LMFDB - Embedded newform 108.4.e.a.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(37,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("108.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.6831243.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 49x^{2} + 48 $ x^6 + 13x^4 + 49x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.2
Root			$2.63162i$ of defining polynomial
Character	$\chi$	$=$	108.37
Dual form			108.4.e.a.73.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44901 + 4.24182i) q^{5} +(5.32725 + 9.22708i) q^{7} +O(q^{10})$	$q+(-2.44901 + 4.24182i) q^{5} +(5.32725 + 9.22708i) q^{7} +(17.7268 + 30.7038i) q^{11} +(-36.3501 + 62.9603i) q^{13} +127.417 q^{17} -46.3913 q^{19} +(-65.5739 + 113.577i) q^{23} +(50.5047 + 87.4766i) q^{25} +(-68.7549 - 119.087i) q^{29} +(53.0640 - 91.9096i) q^{31} -52.1861 q^{35} +137.401 q^{37} +(-35.8986 + 62.1782i) q^{41} +(-188.459 - 326.421i) q^{43} +(-306.813 - 531.416i) q^{47} +(114.741 - 198.737i) q^{49} +431.757 q^{53} -173.653 q^{55} +(-142.878 + 247.471i) q^{59} +(21.9682 + 38.0500i) q^{61} +(-178.044 - 308.381i) q^{65} +(22.6052 - 39.1533i) q^{67} -357.328 q^{71} +530.718 q^{73} +(-188.871 + 327.134i) q^{77} +(97.5540 + 168.969i) q^{79} +(380.352 + 658.789i) q^{83} +(-312.047 + 540.480i) q^{85} +1214.67 q^{89} -774.586 q^{91} +(113.613 - 196.783i) q^{95} +(552.402 + 956.788i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} - 6 q^{7}+O(q^{10}) $ 6 * q - 6 * q^5 - 6 * q^7	$ 6 q - 6 q^{5} - 6 q^{7} - 51 q^{11} + 12 q^{13} + 222 q^{17} + 30 q^{19} - 210 q^{23} - 3 q^{25} - 456 q^{29} + 48 q^{31} + 1104 q^{35} - 96 q^{37} - 897 q^{41} + 129 q^{43} - 522 q^{47} - 225 q^{49} + 2208 q^{53} - 216 q^{55} - 453 q^{59} - 402 q^{61} - 1110 q^{65} - 213 q^{67} - 120 q^{71} + 750 q^{73} - 1128 q^{77} + 552 q^{79} + 612 q^{83} + 1188 q^{85} + 924 q^{89} - 264 q^{91} + 2184 q^{95} + 93 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 - 6 * q^7 - 51 * q^11 + 12 * q^13 + 222 * q^17 + 30 * q^19 - 210 * q^23 - 3 * q^25 - 456 * q^29 + 48 * q^31 + 1104 * q^35 - 96 * q^37 - 897 * q^41 + 129 * q^43 - 522 * q^47 - 225 * q^49 + 2208 * q^53 - 216 * q^55 - 453 * q^59 - 402 * q^61 - 1110 * q^65 - 213 * q^67 - 120 * q^71 + 750 * q^73 - 1128 * q^77 + 552 * q^79 + 612 * q^83 + 1188 * q^85 + 924 * q^89 - 264 * q^91 + 2184 * q^95 + 93 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.44901	+	4.24182i	−0.219047	+	0.379400i	−0.954517	−	0.298157i	$-0.903628\pi$
$5$	−2.44901	+	4.24182i	−0.219047	+	0.379400i	0.735470	+	0.677557i	$0.236961\pi$
$6$		0			0
$7$	5.32725	+	9.22708i	0.287645	+	0.498215i	0.973247	−	0.229761i	$-0.0737944\pi$
$7$	5.32725	+	9.22708i	0.287645	+	0.498215i	−0.685602	+	0.727976i	$0.740461\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	17.7268	+	30.7038i	0.485895	+	0.841594i	0.999869	−	0.0162115i	$-0.00516051\pi$
$11$	17.7268	+	30.7038i	0.485895	+	0.841594i	−0.513974	+	0.857806i	$0.671827\pi$
$12$		0			0
$13$	−36.3501	+	62.9603i	−0.775517	+	1.34323i	0.158987	+	0.987281i	$0.449177\pi$
$13$	−36.3501	+	62.9603i	−0.775517	+	1.34323i	−0.934504	+	0.355954i	$0.884156\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	127.417			1.81784			0.908918	−	0.416975i	$-0.136910\pi$
$17$	127.417			1.81784			0.908918	+	0.416975i	$0.136910\pi$
$18$		0			0
$19$	−46.3913			−0.560152			−0.280076	−	0.959978i	$-0.590360\pi$
$19$	−46.3913			−0.560152			−0.280076	+	0.959978i	$0.590360\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−65.5739	+	113.577i	−0.594483	+	1.02967i	0.399137	+	0.916891i	$0.369310\pi$
$23$	−65.5739	+	113.577i	−0.594483	+	1.02967i	−0.993620	+	0.112783i	$0.964024\pi$
$24$		0			0
$25$	50.5047	+	87.4766i	0.404037	+	0.699813i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−68.7549	−	119.087i	−0.440258	−	0.762548i	0.557451	−	0.830210i	$-0.311780\pi$
$29$	−68.7549	−	119.087i	−0.440258	−	0.762548i	−0.997708	+	0.0676615i	$0.978446\pi$
$30$		0			0
$31$	53.0640	−	91.9096i	0.307438	−	0.532498i	−0.670363	−	0.742033i	$-0.733862\pi$
$31$	53.0640	−	91.9096i	0.307438	−	0.532498i	0.977801	+	0.209535i	$0.0671950\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−52.1861			−0.252030
$36$		0			0
$37$	137.401			0.610500			0.305250	−	0.952272i	$-0.401260\pi$
$37$	137.401			0.610500			0.305250	+	0.952272i	$0.401260\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−35.8986	+	62.1782i	−0.136742	+	0.236844i	−0.926261	−	0.376881i	$-0.876996\pi$
$41$	−35.8986	+	62.1782i	−0.136742	+	0.236844i	0.789520	+	0.613725i	$0.210330\pi$
$42$		0			0
$43$	−188.459	−	326.421i	−0.668367	−	1.15765i	−0.978361	−	0.206907i	$-0.933660\pi$
$43$	−188.459	−	326.421i	−0.668367	−	1.15765i	0.309994	−	0.950739i	$-0.399673\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−306.813	−	531.416i	−0.952198	−	1.64926i	−0.740654	−	0.671887i	$-0.765484\pi$
