LMFDB - Embedded newform 2912.2.h.a.2575.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(2575,2912)]

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([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2912.2575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.h (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.14
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.a.2575.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48862i q^{3} +1.02997 q^{5} +(1.70779 - 2.02076i) q^{7} +0.784007 q^{9} -0.608038 q^{11} -1.00000 q^{13} -1.53323i q^{15} -0.526406i q^{17} -1.94810i q^{19} +(-3.00814 - 2.54225i) q^{21} +0.0153141i q^{23} -3.93917 q^{25} -5.63295i q^{27} -4.64455i q^{29} -4.57506 q^{31} +0.905138i q^{33} +(1.75897 - 2.08131i) q^{35} +2.15479i q^{37} +1.48862i q^{39} -6.24596i q^{41} +3.29851 q^{43} +0.807502 q^{45} +2.60155 q^{47} +(-1.16691 - 6.90205i) q^{49} -0.783619 q^{51} +1.16935i q^{53} -0.626260 q^{55} -2.89999 q^{57} -0.783099i q^{59} +5.32371 q^{61} +(1.33892 - 1.58429i) q^{63} -1.02997 q^{65} +4.18167 q^{67} +0.0227969 q^{69} +14.9594i q^{71} -9.18228i q^{73} +5.86392i q^{75} +(-1.03840 + 1.22870i) q^{77} -9.29066i q^{79} -6.03331 q^{81} -1.45140i q^{83} -0.542182i q^{85} -6.91397 q^{87} -12.0597i q^{89} +(-1.70779 + 2.02076i) q^{91} +6.81053i q^{93} -2.00649i q^{95} -8.46670i q^{97} -0.476706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) $ 48 * q - 48 * q^9 + 4 * q^11 - 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 + 24 * q^45 + 40 * q^51 - 20 * q^63 + 4 * q^67 - 20 * q^77 + 64 * q^81 + 40 * q^87 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$		−	1.48862i		−	0.859456i	−0.902958	−	0.429728i	$-0.858609\pi$
$3$		−	1.48862i		−	0.859456i	0.902958	−	0.429728i	$-0.141391\pi$
$4$		0			0
$5$	1.02997			0.460616			0.230308	−	0.973118i	$-0.426027\pi$
$5$	1.02997			0.460616			0.230308	+	0.973118i	$0.426027\pi$
$6$		0			0
$7$	1.70779	−	2.02076i	0.645484	−	0.763774i
$7$	1.70779	−	2.02076i	0.645484	−	0.763774i
$8$		0			0
$9$	0.784007			0.261336
$10$		0			0
$11$	−0.608038			−0.183330			−0.0916651	−	0.995790i	$-0.529219\pi$
$11$	−0.608038			−0.183330			−0.0916651	+	0.995790i	$0.529219\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$		−	1.53323i		−	0.395879i
$16$		0			0
$17$		−	0.526406i		−	0.127672i	−0.997960	−	0.0638361i	$-0.979667\pi$
$17$		−	0.526406i		−	0.127672i	0.997960	−	0.0638361i	$-0.0203335\pi$
$18$		0			0
$19$		−	1.94810i		−	0.446926i	−0.974712	−	0.223463i	$-0.928264\pi$
$19$		−	1.94810i		−	0.446926i	0.974712	−	0.223463i	$-0.0717361\pi$
$20$		0			0
$21$	−3.00814	−	2.54225i	−0.656430	−	0.554765i
$22$		0			0
$23$			0.0153141i			0.00319321i	0.999999	+	0.00159660i	$0.000508215\pi$
$23$			0.0153141i			0.00319321i	−0.999999	+	0.00159660i	$0.999492\pi$
$24$		0			0
$25$	−3.93917			−0.787833
$26$		0			0
$27$		−	5.63295i		−	1.08406i
$28$		0			0
$29$		−	4.64455i		−	0.862471i	−0.902239	−	0.431236i	$-0.858078\pi$
$29$		−	4.64455i		−	0.862471i	0.902239	−	0.431236i	$-0.141922\pi$
$30$		0			0
$31$	−4.57506			−0.821705			−0.410852	−	0.911702i	$-0.634769\pi$
$31$	−4.57506			−0.821705			−0.410852	+	0.911702i	$0.634769\pi$
$32$		0			0
$33$			0.905138i			0.157564i
$34$		0			0
$35$	1.75897	−	2.08131i	0.297320	−	0.351806i
$36$		0			0
$37$			2.15479i			0.354246i	0.984189	+	0.177123i	$0.0566790\pi$
$37$			2.15479i			0.354246i	−0.984189	+	0.177123i	$0.943321\pi$
$38$		0			0
$39$			1.48862i			0.238370i
$40$		0			0
$41$		−	6.24596i		−	0.975455i	−0.872996	−	0.487727i	$-0.837826\pi$
$41$		−	6.24596i		−	0.975455i	0.872996	−	0.487727i	$-0.162174\pi$
$42$		0			0
$43$	3.29851			0.503019			0.251509	−	0.967855i	$-0.419073\pi$
$43$	3.29851			0.503019			0.251509	+	0.967855i	$0.419073\pi$
$44$		0			0
$45$	0.807502			0.120375
$46$		0			0
$47$	2.60155			0.379475			0.189737	−	0.981835i	$-0.439236\pi$
$47$	2.60155			0.379475			0.189737	+	0.981835i	$0.439236\pi$
$48$		0			0
$49$	−1.16691	−	6.90205i	−0.166701	−	0.986008i
$50$		0			0
$51$	−0.783619			−0.109729
$52$		0			0
$53$			1.16935i			0.160622i	0.996770	+	0.0803111i	$0.0255914\pi$
$53$			1.16935i			0.160622i	−0.996770	+	0.0803111i	$0.974409\pi$
$54$		0			0
$55$	−0.626260			−0.0844448
$56$		0			0
$57$	−2.89999			−0.384113
$58$		0			0
$59$		−	0.783099i		−	0.101951i	−0.998700	−	0.0509754i	$-0.983767\pi$
$59$		−	0.783099i		−	0.101951i	0.998700	−	0.0509754i	$-0.0162330\pi$
$60$		0			0
$61$	5.32371			0.681631			0.340815	−	0.940130i	$-0.389297\pi$
$61$	5.32371			0.681631			0.340815	+	0.940130i	$0.389297\pi$
$62$		0			0
$63$	1.33892	−	1.58429i	0.168688	−	0.199601i
$64$		0			0
$65$	−1.02997			−0.127752
$66$		0			0
$67$	4.18167			0.510872			0.255436	−	0.966826i	$-0.417781\pi$
$67$	4.18167			0.510872			0.255436	+	0.966826i	$0.417781\pi$
$68$		0			0
$69$	0.0227969			0.00274442
$70$		0			0
$71$			14.9594i			1.77535i	0.460470	+	0.887675i	$0.347681\pi$
$71$			14.9594i			1.77535i	−0.460470	+	0.887675i	$0.652319\pi$
$72$		0			0
$73$		−	9.18228i		−	1.07470i	−0.843358	−	0.537352i	$-0.819425\pi$
$73$		−	9.18228i		−	1.07470i	0.843358	−	0.537352i	$-0.180575\pi$
$74$		0			0
$75$			5.86392i			0.677108i
$76$		0			0
$77$	−1.03840	+	1.22870i	−0.118337	+	0.140023i
$78$		0			0
$79$		−	9.29066i		−	1.04528i	−0.852553	−	0.522641i	$-0.824947\pi$
$79$		−	9.29066i		−	1.04528i	0.852553	−	0.522641i	$-0.175053\pi$
$80$		0			0
$81$	−6.03331			−0.670368
