LMFDB - Embedded newform 225.8.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,8,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.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:	$70.2866307339$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{115}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 115 $ x^2 - 115
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.7238$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.4476 q^{2} +332.000 q^{4} +1485.00 q^{7} -4375.31 q^{8} +O(q^{10})$	$q-21.4476 q^{2} +332.000 q^{4} +1485.00 q^{7} -4375.31 q^{8} -4932.95 q^{11} +4415.00 q^{13} -31849.7 q^{14} +51344.0 q^{16} +25780.0 q^{17} -38099.0 q^{19} +105800. q^{22} +36160.7 q^{23} -94691.2 q^{26} +493020. q^{28} +243001. q^{29} +206583. q^{31} -541166. q^{32} -552920. q^{34} -75470.0 q^{37} +817133. q^{38} -756886. q^{41} +450665. q^{43} -1.63774e6 q^{44} -775560. q^{46} +100289. q^{47} +1.38168e6 q^{49} +1.46578e6 q^{52} -301082. q^{53} -6.49734e6 q^{56} -5.21180e6 q^{58} -1.19206e6 q^{59} -257033. q^{61} -4.43071e6 q^{62} +5.03469e6 q^{64} -3.24841e6 q^{67} +8.55897e6 q^{68} +2.59151e6 q^{71} -1.08703e6 q^{73} +1.61865e6 q^{74} -1.26489e7 q^{76} -7.32543e6 q^{77} +540084. q^{79} +1.62334e7 q^{82} +2.58632e6 q^{83} -9.66569e6 q^{86} +2.15832e7 q^{88} +386057. q^{89} +6.55628e6 q^{91} +1.20053e7 q^{92} -2.15096e6 q^{94} -3.74976e6 q^{97} -2.96338e7 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 664 q^{4} + 2970 q^{7}+O(q^{10}) $ 2 * q + 664 * q^4 + 2970 * q^7	$ 2 q + 664 q^{4} + 2970 q^{7} + 8830 q^{13} + 102688 q^{16} - 76198 q^{19} + 211600 q^{22} + 986040 q^{28} + 413166 q^{31} - 1105840 q^{34} - 150940 q^{37} + 901330 q^{43} - 1551120 q^{46} + 2763364 q^{49} + 2931560 q^{52} - 10423600 q^{58} - 514066 q^{61} + 10069376 q^{64} - 6496830 q^{67} - 2174060 q^{73} - 25297736 q^{76} + 1080168 q^{79} + 32466800 q^{82} + 43166400 q^{88} + 13112550 q^{91} - 4301920 q^{94} - 7499530 q^{97}+O(q^{100}) $ 2 * q + 664 * q^4 + 2970 * q^7 + 8830 * q^13 + 102688 * q^16 - 76198 * q^19 + 211600 * q^22 + 986040 * q^28 + 413166 * q^31 - 1105840 * q^34 - 150940 * q^37 + 901330 * q^43 - 1551120 * q^46 + 2763364 * q^49 + 2931560 * q^52 - 10423600 * q^58 - 514066 * q^61 + 10069376 * q^64 - 6496830 * q^67 - 2174060 * q^73 - 25297736 * q^76 + 1080168 * q^79 + 32466800 * q^82 + 43166400 * q^88 + 13112550 * q^91 - 4301920 * q^94 - 7499530 * q^97

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 $

$2$	−21.4476	−1.89572	−0.947859	−	0.318689i	$-0.896758\pi$
$2$	−21.4476	−1.89572	−0.947859	+	0.318689i	$0.896758\pi$
$3$	0	0
$3$	0	0
$4$	332.000	2.59375
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1485.00	1.63638	0.818188	−	0.574950i	$-0.194979\pi$
$7$	1485.00	1.63638	0.818188	+	0.574950i	$0.194979\pi$
$8$	−4375.31	−3.02130
$9$	0	0
$10$	0	0
$11$	−4932.95	−1.11746	−0.558730	−	0.829349i	$-0.688711\pi$
$11$	−4932.95	−1.11746	−0.558730	+	0.829349i	$0.688711\pi$
$12$	0	0
$13$	4415.00	0.557351	0.278676	−	0.960385i	$-0.410105\pi$
$13$	4415.00	0.557351	0.278676	+	0.960385i	$0.410105\pi$
$14$	−31849.7	−3.10211
$15$	0	0
$16$	51344.0	3.13379
$17$	25780.0	1.27266	0.636330	−	0.771417i	$-0.280452\pi$
$17$	25780.0	1.27266	0.636330	+	0.771417i	$0.280452\pi$
$18$	0	0
$19$	−38099.0	−1.27431	−0.637156	−	0.770735i	$-0.719889\pi$
$19$	−38099.0	−1.27431	−0.637156	+	0.770735i	$0.719889\pi$
$20$	0	0
$21$	0	0
$22$	105800.	2.11839
$23$	36160.7	0.619711	0.309855	−	0.950784i	$-0.399719\pi$
$23$	36160.7	0.619711	0.309855	+	0.950784i	$0.399719\pi$
$24$	0	0
$25$	0	0
$26$	−94691.2	−1.05658
$27$	0	0
$28$	493020.	4.24435
$29$	243001.	1.85019	0.925094	−	0.379738i	$-0.123986\pi$
$29$	243001.	1.85019	0.925094	+	0.379738i	$0.123986\pi$
$30$	0	0
$31$	206583.	1.24546	0.622728	−	0.782438i	$-0.286024\pi$
$31$	206583.	1.24546	0.622728	+	0.782438i	$0.286024\pi$
$32$	−541166.	−2.91948
$33$	0	0
$34$	−552920.	−2.41260
$35$	0	0
$36$	0	0
$37$	−75470.0	−0.244945	−0.122472	−	0.992472i	$-0.539082\pi$
$37$	−75470.0	−0.244945	−0.122472	+	0.992472i	$0.539082\pi$
$38$	817133.	2.41574
$39$	0	0
$40$	0	0
$41$	−756886.	−1.71509	−0.857545	−	0.514408i	$-0.828012\pi$
$41$	−756886.	−1.71509	−0.857545	+	0.514408i	$0.828012\pi$
$42$	0	0
$43$	450665.	0.864399	0.432199	−	0.901778i	$-0.357738\pi$
$43$	450665.	0.864399	0.432199	+	0.901778i	$0.357738\pi$
$44$	−1.63774e6	−2.89841
$45$	0	0
$46$	−775560.	−1.17480
$47$	100289.	0.140900	0.0704500	−	0.997515i	$-0.477556\pi$
$47$	100289.	0.140900	0.0704500	+	0.997515i	$0.477556\pi$
$48$	0	0
$49$	1.38168e6	1.67773
$50$	0	0
$51$	0	0
$52$	1.46578e6	1.44563
$53$	−301082.	−0.277791	−0.138896	−	0.990307i	$-0.544355\pi$
$53$	−301082.	−0.277791	−0.138896	+	0.990307i	$0.544355\pi$
$54$	0	0
$55$	0	0
$56$	−6.49734e6	−4.94399
$57$	0	0
$58$	−5.21180e6	−3.50744
$59$	−1.19206e6	−0.755641	−0.377820	−	0.925879i	$-0.623326\pi$
$59$	−1.19206e6	−0.755641	−0.377820	+	0.925879i	$0.623326\pi$
$60$	0	0
$61$	−257033.	−0.144989	−0.0724944	−	0.997369i	$-0.523096\pi$
