LMFDB - Embedded newform 684.3.y.g.145.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
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} - 34x^{6} + 921x^{4} - 7990x^{2} + 55225 $ x^8 - 34x^6 + 921x^4 - 7990*x^2 + 55225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.4
Root			$4.27333 + 2.46721i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.3.y.g.217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.48916 + 6.04340i) q^{5} -3.44949 q^{7} +O(q^{10})$	$q+(3.48916 + 6.04340i) q^{5} -3.44949 q^{7} +10.1150 q^{11} +(-0.151531 - 0.0874863i) q^{13} +(1.56834 + 2.71645i) q^{17} +(11.3485 + 15.2385i) q^{19} +(3.48916 - 6.04340i) q^{23} +(-11.8485 + 20.5222i) q^{25} +(-4.70502 - 2.71645i) q^{29} +36.8017i q^{31} +(-12.0358 - 20.8467i) q^{35} +27.1879i q^{37} +(-51.2800 + 29.6065i) q^{41} +(10.9722 + 19.0044i) q^{43} +(6.27337 - 10.8658i) q^{47} -37.1010 q^{49} +(-36.1075 - 20.8467i) q^{53} +(35.2929 + 61.1290i) q^{55} +(21.9924 - 12.6973i) q^{59} +(-26.1413 + 45.2781i) q^{61} -1.22102i q^{65} +(41.3105 + 23.8506i) q^{67} +(14.1151 - 8.14934i) q^{71} +(14.9495 + 25.8933i) q^{73} -34.8916 q^{77} +(-53.1742 + 30.7002i) q^{79} +148.976 q^{83} +(-10.9444 + 18.9562i) q^{85} +(54.9276 + 31.7124i) q^{89} +(0.522704 + 0.301783i) q^{91} +(-52.4958 + 121.753i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 60 * q^13 + 32 * q^19 - 36 * q^25 - 20 * q^43 - 336 * q^49 + 8 * q^55 + 124 * q^61 - 228 * q^67 + 100 * q^73 - 396 * q^79 + 128 * q^85 - 84 * q^91 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.48916	+	6.04340i	0.697832	+	1.20868i	0.969217	+	0.246209i	$0.0791851\pi$
$5$	3.48916	+	6.04340i	0.697832	+	1.20868i	−0.271385	+	0.962471i	$0.587482\pi$
$6$		0			0
$7$	−3.44949			−0.492784			−0.246392	−	0.969170i	$-0.579245\pi$
$7$	−3.44949			−0.492784			−0.246392	+	0.969170i	$0.579245\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.1150			0.919546			0.459773	−	0.888037i	$-0.347931\pi$
$11$	10.1150			0.919546			0.459773	+	0.888037i	$0.347931\pi$
$12$		0			0
$13$	−0.151531	−	0.0874863i	−0.0116562	−	0.00672972i	0.494161	−	0.869371i	$-0.335475\pi$
$13$	−0.151531	−	0.0874863i	−0.0116562	−	0.00672972i	−0.505817	+	0.862641i	$0.668809\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.56834	+	2.71645i	0.0922554	+	0.159791i	0.908460	−	0.417972i	$-0.137259\pi$
$17$	1.56834	+	2.71645i	0.0922554	+	0.159791i	−0.816204	+	0.577763i	$0.803926\pi$
$18$		0			0
$19$	11.3485	+	15.2385i	0.597288	+	0.802027i
$19$	11.3485	+	15.2385i	0.597288	+	0.802027i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.48916	−	6.04340i	0.151703	−	0.262757i	−0.780151	−	0.625591i	$-0.784858\pi$
$23$	3.48916	−	6.04340i	0.151703	−	0.262757i	0.931853	+	0.362835i	$0.118191\pi$
$24$		0			0
$25$	−11.8485	+	20.5222i	−0.473939	+	0.820886i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.70502	−	2.71645i	−0.162242	−	0.0936706i	0.416681	−	0.909053i	$-0.363193\pi$
$29$	−4.70502	−	2.71645i	−0.162242	−	0.0936706i	−0.578923	+	0.815382i	$0.696527\pi$
$30$		0			0
$31$			36.8017i			1.18715i	0.804779	+	0.593575i	$0.202284\pi$
$31$			36.8017i			1.18715i	−0.804779	+	0.593575i	$0.797716\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−12.0358	−	20.8467i	−0.343881	−	0.595619i
$36$		0			0
$37$			27.1879i			0.734808i	0.930061	+	0.367404i	$0.119753\pi$
$37$			27.1879i			0.734808i	−0.930061	+	0.367404i	$0.880247\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−51.2800	+	29.6065i	−1.25073	+	0.722110i	−0.971255	−	0.238043i	$-0.923494\pi$
$41$	−51.2800	+	29.6065i	−1.25073	+	0.722110i	−0.279476	+	0.960153i	$0.590161\pi$
$42$		0			0
$43$	10.9722	+	19.0044i	0.255167	+	0.441963i	0.964941	−	0.262467i	$-0.0845361\pi$
$43$	10.9722	+	19.0044i	0.255167	+	0.441963i	−0.709774	+	0.704430i	$0.751203\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.27337	−	10.8658i	0.133476	−	0.231187i	−0.791538	−	0.611120i	$-0.790719\pi$
$47$	6.27337	−	10.8658i	0.133476	−	0.231187i	0.925014	+	0.379933i	$0.124053\pi$
$48$		0			0
$49$	−37.1010			−0.757164
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−36.1075	−	20.8467i	−0.681273	−	0.393333i	0.119062	−	0.992887i	$-0.462011\pi$
$53$	−36.1075	−	20.8467i	−0.681273	−	0.393333i	−0.800334	+	0.599554i	$0.795345\pi$
$54$		0			0
$55$	35.2929	+	61.1290i	0.641688	+	1.11144i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	21.9924	−	12.6973i	0.372752	−	0.215209i	−0.301908	−	0.953337i	$-0.597624\pi$
$59$	21.9924	−	12.6973i	0.372752	−	0.215209i	0.674660	+	0.738128i	$0.264290\pi$
$60$		0			0
$61$	−26.1413	+	45.2781i	−0.428546	+	0.742264i	−0.996744	−	0.0806279i	$-0.974307\pi$
$61$	−26.1413	+	45.2781i	−0.428546	+	0.742264i	0.568198	+	0.822892i	$0.307641\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.22102i		−	0.0187848i
$66$		0			0
$67$	41.3105	+	23.8506i	0.616574	+	0.355979i	0.775534	−	0.631306i	$-0.217481\pi$
$67$	41.3105	+	23.8506i	0.616574	+	0.355979i	−0.158960	+	0.987285i	$0.550814\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.1151	−	8.14934i	0.198804	−	0.114779i	−0.397294	−	0.917692i	$-0.630051\pi$
$71$	14.1151	−	8.14934i	0.198804	−	0.114779i	0.596097	+	0.802912i	$0.296717\pi$
$72$		0			0
$73$	14.9495	+	25.8933i	0.204788	+	0.354702i	0.950065	−	0.312052i	$-0.101016\pi$
$73$	14.9495	+	25.8933i	0.204788	+	0.354702i	−0.745277	+	0.666754i	$0.767683\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−34.8916			−0.453138
$78$		0			0
$79$	−53.1742	+	30.7002i	−0.673092	+	0.388610i	−0.797247	−	0.603653i	$-0.793711\pi$
$79$	−53.1742	+	30.7002i	−0.673092	+	0.388610i	0.124155	+	0.992263i	$0.460378\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	148.976			1.79490			0.897448	−	0.441119i	$-0.145419\pi$
