LMFDB - Embedded newform 2450.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.41421 q^{12} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -7.07107 q^{19} +2.00000 q^{22} +4.00000 q^{23} -1.41421 q^{24} -5.65685 q^{27} +2.00000 q^{29} +8.48528 q^{31} -1.00000 q^{32} -2.82843 q^{33} -1.41421 q^{34} -1.00000 q^{36} -10.0000 q^{37} +7.07107 q^{38} -9.89949 q^{41} -2.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -2.82843 q^{47} +1.41421 q^{48} +2.00000 q^{51} +2.00000 q^{53} +5.65685 q^{54} -10.0000 q^{57} -2.00000 q^{58} -1.41421 q^{59} +2.82843 q^{61} -8.48528 q^{62} +1.00000 q^{64} +2.82843 q^{66} -12.0000 q^{67} +1.41421 q^{68} +5.65685 q^{69} -12.0000 q^{71} +1.00000 q^{72} +1.41421 q^{73} +10.0000 q^{74} -7.07107 q^{76} -4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{82} -9.89949 q^{83} +2.00000 q^{86} +2.82843 q^{87} +2.00000 q^{88} -7.07107 q^{89} +4.00000 q^{92} +12.0000 q^{93} +2.82843 q^{94} -1.41421 q^{96} -9.89949 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 4 q^{11} + 2 q^{16} + 2 q^{18} + 4 q^{22} + 8 q^{23} + 4 q^{29} - 2 q^{32} - 2 q^{36} - 20 q^{37} - 4 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{51} + 4 q^{53} - 20 q^{57} - 4 q^{58} + 2 q^{64} - 24 q^{67} - 24 q^{71} + 2 q^{72} + 20 q^{74} - 8 q^{79} - 10 q^{81} + 4 q^{86} + 4 q^{88} + 8 q^{92} + 24 q^{93} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 4 * q^11 + 2 * q^16 + 2 * q^18 + 4 * q^22 + 8 * q^23 + 4 * q^29 - 2 * q^32 - 2 * q^36 - 20 * q^37 - 4 * q^43 - 4 * q^44 - 8 * q^46 + 4 * q^51 + 4 * q^53 - 20 * q^57 - 4 * q^58 + 2 * q^64 - 24 * q^67 - 24 * q^71 + 2 * q^72 + 20 * q^74 - 8 * q^79 - 10 * q^81 + 4 * q^86 + 4 * q^88 + 8 * q^92 + 24 * q^93 + 4 * q^99

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.41421	0.408248
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	1.00000	0.235702
$19$	−7.07107	−1.62221	−0.811107	−	0.584898i	$-0.801135\pi$
$19$	−7.07107	−1.62221	−0.811107	+	0.584898i	$0.801135\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.41421	−0.288675
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	−1.00000	−0.176777
$33$	−2.82843	−0.492366
$34$	−1.41421	−0.242536
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	7.07107	1.14708
$39$	0	0
$40$	0	0
$41$	−9.89949	−1.54604	−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949	−1.54604	−0.773021	+	0.634381i	$0.781255\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	0	0
$57$	−10.0000	−1.32453
$58$	−2.00000	−0.262613
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	−8.48528	−1.07763
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.82843	0.348155
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	1.41421	0.171499
$69$	5.65685	0.681005
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−7.07107	−0.811107
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	9.89949	1.09322
$83$	−9.89949	−1.08661	−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949	−1.08661	−0.543305	+	0.839535i	$0.682827\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	2.82843	0.303239
$88$	2.00000	0.213201
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	12.0000	1.24434
$94$	2.82843	0.291730
$95$	0	0
$96$	−1.41421	−0.144338
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	−2.00000	−0.198030
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−5.65685	−0.544331
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−14.1421	−1.34231
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	10.0000	0.936586
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	1.41421	0.130189
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.82843	−0.256074
