78823	0.188239
$406$	0	0
$407$	22.9145	1.13583
$408$	0	0
$409$	−24.4383	−1.20840	−0.604198	−	0.796834i	$-0.706506\pi$
$409$	−24.4383	−1.20840	−0.604198	+	0.796834i	$0.706506\pi$
$410$	0	0
$411$	2.11021	0.104089
$412$	0	0
$413$	−5.83536	−0.287140
$414$	0	0
$415$	−7.93884	−0.389702
$416$	0	0
$417$	−0.375094	−0.0183684
$418$	0	0
$419$	−37.5863	−1.83621	−0.918106	−	0.396334i	$-0.870282\pi$
$419$	−37.5863	−1.83621	−0.918106	+	0.396334i	$0.870282\pi$
$420$	0	0
$421$	−35.6890	−1.73937	−0.869687	−	0.493603i	$-0.835679\pi$
$421$	−35.6890	−1.73937	−0.869687	+	0.493603i	$0.835679\pi$
$422$	0	0
$423$	−14.6995	−0.714714
$424$	0	0
$425$	−30.4373	−1.47642
$426$	0	0
$427$	2.83558	0.137224
$428$	0	0
$429$	−3.89471	−0.188038
$430$	0	0
$431$	4.65716	0.224327	0.112164	−	0.993690i	$-0.464222\pi$
$431$	4.65716	0.224327	0.112164	+	0.993690i	$0.464222\pi$
$432$	0	0
$433$	6.76438	0.325076	0.162538	−	0.986702i	$-0.448032\pi$
$433$	6.76438	0.325076	0.162538	+	0.986702i	$0.448032\pi$
$434$	0	0
$435$	−0.765239	−0.0366904
$436$	0	0
$437$	7.05653	0.337560
$438$	0	0
$439$	12.4235	0.592941	0.296471	−	0.955042i	$-0.404190\pi$
$439$	12.4235	0.592941	0.296471	+	0.955042i	$0.404190\pi$
$440$	0	0
$441$	−2.45865	−0.117079
$442$	0	0
$443$	−38.8193	−1.84436	−0.922181	−	0.386758i	$-0.873595\pi$
$443$	−38.8193	−1.84436	−0.922181	+	0.386758i	$0.873595\pi$
$444$	0	0
$445$	6.58058	0.311949
$446$	0	0
$447$	4.63794	0.219367
$448$	0	0
$449$	4.51542	0.213096	0.106548	−	0.994308i	$-0.466020\pi$
$449$	4.51542	0.213096	0.106548	+	0.994308i	$0.466020\pi$
$450$	0	0
$451$	−44.8624	−2.11249
$452$	0	0
$453$	−0.693103	−0.0325648
$454$	0	0
$455$	0.856892	0.0401717
$456$	0	0
$457$	−13.9847	−0.654175	−0.327088	−	0.944994i	$-0.606067\pi$
$457$	−13.9847	−0.654175	−0.327088	+	0.944994i	$0.606067\pi$
$458$	0	0
$459$	28.6574	1.33761
$460$	0	0
$461$	42.4929	1.97909	0.989545	−	0.144222i	$-0.0460678\pi$
$461$	42.4929	1.97909	0.989545	+	0.144222i	$0.0460678\pi$
$462$	0	0
$463$	22.1367	1.02878	0.514389	−	0.857557i	$-0.328019\pi$
$463$	22.1367	1.02878	0.514389	+	0.857557i	$0.328019\pi$
$464$	0	0
$465$	−3.01013	−0.139591
$466$	0	0
$467$	6.08560	0.281608	0.140804	−	0.990037i	$-0.455031\pi$
$467$	6.08560	0.281608	0.140804	+	0.990037i	$0.455031\pi$
$468$	0	0
$469$	10.0391	0.463562
$470$	0	0
$471$	2.09145	0.0963691
$472$	0	0
$473$	−45.5036	−2.09226
$474$	0	0
$475$	4.87620	0.223735
$476$	0	0
$477$	−26.5703	−1.21657
$478$	0	0
$479$	24.0642	1.09952	0.549760	−	0.835323i	$-0.314719\pi$
$479$	24.0642	1.09952	0.549760	+	0.835323i	$0.314719\pi$
$480$	0	0
$481$	4.32888	0.197380
$482$	0	0
$483$	−4.54196	−0.206666
$484$	0	0
