LMFDB - Embedded newform 2912.2.h.a.2575.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(2575,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [48,0,0,0,0,0,0,0,-48,0,4,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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			2575.5
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.a.2575.44

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89173i q^{3} -2.33723 q^{5} +(1.68858 + 2.03684i) q^{7} -5.36210 q^{9} -0.460555 q^{11} -1.00000 q^{13} +6.75863i q^{15} -5.97523i q^{17} -7.60833i q^{19} +(5.88998 - 4.88291i) q^{21} -4.22795i q^{23} +0.462642 q^{25} +6.83054i q^{27} +3.86435i q^{29} -5.75982 q^{31} +1.33180i q^{33} +(-3.94659 - 4.76056i) q^{35} -2.44158i q^{37} +2.89173i q^{39} +8.80574i q^{41} +5.08694 q^{43} +12.5325 q^{45} +2.29606 q^{47} +(-1.29742 + 6.87871i) q^{49} -17.2788 q^{51} +8.79106i q^{53} +1.07642 q^{55} -22.0012 q^{57} +7.52488i q^{59} -0.719026 q^{61} +(-9.05431 - 10.9217i) q^{63} +2.33723 q^{65} -14.3569 q^{67} -12.2261 q^{69} +12.9030i q^{71} -4.52589i q^{73} -1.33784i q^{75} +(-0.777683 - 0.938076i) q^{77} -10.7927i q^{79} +3.66579 q^{81} +5.97421i q^{83} +13.9655i q^{85} +11.1746 q^{87} -2.78912i q^{89} +(-1.68858 - 2.03684i) q^{91} +16.6558i q^{93} +17.7824i q^{95} +9.23618i q^{97} +2.46954 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.89173i		−	1.66954i	−0.550598	−	0.834770i	$-0.685600\pi$
$3$		−	2.89173i		−	1.66954i	0.550598	−	0.834770i	$-0.314400\pi$
$4$		0			0
$5$	−2.33723			−1.04524			−0.522620	−	0.852565i	$-0.675045\pi$
$5$	−2.33723			−1.04524			−0.522620	+	0.852565i	$0.675045\pi$
$6$		0			0
$7$	1.68858	+	2.03684i	0.638222	+	0.769852i
$7$	1.68858	+	2.03684i	0.638222	+	0.769852i
$8$		0			0
$9$	−5.36210			−1.78737
$10$		0			0
$11$	−0.460555			−0.138863			−0.0694313	−	0.997587i	$-0.522118\pi$
$11$	−0.460555			−0.138863			−0.0694313	+	0.997587i	$0.522118\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$			6.75863i			1.74507i
$16$		0			0
$17$		−	5.97523i		−	1.44921i	−0.689166	−	0.724603i	$-0.742023\pi$
$17$		−	5.97523i		−	1.44921i	0.689166	−	0.724603i	$-0.257977\pi$
$18$		0			0
$19$		−	7.60833i		−	1.74547i	−0.488195	−	0.872735i	$-0.662344\pi$
$19$		−	7.60833i		−	1.74547i	0.488195	−	0.872735i	$-0.337656\pi$
$20$		0			0
$21$	5.88998	−	4.88291i	1.28530	−	1.06554i
$22$		0			0
$23$		−	4.22795i		−	0.881589i	−0.897608	−	0.440795i	$-0.854697\pi$
$23$		−	4.22795i		−	0.881589i	0.897608	−	0.440795i	$-0.145303\pi$
$24$		0			0
$25$	0.462642			0.0925284
$26$		0			0
$27$			6.83054i			1.31454i
$28$		0			0
$29$			3.86435i			0.717591i	0.933416	+	0.358795i	$0.116812\pi$
$29$			3.86435i			0.717591i	−0.933416	+	0.358795i	$0.883188\pi$
$30$		0			0
$31$	−5.75982			−1.03449			−0.517247	−	0.855836i	$-0.673043\pi$
$31$	−5.75982			−1.03449			−0.517247	+	0.855836i	$0.673043\pi$
$32$		0			0
$33$			1.33180i			0.231837i
$34$		0			0
$35$	−3.94659	−	4.76056i	−0.667096	−	0.804681i
$36$		0			0
$37$		−	2.44158i		−	0.401394i	−0.979653	−	0.200697i	$-0.935679\pi$
$37$		−	2.44158i		−	0.401394i	0.979653	−	0.200697i	$-0.0643206\pi$
$38$		0			0
$39$			2.89173i			0.463047i
$40$		0			0
$41$			8.80574i			1.37523i	0.726077	+	0.687613i	$0.241341\pi$
$41$			8.80574i			1.37523i	−0.726077	+	0.687613i	$0.758659\pi$
$42$		0			0
$43$	5.08694			0.775751			0.387875	−	0.921712i	$-0.373209\pi$
$43$	5.08694			0.775751			0.387875	+	0.921712i	$0.373209\pi$
$44$		0			0
$45$	12.5325			1.86823
$46$		0			0
$47$	2.29606			0.334914			0.167457	−	0.985879i	$-0.446444\pi$
$47$	2.29606			0.334914			0.167457	+	0.985879i	$0.446444\pi$
$48$		0			0
$49$	−1.29742	+	6.87871i	−0.185345	+	0.982673i
$50$		0			0
$51$	−17.2788			−2.41951
$52$		0			0
$53$			8.79106i			1.20754i	0.797157	+	0.603772i	$0.206336\pi$
$53$			8.79106i			1.20754i	−0.797157	+	0.603772i	$0.793664\pi$
$54$		0			0
$55$	1.07642			0.145145
$56$		0			0
$57$	−22.0012			−2.91413
$58$		0			0
$59$			7.52488i			0.979657i	0.871819	+	0.489828i	$0.162941\pi$
$59$			7.52488i			0.979657i	−0.871819	+	0.489828i	$0.837059\pi$
$60$		0			0
$61$	−0.719026			−0.0920619			−0.0460310	−	0.998940i	$-0.514657\pi$
$61$	−0.719026			−0.0920619			−0.0460310	+	0.998940i	$0.514657\pi$
$62$		0			0
$63$	−9.05431	−	10.9217i	−1.14074	−	1.37601i
$64$		0			0
$65$	2.33723			0.289898
$66$		0			0
$67$	−14.3569			−1.75397			−0.876985	−	0.480518i	$-0.840449\pi$
$67$	−14.3569			−1.75397			−0.876985	+	0.480518i	$0.840449\pi$
$68$		0			0
$69$	−12.2261			−1.47185
$70$		0			0
$71$			12.9030i			1.53131i	0.643253	+	0.765654i	$0.277584\pi$
$71$			12.9030i			1.53131i	−0.643253	+	0.765654i	$0.722416\pi$
$72$		0			0
$73$		−	4.52589i		−	0.529716i	−0.964287	−	0.264858i	$-0.914675\pi$
$73$		−	4.52589i		−	0.529716i	0.964287	−	0.264858i	$-0.0853250\pi$
$74$		0			0
$75$		−	1.33784i		−	0.154480i
$76$		0			0
$77$	−0.777683	−	0.938076i	−0.0886252	−	0.106904i
$78$		0			0
$79$		−	10.7927i		−	1.21427i	−0.794597	−	0.607137i	$-0.792318\pi$
$79$		−	10.7927i		−	1.21427i	0.794597	−	0.607137i	$-0.207682\pi$
$80$		0			0
$81$	3.66579			0.407310
$82$		0			0
$83$			5.97421i			0.655754i	0.944720	+	0.327877i	$0.106333\pi$
$83$			5.97421i			0.655754i	−0.944720	+	0.327877i	$0.893667\pi$
$84$		0			0
$85$			13.9655i			1.51477i
$86$		0			0
$87$	11.1746			1.19805
$88$		0			0
