LMFDB - Embedded newform 225.12.b.f.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,Mod(199,225)]

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, 1]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$172.877215626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{151})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 75x^{2} + 1444 $ x^4 - 75*x^2 + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$6.14410 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.12.b.f.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+63.7292i q^{2} -2013.42 q^{4} -41926.3i q^{7} +2204.06i q^{8} +O(q^{10})$	$q+63.7292i q^{2} -2013.42 q^{4} -41926.3i q^{7} +2204.06i q^{8} +957905. q^{11} +1.39098e6i q^{13} +2.67193e6 q^{14} -4.26394e6 q^{16} +3.76857e6i q^{17} -9.41036e6 q^{19} +6.10466e7i q^{22} -3.02942e7i q^{23} -8.86460e7 q^{26} +8.44151e7i q^{28} +1.03553e8 q^{29} -5.48554e7 q^{31} -2.67224e8i q^{32} -2.40168e8 q^{34} -4.78282e8i q^{37} -5.99715e8i q^{38} +9.29557e8 q^{41} -2.68608e7i q^{43} -1.92866e9 q^{44} +1.93063e9 q^{46} -1.20497e9i q^{47} +2.19508e8 q^{49} -2.80062e9i q^{52} +4.02058e9i q^{53} +9.24080e7 q^{56} +6.59933e9i q^{58} +7.97972e9 q^{59} -2.07472e9 q^{61} -3.49589e9i q^{62} +8.29741e9 q^{64} -5.61370e9i q^{67} -7.58770e9i q^{68} -1.51224e10 q^{71} -6.64484e9i q^{73} +3.04806e10 q^{74} +1.89470e10 q^{76} -4.01615e10i q^{77} +1.57985e10 q^{79} +5.92399e10i q^{82} +2.04046e10i q^{83} +1.71182e9 q^{86} +2.11128e9i q^{88} -4.21030e10 q^{89} +5.83187e10 q^{91} +6.09948e10i q^{92} +7.67919e10 q^{94} -1.10181e11i q^{97} +1.39891e10i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 13952 q^{4}+O(q^{10}) $ 4 * q - 13952 * q^4	$ 4 q - 13952 q^{4} + 1236352 q^{11} + 2668944 q^{14} + 12011264 q^{16} - 10650640 q^{19} + 161523472 q^{26} + 188280760 q^{29} + 489086928 q^{31} - 890716144 q^{34} + 1491486632 q^{41} - 485451776 q^{44} - 936389712 q^{46} + 3883354828 q^{49} + 7981107840 q^{56} + 14635031120 q^{59} - 3032851352 q^{61} - 1639063552 q^{64} - 65876943088 q^{71} + 137536396144 q^{74} - 2650785280 q^{76} + 6605646240 q^{79} + 113412187792 q^{86} - 25349541720 q^{89} + 181275718128 q^{91} + 120579531056 q^{94}+O(q^{100}) $ 4 * q - 13952 * q^4 + 1236352 * q^11 + 2668944 * q^14 + 12011264 * q^16 - 10650640 * q^19 + 161523472 * q^26 + 188280760 * q^29 + 489086928 * q^31 - 890716144 * q^34 + 1491486632 * q^41 - 485451776 * q^44 - 936389712 * q^46 + 3883354828 * q^49 + 7981107840 * q^56 + 14635031120 * q^59 - 3032851352 * q^61 - 1639063552 * q^64 - 65876943088 * q^71 + 137536396144 * q^74 - 2650785280 * q^76 + 6605646240 * q^79 + 113412187792 * q^86 - 25349541720 * q^89 + 181275718128 * q^91 + 120579531056 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	63.7292i	1.40823i	0.710086	+	0.704115i	$0.248656\pi$
$2$	63.7292i	1.40823i	−0.710086	+	0.704115i	$0.751344\pi$
$3$	0	0
$3$	0	0
$4$	−2013.42	−0.983113
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 41926.3i	− 0.942861i	−0.881903	−	0.471431i	$-0.843738\pi$
$7$	− 41926.3i	− 0.942861i	0.881903	−	0.471431i	$-0.156262\pi$
$8$	2204.06i	0.0237809i
$9$	0	0
$10$	0	0
$11$	957905.	1.79334	0.896670	−	0.442699i	$-0.145979\pi$
$11$	957905.	1.79334	0.896670	+	0.442699i	$0.145979\pi$
$12$	0	0
$13$	1.39098e6i	1.03904i	0.854458	+	0.519520i	$0.173889\pi$
$13$	1.39098e6i	1.03904i	−0.854458	+	0.519520i	$0.826111\pi$
$14$	2.67193e6	1.32777
$15$	0	0
$16$	−4.26394e6	−1.01660
$17$	3.76857e6i	0.643736i	0.946785	+	0.321868i	$0.104311\pi$
$17$	3.76857e6i	0.643736i	−0.946785	+	0.321868i	$0.895689\pi$
$18$	0	0
$19$	−9.41036e6	−0.871889	−0.435945	−	0.899973i	$-0.643586\pi$
$19$	−9.41036e6	−0.871889	−0.435945	+	0.899973i	$0.643586\pi$
$20$	0	0
$21$	0	0
$22$	6.10466e7i	2.52544i
$23$	− 3.02942e7i	− 0.981423i	−0.871322	−	0.490712i	$-0.836737\pi$
$23$	− 3.02942e7i	− 0.981423i	0.871322	−	0.490712i	$-0.163263\pi$
$24$	0	0
$25$	0	0
$26$	−8.86460e7	−1.46321
$27$	0	0
$28$	8.44151e7i	0.926939i
$29$	1.03553e8	0.937502	0.468751	−	0.883330i	$-0.344704\pi$
$29$	1.03553e8	0.937502	0.468751	+	0.883330i	$0.344704\pi$
$30$	0	0
$31$	−5.48554e7	−0.344136	−0.172068	−	0.985085i	$-0.555045\pi$
$31$	−5.48554e7	−0.344136	−0.172068	+	0.985085i	$0.555045\pi$
$32$	− 2.67224e8i	− 1.40783i
$33$	0	0
$34$	−2.40168e8	−0.906529
$35$	0	0
$36$	0	0
$37$	− 4.78282e8i	− 1.13390i	−0.823752	−	0.566950i	$-0.808123\pi$
$37$	− 4.78282e8i	− 1.13390i	0.823752	−	0.566950i	$-0.191877\pi$
$38$	− 5.99715e8i	− 1.22782i
$39$	0	0
$40$	0	0
$41$	9.29557e8	1.25304	0.626520	−	0.779406i	$-0.284479\pi$
$41$	9.29557e8	1.25304	0.626520	+	0.779406i	$0.284479\pi$
$42$	0	0
$43$	− 2.68608e7i	− 0.0278639i	−0.999903	−	0.0139320i	$-0.995565\pi$
$43$	− 2.68608e7i	− 0.0278639i	0.999903	−	0.0139320i	$-0.00443483\pi$
$44$	−1.92866e9	−1.76306
$45$	0	0
$46$	1.93063e9	1.38207
$47$	− 1.20497e9i	− 0.766370i	−0.923672	−	0.383185i	$-0.874827\pi$
$47$	− 1.20497e9i	− 0.766370i	0.923672	−	0.383185i	$-0.125173\pi$
$48$	0	0
$49$	2.19508e8	0.111013
$50$	0	0
$51$	0	0
$52$	− 2.80062e9i	− 1.02149i
$53$	4.02058e9i	1.32060i	0.751001	+	0.660301i	$0.229571\pi$
$53$	4.02058e9i	1.32060i	−0.751001	+	0.660301i	$0.770429\pi$
$54$	0	0
$55$	0	0
$56$	9.24080e7	0.0224221
$57$	0	0
$58$	6.59933e9i	1.32022i
$59$	7.97972e9	1.45312	0.726560	−	0.687103i	$-0.241118\pi$
$59$	7.97972e9	1.45312	0.726560	+	0.687103i	$0.241118\pi$