$485$	−2.10856	−0.0957446
$486$	0	0
$487$	21.9199	0.993286	0.496643	−	0.867955i	$-0.334566\pi$
$487$	21.9199	0.993286	0.496643	+	0.867955i	$0.334566\pi$
$488$	0	0
$489$	2.45978	0.111235
$490$	0	0
$491$	−1.92360	−0.0868109	−0.0434055	−	0.999058i	$-0.513821\pi$
$491$	−1.92360	−0.0868109	−0.0434055	+	0.999058i	$0.513821\pi$
$492$	0	0
$493$	8.66050	0.390049
$494$	0	0
$495$	11.1521	0.501251
$496$	0	0
$497$	6.12899	0.274923
$498$	0	0
$499$	28.8472	1.29138	0.645690	−	0.763600i	$-0.276570\pi$
$499$	28.8472	1.29138	0.645690	+	0.763600i	$0.276570\pi$
$500$	0	0
$501$	0.470867	0.0210368
$502$	0	0
$503$	20.5272	0.915263	0.457632	−	0.889142i	$-0.348698\pi$
$503$	20.5272	0.915263	0.457632	+	0.889142i	$0.348698\pi$
$504$	0	0
$505$	9.78141	0.435267
$506$	0	0
$507$	−0.735766	−0.0326765
$508$	0	0
$509$	−19.0544	−0.844572	−0.422286	−	0.906463i	$-0.638772\pi$
$509$	−19.0544	−0.844572	−0.422286	+	0.906463i	$0.638772\pi$
$510$	0	0
$511$	−10.0079	−0.442722
$512$	0	0
$513$	−4.59105	−0.202700
$514$	0	0
$515$	5.91294	0.260555
$516$	0	0
$517$	−31.6476	−1.39186
$518$	0	0
$519$	−7.36279	−0.323190
$520$	0	0
$521$	−24.9647	−1.09372	−0.546861	−	0.837223i	$-0.684178\pi$
$521$	−24.9647	−1.09372	−0.546861	+	0.837223i	$0.684178\pi$
$522$	0	0
$523$	11.2523	0.492027	0.246013	−	0.969266i	$-0.420879\pi$
$523$	11.2523	0.492027	0.246013	+	0.969266i	$0.420879\pi$
$524$	0	0
$525$	−3.13858	−0.136979
$526$	0	0
$527$	34.0668	1.48397
$528$	0	0
$529$	15.1072	0.656837
$530$	0	0
$531$	−14.3471	−0.622612
$532$	0	0
$533$	−8.47515	−0.367100
$534$	0	0
$535$	−8.51435	−0.368108
$536$	0	0
$537$	3.41314	0.147288
$538$	0	0
$539$	−5.29340	−0.228003
$540$	0	0
$541$	20.5501	0.883517	0.441759	−	0.897134i	$-0.354355\pi$
$541$	20.5501	0.883517	0.441759	+	0.897134i	$0.354355\pi$
$542$	0	0
$543$	−0.168314	−0.00722303
$544$	0	0
$545$	13.0139	0.557452
$546$	0	0
$547$	23.3560	0.998629	0.499314	−	0.866421i	$-0.333585\pi$
$547$	23.3560	0.998629	0.499314	+	0.866421i	$0.333585\pi$
$548$	0	0
$549$	6.97170	0.297545
$550$	0	0
$551$	−1.38745	−0.0591075
$552$	0	0
$553$	−8.70793	−0.370299
$554$	0	0
$555$	2.72924	0.115850
$556$	0	0
$557$	−26.7694	−1.13426	−0.567128	−	0.823630i	$-0.691946\pi$
$557$	−26.7694	−1.13426	−0.567128	+	0.823630i	$0.691946\pi$
$558$	0	0
$559$	−8.59628	−0.363584
$560$	0	0
$561$	27.7899	1.17329
$562$	0	0
$563$	−36.3128	−1.53040	−0.765200	−	0.643793i	$-0.777360\pi$
$563$	−36.3128	−1.53040	−0.765200	+	0.643793i	$0.777360\pi$
$564$	0	0
$565$	−11.2421	−0.472957
$566$	0	0
$567$	−4.42090	−0.185660
$568$	0	0
$569$	−26.7239	−1.12033	−0.560163	−	0.828382i	$-0.689262\pi$
$569$	−26.7239	−1.12033	−0.560163	+	0.828382i	$0.689262\pi$
