LMFDB - Embedded newform 1350.2.f.c.107.4

Error: no document with id 271630824 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,2,Mod(107,1350)] 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("1350.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0, -40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1350.107
Dual form			1350.2.f.c.593.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{8} +5.19615i q^{11} +(-3.67423 + 3.67423i) q^{13} -1.00000 q^{16} +(-2.12132 + 2.12132i) q^{17} +2.00000i q^{19} +(3.67423 + 3.67423i) q^{22} +(2.12132 + 2.12132i) q^{23} +5.19615i q^{26} +5.19615 q^{29} -5.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} +3.00000i q^{34} +(1.41421 + 1.41421i) q^{38} +10.3923i q^{41} +(-3.67423 + 3.67423i) q^{43} +5.19615 q^{44} +3.00000 q^{46} +(6.36396 - 6.36396i) q^{47} -7.00000i q^{49} +(3.67423 + 3.67423i) q^{52} +(-8.48528 - 8.48528i) q^{53} +(3.67423 - 3.67423i) q^{58} +10.3923 q^{59} +8.00000 q^{61} +(-3.53553 + 3.53553i) q^{62} +1.00000i q^{64} +(-7.34847 - 7.34847i) q^{67} +(2.12132 + 2.12132i) q^{68} +10.3923i q^{71} +(-7.34847 + 7.34847i) q^{73} +2.00000 q^{76} +1.00000i q^{79} +(7.34847 + 7.34847i) q^{82} +(8.48528 + 8.48528i) q^{83} +5.19615i q^{86} +(3.67423 - 3.67423i) q^{88} +10.3923 q^{89} +(2.12132 - 2.12132i) q^{92} -9.00000i q^{94} +(-7.34847 - 7.34847i) q^{97} +(-4.94975 - 4.94975i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{16} - 40 q^{31} + 24 q^{46} + 64 q^{61} + 16 q^{76}+O(q^{100}) $ 8 * q - 8 * q^16 - 40 * q^31 + 24 * q^46 + 64 * q^61 + 16 * q^76

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	0.707107	−	0.707107i	0.500000	−	0.500000i
$2$	0.707107	−	0.707107i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.707107	−	0.707107i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$		0			0
$13$	−3.67423	+	3.67423i	−1.01905	+	1.01905i	−0.0192343	+	0.999815i	$0.506123\pi$
$13$	−3.67423	+	3.67423i	−1.01905	+	1.01905i	−0.999815	+	0.0192343i	$0.993877\pi$
$14$		0			0
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−2.12132	+	2.12132i	−0.514496	+	0.514496i	−0.915901	−	0.401405i	$-0.868522\pi$
$17$	−2.12132	+	2.12132i	−0.514496	+	0.514496i	0.401405	+	0.915901i	$0.368522\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$	3.67423	+	3.67423i	0.783349	+	0.783349i
$23$	2.12132	+	2.12132i	0.442326	+	0.442326i	0.892793	−	0.450467i	$-0.148743\pi$
$23$	2.12132	+	2.12132i	0.442326	+	0.442326i	−0.450467	+	0.892793i	$0.648743\pi$
$24$		0			0
$25$		0			0
$26$			5.19615i			1.01905i
$27$		0			0
$28$		0			0
$29$	5.19615			0.964901			0.482451	−	0.875923i	$-0.339747\pi$
$29$	5.19615			0.964901			0.482451	+	0.875923i	$0.339747\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$	−0.707107	+	0.707107i	−0.125000	+	0.125000i
$33$		0			0
$34$			3.00000i			0.514496i
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	1.41421	+	1.41421i	0.229416	+	0.229416i
$39$		0			0
$40$		0			0
$41$			10.3923i			1.62301i	0.584349	+	0.811503i	$0.301350\pi$
$41$			10.3923i			1.62301i	−0.584349	+	0.811503i	$0.698650\pi$
$42$		0			0
$43$	−3.67423	+	3.67423i	−0.560316	+	0.560316i	−0.929397	−	0.369082i	$-0.879672\pi$
$43$	−3.67423	+	3.67423i	−0.560316	+	0.560316i	0.369082	+	0.929397i	$0.379672\pi$
$44$	5.19615			0.783349
$45$		0			0
$46$	3.00000			0.442326
$47$	6.36396	−	6.36396i	0.928279	−	0.928279i	−0.0693157	−	0.997595i	$-0.522082\pi$
$47$	6.36396	−	6.36396i	0.928279	−	0.928279i	0.997595	+	0.0693157i	$0.0220816\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$	3.67423	+	3.67423i	0.509525	+	0.509525i
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.983243	−	0.182300i	$-0.941646\pi$
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.182300	−	0.983243i	$-0.558354\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	3.67423	−	3.67423i	0.482451	−	0.482451i
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$	−3.53553	+	3.53553i	−0.449013	+	0.449013i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−7.34847	−	7.34847i	−0.897758	−	0.897758i	0.0974792	−	0.995238i	$-0.468922\pi$
$67$	−7.34847	−	7.34847i	−0.897758	−	0.897758i	−0.995238	+	0.0974792i	$0.968922\pi$
$68$	2.12132	+	2.12132i	0.257248	+	0.257248i
$69$		0			0
$70$		0			0
$71$			10.3923i			1.23334i	0.787222	+	0.616670i	$0.211519\pi$
$71$			10.3923i			1.23334i	−0.787222	+	0.616670i	$0.788481\pi$
$72$		0			0
$73$	−7.34847	+	7.34847i	−0.860073	+	0.860073i	−0.991346	−	0.131273i	$-0.958094\pi$
$73$	−7.34847	+	7.34847i	−0.860073	+	0.860073i	0.131273	+	0.991346i	$0.458094\pi$
$74$		0			0
$75$		0			0
$76$	2.00000			0.229416
$77$		0			0
$78$		0			0
$79$			1.00000i			0.112509i	0.998416	+	0.0562544i	$0.0179158\pi$
$79$			1.00000i			0.112509i	−0.998416	+	0.0562544i	$0.982084\pi$
$80$		0			0
$81$		0			0
$82$	7.34847	+	7.34847i	0.811503	+	0.811503i
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	0.997792	−	0.0664117i	$-0.0211551\pi$
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	−0.0664117	+	0.997792i	$0.521155\pi$
$84$		0			0
$85$		0			0
$86$			5.19615i			0.560316i
$87$		0			0
$88$	3.67423	−	3.67423i	0.391675	−	0.391675i
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$		0			0
$92$	2.12132	−	2.12132i	0.221163	−	0.221163i
$93$		0			0
$94$		−	9.00000i		−	0.928279i
$95$		0			0
$96$		0			0
$97$	−7.34847	−	7.34847i	−0.746124	−	0.746124i	0.227625	−	0.973749i	$-0.426904\pi$
