LMFDB - Embedded newform 2912.2.h.a.2575.15

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			2575.15
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.a.2575.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40360i q^{3} -3.11493 q^{5} +(2.64532 - 0.0479799i) q^{7} +1.02989 q^{9} +0.711549 q^{11} -1.00000 q^{13} +4.37213i q^{15} -0.128191i q^{17} +5.49116i q^{19} +(-0.0673448 - 3.71298i) q^{21} +5.12058i q^{23} +4.70280 q^{25} -5.65638i q^{27} +7.07307i q^{29} -10.3051 q^{31} -0.998733i q^{33} +(-8.23998 + 0.149454i) q^{35} -8.15798i q^{37} +1.40360i q^{39} -0.100841i q^{41} -8.73959 q^{43} -3.20805 q^{45} -2.81273 q^{47} +(6.99540 - 0.253844i) q^{49} -0.179929 q^{51} +8.53210i q^{53} -2.21643 q^{55} +7.70742 q^{57} +7.38024i q^{59} -11.3324 q^{61} +(2.72440 - 0.0494143i) q^{63} +3.11493 q^{65} +14.5057 q^{67} +7.18727 q^{69} +8.43273i q^{71} -8.62895i q^{73} -6.60088i q^{75} +(1.88227 - 0.0341401i) q^{77} +16.3766i q^{79} -4.84963 q^{81} +4.06811i q^{83} +0.399306i q^{85} +9.92780 q^{87} +2.25678i q^{89} +(-2.64532 + 0.0479799i) q^{91} +14.4643i q^{93} -17.1046i q^{95} +5.13148i q^{97} +0.732821 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.40360i		−	0.810371i	−0.914234	−	0.405186i	$-0.867207\pi$
$3$		−	1.40360i		−	0.810371i	0.914234	−	0.405186i	$-0.132793\pi$
$4$		0			0
$5$	−3.11493			−1.39304			−0.696520	−	0.717537i	$-0.745269\pi$
$5$	−3.11493			−1.39304			−0.696520	+	0.717537i	$0.745269\pi$
$6$		0			0
$7$	2.64532	−	0.0479799i	0.999836	−	0.0181347i
$7$	2.64532	−	0.0479799i	0.999836	−	0.0181347i
$8$		0			0
$9$	1.02989			0.343298
$10$		0			0
$11$	0.711549			0.214540			0.107270	−	0.994230i	$-0.465789\pi$
$11$	0.711549			0.214540			0.107270	+	0.994230i	$0.465789\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$			4.37213i			1.12888i
$16$		0			0
$17$		−	0.128191i		−	0.0310908i	−0.999879	−	0.0155454i	$-0.995052\pi$
$17$		−	0.128191i		−	0.0310908i	0.999879	−	0.0155454i	$-0.00494846\pi$
$18$		0			0
$19$			5.49116i			1.25976i	0.776693	+	0.629879i	$0.216896\pi$
$19$			5.49116i			1.25976i	−0.776693	+	0.629879i	$0.783104\pi$
$20$		0			0
$21$	−0.0673448	−	3.71298i	−0.0146958	−	0.810238i
$22$		0			0
$23$			5.12058i			1.06771i	0.845574	+	0.533857i	$0.179258\pi$
$23$			5.12058i			1.06771i	−0.845574	+	0.533857i	$0.820742\pi$
$24$		0			0
$25$	4.70280			0.940561
$26$		0			0
$27$		−	5.65638i		−	1.08857i
$28$		0			0
$29$			7.07307i			1.31344i	0.754136	+	0.656718i	$0.228056\pi$
$29$			7.07307i			1.31344i	−0.754136	+	0.656718i	$0.771944\pi$
$30$		0			0
$31$	−10.3051			−1.85085			−0.925424	−	0.378934i	$-0.876291\pi$
$31$	−10.3051			−1.85085			−0.925424	+	0.378934i	$0.876291\pi$
$32$		0			0
$33$		−	0.998733i		−	0.173857i
$34$		0			0
$35$	−8.23998	+	0.149454i	−1.39281	+	0.0252624i
$36$		0			0
$37$		−	8.15798i		−	1.34116i	−0.741836	−	0.670582i	$-0.766045\pi$
$37$		−	8.15798i		−	1.34116i	0.741836	−	0.670582i	$-0.233955\pi$
$38$		0			0
$39$			1.40360i			0.224757i
$40$		0			0
$41$		−	0.100841i		−	0.0157488i	−0.999969	−	0.00787439i	$-0.997493\pi$
$41$		−	0.100841i		−	0.0157488i	0.999969	−	0.00787439i	$-0.00250652\pi$
$42$		0			0
$43$	−8.73959			−1.33278			−0.666388	−	0.745605i	$-0.732160\pi$
$43$	−8.73959			−1.33278			−0.666388	+	0.745605i	$0.732160\pi$
$44$		0			0
$45$	−3.20805			−0.478228
$46$		0			0
$47$	−2.81273			−0.410278			−0.205139	−	0.978733i	$-0.565765\pi$
$47$	−2.81273			−0.410278			−0.205139	+	0.978733i	$0.565765\pi$
$48$		0			0
$49$	6.99540	−	0.253844i	0.999342	−	0.0362635i
$50$		0			0
$51$	−0.179929			−0.0251951
$52$		0			0
$53$			8.53210i			1.17197i	0.810321	+	0.585987i	$0.199293\pi$
$53$			8.53210i			1.17197i	−0.810321	+	0.585987i	$0.800707\pi$
$54$		0			0
$55$	−2.21643			−0.298863
$56$		0			0
$57$	7.70742			1.02087
$58$		0			0
$59$			7.38024i			0.960825i	0.877043	+	0.480413i	$0.159513\pi$
$59$			7.38024i			0.960825i	−0.877043	+	0.480413i	$0.840487\pi$
$60$		0			0
$61$	−11.3324			−1.45097			−0.725485	−	0.688238i	$-0.758384\pi$
$61$	−11.3324			−1.45097			−0.725485	+	0.688238i	$0.758384\pi$
$62$		0			0
$63$	2.72440	−	0.0494143i	0.343242	−	0.00622562i
$64$		0			0
$65$	3.11493			0.386360
$66$		0			0
$67$	14.5057			1.77215			0.886077	−	0.463539i	$-0.153420\pi$
$67$	14.5057			1.77215			0.886077	+	0.463539i	$0.153420\pi$
$68$		0			0
$69$	7.18727			0.865246
$70$		0			0
$71$			8.43273i			1.00078i	0.865800	+	0.500390i	$0.166810\pi$
$71$			8.43273i			1.00078i	−0.865800	+	0.500390i	$0.833190\pi$
$72$		0			0
$73$		−	8.62895i		−	1.00994i	−0.863136	−	0.504971i	$-0.831503\pi$
$73$		−	8.62895i		−	1.00994i	0.863136	−	0.504971i	$-0.168497\pi$
$74$		0			0
$75$		−	6.60088i		−	0.762204i
$76$		0			0
$77$	1.88227	−	0.0341401i	0.214505	−	0.00389062i
$78$		0			0
$79$			16.3766i			1.84251i	0.388957	+	0.921256i	$0.372835\pi$
$79$			16.3766i			1.84251i	−0.388957	+	0.921256i	$0.627165\pi$
$80$		0			0
$81$	−4.84963			−0.538848
$82$		0			0
$83$			4.06811i			0.446533i	0.974757	+	0.223266i	$0.0716720\pi$
$83$			4.06811i			0.446533i	−0.974757	+	0.223266i	$0.928328\pi$
$84$		0			0
$85$			0.399306i			0.0433108i
$86$		0			0
$87$	9.92780			1.06437
$88$		0			0
$89$			2.25678i			0.239218i	0.992821	+	0.119609i	$0.0381641\pi$
$89$			2.25678i			0.239218i	−0.992821	+	0.119609i	$0.961836\pi$
$90$		0			0
$91$	−2.64532	+	0.0479799i	−0.277304	+	0.00502966i