$123$	−14.0000	−1.26234
$124$	8.48528	0.762001
$125$	0	0
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	−2.82843	−0.249029
$130$	0	0
$131$	12.7279	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$132$	−2.82843	−0.246183
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	−1.41421	−0.121268
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−5.65685	−0.481543
$139$	9.89949	0.839664	0.419832	−	0.907602i	$-0.362089\pi$
$139$	9.89949	0.839664	0.419832	+	0.907602i	$0.362089\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	12.0000	1.00702
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−1.41421	−0.117041
$147$	0	0
$148$	−10.0000	−0.821995
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	7.07107	0.573539
$153$	−1.41421	−0.114332
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.3137	0.902932	0.451466	−	0.892288i	$-0.350901\pi$
$157$	11.3137	0.902932	0.451466	+	0.892288i	$0.350901\pi$
$158$	4.00000	0.318223
$159$	2.82843	0.224309
$160$	0	0
$161$	0	0
$162$	5.00000	0.392837
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−9.89949	−0.773021
$165$	0	0
$166$	9.89949	0.768350
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	7.07107	0.540738
$172$	−2.00000	−0.152499
$173$	16.9706	1.29025	0.645124	−	0.764078i	$-0.276806\pi$
$173$	16.9706	1.29025	0.645124	+	0.764078i	$0.276806\pi$
$174$	−2.82843	−0.214423
$175$	0	0
$176$	−2.00000	−0.150756
$177$	−2.00000	−0.150329
$178$	7.07107	0.529999
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−12.0000	−0.879883
$187$	−2.82843	−0.206835
$188$	−2.82843	−0.206284
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	1.41421	0.102062
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	9.89949	0.710742
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−2.00000	−0.142134
$199$	8.48528	0.601506	0.300753	−	0.953702i	$-0.402762\pi$
$199$	8.48528	0.601506	0.300753	+	0.953702i	$0.402762\pi$
$200$	0	0
$201$	−16.9706	−1.19701
$202$	−8.48528	−0.597022
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	2.82843	0.197066
$207$	−4.00000	−0.278019
$208$	0	0
$209$	14.1421	0.978232
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	2.00000	0.137361
$213$	−16.9706	−1.16280
$214$	−4.00000	−0.273434
$215$	0	0
$216$	5.65685	0.384900
$217$	0	0
$218$	2.00000	0.135457
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	14.1421	0.949158
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−12.0000	−0.798228
$227$	21.2132	1.40797	0.703985	−	0.710215i	$-0.251402\pi$
$227$	21.2132	1.40797	0.703985	+	0.710215i	$0.251402\pi$
$228$	−10.0000	−0.662266
$229$	−16.9706	−1.12145	−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706	−1.12145	−0.560723	+	0.828003i	$0.689477\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	−1.41421	−0.0920575
$237$	−5.65685	−0.367452
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−21.2132	−1.36646	−0.683231	−	0.730202i	$-0.739426\pi$
$241$	−21.2132	−1.36646	−0.683231	+	0.730202i	$0.739426\pi$
$242$	7.00000	0.449977
$243$	9.89949	0.635053
$244$	2.82843	0.181071
$245$	0	0
$246$	14.0000	0.892607
$247$	0	0
$248$	−8.48528	−0.538816
$249$	−14.0000	−0.887214
$250$	0	0
$251$	9.89949	0.624851	0.312425	−	0.949942i	$-0.398859\pi$
$251$	9.89949	0.624851	0.312425	+	0.949942i	$0.398859\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.7279	−0.793946	−0.396973	−	0.917830i	$-0.629939\pi$
$257$	−12.7279	−0.793946	−0.396973	+	0.917830i	$0.629939\pi$
$258$	2.82843	0.176090
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−12.7279	−0.786334
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	2.82843	0.174078
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−12.0000	−0.733017
