LMFDB - Embedded newform 684.5.y.f.145.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,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, 5, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.6
Character	$\chi$	$=$	684.145
Dual form			684.5.y.f.217.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.78239 - 15.2115i) q^{5} +69.3595 q^{7} +O(q^{10})$	$q+(-8.78239 - 15.2115i) q^{5} +69.3595 q^{7} +198.542 q^{11} +(215.053 + 124.161i) q^{13} +(112.275 + 194.466i) q^{17} +(240.948 - 268.822i) q^{19} +(-462.386 + 800.876i) q^{23} +(158.239 - 274.079i) q^{25} +(-964.886 - 557.077i) q^{29} +1357.53i q^{31} +(-609.142 - 1055.06i) q^{35} +917.274i q^{37} +(620.986 - 358.526i) q^{41} +(-806.604 - 1397.08i) q^{43} +(-1967.20 + 3407.29i) q^{47} +2409.74 q^{49} +(1559.52 + 900.392i) q^{53} +(-1743.67 - 3020.13i) q^{55} +(3318.09 - 1915.70i) q^{59} +(239.475 - 414.783i) q^{61} -4361.71i q^{65} +(3194.82 + 1844.53i) q^{67} +(-4286.83 + 2475.00i) q^{71} +(1556.19 + 2695.40i) q^{73} +13770.8 q^{77} +(-4956.75 + 2861.78i) q^{79} +6050.14 q^{83} +(1972.09 - 3415.76i) q^{85} +(-11990.3 - 6922.59i) q^{89} +(14915.9 + 8611.72i) q^{91} +(-6205.29 - 1304.30i) q^{95} +(15083.0 - 8708.20i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 304 q^{7}+O(q^{10}) $ 24 * q + 304 * q^7	$ 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) $ 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * 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
$5$	−8.78239	−	15.2115i	−0.351295	−	0.608462i	0.635181	−	0.772363i	$-0.280925\pi$
$5$	−8.78239	−	15.2115i	−0.351295	−	0.608462i	−0.986477	+	0.163902i	$0.947592\pi$
$6$		0			0
$7$	69.3595			1.41550			0.707750	−	0.706463i	$-0.249710\pi$
$7$	69.3595			1.41550			0.707750	+	0.706463i	$0.249710\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	198.542			1.64084			0.820422	−	0.571759i	$-0.193739\pi$
$11$	198.542			1.64084			0.820422	+	0.571759i	$0.193739\pi$
$12$		0			0
$13$	215.053	+	124.161i	1.27250	+	0.734678i	0.975458	−	0.220186i	$-0.0706664\pi$
$13$	215.053	+	124.161i	1.27250	+	0.734678i	0.297042	+	0.954864i	$0.404000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	112.275	+	194.466i	0.388496	+	0.672894i	0.992247	−	0.124278i	$-0.0396615\pi$
$17$	112.275	+	194.466i	0.388496	+	0.672894i	−0.603752	+	0.797172i	$0.706328\pi$
$18$		0			0
$19$	240.948	−	268.822i	0.667446	−	0.744658i
$19$	240.948	−	268.822i	0.667446	−	0.744658i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−462.386	+	800.876i	−0.874075	+	1.51394i	−0.0163304	+	0.999867i	$0.505198\pi$
$23$	−462.386	+	800.876i	−0.874075	+	1.51394i	−0.857745	+	0.514076i	$0.828135\pi$
$24$		0			0
$25$	158.239	−	274.079i	0.253183	−	0.438526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−964.886	−	557.077i	−1.14731	−	0.662398i	−0.199078	−	0.979984i	$-0.563795\pi$
$29$	−964.886	−	557.077i	−1.14731	−	0.662398i	−0.948230	+	0.317585i	$0.897128\pi$
$30$		0			0
$31$			1357.53i			1.41262i	0.707903	+	0.706310i	$0.249641\pi$
$31$			1357.53i			1.41262i	−0.707903	+	0.706310i	$0.750359\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−609.142	−	1055.06i	−0.497259	−	0.861277i
$36$		0			0
$37$			917.274i			0.670032i	0.942212	+	0.335016i	$0.108742\pi$
$37$			917.274i			0.670032i	−0.942212	+	0.335016i	$0.891258\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	620.986	−	358.526i	0.369415	−	0.213282i	−0.303788	−	0.952740i	$-0.598251\pi$
$41$	620.986	−	358.526i	0.369415	−	0.213282i	0.673203	+	0.739458i	$0.264918\pi$
$42$		0			0
$43$	−806.604	−	1397.08i	−0.436238	−	0.755586i	0.561158	−	0.827709i	$-0.310356\pi$
$43$	−806.604	−	1397.08i	−0.436238	−	0.755586i	−0.997396	+	0.0721225i	$0.977023\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1967.20	+	3407.29i	−0.890539	+	1.54246i	−0.0513079	+	0.998683i	$0.516339\pi$
$47$	−1967.20	+	3407.29i	−0.890539	+	1.54246i	−0.839231	+	0.543775i	$0.816994\pi$
$48$		0			0
$49$	2409.74			1.00364
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1559.52	+	900.392i	0.555189	+	0.320538i	0.751212	−	0.660061i	$-0.229470\pi$
$53$	1559.52	+	900.392i	0.555189	+	0.320538i	−0.196024	+	0.980599i	$0.562803\pi$
$54$		0			0
$55$	−1743.67	−	3020.13i	−0.576421	−	0.998390i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3318.09	−	1915.70i	0.953201	−	0.550331i	0.0591269	−	0.998250i	$-0.481168\pi$
$59$	3318.09	−	1915.70i	0.953201	−	0.550331i	0.894074	+	0.447920i	$0.147835\pi$
$60$		0			0
$61$	239.475	−	414.783i	0.0643577	−	0.111471i	−0.832051	−	0.554699i	$-0.812833\pi$
$61$	239.475	−	414.783i	0.0643577	−	0.111471i	0.896409	+	0.443228i	$0.146167\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4361.71i		−	1.03236i
$66$		0			0
$67$	3194.82	+	1844.53i	0.711699	+	0.410900i	0.811690	−	0.584089i	$-0.198548\pi$
$67$	3194.82	+	1844.53i	0.711699	+	0.410900i	−0.0999909	+	0.994988i	$0.531881\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4286.83	+	2475.00i	−0.850392	+	0.490974i	−0.860783	−	0.508972i	$-0.830026\pi$
$71$	−4286.83	+	2475.00i	−0.850392	+	0.490974i	0.0103912	+	0.999946i	$0.496692\pi$
$72$		0			0
$73$	1556.19	+	2695.40i	0.292023	+	0.505798i	0.974288	−	0.225307i	$-0.0723383\pi$
$73$	1556.19	+	2695.40i	0.292023	+	0.505798i	−0.682265	+	0.731105i	$0.739005\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13770.8			2.32261
$78$		0			0
$79$	−4956.75	+	2861.78i	−0.794223	+	0.458545i	−0.841447	−	0.540339i	$-0.818296\pi$
$79$	−4956.75	+	2861.78i	−0.794223	+	0.458545i	0.0472240	+	0.998884i	$0.484963\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6050.14			0.878232			0.439116	−	0.898430i	$-0.355292\pi$
$83$	6050.14			0.878232			0.439116	+	0.898430i	$0.355292\pi$
$84$		0			0
