LMFDB - Embedded newform 2912.2.h.b.2575.19

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			2575.19
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.b.2575.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.788146i q^{3} +0.604005 q^{5} +(-2.57195 - 0.620533i) q^{7} +2.37883 q^{9} -5.13402 q^{11} +1.00000 q^{13} -0.476044i q^{15} -5.35639i q^{17} +5.97202i q^{19} +(-0.489070 + 2.02707i) q^{21} +1.05537i q^{23} -4.63518 q^{25} -4.23930i q^{27} +7.55238i q^{29} -2.56972 q^{31} +4.04636i q^{33} +(-1.55347 - 0.374805i) q^{35} +7.85789i q^{37} -0.788146i q^{39} -1.72114i q^{41} +6.68005 q^{43} +1.43682 q^{45} +11.8901 q^{47} +(6.22988 + 3.19196i) q^{49} -4.22162 q^{51} +4.72464i q^{53} -3.10097 q^{55} +4.70683 q^{57} -1.93446i q^{59} -13.1193 q^{61} +(-6.11823 - 1.47614i) q^{63} +0.604005 q^{65} -7.82402 q^{67} +0.831788 q^{69} +9.20651i q^{71} -16.1420i q^{73} +3.65320i q^{75} +(13.2045 + 3.18583i) q^{77} +8.46903i q^{79} +3.79529 q^{81} +10.2816i q^{83} -3.23528i q^{85} +5.95238 q^{87} +12.9348i q^{89} +(-2.57195 - 0.620533i) q^{91} +2.02531i q^{93} +3.60713i q^{95} +14.1965i q^{97} -12.2129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.788146i		−	0.455036i	−0.973774	−	0.227518i	$-0.926939\pi$
$3$		−	0.788146i		−	0.455036i	0.973774	−	0.227518i	$-0.0730611\pi$
$4$		0			0
$5$	0.604005			0.270119			0.135060	−	0.990837i	$-0.456877\pi$
$5$	0.604005			0.270119			0.135060	+	0.990837i	$0.456877\pi$
$6$		0			0
$7$	−2.57195	−	0.620533i	−0.972107	−	0.234539i
$7$	−2.57195	−	0.620533i	−0.972107	−	0.234539i
$8$		0			0
$9$	2.37883			0.792942
$10$		0			0
$11$	−5.13402			−1.54797			−0.773983	−	0.633206i	$-0.781738\pi$
$11$	−5.13402			−1.54797			−0.773983	+	0.633206i	$0.781738\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$		−	0.476044i		−	0.122914i
$16$		0			0
$17$		−	5.35639i		−	1.29911i	−0.760313	−	0.649557i	$-0.774954\pi$
$17$		−	5.35639i		−	1.29911i	0.760313	−	0.649557i	$-0.225046\pi$
$18$		0			0
$19$			5.97202i			1.37008i	0.728507	+	0.685038i	$0.240215\pi$
$19$			5.97202i			1.37008i	−0.728507	+	0.685038i	$0.759785\pi$
$20$		0			0
$21$	−0.489070	+	2.02707i	−0.106724	+	0.442344i
$22$		0			0
$23$			1.05537i			0.220061i	0.993928	+	0.110030i	$0.0350948\pi$
$23$			1.05537i			0.220061i	−0.993928	+	0.110030i	$0.964905\pi$
$24$		0			0
$25$	−4.63518			−0.927036
$26$		0			0
$27$		−	4.23930i		−	0.815854i
$28$		0			0
$29$			7.55238i			1.40244i	0.712944	+	0.701221i	$0.247361\pi$
$29$			7.55238i			1.40244i	−0.712944	+	0.701221i	$0.752639\pi$
$30$		0			0
$31$	−2.56972			−0.461535			−0.230767	−	0.973009i	$-0.574124\pi$
$31$	−2.56972			−0.461535			−0.230767	+	0.973009i	$0.574124\pi$
$32$		0			0
$33$			4.04636i			0.704381i
$34$		0			0
$35$	−1.55347	−	0.374805i	−0.262585	−	0.0633536i
$36$		0			0
$37$			7.85789i			1.29183i	0.763410	+	0.645914i	$0.223524\pi$
$37$			7.85789i			1.29183i	−0.763410	+	0.645914i	$0.776476\pi$
$38$		0			0
$39$		−	0.788146i		−	0.126204i
$40$		0			0
$41$		−	1.72114i		−	0.268797i	−0.990927	−	0.134398i	$-0.957090\pi$
$41$		−	1.72114i		−	0.268797i	0.990927	−	0.134398i	$-0.0429102\pi$
$42$		0			0
$43$	6.68005			1.01870			0.509349	−	0.860560i	$-0.329886\pi$
$43$	6.68005			1.01870			0.509349	+	0.860560i	$0.329886\pi$
$44$		0			0
$45$	1.43682			0.214189
$46$		0			0
$47$	11.8901			1.73435			0.867177	−	0.498000i	$-0.165932\pi$
$47$	11.8901			1.73435			0.867177	+	0.498000i	$0.165932\pi$
$48$		0			0
$49$	6.22988	+	3.19196i	0.889983	+	0.455994i
$50$		0			0
$51$	−4.22162			−0.591144
$52$		0			0
$53$			4.72464i			0.648979i	0.945889	+	0.324489i	$0.105192\pi$
$53$			4.72464i			0.648979i	−0.945889	+	0.324489i	$0.894808\pi$
$54$		0			0
$55$	−3.10097			−0.418135
$56$		0			0
$57$	4.70683			0.623435
$58$		0			0
$59$		−	1.93446i		−	0.251845i	−0.992040	−	0.125922i	$-0.959811\pi$
$59$		−	1.93446i		−	0.251845i	0.992040	−	0.125922i	$-0.0401890\pi$
$60$		0			0
$61$	−13.1193			−1.67975			−0.839877	−	0.542777i	$-0.817373\pi$
$61$	−13.1193			−1.67975			−0.839877	+	0.542777i	$0.817373\pi$
$62$		0			0
$63$	−6.11823	−	1.47614i	−0.770824	−	0.185976i
$64$		0			0
$65$	0.604005			0.0749176
$66$		0			0
$67$	−7.82402			−0.955856			−0.477928	−	0.878399i	$-0.658612\pi$
$67$	−7.82402			−0.955856			−0.477928	+	0.878399i	$0.658612\pi$
$68$		0			0
$69$	0.831788			0.100136
$70$		0			0
$71$			9.20651i			1.09261i	0.837586	+	0.546306i	$0.183966\pi$
$71$			9.20651i			1.09261i	−0.837586	+	0.546306i	$0.816034\pi$
$72$		0			0
$73$		−	16.1420i		−	1.88927i	−0.328119	−	0.944636i	$-0.606415\pi$
$73$		−	16.1420i		−	1.88927i	0.328119	−	0.944636i	$-0.393585\pi$
$74$		0			0
$75$			3.65320i			0.421835i
$76$		0			0
$77$	13.2045	+	3.18583i	1.50479	+	0.363059i
$78$		0			0
$79$			8.46903i			0.952840i	0.879218	+	0.476420i	$0.158066\pi$
$79$			8.46903i			0.952840i	−0.879218	+	0.476420i	$0.841934\pi$
$80$		0			0
$81$	3.79529			0.421699
$82$		0			0
$83$			10.2816i			1.12856i	0.825585	+	0.564278i	$0.190846\pi$
$83$			10.2816i			1.12856i	−0.825585	+	0.564278i	$0.809154\pi$
$84$		0			0
$85$		−	3.23528i		−	0.350916i
$86$		0			0
$87$	5.95238			0.638162
$88$		0			0
$89$			12.9348i			1.37109i	0.728031	+	0.685544i	$0.240435\pi$
$89$			12.9348i			1.37109i	−0.728031	+	0.685544i	$0.759565\pi$
$90$		0			0
$91$	−2.57195	−	0.620533i	−0.269614	−	0.0650495i
$92$		0			0
