LMFDB - Embedded newform 225.12.b.a.199.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.12.b.a.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+78.0000i q^{2} -4036.00 q^{4} +27760.0i q^{7} -155064. i q^{8} +O(q^{10})$	$q+78.0000i q^{2} -4036.00 q^{4} +27760.0i q^{7} -155064. i q^{8} -637836. q^{11} +766214. i q^{13} -2.16528e6 q^{14} +3.82926e6 q^{16} +3.08435e6i q^{17} +1.95114e7 q^{19} -4.97512e7i q^{22} -1.53124e7i q^{23} -5.97647e7 q^{26} -1.12039e8i q^{28} +1.07513e7 q^{29} -5.09374e7 q^{31} -1.88885e7i q^{32} -2.40580e8 q^{34} -6.64741e8i q^{37} +1.52189e9i q^{38} -8.98833e8 q^{41} -9.57947e8i q^{43} +2.57431e9 q^{44} +1.19436e9 q^{46} -1.55574e9i q^{47} +1.20671e9 q^{49} -3.09244e9i q^{52} -3.79242e9i q^{53} +4.30458e9 q^{56} +8.38598e8i q^{58} +5.55307e8 q^{59} +4.95042e9 q^{61} -3.97312e9i q^{62} +9.31563e9 q^{64} -5.29240e9i q^{67} -1.24485e10i q^{68} +1.48311e10 q^{71} +1.39710e10i q^{73} +5.18498e10 q^{74} -7.87480e10 q^{76} -1.77063e10i q^{77} -3.72054e9 q^{79} -7.01090e10i q^{82} -8.76845e9i q^{83} +7.47199e10 q^{86} +9.89054e10i q^{88} -2.54728e10 q^{89} -2.12701e10 q^{91} +6.18007e10i q^{92} +1.21348e11 q^{94} +3.90925e10i q^{97} +9.41233e10i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8072 q^{4}+O(q^{10}) $ 2 * q - 8072 * q^4	$ 2 q - 8072 q^{4} - 1275672 q^{11} - 4330560 q^{14} + 7658528 q^{16} + 39022808 q^{19} - 119529384 q^{26} + 21502524 q^{29} - 101874800 q^{31} - 481159224 q^{34} - 1797666900 q^{41} + 5148612192 q^{44} + 2388728160 q^{46} + 2413418286 q^{49} + 8609153280 q^{56} + 1110613848 q^{59} + 9900841996 q^{61} + 18631268224 q^{64} + 29662172496 q^{71} + 103699569480 q^{74} - 157496053088 q^{76} - 7441084720 q^{79} + 149439761328 q^{86} - 50945538348 q^{89} - 42540201280 q^{91} + 242695649664 q^{94}+O(q^{100}) $ 2 * q - 8072 * q^4 - 1275672 * q^11 - 4330560 * q^14 + 7658528 * q^16 + 39022808 * q^19 - 119529384 * q^26 + 21502524 * q^29 - 101874800 * q^31 - 481159224 * q^34 - 1797666900 * q^41 + 5148612192 * q^44 + 2388728160 * q^46 + 2413418286 * q^49 + 8609153280 * q^56 + 1110613848 * q^59 + 9900841996 * q^61 + 18631268224 * q^64 + 29662172496 * q^71 + 103699569480 * q^74 - 157496053088 * q^76 - 7441084720 * q^79 + 149439761328 * q^86 - 50945538348 * q^89 - 42540201280 * q^91 + 242695649664 * 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 $

$2$	78.0000i	1.72357i	0.507271	+	0.861786i	$0.330654\pi$
$2$	78.0000i	1.72357i	−0.507271	+	0.861786i	$0.669346\pi$
$3$	0	0
$3$	0	0
$4$	−4036.00	−1.97070
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27760.0i	0.624281i	0.950036	+	0.312141i	$0.101046\pi$
$7$	27760.0i	0.624281i	−0.950036	+	0.312141i	$0.898954\pi$
$8$	− 155064.i	− 1.67308i
$9$	0	0
$10$	0	0
$11$	−637836.	−1.19412	−0.597062	−	0.802195i	$-0.703665\pi$
$11$	−637836.	−1.19412	−0.597062	+	0.802195i	$0.703665\pi$
$12$	0	0
$13$	766214.i	0.572350i	0.958177	+	0.286175i	$0.0923839\pi$
$13$	766214.i	0.572350i	−0.958177	+	0.286175i	$0.907616\pi$
$14$	−2.16528e6	−1.07599
$15$	0	0
$16$	3.82926e6	0.912968
$17$	3.08435e6i	0.526860i	0.964679	+	0.263430i	$0.0848538\pi$
$17$	3.08435e6i	0.526860i	−0.964679	+	0.263430i	$0.915146\pi$
$18$	0	0
$19$	1.95114e7	1.80777	0.903886	−	0.427773i	$-0.140702\pi$
$19$	1.95114e7	1.80777	0.903886	+	0.427773i	$0.140702\pi$
$20$	0	0
$21$	0	0
$22$	− 4.97512e7i	− 2.05816i
$23$	− 1.53124e7i	− 0.496066i	−0.968752	−	0.248033i	$-0.920216\pi$
$23$	− 1.53124e7i	− 0.496066i	0.968752	−	0.248033i	$-0.0797841\pi$
$24$	0	0
$25$	0	0
$26$	−5.97647e7	−0.986487
$27$	0	0
$28$	− 1.12039e8i	− 1.23027i
$29$	1.07513e7	0.0973353	0.0486677	−	0.998815i	$-0.484502\pi$
$29$	1.07513e7	0.0973353	0.0486677	+	0.998815i	$0.484502\pi$
$30$	0	0
$31$	−5.09374e7	−0.319556	−0.159778	−	0.987153i	$-0.551078\pi$
$31$	−5.09374e7	−0.319556	−0.159778	+	0.987153i	$0.551078\pi$
$32$	− 1.88885e7i	− 0.0995112i
$33$	0	0
$34$	−2.40580e8	−0.908081
$35$	0	0
$36$	0	0
$37$	− 6.64741e8i	− 1.57595i	−0.615706	−	0.787976i	$-0.711129\pi$
$37$	− 6.64741e8i	− 1.57595i	0.615706	−	0.787976i	$-0.288871\pi$
$38$	1.52189e9i	3.11583i
$39$	0	0
$40$	0	0
$41$	−8.98833e8	−1.21162	−0.605812	−	0.795608i	$-0.707152\pi$
$41$	−8.98833e8	−1.21162	−0.605812	+	0.795608i	$0.707152\pi$
$42$	0	0
$43$	− 9.57947e8i	− 0.993722i	−0.867830	−	0.496861i	$-0.834486\pi$
$43$	− 9.57947e8i	− 0.993722i	0.867830	−	0.496861i	$-0.165514\pi$
$44$	2.57431e9	2.35326
$45$	0	0
$46$	1.19436e9	0.855005
$47$	− 1.55574e9i	− 0.989462i	−0.869046	−	0.494731i	$-0.835267\pi$
$47$	− 1.55574e9i	− 0.989462i	0.869046	−	0.494731i	$-0.164733\pi$
$48$	0	0
$49$	1.20671e9	0.610273
$50$	0	0
$51$	0	0
$52$	− 3.09244e9i	− 1.12793i
$53$	− 3.79242e9i	− 1.24566i	−0.782358	−	0.622829i	$-0.785983\pi$
$53$	− 3.79242e9i	− 1.24566i	0.782358	−	0.622829i	$-0.214017\pi$
$54$	0	0
$55$	0	0
$56$	4.30458e9	1.04447
$57$	0	0
$58$	8.38598e8i	0.167765i
$59$	5.55307e8	0.101122	0.0505612	−	0.998721i	$-0.483899\pi$
$59$	5.55307e8	0.101122	0.0505612	+	0.998721i	$0.483899\pi$
$60$	0	0
$61$	4.95042e9	0.750461	0.375230	−	0.926932i	$-0.377564\pi$
$61$	4.95042e9	0.750461	0.375230	+	0.926932i	$0.377564\pi$
$62$	− 3.97312e9i	− 0.550779i
$63$	0	0
$64$	9.31563e9	1.08448
$65$	0	0
$66$	0	0
$67$	− 5.29240e9i	− 0.478896i	−0.970909	−	0.239448i	$-0.923034\pi$
$67$	− 5.29240e9i	− 0.478896i	0.970909	−	0.239448i	$-0.0769665\pi$
