LMFDB - Embedded newform 1152.4.l.b.863.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(287,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.287"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			863.9
Character	$\chi$	$=$	1152.863
Dual form			1152.4.l.b.287.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.62348 - 3.62348i) q^{5} -33.7361 q^{7} +(-44.1694 + 44.1694i) q^{11} +(-42.1143 - 42.1143i) q^{13} -17.6392i q^{17} +(-99.2533 + 99.2533i) q^{19} -83.8215i q^{23} -98.7408i q^{25} +(-89.7300 + 89.7300i) q^{29} +46.2516i q^{31} +(122.242 + 122.242i) q^{35} +(7.47930 - 7.47930i) q^{37} -299.247 q^{41} +(56.3439 + 56.3439i) q^{43} +280.235 q^{47} +795.127 q^{49} +(8.15996 + 8.15996i) q^{53} +320.094 q^{55} +(-193.284 + 193.284i) q^{59} +(-127.057 - 127.057i) q^{61} +305.201i q^{65} +(110.221 - 110.221i) q^{67} -1070.35i q^{71} +1015.84i q^{73} +(1490.11 - 1490.11i) q^{77} +161.531i q^{79} +(-64.4215 - 64.4215i) q^{83} +(-63.9151 + 63.9151i) q^{85} +177.502 q^{89} +(1420.77 + 1420.77i) q^{91} +719.285 q^{95} +559.412 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} - 576 q^{55} - 1824 q^{61} - 816 q^{67} + 480 q^{85} + 3600 q^{91}+O(q^{100}) $ 48 * q + 48 * q^19 - 864 * q^43 + 2352 * q^49 - 576 * q^55 - 1824 * q^61 - 816 * q^67 + 480 * q^85 + 3600 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	−3.62348	−	3.62348i	−0.324094	−	0.324094i	0.526241	−	0.850335i	$-0.323601\pi$
$5$	−3.62348	−	3.62348i	−0.324094	−	0.324094i	−0.850335	+	0.526241i	$0.823601\pi$
$6$		0			0
$7$	−33.7361			−1.82158			−0.910790	−	0.412869i	$-0.864527\pi$
$7$	−33.7361			−1.82158			−0.910790	+	0.412869i	$0.864527\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−44.1694	+	44.1694i	−1.21069	+	1.21069i	−0.239889	+	0.970800i	$0.577111\pi$
$11$	−44.1694	+	44.1694i	−1.21069	+	1.21069i	−0.970800	+	0.239889i	$0.922889\pi$
$12$		0			0
$13$	−42.1143	−	42.1143i	−0.898493	−	0.898493i	0.0968099	−	0.995303i	$-0.469136\pi$
$13$	−42.1143	−	42.1143i	−0.898493	−	0.898493i	−0.995303	+	0.0968099i	$0.969136\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	17.6392i		−	0.251654i	−0.992052	−	0.125827i	$-0.959842\pi$
$17$		−	17.6392i		−	0.251654i	0.992052	−	0.125827i	$-0.0401585\pi$
$18$		0			0
$19$	−99.2533	+	99.2533i	−1.19844	+	1.19844i	−0.223801	+	0.974635i	$0.571847\pi$
$19$	−99.2533	+	99.2533i	−1.19844	+	1.19844i	−0.974635	+	0.223801i	$0.928153\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	83.8215i		−	0.759913i	−0.925004	−	0.379956i	$-0.875939\pi$
$23$		−	83.8215i		−	0.759913i	0.925004	−	0.379956i	$-0.124061\pi$
$24$		0			0
$25$		−	98.7408i		−	0.789926i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−89.7300	+	89.7300i	−0.574567	+	0.574567i	−0.933401	−	0.358834i	$-0.883174\pi$
$29$	−89.7300	+	89.7300i	−0.574567	+	0.574567i	0.358834	+	0.933401i	$0.383174\pi$
$30$		0			0
$31$			46.2516i			0.267969i	0.990983	+	0.133984i	$0.0427772\pi$
$31$			46.2516i			0.267969i	−0.990983	+	0.133984i	$0.957223\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	122.242	+	122.242i	0.590363	+	0.590363i
$36$		0			0
$37$	7.47930	−	7.47930i	0.0332321	−	0.0332321i	−0.690295	−	0.723528i	$-0.742519\pi$
$37$	7.47930	−	7.47930i	0.0332321	−	0.0332321i	0.723528	+	0.690295i	$0.242519\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−299.247			−1.13986			−0.569932	−	0.821692i	$-0.693031\pi$
$41$	−299.247			−1.13986			−0.569932	+	0.821692i	$0.693031\pi$
$42$		0			0
$43$	56.3439	+	56.3439i	0.199823	+	0.199823i	0.799924	−	0.600101i	$-0.204873\pi$
$43$	56.3439	+	56.3439i	0.199823	+	0.199823i	−0.600101	+	0.799924i	$0.704873\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	280.235			0.869711			0.434855	−	0.900500i	$-0.356799\pi$
$47$	280.235			0.869711			0.434855	+	0.900500i	$0.356799\pi$
$48$		0			0
$49$	795.127			2.31816
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.15996	+	8.15996i	0.0211482	+	0.0211482i	0.717602	−	0.696454i	$-0.245240\pi$
$53$	8.15996	+	8.15996i	0.0211482	+	0.0211482i	−0.696454	+	0.717602i	$0.745240\pi$
$54$		0			0
$55$	320.094			0.784754
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−193.284	+	193.284i	−0.426499	+	0.426499i	−0.887434	−	0.460935i	$-0.847514\pi$
$59$	−193.284	+	193.284i	−0.426499	+	0.426499i	0.460935	+	0.887434i	$0.347514\pi$
$60$		0			0
$61$	−127.057	−	127.057i	−0.266688	−	0.266688i	0.561076	−	0.827764i	$-0.310388\pi$
$61$	−127.057	−	127.057i	−0.266688	−	0.266688i	−0.827764	+	0.561076i	$0.810388\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			305.201i			0.582392i
$66$		0			0
$67$	110.221	−	110.221i	0.200980	−	0.200980i	−0.599440	−	0.800420i	$-0.704610\pi$
$67$	110.221	−	110.221i	0.200980	−	0.200980i	0.800420	+	0.599440i	$0.204610\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1070.35i		−	1.78911i	−0.446955	−	0.894556i	$-0.647492\pi$
$71$		−	1070.35i		−	1.78911i	0.446955	−	0.894556i	$-0.352508\pi$
$72$		0			0
$73$			1015.84i			1.62870i	0.580373	+	0.814351i	$0.302907\pi$
$73$			1015.84i			1.62870i	−0.580373	+	0.814351i	$0.697093\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1490.11	−	1490.11i	2.20537	−	2.20537i
$78$		0			0
$79$			161.531i			0.230047i	0.993363	+	0.115023i	$0.0366943\pi$
$79$			161.531i			0.230047i	−0.993363	+	0.115023i	$0.963306\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−64.4215	−	64.4215i	−0.0851950	−	0.0851950i	0.663225	−	0.748420i	$-0.269187\pi$
$83$	−64.4215	−	64.4215i	−0.0851950	−	0.0851950i	−0.748420	+	0.663225i	$0.769187\pi$
$84$		0			0