$570$	0	0
$571$	15.1208	0.632787	0.316394	−	0.948628i	$-0.397528\pi$
$571$	15.1208	0.632787	0.316394	+	0.948628i	$0.397528\pi$
$572$	0	0
$573$	−18.0218	−0.752869
$574$	0	0
$575$	26.3328	1.09816
$576$	0	0
$577$	−19.0625	−0.793583	−0.396792	−	0.917909i	$-0.629876\pi$
$577$	−19.0625	−0.793583	−0.396792	+	0.917909i	$0.629876\pi$
$578$	0	0
$579$	−8.76526	−0.364272
$580$	0	0
$581$	9.26469	0.384364
$582$	0	0
$583$	−57.2051	−2.36919
$584$	0	0
$585$	2.10680	0.0871053
$586$	0	0
$587$	−17.1581	−0.708189	−0.354094	−	0.935210i	$-0.615211\pi$
$587$	−17.1581	−0.708189	−0.354094	+	0.935210i	$0.615211\pi$
$588$	0	0
$589$	−5.45766	−0.224879
$590$	0	0
$591$	4.42247	0.181916
$592$	0	0
$593$	−47.5299	−1.95182	−0.975910	−	0.218172i	$-0.929991\pi$
$593$	−47.5299	−1.95182	−0.975910	+	0.218172i	$0.929991\pi$
$594$	0	0
$595$	−6.11417	−0.250657
$596$	0	0
$597$	−9.24310	−0.378295
$598$	0	0
$599$	−45.2109	−1.84727	−0.923633	−	0.383279i	$-0.874795\pi$
$599$	−45.2109	−1.84727	−0.923633	+	0.383279i	$0.874795\pi$
$600$	0	0
$601$	6.47616	0.264168	0.132084	−	0.991239i	$-0.457833\pi$
$601$	6.47616	0.264168	0.132084	+	0.991239i	$0.457833\pi$
$602$	0	0
$603$	24.6826	1.00515
$604$	0	0
$605$	14.5844	0.592941
$606$	0	0
$607$	−24.5572	−0.996746	−0.498373	−	0.866963i	$-0.666069\pi$
$607$	−24.5572	−0.996746	−0.498373	+	0.866963i	$0.666069\pi$
$608$	0	0
$609$	0.893040	0.0361878
$610$	0	0
$611$	−5.97869	−0.241872
$612$	0	0
$613$	−15.8984	−0.642128	−0.321064	−	0.947057i	$-0.604041\pi$
$613$	−15.8984	−0.642128	−0.321064	+	0.947057i	$0.604041\pi$
$614$	0	0
$615$	−5.34335	−0.215464
$616$	0	0
$617$	−1.87974	−0.0756756	−0.0378378	−	0.999284i	$-0.512047\pi$
$617$	−1.87974	−0.0756756	−0.0378378	+	0.999284i	$0.512047\pi$
$618$	0	0
$619$	−32.1905	−1.29385	−0.646923	−	0.762555i	$-0.723944\pi$
$619$	−32.1905	−1.29385	−0.646923	+	0.762555i	$0.723944\pi$
$620$	0	0
$621$	−24.7930	−0.994908
$622$	0	0
$623$	−7.67959	−0.307676
$624$	0	0
$625$	14.5252	0.581007
$626$	0	0
$627$	−4.45207	−0.177799
$628$	0	0
$629$	−30.8878	−1.23158
$630$	0	0
$631$	21.0326	0.837296	0.418648	−	0.908149i	$-0.362504\pi$
$631$	21.0326	0.837296	0.418648	+	0.908149i	$0.362504\pi$
$632$	0	0
$633$	10.6183	0.422040
$634$	0	0
$635$	7.46301	0.296160
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	15.0690	0.596122
$640$	0	0
$641$	−48.3515	−1.90977	−0.954885	−	0.296976i	$-0.904022\pi$
$641$	−48.3515	−1.90977	−0.954885	+	0.296976i	$0.904022\pi$
$642$	0	0
$643$	17.9445	0.707661	0.353830	−	0.935310i	$-0.384879\pi$
$643$	17.9445	0.707661	0.353830	+	0.935310i	$0.384879\pi$
$644$	0	0
$645$	−5.41971	−0.213401
$646$	0	0