$92$		0			0
$93$			14.4643i			1.49987i
$94$		0			0
$95$		−	17.1046i		−	1.75489i
$96$		0			0
$97$			5.13148i			0.521023i	0.965471	+	0.260511i	$0.0838912\pi$
$97$			5.13148i			0.521023i	−0.965471	+	0.260511i	$0.916109\pi$
$98$		0			0
$99$	0.732821			0.0736513
$100$		0			0
$101$	16.0644			1.59846			0.799231	−	0.601023i	$-0.205240\pi$
$101$	16.0644			1.59846			0.799231	+	0.601023i	$0.205240\pi$
$102$		0			0
$103$	−4.38089			−0.431662			−0.215831	−	0.976431i	$-0.569246\pi$
$103$	−4.38089			−0.431662			−0.215831	+	0.976431i	$0.569246\pi$
$104$		0			0
$105$	0.209775	+	11.5657i	0.0204719	+	1.12869i
$106$		0			0
$107$	2.75910			0.266732			0.133366	−	0.991067i	$-0.457421\pi$
$107$	2.75910			0.266732			0.133366	+	0.991067i	$0.457421\pi$
$108$		0			0
$109$		−	14.4976i		−	1.38861i	−0.719679	−	0.694307i	$-0.755711\pi$
$109$		−	14.4976i		−	1.38861i	0.719679	−	0.694307i	$-0.244289\pi$
$110$		0			0
$111$	−11.4506			−1.08684
$112$		0			0
$113$	9.63185			0.906088			0.453044	−	0.891488i	$-0.350338\pi$
$113$	9.63185			0.906088			0.453044	+	0.891488i	$0.350338\pi$
$114$		0			0
$115$		−	15.9503i		−	1.48737i
$116$		0			0
$117$	−1.02989			−0.0952138
$118$		0			0
$119$	−0.00615058	−	0.339105i	−0.000563823	−	0.0310857i
$120$		0			0
$121$	−10.4937			−0.953973
$122$		0			0
$123$	−0.141541			−0.0127624
$124$		0			0
$125$	0.925743			0.0828010
$126$		0			0
$127$			12.5244i			1.11136i	0.831396	+	0.555681i	$0.187542\pi$
$127$			12.5244i			1.11136i	−0.831396	+	0.555681i	$0.812458\pi$
$128$		0			0
$129$			12.2669i			1.08004i
$130$		0			0
$131$			9.32611i			0.814826i	0.913244	+	0.407413i	$0.133569\pi$
$131$			9.32611i			0.814826i	−0.913244	+	0.407413i	$0.866431\pi$
$132$		0			0
$133$	0.263466	+	14.5259i	0.0228454	+	1.25955i
$134$		0			0
$135$			17.6192i			1.51642i
$136$		0			0
$137$	3.48572			0.297806			0.148903	−	0.988852i	$-0.452426\pi$
$137$	3.48572			0.297806			0.148903	+	0.988852i	$0.452426\pi$
$138$		0			0
$139$			13.7408i			1.16548i	0.812659	+	0.582740i	$0.198019\pi$
$139$			13.7408i			1.16548i	−0.812659	+	0.582740i	$0.801981\pi$
$140$		0			0
$141$			3.94795i			0.332478i
$142$		0			0
$143$	−0.711549			−0.0595027
$144$		0			0
$145$		−	22.0321i		−	1.82967i
$146$		0			0
$147$	−0.356297	−	9.81877i	−0.0293869	−	0.809838i
$148$		0			0
$149$		−	0.0763024i		−	0.00625094i	−0.999995	−	0.00312547i	$-0.999005\pi$
$149$		−	0.0763024i		−	0.00625094i	0.999995	−	0.00312547i	$-0.000994869\pi$
$150$		0			0
$151$			1.78506i			0.145266i	0.997359	+	0.0726329i	$0.0231402\pi$
$151$			1.78506i			0.145266i	−0.997359	+	0.0726329i	$0.976860\pi$
$152$		0			0
$153$		−	0.132023i		−	0.0106734i
$154$		0			0
$155$	32.0996			2.57831
$156$		0			0
$157$	−7.52175			−0.600301			−0.300151	−	0.953892i	$-0.597037\pi$
$157$	−7.52175			−0.600301			−0.300151	+	0.953892i	$0.597037\pi$
$158$		0			0
$159$	11.9757			0.949734
$160$		0			0
$161$	0.245685	+	13.5456i	0.0193627	+	1.06754i
$162$		0			0
$163$	5.68209			0.445056			0.222528	−	0.974926i	$-0.428569\pi$
$163$	5.68209			0.445056			0.222528	+	0.974926i	$0.428569\pi$
$164$		0			0
$165$			3.11099i			0.242190i
$166$		0			0
$167$	9.75242			0.754665			0.377332	−	0.926078i	$-0.376841\pi$
$167$	9.75242			0.754665			0.377332	+	0.926078i	$0.376841\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$			5.65532i			0.432473i
$172$		0			0
$173$	−2.54932			−0.193821			−0.0969106	−	0.995293i	$-0.530896\pi$
$173$	−2.54932			−0.193821			−0.0969106	+	0.995293i	$0.530896\pi$
$174$		0			0
$175$	12.4404	−	0.225640i	0.940406	−	0.0170568i
$176$		0			0
$177$	10.3589			0.778625
$178$		0			0
$179$	−6.78232			−0.506934			−0.253467	−	0.967344i	$-0.581571\pi$
$179$	−6.78232			−0.506934			−0.253467	+	0.967344i	$0.581571\pi$
$180$		0			0
$181$	2.16014			0.160562			0.0802810	−	0.996772i	$-0.474418\pi$
$181$	2.16014			0.160562			0.0802810	+	0.996772i	$0.474418\pi$
$182$		0			0
$183$			15.9063i			1.17582i
$184$		0			0
$185$			25.4116i			1.86829i
$186$		0			0
$187$		−	0.0912140i		−	0.00667023i
$188$		0			0
$189$	−0.271393	−	14.9629i	−0.0197409	−	1.08839i
$190$		0			0
$191$			9.22106i			0.667212i	0.942713	+	0.333606i	$0.108266\pi$
$191$			9.22106i			0.667212i	−0.942713	+	0.333606i	$0.891734\pi$
$192$		0			0
$193$	−13.3511			−0.961036			−0.480518	−	0.876985i	$-0.659551\pi$
$193$	−13.3511			−0.961036			−0.480518	+	0.876985i	$0.659551\pi$
$194$		0			0
$195$		−	4.37213i		−	0.313095i
$196$		0			0
$197$			24.6758i			1.75808i	0.476751	+	0.879038i	$0.341814\pi$
$197$			24.6758i			1.75808i	−0.476751	+	0.879038i	$0.658186\pi$
$198$		0			0
$199$	−16.0513			−1.13785			−0.568924	−	0.822390i	$-0.692640\pi$
$199$	−16.0513			−1.13785			−0.568924	+	0.822390i	$0.692640\pi$
$200$		0			0
$201$		−	20.3603i		−	1.43610i
$202$		0			0
$203$	0.339366	+	18.7105i	0.0238188	+	1.31322i
$204$		0			0
$205$			0.314114i			0.0219387i
$206$		0			0
$207$			5.27366i			0.366545i
$208$		0			0
$209$			3.90723i			0.270269i
$210$		0			0
$211$	8.24694			0.567743			0.283871	−	0.958862i	$-0.408381\pi$
$211$	8.24694			0.567743			0.283871	+	0.958862i	$0.408381\pi$
$212$		0			0
$213$	11.8362			0.811004
$214$		0			0
$215$	27.2232			1.85661
$216$		0			0
$217$	−27.2602	+	0.494437i	−1.85054	+	0.0335646i
$218$		0			0
$219$	−12.1116			−0.818429
$220$		0			0
$221$			0.128191i			0.00862304i
$222$		0			0
