LMFDB - Embedded newform 2912.2.c.b.1457.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(1457,2912)]

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([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2912.1457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.c (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:	$38$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.14
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.b.1457.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.07539i q^{3} -3.29373i q^{5} -1.00000 q^{7} +1.84353 q^{9} -3.53548i q^{11} +1.00000i q^{13} -3.54204 q^{15} +7.07564 q^{17} -8.24042i q^{19} +1.07539i q^{21} +3.91963 q^{23} -5.84864 q^{25} -5.20869i q^{27} +4.44254i q^{29} -2.05760 q^{31} -3.80202 q^{33} +3.29373i q^{35} -9.27443i q^{37} +1.07539 q^{39} +7.97166 q^{41} +10.9281i q^{43} -6.07210i q^{45} +2.18113 q^{47} +1.00000 q^{49} -7.60908i q^{51} +7.21685i q^{53} -11.6449 q^{55} -8.86167 q^{57} -4.70334i q^{59} -11.2488i q^{61} -1.84353 q^{63} +3.29373 q^{65} +5.10411i q^{67} -4.21514i q^{69} +2.70611 q^{71} -2.30012 q^{73} +6.28957i q^{75} +3.53548i q^{77} +10.5704 q^{79} -0.0707804 q^{81} +4.62086i q^{83} -23.3052i q^{85} +4.77747 q^{87} -17.8295 q^{89} -1.00000i q^{91} +2.21272i q^{93} -27.1417 q^{95} +3.39038 q^{97} -6.51778i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 38 q^{7} - 46 q^{9} + 8 q^{15} + 20 q^{17} - 12 q^{23} - 50 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 38 q^{49} + 8 q^{57} + 46 q^{63} + 20 q^{65} + 12 q^{79} + 62 q^{81} + 56 q^{87}+O(q^{100}) $ 38 * q - 38 * q^7 - 46 * q^9 + 8 * q^15 + 20 * q^17 - 12 * q^23 - 50 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 38 * q^49 + 8 * q^57 + 46 * q^63 + 20 * q^65 + 12 * q^79 + 62 * q^81 + 56 * q^87

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
$3$	− 1.07539i	− 0.620877i	−0.950593	−	0.310439i	$-0.899524\pi$
$3$	− 1.07539i	− 0.620877i	0.950593	−	0.310439i	$-0.100476\pi$
$4$	0	0
$5$	− 3.29373i	− 1.47300i	−0.676438	−	0.736500i	$-0.736477\pi$
$5$	− 3.29373i	− 1.47300i	0.676438	−	0.736500i	$-0.263523\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.84353	0.614511
$10$	0	0
$11$	− 3.53548i	− 1.06599i	−0.846119	−	0.532994i	$-0.821067\pi$
$11$	− 3.53548i	− 1.06599i	0.846119	−	0.532994i	$-0.178933\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−3.54204	−0.914552
$16$	0	0
$17$	7.07564	1.71610	0.858048	−	0.513570i	$-0.171677\pi$
$17$	7.07564	1.71610	0.858048	+	0.513570i	$0.171677\pi$
$18$	0	0
$19$	− 8.24042i	− 1.89048i	−0.326374	−	0.945241i	$-0.605827\pi$
$19$	− 8.24042i	− 1.89048i	0.326374	−	0.945241i	$-0.394173\pi$
$20$	0	0
$21$	1.07539i	0.234670i
$22$	0	0
$23$	3.91963	0.817300	0.408650	−	0.912691i	$-0.366000\pi$
$23$	3.91963	0.817300	0.408650	+	0.912691i	$0.366000\pi$
$24$	0	0
$25$	−5.84864	−1.16973
$26$	0	0
$27$	− 5.20869i	− 1.00241i
$28$	0	0
$29$	4.44254i	0.824959i	0.910967	+	0.412479i	$0.135337\pi$
$29$	4.44254i	0.824959i	−0.910967	+	0.412479i	$0.864663\pi$
$30$	0	0
$31$	−2.05760	−0.369556	−0.184778	−	0.982780i	$-0.559157\pi$
$31$	−2.05760	−0.369556	−0.184778	+	0.982780i	$0.559157\pi$
$32$	0	0
$33$	−3.80202	−0.661847
$34$	0	0
$35$	3.29373i	0.556741i
$36$	0	0
$37$	− 9.27443i	− 1.52471i	−0.647161	−	0.762353i	$-0.724044\pi$
$37$	− 9.27443i	− 1.52471i	0.647161	−	0.762353i	$-0.275956\pi$
$38$	0	0
$39$	1.07539	0.172200
$40$	0	0
$41$	7.97166	1.24496	0.622482	−	0.782634i	$-0.286124\pi$
$41$	7.97166	1.24496	0.622482	+	0.782634i	$0.286124\pi$
$42$	0	0
$43$	10.9281i	1.66651i	0.552887	+	0.833256i	$0.313526\pi$
$43$	10.9281i	1.66651i	−0.552887	+	0.833256i	$0.686474\pi$
$44$	0	0
$45$	− 6.07210i	− 0.905175i
$46$	0	0
$47$	2.18113	0.318150	0.159075	−	0.987267i	$-0.449149\pi$
$47$	2.18113	0.318150	0.159075	+	0.987267i	$0.449149\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 7.60908i	− 1.06548i
$52$	0	0
$53$	7.21685i	0.991311i	0.868519	+	0.495655i	$0.165072\pi$
$53$	7.21685i	0.991311i	−0.868519	+	0.495655i	$0.834928\pi$
$54$	0	0
$55$	−11.6449	−1.57020
$56$	0	0
$57$	−8.86167	−1.17376
$58$	0	0
$59$	− 4.70334i	− 0.612322i	−0.951980	−	0.306161i	$-0.900955\pi$
$59$	− 4.70334i	− 0.612322i	0.951980	−	0.306161i	$-0.0990446\pi$
$60$	0	0
$61$	− 11.2488i	− 1.44026i	−0.693838	−	0.720131i	$-0.744082\pi$
$61$	− 11.2488i	− 1.44026i	0.693838	−	0.720131i	$-0.255918\pi$
$62$	0	0
$63$	−1.84353	−0.232263
$64$	0	0
$65$	3.29373	0.408537
$66$	0	0
$67$	5.10411i	0.623566i	0.950153	+	0.311783i	$0.100926\pi$
$67$	5.10411i	0.623566i	−0.950153	+	0.311783i	$0.899074\pi$
$68$	0	0
$69$	− 4.21514i	− 0.507443i
$70$	0	0
$71$	2.70611	0.321156	0.160578	−	0.987023i	$-0.448664\pi$
$71$	2.70611	0.321156	0.160578	+	0.987023i	$0.448664\pi$
$72$	0	0