$60$	0	0
$61$	−2.07472e9	−0.314518	−0.157259	−	0.987557i	$-0.550266\pi$
$61$	−2.07472e9	−0.314518	−0.157259	+	0.987557i	$0.550266\pi$
$62$	− 3.49589e9i	− 0.484623i
$63$	0	0
$64$	8.29741e9	0.965946
$65$	0	0
$66$	0	0
$67$	− 5.61370e9i	− 0.507969i	−0.967208	−	0.253985i	$-0.918259\pi$
$67$	− 5.61370e9i	− 0.507969i	0.967208	−	0.253985i	$-0.0817413\pi$
$68$	− 7.58770e9i	− 0.632865i
$69$	0	0
$70$	0	0
$71$	−1.51224e10	−0.994714	−0.497357	−	0.867546i	$-0.665696\pi$
$71$	−1.51224e10	−0.994714	−0.497357	+	0.867546i	$0.665696\pi$
$72$	0	0
$73$	− 6.64484e9i	− 0.375153i	−0.982250	−	0.187577i	$-0.939937\pi$
$73$	− 6.64484e9i	− 0.375153i	0.982250	−	0.187577i	$-0.0600633\pi$
$74$	3.04806e10	1.59679
$75$	0	0
$76$	1.89470e10	0.857166
$77$	− 4.01615e10i	− 1.69087i
$78$	0	0
$79$	1.57985e10	0.577652	0.288826	−	0.957382i	$-0.406735\pi$
$79$	1.57985e10	0.577652	0.288826	+	0.957382i	$0.406735\pi$
$80$	0	0
$81$	0	0
$82$	5.92399e10i	1.76457i
$83$	2.04046e10i	0.568589i	0.958737	+	0.284295i	$0.0917594\pi$
$83$	2.04046e10i	0.568589i	−0.958737	+	0.284295i	$0.908241\pi$
$84$	0	0
$85$	0	0
$86$	1.71182e9	0.0392389
$87$	0	0
$88$	2.11128e9i	0.0426472i
$89$	−4.21030e10	−0.799223	−0.399611	−	0.916685i	$-0.630855\pi$
$89$	−4.21030e10	−0.799223	−0.399611	+	0.916685i	$0.630855\pi$
$90$	0	0
$91$	5.83187e10	0.979670
$92$	6.09948e10i	0.964850i
$93$	0	0
$94$	7.67919e10	1.07923
$95$	0	0
$96$	0	0
$97$	− 1.10181e11i	− 1.30275i	−0.758755	−	0.651376i	$-0.774192\pi$
$97$	− 1.10181e11i	− 1.30275i	0.758755	−	0.651376i	$-0.225808\pi$
$98$	1.39891e10i	0.156331i
$99$	0	0
$100$	0	0
$101$	3.33271e10	0.315522	0.157761	−	0.987477i	$-0.449572\pi$
$101$	3.33271e10	0.315522	0.157761	+	0.987477i	$0.449572\pi$
$102$	0	0
$103$	8.49419e10i	0.721966i	0.932572	+	0.360983i	$0.117559\pi$
$103$	8.49419e10i	0.721966i	−0.932572	+	0.360983i	$0.882441\pi$
$104$	−3.06580e9	−0.0247093
$105$	0	0
$106$	−2.56229e11	−1.85971
$107$	1.43774e11i	0.990989i	0.868611	+	0.495494i	$0.165013\pi$
$107$	1.43774e11i	0.990989i	−0.868611	+	0.495494i	$0.834987\pi$
$108$	0	0
$109$	3.01296e11	1.87563	0.937816	−	0.347134i	$-0.112845\pi$
$109$	3.01296e11	1.87563	0.937816	+	0.347134i	$0.112845\pi$
$110$	0	0
$111$	0	0
$112$	1.78771e11i	0.958515i
$113$	2.22891e11i	1.13805i	0.822320	+	0.569026i	$0.192680\pi$
$113$	2.22891e11i	1.13805i	−0.822320	+	0.569026i	$0.807320\pi$
$114$	0	0
$115$	0	0
$116$	−2.08495e11	−0.921671
$117$	0	0
$118$	5.08541e11i	2.04633i
$119$	1.58003e11	0.606954
$120$	0	0
$121$	6.32271e11	2.21607
$122$	− 1.32220e11i	− 0.442914i
$123$	0	0
$124$	1.10447e11	0.338324
$125$	0	0
$126$	0	0
$127$	1.79939e11i	0.483287i	0.970365	+	0.241644i	$0.0776865\pi$
$127$	1.79939e11i	0.483287i	−0.970365	+	0.241644i	$0.922313\pi$
$128$	− 1.84863e10i	− 0.0475549i
$129$	0	0
$130$	0	0
$131$	5.65153e11	1.27989	0.639946	−	0.768420i	$-0.278957\pi$
$131$	5.65153e11	1.27989	0.639946	+	0.768420i	$0.278957\pi$
$132$	0	0
$133$	3.94542e11i	0.822071i
$134$	3.57757e11	0.715338
$135$	0	0
$136$	−8.30615e9	−0.0153086
$137$	1.24499e9i	0.00220396i	0.999999	+	0.00110198i	$0.000350771\pi$
$137$	1.24499e9i	0.00220396i	−0.999999	+	0.00110198i	$0.999649\pi$
$138$	0	0
$139$	−4.68084e11	−0.765142	−0.382571	−	0.923926i	$-0.624961\pi$
$139$	−4.68084e11	−0.765142	−0.382571	+	0.923926i	$0.624961\pi$
$140$	0	0
$141$	0	0
$142$	− 9.63736e11i	− 1.40079i
$143$	1.33243e12i	1.86335i
$144$	0	0
$145$	0	0
$146$	4.23470e11	0.528302
$147$	0	0
$148$	9.62981e11i	1.11475i
$149$	1.05439e12	1.17619	0.588096	−	0.808791i	$-0.299878\pi$
$149$	1.05439e12	1.17619	0.588096	+	0.808791i	$0.299878\pi$
$150$	0	0
$151$	4.91301e11	0.509301	0.254651	−	0.967033i	$-0.418040\pi$
$151$	4.91301e11	0.509301	0.254651	+	0.967033i	$0.418040\pi$
$152$	− 2.07410e10i	− 0.0207343i
$153$	0	0
$154$	2.55946e12	2.38114
$155$	0	0
$156$	0	0
$157$	− 6.17375e11i	− 0.516536i	−0.966073	−	0.258268i	$-0.916848\pi$
$157$	− 6.17375e11i	− 0.516536i	0.966073	−	0.258268i	$-0.0831518\pi$
$158$	1.00683e12i	0.813467i
$159$	0	0
$160$	0	0
$161$	−1.27013e12	−0.925346
$162$	0	0
$163$	2.97234e11i	0.202333i	0.994870	+	0.101166i	$0.0322574\pi$
$163$	2.97234e11i	0.202333i	−0.994870	+	0.101166i	$0.967743\pi$
$164$	−1.87158e12	−1.23188
$165$	0	0
$166$	−1.30037e12	−0.800705
$167$	− 2.53524e11i	− 0.151035i	−0.997144	−	0.0755177i	$-0.975939\pi$
$167$	− 2.53524e11i	− 0.151035i	0.997144	−	0.0755177i	$-0.0240609\pi$
$168$	0	0
$169$	−1.42662e11	−0.0796035
$170$	0	0
$171$	0	0
$172$	5.40820e10i	0.0273934i
$173$	− 1.80555e12i	− 0.885843i	−0.896560	−	0.442922i	$-0.853942\pi$
$173$	− 1.80555e12i	− 0.885843i	0.896560	−	0.442922i	$-0.146058\pi$
$174$	0	0
$175$	0	0
$176$	−4.08445e12	−1.82311
$177$	0	0
$178$	− 2.68319e12i	− 1.12549i
$179$	1.63020e12	0.663054	0.331527	−	0.943446i	$-0.392436\pi$
$179$	1.63020e12	0.663054	0.331527	+	0.943446i	$0.392436\pi$
$180$	0	0
$181$	−4.16028e12	−1.59181	−0.795904	−	0.605423i	$-0.793004\pi$
$181$	−4.16028e12	−1.59181	−0.795904	+	0.605423i	$0.793004\pi$
$182$	3.71660e12i	1.37960i
$183$	0	0
$184$	6.67701e10	0.0233391
$185$	0	0
$186$	0	0
$187$	3.60994e12i	1.15444i
$188$	2.42611e12i	0.753429i
$189$	0	0
$190$	0	0
$191$	3.03600e12	0.864208	0.432104	−	0.901824i	$-0.357771\pi$
$191$	3.03600e12	0.864208	0.432104	+	0.901824i	$0.357771\pi$
$192$	0	0
$193$	3.77397e12i	1.01445i	0.861812	+	0.507227i	$0.169330\pi$