$61$	−257033.	−0.144989	−0.0724944	+	0.997369i	$0.523096\pi$
$62$	−4.43071e6	−2.36104
$63$	0	0
$64$	5.03469e6	2.40073
$65$	0	0
$66$	0	0
$67$	−3.24841e6	−1.31950	−0.659750	−	0.751485i	$-0.729338\pi$
$67$	−3.24841e6	−1.31950	−0.659750	+	0.751485i	$0.729338\pi$
$68$	8.55897e6	3.30096
$69$	0	0
$70$	0	0
$71$	2.59151e6	0.859309	0.429655	−	0.902993i	$-0.358635\pi$
$71$	2.59151e6	0.859309	0.429655	+	0.902993i	$0.358635\pi$
$72$	0	0
$73$	−1.08703e6	−0.327048	−0.163524	−	0.986539i	$-0.552286\pi$
$73$	−1.08703e6	−0.327048	−0.163524	+	0.986539i	$0.552286\pi$
$74$	1.61865e6	0.464347
$75$	0	0
$76$	−1.26489e7	−3.30525
$77$	−7.32543e6	−1.82859
$78$	0	0
$79$	540084.	0.123244	0.0616221	−	0.998100i	$-0.480373\pi$
$79$	540084.	0.123244	0.0616221	+	0.998100i	$0.480373\pi$
$80$	0	0
$81$	0	0
$82$	1.62334e7	3.25133
$83$	2.58632e6	0.496489	0.248245	−	0.968697i	$-0.420146\pi$
$83$	2.58632e6	0.496489	0.248245	+	0.968697i	$0.420146\pi$
$84$	0	0
$85$	0	0
$86$	−9.66569e6	−1.63866
$87$	0	0
$88$	2.15832e7	3.37619
$89$	386057.	0.0580479	0.0290239	−	0.999579i	$-0.490760\pi$
$89$	386057.	0.0580479	0.0290239	+	0.999579i	$0.490760\pi$
$90$	0	0
$91$	6.55628e6	0.912037
$92$	1.20053e7	1.60737
$93$	0	0
$94$	−2.15096e6	−0.267107
$95$	0	0
$96$	0	0
$97$	−3.74976e6	−0.417160	−0.208580	−	0.978005i	$-0.566884\pi$
$97$	−3.74976e6	−0.417160	−0.208580	+	0.978005i	$0.566884\pi$
$98$	−2.96338e7	−3.18050
$99$	0	0
$100$	0	0
$101$	584876.	0.0564858	0.0282429	−	0.999601i	$-0.491009\pi$
$101$	584876.	0.0564858	0.0282429	+	0.999601i	$0.491009\pi$
$102$	0	0
$103$	1.47659e7	1.33146	0.665731	−	0.746192i	$-0.268120\pi$
$103$	1.47659e7	1.33146	0.665731	+	0.746192i	$0.268120\pi$
$104$	−1.93170e7	−1.68393
$105$	0	0
$106$	6.45748e6	0.526614
$107$	2.08951e7	1.64892	0.824462	−	0.565917i	$-0.191478\pi$
$107$	2.08951e7	1.64892	0.824462	+	0.565917i	$0.191478\pi$
$108$	0	0
$109$	−6.32319e6	−0.467674	−0.233837	−	0.972276i	$-0.575128\pi$
$109$	−6.32319e6	−0.467674	−0.233837	+	0.972276i	$0.575128\pi$
$110$	0	0
$111$	0	0
$112$	7.62458e7	5.12806
$113$	1.63952e7	1.06892	0.534458	−	0.845195i	$-0.320516\pi$
$113$	1.63952e7	1.06892	0.534458	+	0.845195i	$0.320516\pi$
$114$	0	0
$115$	0	0
$116$	8.06765e7	4.79892
$117$	0	0
$118$	2.55668e7	1.43248
$119$	3.82833e7	2.08255
$120$	0	0
$121$	4.84683e6	0.248719
$122$	5.51274e6	0.274858
$123$	0	0
$124$	6.85856e7	3.23040
$125$	0	0
$126$	0	0
$127$	5.54354e6	0.240145	0.120073	−	0.992765i	$-0.461687\pi$
$127$	5.54354e6	0.240145	0.120073	+	0.992765i	$0.461687\pi$
$128$	−3.87128e7	−1.63162
$129$	0	0
$130$	0	0
$131$	4.89070e6	0.190073	0.0950367	−	0.995474i	$-0.469703\pi$
$131$	4.89070e6	0.190073	0.0950367	+	0.995474i	$0.469703\pi$
$132$	0	0
$133$	−5.65770e7	−2.08526
$134$	6.96707e7	2.50140
$135$	0	0
$136$	−1.12796e8	−3.84509
$137$	−3.19411e7	−1.06127	−0.530637	−	0.847599i	$-0.678047\pi$
$137$	−3.19411e7	−1.06127	−0.530637	+	0.847599i	$0.678047\pi$
$138$	0	0
$139$	−3.09985e7	−0.979013	−0.489506	−	0.872000i	$-0.662823\pi$
$139$	−3.09985e7	−0.979013	−0.489506	+	0.872000i	$0.662823\pi$
$140$	0	0
$141$	0	0
$142$	−5.55818e7	−1.62901
$143$	−2.17790e7	−0.622818
$144$	0	0
$145$	0	0
$146$	2.33142e7	0.619991
$147$	0	0
$148$	−2.50560e7	−0.635326
$149$	1.21007e6	0.0299682	0.0149841	−	0.999888i	$-0.495230\pi$
$149$	1.21007e6	0.0299682	0.0149841	+	0.999888i	$0.495230\pi$
$150$	0	0
$151$	−4.32972e7	−1.02339	−0.511695	−	0.859167i	$-0.670982\pi$
$151$	−4.32972e7	−1.02339	−0.511695	+	0.859167i	$0.670982\pi$
$152$	1.66695e8	3.85008
$153$	0	0
$154$	1.57113e8	3.46649
$155$	0	0
$156$	0	0
$157$	2.90235e7	0.598551	0.299276	−	0.954167i	$-0.403255\pi$
$157$	2.90235e7	0.598551	0.299276	+	0.954167i	$0.403255\pi$
$158$	−1.15835e7	−0.233636
$159$	0	0
$160$	0	0
$161$	5.36986e7	1.01408
$162$	0	0
$163$	9.00528e6	0.162870	0.0814349	−	0.996679i	$-0.474050\pi$
$163$	9.00528e6	0.162870	0.0814349	+	0.996679i	$0.474050\pi$
$164$	−2.51286e8	−4.44852
$165$	0	0
$166$	−5.54705e7	−0.941204
$167$	−8.92498e7	−1.48286	−0.741429	−	0.671031i	$-0.765852\pi$
$167$	−8.92498e7	−1.48286	−0.741429	+	0.671031i	$0.765852\pi$
$168$	0	0
$169$	−4.32563e7	−0.689360
$170$	0	0
$171$	0	0
$172$	1.49621e8	2.24203
$173$	3.01015e7	0.442005	0.221002	−	0.975273i	$-0.429067\pi$
$173$	3.01015e7	0.442005	0.221002	+	0.975273i	$0.429067\pi$
$174$	0	0
$175$	0	0
$176$	−2.53277e8	−3.50189
$177$	0	0
$178$	−8.28000e6	−0.110042
$179$	1.30890e8	1.70577	0.852883	−	0.522102i	$-0.174852\pi$
$179$	1.30890e8	1.70577	0.852883	+	0.522102i	$0.174852\pi$
$180$	0	0
$181$	1.13498e8	1.42270	0.711350	−	0.702838i	$-0.248084\pi$
$181$	1.13498e8	1.42270	0.711350	+	0.702838i	$0.248084\pi$
$182$	−1.40616e8	−1.72897
$183$	0	0
$184$	−1.58214e8	−1.87233
$185$	0	0
$186$	0	0
$187$	−1.27172e8	−1.42215
$188$	3.32960e7	0.365459
$189$	0	0
$190$	0	0
$191$	1.17308e7	0.121818	0.0609088	−	0.998143i	$-0.480600\pi$
$191$	1.17308e7	0.121818	0.0609088	+	0.998143i	$0.480600\pi$
$192$	0	0
$193$	1.25584e8	1.25743	0.628713	−	0.777637i	$-0.283582\pi$
$193$	1.25584e8	1.25743	0.628713	+	0.777637i	$0.283582\pi$
$194$	8.04235e7	0.790819
