LMFDB - Embedded newform 2912.2.a.q.1.3

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:	$1$
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			1.3
Root			$0.156184$ 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)

$f(q)$	$=$	$q-1.15618 q^{3} +2.52637 q^{5} +1.00000 q^{7} -1.66324 q^{9} -0.379076 q^{11} -1.00000 q^{13} -2.92094 q^{15} +3.35468 q^{17} -6.08058 q^{19} -1.15618 q^{21} -1.80392 q^{23} +1.38253 q^{25} +5.39156 q^{27} -7.00153 q^{29} -7.84534 q^{31} +0.438281 q^{33} +2.52637 q^{35} +8.22787 q^{37} +1.15618 q^{39} +1.40935 q^{41} -8.46716 q^{43} -4.20195 q^{45} -7.03723 q^{47} +1.00000 q^{49} -3.87863 q^{51} -9.20753 q^{53} -0.957685 q^{55} +7.03027 q^{57} -4.15237 q^{59} +1.40935 q^{61} -1.66324 q^{63} -2.52637 q^{65} +8.21934 q^{67} +2.08566 q^{69} -2.92094 q^{71} +8.18871 q^{73} -1.59846 q^{75} -0.379076 q^{77} -3.61293 q^{79} -1.24391 q^{81} -9.04474 q^{83} +8.47516 q^{85} +8.09505 q^{87} +5.58484 q^{89} -1.00000 q^{91} +9.07066 q^{93} -15.3618 q^{95} +1.18961 q^{97} +0.630494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.15618	−0.667523	−0.333761	−	0.942658i	$-0.608318\pi$
$3$	−1.15618	−0.667523	−0.333761	+	0.942658i	$0.608318\pi$
$4$	0	0
$5$	2.52637	1.12983	0.564913	−	0.825151i	$-0.308910\pi$
$5$	2.52637	1.12983	0.564913	+	0.825151i	$0.308910\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.66324	−0.554413
$10$	0	0
$11$	−0.379076	−0.114296	−0.0571478	−	0.998366i	$-0.518201\pi$
$11$	−0.379076	−0.114296	−0.0571478	+	0.998366i	$0.518201\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.92094	−0.754185
$16$	0	0
$17$	3.35468	0.813630	0.406815	−	0.913511i	$-0.366639\pi$
$17$	3.35468	0.813630	0.406815	+	0.913511i	$0.366639\pi$
$18$	0	0
$19$	−6.08058	−1.39498	−0.697491	−	0.716594i	$-0.745700\pi$
$19$	−6.08058	−1.39498	−0.697491	+	0.716594i	$0.745700\pi$
$20$	0	0
$21$	−1.15618	−0.252300
$22$	0	0
$23$	−1.80392	−0.376144	−0.188072	−	0.982155i	$-0.560224\pi$
$23$	−1.80392	−0.376144	−0.188072	+	0.982155i	$0.560224\pi$
$24$	0	0
$25$	1.38253	0.276506
$26$	0	0
$27$	5.39156	1.03761
$28$	0	0
$29$	−7.00153	−1.30015	−0.650075	−	0.759870i	$-0.725263\pi$
$29$	−7.00153	−1.30015	−0.650075	+	0.759870i	$0.725263\pi$
$30$	0	0
$31$	−7.84534	−1.40907	−0.704533	−	0.709672i	$-0.748843\pi$
$31$	−7.84534	−1.40907	−0.704533	+	0.709672i	$0.748843\pi$
$32$	0	0
$33$	0.438281	0.0762950
$34$	0	0
$35$	2.52637	0.427034
$36$	0	0
$37$	8.22787	1.35265	0.676327	−	0.736601i	$-0.263571\pi$
$37$	8.22787	1.35265	0.676327	+	0.736601i	$0.263571\pi$
$38$	0	0
$39$	1.15618	0.185138
$40$	0	0
$41$	1.40935	0.220103	0.110051	−	0.993926i	$-0.464898\pi$
$41$	1.40935	0.220103	0.110051	+	0.993926i	$0.464898\pi$
$42$	0	0
$43$	−8.46716	−1.29123	−0.645615	−	0.763663i	$-0.723399\pi$
$43$	−8.46716	−1.29123	−0.645615	+	0.763663i	$0.723399\pi$
$44$	0	0
$45$	−4.20195	−0.626390
$46$	0	0
$47$	−7.03723	−1.02649	−0.513243	−	0.858243i	$-0.671556\pi$
$47$	−7.03723	−1.02649	−0.513243	+	0.858243i	$0.671556\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.87863	−0.543117
$52$	0	0
$53$	−9.20753	−1.26475	−0.632376	−	0.774662i	$-0.717920\pi$
$53$	−9.20753	−1.26475	−0.632376	+	0.774662i	$0.717920\pi$
$54$	0	0
$55$	−0.957685	−0.129134
$56$	0	0
$57$	7.03027	0.931182
$58$	0	0
$59$	−4.15237	−0.540593	−0.270297	−	0.962777i	$-0.587122\pi$
$59$	−4.15237	−0.540593	−0.270297	+	0.962777i	$0.587122\pi$
$60$	0	0
$61$	1.40935	0.180448	0.0902241	−	0.995921i	$-0.471242\pi$
$61$	1.40935	0.180448	0.0902241	+	0.995921i	$0.471242\pi$
$62$	0	0
$63$	−1.66324	−0.209549
$64$	0	0
$65$	−2.52637	−0.313357
$66$	0	0
$67$	8.21934	1.00415	0.502076	−	0.864824i	$-0.332570\pi$
$67$	8.21934	1.00415	0.502076	+	0.864824i	$0.332570\pi$
$68$	0	0
$69$	2.08566	0.251085
$70$	0	0
$71$	−2.92094	−0.346652	−0.173326	−	0.984864i	$-0.555451\pi$
$71$	−2.92094	−0.346652	−0.173326	+	0.984864i	$0.555451\pi$
$72$	0	0
$73$	8.18871	0.958416	0.479208	−	0.877701i	$-0.340924\pi$
$73$	8.18871	0.958416	0.479208	+	0.877701i	$0.340924\pi$
$74$	0	0
$75$	−1.59846	−0.184574
$76$	0	0
$77$	−0.379076	−0.0431997
$78$	0	0
$79$	−3.61293	−0.406486	−0.203243	−	0.979128i	$-0.565148\pi$
$79$	−3.61293	−0.406486	−0.203243	+	0.979128i	$0.565148\pi$
$80$	0	0
$81$	−1.24391	−0.138213
$82$	0	0
$83$	−9.04474	−0.992789	−0.496394	−	0.868097i	$-0.665343\pi$
$83$	−9.04474	−0.992789	−0.496394	+	0.868097i	$0.665343\pi$
$84$	0	0
$85$	8.47516	0.919260
$86$	0	0
$87$	8.09505	0.867880
$88$	0	0
$89$	5.58484	0.591992	0.295996	−	0.955189i	$-0.404349\pi$
