LMFDB - Embedded newform 4050.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(1,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.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:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 450)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	4050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} +1.00000 q^{8} -4.89898 q^{11} +4.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +4.89898 q^{17} +2.55051 q^{19} -4.89898 q^{22} -2.44949 q^{23} +4.44949 q^{26} +4.44949 q^{28} -2.44949 q^{29} -0.449490 q^{31} +1.00000 q^{32} +4.89898 q^{34} +3.34847 q^{37} +2.55051 q^{38} -9.00000 q^{41} +7.44949 q^{43} -4.89898 q^{44} -2.44949 q^{46} +1.10102 q^{47} +12.7980 q^{49} +4.44949 q^{52} +8.44949 q^{53} +4.44949 q^{56} -2.44949 q^{58} +0.550510 q^{59} +8.00000 q^{61} -0.449490 q^{62} +1.00000 q^{64} -14.3485 q^{67} +4.89898 q^{68} +1.34847 q^{71} -1.00000 q^{73} +3.34847 q^{74} +2.55051 q^{76} -21.7980 q^{77} -12.6969 q^{79} -9.00000 q^{82} +0.550510 q^{83} +7.44949 q^{86} -4.89898 q^{88} +9.00000 q^{89} +19.7980 q^{91} -2.44949 q^{92} +1.10102 q^{94} -10.7980 q^{97} +12.7980 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 4 q^{13} + 4 q^{14} + 2 q^{16} + 10 q^{19} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} - 8 q^{37} + 10 q^{38} - 18 q^{41} + 10 q^{43} + 12 q^{47} + 6 q^{49} + 4 q^{52} + 12 q^{53} + 4 q^{56} + 6 q^{59} + 16 q^{61} + 4 q^{62} + 2 q^{64} - 14 q^{67} - 12 q^{71} - 2 q^{73} - 8 q^{74} + 10 q^{76} - 24 q^{77} + 4 q^{79} - 18 q^{82} + 6 q^{83} + 10 q^{86} + 18 q^{89} + 20 q^{91} + 12 q^{94} - 2 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 4 * q^13 + 4 * q^14 + 2 * q^16 + 10 * q^19 + 4 * q^26 + 4 * q^28 + 4 * q^31 + 2 * q^32 - 8 * q^37 + 10 * q^38 - 18 * q^41 + 10 * q^43 + 12 * q^47 + 6 * q^49 + 4 * q^52 + 12 * q^53 + 4 * q^56 + 6 * q^59 + 16 * q^61 + 4 * q^62 + 2 * q^64 - 14 * q^67 - 12 * q^71 - 2 * q^73 - 8 * q^74 + 10 * q^76 - 24 * q^77 + 4 * q^79 - 18 * q^82 + 6 * q^83 + 10 * q^86 + 18 * q^89 + 20 * q^91 + 12 * q^94 - 2 * q^97 + 6 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.44949	1.68175	0.840875	−	0.541230i	$-0.182041\pi$
$7$	4.44949	1.68175	0.840875	+	0.541230i	$0.182041\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	4.44949	1.23407	0.617033	−	0.786937i	$-0.288334\pi$
$13$	4.44949	1.23407	0.617033	+	0.786937i	$0.288334\pi$
$14$	4.44949	1.18918
$15$	0	0
$16$	1.00000	0.250000
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	2.55051	0.585127	0.292564	−	0.956246i	$-0.405492\pi$
$19$	2.55051	0.585127	0.292564	+	0.956246i	$0.405492\pi$
$20$	0	0
$21$	0	0
$22$	−4.89898	−1.04447
$23$	−2.44949	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$23$	−2.44949	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$24$	0	0
$25$	0	0
$26$	4.44949	0.872617
$27$	0	0
$28$	4.44949	0.840875
$29$	−2.44949	−0.454859	−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949	−0.454859	−0.227429	+	0.973795i	$0.573032\pi$
$30$	0	0
$31$	−0.449490	−0.0807307	−0.0403654	−	0.999185i	$-0.512852\pi$
$31$	−0.449490	−0.0807307	−0.0403654	+	0.999185i	$0.512852\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.89898	0.840168
$35$	0	0
$36$	0	0
$37$	3.34847	0.550485	0.275242	−	0.961375i	$-0.411242\pi$
$37$	3.34847	0.550485	0.275242	+	0.961375i	$0.411242\pi$
$38$	2.55051	0.413747
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	7.44949	1.13604	0.568018	−	0.823016i	$-0.307710\pi$
$43$	7.44949	1.13604	0.568018	+	0.823016i	$0.307710\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	−2.44949	−0.361158
$47$	1.10102	0.160600	0.0803002	−	0.996771i	$-0.474412\pi$
$47$	1.10102	0.160600	0.0803002	+	0.996771i	$0.474412\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	0	0
$51$	0	0
$52$	4.44949	0.617033
$53$	8.44949	1.16063	0.580313	−	0.814393i	$-0.302930\pi$
$53$	8.44949	1.16063	0.580313	+	0.814393i	$0.302930\pi$
$54$	0	0
$55$	0	0
$56$	4.44949	0.594588
$57$	0	0
$58$	−2.44949	−0.321634
$59$	0.550510	0.0716703	0.0358352	−	0.999358i	$-0.488591\pi$
$59$	0.550510	0.0716703	0.0358352	+	0.999358i	$0.488591\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−0.449490	−0.0570853
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−14.3485	−1.75294	−0.876472	−	0.481452i	$-0.840109\pi$
$67$	−14.3485	−1.75294	−0.876472	+	0.481452i	$0.840109\pi$
$68$	4.89898	0.594089
$69$	0	0
$70$	0	0
$71$	1.34847	0.160034	0.0800169	−	0.996794i	$-0.474503\pi$
$71$	1.34847	0.160034	0.0800169	+	0.996794i	$0.474503\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	3.34847	0.389252
$75$	0	0
$76$	2.55051	0.292564
$77$	−21.7980	−2.48411
$78$	0	0
