LMFDB - Embedded newform 7623.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.a (trivial)

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:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.05896$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32691 q^{2} -0.239314 q^{4} -2.32691 q^{5} +1.00000 q^{7} +2.97136 q^{8} +O(q^{10})$	$q-1.32691 q^{2} -0.239314 q^{4} -2.32691 q^{5} +1.00000 q^{7} +2.97136 q^{8} +3.08759 q^{10} +4.53759 q^{13} -1.32691 q^{14} -3.46410 q^{16} +3.29827 q^{17} +1.08759 q^{19} +0.556861 q^{20} -6.29827 q^{23} +0.414503 q^{25} -6.02096 q^{26} -0.239314 q^{28} -3.16583 q^{29} -2.97136 q^{31} -1.34618 q^{32} -4.37651 q^{34} -2.32691 q^{35} +1.70342 q^{37} -1.44314 q^{38} -6.91409 q^{40} -4.90724 q^{41} +8.38587 q^{43} +8.35723 q^{46} +4.69825 q^{47} +1.00000 q^{49} -0.550008 q^{50} -1.08591 q^{52} +4.98547 q^{53} +2.97136 q^{56} +4.20077 q^{58} -10.0654 q^{59} +1.74141 q^{61} +3.94273 q^{62} +8.71446 q^{64} -10.5585 q^{65} -9.44958 q^{67} -0.789322 q^{68} +3.08759 q^{70} -16.1593 q^{71} -14.4700 q^{73} -2.26028 q^{74} -0.260276 q^{76} +12.8359 q^{79} +8.06065 q^{80} +6.51146 q^{82} +0.205514 q^{83} -7.67478 q^{85} -11.1273 q^{86} -3.93798 q^{89} +4.53759 q^{91} +1.50726 q^{92} -6.23415 q^{94} -2.53073 q^{95} +13.0444 q^{97} -1.32691 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 6 * q^5 + 4 * q^7	$ 4 q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 4 q^{7} + 14 q^{10} - 2 q^{13} - 2 q^{14} - 2 q^{17} + 6 q^{19} - 8 q^{20} - 10 q^{23} + 4 q^{28} - 14 q^{29} - 12 q^{32} - 2 q^{34} - 6 q^{35} - 12 q^{37} - 16 q^{38} + 8 q^{40} - 16 q^{41} + 20 q^{43} + 8 q^{46} - 2 q^{47} + 4 q^{49} - 24 q^{50} - 40 q^{52} + 16 q^{53} - 8 q^{58} - 2 q^{59} + 2 q^{61} - 8 q^{62} - 16 q^{64} + 2 q^{65} - 20 q^{67} - 20 q^{68} + 14 q^{70} + 10 q^{71} - 10 q^{73} + 20 q^{74} + 28 q^{76} + 16 q^{79} - 12 q^{80} + 28 q^{82} - 18 q^{83} - 26 q^{86} + 14 q^{89} - 2 q^{91} + 8 q^{92} - 18 q^{94} - 22 q^{95} + 38 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 6 * q^5 + 4 * q^7 + 14 * q^10 - 2 * q^13 - 2 * q^14 - 2 * q^17 + 6 * q^19 - 8 * q^20 - 10 * q^23 + 4 * q^28 - 14 * q^29 - 12 * q^32 - 2 * q^34 - 6 * q^35 - 12 * q^37 - 16 * q^38 + 8 * q^40 - 16 * q^41 + 20 * q^43 + 8 * q^46 - 2 * q^47 + 4 * q^49 - 24 * q^50 - 40 * q^52 + 16 * q^53 - 8 * q^58 - 2 * q^59 + 2 * q^61 - 8 * q^62 - 16 * q^64 + 2 * q^65 - 20 * q^67 - 20 * q^68 + 14 * q^70 + 10 * q^71 - 10 * q^73 + 20 * q^74 + 28 * q^76 + 16 * q^79 - 12 * q^80 + 28 * q^82 - 18 * q^83 - 26 * q^86 + 14 * q^89 - 2 * q^91 + 8 * q^92 - 18 * q^94 - 22 * q^95 + 38 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.32691	−0.938266	−0.469133	−	0.883128i	$-0.655434\pi$
$2$	−1.32691	−0.938266	−0.469133	+	0.883128i	$0.655434\pi$
$3$	0	0
$3$	0	0
$4$	−0.239314	−0.119657
$5$	−2.32691	−1.04063	−0.520313	−	0.853976i	$-0.674185\pi$
$5$	−2.32691	−1.04063	−0.520313	+	0.853976i	$0.674185\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.97136	1.05054
$9$	0	0
$10$	3.08759	0.976383
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.53759	1.25850	0.629250	−	0.777203i	$-0.283362\pi$
$13$	4.53759	1.25850	0.629250	+	0.777203i	$0.283362\pi$
$14$	−1.32691	−0.354631
$15$	0	0
$16$	−3.46410	−0.866025
$17$	3.29827	0.799949	0.399974	−	0.916526i	$-0.369019\pi$
$17$	3.29827	0.799949	0.399974	+	0.916526i	$0.369019\pi$
$18$	0	0
$19$	1.08759	0.249511	0.124756	−	0.992187i	$-0.460185\pi$
$19$	1.08759	0.249511	0.124756	+	0.992187i	$0.460185\pi$
$20$	0.556861	0.124518
$21$	0	0
$22$	0	0
$23$	−6.29827	−1.31328	−0.656640	−	0.754204i	$-0.728023\pi$
$23$	−6.29827	−1.31328	−0.656640	+	0.754204i	$0.728023\pi$
$24$	0	0
$25$	0.414503	0.0829007
$26$	−6.02096	−1.18081
$27$	0	0
$28$	−0.239314	−0.0452260
$29$	−3.16583	−0.587880	−0.293940	−	0.955824i	$-0.594967\pi$
$29$	−3.16583	−0.587880	−0.293940	+	0.955824i	$0.594967\pi$
$30$	0	0
$31$	−2.97136	−0.533673	−0.266836	−	0.963742i	$-0.585978\pi$
$31$	−2.97136	−0.533673	−0.266836	+	0.963742i	$0.585978\pi$
$32$	−1.34618	−0.237974
$33$	0	0
$34$	−4.37651	−0.750565
$35$	−2.32691	−0.393319
$36$	0	0
$37$	1.70342	0.280040	0.140020	−	0.990149i	$-0.455283\pi$
$37$	1.70342	0.280040	0.140020	+	0.990149i	$0.455283\pi$
$38$	−1.44314	−0.234108
$39$	0	0
$40$	−6.91409	−1.09321
$41$	−4.90724	−0.766382	−0.383191	−	0.923669i	$-0.625175\pi$
$41$	−4.90724	−0.766382	−0.383191	+	0.923669i	$0.625175\pi$
$42$	0	0
$43$	8.38587	1.27883	0.639416	−	0.768861i	$-0.279176\pi$
$43$	8.38587	1.27883	0.639416	+	0.768861i	$0.279176\pi$
$44$	0	0
$45$	0	0
$46$	8.35723	1.23221
$47$	4.69825	0.685310	0.342655	−	0.939461i	$-0.388674\pi$
$47$	4.69825	0.685310	0.342655	+	0.939461i	$0.388674\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.550008	−0.0777829
$51$	0	0
$52$	−1.08591	−0.150588
$53$	4.98547	0.684808	0.342404	−	0.939553i	$-0.388759\pi$
$53$	4.98547	0.684808	0.342404	+	0.939553i	$0.388759\pi$
$54$	0	0
$55$	0	0
$56$	2.97136	0.397065
$57$	0	0
$58$	4.20077	0.551587
$59$	−10.0654	−1.31040	−0.655201	−	0.755454i	$-0.727416\pi$