$97$	−7.34847	−	7.34847i	−0.746124	−	0.746124i	−0.973749	+	0.227625i	$0.926904\pi$
$98$	−4.94975	−	4.94975i	−0.500000	−	0.500000i
$99$		0			0
$100$		0			0
$101$			5.19615i			0.517036i	0.966006	+	0.258518i	$0.0832342\pi$
$101$			5.19615i			0.517036i	−0.966006	+	0.258518i	$0.916766\pi$
$102$		0			0
$103$	7.34847	−	7.34847i	0.724066	−	0.724066i	−0.245365	−	0.969431i	$-0.578908\pi$
$103$	7.34847	−	7.34847i	0.724066	−	0.724066i	0.969431	+	0.245365i	$0.0789077\pi$
$104$	5.19615			0.509525
$105$		0			0
$106$	−12.0000			−1.16554
$107$	−12.7279	+	12.7279i	−1.23045	+	1.23045i	−0.266666	+	0.963789i	$0.585922\pi$
$107$	−12.7279	+	12.7279i	−1.23045	+	1.23045i	−0.963789	+	0.266666i	$0.914078\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.36396	+	6.36396i	0.598671	+	0.598671i	0.939959	−	0.341288i	$-0.110863\pi$
$113$	6.36396	+	6.36396i	0.598671	+	0.598671i	−0.341288	+	0.939959i	$0.610863\pi$
$114$		0			0
$115$		0			0
$116$		−	5.19615i		−	0.482451i
$117$		0			0
$118$	7.34847	−	7.34847i	0.676481	−	0.676481i
$119$		0			0
$120$		0			0
$121$	−16.0000			−1.45455
$122$	5.65685	−	5.65685i	0.512148	−	0.512148i
$123$		0			0
$124$			5.00000i			0.449013i
$125$		0			0
$126$		0			0
$127$	−14.6969	−	14.6969i	−1.30414	−	1.30414i	−0.925573	−	0.378570i	$-0.876416\pi$
$127$	−14.6969	−	14.6969i	−1.30414	−	1.30414i	−0.378570	−	0.925573i	$-0.623584\pi$
$128$	0.707107	+	0.707107i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$		−	5.19615i		−	0.453990i	−0.973896	−	0.226995i	$-0.927110\pi$
$131$		−	5.19615i		−	0.453990i	0.973896	−	0.226995i	$-0.0728901\pi$
$132$		0			0
$133$		0			0
$134$	−10.3923			−0.897758
$135$		0			0
$136$	3.00000			0.257248
$137$	4.24264	−	4.24264i	0.362473	−	0.362473i	−0.502249	−	0.864723i	$-0.667494\pi$
$137$	4.24264	−	4.24264i	0.362473	−	0.362473i	0.864723	+	0.502249i	$0.167494\pi$
$138$		0			0
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$		0			0
$142$	7.34847	+	7.34847i	0.616670	+	0.616670i
$143$	−19.0919	−	19.0919i	−1.59654	−	1.59654i
$144$		0			0
$145$		0			0
$146$			10.3923i			0.860073i
$147$		0			0
$148$		0			0
$149$	−15.5885			−1.27706			−0.638528	−	0.769599i	$-0.720456\pi$
$149$	−15.5885			−1.27706			−0.638528	+	0.769599i	$0.720456\pi$
$150$		0			0
$151$	19.0000			1.54620			0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000			1.54620			0.773099	+	0.634285i	$0.218706\pi$
$152$	1.41421	−	1.41421i	0.114708	−	0.114708i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.67423	+	3.67423i	0.293236	+	0.293236i	0.838357	−	0.545121i	$-0.183516\pi$
$157$	3.67423	+	3.67423i	0.293236	+	0.293236i	−0.545121	+	0.838357i	$0.683516\pi$
$158$	0.707107	+	0.707107i	0.0562544	+	0.0562544i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.0227	−	11.0227i	0.863365	−	0.863365i	−0.128363	−	0.991727i	$-0.540972\pi$
$163$	11.0227	−	11.0227i	0.863365	−	0.863365i	0.991727	+	0.128363i	$0.0409721\pi$
$164$	10.3923			0.811503
$165$		0			0
$166$	12.0000			0.931381
$167$	8.48528	−	8.48528i	0.656611	−	0.656611i	−0.297966	−	0.954577i	$-0.596308\pi$
$167$	8.48528	−	8.48528i	0.656611	−	0.656611i	0.954577	+	0.297966i	$0.0963081\pi$
$168$		0			0
$169$		−	14.0000i		−	1.07692i
$170$		0			0
$171$		0			0
$172$	3.67423	+	3.67423i	0.280158	+	0.280158i
$173$	16.9706	+	16.9706i	1.29025	+	1.29025i	0.934632	+	0.355616i	$0.115729\pi$
$173$	16.9706	+	16.9706i	1.29025	+	1.29025i	0.355616	+	0.934632i	$0.384271\pi$
$174$		0			0
$175$		0			0
$176$		−	5.19615i		−	0.391675i
$177$		0			0
$178$	7.34847	−	7.34847i	0.550791	−	0.550791i
$179$	−10.3923			−0.776757			−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923			−0.776757			−0.388379	+	0.921500i	$0.626965\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		−	3.00000i		−	0.221163i
$185$		0			0
$186$		0			0
$187$	−11.0227	−	11.0227i	−0.806060	−	0.806060i
$188$	−6.36396	−	6.36396i	−0.464140	−	0.464140i
$189$		0			0
$190$		0			0
$191$		−	10.3923i		−	0.751961i	−0.926628	−	0.375980i	$-0.877306\pi$
$191$		−	10.3923i		−	0.751961i	0.926628	−	0.375980i	$-0.122694\pi$
$192$		0			0
$193$	−14.6969	+	14.6969i	−1.05791	+	1.05791i	−0.0596919	+	0.998217i	$0.519012\pi$
$193$	−14.6969	+	14.6969i	−1.05791	+	1.05791i	−0.998217	+	0.0596919i	$0.980988\pi$
$194$	−10.3923			−0.746124
$195$		0			0
$196$	−7.00000			−0.500000
$197$	−4.24264	+	4.24264i	−0.302276	+	0.302276i	−0.841904	−	0.539628i	$-0.818565\pi$
$197$	−4.24264	+	4.24264i	−0.302276	+	0.302276i	0.539628	+	0.841904i	$0.318565\pi$
$198$		0			0
$199$		−	11.0000i		−	0.779769i	−0.920864	−	0.389885i	$-0.872515\pi$
$199$		−	11.0000i		−	0.779769i	0.920864	−	0.389885i	$-0.127485\pi$
$200$		0			0
$201$		0			0
$202$	3.67423	+	3.67423i	0.258518	+	0.258518i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		−	10.3923i		−	0.724066i
$207$		0			0
$208$	3.67423	−	3.67423i	0.254762	−	0.254762i
$209$	−10.3923			−0.718851
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$	−8.48528	+	8.48528i	−0.582772	+	0.582772i
$213$		0			0
$214$			18.0000i			1.23045i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	11.3137	+	11.3137i	0.766261	+	0.766261i
$219$		0			0
$220$		0			0
$221$		−	15.5885i		−	1.04859i
$222$		0			0
$223$	7.34847	−	7.34847i	0.492090	−	0.492090i	−0.416874	−	0.908964i	$-0.636874\pi$
$223$	7.34847	−	7.34847i	0.492090	−	0.492090i	0.908964	+	0.416874i	$0.136874\pi$
$224$		0			0
$225$		0			0
$226$	9.00000			0.598671
