LMFDB - Embedded newform 225.12.b.i.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:	$\Q(i, \sqrt{1801})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 901x^{2} + 202500 $ x^4 + 901*x^2 + 202500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$20.7191i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.12.b.i.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.7191i q^{2} +1831.35 q^{4} -79941.2i q^{7} +57100.5i q^{8} -805067. q^{11} -1.19767e6i q^{13} +1.17666e6 q^{14} +2.91013e6 q^{16} +2.63319e6i q^{17} -1.16061e7 q^{19} -1.18499e7i q^{22} -1.84216e7i q^{23} +1.76286e7 q^{26} -1.46400e8i q^{28} -1.90527e8 q^{29} +1.01127e8 q^{31} +1.59776e8i q^{32} -3.87581e7 q^{34} -8.06675e7i q^{37} -1.70831e8i q^{38} -2.26316e8 q^{41} +1.67149e9i q^{43} -1.47436e9 q^{44} +2.71150e8 q^{46} +8.58507e8i q^{47} -4.41327e9 q^{49} -2.19335e9i q^{52} +3.52750e9i q^{53} +4.56468e9 q^{56} -2.80438e9i q^{58} +4.35760e9 q^{59} -1.65393e9 q^{61} +1.48849e9i q^{62} +3.60819e9 q^{64} -7.58610e9i q^{67} +4.82228e9i q^{68} +2.75809e10 q^{71} +3.22368e10i q^{73} +1.18735e9 q^{74} -2.12548e10 q^{76} +6.43580e10i q^{77} +2.43149e9 q^{79} -3.33116e9i q^{82} -1.20729e10i q^{83} -2.46028e10 q^{86} -4.59697e10i q^{88} +4.44073e9 q^{89} -9.57433e10 q^{91} -3.37364e10i q^{92} -1.26364e10 q^{94} +2.04453e10i q^{97} -6.49594e10i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6222 q^{4} - 591136 q^{11} + 6353592 q^{14} + 5948130 q^{16} - 35255952 q^{19} + 138104132 q^{26} - 403763896 q^{29} - 142114016 q^{31} + 252161404 q^{34} + 655310296 q^{41} - 1644753136 q^{44} + 3217895664 q^{46}+ \cdots + 40951225552 q^{94}+O(q^{100}) $ 4 * q + 6222 * q^4 - 591136 * q^11 + 6353592 * q^14 + 5948130 * q^16 - 35255952 * q^19 + 138104132 * q^26 - 403763896 * q^29 - 142114016 * q^31 + 252161404 * q^34 + 655310296 * q^41 - 1644753136 * q^44 + 3217895664 * q^46 - 15285229540 * q^49 + 22440862920 * q^56 - 2957541152 * q^59 - 16529782920 * q^61 - 3092633362 * q^64 + 40437776512 * q^71 - 32284748068 * q^74 - 57921409528 * q^76 - 44649990880 * q^79 + 35119396184 * q^86 + 158419366152 * q^89 - 459213424096 * q^91 + 40951225552 * q^94

Character values

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

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

Coefficient data

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

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

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

$2$	14.7191i	0.325249i	0.986688	+	0.162625i	$0.0519959\pi$
$2$	14.7191i	0.325249i	−0.986688	+	0.162625i	$0.948004\pi$
$3$	0	0
$3$	0	0
$4$	1831.35	0.894213
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 79941.2i	− 1.79776i	−0.438195	−	0.898880i	$-0.644382\pi$
$7$	− 79941.2i	− 1.79776i	0.438195	−	0.898880i	$-0.355618\pi$
$8$	57100.5i	0.616091i
$9$	0	0
$10$	0	0
$11$	−805067.	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$11$	−805067.	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$12$	0	0
$13$	− 1.19767e6i	− 0.894641i	−0.894374	−	0.447321i	$-0.852378\pi$
$13$	− 1.19767e6i	− 0.894641i	0.894374	−	0.447321i	$-0.147622\pi$
$14$	1.17666e6	0.584720
$15$	0	0
$16$	2.91013e6	0.693830
$17$	2.63319e6i	0.449793i	0.974383	+	0.224896i	$0.0722044\pi$
$17$	2.63319e6i	0.449793i	−0.974383	+	0.224896i	$0.927796\pi$
$18$	0	0
$19$	−1.16061e7	−1.07533	−0.537664	−	0.843159i	$-0.680693\pi$
$19$	−1.16061e7	−1.07533	−0.537664	+	0.843159i	$0.680693\pi$
$20$	0	0
$21$	0	0
$22$	− 1.18499e7i	− 0.490217i
$23$	− 1.84216e7i	− 0.596794i	−0.954442	−	0.298397i	$-0.903548\pi$
$23$	− 1.84216e7i	− 0.596794i	0.954442	−	0.298397i	$-0.0964520\pi$
$24$	0	0
$25$	0	0
$26$	1.76286e7	0.290981
$27$	0	0
$28$	− 1.46400e8i	− 1.60758i