$85$	1972.09	−	3415.76i	0.272954	−	0.472769i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11990.3	−	6922.59i	−1.51373	−	0.873955i	−0.999871	−	0.0160824i	$-0.994881\pi$
$89$	−11990.3	−	6922.59i	−1.51373	−	0.873955i	−0.513863	−	0.857872i	$-0.671786\pi$
$90$		0			0
$91$	14915.9	+	8611.72i	1.80122	+	1.03994i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6205.29	−	1304.30i	−0.687567	−	0.144520i
$96$		0			0
$97$	15083.0	−	8708.20i	1.60304	−	0.925518i	0.612170	−	0.790726i	$-0.290297\pi$
$97$	15083.0	−	8708.20i	1.60304	−	0.925518i	0.990874	−	0.134792i	$-0.0430366\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1364.20	+	2362.87i	−0.133732	+	0.231631i	−0.925113	−	0.379693i	$-0.876030\pi$
$101$	−1364.20	+	2362.87i	−0.133732	+	0.231631i	0.791380	+	0.611324i	$0.209363\pi$
$102$		0			0
$103$			12967.2i			1.22228i	0.791523	+	0.611140i	$0.209289\pi$
$103$			12967.2i			1.22228i	−0.791523	+	0.611140i	$0.790711\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9578.86i			0.836655i	0.908296	+	0.418327i	$0.137384\pi$
$107$			9578.86i			0.836655i	−0.908296	+	0.418327i	$0.862616\pi$
$108$		0			0
$109$	−663.160	+	382.876i	−0.0558169	+	0.0322259i	−0.527649	−	0.849463i	$-0.676926\pi$
$109$	−663.160	+	382.876i	−0.0558169	+	0.0322259i	0.471832	+	0.881689i	$0.343593\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	17088.1i		−	1.33825i	−0.743152	−	0.669123i	$-0.766670\pi$
$113$		−	17088.1i		−	1.33825i	0.743152	−	0.669123i	$-0.233330\pi$
$114$		0			0
$115$	16243.4			1.22823
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7787.35	+	13488.1i	0.549916	+	0.952482i
$120$		0			0
$121$	24778.0			1.69237
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−16536.9			−1.05836
$126$		0			0
$127$	17988.9	+	10385.9i	1.11531	+	0.643925i	0.940200	−	0.340623i	$-0.110638\pi$
$127$	17988.9	+	10385.9i	1.11531	+	0.643925i	0.175111	+	0.984549i	$0.443971\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14270.7	−	24717.7i	−0.831580	−	1.44034i	−0.896785	−	0.442467i	$-0.854103\pi$
$131$	−14270.7	−	24717.7i	−0.831580	−	1.44034i	0.0652043	−	0.997872i	$-0.479230\pi$
$132$		0			0
$133$	16712.0	−	18645.3i	0.944770	−	1.05406i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9914.08	+	17171.7i	−0.528216	+	0.914896i	0.471243	+	0.882003i	$0.343805\pi$
$137$	−9914.08	+	17171.7i	−0.528216	+	0.914896i	−0.999459	+	0.0328931i	$0.989528\pi$
$138$		0			0
$139$	4474.56	−	7750.17i	0.231591	−	0.401127i	−0.726686	−	0.686970i	$-0.758940\pi$
$139$	4474.56	−	7750.17i	0.231591	−	0.401127i	0.958276	+	0.285843i	$0.0922737\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	42697.0	+	24651.1i	2.08797	+	1.20549i
$144$		0			0
$145$			19569.9i			0.930790i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1035.41	−	1793.38i	−0.0466380	−	0.0807793i	0.841764	−	0.539846i	$-0.181517\pi$
$149$	−1035.41	−	1793.38i	−0.0466380	−	0.0807793i	−0.888402	+	0.459066i	$0.848184\pi$
$150$		0			0
$151$		−	3480.66i		−	0.152654i	−0.997083	−	0.0763269i	$-0.975681\pi$
$151$		−	3480.66i		−	0.152654i	0.997083	−	0.0763269i	$-0.0243192\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20650.1	−	11922.3i	0.859524	−	0.496247i
$156$		0			0
$157$	6672.92	+	11557.8i	0.270717	+	0.468896i	0.969046	−	0.246881i	$-0.0794057\pi$
$157$	6672.92	+	11557.8i	0.270717	+	0.468896i	−0.698328	+	0.715778i	$0.746072\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−32070.8	+	55548.3i	−1.23725	+	2.14298i
$162$		0			0
$163$	−34167.6			−1.28599			−0.642997	−	0.765869i	$-0.722309\pi$
$163$	−34167.6			−1.28599			−0.642997	+	0.765869i	$0.722309\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2263.68	+	1306.93i	0.0811674	+	0.0468620i	0.540034	−	0.841643i	$-0.318411\pi$
$167$	2263.68	+	1306.93i	0.0811674	+	0.0468620i	−0.458867	+	0.888505i	$0.651745\pi$
$168$		0			0
$169$	16551.2	+	28667.6i	0.579505	+	1.00373i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	26173.6	−	15111.4i	0.874524	−	0.504907i	0.00567514	−	0.999984i	$-0.498194\pi$
$173$	26173.6	−	15111.4i	0.874524	−	0.504907i	0.868849	+	0.495077i	$0.164860\pi$
$174$		0			0
$175$	10975.4	−	19010.0i	0.358380	−	0.620733i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	30711.0i		−	0.958491i	−0.877681	−	0.479245i	$-0.840910\pi$
$179$		−	30711.0i		−	0.958491i	0.877681	−	0.479245i	$-0.159090\pi$
$180$		0			0
$181$	−19402.1	−	11201.8i	−0.592230	−	0.341924i	0.173749	−	0.984790i	$-0.444412\pi$
$181$	−19402.1	−	11201.8i	−0.592230	−	0.341924i	−0.765979	+	0.642866i	$0.777745\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13953.2	−	8055.86i	0.407689	−	0.235379i
$186$		0			0
$187$	22291.4	+	38609.8i	0.637461	+	1.10411i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−53559.7			−1.46815			−0.734077	−	0.679066i	$-0.762385\pi$
$191$	−53559.7			−1.46815			−0.734077	+	0.679066i	$0.762385\pi$
$192$		0			0
$193$	386.993	−	223.431i	0.0103894	−	0.00599830i	−0.494796	−	0.869009i	$-0.664757\pi$
$193$	386.993	−	223.431i	0.0103894	−	0.00599830i	0.505186	+	0.863011i	$0.331424\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1325.98			0.0341669			0.0170835	−	0.999854i	$-0.494562\pi$
$197$	1325.98			0.0341669			0.0170835	+	0.999854i	$0.494562\pi$
$198$		0			0
$199$	20670.0	−	35801.5i	0.521956	−	0.904054i	−0.477718	−	0.878513i	$-0.658536\pi$
$199$	20670.0	−	35801.5i	0.521956	−	0.904054i	0.999674	−	0.0255411i	$-0.00813085\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−66924.0	−	38638.6i	−1.62401	−	0.937625i
$204$		0			0
$205$	−10907.5	−	6297.43i	−0.259547	−	0.149850i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	47838.3	−	53372.4i	1.09517	−	1.22187i
$210$		0			0
$211$	60770.8	−	35086.0i	1.36499	−	0.788078i	0.374708	−	0.927143i	$-0.377743\pi$
