LMFDB - Embedded newform 572.2.f.c.441.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (of order $2$, degree $1$, 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:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 $ x^8 + 21x^6 + 136x^4 + 309*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.4
Root			$1.43491i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.c.441.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43491 q^{3} +4.37595i q^{5} +2.43491i q^{7} -0.941037 q^{9} +O(q^{10})$	$q-1.43491 q^{3} +4.37595i q^{5} +2.43491i q^{7} -0.941037 q^{9} +1.00000i q^{11} +(-0.222720 - 3.59867i) q^{13} -6.27908i q^{15} -5.19733 q^{17} -4.25629i q^{19} -3.49387i q^{21} +6.81085 q^{23} -14.1489 q^{25} +5.65503 q^{27} -4.42438 q^{29} -1.62405i q^{31} -1.43491i q^{33} -10.6550 q^{35} +3.50613i q^{37} +(0.319583 + 5.16376i) q^{39} +9.53538i q^{41} -6.42438 q^{43} -4.11793i q^{45} +1.13018i q^{47} +1.07122 q^{49} +7.45770 q^{51} -1.71399 q^{53} -4.37595 q^{55} +6.10739i q^{57} -3.60299i q^{59} -8.42438 q^{61} -2.29134i q^{63} +(15.7476 - 0.974611i) q^{65} +8.01872i q^{67} -9.77295 q^{69} +0.279083i q^{71} +6.07122i q^{73} +20.3024 q^{75} -2.43491 q^{77} -3.10047 q^{79} -5.29134 q^{81} +2.88035i q^{83} -22.7432i q^{85} +6.34858 q^{87} +6.11558i q^{89} +(8.76242 - 0.542303i) q^{91} +2.33037i q^{93} +18.6253 q^{95} -12.3759i q^{97} -0.941037i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 + 18 * q^9	$ 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) $ 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.43491			−0.828445			−0.414222	−	0.910176i	$-0.635946\pi$
$3$	−1.43491			−0.828445			−0.414222	+	0.910176i	$0.635946\pi$
$4$		0			0
$5$			4.37595i			1.95698i	0.206289	+	0.978491i	$0.433861\pi$
$5$			4.37595i			1.95698i	−0.206289	+	0.978491i	$0.566139\pi$
$6$		0			0
$7$			2.43491i			0.920309i	0.887839	+	0.460155i	$0.152206\pi$
$7$			2.43491i			0.920309i	−0.887839	+	0.460155i	$0.847794\pi$
$8$		0			0
$9$	−0.941037			−0.313679
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−0.222720	−	3.59867i	−0.0617715	−	0.998090i
$13$	−0.222720	−	3.59867i	−0.0617715	−	0.998090i
$14$		0			0
$15$		−	6.27908i		−	1.62125i
$16$		0			0
$17$	−5.19733			−1.26054			−0.630269	−	0.776377i	$-0.717055\pi$
$17$	−5.19733			−1.26054			−0.630269	+	0.776377i	$0.717055\pi$
$18$		0			0
$19$		−	4.25629i		−	0.976461i	−0.872715	−	0.488230i	$-0.837643\pi$
$19$		−	4.25629i		−	0.976461i	0.872715	−	0.488230i	$-0.162357\pi$
$20$		0			0
$21$		−	3.49387i		−	0.762425i
$22$		0			0
$23$	6.81085			1.42016			0.710081	−	0.704120i	$-0.248658\pi$
$23$	6.81085			1.42016			0.710081	+	0.704120i	$0.248658\pi$
$24$		0			0
$25$	−14.1489			−2.82978
$26$		0			0
$27$	5.65503			1.08831
$28$		0			0
$29$	−4.42438			−0.821586			−0.410793	−	0.911729i	$-0.634748\pi$
$29$	−4.42438			−0.821586			−0.410793	+	0.911729i	$0.634748\pi$
$30$		0			0
$31$		−	1.62405i		−	0.291689i	−0.989308	−	0.145844i	$-0.953410\pi$
$31$		−	1.62405i		−	0.291689i	0.989308	−	0.145844i	$-0.0465899\pi$
$32$		0			0
$33$		−	1.43491i		−	0.249786i
$34$		0			0
$35$	−10.6550			−1.80103
$36$		0			0
$37$			3.50613i			0.576404i	0.957570	+	0.288202i	$0.0930574\pi$
$37$			3.50613i			0.576404i	−0.957570	+	0.288202i	$0.906943\pi$
$38$		0			0
$39$	0.319583	+	5.16376i	0.0511743	+	0.826863i
$40$		0			0
$41$			9.53538i			1.48918i	0.667524	+	0.744588i	$0.267354\pi$
$41$			9.53538i			1.48918i	−0.667524	+	0.744588i	$0.732646\pi$
$42$		0			0
$43$	−6.42438			−0.979708			−0.489854	−	0.871804i	$-0.662950\pi$
$43$	−6.42438			−0.979708			−0.489854	+	0.871804i	$0.662950\pi$
$44$		0			0
$45$		−	4.11793i		−	0.613864i
$46$		0			0
$47$			1.13018i			0.164854i	0.996597	+	0.0824270i	$0.0262671\pi$
$47$			1.13018i			0.164854i	−0.996597	+	0.0824270i	$0.973733\pi$
$48$		0			0
$49$	1.07122			0.153031
$50$		0			0
$51$	7.45770			1.04429
$52$		0			0
$53$	−1.71399			−0.235435			−0.117717	−	0.993047i	$-0.537558\pi$
$53$	−1.71399			−0.235435			−0.117717	+	0.993047i	$0.537558\pi$
$54$		0			0
$55$	−4.37595			−0.590052
$56$		0			0
$57$			6.10739i			0.808944i
$58$		0			0
$59$		−	3.60299i		−	0.469069i	−0.972108	−	0.234535i	$-0.924643\pi$
$59$		−	3.60299i		−	0.469069i	0.972108	−	0.234535i	$-0.0753567\pi$
$60$		0			0
$61$	−8.42438			−1.07863			−0.539315	−	0.842104i	$-0.681317\pi$
$61$	−8.42438			−1.07863			−0.539315	+	0.842104i	$0.681317\pi$
$62$		0			0
$63$		−	2.29134i		−	0.288682i
$64$		0			0
$65$	15.7476	−	0.974611i	1.95325	−	0.120886i
$66$		0			0
$67$			8.01872i			0.979642i	0.871823	+	0.489821i	$0.162938\pi$
$67$			8.01872i			0.979642i	−0.871823	+	0.489821i	$0.837062\pi$
$68$		0			0
$69$	−9.77295			−1.17653
$70$		0			0
$71$			0.279083i			0.0331210i	0.999863	+	0.0165605i	$0.00527161\pi$
$71$			0.279083i			0.0331210i	−0.999863	+	0.0165605i	$0.994728\pi$
$72$		0			0
$73$			6.07122i			0.710582i	0.934756	+	0.355291i	$0.115618\pi$
$73$			6.07122i			0.710582i	−0.934756	+	0.355291i	$0.884382\pi$
$74$		0			0
$75$	20.3024			2.34432
$76$		0			0
$77$	−2.43491			−0.277484
$78$		0			0
$79$	−3.10047			−0.348830			−0.174415	−	0.984672i	$-0.555803\pi$
$79$	−3.10047			−0.348830			−0.174415	+	0.984672i	$0.555803\pi$
$80$		0			0
$81$	−5.29134			−0.587927
$82$		0			0