$195$	0	0
$196$	4.58718e8	4.35161
$197$	−1.73136e8	−1.61345	−0.806725	−	0.590927i	$-0.798762\pi$
$197$	−1.73136e8	−1.61345	−0.806725	+	0.590927i	$0.798762\pi$
$198$	0	0
$199$	4.73931e7	0.426314	0.213157	−	0.977018i	$-0.431625\pi$
$199$	4.73931e7	0.426314	0.213157	+	0.977018i	$0.431625\pi$
$200$	0	0
$201$	0	0
$202$	−1.25442e7	−0.107081
$203$	3.60857e8	3.02760
$204$	0	0
$205$	0	0
$206$	−3.16692e8	−2.52408
$207$	0	0
$208$	2.26684e8	1.74662
$209$	1.87940e8	1.42399
$210$	0	0
$211$	−1.28295e8	−0.940204	−0.470102	−	0.882612i	$-0.655783\pi$
$211$	−1.28295e8	−0.940204	−0.470102	+	0.882612i	$0.655783\pi$
$212$	−9.99591e7	−0.720521
$213$	0	0
$214$	−4.48149e8	−3.12590
$215$	0	0
$216$	0	0
$217$	3.06776e8	2.03804
$218$	1.35617e8	0.886579
$219$	0	0
$220$	0	0
$221$	1.13819e8	0.709318
$222$	0	0
$223$	1.76978e8	1.06869	0.534345	−	0.845267i	$-0.320558\pi$
$223$	1.76978e8	1.06869	0.534345	+	0.845267i	$0.320558\pi$
$224$	−8.03632e8	−4.77737
$225$	0	0
$226$	−3.51639e8	−2.02636
$227$	1.77436e8	1.00682	0.503408	−	0.864049i	$-0.332079\pi$
$227$	1.77436e8	1.00682	0.503408	+	0.864049i	$0.332079\pi$
$228$	0	0
$229$	1.78476e8	0.982102	0.491051	−	0.871131i	$-0.336613\pi$
$229$	1.78476e8	0.982102	0.491051	+	0.871131i	$0.336613\pi$
$230$	0	0
$231$	0	0
$232$	−1.06321e9	−5.58998
$233$	−3.32762e7	−0.172341	−0.0861703	−	0.996280i	$-0.527463\pi$
$233$	−3.32762e7	−0.172341	−0.0861703	+	0.996280i	$0.527463\pi$
$234$	0	0
$235$	0	0
$236$	−3.95763e8	−1.95994
$237$	0	0
$238$	−8.21086e8	−3.94793
$239$	2.18152e8	1.03363	0.516817	−	0.856096i	$-0.327117\pi$
$239$	2.18152e8	1.03363	0.516817	+	0.856096i	$0.327117\pi$
$240$	0	0
$241$	3.30190e8	1.51951	0.759756	−	0.650208i	$-0.225318\pi$
$241$	3.30190e8	1.51951	0.759756	+	0.650208i	$0.225318\pi$
$242$	−1.03953e8	−0.471501
$243$	0	0
$244$	−8.53350e7	−0.376065
$245$	0	0
$246$	0	0
$247$	−1.68207e8	−0.710240
$248$	−9.03865e8	−3.76290
$249$	0	0
$250$	0	0
$251$	−1.66829e8	−0.665908	−0.332954	−	0.942943i	$-0.608045\pi$
$251$	−1.66829e8	−0.665908	−0.332954	+	0.942943i	$0.608045\pi$
$252$	0	0
$253$	−1.78379e8	−0.692503
$254$	−1.18896e8	−0.455248
$255$	0	0
$256$	1.85856e8	0.692369
$257$	−1.45032e8	−0.532964	−0.266482	−	0.963840i	$-0.585861\pi$
$257$	−1.45032e8	−0.532964	−0.266482	+	0.963840i	$0.585861\pi$
$258$	0	0
$259$	−1.12073e8	−0.400822
$260$	0	0
$261$	0	0
$262$	−1.04894e8	−0.360326
$263$	1.49106e8	0.505419	0.252709	−	0.967542i	$-0.418678\pi$
$263$	1.49106e8	0.505419	0.252709	+	0.967542i	$0.418678\pi$
$264$	0	0
$265$	0	0
$266$	1.21344e9	3.95306
$267$	0	0
$268$	−1.07847e9	−3.42246
$269$	5.69895e8	1.78510	0.892549	−	0.450951i	$-0.148915\pi$
$269$	5.69895e8	1.78510	0.892549	+	0.450951i	$0.148915\pi$
$270$	0	0
$271$	−3.87621e8	−1.18308	−0.591542	−	0.806275i	$-0.701480\pi$
$271$	−3.87621e8	−1.18308	−0.591542	+	0.806275i	$0.701480\pi$
$272$	1.32365e9	3.98825
$273$	0	0
$274$	6.85060e8	2.01188
$275$	0	0
$276$	0	0
$277$	−1.48585e8	−0.420045	−0.210023	−	0.977697i	$-0.567354\pi$
$277$	−1.48585e8	−0.420045	−0.210023	+	0.977697i	$0.567354\pi$
$278$	6.64843e8	1.85593
$279$	0	0
$280$	0	0
$281$	5.25272e8	1.41225	0.706126	−	0.708086i	$-0.250441\pi$
$281$	5.25272e8	1.41225	0.706126	+	0.708086i	$0.250441\pi$
$282$	0	0
$283$	5.27939e8	1.38462	0.692312	−	0.721599i	$-0.256592\pi$
$283$	5.27939e8	1.38462	0.692312	+	0.721599i	$0.256592\pi$
$284$	8.60383e8	2.22883
$285$	0	0
$286$	4.67107e8	1.18069
$287$	−1.12398e9	−2.80653
$288$	0	0
$289$	2.54271e8	0.619662
$290$	0	0
$291$	0	0
$292$	−3.60894e8	−0.848281
$293$	3.97173e8	0.922451	0.461226	−	0.887283i	$-0.347410\pi$
$293$	3.97173e8	0.922451	0.461226	+	0.887283i	$0.347410\pi$
$294$	0	0
$295$	0	0
$296$	3.30205e8	0.740052
$297$	0	0
$298$	−2.59532e7	−0.0568112
$299$	1.59649e8	0.345397
$300$	0	0
$301$	6.69238e8	1.41448
$302$	9.28622e8	1.94006
$303$	0	0
$304$	−1.95616e9	−3.99343
$305$	0	0
$306$	0	0
$307$	6.48511e8	1.27918	0.639592	−	0.768715i	$-0.279103\pi$
$307$	6.48511e8	1.27918	0.639592	+	0.768715i	$0.279103\pi$
$308$	−2.43204e9	−4.74290
$309$	0	0
$310$	0	0
$311$	8.54887e8	1.61156	0.805782	−	0.592213i	$-0.201745\pi$
$311$	8.54887e8	1.61156	0.805782	+	0.592213i	$0.201745\pi$
$312$	0	0
$313$	1.96455e8	0.362125	0.181062	−	0.983472i	$-0.442046\pi$
$313$	1.96455e8	0.362125	0.181062	+	0.983472i	$0.442046\pi$
$314$	−6.22485e8	−1.13468
$315$	0	0
$316$	1.79308e8	0.319665
$317$	1.31061e8	0.231082	0.115541	−	0.993303i	$-0.463140\pi$
$317$	1.31061e8	0.231082	0.115541	+	0.993303i	$0.463140\pi$
$318$	0	0
$319$	−1.19871e9	−2.06751
$320$	0	0
$321$	0	0
$322$	−1.15171e9	−1.92241
$323$	−9.82193e8	−1.62177
$324$	0	0
$325$	0	0
$326$	−1.93142e8	−0.308755
$327$	0	0
$328$	3.31161e9	5.18181
$329$	1.48929e8	0.230565
$330$	0	0
$331$	−3.77233e8	−0.571757	−0.285878	−	0.958266i	$-0.592285\pi$
$331$	−3.77233e8	−0.571757	−0.285878	+	0.958266i	$0.592285\pi$
$332$	8.58660e8	1.28777
$333$	0	0
$334$	1.91420e9	2.81108
$335$	0	0
$336$	0	0
$337$	1.01606e9	1.44615	0.723074	−	0.690771i	$-0.242729\pi$
$337$	1.01606e9	1.44615	0.723074	+	0.690771i	$0.242729\pi$
$338$	9.27744e8	1.30683