$68$	− 1.24485e10i	− 1.03828i
$69$	0	0
$70$	0	0
$71$	1.48311e10	0.975556	0.487778	−	0.872968i	$-0.337808\pi$
$71$	1.48311e10	0.975556	0.487778	+	0.872968i	$0.337808\pi$
$72$	0	0
$73$	1.39710e10i	0.788773i	0.918945	+	0.394386i	$0.129043\pi$
$73$	1.39710e10i	0.788773i	−0.918945	+	0.394386i	$0.870957\pi$
$74$	5.18498e10	2.71627
$75$	0	0
$76$	−7.87480e10	−3.56258
$77$	− 1.77063e10i	− 0.745469i
$78$	0	0
$79$	−3.72054e9	−0.136037	−0.0680185	−	0.997684i	$-0.521668\pi$
$79$	−3.72054e9	−0.136037	−0.0680185	+	0.997684i	$0.521668\pi$
$80$	0	0
$81$	0	0
$82$	− 7.01090e10i	− 2.08832i
$83$	− 8.76845e9i	− 0.244339i	−0.992509	−	0.122170i	$-0.961015\pi$
$83$	− 8.76845e9i	− 0.244339i	0.992509	−	0.122170i	$-0.0389852\pi$
$84$	0	0
$85$	0	0
$86$	7.47199e10	1.71275
$87$	0	0
$88$	9.89054e10i	1.99786i
$89$	−2.54728e10	−0.483539	−0.241769	−	0.970334i	$-0.577728\pi$
$89$	−2.54728e10	−0.483539	−0.241769	+	0.970334i	$0.577728\pi$
$90$	0	0
$91$	−2.12701e10	−0.357307
$92$	6.18007e10i	0.977598i
$93$	0	0
$94$	1.21348e11	1.70541
$95$	0	0
$96$	0	0
$97$	3.90925e10i	0.462220i	0.972928	+	0.231110i	$0.0742357\pi$
$97$	3.90925e10i	0.462220i	−0.972928	+	0.231110i	$0.925764\pi$
$98$	9.41233e10i	1.05185i
$99$	0	0
$100$	0	0
$101$	−9.31078e9	−0.0881492	−0.0440746	−	0.999028i	$-0.514034\pi$
$101$	−9.31078e9	−0.0881492	−0.0440746	+	0.999028i	$0.514034\pi$
$102$	0	0
$103$	4.85751e10i	0.412865i	0.978461	+	0.206433i	$0.0661855\pi$
$103$	4.85751e10i	0.412865i	−0.978461	+	0.206433i	$0.933815\pi$
$104$	1.18812e11	0.957586
$105$	0	0
$106$	2.95809e11	2.14698
$107$	2.25596e11i	1.55496i	0.628906	+	0.777482i	$0.283503\pi$
$107$	2.25596e11i	1.55496i	−0.628906	+	0.777482i	$0.716497\pi$
$108$	0	0
$109$	6.94512e10	0.432348	0.216174	−	0.976355i	$-0.430642\pi$
$109$	6.94512e10	0.432348	0.216174	+	0.976355i	$0.430642\pi$
$110$	0	0
$111$	0	0
$112$	1.06300e11i	0.569949i
$113$	3.59665e11i	1.83640i	0.396122	+	0.918198i	$0.370356\pi$
$113$	3.59665e11i	1.83640i	−0.396122	+	0.918198i	$0.629644\pi$
$114$	0	0
$115$	0	0
$116$	−4.33921e10	−0.191819
$117$	0	0
$118$	4.33139e10i	0.174292i
$119$	−8.56217e10	−0.328909
$120$	0	0
$121$	1.21523e11	0.425931
$122$	3.86133e11i	1.29347i
$123$	0	0
$124$	2.05583e11	0.629751
$125$	0	0
$126$	0	0
$127$	− 2.50273e11i	− 0.672192i	−0.941828	−	0.336096i	$-0.890893\pi$
$127$	− 2.50273e11i	− 0.672192i	0.941828	−	0.336096i	$-0.109107\pi$
$128$	6.87936e11i	1.76967i
$129$	0	0
$130$	0	0
$131$	9.78918e10	0.221694	0.110847	−	0.993837i	$-0.464644\pi$
$131$	9.78918e10	0.221694	0.110847	+	0.993837i	$0.464644\pi$
$132$	0	0
$133$	5.41637e11i	1.12856i
$134$	4.12807e11	0.825412
$135$	0	0
$136$	4.78272e11	0.881477
$137$	− 1.55015e11i	− 0.274416i	−0.990542	−	0.137208i	$-0.956187\pi$
$137$	− 1.55015e11i	− 0.274416i	0.990542	−	0.137208i	$-0.0438129\pi$
$138$	0	0
$139$	1.12627e12	1.84102	0.920512	−	0.390715i	$-0.127772\pi$
$139$	1.12627e12	1.84102	0.920512	+	0.390715i	$0.127772\pi$
$140$	0	0
$141$	0	0
$142$	1.15682e12i	1.68144i
$143$	− 4.88719e11i	− 0.683457i
$144$	0	0
$145$	0	0
$146$	−1.08974e12	−1.35951
$147$	0	0
$148$	2.68289e12i	3.10573i
$149$	−1.38458e12	−1.54452	−0.772261	−	0.635306i	$-0.780874\pi$
$149$	−1.38458e12	−1.54452	−0.772261	+	0.635306i	$0.780874\pi$
$150$	0	0
$151$	−6.98601e11	−0.724196	−0.362098	−	0.932140i	$-0.617939\pi$
$151$	−6.98601e11	−0.724196	−0.362098	+	0.932140i	$0.617939\pi$
$152$	− 3.02552e12i	− 3.02454i
$153$	0	0
$154$	1.38109e12	1.28487
$155$	0	0
$156$	0	0
$157$	− 2.13127e12i	− 1.78316i	−0.452863	−	0.891580i	$-0.649597\pi$
$157$	− 2.13127e12i	− 1.78316i	0.452863	−	0.891580i	$-0.350403\pi$
$158$	− 2.90202e11i	− 0.234470i
$159$	0	0
$160$	0	0
$161$	4.25071e11	0.309684
$162$	0	0
$163$	− 1.63564e11i	− 0.111342i	−0.998449	−	0.0556708i	$-0.982270\pi$
$163$	− 1.63564e11i	− 0.111342i	0.998449	−	0.0556708i	$-0.0177297\pi$
$164$	3.62769e12	2.38775
$165$	0	0
$166$	6.83939e11	0.421137
$167$	− 8.80943e9i	− 0.00524816i	−0.999997	−	0.00262408i	$-0.999165\pi$
$167$	− 8.80943e9i	− 0.00524816i	0.999997	−	0.00262408i	$-0.000835271\pi$
$168$	0	0
$169$	1.20508e12	0.672416
$170$	0	0
$171$	0	0
$172$	3.86627e12i	1.95833i
$173$	7.30852e11i	0.358571i	0.983797	+	0.179286i	$0.0573786\pi$
$173$	7.30852e11i	0.358571i	−0.983797	+	0.179286i	$0.942621\pi$
$174$	0	0
$175$	0	0
$176$	−2.44244e12	−1.09020
$177$	0	0
$178$	− 1.98688e12i	− 0.833414i
$179$	3.92371e12	1.59590	0.797950	−	0.602724i	$-0.205918\pi$
$179$	3.92371e12	1.59590	0.797950	+	0.602724i	$0.205918\pi$
$180$	0	0
$181$	2.27931e12	0.872110	0.436055	−	0.899920i	$-0.356375\pi$
$181$	2.27931e12	0.872110	0.436055	+	0.899920i	$0.356375\pi$
$182$	− 1.65907e12i	− 0.615845i
$183$	0	0
$184$	−2.37440e12	−0.829956
$185$	0	0
$186$	0	0
$187$	− 1.96731e12i	− 0.629136i
$188$	6.27897e12i	1.94994i
$189$	0	0
$190$	0	0
$191$	−3.38709e12	−0.964147	−0.482073	−	0.876131i	$-0.660116\pi$
$191$	−3.38709e12	−0.964147	−0.482073	+	0.876131i	$0.660116\pi$
$192$	0	0
$193$	− 4.92921e12i	− 1.32499i	−0.749067	−	0.662494i	$-0.769498\pi$
$193$	− 4.92921e12i	− 1.32499i	0.749067	−	0.662494i	$-0.230502\pi$
$194$	−3.04921e12	−0.796670
$195$	0	0
$196$	−4.87028e12	−1.20267
$197$	6.35785e12i	1.52667i	0.646001	+	0.763337i	$0.276440\pi$
$197$	6.35785e12i	1.52667i	−0.646001	+	0.763337i	$0.723560\pi$
$198$	0	0
$199$	3.78554e12	0.859875	0.429938	−	0.902859i	$-0.358536\pi$
$199$	3.78554e12	0.859875	0.429938	+	0.902859i	$0.358536\pi$