$73$	−2.30012	−0.269209	−0.134604	−	0.990899i	$-0.542976\pi$
$73$	−2.30012	−0.269209	−0.134604	+	0.990899i	$0.542976\pi$
$74$	0	0
$75$	6.28957i	0.726257i
$76$	0	0
$77$	3.53548i	0.402905i
$78$	0	0
$79$	10.5704	1.18926	0.594630	−	0.803999i	$-0.297299\pi$
$79$	10.5704	1.18926	0.594630	+	0.803999i	$0.297299\pi$
$80$	0	0
$81$	−0.0707804	−0.00786449
$82$	0	0
$83$	4.62086i	0.507205i	0.967309	+	0.253602i	$0.0816155\pi$
$83$	4.62086i	0.507205i	−0.967309	+	0.253602i	$0.918385\pi$
$84$	0	0
$85$	− 23.3052i	− 2.52781i
$86$	0	0
$87$	4.77747	0.512198
$88$	0	0
$89$	−17.8295	−1.88992	−0.944961	−	0.327184i	$-0.893900\pi$
$89$	−17.8295	−1.88992	−0.944961	+	0.327184i	$0.893900\pi$
$90$	0	0
$91$	− 1.00000i	− 0.104828i
$92$	0	0
$93$	2.21272i	0.229449i
$94$	0	0
$95$	−27.1417	−2.78468
$96$	0	0
$97$	3.39038	0.344241	0.172121	−	0.985076i	$-0.444938\pi$
$97$	3.39038	0.344241	0.172121	+	0.985076i	$0.444938\pi$
$98$	0	0
$99$	− 6.51778i	− 0.655061i
$100$	0	0
$101$	1.36196i	0.135520i	0.997702	+	0.0677601i	$0.0215852\pi$
$101$	1.36196i	0.135520i	−0.997702	+	0.0677601i	$0.978415\pi$
$102$	0	0
$103$	−14.1718	−1.39639	−0.698195	−	0.715908i	$-0.746013\pi$
$103$	−14.1718	−1.39639	−0.698195	+	0.715908i	$0.746013\pi$
$104$	0	0
$105$	3.54204	0.345668
$106$	0	0
$107$	0.539384i	0.0521442i	0.999660	+	0.0260721i	$0.00829995\pi$
$107$	0.539384i	0.0521442i	−0.999660	+	0.0260721i	$0.991700\pi$
$108$	0	0
$109$	13.6657i	1.30894i	0.756088	+	0.654470i	$0.227108\pi$
$109$	13.6657i	1.30894i	−0.756088	+	0.654470i	$0.772892\pi$
$110$	0	0
$111$	−9.97364	−0.946656
$112$	0	0
$113$	−1.23174	−0.115873	−0.0579363	−	0.998320i	$-0.518452\pi$
$113$	−1.23174	−0.115873	−0.0579363	+	0.998320i	$0.518452\pi$
$114$	0	0
$115$	− 12.9102i	− 1.20388i
$116$	0	0
$117$	1.84353i	0.170435i
$118$	0	0
$119$	−7.07564	−0.648623
$120$	0	0
$121$	−1.49961	−0.136328
$122$	0	0
$123$	− 8.57265i	− 0.772970i
$124$	0	0
$125$	2.79518i	0.250008i
$126$	0	0
$127$	−7.07896	−0.628156	−0.314078	−	0.949397i	$-0.601695\pi$
$127$	−7.07896	−0.628156	−0.314078	+	0.949397i	$0.601695\pi$
$128$	0	0
$129$	11.7519	1.03470
$130$	0	0
$131$	11.0564i	0.966003i	0.875620	+	0.483001i	$0.160453\pi$
$131$	11.0564i	0.966003i	−0.875620	+	0.483001i	$0.839547\pi$
$132$	0	0
$133$	8.24042i	0.714535i
$134$	0	0
$135$	−17.1560	−1.47655
$136$	0	0
$137$	−15.4186	−1.31730	−0.658652	−	0.752448i	$-0.728873\pi$
$137$	−15.4186	−1.31730	−0.658652	+	0.752448i	$0.728873\pi$
$138$	0	0
$139$	19.6410i	1.66593i	0.553327	+	0.832964i	$0.313358\pi$
$139$	19.6410i	1.66593i	−0.553327	+	0.832964i	$0.686642\pi$
$140$	0	0
$141$	− 2.34556i	− 0.197532i
$142$	0	0
$143$	3.53548	0.295652
$144$	0	0
$145$	14.6325	1.21516
$146$	0	0
$147$	− 1.07539i	− 0.0886968i
$148$	0	0
$149$	− 8.12461i	− 0.665594i	−0.942999	−	0.332797i	$-0.892008\pi$
$149$	− 8.12461i	− 0.665594i	0.942999	−	0.332797i	$-0.107992\pi$
$150$	0	0
$151$	12.1005	0.984729	0.492364	−	0.870389i	$-0.336133\pi$
$151$	12.1005	0.984729	0.492364	+	0.870389i	$0.336133\pi$
$152$	0	0
$153$	13.0442	1.05456
$154$	0	0
$155$	6.77717i	0.544355i
$156$	0	0
$157$	1.92301i	0.153473i	0.997051	+	0.0767364i	$0.0244500\pi$
$157$	1.92301i	0.153473i	−0.997051	+	0.0767364i	$0.975550\pi$
$158$	0	0
$159$	7.76094	0.615482
$160$	0	0
$161$	−3.91963	−0.308910
$162$	0	0
$163$	6.46440i	0.506331i	0.967423	+	0.253165i	$0.0814717\pi$
$163$	6.46440i	0.506331i	−0.967423	+	0.253165i	$0.918528\pi$
$164$	0	0
$165$	12.5228i	0.974901i
$166$	0	0
$167$	−6.58375	−0.509466	−0.254733	−	0.967011i	$-0.581988\pi$
$167$	−6.58375	−0.509466	−0.254733	+	0.967011i	$0.581988\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 15.1915i	− 1.16172i
$172$	0	0
$173$	6.25239i	0.475360i	0.971343	+	0.237680i	$0.0763870\pi$
$173$	6.25239i	0.475360i	−0.971343	+	0.237680i	$0.923613\pi$
$174$	0	0
$175$	5.84864	0.442115
$176$	0	0
$177$	−5.05793	−0.380177
$178$	0	0
$179$	3.50515i	0.261988i	0.991383	+	0.130994i	$0.0418168\pi$
$179$	3.50515i	0.261988i	−0.991383	+	0.130994i	$0.958183\pi$
$180$	0	0
$181$	6.61055i	0.491358i	0.969351	+	0.245679i	$0.0790109\pi$
$181$	6.61055i	0.491358i	−0.969351	+	0.245679i	$0.920989\pi$
$182$	0	0
$183$	−12.0969	−0.894226
$184$	0	0
$185$	−30.5474	−2.24589
$186$	0	0
$187$	− 25.0158i	− 1.82934i
$188$	0	0
$189$	5.20869i	0.378877i
$190$	0	0
$191$	−21.5534	−1.55955	−0.779773	−	0.626062i	$-0.784666\pi$
$191$	−21.5534	−1.55955	−0.779773	+	0.626062i	$0.784666\pi$
$192$	0	0
$193$	−17.1582	−1.23508	−0.617538	−	0.786541i	$-0.711870\pi$
$193$	−17.1582	−1.23508	−0.617538	+	0.786541i	$0.711870\pi$
$194$	0	0
$195$	− 3.54204i	− 0.253651i
$196$	0	0
$197$	− 14.4506i	− 1.02956i	−0.857322	−	0.514780i	$-0.827873\pi$
$197$	− 14.4506i	− 1.02956i	0.857322	−	0.514780i	$-0.172127\pi$
$198$	0	0
