LMFDB - Embedded newform 648.4.i.m.433.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,4,Mod(217,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.217"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-8,0,-24,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.2
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.m.217.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.61684 - 11.4607i) q^{5} +(2.61684 + 4.53251i) q^{7} +(-0.500000 - 0.866025i) q^{11} +(-42.4674 + 73.5557i) q^{13} +40.9348 q^{17} +57.5326 q^{19} +(-57.4674 + 99.5364i) q^{23} +(-25.0652 - 43.4143i) q^{25} +(101.234 + 175.342i) q^{29} +(-137.084 + 237.437i) q^{31} +69.2610 q^{35} +242.935 q^{37} +(164.168 - 284.348i) q^{41} +(-140.636 - 243.588i) q^{43} +(-11.9348 - 20.6716i) q^{47} +(157.804 - 273.325i) q^{49} +300.038 q^{53} -13.2337 q^{55} +(-376.804 + 652.644i) q^{59} +(-247.739 - 429.097i) q^{61} +(562.000 + 973.413i) q^{65} +(204.973 - 355.023i) q^{67} +1114.80 q^{71} -287.000 q^{73} +(2.61684 - 4.53251i) q^{77} +(615.739 + 1066.49i) q^{79} +(471.239 + 816.210i) q^{83} +(270.859 - 469.141i) q^{85} -190.206 q^{89} -444.522 q^{91} +(380.684 - 659.365i) q^{95} +(153.435 + 265.757i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} - 24 q^{7} - 2 q^{11} - 32 q^{13} - 112 q^{17} + 368 q^{19} - 92 q^{23} - 376 q^{25} + 336 q^{29} - 376 q^{31} + 1380 q^{35} + 696 q^{37} + 312 q^{41} - 80 q^{43} + 228 q^{47} - 196 q^{49}+ \cdots + 338 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 - 24 * q^7 - 2 * q^11 - 32 * q^13 - 112 * q^17 + 368 * q^19 - 92 * q^23 - 376 * q^25 + 336 * q^29 - 376 * q^31 + 1380 * q^35 + 696 * q^37 + 312 * q^41 - 80 * q^43 + 228 * q^47 - 196 * q^49 + 304 * q^53 + 16 * q^55 - 680 * q^59 + 112 * q^61 + 2248 * q^65 - 352 * q^67 + 3632 * q^71 - 1148 * q^73 - 24 * q^77 + 1360 * q^79 + 782 * q^83 + 2600 * q^85 + 480 * q^89 - 3984 * q^91 - 1924 * q^95 + 338 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	6.61684	−	11.4607i	0.591829	−	1.02508i	−0.402158	−	0.915570i	$-0.631740\pi$
$5$	6.61684	−	11.4607i	0.591829	−	1.02508i	0.993986	−	0.109507i	$-0.0349271\pi$
$6$		0			0
$7$	2.61684	+	4.53251i	0.141296	+	0.244732i	0.927985	−	0.372617i	$-0.121540\pi$
$7$	2.61684	+	4.53251i	0.141296	+	0.244732i	−0.786689	+	0.617350i	$0.788206\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.0137051	−	0.0237379i	0.859092	−	0.511822i	$-0.171029\pi$
$11$	−0.500000	−	0.866025i	−0.0137051	−	0.0237379i	−0.872797	+	0.488084i	$0.837696\pi$
$12$		0			0
$13$	−42.4674	+	73.5557i	−0.906025	+	1.56928i	−0.0864907	+	0.996253i	$0.527565\pi$
$13$	−42.4674	+	73.5557i	−0.906025	+	1.56928i	−0.819535	+	0.573029i	$0.805768\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	40.9348			0.584008			0.292004	−	0.956417i	$-0.405678\pi$
$17$	40.9348			0.584008			0.292004	+	0.956417i	$0.405678\pi$
$18$		0			0
$19$	57.5326			0.694678			0.347339	−	0.937740i	$-0.387085\pi$
$19$	57.5326			0.694678			0.347339	+	0.937740i	$0.387085\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−57.4674	+	99.5364i	−0.520990	+	0.902382i	0.478712	+	0.877972i	$0.341104\pi$
$23$	−57.4674	+	99.5364i	−0.520990	+	0.902382i	−0.999702	+	0.0244095i	$0.992229\pi$
$24$		0			0
$25$	−25.0652	−	43.4143i	−0.200522	−	0.347314i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	101.234	+	175.342i	0.648228	+	1.12276i	0.983546	+	0.180659i	$0.0578231\pi$
$29$	101.234	+	175.342i	0.648228	+	1.12276i	−0.335317	+	0.942105i	$0.608844\pi$
$30$		0			0
$31$	−137.084	+	237.437i	−0.794228	+	1.37564i	0.129101	+	0.991631i	$0.458791\pi$
$31$	−137.084	+	237.437i	−0.794228	+	1.37564i	−0.923329	+	0.384011i	$0.874542\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	69.2610			0.334493
$36$		0			0
$37$	242.935			1.07941			0.539706	−	0.841854i	$-0.318535\pi$
$37$	242.935			1.07941			0.539706	+	0.841854i	$0.318535\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	164.168	−	284.348i	0.625337	−	1.08311i	−0.363139	−	0.931735i	$-0.618295\pi$
$41$	164.168	−	284.348i	0.625337	−	1.08311i	0.988476	−	0.151380i	$-0.0483717\pi$
$42$		0			0
$43$	−140.636	−	243.588i	−0.498762	−	0.863881i	0.501237	−	0.865310i	$-0.332878\pi$
$43$	−140.636	−	243.588i	−0.498762	−	0.863881i	−0.999999	+	0.00142907i	$0.999545\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.9348	−	20.6716i	−0.0370396	−	0.0641545i	0.846911	−	0.531734i	$-0.178459\pi$
$47$	−11.9348	−	20.6716i	−0.0370396	−	0.0641545i	−0.883951	+	0.467580i	$0.845126\pi$
$48$		0			0
$49$	157.804	−	273.325i	0.460071	−	0.796866i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	300.038			0.777611			0.388805	−	0.921320i	$-0.372888\pi$
$53$	300.038			0.777611			0.388805	+	0.921320i	$0.372888\pi$
$54$		0			0
$55$	−13.2337			−0.0324442
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−376.804	+	652.644i	−0.831453	+	1.44012i	0.0654327	+	0.997857i	$0.479157\pi$
$59$	−376.804	+	652.644i	−0.831453	+	1.44012i	−0.896886	+	0.442262i	$0.854176\pi$
$60$		0			0
$61$	−247.739	−	429.097i	−0.519996	−	0.900659i	−0.999730	−	0.0232449i	$-0.992600\pi$
$61$	−247.739	−	429.097i	−0.519996	−	0.900659i	0.479734	−	0.877414i	$-0.340733\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	562.000	+	973.413i	1.07242	+	1.85749i
$66$		0			0
$67$	204.973	−	355.023i	0.373752	−	0.647358i	−0.616387	−	0.787443i	$-0.711404\pi$
$67$	204.973	−	355.023i	0.373752	−	0.647358i	0.990139	+	0.140085i	$0.0447377\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1114.80			1.86342			0.931711	−	0.363201i	$-0.118316\pi$
$71$	1114.80			1.86342			0.931711	+	0.363201i	$0.118316\pi$
$72$		0			0
$73$	−287.000			−0.460148			−0.230074	−	0.973173i	$-0.573897\pi$
$73$	−287.000			−0.460148			−0.230074	+	0.973173i	$0.573897\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.61684	−	4.53251i	0.00387295	−	0.00670814i
$78$		0			0
