LMFDB - Embedded newform 2450.2.c.x.99.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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([1, 0]))

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

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

chi := DirichletCharacter("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.8
Root			$-0.819051 - 1.52616i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.x.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +3.05231i q^{3} -1.00000 q^{4} -3.05231 q^{6} -1.00000i q^{8} -6.31662 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +3.05231i q^{3} -1.00000 q^{4} -3.05231 q^{6} -1.00000i q^{8} -6.31662 q^{9} +5.31662 q^{11} -3.05231i q^{12} +3.27620i q^{13} +1.00000 q^{16} +7.29496i q^{17} -6.31662i q^{18} -1.63810 q^{19} +5.31662i q^{22} +8.63325i q^{23} +3.05231 q^{24} -3.27620 q^{26} -10.1234i q^{27} +2.63325 q^{29} -5.65685 q^{31} +1.00000i q^{32} +16.2280i q^{33} -7.29496 q^{34} +6.31662 q^{36} -4.63325i q^{37} -1.63810i q^{38} -10.0000 q^{39} +2.60454 q^{41} -6.63325i q^{43} -5.31662 q^{44} -8.63325 q^{46} +6.10463i q^{47} +3.05231i q^{48} -22.2665 q^{51} -3.27620i q^{52} -8.00000i q^{53} +10.1234 q^{54} -5.00000i q^{57} +2.63325i q^{58} -1.86199 q^{59} +6.10463 q^{61} -5.65685i q^{62} -1.00000 q^{64} -16.2280 q^{66} +1.63325i q^{67} -7.29496i q^{68} -26.3514 q^{69} -2.00000 q^{71} +6.31662i q^{72} -7.29496i q^{73} +4.63325 q^{74} +1.63810 q^{76} -10.0000i q^{78} -1.36675 q^{79} +11.9499 q^{81} +2.60454i q^{82} -8.26139i q^{83} +6.63325 q^{86} +8.03751i q^{87} -5.31662i q^{88} -4.91430 q^{89} -8.63325i q^{92} -17.2665i q^{93} -6.10463 q^{94} -3.05231 q^{96} -4.24264i q^{97} -33.5831 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 24 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 - 24 * q^9	$ 8 q - 8 q^{4} - 24 q^{9} + 16 q^{11} + 8 q^{16} - 32 q^{29} + 24 q^{36} - 80 q^{39} - 16 q^{44} - 16 q^{46} - 72 q^{51} - 8 q^{64} - 16 q^{71} - 16 q^{74} - 64 q^{79} + 16 q^{81} - 136 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 - 24 * q^9 + 16 * q^11 + 8 * q^16 - 32 * q^29 + 24 * q^36 - 80 * q^39 - 16 * q^44 - 16 * q^46 - 72 * q^51 - 8 * q^64 - 16 * q^71 - 16 * q^74 - 64 * q^79 + 16 * q^81 - 136 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	3.05231i	1.76225i	0.472879	+	0.881127i	$0.343215\pi$
$3$	3.05231i	1.76225i	−0.472879	+	0.881127i	$0.656785\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.05231	−1.24610
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	−6.31662	−2.10554
$10$	0	0
$11$	5.31662	1.60302	0.801511	−	0.597980i	$-0.204030\pi$
$11$	5.31662	1.60302	0.801511	+	0.597980i	$0.204030\pi$
$12$	− 3.05231i	− 0.881127i
$13$	3.27620i	0.908655i	0.890835	+	0.454328i	$0.150120\pi$
$13$	3.27620i	0.908655i	−0.890835	+	0.454328i	$0.849880\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.29496i	1.76929i	0.466268	+	0.884643i	$0.345598\pi$
$17$	7.29496i	1.76929i	−0.466268	+	0.884643i	$0.654402\pi$
$18$	− 6.31662i	− 1.48884i
$19$	−1.63810	−0.375806	−0.187903	−	0.982188i	$-0.560169\pi$
$19$	−1.63810	−0.375806	−0.187903	+	0.982188i	$0.560169\pi$
$20$	0	0
$21$	0	0
$22$	5.31662i	1.13351i
$23$	8.63325i	1.80016i	0.435728	+	0.900078i	$0.356491\pi$
$23$	8.63325i	1.80016i	−0.435728	+	0.900078i	$0.643509\pi$
$24$	3.05231	0.623051
$25$	0	0
$26$	−3.27620	−0.642516
$27$	− 10.1234i	− 1.94825i
$28$	0	0
$29$	2.63325	0.488982	0.244491	−	0.969652i	$-0.421379\pi$
$29$	2.63325	0.488982	0.244491	+	0.969652i	$0.421379\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	1.00000i	0.176777i
$33$	16.2280i	2.82493i
$34$	−7.29496	−1.25107
$35$	0	0
$36$	6.31662	1.05277
$37$	− 4.63325i	− 0.761702i	−0.924637	−	0.380851i	$-0.875631\pi$
$37$	− 4.63325i	− 0.761702i	0.924637	−	0.380851i	$-0.124369\pi$
$38$	− 1.63810i	− 0.265735i
$39$	−10.0000	−1.60128
$40$	0	0
$41$	2.60454	0.406761	0.203380	−	0.979100i	$-0.434807\pi$
$41$	2.60454	0.406761	0.203380	+	0.979100i	$0.434807\pi$
$42$	0	0
$43$	− 6.63325i	− 1.01156i	−0.862662	−	0.505781i	$-0.831205\pi$
$43$	− 6.63325i	− 1.01156i	0.862662	−	0.505781i	$-0.168795\pi$
$44$	−5.31662	−0.801511
$45$	0	0
$46$	−8.63325	−1.27290
$47$	6.10463i	0.890452i	0.895418	+	0.445226i	$0.146877\pi$
$47$	6.10463i	0.890452i	−0.895418	+	0.445226i	$0.853123\pi$
$48$	3.05231i	0.440564i
$49$	0	0
$50$	0	0
$51$	−22.2665	−3.11793
$52$	− 3.27620i	− 0.454328i
$53$	− 8.00000i	− 1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$	− 8.00000i	− 1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$	10.1234	1.37762
$55$	0	0
$56$	0	0
$57$	− 5.00000i	− 0.662266i
$58$	2.63325i	0.345763i
$59$	−1.86199	−0.242410	−0.121205	−	0.992627i	$-0.538676\pi$
$59$	−1.86199	−0.242410	−0.121205	+	0.992627i	$0.538676\pi$
$60$	0	0
$61$	6.10463	0.781618	0.390809	−	0.920472i	$-0.372195\pi$
$61$	6.10463	0.781618	0.390809	+	0.920472i	$0.372195\pi$
$62$	− 5.65685i	− 0.718421i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−16.2280	−1.99753
$67$	1.63325i	0.199533i	0.995011	+	0.0997666i	$0.0318096\pi$
$67$	1.63325i	0.199533i	−0.995011	+	0.0997666i	$0.968190\pi$