$223$	11.0819			0.742096			0.371048	−	0.928614i	$-0.378999\pi$
$223$	11.0819			0.742096			0.371048	+	0.928614i	$0.378999\pi$
$224$		0			0
$225$	4.84339			0.322893
$226$		0			0
$227$		−	13.0912i		−	0.868896i	−0.900697	−	0.434448i	$-0.856944\pi$
$227$		−	13.0912i		−	0.868896i	0.900697	−	0.434448i	$-0.143056\pi$
$228$		0			0
$229$	26.7429			1.76722			0.883610	−	0.468224i	$-0.155106\pi$
$229$	26.7429			1.76722			0.883610	+	0.468224i	$0.155106\pi$
$230$		0			0
$231$	−0.0479192	−	2.64197i	−0.00315285	−	0.173829i
$232$		0			0
$233$	−20.2305			−1.32535			−0.662673	−	0.748909i	$-0.730578\pi$
$233$	−20.2305			−1.32535			−0.662673	+	0.748909i	$0.730578\pi$
$234$		0			0
$235$	8.76145			0.571534
$236$		0			0
$237$	22.9863			1.49312
$238$		0			0
$239$		−	13.7831i		−	0.891554i	−0.895144	−	0.445777i	$-0.852927\pi$
$239$		−	13.7831i		−	0.891554i	0.895144	−	0.445777i	$-0.147073\pi$
$240$		0			0
$241$		−	12.7451i		−	0.820981i	−0.911865	−	0.410490i	$-0.865358\pi$
$241$		−	12.7451i		−	0.820981i	0.911865	−	0.410490i	$-0.134642\pi$
$242$		0			0
$243$		−	10.1622i		−	0.651903i
$244$		0			0
$245$	−21.7902	+	0.790708i	−1.39212	+	0.0505165i
$246$		0			0
$247$		−	5.49116i		−	0.349394i
$248$		0			0
$249$	5.71001			0.361857
$250$		0			0
$251$			22.4676i			1.41814i	0.705137	+	0.709071i	$0.250886\pi$
$251$			22.4676i			1.41814i	−0.705137	+	0.709071i	$0.749114\pi$
$252$		0			0
$253$			3.64354i			0.229068i
$254$		0			0
$255$	0.560467			0.0350978
$256$		0			0
$257$		−	14.3250i		−	0.893568i	−0.894642	−	0.446784i	$-0.852569\pi$
$257$		−	14.3250i		−	0.893568i	0.894642	−	0.446784i	$-0.147431\pi$
$258$		0			0
$259$	−0.391419	−	21.5804i	−0.0243216	−	1.34094i
$260$		0			0
$261$			7.28452i			0.450901i
$262$		0			0
$263$		−	13.3682i		−	0.824317i	−0.911112	−	0.412158i	$-0.864775\pi$
$263$		−	13.3682i		−	0.824317i	0.911112	−	0.412158i	$-0.135225\pi$
$264$		0			0
$265$		−	26.5769i		−	1.63261i
$266$		0			0
$267$	3.16762			0.193855
$268$		0			0
$269$	−13.2650			−0.808782			−0.404391	−	0.914586i	$-0.632517\pi$
$269$	−13.2650			−0.808782			−0.404391	+	0.914586i	$0.632517\pi$
$270$		0			0
$271$	−14.9368			−0.907348			−0.453674	−	0.891168i	$-0.649887\pi$
$271$	−14.9368			−0.907348			−0.453674	+	0.891168i	$0.649887\pi$
$272$		0			0
$273$	0.0673448	+	3.71298i	0.00407590	+	0.224720i
$274$		0			0
$275$	3.34628			0.201788
$276$		0			0
$277$		−	2.07792i		−	0.124850i	−0.998050	−	0.0624251i	$-0.980117\pi$
$277$		−	2.07792i		−	0.124850i	0.998050	−	0.0624251i	$-0.0198834\pi$
$278$		0			0
$279$	−10.6132			−0.635393
$280$		0			0
$281$	7.66033			0.456977			0.228488	−	0.973547i	$-0.426622\pi$
$281$	7.66033			0.456977			0.228488	+	0.973547i	$0.426622\pi$
$282$		0			0
$283$			4.75444i			0.282622i	0.989965	+	0.141311i	$0.0451318\pi$
$283$			4.75444i			0.282622i	−0.989965	+	0.141311i	$0.954868\pi$
$284$		0			0
$285$	−24.0081			−1.42212
$286$		0			0
$287$	−0.00483836	−	0.266757i	−0.000285599	−	0.0157462i
$288$		0			0
$289$	16.9836			0.999033
$290$		0			0
$291$	7.20257			0.422222
$292$		0			0
$293$	20.4685			1.19578			0.597890	−	0.801578i	$-0.296006\pi$
$293$	20.4685			1.19578			0.597890	+	0.801578i	$0.296006\pi$
$294$		0			0
$295$		−	22.9889i		−	1.33847i
$296$		0			0
$297$		−	4.02479i		−	0.233542i
$298$		0			0
$299$		−	5.12058i		−	0.296131i
$300$		0			0
$301$	−23.1190	+	0.419325i	−1.33256	+	0.0241695i
$302$		0			0
$303$		−	22.5480i		−	1.29535i
$304$		0			0
$305$	35.2998			2.02126
$306$		0			0
$307$		−	0.696807i		−	0.0397689i	−0.999802	−	0.0198844i	$-0.993670\pi$
$307$		−	0.696807i		−	0.0397689i	0.999802	−	0.0198844i	$-0.00632983\pi$
$308$		0			0
$309$			6.14903i			0.349806i
$310$		0			0
$311$	21.4020			1.21360			0.606798	−	0.794856i	$-0.292454\pi$
$311$	21.4020			1.21360			0.606798	+	0.794856i	$0.292454\pi$
$312$		0			0
$313$		−	3.05480i		−	0.172667i	−0.996266	−	0.0863337i	$-0.972485\pi$
$313$		−	3.05480i		−	0.172667i	0.996266	−	0.0863337i	$-0.0275151\pi$
$314$		0			0
$315$	−8.48632	+	0.153922i	−0.478150	+	0.00867253i
$316$		0			0
$317$			26.8368i			1.50730i	0.657274	+	0.753652i	$0.271709\pi$
$317$			26.8368i			1.50730i	−0.657274	+	0.753652i	$0.728291\pi$
$318$		0			0
$319$			5.03284i			0.281785i
$320$		0			0
$321$		−	3.87268i		−	0.216152i
$322$		0			0
$323$	0.703916			0.0391669
$324$		0			0
$325$	−4.70280			−0.260865
$326$		0			0
$327$	−20.3488			−1.12529
$328$		0			0
$329$	−7.44055	+	0.134954i	−0.410211	+	0.00744028i
$330$		0			0
$331$	17.6603			0.970696			0.485348	−	0.874321i	$-0.338693\pi$
$331$	17.6603			0.970696			0.485348	+	0.874321i	$0.338693\pi$
$332$		0			0
$333$		−	8.40186i		−	0.460419i
$334$		0			0
$335$	−45.1843			−2.46868
$336$		0			0
$337$	1.87767			0.102283			0.0511416	−	0.998691i	$-0.483714\pi$
$337$	1.87767			0.102283			0.0511416	+	0.998691i	$0.483714\pi$
$338$		0			0
$339$		−	13.5193i		−	0.734268i
$340$		0			0
$341$	−7.33257			−0.397081
$342$		0			0
$343$	18.4929	−	1.00714i	0.998520	−	0.0543803i
$344$		0			0
$345$	−22.3879			−1.20532
$346$		0			0
$347$	−14.7147			−0.789924			−0.394962	−	0.918697i	$-0.629242\pi$
$347$	−14.7147			−0.789924			−0.394962	+	0.918697i	$0.629242\pi$
$348$		0			0
$349$	−1.50124			−0.0803598			−0.0401799	−	0.999192i	$-0.512793\pi$
$349$	−1.50124			−0.0803598			−0.0401799	+	0.999192i	$0.512793\pi$
$350$		0			0
$351$			5.65638i			0.301915i
$352$		0			0