$89$		−	2.78912i		−	0.295646i	−0.989014	−	0.147823i	$-0.952773\pi$
$89$		−	2.78912i		−	0.295646i	0.989014	−	0.147823i	$-0.0472266\pi$
$90$		0			0
$91$	−1.68858	−	2.03684i	−0.177011	−	0.213519i
$92$		0			0
$93$			16.6558i			1.72713i
$94$		0			0
$95$			17.7824i			1.82444i
$96$		0			0
$97$			9.23618i			0.937792i	0.883253	+	0.468896i	$0.155348\pi$
$97$			9.23618i			0.937792i	−0.883253	+	0.468896i	$0.844652\pi$
$98$		0			0
$99$	2.46954			0.248198
$100$		0			0
$101$	2.22913			0.221807			0.110903	−	0.993831i	$-0.464626\pi$
$101$	2.22913			0.221807			0.110903	+	0.993831i	$0.464626\pi$
$102$		0			0
$103$	−4.08568			−0.402574			−0.201287	−	0.979532i	$-0.564512\pi$
$103$	−4.08568			−0.402574			−0.201287	+	0.979532i	$0.564512\pi$
$104$		0			0
$105$	−13.7662	+	11.4125i	−1.34345	+	1.11374i
$106$		0			0
$107$	−5.04195			−0.487424			−0.243712	−	0.969848i	$-0.578365\pi$
$107$	−5.04195			−0.487424			−0.243712	+	0.969848i	$0.578365\pi$
$108$		0			0
$109$			9.75771i			0.934619i	0.884094	+	0.467310i	$0.154777\pi$
$109$			9.75771i			0.934619i	−0.884094	+	0.467310i	$0.845223\pi$
$110$		0			0
$111$	−7.06039			−0.670143
$112$		0			0
$113$	−13.8892			−1.30658			−0.653292	−	0.757106i	$-0.726613\pi$
$113$	−13.8892			−1.30658			−0.653292	+	0.757106i	$0.726613\pi$
$114$		0			0
$115$			9.88170i			0.921473i
$116$		0			0
$117$	5.36210			0.495726
$118$		0			0
$119$	12.1706	−	10.0896i	1.11568	−	0.924916i
$120$		0			0
$121$	−10.7879			−0.980717
$122$		0			0
$123$	25.4638			2.29600
$124$		0			0
$125$	10.6048			0.948526
$126$		0			0
$127$		−	8.16487i		−	0.724515i	−0.932078	−	0.362258i	$-0.882006\pi$
$127$		−	8.16487i		−	0.724515i	0.932078	−	0.362258i	$-0.117994\pi$
$128$		0			0
$129$		−	14.7100i		−	1.29515i
$130$		0			0
$131$			8.02619i			0.701251i	0.936516	+	0.350626i	$0.114031\pi$
$131$			8.02619i			0.701251i	−0.936516	+	0.350626i	$0.885969\pi$
$132$		0			0
$133$	15.4969	−	12.8472i	1.34375	−	1.11400i
$134$		0			0
$135$		−	15.9645i		−	1.37401i
$136$		0			0
$137$	19.9925			1.70807			0.854036	−	0.520213i	$-0.174147\pi$
$137$	19.9925			1.70807			0.854036	+	0.520213i	$0.174147\pi$
$138$		0			0
$139$		−	11.4025i		−	0.967151i	−0.875303	−	0.483575i	$-0.839338\pi$
$139$		−	11.4025i		−	0.967151i	0.875303	−	0.483575i	$-0.160662\pi$
$140$		0			0
$141$		−	6.63957i		−	0.559153i
$142$		0			0
$143$	0.460555			0.0385136
$144$		0			0
$145$		−	9.03186i		−	0.750055i
$146$		0			0
$147$	19.8914	+	3.75178i	1.64061	+	0.309441i
$148$		0			0
$149$			18.0784i			1.48104i	0.672034	+	0.740520i	$0.265421\pi$
$149$			18.0784i			1.48104i	−0.672034	+	0.740520i	$0.734579\pi$
$150$		0			0
$151$		−	11.3424i		−	0.923034i	−0.887131	−	0.461517i	$-0.847305\pi$
$151$		−	11.3424i		−	0.923034i	0.887131	−	0.461517i	$-0.152695\pi$
$152$		0			0
$153$			32.0398i			2.59026i
$154$		0			0
$155$	13.4620			1.08129
$156$		0			0
$157$	−6.53188			−0.521300			−0.260650	−	0.965433i	$-0.583937\pi$
$157$	−6.53188			−0.521300			−0.260650	+	0.965433i	$0.583937\pi$
$158$		0			0
$159$	25.4214			2.01604
$160$		0			0
$161$	8.61165	−	7.13922i	0.678693	−	0.562650i
$162$		0			0
$163$	−0.694265			−0.0543790			−0.0271895	−	0.999630i	$-0.508656\pi$
$163$	−0.694265			−0.0543790			−0.0271895	+	0.999630i	$0.508656\pi$
$164$		0			0
$165$		−	3.11272i		−	0.242325i
$166$		0			0
$167$	−24.4164			−1.88940			−0.944698	−	0.327943i	$-0.893645\pi$
$167$	−24.4164			−1.88940			−0.944698	+	0.327943i	$0.893645\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$			40.7966i			3.11979i
$172$		0			0
$173$	−11.0202			−0.837854			−0.418927	−	0.908020i	$-0.637594\pi$
$173$	−11.0202			−0.837854			−0.418927	+	0.908020i	$0.637594\pi$
$174$		0			0
$175$	0.781207	+	0.942327i	0.0590537	+	0.0712332i
$176$		0			0
$177$	21.7599			1.63558
$178$		0			0
$179$	−3.68730			−0.275602			−0.137801	−	0.990460i	$-0.544003\pi$
$179$	−3.68730			−0.275602			−0.137801	+	0.990460i	$0.544003\pi$
$180$		0			0
$181$	−0.0892583			−0.00663452			−0.00331726	−	0.999994i	$-0.501056\pi$
$181$	−0.0892583			−0.00663452			−0.00331726	+	0.999994i	$0.501056\pi$
$182$		0			0
$183$			2.07923i			0.153701i
$184$		0			0
$185$			5.70654i			0.419553i
$186$		0			0
$187$			2.75192i			0.201241i
$188$		0			0
$189$	−13.9127	+	11.5339i	−1.01200	+	0.838967i
$190$		0			0
$191$			6.12726i			0.443353i	0.975120	+	0.221676i	$0.0711528\pi$
$191$			6.12726i			0.443353i	−0.975120	+	0.221676i	$0.928847\pi$
$192$		0			0
$193$	23.5775			1.69715			0.848573	−	0.529078i	$-0.177462\pi$
$193$	23.5775			1.69715			0.848573	+	0.529078i	$0.177462\pi$
$194$		0			0
$195$		−	6.75863i		−	0.483996i
$196$		0			0
$197$		−	19.7177i		−	1.40483i	−0.711769	−	0.702414i	$-0.752106\pi$
$197$		−	19.7177i		−	1.40483i	0.711769	−	0.702414i	$-0.247894\pi$
$198$		0			0
$199$	−11.4515			−0.811773			−0.405886	−	0.913924i	$-0.633037\pi$
$199$	−11.4515			−0.811773			−0.405886	+	0.913924i	$0.633037\pi$
$200$		0			0
$201$			41.5162i			2.92832i
$202$		0			0
$203$	−7.87104	+	6.52524i	−0.552439	+	0.457982i
$204$		0			0
$205$		−	20.5810i		−	1.43744i
$206$		0			0
$207$			22.6707i			1.57572i
$208$		0			0
$209$			3.50405i			0.242380i
$210$		0			0
$211$	−17.9521			−1.23588			−0.617938	−	0.786227i	$-0.712032\pi$
$211$	−17.9521			−1.23588			−0.617938	+	0.786227i	$0.712032\pi$
$212$		0			0
$213$	37.3120			2.55658
$214$		0			0
$215$	−11.8893			−0.810846