$85$	−63.9151	+	63.9151i	−0.0815596	+	0.0815596i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	177.502			0.211406			0.105703	−	0.994398i	$-0.466291\pi$
$89$	177.502			0.211406			0.105703	+	0.994398i	$0.466291\pi$
$90$		0			0
$91$	1420.77	+	1420.77i	1.63668	+	1.63668i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	719.285			0.776811
$96$		0			0
$97$	559.412			0.585564			0.292782	−	0.956179i	$-0.405419\pi$
$97$	559.412			0.585564			0.292782	+	0.956179i	$0.405419\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−850.608	−	850.608i	−0.838006	−	0.838006i	0.150590	−	0.988596i	$-0.451883\pi$
$101$	−850.608	−	850.608i	−0.838006	−	0.838006i	−0.988596	+	0.150590i	$0.951883\pi$
$102$		0			0
$103$	−757.707			−0.724845			−0.362423	−	0.932014i	$-0.618050\pi$
$103$	−757.707			−0.724845			−0.362423	+	0.932014i	$0.618050\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	364.469	−	364.469i	0.329295	−	0.329295i	−0.523024	−	0.852318i	$-0.675196\pi$
$107$	364.469	−	364.469i	0.329295	−	0.329295i	0.852318	+	0.523024i	$0.175196\pi$
$108$		0			0
$109$	206.900	+	206.900i	0.181812	+	0.181812i	0.792145	−	0.610333i	$-0.208964\pi$
$109$	206.900	+	206.900i	0.181812	+	0.181812i	−0.610333	+	0.792145i	$0.708964\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1379.24i		−	1.14821i	−0.818781	−	0.574105i	$-0.805350\pi$
$113$		−	1379.24i		−	1.14821i	0.818781	−	0.574105i	$-0.194650\pi$
$114$		0			0
$115$	−303.726	+	303.726i	−0.246283	+	0.246283i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			595.077i			0.458409i
$120$		0			0
$121$		−	2570.88i		−	1.93154i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−810.720	+	810.720i	−0.580104	+	0.580104i
$126$		0			0
$127$			1260.84i			0.880958i	0.897763	+	0.440479i	$0.145191\pi$
$127$			1260.84i			0.880958i	−0.897763	+	0.440479i	$0.854809\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1015.28	+	1015.28i	0.677138	+	0.677138i	0.959352	−	0.282214i	$-0.0910687\pi$
$131$	1015.28	+	1015.28i	0.677138	+	0.677138i	−0.282214	+	0.959352i	$0.591069\pi$
$132$		0			0
$133$	3348.43	−	3348.43i	2.18305	−	2.18305i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1875.46			−1.16957			−0.584785	−	0.811189i	$-0.698821\pi$
$137$	−1875.46			−1.16957			−0.584785	+	0.811189i	$0.698821\pi$
$138$		0			0
$139$	1048.87	+	1048.87i	0.640031	+	0.640031i	0.950563	−	0.310532i	$-0.100507\pi$
$139$	1048.87	+	1048.87i	0.640031	+	0.640031i	−0.310532	+	0.950563i	$0.600507\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3720.33			2.17559
$144$		0			0
$145$	650.270			0.372427
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1943.41	−	1943.41i	−1.06852	−	1.06852i	−0.997473	−	0.0710521i	$-0.977364\pi$
$149$	−1943.41	−	1943.41i	−1.06852	−	1.06852i	−0.0710521	−	0.997473i	$-0.522636\pi$
$150$		0			0
$151$	−1790.47			−0.964945			−0.482472	−	0.875911i	$-0.660261\pi$
$151$	−1790.47			−0.964945			−0.482472	+	0.875911i	$0.660261\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	167.592	−	167.592i	0.0868470	−	0.0868470i
$156$		0			0
$157$	872.163	+	872.163i	0.443351	+	0.443351i	0.893137	−	0.449785i	$-0.148500\pi$
$157$	872.163	+	872.163i	0.443351	+	0.443351i	−0.449785	+	0.893137i	$0.648500\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2827.82i			1.38424i
$162$		0			0
$163$	−541.663	+	541.663i	−0.260284	+	0.260284i	−0.825169	−	0.564885i	$-0.808920\pi$
$163$	−541.663	+	541.663i	−0.260284	+	0.260284i	0.564885	+	0.825169i	$0.308920\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			271.228i			0.125678i	0.998024	+	0.0628390i	$0.0200155\pi$
$167$			271.228i			0.125678i	−0.998024	+	0.0628390i	$0.979985\pi$
$168$		0			0
$169$			1350.23i			0.614579i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1396.64	−	1396.64i	0.613783	−	0.613783i	−0.330147	−	0.943930i	$-0.607098\pi$
$173$	1396.64	−	1396.64i	0.613783	−	0.613783i	0.943930	+	0.330147i	$0.107098\pi$
$174$		0			0
$175$			3331.13i			1.43891i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	675.124	+	675.124i	0.281906	+	0.281906i	0.833869	−	0.551963i	$-0.186121\pi$
$179$	675.124	+	675.124i	0.281906	+	0.281906i	−0.551963	+	0.833869i	$0.686121\pi$
$180$		0			0
$181$	−1306.81	+	1306.81i	−0.536655	+	0.536655i	−0.922545	−	0.385890i	$-0.873894\pi$
$181$	−1306.81	+	1306.81i	−0.536655	+	0.536655i	0.385890	+	0.922545i	$0.373894\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−54.2021			−0.0215407
$186$		0			0
$187$	779.111	+	779.111i	0.304675	+	0.304675i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1942.70			−0.735961			−0.367980	−	0.929834i	$-0.619951\pi$
$191$	−1942.70			−0.735961			−0.367980	+	0.929834i	$0.619951\pi$
$192$		0			0
$193$	4145.80			1.54623			0.773113	−	0.634269i	$-0.218699\pi$
$193$	4145.80			1.54623			0.773113	+	0.634269i	$0.218699\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1926.84	+	1926.84i	0.696860	+	0.696860i	0.963732	−	0.266872i	$-0.0859902\pi$
$197$	1926.84	+	1926.84i	0.696860	+	0.696860i	−0.266872	+	0.963732i	$0.585990\pi$
$198$		0			0
$199$	3804.49			1.35524			0.677622	−	0.735411i	$-0.263011\pi$
$199$	3804.49			1.35524			0.677622	+	0.735411i	$0.263011\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3027.15	−	3027.15i	1.04662	−	1.04662i
$204$		0			0
$205$	1084.31	+	1084.31i	0.369423	+	0.369423i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8767.93i		−	2.90187i
$210$		0			0
$211$	−2274.25	+	2274.25i	−0.742019	+	0.742019i	−0.972966	−	0.230947i	$-0.925818\pi$
$211$	−2274.25	+	2274.25i	−0.742019	+	0.742019i	0.230947	+	0.972966i	$0.425818\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	408.322i		−	0.129523i
$216$		0			0