$353$			2.52907i			0.134609i	0.997732	+	0.0673043i	$0.0214398\pi$
$353$			2.52907i			0.134609i	−0.997732	+	0.0673043i	$0.978560\pi$
$354$		0			0
$355$		−	26.2674i		−	1.39413i
$356$		0			0
$357$	−0.475969	+	0.00863299i	−0.0251910	+	0.000456906i
$358$		0			0
$359$		−	27.9988i		−	1.47772i	−0.673858	−	0.738861i	$-0.735364\pi$
$359$		−	27.9988i		−	1.47772i	0.673858	−	0.738861i	$-0.264636\pi$
$360$		0			0
$361$	−11.1528			−0.586992
$362$		0			0
$363$			14.7290i			0.773072i
$364$		0			0
$365$			26.8786i			1.40689i
$366$		0			0
$367$	−13.6580			−0.712944			−0.356472	−	0.934306i	$-0.616020\pi$
$367$	−13.6580			−0.712944			−0.356472	+	0.934306i	$0.616020\pi$
$368$		0			0
$369$		−	0.103856i		−	0.00540653i
$370$		0			0
$371$	0.409369	+	22.5701i	0.0212534	+	1.17178i
$372$		0			0
$373$		−	12.4286i		−	0.643528i	−0.946820	−	0.321764i	$-0.895724\pi$
$373$		−	12.4286i		−	0.643528i	0.946820	−	0.321764i	$-0.104276\pi$
$374$		0			0
$375$		−	1.29938i		−	0.0670995i
$376$		0			0
$377$		−	7.07307i		−	0.364282i
$378$		0			0
$379$	−10.5288			−0.540826			−0.270413	−	0.962744i	$-0.587160\pi$
$379$	−10.5288			−0.540826			−0.270413	+	0.962744i	$0.587160\pi$
$380$		0			0
$381$	17.5793			0.900616
$382$		0			0
$383$	7.90669			0.404013			0.202007	−	0.979384i	$-0.435254\pi$
$383$	7.90669			0.404013			0.202007	+	0.979384i	$0.435254\pi$
$384$		0			0
$385$	−5.86315	+	0.106344i	−0.298814	+	0.00541979i
$386$		0			0
$387$	−9.00086			−0.457540
$388$		0			0
$389$		−	9.18004i		−	0.465446i	−0.972543	−	0.232723i	$-0.925236\pi$
$389$		−	9.18004i		−	0.465446i	0.972543	−	0.232723i	$-0.0747636\pi$
$390$		0			0
$391$	0.656411			0.0331961
$392$		0			0
$393$	13.0902			0.660312
$394$		0			0
$395$		−	51.0120i		−	2.56669i
$396$		0			0
$397$	−24.0822			−1.20865			−0.604325	−	0.796738i	$-0.706557\pi$
$397$	−24.0822			−1.20865			−0.604325	+	0.796738i	$0.706557\pi$
$398$		0			0
$399$	20.3886	−	0.369801i	1.02070	−	0.0185132i
$400$		0			0
$401$	−28.7217			−1.43429			−0.717146	−	0.696923i	$-0.754552\pi$
$401$	−28.7217			−1.43429			−0.717146	+	0.696923i	$0.754552\pi$
$402$		0			0
$403$	10.3051			0.513333
$404$		0			0
$405$	15.1063			0.750637
$406$		0			0
$407$		−	5.80480i		−	0.287733i
$408$		0			0
$409$			17.5653i			0.868550i	0.900780	+	0.434275i	$0.142995\pi$
$409$			17.5653i			0.868550i	−0.900780	+	0.434275i	$0.857005\pi$
$410$		0			0
$411$		−	4.89258i		−	0.241333i
$412$		0			0
$413$	0.354103	+	19.5231i	0.0174243	+	0.960667i
$414$		0			0
$415$		−	12.6719i		−	0.622038i
$416$		0			0
$417$	19.2867			0.944471
$418$		0			0
$419$			30.2826i			1.47940i	0.672935	+	0.739702i	$0.265033\pi$
$419$			30.2826i			1.47940i	−0.672935	+	0.739702i	$0.734967\pi$
$420$		0			0
$421$		−	29.1955i		−	1.42290i	−0.702736	−	0.711450i	$-0.748039\pi$
$421$		−	29.1955i		−	1.42290i	0.702736	−	0.711450i	$-0.251961\pi$
$422$		0			0
$423$	−2.89681			−0.140848
$424$		0			0
$425$		−	0.602856i		−	0.0292428i
$426$		0			0
$427$	−29.9779	+	0.543730i	−1.45073	+	0.0263129i
$428$		0			0
$429$			0.998733i			0.0482193i
$430$		0			0
$431$			25.5623i			1.23129i	0.788023	+	0.615645i	$0.211105\pi$
$431$			25.5623i			1.23129i	−0.788023	+	0.615645i	$0.788895\pi$
$432$		0			0
$433$			22.4517i			1.07896i	0.841998	+	0.539481i	$0.181379\pi$
$433$			22.4517i			1.07896i	−0.841998	+	0.539481i	$0.818621\pi$
$434$		0			0
$435$	−30.9244			−1.48271
$436$		0			0
$437$	−28.1179			−1.34506
$438$		0			0
$439$	−22.3455			−1.06649			−0.533245	−	0.845961i	$-0.679028\pi$
$439$	−22.3455			−1.06649			−0.533245	+	0.845961i	$0.679028\pi$
$440$		0			0
$441$	7.20452	−	0.261433i	0.343073	−	0.0124492i
$442$		0			0
$443$	−21.6237			−1.02737			−0.513686	−	0.857978i	$-0.671721\pi$
$443$	−21.6237			−1.02737			−0.513686	+	0.857978i	$0.671721\pi$
$444$		0			0
$445$		−	7.02971i		−	0.333240i
$446$		0			0
$447$	−0.107098			−0.00506558
$448$		0			0
$449$	−2.89960			−0.136840			−0.0684202	−	0.997657i	$-0.521796\pi$
$449$	−2.89960			−0.136840			−0.0684202	+	0.997657i	$0.521796\pi$
$450$		0			0
$451$		−	0.0717536i		−	0.00337874i
$452$		0			0
$453$	2.50551			0.117719
$454$		0			0
$455$	8.23998	−	0.149454i	0.386296	−	0.00700652i
$456$		0			0
$457$	10.5445			0.493253			0.246626	−	0.969111i	$-0.420678\pi$
$457$	10.5445			0.493253			0.246626	+	0.969111i	$0.420678\pi$
$458$		0			0
$459$	−0.725095			−0.0338445
$460$		0			0
$461$	−14.9774			−0.697566			−0.348783	−	0.937203i	$-0.613405\pi$
$461$	−14.9774			−0.697566			−0.348783	+	0.937203i	$0.613405\pi$
$462$		0			0
$463$		−	7.38989i		−	0.343438i	−0.985146	−	0.171719i	$-0.945068\pi$
$463$		−	7.38989i		−	0.343438i	0.985146	−	0.171719i	$-0.0549320\pi$
$464$		0			0
$465$		−	45.0552i		−	2.08938i
$466$		0			0
$467$		−	19.0398i		−	0.881058i	−0.897738	−	0.440529i	$-0.854791\pi$
$467$		−	19.0398i		−	0.881058i	0.897738	−	0.440529i	$-0.145209\pi$
$468$		0			0
$469$	38.3722	−	0.695983i	1.77186	−	0.0321375i
$470$		0			0
$471$			10.5576i			0.486467i
$472$		0			0
$473$	−6.21865			−0.285934
$474$		0			0
$475$			25.8239i			1.18488i
$476$		0			0
$477$			8.78716i			0.402336i
$478$		0			0
$479$	−6.93161			−0.316713			−0.158357	−	0.987382i	$-0.550620\pi$
$479$	−6.93161			−0.316713			−0.158357	+	0.987382i	$0.550620\pi$
$480$		0			0
$481$			8.15798i			0.371972i
$482$		0			0
$483$	19.0126	−	0.344845i	0.865103	−	0.0156910i
$484$		0			0
$485$		−	15.9842i		−	0.725806i