$68$	− 7.29496i	− 0.884643i
$69$	−26.3514	−3.17234
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	6.31662i	0.744421i
$73$	− 7.29496i	− 0.853810i	−0.904296	−	0.426905i	$-0.859604\pi$
$73$	− 7.29496i	− 0.853810i	0.904296	−	0.426905i	$-0.140396\pi$
$74$	4.63325	0.538604
$75$	0	0
$76$	1.63810	0.187903
$77$	0	0
$78$	− 10.0000i	− 1.13228i
$79$	−1.36675	−0.153771	−0.0768857	−	0.997040i	$-0.524498\pi$
$79$	−1.36675	−0.153771	−0.0768857	+	0.997040i	$0.524498\pi$
$80$	0	0
$81$	11.9499	1.32776
$82$	2.60454i	0.287623i
$83$	− 8.26139i	− 0.906806i	−0.891306	−	0.453403i	$-0.850210\pi$
$83$	− 8.26139i	− 0.906806i	0.891306	−	0.453403i	$-0.149790\pi$
$84$	0	0
$85$	0	0
$86$	6.63325	0.715282
$87$	8.03751i	0.861711i
$88$	− 5.31662i	− 0.566754i
$89$	−4.91430	−0.520915	−0.260458	−	0.965485i	$-0.583873\pi$
$89$	−4.91430	−0.520915	−0.260458	+	0.965485i	$0.583873\pi$
$90$	0	0
$91$	0	0
$92$	− 8.63325i	− 0.900078i
$93$	− 17.2665i	− 1.79045i
$94$	−6.10463	−0.629644
$95$	0	0
$96$	−3.05231	−0.311526
$97$	− 4.24264i	− 0.430775i	−0.976529	−	0.215387i	$-0.930899\pi$
$97$	− 4.24264i	− 0.430775i	0.976529	−	0.215387i	$-0.0691014\pi$
$98$	0	0
$99$	−33.5831	−3.37523
$100$	0	0
$101$	15.0377	1.49631	0.748153	−	0.663526i	$-0.230941\pi$
$101$	15.0377	1.49631	0.748153	+	0.663526i	$0.230941\pi$
$102$	− 22.2665i	− 2.20471i
$103$	11.3137i	1.11477i	0.830253	+	0.557386i	$0.188196\pi$
$103$	11.3137i	1.11477i	−0.830253	+	0.557386i	$0.811804\pi$
$104$	3.27620	0.321258
$105$	0	0
$106$	8.00000	0.777029
$107$	5.63325i	0.544587i	0.962214	+	0.272293i	$0.0877821\pi$
$107$	5.63325i	0.544587i	−0.962214	+	0.272293i	$0.912218\pi$
$108$	10.1234i	0.974123i
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	14.1421	1.34231
$112$	0	0
$113$	− 0.683375i	− 0.0642865i	−0.999483	−	0.0321433i	$-0.989767\pi$
$113$	− 0.683375i	− 0.0642865i	0.999483	−	0.0321433i	$-0.0102333\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	−2.63325	−0.244491
$117$	− 20.6945i	− 1.91321i
$118$	− 1.86199i	− 0.171410i
$119$	0	0
$120$	0	0
$121$	17.2665	1.56968
$122$	6.10463i	0.552687i
$123$	7.94987i	0.716816i
$124$	5.65685	0.508001
$125$	0	0
$126$	0	0
$127$	− 18.6332i	− 1.65343i	−0.562618	−	0.826717i	$-0.690206\pi$
$127$	− 18.6332i	− 1.65343i	0.562618	−	0.826717i	$-0.309794\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	20.2468	1.78263
$130$	0	0
$131$	−12.2801	−1.07292	−0.536461	−	0.843925i	$-0.680239\pi$
$131$	−12.2801	−1.07292	−0.536461	+	0.843925i	$0.680239\pi$
$132$	− 16.2280i	− 1.41247i
$133$	0	0
$134$	−1.63325	−0.141091
$135$	0	0
$136$	7.29496	0.625537
$137$	13.9499i	1.19182i	0.803052	+	0.595909i	$0.203208\pi$
$137$	13.9499i	1.19182i	−0.803052	+	0.595909i	$0.796792\pi$
$138$	− 26.3514i	− 2.24318i
$139$	15.7093	1.33245	0.666225	−	0.745751i	$-0.267909\pi$
$139$	15.7093	1.33245	0.666225	+	0.745751i	$0.267909\pi$
$140$	0	0
$141$	−18.6332	−1.56920
$142$	− 2.00000i	− 0.167836i
$143$	17.4183i	1.45659i
$144$	−6.31662	−0.526385
$145$	0	0
$146$	7.29496	0.603735
$147$	0	0
$148$	4.63325i	0.380851i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−0.633250	−0.0515331	−0.0257666	−	0.999668i	$-0.508203\pi$
$151$	−0.633250	−0.0515331	−0.0257666	+	0.999668i	$0.508203\pi$
$152$	1.63810i	0.132868i
$153$	− 46.0795i	− 3.72531i
$154$	0	0
$155$	0	0
$156$	10.0000	0.800641
$157$	9.38083i	0.748672i	0.927293	+	0.374336i	$0.122129\pi$
$157$	9.38083i	0.748672i	−0.927293	+	0.374336i	$0.877871\pi$
$158$	− 1.36675i	− 0.108733i
$159$	24.4185	1.93651
$160$	0	0
$161$	0	0
$162$	11.9499i	0.938871i
$163$	2.68338i	0.210178i	0.994463	+	0.105089i	$0.0335128\pi$
$163$	2.68338i	0.210178i	−0.994463	+	0.105089i	$0.966487\pi$
$164$	−2.60454	−0.203380
$165$	0	0
$166$	8.26139	0.641209
$167$	21.5901i	1.67069i	0.549725	+	0.835346i	$0.314732\pi$
$167$	21.5901i	1.67069i	−0.549725	+	0.835346i	$0.685268\pi$
$168$	0	0
$169$	2.26650	0.174346
$170$	0	0
$171$	10.3473	0.791276
$172$	6.63325i	0.505781i
$173$	− 7.00018i	− 0.532214i	−0.963944	−	0.266107i	$-0.914263\pi$
$173$	− 7.00018i	− 0.532214i	0.963944	−	0.266107i	$-0.0857374\pi$
$174$	−8.03751	−0.609322
$175$	0	0
$176$	5.31662	0.400756
$177$	− 5.68338i	− 0.427189i
$178$	− 4.91430i	− 0.368343i
$179$	−20.8997	−1.56212	−0.781060	−	0.624456i	$-0.785321\pi$
$179$	−20.8997	−1.56212	−0.781060	+	0.624456i	$0.785321\pi$
$180$	0	0
$181$	9.82861	0.730555	0.365277	−	0.930899i	$-0.380974\pi$
$181$	9.82861	0.730555	0.365277	+	0.930899i	$0.380974\pi$
$182$	0	0
$183$	18.6332i	1.37741i
$184$	8.63325	0.636452
$185$	0	0
$186$	17.2665	1.26604
$187$	38.7845i	2.83621i
$188$	− 6.10463i	− 0.445226i
$189$	0	0
$190$	0	0
$191$	15.2665	1.10465	0.552323	−	0.833630i	$-0.313742\pi$
$191$	15.2665	1.10465	0.552323	+	0.833630i	$0.313742\pi$
$192$	− 3.05231i	− 0.220282i
$193$	3.31662i	0.238736i	0.992850	+	0.119368i	$0.0380868\pi$
$193$	3.31662i	0.238736i	−0.992850	+	0.119368i	$0.961913\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	0	0
$197$	6.63325i	0.472599i	0.971680	+	0.236300i	$0.0759347\pi$
$197$	6.63325i	0.472599i	−0.971680	+	0.236300i	$0.924065\pi$
$198$	− 33.5831i	− 2.38665i
$199$	8.93306	0.633248	0.316624	−	0.948551i	$-0.397451\pi$