$93$			2.02531i			0.210015i
$94$		0			0
$95$			3.60713i			0.370084i
$96$		0			0
$97$			14.1965i			1.44143i	0.693231	+	0.720716i	$0.256187\pi$
$97$			14.1965i			1.44143i	−0.693231	+	0.720716i	$0.743813\pi$
$98$		0			0
$99$	−12.2129			−1.22745
$100$		0			0
$101$	−1.54943			−0.154174			−0.0770872	−	0.997024i	$-0.524562\pi$
$101$	−1.54943			−0.154174			−0.0770872	+	0.997024i	$0.524562\pi$
$102$		0			0
$103$	−6.73825			−0.663939			−0.331970	−	0.943290i	$-0.607713\pi$
$103$	−6.73825			−0.663939			−0.331970	+	0.943290i	$0.607713\pi$
$104$		0			0
$105$	−0.295401	+	1.22436i	−0.0288282	+	0.119486i
$106$		0			0
$107$	14.0761			1.36079			0.680396	−	0.732844i	$-0.261808\pi$
$107$	14.0761			1.36079			0.680396	+	0.732844i	$0.261808\pi$
$108$		0			0
$109$			8.86808i			0.849408i	0.905332	+	0.424704i	$0.139622\pi$
$109$			8.86808i			0.849408i	−0.905332	+	0.424704i	$0.860378\pi$
$110$		0			0
$111$	6.19316			0.587829
$112$		0			0
$113$	−6.37194			−0.599422			−0.299711	−	0.954030i	$-0.596890\pi$
$113$	−6.37194			−0.599422			−0.299711	+	0.954030i	$0.596890\pi$
$114$		0			0
$115$			0.637451i			0.0594426i
$116$		0			0
$117$	2.37883			0.219923
$118$		0			0
$119$	−3.32381	+	13.7764i	−0.304693	+	1.26288i
$120$		0			0
$121$	15.3582			1.39620
$122$		0			0
$123$	−1.35651			−0.122312
$124$		0			0
$125$	−5.81970			−0.520529
$126$		0			0
$127$			4.16676i			0.369740i	0.982763	+	0.184870i	$0.0591864\pi$
$127$			4.16676i			0.369740i	−0.982763	+	0.184870i	$0.940814\pi$
$128$		0			0
$129$		−	5.26485i		−	0.463545i
$130$		0			0
$131$		−	11.7402i		−	1.02575i	−0.858464	−	0.512874i	$-0.828581\pi$
$131$		−	11.7402i		−	1.02575i	0.858464	−	0.512874i	$-0.171419\pi$
$132$		0			0
$133$	3.70584	−	15.3598i	0.321337	−	1.33186i
$134$		0			0
$135$		−	2.56056i		−	0.220378i
$136$		0			0
$137$	−12.7548			−1.08972			−0.544859	−	0.838527i	$-0.683417\pi$
$137$	−12.7548			−1.08972			−0.544859	+	0.838527i	$0.683417\pi$
$138$		0			0
$139$		−	10.8552i		−	0.920729i	−0.887730	−	0.460364i	$-0.847719\pi$
$139$		−	10.8552i		−	0.920729i	0.887730	−	0.460364i	$-0.152281\pi$
$140$		0			0
$141$		−	9.37116i		−	0.789194i
$142$		0			0
$143$	−5.13402			−0.429329
$144$		0			0
$145$			4.56167i			0.378826i
$146$		0			0
$147$	2.51573	−	4.91005i	0.207494	−	0.404974i
$148$		0			0
$149$			10.1922i			0.834976i	0.908682	+	0.417488i	$0.137090\pi$
$149$			10.1922i			0.834976i	−0.908682	+	0.417488i	$0.862910\pi$
$150$		0			0
$151$			5.82290i			0.473861i	0.971527	+	0.236931i	$0.0761414\pi$
$151$			5.82290i			0.473861i	−0.971527	+	0.236931i	$0.923859\pi$
$152$		0			0
$153$		−	12.7419i		−	1.03012i
$154$		0			0
$155$	−1.55212			−0.124669
$156$		0			0
$157$	17.2526			1.37691			0.688455	−	0.725279i	$-0.258289\pi$
$157$	17.2526			1.37691			0.688455	+	0.725279i	$0.258289\pi$
$158$		0			0
$159$	3.72370			0.295309
$160$		0			0
$161$	0.654894	−	2.71437i	0.0516128	−	0.213922i
$162$		0			0
$163$	−9.26702			−0.725849			−0.362924	−	0.931819i	$-0.618222\pi$
$163$	−9.26702			−0.725849			−0.362924	+	0.931819i	$0.618222\pi$
$164$		0			0
$165$			2.44402i			0.190267i
$166$		0			0
$167$	8.08007			0.625255			0.312627	−	0.949876i	$-0.398791\pi$
$167$	8.08007			0.625255			0.312627	+	0.949876i	$0.398791\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$			14.2064i			1.08639i
$172$		0			0
$173$	−1.96960			−0.149746			−0.0748730	−	0.997193i	$-0.523855\pi$
$173$	−1.96960			−0.149746			−0.0748730	+	0.997193i	$0.523855\pi$
$174$		0			0
$175$	11.9215	+	2.87628i	0.901177	+	0.217426i
$176$		0			0
$177$	−1.52463			−0.114599
$178$		0			0
$179$	6.88265			0.514434			0.257217	−	0.966354i	$-0.417195\pi$
$179$	6.88265			0.514434			0.257217	+	0.966354i	$0.417195\pi$
$180$		0			0
$181$	−5.06451			−0.376442			−0.188221	−	0.982127i	$-0.560272\pi$
$181$	−5.06451			−0.376442			−0.188221	+	0.982127i	$0.560272\pi$
$182$		0			0
$183$			10.3399i			0.764349i
$184$		0			0
$185$			4.74620i			0.348948i
$186$		0			0
$187$			27.4998i			2.01099i
$188$		0			0
$189$	−2.63062	+	10.9033i	−0.191350	+	0.793097i
$190$		0			0
$191$		−	3.97407i		−	0.287554i	−0.989610	−	0.143777i	$-0.954075\pi$
$191$		−	3.97407i		−	0.287554i	0.989610	−	0.143777i	$-0.0459248\pi$
$192$		0			0
$193$	12.6351			0.909495			0.454748	−	0.890620i	$-0.349729\pi$
$193$	12.6351			0.909495			0.454748	+	0.890620i	$0.349729\pi$
$194$		0			0
$195$		−	0.476044i		−	0.0340902i
$196$		0			0
$197$			24.3796i			1.73697i	0.495711	+	0.868487i	$0.334907\pi$
$197$			24.3796i			1.73697i	−0.495711	+	0.868487i	$0.665093\pi$
$198$		0			0
$199$	−7.03688			−0.498831			−0.249416	−	0.968397i	$-0.580239\pi$
$199$	−7.03688			−0.498831			−0.249416	+	0.968397i	$0.580239\pi$
$200$		0			0
$201$			6.16647i			0.434949i
$202$		0			0
$203$	4.68650	−	19.4244i	0.328928	−	1.36332i
$204$		0			0
$205$		−	1.03958i		−	0.0726072i
$206$		0			0
$207$			2.51055i			0.174495i
$208$		0			0
$209$		−	30.6605i		−	2.12083i
$210$		0			0
$211$	−19.1341			−1.31725			−0.658623	−	0.752473i	$-0.728861\pi$
$211$	−19.1341			−1.31725			−0.658623	+	0.752473i	$0.728861\pi$
$212$		0			0
$213$	7.25607			0.497178
$214$		0			0
$215$	4.03478			0.275170
$216$		0			0
$217$	6.60919	+	1.59459i	0.448661	+	0.108248i
$218$		0			0
$219$	−12.7222			−0.859688
$220$		0			0
$221$		−	5.35639i		−	0.360310i
$222$		0			0
$223$	27.2121			1.82226			0.911130	−	0.412120i	$-0.135211\pi$