$486$		0			0
$487$		−	36.0554i		−	1.63383i	−0.576760	−	0.816914i	$-0.695683\pi$
$487$		−	36.0554i		−	1.63383i	0.576760	−	0.816914i	$-0.304317\pi$
$488$		0			0
$489$		−	7.97540i		−	0.360660i
$490$		0			0
$491$	28.4173			1.28245			0.641227	−	0.767352i	$-0.278426\pi$
$491$	28.4173			1.28245			0.641227	+	0.767352i	$0.278426\pi$
$492$		0			0
$493$	0.906702			0.0408358
$494$		0			0
$495$	−2.28269			−0.102599
$496$		0			0
$497$	0.404602	+	22.3072i	0.0181489	+	1.00062i
$498$		0			0
$499$	−16.1378			−0.722429			−0.361214	−	0.932483i	$-0.617638\pi$
$499$	−16.1378			−0.722429			−0.361214	+	0.932483i	$0.617638\pi$
$500$		0			0
$501$		−	13.6885i		−	0.611559i
$502$		0			0
$503$	15.8090			0.704888			0.352444	−	0.935833i	$-0.385351\pi$
$503$	15.8090			0.704888			0.352444	+	0.935833i	$0.385351\pi$
$504$		0			0
$505$	−50.0394			−2.22672
$506$		0			0
$507$		−	1.40360i		−	0.0623363i
$508$		0			0
$509$	12.3255			0.546320			0.273160	−	0.961969i	$-0.411931\pi$
$509$	12.3255			0.546320			0.273160	+	0.961969i	$0.411931\pi$
$510$		0			0
$511$	−0.414017	−	22.8263i	−0.0183150	−	1.00978i
$512$		0			0
$513$	31.0601			1.37134
$514$		0			0
$515$	13.6462			0.601322
$516$		0			0
$517$	−2.00139			−0.0880211
$518$		0			0
$519$			3.57824i			0.157067i
$520$		0			0
$521$			32.9577i			1.44390i	0.691945	+	0.721950i	$0.256754\pi$
$521$			32.9577i			1.44390i	−0.691945	+	0.721950i	$0.743246\pi$
$522$		0			0
$523$			2.21195i			0.0967220i	0.998830	+	0.0483610i	$0.0153998\pi$
$523$			2.21195i			0.0967220i	−0.998830	+	0.0483610i	$0.984600\pi$
$524$		0			0
$525$	−0.316710	−	17.4614i	−0.0138223	−	0.762078i
$526$		0			0
$527$			1.32102i			0.0575444i
$528$		0			0
$529$	−3.22035			−0.140015
$530$		0			0
$531$			7.60087i			0.329850i
$532$		0			0
$533$			0.100841i			0.00436792i
$534$		0			0
$535$	−8.59440			−0.371568
$536$		0			0
$537$			9.51969i			0.410805i
$538$		0			0
$539$	4.97757	−	0.180623i	0.214399	−	0.00777997i
$540$		0			0
$541$		−	11.3965i		−	0.489976i	−0.969526	−	0.244988i	$-0.921216\pi$
$541$		−	11.3965i		−	0.489976i	0.969526	−	0.244988i	$-0.0787840\pi$
$542$		0			0
$543$		−	3.03198i		−	0.130115i
$544$		0			0
$545$			45.1589i			1.93440i
$546$		0			0
$547$	−20.8868			−0.893053			−0.446527	−	0.894770i	$-0.647339\pi$
$547$	−20.8868			−0.893053			−0.446527	+	0.894770i	$0.647339\pi$
$548$		0			0
$549$	−11.6712			−0.498115
$550$		0			0
$551$	−38.8394			−1.65461
$552$		0			0
$553$	0.785748	+	43.3213i	0.0334134	+	1.84221i
$554$		0			0
$555$	35.6678			1.51401
$556$		0			0
$557$		−	36.5235i		−	1.54755i	−0.633460	−	0.773776i	$-0.718366\pi$
$557$		−	36.5235i		−	1.54755i	0.633460	−	0.773776i	$-0.281634\pi$
$558$		0			0
$559$	8.73959			0.369645
$560$		0			0
$561$	−0.128028			−0.00540536
$562$		0			0
$563$			21.2293i			0.894707i	0.894357	+	0.447354i	$0.147633\pi$
$563$			21.2293i			0.894707i	−0.894357	+	0.447354i	$0.852367\pi$
$564$		0			0
$565$	−30.0025			−1.26222
$566$		0			0
$567$	−12.8288	+	0.232685i	−0.538759	+	0.00977185i
$568$		0			0
$569$	−18.2538			−0.765239			−0.382619	−	0.923906i	$-0.624978\pi$
$569$	−18.2538			−0.765239			−0.382619	+	0.923906i	$0.624978\pi$
$570$		0			0
$571$	−4.06655			−0.170180			−0.0850899	−	0.996373i	$-0.527118\pi$
$571$	−4.06655			−0.170180			−0.0850899	+	0.996373i	$0.527118\pi$
$572$		0			0
$573$	12.9427			0.540690
$574$		0			0
$575$			24.0811i			1.00425i
$576$		0			0
$577$			32.8326i			1.36684i	0.730025	+	0.683420i	$0.239508\pi$
$577$			32.8326i			1.36684i	−0.730025	+	0.683420i	$0.760492\pi$
$578$		0			0
$579$			18.7397i			0.778796i
$580$		0			0
$581$	0.195187	+	10.7614i	0.00809774	+	0.446459i
$582$		0			0
$583$			6.07100i			0.251435i
$584$		0			0
$585$	3.20805			0.132637
$586$		0			0
$587$			34.6043i			1.42827i	0.700008	+	0.714135i	$0.253180\pi$
$587$			34.6043i			1.42827i	−0.700008	+	0.714135i	$0.746820\pi$
$588$		0			0
$589$		−	56.5869i		−	2.33162i
$590$		0			0
$591$	34.6350			1.42469
$592$		0			0
$593$		−	30.7831i		−	1.26411i	−0.774923	−	0.632056i	$-0.782211\pi$
$593$		−	30.7831i		−	1.26411i	0.774923	−	0.632056i	$-0.217789\pi$
$594$		0			0
$595$	0.0191587	+	1.05629i	0.000785428	+	0.0433036i
$596$		0			0
$597$			22.5297i			0.922080i
$598$		0			0
$599$			18.1200i			0.740363i	0.928959	+	0.370181i	$0.120704\pi$
$599$			18.1200i			0.740363i	−0.928959	+	0.370181i	$0.879296\pi$
$600$		0			0
$601$			2.44947i			0.0999158i	0.998751	+	0.0499579i	$0.0159087\pi$
$601$			2.44947i			0.0999158i	−0.998751	+	0.0499579i	$0.984091\pi$
$602$		0			0
$603$	14.9393			0.608377
$604$		0			0
$605$	32.6872			1.32892
$606$		0			0
$607$	17.5171			0.710998			0.355499	−	0.934677i	$-0.384311\pi$
$607$	17.5171			0.710998			0.355499	+	0.934677i	$0.384311\pi$
$608$		0			0
$609$	26.2622	−	0.476335i	1.06420	−	0.0193021i
$610$		0			0
$611$	2.81273			0.113791
$612$		0			0
$613$		−	12.6633i		−	0.511465i	−0.966748	−	0.255732i	$-0.917683\pi$
$613$		−	12.6633i		−	0.511465i	0.966748	−	0.255732i	$-0.0823166\pi$
$614$		0			0
$615$	0.440892			0.0177785
$616$		0			0
$617$	−39.5883			−1.59376			−0.796882	−	0.604135i	$-0.793519\pi$
$617$	−39.5883			−1.59376			−0.796882	+	0.604135i	$0.793519\pi$
$618$		0			0
$619$			14.7549i			0.593049i	0.955025	+	0.296525i	$0.0958277\pi$
$619$			14.7549i			0.593049i	−0.955025	+	0.296525i	$0.904172\pi$
$620$		0			0
$621$	28.9639			1.16228
$622$		0			0