$29$	−1.90527e8	−1.72491	−0.862457	−	0.506130i	$-0.831076\pi$
$29$	−1.90527e8	−1.72491	−0.862457	+	0.506130i	$0.831076\pi$
$30$	0	0
$31$	1.01127e8	0.634419	0.317209	−	0.948356i	$-0.397254\pi$
$31$	1.01127e8	0.634419	0.317209	+	0.948356i	$0.397254\pi$
$32$	1.59776e8i	0.841759i
$33$	0	0
$34$	−3.87581e7	−0.146295
$35$	0	0
$36$	0	0
$37$	− 8.06675e7i	− 0.191245i	−0.995418	−	0.0956223i	$-0.969516\pi$
$37$	− 8.06675e7i	− 0.191245i	0.995418	−	0.0956223i	$-0.0304841\pi$
$38$	− 1.70831e8i	− 0.349749i
$39$	0	0
$40$	0	0
$41$	−2.26316e8	−0.305073	−0.152537	−	0.988298i	$-0.548744\pi$
$41$	−2.26316e8	−0.305073	−0.152537	+	0.988298i	$0.548744\pi$
$42$	0	0
$43$	1.67149e9i	1.73391i	0.498385	+	0.866956i	$0.333927\pi$
$43$	1.67149e9i	1.73391i	−0.498385	+	0.866956i	$0.666073\pi$
$44$	−1.47436e9	−1.34776
$45$	0	0
$46$	2.71150e8	0.194107
$47$	8.58507e8i	0.546016i	0.962012	+	0.273008i	$0.0880186\pi$
$47$	8.58507e8i	0.546016i	−0.962012	+	0.273008i	$0.911981\pi$
$48$	0	0
$49$	−4.41327e9	−2.23194
$50$	0	0
$51$	0	0
$52$	− 2.19335e9i	− 0.800000i
$53$	3.52750e9i	1.15864i	0.815099	+	0.579322i	$0.196683\pi$
$53$	3.52750e9i	1.15864i	−0.815099	+	0.579322i	$0.803317\pi$
$54$	0	0
$55$	0	0
$56$	4.56468e9	1.10758
$57$	0	0
$58$	− 2.80438e9i	− 0.561027i
$59$	4.35760e9	0.793527	0.396763	−	0.917921i	$-0.370133\pi$
$59$	4.35760e9	0.793527	0.396763	+	0.917921i	$0.370133\pi$
$60$	0	0
$61$	−1.65393e9	−0.250727	−0.125364	−	0.992111i	$-0.540010\pi$
$61$	−1.65393e9	−0.250727	−0.125364	+	0.992111i	$0.540010\pi$
$62$	1.48849e9i	0.206344i
$63$	0	0
$64$	3.60819e9	0.420049
$65$	0	0
$66$	0	0
$67$	− 7.58610e9i	− 0.686447i	−0.939254	−	0.343224i	$-0.888481\pi$
$67$	− 7.58610e9i	− 0.686447i	0.939254	−	0.343224i	$-0.111519\pi$
$68$	4.82228e9i	0.402211i
$69$	0	0
$70$	0	0
$71$	2.75809e10	1.81421	0.907104	−	0.420905i	$-0.138288\pi$
$71$	2.75809e10	1.81421	0.907104	+	0.420905i	$0.138288\pi$
$72$	0	0
$73$	3.22368e10i	1.82002i	0.414584	+	0.910011i	$0.363927\pi$
$73$	3.22368e10i	1.82002i	−0.414584	+	0.910011i	$0.636073\pi$
$74$	1.18735e9	0.0622022
$75$	0	0
$76$	−2.12548e10	−0.961572
$77$	6.43580e10i	2.70959i
$78$	0	0
$79$	2.43149e9	0.0889046	0.0444523	−	0.999012i	$-0.485846\pi$
$79$	2.43149e9	0.0889046	0.0444523	+	0.999012i	$0.485846\pi$
$80$	0	0
$81$	0	0
$82$	− 3.33116e9i	− 0.0992247i
$83$	− 1.20729e10i	− 0.336421i	−0.985751	−	0.168210i	$-0.946201\pi$
$83$	− 1.20729e10i	− 0.336421i	0.985751	−	0.168210i	$-0.0537988\pi$
$84$	0	0
$85$	0	0
$86$	−2.46028e10	−0.563953
$87$	0	0
$88$	− 4.59697e10i	− 0.928575i
$89$	4.44073e9	0.0842964	0.0421482	−	0.999111i	$-0.486580\pi$
$89$	4.44073e9	0.0842964	0.0421482	+	0.999111i	$0.486580\pi$
$90$	0	0
$91$	−9.57433e10	−1.60835
$92$	− 3.37364e10i	− 0.533661i
$93$	0	0
$94$	−1.26364e10	−0.177591
$95$	0	0
$96$	0	0
$97$	2.04453e10i	0.241740i	0.992668	+	0.120870i	$0.0385684\pi$
$97$	2.04453e10i	0.241740i	−0.992668	+	0.120870i	$0.961432\pi$
$98$	− 6.49594e10i	− 0.725937i
$99$	0	0
$100$	0	0
$101$	1.55947e11	1.47642	0.738209	−	0.674572i	$-0.235672\pi$
$101$	1.55947e11	1.47642	0.738209	+	0.674572i	$0.235672\pi$
$102$	0	0
$103$	− 5.10325e10i	− 0.433752i	−0.976199	−	0.216876i	$-0.930413\pi$
$103$	− 5.10325e10i	− 0.433752i	0.976199	−	0.216876i	$-0.0695868\pi$
$104$	6.83876e10	0.551181
$105$	0	0
$106$	−5.19217e10	−0.376848
$107$	1.06665e10i	0.0735207i	0.999324	+	0.0367603i	$0.0117038\pi$
$107$	1.06665e10i	0.0735207i	−0.999324	+	0.0367603i	$0.988296\pi$
$108$	0	0
$109$	3.12513e10	0.194546	0.0972729	−	0.995258i	$-0.468988\pi$
$109$	3.12513e10	0.194546	0.0972729	+	0.995258i	$0.468988\pi$
$110$	0	0
$111$	0	0
$112$	− 2.32640e11i	− 1.24734i
$113$	3.96346e10i	0.202369i	0.994868	+	0.101184i	$0.0322632\pi$
$113$	3.96346e10i	0.202369i	−0.994868	+	0.101184i	$0.967737\pi$
$114$	0	0
$115$	0	0
$116$	−3.48921e11	−1.54244