$193$	3.77397e12i	1.01445i	−0.861812	+	0.507227i	$0.830670\pi$
$194$	7.02174e12	1.83457
$195$	0	0
$196$	−4.41961e11	−0.109138
$197$	− 1.03079e12i	− 0.247518i	−0.992312	−	0.123759i	$-0.960505\pi$
$197$	− 1.03079e12i	− 0.247518i	0.992312	−	0.123759i	$-0.0394950\pi$
$198$	0	0
$199$	−1.14293e12	−0.259614	−0.129807	−	0.991539i	$-0.541436\pi$
$199$	−1.14293e12	−0.259614	−0.129807	+	0.991539i	$0.541436\pi$
$200$	0	0
$201$	0	0
$202$	2.12391e12i	0.444328i
$203$	− 4.34159e12i	− 0.883935i
$204$	0	0
$205$	0	0
$206$	−5.41328e12	−1.01670
$207$	0	0
$208$	− 5.93105e12i	− 1.05629i
$209$	−9.01423e12	−1.56359
$210$	0	0
$211$	6.96492e12	1.14647	0.573235	−	0.819391i	$-0.305688\pi$
$211$	6.96492e12	1.14647	0.573235	+	0.819391i	$0.305688\pi$
$212$	− 8.09510e12i	− 1.29830i
$213$	0	0
$214$	−9.16259e12	−1.39554
$215$	0	0
$216$	0	0
$217$	2.29989e12i	0.324472i
$218$	1.92014e13i	2.64132i
$219$	0	0
$220$	0	0
$221$	−5.24201e12	−0.668868
$222$	0	0
$223$	6.80403e12i	0.826208i	0.910684	+	0.413104i	$0.135555\pi$
$223$	6.80403e12i	0.826208i	−0.910684	+	0.413104i	$0.864445\pi$
$224$	−1.12037e13	−1.32739
$225$	0	0
$226$	−1.42047e13	−1.60264
$227$	8.00368e12i	0.881348i	0.897667	+	0.440674i	$0.145261\pi$
$227$	8.00368e12i	0.881348i	−0.897667	+	0.440674i	$0.854739\pi$
$228$	0	0
$229$	−1.20624e13	−1.26572	−0.632862	−	0.774265i	$-0.718120\pi$
$229$	−1.20624e13	−1.26572	−0.632862	+	0.774265i	$0.718120\pi$
$230$	0	0
$231$	0	0
$232$	2.28236e11i	0.0222946i
$233$	1.05744e13i	1.00878i	0.863476	+	0.504390i	$0.168283\pi$
$233$	1.05744e13i	1.00878i	−0.863476	+	0.504390i	$0.831717\pi$
$234$	0	0
$235$	0	0
$236$	−1.60665e13	−1.42858
$237$	0	0
$238$	1.00694e13i	0.854731i
$239$	−4.81329e12	−0.399258	−0.199629	−	0.979872i	$-0.563974\pi$
$239$	−4.81329e12	−0.399258	−0.199629	+	0.979872i	$0.563974\pi$
$240$	0	0
$241$	−1.29674e13	−1.02745	−0.513724	−	0.857956i	$-0.671734\pi$
$241$	−1.29674e13	−1.02745	−0.513724	+	0.857956i	$0.671734\pi$
$242$	4.02941e13i	3.12074i
$243$	0	0
$244$	4.17728e12	0.309207
$245$	0	0
$246$	0	0
$247$	− 1.30896e13i	− 0.905928i
$248$	− 1.20904e11i	− 0.00818385i
$249$	0	0
$250$	0	0
$251$	−5.10147e12	−0.323214	−0.161607	−	0.986855i	$-0.551668\pi$
$251$	−5.10147e12	−0.323214	−0.161607	+	0.986855i	$0.551668\pi$
$252$	0	0
$253$	− 2.90190e13i	− 1.76003i
$254$	−1.14674e13	−0.680580
$255$	0	0
$256$	1.81712e13	1.03291
$257$	− 1.20976e13i	− 0.673079i	−0.941669	−	0.336539i	$-0.890744\pi$
$257$	− 1.20976e13i	− 0.673079i	0.941669	−	0.336539i	$-0.109256\pi$
$258$	0	0
$259$	−2.00526e13	−1.06911
$260$	0	0
$261$	0	0
$262$	3.60167e13i	1.80238i
$263$	1.09431e13i	0.536271i	0.963381	+	0.268135i	$0.0864074\pi$
$263$	1.09431e13i	0.536271i	−0.963381	+	0.268135i	$0.913593\pi$
$264$	0	0
$265$	0	0
$266$	−2.51439e13	−1.15767
$267$	0	0
$268$	1.13027e13i	0.499391i
$269$	1.23621e13	0.535125	0.267562	−	0.963541i	$-0.413782\pi$
$269$	1.23621e13	0.535125	0.267562	+	0.963541i	$0.413782\pi$
$270$	0	0
$271$	1.34683e13	0.559734	0.279867	−	0.960039i	$-0.409710\pi$
$271$	1.34683e13	0.559734	0.279867	+	0.960039i	$0.409710\pi$
$272$	− 1.60690e13i	− 0.654424i
$273$	0	0
$274$	−7.93424e10	−0.00310368
$275$	0	0
$276$	0	0
$277$	− 2.20976e13i	− 0.814153i	−0.913394	−	0.407077i	$-0.866548\pi$
$277$	− 2.20976e13i	− 0.814153i	0.913394	−	0.407077i	$-0.133452\pi$
$278$	− 2.98306e13i	− 1.07750i
$279$	0	0
$280$	0	0
$281$	−7.15247e12	−0.243541	−0.121770	−	0.992558i	$-0.538857\pi$
$281$	−7.15247e12	−0.243541	−0.121770	+	0.992558i	$0.538857\pi$
$282$	0	0
$283$	− 9.90996e12i	− 0.324524i	−0.986748	−	0.162262i	$-0.948121\pi$
$283$	− 9.90996e12i	− 0.324524i	0.986748	−	0.162262i	$-0.0518789\pi$
$284$	3.04476e13	0.977917
$285$	0	0
$286$	−8.49145e13	−2.62403
$287$	− 3.89729e13i	− 1.18144i
$288$	0	0
$289$	2.00697e13	0.585604
$290$	0	0
$291$	0	0
$292$	1.33788e13i	0.368818i
$293$	6.76585e13i	1.83042i	0.402980	+	0.915209i	$0.367974\pi$
$293$	6.76585e13i	1.83042i	−0.402980	+	0.915209i	$0.632026\pi$
$294$	0	0
$295$	0	0
$296$	1.05416e12	0.0269651
$297$	0	0
$298$	6.71957e13i	1.65635i
$299$	4.21386e13	1.01974
$300$	0	0
$301$	−1.12618e12	−0.0262718
$302$	3.13103e13i	0.717213i
$303$	0	0
$304$	4.01252e13	0.886364
$305$	0	0
$306$	0	0
$307$	2.76335e13i	0.578328i	0.957280	+	0.289164i	$0.0933773\pi$
$307$	2.76335e13i	0.578328i	−0.957280	+	0.289164i	$0.906623\pi$
$308$	8.08617e13i	1.66232i
$309$	0	0
$310$	0	0
$311$	1.21728e13	0.237251	0.118626	−	0.992939i	$-0.462151\pi$
$311$	1.21728e13	0.237251	0.118626	+	0.992939i	$0.462151\pi$
$312$	0	0
$313$	8.17271e13i	1.53770i	0.639428	+	0.768851i	$0.279171\pi$
$313$	8.17271e13i	1.53770i	−0.639428	+	0.768851i	$0.720829\pi$
$314$	3.93448e13	0.727402
$315$	0	0
$316$	−3.18089e13	−0.567897
$317$	7.61714e13i	1.33649i	0.743941	+	0.668246i	$0.232955\pi$
$317$	7.61714e13i	1.33649i	−0.743941	+	0.668246i	$0.767045\pi$
$318$	0	0
$319$	9.91937e13	1.68126
$320$	0	0
$321$	0	0
$322$	− 8.09441e13i	− 1.30310i
$323$	− 3.54636e13i	− 0.561267i
$324$	0	0
$325$	0	0
$326$	−1.89425e13	−0.284931
$327$	0	0
$328$	2.04880e12i	0.0297984i
$329$	−5.05201e13	−0.722581
$330$	0	0
$331$	8.05093e13	1.11376	0.556881	−	0.830592i	$-0.311998\pi$
$331$	8.05093e13	1.11376	0.556881	+	0.830592i	$0.311998\pi$
$332$	− 4.10830e13i	− 0.558988i
$333$	0	0
$334$	1.61569e13	0.212693
$335$	0	0
$336$	0	0
$337$	7.90669e13i	0.990901i	0.868636	+	0.495451i	$0.164997\pi$
$337$	7.90669e13i	0.990901i	−0.868636	+	0.495451i	$0.835003\pi$