$82$		0			0
$83$		−	1.45140i		−	0.159311i	−0.996822	−	0.0796557i	$-0.974618\pi$
$83$		−	1.45140i		−	0.159311i	0.996822	−	0.0796557i	$-0.0253821\pi$
$84$		0			0
$85$		−	0.542182i		−	0.0588078i
$86$		0			0
$87$	−6.91397			−0.741256
$88$		0			0
$89$		−	12.0597i		−	1.27832i	−0.769073	−	0.639162i	$-0.779282\pi$
$89$		−	12.0597i		−	1.27832i	0.769073	−	0.639162i	$-0.220718\pi$
$90$		0			0
$91$	−1.70779	+	2.02076i	−0.179025	+	0.211833i
$92$		0			0
$93$			6.81053i			0.706219i
$94$		0			0
$95$		−	2.00649i		−	0.205861i
$96$		0			0
$97$		−	8.46670i		−	0.859663i	−0.902909	−	0.429831i	$-0.858573\pi$
$97$		−	8.46670i		−	0.859663i	0.902909	−	0.429831i	$-0.141427\pi$
$98$		0			0
$99$	−0.476706			−0.0479107
$100$		0			0
$101$	7.70127			0.766305			0.383153	−	0.923685i	$-0.374838\pi$
$101$	7.70127			0.766305			0.383153	+	0.923685i	$0.374838\pi$
$102$		0			0
$103$	4.43728			0.437218			0.218609	−	0.975813i	$-0.429848\pi$
$103$	4.43728			0.437218			0.218609	+	0.975813i	$0.429848\pi$
$104$		0			0
$105$	−3.09829	−	2.61844i	−0.302362	−	0.255534i
$106$		0			0
$107$	−6.18159			−0.597597			−0.298798	−	0.954316i	$-0.596586\pi$
$107$	−6.18159			−0.597597			−0.298798	+	0.954316i	$0.596586\pi$
$108$		0			0
$109$			16.1401i			1.54594i	0.634441	+	0.772971i	$0.281230\pi$
$109$			16.1401i			1.54594i	−0.634441	+	0.772971i	$0.718770\pi$
$110$		0			0
$111$	3.20767			0.304459
$112$		0			0
$113$	−2.76655			−0.260255			−0.130128	−	0.991497i	$-0.541539\pi$
$113$	−2.76655			−0.260255			−0.130128	+	0.991497i	$0.541539\pi$
$114$		0			0
$115$			0.0157730i			0.00147084i
$116$		0			0
$117$	−0.784007			−0.0724815
$118$		0			0
$119$	−1.06374	−	0.898991i	−0.0975127	−	0.0824104i
$120$		0			0
$121$	−10.6303			−0.966390
$122$		0			0
$123$	−9.29787			−0.838360
$124$		0			0
$125$	−9.20706			−0.823504
$126$		0			0
$127$			11.5046i			1.02087i	0.859917	+	0.510434i	$0.170515\pi$
$127$			11.5046i			1.02087i	−0.859917	+	0.510434i	$0.829485\pi$
$128$		0			0
$129$		−	4.91024i		−	0.432322i
$130$		0			0
$131$		−	10.8481i		−	0.947800i	−0.880579	−	0.473900i	$-0.842846\pi$
$131$		−	10.8481i		−	0.947800i	0.880579	−	0.473900i	$-0.157154\pi$
$132$		0			0
$133$	−3.93664	−	3.32695i	−0.341350	−	0.288483i
$134$		0			0
$135$		−	5.80176i		−	0.499336i
$136$		0			0
$137$	−3.67475			−0.313955			−0.156977	−	0.987602i	$-0.550175\pi$
$137$	−3.67475			−0.313955			−0.156977	+	0.987602i	$0.550175\pi$
$138$		0			0
$139$			14.5439i			1.23360i	0.787121	+	0.616798i	$0.211571\pi$
$139$			14.5439i			1.23360i	−0.787121	+	0.616798i	$0.788429\pi$
$140$		0			0
$141$		−	3.87272i		−	0.326142i
$142$		0			0
$143$	0.608038			0.0508467
$144$		0			0
$145$		−	4.78374i		−	0.397268i
$146$		0			0
$147$	−10.2745	+	1.73708i	−0.847430	+	0.143272i
$148$		0			0
$149$		−	15.7637i		−	1.29142i	−0.763584	−	0.645708i	$-0.776562\pi$
$149$		−	15.7637i		−	1.29142i	0.763584	−	0.645708i	$-0.223438\pi$
$150$		0			0
$151$			8.76049i			0.712919i	0.934311	+	0.356459i	$0.116016\pi$
$151$			8.76049i			0.712919i	−0.934311	+	0.356459i	$0.883984\pi$
$152$		0			0
$153$		−	0.412706i		−	0.0333653i
$154$		0			0
$155$	−4.71216			−0.378490
$156$		0			0
$157$	13.6353			1.08821			0.544106	−	0.839016i	$-0.316869\pi$
$157$	13.6353			1.08821			0.544106	+	0.839016i	$0.316869\pi$
$158$		0			0
$159$	1.74071			0.138048
$160$		0			0
$161$	0.0309460	+	0.0261533i	0.00243889	+	0.00206117i
$162$		0			0
$163$	5.71060			0.447288			0.223644	−	0.974671i	$-0.428205\pi$
$163$	5.71060			0.447288			0.223644	+	0.974671i	$0.428205\pi$
$164$		0			0
$165$			0.932263i			0.0725766i
$166$		0			0
$167$	16.0117			1.23902			0.619512	−	0.784987i	$-0.287331\pi$
$167$	16.0117			1.23902			0.619512	+	0.784987i	$0.287331\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		−	1.52733i		−	0.116798i
$172$		0			0
$173$	13.1654			1.00095			0.500475	−	0.865751i	$-0.333159\pi$
$173$	13.1654			1.00095			0.500475	+	0.865751i	$0.333159\pi$
$174$		0			0
$175$	−6.72727	+	7.96009i	−0.508534	+	0.601726i
$176$		0			0
$177$	−1.16574			−0.0876223
$178$		0			0
$179$	−3.27796			−0.245006			−0.122503	−	0.992468i	$-0.539092\pi$
$179$	−3.27796			−0.245006			−0.122503	+	0.992468i	$0.539092\pi$
$180$		0			0
$181$	−13.4214			−0.997606			−0.498803	−	0.866715i	$-0.666227\pi$
$181$	−13.4214			−0.997606			−0.498803	+	0.866715i	$0.666227\pi$
$182$		0			0
$183$		−	7.92498i		−	0.585831i
$184$		0			0
$185$			2.21937i			0.163171i
$186$		0			0
$187$			0.320075i			0.0234062i
$188$		0			0
$189$	−11.3828	−	9.61990i	−0.827978	−	0.699745i
$190$		0			0
$191$		−	13.3702i		−	0.967437i	−0.875224	−	0.483719i	$-0.839286\pi$
$191$		−	13.3702i		−	0.967437i	0.875224	−	0.483719i	$-0.160714\pi$
$192$		0			0
$193$	−14.0635			−1.01231			−0.506157	−	0.862441i	$-0.668934\pi$
$193$	−14.0635			−1.01231			−0.506157	+	0.862441i	$0.668934\pi$
$194$		0			0
$195$			1.53323i			0.109797i
$196$		0			0
$197$			7.40334i			0.527466i	0.964596	+	0.263733i	$0.0849538\pi$
$197$			7.40334i			0.527466i	−0.964596	+	0.263733i	$0.915046\pi$
$198$		0			0
$199$	−21.2213			−1.50434			−0.752170	−	0.658969i	$-0.770993\pi$
$199$	−21.2213			−1.50434			−0.752170	+	0.658969i	$0.770993\pi$
$200$		0			0
$201$		−	6.22492i		−	0.439072i
$202$		0			0
$203$	−9.38550	−	7.93191i	−0.658733	−	0.556711i
$204$		0			0
$205$		−	6.43314i		−	0.449310i
$206$		0			0
$207$			0.0120064i			0.000834499i
$208$		0			0
$209$			1.18452i			0.0819350i