$59$	−10.0654	−1.31040	−0.655201	+	0.755454i	$0.727416\pi$
$60$	0	0
$61$	1.74141	0.222965	0.111482	−	0.993766i	$-0.464440\pi$
$61$	1.74141	0.222965	0.111482	+	0.993766i	$0.464440\pi$
$62$	3.94273	0.500727
$63$	0	0
$64$	8.71446	1.08931
$65$	−10.5585	−1.30963
$66$	0	0
$67$	−9.44958	−1.15445	−0.577225	−	0.816585i	$-0.695864\pi$
$67$	−9.44958	−1.15445	−0.577225	+	0.816585i	$0.695864\pi$
$68$	−0.789322	−0.0957193
$69$	0	0
$70$	3.08759	0.369038
$71$	−16.1593	−1.91776	−0.958878	−	0.283820i	$-0.908398\pi$
$71$	−16.1593	−1.91776	−0.958878	+	0.283820i	$0.908398\pi$
$72$	0	0
$73$	−14.4700	−1.69358	−0.846792	−	0.531924i	$-0.821469\pi$
$73$	−14.4700	−1.69358	−0.846792	+	0.531924i	$0.821469\pi$
$74$	−2.26028	−0.262752
$75$	0	0
$76$	−0.260276	−0.0298557
$77$	0	0
$78$	0	0
$79$	12.8359	1.44415	0.722074	−	0.691816i	$-0.243189\pi$
$79$	12.8359	1.44415	0.722074	+	0.691816i	$0.243189\pi$
$80$	8.06065	0.901208
$81$	0	0
$82$	6.51146	0.719070
$83$	0.205514	0.0225581	0.0112790	−	0.999936i	$-0.496410\pi$
$83$	0.205514	0.0225581	0.0112790	+	0.999936i	$0.496410\pi$
$84$	0	0
$85$	−7.67478	−0.832447
$86$	−11.1273	−1.19989
$87$	0	0
$88$	0	0
$89$	−3.93798	−0.417425	−0.208713	−	0.977977i	$-0.566927\pi$
$89$	−3.93798	−0.417425	−0.208713	+	0.977977i	$0.566927\pi$
$90$	0	0
$91$	4.53759	0.475668
$92$	1.50726	0.157143
$93$	0	0
$94$	−6.23415	−0.643003
$95$	−2.53073	−0.259648
$96$	0	0
$97$	13.0444	1.32446	0.662231	−	0.749300i	$-0.269610\pi$
$97$	13.0444	1.32446	0.662231	+	0.749300i	$0.269610\pi$
$98$	−1.32691	−0.134038
$99$	0	0
$100$	−0.0991963	−0.00991963
$101$	−7.01580	−0.698098	−0.349049	−	0.937104i	$-0.613495\pi$
$101$	−7.01580	−0.698098	−0.349049	+	0.937104i	$0.613495\pi$
$102$	0	0
$103$	7.36659	0.725852	0.362926	−	0.931818i	$-0.381778\pi$
$103$	7.36659	0.725852	0.362926	+	0.931818i	$0.381778\pi$
$104$	13.4828	1.32210
$105$	0	0
$106$	−6.61527	−0.642532
$107$	−1.20899	−0.116877	−0.0584387	−	0.998291i	$-0.518612\pi$
$107$	−1.20899	−0.116877	−0.0584387	+	0.998291i	$0.518612\pi$
$108$	0	0
$109$	−15.8465	−1.51782	−0.758909	−	0.651196i	$-0.774267\pi$
$109$	−15.8465	−1.51782	−0.758909	+	0.651196i	$0.774267\pi$
$110$	0	0
$111$	0	0
$112$	−3.46410	−0.327327
$113$	18.4717	1.73767	0.868835	−	0.495103i	$-0.164870\pi$
$113$	18.4717	1.73767	0.868835	+	0.495103i	$0.164870\pi$
$114$	0	0
$115$	14.6555	1.36663
$116$	0.757626	0.0703438
$117$	0	0
$118$	13.3559	1.22951
$119$	3.29827	0.302352
$120$	0	0
$121$	0	0
$122$	−2.31069	−0.209200
$123$	0	0
$124$	0.711088	0.0638576
$125$	10.6700	0.954357
$126$	0	0
$127$	16.0576	1.42488	0.712440	−	0.701733i	$-0.247590\pi$
$127$	16.0576	1.42488	0.712440	+	0.701733i	$0.247590\pi$
$128$	−8.87093	−0.784087
$129$	0	0
$130$	14.0102	1.22878
$131$	−19.2597	−1.68273	−0.841365	−	0.540467i	$-0.818247\pi$
$131$	−19.2597	−1.68273	−0.841365	+	0.540467i	$0.818247\pi$
$132$	0	0
$133$	1.08759	0.0943064
$134$	12.5387	1.08318
$135$	0	0
$136$	9.80037	0.840375
$137$	−5.24059	−0.447733	−0.223867	−	0.974620i	$-0.571868\pi$
$137$	−5.24059	−0.447733	−0.223867	+	0.974620i	$0.571868\pi$
$138$	0	0
$139$	22.9872	1.94975	0.974873	−	0.222762i	$-0.0715073\pi$
$139$	22.9872	1.94975	0.974873	+	0.222762i	$0.0715073\pi$
$140$	0.556861	0.0470633
$141$	0	0
$142$	21.4419	1.79936
$143$	0	0
$144$	0	0
$145$	7.36659	0.611762
$146$	19.2003	1.58903
$147$	0	0
$148$	−0.407651	−0.0335087
$149$	3.47305	0.284523	0.142262	−	0.989829i	$-0.454563\pi$
$149$	3.47305	0.284523	0.142262	+	0.989829i	$0.454563\pi$
$150$	0	0
$151$	−3.15760	−0.256962	−0.128481	−	0.991712i	$-0.541010\pi$
$151$	−3.15760	−0.256962	−0.128481	+	0.991712i	$0.541010\pi$
$152$	3.23164	0.262121
$153$	0	0
$154$	0	0
$155$	6.91409	0.555353
$156$	0	0
$157$	7.50726	0.599145	0.299572	−	0.954074i	$-0.403156\pi$
$157$	7.50726	0.599145	0.299572	+	0.954074i	$0.403156\pi$
$158$	−17.0320	−1.35499
$159$	0	0
$160$	3.13244	0.247641
$161$	−6.29827	−0.496373
$162$	0	0
$163$	−14.3187	−1.12153	−0.560763	−	0.827976i	$-0.689492\pi$
$163$	−14.3187	−1.12153	−0.560763	+	0.827976i	$0.689492\pi$
$164$	1.17437	0.0917029
$165$	0	0
$166$	−0.272698	−0.0211655
$167$	−13.1628	−1.01857	−0.509283	−	0.860599i	$-0.670089\pi$
$167$	−13.1628	−1.01857	−0.509283	+	0.860599i	$0.670089\pi$
$168$	0	0
$169$	7.58969	0.583823
$170$	10.1837	0.781057
$171$	0	0
$172$	−2.00685	−0.153021
$173$	−12.9479	−0.984410	−0.492205	−	0.870479i	$-0.663809\pi$
$173$	−12.9479	−0.984410	−0.492205	+	0.870479i	$0.663809\pi$
$174$	0	0
$175$	0.414503	0.0313335
$176$	0	0
$177$	0	0
$178$	5.22534	0.391656
$179$	18.7751	1.40332	0.701659	−	0.712513i	$-0.252443\pi$
$179$	18.7751	1.40332	0.701659	+	0.712513i	$0.252443\pi$
$180$	0	0
$181$	−7.59696	−0.564678	−0.282339	−	0.959315i	$-0.591110\pi$
$181$	−7.59696	−0.564678	−0.282339	+	0.959315i	$0.591110\pi$
$182$	−6.02096	−0.446303
$183$	0	0
$184$	−18.7145	−1.37965
$185$	−3.96369	−0.291416
$186$	0	0
$187$	0	0
$188$	−1.12436	−0.0820021
$189$	0	0
$190$	3.35805	0.243619
$191$	21.6059	1.56335	0.781674	−	0.623687i	$-0.214366\pi$
$191$	21.6059	1.56335	0.781674	+	0.623687i	$0.214366\pi$
$192$	0	0