$338$	− 9.09176e12i	− 0.112100i
$339$	0	0
$340$	0	0
$341$	−5.25463e13	−0.617153
$342$	0	0
$343$	− 9.21053e13i	− 1.04753i
$344$	5.92027e10	0.000662629 0
$345$	0	0
$346$	1.15067e14	1.24747
$347$	1.40156e14i	1.49554i	0.663956	+	0.747772i	$0.268876\pi$
$347$	1.40156e14i	1.49554i	−0.663956	+	0.747772i	$0.731124\pi$
$348$	0	0
$349$	9.88347e13	1.02181	0.510905	−	0.859637i	$-0.329311\pi$
$349$	9.88347e13	1.02181	0.510905	+	0.859637i	$0.329311\pi$
$350$	0	0
$351$	0	0
$352$	− 2.55975e14i	− 2.52472i
$353$	2.74254e13i	0.266313i	0.991095	+	0.133156i	$0.0425113\pi$
$353$	2.74254e13i	0.266313i	−0.991095	+	0.133156i	$0.957489\pi$
$354$	0	0
$355$	0	0
$356$	8.47708e13	0.785726
$357$	0	0
$358$	1.03891e14i	0.933733i
$359$	1.57856e14	1.39715	0.698574	−	0.715538i	$-0.253818\pi$
$359$	1.57856e14	1.39715	0.698574	+	0.715538i	$0.253818\pi$
$360$	0	0
$361$	−2.79354e13	−0.239809
$362$	− 2.65132e14i	− 2.24163i
$363$	0	0
$364$	−1.17420e14	−0.963127
$365$	0	0
$366$	0	0
$367$	1.95534e14i	1.53306i	0.642210	+	0.766529i	$0.278018\pi$
$367$	1.95534e14i	1.53306i	−0.642210	+	0.766529i	$0.721982\pi$
$368$	1.29173e14i	0.997717i
$369$	0	0
$370$	0	0
$371$	1.68568e14	1.24514
$372$	0	0
$373$	8.35940e13i	0.599482i	0.954021	+	0.299741i	$0.0969003\pi$
$373$	8.35940e13i	0.599482i	−0.954021	+	0.299741i	$0.903100\pi$
$374$	−2.30059e14	−1.62572
$375$	0	0
$376$	2.65583e12	0.0182249
$377$	1.44040e14i	0.974102i
$378$	0	0
$379$	−2.81905e14	−1.85177	−0.925887	−	0.377802i	$-0.876680\pi$
$379$	−2.81905e14	−1.85177	−0.925887	+	0.377802i	$0.876680\pi$
$380$	0	0
$381$	0	0
$382$	1.93482e14i	1.21700i
$383$	− 2.51689e13i	− 0.156053i	−0.996951	−	0.0780264i	$-0.975138\pi$
$383$	− 2.51689e13i	− 0.156053i	0.996951	−	0.0780264i	$-0.0248618\pi$
$384$	0	0
$385$	0	0
$386$	−2.40512e14	−1.42859
$387$	0	0
$388$	2.21840e14i	1.28075i
$389$	−1.44036e14	−0.819879	−0.409939	−	0.912113i	$-0.634450\pi$
$389$	−1.44036e14	−0.819879	−0.409939	+	0.912113i	$0.634450\pi$
$390$	0	0
$391$	1.14166e14	0.631778
$392$	4.83809e11i	0.00263998i
$393$	0	0
$394$	6.56916e13	0.348563
$395$	0	0
$396$	0	0
$397$	− 3.78848e13i	− 0.192804i	−0.995342	−	0.0964021i	$-0.969267\pi$
$397$	− 3.78848e13i	− 0.192804i	0.995342	−	0.0964021i	$-0.0307335\pi$
$398$	− 7.28380e13i	− 0.365596i
$399$	0	0
$400$	0	0
$401$	−1.24111e13	−0.0597747	−0.0298874	−	0.999553i	$-0.509515\pi$
$401$	−1.24111e13	−0.0597747	−0.0298874	+	0.999553i	$0.509515\pi$
$402$	0	0
$403$	− 7.63027e13i	− 0.357571i
$404$	−6.71012e13	−0.310194
$405$	0	0
$406$	2.76686e14	1.24478
$407$	− 4.58149e14i	− 2.03347i
$408$	0	0
$409$	4.37138e14	1.88860	0.944301	−	0.329082i	$-0.106739\pi$
$409$	4.37138e14	1.88860	0.944301	+	0.329082i	$0.106739\pi$
$410$	0	0
$411$	0	0
$412$	− 1.71023e14i	− 0.709775i
$413$	− 3.34560e14i	− 1.37009i
$414$	0	0
$415$	0	0
$416$	3.71702e14	1.46279
$417$	0	0
$418$	− 5.74470e14i	− 2.20190i
$419$	5.14822e14	1.94751	0.973755	−	0.227598i	$-0.0730871\pi$
$419$	5.14822e14	1.94751	0.973755	+	0.227598i	$0.0730871\pi$
$420$	0	0
$421$	−3.90682e14	−1.43970	−0.719850	−	0.694130i	$-0.755789\pi$
$421$	−3.90682e14	−1.43970	−0.719850	+	0.694130i	$0.755789\pi$
$422$	4.43869e14i	1.61449i
$423$	0	0
$424$	−8.86159e12	−0.0314050
$425$	0	0
$426$	0	0
$427$	8.69855e13i	0.296547i
$428$	− 2.89476e14i	− 0.974254i
$429$	0	0
$430$	0	0
$431$	3.29050e14	1.06571	0.532853	−	0.846208i	$-0.321120\pi$
$431$	3.29050e14	1.06571	0.532853	+	0.846208i	$0.321120\pi$
$432$	0	0
$433$	− 5.59793e14i	− 1.76744i	−0.468019	−	0.883718i	$-0.655032\pi$
$433$	− 5.59793e14i	− 1.76744i	0.468019	−	0.883718i	$-0.344968\pi$
$434$	−1.46570e14	−0.456932
$435$	0	0
$436$	−6.06634e14	−1.84396
$437$	2.85079e14i	0.855693i
$438$	0	0
$439$	−1.25752e14	−0.368095	−0.184047	−	0.982917i	$-0.558920\pi$
$439$	−1.25752e14	−0.368095	−0.184047	+	0.982917i	$0.558920\pi$
$440$	0	0
$441$	0	0
$442$	− 3.34069e14i	− 0.941920i
$443$	− 3.33211e14i	− 0.927894i	−0.885863	−	0.463947i	$-0.846433\pi$
$443$	− 3.33211e14i	− 0.927894i	0.885863	−	0.463947i	$-0.153567\pi$
$444$	0	0
$445$	0	0
$446$	−4.33616e14	−1.16349
$447$	0	0
$448$	− 3.47880e14i	− 0.910753i
$449$	−1.08196e14	−0.279804	−0.139902	−	0.990165i	$-0.544679\pi$
$449$	−1.08196e14	−0.279804	−0.139902	+	0.990165i	$0.544679\pi$
$450$	0	0
$451$	8.90427e14	2.24713
$452$	− 4.48773e14i	− 1.11883i
$453$	0	0
$454$	−5.10068e14	−1.24114
$455$	0	0
$456$	0	0
$457$	1.81057e14i	0.424890i	0.977173	+	0.212445i	$0.0681426\pi$
$457$	1.81057e14i	0.424890i	−0.977173	+	0.212445i	$0.931857\pi$
$458$	− 7.68728e14i	− 1.78243i
$459$	0	0
$460$	0	0
$461$	3.74610e14	0.837962	0.418981	−	0.907995i	$-0.362387\pi$
$461$	3.74610e14	0.837962	0.418981	+	0.907995i	$0.362387\pi$
$462$	0	0
$463$	2.33341e13i	0.0509678i	0.999675	+	0.0254839i	$0.00811266\pi$
$463$	2.33341e13i	0.0509678i	−0.999675	+	0.0254839i	$0.991887\pi$
$464$	−4.41542e14	−0.953067
$465$	0	0
$466$	−6.73896e14	−1.42060
$467$	− 7.37382e14i	− 1.53621i	−0.640326	−	0.768103i	$-0.721201\pi$
$467$	− 7.37382e14i	− 1.53621i	0.640326	−	0.768103i	$-0.278799\pi$
$468$	0	0
$469$	−2.35362e14	−0.478945
$470$	0	0
$471$	0	0
$472$	1.75877e13i	0.0345564i
$473$	− 2.57301e13i	− 0.0499695i
$474$	0	0
$475$	0	0
$476$	−3.18125e14	−0.596704
$477$	0	0
$478$	− 3.06747e14i	− 0.562247i
$479$	−3.39424e14	−0.615030	−0.307515	−	0.951543i	$-0.599497\pi$
$479$	−3.39424e14	−0.615030	−0.307515	+	0.951543i	$0.599497\pi$