$79$	−12.6969	−1.42852	−0.714259	−	0.699882i	$-0.753236\pi$
$79$	−12.6969	−1.42852	−0.714259	+	0.699882i	$0.753236\pi$
$80$	0	0
$81$	0	0
$82$	−9.00000	−0.993884
$83$	0.550510	0.0604264	0.0302132	−	0.999543i	$-0.490381\pi$
$83$	0.550510	0.0604264	0.0302132	+	0.999543i	$0.490381\pi$
$84$	0	0
$85$	0	0
$86$	7.44949	0.803299
$87$	0	0
$88$	−4.89898	−0.522233
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	19.7980	2.07539
$92$	−2.44949	−0.255377
$93$	0	0
$94$	1.10102	0.113562
$95$	0	0
$96$	0	0
$97$	−10.7980	−1.09637	−0.548183	−	0.836358i	$-0.684680\pi$
$97$	−10.7980	−1.09637	−0.548183	+	0.836358i	$0.684680\pi$
$98$	12.7980	1.29279
$99$	0	0
$100$	0	0
$101$	−3.55051	−0.353289	−0.176644	−	0.984275i	$-0.556524\pi$
$101$	−3.55051	−0.353289	−0.176644	+	0.984275i	$0.556524\pi$
$102$	0	0
$103$	−12.6969	−1.25107	−0.625533	−	0.780198i	$-0.715119\pi$
$103$	−12.6969	−1.25107	−0.625533	+	0.780198i	$0.715119\pi$
$104$	4.44949	0.436308
$105$	0	0
$106$	8.44949	0.820687
$107$	15.2474	1.47403	0.737013	−	0.675878i	$-0.236236\pi$
$107$	15.2474	1.47403	0.737013	+	0.675878i	$0.236236\pi$
$108$	0	0
$109$	10.4495	1.00088	0.500440	−	0.865771i	$-0.333172\pi$
$109$	10.4495	1.00088	0.500440	+	0.865771i	$0.333172\pi$
$110$	0	0
$111$	0	0
$112$	4.44949	0.420437
$113$	13.8990	1.30751	0.653753	−	0.756708i	$-0.273194\pi$
$113$	13.8990	1.30751	0.653753	+	0.756708i	$0.273194\pi$
$114$	0	0
$115$	0	0
$116$	−2.44949	−0.227429
$117$	0	0
$118$	0.550510	0.0506786
$119$	21.7980	1.99822
$120$	0	0
$121$	13.0000	1.18182
$122$	8.00000	0.724286
$123$	0	0
$124$	−0.449490	−0.0403654
$125$	0	0
$126$	0	0
$127$	−11.3485	−1.00701	−0.503507	−	0.863991i	$-0.667957\pi$
$127$	−11.3485	−1.00701	−0.503507	+	0.863991i	$0.667957\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−15.7980	−1.38027	−0.690137	−	0.723679i	$-0.742450\pi$
$131$	−15.7980	−1.38027	−0.690137	+	0.723679i	$0.742450\pi$
$132$	0	0
$133$	11.3485	0.984037
$134$	−14.3485	−1.23952
$135$	0	0
$136$	4.89898	0.420084
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	1.34847	0.113161
$143$	−21.7980	−1.82284
$144$	0	0
$145$	0	0
$146$	−1.00000	−0.0827606
$147$	0	0
$148$	3.34847	0.275242
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	2.55051	0.206874
$153$	0	0
$154$	−21.7980	−1.75653
$155$	0	0
$156$	0	0
$157$	−0.202041	−0.0161246	−0.00806231	−	0.999967i	$-0.502566\pi$
$157$	−0.202041	−0.0161246	−0.00806231	+	0.999967i	$0.502566\pi$
$158$	−12.6969	−1.01011
$159$	0	0
$160$	0	0
$161$	−10.8990	−0.858960
$162$	0	0
$163$	2.55051	0.199771	0.0998857	−	0.994999i	$-0.468152\pi$
$163$	2.55051	0.199771	0.0998857	+	0.994999i	$0.468152\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	0.550510	0.0427279
$167$	19.5959	1.51638	0.758189	−	0.652035i	$-0.226085\pi$
$167$	19.5959	1.51638	0.758189	+	0.652035i	$0.226085\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	0	0
$172$	7.44949	0.568018
$173$	9.79796	0.744925	0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796	0.744925	0.372463	+	0.928047i	$0.378514\pi$
$174$	0	0
$175$	0	0
$176$	−4.89898	−0.369274
$177$	0	0
$178$	9.00000	0.674579
$179$	15.2474	1.13965	0.569824	−	0.821767i	$-0.307011\pi$
$179$	15.2474	1.13965	0.569824	+	0.821767i	$0.307011\pi$
$180$	0	0
$181$	−1.79796	−0.133641	−0.0668206	−	0.997765i	$-0.521286\pi$
$181$	−1.79796	−0.133641	−0.0668206	+	0.997765i	$0.521286\pi$
$182$	19.7980	1.46752
$183$	0	0
$184$	−2.44949	−0.180579
$185$	0	0
$186$	0	0
$187$	−24.0000	−1.75505
$188$	1.10102	0.0803002
$189$	0	0
$190$	0	0
$191$	10.8990	0.788622	0.394311	−	0.918977i	$-0.370983\pi$
$191$	10.8990	0.788622	0.394311	+	0.918977i	$0.370983\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	−10.7980	−0.775248
$195$	0	0
$196$	12.7980	0.914140
$197$	−24.2474	−1.72756	−0.863780	−	0.503870i	$-0.831909\pi$
$197$	−24.2474	−1.72756	−0.863780	+	0.503870i	$0.831909\pi$
$198$	0	0
$199$	5.79796	0.411006	0.205503	−	0.978656i	$-0.434117\pi$
$199$	5.79796	0.411006	0.205503	+	0.978656i	$0.434117\pi$
$200$	0	0
$201$	0	0
$202$	−3.55051	−0.249813
$203$	−10.8990	−0.764958
$204$	0	0
$205$	0	0
$206$	−12.6969	−0.884638
$207$	0	0
$208$	4.44949	0.308517
$209$	−12.4949	−0.864290
$210$	0	0
$211$	1.44949	0.0997870	0.0498935	−	0.998755i	$-0.484112\pi$
$211$	1.44949	0.0997870	0.0498935	+	0.998755i	$0.484112\pi$
$212$	8.44949	0.580313
$213$	0	0
$214$	15.2474	1.04229
$215$	0	0
$216$	0	0
$217$	−2.00000	−0.135769
$218$	10.4495	0.707729
$219$	0	0
$220$	0	0
$221$	21.7980	1.46629
$222$	0	0
$223$	−1.79796	−0.120400	−0.0602001	−	0.998186i	$-0.519174\pi$
$223$	−1.79796	−0.120400	−0.0602001	+	0.998186i	$0.519174\pi$
$224$	4.44949	0.297294
$225$	0	0
$226$	13.8990	0.924546
$227$	−28.3485	−1.88155	−0.940777	−	0.339026i	$-0.889903\pi$