$47$	−306.813	−	531.416i	−0.952198	−	1.64926i	−0.211544	−	0.977368i	$-0.567849\pi$
$48$		0			0
$49$	114.741	−	198.737i	0.334521	−	0.579407i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	431.757			1.11899			0.559494	−	0.828835i	$-0.310996\pi$
$53$	431.757			1.11899			0.559494	+	0.828835i	$0.310996\pi$
$54$		0			0
$55$	−173.653			−0.425734
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−142.878	+	247.471i	−0.315272	+	0.546068i	−0.979495	−	0.201467i	$-0.935429\pi$
$59$	−142.878	+	247.471i	−0.315272	+	0.546068i	0.664223	+	0.747535i	$0.268763\pi$
$60$		0			0
$61$	21.9682	+	38.0500i	0.0461104	+	0.0798656i	0.888159	−	0.459535i	$-0.151984\pi$
$61$	21.9682	+	38.0500i	0.0461104	+	0.0798656i	−0.842049	+	0.539401i	$0.818651\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−178.044	−	308.381i	−0.339748	−	0.588462i
$66$		0			0
$67$	22.6052	−	39.1533i	0.0412188	−	0.0713930i	−0.844680	−	0.535272i	$-0.820209\pi$
$67$	22.6052	−	39.1533i	0.0412188	−	0.0713930i	0.885899	+	0.463879i	$0.153543\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−357.328			−0.597282			−0.298641	−	0.954366i	$-0.596533\pi$
$71$	−357.328			−0.597282			−0.298641	+	0.954366i	$0.596533\pi$
$72$		0			0
$73$	530.718			0.850901			0.425451	−	0.904982i	$-0.360116\pi$
$73$	530.718			0.850901			0.425451	+	0.904982i	$0.360116\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−188.871	+	327.134i	−0.279530	+	0.484160i
$78$		0			0
$79$	97.5540	+	168.969i	0.138933	+	0.240639i	0.927093	−	0.374832i	$-0.122299\pi$
$79$	97.5540	+	168.969i	0.138933	+	0.240639i	−0.788160	+	0.615470i	$0.788966\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	380.352	+	658.789i	0.503001	+	0.871223i	0.999994	+	0.00346848i	$0.00110405\pi$
$83$	380.352	+	658.789i	0.503001	+	0.871223i	−0.496993	+	0.867754i	$0.665563\pi$
$84$		0			0
$85$	−312.047	+	540.480i	−0.398191	+	0.689686i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1214.67			1.44668			0.723339	−	0.690493i	$-0.242607\pi$
$89$	1214.67			1.44668			0.723339	+	0.690493i	$0.242607\pi$
$90$		0			0
$91$	−774.586			−0.892293
$92$		0			0
$93$		0			0
$94$		0			0
$95$	113.613	−	196.783i	0.122699	−	0.212522i
$96$		0			0
$97$	552.402	+	956.788i	0.578226	+	1.00152i	0.995683	+	0.0928202i	$0.0295882\pi$
$97$	552.402	+	956.788i	0.578226	+	1.00152i	−0.417457	+	0.908697i	$0.637078\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	508.962	+	881.547i	0.501421	+	0.868487i	0.999999	+	0.00164208i	$0.000522691\pi$
$101$	508.962	+	881.547i	0.501421	+	0.868487i	−0.498577	+	0.866845i	$0.666144\pi$
$102$		0			0
$103$	−416.128	+	720.756i	−0.398081	+	0.689497i	−0.993489	−	0.113927i	$-0.963657\pi$
$103$	−416.128	+	720.756i	−0.398081	+	0.689497i	0.595408	+	0.803423i	$0.296990\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−481.992			−0.435476			−0.217738	−	0.976007i	$-0.569868\pi$
$107$	−481.992			−0.435476			−0.217738	+	0.976007i	$0.569868\pi$
$108$		0			0
$109$	−904.531			−0.794847			−0.397424	−	0.917635i	$-0.630096\pi$
$109$	−904.531			−0.794847			−0.397424	+	0.917635i	$0.630096\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	400.494	−	693.675i	0.333410	−	0.577482i	−0.649768	−	0.760132i	$-0.725134\pi$
$113$	400.494	−	693.675i	0.333410	−	0.577482i	0.983178	+	0.182650i	$0.0584675\pi$
$114$		0			0
$115$	−321.183	−	556.305i	−0.260439	−	0.451093i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	678.784	+	1175.69i	0.522891	+	0.905673i
$120$		0			0
$121$	37.0186	−	64.1181i	0.0278126	−	0.0481729i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1107.00			−0.792105
$126$		0			0
$127$	1755.04			1.22626			0.613129	−	0.789982i	$-0.289910\pi$
$127$	1755.04			1.22626			0.613129	+	0.789982i	$0.289910\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	551.870	−	955.868i	0.368070	−	0.637516i	−0.621194	−	0.783657i	$-0.713352\pi$
$131$	551.870	−	955.868i	0.368070	−	0.637516i	0.989264	+	0.146141i	$0.0466854\pi$
$132$		0			0
$133$	−247.138	−	428.056i	−0.161125	−	0.279076i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1063.25	−	1841.61i	−0.663064	−	1.14846i	−0.979806	−	0.199949i	$-0.935922\pi$
$137$	−1063.25	−	1841.61i	−0.663064	−	1.14846i	0.316742	−	0.948512i	$-0.397411\pi$
$138$		0			0
$139$	1127.73	−	1953.29i	0.688151	−	1.19191i	−0.284284	−	0.958740i	$-0.591756\pi$
$139$	1127.73	−	1953.29i	0.688151	−	1.19191i	0.972435	−	0.233173i	$-0.0749108\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2577.49			−1.50728
$144$		0			0
$145$	673.527			0.385748
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−342.725	+	593.617i	−0.188437	+	0.326383i	−0.944729	−	0.327851i	$-0.893676\pi$
$149$	−342.725	+	593.617i	−0.188437	+	0.326383i	0.756292	+	0.654234i	$0.227009\pi$
$150$		0			0
$151$	−270.165	−	467.939i	−0.145601	−	0.252188i	0.783996	−	0.620766i	$-0.213178\pi$