$83$			2.88035i			0.316159i	0.987426	+	0.158080i	$0.0505303\pi$
$83$			2.88035i			0.316159i	−0.987426	+	0.158080i	$0.949470\pi$
$84$		0			0
$85$		−	22.7432i		−	2.46685i
$86$		0			0
$87$	6.34858			0.680639
$88$		0			0
$89$			6.11558i			0.648250i	0.946014	+	0.324125i	$0.105070\pi$
$89$			6.11558i			0.648250i	−0.946014	+	0.324125i	$0.894930\pi$
$90$		0			0
$91$	8.76242	−	0.542303i	0.918552	−	0.0568488i
$92$		0			0
$93$			2.33037i			0.241648i
$94$		0			0
$95$	18.6253			1.91092
$96$		0			0
$97$		−	12.3759i		−	1.25659i	−0.777976	−	0.628293i	$-0.783754\pi$
$97$		−	12.3759i		−	1.25659i	0.777976	−	0.628293i	$-0.216246\pi$
$98$		0			0
$99$		−	0.941037i		−	0.0945778i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	18.7327			1.84579			0.922894	−	0.385053i	$-0.125817\pi$
$103$	18.7327			1.84579			0.922894	+	0.385053i	$0.125817\pi$
$104$		0			0
$105$	15.2890			1.49205
$106$		0			0
$107$	−8.32751			−0.805051			−0.402526	−	0.915409i	$-0.631868\pi$
$107$	−8.32751			−0.805051			−0.402526	+	0.915409i	$0.631868\pi$
$108$		0			0
$109$		−	1.38648i		−	0.132800i	−0.997793	−	0.0664002i	$-0.978849\pi$
$109$		−	1.38648i		−	0.132800i	0.997793	−	0.0664002i	$-0.0211514\pi$
$110$		0			0
$111$		−	5.03097i		−	0.477519i
$112$		0			0
$113$	12.4472			1.17093			0.585465	−	0.810697i	$-0.300912\pi$
$113$	12.4472			1.17093			0.585465	+	0.810697i	$0.300912\pi$
$114$		0			0
$115$			29.8039i			2.77923i
$116$		0			0
$117$	0.209588	+	3.38648i	0.0193764	+	0.313080i
$118$		0			0
$119$		−	12.6550i		−	1.16008i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	13.6824i		−	1.23370i
$124$		0			0
$125$		−	40.0351i		−	3.58085i
$126$		0			0
$127$	−0.890881			−0.0790529			−0.0395264	−	0.999219i	$-0.512585\pi$
$127$	−0.890881			−0.0790529			−0.0395264	+	0.999219i	$0.512585\pi$
$128$		0			0
$129$	9.21839			0.811634
$130$		0			0
$131$	−11.0123			−0.962145			−0.481073	−	0.876681i	$-0.659753\pi$
$131$	−11.0123			−0.962145			−0.481073	+	0.876681i	$0.659753\pi$
$132$		0			0
$133$	10.3637			0.898646
$134$		0			0
$135$			24.7461i			2.12980i
$136$		0			0
$137$			22.5769i			1.92887i	0.264314	+	0.964437i	$0.414854\pi$
$137$			22.5769i			1.92887i	−0.264314	+	0.964437i	$0.585146\pi$
$138$		0			0
$139$	−18.6550			−1.58230			−0.791149	−	0.611623i	$-0.790517\pi$
$139$	−18.6550			−1.58230			−0.791149	+	0.611623i	$0.790517\pi$
$140$		0			0
$141$		−	1.62171i		−	0.136573i
$142$		0			0
$143$	3.59867	−	0.222720i	0.300936	−	0.0186248i
$144$		0			0
$145$		−	19.3608i		−	1.60783i
$146$		0			0
$147$	−1.53710			−0.126778
$148$		0			0
$149$			18.0566i			1.47926i	0.673016	+	0.739628i	$0.264998\pi$
$149$			18.0566i			1.47926i	−0.673016	+	0.739628i	$0.735002\pi$
$150$		0			0
$151$			9.00361i			0.732703i	0.930477	+	0.366352i	$0.119393\pi$
$151$			9.00361i			0.732703i	−0.930477	+	0.366352i	$0.880607\pi$
$152$		0			0
$153$	4.89088			0.395404
$154$		0			0
$155$	7.10677			0.570830
$156$		0			0
$157$	−3.84417			−0.306798			−0.153399	−	0.988164i	$-0.549022\pi$
$157$	−3.84417			−0.306798			−0.153399	+	0.988164i	$0.549022\pi$
$158$		0			0
$159$	2.45942			0.195045
$160$		0			0
$161$			16.5838i			1.30699i
$162$		0			0
$163$			12.3947i			0.970825i	0.874285	+	0.485412i	$0.161331\pi$
$163$			12.3947i			0.970825i	−0.874285	+	0.485412i	$0.838669\pi$
$164$		0			0
$165$	6.27908			0.488826
$166$		0			0
$167$			11.6171i			0.898960i	0.893290	+	0.449480i	$0.148391\pi$
$167$			11.6171i			0.898960i	−0.893290	+	0.449480i	$0.851609\pi$
$168$		0			0
$169$	−12.9008	+	1.60299i	−0.992369	+	0.123307i
$170$		0			0
$171$			4.00533i			0.306295i
$172$		0			0
$173$	18.9370			1.43975			0.719875	−	0.694103i	$-0.244199\pi$
$173$	18.9370			1.43975			0.719875	+	0.694103i	$0.244199\pi$
$174$		0			0
$175$		−	34.4513i		−	2.60427i
$176$		0			0
$177$			5.16996i			0.388598i
$178$		0			0
$179$	20.7917			1.55404			0.777021	−	0.629474i	$-0.216730\pi$
$179$	20.7917			1.55404			0.777021	+	0.629474i	$0.216730\pi$
$180$		0			0
$181$	−20.4846			−1.52261			−0.761304	−	0.648395i	$-0.775441\pi$
$181$	−20.4846			−1.52261			−0.761304	+	0.648395i	$0.775441\pi$
$182$		0			0
$183$	12.0882			0.893586
$184$		0			0
$185$	−15.3426			−1.12801
$186$		0			0
$187$		−	5.19733i		−	0.380067i
$188$		0			0
$189$			13.7695i			1.00158i
$190$		0			0
$191$	4.95975			0.358875			0.179438	−	0.983769i	$-0.442572\pi$
$191$	4.95975			0.358875			0.179438	+	0.983769i	$0.442572\pi$
$192$		0			0
$193$			13.2837i			0.956179i	0.878311	+	0.478089i	$0.158671\pi$
$193$			13.2837i			0.956179i	−0.878311	+	0.478089i	$0.841329\pi$
$194$		0			0
$195$	−22.5963	+	1.39848i	−1.61816	+	0.100147i
$196$		0			0
$197$		−	13.5961i		−	0.968680i	−0.874880	−	0.484340i	$-0.839060\pi$
$197$		−	13.5961i		−	0.968680i	0.874880	−	0.484340i	$-0.160940\pi$
$198$		0			0
$199$	−22.9785			−1.62890			−0.814450	−	0.580233i	$-0.802961\pi$
$199$	−22.9785			−1.62890			−0.814450	+	0.580233i	$0.802961\pi$
$200$		0			0
$201$		−	11.5061i		−	0.811580i
$202$		0			0
$203$		−	10.7730i		−	0.756113i
$204$		0			0
$205$	−41.7263			−2.91429
$206$		0			0
$207$	−6.40927			−0.445475
$208$		0			0
$209$	4.25629			0.294414
$210$		0			0
$211$	21.1466			1.45579			0.727894	−	0.685689i	$-0.240499\pi$
$211$	21.1466			1.45579			0.727894	+	0.685689i	$0.240499\pi$
$212$		0			0
$213$		−	0.400458i		−	0.0274389i