$227$	−28.3485	−1.88155	−0.940777	+	0.339026i	$0.889903\pi$
$228$	0	0
$229$	−21.1464	−1.39740	−0.698698	−	0.715417i	$-0.746237\pi$
$229$	−21.1464	−1.39740	−0.698698	+	0.715417i	$0.746237\pi$
$230$	0	0
$231$	0	0
$232$	−2.44949	−0.160817
$233$	−5.69694	−0.373219	−0.186609	−	0.982434i	$-0.559750\pi$
$233$	−5.69694	−0.373219	−0.186609	+	0.982434i	$0.559750\pi$
$234$	0	0
$235$	0	0
$236$	0.550510	0.0358352
$237$	0	0
$238$	21.7980	1.41295
$239$	−9.55051	−0.617771	−0.308886	−	0.951099i	$-0.599956\pi$
$239$	−9.55051	−0.617771	−0.308886	+	0.951099i	$0.599956\pi$
$240$	0	0
$241$	−22.7980	−1.46855	−0.734273	−	0.678854i	$-0.762477\pi$
$241$	−22.7980	−1.46855	−0.734273	+	0.678854i	$0.762477\pi$
$242$	13.0000	0.835672
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	11.3485	0.722086
$248$	−0.449490	−0.0285426
$249$	0	0
$250$	0	0
$251$	5.44949	0.343969	0.171984	−	0.985100i	$-0.444982\pi$
$251$	5.44949	0.343969	0.171984	+	0.985100i	$0.444982\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	−11.3485	−0.712066
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.79796	−0.424045	−0.212023	−	0.977265i	$-0.568005\pi$
$257$	−6.79796	−0.424045	−0.212023	+	0.977265i	$0.568005\pi$
$258$	0	0
$259$	14.8990	0.925778
$260$	0	0
$261$	0	0
$262$	−15.7980	−0.976001
$263$	−15.5505	−0.958886	−0.479443	−	0.877573i	$-0.659161\pi$
$263$	−15.5505	−0.958886	−0.479443	+	0.877573i	$0.659161\pi$
$264$	0	0
$265$	0	0
$266$	11.3485	0.695819
$267$	0	0
$268$	−14.3485	−0.876472
$269$	−9.55051	−0.582305	−0.291152	−	0.956677i	$-0.594039\pi$
$269$	−9.55051	−0.582305	−0.291152	+	0.956677i	$0.594039\pi$
$270$	0	0
$271$	0.651531	0.0395777	0.0197888	−	0.999804i	$-0.493701\pi$
$271$	0.651531	0.0395777	0.0197888	+	0.999804i	$0.493701\pi$
$272$	4.89898	0.297044
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	0	0
$277$	−6.44949	−0.387512	−0.193756	−	0.981050i	$-0.562067\pi$
$277$	−6.44949	−0.387512	−0.193756	+	0.981050i	$0.562067\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	−28.8990	−1.72397	−0.861984	−	0.506935i	$-0.830778\pi$
$281$	−28.8990	−1.72397	−0.861984	+	0.506935i	$0.830778\pi$
$282$	0	0
$283$	11.2474	0.668591	0.334296	−	0.942468i	$-0.391502\pi$
$283$	11.2474	0.668591	0.334296	+	0.942468i	$0.391502\pi$
$284$	1.34847	0.0800169
$285$	0	0
$286$	−21.7980	−1.28894
$287$	−40.0454	−2.36381
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	28.0454	1.63843	0.819215	−	0.573486i	$-0.194409\pi$
$293$	28.0454	1.63843	0.819215	+	0.573486i	$0.194409\pi$
$294$	0	0
$295$	0	0
$296$	3.34847	0.194626
$297$	0	0
$298$	6.00000	0.347571
$299$	−10.8990	−0.630304
$300$	0	0
$301$	33.1464	1.91053
$302$	20.0000	1.15087
$303$	0	0
$304$	2.55051	0.146282
$305$	0	0
$306$	0	0
$307$	−6.69694	−0.382214	−0.191107	−	0.981569i	$-0.561208\pi$
$307$	−6.69694	−0.382214	−0.191107	+	0.981569i	$0.561208\pi$
$308$	−21.7980	−1.24205
$309$	0	0
$310$	0	0
$311$	10.8990	0.618024	0.309012	−	0.951058i	$-0.400002\pi$
$311$	10.8990	0.618024	0.309012	+	0.951058i	$0.400002\pi$
$312$	0	0
$313$	3.89898	0.220383	0.110192	−	0.993910i	$-0.464854\pi$
$313$	3.89898	0.220383	0.110192	+	0.993910i	$0.464854\pi$
$314$	−0.202041	−0.0114018
$315$	0	0
$316$	−12.6969	−0.714259
$317$	17.1464	0.963039	0.481520	−	0.876435i	$-0.340085\pi$
$317$	17.1464	0.963039	0.481520	+	0.876435i	$0.340085\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	0	0
$322$	−10.8990	−0.607376
$323$	12.4949	0.695235
$324$	0	0
$325$	0	0
$326$	2.55051	0.141260
$327$	0	0
$328$	−9.00000	−0.496942
$329$	4.89898	0.270089
$330$	0	0
$331$	−8.34847	−0.458873	−0.229437	−	0.973324i	$-0.573688\pi$
$331$	−8.34847	−0.458873	−0.229437	+	0.973324i	$0.573688\pi$
$332$	0.550510	0.0302132
$333$	0	0
$334$	19.5959	1.07224
$335$	0	0
$336$	0	0
$337$	−11.1010	−0.604711	−0.302356	−	0.953195i	$-0.597773\pi$
$337$	−11.1010	−0.604711	−0.302356	+	0.953195i	$0.597773\pi$
$338$	6.79796	0.369760
$339$	0	0
$340$	0	0
$341$	2.20204	0.119247
$342$	0	0
$343$	25.7980	1.39296
$344$	7.44949	0.401650
$345$	0	0
$346$	9.79796	0.526742
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	−4.89898	−0.261116
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	0	0
$356$	9.00000	0.476999
$357$	0	0
$358$	15.2474	0.805853
$359$	33.7980	1.78379	0.891894	−	0.452244i	$-0.149377\pi$
$359$	33.7980	1.78379	0.891894	+	0.452244i	$0.149377\pi$
$360$	0	0
$361$	−12.4949	−0.657626
$362$	−1.79796	−0.0944986
$363$	0	0
$364$	19.7980	1.03770
$365$	0	0
$366$	0	0
$367$	16.6969	0.871573	0.435787	−	0.900050i	$-0.356470\pi$
$367$	16.6969	0.871573	0.435787	+	0.900050i	$0.356470\pi$
$368$	−2.44949	−0.127688
$369$	0	0
$370$	0	0
$371$	37.5959	1.95188
$372$	0	0
$373$	15.5959	0.807526	0.403763	−	0.914864i	$-0.367702\pi$