$223$	27.2121			1.82226			0.911130	+	0.412120i	$0.135211\pi$
$224$		0			0
$225$	−11.0263			−0.735085
$226$		0			0
$227$		−	7.73995i		−	0.513718i	−0.966449	−	0.256859i	$-0.917312\pi$
$227$		−	7.73995i		−	0.513718i	0.966449	−	0.256859i	$-0.0826876\pi$
$228$		0			0
$229$	−4.25200			−0.280980			−0.140490	−	0.990082i	$-0.544868\pi$
$229$	−4.25200			−0.280980			−0.140490	+	0.990082i	$0.544868\pi$
$230$		0			0
$231$	2.51090	−	10.4070i	0.165205	−	0.684733i
$232$		0			0
$233$	−24.4234			−1.60003			−0.800016	−	0.599979i	$-0.795176\pi$
$233$	−24.4234			−1.60003			−0.800016	+	0.599979i	$0.795176\pi$
$234$		0			0
$235$	7.18170			0.468482
$236$		0			0
$237$	6.67483			0.433577
$238$		0			0
$239$			14.0846i			0.911058i	0.890221	+	0.455529i	$0.150550\pi$
$239$			14.0846i			0.911058i	−0.890221	+	0.455529i	$0.849450\pi$
$240$		0			0
$241$		−	3.77081i		−	0.242899i	−0.992598	−	0.121449i	$-0.961246\pi$
$241$		−	3.77081i		−	0.242899i	0.992598	−	0.121449i	$-0.0387542\pi$
$242$		0			0
$243$		−	15.7091i		−	1.00774i
$244$		0			0
$245$	3.76288	+	1.92796i	0.240401	+	0.123173i
$246$		0			0
$247$			5.97202i			0.379991i
$248$		0			0
$249$	8.10344			0.513534
$250$		0			0
$251$			12.0792i			0.762432i	0.924486	+	0.381216i	$0.124495\pi$
$251$			12.0792i			0.762432i	−0.924486	+	0.381216i	$0.875505\pi$
$252$		0			0
$253$		−	5.41831i		−	0.340646i
$254$		0			0
$255$	−2.54988			−0.159679
$256$		0			0
$257$			5.97005i			0.372402i	0.982512	+	0.186201i	$0.0596175\pi$
$257$			5.97005i			0.372402i	−0.982512	+	0.186201i	$0.940382\pi$
$258$		0			0
$259$	4.87607	−	20.2101i	0.302984	−	1.25580i
$260$		0			0
$261$			17.9658i			1.11205i
$262$		0			0
$263$			24.4407i			1.50708i	0.657403	+	0.753540i	$0.271655\pi$
$263$			24.4407i			1.50708i	−0.657403	+	0.753540i	$0.728345\pi$
$264$		0			0
$265$			2.85370i			0.175302i
$266$		0			0
$267$	10.1945			0.623895
$268$		0			0
$269$	15.0253			0.916107			0.458054	−	0.888925i	$-0.348547\pi$
$269$	15.0253			0.916107			0.458054	+	0.888925i	$0.348547\pi$
$270$		0			0
$271$	2.88011			0.174954			0.0874772	−	0.996167i	$-0.472120\pi$
$271$	2.88011			0.174954			0.0874772	+	0.996167i	$0.472120\pi$
$272$		0			0
$273$	−0.489070	+	2.02707i	−0.0295999	+	0.122684i
$274$		0			0
$275$	23.7971			1.43502
$276$		0			0
$277$		−	26.4457i		−	1.58897i	−0.607285	−	0.794484i	$-0.707742\pi$
$277$		−	26.4457i		−	1.58897i	0.607285	−	0.794484i	$-0.292258\pi$
$278$		0			0
$279$	−6.11291			−0.365970
$280$		0			0
$281$	−16.6213			−0.991544			−0.495772	−	0.868453i	$-0.665115\pi$
$281$	−16.6213			−0.991544			−0.495772	+	0.868453i	$0.665115\pi$
$282$		0			0
$283$			24.7211i			1.46952i	0.678328	+	0.734760i	$0.262705\pi$
$283$			24.7211i			1.46952i	−0.678328	+	0.734760i	$0.737295\pi$
$284$		0			0
$285$	2.84295			0.168402
$286$		0			0
$287$	−1.06802	+	4.42669i	−0.0630434	+	0.261299i
$288$		0			0
$289$	−11.6909			−0.687699
$290$		0			0
$291$	11.1889			0.655904
$292$		0			0
$293$	−24.8485			−1.45166			−0.725831	−	0.687873i	$-0.758545\pi$
$293$	−24.8485			−1.45166			−0.725831	+	0.687873i	$0.758545\pi$
$294$		0			0
$295$		−	1.16842i		−	0.0680281i
$296$		0			0
$297$			21.7647i			1.26291i
$298$		0			0
$299$			1.05537i			0.0610338i
$300$		0			0
$301$	−17.1808	−	4.14519i	−0.990283	−	0.238925i
$302$		0			0
$303$			1.22118i			0.0701550i
$304$		0			0
$305$	−7.92412			−0.453734
$306$		0			0
$307$			6.93218i			0.395640i	0.980238	+	0.197820i	$0.0633862\pi$
$307$			6.93218i			0.395640i	−0.980238	+	0.197820i	$0.936614\pi$
$308$		0			0
$309$			5.31072i			0.302116i
$310$		0			0
$311$	−25.0732			−1.42177			−0.710884	−	0.703309i	$-0.751705\pi$
$311$	−25.0732			−1.42177			−0.710884	+	0.703309i	$0.751705\pi$
$312$		0			0
$313$			10.5548i			0.596594i	0.954473	+	0.298297i	$0.0964187\pi$
$313$			10.5548i			0.596594i	−0.954473	+	0.298297i	$0.903581\pi$
$314$		0			0
$315$	−3.69544	−	0.891595i	−0.208214	−	0.0502357i
$316$		0			0
$317$		−	10.3502i		−	0.581327i	−0.956825	−	0.290663i	$-0.906124\pi$
$317$		−	10.3502i		−	0.581327i	0.956825	−	0.290663i	$-0.0938760\pi$
$318$		0			0
$319$		−	38.7741i		−	2.17093i
$320$		0			0
$321$		−	11.0941i		−	0.619210i
$322$		0			0
$323$	31.9885			1.77989
$324$		0			0
$325$	−4.63518			−0.257113
$326$		0			0
$327$	6.98935			0.386512
$328$		0			0
$329$	−30.5808	−	7.37821i	−1.68598	−	0.406774i
$330$		0			0
$331$	−0.259825			−0.0142813			−0.00714063	−	0.999975i	$-0.502273\pi$
$331$	−0.259825			−0.0142813			−0.00714063	+	0.999975i	$0.502273\pi$
$332$		0			0
$333$			18.6925i			1.02435i
$334$		0			0
$335$	−4.72575			−0.258195
$336$		0			0
$337$	15.7576			0.858373			0.429186	−	0.903216i	$-0.358800\pi$
$337$	15.7576			0.858373			0.429186	+	0.903216i	$0.358800\pi$
$338$		0			0
$339$			5.02202i			0.272759i
$340$		0			0
$341$	13.1930			0.714440
$342$		0			0
$343$	−14.0422	−	12.0754i	−0.758210	−	0.652011i
$344$		0			0
$345$	0.502404			0.0270485
$346$		0			0
$347$	5.05895			0.271579			0.135789	−	0.990738i	$-0.456643\pi$
$347$	5.05895			0.271579			0.135789	+	0.990738i	$0.456643\pi$
$348$		0			0
$349$	11.5491			0.618207			0.309104	−	0.951028i	$-0.399971\pi$
$349$	11.5491			0.618207			0.309104	+	0.951028i	$0.399971\pi$
$350$		0			0
$351$		−	4.23930i		−	0.226277i
$352$		0			0
$353$			19.9048i			1.05943i	0.848177	+	0.529713i	$0.177700\pi$