$210$		0			0
$211$	9.27855			0.638762			0.319381	−	0.947626i	$-0.396525\pi$
$211$	9.27855			0.638762			0.319381	+	0.947626i	$0.396525\pi$
$212$		0			0
$213$	22.2688			1.52584
$214$		0			0
$215$	3.39736			0.231698
$216$		0			0
$217$	−7.81324	+	9.24507i	−0.530397	+	0.627597i
$218$		0			0
$219$	−13.6689			−0.923661
$220$		0			0
$221$			0.526406i			0.0354099i
$222$		0			0
$223$	0.650804			0.0435811			0.0217905	−	0.999763i	$-0.493063\pi$
$223$	0.650804			0.0435811			0.0217905	+	0.999763i	$0.493063\pi$
$224$		0			0
$225$	−3.08833			−0.205889
$226$		0			0
$227$		−	22.9227i		−	1.52143i	−0.649084	−	0.760717i	$-0.724848\pi$
$227$		−	22.9227i		−	1.52143i	0.649084	−	0.760717i	$-0.275152\pi$
$228$		0			0
$229$	−9.33589			−0.616933			−0.308467	−	0.951235i	$-0.599816\pi$
$229$	−9.33589			−0.616933			−0.308467	+	0.951235i	$0.599816\pi$
$230$		0			0
$231$	1.82906	+	1.54579i	0.120343	+	0.101705i
$232$		0			0
$233$	−9.56308			−0.626498			−0.313249	−	0.949671i	$-0.601417\pi$
$233$	−9.56308			−0.626498			−0.313249	+	0.949671i	$0.601417\pi$
$234$		0			0
$235$	2.67951			0.174792
$236$		0			0
$237$	−13.8303			−0.898373
$238$		0			0
$239$		−	16.2651i		−	1.05210i	−0.850453	−	0.526051i	$-0.823672\pi$
$239$		−	16.2651i		−	1.05210i	0.850453	−	0.526051i	$-0.176328\pi$
$240$		0			0
$241$			17.9771i			1.15800i	0.815326	+	0.579002i	$0.196558\pi$
$241$			17.9771i			1.15800i	−0.815326	+	0.579002i	$0.803442\pi$
$242$		0			0
$243$		−	7.91754i		−	0.507911i
$244$		0			0
$245$	−1.20188	−	7.10890i	−0.0767851	−	0.454171i
$246$		0			0
$247$			1.94810i			0.123955i
$248$		0			0
$249$	−2.16058			−0.136921
$250$		0			0
$251$		−	16.3892i		−	1.03447i	−0.855842	−	0.517237i	$-0.826961\pi$
$251$		−	16.3892i		−	1.03447i	0.855842	−	0.517237i	$-0.173039\pi$
$252$		0			0
$253$		−	0.00931155i		−	0.000585412i
$254$		0			0
$255$	−0.807103			−0.0505428
$256$		0			0
$257$			27.4462i			1.71205i	0.516937	+	0.856023i	$0.327072\pi$
$257$			27.4462i			1.71205i	−0.516937	+	0.856023i	$0.672928\pi$
$258$		0			0
$259$	4.35431	+	3.67993i	0.270564	+	0.228660i
$260$		0			0
$261$		−	3.64136i		−	0.225394i
$262$		0			0
$263$			4.35567i			0.268582i	0.990942	+	0.134291i	$0.0428757\pi$
$263$			4.35567i			0.268582i	−0.990942	+	0.134291i	$0.957124\pi$
$264$		0			0
$265$			1.20439i			0.0739851i
$266$		0			0
$267$	−17.9523			−1.09866
$268$		0			0
$269$	20.3914			1.24328			0.621642	−	0.783302i	$-0.286466\pi$
$269$	20.3914			1.24328			0.621642	+	0.783302i	$0.286466\pi$
$270$		0			0
$271$	31.3447			1.90405			0.952027	−	0.306015i	$-0.0989957\pi$
$271$	31.3447			1.90405			0.952027	+	0.306015i	$0.0989957\pi$
$272$		0			0
$273$	3.00814	+	2.54225i	0.182061	+	0.153864i
$274$		0			0
$275$	2.39516			0.144434
$276$		0			0
$277$			5.40265i			0.324614i	0.986740	+	0.162307i	$0.0518934\pi$
$277$			5.40265i			0.324614i	−0.986740	+	0.162307i	$0.948107\pi$
$278$		0			0
$279$	−3.58688			−0.214741
$280$		0			0
$281$	0.635876			0.0379332			0.0189666	−	0.999820i	$-0.493962\pi$
$281$	0.635876			0.0379332			0.0189666	+	0.999820i	$0.493962\pi$
$282$		0			0
$283$			6.06925i			0.360779i	0.983595	+	0.180390i	$0.0577359\pi$
$283$			6.06925i			0.360779i	−0.983595	+	0.180390i	$0.942264\pi$
$284$		0			0
$285$	−2.98690			−0.176928
$286$		0			0
$287$	−12.6216	−	10.6668i	−0.745027	−	0.629640i
$288$		0			0
$289$	16.7229			0.983700
$290$		0			0
$291$	−12.6037			−0.738842
$292$		0			0
$293$	−29.8751			−1.74532			−0.872660	−	0.488328i	$-0.837607\pi$
$293$	−29.8751			−1.74532			−0.872660	+	0.488328i	$0.837607\pi$
$294$		0			0
$295$		−	0.806568i		−	0.0469602i
$296$		0			0
$297$			3.42505i			0.198741i
$298$		0			0
$299$		−	0.0153141i		−	0.000885637i
$300$		0			0
$301$	5.63317	−	6.66549i	0.324690	−	0.384192i
$302$		0			0
$303$		−	11.4643i		−	0.658606i
$304$		0			0
$305$	5.48325			0.313970
$306$		0			0
$307$		−	22.3570i		−	1.27598i	−0.770044	−	0.637991i	$-0.779766\pi$
$307$		−	22.3570i		−	1.27598i	0.770044	−	0.637991i	$-0.220234\pi$
$308$		0			0
$309$		−	6.60543i		−	0.375770i
$310$		0			0
$311$	−21.5289			−1.22079			−0.610396	−	0.792096i	$-0.708990\pi$
$311$	−21.5289			−1.22079			−0.610396	+	0.792096i	$0.708990\pi$
$312$		0			0
$313$		−	2.06304i		−	0.116610i	−0.998299	−	0.0583050i	$-0.981430\pi$
$313$		−	2.06304i		−	0.116610i	0.998299	−	0.0583050i	$-0.0185696\pi$
$314$		0			0
$315$	1.37904	−	1.63176i	0.0777003	−	0.0919395i
$316$		0			0
$317$			32.5712i			1.82938i	0.404156	+	0.914690i	$0.367565\pi$
$317$			32.5712i			1.82938i	−0.404156	+	0.914690i	$0.632435\pi$
$318$		0			0
$319$			2.82406i			0.158117i
$320$		0			0
$321$			9.20205i			0.513608i
$322$		0			0
$323$	−1.02549			−0.0570600
$324$		0			0
$325$	3.93917			0.218506
$326$		0			0
$327$	24.0265			1.32867
$328$		0			0
$329$	4.44290	−	5.25709i	0.244945	−	0.289833i
$330$		0			0
$331$	14.1674			0.778710			0.389355	−	0.921088i	$-0.372698\pi$
$331$	14.1674			0.778710			0.389355	+	0.921088i	$0.372698\pi$
$332$		0			0
$333$			1.68937i			0.0925770i
$334$		0			0
$335$	4.30699			0.235316
$336$		0			0
$337$	23.0042			1.25312			0.626561	−	0.779373i	$-0.284462\pi$
$337$	23.0042			1.25312			0.626561	+	0.779373i	$0.284462\pi$
$338$		0			0
$339$			4.11834i			0.223678i
$340$		0			0
$341$	2.78181			0.150643
$342$		0			0
$343$	−15.9402	−	9.42922i	−0.860689	−	0.509130i
$344$		0			0
$345$	0.0234801			0.00126412
$346$		0			0
$347$	−17.8224			−0.956756			−0.478378	−	0.878154i	$-0.658775\pi$