$373$	15.5959	0.807526	0.403763	+	0.914864i	$0.367702\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	1.10102	0.0567808
$377$	−10.8990	−0.561326
$378$	0	0
$379$	0.898979	0.0461775	0.0230887	−	0.999733i	$-0.492650\pi$
$379$	0.898979	0.0461775	0.0230887	+	0.999733i	$0.492650\pi$
$380$	0	0
$381$	0	0
$382$	10.8990	0.557640
$383$	−26.4495	−1.35151	−0.675753	−	0.737128i	$-0.736181\pi$
$383$	−26.4495	−1.35151	−0.675753	+	0.737128i	$0.736181\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	−10.7980	−0.548183
$389$	13.5959	0.689340	0.344670	−	0.938724i	$-0.387991\pi$
$389$	13.5959	0.689340	0.344670	+	0.938724i	$0.387991\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	12.7980	0.646395
$393$	0	0
$394$	−24.2474	−1.22157
$395$	0	0
$396$	0	0
$397$	21.5959	1.08387	0.541934	−	0.840421i	$-0.317692\pi$
$397$	21.5959	1.08387	0.541934	+	0.840421i	$0.317692\pi$
$398$	5.79796	0.290625
$399$	0	0
$400$	0	0
$401$	−9.30306	−0.464573	−0.232286	−	0.972647i	$-0.574621\pi$
$401$	−9.30306	−0.464573	−0.232286	+	0.972647i	$0.574621\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	−3.55051	−0.176644
$405$	0	0
$406$	−10.8990	−0.540907
$407$	−16.4041	−0.813120
$408$	0	0
$409$	9.89898	0.489473	0.244737	−	0.969590i	$-0.421299\pi$
$409$	9.89898	0.489473	0.244737	+	0.969590i	$0.421299\pi$
$410$	0	0
$411$	0	0
$412$	−12.6969	−0.625533
$413$	2.44949	0.120532
$414$	0	0
$415$	0	0
$416$	4.44949	0.218154
$417$	0	0
$418$	−12.4949	−0.611145
$419$	−8.14643	−0.397979	−0.198990	−	0.980002i	$-0.563766\pi$
$419$	−8.14643	−0.397979	−0.198990	+	0.980002i	$0.563766\pi$
$420$	0	0
$421$	18.0454	0.879479	0.439740	−	0.898125i	$-0.355071\pi$
$421$	18.0454	0.879479	0.439740	+	0.898125i	$0.355071\pi$
$422$	1.44949	0.0705601
$423$	0	0
$424$	8.44949	0.410343
$425$	0	0
$426$	0	0
$427$	35.5959	1.72261
$428$	15.2474	0.737013
$429$	0	0
$430$	0	0
$431$	−10.6515	−0.513066	−0.256533	−	0.966535i	$-0.582580\pi$
$431$	−10.6515	−0.513066	−0.256533	+	0.966535i	$0.582580\pi$
$432$	0	0
$433$	−29.5959	−1.42229	−0.711145	−	0.703046i	$-0.751823\pi$
$433$	−29.5959	−1.42229	−0.711145	+	0.703046i	$0.751823\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	10.4495	0.500440
$437$	−6.24745	−0.298856
$438$	0	0
$439$	−11.3485	−0.541633	−0.270816	−	0.962631i	$-0.587294\pi$
$439$	−11.3485	−0.541633	−0.270816	+	0.962631i	$0.587294\pi$
$440$	0	0
$441$	0	0
$442$	21.7980	1.03682
$443$	−27.7980	−1.32072	−0.660360	−	0.750949i	$-0.729596\pi$
$443$	−27.7980	−1.32072	−0.660360	+	0.750949i	$0.729596\pi$
$444$	0	0
$445$	0	0
$446$	−1.79796	−0.0851358
$447$	0	0
$448$	4.44949	0.210219
$449$	−10.5959	−0.500052	−0.250026	−	0.968239i	$-0.580439\pi$
$449$	−10.5959	−0.500052	−0.250026	+	0.968239i	$0.580439\pi$
$450$	0	0
$451$	44.0908	2.07616
$452$	13.8990	0.653753
$453$	0	0
$454$	−28.3485	−1.33046
$455$	0	0
$456$	0	0
$457$	−15.6969	−0.734272	−0.367136	−	0.930167i	$-0.619662\pi$
$457$	−15.6969	−0.734272	−0.367136	+	0.930167i	$0.619662\pi$
$458$	−21.1464	−0.988108
$459$	0	0
$460$	0	0
$461$	−19.3485	−0.901148	−0.450574	−	0.892739i	$-0.648781\pi$
$461$	−19.3485	−0.901148	−0.450574	+	0.892739i	$0.648781\pi$
$462$	0	0
$463$	9.34847	0.434460	0.217230	−	0.976120i	$-0.430298\pi$
$463$	9.34847	0.434460	0.217230	+	0.976120i	$0.430298\pi$
$464$	−2.44949	−0.113715
$465$	0	0
$466$	−5.69694	−0.263906
$467$	−4.34847	−0.201223	−0.100612	−	0.994926i	$-0.532080\pi$
$467$	−4.34847	−0.201223	−0.100612	+	0.994926i	$0.532080\pi$
$468$	0	0
$469$	−63.8434	−2.94801
$470$	0	0
$471$	0	0
$472$	0.550510	0.0253393
$473$	−36.4949	−1.67804
$474$	0	0
$475$	0	0
$476$	21.7980	0.999108
$477$	0	0
$478$	−9.55051	−0.436830
$479$	24.2474	1.10789	0.553947	−	0.832552i	$-0.313121\pi$
$479$	24.2474	1.10789	0.553947	+	0.832552i	$0.313121\pi$
$480$	0	0
$481$	14.8990	0.679335
$482$	−22.7980	−1.03842
$483$	0	0
$484$	13.0000	0.590909
$485$	0	0
$486$	0	0
$487$	−19.5505	−0.885918	−0.442959	−	0.896542i	$-0.646071\pi$
$487$	−19.5505	−0.885918	−0.442959	+	0.896542i	$0.646071\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	0	0
$491$	2.75255	0.124221	0.0621105	−	0.998069i	$-0.480217\pi$
$491$	2.75255	0.124221	0.0621105	+	0.998069i	$0.480217\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	11.3485	0.510592
$495$	0	0
$496$	−0.449490	−0.0201827
$497$	6.00000	0.269137
$498$	0	0
$499$	6.34847	0.284197	0.142098	−	0.989853i	$-0.454615\pi$
$499$	6.34847	0.284197	0.142098	+	0.989853i	$0.454615\pi$
$500$	0	0
$501$	0	0
$502$	5.44949	0.243223
$503$	−26.4495	−1.17932	−0.589662	−	0.807650i	$-0.700739\pi$
$503$	−26.4495	−1.17932	−0.589662	+	0.807650i	$0.700739\pi$
$504$	0	0
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	−11.3485	−0.503507