$353$			19.9048i			1.05943i	−0.848177	+	0.529713i	$0.822300\pi$
$354$		0			0
$355$			5.56078i			0.295135i
$356$		0			0
$357$	10.8578	+	2.61965i	0.574655	+	0.138647i
$358$		0			0
$359$		−	13.0862i		−	0.690662i	−0.938481	−	0.345331i	$-0.887767\pi$
$359$		−	13.0862i		−	0.690662i	0.938481	−	0.345331i	$-0.112233\pi$
$360$		0			0
$361$	−16.6651			−0.877109
$362$		0			0
$363$		−	12.1045i		−	0.635321i
$364$		0			0
$365$		−	9.74982i		−	0.510329i
$366$		0			0
$367$	−22.9170			−1.19626			−0.598128	−	0.801401i	$-0.704089\pi$
$367$	−22.9170			−1.19626			−0.598128	+	0.801401i	$0.704089\pi$
$368$		0			0
$369$		−	4.09429i		−	0.213140i
$370$		0			0
$371$	2.93179	−	12.1515i	0.152211	−	0.630877i
$372$		0			0
$373$		−	30.1413i		−	1.56066i	−0.625369	−	0.780329i	$-0.715052\pi$
$373$		−	30.1413i		−	1.56066i	0.625369	−	0.780329i	$-0.284948\pi$
$374$		0			0
$375$			4.58677i			0.236860i
$376$		0			0
$377$			7.55238i			0.388967i
$378$		0			0
$379$	−5.38785			−0.276755			−0.138378	−	0.990380i	$-0.544189\pi$
$379$	−5.38785			−0.276755			−0.138378	+	0.990380i	$0.544189\pi$
$380$		0			0
$381$	3.28401			0.168245
$382$		0			0
$383$	1.92001			0.0981080			0.0490540	−	0.998796i	$-0.484379\pi$
$383$	1.92001			0.0981080			0.0490540	+	0.998796i	$0.484379\pi$
$384$		0			0
$385$	7.97556	+	1.92426i	0.406472	+	0.0980692i
$386$		0			0
$387$	15.8907			0.807768
$388$		0			0
$389$		−	1.39413i		−	0.0706852i	−0.999375	−	0.0353426i	$-0.988748\pi$
$389$		−	1.39413i		−	0.0706852i	0.999375	−	0.0353426i	$-0.0112522\pi$
$390$		0			0
$391$	5.65299			0.285884
$392$		0			0
$393$	−9.25301			−0.466753
$394$		0			0
$395$			5.11533i			0.257380i
$396$		0			0
$397$	−16.7676			−0.841540			−0.420770	−	0.907167i	$-0.638240\pi$
$397$	−16.7676			−0.841540			−0.420770	+	0.907167i	$0.638240\pi$
$398$		0			0
$399$	−12.1057	−	2.92074i	−0.606045	−	0.146220i
$400$		0			0
$401$	28.1223			1.40436			0.702179	−	0.712000i	$-0.252211\pi$
$401$	28.1223			1.40436			0.702179	+	0.712000i	$0.252211\pi$
$402$		0			0
$403$	−2.56972			−0.128007
$404$		0			0
$405$	2.29237			0.113909
$406$		0			0
$407$		−	40.3426i		−	1.99971i
$408$		0			0
$409$			11.4027i			0.563825i	0.959440	+	0.281912i	$0.0909688\pi$
$409$			11.4027i			0.563825i	−0.959440	+	0.281912i	$0.909031\pi$
$410$		0			0
$411$			10.0527i			0.495862i
$412$		0			0
$413$	−1.20039	+	4.97533i	−0.0590675	+	0.244820i
$414$		0			0
$415$			6.21016i			0.304845i
$416$		0			0
$417$	−8.55551			−0.418965
$418$		0			0
$419$		−	17.5806i		−	0.858870i	−0.903098	−	0.429435i	$-0.858713\pi$
$419$		−	17.5806i		−	0.858870i	0.903098	−	0.429435i	$-0.141287\pi$
$420$		0			0
$421$		−	9.28914i		−	0.452725i	−0.974043	−	0.226363i	$-0.927317\pi$
$421$		−	9.28914i		−	0.452725i	0.974043	−	0.226363i	$-0.0726834\pi$
$422$		0			0
$423$	28.2845			1.37524
$424$		0			0
$425$			24.8278i			1.20433i
$426$		0			0
$427$	33.7422	+	8.14095i	1.63290	+	0.393968i
$428$		0			0
$429$			4.04636i			0.195360i
$430$		0			0
$431$		−	28.3363i		−	1.36491i	−0.730928	−	0.682455i	$-0.760912\pi$
$431$		−	28.3363i		−	1.36491i	0.730928	−	0.682455i	$-0.239088\pi$
$432$		0			0
$433$		−	27.0957i		−	1.30213i	−0.759020	−	0.651067i	$-0.774322\pi$
$433$		−	27.0957i		−	1.30213i	0.759020	−	0.651067i	$-0.225678\pi$
$434$		0			0
$435$	3.59527			0.172380
$436$		0			0
$437$	−6.30272			−0.301500
$438$		0			0
$439$	28.2082			1.34630			0.673152	−	0.739504i	$-0.264940\pi$
$439$	28.2082			1.34630			0.673152	+	0.739504i	$0.264940\pi$
$440$		0			0
$441$	14.8198	+	7.59312i	0.705705	+	0.361577i
$442$		0			0
$443$	−3.44165			−0.163518			−0.0817589	−	0.996652i	$-0.526054\pi$
$443$	−3.44165			−0.163518			−0.0817589	+	0.996652i	$0.526054\pi$
$444$		0			0
$445$			7.81269i			0.370357i
$446$		0			0
$447$	8.03293			0.379945
$448$		0			0
$449$	24.1129			1.13796			0.568980	−	0.822352i	$-0.307338\pi$
$449$	24.1129			1.13796			0.568980	+	0.822352i	$0.307338\pi$
$450$		0			0
$451$			8.83637i			0.416088i
$452$		0			0
$453$	4.58930			0.215624
$454$		0			0
$455$	−1.55347	−	0.374805i	−0.0728279	−	0.0175711i
$456$		0			0
$457$	−28.6440			−1.33991			−0.669954	−	0.742403i	$-0.733686\pi$
$457$	−28.6440			−1.33991			−0.669954	+	0.742403i	$0.733686\pi$
$458$		0			0
$459$	−22.7073			−1.05989
$460$		0			0
$461$	−33.8682			−1.57740			−0.788699	−	0.614779i	$-0.789245\pi$
$461$	−33.8682			−1.57740			−0.788699	+	0.614779i	$0.789245\pi$
$462$		0			0
$463$		−	8.87772i		−	0.412582i	−0.978491	−	0.206291i	$-0.933861\pi$
$463$		−	8.87772i		−	0.412582i	0.978491	−	0.206291i	$-0.0661394\pi$
$464$		0			0
$465$			1.22330i			0.0567291i
$466$		0			0
$467$			13.3956i			0.619875i	0.950757	+	0.309937i	$0.100308\pi$
$467$			13.3956i			0.619875i	−0.950757	+	0.309937i	$0.899692\pi$
$468$		0			0
$469$	20.1230	+	4.85506i	0.929194	+	0.224186i
$470$		0			0
$471$		−	13.5976i		−	0.626544i
$472$		0			0
$473$	−34.2955			−1.57691
$474$		0			0
$475$		−	27.6814i		−	1.27011i
$476$		0			0
$477$			11.2391i			0.514603i
$478$		0			0
$479$	−40.5532			−1.85292			−0.926462	−	0.376387i	$-0.877166\pi$
$479$	−40.5532			−1.85292			−0.926462	+	0.376387i	$0.877166\pi$
$480$		0			0
$481$			7.85789i			0.358289i
$482$		0			0
$483$	−2.13932	−	0.516152i	−0.0973425	−	0.0234857i
$484$		0			0
$485$			8.57473i			0.389358i
$486$		0			0
$487$			11.2722i			0.510790i	0.966837	+	0.255395i	$0.0822055\pi$
