LMFDB - Embedded newform 2912.2.h.b.2575.33

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			2575.33
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.b.2575.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35635i q^{3} -2.01767 q^{5} +(-2.18534 + 1.49140i) q^{7} +1.16033 q^{9} +4.92824 q^{11} +1.00000 q^{13} -2.73666i q^{15} -4.50218i q^{17} +4.26085i q^{19} +(-2.02285 - 2.96408i) q^{21} -9.28723i q^{23} -0.929012 q^{25} +5.64284i q^{27} -6.03717i q^{29} +2.67926 q^{31} +6.68440i q^{33} +(4.40930 - 3.00915i) q^{35} +1.91540i q^{37} +1.35635i q^{39} -9.84828i q^{41} +1.98858 q^{43} -2.34116 q^{45} +6.61191 q^{47} +(2.55145 - 6.51844i) q^{49} +6.10651 q^{51} +7.90627i q^{53} -9.94356 q^{55} -5.77918 q^{57} -9.77831i q^{59} -10.0807 q^{61} +(-2.53571 + 1.73051i) q^{63} -2.01767 q^{65} +15.8803 q^{67} +12.5967 q^{69} -2.17824i q^{71} +7.87181i q^{73} -1.26006i q^{75} +(-10.7699 + 7.34998i) q^{77} +6.79571i q^{79} -4.17266 q^{81} +15.5710i q^{83} +9.08390i q^{85} +8.18848 q^{87} -5.33818i q^{89} +(-2.18534 + 1.49140i) q^{91} +3.63400i q^{93} -8.59698i q^{95} +5.40946i q^{97} +5.71838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			1.35635i			0.783086i	0.920160	+	0.391543i	$0.128059\pi$
$3$			1.35635i			0.783086i	−0.920160	+	0.391543i	$0.871941\pi$
$4$		0			0
$5$	−2.01767			−0.902329			−0.451164	−	0.892441i	$-0.648991\pi$
$5$	−2.01767			−0.902329			−0.451164	+	0.892441i	$0.648991\pi$
$6$		0			0
$7$	−2.18534	+	1.49140i	−0.825982	+	0.563696i
$7$	−2.18534	+	1.49140i	−0.825982	+	0.563696i
$8$		0			0
$9$	1.16033			0.386776
$10$		0			0
$11$	4.92824			1.48592			0.742961	−	0.669335i	$-0.233421\pi$
$11$	4.92824			1.48592			0.742961	+	0.669335i	$0.233421\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$		−	2.73666i		−	0.706601i
$16$		0			0
$17$		−	4.50218i		−	1.09194i	−0.837805	−	0.545969i	$-0.816162\pi$
$17$		−	4.50218i		−	1.09194i	0.837805	−	0.545969i	$-0.183838\pi$
$18$		0			0
$19$			4.26085i			0.977506i	0.872422	+	0.488753i	$0.162548\pi$
$19$			4.26085i			0.977506i	−0.872422	+	0.488753i	$0.837452\pi$
$20$		0			0
$21$	−2.02285	−	2.96408i	−0.441423	−	0.646815i
$22$		0			0
$23$		−	9.28723i		−	1.93652i	−0.249944	−	0.968260i	$-0.580412\pi$
$23$		−	9.28723i		−	1.93652i	0.249944	−	0.968260i	$-0.419588\pi$
$24$		0			0
$25$	−0.929012			−0.185802
$26$		0			0
$27$			5.64284i			1.08597i
$28$		0			0
$29$		−	6.03717i		−	1.12107i	−0.828130	−	0.560537i	$-0.810595\pi$
$29$		−	6.03717i		−	1.12107i	0.828130	−	0.560537i	$-0.189405\pi$
$30$		0			0
$31$	2.67926			0.481209			0.240605	−	0.970623i	$-0.422654\pi$
$31$	2.67926			0.481209			0.240605	+	0.970623i	$0.422654\pi$
$32$		0			0
$33$			6.68440i			1.16360i
$34$		0			0
$35$	4.40930	−	3.00915i	0.745308	−	0.508639i
$36$		0			0
$37$			1.91540i			0.314890i	0.987528	+	0.157445i	$0.0503257\pi$
$37$			1.91540i			0.314890i	−0.987528	+	0.157445i	$0.949674\pi$
$38$		0			0
$39$			1.35635i			0.217189i
$40$		0			0
$41$		−	9.84828i		−	1.53804i	−0.639223	−	0.769022i	$-0.720744\pi$
$41$		−	9.84828i		−	1.53804i	0.639223	−	0.769022i	$-0.279256\pi$
$42$		0			0
$43$	1.98858			0.303255			0.151628	−	0.988438i	$-0.451549\pi$
$43$	1.98858			0.303255			0.151628	+	0.988438i	$0.451549\pi$
$44$		0			0
$45$	−2.34116			−0.348999
$46$		0			0
$47$	6.61191			0.964446			0.482223	−	0.876048i	$-0.339829\pi$
$47$	6.61191			0.964446			0.482223	+	0.876048i	$0.339829\pi$
$48$		0			0
$49$	2.55145	−	6.51844i	0.364493	−	0.931206i
$50$		0			0
$51$	6.10651			0.855082
$52$		0			0
$53$			7.90627i			1.08601i	0.839730	+	0.543005i	$0.182713\pi$
$53$			7.90627i			1.08601i	−0.839730	+	0.543005i	$0.817287\pi$
$54$		0			0
$55$	−9.94356			−1.34079
$56$		0			0
$57$	−5.77918			−0.765471
$58$		0			0
$59$		−	9.77831i		−	1.27303i	−0.771265	−	0.636514i	$-0.780376\pi$
$59$		−	9.77831i		−	1.27303i	0.771265	−	0.636514i	$-0.219624\pi$
$60$		0			0
$61$	−10.0807			−1.29070			−0.645352	−	0.763885i	$-0.723289\pi$
$61$	−10.0807			−1.29070			−0.645352	+	0.763885i	$0.723289\pi$
$62$		0			0
$63$	−2.53571	+	1.73051i	−0.319470	+	0.218024i
$64$		0			0
$65$	−2.01767			−0.250261
$66$		0			0
$67$	15.8803			1.94008			0.970041	−	0.242942i	$-0.0781125\pi$
$67$	15.8803			1.94008			0.970041	+	0.242942i	$0.0781125\pi$
$68$		0			0
$69$	12.5967			1.51646
$70$		0			0
$71$		−	2.17824i		−	0.258510i	−0.991611	−	0.129255i	$-0.958741\pi$
$71$		−	2.17824i		−	0.258510i	0.991611	−	0.129255i	$-0.0412586\pi$
$72$		0			0
$73$			7.87181i			0.921326i	0.887575	+	0.460663i	$0.152388\pi$
$73$			7.87181i			0.921326i	−0.887575	+	0.460663i	$0.847612\pi$
$74$		0			0
$75$		−	1.26006i		−	0.145499i
$76$		0			0
$77$	−10.7699	+	7.34998i	−1.22734	+	0.837608i
$78$		0			0
$79$			6.79571i			0.764577i	0.924043	+	0.382289i	$0.124864\pi$
$79$			6.79571i			0.764577i	−0.924043	+	0.382289i	$0.875136\pi$
$80$		0			0
$81$	−4.17266			−0.463629
$82$		0			0
$83$			15.5710i			1.70913i	0.519340	+	0.854567i	$0.326178\pi$
$83$			15.5710i			1.70913i	−0.519340	+	0.854567i	$0.673822\pi$
$84$		0			0
$85$			9.08390i			0.985288i
$86$		0			0
$87$	8.18848			0.877897
$88$		0			0
$89$		−	5.33818i		−	0.565846i	−0.959143	−	0.282923i	$-0.908696\pi$
$89$		−	5.33818i		−	0.565846i	0.959143	−	0.282923i	$-0.0913041\pi$
$90$		0			0
$91$	−2.18534	+	1.49140i	−0.229086	+	0.156341i
$92$		0			0