$217$		−	1560.35i		−	0.488126i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−742.861	+	742.861i	−0.226110	+	0.226110i
$222$		0			0
$223$			2370.49i			0.711836i	0.934517	+	0.355918i	$0.115832\pi$
$223$			2370.49i			0.711836i	−0.934517	+	0.355918i	$0.884168\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3494.40	−	3494.40i	−1.02172	−	1.02172i	−0.999759	−	0.0219659i	$-0.993007\pi$
$227$	−3494.40	−	3494.40i	−1.02172	−	1.02172i	−0.0219659	−	0.999759i	$-0.506993\pi$
$228$		0			0
$229$	2251.16	−	2251.16i	0.649611	−	0.649611i	−0.303288	−	0.952899i	$-0.598084\pi$
$229$	2251.16	−	2251.16i	0.649611	−	0.649611i	0.952899	+	0.303288i	$0.0980844\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3104.86			0.872989			0.436494	−	0.899707i	$-0.356220\pi$
$233$	3104.86			0.872989			0.436494	+	0.899707i	$0.356220\pi$
$234$		0			0
$235$	−1015.42	−	1015.42i	−0.281868	−	0.281868i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5602.27			−1.51624			−0.758119	−	0.652117i	$-0.773881\pi$
$239$	−5602.27			−1.51624			−0.758119	+	0.652117i	$0.773881\pi$
$240$		0			0
$241$	−278.260			−0.0743748			−0.0371874	−	0.999308i	$-0.511840\pi$
$241$	−278.260			−0.0743748			−0.0371874	+	0.999308i	$0.511840\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2881.13	−	2881.13i	−0.751300	−	0.751300i
$246$		0			0
$247$	8359.97			2.15357
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4432.83	−	4432.83i	1.11473	−	1.11473i	0.122231	−	0.992502i	$-0.460995\pi$
$251$	4432.83	−	4432.83i	1.11473	−	1.11473i	0.992502	−	0.122231i	$-0.0390048\pi$
$252$		0			0
$253$	3702.35	+	3702.35i	0.920019	+	0.920019i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	3828.87i		−	0.929332i	−0.885486	−	0.464666i	$-0.846174\pi$
$257$		−	3828.87i		−	0.929332i	0.885486	−	0.464666i	$-0.153826\pi$
$258$		0			0
$259$	−252.323	+	252.323i	−0.0605350	+	0.0605350i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2299.83i		−	0.539216i	−0.962970	−	0.269608i	$-0.913106\pi$
$263$		−	2299.83i		−	0.539216i	0.962970	−	0.269608i	$-0.0868941\pi$
$264$		0			0
$265$		−	59.1349i		−	0.0137080i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3350.78	+	3350.78i	−0.759481	+	0.759481i	−0.976228	−	0.216747i	$-0.930455\pi$
$269$	−3350.78	+	3350.78i	−0.759481	+	0.759481i	0.216747	+	0.976228i	$0.430455\pi$
$270$		0			0
$271$			3778.66i			0.847000i	0.905896	+	0.423500i	$0.139199\pi$
$271$			3778.66i			0.847000i	−0.905896	+	0.423500i	$0.860801\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4361.32	+	4361.32i	0.956356	+	0.956356i
$276$		0			0
$277$	161.818	−	161.818i	0.0351000	−	0.0351000i	−0.689339	−	0.724439i	$-0.742099\pi$
$277$	161.818	−	161.818i	0.0351000	−	0.0351000i	0.724439	+	0.689339i	$0.242099\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4259.63			0.904299			0.452149	−	0.891942i	$-0.350657\pi$
$281$	4259.63			0.904299			0.452149	+	0.891942i	$0.350657\pi$
$282$		0			0
$283$	769.856	+	769.856i	0.161707	+	0.161707i	0.783323	−	0.621615i	$-0.213523\pi$
$283$	769.856	+	769.856i	0.161707	+	0.161707i	−0.621615	+	0.783323i	$0.713523\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10095.4			2.07636
$288$		0			0
$289$	4601.86			0.936670
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4375.88	+	4375.88i	0.872497	+	0.872497i	0.992744	−	0.120247i	$-0.0383685\pi$
$293$	4375.88	+	4375.88i	0.872497	+	0.872497i	−0.120247	+	0.992744i	$0.538369\pi$
$294$		0			0
$295$	1400.72			0.276451
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3530.09	+	3530.09i	−0.682776	+	0.682776i
$300$		0			0
$301$	−1900.83	−	1900.83i	−0.363993	−	0.363993i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			920.776i			0.172864i
$306$		0			0
$307$	1892.89	−	1892.89i	0.351899	−	0.351899i	−0.508917	−	0.860816i	$-0.669954\pi$
$307$	1892.89	−	1892.89i	0.351899	−	0.351899i	0.860816	+	0.508917i	$0.169954\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4969.36i			0.906067i	0.891494	+	0.453033i	$0.149658\pi$
$311$			4969.36i			0.906067i	−0.891494	+	0.453033i	$0.850342\pi$
$312$		0			0
$313$		−	431.319i		−	0.0778901i	−0.999241	−	0.0389450i	$-0.987600\pi$
$313$		−	431.319i		−	0.0778901i	0.999241	−	0.0389450i	$-0.0123997\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4151.14	+	4151.14i	−0.735493	+	0.735493i	−0.971702	−	0.236209i	$-0.924095\pi$
$317$	−4151.14	+	4151.14i	−0.735493	+	0.735493i	0.236209	+	0.971702i	$0.424095\pi$
$318$		0			0
$319$		−	7926.65i		−	1.39125i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1750.74	+	1750.74i	0.301591	+	0.301591i
$324$		0			0
$325$	−4158.40	+	4158.40i	−0.709743	+	0.709743i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9454.03			−1.58425
$330$		0			0
$331$	−3350.31	−	3350.31i	−0.556343	−	0.556343i	0.371921	−	0.928264i	$-0.378699\pi$
$331$	−3350.31	−	3350.31i	−0.556343	−	0.556343i	−0.928264	+	0.371921i	$0.878699\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−798.770			−0.130273
$336$		0			0
$337$	−9901.48			−1.60050			−0.800249	−	0.599668i	$-0.795299\pi$
$337$	−9901.48			−1.60050			−0.800249	+	0.599668i	$0.795299\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2042.90	−	2042.90i	−0.324427	−	0.324427i
$342$		0			0
$343$	−15253.0			−2.40113
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1244.90	+	1244.90i	−0.192593	+	0.192593i	−0.796816	−	0.604222i	$-0.793484\pi$
$347$	−1244.90	+	1244.90i	−0.192593	+	0.192593i	0.604222	+	0.796816i	$0.293484\pi$
$348$		0			0
$349$	−84.2939	−	84.2939i	−0.0129288	−	0.0129288i	0.700613	−	0.713542i	$-0.252910\pi$