$216$		0			0
$217$	−9.72589	−	11.7318i	−0.660236	−	0.796407i
$218$		0			0
$219$	−13.0877			−0.884382
$220$		0			0
$221$			5.97523i			0.401938i
$222$		0			0
$223$	−9.20542			−0.616440			−0.308220	−	0.951315i	$-0.599733\pi$
$223$	−9.20542			−0.616440			−0.308220	+	0.951315i	$0.599733\pi$
$224$		0			0
$225$	−2.48073			−0.165382
$226$		0			0
$227$		−	21.3197i		−	1.41504i	−0.706694	−	0.707520i	$-0.749814\pi$
$227$		−	21.3197i		−	1.41504i	0.706694	−	0.707520i	$-0.250186\pi$
$228$		0			0
$229$	18.3650			1.21359			0.606796	−	0.794858i	$-0.292455\pi$
$229$	18.3650			1.21359			0.606796	+	0.794858i	$0.292455\pi$
$230$		0			0
$231$	−2.71266	+	2.24885i	−0.178480	+	0.147963i
$232$		0			0
$233$	−5.80479			−0.380284			−0.190142	−	0.981757i	$-0.560895\pi$
$233$	−5.80479			−0.380284			−0.190142	+	0.981757i	$0.560895\pi$
$234$		0			0
$235$	−5.36641			−0.350066
$236$		0			0
$237$	−31.2096			−2.02728
$238$		0			0
$239$			22.3151i			1.44344i	0.692184	+	0.721721i	$0.256649\pi$
$239$			22.3151i			1.44344i	−0.692184	+	0.721721i	$0.743351\pi$
$240$		0			0
$241$		−	2.48560i		−	0.160112i	−0.996790	−	0.0800558i	$-0.974490\pi$
$241$		−	2.48560i		−	0.160112i	0.996790	−	0.0800558i	$-0.0255098\pi$
$242$		0			0
$243$			9.89116i			0.634518i
$244$		0			0
$245$	3.03236	−	16.0771i	0.193730	−	1.02713i
$246$		0			0
$247$			7.60833i			0.484106i
$248$		0			0
$249$	17.2758			1.09481
$250$		0			0
$251$			6.05862i			0.382417i	0.981549	+	0.191208i	$0.0612406\pi$
$251$			6.05862i			0.382417i	−0.981549	+	0.191208i	$0.938759\pi$
$252$		0			0
$253$			1.94721i			0.122420i
$254$		0			0
$255$	40.3844			2.52897
$256$		0			0
$257$		−	13.5211i		−	0.843423i	−0.906730	−	0.421712i	$-0.861429\pi$
$257$		−	13.5211i		−	0.843423i	0.906730	−	0.421712i	$-0.138571\pi$
$258$		0			0
$259$	4.97311	−	4.12280i	0.309014	−	0.256178i
$260$		0			0
$261$		−	20.7210i		−	1.28260i
$262$		0			0
$263$			17.7537i			1.09474i	0.836891	+	0.547369i	$0.184371\pi$
$263$			17.7537i			1.09474i	−0.836891	+	0.547369i	$0.815629\pi$
$264$		0			0
$265$		−	20.5467i		−	1.26218i
$266$		0			0
$267$	−8.06537			−0.493593
$268$		0			0
$269$	−19.6335			−1.19708			−0.598539	−	0.801093i	$-0.704252\pi$
$269$	−19.6335			−1.19708			−0.598539	+	0.801093i	$0.704252\pi$
$270$		0			0
$271$	20.4513			1.24233			0.621164	−	0.783681i	$-0.286660\pi$
$271$	20.4513			1.24233			0.621164	+	0.783681i	$0.286660\pi$
$272$		0			0
$273$	−5.88998	+	4.88291i	−0.356478	+	0.295527i
$274$		0			0
$275$	−0.213072			−0.0128487
$276$		0			0
$277$		−	8.40932i		−	0.505267i	−0.967562	−	0.252633i	$-0.918703\pi$
$277$		−	8.40932i		−	0.505267i	0.967562	−	0.252633i	$-0.0812967\pi$
$278$		0			0
$279$	30.8847			1.84902
$280$		0			0
$281$	−3.68325			−0.219724			−0.109862	−	0.993947i	$-0.535041\pi$
$281$	−3.68325			−0.219724			−0.109862	+	0.993947i	$0.535041\pi$
$282$		0			0
$283$		−	18.7946i		−	1.11723i	−0.829428	−	0.558613i	$-0.811334\pi$
$283$		−	18.7946i		−	1.11723i	0.829428	−	0.558613i	$-0.188666\pi$
$284$		0			0
$285$	51.4219			3.04597
$286$		0			0
$287$	−17.9359	+	14.8692i	−1.05872	+	0.877700i
$288$		0			0
$289$	−18.7034			−1.10020
$290$		0			0
$291$	26.7085			1.56568
$292$		0			0
$293$	−1.11257			−0.0649972			−0.0324986	−	0.999472i	$-0.510346\pi$
$293$	−1.11257			−0.0649972			−0.0324986	+	0.999472i	$0.510346\pi$
$294$		0			0
$295$		−	17.5874i		−	1.02398i
$296$		0			0
$297$		−	3.14584i		−	0.182540i
$298$		0			0
$299$			4.22795i			0.244509i
$300$		0			0
$301$	8.58969	+	10.3613i	0.495101	+	0.597214i
$302$		0			0
$303$		−	6.44604i		−	0.370315i
$304$		0			0
$305$	1.68053			0.0962269
$306$		0			0
$307$		−	22.3431i		−	1.27519i	−0.770373	−	0.637594i	$-0.779930\pi$
$307$		−	22.3431i		−	1.27519i	0.770373	−	0.637594i	$-0.220070\pi$
$308$		0			0
$309$			11.8147i			0.672113i
$310$		0			0
$311$	22.2225			1.26012			0.630060	−	0.776546i	$-0.283030\pi$
$311$	22.2225			1.26012			0.630060	+	0.776546i	$0.283030\pi$
$312$		0			0
$313$			31.2412i			1.76586i	0.469507	+	0.882929i	$0.344432\pi$
$313$			31.2412i			1.76586i	−0.469507	+	0.882929i	$0.655568\pi$
$314$		0			0
$315$	21.1620	+	25.5266i	1.19234	+	1.43826i
$316$		0			0
$317$		−	26.9162i		−	1.51177i	−0.654706	−	0.755883i	$-0.727208\pi$
$317$		−	26.9162i		−	1.51177i	0.654706	−	0.755883i	$-0.272792\pi$
$318$		0			0
$319$		−	1.77974i		−	0.0996465i
$320$		0			0
$321$			14.5800i			0.813774i
$322$		0			0
$323$	−45.4615			−2.52955
$324$		0			0
$325$	−0.462642			−0.0256628
$326$		0			0
$327$	28.2167			1.56038
$328$		0			0
$329$	3.87707	+	4.67669i	0.213750	+	0.257834i
$330$		0			0
$331$	−9.05091			−0.497483			−0.248741	−	0.968570i	$-0.580017\pi$
$331$	−9.05091			−0.497483			−0.248741	+	0.968570i	$0.580017\pi$
$332$		0			0
$333$			13.0920i			0.717437i
$334$		0			0
$335$	33.5553			1.83332
$336$		0			0
$337$	23.7041			1.29124			0.645622	−	0.763658i	$-0.276598\pi$
$337$	23.7041			1.29124			0.645622	+	0.763658i	$0.276598\pi$
$338$		0			0
$339$			40.1638i			2.18140i
$340$		0			0
$341$	2.65271			0.143652
$342$		0			0
$343$	−16.2016	+	8.97261i	−0.874805	+	0.484476i
$344$		0			0
$345$	28.5752			1.53844
$346$		0			0
$347$	2.91046			0.156242			0.0781209	−	0.996944i	$-0.475108\pi$
$347$	2.91046			0.156242			0.0781209	+	0.996944i	$0.475108\pi$
$348$		0			0
$349$	−6.80639			−0.364338			−0.182169	−	0.983267i	$-0.558312\pi$