$227$	−4.24264	+	4.24264i	−0.281594	+	0.281594i	−0.833744	−	0.552151i	$-0.813807\pi$
$227$	−4.24264	+	4.24264i	−0.281594	+	0.281594i	0.552151	+	0.833744i	$0.313807\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$		0			0
$232$	−3.67423	−	3.67423i	−0.241225	−	0.241225i
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	0.154205	−	0.988039i	$-0.450718\pi$
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	−0.988039	+	0.154205i	$0.950718\pi$
$234$		0			0
$235$		0			0
$236$		−	10.3923i		−	0.676481i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1.00000			−0.0644157			−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000			−0.0644157			−0.0322078	+	0.999481i	$0.510254\pi$
$242$	−11.3137	+	11.3137i	−0.727273	+	0.727273i
$243$		0			0
$244$		−	8.00000i		−	0.512148i
$245$		0			0
$246$		0			0
$247$	−7.34847	−	7.34847i	−0.467572	−	0.467572i
$248$	3.53553	+	3.53553i	0.224507	+	0.224507i
$249$		0			0
$250$		0			0
$251$		−	15.5885i		−	0.983935i	−0.870614	−	0.491967i	$-0.836278\pi$
$251$		−	15.5885i		−	0.983935i	0.870614	−	0.491967i	$-0.163722\pi$
$252$		0			0
$253$	−11.0227	+	11.0227i	−0.692991	+	0.692991i
$254$	−20.7846			−1.30414
$255$		0			0
$256$	1.00000			0.0625000
$257$	10.6066	−	10.6066i	0.661622	−	0.661622i	−0.294141	−	0.955762i	$-0.595033\pi$
$257$	10.6066	−	10.6066i	0.661622	−	0.661622i	0.955762	+	0.294141i	$0.0950334\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−3.67423	−	3.67423i	−0.226995	−	0.226995i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−7.34847	+	7.34847i	−0.448879	+	0.448879i
$269$	−5.19615			−0.316815			−0.158408	−	0.987374i	$-0.550636\pi$
$269$	−5.19615			−0.316815			−0.158408	+	0.987374i	$0.550636\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	2.12132	−	2.12132i	0.128624	−	0.128624i
$273$		0			0
$274$		−	6.00000i		−	0.362473i
$275$		0			0
$276$		0			0
$277$	14.6969	+	14.6969i	0.883053	+	0.883053i	0.993844	−	0.110790i	$-0.0353382\pi$
$277$	14.6969	+	14.6969i	0.883053	+	0.883053i	−0.110790	+	0.993844i	$0.535338\pi$
$278$	−2.82843	−	2.82843i	−0.169638	−	0.169638i
$279$		0			0
$280$		0			0
$281$		−	31.1769i		−	1.85986i	−0.367738	−	0.929929i	$-0.619868\pi$
$281$		−	31.1769i		−	1.85986i	0.367738	−	0.929929i	$-0.380132\pi$
$282$		0			0
$283$	7.34847	−	7.34847i	0.436821	−	0.436821i	−0.454120	−	0.890941i	$-0.650046\pi$
$283$	7.34847	−	7.34847i	0.436821	−	0.436821i	0.890941	+	0.454120i	$0.150046\pi$
$284$	10.3923			0.616670
$285$		0			0
$286$	−27.0000			−1.59654
$287$		0			0
$288$		0			0
$289$			8.00000i			0.470588i
$290$		0			0
$291$		0			0
$292$	7.34847	+	7.34847i	0.430037	+	0.430037i
$293$	4.24264	+	4.24264i	0.247858	+	0.247858i	0.820091	−	0.572233i	$-0.193923\pi$
$293$	4.24264	+	4.24264i	0.247858	+	0.247858i	−0.572233	+	0.820091i	$0.693923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−11.0227	+	11.0227i	−0.638528	+	0.638528i
$299$	−15.5885			−0.901504
$300$		0			0
$301$		0			0
$302$	13.4350	−	13.4350i	0.773099	−	0.773099i
$303$		0			0
$304$		−	2.00000i		−	0.114708i
$305$		0			0
$306$		0			0
$307$	11.0227	+	11.0227i	0.629099	+	0.629099i	0.947841	−	0.318742i	$-0.103260\pi$
$307$	11.0227	+	11.0227i	0.629099	+	0.629099i	−0.318742	+	0.947841i	$0.603260\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	10.3923i		−	0.589294i	−0.955606	−	0.294647i	$-0.904798\pi$
$311$		−	10.3923i		−	0.589294i	0.955606	−	0.294647i	$-0.0952020\pi$
$312$		0			0
$313$	−22.0454	+	22.0454i	−1.24608	+	1.24608i	−0.288643	+	0.957437i	$0.593204\pi$
$313$	−22.0454	+	22.0454i	−1.24608	+	1.24608i	−0.957437	+	0.288643i	$0.906796\pi$
$314$	5.19615			0.293236
$315$		0			0
$316$	1.00000			0.0562544
$317$	8.48528	−	8.48528i	0.476581	−	0.476581i	−0.427456	−	0.904036i	$-0.640590\pi$
$317$	8.48528	−	8.48528i	0.476581	−	0.476581i	0.904036	+	0.427456i	$0.140590\pi$
$318$		0			0
$319$			27.0000i			1.51171i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.24264	−	4.24264i	−0.236067	−	0.236067i
$324$		0			0
$325$		0			0
$326$		−	15.5885i		−	0.863365i
$327$		0			0
$328$	7.34847	−	7.34847i	0.405751	−	0.405751i
$329$		0			0
$330$		0			0
$331$	26.0000			1.42909			0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000			1.42909			0.714545	+	0.699590i	$0.246634\pi$
$332$	8.48528	−	8.48528i	0.465690	−	0.465690i
$333$		0			0
$334$		−	12.0000i		−	0.656611i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−9.89949	−	9.89949i	−0.538462	−	0.538462i
$339$		0			0
$340$		0			0
$341$		−	25.9808i		−	1.40694i
$342$		0			0
$343$		0			0
$344$	5.19615			0.280158
$345$		0			0
$346$	24.0000			1.29025
$347$	−12.7279	+	12.7279i	−0.683271	+	0.683271i	−0.960736	−	0.277465i	$-0.910506\pi$
$347$	−12.7279	+	12.7279i	−0.683271	+	0.683271i	0.277465	+	0.960736i	$0.410506\pi$
$348$		0			0
$349$		−	28.0000i		−	1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$		−	28.0000i		−	1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$		0			0
$351$		0			0
$352$	−3.67423	−	3.67423i	−0.195837	−	0.195837i
$353$	6.36396	+	6.36396i	0.338719	+	0.338719i	0.855885	−	0.517166i	$-0.173013\pi$
$353$	6.36396	+	6.36396i	0.338719	+	0.338719i	−0.517166	+	0.855885i	$0.673013\pi$
$354$		0			0
$355$		0			0
$356$		−	10.3923i		−	0.550791i
$357$		0			0
$358$	−7.34847	+	7.34847i	−0.388379	+	0.388379i
