LMFDB - Embedded newform 225.8.b.n.199.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,8,Mod(199,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.199"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-138,0,0,0,0,0,0,-6896] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$70.2866307339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{601})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 301x^{2} + 22500 $ x^4 + 301*x^2 + 22500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$12.7577i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.8.b.n.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+15.7577i q^{2} -120.304 q^{4} +34.4284i q^{7} +121.278i q^{8} +3963.55 q^{11} +5606.30i q^{13} -542.511 q^{14} -17309.9 q^{16} -19906.7i q^{17} +49993.7 q^{19} +62456.2i q^{22} -109762. i q^{23} -88342.2 q^{26} -4141.86i q^{28} +192477. q^{29} +125541. q^{31} -257240. i q^{32} +313683. q^{34} +74353.6i q^{37} +787784. i q^{38} -577802. q^{41} +264291. i q^{43} -476829. q^{44} +1.72959e6 q^{46} -306207. i q^{47} +822358. q^{49} -674458. i q^{52} +446219. i q^{53} -4175.41 q^{56} +3.03299e6i q^{58} +1.97951e6 q^{59} -1.27494e6 q^{61} +1.97824e6i q^{62} +1.83783e6 q^{64} +4.12943e6i q^{67} +2.39485e6i q^{68} +2.81187e6 q^{71} +4.01991e6i q^{73} -1.17164e6 q^{74} -6.01442e6 q^{76} +136459. i q^{77} +1.32785e6 q^{79} -9.10480e6i q^{82} +1.91033e6i q^{83} -4.16460e6 q^{86} +480691. i q^{88} +8.00695e6 q^{89} -193016. q^{91} +1.32047e7i q^{92} +4.82511e6 q^{94} +3.89241e6i q^{97} +1.29584e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 138 q^{4} - 6896 q^{11} - 24528 q^{14} - 48990 q^{16} + 99168 q^{19} - 432308 q^{26} + 363544 q^{29} + 608464 q^{31} + 879844 q^{34} - 1262344 q^{41} - 1714136 q^{44} + 3772944 q^{46} - 290980 q^{49}+ \cdots + 6825472 q^{94}+O(q^{100}) $ 4 * q - 138 * q^4 - 6896 * q^11 - 24528 * q^14 - 48990 * q^16 + 99168 * q^19 - 432308 * q^26 + 363544 * q^29 + 608464 * q^31 + 879844 * q^34 - 1262344 * q^41 - 1714136 * q^44 + 3772944 * q^46 - 290980 * q^49 - 4211760 * q^56 + 574448 * q^59 - 5028360 * q^61 - 582082 * q^64 + 7497632 * q^71 + 5152612 * q^74 - 12070888 * q^76 + 9255440 * q^79 - 6764216 * q^86 + 33032712 * q^89 - 39452896 * q^91 + 6825472 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	15.7577i	1.39279i	0.717657	+	0.696396i	$0.245214\pi$
$2$	15.7577i	1.39279i	−0.717657	+	0.696396i	$0.754786\pi$
$3$	0	0
$3$	0	0
$4$	−120.304	−0.939872
$5$	0	0
$5$	0	0
$6$	0	0
$7$	34.4284i	0.0379380i	0.999820	+	0.0189690i	$0.00603838\pi$
$7$	34.4284i	0.0379380i	−0.999820	+	0.0189690i	$0.993962\pi$
$8$	121.278i	0.0837465i
$9$	0	0
$10$	0	0
$11$	3963.55	0.897863	0.448931	−	0.893566i	$-0.351805\pi$
$11$	3963.55	0.897863	0.448931	+	0.893566i	$0.351805\pi$
$12$	0	0
$13$	5606.30i	0.707742i	0.935294	+	0.353871i	$0.115135\pi$
$13$	5606.30i	0.707742i	−0.935294	+	0.353871i	$0.884865\pi$
$14$	−542.511	−0.0528397
$15$	0	0
$16$	−17309.9	−1.05651
$17$	− 19906.7i	− 0.982717i	−0.870958	−	0.491358i	$-0.836501\pi$
$17$	− 19906.7i	− 0.982717i	0.870958	−	0.491358i	$-0.163499\pi$
$18$	0	0
$19$	49993.7	1.67216	0.836080	−	0.548607i	$-0.184842\pi$
$19$	49993.7	1.67216	0.836080	+	0.548607i	$0.184842\pi$
$20$	0	0
$21$	0	0
$22$	62456.2i	1.25054i
$23$	− 109762.i	− 1.88107i	−0.339703	−	0.940533i	$-0.610326\pi$
$23$	− 109762.i	− 1.88107i	0.339703	−	0.940533i	$-0.389674\pi$
$24$	0	0
$25$	0	0
$26$	−88342.2	−0.985738
$27$	0	0
$28$	− 4141.86i	− 0.0356568i
$29$	192477.	1.46550	0.732752	−	0.680496i	$-0.238236\pi$
$29$	192477.	1.46550	0.732752	+	0.680496i	$0.238236\pi$
$30$	0	0
$31$	125541.	0.756870	0.378435	−	0.925628i	$-0.376462\pi$
$31$	125541.	0.756870	0.378435	+	0.925628i	$0.376462\pi$
$32$	− 257240.i	− 1.38776i
$33$	0	0
$34$	313683.	1.36872
$35$	0	0
$36$	0	0
$37$	74353.6i	0.241322i	0.992694	+	0.120661i	$0.0385013\pi$
$37$	74353.6i	0.241322i	−0.992694	+	0.120661i	$0.961499\pi$
$38$	787784.i	2.32897i
$39$	0	0
$40$	0	0
$41$	−577802.	−1.30929	−0.654644	−	0.755937i	$-0.727182\pi$
$41$	−577802.	−1.30929	−0.654644	+	0.755937i	$0.727182\pi$
$42$	0	0
$43$	264291.i	0.506923i	0.967345	+	0.253462i	$0.0815691\pi$
$43$	264291.i	0.506923i	−0.967345	+	0.253462i	$0.918431\pi$
$44$	−476829.	−0.843876
$45$	0	0
$46$	1.72959e6	2.61993
$47$	− 306207.i	− 0.430203i	−0.976592	−	0.215101i	$-0.930992\pi$
$47$	− 306207.i	− 0.430203i	0.976592	−	0.215101i	$-0.0690082\pi$
$48$	0	0
$49$	822358.	0.998561
$50$	0	0
$51$	0	0
$52$	− 674458.i	− 0.665186i
$53$	446219.i	0.411702i	0.978583	+	0.205851i	$0.0659962\pi$
$53$	446219.i	0.411702i	−0.978583	+	0.205851i	$0.934004\pi$
$54$	0	0
$55$	0	0
$56$	−4175.41	−0.00317717
$57$	0	0
$58$	3.03299e6i	2.04114i
$59$	1.97951e6	1.25481	0.627403	−	0.778695i	$-0.284118\pi$
$59$	1.97951e6	1.25481	0.627403	+	0.778695i	$0.284118\pi$
$60$	0	0
$61$	−1.27494e6	−0.719175	−0.359587	−	0.933111i	$-0.617083\pi$
$61$	−1.27494e6	−0.719175	−0.359587	+	0.933111i	$0.617083\pi$
$62$	1.97824e6i	1.05416i
$63$	0	0
$64$	1.83783e6	0.876345
$65$	0	0
$66$	0	0
$67$	4.12943e6i	1.67737i	0.544619	+	0.838684i	$0.316674\pi$
$67$	4.12943e6i	1.67737i	−0.544619	+	0.838684i	$0.683326\pi$