$83$	148.976			1.79490			0.897448	+	0.441119i	$0.145419\pi$
$84$		0			0
$85$	−10.9444	+	18.9562i	−0.128757	+	0.223015i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	54.9276	+	31.7124i	0.617164	+	0.356320i	0.775764	−	0.631023i	$-0.217365\pi$
$89$	54.9276	+	31.7124i	0.617164	+	0.356320i	−0.158600	+	0.987343i	$0.550698\pi$
$90$		0			0
$91$	0.522704	+	0.301783i	0.00574400	+	0.00331630i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−52.4958	+	121.753i	−0.552588	+	1.28161i
$96$		0			0
$97$	27.0000	−	15.5885i	0.278351	−	0.160706i	−0.354326	−	0.935122i	$-0.615290\pi$
$97$	27.0000	−	15.5885i	0.278351	−	0.160706i	0.632676	+	0.774416i	$0.281956\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	49.0067	−	84.8820i	0.485215	−	0.840416i	−0.514641	−	0.857406i	$-0.672075\pi$
$101$	49.0067	−	84.8820i	0.485215	−	0.840416i	0.999856	+	0.0169894i	$0.00540817\pi$
$102$		0			0
$103$			100.713i			0.977792i	0.872342	+	0.488896i	$0.162600\pi$
$103$			100.713i			0.977792i	−0.872342	+	0.488896i	$0.837400\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.43289i		−	0.0507747i	−0.999678	−	0.0253874i	$-0.991918\pi$
$107$		−	5.43289i		−	0.0507747i	0.999678	−	0.0253874i	$-0.00808191\pi$
$108$		0			0
$109$	−59.6969	+	34.4660i	−0.547678	+	0.316202i	−0.748185	−	0.663490i	$-0.769074\pi$
$109$	−59.6969	+	34.4660i	−0.547678	+	0.316202i	0.200507	+	0.979692i	$0.435741\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			200.043i			1.77029i	0.465315	+	0.885145i	$0.345941\pi$
$113$			200.043i			1.77029i	−0.465315	+	0.885145i	$0.654059\pi$
$114$		0			0
$115$	48.6969			0.423452
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.40998	−	9.37036i	−0.0454620	−	0.0787425i
$120$		0			0
$121$	−18.6867			−0.154436
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.09318			0.0727454
$126$		0			0
$127$	99.5755	+	57.4899i	0.784059	+	0.452677i	0.837867	−	0.545875i	$-0.183802\pi$
$127$	99.5755	+	57.4899i	0.784059	+	0.452677i	−0.0538078	+	0.998551i	$0.517136\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−51.8265	−	89.7661i	−0.395622	−	0.685237i	0.597558	−	0.801825i	$-0.296138\pi$
$131$	−51.8265	−	89.7661i	−0.395622	−	0.685237i	−0.993180	+	0.116588i	$0.962804\pi$
$132$		0			0
$133$	−39.1464	−	52.5651i	−0.294334	−	0.395226i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	64.3732	−	111.498i	0.469877	−	0.813852i	−0.529529	−	0.848292i	$-0.677631\pi$
$137$	64.3732	−	111.498i	0.469877	−	0.813852i	0.999407	+	0.0344400i	$0.0109648\pi$
$138$		0			0
$139$	−9.17423	+	15.8902i	−0.0660017	+	0.114318i	−0.897138	−	0.441751i	$-0.854358\pi$
$139$	−9.17423	+	15.8902i	−0.0660017	+	0.114318i	0.831136	+	0.556069i	$0.187691\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.53273	−	0.884924i	−0.0107184	−	0.00618828i
$144$		0			0
$145$		−	37.9125i		−	0.261465i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−114.437	−	198.211i	−0.768036	−	1.33028i	−0.938627	−	0.344935i	$-0.887901\pi$
$149$	−114.437	−	198.211i	−0.768036	−	1.33028i	0.170591	−	0.985342i	$-0.445432\pi$
$150$		0			0
$151$		−	161.063i		−	1.06664i	−0.845913	−	0.533321i	$-0.820944\pi$
$151$		−	161.063i		−	1.06664i	0.845913	−	0.533321i	$-0.179056\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−222.407	+	128.407i	−1.43489	+	0.828431i
$156$		0			0
$157$	−131.030	−	226.951i	−0.834587	−	1.44555i	−0.894367	−	0.447335i	$-0.852373\pi$
$157$	−131.030	−	226.951i	−0.834587	−	1.44555i	0.0597799	−	0.998212i	$-0.480960\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0358	+	20.8467i	−0.0747566	+	0.129482i
$162$		0			0
$163$	109.833			0.673823			0.336912	−	0.941536i	$-0.390618\pi$
$163$	109.833			0.673823			0.336912	+	0.941536i	$0.390618\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−198.882	−	114.825i	−1.19091	−	0.687573i	−0.232398	−	0.972621i	$-0.574657\pi$
$167$	−198.882	−	114.825i	−1.19091	−	0.687573i	−0.958513	+	0.285048i	$0.907990\pi$
$168$		0			0
$169$	−84.4847	−	146.332i	−0.499909	−	0.865869i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	244.400	−	141.104i	1.41271	−	0.815631i	0.417071	−	0.908874i	$-0.363057\pi$
$173$	244.400	−	141.104i	1.41271	−	0.815631i	0.995643	+	0.0932428i	$0.0297233\pi$
$174$		0			0
$175$	40.8712	−	70.7909i	0.233550	−	0.404520i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	22.9526i		−	0.128227i	−0.997943	−	0.0641134i	$-0.979578\pi$
$179$		−	22.9526i		−	0.128227i	0.997943	−	0.0641134i	$-0.0204219\pi$
$180$		0			0
$181$	27.4699	+	15.8598i	0.151767	+	0.0876230i	0.573961	−	0.818883i	$-0.305406\pi$
$181$	27.4699	+	15.8598i	0.151767	+	0.0876230i	−0.422193	+	0.906506i	$0.638740\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−164.307	+	94.8629i	−0.888148	+	0.512772i
$186$		0			0
$187$	15.8638	+	27.4769i	0.0848330	+	0.146935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−96.6746			−0.506150			−0.253075	−	0.967447i	$-0.581442\pi$
$191$	−96.6746			−0.506150			−0.253075	+	0.967447i	$0.581442\pi$
$192$		0			0
$193$	92.5454	−	53.4311i	0.479510	−	0.276845i	−0.240702	−	0.970599i	$-0.577378\pi$
$193$	92.5454	−	53.4311i	0.479510	−	0.276845i	0.720212	+	0.693754i	$0.244044\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	253.651			1.28757			0.643785	−	0.765207i	$-0.277363\pi$
$197$	253.651			1.28757			0.643785	+	0.765207i	$0.277363\pi$
$198$		0			0
$199$	89.4115	−	154.865i	0.449304	−	0.778217i	−0.549037	−	0.835798i	$-0.685005\pi$
$199$	89.4115	−	154.865i	0.449304	−	0.778217i	0.998341	+	0.0575809i	$0.0183387\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.2299	+	9.37036i	0.0799504	+	0.0461594i
$204$		0			0
$205$	−357.848	−	206.604i	−1.74560	−	1.00782i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	114.790	+	154.138i	0.549233	+	0.737500i
$210$		0			0