$117$	0	0
$118$	6.41400e10i	0.258094i
$119$	2.10500e11	0.808620
$120$	0	0
$121$	3.62821e11	1.27166
$122$	− 2.43443e10i	− 0.0815489i
$123$	0	0
$124$	1.85198e11	0.567305
$125$	0	0
$126$	0	0
$127$	− 2.63460e10i	− 0.0707611i	−0.999374	−	0.0353806i	$-0.988736\pi$
$127$	− 2.63460e10i	− 0.0707611i	0.999374	−	0.0353806i	$-0.0112643\pi$
$128$	3.80331e11i	0.978379i
$129$	0	0
$130$	0	0
$131$	2.19917e11	0.498044	0.249022	−	0.968498i	$-0.419891\pi$
$131$	2.19917e11	0.498044	0.249022	+	0.968498i	$0.419891\pi$
$132$	0	0
$133$	9.27805e11i	1.93318i
$134$	1.11661e11	0.223266
$135$	0	0
$136$	−1.50356e11	−0.277113
$137$	− 5.54041e11i	− 0.980795i	−0.871499	−	0.490398i	$-0.836852\pi$
$137$	− 5.54041e11i	− 0.980795i	0.871499	−	0.490398i	$-0.163148\pi$
$138$	0	0
$139$	6.17540e11	1.00945	0.504723	−	0.863281i	$-0.331594\pi$
$139$	6.17540e11	1.00945	0.504723	+	0.863281i	$0.331594\pi$
$140$	0	0
$141$	0	0
$142$	4.05966e11i	0.590070i
$143$	9.64205e11i	1.34841i
$144$	0	0
$145$	0	0
$146$	−4.74497e11	−0.591961
$147$	0	0
$148$	− 1.47730e11i	− 0.171013i
$149$	−6.68742e11	−0.745992	−0.372996	−	0.927833i	$-0.621670\pi$
$149$	−6.68742e11	−0.745992	−0.372996	+	0.927833i	$0.621670\pi$
$150$	0	0
$151$	1.38243e12	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$151$	1.38243e12	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$152$	− 6.62713e11i	− 0.662500i
$153$	0	0
$154$	−9.47292e11	−0.881292
$155$	0	0
$156$	0	0
$157$	7.93661e11i	0.664029i	0.943274	+	0.332014i	$0.107728\pi$
$157$	7.93661e11i	0.664029i	−0.943274	+	0.332014i	$0.892272\pi$
$158$	3.57894e10i	0.0289161i
$159$	0	0
$160$	0	0
$161$	−1.47265e12	−1.07289
$162$	0	0
$163$	− 4.67399e11i	− 0.318168i	−0.987265	−	0.159084i	$-0.949146\pi$
$163$	− 4.67399e11i	− 0.318168i	0.987265	−	0.159084i	$-0.0508540\pi$
$164$	−4.14463e11	−0.272800
$165$	0	0
$166$	1.77702e11	0.109421
$167$	2.87482e12i	1.71266i	0.516432	+	0.856328i	$0.327260\pi$
$167$	2.87482e12i	1.71266i	−0.516432	+	0.856328i	$0.672740\pi$
$168$	0	0
$169$	3.57745e11	0.199617
$170$	0	0
$171$	0	0
$172$	3.06108e12i	1.55049i
$173$	− 1.53085e12i	− 0.751068i	−0.926809	−	0.375534i	$-0.877459\pi$
$173$	− 1.53085e12i	− 0.751068i	0.926809	−	0.375534i	$-0.122541\pi$
$174$	0	0
$175$	0	0
$176$	−2.34285e12	−1.04574
$177$	0	0
$178$	6.53635e10i	0.0274173i
$179$	−1.91337e12	−0.778228	−0.389114	−	0.921190i	$-0.627219\pi$
$179$	−1.91337e12	−0.778228	−0.389114	+	0.921190i	$0.627219\pi$
$180$	0	0
$181$	−1.70819e12	−0.653587	−0.326794	−	0.945096i	$-0.605968\pi$
$181$	−1.70819e12	−0.653587	−0.326794	+	0.945096i	$0.605968\pi$
$182$	− 1.40925e12i	− 0.523115i
$183$	0	0
$184$	1.05188e12	0.367680
$185$	0	0
$186$	0	0
$187$	− 2.11989e12i	− 0.677930i
$188$	1.57223e12i	0.488255i
$189$	0	0
$190$	0	0
$191$	−3.16020e12	−0.899562	−0.449781	−	0.893139i	$-0.648498\pi$
$191$	−3.16020e12	−0.899562	−0.449781	+	0.893139i	$0.648498\pi$
$192$	0	0
$193$	2.77296e12i	0.745382i	0.927956	+	0.372691i	$0.121565\pi$
$193$	2.77296e12i	0.745382i	−0.927956	+	0.372691i	$0.878435\pi$
$194$	−3.00936e11	−0.0786256
$195$	0	0
$196$	−8.08224e12	−1.99583
$197$	3.86504e12i	0.928089i	0.885812	+	0.464044i	$0.153602\pi$
$197$	3.86504e12i	0.928089i	−0.885812	+	0.464044i	$0.846398\pi$
$198$	0	0
$199$	1.01839e12	0.231325	0.115663	−	0.993289i	$-0.463101\pi$
$199$	1.01839e12	0.231325	0.115663	+	0.993289i	$0.463101\pi$
$200$	0	0
$201$	0	0
$202$	2.29540e12i	0.480204i
$203$	1.52310e13i	3.10098i
$204$	0	0
$205$	0	0
$206$	7.51152e11	0.141078
$207$	0	0
$208$	− 3.48538e12i	− 0.620729i
$209$	9.34367e12	1.62074
$210$	0	0
$211$	2.05668e12	0.338543	0.169271	−	0.985569i	$-0.445859\pi$
$211$	2.05668e12	0.338543	0.169271	+	0.985569i	$0.445859\pi$
$212$	6.46009e12i	1.03608i
$213$	0	0
$214$	−1.57001e11	−0.0239125
$215$	0	0
$216$	0	0