$93$			3.63400i			0.376828i
$94$		0			0
$95$		−	8.59698i		−	0.882032i
$96$		0			0
$97$			5.40946i			0.549248i	0.961552	+	0.274624i	$0.0885533\pi$
$97$			5.40946i			0.549248i	−0.961552	+	0.274624i	$0.911447\pi$
$98$		0			0
$99$	5.71838			0.574718
$100$		0			0
$101$	14.7053			1.46323			0.731617	−	0.681716i	$-0.238766\pi$
$101$	14.7053			1.46323			0.731617	+	0.681716i	$0.238766\pi$
$102$		0			0
$103$	15.8101			1.55782			0.778909	−	0.627136i	$-0.215773\pi$
$103$	15.8101			1.55782			0.778909	+	0.627136i	$0.215773\pi$
$104$		0			0
$105$	4.08145	+	5.98053i	0.398309	+	0.583640i
$106$		0			0
$107$	5.79170			0.559904			0.279952	−	0.960014i	$-0.409681\pi$
$107$	5.79170			0.559904			0.279952	+	0.960014i	$0.409681\pi$
$108$		0			0
$109$		−	14.0192i		−	1.34279i	−0.741099	−	0.671396i	$-0.765695\pi$
$109$		−	14.0192i		−	1.34279i	0.741099	−	0.671396i	$-0.234305\pi$
$110$		0			0
$111$	−2.59795			−0.246586
$112$		0			0
$113$	−2.00196			−0.188328			−0.0941641	−	0.995557i	$-0.530018\pi$
$113$	−2.00196			−0.188328			−0.0941641	+	0.995557i	$0.530018\pi$
$114$		0			0
$115$			18.7385i			1.74738i
$116$		0			0
$117$	1.16033			0.107272
$118$		0			0
$119$	6.71455	+	9.83880i	0.615522	+	0.901922i
$120$		0			0
$121$	13.2876			1.20796
$122$		0			0
$123$	13.3577			1.20442
$124$		0			0
$125$	11.9628			1.06998
$126$		0			0
$127$			13.9958i			1.24193i	0.783839	+	0.620964i	$0.213259\pi$
$127$			13.9958i			1.24193i	−0.783839	+	0.620964i	$0.786741\pi$
$128$		0			0
$129$			2.69720i			0.237475i
$130$		0			0
$131$			0.946942i			0.0827347i	0.999144	+	0.0413674i	$0.0131714\pi$
$131$			0.946942i			0.0827347i	−0.999144	+	0.0413674i	$0.986829\pi$
$132$		0			0
$133$	−6.35463	−	9.31142i	−0.551016	−	0.807402i
$134$		0			0
$135$		−	11.3854i		−	0.979898i
$136$		0			0
$137$	13.4840			1.15201			0.576007	−	0.817445i	$-0.304610\pi$
$137$	13.4840			1.15201			0.576007	+	0.817445i	$0.304610\pi$
$138$		0			0
$139$		−	10.4050i		−	0.882544i	−0.897373	−	0.441272i	$-0.854527\pi$
$139$		−	10.4050i		−	0.882544i	0.897373	−	0.441272i	$-0.145473\pi$
$140$		0			0
$141$			8.96803i			0.755245i
$142$		0			0
$143$	4.92824			0.412120
$144$		0			0
$145$			12.1810i			1.01158i
$146$		0			0
$147$	8.84126	+	3.46065i	0.729215	+	0.285430i
$148$		0			0
$149$			4.85942i			0.398100i	0.979989	+	0.199050i	$0.0637855\pi$
$149$			4.85942i			0.398100i	−0.979989	+	0.199050i	$0.936214\pi$
$150$		0			0
$151$			1.05783i			0.0860854i	0.999073	+	0.0430427i	$0.0137051\pi$
$151$			1.05783i			0.0860854i	−0.999073	+	0.0430427i	$0.986295\pi$
$152$		0			0
$153$		−	5.22400i		−	0.422335i
$154$		0			0
$155$	−5.40586			−0.434209
$156$		0			0
$157$	8.12606			0.648531			0.324265	−	0.945966i	$-0.394883\pi$
$157$	8.12606			0.648531			0.324265	+	0.945966i	$0.394883\pi$
$158$		0			0
$159$	−10.7236			−0.850439
$160$		0			0
$161$	13.8510	+	20.2958i	1.09161	+	1.59953i
$162$		0			0
$163$	−9.36955			−0.733880			−0.366940	−	0.930245i	$-0.619595\pi$
$163$	−9.36955			−0.733880			−0.366940	+	0.930245i	$0.619595\pi$
$164$		0			0
$165$		−	13.4869i		−	1.04995i
$166$		0			0
$167$	−15.0871			−1.16747			−0.583737	−	0.811942i	$-0.698410\pi$
$167$	−15.0871			−1.16747			−0.583737	+	0.811942i	$0.698410\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$			4.94398i			0.378075i
$172$		0			0
$173$	9.07705			0.690116			0.345058	−	0.938581i	$-0.387859\pi$
$173$	9.07705			0.690116			0.345058	+	0.938581i	$0.387859\pi$
$174$		0			0
$175$	2.03021	−	1.38553i	0.153470	−	0.104736i
$176$		0			0
$177$	13.2628			0.996891
$178$		0			0
$179$	−16.7901			−1.25495			−0.627476	−	0.778636i	$-0.715912\pi$
$179$	−16.7901			−1.25495			−0.627476	+	0.778636i	$0.715912\pi$
$180$		0			0
$181$	3.66980			0.272774			0.136387	−	0.990656i	$-0.456451\pi$
$181$	3.66980			0.272774			0.136387	+	0.990656i	$0.456451\pi$
$182$		0			0
$183$		−	13.6729i		−	1.01073i
$184$		0			0
$185$		−	3.86465i		−	0.284135i
$186$		0			0
$187$		−	22.1878i		−	1.62253i
$188$		0			0
$189$	−8.41573	−	12.3315i	−0.612155	−	0.896988i
$190$		0			0
$191$			25.9830i			1.88007i	0.341082	+	0.940033i	$0.389206\pi$
$191$			25.9830i			1.88007i	−0.341082	+	0.940033i	$0.610794\pi$
$192$		0			0
$193$	18.2177			1.31134			0.655669	−	0.755049i	$-0.272387\pi$
$193$	18.2177			1.31134			0.655669	+	0.755049i	$0.272387\pi$
$194$		0			0
$195$		−	2.73666i		−	0.195976i
$196$		0			0
$197$		−	6.35885i		−	0.453049i	−0.974005	−	0.226525i	$-0.927264\pi$
$197$		−	6.35885i		−	0.453049i	0.974005	−	0.226525i	$-0.0727363\pi$
$198$		0			0
$199$	−7.66867			−0.543617			−0.271809	−	0.962351i	$-0.587622\pi$
$199$	−7.66867			−0.543617			−0.271809	+	0.962351i	$0.587622\pi$
$200$		0			0
$201$			21.5391i			1.51925i
$202$		0			0
$203$	9.00383	+	13.1933i	0.631945	+	0.925987i
$204$		0			0
$205$			19.8706i			1.38782i
$206$		0			0
$207$		−	10.7762i		−	0.748999i
$208$		0			0
$209$			20.9985i			1.45250i
$210$		0			0
$211$	−18.9359			−1.30360			−0.651801	−	0.758390i	$-0.725986\pi$
$211$	−18.9359			−1.30360			−0.651801	+	0.758390i	$0.725986\pi$
$212$		0			0
$213$	2.95445			0.202436
$214$		0			0
$215$	−4.01229			−0.273636
$216$		0			0
$217$	−5.85510	+	3.99585i	−0.397470	+	0.271256i
$218$		0			0
$219$	−10.6769			−0.721478
$220$		0			0
$221$		−	4.50218i		−	0.302849i
$222$		0			0
$223$	−0.523181			−0.0350348			−0.0175174	−	0.999847i	$-0.505576\pi$