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−4.44949	−0.196834
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.79796	−0.299845
$515$	0	0
$516$	0	0
$517$	−5.39388	−0.237222
$518$	14.8990	0.654624
$519$	0	0
$520$	0	0
$521$	−29.3939	−1.28777	−0.643885	−	0.765123i	$-0.722678\pi$
$521$	−29.3939	−1.28777	−0.643885	+	0.765123i	$0.722678\pi$
$522$	0	0
$523$	−5.65153	−0.247124	−0.123562	−	0.992337i	$-0.539432\pi$
$523$	−5.65153	−0.247124	−0.123562	+	0.992337i	$0.539432\pi$
$524$	−15.7980	−0.690137
$525$	0	0
$526$	−15.5505	−0.678034
$527$	−2.20204	−0.0959224
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	0	0
$532$	11.3485	0.492019
$533$	−40.0454	−1.73456
$534$	0	0
$535$	0	0
$536$	−14.3485	−0.619759
$537$	0	0
$538$	−9.55051	−0.411752
$539$	−62.6969	−2.70055
$540$	0	0
$541$	−18.2020	−0.782567	−0.391283	−	0.920270i	$-0.627969\pi$
$541$	−18.2020	−0.782567	−0.391283	+	0.920270i	$0.627969\pi$
$542$	0.651531	0.0279856
$543$	0	0
$544$	4.89898	0.210042
$545$	0	0
$546$	0	0
$547$	30.3485	1.29761	0.648803	−	0.760956i	$-0.275270\pi$
$547$	30.3485	1.29761	0.648803	+	0.760956i	$0.275270\pi$
$548$	−3.00000	−0.128154
$549$	0	0
$550$	0	0
$551$	−6.24745	−0.266150
$552$	0	0
$553$	−56.4949	−2.40241
$554$	−6.44949	−0.274013
$555$	0	0
$556$	−4.00000	−0.169638
$557$	17.3939	0.737002	0.368501	−	0.929627i	$-0.379871\pi$
$557$	17.3939	0.737002	0.368501	+	0.929627i	$0.379871\pi$
$558$	0	0
$559$	33.1464	1.40194
$560$	0	0
$561$	0	0
$562$	−28.8990	−1.21903
$563$	−29.9444	−1.26201	−0.631003	−	0.775781i	$-0.717356\pi$
$563$	−29.9444	−1.26201	−0.631003	+	0.775781i	$0.717356\pi$
$564$	0	0
$565$	0	0
$566$	11.2474	0.472766
$567$	0	0
$568$	1.34847	0.0565805
$569$	−14.2020	−0.595381	−0.297690	−	0.954663i	$-0.596216\pi$
$569$	−14.2020	−0.595381	−0.297690	+	0.954663i	$0.596216\pi$
$570$	0	0
$571$	−27.9444	−1.16944	−0.584718	−	0.811237i	$-0.698795\pi$
$571$	−27.9444	−1.16944	−0.584718	+	0.811237i	$0.698795\pi$
$572$	−21.7980	−0.911418
$573$	0	0
$574$	−40.0454	−1.67146
$575$	0	0
$576$	0	0
$577$	−18.3939	−0.765747	−0.382874	−	0.923801i	$-0.625066\pi$
$577$	−18.3939	−0.765747	−0.382874	+	0.923801i	$0.625066\pi$
$578$	7.00000	0.291162
$579$	0	0
$580$	0	0
$581$	2.44949	0.101622
$582$	0	0
$583$	−41.3939	−1.71436
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	28.0454	1.15855
$587$	−2.69694	−0.111315	−0.0556573	−	0.998450i	$-0.517725\pi$
$587$	−2.69694	−0.111315	−0.0556573	+	0.998450i	$0.517725\pi$
$588$	0	0
$589$	−1.14643	−0.0472378
$590$	0	0
$591$	0	0
$592$	3.34847	0.137621
$593$	−1.89898	−0.0779817	−0.0389909	−	0.999240i	$-0.512414\pi$
$593$	−1.89898	−0.0779817	−0.0389909	+	0.999240i	$0.512414\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	−10.8990	−0.445692
$599$	37.3485	1.52602	0.763009	−	0.646388i	$-0.223721\pi$
$599$	37.3485	1.52602	0.763009	+	0.646388i	$0.223721\pi$
$600$	0	0
$601$	32.4949	1.32549	0.662747	−	0.748843i	$-0.269390\pi$
$601$	32.4949	1.32549	0.662747	+	0.748843i	$0.269390\pi$
$602$	33.1464	1.35095
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	−11.3485	−0.460620	−0.230310	−	0.973117i	$-0.573974\pi$
$607$	−11.3485	−0.460620	−0.230310	+	0.973117i	$0.573974\pi$
$608$	2.55051	0.103437
$609$	0	0
$610$	0	0
$611$	4.89898	0.198191
$612$	0	0
$613$	−32.0454	−1.29430	−0.647151	−	0.762362i	$-0.724040\pi$
$613$	−32.0454	−1.29430	−0.647151	+	0.762362i	$0.724040\pi$
$614$	−6.69694	−0.270266
$615$	0	0
$616$	−21.7980	−0.878265
$617$	14.3939	0.579476	0.289738	−	0.957106i	$-0.406432\pi$
$617$	14.3939	0.579476	0.289738	+	0.957106i	$0.406432\pi$
$618$	0	0
$619$	41.7423	1.67777	0.838883	−	0.544311i	$-0.183209\pi$
$619$	41.7423	1.67777	0.838883	+	0.544311i	$0.183209\pi$
$620$	0	0
$621$	0	0
$622$	10.8990	0.437009
$623$	40.0454	1.60439
$624$	0	0
$625$	0	0
$626$	3.89898	0.155835
$627$	0	0
$628$	−0.202041	−0.00806231
$629$	16.4041	0.654074
$630$	0	0
$631$	−25.7980	−1.02700	−0.513500	−	0.858089i	$-0.671651\pi$
$631$	−25.7980	−1.02700	−0.513500	+	0.858089i	$0.671651\pi$
$632$	−12.6969	−0.505057
$633$	0	0
$634$	17.1464	0.680972
$635$	0	0
$636$	0	0
$637$	56.9444	2.25622
$638$	12.0000	0.475085
$639$	0	0
$640$	0	0
$641$	−44.3939	−1.75345	−0.876726	−	0.480989i	$-0.840278\pi$
$641$	−44.3939	−1.75345	−0.876726	+	0.480989i	$0.840278\pi$
$642$	0	0
$643$	−32.3485	−1.27570	−0.637850	−	0.770161i	$-0.720176\pi$
$643$	−32.3485	−1.27570	−0.637850	+	0.770161i	$0.720176\pi$
$644$	−10.8990	−0.429480
$645$	0	0
$646$	12.4949	0.491605
$647$	0.247449	0.00972821	0.00486411	−	0.999988i	$-0.498452\pi$
$647$	0.247449	0.00972821	0.00486411	+	0.999988i	$0.498452\pi$
$648$	0	0
$649$	−2.69694	−0.105864
$650$	0	0
$651$	0	0
$652$	2.55051	0.0998857