$349$	−6.80639			−0.364338			−0.182169	+	0.983267i	$0.558312\pi$
$350$		0			0
$351$		−	6.83054i		−	0.364587i
$352$		0			0
$353$			9.08139i			0.483354i	0.970357	+	0.241677i	$0.0776974\pi$
$353$			9.08139i			0.483354i	−0.970357	+	0.241677i	$0.922303\pi$
$354$		0			0
$355$		−	30.1573i		−	1.60058i
$356$		0			0
$357$	−29.1765	−	35.1940i	−1.54418	−	1.86266i
$358$		0			0
$359$		−	16.2976i		−	0.860157i	−0.902792	−	0.430078i	$-0.858486\pi$
$359$		−	16.2976i		−	0.860157i	0.902792	−	0.430078i	$-0.141514\pi$
$360$		0			0
$361$	−38.8866			−2.04666
$362$		0			0
$363$			31.1957i			1.63735i
$364$		0			0
$365$			10.5780i			0.553680i
$366$		0			0
$367$	−33.9970			−1.77463			−0.887314	−	0.461166i	$-0.847431\pi$
$367$	−33.9970			−1.77463			−0.887314	+	0.461166i	$0.847431\pi$
$368$		0			0
$369$		−	47.2172i		−	2.45803i
$370$		0			0
$371$	−17.9060	+	14.8444i	−0.929631	+	0.770682i
$372$		0			0
$373$		−	25.0611i		−	1.29762i	−0.760953	−	0.648808i	$-0.775268\pi$
$373$		−	25.0611i		−	1.29762i	0.760953	−	0.648808i	$-0.224732\pi$
$374$		0			0
$375$		−	30.6663i		−	1.58360i
$376$		0			0
$377$		−	3.86435i		−	0.199024i
$378$		0			0
$379$	29.6588			1.52347			0.761736	−	0.647887i	$-0.224347\pi$
$379$	29.6588			1.52347			0.761736	+	0.647887i	$0.224347\pi$
$380$		0			0
$381$	−23.6106			−1.20961
$382$		0			0
$383$	20.0478			1.02439			0.512197	−	0.858868i	$-0.328832\pi$
$383$	20.0478			1.02439			0.512197	+	0.858868i	$0.328832\pi$
$384$		0			0
$385$	1.81762	+	2.19250i	0.0926347	+	0.111740i
$386$		0			0
$387$	−27.2767			−1.38655
$388$		0			0
$389$			18.6383i			0.944999i	0.881331	+	0.472500i	$0.156648\pi$
$389$			18.6383i			0.944999i	−0.881331	+	0.472500i	$0.843352\pi$
$390$		0			0
$391$	−25.2630			−1.27760
$392$		0			0
$393$	23.2096			1.17077
$394$		0			0
$395$			25.2250i			1.26921i
$396$		0			0
$397$	−11.1794			−0.561079			−0.280539	−	0.959842i	$-0.590513\pi$
$397$	−11.1794			−0.561079			−0.280539	+	0.959842i	$0.590513\pi$
$398$		0			0
$399$	−37.1507	−	44.8129i	−1.85986	−	2.24345i
$400$		0			0
$401$	−15.4601			−0.772041			−0.386020	−	0.922490i	$-0.626151\pi$
$401$	−15.4601			−0.772041			−0.386020	+	0.922490i	$0.626151\pi$
$402$		0			0
$403$	5.75982			0.286917
$404$		0			0
$405$	−8.56779			−0.425737
$406$		0			0
$407$			1.12448i			0.0557385i
$408$		0			0
$409$		−	36.8136i		−	1.82032i	−0.414262	−	0.910158i	$-0.635960\pi$
$409$		−	36.8136i		−	1.82032i	0.414262	−	0.910158i	$-0.364040\pi$
$410$		0			0
$411$		−	57.8128i		−	2.85170i
$412$		0			0
$413$	−15.3270	+	12.7063i	−0.754191	+	0.625238i
$414$		0			0
$415$		−	13.9631i		−	0.685421i
$416$		0			0
$417$	−32.9731			−1.61470
$418$		0			0
$419$		−	27.2989i		−	1.33364i	−0.745220	−	0.666819i	$-0.767656\pi$
$419$		−	27.2989i		−	1.33364i	0.745220	−	0.666819i	$-0.232344\pi$
$420$		0			0
$421$			30.8692i			1.50447i	0.658893	+	0.752237i	$0.271025\pi$
$421$			30.8692i			1.50447i	−0.658893	+	0.752237i	$0.728975\pi$
$422$		0			0
$423$	−12.3117			−0.598614
$424$		0			0
$425$		−	2.76439i		−	0.134093i
$426$		0			0
$427$	−1.21413	−	1.46454i	−0.0587559	−	0.0708741i
$428$		0			0
$429$		−	1.33180i		−	0.0642999i
$430$		0			0
$431$			33.7399i			1.62519i	0.582827	+	0.812596i	$0.301946\pi$
$431$			33.7399i			1.62519i	−0.582827	+	0.812596i	$0.698054\pi$
$432$		0			0
$433$		−	9.29375i		−	0.446629i	−0.974746	−	0.223315i	$-0.928312\pi$
$433$		−	9.29375i		−	0.446629i	0.974746	−	0.223315i	$-0.0716877\pi$
$434$		0			0
$435$	−26.1177			−1.25225
$436$		0			0
$437$	−32.1676			−1.53879
$438$		0			0
$439$	−19.5689			−0.933975			−0.466987	−	0.884264i	$-0.654661\pi$
$439$	−19.5689			−0.933975			−0.466987	+	0.884264i	$0.654661\pi$
$440$		0			0
$441$	6.95687	−	36.8843i	0.331279	−	1.75640i
$442$		0			0
$443$	11.3511			0.539305			0.269653	−	0.962958i	$-0.413091\pi$
$443$	11.3511			0.539305			0.269653	+	0.962958i	$0.413091\pi$
$444$		0			0
$445$			6.51881i			0.309021i
$446$		0			0
$447$	52.2779			2.47266
$448$		0			0
$449$	−4.08652			−0.192855			−0.0964273	−	0.995340i	$-0.530742\pi$
$449$	−4.08652			−0.192855			−0.0964273	+	0.995340i	$0.530742\pi$
$450$		0			0
$451$		−	4.05553i		−	0.190967i
$452$		0			0
$453$	−32.7992			−1.54104
$454$		0			0
$455$	3.94659	+	4.76056i	0.185019	+	0.223178i
$456$		0			0
$457$	−22.4429			−1.04984			−0.524918	−	0.851153i	$-0.675904\pi$
$457$	−22.4429			−1.04984			−0.524918	+	0.851153i	$0.675904\pi$
$458$		0			0
$459$	40.8141			1.90504
$460$		0			0
$461$	−20.7512			−0.966480			−0.483240	−	0.875488i	$-0.660540\pi$
$461$	−20.7512			−0.966480			−0.483240	+	0.875488i	$0.660540\pi$
$462$		0			0
$463$			27.6171i			1.28347i	0.766925	+	0.641737i	$0.221786\pi$
$463$			27.6171i			1.28347i	−0.766925	+	0.641737i	$0.778214\pi$
$464$		0			0
$465$		−	38.9285i		−	1.80527i
$466$		0			0
$467$			2.28693i			0.105827i	0.998599	+	0.0529133i	$0.0168507\pi$
$467$			2.28693i			0.105827i	−0.998599	+	0.0529133i	$0.983149\pi$
$468$		0			0
$469$	−24.2427	−	29.2426i	−1.11942	−	1.35030i
$470$		0			0
$471$			18.8884i			0.870332i
$472$		0			0
$473$	−2.34282			−0.107723
$474$		0			0
$475$		−	3.51993i		−	0.161506i
$476$		0			0
$477$		−	47.1385i		−	2.15832i
$478$		0			0
$479$	−37.3165			−1.70503			−0.852517	−	0.522699i	$-0.824925\pi$
$479$	−37.3165			−1.70503			−0.852517	+	0.522699i	$0.824925\pi$
$480$		0			0
$481$			2.44158i			0.111327i