$487$			11.2722i			0.510790i	−0.966837	+	0.255395i	$0.917794\pi$
$488$		0			0
$489$			7.30376i			0.330288i
$490$		0			0
$491$	−9.12152			−0.411649			−0.205824	−	0.978589i	$-0.565988\pi$
$491$	−9.12152			−0.411649			−0.205824	+	0.978589i	$0.565988\pi$
$492$		0			0
$493$	40.4535			1.82193
$494$		0			0
$495$	−7.37668			−0.331557
$496$		0			0
$497$	5.71294	−	23.6787i	0.256260	−	1.06213i
$498$		0			0
$499$	−7.93384			−0.355168			−0.177584	−	0.984106i	$-0.556828\pi$
$499$	−7.93384			−0.355168			−0.177584	+	0.984106i	$0.556828\pi$
$500$		0			0
$501$		−	6.36828i		−	0.284514i
$502$		0			0
$503$	12.2543			0.546393			0.273197	−	0.961958i	$-0.411919\pi$
$503$	12.2543			0.546393			0.273197	+	0.961958i	$0.411919\pi$
$504$		0			0
$505$	−0.935866			−0.0416455
$506$		0			0
$507$		−	0.788146i		−	0.0350028i
$508$		0			0
$509$	−17.4470			−0.773326			−0.386663	−	0.922221i	$-0.626372\pi$
$509$	−17.4470			−0.773326			−0.386663	+	0.922221i	$0.626372\pi$
$510$		0			0
$511$	−10.0166	+	41.5163i	−0.443109	+	1.83657i
$512$		0			0
$513$	25.3172			1.11778
$514$		0			0
$515$	−4.06993			−0.179343
$516$		0			0
$517$	−61.0442			−2.68472
$518$		0			0
$519$			1.55233i			0.0681399i
$520$		0			0
$521$		−	20.8509i		−	0.913496i	−0.889596	−	0.456748i	$-0.849014\pi$
$521$		−	20.8509i		−	0.913496i	0.889596	−	0.456748i	$-0.150986\pi$
$522$		0			0
$523$		−	12.5142i		−	0.547206i	−0.961843	−	0.273603i	$-0.911785\pi$
$523$		−	12.5142i		−	0.547206i	0.961843	−	0.273603i	$-0.0882154\pi$
$524$		0			0
$525$	2.26693	−	9.39585i	0.0989368	−	0.410068i
$526$		0			0
$527$			13.7644i			0.599587i
$528$		0			0
$529$	21.8862			0.951573
$530$		0			0
$531$		−	4.60173i		−	0.199698i
$532$		0			0
$533$		−	1.72114i		−	0.0745508i
$534$		0			0
$535$	8.50206			0.367576
$536$		0			0
$537$		−	5.42454i		−	0.234086i
$538$		0			0
$539$	−31.9843	−	16.3876i	−1.37766	−	0.705864i
$540$		0			0
$541$		−	25.7050i		−	1.10514i	−0.833465	−	0.552572i	$-0.813647\pi$
$541$		−	25.7050i		−	1.10514i	0.833465	−	0.552572i	$-0.186353\pi$
$542$		0			0
$543$			3.99158i			0.171295i
$544$		0			0
$545$			5.35637i			0.229442i
$546$		0			0
$547$	−5.61543			−0.240099			−0.120049	−	0.992768i	$-0.538305\pi$
$547$	−5.61543			−0.240099			−0.120049	+	0.992768i	$0.538305\pi$
$548$		0			0
$549$	−31.2085			−1.33195
$550$		0			0
$551$	−45.1030			−1.92145
$552$		0			0
$553$	5.25531	−	21.7819i	0.223478	−	0.926262i
$554$		0			0
$555$	3.74070			0.158784
$556$		0			0
$557$			9.65380i			0.409045i	0.978862	+	0.204522i	$0.0655641\pi$
$557$			9.65380i			0.409045i	−0.978862	+	0.204522i	$0.934436\pi$
$558$		0			0
$559$	6.68005			0.282536
$560$		0			0
$561$	21.6739			0.915072
$562$		0			0
$563$			19.3230i			0.814368i	0.913346	+	0.407184i	$0.133489\pi$
$563$			19.3230i			0.814368i	−0.913346	+	0.407184i	$0.866511\pi$
$564$		0			0
$565$	−3.84868			−0.161915
$566$		0			0
$567$	−9.76131	−	2.35510i	−0.409936	−	0.0989049i
$568$		0			0
$569$	5.19422			0.217753			0.108876	−	0.994055i	$-0.465275\pi$
$569$	5.19422			0.217753			0.108876	+	0.994055i	$0.465275\pi$
$570$		0			0
$571$	−9.89370			−0.414038			−0.207019	−	0.978337i	$-0.566376\pi$
$571$	−9.89370			−0.414038			−0.207019	+	0.978337i	$0.566376\pi$
$572$		0			0
$573$	−3.13215			−0.130847
$574$		0			0
$575$		−	4.89184i		−	0.204004i
$576$		0			0
$577$			3.62103i			0.150745i	0.997155	+	0.0753726i	$0.0240146\pi$
$577$			3.62103i			0.150745i	−0.997155	+	0.0753726i	$0.975985\pi$
$578$		0			0
$579$		−	9.95831i		−	0.413853i
$580$		0			0
$581$	6.38009	−	26.4439i	0.264691	−	1.09708i
$582$		0			0
$583$		−	24.2564i		−	1.00460i
$584$		0			0
$585$	1.43682			0.0594053
$586$		0			0
$587$			3.04130i			0.125528i	0.998028	+	0.0627640i	$0.0199915\pi$
$587$			3.04130i			0.125528i	−0.998028	+	0.0627640i	$0.980008\pi$
$588$		0			0
$589$		−	15.3464i		−	0.632338i
$590$		0			0
$591$	19.2147			0.790387
$592$		0			0
$593$			32.2710i			1.32521i	0.748969	+	0.662606i	$0.230549\pi$
$593$			32.2710i			1.32521i	−0.748969	+	0.662606i	$0.769451\pi$
$594$		0			0
$595$	−2.00760	+	8.32100i	−0.0823036	+	0.341128i
$596$		0			0
$597$			5.54609i			0.226986i
$598$		0			0
$599$		−	13.7558i		−	0.562047i	−0.959701	−	0.281023i	$-0.909326\pi$
$599$		−	13.7558i		−	0.562047i	0.959701	−	0.281023i	$-0.0906739\pi$
$600$		0			0
$601$		−	46.7682i		−	1.90771i	−0.300260	−	0.953857i	$-0.597073\pi$
$601$		−	46.7682i		−	1.90771i	0.300260	−	0.953857i	$-0.402927\pi$
$602$		0			0
$603$	−18.6120			−0.757939
$604$		0			0
$605$	9.27642			0.377140
$606$		0			0
$607$	−41.2226			−1.67317			−0.836587	−	0.547835i	$-0.815452\pi$
$607$	−41.2226			−1.67317			−0.836587	+	0.547835i	$0.815452\pi$
$608$		0			0
$609$	−15.3092	−	3.69364i	−0.620361	−	0.149674i
$610$		0			0
$611$	11.8901			0.481023
$612$		0			0
$613$		−	11.7561i		−	0.474825i	−0.971409	−	0.237412i	$-0.923701\pi$
$613$		−	11.7561i		−	0.474825i	0.971409	−	0.237412i	$-0.0762992\pi$
$614$		0			0
$615$	−0.819338			−0.0330389
$616$		0			0
$617$	−33.8298			−1.36194			−0.680969	−	0.732312i	$-0.738441\pi$
$617$	−33.8298			−1.36194			−0.680969	+	0.732312i	$0.738441\pi$
$618$		0			0
$619$			24.2768i			0.975768i	0.872909	+	0.487884i	$0.162231\pi$
$619$			24.2768i			0.975768i	−0.872909	+	0.487884i	$0.837769\pi$
$620$		0			0
$621$	4.47405			0.179537
$622$		0			0
$623$	8.02647	−	33.2677i	0.321574	−	1.33284i
$624$		0			0