$214$		0			0
$215$		−	28.1127i		−	1.91727i
$216$		0			0
$217$	3.95442			0.268444
$218$		0			0
$219$		−	8.71165i		−	0.588678i
$220$		0			0
$221$	1.15755	+	18.7035i	0.0778653	+	1.25813i
$222$		0			0
$223$			23.4256i			1.56870i	0.620321	+	0.784348i	$0.287002\pi$
$223$			23.4256i			1.56870i	−0.620321	+	0.784348i	$0.712998\pi$
$224$		0			0
$225$	13.3146			0.887642
$226$		0			0
$227$		−	29.0082i		−	1.92534i	−0.270676	−	0.962670i	$-0.587247\pi$
$227$		−	29.0082i		−	1.92534i	0.270676	−	0.962670i	$-0.412753\pi$
$228$		0			0
$229$			9.58193i			0.633192i	0.948561	+	0.316596i	$0.102540\pi$
$229$			9.58193i			0.633192i	−0.948561	+	0.316596i	$0.897460\pi$
$230$		0			0
$231$	3.49387			0.229880
$232$		0			0
$233$	23.2645			1.52411			0.762053	−	0.647514i	$-0.224191\pi$
$233$	23.2645			1.52411			0.762053	+	0.647514i	$0.224191\pi$
$234$		0			0
$235$	−4.94562			−0.322616
$236$		0			0
$237$	4.44889			0.288986
$238$		0			0
$239$		−	8.22658i		−	0.532133i	−0.963955	−	0.266067i	$-0.914276\pi$
$239$		−	8.22658i		−	0.532133i	0.963955	−	0.266067i	$-0.0857241\pi$
$240$		0			0
$241$		−	6.40519i		−	0.412595i	−0.978489	−	0.206297i	$-0.933859\pi$
$241$		−	6.40519i		−	0.412595i	0.978489	−	0.206297i	$-0.0661414\pi$
$242$		0			0
$243$	−9.37250			−0.601246
$244$		0			0
$245$			4.68760i			0.299480i
$246$		0			0
$247$	−15.3170	+	0.947963i	−0.974596	+	0.0603174i
$248$		0			0
$249$		−	4.13304i		−	0.261921i
$250$		0			0
$251$	−10.8319			−0.683705			−0.341852	−	0.939754i	$-0.611054\pi$
$251$	−10.8319			−0.683705			−0.341852	+	0.939754i	$0.611054\pi$
$252$		0			0
$253$			6.81085i			0.428195i
$254$		0			0
$255$			32.6345i			2.04365i
$256$		0			0
$257$	−18.2959			−1.14127			−0.570634	−	0.821204i	$-0.693303\pi$
$257$	−18.2959			−1.14127			−0.570634	+	0.821204i	$0.693303\pi$
$258$		0			0
$259$	−8.53710			−0.530470
$260$		0			0
$261$	4.16350			0.257714
$262$		0			0
$263$	8.91539			0.549747			0.274873	−	0.961480i	$-0.411364\pi$
$263$	8.91539			0.549747			0.274873	+	0.961480i	$0.411364\pi$
$264$		0			0
$265$		−	7.50033i		−	0.460742i
$266$		0			0
$267$		−	8.77530i		−	0.537040i
$268$		0			0
$269$	1.64089			0.100047			0.0500234	−	0.998748i	$-0.484070\pi$
$269$	1.64089			0.100047			0.0500234	+	0.998748i	$0.484070\pi$
$270$		0			0
$271$		−	25.6335i		−	1.55712i	−0.627568	−	0.778562i	$-0.715949\pi$
$271$		−	25.6335i		−	1.55712i	0.627568	−	0.778562i	$-0.284051\pi$
$272$		0			0
$273$	−12.5733	+	0.778156i	−0.760969	+	0.0470961i
$274$		0			0
$275$		−	14.1489i		−	0.853211i
$276$		0			0
$277$	20.5582			1.23522			0.617610	−	0.786484i	$-0.288101\pi$
$277$	20.5582			1.23522			0.617610	+	0.786484i	$0.288101\pi$
$278$		0			0
$279$			1.52830i			0.0914966i
$280$		0			0
$281$			5.49575i			0.327849i	0.986473	+	0.163925i	$0.0524154\pi$
$281$			5.49575i			0.327849i	−0.986473	+	0.163925i	$0.947585\pi$
$282$		0			0
$283$	−11.4121			−0.678380			−0.339190	−	0.940718i	$-0.610153\pi$
$283$	−11.4121			−0.678380			−0.339190	+	0.940718i	$0.610153\pi$
$284$		0			0
$285$	−26.7256			−1.58309
$286$		0			0
$287$	−23.2178			−1.37050
$288$		0			0
$289$	10.0123			0.588956
$290$		0			0
$291$			17.7584i			1.04101i
$292$		0			0
$293$			3.39106i			0.198108i	0.995082	+	0.0990538i	$0.0315816\pi$
$293$			3.39106i			0.198108i	−0.995082	+	0.0990538i	$0.968418\pi$
$294$		0			0
$295$	15.7665			0.917961
$296$		0			0
$297$			5.65503i			0.328138i
$298$		0			0
$299$	−1.51691	−	24.5100i	−0.0877254	−	1.41745i
$300$		0			0
$301$		−	15.6428i		−	0.901634i
$302$		0			0
$303$	8.60945			0.494600
$304$		0			0
$305$		−	36.8646i		−	2.11086i
$306$		0			0
$307$			6.45409i			0.368354i	0.982893	+	0.184177i	$0.0589620\pi$
$307$			6.45409i			0.368354i	−0.982893	+	0.184177i	$0.941038\pi$
$308$		0			0
$309$	−26.8797			−1.52913
$310$		0			0
$311$	13.5750			0.769768			0.384884	−	0.922965i	$-0.374241\pi$
$311$	13.5750			0.769768			0.384884	+	0.922965i	$0.374241\pi$
$312$		0			0
$313$	11.8989			0.672566			0.336283	−	0.941761i	$-0.390830\pi$
$313$	11.8989			0.672566			0.336283	+	0.941761i	$0.390830\pi$
$314$		0			0
$315$	10.0268			0.564945
$316$		0			0
$317$			12.5394i			0.704286i	0.935946	+	0.352143i	$0.114547\pi$
$317$			12.5394i			0.704286i	−0.935946	+	0.352143i	$0.885453\pi$
$318$		0			0
$319$		−	4.42438i		−	0.247718i
$320$		0			0
$321$	11.9492			0.666941
$322$		0			0
$323$			22.1214i			1.23087i
$324$		0			0
$325$	3.15125	+	50.9172i	0.174800	+	2.82438i
$326$		0			0
$327$			1.98947i			0.110018i
$328$		0			0
$329$	−2.75189			−0.151717
$330$		0			0
$331$			14.8675i			0.817190i	0.912716	+	0.408595i	$0.133981\pi$
$331$			14.8675i			0.817190i	−0.912716	+	0.408595i	$0.866019\pi$
$332$		0			0
$333$		−	3.29940i		−	0.180806i
$334$		0			0
$335$	−35.0895			−1.91714
$336$		0			0
$337$	−14.0882			−0.767434			−0.383717	−	0.923451i	$-0.625356\pi$
$337$	−14.0882			−0.767434			−0.383717	+	0.923451i	$0.625356\pi$
$338$		0			0
$339$	−17.8605			−0.970052
$340$		0			0
$341$	1.62405			0.0879475
$342$		0			0
$343$			19.6527i			1.06115i
$344$		0			0
$345$		−	42.7659i		−	2.30244i
$346$		0			0
$347$	16.4244			0.881707			0.440853	−	0.897579i	$-0.354676\pi$
$347$	16.4244			0.881707			0.440853	+	0.897579i	$0.354676\pi$
$348$		0			0
$349$			14.2662i			0.763652i	0.924234	+	0.381826i	$0.124705\pi$