$217$	− 8.08418e12i	− 1.14053i
$218$	4.59990e11i	0.0632759i
$219$	0	0
$220$	0	0
$221$	3.15369e12	0.402403
$222$	0	0
$223$	9.99986e12i	1.21428i	0.794597	+	0.607138i	$0.207682\pi$
$223$	9.99986e12i	1.21428i	−0.794597	+	0.607138i	$0.792318\pi$
$224$	1.27727e13	1.51328
$225$	0	0
$226$	−5.83386e11	−0.0658202
$227$	− 6.77123e12i	− 0.745634i	−0.927905	−	0.372817i	$-0.878392\pi$
$227$	− 6.77123e12i	− 0.745634i	0.927905	−	0.372817i	$-0.121608\pi$
$228$	0	0
$229$	−1.01933e13	−1.06959	−0.534796	−	0.844981i	$-0.679611\pi$
$229$	−1.01933e13	−1.06959	−0.534796	+	0.844981i	$0.679611\pi$
$230$	0	0
$231$	0	0
$232$	− 1.08792e13i	− 1.06270i
$233$	− 6.74446e12i	− 0.643412i	−0.946840	−	0.321706i	$-0.895744\pi$
$233$	− 6.74446e12i	− 0.643412i	0.946840	−	0.321706i	$-0.104256\pi$
$234$	0	0
$235$	0	0
$236$	7.98029e12	0.709582
$237$	0	0
$238$	3.09837e12i	0.263003i
$239$	−1.76501e13	−1.46406	−0.732030	−	0.681272i	$-0.761427\pi$
$239$	−1.76501e13	−1.46406	−0.732030	+	0.681272i	$0.761427\pi$
$240$	0	0
$241$	−1.37662e13	−1.09074	−0.545369	−	0.838196i	$-0.683611\pi$
$241$	−1.37662e13	−1.09074	−0.545369	+	0.838196i	$0.683611\pi$
$242$	5.34039e12i	0.413608i
$243$	0	0
$244$	−3.02891e12	−0.224204
$245$	0	0
$246$	0	0
$247$	1.39003e13i	0.962033i
$248$	5.77438e12i	0.390860i
$249$	0	0
$250$	0	0
$251$	−3.02252e13	−1.91498	−0.957490	−	0.288467i	$-0.906855\pi$
$251$	−3.02252e13	−1.91498	−0.957490	+	0.288467i	$0.906855\pi$
$252$	0	0
$253$	1.48306e13i	0.899491i
$254$	3.87790e11	0.0230150
$255$	0	0
$256$	1.79144e12	0.101832
$257$	6.88263e12i	0.382933i	0.981499	+	0.191466i	$0.0613243\pi$
$257$	6.88263e12i	0.382933i	−0.981499	+	0.191466i	$0.938676\pi$
$258$	0	0
$259$	−6.44866e12	−0.343812
$260$	0	0
$261$	0	0
$262$	3.23698e12i	0.161988i
$263$	3.29948e13i	1.61692i	0.588549	+	0.808462i	$0.299699\pi$
$263$	3.29948e13i	1.61692i	−0.588549	+	0.808462i	$0.700301\pi$
$264$	0	0
$265$	0	0
$266$	−1.36564e13	−0.628765
$267$	0	0
$268$	− 1.38928e13i	− 0.613830i
$269$	−2.97478e13	−1.28771	−0.643855	−	0.765148i	$-0.722666\pi$
$269$	−2.97478e13	−1.28771	−0.643855	+	0.765148i	$0.722666\pi$
$270$	0	0
$271$	−4.30410e13	−1.78876	−0.894378	−	0.447312i	$-0.852381\pi$
$271$	−4.30410e13	−1.78876	−0.894378	+	0.447312i	$0.852381\pi$
$272$	7.66293e12i	0.312080i
$273$	0	0
$274$	8.15498e12	0.319003
$275$	0	0
$276$	0	0
$277$	− 1.46706e13i	− 0.540518i	−0.962788	−	0.270259i	$-0.912891\pi$
$277$	− 1.46706e13i	− 0.540518i	0.962788	−	0.270259i	$-0.0871094\pi$
$278$	9.08963e12i	0.328322i
$279$	0	0
$280$	0	0
$281$	−4.18378e13	−1.42457	−0.712286	−	0.701890i	$-0.752340\pi$
$281$	−4.18378e13	−1.42457	−0.712286	+	0.701890i	$0.752340\pi$
$282$	0	0
$283$	2.34242e13i	0.767078i	0.923525	+	0.383539i	$0.125295\pi$
$283$	2.34242e13i	0.767078i	−0.923525	+	0.383539i	$0.874705\pi$
$284$	5.05102e13	1.62229
$285$	0	0
$286$	−1.41922e13	−0.438568
$287$	1.80920e13i	0.548448i
$288$	0	0
$289$	2.73382e13	0.797686
$290$	0	0
$291$	0	0
$292$	5.90369e13i	1.62749i
$293$	7.31258e12i	0.197833i	0.995096	+	0.0989166i	$0.0315377\pi$
$293$	7.31258e12i	0.197833i	−0.995096	+	0.0989166i	$0.968462\pi$
$294$	0	0
$295$	0	0
$296$	4.60616e12	0.117824
$297$	0	0
$298$	− 9.84328e12i	− 0.242633i
$299$	−2.20630e13	−0.533917
$300$	0	0
$301$	1.33621e14	3.11716
$302$	2.03482e13i	0.466109i
$303$	0	0
$304$	−3.37752e13	−0.746094
$305$	0	0
$306$	0	0
$307$	5.37239e13i	1.12436i	0.827014	+	0.562181i	$0.190038\pi$
$307$	5.37239e13i	1.12436i	−0.827014	+	0.562181i	$0.809962\pi$
$308$	1.17862e14i	2.42295i
$309$	0	0
$310$	0	0
$311$	−5.41406e13	−1.05521	−0.527607	−	0.849489i	$-0.676911\pi$
$311$	−5.41406e13	−1.05521	−0.527607	+	0.849489i	$0.676911\pi$
$312$	0	0
$313$	− 4.29721e12i	− 0.0808524i	−0.999183	−	0.0404262i	$-0.987128\pi$
$313$	− 4.29721e12i	− 0.0808524i	0.999183	−	0.0404262i	$-0.0128716\pi$