$625$	19.6608			0.786431
$626$		0			0
$627$	−24.1650			−0.965055
$628$		0			0
$629$	42.0899			1.67823
$630$		0			0
$631$			10.6353i			0.423385i	0.977336	+	0.211693i	$0.0678976\pi$
$631$			10.6353i			0.423385i	−0.977336	+	0.211693i	$0.932102\pi$
$632$		0			0
$633$			15.0805i			0.599395i
$634$		0			0
$635$			2.51674i			0.0998738i
$636$		0			0
$637$	6.22988	+	3.19196i	0.246837	+	0.126470i
$638$		0			0
$639$			21.9007i			0.866378i
$640$		0			0
$641$	15.9566			0.630247			0.315124	−	0.949051i	$-0.397954\pi$
$641$	15.9566			0.630247			0.315124	+	0.949051i	$0.397954\pi$
$642$		0			0
$643$		−	44.0339i		−	1.73653i	−0.496103	−	0.868264i	$-0.665236\pi$
$643$		−	44.0339i		−	1.73653i	0.496103	−	0.868264i	$-0.334764\pi$
$644$		0			0
$645$		−	3.18000i		−	0.125212i
$646$		0			0
$647$	16.6254			0.653611			0.326805	−	0.945092i	$-0.394028\pi$
$647$	16.6254			0.653611			0.326805	+	0.945092i	$0.394028\pi$
$648$		0			0
$649$			9.93154i			0.389847i
$650$		0			0
$651$	1.25677	−	5.20901i	0.0492568	−	0.204157i
$652$		0			0
$653$		−	20.1847i		−	0.789889i	−0.918705	−	0.394944i	$-0.870764\pi$
$653$		−	20.1847i		−	0.789889i	0.918705	−	0.394944i	$-0.129236\pi$
$654$		0			0
$655$		−	7.09116i		−	0.277074i
$656$		0			0
$657$		−	38.3989i		−	1.49808i
$658$		0			0
$659$	10.9591			0.426906			0.213453	−	0.976953i	$-0.431529\pi$
$659$	10.9591			0.426906			0.213453	+	0.976953i	$0.431529\pi$
$660$		0			0
$661$	−32.7400			−1.27344			−0.636718	−	0.771096i	$-0.719709\pi$
$661$	−32.7400			−1.27344			−0.636718	+	0.771096i	$0.719709\pi$
$662$		0			0
$663$	−4.22162			−0.163954
$664$		0			0
$665$	2.23834	−	9.27737i	0.0867992	−	0.359761i
$666$		0			0
$667$	−7.97058			−0.308622
$668$		0			0
$669$		−	21.4471i		−	0.829194i
$670$		0			0
$671$	67.3548			2.60020
$672$		0			0
$673$	21.9038			0.844329			0.422165	−	0.906519i	$-0.361270\pi$
$673$	21.9038			0.844329			0.422165	+	0.906519i	$0.361270\pi$
$674$		0			0
$675$			19.6499i			0.756325i
$676$		0			0
$677$	1.86436			0.0716533			0.0358267	−	0.999358i	$-0.488594\pi$
$677$	1.86436			0.0716533			0.0358267	+	0.999358i	$0.488594\pi$
$678$		0			0
$679$	8.80936	−	36.5126i	0.338072	−	1.40122i
$680$		0			0
$681$	−6.10021			−0.233760
$682$		0			0
$683$	−21.5663			−0.825211			−0.412605	−	0.910910i	$-0.635381\pi$
$683$	−21.5663			−0.825211			−0.412605	+	0.910910i	$0.635381\pi$
$684$		0			0
$685$	−7.70398			−0.294354
$686$		0			0
$687$			3.35119i			0.127856i
$688$		0			0
$689$			4.72464i			0.179994i
$690$		0			0
$691$			20.4851i			0.779289i	0.920965	+	0.389645i	$0.127402\pi$
$691$			20.4851i			0.779289i	−0.920965	+	0.389645i	$0.872598\pi$
$692$		0			0
$693$	31.4111	+	7.57853i	1.19321	+	0.287885i
$694$		0			0
$695$		−	6.55661i		−	0.248707i
$696$		0			0
$697$	−9.21909			−0.349198
$698$		0			0
$699$			19.2492i			0.728072i
$700$		0			0
$701$		−	24.2801i		−	0.917048i	−0.888682	−	0.458524i	$-0.848378\pi$
$701$		−	24.2801i		−	0.917048i	0.888682	−	0.458524i	$-0.151622\pi$
$702$		0			0
$703$	−46.9275			−1.76990
$704$		0			0
$705$		−	5.66023i		−	0.213176i
$706$		0			0
$707$	3.98507	+	0.961474i	0.149874	+	0.0361600i
$708$		0			0
$709$		−	10.2080i		−	0.383368i	−0.981457	−	0.191684i	$-0.938605\pi$
$709$		−	10.2080i		−	0.383368i	0.981457	−	0.191684i	$-0.0613949\pi$
$710$		0			0
$711$			20.1463i			0.755547i
$712$		0			0
$713$		−	2.71201i		−	0.101566i
$714$		0			0
$715$	−3.10097			−0.115970
$716$		0			0
$717$	11.1007			0.414564
$718$		0			0
$719$	−0.929036			−0.0346472			−0.0173236	−	0.999850i	$-0.505515\pi$
$719$	−0.929036			−0.0346472			−0.0173236	+	0.999850i	$0.505515\pi$
$720$		0			0
$721$	17.3304	+	4.18130i	0.645420	+	0.155720i
$722$		0			0
$723$	−2.97195			−0.110528
$724$		0			0
$725$		−	35.0066i		−	1.30011i
$726$		0			0
$727$	−24.9639			−0.925859			−0.462930	−	0.886395i	$-0.653202\pi$
$727$	−24.9639			−0.925859			−0.462930	+	0.886395i	$0.653202\pi$
$728$		0			0
$729$	−0.995229			−0.0368603
$730$		0			0
$731$		−	35.7809i		−	1.32341i
$732$		0			0
$733$	−3.77004			−0.139250			−0.0696249	−	0.997573i	$-0.522180\pi$
$733$	−3.77004			−0.139250			−0.0696249	+	0.997573i	$0.522180\pi$
$734$		0			0
$735$	1.51951	−	2.96570i	0.0560481	−	0.109391i
$736$		0			0
$737$	40.1687			1.47963
$738$		0			0
$739$	−0.387742			−0.0142633			−0.00713166	−	0.999975i	$-0.502270\pi$
$739$	−0.387742			−0.0142633			−0.00713166	+	0.999975i	$0.502270\pi$
$740$		0			0
$741$	4.70683			0.172910
$742$		0			0
$743$			13.3639i			0.490274i	0.969488	+	0.245137i	$0.0788330\pi$
$743$			13.3639i			0.490274i	−0.969488	+	0.245137i	$0.921167\pi$
$744$		0			0
$745$			6.15613i			0.225543i
$746$		0			0
$747$			24.4582i			0.894880i
$748$		0			0
$749$	−36.2032	−	8.73471i	−1.32284	−	0.319159i
$750$		0			0
$751$		−	31.4243i		−	1.14669i	−0.819314	−	0.573345i	$-0.805646\pi$
$751$		−	31.4243i		−	1.14669i	0.819314	−	0.573345i	$-0.194354\pi$
$752$		0			0
$753$	9.52017			0.346934
$754$		0			0
$755$			3.51706i			0.127999i
$756$		0			0
$757$			15.2414i			0.553958i	0.960876	+	0.276979i	$0.0893333\pi$
$757$			15.2414i			0.553958i	−0.960876	+	0.276979i	$0.910667\pi$
$758$		0			0
$759$	−4.27042			−0.155006
$760$		0			0
$761$		−	17.4267i		−	0.631717i	−0.948806	−	0.315858i	$-0.897708\pi$
$761$		−	17.4267i		−	0.631717i	0.948806	−	0.315858i	$-0.102292\pi$
$762$		0			0
$763$	5.50293	−	22.8083i	0.199220	−	0.825716i