$151$	−270.165	−	467.939i	−0.145601	−	0.252188i	−0.929597	+	0.368578i	$0.879845\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	259.909	+	450.176i	0.134686	+	0.233284i
$156$		0			0
$157$	236.664	−	409.913i	0.120304	−	0.208373i	−0.799583	−	0.600555i	$-0.794946\pi$
$157$	236.664	−	409.913i	0.120304	−	0.208373i	0.919888	+	0.392182i	$0.128280\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1397.31			−0.683999
$162$		0			0
$163$	−198.981			−0.0956160			−0.0478080	−	0.998857i	$-0.515224\pi$
$163$	−198.981			−0.0956160			−0.0478080	+	0.998857i	$0.515224\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2137.82	−	3702.82i	0.990598	−	1.71577i	0.376819	−	0.926287i	$-0.377018\pi$
$167$	2137.82	−	3702.82i	0.990598	−	1.71577i	0.613778	−	0.789479i	$-0.289649\pi$
$168$		0			0
$169$	−1544.17	−	2674.57i	−0.702852	−	1.21738i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.55237	+	13.0811i	0.00331905	+	0.00574877i	0.867680	−	0.497123i	$-0.165610\pi$
$173$	7.55237	+	13.0811i	0.00331905	+	0.00574877i	−0.864361	+	0.502872i	$0.832277\pi$
$174$		0			0
$175$	−538.102	+	932.021i	−0.232438	+	0.402595i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	309.915			0.129408			0.0647042	−	0.997904i	$-0.479390\pi$
$179$	309.915			0.129408			0.0647042	+	0.997904i	$0.479390\pi$
$180$		0			0
$181$	−2253.32			−0.925348			−0.462674	−	0.886529i	$-0.653110\pi$
$181$	−2253.32			−0.925348			−0.462674	+	0.886529i	$0.653110\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−336.496	+	582.828i	−0.133728	+	0.231624i
$186$		0			0
$187$	2258.70	+	3912.19i	0.883277	+	1.52988i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1849.28	+	3203.04i	0.700571	+	1.21342i	0.968266	+	0.249921i	$0.0804046\pi$
$191$	1849.28	+	3203.04i	0.700571	+	1.21342i	−0.267695	+	0.963504i	$0.586262\pi$
$192$		0			0
$193$	1781.64	−	3085.89i	0.664482	−	1.15092i	−0.314943	−	0.949111i	$-0.601985\pi$
$193$	1781.64	−	3085.89i	0.664482	−	1.15092i	0.979425	−	0.201807i	$-0.0646813\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−89.0014			−0.0321883			−0.0160941	−	0.999870i	$-0.505123\pi$
$197$	−89.0014			−0.0321883			−0.0160941	+	0.999870i	$0.505123\pi$
$198$		0			0
$199$	287.103			0.102272			0.0511362	−	0.998692i	$-0.483716\pi$
$199$	287.103			0.102272			0.0511362	+	0.998692i	$0.483716\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	732.550	−	1268.81i	0.253276	−	0.438686i
$204$		0			0
$205$	−175.832	−	304.550i	−0.0599056	−	0.103760i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−822.371	−	1424.39i	−0.272175	−	0.471421i
$210$		0			0
$211$	−2529.93	+	4381.97i	−0.825438	+	1.42970i	0.0761454	+	0.997097i	$0.475739\pi$
$211$	−2529.93	+	4381.97i	−0.825438	+	1.42970i	−0.901584	+	0.432605i	$0.857595\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1846.16			0.585614
$216$		0			0
$217$	1130.74			0.353732
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4631.63	+	8022.23i	−1.40976	+	2.44178i
$222$		0			0
$223$	199.857	+	346.162i	0.0600153	+	0.103949i	0.894472	−	0.447124i	$-0.147552\pi$
$223$	199.857	+	346.162i	0.0600153	+	0.103949i	−0.834457	+	0.551073i	$0.814218\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−187.709	−	325.121i	−0.0548840	−	0.0950619i	0.837278	−	0.546777i	$-0.184146\pi$
$227$	−187.709	−	325.121i	−0.0548840	−	0.0950619i	−0.892162	+	0.451715i	$0.850812\pi$
$228$		0			0
$229$	−2413.28	+	4179.93i	−0.696394	+	1.20619i	0.273315	+	0.961925i	$0.411880\pi$
$229$	−2413.28	+	4179.93i	−0.696394	+	1.20619i	−0.969709	+	0.244265i	$0.921453\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3858.19			1.08480			0.542401	−	0.840120i	$-0.317516\pi$
$233$	3858.19			1.08480			0.542401	+	0.840120i	$0.317516\pi$
$234$		0			0
$235$	3005.56			0.834303
$236$		0			0
$237$		0			0
$238$		0			0
$239$	652.603	−	1130.34i	0.176625	−	0.305924i	−0.764097	−	0.645101i	$-0.776815\pi$
$239$	652.603	−	1130.34i	0.176625	−	0.305924i	0.940722	+	0.339177i	$0.110149\pi$
$240$		0			0
$241$	1108.07	+	1919.23i	0.296170	+	0.512982i	0.975256	−	0.221077i	$-0.0709572\pi$
$241$	1108.07	+	1919.23i	0.296170	+	0.512982i	−0.679086	+	0.734058i	$0.737624\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	562.003	+	973.418i	0.146551	+	0.253834i
$246$		0			0
$247$	1686.33	−	2920.81i	0.434407	−	0.752415i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2993.80			0.752856			0.376428	−	0.926446i	$-0.377152\pi$
$251$	2993.80			0.752856			0.376428	+	0.926446i	$0.377152\pi$
$252$		0			0
$253$	−4649.67			−1.15542
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1467.52	−	2541.81i	0.356191	−	0.616941i	−0.631130	−	0.775677i	$-0.717409\pi$
$257$	1467.52	−	2541.81i	0.356191	−	0.616941i	0.987321	+	0.158736i	$0.0507419\pi$
$258$		0			0