$349$			14.2662i			0.763652i	−0.924234	+	0.381826i	$0.875295\pi$
$350$		0			0
$351$	−1.25949	−	20.3506i	−0.0672265	−	1.08623i
$352$		0			0
$353$		−	3.29134i		−	0.175180i	−0.996157	−	0.0875901i	$-0.972083\pi$
$353$		−	3.29134i		−	0.175180i	0.996157	−	0.0875901i	$-0.0279166\pi$
$354$		0			0
$355$	−1.22125			−0.0648173
$356$		0			0
$357$			18.1588i			0.961066i
$358$		0			0
$359$			30.0058i			1.58365i	0.610750	+	0.791824i	$0.290868\pi$
$359$			30.0058i			1.58365i	−0.610750	+	0.791824i	$0.709132\pi$
$360$		0			0
$361$	0.883955			0.0465239
$362$		0			0
$363$	1.43491			0.0753132
$364$		0			0
$365$	−26.5673			−1.39060
$366$		0			0
$367$	28.3877			1.48183			0.740914	−	0.671600i	$-0.234393\pi$
$367$	28.3877			1.48183			0.740914	+	0.671600i	$0.234393\pi$
$368$		0			0
$369$		−	8.97314i		−	0.467123i
$370$		0			0
$371$		−	4.17341i		−	0.216673i
$372$		0			0
$373$	−36.0164			−1.86486			−0.932429	−	0.361354i	$-0.882314\pi$
$373$	−36.0164			−1.86486			−0.932429	+	0.361354i	$0.882314\pi$
$374$		0			0
$375$			57.4467i			2.96653i
$376$		0			0
$377$	0.985398	+	15.9219i	0.0507506	+	0.820017i
$378$		0			0
$379$		−	20.0643i		−	1.03063i	−0.857000	−	0.515317i	$-0.827674\pi$
$379$		−	20.0643i		−	1.03063i	0.857000	−	0.515317i	$-0.172326\pi$
$380$		0			0
$381$	1.27833			0.0654910
$382$		0			0
$383$		−	6.35488i		−	0.324719i	−0.986732	−	0.162360i	$-0.948090\pi$
$383$		−	6.35488i		−	0.324719i	0.986732	−	0.162360i	$-0.0519105\pi$
$384$		0			0
$385$		−	10.6550i		−	0.543031i
$386$		0			0
$387$	6.04558			0.307314
$388$		0			0
$389$	−4.30473			−0.218258			−0.109129	−	0.994028i	$-0.534806\pi$
$389$	−4.30473			−0.218258			−0.109129	+	0.994028i	$0.534806\pi$
$390$		0			0
$391$	−35.3983			−1.79017
$392$		0			0
$393$	15.8016			0.797084
$394$		0			0
$395$		−	13.5675i		−	0.682654i
$396$		0			0
$397$		−	6.45409i		−	0.323922i	−0.986797	−	0.161961i	$-0.948218\pi$
$397$		−	6.45409i		−	0.323922i	0.986797	−	0.161961i	$-0.0517818\pi$
$398$		0			0
$399$	−14.8709			−0.744479
$400$		0			0
$401$			19.9406i			0.995785i	0.867239	+	0.497892i	$0.165893\pi$
$401$			19.9406i			0.995785i	−0.867239	+	0.497892i	$0.834107\pi$
$402$		0			0
$403$	−5.84443	+	0.361710i	−0.291132	+	0.0180180i
$404$		0			0
$405$		−	23.1546i		−	1.15056i
$406$		0			0
$407$	−3.50613			−0.173792
$408$		0			0
$409$		−	9.10912i		−	0.450417i	−0.974311	−	0.225208i	$-0.927694\pi$
$409$		−	9.10912i		−	0.450417i	0.974311	−	0.225208i	$-0.0723063\pi$
$410$		0			0
$411$		−	32.3958i		−	1.59797i
$412$		0			0
$413$	8.77295			0.431689
$414$		0			0
$415$	−12.6043			−0.618718
$416$		0			0
$417$	26.7683			1.31085
$418$		0			0
$419$	−27.3667			−1.33695			−0.668475	−	0.743735i	$-0.733053\pi$
$419$	−27.3667			−1.33695			−0.668475	+	0.743735i	$0.733053\pi$
$420$		0			0
$421$			21.7641i			1.06072i	0.847773	+	0.530360i	$0.177943\pi$
$421$			21.7641i			1.06072i	−0.847773	+	0.530360i	$0.822057\pi$
$422$		0			0
$423$		−	1.06354i		−	0.0517113i
$424$		0			0
$425$	73.5365			3.56705
$426$		0			0
$427$		−	20.5126i		−	0.992674i
$428$		0			0
$429$	−5.16376	+	0.319583i	−0.249309	+	0.0154296i
$430$		0			0
$431$		−	18.1238i		−	0.872991i	−0.899706	−	0.436496i	$-0.856219\pi$
$431$		−	18.1238i		−	0.872991i	0.899706	−	0.436496i	$-0.143781\pi$
$432$		0			0
$433$	19.1607			0.920804			0.460402	−	0.887711i	$-0.347705\pi$
$433$	19.1607			0.920804			0.460402	+	0.887711i	$0.347705\pi$
$434$		0			0
$435$			27.7810i			1.33200i
$436$		0			0
$437$		−	28.9890i		−	1.38673i
$438$		0			0
$439$	−5.60014			−0.267280			−0.133640	−	0.991030i	$-0.542667\pi$
$439$	−5.60014			−0.267280			−0.133640	+	0.991030i	$0.542667\pi$
$440$		0			0
$441$	−1.00806			−0.0480027
$442$		0			0
$443$	14.2143			0.675343			0.337671	−	0.941264i	$-0.390361\pi$
$443$	14.2143			0.675343			0.337671	+	0.941264i	$0.390361\pi$
$444$		0			0
$445$	−26.7614			−1.26861
$446$		0			0
$447$		−	25.9096i		−	1.22548i
$448$		0			0
$449$			25.3426i			1.19599i	0.801499	+	0.597996i	$0.204036\pi$
$449$			25.3426i			1.19599i	−0.801499	+	0.597996i	$0.795964\pi$
$450$		0			0
$451$	−9.53538			−0.449003
$452$		0			0
$453$		−	12.9194i		−	0.607004i
$454$		0			0
$455$	2.37309	+	38.3439i	0.111252	+	1.79759i
$456$		0			0
$457$		−	25.5505i		−	1.19520i	−0.801794	−	0.597601i	$-0.796121\pi$
$457$		−	25.5505i		−	1.19520i	0.801794	−	0.597601i	$-0.203879\pi$
$458$		0			0
$459$	−29.3911			−1.37186
$460$		0			0
$461$		−	31.3654i		−	1.46083i	−0.683002	−	0.730416i	$-0.739326\pi$
$461$		−	31.3654i		−	1.46083i	0.683002	−	0.730416i	$-0.260674\pi$
$462$		0			0
$463$		−	18.0643i		−	0.839519i	−0.907635	−	0.419759i	$-0.862114\pi$
$463$		−	18.0643i		−	0.839519i	0.907635	−	0.419759i	$-0.137886\pi$
$464$		0			0
$465$	−10.1976			−0.472901
$466$		0			0
$467$	27.8039			1.28661			0.643306	−	0.765609i	$-0.277562\pi$
$467$	27.8039			1.28661			0.643306	+	0.765609i	$0.277562\pi$
$468$		0			0
$469$	−19.5248			−0.901574
$470$		0			0
$471$	5.51604			0.254166
$472$		0			0
$473$		−	6.42438i		−	0.295393i
$474$		0			0
$475$			60.2219i			2.76317i
$476$		0			0
$477$	1.61293			0.0738510
$478$		0			0
$479$			18.0245i			0.823561i	0.911283	+	0.411780i	$0.135093\pi$
$479$			18.0245i			0.823561i	−0.911283	+	0.411780i	$0.864907\pi$
$480$		0			0
$481$	12.6174	−	0.780886i	0.575303	−	0.0356053i
$482$		0			0