$199$	8.93306	0.633248	0.316624	+	0.948551i	$0.397451\pi$
$200$	0	0
$201$	−4.98519	−0.351628
$202$	15.0377i	1.05805i
$203$	0	0
$204$	22.2665	1.55897
$205$	0	0
$206$	−11.3137	−0.788263
$207$	− 54.5330i	− 3.79031i
$208$	3.27620i	0.227164i
$209$	−8.70917	−0.602426
$210$	0	0
$211$	−1.63325	−0.112438	−0.0562188	−	0.998418i	$-0.517904\pi$
$211$	−1.63325	−0.112438	−0.0562188	+	0.998418i	$0.517904\pi$
$212$	8.00000i	0.549442i
$213$	− 6.10463i	− 0.418282i
$214$	−5.63325	−0.385081
$215$	0	0
$216$	−10.1234	−0.688809
$217$	0	0
$218$	− 18.0000i	− 1.21911i
$219$	22.2665	1.50463
$220$	0	0
$221$	−23.8997	−1.60767
$222$	14.1421i	0.949158i
$223$	3.27620i	0.219391i	0.993965	+	0.109695i	$0.0349875\pi$
$223$	3.27620i	0.219391i	−0.993965	+	0.109695i	$0.965012\pi$
$224$	0	0
$225$	0	0
$226$	0.683375	0.0454574
$227$	− 3.79487i	− 0.251874i	−0.992038	−	0.125937i	$-0.959806\pi$
$227$	− 3.79487i	− 0.251874i	0.992038	−	0.125937i	$-0.0401937\pi$
$228$	5.00000i	0.331133i
$229$	−14.5899	−0.964128	−0.482064	−	0.876136i	$-0.660113\pi$
$229$	−14.5899	−0.964128	−0.482064	+	0.876136i	$0.660113\pi$
$230$	0	0
$231$	0	0
$232$	− 2.63325i	− 0.172881i
$233$	− 13.2665i	− 0.869117i	−0.900644	−	0.434559i	$-0.856904\pi$
$233$	− 13.2665i	− 0.869117i	0.900644	−	0.434559i	$-0.143096\pi$
$234$	20.6945	1.35284
$235$	0	0
$236$	1.86199	0.121205
$237$	− 4.17175i	− 0.270984i
$238$	0	0
$239$	16.6332	1.07592	0.537958	−	0.842972i	$-0.319196\pi$
$239$	16.6332	1.07592	0.537958	+	0.842972i	$0.319196\pi$
$240$	0	0
$241$	−21.8140	−1.40516	−0.702581	−	0.711604i	$-0.747969\pi$
$241$	−21.8140	−1.40516	−0.702581	+	0.711604i	$0.747969\pi$
$242$	17.2665i	1.10993i
$243$	6.10463i	0.391612i
$244$	−6.10463	−0.390809
$245$	0	0
$246$	−7.94987	−0.506865
$247$	− 5.36675i	− 0.341478i
$248$	5.65685i	0.359211i
$249$	25.2164	1.59802
$250$	0	0
$251$	18.6087	1.17457	0.587284	−	0.809381i	$-0.300197\pi$
$251$	18.6087	1.17457	0.587284	+	0.809381i	$0.300197\pi$
$252$	0	0
$253$	45.8997i	2.88569i
$254$	18.6332	1.16915
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 1.41421i	− 0.0882162i	−0.999027	−	0.0441081i	$-0.985955\pi$
$257$	− 1.41421i	− 0.0882162i	0.999027	−	0.0441081i	$-0.0140446\pi$
$258$	20.2468i	1.26051i
$259$	0	0
$260$	0	0
$261$	−16.6332	−1.02957
$262$	− 12.2801i	− 0.758670i
$263$	− 15.8997i	− 0.980421i	−0.871604	−	0.490210i	$-0.836920\pi$
$263$	− 15.8997i	− 0.980421i	0.871604	−	0.490210i	$-0.163080\pi$
$264$	16.2280	0.998765
$265$	0	0
$266$	0	0
$267$	− 15.0000i	− 0.917985i
$268$	− 1.63325i	− 0.0997666i
$269$	8.03751	0.490055	0.245028	−	0.969516i	$-0.421203\pi$
$269$	8.03751	0.490055	0.245028	+	0.969516i	$0.421203\pi$
$270$	0	0
$271$	−1.34333	−0.0816012	−0.0408006	−	0.999167i	$-0.512991\pi$
$271$	−1.34333	−0.0816012	−0.0408006	+	0.999167i	$0.512991\pi$
$272$	7.29496i	0.442322i
$273$	0	0
$274$	−13.9499	−0.842743
$275$	0	0
$276$	26.3514	1.58617
$277$	− 6.63325i	− 0.398553i	−0.979943	−	0.199277i	$-0.936141\pi$
$277$	− 6.63325i	− 0.398553i	0.979943	−	0.199277i	$-0.0638592\pi$
$278$	15.7093i	0.942184i
$279$	35.7322	2.13923
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	− 18.6332i	− 1.10959i
$283$	− 25.2320i	− 1.49988i	−0.661504	−	0.749942i	$-0.730081\pi$
$283$	− 25.2320i	− 1.49988i	0.661504	−	0.749942i	$-0.269919\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	−17.4183	−1.02997
$287$	0	0
$288$	− 6.31662i	− 0.372211i
$289$	−36.2164	−2.13037
$290$	0	0
$291$	12.9499	0.759135
$292$	7.29496i	0.426905i
$293$	− 5.20908i	− 0.304318i	−0.988356	−	0.152159i	$-0.951377\pi$
$293$	− 5.20908i	− 0.304318i	0.988356	−	0.152159i	$-0.0486225\pi$
$294$	0	0
$295$	0	0
$296$	−4.63325	−0.269302
$297$	− 53.8222i	− 3.12308i
$298$	0	0
$299$	−28.2843	−1.63572
$300$	0	0
$301$	0	0
$302$	− 0.633250i	− 0.0364394i
$303$	45.8997i	2.63687i
$304$	−1.63810	−0.0939515
$305$	0	0
$306$	46.0795	2.63419
$307$	− 15.3325i	− 0.875070i	−0.899201	−	0.437535i	$-0.855852\pi$
$307$	− 15.3325i	− 0.875070i	0.899201	−	0.437535i	$-0.144148\pi$
$308$	0	0
$309$	−34.5330	−1.96451
$310$	0	0
$311$	−27.6947	−1.57042	−0.785212	−	0.619227i	$-0.787446\pi$
$311$	−27.6947	−1.57042	−0.785212	+	0.619227i	$0.787446\pi$
$312$	10.0000i	0.566139i
$313$	− 11.6906i	− 0.660792i	−0.943842	−	0.330396i	$-0.892818\pi$
$313$	− 11.6906i	− 0.660792i	0.943842	−	0.330396i	$-0.107182\pi$
$314$	−9.38083	−0.529391
$315$	0	0
$316$	1.36675	0.0768857
$317$	− 33.8997i	− 1.90400i	−0.306099	−	0.952000i	$-0.599024\pi$
$317$	− 33.8997i	− 1.90400i	0.306099	−	0.952000i	$-0.400976\pi$
$318$	24.4185i	1.36932i
$319$	14.0000	0.783850
$320$	0	0
$321$	−17.1945	−0.959701
$322$	0	0
$323$	− 11.9499i	− 0.664909i
$324$	−11.9499	−0.663882
$325$	0	0
$326$	−2.68338	−0.148618
$327$	− 54.9417i	− 3.03828i
$328$	− 2.60454i	− 0.143812i
$329$	0	0
$330$	0	0
$331$	22.6834	1.24679	0.623396	−	0.781907i	$-0.285753\pi$
$331$	22.6834	1.24679	0.623396	+	0.781907i	$0.285753\pi$
$332$	8.26139i	0.453403i
$333$	29.2665i	1.60379i
$334$	−21.5901	−1.18136
$335$	0	0
$336$	0	0
$337$	27.0000i	1.47078i	0.677642	+	0.735392i	$0.263002\pi$
$337$	27.0000i	1.47078i	−0.677642	+	0.735392i	$0.736998\pi$
$338$	2.26650i	0.123281i