$68$	2.39485e6i	0.923627i
$69$	0	0
$70$	0	0
$71$	2.81187e6	0.932377	0.466188	−	0.884685i	$-0.345627\pi$
$71$	2.81187e6	0.932377	0.466188	+	0.884685i	$0.345627\pi$
$72$	0	0
$73$	4.01991e6i	1.20945i	0.796436	+	0.604723i	$0.206716\pi$
$73$	4.01991e6i	1.20945i	−0.796436	+	0.604723i	$0.793284\pi$
$74$	−1.17164e6	−0.336111
$75$	0	0
$76$	−6.01442e6	−1.57162
$77$	136459.i	0.0340631i
$78$	0	0
$79$	1.32785e6	0.303009	0.151505	−	0.988457i	$-0.451588\pi$
$79$	1.32785e6	0.303009	0.151505	+	0.988457i	$0.451588\pi$
$80$	0	0
$81$	0	0
$82$	− 9.10480e6i	− 1.82357i
$83$	1.91033e6i	0.366721i	0.983046	+	0.183361i	$0.0586976\pi$
$83$	1.91033e6i	0.366721i	−0.983046	+	0.183361i	$0.941302\pi$
$84$	0	0
$85$	0	0
$86$	−4.16460e6	−0.706039
$87$	0	0
$88$	480691.i	0.0751929i
$89$	8.00695e6	1.20393	0.601966	−	0.798522i	$-0.294384\pi$
$89$	8.00695e6	1.20393	0.601966	+	0.798522i	$0.294384\pi$
$90$	0	0
$91$	−193016.	−0.0268503
$92$	1.32047e7i	1.76796i
$93$	0	0
$94$	4.82511e6	0.599183
$95$	0	0
$96$	0	0
$97$	3.89241e6i	0.433029i	0.976279	+	0.216515i	$0.0694689\pi$
$97$	3.89241e6i	0.433029i	−0.976279	+	0.216515i	$0.930531\pi$
$98$	1.29584e7i	1.39079i
$99$	0	0
$100$	0	0
$101$	−1.07779e7	−1.04090	−0.520450	−	0.853892i	$-0.674236\pi$
$101$	−1.07779e7	−1.04090	−0.520450	+	0.853892i	$0.674236\pi$
$102$	0	0
$103$	− 1.93103e7i	− 1.74124i	−0.491952	−	0.870622i	$-0.663717\pi$
$103$	− 1.93103e7i	− 1.74124i	0.491952	−	0.870622i	$-0.336283\pi$
$104$	−679921.	−0.0592709
$105$	0	0
$106$	−7.03137e6	−0.573415
$107$	− 6.90473e6i	− 0.544883i	−0.962172	−	0.272442i	$-0.912169\pi$
$107$	− 6.90473e6i	− 0.544883i	0.962172	−	0.272442i	$-0.0878312\pi$
$108$	0	0
$109$	−2.39533e7	−1.77163	−0.885814	−	0.464040i	$-0.846399\pi$
$109$	−2.39533e7	−1.77163	−0.885814	+	0.464040i	$0.846399\pi$
$110$	0	0
$111$	0	0
$112$	− 595953.i	− 0.0400820i
$113$	9.56587e6i	0.623663i	0.950137	+	0.311831i	$0.100942\pi$
$113$	9.56587e6i	0.623663i	−0.950137	+	0.311831i	$0.899058\pi$
$114$	0	0
$115$	0	0
$116$	−2.31557e7	−1.37738
$117$	0	0
$118$	3.11925e7i	1.74768i
$119$	685357.	0.0372823
$120$	0	0
$121$	−3.77744e6	−0.193843
$122$	− 2.00900e7i	− 1.00166i
$123$	0	0
$124$	−1.51031e7	−0.711360
$125$	0	0
$126$	0	0
$127$	1.19412e7i	0.517290i	0.965972	+	0.258645i	$0.0832759\pi$
$127$	1.19412e7i	0.517290i	−0.965972	+	0.258645i	$0.916724\pi$
$128$	− 3.96685e6i	− 0.167190i
$129$	0	0
$130$	0	0
$131$	1.60064e6	0.0622077	0.0311039	−	0.999516i	$-0.490098\pi$
$131$	1.60064e6	0.0622077	0.0311039	+	0.999516i	$0.490098\pi$
$132$	0	0
$133$	1.72121e6i	0.0634384i
$134$	−6.50701e7	−2.33623
$135$	0	0
$136$	2.41424e6	0.0822991
$137$	2.88030e7i	0.957008i	0.878085	+	0.478504i	$0.158821\pi$
$137$	2.88030e7i	0.957008i	−0.878085	+	0.478504i	$0.841179\pi$
$138$	0	0
$139$	2.14740e6	0.0678204	0.0339102	−	0.999425i	$-0.489204\pi$
$139$	2.14740e6	0.0678204	0.0339102	+	0.999425i	$0.489204\pi$
$140$	0	0
$141$	0	0
$142$	4.43085e7i	1.29861i
$143$	2.22209e7i	0.635455i
$144$	0	0
$145$	0	0
$146$	−6.33444e7	−1.68451
$147$	0	0
$148$	− 8.94501e6i	− 0.226811i
$149$	4.96813e6	0.123038	0.0615192	−	0.998106i	$-0.480405\pi$
$149$	4.96813e6	0.123038	0.0615192	+	0.998106i	$0.480405\pi$
$150$	0	0
$151$	−2.10485e7	−0.497511	−0.248756	−	0.968566i	$-0.580022\pi$
$151$	−2.10485e7	−0.497511	−0.248756	+	0.968566i	$0.580022\pi$
$152$	6.06313e6i	0.140038i
$153$	0	0
$154$	−2.15027e6	−0.0474428
$155$	0	0
$156$	0	0
$157$	2.61392e7i	0.539069i	0.962991	+	0.269534i	$0.0868698\pi$
$157$	2.61392e7i	0.539069i	−0.962991	+	0.269534i	$0.913130\pi$
$158$	2.09239e7i	0.422029i
$159$	0	0
$160$	0	0
$161$	3.77893e6	0.0713638
$162$	0	0
$163$	− 1.01817e8i	− 1.84147i	−0.390194	−	0.920733i	$-0.627592\pi$
$163$	− 1.01817e8i	− 1.84147i	0.390194	−	0.920733i	$-0.372408\pi$
$164$	6.95116e7	1.23056
$165$	0	0
$166$	−3.01024e7	−0.510766
$167$	6.14576e7i	1.02110i	0.859848	+	0.510550i	$0.170558\pi$
$167$	6.14576e7i	1.02110i	−0.859848	+	0.510550i	$0.829442\pi$
$168$	0	0
$169$	3.13179e7	0.499101
$170$	0	0
$171$	0	0
$172$	− 3.17951e7i	− 0.476443i
$173$	2.74394e7i	0.402914i	0.979497	+	0.201457i	$0.0645677\pi$
$173$	2.74394e7i	0.402914i	−0.979497	+	0.201457i	$0.935432\pi$
$174$	0	0
$175$	0	0
$176$	−6.86087e7	−0.948604
$177$	0	0
$178$	1.26171e8i	1.67683i
$179$	4.05126e7	0.527964	0.263982	−	0.964528i	$-0.414964\pi$
$179$	4.05126e7	0.527964	0.263982	+	0.964528i	$0.414964\pi$
$180$	0	0
$181$	9.66189e7	1.21112	0.605560	−	0.795800i	$-0.292949\pi$
$181$	9.66189e7	1.21112	0.605560	+	0.795800i	$0.292949\pi$
$182$	− 3.04148e6i	− 0.0373969i
$183$	0	0
$184$	1.33117e7	0.157533
$185$	0	0
$186$	0	0
$187$	− 7.89012e7i	− 0.882345i
$188$	3.68378e7i	0.404335i
$189$	0	0
$190$	0	0
$191$	1.42378e8	1.47852	0.739260	−	0.673420i	$-0.235175\pi$
$191$	1.42378e8	1.47852	0.739260	+	0.673420i	$0.235175\pi$
$192$	0	0
$193$	1.27527e8i	1.27688i	0.769671	+	0.638441i	$0.220420\pi$
$193$	1.27527e8i	1.27688i	−0.769671	+	0.638441i	$0.779580\pi$
$194$	−6.13352e7	−0.603120
$195$	0	0
$196$	−9.89326e7	−0.938519
$197$	− 1.23507e8i	− 1.15096i	−0.817816	−	0.575479i	$-0.804816\pi$
$197$	− 1.23507e8i	− 1.15096i	0.817816	−	0.575479i	$-0.195184\pi$
$198$	0	0
