LMFDB - Embedded newform 2912.2.h.a.2575.11

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)

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")

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);

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			2575.11
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.a.2575.38

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.92788i q^{3} -2.16680 q^{5} +(-1.74001 + 1.99308i) q^{7} -0.716713 q^{9} +5.04203 q^{11} -1.00000 q^{13} +4.17733i q^{15} +0.687834i q^{17} +1.09617i q^{19} +(3.84242 + 3.35452i) q^{21} +3.80258i q^{23} -0.304972 q^{25} -4.40190i q^{27} -6.62535i q^{29} -8.65921 q^{31} -9.72042i q^{33} +(3.77025 - 4.31861i) q^{35} -9.25001i q^{37} +1.92788i q^{39} -6.30848i q^{41} +3.27423 q^{43} +1.55297 q^{45} +2.09029 q^{47} +(-0.944747 - 6.93595i) q^{49} +1.32606 q^{51} -2.89312i q^{53} -10.9251 q^{55} +2.11329 q^{57} +11.6482i q^{59} -1.96479 q^{61} +(1.24709 - 1.42847i) q^{63} +2.16680 q^{65} -13.5558 q^{67} +7.33091 q^{69} -5.07363i q^{71} +2.08832i q^{73} +0.587948i q^{75} +(-8.77317 + 10.0492i) q^{77} +0.772654i q^{79} -10.6365 q^{81} +5.41871i q^{83} -1.49040i q^{85} -12.7729 q^{87} +3.65311i q^{89} +(1.74001 - 1.99308i) q^{91} +16.6939i q^{93} -2.37519i q^{95} +2.48774i q^{97} -3.61369 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.92788i		−	1.11306i	−0.830827	−	0.556530i	$-0.812132\pi$
$3$		−	1.92788i		−	1.11306i	0.830827	−	0.556530i	$-0.187868\pi$
$4$		0			0
$5$	−2.16680			−0.969023			−0.484512	−	0.874785i	$-0.661003\pi$
$5$	−2.16680			−0.969023			−0.484512	+	0.874785i	$0.661003\pi$
$6$		0			0
$7$	−1.74001	+	1.99308i	−0.657661	+	0.753314i
$7$	−1.74001	+	1.99308i	−0.657661	+	0.753314i
$8$		0			0
$9$	−0.716713			−0.238904
$10$		0			0
$11$	5.04203			1.52023			0.760115	−	0.649789i	$-0.225143\pi$
$11$	5.04203			1.52023			0.760115	+	0.649789i	$0.225143\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$			4.17733i			1.07858i
$16$		0			0
$17$			0.687834i			0.166824i	0.996515	+	0.0834121i	$0.0265818\pi$
$17$			0.687834i			0.166824i	−0.996515	+	0.0834121i	$0.973418\pi$
$18$		0			0
$19$			1.09617i			0.251479i	0.992063	+	0.125740i	$0.0401304\pi$
$19$			1.09617i			0.251479i	−0.992063	+	0.125740i	$0.959870\pi$
$20$		0			0
$21$	3.84242	+	3.35452i	0.838484	+	0.732017i
$22$		0			0
$23$			3.80258i			0.792892i	0.918058	+	0.396446i	$0.129757\pi$
$23$			3.80258i			0.792892i	−0.918058	+	0.396446i	$0.870243\pi$
$24$		0			0
$25$	−0.304972			−0.0609943
$26$		0			0
$27$		−	4.40190i		−	0.847146i
$28$		0			0
$29$		−	6.62535i		−	1.23030i	−0.788411	−	0.615148i	$-0.789096\pi$
$29$		−	6.62535i		−	1.23030i	0.788411	−	0.615148i	$-0.210904\pi$
$30$		0			0
$31$	−8.65921			−1.55524			−0.777620	−	0.628734i	$-0.783573\pi$
$31$	−8.65921			−1.55524			−0.777620	+	0.628734i	$0.783573\pi$
$32$		0			0
$33$		−	9.72042i		−	1.69211i
$34$		0			0
$35$	3.77025	−	4.31861i	0.637289	−	0.729979i
$36$		0			0
$37$		−	9.25001i		−	1.52069i	−0.649518	−	0.760346i	$-0.725029\pi$
$37$		−	9.25001i		−	1.52069i	0.649518	−	0.760346i	$-0.274971\pi$
$38$		0			0
$39$			1.92788i			0.308708i
$40$		0			0
$41$		−	6.30848i		−	0.985218i	−0.870251	−	0.492609i	$-0.836043\pi$
$41$		−	6.30848i		−	0.985218i	0.870251	−	0.492609i	$-0.163957\pi$
$42$		0			0
$43$	3.27423			0.499315			0.249657	−	0.968334i	$-0.419682\pi$
$43$	3.27423			0.499315			0.249657	+	0.968334i	$0.419682\pi$
$44$		0			0
$45$	1.55297			0.231504
$46$		0			0
$47$	2.09029			0.304900			0.152450	−	0.988311i	$-0.451284\pi$
$47$	2.09029			0.304900			0.152450	+	0.988311i	$0.451284\pi$
$48$		0			0
$49$	−0.944747	−	6.93595i	−0.134964	−	0.990851i
$50$		0			0
$51$	1.32606			0.185685
$52$		0			0
$53$		−	2.89312i		−	0.397400i	−0.980060	−	0.198700i	$-0.936328\pi$
$53$		−	2.89312i		−	0.397400i	0.980060	−	0.198700i	$-0.0636720\pi$
$54$		0			0
$55$	−10.9251			−1.47314
$56$		0			0
$57$	2.11329			0.279912
$58$		0			0
$59$			11.6482i			1.51647i	0.651980	+	0.758236i	$0.273939\pi$
$59$			11.6482i			1.51647i	−0.651980	+	0.758236i	$0.726061\pi$
$60$		0			0
$61$	−1.96479			−0.251566			−0.125783	−	0.992058i	$-0.540144\pi$
$61$	−1.96479			−0.251566			−0.125783	+	0.992058i	$0.540144\pi$
$62$		0			0
$63$	1.24709	−	1.42847i	0.157118	−	0.179970i
$64$		0			0
$65$	2.16680			0.268759
$66$		0			0
$67$	−13.5558			−1.65611			−0.828053	−	0.560650i	$-0.810551\pi$
$67$	−13.5558			−1.65611			−0.828053	+	0.560650i	$0.810551\pi$
$68$		0			0
$69$	7.33091			0.882537
$70$		0			0
$71$		−	5.07363i		−	0.602129i	−0.953604	−	0.301064i	$-0.902658\pi$
$71$		−	5.07363i		−	0.602129i	0.953604	−	0.301064i	$-0.0973419\pi$
$72$		0			0
$73$			2.08832i			0.244420i	0.992504	+	0.122210i	$0.0389981\pi$
$73$			2.08832i			0.244420i	−0.992504	+	0.122210i	$0.961002\pi$
$74$		0			0
$75$			0.587948i			0.0678904i
$76$		0			0
$77$	−8.77317	+	10.0492i	−0.999796	+	1.14521i
$78$		0			0
$79$			0.772654i			0.0869303i	0.999055	+	0.0434652i	$0.0138398\pi$
$79$			0.772654i			0.0869303i	−0.999055	+	0.0434652i	$0.986160\pi$
$80$		0			0
$81$	−10.6365			−1.18183
$82$		0			0
$83$			5.41871i			0.594781i	0.954756	+	0.297390i	$0.0961163\pi$
$83$			5.41871i			0.594781i	−0.954756	+	0.297390i	$0.903884\pi$
$84$		0			0
$85$		−	1.49040i		−	0.161656i
$86$		0			0
$87$	−12.7729			−1.36939
$88$		0			0