$339$	0	0
$340$	0	0
$341$	−1.01906e9	−1.39175
$342$	0	0
$343$	8.28836e8	1.10902
$344$	−1.97180e9	−2.61161
$345$	0	0
$346$	−6.45605e8	−0.837917
$347$	6.03712e8	0.775669	0.387835	−	0.921729i	$-0.373223\pi$
$347$	6.03712e8	0.775669	0.387835	+	0.921729i	$0.373223\pi$
$348$	0	0
$349$	−1.09383e9	−1.37740	−0.688700	−	0.725047i	$-0.741818\pi$
$349$	−1.09383e9	−1.37740	−0.688700	+	0.725047i	$0.741818\pi$
$350$	0	0
$351$	0	0
$352$	2.66955e9	3.26241
$353$	1.16369e9	1.40808	0.704038	−	0.710162i	$-0.251378\pi$
$353$	1.16369e9	1.40808	0.704038	+	0.710162i	$0.251378\pi$
$354$	0	0
$355$	0	0
$356$	1.28171e8	0.150562
$357$	0	0
$358$	−2.80727e9	−3.23365
$359$	3.55392e8	0.405394	0.202697	−	0.979242i	$-0.435029\pi$
$359$	3.55392e8	0.405394	0.202697	+	0.979242i	$0.435029\pi$
$360$	0	0
$361$	5.57662e8	0.623873
$362$	−2.43426e9	−2.69704
$363$	0	0
$364$	2.17668e9	2.36560
$365$	0	0
$366$	0	0
$367$	−6.34380e8	−0.669913	−0.334956	−	0.942234i	$-0.608722\pi$
$367$	−6.34380e8	−0.669913	−0.334956	+	0.942234i	$0.608722\pi$
$368$	1.85663e9	1.94204
$369$	0	0
$370$	0	0
$371$	−4.47106e8	−0.454571
$372$	0	0
$373$	−7.81203e8	−0.779441	−0.389721	−	0.920933i	$-0.627428\pi$
$373$	−7.81203e8	−0.779441	−0.389721	+	0.920933i	$0.627428\pi$
$374$	2.72753e9	2.69599
$375$	0	0
$376$	−4.38796e8	−0.425701
$377$	1.07285e9	1.03120
$378$	0	0
$379$	−6.84215e8	−0.645588	−0.322794	−	0.946469i	$-0.604622\pi$
$379$	−6.84215e8	−0.645588	−0.322794	+	0.946469i	$0.604622\pi$
$380$	0	0
$381$	0	0
$382$	−2.51597e8	−0.230932
$383$	1.23598e9	1.12413	0.562065	−	0.827093i	$-0.310007\pi$
$383$	1.23598e9	1.12413	0.562065	+	0.827093i	$0.310007\pi$
$384$	0	0
$385$	0	0
$386$	−2.69347e9	−2.38373
$387$	0	0
$388$	−1.24492e9	−1.08201
$389$	−7.26058e8	−0.625385	−0.312693	−	0.949854i	$-0.601231\pi$
$389$	−7.26058e8	−0.625385	−0.312693	+	0.949854i	$0.601231\pi$
$390$	0	0
$391$	9.32223e8	0.788681
$392$	−6.04529e9	−5.06893
$393$	0	0
$394$	3.71335e9	3.05865
$395$	0	0
$396$	0	0
$397$	−5.48794e8	−0.440192	−0.220096	−	0.975478i	$-0.570637\pi$
$397$	−5.48794e8	−0.440192	−0.220096	+	0.975478i	$0.570637\pi$
$398$	−1.01647e9	−0.808172
$399$	0	0
$400$	0	0
$401$	−8.37217e8	−0.648384	−0.324192	−	0.945991i	$-0.605092\pi$
$401$	−8.37217e8	−0.648384	−0.324192	+	0.945991i	$0.605092\pi$
$402$	0	0
$403$	9.12064e8	0.694157
$404$	1.94179e8	0.146510
$405$	0	0
$406$	−7.73952e9	−5.73949
$407$	3.72290e8	0.273716
$408$	0	0
$409$	1.54630e9	1.11754	0.558769	−	0.829323i	$-0.311274\pi$
$409$	1.54630e9	1.11754	0.558769	+	0.829323i	$0.311274\pi$
$410$	0	0
$411$	0	0
$412$	4.90227e9	3.45348
$413$	−1.77021e9	−1.23651
$414$	0	0
$415$	0	0
$416$	−2.38925e9	−1.62718
$417$	0	0
$418$	−4.03087e9	−2.69949
$419$	−2.38498e9	−1.58393	−0.791965	−	0.610567i	$-0.790942\pi$
$419$	−2.38498e9	−1.58393	−0.791965	+	0.610567i	$0.790942\pi$
$420$	0	0
$421$	6.06241e8	0.395966	0.197983	−	0.980205i	$-0.436561\pi$
$421$	6.06241e8	0.395966	0.197983	+	0.980205i	$0.436561\pi$
$422$	2.75163e9	1.78236
$423$	0	0
$424$	1.31733e9	0.839291
$425$	0	0
$426$	0	0
$427$	−3.81694e8	−0.237256
$428$	6.93717e9	4.27690
$429$	0	0
$430$	0	0
$431$	1.49694e9	0.900601	0.450301	−	0.892877i	$-0.351317\pi$
$431$	1.49694e9	0.900601	0.450301	+	0.892877i	$0.351317\pi$
$432$	0	0
$433$	2.70565e9	1.60164	0.800818	−	0.598907i	$-0.204398\pi$
$433$	2.70565e9	1.60164	0.800818	+	0.598907i	$0.204398\pi$
$434$	−6.57961e9	−3.86354
$435$	0	0
$436$	−2.09930e9	−1.21303
$437$	−1.37769e9	−0.789705
$438$	0	0
$439$	1.72333e8	0.0972172	0.0486086	−	0.998818i	$-0.484521\pi$
$439$	1.72333e8	0.0972172	0.0486086	+	0.998818i	$0.484521\pi$
$440$	0	0
$441$	0	0
$442$	−2.44114e9	−1.34467
$443$	−2.37901e9	−1.30012	−0.650061	−	0.759882i	$-0.725257\pi$
$443$	−2.37901e9	−1.30012	−0.650061	+	0.759882i	$0.725257\pi$
$444$	0	0
$445$	0	0
$446$	−3.79575e9	−2.02594
$447$	0	0
$448$	7.47651e9	3.92849
$449$	−2.03909e9	−1.06310	−0.531550	−	0.847027i	$-0.678390\pi$
$449$	−2.03909e9	−1.06310	−0.531550	+	0.847027i	$0.678390\pi$
$450$	0	0
$451$	3.73368e9	1.91655
$452$	5.44322e9	2.77250
$453$	0	0
$454$	−3.80557e9	−1.90864
$455$	0	0
$456$	0	0
$457$	1.56592e8	0.0767470	0.0383735	−	0.999263i	$-0.487782\pi$
$457$	1.56592e8	0.0767470	0.0383735	+	0.999263i	$0.487782\pi$
$458$	−3.82789e9	−1.86179
$459$	0	0
$460$	0	0
$461$	−2.00665e9	−0.953932	−0.476966	−	0.878922i	$-0.658263\pi$
$461$	−2.00665e9	−0.953932	−0.476966	+	0.878922i	$0.658263\pi$
$462$	0	0
$463$	1.60593e9	0.751955	0.375978	−	0.926629i	$-0.377307\pi$
$463$	1.60593e9	0.751955	0.375978	+	0.926629i	$0.377307\pi$
$464$	1.24767e10	5.79810
$465$	0	0
$466$	7.13695e8	0.326709
$467$	−5.22554e8	−0.237423	−0.118711	−	0.992929i	$-0.537876\pi$
$467$	−5.22554e8	−0.237423	−0.118711	+	0.992929i	$0.537876\pi$
$468$	0	0
$469$	−4.82390e9	−2.15920
$470$	0	0
$471$	0	0
$472$	5.21563e9	2.28302
$473$	−2.22311e9	−0.965932
$474$	0	0
$475$	0	0
$476$	1.27101e10	5.40161
$477$	0	0
$478$	−4.67884e9	−1.95948
$479$	−9.31451e8	−0.387245	−0.193623	−	0.981076i	$-0.562024\pi$
$479$	−9.31451e8	−0.387245	−0.193623	+	0.981076i	$0.562024\pi$