$314$	−1.16820e13	−0.215975
$315$	0	0
$316$	4.45291e12	0.0794996
$317$	− 2.81928e13i	− 0.494666i	−0.968931	−	0.247333i	$-0.920446\pi$
$317$	− 2.81928e13i	− 0.494666i	0.968931	−	0.247333i	$-0.0795541\pi$
$318$	0	0
$319$	1.53387e14	2.59980
$320$	0	0
$321$	0	0
$322$	− 2.16760e13i	− 0.348958i
$323$	− 3.05610e13i	− 0.483675i
$324$	0	0
$325$	0	0
$326$	6.87969e12	0.103484
$327$	0	0
$328$	− 1.29227e13i	− 0.187953i
$329$	6.86301e13	0.981606
$330$	0	0
$331$	−9.95469e13	−1.37713	−0.688563	−	0.725176i	$-0.741758\pi$
$331$	−9.95469e13	−1.37713	−0.688563	+	0.725176i	$0.741758\pi$
$332$	− 2.21097e13i	− 0.300832i
$333$	0	0
$334$	−4.23148e13	−0.557040
$335$	0	0
$336$	0	0
$337$	9.05175e13i	1.13440i	0.823579	+	0.567202i	$0.191974\pi$
$337$	9.05175e13i	1.13440i	−0.823579	+	0.567202i	$0.808026\pi$
$338$	5.26568e12i	0.0649251i
$339$	0	0
$340$	0	0
$341$	−8.14136e13	−0.956198
$342$	0	0
$343$	1.94733e14i	2.21473i
$344$	−9.54428e13	−1.06825
$345$	0	0
$346$	2.25327e13	0.244284
$347$	− 6.36892e13i	− 0.679601i	−0.940498	−	0.339801i	$-0.889640\pi$
$347$	− 6.36892e13i	− 0.679601i	0.940498	−	0.339801i	$-0.110360\pi$
$348$	0	0
$349$	−7.08598e13	−0.732588	−0.366294	−	0.930499i	$-0.619374\pi$
$349$	−7.08598e13	−0.732588	−0.366294	+	0.930499i	$0.619374\pi$
$350$	0	0
$351$	0	0
$352$	− 1.28631e14i	− 1.26870i
$353$	− 6.62893e13i	− 0.643699i	−0.946791	−	0.321849i	$-0.895696\pi$
$353$	− 6.62893e13i	− 0.643699i	0.946791	−	0.321849i	$-0.104304\pi$
$354$	0	0
$355$	0	0
$356$	8.13252e12	0.0753790
$357$	0	0
$358$	− 2.81630e13i	− 0.253118i
$359$	6.80217e13	0.602044	0.301022	−	0.953617i	$-0.402672\pi$
$359$	6.80217e13	0.602044	0.301022	+	0.953617i	$0.402672\pi$
$360$	0	0
$361$	1.82109e13	0.156330
$362$	− 2.51430e13i	− 0.212579i
$363$	0	0
$364$	−1.75339e14	−1.43821
$365$	0	0
$366$	0	0
$367$	− 2.19771e14i	− 1.72309i	−0.507685	−	0.861543i	$-0.669499\pi$
$367$	− 2.19771e14i	− 1.72309i	0.507685	−	0.861543i	$-0.330501\pi$
$368$	− 5.36094e13i	− 0.414074i
$369$	0	0
$370$	0	0
$371$	2.81993e14	2.08296
$372$	0	0
$373$	− 1.67232e14i	− 1.19928i	−0.800271	−	0.599639i	$-0.795311\pi$
$373$	− 1.67232e14i	− 1.19928i	0.800271	−	0.599639i	$-0.204689\pi$
$374$	3.12029e13	0.220496
$375$	0	0
$376$	−4.90212e13	−0.336396
$377$	2.28189e14i	1.54318i
$378$	0	0
$379$	1.57778e14	1.03641	0.518205	−	0.855256i	$-0.326600\pi$
$379$	1.57778e14	1.03641	0.518205	+	0.855256i	$0.326600\pi$
$380$	0	0
$381$	0	0
$382$	− 4.65153e13i	− 0.292582i
$383$	− 5.49255e13i	− 0.340550i	−0.985397	−	0.170275i	$-0.945534\pi$
$383$	− 5.49255e13i	− 0.340550i	0.985397	−	0.170275i	$-0.0544655\pi$
$384$	0	0
$385$	0	0
$386$	−4.08155e13	−0.242435
$387$	0	0
$388$	3.74424e13i	0.216167i
$389$	−2.32940e14	−1.32593	−0.662965	−	0.748650i	$-0.730702\pi$
$389$	−2.32940e14	−1.32593	−0.662965	+	0.748650i	$0.730702\pi$
$390$	0	0
$391$	4.85076e13	0.268434
$392$	− 2.52000e14i	− 1.37508i
$393$	0	0
$394$	−5.68899e13	−0.301860
$395$	0	0
$396$	0	0
$397$	4.76134e13i	0.242316i	0.992633	+	0.121158i	$0.0386607\pi$
$397$	4.76134e13i	0.242316i	−0.992633	+	0.121158i	$0.961339\pi$
$398$	1.49898e13i	0.0752383i
$399$	0	0
$400$	0	0
$401$	2.99074e14	1.44040	0.720202	−	0.693764i	$-0.244049\pi$
$401$	2.99074e14	1.44040	0.720202	+	0.693764i	$0.244049\pi$
$402$	0	0
$403$	− 1.21116e14i	− 0.567577i
$404$	2.85593e14	1.32023
$405$	0	0
$406$	−2.24186e14	−1.00859
$407$	6.49428e13i	0.288245i
$408$	0	0
$409$	2.55879e14	1.10550	0.552748	−	0.833349i	$-0.313579\pi$
$409$	2.55879e14	1.10550	0.552748	+	0.833349i	$0.313579\pi$
$410$	0	0
$411$	0	0
$412$	− 9.34582e13i	− 0.387867i
$413$	− 3.48352e14i	− 1.42657i
$414$	0	0
$415$	0	0
$416$	1.91359e14	0.753072
$417$	0	0
$418$	1.37530e14i	0.527144i
$419$	2.21825e14	0.839139	0.419570	−	0.907723i	$-0.362181\pi$