$764$		0			0
$765$		−	7.69618i		−	0.278256i
$766$		0			0
$767$		−	1.93446i		−	0.0698492i
$768$		0			0
$769$		−	2.05841i		−	0.0742280i	−0.999311	−	0.0371140i	$-0.988184\pi$
$769$		−	2.05841i		−	0.0742280i	0.999311	−	0.0371140i	$-0.0118165\pi$
$770$		0			0
$771$	4.70527			0.169456
$772$		0			0
$773$	19.4738			0.700424			0.350212	−	0.936671i	$-0.386110\pi$
$773$	19.4738			0.700424			0.350212	+	0.936671i	$0.386110\pi$
$774$		0			0
$775$	11.9111			0.427859
$776$		0			0
$777$	−15.9285	−	3.84306i	−0.571432	−	0.137869i
$778$		0			0
$779$	10.2787			0.368272
$780$		0			0
$781$		−	47.2664i		−	1.69133i
$782$		0			0
$783$	32.0168			1.14419
$784$		0			0
$785$	10.4207			0.371930
$786$		0			0
$787$		−	52.1659i		−	1.85951i	−0.368174	−	0.929757i	$-0.620017\pi$
$787$		−	52.1659i		−	1.85951i	0.368174	−	0.929757i	$-0.379983\pi$
$788$		0			0
$789$	19.2629			0.685776
$790$		0			0
$791$	16.3883	+	3.95400i	0.582702	+	0.140588i
$792$		0			0
$793$	−13.1193			−0.465880
$794$		0			0
$795$	2.24914			0.0797686
$796$		0			0
$797$	−30.1064			−1.06642			−0.533212	−	0.845982i	$-0.679015\pi$
$797$	−30.1064			−1.06642			−0.533212	+	0.845982i	$0.679015\pi$
$798$		0			0
$799$		−	63.6881i		−	2.25312i
$800$		0			0
$801$			30.7697i			1.08719i
$802$		0			0
$803$			82.8732i			2.92453i
$804$		0			0
$805$	0.395559	−	1.63949i	0.0139416	−	0.0577845i
$806$		0			0
$807$		−	11.8421i		−	0.416862i
$808$		0			0
$809$	8.47468			0.297954			0.148977	−	0.988841i	$-0.452402\pi$
$809$	8.47468			0.297954			0.148977	+	0.988841i	$0.452402\pi$
$810$		0			0
$811$		−	13.2805i		−	0.466340i	−0.972436	−	0.233170i	$-0.925090\pi$
$811$		−	13.2805i		−	0.466340i	0.972436	−	0.233170i	$-0.0749099\pi$
$812$		0			0
$813$		−	2.26995i		−	0.0796106i
$814$		0			0
$815$	−5.59732			−0.196066
$816$		0			0
$817$			39.8934i			1.39569i
$818$		0			0
$819$	−6.11823	−	1.47614i	−0.213788	−	0.0515805i
$820$		0			0
$821$		−	4.19157i		−	0.146287i	−0.997321	−	0.0731434i	$-0.976697\pi$
$821$		−	4.19157i		−	0.146287i	0.997321	−	0.0731434i	$-0.0233031\pi$
$822$		0			0
$823$		−	38.0868i		−	1.32762i	−0.747900	−	0.663812i	$-0.768938\pi$
$823$		−	38.0868i		−	1.32762i	0.747900	−	0.663812i	$-0.231062\pi$
$824$		0			0
$825$		−	18.7556i		−	0.652986i
$826$		0			0
$827$	19.5284			0.679069			0.339534	−	0.940594i	$-0.389731\pi$
$827$	19.5284			0.679069			0.339534	+	0.940594i	$0.389731\pi$
$828$		0			0
$829$	−32.1858			−1.11786			−0.558930	−	0.829215i	$-0.688788\pi$
$829$	−32.1858			−1.11786			−0.558930	+	0.829215i	$0.688788\pi$
$830$		0			0
$831$	−20.8431			−0.723038
$832$		0			0
$833$	17.0974	−	33.3696i	0.592389	−	1.15619i
$834$		0			0
$835$	4.88040			0.168893
$836$		0			0
$837$			10.8938i			0.376545i
$838$		0			0
$839$	5.30188			0.183041			0.0915206	−	0.995803i	$-0.470827\pi$
$839$	5.30188			0.183041			0.0915206	+	0.995803i	$0.470827\pi$
$840$		0			0
$841$	−28.0384			−0.966842
$842$		0			0
$843$			13.1000i			0.451188i
$844$		0			0
$845$	0.604005			0.0207784
$846$		0			0
$847$	−39.5005	−	9.53025i	−1.35725	−	0.327463i
$848$		0			0
$849$	19.4839			0.668685
$850$		0			0
$851$	−8.29301			−0.284281
$852$		0			0
$853$	−26.8385			−0.918934			−0.459467	−	0.888195i	$-0.651960\pi$
$853$	−26.8385			−0.918934			−0.459467	+	0.888195i	$0.651960\pi$
$854$		0			0
$855$			8.58074i			0.293455i
$856$		0			0
$857$			28.6223i			0.977718i	0.872363	+	0.488859i	$0.162587\pi$
$857$			28.6223i			0.977718i	−0.872363	+	0.488859i	$0.837413\pi$
$858$		0			0
$859$			21.8562i			0.745726i	0.927886	+	0.372863i	$0.121624\pi$
$859$			21.8562i			0.745726i	−0.927886	+	0.372863i	$0.878376\pi$
$860$		0			0
$861$	3.48888	+	0.841758i	0.118901	+	0.0286870i
$862$		0			0
$863$			40.7632i			1.38760i	0.720170	+	0.693798i	$0.244064\pi$
$863$			40.7632i			1.38760i	−0.720170	+	0.693798i	$0.755936\pi$
$864$		0			0
$865$	−1.18965			−0.0404493
$866$		0			0
$867$			9.21413i			0.312928i
$868$		0			0
$869$		−	43.4802i		−	1.47496i
$870$		0			0
$871$	−7.82402			−0.265107
$872$		0			0
$873$			33.7709i			1.14297i
$874$		0			0
$875$	14.9680	+	3.61131i	0.506010	+	0.122085i
$876$		0			0
$877$		−	10.3771i		−	0.350408i	−0.984532	−	0.175204i	$-0.943941\pi$
$877$		−	10.3771i		−	0.350408i	0.984532	−	0.175204i	$-0.0560586\pi$
$878$		0			0
$879$			19.5842i			0.660559i
$880$		0			0
$881$			36.0522i			1.21463i	0.794461	+	0.607316i	$0.207754\pi$
$881$			36.0522i			1.21463i	−0.794461	+	0.607316i	$0.792246\pi$
$882$		0			0
$883$	−44.0134			−1.48117			−0.740585	−	0.671963i	$-0.765451\pi$
$883$	−44.0134			−1.48117			−0.740585	+	0.671963i	$0.765451\pi$
$884$		0			0
$885$	−0.920886			−0.0309553
$886$		0			0
$887$	24.5694			0.824960			0.412480	−	0.910967i	$-0.364663\pi$
$887$	24.5694			0.824960			0.412480	+	0.910967i	$0.364663\pi$
$888$		0			0
$889$	2.58561	−	10.7167i	0.0867185	−	0.359427i
$890$		0			0
$891$	−19.4851			−0.652776
$892$		0			0
$893$			71.0082i			2.37620i
$894$		0			0
$895$	4.15716			0.138958
$896$		0			0
$897$	0.831788			0.0277726
$898$		0			0
$899$		−	19.4075i		−	0.647275i
$900$		0			0
$901$	25.3070			0.843098
$902$		0			0
$903$	−3.26701	+	13.5410i	−0.108719	+	0.450615i
$904$		0			0
$905$	−3.05899			−0.101684
$906$		0			0
$907$	24.3438			0.808322			0.404161	−	0.914688i	$-0.367564\pi$
$907$	24.3438			0.808322			0.404161	+	0.914688i	$0.367564\pi$
$908$		0			0