$211$	211.098	−	121.878i	1.00047	−	0.577619i	0.0920795	−	0.995752i	$-0.470649\pi$
$211$	211.098	−	121.878i	1.00047	−	0.577619i	0.908386	+	0.418133i	$0.137315\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−76.5675	+	132.619i	−0.356128	+	0.616831i
$216$		0			0
$217$		−	126.947i		−	0.585009i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.548834i		−	0.00248341i
$222$		0			0
$223$	12.3559	−	7.13366i	0.0554075	−	0.0319895i	−0.472040	−	0.881577i	$-0.656482\pi$
$223$	12.3559	−	7.13366i	0.0554075	−	0.0319895i	0.527448	+	0.849587i	$0.323149\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			259.256i			1.14210i	0.820917	+	0.571048i	$0.193463\pi$
$227$			259.256i			1.14210i	−0.820917	+	0.571048i	$0.806537\pi$
$228$		0			0
$229$	56.4143			0.246351			0.123175	−	0.992385i	$-0.460692\pi$
$229$	56.4143			0.246351			0.123175	+	0.992385i	$0.460692\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	54.5751	+	94.5268i	0.234228	+	0.405694i	0.959048	−	0.283244i	$-0.0914105\pi$
$233$	54.5751	+	94.5268i	0.234228	+	0.405694i	−0.724820	+	0.688938i	$0.758077\pi$
$234$		0			0
$235$	87.5551			0.372575
$236$		0			0
$237$		0			0
$238$		0			0
$239$	405.764			1.69776			0.848879	−	0.528587i	$-0.177278\pi$
$239$	405.764			1.69776			0.848879	+	0.528587i	$0.177278\pi$
$240$		0			0
$241$	−245.787	−	141.905i	−1.01986	−	0.588819i	−0.105799	−	0.994387i	$-0.533740\pi$
$241$	−245.787	−	141.905i	−1.01986	−	0.588819i	−0.914065	+	0.405569i	$0.867074\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−129.451	−	224.216i	−0.528373	−	0.915169i
$246$		0			0
$247$	−0.386481	−	3.30194i	−0.00156470	−	0.0133682i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	139.566	−	241.736i	0.556041	−	0.963092i	−0.441780	−	0.897123i	$-0.645653\pi$
$251$	139.566	−	241.736i	0.556041	−	0.963092i	0.997822	−	0.0659686i	$-0.0210137\pi$
$252$		0			0
$253$	35.2929	−	61.1290i	0.139497	−	0.241617i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	278.868	+	161.004i	1.08509	+	0.626476i	0.932264	−	0.361778i	$-0.117830\pi$
$257$	278.868	+	161.004i	1.08509	+	0.626476i	0.152823	+	0.988254i	$0.451163\pi$
$258$		0			0
$259$		−	93.7844i		−	0.362102i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−44.1432	−	76.4583i	−0.167845	−	0.290716i	0.769817	−	0.638265i	$-0.220347\pi$
$263$	−44.1432	−	76.4583i	−0.167845	−	0.290716i	−0.937662	+	0.347549i	$0.887014\pi$
$264$		0			0
$265$		−	290.949i		−	1.09792i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−198.407	+	114.550i	−0.737572	+	0.425837i	−0.821186	−	0.570661i	$-0.806687\pi$
$269$	−198.407	+	114.550i	−0.737572	+	0.425837i	0.0836140	+	0.996498i	$0.473354\pi$
$270$		0			0
$271$	25.0908	+	43.4586i	0.0925860	+	0.160364i	0.908599	−	0.417670i	$-0.137153\pi$
$271$	25.0908	+	43.4586i	0.0925860	+	0.160364i	−0.816013	+	0.578034i	$0.803820\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−119.847	+	207.582i	−0.435808	+	0.754842i
$276$		0			0
$277$	−323.918			−1.16938			−0.584690	−	0.811257i	$-0.698784\pi$
$277$	−323.918			−1.16938			−0.584690	+	0.811257i	$0.698784\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.05743	−	0.610508i	−0.00376310	−	0.00217262i	0.498117	−	0.867110i	$-0.334025\pi$
$281$	−1.05743	−	0.610508i	−0.00376310	−	0.00217262i	−0.501880	+	0.864937i	$0.667358\pi$
$282$		0			0
$283$	203.247	+	352.035i	0.718189	+	1.24394i	0.961717	+	0.274046i	$0.0883620\pi$
$283$	203.247	+	352.035i	0.718189	+	1.24394i	−0.243528	+	0.969894i	$0.578305\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	176.890	−	102.127i	0.616340	−	0.355844i
$288$		0			0
$289$	139.581	−	241.761i	0.482978	−	0.836542i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	362.604i		−	1.23756i	−0.785566	−	0.618778i	$-0.787628\pi$
$293$		−	362.604i		−	1.23756i	0.785566	−	0.618778i	$-0.212372\pi$
$294$		0			0
$295$	153.470	+	88.6059i	0.520237	+	0.300359i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.05743	+	0.610508i	−0.00353656	+	0.00204183i
$300$		0			0
$301$	−37.8485	−	65.5555i	−0.125742	−	0.217792i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−364.845			−1.19621
$306$		0			0
$307$	−54.3031	+	31.3519i	−0.176883	+	0.102123i	−0.585827	−	0.810436i	$-0.699230\pi$
$307$	−54.3031	+	31.3519i	−0.176883	+	0.102123i	0.408944	+	0.912559i	$0.365897\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	248.471			0.798942			0.399471	−	0.916746i	$-0.369194\pi$
$311$	248.471			0.798942			0.399471	+	0.916746i	$0.369194\pi$
$312$		0			0
$313$	164.000	−	284.056i	0.523962	−	0.907528i	−0.475649	−	0.879635i	$-0.657787\pi$
$313$	164.000	−	284.056i	0.523962	−	0.907528i	0.999611	−	0.0278932i	$-0.00887983\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−292.983	−	169.154i	−0.924235	−	0.533607i	−0.0392515	−	0.999229i	$-0.512497\pi$
$317$	−292.983	−	169.154i	−0.924235	−	0.533607i	−0.884984	+	0.465622i	$0.845831\pi$
$318$		0			0
$319$	−47.5913	−	27.4769i	−0.149189	−	0.0861344i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−23.5963	+	54.7267i	−0.0730537	+	0.169433i
$324$		0			0
$325$	3.59082	−	2.07316i	0.0110487	−	0.00637895i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−21.6399	+	37.4814i	−0.0657748	+	0.113925i
$330$		0			0
$331$		−	197.707i		−	0.597303i	−0.954362	−	0.298652i	$-0.903463\pi$
$331$		−	197.707i		−	0.597303i	0.954362	−	0.298652i	$-0.0965369\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			332.874i			0.993654i
$336$		0			0
$337$	−288.591	+	166.618i	−0.856353	+	0.494415i	−0.862789	−	0.505564i	$-0.831284\pi$
$337$	−288.591	+	166.618i	−0.856353	+	0.494415i	0.00643660	+	0.999979i	$0.497951\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			372.249i			1.09164i
$342$		0			0
$343$	297.005			0.865903