$211$	60770.8	−	35086.0i	1.36499	−	0.788078i	0.990283	+	0.139065i	$0.0444096\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14167.8	+	24539.4i	−0.306497	+	0.530868i
$216$		0			0
$217$			94157.4i			1.99956i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			55760.7i			1.14168i
$222$		0			0
$223$	83511.7	−	48215.5i	1.67934	−	0.969565i	0.717250	−	0.696816i	$-0.245400\pi$
$223$	83511.7	−	48215.5i	1.67934	−	0.969565i	0.962085	−	0.272749i	$-0.0879330\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			61235.8i			1.18837i	0.804327	+	0.594187i	$0.202526\pi$
$227$			61235.8i			1.18837i	−0.804327	+	0.594187i	$0.797474\pi$
$228$		0			0
$229$	63404.2			1.20906			0.604529	−	0.796583i	$-0.293361\pi$
$229$	63404.2			1.20906			0.604529	+	0.796583i	$0.293361\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−25896.2	−	44853.6i	−0.477007	−	0.826200i	0.522646	−	0.852550i	$-0.324945\pi$
$233$	−25896.2	−	44853.6i	−0.477007	−	0.826200i	−0.999653	+	0.0263499i	$0.991612\pi$
$234$		0			0
$235$	69106.8			1.25137
$236$		0			0
$237$		0			0
$238$		0			0
$239$	72920.0			1.27659			0.638294	−	0.769793i	$-0.279640\pi$
$239$	72920.0			1.27659			0.638294	+	0.769793i	$0.279640\pi$
$240$		0			0
$241$	−12718.6	−	7343.10i	−0.218981	−	0.126429i	0.386497	−	0.922290i	$-0.373685\pi$
$241$	−12718.6	−	7343.10i	−0.218981	−	0.126429i	−0.605478	+	0.795862i	$0.707018\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21163.2	−	36655.8i	−0.352574	−	0.610676i
$246$		0			0
$247$	85193.5	−	27894.5i	1.39641	−	0.457219i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	29165.7	−	50516.5i	0.462940	−	0.801836i	−0.536166	−	0.844113i	$-0.680128\pi$
$251$	29165.7	−	50516.5i	0.462940	−	0.801836i	0.999106	+	0.0422769i	$0.0134612\pi$
$252$		0			0
$253$	−91803.0	+	159008.i	−1.43422	+	2.48414i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	37732.1	+	21784.6i	0.571274	+	0.329825i	0.757658	−	0.652652i	$-0.226344\pi$
$257$	37732.1	+	21784.6i	0.571274	+	0.329825i	−0.186384	+	0.982477i	$0.559677\pi$
$258$		0			0
$259$			63621.7i			0.948430i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	57099.3	+	98898.8i	0.825504	+	1.42981i	0.901534	+	0.432709i	$0.142442\pi$
$263$	57099.3	+	98898.8i	0.825504	+	1.42981i	−0.0760299	+	0.997106i	$0.524224\pi$
$264$		0			0
$265$		−	31630.4i		−	0.450415i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	78008.2	−	45038.1i	1.07804	−	0.622408i	0.147675	−	0.989036i	$-0.452821\pi$
$269$	78008.2	−	45038.1i	1.07804	−	0.622408i	0.930368	+	0.366628i	$0.119488\pi$
$270$		0			0
$271$	−60721.0	−	105172.i	−0.826800	−	1.43206i	−0.900536	−	0.434781i	$-0.856826\pi$
$271$	−60721.0	−	105172.i	−0.826800	−	1.43206i	0.0737362	−	0.997278i	$-0.476508\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	31417.2	−	54416.1i	0.415434	−	0.719552i
$276$		0			0
$277$	−103007.			−1.34247			−0.671237	−	0.741243i	$-0.734237\pi$
$277$	−103007.			−1.34247			−0.671237	+	0.741243i	$0.734237\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22444.9	−	12958.6i	−0.284253	−	0.164114i	0.351094	−	0.936340i	$-0.385810\pi$
$281$	−22444.9	−	12958.6i	−0.284253	−	0.164114i	−0.635347	+	0.772227i	$0.719143\pi$
$282$		0			0
$283$	−12448.4	−	21561.2i	−0.155432	−	0.269216i	0.777784	−	0.628531i	$-0.216344\pi$
$283$	−12448.4	−	21561.2i	−0.155432	−	0.269216i	−0.933216	+	0.359315i	$0.883010\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	43071.3	−	24867.2i	0.522906	−	0.301900i
$288$		0			0
$289$	16549.0	−	28663.8i	0.198142	−	0.343192i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	63231.8i		−	0.736547i	−0.929718	−	0.368273i	$-0.879949\pi$
$293$		−	63231.8i		−	0.736547i	0.929718	−	0.368273i	$-0.120051\pi$
$294$		0			0
$295$	−58281.5	−	33648.8i	−0.669710	−	0.386657i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−198874.	+	114820.i	−2.22452	+	1.28433i
$300$		0			0
$301$	−55945.6	−	96900.7i	−0.617495	−	1.06953i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8412.65			−0.0904343
$306$		0			0
$307$	−61727.4	+	35638.3i	−0.654940	+	0.378129i	−0.790346	−	0.612661i	$-0.790099\pi$
$307$	−61727.4	+	35638.3i	−0.654940	+	0.378129i	0.135407	+	0.990790i	$0.456766\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−114027.			−1.17893			−0.589464	−	0.807795i	$-0.700661\pi$
$311$	−114027.			−1.17893			−0.589464	+	0.807795i	$0.700661\pi$
$312$		0			0
$313$	−75917.5	+	131493.i	−0.774914	+	1.34219i	0.159929	+	0.987129i	$0.448873\pi$
$313$	−75917.5	+	131493.i	−0.774914	+	1.34219i	−0.934843	+	0.355062i	$0.884460\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27026.5	−	15603.8i	−0.268950	−	0.155278i	0.359460	−	0.933160i	$-0.382961\pi$
$317$	−27026.5	−	15603.8i	−0.268950	−	0.155278i	−0.628410	+	0.777882i	$0.716294\pi$
$318$		0			0
$319$	−191570.	−	110603.i	−1.88255	−	1.08689i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	79329.3	+	16674.3i	0.760376	+	0.159824i
$324$		0			0
$325$	68059.6	−	39294.2i	0.644351	−	0.372016i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−136444.	+	236328.i	−1.26056	+	2.18335i
$330$		0			0
$331$			59754.0i			0.545395i	0.962100	+	0.272697i	$0.0879158\pi$
$331$			59754.0i			0.545395i	−0.962100	+	0.272697i	$0.912084\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	64797.4i		−	0.577389i
$336$		0			0
$337$	−114120.	+	65887.0i	−1.00485	+	0.580150i	−0.909679	−	0.415311i	$-0.863673\pi$
$337$	−114120.	+	65887.0i	−1.00485	+	0.580150i	−0.0951696	+	0.995461i	$0.530339\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			269526.i			2.31789i
$342$		0			0
$343$	606.009			0.00515099
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−85828.2	−	148659.i	−0.712806	−	1.23462i	−0.963800	−	0.266627i	$-0.914091\pi$