$480$	0	0
$481$	6.65281e14	1.17817
$482$	− 8.26404e14i	− 1.44688i
$483$	0	0
$484$	−1.27302e15	−2.17865
$485$	0	0
$486$	0	0
$487$	3.27468e14i	0.541700i	0.962622	+	0.270850i	$0.0873048\pi$
$487$	3.27468e14i	0.541700i	−0.962622	+	0.270850i	$0.912695\pi$
$488$	− 4.57280e12i	− 0.00747952i
$489$	0	0
$490$	0	0
$491$	3.28187e14	0.519006	0.259503	−	0.965742i	$-0.416441\pi$
$491$	3.28187e14	0.519006	0.259503	+	0.965742i	$0.416441\pi$
$492$	0	0
$493$	3.90246e14i	0.603504i
$494$	8.34191e14	1.27576
$495$	0	0
$496$	2.33900e14	0.349849
$497$	6.34025e14i	0.937878i
$498$	0	0
$499$	−2.84812e14	−0.412103	−0.206052	−	0.978541i	$-0.566061\pi$
$499$	−2.84812e14	−0.412103	−0.206052	+	0.978541i	$0.566061\pi$
$500$	0	0
$501$	0	0
$502$	− 3.25112e14i	− 0.455159i
$503$	2.01563e13i	0.0279117i	0.999903	+	0.0139559i	$0.00444243\pi$
$503$	2.01563e13i	0.0279117i	−0.999903	+	0.0139559i	$0.995558\pi$
$504$	0	0
$505$	0	0
$506$	1.84936e15	2.47852
$507$	0	0
$508$	− 3.62292e14i	− 0.475126i
$509$	−6.91697e14	−0.897362	−0.448681	−	0.893692i	$-0.648106\pi$
$509$	−6.91697e14	−0.897362	−0.448681	+	0.893692i	$0.648106\pi$
$510$	0	0
$511$	−2.78594e14	−0.353717
$512$	1.12018e15i	1.40703i
$513$	0	0
$514$	7.70969e14	0.947850
$515$	0	0
$516$	0	0
$517$	− 1.15425e15i	− 1.37436i
$518$	− 1.27794e15i	− 1.50555i
$519$	0	0
$520$	0	0
$521$	−5.00013e14	−0.570656	−0.285328	−	0.958430i	$-0.592102\pi$
$521$	−5.00013e14	−0.570656	−0.285328	+	0.958430i	$0.592102\pi$
$522$	0	0
$523$	− 2.89272e14i	− 0.323256i	−0.986852	−	0.161628i	$-0.948326\pi$
$523$	− 2.89272e14i	− 0.323256i	0.986852	−	0.161628i	$-0.0516745\pi$
$524$	−1.13789e15	−1.25828
$525$	0	0
$526$	−6.97396e14	−0.755193
$527$	− 2.06727e14i	− 0.221533i
$528$	0	0
$529$	3.50713e13	0.0368083
$530$	0	0
$531$	0	0
$532$	− 7.94377e14i	− 0.808188i
$533$	1.29299e15i	1.30196i
$534$	0	0
$535$	0	0
$536$	1.23729e13	0.0120800
$537$	0	0
$538$	7.87828e14i	0.753579i
$539$	2.10268e14	0.199084
$540$	0	0
$541$	1.20798e15	1.12066	0.560331	−	0.828269i	$-0.310674\pi$
$541$	1.20798e15	1.12066	0.560331	+	0.828269i	$0.310674\pi$
$542$	8.58325e14i	0.788235i
$543$	0	0
$544$	1.00705e15	0.906271
$545$	0	0
$546$	0	0
$547$	1.74686e15i	1.52521i	0.646867	+	0.762603i	$0.276079\pi$
$547$	1.74686e15i	1.52521i	−0.646867	+	0.762603i	$0.723921\pi$
$548$	− 2.50668e12i	− 0.00216674i
$549$	0	0
$550$	0	0
$551$	−9.74468e14	−0.817398
$552$	0	0
$553$	− 6.62373e14i	− 0.544646i
$554$	1.40826e15	1.14652
$555$	0	0
$556$	9.42447e14	0.752221
$557$	8.14516e14i	0.643719i	0.946787	+	0.321859i	$0.104308\pi$
$557$	8.14516e14i	0.643719i	−0.946787	+	0.321859i	$0.895692\pi$
$558$	0	0
$559$	3.73628e13	0.0289517
$560$	0	0
$561$	0	0
$562$	− 4.55821e14i	− 0.342961i
$563$	8.94179e14i	0.666237i	0.942885	+	0.333118i	$0.108101\pi$
$563$	8.94179e14i	0.666237i	−0.942885	+	0.333118i	$0.891899\pi$
$564$	0	0
$565$	0	0
$566$	6.31554e14	0.457004
$567$	0	0
$568$	− 3.33305e13i	− 0.0236552i
$569$	5.58478e14	0.392544	0.196272	−	0.980549i	$-0.437116\pi$
$569$	5.58478e14	0.392544	0.196272	+	0.980549i	$0.437116\pi$
$570$	0	0
$571$	1.19826e15	0.826136	0.413068	−	0.910700i	$-0.364457\pi$
$571$	1.19826e15	0.826136	0.413068	+	0.910700i	$0.364457\pi$
$572$	− 2.68273e15i	− 1.83189i
$573$	0	0
$574$	2.48371e15	1.66374
$575$	0	0
$576$	0	0
$577$	1.70453e15i	1.10953i	0.832008	+	0.554764i	$0.187192\pi$
$577$	1.70453e15i	1.10953i	−0.832008	+	0.554764i	$0.812808\pi$
$578$	1.27903e15i	0.824665i
$579$	0	0
$580$	0	0
$581$	8.55491e14	0.536101
$582$	0	0
$583$	3.85134e15i	2.36829i
$584$	1.46456e13	0.00892147
$585$	0	0
$586$	−4.31182e15	−2.57765
$587$	8.77553e14i	0.519713i	0.965647	+	0.259857i	$0.0836753\pi$
$587$	8.77553e14i	0.519713i	−0.965647	+	0.259857i	$0.916325\pi$
$588$	0	0
$589$	5.16209e14	0.300048
$590$	0	0
$591$	0	0
$592$	2.03937e15i	1.15273i
$593$	− 3.16216e15i	− 1.77086i	−0.464776	−	0.885428i	$-0.653865\pi$
$593$	− 3.16216e15i	− 1.77086i	0.464776	−	0.885428i	$-0.346135\pi$
$594$	0	0
$595$	0	0
$596$	−2.12293e15	−1.15633
$597$	0	0
$598$	2.68546e15i	1.43603i
$599$	1.86633e15	0.988873	0.494437	−	0.869214i	$-0.335374\pi$
$599$	1.86633e15	0.988873	0.494437	+	0.869214i	$0.335374\pi$
$600$	0	0
$601$	8.57728e14	0.446211	0.223105	−	0.974794i	$-0.428381\pi$
$601$	8.57728e14	0.446211	0.223105	+	0.974794i	$0.428381\pi$
$602$	− 7.17703e13i	− 0.0369968i
$603$	0	0
$604$	−9.89193e14	−0.500701
$605$	0	0
$606$	0	0
$607$	− 2.90938e15i	− 1.43306i	−0.697558	−	0.716528i	$-0.745730\pi$
$607$	− 2.90938e15i	− 1.43306i	0.697558	−	0.716528i	$-0.254270\pi$
$608$	2.51467e15i	1.22747i
$609$	0	0
$610$	0	0
$611$	1.67609e15	0.796289
$612$	0	0
$613$	− 3.95415e15i	− 1.84510i	−0.385875	−	0.922551i	$-0.626100\pi$
$613$	− 3.95415e15i	− 1.84510i	0.385875	−	0.922551i	$-0.373900\pi$
$614$	−1.76106e15	−0.814419
$615$	0	0
$616$	8.85181e13	0.0402104
$617$	− 1.44545e15i	− 0.650778i	−0.945580	−	0.325389i	$-0.894505\pi$
$617$	− 1.44545e15i	− 0.650778i	0.945580	−	0.325389i	$-0.105495\pi$
$618$	0	0
$619$	4.27132e15	1.88914	0.944569	−	0.328314i	$-0.106481\pi$
$619$	4.27132e15	1.88914	0.944569	+	0.328314i	$0.106481\pi$
$620$	0	0
$621$	0	0
$622$	7.75764e14i	0.334105i
$623$	1.76522e15i	0.753556i
$624$	0	0
$625$	0	0
$626$	−5.20840e15	−2.16544
$627$	0	0
$628$	1.24303e15i	0.507813i
$629$	1.80244e15	0.729933
$630$	0	0
$631$	2.19799e15	0.874711	0.437355	−	0.899289i	$-0.355915\pi$