$259$	731.968	+	1267.81i	0.175607	+	0.304161i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	776.883	+	1345.60i	0.182147	+	0.315488i	0.942611	−	0.333892i	$-0.108362\pi$
$263$	776.883	+	1345.60i	0.182147	+	0.315488i	−0.760464	+	0.649380i	$0.775029\pi$
$264$		0			0
$265$	−1057.38	+	1831.43i	−0.245110	+	0.424543i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1762.62			0.399513			0.199756	−	0.979846i	$-0.435985\pi$
$269$	1762.62			0.399513			0.199756	+	0.979846i	$0.435985\pi$
$270$		0			0
$271$	−6924.63			−1.55218			−0.776091	−	0.630621i	$-0.782800\pi$
$271$	−6924.63			−1.55218			−0.776091	+	0.630621i	$0.782800\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1790.58	+	3101.37i	−0.392639	+	0.680071i
$276$		0			0
$277$	−3012.48	−	5217.76i	−0.653437	−	1.13179i	−0.982283	−	0.187403i	$-0.939993\pi$
$277$	−3012.48	−	5217.76i	−0.653437	−	1.13179i	0.328846	−	0.944384i	$-0.393340\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3168.74	−	5488.42i	−0.672709	−	1.16517i	−0.977133	−	0.212629i	$-0.931797\pi$
$281$	−3168.74	−	5488.42i	−0.672709	−	1.16517i	0.304424	−	0.952537i	$-0.401536\pi$
$282$		0			0
$283$	984.338	−	1704.92i	0.206759	−	0.358117i	−0.743933	−	0.668254i	$-0.767042\pi$
$283$	984.338	−	1704.92i	0.206759	−	0.358117i	0.950692	+	0.310137i	$0.100375\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−764.963			−0.157332
$288$		0			0
$289$	11322.1			2.30453
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−587.208	+	1017.07i	−0.117082	+	0.202792i	−0.918610	−	0.395165i	$-0.870687\pi$
$293$	−587.208	+	1017.07i	−0.117082	+	0.202792i	0.801528	+	0.597957i	$0.204021\pi$
$294$		0			0
$295$	−699.818	−	1212.12i	−0.138119	−	0.239228i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4767.24	−	8257.10i	−0.922062	−	1.59706i
$300$		0			0
$301$	2007.94	−	3477.86i	0.384504	−	0.665981i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−215.202			−0.0404013
$306$		0			0
$307$	3258.92			0.605851			0.302926	−	0.953014i	$-0.402037\pi$
$307$	3258.92			0.605851			0.302926	+	0.953014i	$0.402037\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3979.01	+	6891.84i	−0.725495	+	1.25659i	0.233275	+	0.972411i	$0.425056\pi$
$311$	−3979.01	+	6891.84i	−0.725495	+	1.25659i	−0.958770	+	0.284183i	$0.908278\pi$
$312$		0			0
$313$	2098.13	+	3634.06i	0.378892	+	0.656260i	0.990901	−	0.134590i	$-0.0429719\pi$
$313$	2098.13	+	3634.06i	0.378892	+	0.656260i	−0.612009	+	0.790850i	$0.709639\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2416.78	−	4185.99i	−0.428202	−	0.741667i	0.568512	−	0.822675i	$-0.307519\pi$
$317$	−2416.78	−	4185.99i	−0.428202	−	0.741667i	−0.996713	+	0.0810079i	$0.974186\pi$
$318$		0			0
$319$	2437.61	−	4222.07i	0.427838	−	0.741037i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5911.05			−1.01826
$324$		0			0
$325$	−7343.41			−1.25335
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3268.94	−	5661.98i	0.547789	−	0.948799i
$330$		0			0
$331$	−5270.34	−	9128.50i	−0.875179	−	1.51585i	−0.856572	−	0.516027i	$-0.827410\pi$
$331$	−5270.34	−	9128.50i	−0.875179	−	1.51585i	−0.0186067	−	0.999827i	$-0.505923\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	110.721	+	191.774i	0.0180577	+	0.0312768i
$336$		0			0
$337$	2437.45	−	4221.79i	0.393995	−	0.682420i	−0.598977	−	0.800766i	$-0.704426\pi$
$337$	2437.45	−	4221.79i	0.393995	−	0.682420i	0.992972	+	0.118346i	$0.0377593\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3762.63			0.597530
$342$		0			0
$343$	6099.51			0.960182
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−135.378	+	234.482i	−0.0209437	+	0.0362756i	−0.876307	−	0.481753i	$-0.840000\pi$
$347$	−135.378	+	234.482i	−0.0209437	+	0.0362756i	0.855364	+	0.518028i	$0.173334\pi$
$348$		0			0
$349$	5219.99	+	9041.30i	0.800630	+	1.38673i	0.919202	+	0.393786i	$0.128835\pi$
$349$	5219.99	+	9041.30i	0.800630	+	1.38673i	−0.118572	+	0.992945i	$0.537832\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	32.9627	+	57.0931i	0.00497005	+	0.00860838i	0.868500	−	0.495690i	$-0.165085\pi$
$353$	32.9627	+	57.0931i	0.00497005	+	0.00860838i	−0.863530	+	0.504298i	$0.831751\pi$
$354$		0			0
$355$	875.101	−	1515.72i	0.130833	−	0.226609i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1973.98			−0.290203			−0.145101	−	0.989417i	$-0.546351\pi$
$359$	−1973.98			−0.290203			−0.145101	+	0.989417i	$0.546351\pi$
$360$		0			0
$361$	−4706.85			−0.686230
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1299.73	+	2251.21i	−0.186387	+	0.322832i
$366$		0			0
$367$	−2187.97	−	3789.68i	−0.311202	−	0.539018i	0.667421	−	0.744681i	$-0.267398\pi$
$367$	−2187.97	−	3789.68i	−0.311202	−	0.539018i	−0.978623	+	0.205663i	$0.934065\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2300.08	+	3983.85i	0.321871	+	0.557497i
$372$		0			0
$373$	−5417.10	+	9382.69i	−0.751975	+	1.30246i	0.194890	+	0.980825i	$0.437565\pi$