$347$	−85828.2	−	148659.i	−0.712806	−	1.23462i	0.250994	−	0.967989i	$-0.419242\pi$
$348$		0			0
$349$	−38126.1			−0.313019			−0.156510	−	0.987676i	$-0.550024\pi$
$349$	−38126.1			−0.313019			−0.156510	+	0.987676i	$0.550024\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	137522.			1.10363			0.551815	−	0.833966i	$-0.313935\pi$
$353$	137522.			1.10363			0.551815	+	0.833966i	$0.313935\pi$
$354$		0			0
$355$	75297.1	+	43472.8i	0.597478	+	0.344954i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−104262.	−	180588.i	−0.808983	−	1.40120i	−0.913569	−	0.406683i	$-0.866685\pi$
$359$	−104262.	−	180588.i	−0.808983	−	1.40120i	0.104587	−	0.994516i	$-0.466648\pi$
$360$		0			0
$361$	−14209.1	−	129544.i	−0.109031	−	0.994038i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	27334.1	−	47344.1i	0.205173	−	0.355369i
$366$		0			0
$367$	23137.6	−	40075.4i	0.171785	−	0.297540i	−0.767259	−	0.641337i	$-0.778380\pi$
$367$	23137.6	−	40075.4i	0.171785	−	0.297540i	0.939044	+	0.343797i	$0.111713\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	108168.	+	62450.7i	0.785869	+	0.453722i
$372$		0			0
$373$		−	51521.8i		−	0.370317i	−0.982709	−	0.185158i	$-0.940720\pi$
$373$		−	51521.8i		−	0.370317i	0.982709	−	0.185158i	$-0.0592798\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−138334.	−	239602.i	−0.973300	−	1.68580i
$378$		0			0
$379$		−	32907.0i		−	0.229092i	−0.993418	−	0.114546i	$-0.963459\pi$
$379$		−	32907.0i		−	0.229092i	0.993418	−	0.114546i	$-0.0365414\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	71494.5	−	41277.4i	0.487388	−	0.281394i	−0.236102	−	0.971728i	$-0.575870\pi$
$383$	71494.5	−	41277.4i	0.487388	−	0.281394i	0.723490	+	0.690335i	$0.242537\pi$
$384$		0			0
$385$	−120940.	−	209475.i	−0.815924	−	1.41322i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−45414.6	+	78660.4i	−0.300121	+	0.519825i	−0.976163	−	0.217038i	$-0.930360\pi$
$389$	−45414.6	+	78660.4i	−0.300121	+	0.519825i	0.676042	+	0.736863i	$0.263694\pi$
$390$		0			0
$391$	−207658.			−1.35830
$392$		0			0
$393$		0			0
$394$		0			0
$395$	87064.1	+	50266.5i	0.558014	+	0.322170i
$396$		0			0
$397$	−14737.1	−	25525.4i	−0.0935041	−	0.161954i	0.815479	−	0.578786i	$-0.196473\pi$
$397$	−14737.1	−	25525.4i	−0.0935041	−	0.161954i	−0.908983	+	0.416833i	$0.863140\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8870.42	−	5121.34i	0.0551639	−	0.0318489i	−0.472164	−	0.881511i	$-0.656527\pi$
$401$	8870.42	−	5121.34i	0.0551639	−	0.0318489i	0.527328	+	0.849662i	$0.323194\pi$
$402$		0			0
$403$	−168551.	+	291940.i	−1.03782	+	1.79756i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			182118.i			1.09942i
$408$		0			0
$409$	91906.5	+	53062.3i	0.549414	+	0.317204i	0.748886	−	0.662699i	$-0.230589\pi$
$409$	91906.5	+	53062.3i	0.549414	+	0.317204i	−0.199472	+	0.979904i	$0.563923\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	230141.	−	132872.i	1.34925	−	0.778993i
$414$		0			0
$415$	−53134.7	−	92032.0i	−0.308519	−	0.534371i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−62121.1			−0.353843			−0.176922	−	0.984225i	$-0.556614\pi$
$419$	−62121.1			−0.353843			−0.176922	+	0.984225i	$0.556614\pi$
$420$		0			0
$421$	−243094.	+	140351.i	−1.37155	+	0.791863i	−0.991123	−	0.132949i	$-0.957555\pi$
$421$	−243094.	+	140351.i	−1.37155	+	0.791863i	−0.380424	+	0.924812i	$0.624222\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	71065.5			0.393442
$426$		0			0
$427$	16609.9	−	28769.1i	0.0910984	−	0.157787i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	143750.	+	82993.8i	0.773841	+	0.446777i	0.834243	−	0.551397i	$-0.185905\pi$
$431$	143750.	+	82993.8i	0.773841	+	0.446777i	−0.0604020	+	0.998174i	$0.519238\pi$
$432$		0			0
$433$	−119017.	−	68714.6i	−0.634795	−	0.366499i	0.147811	−	0.989016i	$-0.452777\pi$
$433$	−119017.	−	68714.6i	−0.634795	−	0.366499i	−0.782607	+	0.622516i	$0.786110\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	103882.	+	317269.i	0.543972	+	1.66136i
$438$		0			0
$439$	270999.	−	156461.i	1.40617	−	0.811854i	0.411156	−	0.911565i	$-0.365125\pi$
$439$	270999.	−	156461.i	1.40617	−	0.811854i	0.995016	+	0.0997110i	$0.0317918\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	114383.	−	198117.i	0.582846	−	1.00952i	−0.412294	−	0.911051i	$-0.635272\pi$
$443$	114383.	−	198117.i	0.582846	−	1.00952i	0.995140	−	0.0984684i	$-0.0313943\pi$
$444$		0			0
$445$			243188.i			1.22807i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			185010.i			0.917703i	0.888513	+	0.458852i	$0.151739\pi$
$449$			185010.i			0.917703i	−0.888513	+	0.458852i	$0.848261\pi$
$450$		0			0
$451$	123292.	−	71182.6i	0.606152	−	0.349962i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	302526.i		−	1.46130i
$456$		0			0
$457$	−285673.			−1.36784			−0.683922	−	0.729555i	$-0.739727\pi$
$457$	−285673.			−1.36784			−0.683922	+	0.729555i	$0.739727\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5868.61	+	10164.7i	0.0276143	+	0.0478293i	0.879502	−	0.475895i	$-0.157876\pi$
$461$	5868.61	+	10164.7i	0.0276143	+	0.0478293i	−0.851888	+	0.523724i	$0.824542\pi$
$462$		0			0
$463$	−204761.			−0.955181			−0.477590	−	0.878583i	$-0.658490\pi$
$463$	−204761.			−0.955181			−0.477590	+	0.878583i	$0.658490\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−216706.			−0.993660			−0.496830	−	0.867848i	$-0.665503\pi$
$467$	−216706.			−0.993660			−0.496830	+	0.867848i	$0.665503\pi$
$468$		0			0
$469$	221591.	+	127935.i	1.00741	+	0.581628i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−160145.	−	277379.i	−0.715798	−	1.23980i
$474$		0			0
$475$	−35550.8	−	108577.i	−0.157566	−	0.481227i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	133293.	−	230870.i	0.580947	−	1.00623i	−0.414420	−	0.910086i	$-0.636016\pi$