$631$	2.19799e15	0.874711	0.437355	+	0.899289i	$0.355915\pi$
$632$	3.48207e13i	0.0137371i
$633$	0	0
$634$	−4.85435e15	−1.88209
$635$	0	0
$636$	0	0
$637$	3.05331e14i	0.115347i
$638$	6.32154e15i	2.36760i
$639$	0	0
$640$	0	0
$641$	6.69744e14	0.244450	0.122225	−	0.992502i	$-0.460997\pi$
$641$	6.69744e14	0.244450	0.122225	+	0.992502i	$0.460997\pi$
$642$	0	0
$643$	3.43086e15i	1.23096i	0.788154	+	0.615478i	$0.211037\pi$
$643$	3.43086e15i	1.23096i	−0.788154	+	0.615478i	$0.788963\pi$
$644$	2.55729e15	0.909720
$645$	0	0
$646$	2.26007e15	0.790393
$647$	− 2.12502e15i	− 0.736869i	−0.929654	−	0.368435i	$-0.879894\pi$
$647$	− 2.12502e15i	− 0.736869i	0.929654	−	0.368435i	$-0.120106\pi$
$648$	0	0
$649$	7.64381e15	2.60594
$650$	0	0
$651$	0	0
$652$	− 5.98455e14i	− 0.198916i
$653$	− 4.76025e15i	− 1.56894i	−0.620166	−	0.784471i	$-0.712935\pi$
$653$	− 4.76025e15i	− 1.56894i	0.620166	−	0.784471i	$-0.287065\pi$
$654$	0	0
$655$	0	0
$656$	−3.96357e15	−1.27384
$657$	0	0
$658$	− 3.21961e15i	− 1.01756i
$659$	1.31734e15	0.412886	0.206443	−	0.978459i	$-0.433811\pi$
$659$	1.31734e15	0.412886	0.206443	+	0.978459i	$0.433811\pi$
$660$	0	0
$661$	−2.45528e15	−0.756822	−0.378411	−	0.925638i	$-0.623529\pi$
$661$	−2.45528e15	−0.756822	−0.378411	+	0.925638i	$0.623529\pi$
$662$	5.13080e15i	1.56843i
$663$	0	0
$664$	−4.49729e13	−0.0135215
$665$	0	0
$666$	0	0
$667$	− 3.13705e15i	− 0.920087i
$668$	5.10449e14i	0.148485i
$669$	0	0
$670$	0	0
$671$	−1.98739e15	−0.564038
$672$	0	0
$673$	− 8.60705e14i	− 0.240310i	−0.992755	−	0.120155i	$-0.961661\pi$
$673$	− 8.60705e14i	− 0.240310i	0.992755	−	0.120155i	$-0.0383391\pi$
$674$	−5.03888e15	−1.39542
$675$	0	0
$676$	2.87238e14	0.0782593
$677$	1.90003e15i	0.513479i	0.966481	+	0.256739i	$0.0826482\pi$
$677$	1.90003e15i	0.513479i	−0.966481	+	0.256739i	$0.917352\pi$
$678$	0	0
$679$	−4.61948e15	−1.22831
$680$	0	0
$681$	0	0
$682$	− 3.34873e15i	− 0.869093i
$683$	2.35084e15i	0.605214i	0.953115	+	0.302607i	$0.0978570\pi$
$683$	2.35084e15i	0.605214i	−0.953115	+	0.302607i	$0.902143\pi$
$684$	0	0
$685$	0	0
$686$	5.86980e15	1.47516
$687$	0	0
$688$	1.14533e14i	0.0283265i
$689$	−5.59255e15	−1.37216
$690$	0	0
$691$	−6.60668e15	−1.59534	−0.797672	−	0.603092i	$-0.793935\pi$
$691$	−6.60668e15	−1.59534	−0.797672	+	0.603092i	$0.793935\pi$
$692$	3.63533e15i	0.870884i
$693$	0	0
$694$	−8.93203e15	−2.10607
$695$	0	0
$696$	0	0
$697$	3.50310e15i	0.806627i
$698$	6.29866e15i	1.43894i
$699$	0	0
$700$	0	0
$701$	8.01300e14	0.178791	0.0893956	−	0.995996i	$-0.471506\pi$
$701$	8.01300e14	0.178791	0.0893956	+	0.995996i	$0.471506\pi$
$702$	0	0
$703$	4.50081e15i	0.988636i
$704$	7.94813e15	1.73227
$705$	0	0
$706$	−1.74780e15	−0.375030
$707$	− 1.39728e15i	− 0.297493i
$708$	0	0
$709$	−4.10789e15	−0.861121	−0.430560	−	0.902562i	$-0.641684\pi$
$709$	−4.10789e15	−0.861121	−0.430560	+	0.902562i	$0.641684\pi$
$710$	0	0
$711$	0	0
$712$	− 9.27973e13i	− 0.0190062i
$713$	1.66180e15i	0.337743i
$714$	0	0
$715$	0	0
$716$	−3.28227e15	−0.651857
$717$	0	0
$718$	1.00601e16i	1.96751i
$719$	2.12875e15	0.413157	0.206578	−	0.978430i	$-0.433767\pi$
$719$	2.12875e15	0.413157	0.206578	+	0.978430i	$0.433767\pi$
$720$	0	0
$721$	3.56130e15	0.680714
$722$	− 1.78030e15i	− 0.337706i
$723$	0	0
$724$	8.37637e15	1.56493
$725$	0	0
$726$	0	0
$727$	4.60450e15i	0.840898i	0.907316	+	0.420449i	$0.138127\pi$
$727$	4.60450e15i	0.840898i	−0.907316	+	0.420449i	$0.861873\pi$
$728$	1.28538e14i	0.0232974i
$729$	0	0
$730$	0	0
$731$	1.01227e14	0.0179370
$732$	0	0
$733$	− 3.94830e15i	− 0.689190i	−0.938752	−	0.344595i	$-0.888016\pi$
$733$	− 3.94830e15i	− 0.689190i	0.938752	−	0.344595i	$-0.111984\pi$
$734$	−1.24612e16	−2.15890
$735$	0	0
$736$	−8.09532e15	−1.38168
$737$	− 5.37739e15i	− 0.910962i
$738$	0	0
$739$	−2.01640e15	−0.336537	−0.168269	−	0.985741i	$-0.553818\pi$
$739$	−2.01640e15	−0.336537	−0.168269	+	0.985741i	$0.553818\pi$
$740$	0	0
$741$	0	0
$742$	1.07427e16i	1.75345i
$743$	7.15627e15i	1.15944i	0.814816	+	0.579720i	$0.196838\pi$
$743$	7.15627e15i	1.15944i	−0.814816	+	0.579720i	$0.803162\pi$
$744$	0	0
$745$	0	0
$746$	−5.32738e15	−0.844209
$747$	0	0
$748$	− 7.26830e15i	− 1.13494i
$749$	6.02791e15	0.934365
$750$	0	0
$751$	−2.55257e14	−0.0389904	−0.0194952	−	0.999810i	$-0.506206\pi$
$751$	−2.55257e14	−0.0389904	−0.0194952	+	0.999810i	$0.506206\pi$
$752$	5.13792e15i	0.779094i
$753$	0	0
$754$	−9.17953e15	−1.37176
$755$	0	0
$756$	0	0
$757$	5.67357e15i	0.829524i	0.909930	+	0.414762i	$0.136135\pi$
$757$	5.67357e15i	0.829524i	−0.909930	+	0.414762i	$0.863865\pi$
$758$	− 1.79656e16i	− 2.60772i
$759$	0	0
$760$	0	0
$761$	−8.59220e15	−1.22036	−0.610181	−	0.792262i	$-0.708903\pi$
$761$	−8.59220e15	−1.22036	−0.610181	+	0.792262i	$0.708903\pi$
$762$	0	0
$763$	− 1.26322e16i	− 1.76846i
$764$	−6.11273e15	−0.849614
$765$	0	0
$766$	1.60400e15	0.219758
$767$	1.10996e16i	1.50985i
$768$	0	0
$769$	−1.23682e16	−1.65849	−0.829243	−	0.558888i	$-0.811228\pi$
$769$	−1.23682e16	−1.65849	−0.829243	+	0.558888i	$0.811228\pi$
$770$	0	0
$771$	0	0
$772$	− 7.59856e15i	− 0.997324i
$773$	− 8.62674e15i	− 1.12424i	−0.827055	−	0.562120i	$-0.809986\pi$
$773$	− 8.62674e15i	− 1.12424i	0.827055	−	0.562120i	$-0.190014\pi$
$774$	0	0
$775$	0	0
$776$	2.42845e14	0.0309806
$777$	0	0
$778$	− 9.17933e15i	− 1.15458i
$779$	−8.74746e15	−1.09251
$780$	0	0
$781$	−1.44858e16	−1.78386
$782$	7.27571e15i	0.889689i