$199$	7.46881e7	0.671839	0.335920	−	0.941891i	$-0.390953\pi$
$199$	7.46881e7	0.671839	0.335920	+	0.941891i	$0.390953\pi$
$200$	0	0
$201$	0	0
$202$	− 1.69834e8i	− 1.44976i
$203$	6.62670e6i	0.0555982i
$204$	0	0
$205$	0	0
$206$	3.04286e8	2.42519
$207$	0	0
$208$	− 9.70446e7i	− 0.747739i
$209$	1.98153e8	1.50137
$210$	0	0
$211$	1.24482e8	0.912257	0.456128	−	0.889914i	$-0.349236\pi$
$211$	1.24482e8	0.912257	0.456128	+	0.889914i	$0.349236\pi$
$212$	− 5.36818e7i	− 0.386947i
$213$	0	0
$214$	1.08802e8	0.758910
$215$	0	0
$216$	0	0
$217$	4.32219e6i	0.0287141i
$218$	− 3.77448e8i	− 2.46751i
$219$	0	0
$220$	0	0
$221$	1.11603e8	0.695510
$222$	0	0
$223$	2.17519e8i	1.31350i	0.754109	+	0.656749i	$0.228069\pi$
$223$	2.17519e8i	1.31350i	−0.754109	+	0.656749i	$0.771931\pi$
$224$	8.85637e6	0.0526487
$225$	0	0
$226$	−1.50736e8	−0.868633
$227$	1.39021e8i	0.788840i	0.918930	+	0.394420i	$0.129054\pi$
$227$	1.39021e8i	0.788840i	−0.918930	+	0.394420i	$0.870946\pi$
$228$	0	0
$229$	−1.06507e8	−0.586076	−0.293038	−	0.956101i	$-0.594666\pi$
$229$	−1.06507e8	−0.586076	−0.293038	+	0.956101i	$0.594666\pi$
$230$	0	0
$231$	0	0
$232$	2.33433e7i	0.122731i
$233$	3.55655e7i	0.184197i	0.995750	+	0.0920986i	$0.0293575\pi$
$233$	3.55655e7i	0.184197i	−0.995750	+	0.0920986i	$0.970642\pi$
$234$	0	0
$235$	0	0
$236$	−2.38143e8	−1.17936
$237$	0	0
$238$	1.07996e7i	0.0519265i
$239$	−8.12794e7	−0.385113	−0.192556	−	0.981286i	$-0.561678\pi$
$239$	−8.12794e7	−0.385113	−0.192556	+	0.981286i	$0.561678\pi$
$240$	0	0
$241$	1.78776e8	0.822717	0.411359	−	0.911474i	$-0.365054\pi$
$241$	1.78776e8	0.822717	0.411359	+	0.911474i	$0.365054\pi$
$242$	− 5.95236e7i	− 0.269983i
$243$	0	0
$244$	1.53379e8	0.675932
$245$	0	0
$246$	0	0
$247$	2.80280e8i	1.18346i
$248$	1.52254e7i	0.0633852i
$249$	0	0
$250$	0	0
$251$	3.53343e7	0.141039	0.0705195	−	0.997510i	$-0.477534\pi$
$251$	3.53343e7	0.141039	0.0705195	+	0.997510i	$0.477534\pi$
$252$	0	0
$253$	− 4.35047e8i	− 1.68894i
$254$	−1.88165e8	−0.720477
$255$	0	0
$256$	2.97750e8	1.10921
$257$	− 4.18029e8i	− 1.53617i	−0.640345	−	0.768087i	$-0.721209\pi$
$257$	− 4.18029e8i	− 1.53617i	0.640345	−	0.768087i	$-0.278791\pi$
$258$	0	0
$259$	−2.55988e6	−0.00915525
$260$	0	0
$261$	0	0
$262$	2.52223e7i	0.0866425i
$263$	− 2.02813e8i	− 0.687466i	−0.939067	−	0.343733i	$-0.888308\pi$
$263$	− 2.02813e8i	− 0.687466i	0.939067	−	0.343733i	$-0.111692\pi$
$264$	0	0
$265$	0	0
$266$	−2.71222e7	−0.0883565
$267$	0	0
$268$	− 4.96785e8i	− 1.57651i
$269$	−2.88920e8	−0.904993	−0.452496	−	0.891766i	$-0.649466\pi$
$269$	−2.88920e8	−0.904993	−0.452496	+	0.891766i	$0.649466\pi$
$270$	0	0
$271$	3.61648e8	1.10381	0.551904	−	0.833908i	$-0.313902\pi$
$271$	3.61648e8	1.10381	0.551904	+	0.833908i	$0.313902\pi$
$272$	3.44583e8i	1.03825i
$273$	0	0
$274$	−4.53867e8	−1.33291
$275$	0	0
$276$	0	0
$277$	3.16424e8i	0.894522i	0.894404	+	0.447261i	$0.147600\pi$
$277$	3.16424e8i	0.894522i	−0.894404	+	0.447261i	$0.852400\pi$
$278$	3.38379e7i	0.0944597i
$279$	0	0
$280$	0	0
$281$	3.81619e7	0.102603	0.0513013	−	0.998683i	$-0.483663\pi$
$281$	3.81619e7	0.102603	0.0513013	+	0.998683i	$0.483663\pi$
$282$	0	0
$283$	− 5.36394e8i	− 1.40680i	−0.710796	−	0.703398i	$-0.751665\pi$
$283$	− 5.36394e8i	− 1.40680i	0.710796	−	0.703398i	$-0.248335\pi$
$284$	−3.38278e8	−0.876314
$285$	0	0
$286$	−3.50149e8	−0.885057
$287$	− 1.98928e7i	− 0.0496718i
$288$	0	0
$289$	1.40615e7	0.0342681
$290$	0	0
$291$	0	0
$292$	− 4.83610e8i	− 1.13672i
$293$	− 5.38585e8i	− 1.25088i	−0.780270	−	0.625442i	$-0.784919\pi$
$293$	− 5.38585e8i	− 1.25088i	0.780270	−	0.625442i	$-0.215081\pi$
$294$	0	0
$295$	0	0
$296$	−9.01745e6	−0.0202098
$297$	0	0
$298$	7.82860e7i	0.171367i
$299$	6.15358e8	1.33131
$300$	0	0
$301$	−9.09911e6	−0.0192316
$302$	− 3.31676e8i	− 0.692930i
$303$	0	0
$304$	−8.65387e8	−1.76666
$305$	0	0
$306$	0	0
$307$	− 6.10303e7i	− 0.120382i	−0.998187	−	0.0601910i	$-0.980829\pi$
$307$	− 6.10303e7i	− 0.120382i	0.998187	−	0.0601910i	$-0.0191710\pi$
$308$	− 1.64165e7i	− 0.0320149i
$309$	0	0
$310$	0	0
$311$	−5.75355e8	−1.08461	−0.542306	−	0.840181i	$-0.682449\pi$
$311$	−5.75355e8	−1.08461	−0.542306	+	0.840181i	$0.682449\pi$
$312$	0	0
$313$	7.32396e8i	1.35002i	0.737807	+	0.675011i	$0.235861\pi$
$313$	7.32396e8i	1.35002i	−0.737807	+	0.675011i	$0.764139\pi$
$314$	−4.11893e8	−0.750811
$315$	0	0
$316$	−1.59746e8	−0.284790
$317$	− 1.67850e7i	− 0.0295947i	−0.999891	−	0.0147973i	$-0.995290\pi$
$317$	− 1.67850e7i	− 0.0295947i	0.999891	−	0.0147973i	$-0.00471031\pi$
$318$	0	0
$319$	7.62894e8	1.31582
$320$	0	0
$321$	0	0
$322$	5.95470e7i	0.0993950i
$323$	− 9.95211e8i	− 1.64326i
$324$	0	0
$325$	0	0
$326$	1.60440e9	2.56478
$327$	0	0
$328$	− 7.00746e7i	− 0.109648i
$329$	1.05422e7	0.0163210
$330$	0	0
$331$	−1.08406e9	−1.64306	−0.821532	−	0.570163i	$-0.806880\pi$
$331$	−1.08406e9	−1.64306	−0.821532	+	0.570163i	$0.806880\pi$
$332$	− 2.29820e8i	− 0.344671i
$333$	0	0
$334$	−9.68427e8	−1.42218
$335$	0	0
$336$	0	0
$337$	− 7.36350e7i	− 0.104804i	−0.998626	−	0.0524022i	$-0.983312\pi$
$337$	− 7.36350e7i	− 0.104804i	0.998626	−	0.0524022i	$-0.0166878\pi$
$338$	4.93496e8i	0.695145i
$339$	0	0
$340$	0	0