$623$	0.108280	+	5.96989i	0.00433815	+	0.239179i
$624$		0			0
$625$	−26.3977			−1.05591
$626$		0			0
$627$	5.48420			0.219018
$628$		0			0
$629$	−1.04578			−0.0416979
$630$		0			0
$631$		−	24.2732i		−	0.966303i	−0.875537	−	0.483151i	$-0.839492\pi$
$631$		−	24.2732i		−	0.966303i	0.875537	−	0.483151i	$-0.160508\pi$
$632$		0			0
$633$		−	11.5754i		−	0.460082i
$634$		0			0
$635$		−	39.0127i		−	1.54817i
$636$		0			0
$637$	−6.99540	+	0.253844i	−0.277168	+	0.0100577i
$638$		0			0
$639$			8.68482i			0.343566i
$640$		0			0
$641$	−7.40358			−0.292424			−0.146212	−	0.989253i	$-0.546708\pi$
$641$	−7.40358			−0.292424			−0.146212	+	0.989253i	$0.546708\pi$
$642$		0			0
$643$			33.4236i			1.31810i	0.752101	+	0.659048i	$0.229041\pi$
$643$			33.4236i			1.31810i	−0.752101	+	0.659048i	$0.770959\pi$
$644$		0			0
$645$		−	38.2107i		−	1.50454i
$646$		0			0
$647$	−34.4335			−1.35372			−0.676861	−	0.736111i	$-0.736660\pi$
$647$	−34.4335			−1.35372			−0.676861	+	0.736111i	$0.736660\pi$
$648$		0			0
$649$			5.25140i			0.206136i
$650$		0			0
$651$	0.693994	+	38.2625i	0.0271998	+	1.49963i
$652$		0			0
$653$			3.01244i			0.117886i	0.998261	+	0.0589430i	$0.0187730\pi$
$653$			3.01244i			0.117886i	−0.998261	+	0.0589430i	$0.981227\pi$
$654$		0			0
$655$		−	29.0502i		−	1.13509i
$656$		0			0
$657$		−	8.88692i		−	0.346712i
$658$		0			0
$659$	−49.8643			−1.94244			−0.971218	−	0.238191i	$-0.923446\pi$
$659$	−49.8643			−1.94244			−0.971218	+	0.238191i	$0.923446\pi$
$660$		0			0
$661$	−13.2329			−0.514701			−0.257350	−	0.966318i	$-0.582849\pi$
$661$	−13.2329			−0.514701			−0.257350	+	0.966318i	$0.582849\pi$
$662$		0			0
$663$	0.179929			0.00698787
$664$		0			0
$665$	−0.820677	−	45.2471i	−0.0318245	−	1.75461i
$666$		0			0
$667$	−36.2182			−1.40238
$668$		0			0
$669$		−	15.5545i		−	0.601373i
$670$		0			0
$671$	−8.06358			−0.311291
$672$		0			0
$673$	44.0069			1.69634			0.848170	−	0.529724i	$-0.177704\pi$
$673$	44.0069			1.69634			0.848170	+	0.529724i	$0.177704\pi$
$674$		0			0
$675$		−	26.6008i		−	1.02387i
$676$		0			0
$677$	−5.69662			−0.218939			−0.109470	−	0.993990i	$-0.534915\pi$
$677$	−5.69662			−0.218939			−0.109470	+	0.993990i	$0.534915\pi$
$678$		0			0
$679$	0.246208	+	13.5744i	0.00944860	+	0.520937i
$680$		0			0
$681$	−18.3749			−0.704128
$682$		0			0
$683$	−21.1458			−0.809121			−0.404560	−	0.914511i	$-0.632575\pi$
$683$	−21.1458			−0.809121			−0.404560	+	0.914511i	$0.632575\pi$
$684$		0			0
$685$	−10.8578			−0.414855
$686$		0			0
$687$		−	37.5364i		−	1.43210i
$688$		0			0
$689$		−	8.53210i		−	0.325047i
$690$		0			0
$691$		−	32.1264i		−	1.22215i	−0.791575	−	0.611073i	$-0.790738\pi$
$691$		−	32.1264i		−	1.22215i	0.791575	−	0.611073i	$-0.209262\pi$
$692$		0			0
$693$	1.93854	−	0.0351607i	0.0736391	−	0.00133564i
$694$		0			0
$695$		−	42.8017i		−	1.62356i
$696$		0			0
$697$	−0.0129269			−0.000489642
$698$		0			0
$699$			28.3956i			1.07402i
$700$		0			0
$701$		−	27.9007i		−	1.05379i	−0.849929	−	0.526897i	$-0.823355\pi$
$701$		−	27.9007i		−	1.05379i	0.849929	−	0.526897i	$-0.176645\pi$
$702$		0			0
$703$	44.7968			1.68954
$704$		0			0
$705$		−	12.2976i		−	0.463155i
$706$		0			0
$707$	42.4953	−	0.770767i	1.59820	−	0.0289877i
$708$		0			0
$709$		−	9.64782i		−	0.362331i	−0.983453	−	0.181166i	$-0.942013\pi$
$709$		−	9.64782i		−	0.362331i	0.983453	−	0.181166i	$-0.0579870\pi$
$710$		0			0
$711$			16.8662i			0.632531i
$712$		0			0
$713$		−	52.7680i		−	1.97618i
$714$		0			0
$715$	2.21643			0.0828897
$716$		0			0
$717$	−19.3460			−0.722490
$718$		0			0
$719$	7.10932			0.265133			0.132566	−	0.991174i	$-0.457678\pi$
$719$	7.10932			0.265133			0.132566	+	0.991174i	$0.457678\pi$
$720$		0			0
$721$	−11.5888	+	0.210195i	−0.431591	+	0.00782806i
$722$		0			0
$723$	−17.8890			−0.665299
$724$		0			0
$725$			33.2633i			1.23537i
$726$		0			0
$727$	31.7118			1.17613			0.588063	−	0.808815i	$-0.299891\pi$
$727$	31.7118			1.17613			0.588063	+	0.808815i	$0.299891\pi$
$728$		0			0
$729$	−28.8126			−1.06713
$730$		0			0
$731$			1.12033i			0.0414371i
$732$		0			0
$733$	3.95576			0.146109			0.0730546	−	0.997328i	$-0.476725\pi$
$733$	3.95576			0.146109			0.0730546	+	0.997328i	$0.476725\pi$
$734$		0			0
$735$	1.10984	+	30.5848i	0.0409371	+	1.12814i
$736$		0			0
$737$	10.3215			0.380198
$738$		0			0
$739$	−4.79871			−0.176523			−0.0882617	−	0.996097i	$-0.528131\pi$
$739$	−4.79871			−0.176523			−0.0882617	+	0.996097i	$0.528131\pi$
$740$		0			0
$741$	−7.70742			−0.283139
$742$		0			0
$743$			11.6943i			0.429021i	0.976722	+	0.214511i	$0.0688157\pi$
$743$			11.6943i			0.429021i	−0.976722	+	0.214511i	$0.931184\pi$
$744$		0			0
$745$			0.237677i			0.00870781i
$746$		0			0
$747$			4.18972i			0.153294i
$748$		0			0
$749$	7.29868	−	0.132381i	0.266688	−	0.00483710i
$750$		0			0
$751$			4.77337i			0.174183i	0.996200	+	0.0870914i	$0.0277572\pi$
$751$			4.77337i			0.174183i	−0.996200	+	0.0870914i	$0.972243\pi$
$752$		0			0
$753$	31.5356			1.14922
$754$		0			0
$755$		−	5.56033i		−	0.202361i
$756$		0			0
$757$		−	44.2310i		−	1.60760i	−0.594898	−	0.803801i	$-0.702808\pi$
$757$		−	44.2310i		−	1.60760i	0.594898	−	0.803801i	$-0.297192\pi$
$758$		0			0
$759$	5.11409			0.185630
$760$		0			0
$761$			19.9559i			0.723401i	0.932294	+	0.361700i	$0.117804\pi$
$761$			19.9559i			0.723401i	−0.932294	+	0.361700i	$0.882196\pi$