$482$		0			0
$483$	−20.6447	−	24.9026i	−0.939366	−	1.13311i
$484$		0			0
$485$		−	21.5871i		−	0.980219i
$486$		0			0
$487$		−	0.772784i		−	0.0350182i	−0.999847	−	0.0175091i	$-0.994426\pi$
$487$		−	0.772784i		−	0.0350182i	0.999847	−	0.0175091i	$-0.00557360\pi$
$488$		0			0
$489$			2.00763i			0.0907880i
$490$		0			0
$491$	−10.9597			−0.494603			−0.247302	−	0.968939i	$-0.579544\pi$
$491$	−10.9597			−0.494603			−0.247302	+	0.968939i	$0.579544\pi$
$492$		0			0
$493$	23.0904			1.03994
$494$		0			0
$495$	−5.77188			−0.259427
$496$		0			0
$497$	−26.2814	+	21.7877i	−1.17888	+	0.977314i
$498$		0			0
$499$	2.70238			0.120975			0.0604876	−	0.998169i	$-0.480734\pi$
$499$	2.70238			0.120975			0.0604876	+	0.998169i	$0.480734\pi$
$500$		0			0
$501$			70.6055i			3.15442i
$502$		0			0
$503$	−9.31407			−0.415294			−0.207647	−	0.978204i	$-0.566581\pi$
$503$	−9.31407			−0.415294			−0.207647	+	0.978204i	$0.566581\pi$
$504$		0			0
$505$	−5.20999			−0.231841
$506$		0			0
$507$		−	2.89173i		−	0.128426i
$508$		0			0
$509$	−15.2303			−0.675069			−0.337535	−	0.941313i	$-0.609593\pi$
$509$	−15.2303			−0.675069			−0.337535	+	0.941313i	$0.609593\pi$
$510$		0			0
$511$	9.21851	−	7.64232i	0.407803	−	0.338076i
$512$		0			0
$513$	51.9690			2.29449
$514$		0			0
$515$	9.54917			0.420787
$516$		0			0
$517$	−1.05746			−0.0465070
$518$		0			0
$519$			31.8676i			1.39883i
$520$		0			0
$521$		−	20.8460i		−	0.913281i	−0.889651	−	0.456640i	$-0.849053\pi$
$521$		−	20.8460i		−	0.913281i	0.889651	−	0.456640i	$-0.150947\pi$
$522$		0			0
$523$			13.6269i			0.595861i	0.954588	+	0.297931i	$0.0962964\pi$
$523$			13.6269i			0.595861i	−0.954588	+	0.297931i	$0.903704\pi$
$524$		0			0
$525$	2.72495	−	2.25904i	0.118927	−	0.0985925i
$526$		0			0
$527$			34.4162i			1.49919i
$528$		0			0
$529$	5.12442			0.222801
$530$		0			0
$531$		−	40.3492i		−	1.75100i
$532$		0			0
$533$		−	8.80574i		−	0.381419i
$534$		0			0
$535$	11.7842			0.509475
$536$		0			0
$537$			10.6627i			0.460129i
$538$		0			0
$539$	0.597532	−	3.16803i	0.0257375	−	0.136457i
$540$		0			0
$541$			23.8219i			1.02418i	0.858931	+	0.512092i	$0.171129\pi$
$541$			23.8219i			1.02418i	−0.858931	+	0.512092i	$0.828871\pi$
$542$		0			0
$543$			0.258111i			0.0110766i
$544$		0			0
$545$		−	22.8060i		−	0.976902i
$546$		0			0
$547$	44.7381			1.91286			0.956432	−	0.291956i	$-0.0943061\pi$
$547$	44.7381			1.91286			0.956432	+	0.291956i	$0.0943061\pi$
$548$		0			0
$549$	3.85549			0.164548
$550$		0			0
$551$	29.4012			1.25253
$552$		0			0
$553$	21.9830	−	18.2243i	0.934811	−	0.774976i
$554$		0			0
$555$	16.5018			0.700461
$556$		0			0
$557$		−	29.8939i		−	1.26665i	−0.773888	−	0.633323i	$-0.781691\pi$
$557$		−	29.8939i		−	1.26665i	0.773888	−	0.633323i	$-0.218309\pi$
$558$		0			0
$559$	−5.08694			−0.215155
$560$		0			0
$561$	7.95782			0.335979
$562$		0			0
$563$		−	17.2575i		−	0.727318i	−0.931532	−	0.363659i	$-0.881527\pi$
$563$		−	17.2575i		−	0.727318i	0.931532	−	0.363659i	$-0.118473\pi$
$564$		0			0
$565$	32.4622			1.36570
$566$		0			0
$567$	6.18996	+	7.46661i	0.259954	+	0.313568i
$568$		0			0
$569$	−3.10404			−0.130128			−0.0650641	−	0.997881i	$-0.520725\pi$
$569$	−3.10404			−0.130128			−0.0650641	+	0.997881i	$0.520725\pi$
$570$		0			0
$571$	−5.07780			−0.212499			−0.106250	−	0.994339i	$-0.533884\pi$
$571$	−5.07780			−0.212499			−0.106250	+	0.994339i	$0.533884\pi$
$572$		0			0
$573$	17.7184			0.740195
$574$		0			0
$575$		−	1.95603i		−	0.0815720i
$576$		0			0
$577$		−	12.0624i		−	0.502166i	−0.967966	−	0.251083i	$-0.919213\pi$
$577$		−	12.0624i		−	0.502166i	0.967966	−	0.251083i	$-0.0807867\pi$
$578$		0			0
$579$		−	68.1798i		−	2.83345i
$580$		0			0
$581$	−12.1685	+	10.0879i	−0.504834	+	0.418517i
$582$		0			0
$583$		−	4.04877i		−	0.167683i
$584$		0			0
$585$	−12.5325			−0.518153
$586$		0			0
$587$		−	14.4899i		−	0.598063i	−0.954243	−	0.299032i	$-0.903336\pi$
$587$		−	14.4899i		−	0.598063i	0.954243	−	0.299032i	$-0.0966636\pi$
$588$		0			0
$589$			43.8225i			1.80568i
$590$		0			0
$591$	−57.0182			−2.34542
$592$		0			0
$593$		−	30.7710i		−	1.26361i	−0.775126	−	0.631807i	$-0.782314\pi$
$593$		−	30.7710i		−	1.26361i	0.775126	−	0.631807i	$-0.217686\pi$
$594$		0			0
$595$	−28.4454	+	23.5818i	−1.16615	+	0.966760i
$596$		0			0
$597$			33.1145i			1.35529i
$598$		0			0
$599$		−	14.9116i		−	0.609271i	−0.952469	−	0.304635i	$-0.901465\pi$
$599$		−	14.9116i		−	0.609271i	0.952469	−	0.304635i	$-0.0985346\pi$
$600$		0			0
$601$			8.78840i			0.358486i	0.983805	+	0.179243i	$0.0573649\pi$
$601$			8.78840i			0.358486i	−0.983805	+	0.179243i	$0.942635\pi$
$602$		0			0
$603$	76.9829			3.13499
$604$		0			0
$605$	25.2138			1.02509
$606$		0			0
$607$	−33.8093			−1.37228			−0.686140	−	0.727470i	$-0.740696\pi$
$607$	−33.8093			−1.37228			−0.686140	+	0.727470i	$0.740696\pi$
$608$		0			0
$609$	18.8692	+	22.7609i	0.764620	+	0.922319i
$610$		0			0
$611$	−2.29606			−0.0928884
$612$		0			0
$613$		−	19.9650i		−	0.806380i	−0.915116	−	0.403190i	$-0.867901\pi$
$613$		−	19.9650i		−	0.806380i	0.915116	−	0.403190i	$-0.132099\pi$
$614$		0			0
$615$	−59.5148			−2.39987
$616$		0			0
$617$	−0.222709			−0.00896592			−0.00448296	−	0.999990i	$-0.501427\pi$
$617$	−0.222709			−0.00896592			−0.00448296	+	0.999990i	$0.501427\pi$
$618$		0			0
$619$		−	13.1069i		−	0.526809i	−0.964685	−	0.263405i	$-0.915155\pi$