$783$	0	0
$784$	−9.35970e14	−0.112856
$785$	0	0
$786$	0	0
$787$	− 1.18931e16i	− 1.40421i	−0.712073	−	0.702106i	$-0.752243\pi$
$787$	− 1.18931e16i	− 1.40421i	0.712073	−	0.702106i	$-0.247757\pi$
$788$	2.07541e15i	0.243338i
$789$	0	0
$790$	0	0
$791$	9.34502e15	1.07302
$792$	0	0
$793$	− 2.88589e15i	− 0.326797i
$794$	2.41437e15	0.271513
$795$	0	0
$796$	2.30119e15	0.255230
$797$	− 6.33100e15i	− 0.697351i	−0.937243	−	0.348676i	$-0.886631\pi$
$797$	− 6.33100e15i	− 0.697351i	0.937243	−	0.348676i	$-0.113369\pi$
$798$	0	0
$799$	4.54103e15	0.493340
$800$	0	0
$801$	0	0
$802$	− 7.90952e14i	− 0.0841766i
$803$	− 6.36512e15i	− 0.672777i
$804$	0	0
$805$	0	0
$806$	4.86271e15	0.503542
$807$	0	0
$808$	7.34547e13i	0.00750339i
$809$	1.06496e16	1.08048	0.540238	−	0.841512i	$-0.318334\pi$
$809$	1.06496e16	1.08048	0.540238	+	0.841512i	$0.318334\pi$
$810$	0	0
$811$	6.79444e15	0.680047	0.340024	−	0.940417i	$-0.389565\pi$
$811$	6.79444e15	0.680047	0.340024	+	0.940417i	$0.389565\pi$
$812$	8.74141e15i	0.869008i
$813$	0	0
$814$	2.91975e16	2.86359
$815$	0	0
$816$	0	0
$817$	2.52770e14i	0.0242943i
$818$	2.78585e16i	2.65959i
$819$	0	0
$820$	0	0
$821$	−2.03099e16	−1.90029	−0.950145	−	0.311807i	$-0.899066\pi$
$821$	−2.03099e16	−1.90029	−0.950145	+	0.311807i	$0.899066\pi$
$822$	0	0
$823$	− 1.88821e15i	− 0.174322i	−0.996194	−	0.0871609i	$-0.972221\pi$
$823$	− 1.88821e15i	− 0.174322i	0.996194	−	0.0871609i	$-0.0277794\pi$
$824$	−1.87217e14	−0.0171690
$825$	0	0
$826$	2.13213e16	1.92940
$827$	− 1.10204e16i	− 0.990644i	−0.868710	−	0.495322i	$-0.835050\pi$
$827$	− 1.10204e16i	− 0.990644i	0.868710	−	0.495322i	$-0.164950\pi$
$828$	0	0
$829$	8.45180e15	0.749720	0.374860	−	0.927081i	$-0.377691\pi$
$829$	8.45180e15	0.749720	0.374860	+	0.927081i	$0.377691\pi$
$830$	0	0
$831$	0	0
$832$	1.15415e16i	1.00366i
$833$	8.27233e14i	0.0714629i
$834$	0	0
$835$	0	0
$836$	1.81494e16	1.53719
$837$	0	0
$838$	3.28092e16i	2.74254i
$839$	5.30367e15	0.440439	0.220219	−	0.975450i	$-0.429323\pi$
$839$	5.30367e15	0.440439	0.220219	+	0.975450i	$0.429323\pi$
$840$	0	0
$841$	−1.47735e15	−0.121089
$842$	− 2.48979e16i	− 2.02743i
$843$	0	0
$844$	−1.40233e16	−1.12711
$845$	0	0
$846$	0	0
$847$	− 2.65088e16i	− 2.08945i
$848$	− 1.71435e16i	− 1.34253i
$849$	0	0
$850$	0	0
$851$	−1.44892e16	−1.11284
$852$	0	0
$853$	− 4.43767e15i	− 0.336462i	−0.985748	−	0.168231i	$-0.946195\pi$
$853$	− 4.43767e15i	− 0.336462i	0.985748	−	0.168231i	$-0.0538054\pi$
$854$	−5.54352e15	−0.417607
$855$	0	0
$856$	−3.16885e14	−0.0235666
$857$	− 2.12803e16i	− 1.57247i	−0.617926	−	0.786236i	$-0.712027\pi$
$857$	− 2.12803e16i	− 1.57247i	0.617926	−	0.786236i	$-0.287973\pi$
$858$	0	0
$859$	4.61602e15	0.336748	0.168374	−	0.985723i	$-0.446148\pi$
$859$	4.61602e15	0.336748	0.168374	+	0.985723i	$0.446148\pi$
$860$	0	0
$861$	0	0
$862$	2.09701e16i	1.50076i
$863$	− 6.10081e15i	− 0.433839i	−0.976190	−	0.216919i	$-0.930399\pi$
$863$	− 6.10081e15i	− 0.433839i	0.976190	−	0.216919i	$-0.0696009\pi$
$864$	0	0
$865$	0	0
$866$	3.56752e16	2.48896
$867$	0	0
$868$	− 4.63062e15i	− 0.318993i
$869$	1.51334e16	1.03593
$870$	0	0
$871$	7.80854e15	0.527800
$872$	6.64073e14i	0.0446041i
$873$	0	0
$874$	−1.81679e16	−1.20501
$875$	0	0
$876$	0	0
$877$	− 9.32501e14i	− 0.0606948i	−0.999539	−	0.0303474i	$-0.990339\pi$
$877$	− 9.32501e14i	− 0.0606948i	0.999539	−	0.0303474i	$-0.00966137\pi$
$878$	− 8.01407e15i	− 0.518362i
$879$	0	0
$880$	0	0
$881$	1.26049e16	0.800150	0.400075	−	0.916482i	$-0.368984\pi$
$881$	1.26049e16	0.800150	0.400075	+	0.916482i	$0.368984\pi$
$882$	0	0
$883$	2.18236e16i	1.36818i	0.729397	+	0.684090i	$0.239801\pi$
$883$	2.18236e16i	1.36818i	−0.729397	+	0.684090i	$0.760199\pi$
$884$	1.05543e16	0.657572
$885$	0	0
$886$	2.12353e16	1.30669
$887$	− 2.71175e16i	− 1.65832i	−0.559008	−	0.829162i	$-0.688818\pi$
$887$	− 2.71175e16i	− 1.65832i	0.559008	−	0.829162i	$-0.311182\pi$
$888$	0	0
$889$	7.54420e15	0.455673
$890$	0	0
$891$	0	0
$892$	− 1.36993e16i	− 0.812256i
$893$	1.13392e16i	0.668190i
$894$	0	0
$895$	0	0
$896$	−7.75064e14	−0.0448377
$897$	0	0
$898$	− 6.89522e15i	− 0.394029i
$899$	−5.68042e15	−0.322628
$900$	0	0
$901$	−1.51519e16	−0.850119
$902$	5.67463e16i	3.16447i
$903$	0	0
$904$	−4.91265e14	−0.0270638
$905$	0	0
$906$	0	0
$907$	− 2.41689e16i	− 1.30742i	−0.756744	−	0.653712i	$-0.773211\pi$
$907$	− 2.41689e16i	− 1.30742i	0.756744	−	0.653712i	$-0.226789\pi$
$908$	− 1.61147e16i	− 0.866465i
$909$	0	0
$910$	0	0
$911$	2.08024e16	1.09841	0.549203	−	0.835689i	$-0.314931\pi$
$911$	2.08024e16	1.09841	0.549203	+	0.835689i	$0.314931\pi$
$912$	0	0
$913$	1.95457e16i	1.01967i
$914$	−1.15386e16	−0.598342
$915$	0	0
$916$	2.42866e16	1.24435
$917$	− 2.36948e16i	− 1.20676i
$918$	0	0
$919$	1.27606e16	0.642151	0.321076	−	0.947054i	$-0.395956\pi$
$919$	1.27606e16	0.642151	0.321076	+	0.947054i	$0.395956\pi$
$920$	0	0
$921$	0	0
$922$	2.38736e16i	1.18004i
$923$	− 2.10349e16i	− 1.03355i
$924$	0	0
$925$	0	0
$926$	−1.48707e15	−0.0717744
$927$	0	0
$928$	− 2.76717e16i	− 1.31984i
$929$	2.66057e16	1.26150	0.630751	−	0.775985i	$-0.282747\pi$
$929$	2.66057e16	1.26150	0.630751	+	0.775985i	$0.282747\pi$
$930$	0	0
$931$	−2.06565e15	−0.0967908
$932$	− 2.12906e16i	− 0.991745i
$933$	0	0
$934$	4.69928e16	2.16333
$935$	0	0
$936$	0	0
$937$	1.35312e16i	0.612023i	0.952028	+	0.306012i	$0.0989946\pi$