$193$	25.3968	1.82810	0.914051	−	0.405598i	$-0.132937\pi$
$193$	25.3968	1.82810	0.914051	+	0.405598i	$0.132937\pi$
$194$	−17.3088	−1.24270
$195$	0	0
$196$	−0.239314	−0.0170938
$197$	6.79617	0.484207	0.242104	−	0.970250i	$-0.422163\pi$
$197$	6.79617	0.484207	0.242104	+	0.970250i	$0.422163\pi$
$198$	0	0
$199$	−19.2276	−1.36301	−0.681505	−	0.731814i	$-0.738674\pi$
$199$	−19.2276	−1.36301	−0.681505	+	0.731814i	$0.738674\pi$
$200$	1.23164	0.0870902
$201$	0	0
$202$	9.30932	0.655002
$203$	−3.16583	−0.222198
$204$	0	0
$205$	11.4187	0.797517
$206$	−9.77480	−0.681042
$207$	0	0
$208$	−15.7187	−1.08989
$209$	0	0
$210$	0	0
$211$	−12.1499	−0.836436	−0.418218	−	0.908347i	$-0.637345\pi$
$211$	−12.1499	−0.836436	−0.418218	+	0.908347i	$0.637345\pi$
$212$	−1.19309	−0.0819419
$213$	0	0
$214$	1.60422	0.109662
$215$	−19.5131	−1.33079
$216$	0	0
$217$	−2.97136	−0.201709
$218$	21.0268	1.42412
$219$	0	0
$220$	0	0
$221$	14.9662	1.00674
$222$	0	0
$223$	13.9217	0.932264	0.466132	−	0.884715i	$-0.345647\pi$
$223$	13.9217	0.932264	0.466132	+	0.884715i	$0.345647\pi$
$224$	−1.34618	−0.0899456
$225$	0	0
$226$	−24.5102	−1.63040
$227$	10.5593	0.700843	0.350422	−	0.936592i	$-0.386038\pi$
$227$	10.5593	0.700843	0.350422	+	0.936592i	$0.386038\pi$
$228$	0	0
$229$	−28.9118	−1.91054	−0.955271	−	0.295731i	$-0.904437\pi$
$229$	−28.9118	−1.91054	−0.955271	+	0.295731i	$0.904437\pi$
$230$	−19.4465	−1.28227
$231$	0	0
$232$	−9.40683	−0.617589
$233$	18.2551	1.19593	0.597966	−	0.801521i	$-0.295976\pi$
$233$	18.2551	1.19593	0.597966	+	0.801521i	$0.295976\pi$
$234$	0	0
$235$	−10.9324	−0.713151
$236$	2.40879	0.156799
$237$	0	0
$238$	−4.37651	−0.283687
$239$	2.03621	0.131711	0.0658557	−	0.997829i	$-0.479022\pi$
$239$	2.03621	0.131711	0.0658557	+	0.997829i	$0.479022\pi$
$240$	0	0
$241$	−28.9345	−1.86384	−0.931918	−	0.362670i	$-0.881865\pi$
$241$	−28.9345	−1.86384	−0.931918	+	0.362670i	$0.881865\pi$
$242$	0	0
$243$	0	0
$244$	−0.416744	−0.0266793
$245$	−2.32691	−0.148661
$246$	0	0
$247$	4.93506	0.314010
$248$	−8.82901	−0.560642
$249$	0	0
$250$	−14.1582	−0.895440
$251$	27.7896	1.75407	0.877033	−	0.480430i	$-0.159519\pi$
$251$	27.7896	1.75407	0.877033	+	0.480430i	$0.159519\pi$
$252$	0	0
$253$	0	0
$254$	−21.3070	−1.33692
$255$	0	0
$256$	−5.65801	−0.353626
$257$	−12.8265	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$257$	−12.8265	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$258$	0	0
$259$	1.70342	0.105845
$260$	2.52681	0.156706
$261$	0	0
$262$	25.5559	1.57885
$263$	2.55283	0.157414	0.0787072	−	0.996898i	$-0.474921\pi$
$263$	2.55283	0.157414	0.0787072	+	0.996898i	$0.474921\pi$
$264$	0	0
$265$	−11.6007	−0.712628
$266$	−1.44314	−0.0884845
$267$	0	0
$268$	2.26141	0.138138
$269$	−19.0274	−1.16012	−0.580061	−	0.814573i	$-0.696971\pi$
$269$	−19.0274	−1.16012	−0.580061	+	0.814573i	$0.696971\pi$
$270$	0	0
$271$	10.3268	0.627307	0.313653	−	0.949538i	$-0.398447\pi$
$271$	10.3268	0.627307	0.313653	+	0.949538i	$0.398447\pi$
$272$	−11.4256	−0.692776
$273$	0	0
$274$	6.95378	0.420093
$275$	0	0
$276$	0	0
$277$	2.89957	0.174218	0.0871091	−	0.996199i	$-0.472237\pi$
$277$	2.89957	0.174218	0.0871091	+	0.996199i	$0.472237\pi$
$278$	−30.5019	−1.82938
$279$	0	0
$280$	−6.91409	−0.413196
$281$	−17.4209	−1.03925	−0.519623	−	0.854396i	$-0.673928\pi$
$281$	−17.4209	−1.03925	−0.519623	+	0.854396i	$0.673928\pi$
$282$	0	0
$283$	9.49232	0.564260	0.282130	−	0.959376i	$-0.408959\pi$
$283$	9.49232	0.564260	0.282130	+	0.959376i	$0.408959\pi$
$284$	3.86714	0.229473
$285$	0	0
$286$	0	0
$287$	−4.90724	−0.289665
$288$	0	0
$289$	−6.12139	−0.360082
$290$	−9.77480	−0.573996
$291$	0	0
$292$	3.46287	0.202649
$293$	−3.27005	−0.191039	−0.0955193	−	0.995428i	$-0.530451\pi$
$293$	−3.27005	−0.191039	−0.0955193	+	0.995428i	$0.530451\pi$
$294$	0	0
$295$	23.4213	1.36364
$296$	5.06147	0.294192
$297$	0	0
$298$	−4.60842	−0.266958
$299$	−28.5790	−1.65276
$300$	0	0
$301$	8.38587	0.483353
$302$	4.18985	0.241099
$303$	0	0
$304$	−3.76754	−0.216083
$305$	−4.05211	−0.232023
$306$	0	0
$307$	−16.5841	−0.946506	−0.473253	−	0.880927i	$-0.656920\pi$
$307$	−16.5841	−0.946506	−0.473253	+	0.880927i	$0.656920\pi$
$308$	0	0
$309$	0	0
$310$	−9.17437	−0.521069
$311$	6.64098	0.376575	0.188288	−	0.982114i	$-0.439706\pi$
$311$	6.64098	0.376575	0.188288	+	0.982114i	$0.439706\pi$
$312$	0	0
$313$	−3.58663	−0.202728	−0.101364	−	0.994849i	$-0.532321\pi$
$313$	−3.58663	−0.202728	−0.101364	+	0.994849i	$0.532321\pi$
$314$	−9.96145	−0.562157
$315$	0	0
$316$	−3.07180	−0.172802
$317$	11.4017	0.640381	0.320191	−	0.947353i	$-0.396253\pi$
$317$	11.4017	0.640381	0.320191	+	0.947353i	$0.396253\pi$
$318$	0	0
$319$	0	0
$320$	−20.2778	−1.13356
$321$	0	0
$322$	8.35723	0.465730
$323$	3.58718	0.199596
$324$	0	0
$325$	1.88085	0.104331
$326$	18.9996	1.05229
$327$	0	0
$328$	−14.5812	−0.805112
$329$	4.69825	0.259023
$330$	0	0
$331$	−6.84006	−0.375964	−0.187982	−	0.982173i	$-0.560195\pi$
$331$	−6.84006	−0.375964	−0.187982	+	0.982173i	$0.560195\pi$
$332$	−0.0491822	−0.00269922
$333$	0	0
$334$	17.4658	0.955686
$335$	21.9883	1.20135