$269$	−11.3137	−0.689809	−0.344904	−	0.938638i	$-0.612089\pi$
$269$	−11.3137	−0.689809	−0.344904	+	0.938638i	$0.612089\pi$
$270$	0	0
$271$	22.6274	1.37452	0.687259	−	0.726413i	$-0.258814\pi$
$271$	22.6274	1.37452	0.687259	+	0.726413i	$0.258814\pi$
$272$	1.41421	0.0857493
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	5.65685	0.340503
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−9.89949	−0.593732
$279$	−8.48528	−0.508001
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	4.00000	0.238197
$283$	1.41421	0.0840663	0.0420331	−	0.999116i	$-0.486616\pi$
$283$	1.41421	0.0840663	0.0420331	+	0.999116i	$0.486616\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−14.0000	−0.820695
$292$	1.41421	0.0827606
$293$	19.7990	1.15667	0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990	1.15667	0.578335	+	0.815800i	$0.303703\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	11.3137	0.656488
$298$	−10.0000	−0.579284
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	12.0000	0.689382
$304$	−7.07107	−0.405554
$305$	0	0
$306$	1.41421	0.0808452
$307$	9.89949	0.564994	0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949	0.564994	0.282497	+	0.959268i	$0.408837\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	−11.3137	−0.638470
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	−2.82843	−0.158610
$319$	−4.00000	−0.223957
$320$	0	0
$321$	5.65685	0.315735
$322$	0	0
$323$	−10.0000	−0.556415
$324$	−5.00000	−0.277778
$325$	0	0
$326$	10.0000	0.553849
$327$	−2.82843	−0.156412
$328$	9.89949	0.546608
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−9.89949	−0.543305
$333$	10.0000	0.547997
$334$	−19.7990	−1.08335
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	13.0000	0.707107
$339$	16.9706	0.921714
$340$	0	0
$341$	−16.9706	−0.919007
$342$	−7.07107	−0.382360
$343$	0	0
$344$	2.00000	0.107833
$345$	0	0
$346$	−16.9706	−0.912343
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	2.82843	0.151620
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	2.00000	0.106299
$355$	0	0
$356$	−7.07107	−0.374766
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	−9.89949	−0.519589
$364$	0	0
$365$	0	0
$366$	−4.00000	−0.209083
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	4.00000	0.208514
$369$	9.89949	0.515347
$370$	0	0
$371$	0	0
$372$	12.0000	0.622171
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	2.82843	0.146254
$375$	0	0
$376$	2.82843	0.145865
$377$	0	0
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	−22.6274	−1.15924
$382$	4.00000	0.204658
$383$	36.7696	1.87884	0.939418	−	0.342773i	$-0.111366\pi$
$383$	36.7696	1.87884	0.939418	+	0.342773i	$0.111366\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	−16.0000	−0.814379
$387$	2.00000	0.101666
$388$	−9.89949	−0.502571
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	18.0000	0.907980
$394$	2.00000	0.100759
$395$	0	0
$396$	2.00000	0.100504
$397$	−22.6274	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$397$	−22.6274	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$398$	−8.48528	−0.425329
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	16.9706	0.846415
$403$	0	0
$404$	8.48528	0.422159
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	−2.00000	−0.0990148
$409$	38.1838	1.88807	0.944033	−	0.329851i	$-0.106999\pi$
$409$	38.1838	1.88807	0.944033	+	0.329851i	$0.106999\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	−2.82843	−0.139347
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	0	0
$417$	14.0000	0.685583
$418$	−14.1421	−0.691714
$419$	−9.89949	−0.483622	−0.241811	−	0.970323i	$-0.577741\pi$