$79$	615.739	+	1066.49i	0.876912	+	1.51886i	0.854712	+	0.519103i	$0.173734\pi$
$79$	615.739	+	1066.49i	0.876912	+	1.51886i	0.0222001	+	0.999754i	$0.492933\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	471.239	+	816.210i	0.623195	+	1.07941i	0.988887	+	0.148670i	$0.0474991\pi$
$83$	471.239	+	816.210i	0.623195	+	1.07941i	−0.365692	+	0.930736i	$0.619168\pi$
$84$		0			0
$85$	270.859	−	469.141i	0.345633	−	0.598653i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−190.206			−0.226537			−0.113269	−	0.993564i	$-0.536132\pi$
$89$	−190.206			−0.226537			−0.113269	+	0.993564i	$0.536132\pi$
$90$		0			0
$91$	−444.522			−0.512072
$92$		0			0
$93$		0			0
$94$		0			0
$95$	380.684	−	659.365i	0.411130	−	0.712099i
$96$		0			0
$97$	153.435	+	265.757i	0.160608	+	0.278181i	0.935087	−	0.354419i	$-0.115321\pi$
$97$	153.435	+	265.757i	0.160608	+	0.278181i	−0.774479	+	0.632599i	$0.781988\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	336.448	+	582.746i	0.331464	+	0.574113i	0.982799	−	0.184678i	$-0.0591241\pi$
$101$	336.448	+	582.746i	0.331464	+	0.574113i	−0.651335	+	0.758790i	$0.725791\pi$
$102$		0			0
$103$	−420.076	+	727.593i	−0.401857	+	0.696037i	−0.993950	−	0.109832i	$-0.964969\pi$
$103$	−420.076	+	727.593i	−0.401857	+	0.696037i	0.592093	+	0.805870i	$0.298302\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1658.09			1.49807			0.749034	−	0.662532i	$-0.230518\pi$
$107$	1658.09			1.49807			0.749034	+	0.662532i	$0.230518\pi$
$108$		0			0
$109$	−1263.76			−1.11052			−0.555258	−	0.831678i	$-0.687381\pi$
$109$	−1263.76			−1.11052			−0.555258	+	0.831678i	$0.687381\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−237.179	+	410.806i	−0.197451	+	0.341995i	−0.947701	−	0.319159i	$-0.896600\pi$
$113$	−237.179	+	410.806i	−0.197451	+	0.341995i	0.750250	+	0.661154i	$0.229933\pi$
$114$		0			0
$115$	760.505	+	1317.23i	0.616674	+	1.06811i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	107.120	+	185.537i	0.0825182	+	0.142926i
$120$		0			0
$121$	665.000	−	1151.81i	0.499624	−	0.865375i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	990.800			0.708959
$126$		0			0
$127$	−560.375			−0.391537			−0.195769	−	0.980650i	$-0.562720\pi$
$127$	−560.375			−0.391537			−0.195769	+	0.980650i	$0.562720\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−394.652	+	683.557i	−0.263213	+	0.455898i	−0.967094	−	0.254420i	$-0.918115\pi$
$131$	−394.652	+	683.557i	−0.263213	+	0.455898i	0.703881	+	0.710318i	$0.251449\pi$
$132$		0			0
$133$	150.554	+	260.767i	0.0981555	+	0.170010i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	665.234	+	1152.22i	0.414852	+	0.718545i	0.995413	−	0.0956719i	$-0.0305000\pi$
$137$	665.234	+	1152.22i	0.414852	+	0.718545i	−0.580561	+	0.814217i	$0.697167\pi$
$138$		0			0
$139$	−1569.68	+	2718.77i	−0.957834	+	1.65902i	−0.230088	+	0.973170i	$0.573901\pi$
$139$	−1569.68	+	2718.77i	−0.957834	+	1.65902i	−0.727746	+	0.685847i	$0.759432\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	84.9348			0.0496685
$144$		0			0
$145$	2679.39			1.53456
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1333.12	+	2309.04i	−0.732977	+	1.26955i	0.222628	+	0.974904i	$0.428537\pi$
$149$	−1333.12	+	2309.04i	−0.732977	+	1.26955i	−0.955605	+	0.294650i	$0.904797\pi$
$150$		0			0
$151$	−239.703	−	415.178i	−0.129184	−	0.223753i	0.794177	−	0.607687i	$-0.207902\pi$
$151$	−239.703	−	415.178i	−0.129184	−	0.223753i	−0.923361	+	0.383934i	$0.874569\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1814.13	+	3142.16i	0.940093	+	1.62829i
$156$		0			0
$157$	1002.41	−	1736.23i	0.509562	−	0.882587i	−0.490377	−	0.871511i	$-0.663141\pi$
$157$	1002.41	−	1736.23i	0.509562	−	0.882587i	0.999939	−	0.0110768i	$-0.00352592\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−601.533			−0.294456
$162$		0			0
$163$	−832.043			−0.399820			−0.199910	−	0.979814i	$-0.564065\pi$
$163$	−832.043			−0.399820			−0.199910	+	0.979814i	$0.564065\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1435.83	−	2486.92i	0.665314	−	1.15236i	−0.313885	−	0.949461i	$-0.601631\pi$
$167$	1435.83	−	2486.92i	0.665314	−	1.15236i	0.979200	−	0.202898i	$-0.0650359\pi$
$168$		0			0
$169$	−2508.46	−	4344.77i	−1.14176	−	1.97759i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1789.72	−	3099.89i	−0.786531	−	1.36231i	−0.928080	−	0.372381i	$-0.878542\pi$
$173$	−1789.72	−	3099.89i	−0.786531	−	1.36231i	0.141549	−	0.989931i	$-0.454792\pi$
$174$		0			0
$175$	131.184	−	227.217i	0.0566660	−	0.0981484i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2162.09			−0.902804			−0.451402	−	0.892321i	$-0.649076\pi$
$179$	−2162.09			−0.902804			−0.451402	+	0.892321i	$0.649076\pi$
$180$		0			0
$181$	1740.26			0.714655			0.357328	−	0.933979i	$-0.383688\pi$
$181$	1740.26			0.714655			0.357328	+	0.933979i	$0.383688\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1607.46	−	2784.20i	0.638827	−	1.10648i
$186$		0			0
$187$	−20.4674	−	35.4505i	−0.00800387	−	0.0138631i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1268.90	−	2197.80i	−0.480704	−	0.832604i	0.519051	−	0.854743i	$-0.326286\pi$
$191$	−1268.90	−	2197.80i	−0.480704	−	0.832604i	−0.999755	+	0.0221395i	$0.992952\pi$
$192$		0			0
$193$	−95.1738	+	164.846i	−0.0354962	+	0.0614811i	−0.883228	−	0.468944i	$-0.844634\pi$
$193$	−95.1738	+	164.846i	−0.0354962	+	0.0614811i	0.847732	+	0.530425i	$0.177968\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5314.25			1.92195			0.960977	−	0.276629i	$-0.0892172\pi$
$197$	5314.25			1.92195			0.960977	+	0.276629i	$0.0892172\pi$
$198$		0			0
$199$	3787.15			1.34906			0.674532	−	0.738246i	$-0.264346\pi$
$199$	3787.15			1.34906			0.674532	+	0.738246i	$0.264346\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−529.826	+	917.685i	−0.183185	+	0.317285i
$204$		0			0