$480$	0	0
$481$	−3.33200e8	−0.136520
$482$	−7.08179e9	−2.88057
$483$	0	0
$484$	1.60915e9	0.645115
$485$	0	0
$486$	0	0
$487$	2.33272e9	0.915189	0.457594	−	0.889161i	$-0.348711\pi$
$487$	2.33272e9	0.915189	0.457594	+	0.889161i	$0.348711\pi$
$488$	1.12460e9	0.438055
$489$	0	0
$490$	0	0
$491$	−1.13441e9	−0.432498	−0.216249	−	0.976338i	$-0.569382\pi$
$491$	−1.13441e9	−0.432498	−0.216249	+	0.976338i	$0.569382\pi$
$492$	0	0
$493$	6.26458e9	2.35466
$494$	3.60764e9	1.34641
$495$	0	0
$496$	1.06068e10	3.90300
$497$	3.84840e9	1.40615
$498$	0	0
$499$	2.23935e9	0.806806	0.403403	−	0.915022i	$-0.367827\pi$
$499$	2.23935e9	0.806806	0.403403	+	0.915022i	$0.367827\pi$
$500$	0	0
$501$	0	0
$502$	3.57809e9	1.26238
$503$	−4.71194e8	−0.165087	−0.0825433	−	0.996587i	$-0.526304\pi$
$503$	−4.71194e8	−0.165087	−0.0825433	+	0.996587i	$0.526304\pi$
$504$	0	0
$505$	0	0
$506$	3.82580e9	1.31279
$507$	0	0
$508$	1.84046e9	0.622877
$509$	−1.61910e8	−0.0544204	−0.0272102	−	0.999630i	$-0.508662\pi$
$509$	−1.61910e8	−0.0544204	−0.0272102	+	0.999630i	$0.508662\pi$
$510$	0	0
$511$	−1.61424e9	−0.535174
$512$	9.69061e8	0.319085
$513$	0	0
$514$	3.11059e9	1.01035
$515$	0	0
$516$	0	0
$517$	−4.94721e8	−0.157450
$518$	2.40370e9	0.759846
$519$	0	0
$520$	0	0
$521$	2.89079e9	0.895539	0.447769	−	0.894149i	$-0.352219\pi$
$521$	2.89079e9	0.895539	0.447769	+	0.894149i	$0.352219\pi$
$522$	0	0
$523$	−2.26583e9	−0.692582	−0.346291	−	0.938127i	$-0.612559\pi$
$523$	−2.26583e9	−0.692582	−0.346291	+	0.938127i	$0.612559\pi$
$524$	1.62371e9	0.493003
$525$	0	0
$526$	−3.19798e9	−0.958132
$527$	5.32572e9	1.58504
$528$	0	0
$529$	−2.09723e9	−0.615959
$530$	0	0
$531$	0	0
$532$	−1.87836e10	−5.40863
$533$	−3.34165e9	−0.955908
$534$	0	0
$535$	0	0
$536$	1.42128e10	3.98661
$537$	0	0
$538$	−1.22229e10	−3.38404
$539$	−6.81577e9	−1.87480
$540$	0	0
$541$	2.25911e9	0.613405	0.306702	−	0.951805i	$-0.400774\pi$
$541$	2.25911e9	0.613405	0.306702	+	0.951805i	$0.400774\pi$
$542$	8.31355e9	2.24279
$543$	0	0
$544$	−1.39513e10	−3.71550
$545$	0	0
$546$	0	0
$547$	3.32990e9	0.869913	0.434957	−	0.900451i	$-0.356764\pi$
$547$	3.32990e9	0.869913	0.434957	+	0.900451i	$0.356764\pi$
$548$	−1.06044e10	−2.75268
$549$	0	0
$550$	0	0
$551$	−9.25811e9	−2.35772
$552$	0	0
$553$	8.02025e8	0.201674
$554$	3.18679e9	0.796287
$555$	0	0
$556$	−1.02915e10	−2.53931
$557$	4.84555e9	1.18809	0.594046	−	0.804431i	$-0.297530\pi$
$557$	4.84555e9	1.18809	0.594046	+	0.804431i	$0.297530\pi$
$558$	0	0
$559$	1.98969e9	0.481774
$560$	0	0
$561$	0	0
$562$	−1.12658e10	−2.67723
$563$	−1.98952e9	−0.469861	−0.234931	−	0.972012i	$-0.575486\pi$
$563$	−1.98952e9	−0.469861	−0.234931	+	0.972012i	$0.575486\pi$
$564$	0	0
$565$	0	0
$566$	−1.13230e10	−2.62486
$567$	0	0
$568$	−1.13387e10	−2.59623
$569$	−6.31733e9	−1.43761	−0.718804	−	0.695213i	$-0.755310\pi$
$569$	−6.31733e9	−1.43761	−0.718804	+	0.695213i	$0.755310\pi$
$570$	0	0
$571$	5.74521e9	1.29146	0.645728	−	0.763568i	$-0.276554\pi$
$571$	5.74521e9	1.29146	0.645728	+	0.763568i	$0.276554\pi$
$572$	−7.23062e9	−1.61543
$573$	0	0
$574$	2.41066e10	5.32040
$575$	0	0
$576$	0	0
$577$	−2.23701e9	−0.484788	−0.242394	−	0.970178i	$-0.577933\pi$
$577$	−2.23701e9	−0.484788	−0.242394	+	0.970178i	$0.577933\pi$
$578$	−5.45351e9	−1.17470
$579$	0	0
$580$	0	0
$581$	3.84069e9	0.812443
$582$	0	0
$583$	1.48522e9	0.310421
$584$	4.75610e9	0.988111
$585$	0	0
$586$	−8.51842e9	−1.74871
$587$	2.59828e9	0.530216	0.265108	−	0.964219i	$-0.414592\pi$
$587$	2.59828e9	0.530216	0.265108	+	0.964219i	$0.414592\pi$
$588$	0	0
$589$	−7.87061e9	−1.58710
$590$	0	0
$591$	0	0
$592$	−3.87493e9	−0.767605
$593$	7.18698e9	1.41532	0.707661	−	0.706552i	$-0.249751\pi$
$593$	7.18698e9	1.41532	0.707661	+	0.706552i	$0.249751\pi$
$594$	0	0
$595$	0	0
$596$	4.01745e8	0.0777299
$597$	0	0
$598$	−3.42410e9	−0.654775
$599$	8.78069e9	1.66930	0.834651	−	0.550780i	$-0.185670\pi$
$599$	8.78069e9	1.66930	0.834651	+	0.550780i	$0.185670\pi$
$600$	0	0
$601$	−1.04187e10	−1.95773	−0.978866	−	0.204504i	$-0.934442\pi$
$601$	−1.04187e10	−1.95773	−0.978866	+	0.204504i	$0.934442\pi$
$602$	−1.43535e10	−2.68146
$603$	0	0
$604$	−1.43747e10	−2.65442
$605$	0	0
$606$	0	0
$607$	7.93995e9	1.44098	0.720489	−	0.693466i	$-0.243917\pi$
$607$	7.93995e9	1.44098	0.720489	+	0.693466i	$0.243917\pi$
$608$	2.06179e10	3.72033
$609$	0	0
$610$	0	0
$611$	4.42776e8	0.0785308
$612$	0	0
$613$	−2.87459e9	−0.504040	−0.252020	−	0.967722i	$-0.581095\pi$
$613$	−2.87459e9	−0.504040	−0.252020	+	0.967722i	$0.581095\pi$
$614$	−1.39090e10	−2.42497
$615$	0	0
$616$	3.20511e10	5.52471
$617$	−9.76516e9	−1.67371	−0.836857	−	0.547422i	$-0.815609\pi$
$617$	−9.76516e9	−1.67371	−0.836857	+	0.547422i	$0.815609\pi$
$618$	0	0
$619$	−5.15562e9	−0.873703	−0.436852	−	0.899534i	$-0.643906\pi$
$619$	−5.15562e9	−0.873703	−0.436852	+	0.899534i	$0.643906\pi$
$620$	0	0
$621$	0	0
$622$	−1.83353e10	−3.05507
$623$	5.73295e8	0.0949882
$624$	0	0
$625$	0	0
$626$	−4.21349e9	−0.686487
$627$	0	0
$628$	9.63581e9	1.55249
$629$	−1.94562e9	−0.311731
$630$	0	0