$89$	5.58484	0.591992	0.295996	+	0.955189i	$0.404349\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	9.07066	0.940583
$94$	0	0
$95$	−15.3618	−1.57609
$96$	0	0
$97$	1.18961	0.120786	0.0603931	−	0.998175i	$-0.480765\pi$
$97$	1.18961	0.120786	0.0603931	+	0.998175i	$0.480765\pi$
$98$	0	0
$99$	0.630494	0.0633670
$100$	0	0
$101$	16.2448	1.61641	0.808207	−	0.588898i	$-0.200438\pi$
$101$	16.2448	1.61641	0.808207	+	0.588898i	$0.200438\pi$
$102$	0	0
$103$	6.37921	0.628563	0.314281	−	0.949330i	$-0.398236\pi$
$103$	6.37921	0.628563	0.314281	+	0.949330i	$0.398236\pi$
$104$	0	0
$105$	−2.92094	−0.285055
$106$	0	0
$107$	−7.25389	−0.701260	−0.350630	−	0.936514i	$-0.614033\pi$
$107$	−7.25389	−0.701260	−0.350630	+	0.936514i	$0.614033\pi$
$108$	0	0
$109$	−18.2373	−1.74681	−0.873406	−	0.486992i	$-0.838094\pi$
$109$	−18.2373	−1.74681	−0.873406	+	0.486992i	$0.838094\pi$
$110$	0	0
$111$	−9.51293	−0.902928
$112$	0	0
$113$	−0.307285	−0.0289069	−0.0144535	−	0.999896i	$-0.504601\pi$
$113$	−0.307285	−0.0289069	−0.0144535	+	0.999896i	$0.504601\pi$
$114$	0	0
$115$	−4.55737	−0.424977
$116$	0	0
$117$	1.66324	0.153767
$118$	0	0
$119$	3.35468	0.307523
$120$	0	0
$121$	−10.8563	−0.986937
$122$	0	0
$123$	−1.62946	−0.146924
$124$	0	0
$125$	−9.13906	−0.817422
$126$	0	0
$127$	17.9618	1.59385	0.796925	−	0.604079i	$-0.206459\pi$
$127$	17.9618	1.59385	0.796925	+	0.604079i	$0.206459\pi$
$128$	0	0
$129$	9.78959	0.861926
$130$	0	0
$131$	−7.21085	−0.630015	−0.315007	−	0.949089i	$-0.602007\pi$
$131$	−7.21085	−0.630015	−0.315007	+	0.949089i	$0.602007\pi$
$132$	0	0
$133$	−6.08058	−0.527253
$134$	0	0
$135$	13.6211	1.17231
$136$	0	0
$137$	2.21158	0.188948	0.0944740	−	0.995527i	$-0.469883\pi$
$137$	2.21158	0.188948	0.0944740	+	0.995527i	$0.469883\pi$
$138$	0	0
$139$	−6.61432	−0.561019	−0.280509	−	0.959851i	$-0.590503\pi$
$139$	−6.61432	−0.561019	−0.280509	+	0.959851i	$0.590503\pi$
$140$	0	0
$141$	8.13633	0.685203
$142$	0	0
$143$	0.379076	0.0316999
$144$	0	0
$145$	−17.6884	−1.46894
$146$	0	0
$147$	−1.15618	−0.0953604
$148$	0	0
$149$	−13.0239	−1.06696	−0.533481	−	0.845812i	$-0.679117\pi$
$149$	−13.0239	−1.06696	−0.533481	+	0.845812i	$0.679117\pi$
$150$	0	0
$151$	8.84116	0.719483	0.359742	−	0.933052i	$-0.382865\pi$
$151$	8.84116	0.719483	0.359742	+	0.933052i	$0.382865\pi$
$152$	0	0
$153$	−5.57964	−0.451087
$154$	0	0
$155$	−19.8202	−1.59200
$156$	0	0
$157$	2.00588	0.160086	0.0800432	−	0.996791i	$-0.474494\pi$
$157$	2.00588	0.160086	0.0800432	+	0.996791i	$0.474494\pi$
$158$	0	0
$159$	10.6456	0.844250
$160$	0	0
$161$	−1.80392	−0.142169
$162$	0	0
$163$	−0.768316	−0.0601792	−0.0300896	−	0.999547i	$-0.509579\pi$
$163$	−0.768316	−0.0601792	−0.0300896	+	0.999547i	$0.509579\pi$
$164$	0	0
$165$	1.10726	0.0862000
$166$	0	0
$167$	10.8847	0.842287	0.421143	−	0.906994i	$-0.361629\pi$
$167$	10.8847	0.842287	0.421143	+	0.906994i	$0.361629\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.1135	0.773396
$172$	0	0
$173$	−11.2213	−0.853137	−0.426568	−	0.904455i	$-0.640278\pi$
$173$	−11.2213	−0.853137	−0.426568	+	0.904455i	$0.640278\pi$
$174$	0	0
$175$	1.38253	0.104510
$176$	0	0
$177$	4.80091	0.360858
$178$	0	0
$179$	−10.2940	−0.769413	−0.384706	−	0.923039i	$-0.625697\pi$
$179$	−10.2940	−0.769413	−0.384706	+	0.923039i	$0.625697\pi$
$180$	0	0
$181$	−23.6259	−1.75610	−0.878048	−	0.478572i	$-0.841154\pi$
$181$	−23.6259	−1.75610	−0.878048	+	0.478572i	$0.841154\pi$
$182$	0	0
$183$	−1.62946	−0.120453
$184$	0	0
$185$	20.7866	1.52826
$186$	0	0
$187$	−1.27168	−0.0929944
$188$	0	0
$189$	5.39156	0.392178
$190$	0	0
$191$	18.7332	1.35548	0.677742	−	0.735300i	$-0.262959\pi$
$191$	18.7332	1.35548	0.677742	+	0.735300i	$0.262959\pi$
$192$	0	0
$193$	−3.20215	−0.230496	−0.115248	−	0.993337i	$-0.536766\pi$
$193$	−3.20215	−0.230496	−0.115248	+	0.993337i	$0.536766\pi$
$194$	0	0
$195$	2.92094	0.209173
$196$	0	0
$197$	1.40450	0.100067	0.0500334	−	0.998748i	$-0.484067\pi$
$197$	1.40450	0.100067	0.0500334	+	0.998748i	$0.484067\pi$
$198$	0	0
$199$	−10.8044	−0.765905	−0.382952	−	0.923768i	$-0.625093\pi$
$199$	−10.8044	−0.765905	−0.382952	+	0.923768i	$0.625093\pi$
$200$	0	0
$201$	−9.50306	−0.670294
$202$	0	0
$203$	−7.00153	−0.491411
$204$	0	0
$205$	3.56052	0.248678
$206$	0	0
$207$	3.00035	0.208539
$208$	0	0
$209$	2.30500	0.159440
$210$	0	0
$211$	−14.9009	−1.02582	−0.512910	−	0.858442i	$-0.671433\pi$
$211$	−14.9009	−1.02582	−0.512910	+	0.858442i	$0.671433\pi$
$212$	0	0
$213$	3.37715	0.231398
$214$	0	0
$215$	−21.3912	−1.45886