$347$	−17.8224			−0.956756			−0.478378	+	0.878154i	$0.658775\pi$
$348$		0			0
$349$	0.0209269			0.00112019			0.000560096	−	1.00000i	$-0.499822\pi$
$349$	0.0209269			0.00112019			0.000560096		1.00000i	$0.499822\pi$
$350$		0			0
$351$			5.63295i			0.300665i
$352$		0			0
$353$			8.02237i			0.426988i	0.976944	+	0.213494i	$0.0684844\pi$
$353$			8.02237i			0.426988i	−0.976944	+	0.213494i	$0.931516\pi$
$354$		0			0
$355$			15.4077i			0.817755i
$356$		0			0
$357$	−1.33826	+	1.58350i	−0.0708281	+	0.0838079i
$358$		0			0
$359$			14.8589i			0.784223i	0.919918	+	0.392112i	$0.128255\pi$
$359$			14.8589i			0.784223i	−0.919918	+	0.392112i	$0.871745\pi$
$360$		0			0
$361$	15.2049			0.800257
$362$		0			0
$363$			15.8245i			0.830570i
$364$		0			0
$365$		−	9.45746i		−	0.495026i
$366$		0			0
$367$	−20.8036			−1.08594			−0.542970	−	0.839752i	$-0.682700\pi$
$367$	−20.8036			−1.08594			−0.542970	+	0.839752i	$0.682700\pi$
$368$		0			0
$369$		−	4.89687i		−	0.254921i
$370$		0			0
$371$	2.36296	+	1.99700i	0.122679	+	0.103679i
$372$		0			0
$373$		−	19.1563i		−	0.991875i	−0.868358	−	0.495937i	$-0.834825\pi$
$373$		−	19.1563i		−	0.991875i	0.868358	−	0.495937i	$-0.165175\pi$
$374$		0			0
$375$			13.7058i			0.707766i
$376$		0			0
$377$			4.64455i			0.239206i
$378$		0			0
$379$	−1.31524			−0.0675596			−0.0337798	−	0.999429i	$-0.510754\pi$
$379$	−1.31524			−0.0675596			−0.0337798	+	0.999429i	$0.510754\pi$
$380$		0			0
$381$	17.1260			0.877392
$382$		0			0
$383$	8.37026			0.427700			0.213850	−	0.976866i	$-0.431400\pi$
$383$	8.37026			0.427700			0.213850	+	0.976866i	$0.431400\pi$
$384$		0			0
$385$	−1.06952	+	1.26552i	−0.0545078	+	0.0644968i
$386$		0			0
$387$	2.58606			0.131457
$388$		0			0
$389$			29.5123i			1.49634i	0.663510	+	0.748168i	$0.269066\pi$
$389$			29.5123i			1.49634i	−0.663510	+	0.748168i	$0.730934\pi$
$390$		0			0
$391$	0.00806143			0.000407684
$392$		0			0
$393$	−16.1487			−0.814592
$394$		0			0
$395$		−	9.56909i		−	0.481473i
$396$		0			0
$397$	−18.5170			−0.929339			−0.464670	−	0.885484i	$-0.653827\pi$
$397$	−18.5170			−0.929339			−0.464670	+	0.885484i	$0.653827\pi$
$398$		0			0
$399$	−4.95257	+	5.86017i	−0.247939	+	0.293375i
$400$		0			0
$401$	23.4394			1.17051			0.585253	−	0.810850i	$-0.300995\pi$
$401$	23.4394			1.17051			0.585253	+	0.810850i	$0.300995\pi$
$402$		0			0
$403$	4.57506			0.227900
$404$		0			0
$405$	−6.21412			−0.308782
$406$		0			0
$407$		−	1.31019i		−	0.0649440i
$408$		0			0
$409$		−	1.04236i		−	0.0515414i	−0.999668	−	0.0257707i	$-0.991796\pi$
$409$		−	1.04236i		−	0.0515414i	0.999668	−	0.0257707i	$-0.00820398\pi$
$410$		0			0
$411$			5.47031i			0.269830i
$412$		0			0
$413$	−1.58245	−	1.33737i	−0.0778674	−	0.0658076i
$414$		0			0
$415$		−	1.49489i		−	0.0733813i
$416$		0			0
$417$	21.6503			1.06022
$418$		0			0
$419$			5.33068i			0.260421i	0.991486	+	0.130210i	$0.0415653\pi$
$419$			5.33068i			0.260421i	−0.991486	+	0.130210i	$0.958435\pi$
$420$		0			0
$421$			35.9690i			1.75302i	0.481381	+	0.876511i	$0.340135\pi$
$421$			35.9690i			1.75302i	−0.481381	+	0.876511i	$0.659865\pi$
$422$		0			0
$423$	2.03963			0.0991703
$424$		0			0
$425$			2.07360i			0.100584i
$426$		0			0
$427$	9.09177	−	10.7579i	0.439982	−	0.520612i
$428$		0			0
$429$		−	0.905138i		−	0.0437005i
$430$		0			0
$431$			22.1177i			1.06537i	0.846312	+	0.532687i	$0.178818\pi$
$431$			22.1177i			1.06537i	−0.846312	+	0.532687i	$0.821182\pi$
$432$		0			0
$433$			11.5805i			0.556523i	0.960505	+	0.278261i	$0.0897581\pi$
$433$			11.5805i			0.556523i	−0.960505	+	0.278261i	$0.910242\pi$
$434$		0			0
$435$	−7.12117			−0.341434
$436$		0			0
$437$	0.0298334			0.00142713
$438$		0			0
$439$	38.2244			1.82435			0.912175	−	0.409801i	$-0.134402\pi$
$439$	38.2244			1.82435			0.912175	+	0.409801i	$0.134402\pi$
$440$		0			0
$441$	−0.914863	−	5.41126i	−0.0435649	−	0.257679i
$442$		0			0
$443$	−15.3359			−0.728629			−0.364315	−	0.931276i	$-0.618697\pi$
$443$	−15.3359			−0.728629			−0.364315	+	0.931276i	$0.618697\pi$
$444$		0			0
$445$		−	12.4211i		−	0.588816i
$446$		0			0
$447$	−23.4662			−1.10992
$448$		0			0
$449$	23.3025			1.09971			0.549857	−	0.835259i	$-0.314682\pi$
$449$	23.3025			1.09971			0.549857	+	0.835259i	$0.314682\pi$
$450$		0			0
$451$			3.79778i			0.178830i
$452$		0			0
$453$	13.0411			0.612722
$454$		0			0
$455$	−1.75897	+	2.08131i	−0.0824618	+	0.0975735i
$456$		0			0
$457$	7.74554			0.362321			0.181161	−	0.983454i	$-0.442015\pi$
$457$	7.74554			0.362321			0.181161	+	0.983454i	$0.442015\pi$
$458$		0			0
$459$	−2.96522			−0.138405
$460$		0			0
$461$	−22.6982			−1.05716			−0.528581	−	0.848883i	$-0.677276\pi$
$461$	−22.6982			−1.05716			−0.528581	+	0.848883i	$0.677276\pi$
$462$		0			0
$463$			15.2556i			0.708988i	0.935058	+	0.354494i	$0.115347\pi$
$463$			15.2556i			0.708988i	−0.935058	+	0.354494i	$0.884653\pi$
$464$		0			0
$465$			7.01463i			0.325296i
$466$		0			0
$467$		−	33.7119i		−	1.56000i	−0.625779	−	0.780000i	$-0.715219\pi$
$467$		−	33.7119i		−	1.56000i	0.625779	−	0.780000i	$-0.284781\pi$
$468$		0			0
$469$	7.14141	−	8.45013i	0.329760	−	0.390191i
$470$		0			0
$471$		−	20.2977i		−	0.935271i
$472$		0			0
$473$	−2.00562			−0.0922185
$474$		0			0
$475$			7.67390i			0.352103i
$476$		0			0
$477$			0.916776i			0.0419763i
$478$		0			0
$479$	27.5473			1.25867			0.629333	−	0.777136i	$-0.283328\pi$