$647$	38.3527	1.50780	0.753901	−	0.656988i	$-0.228170\pi$
$647$	38.3527	1.50780	0.753901	+	0.656988i	$0.228170\pi$
$648$	0	0
$649$	−30.8889	−1.21250
$650$	0	0
$651$	3.51285	0.137679
$652$	0	0
$653$	4.72534	0.184917	0.0924585	−	0.995717i	$-0.470527\pi$
$653$	4.72534	0.184917	0.0924585	+	0.995717i	$0.470527\pi$
$654$	0	0
$655$	−6.00910	−0.234795
$656$	0	0
$657$	−24.6058	−0.959964
$658$	0	0
$659$	34.9188	1.36025	0.680123	−	0.733098i	$-0.261927\pi$
$659$	34.9188	1.36025	0.680123	+	0.733098i	$0.261927\pi$
$660$	0	0
$661$	23.5880	0.917467	0.458733	−	0.888574i	$-0.348303\pi$
$661$	23.5880	0.917467	0.458733	+	0.888574i	$0.348303\pi$
$662$	0	0
$663$	5.24990	0.203889
$664$	0	0
$665$	0.979520	0.0379842
$666$	0	0
$667$	−7.49264	−0.290116
$668$	0	0
$669$	−5.14031	−0.198736
$670$	0	0
$671$	15.0099	0.579450
$672$	0	0
$673$	−0.165769	−0.00638993	−0.00319496	−	0.999995i	$-0.501017\pi$
$673$	−0.165769	−0.00638993	−0.00319496	+	0.999995i	$0.501017\pi$
$674$	0	0
$675$	−17.1324	−0.659427
$676$	0	0
$677$	29.6955	1.14129	0.570646	−	0.821196i	$-0.306693\pi$
$677$	29.6955	1.14129	0.570646	+	0.821196i	$0.306693\pi$
$678$	0	0
$679$	2.46070	0.0944331
$680$	0	0
$681$	−3.54994	−0.136034
$682$	0	0
$683$	25.2448	0.965966	0.482983	−	0.875630i	$-0.339553\pi$
$683$	25.2448	0.965966	0.482983	+	0.875630i	$0.339553\pi$
$684$	0	0
$685$	−2.45760	−0.0939002
$686$	0	0
$687$	16.8066	0.641213
$688$	0	0
$689$	−10.8069	−0.411709
$690$	0	0
$691$	−21.8769	−0.832238	−0.416119	−	0.909310i	$-0.636610\pi$
$691$	−21.8769	−0.832238	−0.416119	+	0.909310i	$0.636610\pi$
$692$	0	0
$693$	−13.0146	−0.494385
$694$	0	0
$695$	0.436844	0.0165704
$696$	0	0
$697$	60.4727	2.29057
$698$	0	0
$699$	−16.9224	−0.640062
$700$	0	0
$701$	−7.84464	−0.296288	−0.148144	−	0.988966i	$-0.547330\pi$
$701$	−7.84464	−0.296288	−0.148144	+	0.988966i	$0.547330\pi$
$702$	0	0
$703$	4.94838	0.186632
$704$	0	0
$705$	−3.76940	−0.141964
$706$	0	0
$707$	−11.4150	−0.429305
$708$	0	0
$709$	−26.6225	−0.999827	−0.499914	−	0.866075i	$-0.666635\pi$
$709$	−26.6225	−0.999827	−0.499914	+	0.866075i	$0.666635\pi$
$710$	0	0
$711$	−21.4098	−0.802928
$712$	0	0
$713$	−29.4729	−1.10377
$714$	0	0
$715$	4.53588	0.169632
$716$	0	0
$717$	−14.1200	−0.527322
$718$	0	0
$719$	−7.77031	−0.289784	−0.144892	−	0.989448i	$-0.546283\pi$
$719$	−7.77031	−0.289784	−0.144892	+	0.989448i	$0.546283\pi$
$720$	0	0
$721$	−6.90045	−0.256986
$722$	0	0
$723$	−18.2952	−0.680405
$724$	0	0
$725$	−5.17756	−0.192290
$726$	0	0
$727$	−32.0853	−1.18998	−0.594989	−	0.803734i	$-0.702844\pi$
$727$	−32.0853	−1.18998	−0.594989	+	0.803734i	$0.702844\pi$
$728$	0	0
$729$	−2.00430	−0.0742332
$730$	0	0