$653$	6.24745	0.244482	0.122241	−	0.992500i	$-0.460992\pi$
$653$	6.24745	0.244482	0.122241	+	0.992500i	$0.460992\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	0	0
$658$	4.89898	0.190982
$659$	−44.1464	−1.71970	−0.859850	−	0.510546i	$-0.829443\pi$
$659$	−44.1464	−1.71970	−0.859850	+	0.510546i	$0.829443\pi$
$660$	0	0
$661$	−51.3939	−1.99899	−0.999495	−	0.0317744i	$-0.989884\pi$
$661$	−51.3939	−1.99899	−0.999495	+	0.0317744i	$0.989884\pi$
$662$	−8.34847	−0.324472
$663$	0	0
$664$	0.550510	0.0213639
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	19.5959	0.758189
$669$	0	0
$670$	0	0
$671$	−39.1918	−1.51298
$672$	0	0
$673$	−6.69694	−0.258148	−0.129074	−	0.991635i	$-0.541200\pi$
$673$	−6.69694	−0.258148	−0.129074	+	0.991635i	$0.541200\pi$
$674$	−11.1010	−0.427595
$675$	0	0
$676$	6.79796	0.261460
$677$	−37.8434	−1.45444	−0.727219	−	0.686405i	$-0.759188\pi$
$677$	−37.8434	−1.45444	−0.727219	+	0.686405i	$0.759188\pi$
$678$	0	0
$679$	−48.0454	−1.84381
$680$	0	0
$681$	0	0
$682$	2.20204	0.0843205
$683$	11.9444	0.457039	0.228520	−	0.973539i	$-0.426611\pi$
$683$	11.9444	0.457039	0.228520	+	0.973539i	$0.426611\pi$
$684$	0	0
$685$	0	0
$686$	25.7980	0.984971
$687$	0	0
$688$	7.44949	0.284009
$689$	37.5959	1.43229
$690$	0	0
$691$	−5.04541	−0.191936	−0.0959682	−	0.995384i	$-0.530595\pi$
$691$	−5.04541	−0.191936	−0.0959682	+	0.995384i	$0.530595\pi$
$692$	9.79796	0.372463
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	−44.0908	−1.67006
$698$	14.0000	0.529908
$699$	0	0
$700$	0	0
$701$	−14.2020	−0.536404	−0.268202	−	0.963363i	$-0.586429\pi$
$701$	−14.2020	−0.536404	−0.268202	+	0.963363i	$0.586429\pi$
$702$	0	0
$703$	8.54031	0.322104
$704$	−4.89898	−0.184637
$705$	0	0
$706$	9.00000	0.338719
$707$	−15.7980	−0.594143
$708$	0	0
$709$	−0.449490	−0.0168809	−0.00844047	−	0.999964i	$-0.502687\pi$
$709$	−0.449490	−0.0168809	−0.00844047	+	0.999964i	$0.502687\pi$
$710$	0	0
$711$	0	0
$712$	9.00000	0.337289
$713$	1.10102	0.0412335
$714$	0	0
$715$	0	0
$716$	15.2474	0.569824
$717$	0	0
$718$	33.7980	1.26133
$719$	−52.0454	−1.94097	−0.970483	−	0.241169i	$-0.922469\pi$
$719$	−52.0454	−1.94097	−0.970483	+	0.241169i	$0.922469\pi$
$720$	0	0
$721$	−56.4949	−2.10398
$722$	−12.4949	−0.465012
$723$	0	0
$724$	−1.79796	−0.0668206
$725$	0	0
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	19.7980	0.733761
$729$	0	0
$730$	0	0
$731$	36.4949	1.34981
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	16.6969	0.616295
$735$	0	0
$736$	−2.44949	−0.0902894
$737$	70.2929	2.58927
$738$	0	0
$739$	−41.0454	−1.50988	−0.754940	−	0.655794i	$-0.772334\pi$
$739$	−41.0454	−1.50988	−0.754940	+	0.655794i	$0.772334\pi$
$740$	0	0
$741$	0	0
$742$	37.5959	1.38019
$743$	−34.6515	−1.27124	−0.635621	−	0.772002i	$-0.719256\pi$
$743$	−34.6515	−1.27124	−0.635621	+	0.772002i	$0.719256\pi$
$744$	0	0
$745$	0	0
$746$	15.5959	0.571007
$747$	0	0
$748$	−24.0000	−0.877527
$749$	67.8434	2.47894
$750$	0	0
$751$	−50.0454	−1.82618	−0.913091	−	0.407755i	$-0.866312\pi$
$751$	−50.0454	−1.82618	−0.913091	+	0.407755i	$0.866312\pi$
$752$	1.10102	0.0401501
$753$	0	0
$754$	−10.8990	−0.396917
$755$	0	0
$756$	0	0
$757$	6.04541	0.219724	0.109862	−	0.993947i	$-0.464959\pi$
$757$	6.04541	0.219724	0.109862	+	0.993947i	$0.464959\pi$
$758$	0.898979	0.0326524
$759$	0	0
$760$	0	0
$761$	−4.10102	−0.148662	−0.0743309	−	0.997234i	$-0.523682\pi$
$761$	−4.10102	−0.148662	−0.0743309	+	0.997234i	$0.523682\pi$
$762$	0	0
$763$	46.4949	1.68323
$764$	10.8990	0.394311
$765$	0	0
$766$	−26.4495	−0.955659
$767$	2.44949	0.0884459
$768$	0	0
$769$	50.1918	1.80996	0.904982	−	0.425450i	$-0.139884\pi$
$769$	50.1918	1.80996	0.904982	+	0.425450i	$0.139884\pi$
$770$	0	0
$771$	0	0
$772$	20.0000	0.719816
$773$	49.5959	1.78384	0.891921	−	0.452192i	$-0.149358\pi$
$773$	49.5959	1.78384	0.891921	+	0.452192i	$0.149358\pi$
$774$	0	0
$775$	0	0
$776$	−10.7980	−0.387624
$777$	0	0
$778$	13.5959	0.487437
$779$	−22.9546	−0.822434
$780$	0	0
$781$	−6.60612	−0.236386
$782$	−12.0000	−0.429119
$783$	0	0
$784$	12.7980	0.457070
$785$	0	0
$786$	0	0
$787$	5.30306	0.189034	0.0945169	−	0.995523i	$-0.469869\pi$
$787$	5.30306	0.189034	0.0945169	+	0.995523i	$0.469869\pi$
$788$	−24.2474	−0.863780
$789$	0	0
$790$	0	0
$791$	61.8434	2.19890
$792$	0	0
$793$	35.5959	1.26405
$794$	21.5959	0.766410
$795$	0	0
$796$	5.79796	0.205503
$797$	5.75255	0.203766	0.101883	−	0.994796i	$-0.467513\pi$
$797$	5.75255	0.203766	0.101883	+	0.994796i	$0.467513\pi$
$798$	0	0
$799$	5.39388	0.190822
$800$	0	0
$801$	0	0
$802$	−9.30306	−0.328503
$803$	4.89898	0.172881
$804$	0	0
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	−3.55051	−0.124907
$809$	35.6969	1.25504	0.627519	−	0.778601i	$-0.284071\pi$