$762$		0			0
$763$	−0.695592	−	38.3506i	−0.0251821	−	1.38839i
$764$		0			0
$765$			0.411243i			0.0148685i
$766$		0			0
$767$		−	7.38024i		−	0.266485i
$768$		0			0
$769$			12.6775i			0.457162i	0.973525	+	0.228581i	$0.0734086\pi$
$769$			12.6775i			0.457162i	−0.973525	+	0.228581i	$0.926591\pi$
$770$		0			0
$771$	−20.1066			−0.724122
$772$		0			0
$773$	−7.32173			−0.263344			−0.131672	−	0.991293i	$-0.542035\pi$
$773$	−7.32173			−0.263344			−0.131672	+	0.991293i	$0.542035\pi$
$774$		0			0
$775$	−48.4628			−1.74084
$776$		0			0
$777$	−30.2904	+	0.549398i	−1.08666	+	0.0197095i
$778$		0			0
$779$	0.553736			0.0198397
$780$		0			0
$781$			6.00030i			0.214708i
$782$		0			0
$783$	40.0080			1.42977
$784$		0			0
$785$	23.4297			0.836244
$786$		0			0
$787$			50.2628i			1.79168i	0.444381	+	0.895838i	$0.353424\pi$
$787$			50.2628i			1.79168i	−0.444381	+	0.895838i	$0.646576\pi$
$788$		0			0
$789$	−18.7636			−0.668003
$790$		0			0
$791$	25.4793	−	0.462135i	0.905939	−	0.0164316i
$792$		0			0
$793$	11.3324			0.402427
$794$		0			0
$795$	−37.3035			−1.32302
$796$		0			0
$797$	50.0051			1.77127			0.885636	−	0.464381i	$-0.153723\pi$
$797$	50.0051			1.77127			0.885636	+	0.464381i	$0.153723\pi$
$798$		0			0
$799$			0.360565i			0.0127559i
$800$		0			0
$801$			2.32424i			0.0821231i
$802$		0			0
$803$		−	6.13992i		−	0.216673i
$804$		0			0
$805$	−0.765293	−	42.1935i	−0.0269730	−	1.48713i
$806$		0			0
$807$			18.6188i			0.655414i
$808$		0			0
$809$	−18.2457			−0.641483			−0.320742	−	0.947167i	$-0.603932\pi$
$809$	−18.2457			−0.641483			−0.320742	+	0.947167i	$0.603932\pi$
$810$		0			0
$811$		−	49.8112i		−	1.74911i	−0.484929	−	0.874554i	$-0.661154\pi$
$811$		−	49.8112i		−	1.74911i	0.484929	−	0.874554i	$-0.338846\pi$
$812$		0			0
$813$			20.9654i			0.735289i
$814$		0			0
$815$	−17.6993			−0.619980
$816$		0			0
$817$		−	47.9905i		−	1.67898i
$818$		0			0
$819$	−2.72440	+	0.0494143i	−0.0951982	+	0.00172668i
$820$		0			0
$821$		−	23.2872i		−	0.812727i	−0.913711	−	0.406364i	$-0.866797\pi$
$821$		−	23.2872i		−	0.812727i	0.913711	−	0.406364i	$-0.133203\pi$
$822$		0			0
$823$			46.7989i			1.63131i	0.578540	+	0.815654i	$0.303623\pi$
$823$			46.7989i			1.63131i	−0.578540	+	0.815654i	$0.696377\pi$
$824$		0			0
$825$		−	4.69685i		−	0.163523i
$826$		0			0
$827$	33.8764			1.17800			0.588998	−	0.808134i	$-0.299522\pi$
$827$	33.8764			1.17800			0.588998	+	0.808134i	$0.299522\pi$
$828$		0			0
$829$	1.54700			0.0537296			0.0268648	−	0.999639i	$-0.491448\pi$
$829$	1.54700			0.0537296			0.0268648	+	0.999639i	$0.491448\pi$
$830$		0			0
$831$	−2.91658			−0.101175
$832$		0			0
$833$	−0.0325405	−	0.896745i	−0.00112746	−	0.0310704i
$834$		0			0
$835$	−30.3781			−1.05128
$836$		0			0
$837$			58.2895i			2.01478i
$838$		0			0
$839$	−12.5119			−0.431959			−0.215980	−	0.976398i	$-0.569295\pi$
$839$	−12.5119			−0.431959			−0.215980	+	0.976398i	$0.569295\pi$
$840$		0			0
$841$	−21.0284			−0.725116
$842$		0			0
$843$		−	10.7521i		−	0.370321i
$844$		0			0
$845$	−3.11493			−0.107157
$846$		0			0
$847$	−27.7591	+	0.503487i	−0.953816	+	0.0173000i
$848$		0			0
$849$	6.67336			0.229029
$850$		0			0
$851$	41.7736			1.43198
$852$		0			0
$853$	−3.10292			−0.106242			−0.0531210	−	0.998588i	$-0.516917\pi$
$853$	−3.10292			−0.106242			−0.0531210	+	0.998588i	$0.516917\pi$
$854$		0			0
$855$		−	17.6159i		−	0.602452i
$856$		0			0
$857$		−	35.6654i		−	1.21831i	−0.793052	−	0.609154i	$-0.791509\pi$
$857$		−	35.6654i		−	1.21831i	0.793052	−	0.609154i	$-0.208491\pi$
$858$		0			0
$859$		−	0.510482i		−	0.0174174i	−0.999962	−	0.00870870i	$-0.997228\pi$
$859$		−	0.510482i		−	0.0174174i	0.999962	−	0.00870870i	$-0.00277210\pi$
$860$		0			0
$861$	−0.374422	+	0.00679115i	−0.0127603	+	0.000231442i
$862$		0			0
$863$		−	36.6222i		−	1.24663i	−0.781969	−	0.623317i	$-0.785785\pi$
$863$		−	36.6222i		−	1.24663i	0.781969	−	0.623317i	$-0.214215\pi$
$864$		0			0
$865$	7.94096			0.270001
$866$		0			0
$867$		−	23.8382i		−	0.809588i
$868$		0			0
$869$			11.6528i			0.395293i
$870$		0			0
$871$	−14.5057			−0.491507
$872$		0			0
$873$			5.28489i			0.178866i
$874$		0			0
$875$	2.44888	−	0.0444171i	0.0827874	−	0.00150157i
$876$		0			0
$877$		−	1.65400i		−	0.0558515i	−0.999610	−	0.0279257i	$-0.991110\pi$
$877$		−	1.65400i		−	0.0558515i	0.999610	−	0.0279257i	$-0.00889020\pi$
$878$		0			0
$879$		−	28.7296i		−	0.969026i
$880$		0			0
$881$		−	17.3400i		−	0.584199i	−0.956388	−	0.292099i	$-0.905646\pi$
$881$		−	17.3400i		−	0.584199i	0.956388	−	0.292099i	$-0.0943538\pi$
$882$		0			0
$883$	−18.7952			−0.632510			−0.316255	−	0.948674i	$-0.602426\pi$
$883$	−18.7952			−0.632510			−0.316255	+	0.948674i	$0.602426\pi$
$884$		0			0
$885$	−32.2674			−1.08466
$886$		0			0
$887$	−1.74179			−0.0584836			−0.0292418	−	0.999572i	$-0.509309\pi$
$887$	−1.74179			−0.0584836			−0.0292418	+	0.999572i	$0.509309\pi$
$888$		0			0
$889$	0.600921	+	33.1310i	0.0201542	+	1.11118i
$890$		0			0
$891$	−3.45075			−0.115604
$892$		0			0
$893$		−	15.4451i		−	0.516851i
$894$		0			0
$895$	21.1265			0.706180
$896$		0			0
$897$	−7.18727			−0.239976
$898$		0			0
$899$		−	72.8886i		−	2.43097i
$900$		0			0
$901$	1.09374			0.0364376
$902$		0			0
$903$	0.588567	+	32.4499i	0.0195863	+	1.07987i
$904$		0			0
$905$	−6.72869			−0.223669
$906$		0			0
$907$	6.46569			0.214690			0.107345	−	0.994222i	$-0.465765\pi$