$731$	61.3370	2.26863
$732$	0	0
$733$	46.7875	1.72814	0.864068	−	0.503375i	$-0.167909\pi$
$733$	46.7875	1.72814	0.864068	+	0.503375i	$0.167909\pi$
$734$	0	0
$735$	−0.630472	−0.0232553
$736$	0	0
$737$	53.1410	1.95747
$738$	0	0
$739$	40.5225	1.49064	0.745322	−	0.666705i	$-0.232296\pi$
$739$	40.5225	1.49064	0.745322	+	0.666705i	$0.232296\pi$
$740$	0	0
$741$	−0.841060	−0.0308971
$742$	0	0
$743$	−44.5100	−1.63292	−0.816458	−	0.577406i	$-0.804065\pi$
$743$	−44.5100	−1.63292	−0.816458	+	0.577406i	$0.804065\pi$
$744$	0	0
$745$	−5.40146	−0.197894
$746$	0	0
$747$	22.7786	0.833426
$748$	0	0
$749$	9.93632	0.363065
$750$	0	0
$751$	−24.6523	−0.899574	−0.449787	−	0.893136i	$-0.648500\pi$
$751$	−24.6523	−0.899574	−0.449787	+	0.893136i	$0.648500\pi$
$752$	0	0
$753$	6.19825	0.225877
$754$	0	0
$755$	0.807206	0.0293772
$756$	0	0
$757$	−1.58543	−0.0576233	−0.0288116	−	0.999585i	$-0.509172\pi$
$757$	−1.58543	−0.0576233	−0.0288116	+	0.999585i	$0.509172\pi$
$758$	0	0
$759$	−24.0424	−0.872685
$760$	0	0
$761$	45.0122	1.63169	0.815845	−	0.578271i	$-0.196272\pi$
$761$	45.0122	1.63169	0.815845	+	0.578271i	$0.196272\pi$
$762$	0	0
$763$	−15.1873	−0.549816
$764$	0	0
$765$	−15.0326	−0.543505
$766$	0	0
$767$	−5.83536	−0.210703
$768$	0	0
$769$	−10.1507	−0.366042	−0.183021	−	0.983109i	$-0.558588\pi$
$769$	−10.1507	−0.366042	−0.183021	+	0.983109i	$0.558588\pi$
$770$	0	0
$771$	8.86666	0.319325
$772$	0	0
$773$	−4.37831	−0.157477	−0.0787385	−	0.996895i	$-0.525089\pi$
$773$	−4.37831	−0.157477	−0.0787385	+	0.996895i	$0.525089\pi$
$774$	0	0
$775$	−20.3664	−0.731581
$776$	0	0
$777$	−3.18504	−0.114263
$778$	0	0
$779$	−9.68802	−0.347109
$780$	0	0
$781$	32.4432	1.16091
$782$	0	0
$783$	4.87479	0.174211
$784$	0	0
$785$	−2.43576	−0.0869360
$786$	0	0
$787$	4.88913	0.174279	0.0871394	−	0.996196i	$-0.472227\pi$
$787$	4.88913	0.174279	0.0871394	+	0.996196i	$0.472227\pi$
$788$	0	0
$789$	−19.7165	−0.701926
$790$	0	0
$791$	13.1196	0.466478
$792$	0	0
$793$	2.83558	0.100695
$794$	0	0
$795$	−6.81343	−0.241647
$796$	0	0
$797$	−7.29371	−0.258356	−0.129178	−	0.991621i	$-0.541234\pi$
$797$	−7.29371	−0.258356	−0.129178	+	0.991621i	$0.541234\pi$
$798$	0	0
$799$	42.6597	1.50919
$800$	0	0
$801$	−18.8814	−0.667142
$802$	0	0
$803$	−52.9757	−1.86947
$804$	0	0
$805$	5.28968	0.186437
$806$	0	0
$807$	−11.7886	−0.414979
$808$	0	0
$809$	40.6990	1.43090	0.715451	−	0.698663i	$-0.246221\pi$
$809$	40.6990	1.43090	0.715451	+	0.698663i	$0.246221\pi$
$810$	0	0
$811$	31.3590	1.10116	0.550581	−	0.834782i	$-0.314406\pi$
$811$	31.3590	1.10116	0.550581	+	0.834782i	$0.314406\pi$
$812$	0	0
$813$	8.25886	0.289651
$814$	0	0