$339$	2.08588	0.113289
$340$	0	0
$341$	−30.0754	−1.62867
$342$	10.3473i	0.559516i
$343$	0	0
$344$	−6.63325	−0.357641
$345$	0	0
$346$	7.00018	0.376332
$347$	− 2.68338i	− 0.144051i	−0.997403	−	0.0720256i	$-0.977054\pi$
$347$	− 2.68338i	− 0.144051i	0.997403	−	0.0720256i	$-0.0229463\pi$
$348$	− 8.03751i	− 0.430856i
$349$	−19.6572	−1.05223	−0.526113	−	0.850415i	$-0.676351\pi$
$349$	−19.6572	−1.05223	−0.526113	+	0.850415i	$0.676351\pi$
$350$	0	0
$351$	33.1662	1.77028
$352$	5.31662i	0.283377i
$353$	9.00394i	0.479232i	0.970868	+	0.239616i	$0.0770215\pi$
$353$	9.00394i	0.479232i	−0.970868	+	0.239616i	$0.922979\pi$
$354$	5.68338	0.302068
$355$	0	0
$356$	4.91430	0.260458
$357$	0	0
$358$	− 20.8997i	− 1.10459i
$359$	31.2665	1.65018	0.825091	−	0.564999i	$-0.191124\pi$
$359$	31.2665	1.65018	0.825091	+	0.564999i	$0.191124\pi$
$360$	0	0
$361$	−16.3166	−0.858770
$362$	9.82861i	0.516580i
$363$	52.7028i	2.76618i
$364$	0	0
$365$	0	0
$366$	−18.6332	−0.973976
$367$	27.2469i	1.42228i	0.703051	+	0.711139i	$0.251821\pi$
$367$	27.2469i	1.42228i	−0.703051	+	0.711139i	$0.748179\pi$
$368$	8.63325i	0.450039i
$369$	−16.4519	−0.856452
$370$	0	0
$371$	0	0
$372$	17.2665i	0.895226i
$373$	33.8997i	1.75526i	0.479336	+	0.877631i	$0.340877\pi$
$373$	33.8997i	1.75526i	−0.479336	+	0.877631i	$0.659123\pi$
$374$	−38.7845	−2.00550
$375$	0	0
$376$	6.10463	0.314822
$377$	8.62706i	0.444316i
$378$	0	0
$379$	7.94987	0.408358	0.204179	−	0.978934i	$-0.434548\pi$
$379$	7.94987	0.408358	0.204179	+	0.978934i	$0.434548\pi$
$380$	0	0
$381$	56.8745	2.91377
$382$	15.2665i	0.781102i
$383$	18.3139i	0.935796i	0.883782	+	0.467898i	$0.154989\pi$
$383$	18.3139i	0.935796i	−0.883782	+	0.467898i	$0.845011\pi$
$384$	3.05231	0.155763
$385$	0	0
$386$	−3.31662	−0.168812
$387$	41.8997i	2.12988i
$388$	4.24264i	0.215387i
$389$	28.6332	1.45176	0.725882	−	0.687820i	$-0.241432\pi$
$389$	28.6332	1.45176	0.725882	+	0.687820i	$0.241432\pi$
$390$	0	0
$391$	−62.9792	−3.18499
$392$	0	0
$393$	− 37.4829i	− 1.89076i
$394$	−6.63325	−0.334178
$395$	0	0
$396$	33.5831	1.68762
$397$	11.7615i	0.590292i	0.955452	+	0.295146i	$0.0953683\pi$
$397$	11.7615i	0.590292i	−0.955452	+	0.295146i	$0.904632\pi$
$398$	8.93306i	0.447774i
$399$	0	0
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	− 4.98519i	− 0.248639i
$403$	− 18.5330i	− 0.923194i
$404$	−15.0377	−0.748153
$405$	0	0
$406$	0	0
$407$	− 24.6332i	− 1.22102i
$408$	22.2665i	1.10236i
$409$	−38.7845	−1.91777	−0.958886	−	0.283791i	$-0.908408\pi$
$409$	−38.7845	−1.91777	−0.958886	+	0.283791i	$0.908408\pi$
$410$	0	0
$411$	−42.5794	−2.10029
$412$	− 11.3137i	− 0.557386i
$413$	0	0
$414$	54.5330	2.68015
$415$	0	0
$416$	−3.27620	−0.160629
$417$	47.9499i	2.34812i
$418$	− 8.70917i	− 0.425979i
$419$	19.1273	0.934431	0.467216	−	0.884143i	$-0.345257\pi$
$419$	19.1273	0.934431	0.467216	+	0.884143i	$0.345257\pi$
$420$	0	0
$421$	38.5330	1.87798	0.938992	−	0.343940i	$-0.111762\pi$
$421$	38.5330	1.87798	0.938992	+	0.343940i	$0.111762\pi$
$422$	− 1.63325i	− 0.0795053i
$423$	− 38.5607i	− 1.87488i
$424$	−8.00000	−0.388514
$425$	0	0
$426$	6.10463	0.295770
$427$	0	0
$428$	− 5.63325i	− 0.272293i
$429$	−53.1662	−2.56689
$430$	0	0
$431$	7.36675	0.354844	0.177422	−	0.984135i	$-0.443224\pi$
$431$	7.36675	0.354844	0.177422	+	0.984135i	$0.443224\pi$
$432$	− 10.1234i	− 0.487061i
$433$	− 3.50009i	− 0.168204i	−0.996457	−	0.0841018i	$-0.973198\pi$
$433$	− 3.50009i	− 0.168204i	0.996457	−	0.0841018i	$-0.0268021\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	− 14.1421i	− 0.676510i
$438$	22.2665i	1.06393i
$439$	9.38083	0.447723	0.223861	−	0.974621i	$-0.428134\pi$
$439$	9.38083	0.447723	0.223861	+	0.974621i	$0.428134\pi$
$440$	0	0
$441$	0	0
$442$	− 23.8997i	− 1.13680i
$443$	12.8997i	0.612886i	0.951889	+	0.306443i	$0.0991388\pi$
$443$	12.8997i	0.612886i	−0.951889	+	0.306443i	$0.900861\pi$
$444$	−14.1421	−0.671156
$445$	0	0
$446$	−3.27620	−0.155133
$447$	0	0
$448$	0	0
$449$	34.2665	1.61714	0.808568	−	0.588403i	$-0.200243\pi$
$449$	34.2665	1.61714	0.808568	+	0.588403i	$0.200243\pi$
$450$	0	0
$451$	13.8474	0.652047
$452$	0.683375i	0.0321433i
$453$	− 1.93288i	− 0.0908145i
$454$	3.79487	0.178102
$455$	0	0
$456$	−5.00000	−0.234146
$457$	16.6834i	0.780415i	0.920727	+	0.390208i	$0.127597\pi$
$457$	16.6834i	0.780415i	−0.920727	+	0.390208i	$0.872403\pi$
$458$	− 14.5899i	− 0.681742i
$459$	73.8496	3.44701
$460$	0	0
$461$	−6.10463	−0.284321	−0.142160	−	0.989844i	$-0.545405\pi$
$461$	−6.10463	−0.284321	−0.142160	+	0.989844i	$0.545405\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	2.63325	0.122246
$465$	0	0
$466$	13.2665	0.614559
$467$	− 0.0708883i	− 0.00328032i	−0.999999	−	0.00164016i	$-0.999478\pi$
$467$	− 0.0708883i	− 0.00328032i	0.999999	−	0.00164016i	$-0.000522080\pi$
$468$	20.6945i	0.956605i
$469$	0	0
$470$	0	0
$471$	−28.6332	−1.31935
$472$	1.86199i	0.0857050i
$473$	− 35.2665i	− 1.62156i
$474$	4.17175	0.191615
$475$	0	0
$476$	0	0
$477$	50.5330i	2.31375i
$478$	16.6332i	0.760787i
$479$	5.06730	0.231531	0.115765	−	0.993277i	$-0.463068\pi$
$479$	5.06730	0.231531	0.115765	+	0.993277i	$0.463068\pi$
$480$	0	0
$481$	15.1795	0.692124