$349$	−84.2939	−	84.2939i	−0.0129288	−	0.0129288i	−0.713542	+	0.700613i	$0.752910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			366.806i			0.0553063i	0.999618	+	0.0276531i	$0.00880339\pi$
$353$			366.806i			0.0553063i	−0.999618	+	0.0276531i	$0.991197\pi$
$354$		0			0
$355$	−3878.39	+	3878.39i	−0.579840	+	0.579840i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6841.31i			1.00577i	0.864354	+	0.502884i	$0.167728\pi$
$359$			6841.31i			1.00577i	−0.864354	+	0.502884i	$0.832272\pi$
$360$		0			0
$361$		−	12843.5i		−	1.87250i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3680.88	−	3680.88i	0.527852	−	0.527852i
$366$		0			0
$367$			1916.32i			0.272564i	0.990670	+	0.136282i	$0.0435153\pi$
$367$			1916.32i			0.272564i	−0.990670	+	0.136282i	$0.956485\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−275.286	−	275.286i	−0.0385232	−	0.0385232i
$372$		0			0
$373$	7158.65	−	7158.65i	0.993728	−	0.993728i	−0.00625218	−	0.999980i	$-0.501990\pi$
$373$	7158.65	−	7158.65i	0.993728	−	0.993728i	0.999980	+	0.00625218i	$0.00199014\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7557.84			1.03249
$378$		0			0
$379$	−5108.00	−	5108.00i	−0.692297	−	0.692297i	0.270440	−	0.962737i	$-0.412831\pi$
$379$	−5108.00	−	5108.00i	−0.692297	−	0.692297i	−0.962737	+	0.270440i	$0.912831\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10334.0			−1.37870			−0.689349	−	0.724429i	$-0.742103\pi$
$383$	−10334.0			−1.37870			−0.689349	+	0.724429i	$0.742103\pi$
$384$		0			0
$385$	−10798.7			−1.42949
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4119.25	−	4119.25i	−0.536900	−	0.536900i	0.385717	−	0.922617i	$-0.373954\pi$
$389$	−4119.25	−	4119.25i	−0.536900	−	0.536900i	−0.922617	+	0.385717i	$0.873954\pi$
$390$		0			0
$391$	−1478.54			−0.191235
$392$		0			0
$393$		0			0
$394$		0			0
$395$	585.306	−	585.306i	0.0745568	−	0.0745568i
$396$		0			0
$397$	−6024.12	−	6024.12i	−0.761566	−	0.761566i	0.215039	−	0.976605i	$-0.431012\pi$
$397$	−6024.12	−	6024.12i	−0.761566	−	0.761566i	−0.976605	+	0.215039i	$0.931012\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			11848.6i			1.47554i	0.675050	+	0.737772i	$0.264122\pi$
$401$			11848.6i			1.47554i	−0.675050	+	0.737772i	$0.735878\pi$
$402$		0			0
$403$	1947.85	−	1947.85i	0.240768	−	0.240768i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			660.713i			0.0804676i
$408$		0			0
$409$			12692.3i			1.53446i	0.641374	+	0.767228i	$0.278365\pi$
$409$			12692.3i			1.53446i	−0.641374	+	0.767228i	$0.721635\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6520.66	−	6520.66i	0.776903	−	0.776903i
$414$		0			0
$415$			466.860i			0.0552223i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9974.97	−	9974.97i	−1.16303	−	1.16303i	−0.983809	−	0.179220i	$-0.942643\pi$
$419$	−9974.97	−	9974.97i	−1.16303	−	1.16303i	−0.179220	−	0.983809i	$-0.557357\pi$
$420$		0			0
$421$	−6979.64	+	6979.64i	−0.807997	+	0.807997i	−0.984331	−	0.176333i	$-0.943576\pi$
$421$	−6979.64	+	6979.64i	−0.807997	+	0.807997i	0.176333	+	0.984331i	$0.443576\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1741.70			−0.198788
$426$		0			0
$427$	4286.41	+	4286.41i	0.485794	+	0.485794i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6433.80			−0.719038			−0.359519	−	0.933138i	$-0.617059\pi$
$431$	−6433.80			−0.719038			−0.359519	+	0.933138i	$0.617059\pi$
$432$		0			0
$433$	2276.81			0.252694			0.126347	−	0.991986i	$-0.459675\pi$
$433$	2276.81			0.252694			0.126347	+	0.991986i	$0.459675\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8319.57	+	8319.57i	0.910707	+	0.910707i
$438$		0			0
$439$	8874.90			0.964866			0.482433	−	0.875933i	$-0.339753\pi$
$439$	8874.90			0.964866			0.482433	+	0.875933i	$0.339753\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−801.170	+	801.170i	−0.0859249	+	0.0859249i	−0.748763	−	0.662838i	$-0.769352\pi$
$443$	−801.170	+	801.170i	−0.0859249	+	0.0859249i	0.662838	+	0.748763i	$0.269352\pi$
$444$		0			0
$445$	−643.173	−	643.173i	−0.0685154	−	0.0685154i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	453.819i		−	0.0476995i	−0.999716	−	0.0238497i	$-0.992408\pi$
$449$		−	453.819i		−	0.0476995i	0.999716	−	0.0238497i	$-0.00759232\pi$
$450$		0			0
$451$	13217.6	−	13217.6i	1.38002	−	1.38002i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	10296.3i		−	1.06087i
$456$		0			0
$457$			17430.4i			1.78416i	0.451876	+	0.892081i	$0.350755\pi$
$457$			17430.4i			1.78416i	−0.451876	+	0.892081i	$0.649245\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2812.24	+	2812.24i	−0.284119	+	0.284119i	−0.834749	−	0.550630i	$-0.814387\pi$
$461$	−2812.24	+	2812.24i	−0.284119	+	0.284119i	0.550630	+	0.834749i	$0.314387\pi$
$462$		0			0
$463$		−	8580.37i		−	0.861260i	−0.902528	−	0.430630i	$-0.858291\pi$
$463$		−	8580.37i		−	0.861260i	0.902528	−	0.430630i	$-0.141709\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4028.61	−	4028.61i	−0.399190	−	0.399190i	0.478757	−	0.877947i	$-0.341087\pi$
$467$	−4028.61	−	4028.61i	−0.399190	−	0.399190i	−0.877947	+	0.478757i	$0.841087\pi$
$468$		0			0
$469$	−3718.45	+	3718.45i	−0.366102	+	0.366102i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4977.36			−0.483846
$474$		0			0
$475$	9800.35	+	9800.35i	0.946676	+	0.946676i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7073.65			0.674746			0.337373	−	0.941371i	$-0.390462\pi$
$479$	7073.65			0.674746			0.337373	+	0.941371i	$0.390462\pi$
$480$		0			0
$481$	−629.971			−0.0597177
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2027.02	−	2027.02i	−0.189778	−	0.189778i
$486$		0			0
$487$	−16066.6			−1.49497			−0.747483	−	0.664281i	$-0.768738\pi$