$205$	−2172.55	−	3762.97i	−0.740184	−	1.28204i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−28.7663	−	49.8247i	−0.00952061	−	0.0164902i
$210$		0			0
$211$	−1221.03	+	2114.88i	−0.398384	+	0.690021i	−0.993527	−	0.113599i	$-0.963762\pi$
$211$	−1221.03	+	2114.88i	−0.398384	+	0.690021i	0.595143	+	0.803620i	$0.297095\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3722.26			−1.18073
$216$		0			0
$217$	−1434.91			−0.448886
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1738.39	+	3010.98i	−0.529126	+	0.916473i
$222$		0			0
$223$	1164.93	+	2017.73i	0.349820	+	0.605906i	0.986217	−	0.165456i	$-0.0529095\pi$
$223$	1164.93	+	2017.73i	0.349820	+	0.605906i	−0.636397	+	0.771361i	$0.719576\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−663.196	−	1148.69i	−0.193911	−	0.335864i	0.752632	−	0.658442i	$-0.228784\pi$
$227$	−663.196	−	1148.69i	−0.193911	−	0.335864i	−0.946543	+	0.322577i	$0.895451\pi$
$228$		0			0
$229$	772.065	−	1337.26i	0.222793	−	0.385888i	−0.732862	−	0.680377i	$-0.761816\pi$
$229$	772.065	−	1337.26i	0.222793	−	0.385888i	0.955655	+	0.294489i	$0.0951494\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2266.32			−0.637216			−0.318608	−	0.947887i	$-0.603215\pi$
$233$	−2266.32			−0.637216			−0.318608	+	0.947887i	$0.603215\pi$
$234$		0			0
$235$	−315.882			−0.0876844
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.8376	−	44.7520i	0.00699286	−	0.0121120i	−0.862508	−	0.506044i	$-0.831107\pi$
$239$	25.8376	−	44.7520i	0.00699286	−	0.0121120i	0.869501	+	0.493932i	$0.164441\pi$
$240$		0			0
$241$	1161.98	+	2012.60i	0.310579	+	0.537939i	0.978488	−	0.206304i	$-0.0661437\pi$
$241$	1161.98	+	2012.60i	0.310579	+	0.537939i	−0.667909	+	0.744243i	$0.732810\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2088.33	−	3617.10i	−0.544566	−	0.943216i
$246$		0			0
$247$	−2443.26	+	4231.85i	−0.629396	+	1.09015i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6958.52			−1.74987			−0.874936	−	0.484239i	$-0.839097\pi$
$251$	−6958.52			−1.74987			−0.874936	+	0.484239i	$0.839097\pi$
$252$		0			0
$253$	114.935			0.0285608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1487.46	−	2576.35i	0.361031	−	0.625324i	−0.627100	−	0.778939i	$-0.715758\pi$
$257$	1487.46	−	2576.35i	0.361031	−	0.625324i	0.988131	+	0.153615i	$0.0490915\pi$
$258$		0			0
$259$	635.722	+	1101.10i	0.152517	+	0.264167i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	58.4234	+	101.192i	0.0136979	+	0.0237254i	0.872793	−	0.488090i	$-0.162306\pi$
$263$	58.4234	+	101.192i	0.0136979	+	0.0237254i	−0.859095	+	0.511816i	$0.828973\pi$
$264$		0			0
$265$	1985.30	−	3438.65i	0.460212	−	0.797111i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3334.81			−0.755863			−0.377932	−	0.925834i	$-0.623365\pi$
$269$	−3334.81			−0.755863			−0.377932	+	0.925834i	$0.623365\pi$
$270$		0			0
$271$	−3120.23			−0.699412			−0.349706	−	0.936860i	$-0.613718\pi$
$271$	−3120.23			−0.699412			−0.349706	+	0.936860i	$0.613718\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−25.0652	+	43.4143i	−0.00549633	+	0.00951993i
$276$		0			0
$277$	1297.20	+	2246.81i	0.281375	+	0.487356i	0.971724	−	0.236121i	$-0.0758761\pi$
$277$	1297.20	+	2246.81i	0.281375	+	0.487356i	−0.690348	+	0.723477i	$0.742543\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1141.55	−	1977.23i	−0.242347	−	0.419757i	0.719036	−	0.694973i	$-0.244584\pi$
$281$	−1141.55	−	1977.23i	−0.242347	−	0.419757i	−0.961382	+	0.275217i	$0.911250\pi$
$282$		0			0
$283$	−823.798	+	1426.86i	−0.173038	+	0.299710i	−0.939480	−	0.342602i	$-0.888692\pi$
$283$	−823.798	+	1426.86i	−0.173038	+	0.299710i	0.766443	+	0.642313i	$0.222025\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1718.41			0.353431
$288$		0			0
$289$	−3237.35			−0.658935
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1466.66	−	2540.32i	0.292434	−	0.506510i	−0.681951	−	0.731398i	$-0.738868\pi$
$293$	1466.66	−	2540.32i	0.292434	−	0.506510i	0.974385	+	0.224888i	$0.0722016\pi$
$294$		0			0
$295$	4986.51	+	8636.89i	0.984155	+	1.70461i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4880.98	−	8454.10i	−0.944061	−	1.63516i
$300$		0			0
$301$	736.044	−	1274.87i	0.140946	−	0.244126i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6557.00			−1.23099
$306$		0			0
$307$	−8216.93			−1.52757			−0.763787	−	0.645468i	$-0.776662\pi$
$307$	−8216.93			−1.52757			−0.763787	+	0.645468i	$0.776662\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	168.815	−	292.396i	0.0307801	−	0.0533127i	−0.850225	−	0.526419i	$-0.823534\pi$
$311$	168.815	−	292.396i	0.0307801	−	0.0533127i	0.881005	+	0.473107i	$0.156868\pi$
$312$		0			0
$313$	386.283	+	669.062i	0.0697572	+	0.120823i	0.898794	−	0.438370i	$-0.144444\pi$
$313$	386.283	+	669.062i	0.0697572	+	0.120823i	−0.829037	+	0.559194i	$0.811111\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3886.96	−	6732.42i	−0.688686	−	1.19284i	−0.972263	−	0.233890i	$-0.924854\pi$
$317$	−3886.96	−	6732.42i	−0.688686	−	1.19284i	0.283577	−	0.958950i	$-0.408479\pi$
$318$		0			0
$319$	101.234	−	175.342i	0.0177680	−	0.0307751i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2355.08			0.405698
$324$		0			0
$325$	4257.82			0.726712
$326$		0			0
$327$		0			0
$328$		0			0
$329$	62.4628	−	108.189i	0.0104671	−	0.0181296i
$330$		0			0
$331$	−2130.66	−	3690.41i	−0.353811	−	0.612819i	0.633103	−	0.774068i	$-0.281781\pi$
$331$	−2130.66	−	3690.41i	−0.353811	−	0.612819i	−0.986914	+	0.161249i	$0.948448\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2712.54	−	4698.27i	−0.442394	−	0.766250i
$336$		0			0
$337$	2364.46	−	4095.36i	0.382196	−	0.661983i	−0.609180	−	0.793032i	$-0.708501\pi$
$337$	2364.46	−	4095.36i	0.382196	−	0.661983i	0.991376	+	0.131049i	$0.0418345\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	274.168			0.0435397
$342$		0			0
$343$	3446.95			0.542618