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−204.221	−	353.721i	−0.588533	−	1.01937i	−0.994425	−	0.105448i	$-0.966372\pi$
$347$	−204.221	−	353.721i	−0.588533	−	1.01937i	0.405892	−	0.913921i	$-0.366961\pi$
$348$		0			0
$349$	−378.444			−1.08437			−0.542183	−	0.840260i	$-0.682402\pi$
$349$	−378.444			−1.08437			−0.542183	+	0.840260i	$0.682402\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−356.528			−1.00999			−0.504997	−	0.863121i	$-0.668506\pi$
$353$	−356.528			−1.00999			−0.504997	+	0.863121i	$0.668506\pi$
$354$		0			0
$355$	98.4995	+	56.8687i	0.277463	+	0.160194i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−305.248	−	528.705i	−0.850273	−	1.47272i	−0.880962	−	0.473187i	$-0.843103\pi$
$359$	−305.248	−	528.705i	−0.850273	−	1.47272i	0.0306887	−	0.999529i	$-0.490230\pi$
$360$		0			0
$361$	−103.424	+	345.868i	−0.286494	+	0.958082i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−104.322	+	180.692i	−0.285815	+	0.495045i
$366$		0			0
$367$	−12.7497	+	22.0832i	−0.0347404	+	0.0601722i	−0.882873	−	0.469612i	$-0.844394\pi$
$367$	−12.7497	+	22.0832i	−0.0347404	+	0.0601722i	0.848132	+	0.529784i	$0.177727\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	124.552	+	71.9103i	0.335721	+	0.193828i
$372$		0			0
$373$			443.527i			1.18908i	0.804066	+	0.594540i	$0.202666\pi$
$373$			443.527i			1.18908i	−0.804066	+	0.594540i	$0.797334\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.475304	+	0.823251i	0.00126075	+	0.00218369i
$378$		0			0
$379$			146.375i			0.386214i	0.981178	+	0.193107i	$0.0618564\pi$
$379$			146.375i			0.386214i	−0.981178	+	0.193107i	$0.938144\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	434.822	−	251.045i	1.13531	−	0.655469i	0.190042	−	0.981776i	$-0.439138\pi$
$383$	434.822	−	251.045i	1.13531	−	0.655469i	0.945264	+	0.326307i	$0.105804\pi$
$384$		0			0
$385$	−121.742	−	210.864i	−0.316214	−	0.547699i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−38.8561	+	67.3007i	−0.0998870	+	0.173009i	−0.911638	−	0.410995i	$-0.865181\pi$
$389$	−38.8561	+	67.3007i	−0.0998870	+	0.173009i	0.811751	+	0.584004i	$0.198515\pi$
$390$		0			0
$391$	21.8888			0.0559815
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−371.067	−	214.236i	−0.939410	−	0.542368i
$396$		0			0
$397$	99.3332	+	172.050i	0.250209	+	0.433376i	0.963583	−	0.267408i	$-0.0861672\pi$
$397$	99.3332	+	172.050i	0.250209	+	0.433376i	−0.713374	+	0.700784i	$0.752834\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	329.197	−	190.062i	0.820940	−	0.473970i	−0.0298006	−	0.999556i	$-0.509487\pi$
$401$	329.197	−	190.062i	0.820940	−	0.473970i	0.850740	+	0.525586i	$0.176154\pi$
$402$		0			0
$403$	3.21964	−	5.57658i	0.00798919	−	0.0138377i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			275.006i			0.675689i
$408$		0			0
$409$	384.909	+	222.227i	0.941098	+	0.543343i	0.890304	−	0.455366i	$-0.150492\pi$
$409$	384.909	+	222.227i	0.941098	+	0.543343i	0.0507938	+	0.998709i	$0.483825\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−75.8625	+	43.7992i	−0.183686	+	0.106051i
$414$		0			0
$415$	519.803	+	900.324i	1.25254	+	2.16946i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	492.182			1.17466			0.587329	−	0.809348i	$-0.300180\pi$
$419$	492.182			1.17466			0.587329	+	0.809348i	$0.300180\pi$
$420$		0			0
$421$	−435.681	+	251.541i	−1.03487	+	0.597484i	−0.918377	−	0.395708i	$-0.870499\pi$
$421$	−435.681	+	251.541i	−1.03487	+	0.597484i	−0.116495	+	0.993191i	$0.537166\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−74.3298			−0.174894
$426$		0			0
$427$	90.1742	−	156.186i	0.211181	−	0.365776i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	79.5101	+	45.9052i	0.184478	+	0.106509i	0.589395	−	0.807845i	$-0.299366\pi$
$431$	79.5101	+	45.9052i	0.184478	+	0.106509i	−0.404917	+	0.914354i	$0.632700\pi$
$432$		0			0
$433$	268.847	+	155.219i	0.620895	+	0.358474i	0.777217	−	0.629232i	$-0.216631\pi$
$433$	268.847	+	155.219i	0.620895	+	0.358474i	−0.156323	+	0.987706i	$0.549964\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	131.689	−	15.4138i	0.301348	−	0.0352718i
$438$		0			0
$439$	−195.901	+	113.103i	−0.446243	+	0.257639i	−0.706242	−	0.707970i	$-0.749611\pi$
$439$	−195.901	+	113.103i	−0.446243	+	0.257639i	0.259999	+	0.965609i	$0.416278\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−49.3235	+	85.4309i	−0.111340	+	0.192846i	−0.916311	−	0.400468i	$-0.868848\pi$
$443$	−49.3235	+	85.4309i	−0.111340	+	0.192846i	0.804971	+	0.593314i	$0.202181\pi$
$444$		0			0
$445$			442.599i			0.994605i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			258.035i			0.574688i	0.957828	+	0.287344i	$0.0927722\pi$
$449$			258.035i			0.574688i	−0.957828	+	0.287344i	$0.907228\pi$
$450$		0			0
$451$	−518.697	+	299.470i	−1.15010	+	0.664013i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.21188i			0.00925688i
$456$		0			0
$457$	102.212			0.223659			0.111830	−	0.993727i	$-0.464329\pi$
$457$	102.212			0.223659			0.111830	+	0.993727i	$0.464329\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	53.9057	+	93.3675i	0.116932	+	0.202532i	0.918550	−	0.395304i	$-0.129361\pi$
$461$	53.9057	+	93.3675i	0.116932	+	0.202532i	−0.801618	+	0.597836i	$0.796027\pi$
$462$		0			0
$463$	−271.692			−0.586809			−0.293404	−	0.955988i	$-0.594788\pi$
$463$	−271.692			−0.586809			−0.293404	+	0.955988i	$0.594788\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	617.403			1.32206			0.661031	−	0.750358i	$-0.270119\pi$
$467$	617.403			1.32206			0.661031	+	0.750358i	$0.270119\pi$
$468$		0			0
$469$	−142.500	−	82.2724i	−0.303838	−	0.175421i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	110.984	+	192.230i	0.234638	+	0.406405i
$474$		0			0