$223$	−0.523181			−0.0350348			−0.0175174	+	0.999847i	$0.505576\pi$
$224$		0			0
$225$	−1.07796			−0.0718639
$226$		0			0
$227$			6.25764i			0.415334i	0.978200	+	0.207667i	$0.0665870\pi$
$227$			6.25764i			0.415334i	−0.978200	+	0.207667i	$0.933413\pi$
$228$		0			0
$229$	9.29723			0.614379			0.307189	−	0.951648i	$-0.400612\pi$
$229$	9.29723			0.614379			0.307189	+	0.951648i	$0.400612\pi$
$230$		0			0
$231$	−9.96912	−	14.6077i	−0.655920	−	0.961117i
$232$		0			0
$233$	−0.709761			−0.0464980			−0.0232490	−	0.999730i	$-0.507401\pi$
$233$	−0.709761			−0.0464980			−0.0232490	+	0.999730i	$0.507401\pi$
$234$		0			0
$235$	−13.3406			−0.870248
$236$		0			0
$237$	−9.21733			−0.598730
$238$		0			0
$239$		−	3.28084i		−	0.212220i	−0.994354	−	0.106110i	$-0.966160\pi$
$239$		−	3.28084i		−	0.212220i	0.994354	−	0.106110i	$-0.0338395\pi$
$240$		0			0
$241$		−	10.9276i		−	0.703907i	−0.936017	−	0.351954i	$-0.885517\pi$
$241$		−	10.9276i		−	0.703907i	0.936017	−	0.351954i	$-0.114483\pi$
$242$		0			0
$243$			11.2690i			0.722904i
$244$		0			0
$245$	−5.14798	+	13.1521i	−0.328893	+	0.840254i
$246$		0			0
$247$			4.26085i			0.271111i
$248$		0			0
$249$	−21.1196			−1.33840
$250$		0			0
$251$			11.2022i			0.707078i	0.935420	+	0.353539i	$0.115022\pi$
$251$			11.2022i			0.707078i	−0.935420	+	0.353539i	$0.884978\pi$
$252$		0			0
$253$		−	45.7697i		−	2.87752i
$254$		0			0
$255$	−12.3209			−0.771565
$256$		0			0
$257$		−	19.9074i		−	1.24179i	−0.783894	−	0.620894i	$-0.786770\pi$
$257$		−	19.9074i		−	1.24179i	0.783894	−	0.620894i	$-0.213230\pi$
$258$		0			0
$259$	−2.85663	−	4.18581i	−0.177502	−	0.260094i
$260$		0			0
$261$		−	7.00509i		−	0.433604i
$262$		0			0
$263$		−	25.7281i		−	1.58646i	−0.608919	−	0.793232i	$-0.708397\pi$
$263$		−	25.7281i		−	1.58646i	0.608919	−	0.793232i	$-0.291603\pi$
$264$		0			0
$265$		−	15.9522i		−	0.979938i
$266$		0			0
$267$	7.24042			0.443106
$268$		0			0
$269$	−22.5092			−1.37241			−0.686206	−	0.727407i	$-0.740725\pi$
$269$	−22.5092			−1.37241			−0.686206	+	0.727407i	$0.740725\pi$
$270$		0			0
$271$	26.5569			1.61322			0.806608	−	0.591086i	$-0.201301\pi$
$271$	26.5569			1.61322			0.806608	+	0.591086i	$0.201301\pi$
$272$		0			0
$273$	−2.02285	−	2.96408i	−0.122429	−	0.179394i
$274$		0			0
$275$	−4.57840			−0.276088
$276$		0			0
$277$		−	0.199021i		−	0.0119580i	−0.999982	−	0.00597902i	$-0.998097\pi$
$277$		−	0.199021i		−	0.0119580i	0.999982	−	0.00597902i	$-0.00190319\pi$
$278$		0			0
$279$	3.10882			0.186120
$280$		0			0
$281$	−2.06373			−0.123112			−0.0615558	−	0.998104i	$-0.519606\pi$
$281$	−2.06373			−0.123112			−0.0615558	+	0.998104i	$0.519606\pi$
$282$		0			0
$283$			4.87971i			0.290069i	0.989427	+	0.145034i	$0.0463293\pi$
$283$			4.87971i			0.290069i	−0.989427	+	0.145034i	$0.953671\pi$
$284$		0			0
$285$	11.6605			0.690707
$286$		0			0
$287$	14.6877	+	21.5219i	0.866989	+	1.27040i
$288$		0			0
$289$	−3.26961			−0.192330
$290$		0			0
$291$	−7.33710			−0.430108
$292$		0			0
$293$	−24.0764			−1.40656			−0.703280	−	0.710913i	$-0.748282\pi$
$293$	−24.0764			−1.40656			−0.703280	+	0.710913i	$0.748282\pi$
$294$		0			0
$295$			19.7294i			1.14869i
$296$		0			0
$297$			27.8093i			1.61366i
$298$		0			0
$299$		−	9.28723i		−	0.537094i
$300$		0			0
$301$	−4.34573	+	2.96577i	−0.250484	+	0.170944i
$302$		0			0
$303$			19.9455i			1.14584i
$304$		0			0
$305$	20.3396			1.16464
$306$		0			0
$307$		−	13.1626i		−	0.751229i	−0.926776	−	0.375615i	$-0.877432\pi$
$307$		−	13.1626i		−	0.751229i	0.926776	−	0.375615i	$-0.122568\pi$
$308$		0			0
$309$			21.4440i			1.21991i
$310$		0			0
$311$	11.0951			0.629146			0.314573	−	0.949233i	$-0.398139\pi$
$311$	11.0951			0.629146			0.314573	+	0.949233i	$0.398139\pi$
$312$		0			0
$313$		−	11.4566i		−	0.647563i	−0.946132	−	0.323782i	$-0.895046\pi$
$313$		−	11.4566i		−	0.647563i	0.946132	−	0.323782i	$-0.104954\pi$
$314$		0			0
$315$	5.11623	−	3.49160i	0.288267	−	0.196729i
$316$		0			0
$317$			4.30825i			0.241975i	0.992654	+	0.120988i	$0.0386061\pi$
$317$			4.30825i			0.241975i	−0.992654	+	0.120988i	$0.961394\pi$
$318$		0			0
$319$		−	29.7526i		−	1.66583i
$320$		0			0
$321$			7.85554i			0.438453i
$322$		0			0
$323$	19.1831			1.06738
$324$		0			0
$325$	−0.929012			−0.0515323
$326$		0			0
$327$	19.0148			1.05152
$328$		0			0
$329$	−14.4493	+	9.86100i	−0.796615	+	0.543655i
$330$		0			0
$331$	1.04258			0.0573055			0.0286527	−	0.999589i	$-0.490878\pi$
$331$	1.04258			0.0573055			0.0286527	+	0.999589i	$0.490878\pi$
$332$		0			0
$333$			2.22249i			0.121792i
$334$		0			0
$335$	−32.0411			−1.75059
$336$		0			0
$337$	−3.99839			−0.217806			−0.108903	−	0.994052i	$-0.534734\pi$
$337$	−3.99839			−0.217806			−0.108903	+	0.994052i	$0.534734\pi$
$338$		0			0
$339$		−	2.71534i		−	0.147477i
$340$		0			0
$341$	13.2040			0.715039
$342$		0			0
$343$	4.14581	+	18.0503i	0.223853	+	0.974623i
$344$		0			0
$345$	−25.4159			−1.36835
$346$		0			0
$347$	3.26373			0.175206			0.0876030	−	0.996155i	$-0.472079\pi$
$347$	3.26373			0.175206			0.0876030	+	0.996155i	$0.472079\pi$
$348$		0			0
$349$	−3.99174			−0.213673			−0.106837	−	0.994277i	$-0.534072\pi$
$349$	−3.99174			−0.213673			−0.106837	+	0.994277i	$0.534072\pi$
$350$		0			0
$351$			5.64284i			0.301193i
$352$		0			0
$353$			4.75882i			0.253286i	0.991948	+	0.126643i	$0.0404203\pi$