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1292.02	−	2237.85i	0.199883	−	0.346207i	−0.748608	−	0.663013i	$-0.769277\pi$
$347$	1292.02	−	2237.85i	0.199883	−	0.346207i	0.948490	+	0.316806i	$0.102611\pi$
$348$		0			0
$349$	−2061.23	−	3570.15i	−0.316146	−	0.547581i	0.663534	−	0.748146i	$-0.269056\pi$
$349$	−2061.23	−	3570.15i	−0.316146	−	0.547581i	−0.979680	+	0.200565i	$0.935722\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4762.46	+	8248.82i	0.718074	+	1.24374i	0.961762	+	0.273887i	$0.0883095\pi$
$353$	4762.46	+	8248.82i	0.718074	+	1.24374i	−0.243688	+	0.969854i	$0.578357\pi$
$354$		0			0
$355$	7376.49	−	12776.4i	1.10283	−	1.91015i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3048.39			−0.448156			−0.224078	−	0.974571i	$-0.571937\pi$
$359$	−3048.39			−0.448156			−0.224078	+	0.974571i	$0.571937\pi$
$360$		0			0
$361$	−3549.00			−0.517422
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1899.03	+	3289.22i	−0.272329	+	0.471687i
$366$		0			0
$367$	2618.95	+	4536.15i	0.372501	+	0.645191i	0.989950	−	0.141420i	$-0.0451668\pi$
$367$	2618.95	+	4536.15i	0.372501	+	0.645191i	−0.617448	+	0.786612i	$0.711833\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	785.152	+	1359.92i	0.109874	+	0.190307i
$372$		0			0
$373$	−1435.75	+	2486.79i	−0.199304	+	0.345205i	−0.948303	−	0.317367i	$-0.897201\pi$
$373$	−1435.75	+	2486.79i	−0.199304	+	0.345205i	0.748999	+	0.662571i	$0.230535\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−17196.5			−2.34925
$378$		0			0
$379$	1649.20			0.223518			0.111759	−	0.993735i	$-0.464351\pi$
$379$	1649.20			0.223518			0.111759	+	0.993735i	$0.464351\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4091.38	+	7086.48i	−0.545848	+	0.945437i	0.452705	+	0.891660i	$0.350459\pi$
$383$	−4091.38	+	7086.48i	−0.545848	+	0.945437i	−0.998553	+	0.0537762i	$0.982874\pi$
$384$		0			0
$385$	−34.6305	−	59.9818i	−0.00458424	−	0.00794014i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	690.289	+	1195.62i	0.0899719	+	0.155836i	0.907499	−	0.420054i	$-0.137989\pi$
$389$	690.289	+	1195.62i	0.0899719	+	0.155836i	−0.817527	+	0.575890i	$0.804656\pi$
$390$		0			0
$391$	−2352.41	+	4074.50i	−0.304262	+	0.526998i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16297.0			2.07593
$396$		0			0
$397$	−5772.41			−0.729746			−0.364873	−	0.931057i	$-0.618888\pi$
$397$	−5772.41			−0.729746			−0.364873	+	0.931057i	$0.618888\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4689.42	−	8122.32i	0.583987	−	1.01149i	−0.411014	−	0.911629i	$-0.634825\pi$
$401$	4689.42	−	8122.32i	0.583987	−	1.01149i	0.995001	−	0.0998656i	$-0.0318413\pi$
$402$		0			0
$403$	−11643.2	−	20166.6i	−1.43918	−	2.49273i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−121.467	−	210.388i	−0.0147934	−	0.0256229i
$408$		0			0
$409$	−3536.54	+	6125.47i	−0.427557	+	0.740551i	−0.996655	−	0.0817189i	$-0.973959\pi$
$409$	−3536.54	+	6125.47i	−0.427557	+	0.740551i	0.569098	+	0.822269i	$0.307292\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3944.15			−0.469925
$414$		0			0
$415$	12472.5			1.47530
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4717.82	−	8171.51i	0.550073	−	0.952755i	−0.448195	−	0.893936i	$-0.647933\pi$
$419$	4717.82	−	8171.51i	0.550073	−	0.952755i	0.998269	−	0.0588195i	$-0.0187336\pi$
$420$		0			0
$421$	1310.06	+	2269.10i	0.151659	+	0.262682i	0.931838	−	0.362876i	$-0.118205\pi$
$421$	1310.06	+	2269.10i	0.151659	+	0.262682i	−0.780178	+	0.625557i	$0.784872\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1026.04	−	1777.15i	−0.117106	−	0.202834i
$426$		0			0
$427$	1296.59	−	2245.76i	0.146947	−	0.254519i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3485.65			0.389555			0.194777	−	0.980847i	$-0.437602\pi$
$431$	3485.65			0.389555			0.194777	+	0.980847i	$0.437602\pi$
$432$		0			0
$433$	9818.69			1.08974			0.544869	−	0.838521i	$-0.316579\pi$
$433$	9818.69			1.08974			0.544869	+	0.838521i	$0.316579\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3306.25	+	5726.59i	−0.361921	+	0.626865i
$438$		0			0
$439$	617.502	+	1069.54i	0.0671338	+	0.116279i	0.897639	−	0.440732i	$-0.145281\pi$
$439$	617.502	+	1069.54i	0.0671338	+	0.116279i	−0.830505	+	0.557012i	$0.811948\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5372.17	−	9304.88i	−0.576162	−	0.997941i	−0.995914	−	0.0903028i	$-0.971217\pi$
$443$	−5372.17	−	9304.88i	−0.576162	−	0.997941i	0.419753	−	0.907639i	$-0.362117\pi$
$444$		0			0
$445$	−1258.57	+	2179.90i	−0.134071	+	0.232218i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16166.3			1.69919			0.849595	−	0.527435i	$-0.176846\pi$
$449$	16166.3			1.69919			0.849595	+	0.527435i	$0.176846\pi$
$450$		0			0
$451$	−328.337			−0.0342811
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2941.33	+	5094.54i	−0.303059	+	0.524913i
$456$		0			0
$457$	1061.13	+	1837.93i	0.108616	+	0.188129i	0.915210	−	0.402977i	$-0.132025\pi$
$457$	1061.13	+	1837.93i	0.108616	+	0.188129i	−0.806594	+	0.591106i	$0.798691\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1736.70	+	3008.06i	0.175459	+	0.303903i	0.940320	−	0.340292i	$-0.110526\pi$
$461$	1736.70	+	3008.06i	0.175459	+	0.303903i	−0.764861	+	0.644195i	$0.777193\pi$
$462$		0			0
$463$	−1787.85	+	3096.65i	−0.179457	+	0.310828i	−0.941695	−	0.336469i	$-0.890767\pi$
$463$	−1787.85	+	3096.65i	−0.179457	+	0.310828i	0.762238	+	0.647297i	$0.224101\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9406.95			−0.932124			−0.466062	−	0.884752i	$-0.654328\pi$
$467$	−9406.95			−0.932124			−0.466062	+	0.884752i	$0.654328\pi$
$468$		0			0
$469$	2145.53			0.211239
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−140.636	+	243.588i	−0.0136711	+	0.0236791i
$474$		0			0
$475$	−1442.07	−	2497.74i	−0.139298	−	0.241272i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9262.02	+	16042.3i	0.883492	+	1.53025i	0.847433	+	0.530903i	$0.178147\pi$