$199$	−10.2918	−0.729570	−0.364785	−	0.931092i	$-0.618857\pi$
$199$	−10.2918	−0.729570	−0.364785	+	0.931092i	$0.618857\pi$
$200$	0	0
$201$	5.48891	0.387158
$202$	0	0
$203$	− 4.44254i	− 0.311805i
$204$	0	0
$205$	− 26.2565i	− 1.83383i
$206$	0	0
$207$	7.22597	0.502240
$208$	0	0
$209$	−29.1338	−2.01523
$210$	0	0
$211$	14.4148i	0.992354i	0.868221	+	0.496177i	$0.165263\pi$
$211$	14.4148i	0.992354i	−0.868221	+	0.496177i	$0.834737\pi$
$212$	0	0
$213$	− 2.91013i	− 0.199399i
$214$	0	0
$215$	35.9940	2.45477
$216$	0	0
$217$	2.05760	0.139679
$218$	0	0
$219$	2.47353i	0.167146i
$220$	0	0
$221$	7.07564i	0.475959i
$222$	0	0
$223$	15.5235	1.03953	0.519766	−	0.854309i	$-0.326019\pi$
$223$	15.5235	1.03953	0.519766	+	0.854309i	$0.326019\pi$
$224$	0	0
$225$	−10.7822	−0.718811
$226$	0	0
$227$	24.4833i	1.62502i	0.582950	+	0.812508i	$0.301898\pi$
$227$	24.4833i	1.62502i	−0.582950	+	0.812508i	$0.698102\pi$
$228$	0	0
$229$	− 4.71799i	− 0.311774i	−0.987775	−	0.155887i	$-0.950176\pi$
$229$	− 4.71799i	− 0.311774i	0.987775	−	0.155887i	$-0.0498235\pi$
$230$	0	0
$231$	3.80202	0.250155
$232$	0	0
$233$	−8.58327	−0.562309	−0.281154	−	0.959663i	$-0.590717\pi$
$233$	−8.58327	−0.562309	−0.281154	+	0.959663i	$0.590717\pi$
$234$	0	0
$235$	− 7.18403i	− 0.468635i
$236$	0	0
$237$	− 11.3673i	− 0.738385i
$238$	0	0
$239$	−3.56238	−0.230431	−0.115216	−	0.993340i	$-0.536756\pi$
$239$	−3.56238	−0.230431	−0.115216	+	0.993340i	$0.536756\pi$
$240$	0	0
$241$	5.93125	0.382065	0.191033	−	0.981584i	$-0.438816\pi$
$241$	5.93125	0.382065	0.191033	+	0.981584i	$0.438816\pi$
$242$	0	0
$243$	− 15.5500i	− 0.997531i
$244$	0	0
$245$	− 3.29373i	− 0.210428i
$246$	0	0
$247$	8.24042	0.524325
$248$	0	0
$249$	4.96923	0.314912
$250$	0	0
$251$	− 21.0334i	− 1.32762i	−0.747902	−	0.663809i	$-0.768939\pi$
$251$	− 21.0334i	− 1.32762i	0.747902	−	0.663809i	$-0.231061\pi$
$252$	0	0
$253$	− 13.8578i	− 0.871231i
$254$	0	0
$255$	−25.0622	−1.56946
$256$	0	0
$257$	20.8969	1.30351	0.651756	−	0.758429i	$-0.274033\pi$
$257$	20.8969	1.30351	0.651756	+	0.758429i	$0.274033\pi$
$258$	0	0
$259$	9.27443i	0.576285i
$260$	0	0
$261$	8.18997i	0.506946i
$262$	0	0
$263$	10.8422	0.668556	0.334278	−	0.942475i	$-0.391508\pi$
$263$	10.8422	0.668556	0.334278	+	0.942475i	$0.391508\pi$
$264$	0	0
$265$	23.7703	1.46020
$266$	0	0
$267$	19.1737i	1.17341i
$268$	0	0
$269$	− 12.2396i	− 0.746259i	−0.927779	−	0.373129i	$-0.878285\pi$
$269$	− 12.2396i	− 0.746259i	0.927779	−	0.373129i	$-0.121715\pi$
$270$	0	0
$271$	12.5722	0.763710	0.381855	−	0.924222i	$-0.375285\pi$
$271$	12.5722	0.763710	0.381855	+	0.924222i	$0.375285\pi$
$272$	0	0
$273$	−1.07539	−0.0650856
$274$	0	0
$275$	20.6777i	1.24691i
$276$	0	0
$277$	− 20.6185i	− 1.23885i	−0.785057	−	0.619423i	$-0.787367\pi$
$277$	− 20.6185i	− 1.23885i	0.785057	−	0.619423i	$-0.212633\pi$
$278$	0	0
$279$	−3.79325	−0.227096
$280$	0	0
$281$	−23.8821	−1.42468	−0.712342	−	0.701832i	$-0.752366\pi$
$281$	−23.8821	−1.42468	−0.712342	+	0.701832i	$0.752366\pi$
$282$	0	0
$283$	− 8.59400i	− 0.510861i	−0.966827	−	0.255430i	$-0.917783\pi$
$283$	− 8.59400i	− 0.510861i	0.966827	−	0.255430i	$-0.0822171\pi$
$284$	0	0
$285$	29.1879i	1.72894i
$286$	0	0
$287$	−7.97166	−0.470552
$288$	0	0
$289$	33.0647	1.94498
$290$	0	0
$291$	− 3.64599i	− 0.213731i
$292$	0	0
$293$	29.5096i	1.72397i	0.506933	+	0.861986i	$0.330779\pi$
$293$	29.5096i	1.72397i	−0.506933	+	0.861986i	$0.669221\pi$
$294$	0	0
$295$	−15.4915	−0.901950
$296$	0	0
$297$	−18.4152	−1.06856
$298$	0	0
$299$	3.91963i	0.226678i
$300$	0	0
$301$	− 10.9281i	− 0.629883i
$302$	0	0
$303$	1.46464	0.0841414
$304$	0	0
$305$	−37.0505	−2.12150
$306$	0	0
$307$	15.3024i	0.873355i	0.899618	+	0.436677i	$0.143845\pi$
$307$	15.3024i	0.873355i	−0.899618	+	0.436677i	$0.856155\pi$
$308$	0	0
$309$	15.2402i	0.866986i
$310$	0	0
$311$	20.3603	1.15453	0.577264	−	0.816557i	$-0.304120\pi$
$311$	20.3603	1.15453	0.577264	+	0.816557i	$0.304120\pi$
$312$	0	0
$313$	19.7452	1.11607	0.558033	−	0.829819i	$-0.311556\pi$
$313$	19.7452	1.11607	0.558033	+	0.829819i	$0.311556\pi$
$314$	0	0
$315$	6.07210i	0.342124i
$316$	0	0
$317$	22.9991i	1.29176i	0.763439	+	0.645880i	$0.223509\pi$
$317$	22.9991i	1.29176i	−0.763439	+	0.645880i	$0.776491\pi$
$318$	0	0
$319$	15.7065	0.879395
$320$	0	0
$321$	0.580049	0.0323752
$322$	0	0
$323$	− 58.3062i	− 3.24425i
$324$	0	0
$325$	− 5.84864i	− 0.324424i
$326$	0	0
$327$	14.6960	0.812691
$328$	0	0
$329$	−2.18113	−0.120249
$330$	0	0
$331$	8.59180i	0.472248i	0.971723	+	0.236124i	$0.0758771\pi$
$331$	8.59180i	0.472248i	−0.971723	+	0.236124i	$0.924123\pi$
$332$	0	0
$333$	− 17.0977i	− 0.936950i
$334$	0	0
$335$	16.8115	0.918513
$336$	0	0
$337$	13.5694	0.739175	0.369587	−	0.929196i	$-0.379499\pi$
$337$	13.5694	0.739175	0.369587	+	0.929196i	$0.379499\pi$