$359$	−10.3923			−0.548485			−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923			−0.548485			−0.274242	+	0.961661i	$0.588427\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$	−1.41421	+	1.41421i	−0.0743294	+	0.0743294i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.34847	+	7.34847i	0.383587	+	0.383587i	0.872393	−	0.488806i	$-0.162567\pi$
$367$	7.34847	+	7.34847i	0.383587	+	0.383587i	−0.488806	+	0.872393i	$0.662567\pi$
$368$	−2.12132	−	2.12132i	−0.110581	−	0.110581i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−3.67423	+	3.67423i	−0.190245	+	0.190245i	−0.795802	−	0.605557i	$-0.792950\pi$
$373$	−3.67423	+	3.67423i	−0.190245	+	0.190245i	0.605557	+	0.795802i	$0.292950\pi$
$374$	−15.5885			−0.806060
$375$		0			0
$376$	−9.00000			−0.464140
$377$	−19.0919	+	19.0919i	−0.983282	+	0.983282i
$378$		0			0
$379$		−	16.0000i		−	0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$		−	16.0000i		−	0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$		0			0
$381$		0			0
$382$	−7.34847	−	7.34847i	−0.375980	−	0.375980i
$383$	10.6066	+	10.6066i	0.541972	+	0.541972i	0.924107	−	0.382135i	$-0.124811\pi$
$383$	10.6066	+	10.6066i	0.541972	+	0.541972i	−0.382135	+	0.924107i	$0.624811\pi$
$384$		0			0
$385$		0			0
$386$			20.7846i			1.05791i
$387$		0			0
$388$	−7.34847	+	7.34847i	−0.373062	+	0.373062i
$389$	15.5885			0.790366			0.395183	−	0.918602i	$-0.370681\pi$
$389$	15.5885			0.790366			0.395183	+	0.918602i	$0.370681\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$	−4.94975	+	4.94975i	−0.250000	+	0.250000i
$393$		0			0
$394$			6.00000i			0.302276i
$395$		0			0
$396$		0			0
$397$	25.7196	+	25.7196i	1.29083	+	1.29083i	0.934273	+	0.356559i	$0.116050\pi$
$397$	25.7196	+	25.7196i	1.29083	+	1.29083i	0.356559	+	0.934273i	$0.383950\pi$
$398$	−7.77817	−	7.77817i	−0.389885	−	0.389885i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	18.3712	−	18.3712i	0.915133	−	0.915133i
$404$	5.19615			0.258518
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			5.00000i			0.247234i	0.992330	+	0.123617i	$0.0394494\pi$
$409$			5.00000i			0.247234i	−0.992330	+	0.123617i	$0.960551\pi$
$410$		0			0
$411$		0			0
$412$	−7.34847	−	7.34847i	−0.362033	−	0.362033i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		−	5.19615i		−	0.254762i
$417$		0			0
$418$	−7.34847	+	7.34847i	−0.359425	+	0.359425i
$419$	−15.5885			−0.761546			−0.380773	−	0.924669i	$-0.624342\pi$
$419$	−15.5885			−0.761546			−0.380773	+	0.924669i	$0.624342\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$	−2.82843	+	2.82843i	−0.137686	+	0.137686i
$423$		0			0
$424$			12.0000i			0.582772i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	12.7279	+	12.7279i	0.615227	+	0.615227i
$429$		0			0
$430$		0			0
$431$		−	20.7846i		−	1.00116i	−0.865690	−	0.500580i	$-0.833120\pi$
$431$		−	20.7846i		−	1.00116i	0.865690	−	0.500580i	$-0.166880\pi$
$432$		0			0
$433$	−7.34847	+	7.34847i	−0.353145	+	0.353145i	−0.861278	−	0.508133i	$-0.830336\pi$
$433$	−7.34847	+	7.34847i	−0.353145	+	0.353145i	0.508133	+	0.861278i	$0.330336\pi$
$434$		0			0
$435$		0			0
$436$	16.0000			0.766261
$437$	−4.24264	+	4.24264i	−0.202953	+	0.202953i
$438$		0			0
$439$		−	8.00000i		−	0.381819i	−0.981608	−	0.190910i	$-0.938856\pi$
$439$		−	8.00000i		−	0.381819i	0.981608	−	0.190910i	$-0.0611437\pi$
$440$		0			0
$441$		0			0
$442$	−11.0227	−	11.0227i	−0.524297	−	0.524297i
$443$	4.24264	+	4.24264i	0.201574	+	0.201574i	0.800674	−	0.599100i	$-0.204475\pi$
$443$	4.24264	+	4.24264i	0.201574	+	0.201574i	−0.599100	+	0.800674i	$0.704475\pi$
$444$		0			0
$445$		0			0
$446$		−	10.3923i		−	0.492090i
$447$		0			0
$448$		0			0
$449$	−20.7846			−0.980886			−0.490443	−	0.871473i	$-0.663165\pi$
$449$	−20.7846			−0.980886			−0.490443	+	0.871473i	$0.663165\pi$
$450$		0			0
$451$	−54.0000			−2.54276
$452$	6.36396	−	6.36396i	0.299336	−	0.299336i
$453$		0			0
$454$			6.00000i			0.281594i
$455$		0			0
$456$		0			0
$457$	14.6969	+	14.6969i	0.687494	+	0.687494i	0.961677	−	0.274184i	$-0.0884076\pi$
$457$	14.6969	+	14.6969i	0.687494	+	0.687494i	−0.274184	+	0.961677i	$0.588408\pi$
$458$	9.89949	+	9.89949i	0.462573	+	0.462573i
$459$		0			0
$460$		0			0
$461$			41.5692i			1.93607i	0.250812	+	0.968036i	$0.419302\pi$
$461$			41.5692i			1.93607i	−0.250812	+	0.968036i	$0.580698\pi$
$462$		0			0
$463$	7.34847	−	7.34847i	0.341512	−	0.341512i	−0.515423	−	0.856936i	$-0.672365\pi$
$463$	7.34847	−	7.34847i	0.341512	−	0.341512i	0.856936	+	0.515423i	$0.172365\pi$
$464$	−5.19615			−0.241225
$465$		0			0
$466$	−18.0000			−0.833834
$467$	21.2132	−	21.2132i	0.981630	−	0.981630i	−0.0182043	−	0.999834i	$-0.505795\pi$
$467$	21.2132	−	21.2132i	0.981630	−	0.981630i	0.999834	+	0.0182043i	$0.00579493\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−7.34847	−	7.34847i	−0.338241	−	0.338241i
$473$	−19.0919	−	19.0919i	−0.877846	−	0.877846i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.3923			0.474837			0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923			0.474837			0.237418	+	0.971408i	$0.423699\pi$
$480$		0			0
$481$		0			0
$482$	−0.707107	+	0.707107i	−0.0322078	+	0.0322078i
$483$		0			0
$484$			16.0000i			0.727273i
$485$		0			0
$486$		0			0
$487$	−14.6969	−	14.6969i	−0.665982	−	0.665982i	0.290802	−	0.956783i	$-0.406078\pi$
$487$	−14.6969	−	14.6969i	−0.665982	−	0.665982i	−0.956783	+	0.290802i	$0.906078\pi$
$488$	−5.65685	−	5.65685i	−0.256074	−	0.256074i
$489$		0			0
$490$		0			0
$491$			10.3923i			0.468998i	0.972116	+	0.234499i	$0.0753450\pi$