$200$	0	0
$201$	0	0
$202$	− 7.26241e11i	− 0.151932i
$203$	2.98455e11i	0.0607646i
$204$	0	0
$205$	0	0
$206$	−3.78885e12	−0.711604
$207$	0	0
$208$	2.93404e12i	0.522537i
$209$	−1.24451e13	−2.15870
$210$	0	0
$211$	1.79494e11	0.0295458	0.0147729	−	0.999891i	$-0.495297\pi$
$211$	1.79494e11	0.0295458	0.0147729	+	0.999891i	$0.495297\pi$
$212$	1.53062e13i	2.45482i
$213$	0	0
$214$	−1.75965e13	−2.68009
$215$	0	0
$216$	0	0
$217$	− 1.41402e12i	− 0.199493i
$218$	5.41719e12i	0.745184i
$219$	0	0
$220$	0	0
$221$	−2.36328e12	−0.301548
$222$	0	0
$223$	2.22568e12i	0.270263i	0.990828	+	0.135132i	$0.0431457\pi$
$223$	2.22568e12i	0.270263i	−0.990828	+	0.135132i	$0.956854\pi$
$224$	5.24344e11	0.0621230
$225$	0	0
$226$	−2.80538e13	−3.16516
$227$	− 4.15848e11i	− 0.0457923i	−0.999738	−	0.0228961i	$-0.992711\pi$
$227$	− 4.15848e11i	− 0.0457923i	0.999738	−	0.0228961i	$-0.00728870\pi$
$228$	0	0
$229$	−1.81248e13	−1.90186	−0.950928	−	0.309414i	$-0.899867\pi$
$229$	−1.81248e13	−1.90186	−0.950928	+	0.309414i	$0.899867\pi$
$230$	0	0
$231$	0	0
$232$	− 1.66713e12i	− 0.162850i
$233$	1.87641e10i	0.00179007i	1.00000		0.000895033i	$0.000284898\pi$
$233$	1.87641e10i	0.00179007i	−1.00000		0.000895033i	$0.999715\pi$
$234$	0	0
$235$	0	0
$236$	−2.24122e12	−0.199282
$237$	0	0
$238$	− 6.67849e12i	− 0.566898i
$239$	−1.76252e13	−1.46200	−0.730999	−	0.682379i	$-0.760946\pi$
$239$	−1.76252e13	−1.46200	−0.730999	+	0.682379i	$0.760946\pi$
$240$	0	0
$241$	−8.90117e11	−0.0705267	−0.0352633	−	0.999378i	$-0.511227\pi$
$241$	−8.90117e11	−0.0705267	−0.0352633	+	0.999378i	$0.511227\pi$
$242$	9.47880e12i	0.734123i
$243$	0	0
$244$	−1.99799e13	−1.47894
$245$	0	0
$246$	0	0
$247$	1.49499e13i	1.03468i
$248$	7.89856e12i	0.534643i
$249$	0	0
$250$	0	0
$251$	2.42280e13	1.53502	0.767508	−	0.641040i	$-0.221497\pi$
$251$	2.42280e13	1.53502	0.767508	+	0.641040i	$0.221497\pi$
$252$	0	0
$253$	9.76677e12i	0.592364i
$254$	1.95213e13	1.15857
$255$	0	0
$256$	−3.45806e13	−1.96568
$257$	7.80492e12i	0.434246i	0.976144	+	0.217123i	$0.0696673\pi$
$257$	7.80492e12i	0.434246i	−0.976144	+	0.217123i	$0.930333\pi$
$258$	0	0
$259$	1.84532e13	0.983837
$260$	0	0
$261$	0	0
$262$	7.63556e12i	0.382106i
$263$	− 1.65956e13i	− 0.813272i	−0.913590	−	0.406636i	$-0.866702\pi$
$263$	− 1.65956e13i	− 0.813272i	0.913590	−	0.406636i	$-0.133298\pi$
$264$	0	0
$265$	0	0
$266$	−4.22477e13	−1.94515
$267$	0	0
$268$	2.13601e13i	0.943762i
$269$	2.85236e13	1.23471	0.617357	−	0.786683i	$-0.288203\pi$
$269$	2.85236e13	1.23471	0.617357	+	0.786683i	$0.288203\pi$
$270$	0	0
$271$	2.33800e13	0.971658	0.485829	−	0.874054i	$-0.338518\pi$
$271$	2.33800e13	0.971658	0.485829	+	0.874054i	$0.338518\pi$
$272$	1.18108e13i	0.481006i
$273$	0	0
$274$	1.20911e13	0.472976
$275$	0	0
$276$	0	0
$277$	3.03641e13i	1.11872i	0.828924	+	0.559361i	$0.188953\pi$
$277$	3.03641e13i	1.11872i	−0.828924	+	0.559361i	$0.811047\pi$
$278$	8.78487e13i	3.17314i
$279$	0	0
$280$	0	0
$281$	−1.59749e13	−0.543942	−0.271971	−	0.962306i	$-0.587675\pi$
$281$	−1.59749e13	−0.543942	−0.271971	+	0.962306i	$0.587675\pi$
$282$	0	0
$283$	2.87045e13i	0.939993i	0.882668	+	0.469997i	$0.155745\pi$
$283$	2.87045e13i	0.939993i	−0.882668	+	0.469997i	$0.844255\pi$
$284$	−5.98583e13	−1.92253
$285$	0	0
$286$	3.81201e13	1.17799
$287$	− 2.49516e13i	− 0.756394i
$288$	0	0
$289$	2.47587e13	0.722419
$290$	0	0
$291$	0	0
$292$	− 5.63870e13i	− 1.55444i
$293$	− 4.81754e13i	− 1.30333i	−0.758508	−	0.651663i	$-0.774071\pi$
$293$	− 4.81754e13i	− 1.30333i	0.758508	−	0.651663i	$-0.225929\pi$
$294$	0	0
$295$	0	0
$296$	−1.03077e14	−2.63669
$297$	0	0
$298$	− 1.07997e14i	− 2.66209i
$299$	1.17325e13	0.283923
$300$	0	0
$301$	2.65926e13	0.620362
$302$	− 5.44909e13i	− 1.24820i
$303$	0	0
$304$	7.47143e13	1.65044
$305$	0	0
$306$	0	0
$307$	− 2.57350e13i	− 0.538597i	−0.963057	−	0.269298i	$-0.913208\pi$
$307$	− 2.57350e13i	− 0.538597i	0.963057	−	0.269298i	$-0.0867918\pi$
$308$	7.14627e13i	1.46910i
$309$	0	0
$310$	0	0
$311$	−3.46043e13	−0.674446	−0.337223	−	0.941425i	$-0.609488\pi$
$311$	−3.46043e13	−0.674446	−0.337223	+	0.941425i	$0.609488\pi$
$312$	0	0
$313$	1.26066e13i	0.237194i	0.992942	+	0.118597i	$0.0378397\pi$
$313$	1.26066e13i	0.237194i	−0.992942	+	0.118597i	$0.962160\pi$
$314$	1.66239e14	3.07341
$315$	0	0
$316$	1.50161e13	0.268089
$317$	− 8.18243e13i	− 1.43568i	−0.696210	−	0.717838i	$-0.745132\pi$
$317$	− 8.18243e13i	− 1.43568i	0.696210	−	0.717838i	$-0.254868\pi$
$318$	0	0
$319$	−6.85754e12	−0.116230
$320$	0	0
$321$	0	0
$322$	3.31555e13i	0.533764i
$323$	6.01801e13i	0.952443i
$324$	0	0
$325$	0	0
$326$	1.27580e13	0.191905
$327$	0	0
$328$	1.39377e14i	2.02714i
$329$	4.31874e13	0.617703
$330$	0	0
$331$	−3.40115e13	−0.470513	−0.235256	−	0.971933i	$-0.575593\pi$
$331$	−3.40115e13	−0.470513	−0.235256	+	0.971933i	$0.575593\pi$
$332$	3.53895e13i	0.481520i
$333$	0	0
$334$	6.87135e11	0.00904558
$335$	0	0
$336$	0	0
$337$	5.99439e13i	0.751244i	0.926773	+	0.375622i	$0.122571\pi$
$337$	5.99439e13i	0.751244i	−0.926773	+	0.375622i	$0.877429\pi$
$338$	9.39960e13i	1.15896i
$339$	0	0
$340$	0	0
$341$	3.24897e13	0.381590
$342$	0	0
$343$	8.83888e13i	1.00526i
$344$	−1.48543e14	−1.66257
$345$	0	0
$346$	−5.70064e13	−0.618024
$347$	9.78685e13i	1.04431i	0.852850	+	0.522157i	$0.174872\pi$
$347$	9.78685e13i	1.04431i	−0.852850	+	0.522157i	$0.825128\pi$