$89$			3.65311i			0.387229i	0.981078	+	0.193615i	$0.0620212\pi$
$89$			3.65311i			0.387229i	−0.981078	+	0.193615i	$0.937979\pi$
$90$		0			0
$91$	1.74001	−	1.99308i	0.182402	−	0.208932i
$92$		0			0
$93$			16.6939i			1.73108i
$94$		0			0
$95$		−	2.37519i		−	0.243689i
$96$		0			0
$97$			2.48774i			0.252592i	0.991993	+	0.126296i	$0.0403088\pi$
$97$			2.48774i			0.252592i	−0.991993	+	0.126296i	$0.959691\pi$
$98$		0			0
$99$	−3.61369			−0.363189
$100$		0			0
$101$	−17.2080			−1.71226			−0.856131	−	0.516759i	$-0.827138\pi$
$101$	−17.2080			−1.71226			−0.856131	+	0.516759i	$0.827138\pi$
$102$		0			0
$103$	7.58040			0.746919			0.373460	−	0.927646i	$-0.378171\pi$
$103$	7.58040			0.746919			0.373460	+	0.927646i	$0.378171\pi$
$104$		0			0
$105$	−8.32576	−	7.26858i	−0.812511	−	0.709341i
$106$		0			0
$107$	−3.55104			−0.343292			−0.171646	−	0.985159i	$-0.554908\pi$
$107$	−3.55104			−0.343292			−0.171646	+	0.985159i	$0.554908\pi$
$108$		0			0
$109$		−	8.44828i		−	0.809199i	−0.914494	−	0.404599i	$-0.867411\pi$
$109$		−	8.44828i		−	0.809199i	0.914494	−	0.404599i	$-0.132589\pi$
$110$		0			0
$111$	−17.8329			−1.69262
$112$		0			0
$113$	−15.2181			−1.43159			−0.715797	−	0.698308i	$-0.753936\pi$
$113$	−15.2181			−1.43159			−0.715797	+	0.698308i	$0.753936\pi$
$114$		0			0
$115$		−	8.23943i		−	0.768331i
$116$		0			0
$117$	0.716713			0.0662601
$118$		0			0
$119$	−1.37091	−	1.19684i	−0.125671	−	0.109714i
$120$		0			0
$121$	14.4221			1.31110
$122$		0			0
$123$	−12.1620			−1.09661
$124$		0			0
$125$	11.4948			1.02813
$126$		0			0
$127$			4.05535i			0.359854i	0.983680	+	0.179927i	$0.0575862\pi$
$127$			4.05535i			0.359854i	−0.983680	+	0.179927i	$0.942414\pi$
$128$		0			0
$129$		−	6.31231i		−	0.555768i
$130$		0			0
$131$		−	12.3281i		−	1.07711i	−0.842591	−	0.538554i	$-0.818971\pi$
$131$		−	12.3281i		−	1.07711i	0.842591	−	0.538554i	$-0.181029\pi$
$132$		0			0
$133$	−2.18476	−	1.90735i	−0.189443	−	0.165388i
$134$		0			0
$135$			9.53804i			0.820904i
$136$		0			0
$137$	−10.0410			−0.857859			−0.428929	−	0.903338i	$-0.641109\pi$
$137$	−10.0410			−0.857859			−0.428929	+	0.903338i	$0.641109\pi$
$138$		0			0
$139$		−	22.5954i		−	1.91651i	−0.285909	−	0.958257i	$-0.592295\pi$
$139$		−	22.5954i		−	1.91651i	0.285909	−	0.958257i	$-0.407705\pi$
$140$		0			0
$141$		−	4.02983i		−	0.339373i
$142$		0			0
$143$	−5.04203			−0.421636
$144$		0			0
$145$			14.3558i			1.19219i
$146$		0			0
$147$	−13.3717	+	1.82136i	−1.10288	+	0.150223i
$148$		0			0
$149$		−	0.279812i		−	0.0229231i	−0.999934	−	0.0114615i	$-0.996352\pi$
$149$		−	0.279812i		−	0.0229231i	0.999934	−	0.0114615i	$-0.00364840\pi$
$150$		0			0
$151$			19.0561i			1.55076i	0.631493	+	0.775382i	$0.282443\pi$
$151$			19.0561i			1.55076i	−0.631493	+	0.775382i	$0.717557\pi$
$152$		0			0
$153$		−	0.492979i		−	0.0398550i
$154$		0			0
$155$	18.7628			1.50706
$156$		0			0
$157$	12.5958			1.00525			0.502626	−	0.864504i	$-0.332367\pi$
$157$	12.5958			1.00525			0.502626	+	0.864504i	$0.332367\pi$
$158$		0			0
$159$	−5.57758			−0.442331
$160$		0			0
$161$	−7.57885	−	6.61652i	−0.597297	−	0.521454i
$162$		0			0
$163$	−14.7205			−1.15300			−0.576498	−	0.817099i	$-0.695581\pi$
$163$	−14.7205			−1.15300			−0.576498	+	0.817099i	$0.695581\pi$
$164$		0			0
$165$			21.0622i			1.63969i
$166$		0			0
$167$	6.51701			0.504302			0.252151	−	0.967688i	$-0.418862\pi$
$167$	6.51701			0.504302			0.252151	+	0.967688i	$0.418862\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		−	0.785642i		−	0.0600795i
$172$		0			0
$173$	−21.8519			−1.66137			−0.830686	−	0.556742i	$-0.812051\pi$
$173$	−21.8519			−1.66137			−0.830686	+	0.556742i	$0.812051\pi$
$174$		0			0
$175$	0.530653	−	0.607833i	0.0401136	−	0.0459479i
$176$		0			0
$177$	22.4564			1.68793
$178$		0			0
$179$	−12.9272			−0.966228			−0.483114	−	0.875557i	$-0.660494\pi$
$179$	−12.9272			−0.966228			−0.483114	+	0.875557i	$0.660494\pi$
$180$		0			0
$181$	−9.89018			−0.735132			−0.367566	−	0.929998i	$-0.619809\pi$
$181$	−9.89018			−0.735132			−0.367566	+	0.929998i	$0.619809\pi$
$182$		0			0
$183$			3.78788i			0.280008i
$184$		0			0
$185$			20.0429i			1.47359i
$186$		0			0
$187$			3.46808i			0.253611i
$188$		0			0
$189$	8.77334	+	7.65934i	0.638167	+	0.557135i
$190$		0			0
$191$		−	13.5574i		−	0.980978i	−0.871448	−	0.490489i	$-0.836818\pi$
$191$		−	13.5574i		−	0.980978i	0.871448	−	0.490489i	$-0.163182\pi$
$192$		0			0
$193$	−23.1425			−1.66584			−0.832919	−	0.553396i	$-0.813332\pi$
$193$	−23.1425			−1.66584			−0.832919	+	0.553396i	$0.813332\pi$
$194$		0			0
$195$		−	4.17733i		−	0.299145i
$196$		0			0
$197$		−	0.809664i		−	0.0576862i	−0.999584	−	0.0288431i	$-0.990818\pi$
$197$		−	0.809664i		−	0.0576862i	0.999584	−	0.0288431i	$-0.00918232\pi$
$198$		0			0
$199$	13.7274			0.973109			0.486554	−	0.873650i	$-0.338254\pi$
$199$	13.7274			0.973109			0.486554	+	0.873650i	$0.338254\pi$
$200$		0			0
$201$			26.1339i			1.84335i
$202$		0			0
$203$	13.2049	+	11.5282i	0.926800	+	0.809118i
$204$		0			0
$205$			13.6692i			0.954699i
$206$		0			0
$207$		−	2.72536i		−	0.189425i
$208$		0			0
$209$			5.52694i			0.382306i
$210$		0			0
$211$	−22.1210			−1.52288			−0.761438	−	0.648238i	$-0.775506\pi$
$211$	−22.1210			−1.52288			−0.761438	+	0.648238i	$0.775506\pi$
$212$		0			0
$213$	−9.78134			−0.670206
$214$		0			0