$216$	0	0
$217$	−7.84534	−0.532577
$218$	0	0
$219$	−9.46765	−0.639765
$220$	0	0
$221$	−3.35468	−0.225660
$222$	0	0
$223$	−17.9369	−1.20114	−0.600571	−	0.799571i	$-0.705060\pi$
$223$	−17.9369	−1.20114	−0.600571	+	0.799571i	$0.705060\pi$
$224$	0	0
$225$	−2.29948	−0.153299
$226$	0	0
$227$	−3.23978	−0.215032	−0.107516	−	0.994203i	$-0.534290\pi$
$227$	−3.23978	−0.215032	−0.107516	+	0.994203i	$0.534290\pi$
$228$	0	0
$229$	−13.7292	−0.907252	−0.453626	−	0.891192i	$-0.649870\pi$
$229$	−13.7292	−0.907252	−0.453626	+	0.891192i	$0.649870\pi$
$230$	0	0
$231$	0.438281	0.0288368
$232$	0	0
$233$	−10.2187	−0.669450	−0.334725	−	0.942316i	$-0.608643\pi$
$233$	−10.2187	−0.669450	−0.334725	+	0.942316i	$0.608643\pi$
$234$	0	0
$235$	−17.7786	−1.15975
$236$	0	0
$237$	4.17721	0.271339
$238$	0	0
$239$	8.67983	0.561451	0.280726	−	0.959788i	$-0.409425\pi$
$239$	8.67983	0.561451	0.280726	+	0.959788i	$0.409425\pi$
$240$	0	0
$241$	3.33053	0.214538	0.107269	−	0.994230i	$-0.465789\pi$
$241$	3.33053	0.214538	0.107269	+	0.994230i	$0.465789\pi$
$242$	0	0
$243$	−14.7365	−0.945346
$244$	0	0
$245$	2.52637	0.161404
$246$	0	0
$247$	6.08058	0.386898
$248$	0	0
$249$	10.4574	0.662709
$250$	0	0
$251$	−28.3764	−1.79110	−0.895551	−	0.444959i	$-0.853218\pi$
$251$	−28.3764	−1.79110	−0.895551	+	0.444959i	$0.853218\pi$
$252$	0	0
$253$	0.683823	0.0429916
$254$	0	0
$255$	−9.79884	−0.613627
$256$	0	0
$257$	−5.43931	−0.339295	−0.169648	−	0.985505i	$-0.554263\pi$
$257$	−5.43931	−0.339295	−0.169648	+	0.985505i	$0.554263\pi$
$258$	0	0
$259$	8.22787	0.511255
$260$	0	0
$261$	11.6452	0.720821
$262$	0	0
$263$	29.0772	1.79297	0.896487	−	0.443069i	$-0.146110\pi$
$263$	29.0772	1.79297	0.896487	+	0.443069i	$0.146110\pi$
$264$	0	0
$265$	−23.2616	−1.42895
$266$	0	0
$267$	−6.45710	−0.395168
$268$	0	0
$269$	16.9618	1.03418	0.517089	−	0.855932i	$-0.327016\pi$
$269$	16.9618	1.03418	0.517089	+	0.855932i	$0.327016\pi$
$270$	0	0
$271$	−25.3079	−1.53735	−0.768673	−	0.639642i	$-0.779083\pi$
$271$	−25.3079	−1.53735	−0.768673	+	0.639642i	$0.779083\pi$
$272$	0	0
$273$	1.15618	0.0699754
$274$	0	0
$275$	−0.524084	−0.0316035
$276$	0	0
$277$	26.7705	1.60849	0.804243	−	0.594301i	$-0.202571\pi$
$277$	26.7705	1.60849	0.804243	+	0.594301i	$0.202571\pi$
$278$	0	0
$279$	13.0487	0.781204
$280$	0	0
$281$	−16.2828	−0.971350	−0.485675	−	0.874139i	$-0.661426\pi$
$281$	−16.2828	−0.971350	−0.485675	+	0.874139i	$0.661426\pi$
$282$	0	0
$283$	31.8908	1.89571	0.947855	−	0.318702i	$-0.103247\pi$
$283$	31.8908	1.89571	0.947855	+	0.318702i	$0.103247\pi$
$284$	0	0
$285$	17.7610	1.05207
$286$	0	0
$287$	1.40935	0.0831910
$288$	0	0
$289$	−5.74611	−0.338006
$290$	0	0
$291$	−1.37540	−0.0806276
$292$	0	0
$293$	−7.84989	−0.458595	−0.229298	−	0.973356i	$-0.573643\pi$
$293$	−7.84989	−0.458595	−0.229298	+	0.973356i	$0.573643\pi$
$294$	0	0
$295$	−10.4904	−0.610776
$296$	0	0
$297$	−2.04381	−0.118594
$298$	0	0
$299$	1.80392	0.104324
$300$	0	0
$301$	−8.46716	−0.488039
$302$	0	0
$303$	−18.7819	−1.07899
$304$	0	0
$305$	3.56052	0.203875
$306$	0	0
$307$	−23.5081	−1.34168	−0.670839	−	0.741603i	$-0.734066\pi$
$307$	−23.5081	−1.34168	−0.670839	+	0.741603i	$0.734066\pi$
$308$	0	0
$309$	−7.37554	−0.419580
$310$	0	0
$311$	21.2731	1.20629	0.603144	−	0.797633i	$-0.293915\pi$
$311$	21.2731	1.20629	0.603144	+	0.797633i	$0.293915\pi$
$312$	0	0
$313$	0.901988	0.0509834	0.0254917	−	0.999675i	$-0.491885\pi$
$313$	0.901988	0.0509834	0.0254917	+	0.999675i	$0.491885\pi$
$314$	0	0
$315$	−4.20195	−0.236753
$316$	0	0
$317$	2.30091	0.129232	0.0646161	−	0.997910i	$-0.479418\pi$
$317$	2.30091	0.129232	0.0646161	+	0.997910i	$0.479418\pi$
$318$	0	0
$319$	2.65411	0.148602
$320$	0	0
$321$	8.38683	0.468107
$322$	0	0
$323$	−20.3984	−1.13500
$324$	0	0
$325$	−1.38253	−0.0766890
$326$	0	0
$327$	21.0856	1.16604
$328$	0	0
$329$	−7.03723	−0.387975
$330$	0	0
$331$	−4.90024	−0.269342	−0.134671	−	0.990890i	$-0.542998\pi$
$331$	−4.90024	−0.269342	−0.134671	+	0.990890i	$0.542998\pi$
$332$	0	0
$333$	−13.6849	−0.749929
$334$	0	0
$335$	20.7651	1.13452
$336$	0	0
$337$	−7.92204	−0.431541	−0.215770	−	0.976444i	$-0.569226\pi$
$337$	−7.92204	−0.431541	−0.215770	+	0.976444i	$0.569226\pi$
$338$	0	0
$339$	0.355278	0.0192960
$340$	0	0
$341$	2.97398	0.161050
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	5.26916	0.283682
$346$	0	0
$347$	−18.3241	−0.983690	−0.491845	−	0.870683i	$-0.663677\pi$
$347$	−18.3241	−0.983690	−0.491845	+	0.870683i	$0.663677\pi$
$348$	0	0
$349$	−16.4231	−0.879106	−0.439553	−	0.898217i	$-0.644863\pi$