$815$	−2.86472	−0.100347
$816$	0	0
$817$	−9.82648	−0.343785
$818$	0	0
$819$	−2.45865	−0.0859121
$820$	0	0
$821$	−29.0879	−1.01517	−0.507587	−	0.861601i	$-0.669462\pi$
$821$	−29.0879	−1.01517	−0.507587	+	0.861601i	$0.669462\pi$
$822$	0	0
$823$	−13.9004	−0.484538	−0.242269	−	0.970209i	$-0.577892\pi$
$823$	−13.9004	−0.484538	−0.242269	+	0.970209i	$0.577892\pi$
$824$	0	0
$825$	−16.6138	−0.578418
$826$	0	0
$827$	17.2293	0.599121	0.299560	−	0.954077i	$-0.403160\pi$
$827$	17.2293	0.599121	0.299560	+	0.954077i	$0.403160\pi$
$828$	0	0
$829$	−48.7780	−1.69413	−0.847065	−	0.531489i	$-0.821633\pi$
$829$	−48.7780	−1.69413	−0.847065	+	0.531489i	$0.821633\pi$
$830$	0	0
$831$	−0.503220	−0.0174565
$832$	0	0
$833$	7.13529	0.247223
$834$	0	0
$835$	−0.548383	−0.0189776
$836$	0	0
$837$	19.1754	0.662799
$838$	0	0
$839$	3.73303	0.128878	0.0644392	−	0.997922i	$-0.479474\pi$
$839$	3.73303	0.128878	0.0644392	+	0.997922i	$0.479474\pi$
$840$	0	0
$841$	−27.5268	−0.949200
$842$	0	0
$843$	2.46032	0.0847380
$844$	0	0
$845$	0.856892	0.0294780
$846$	0	0
$847$	−17.0201	−0.584819
$848$	0	0
$849$	−15.9516	−0.547459
$850$	0	0
$851$	26.7226	0.916040
$852$	0	0
$853$	−29.9665	−1.02603	−0.513017	−	0.858379i	$-0.671472\pi$
$853$	−29.9665	−1.02603	−0.513017	+	0.858379i	$0.671472\pi$
$854$	0	0
$855$	2.40830	0.0823620
$856$	0	0
$857$	43.3076	1.47936	0.739679	−	0.672960i	$-0.234977\pi$
$857$	43.3076	1.47936	0.739679	+	0.672960i	$0.234977\pi$
$858$	0	0
$859$	25.6969	0.876766	0.438383	−	0.898788i	$-0.355551\pi$
$859$	25.6969	0.876766	0.438383	+	0.898788i	$0.355551\pi$
$860$	0	0
$861$	6.23573	0.212513
$862$	0	0
$863$	27.4190	0.933354	0.466677	−	0.884428i	$-0.345451\pi$
$863$	27.4190	0.933354	0.466677	+	0.884428i	$0.345451\pi$
$864$	0	0
$865$	8.57489	0.291555
$866$	0	0
$867$	−24.9516	−0.847400
$868$	0	0
$869$	−46.0946	−1.56365
$870$	0	0
$871$	10.0391	0.340162
$872$	0	0
$873$	6.05000	0.204762
$874$	0	0
$875$	7.93973	0.268412
$876$	0	0
$877$	−57.7993	−1.95174	−0.975871	−	0.218346i	$-0.929934\pi$
$877$	−57.7993	−1.95174	−0.975871	+	0.218346i	$0.929934\pi$
$878$	0	0
$879$	−15.4585	−0.521404
$880$	0	0
$881$	26.2663	0.884935	0.442467	−	0.896785i	$-0.354103\pi$
$881$	26.2663	0.884935	0.442467	+	0.896785i	$0.354103\pi$
$882$	0	0
$883$	53.4865	1.79996	0.899982	−	0.435928i	$-0.143580\pi$
$883$	53.4865	1.79996	0.899982	+	0.435928i	$0.143580\pi$
$884$	0	0
$885$	−3.67903	−0.123669
$886$	0	0
$887$	6.48554	0.217763	0.108882	−	0.994055i	$-0.465273\pi$
$887$	6.48554	0.217763	0.108882	+	0.994055i	$0.465273\pi$
$888$	0	0
$889$	−8.70939	−0.292104
$890$	0	0
$891$	−23.4016	−0.783983
$892$	0	0
$893$	−6.83429	−0.228701
$894$	0	0