$479$	27.5473			1.25867			0.629333	+	0.777136i	$0.283328\pi$
$480$		0			0
$481$		−	2.15479i		−	0.0982501i
$482$		0			0
$483$	0.0389323	−	0.0460669i	0.00177148	−	0.00209612i
$484$		0			0
$485$		−	8.72043i		−	0.395974i
$486$		0			0
$487$			13.1983i			0.598073i	0.954242	+	0.299037i	$0.0966653\pi$
$487$			13.1983i			0.598073i	−0.954242	+	0.299037i	$0.903335\pi$
$488$		0			0
$489$		−	8.50091i		−	0.384425i
$490$		0			0
$491$	8.07921			0.364609			0.182305	−	0.983242i	$-0.441644\pi$
$491$	8.07921			0.364609			0.182305	+	0.983242i	$0.441644\pi$
$492$		0			0
$493$	−2.44492			−0.110114
$494$		0			0
$495$	−0.490992			−0.0220684
$496$		0			0
$497$	30.2292	+	25.5475i	1.35597	+	1.14596i
$498$		0			0
$499$	26.0339			1.16544			0.582720	−	0.812673i	$-0.301989\pi$
$499$	26.0339			1.16544			0.582720	+	0.812673i	$0.301989\pi$
$500$		0			0
$501$		−	23.8354i		−	1.06489i
$502$		0			0
$503$	5.46230			0.243552			0.121776	−	0.992558i	$-0.461141\pi$
$503$	5.46230			0.243552			0.121776	+	0.992558i	$0.461141\pi$
$504$		0			0
$505$	7.93207			0.352972
$506$		0			0
$507$		−	1.48862i		−	0.0661120i
$508$		0			0
$509$	18.6505			0.826670			0.413335	−	0.910579i	$-0.364364\pi$
$509$	18.6505			0.826670			0.413335	+	0.910579i	$0.364364\pi$
$510$		0			0
$511$	−18.5551	−	15.6814i	−0.820831	−	0.693705i
$512$		0			0
$513$	−10.9736			−0.484495
$514$		0			0
$515$	4.57026			0.201390
$516$		0			0
$517$	−1.58184			−0.0695692
$518$		0			0
$519$		−	19.5984i		−	0.860273i
$520$		0			0
$521$		−	16.3034i		−	0.714267i	−0.934053	−	0.357134i	$-0.883754\pi$
$521$		−	16.3034i		−	0.714267i	0.934053	−	0.357134i	$-0.116246\pi$
$522$		0			0
$523$			19.4624i			0.851031i	0.904951	+	0.425516i	$0.139907\pi$
$523$			19.4624i			0.851031i	−0.904951	+	0.425516i	$0.860093\pi$
$524$		0			0
$525$	11.8496	+	10.0144i	0.517157	+	0.437062i
$526$		0			0
$527$			2.40834i			0.104909i
$528$		0			0
$529$	22.9998			0.999990
$530$		0			0
$531$		−	0.613955i		−	0.0266434i
$532$		0			0
$533$			6.24596i			0.270543i
$534$		0			0
$535$	−6.36684			−0.275263
$536$		0			0
$537$			4.87964i			0.210572i
$538$		0			0
$539$	0.709523	+	4.19671i	0.0305613	+	0.180765i
$540$		0			0
$541$		−	10.8033i		−	0.464471i	−0.972660	−	0.232236i	$-0.925396\pi$
$541$		−	10.8033i		−	0.464471i	0.972660	−	0.232236i	$-0.0746040\pi$
$542$		0			0
$543$			19.9794i			0.857398i
$544$		0			0
$545$			16.6238i			0.712086i
$546$		0			0
$547$	32.4266			1.38646			0.693230	−	0.720717i	$-0.256187\pi$
$547$	32.4266			1.38646			0.693230	+	0.720717i	$0.256187\pi$
$548$		0			0
$549$	4.17382			0.178134
$550$		0			0
$551$	−9.04806			−0.385460
$552$		0			0
$553$	−18.7742	−	15.8665i	−0.798358	−	0.674712i
$554$		0			0
$555$	3.30380			0.140238
$556$		0			0
$557$		−	4.44482i		−	0.188333i	−0.995556	−	0.0941666i	$-0.969981\pi$
$557$		−	4.44482i		−	0.188333i	0.995556	−	0.0941666i	$-0.0300186\pi$
$558$		0			0
$559$	−3.29851			−0.139512
$560$		0			0
$561$	0.476470			0.0201166
$562$		0			0
$563$		−	9.65256i		−	0.406807i	−0.979095	−	0.203403i	$-0.934800\pi$
$563$		−	9.65256i		−	0.406807i	0.979095	−	0.203403i	$-0.0652003\pi$
$564$		0			0
$565$	−2.84946			−0.119878
$566$		0			0
$567$	−10.3036	+	12.1919i	−0.432712	+	0.512010i
$568$		0			0
$569$	12.1247			0.508296			0.254148	−	0.967165i	$-0.418205\pi$
$569$	12.1247			0.508296			0.254148	+	0.967165i	$0.418205\pi$
$570$		0			0
$571$	33.2882			1.39307			0.696534	−	0.717524i	$-0.254724\pi$
$571$	33.2882			1.39307			0.696534	+	0.717524i	$0.254724\pi$
$572$		0			0
$573$	−19.9032			−0.831469
$574$		0			0
$575$		−	0.0603247i		−	0.00251572i
$576$		0			0
$577$			27.7175i			1.15390i	0.816781	+	0.576948i	$0.195757\pi$
$577$			27.7175i			1.15390i	−0.816781	+	0.576948i	$0.804243\pi$
$578$		0			0
$579$			20.9353i			0.870039i
$580$		0			0
$581$	−2.93291	−	2.47868i	−0.121678	−	0.102833i
$582$		0			0
$583$		−	0.711007i		−	0.0294469i
$584$		0			0
$585$	−0.807502			−0.0333861
$586$		0			0
$587$		−	21.9684i		−	0.906732i	−0.891324	−	0.453366i	$-0.850223\pi$
$587$		−	21.9684i		−	0.906732i	0.891324	−	0.453366i	$-0.149777\pi$
$588$		0			0
$589$			8.91269i			0.367241i
$590$		0			0
$591$	11.0208			0.453334
$592$		0			0
$593$			16.3479i			0.671327i	0.941982	+	0.335663i	$0.108960\pi$
$593$			16.3479i			0.671327i	−0.941982	+	0.335663i	$0.891040\pi$
$594$		0			0
$595$	−1.09562	−	0.925932i	−0.0449159	−	0.0379595i
$596$		0			0
$597$			31.5905i			1.29291i
$598$		0			0
$599$		−	36.2459i		−	1.48097i	−0.672075	−	0.740483i	$-0.734597\pi$
$599$		−	36.2459i		−	1.48097i	0.672075	−	0.740483i	$-0.265403\pi$
$600$		0			0
$601$		−	31.2461i		−	1.27455i	−0.770635	−	0.637277i	$-0.780061\pi$
$601$		−	31.2461i		−	1.27455i	0.770635	−	0.637277i	$-0.219939\pi$
$602$		0			0
$603$	3.27846			0.133509
$604$		0			0
$605$	−10.9489			−0.445135
$606$		0			0
$607$	41.8217			1.69749			0.848745	−	0.528803i	$-0.177359\pi$
$607$	41.8217			1.69749			0.848745	+	0.528803i	$0.177359\pi$
$608$		0			0
$609$	−11.8076	+	13.9715i	−0.478469	+	0.566152i
$610$		0			0
$611$	−2.60155			−0.105247
$612$		0			0
$613$			4.71554i			0.190459i	0.995455	+	0.0952293i	$0.0303584\pi$
$613$			4.71554i			0.190459i	−0.995455	+	0.0952293i	$0.969642\pi$
$614$		0			0
$615$	−9.57651			−0.386162
$616$		0			0
$617$	7.74058			0.311624			0.155812	−	0.987787i	$-0.450201\pi$
$617$	7.74058			0.311624			0.155812	+	0.987787i	$0.450201\pi$
$618$		0			0
$619$			44.8288i			1.80182i	0.434004	+	0.900911i	$0.357100\pi$