$336$	0	0
$337$	−8.40501	−0.457850	−0.228925	−	0.973444i	$-0.573521\pi$
$337$	−8.40501	−0.457850	−0.228925	+	0.973444i	$0.573521\pi$
$338$	−10.0708	−0.547781
$339$	0	0
$340$	1.83668	0.0996079
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	24.9175	1.34346
$345$	0	0
$346$	17.1807	0.923639
$347$	−8.47346	−0.454879	−0.227440	−	0.973792i	$-0.573035\pi$
$347$	−8.47346	−0.454879	−0.227440	+	0.973792i	$0.573035\pi$
$348$	0	0
$349$	16.0748	0.860462	0.430231	−	0.902719i	$-0.358432\pi$
$349$	16.0748	0.860462	0.430231	+	0.902719i	$0.358432\pi$
$350$	−0.550008	−0.0293992
$351$	0	0
$352$	0	0
$353$	13.3786	0.712072	0.356036	−	0.934472i	$-0.384128\pi$
$353$	13.3786	0.712072	0.356036	+	0.934472i	$0.384128\pi$
$354$	0	0
$355$	37.6012	1.99566
$356$	0.942413	0.0499478
$357$	0	0
$358$	−24.9129	−1.31669
$359$	−27.1016	−1.43037	−0.715184	−	0.698936i	$-0.753657\pi$
$359$	−27.1016	−1.43037	−0.715184	+	0.698936i	$0.753657\pi$
$360$	0	0
$361$	−17.8171	−0.937744
$362$	10.0805	0.529818
$363$	0	0
$364$	−1.08591	−0.0569170
$365$	33.6703	1.76239
$366$	0	0
$367$	−0.0252953	−0.00132040	−0.000660202	−	1.00000i	$-0.500210\pi$
$367$	−0.0252953	−0.00132040	−0.000660202		1.00000i	$0.500210\pi$
$368$	21.8179	1.13733
$369$	0	0
$370$	5.25946	0.273426
$371$	4.98547	0.258833
$372$	0	0
$373$	−23.7038	−1.22734	−0.613669	−	0.789563i	$-0.710307\pi$
$373$	−23.7038	−1.22734	−0.613669	+	0.789563i	$0.710307\pi$
$374$	0	0
$375$	0	0
$376$	13.9602	0.719943
$377$	−14.3652	−0.739847
$378$	0	0
$379$	−29.0675	−1.49310	−0.746549	−	0.665331i	$-0.768290\pi$
$379$	−29.0675	−1.49310	−0.746549	+	0.665331i	$0.768290\pi$
$380$	0.605639	0.0310686
$381$	0	0
$382$	−28.6691	−1.46684
$383$	−16.8650	−0.861764	−0.430882	−	0.902408i	$-0.641797\pi$
$383$	−16.8650	−0.861764	−0.430882	+	0.902408i	$0.641797\pi$
$384$	0	0
$385$	0	0
$386$	−33.6993	−1.71525
$387$	0	0
$388$	−3.12171	−0.158481
$389$	−13.0752	−0.662938	−0.331469	−	0.943466i	$-0.607544\pi$
$389$	−13.0752	−0.662938	−0.331469	+	0.943466i	$0.607544\pi$
$390$	0	0
$391$	−20.7734	−1.05056
$392$	2.97136	0.150077
$393$	0	0
$394$	−9.01790	−0.454315
$395$	−29.8679	−1.50282
$396$	0	0
$397$	1.28891	0.0646886	0.0323443	−	0.999477i	$-0.489703\pi$
$397$	1.28891	0.0646886	0.0323443	+	0.999477i	$0.489703\pi$
$398$	25.5133	1.27887
$399$	0	0
$400$	−1.43588	−0.0717941
$401$	−7.50643	−0.374853	−0.187427	−	0.982279i	$-0.560015\pi$
$401$	−7.50643	−0.374853	−0.187427	+	0.982279i	$0.560015\pi$
$402$	0	0
$403$	−13.4828	−0.671627
$404$	1.67898	0.0835322
$405$	0	0
$406$	4.20077	0.208480
$407$	0	0
$408$	0	0
$409$	−37.3746	−1.84805	−0.924027	−	0.382327i	$-0.875123\pi$
$409$	−37.3746	−1.84805	−0.924027	+	0.382327i	$0.875123\pi$
$410$	−15.1516	−0.748283
$411$	0	0
$412$	−1.76293	−0.0868532
$413$	−10.0654	−0.495286
$414$	0	0
$415$	−0.478211	−0.0234745
$416$	−6.10842	−0.299490
$417$	0	0
$418$	0	0
$419$	−13.0696	−0.638491	−0.319246	−	0.947672i	$-0.603430\pi$
$419$	−13.0696	−0.638491	−0.319246	+	0.947672i	$0.603430\pi$
$420$	0	0
$421$	−21.3443	−1.04026	−0.520128	−	0.854088i	$-0.674116\pi$
$421$	−21.3443	−1.04026	−0.520128	+	0.854088i	$0.674116\pi$
$422$	16.1218	0.784799
$423$	0	0
$424$	14.8137	0.719415
$425$	1.36715	0.0663163
$426$	0	0
$427$	1.74141	0.0842728
$428$	0.289328	0.0139852
$429$	0	0
$430$	25.8922	1.24863
$431$	24.8042	1.19477	0.597387	−	0.801953i	$-0.296206\pi$
$431$	24.8042	1.19477	0.597387	+	0.801953i	$0.296206\pi$
$432$	0	0
$433$	−25.9947	−1.24923	−0.624614	−	0.780934i	$-0.714744\pi$
$433$	−25.9947	−1.24923	−0.624614	+	0.780934i	$0.714744\pi$
$434$	3.94273	0.189257
$435$	0	0
$436$	3.79228	0.181617
$437$	−6.84997	−0.327678
$438$	0	0
$439$	−0.615268	−0.0293652	−0.0146826	−	0.999892i	$-0.504674\pi$
$439$	−0.615268	−0.0293652	−0.0146826	+	0.999892i	$0.504674\pi$
$440$	0	0
$441$	0	0
$442$	−19.8588	−0.944586
$443$	36.3307	1.72612	0.863062	−	0.505098i	$-0.168544\pi$
$443$	36.3307	1.72612	0.863062	+	0.505098i	$0.168544\pi$
$444$	0	0
$445$	9.16332	0.434383
$446$	−18.4728	−0.874711
$447$	0	0
$448$	8.71446	0.411720
$449$	−28.5858	−1.34905	−0.674524	−	0.738253i	$-0.735651\pi$
$449$	−28.5858	−1.34905	−0.674524	+	0.738253i	$0.735651\pi$
$450$	0	0
$451$	0	0
$452$	−4.42052	−0.207924
$453$	0	0
$454$	−14.0112	−0.657578
$455$	−10.5585	−0.494992
$456$	0	0
$457$	−39.2884	−1.83783	−0.918917	−	0.394451i	$-0.870935\pi$
$457$	−39.2884	−1.83783	−0.918917	+	0.394451i	$0.870935\pi$
$458$	38.3633	1.79260
$459$	0	0
$460$	−3.50726	−0.163527
$461$	−0.411163	−0.0191498	−0.00957489	−	0.999954i	$-0.503048\pi$
$461$	−0.411163	−0.0191498	−0.00957489	+	0.999954i	$0.503048\pi$
$462$	0	0
$463$	−17.6179	−0.818774	−0.409387	−	0.912361i	$-0.634257\pi$
$463$	−17.6179	−0.818774	−0.409387	+	0.912361i	$0.634257\pi$
$464$	10.9668	0.509119
$465$	0	0
$466$	−24.2229	−1.12210
$467$	7.61248	0.352264	0.176132	−	0.984367i	$-0.443642\pi$
$467$	7.61248	0.352264	0.176132	+	0.984367i	$0.443642\pi$
$468$	0	0
$469$	−9.44958	−0.436341
$470$	14.5063	0.669125
$471$	0	0
$472$	−29.9080	−1.37663
$473$	0	0
$474$	0	0
$475$	0.450812	0.0206847
$476$	−0.789322	−0.0361785