$338$	0	0
$339$	1.32461i	0.0719427i
$340$	0	0
$341$	7.27460i	0.393942i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.8835	−0.747463
$346$	0	0
$347$	− 7.08248i	− 0.380207i	−0.981764	−	0.190104i	$-0.939118\pi$
$347$	− 7.08248i	− 0.380207i	0.981764	−	0.190104i	$-0.0608824\pi$
$348$	0	0
$349$	− 25.7646i	− 1.37915i	−0.724216	−	0.689574i	$-0.757798\pi$
$349$	− 25.7646i	− 1.37915i	0.724216	−	0.689574i	$-0.242202\pi$
$350$	0	0
$351$	5.20869	0.278019
$352$	0	0
$353$	20.2468	1.07763	0.538814	−	0.842425i	$-0.318873\pi$
$353$	20.2468	1.07763	0.538814	+	0.842425i	$0.318873\pi$
$354$	0	0
$355$	− 8.91319i	− 0.473063i
$356$	0	0
$357$	7.60908i	0.402715i
$358$	0	0
$359$	27.8989	1.47245	0.736223	−	0.676739i	$-0.236607\pi$
$359$	27.8989	1.47245	0.736223	+	0.676739i	$0.236607\pi$
$360$	0	0
$361$	−48.9045	−2.57392
$362$	0	0
$363$	1.61267i	0.0846433i
$364$	0	0
$365$	7.57597i	0.396545i
$366$	0	0
$367$	−14.4161	−0.752517	−0.376258	−	0.926515i	$-0.622789\pi$
$367$	−14.4161	−0.752517	−0.376258	+	0.926515i	$0.622789\pi$
$368$	0	0
$369$	14.6960	0.765045
$370$	0	0
$371$	− 7.21685i	− 0.374680i
$372$	0	0
$373$	5.84685i	0.302738i	0.988477	+	0.151369i	$0.0483682\pi$
$373$	5.84685i	0.302738i	−0.988477	+	0.151369i	$0.951632\pi$
$374$	0	0
$375$	3.00591	0.155224
$376$	0	0
$377$	−4.44254	−0.228802
$378$	0	0
$379$	− 3.49300i	− 0.179423i	−0.995968	−	0.0897117i	$-0.971405\pi$
$379$	− 3.49300i	− 0.179423i	0.995968	−	0.0897117i	$-0.0285946\pi$
$380$	0	0
$381$	7.61265i	0.390008i
$382$	0	0
$383$	11.9167	0.608917	0.304459	−	0.952526i	$-0.401524\pi$
$383$	11.9167	0.608917	0.304459	+	0.952526i	$0.401524\pi$
$384$	0	0
$385$	11.6449	0.593479
$386$	0	0
$387$	20.1462i	1.02409i
$388$	0	0
$389$	− 16.2782i	− 0.825337i	−0.910881	−	0.412669i	$-0.864597\pi$
$389$	− 16.2782i	− 0.825337i	0.910881	−	0.412669i	$-0.135403\pi$
$390$	0	0
$391$	27.7339	1.40256
$392$	0	0
$393$	11.8900	0.599769
$394$	0	0
$395$	− 34.8159i	− 1.75178i
$396$	0	0
$397$	20.7126i	1.03953i	0.854308	+	0.519767i	$0.173981\pi$
$397$	20.7126i	1.03953i	−0.854308	+	0.519767i	$0.826019\pi$
$398$	0	0
$399$	8.86167	0.443639
$400$	0	0
$401$	−8.10034	−0.404512	−0.202256	−	0.979333i	$-0.564827\pi$
$401$	−8.10034	−0.404512	−0.202256	+	0.979333i	$0.564827\pi$
$402$	0	0
$403$	− 2.05760i	− 0.102496i
$404$	0	0
$405$	0.233131i	0.0115844i
$406$	0	0
$407$	−32.7896	−1.62532
$408$	0	0
$409$	3.28603	0.162484	0.0812419	−	0.996694i	$-0.474111\pi$
$409$	3.28603	0.162484	0.0812419	+	0.996694i	$0.474111\pi$
$410$	0	0
$411$	16.5811i	0.817884i
$412$	0	0
$413$	4.70334i	0.231436i
$414$	0	0
$415$	15.2198	0.747112
$416$	0	0
$417$	21.1218	1.03434
$418$	0	0
$419$	− 10.5245i	− 0.514154i	−0.966391	−	0.257077i	$-0.917241\pi$
$419$	− 10.5245i	− 0.514154i	0.966391	−	0.257077i	$-0.0827594\pi$
$420$	0	0
$421$	− 1.95938i	− 0.0954943i	−0.998859	−	0.0477471i	$-0.984796\pi$
$421$	− 1.95938i	− 0.0954943i	0.998859	−	0.0477471i	$-0.0152042\pi$
$422$	0	0
$423$	4.02098	0.195507
$424$	0	0
$425$	−41.3829	−2.00736
$426$	0	0
$427$	11.2488i	0.544368i
$428$	0	0
$429$	− 3.80202i	− 0.183563i
$430$	0	0
$431$	17.2196	0.829437	0.414719	−	0.909950i	$-0.363880\pi$
$431$	17.2196	0.829437	0.414719	+	0.909950i	$0.363880\pi$
$432$	0	0
$433$	27.4973	1.32144	0.660719	−	0.750634i	$-0.270252\pi$
$433$	27.4973	1.32144	0.660719	+	0.750634i	$0.270252\pi$
$434$	0	0
$435$	− 15.7357i	− 0.754467i
$436$	0	0
$437$	− 32.2994i	− 1.54509i
$438$	0	0
$439$	11.2679	0.537785	0.268893	−	0.963170i	$-0.413342\pi$
$439$	11.2679	0.537785	0.268893	+	0.963170i	$0.413342\pi$
$440$	0	0
$441$	1.84353	0.0877873
$442$	0	0
$443$	− 6.61429i	− 0.314255i	−0.987578	−	0.157127i	$-0.949777\pi$
$443$	− 6.61429i	− 0.314255i	0.987578	−	0.157127i	$-0.0502233\pi$
$444$	0	0
$445$	58.7255i	2.78385i
$446$	0	0
$447$	−8.73713	−0.413252
$448$	0	0
$449$	−30.8166	−1.45433	−0.727164	−	0.686464i	$-0.759162\pi$
$449$	−30.8166	−1.45433	−0.727164	+	0.686464i	$0.759162\pi$
$450$	0	0
$451$	− 28.1836i	− 1.32712i
$452$	0	0
$453$	− 13.0128i	− 0.611396i
$454$	0	0
$455$	−3.29373	−0.154412
$456$	0	0
$457$	−0.152457	−0.00713164	−0.00356582	−	0.999994i	$-0.501135\pi$
$457$	−0.152457	−0.00713164	−0.00356582	+	0.999994i	$0.501135\pi$
$458$	0	0
$459$	− 36.8548i	− 1.72024i
$460$	0	0
$461$	8.56283i	0.398811i	0.979917	+	0.199405i	$0.0639010\pi$
$461$	8.56283i	0.398811i	−0.979917	+	0.199405i	$0.936099\pi$
$462$	0	0
$463$	−2.68222	−0.124653	−0.0623267	−	0.998056i	$-0.519852\pi$
$463$	−2.68222	−0.124653	−0.0623267	+	0.998056i	$0.519852\pi$
$464$	0	0
$465$	7.28811	0.337978
$466$	0	0
$467$	− 34.2546i	− 1.58511i	−0.609797	−	0.792557i	$-0.708749\pi$
$467$	− 34.2546i	− 1.58511i	0.609797	−	0.792557i	$-0.291251\pi$
$468$	0	0
$469$	− 5.10411i	− 0.235686i
$470$	0	0
$471$	2.06799	0.0952878
$472$	0	0