$475$	−447.189	+	52.3420i	−0.941451	+	0.110194i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	53.2364	−	92.2081i	0.111141	−	0.192501i	−0.805090	−	0.593153i	$-0.797883\pi$
$479$	53.2364	−	92.2081i	0.111141	−	0.192501i	0.916230	+	0.400652i	$0.131216\pi$
$480$		0			0
$481$	2.37857	−	4.11980i	0.00494505	−	0.00856508i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	188.415	+	108.781i	0.388484	+	0.224291i
$486$		0			0
$487$			719.920i			1.47827i	0.673555	+	0.739137i	$0.264767\pi$
$487$			719.920i			1.47827i	−0.673555	+	0.739137i	$0.735233\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	43.8263	+	75.9094i	0.0892593	+	0.154602i	0.907198	−	0.420703i	$-0.138217\pi$
$491$	43.8263	+	75.9094i	0.0892593	+	0.154602i	−0.817939	+	0.575305i	$0.804883\pi$
$492$		0			0
$493$		−	17.0413i		−	0.0345665i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−48.6898	+	28.1111i	−0.0979674	+	0.0565615i
$498$		0			0
$499$	78.7293	+	136.363i	0.157774	+	0.273273i	0.934066	−	0.357101i	$-0.116235\pi$
$499$	78.7293	+	136.363i	0.157774	+	0.273273i	−0.776292	+	0.630374i	$0.782902\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−141.523	+	245.125i	−0.281357	+	0.487325i	−0.971719	−	0.236139i	$-0.924118\pi$
$503$	−141.523	+	245.125i	−0.281357	+	0.487325i	0.690362	+	0.723464i	$0.257451\pi$
$504$		0			0
$505$	683.968			1.35439
$506$		0			0
$507$		0			0
$508$		0			0
$509$	747.314	+	431.462i	1.46820	+	0.847666i	0.999365	−	0.0356201i	$-0.0113406\pi$
$509$	747.314	+	431.462i	1.46820	+	0.847666i	0.468835	+	0.883286i	$0.344674\pi$
$510$		0			0
$511$	−51.5681	−	89.3186i	−0.100916	−	0.174792i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−608.647	+	351.402i	−1.18184	+	0.682334i
$516$		0			0
$517$	63.4551	−	109.907i	0.122737	−	0.212587i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	426.578i		−	0.818767i	−0.912362	−	0.409384i	$-0.865744\pi$
$521$		−	426.578i		−	0.818767i	0.912362	−	0.409384i	$-0.134256\pi$
$522$		0			0
$523$	703.309	+	406.056i	1.34476	+	0.776397i	0.987502	−	0.157608i	$-0.0503783\pi$
$523$	703.309	+	406.056i	1.34476	+	0.776397i	0.357258	+	0.934006i	$0.383712\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−99.9698	+	57.7176i	−0.189696	+	0.109521i
$528$		0			0
$529$	240.152	+	415.955i	0.453973	+	0.786304i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.3607			0.0194384
$534$		0			0
$535$	32.8332	−	18.9562i	0.0613704	−	0.0354322i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−375.277			−0.696247
$540$		0			0
$541$	49.7429	−	86.1572i	0.0919461	−	0.159255i	−0.816384	−	0.577510i	$-0.804025\pi$
$541$	49.7429	−	86.1572i	0.0919461	−	0.159255i	0.908330	+	0.418254i	$0.137358\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−416.584	−	240.515i	−0.764375	−	0.441312i
$546$		0			0
$547$	−552.901	−	319.217i	−1.01079	−	0.583578i	−0.0993654	−	0.995051i	$-0.531681\pi$
$547$	−552.901	−	319.217i	−1.01079	−	0.583578i	−0.911422	+	0.411473i	$0.865015\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0002	−	102.525i	−0.0217790	−	0.186071i
$552$		0			0
$553$	183.424	−	105.900i	0.331689	−	0.191501i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.18040	+	15.9009i	−0.0164819	+	0.0285474i	−0.874149	−	0.485658i	$-0.838580\pi$
$557$	−9.18040	+	15.9009i	−0.0164819	+	0.0285474i	0.857667	+	0.514206i	$0.171913\pi$
$558$		0			0
$559$		−	3.83967i		−	0.00686882i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	579.618i		−	1.02952i	−0.857335	−	0.514758i	$-0.827882\pi$
$563$		−	579.618i		−	1.02952i	0.857335	−	0.514758i	$-0.172118\pi$
$564$		0			0
$565$	−1208.94	+	697.981i	−2.13971	+	1.23536i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	785.519i		−	1.38053i	−0.723559	−	0.690263i	$-0.757495\pi$
$569$		−	785.519i		−	1.38053i	0.723559	−	0.690263i	$-0.242505\pi$
$570$		0			0
$571$	37.6219			0.0658878			0.0329439	−	0.999457i	$-0.489512\pi$
$571$	37.6219			0.0658878			0.0329439	+	0.999457i	$0.489512\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	82.6824	+	143.210i	0.143795	+	0.249061i
$576$		0			0
$577$	720.504			1.24871			0.624354	−	0.781142i	$-0.285362\pi$
$577$	720.504			1.24871			0.624354	+	0.781142i	$0.285362\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−513.893			−0.884497
$582$		0			0
$583$	−365.227	−	210.864i	−0.626461	−	0.361688i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	100.869	+	174.710i	0.171838	+	0.297632i	0.939062	−	0.343747i	$-0.111696\pi$
$587$	100.869	+	174.710i	0.171838	+	0.297632i	−0.767225	+	0.641378i	$0.778363\pi$
$588$		0			0
$589$	−560.803	+	417.643i	−0.952127	+	0.709070i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−70.3653	+	121.876i	−0.118660	+	0.205525i	−0.919237	−	0.393705i	$-0.871193\pi$
$593$	−70.3653	+	121.876i	−0.118660	+	0.205525i	0.800577	+	0.599230i	$0.204526\pi$
$594$		0			0
$595$	37.7526	−	65.3893i	0.0634497	−	0.109898i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−841.997	−	486.127i	−1.40567	−	0.811564i	−0.410703	−	0.911769i	$-0.634717\pi$
$599$	−841.997	−	486.127i	−1.40567	−	0.811564i	−0.994967	+	0.100205i	$0.968050\pi$
$600$		0			0
$601$		−	486.915i		−	0.810174i	−0.914278	−	0.405087i	$-0.867241\pi$
$601$		−	486.915i		−	0.810174i	0.914278	−	0.405087i	$-0.132759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−65.2010	−	112.931i	−0.107770	−	0.186664i
$606$		0			0
$607$		−	819.811i		−	1.35059i	−0.737546	−	0.675297i	$-0.764015\pi$
$607$		−	819.811i		−	1.35059i	0.737546	−	0.675297i	$-0.235985\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.90122	+	1.09767i	−0.00311165	+	0.00179651i
$612$		0			0
$613$	262.388	+	454.470i	0.428040	+	0.741386i	0.996699	−	0.0811869i	$-0.0258711\pi$
$613$	262.388	+	454.470i	0.428040	+	0.741386i	−0.568659	+	0.822573i	$0.692538\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	420.462	−	728.261i	0.681461	−	1.18033i	−0.293074	−	0.956090i	$-0.594678\pi$