$341$	4.97590e8	0.679565
$342$	0	0
$343$	5.66658e7i	0.0758214i
$344$	−3.20526e7	−0.0424530
$345$	0	0
$346$	−4.32380e8	−0.561176
$347$	− 2.99713e8i	− 0.385082i	−0.981289	−	0.192541i	$-0.938327\pi$
$347$	− 2.99713e8i	− 0.385082i	0.981289	−	0.192541i	$-0.0616728\pi$
$348$	0	0
$349$	4.00569e8	0.504415	0.252208	−	0.967673i	$-0.418843\pi$
$349$	4.00569e8	0.504415	0.252208	+	0.967673i	$0.418843\pi$
$350$	0	0
$351$	0	0
$352$	− 1.01958e9i	− 1.24602i
$353$	5.93100e8i	0.717657i	0.933404	+	0.358828i	$0.116824\pi$
$353$	5.93100e8i	0.717657i	−0.933404	+	0.358828i	$0.883176\pi$
$354$	0	0
$355$	0	0
$356$	−9.63264e8	−1.13154
$357$	0	0
$358$	6.38383e8i	0.735344i
$359$	8.77947e8	1.00147	0.500735	−	0.865601i	$-0.333063\pi$
$359$	8.77947e8	1.00147	0.500735	+	0.865601i	$0.333063\pi$
$360$	0	0
$361$	1.60550e9	1.79612
$362$	1.52249e9i	1.68684i
$363$	0	0
$364$	2.32205e7	0.0252358
$365$	0	0
$366$	0	0
$367$	− 4.65347e8i	− 0.491412i	−0.969344	−	0.245706i	$-0.920980\pi$
$367$	− 4.65347e8i	− 0.491412i	0.969344	−	0.245706i	$-0.0790198\pi$
$368$	1.89997e9i	1.98737i
$369$	0	0
$370$	0	0
$371$	−1.53626e7	−0.0156191
$372$	0	0
$373$	− 2.88096e8i	− 0.287446i	−0.989618	−	0.143723i	$-0.954093\pi$
$373$	− 2.88096e8i	− 0.287446i	0.989618	−	0.143723i	$-0.0459074\pi$
$374$	1.24330e9	1.22892
$375$	0	0
$376$	3.71362e7	0.0360280
$377$	1.07909e9i	1.03720i
$378$	0	0
$379$	8.21572e8	0.775190	0.387595	−	0.921830i	$-0.373306\pi$
$379$	8.21572e8	0.775190	0.387595	+	0.921830i	$0.373306\pi$
$380$	0	0
$381$	0	0
$382$	2.24355e9i	2.05927i
$383$	6.41573e8i	0.583513i	0.956493	+	0.291757i	$0.0942397\pi$
$383$	6.41573e8i	0.583513i	−0.956493	+	0.291757i	$0.905760\pi$
$384$	0	0
$385$	0	0
$386$	−2.00952e9	−1.77843
$387$	0	0
$388$	− 4.68270e8i	− 0.406992i
$389$	−1.25481e9	−1.08082	−0.540411	−	0.841401i	$-0.681731\pi$
$389$	−1.25481e9	−1.08082	−0.540411	+	0.841401i	$0.681731\pi$
$390$	0	0
$391$	−2.18500e9	−1.84855
$392$	9.97338e7i	0.0836260i
$393$	0	0
$394$	1.94618e9	1.60305
$395$	0	0
$396$	0	0
$397$	− 1.29837e9i	− 1.04144i	−0.853729	−	0.520718i	$-0.825664\pi$
$397$	− 1.29837e9i	− 1.04144i	0.853729	−	0.520718i	$-0.174336\pi$
$398$	1.17691e9i	0.935733i
$399$	0	0
$400$	0	0
$401$	−2.16310e9	−1.67522	−0.837610	−	0.546268i	$-0.816048\pi$
$401$	−2.16310e9	−1.67522	−0.837610	+	0.546268i	$0.816048\pi$
$402$	0	0
$403$	7.03823e8i	0.535668i
$404$	1.29662e9	0.978313
$405$	0	0
$406$	−1.04421e8	−0.0774368
$407$	2.94704e8i	0.216674i
$408$	0	0
$409$	1.26448e9	0.913864	0.456932	−	0.889502i	$-0.348948\pi$
$409$	1.26448e9	0.913864	0.456932	+	0.889502i	$0.348948\pi$
$410$	0	0
$411$	0	0
$412$	2.32310e9i	1.63655i
$413$	6.81516e7i	0.0476048i
$414$	0	0
$415$	0	0
$416$	1.44217e9	0.982174
$417$	0	0
$418$	3.12242e9i	2.09110i
$419$	−1.84627e9	−1.22616	−0.613080	−	0.790021i	$-0.710070\pi$
$419$	−1.84627e9	−1.22616	−0.613080	+	0.790021i	$0.710070\pi$
$420$	0	0
$421$	2.24531e9	1.46652	0.733262	−	0.679947i	$-0.237997\pi$
$421$	2.24531e9	1.46652	0.733262	+	0.679947i	$0.237997\pi$
$422$	1.96154e9i	1.27058i
$423$	0	0
$424$	−5.41165e7	−0.0344786
$425$	0	0
$426$	0	0
$427$	− 4.38941e7i	− 0.0272840i
$428$	8.30664e8i	0.512120i
$429$	0	0
$430$	0	0
$431$	1.63983e9	0.986568	0.493284	−	0.869868i	$-0.335796\pi$
$431$	1.63983e9	0.986568	0.493284	+	0.869868i	$0.335796\pi$
$432$	0	0
$433$	2.53503e9i	1.50063i	0.661078	+	0.750317i	$0.270099\pi$
$433$	2.53503e9i	1.50063i	−0.661078	+	0.750317i	$0.729901\pi$
$434$	−6.81076e7	−0.0399928
$435$	0	0
$436$	2.88167e9	1.66510
$437$	− 5.48740e9i	− 3.14544i
$438$	0	0
$439$	−1.74546e8	−0.0984655	−0.0492327	−	0.998787i	$-0.515678\pi$
$439$	−1.74546e8	−0.0984655	−0.0492327	+	0.998787i	$0.515678\pi$
$440$	0	0
$441$	0	0
$442$	1.75860e9i	0.968701i
$443$	1.13523e9i	0.620400i	0.950671	+	0.310200i	$0.100396\pi$
$443$	1.13523e9i	0.620400i	−0.950671	+	0.310200i	$0.899604\pi$
$444$	0	0
$445$	0	0
$446$	−3.42758e9	−1.82943
$447$	0	0
$448$	6.32736e7i	0.0332468i
$449$	2.28121e9	1.18933	0.594665	−	0.803973i	$-0.297285\pi$
$449$	2.28121e9	1.18933	0.594665	+	0.803973i	$0.297285\pi$
$450$	0	0
$451$	−2.29015e9	−1.17556
$452$	− 1.15081e9i	− 0.586163i
$453$	0	0
$454$	−2.19064e9	−1.09869
$455$	0	0
$456$	0	0
$457$	− 2.93541e9i	− 1.43867i	−0.694663	−	0.719335i	$-0.744447\pi$
$457$	− 2.93541e9i	− 1.43867i	0.694663	−	0.719335i	$-0.255553\pi$
$458$	− 1.67830e9i	− 0.816282i
$459$	0	0
$460$	0	0
$461$	6.32918e8	0.300881	0.150440	−	0.988619i	$-0.451931\pi$
$461$	6.32918e8	0.300881	0.150440	+	0.988619i	$0.451931\pi$
$462$	0	0
$463$	− 1.41260e9i	− 0.661431i	−0.943730	−	0.330716i	$-0.892710\pi$
$463$	− 1.41260e9i	− 0.661431i	0.943730	−	0.330716i	$-0.107290\pi$
$464$	−3.33177e9	−1.54832
$465$	0	0
$466$	−5.60429e8	−0.256549
$467$	− 3.77542e8i	− 0.171536i	−0.996315	−	0.0857682i	$-0.972666\pi$
$467$	− 3.77542e8i	− 0.171536i	0.996315	−	0.0857682i	$-0.0273344\pi$
$468$	0	0
$469$	−1.42170e8	−0.0636359
$470$	0	0
$471$	0	0
$472$	2.40071e8i	0.105086i
$473$	1.04753e9i	0.455147i
$474$	0	0
$475$	0	0
$476$	−8.24509e7	−0.0350406
$477$	0	0
$478$	− 1.28077e9i	− 0.536382i
$479$	4.30550e9	1.78999	0.894993	−	0.446080i	$-0.147180\pi$
$479$	4.30550e9	1.78999	0.894993	+	0.446080i	$0.147180\pi$