$215$	−7.09460			−0.483848
$216$		0			0
$217$	15.0671	−	17.2585i	1.02282	−	1.17158i
$218$		0			0
$219$	4.02603			0.272054
$220$		0			0
$221$		−	0.687834i		−	0.0462687i
$222$		0			0
$223$	10.5227			0.704650			0.352325	−	0.935878i	$-0.385391\pi$
$223$	10.5227			0.704650			0.352325	+	0.935878i	$0.385391\pi$
$224$		0			0
$225$	0.218577			0.0145718
$226$		0			0
$227$			15.9121i			1.05612i	0.849206	+	0.528062i	$0.177081\pi$
$227$			15.9121i			1.05612i	−0.849206	+	0.528062i	$0.822919\pi$
$228$		0			0
$229$	−10.2718			−0.678779			−0.339390	−	0.940646i	$-0.610221\pi$
$229$	−10.2718			−0.678779			−0.339390	+	0.940646i	$0.610221\pi$
$230$		0			0
$231$	19.3736	+	16.9136i	1.27469	+	1.11283i
$232$		0			0
$233$	−4.62581			−0.303047			−0.151524	−	0.988454i	$-0.548418\pi$
$233$	−4.62581			−0.303047			−0.151524	+	0.988454i	$0.548418\pi$
$234$		0			0
$235$	−4.52925			−0.295456
$236$		0			0
$237$	1.48958			0.0967588
$238$		0			0
$239$		−	5.81217i		−	0.375958i	−0.982173	−	0.187979i	$-0.939806\pi$
$239$		−	5.81217i		−	0.375958i	0.982173	−	0.187979i	$-0.0601937\pi$
$240$		0			0
$241$			0.550538i			0.0354633i	0.999843	+	0.0177316i	$0.00564445\pi$
$241$			0.550538i			0.0354633i	−0.999843	+	0.0177316i	$0.994356\pi$
$242$		0			0
$243$			7.30010i			0.468302i
$244$		0			0
$245$	2.04708	+	15.0288i	0.130783	+	0.960157i
$246$		0			0
$247$		−	1.09617i		−	0.0697479i
$248$		0			0
$249$	10.4466			0.662027
$250$		0			0
$251$		−	26.0984i		−	1.64731i	−0.567088	−	0.823657i	$-0.691930\pi$
$251$		−	26.0984i		−	1.64731i	0.567088	−	0.823657i	$-0.308070\pi$
$252$		0			0
$253$			19.1727i			1.20538i
$254$		0			0
$255$	−2.87331			−0.179933
$256$		0			0
$257$		−	23.7161i		−	1.47937i	−0.672952	−	0.739686i	$-0.734974\pi$
$257$		−	23.7161i		−	1.47937i	0.672952	−	0.739686i	$-0.265026\pi$
$258$		0			0
$259$	18.4360	+	16.0951i	1.14556	+	1.00010i
$260$		0			0
$261$			4.74847i			0.293923i
$262$		0			0
$263$			15.7303i			0.969970i	0.874522	+	0.484985i	$0.161175\pi$
$263$			15.7303i			0.969970i	−0.874522	+	0.484985i	$0.838825\pi$
$264$		0			0
$265$			6.26881i			0.385090i
$266$		0			0
$267$	7.04276			0.431010
$268$		0			0
$269$	26.9053			1.64045			0.820223	−	0.572043i	$-0.193849\pi$
$269$	26.9053			1.64045			0.820223	+	0.572043i	$0.193849\pi$
$270$		0			0
$271$	23.7291			1.44144			0.720720	−	0.693226i	$-0.243811\pi$
$271$	23.7291			1.44144			0.720720	+	0.693226i	$0.243811\pi$
$272$		0			0
$273$	−3.84242	−	3.35452i	−0.232554	−	0.203025i
$274$		0			0
$275$	−1.53768			−0.0927254
$276$		0			0
$277$		−	29.3334i		−	1.76248i	−0.472673	−	0.881238i	$-0.656711\pi$
$277$		−	29.3334i		−	1.76248i	0.472673	−	0.881238i	$-0.343289\pi$
$278$		0			0
$279$	6.20617			0.371554
$280$		0			0
$281$	9.82013			0.585820			0.292910	−	0.956140i	$-0.405376\pi$
$281$	9.82013			0.585820			0.292910	+	0.956140i	$0.405376\pi$
$282$		0			0
$283$		−	18.2222i		−	1.08320i	−0.840637	−	0.541599i	$-0.817819\pi$
$283$		−	18.2222i		−	1.08320i	0.840637	−	0.541599i	$-0.182181\pi$
$284$		0			0
$285$	−4.57908			−0.271241
$286$		0			0
$287$	12.5733	+	10.9768i	0.742179	+	0.647940i
$288$		0			0
$289$	16.5269			0.972170
$290$		0			0
$291$	4.79606			0.281150
$292$		0			0
$293$	−10.0394			−0.586508			−0.293254	−	0.956034i	$-0.594738\pi$
$293$	−10.0394			−0.586508			−0.293254	+	0.956034i	$0.594738\pi$
$294$		0			0
$295$		−	25.2394i		−	1.46950i
$296$		0			0
$297$		−	22.1945i		−	1.28786i
$298$		0			0
$299$		−	3.80258i		−	0.219909i
$300$		0			0
$301$	−5.69718	+	6.52580i	−0.328380	+	0.376141i
$302$		0			0
$303$			33.1750i			1.90585i
$304$		0			0
$305$	4.25732			0.243773
$306$		0			0
$307$			3.94497i			0.225151i	0.993643	+	0.112576i	$0.0359101\pi$
$307$			3.94497i			0.225151i	−0.993643	+	0.112576i	$0.964090\pi$
$308$		0			0
$309$		−	14.6141i		−	0.831367i
$310$		0			0
$311$	−17.3290			−0.982640			−0.491320	−	0.870979i	$-0.663485\pi$
$311$	−17.3290			−0.982640			−0.491320	+	0.870979i	$0.663485\pi$
$312$		0			0
$313$			10.1877i			0.575844i	0.957654	+	0.287922i	$0.0929645\pi$
$313$			10.1877i			0.575844i	−0.957654	+	0.287922i	$0.907036\pi$
$314$		0			0
$315$	−2.70219	+	3.09520i	−0.152251	+	0.174395i
$316$		0			0
$317$			5.12951i			0.288102i	0.989570	+	0.144051i	$0.0460130\pi$
$317$			5.12951i			0.288102i	−0.989570	+	0.144051i	$0.953987\pi$
$318$		0			0
$319$		−	33.4052i		−	1.87033i
$320$		0			0
$321$			6.84596i			0.382105i
$322$		0			0
$323$	−0.753985			−0.0419529
$324$		0			0
$325$	0.304972			0.0169168
$326$		0			0
$327$	−16.2873			−0.900688
$328$		0			0
$329$	−3.63712	+	4.16612i	−0.200521	+	0.229686i
$330$		0			0
$331$	10.6345			0.584527			0.292263	−	0.956338i	$-0.405592\pi$
$331$	10.6345			0.584527			0.292263	+	0.956338i	$0.405592\pi$
$332$		0			0
$333$			6.62960i			0.363300i
$334$		0			0
$335$	29.3727			1.60480
$336$		0			0
$337$	−19.4556			−1.05981			−0.529906	−	0.848057i	$-0.677773\pi$
$337$	−19.4556			−1.05981			−0.529906	+	0.848057i	$0.677773\pi$
$338$		0			0
$339$			29.3386i			1.59345i
$340$		0			0
$341$	−43.6600			−2.36432
$342$		0			0
$343$	15.4678	+	10.1857i	0.835182	+	0.549974i
$344$		0			0
$345$	−15.8846			−0.855199
$346$		0			0
$347$	−27.4986			−1.47620			−0.738102	−	0.674690i	$-0.764277\pi$
$347$	−27.4986			−1.47620			−0.738102	+	0.674690i	$0.764277\pi$
$348$		0			0