$479$	9262.02	+	16042.3i	0.883492	+	1.53025i	0.0360589	+	0.999350i	$0.488520\pi$
$480$		0			0
$481$	−10316.8	+	17869.2i	−0.977974	+	1.69390i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4061.02			0.380209
$486$		0			0
$487$	−825.955			−0.0768533			−0.0384267	−	0.999261i	$-0.512235\pi$
$487$	−825.955			−0.0768533			−0.0384267	+	0.999261i	$0.512235\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4459.18	+	7723.52i	−0.409857	+	0.709893i	−0.994873	−	0.101128i	$-0.967755\pi$
$491$	−4459.18	+	7723.52i	−0.409857	+	0.709893i	0.585016	+	0.811021i	$0.301088\pi$
$492$		0			0
$493$	4143.98	+	7177.58i	0.378571	+	0.655703i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2917.27	+	5052.86i	0.263295	+	0.456040i
$498$		0			0
$499$	−9107.20	+	15774.1i	−0.817022	+	1.41512i	0.0908443	+	0.995865i	$0.471043\pi$
$499$	−9107.20	+	15774.1i	−0.817022	+	1.41512i	−0.907867	+	0.419259i	$0.862290\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2264.42			0.200726			0.100363	−	0.994951i	$-0.468000\pi$
$503$	2264.42			0.200726			0.100363	+	0.994951i	$0.468000\pi$
$504$		0			0
$505$	8904.91			0.784679
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7117.59	−	12328.0i	0.619806	−	1.07354i	−0.369714	−	0.929145i	$-0.620544\pi$
$509$	7117.59	−	12328.0i	0.619806	−	1.07354i	0.989521	−	0.144391i	$-0.0461222\pi$
$510$		0			0
$511$	−751.034	−	1300.83i	−0.0650172	−	0.112613i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5559.15	+	9628.74i	0.475661	+	0.823870i
$516$		0			0
$517$	−11.9348	+	20.6716i	−0.00101526	+	0.00175848i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11343.3			0.953856			0.476928	−	0.878942i	$-0.341750\pi$
$521$	11343.3			0.953856			0.476928	+	0.878942i	$0.341750\pi$
$522$		0			0
$523$	−20464.1			−1.71096			−0.855482	−	0.517832i	$-0.826739\pi$
$523$	−20464.1			−1.71096			−0.855482	+	0.517832i	$0.826739\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5611.51	+	9719.42i	−0.463835	+	0.803386i
$528$		0			0
$529$	−521.499	−	903.262i	−0.0428617	−	0.0742387i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13943.6	+	24151.0i	1.13314	+	1.96266i
$534$		0			0
$535$	10971.3	−	19002.8i	0.886599	−	1.53564i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−315.609			−0.0252212
$540$		0			0
$541$	−1993.69			−0.158439			−0.0792196	−	0.996857i	$-0.525243\pi$
$541$	−1993.69			−0.158439			−0.0792196	+	0.996857i	$0.525243\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8362.10	+	14483.6i	−0.657235	+	1.13836i
$546$		0			0
$547$	1362.01	+	2359.07i	0.106463	+	0.184399i	0.914335	−	0.404959i	$-0.132714\pi$
$547$	1362.01	+	2359.07i	0.106463	+	0.184399i	−0.807872	+	0.589358i	$0.799381\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5824.24	+	10087.9i	0.450310	+	0.779960i
$552$		0			0
$553$	−3222.59	+	5581.68i	−0.247809	+	0.429218i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15046.5			1.14460			0.572299	−	0.820045i	$-0.306052\pi$
$557$	15046.5			1.14460			0.572299	+	0.820045i	$0.306052\pi$
$558$		0			0
$559$	23889.7			1.80756
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9804.28	−	16981.5i	0.733928	−	1.27120i	−0.221265	−	0.975214i	$-0.571018\pi$
$563$	9804.28	−	16981.5i	0.733928	−	1.27120i	0.955192	−	0.295986i	$-0.0956482\pi$
$564$		0			0
$565$	3138.75	+	5436.48i	0.233714	+	0.404804i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6984.27	−	12097.1i	−0.514580	−	0.891278i	−0.999857	−	0.0169178i	$-0.994615\pi$
$569$	−6984.27	−	12097.1i	−0.514580	−	0.891278i	0.485277	−	0.874360i	$-0.338719\pi$
$570$		0			0
$571$	11807.0	−	20450.4i	0.865339	−	1.49881i	−0.00137176	−	0.999999i	$-0.500437\pi$
$571$	11807.0	−	20450.4i	0.865339	−	1.49881i	0.866710	−	0.498812i	$-0.166230\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5761.74			0.417880
$576$		0			0
$577$	−23002.4			−1.65962			−0.829811	−	0.558044i	$-0.811552\pi$
$577$	−23002.4			−1.65962			−0.829811	+	0.558044i	$0.811552\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2466.32	+	4271.79i	−0.176110	+	0.305032i
$582$		0			0
$583$	−150.019	−	259.840i	−0.0106572	−	0.0184588i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4050.22	+	7015.19i	0.284788	+	0.493267i	0.972558	−	0.232662i	$-0.0747435\pi$
$587$	4050.22	+	7015.19i	0.284788	+	0.493267i	−0.687770	+	0.725929i	$0.741410\pi$
$588$		0			0
$589$	−7886.81	+	13660.4i	−0.551733	+	0.955629i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22256.2			1.54124			0.770618	−	0.637298i	$-0.219948\pi$
$593$	22256.2			1.54124			0.770618	+	0.637298i	$0.219948\pi$
$594$		0			0
$595$	2835.18			0.195346
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4193.15	−	7262.75i	0.286023	−	0.495406i	−0.686834	−	0.726814i	$-0.741000\pi$
$599$	4193.15	−	7262.75i	0.286023	−	0.495406i	0.972857	+	0.231409i	$0.0743334\pi$
$600$		0			0
$601$	4907.11	+	8499.36i	0.333053	+	0.576865i	0.983109	−	0.183021i	$-0.0585877\pi$
$601$	4907.11	+	8499.36i	0.333053	+	0.576865i	−0.650056	+	0.759887i	$0.725254\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8800.40	−	15242.7i	−0.591384	−	1.02431i
$606$		0			0
$607$	12038.8	−	20851.9i	0.805011	−	1.39432i	−0.111273	−	0.993790i	$-0.535493\pi$
$607$	12038.8	−	20851.9i	0.805011	−	1.39432i	0.916284	−	0.400530i	$-0.131174\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2027.35			0.134235
$612$		0			0
$613$	−18679.6			−1.23077			−0.615386	−	0.788226i	$-0.711000\pi$
$613$	−18679.6			−1.23077			−0.615386	+	0.788226i	$0.711000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	825.440	−	1429.70i	0.0538589	−	0.0932864i	−0.837839	−	0.545917i	$-0.816181\pi$
$617$	825.440	−	1429.70i	0.0538589	−	0.0932864i	0.891698	+	0.452631i	$0.149515\pi$
$618$		0			0
$619$	−12516.4	−	21679.1i	−0.812727	−	1.40768i	−0.910949	−	0.412520i	$-0.864649\pi$