$617$	420.462	−	728.261i	0.681461	−	1.18033i	0.974535	−	0.224236i	$-0.0719885\pi$
$618$		0			0
$619$	−420.196			−0.678831			−0.339416	−	0.940637i	$-0.610229\pi$
$619$	−420.196			−0.678831			−0.339416	+	0.940637i	$0.610229\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−189.472	−	109.392i	−0.304128	−	0.175589i
$624$		0			0
$625$	327.939	+	568.008i	0.524703	+	0.908812i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−73.8545	+	42.6399i	−0.117416	+	0.0677900i
$630$		0			0
$631$	472.805	−	818.923i	0.749295	−	1.29782i	−0.198866	−	0.980027i	$-0.563726\pi$
$631$	472.805	−	818.923i	0.749295	−	1.29782i	0.948161	−	0.317791i	$-0.102941\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			802.366i			1.26357i
$636$		0			0
$637$	5.62195	+	3.24583i	0.00882566	+	0.00509550i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	244.875	−	141.379i	0.382020	−	0.220559i	−0.296677	−	0.954978i	$-0.595878\pi$
$641$	244.875	−	141.379i	0.382020	−	0.220559i	0.678697	+	0.734419i	$0.262545\pi$
$642$		0			0
$643$	575.275	+	996.406i	0.894674	+	1.54962i	0.834208	+	0.551450i	$0.185925\pi$
$643$	575.275	+	996.406i	0.894674	+	1.54962i	0.0604660	+	0.998170i	$0.480741\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−366.501			−0.566461			−0.283231	−	0.959052i	$-0.591406\pi$
$647$	−366.501			−0.566461			−0.283231	+	0.959052i	$0.591406\pi$
$648$		0			0
$649$	222.453	−	128.433i	0.342763	−	0.197894i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−712.248			−1.09073			−0.545366	−	0.838198i	$-0.683609\pi$
$653$	−712.248			−1.09073			−0.545366	+	0.838198i	$0.683609\pi$
$654$		0			0
$655$	361.662	−	626.417i	0.552155	−	0.956361i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	804.356	+	464.395i	1.22057	+	0.704697i	0.965040	−	0.262104i	$-0.0844163\pi$
$659$	804.356	+	464.395i	1.22057	+	0.704697i	0.255531	+	0.966801i	$0.417750\pi$
$660$		0			0
$661$	−788.697	−	455.354i	−1.19319	−	0.688887i	−0.234159	−	0.972198i	$-0.575234\pi$
$661$	−788.697	−	455.354i	−1.19319	−	0.688887i	−0.959028	+	0.283311i	$0.908567\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	181.084	−	419.986i	0.272306	−	0.631557i
$666$		0			0
$667$	−32.8332	+	18.9562i	−0.0492251	+	0.0284201i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−264.420	+	457.988i	−0.394068	+	0.682546i
$672$		0			0
$673$		−	197.173i		−	0.292976i	−0.989212	−	0.146488i	$-0.953203\pi$
$673$		−	197.173i		−	0.292976i	0.989212	−	0.146488i	$-0.0467970\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	38.5791i		−	0.0569854i	−0.999594	−	0.0284927i	$-0.990929\pi$
$677$		−	38.5791i		−	0.0569854i	0.999594	−	0.0284927i	$-0.00907073\pi$
$678$		0			0
$679$	−93.1362	+	53.7722i	−0.137167	+	0.0791933i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1073.83i		−	1.57223i	−0.618081	−	0.786114i	$-0.712090\pi$
$683$		−	1073.83i		−	1.57223i	0.618081	−	0.786114i	$-0.287910\pi$
$684$		0			0
$685$	898.434			1.31158
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.64759	+	6.31782i	0.00529404	+	0.00916955i
$690$		0			0
$691$	−966.786			−1.39911			−0.699555	−	0.714578i	$-0.746619\pi$
$691$	−966.786			−1.39911			−0.699555	+	0.714578i	$0.746619\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−128.041			−0.184232
$696$		0			0
$697$	−160.849	−	92.8662i	−0.230773	−	0.133237i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	691.064	+	1196.96i	0.985825	+	1.70750i	0.638209	+	0.769863i	$0.279675\pi$
$701$	691.064	+	1196.96i	0.985825	+	1.70750i	0.347616	+	0.937637i	$0.386991\pi$
$702$		0			0
$703$	−414.303	+	308.541i	−0.589336	+	0.438892i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−169.048	+	292.800i	−0.239106	+	0.414144i
$708$		0			0
$709$	183.197	−	317.306i	0.258388	−	0.447541i	−0.707422	−	0.706791i	$-0.750142\pi$
$709$	183.197	−	317.306i	0.258388	−	0.447541i	0.965810	+	0.259250i	$0.0834754\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	222.407	+	128.407i	0.311932	+	0.180094i
$714$		0			0
$715$		−	12.3506i		−	0.0172735i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	674.358	+	1168.02i	0.937912	+	1.62451i	0.769358	+	0.638818i	$0.220576\pi$
$719$	674.358	+	1168.02i	0.937912	+	1.62451i	0.168554	+	0.985692i	$0.446090\pi$
$720$		0			0
$721$		−	347.407i		−	0.481840i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	111.495	−	64.3715i	0.153786	−	0.0887882i
$726$		0			0
$727$	−594.082	−	1028.98i	−0.817170	−	1.41538i	−0.907759	−	0.419491i	$-0.862209\pi$
$727$	−594.082	−	1028.98i	−0.817170	−	1.41538i	0.0905897	−	0.995888i	$-0.471125\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.4163	+	59.6108i	−0.0470811	+	0.0815469i
$732$		0			0
$733$	47.5143			0.0648217			0.0324108	−	0.999475i	$-0.489682\pi$
$733$	47.5143			0.0648217			0.0324108	+	0.999475i	$0.489682\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	417.855	+	241.249i	0.566968	+	0.327339i
$738$		0			0
$739$	25.3717	+	43.9451i	0.0343325	+	0.0594656i	0.882681	−	0.469972i	$-0.155736\pi$
$739$	25.3717	+	43.9451i	0.0343325	+	0.0594656i	−0.848349	+	0.529438i	$0.822403\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−712.157	+	411.164i	−0.958489	+	0.553384i	−0.895708	−	0.444644i	$-0.853330\pi$
$743$	−712.157	+	411.164i	−0.958489	+	0.553384i	−0.0627811	+	0.998027i	$0.519997\pi$
$744$		0			0
$745$	798.580	−	1383.18i	1.07192	−	1.85662i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			18.7407i			0.0250210i
$750$		0			0
$751$	33.3115	+	19.2324i	0.0443562	+	0.0256091i	0.522014	−	0.852937i	$-0.325181\pi$
$751$	33.3115	+	19.2324i	0.0443562	+	0.0256091i	−0.477658	+	0.878546i	$0.658514\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	973.369	−	561.975i	1.28923	−	0.744337i