$419$	2.21825e14	0.839139	0.419570	+	0.907723i	$0.362181\pi$
$420$	0	0
$421$	9.78949e13	0.360752	0.180376	−	0.983598i	$-0.442269\pi$
$421$	9.78949e13	0.360752	0.180376	+	0.983598i	$0.442269\pi$
$422$	3.02725e13i	0.110111i
$423$	0	0
$424$	−2.01422e14	−0.713831
$425$	0	0
$426$	0	0
$427$	1.32217e14i	0.450748i
$428$	1.95340e13i	0.0657432i
$429$	0	0
$430$	0	0
$431$	−1.11415e14	−0.360842	−0.180421	−	0.983589i	$-0.557746\pi$
$431$	−1.11415e14	−0.360842	−0.180421	+	0.983589i	$0.557746\pi$
$432$	0	0
$433$	− 3.42045e14i	− 1.07994i	−0.841684	−	0.539970i	$-0.818436\pi$
$433$	− 3.42045e14i	− 1.07994i	0.841684	−	0.539970i	$-0.181564\pi$
$434$	1.18992e14	0.370957
$435$	0	0
$436$	5.72320e13	0.173965
$437$	2.13803e14i	0.641750i
$438$	0	0
$439$	−1.20947e14	−0.354029	−0.177015	−	0.984208i	$-0.556644\pi$
$439$	−1.20947e14	−0.354029	−0.177015	+	0.984208i	$0.556644\pi$
$440$	0	0
$441$	0	0
$442$	4.64195e13i	0.130881i
$443$	1.27316e14i	0.354539i	0.984162	+	0.177270i	$0.0567264\pi$
$443$	1.27316e14i	0.354539i	−0.984162	+	0.177270i	$0.943274\pi$
$444$	0	0
$445$	0	0
$446$	−1.47189e14	−0.394942
$447$	0	0
$448$	− 2.88443e14i	− 0.755147i
$449$	−1.61837e14	−0.418526	−0.209263	−	0.977859i	$-0.567106\pi$
$449$	−1.61837e14	−0.418526	−0.209263	+	0.977859i	$0.567106\pi$
$450$	0	0
$451$	1.82199e14	0.459807
$452$	7.25848e13i	0.180961i
$453$	0	0
$454$	9.96664e13	0.242517
$455$	0	0
$456$	0	0
$457$	3.49165e14i	0.819391i	0.912222	+	0.409696i	$0.134365\pi$
$457$	3.49165e14i	0.819391i	−0.912222	+	0.409696i	$0.865635\pi$
$458$	− 1.50036e14i	− 0.347884i
$459$	0	0
$460$	0	0
$461$	4.49619e14	1.00575	0.502875	−	0.864359i	$-0.332276\pi$
$461$	4.49619e14	1.00575	0.502875	+	0.864359i	$0.332276\pi$
$462$	0	0
$463$	4.21098e13i	0.0919787i	0.998942	+	0.0459894i	$0.0146440\pi$
$463$	4.21098e13i	0.0919787i	−0.998942	+	0.0459894i	$0.985356\pi$
$464$	−5.54459e14	−1.19680
$465$	0	0
$466$	9.92723e13	0.209269
$467$	− 7.91358e13i	− 0.164866i	−0.996597	−	0.0824328i	$-0.973731\pi$
$467$	− 7.91358e13i	− 0.164866i	0.996597	−	0.0824328i	$-0.0262690\pi$
$468$	0	0
$469$	−6.06442e14	−1.23407
$470$	0	0
$471$	0	0
$472$	2.48821e14i	0.488885i
$473$	− 1.34566e15i	− 2.61336i
$474$	0	0
$475$	0	0
$476$	3.85499e14	0.723078
$477$	0	0
$478$	− 2.59794e14i	− 0.476184i
$479$	−9.62318e14	−1.74371	−0.871853	−	0.489768i	$-0.837081\pi$
$479$	−9.62318e14	−1.74371	−0.871853	+	0.489768i	$0.837081\pi$
$480$	0	0
$481$	−9.66132e13	−0.171095
$482$	− 2.02626e14i	− 0.354762i
$483$	0	0
$484$	6.64451e14	1.13714
$485$	0	0
$486$	0	0
$487$	− 4.57117e14i	− 0.756168i	−0.925771	−	0.378084i	$-0.876583\pi$
$487$	− 4.57117e14i	− 0.756168i	0.925771	−	0.378084i	$-0.123417\pi$
$488$	− 9.44399e13i	− 0.154471i
$489$	0	0
$490$	0	0
$491$	−4.40122e14	−0.696025	−0.348013	−	0.937490i	$-0.613143\pi$
$491$	−4.40122e14	−0.696025	−0.348013	+	0.937490i	$0.613143\pi$
$492$	0	0
$493$	− 5.01693e14i	− 0.775854i
$494$	−2.04599e14	−0.312900
$495$	0	0
$496$	2.94292e14	0.440179
$497$	− 2.20485e15i	− 3.26151i
$498$	0	0
$499$	−6.36085e14	−0.920369	−0.460185	−	0.887823i	$-0.652217\pi$
$499$	−6.36085e14	−0.920369	−0.460185	+	0.887823i	$0.652217\pi$
$500$	0	0
$501$	0	0
$502$	− 4.44888e14i	− 0.622846i
$503$	4.81639e12i	0.00666957i	0.999994	+	0.00333478i	$0.00106150\pi$
$503$	4.81639e12i	0.00666957i	−0.999994	+	0.00333478i	$0.998939\pi$
$504$	0	0
$505$	0	0
$506$	−2.18294e14	−0.292559
$507$	0	0
$508$	− 4.82488e13i	− 0.0632755i
$509$	−5.73829e14	−0.744449	−0.372224	−	0.928143i	$-0.621405\pi$
$509$	−5.73829e14	−0.744449	−0.372224	+	0.928143i	$0.621405\pi$
$510$	0	0
$511$	2.57705e15	3.27196
$512$	8.05287e14i	1.01150i
$513$	0	0
$514$	−1.01306e14	−0.124548
$515$	0	0
$516$	0	0
$517$	− 6.91155e14i	− 0.822958i
$518$	− 9.49185e13i	− 0.111825i
$519$	0	0
$520$	0	0