$895$	−3.97503	−0.132871
$896$	0	0
$897$	−4.54196	−0.151652
$898$	0	0
$899$	5.79497	0.193273
$900$	0	0
$901$	77.1102	2.56891
$902$	0	0
$903$	6.32485	0.210478
$904$	0	0
$905$	0.196022	0.00651601
$906$	0	0
$907$	9.58107	0.318134	0.159067	−	0.987268i	$-0.449151\pi$
$907$	9.58107	0.318134	0.159067	+	0.987268i	$0.449151\pi$
$908$	0	0
$909$	−28.0654	−0.930872
$910$	0	0
$911$	−23.4620	−0.777330	−0.388665	−	0.921379i	$-0.627064\pi$
$911$	−23.4620	−0.777330	−0.388665	+	0.921379i	$0.627064\pi$
$912$	0	0
$913$	49.0417	1.62304
$914$	0	0
$915$	1.78776	0.0591014
$916$	0	0
$917$	7.01266	0.231579
$918$	0	0
$919$	29.3286	0.967462	0.483731	−	0.875217i	$-0.339281\pi$
$919$	29.3286	0.967462	0.483731	+	0.875217i	$0.339281\pi$
$920$	0	0
$921$	−13.2625	−0.437016
$922$	0	0
$923$	6.12899	0.201738
$924$	0	0
$925$	18.4659	0.607154
$926$	0	0
$927$	−16.9658	−0.557229
$928$	0	0
$929$	6.74246	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$929$	6.74246	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$930$	0	0
$931$	−1.14311	−0.0374639
$932$	0	0
$933$	−19.5450	−0.639876
$934$	0	0
$935$	−32.3648	−1.05844
$936$	0	0
$937$	54.6764	1.78620	0.893100	−	0.449858i	$-0.148525\pi$
$937$	54.6764	1.78620	0.893100	+	0.449858i	$0.148525\pi$
$938$	0	0
$939$	−1.62569	−0.0530523
$940$	0	0
$941$	4.37208	0.142526	0.0712629	−	0.997458i	$-0.477297\pi$
$941$	4.37208	0.142526	0.0712629	+	0.997458i	$0.477297\pi$
$942$	0	0
$943$	−52.3180	−1.70371
$944$	0	0
$945$	−3.44152	−0.111953
$946$	0	0
$947$	−40.2296	−1.30729	−0.653644	−	0.756803i	$-0.726760\pi$
$947$	−40.2296	−1.30729	−0.653644	+	0.756803i	$0.726760\pi$
$948$	0	0
$949$	−10.0079	−0.324869
$950$	0	0
$951$	15.6099	0.506186
$952$	0	0
$953$	50.2547	1.62791	0.813954	−	0.580929i	$-0.197311\pi$
$953$	50.2547	1.62791	0.813954	+	0.580929i	$0.197311\pi$
$954$	0	0
$955$	20.9886	0.679175
$956$	0	0
$957$	4.72722	0.152809
$958$	0	0
$959$	2.86804	0.0926140
$960$	0	0
$961$	−8.20503	−0.264678
$962$	0	0
$963$	24.4299	0.787243
$964$	0	0
$965$	10.2082	0.328615
$966$	0	0
$967$	36.4126	1.17095	0.585475	−	0.810690i	$-0.300908\pi$
$967$	36.4126	1.17095	0.585475	+	0.810690i	$0.300908\pi$
$968$	0	0
$969$	6.00121	0.192787
$970$	0	0
$971$	6.09161	0.195489	0.0977445	−	0.995212i	$-0.468837\pi$
$971$	6.09161	0.195489	0.0977445	+	0.995212i	$0.468837\pi$
$972$	0	0
$973$	−0.509801	−0.0163435
$974$	0	0
$975$	−3.13858	−0.100515
$976$	0	0
$977$	−45.7294	−1.46301	−0.731506	−	0.681835i	$-0.761182\pi$
$977$	−45.7294	−1.46301	−0.731506	+	0.681835i	$0.761182\pi$
$978$	0	0
$979$	−40.6512	−1.29922
$980$	0	0
$981$	−37.3402	−1.19218
$982$	0	0
$983$	43.9546	1.40193	0.700966	−	0.713194i	$-0.252752\pi$