$348$	0	0
$349$	1.42790e14	1.47624	0.738120	−	0.674670i	$-0.235714\pi$
$349$	1.42790e14	1.47624	0.738120	+	0.674670i	$0.235714\pi$
$350$	0	0
$351$	0	0
$352$	1.20478e13i	0.118829i
$353$	− 1.44246e14i	− 1.40069i	−0.713804	−	0.700346i	$-0.753029\pi$
$353$	− 1.44246e14i	− 1.40069i	0.713804	−	0.700346i	$-0.246971\pi$
$354$	0	0
$355$	0	0
$356$	1.02808e14	0.952911
$357$	0	0
$358$	3.06050e14i	2.75065i
$359$	−1.24349e14	−1.10059	−0.550293	−	0.834972i	$-0.685484\pi$
$359$	−1.24349e14	−1.10059	−0.550293	+	0.834972i	$0.685484\pi$
$360$	0	0
$361$	2.64205e14	2.26804
$362$	1.77786e14i	1.50314i
$363$	0	0
$364$	8.58461e13	0.704147
$365$	0	0
$366$	0	0
$367$	1.60110e14i	1.25532i	0.778488	+	0.627659i	$0.215987\pi$
$367$	1.60110e14i	1.25532i	−0.778488	+	0.627659i	$0.784013\pi$
$368$	− 5.86351e13i	− 0.452892i
$369$	0	0
$370$	0	0
$371$	1.05277e14	0.777641
$372$	0	0
$373$	− 3.47258e13i	− 0.249031i	−0.992218	−	0.124516i	$-0.960262\pi$
$373$	− 3.47258e13i	− 0.249031i	0.992218	−	0.124516i	$-0.0397377\pi$
$374$	1.53450e14	1.08436
$375$	0	0
$376$	−2.41239e14	−1.65545
$377$	8.23777e12i	0.0557099i
$378$	0	0
$379$	1.46500e14	0.962327	0.481163	−	0.876631i	$-0.340214\pi$
$379$	1.46500e14	0.962327	0.481163	+	0.876631i	$0.340214\pi$
$380$	0	0
$381$	0	0
$382$	− 2.64193e14i	− 1.66178i
$383$	− 6.43419e13i	− 0.398933i	−0.979905	−	0.199467i	$-0.936079\pi$
$383$	− 6.43419e13i	− 0.398933i	0.979905	−	0.199467i	$-0.0639210\pi$
$384$	0	0
$385$	0	0
$386$	3.84478e14	2.28371
$387$	0	0
$388$	− 1.57777e14i	− 0.910899i
$389$	−3.98900e13	−0.227061	−0.113530	−	0.993535i	$-0.536216\pi$
$389$	−3.98900e13	−0.227061	−0.113530	+	0.993535i	$0.536216\pi$
$390$	0	0
$391$	4.72287e13	0.261357
$392$	− 1.87117e14i	− 1.02103i
$393$	0	0
$394$	−4.95912e14	−2.63133
$395$	0	0
$396$	0	0
$397$	− 1.06552e14i	− 0.542268i	−0.962542	−	0.271134i	$-0.912601\pi$
$397$	− 1.06552e14i	− 0.542268i	0.962542	−	0.271134i	$-0.0873986\pi$
$398$	2.95272e14i	1.48206i
$399$	0	0
$400$	0	0
$401$	−3.41445e13	−0.164447	−0.0822236	−	0.996614i	$-0.526202\pi$
$401$	−3.41445e13	−0.164447	−0.0822236	+	0.996614i	$0.526202\pi$
$402$	0	0
$403$	− 3.90289e13i	− 0.182898i
$404$	3.75783e13	0.173716
$405$	0	0
$406$	−2.32795e13	−0.104732
$407$	4.23996e14i	1.88188i
$408$	0	0
$409$	−5.33349e13	−0.230427	−0.115213	−	0.993341i	$-0.536755\pi$
$409$	−5.33349e13	−0.230427	−0.115213	+	0.993341i	$0.536755\pi$
$410$	0	0
$411$	0	0
$412$	− 1.96049e14i	− 0.813635i
$413$	1.54153e13i	0.0631288i
$414$	0	0
$415$	0	0
$416$	1.44726e13	0.0569553
$417$	0	0
$418$	− 9.70716e14i	− 3.72068i
$419$	−1.01288e14	−0.383159	−0.191580	−	0.981477i	$-0.561361\pi$
$419$	−1.01288e14	−0.383159	−0.191580	+	0.981477i	$0.561361\pi$
$420$	0	0
$421$	−1.57928e14	−0.581981	−0.290991	−	0.956726i	$-0.593985\pi$
$421$	−1.57928e14	−0.581981	−0.290991	+	0.956726i	$0.593985\pi$
$422$	1.40005e13i	0.0509244i
$423$	0	0
$424$	−5.88067e14	−2.08408
$425$	0	0
$426$	0	0
$427$	1.37424e14i	0.468499i
$428$	− 9.10504e14i	− 3.06437i
$429$	0	0
$430$	0	0
$431$	−5.13171e14	−1.66202	−0.831012	−	0.556254i	$-0.812238\pi$
$431$	−5.13171e14	−1.66202	−0.831012	+	0.556254i	$0.812238\pi$
$432$	0	0
$433$	− 7.49248e13i	− 0.236560i	−0.992980	−	0.118280i	$-0.962262\pi$
$433$	− 7.49248e13i	− 0.236560i	0.992980	−	0.118280i	$-0.0377381\pi$
$434$	1.10294e14	0.343841
$435$	0	0
$436$	−2.80305e14	−0.852030
$437$	− 2.98766e14i	− 0.896773i
$438$	0	0
$439$	3.68335e14	1.07817	0.539086	−	0.842250i	$-0.318770\pi$
$439$	3.68335e14	1.07817	0.539086	+	0.842250i	$0.318770\pi$
$440$	0	0
$441$	0	0
$442$	− 1.84335e14i	− 0.519740i
$443$	1.11248e14i	0.309793i	0.987931	+	0.154896i	$0.0495043\pi$
$443$	1.11248e14i	0.309793i	−0.987931	+	0.154896i	$0.950496\pi$
$444$	0	0
$445$	0	0
$446$	−1.73603e14	−0.465818
$447$	0	0
$448$	2.58602e14i	0.677022i
$449$	−8.83314e13	−0.228434	−0.114217	−	0.993456i	$-0.536436\pi$
$449$	−8.83314e13	−0.228434	−0.114217	+	0.993456i	$0.536436\pi$
$450$	0	0
$451$	5.73308e14	1.44683
$452$	− 1.45161e15i	− 3.61899i
$453$	0	0
$454$	3.24361e13	0.0789263
$455$	0	0
$456$	0	0
$457$	9.42094e12i	0.0221083i	0.999939	+	0.0110541i	$0.00351871\pi$
$457$	9.42094e12i	0.0221083i	−0.999939	+	0.0110541i	$0.996481\pi$
$458$	− 1.41373e15i	− 3.27799i
$459$	0	0
$460$	0	0
$461$	6.97134e14	1.55941	0.779706	−	0.626146i	$-0.215369\pi$
$461$	6.97134e14	1.55941	0.779706	+	0.626146i	$0.215369\pi$
$462$	0	0
$463$	− 1.87941e14i	− 0.410513i	−0.978708	−	0.205256i	$-0.934197\pi$
$463$	− 1.87941e14i	− 0.410513i	0.978708	−	0.205256i	$-0.0658028\pi$
$464$	4.11694e13	0.0888640
$465$	0	0
$466$	−1.46360e12	−0.00308531
$467$	3.20007e14i	0.666678i	0.942807	+	0.333339i	$0.108175\pi$
$467$	3.20007e14i	0.666678i	−0.942807	+	0.333339i	$0.891825\pi$
$468$	0	0
$469$	1.46917e14	0.298966
$470$	0	0
$471$	0	0
$472$	− 8.61081e13i	− 0.169185i
$473$	6.11013e14i	1.18663i
$474$	0	0
$475$	0	0
$476$	3.45569e14	0.648181
$477$	0	0
$478$	− 1.37477e15i	− 2.51986i
$479$	−1.50382e14	−0.272491	−0.136245	−	0.990675i	$-0.543504\pi$
$479$	−1.50382e14	−0.272491	−0.136245	+	0.990675i	$0.543504\pi$
$480$	0	0
$481$	5.09334e14	0.901996
$482$	− 6.94291e13i	− 0.121558i
$483$	0	0
$484$	−4.90467e14	−0.839384
$485$	0	0
$486$	0	0
$487$	− 1.76546e14i	− 0.292045i	−0.989281	−	0.146022i	$-0.953353\pi$
$487$	− 1.76546e14i	− 0.292045i	0.989281	−	0.146022i	$-0.0466471\pi$
$488$	− 7.67632e14i	− 1.25558i
$489$	0	0