$487$	−16066.6			−1.49497			−0.747483	+	0.664281i	$0.768738\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4947.42	+	4947.42i	−0.454733	+	0.454733i	−0.896922	−	0.442189i	$-0.854202\pi$
$491$	−4947.42	+	4947.42i	−0.454733	+	0.454733i	0.442189	+	0.896922i	$0.354202\pi$
$492$		0			0
$493$	1582.76	+	1582.76i	0.144592	+	0.144592i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			36109.4i			3.25901i
$498$		0			0
$499$	10229.8	−	10229.8i	0.917733	−	0.917733i	−0.0791313	−	0.996864i	$-0.525215\pi$
$499$	10229.8	−	10229.8i	0.917733	−	0.917733i	0.996864	+	0.0791313i	$0.0252146\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	6091.42i		−	0.539966i	−0.962865	−	0.269983i	$-0.912982\pi$
$503$		−	6091.42i		−	0.539966i	0.962865	−	0.269983i	$-0.0870181\pi$
$504$		0			0
$505$			6164.32i			0.543185i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10933.5	−	10933.5i	0.952097	−	0.952097i	−0.0468068	−	0.998904i	$-0.514904\pi$
$509$	10933.5	−	10933.5i	0.952097	−	0.952097i	0.998904	+	0.0468068i	$0.0149045\pi$
$510$		0			0
$511$		−	34270.6i		−	2.96681i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2745.53	+	2745.53i	0.234918	+	0.234918i
$516$		0			0
$517$	−12377.8	+	12377.8i	−1.05295	+	1.05295i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8966.41			0.753984			0.376992	−	0.926217i	$-0.376958\pi$
$521$	8966.41			0.753984			0.376992	+	0.926217i	$0.376958\pi$
$522$		0			0
$523$	−14085.5	−	14085.5i	−1.17766	−	1.17766i	−0.980340	−	0.197318i	$-0.936777\pi$
$523$	−14085.5	−	14085.5i	−1.17766	−	1.17766i	−0.197318	−	0.980340i	$-0.563223\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	815.838			0.0674354
$528$		0			0
$529$	5140.95			0.422532
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12602.6	+	12602.6i	1.02416	+	1.02416i
$534$		0			0
$535$	−2641.29			−0.213445
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−35120.3	+	35120.3i	−2.80657	+	2.80657i
$540$		0			0
$541$	−3398.59	−	3398.59i	−0.270087	−	0.270087i	0.559048	−	0.829135i	$-0.311167\pi$
$541$	−3398.59	−	3398.59i	−0.270087	−	0.270087i	−0.829135	+	0.559048i	$0.811167\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1499.40i		−	0.117848i
$546$		0			0
$547$	−15912.7	+	15912.7i	−1.24383	+	1.24383i	−0.285438	+	0.958397i	$0.592139\pi$
$547$	−15912.7	+	15912.7i	−1.24383	+	1.24383i	−0.958397	+	0.285438i	$0.907861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	17812.0i		−	1.37716i
$552$		0			0
$553$		−	5449.45i		−	0.419049i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12309.2	+	12309.2i	−0.936371	+	0.936371i	−0.998093	−	0.0617225i	$-0.980341\pi$
$557$	−12309.2	+	12309.2i	−0.936371	+	0.936371i	0.0617225	+	0.998093i	$0.480341\pi$
$558$		0			0
$559$		−	4745.77i		−	0.359078i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13010.7	+	13010.7i	0.973955	+	0.973955i	0.999669	−	0.0257144i	$-0.00818604\pi$
$563$	13010.7	+	13010.7i	0.973955	+	0.973955i	−0.0257144	+	0.999669i	$0.508186\pi$
$564$		0			0
$565$	−4997.64	+	4997.64i	−0.372128	+	0.372128i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20663.0			−1.52238			−0.761192	−	0.648527i	$-0.775385\pi$
$569$	−20663.0			−1.52238			−0.761192	+	0.648527i	$0.775385\pi$
$570$		0			0
$571$	−6847.53	−	6847.53i	−0.501857	−	0.501857i	0.410158	−	0.912015i	$-0.365474\pi$
$571$	−6847.53	−	6847.53i	−0.501857	−	0.501857i	−0.912015	+	0.410158i	$0.865474\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8276.60			−0.600275
$576$		0			0
$577$	−12587.4			−0.908178			−0.454089	−	0.890956i	$-0.650035\pi$
$577$	−12587.4			−0.908178			−0.454089	+	0.890956i	$0.650035\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2173.33	+	2173.33i	0.155189	+	0.155189i
$582$		0			0
$583$	−720.841			−0.0512079
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10285.9	−	10285.9i	0.723241	−	0.723241i	−0.246023	−	0.969264i	$-0.579124\pi$
$587$	10285.9	−	10285.9i	0.723241	−	0.723241i	0.969264	+	0.246023i	$0.0791238\pi$
$588$		0			0
$589$	−4590.62	−	4590.62i	−0.321143	−	0.321143i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19189.3i		−	1.32886i	−0.747353	−	0.664428i	$-0.768675\pi$
$593$		−	19189.3i		−	1.32886i	0.747353	−	0.664428i	$-0.231325\pi$
$594$		0			0
$595$	2156.25	−	2156.25i	0.148567	−	0.148567i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			25709.1i			1.75366i	0.480796	+	0.876832i	$0.340348\pi$
$599$			25709.1i			1.75366i	−0.480796	+	0.876832i	$0.659652\pi$
$600$		0			0
$601$		−	6062.56i		−	0.411476i	−0.978607	−	0.205738i	$-0.934041\pi$
$601$		−	6062.56i		−	0.411476i	0.978607	−	0.205738i	$-0.0659595\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9315.52	+	9315.52i	−0.625999	+	0.625999i
$606$		0			0
$607$			21062.9i			1.40843i	0.709989	+	0.704213i	$0.248700\pi$
$607$			21062.9i			1.40843i	−0.709989	+	0.704213i	$0.751300\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11801.9	−	11801.9i	−0.781429	−	0.781429i
$612$		0			0
$613$	−5955.84	+	5955.84i	−0.392421	+	0.392421i	−0.875550	−	0.483128i	$-0.839500\pi$
$613$	−5955.84	+	5955.84i	−0.392421	+	0.392421i	0.483128	+	0.875550i	$0.339500\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	28289.5			1.84586			0.922929	−	0.384970i	$-0.125788\pi$
$617$	28289.5			1.84586			0.922929	+	0.384970i	$0.125788\pi$
$618$		0			0
$619$	−9943.57	−	9943.57i	−0.645664	−	0.645664i	0.306278	−	0.951942i	$-0.400916\pi$
$619$	−9943.57	−	9943.57i	−0.645664	−	0.645664i	−0.951942	+	0.306278i	$0.900916\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5988.22			−0.385093
$624$		0			0
$625$	−6467.35			−0.413910