$349$	23.0163			1.23203			0.616017	−	0.787733i	$-0.288745\pi$
$349$	23.0163			1.23203			0.616017	+	0.787733i	$0.288745\pi$
$350$		0			0
$351$			4.40190i			0.234956i
$352$		0			0
$353$		−	32.6293i		−	1.73668i	−0.495967	−	0.868342i	$-0.665186\pi$
$353$		−	32.6293i		−	1.73668i	0.495967	−	0.868342i	$-0.334814\pi$
$354$		0			0
$355$			10.9935i			0.583477i
$356$		0			0
$357$	−2.30735	+	2.64294i	−0.122118	+	0.139879i
$358$		0			0
$359$			14.0579i			0.741947i	0.928643	+	0.370973i	$0.120976\pi$
$359$			14.0579i			0.741947i	−0.928643	+	0.370973i	$0.879024\pi$
$360$		0			0
$361$	17.7984			0.936758
$362$		0			0
$363$		−	27.8040i		−	1.45933i
$364$		0			0
$365$		−	4.52498i		−	0.236849i
$366$		0			0
$367$	13.8941			0.725266			0.362633	−	0.931932i	$-0.381878\pi$
$367$	13.8941			0.725266			0.362633	+	0.931932i	$0.381878\pi$
$368$		0			0
$369$			4.52137i			0.235373i
$370$		0			0
$371$	5.76622	+	5.03405i	0.299367	+	0.261355i
$372$		0			0
$373$		−	9.46915i		−	0.490294i	−0.969486	−	0.245147i	$-0.921164\pi$
$373$		−	9.46915i		−	0.490294i	0.969486	−	0.245147i	$-0.0788362\pi$
$374$		0			0
$375$		−	22.1606i		−	1.14437i
$376$		0			0
$377$			6.62535i			0.341223i
$378$		0			0
$379$	29.3716			1.50872			0.754359	−	0.656462i	$-0.227948\pi$
$379$	29.3716			1.50872			0.754359	+	0.656462i	$0.227948\pi$
$380$		0			0
$381$	7.81822			0.400540
$382$		0			0
$383$	−23.8795			−1.22019			−0.610093	−	0.792330i	$-0.708868\pi$
$383$	−23.8795			−1.22019			−0.610093	+	0.792330i	$0.708868\pi$
$384$		0			0
$385$	19.0097	−	21.7746i	0.968825	−	1.10973i
$386$		0			0
$387$	−2.34668			−0.119288
$388$		0			0
$389$			24.2232i			1.22816i	0.789242	+	0.614082i	$0.210474\pi$
$389$			24.2232i			1.22816i	−0.789242	+	0.614082i	$0.789526\pi$
$390$		0			0
$391$	−2.61554			−0.132274
$392$		0			0
$393$	−23.7670			−1.19889
$394$		0			0
$395$		−	1.67419i		−	0.0842375i
$396$		0			0
$397$	−24.1030			−1.20969			−0.604847	−	0.796342i	$-0.706766\pi$
$397$	−24.1030			−1.20969			−0.604847	+	0.796342i	$0.706766\pi$
$398$		0			0
$399$	−3.67714	+	4.21196i	−0.184087	+	0.210862i
$400$		0			0
$401$	−21.2274			−1.06004			−0.530022	−	0.847984i	$-0.677817\pi$
$401$	−21.2274			−1.06004			−0.530022	+	0.847984i	$0.677817\pi$
$402$		0			0
$403$	8.65921			0.431346
$404$		0			0
$405$	23.0471			1.14522
$406$		0			0
$407$		−	46.6388i		−	2.31180i
$408$		0			0
$409$			15.1763i			0.750420i	0.926940	+	0.375210i	$0.122429\pi$
$409$			15.1763i			0.750420i	−0.926940	+	0.375210i	$0.877571\pi$
$410$		0			0
$411$			19.3578i			0.954849i
$412$		0			0
$413$	−23.2159	−	20.2680i	−1.14238	−	0.997325i
$414$		0			0
$415$		−	11.7413i		−	0.576356i
$416$		0			0
$417$	−43.5611			−2.13320
$418$		0			0
$419$		−	22.3917i		−	1.09391i	−0.837163	−	0.546953i	$-0.815788\pi$
$419$		−	22.3917i		−	1.09391i	0.837163	−	0.546953i	$-0.184212\pi$
$420$		0			0
$421$			6.95017i			0.338731i	0.985553	+	0.169365i	$0.0541718\pi$
$421$			6.95017i			0.338731i	−0.985553	+	0.169365i	$0.945828\pi$
$422$		0			0
$423$	−1.49814			−0.0728420
$424$		0			0
$425$		−	0.209770i		−	0.0101753i
$426$		0			0
$427$	3.41876	−	3.91599i	0.165445	−	0.189508i
$428$		0			0
$429$			9.72042i			0.469306i
$430$		0			0
$431$		−	17.5920i		−	0.847378i	−0.905808	−	0.423689i	$-0.860735\pi$
$431$		−	17.5920i		−	0.847378i	0.905808	−	0.423689i	$-0.139265\pi$
$432$		0			0
$433$			12.7562i			0.613025i	0.951867	+	0.306512i	$0.0991620\pi$
$433$			12.7562i			0.613025i	−0.951867	+	0.306512i	$0.900838\pi$
$434$		0			0
$435$	27.6763			1.32698
$436$		0			0
$437$	−4.16829			−0.199396
$438$		0			0
$439$	−39.8195			−1.90048			−0.950240	−	0.311518i	$-0.899162\pi$
$439$	−39.8195			−1.90048			−0.950240	+	0.311518i	$0.899162\pi$
$440$		0			0
$441$	0.677112	+	4.97109i	0.0322434	+	0.236718i
$442$		0			0
$443$	2.94265			0.139810			0.0699049	−	0.997554i	$-0.477730\pi$
$443$	2.94265			0.139810			0.0699049	+	0.997554i	$0.477730\pi$
$444$		0			0
$445$		−	7.91557i		−	0.375234i
$446$		0			0
$447$	−0.539443			−0.0255148
$448$		0			0
$449$	12.7756			0.602918			0.301459	−	0.953479i	$-0.402526\pi$
$449$	12.7756			0.602918			0.301459	+	0.953479i	$0.402526\pi$
$450$		0			0
$451$		−	31.8075i		−	1.49776i
$452$		0			0
$453$	36.7378			1.72609
$454$		0			0
$455$	−3.77025	+	4.31861i	−0.176752	+	0.202460i
$456$		0			0
$457$	−6.05610			−0.283293			−0.141646	−	0.989917i	$-0.545240\pi$
$457$	−6.05610			−0.283293			−0.141646	+	0.989917i	$0.545240\pi$
$458$		0			0
$459$	3.02777			0.141324
$460$		0			0
$461$	13.5455			0.630878			0.315439	−	0.948946i	$-0.397848\pi$
$461$	13.5455			0.630878			0.315439	+	0.948946i	$0.397848\pi$
$462$		0			0
$463$		−	24.3379i		−	1.13108i	−0.824721	−	0.565539i	$-0.808668\pi$
$463$		−	24.3379i		−	1.13108i	0.824721	−	0.565539i	$-0.191332\pi$
$464$		0			0
$465$		−	36.1724i		−	1.67745i
$466$		0			0
$467$			29.3776i			1.35943i	0.733476	+	0.679716i	$0.237897\pi$
$467$			29.3776i			1.35943i	−0.733476	+	0.679716i	$0.762103\pi$
$468$		0			0
$469$	23.5872	−	27.0178i	1.08916	−	1.24757i
$470$		0			0
$471$		−	24.2831i		−	1.11891i
$472$		0			0
$473$	16.5088			0.759073
$474$		0			0
$475$		−	0.334302i		−	0.0153388i
$476$		0			0
$477$			2.07353i			0.0949406i
$478$		0			0
$479$	2.86853			0.131066			0.0655332	−	0.997850i	$-0.479125\pi$
$479$	2.86853			0.131066			0.0655332	+	0.997850i	$0.479125\pi$
$480$		0			0
$481$			9.25001i			0.421764i