$490$	0	0
$491$	8.60958e14	1.36155	0.680775	−	0.732492i	$-0.261643\pi$
$491$	8.60958e14	1.36155	0.680775	+	0.732492i	$0.261643\pi$
$492$	0	0
$493$	3.31607e13i	0.0512821i
$494$	−1.16609e15	−1.78334
$495$	0	0
$496$	−1.95053e14	−0.291745
$497$	4.11711e14i	0.609021i
$498$	0	0
$499$	−6.01209e14	−0.869907	−0.434953	−	0.900453i	$-0.643235\pi$
$499$	−6.01209e14	−0.869907	−0.434953	+	0.900453i	$0.643235\pi$
$500$	0	0
$501$	0	0
$502$	1.88979e15i	2.64571i
$503$	− 1.09203e15i	− 1.51221i	−0.654453	−	0.756103i	$-0.727101\pi$
$503$	− 1.09203e15i	− 1.51221i	0.654453	−	0.756103i	$-0.272899\pi$
$504$	0	0
$505$	0	0
$506$	−7.61808e14	−1.02098
$507$	0	0
$508$	1.01010e15i	1.32469i
$509$	8.76371e14	1.13695	0.568474	−	0.822702i	$-0.307534\pi$
$509$	8.76371e14	1.13695	0.568474	+	0.822702i	$0.307534\pi$
$510$	0	0
$511$	−3.87835e14	−0.492416
$512$	− 1.28839e15i	− 1.61832i
$513$	0	0
$514$	−6.08784e14	−0.748455
$515$	0	0
$516$	0	0
$517$	9.92308e14i	1.18154i
$518$	1.43935e15i	1.69571i
$519$	0	0
$520$	0	0
$521$	3.71989e14	0.424544	0.212272	−	0.977211i	$-0.431914\pi$
$521$	3.71989e14	0.424544	0.212272	+	0.977211i	$0.431914\pi$
$522$	0	0
$523$	− 9.73507e14i	− 1.08788i	−0.839125	−	0.543939i	$-0.816932\pi$
$523$	− 9.73507e14i	− 1.08788i	0.839125	−	0.543939i	$-0.183068\pi$
$524$	−3.95091e14	−0.436893
$525$	0	0
$526$	1.29445e15	1.40173
$527$	− 1.57109e14i	− 0.168361i
$528$	0	0
$529$	7.18341e14	0.753919
$530$	0	0
$531$	0	0
$532$	− 2.18605e15i	− 2.22405i
$533$	− 6.88699e14i	− 0.693473i
$534$	0	0
$535$	0	0
$536$	−8.20661e14	−0.801230
$537$	0	0
$538$	2.22484e15i	2.12812i
$539$	−7.69683e14	−0.728741
$540$	0	0
$541$	1.74606e15	1.61985	0.809925	−	0.586533i	$-0.199508\pi$
$541$	1.74606e15	1.61985	0.809925	+	0.586533i	$0.199508\pi$
$542$	1.82364e15i	1.67472i
$543$	0	0
$544$	5.82588e13	0.0524285
$545$	0	0
$546$	0	0
$547$	− 1.74624e14i	− 0.152467i	−0.997090	−	0.0762333i	$-0.975711\pi$
$547$	− 1.74624e14i	− 0.152467i	0.997090	−	0.0762333i	$-0.0242894\pi$
$548$	6.25639e14i	0.540793i
$549$	0	0
$550$	0	0
$551$	2.09772e14	0.175960
$552$	0	0
$553$	− 1.03282e14i	− 0.0849254i
$554$	−2.36840e15	−1.92820
$555$	0	0
$556$	−4.54561e15	−3.62811
$557$	1.58365e15i	1.25157i	0.779995	+	0.625785i	$0.215221\pi$
$557$	1.58365e15i	1.25157i	−0.779995	+	0.625785i	$0.784779\pi$
$558$	0	0
$559$	7.33993e14	0.568757
$560$	0	0
$561$	0	0
$562$	− 1.24604e15i	− 0.937523i
$563$	9.75798e14i	0.727049i	0.931585	+	0.363525i	$0.118427\pi$
$563$	9.75798e14i	0.727049i	−0.931585	+	0.363525i	$0.881573\pi$
$564$	0	0
$565$	0	0
$566$	−2.23895e15	−1.62015
$567$	0	0
$568$	− 2.29977e15i	− 1.63218i
$569$	1.92427e15	1.35254	0.676269	−	0.736655i	$-0.263596\pi$
$569$	1.92427e15	1.35254	0.676269	+	0.736655i	$0.263596\pi$
$570$	0	0
$571$	1.61132e15	1.11092	0.555461	−	0.831543i	$-0.312542\pi$
$571$	1.61132e15	1.11092	0.555461	+	0.831543i	$0.312542\pi$
$572$	1.97247e15i	1.34689i
$573$	0	0
$574$	1.94623e15	1.30370
$575$	0	0
$576$	0	0
$577$	2.53250e15i	1.64848i	0.566242	+	0.824239i	$0.308397\pi$
$577$	2.53250e15i	1.64848i	−0.566242	+	0.824239i	$0.691603\pi$
$578$	1.93118e15i	1.24514i
$579$	0	0
$580$	0	0
$581$	2.43412e14	0.152536
$582$	0	0
$583$	2.41894e15i	1.48747i
$584$	2.16640e15	1.31968
$585$	0	0
$586$	3.75768e15	2.24638
$587$	− 3.11508e15i	− 1.84484i	−0.386186	−	0.922421i	$-0.626208\pi$
$587$	− 3.11508e15i	− 1.84484i	0.386186	−	0.922421i	$-0.373792\pi$
$588$	0	0
$589$	−9.93860e14	−0.577685
$590$	0	0
$591$	0	0
$592$	− 2.54547e15i	− 1.43879i
$593$	2.20723e15i	1.23608i	0.786146	+	0.618040i	$0.212073\pi$
$593$	2.20723e15i	1.23608i	−0.786146	+	0.618040i	$0.787927\pi$
$594$	0	0
$595$	0	0
$596$	5.58817e15	3.04379
$597$	0	0
$598$	9.15138e14i	0.489362i
$599$	3.41159e15	1.80763	0.903813	−	0.427927i	$-0.140756\pi$
$599$	3.41159e15	1.80763	0.903813	+	0.427927i	$0.140756\pi$
$600$	0	0
$601$	−1.57545e15	−0.819588	−0.409794	−	0.912178i	$-0.634399\pi$
$601$	−1.57545e15	−0.819588	−0.409794	+	0.912178i	$0.634399\pi$
$602$	2.07422e15i	1.06924i
$603$	0	0
$604$	2.81956e15	1.42718
$605$	0	0
$606$	0	0
$607$	5.05592e14i	0.249036i	0.992217	+	0.124518i	$0.0397385\pi$
$607$	5.05592e14i	0.249036i	−0.992217	+	0.124518i	$0.960262\pi$
$608$	− 3.68541e14i	− 0.179894i
$609$	0	0
$610$	0	0
$611$	1.19203e15	0.566319
$612$	0	0
$613$	− 1.98375e15i	− 0.925665i	−0.886446	−	0.462832i	$-0.846833\pi$
$613$	− 1.98375e15i	− 0.925665i	0.886446	−	0.462832i	$-0.153167\pi$
$614$	2.00733e15	0.928310
$615$	0	0
$616$	−2.74561e15	−1.24723
$617$	− 1.93099e15i	− 0.869382i	−0.900580	−	0.434691i	$-0.856858\pi$
$617$	− 1.93099e15i	− 0.869382i	0.900580	−	0.434691i	$-0.143142\pi$
$618$	0	0
$619$	−9.47689e14	−0.419148	−0.209574	−	0.977793i	$-0.567208\pi$
$619$	−9.47689e14	−0.419148	−0.209574	+	0.977793i	$0.567208\pi$
$620$	0	0
$621$	0	0
$622$	− 2.69913e15i	− 1.16246i
$623$	− 7.07124e14i	− 0.301864i
$624$	0	0
$625$	0	0
$626$	−9.83314e14	−0.408821
$627$	0	0
$628$	8.60181e15i	3.51408i
$629$	2.05030e15	0.830306
$630$	0	0
$631$	1.00593e15	0.400318	0.200159	−	0.979763i	$-0.435854\pi$
$631$	1.00593e15	0.400318	0.200159	+	0.979763i	$0.435854\pi$
$632$	5.76922e14i	0.227601i
$633$	0	0
$634$	6.38230e15	2.47449
$635$	0	0
$636$	0	0
$637$	9.24597e14i	0.349290i
$638$	− 5.34888e14i	− 0.200332i
$639$	0	0
$640$	0	0
$641$	−1.13144e15	−0.412966	−0.206483	−	0.978450i	$-0.566202\pi$
$641$	−1.13144e15	−0.412966	−0.206483	+	0.978450i	$0.566202\pi$