$619$	−12516.4	−	21679.1i	−0.812727	−	1.40768i	0.0982217	−	0.995165i	$-0.468685\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−497.740	−	862.112i	−0.0320089	−	0.0554411i
$624$		0			0
$625$	9689.12	−	16782.1i	0.620104	−	1.07405i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9944.47			0.630385
$630$		0			0
$631$	1666.38			0.105131			0.0525653	−	0.998617i	$-0.483260\pi$
$631$	1666.38			0.105131			0.0525653	+	0.998617i	$0.483260\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3707.91	+	6422.29i	−0.231723	+	0.401356i
$636$		0			0
$637$	13403.1	+	23214.8i	0.833672	+	1.44396i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12013.3	+	20807.6i	0.740243	+	1.28214i	0.952385	+	0.304900i	$0.0986229\pi$
$641$	12013.3	+	20807.6i	0.740243	+	1.28214i	−0.212142	+	0.977239i	$0.568044\pi$
$642$		0			0
$643$	6386.25	−	11061.3i	0.391678	−	0.678407i	−0.600993	−	0.799254i	$-0.705228\pi$
$643$	6386.25	−	11061.3i	0.391678	−	0.678407i	0.992671	+	0.120848i	$0.0385613\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19420.6			−1.18007			−0.590034	−	0.807378i	$-0.700886\pi$
$647$	−19420.6			−1.18007			−0.590034	+	0.807378i	$0.700886\pi$
$648$		0			0
$649$	753.609			0.0455805
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−405.872	+	702.991i	−0.0243231	+	0.0421289i	−0.877931	−	0.478788i	$-0.841076\pi$
$653$	−405.872	+	702.991i	−0.0243231	+	0.0421289i	0.853608	+	0.520917i	$0.174410\pi$
$654$		0			0
$655$	5222.70	+	9045.98i	0.311554	+	0.539627i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6082.89	+	10535.9i	0.359569	+	0.622791i	0.987889	−	0.155164i	$-0.0495905\pi$
$659$	6082.89	+	10535.9i	0.359569	+	0.622791i	−0.628320	+	0.777955i	$0.716257\pi$
$660$		0			0
$661$	1502.17	−	2601.83i	0.0883925	−	0.153100i	−0.818439	−	0.574593i	$-0.805160\pi$
$661$	1502.17	−	2601.83i	0.0883925	−	0.153100i	0.906832	+	0.421493i	$0.138494\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3984.77			0.232365
$666$		0			0
$667$	−23270.5			−1.35088
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−247.739	+	429.097i	−0.0142531	+	0.0246872i
$672$		0			0
$673$	636.413	+	1102.30i	0.0364516	+	0.0631360i	0.883676	−	0.468100i	$-0.155061\pi$
$673$	636.413	+	1102.30i	0.0364516	+	0.0631360i	−0.847224	+	0.531236i	$0.821728\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5519.94	+	9560.81i	0.313366	+	0.542765i	0.979089	−	0.203434i	$-0.0652102\pi$
$677$	5519.94	+	9560.81i	0.313366	+	0.542765i	−0.665723	+	0.746199i	$0.731877\pi$
$678$		0			0
$679$	−803.030	+	1390.89i	−0.0453865	+	0.0786118i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6103.65			0.341947			0.170974	−	0.985276i	$-0.445309\pi$
$683$	6103.65			0.341947			0.170974	+	0.985276i	$0.445309\pi$
$684$		0			0
$685$	17607.0			0.982085
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12741.8	+	22069.5i	−0.704535	+	1.22029i
$690$		0			0
$691$	−15379.6	−	26638.3i	−0.846698	−	1.46652i	−0.884139	−	0.467225i	$-0.845254\pi$
$691$	−15379.6	−	26638.3i	−0.846698	−	1.46652i	0.0374409	−	0.999299i	$-0.488079\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	20772.7	+	35979.4i	1.13375	+	1.96371i
$696$		0			0
$697$	6720.19	−	11639.7i	0.365202	−	0.632548i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30619.5			−1.64976			−0.824880	−	0.565308i	$-0.808757\pi$
$701$	−30619.5			−1.64976			−0.824880	+	0.565308i	$0.808757\pi$
$702$		0			0
$703$	13976.7			0.749844
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1760.87	+	3049.91i	−0.0936693	+	0.162240i
$708$		0			0
$709$	−14325.5	−	24812.5i	−0.758822	−	1.31432i	−0.943452	−	0.331510i	$-0.892442\pi$
$709$	−14325.5	−	24812.5i	−0.758822	−	1.31432i	0.184630	−	0.982808i	$-0.440891\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15755.7	−	27289.7i	−0.827570	−	1.43339i
$714$		0			0
$715$	562.000	−	973.413i	0.0293953	−	0.0509141i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24899.6			1.29151			0.645756	−	0.763544i	$-0.276542\pi$
$719$	24899.6			1.29151			0.645756	+	0.763544i	$0.276542\pi$
$720$		0			0
$721$	−4397.09			−0.227124
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5074.90	−	8789.98i	0.259968	−	0.450278i
$726$		0			0
$727$	17159.0	+	29720.3i	0.875369	+	1.51618i	0.856370	+	0.516363i	$0.172715\pi$
$727$	17159.0	+	29720.3i	0.875369	+	1.51618i	0.0189990	+	0.999820i	$0.493952\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5756.89	−	9971.23i	−0.291281	−	0.504513i
$732$		0			0
$733$	−13876.0	+	24033.9i	−0.699209	+	1.21107i	0.269532	+	0.962992i	$0.413131\pi$
$733$	−13876.0	+	24033.9i	−0.699209	+	1.21107i	−0.968741	+	0.248074i	$0.920202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−409.945			−0.0204892
$738$		0			0
$739$	−24701.0			−1.22955			−0.614777	−	0.788701i	$-0.710754\pi$
$739$	−24701.0			−1.22955			−0.614777	+	0.788701i	$0.710754\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11593.4	−	20080.4i	0.572438	−	0.991491i	−0.423877	−	0.905720i	$-0.639331\pi$
$743$	11593.4	−	20080.4i	0.572438	−	0.991491i	0.996315	−	0.0857715i	$-0.0273355\pi$
$744$		0			0
$745$	17642.1	+	30557.1i	0.867594	+	1.50272i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4338.95	+	7515.29i	0.211671	+	0.366626i
$750$		0			0
$751$	−12455.0	+	21572.6i	−0.605177	+	1.04820i	0.386846	+	0.922144i	$0.373564\pi$
$751$	−12455.0	+	21572.6i	−0.605177	+	1.04820i	−0.992023	+	0.126053i	$0.959769\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6344.32			−0.305819
$756$		0			0
$757$	2659.35			0.127682			0.0638412	−	0.997960i	$-0.479665\pi$
$757$	2659.35			0.127682			0.0638412	+	0.997960i	$0.479665\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12707.6	+	22010.3i	−0.605324	+	1.04845i	0.386677	+	0.922215i	$0.373623\pi$
$761$	−12707.6	+	22010.3i	−0.605324	+	1.04845i	−0.992000	+	0.126236i	$0.959710\pi$
$762$		0			0
$763$	−3307.06	−	5728.00i	−0.156912	−	0.271779i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−32003.8	−	55432.2i	−1.50664	−	2.60957i