$631$	7.85202e9	1.24417	0.622083	−	0.782951i	$-0.286286\pi$
$631$	7.85202e9	1.24417	0.622083	+	0.782951i	$0.286286\pi$
$632$	−2.36304e9	−0.372358
$633$	0	0
$634$	−2.81094e9	−0.438066
$635$	0	0
$636$	0	0
$637$	6.10013e9	0.935084
$638$	2.57096e10	3.91942
$639$	0	0
$640$	0	0
$641$	4.26696e9	0.639905	0.319953	−	0.947434i	$-0.396333\pi$
$641$	4.26696e9	0.639905	0.319953	+	0.947434i	$0.396333\pi$
$642$	0	0
$643$	3.18717e9	0.472788	0.236394	−	0.971657i	$-0.424034\pi$
$643$	3.18717e9	0.472788	0.236394	+	0.971657i	$0.424034\pi$
$644$	1.78279e10	2.63027
$645$	0	0
$646$	2.10657e10	3.07441
$647$	−2.97772e9	−0.432233	−0.216117	−	0.976368i	$-0.569339\pi$
$647$	−2.97772e9	−0.432233	−0.216117	+	0.976368i	$0.569339\pi$
$648$	0	0
$649$	5.88036e9	0.844399
$650$	0	0
$651$	0	0
$652$	2.98975e9	0.422443
$653$	−1.28757e10	−1.80957	−0.904785	−	0.425868i	$-0.859969\pi$
$653$	−1.28757e10	−1.80957	−0.904785	+	0.425868i	$0.859969\pi$
$654$	0	0
$655$	0	0
$656$	−3.88616e10	−5.37473
$657$	0	0
$658$	−3.19418e9	−0.437087
$659$	−5.42800e9	−0.738824	−0.369412	−	0.929266i	$-0.620441\pi$
$659$	−5.42800e9	−0.738824	−0.369412	+	0.929266i	$0.620441\pi$
$660$	0	0
$661$	−7.71802e9	−1.03944	−0.519721	−	0.854336i	$-0.673964\pi$
$661$	−7.71802e9	−1.03944	−0.519721	+	0.854336i	$0.673964\pi$
$662$	8.09074e9	1.08389
$663$	0	0
$664$	−1.13160e10	−1.50004
$665$	0	0
$666$	0	0
$667$	8.78709e9	1.14658
$668$	−2.96309e10	−3.84616
$669$	0	0
$670$	0	0
$671$	1.26793e9	0.162019
$672$	0	0
$673$	1.38851e10	1.75588	0.877940	−	0.478770i	$-0.158917\pi$
$673$	1.38851e10	1.75588	0.877940	+	0.478770i	$0.158917\pi$
$674$	−2.17920e10	−2.74149
$675$	0	0
$676$	−1.43611e10	−1.78803
$677$	−1.12086e10	−1.38833	−0.694164	−	0.719817i	$-0.744226\pi$
$677$	−1.12086e10	−1.38833	−0.694164	+	0.719817i	$0.744226\pi$
$678$	0	0
$679$	−5.56840e9	−0.682631
$680$	0	0
$681$	0	0
$682$	2.18565e10	2.63837
$683$	7.53971e9	0.905487	0.452744	−	0.891641i	$-0.350445\pi$
$683$	7.53971e9	0.905487	0.452744	+	0.891641i	$0.350445\pi$
$684$	0	0
$685$	0	0
$686$	−1.77766e10	−2.10239
$687$	0	0
$688$	2.31389e10	2.70884
$689$	−1.32928e9	−0.154827
$690$	0	0
$691$	4.23931e9	0.488790	0.244395	−	0.969676i	$-0.421411\pi$
$691$	4.23931e9	0.488790	0.244395	+	0.969676i	$0.421411\pi$
$692$	9.99370e9	1.14645
$693$	0	0
$694$	−1.29482e10	−1.47045
$695$	0	0
$696$	0	0
$697$	−1.95125e10	−2.18273
$698$	2.34600e10	2.61116
$699$	0	0
$700$	0	0
$701$	−1.37775e10	−1.51063	−0.755315	−	0.655361i	$-0.772516\pi$
$701$	−1.37775e10	−1.51063	−0.755315	+	0.655361i	$0.772516\pi$
$702$	0	0
$703$	2.87533e9	0.312136
$704$	−2.48359e10	−2.68272
$705$	0	0
$706$	−2.49584e10	−2.66932
$707$	8.68541e8	0.0924320
$708$	0	0
$709$	3.52673e9	0.371630	0.185815	−	0.982585i	$-0.440508\pi$
$709$	3.52673e9	0.371630	0.185815	+	0.982585i	$0.440508\pi$
$710$	0	0
$711$	0	0
$712$	−1.68912e9	−0.175380
$713$	7.47018e9	0.771823
$714$	0	0
$715$	0	0
$716$	4.34554e10	4.42433
$717$	0	0
$718$	−7.62231e9	−0.768512
$719$	6.74410e9	0.676664	0.338332	−	0.941027i	$-0.390137\pi$
$719$	6.74410e9	0.676664	0.338332	+	0.941027i	$0.390137\pi$
$720$	0	0
$721$	2.19273e10	2.17877
$722$	−1.19605e10	−1.18269
$723$	0	0
$724$	3.76814e10	3.69013
$725$	0	0
$726$	0	0
$727$	−1.77504e10	−1.71332	−0.856659	−	0.515882i	$-0.827464\pi$
$727$	−1.77504e10	−1.71332	−0.856659	+	0.515882i	$0.827464\pi$
$728$	−2.86858e10	−2.75554
$729$	0	0
$730$	0	0
$731$	1.16182e10	1.10009
$732$	0	0
$733$	−1.03364e10	−0.969408	−0.484704	−	0.874678i	$-0.661073\pi$
$733$	−1.03364e10	−0.969408	−0.484704	+	0.874678i	$0.661073\pi$
$734$	1.36059e10	1.26997
$735$	0	0
$736$	−1.95689e10	−1.80923
$737$	1.60243e10	1.47449
$738$	0	0
$739$	1.19216e10	1.08662	0.543310	−	0.839532i	$-0.317171\pi$
$739$	1.19216e10	1.08662	0.543310	+	0.839532i	$0.317171\pi$
$740$	0	0
$741$	0	0
$742$	9.58936e9	0.861739
$743$	2.03848e9	0.182325	0.0911623	−	0.995836i	$-0.470942\pi$
$743$	2.03848e9	0.182325	0.0911623	+	0.995836i	$0.470942\pi$
$744$	0	0
$745$	0	0
$746$	1.67549e10	1.47760
$747$	0	0
$748$	−4.22210e10	−3.68869
$749$	3.10292e10	2.69826
$750$	0	0
$751$	−1.19791e10	−1.03201	−0.516005	−	0.856585i	$-0.672582\pi$
$751$	−1.19791e10	−1.03201	−0.516005	+	0.856585i	$0.672582\pi$
$752$	5.14924e9	0.441551
$753$	0	0
$754$	−2.30101e10	−1.95487
$755$	0	0
$756$	0	0
$757$	3.28730e9	0.275425	0.137713	−	0.990472i	$-0.456025\pi$
$757$	3.28730e9	0.275425	0.137713	+	0.990472i	$0.456025\pi$
$758$	1.46748e10	1.22385
$759$	0	0
$760$	0	0
$761$	−9.23337e9	−0.759476	−0.379738	−	0.925094i	$-0.623986\pi$
$761$	−9.23337e9	−0.759476	−0.379738	+	0.925094i	$0.623986\pi$
$762$	0	0
$763$	−9.38994e9	−0.765291
$764$	3.89462e9	0.315964
$765$	0	0
$766$	−2.65089e10	−2.13104
$767$	−5.26294e9	−0.421157
$768$	0	0
$769$	1.22423e10	0.970780	0.485390	−	0.874298i	$-0.338678\pi$
$769$	1.22423e10	0.970780	0.485390	+	0.874298i	$0.338678\pi$
$770$	0	0
$771$	0	0
$772$	4.16938e10	3.26145
$773$	7.44597e9	0.579820	0.289910	−	0.957054i	$-0.406375\pi$
$773$	7.44597e9	0.579820	0.289910	+	0.957054i	$0.406375\pi$
$774$	0	0
$775$	0	0
$776$	1.64064e10	1.26037