$349$	−16.4231	−0.879106	−0.439553	+	0.898217i	$0.644863\pi$
$350$	0	0
$351$	−5.39156	−0.287780
$352$	0	0
$353$	−12.3862	−0.659253	−0.329627	−	0.944111i	$-0.606923\pi$
$353$	−12.3862	−0.659253	−0.329627	+	0.944111i	$0.606923\pi$
$354$	0	0
$355$	−7.37938	−0.391657
$356$	0	0
$357$	−3.87863	−0.205279
$358$	0	0
$359$	18.6782	0.985798	0.492899	−	0.870087i	$-0.335937\pi$
$359$	18.6782	0.985798	0.492899	+	0.870087i	$0.335937\pi$
$360$	0	0
$361$	17.9735	0.945973
$362$	0	0
$363$	12.5519	0.658803
$364$	0	0
$365$	20.6877	1.08284
$366$	0	0
$367$	−3.13179	−0.163478	−0.0817391	−	0.996654i	$-0.526047\pi$
$367$	−3.13179	−0.163478	−0.0817391	+	0.996654i	$0.526047\pi$
$368$	0	0
$369$	−2.34408	−0.122028
$370$	0	0
$371$	−9.20753	−0.478031
$372$	0	0
$373$	34.1369	1.76754	0.883770	−	0.467921i	$-0.154997\pi$
$373$	34.1369	1.76754	0.883770	+	0.467921i	$0.154997\pi$
$374$	0	0
$375$	10.5664	0.545648
$376$	0	0
$377$	7.00153	0.360597
$378$	0	0
$379$	17.4384	0.895751	0.447876	−	0.894096i	$-0.352181\pi$
$379$	17.4384	0.895751	0.447876	+	0.894096i	$0.352181\pi$
$380$	0	0
$381$	−20.7671	−1.06393
$382$	0	0
$383$	−33.4022	−1.70677	−0.853387	−	0.521278i	$-0.825456\pi$
$383$	−33.4022	−1.70677	−0.853387	+	0.521278i	$0.825456\pi$
$384$	0	0
$385$	−0.957685	−0.0488081
$386$	0	0
$387$	14.0829	0.715875
$388$	0	0
$389$	−15.5630	−0.789076	−0.394538	−	0.918880i	$-0.629095\pi$
$389$	−15.5630	−0.789076	−0.394538	+	0.918880i	$0.629095\pi$
$390$	0	0
$391$	−6.05159	−0.306042
$392$	0	0
$393$	8.33706	0.420549
$394$	0	0
$395$	−9.12758	−0.459258
$396$	0	0
$397$	−22.8330	−1.14596	−0.572978	−	0.819571i	$-0.694212\pi$
$397$	−22.8330	−1.14596	−0.572978	+	0.819571i	$0.694212\pi$
$398$	0	0
$399$	7.03027	0.351954
$400$	0	0
$401$	−17.0500	−0.851434	−0.425717	−	0.904856i	$-0.639978\pi$
$401$	−17.0500	−0.851434	−0.425717	+	0.904856i	$0.639978\pi$
$402$	0	0
$403$	7.84534	0.390804
$404$	0	0
$405$	−3.14259	−0.156156
$406$	0	0
$407$	−3.11899	−0.154602
$408$	0	0
$409$	17.9104	0.885614	0.442807	−	0.896617i	$-0.353983\pi$
$409$	17.9104	0.885614	0.442807	+	0.896617i	$0.353983\pi$
$410$	0	0
$411$	−2.55699	−0.126127
$412$	0	0
$413$	−4.15237	−0.204325
$414$	0	0
$415$	−22.8503	−1.12168
$416$	0	0
$417$	7.64736	0.374493
$418$	0	0
$419$	1.29444	0.0632377	0.0316188	−	0.999500i	$-0.489934\pi$
$419$	1.29444	0.0632377	0.0316188	+	0.999500i	$0.489934\pi$
$420$	0	0
$421$	27.3342	1.33219	0.666095	−	0.745867i	$-0.267965\pi$
$421$	27.3342	1.33219	0.666095	+	0.745867i	$0.267965\pi$
$422$	0	0
$423$	11.7046	0.569097
$424$	0	0
$425$	4.63795	0.224974
$426$	0	0
$427$	1.40935	0.0682030
$428$	0	0
$429$	−0.438281	−0.0211604
$430$	0	0
$431$	38.9603	1.87665	0.938325	−	0.345753i	$-0.112377\pi$
$431$	38.9603	1.87665	0.938325	+	0.345753i	$0.112377\pi$
$432$	0	0
$433$	26.5744	1.27709	0.638543	−	0.769586i	$-0.279537\pi$
$433$	26.5744	1.27709	0.638543	+	0.769586i	$0.279537\pi$
$434$	0	0
$435$	20.4511	0.980554
$436$	0	0
$437$	10.9689	0.524714
$438$	0	0
$439$	−7.16511	−0.341972	−0.170986	−	0.985273i	$-0.554695\pi$
$439$	−7.16511	−0.341972	−0.170986	+	0.985273i	$0.554695\pi$
$440$	0	0
$441$	−1.66324	−0.0792019
$442$	0	0
$443$	39.7320	1.88772	0.943862	−	0.330339i	$-0.107163\pi$
$443$	39.7320	1.88772	0.943862	+	0.330339i	$0.107163\pi$
$444$	0	0
$445$	14.1094	0.668848
$446$	0	0
$447$	15.0581	0.712222
$448$	0	0
$449$	12.7473	0.601581	0.300791	−	0.953690i	$-0.402749\pi$
$449$	12.7473	0.601581	0.300791	+	0.953690i	$0.402749\pi$
$450$	0	0
$451$	−0.534249	−0.0251568
$452$	0	0
$453$	−10.2220	−0.480271
$454$	0	0
$455$	−2.52637	−0.118438
$456$	0	0
$457$	−9.54084	−0.446301	−0.223151	−	0.974784i	$-0.571634\pi$
$457$	−9.54084	−0.446301	−0.223151	+	0.974784i	$0.571634\pi$
$458$	0	0
$459$	18.0870	0.844228
$460$	0	0
$461$	6.57416	0.306189	0.153095	−	0.988212i	$-0.451076\pi$
$461$	6.57416	0.306189	0.153095	+	0.988212i	$0.451076\pi$
$462$	0	0
$463$	9.93180	0.461570	0.230785	−	0.973005i	$-0.425871\pi$
$463$	9.93180	0.461570	0.230785	+	0.973005i	$0.425871\pi$
$464$	0	0
$465$	22.9158	1.06270
$466$	0	0
$467$	9.25918	0.428464	0.214232	−	0.976783i	$-0.431275\pi$
$467$	9.25918	0.428464	0.214232	+	0.976783i	$0.431275\pi$
$468$	0	0
$469$	8.21934	0.379534
$470$	0	0
$471$	−2.31916	−0.106861
$472$	0	0
$473$	3.20970	0.147582
$474$	0	0
$475$	−8.40659	−0.385721
$476$	0	0
$477$	15.3143	0.701195
$478$	0	0
$479$	16.8996	0.772161	0.386080	−	0.922465i	$-0.373829\pi$
$479$	16.8996	0.772161	0.386080	+	0.922465i	$0.373829\pi$
$480$	0	0
$481$	−8.22787	−0.375159