$491$			10.3923i			0.468998i	−0.972116	+	0.234499i	$0.924655\pi$
$492$		0			0
$493$	−11.0227	+	11.0227i	−0.496438	+	0.496438i
$494$	−10.3923			−0.467572
$495$		0			0
$496$	5.00000			0.224507
$497$		0			0
$498$		0			0
$499$			40.0000i			1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$			40.0000i			1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$		0			0
$501$		0			0
$502$	−11.0227	−	11.0227i	−0.491967	−	0.491967i
$503$	14.8492	+	14.8492i	0.662095	+	0.662095i	0.955874	−	0.293779i	$-0.0949128\pi$
$503$	14.8492	+	14.8492i	0.662095	+	0.662095i	−0.293779	+	0.955874i	$0.594913\pi$
$504$		0			0
$505$		0			0
$506$			15.5885i			0.692991i
$507$		0			0
$508$	−14.6969	+	14.6969i	−0.652071	+	0.652071i
$509$	15.5885			0.690946			0.345473	−	0.938429i	$-0.387718\pi$
$509$	15.5885			0.690946			0.345473	+	0.938429i	$0.387718\pi$
$510$		0			0
$511$		0			0
$512$	0.707107	−	0.707107i	0.0312500	−	0.0312500i
$513$		0			0
$514$		−	15.0000i		−	0.661622i
$515$		0			0
$516$		0			0
$517$	33.0681	+	33.0681i	1.45433	+	1.45433i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			31.1769i			1.36589i	0.730472	+	0.682943i	$0.239300\pi$
$521$			31.1769i			1.36589i	−0.730472	+	0.682943i	$0.760700\pi$
$522$		0			0
$523$	25.7196	−	25.7196i	1.12464	−	1.12464i	0.133607	−	0.991034i	$-0.457344\pi$
$523$	25.7196	−	25.7196i	1.12464	−	1.12464i	0.991034	−	0.133607i	$-0.0426560\pi$
$524$	−5.19615			−0.226995
$525$		0			0
$526$		0			0
$527$	10.6066	−	10.6066i	0.462031	−	0.462031i
$528$		0			0
$529$		−	14.0000i		−	0.608696i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−38.1838	−	38.1838i	−1.65392	−	1.65392i
$534$		0			0
$535$		0			0
$536$			10.3923i			0.448879i
$537$		0			0
$538$	−3.67423	+	3.67423i	−0.158408	+	0.158408i
$539$	36.3731			1.56670
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$	−11.3137	+	11.3137i	−0.485965	+	0.485965i
$543$		0			0
$544$		−	3.00000i		−	0.128624i
$545$		0			0
$546$		0			0
$547$	−25.7196	−	25.7196i	−1.09969	−	1.09969i	−0.994446	−	0.105246i	$-0.966437\pi$
$547$	−25.7196	−	25.7196i	−1.09969	−	1.09969i	−0.105246	−	0.994446i	$-0.533563\pi$
$548$	−4.24264	−	4.24264i	−0.181237	−	0.181237i
$549$		0			0
$550$		0			0
$551$			10.3923i			0.442727i
$552$		0			0
$553$		0			0
$554$	20.7846			0.883053
$555$		0			0
$556$	−4.00000			−0.169638
$557$	12.7279	−	12.7279i	0.539299	−	0.539299i	−0.384024	−	0.923323i	$-0.625462\pi$
$557$	12.7279	−	12.7279i	0.539299	−	0.539299i	0.923323	+	0.384024i	$0.125462\pi$
$558$		0			0
$559$		−	27.0000i		−	1.14198i
$560$		0			0
$561$		0			0
$562$	−22.0454	−	22.0454i	−0.929929	−	0.929929i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		−	10.3923i		−	0.436821i
$567$		0			0
$568$	7.34847	−	7.34847i	0.308335	−	0.308335i
$569$	−20.7846			−0.871336			−0.435668	−	0.900107i	$-0.643488\pi$
$569$	−20.7846			−0.871336			−0.435668	+	0.900107i	$0.643488\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	−19.0919	+	19.0919i	−0.798272	+	0.798272i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−14.6969	−	14.6969i	−0.611842	−	0.611842i	0.331584	−	0.943426i	$-0.392417\pi$
$577$	−14.6969	−	14.6969i	−0.611842	−	0.611842i	−0.943426	+	0.331584i	$0.892417\pi$
$578$	5.65685	+	5.65685i	0.235294	+	0.235294i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	44.0908	−	44.0908i	1.82605	−	1.82605i
$584$	10.3923			0.430037
$585$		0			0
$586$	6.00000			0.247858
$587$	16.9706	−	16.9706i	0.700450	−	0.700450i	−0.264057	−	0.964507i	$-0.585061\pi$
$587$	16.9706	−	16.9706i	0.700450	−	0.700450i	0.964507	+	0.264057i	$0.0850607\pi$
$588$		0			0
$589$		−	10.0000i		−	0.412043i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	27.5772	+	27.5772i	1.13246	+	1.13246i	0.989767	+	0.142691i	$0.0455756\pi$
$593$	27.5772	+	27.5772i	1.13246	+	1.13246i	0.142691	+	0.989767i	$0.454424\pi$
$594$		0			0
$595$		0			0
$596$			15.5885i			0.638528i
$597$		0			0
$598$	−11.0227	+	11.0227i	−0.450752	+	0.450752i
$599$	−10.3923			−0.424618			−0.212309	−	0.977203i	$-0.568098\pi$
$599$	−10.3923			−0.424618			−0.212309	+	0.977203i	$0.568098\pi$
$600$		0			0
$601$	19.0000			0.775026			0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000			0.775026			0.387513	+	0.921864i	$0.373334\pi$
$602$		0			0
$603$		0			0
$604$		−	19.0000i		−	0.773099i
$605$		0			0
$606$		0			0
$607$	22.0454	+	22.0454i	0.894795	+	0.894795i	0.994970	−	0.100174i	$-0.0319401\pi$
$607$	22.0454	+	22.0454i	0.894795	+	0.894795i	−0.100174	+	0.994970i	$0.531940\pi$
$608$	−1.41421	−	1.41421i	−0.0573539	−	0.0573539i
$609$		0			0
$610$		0			0
$611$			46.7654i			1.89192i
$612$		0			0
$613$	18.3712	−	18.3712i	0.742005	−	0.742005i	−0.230959	−	0.972964i	$-0.574186\pi$
$613$	18.3712	−	18.3712i	0.742005	−	0.742005i	0.972964	+	0.230959i	$0.0741863\pi$
$614$	15.5885			0.629099
$615$		0			0
$616$		0			0
$617$	−27.5772	+	27.5772i	−1.11021	+	1.11021i	−0.117094	+	0.993121i	$0.537358\pi$
$617$	−27.5772	+	27.5772i	−1.11021	+	1.11021i	−0.993121	+	0.117094i	$0.962642\pi$
$618$		0			0
$619$			10.0000i			0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$			10.0000i			0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$		0			0
$621$		0			0
$622$	−7.34847	−	7.34847i	−0.294647	−	0.294647i
$623$		0			0
$624$		0			0
$625$		0			0
$626$			31.1769i			1.24608i
$627$		0			0
$628$	3.67423	−	3.67423i	0.146618	−	0.146618i
$629$		0			0