$521$	−5.14705e14	−0.587422	−0.293711	−	0.955894i	$-0.594890\pi$
$521$	−5.14705e14	−0.587422	−0.293711	+	0.955894i	$0.594890\pi$
$522$	0	0
$523$	1.20146e15i	1.34261i	0.741182	+	0.671304i	$0.234265\pi$
$523$	1.20146e15i	1.34261i	−0.741182	+	0.671304i	$0.765735\pi$
$524$	4.02745e14	0.445357
$525$	0	0
$526$	−4.85654e14	−0.525903
$527$	2.66285e14i	0.285357i
$528$	0	0
$529$	6.13454e14	0.643836
$530$	0	0
$531$	0	0
$532$	1.69913e15i	1.72868i
$533$	2.71052e14i	0.272931i
$534$	0	0
$535$	0	0
$536$	4.33170e14	0.422914
$537$	0	0
$538$	− 4.37861e14i	− 0.418826i
$539$	3.55298e15	3.36399
$540$	0	0
$541$	−1.89107e15	−1.75438	−0.877190	−	0.480143i	$-0.840585\pi$
$541$	−1.89107e15	−1.75438	−0.877190	+	0.480143i	$0.840585\pi$
$542$	− 6.33524e14i	− 0.581791i
$543$	0	0
$544$	−4.20721e14	−0.378617
$545$	0	0
$546$	0	0
$547$	− 1.37400e14i	− 0.119966i	−0.998199	−	0.0599830i	$-0.980895\pi$
$547$	− 1.37400e14i	− 0.119966i	0.998199	−	0.0599830i	$-0.0191046\pi$
$548$	− 1.01464e15i	− 0.877040i
$549$	0	0
$550$	0	0
$551$	2.21127e15	1.85485
$552$	0	0
$553$	− 1.94377e14i	− 0.159829i
$554$	2.15939e14	0.175803
$555$	0	0
$556$	1.13093e15	0.902661
$557$	− 1.18622e15i	− 0.937483i	−0.883335	−	0.468742i	$-0.844708\pi$
$557$	− 1.18622e15i	− 0.937483i	0.883335	−	0.468742i	$-0.155292\pi$
$558$	0	0
$559$	2.00189e15	1.55123
$560$	0	0
$561$	0	0
$562$	− 6.15815e14i	− 0.463340i
$563$	2.58208e15i	1.92386i	0.273298	+	0.961929i	$0.411885\pi$
$563$	2.58208e15i	1.92386i	−0.273298	+	0.961929i	$0.588115\pi$
$564$	0	0
$565$	0	0
$566$	−3.44783e14	−0.249491
$567$	0	0
$568$	1.57488e15i	1.11772i
$569$	6.86455e13	0.0482497	0.0241249	−	0.999709i	$-0.492320\pi$
$569$	6.86455e13	0.0482497	0.0241249	+	0.999709i	$0.492320\pi$
$570$	0	0
$571$	−1.48521e15	−1.02397	−0.511986	−	0.858994i	$-0.671090\pi$
$571$	−1.48521e15	−1.02397	−0.511986	+	0.858994i	$0.671090\pi$
$572$	1.76579e15i	1.20576i
$573$	0	0
$574$	−2.66297e14	−0.178382
$575$	0	0
$576$	0	0
$577$	1.11682e14i	0.0726966i	0.999339	+	0.0363483i	$0.0115726\pi$
$577$	1.11682e14i	0.0726966i	−0.999339	+	0.0363483i	$0.988427\pi$
$578$	4.02394e14i	0.259447i
$579$	0	0
$580$	0	0
$581$	−9.65124e14	−0.604803
$582$	0	0
$583$	− 2.83988e15i	− 1.74631i
$584$	−1.84074e15	−1.12130
$585$	0	0
$586$	−1.07635e14	−0.0643450
$587$	1.03228e15i	0.611347i	0.952136	+	0.305673i	$0.0988815\pi$
$587$	1.03228e15i	0.611347i	−0.952136	+	0.305673i	$0.901118\pi$
$588$	0	0
$589$	−1.17368e15	−0.682208
$590$	0	0
$591$	0	0
$592$	− 2.34753e14i	− 0.132691i
$593$	− 5.05946e14i	− 0.283337i	−0.989914	−	0.141668i	$-0.954753\pi$
$593$	− 5.05946e14i	− 0.283337i	0.989914	−	0.141668i	$-0.0452467\pi$
$594$	0	0
$595$	0	0
$596$	−1.22470e15	−0.667076
$597$	0	0
$598$	− 3.24748e14i	− 0.173656i
$599$	−1.86439e14	−0.0987846	−0.0493923	−	0.998779i	$-0.515728\pi$
$599$	−1.86439e14	−0.0987846	−0.0493923	+	0.998779i	$0.515728\pi$
$600$	0	0
$601$	−6.39018e14	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$601$	−6.39018e14	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$602$	1.96678e15i	1.01385i
$603$	0	0
$604$	2.53172e15	1.28148
$605$	0	0
$606$	0	0
$607$	− 3.07440e15i	− 1.51434i	−0.653219	−	0.757169i	$-0.726582\pi$
$607$	− 3.07440e15i	− 1.51434i	0.653219	−	0.757169i	$-0.273418\pi$
$608$	− 1.85438e15i	− 0.905166i
$609$	0	0
$610$	0	0
$611$	1.02821e15	0.488489
$612$	0	0
$613$	1.45823e15i	0.680447i	0.940345	+	0.340223i	$0.110503\pi$
$613$	1.45823e15i	0.680447i	−0.940345	+	0.340223i	$0.889497\pi$
$614$	−7.90767e14	−0.365698
$615$	0	0
$616$	−3.67488e15	−1.66935
$617$	2.52038e15i	1.13474i	0.823462	+	0.567372i	$0.192040\pi$
$617$	2.52038e15i	1.13474i	−0.823462	+	0.567372i	$0.807960\pi$
$618$	0	0
$619$	−2.76117e15	−1.22122	−0.610612	−	0.791930i	$-0.709076\pi$
$619$	−2.76117e15	−1.22122	−0.610612	+	0.791930i	$0.709076\pi$