$473$	38.6359	1.77648
$474$	0	0
$475$	48.1952i	2.21135i
$476$	0	0
$477$	13.3045i	0.609172i
$478$	0	0
$479$	−20.9436	−0.956939	−0.478469	−	0.878104i	$-0.658808\pi$
$479$	−20.9436	−0.956939	−0.478469	+	0.878104i	$0.658808\pi$
$480$	0	0
$481$	9.27443	0.422878
$482$	0	0
$483$	4.21514i	0.191795i
$484$	0	0
$485$	− 11.1670i	− 0.507067i
$486$	0	0
$487$	−25.8592	−1.17179	−0.585897	−	0.810386i	$-0.699258\pi$
$487$	−25.8592	−1.17179	−0.585897	+	0.810386i	$0.699258\pi$
$488$	0	0
$489$	6.95176	0.314369
$490$	0	0
$491$	− 15.7060i	− 0.708802i	−0.935094	−	0.354401i	$-0.884685\pi$
$491$	− 15.7060i	− 0.708802i	0.935094	−	0.354401i	$-0.115315\pi$
$492$	0	0
$493$	31.4338i	1.41571i
$494$	0	0
$495$	−21.4678	−0.964905
$496$	0	0
$497$	−2.70611	−0.121386
$498$	0	0
$499$	7.09732i	0.317720i	0.987301	+	0.158860i	$0.0507818\pi$
$499$	7.09732i	0.317720i	−0.987301	+	0.158860i	$0.949218\pi$
$500$	0	0
$501$	7.08010i	0.316316i
$502$	0	0
$503$	−4.14825	−0.184961	−0.0924806	−	0.995714i	$-0.529480\pi$
$503$	−4.14825	−0.184961	−0.0924806	+	0.995714i	$0.529480\pi$
$504$	0	0
$505$	4.48593	0.199621
$506$	0	0
$507$	1.07539i	0.0477598i
$508$	0	0
$509$	− 4.40744i	− 0.195356i	−0.995218	−	0.0976781i	$-0.968858\pi$
$509$	− 4.40744i	− 0.195356i	0.995218	−	0.0976781i	$-0.0311416\pi$
$510$	0	0
$511$	2.30012	0.101751
$512$	0	0
$513$	−42.9218	−1.89504
$514$	0	0
$515$	46.6780i	2.05688i
$516$	0	0
$517$	− 7.71133i	− 0.339144i
$518$	0	0
$519$	6.72377	0.295141
$520$	0	0
$521$	23.9822	1.05068	0.525340	−	0.850893i	$-0.323938\pi$
$521$	23.9822	1.05068	0.525340	+	0.850893i	$0.323938\pi$
$522$	0	0
$523$	− 22.9106i	− 1.00181i	−0.865503	−	0.500904i	$-0.833001\pi$
$523$	− 22.9106i	− 1.00181i	0.865503	−	0.500904i	$-0.166999\pi$
$524$	0	0
$525$	− 6.28957i	− 0.274499i
$526$	0	0
$527$	−14.5588	−0.634193
$528$	0	0
$529$	−7.63649	−0.332021
$530$	0	0
$531$	− 8.67076i	− 0.376279i
$532$	0	0
$533$	7.97166i	0.345291i
$534$	0	0
$535$	1.77658	0.0768084
$536$	0	0
$537$	3.76941	0.162662
$538$	0	0
$539$	− 3.53548i	− 0.152284i
$540$	0	0
$541$	29.7802i	1.28035i	0.768228	+	0.640176i	$0.221139\pi$
$541$	29.7802i	1.28035i	−0.768228	+	0.640176i	$0.778861\pi$
$542$	0	0
$543$	7.10892	0.305073
$544$	0	0
$545$	45.0112	1.92807
$546$	0	0
$547$	3.91692i	0.167476i	0.996488	+	0.0837378i	$0.0266858\pi$
$547$	3.91692i	0.167476i	−0.996488	+	0.0837378i	$0.973314\pi$
$548$	0	0
$549$	− 20.7375i	− 0.885057i
$550$	0	0
$551$	36.6084	1.55957
$552$	0	0
$553$	−10.5704	−0.449498
$554$	0	0
$555$	32.8504i	1.39442i
$556$	0	0
$557$	39.7322i	1.68351i	0.539863	+	0.841753i	$0.318476\pi$
$557$	39.7322i	1.68351i	−0.539863	+	0.841753i	$0.681524\pi$
$558$	0	0
$559$	−10.9281	−0.462207
$560$	0	0
$561$	−26.9018	−1.13579
$562$	0	0
$563$	4.85396i	0.204570i	0.994755	+	0.102285i	$0.0326154\pi$
$563$	4.85396i	0.204570i	−0.994755	+	0.102285i	$0.967385\pi$
$564$	0	0
$565$	4.05702i	0.170680i
$566$	0	0
$567$	0.0707804	0.00297250
$568$	0	0
$569$	43.2659	1.81380	0.906901	−	0.421343i	$-0.138441\pi$
$569$	43.2659	1.81380	0.906901	+	0.421343i	$0.138441\pi$
$570$	0	0
$571$	3.98213i	0.166647i	0.996523	+	0.0833234i	$0.0265534\pi$
$571$	3.98213i	0.166647i	−0.996523	+	0.0833234i	$0.973447\pi$
$572$	0	0
$573$	23.1783i	0.968287i
$574$	0	0
$575$	−22.9245	−0.956018
$576$	0	0
$577$	22.1093	0.920424	0.460212	−	0.887809i	$-0.347773\pi$
$577$	22.1093	0.920424	0.460212	+	0.887809i	$0.347773\pi$
$578$	0	0
$579$	18.4518i	0.766831i
$580$	0	0
$581$	− 4.62086i	− 0.191705i
$582$	0	0
$583$	25.5150	1.05672
$584$	0	0
$585$	6.07210	0.251050
$586$	0	0
$587$	43.8225i	1.80875i	0.426740	+	0.904374i	$0.359662\pi$
$587$	43.8225i	1.80875i	−0.426740	+	0.904374i	$0.640338\pi$
$588$	0	0
$589$	16.9555i	0.698638i
$590$	0	0
$591$	−15.5400	−0.639231
$592$	0	0
$593$	−14.1537	−0.581224	−0.290612	−	0.956841i	$-0.593859\pi$
$593$	−14.1537	−0.581224	−0.290612	+	0.956841i	$0.593859\pi$
$594$	0	0
$595$	23.3052i	0.955421i
$596$	0	0
$597$	11.0678i	0.452973i
$598$	0	0
$599$	12.0963	0.494240	0.247120	−	0.968985i	$-0.420516\pi$
$599$	12.0963	0.494240	0.247120	+	0.968985i	$0.420516\pi$
$600$	0	0
$601$	18.9414	0.772637	0.386318	−	0.922366i	$-0.373747\pi$
$601$	18.9414	0.772637	0.386318	+	0.922366i	$0.373747\pi$
$602$	0	0
$603$	9.40960i	0.383189i
$604$	0	0
$605$	4.93932i	0.200812i
$606$	0	0
$607$	13.8448	0.561945	0.280972	−	0.959716i	$-0.409343\pi$
$607$	13.8448	0.561945	0.280972	+	0.959716i	$0.409343\pi$
$608$	0	0
$609$	−4.77747	−0.193593
$610$	0	0
$611$	2.18113i	0.0882389i
$612$	0	0
$613$	− 27.5703i	− 1.11355i	−0.830662	−	0.556777i	$-0.812038\pi$
$613$	− 27.5703i	− 1.11355i	0.830662	−	0.556777i	$-0.187962\pi$
$614$	0	0
$615$	−28.2360	−1.13858
$616$	0	0
$617$	39.6703	1.59706	0.798532	−	0.601952i	$-0.205610\pi$
$617$	39.6703	1.59706	0.798532	+	0.601952i	$0.205610\pi$