$373$	−5417.10	+	9382.69i	−0.751975	+	1.30246i	−0.946864	+	0.321633i	$0.895768\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9997.01			1.36571
$378$		0			0
$379$	−9390.04			−1.27265			−0.636325	−	0.771421i	$-0.719546\pi$
$379$	−9390.04			−1.27265			−0.636325	+	0.771421i	$0.719546\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3566.61	−	6177.55i	0.475836	−	0.824173i	−0.523781	−	0.851853i	$-0.675479\pi$
$383$	3566.61	−	6177.55i	0.475836	−	0.824173i	0.999617	+	0.0276806i	$0.00881215\pi$
$384$		0			0
$385$	−925.094	−	1602.31i	−0.122460	−	0.212107i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4243.29	+	7349.59i	0.553068	+	0.957941i	0.998051	+	0.0624026i	$0.0198763\pi$
$389$	4243.29	+	7349.59i	0.553068	+	0.957941i	−0.444983	+	0.895539i	$0.646790\pi$
$390$		0			0
$391$	−8355.24	+	14471.7i	−1.08067	+	1.87178i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−955.645			−0.121731
$396$		0			0
$397$	13046.4			1.64932			0.824660	−	0.565628i	$-0.191366\pi$
$397$	13046.4			1.64932			0.824660	+	0.565628i	$0.191366\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1302.03	+	2255.19i	−0.162146	+	0.280844i	−0.935638	−	0.352961i	$-0.885175\pi$
$401$	−1302.03	+	2255.19i	−0.162146	+	0.280844i	0.773492	+	0.633806i	$0.218508\pi$
$402$		0			0
$403$	3857.77	+	6681.85i	0.476847	+	0.825923i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2435.68	+	4218.72i	0.296639	+	0.513794i
$408$		0			0
$409$	3996.52	−	6922.18i	0.483167	−	0.836870i	−0.516646	−	0.856199i	$-0.672820\pi$
$409$	3996.52	−	6922.18i	0.483167	−	0.836870i	0.999813	+	0.0193290i	$0.00615301\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3044.58			−0.362746
$414$		0			0
$415$	−3725.95			−0.440722
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−139.200	+	241.102i	−0.0162300	+	0.0281112i	−0.874026	−	0.485879i	$-0.838500\pi$
$419$	−139.200	+	241.102i	−0.0162300	+	0.0281112i	0.857796	+	0.513990i	$0.171833\pi$
$420$		0			0
$421$	429.620	+	744.124i	0.0497350	+	0.0861435i	0.889821	−	0.456309i	$-0.150829\pi$
$421$	429.620	+	744.124i	0.0497350	+	0.0861435i	−0.840086	+	0.542453i	$0.817496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6435.16	+	11146.0i	0.734473	+	1.27215i
$426$		0			0
$427$	−234.060	+	405.404i	−0.0265269	+	0.0459459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9455.34			−1.05672			−0.528362	−	0.849019i	$-0.677193\pi$
$431$	−9455.34			−1.05672			−0.528362	+	0.849019i	$0.677193\pi$
$432$		0			0
$433$	1527.61			0.169544			0.0847718	−	0.996400i	$-0.472984\pi$
$433$	1527.61			0.169544			0.0847718	+	0.996400i	$0.472984\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3042.06	−	5269.00i	0.333001	−	0.576774i
$438$		0			0
$439$	1303.03	+	2256.91i	0.141663	+	0.245368i	0.928123	−	0.372274i	$-0.121422\pi$
$439$	1303.03	+	2256.91i	0.141663	+	0.245368i	−0.786460	+	0.617641i	$0.788088\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	726.222	+	1257.85i	0.0778868	+	0.134904i	0.902338	−	0.431029i	$-0.141849\pi$
$443$	726.222	+	1257.85i	0.0778868	+	0.134904i	−0.824451	+	0.565933i	$0.808516\pi$
$444$		0			0
$445$	−2974.73	+	5152.39i	−0.316890	+	0.548869i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12241.5			−1.28666			−0.643331	−	0.765589i	$-0.722448\pi$
$449$	−12241.5			−1.28666			−0.643331	+	0.765589i	$0.722448\pi$
$450$		0			0
$451$	−2545.47			−0.265769
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1896.97	−	3285.65i	0.195454	−	0.338536i
$456$		0			0
$457$	4878.13	+	8449.16i	0.499320	+	0.864847i	1.00000		0.000785332i	$-0.000249979\pi$
$457$	4878.13	+	8449.16i	0.499320	+	0.864847i	−0.500680	+	0.865632i	$0.666917\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5122.00	+	8871.56i	0.517474	+	0.896290i	0.999794	+	0.0202956i	$0.00646074\pi$
$461$	5122.00	+	8871.56i	0.517474	+	0.896290i	−0.482321	+	0.875995i	$0.660206\pi$
$462$		0			0
$463$	6943.67	−	12026.8i	0.696976	−	1.20720i	−0.272534	−	0.962146i	$-0.587862\pi$
$463$	6943.67	−	12026.8i	0.696976	−	1.20720i	0.969510	−	0.245051i	$-0.0788048\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14367.2			1.42363			0.711815	−	0.702367i	$-0.247874\pi$
$467$	14367.2			1.42363			0.711815	+	0.702367i	$0.247874\pi$
$468$		0			0
$469$	481.694			0.0474255
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6681.57	−	11572.8i	0.649512	−	1.12499i
$474$		0			0
$475$	−2342.98	−	4058.15i	−0.226322	−	0.392002i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5185.59	−	8981.71i	−0.494647	−	0.856753i	0.505334	−	0.862924i	$-0.331369\pi$
$479$	−5185.59	−	8981.71i	−0.494647	−	0.856753i	−0.999981	+	0.00617045i	$0.998036\pi$
$480$		0			0
$481$	−4994.53	+	8650.78i	−0.473453	+	0.820045i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5411.36			−0.506634
$486$		0			0
$487$	−4805.47			−0.447139			−0.223569	−	0.974688i	$-0.571771\pi$