$479$	133293.	−	230870.i	0.580947	−	1.00623i	0.995367	−	0.0961443i	$-0.0306510\pi$
$480$		0			0
$481$	−113889.	+	197262.i	−0.492258	+	0.852616i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−264930.	−	152958.i	−1.12628	−	0.650261i
$486$		0			0
$487$		−	370386.i		−	1.56170i	−0.624720	−	0.780849i	$-0.714787\pi$
$487$		−	370386.i		−	1.56170i	0.624720	−	0.780849i	$-0.285213\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	92032.0	+	159404.i	0.381747	+	0.661205i	0.991312	−	0.131531i	$-0.0419892\pi$
$491$	92032.0	+	159404.i	0.381747	+	0.661205i	−0.609565	+	0.792736i	$0.708656\pi$
$492$		0			0
$493$		−	250184.i		−	1.02936i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−297332.	+	171665.i	−1.20373	+	0.694973i
$498$		0			0
$499$	−159742.	−	276681.i	−0.641530	−	1.11116i	−0.985091	−	0.172032i	$-0.944967\pi$
$499$	−159742.	−	276681.i	−0.641530	−	1.11116i	0.343562	−	0.939130i	$-0.388367\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−122381.	+	211970.i	−0.483701	+	0.837795i	−0.999825	−	0.0187194i	$-0.994041\pi$
$503$	−122381.	+	211970.i	−0.483701	+	0.837795i	0.516124	+	0.856514i	$0.327374\pi$
$504$		0			0
$505$	47923.9			0.187918
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−133437.	−	77039.8i	−0.515039	−	0.297358i	0.219864	−	0.975531i	$-0.429439\pi$
$509$	−133437.	−	77039.8i	−0.515039	−	0.297358i	−0.734902	+	0.678173i	$0.762772\pi$
$510$		0			0
$511$	107936.	+	186951.i	0.413358	+	0.715957i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	197251.	−	113883.i	0.743710	−	0.429381i
$516$		0			0
$517$	−390572.	+	676491.i	−1.46123	+	2.53093i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	263048.i		−	0.969080i	−0.874769	−	0.484540i	$-0.838987\pi$
$521$		−	263048.i		−	0.969080i	0.874769	−	0.484540i	$-0.161013\pi$
$522$		0			0
$523$	12750.2	+	7361.33i	0.0466137	+	0.0269124i	0.523126	−	0.852256i	$-0.324766\pi$
$523$	12750.2	+	7361.33i	0.0466137	+	0.0269124i	−0.476512	+	0.879168i	$0.658099\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−263993.	+	152417.i	−0.950543	+	0.548796i
$528$		0			0
$529$	−287681.	−	498277.i	−1.02801	−	1.78057i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	178059.			0.626774
$534$		0			0
$535$	145709.	−	84125.3i	0.509072	−	0.293913i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	478434.			1.64681
$540$		0			0
$541$	63018.2	−	109151.i	0.215314	−	0.372934i	−0.738056	−	0.674739i	$-0.764256\pi$
$541$	63018.2	−	109151.i	0.215314	−	0.372934i	0.953370	+	0.301805i	$0.0975893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11648.3	+	6725.12i	0.0392164	+	0.0226416i
$546$		0			0
$547$	−111488.	−	64367.5i	−0.372608	−	0.215126i	0.301989	−	0.953311i	$-0.402349\pi$
$547$	−111488.	−	64367.5i	−0.372608	−	0.215126i	−0.674597	+	0.738186i	$0.735683\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−382242.	+	125156.i	−1.25903	+	0.412237i
$552$		0			0
$553$	−343797.	+	198491.i	−1.12422	+	0.649070i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	139231.	−	241155.i	0.448772	−	0.777296i	−0.549534	−	0.835471i	$-0.685195\pi$
$557$	139231.	−	241155.i	0.448772	−	0.777296i	0.998306	+	0.0581752i	$0.0185282\pi$
$558$		0			0
$559$		−	400594.i		−	1.28198i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			266872.i			0.841951i	0.907072	+	0.420976i	$0.138312\pi$
$563$			266872.i			0.841951i	−0.907072	+	0.420976i	$0.861688\pi$
$564$		0			0
$565$	−259936.	+	150074.i	−0.814271	+	0.470120i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			144616.i			0.446676i	0.974741	+	0.223338i	$0.0716953\pi$
$569$			144616.i			0.446676i	−0.974741	+	0.223338i	$0.928305\pi$
$570$		0			0
$571$	210006.			0.644108			0.322054	−	0.946721i	$-0.395627\pi$
$571$	210006.			0.644108			0.322054	+	0.946721i	$0.395627\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	146335.	+	253460.i	0.442602	+	0.766609i
$576$		0			0
$577$	−284942.			−0.855865			−0.427933	−	0.903811i	$-0.640758\pi$
$577$	−284942.			−0.855865			−0.427933	+	0.903811i	$0.640758\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	419635.			1.24314
$582$		0			0
$583$	309631.	+	178766.i	0.910978	+	0.525953i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	194123.	+	336232.i	0.563380	+	0.975803i	0.997198	+	0.0748029i	$0.0238328\pi$
$587$	194123.	+	336232.i	0.563380	+	0.975803i	−0.433818	+	0.901001i	$0.642834\pi$
$588$		0			0
$589$	364933.	+	327093.i	1.05192	+	0.942847i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−200973.	+	348096.i	−0.571516	+	0.989895i	0.424894	+	0.905243i	$0.360311\pi$
$593$	−200973.	+	348096.i	−0.571516	+	0.989895i	−0.996411	+	0.0846520i	$0.973022\pi$
$594$		0			0
$595$	136783.	−	236915.i	0.386366	−	0.669205i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	417883.	+	241265.i	1.16467	+	0.672420i	0.952418	−	0.304796i	$-0.0985884\pi$
$599$	417883.	+	241265.i	1.16467	+	0.672420i	0.212247	+	0.977216i	$0.431922\pi$
$600$		0			0
$601$			590266.i			1.63418i	0.576514	+	0.817088i	$0.304413\pi$
$601$			590266.i			1.63418i	−0.576514	+	0.817088i	$0.695587\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−217610.	−	376911.i	−0.594521	−	1.02974i
$606$		0			0
$607$		−	105690.i		−	0.286852i	−0.989661	−	0.143426i	$-0.954188\pi$
$607$		−	105690.i		−	0.286852i	0.989661	−	0.143426i	$-0.0458119\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−846103.	+	488498.i	−2.26642	+	1.30852i
$612$		0			0
$613$	62034.8	+	107447.i	0.165088	+	0.285940i	0.936686	−	0.350169i	$-0.113876\pi$
$613$	62034.8	+	107447.i	0.165088	+	0.285940i	−0.771599	+	0.636109i	$0.780543\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	176806.	−	306238.i	0.464438	−	0.804430i	−0.534738	−	0.845018i	$-0.679590\pi$