$477$	0	0
$478$	−2.70186	−0.123580
$479$	−34.7448	−1.58753	−0.793765	−	0.608225i	$-0.791882\pi$
$479$	−34.7448	−1.58753	−0.793765	+	0.608225i	$0.791882\pi$
$480$	0	0
$481$	7.72939	0.352430
$482$	38.3934	1.74877
$483$	0	0
$484$	0	0
$485$	−30.3532	−1.37827
$486$	0	0
$487$	21.6667	0.981811	0.490906	−	0.871213i	$-0.336666\pi$
$487$	21.6667	0.981811	0.490906	+	0.871213i	$0.336666\pi$
$488$	5.17437	0.234233
$489$	0	0
$490$	3.08759	0.139483
$491$	−13.3539	−0.602653	−0.301326	−	0.953521i	$-0.597429\pi$
$491$	−13.3539	−0.602653	−0.301326	+	0.953521i	$0.597429\pi$
$492$	0	0
$493$	−10.4418	−0.470274
$494$	−6.54837	−0.294625
$495$	0	0
$496$	10.2931	0.462174
$497$	−16.1593	−0.724843
$498$	0	0
$499$	−19.5813	−0.876579	−0.438290	−	0.898834i	$-0.644416\pi$
$499$	−19.5813	−0.876579	−0.438290	+	0.898834i	$0.644416\pi$
$500$	−2.55348	−0.114195
$501$	0	0
$502$	−36.8743	−1.64578
$503$	16.7345	0.746153	0.373076	−	0.927801i	$-0.378303\pi$
$503$	16.7345	0.746153	0.373076	+	0.927801i	$0.378303\pi$
$504$	0	0
$505$	16.3251	0.726458
$506$	0	0
$507$	0	0
$508$	−3.84280	−0.170497
$509$	4.40431	0.195218	0.0976088	−	0.995225i	$-0.468881\pi$
$509$	4.40431	0.195218	0.0976088	+	0.995225i	$0.468881\pi$
$510$	0	0
$511$	−14.4700	−0.640115
$512$	25.2495	1.11588
$513$	0	0
$514$	17.0196	0.750702
$515$	−17.1414	−0.755340
$516$	0	0
$517$	0	0
$518$	−2.26028	−0.0993108
$519$	0	0
$520$	−31.3733	−1.37581
$521$	42.9199	1.88035	0.940177	−	0.340686i	$-0.110659\pi$
$521$	42.9199	1.88035	0.940177	+	0.340686i	$0.110659\pi$
$522$	0	0
$523$	−9.54178	−0.417233	−0.208617	−	0.977998i	$-0.566896\pi$
$523$	−9.54178	−0.417233	−0.208617	+	0.977998i	$0.566896\pi$
$524$	4.60912	0.201350
$525$	0	0
$526$	−3.38738	−0.147697
$527$	−9.80037	−0.426911
$528$	0	0
$529$	16.6682	0.724706
$530$	15.3931	0.668635
$531$	0	0
$532$	−0.260276	−0.0112844
$533$	−22.2670	−0.964492
$534$	0	0
$535$	2.81321	0.121626
$536$	−28.0781	−1.21279
$537$	0	0
$538$	25.2476	1.08850
$539$	0	0
$540$	0	0
$541$	−12.3576	−0.531297	−0.265648	−	0.964070i	$-0.585586\pi$
$541$	−12.3576	−0.531297	−0.265648	+	0.964070i	$0.585586\pi$
$542$	−13.7027	−0.588581
$543$	0	0
$544$	−4.44008	−0.190367
$545$	36.8733	1.57948
$546$	0	0
$547$	−10.6187	−0.454025	−0.227012	−	0.973892i	$-0.572896\pi$
$547$	−10.6187	−0.454025	−0.227012	+	0.973892i	$0.572896\pi$
$548$	1.25414	0.0535744
$549$	0	0
$550$	0	0
$551$	−3.44314	−0.146683
$552$	0	0
$553$	12.8359	0.545836
$554$	−3.84746	−0.163463
$555$	0	0
$556$	−5.50114	−0.233300
$557$	17.2434	0.730627	0.365313	−	0.930885i	$-0.380962\pi$
$557$	17.2434	0.730627	0.365313	+	0.930885i	$0.380962\pi$
$558$	0	0
$559$	38.0516	1.60941
$560$	8.06065	0.340625
$561$	0	0
$562$	23.1160	0.975089
$563$	−18.3349	−0.772724	−0.386362	−	0.922347i	$-0.626268\pi$
$563$	−18.3349	−0.772724	−0.386362	+	0.922347i	$0.626268\pi$
$564$	0	0
$565$	−42.9819	−1.80826
$566$	−12.5954	−0.529426
$567$	0	0
$568$	−48.0151	−2.01467
$569$	−13.3639	−0.560246	−0.280123	−	0.959964i	$-0.590375\pi$
$569$	−13.3639	−0.560246	−0.280123	+	0.959964i	$0.590375\pi$
$570$	0	0
$571$	8.21601	0.343829	0.171915	−	0.985112i	$-0.445005\pi$
$571$	8.21601	0.343829	0.171915	+	0.985112i	$0.445005\pi$
$572$	0	0
$573$	0	0
$574$	6.51146	0.271783
$575$	−2.61066	−0.108872
$576$	0	0
$577$	33.8566	1.40947	0.704734	−	0.709472i	$-0.251066\pi$
$577$	33.8566	1.40947	0.704734	+	0.709472i	$0.251066\pi$
$578$	8.12253	0.337853
$579$	0	0
$580$	−1.76293	−0.0732015
$581$	0.205514	0.00852614
$582$	0	0
$583$	0	0
$584$	−42.9956	−1.77917
$585$	0	0
$586$	4.33906	0.179245
$587$	2.51243	0.103699	0.0518495	−	0.998655i	$-0.483488\pi$
$587$	2.51243	0.103699	0.0518495	+	0.998655i	$0.483488\pi$
$588$	0	0
$589$	−3.23164	−0.133157
$590$	−31.0779	−1.27946
$591$	0	0
$592$	−5.90080	−0.242522
$593$	24.5112	1.00655	0.503277	−	0.864125i	$-0.332128\pi$
$593$	24.5112	1.00655	0.503277	+	0.864125i	$0.332128\pi$
$594$	0	0
$595$	−7.67478	−0.314635
$596$	−0.831148	−0.0340451
$597$	0	0
$598$	37.9217	1.55073
$599$	−32.2579	−1.31802	−0.659012	−	0.752133i	$-0.729025\pi$
$599$	−32.2579	−1.31802	−0.659012	+	0.752133i	$0.729025\pi$
$600$	0	0
$601$	34.3485	1.40110	0.700552	−	0.713601i	$-0.252937\pi$
$601$	34.3485	1.40110	0.700552	+	0.713601i	$0.252937\pi$
$602$	−11.1273	−0.453514
$603$	0	0
$604$	0.755658	0.0307473
$605$	0	0
$606$	0	0
$607$	16.6066	0.674042	0.337021	−	0.941497i	$-0.390581\pi$
$607$	16.6066	0.674042	0.337021	+	0.941497i	$0.390581\pi$
$608$	−1.46410	−0.0593772
$609$	0	0
$610$	5.37677	0.217699
$611$	21.3187	0.862463
$612$	0	0
$613$	−12.8483	−0.518939	−0.259469	−	0.965751i	$-0.583548\pi$
$613$	−12.8483	−0.518939	−0.259469	+	0.965751i	$0.583548\pi$
$614$	22.0056	0.888074
$615$	0	0
$616$	0	0
$617$	18.5521	0.746881	0.373441	−	0.927654i	$-0.378178\pi$
$617$	18.5521	0.746881	0.373441	+	0.927654i	$0.378178\pi$
$618$	0	0
$619$	24.5919	0.988433	0.494217	−	0.869339i	$-0.335455\pi$
$619$	24.5919	0.988433	0.494217	+	0.869339i	$0.335455\pi$
$620$	−1.65464	−0.0664518
$621$	0	0
$622$	−8.81197	−0.353328
$623$	−3.93798	−0.157772
$624$	0	0
$625$	−26.9007	−1.07603