$480$	0	0
$481$	−4.16849e8	−0.170793
$482$	2.81710e9i	1.14587i
$483$	0	0
$484$	4.54440e8	0.182187
$485$	0	0
$486$	0	0
$487$	− 8.36858e8i	− 0.328322i	−0.986434	−	0.164161i	$-0.947508\pi$
$487$	− 8.36858e8i	− 0.328322i	0.986434	−	0.164161i	$-0.0524917\pi$
$488$	− 1.54622e8i	− 0.0602284i
$489$	0	0
$490$	0	0
$491$	−3.30420e9	−1.25974	−0.629870	−	0.776700i	$-0.716892\pi$
$491$	−3.30420e9	−1.25974	−0.629870	+	0.776700i	$0.716892\pi$
$492$	0	0
$493$	− 3.83159e9i	− 1.44017i
$494$	−4.41656e9	−1.64831
$495$	0	0
$496$	−2.17311e9	−0.799643
$497$	9.68084e7i	0.0353725i
$498$	0	0
$499$	−3.42619e9	−1.23441	−0.617205	−	0.786802i	$-0.711735\pi$
$499$	−3.42619e9	−1.23441	−0.617205	+	0.786802i	$0.711735\pi$
$500$	0	0
$501$	0	0
$502$	5.56786e8i	0.196438i
$503$	− 2.20556e9i	− 0.772734i	−0.922345	−	0.386367i	$-0.873730\pi$
$503$	− 2.20556e9i	− 0.772734i	0.922345	−	0.386367i	$-0.126270\pi$
$504$	0	0
$505$	0	0
$506$	6.85531e9	2.35234
$507$	0	0
$508$	− 1.43656e9i	− 0.486186i
$509$	1.55340e9	0.522122	0.261061	−	0.965322i	$-0.415928\pi$
$509$	1.55340e9	0.522122	0.261061	+	0.965322i	$0.415928\pi$
$510$	0	0
$511$	−1.38399e8	−0.0458839
$512$	4.18409e9i	1.37770i
$513$	0	0
$514$	6.58715e9	2.13957
$515$	0	0
$516$	0	0
$517$	− 1.21367e9i	− 0.386263i
$518$	− 4.03377e7i	− 0.0127514i
$519$	0	0
$520$	0	0
$521$	−5.82978e9	−1.80601	−0.903005	−	0.429629i	$-0.858644\pi$
$521$	−5.82978e9	−1.80601	−0.903005	+	0.429629i	$0.858644\pi$
$522$	0	0
$523$	− 1.98399e9i	− 0.606433i	−0.952922	−	0.303216i	$-0.901940\pi$
$523$	− 1.98399e9i	− 0.606433i	0.952922	−	0.303216i	$-0.0980605\pi$
$524$	−1.92563e8	−0.0584673
$525$	0	0
$526$	3.19586e9	0.957498
$527$	− 2.49912e9i	− 0.743788i
$528$	0	0
$529$	−8.64284e9	−2.53841
$530$	0	0
$531$	0	0
$532$	− 2.07067e8i	− 0.0596239i
$533$	− 3.23933e9i	− 0.926638i
$534$	0	0
$535$	0	0
$536$	−5.00809e8	−0.140474
$537$	0	0
$538$	− 4.55271e9i	− 1.26047i
$539$	3.25946e9	0.896570
$540$	0	0
$541$	−7.16394e9	−1.94519	−0.972594	−	0.232511i	$-0.925306\pi$
$541$	−7.16394e9	−1.94519	−0.972594	+	0.232511i	$0.925306\pi$
$542$	5.69872e9i	1.53737i
$543$	0	0
$544$	−5.12080e9	−1.36377
$545$	0	0
$546$	0	0
$547$	− 1.97289e9i	− 0.515403i	−0.966225	−	0.257702i	$-0.917035\pi$
$547$	− 1.97289e9i	− 0.515403i	0.966225	−	0.257702i	$-0.0829652\pi$
$548$	− 3.46510e9i	− 0.899464i
$549$	0	0
$550$	0	0
$551$	9.62266e9	2.45056
$552$	0	0
$553$	4.57160e7i	0.0114956i
$554$	−4.98611e9	−1.24588
$555$	0	0
$556$	−2.58339e8	−0.0637424
$557$	− 6.01464e9i	− 1.47474i	−0.675487	−	0.737372i	$-0.736067\pi$
$557$	− 6.01464e9i	− 1.47474i	0.675487	−	0.737372i	$-0.263933\pi$
$558$	0	0
$559$	−1.48169e9	−0.358771
$560$	0	0
$561$	0	0
$562$	6.01342e8i	0.142904i
$563$	− 1.43964e9i	− 0.339996i	−0.985444	−	0.169998i	$-0.945624\pi$
$563$	− 1.43964e9i	− 0.339996i	0.985444	−	0.169998i	$-0.0543762\pi$
$564$	0	0
$565$	0	0
$566$	8.45231e9	1.95938
$567$	0	0
$568$	3.41018e8i	0.0780833i
$569$	−4.34990e9	−0.989889	−0.494944	−	0.868925i	$-0.664812\pi$
$569$	−4.34990e9	−0.989889	−0.494944	+	0.868925i	$0.664812\pi$
$570$	0	0
$571$	−3.91778e9	−0.880671	−0.440336	−	0.897833i	$-0.645141\pi$
$571$	−3.91778e9	−0.880671	−0.440336	+	0.897833i	$0.645141\pi$
$572$	− 2.67325e9i	− 0.597246i
$573$	0	0
$574$	3.13464e8	0.0691825
$575$	0	0
$576$	0	0
$577$	5.57477e9i	1.20812i	0.796938	+	0.604062i	$0.206452\pi$
$577$	5.57477e9i	1.20812i	−0.796938	+	0.604062i	$0.793548\pi$
$578$	2.21577e8i	0.0477284i
$579$	0	0
$580$	0	0
$581$	−6.57698e7	−0.0139127
$582$	0	0
$583$	1.76861e9i	0.369652i
$584$	−4.87526e8	−0.101287
$585$	0	0
$586$	8.48683e9	1.74222
$587$	3.96579e9i	0.809275i	0.914477	+	0.404638i	$0.132602\pi$
$587$	3.96579e9i	0.809275i	−0.914477	+	0.404638i	$0.867398\pi$
$588$	0	0
$589$	6.27628e9	1.26561
$590$	0	0
$591$	0	0
$592$	− 1.28705e9i	− 0.254959i
$593$	− 1.59443e9i	− 0.313988i	−0.987600	−	0.156994i	$-0.949820\pi$
$593$	− 1.59443e9i	− 0.313988i	0.987600	−	0.156994i	$-0.0501804\pi$
$594$	0	0
$595$	0	0
$596$	−5.97683e8	−0.115640
$597$	0	0
$598$	9.69660e9i	1.85424i
$599$	1.99503e9	0.379276	0.189638	−	0.981854i	$-0.439269\pi$
$599$	1.99503e9	0.379276	0.189638	+	0.981854i	$0.439269\pi$
$600$	0	0
$601$	5.69444e9	1.07002	0.535008	−	0.844847i	$-0.320308\pi$
$601$	5.69444e9	1.07002	0.535008	+	0.844847i	$0.320308\pi$
$602$	− 1.43381e8i	− 0.0267857i
$603$	0	0
$604$	2.53221e9	0.467597
$605$	0	0
$606$	0	0
$607$	− 1.61948e9i	− 0.293910i	−0.989143	−	0.146955i	$-0.953053\pi$
$607$	− 1.61948e9i	− 0.293910i	0.989143	−	0.146955i	$-0.0469472\pi$
$608$	− 1.28604e10i	− 2.32055i
$609$	0	0
$610$	0	0
$611$	1.71669e9	0.304472
$612$	0	0
$613$	− 8.66201e8i	− 0.151882i	−0.997112	−	0.0759412i	$-0.975804\pi$
$613$	− 8.66201e8i	− 0.151882i	0.997112	−	0.0759412i	$-0.0241961\pi$
$614$	9.61694e8	0.167667
$615$	0	0
$616$	−1.65494e7	−0.00285267
$617$	− 8.93265e9i	− 1.53102i	−0.643421	−	0.765512i	$-0.722486\pi$
$617$	− 8.93265e9i	− 1.53102i	0.643421	−	0.765512i	$-0.277514\pi$
$618$	0	0
$619$	3.70947e9	0.628630	0.314315	−	0.949319i	$-0.398225\pi$
$619$	3.70947e9	0.628630	0.314315	+	0.949319i	$0.398225\pi$
$620$	0	0
$621$	0	0
$622$	− 9.06625e9i	− 1.51064i
$623$	2.75667e8i	0.0456747i