$482$	− 21.8140i	− 0.993599i
$483$	0	0
$484$	−17.2665	−0.784841
$485$	0	0
$486$	−6.10463	−0.276912
$487$	31.2665i	1.41682i	0.705801	+	0.708410i	$0.250587\pi$
$487$	31.2665i	1.41682i	−0.705801	+	0.708410i	$0.749413\pi$
$488$	− 6.10463i	− 0.276344i
$489$	−8.19051	−0.370387
$490$	0	0
$491$	−21.2665	−0.959744	−0.479872	−	0.877339i	$-0.659317\pi$
$491$	−21.2665	−0.959744	−0.479872	+	0.877339i	$0.659317\pi$
$492$	− 7.94987i	− 0.358408i
$493$	19.2094i	0.865150i
$494$	5.36675	0.241462
$495$	0	0
$496$	−5.65685	−0.254000
$497$	0	0
$498$	25.2164i	1.12997i
$499$	13.2665	0.593890	0.296945	−	0.954895i	$-0.404032\pi$
$499$	13.2665	0.593890	0.296945	+	0.954895i	$0.404032\pi$
$500$	0	0
$501$	−65.8997	−2.94418
$502$	18.6087i	0.830545i
$503$	12.6570i	0.564349i	0.959363	+	0.282175i	$0.0910558\pi$
$503$	12.6570i	0.564349i	−0.959363	+	0.282175i	$0.908944\pi$
$504$	0	0
$505$	0	0
$506$	−45.8997	−2.04049
$507$	6.91807i	0.307242i
$508$	18.6332i	0.826717i
$509$	2.38065	0.105521	0.0527603	−	0.998607i	$-0.483198\pi$
$509$	2.38065	0.105521	0.0527603	+	0.998607i	$0.483198\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	16.5831i	0.732163i
$514$	1.41421	0.0623783
$515$	0	0
$516$	−20.2468	−0.891314
$517$	32.4560i	1.42741i
$518$	0	0
$519$	21.3668	0.937896
$520$	0	0
$521$	−40.1279	−1.75803	−0.879017	−	0.476791i	$-0.841800\pi$
$521$	−40.1279	−1.75803	−0.879017	+	0.476791i	$0.841800\pi$
$522$	− 16.6332i	− 0.728018i
$523$	34.0941i	1.49083i	0.666600	+	0.745416i	$0.267749\pi$
$523$	34.0941i	1.49083i	−0.666600	+	0.745416i	$0.732251\pi$
$524$	12.2801	0.536461
$525$	0	0
$526$	15.8997	0.693262
$527$	− 41.2665i	− 1.79760i
$528$	16.2280i	0.706234i
$529$	−51.5330	−2.24057
$530$	0	0
$531$	11.7615	0.510405
$532$	0	0
$533$	8.53300i	0.369605i
$534$	15.0000	0.649113
$535$	0	0
$536$	1.63325	0.0705456
$537$	− 63.7926i	− 2.75285i
$538$	8.03751i	0.346522i
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 1.34333i	− 0.0577008i
$543$	30.0000i	1.28742i
$544$	−7.29496	−0.312769
$545$	0	0
$546$	0	0
$547$	35.9499i	1.53711i	0.639786	+	0.768553i	$0.279023\pi$
$547$	35.9499i	1.53711i	−0.639786	+	0.768553i	$0.720977\pi$
$548$	− 13.9499i	− 0.595909i
$549$	−38.5607	−1.64573
$550$	0	0
$551$	−4.31353	−0.183763
$552$	26.3514i	1.12159i
$553$	0	0
$554$	6.63325	0.281820
$555$	0	0
$556$	−15.7093	−0.666225
$557$	10.7335i	0.454793i	0.973802	+	0.227397i	$0.0730213\pi$
$557$	10.7335i	0.454793i	−0.973802	+	0.227397i	$0.926979\pi$
$558$	35.7322i	1.51267i
$559$	21.7319	0.919160
$560$	0	0
$561$	−118.383	−4.99812
$562$	− 4.00000i	− 0.168730i
$563$	− 9.45172i	− 0.398342i	−0.979965	−	0.199171i	$-0.936175\pi$
$563$	− 9.45172i	− 0.398342i	0.979965	−	0.199171i	$-0.0638250\pi$
$564$	18.6332	0.784601
$565$	0	0
$566$	25.2320	1.06058
$567$	0	0
$568$	2.00000i	0.0839181i
$569$	−4.26650	−0.178861	−0.0894305	−	0.995993i	$-0.528505\pi$
$569$	−4.26650	−0.178861	−0.0894305	+	0.995993i	$0.528505\pi$
$570$	0	0
$571$	2.63325	0.110198	0.0550990	−	0.998481i	$-0.482453\pi$
$571$	2.63325	0.110198	0.0550990	+	0.998481i	$0.482453\pi$
$572$	− 17.4183i	− 0.728297i
$573$	46.5982i	1.94667i
$574$	0	0
$575$	0	0
$576$	6.31662	0.263193
$577$	− 36.4748i	− 1.51846i	−0.650820	−	0.759232i	$-0.725575\pi$
$577$	− 36.4748i	− 1.51846i	0.650820	−	0.759232i	$-0.274425\pi$
$578$	− 36.2164i	− 1.50640i
$579$	−10.1234	−0.420713
$580$	0	0
$581$	0	0
$582$	12.9499i	0.536790i
$583$	− 42.5330i	− 1.76154i
$584$	−7.29496	−0.301867
$585$	0	0
$586$	5.20908	0.215185
$587$	17.1236i	0.706765i	0.935479	+	0.353383i	$0.114969\pi$
$587$	17.1236i	0.706765i	−0.935479	+	0.353383i	$0.885031\pi$
$588$	0	0
$589$	9.26650	0.381819
$590$	0	0
$591$	−20.2468	−0.832841
$592$	− 4.63325i	− 0.190425i
$593$	− 29.9224i	− 1.22876i	−0.789009	−	0.614382i	$-0.789405\pi$
$593$	− 29.9224i	− 1.22876i	0.789009	−	0.614382i	$-0.210595\pi$
$594$	53.8222	2.20835
$595$	0	0
$596$	0	0
$597$	27.2665i	1.11594i
$598$	− 28.2843i	− 1.15663i
$599$	6.73350	0.275123	0.137562	−	0.990493i	$-0.456073\pi$
$599$	6.73350	0.275123	0.137562	+	0.990493i	$0.456073\pi$
$600$	0	0
$601$	−1.26121	−0.0514460	−0.0257230	−	0.999669i	$-0.508189\pi$
$601$	−1.26121	−0.0514460	−0.0257230	+	0.999669i	$0.508189\pi$
$602$	0	0
$603$	− 10.3166i	− 0.420125i
$604$	0.633250	0.0257666
$605$	0	0
$606$	−45.8997	−1.86455
$607$	11.3137i	0.459209i	0.973284	+	0.229605i	$0.0737433\pi$
$607$	11.3137i	0.459209i	−0.973284	+	0.229605i	$0.926257\pi$
$608$	− 1.63810i	− 0.0664338i
$609$	0	0
$610$	0	0
$611$	−20.0000	−0.809113
$612$	46.0795i	1.86265i
$613$	5.36675i	0.216761i	0.994109	+	0.108381i	$0.0345665\pi$
$613$	5.36675i	0.216761i	−0.994109	+	0.108381i	$0.965434\pi$
$614$	15.3325	0.618768
$615$	0	0
$616$	0	0
$617$	− 15.2665i	− 0.614606i	−0.951612	−	0.307303i	$-0.900573\pi$
$617$	− 15.2665i	− 0.614606i	0.951612	−	0.307303i	$-0.0994265\pi$
$618$	− 34.5330i	− 1.38912i
$619$	26.2805	1.05630	0.528151	−	0.849150i	$-0.322885\pi$
$619$	26.2805	1.05630	0.528151	+	0.849150i	$0.322885\pi$
$620$	0	0
$621$	87.3977	3.50715
$622$	− 27.6947i	− 1.11046i
$623$	0	0
$624$	−10.0000	−0.400320
$625$	0	0
$626$	11.6906	0.467250
$627$	− 26.5831i	− 1.06163i