$353$			4.75882i			0.253286i	−0.991948	+	0.126643i	$0.959580\pi$
$354$		0			0
$355$			4.39497i			0.233261i
$356$		0			0
$357$	−13.3448	+	9.10725i	−0.706283	+	0.482007i
$358$		0			0
$359$		−	15.2984i		−	0.807418i	−0.914887	−	0.403709i	$-0.867721\pi$
$359$		−	15.2984i		−	0.807418i	0.914887	−	0.403709i	$-0.132279\pi$
$360$		0			0
$361$	0.845169			0.0444826
$362$		0			0
$363$			18.0226i			0.945939i
$364$		0			0
$365$		−	15.8827i		−	0.831339i
$366$		0			0
$367$	−24.0456			−1.25517			−0.627585	−	0.778548i	$-0.715956\pi$
$367$	−24.0456			−1.25517			−0.627585	+	0.778548i	$0.715956\pi$
$368$		0			0
$369$		−	11.4272i		−	0.594878i
$370$		0			0
$371$	−11.7914	−	17.2779i	−0.612180	−	0.897025i
$372$		0			0
$373$			7.07711i			0.366439i	0.983072	+	0.183219i	$0.0586519\pi$
$373$			7.07711i			0.366439i	−0.983072	+	0.183219i	$0.941348\pi$
$374$		0			0
$375$			16.2257i			0.837890i
$376$		0			0
$377$		−	6.03717i		−	0.310930i
$378$		0			0
$379$	−19.0848			−0.980322			−0.490161	−	0.871632i	$-0.663062\pi$
$379$	−19.0848			−0.980322			−0.490161	+	0.871632i	$0.663062\pi$
$380$		0			0
$381$	−18.9832			−0.972536
$382$		0			0
$383$	16.0328			0.819236			0.409618	−	0.912257i	$-0.365662\pi$
$383$	16.0328			0.819236			0.409618	+	0.912257i	$0.365662\pi$
$384$		0			0
$385$	21.7301	−	14.8298i	1.10747	−	0.755798i
$386$		0			0
$387$	2.30740			0.117292
$388$		0			0
$389$		−	23.9199i		−	1.21279i	−0.795164	−	0.606395i	$-0.792615\pi$
$389$		−	23.9199i		−	1.21279i	0.795164	−	0.606395i	$-0.207385\pi$
$390$		0			0
$391$	−41.8128			−2.11456
$392$		0			0
$393$	−1.28438			−0.0647884
$394$		0			0
$395$		−	13.7115i		−	0.689900i
$396$		0			0
$397$	20.5439			1.03107			0.515535	−	0.856869i	$-0.327593\pi$
$397$	20.5439			1.03107			0.515535	+	0.856869i	$0.327593\pi$
$398$		0			0
$399$	12.6295	−	8.61907i	0.632266	−	0.431493i
$400$		0			0
$401$	31.3007			1.56308			0.781542	−	0.623852i	$-0.214433\pi$
$401$	31.3007			1.56308			0.781542	+	0.623852i	$0.214433\pi$
$402$		0			0
$403$	2.67926			0.133463
$404$		0			0
$405$	8.41904			0.418346
$406$		0			0
$407$			9.43957i			0.467902i
$408$		0			0
$409$		−	15.6476i		−	0.773722i	−0.922138	−	0.386861i	$-0.873559\pi$
$409$		−	15.6476i		−	0.773722i	0.922138	−	0.386861i	$-0.126441\pi$
$410$		0			0
$411$			18.2889i			0.902126i
$412$		0			0
$413$	14.5834	+	21.3690i	0.717601	+	1.05150i
$414$		0			0
$415$		−	31.4170i		−	1.54220i
$416$		0			0
$417$	14.1128			0.691108
$418$		0			0
$419$			20.5740i			1.00511i	0.864546	+	0.502553i	$0.167606\pi$
$419$			20.5740i			1.00511i	−0.864546	+	0.502553i	$0.832394\pi$
$420$		0			0
$421$			21.4079i			1.04336i	0.853141	+	0.521680i	$0.174694\pi$
$421$			21.4079i			1.04336i	−0.853141	+	0.521680i	$0.825306\pi$
$422$		0			0
$423$	7.67198			0.373024
$424$		0			0
$425$			4.18258i			0.202885i
$426$		0			0
$427$	22.0298	−	15.0344i	1.06610	−	0.727565i
$428$		0			0
$429$			6.68440i			0.322726i
$430$		0			0
$431$			10.3475i			0.498422i	0.968449	+	0.249211i	$0.0801712\pi$
$431$			10.3475i			0.498422i	−0.968449	+	0.249211i	$0.919829\pi$
$432$		0			0
$433$			9.33066i			0.448403i	0.974543	+	0.224201i	$0.0719773\pi$
$433$			9.33066i			0.448403i	−0.974543	+	0.224201i	$0.928023\pi$
$434$		0			0
$435$	−16.5216			−0.792152
$436$		0			0
$437$	39.5715			1.89296
$438$		0			0
$439$	−1.46740			−0.0700351			−0.0350176	−	0.999387i	$-0.511149\pi$
$439$	−1.46740			−0.0700351			−0.0350176	+	0.999387i	$0.511149\pi$
$440$		0			0
$441$	2.96052	−	7.56353i	0.140977	−	0.360168i
$442$		0			0
$443$	15.1974			0.722049			0.361024	−	0.932556i	$-0.382427\pi$
$443$	15.1974			0.722049			0.361024	+	0.932556i	$0.382427\pi$
$444$		0			0
$445$			10.7707i			0.510579i
$446$		0			0
$447$	−6.59106			−0.311746
$448$		0			0
$449$	2.85625			0.134795			0.0673974	−	0.997726i	$-0.478530\pi$
$449$	2.85625			0.134795			0.0673974	+	0.997726i	$0.478530\pi$
$450$		0			0
$451$		−	48.5347i		−	2.28541i
$452$		0			0
$453$	−1.43479			−0.0674123
$454$		0			0
$455$	4.40930	−	3.00915i	0.206711	−	0.141071i
$456$		0			0
$457$	−18.5765			−0.868972			−0.434486	−	0.900679i	$-0.643070\pi$
$457$	−18.5765			−0.868972			−0.434486	+	0.900679i	$0.643070\pi$
$458$		0			0
$459$	25.4051			1.18581
$460$		0			0
$461$	15.7972			0.735751			0.367875	−	0.929875i	$-0.380085\pi$
$461$	15.7972			0.735751			0.367875	+	0.929875i	$0.380085\pi$
$462$		0			0
$463$		−	29.8412i		−	1.38684i	−0.720534	−	0.693420i	$-0.756103\pi$
$463$		−	29.8412i		−	1.38684i	0.720534	−	0.693420i	$-0.243897\pi$
$464$		0			0
$465$		−	7.33221i		−	0.340023i
$466$		0			0
$467$		−	28.2989i		−	1.30952i	−0.755838	−	0.654759i	$-0.772770\pi$
$467$		−	28.2989i		−	1.30952i	0.755838	−	0.654759i	$-0.227230\pi$
$468$		0			0
$469$	−34.7038	+	23.6838i	−1.60247	+	1.09362i
$470$		0			0
$471$			11.0218i			0.507855i
$472$		0			0
$473$	9.80020			0.450614
$474$		0			0
$475$		−	3.95838i		−	0.181623i
$476$		0			0
$477$			9.17386i			0.420042i
$478$		0			0
$479$	34.1779			1.56163			0.780814	−	0.624763i	$-0.214804\pi$
$479$	34.1779			1.56163			0.780814	+	0.624763i	$0.214804\pi$
$480$		0			0
$481$			1.91540i			0.0873349i
$482$		0			0
$483$	−27.5281	+	18.7867i	−1.25257	+	0.854825i
$484$		0			0
$485$		−	10.9145i		−	0.495602i
$486$		0			0
$487$			19.3724i			0.877846i	0.898525	+	0.438923i	$0.144640\pi$