$624$	0	0
$625$	0	0
$626$	−1.15408e10	−1.88030
$627$	0	0
$628$	− 3.14464e9i	− 0.506656i
$629$	1.48014e9	0.237151
$630$	0	0
$631$	1.11559e9	0.176767	0.0883834	−	0.996087i	$-0.471830\pi$
$631$	1.11559e9	0.176767	0.0883834	+	0.996087i	$0.471830\pi$
$632$	1.61039e8i	0.0253760i
$633$	0	0
$634$	2.64492e8	0.0412192
$635$	0	0
$636$	0	0
$637$	4.61039e9i	0.706723i
$638$	1.20214e10i	1.83267i
$639$	0	0
$640$	0	0
$641$	−4.96276e8	−0.0744252	−0.0372126	−	0.999307i	$-0.511848\pi$
$641$	−4.96276e8	−0.0744252	−0.0372126	+	0.999307i	$0.511848\pi$
$642$	0	0
$643$	− 8.93408e8i	− 0.132529i	−0.997802	−	0.0662646i	$-0.978892\pi$
$643$	− 8.93408e8i	− 0.132529i	0.997802	−	0.0662646i	$-0.0211081\pi$
$644$	−4.54619e8	−0.0670728
$645$	0	0
$646$	1.56822e10	2.28872
$647$	3.05033e9i	0.442774i	0.975186	+	0.221387i	$0.0710584\pi$
$647$	3.05033e9i	0.442774i	−0.975186	+	0.221387i	$0.928942\pi$
$648$	0	0
$649$	7.84590e9	1.12664
$650$	0	0
$651$	0	0
$652$	1.22489e10i	1.73074i
$653$	1.29545e10i	1.82064i	0.413908	+	0.910319i	$0.364164\pi$
$653$	1.29545e10i	1.82064i	−0.413908	+	0.910319i	$0.635836\pi$
$654$	0	0
$655$	0	0
$656$	1.00017e10	1.38328
$657$	0	0
$658$	1.66121e8i	0.0227318i
$659$	8.43740e9	1.14844	0.574222	−	0.818700i	$-0.305305\pi$
$659$	8.43740e9	1.14844	0.574222	+	0.818700i	$0.305305\pi$
$660$	0	0
$661$	7.97949e9	1.07466	0.537328	−	0.843373i	$-0.319434\pi$
$661$	7.97949e9	1.07466	0.537328	+	0.843373i	$0.319434\pi$
$662$	− 1.70822e10i	− 2.28845i
$663$	0	0
$664$	−2.31681e8	−0.0307116
$665$	0	0
$666$	0	0
$667$	− 2.11267e10i	− 2.75671i
$668$	− 7.39357e9i	− 0.959702i
$669$	0	0
$670$	0	0
$671$	−5.05328e9	−0.645720
$672$	0	0
$673$	− 4.53364e9i	− 0.573317i	−0.958033	−	0.286658i	$-0.907456\pi$
$673$	− 4.53364e9i	− 0.573317i	0.958033	−	0.286658i	$-0.0925444\pi$
$674$	1.16032e9	0.145971
$675$	0	0
$676$	−3.76765e9	−0.469091
$677$	1.06085e9i	0.131400i	0.997839	+	0.0656998i	$0.0209280\pi$
$677$	1.06085e9i	0.131400i	−0.997839	+	0.0656998i	$0.979072\pi$
$678$	0	0
$679$	−1.34009e8	−0.0164282
$680$	0	0
$681$	0	0
$682$	7.84084e9i	0.946493i
$683$	− 1.53865e9i	− 0.184785i	−0.995723	−	0.0923924i	$-0.970549\pi$
$683$	− 1.53865e9i	− 0.184785i	0.995723	−	0.0923924i	$-0.0294514\pi$
$684$	0	0
$685$	0	0
$686$	−8.92920e8	−0.105603
$687$	0	0
$688$	− 4.57485e9i	− 0.535571i
$689$	−2.50164e9	−0.291379
$690$	0	0
$691$	−5.67788e9	−0.654656	−0.327328	−	0.944911i	$-0.606148\pi$
$691$	−5.67788e9	−0.654656	−0.327328	+	0.944911i	$0.606148\pi$
$692$	− 3.30105e9i	− 0.378688i
$693$	0	0
$694$	4.72278e9	0.536339
$695$	0	0
$696$	0	0
$697$	1.15021e10i	1.28666i
$698$	6.31202e9i	0.702546i
$699$	0	0
$700$	0	0
$701$	−3.43469e9	−0.376595	−0.188298	−	0.982112i	$-0.560297\pi$
$701$	−3.43469e9	−0.376595	−0.188298	+	0.982112i	$0.560297\pi$
$702$	0	0
$703$	3.71722e9i	0.403528i
$704$	7.28433e9	0.786837
$705$	0	0
$706$	−9.34587e9	−0.999547
$707$	− 3.71066e8i	− 0.0394897i
$708$	0	0
$709$	1.79411e10	1.89055	0.945273	−	0.326281i	$-0.105796\pi$
$709$	1.79411e10	1.89055	0.945273	+	0.326281i	$0.105796\pi$
$710$	0	0
$711$	0	0
$712$	9.71065e8i	0.100825i
$713$	− 1.37797e10i	− 1.42372i
$714$	0	0
$715$	0	0
$716$	−4.87381e9	−0.496218
$717$	0	0
$718$	1.38344e10i	1.39484i
$719$	−5.25757e9	−0.527514	−0.263757	−	0.964589i	$-0.584962\pi$
$719$	−5.25757e9	−0.527514	−0.263757	+	0.964589i	$0.584962\pi$
$720$	0	0
$721$	6.64825e8	0.0660593
$722$	2.52989e10i	2.50162i
$723$	0	0
$724$	−1.16236e10	−1.13830
$725$	0	0
$726$	0	0
$727$	− 7.82419e9i	− 0.755212i	−0.925966	−	0.377606i	$-0.876747\pi$
$727$	− 7.82419e9i	− 0.755212i	0.925966	−	0.377606i	$-0.123253\pi$
$728$	− 2.34086e7i	− 0.00224862i
$729$	0	0
$730$	0	0
$731$	5.26116e9	0.498162
$732$	0	0
$733$	4.35449e9i	0.408388i	0.978930	+	0.204194i	$0.0654574\pi$
$733$	4.35449e9i	0.408388i	−0.978930	+	0.204194i	$0.934543\pi$
$734$	7.33278e9	0.684435
$735$	0	0
$736$	−2.82351e10	−2.61046
$737$	1.63672e10i	1.50605i
$738$	0	0
$739$	−4.46423e9	−0.406903	−0.203452	−	0.979085i	$-0.565216\pi$
$739$	−4.46423e9	−0.406903	−0.203452	+	0.979085i	$0.565216\pi$
$740$	0	0
$741$	0	0
$742$	− 2.42079e8i	− 0.0217542i
$743$	− 1.19060e9i	− 0.106489i	−0.998581	−	0.0532447i	$-0.983044\pi$
$743$	− 1.19060e9i	− 0.106489i	0.998581	−	0.0532447i	$-0.0169563\pi$
$744$	0	0
$745$	0	0
$746$	4.53971e9	0.400352
$747$	0	0
$748$	9.49210e9i	0.829291i
$749$	2.37719e8	0.0206718
$750$	0	0
$751$	−1.78291e9	−0.153599	−0.0767997	−	0.997047i	$-0.524470\pi$
$751$	−1.78291e9	−0.153599	−0.0767997	+	0.997047i	$0.524470\pi$
$752$	5.30042e9i	0.454515i
$753$	0	0
$754$	−1.70039e10	−1.44460
$755$	0	0
$756$	0	0
$757$	5.28551e9i	0.442844i	0.975178	+	0.221422i	$0.0710699\pi$
$757$	5.28551e9i	0.442844i	−0.975178	+	0.221422i	$0.928930\pi$
$758$	1.29460e10i	1.07968i
$759$	0	0
$760$	0	0
$761$	6.71171e9	0.552061	0.276030	−	0.961149i	$-0.410981\pi$
$761$	6.71171e9	0.552061	0.276030	+	0.961149i	$0.410981\pi$
$762$	0	0
$763$	− 8.24675e8i	− 0.0672120i
$764$	−1.71286e10	−1.38962
$765$	0	0
$766$	−1.01097e10	−0.812713
$767$	1.10978e10i	0.888079i
$768$	0	0
$769$	−1.25190e10	−0.992725	−0.496363	−	0.868115i	$-0.665331\pi$
$769$	−1.25190e10	−0.992725	−0.496363	+	0.868115i	$0.665331\pi$