$626$	4.75913	0.190213
$627$	0	0
$628$	−1.79659	−0.0716918
$629$	5.61833	0.224017
$630$	0	0
$631$	−41.8792	−1.66719	−0.833593	−	0.552379i	$-0.813720\pi$
$631$	−41.8792	−1.66719	−0.833593	+	0.552379i	$0.813720\pi$
$632$	38.1400	1.51713
$633$	0	0
$634$	−15.1290	−0.600848
$635$	−37.3645	−1.48277
$636$	0	0
$637$	4.53759	0.179786
$638$	0	0
$639$	0	0
$640$	20.6418	0.815941
$641$	2.23529	0.0882885	0.0441442	−	0.999025i	$-0.485944\pi$
$641$	2.23529	0.0882885	0.0441442	+	0.999025i	$0.485944\pi$
$642$	0	0
$643$	−13.8690	−0.546940	−0.273470	−	0.961880i	$-0.588171\pi$
$643$	−13.8690	−0.546940	−0.273470	+	0.961880i	$0.588171\pi$
$644$	1.50726	0.0593945
$645$	0	0
$646$	−4.75987	−0.187274
$647$	8.04106	0.316127	0.158063	−	0.987429i	$-0.449475\pi$
$647$	8.04106	0.316127	0.158063	+	0.987429i	$0.449475\pi$
$648$	0	0
$649$	0	0
$650$	−2.49571	−0.0978898
$651$	0	0
$652$	3.42666	0.134198
$653$	2.75480	0.107804	0.0539019	−	0.998546i	$-0.482834\pi$
$653$	2.75480	0.107804	0.0539019	+	0.998546i	$0.482834\pi$
$654$	0	0
$655$	44.8156	1.75109
$656$	16.9992	0.663706
$657$	0	0
$658$	−6.23415	−0.243032
$659$	37.1797	1.44832	0.724159	−	0.689634i	$-0.242228\pi$
$659$	37.1797	1.44832	0.724159	+	0.689634i	$0.242228\pi$
$660$	0	0
$661$	−2.19963	−0.0855556	−0.0427778	−	0.999085i	$-0.513621\pi$
$661$	−2.19963	−0.0855556	−0.0427778	+	0.999085i	$0.513621\pi$
$662$	9.07613	0.352754
$663$	0	0
$664$	0.610656	0.0236980
$665$	−2.53073	−0.0981376
$666$	0	0
$667$	19.9393	0.772051
$668$	3.15003	0.121878
$669$	0	0
$670$	−29.1765	−1.12719
$671$	0	0
$672$	0	0
$673$	−28.2877	−1.09041	−0.545205	−	0.838303i	$-0.683548\pi$
$673$	−28.2877	−1.09041	−0.545205	+	0.838303i	$0.683548\pi$
$674$	11.1527	0.429585
$675$	0	0
$676$	−1.81632	−0.0698584
$677$	11.1440	0.428300	0.214150	−	0.976801i	$-0.431302\pi$
$677$	11.1440	0.428300	0.214150	+	0.976801i	$0.431302\pi$
$678$	0	0
$679$	13.0444	0.500599
$680$	−22.8046	−0.874515
$681$	0	0
$682$	0	0
$683$	−45.6300	−1.74598	−0.872992	−	0.487735i	$-0.837823\pi$
$683$	−45.6300	−1.74598	−0.872992	+	0.487735i	$0.837823\pi$
$684$	0	0
$685$	12.1944	0.465923
$686$	−1.32691	−0.0506616
$687$	0	0
$688$	−29.0495	−1.10750
$689$	22.6220	0.861830
$690$	0	0
$691$	11.1889	0.425645	0.212823	−	0.977091i	$-0.431734\pi$
$691$	11.1889	0.425645	0.212823	+	0.977091i	$0.431734\pi$
$692$	3.09861	0.117791
$693$	0	0
$694$	11.2435	0.426798
$695$	−53.4890	−2.02895
$696$	0	0
$697$	−16.1854	−0.613066
$698$	−21.3297	−0.807342
$699$	0	0
$700$	−0.0991963	−0.00374927
$701$	−1.02581	−0.0387443	−0.0193722	−	0.999812i	$-0.506167\pi$
$701$	−1.02581	−0.0387443	−0.0193722	+	0.999812i	$0.506167\pi$
$702$	0	0
$703$	1.85263	0.0698731
$704$	0	0
$705$	0	0
$706$	−17.7522	−0.668113
$707$	−7.01580	−0.263856
$708$	0	0
$709$	−20.4828	−0.769249	−0.384624	−	0.923073i	$-0.625669\pi$
$709$	−20.4828	−0.769249	−0.384624	+	0.923073i	$0.625669\pi$
$710$	−49.8933	−1.87246
$711$	0	0
$712$	−11.7012	−0.438520
$713$	18.7145	0.700862
$714$	0	0
$715$	0	0
$716$	−4.49314	−0.167917
$717$	0	0
$718$	35.9614	1.34207
$719$	−32.9187	−1.22766	−0.613830	−	0.789438i	$-0.710372\pi$
$719$	−32.9187	−1.22766	−0.613830	+	0.789438i	$0.710372\pi$
$720$	0	0
$721$	7.36659	0.274346
$722$	23.6417	0.879853
$723$	0	0
$724$	1.81806	0.0675676
$725$	−1.31225	−0.0487356
$726$	0	0
$727$	31.1632	1.15578	0.577889	−	0.816115i	$-0.303877\pi$
$727$	31.1632	1.15578	0.577889	+	0.816115i	$0.303877\pi$
$728$	13.4828	0.499707
$729$	0	0
$730$	−44.6775	−1.65359
$731$	27.6589	1.02300
$732$	0	0
$733$	9.48073	0.350179	0.175089	−	0.984553i	$-0.443979\pi$
$733$	9.48073	0.350179	0.175089	+	0.984553i	$0.443979\pi$
$734$	0.0335645	0.00123889
$735$	0	0
$736$	8.47863	0.312526
$737$	0	0
$738$	0	0
$739$	37.9205	1.39493	0.697465	−	0.716619i	$-0.254311\pi$
$739$	37.9205	1.39493	0.697465	+	0.716619i	$0.254311\pi$
$740$	0.948566	0.0348700
$741$	0	0
$742$	−6.61527	−0.242854
$743$	−21.0795	−0.773332	−0.386666	−	0.922220i	$-0.626373\pi$
$743$	−21.0795	−0.773332	−0.386666	+	0.922220i	$0.626373\pi$
$744$	0	0
$745$	−8.08146	−0.296082
$746$	31.4528	1.15157
$747$	0	0
$748$	0	0
$749$	−1.20899	−0.0441755
$750$	0	0
$751$	−49.9699	−1.82343	−0.911713	−	0.410828i	$-0.865240\pi$
$751$	−49.9699	−1.82343	−0.911713	+	0.410828i	$0.865240\pi$
$752$	−16.2752	−0.593496
$753$	0	0
$754$	19.0613	0.694173
$755$	7.34746	0.267401
$756$	0	0
$757$	33.8732	1.23114	0.615571	−	0.788081i	$-0.288925\pi$
$757$	33.8732	1.23114	0.615571	+	0.788081i	$0.288925\pi$
$758$	38.5699	1.40092
$759$	0	0
$760$	−7.51973	−0.272769
$761$	−1.57878	−0.0572307	−0.0286154	−	0.999590i	$-0.509110\pi$
$761$	−1.57878	−0.0572307	−0.0286154	+	0.999590i	$0.509110\pi$
$762$	0	0
$763$	−15.8465	−0.573682
$764$	−5.17059	−0.187065
$765$	0	0
$766$	22.3784	0.808564
$767$	−45.6726	−1.64914
$768$	0	0
$769$	17.0786	0.615868	0.307934	−	0.951408i	$-0.400362\pi$
$769$	17.0786	0.615868	0.307934	+	0.951408i	$0.400362\pi$
$770$	0	0
$771$	0	0
$772$	−6.07781	−0.218745
$773$	−33.4826	−1.20429	−0.602143	−	0.798389i	$-0.705686\pi$
$773$	−33.4826	−1.20429	−0.602143	+	0.798389i	$0.705686\pi$
$774$	0	0