$487$	−4805.47			−0.447139			−0.223569	+	0.974688i	$0.571771\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8774.67	+	15198.2i	−0.806508	+	1.39691i	0.108760	+	0.994068i	$0.465312\pi$
$491$	−8774.67	+	15198.2i	−0.806508	+	1.39691i	−0.915268	+	0.402845i	$0.868021\pi$
$492$		0			0
$493$	−8760.56	−	15173.7i	−0.800316	−	1.38619i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1903.58	−	3297.09i	−0.171805	−	0.297575i
$498$		0			0
$499$	397.080	−	687.763i	0.0356227	−	0.0617004i	−0.847664	−	0.530533i	$-0.821992\pi$
$499$	397.080	−	687.763i	0.0356227	−	0.0617004i	0.883287	+	0.468832i	$0.155325\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6389.32			−0.566374			−0.283187	−	0.959065i	$-0.591392\pi$
$503$	−6389.32			−0.566374			−0.283187	+	0.959065i	$0.591392\pi$
$504$		0			0
$505$	−4985.82			−0.439338
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6935.53	−	12012.7i	0.603953	−	1.04608i	−0.388263	−	0.921549i	$-0.626925\pi$
$509$	6935.53	−	12012.7i	0.603953	−	1.04608i	0.992216	−	0.124529i	$-0.0397419\pi$
$510$		0			0
$511$	2827.27	+	4896.97i	0.244757	+	0.423932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2038.21	−	3530.28i	−0.174397	−	0.302064i
$516$		0			0
$517$	10877.7	−	18840.7i	0.925336	−	1.60273i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13682.5			−1.15056			−0.575280	−	0.817956i	$-0.695107\pi$
$521$	−13682.5			−1.15056			−0.575280	+	0.817956i	$0.695107\pi$
$522$		0			0
$523$	8390.18			0.701485			0.350743	−	0.936472i	$-0.385929\pi$
$523$	8390.18			0.701485			0.350743	+	0.936472i	$0.385929\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6761.27	−	11710.9i	0.558872	−	0.967995i
$528$		0			0
$529$	−2516.37	−	4358.48i	−0.206819	−	0.358221i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2609.84	−	4520.37i	−0.212091	−	0.367353i
$534$		0			0
$535$	1180.40	−	2044.52i	0.0953894	−	0.165219i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8135.96			0.650168
$540$		0			0
$541$	−3447.33			−0.273960			−0.136980	−	0.990574i	$-0.543740\pi$
$541$	−3447.33			−0.273960			−0.136980	+	0.990574i	$0.543740\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2215.21	−	3836.85i	0.174108	−	0.301565i
$546$		0			0
$547$	−7269.23	−	12590.7i	−0.568208	−	0.984166i	−0.996743	−	0.0806404i	$-0.974303\pi$
$547$	−7269.23	−	12590.7i	−0.568208	−	0.984166i	0.428535	−	0.903525i	$-0.359030\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3189.63	+	5524.60i	0.246611	+	0.427143i
$552$		0			0
$553$	−1039.39	+	1800.28i	−0.0799265	+	0.138437i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14894.7			−1.13305			−0.566525	−	0.824044i	$-0.691713\pi$
$557$	−14894.7			−1.13305			−0.566525	+	0.824044i	$0.691713\pi$
$558$		0			0
$559$	27402.1			2.07332
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6068.66	+	10511.2i	−0.454287	+	0.786848i	−0.998647	−	0.0520035i	$-0.983439\pi$
$563$	−6068.66	+	10511.2i	−0.454287	+	0.786848i	0.544360	+	0.838852i	$0.316773\pi$
$564$		0			0
$565$	1961.63	+	3397.64i	0.146064	+	0.252991i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−260.242	−	450.752i	−0.0191738	−	0.0332100i	0.856279	−	0.516513i	$-0.172770\pi$
$569$	−260.242	−	450.752i	−0.0191738	−	0.0332100i	−0.875453	+	0.483303i	$0.839437\pi$
$570$		0			0
$571$	2295.73	−	3976.33i	0.168255	−	0.291426i	−0.769552	−	0.638585i	$-0.779520\pi$
$571$	2295.73	−	3976.33i	0.168255	−	0.291426i	0.937806	+	0.347159i	$0.112854\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13247.1			−0.960772
$576$		0			0
$577$	−5429.84			−0.391763			−0.195881	−	0.980628i	$-0.562757\pi$
$577$	−5429.84			−0.391763			−0.195881	+	0.980628i	$0.562757\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4052.46	+	7019.07i	−0.289371	+	0.501205i
$582$		0			0
$583$	7653.68	+	13256.6i	0.543710	+	0.941734i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9893.33	−	17135.7i	−0.695641	−	1.20489i	−0.969964	−	0.243248i	$-0.921787\pi$
$587$	−9893.33	−	17135.7i	−0.695641	−	1.20489i	0.274323	−	0.961638i	$-0.411546\pi$
$588$		0			0
$589$	−2461.71	+	4263.80i	−0.172212	+	0.298280i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2181.13			−0.151042			−0.0755212	−	0.997144i	$-0.524062\pi$
$593$	−2181.13			−0.151042			−0.0755212	+	0.997144i	$0.524062\pi$
$594$		0			0
$595$	−6649.41			−0.458150
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2043.62	−	3539.65i	0.139399	−	0.241446i	−0.787870	−	0.615841i	$-0.788816\pi$
$599$	2043.62	−	3539.65i	0.139399	−	0.241446i	0.927269	+	0.374395i	$0.122150\pi$
$600$		0			0
$601$	−5842.45	−	10119.4i	−0.396537	−	0.686822i	0.596759	−	0.802420i	$-0.296455\pi$
$601$	−5842.45	−	10119.4i	−0.396537	−	0.686822i	−0.993296	+	0.115598i	$0.963121\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	181.318	+	314.052i	0.0121845	+	0.0211042i