$628$	− 9.38083i	− 0.374336i
$629$	33.7993	1.34767
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	1.36675i	0.0543664i
$633$	− 4.98519i	− 0.198144i
$634$	33.8997	1.34633
$635$	0	0
$636$	−24.4185	−0.968257
$637$	0	0
$638$	14.0000i	0.554265i
$639$	12.6332	0.499764
$640$	0	0
$641$	24.5330	0.968995	0.484498	−	0.874793i	$-0.339002\pi$
$641$	24.5330	0.968995	0.484498	+	0.874793i	$0.339002\pi$
$642$	− 17.1945i	− 0.678611i
$643$	29.2507i	1.15354i	0.816908	+	0.576768i	$0.195686\pi$
$643$	29.2507i	1.15354i	−0.816908	+	0.576768i	$0.804314\pi$
$644$	0	0
$645$	0	0
$646$	11.9499	0.470162
$647$	21.5901i	0.848794i	0.905476	+	0.424397i	$0.139514\pi$
$647$	21.5901i	0.848794i	−0.905476	+	0.424397i	$0.860486\pi$
$648$	− 11.9499i	− 0.469435i
$649$	−9.89949	−0.388589
$650$	0	0
$651$	0	0
$652$	− 2.68338i	− 0.105089i
$653$	− 35.1662i	− 1.37616i	−0.725634	−	0.688081i	$-0.758453\pi$
$653$	− 35.1662i	− 1.37616i	0.725634	−	0.688081i	$-0.241547\pi$
$654$	54.9417	2.14839
$655$	0	0
$656$	2.60454	0.101690
$657$	46.0795i	1.79773i
$658$	0	0
$659$	13.4169	0.522647	0.261324	−	0.965251i	$-0.415841\pi$
$659$	13.4169	0.522647	0.261324	+	0.965251i	$0.415841\pi$
$660$	0	0
$661$	−23.0752	−0.897521	−0.448760	−	0.893652i	$-0.648134\pi$
$661$	−23.0752	−0.897521	−0.448760	+	0.893652i	$0.648134\pi$
$662$	22.6834i	0.881614i
$663$	− 72.9496i	− 2.83313i
$664$	−8.26139	−0.320604
$665$	0	0
$666$	−29.2665	−1.13405
$667$	22.7335i	0.880245i
$668$	− 21.5901i	− 0.835346i
$669$	−10.0000	−0.386622
$670$	0	0
$671$	32.4560	1.25295
$672$	0	0
$673$	12.0000i	0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$	12.0000i	0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$	−27.0000	−1.04000
$675$	0	0
$676$	−2.26650	−0.0871730
$677$	10.2764i	0.394953i	0.980308	+	0.197477i	$0.0632747\pi$
$677$	10.2764i	0.394953i	−0.980308	+	0.197477i	$0.936725\pi$
$678$	2.08588i	0.0801076i
$679$	0	0
$680$	0	0
$681$	11.5831	0.443866
$682$	− 30.0754i	− 1.15165i
$683$	16.1662i	0.618584i	0.950967	+	0.309292i	$0.100092\pi$
$683$	16.1662i	0.618584i	−0.950967	+	0.309292i	$0.899908\pi$
$684$	−10.3473	−0.395638
$685$	0	0
$686$	0	0
$687$	− 44.5330i	− 1.69904i
$688$	− 6.63325i	− 0.252890i
$689$	26.2096	0.998507
$690$	0	0
$691$	38.8554	1.47813	0.739065	−	0.673634i	$-0.235268\pi$
$691$	38.8554	1.47813	0.739065	+	0.673634i	$0.235268\pi$
$692$	7.00018i	0.266107i
$693$	0	0
$694$	2.68338	0.101860
$695$	0	0
$696$	8.03751	0.304661
$697$	19.0000i	0.719676i
$698$	− 19.6572i	− 0.744036i
$699$	40.4935	1.53161
$700$	0	0
$701$	27.7995	1.04997	0.524986	−	0.851111i	$-0.324070\pi$
$701$	27.7995	1.04997	0.524986	+	0.851111i	$0.324070\pi$
$702$	33.1662i	1.25178i
$703$	7.58973i	0.286252i
$704$	−5.31662	−0.200378
$705$	0	0
$706$	−9.00394	−0.338868
$707$	0	0
$708$	5.68338i	0.213594i
$709$	18.5330	0.696021	0.348011	−	0.937491i	$-0.386857\pi$
$709$	18.5330	0.696021	0.348011	+	0.937491i	$0.386857\pi$
$710$	0	0
$711$	8.63325	0.323772
$712$	4.91430i	0.184171i
$713$	− 48.8370i	− 1.82896i
$714$	0	0
$715$	0	0
$716$	20.8997	0.781060
$717$	50.7699i	1.89604i
$718$	31.2665i	1.16686i
$719$	17.8661	0.666294	0.333147	−	0.942875i	$-0.391890\pi$
$719$	17.8661	0.666294	0.333147	+	0.942875i	$0.391890\pi$
$720$	0	0
$721$	0	0
$722$	− 16.3166i	− 0.607242i
$723$	− 66.5831i	− 2.47625i
$724$	−9.82861	−0.365277
$725$	0	0
$726$	−52.7028	−1.95598
$727$	− 43.7697i	− 1.62333i	−0.584124	−	0.811665i	$-0.698562\pi$
$727$	− 43.7697i	− 1.62333i	0.584124	−	0.811665i	$-0.301438\pi$
$728$	0	0
$729$	17.2164	0.637643
$730$	0	0
$731$	48.3893	1.78974
$732$	− 18.6332i	− 0.688705i
$733$	7.58973i	0.280333i	0.990128	+	0.140167i	$0.0447638\pi$
$733$	7.58973i	0.280333i	−0.990128	+	0.140167i	$0.955236\pi$
$734$	−27.2469	−1.00570
$735$	0	0
$736$	−8.63325	−0.318226
$737$	8.68338i	0.319856i
$738$	− 16.4519i	− 0.605603i
$739$	−33.1662	−1.22004	−0.610020	−	0.792386i	$-0.708839\pi$
$739$	−33.1662	−1.22004	−0.610020	+	0.792386i	$0.708839\pi$
$740$	0	0
$741$	16.3810	0.601771
$742$	0	0
$743$	8.63325i	0.316723i	0.987381	+	0.158362i	$0.0506212\pi$
$743$	8.63325i	0.316723i	−0.987381	+	0.158362i	$0.949379\pi$
$744$	−17.2665	−0.633021
$745$	0	0
$746$	−33.8997	−1.24116
$747$	52.1841i	1.90932i
$748$	− 38.7845i	− 1.41810i
$749$	0	0
$750$	0	0
$751$	1.36675	0.0498734	0.0249367	−	0.999689i	$-0.492062\pi$
$751$	1.36675	0.0498734	0.0249367	+	0.999689i	$0.492062\pi$
$752$	6.10463i	0.222613i
$753$	56.7995i	2.06989i
$754$	−8.62706	−0.314179
$755$	0	0
$756$	0	0
$757$	26.6332i	0.968002i	0.875067	+	0.484001i	$0.160817\pi$
$757$	26.6332i	0.968002i	−0.875067	+	0.484001i	$0.839183\pi$
$758$	7.94987i	0.288752i
$759$	−140.100	−5.08533
$760$	0	0
$761$	−32.6799	−1.18465	−0.592323	−	0.805701i	$-0.701789\pi$
$761$	−32.6799	−1.18465	−0.592323	+	0.805701i	$0.701789\pi$
$762$	56.8745i	2.06035i
$763$	0	0
$764$	−15.2665	−0.552323
$765$	0	0
$766$	−18.3139	−0.661708
$767$	− 6.10025i	− 0.220267i
$768$	3.05231i	0.110141i
$769$	36.7808	1.32635	0.663174	−	0.748465i	$-0.269209\pi$
$769$	36.7808	1.32635	0.663174	+	0.748465i	$0.269209\pi$
$770$	0	0
$771$	4.31662	0.155459
$772$	− 3.31662i	− 0.119368i
$773$	35.7322i	1.28520i	0.766202	+	0.642599i	$0.222144\pi$