$487$			19.3724i			0.877846i	−0.898525	+	0.438923i	$0.855360\pi$
$488$		0			0
$489$		−	12.7083i		−	0.574691i
$490$		0			0
$491$	−7.16183			−0.323209			−0.161604	−	0.986856i	$-0.551667\pi$
$491$	−7.16183			−0.323209			−0.161604	+	0.986856i	$0.551667\pi$
$492$		0			0
$493$	−27.1804			−1.22414
$494$		0			0
$495$	−11.5378			−0.518585
$496$		0			0
$497$	3.24863	+	4.76021i	0.145721	+	0.213525i
$498$		0			0
$499$	3.75215			0.167969			0.0839846	−	0.996467i	$-0.473235\pi$
$499$	3.75215			0.167969			0.0839846	+	0.996467i	$0.473235\pi$
$500$		0			0
$501$		−	20.4633i		−	0.914234i
$502$		0			0
$503$	−16.8477			−0.751200			−0.375600	−	0.926782i	$-0.622563\pi$
$503$	−16.8477			−0.751200			−0.375600	+	0.926782i	$0.622563\pi$
$504$		0			0
$505$	−29.6705			−1.32032
$506$		0			0
$507$			1.35635i			0.0602374i
$508$		0			0
$509$	27.6736			1.22661			0.613305	−	0.789846i	$-0.289840\pi$
$509$	27.6736			1.22661			0.613305	+	0.789846i	$0.289840\pi$
$510$		0			0
$511$	−11.7400	−	17.2026i	−0.519348	−	0.760999i
$512$		0			0
$513$	−24.0433			−1.06154
$514$		0			0
$515$	−31.8996			−1.40567
$516$		0			0
$517$	32.5851			1.43309
$518$		0			0
$519$			12.3116i			0.540420i
$520$		0			0
$521$		−	17.6143i		−	0.771696i	−0.922562	−	0.385848i	$-0.873909\pi$
$521$		−	17.6143i		−	0.771696i	0.922562	−	0.385848i	$-0.126091\pi$
$522$		0			0
$523$		−	11.0393i		−	0.482714i	−0.970436	−	0.241357i	$-0.922407\pi$
$523$		−	11.0393i		−	0.482714i	0.970436	−	0.241357i	$-0.0775926\pi$
$524$		0			0
$525$	1.87926	+	2.75367i	0.0820175	+	0.120180i
$526$		0			0
$527$		−	12.0625i		−	0.525451i
$528$		0			0
$529$	−63.2526			−2.75011
$530$		0			0
$531$		−	11.3460i		−	0.492376i
$532$		0			0
$533$		−	9.84828i		−	0.426576i
$534$		0			0
$535$	−11.6857			−0.505218
$536$		0			0
$537$		−	22.7732i		−	0.982736i
$538$		0			0
$539$	12.5742	−	32.1245i	0.541608	−	1.38370i
$540$		0			0
$541$		−	2.04925i		−	0.0881043i	−0.999029	−	0.0440522i	$-0.985973\pi$
$541$		−	2.04925i		−	0.0881043i	0.999029	−	0.0440522i	$-0.0140268\pi$
$542$		0			0
$543$			4.97751i			0.213606i
$544$		0			0
$545$			28.2860i			1.21164i
$546$		0			0
$547$	4.74971			0.203083			0.101542	−	0.994831i	$-0.467623\pi$
$547$	4.74971			0.203083			0.101542	+	0.994831i	$0.467623\pi$
$548$		0			0
$549$	−11.6969			−0.499213
$550$		0			0
$551$	25.7234			1.09586
$552$		0			0
$553$	−10.1351	−	14.8510i	−0.430989	−	0.631527i
$554$		0			0
$555$	5.24180			0.222502
$556$		0			0
$557$		−	3.12828i		−	0.132550i	−0.997801	−	0.0662748i	$-0.978889\pi$
$557$		−	3.12828i		−	0.132550i	0.997801	−	0.0662748i	$-0.0211114\pi$
$558$		0			0
$559$	1.98858			0.0841079
$560$		0			0
$561$	30.0944			1.27058
$562$		0			0
$563$			3.85763i			0.162580i	0.996691	+	0.0812898i	$0.0259039\pi$
$563$			3.85763i			0.162580i	−0.996691	+	0.0812898i	$0.974096\pi$
$564$		0			0
$565$	4.03929			0.169934
$566$		0			0
$567$	9.11869	−	6.22310i	0.382949	−	0.261346i
$568$		0			0
$569$	−31.1969			−1.30784			−0.653922	−	0.756562i	$-0.726877\pi$
$569$	−31.1969			−1.30784			−0.653922	+	0.756562i	$0.726877\pi$
$570$		0			0
$571$	−26.0516			−1.09023			−0.545113	−	0.838363i	$-0.683513\pi$
$571$	−26.0516			−1.09023			−0.545113	+	0.838363i	$0.683513\pi$
$572$		0			0
$573$	−35.2420			−1.47225
$574$		0			0
$575$			8.62795i			0.359810i
$576$		0			0
$577$		−	3.98708i		−	0.165984i	−0.996550	−	0.0829921i	$-0.973552\pi$
$577$		−	3.98708i		−	0.165984i	0.996550	−	0.0829921i	$-0.0264476\pi$
$578$		0			0
$579$			24.7095i			1.02689i
$580$		0			0
$581$	−23.2225	−	34.0279i	−0.963433	−	1.41171i
$582$		0			0
$583$			38.9640i			1.61373i
$584$		0			0
$585$	−2.34116			−0.0967949
$586$		0			0
$587$		−	27.6198i		−	1.13999i	−0.821648	−	0.569996i	$-0.806945\pi$
$587$		−	27.6198i		−	1.13999i	0.821648	−	0.569996i	$-0.193055\pi$
$588$		0			0
$589$			11.4159i			0.470385i
$590$		0			0
$591$	8.62479			0.354777
$592$		0			0
$593$			28.0907i			1.15355i	0.816904	+	0.576774i	$0.195689\pi$
$593$			28.0907i			1.15355i	−0.816904	+	0.576774i	$0.804311\pi$
$594$		0			0
$595$	−13.5477	−	19.8515i	−0.555403	−	0.813830i
$596$		0			0
$597$		−	10.4014i		−	0.425699i
$598$		0			0
$599$			26.7663i			1.09364i	0.837250	+	0.546821i	$0.184162\pi$
$599$			26.7663i			1.09364i	−0.837250	+	0.546821i	$0.815838\pi$
$600$		0			0
$601$		−	35.5607i		−	1.45055i	−0.688458	−	0.725276i	$-0.741712\pi$
$601$		−	35.5607i		−	1.45055i	0.688458	−	0.725276i	$-0.258288\pi$
$602$		0			0
$603$	18.4263			0.750377
$604$		0			0
$605$	−26.8099			−1.08998
$606$		0			0
$607$	33.7272			1.36895			0.684473	−	0.729038i	$-0.260032\pi$
$607$	33.7272			1.36895			0.684473	+	0.729038i	$0.260032\pi$
$608$		0			0
$609$	−17.8946	+	12.2123i	−0.725128	+	0.494868i
$610$		0			0
$611$	6.61191			0.267489
$612$		0			0
$613$			40.6244i			1.64080i	0.571788	+	0.820401i	$0.306250\pi$
$613$			40.6244i			1.64080i	−0.571788	+	0.820401i	$0.693750\pi$
$614$		0			0
$615$	−26.9514			−1.08678
$616$		0			0
$617$	16.6113			0.668745			0.334372	−	0.942441i	$-0.391476\pi$
$617$	16.6113			0.668745			0.334372	+	0.942441i	$0.391476\pi$
$618$		0			0
$619$			7.83199i			0.314794i	0.987535	+	0.157397i	$0.0503103\pi$
$619$			7.83199i			0.314794i	−0.987535	+	0.157397i	$0.949690\pi$
$620$		0			0
$621$	52.4063			2.10299
$622$		0			0
$623$	7.96136	+	11.6658i	0.318965	+	0.467379i
$624$		0			0
$625$	−19.4919			−0.779675