$606$		0			0
$607$	−1865.37	+	3230.92i	−0.124733	+	0.216045i	−0.921629	−	0.388073i	$-0.873141\pi$
$607$	−1865.37	+	3230.92i	−0.124733	+	0.216045i	0.796895	+	0.604117i	$0.206474\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	44610.8			2.95378
$612$		0			0
$613$	−2259.79			−0.148894			−0.0744470	−	0.997225i	$-0.523719\pi$
$613$	−2259.79			−0.148894			−0.0744470	+	0.997225i	$0.523719\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11613.8	+	20115.7i	−0.757787	+	1.31253i	0.186190	+	0.982514i	$0.440386\pi$
$617$	−11613.8	+	20115.7i	−0.757787	+	1.31253i	−0.943977	+	0.330012i	$0.892947\pi$
$618$		0			0
$619$	4199.81	+	7274.28i	0.272705	+	0.472339i	0.969554	−	0.244879i	$-0.0787483\pi$
$619$	4199.81	+	7274.28i	0.272705	+	0.472339i	−0.696848	+	0.717218i	$0.745415\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6470.83	+	11207.8i	0.416129	+	0.720757i
$624$		0			0
$625$	−3602.02	+	6238.89i	−0.230529	+	0.399289i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17507.2			1.10979
$630$		0			0
$631$	−22475.2			−1.41795			−0.708973	−	0.705236i	$-0.750841\pi$
$631$	−22475.2			−1.41795			−0.708973	+	0.705236i	$0.750841\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4298.13	+	7444.57i	−0.268608	+	0.465242i
$636$		0			0
$637$	8341.68	+	14448.2i	0.518853	+	0.898680i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4437.04	+	7685.17i	0.273405	+	0.473551i	0.969731	−	0.244174i	$-0.0785168\pi$
$641$	4437.04	+	7685.17i	0.273405	+	0.473551i	−0.696327	+	0.717725i	$0.745184\pi$
$642$		0			0
$643$	−10676.6	+	18492.4i	−0.654810	+	1.13416i	0.327132	+	0.944979i	$0.393918\pi$
$643$	−10676.6	+	18492.4i	−0.654810	+	1.13416i	−0.981941	+	0.189185i	$0.939415\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24145.6			1.46718			0.733588	−	0.679595i	$-0.237844\pi$
$647$	24145.6			1.46718			0.733588	+	0.679595i	$0.237844\pi$
$648$		0			0
$649$	−10131.1			−0.612757
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5847.64	−	10128.4i	0.350438	−	0.606976i	−0.635889	−	0.771781i	$-0.719366\pi$
$653$	5847.64	−	10128.4i	0.350438	−	0.606976i	0.986326	+	0.164805i	$0.0526995\pi$
$654$		0			0
$655$	2703.08	+	4681.87i	0.161249	+	0.279291i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2914.45	−	5047.97i	−0.172277	−	0.298393i	0.766938	−	0.641721i	$-0.221779\pi$
$659$	−2914.45	−	5047.97i	−0.172277	−	0.298393i	−0.939216	+	0.343328i	$0.888446\pi$
$660$		0			0
$661$	810.655	−	1404.10i	0.0477017	−	0.0826218i	−0.841189	−	0.540742i	$-0.818144\pi$
$661$	810.655	−	1404.10i	0.0477017	−	0.0826218i	0.888890	+	0.458120i	$0.151477\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2420.98			0.141175
$666$		0			0
$667$	18034.1			1.04690
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−778.853	+	1349.01i	−0.0448096	+	0.0776126i
$672$		0			0
$673$	−10188.2	−	17646.5i	−0.583545	−	1.01073i	−0.995055	−	0.0993246i	$-0.968332\pi$
$673$	−10188.2	−	17646.5i	−0.583545	−	1.01073i	0.411510	−	0.911405i	$-0.365002\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5675.10	−	9829.57i	−0.322174	−	0.558022i	0.658762	−	0.752351i	$-0.271080\pi$
$677$	−5675.10	−	9829.57i	−0.322174	−	0.558022i	−0.980936	+	0.194329i	$0.937747\pi$
$678$		0			0
$679$	−5885.57	+	10194.1i	−0.332647	+	0.576162i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3508.08			0.196534			0.0982670	−	0.995160i	$-0.468670\pi$
$683$	3508.08			0.196534			0.0982670	+	0.995160i	$0.468670\pi$
$684$		0			0
$685$	10415.7			0.580968
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15694.4	+	27183.5i	−0.867793	+	1.50306i
$690$		0			0
$691$	8769.38	+	15189.0i	0.482783	+	0.836204i	0.999805	−	0.0197680i	$-0.00629275\pi$
$691$	8769.38	+	15189.0i	0.482783	+	0.836204i	−0.517022	+	0.855972i	$0.672959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5523.67	+	9567.27i	0.301474	+	0.522169i
$696$		0			0
$697$	−4574.10	+	7922.57i	−0.248574	+	0.430543i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15208.5			−0.819428			−0.409714	−	0.912214i	$-0.634371\pi$
$701$	−15208.5			−0.819428			−0.409714	+	0.912214i	$0.634371\pi$
$702$		0			0
$703$	−6374.19			−0.341973
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5422.73	+	9392.45i	−0.288462	+	0.499632i
$708$		0			0
$709$	−97.2400	−	168.425i	−0.00515081	−	0.00892146i	0.863439	−	0.504454i	$-0.168306\pi$
$709$	−97.2400	−	168.425i	−0.00515081	−	0.00892146i	−0.868589	+	0.495533i	$0.834973\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6959.23	+	12053.7i	0.365533	+	0.633122i
$714$		0			0
$715$	6312.32	−	10933.3i	0.330164	−	0.571861i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1001.11			−0.0519266			−0.0259633	−	0.999663i	$-0.508265\pi$
$719$	−1001.11			−0.0519266			−0.0259633	+	0.999663i	$0.508265\pi$
$720$		0			0
$721$	−8867.29			−0.458024