$619$		−	13.1069i		−	0.526809i	0.964685	−	0.263405i	$-0.0848454\pi$
$620$		0			0
$621$	28.8792			1.15888
$622$		0			0
$623$	5.68098	−	4.70964i	0.227604	−	0.188688i
$624$		0			0
$625$	−27.0992			−1.08397
$626$		0			0
$627$	10.1328			0.404664
$628$		0			0
$629$	−14.5890			−0.581702
$630$		0			0
$631$			0.405159i			0.0161291i	0.999967	+	0.00806457i	$0.00256706\pi$
$631$			0.405159i			0.0161291i	−0.999967	+	0.00806457i	$0.997433\pi$
$632$		0			0
$633$			51.9127i			2.06335i
$634$		0			0
$635$			19.0832i			0.757293i
$636$		0			0
$637$	1.29742	−	6.87871i	0.0514055	−	0.272545i
$638$		0			0
$639$		−	69.1873i		−	2.73701i
$640$		0			0
$641$	30.4895			1.20426			0.602131	−	0.798397i	$-0.294318\pi$
$641$	30.4895			1.20426			0.602131	+	0.798397i	$0.294318\pi$
$642$		0			0
$643$			10.3645i			0.408735i	0.978894	+	0.204367i	$0.0655137\pi$
$643$			10.3645i			0.408735i	−0.978894	+	0.204367i	$0.934486\pi$
$644$		0			0
$645$			34.3808i			1.35374i
$646$		0			0
$647$	3.23095			0.127022			0.0635108	−	0.997981i	$-0.479770\pi$
$647$	3.23095			0.127022			0.0635108	+	0.997981i	$0.479770\pi$
$648$		0			0
$649$		−	3.46562i		−	0.136038i
$650$		0			0
$651$	−33.9252	+	28.1246i	−1.32963	+	1.10229i
$652$		0			0
$653$			18.0625i			0.706841i	0.935465	+	0.353420i	$0.114982\pi$
$653$			18.0625i			0.706841i	−0.935465	+	0.353420i	$0.885018\pi$
$654$		0			0
$655$		−	18.7590i		−	0.732976i
$656$		0			0
$657$			24.2683i			0.946795i
$658$		0			0
$659$	−20.1619			−0.785394			−0.392697	−	0.919668i	$-0.628458\pi$
$659$	−20.1619			−0.785394			−0.392697	+	0.919668i	$0.628458\pi$
$660$		0			0
$661$	−4.75414			−0.184915			−0.0924573	−	0.995717i	$-0.529472\pi$
$661$	−4.75414			−0.184915			−0.0924573	+	0.995717i	$0.529472\pi$
$662$		0			0
$663$	17.2788			0.671051
$664$		0			0
$665$	−36.2199	+	30.0270i	−1.40455	+	1.16440i
$666$		0			0
$667$	16.3383			0.632620
$668$		0			0
$669$			26.6196i			1.02917i
$670$		0			0
$671$	0.331151			0.0127840
$672$		0			0
$673$	36.7876			1.41806			0.709029	−	0.705180i	$-0.249134\pi$
$673$	36.7876			1.41806			0.709029	+	0.705180i	$0.249134\pi$
$674$		0			0
$675$			3.16010i			0.121632i
$676$		0			0
$677$	−12.4785			−0.479586			−0.239793	−	0.970824i	$-0.577080\pi$
$677$	−12.4785			−0.479586			−0.239793	+	0.970824i	$0.577080\pi$
$678$		0			0
$679$	−18.8126	+	15.5960i	−0.721961	+	0.598520i
$680$		0			0
$681$	−61.6508			−2.36247
$682$		0			0
$683$	−9.99053			−0.382277			−0.191138	−	0.981563i	$-0.561218\pi$
$683$	−9.99053			−0.382277			−0.191138	+	0.981563i	$0.561218\pi$
$684$		0			0
$685$	−46.7270			−1.78535
$686$		0			0
$687$		−	53.1065i		−	2.02614i
$688$		0			0
$689$		−	8.79106i		−	0.334913i
$690$		0			0
$691$		−	46.3271i		−	1.76236i	−0.472777	−	0.881182i	$-0.656748\pi$
$691$		−	46.3271i		−	1.76236i	0.472777	−	0.881182i	$-0.343252\pi$
$692$		0			0
$693$	4.17001	+	5.03005i	0.158406	+	0.191076i
$694$		0			0
$695$			26.6504i			1.01091i
$696$		0			0
$697$	52.6164			1.99299
$698$		0			0
$699$			16.7859i			0.634900i
$700$		0			0
$701$		−	13.7321i		−	0.518654i	−0.965790	−	0.259327i	$-0.916499\pi$
$701$		−	13.7321i		−	0.518654i	0.965790	−	0.259327i	$-0.0835007\pi$
$702$		0			0
$703$	−18.5763			−0.700620
$704$		0			0
$705$			15.5182i			0.584449i
$706$		0			0
$707$	3.76406	+	4.54038i	0.141562	+	0.170758i
$708$		0			0
$709$		−	3.41445i		−	0.128232i	−0.997942	−	0.0641162i	$-0.979577\pi$
$709$		−	3.41445i		−	0.128232i	0.997942	−	0.0641162i	$-0.0204228\pi$
$710$		0			0
$711$			57.8715i			2.17035i
$712$		0			0
$713$			24.3522i			0.911998i
$714$		0			0
$715$	−1.07642			−0.0402559
$716$		0			0
$717$	64.5291			2.40988
$718$		0			0
$719$	30.3286			1.13107			0.565534	−	0.824725i	$-0.308670\pi$
$719$	30.3286			1.13107			0.565534	+	0.824725i	$0.308670\pi$
$720$		0			0
$721$	−6.89898	−	8.32187i	−0.256932	−	0.309922i
$722$		0			0
$723$	−7.18768			−0.267313
$724$		0			0
$725$			1.78781i			0.0663976i
$726$		0			0
$727$	6.20103			0.229983			0.114992	−	0.993366i	$-0.463316\pi$
$727$	6.20103			0.229983			0.114992	+	0.993366i	$0.463316\pi$
$728$		0			0
$729$	39.5999			1.46666
$730$		0			0
$731$		−	30.3956i		−	1.12422i
$732$		0			0
$733$	27.5049			1.01592			0.507958	−	0.861382i	$-0.330401\pi$
$733$	27.5049			1.01592			0.507958	+	0.861382i	$0.330401\pi$
$734$		0			0
$735$	−46.4907	−	8.76876i	−1.71484	−	0.323441i
$736$		0			0
$737$	6.61213			0.243561
$738$		0			0
$739$	14.2957			0.525877			0.262939	−	0.964813i	$-0.415308\pi$
$739$	14.2957			0.525877			0.262939	+	0.964813i	$0.415308\pi$
$740$		0			0
$741$	22.0012			0.808235
$742$		0			0
$743$		−	4.77889i		−	0.175320i	−0.996150	−	0.0876602i	$-0.972061\pi$
$743$		−	4.77889i		−	0.175320i	0.996150	−	0.0876602i	$-0.0279390\pi$
$744$		0			0
$745$		−	42.2534i		−	1.54804i
$746$		0			0
$747$		−	32.0343i		−	1.17207i
$748$		0			0
$749$	−8.51372	−	10.2696i	−0.311085	−	0.375244i
$750$		0			0
$751$		−	37.9486i		−	1.38476i	−0.721531	−	0.692382i	$-0.756561\pi$
$751$		−	37.9486i		−	1.38476i	0.721531	−	0.692382i	$-0.243439\pi$
$752$		0			0
$753$	17.5199			0.638460
$754$		0			0
$755$			26.5099i			0.964793i
$756$		0			0
$757$			20.1135i			0.731038i	0.930804	+	0.365519i	$0.119108\pi$
$757$			20.1135i			0.731038i	−0.930804	+	0.365519i	$0.880892\pi$
$758$		0			0
$759$	5.63079			0.204385
$760$		0			0