$626$		0			0
$627$	−28.4812			−1.13743
$628$		0			0
$629$	8.62348			0.343841
$630$		0			0
$631$		−	23.8991i		−	0.951407i	−0.879606	−	0.475703i	$-0.842194\pi$
$631$		−	23.8991i		−	0.951407i	0.879606	−	0.475703i	$-0.157806\pi$
$632$		0			0
$633$		−	25.6836i		−	1.02083i
$634$		0			0
$635$		−	28.2389i		−	1.12063i
$636$		0			0
$637$	2.55145	−	6.51844i	0.101092	−	0.258270i
$638$		0			0
$639$		−	2.52747i		−	0.0999854i
$640$		0			0
$641$	43.1306			1.70356			0.851778	−	0.523902i	$-0.175524\pi$
$641$	43.1306			1.70356			0.851778	+	0.523902i	$0.175524\pi$
$642$		0			0
$643$			27.6333i			1.08975i	0.838517	+	0.544876i	$0.183423\pi$
$643$			27.6333i			1.08975i	−0.838517	+	0.544876i	$0.816577\pi$
$644$		0			0
$645$		−	5.44206i		−	0.214281i
$646$		0			0
$647$	1.50328			0.0591000			0.0295500	−	0.999563i	$-0.490593\pi$
$647$	1.50328			0.0591000			0.0295500	+	0.999563i	$0.490593\pi$
$648$		0			0
$649$		−	48.1899i		−	1.89162i
$650$		0			0
$651$	−5.41975	−	7.94154i	−0.212417	−	0.311254i
$652$		0			0
$653$		−	13.8056i		−	0.540254i	−0.962825	−	0.270127i	$-0.912934\pi$
$653$		−	13.8056i		−	0.540254i	0.962825	−	0.270127i	$-0.0870658\pi$
$654$		0			0
$655$		−	1.91062i		−	0.0746539i
$656$		0			0
$657$			9.13388i			0.356346i
$658$		0			0
$659$	−7.47530			−0.291197			−0.145598	−	0.989344i	$-0.546511\pi$
$659$	−7.47530			−0.291197			−0.145598	+	0.989344i	$0.546511\pi$
$660$		0			0
$661$	3.90310			0.151813			0.0759065	−	0.997115i	$-0.475815\pi$
$661$	3.90310			0.151813			0.0759065	+	0.997115i	$0.475815\pi$
$662$		0			0
$663$	6.10651			0.237157
$664$		0			0
$665$	12.8215	+	18.7874i	0.497198	+	0.728542i
$666$		0			0
$667$	−56.0685			−2.17098
$668$		0			0
$669$		−	0.709614i		−	0.0274353i
$670$		0			0
$671$	−49.6803			−1.91789
$672$		0			0
$673$	23.7338			0.914871			0.457436	−	0.889243i	$-0.348768\pi$
$673$	23.7338			0.914871			0.457436	+	0.889243i	$0.348768\pi$
$674$		0			0
$675$		−	5.24227i		−	0.201775i
$676$		0			0
$677$	−36.7018			−1.41056			−0.705282	−	0.708927i	$-0.749179\pi$
$677$	−36.7018			−1.41056			−0.705282	+	0.708927i	$0.749179\pi$
$678$		0			0
$679$	−8.06767	−	11.8215i	−0.309609	−	0.453669i
$680$		0			0
$681$	−8.48752			−0.325242
$682$		0			0
$683$	6.55245			0.250722			0.125361	−	0.992111i	$-0.459991\pi$
$683$	6.55245			0.250722			0.125361	+	0.992111i	$0.459991\pi$
$684$		0			0
$685$	−27.2062			−1.03950
$686$		0			0
$687$			12.6103i			0.481111i
$688$		0			0
$689$			7.90627i			0.301205i
$690$		0			0
$691$		−	30.3755i		−	1.15554i	−0.816200	−	0.577770i	$-0.803923\pi$
$691$		−	30.3755i		−	1.15554i	0.816200	−	0.577770i	$-0.196077\pi$
$692$		0			0
$693$	−12.4966	+	8.52839i	−0.474707	+	0.323967i
$694$		0			0
$695$			20.9939i			0.796345i
$696$		0			0
$697$	−44.3387			−1.67945
$698$		0			0
$699$		−	0.962681i		−	0.0364119i
$700$		0			0
$701$		−	14.6155i		−	0.552021i	−0.961155	−	0.276010i	$-0.910988\pi$
$701$		−	14.6155i		−	0.552021i	0.961155	−	0.276010i	$-0.0890124\pi$
$702$		0			0
$703$	−8.16124			−0.307807
$704$		0			0
$705$		−	18.0945i		−	0.681479i
$706$		0			0
$707$	−32.1362	+	21.9315i	−1.20861	+	0.824820i
$708$		0			0
$709$		−	11.8485i		−	0.444981i	−0.974935	−	0.222490i	$-0.928581\pi$
$709$		−	11.8485i		−	0.444981i	0.974935	−	0.222490i	$-0.0714186\pi$
$710$		0			0
$711$			7.88525i			0.295720i
$712$		0			0
$713$		−	24.8829i		−	0.931872i
$714$		0			0
$715$	−9.94356			−0.371868
$716$		0			0
$717$	4.44995			0.166186
$718$		0			0
$719$	−28.6394			−1.06807			−0.534035	−	0.845463i	$-0.679325\pi$
$719$	−28.6394			−1.06807			−0.534035	+	0.845463i	$0.679325\pi$
$720$		0			0
$721$	−34.5506	+	23.5792i	−1.28673	+	0.878137i
$722$		0			0
$723$	14.8216			0.551220
$724$		0			0
$725$			5.60860i			0.208298i
$726$		0			0
$727$	23.6706			0.877896			0.438948	−	0.898513i	$-0.355351\pi$
$727$	23.6706			0.877896			0.438948	+	0.898513i	$0.355351\pi$
$728$		0			0
$729$	−27.8026			−1.02972
$730$		0			0
$731$		−	8.95294i		−	0.331136i
$732$		0			0
$733$	33.9964			1.25569			0.627844	−	0.778340i	$-0.283938\pi$
$733$	33.9964			1.25569			0.627844	+	0.778340i	$0.283938\pi$
$734$		0			0
$735$	−17.8387	−	6.98244i	−0.657992	−	0.257551i
$736$		0			0
$737$	78.2618			2.88281
$738$		0			0
$739$	−17.0248			−0.626267			−0.313134	−	0.949709i	$-0.601379\pi$
$739$	−17.0248			−0.626267			−0.313134	+	0.949709i	$0.601379\pi$
$740$		0			0
$741$	−5.77918			−0.212304
$742$		0			0
$743$		−	34.0743i		−	1.25006i	−0.780599	−	0.625032i	$-0.785086\pi$
$743$		−	34.0743i		−	1.25006i	0.780599	−	0.625032i	$-0.214914\pi$
$744$		0			0
$745$		−	9.80471i		−	0.359217i
$746$		0			0
$747$			18.0674i			0.661052i
$748$		0			0
$749$	−12.6568	+	8.63774i	−0.462471	+	0.315616i
$750$		0			0
$751$			21.2437i			0.775193i	0.921829	+	0.387596i	$0.126695\pi$
$751$			21.2437i			0.775193i	−0.921829	+	0.387596i	$0.873305\pi$
$752$		0			0
$753$	−15.1941			−0.553703
$754$		0			0
$755$		−	2.13436i		−	0.0776773i
$756$		0			0
$757$		−	12.9931i		−	0.472243i	−0.971724	−	0.236122i	$-0.924124\pi$
$757$		−	12.9931i		−	0.472243i	0.971724	−	0.236122i	$-0.0758764\pi$
$758$		0			0
$759$	62.0795			2.25334
$760$		0			0
$761$		−	7.45111i		−	0.270103i	−0.990839	−	0.135051i	$-0.956880\pi$
$761$		−	7.45111i		−	0.270103i	0.990839	−	0.135051i	$-0.0431199\pi$
$762$		0			0
$763$	20.9082	+	30.6367i	0.756926	+	1.10912i
$764$		0			0