$618$	0	0
$619$	5.27718i	0.212108i	0.994360	+	0.106054i	$0.0338216\pi$
$619$	5.27718i	0.212108i	−0.994360	+	0.106054i	$0.966178\pi$
$620$	0	0
$621$	− 20.4162i	− 0.819272i
$622$	0	0
$623$	17.8295	0.714323
$624$	0	0
$625$	−20.0366	−0.801465
$626$	0	0
$627$	31.3303i	1.25121i
$628$	0	0
$629$	− 65.6225i	− 2.61654i
$630$	0	0
$631$	−40.3059	−1.60455	−0.802277	−	0.596953i	$-0.796378\pi$
$631$	−40.3059	−1.60455	−0.802277	+	0.596953i	$0.796378\pi$
$632$	0	0
$633$	15.5015	0.616130
$634$	0	0
$635$	23.3162i	0.925274i
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	0	0
$639$	4.98881	0.197354
$640$	0	0
$641$	−28.4345	−1.12309	−0.561547	−	0.827445i	$-0.689793\pi$
$641$	−28.4345	−1.12309	−0.561547	+	0.827445i	$0.689793\pi$
$642$	0	0
$643$	27.4356i	1.08196i	0.841037	+	0.540978i	$0.181946\pi$
$643$	27.4356i	1.08196i	−0.841037	+	0.540978i	$0.818054\pi$
$644$	0	0
$645$	− 38.7077i	− 1.52411i
$646$	0	0
$647$	27.2040	1.06950	0.534751	−	0.845010i	$-0.320406\pi$
$647$	27.2040	1.06950	0.534751	+	0.845010i	$0.320406\pi$
$648$	0	0
$649$	−16.6286	−0.652728
$650$	0	0
$651$	− 2.21272i	− 0.0867235i
$652$	0	0
$653$	23.2488i	0.909794i	0.890544	+	0.454897i	$0.150324\pi$
$653$	23.2488i	0.909794i	−0.890544	+	0.454897i	$0.849676\pi$
$654$	0	0
$655$	36.4168	1.42292
$656$	0	0
$657$	−4.24035	−0.165432
$658$	0	0
$659$	− 20.4720i	− 0.797476i	−0.917065	−	0.398738i	$-0.869448\pi$
$659$	− 20.4720i	− 0.797476i	0.917065	−	0.398738i	$-0.130552\pi$
$660$	0	0
$661$	15.2233i	0.592116i	0.955170	+	0.296058i	$0.0956722\pi$
$661$	15.2233i	0.592116i	−0.955170	+	0.296058i	$0.904328\pi$
$662$	0	0
$663$	7.60908	0.295512
$664$	0	0
$665$	27.1417	1.05251
$666$	0	0
$667$	17.4131i	0.674238i
$668$	0	0
$669$	− 16.6939i	− 0.645422i
$670$	0	0
$671$	−39.7699	−1.53530
$672$	0	0
$673$	36.5446	1.40869	0.704346	−	0.709857i	$-0.251240\pi$
$673$	36.5446	1.40869	0.704346	+	0.709857i	$0.251240\pi$
$674$	0	0
$675$	30.4638i	1.17255i
$676$	0	0
$677$	5.69044i	0.218702i	0.994003	+	0.109351i	$0.0348772\pi$
$677$	5.69044i	0.218702i	−0.994003	+	0.109351i	$0.965123\pi$
$678$	0	0
$679$	−3.39038	−0.130111
$680$	0	0
$681$	26.3292	1.00894
$682$	0	0
$683$	34.9695i	1.33807i	0.743231	+	0.669035i	$0.233292\pi$
$683$	34.9695i	1.33807i	−0.743231	+	0.669035i	$0.766708\pi$
$684$	0	0
$685$	50.7848i	1.94039i
$686$	0	0
$687$	−5.07369	−0.193573
$688$	0	0
$689$	−7.21685	−0.274940
$690$	0	0
$691$	16.4333i	0.625153i	0.949893	+	0.312577i	$0.101192\pi$
$691$	16.4333i	0.625153i	−0.949893	+	0.312577i	$0.898808\pi$
$692$	0	0
$693$	6.51778i	0.247590i
$694$	0	0
$695$	64.6921	2.45391
$696$	0	0
$697$	56.4046	2.13648
$698$	0	0
$699$	9.23037i	0.349125i
$700$	0	0
$701$	− 1.51469i	− 0.0572091i	−0.999591	−	0.0286046i	$-0.990894\pi$
$701$	− 1.51469i	− 0.0572091i	0.999591	−	0.0286046i	$-0.00910635\pi$
$702$	0	0
$703$	−76.4252	−2.88243
$704$	0	0
$705$	−7.72565	−0.290965
$706$	0	0
$707$	− 1.36196i	− 0.0512218i
$708$	0	0
$709$	− 0.866021i	− 0.0325241i	−0.999868	−	0.0162621i	$-0.994823\pi$
$709$	− 0.866021i	− 0.0325241i	0.999868	−	0.0162621i	$-0.00517660\pi$
$710$	0	0
$711$	19.4868	0.730814
$712$	0	0
$713$	−8.06503	−0.302038
$714$	0	0
$715$	− 11.6449i	− 0.435495i
$716$	0	0
$717$	3.83095i	0.143070i
$718$	0	0
$719$	−7.35218	−0.274190	−0.137095	−	0.990558i	$-0.543777\pi$
$719$	−7.35218	−0.274190	−0.137095	+	0.990558i	$0.543777\pi$
$720$	0	0
$721$	14.1718	0.527786
$722$	0	0
$723$	− 6.37841i	− 0.237216i
$724$	0	0
$725$	− 25.9828i	− 0.964977i
$726$	0	0
$727$	35.8895	1.33107	0.665534	−	0.746367i	$-0.268204\pi$
$727$	35.8895	1.33107	0.665534	+	0.746367i	$0.268204\pi$
$728$	0	0
$729$	−16.9346	−0.627209
$730$	0	0
$731$	77.3230i	2.85989i
$732$	0	0
$733$	12.9753i	0.479255i	0.970865	+	0.239627i	$0.0770253\pi$
$733$	12.9753i	0.479255i	−0.970865	+	0.239627i	$0.922975\pi$
$734$	0	0
$735$	−3.54204	−0.130650
$736$	0	0
$737$	18.0455	0.664714
$738$	0	0
$739$	− 19.0725i	− 0.701593i	−0.936452	−	0.350796i	$-0.885911\pi$
$739$	− 19.0725i	− 0.701593i	0.936452	−	0.350796i	$-0.114089\pi$
$740$	0	0
$741$	− 8.86167i	− 0.325542i
$742$	0	0
$743$	32.0906	1.17729	0.588645	−	0.808392i	$-0.299662\pi$
$743$	32.0906	1.17729	0.588645	+	0.808392i	$0.299662\pi$
$744$	0	0
$745$	−26.7602	−0.980419
$746$	0	0
$747$	8.51870i	0.311683i
$748$	0	0
$749$	− 0.539384i	− 0.0197087i
$750$	0	0
$751$	0.631312	0.0230369	0.0115185	−	0.999934i	$-0.496333\pi$
$751$	0.631312	0.0230369	0.0115185	+	0.999934i	$0.496333\pi$
$752$	0	0
$753$	−22.6192	−0.824288
$754$	0	0
$755$	− 39.8559i	− 1.45050i
$756$	0	0
$757$	− 48.2035i	− 1.75199i	−0.482324	−	0.875993i	$-0.660207\pi$
$757$	− 48.2035i	− 1.75199i	0.482324	−	0.875993i	$-0.339793\pi$
$758$	0	0
$759$	−14.9025	−0.540928
$760$	0	0
$761$	19.3671	0.702057	0.351028	−	0.936365i	$-0.385832\pi$