$619$			44.8288i			1.80182i	−0.434004	+	0.900911i	$0.642900\pi$
$620$		0			0
$621$	0.0862636			0.00346164
$622$		0			0
$623$	−24.3697	−	20.5954i	−0.976350	−	0.825137i
$624$		0			0
$625$	10.2128			0.408514
$626$		0			0
$627$	1.76330			0.0704195
$628$		0			0
$629$	1.13430			0.0452273
$630$		0			0
$631$		−	17.8015i		−	0.708666i	−0.935119	−	0.354333i	$-0.884708\pi$
$631$		−	17.8015i		−	0.708666i	0.935119	−	0.354333i	$-0.115292\pi$
$632$		0			0
$633$		−	13.8122i		−	0.548988i
$634$		0			0
$635$			11.8494i			0.470228i
$636$		0			0
$637$	1.16691	+	6.90205i	0.0462345	+	0.273469i
$638$		0			0
$639$			11.7282i			0.463962i
$640$		0			0
$641$	13.9549			0.551187			0.275594	−	0.961274i	$-0.411126\pi$
$641$	13.9549			0.551187			0.275594	+	0.961274i	$0.411126\pi$
$642$		0			0
$643$			42.3514i			1.67017i	0.550117	+	0.835087i	$0.314583\pi$
$643$			42.3514i			1.67017i	−0.550117	+	0.835087i	$0.685417\pi$
$644$		0			0
$645$		−	5.05739i		−	0.199134i
$646$		0			0
$647$	28.5451			1.12222			0.561111	−	0.827741i	$-0.310374\pi$
$647$	28.5451			1.12222			0.561111	+	0.827741i	$0.310374\pi$
$648$		0			0
$649$			0.476154i			0.0186907i
$650$		0			0
$651$	13.7624	+	11.6310i	0.539392	+	0.455853i
$652$		0			0
$653$		−	35.7866i		−	1.40044i	−0.713927	−	0.700220i	$-0.753085\pi$
$653$		−	35.7866i		−	1.40044i	0.713927	−	0.700220i	$-0.246915\pi$
$654$		0			0
$655$		−	11.1732i		−	0.436572i
$656$		0			0
$657$		−	7.19897i		−	0.280859i
$658$		0			0
$659$	9.09685			0.354363			0.177182	−	0.984178i	$-0.443302\pi$
$659$	9.09685			0.354363			0.177182	+	0.984178i	$0.443302\pi$
$660$		0			0
$661$	31.3058			1.21765			0.608827	−	0.793303i	$-0.291640\pi$
$661$	31.3058			1.21765			0.608827	+	0.793303i	$0.291640\pi$
$662$		0			0
$663$	0.783619			0.0304332
$664$		0			0
$665$	−4.05462	−	3.42666i	−0.157231	−	0.132880i
$666$		0			0
$667$	0.0711270			0.00275405
$668$		0			0
$669$		−	0.968800i		−	0.0374560i
$670$		0			0
$671$	−3.23701			−0.124964
$672$		0			0
$673$	−35.1110			−1.35343			−0.676715	−	0.736245i	$-0.736597\pi$
$673$	−35.1110			−1.35343			−0.676715	+	0.736245i	$0.736597\pi$
$674$		0			0
$675$			22.1891i			0.854060i
$676$		0			0
$677$	37.3515			1.43554			0.717768	−	0.696283i	$-0.245164\pi$
$677$	37.3515			1.43554			0.717768	+	0.696283i	$0.245164\pi$
$678$		0			0
$679$	−17.1091	−	14.4593i	−0.656588	−	0.554899i
$680$		0			0
$681$	−34.1232			−1.30760
$682$		0			0
$683$	18.2389			0.697893			0.348947	−	0.937143i	$-0.386539\pi$
$683$	18.2389			0.697893			0.348947	+	0.937143i	$0.386539\pi$
$684$		0			0
$685$	−3.78487			−0.144613
$686$		0			0
$687$			13.8976i			0.530227i
$688$		0			0
$689$		−	1.16935i		−	0.0445486i
$690$		0			0
$691$			30.2469i			1.15065i	0.817926	+	0.575323i	$0.195124\pi$
$691$			30.2469i			1.15065i	−0.817926	+	0.575323i	$0.804876\pi$
$692$		0			0
$693$	−0.814113	+	0.963306i	−0.0309256	+	0.0365930i
$694$		0			0
$695$			14.9797i			0.568214i
$696$		0			0
$697$	−3.28791			−0.124538
$698$		0			0
$699$			14.2358i			0.538447i
$700$		0			0
$701$			34.8391i			1.31586i	0.753081	+	0.657928i	$0.228567\pi$
$701$			34.8391i			1.31586i	−0.753081	+	0.657928i	$0.771433\pi$
$702$		0			0
$703$	4.19776			0.158321
$704$		0			0
$705$		−	3.98878i		−	0.150226i
$706$		0			0
$707$	13.1522	−	15.5624i	0.494638	−	0.585284i
$708$		0			0
$709$		−	39.3740i		−	1.47872i	−0.673309	−	0.739361i	$-0.735128\pi$
$709$		−	39.3740i		−	1.47872i	0.673309	−	0.739361i	$-0.264872\pi$
$710$		0			0
$711$		−	7.28394i		−	0.273169i
$712$		0			0
$713$		−	0.0700629i		−	0.00262387i
$714$		0			0
$715$	0.626260			0.0234208
$716$		0			0
$717$	−24.2126			−0.904235
$718$		0			0
$719$	4.48168			0.167139			0.0835693	−	0.996502i	$-0.473368\pi$
$719$	4.48168			0.167139			0.0835693	+	0.996502i	$0.473368\pi$
$720$		0			0
$721$	7.57794	−	8.96665i	0.282217	−	0.333936i
$722$		0			0
$723$	26.7610			0.995253
$724$		0			0
$725$			18.2956i			0.679483i
$726$		0			0
$727$	−16.5368			−0.613315			−0.306657	−	0.951820i	$-0.599211\pi$
$727$	−16.5368			−0.613315			−0.306657	+	0.951820i	$0.599211\pi$
$728$		0			0
$729$	−29.8862			−1.10689
$730$		0			0
$731$		−	1.73636i		−	0.0642215i
$732$		0			0
$733$	30.2572			1.11757			0.558787	−	0.829311i	$-0.311267\pi$
$733$	30.2572			1.11757			0.558787	+	0.829311i	$0.311267\pi$
$734$		0			0
$735$	−10.5825	+	1.78914i	−0.390340	+	0.0659934i
$736$		0			0
$737$	−2.54261			−0.0936583
$738$		0			0
$739$	32.9352			1.21154			0.605771	−	0.795639i	$-0.292865\pi$
$739$	32.9352			1.21154			0.605771	+	0.795639i	$0.292865\pi$
$740$		0			0
$741$	2.89999			0.106534
$742$		0			0
$743$			26.2593i			0.963360i	0.876347	+	0.481680i	$0.159973\pi$
$743$			26.2593i			0.963360i	−0.876347	+	0.481680i	$0.840027\pi$
$744$		0			0
$745$		−	16.2362i		−	0.594847i
$746$		0			0
$747$		−	1.13790i		−	0.0416337i
$748$		0			0
$749$	−10.5569	+	12.4915i	−0.385739	+	0.456429i
$750$		0			0
$751$			43.7992i			1.59826i	0.601161	+	0.799128i	$0.294705\pi$
$751$			43.7992i			1.59826i	−0.601161	+	0.799128i	$0.705295\pi$
$752$		0			0
$753$	−24.3972			−0.889085
$754$		0			0
$755$			9.02303i			0.328382i
$756$		0			0
$757$		−	18.1033i		−	0.657974i	−0.944334	−	0.328987i	$-0.893293\pi$
$757$		−	18.1033i		−	0.657974i	0.944334	−	0.328987i	$-0.106707\pi$
$758$		0			0
$759$	−0.0138614			−0.000503136
$760$		0			0
$761$		−	23.9490i		−	0.868152i	−0.900876	−	0.434076i	$-0.857075\pi$