$642$	0	0
$643$	− 8.68306e14i	− 0.311539i	−0.987793	−	0.155769i	$-0.950214\pi$
$643$	− 8.68306e14i	− 0.311539i	0.987793	−	0.155769i	$-0.0497857\pi$
$644$	−1.71559e15	−0.610296
$645$	0	0
$646$	−4.69405e15	−1.64160
$647$	− 1.37636e15i	− 0.477265i	−0.971110	−	0.238633i	$-0.923301\pi$
$647$	− 1.37636e15i	− 0.477265i	0.971110	−	0.238633i	$-0.0766992\pi$
$648$	0	0
$649$	−3.54195e14	−0.120753
$650$	0	0
$651$	0	0
$652$	6.60146e14i	0.219421i
$653$	− 2.55276e15i	− 0.841370i	−0.907207	−	0.420685i	$-0.861790\pi$
$653$	− 2.55276e15i	− 0.841370i	0.907207	−	0.420685i	$-0.138210\pi$
$654$	0	0
$655$	0	0
$656$	−3.44187e15	−1.10617
$657$	0	0
$658$	3.36862e15i	1.06466i
$659$	9.26792e14	0.290478	0.145239	−	0.989397i	$-0.453605\pi$
$659$	9.26792e14	0.290478	0.145239	+	0.989397i	$0.453605\pi$
$660$	0	0
$661$	−1.90332e15	−0.586683	−0.293342	−	0.956008i	$-0.594767\pi$
$661$	−1.90332e15	−0.586683	−0.293342	+	0.956008i	$0.594767\pi$
$662$	− 2.65289e15i	− 0.810963i
$663$	0	0
$664$	−1.35967e15	−0.408799
$665$	0	0
$666$	0	0
$667$	− 1.64627e14i	− 0.0482847i
$668$	3.55548e13i	0.0103426i
$669$	0	0
$670$	0	0
$671$	−3.15756e15	−0.896143
$672$	0	0
$673$	4.92990e15i	1.37643i	0.725505	+	0.688217i	$0.241606\pi$
$673$	4.92990e15i	1.37643i	−0.725505	+	0.688217i	$0.758394\pi$
$674$	−4.67563e15	−1.29482
$675$	0	0
$676$	−4.86369e15	−1.32513
$677$	4.30293e14i	0.116286i	0.998308	+	0.0581429i	$0.0185179\pi$
$677$	4.30293e14i	0.116286i	−0.998308	+	0.0581429i	$0.981482\pi$
$678$	0	0
$679$	−1.08521e15	−0.288555
$680$	0	0
$681$	0	0
$682$	2.53420e15i	0.657698i
$683$	3.81244e15i	0.981498i	0.871301	+	0.490749i	$0.163277\pi$
$683$	3.81244e15i	0.981498i	−0.871301	+	0.490749i	$0.836723\pi$
$684$	0	0
$685$	0	0
$686$	−6.89433e15	−1.73264
$687$	0	0
$688$	− 3.66823e15i	− 0.907236i
$689$	2.90580e15	0.712952
$690$	0	0
$691$	4.03729e15	0.974901	0.487450	−	0.873151i	$-0.337927\pi$
$691$	4.03729e15	0.974901	0.487450	+	0.873151i	$0.337927\pi$
$692$	− 2.94972e15i	− 0.706638i
$693$	0	0
$694$	−7.63374e15	−1.79995
$695$	0	0
$696$	0	0
$697$	− 2.77232e15i	− 0.638356i
$698$	1.11376e16i	2.54441i
$699$	0	0
$700$	0	0
$701$	4.71267e15	1.05152	0.525761	−	0.850632i	$-0.323781\pi$
$701$	4.71267e15	1.05152	0.525761	+	0.850632i	$0.323781\pi$
$702$	0	0
$703$	− 1.29700e16i	− 2.84896i
$704$	−5.94185e15	−1.29501
$705$	0	0
$706$	1.12512e16	2.41419
$707$	− 2.58467e14i	− 0.0550299i
$708$	0	0
$709$	−8.66706e14	−0.181684	−0.0908422	−	0.995865i	$-0.528956\pi$
$709$	−8.66706e14	−0.181684	−0.0908422	+	0.995865i	$0.528956\pi$
$710$	0	0
$711$	0	0
$712$	3.94991e15i	0.808997i
$713$	7.79972e14i	0.158521i
$714$	0	0
$715$	0	0
$716$	−1.58361e16	−3.14505
$717$	0	0
$718$	− 9.69924e15i	− 1.89694i
$719$	2.62666e13	0.00509795	0.00254897	−	0.999997i	$-0.499189\pi$
$719$	2.62666e13	0.00509795	0.00254897	+	0.999997i	$0.499189\pi$
$720$	0	0
$721$	−1.34844e15	−0.257744
$722$	2.06080e16i	3.90913i
$723$	0	0
$724$	−9.19929e15	−1.71867
$725$	0	0
$726$	0	0
$727$	5.68018e15i	1.03734i	0.854973	+	0.518672i	$0.173573\pi$
$727$	5.68018e15i	1.03734i	−0.854973	+	0.518672i	$0.826427\pi$
$728$	3.29823e15i	0.597803i
$729$	0	0
$730$	0	0
$731$	2.95465e15	0.523552
$732$	0	0
$733$	− 1.62312e15i	− 0.283320i	−0.989915	−	0.141660i	$-0.954756\pi$
$733$	− 1.62312e15i	− 0.283320i	0.989915	−	0.141660i	$-0.0452440\pi$
$734$	−1.24885e16	−2.16363
$735$	0	0
$736$	−2.89227e14	−0.0493641
$737$	3.37568e15i	0.571861i
$738$	0	0
$739$	−3.64794e15	−0.608840	−0.304420	−	0.952538i	$-0.598463\pi$
$739$	−3.64794e15	−0.608840	−0.304420	+	0.952538i	$0.598463\pi$
$740$	0	0
$741$	0	0
$742$	8.21164e15i	1.34032i
$743$	− 6.76045e15i	− 1.09531i	−0.836704	−	0.547655i	$-0.815521\pi$
$743$	− 6.76045e15i	− 1.09531i	0.836704	−	0.547655i	$-0.184479\pi$
$744$	0	0
$745$	0	0
$746$	2.70861e15	0.429224
$747$	0	0
$748$	7.94007e15i	1.23984i
$749$	−6.26254e15	−0.970734
$750$	0	0
$751$	2.64833e13	0.00404532	0.00202266	−	0.999998i	$-0.499356\pi$
$751$	2.64833e13	0.00404532	0.00202266	+	0.999998i	$0.499356\pi$
$752$	− 5.95734e15i	− 0.903347i
$753$	0	0
$754$	−6.42546e14	−0.0960200
$755$	0	0
$756$	0	0
$757$	− 7.37364e15i	− 1.07809i	−0.842277	−	0.539045i	$-0.818785\pi$
$757$	− 7.37364e15i	− 1.07809i	0.842277	−	0.539045i	$-0.181215\pi$
$758$	1.14270e16i	1.65864i
$759$	0	0
$760$	0	0
$761$	−6.20983e15	−0.881991	−0.440996	−	0.897509i	$-0.645375\pi$
$761$	−6.20983e15	−0.881991	−0.440996	+	0.897509i	$0.645375\pi$
$762$	0	0
$763$	1.92796e15i	0.269907i
$764$	1.36703e16	1.90005
$765$	0	0
$766$	5.01866e15	0.687591
$767$	4.25484e14i	0.0578774i
$768$	0	0
$769$	8.86195e15	1.18832	0.594162	−	0.804346i	$-0.297484\pi$
$769$	8.86195e15	1.18832	0.594162	+	0.804346i	$0.297484\pi$
$770$	0	0
$771$	0	0
$772$	1.98943e16i	2.61116i
$773$	2.95047e15i	0.384507i	0.981345	+	0.192254i	$0.0615796\pi$
$773$	2.95047e15i	0.384507i	−0.981345	+	0.192254i	$0.938420\pi$
$774$	0	0
$775$	0	0
$776$	6.06184e15	0.773330
$777$	0	0
$778$	− 3.11142e15i	− 0.391355i
$779$	−1.75375e16	−2.19034
$780$	0	0
$781$	−9.45980e15	−1.16493
$782$	3.68384e15i	0.450468i
$783$	0	0
$784$	4.62081e15	0.557160
$785$	0	0
$786$	0	0
$787$	− 1.35545e16i	− 1.60038i	−0.599746	−	0.800190i	$-0.704732\pi$
$787$	− 1.35545e16i	− 1.60038i	0.599746	−	0.800190i	$-0.295268\pi$
$788$	− 2.56603e16i	− 3.00862i
$789$	0	0
$790$	0	0
$791$	−9.98429e15	−1.14643
$792$	0	0
$793$	3.79308e15i	0.429526i