$756$		0			0
$757$	155.046	+	268.547i	0.204816	+	0.354752i	0.950074	−	0.312024i	$-0.101007\pi$
$757$	155.046	+	268.547i	0.204816	+	0.354752i	−0.745258	+	0.666776i	$0.767674\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	923.570			1.21363			0.606813	−	0.794844i	$-0.292448\pi$
$761$	923.570			1.21363			0.606813	+	0.794844i	$0.292448\pi$
$762$		0			0
$763$	205.924	−	118.890i	0.269887	−	0.155819i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.44336			−0.00579317
$768$		0			0
$769$	749.307	−	1297.84i	0.974392	−	1.68770i	0.292463	−	0.956277i	$-0.405525\pi$
$769$	749.307	−	1297.84i	0.974392	−	1.68770i	0.681928	−	0.731419i	$-0.261142\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	588.080	+	339.528i	0.760776	+	0.439234i	0.829574	−	0.558396i	$-0.188583\pi$
$773$	588.080	+	339.528i	0.760776	+	0.439234i	−0.0687981	+	0.997631i	$0.521916\pi$
$774$		0			0
$775$	−755.249	−	436.043i	−0.974515	−	0.562637i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1033.11	−	445.442i	−1.32620	−	0.571812i
$780$		0			0
$781$	142.774	−	82.4306i	0.182809	−	0.105545i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	914.370	−	1583.74i	1.16480	−	2.01750i
$786$		0			0
$787$		−	1166.31i		−	1.48197i	−0.671524	−	0.740983i	$-0.734360\pi$
$787$		−	1166.31i		−	1.48197i	0.671524	−	0.740983i	$-0.265640\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	690.045i		−	0.872371i
$792$		0			0
$793$	7.92243	−	4.57402i	0.00999045	−	0.00576799i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			771.416i			0.967899i	0.875096	+	0.483950i	$0.160798\pi$
$797$			771.416i			0.967899i	−0.875096	+	0.483950i	$0.839202\pi$
$798$		0			0
$799$	39.3551			0.0492555
$800$		0			0
$801$		0			0
$802$		0			0
$803$	151.214	+	261.911i	0.188311	+	0.326165i
$804$		0			0
$805$	−167.980			−0.208670
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1399.05			−1.72935			−0.864676	−	0.502330i	$-0.832476\pi$
$809$	−1399.05			−1.72935			−0.864676	+	0.502330i	$0.832476\pi$
$810$		0			0
$811$	−886.665	−	511.916i	−1.09330	−	0.631216i	−0.158846	−	0.987303i	$-0.550777\pi$
$811$	−886.665	−	511.916i	−1.09330	−	0.631216i	−0.934453	+	0.356087i	$0.884111\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	383.225	+	663.766i	0.470215	+	0.814437i
$816$		0			0
$817$	−165.081	+	382.871i	−0.202058	+	0.468630i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−351.435	+	608.703i	−0.428057	+	0.741417i	−0.996700	−	0.0811675i	$-0.974135\pi$
$821$	−351.435	+	608.703i	−0.428057	+	0.741417i	0.568643	+	0.822584i	$0.307468\pi$
$822$		0			0
$823$	44.2418	−	76.6291i	0.0537568	−	0.0931095i	−0.837895	−	0.545832i	$-0.816214\pi$
$823$	44.2418	−	76.6291i	0.0537568	−	0.0931095i	0.891652	+	0.452722i	$0.149547\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−616.049	−	355.676i	−0.744920	−	0.430080i	0.0789357	−	0.996880i	$-0.474848\pi$
$827$	−616.049	−	355.676i	−0.744920	−	0.430080i	−0.823855	+	0.566800i	$0.808181\pi$
$828$		0			0
$829$			763.135i			0.920548i	0.887777	+	0.460274i	$0.152249\pi$
$829$			763.135i			0.920548i	−0.887777	+	0.460274i	$0.847751\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−58.1871	−	100.783i	−0.0698524	−	0.120988i
$834$		0			0
$835$		−	1602.57i		−	1.91924i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1136.25	−	656.014i	1.35429	−	0.781900i	0.365444	−	0.930833i	$-0.380917\pi$
$839$	1136.25	−	656.014i	1.35429	−	0.781900i	0.988847	+	0.148933i	$0.0475839\pi$
$840$		0			0
$841$	−405.742	−	702.765i	−0.482452	−	0.835631i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	589.561	−	1021.15i	0.697706	−	1.20846i
$846$		0			0
$847$	64.4597			0.0761035
$848$		0			0
$849$		0			0
$850$		0			0
$851$	164.307	+	94.8629i	0.193076	+	0.111472i
$852$		0			0
$853$	231.166	+	400.392i	0.271004	+	0.469393i	0.969119	−	0.246593i	$-0.0793110\pi$
$853$	231.166	+	400.392i	0.271004	+	0.469393i	−0.698115	+	0.715985i	$0.745978\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−881.383	+	508.867i	−1.02845	+	0.593777i	−0.916541	−	0.399941i	$-0.869030\pi$
$857$	−881.383	+	508.867i	−1.02845	+	0.593777i	−0.111911	+	0.993718i	$0.535697\pi$
$858$		0			0
$859$	315.957	−	547.254i	0.367820	−	0.637083i	−0.621404	−	0.783490i	$-0.713438\pi$
$859$	315.957	−	547.254i	0.367820	−	0.637083i	0.989224	+	0.146407i	$0.0467709\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			436.469i			0.505758i	0.967498	+	0.252879i	$0.0813775\pi$
$863$			436.469i			0.505758i	−0.967498	+	0.252879i	$0.918623\pi$
$864$		0			0
$865$	1705.50	+	984.670i	1.97167	+	1.13835i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−537.857	+	310.532i	−0.618938	+	0.357344i
$870$		0			0
$871$	−4.17320	−	7.22820i	−0.00479128	−	0.00829874i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−31.3668			−0.0358478
$876$		0			0
$877$	−515.348	+	297.536i	−0.587626	+	0.339266i	−0.764158	−	0.645029i	$-0.776845\pi$
$877$	−515.348	+	297.536i	−0.587626	+	0.339266i	0.176532	+	0.984295i	$0.443512\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1532.34			−1.73932			−0.869659	−	0.493653i	$-0.835661\pi$
$881$	−1532.34			−1.73932			−0.869659	+	0.493653i	$0.835661\pi$
$882$		0			0
$883$	623.164	−	1079.35i	0.705735	−	1.22237i	−0.260691	−	0.965422i	$-0.583950\pi$
$883$	623.164	−	1079.35i	0.705735	−	1.22237i	0.966426	−	0.256947i	$-0.0827164\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−532.677	−	307.541i	−0.600538	−	0.346721i	0.168715	−	0.985665i	$-0.446038\pi$
$887$	−532.677	−	307.541i	−0.600538	−	0.346721i	−0.769253	+	0.638944i	$0.779372\pi$
$888$		0			0
$889$	−343.485	−	198.311i	−0.386372	−	0.223072i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	236.772	−	27.7133i	0.265142	−	0.0310339i