$768$		0			0
$769$	−15026.6	+	26026.8i	−0.704645	+	1.22048i	0.262175	+	0.965020i	$0.415560\pi$
$769$	−15026.6	+	26026.8i	−0.704645	+	1.22048i	−0.966820	+	0.255460i	$0.917773\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6066.30			−0.282263			−0.141132	−	0.989991i	$-0.545074\pi$
$773$	−6066.30			−0.282263			−0.141132	+	0.989991i	$0.545074\pi$
$774$		0			0
$775$	13744.2			0.637040
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9445.04	−	16359.3i	0.434408	−	0.752416i
$780$		0			0
$781$	−557.402	−	965.449i	−0.0255383	−	0.0442336i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13265.6	−	22976.7i	−0.603147	−	1.04468i
$786$		0			0
$787$	2748.09	−	4759.83i	0.124471	−	0.215590i	−0.797055	−	0.603907i	$-0.793610\pi$
$787$	2748.09	−	4759.83i	0.124471	−	0.215590i	0.921526	+	0.388316i	$0.126943\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2482.64			−0.111596
$792$		0			0
$793$	42083.3			1.88452
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13904.0	−	24082.5i	0.617949	−	1.07032i	−0.371910	−	0.928269i	$-0.621297\pi$
$797$	13904.0	−	24082.5i	0.617949	−	1.07032i	0.989859	−	0.142051i	$-0.0453696\pi$
$798$		0			0
$799$	−488.546	−	846.187i	−0.0216314	−	0.0374667i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	143.500	+	248.549i	0.00630636	+	0.0109229i
$804$		0			0
$805$	−3980.25	+	6893.99i	−0.174267	+	0.301840i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6720.58			−0.292068			−0.146034	−	0.989280i	$-0.546651\pi$
$809$	−6720.58			−0.292068			−0.146034	+	0.989280i	$0.546651\pi$
$810$		0			0
$811$	9923.66			0.429675			0.214838	−	0.976650i	$-0.431078\pi$
$811$	9923.66			0.429675			0.214838	+	0.976650i	$0.431078\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5505.50	+	9535.80i	−0.236625	+	0.409846i
$816$		0			0
$817$	−8091.15	−	14014.3i	−0.346479	−	0.600119i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9559.38	+	16557.3i	0.406364	+	0.703843i	0.994479	−	0.104934i	$-0.0334632\pi$
$821$	9559.38	+	16557.3i	0.406364	+	0.703843i	−0.588115	+	0.808777i	$0.700130\pi$
$822$		0			0
$823$	−14171.6	+	24546.0i	−0.600234	+	1.03963i	0.392552	+	0.919730i	$0.371592\pi$
$823$	−14171.6	+	24546.0i	−0.600234	+	1.03963i	−0.992785	+	0.119905i	$0.961741\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25909.2			1.08942			0.544711	−	0.838624i	$-0.316639\pi$
$827$	25909.2			1.08942			0.544711	+	0.838624i	$0.316639\pi$
$828$		0			0
$829$	−3137.00			−0.131426			−0.0657132	−	0.997839i	$-0.520932\pi$
$829$	−3137.00			−0.131426			−0.0657132	+	0.997839i	$0.520932\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6459.68	−	11188.5i	0.268685	−	0.465376i
$834$		0			0
$835$	−19001.3	−	32911.2i	−0.787504	−	1.36400i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4055.31	−	7024.01i	−0.166871	−	0.289030i	0.770447	−	0.637504i	$-0.220033\pi$
$839$	−4055.31	−	7024.01i	−0.166871	−	0.289030i	−0.937318	+	0.348475i	$0.886700\pi$
$840$		0			0
$841$	−8302.02	+	14379.5i	−0.340400	+	0.589590i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−66392.2			−2.70291
$846$		0			0
$847$	6960.80			0.282380
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13960.8	+	24180.9i	−0.562363	+	0.974041i
$852$		0			0
$853$	−10158.7	−	17595.3i	−0.407768	−	0.706275i	0.586871	−	0.809680i	$-0.300360\pi$
$853$	−10158.7	−	17595.3i	−0.407768	−	0.706275i	−0.994639	+	0.103405i	$0.967026\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1469.32	−	2544.93i	−0.0585658	−	0.101439i	0.835256	−	0.549861i	$-0.185319\pi$
$857$	−1469.32	−	2544.93i	−0.0585658	−	0.101439i	−0.893822	+	0.448422i	$0.851986\pi$
$858$		0			0
$859$	5355.13	−	9275.35i	0.212706	−	0.368418i	−0.739854	−	0.672767i	$-0.765106\pi$
$859$	5355.13	−	9275.35i	0.212706	−	0.368418i	0.952561	+	0.304349i	$0.0984390\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12400.3			−0.489122			−0.244561	−	0.969634i	$-0.578644\pi$
$863$	−12400.3			−0.489122			−0.244561	+	0.969634i	$0.578644\pi$
$864$		0			0
$865$	−47369.2			−1.86197
$866$		0			0
$867$		0			0
$868$		0			0
$869$	615.739	−	1066.49i	0.0240363	−	0.0416320i
$870$		0			0
$871$	17409.3	+	30153.8i	0.677258	+	1.17305i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2592.77	+	4490.81i	0.100173	+	0.173505i
$876$		0			0
$877$	15395.1	−	26665.1i	0.592765	−	1.02670i	−0.401093	−	0.916037i	$-0.631370\pi$
$877$	15395.1	−	26665.1i	0.592765	−	1.02670i	0.993858	−	0.110662i	$-0.0352970\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20320.3			0.777082			0.388541	−	0.921431i	$-0.372979\pi$
$881$	20320.3			0.777082			0.388541	+	0.921431i	$0.372979\pi$
$882$		0			0
$883$	27680.1			1.05494			0.527468	−	0.849575i	$-0.323141\pi$
$883$	27680.1			1.05494			0.527468	+	0.849575i	$0.323141\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20562.4	−	35615.1i	0.778374	−	1.34818i	−0.154504	−	0.987992i	$-0.549378\pi$
$887$	20562.4	−	35615.1i	0.778374	−	1.34818i	0.932878	−	0.360191i	$-0.117289\pi$
$888$		0			0
$889$	−1466.41	−	2539.90i	−0.0553228	−	0.0958218i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−686.638	−	1189.29i	−0.0257306	−	0.0445668i
$894$		0			0
$895$	−14306.2	+	24779.0i	−0.534305	+	0.925444i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−55510.2			−2.05936
$900$		0			0
$901$	12282.0			0.454131
$902$		0			0
$903$		0			0
$904$		0			0
$905$	11515.0	−	19944.6i	0.422953	−	0.732577i
$906$		0			0
$907$	−13485.7	−	23357.9i	−0.493698	−	0.855111i	0.506275	−	0.862372i	$-0.331022\pi$
$907$	−13485.7	−	23357.9i	−0.493698	−	0.855111i	−0.999974	+	0.00726135i	$0.997689\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23141.0	−	40081.4i	−0.841597	−	1.45769i	−0.888544	−	0.458792i	$-0.848282\pi$
$911$	−23141.0	−	40081.4i	−0.841597	−	1.45769i	0.0469462	−	0.998897i	$-0.485051\pi$
$912$		0			0