$761$		−	9.87284i		−	0.357890i	−0.983859	−	0.178945i	$-0.942732\pi$
$761$		−	9.87284i		−	0.357890i	0.983859	−	0.178945i	$-0.0572685\pi$
$762$		0			0
$763$	−19.8749	+	16.4766i	−0.719519	+	0.596495i
$764$		0			0
$765$		−	74.8843i		−	2.70745i
$766$		0			0
$767$		−	7.52488i		−	0.271708i
$768$		0			0
$769$			25.4391i			0.917358i	0.888602	+	0.458679i	$0.151677\pi$
$769$			25.4391i			0.917358i	−0.888602	+	0.458679i	$0.848323\pi$
$770$		0			0
$771$	−39.0994			−1.40813
$772$		0			0
$773$	46.2125			1.66215			0.831073	−	0.556163i	$-0.187727\pi$
$773$	46.2125			1.66215			0.831073	+	0.556163i	$0.187727\pi$
$774$		0			0
$775$	−2.66473			−0.0957200
$776$		0			0
$777$	−11.9220	−	14.3809i	−0.427700	−	0.515911i
$778$		0			0
$779$	66.9970			2.40042
$780$		0			0
$781$		−	5.94255i		−	0.212641i
$782$		0			0
$783$	−26.3956			−0.943301
$784$		0			0
$785$	15.2665			0.544884
$786$		0			0
$787$			16.0367i			0.571647i	0.958282	+	0.285823i	$0.0922671\pi$
$787$			16.0367i			0.571647i	−0.958282	+	0.285823i	$0.907733\pi$
$788$		0			0
$789$	51.3388			1.82771
$790$		0			0
$791$	−23.4530	−	28.2900i	−0.833891	−	1.00588i
$792$		0			0
$793$	0.719026			0.0255334
$794$		0			0
$795$	−59.4156			−2.10725
$796$		0			0
$797$	12.9270			0.457896			0.228948	−	0.973439i	$-0.426471\pi$
$797$	12.9270			0.457896			0.228948	+	0.973439i	$0.426471\pi$
$798$		0			0
$799$		−	13.7195i		−	0.485360i
$800$		0			0
$801$			14.9555i			0.528427i
$802$		0			0
$803$			2.08442i			0.0735577i
$804$		0			0
$805$	−20.1274	+	16.6860i	−0.709398	+	0.588104i
$806$		0			0
$807$			56.7749i			1.99857i
$808$		0			0
$809$	−16.2943			−0.572876			−0.286438	−	0.958099i	$-0.592471\pi$
$809$	−16.2943			−0.572876			−0.286438	+	0.958099i	$0.592471\pi$
$810$		0			0
$811$		−	30.8923i		−	1.08477i	−0.840129	−	0.542387i	$-0.817521\pi$
$811$		−	30.8923i		−	1.08477i	0.840129	−	0.542387i	$-0.182479\pi$
$812$		0			0
$813$		−	59.1396i		−	2.07412i
$814$		0			0
$815$	1.62266			0.0568392
$816$		0			0
$817$		−	38.7031i		−	1.35405i
$818$		0			0
$819$	9.05431	+	10.9217i	0.316383	+	0.381636i
$820$		0			0
$821$		−	7.06460i		−	0.246556i	−0.992372	−	0.123278i	$-0.960659\pi$
$821$		−	7.06460i		−	0.246556i	0.992372	−	0.123278i	$-0.0393407\pi$
$822$		0			0
$823$		−	17.2933i		−	0.602806i	−0.953497	−	0.301403i	$-0.902545\pi$
$823$		−	17.2933i		−	0.602806i	0.953497	−	0.301403i	$-0.0974549\pi$
$824$		0			0
$825$			0.616147i			0.0214515i
$826$		0			0
$827$	−46.0931			−1.60282			−0.801408	−	0.598119i	$-0.795915\pi$
$827$	−46.0931			−1.60282			−0.801408	+	0.598119i	$0.795915\pi$
$828$		0			0
$829$	−54.7609			−1.90192			−0.950962	−	0.309307i	$-0.899903\pi$
$829$	−54.7609			−1.90192			−0.950962	+	0.309307i	$0.899903\pi$
$830$		0			0
$831$	−24.3175			−0.843564
$832$		0			0
$833$	41.1019	+	7.75236i	1.42410	+	0.268603i
$834$		0			0
$835$	57.0667			1.97487
$836$		0			0
$837$		−	39.3427i		−	1.35988i
$838$		0			0
$839$	28.0831			0.969538			0.484769	−	0.874642i	$-0.338904\pi$
$839$	28.0831			0.969538			0.484769	+	0.874642i	$0.338904\pi$
$840$		0			0
$841$	14.0668			0.485063
$842$		0			0
$843$			10.6510i			0.366838i
$844$		0			0
$845$	−2.33723			−0.0804031
$846$		0			0
$847$	−18.2162	−	21.9732i	−0.625915	−	0.755007i
$848$		0			0
$849$	−54.3490			−1.86525
$850$		0			0
$851$	−10.3229			−0.353864
$852$		0			0
$853$	16.1139			0.551730			0.275865	−	0.961196i	$-0.411036\pi$
$853$	16.1139			0.551730			0.275865	+	0.961196i	$0.411036\pi$
$854$		0			0
$855$		−	95.3510i		−	3.26093i
$856$		0			0
$857$			9.70825i			0.331628i	0.986157	+	0.165814i	$0.0530251\pi$
$857$			9.70825i			0.331628i	−0.986157	+	0.165814i	$0.946975\pi$
$858$		0			0
$859$		−	21.0569i		−	0.718452i	−0.933251	−	0.359226i	$-0.883041\pi$
$859$		−	21.0569i		−	0.718452i	0.933251	−	0.359226i	$-0.116959\pi$
$860$		0			0
$861$	42.9976	+	51.8657i	1.46536	+	1.76758i
$862$		0			0
$863$		−	8.82880i		−	0.300536i	−0.988645	−	0.150268i	$-0.951986\pi$
$863$		−	8.82880i		−	0.300536i	0.988645	−	0.150268i	$-0.0480136\pi$
$864$		0			0
$865$	25.7568			0.875759
$866$		0			0
$867$			54.0852i			1.83683i
$868$		0			0
$869$			4.97063i			0.168617i
$870$		0			0
$871$	14.3569			0.486464
$872$		0			0
$873$		−	49.5253i		−	1.67618i
$874$		0			0
$875$	17.9071	+	21.6004i	0.605370	+	0.730225i
$876$		0			0
$877$			47.6620i			1.60943i	0.593659	+	0.804716i	$0.297683\pi$
$877$			47.6620i			1.60943i	−0.593659	+	0.804716i	$0.702317\pi$
$878$		0			0
$879$			3.21726i			0.108516i
$880$		0			0
$881$			9.48773i			0.319650i	0.987145	+	0.159825i	$0.0510929\pi$
$881$			9.48773i			0.319650i	−0.987145	+	0.159825i	$0.948907\pi$
$882$		0			0
$883$	41.1681			1.38542			0.692709	−	0.721217i	$-0.256417\pi$
$883$	41.1681			1.38542			0.692709	+	0.721217i	$0.256417\pi$
$884$		0			0
$885$	−50.8579			−1.70957
$886$		0			0
$887$	−16.6945			−0.560545			−0.280273	−	0.959920i	$-0.590425\pi$
$887$	−16.6945			−0.560545			−0.280273	+	0.959920i	$0.590425\pi$
$888$		0			0
$889$	16.6305	−	13.7870i	0.557770	−	0.462402i
$890$		0			0
$891$	−1.68830			−0.0565601
$892$		0			0
$893$		−	17.4691i		−	0.584582i
$894$		0			0
$895$	8.61808			0.288070
$896$		0			0
$897$	12.2261			0.408217
$898$		0			0
$899$		−	22.2579i		−	0.742343i
$900$		0			0
$901$	52.5286			1.74998
$902$		0			0
$903$	29.9620	−	24.8390i	0.997072	−	0.826592i
$904$		0			0
$905$	0.208617			0.00693467