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−131.928	−	131.928i	−0.00836301	−	0.00836301i
$630$		0			0
$631$	9728.67			0.613775			0.306888	−	0.951746i	$-0.400712\pi$
$631$	9728.67			0.613775			0.306888	+	0.951746i	$0.400712\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4568.64	−	4568.64i	0.285513	−	0.285513i
$636$		0			0
$637$	−33486.2	−	33486.2i	−2.08285	−	2.08285i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			445.753i			0.0274667i	0.999906	+	0.0137334i	$0.00437160\pi$
$641$			445.753i			0.0274667i	−0.999906	+	0.0137334i	$0.995628\pi$
$642$		0			0
$643$	−879.050	+	879.050i	−0.0539134	+	0.0539134i	−0.733549	−	0.679636i	$-0.762138\pi$
$643$	−879.050	+	879.050i	−0.0539134	+	0.0539134i	0.679636	+	0.733549i	$0.262138\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	4067.41i		−	0.247150i	−0.992335	−	0.123575i	$-0.960564\pi$
$647$		−	4067.41i		−	0.247150i	0.992335	−	0.123575i	$-0.0394360\pi$
$648$		0			0
$649$		−	17074.5i		−	1.03272i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	595.450	−	595.450i	0.0356842	−	0.0356842i	−0.689040	−	0.724724i	$-0.741967\pi$
$653$	595.450	−	595.450i	0.0356842	−	0.0356842i	0.724724	+	0.689040i	$0.241967\pi$
$654$		0			0
$655$		−	7357.66i		−	0.438912i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9075.08	−	9075.08i	−0.536441	−	0.536441i	0.386041	−	0.922482i	$-0.373842\pi$
$659$	−9075.08	−	9075.08i	−0.536441	−	0.536441i	−0.922482	+	0.386041i	$0.873842\pi$
$660$		0			0
$661$	−1351.64	+	1351.64i	−0.0795353	+	0.0795353i	−0.745755	−	0.666220i	$-0.767911\pi$
$661$	−1351.64	+	1351.64i	−0.0795353	+	0.0795353i	0.666220	+	0.745755i	$0.267911\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−24265.9			−1.41502
$666$		0			0
$667$	7521.31	+	7521.31i	0.436621	+	0.436621i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11224.1			0.645753
$672$		0			0
$673$	−10501.3			−0.601482			−0.300741	−	0.953706i	$-0.597234\pi$
$673$	−10501.3			−0.601482			−0.300741	+	0.953706i	$0.597234\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10590.4	−	10590.4i	−0.601215	−	0.601215i	0.339420	−	0.940635i	$-0.389769\pi$
$677$	−10590.4	−	10590.4i	−0.601215	−	0.601215i	−0.940635	+	0.339420i	$0.889769\pi$
$678$		0			0
$679$	−18872.4			−1.06665
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2888.12	+	2888.12i	−0.161802	+	0.161802i	−0.783365	−	0.621563i	$-0.786498\pi$
$683$	−2888.12	+	2888.12i	−0.161802	+	0.161802i	0.621563	+	0.783365i	$0.286498\pi$
$684$		0			0
$685$	6795.68	+	6795.68i	0.379050	+	0.379050i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	687.302i		−	0.0380031i
$690$		0			0
$691$	−12657.6	+	12657.6i	−0.696844	+	0.696844i	−0.963728	−	0.266885i	$-0.914006\pi$
$691$	−12657.6	+	12657.6i	−0.696844	+	0.696844i	0.266885	+	0.963728i	$0.414006\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7601.14i		−	0.414860i
$696$		0			0
$697$			5278.46i			0.286852i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18873.8	−	18873.8i	1.01691	−	1.01691i	0.0170555	−	0.999855i	$-0.494571\pi$
$701$	18873.8	−	18873.8i	1.01691	−	1.01691i	0.999855	−	0.0170555i	$-0.00542920\pi$
$702$		0			0
$703$			1484.69i			0.0796531i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	28696.2	+	28696.2i	1.52650	+	1.52650i
$708$		0			0
$709$	21572.6	−	21572.6i	1.14270	−	1.14270i	0.154747	−	0.987954i	$-0.450544\pi$
$709$	21572.6	−	21572.6i	1.14270	−	1.14270i	0.987954	−	0.154747i	$-0.0494561\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3876.88			0.203633
$714$		0			0
$715$	−13480.5	−	13480.5i	−0.705096	−	0.705096i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29101.6			−1.50947			−0.754733	−	0.656032i	$-0.772234\pi$
$719$	−29101.6			−1.50947			−0.754733	+	0.656032i	$0.772234\pi$
$720$		0			0
$721$	25562.1			1.32036
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8860.02	+	8860.02i	0.453866	+	0.453866i
$726$		0			0
$727$	23683.7			1.20822			0.604112	−	0.796899i	$-0.293528\pi$
$727$	23683.7			1.20822			0.604112	+	0.796899i	$0.293528\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	993.859	−	993.859i	0.0502862	−	0.0502862i
$732$		0			0
$733$	1650.17	+	1650.17i	0.0831521	+	0.0831521i	0.747460	−	0.664307i	$-0.231273\pi$
$733$	1650.17	+	1650.17i	0.0831521	+	0.0831521i	−0.664307	+	0.747460i	$0.731273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			9736.83i			0.486650i
$738$		0			0
$739$	9966.72	−	9966.72i	0.496119	−	0.496119i	−0.414109	−	0.910227i	$-0.635907\pi$
$739$	9966.72	−	9966.72i	0.496119	−	0.496119i	0.910227	+	0.414109i	$0.135907\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			13180.6i			0.650806i	0.945576	+	0.325403i	$0.105500\pi$
$743$			13180.6i			0.650806i	−0.945576	+	0.325403i	$0.894500\pi$
$744$		0			0
$745$			14083.8i			0.692605i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12295.8	+	12295.8i	−0.599837	+	0.599837i
$750$		0			0
$751$		−	18926.8i		−	0.919641i	−0.888012	−	0.459821i	$-0.847914\pi$
$751$		−	18926.8i		−	0.919641i	0.888012	−	0.459821i	$-0.152086\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6487.74	+	6487.74i	0.312733	+	0.312733i
$756$		0			0
$757$	−23156.4	+	23156.4i	−1.11180	+	1.11180i	−0.118896	+	0.992907i	$0.537935\pi$
$757$	−23156.4	+	23156.4i	−1.11180	+	1.11180i	−0.992907	+	0.118896i	$0.962065\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6699.86			0.319146			0.159573	−	0.987186i	$-0.448988\pi$
$761$	6699.86			0.319146			0.159573	+	0.987186i	$0.448988\pi$
$762$		0			0
$763$	−6980.02	−	6980.02i	−0.331184	−	0.331184i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16280.1			0.766413
$768$		0			0