$419$	−9.89949	−0.483622	−0.241811	+	0.970323i	$0.577741\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	12.0000	0.584151
$423$	2.82843	0.137523
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	16.9706	0.822226
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−5.65685	−0.272166
$433$	29.6985	1.42722	0.713609	−	0.700544i	$-0.247059\pi$
$433$	29.6985	1.42722	0.713609	+	0.700544i	$0.247059\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−28.2843	−1.35302
$438$	−2.00000	−0.0955637
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−14.1421	−0.671156
$445$	0	0
$446$	0	0
$447$	14.1421	0.668900
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	19.7990	0.932298
$452$	12.0000	0.564433
$453$	−22.6274	−1.06313
$454$	−21.2132	−0.995585
$455$	0	0
$456$	10.0000	0.468293
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	16.9706	0.792982
$459$	−8.00000	−0.373408
$460$	0	0
$461$	39.5980	1.84426	0.922131	−	0.386878i	$-0.126447\pi$
$461$	39.5980	1.84426	0.922131	+	0.386878i	$0.126447\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	24.0000	1.11178
$467$	−32.5269	−1.50517	−0.752583	−	0.658497i	$-0.771192\pi$
$467$	−32.5269	−1.50517	−0.752583	+	0.658497i	$0.771192\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.0000	0.737241
$472$	1.41421	0.0650945
$473$	4.00000	0.183920
$474$	5.65685	0.259828
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	12.0000	0.548867
$479$	−31.1127	−1.42158	−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127	−1.42158	−0.710788	+	0.703407i	$0.751661\pi$
$480$	0	0
$481$	0	0
$482$	21.2132	0.966235
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−9.89949	−0.449050
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−2.82843	−0.128037
$489$	−14.1421	−0.639529
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−14.0000	−0.631169
$493$	2.82843	0.127386
$494$	0	0
$495$	0	0
$496$	8.48528	0.381000
$497$	0	0
$498$	14.0000	0.627355
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	28.0000	1.25095
$502$	−9.89949	−0.441836
$503$	−39.5980	−1.76559	−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980	−1.76559	−0.882793	+	0.469762i	$0.844340\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	−18.3848	−0.816497
$508$	−16.0000	−0.709885
$509$	22.6274	1.00294	0.501471	−	0.865174i	$-0.332792\pi$
$509$	22.6274	1.00294	0.501471	+	0.865174i	$0.332792\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	40.0000	1.76604
$514$	12.7279	0.561405
$515$	0	0
$516$	−2.82843	−0.124515
$517$	5.65685	0.248788
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	2.00000	0.0875376
$523$	−12.7279	−0.556553	−0.278277	−	0.960501i	$-0.589763\pi$
$523$	−12.7279	−0.556553	−0.278277	+	0.960501i	$0.589763\pi$
$524$	12.7279	0.556022
$525$	0	0
$526$	12.0000	0.523225
$527$	12.0000	0.522728
$528$	−2.82843	−0.123091
$529$	−7.00000	−0.304348
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	0	0
$534$	10.0000	0.432742
$535$	0	0
$536$	12.0000	0.518321
$537$	16.9706	0.732334
$538$	11.3137	0.487769
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−22.6274	−0.971931
$543$	0	0
$544$	−1.41421	−0.0606339
$545$	0	0
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−12.0000	−0.512615
$549$	−2.82843	−0.120714
$550$	0	0
$551$	−14.1421	−0.602475
$552$	−5.65685	−0.240772
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	9.89949	0.419832
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	8.48528	0.359211
$559$	0	0
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−16.0000	−0.674919
$563$	1.41421	0.0596020	0.0298010	−	0.999556i	$-0.490513\pi$
$563$	1.41421	0.0596020	0.0298010	+	0.999556i	$0.490513\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−1.41421	−0.0594438