$482$	0	0
$483$	2.08566	0.0949010
$484$	0	0
$485$	3.00538	0.136467
$486$	0	0
$487$	−21.4625	−0.972557	−0.486279	−	0.873804i	$-0.661646\pi$
$487$	−21.4625	−0.972557	−0.486279	+	0.873804i	$0.661646\pi$
$488$	0	0
$489$	0.888315	0.0401710
$490$	0	0
$491$	24.4322	1.10261	0.551305	−	0.834303i	$-0.314130\pi$
$491$	24.4322	1.10261	0.551305	+	0.834303i	$0.314130\pi$
$492$	0	0
$493$	−23.4879	−1.05784
$494$	0	0
$495$	1.59286	0.0715937
$496$	0	0
$497$	−2.92094	−0.131022
$498$	0	0
$499$	38.3086	1.71493	0.857465	−	0.514542i	$-0.172038\pi$
$499$	38.3086	1.71493	0.857465	+	0.514542i	$0.172038\pi$
$500$	0	0
$501$	−12.5848	−0.562246
$502$	0	0
$503$	38.9983	1.73885	0.869424	−	0.494066i	$-0.164490\pi$
$503$	38.9983	1.73885	0.869424	+	0.494066i	$0.164490\pi$
$504$	0	0
$505$	41.0402	1.82627
$506$	0	0
$507$	−1.15618	−0.0513479
$508$	0	0
$509$	21.3114	0.944610	0.472305	−	0.881435i	$-0.343422\pi$
$509$	21.3114	0.944610	0.472305	+	0.881435i	$0.343422\pi$
$510$	0	0
$511$	8.18871	0.362247
$512$	0	0
$513$	−32.7838	−1.44744
$514$	0	0
$515$	16.1162	0.710166
$516$	0	0
$517$	2.66764	0.117323
$518$	0	0
$519$	12.9738	0.569488
$520$	0	0
$521$	42.7173	1.87148	0.935739	−	0.352694i	$-0.114734\pi$
$521$	42.7173	1.87148	0.935739	+	0.352694i	$0.114734\pi$
$522$	0	0
$523$	−41.3229	−1.80693	−0.903463	−	0.428667i	$-0.858983\pi$
$523$	−41.3229	−1.80693	−0.903463	+	0.428667i	$0.858983\pi$
$524$	0	0
$525$	−1.59846	−0.0697625
$526$	0	0
$527$	−26.3186	−1.14646
$528$	0	0
$529$	−19.7459	−0.858516
$530$	0	0
$531$	6.90639	0.299712
$532$	0	0
$533$	−1.40935	−0.0610455
$534$	0	0
$535$	−18.3260	−0.792302
$536$	0	0
$537$	11.9018	0.513601
$538$	0	0
$539$	−0.379076	−0.0163279
$540$	0	0
$541$	−5.81736	−0.250108	−0.125054	−	0.992150i	$-0.539910\pi$
$541$	−5.81736	−0.250108	−0.125054	+	0.992150i	$0.539910\pi$
$542$	0	0
$543$	27.3158	1.17223
$544$	0	0
$545$	−46.0740	−1.97359
$546$	0	0
$547$	13.1777	0.563438	0.281719	−	0.959497i	$-0.409095\pi$
$547$	13.1777	0.563438	0.281719	+	0.959497i	$0.409095\pi$
$548$	0	0
$549$	−2.34408	−0.100043
$550$	0	0
$551$	42.5734	1.81369
$552$	0	0
$553$	−3.61293	−0.153637
$554$	0	0
$555$	−24.0332	−1.02015
$556$	0	0
$557$	25.9250	1.09848	0.549239	−	0.835665i	$-0.314918\pi$
$557$	25.9250	1.09848	0.549239	+	0.835665i	$0.314918\pi$
$558$	0	0
$559$	8.46716	0.358123
$560$	0	0
$561$	1.47029	0.0620759
$562$	0	0
$563$	9.72655	0.409925	0.204963	−	0.978770i	$-0.434293\pi$
$563$	9.72655	0.409925	0.204963	+	0.978770i	$0.434293\pi$
$564$	0	0
$565$	−0.776314	−0.0326598
$566$	0	0
$567$	−1.24391	−0.0522395
$568$	0	0
$569$	−29.3247	−1.22936	−0.614679	−	0.788778i	$-0.710714\pi$
$569$	−29.3247	−1.22936	−0.614679	+	0.788778i	$0.710714\pi$
$570$	0	0
$571$	−5.90207	−0.246994	−0.123497	−	0.992345i	$-0.539411\pi$
$571$	−5.90207	−0.246994	−0.123497	+	0.992345i	$0.539411\pi$
$572$	0	0
$573$	−21.6590	−0.904817
$574$	0	0
$575$	−2.49398	−0.104006
$576$	0	0
$577$	−17.2365	−0.717566	−0.358783	−	0.933421i	$-0.616808\pi$
$577$	−17.2365	−0.717566	−0.358783	+	0.933421i	$0.616808\pi$
$578$	0	0
$579$	3.70227	0.153861
$580$	0	0
$581$	−9.04474	−0.375239
$582$	0	0
$583$	3.49035	0.144556
$584$	0	0
$585$	4.20195	0.173729
$586$	0	0
$587$	1.13540	0.0468630	0.0234315	−	0.999725i	$-0.492541\pi$
$587$	1.13540	0.0468630	0.0234315	+	0.999725i	$0.492541\pi$
$588$	0	0
$589$	47.7043	1.96562
$590$	0	0
$591$	−1.62386	−0.0667968
$592$	0	0
$593$	−38.5958	−1.58494	−0.792470	−	0.609911i	$-0.791205\pi$
$593$	−38.5958	−1.58494	−0.792470	+	0.609911i	$0.791205\pi$
$594$	0	0
$595$	8.47516	0.347448
$596$	0	0
$597$	12.4919	0.511259
$598$	0	0
$599$	2.38561	0.0974733	0.0487366	−	0.998812i	$-0.484480\pi$
$599$	2.38561	0.0974733	0.0487366	+	0.998812i	$0.484480\pi$
$600$	0	0
$601$	−30.1337	−1.22918	−0.614589	−	0.788847i	$-0.710678\pi$
$601$	−30.1337	−1.22918	−0.614589	+	0.788847i	$0.710678\pi$
$602$	0	0
$603$	−13.6707	−0.556715
$604$	0	0
$605$	−27.4270	−1.11507
$606$	0	0
$607$	25.7888	1.04673	0.523367	−	0.852107i	$-0.324676\pi$
$607$	25.7888	1.04673	0.523367	+	0.852107i	$0.324676\pi$
$608$	0	0
$609$	8.09505	0.328028
$610$	0	0
$611$	7.03723	0.284696
$612$	0	0
$613$	5.25885	0.212403	0.106202	−	0.994345i	$-0.466131\pi$
$613$	5.25885	0.212403	0.106202	+	0.994345i	$0.466131\pi$
$614$	0	0
$615$	−4.11662	−0.165998
$616$	0	0
$617$	−2.59401	−0.104431	−0.0522154	−	0.998636i	$-0.516628\pi$
$617$	−2.59401	−0.104431	−0.0522154	+	0.998636i	$0.516628\pi$
$618$	0	0
$619$	−9.43016	−0.379030	−0.189515	−	0.981878i	$-0.560692\pi$