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	0.707107	−	0.707107i	0.0281272	−	0.0281272i
$633$		0			0
$634$		−	12.0000i		−	0.476581i
$635$		0			0
$636$		0			0
$637$	25.7196	+	25.7196i	1.01905	+	1.01905i
$638$	19.0919	+	19.0919i	0.755855	+	0.755855i
$639$		0			0
$640$		0			0
$641$		−	20.7846i		−	0.820943i	−0.911873	−	0.410471i	$-0.865364\pi$
$641$		−	20.7846i		−	0.820943i	0.911873	−	0.410471i	$-0.134636\pi$
$642$		0			0
$643$	11.0227	−	11.0227i	0.434693	−	0.434693i	−0.455528	−	0.890221i	$-0.650550\pi$
$643$	11.0227	−	11.0227i	0.434693	−	0.434693i	0.890221	+	0.455528i	$0.150550\pi$
$644$		0			0
$645$		0			0
$646$	−6.00000			−0.236067
$647$	−16.9706	+	16.9706i	−0.667182	+	0.667182i	−0.957063	−	0.289881i	$-0.906384\pi$
$647$	−16.9706	+	16.9706i	−0.667182	+	0.667182i	0.289881	+	0.957063i	$0.406384\pi$
$648$		0			0
$649$			54.0000i			2.11969i
$650$		0			0
$651$		0			0
$652$	−11.0227	−	11.0227i	−0.431682	−	0.431682i
$653$	4.24264	+	4.24264i	0.166027	+	0.166027i	0.785231	−	0.619203i	$-0.212544\pi$
$653$	4.24264	+	4.24264i	0.166027	+	0.166027i	−0.619203	+	0.785231i	$0.712544\pi$
$654$		0			0
$655$		0			0
$656$		−	10.3923i		−	0.405751i
$657$		0			0
$658$		0			0
$659$	31.1769			1.21448			0.607240	−	0.794518i	$-0.292277\pi$
$659$	31.1769			1.21448			0.607240	+	0.794518i	$0.292277\pi$
$660$		0			0
$661$	−32.0000			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$662$	18.3848	−	18.3848i	0.714545	−	0.714545i
$663$		0			0
$664$		−	12.0000i		−	0.465690i
$665$		0			0
$666$		0			0
$667$	11.0227	+	11.0227i	0.426801	+	0.426801i
$668$	−8.48528	−	8.48528i	−0.328305	−	0.328305i
$669$		0			0
$670$		0			0
$671$			41.5692i			1.60476i
$672$		0			0
$673$	−14.6969	+	14.6969i	−0.566525	+	0.566525i	−0.931153	−	0.364628i	$-0.881196\pi$
$673$	−14.6969	+	14.6969i	−0.566525	+	0.566525i	0.364628	+	0.931153i	$0.381196\pi$
$674$		0			0
$675$		0			0
$676$	−14.0000			−0.538462
$677$	−25.4558	+	25.4558i	−0.978348	+	0.978348i	−0.999771	−	0.0214229i	$-0.993180\pi$
$677$	−25.4558	+	25.4558i	−0.978348	+	0.978348i	0.0214229	+	0.999771i	$0.493180\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−18.3712	−	18.3712i	−0.703469	−	0.703469i
$683$	16.9706	+	16.9706i	0.649361	+	0.649361i	0.952838	−	0.303478i	$-0.0981479\pi$
$683$	16.9706	+	16.9706i	0.649361	+	0.649361i	−0.303478	+	0.952838i	$0.598148\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	3.67423	−	3.67423i	0.140079	−	0.140079i
$689$	62.3538			2.37549
$690$		0			0
$691$	10.0000			0.380418			0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000			0.380418			0.190209	+	0.981744i	$0.439083\pi$
$692$	16.9706	−	16.9706i	0.645124	−	0.645124i
$693$		0			0
$694$			18.0000i			0.683271i
$695$		0			0
$696$		0			0
$697$	−22.0454	−	22.0454i	−0.835029	−	0.835029i
$698$	−19.7990	−	19.7990i	−0.749403	−	0.749403i
$699$		0			0
$700$		0			0
$701$			36.3731i			1.37379i	0.726756	+	0.686896i	$0.241027\pi$
$701$			36.3731i			1.37379i	−0.726756	+	0.686896i	$0.758973\pi$
$702$		0			0
$703$		0			0
$704$	−5.19615			−0.195837
$705$		0			0
$706$	9.00000			0.338719
$707$		0			0
$708$		0			0
$709$		−	28.0000i		−	1.05156i	−0.850620	−	0.525781i	$-0.823773\pi$
$709$		−	28.0000i		−	1.05156i	0.850620	−	0.525781i	$-0.176227\pi$
$710$		0			0
$711$		0			0
$712$	−7.34847	−	7.34847i	−0.275396	−	0.275396i
$713$	−10.6066	−	10.6066i	−0.397220	−	0.397220i
$714$		0			0
$715$		0			0
$716$			10.3923i			0.388379i
$717$		0			0
$718$	−7.34847	+	7.34847i	−0.274242	+	0.274242i
$719$	41.5692			1.55027			0.775135	−	0.631795i	$-0.217682\pi$
$719$	41.5692			1.55027			0.775135	+	0.631795i	$0.217682\pi$
$720$		0			0
$721$		0			0
$722$	10.6066	−	10.6066i	0.394737	−	0.394737i
$723$		0			0
$724$			2.00000i			0.0743294i
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	15.5885i		−	0.576560i
$732$		0			0
$733$	29.3939	−	29.3939i	1.08569	−	1.08569i	0.0897206	−	0.995967i	$-0.471403\pi$
$733$	29.3939	−	29.3939i	1.08569	−	1.08569i	0.995967	−	0.0897206i	$-0.0285974\pi$
$734$	10.3923			0.383587
$735$		0			0
$736$	−3.00000			−0.110581
$737$	38.1838	−	38.1838i	1.40652	−	1.40652i
$738$		0			0
$739$		−	34.0000i		−	1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$		−	34.0000i		−	1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.0919	−	19.0919i	−0.700413	−	0.700413i	0.264086	−	0.964499i	$-0.414930\pi$
$743$	−19.0919	−	19.0919i	−0.700413	−	0.700413i	−0.964499	+	0.264086i	$0.914930\pi$
$744$		0			0
$745$		0			0
$746$			5.19615i			0.190245i
$747$		0			0
$748$	−11.0227	+	11.0227i	−0.403030	+	0.403030i
$749$		0			0
$750$		0			0
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$	−6.36396	+	6.36396i	−0.232070	+	0.232070i
$753$		0			0
$754$			27.0000i			0.983282i
$755$		0			0
$756$		0			0
$757$	25.7196	+	25.7196i	0.934796	+	0.934796i	0.998001	−	0.0632043i	$-0.0201320\pi$
$757$	25.7196	+	25.7196i	0.934796	+	0.934796i	−0.0632043	+	0.998001i	$0.520132\pi$
$758$	−11.3137	−	11.3137i	−0.410932	−	0.410932i
$759$		0			0
$760$		0			0
$761$			20.7846i			0.753442i	0.926327	+	0.376721i	$0.122948\pi$
$761$			20.7846i			0.753442i	−0.926327	+	0.376721i	$0.877052\pi$
$762$		0			0
$763$		0			0
$764$	−10.3923			−0.375980
$765$		0			0
$766$	15.0000			0.541972
$767$	−38.1838	+	38.1838i	−1.37874	+	1.37874i
$768$		0			0
$769$			13.0000i			0.468792i	0.972141	+	0.234396i	$0.0753112\pi$
$769$			13.0000i			0.468792i	−0.972141	+	0.234396i	$0.924689\pi$
$770$		0			0
$771$		0			0