$567$	0	0
$568$	12.0000	0.503509
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	−5.65685	−0.236318
$574$	0	0
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	21.2132	0.883117	0.441559	−	0.897232i	$-0.354426\pi$
$577$	21.2132	0.883117	0.441559	+	0.897232i	$0.354426\pi$
$578$	15.0000	0.623918
$579$	22.6274	0.940363
$580$	0	0
$581$	0	0
$582$	14.0000	0.580319
$583$	−4.00000	−0.165663
$584$	−1.41421	−0.0585206
$585$	0	0
$586$	−19.7990	−0.817889
$587$	29.6985	1.22579	0.612894	−	0.790165i	$-0.290005\pi$
$587$	29.6985	1.22579	0.612894	+	0.790165i	$0.290005\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	0	0
$591$	−2.82843	−0.116346
$592$	−10.0000	−0.410997
$593$	7.07107	0.290374	0.145187	−	0.989404i	$-0.453622\pi$
$593$	7.07107	0.290374	0.145187	+	0.989404i	$0.453622\pi$
$594$	−11.3137	−0.464207
$595$	0	0
$596$	10.0000	0.409616
$597$	12.0000	0.491127
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	29.6985	1.21143	0.605713	−	0.795683i	$-0.292888\pi$
$601$	29.6985	1.21143	0.605713	+	0.795683i	$0.292888\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−12.0000	−0.487467
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	7.07107	0.286770
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.41421	−0.0571662
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	−9.89949	−0.399511
$615$	0	0
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	4.00000	0.160904
$619$	18.3848	0.738947	0.369473	−	0.929241i	$-0.379538\pi$
$619$	18.3848	0.738947	0.369473	+	0.929241i	$0.379538\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	11.3137	0.453638
$623$	0	0
$624$	0	0
$625$	0	0
$626$	12.7279	0.508710
$627$	20.0000	0.798723
$628$	11.3137	0.451466
$629$	−14.1421	−0.563884
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	4.00000	0.159111
$633$	−16.9706	−0.674519
$634$	10.0000	0.397151
$635$	0	0
$636$	2.82843	0.112154
$637$	0	0
$638$	4.00000	0.158362
$639$	12.0000	0.474713
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	−5.65685	−0.223258
$643$	−9.89949	−0.390398	−0.195199	−	0.980764i	$-0.562535\pi$
$643$	−9.89949	−0.390398	−0.195199	+	0.980764i	$0.562535\pi$
$644$	0	0
$645$	0	0
$646$	10.0000	0.393445
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	5.00000	0.196419
$649$	2.82843	0.111025
$650$	0	0
$651$	0	0
$652$	−10.0000	−0.391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	2.82843	0.110600
$655$	0	0
$656$	−9.89949	−0.386510
$657$	−1.41421	−0.0551737
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	9.89949	0.384175
$665$	0	0
$666$	−10.0000	−0.387492
$667$	8.00000	0.309761
$668$	19.7990	0.766046
$669$	0	0
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	−16.9706	−0.651751
$679$	0	0
$680$	0	0
$681$	30.0000	1.14960
$682$	16.9706	0.649836
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	7.07107	0.270369
$685$	0	0
$686$	0	0
$687$	−24.0000	−0.915657
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	0	0
$691$	12.7279	0.484193	0.242096	−	0.970252i	$-0.422165\pi$
$691$	12.7279	0.484193	0.242096	+	0.970252i	$0.422165\pi$
$692$	16.9706	0.645124
$693$	0	0
$694$	−30.0000	−1.13878
$695$	0	0
$696$	−2.82843	−0.107211
$697$	−14.0000	−0.530288
$698$	0	0
$699$	−33.9411	−1.28377
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	70.7107	2.66690
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−1.41421	−0.0532246
$707$	0	0
$708$	−2.00000	−0.0751646
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	7.07107	0.264999
$713$	33.9411	1.27111
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	−16.9706	−0.633777
$718$	32.0000	1.19423
$719$	2.82843	0.105483	0.0527413	−	0.998608i	$-0.483204\pi$