$761$		−	23.9490i		−	0.868152i	0.900876	−	0.434076i	$-0.142925\pi$
$762$		0			0
$763$	32.6152	+	27.5639i	1.18075	+	0.997881i
$764$		0			0
$765$		−	0.425074i		−	0.0153686i
$766$		0			0
$767$			0.783099i			0.0282761i
$768$		0			0
$769$			39.8803i			1.43812i	0.694948	+	0.719060i	$0.255427\pi$
$769$			39.8803i			1.43812i	−0.694948	+	0.719060i	$0.744573\pi$
$770$		0			0
$771$	40.8570			1.47143
$772$		0			0
$773$	−17.7022			−0.636704			−0.318352	−	0.947973i	$-0.603129\pi$
$773$	−17.7022			−0.636704			−0.318352	+	0.947973i	$0.603129\pi$
$774$		0			0
$775$	18.0219			0.647366
$776$		0			0
$777$	5.47803	−	6.48192i	0.196523	−	0.232537i
$778$		0			0
$779$	−12.1678			−0.435956
$780$		0			0
$781$		−	9.09586i		−	0.325476i
$782$		0			0
$783$	−26.1625			−0.934972
$784$		0			0
$785$	14.0439			0.501248
$786$		0			0
$787$			20.7451i			0.739482i	0.929135	+	0.369741i	$0.120554\pi$
$787$			20.7451i			0.739482i	−0.929135	+	0.369741i	$0.879446\pi$
$788$		0			0
$789$	6.48394			0.230835
$790$		0			0
$791$	−4.72468	+	5.59052i	−0.167990	+	0.198776i
$792$		0			0
$793$	−5.32371			−0.189050
$794$		0			0
$795$	1.79288			0.0635869
$796$		0			0
$797$	−11.6215			−0.411654			−0.205827	−	0.978588i	$-0.565988\pi$
$797$	−11.6215			−0.411654			−0.205827	+	0.978588i	$0.565988\pi$
$798$		0			0
$799$		−	1.36947i		−	0.0484484i
$800$		0			0
$801$		−	9.45487i		−	0.334071i
$802$		0			0
$803$			5.58317i			0.197026i
$804$		0			0
$805$	0.0318734	+	0.0269370i	0.00112339	+	0.000949405i
$806$		0			0
$807$		−	30.3550i		−	1.06855i
$808$		0			0
$809$	−26.3459			−0.926271			−0.463136	−	0.886287i	$-0.653276\pi$
$809$	−26.3459			−0.926271			−0.463136	+	0.886287i	$0.653276\pi$
$810$		0			0
$811$			31.7543i			1.11504i	0.830163	+	0.557521i	$0.188248\pi$
$811$			31.7543i			1.11504i	−0.830163	+	0.557521i	$0.811752\pi$
$812$		0			0
$813$		−	46.6604i		−	1.63645i
$814$		0			0
$815$	5.88173			0.206028
$816$		0			0
$817$		−	6.42585i		−	0.224812i
$818$		0			0
$819$	−1.33892	+	1.58429i	−0.0467856	+	0.0553594i
$820$		0			0
$821$		−	32.1508i		−	1.12207i	−0.827791	−	0.561036i	$-0.810403\pi$
$821$		−	32.1508i		−	1.12207i	0.827791	−	0.561036i	$-0.189597\pi$
$822$		0			0
$823$		−	26.3374i		−	0.918065i	−0.888419	−	0.459033i	$-0.848196\pi$
$823$		−	26.3374i		−	0.918065i	0.888419	−	0.459033i	$-0.151804\pi$
$824$		0			0
$825$		−	3.56549i		−	0.124134i
$826$		0			0
$827$	−28.2351			−0.981832			−0.490916	−	0.871207i	$-0.663338\pi$
$827$	−28.2351			−0.981832			−0.490916	+	0.871207i	$0.663338\pi$
$828$		0			0
$829$	7.20127			0.250110			0.125055	−	0.992150i	$-0.460089\pi$
$829$	7.20127			0.250110			0.125055	+	0.992150i	$0.460089\pi$
$830$		0			0
$831$	8.04249			0.278991
$832$		0			0
$833$	−3.63328	+	0.614267i	−0.125886	+	0.0212831i
$834$		0			0
$835$	16.4916			0.570714
$836$		0			0
$837$			25.7711i			0.890779i
$838$		0			0
$839$	−14.9034			−0.514524			−0.257262	−	0.966342i	$-0.582820\pi$
$839$	−14.9034			−0.514524			−0.257262	+	0.966342i	$0.582820\pi$
$840$		0			0
$841$	7.42817			0.256144
$842$		0			0
$843$		−	0.946579i		−	0.0326019i
$844$		0			0
$845$	1.02997			0.0354320
$846$		0			0
$847$	−18.1543	+	21.4812i	−0.623789	+	0.738103i
$848$		0			0
$849$	9.03481			0.310074
$850$		0			0
$851$	−0.0329987			−0.00113118
$852$		0			0
$853$	−49.1718			−1.68361			−0.841804	−	0.539783i	$-0.818506\pi$
$853$	−49.1718			−1.68361			−0.841804	+	0.539783i	$0.818506\pi$
$854$		0			0
$855$		−	1.57310i		−	0.0537988i
$856$		0			0
$857$			20.6632i			0.705843i	0.935653	+	0.352922i	$0.114812\pi$
$857$			20.6632i			0.705843i	−0.935653	+	0.352922i	$0.885188\pi$
$858$		0			0
$859$			20.8182i			0.710309i	0.934808	+	0.355154i	$0.115572\pi$
$859$			20.8182i			0.710309i	−0.934808	+	0.355154i	$0.884428\pi$
$860$		0			0
$861$	−15.8788	+	18.7887i	−0.541148	+	0.640318i
$862$		0			0
$863$			30.4131i			1.03527i	0.855601	+	0.517637i	$0.173188\pi$
$863$			30.4131i			1.03527i	−0.855601	+	0.517637i	$0.826812\pi$
$864$		0			0
$865$	13.5600			0.461054
$866$		0			0
$867$		−	24.8941i		−	0.845447i
$868$		0			0
$869$			5.64907i			0.191632i
$870$		0			0
$871$	−4.18167			−0.141690
$872$		0			0
$873$		−	6.63795i		−	0.224661i
$874$		0			0
$875$	−15.7237	+	18.6052i	−0.531559	+	0.628971i
$876$		0			0
$877$			8.60597i			0.290603i	0.989387	+	0.145301i	$0.0464152\pi$
$877$			8.60597i			0.290603i	−0.989387	+	0.145301i	$0.953585\pi$
$878$		0			0
$879$			44.4727i			1.50003i
$880$		0			0
$881$		−	22.7259i		−	0.765657i	−0.923820	−	0.382828i	$-0.874950\pi$
$881$		−	22.7259i		−	0.765657i	0.923820	−	0.382828i	$-0.125050\pi$
$882$		0			0
$883$	−26.0737			−0.877449			−0.438724	−	0.898622i	$-0.644570\pi$
$883$	−26.0737			−0.877449			−0.438724	+	0.898622i	$0.644570\pi$
$884$		0			0
$885$	−1.20067			−0.0403602
$886$		0			0
$887$	−18.7828			−0.630664			−0.315332	−	0.948981i	$-0.602116\pi$
$887$	−18.7828			−0.630664			−0.315332	+	0.948981i	$0.602116\pi$
$888$		0			0
$889$	23.2480	+	19.6475i	0.779713	+	0.658954i
$890$		0			0
$891$	3.66848			0.122899
$892$		0			0
$893$		−	5.06809i		−	0.169597i
$894$		0			0
$895$	−3.37620			−0.112854
$896$		0			0
$897$	−0.0227969			−0.000761166
$898$		0			0
$899$			21.2491i			0.708696i
$900$		0			0
$901$	0.615551			0.0205070
$902$		0			0
$903$	−9.92239	−	8.38565i	−0.330196	−	0.279057i
$904$		0			0
$905$	−13.8236			−0.459513
$906$		0			0
$907$	2.93997			0.0976201			0.0488101	−	0.998808i	$-0.484457\pi$
$907$	2.93997			0.0976201			0.0488101	+	0.998808i	$0.484457\pi$