$909$	−3.68583			−0.122251
$910$		0			0
$911$			6.47030i			0.214371i	0.994239	+	0.107185i	$0.0341838\pi$
$911$			6.47030i			0.214371i	−0.994239	+	0.107185i	$0.965816\pi$
$912$		0			0
$913$		−	52.7862i		−	1.74697i
$914$		0			0
$915$			6.24537i			0.206465i
$916$		0			0
$917$	−7.28519	+	30.1953i	−0.240578	+	0.997137i
$918$		0			0
$919$			2.05771i			0.0678775i	0.999424	+	0.0339388i	$0.0108051\pi$
$919$			2.05771i			0.0678775i	−0.999424	+	0.0339388i	$0.989195\pi$
$920$		0			0
$921$	5.46357			0.180031
$922$		0			0
$923$			9.20651i			0.303036i
$924$		0			0
$925$		−	36.4227i		−	1.19757i
$926$		0			0
$927$	−16.0291			−0.526465
$928$		0			0
$929$		−	20.0057i		−	0.656367i	−0.944614	−	0.328184i	$-0.893564\pi$
$929$		−	20.0057i		−	0.656367i	0.944614	−	0.328184i	$-0.106436\pi$
$930$		0			0
$931$	−19.0625	+	37.2050i	−0.624747	+	1.21934i
$932$		0			0
$933$			19.7613i			0.646956i
$934$		0			0
$935$			16.6100i			0.543206i
$936$		0			0
$937$			19.3261i			0.631355i	0.948867	+	0.315677i	$0.102232\pi$
$937$			19.3261i			0.631355i	−0.948867	+	0.315677i	$0.897768\pi$
$938$		0			0
$939$	8.31875			0.271472
$940$		0			0
$941$	54.5495			1.77826			0.889132	−	0.457651i	$-0.151309\pi$
$941$	54.5495			1.77826			0.889132	+	0.457651i	$0.151309\pi$
$942$		0			0
$943$	1.81645			0.0591516
$944$		0			0
$945$	−1.58891	+	6.58563i	−0.0516872	+	0.214231i
$946$		0			0
$947$	−41.5250			−1.34938			−0.674691	−	0.738100i	$-0.735723\pi$
$947$	−41.5250			−1.34938			−0.674691	+	0.738100i	$0.735723\pi$
$948$		0			0
$949$		−	16.1420i		−	0.523990i
$950$		0			0
$951$	−8.15749			−0.264525
$952$		0			0
$953$	8.36225			0.270880			0.135440	−	0.990786i	$-0.456755\pi$
$953$	8.36225			0.270880			0.135440	+	0.990786i	$0.456755\pi$
$954$		0			0
$955$		−	2.40036i		−	0.0776738i
$956$		0			0
$957$	−30.5596			−0.987853
$958$		0			0
$959$	32.8048	+	7.91479i	1.05932	+	0.255582i
$960$		0			0
$961$	−24.3966			−0.786986
$962$		0			0
$963$	33.4847			1.07903
$964$		0			0
$965$	7.63167			0.245672
$966$		0			0
$967$		−	52.3203i		−	1.68251i	−0.540639	−	0.841254i	$-0.681818\pi$
$967$		−	52.3203i		−	1.68251i	0.540639	−	0.841254i	$-0.318182\pi$
$968$		0			0
$969$		−	25.2116i		−	0.809913i
$970$		0			0
$971$			41.0903i			1.31865i	0.751857	+	0.659326i	$0.229158\pi$
$971$			41.0903i			1.31865i	−0.751857	+	0.659326i	$0.770842\pi$
$972$		0			0
$973$	−6.73602	+	27.9191i	−0.215947	+	0.895047i
$974$		0			0
$975$			3.65320i			0.116996i
$976$		0			0
$977$	4.44196			0.142111			0.0710555	−	0.997472i	$-0.477363\pi$
$977$	4.44196			0.142111			0.0710555	+	0.997472i	$0.477363\pi$
$978$		0			0
$979$		−	66.4076i		−	2.12240i
$980$		0			0
$981$			21.0956i			0.673532i
$982$		0			0
$983$	3.41333			0.108868			0.0544342	−	0.998517i	$-0.482664\pi$
$983$	3.41333			0.108868			0.0544342	+	0.998517i	$0.482664\pi$
$984$		0			0
$985$			14.7254i			0.469190i
$986$		0			0
$987$	−5.81511	+	24.1022i	−0.185097	+	0.767181i
$988$		0			0
$989$			7.04995i			0.224175i
$990$		0			0
$991$			46.3067i			1.47098i	0.677535	+	0.735490i	$0.263048\pi$
$991$			46.3067i			1.47098i	−0.677535	+	0.735490i	$0.736952\pi$
$992$		0			0
$993$			0.204780i			0.00649849i
$994$		0			0
$995$	−4.25031			−0.134744
$996$		0			0
$997$	38.1823			1.20924			0.604622	−	0.796512i	$-0.293324\pi$
$997$	38.1823			1.20924			0.604622	+	0.796512i	$0.293324\pi$
$998$		0			0
$999$	33.3119			1.05394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.b.2575.19		48
4.3	odd	2		728.2.h.b.27.26	yes	48
7.6	odd	2		2912.2.h.a.2575.30		48
8.3	odd	2		2912.2.h.a.2575.19		48
8.5	even	2		728.2.h.a.27.25	✓	48
28.27	even	2		728.2.h.a.27.26	yes	48
56.13	odd	2		728.2.h.b.27.25	yes	48
56.27	even	2	inner	2912.2.h.b.2575.30		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.25	✓	48	8.5	even	2
728.2.h.a.27.26	yes	48	28.27	even	2
728.2.h.b.27.25	yes	48	56.13	odd	2
728.2.h.b.27.26	yes	48	4.3	odd	2
2912.2.h.a.2575.19		48	8.3	odd	2
2912.2.h.a.2575.30		48	7.6	odd	2
2912.2.h.b.2575.19		48	1.1	even	1	trivial
2912.2.h.b.2575.30		48	56.27	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.788146i q^{3} +0.604005 q^{5} +(-2.57195 - 0.620533i) q^{7} +2.37883 q^{9} -5.13402 q^{11} +1.00000 q^{13} -0.476044i q^{15} -5.35639i q^{17} +5.97202i q^{19} +(-0.489070 + 2.02707i) q^{21} +1.05537i q^{23} -4.63518 q^{25} -4.23930i q^{27} +7.55238i q^{29} -2.56972 q^{31} +4.04636i q^{33} +(-1.55347 - 0.374805i) q^{35} +7.85789i q^{37} -0.788146i q^{39} -1.72114i q^{41} +6.68005 q^{43} +1.43682 q^{45} +11.8901 q^{47} +(6.22988 + 3.19196i) q^{49} -4.22162 q^{51} +4.72464i q^{53} -3.10097 q^{55} +4.70683 q^{57} -1.93446i q^{59} -13.1193 q^{61} +(-6.11823 - 1.47614i) q^{63} +0.604005 q^{65} -7.82402 q^{67} +0.831788 q^{69} +9.20651i q^{71} -16.1420i q^{73} +3.65320i q^{75} +(13.2045 + 3.18583i) q^{77} +8.46903i q^{79} +3.79529 q^{81} +10.2816i q^{83} -3.23528i q^{85} +5.95238 q^{87} +12.9348i q^{89} +(-2.57195 - 0.620533i) q^{91} +2.02531i q^{93} +3.60713i q^{95} +14.1965i q^{97} -12.2129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} + 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} - 24 q^{45} + 40 q^{51} + 20 q^{63} + 4 q^{67} + 20 q^{77} + 64 q^{81} - 40 q^{87} - 20 q^{99}+O(q^{100}) \) 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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