$719$	2.82843	0.105483	0.0527413	+	0.998608i	$0.483204\pi$
$720$	0	0
$721$	0	0
$722$	−31.0000	−1.15370
$723$	−30.0000	−1.11571
$724$	0	0
$725$	0	0
$726$	9.89949	0.367405
$727$	19.7990	0.734304	0.367152	−	0.930161i	$-0.380333\pi$
$727$	19.7990	0.734304	0.367152	+	0.930161i	$0.380333\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−2.82843	−0.104613
$732$	4.00000	0.147844
$733$	−42.4264	−1.56706	−0.783528	−	0.621357i	$-0.786582\pi$
$733$	−42.4264	−1.56706	−0.783528	+	0.621357i	$0.786582\pi$
$734$	28.2843	1.04399
$735$	0	0
$736$	−4.00000	−0.147442
$737$	24.0000	0.884051
$738$	−9.89949	−0.364405
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	10.0000	0.366126
$747$	9.89949	0.362204
$748$	−2.82843	−0.103418
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−2.82843	−0.103142
$753$	14.0000	0.510188
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	26.0000	0.944363
$759$	−11.3137	−0.410662
$760$	0	0
$761$	−7.07107	−0.256326	−0.128163	−	0.991753i	$-0.540908\pi$
$761$	−7.07107	−0.256326	−0.128163	+	0.991753i	$0.540908\pi$
$762$	22.6274	0.819705
$763$	0	0
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−36.7696	−1.32854
$767$	0	0
$768$	1.41421	0.0510310
$769$	29.6985	1.07095	0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985	1.07095	0.535477	+	0.844550i	$0.320132\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	16.0000	0.575853
$773$	−48.0833	−1.72943	−0.864717	−	0.502259i	$-0.832502\pi$
$773$	−48.0833	−1.72943	−0.864717	+	0.502259i	$0.832502\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	9.89949	0.355371
$777$	0	0
$778$	−26.0000	−0.932145
$779$	70.0000	2.50801
$780$	0	0
$781$	24.0000	0.858788
$782$	−5.65685	−0.202289
$783$	−11.3137	−0.404319
$784$	0	0
$785$	0	0
$786$	−18.0000	−0.642039
$787$	1.41421	0.0504113	0.0252056	−	0.999682i	$-0.491976\pi$
$787$	1.41421	0.0504113	0.0252056	+	0.999682i	$0.491976\pi$
$788$	−2.00000	−0.0712470
$789$	−16.9706	−0.604168
$790$	0	0
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	0	0
$794$	22.6274	0.803017
$795$	0	0
$796$	8.48528	0.300753
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	7.07107	0.249844
$802$	18.0000	0.635602
$803$	−2.82843	−0.0998130
$804$	−16.9706	−0.598506
$805$	0	0
$806$	0	0
$807$	−16.0000	−0.563227
$808$	−8.48528	−0.298511
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	−29.6985	−1.04285	−0.521427	−	0.853296i	$-0.674600\pi$
$811$	−29.6985	−1.04285	−0.521427	+	0.853296i	$0.674600\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	−20.0000	−0.701000
$815$	0	0
$816$	2.00000	0.0700140
$817$	14.1421	0.494771
$818$	−38.1838	−1.33506
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	16.9706	0.591916
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	2.82843	0.0985329
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−4.00000	−0.139010
$829$	−31.1127	−1.08059	−0.540294	−	0.841476i	$-0.681687\pi$
$829$	−31.1127	−1.08059	−0.540294	+	0.841476i	$0.681687\pi$
$830$	0	0
$831$	2.82843	0.0981170
$832$	0	0
$833$	0	0
$834$	−14.0000	−0.484780
$835$	0	0
$836$	14.1421	0.489116
$837$	−48.0000	−1.65912
$838$	9.89949	0.341972
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−30.0000	−1.03387
$843$	22.6274	0.779330
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−2.82843	−0.0972433
$847$	0	0
$848$	2.00000	0.0686803
$849$	2.00000	0.0686398
$850$	0	0
$851$	−40.0000	−1.37118
$852$	−16.9706	−0.581402
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−18.3848	−0.628012	−0.314006	−	0.949421i	$-0.601671\pi$
$857$	−18.3848	−0.628012	−0.314006	+	0.949421i	$0.601671\pi$
$858$	0	0
$859$	−26.8701	−0.916795	−0.458397	−	0.888747i	$-0.651576\pi$