$619$	−9.43016	−0.379030	−0.189515	+	0.981878i	$0.560692\pi$
$620$	0	0
$621$	−9.72596	−0.390289
$622$	0	0
$623$	5.58484	0.223752
$624$	0	0
$625$	−30.0013	−1.20005
$626$	0	0
$627$	−2.66500	−0.106430
$628$	0	0
$629$	27.6019	1.10056
$630$	0	0
$631$	−3.59402	−0.143076	−0.0715379	−	0.997438i	$-0.522791\pi$
$631$	−3.59402	−0.143076	−0.0715379	+	0.997438i	$0.522791\pi$
$632$	0	0
$633$	17.2282	0.684758
$634$	0	0
$635$	45.3780	1.80077
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	4.85823	0.192189
$640$	0	0
$641$	44.6709	1.76439	0.882197	−	0.470881i	$-0.156064\pi$
$641$	44.6709	1.76439	0.882197	+	0.470881i	$0.156064\pi$
$642$	0	0
$643$	29.4073	1.15971	0.579854	−	0.814720i	$-0.303109\pi$
$643$	29.4073	1.15971	0.579854	+	0.814720i	$0.303109\pi$
$644$	0	0
$645$	24.7321	0.973826
$646$	0	0
$647$	21.8712	0.859844	0.429922	−	0.902866i	$-0.358541\pi$
$647$	21.8712	0.859844	0.429922	+	0.902866i	$0.358541\pi$
$648$	0	0
$649$	1.57406	0.0617874
$650$	0	0
$651$	9.07066	0.355507
$652$	0	0
$653$	−26.2583	−1.02757	−0.513783	−	0.857920i	$-0.671756\pi$
$653$	−26.2583	−1.02757	−0.513783	+	0.857920i	$0.671756\pi$
$654$	0	0
$655$	−18.2172	−0.711807
$656$	0	0
$657$	−13.6198	−0.531359
$658$	0	0
$659$	9.21534	0.358978	0.179489	−	0.983760i	$-0.442555\pi$
$659$	9.21534	0.358978	0.179489	+	0.983760i	$0.442555\pi$
$660$	0	0
$661$	20.5617	0.799759	0.399879	−	0.916568i	$-0.369052\pi$
$661$	20.5617	0.799759	0.399879	+	0.916568i	$0.369052\pi$
$662$	0	0
$663$	3.87863	0.150633
$664$	0	0
$665$	−15.3618	−0.595704
$666$	0	0
$667$	12.6302	0.489044
$668$	0	0
$669$	20.7383	0.801790
$670$	0	0
$671$	−0.534249	−0.0206244
$672$	0	0
$673$	17.6236	0.679340	0.339670	−	0.940545i	$-0.389685\pi$
$673$	17.6236	0.679340	0.339670	+	0.940545i	$0.389685\pi$
$674$	0	0
$675$	7.45400	0.286905
$676$	0	0
$677$	−24.3920	−0.937461	−0.468730	−	0.883341i	$-0.655288\pi$
$677$	−24.3920	−0.937461	−0.468730	+	0.883341i	$0.655288\pi$
$678$	0	0
$679$	1.18961	0.0456529
$680$	0	0
$681$	3.74578	0.143539
$682$	0	0
$683$	−36.0699	−1.38018	−0.690089	−	0.723725i	$-0.742428\pi$
$683$	−36.0699	−1.38018	−0.690089	+	0.723725i	$0.742428\pi$
$684$	0	0
$685$	5.58726	0.213478
$686$	0	0
$687$	15.8735	0.605612
$688$	0	0
$689$	9.20753	0.350779
$690$	0	0
$691$	44.9086	1.70840	0.854202	−	0.519941i	$-0.174046\pi$
$691$	44.9086	1.70840	0.854202	+	0.519941i	$0.174046\pi$
$692$	0	0
$693$	0.630494	0.0239505
$694$	0	0
$695$	−16.7102	−0.633854
$696$	0	0
$697$	4.72791	0.179082
$698$	0	0
$699$	11.8147	0.446873
$700$	0	0
$701$	42.3171	1.59830	0.799148	−	0.601134i	$-0.205284\pi$
$701$	42.3171	1.59830	0.799148	+	0.601134i	$0.205284\pi$
$702$	0	0
$703$	−50.0303	−1.88693
$704$	0	0
$705$	20.5554	0.774160
$706$	0	0
$707$	16.2448	0.610947
$708$	0	0
$709$	15.2915	0.574284	0.287142	−	0.957888i	$-0.407295\pi$
$709$	15.2915	0.574284	0.287142	+	0.957888i	$0.407295\pi$
$710$	0	0
$711$	6.00916	0.225361
$712$	0	0
$713$	14.1524	0.530011
$714$	0	0
$715$	0.957685	0.0358154
$716$	0	0
$717$	−10.0355	−0.374782
$718$	0	0
$719$	7.84648	0.292624	0.146312	−	0.989238i	$-0.453260\pi$
$719$	7.84648	0.292624	0.146312	+	0.989238i	$0.453260\pi$
$720$	0	0
$721$	6.37921	0.237574
$722$	0	0
$723$	−3.85070	−0.143209
$724$	0	0
$725$	−9.67983	−0.359500
$726$	0	0
$727$	−9.52806	−0.353376	−0.176688	−	0.984267i	$-0.556538\pi$
$727$	−9.52806	−0.353376	−0.176688	+	0.984267i	$0.556538\pi$
$728$	0	0
$729$	20.7698	0.769253
$730$	0	0
$731$	−28.4046	−1.05058
$732$	0	0
$733$	−45.6515	−1.68618	−0.843088	−	0.537775i	$-0.819265\pi$
$733$	−45.6515	−1.68618	−0.843088	+	0.537775i	$0.819265\pi$
$734$	0	0
$735$	−2.92094	−0.107741
$736$	0	0
$737$	−3.11575	−0.114770
$738$	0	0
$739$	48.2845	1.77618	0.888088	−	0.459674i	$-0.152034\pi$
$739$	48.2845	1.77618	0.888088	+	0.459674i	$0.152034\pi$
$740$	0	0
$741$	−7.03027	−0.258263
$742$	0	0
$743$	10.5289	0.386270	0.193135	−	0.981172i	$-0.438135\pi$
$743$	10.5289	0.386270	0.193135	+	0.981172i	$0.438135\pi$
$744$	0	0
$745$	−32.9032	−1.20548
$746$	0	0
$747$	15.0436	0.550415
$748$	0	0
$749$	−7.25389	−0.265052
$750$	0	0
$751$	46.3573	1.69160	0.845802	−	0.533498i	$-0.179123\pi$
$751$	46.3573	1.69160	0.845802	+	0.533498i	$0.179123\pi$
$752$	0	0
$753$	32.8083	1.19560
$754$	0	0
$755$	22.3360	0.812890
$756$	0	0
$757$	−6.63021	−0.240979	−0.120490	−	0.992715i	$-0.538446\pi$
$757$	−6.63021	−0.240979	−0.120490	+	0.992715i	$0.538446\pi$
$758$	0	0
$759$	−0.790625	−0.0286979
$760$	0	0
$761$	−14.2846	−0.517818	−0.258909	−	0.965902i	$-0.583363\pi$
$761$	−14.2846	−0.517818	−0.258909	+	0.965902i	$0.583363\pi$