$617$	176806.	−	306238.i	0.464438	−	0.804430i	0.999176	+	0.0405878i	$0.0129230\pi$
$618$		0			0
$619$	577689.			1.50769			0.753846	−	0.657051i	$-0.228196\pi$
$619$	577689.			1.50769			0.753846	+	0.657051i	$0.228196\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−831640.	−	480148.i	−2.14269	−	1.23708i
$624$		0			0
$625$	46333.5	+	80252.0i	0.118614	+	0.205445i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−178379.	+	102987.i	−0.450861	+	0.260305i
$630$		0			0
$631$	13672.7	−	23681.8i	0.0343396	−	0.0594779i	−0.848345	−	0.529444i	$-0.822401\pi$
$631$	13672.7	−	23681.8i	0.0343396	−	0.0594779i	0.882684	+	0.469966i	$0.155734\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	364851.i		−	0.904832i
$636$		0			0
$637$	518220.	+	299195.i	1.27713	+	0.737352i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	658671.	−	380284.i	1.60307	−	0.925534i	0.612203	−	0.790701i	$-0.290284\pi$
$641$	658671.	−	380284.i	1.60307	−	0.925534i	0.990868	−	0.134833i	$-0.0430497\pi$
$642$		0			0
$643$	91546.7	+	158563.i	0.221422	+	0.383514i	0.955240	−	0.295832i	$-0.0955969\pi$
$643$	91546.7	+	158563.i	0.221422	+	0.383514i	−0.733818	+	0.679346i	$0.762264\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	110622.			0.264261			0.132131	−	0.991232i	$-0.457818\pi$
$647$	110622.			0.264261			0.132131	+	0.991232i	$0.457818\pi$
$648$		0			0
$649$	658781.	−	380347.i	1.56405	−	0.903007i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	531577.			1.24664			0.623318	−	0.781969i	$-0.285784\pi$
$653$	531577.			1.24664			0.623318	+	0.781969i	$0.285784\pi$
$654$		0			0
$655$	−250662.	+	434160.i	−0.584261	+	1.01197i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	177115.	+	102257.i	0.407835	+	0.235464i	0.689859	−	0.723944i	$-0.257672\pi$
$659$	177115.	+	102257.i	0.407835	+	0.235464i	−0.282024	+	0.959407i	$0.591006\pi$
$660$		0			0
$661$	−131792.	−	76090.3i	−0.301639	−	0.174151i	0.341540	−	0.939867i	$-0.389051\pi$
$661$	−131792.	−	76090.3i	−0.301639	−	0.174151i	−0.643179	+	0.765716i	$0.722385\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−430396.	−	90465.3i	−0.973250	−	0.204568i
$666$		0			0
$667$	892299.	−	515169.i	2.00567	−	1.15797i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	47545.9	−	82351.9i	0.105601	−	0.182906i
$672$		0			0
$673$		−	115954.i		−	0.256008i	−0.991774	−	0.128004i	$-0.959143\pi$
$673$		−	115954.i		−	0.256008i	0.991774	−	0.128004i	$-0.0408571\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	277521.i		−	0.605505i	−0.953069	−	0.302753i	$-0.902094\pi$
$677$		−	277521.i		−	0.605505i	0.953069	−	0.302753i	$-0.0979056\pi$
$678$		0			0
$679$	1.04615e6	−	603996.i	2.26911	−	1.31007i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			472761.i			1.01344i	0.862109	+	0.506722i	$0.169143\pi$
$683$			472761.i			1.01344i	−0.862109	+	0.506722i	$0.830857\pi$
$684$		0			0
$685$	348277.			0.742239
$686$		0			0
$687$		0			0
$688$		0			0
$689$	223586.	+	387263.i	0.470985	+	0.815770i
$690$		0			0
$691$	154927.			0.324468			0.162234	−	0.986752i	$-0.448130\pi$
$691$	154927.			0.324468			0.162234	+	0.986752i	$0.448130\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−157189.			−0.325427
$696$		0			0
$697$	139443.	+	80507.3i	0.287032	+	0.165718i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	210714.	+	364967.i	0.428802	+	0.742708i	0.996767	−	0.0803454i	$-0.0256023\pi$
$701$	210714.	+	364967.i	0.428802	+	0.742708i	−0.567965	+	0.823053i	$0.692269\pi$
$702$		0			0
$703$	246583.	+	221015.i	0.498945	+	0.447210i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−94620.5	+	163887.i	−0.189298	+	0.327874i
$708$		0			0
$709$	−275405.	+	477016.i	−0.547873	+	0.948943i	0.450547	+	0.892753i	$0.351229\pi$
$709$	−275405.	+	477016.i	−0.547873	+	0.948943i	−0.998420	+	0.0561908i	$0.982104\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.08721e6	−	627701.i	−2.13862	−	1.23474i
$714$		0			0
$715$		−	865982.i		−	1.69394i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−448073.	−	776085.i	−0.866744	−	1.50124i	−0.865305	−	0.501246i	$-0.832875\pi$
$719$	−448073.	−	776085.i	−0.866744	−	1.50124i	−0.00143881	−	0.999999i	$-0.500458\pi$
$720$		0			0
$721$			899396.i			1.73014i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−305366.	+	176303.i	−0.580958	+	0.335416i
$726$		0			0
$727$	252853.	+	437953.i	0.478408	+	0.828627i	0.999694	−	0.0247554i	$-0.00788068\pi$
$727$	252853.	+	437953.i	0.478408	+	0.828627i	−0.521286	+	0.853382i	$0.674547\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	181123.	−	313715.i	0.338953	−	0.587084i
$732$		0			0
$733$	−527851.			−0.982435			−0.491217	−	0.871037i	$-0.663448\pi$
$733$	−527851.			−0.982435			−0.491217	+	0.871037i	$0.663448\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	634305.	+	366216.i	1.16779	+	0.674222i
$738$		0			0
$739$	214471.	+	371475.i	0.392717	+	0.680206i	0.992807	−	0.119727i	$-0.0382018\pi$
$739$	214471.	+	371475.i	0.392717	+	0.680206i	−0.600090	+	0.799933i	$0.704868\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	818250.	−	472417.i	1.48221	−	0.855752i	0.482410	−	0.875946i	$-0.339762\pi$
$743$	818250.	−	472417.i	1.48221	−	0.855752i	0.999796	+	0.0201940i	$0.00642839\pi$
$744$		0			0
$745$	−18186.7	+	31500.3i	−0.0327674	+	0.0567548i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			664385.i			1.18428i
$750$		0			0
$751$	308732.	+	178246.i	0.547396	+	0.316039i	0.748071	−	0.663619i	$-0.230980\pi$
$751$	308732.	+	178246.i	0.547396	+	0.316039i	−0.200675	+	0.979658i	$0.564314\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−52946.2	+	30568.5i	−0.0928839	+	0.0536266i
$756$		0			0
$757$	−279387.	−	483913.i	−0.487545	−	0.844453i	0.512352	−	0.858776i	$-0.328774\pi$