$769$	−13009.6			−0.610064			−0.305032	−	0.952342i	$-0.598667\pi$
$769$	−13009.6			−0.610064			−0.305032	+	0.952342i	$0.598667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−991.579	−	991.579i	−0.0461379	−	0.0461379i	0.683661	−	0.729799i	$-0.260387\pi$
$773$	−991.579	−	991.579i	−0.0461379	−	0.0461379i	−0.729799	+	0.683661i	$0.760387\pi$
$774$		0			0
$775$	4566.92			0.211675
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29701.2	−	29701.2i	1.36605	−	1.36605i
$780$		0			0
$781$	47276.7	+	47276.7i	2.16606	+	2.16606i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	6320.53i		−	0.287375i
$786$		0			0
$787$	19796.5	−	19796.5i	0.896659	−	0.896659i	−0.0984801	−	0.995139i	$-0.531398\pi$
$787$	19796.5	−	19796.5i	0.896659	−	0.896659i	0.995139	+	0.0984801i	$0.0313981\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			46530.2i			2.09156i
$792$		0			0
$793$			10701.8i			0.479235i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13170.1	+	13170.1i	−0.585331	+	0.585331i	−0.936363	−	0.351032i	$-0.885831\pi$
$797$	−13170.1	+	13170.1i	−0.585331	+	0.585331i	0.351032	+	0.936363i	$0.385831\pi$
$798$		0			0
$799$		−	4943.10i		−	0.218866i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−44869.1	−	44869.1i	−1.97185	−	1.97185i
$804$		0			0
$805$	10246.5	−	10246.5i	0.448624	−	0.448624i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19658.9			0.854352			0.427176	−	0.904169i	$-0.359509\pi$
$809$	19658.9			0.854352			0.427176	+	0.904169i	$0.359509\pi$
$810$		0			0
$811$	26123.4	+	26123.4i	1.13109	+	1.13109i	0.989996	+	0.141099i	$0.0450636\pi$
$811$	26123.4	+	26123.4i	1.13109	+	1.13109i	0.141099	+	0.989996i	$0.454936\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3925.41			0.168713
$816$		0			0
$817$	−11184.6			−0.478949
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8922.97	+	8922.97i	0.379310	+	0.379310i	0.870853	−	0.491543i	$-0.163567\pi$
$821$	8922.97	+	8922.97i	0.379310	+	0.379310i	−0.491543	+	0.870853i	$0.663567\pi$
$822$		0			0
$823$	19943.0			0.844677			0.422339	−	0.906438i	$-0.361209\pi$
$823$	19943.0			0.844677			0.422339	+	0.906438i	$0.361209\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18332.6	−	18332.6i	0.770843	−	0.770843i	−0.207411	−	0.978254i	$-0.566504\pi$
$827$	18332.6	−	18332.6i	0.770843	−	0.770843i	0.978254	+	0.207411i	$0.0665037\pi$
$828$		0			0
$829$	26778.5	+	26778.5i	1.12190	+	1.12190i	0.991455	+	0.130446i	$0.0416408\pi$
$829$	26778.5	+	26778.5i	1.12190	+	1.12190i	0.130446	+	0.991455i	$0.458359\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	14025.4i		−	0.583374i
$834$		0			0
$835$	982.788	−	982.788i	0.0407315	−	0.0407315i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	28064.1i		−	1.15480i	−0.816461	−	0.577401i	$-0.804067\pi$
$839$		−	28064.1i		−	1.15480i	0.816461	−	0.577401i	$-0.195933\pi$
$840$		0			0
$841$			8286.04i			0.339745i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4892.53	−	4892.53i	0.199181	−	0.199181i
$846$		0			0
$847$			86731.5i			3.51845i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−626.926	−	626.926i	−0.0252535	−	0.0252535i
$852$		0			0
$853$	25918.5	−	25918.5i	1.04037	−	1.04037i	0.0412172	−	0.999150i	$-0.486876\pi$
$853$	25918.5	−	25918.5i	1.04037	−	1.04037i	0.999150	−	0.0412172i	$-0.0131236\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23507.5			0.936992			0.468496	−	0.883466i	$-0.344796\pi$
$857$	23507.5			0.936992			0.468496	+	0.883466i	$0.344796\pi$
$858$		0			0
$859$	11981.1	+	11981.1i	0.475890	+	0.475890i	0.903814	−	0.427925i	$-0.140755\pi$
$859$	11981.1	+	11981.1i	0.475890	+	0.475890i	−0.427925	+	0.903814i	$0.640755\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8191.53			0.323109			0.161554	−	0.986864i	$-0.448349\pi$
$863$	8191.53			0.323109			0.161554	+	0.986864i	$0.448349\pi$
$864$		0			0
$865$	−10121.4			−0.397846
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7134.75	−	7134.75i	−0.278515	−	0.278515i
$870$		0			0
$871$	−9283.80			−0.361159
$872$		0			0
$873$		0			0
$874$		0			0
$875$	27350.6	−	27350.6i	1.05671	−	1.05671i
$876$		0			0
$877$	7896.06	+	7896.06i	0.304026	+	0.304026i	0.842587	−	0.538561i	$-0.181032\pi$
$877$	7896.06	+	7896.06i	0.304026	+	0.304026i	−0.538561	+	0.842587i	$0.681032\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	43097.6i		−	1.64812i	−0.566501	−	0.824061i	$-0.691703\pi$
$881$		−	43097.6i		−	1.64812i	0.566501	−	0.824061i	$-0.308297\pi$
$882$		0			0
$883$	21106.8	−	21106.8i	0.804418	−	0.804418i	−0.179364	−	0.983783i	$-0.557404\pi$
$883$	21106.8	−	21106.8i	0.804418	−	0.804418i	0.983783	+	0.179364i	$0.0574041\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	5713.71i		−	0.216288i	−0.994135	−	0.108144i	$-0.965509\pi$
$887$		−	5713.71i		−	0.216288i	0.994135	−	0.108144i	$-0.0344908\pi$
$888$		0			0
$889$		−	42536.0i		−	1.60474i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−27814.2	+	27814.2i	−1.04229	+	1.04229i
$894$		0			0
$895$		−	4892.59i		−	0.182728i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4150.15	−	4150.15i	−0.153966	−	0.153966i
$900$		0			0
$901$	143.935	−	143.935i	0.00532204	−	0.00532204i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9470.40			0.347853
$906$		0			0
$907$	2440.45	+	2440.45i	0.0893426	+	0.0893426i	0.750366	−	0.661023i	$-0.229877\pi$
$907$	2440.45	+	2440.45i	0.0893426	+	0.0893426i	−0.661023	+	0.750366i	$0.729877\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−41215.3			−1.49893			−0.749463	−	0.662046i	$-0.769688\pi$
$911$	−41215.3			−1.49893			−0.749463	+	0.662046i	$0.769688\pi$
$912$		0			0