$482$		0			0
$483$	−12.7558	+	14.6111i	−0.580410	+	0.664828i
$484$		0			0
$485$		−	5.39044i		−	0.244767i
$486$		0			0
$487$		−	24.5884i		−	1.11421i	−0.830443	−	0.557104i	$-0.811912\pi$
$487$		−	24.5884i		−	1.11421i	0.830443	−	0.557104i	$-0.188088\pi$
$488$		0			0
$489$			28.3792i			1.28335i
$490$		0			0
$491$	−40.7235			−1.83783			−0.918913	−	0.394460i	$-0.870932\pi$
$491$	−40.7235			−1.83783			−0.918913	+	0.394460i	$0.870932\pi$
$492$		0			0
$493$	4.55714			0.205243
$494$		0			0
$495$	7.83014			0.351939
$496$		0			0
$497$	10.1122	+	8.82815i	0.453592	+	0.395997i
$498$		0			0
$499$	37.7172			1.68845			0.844226	−	0.535987i	$-0.180060\pi$
$499$	37.7172			1.68845			0.844226	+	0.535987i	$0.180060\pi$
$500$		0			0
$501$		−	12.5640i		−	0.561318i
$502$		0			0
$503$	2.68750			0.119830			0.0599148	−	0.998203i	$-0.480917\pi$
$503$	2.68750			0.119830			0.0599148	+	0.998203i	$0.480917\pi$
$504$		0			0
$505$	37.2864			1.65922
$506$		0			0
$507$		−	1.92788i		−	0.0856201i
$508$		0			0
$509$	−5.67996			−0.251760			−0.125880	−	0.992045i	$-0.540175\pi$
$509$	−5.67996			−0.251760			−0.125880	+	0.992045i	$0.540175\pi$
$510$		0			0
$511$	−4.16220	−	3.63370i	−0.184125	−	0.160745i
$512$		0			0
$513$	4.82524			0.213040
$514$		0			0
$515$	−16.4252			−0.723782
$516$		0			0
$517$	10.5393			0.463519
$518$		0			0
$519$			42.1279i			1.84921i
$520$		0			0
$521$		−	18.1395i		−	0.794706i	−0.917666	−	0.397353i	$-0.869929\pi$
$521$		−	18.1395i		−	0.794706i	0.917666	−	0.397353i	$-0.130071\pi$
$522$		0			0
$523$			26.7276i			1.16872i	0.811496	+	0.584358i	$0.198654\pi$
$523$			26.7276i			1.16872i	−0.811496	+	0.584358i	$0.801346\pi$
$524$		0			0
$525$	−1.17183	−	1.02303i	−0.0511428	−	0.0446489i
$526$		0			0
$527$		−	5.95610i		−	0.259452i
$528$		0			0
$529$	8.54040			0.371322
$530$		0			0
$531$		−	8.34845i		−	0.362292i
$532$		0			0
$533$			6.30848i			0.273250i
$534$		0			0
$535$	7.69439			0.332658
$536$		0			0
$537$			24.9222i			1.07547i
$538$		0			0
$539$	−4.76344	−	34.9713i	−0.205176	−	1.50632i
$540$		0			0
$541$		−	32.6687i		−	1.40454i	−0.711912	−	0.702268i	$-0.752171\pi$
$541$		−	32.6687i		−	1.40454i	0.711912	−	0.702268i	$-0.247829\pi$
$542$		0			0
$543$			19.0671i			0.818246i
$544$		0			0
$545$			18.3058i			0.784132i
$546$		0			0
$547$	−33.9785			−1.45281			−0.726407	−	0.687265i	$-0.758811\pi$
$547$	−33.9785			−1.45281			−0.726407	+	0.687265i	$0.758811\pi$
$548$		0			0
$549$	1.40819			0.0601002
$550$		0			0
$551$	7.26253			0.309394
$552$		0			0
$553$	−1.53996	−	1.34442i	−0.0654858	−	0.0571707i
$554$		0			0
$555$	38.6403			1.64019
$556$		0			0
$557$		−	12.1520i		−	0.514899i	−0.966292	−	0.257449i	$-0.917118\pi$
$557$		−	12.1520i		−	0.514899i	0.966292	−	0.257449i	$-0.0828821\pi$
$558$		0			0
$559$	−3.27423			−0.138485
$560$		0			0
$561$	6.68603			0.282284
$562$		0			0
$563$			35.0006i			1.47510i	0.675293	+	0.737549i	$0.264017\pi$
$563$			35.0006i			1.47510i	−0.675293	+	0.737549i	$0.735983\pi$
$564$		0			0
$565$	32.9745			1.38725
$566$		0			0
$567$	18.5075	−	21.1993i	0.777243	−	0.890288i
$568$		0			0
$569$	41.2142			1.72779			0.863894	−	0.503673i	$-0.168018\pi$
$569$	41.2142			1.72779			0.863894	+	0.503673i	$0.168018\pi$
$570$		0			0
$571$	19.0805			0.798496			0.399248	−	0.916843i	$-0.369271\pi$
$571$	19.0805			0.798496			0.399248	+	0.916843i	$0.369271\pi$
$572$		0			0
$573$	−26.1370			−1.09189
$574$		0			0
$575$		−	1.15968i		−	0.0483619i
$576$		0			0
$577$			14.5431i			0.605438i	0.953080	+	0.302719i	$0.0978944\pi$
$577$			14.5431i			0.605438i	−0.953080	+	0.302719i	$0.902106\pi$
$578$		0			0
$579$			44.6160i			1.85418i
$580$		0			0
$581$	−10.7999	−	9.42860i	−0.448057	−	0.391164i
$582$		0			0
$583$		−	14.5872i		−	0.604140i
$584$		0			0
$585$	−1.55297			−0.0642076
$586$		0			0
$587$		−	21.4770i		−	0.886451i	−0.896410	−	0.443226i	$-0.853834\pi$
$587$		−	21.4770i		−	0.886451i	0.896410	−	0.443226i	$-0.146166\pi$
$588$		0			0
$589$		−	9.49200i		−	0.391111i
$590$		0			0
$591$	−1.56093			−0.0642083
$592$		0			0
$593$			3.89623i			0.159999i	0.996795	+	0.0799994i	$0.0254919\pi$
$593$			3.89623i			0.159999i	−0.996795	+	0.0799994i	$0.974508\pi$
$594$		0			0
$595$	2.97049	+	2.59331i	0.121778	+	0.106315i
$596$		0			0
$597$		−	26.4647i		−	1.08313i
$598$		0			0
$599$		−	36.2640i		−	1.48171i	−0.671667	−	0.740853i	$-0.734422\pi$
$599$		−	36.2640i		−	1.48171i	0.671667	−	0.740853i	$-0.265578\pi$
$600$		0			0
$601$		−	25.1481i		−	1.02581i	−0.858445	−	0.512906i	$-0.828569\pi$
$601$		−	25.1481i		−	1.02581i	0.858445	−	0.512906i	$-0.171431\pi$
$602$		0			0
$603$	9.71562			0.395651
$604$		0			0
$605$	−31.2498			−1.27048
$606$		0			0
$607$	−15.8233			−0.642247			−0.321123	−	0.947037i	$-0.604060\pi$
$607$	−15.8233			−0.642247			−0.321123	+	0.947037i	$0.604060\pi$
$608$		0			0
$609$	22.2249	−	25.4574i	0.900598	−	1.03158i
$610$		0			0
$611$	−2.09029			−0.0845642
$612$		0			0
$613$		−	12.6133i		−	0.509447i	−0.967014	−	0.254723i	$-0.918016\pi$
$613$		−	12.6133i		−	0.509447i	0.967014	−	0.254723i	$-0.0819844\pi$
$614$		0			0
$615$	26.3526			1.06264
$616$		0			0
$617$	33.7479			1.35864			0.679319	−	0.733843i	$-0.262275\pi$
$617$	33.7479			1.35864			0.679319	+	0.733843i	$0.262275\pi$
$618$		0			0
$619$			47.3965i			1.90502i	0.304500	+	0.952512i	$0.401511\pi$