$757$	−279387.	−	483913.i	−0.487545	−	0.844453i	−0.999897	+	0.0143221i	$0.995441\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−702384.			−1.21284			−0.606422	−	0.795143i	$-0.707396\pi$
$761$	−702384.			−1.21284			−0.606422	+	0.795143i	$0.707396\pi$
$762$		0			0
$763$	−45996.4	+	26556.1i	−0.0790087	+	0.0456157i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	951419.			1.61726
$768$		0			0
$769$	−33435.8	+	57912.5i	−0.0565404	+	0.0979309i	−0.892910	−	0.450235i	$-0.851340\pi$
$769$	−33435.8	+	57912.5i	−0.0565404	+	0.0979309i	0.836370	+	0.548166i	$0.184674\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−103844.	−	59954.4i	−0.173789	−	0.100337i	0.410582	−	0.911824i	$-0.365325\pi$
$773$	−103844.	−	59954.4i	−0.173789	−	0.100337i	−0.584371	+	0.811486i	$0.698659\pi$
$774$		0			0
$775$	372069.	+	214814.i	0.619470	+	0.357651i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	53245.7	−	253321.i	0.0877424	−	0.417442i
$780$		0			0
$781$	−851115.	+	491392.i	−1.39536	+	0.805612i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	117208.	−	203011.i	0.190204	−	0.329442i
$786$		0			0
$787$		−	388902.i		−	0.627900i	−0.949440	−	0.313950i	$-0.898348\pi$
$787$		−	388902.i		−	0.627900i	0.949440	−	0.313950i	$-0.101652\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.18522e6i		−	1.89429i
$792$		0			0
$793$	102999.	−	59466.8i	0.163790	−	0.0945645i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			506429.i			0.797263i	0.917111	+	0.398632i	$0.130515\pi$
$797$			506429.i			0.797263i	−0.917111	+	0.398632i	$0.869485\pi$
$798$		0			0
$799$	−883472.			−1.38388
$800$		0			0
$801$		0			0
$802$		0			0
$803$	308969.	+	535150.i	0.479164	+	0.829936i
$804$		0			0
$805$	1.12663e6			1.73857
$806$		0			0
$807$		0			0
$808$		0			0
$809$	213567.			0.326315			0.163158	−	0.986600i	$-0.447832\pi$
$809$	213567.			0.326315			0.163158	+	0.986600i	$0.447832\pi$
$810$		0			0
$811$	−259874.	−	150038.i	−0.395113	−	0.228119i	0.289260	−	0.957251i	$-0.406591\pi$
$811$	−259874.	−	150038.i	−0.395113	−	0.228119i	−0.684373	+	0.729132i	$0.739924\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	300073.	+	519741.i	0.451764	+	0.782477i
$816$		0			0
$817$	−569915.	−	119791.i	−0.853819	−	0.179465i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	458968.	−	794956.i	0.680920	−	1.17939i	−0.293780	−	0.955873i	$-0.594913\pi$
$821$	458968.	−	794956.i	0.680920	−	1.17939i	0.974700	−	0.223515i	$-0.0717532\pi$
$822$		0			0
$823$	254126.	−	440159.i	0.375188	−	0.649845i	−0.615167	−	0.788397i	$-0.710911\pi$
$823$	254126.	−	440159.i	0.375188	−	0.649845i	0.990355	+	0.138552i	$0.0442447\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	923604.	+	533243.i	1.35044	+	0.779676i	0.988311	−	0.152453i	$-0.0487172\pi$
$827$	923604.	+	533243.i	1.35044	+	0.779676i	0.362127	+	0.932129i	$0.382051\pi$
$828$		0			0
$829$		−	170275.i		−	0.247766i	−0.992297	−	0.123883i	$-0.960465\pi$
$829$		−	170275.i		−	0.247766i	0.992297	−	0.123883i	$-0.0395347\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	270554.	+	468613.i	0.389909	+	0.675343i
$834$		0			0
$835$		−	45912.0i		−	0.0658496i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−68969.0	+	39819.3i	−0.0979784	+	0.0565678i	−0.548189	−	0.836355i	$-0.684682\pi$
$839$	−68969.0	+	39819.3i	−0.0979784	+	0.0565678i	0.450210	+	0.892923i	$0.351349\pi$
$840$		0			0
$841$	267029.	+	462508.i	0.377543	+	0.653924i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	290719.	−	503539.i	0.407155	−	0.705213i
$846$		0			0
$847$	1.71859e6			2.39555
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−734623.	−	424135.i	−1.01439	−	0.585659i
$852$		0			0
$853$	21669.1	+	37531.9i	0.0297812	+	0.0515826i	0.880532	−	0.473987i	$-0.157186\pi$
$853$	21669.1	+	37531.9i	0.0297812	+	0.0515826i	−0.850751	+	0.525569i	$0.823852\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−584854.	+	337665.i	−0.796316	+	0.459753i	−0.842181	−	0.539194i	$-0.818729\pi$
$857$	−584854.	+	337665.i	−0.796316	+	0.459753i	0.0458651	+	0.998948i	$0.485396\pi$
$858$		0			0
$859$	586605.	−	1.01603e6i	0.794986	−	1.37696i	−0.127862	−	0.991792i	$-0.540812\pi$
$859$	586605.	−	1.01603e6i	0.794986	−	1.37696i	0.922848	−	0.385164i	$-0.125855\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.11497e6i			1.49707i	0.663095	+	0.748535i	$0.269243\pi$
$863$			1.11497e6i			1.49707i	−0.663095	+	0.748535i	$0.730757\pi$
$864$		0			0
$865$	−459734.	−	265428.i	−0.614433	−	0.354743i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−984123.	+	568184.i	−1.30320	+	0.752401i
$870$		0			0
$871$	458036.	+	793341.i	0.603758	+	1.04574i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.14699e6			−1.49811
$876$		0			0
$877$	128603.	−	74248.9i	0.167206	−	0.0965363i	−0.414062	−	0.910249i	$-0.635890\pi$
$877$	128603.	−	74248.9i	0.167206	−	0.0965363i	0.581268	+	0.813712i	$0.302557\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−480554.			−0.619142			−0.309571	−	0.950876i	$-0.600185\pi$
$881$	−480554.			−0.619142			−0.309571	+	0.950876i	$0.600185\pi$
$882$		0			0
$883$	−66966.9	+	115990.i	−0.0858893	+	0.148765i	−0.905770	−	0.423770i	$-0.860706\pi$
$883$	−66966.9	+	115990.i	−0.0858893	+	0.148765i	0.819881	+	0.572535i	$0.194040\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−492200.	−	284172.i	−0.625596	−	0.361188i	0.153448	−	0.988157i	$-0.450962\pi$
$887$	−492200.	−	284172.i	−0.625596	−	0.361188i	−0.779045	+	0.626969i	$0.784295\pi$
$888$		0			0
$889$	1.24770e6	+	720359.i	1.57872	+	0.911476i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	441960.	+	1.34981e6i	0.554218	+	1.69265i
$894$		0			0