$762$	0	0
$763$	−18.2373	−0.660233
$764$	0	0
$765$	−14.0962	−0.509650
$766$	0	0
$767$	4.15237	0.149934
$768$	0	0
$769$	−36.2588	−1.30752	−0.653762	−	0.756700i	$-0.726810\pi$
$769$	−36.2588	−1.30752	−0.653762	+	0.756700i	$0.726810\pi$
$770$	0	0
$771$	6.28884	0.226487
$772$	0	0
$773$	−23.8145	−0.856550	−0.428275	−	0.903649i	$-0.640878\pi$
$773$	−23.8145	−0.856550	−0.428275	+	0.903649i	$0.640878\pi$
$774$	0	0
$775$	−10.8464	−0.389615
$776$	0	0
$777$	−9.51293	−0.341275
$778$	0	0
$779$	−8.56964	−0.307039
$780$	0	0
$781$	1.10726	0.0396208
$782$	0	0
$783$	−37.7492	−1.34904
$784$	0	0
$785$	5.06758	0.180870
$786$	0	0
$787$	−14.5618	−0.519072	−0.259536	−	0.965733i	$-0.583570\pi$
$787$	−14.5618	−0.519072	−0.259536	+	0.965733i	$0.583570\pi$
$788$	0	0
$789$	−33.6185	−1.19685
$790$	0	0
$791$	−0.307285	−0.0109258
$792$	0	0
$793$	−1.40935	−0.0500473
$794$	0	0
$795$	26.8947	0.953856
$796$	0	0
$797$	−44.8286	−1.58791	−0.793956	−	0.607976i	$-0.791982\pi$
$797$	−44.8286	−1.58791	−0.793956	+	0.607976i	$0.791982\pi$
$798$	0	0
$799$	−23.6077	−0.835180
$800$	0	0
$801$	−9.28893	−0.328208
$802$	0	0
$803$	−3.10414	−0.109543
$804$	0	0
$805$	−4.55737	−0.160626
$806$	0	0
$807$	−19.6109	−0.690337
$808$	0	0
$809$	0.0588273	0.00206826	0.00103413	−	0.999999i	$-0.499671\pi$
$809$	0.0588273	0.00206826	0.00103413	+	0.999999i	$0.499671\pi$
$810$	0	0
$811$	−16.3293	−0.573401	−0.286700	−	0.958020i	$-0.592558\pi$
$811$	−16.3293	−0.573401	−0.286700	+	0.958020i	$0.592558\pi$
$812$	0	0
$813$	29.2606	1.02621
$814$	0	0
$815$	−1.94105	−0.0679920
$816$	0	0
$817$	51.4853	1.80124
$818$	0	0
$819$	1.66324	0.0581183
$820$	0	0
$821$	33.2365	1.15996	0.579981	−	0.814630i	$-0.303060\pi$
$821$	33.2365	1.15996	0.579981	+	0.814630i	$0.303060\pi$
$822$	0	0
$823$	−40.5997	−1.41522	−0.707609	−	0.706605i	$-0.750226\pi$
$823$	−40.5997	−1.41522	−0.707609	+	0.706605i	$0.750226\pi$
$824$	0	0
$825$	0.605937	0.0210960
$826$	0	0
$827$	−24.8197	−0.863065	−0.431532	−	0.902097i	$-0.642027\pi$
$827$	−24.8197	−0.863065	−0.431532	+	0.902097i	$0.642027\pi$
$828$	0	0
$829$	−19.6000	−0.680736	−0.340368	−	0.940292i	$-0.610552\pi$
$829$	−19.6000	−0.680736	−0.340368	+	0.940292i	$0.610552\pi$
$830$	0	0
$831$	−30.9517	−1.07370
$832$	0	0
$833$	3.35468	0.116233
$834$	0	0
$835$	27.4989	0.951637
$836$	0	0
$837$	−42.2986	−1.46206
$838$	0	0
$839$	−10.1625	−0.350847	−0.175423	−	0.984493i	$-0.556129\pi$
$839$	−10.1625	−0.350847	−0.175423	+	0.984493i	$0.556129\pi$
$840$	0	0
$841$	20.0214	0.690392
$842$	0	0
$843$	18.8259	0.648398
$844$	0	0
$845$	2.52637	0.0869097
$846$	0	0
$847$	−10.8563	−0.373027
$848$	0	0
$849$	−36.8716	−1.26543
$850$	0	0
$851$	−14.8424	−0.508792
$852$	0	0
$853$	28.7239	0.983486	0.491743	−	0.870740i	$-0.336360\pi$
$853$	28.7239	0.983486	0.491743	+	0.870740i	$0.336360\pi$
$854$	0	0
$855$	25.5503	0.873803
$856$	0	0
$857$	−10.4133	−0.355710	−0.177855	−	0.984057i	$-0.556916\pi$
$857$	−10.4133	−0.355710	−0.177855	+	0.984057i	$0.556916\pi$
$858$	0	0
$859$	−26.3154	−0.897870	−0.448935	−	0.893564i	$-0.648196\pi$
$859$	−26.3154	−0.897870	−0.448935	+	0.893564i	$0.648196\pi$
$860$	0	0
$861$	−1.62946	−0.0555319
$862$	0	0
$863$	7.96335	0.271076	0.135538	−	0.990772i	$-0.456724\pi$
$863$	7.96335	0.271076	0.135538	+	0.990772i	$0.456724\pi$
$864$	0	0
$865$	−28.3490	−0.963896
$866$	0	0
$867$	6.64355	0.225627
$868$	0	0
$869$	1.36957	0.0464596
$870$	0	0
$871$	−8.21934	−0.278502
$872$	0	0
$873$	−1.97860	−0.0669655
$874$	0	0
$875$	−9.13906	−0.308956
$876$	0	0
$877$	3.97446	0.134208	0.0671040	−	0.997746i	$-0.478624\pi$
$877$	3.97446	0.134208	0.0671040	+	0.997746i	$0.478624\pi$
$878$	0	0
$879$	9.07591	0.306123
$880$	0	0
$881$	24.1022	0.812024	0.406012	−	0.913868i	$-0.366919\pi$
$881$	24.1022	0.812024	0.406012	+	0.913868i	$0.366919\pi$
$882$	0	0
$883$	−36.5212	−1.22904	−0.614518	−	0.788902i	$-0.710650\pi$
$883$	−36.5212	−1.22904	−0.614518	+	0.788902i	$0.710650\pi$
$884$	0	0
$885$	12.1289	0.407707
$886$	0	0
$887$	13.7431	0.461449	0.230725	−	0.973019i	$-0.425890\pi$
$887$	13.7431	0.461449	0.230725	+	0.973019i	$0.425890\pi$
$888$	0	0
$889$	17.9618	0.602418
$890$	0	0
$891$	0.471538	0.0157971
$892$	0	0
$893$	42.7905	1.43193
$894$	0	0
$895$	−26.0065	−0.869302
$896$	0	0
$897$	−2.08566	−0.0696383
$898$	0	0
$899$	54.9294	1.83200
$900$	0	0
$901$	−30.8883	−1.02904
$902$	0	0
$903$	9.78959	0.325777
$904$	0	0
$905$	−59.6876	−1.98408
$906$	0	0
$907$	−51.9137	−1.72377	−0.861884	−	0.507106i	$-0.830715\pi$