$770$	0	0
$771$	0	0
$772$	− 1.53419e10i	− 1.20011i
$773$	− 1.89490e9i	− 0.147556i	−0.997275	−	0.0737781i	$-0.976494\pi$
$773$	− 1.89490e9i	− 0.147556i	0.997275	−	0.0737781i	$-0.0235056\pi$
$774$	0	0
$775$	0	0
$776$	−4.72063e8	−0.0362647
$777$	0	0
$778$	− 1.97729e10i	− 1.50536i
$779$	−2.88865e10	−2.18934
$780$	0	0
$781$	1.11450e10	0.837146
$782$	− 3.44304e10i	− 2.57465i
$783$	0	0
$784$	−1.42349e10	−1.05499
$785$	0	0
$786$	0	0
$787$	2.12095e10i	1.55103i	0.631331	+	0.775513i	$0.282509\pi$
$787$	2.12095e10i	1.55103i	−0.631331	+	0.775513i	$0.717491\pi$
$788$	1.48583e10i	1.08175i
$789$	0	0
$790$	0	0
$791$	−3.29338e8	−0.0236605
$792$	0	0
$793$	− 7.14769e9i	− 0.508990i
$794$	2.04593e10	1.45050
$795$	0	0
$796$	−8.98524e9	−0.631443
$797$	5.56395e9i	0.389295i	0.980873	+	0.194647i	$0.0623563\pi$
$797$	5.56395e9i	0.389295i	−0.980873	+	0.194647i	$0.937644\pi$
$798$	0	0
$799$	−6.09558e9	−0.422767
$800$	0	0
$801$	0	0
$802$	− 3.40854e10i	− 2.33323i
$803$	1.59331e10i	1.08592i
$804$	0	0
$805$	0	0
$806$	−1.10906e10	−0.746075
$807$	0	0
$808$	− 1.30712e9i	− 0.0871718i
$809$	6.34040e9	0.421014	0.210507	−	0.977592i	$-0.432488\pi$
$809$	6.34040e9	0.421014	0.210507	+	0.977592i	$0.432488\pi$
$810$	0	0
$811$	1.06272e10	0.699597	0.349798	−	0.936825i	$-0.386250\pi$
$811$	1.06272e10	0.699597	0.349798	+	0.936825i	$0.386250\pi$
$812$	− 7.97215e8i	− 0.0522552i
$813$	0	0
$814$	−4.64385e9	−0.301782
$815$	0	0
$816$	0	0
$817$	1.32129e10i	0.847657i
$818$	1.99253e10i	1.27282i
$819$	0	0
$820$	0	0
$821$	−2.17793e10	−1.37355	−0.686773	−	0.726872i	$-0.740973\pi$
$821$	−2.17793e10	−1.37355	−0.686773	+	0.726872i	$0.740973\pi$
$822$	0	0
$823$	2.93033e10i	1.83239i	0.400734	+	0.916194i	$0.368755\pi$
$823$	2.93033e10i	1.83239i	−0.400734	+	0.916194i	$0.631245\pi$
$824$	2.34192e9	0.145823
$825$	0	0
$826$	−1.07391e9	−0.0663036
$827$	1.35378e10i	0.832301i	0.909296	+	0.416150i	$0.136621\pi$
$827$	1.35378e10i	0.832301i	−0.909296	+	0.416150i	$0.863379\pi$
$828$	0	0
$829$	5.06434e9	0.308732	0.154366	−	0.988014i	$-0.450666\pi$
$829$	5.06434e9	0.308732	0.154366	+	0.988014i	$0.450666\pi$
$830$	0	0
$831$	0	0
$832$	1.03034e10i	0.620226i
$833$	− 1.63704e10i	− 0.981302i
$834$	0	0
$835$	0	0
$836$	−2.38385e10	−1.41110
$837$	0	0
$838$	− 2.90929e10i	− 1.70779i
$839$	−7.42054e9	−0.433779	−0.216890	−	0.976196i	$-0.569591\pi$
$839$	−7.42054e9	−0.433779	−0.216890	+	0.976196i	$0.569591\pi$
$840$	0	0
$841$	1.97977e10	1.14770
$842$	3.53808e10i	2.04256i
$843$	0	0
$844$	−1.49756e10	−0.857404
$845$	0	0
$846$	0	0
$847$	− 1.30051e8i	− 0.00735399i
$848$	− 7.72401e9i	− 0.434968i
$849$	0	0
$850$	0	0
$851$	8.16119e9	0.453942
$852$	0	0
$853$	− 5.19233e9i	− 0.286444i	−0.989691	−	0.143222i	$-0.954254\pi$
$853$	− 5.19233e9i	− 0.286444i	0.989691	−	0.143222i	$-0.0457464\pi$
$854$	6.91668e8	0.0380010
$855$	0	0
$856$	8.37391e8	0.0456321
$857$	− 3.38094e10i	− 1.83486i	−0.397892	−	0.917432i	$-0.630258\pi$
$857$	− 3.38094e10i	− 1.83486i	0.397892	−	0.917432i	$-0.369742\pi$
$858$	0	0
$859$	−1.19130e10	−0.641277	−0.320639	−	0.947202i	$-0.603897\pi$
$859$	−1.19130e10	−0.641277	−0.320639	+	0.947202i	$0.603897\pi$
$860$	0	0
$861$	0	0
$862$	2.58398e10i	1.37408i
$863$	1.42156e10i	0.752882i	0.926441	+	0.376441i	$0.122852\pi$
$863$	1.42156e10i	0.752882i	−0.926441	+	0.376441i	$0.877148\pi$
$864$	0	0
$865$	0	0
$866$	−3.99461e10	−2.09007
$867$	0	0
$868$	− 5.19975e8i	− 0.0269876i
$869$	5.26302e9	0.272061
$870$	0	0
$871$	−2.31508e10	−1.18714
$872$	− 2.90501e9i	− 0.148368i
$873$	0	0
$874$	8.64686e10	4.38095
$875$	0	0
$876$	0	0
$877$	2.34323e9i	0.117305i	0.998278	+	0.0586525i	$0.0186804\pi$
$877$	2.34323e9i	0.117305i	−0.998278	+	0.0586525i	$0.981320\pi$
$878$	− 2.75043e9i	− 0.137142i
$879$	0	0
$880$	0	0
$881$	−1.08237e9	−0.0533288	−0.0266644	−	0.999644i	$-0.508489\pi$
$881$	−1.08237e9	−0.0533288	−0.0266644	+	0.999644i	$0.508489\pi$
$882$	0	0
$883$	− 1.55829e10i	− 0.761705i	−0.924636	−	0.380852i	$-0.875631\pi$
$883$	− 1.55829e10i	− 0.761705i	0.924636	−	0.380852i	$-0.124369\pi$
$884$	−1.34262e10	−0.653690
$885$	0	0
$886$	−1.78886e10	−0.864088
$887$	− 1.48785e10i	− 0.715858i	−0.933749	−	0.357929i	$-0.883483\pi$
$887$	− 1.48785e10i	− 0.715858i	0.933749	−	0.357929i	$-0.116517\pi$
$888$	0	0
$889$	−4.11116e8	−0.0196249
$890$	0	0
$891$	0	0
$892$	− 2.61683e10i	− 1.23452i
$893$	− 1.53084e10i	− 0.719368i
$894$	0	0
$895$	0	0
$896$	1.36572e8	0.00634286
$897$	0	0
$898$	3.59465e10i	1.65649i
$899$	2.41639e10	1.10919
$900$	0	0
$901$	8.88276e9	0.404586
$902$	− 3.60873e10i	− 1.63731i
$903$	0	0
$904$	−1.16013e9	−0.0522296
$905$	0	0
$906$	0	0
$907$	3.01866e10i	1.34335i	0.740848	+	0.671673i	$0.234424\pi$
$907$	3.01866e10i	1.34335i	−0.740848	+	0.671673i	$0.765576\pi$
$908$	− 1.67247e10i	− 0.741408i
$909$	0	0
$910$	0	0
$911$	3.12329e10	1.36867	0.684334	−	0.729168i	$-0.260093\pi$
$911$	3.12329e10	1.36867	0.684334	+	0.729168i	$0.260093\pi$
$912$	0	0
$913$	7.57170e9i	0.329265i
$914$	4.62551e10	2.00377
$915$	0	0
$916$	1.28132e10	0.550836
$917$	5.51076e7i	0.00236004i
$918$	0	0
$919$	1.27737e10	0.542891	0.271446	−	0.962454i	$-0.412498\pi$
$919$	1.27737e10	0.542891	0.271446	+	0.962454i	$0.412498\pi$