$765$			10.5403i			0.381085i
$766$		0			0
$767$		−	9.77831i		−	0.353074i
$768$		0			0
$769$		−	41.8737i		−	1.51000i	−0.655723	−	0.755001i	$-0.727636\pi$
$769$		−	41.8737i		−	1.51000i	0.655723	−	0.755001i	$-0.272364\pi$
$770$		0			0
$771$	27.0013			0.972427
$772$		0			0
$773$	40.7234			1.46472			0.732360	−	0.680918i	$-0.238419\pi$
$773$	40.7234			1.46472			0.732360	+	0.680918i	$0.238419\pi$
$774$		0			0
$775$	−2.48907			−0.0894099
$776$		0			0
$777$	5.67741	−	3.87458i	0.203676	−	0.139000i
$778$		0			0
$779$	41.9620			1.50345
$780$		0			0
$781$		−	10.7349i		−	0.384125i
$782$		0			0
$783$	34.0668			1.21745
$784$		0			0
$785$	−16.3957			−0.585188
$786$		0			0
$787$			53.4793i			1.90633i	0.302447	+	0.953166i	$0.402196\pi$
$787$			53.4793i			1.90633i	−0.302447	+	0.953166i	$0.597804\pi$
$788$		0			0
$789$	34.8962			1.24234
$790$		0			0
$791$	4.37496	−	2.98572i	0.155556	−	0.106160i
$792$		0			0
$793$	−10.0807			−0.357977
$794$		0			0
$795$	21.6367			0.767376
$796$		0			0
$797$	−31.1556			−1.10359			−0.551793	−	0.833981i	$-0.686056\pi$
$797$	−31.1556			−1.10359			−0.551793	+	0.833981i	$0.686056\pi$
$798$		0			0
$799$		−	29.7680i		−	1.05312i
$800$		0			0
$801$		−	6.19404i		−	0.218856i
$802$		0			0
$803$			38.7942i			1.36902i
$804$		0			0
$805$	−27.9467	−	40.9502i	−0.984991	−	1.44330i
$806$		0			0
$807$		−	30.5303i		−	1.07472i
$808$		0			0
$809$	−36.8766			−1.29651			−0.648256	−	0.761423i	$-0.724501\pi$
$809$	−36.8766			−1.29651			−0.648256	+	0.761423i	$0.724501\pi$
$810$		0			0
$811$		−	2.50108i		−	0.0878247i	−0.999035	−	0.0439123i	$-0.986018\pi$
$811$		−	2.50108i		−	0.0878247i	0.999035	−	0.0439123i	$-0.0139822\pi$
$812$		0			0
$813$			36.0203i			1.26329i
$814$		0			0
$815$	18.9047			0.662201
$816$		0			0
$817$			8.47303i			0.296434i
$818$		0			0
$819$	−2.53571	+	1.73051i	−0.0886050	+	0.0604690i
$820$		0			0
$821$		−	0.865705i		−	0.0302133i	−0.999886	−	0.0151067i	$-0.995191\pi$
$821$		−	0.865705i		−	0.0302133i	0.999886	−	0.0151067i	$-0.00480878\pi$
$822$		0			0
$823$		−	32.1222i		−	1.11971i	−0.828590	−	0.559856i	$-0.810857\pi$
$823$		−	32.1222i		−	1.11971i	0.828590	−	0.559856i	$-0.189143\pi$
$824$		0			0
$825$		−	6.20989i		−	0.216201i
$826$		0			0
$827$	−34.3275			−1.19369			−0.596843	−	0.802358i	$-0.703578\pi$
$827$	−34.3275			−1.19369			−0.596843	+	0.802358i	$0.703578\pi$
$828$		0			0
$829$	−2.07242			−0.0719781			−0.0359891	−	0.999352i	$-0.511458\pi$
$829$	−2.07242			−0.0719781			−0.0359891	+	0.999352i	$0.511458\pi$
$830$		0			0
$831$	0.269942			0.00936417
$832$		0			0
$833$	−29.3472	−	11.4871i	−1.01682	−	0.398004i
$834$		0			0
$835$	30.4408			1.05345
$836$		0			0
$837$			15.1186i			0.522577i
$838$		0			0
$839$	−56.4106			−1.94751			−0.973754	−	0.227602i	$-0.926912\pi$
$839$	−56.4106			−1.94751			−0.973754	+	0.227602i	$0.926912\pi$
$840$		0			0
$841$	−7.44737			−0.256806
$842$		0			0
$843$		−	2.79913i		−	0.0964070i
$844$		0			0
$845$	−2.01767			−0.0694099
$846$		0			0
$847$	−29.0379	+	19.8171i	−0.997755	+	0.680924i
$848$		0			0
$849$	−6.61857			−0.227149
$850$		0			0
$851$	17.7888			0.609792
$852$		0			0
$853$	−13.1684			−0.450877			−0.225439	−	0.974257i	$-0.572381\pi$
$853$	−13.1684			−0.450877			−0.225439	+	0.974257i	$0.572381\pi$
$854$		0			0
$855$		−	9.97531i		−	0.341148i
$856$		0			0
$857$			40.7607i			1.39236i	0.717867	+	0.696180i	$0.245118\pi$
$857$			40.7607i			1.39236i	−0.717867	+	0.696180i	$0.754882\pi$
$858$		0			0
$859$			13.0322i			0.444653i	0.974972	+	0.222327i	$0.0713651\pi$
$859$			13.0322i			0.444653i	−0.974972	+	0.222327i	$0.928635\pi$
$860$		0			0
$861$	−29.1911	+	19.9216i	−0.994830	+	0.678927i
$862$		0			0
$863$			1.78583i			0.0607905i	0.999538	+	0.0303952i	$0.00967659\pi$
$863$			1.78583i			0.0607905i	−0.999538	+	0.0303952i	$0.990323\pi$
$864$		0			0
$865$	−18.3145			−0.622711
$866$		0			0
$867$		−	4.43472i		−	0.150611i
$868$		0			0
$869$			33.4909i			1.13610i
$870$		0			0
$871$	15.8803			0.538082
$872$		0			0
$873$			6.27675i			0.212436i
$874$		0			0
$875$	−26.1428	+	17.8413i	−0.883788	+	0.603146i
$876$		0			0
$877$			14.4070i			0.486488i	0.969965	+	0.243244i	$0.0782116\pi$
$877$			14.4070i			0.486488i	−0.969965	+	0.243244i	$0.921788\pi$
$878$		0			0
$879$		−	32.6559i		−	1.10146i
$880$		0			0
$881$			10.1248i			0.341112i	0.985348	+	0.170556i	$0.0545564\pi$
$881$			10.1248i			0.341112i	−0.985348	+	0.170556i	$0.945444\pi$
$882$		0			0
$883$	0.613830			0.0206570			0.0103285	−	0.999947i	$-0.496712\pi$
$883$	0.613830			0.0206570			0.0103285	+	0.999947i	$0.496712\pi$
$884$		0			0
$885$	−26.7599			−0.899523
$886$		0			0
$887$	4.17783			0.140278			0.0701389	−	0.997537i	$-0.477656\pi$
$887$	4.17783			0.140278			0.0701389	+	0.997537i	$0.477656\pi$
$888$		0			0
$889$	−20.8734	−	30.5857i	−0.700070	−	1.02581i
$890$		0			0
$891$	−20.5639			−0.688916
$892$		0			0
$893$			28.1724i			0.942752i
$894$		0			0
$895$	33.8769			1.13238
$896$		0			0
$897$	12.5967			0.420591
$898$		0			0
$899$		−	16.1751i		−	0.539471i
$900$		0			0
$901$	35.5954			1.18586
$902$		0			0
$903$	−4.02260	−	5.89431i	−0.133864	−	0.196150i
$904$		0			0
$905$	−7.40444			−0.246132
$906$		0			0
$907$	−17.4110			−0.578123			−0.289061	−	0.957311i	$-0.593343\pi$
$907$	−17.4110			−0.578123			−0.289061	+	0.957311i	$0.593343\pi$
$908$		0			0