$619$			47.3965i			1.90502i	−0.304500	+	0.952512i	$0.598489\pi$
$620$		0			0
$621$	16.7386			0.671695
$622$		0			0
$623$	−7.28095	−	6.35645i	−0.291705	−	0.254666i
$624$		0			0
$625$	−23.3821			−0.935285
$626$		0			0
$627$	10.6553			0.425530
$628$		0			0
$629$	6.36247			0.253688
$630$		0			0
$631$			10.6781i			0.425088i	0.977151	+	0.212544i	$0.0681749\pi$
$631$			10.6781i			0.425088i	−0.977151	+	0.212544i	$0.931825\pi$
$632$		0			0
$633$			42.6467i			1.69505i
$634$		0			0
$635$		−	8.78714i		−	0.348707i
$636$		0			0
$637$	0.944747	+	6.93595i	0.0374322	+	0.274812i
$638$		0			0
$639$			3.63633i			0.143851i
$640$		0			0
$641$	28.3033			1.11791			0.558956	−	0.829197i	$-0.311202\pi$
$641$	28.3033			1.11791			0.558956	+	0.829197i	$0.311202\pi$
$642$		0			0
$643$			13.4258i			0.529462i	0.964322	+	0.264731i	$0.0852831\pi$
$643$			13.4258i			0.529462i	−0.964322	+	0.264731i	$0.914717\pi$
$644$		0			0
$645$			13.6775i			0.538552i
$646$		0			0
$647$	−17.7213			−0.696698			−0.348349	−	0.937365i	$-0.613258\pi$
$647$	−17.7213			−0.696698			−0.348349	+	0.937365i	$0.613258\pi$
$648$		0			0
$649$			58.7308i			2.30539i
$650$		0			0
$651$	−33.2723	−	29.0475i	−1.30404	−	1.13846i
$652$		0			0
$653$			18.2377i			0.713695i	0.934163	+	0.356847i	$0.116148\pi$
$653$			18.2377i			0.713695i	−0.934163	+	0.356847i	$0.883852\pi$
$654$		0			0
$655$			26.7125i			1.04374i
$656$		0			0
$657$		−	1.49673i		−	0.0583930i
$658$		0			0
$659$	−11.1172			−0.433064			−0.216532	−	0.976276i	$-0.569475\pi$
$659$	−11.1172			−0.433064			−0.216532	+	0.976276i	$0.569475\pi$
$660$		0			0
$661$	10.0383			0.390444			0.195222	−	0.980759i	$-0.437457\pi$
$661$	10.0383			0.390444			0.195222	+	0.980759i	$0.437457\pi$
$662$		0			0
$663$	−1.32606			−0.0514999
$664$		0			0
$665$	4.73395	+	4.13285i	0.183575	+	0.160265i
$666$		0			0
$667$	25.1934			0.975493
$668$		0			0
$669$		−	20.2864i		−	0.784318i
$670$		0			0
$671$	−9.90655			−0.382438
$672$		0			0
$673$	18.9917			0.732077			0.366039	−	0.930600i	$-0.380714\pi$
$673$	18.9917			0.732077			0.366039	+	0.930600i	$0.380714\pi$
$674$		0			0
$675$			1.34245i			0.0516711i
$676$		0			0
$677$	−11.8708			−0.456231			−0.228115	−	0.973634i	$-0.573256\pi$
$677$	−11.8708			−0.456231			−0.228115	+	0.973634i	$0.573256\pi$
$678$		0			0
$679$	−4.95826	−	4.32868i	−0.190281	−	0.166120i
$680$		0			0
$681$	30.6766			1.17553
$682$		0			0
$683$	−24.1203			−0.922937			−0.461469	−	0.887156i	$-0.652677\pi$
$683$	−24.1203			−0.922937			−0.461469	+	0.887156i	$0.652677\pi$
$684$		0			0
$685$	21.7568			0.831285
$686$		0			0
$687$			19.8028i			0.755523i
$688$		0			0
$689$			2.89312i			0.110219i
$690$		0			0
$691$		−	15.6365i		−	0.594842i	−0.954746	−	0.297421i	$-0.903874\pi$
$691$		−	15.6365i		−	0.594842i	0.954746	−	0.297421i	$-0.0961265\pi$
$692$		0			0
$693$	6.28784	−	7.20237i	0.238855	−	0.273596i
$694$		0			0
$695$			48.9597i			1.85715i
$696$		0			0
$697$	4.33918			0.164358
$698$		0			0
$699$			8.91800i			0.337310i
$700$		0			0
$701$		−	6.05523i		−	0.228703i	−0.993440	−	0.114351i	$-0.963521\pi$
$701$		−	6.05523i		−	0.228703i	0.993440	−	0.114351i	$-0.0364790\pi$
$702$		0			0
$703$	10.1396			0.382423
$704$		0			0
$705$			8.73184i			0.328860i
$706$		0			0
$707$	29.9421	−	34.2970i	1.12609	−	1.28987i
$708$		0			0
$709$		−	10.6349i		−	0.399401i	−0.979857	−	0.199701i	$-0.936003\pi$
$709$		−	10.6349i		−	0.399401i	0.979857	−	0.199701i	$-0.0639969\pi$
$710$		0			0
$711$		−	0.553771i		−	0.0207680i
$712$		0			0
$713$		−	32.9273i		−	1.23314i
$714$		0			0
$715$	10.9251			0.408575
$716$		0			0
$717$	−11.2052			−0.418464
$718$		0			0
$719$	−6.21956			−0.231950			−0.115975	−	0.993252i	$-0.536999\pi$
$719$	−6.21956			−0.231950			−0.115975	+	0.993252i	$0.536999\pi$
$720$		0			0
$721$	−13.1900	+	15.1084i	−0.491220	+	0.562665i
$722$		0			0
$723$	1.06137			0.0394728
$724$		0			0
$725$			2.02054i			0.0750411i
$726$		0			0
$727$	26.7235			0.991118			0.495559	−	0.868574i	$-0.334963\pi$
$727$	26.7235			0.991118			0.495559	+	0.868574i	$0.334963\pi$
$728$		0			0
$729$	−17.8357			−0.660581
$730$		0			0
$731$			2.25212i			0.0832978i
$732$		0			0
$733$	21.9273			0.809905			0.404953	−	0.914338i	$-0.367288\pi$
$733$	21.9273			0.809905			0.404953	+	0.914338i	$0.367288\pi$
$734$		0			0
$735$	28.9738	−	3.94652i	1.06871	−	0.145569i
$736$		0			0
$737$	−68.3488			−2.51766
$738$		0			0
$739$	29.1929			1.07388			0.536938	−	0.843621i	$-0.319581\pi$
$739$	29.1929			1.07388			0.536938	+	0.843621i	$0.319581\pi$
$740$		0			0
$741$	−2.11329			−0.0776336
$742$		0			0
$743$			43.4204i			1.59294i	0.604678	+	0.796470i	$0.293302\pi$
$743$			43.4204i			1.59294i	−0.604678	+	0.796470i	$0.706698\pi$
$744$		0			0
$745$			0.606296i			0.0222130i
$746$		0			0
$747$		−	3.88366i		−	0.142096i
$748$		0			0
$749$	6.17883	−	7.07751i	0.225770	−	0.258606i
$750$		0			0
$751$			11.5694i			0.422174i	0.977467	+	0.211087i	$0.0677003\pi$
$751$			11.5694i			0.422174i	−0.977467	+	0.211087i	$0.932300\pi$
$752$		0			0
$753$	−50.3145			−1.83356
$754$		0			0
$755$		−	41.2908i		−	1.50273i
$756$		0			0
$757$		−	3.36324i		−	0.122239i	−0.998130	−	0.0611196i	$-0.980533\pi$
$757$		−	3.36324i		−	0.122239i	0.998130	−	0.0611196i	$-0.0194671\pi$
$758$		0			0
$759$	36.9627			1.34166
$760$		0			0
$761$			28.7317i			1.04152i	0.853702	+	0.520761i	$0.174352\pi$