$894$		0			0
$895$	138.712	−	80.0853i	0.154985	−	0.0894807i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	99.9698	−	173.153i	0.111201	−	0.192606i
$900$		0			0
$901$		−	130.779i		−	0.145148i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			221.349i			0.244584i
$906$		0			0
$907$	−1373.26	+	792.851i	−1.51407	+	0.874146i	−0.514202	+	0.857669i	$0.671912\pi$
$907$	−1373.26	+	792.851i	−1.51407	+	0.874146i	−0.999864	+	0.0164769i	$0.994755\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1608.77i		−	1.76593i	−0.469435	−	0.882967i	$-0.655542\pi$
$911$		−	1608.77i		−	1.76593i	0.469435	−	0.882967i	$-0.344458\pi$
$912$		0			0
$913$	1506.90			1.65049
$914$		0			0
$915$		0			0
$916$		0			0
$917$	178.775	+	309.647i	0.194956	+	0.337674i
$918$		0			0
$919$	−1057.74			−1.15097			−0.575485	−	0.817812i	$-0.695187\pi$
$919$	−1057.74			−1.15097			−0.575485	+	0.817812i	$0.695187\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.85182			−0.00308973
$924$		0			0
$925$	−557.954	−	322.135i	−0.603194	−	0.348254i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	362.449	+	627.780i	0.390150	+	0.675759i	0.992469	−	0.122496i	$-0.0390900\pi$
$929$	362.449	+	627.780i	0.390150	+	0.675759i	−0.602319	+	0.798255i	$0.705757\pi$
$930$		0			0
$931$	−421.040	−	565.364i	−0.452245	−	0.607266i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−110.702	+	191.742i	−0.118398	+	0.205072i
$936$		0			0
$937$	376.045	−	651.329i	0.401329	−	0.695122i	−0.592558	−	0.805528i	$-0.701882\pi$
$937$	376.045	−	651.329i	0.401329	−	0.695122i	0.993887	+	0.110406i	$0.0352152\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1593.91	−	920.244i	−1.69385	−	0.977942i	−0.951361	−	0.308079i	$-0.900314\pi$
$941$	−1593.91	−	920.244i	−1.69385	−	0.977942i	−0.742485	−	0.669863i	$-0.766353\pi$
$942$		0			0
$943$			413.207i			0.438184i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	38.6460	+	66.9369i	0.0408089	+	0.0706831i	0.885708	−	0.464242i	$-0.153673\pi$
$947$	38.6460	+	66.9369i	0.0408089	+	0.0706831i	−0.844900	+	0.534925i	$0.820340\pi$
$948$		0			0
$949$		−	5.23150i		−	0.00551265i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−465.643	+	268.839i	−0.488607	+	0.282097i	−0.723996	−	0.689804i	$-0.757697\pi$
$953$	−465.643	+	268.839i	−0.488607	+	0.282097i	0.235389	+	0.971901i	$0.424363\pi$
$954$		0			0
$955$	−337.313	−	584.244i	−0.353208	−	0.611774i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−222.055	+	384.610i	−0.231548	+	0.401053i
$960$		0			0
$961$	−393.362			−0.409326
$962$		0			0
$963$		0			0
$964$		0			0
$965$	645.811	+	372.859i	0.669235	+	0.386383i
$966$		0			0
$967$	195.326	+	338.315i	0.201992	+	0.349860i	0.949170	−	0.314763i	$-0.101925\pi$
$967$	195.326	+	338.315i	0.201992	+	0.349860i	−0.747178	+	0.664624i	$0.768592\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−614.623	+	354.853i	−0.632979	+	0.365451i	−0.781905	−	0.623398i	$-0.785752\pi$
$971$	−614.623	+	354.853i	−0.632979	+	0.365451i	0.148926	+	0.988848i	$0.452418\pi$
$972$		0			0
$973$	31.6464	−	54.8132i	0.0325246	−	0.0563342i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	5.13076i		−	0.00525154i	−0.999997	−	0.00262577i	$-0.999164\pi$
$977$		−	5.13076i		−	0.00525154i	0.999997	−	0.00262577i	$-0.000835810\pi$
$978$		0			0
$979$	555.592	+	320.771i	0.567510	+	0.327652i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	209.932	−	121.204i	0.213562	−	0.123300i	−0.389404	−	0.921067i	$-0.627319\pi$
$983$	209.932	−	121.204i	0.213562	−	0.123300i	0.602966	+	0.797767i	$0.293986\pi$
$984$		0			0
$985$	885.030	+	1532.92i	0.898507	+	1.55626i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	153.135			0.154838
$990$		0			0
$991$	−1052.70	+	607.778i	−1.06226	+	0.613298i	−0.926057	−	0.377383i	$-0.876824\pi$
$991$	−1052.70	+	607.778i	−1.06226	+	0.613298i	−0.136205	+	0.990681i	$0.543491\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1247.88			1.25415
$996$		0			0
$997$	134.788	−	233.460i	0.135194	−	0.234163i	−0.790478	−	0.612491i	$-0.790168\pi$
$997$	134.788	−	233.460i	0.135194	−	0.234163i	0.925671	+	0.378328i	$0.123501\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.y.g.145.4	yes	8
3.2	odd	2	inner	684.3.y.g.145.1	✓	8
19.8	odd	6	inner	684.3.y.g.217.4	yes	8
57.8	even	6	inner	684.3.y.g.217.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.y.g.145.1	✓	8	3.2	odd	2	inner
684.3.y.g.145.4	yes	8	1.1	even	1	trivial
684.3.y.g.217.1	yes	8	57.8	even	6	inner
684.3.y.g.217.4	yes	8	19.8	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.48916 + 6.04340i) q^{5} -3.44949 q^{7} +O(q^{10})\)	\(q+(3.48916 + 6.04340i) q^{5} -3.44949 q^{7} +10.1150 q^{11} +(-0.151531 - 0.0874863i) q^{13} +(1.56834 + 2.71645i) q^{17} +(11.3485 + 15.2385i) q^{19} +(3.48916 - 6.04340i) q^{23} +(-11.8485 + 20.5222i) q^{25} +(-4.70502 - 2.71645i) q^{29} +36.8017i q^{31} +(-12.0358 - 20.8467i) q^{35} +27.1879i q^{37} +(-51.2800 + 29.6065i) q^{41} +(10.9722 + 19.0044i) q^{43} +(6.27337 - 10.8658i) q^{47} -37.1010 q^{49} +(-36.1075 - 20.8467i) q^{53} +(35.2929 + 61.1290i) q^{55} +(21.9924 - 12.6973i) q^{59} +(-26.1413 + 45.2781i) q^{61} -1.22102i q^{65} +(41.3105 + 23.8506i) q^{67} +(14.1151 - 8.14934i) q^{71} +(14.9495 + 25.8933i) q^{73} -34.8916 q^{77} +(-53.1742 + 30.7002i) q^{79} +148.976 q^{83} +(-10.9444 + 18.9562i) q^{85} +(54.9276 + 31.7124i) q^{89} +(0.522704 + 0.301783i) q^{91} +(-52.4958 + 121.753i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 60 * q^13 + 32 * q^19 - 36 * q^25 - 20 * q^43 - 336 * q^49 + 8 * q^55 + 124 * q^61 - 228 * q^67 + 100 * q^73 - 396 * q^79 + 128 * q^85 - 84 * q^91 + 216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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