$772$	14.6969	+	14.6969i	0.528954	+	0.528954i
$773$	21.2132	+	21.2132i	0.762986	+	0.762986i	0.976861	−	0.213875i	$-0.0686086\pi$
$773$	21.2132	+	21.2132i	0.762986	+	0.762986i	−0.213875	+	0.976861i	$0.568609\pi$
$774$		0			0
$775$		0			0
$776$			10.3923i			0.373062i
$777$		0			0
$778$	11.0227	−	11.0227i	0.395183	−	0.395183i
$779$	−20.7846			−0.744686
$780$		0			0
$781$	−54.0000			−1.93227
$782$	−6.36396	+	6.36396i	−0.227575	+	0.227575i
$783$		0			0
$784$			7.00000i			0.250000i
$785$		0			0
$786$		0			0
$787$	−18.3712	−	18.3712i	−0.654862	−	0.654862i	0.299298	−	0.954160i	$-0.403248\pi$
$787$	−18.3712	−	18.3712i	−0.654862	−	0.654862i	−0.954160	+	0.299298i	$0.903248\pi$
$788$	4.24264	+	4.24264i	0.151138	+	0.151138i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−29.3939	+	29.3939i	−1.04381	+	1.04381i
$794$	36.3731			1.29083
$795$		0			0
$796$	−11.0000			−0.389885
$797$	21.2132	−	21.2132i	0.751410	−	0.751410i	−0.223332	−	0.974742i	$-0.571693\pi$
$797$	21.2132	−	21.2132i	0.751410	−	0.751410i	0.974742	+	0.223332i	$0.0716935\pi$
$798$		0			0
$799$			27.0000i			0.955191i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−38.1838	−	38.1838i	−1.34748	−	1.34748i
$804$		0			0
$805$		0			0
$806$		−	25.9808i		−	0.915133i
$807$		0			0
$808$	3.67423	−	3.67423i	0.129259	−	0.129259i
$809$	20.7846			0.730748			0.365374	−	0.930861i	$-0.380941\pi$
$809$	20.7846			0.730748			0.365374	+	0.930861i	$0.380941\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−7.34847	−	7.34847i	−0.257090	−	0.257090i
$818$	3.53553	+	3.53553i	0.123617	+	0.123617i
$819$		0			0
$820$		0			0
$821$			41.5692i			1.45078i	0.688340	+	0.725388i	$0.258340\pi$
$821$			41.5692i			1.45078i	−0.688340	+	0.725388i	$0.741660\pi$
$822$		0			0
$823$	14.6969	−	14.6969i	0.512303	−	0.512303i	−0.402928	−	0.915231i	$-0.632008\pi$
$823$	14.6969	−	14.6969i	0.512303	−	0.512303i	0.915231	+	0.402928i	$0.132008\pi$
$824$	−10.3923			−0.362033
$825$		0			0
$826$		0			0
$827$	29.6985	−	29.6985i	1.03272	−	1.03272i	0.0332711	−	0.999446i	$-0.489408\pi$
$827$	29.6985	−	29.6985i	1.03272	−	1.03272i	0.999446	−	0.0332711i	$-0.0105925\pi$
$828$		0			0
$829$			20.0000i			0.694629i	0.937749	+	0.347314i	$0.112906\pi$
$829$			20.0000i			0.694629i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$	−3.67423	−	3.67423i	−0.127381	−	0.127381i
$833$	14.8492	+	14.8492i	0.514496	+	0.514496i
$834$		0			0
$835$		0			0
$836$			10.3923i			0.359425i
$837$		0			0
$838$	−11.0227	+	11.0227i	−0.380773	+	0.380773i
$839$	10.3923			0.358782			0.179391	−	0.983778i	$-0.442587\pi$
$839$	10.3923			0.358782			0.179391	+	0.983778i	$0.442587\pi$
$840$		0			0
$841$	−2.00000			−0.0689655
$842$	7.07107	−	7.07107i	0.243685	−	0.243685i
$843$		0			0
$844$			4.00000i			0.137686i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	8.48528	+	8.48528i	0.291386	+	0.291386i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	3.67423	−	3.67423i	0.125803	−	0.125803i	−0.641402	−	0.767205i	$-0.721647\pi$
$853$	3.67423	−	3.67423i	0.125803	−	0.125803i	0.767205	+	0.641402i	$0.221647\pi$
$854$		0			0
$855$		0			0
$856$	18.0000			0.615227
$857$	4.24264	−	4.24264i	0.144926	−	0.144926i	−0.630921	−	0.775847i	$-0.717323\pi$
$857$	4.24264	−	4.24264i	0.144926	−	0.144926i	0.775847	+	0.630921i	$0.217323\pi$
$858$		0			0
$859$		−	50.0000i		−	1.70598i	−0.521929	−	0.852989i	$-0.674787\pi$
$859$		−	50.0000i		−	1.70598i	0.521929	−	0.852989i	$-0.325213\pi$
$860$		0			0
$861$		0			0
$862$	−14.6969	−	14.6969i	−0.500580	−	0.500580i
$863$	36.0624	+	36.0624i	1.22758	+	1.22758i	0.964877	+	0.262703i	$0.0846140\pi$
$863$	36.0624	+	36.0624i	1.22758	+	1.22758i	0.262703	+	0.964877i	$0.415386\pi$
$864$		0			0
$865$		0			0
$866$			10.3923i			0.353145i
$867$		0			0
$868$		0			0
$869$	−5.19615			−0.176267
$870$		0			0
$871$	54.0000			1.82972
$872$	11.3137	−	11.3137i	0.383131	−	0.383131i
$873$		0			0
$874$			6.00000i			0.202953i
$875$		0			0
$876$		0			0
$877$	11.0227	+	11.0227i	0.372210	+	0.372210i	0.868282	−	0.496071i	$-0.165225\pi$
$877$	11.0227	+	11.0227i	0.372210	+	0.372210i	−0.496071	+	0.868282i	$0.665225\pi$
$878$	−5.65685	−	5.65685i	−0.190910	−	0.190910i
$879$		0			0
$880$		0			0
$881$		−	10.3923i		−	0.350126i	−0.984557	−	0.175063i	$-0.943987\pi$
$881$		−	10.3923i		−	0.350126i	0.984557	−	0.175063i	$-0.0560129\pi$
$882$		0			0
$883$	7.34847	−	7.34847i	0.247296	−	0.247296i	−0.572564	−	0.819860i	$-0.694051\pi$
$883$	7.34847	−	7.34847i	0.247296	−	0.247296i	0.819860	+	0.572564i	$0.194051\pi$
$884$	−15.5885			−0.524297
$885$		0			0
$886$	6.00000			0.201574
$887$	−40.3051	+	40.3051i	−1.35331	+	1.35331i	−0.471385	+	0.881928i	$0.656246\pi$
$887$	−40.3051	+	40.3051i	−1.35331	+	1.35331i	−0.881928	+	0.471385i	$0.843754\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−7.34847	−	7.34847i	−0.246045	−	0.246045i
$893$	12.7279	+	12.7279i	0.425924	+	0.425924i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−14.6969	+	14.6969i	−0.490443	+	0.490443i
$899$	−25.9808			−0.866507
$900$		0			0
$901$	36.0000			1.19933
$902$	−38.1838	+	38.1838i	−1.27138	+	1.27138i
$903$		0			0
$904$		−	9.00000i		−	0.299336i
$905$		0			0
$906$		0			0
$907$	18.3712	+	18.3712i	0.610005	+	0.610005i	0.942947	−	0.332942i	$-0.108041\pi$
$907$	18.3712	+	18.3712i	0.610005	+	0.610005i	−0.332942	+	0.942947i	$0.608041\pi$
$908$	4.24264	+	4.24264i	0.140797	+	0.140797i
$909$		0			0