$809$	35.6969	1.25504	0.627519	+	0.778601i	$0.284071\pi$
$810$	0	0
$811$	−33.4495	−1.17457	−0.587285	−	0.809380i	$-0.699803\pi$
$811$	−33.4495	−1.17457	−0.587285	+	0.809380i	$0.699803\pi$
$812$	−10.8990	−0.382479
$813$	0	0
$814$	−16.4041	−0.574963
$815$	0	0
$816$	0	0
$817$	19.0000	0.664726
$818$	9.89898	0.346110
$819$	0	0
$820$	0	0
$821$	51.1918	1.78661	0.893304	−	0.449454i	$-0.148381\pi$
$821$	51.1918	1.78661	0.893304	+	0.449454i	$0.148381\pi$
$822$	0	0
$823$	−26.8990	−0.937639	−0.468820	−	0.883294i	$-0.655321\pi$
$823$	−26.8990	−0.937639	−0.468820	+	0.883294i	$0.655321\pi$
$824$	−12.6969	−0.442319
$825$	0	0
$826$	2.44949	0.0852286
$827$	−17.9444	−0.623987	−0.311994	−	0.950084i	$-0.600997\pi$
$827$	−17.9444	−0.623987	−0.311994	+	0.950084i	$0.600997\pi$
$828$	0	0
$829$	26.7423	0.928800	0.464400	−	0.885626i	$-0.346270\pi$
$829$	26.7423	0.928800	0.464400	+	0.885626i	$0.346270\pi$
$830$	0	0
$831$	0	0
$832$	4.44949	0.154258
$833$	62.6969	2.17232
$834$	0	0
$835$	0	0
$836$	−12.4949	−0.432145
$837$	0	0
$838$	−8.14643	−0.281414
$839$	−22.6515	−0.782018	−0.391009	−	0.920387i	$-0.627874\pi$
$839$	−22.6515	−0.782018	−0.391009	+	0.920387i	$0.627874\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	18.0454	0.621886
$843$	0	0
$844$	1.44949	0.0498935
$845$	0	0
$846$	0	0
$847$	57.8434	1.98752
$848$	8.44949	0.290157
$849$	0	0
$850$	0	0
$851$	−8.20204	−0.281162
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	35.5959	1.21807
$855$	0	0
$856$	15.2474	0.521147
$857$	40.1010	1.36982	0.684912	−	0.728625i	$-0.259840\pi$
$857$	40.1010	1.36982	0.684912	+	0.728625i	$0.259840\pi$
$858$	0	0
$859$	−5.65153	−0.192828	−0.0964139	−	0.995341i	$-0.530737\pi$
$859$	−5.65153	−0.192828	−0.0964139	+	0.995341i	$0.530737\pi$
$860$	0	0
$861$	0	0
$862$	−10.6515	−0.362793
$863$	−19.8434	−0.675476	−0.337738	−	0.941240i	$-0.609662\pi$
$863$	−19.8434	−0.675476	−0.337738	+	0.941240i	$0.609662\pi$
$864$	0	0
$865$	0	0
$866$	−29.5959	−1.00571
$867$	0	0
$868$	−2.00000	−0.0678844
$869$	62.2020	2.11006
$870$	0	0
$871$	−63.8434	−2.16325
$872$	10.4495	0.353864
$873$	0	0
$874$	−6.24745	−0.211323
$875$	0	0
$876$	0	0
$877$	−12.2020	−0.412034	−0.206017	−	0.978548i	$-0.566050\pi$
$877$	−12.2020	−0.412034	−0.206017	+	0.978548i	$0.566050\pi$
$878$	−11.3485	−0.382992
$879$	0	0
$880$	0	0
$881$	−9.30306	−0.313428	−0.156714	−	0.987644i	$-0.550090\pi$
$881$	−9.30306	−0.313428	−0.156714	+	0.987644i	$0.550090\pi$
$882$	0	0
$883$	28.2020	0.949074	0.474537	−	0.880235i	$-0.342615\pi$
$883$	28.2020	0.949074	0.474537	+	0.880235i	$0.342615\pi$
$884$	21.7980	0.733145
$885$	0	0
$886$	−27.7980	−0.933891
$887$	−54.4949	−1.82976	−0.914880	−	0.403726i	$-0.867715\pi$
$887$	−54.4949	−1.82976	−0.914880	+	0.403726i	$0.867715\pi$
$888$	0	0
$889$	−50.4949	−1.69354
$890$	0	0
$891$	0	0
$892$	−1.79796	−0.0602001
$893$	2.80816	0.0939716
$894$	0	0
$895$	0	0
$896$	4.44949	0.148647
$897$	0	0
$898$	−10.5959	−0.353590
$899$	1.10102	0.0367211
$900$	0	0
$901$	41.3939	1.37903
$902$	44.0908	1.46806
$903$	0	0
$904$	13.8990	0.462273
$905$	0	0
$906$	0	0
$907$	21.6515	0.718927	0.359464	−	0.933159i	$-0.382960\pi$
$907$	21.6515	0.718927	0.359464	+	0.933159i	$0.382960\pi$
$908$	−28.3485	−0.940777
$909$	0	0
$910$	0	0
$911$	−7.34847	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$911$	−7.34847	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$912$	0	0
$913$	−2.69694	−0.0892556
$914$	−15.6969	−0.519209
$915$	0	0
$916$	−21.1464	−0.698698
$917$	−70.2929	−2.32127
$918$	0	0
$919$	−11.3485	−0.374351	−0.187176	−	0.982326i	$-0.559933\pi$
$919$	−11.3485	−0.374351	−0.187176	+	0.982326i	$0.559933\pi$
$920$	0	0
$921$	0	0
$922$	−19.3485	−0.637208
$923$	6.00000	0.197492
$924$	0	0
$925$	0	0
$926$	9.34847	0.307210
$927$	0	0
$928$	−2.44949	−0.0804084
$929$	−27.1918	−0.892135	−0.446068	−	0.894999i	$-0.647176\pi$
$929$	−27.1918	−0.892135	−0.446068	+	0.894999i	$0.647176\pi$
$930$	0	0
$931$	32.6413	1.06978
$932$	−5.69694	−0.186609
$933$	0	0
$934$	−4.34847	−0.142286
$935$	0	0
$936$	0	0
$937$	26.7980	0.875451	0.437726	−	0.899109i	$-0.355784\pi$
$937$	26.7980	0.875451	0.437726	+	0.899109i	$0.355784\pi$
$938$	−63.8434	−2.08456
$939$	0	0
$940$	0	0
$941$	29.6413	0.966280	0.483140	−	0.875543i	$-0.339496\pi$
$941$	29.6413	0.966280	0.483140	+	0.875543i	$0.339496\pi$
$942$	0	0
$943$	22.0454	0.717897
$944$	0.550510	0.0179176
$945$	0	0
$946$	−36.4949	−1.18655
$947$	8.14643	0.264723	0.132362	−	0.991201i	$-0.457744\pi$
$947$	8.14643	0.264723	0.132362	+	0.991201i	$0.457744\pi$
$948$	0	0
$949$	−4.44949	−0.144437
$950$	0	0
$951$	0	0
$952$	21.7980	0.706476
$953$	−21.7980	−0.706105	−0.353053	−	0.935603i	$-0.614856\pi$