$909$	17.0630			0.565943
$910$		0			0
$911$		−	15.7090i		−	0.520462i	−0.965546	−	0.260231i	$-0.916201\pi$
$911$		−	15.7090i		−	0.520462i	0.965546	−	0.260231i	$-0.0837987\pi$
$912$		0			0
$913$			76.7375i			2.53964i
$914$		0			0
$915$			27.5875i			0.912014i
$916$		0			0
$917$	−1.41227	−	2.06939i	−0.0466373	−	0.0683374i
$918$		0			0
$919$			42.1694i			1.39104i	0.718507	+	0.695520i	$0.244826\pi$
$919$			42.1694i			1.39104i	−0.718507	+	0.695520i	$0.755174\pi$
$920$		0			0
$921$	17.8530			0.588278
$922$		0			0
$923$		−	2.17824i		−	0.0716977i
$924$		0			0
$925$		−	1.77943i		−	0.0585074i
$926$		0			0
$927$	18.3449			0.602527
$928$		0			0
$929$		−	52.3499i		−	1.71754i	−0.512359	−	0.858772i	$-0.671228\pi$
$929$		−	52.3499i		−	1.71754i	0.512359	−	0.858772i	$-0.328772\pi$
$930$		0			0
$931$	27.7741	+	10.8713i	0.910259	+	0.356294i
$932$		0			0
$933$			15.0488i			0.492676i
$934$		0			0
$935$			44.7677i			1.46406i
$936$		0			0
$937$			8.53783i			0.278919i	0.990228	+	0.139459i	$0.0445365\pi$
$937$			8.53783i			0.278919i	−0.990228	+	0.139459i	$0.955464\pi$
$938$		0			0
$939$	15.5391			0.507098
$940$		0			0
$941$	35.4337			1.15511			0.577553	−	0.816353i	$-0.304008\pi$
$941$	35.4337			1.15511			0.577553	+	0.816353i	$0.304008\pi$
$942$		0			0
$943$	−91.4632			−2.97845
$944$		0			0
$945$	16.9802	+	24.8810i	0.552365	+	0.809378i
$946$		0			0
$947$	−12.5985			−0.409397			−0.204698	−	0.978825i	$-0.565621\pi$
$947$	−12.5985			−0.409397			−0.204698	+	0.978825i	$0.565621\pi$
$948$		0			0
$949$			7.87181i			0.255530i
$950$		0			0
$951$	−5.84347			−0.189488
$952$		0			0
$953$	−21.7924			−0.705927			−0.352963	−	0.935637i	$-0.614826\pi$
$953$	−21.7924			−0.705927			−0.352963	+	0.935637i	$0.614826\pi$
$954$		0			0
$955$		−	52.4252i		−	1.69644i
$956$		0			0
$957$	40.3548			1.30449
$958$		0			0
$959$	−29.4671	+	20.1100i	−0.951543	+	0.649386i
$960$		0			0
$961$	−23.8216			−0.768438
$962$		0			0
$963$	6.72026			0.216557
$964$		0			0
$965$	−36.7572			−1.18326
$966$		0			0
$967$			30.2075i			0.971408i	0.874123	+	0.485704i	$0.161437\pi$
$967$			30.2075i			0.971408i	−0.874123	+	0.485704i	$0.838563\pi$
$968$		0			0
$969$			26.0189i			0.835848i
$970$		0			0
$971$			26.3321i			0.845037i	0.906354	+	0.422518i	$0.138854\pi$
$971$			26.3321i			0.845037i	−0.906354	+	0.422518i	$0.861146\pi$
$972$		0			0
$973$	15.5181	+	22.7386i	0.497487	+	0.728966i
$974$		0			0
$975$		−	1.26006i		−	0.0403543i
$976$		0			0
$977$	24.2718			0.776523			0.388261	−	0.921549i	$-0.373076\pi$
$977$	24.2718			0.776523			0.388261	+	0.921549i	$0.373076\pi$
$978$		0			0
$979$		−	26.3079i		−	0.840803i
$980$		0			0
$981$		−	16.2668i		−	0.519359i
$982$		0			0
$983$	50.3405			1.60561			0.802806	−	0.596240i	$-0.203339\pi$
$983$	50.3405			1.60561			0.802806	+	0.596240i	$0.203339\pi$
$984$		0			0
$985$			12.8300i			0.408799i
$986$		0			0
$987$	−13.3749	−	19.5982i	−0.425729	−	0.623819i
$988$		0			0
$989$		−	18.4684i		−	0.587260i
$990$		0			0
$991$			32.0033i			1.01662i	0.861175	+	0.508309i	$0.169729\pi$
$991$			32.0033i			1.01662i	−0.861175	+	0.508309i	$0.830271\pi$
$992$		0			0
$993$			1.41410i			0.0448752i
$994$		0			0
$995$	15.4728			0.490522
$996$		0			0
$997$	−15.9693			−0.505752			−0.252876	−	0.967499i	$-0.581376\pi$
$997$	−15.9693			−0.505752			−0.252876	+	0.967499i	$0.581376\pi$
$998$		0			0
$999$	−10.8083			−0.341960

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.b.2575.33		48
4.3	odd	2		728.2.h.b.27.3	yes	48
7.6	odd	2		2912.2.h.a.2575.16		48
8.3	odd	2		2912.2.h.a.2575.33		48
8.5	even	2		728.2.h.a.27.4	yes	48
28.27	even	2		728.2.h.a.27.3	✓	48
56.13	odd	2		728.2.h.b.27.4	yes	48
56.27	even	2	inner	2912.2.h.b.2575.16		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.3	✓	48	28.27	even	2
728.2.h.a.27.4	yes	48	8.5	even	2
728.2.h.b.27.3	yes	48	4.3	odd	2
728.2.h.b.27.4	yes	48	56.13	odd	2
2912.2.h.a.2575.16		48	7.6	odd	2
2912.2.h.a.2575.33		48	8.3	odd	2
2912.2.h.b.2575.16		48	56.27	even	2	inner
2912.2.h.b.2575.33		48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.35635i q^{3} -2.01767 q^{5} +(-2.18534 + 1.49140i) q^{7} +1.16033 q^{9} +4.92824 q^{11} +1.00000 q^{13} -2.73666i q^{15} -4.50218i q^{17} +4.26085i q^{19} +(-2.02285 - 2.96408i) q^{21} -9.28723i q^{23} -0.929012 q^{25} +5.64284i q^{27} -6.03717i q^{29} +2.67926 q^{31} +6.68440i q^{33} +(4.40930 - 3.00915i) q^{35} +1.91540i q^{37} +1.35635i q^{39} -9.84828i q^{41} +1.98858 q^{43} -2.34116 q^{45} +6.61191 q^{47} +(2.55145 - 6.51844i) q^{49} +6.10651 q^{51} +7.90627i q^{53} -9.94356 q^{55} -5.77918 q^{57} -9.77831i q^{59} -10.0807 q^{61} +(-2.53571 + 1.73051i) q^{63} -2.01767 q^{65} +15.8803 q^{67} +12.5967 q^{69} -2.17824i q^{71} +7.87181i q^{73} -1.26006i q^{75} +(-10.7699 + 7.34998i) q^{77} +6.79571i q^{79} -4.17266 q^{81} +15.5710i q^{83} +9.08390i q^{85} +8.18848 q^{87} -5.33818i q^{89} +(-2.18534 + 1.49140i) q^{91} +3.63400i q^{93} -8.59698i q^{95} +5.40946i q^{97} +5.71838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} + 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} - 24 q^{45} + 40 q^{51} + 20 q^{63} + 4 q^{67} + 20 q^{77} + 64 q^{81} - 40 q^{87} - 20 q^{99}+O(q^{100}) \) 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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