$983$	43.9546	1.40193	0.700966	+	0.713194i	$0.252752\pi$
$984$	0	0
$985$	−5.15052	−0.164109
$986$	0	0
$987$	4.39892	0.140019
$988$	0	0
$989$	−53.0658	−1.68739
$990$	0	0
$991$	33.6773	1.06979	0.534897	−	0.844917i	$-0.320350\pi$
$991$	33.6773	1.06979	0.534897	+	0.844917i	$0.320350\pi$
$992$	0	0
$993$	2.73794	0.0868860
$994$	0	0
$995$	10.7647	0.341265
$996$	0	0
$997$	41.4863	1.31389	0.656943	−	0.753941i	$-0.271849\pi$
$997$	41.4863	1.31389	0.656943	+	0.753941i	$0.271849\pi$
$998$	0	0
$999$	−17.3860	−0.550069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.a.u.1.3	✓	6
4.3	odd	2		2912.2.a.v.1.4	yes	6
8.3	odd	2		5824.2.a.cm.1.3		6
8.5	even	2		5824.2.a.cn.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.a.u.1.3	✓	6	1.1	even	1	trivial
2912.2.a.v.1.4	yes	6	4.3	odd	2
5824.2.a.cm.1.3		6	8.3	odd	2
5824.2.a.cn.1.4		6	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.735766 q^{3} +0.856892 q^{5} -1.00000 q^{7} -2.45865 q^{9} -5.29340 q^{11} -1.00000 q^{13} -0.630472 q^{15} +7.13529 q^{17} -1.14311 q^{19} +0.735766 q^{21} -6.17311 q^{23} -4.26574 q^{25} +4.01629 q^{27} +1.21376 q^{29} +4.77441 q^{31} +3.89471 q^{33} -0.856892 q^{35} -4.32888 q^{37} +0.735766 q^{39} +8.47515 q^{41} +8.59628 q^{43} -2.10680 q^{45} +5.97869 q^{47} +1.00000 q^{49} -5.24990 q^{51} +10.8069 q^{53} -4.53588 q^{55} +0.841060 q^{57} +5.83536 q^{59} -2.83558 q^{61} +2.45865 q^{63} -0.856892 q^{65} -10.0391 q^{67} +4.54196 q^{69} -6.12899 q^{71} +10.0079 q^{73} +3.13858 q^{75} +5.29340 q^{77} +8.70793 q^{79} +4.42090 q^{81} -9.26469 q^{83} +6.11417 q^{85} -0.893040 q^{87} +7.67959 q^{89} +1.00000 q^{91} -3.51285 q^{93} -0.979520 q^{95} -2.46070 q^{97} +13.0146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 3 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} - 9 q^{19} + 4 q^{21} + 11 q^{23} + 21 q^{25} - 22 q^{27} + 7 q^{29} - q^{31} + 18 q^{33} - 3 q^{35} + 16 q^{37}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 3 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 - 6 * q^13 + 2 * q^15 - 2 * q^17 - 9 * q^19 + 4 * q^21 + 11 * q^23 + 21 * q^25 - 22 * q^27 + 7 * q^29 - q^31 + 18 * q^33 - 3 * q^35 + 16 * q^37 + 4 * q^39 - q^43 + 29 * q^45 + 11 * q^47 + 6 * q^49 + 8 * q^51 + 23 * q^53 - 18 * q^55 + 10 * q^57 - 10 * q^59 + 10 * q^61 - 8 * q^63 - 3 * q^65 + 8 * q^67 + 6 * q^69 + 14 * q^71 + 11 * q^73 - 14 * q^75 + 8 * q^77 + 3 * q^79 + 30 * q^81 - 17 * q^83 + 6 * q^85 + 36 * q^87 - 5 * q^89 + 6 * q^91 - 16 * q^93 + 45 * q^95 + 27 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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