$907$	−51.9137	−1.72377	−0.861884	+	0.507106i	$0.830715\pi$
$908$	0	0
$909$	−27.0189	−0.896161
$910$	0	0
$911$	−41.7238	−1.38237	−0.691186	−	0.722677i	$-0.742911\pi$
$911$	−41.7238	−1.38237	−0.691186	+	0.722677i	$0.742911\pi$
$912$	0	0
$913$	3.42864	0.113471
$914$	0	0
$915$	−4.11662	−0.136091
$916$	0	0
$917$	−7.21085	−0.238123
$918$	0	0
$919$	−2.14840	−0.0708692	−0.0354346	−	0.999372i	$-0.511282\pi$
$919$	−2.14840	−0.0708692	−0.0354346	+	0.999372i	$0.511282\pi$
$920$	0	0
$921$	27.1797	0.895601
$922$	0	0
$923$	2.92094	0.0961440
$924$	0	0
$925$	11.3753	0.374017
$926$	0	0
$927$	−10.6102	−0.348483
$928$	0	0
$929$	−18.9288	−0.621033	−0.310517	−	0.950568i	$-0.600502\pi$
$929$	−18.9288	−0.621033	−0.310517	+	0.950568i	$0.600502\pi$
$930$	0	0
$931$	−6.08058	−0.199283
$932$	0	0
$933$	−24.5956	−0.805224
$934$	0	0
$935$	−3.21273	−0.105067
$936$	0	0
$937$	17.9584	0.586675	0.293337	−	0.956009i	$-0.405234\pi$
$937$	17.9584	0.586675	0.293337	+	0.956009i	$0.405234\pi$
$938$	0	0
$939$	−1.04286	−0.0340326
$940$	0	0
$941$	9.60380	0.313075	0.156537	−	0.987672i	$-0.449967\pi$
$941$	9.60380	0.313075	0.156537	+	0.987672i	$0.449967\pi$
$942$	0	0
$943$	−2.54235	−0.0827903
$944$	0	0
$945$	13.6211	0.443093
$946$	0	0
$947$	13.1503	0.427328	0.213664	−	0.976907i	$-0.431460\pi$
$947$	13.1503	0.427328	0.213664	+	0.976907i	$0.431460\pi$
$948$	0	0
$949$	−8.18871	−0.265817
$950$	0	0
$951$	−2.66028	−0.0862655
$952$	0	0
$953$	−17.4224	−0.564367	−0.282184	−	0.959360i	$-0.591059\pi$
$953$	−17.4224	−0.564367	−0.282184	+	0.959360i	$0.591059\pi$
$954$	0	0
$955$	47.3268	1.53146
$956$	0	0
$957$	−3.06864	−0.0991950
$958$	0	0
$959$	2.21158	0.0714156
$960$	0	0
$961$	30.5494	0.985465
$962$	0	0
$963$	12.0650	0.388788
$964$	0	0
$965$	−8.08980	−0.260420
$966$	0	0
$967$	27.2914	0.877631	0.438815	−	0.898577i	$-0.355398\pi$
$967$	27.2914	0.877631	0.438815	+	0.898577i	$0.355398\pi$
$968$	0	0
$969$	23.5843	0.757638
$970$	0	0
$971$	−15.2896	−0.490667	−0.245334	−	0.969439i	$-0.578898\pi$
$971$	−15.2896	−0.490667	−0.245334	+	0.969439i	$0.578898\pi$
$972$	0	0
$973$	−6.61432	−0.212045
$974$	0	0
$975$	1.59846	0.0511917
$976$	0	0
$977$	47.1962	1.50994	0.754970	−	0.655760i	$-0.227651\pi$
$977$	47.1962	1.50994	0.754970	+	0.655760i	$0.227651\pi$
$978$	0	0
$979$	−2.11708	−0.0676621
$980$	0	0
$981$	30.3329	0.968456
$982$	0	0
$983$	21.2367	0.677346	0.338673	−	0.940904i	$-0.390022\pi$
$983$	21.2367	0.677346	0.338673	+	0.940904i	$0.390022\pi$
$984$	0	0
$985$	3.54829	0.113058
$986$	0	0
$987$	8.13633	0.258982
$988$	0	0
$989$	15.2741	0.485688
$990$	0	0
$991$	−31.0244	−0.985522	−0.492761	−	0.870165i	$-0.664012\pi$
$991$	−31.0244	−0.985522	−0.492761	+	0.870165i	$0.664012\pi$
$992$	0	0
$993$	5.66558	0.179792
$994$	0	0
$995$	−27.2959	−0.865339
$996$	0	0
$997$	−46.1666	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$997$	−46.1666	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$998$	0	0
$999$	44.3611	1.40352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.a.q.1.3	✓	5
4.3	odd	2		2912.2.a.t.1.3	yes	5
8.3	odd	2		5824.2.a.ci.1.3		5
8.5	even	2		5824.2.a.cl.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.a.q.1.3	✓	5	1.1	even	1	trivial
2912.2.a.t.1.3	yes	5	4.3	odd	2
5824.2.a.ci.1.3		5	8.3	odd	2
5824.2.a.cl.1.3		5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.15618 q^{3} +2.52637 q^{5} +1.00000 q^{7} -1.66324 q^{9} -0.379076 q^{11} -1.00000 q^{13} -2.92094 q^{15} +3.35468 q^{17} -6.08058 q^{19} -1.15618 q^{21} -1.80392 q^{23} +1.38253 q^{25} +5.39156 q^{27} -7.00153 q^{29} -7.84534 q^{31} +0.438281 q^{33} +2.52637 q^{35} +8.22787 q^{37} +1.15618 q^{39} +1.40935 q^{41} -8.46716 q^{43} -4.20195 q^{45} -7.03723 q^{47} +1.00000 q^{49} -3.87863 q^{51} -9.20753 q^{53} -0.957685 q^{55} +7.03027 q^{57} -4.15237 q^{59} +1.40935 q^{61} -1.66324 q^{63} -2.52637 q^{65} +8.21934 q^{67} +2.08566 q^{69} -2.92094 q^{71} +8.18871 q^{73} -1.59846 q^{75} -0.379076 q^{77} -3.61293 q^{79} -1.24391 q^{81} -9.04474 q^{83} +8.47516 q^{85} +8.09505 q^{87} +5.58484 q^{89} -1.00000 q^{91} +9.07066 q^{93} -15.3618 q^{95} +1.18961 q^{97} +0.630494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 3 q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} + 4 q^{17} - 7 q^{19} - 5 q^{21} + 4 q^{23} + 2 q^{25} - 23 q^{27} + 3 q^{29} - 2 q^{31} + 17 q^{33} - 3 q^{35} - q^{37} + 5 q^{39} - 7 q^{41}+ \cdots - 40 q^{99}+O(q^{100}) \) 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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