$908$		0			0
$909$	6.03785			0.200263
$910$		0			0
$911$		−	37.3247i		−	1.23662i	−0.785934	−	0.618311i	$-0.787817\pi$
$911$		−	37.3247i		−	1.23662i	0.785934	−	0.618311i	$-0.212183\pi$
$912$		0			0
$913$			0.882503i			0.0292066i
$914$		0			0
$915$		−	8.16248i		−	0.269843i
$916$		0			0
$917$	−21.9213	−	18.5262i	−0.723905	−	0.611790i
$918$		0			0
$919$			8.61037i			0.284030i	0.989865	+	0.142015i	$0.0453581\pi$
$919$			8.61037i			0.284030i	−0.989865	+	0.142015i	$0.954642\pi$
$920$		0			0
$921$	−33.2811			−1.09665
$922$		0			0
$923$		−	14.9594i		−	0.492394i
$924$		0			0
$925$		−	8.48808i		−	0.279086i
$926$		0			0
$927$	3.47886			0.114261
$928$		0			0
$929$			55.5422i			1.82228i	0.412096	+	0.911140i	$0.364797\pi$
$929$			55.5422i			1.82228i	−0.412096	+	0.911140i	$0.635203\pi$
$930$		0			0
$931$	−13.4459	+	2.27326i	−0.440672	+	0.0745029i
$932$		0			0
$933$			32.0484i			1.04922i
$934$		0			0
$935$			0.329667i			0.0107813i
$936$		0			0
$937$			29.4465i			0.961974i	0.876728	+	0.480987i	$0.159722\pi$
$937$			29.4465i			0.961974i	−0.876728	+	0.480987i	$0.840278\pi$
$938$		0			0
$939$	−3.07109			−0.100221
$940$		0			0
$941$	14.0302			0.457371			0.228686	−	0.973500i	$-0.426557\pi$
$941$	14.0302			0.457371			0.228686	+	0.973500i	$0.426557\pi$
$942$		0			0
$943$	0.0956512			0.00311483
$944$		0			0
$945$	−11.7239	−	9.90819i	−0.381380	−	0.322314i
$946$		0			0
$947$	−46.5895			−1.51396			−0.756978	−	0.653440i	$-0.773325\pi$
$947$	−46.5895			−1.51396			−0.756978	+	0.653440i	$0.773325\pi$
$948$		0			0
$949$			9.18228i			0.298069i
$950$		0			0
$951$	48.4862			1.57227
$952$		0			0
$953$	−42.9548			−1.39144			−0.695721	−	0.718312i	$-0.744915\pi$
$953$	−42.9548			−1.39144			−0.695721	+	0.718312i	$0.744915\pi$
$954$		0			0
$955$		−	13.7709i		−	0.445617i
$956$		0			0
$957$	4.20396			0.135895
$958$		0			0
$959$	−6.27570	+	7.42577i	−0.202653	+	0.239790i
$960$		0			0
$961$	−10.0688			−0.324801
$962$		0			0
$963$	−4.84641			−0.156173
$964$		0			0
$965$	−14.4850			−0.466288
$966$		0			0
$967$			13.9605i			0.448940i	0.974481	+	0.224470i	$0.0720652\pi$
$967$			13.9605i			0.448940i	−0.974481	+	0.224470i	$0.927935\pi$
$968$		0			0
$969$			1.52657i			0.0490406i
$970$		0			0
$971$		−	12.7064i		−	0.407767i	−0.978995	−	0.203884i	$-0.934644\pi$
$971$		−	12.7064i		−	0.407767i	0.978995	−	0.203884i	$-0.0653564\pi$
$972$		0			0
$973$	29.3896	+	24.8379i	0.942189	+	0.796267i
$974$		0			0
$975$		−	5.86392i		−	0.187796i
$976$		0			0
$977$	45.8920			1.46821			0.734107	−	0.679034i	$-0.237601\pi$
$977$	45.8920			1.46821			0.734107	+	0.679034i	$0.237601\pi$
$978$		0			0
$979$			7.33274i			0.234355i
$980$		0			0
$981$			12.6540i			0.404010i
$982$		0			0
$983$	34.2113			1.09117			0.545586	−	0.838055i	$-0.316307\pi$
$983$	34.2113			1.09117			0.545586	+	0.838055i	$0.316307\pi$
$984$		0			0
$985$			7.62520i			0.242959i
$986$		0			0
$987$	−7.82582	−	6.61379i	−0.249099	−	0.210519i
$988$		0			0
$989$			0.0505137i			0.00160624i
$990$		0			0
$991$		−	51.9670i		−	1.65079i	−0.564559	−	0.825393i	$-0.690954\pi$
$991$		−	51.9670i		−	1.65079i	0.564559	−	0.825393i	$-0.309046\pi$
$992$		0			0
$993$		−	21.0899i		−	0.669266i
$994$		0			0
$995$	−21.8573			−0.692923
$996$		0			0
$997$	6.95041			0.220122			0.110061	−	0.993925i	$-0.464895\pi$
$997$	6.95041			0.220122			0.110061	+	0.993925i	$0.464895\pi$
$998$		0			0
$999$	12.1378			0.384024

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.a.2575.14		48
4.3	odd	2		728.2.h.a.27.47	✓	48
7.6	odd	2		2912.2.h.b.2575.35		48
8.3	odd	2		2912.2.h.b.2575.14		48
8.5	even	2		728.2.h.b.27.48	yes	48
28.27	even	2		728.2.h.b.27.47	yes	48
56.13	odd	2		728.2.h.a.27.48	yes	48
56.27	even	2	inner	2912.2.h.a.2575.35		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.47	✓	48	4.3	odd	2
728.2.h.a.27.48	yes	48	56.13	odd	2
728.2.h.b.27.47	yes	48	28.27	even	2
728.2.h.b.27.48	yes	48	8.5	even	2
2912.2.h.a.2575.14		48	1.1	even	1	trivial
2912.2.h.a.2575.35		48	56.27	even	2	inner
2912.2.h.b.2575.14		48	8.3	odd	2
2912.2.h.b.2575.35		48	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.48862i q^{3} +1.02997 q^{5} +(1.70779 - 2.02076i) q^{7} +0.784007 q^{9} -0.608038 q^{11} -1.00000 q^{13} -1.53323i q^{15} -0.526406i q^{17} -1.94810i q^{19} +(-3.00814 - 2.54225i) q^{21} +0.0153141i q^{23} -3.93917 q^{25} -5.63295i q^{27} -4.64455i q^{29} -4.57506 q^{31} +0.905138i q^{33} +(1.75897 - 2.08131i) q^{35} +2.15479i q^{37} +1.48862i q^{39} -6.24596i q^{41} +3.29851 q^{43} +0.807502 q^{45} +2.60155 q^{47} +(-1.16691 - 6.90205i) q^{49} -0.783619 q^{51} +1.16935i q^{53} -0.626260 q^{55} -2.89999 q^{57} -0.783099i q^{59} +5.32371 q^{61} +(1.33892 - 1.58429i) q^{63} -1.02997 q^{65} +4.18167 q^{67} +0.0227969 q^{69} +14.9594i q^{71} -9.18228i q^{73} +5.86392i q^{75} +(-1.03840 + 1.22870i) q^{77} -9.29066i q^{79} -6.03331 q^{81} -1.45140i q^{83} -0.542182i q^{85} -6.91397 q^{87} -12.0597i q^{89} +(-1.70779 + 2.02076i) q^{91} +6.81053i q^{93} -2.00649i q^{95} -8.46670i q^{97} -0.476706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) \) 48 * q - 48 * q^9 + 4 * q^11 - 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 + 24 * q^45 + 40 * q^51 - 20 * q^63 + 4 * q^67 - 20 * q^77 + 64 * q^81 + 40 * q^87 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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