$761$	19.3671	0.702057	0.351028	+	0.936365i	$0.385832\pi$
$762$	0	0
$763$	− 13.6657i	− 0.494733i
$764$	0	0
$765$	− 42.9640i	− 1.55337i
$766$	0	0
$767$	4.70334	0.169828
$768$	0	0
$769$	−35.3325	−1.27412	−0.637062	−	0.770813i	$-0.719850\pi$
$769$	−35.3325	−1.27412	−0.637062	+	0.770813i	$0.719850\pi$
$770$	0	0
$771$	− 22.4723i	− 0.809321i
$772$	0	0
$773$	− 6.73815i	− 0.242354i	−0.992631	−	0.121177i	$-0.961333\pi$
$773$	− 6.73815i	− 0.242354i	0.992631	−	0.121177i	$-0.0386669\pi$
$774$	0	0
$775$	12.0342	0.432279
$776$	0	0
$777$	9.97364	0.357802
$778$	0	0
$779$	− 65.6898i	− 2.35358i
$780$	0	0
$781$	− 9.56740i	− 0.342348i
$782$	0	0
$783$	23.1398	0.826950
$784$	0	0
$785$	6.33386	0.226065
$786$	0	0
$787$	14.0496i	0.500815i	0.968141	+	0.250407i	$0.0805646\pi$
$787$	14.0496i	0.500815i	−0.968141	+	0.250407i	$0.919435\pi$
$788$	0	0
$789$	− 11.6596i	− 0.415091i
$790$	0	0
$791$	1.23174	0.0437957
$792$	0	0
$793$	11.2488	0.399457
$794$	0	0
$795$	− 25.5624i	− 0.906605i
$796$	0	0
$797$	− 5.17217i	− 0.183208i	−0.995796	−	0.0916038i	$-0.970801\pi$
$797$	− 5.17217i	− 0.183208i	0.995796	−	0.0916038i	$-0.0291993\pi$
$798$	0	0
$799$	15.4329	0.545976
$800$	0	0
$801$	−32.8693	−1.16138
$802$	0	0
$803$	8.13203i	0.286973i
$804$	0	0
$805$	12.9102i	0.455025i
$806$	0	0
$807$	−13.1623	−0.463335
$808$	0	0
$809$	−39.2545	−1.38012	−0.690058	−	0.723754i	$-0.742415\pi$
$809$	−39.2545	−1.38012	−0.690058	+	0.723754i	$0.742415\pi$
$810$	0	0
$811$	− 28.7996i	− 1.01129i	−0.862741	−	0.505646i	$-0.831254\pi$
$811$	− 28.7996i	− 1.01129i	0.862741	−	0.505646i	$-0.168746\pi$
$812$	0	0
$813$	− 13.5201i	− 0.474170i
$814$	0	0
$815$	21.2920	0.745825
$816$	0	0
$817$	90.0517	3.15051
$818$	0	0
$819$	− 1.84353i	− 0.0644183i
$820$	0	0
$821$	8.35801i	0.291697i	0.989307	+	0.145848i	$0.0465911\pi$
$821$	8.35801i	0.291697i	−0.989307	+	0.145848i	$0.953409\pi$
$822$	0	0
$823$	27.4158	0.955653	0.477827	−	0.878454i	$-0.341425\pi$
$823$	27.4158	0.955653	0.477827	+	0.878454i	$0.341425\pi$
$824$	0	0
$825$	22.2366	0.774181
$826$	0	0
$827$	− 22.7213i	− 0.790099i	−0.918660	−	0.395049i	$-0.870727\pi$
$827$	− 22.7213i	− 0.790099i	0.918660	−	0.395049i	$-0.129273\pi$
$828$	0	0
$829$	− 24.3132i	− 0.844433i	−0.906495	−	0.422217i	$-0.861252\pi$
$829$	− 24.3132i	− 0.844433i	0.906495	−	0.422217i	$-0.138748\pi$
$830$	0	0
$831$	−22.1730	−0.769172
$832$	0	0
$833$	7.07564	0.245156
$834$	0	0
$835$	21.6851i	0.750443i
$836$	0	0
$837$	10.7174i	0.370448i
$838$	0	0
$839$	−4.37406	−0.151009	−0.0755047	−	0.997145i	$-0.524057\pi$
$839$	−4.37406	−0.151009	−0.0755047	+	0.997145i	$0.524057\pi$
$840$	0	0
$841$	9.26386	0.319443
$842$	0	0
$843$	25.6826i	0.884555i
$844$	0	0
$845$	3.29373i	0.113308i
$846$	0	0
$847$	1.49961	0.0515273
$848$	0	0
$849$	−9.24192	−0.317182
$850$	0	0
$851$	− 36.3524i	− 1.24614i
$852$	0	0
$853$	− 40.2813i	− 1.37920i	−0.724189	−	0.689602i	$-0.757786\pi$
$853$	− 40.2813i	− 1.37920i	0.724189	−	0.689602i	$-0.242214\pi$
$854$	0	0
$855$	−50.0366	−1.71122
$856$	0	0
$857$	2.44368	0.0834745	0.0417372	−	0.999129i	$-0.486711\pi$
$857$	2.44368	0.0834745	0.0417372	+	0.999129i	$0.486711\pi$
$858$	0	0
$859$	35.2686i	1.20335i	0.798741	+	0.601675i	$0.205500\pi$
$859$	35.2686i	1.20335i	−0.798741	+	0.601675i	$0.794500\pi$
$860$	0	0
$861$	8.57265i	0.292155i
$862$	0	0
$863$	−47.1579	−1.60527	−0.802637	−	0.596468i	$-0.796570\pi$
$863$	−47.1579	−1.60527	−0.802637	+	0.596468i	$0.796570\pi$
$864$	0	0
$865$	20.5937	0.700206
$866$	0	0
$867$	− 35.5575i	− 1.20760i
$868$	0	0
$869$	− 37.3713i	− 1.26774i
$870$	0	0
$871$	−5.10411	−0.172946
$872$	0	0
$873$	6.25028	0.211540
$874$	0	0
$875$	− 2.79518i	− 0.0944942i
$876$	0	0
$877$	41.3342i	1.39576i	0.716216	+	0.697878i	$0.245872\pi$
$877$	41.3342i	1.39576i	−0.716216	+	0.697878i	$0.754128\pi$
$878$	0	0
$879$	31.7344	1.07037
$880$	0	0
$881$	16.2539	0.547608	0.273804	−	0.961785i	$-0.411718\pi$
$881$	16.2539	0.547608	0.273804	+	0.961785i	$0.411718\pi$
$882$	0	0
$883$	− 8.18469i	− 0.275437i	−0.990471	−	0.137718i	$-0.956023\pi$
$883$	− 8.18469i	− 0.275437i	0.990471	−	0.137718i	$-0.0439769\pi$
$884$	0	0
$885$	16.6594i	0.560001i
$886$	0	0
$887$	−13.9240	−0.467521	−0.233760	−	0.972294i	$-0.575103\pi$
$887$	−13.9240	−0.467521	−0.233760	+	0.972294i	$0.575103\pi$
$888$	0	0
$889$	7.07896	0.237421
$890$	0	0
$891$	0.250243i	0.00838345i
$892$	0	0
$893$	− 17.9734i	− 0.601457i
$894$	0	0
$895$	11.5450	0.385908
$896$	0	0
$897$	4.21514	0.140739
$898$	0	0
$899$	− 9.14096i	− 0.304868i
$900$	0	0
$901$	51.0639i	1.70118i
$902$	0	0
$903$	−11.7519	−0.391080
$904$	0	0
$905$	21.7733	0.723770
$906$	0	0
$907$	− 4.38627i	− 0.145644i	−0.997345	−	0.0728219i	$-0.976800\pi$
$907$	− 4.38627i	− 0.145644i	0.997345	−	0.0728219i	$-0.0232005\pi$