$920$	0	0
$921$	0	0
$922$	9.97330e9i	0.419064i
$923$	1.57642e10i	0.659882i
$924$	0	0
$925$	0	0
$926$	2.22592e10	0.921237
$927$	0	0
$928$	− 4.95129e10i	− 2.03376i
$929$	−3.36628e10	−1.37751	−0.688755	−	0.724994i	$-0.741843\pi$
$929$	−3.36628e10	−1.37751	−0.688755	+	0.724994i	$0.741843\pi$
$930$	0	0
$931$	4.11127e10	1.66975
$932$	− 4.27866e9i	− 0.173122i
$933$	0	0
$934$	5.94917e9	0.238915
$935$	0	0
$936$	0	0
$937$	− 3.94639e9i	− 0.156715i	−0.996925	−	0.0783577i	$-0.975032\pi$
$937$	− 3.94639e9i	− 0.156715i	0.996925	−	0.0783577i	$-0.0249676\pi$
$938$	− 2.24026e9i	− 0.0886317i
$939$	0	0
$940$	0	0
$941$	1.71604e10	0.671373	0.335686	−	0.941974i	$-0.391032\pi$
$941$	1.71604e10	0.671373	0.335686	+	0.941974i	$0.391032\pi$
$942$	0	0
$943$	6.34206e10i	2.46286i
$944$	−3.42652e10	−1.32572
$945$	0	0
$946$	−1.65066e10	−0.633926
$947$	2.96234e10i	1.13347i	0.823900	+	0.566735i	$0.191794\pi$
$947$	2.96234e10i	1.13347i	−0.823900	+	0.566735i	$0.808206\pi$
$948$	0	0
$949$	−2.25368e10	−0.855976
$950$	0	0
$951$	0	0
$952$	8.31186e7i	0.00312226i
$953$	2.70351e9i	0.101182i	0.998719	+	0.0505910i	$0.0161105\pi$
$953$	2.70351e9i	0.101182i	−0.998719	+	0.0505910i	$0.983889\pi$
$954$	0	0
$955$	0	0
$956$	9.77821e9	0.361957
$957$	0	0
$958$	6.78446e10i	2.49308i
$959$	−9.91642e8	−0.0363069
$960$	0	0
$961$	−1.17520e10	−0.427148
$962$	− 6.56856e9i	− 0.237880i
$963$	0	0
$964$	−2.15074e10	−0.773248
$965$	0	0
$966$	0	0
$967$	− 3.15584e10i	− 1.12233i	−0.827703	−	0.561167i	$-0.810353\pi$
$967$	− 3.15584e10i	− 1.12233i	0.827703	−	0.561167i	$-0.189647\pi$
$968$	− 4.58120e8i	− 0.0162336i
$969$	0	0
$970$	0	0
$971$	−2.06410e10	−0.723540	−0.361770	−	0.932267i	$-0.617827\pi$
$971$	−2.06410e10	−0.723540	−0.361770	+	0.932267i	$0.617827\pi$
$972$	0	0
$973$	7.39315e7i	0.00257297i
$974$	1.31869e10	0.457285
$975$	0	0
$976$	2.20690e10	0.759817
$977$	3.75871e10i	1.28946i	0.764410	+	0.644731i	$0.223030\pi$
$977$	3.75871e10i	1.28946i	−0.764410	+	0.644731i	$0.776970\pi$
$978$	0	0
$979$	3.17359e10	1.08097
$980$	0	0
$981$	0	0
$982$	− 5.20664e10i	− 1.75456i
$983$	6.64465e9i	0.223118i	0.993758	+	0.111559i	$0.0355844\pi$
$983$	6.64465e9i	0.223118i	−0.993758	+	0.111559i	$0.964416\pi$
$984$	0	0
$985$	0	0
$986$	6.03769e10	2.00586
$987$	0	0
$988$	− 3.37187e10i	− 1.11230i
$989$	2.90090e10	0.953556
$990$	0	0
$991$	1.79433e10	0.585657	0.292829	−	0.956165i	$-0.405403\pi$
$991$	1.79433e10	0.585657	0.292829	+	0.956165i	$0.405403\pi$
$992$	− 3.22943e10i	− 1.05035i
$993$	0	0
$994$	−1.52547e9	−0.0492666
$995$	0	0
$996$	0	0
$997$	9.85820e8i	0.0315039i	0.999876	+	0.0157520i	$0.00501421\pi$
$997$	9.85820e8i	0.0315039i	−0.999876	+	0.0157520i	$0.994986\pi$
$998$	− 5.39887e10i	− 1.71928i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.b.n.199.4		4
3.2	odd	2		75.8.b.d.49.1		4
5.2	odd	4		45.8.a.i.1.1		2
5.3	odd	4		225.8.a.t.1.2		2
5.4	even	2	inner	225.8.b.n.199.1		4
15.2	even	4		15.8.a.c.1.2	✓	2
15.8	even	4		75.8.a.e.1.1		2
15.14	odd	2		75.8.b.d.49.4		4
60.47	odd	4		240.8.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.a.c.1.2	✓	2	15.2	even	4
45.8.a.i.1.1		2	5.2	odd	4
75.8.a.e.1.1		2	15.8	even	4
75.8.b.d.49.1		4	3.2	odd	2
75.8.b.d.49.4		4	15.14	odd	2
225.8.a.t.1.2		2	5.3	odd	4
225.8.b.n.199.1		4	5.4	even	2	inner
225.8.b.n.199.4		4	1.1	even	1	trivial
240.8.a.p.1.2		2	60.47	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.7577i q^{2} -120.304 q^{4} +34.4284i q^{7} +121.278i q^{8} +3963.55 q^{11} +5606.30i q^{13} -542.511 q^{14} -17309.9 q^{16} -19906.7i q^{17} +49993.7 q^{19} +62456.2i q^{22} -109762. i q^{23} -88342.2 q^{26} -4141.86i q^{28} +192477. q^{29} +125541. q^{31} -257240. i q^{32} +313683. q^{34} +74353.6i q^{37} +787784. i q^{38} -577802. q^{41} +264291. i q^{43} -476829. q^{44} +1.72959e6 q^{46} -306207. i q^{47} +822358. q^{49} -674458. i q^{52} +446219. i q^{53} -4175.41 q^{56} +3.03299e6i q^{58} +1.97951e6 q^{59} -1.27494e6 q^{61} +1.97824e6i q^{62} +1.83783e6 q^{64} +4.12943e6i q^{67} +2.39485e6i q^{68} +2.81187e6 q^{71} +4.01991e6i q^{73} -1.17164e6 q^{74} -6.01442e6 q^{76} +136459. i q^{77} +1.32785e6 q^{79} -9.10480e6i q^{82} +1.91033e6i q^{83} -4.16460e6 q^{86} +480691. i q^{88} +8.00695e6 q^{89} -193016. q^{91} +1.32047e7i q^{92} +4.82511e6 q^{94} +3.89241e6i q^{97} +1.29584e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 138 q^{4} - 6896 q^{11} - 24528 q^{14} - 48990 q^{16} + 99168 q^{19} - 432308 q^{26} + 363544 q^{29} + 608464 q^{31} + 879844 q^{34} - 1262344 q^{41} - 1714136 q^{44} + 3772944 q^{46} - 290980 q^{49}+ \cdots + 6825472 q^{94}+O(q^{100}) \) 4 * q - 138 * q^4 - 6896 * q^11 - 24528 * q^14 - 48990 * q^16 + 99168 * q^19 - 432308 * q^26 + 363544 * q^29 + 608464 * q^31 + 879844 * q^34 - 1262344 * q^41 - 1714136 * q^44 + 3772944 * q^46 - 290980 * q^49 - 4211760 * q^56 + 574448 * q^59 - 5028360 * q^61 - 582082 * q^64 + 7497632 * q^71 + 5152612 * q^74 - 12070888 * q^76 + 9255440 * q^79 - 6764216 * q^86 + 33032712 * q^89 - 39452896 * q^91 + 6825472 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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