$483$		−	23.7963i		−	1.08277i
$484$		0			0
$485$	54.1565			2.45912
$486$		0			0
$487$		−	16.9854i		−	0.769682i	−0.922983	−	0.384841i	$-0.874256\pi$
$487$		−	16.9854i		−	0.769682i	0.922983	−	0.384841i	$-0.125744\pi$
$488$		0			0
$489$		−	17.7852i		−	0.804275i
$490$		0			0
$491$	23.6638			1.06793			0.533967	−	0.845505i	$-0.320701\pi$
$491$	23.6638			1.06793			0.533967	+	0.845505i	$0.320701\pi$
$492$		0			0
$493$	22.9950			1.03564
$494$		0			0
$495$	4.11793			0.185087
$496$		0			0
$497$	−0.679541			−0.0304816
$498$		0			0
$499$		−	22.1343i		−	0.990867i	−0.868646	−	0.495434i	$-0.835009\pi$
$499$		−	22.1343i		−	0.990867i	0.868646	−	0.495434i	$-0.164991\pi$
$500$		0			0
$501$		−	16.6695i		−	0.744739i
$502$		0			0
$503$	36.9283			1.64655			0.823276	−	0.567641i	$-0.192144\pi$
$503$	36.9283			1.64655			0.823276	+	0.567641i	$0.192144\pi$
$504$		0			0
$505$		−	26.2557i		−	1.16836i
$506$		0			0
$507$	18.5115	−	2.30015i	0.822123	−	0.102153i
$508$		0			0
$509$		−	3.74543i		−	0.166013i	−0.996549	−	0.0830066i	$-0.973548\pi$
$509$		−	3.74543i		−	0.166013i	0.996549	−	0.0830066i	$-0.0264523\pi$
$510$		0			0
$511$	−14.7829			−0.653955
$512$		0			0
$513$		−	24.0695i		−	1.06269i
$514$		0			0
$515$			81.9733i			3.61218i
$516$		0			0
$517$	−1.13018			−0.0497054
$518$		0			0
$519$	−27.1728			−1.19275
$520$		0			0
$521$	−15.9785			−0.700030			−0.350015	−	0.936744i	$-0.613823\pi$
$521$	−15.9785			−0.700030			−0.350015	+	0.936744i	$0.613823\pi$
$522$		0			0
$523$	30.5621			1.33639			0.668194	−	0.743987i	$-0.267068\pi$
$523$	30.5621			1.33639			0.668194	+	0.743987i	$0.267068\pi$
$524$		0			0
$525$			49.4344i			2.15750i
$526$		0			0
$527$			8.44075i			0.367685i
$528$		0			0
$529$	23.3877			1.01686
$530$		0			0
$531$			3.39055i			0.147137i
$532$		0			0
$533$	34.3146	−	2.12372i	1.48633	−	0.0919886i
$534$		0			0
$535$		−	36.4407i		−	1.57547i
$536$		0			0
$537$	−29.8342			−1.28744
$538$		0			0
$539$			1.07122i			0.0461407i
$540$		0			0
$541$			29.2785i			1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$541$			29.2785i			1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$542$		0			0
$543$	29.3935			1.26140
$544$		0			0
$545$	6.06715			0.259888
$546$		0			0
$547$	17.8610			0.763682			0.381841	−	0.924228i	$-0.375290\pi$
$547$	17.8610			0.763682			0.381841	+	0.924228i	$0.375290\pi$
$548$		0			0
$549$	7.92765			0.338344
$550$		0			0
$551$			18.8315i			0.802247i
$552$		0			0
$553$		−	7.54936i		−	0.321031i
$554$		0			0
$555$	22.0153			0.934496
$556$		0			0
$557$		−	3.87154i		−	0.164043i	−0.996631	−	0.0820213i	$-0.973862\pi$
$557$		−	3.87154i		−	0.164043i	0.996631	−	0.0820213i	$-0.0261375\pi$
$558$		0			0
$559$	1.43084	+	23.1192i	0.0605180	+	0.977837i
$560$		0			0
$561$			7.45770i			0.314864i
$562$		0			0
$563$	−35.0866			−1.47872			−0.739362	−	0.673308i	$-0.764873\pi$
$563$	−35.0866			−1.47872			−0.739362	+	0.673308i	$0.764873\pi$
$564$		0			0
$565$			54.4681i			2.29149i
$566$		0			0
$567$		−	12.8839i		−	0.541074i
$568$		0			0
$569$	8.37829			0.351236			0.175618	−	0.984458i	$-0.443808\pi$
$569$	8.37829			0.351236			0.175618	+	0.984458i	$0.443808\pi$
$570$		0			0
$571$	17.3824			0.727431			0.363716	−	0.931510i	$-0.381508\pi$
$571$	17.3824			0.727431			0.363716	+	0.931510i	$0.381508\pi$
$572$		0			0
$573$	−7.11680			−0.297308
$574$		0			0
$575$	−96.3661			−4.01874
$576$		0			0
$577$		−	8.65268i		−	0.360216i	−0.983647	−	0.180108i	$-0.942355\pi$
$577$		−	8.65268i		−	0.360216i	0.983647	−	0.180108i	$-0.0576447\pi$
$578$		0			0
$579$		−	19.0608i		−	0.792142i
$580$		0			0
$581$	−7.01339			−0.290964
$582$		0			0
$583$		−	1.71399i		−	0.0709863i
$584$		0			0
$585$	−14.8190	+	0.917145i	−0.612692	+	0.0379193i
$586$		0			0
$587$			32.2522i			1.33119i	0.746312	+	0.665596i	$0.231823\pi$
$587$			32.2522i			1.33119i	−0.746312	+	0.665596i	$0.768177\pi$
$588$		0			0
$589$	−6.91245			−0.284823
$590$		0			0
$591$			19.5091i			0.802498i
$592$		0			0
$593$			27.4846i			1.12866i	0.825550	+	0.564329i	$0.190865\pi$
$593$			27.4846i			1.12866i	−0.825550	+	0.564329i	$0.809135\pi$
$594$		0			0
$595$	55.3777			2.27027
$596$		0			0
$597$	32.9720			1.34945
$598$		0			0
$599$	−2.84417			−0.116210			−0.0581049	−	0.998310i	$-0.518506\pi$
$599$	−2.84417			−0.116210			−0.0581049	+	0.998310i	$0.518506\pi$
$600$		0			0
$601$	8.63052			0.352046			0.176023	−	0.984386i	$-0.443677\pi$
$601$	8.63052			0.352046			0.176023	+	0.984386i	$0.443677\pi$
$602$		0			0
$603$		−	7.54591i		−	0.307293i
$604$		0			0
$605$		−	4.37595i		−	0.177907i
$606$		0			0
$607$	17.7771			0.721549			0.360775	−	0.932653i	$-0.382512\pi$
$607$	17.7771			0.721549			0.360775	+	0.932653i	$0.382512\pi$
$608$		0			0
$609$			15.4582i			0.626398i
$610$		0			0
$611$	4.06715	−	0.251714i	0.164539	−	0.0101833i
$612$		0			0
$613$		−	16.2142i		−	0.654884i	−0.944871	−	0.327442i	$-0.893813\pi$
$613$		−	16.2142i		−	0.654884i	0.944871	−	0.327442i	$-0.106187\pi$
$614$		0			0
$615$	59.8734			2.41433
$616$		0			0
$617$			30.1799i			1.21500i	0.794321	+	0.607498i	$0.207827\pi$
$617$			30.1799i			1.21500i	−0.794321	+	0.607498i	$0.792173\pi$
$618$		0			0
$619$			37.3801i			1.50243i	0.660057	+	0.751216i	$0.270532\pi$
$619$			37.3801i			1.50243i	−0.660057	+	0.751216i	$0.729468\pi$
$620$		0			0