$913$	471.239	−	816.210i	0.0170819	−	0.0295866i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4130.97			−0.148764
$918$		0			0
$919$	36864.3			1.32322			0.661611	−	0.749847i	$-0.269873\pi$
$919$	36864.3			1.32322			0.661611	+	0.749847i	$0.269873\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−47342.8	+	82000.2i	−1.68831	+	2.92423i
$924$		0			0
$925$	−6089.22	−	10546.8i	−0.216446	−	0.374895i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17139.3	−	29686.2i	−0.605299	−	1.04841i	−0.992004	−	0.126205i	$-0.959720\pi$
$929$	−17139.3	−	29686.2i	−0.605299	−	1.04841i	0.386705	−	0.922203i	$-0.373613\pi$
$930$		0			0
$931$	9078.89	−	15725.1i	0.319601	−	0.553565i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−541.718			−0.0189477
$936$		0			0
$937$	450.515			0.0157072			0.00785362	−	0.999969i	$-0.497500\pi$
$937$	450.515			0.0157072			0.00785362	+	0.999969i	$0.497500\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3549.44	−	6147.80i	0.122963	−	0.212978i	−0.797972	−	0.602695i	$-0.794094\pi$
$941$	3549.44	−	6147.80i	0.122963	−	0.212978i	0.920935	+	0.389716i	$0.127427\pi$
$942$		0			0
$943$	18868.7	+	32681.5i	0.651589	+	1.12858i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4916.68	−	8515.93i	−0.168712	−	0.292218i	0.769255	−	0.638942i	$-0.220628\pi$
$947$	−4916.68	−	8515.93i	−0.168712	−	0.292218i	−0.937967	+	0.346724i	$0.887294\pi$
$948$		0			0
$949$	12188.1	−	21110.5i	0.416906	−	0.722102i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41260.1			−1.40246			−0.701231	−	0.712934i	$-0.747366\pi$
$953$	−41260.1			−1.40246			−0.701231	+	0.712934i	$0.747366\pi$
$954$		0			0
$955$	−33584.5			−1.13798
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3481.63	+	6030.35i	−0.117234	+	0.203056i
$960$		0			0
$961$	−22688.7	−	39297.9i	−0.761595	−	1.31912i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1259.50	+	2181.52i	0.0420153	+	0.0727726i
$966$		0			0
$967$	11321.4	−	19609.2i	0.376496	−	0.652110i	−0.614054	−	0.789264i	$-0.710462\pi$
$967$	11321.4	−	19609.2i	0.376496	−	0.652110i	0.990550	+	0.137154i	$0.0437955\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3166.82			−0.104663			−0.0523316	−	0.998630i	$-0.516665\pi$
$971$	−3166.82			−0.104663			−0.0523316	+	0.998630i	$0.516665\pi$
$972$		0			0
$973$	−16430.5			−0.541353
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10321.5	−	17877.4i	0.337988	−	0.585413i	−0.646066	−	0.763282i	$-0.723587\pi$
$977$	10321.5	−	17877.4i	0.337988	−	0.585413i	0.984054	+	0.177869i	$0.0569203\pi$
$978$		0			0
$979$	95.1032	+	164.724i	0.00310471	+	0.00537752i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29644.7	−	51346.2i	−0.961872	−	1.66601i	−0.717793	−	0.696257i	$-0.754847\pi$
$983$	−29644.7	−	51346.2i	−0.961872	−	1.66601i	−0.244080	−	0.969755i	$-0.578486\pi$
$984$		0			0
$985$	35163.6	−	60905.1i	1.13747	−	1.97015i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32327.9			1.03940
$990$		0			0
$991$	−14404.0			−0.461712			−0.230856	−	0.972988i	$-0.574153\pi$
$991$	−14404.0			−0.461712			−0.230856	+	0.972988i	$0.574153\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25059.0	−	43403.4i	0.798414	−	1.38289i
$996$		0			0
$997$	24447.4	+	42344.2i	0.776588	+	1.34509i	0.933898	+	0.357541i	$0.116385\pi$
$997$	24447.4	+	42344.2i	0.776588	+	1.34509i	−0.157309	+	0.987549i	$0.550282\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.m.433.2		4
3.2	odd	2		648.4.i.s.433.1		4
9.2	odd	6		648.4.i.s.217.1		4
9.4	even	3		216.4.a.h.1.1	yes	2
9.5	odd	6		216.4.a.e.1.2	✓	2
9.7	even	3	inner	648.4.i.m.217.2		4
36.23	even	6		432.4.a.o.1.2		2
36.31	odd	6		432.4.a.s.1.1		2
72.5	odd	6		1728.4.a.bt.1.1		2
72.13	even	6		1728.4.a.bh.1.2		2
72.59	even	6		1728.4.a.bs.1.1		2
72.67	odd	6		1728.4.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.a.e.1.2	✓	2	9.5	odd	6
216.4.a.h.1.1	yes	2	9.4	even	3
432.4.a.o.1.2		2	36.23	even	6
432.4.a.s.1.1		2	36.31	odd	6
648.4.i.m.217.2		4	9.7	even	3	inner
648.4.i.m.433.2		4	1.1	even	1	trivial
648.4.i.s.217.1		4	9.2	odd	6
648.4.i.s.433.1		4	3.2	odd	2
1728.4.a.bg.1.2		2	72.67	odd	6
1728.4.a.bh.1.2		2	72.13	even	6
1728.4.a.bs.1.1		2	72.59	even	6
1728.4.a.bt.1.1		2	72.5	odd	6

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.61684 - 11.4607i) q^{5} +(2.61684 + 4.53251i) q^{7} +(-0.500000 - 0.866025i) q^{11} +(-42.4674 + 73.5557i) q^{13} +40.9348 q^{17} +57.5326 q^{19} +(-57.4674 + 99.5364i) q^{23} +(-25.0652 - 43.4143i) q^{25} +(101.234 + 175.342i) q^{29} +(-137.084 + 237.437i) q^{31} +69.2610 q^{35} +242.935 q^{37} +(164.168 - 284.348i) q^{41} +(-140.636 - 243.588i) q^{43} +(-11.9348 - 20.6716i) q^{47} +(157.804 - 273.325i) q^{49} +300.038 q^{53} -13.2337 q^{55} +(-376.804 + 652.644i) q^{59} +(-247.739 - 429.097i) q^{61} +(562.000 + 973.413i) q^{65} +(204.973 - 355.023i) q^{67} +1114.80 q^{71} -287.000 q^{73} +(2.61684 - 4.53251i) q^{77} +(615.739 + 1066.49i) q^{79} +(471.239 + 816.210i) q^{83} +(270.859 - 469.141i) q^{85} -190.206 q^{89} -444.522 q^{91} +(380.684 - 659.365i) q^{95} +(153.435 + 265.757i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} - 24 q^{7} - 2 q^{11} - 32 q^{13} - 112 q^{17} + 368 q^{19} - 92 q^{23} - 376 q^{25} + 336 q^{29} - 376 q^{31} + 1380 q^{35} + 696 q^{37} + 312 q^{41} - 80 q^{43} + 228 q^{47} - 196 q^{49}+ \cdots + 338 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 - 24 * q^7 - 2 * q^11 - 32 * q^13 - 112 * q^17 + 368 * q^19 - 92 * q^23 - 376 * q^25 + 336 * q^29 - 376 * q^31 + 1380 * q^35 + 696 * q^37 + 312 * q^41 - 80 * q^43 + 228 * q^47 - 196 * q^49 + 304 * q^53 + 16 * q^55 - 680 * q^59 + 112 * q^61 + 2248 * q^65 - 352 * q^67 + 3632 * q^71 - 1148 * q^73 - 24 * q^77 + 1360 * q^79 + 782 * q^83 + 2600 * q^85 + 480 * q^89 - 3984 * q^91 - 1924 * q^95 + 338 * q^97

Introduction

L-functions

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