$907$	6.46569			0.214690			0.107345	+	0.994222i	$0.465765\pi$
$908$		0			0
$909$	16.5446			0.548750
$910$		0			0
$911$		−	39.3511i		−	1.30376i	−0.758321	−	0.651881i	$-0.773980\pi$
$911$		−	39.3511i		−	1.30376i	0.758321	−	0.651881i	$-0.226020\pi$
$912$		0			0
$913$			2.89466i			0.0957991i
$914$		0			0
$915$		−	49.5469i		−	1.63797i
$916$		0			0
$917$	0.447466	+	24.6705i	0.0147766	+	0.814692i
$918$		0			0
$919$		−	1.51489i		−	0.0499716i	−0.999688	−	0.0249858i	$-0.992046\pi$
$919$		−	1.51489i		−	0.0499716i	0.999688	−	0.0249858i	$-0.00795405\pi$
$920$		0			0
$921$	−0.978041			−0.0322275
$922$		0			0
$923$		−	8.43273i		−	0.277567i
$924$		0			0
$925$		−	38.3654i		−	1.26145i
$926$		0			0
$927$	−4.51185			−0.148189
$928$		0			0
$929$		−	29.8691i		−	0.979975i	−0.871729	−	0.489987i	$-0.837001\pi$
$929$		−	29.8691i		−	0.979975i	0.871729	−	0.489987i	$-0.162999\pi$
$930$		0			0
$931$	1.39390	+	38.4128i	0.0456832	+	1.25893i
$932$		0			0
$933$		−	30.0400i		−	0.983464i
$934$		0			0
$935$			0.284125i			0.00929190i
$936$		0			0
$937$			35.3032i			1.15330i	0.816990	+	0.576652i	$0.195641\pi$
$937$			35.3032i			1.15330i	−0.816990	+	0.576652i	$0.804359\pi$
$938$		0			0
$939$	−4.28773			−0.139925
$940$		0			0
$941$	−40.5751			−1.32271			−0.661355	−	0.750073i	$-0.730018\pi$
$941$	−40.5751			−1.32271			−0.661355	+	0.750073i	$0.730018\pi$
$942$		0			0
$943$	0.516366			0.0168152
$944$		0			0
$945$	0.845370	+	46.6084i	0.0274999	+	1.51617i
$946$		0			0
$947$	−2.97693			−0.0967373			−0.0483687	−	0.998830i	$-0.515402\pi$
$947$	−2.97693			−0.0967373			−0.0483687	+	0.998830i	$0.515402\pi$
$948$		0			0
$949$			8.62895i			0.280108i
$950$		0			0
$951$	37.6682			1.22148
$952$		0			0
$953$	31.5607			1.02235			0.511175	−	0.859477i	$-0.329210\pi$
$953$	31.5607			1.02235			0.511175	+	0.859477i	$0.329210\pi$
$954$		0			0
$955$		−	28.7230i		−	0.929453i
$956$		0			0
$957$	7.06411			0.228350
$958$		0			0
$959$	9.22084	−	0.167245i	0.297757	−	0.00540062i
$960$		0			0
$961$	75.1948			2.42564
$962$		0			0
$963$	2.84158			0.0915686
$964$		0			0
$965$	41.5879			1.33876
$966$		0			0
$967$			16.6169i			0.534365i	0.963646	+	0.267182i	$0.0860926\pi$
$967$			16.6169i			0.534365i	−0.963646	+	0.267182i	$0.913907\pi$
$968$		0			0
$969$		−	0.988019i		−	0.0317398i
$970$		0			0
$971$		−	22.3652i		−	0.717733i	−0.933389	−	0.358866i	$-0.883163\pi$
$971$		−	22.3652i		−	0.717733i	0.933389	−	0.358866i	$-0.116837\pi$
$972$		0			0
$973$	0.659283	+	36.3488i	0.0211356	+	1.16529i
$974$		0			0
$975$			6.60088i			0.211397i
$976$		0			0
$977$	46.7904			1.49696			0.748480	−	0.663158i	$-0.230784\pi$
$977$	46.7904			1.49696			0.748480	+	0.663158i	$0.230784\pi$
$978$		0			0
$979$			1.60581i			0.0513218i
$980$		0			0
$981$		−	14.9310i		−	0.476709i
$982$		0			0
$983$	9.87533			0.314974			0.157487	−	0.987521i	$-0.449661\pi$
$983$	9.87533			0.314974			0.157487	+	0.987521i	$0.449661\pi$
$984$		0			0
$985$		−	76.8634i		−	2.44907i
$986$		0			0
$987$	0.189423	+	10.4436i	0.00602939	+	0.332423i
$988$		0			0
$989$		−	44.7518i		−	1.42302i
$990$		0			0
$991$			6.15361i			0.195476i	0.995212	+	0.0977379i	$0.0311607\pi$
$991$			6.15361i			0.195476i	−0.995212	+	0.0977379i	$0.968839\pi$
$992$		0			0
$993$		−	24.7880i		−	0.786624i
$994$		0			0
$995$	49.9988			1.58507
$996$		0			0
$997$	−5.86468			−0.185736			−0.0928682	−	0.995678i	$-0.529604\pi$
$997$	−5.86468			−0.185736			−0.0928682	+	0.995678i	$0.529604\pi$
$998$		0			0
$999$	−46.1446			−1.45995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.a.2575.15		48
4.3	odd	2		728.2.h.a.27.16	yes	48
7.6	odd	2		2912.2.h.b.2575.34		48
8.3	odd	2		2912.2.h.b.2575.15		48
8.5	even	2		728.2.h.b.27.15	yes	48
28.27	even	2		728.2.h.b.27.16	yes	48
56.13	odd	2		728.2.h.a.27.15	✓	48
56.27	even	2	inner	2912.2.h.a.2575.34		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.15	✓	48	56.13	odd	2
728.2.h.a.27.16	yes	48	4.3	odd	2
728.2.h.b.27.15	yes	48	8.5	even	2
728.2.h.b.27.16	yes	48	28.27	even	2
2912.2.h.a.2575.15		48	1.1	even	1	trivial
2912.2.h.a.2575.34		48	56.27	even	2	inner
2912.2.h.b.2575.15		48	8.3	odd	2
2912.2.h.b.2575.34		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)

\(f(q)\)	\(=\)	\(q-1.40360i q^{3} -3.11493 q^{5} +(2.64532 - 0.0479799i) q^{7} +1.02989 q^{9} +0.711549 q^{11} -1.00000 q^{13} +4.37213i q^{15} -0.128191i q^{17} +5.49116i q^{19} +(-0.0673448 - 3.71298i) q^{21} +5.12058i q^{23} +4.70280 q^{25} -5.65638i q^{27} +7.07307i q^{29} -10.3051 q^{31} -0.998733i q^{33} +(-8.23998 + 0.149454i) q^{35} -8.15798i q^{37} +1.40360i q^{39} -0.100841i q^{41} -8.73959 q^{43} -3.20805 q^{45} -2.81273 q^{47} +(6.99540 - 0.253844i) q^{49} -0.179929 q^{51} +8.53210i q^{53} -2.21643 q^{55} +7.70742 q^{57} +7.38024i q^{59} -11.3324 q^{61} +(2.72440 - 0.0494143i) q^{63} +3.11493 q^{65} +14.5057 q^{67} +7.18727 q^{69} +8.43273i q^{71} -8.62895i q^{73} -6.60088i q^{75} +(1.88227 - 0.0341401i) q^{77} +16.3766i q^{79} -4.84963 q^{81} +4.06811i q^{83} +0.399306i q^{85} +9.92780 q^{87} +2.25678i q^{89} +(-2.64532 + 0.0479799i) q^{91} +14.4643i q^{93} -17.1046i q^{95} +5.13148i q^{97} +0.732821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) \) 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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