$621$	38.5156			1.54558
$622$		0			0
$623$	−14.8909			−0.596591
$624$		0			0
$625$	104.447			4.17788
$626$		0			0
$627$	−6.10739			−0.243906
$628$		0			0
$629$		−	18.2225i		−	0.726579i
$630$		0			0
$631$		−	39.0099i		−	1.55296i	−0.630142	−	0.776480i	$-0.717003\pi$
$631$		−	39.0099i		−	1.55296i	0.630142	−	0.776480i	$-0.282997\pi$
$632$		0			0
$633$	−30.3434			−1.20604
$634$		0			0
$635$		−	3.89845i		−	0.154705i
$636$		0			0
$637$	−0.238582	−	3.85496i	−0.00945297	−	0.152739i
$638$		0			0
$639$		−	0.262627i		−	0.0103894i
$640$		0			0
$641$	9.62594			0.380202			0.190101	−	0.981765i	$-0.439119\pi$
$641$	9.62594			0.380202			0.190101	+	0.981765i	$0.439119\pi$
$642$		0			0
$643$		−	29.9977i		−	1.18299i	−0.806308	−	0.591496i	$-0.798538\pi$
$643$		−	29.9977i		−	1.18299i	0.806308	−	0.591496i	$-0.201462\pi$
$644$		0			0
$645$			40.3392i			1.58835i
$646$		0			0
$647$	−32.4846			−1.27710			−0.638551	−	0.769580i	$-0.720466\pi$
$647$	−32.4846			−1.27710			−0.638551	+	0.769580i	$0.720466\pi$
$648$		0			0
$649$	3.60299			0.141430
$650$		0			0
$651$	−5.67424			−0.222391
$652$		0			0
$653$	15.5857			0.609915			0.304958	−	0.952366i	$-0.401358\pi$
$653$	15.5857			0.609915			0.304958	+	0.952366i	$0.401358\pi$
$654$		0			0
$655$		−	48.1890i		−	1.88290i
$656$		0			0
$657$		−	5.71324i		−	0.222895i
$658$		0			0
$659$	−47.2137			−1.83918			−0.919592	−	0.392874i	$-0.871481\pi$
$659$	−47.2137			−1.83918			−0.919592	+	0.392874i	$0.871481\pi$
$660$		0			0
$661$		−	31.6826i		−	1.23231i	−0.787626	−	0.616154i	$-0.788690\pi$
$661$		−	31.6826i		−	1.23231i	0.787626	−	0.616154i	$-0.211310\pi$
$662$		0			0
$663$	−1.66098	−	26.8378i	−0.0645071	−	1.04229i
$664$		0			0
$665$			45.3509i			1.75863i
$666$		0			0
$667$	−30.1338			−1.16678
$668$		0			0
$669$		−	33.6137i		−	1.29958i
$670$		0			0
$671$		−	8.42438i		−	0.325219i
$672$		0			0
$673$	9.69700			0.373792			0.186896	−	0.982380i	$-0.440157\pi$
$673$	9.69700			0.373792			0.186896	+	0.982380i	$0.440157\pi$
$674$		0			0
$675$	−80.0124			−3.07968
$676$		0			0
$677$	−14.1937			−0.545509			−0.272755	−	0.962084i	$-0.587935\pi$
$677$	−14.1937			−0.545509			−0.272755	+	0.962084i	$0.587935\pi$
$678$		0			0
$679$	30.1343			1.15645
$680$		0			0
$681$			41.6241i			1.59504i
$682$		0			0
$683$		−	7.82013i		−	0.299229i	−0.988744	−	0.149614i	$-0.952197\pi$
$683$		−	7.82013i		−	0.299229i	0.988744	−	0.149614i	$-0.0478032\pi$
$684$		0			0
$685$	−98.7952			−3.77477
$686$		0			0
$687$		−	13.7492i		−	0.524564i
$688$		0			0
$689$	0.381740	+	6.16808i	0.0145432	+	0.234985i
$690$		0			0
$691$		−	16.0643i		−	0.611115i	−0.952174	−	0.305557i	$-0.901157\pi$
$691$		−	16.0643i		−	0.611115i	0.952174	−	0.305557i	$-0.0988427\pi$
$692$		0			0
$693$	2.29134			0.0870408
$694$		0			0
$695$		−	81.6334i		−	3.09653i
$696$		0			0
$697$		−	49.5585i		−	1.87716i
$698$		0			0
$699$	−33.3824			−1.26264
$700$		0			0
$701$	−8.06664			−0.304673			−0.152336	−	0.988329i	$-0.548680\pi$
$701$	−8.06664			−0.304673			−0.152336	+	0.988329i	$0.548680\pi$
$702$		0			0
$703$	14.9231			0.562836
$704$		0			0
$705$	7.09651			0.267270
$706$		0			0
$707$		−	14.6095i		−	0.549445i
$708$		0			0
$709$		−	6.34572i		−	0.238319i	−0.992875	−	0.119159i	$-0.961980\pi$
$709$		−	6.34572i		−	0.238319i	0.992875	−	0.119159i	$-0.0380199\pi$
$710$		0			0
$711$	2.91766			0.109421
$712$		0			0
$713$		−	11.0612i		−	0.414245i
$714$		0			0
$715$	0.974611	+	15.7476i	0.0364484	+	0.588926i
$716$		0			0
$717$			11.8044i			0.440843i
$718$		0			0
$719$	−32.7917			−1.22292			−0.611462	−	0.791274i	$-0.709418\pi$
$719$	−32.7917			−1.22292			−0.611462	+	0.791274i	$0.709418\pi$
$720$		0			0
$721$			45.6124i			1.69870i
$722$		0			0
$723$			9.19087i			0.341812i
$724$		0			0
$725$	62.6001			2.32491
$726$		0			0
$727$	2.27375			0.0843288			0.0421644	−	0.999111i	$-0.486575\pi$
$727$	2.27375			0.0843288			0.0421644	+	0.999111i	$0.486575\pi$
$728$		0			0
$729$	29.3227			1.08603
$730$		0			0
$731$	33.3896			1.23496
$732$		0			0
$733$			31.5599i			1.16569i	0.812583	+	0.582846i	$0.198061\pi$
$733$			31.5599i			1.16569i	−0.812583	+	0.582846i	$0.801939\pi$
$734$		0			0
$735$		−	6.72628i		−	0.248102i
$736$		0			0
$737$	−8.01872			−0.295373
$738$		0			0
$739$		−	13.7205i		−	0.504715i	−0.967634	−	0.252358i	$-0.918794\pi$
$739$		−	13.7205i		−	0.504715i	0.967634	−	0.252358i	$-0.0812059\pi$
$740$		0			0
$741$	21.9785	−	1.36024i	0.807399	−	0.0499697i
$742$		0			0
$743$		−	9.66435i		−	0.354550i	−0.984161	−	0.177275i	$-0.943272\pi$
$743$		−	9.66435i		−	0.354550i	0.984161	−	0.177275i	$-0.0567283\pi$
$744$		0			0
$745$	−79.0148			−2.89488
$746$		0			0
$747$		−	2.71051i		−	0.0991725i
$748$		0			0
$749$		−	20.2767i		−	0.740896i
$750$		0			0
$751$	9.66607			0.352720			0.176360	−	0.984326i	$-0.443568\pi$
$751$	9.66607			0.352720			0.176360	+	0.984326i	$0.443568\pi$
$752$		0			0
$753$	15.5428			0.566412
$754$		0			0
$755$	−39.3993			−1.43389
$756$		0			0
$757$	−12.5179			−0.454972			−0.227486	−	0.973781i	$-0.573051\pi$
$757$	−12.5179			−0.454972			−0.227486	+	0.973781i	$0.573051\pi$
$758$		0			0
$759$		−	9.77295i		−	0.354736i
$760$		0			0
$761$			7.52817i			0.272896i	0.990647	+	0.136448i	$0.0435686\pi$
$761$			7.52817i			0.272896i	−0.990647	+	0.136448i	$0.956431\pi$