$913$	5690.92			0.206289
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−34251.5	−	34251.5i	−1.23346	−	1.23346i
$918$		0			0
$919$	4600.68			0.165139			0.0825694	−	0.996585i	$-0.473687\pi$
$919$	4600.68			0.165139			0.0825694	+	0.996585i	$0.473687\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−45077.0	+	45077.0i	−1.60751	+	1.60751i
$924$		0			0
$925$	−738.512	−	738.512i	−0.0262509	−	0.0262509i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	22590.7i		−	0.797821i	−0.916990	−	0.398911i	$-0.869388\pi$
$929$		−	22590.7i		−	0.797821i	0.916990	−	0.398911i	$-0.130612\pi$
$930$		0			0
$931$	−78919.1	+	78919.1i	−2.77816	+	2.77816i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	5646.19i		−	0.197487i
$936$		0			0
$937$		−	37208.7i		−	1.29728i	−0.761094	−	0.648642i	$-0.775337\pi$
$937$		−	37208.7i		−	1.29728i	0.761094	−	0.648642i	$-0.224663\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29322.0	−	29322.0i	1.01580	−	1.01580i	0.0159300	−	0.999873i	$-0.494929\pi$
$941$	29322.0	−	29322.0i	1.01580	−	1.01580i	0.999873	−	0.0159300i	$-0.00507091\pi$
$942$		0			0
$943$			25083.3i			0.866198i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24343.1	+	24343.1i	0.835315	+	0.835315i	0.988238	−	0.152923i	$-0.0488688\pi$
$947$	24343.1	+	24343.1i	0.835315	+	0.835315i	−0.152923	+	0.988238i	$0.548869\pi$
$948$		0			0
$949$	42781.4	−	42781.4i	1.46338	−	1.46338i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12372.3			−0.420542			−0.210271	−	0.977643i	$-0.567435\pi$
$953$	−12372.3			−0.420542			−0.210271	+	0.977643i	$0.567435\pi$
$954$		0			0
$955$	7039.32	+	7039.32i	0.238520	+	0.238520i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	63270.7			2.13046
$960$		0			0
$961$	27651.8			0.928193
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−15022.2	−	15022.2i	−0.501122	−	0.501122i
$966$		0			0
$967$	−18615.7			−0.619069			−0.309534	−	0.950888i	$-0.600173\pi$
$967$	−18615.7			−0.619069			−0.309534	+	0.950888i	$0.600173\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	23920.0	−	23920.0i	0.790555	−	0.790555i	−0.191029	−	0.981584i	$-0.561183\pi$
$971$	23920.0	−	23920.0i	0.790555	−	0.790555i	0.981584	+	0.191029i	$0.0611825\pi$
$972$		0			0
$973$	−35384.9	−	35384.9i	−1.16587	−	1.16587i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			3892.45i			0.127462i	0.997967	+	0.0637311i	$0.0203000\pi$
$977$			3892.45i			0.127462i	−0.997967	+	0.0637311i	$0.979700\pi$
$978$		0			0
$979$	−7840.14	+	7840.14i	−0.255947	+	0.255947i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1771.35i			0.0574742i	0.999587	+	0.0287371i	$0.00914857\pi$
$983$			1771.35i			0.0574742i	−0.999587	+	0.0287371i	$0.990851\pi$
$984$		0			0
$985$		−	13963.7i		−	0.451696i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4722.84	−	4722.84i	0.151848	−	0.151848i
$990$		0			0
$991$			2828.01i			0.0906504i	0.998972	+	0.0453252i	$0.0144324\pi$
$991$			2828.01i			0.0906504i	−0.998972	+	0.0453252i	$0.985568\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13785.5	−	13785.5i	−0.439226	−	0.439226i
$996$		0			0
$997$	14851.0	−	14851.0i	0.471750	−	0.471750i	−0.430731	−	0.902480i	$-0.641744\pi$
$997$	14851.0	−	14851.0i	0.471750	−	0.471750i	0.902480	+	0.430731i	$0.141744\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.l.b.863.9		48
3.2	odd	2	inner	1152.4.l.b.863.16		48
4.3	odd	2		1152.4.l.a.863.9		48
8.3	odd	2		576.4.l.a.431.16		48
8.5	even	2		144.4.l.a.35.5	✓	48
12.11	even	2		1152.4.l.a.863.16		48
16.3	odd	4		144.4.l.a.107.20	yes	48
16.5	even	4		1152.4.l.a.287.16		48
16.11	odd	4	inner	1152.4.l.b.287.16		48
16.13	even	4		576.4.l.a.143.9		48
24.5	odd	2		144.4.l.a.35.20	yes	48
24.11	even	2		576.4.l.a.431.9		48
48.5	odd	4		1152.4.l.a.287.9		48
48.11	even	4	inner	1152.4.l.b.287.9		48
48.29	odd	4		576.4.l.a.143.16		48
48.35	even	4		144.4.l.a.107.5	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.5	✓	48	8.5	even	2
144.4.l.a.35.20	yes	48	24.5	odd	2
144.4.l.a.107.5	yes	48	48.35	even	4
144.4.l.a.107.20	yes	48	16.3	odd	4
576.4.l.a.143.9		48	16.13	even	4
576.4.l.a.143.16		48	48.29	odd	4
576.4.l.a.431.9		48	24.11	even	2
576.4.l.a.431.16		48	8.3	odd	2
1152.4.l.a.287.9		48	48.5	odd	4
1152.4.l.a.287.16		48	16.5	even	4
1152.4.l.a.863.9		48	4.3	odd	2
1152.4.l.a.863.16		48	12.11	even	2
1152.4.l.b.287.9		48	48.11	even	4	inner
1152.4.l.b.287.16		48	16.11	odd	4	inner
1152.4.l.b.863.9		48	1.1	even	1	trivial
1152.4.l.b.863.16		48	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.62348 - 3.62348i) q^{5} -33.7361 q^{7} +(-44.1694 + 44.1694i) q^{11} +(-42.1143 - 42.1143i) q^{13} -17.6392i q^{17} +(-99.2533 + 99.2533i) q^{19} -83.8215i q^{23} -98.7408i q^{25} +(-89.7300 + 89.7300i) q^{29} +46.2516i q^{31} +(122.242 + 122.242i) q^{35} +(7.47930 - 7.47930i) q^{37} -299.247 q^{41} +(56.3439 + 56.3439i) q^{43} +280.235 q^{47} +795.127 q^{49} +(8.15996 + 8.15996i) q^{53} +320.094 q^{55} +(-193.284 + 193.284i) q^{59} +(-127.057 - 127.057i) q^{61} +305.201i q^{65} +(110.221 - 110.221i) q^{67} -1070.35i q^{71} +1015.84i q^{73} +(1490.11 - 1490.11i) q^{77} +161.531i q^{79} +(-64.4215 - 64.4215i) q^{83} +(-63.9151 + 63.9151i) q^{85} +177.502 q^{89} +(1420.77 + 1420.77i) q^{91} +719.285 q^{95} +559.412 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} - 576 q^{55} - 1824 q^{61} - 816 q^{67} + 480 q^{85} + 3600 q^{91}+O(q^{100}) \) 48 * q + 48 * q^19 - 864 * q^43 + 2352 * q^49 - 576 * q^55 - 1824 * q^61 - 816 * q^67 + 480 * q^85 + 3600 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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