$859$	−26.8701	−0.916795	−0.458397	+	0.888747i	$0.651576\pi$
$860$	0	0
$861$	0	0
$862$	−12.0000	−0.408722
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	5.65685	0.192450
$865$	0	0
$866$	−29.6985	−1.00920
$867$	−21.2132	−0.720438
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	9.89949	0.335047
$874$	28.2843	0.956730
$875$	0	0
$876$	2.00000	0.0675737
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	16.9706	0.572729
$879$	28.0000	0.944417
$880$	0	0
$881$	29.6985	1.00057	0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985	1.00057	0.500284	+	0.865862i	$0.333229\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	−4.00000	−0.134383
$887$	36.7696	1.23460	0.617300	−	0.786728i	$-0.288226\pi$
$887$	36.7696	1.23460	0.617300	+	0.786728i	$0.288226\pi$
$888$	14.1421	0.474579
$889$	0	0
$890$	0	0
$891$	10.0000	0.335013
$892$	0	0
$893$	20.0000	0.669274
$894$	−14.1421	−0.472984
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	16.9706	0.566000
$900$	0	0
$901$	2.82843	0.0942286
$902$	−19.7990	−0.659234
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	22.6274	0.751746
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	21.2132	0.703985
$909$	−8.48528	−0.281439
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−10.0000	−0.331133
$913$	19.7990	0.655251
$914$	24.0000	0.793849
$915$	0	0
$916$	−16.9706	−0.560723
$917$	0	0
$918$	8.00000	0.264039
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	−39.5980	−1.30409
$923$	0	0
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	2.82843	0.0928977
$928$	−2.00000	−0.0656532
$929$	32.5269	1.06717	0.533587	−	0.845745i	$-0.320844\pi$
$929$	32.5269	1.06717	0.533587	+	0.845745i	$0.320844\pi$
$930$	0	0
$931$	0	0
$932$	−24.0000	−0.786146
$933$	−16.0000	−0.523816
$934$	32.5269	1.06431
$935$	0	0
$936$	0	0
$937$	−9.89949	−0.323402	−0.161701	−	0.986840i	$-0.551698\pi$
$937$	−9.89949	−0.323402	−0.161701	+	0.986840i	$0.551698\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	0	0
$941$	−31.1127	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$941$	−31.1127	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$942$	−16.0000	−0.521308
$943$	−39.5980	−1.28949
$944$	−1.41421	−0.0460287
$945$	0	0
$946$	−4.00000	−0.130051
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	−5.65685	−0.183726
$949$	0	0
$950$	0	0
$951$	−14.1421	−0.458590
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−12.0000	−0.388108
$957$	−5.65685	−0.182860
$958$	31.1127	1.00521
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−4.00000	−0.128898
$964$	−21.2132	−0.683231
$965$	0	0
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	7.00000	0.224989
$969$	−14.1421	−0.454311
$970$	0	0
$971$	32.5269	1.04384	0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269	1.04384	0.521919	+	0.852995i	$0.325216\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	12.0000	0.384505
$975$	0	0
$976$	2.82843	0.0905357
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	14.1421	0.452216
$979$	14.1421	0.451985
$980$	0	0
$981$	2.00000	0.0638551
$982$	12.0000	0.382935
$983$	−48.0833	−1.53362	−0.766809	−	0.641875i	$-0.778157\pi$
$983$	−48.0833	−1.53362	−0.766809	+	0.641875i	$0.778157\pi$
$984$	14.0000	0.446304
$985$	0	0
$986$	−2.82843	−0.0900755
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−8.48528	−0.269408
$993$	14.1421	0.448787
$994$	0	0
$995$	0	0
$996$	−14.0000	−0.443607
$997$	31.1127	0.985349	0.492675	−	0.870214i	$-0.336019\pi$
$997$	31.1127	0.985349	0.492675	+	0.870214i	$0.336019\pi$
$998$	4.00000	0.126618
$999$	56.5685	1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bj.1.2		2
5.2	odd	4		2450.2.c.v.99.1		4