$953$	−21.7980	−0.706105	−0.353053	+	0.935603i	$0.614856\pi$
$954$	0	0
$955$	0	0
$956$	−9.55051	−0.308886
$957$	0	0
$958$	24.2474	0.783400
$959$	−13.3485	−0.431045
$960$	0	0
$961$	−30.7980	−0.993483
$962$	14.8990	0.480362
$963$	0	0
$964$	−22.7980	−0.734273
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	13.0000	0.417836
$969$	0	0
$970$	0	0
$971$	−22.8434	−0.733079	−0.366539	−	0.930403i	$-0.619457\pi$
$971$	−22.8434	−0.733079	−0.366539	+	0.930403i	$0.619457\pi$
$972$	0	0
$973$	−17.7980	−0.570576
$974$	−19.5505	−0.626439
$975$	0	0
$976$	8.00000	0.256074
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	0	0
$979$	−44.0908	−1.40915
$980$	0	0
$981$	0	0
$982$	2.75255	0.0878374
$983$	22.8990	0.730364	0.365182	−	0.930936i	$-0.381007\pi$
$983$	22.8990	0.730364	0.365182	+	0.930936i	$0.381007\pi$
$984$	0	0
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	11.3485	0.361043
$989$	−18.2474	−0.580235
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−0.449490	−0.0142713
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	36.0454	1.14157	0.570785	−	0.821100i	$-0.306639\pi$
$997$	36.0454	1.14157	0.570785	+	0.821100i	$0.306639\pi$
$998$	6.34847	0.200957
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.by.1.2		2
3.2	odd	2		4050.2.a.br.1.2		2
5.2	odd	4		4050.2.c.w.649.4		4
5.3	odd	4		4050.2.c.w.649.1		4
5.4	even	2		4050.2.a.bl.1.1		2
9.2	odd	6		450.2.e.m.301.2	yes	4
9.4	even	3		1350.2.e.k.451.1		4
9.5	odd	6		450.2.e.m.151.1	yes	4
9.7	even	3		1350.2.e.k.901.1		4
15.2	even	4		4050.2.c.y.649.2		4
15.8	even	4		4050.2.c.y.649.3		4
15.14	odd	2		4050.2.a.bu.1.1		2
45.2	even	12		450.2.j.f.49.2		8
45.4	even	6		1350.2.e.n.451.2		4
45.7	odd	12		1350.2.j.g.199.4		8
45.13	odd	12		1350.2.j.g.1099.4		8
45.14	odd	6		450.2.e.l.151.2	✓	4
45.22	odd	12		1350.2.j.g.1099.1		8
45.23	even	12		450.2.j.f.349.2		8
45.29	odd	6		450.2.e.l.301.1	yes	4
45.32	even	12		450.2.j.f.349.3		8
45.34	even	6		1350.2.e.n.901.2		4
45.38	even	12		450.2.j.f.49.3		8
45.43	odd	12		1350.2.j.g.199.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.e.l.151.2	✓	4	45.14	odd	6
450.2.e.l.301.1	yes	4	45.29	odd	6
450.2.e.m.151.1	yes	4	9.5	odd	6
450.2.e.m.301.2	yes	4	9.2	odd	6
450.2.j.f.49.2		8	45.2	even	12
450.2.j.f.49.3		8	45.38	even	12
450.2.j.f.349.2		8	45.23	even	12
450.2.j.f.349.3		8	45.32	even	12
1350.2.e.k.451.1		4	9.4	even	3
1350.2.e.k.901.1		4	9.7	even	3
1350.2.e.n.451.2		4	45.4	even	6
1350.2.e.n.901.2		4	45.34	even	6
1350.2.j.g.199.1		8	45.43	odd	12
1350.2.j.g.199.4		8	45.7	odd	12
1350.2.j.g.1099.1		8	45.22	odd	12
1350.2.j.g.1099.4		8	45.13	odd	12
4050.2.a.bl.1.1		2	5.4	even	2
4050.2.a.br.1.2		2	3.2	odd	2
4050.2.a.bu.1.1		2	15.14	odd	2
4050.2.a.by.1.2		2	1.1	even	1	trivial
4050.2.c.w.649.1		4	5.3	odd	4
4050.2.c.w.649.4		4	5.2	odd	4
4050.2.c.y.649.2		4	15.2	even	4
4050.2.c.y.649.3		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.44949 q^{7} +1.00000 q^{8} -4.89898 q^{11} +4.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +4.89898 q^{17} +2.55051 q^{19} -4.89898 q^{22} -2.44949 q^{23} +4.44949 q^{26} +4.44949 q^{28} -2.44949 q^{29} -0.449490 q^{31} +1.00000 q^{32} +4.89898 q^{34} +3.34847 q^{37} +2.55051 q^{38} -9.00000 q^{41} +7.44949 q^{43} -4.89898 q^{44} -2.44949 q^{46} +1.10102 q^{47} +12.7980 q^{49} +4.44949 q^{52} +8.44949 q^{53} +4.44949 q^{56} -2.44949 q^{58} +0.550510 q^{59} +8.00000 q^{61} -0.449490 q^{62} +1.00000 q^{64} -14.3485 q^{67} +4.89898 q^{68} +1.34847 q^{71} -1.00000 q^{73} +3.34847 q^{74} +2.55051 q^{76} -21.7980 q^{77} -12.6969 q^{79} -9.00000 q^{82} +0.550510 q^{83} +7.44949 q^{86} -4.89898 q^{88} +9.00000 q^{89} +19.7980 q^{91} -2.44949 q^{92} +1.10102 q^{94} -10.7980 q^{97} +12.7980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 4 q^{13} + 4 q^{14} + 2 q^{16} + 10 q^{19} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} - 8 q^{37} + 10 q^{38} - 18 q^{41} + 10 q^{43} + 12 q^{47} + 6 q^{49} + 4 q^{52} + 12 q^{53} + 4 q^{56} + 6 q^{59} + 16 q^{61} + 4 q^{62} + 2 q^{64} - 14 q^{67} - 12 q^{71} - 2 q^{73} - 8 q^{74} + 10 q^{76} - 24 q^{77} + 4 q^{79} - 18 q^{82} + 6 q^{83} + 10 q^{86} + 18 q^{89} + 20 q^{91} + 12 q^{94} - 2 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 4 * q^13 + 4 * q^14 + 2 * q^16 + 10 * q^19 + 4 * q^26 + 4 * q^28 + 4 * q^31 + 2 * q^32 - 8 * q^37 + 10 * q^38 - 18 * q^41 + 10 * q^43 + 12 * q^47 + 6 * q^49 + 4 * q^52 + 12 * q^53 + 4 * q^56 + 6 * q^59 + 16 * q^61 + 4 * q^62 + 2 * q^64 - 14 * q^67 - 12 * q^71 - 2 * q^73 - 8 * q^74 + 10 * q^76 - 24 * q^77 + 4 * q^79 - 18 * q^82 + 6 * q^83 + 10 * q^86 + 18 * q^89 + 20 * q^91 + 12 * q^94 - 2 * q^97 + 6 * q^98

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