$777$	0	0
$778$	1.55722e10	1.18555
$779$	2.88366e10	2.18556
$780$	0	0
$781$	−1.27838e10	−0.960244
$782$	−1.99940e10	−1.49512
$783$	0	0
$784$	7.09411e10	5.25765
$785$	0	0
$786$	0	0
$787$	8.57755e9	0.627266	0.313633	−	0.949544i	$-0.398454\pi$
$787$	8.57755e9	0.627266	0.313633	+	0.949544i	$0.398454\pi$
$788$	−5.74812e10	−4.18489
$789$	0	0
$790$	0	0
$791$	2.43469e10	1.74915
$792$	0	0
$793$	−1.13480e9	−0.0808097
$794$	1.17703e10	0.834481
$795$	0	0
$796$	1.57345e10	1.10575
$797$	−4.77411e9	−0.334032	−0.167016	−	0.985954i	$-0.553413\pi$
$797$	−4.77411e9	−0.334032	−0.167016	+	0.985954i	$0.553413\pi$
$798$	0	0
$799$	2.58545e9	0.179318
$800$	0	0
$801$	0	0
$802$	1.79563e10	1.22915
$803$	5.36227e9	0.365463
$804$	0	0
$805$	0	0
$806$	−1.95616e10	−1.31593
$807$	0	0
$808$	−2.55902e9	−0.170661
$809$	2.35228e9	0.156196	0.0780980	−	0.996946i	$-0.475115\pi$
$809$	2.35228e9	0.156196	0.0780980	+	0.996946i	$0.475115\pi$
$810$	0	0
$811$	−1.70112e10	−1.11986	−0.559928	−	0.828541i	$-0.689171\pi$
$811$	−1.70112e10	−1.11986	−0.559928	+	0.828541i	$0.689171\pi$
$812$	1.19805e11	7.85285
$813$	0	0
$814$	−7.98473e9	−0.518889
$815$	0	0
$816$	0	0
$817$	−1.71699e10	−1.10151
$818$	−3.31644e10	−2.11854
$819$	0	0
$820$	0	0
$821$	1.40347e10	0.885122	0.442561	−	0.896738i	$-0.354070\pi$
$821$	1.40347e10	0.885122	0.442561	+	0.896738i	$0.354070\pi$
$822$	0	0
$823$	2.85901e8	0.0178779	0.00893895	−	0.999960i	$-0.497155\pi$
$823$	2.85901e8	0.0178779	0.00893895	+	0.999960i	$0.497155\pi$
$824$	−6.46053e10	−4.02275
$825$	0	0
$826$	3.79667e10	2.34408
$827$	−1.82776e10	−1.12370	−0.561849	−	0.827240i	$-0.689910\pi$
$827$	−1.82776e10	−1.12370	−0.561849	+	0.827240i	$0.689910\pi$
$828$	0	0
$829$	−2.70199e10	−1.64719	−0.823594	−	0.567180i	$-0.808035\pi$
$829$	−2.70199e10	−1.64719	−0.823594	+	0.567180i	$0.808035\pi$
$830$	0	0
$831$	0	0
$832$	2.22281e10	1.33805
$833$	3.56198e10	2.13518
$834$	0	0
$835$	0	0
$836$	6.23962e10	3.69349
$837$	0	0
$838$	5.11521e10	3.00268
$839$	−2.70701e10	−1.58242	−0.791211	−	0.611543i	$-0.790549\pi$
$839$	−2.70701e10	−1.58242	−0.791211	+	0.611543i	$0.790549\pi$
$840$	0	0
$841$	4.17998e10	2.42320
$842$	−1.30024e10	−0.750641
$843$	0	0
$844$	−4.25940e10	−2.43865
$845$	0	0
$846$	0	0
$847$	7.19754e9	0.406998
$848$	−1.54587e10	−0.870539
$849$	0	0
$850$	0	0
$851$	−2.72905e9	−0.151795
$852$	0	0
$853$	−3.03095e10	−1.67208	−0.836040	−	0.548669i	$-0.815135\pi$
$853$	−3.03095e10	−1.67208	−0.836040	+	0.548669i	$0.815135\pi$
$854$	8.18642e9	0.449771
$855$	0	0
$856$	−9.14225e10	−4.98190
$857$	−2.33111e10	−1.26512	−0.632558	−	0.774513i	$-0.717995\pi$
$857$	−2.33111e10	−1.26512	−0.632558	+	0.774513i	$0.717995\pi$
$858$	0	0
$859$	3.83240e9	0.206298	0.103149	−	0.994666i	$-0.467108\pi$
$859$	3.83240e9	0.206298	0.103149	+	0.994666i	$0.467108\pi$
$860$	0	0
$861$	0	0
$862$	−3.21057e10	−1.70729
$863$	−1.55104e10	−0.821460	−0.410730	−	0.911757i	$-0.634726\pi$
$863$	−1.55104e10	−0.821460	−0.410730	+	0.911757i	$0.634726\pi$
$864$	0	0
$865$	0	0
$866$	−5.80297e10	−3.03625
$867$	0	0
$868$	1.01850e11	5.28616
$869$	−2.66421e9	−0.137721
$870$	0	0
$871$	−1.43418e10	−0.735425
$872$	2.76659e10	1.41299
$873$	0	0
$874$	2.95481e10	1.49706
$875$	0	0
$876$	0	0
$877$	2.46944e10	1.23623	0.618115	−	0.786088i	$-0.287897\pi$
$877$	2.46944e10	1.23623	0.618115	+	0.786088i	$0.287897\pi$
$878$	−3.69614e9	−0.184297
$879$	0	0
$880$	0	0
$881$	3.29632e10	1.62410	0.812052	−	0.583586i	$-0.198351\pi$
$881$	3.29632e10	1.62410	0.812052	+	0.583586i	$0.198351\pi$
$882$	0	0
$883$	1.72362e10	0.842520	0.421260	−	0.906940i	$-0.361588\pi$
$883$	1.72362e10	0.842520	0.421260	+	0.906940i	$0.361588\pi$
$884$	3.77878e10	1.83979
$885$	0	0
$886$	5.10242e10	2.46467
$887$	−2.00236e10	−0.963407	−0.481704	−	0.876334i	$-0.659982\pi$
$887$	−2.00236e10	−0.963407	−0.481704	+	0.876334i	$0.659982\pi$
$888$	0	0
$889$	8.23216e9	0.392968
$890$	0	0
$891$	0	0
$892$	5.87566e10	2.77191
$893$	−3.82091e9	−0.179551
$894$	0	0
$895$	0	0
$896$	−5.74885e10	−2.66995
$897$	0	0
$898$	4.37336e10	2.01534
$899$	5.02000e10	2.30433
$900$	0	0
$901$	−7.76189e9	−0.353534
$902$	−8.00786e10	−3.63323
$903$	0	0
$904$	−7.17343e10	−3.22952
$905$	0	0
$906$	0	0
$907$	−2.89364e10	−1.28771	−0.643856	−	0.765147i	$-0.722667\pi$
$907$	−2.89364e10	−1.28771	−0.643856	+	0.765147i	$0.722667\pi$
$908$	5.89086e10	2.61143
$909$	0	0
$910$	0	0
$911$	−1.88416e10	−0.825663	−0.412832	−	0.910807i	$-0.635460\pi$
$911$	−1.88416e10	−0.825663	−0.412832	+	0.910807i	$0.635460\pi$
$912$	0	0
$913$	−1.27582e10	−0.554807
$914$	−3.35851e9	−0.145491
$915$	0	0
$916$	5.92541e10	2.54733
$917$	7.26269e9	0.311032
$918$	0	0
$919$	3.79261e9	0.161189	0.0805943	−	0.996747i	$-0.474318\pi$
$919$	3.79261e9	0.161189	0.0805943	+	0.996747i	$0.474318\pi$
$920$	0	0
$921$	0	0
$922$	4.30377e10	1.80839
$923$	1.14415e10	0.478937
$924$	0	0
$925$	0	0
$926$	−3.44433e10	−1.42550
$927$	0	0
$928$	−1.31504e11	−5.40159
$929$	2.42383e10	0.991854	0.495927	−	0.868364i	$-0.334828\pi$
$929$	2.42383e10	0.991854	0.495927	+	0.868364i	$0.334828\pi$