$937$	1.35312e16i	0.612023i	−0.952028	+	0.306012i	$0.901005\pi$
$938$	− 1.49994e16i	− 0.674465i
$939$	0	0
$940$	0	0
$941$	−2.75735e16	−1.21829	−0.609143	−	0.793061i	$-0.708486\pi$
$941$	−2.75735e16	−1.21829	−0.609143	+	0.793061i	$0.708486\pi$
$942$	0	0
$943$	− 2.81602e16i	− 1.22976i
$944$	−3.40250e16	−1.47724
$945$	0	0
$946$	1.63976e15	0.0703686
$947$	4.64018e16i	1.97975i	0.141955	+	0.989873i	$0.454661\pi$
$947$	4.64018e16i	1.97975i	−0.141955	+	0.989873i	$0.545339\pi$
$948$	0	0
$949$	9.24283e15	0.389799
$950$	0	0
$951$	0	0
$952$	3.48246e14i	0.0144339i
$953$	− 5.27189e15i	− 0.217248i	−0.994083	−	0.108624i	$-0.965356\pi$
$953$	− 5.27189e15i	− 0.217248i	0.994083	−	0.108624i	$-0.0346444\pi$
$954$	0	0
$955$	0	0
$956$	9.69115e15	0.392516
$957$	0	0
$958$	− 2.16312e16i	− 0.866105i
$959$	5.21979e13	0.00207803
$960$	0	0
$961$	−2.23994e16	−0.881571
$962$	4.23978e16i	1.65913i
$963$	0	0
$964$	2.61088e16	1.01010
$965$	0	0
$966$	0	0
$967$	− 3.13722e16i	− 1.19316i	−0.802553	−	0.596581i	$-0.796526\pi$
$967$	− 3.13722e16i	− 1.19316i	0.802553	−	0.596581i	$-0.203474\pi$
$968$	1.39356e15i	0.0527001i
$969$	0	0
$970$	0	0
$971$	4.02261e16	1.49555	0.747776	−	0.663951i	$-0.231122\pi$
$971$	4.02261e16	1.49555	0.747776	+	0.663951i	$0.231122\pi$
$972$	0	0
$973$	1.96250e16i	0.721423i
$974$	−2.08693e16	−0.762839
$975$	0	0
$976$	8.84648e15	0.319740
$977$	− 1.61108e16i	− 0.579025i	−0.957174	−	0.289513i	$-0.906507\pi$
$977$	− 1.61108e16i	− 0.579025i	0.957174	−	0.289513i	$-0.0934932\pi$
$978$	0	0
$979$	−4.03307e16	−1.43328
$980$	0	0
$981$	0	0
$982$	2.09151e16i	0.730880i
$983$	3.39027e16i	1.17812i	0.808089	+	0.589060i	$0.200502\pi$
$983$	3.39027e16i	1.17812i	−0.808089	+	0.589060i	$0.799498\pi$
$984$	0	0
$985$	0	0
$986$	−2.48701e16	−0.849873
$987$	0	0
$988$	2.63548e16i	0.890629i
$989$	−8.13727e14	−0.0273463
$990$	0	0
$991$	−4.18340e16	−1.39035	−0.695174	−	0.718841i	$-0.744673\pi$
$991$	−4.18340e16	−1.39035	−0.695174	+	0.718841i	$0.744673\pi$
$992$	1.46586e16i	0.484484i
$993$	0	0
$994$	−4.04059e16	−1.32075
$995$	0	0
$996$	0	0
$997$	− 3.94947e16i	− 1.26974i	−0.772618	−	0.634871i	$-0.781053\pi$
$997$	− 3.94947e16i	− 1.26974i	0.772618	−	0.634871i	$-0.218947\pi$
$998$	− 1.81509e16i	− 0.580336i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.b.f.199.3		4
3.2	odd	2		25.12.b.c.24.2		4
5.2	odd	4		45.12.a.d.1.1		2
5.3	odd	4		225.12.a.h.1.2		2
5.4	even	2	inner	225.12.b.f.199.2		4
15.2	even	4		5.12.a.b.1.2	✓	2
15.8	even	4		25.12.a.c.1.1		2
15.14	odd	2		25.12.b.c.24.3		4
60.47	odd	4		80.12.a.j.1.1		2
105.62	odd	4		245.12.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.12.a.b.1.2	✓	2	15.2	even	4
25.12.a.c.1.1		2	15.8	even	4
25.12.b.c.24.2		4	3.2	odd	2
25.12.b.c.24.3		4	15.14	odd	2
45.12.a.d.1.1		2	5.2	odd	4
80.12.a.j.1.1		2	60.47	odd	4
225.12.a.h.1.2		2	5.3	odd	4
225.12.b.f.199.2		4	5.4	even	2	inner
225.12.b.f.199.3		4	1.1	even	1	trivial
245.12.a.b.1.2		2	105.62	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+63.7292i q^{2} -2013.42 q^{4} -41926.3i q^{7} +2204.06i q^{8} +O(q^{10})\)	\(q+63.7292i q^{2} -2013.42 q^{4} -41926.3i q^{7} +2204.06i q^{8} +957905. q^{11} +1.39098e6i q^{13} +2.67193e6 q^{14} -4.26394e6 q^{16} +3.76857e6i q^{17} -9.41036e6 q^{19} +6.10466e7i q^{22} -3.02942e7i q^{23} -8.86460e7 q^{26} +8.44151e7i q^{28} +1.03553e8 q^{29} -5.48554e7 q^{31} -2.67224e8i q^{32} -2.40168e8 q^{34} -4.78282e8i q^{37} -5.99715e8i q^{38} +9.29557e8 q^{41} -2.68608e7i q^{43} -1.92866e9 q^{44} +1.93063e9 q^{46} -1.20497e9i q^{47} +2.19508e8 q^{49} -2.80062e9i q^{52} +4.02058e9i q^{53} +9.24080e7 q^{56} +6.59933e9i q^{58} +7.97972e9 q^{59} -2.07472e9 q^{61} -3.49589e9i q^{62} +8.29741e9 q^{64} -5.61370e9i q^{67} -7.58770e9i q^{68} -1.51224e10 q^{71} -6.64484e9i q^{73} +3.04806e10 q^{74} +1.89470e10 q^{76} -4.01615e10i q^{77} +1.57985e10 q^{79} +5.92399e10i q^{82} +2.04046e10i q^{83} +1.71182e9 q^{86} +2.11128e9i q^{88} -4.21030e10 q^{89} +5.83187e10 q^{91} +6.09948e10i q^{92} +7.67919e10 q^{94} -1.10181e11i q^{97} +1.39891e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 13952 q^{4}+O(q^{10}) \) 4 * q - 13952 * q^4	\( 4 q - 13952 q^{4} + 1236352 q^{11} + 2668944 q^{14} + 12011264 q^{16} - 10650640 q^{19} + 161523472 q^{26} + 188280760 q^{29} + 489086928 q^{31} - 890716144 q^{34} + 1491486632 q^{41} - 485451776 q^{44} - 936389712 q^{46} + 3883354828 q^{49} + 7981107840 q^{56} + 14635031120 q^{59} - 3032851352 q^{61} - 1639063552 q^{64} - 65876943088 q^{71} + 137536396144 q^{74} - 2650785280 q^{76} + 6605646240 q^{79} + 113412187792 q^{86} - 25349541720 q^{89} + 181275718128 q^{91} + 120579531056 q^{94}+O(q^{100}) \) 4 * q - 13952 * q^4 + 1236352 * q^11 + 2668944 * q^14 + 12011264 * q^16 - 10650640 * q^19 + 161523472 * q^26 + 188280760 * q^29 + 489086928 * q^31 - 890716144 * q^34 + 1491486632 * q^41 - 485451776 * q^44 - 936389712 * q^46 + 3883354828 * q^49 + 7981107840 * q^56 + 14635031120 * q^59 - 3032851352 * q^61 - 1639063552 * q^64 - 65876943088 * q^71 + 137536396144 * q^74 - 2650785280 * q^76 + 6605646240 * q^79 + 113412187792 * q^86 - 25349541720 * q^89 + 181275718128 * q^91 + 120579531056 * q^94

Introduction

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Modular forms

Varieties

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Database

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