$794$	8.31105e15	0.934638
$795$	0	0
$796$	−1.52784e16	−1.69456
$797$	− 1.59922e16i	− 1.76152i	−0.473558	−	0.880762i	$-0.657031\pi$
$797$	− 1.59922e16i	− 1.76152i	0.473558	−	0.880762i	$-0.342969\pi$
$798$	0	0
$799$	4.79846e15	0.521308
$800$	0	0
$801$	0	0
$802$	− 2.66327e15i	− 0.283437i
$803$	− 8.91121e15i	− 0.941892i
$804$	0	0
$805$	0	0
$806$	3.04426e15	0.315238
$807$	0	0
$808$	1.44377e15i	0.147481i
$809$	9.45859e15	0.959643	0.479821	−	0.877366i	$-0.340701\pi$
$809$	9.45859e15	0.959643	0.479821	+	0.877366i	$0.340701\pi$
$810$	0	0
$811$	−1.07996e16	−1.08092	−0.540461	−	0.841369i	$-0.681750\pi$
$811$	−1.07996e16	−1.08092	−0.540461	+	0.841369i	$0.681750\pi$
$812$	− 1.20456e15i	− 0.119749i
$813$	0	0
$814$	−3.30717e16	−3.24356
$815$	0	0
$816$	0	0
$817$	− 1.86909e16i	− 1.79642i
$818$	− 4.16012e15i	− 0.397158i
$819$	0	0
$820$	0	0
$821$	1.10411e16	1.03306	0.516529	−	0.856270i	$-0.327224\pi$
$821$	1.10411e16	1.03306	0.516529	+	0.856270i	$0.327224\pi$
$822$	0	0
$823$	1.26104e16i	1.16420i	0.813117	+	0.582101i	$0.197769\pi$
$823$	1.26104e16i	1.16420i	−0.813117	+	0.582101i	$0.802231\pi$
$824$	7.53224e15	0.690756
$825$	0	0
$826$	−1.20239e15	−0.108807
$827$	− 7.86172e14i	− 0.0706704i	−0.999376	−	0.0353352i	$-0.988750\pi$
$827$	− 7.86172e14i	− 0.0706704i	0.999376	−	0.0353352i	$-0.0112499\pi$
$828$	0	0
$829$	5.92589e15	0.525659	0.262829	−	0.964842i	$-0.415344\pi$
$829$	5.92589e15	0.525659	0.262829	+	0.964842i	$0.415344\pi$
$830$	0	0
$831$	0	0
$832$	7.13777e15i	0.620704i
$833$	3.72192e15i	0.321528i
$834$	0	0
$835$	0	0
$836$	5.02283e16	4.25416
$837$	0	0
$838$	− 7.90044e15i	− 0.660403i
$839$	2.10209e16	1.74566	0.872831	−	0.488022i	$-0.162282\pi$
$839$	2.10209e16	1.74566	0.872831	+	0.488022i	$0.162282\pi$
$840$	0	0
$841$	−1.20849e16	−0.990526
$842$	− 1.23184e16i	− 1.00309i
$843$	0	0
$844$	−7.24437e14	−0.0582260
$845$	0	0
$846$	0	0
$847$	3.37348e15i	0.265901i
$848$	− 1.45222e16i	− 1.13725i
$849$	0	0
$850$	0	0
$851$	−1.01788e16	−0.781775
$852$	0	0
$853$	− 1.98378e16i	− 1.50409i	−0.659110	−	0.752046i	$-0.729067\pi$
$853$	− 1.98378e16i	− 1.50409i	0.659110	−	0.752046i	$-0.270933\pi$
$854$	−1.07190e16	−0.807491
$855$	0	0
$856$	3.49818e16	2.60157
$857$	7.96639e15i	0.588663i	0.955703	+	0.294332i	$0.0950970\pi$
$857$	7.96639e15i	0.588663i	−0.955703	+	0.294332i	$0.904903\pi$
$858$	0	0
$859$	1.59749e16	1.16540	0.582701	−	0.812687i	$-0.301996\pi$
$859$	1.59749e16	1.16540	0.582701	+	0.812687i	$0.301996\pi$
$860$	0	0
$861$	0	0
$862$	− 4.00274e16i	− 2.86462i
$863$	4.02926e15i	0.286527i	0.989685	+	0.143264i	$0.0457596\pi$
$863$	4.02926e15i	0.286527i	−0.989685	+	0.143264i	$0.954240\pi$
$864$	0	0
$865$	0	0
$866$	5.84413e15	0.407729
$867$	0	0
$868$	5.70699e15i	0.393142i
$869$	2.37310e15	0.162445
$870$	0	0
$871$	4.05511e15	0.274096
$872$	− 1.07694e16i	− 0.723352i
$873$	0	0
$874$	2.33037e16	1.54565
$875$	0	0
$876$	0	0
$877$	− 9.76958e15i	− 0.635884i	−0.948110	−	0.317942i	$-0.897008\pi$
$877$	− 9.76958e15i	− 0.635884i	0.948110	−	0.317942i	$-0.102992\pi$
$878$	2.87301e16i	1.85831i
$879$	0	0
$880$	0	0
$881$	−1.62583e16	−1.03207	−0.516033	−	0.856569i	$-0.672592\pi$
$881$	−1.62583e16	−1.03207	−0.516033	+	0.856569i	$0.672592\pi$
$882$	0	0
$883$	1.66061e16i	1.04108i	0.853837	+	0.520540i	$0.174269\pi$
$883$	1.66061e16i	1.04108i	−0.853837	+	0.520540i	$0.825731\pi$
$884$	9.53818e15	0.594262
$885$	0	0
$886$	−8.67733e15	−0.533950
$887$	− 1.91314e16i	− 1.16995i	−0.811051	−	0.584975i	$-0.801104\pi$
$887$	− 1.91314e16i	− 1.16995i	0.811051	−	0.584975i	$-0.198896\pi$
$888$	0	0
$889$	6.94758e15	0.419637
$890$	0	0
$891$	0	0
$892$	− 8.98286e15i	− 0.532608i
$893$	− 3.03547e16i	− 1.78872i
$894$	0	0
$895$	0	0
$896$	−1.90971e16	−1.10477
$897$	0	0
$898$	− 6.88985e15i	− 0.393722i
$899$	−5.47641e14	−0.0311041
$900$	0	0
$901$	1.16972e16	0.656287
$902$	4.47180e16i	2.49372i
$903$	0	0
$904$	5.57710e16	3.07243
$905$	0	0
$906$	0	0
$907$	1.07252e16i	0.580182i	0.956999	+	0.290091i	$0.0936855\pi$
$907$	1.07252e16i	0.580182i	−0.956999	+	0.290091i	$0.906314\pi$
$908$	1.67836e15i	0.0902430i
$909$	0	0
$910$	0	0
$911$	2.82249e16	1.49032	0.745162	−	0.666883i	$-0.232372\pi$
$911$	2.82249e16	1.49032	0.745162	+	0.666883i	$0.232372\pi$
$912$	0	0
$913$	5.59284e15i	0.291771i
$914$	−7.34833e14	−0.0381052
$915$	0	0
$916$	7.31516e16	3.74799
$917$	2.71748e15i	0.138399i
$918$	0	0
$919$	−1.83551e16	−0.923679	−0.461839	−	0.886964i	$-0.652810\pi$
$919$	−1.83551e16	−0.923679	−0.461839	+	0.886964i	$0.652810\pi$
$920$	0	0
$921$	0	0
$922$	5.43764e16i	2.68776i
$923$	1.13638e16i	0.558359i
$924$	0	0
$925$	0	0
$926$	1.46594e16	0.707549
$927$	0	0
$928$	− 2.03075e14i	− 0.00968596i
$929$	−1.10125e16	−0.522155	−0.261078	−	0.965318i	$-0.584078\pi$
$929$	−1.10125e16	−0.522155	−0.261078	+	0.965318i	$0.584078\pi$
$930$	0	0
$931$	2.35446e16	1.10323
$932$	− 7.57317e13i	− 0.00352769i
$933$	0	0
$934$	−2.49605e16	−1.14907
$935$	0	0
$936$	0	0
$937$	− 8.36357e15i	− 0.378289i	−0.981949	−	0.189145i	$-0.939429\pi$
$937$	− 8.36357e15i	− 0.378289i	0.981949	−	0.189145i	$-0.0605715\pi$
$938$	1.14595e16i	0.515289i
$939$	0	0
$940$	0	0
$941$	−2.22942e16	−0.985028	−0.492514	−	0.870305i	$-0.663922\pi$
$941$	−2.22942e16	−0.985028	−0.492514	+	0.870305i	$0.663922\pi$
$942$	0	0
$943$	1.37633e16i	0.601045i
$944$	2.12642e15	0.0923214
$945$	0	0
$946$	−4.76590e16	−2.04524