$775$	−1.23164	−0.0442418
$776$	38.7598	1.39139
$777$	0	0
$778$	17.3496	0.622012
$779$	−5.33709	−0.191221
$780$	0	0
$781$	0	0
$782$	27.5644	0.985702
$783$	0	0
$784$	−3.46410	−0.123718
$785$	−17.4687	−0.623485
$786$	0	0
$787$	−36.3779	−1.29673	−0.648366	−	0.761328i	$-0.724548\pi$
$787$	−36.3779	−1.29673	−0.648366	+	0.761328i	$0.724548\pi$
$788$	−1.62642	−0.0579387
$789$	0	0
$790$	39.6319	1.41004
$791$	18.4717	0.656777
$792$	0	0
$793$	7.90181	0.280601
$794$	−1.71027	−0.0606951
$795$	0	0
$796$	4.60143	0.163093
$797$	−46.7424	−1.65570	−0.827851	−	0.560948i	$-0.810437\pi$
$797$	−46.7424	−1.65570	−0.827851	+	0.560948i	$0.810437\pi$
$798$	0	0
$799$	15.4961	0.548213
$800$	−0.557997	−0.0197282
$801$	0	0
$802$	9.96035	0.351712
$803$	0	0
$804$	0	0
$805$	14.6555	0.516539
$806$	17.8905	0.630165
$807$	0	0
$808$	−20.8465	−0.733377
$809$	−36.0260	−1.26661	−0.633304	−	0.773903i	$-0.718302\pi$
$809$	−36.0260	−1.26661	−0.633304	+	0.773903i	$0.718302\pi$
$810$	0	0
$811$	−27.6340	−0.970362	−0.485181	−	0.874414i	$-0.661246\pi$
$811$	−27.6340	−0.970362	−0.485181	+	0.874414i	$0.661246\pi$
$812$	0.757626	0.0265875
$813$	0	0
$814$	0	0
$815$	33.3183	1.16709
$816$	0	0
$817$	9.12043	0.319083
$818$	49.5927	1.73397
$819$	0	0
$820$	−2.73265	−0.0954283
$821$	−32.1918	−1.12350	−0.561752	−	0.827306i	$-0.689872\pi$
$821$	−32.1918	−1.12350	−0.561752	+	0.827306i	$0.689872\pi$
$822$	0	0
$823$	19.4232	0.677050	0.338525	−	0.940957i	$-0.390072\pi$
$823$	19.4232	0.677050	0.338525	+	0.940957i	$0.390072\pi$
$824$	21.8888	0.762534
$825$	0	0
$826$	13.3559	0.464710
$827$	−20.7586	−0.721847	−0.360923	−	0.932595i	$-0.617538\pi$
$827$	−20.7586	−0.721847	−0.360923	+	0.932595i	$0.617538\pi$
$828$	0	0
$829$	54.3260	1.88682	0.943410	−	0.331629i	$-0.107598\pi$
$829$	54.3260	1.88682	0.943410	+	0.331629i	$0.107598\pi$
$830$	0.634543	0.0220253
$831$	0	0
$832$	39.5426	1.37089
$833$	3.29827	0.114278
$834$	0	0
$835$	30.6286	1.05994
$836$	0	0
$837$	0	0
$838$	17.3422	0.599075
$839$	−14.2101	−0.490588	−0.245294	−	0.969449i	$-0.578885\pi$
$839$	−14.2101	−0.490588	−0.245294	+	0.969449i	$0.578885\pi$
$840$	0	0
$841$	−18.9775	−0.654398
$842$	28.3219	0.976036
$843$	0	0
$844$	2.90764	0.100085
$845$	−17.6605	−0.607540
$846$	0	0
$847$	0	0
$848$	−17.2702	−0.593061
$849$	0	0
$850$	−1.81408	−0.0622223
$851$	−10.7286	−0.367771
$852$	0	0
$853$	19.5795	0.670388	0.335194	−	0.942149i	$-0.391198\pi$
$853$	19.5795	0.670388	0.335194	+	0.942149i	$0.391198\pi$
$854$	−2.31069	−0.0790703
$855$	0	0
$856$	−3.59235	−0.122784
$857$	18.8814	0.644977	0.322489	−	0.946573i	$-0.395481\pi$
$857$	18.8814	0.644977	0.322489	+	0.946573i	$0.395481\pi$
$858$	0	0
$859$	−7.77215	−0.265182	−0.132591	−	0.991171i	$-0.542330\pi$
$859$	−7.77215	−0.265182	−0.132591	+	0.991171i	$0.542330\pi$
$860$	4.66976	0.159238
$861$	0	0
$862$	−32.9129	−1.12102
$863$	7.94845	0.270568	0.135284	−	0.990807i	$-0.456805\pi$
$863$	7.94845	0.270568	0.135284	+	0.990807i	$0.456805\pi$
$864$	0	0
$865$	30.1286	1.02440
$866$	34.4926	1.17211
$867$	0	0
$868$	0.711088	0.0241359
$869$	0	0
$870$	0	0
$871$	−42.8783	−1.45287
$872$	−47.0857	−1.59452
$873$	0	0
$874$	9.08928	0.307450
$875$	10.6700	0.360713
$876$	0	0
$877$	19.7350	0.666404	0.333202	−	0.942855i	$-0.391871\pi$
$877$	19.7350	0.666404	0.333202	+	0.942855i	$0.391871\pi$
$878$	0.816405	0.0275523
$879$	0	0
$880$	0	0
$881$	32.2300	1.08586	0.542928	−	0.839779i	$-0.317316\pi$
$881$	32.2300	1.08586	0.542928	+	0.839779i	$0.317316\pi$
$882$	0	0
$883$	−18.6496	−0.627610	−0.313805	−	0.949488i	$-0.601604\pi$
$883$	−18.6496	−0.627610	−0.313805	+	0.949488i	$0.601604\pi$
$884$	−3.58162	−0.120463
$885$	0	0
$886$	−48.2075	−1.61956
$887$	37.6169	1.26305	0.631525	−	0.775355i	$-0.282429\pi$
$887$	37.6169	1.26305	0.631525	+	0.775355i	$0.282429\pi$
$888$	0	0
$889$	16.0576	0.538554
$890$	−12.1589	−0.407567
$891$	0	0
$892$	−3.33165	−0.111552
$893$	5.10979	0.170993
$894$	0	0
$895$	−43.6880	−1.46033
$896$	−8.87093	−0.296357
$897$	0	0
$898$	37.9308	1.26577
$899$	9.40683	0.313735
$900$	0	0
$901$	16.4435	0.547811
$902$	0	0
$903$	0	0
$904$	54.8861	1.82548
$905$	17.6774	0.587618
$906$	0	0
$907$	−23.3358	−0.774851	−0.387426	−	0.921901i	$-0.626636\pi$
$907$	−23.3358	−0.774851	−0.387426	+	0.921901i	$0.626636\pi$
$908$	−2.52698	−0.0838607
$909$	0	0
$910$	14.0102	0.464435
$911$	−39.8541	−1.32042	−0.660212	−	0.751079i	$-0.729534\pi$
$911$	−39.8541	−1.32042	−0.660212	+	0.751079i	$0.729534\pi$
$912$	0	0
$913$	0	0
$914$	52.1321	1.72438
$915$	0	0
$916$	6.91898	0.228609
$917$	−19.2597	−0.636012
$918$	0	0
$919$	−7.17929	−0.236823	−0.118411	−	0.992965i	$-0.537780\pi$
$919$	−7.17929	−0.236823	−0.118411	+	0.992965i	$0.537780\pi$
$920$	43.5468	1.43570
$921$	0	0
$922$	0.545576	0.0179676
$923$	−73.3242	−2.41350
$924$	0	0
$925$	0.706072	0.0232155
$926$	23.3774	0.768228
$927$	0	0
$928$	4.26178	0.139900
$929$	−15.3753	−0.504446	−0.252223	−	0.967669i	$-0.581162\pi$
$929$	−15.3753	−0.504446	−0.252223	+	0.967669i	$0.581162\pi$
$930$	0	0
$931$	1.08759	0.0356445
$932$	−4.36870	−0.143101
$933$	0	0