$761$			28.7317i			1.04152i	−0.853702	+	0.520761i	$0.825648\pi$
$762$		0			0
$763$	16.8381	+	14.7001i	0.609581	+	0.532179i
$764$		0			0
$765$			1.06819i			0.0386204i
$766$		0			0
$767$		−	11.6482i		−	0.420594i
$768$		0			0
$769$		−	32.7273i		−	1.18018i	−0.807338	−	0.590089i	$-0.799093\pi$
$769$		−	32.7273i		−	1.18018i	0.807338	−	0.590089i	$-0.200907\pi$
$770$		0			0
$771$	−45.7218			−1.64663
$772$		0			0
$773$	1.68151			0.0604796			0.0302398	−	0.999543i	$-0.490373\pi$
$773$	1.68151			0.0604796			0.0302398	+	0.999543i	$0.490373\pi$
$774$		0			0
$775$	2.64081			0.0948609
$776$		0			0
$777$	31.0294	−	35.5424i	1.11317	−	1.27508i
$778$		0			0
$779$	6.91518			0.247762
$780$		0			0
$781$		−	25.5814i		−	0.915374i
$782$		0			0
$783$	−29.1641			−1.04224
$784$		0			0
$785$	−27.2925			−0.974113
$786$		0			0
$787$			5.75339i			0.205086i	0.994729	+	0.102543i	$0.0326979\pi$
$787$			5.75339i			0.205086i	−0.994729	+	0.102543i	$0.967302\pi$
$788$		0			0
$789$	30.3260			1.07964
$790$		0			0
$791$	26.4795	−	30.3308i	0.941504	−	1.07844i
$792$		0			0
$793$	1.96479			0.0697719
$794$		0			0
$795$	12.0855			0.428629
$796$		0			0
$797$	0.563854			0.0199727			0.00998636	−	0.999950i	$-0.496821\pi$
$797$	0.563854			0.0199727			0.00998636	+	0.999950i	$0.496821\pi$
$798$		0			0
$799$			1.43777i			0.0508648i
$800$		0			0
$801$		−	2.61823i		−	0.0925107i
$802$		0			0
$803$			10.5294i			0.371574i
$804$		0			0
$805$	16.4219	+	14.3367i	0.578794	+	0.505301i
$806$		0			0
$807$		−	51.8702i		−	1.82592i
$808$		0			0
$809$	−40.0661			−1.40865			−0.704325	−	0.709878i	$-0.748750\pi$
$809$	−40.0661			−1.40865			−0.704325	+	0.709878i	$0.748750\pi$
$810$		0			0
$811$			10.3867i			0.364728i	0.983231	+	0.182364i	$0.0583749\pi$
$811$			10.3867i			0.364728i	−0.983231	+	0.182364i	$0.941625\pi$
$812$		0			0
$813$		−	45.7468i		−	1.60441i
$814$		0			0
$815$	31.8963			1.11728
$816$		0			0
$817$			3.58912i			0.125567i
$818$		0			0
$819$	−1.24709	+	1.42847i	−0.0435767	+	0.0499147i
$820$		0			0
$821$			25.9082i			0.904202i	0.891967	+	0.452101i	$0.149325\pi$
$821$			25.9082i			0.904202i	−0.891967	+	0.452101i	$0.850675\pi$
$822$		0			0
$823$			32.7107i			1.14022i	0.821567	+	0.570112i	$0.193100\pi$
$823$			32.7107i			1.14022i	−0.821567	+	0.570112i	$0.806900\pi$
$824$		0			0
$825$			2.96445i			0.103209i
$826$		0			0
$827$	9.60694			0.334066			0.167033	−	0.985951i	$-0.446581\pi$
$827$	9.60694			0.334066			0.167033	+	0.985951i	$0.446581\pi$
$828$		0			0
$829$	55.6806			1.93387			0.966934	−	0.255027i	$-0.0820843\pi$
$829$	55.6806			1.93387			0.966934	+	0.255027i	$0.0820843\pi$
$830$		0			0
$831$	−56.5513			−1.96174
$832$		0			0
$833$	4.77078	−	0.649829i	0.165298	−	0.0225152i
$834$		0			0
$835$	−14.1211			−0.488680
$836$		0			0
$837$			38.1170i			1.31752i
$838$		0			0
$839$	−26.2179			−0.905142			−0.452571	−	0.891728i	$-0.649493\pi$
$839$	−26.2179			−0.905142			−0.452571	+	0.891728i	$0.649493\pi$
$840$		0			0
$841$	−14.8953			−0.513630
$842$		0			0
$843$		−	18.9320i		−	0.652053i
$844$		0			0
$845$	−2.16680			−0.0745402
$846$		0			0
$847$	−25.0945	+	28.7444i	−0.862258	+	0.987668i
$848$		0			0
$849$	−35.1302			−1.20567
$850$		0			0
$851$	35.1739			1.20575
$852$		0			0
$853$	−29.3901			−1.00630			−0.503149	−	0.864200i	$-0.667825\pi$
$853$	−29.3901			−1.00630			−0.503149	+	0.864200i	$0.667825\pi$
$854$		0			0
$855$			1.70233i			0.0582184i
$856$		0			0
$857$			22.3255i			0.762624i	0.924446	+	0.381312i	$0.124528\pi$
$857$			22.3255i			0.762624i	−0.924446	+	0.381312i	$0.875472\pi$
$858$		0			0
$859$			15.7351i			0.536873i	0.963297	+	0.268437i	$0.0865070\pi$
$859$			15.7351i			0.536873i	−0.963297	+	0.268437i	$0.913493\pi$
$860$		0			0
$861$	21.1619	−	24.2398i	0.721196	−	0.826090i
$862$		0			0
$863$			31.8109i			1.08286i	0.840747	+	0.541428i	$0.182116\pi$
$863$			31.8109i			1.08286i	−0.840747	+	0.541428i	$0.817884\pi$
$864$		0			0
$865$	47.3488			1.60991
$866$		0			0
$867$		−	31.8618i		−	1.08208i
$868$		0			0
$869$			3.89574i			0.132154i
$870$		0			0
$871$	13.5558			0.459321
$872$		0			0
$873$		−	1.78299i		−	0.0603452i
$874$		0			0
$875$	−20.0011	+	22.9101i	−0.676160	+	0.774503i
$876$		0			0
$877$			4.71862i			0.159337i	0.996821	+	0.0796683i	$0.0253861\pi$
$877$			4.71862i			0.159337i	−0.996821	+	0.0796683i	$0.974614\pi$
$878$		0			0
$879$			19.3548i			0.652820i
$880$		0			0
$881$		−	34.3311i		−	1.15665i	−0.815808	−	0.578323i	$-0.803707\pi$
$881$		−	34.3311i		−	1.15665i	0.815808	−	0.578323i	$-0.196293\pi$
$882$		0			0
$883$	−13.7434			−0.462503			−0.231252	−	0.972894i	$-0.574282\pi$
$883$	−13.7434			−0.462503			−0.231252	+	0.972894i	$0.574282\pi$
$884$		0			0
$885$	−48.6585			−1.63564
$886$		0			0
$887$	23.5082			0.789328			0.394664	−	0.918825i	$-0.370861\pi$
$887$	23.5082			0.789328			0.394664	+	0.918825i	$0.370861\pi$
$888$		0			0
$889$	−8.08264	−	7.05634i	−0.271083	−	0.236662i
$890$		0			0
$891$	−53.6294			−1.79665
$892$		0			0
$893$			2.29132i			0.0766762i
$894$		0			0
$895$	28.0108			0.936297
$896$		0			0
$897$	−7.33091			−0.244772
$898$		0			0
$899$			57.3703i			1.91341i
$900$		0			0
$901$	1.98998			0.0662960
$902$		0			0
$903$	12.5809	+	10.9835i	0.418668	+	0.365507i
$904$		0			0
$905$	21.4301			0.712359
$906$		0			0