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6944.89	−	12028.9i	0.355761	−	0.616196i
$726$		0			0
$727$	7492.50	+	12977.4i	0.382231	+	0.662043i	0.991381	−	0.131012i	$-0.0418227\pi$
$727$	7492.50	+	12977.4i	0.382231	+	0.662043i	−0.609150	+	0.793055i	$0.708489\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24013.0	−	41591.6i	−1.21498	−	2.10441i
$732$		0			0
$733$	15849.0	−	27451.3i	0.798632	−	1.38327i	−0.121876	−	0.992545i	$-0.538891\pi$
$733$	15849.0	−	27451.3i	0.798632	−	1.38327i	0.920507	−	0.390725i	$-0.127776\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1602.87			0.0801120
$738$		0			0
$739$	−1333.07			−0.0663570			−0.0331785	−	0.999449i	$-0.510563\pi$
$739$	−1333.07			−0.0663570			−0.0331785	+	0.999449i	$0.510563\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1103.33	−	1911.02i	0.0544780	−	0.0943586i	−0.837500	−	0.546437i	$-0.815984\pi$
$743$	1103.33	−	1911.02i	0.0544780	−	0.0943586i	0.891978	+	0.452078i	$0.149317\pi$
$744$		0			0
$745$	−1678.68	−	2907.55i	−0.0825530	−	0.142986i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2567.69	−	4447.37i	−0.125262	−	0.216961i
$750$		0			0
$751$	8344.02	−	14452.3i	0.405430	−	0.702225i	−0.588942	−	0.808175i	$-0.700455\pi$
$751$	8344.02	−	14452.3i	0.405430	−	0.702225i	0.994371	+	0.105951i	$0.0337886\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2646.55			0.127573
$756$		0			0
$757$	17183.8			0.825039			0.412519	−	0.910949i	$-0.364649\pi$
$757$	17183.8			0.825039			0.412519	+	0.910949i	$0.364649\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9476.94	+	16414.5i	−0.451431	+	0.781901i	−0.998475	−	0.0552024i	$-0.982420\pi$
$761$	−9476.94	+	16414.5i	−0.451431	+	0.781901i	0.547044	+	0.837104i	$0.315753\pi$
$762$		0			0
$763$	−4818.67	−	8346.17i	−0.228634	−	0.396005i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10387.2	−	17991.2i	−0.488998	−	0.846969i
$768$		0			0
$769$	−3124.38	+	5411.58i	−0.146512	+	0.253767i	−0.929936	−	0.367721i	$-0.880138\pi$
$769$	−3124.38	+	5411.58i	−0.146512	+	0.253767i	0.783424	+	0.621488i	$0.213471\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4856.58			−0.225975			−0.112988	−	0.993596i	$-0.536042\pi$
$773$	−4856.58			−0.225975			−0.112988	+	0.993596i	$0.536042\pi$
$774$		0			0
$775$	10719.9			0.496866
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1665.38	−	2884.52i	0.0765962	−	0.132669i
$780$		0			0
$781$	−6334.29	−	10971.3i	−0.290216	−	0.502669i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1159.18	+	2007.77i	0.0527046	+	0.0912870i
$786$		0			0
$787$	−8239.01	+	14270.4i	−0.373175	+	0.646359i	−0.990052	−	0.140701i	$-0.955064\pi$
$787$	−8239.01	+	14270.4i	−0.373175	+	0.646359i	0.616877	+	0.787060i	$0.288398\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	8534.13			0.383614
$792$		0			0
$793$	−3194.19			−0.143038
$794$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.44901 + 4.24182i) q^{5} +(5.32725 + 9.22708i) q^{7} +O(q^{10})\)	\(q+(-2.44901 + 4.24182i) q^{5} +(5.32725 + 9.22708i) q^{7} +(17.7268 + 30.7038i) q^{11} +(-36.3501 + 62.9603i) q^{13} +127.417 q^{17} -46.3913 q^{19} +(-65.5739 + 113.577i) q^{23} +(50.5047 + 87.4766i) q^{25} +(-68.7549 - 119.087i) q^{29} +(53.0640 - 91.9096i) q^{31} -52.1861 q^{35} +137.401 q^{37} +(-35.8986 + 62.1782i) q^{41} +(-188.459 - 326.421i) q^{43} +(-306.813 - 531.416i) q^{47} +(114.741 - 198.737i) q^{49} +431.757 q^{53} -173.653 q^{55} +(-142.878 + 247.471i) q^{59} +(21.9682 + 38.0500i) q^{61} +(-178.044 - 308.381i) q^{65} +(22.6052 - 39.1533i) q^{67} -357.328 q^{71} +530.718 q^{73} +(-188.871 + 327.134i) q^{77} +(97.5540 + 168.969i) q^{79} +(380.352 + 658.789i) q^{83} +(-312.047 + 540.480i) q^{85} +1214.67 q^{89} -774.586 q^{91} +(113.613 - 196.783i) q^{95} +(552.402 + 956.788i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} - 6 q^{7}+O(q^{10}) \) 6 * q - 6 * q^5 - 6 * q^7	\( 6 q - 6 q^{5} - 6 q^{7} - 51 q^{11} + 12 q^{13} + 222 q^{17} + 30 q^{19} - 210 q^{23} - 3 q^{25} - 456 q^{29} + 48 q^{31} + 1104 q^{35} - 96 q^{37} - 897 q^{41} + 129 q^{43} - 522 q^{47} - 225 q^{49} + 2208 q^{53} - 216 q^{55} - 453 q^{59} - 402 q^{61} - 1110 q^{65} - 213 q^{67} - 120 q^{71} + 750 q^{73} - 1128 q^{77} + 552 q^{79} + 612 q^{83} + 1188 q^{85} + 924 q^{89} - 264 q^{91} + 2184 q^{95} + 93 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 - 6 * q^7 - 51 * q^11 + 12 * q^13 + 222 * q^17 + 30 * q^19 - 210 * q^23 - 3 * q^25 - 456 * q^29 + 48 * q^31 + 1104 * q^35 - 96 * q^37 - 897 * q^41 + 129 * q^43 - 522 * q^47 - 225 * q^49 + 2208 * q^53 - 216 * q^55 - 453 * q^59 - 402 * q^61 - 1110 * q^65 - 213 * q^67 - 120 * q^71 + 750 * q^73 - 1128 * q^77 + 552 * q^79 + 612 * q^83 + 1188 * q^85 + 924 * q^89 - 264 * q^91 + 2184 * q^95 + 93 * q^97

Introduction

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