$934$	−10.1011	−0.330517
$935$	0	0
$936$	0	0
$937$	−34.8669	−1.13905	−0.569526	−	0.821973i	$-0.692873\pi$
$937$	−34.8669	−1.13905	−0.569526	+	0.821973i	$0.692873\pi$
$938$	12.5387	0.409404
$939$	0	0
$940$	2.61627	0.0853334
$941$	−11.1503	−0.363491	−0.181745	−	0.983346i	$-0.558175\pi$
$941$	−11.1503	−0.363491	−0.181745	+	0.983346i	$0.558175\pi$
$942$	0	0
$943$	30.9071	1.00647
$944$	34.8676	1.13484
$945$	0	0
$946$	0	0
$947$	8.00144	0.260012	0.130006	−	0.991513i	$-0.458500\pi$
$947$	8.00144	0.260012	0.130006	+	0.991513i	$0.458500\pi$
$948$	0	0
$949$	−65.6588	−2.13138
$950$	−0.598186	−0.0194077
$951$	0	0
$952$	9.80037	0.317632
$953$	−50.1472	−1.62443	−0.812213	−	0.583361i	$-0.801737\pi$
$953$	−50.1472	−1.62443	−0.812213	+	0.583361i	$0.801737\pi$
$954$	0	0
$955$	−50.2750	−1.62686
$956$	−0.487293	−0.0157602
$957$	0	0
$958$	46.1032	1.48953
$959$	−5.24059	−0.169227
$960$	0	0
$961$	−22.1710	−0.715193
$962$	−10.2562	−0.330673
$963$	0	0
$964$	6.92442	0.223021
$965$	−59.0961	−1.90237
$966$	0	0
$967$	−12.2748	−0.394731	−0.197366	−	0.980330i	$-0.563239\pi$
$967$	−12.2748	−0.394731	−0.197366	+	0.980330i	$0.563239\pi$
$968$	0	0
$969$	0	0
$970$	40.2759	1.29318
$971$	−31.5666	−1.01302	−0.506511	−	0.862234i	$-0.669065\pi$
$971$	−31.5666	−1.01302	−0.506511	+	0.862234i	$0.669065\pi$
$972$	0	0
$973$	22.9872	0.736935
$974$	−28.7497	−0.921200
$975$	0	0
$976$	−6.03243	−0.193093
$977$	12.1052	0.387279	0.193640	−	0.981073i	$-0.437971\pi$
$977$	12.1052	0.387279	0.193640	+	0.981073i	$0.437971\pi$
$978$	0	0
$979$	0	0
$980$	0.556861	0.0177883
$981$	0	0
$982$	17.7194	0.565449
$983$	5.33084	0.170027	0.0850136	−	0.996380i	$-0.472907\pi$
$983$	5.33084	0.170027	0.0850136	+	0.996380i	$0.472907\pi$
$984$	0	0
$985$	−15.8141	−0.503878
$986$	13.8553	0.441242
$987$	0	0
$988$	−1.18103	−0.0375735
$989$	−52.8165	−1.67947
$990$	0	0
$991$	40.1178	1.27438	0.637192	−	0.770705i	$-0.280096\pi$
$991$	40.1178	1.27438	0.637192	+	0.770705i	$0.280096\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	21.4419	0.680096
$995$	44.7409	1.41838
$996$	0	0
$997$	−15.0344	−0.476143	−0.238071	−	0.971248i	$-0.576515\pi$
$997$	−15.0344	−0.476143	−0.238071	+	0.971248i	$0.576515\pi$
$998$	25.9826	0.822465
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cf.1.2		4
3.2	odd	2		2541.2.a.bo.1.3	yes	4
11.10	odd	2		7623.2.a.cm.1.3		4
33.32	even	2		2541.2.a.bk.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bk.1.2	✓	4	33.32	even	2
2541.2.a.bo.1.3	yes	4	3.2	odd	2
7623.2.a.cf.1.2		4	1.1	even	1	trivial
7623.2.a.cm.1.3		4	11.10	odd	2

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32691 q^{2} -0.239314 q^{4} -2.32691 q^{5} +1.00000 q^{7} +2.97136 q^{8} +O(q^{10})\)	\(q-1.32691 q^{2} -0.239314 q^{4} -2.32691 q^{5} +1.00000 q^{7} +2.97136 q^{8} +3.08759 q^{10} +4.53759 q^{13} -1.32691 q^{14} -3.46410 q^{16} +3.29827 q^{17} +1.08759 q^{19} +0.556861 q^{20} -6.29827 q^{23} +0.414503 q^{25} -6.02096 q^{26} -0.239314 q^{28} -3.16583 q^{29} -2.97136 q^{31} -1.34618 q^{32} -4.37651 q^{34} -2.32691 q^{35} +1.70342 q^{37} -1.44314 q^{38} -6.91409 q^{40} -4.90724 q^{41} +8.38587 q^{43} +8.35723 q^{46} +4.69825 q^{47} +1.00000 q^{49} -0.550008 q^{50} -1.08591 q^{52} +4.98547 q^{53} +2.97136 q^{56} +4.20077 q^{58} -10.0654 q^{59} +1.74141 q^{61} +3.94273 q^{62} +8.71446 q^{64} -10.5585 q^{65} -9.44958 q^{67} -0.789322 q^{68} +3.08759 q^{70} -16.1593 q^{71} -14.4700 q^{73} -2.26028 q^{74} -0.260276 q^{76} +12.8359 q^{79} +8.06065 q^{80} +6.51146 q^{82} +0.205514 q^{83} -7.67478 q^{85} -11.1273 q^{86} -3.93798 q^{89} +4.53759 q^{91} +1.50726 q^{92} -6.23415 q^{94} -2.53073 q^{95} +13.0444 q^{97} -1.32691 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 6 * q^5 + 4 * q^7	\( 4 q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 4 q^{7} + 14 q^{10} - 2 q^{13} - 2 q^{14} - 2 q^{17} + 6 q^{19} - 8 q^{20} - 10 q^{23} + 4 q^{28} - 14 q^{29} - 12 q^{32} - 2 q^{34} - 6 q^{35} - 12 q^{37} - 16 q^{38} + 8 q^{40} - 16 q^{41} + 20 q^{43} + 8 q^{46} - 2 q^{47} + 4 q^{49} - 24 q^{50} - 40 q^{52} + 16 q^{53} - 8 q^{58} - 2 q^{59} + 2 q^{61} - 8 q^{62} - 16 q^{64} + 2 q^{65} - 20 q^{67} - 20 q^{68} + 14 q^{70} + 10 q^{71} - 10 q^{73} + 20 q^{74} + 28 q^{76} + 16 q^{79} - 12 q^{80} + 28 q^{82} - 18 q^{83} - 26 q^{86} + 14 q^{89} - 2 q^{91} + 8 q^{92} - 18 q^{94} - 22 q^{95} + 38 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 6 * q^5 + 4 * q^7 + 14 * q^10 - 2 * q^13 - 2 * q^14 - 2 * q^17 + 6 * q^19 - 8 * q^20 - 10 * q^23 + 4 * q^28 - 14 * q^29 - 12 * q^32 - 2 * q^34 - 6 * q^35 - 12 * q^37 - 16 * q^38 + 8 * q^40 - 16 * q^41 + 20 * q^43 + 8 * q^46 - 2 * q^47 + 4 * q^49 - 24 * q^50 - 40 * q^52 + 16 * q^53 - 8 * q^58 - 2 * q^59 + 2 * q^61 - 8 * q^62 - 16 * q^64 + 2 * q^65 - 20 * q^67 - 20 * q^68 + 14 * q^70 + 10 * q^71 - 10 * q^73 + 20 * q^74 + 28 * q^76 + 16 * q^79 - 12 * q^80 + 28 * q^82 - 18 * q^83 - 26 * q^86 + 14 * q^89 - 2 * q^91 + 8 * q^92 - 18 * q^94 - 22 * q^95 + 38 * q^97 - 2 * q^98

Introduction

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