$947$	− 3.78038e16i	− 1.61291i	−0.591293	−	0.806457i	$-0.701382\pi$
$947$	− 3.78038e16i	− 1.61291i	0.591293	−	0.806457i	$-0.298618\pi$
$948$	0	0
$949$	−1.07048e16	−0.451454
$950$	0	0
$951$	0	0
$952$	1.32768e16i	0.550290i
$953$	− 4.79568e15i	− 0.197624i	−0.995106	−	0.0988119i	$-0.968496\pi$
$953$	− 4.79568e15i	− 0.197624i	0.995106	−	0.0988119i	$-0.0315042\pi$
$954$	0	0
$955$	0	0
$956$	7.11355e16	2.88116
$957$	0	0
$958$	− 1.17298e16i	− 0.469657i
$959$	4.30321e15	0.171313
$960$	0	0
$961$	−2.28139e16	−0.897884
$962$	3.97280e16i	1.55466i
$963$	0	0
$964$	3.59251e15	0.138987
$965$	0	0
$966$	0	0
$967$	− 3.37420e16i	− 1.28329i	−0.767002	−	0.641645i	$-0.778252\pi$
$967$	− 3.37420e16i	− 1.28329i	0.767002	−	0.641645i	$-0.221748\pi$
$968$	− 1.88439e16i	− 0.712616i
$969$	0	0
$970$	0	0
$971$	−3.58587e16	−1.33318	−0.666590	−	0.745425i	$-0.732247\pi$
$971$	−3.58587e16	−1.33318	−0.666590	+	0.745425i	$0.732247\pi$
$972$	0	0
$973$	3.12651e16i	1.14932i
$974$	1.37706e16	0.503360
$975$	0	0
$976$	1.89565e16	0.685147
$977$	− 1.20023e16i	− 0.431366i	−0.976463	−	0.215683i	$-0.930802\pi$
$977$	− 1.20023e16i	− 0.431366i	0.976463	−	0.215683i	$-0.0691977\pi$
$978$	0	0
$979$	1.62474e16	0.577405
$980$	0	0
$981$	0	0
$982$	6.71547e16i	2.34673i
$983$	4.42687e15i	0.153834i	0.997038	+	0.0769170i	$0.0245076\pi$
$983$	4.42687e15i	0.153834i	−0.997038	+	0.0769170i	$0.975492\pi$
$984$	0	0
$985$	0	0
$986$	−2.58653e15	−0.0883884
$987$	0	0
$988$	− 6.03378e16i	− 2.03904i
$989$	−1.46684e16	−0.492951
$990$	0	0
$991$	−3.79167e16	−1.26016	−0.630080	−	0.776530i	$-0.716978\pi$
$991$	−3.79167e16	−1.26016	−0.630080	+	0.776530i	$0.716978\pi$
$992$	9.62130e14i	0.0317994i
$993$	0	0
$994$	−3.21135e16	−1.04969
$995$	0	0
$996$	0	0
$997$	− 5.76003e13i	− 0.00185183i	−1.00000		0.000925915i	$-0.999705\pi$
$997$	− 5.76003e13i	− 0.00185183i	1.00000		0.000925915i	$-0.000294728\pi$
$998$	− 4.68943e16i	− 1.49935i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.b.a.199.2		2
3.2	odd	2		75.12.b.a.49.1		2
5.2	odd	4		9.12.a.a.1.1		1
5.3	odd	4		225.12.a.f.1.1		1
5.4	even	2	inner	225.12.b.a.199.1		2
15.2	even	4		3.12.a.a.1.1	✓	1
15.8	even	4		75.12.a.a.1.1		1
15.14	odd	2		75.12.b.a.49.2		2
20.7	even	4		144.12.a.l.1.1		1
45.2	even	12		81.12.c.a.28.1		2
45.7	odd	12		81.12.c.e.28.1		2
45.22	odd	12		81.12.c.e.55.1		2
45.32	even	12		81.12.c.a.55.1		2
60.47	odd	4		48.12.a.f.1.1		1
105.62	odd	4		147.12.a.c.1.1		1
120.77	even	4		192.12.a.q.1.1		1
120.107	odd	4		192.12.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.12.a.a.1.1	✓	1	15.2	even	4
9.12.a.a.1.1		1	5.2	odd	4
48.12.a.f.1.1		1	60.47	odd	4
75.12.a.a.1.1		1	15.8	even	4
75.12.b.a.49.1		2	3.2	odd	2
75.12.b.a.49.2		2	15.14	odd	2
81.12.c.a.28.1		2	45.2	even	12
81.12.c.a.55.1		2	45.32	even	12
81.12.c.e.28.1		2	45.7	odd	12
81.12.c.e.55.1		2	45.22	odd	12
144.12.a.l.1.1		1	20.7	even	4
147.12.a.c.1.1		1	105.62	odd	4
192.12.a.g.1.1		1	120.107	odd	4
192.12.a.q.1.1		1	120.77	even	4
225.12.a.f.1.1		1	5.3	odd	4
225.12.b.a.199.1		2	5.4	even	2	inner
225.12.b.a.199.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+78.0000i q^{2} -4036.00 q^{4} +27760.0i q^{7} -155064. i q^{8} +O(q^{10})\)	\(q+78.0000i q^{2} -4036.00 q^{4} +27760.0i q^{7} -155064. i q^{8} -637836. q^{11} +766214. i q^{13} -2.16528e6 q^{14} +3.82926e6 q^{16} +3.08435e6i q^{17} +1.95114e7 q^{19} -4.97512e7i q^{22} -1.53124e7i q^{23} -5.97647e7 q^{26} -1.12039e8i q^{28} +1.07513e7 q^{29} -5.09374e7 q^{31} -1.88885e7i q^{32} -2.40580e8 q^{34} -6.64741e8i q^{37} +1.52189e9i q^{38} -8.98833e8 q^{41} -9.57947e8i q^{43} +2.57431e9 q^{44} +1.19436e9 q^{46} -1.55574e9i q^{47} +1.20671e9 q^{49} -3.09244e9i q^{52} -3.79242e9i q^{53} +4.30458e9 q^{56} +8.38598e8i q^{58} +5.55307e8 q^{59} +4.95042e9 q^{61} -3.97312e9i q^{62} +9.31563e9 q^{64} -5.29240e9i q^{67} -1.24485e10i q^{68} +1.48311e10 q^{71} +1.39710e10i q^{73} +5.18498e10 q^{74} -7.87480e10 q^{76} -1.77063e10i q^{77} -3.72054e9 q^{79} -7.01090e10i q^{82} -8.76845e9i q^{83} +7.47199e10 q^{86} +9.89054e10i q^{88} -2.54728e10 q^{89} -2.12701e10 q^{91} +6.18007e10i q^{92} +1.21348e11 q^{94} +3.90925e10i q^{97} +9.41233e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8072 q^{4}+O(q^{10}) \) 2 * q - 8072 * q^4	\( 2 q - 8072 q^{4} - 1275672 q^{11} - 4330560 q^{14} + 7658528 q^{16} + 39022808 q^{19} - 119529384 q^{26} + 21502524 q^{29} - 101874800 q^{31} - 481159224 q^{34} - 1797666900 q^{41} + 5148612192 q^{44} + 2388728160 q^{46} + 2413418286 q^{49} + 8609153280 q^{56} + 1110613848 q^{59} + 9900841996 q^{61} + 18631268224 q^{64} + 29662172496 q^{71} + 103699569480 q^{74} - 157496053088 q^{76} - 7441084720 q^{79} + 149439761328 q^{86} - 50945538348 q^{89} - 42540201280 q^{91} + 242695649664 q^{94}+O(q^{100}) \) 2 * q - 8072 * q^4 - 1275672 * q^11 - 4330560 * q^14 + 7658528 * q^16 + 39022808 * q^19 - 119529384 * q^26 + 21502524 * q^29 - 101874800 * q^31 - 481159224 * q^34 - 1797666900 * q^41 + 5148612192 * q^44 + 2388728160 * q^46 + 2413418286 * q^49 + 8609153280 * q^56 + 1110613848 * q^59 + 9900841996 * q^61 + 18631268224 * q^64 + 29662172496 * q^71 + 103699569480 * q^74 - 157496053088 * q^76 - 7441084720 * q^79 + 149439761328 * q^86 - 50945538348 * q^89 - 42540201280 * q^91 + 242695649664 * q^94

Introduction

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