$773$	35.7322i	1.28520i	−0.766202	+	0.642599i	$0.777856\pi$
$774$	−41.8997	−1.50606
$775$	0	0
$776$	−4.24264	−0.152302
$777$	0	0
$778$	28.6332i	1.02655i
$779$	−4.26650	−0.152863
$780$	0	0
$781$	−10.6332	−0.380488
$782$	− 62.9792i	− 2.25213i
$783$	− 26.6574i	− 0.952657i
$784$	0	0
$785$	0	0
$786$	37.4829	1.33697
$787$	37.7360i	1.34514i	0.740032	+	0.672571i	$0.234810\pi$
$787$	37.7360i	1.34514i	−0.740032	+	0.672571i	$0.765190\pi$
$788$	− 6.63325i	− 0.236300i
$789$	48.5310	1.72775
$790$	0	0
$791$	0	0
$792$	33.5831i	1.19332i
$793$	20.0000i	0.710221i
$794$	−11.7615	−0.417399
$795$	0	0
$796$	−8.93306	−0.316624
$797$	31.5605i	1.11793i	0.829191	+	0.558965i	$0.188801\pi$
$797$	31.5605i	1.11793i	−0.829191	+	0.558965i	$0.811199\pi$
$798$	0	0
$799$	−44.5330	−1.57546
$800$	0	0
$801$	31.0418	1.09681
$802$	− 3.00000i	− 0.105934i
$803$	− 38.7845i	− 1.36868i
$804$	4.98519	0.175814
$805$	0	0
$806$	18.5330	0.652797
$807$	24.5330i	0.863602i
$808$	− 15.0377i	− 0.529024i
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	1.86199	0.0653833	0.0326916	−	0.999465i	$-0.489592\pi$
$811$	1.86199	0.0653833	0.0326916	+	0.999465i	$0.489592\pi$
$812$	0	0
$813$	− 4.10025i	− 0.143802i
$814$	24.6332	0.863395
$815$	0	0
$816$	−22.2665	−0.779483
$817$	10.8659i	0.380151i
$818$	− 38.7845i	− 1.35607i
$819$	0	0
$820$	0	0
$821$	22.7335	0.793405	0.396702	−	0.917947i	$-0.370155\pi$
$821$	22.7335	0.793405	0.396702	+	0.917947i	$0.370155\pi$
$822$	− 42.5794i	− 1.48513i
$823$	48.5330i	1.69175i	0.533378	+	0.845877i	$0.320922\pi$
$823$	48.5330i	1.69175i	−0.533378	+	0.845877i	$0.679078\pi$
$824$	11.3137	0.394132
$825$	0	0
$826$	0	0
$827$	7.63325i	0.265434i	0.991154	+	0.132717i	$0.0423702\pi$
$827$	7.63325i	0.265434i	−0.991154	+	0.132717i	$0.957630\pi$
$828$	54.5330i	1.89515i
$829$	44.2175	1.53574	0.767869	−	0.640607i	$-0.221317\pi$
$829$	44.2175	1.53574	0.767869	+	0.640607i	$0.221317\pi$
$830$	0	0
$831$	20.2468	0.702352
$832$	− 3.27620i	− 0.113582i
$833$	0	0
$834$	−47.9499	−1.66037
$835$	0	0
$836$	8.70917	0.301213
$837$	57.2665i	1.97942i
$838$	19.1273i	0.660743i
$839$	11.7615	0.406052	0.203026	−	0.979173i	$-0.434922\pi$
$839$	11.7615	0.406052	0.203026	+	0.979173i	$0.434922\pi$
$840$	0	0
$841$	−22.0660	−0.760896
$842$	38.5330i	1.32793i
$843$	− 12.2093i	− 0.420509i
$844$	1.63325	0.0562188
$845$	0	0
$846$	38.5607	1.32574
$847$	0	0
$848$	− 8.00000i	− 0.274721i
$849$	77.0159	2.64318
$850$	0	0
$851$	40.0000	1.37118
$852$	6.10463i	0.209141i
$853$	− 45.7026i	− 1.56483i	−0.622759	−	0.782414i	$-0.713988\pi$
$853$	− 45.7026i	− 1.56483i	0.622759	−	0.782414i	$-0.286012\pi$
$854$	0	0
$855$	0	0
$856$	5.63325	0.192541
$857$	4.84341i	0.165448i	0.996573	+	0.0827240i	$0.0263620\pi$
$857$	4.84341i	0.165448i	−0.996573	+	0.0827240i	$0.973638\pi$
$858$	− 53.1662i	− 1.81507i
$859$	−27.0231	−0.922015	−0.461007	−	0.887396i	$-0.652512\pi$
$859$	−27.0231	−0.922015	−0.461007	+	0.887396i	$0.652512\pi$
$860$	0	0
$861$	0	0
$862$	7.36675i	0.250913i
$863$	10.1003i	0.343817i	0.985113	+	0.171908i	$0.0549933\pi$
$863$	10.1003i	0.343817i	−0.985113	+	0.171908i	$0.945007\pi$
$864$	10.1234	0.344404
$865$	0	0
$866$	3.50009	0.118938
$867$	− 110.544i	− 3.75426i
$868$	0	0
$869$	−7.26650	−0.246499
$870$	0	0
$871$	−5.35086	−0.181307
$872$	18.0000i	0.609557i
$873$	26.7992i	0.907014i
$874$	14.1421	0.478365
$875$	0	0
$876$	−22.2665	−0.752315
$877$	− 5.26650i	− 0.177837i	−0.996039	−	0.0889185i	$-0.971659\pi$
$877$	− 5.26650i	− 0.177837i	0.996039	−	0.0889185i	$-0.0283411\pi$
$878$	9.38083i	0.316588i
$879$	15.8997	0.536285
$880$	0	0
$881$	−53.0797	−1.78830	−0.894150	−	0.447768i	$-0.852219\pi$
$881$	−53.0797	−1.78830	−0.894150	+	0.447768i	$0.852219\pi$
$882$	0	0
$883$	15.6332i	0.526101i	0.964782	+	0.263050i	$0.0847285\pi$
$883$	15.6332i	0.526101i	−0.964782	+	0.263050i	$0.915271\pi$
$884$	23.8997	0.803836
$885$	0	0
$886$	−12.8997	−0.433376
$887$	29.6276i	0.994797i	0.867522	+	0.497399i	$0.165711\pi$
$887$	29.6276i	0.994797i	−0.867522	+	0.497399i	$0.834289\pi$
$888$	− 14.1421i	− 0.474579i
$889$	0	0
$890$	0	0
$891$	63.5330	2.12844
$892$	− 3.27620i	− 0.109695i
$893$	− 10.0000i	− 0.334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 86.3325i	− 2.88256i
$898$	34.2665i	1.14349i
$899$	−14.8959	−0.496806
$900$	0	0
$901$	58.3596	1.94424
$902$	13.8474i	0.461067i
$903$	0	0
$904$	−0.683375	−0.0227287
$905$	0	0
$906$	1.93288	0.0642155
$907$	5.26650i	0.174871i	0.996170	+	0.0874356i	$0.0278672\pi$
$907$	5.26650i	0.174871i	−0.996170	+	0.0874356i	$0.972133\pi$
$908$	3.79487i	0.125937i
$909$	−94.9874	−3.15053
$910$	0	0
$911$	−43.1662	−1.43016	−0.715081	−	0.699042i	$-0.753610\pi$
$911$	−43.1662	−1.43016	−0.715081	+	0.699042i	$0.753610\pi$
$912$	− 5.00000i	− 0.165567i
$913$	− 43.9227i	− 1.45363i
$914$	−16.6834	−0.551837
$915$	0	0
$916$	14.5899	0.482064
$917$	0	0
$918$	73.8496i	2.43740i
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	0	0
$921$	46.7995	1.54210
$922$	− 6.10463i	− 0.201045i
$923$	− 6.55240i	− 0.215675i
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	− 71.4645i	− 2.34720i
$928$	2.63325i	0.0864407i