$907$	36.4422			1.21004			0.605021	−	0.796209i	$-0.293165\pi$
$907$	36.4422			1.21004			0.605021	+	0.796209i	$0.293165\pi$
$908$		0			0
$909$	12.3332			0.409067
$910$		0			0
$911$		−	9.46472i		−	0.313580i	−0.987632	−	0.156790i	$-0.949885\pi$
$911$		−	9.46472i		−	0.313580i	0.987632	−	0.156790i	$-0.0501146\pi$
$912$		0			0
$913$			27.3213i			0.904203i
$914$		0			0
$915$		−	8.20759i		−	0.271335i
$916$		0			0
$917$	24.5708	+	21.4509i	0.811400	+	0.708372i
$918$		0			0
$919$		−	43.3424i		−	1.42973i	−0.699261	−	0.714867i	$-0.746487\pi$
$919$		−	43.3424i		−	1.42973i	0.699261	−	0.714867i	$-0.253513\pi$
$920$		0			0
$921$	7.60542			0.250607
$922$		0			0
$923$			5.07363i			0.167001i
$924$		0			0
$925$			2.82099i			0.0927536i
$926$		0			0
$927$	−5.43297			−0.178442
$928$		0			0
$929$		−	39.0336i		−	1.28065i	−0.768104	−	0.640325i	$-0.778799\pi$
$929$		−	39.0336i		−	1.28065i	0.768104	−	0.640325i	$-0.221201\pi$
$930$		0			0
$931$	7.60301	−	1.03561i	0.249179	−	0.0339406i
$932$		0			0
$933$			33.4083i			1.09374i
$934$		0			0
$935$		−	7.51464i		−	0.245755i
$936$		0			0
$937$			53.0178i			1.73202i	0.500030	+	0.866008i	$0.333322\pi$
$937$			53.0178i			1.73202i	−0.500030	+	0.866008i	$0.666678\pi$
$938$		0			0
$939$	19.6407			0.640950
$940$		0			0
$941$	−16.8743			−0.550087			−0.275044	−	0.961432i	$-0.588692\pi$
$941$	−16.8743			−0.550087			−0.275044	+	0.961432i	$0.588692\pi$
$942$		0			0
$943$	23.9885			0.781172
$944$		0			0
$945$	−19.0101	−	16.5963i	−0.618398	−	0.539876i
$946$		0			0
$947$	8.43947			0.274246			0.137123	−	0.990554i	$-0.456214\pi$
$947$	8.43947			0.274246			0.137123	+	0.990554i	$0.456214\pi$
$948$		0			0
$949$		−	2.08832i		−	0.0677899i
$950$		0			0
$951$	9.88907			0.320675
$952$		0			0
$953$	−18.5496			−0.600880			−0.300440	−	0.953801i	$-0.597134\pi$
$953$	−18.5496			−0.600880			−0.300440	+	0.953801i	$0.597134\pi$
$954$		0			0
$955$			29.3761i			0.950590i
$956$		0			0
$957$	−64.4012			−2.08179
$958$		0			0
$959$	17.4714	−	20.0125i	0.564180	−	0.646237i
$960$		0			0
$961$	43.9820			1.41877
$962$		0			0
$963$	2.54507			0.0820138
$964$		0			0
$965$	50.1453			1.61423
$966$		0			0
$967$			7.56789i			0.243367i	0.992569	+	0.121683i	$0.0388293\pi$
$967$			7.56789i			0.243367i	−0.992569	+	0.121683i	$0.961171\pi$
$968$		0			0
$969$			1.45359i			0.0466961i
$970$		0			0
$971$		−	4.56757i		−	0.146580i	−0.997311	−	0.0732901i	$-0.976650\pi$
$971$		−	4.56757i		−	0.146580i	0.997311	−	0.0732901i	$-0.0233499\pi$
$972$		0			0
$973$	45.0344	+	39.3161i	1.44374	+	1.26042i
$974$		0			0
$975$		−	0.587948i		−	0.0188294i
$976$		0			0
$977$	35.0513			1.12139			0.560696	−	0.828022i	$-0.310534\pi$
$977$	35.0513			1.12139			0.560696	+	0.828022i	$0.310534\pi$
$978$		0			0
$979$			18.4191i			0.588677i
$980$		0			0
$981$			6.05499i			0.193321i
$982$		0			0
$983$	−17.4575			−0.556809			−0.278404	−	0.960464i	$-0.589805\pi$
$983$	−17.4575			−0.556809			−0.278404	+	0.960464i	$0.589805\pi$
$984$		0			0
$985$			1.75438i			0.0558993i
$986$		0			0
$987$	8.03178	+	7.01193i	0.255654	+	0.223192i
$988$		0			0
$989$			12.4505i			0.395903i
$990$		0			0
$991$			11.0252i			0.350227i	0.984548	+	0.175114i	$0.0560292\pi$
$991$			11.0252i			0.350227i	−0.984548	+	0.175114i	$0.943971\pi$
$992$		0			0
$993$		−	20.5021i		−	0.650614i
$994$		0			0
$995$	−29.7445			−0.942965
$996$		0			0
$997$	−5.72166			−0.181207			−0.0906033	−	0.995887i	$-0.528880\pi$
$997$	−5.72166			−0.181207			−0.0906033	+	0.995887i	$0.528880\pi$
$998$		0			0
$999$	−40.7176			−1.28825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.a.2575.11		48
4.3	odd	2		728.2.h.a.27.13	✓	48
7.6	odd	2		2912.2.h.b.2575.38		48
8.3	odd	2		2912.2.h.b.2575.11		48
8.5	even	2		728.2.h.b.27.14	yes	48
28.27	even	2		728.2.h.b.27.13	yes	48
56.13	odd	2		728.2.h.a.27.14	yes	48
56.27	even	2	inner	2912.2.h.a.2575.38		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.13	✓	48	4.3	odd	2
728.2.h.a.27.14	yes	48	56.13	odd	2
728.2.h.b.27.13	yes	48	28.27	even	2
728.2.h.b.27.14	yes	48	8.5	even	2
2912.2.h.a.2575.11		48	1.1	even	1	trivial
2912.2.h.a.2575.38		48	56.27	even	2	inner
2912.2.h.b.2575.11		48	8.3	odd	2
2912.2.h.b.2575.38		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)

\(f(q)\)	\(=\)	\(q-1.92788i q^{3} -2.16680 q^{5} +(-1.74001 + 1.99308i) q^{7} -0.716713 q^{9} +5.04203 q^{11} -1.00000 q^{13} +4.17733i q^{15} +0.687834i q^{17} +1.09617i q^{19} +(3.84242 + 3.35452i) q^{21} +3.80258i q^{23} -0.304972 q^{25} -4.40190i q^{27} -6.62535i q^{29} -8.65921 q^{31} -9.72042i q^{33} +(3.77025 - 4.31861i) q^{35} -9.25001i q^{37} +1.92788i q^{39} -6.30848i q^{41} +3.27423 q^{43} +1.55297 q^{45} +2.09029 q^{47} +(-0.944747 - 6.93595i) q^{49} +1.32606 q^{51} -2.89312i q^{53} -10.9251 q^{55} +2.11329 q^{57} +11.6482i q^{59} -1.96479 q^{61} +(1.24709 - 1.42847i) q^{63} +2.16680 q^{65} -13.5558 q^{67} +7.33091 q^{69} -5.07363i q^{71} +2.08832i q^{73} +0.587948i q^{75} +(-8.77317 + 10.0492i) q^{77} +0.772654i q^{79} -10.6365 q^{81} +5.41871i q^{83} -1.49040i q^{85} -12.7729 q^{87} +3.65311i q^{89} +(1.74001 - 1.99308i) q^{91} +16.6939i q^{93} -2.37519i q^{95} +2.48774i q^{97} -3.61369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) \) 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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