$906$		0			0
$907$	−36.0835			−1.19813			−0.599066	−	0.800700i	$-0.704461\pi$
$907$	−36.0835			−1.19813			−0.599066	+	0.800700i	$0.704461\pi$
$908$		0			0
$909$	−11.9528			−0.396450
$910$		0			0
$911$		−	29.0669i		−	0.963031i	−0.876438	−	0.481515i	$-0.840087\pi$
$911$		−	29.0669i		−	0.963031i	0.876438	−	0.481515i	$-0.159913\pi$
$912$		0			0
$913$		−	2.75145i		−	0.0910598i
$914$		0			0
$915$		−	4.85964i		−	0.160655i
$916$		0			0
$917$	−16.3480	+	13.5528i	−0.539860	+	0.447554i
$918$		0			0
$919$			33.6118i			1.10875i	0.832266	+	0.554376i	$0.187043\pi$
$919$			33.6118i			1.10875i	−0.832266	+	0.554376i	$0.812957\pi$
$920$		0			0
$921$	−64.6102			−2.12898
$922$		0			0
$923$		−	12.9030i		−	0.424708i
$924$		0			0
$925$		−	1.12958i		−	0.0371403i
$926$		0			0
$927$	21.9078			0.719547
$928$		0			0
$929$		−	23.6764i		−	0.776796i	−0.921492	−	0.388398i	$-0.873029\pi$
$929$		−	23.6764i		−	0.776796i	0.921492	−	0.388398i	$-0.126971\pi$
$930$		0			0
$931$	52.3355	+	9.87116i	1.71523	+	0.323514i
$932$		0			0
$933$		−	64.2613i		−	2.10382i
$934$		0			0
$935$		−	6.43188i		−	0.210345i
$936$		0			0
$937$		−	22.0243i		−	0.719503i	−0.933048	−	0.359751i	$-0.882862\pi$
$937$		−	22.0243i		−	0.719503i	0.933048	−	0.359751i	$-0.117138\pi$
$938$		0			0
$939$	90.3411			2.94817
$940$		0			0
$941$	33.6076			1.09558			0.547788	−	0.836617i	$-0.315470\pi$
$941$	33.6076			1.09558			0.547788	+	0.836617i	$0.315470\pi$
$942$		0			0
$943$	37.2303			1.21238
$944$		0			0
$945$	32.5172	−	26.9574i	1.05778	−	0.876923i
$946$		0			0
$947$	25.3497			0.823755			0.411877	−	0.911239i	$-0.364873\pi$
$947$	25.3497			0.823755			0.411877	+	0.911239i	$0.364873\pi$
$948$		0			0
$949$			4.52589i			0.146917i
$950$		0			0
$951$	−77.8345			−2.52396
$952$		0			0
$953$	−39.7485			−1.28758			−0.643790	−	0.765203i	$-0.722639\pi$
$953$	−39.7485			−1.28758			−0.643790	+	0.765203i	$0.722639\pi$
$954$		0			0
$955$		−	14.3208i		−	0.463410i
$956$		0			0
$957$	−5.14654			−0.166364
$958$		0			0
$959$	33.7588	+	40.7214i	1.09013	+	1.31496i
$960$		0			0
$961$	2.17547			0.0701764
$962$		0			0
$963$	27.0354			0.871204
$964$		0			0
$965$	−55.1060			−1.77393
$966$		0			0
$967$		−	6.79608i		−	0.218547i	−0.994012	−	0.109274i	$-0.965148\pi$
$967$		−	6.79608i		−	0.218547i	0.994012	−	0.109274i	$-0.0348525\pi$
$968$		0			0
$969$			131.462i			4.22318i
$970$		0			0
$971$			15.5617i			0.499399i	0.968323	+	0.249699i	$0.0803318\pi$
$971$			15.5617i			0.499399i	−0.968323	+	0.249699i	$0.919668\pi$
$972$		0			0
$973$	23.2251	−	19.2541i	0.744563	−	0.617257i
$974$		0			0
$975$			1.33784i			0.0428450i
$976$		0			0
$977$	−33.7979			−1.08129			−0.540645	−	0.841251i	$-0.681820\pi$
$977$	−33.7979			−1.08129			−0.540645	+	0.841251i	$0.681820\pi$
$978$		0			0
$979$			1.28454i			0.0410542i
$980$		0			0
$981$		−	52.3218i		−	1.67051i
$982$		0			0
$983$	20.6785			0.659543			0.329771	−	0.944061i	$-0.393028\pi$
$983$	20.6785			0.659543			0.329771	+	0.944061i	$0.393028\pi$
$984$		0			0
$985$			46.0848i			1.46838i
$986$		0			0
$987$	13.5237	−	11.2114i	0.430465	−	0.356864i
$988$		0			0
$989$		−	21.5073i		−	0.683894i
$990$		0			0
$991$			18.6250i			0.591642i	0.955243	+	0.295821i	$0.0955933\pi$
$991$			18.6250i			0.591642i	−0.955243	+	0.295821i	$0.904407\pi$
$992$		0			0
$993$			26.1728i			0.830568i
$994$		0			0
$995$	26.7647			0.848498
$996$		0			0
$997$	6.17692			0.195625			0.0978125	−	0.995205i	$-0.468815\pi$
$997$	6.17692			0.195625			0.0978125	+	0.995205i	$0.468815\pi$
$998$		0			0
$999$	16.6773			0.527647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.a.2575.5		48
4.3	odd	2		728.2.h.a.27.40	yes	48
7.6	odd	2		2912.2.h.b.2575.44		48
8.3	odd	2		2912.2.h.b.2575.5		48
8.5	even	2		728.2.h.b.27.39	yes	48
28.27	even	2		728.2.h.b.27.40	yes	48
56.13	odd	2		728.2.h.a.27.39	✓	48
56.27	even	2	inner	2912.2.h.a.2575.44		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.39	✓	48	56.13	odd	2
728.2.h.a.27.40	yes	48	4.3	odd	2
728.2.h.b.27.39	yes	48	8.5	even	2
728.2.h.b.27.40	yes	48	28.27	even	2
2912.2.h.a.2575.5		48	1.1	even	1	trivial
2912.2.h.a.2575.44		48	56.27	even	2	inner
2912.2.h.b.2575.5		48	8.3	odd	2
2912.2.h.b.2575.44		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.89173i q^{3} -2.33723 q^{5} +(1.68858 + 2.03684i) q^{7} -5.36210 q^{9} -0.460555 q^{11} -1.00000 q^{13} +6.75863i q^{15} -5.97523i q^{17} -7.60833i q^{19} +(5.88998 - 4.88291i) q^{21} -4.22795i q^{23} +0.462642 q^{25} +6.83054i q^{27} +3.86435i q^{29} -5.75982 q^{31} +1.33180i q^{33} +(-3.94659 - 4.76056i) q^{35} -2.44158i q^{37} +2.89173i q^{39} +8.80574i q^{41} +5.08694 q^{43} +12.5325 q^{45} +2.29606 q^{47} +(-1.29742 + 6.87871i) q^{49} -17.2788 q^{51} +8.79106i q^{53} +1.07642 q^{55} -22.0012 q^{57} +7.52488i q^{59} -0.719026 q^{61} +(-9.05431 - 10.9217i) q^{63} +2.33723 q^{65} -14.3569 q^{67} -12.2261 q^{69} +12.9030i q^{71} -4.52589i q^{73} -1.33784i q^{75} +(-0.777683 - 0.938076i) q^{77} -10.7927i q^{79} +3.66579 q^{81} +5.97421i q^{83} +13.9655i q^{85} +11.1746 q^{87} -2.78912i q^{89} +(-1.68858 - 2.03684i) q^{91} +16.6558i q^{93} +17.7824i q^{95} +9.23618i q^{97} +2.46954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) \) 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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