5.3	odd	4		2450.2.c.v.99.4		4
5.4	even	2		98.2.a.b.1.1	✓	2
7.6	odd	2	inner	2450.2.a.bj.1.1		2
15.14	odd	2		882.2.a.n.1.1		2
20.19	odd	2		784.2.a.l.1.2		2
35.4	even	6		98.2.c.c.79.2		4
35.9	even	6		98.2.c.c.67.2		4
35.13	even	4		2450.2.c.v.99.3		4
35.19	odd	6		98.2.c.c.67.1		4
35.24	odd	6		98.2.c.c.79.1		4
35.27	even	4		2450.2.c.v.99.2		4
35.34	odd	2		98.2.a.b.1.2	yes	2
40.19	odd	2		3136.2.a.bm.1.1		2
40.29	even	2		3136.2.a.bn.1.2		2
60.59	even	2		7056.2.a.cl.1.1		2
105.44	odd	6		882.2.g.l.361.2		4
105.59	even	6		882.2.g.l.667.1		4
105.74	odd	6		882.2.g.l.667.2		4
105.89	even	6		882.2.g.l.361.1		4
105.104	even	2		882.2.a.n.1.2		2
140.19	even	6		784.2.i.m.753.2		4
140.39	odd	6		784.2.i.m.177.1		4
140.59	even	6		784.2.i.m.177.2		4
140.79	odd	6		784.2.i.m.753.1		4
140.139	even	2		784.2.a.l.1.1		2
280.69	odd	2		3136.2.a.bn.1.1		2
280.139	even	2		3136.2.a.bm.1.2		2
420.419	odd	2		7056.2.a.cl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.2.a.b.1.1	✓	2	5.4	even	2
98.2.a.b.1.2	yes	2	35.34	odd	2
98.2.c.c.67.1		4	35.19	odd	6
98.2.c.c.67.2		4	35.9	even	6
98.2.c.c.79.1		4	35.24	odd	6
98.2.c.c.79.2		4	35.4	even	6
784.2.a.l.1.1		2	140.139	even	2
784.2.a.l.1.2		2	20.19	odd	2
784.2.i.m.177.1		4	140.39	odd	6
784.2.i.m.177.2		4	140.59	even	6
784.2.i.m.753.1		4	140.79	odd	6
784.2.i.m.753.2		4	140.19	even	6
882.2.a.n.1.1		2	15.14	odd	2
882.2.a.n.1.2		2	105.104	even	2
882.2.g.l.361.1		4	105.89	even	6
882.2.g.l.361.2		4	105.44	odd	6
882.2.g.l.667.1		4	105.59	even	6
882.2.g.l.667.2		4	105.74	odd	6
2450.2.a.bj.1.1		2	7.6	odd	2	inner
2450.2.a.bj.1.2		2	1.1	even	1	trivial
2450.2.c.v.99.1		4	5.2	odd	4
2450.2.c.v.99.2		4	35.27	even	4
2450.2.c.v.99.3		4	35.13	even	4
2450.2.c.v.99.4		4	5.3	odd	4
3136.2.a.bm.1.1		2	40.19	odd	2
3136.2.a.bm.1.2		2	280.139	even	2
3136.2.a.bn.1.1		2	280.69	odd	2
3136.2.a.bn.1.2		2	40.29	even	2
7056.2.a.cl.1.1		2	60.59	even	2
7056.2.a.cl.1.2		2	420.419	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.41421 q^{12} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -7.07107 q^{19} +2.00000 q^{22} +4.00000 q^{23} -1.41421 q^{24} -5.65685 q^{27} +2.00000 q^{29} +8.48528 q^{31} -1.00000 q^{32} -2.82843 q^{33} -1.41421 q^{34} -1.00000 q^{36} -10.0000 q^{37} +7.07107 q^{38} -9.89949 q^{41} -2.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -2.82843 q^{47} +1.41421 q^{48} +2.00000 q^{51} +2.00000 q^{53} +5.65685 q^{54} -10.0000 q^{57} -2.00000 q^{58} -1.41421 q^{59} +2.82843 q^{61} -8.48528 q^{62} +1.00000 q^{64} +2.82843 q^{66} -12.0000 q^{67} +1.41421 q^{68} +5.65685 q^{69} -12.0000 q^{71} +1.00000 q^{72} +1.41421 q^{73} +10.0000 q^{74} -7.07107 q^{76} -4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{82} -9.89949 q^{83} +2.00000 q^{86} +2.82843 q^{87} +2.00000 q^{88} -7.07107 q^{89} +4.00000 q^{92} +12.0000 q^{93} +2.82843 q^{94} -1.41421 q^{96} -9.89949 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 4 q^{11} + 2 q^{16} + 2 q^{18} + 4 q^{22} + 8 q^{23} + 4 q^{29} - 2 q^{32} - 2 q^{36} - 20 q^{37} - 4 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{51} + 4 q^{53} - 20 q^{57} - 4 q^{58} + 2 q^{64} - 24 q^{67} - 24 q^{71} + 2 q^{72} + 20 q^{74} - 8 q^{79} - 10 q^{81} + 4 q^{86} + 4 q^{88} + 8 q^{92} + 24 q^{93} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 4 * q^11 + 2 * q^16 + 2 * q^18 + 4 * q^22 + 8 * q^23 + 4 * q^29 - 2 * q^32 - 2 * q^36 - 20 * q^37 - 4 * q^43 - 4 * q^44 - 8 * q^46 + 4 * q^51 + 4 * q^53 - 20 * q^57 - 4 * q^58 + 2 * q^64 - 24 * q^67 - 24 * q^71 + 2 * q^72 + 20 * q^74 - 8 * q^79 - 10 * q^81 + 4 * q^86 + 4 * q^88 + 8 * q^92 + 24 * q^93 + 4 * q^99

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