$908$	0	0
$909$	2.51082i	0.0832787i
$910$	0	0
$911$	−40.7194	−1.34909	−0.674547	−	0.738232i	$-0.735661\pi$
$911$	−40.7194	−1.34909	−0.674547	+	0.738232i	$0.735661\pi$
$912$	0	0
$913$	16.3369	0.540674
$914$	0	0
$915$	39.8438i	1.31719i
$916$	0	0
$917$	− 11.0564i	− 0.365115i
$918$	0	0
$919$	−6.58805	−0.217320	−0.108660	−	0.994079i	$-0.534656\pi$
$919$	−6.58805	−0.217320	−0.108660	+	0.994079i	$0.534656\pi$
$920$	0	0
$921$	16.4561	0.542246
$922$	0	0
$923$	2.70611i	0.0890727i
$924$	0	0
$925$	54.2428i	1.78349i
$926$	0	0
$927$	−26.1262	−0.858097
$928$	0	0
$929$	13.8167	0.453312	0.226656	−	0.973975i	$-0.427221\pi$
$929$	13.8167	0.453312	0.226656	+	0.973975i	$0.427221\pi$
$930$	0	0
$931$	− 8.24042i	− 0.270069i
$932$	0	0
$933$	− 21.8953i	− 0.716821i
$934$	0	0
$935$	−82.3952	−2.69461
$936$	0	0
$937$	3.62263	0.118346	0.0591731	−	0.998248i	$-0.481154\pi$
$937$	3.62263	0.118346	0.0591731	+	0.998248i	$0.481154\pi$
$938$	0	0
$939$	− 21.2338i	− 0.692940i
$940$	0	0
$941$	46.8059i	1.52583i	0.646500	+	0.762914i	$0.276232\pi$
$941$	46.8059i	1.52583i	−0.646500	+	0.762914i	$0.723768\pi$
$942$	0	0
$943$	31.2460	1.01751
$944$	0	0
$945$	17.1560	0.558085
$946$	0	0
$947$	− 31.9520i	− 1.03830i	−0.854683	−	0.519151i	$-0.826248\pi$
$947$	− 31.9520i	− 1.03830i	0.854683	−	0.519151i	$-0.173752\pi$
$948$	0	0
$949$	− 2.30012i	− 0.0746651i
$950$	0	0
$951$	24.7331	0.802025
$952$	0	0
$953$	21.8310	0.707175	0.353587	−	0.935402i	$-0.384962\pi$
$953$	21.8310	0.707175	0.353587	+	0.935402i	$0.384962\pi$
$954$	0	0
$955$	70.9909i	2.29721i
$956$	0	0
$957$	− 16.8906i	− 0.545997i
$958$	0	0
$959$	15.4186	0.497894
$960$	0	0
$961$	−26.7663	−0.863429
$962$	0	0
$963$	0.994373i	0.0320432i
$964$	0	0
$965$	56.5145i	1.81927i
$966$	0	0
$967$	−7.00868	−0.225384	−0.112692	−	0.993630i	$-0.535947\pi$
$967$	−7.00868	−0.225384	−0.112692	+	0.993630i	$0.535947\pi$
$968$	0	0
$969$	−62.7020	−2.01428
$970$	0	0
$971$	− 11.1885i	− 0.359057i	−0.983753	−	0.179529i	$-0.942543\pi$
$971$	− 11.1885i	− 0.359057i	0.983753	−	0.179529i	$-0.0574572\pi$
$972$	0	0
$973$	− 19.6410i	− 0.629662i
$974$	0	0
$975$	−6.28957	−0.201427
$976$	0	0
$977$	−44.4907	−1.42338	−0.711691	−	0.702493i	$-0.752070\pi$
$977$	−44.4907	−1.42338	−0.711691	+	0.702493i	$0.752070\pi$
$978$	0	0
$979$	63.0358i	2.01463i
$980$	0	0
$981$	25.1932i	0.804358i
$982$	0	0
$983$	2.17498	0.0693711	0.0346855	−	0.999398i	$-0.488957\pi$
$983$	2.17498	0.0693711	0.0346855	+	0.999398i	$0.488957\pi$
$984$	0	0
$985$	−47.5963	−1.51654
$986$	0	0
$987$	2.34556i	0.0746601i
$988$	0	0
$989$	42.8339i	1.36204i
$990$	0	0
$991$	20.1092	0.638791	0.319396	−	0.947621i	$-0.396520\pi$
$991$	20.1092	0.638791	0.319396	+	0.947621i	$0.396520\pi$
$992$	0	0
$993$	9.23954	0.293208
$994$	0	0
$995$	33.8985i	1.07466i
$996$	0	0
$997$	− 23.3968i	− 0.740985i	−0.928836	−	0.370492i	$-0.879189\pi$
$997$	− 23.3968i	− 0.740985i	0.928836	−	0.370492i	$-0.120811\pi$
$998$	0	0
$999$	−48.3077	−1.52839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.b.1457.14		38
4.3	odd	2		728.2.c.b.365.26	yes	38
8.3	odd	2		728.2.c.b.365.25	✓	38
8.5	even	2	inner	2912.2.c.b.1457.25		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.b.365.25	✓	38	8.3	odd	2
728.2.c.b.365.26	yes	38	4.3	odd	2
2912.2.c.b.1457.14		38	1.1	even	1	trivial
2912.2.c.b.1457.25		38	8.5	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.07539i q^{3} -3.29373i q^{5} -1.00000 q^{7} +1.84353 q^{9} -3.53548i q^{11} +1.00000i q^{13} -3.54204 q^{15} +7.07564 q^{17} -8.24042i q^{19} +1.07539i q^{21} +3.91963 q^{23} -5.84864 q^{25} -5.20869i q^{27} +4.44254i q^{29} -2.05760 q^{31} -3.80202 q^{33} +3.29373i q^{35} -9.27443i q^{37} +1.07539 q^{39} +7.97166 q^{41} +10.9281i q^{43} -6.07210i q^{45} +2.18113 q^{47} +1.00000 q^{49} -7.60908i q^{51} +7.21685i q^{53} -11.6449 q^{55} -8.86167 q^{57} -4.70334i q^{59} -11.2488i q^{61} -1.84353 q^{63} +3.29373 q^{65} +5.10411i q^{67} -4.21514i q^{69} +2.70611 q^{71} -2.30012 q^{73} +6.28957i q^{75} +3.53548i q^{77} +10.5704 q^{79} -0.0707804 q^{81} +4.62086i q^{83} -23.3052i q^{85} +4.77747 q^{87} -17.8295 q^{89} -1.00000i q^{91} +2.21272i q^{93} -27.1417 q^{95} +3.39038 q^{97} -6.51778i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 38 q^{7} - 46 q^{9} + 8 q^{15} + 20 q^{17} - 12 q^{23} - 50 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 38 q^{49} + 8 q^{57} + 46 q^{63} + 20 q^{65} + 12 q^{79} + 62 q^{81} + 56 q^{87}+O(q^{100}) \) 38 * q - 38 * q^7 - 46 * q^9 + 8 * q^15 + 20 * q^17 - 12 * q^23 - 50 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 38 * q^49 + 8 * q^57 + 46 * q^63 + 20 * q^65 + 12 * q^79 + 62 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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