$910$		0			0
$911$			41.5692i			1.37725i	0.725118	+	0.688625i	$0.241785\pi$
$911$			41.5692i			1.37725i	−0.725118	+	0.688625i	$0.758215\pi$
$912$		0			0
$913$	−44.0908	+	44.0908i	−1.45919	+	1.45919i
$914$	20.7846			0.687494
$915$		0			0
$916$	14.0000			0.462573
$917$		0			0
$918$		0			0
$919$		−	25.0000i		−	0.824674i	−0.911031	−	0.412337i	$-0.864713\pi$
$919$		−	25.0000i		−	0.824674i	0.911031	−	0.412337i	$-0.135287\pi$
$920$		0			0
$921$		0			0
$922$	29.3939	+	29.3939i	0.968036	+	0.968036i
$923$	−38.1838	−	38.1838i	−1.25683	−	1.25683i
$924$		0			0
$925$		0			0
$926$		−	10.3923i		−	0.341512i
$927$		0			0
$928$	−3.67423	+	3.67423i	−0.120613	+	0.120613i
$929$	−10.3923			−0.340960			−0.170480	−	0.985361i	$-0.554532\pi$
$929$	−10.3923			−0.340960			−0.170480	+	0.985361i	$0.554532\pi$
$930$		0			0
$931$	14.0000			0.458831
$932$	−12.7279	+	12.7279i	−0.416917	+	0.416917i
$933$		0			0
$934$		−	30.0000i		−	0.981630i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	46.7654i		−	1.52451i	−0.647278	−	0.762254i	$-0.724093\pi$
$941$		−	46.7654i		−	1.52451i	0.647278	−	0.762254i	$-0.275907\pi$
$942$		0			0
$943$	−22.0454	+	22.0454i	−0.717897	+	0.717897i
$944$	−10.3923			−0.338241
$945$		0			0
$946$	−27.0000			−0.877846
$947$	−8.48528	+	8.48528i	−0.275735	+	0.275735i	−0.831404	−	0.555669i	$-0.812462\pi$
$947$	−8.48528	+	8.48528i	−0.275735	+	0.275735i	0.555669	+	0.831404i	$0.312462\pi$
$948$		0			0
$949$		−	54.0000i		−	1.75291i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.36396	+	6.36396i	0.206149	+	0.206149i	0.802628	−	0.596479i	$-0.203434\pi$
$953$	6.36396	+	6.36396i	0.206149	+	0.206149i	−0.596479	+	0.802628i	$0.703434\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$	7.34847	−	7.34847i	0.237418	−	0.237418i
$959$		0			0
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$			1.00000i			0.0322078i
$965$		0			0
$966$		0			0
$967$	−7.34847	−	7.34847i	−0.236311	−	0.236311i	0.579010	−	0.815321i	$-0.303439\pi$
$967$	−7.34847	−	7.34847i	−0.236311	−	0.236311i	−0.815321	+	0.579010i	$0.803439\pi$
$968$	11.3137	+	11.3137i	0.363636	+	0.363636i
$969$		0			0
$970$		0			0
$971$		−	25.9808i		−	0.833762i	−0.908961	−	0.416881i	$-0.863123\pi$
$971$		−	25.9808i		−	0.833762i	0.908961	−	0.416881i	$-0.136877\pi$
$972$		0			0
$973$		0			0
$974$	−20.7846			−0.665982
$975$		0			0
$976$	−8.00000			−0.256074
$977$	19.0919	−	19.0919i	0.610803	−	0.610803i	−0.332352	−	0.943155i	$-0.607842\pi$
$977$	19.0919	−	19.0919i	0.610803	−	0.610803i	0.943155	+	0.332352i	$0.107842\pi$
$978$		0			0
$979$			54.0000i			1.72585i
$980$		0			0
$981$		0			0
$982$	7.34847	+	7.34847i	0.234499	+	0.234499i
$983$	−10.6066	−	10.6066i	−0.338298	−	0.338298i	0.517428	−	0.855726i	$-0.326889\pi$
$983$	−10.6066	−	10.6066i	−0.338298	−	0.338298i	−0.855726	+	0.517428i	$0.826889\pi$
$984$		0			0
$985$		0			0
$986$			15.5885i			0.496438i
$987$		0			0
$988$	−7.34847	+	7.34847i	−0.233786	+	0.233786i
$989$	−15.5885			−0.495684
$990$		0			0
$991$	−25.0000			−0.794151			−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$	3.53553	−	3.53553i	0.112253	−	0.112253i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	11.0227	+	11.0227i	0.349093	+	0.349093i	0.859771	−	0.510679i	$-0.170606\pi$
$997$	11.0227	+	11.0227i	0.349093	+	0.349093i	−0.510679	+	0.859771i	$0.670606\pi$
$998$	28.2843	+	28.2843i	0.895323	+	0.895323i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.f.c.107.4	yes	8
3.2	odd	2	inner	1350.2.f.c.107.1	✓	8
5.2	odd	4	inner	1350.2.f.c.593.4	yes	8
5.3	odd	4	inner	1350.2.f.c.593.2	yes	8
5.4	even	2	inner	1350.2.f.c.107.2	yes	8
15.2	even	4	inner	1350.2.f.c.593.1	yes	8
15.8	even	4	inner	1350.2.f.c.593.3	yes	8
15.14	odd	2	inner	1350.2.f.c.107.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.2.f.c.107.1	✓	8	3.2	odd	2	inner
1350.2.f.c.107.2	yes	8	5.4	even	2	inner
1350.2.f.c.107.3	yes	8	15.14	odd	2	inner
1350.2.f.c.107.4	yes	8	1.1	even	1	trivial
1350.2.f.c.593.1	yes	8	15.2	even	4	inner
1350.2.f.c.593.2	yes	8	5.3	odd	4	inner
1350.2.f.c.593.3	yes	8	15.8	even	4	inner
1350.2.f.c.593.4	yes	8	5.2	odd	4	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{8} +5.19615i q^{11} +(-3.67423 + 3.67423i) q^{13} -1.00000 q^{16} +(-2.12132 + 2.12132i) q^{17} +2.00000i q^{19} +(3.67423 + 3.67423i) q^{22} +(2.12132 + 2.12132i) q^{23} +5.19615i q^{26} +5.19615 q^{29} -5.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} +3.00000i q^{34} +(1.41421 + 1.41421i) q^{38} +10.3923i q^{41} +(-3.67423 + 3.67423i) q^{43} +5.19615 q^{44} +3.00000 q^{46} +(6.36396 - 6.36396i) q^{47} -7.00000i q^{49} +(3.67423 + 3.67423i) q^{52} +(-8.48528 - 8.48528i) q^{53} +(3.67423 - 3.67423i) q^{58} +10.3923 q^{59} +8.00000 q^{61} +(-3.53553 + 3.53553i) q^{62} +1.00000i q^{64} +(-7.34847 - 7.34847i) q^{67} +(2.12132 + 2.12132i) q^{68} +10.3923i q^{71} +(-7.34847 + 7.34847i) q^{73} +2.00000 q^{76} +1.00000i q^{79} +(7.34847 + 7.34847i) q^{82} +(8.48528 + 8.48528i) q^{83} +5.19615i q^{86} +(3.67423 - 3.67423i) q^{88} +10.3923 q^{89} +(2.12132 - 2.12132i) q^{92} -9.00000i q^{94} +(-7.34847 - 7.34847i) q^{97} +(-4.94975 - 4.94975i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{16} - 40 q^{31} + 24 q^{46} + 64 q^{61} + 16 q^{76}+O(q^{100}) \) 8 * q - 8 * q^16 - 40 * q^31 + 24 * q^46 + 64 * q^61 + 16 * q^76

Introduction

L-functions

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