$929$	50.2512	1.64869	0.824345	−	0.566088i	$-0.191544\pi$
$929$	50.2512	1.64869	0.824345	+	0.566088i	$0.191544\pi$
$930$	0	0
$931$	0	0
$932$	13.2665i	0.434559i
$933$	− 84.5330i	− 2.76749i
$934$	0.0708883	0.00231954
$935$	0	0
$936$	−20.6945	−0.676422
$937$	− 26.5753i	− 0.868177i	−0.900870	−	0.434088i	$-0.857071\pi$
$937$	− 26.5753i	− 0.868177i	0.900870	−	0.434088i	$-0.142929\pi$
$938$	0	0
$939$	35.6834	1.16448
$940$	0	0
$941$	52.7028	1.71806	0.859031	−	0.511924i	$-0.171067\pi$
$941$	52.7028	1.71806	0.859031	+	0.511924i	$0.171067\pi$
$942$	− 28.6332i	− 0.932922i
$943$	22.4856i	0.732233i
$944$	−1.86199	−0.0606026
$945$	0	0
$946$	35.2665	1.14661
$947$	− 33.3668i	− 1.08427i	−0.840290	−	0.542137i	$-0.817615\pi$
$947$	− 33.3668i	− 1.08427i	0.840290	−	0.542137i	$-0.182385\pi$
$948$	4.17175i	0.135492i
$949$	23.8997	0.775819
$950$	0	0
$951$	103.473	3.35533
$952$	0	0
$953$	− 58.2665i	− 1.88744i	−0.330750	−	0.943719i	$-0.607302\pi$
$953$	− 58.2665i	− 1.88744i	0.330750	−	0.943719i	$-0.392698\pi$
$954$	−50.5330	−1.63607
$955$	0	0
$956$	−16.6332	−0.537958
$957$	42.7324i	1.38134i
$958$	5.06730i	0.163717i
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	15.1795i	0.489406i
$963$	− 35.5831i	− 1.14665i
$964$	21.8140	0.702581
$965$	0	0
$966$	0	0
$967$	31.8997i	1.02583i	0.858440	+	0.512913i	$0.171434\pi$
$967$	31.8997i	1.02583i	−0.858440	+	0.512913i	$0.828566\pi$
$968$	− 17.2665i	− 0.554966i
$969$	36.4748	1.17174
$970$	0	0
$971$	−15.3325	−0.492042	−0.246021	−	0.969264i	$-0.579123\pi$
$971$	−15.3325	−0.492042	−0.246021	+	0.969264i	$0.579123\pi$
$972$	− 6.10463i	− 0.195806i
$973$	0	0
$974$	−31.2665	−1.00184
$975$	0	0
$976$	6.10463	0.195404
$977$	− 20.6834i	− 0.661720i	−0.943680	−	0.330860i	$-0.892661\pi$
$977$	− 20.6834i	− 0.661720i	0.943680	−	0.330860i	$-0.107339\pi$
$978$	− 8.19051i	− 0.261903i
$979$	−26.1275	−0.835039
$980$	0	0
$981$	113.699	3.63014
$982$	− 21.2665i	− 0.678641i
$983$	49.8744i	1.59075i	0.606121	+	0.795373i	$0.292725\pi$
$983$	49.8744i	1.59075i	−0.606121	+	0.795373i	$0.707275\pi$
$984$	7.94987	0.253433
$985$	0	0
$986$	−19.2094	−0.611753
$987$	0	0
$988$	5.36675i	0.170739i
$989$	57.2665	1.82097
$990$	0	0
$991$	22.5330	0.715784	0.357892	−	0.933763i	$-0.383496\pi$
$991$	22.5330	0.715784	0.357892	+	0.933763i	$0.383496\pi$
$992$	− 5.65685i	− 0.179605i
$993$	69.2368i	2.19716i
$994$	0	0
$995$	0	0
$996$	−25.2164	−0.799011
$997$	− 0.589552i	− 0.0186713i	−0.999956	−	0.00933565i	$-0.997028\pi$
$997$	− 0.589552i	− 0.0186713i	0.999956	−	0.00933565i	$-0.00297167\pi$
$998$	13.2665i	0.419944i
$999$	−46.9042	−1.48398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.x.99.8		8
5.2	odd	4		2450.2.a.bt.1.4	yes	4
5.3	odd	4		2450.2.a.bu.1.1	yes	4
5.4	even	2	inner	2450.2.c.x.99.1		8
7.6	odd	2	inner	2450.2.c.x.99.5		8
35.13	even	4		2450.2.a.bu.1.4	yes	4
35.27	even	4		2450.2.a.bt.1.1	✓	4
35.34	odd	2	inner	2450.2.c.x.99.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.2.a.bt.1.1	✓	4	35.27	even	4
2450.2.a.bt.1.4	yes	4	5.2	odd	4
2450.2.a.bu.1.1	yes	4	5.3	odd	4
2450.2.a.bu.1.4	yes	4	35.13	even	4
2450.2.c.x.99.1		8	5.4	even	2	inner
2450.2.c.x.99.4		8	35.34	odd	2	inner
2450.2.c.x.99.5		8	7.6	odd	2	inner
2450.2.c.x.99.8		8	1.1	even	1	trivial

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +3.05231i q^{3} -1.00000 q^{4} -3.05231 q^{6} -1.00000i q^{8} -6.31662 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +3.05231i q^{3} -1.00000 q^{4} -3.05231 q^{6} -1.00000i q^{8} -6.31662 q^{9} +5.31662 q^{11} -3.05231i q^{12} +3.27620i q^{13} +1.00000 q^{16} +7.29496i q^{17} -6.31662i q^{18} -1.63810 q^{19} +5.31662i q^{22} +8.63325i q^{23} +3.05231 q^{24} -3.27620 q^{26} -10.1234i q^{27} +2.63325 q^{29} -5.65685 q^{31} +1.00000i q^{32} +16.2280i q^{33} -7.29496 q^{34} +6.31662 q^{36} -4.63325i q^{37} -1.63810i q^{38} -10.0000 q^{39} +2.60454 q^{41} -6.63325i q^{43} -5.31662 q^{44} -8.63325 q^{46} +6.10463i q^{47} +3.05231i q^{48} -22.2665 q^{51} -3.27620i q^{52} -8.00000i q^{53} +10.1234 q^{54} -5.00000i q^{57} +2.63325i q^{58} -1.86199 q^{59} +6.10463 q^{61} -5.65685i q^{62} -1.00000 q^{64} -16.2280 q^{66} +1.63325i q^{67} -7.29496i q^{68} -26.3514 q^{69} -2.00000 q^{71} +6.31662i q^{72} -7.29496i q^{73} +4.63325 q^{74} +1.63810 q^{76} -10.0000i q^{78} -1.36675 q^{79} +11.9499 q^{81} +2.60454i q^{82} -8.26139i q^{83} +6.63325 q^{86} +8.03751i q^{87} -5.31662i q^{88} -4.91430 q^{89} -8.63325i q^{92} -17.2665i q^{93} -6.10463 q^{94} -3.05231 q^{96} -4.24264i q^{97} -33.5831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 24 q^{9}+O(q^{10}) \) 8 * q - 8 * q^4 - 24 * q^9	\( 8 q - 8 q^{4} - 24 q^{9} + 16 q^{11} + 8 q^{16} - 32 q^{29} + 24 q^{36} - 80 q^{39} - 16 q^{44} - 16 q^{46} - 72 q^{51} - 8 q^{64} - 16 q^{71} - 16 q^{74} - 64 q^{79} + 16 q^{81} - 136 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 - 24 * q^9 + 16 * q^11 + 8 * q^16 - 32 * q^29 + 24 * q^36 - 80 * q^39 - 16 * q^44 - 16 * q^46 - 72 * q^51 - 8 * q^64 - 16 * q^71 - 16 * q^74 - 64 * q^79 + 16 * q^81 - 136 * q^99

Introduction

L-functions

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