$762$		0			0
$763$	3.37595			0.122217
$764$		0			0
$765$			21.4022i			0.773799i
$766$		0			0
$767$	−12.9660	+	0.802459i	−0.468174	+	0.0289751i
$768$		0			0
$769$		−	12.4770i		−	0.449933i	−0.974366	−	0.224967i	$-0.927773\pi$
$769$		−	12.4770i		−	0.449933i	0.974366	−	0.224967i	$-0.0722273\pi$
$770$		0			0
$771$	26.2530			0.945478
$772$		0			0
$773$		−	23.4069i		−	0.841888i	−0.907087	−	0.420944i	$-0.861699\pi$
$773$		−	23.4069i		−	0.841888i	0.907087	−	0.420944i	$-0.138301\pi$
$774$		0			0
$775$			22.9786i			0.825415i
$776$		0			0
$777$	12.2500			0.439465
$778$		0			0
$779$	40.5854			1.45412
$780$		0			0
$781$	−0.279083			−0.00998636
$782$		0			0
$783$	−25.0200			−0.894141
$784$		0			0
$785$		−	16.8219i		−	0.600399i
$786$		0			0
$787$		−	2.95095i		−	0.105190i	−0.998616	−	0.0525950i	$-0.983251\pi$
$787$		−	2.95095i		−	0.105190i	0.998616	−	0.0525950i	$-0.0167492\pi$
$788$		0			0
$789$	−12.7928			−0.455435
$790$		0			0
$791$			30.3077i			1.07762i
$792$		0			0
$793$	1.87628	+	30.3165i	0.0666286	+	1.07657i
$794$		0			0
$795$			10.7623i			0.381699i
$796$		0			0
$797$	−52.6282			−1.86419			−0.932093	−	0.362220i	$-0.882019\pi$
$797$	−52.6282			−1.86419			−0.932093	+	0.362220i	$0.882019\pi$
$798$		0			0
$799$		−	5.87393i		−	0.207805i
$800$		0			0
$801$		−	5.75499i		−	0.203342i
$802$		0			0
$803$	−6.07122			−0.214249
$804$		0			0
$805$	−72.5698			−2.55775
$806$		0			0
$807$	−2.35453			−0.0828833
$808$		0			0
$809$	−53.4233			−1.87826			−0.939131	−	0.343558i	$-0.888368\pi$
$809$	−53.4233			−1.87826			−0.939131	+	0.343558i	$0.888368\pi$
$810$		0			0
$811$		−	48.6901i		−	1.70974i	−0.518843	−	0.854870i	$-0.673637\pi$
$811$		−	48.6901i		−	1.70974i	0.518843	−	0.854870i	$-0.326363\pi$
$812$		0			0
$813$			36.7817i			1.28999i
$814$		0			0
$815$	−54.2384			−1.89989
$816$		0			0
$817$			27.3440i			0.956647i
$818$		0			0
$819$	−8.24576	+	0.510327i	−0.288130	+	0.0178323i
$820$		0			0
$821$		−	6.96617i		−	0.243121i	−0.992584	−	0.121561i	$-0.961210\pi$
$821$		−	6.96617i		−	0.243121i	0.992584	−	0.121561i	$-0.0387898\pi$
$822$		0			0
$823$	−10.5586			−0.368051			−0.184025	−	0.982921i	$-0.558913\pi$
$823$	−10.5586			−0.368051			−0.184025	+	0.982921i	$0.558913\pi$
$824$		0			0
$825$			20.3024i			0.706838i
$826$		0			0
$827$			45.1459i			1.56988i	0.619573	+	0.784939i	$0.287306\pi$
$827$			45.1459i			1.56988i	−0.619573	+	0.784939i	$0.712694\pi$
$828$		0			0
$829$	14.8190			0.514685			0.257342	−	0.966320i	$-0.417153\pi$
$829$	14.8190			0.514685			0.257342	+	0.966320i	$0.417153\pi$
$830$		0			0
$831$	−29.4991			−1.02331
$832$		0			0
$833$	−5.56748			−0.192902
$834$		0			0
$835$	−50.8359			−1.75925
$836$		0			0
$837$		−	9.18407i		−	0.317448i
$838$		0			0
$839$			35.4338i			1.22331i	0.791125	+	0.611655i	$0.209496\pi$
$839$			35.4338i			1.22331i	−0.791125	+	0.611655i	$0.790504\pi$
$840$		0			0
$841$	−9.42489			−0.324996
$842$		0			0
$843$		−	7.88590i		−	0.271605i
$844$		0			0
$845$	−7.01460	−	56.4532i	−0.241310	−	1.94205i
$846$		0			0
$847$		−	2.43491i		−	0.0836645i
$848$		0			0
$849$	16.3754			0.562001
$850$		0			0
$851$			23.8797i			0.818587i
$852$		0			0
$853$		−	40.5546i		−	1.38856i	−0.719703	−	0.694282i	$-0.755722\pi$
$853$		−	40.5546i		−	1.38856i	0.719703	−	0.694282i	$-0.244278\pi$
$854$		0			0
$855$	−17.5271			−0.599414
$856$		0			0
$857$	30.8349			1.05330			0.526650	−	0.850082i	$-0.323448\pi$
$857$	30.8349			1.05330			0.526650	+	0.850082i	$0.323448\pi$
$858$		0			0
$859$	16.1110			0.549700			0.274850	−	0.961487i	$-0.411372\pi$
$859$	16.1110			0.549700			0.274850	+	0.961487i	$0.411372\pi$
$860$		0			0
$861$	33.3154			1.13539
$862$		0			0
$863$			21.4582i			0.730446i	0.930920	+	0.365223i	$0.119007\pi$
$863$			21.4582i			0.730446i	−0.930920	+	0.365223i	$0.880993\pi$
$864$		0			0
$865$			82.8671i			2.81757i
$866$		0			0
$867$	−14.3667			−0.487918
$868$		0			0
$869$		−	3.10047i		−	0.105176i
$870$		0			0
$871$	28.8567	−	1.78593i	0.977771	−	0.0605139i
$872$		0			0
$873$			11.6462i			0.394165i
$874$		0			0
$875$	97.4818			3.29549
$876$		0			0
$877$		−	3.87800i		−	0.130951i	−0.997854	−	0.0654754i	$-0.979144\pi$
$877$		−	3.87800i		−	0.130951i	0.997854	−	0.0654754i	$-0.0208564\pi$
$878$		0			0
$879$		−	4.86586i		−	0.164121i
$880$		0			0
$881$	−31.8670			−1.07363			−0.536813	−	0.843701i	$-0.680372\pi$
$881$	−31.8670			−1.07363			−0.536813	+	0.843701i	$0.680372\pi$
$882$		0			0
$883$	40.8945			1.37621			0.688105	−	0.725611i	$-0.258443\pi$
$883$	40.8945			1.37621			0.688105	+	0.725611i	$0.258443\pi$
$884$		0			0
$885$	−22.6235			−0.760480
$886$		0			0
$887$	7.30712			0.245349			0.122674	−	0.992447i	$-0.460853\pi$
$887$	7.30712			0.245349			0.122674	+	0.992447i	$0.460853\pi$
$888$		0			0
$889$		−	2.16921i		−	0.0727531i
$890$		0			0
$891$		−	5.29134i		−	0.177267i
$892$		0			0
$893$	4.81039			0.160974
$894$		0			0
$895$			90.9832i			3.04123i
$896$		0			0
$897$	2.17663	+	35.1696i	0.0726757	+	1.17428i
$898$		0			0
$899$			7.18543i			0.239647i
$900$		0			0
$901$	8.90818			0.296775
$902$		0			0
$903$			22.4460i			0.746954i
$904$		0			0
$905$		−	89.6395i		−	2.97972i
$906$		0			0
$907$	19.5057			0.647675			0.323837	−	0.946113i	$-0.395027\pi$
$907$	19.5057			0.647675			0.323837	+	0.946113i	$0.395027\pi$