$895$	−467162.	+	269716.i	−0.583205	+	0.336714i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	756247.	−	1.30986e6i	0.935717	−	1.62071i
$900$		0			0
$901$			404367.i			0.498111i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			393513.i			0.480466i
$906$		0			0
$907$	−337140.	+	194648.i	−0.409822	+	0.236611i	−0.690713	−	0.723129i	$-0.742703\pi$
$907$	−337140.	+	194648.i	−0.409822	+	0.236611i	0.280891	+	0.959740i	$0.409370\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			18843.7i			0.0227054i	0.999936	+	0.0113527i	$0.00361375\pi$
$911$			18843.7i			0.0227054i	−0.999936	+	0.0113527i	$0.996386\pi$
$912$		0			0
$913$	1.20121e6			1.44104
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−989812.	−	1.71440e6i	−1.17710	−	2.03880i
$918$		0			0
$919$	−179230.			−0.212217			−0.106109	−	0.994355i	$-0.533839\pi$
$919$	−179230.			−0.212217			−0.106109	+	0.994355i	$0.533839\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.22919e6			−1.44283
$924$		0			0
$925$	251405.	+	145149.i	0.293826	+	0.169641i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−137548.	−	238241.i	−0.159376	−	0.276048i	0.775268	−	0.631633i	$-0.217615\pi$
$929$	−137548.	−	238241.i	−0.159376	−	0.276048i	−0.934644	+	0.355585i	$0.884282\pi$
$930$		0			0
$931$	580621.	−	647789.i	0.669875	−	0.747368i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	391543.	−	678172.i	0.447874	−	0.775741i
$936$		0			0
$937$	613205.	−	1.06210e6i	0.698436	−	1.20973i	−0.270573	−	0.962700i	$-0.587213\pi$
$937$	613205.	−	1.06210e6i	0.698436	−	1.20973i	0.969009	−	0.247027i	$-0.0794536\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	397493.	+	229493.i	0.448901	+	0.259173i	0.707366	−	0.706848i	$-0.249883\pi$
$941$	397493.	+	229493.i	0.448901	+	0.259173i	−0.258465	+	0.966021i	$0.583217\pi$
$942$		0			0
$943$			663110.i			0.745697i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	224532.	+	388901.i	0.250368	+	0.433649i	0.963627	−	0.267251i	$-0.0861152\pi$
$947$	224532.	+	388901.i	0.250368	+	0.433649i	−0.713259	+	0.700900i	$0.752782\pi$
$948$		0			0
$949$			772870.i			0.858171i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−234976.	+	135663.i	−0.258724	+	0.149375i	−0.623753	−	0.781622i	$-0.714393\pi$
$953$	−234976.	+	135663.i	−0.258724	+	0.149375i	0.365028	+	0.930996i	$0.381059\pi$
$954$		0			0
$955$	470382.	+	814726.i	0.515756	+	0.893315i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−687635.	+	1.19102e6i	−0.747689	+	1.29504i
$960$		0			0
$961$	−919359.			−0.995493
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6797.45	−	3924.51i	−0.00729947	−	0.00421435i
$966$		0			0
$967$	−678130.	−	1.17455e6i	−0.725203	−	1.25609i	−0.958890	−	0.283777i	$-0.908412\pi$
$967$	−678130.	−	1.17455e6i	−0.725203	−	1.25609i	0.233687	−	0.972312i	$-0.424921\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−550345.	+	317742.i	−0.583709	+	0.337005i	−0.762606	−	0.646863i	$-0.776080\pi$
$971$	−550345.	+	317742.i	−0.583709	+	0.337005i	0.178897	+	0.983868i	$0.442747\pi$
$972$		0			0
$973$	310353.	−	537547.i	0.327816	−	0.567794i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.06427e6i		−	1.11497i	−0.830186	−	0.557486i	$-0.811766\pi$
$977$		−	1.06427e6i		−	1.11497i	0.830186	−	0.557486i	$-0.188234\pi$
$978$		0			0
$979$	−2.38058e6	−	1.37443e6i	−2.48380	−	1.43402i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−599079.	+	345878.i	−0.619979	+	0.357945i	−0.776861	−	0.629672i	$-0.783189\pi$
$983$	−599079.	+	345878.i	−0.619979	+	0.357945i	0.156882	+	0.987617i	$0.449856\pi$
$984$		0			0
$985$	−11645.3	−	20170.3i	−0.0120027	−	0.0207893i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.49185e6			1.52522
$990$		0			0
$991$	−742203.	+	428511.i	−0.755745	+	0.436330i	−0.827766	−	0.561073i	$-0.810389\pi$
$991$	−742203.	+	428511.i	−0.755745	+	0.436330i	0.0720208	+	0.997403i	$0.477055\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−726127.			−0.733443
$996$		0			0
$997$	−72330.9	+	125281.i	−0.0727668	+	0.126036i	−0.900113	−	0.435657i	$-0.856516\pi$
$997$	−72330.9	+	125281.i	−0.0727668	+	0.126036i	0.827346	+	0.561692i	$0.189850\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.y.f.145.6	✓	24
3.2	odd	2	inner	684.5.y.f.145.7	yes	24
19.8	odd	6	inner	684.5.y.f.217.6	yes	24
57.8	even	6	inner	684.5.y.f.217.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.5.y.f.145.6	✓	24	1.1	even	1	trivial
684.5.y.f.145.7	yes	24	3.2	odd	2	inner
684.5.y.f.217.6	yes	24	19.8	odd	6	inner
684.5.y.f.217.7	yes	24	57.8	even	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.78239 - 15.2115i) q^{5} +69.3595 q^{7} +O(q^{10})\)	\(q+(-8.78239 - 15.2115i) q^{5} +69.3595 q^{7} +198.542 q^{11} +(215.053 + 124.161i) q^{13} +(112.275 + 194.466i) q^{17} +(240.948 - 268.822i) q^{19} +(-462.386 + 800.876i) q^{23} +(158.239 - 274.079i) q^{25} +(-964.886 - 557.077i) q^{29} +1357.53i q^{31} +(-609.142 - 1055.06i) q^{35} +917.274i q^{37} +(620.986 - 358.526i) q^{41} +(-806.604 - 1397.08i) q^{43} +(-1967.20 + 3407.29i) q^{47} +2409.74 q^{49} +(1559.52 + 900.392i) q^{53} +(-1743.67 - 3020.13i) q^{55} +(3318.09 - 1915.70i) q^{59} +(239.475 - 414.783i) q^{61} -4361.71i q^{65} +(3194.82 + 1844.53i) q^{67} +(-4286.83 + 2475.00i) q^{71} +(1556.19 + 2695.40i) q^{73} +13770.8 q^{77} +(-4956.75 + 2861.78i) q^{79} +6050.14 q^{83} +(1972.09 - 3415.76i) q^{85} +(-11990.3 - 6922.59i) q^{89} +(14915.9 + 8611.72i) q^{91} +(-6205.29 - 1304.30i) q^{95} +(15083.0 - 8708.20i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 304 q^{7}+O(q^{10}) \) 24 * q + 304 * q^7	\( 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) \) 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

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