$930$	0	0
$931$	−5.26407e10	−2.13795
$932$	−1.10477e10	−0.447009
$933$	0	0
$934$	1.12075e10	0.450087
$935$	0	0
$936$	0	0
$937$	−1.22460e10	−0.486300	−0.243150	−	0.969989i	$-0.578181\pi$
$937$	−1.22460e10	−0.486300	−0.243150	+	0.969989i	$0.578181\pi$
$938$	1.03461e11	4.09324
$939$	0	0
$940$	0	0
$941$	−1.48731e10	−0.581886	−0.290943	−	0.956740i	$-0.593969\pi$
$941$	−1.48731e10	−0.581886	−0.290943	+	0.956740i	$0.593969\pi$
$942$	0	0
$943$	−2.73695e10	−1.06286
$944$	−6.12050e10	−2.36802
$945$	0	0
$946$	4.76804e10	1.83114
$947$	−6.54636e9	−0.250481	−0.125241	−	0.992126i	$-0.539970\pi$
$947$	−6.54636e9	−0.250481	−0.125241	+	0.992126i	$0.539970\pi$
$948$	0	0
$949$	−4.79924e9	−0.182281
$950$	0	0
$951$	0	0
$952$	−1.67502e11	−6.29201
$953$	−2.12668e10	−0.795936	−0.397968	−	0.917399i	$-0.630284\pi$
$953$	−2.12668e10	−0.795936	−0.397968	+	0.917399i	$0.630284\pi$
$954$	0	0
$955$	0	0
$956$	7.24265e10	2.68099
$957$	0	0
$958$	1.99774e10	0.734108
$959$	−4.74325e10	−1.73664
$960$	0	0
$961$	1.51639e10	0.551163
$962$	7.14634e9	0.258804
$963$	0	0
$964$	1.09623e11	3.94123
$965$	0	0
$966$	0	0
$967$	−6.11461e9	−0.217458	−0.108729	−	0.994071i	$-0.534678\pi$
$967$	−6.11461e9	−0.217458	−0.108729	+	0.994071i	$0.534678\pi$
$968$	−2.12064e10	−0.751455
$969$	0	0
$970$	0	0
$971$	−3.18636e10	−1.11694	−0.558468	−	0.829526i	$-0.688610\pi$
$971$	−3.18636e10	−1.11694	−0.558468	+	0.829526i	$0.688610\pi$
$972$	0	0
$973$	−4.60327e10	−1.60203
$974$	−5.00312e10	−1.73494
$975$	0	0
$976$	−1.31971e10	−0.454364
$977$	4.38578e10	1.50458	0.752290	−	0.658832i	$-0.228949\pi$
$977$	4.38578e10	1.50458	0.752290	+	0.658832i	$0.228949\pi$
$978$	0	0
$979$	−1.90440e9	−0.0648662
$980$	0	0
$981$	0	0
$982$	2.43304e10	0.819895
$983$	1.64867e10	0.553599	0.276800	−	0.960928i	$-0.410726\pi$
$983$	1.64867e10	0.553599	0.276800	+	0.960928i	$0.410726\pi$
$984$	0	0
$985$	0	0
$986$	−1.34360e11	−4.46377
$987$	0	0
$988$	−5.58448e10	−1.84218
$989$	1.62963e10	0.535677
$990$	0	0
$991$	−4.76549e10	−1.55543	−0.777713	−	0.628619i	$-0.783620\pi$
$991$	−4.76549e10	−1.55543	−0.777713	+	0.628619i	$0.783620\pi$
$992$	−1.11796e11	−3.63609
$993$	0	0
$994$	−8.25390e10	−2.66567
$995$	0	0
$996$	0	0
$997$	3.74257e10	1.19602	0.598008	−	0.801490i	$-0.295959\pi$
$997$	3.74257e10	1.19602	0.598008	+	0.801490i	$0.295959\pi$
$998$	−4.80286e10	−1.52948
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.a.s.1.1	yes	2
3.2	odd	2	inner	225.8.a.s.1.2	yes	2
5.2	odd	4		225.8.b.j.199.2		4
5.3	odd	4		225.8.b.j.199.3		4
5.4	even	2		225.8.a.r.1.2	yes	2
15.2	even	4		225.8.b.j.199.4		4
15.8	even	4		225.8.b.j.199.1		4
15.14	odd	2		225.8.a.r.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.8.a.r.1.1	✓	2	15.14	odd	2
225.8.a.r.1.2	yes	2	5.4	even	2
225.8.a.s.1.1	yes	2	1.1	even	1	trivial
225.8.a.s.1.2	yes	2	3.2	odd	2	inner
225.8.b.j.199.1		4	15.8	even	4
225.8.b.j.199.2		4	5.2	odd	4
225.8.b.j.199.3		4	5.3	odd	4
225.8.b.j.199.4		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-21.4476 q^{2} +332.000 q^{4} +1485.00 q^{7} -4375.31 q^{8} +O(q^{10})\)	\(q-21.4476 q^{2} +332.000 q^{4} +1485.00 q^{7} -4375.31 q^{8} -4932.95 q^{11} +4415.00 q^{13} -31849.7 q^{14} +51344.0 q^{16} +25780.0 q^{17} -38099.0 q^{19} +105800. q^{22} +36160.7 q^{23} -94691.2 q^{26} +493020. q^{28} +243001. q^{29} +206583. q^{31} -541166. q^{32} -552920. q^{34} -75470.0 q^{37} +817133. q^{38} -756886. q^{41} +450665. q^{43} -1.63774e6 q^{44} -775560. q^{46} +100289. q^{47} +1.38168e6 q^{49} +1.46578e6 q^{52} -301082. q^{53} -6.49734e6 q^{56} -5.21180e6 q^{58} -1.19206e6 q^{59} -257033. q^{61} -4.43071e6 q^{62} +5.03469e6 q^{64} -3.24841e6 q^{67} +8.55897e6 q^{68} +2.59151e6 q^{71} -1.08703e6 q^{73} +1.61865e6 q^{74} -1.26489e7 q^{76} -7.32543e6 q^{77} +540084. q^{79} +1.62334e7 q^{82} +2.58632e6 q^{83} -9.66569e6 q^{86} +2.15832e7 q^{88} +386057. q^{89} +6.55628e6 q^{91} +1.20053e7 q^{92} -2.15096e6 q^{94} -3.74976e6 q^{97} -2.96338e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 664 q^{4} + 2970 q^{7}+O(q^{10}) \) 2 * q + 664 * q^4 + 2970 * q^7	\( 2 q + 664 q^{4} + 2970 q^{7} + 8830 q^{13} + 102688 q^{16} - 76198 q^{19} + 211600 q^{22} + 986040 q^{28} + 413166 q^{31} - 1105840 q^{34} - 150940 q^{37} + 901330 q^{43} - 1551120 q^{46} + 2763364 q^{49} + 2931560 q^{52} - 10423600 q^{58} - 514066 q^{61} + 10069376 q^{64} - 6496830 q^{67} - 2174060 q^{73} - 25297736 q^{76} + 1080168 q^{79} + 32466800 q^{82} + 43166400 q^{88} + 13112550 q^{91} - 4301920 q^{94} - 7499530 q^{97}+O(q^{100}) \) 2 * q + 664 * q^4 + 2970 * q^7 + 8830 * q^13 + 102688 * q^16 - 76198 * q^19 + 211600 * q^22 + 986040 * q^28 + 413166 * q^31 - 1105840 * q^34 - 150940 * q^37 + 901330 * q^43 - 1551120 * q^46 + 2763364 * q^49 + 2931560 * q^52 - 10423600 * q^58 - 514066 * q^61 + 10069376 * q^64 - 6496830 * q^67 - 2174060 * q^73 - 25297736 * q^76 + 1080168 * q^79 + 32466800 * q^82 + 43166400 * q^88 + 13112550 * q^91 - 4301920 * q^94 - 7499530 * q^97

Introduction

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