$908$		0			0
$909$	5.64622			0.187273
$910$		0			0
$911$	38.0356			1.26017			0.630087	−	0.776524i	$-0.283019\pi$
$911$	38.0356			1.26017			0.630087	+	0.776524i	$0.283019\pi$
$912$		0			0
$913$	−2.88035			−0.0953256
$914$		0			0
$915$			52.8974i			1.74873i
$916$		0			0
$917$		−	26.8138i		−	0.885471i
$918$		0			0
$919$	−33.8313			−1.11599			−0.557996	−	0.829844i	$-0.688429\pi$
$919$	−33.8313			−1.11599			−0.557996	+	0.829844i	$0.688429\pi$
$920$		0			0
$921$		−	9.26103i		−	0.305161i
$922$		0			0
$923$	1.00433	−	0.0621573i	0.0330578	−	0.00204593i
$924$		0			0
$925$		−	49.6079i		−	1.63110i
$926$		0			0
$927$	−17.6282			−0.578985
$928$		0			0
$929$			20.5673i			0.674792i	0.941363	+	0.337396i	$0.109546\pi$
$929$			20.5673i			0.674792i	−0.941363	+	0.337396i	$0.890454\pi$
$930$		0			0
$931$		−	4.55943i		−	0.149429i
$932$		0			0
$933$	−19.4789			−0.637710
$934$		0			0
$935$	22.7432			0.743784
$936$		0			0
$937$	−6.20908			−0.202842			−0.101421	−	0.994844i	$-0.532339\pi$
$937$	−6.20908			−0.202842			−0.101421	+	0.994844i	$0.532339\pi$
$938$		0			0
$939$	−17.0739			−0.557184
$940$		0			0
$941$		−	24.7550i		−	0.806991i	−0.914982	−	0.403495i	$-0.867795\pi$
$941$		−	24.7550i		−	0.806991i	0.914982	−	0.403495i	$-0.132205\pi$
$942$		0			0
$943$			64.9441i			2.11487i
$944$		0			0
$945$	−60.2545			−1.96008
$946$		0			0
$947$		−	25.5857i		−	0.831423i	−0.909496	−	0.415712i	$-0.863533\pi$
$947$		−	25.5857i		−	0.831423i	0.909496	−	0.415712i	$-0.136467\pi$
$948$		0			0
$949$	21.8483	−	1.35218i	0.709225	−	0.0438937i
$950$		0			0
$951$		−	17.9930i		−	0.583462i
$952$		0			0
$953$	−22.7472			−0.736854			−0.368427	−	0.929657i	$-0.620104\pi$
$953$	−22.7472			−0.736854			−0.368427	+	0.929657i	$0.620104\pi$
$954$		0			0
$955$			21.7036i			0.702313i
$956$		0			0
$957$			6.34858i			0.205220i
$958$		0			0
$959$	−54.9726			−1.77516
$960$		0			0
$961$	28.3624			0.914918
$962$		0			0
$963$	7.83650			0.252528
$964$		0			0
$965$	−58.1286			−1.87123
$966$		0			0
$967$			4.55930i			0.146617i	0.997309	+	0.0733085i	$0.0233558\pi$
$967$			4.55930i			0.146617i	−0.997309	+	0.0733085i	$0.976644\pi$
$968$		0			0
$969$		−	31.7422i		−	1.01970i
$970$		0			0
$971$	−6.89312			−0.221211			−0.110605	−	0.993864i	$-0.535279\pi$
$971$	−6.89312			−0.221211			−0.110605	+	0.993864i	$0.535279\pi$
$972$		0			0
$973$		−	45.4233i		−	1.45620i
$974$		0			0
$975$	−4.52175	−	73.0615i	−0.144812	−	2.33984i
$976$		0			0
$977$			21.7071i			0.694471i	0.937778	+	0.347235i	$0.112879\pi$
$977$			21.7071i			0.694471i	−0.937778	+	0.347235i	$0.887121\pi$
$978$		0			0
$979$	−6.11558			−0.195455
$980$		0			0
$981$			1.30473i			0.0416567i
$982$		0			0
$983$		−	25.7373i		−	0.820892i	−0.911885	−	0.410446i	$-0.865373\pi$
$983$		−	25.7373i		−	0.820892i	0.911885	−	0.410446i	$-0.134627\pi$
$984$		0			0
$985$	59.4956			1.89569
$986$		0			0
$987$	3.94871			0.125689
$988$		0			0
$989$	−43.7555			−1.39134
$990$		0			0
$991$	12.4961			0.396952			0.198476	−	0.980106i	$-0.436401\pi$
$991$	12.4961			0.396952			0.198476	+	0.980106i	$0.436401\pi$
$992$		0			0
$993$		−	21.3335i		−	0.676997i
$994$		0			0
$995$		−	100.553i		−	3.18773i
$996$		0			0
$997$	38.5745			1.22167			0.610834	−	0.791759i	$-0.290834\pi$
$997$	38.5745			1.22167			0.610834	+	0.791759i	$0.290834\pi$
$998$		0			0
$999$			19.8273i			0.627307i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.c.441.4	yes	8
3.2	odd	2		5148.2.e.c.1585.1		8
4.3	odd	2		2288.2.j.i.1585.6		8
13.5	odd	4		7436.2.a.o.1.2		4
13.8	odd	4		7436.2.a.p.1.2		4
13.12	even	2	inner	572.2.f.c.441.3	✓	8
39.38	odd	2		5148.2.e.c.1585.8		8
52.51	odd	2		2288.2.j.i.1585.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.c.441.3	✓	8	13.12	even	2	inner
572.2.f.c.441.4	yes	8	1.1	even	1	trivial
2288.2.j.i.1585.5		8	52.51	odd	2
2288.2.j.i.1585.6		8	4.3	odd	2
5148.2.e.c.1585.1		8	3.2	odd	2
5148.2.e.c.1585.8		8	39.38	odd	2
7436.2.a.o.1.2		4	13.5	odd	4
7436.2.a.p.1.2		4	13.8	odd	4

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:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.43491 q^{3} +4.37595i q^{5} +2.43491i q^{7} -0.941037 q^{9} +O(q^{10})\)	\(q-1.43491 q^{3} +4.37595i q^{5} +2.43491i q^{7} -0.941037 q^{9} +1.00000i q^{11} +(-0.222720 - 3.59867i) q^{13} -6.27908i q^{15} -5.19733 q^{17} -4.25629i q^{19} -3.49387i q^{21} +6.81085 q^{23} -14.1489 q^{25} +5.65503 q^{27} -4.42438 q^{29} -1.62405i q^{31} -1.43491i q^{33} -10.6550 q^{35} +3.50613i q^{37} +(0.319583 + 5.16376i) q^{39} +9.53538i q^{41} -6.42438 q^{43} -4.11793i q^{45} +1.13018i q^{47} +1.07122 q^{49} +7.45770 q^{51} -1.71399 q^{53} -4.37595 q^{55} +6.10739i q^{57} -3.60299i q^{59} -8.42438 q^{61} -2.29134i q^{63} +(15.7476 - 0.974611i) q^{65} +8.01872i q^{67} -9.77295 q^{69} +0.279083i q^{71} +6.07122i q^{73} +20.3024 q^{75} -2.43491 q^{77} -3.10047 q^{79} -5.29134 q^{81} +2.88035i q^{83} -22.7432i q^{85} +6.34858 q^{87} +6.11558i q^{89} +(8.76242 - 0.542303i) q^{91} +2.33037i q^{93} +18.6253 q^{95} -12.3759i q^{97} -0.941037i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 + 18 * q^9	\( 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) \) 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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