$620$	0	0
$621$	0	0
$622$	− 7.96900e14i	− 0.343207i
$623$	− 3.54997e14i	− 0.151545i
$624$	0	0
$625$	0	0
$626$	6.32511e13	0.0262972
$627$	0	0
$628$	1.45347e15i	0.593783i
$629$	2.12413e14	0.0860205
$630$	0	0
$631$	−5.89126e14	−0.234448	−0.117224	−	0.993105i	$-0.537400\pi$
$631$	−5.89126e14	−0.234448	−0.117224	+	0.993105i	$0.537400\pi$
$632$	1.38839e14i	0.0547733i
$633$	0	0
$634$	4.14972e14	0.160890
$635$	0	0
$636$	0	0
$637$	5.28565e15i	1.99679i
$638$	2.25772e15i	0.845582i
$639$	0	0
$640$	0	0
$641$	−1.59787e15	−0.583208	−0.291604	−	0.956539i	$-0.594189\pi$
$641$	−1.59787e15	−0.583208	−0.291604	+	0.956539i	$0.594189\pi$
$642$	0	0
$643$	4.44445e15i	1.59462i	0.603569	+	0.797310i	$0.293745\pi$
$643$	4.44445e15i	1.59462i	−0.603569	+	0.797310i	$0.706255\pi$
$644$	−2.69693e15	−0.959395
$645$	0	0
$646$	4.49830e14	0.157315
$647$	− 8.54247e14i	− 0.296217i	−0.988971	−	0.148108i	$-0.952682\pi$
$647$	− 8.54247e14i	− 0.296217i	0.988971	−	0.148108i	$-0.0473185\pi$
$648$	0	0
$649$	−3.50816e15	−1.19601
$650$	0	0
$651$	0	0
$652$	− 8.55971e14i	− 0.284510i
$653$	2.86428e15i	0.944045i	0.881587	+	0.472022i	$0.156476\pi$
$653$	2.86428e15i	0.944045i	−0.881587	+	0.472022i	$0.843524\pi$
$654$	0	0
$655$	0	0
$656$	−6.58609e14	−0.211669
$657$	0	0
$658$	1.01017e15i	0.319267i
$659$	4.78135e15	1.49858	0.749291	−	0.662241i	$-0.230394\pi$
$659$	4.78135e15	1.49858	0.749291	+	0.662241i	$0.230394\pi$
$660$	0	0
$661$	−5.52275e14	−0.170234	−0.0851172	−	0.996371i	$-0.527126\pi$
$661$	−5.52275e14	−0.170234	−0.0851172	+	0.996371i	$0.527126\pi$
$662$	− 1.46524e15i	− 0.447909i
$663$	0	0
$664$	6.89369e14	0.207266
$665$	0	0
$666$	0	0
$667$	3.50982e15i	1.02942i
$668$	5.26480e15i	1.53148i
$669$	0	0
$670$	0	0
$671$	1.33152e15	0.377897
$672$	0	0
$673$	− 3.31175e15i	− 0.924645i	−0.886712	−	0.462323i	$-0.847016\pi$
$673$	− 3.31175e15i	− 0.924645i	0.886712	−	0.462323i	$-0.152984\pi$
$674$	−1.33234e15	−0.368964
$675$	0	0
$676$	6.55156e14	0.178500
$677$	6.69181e15i	1.80845i	0.427060	+	0.904223i	$0.359549\pi$
$677$	6.69181e15i	1.80845i	−0.427060	+	0.904223i	$0.640451\pi$
$678$	0	0
$679$	1.63442e15	0.434590
$680$	0	0
$681$	0	0
$682$	− 1.19833e15i	− 0.311003i
$683$	7.20810e15i	1.85570i	0.372959	+	0.927848i	$0.378343\pi$
$683$	7.20810e15i	1.85570i	−0.372959	+	0.927848i	$0.621657\pi$
$684$	0	0
$685$	0	0
$686$	−2.86629e15	−0.720340
$687$	0	0
$688$	4.86426e15i	1.20304i
$689$	4.22479e15	1.03657
$690$	0	0
$691$	7.79467e15	1.88221	0.941106	−	0.338112i	$-0.109788\pi$
$691$	7.79467e15	1.88221	0.941106	+	0.338112i	$0.109788\pi$
$692$	− 2.80352e15i	− 0.671615i
$693$	0	0
$694$	9.37448e14	0.221040
$695$	0	0
$696$	0	0
$697$	− 5.95932e14i	− 0.137220i
$698$	− 1.04299e15i	− 0.238274i
$699$	0	0
$700$	0	0
$701$	−6.71290e15	−1.49782	−0.748912	−	0.662669i	$-0.769423\pi$
$701$	−6.71290e15	−1.49782	−0.748912	+	0.662669i	$0.769423\pi$
$702$	0	0
$703$	9.36234e14i	0.205651i
$704$	−2.90483e15	−0.633099
$705$	0	0
$706$	9.75719e14	0.209362
$707$	− 1.24666e16i	− 2.65424i
$708$	0	0
$709$	4.10214e15	0.859917	0.429959	−	0.902849i	$-0.358528\pi$
$709$	4.10214e15	0.859917	0.429959	+	0.902849i	$0.358528\pi$
$710$	0	0
$711$	0	0
$712$	2.53568e14i	0.0519343i
$713$	− 1.86291e15i	− 0.378617i
$714$	0	0
$715$	0	0
$716$	−3.50404e15	−0.695902
$717$	0	0
$718$	1.00122e15i	0.195814i
$719$	−3.31818e15	−0.644008	−0.322004	−	0.946738i	$-0.604356\pi$
$719$	−3.31818e15	−0.644008	−0.322004	+	0.946738i	$0.604356\pi$
$720$	0	0
$721$	−4.07960e15	−0.779783
$722$	2.68047e14i	0.0508460i
$723$	0	0
$724$	−3.12829e15	−0.584446
$725$	0	0
$726$	0	0
$727$	− 3.34273e15i	− 0.610467i	−0.952278	−	0.305233i	$-0.901266\pi$
$727$	− 3.34273e15i	− 0.610467i	0.952278	−	0.305233i	$-0.0987344\pi$
$728$	− 5.46699e15i	− 0.990890i
$729$	0	0
$730$	0	0
$731$	−4.40134e15	−0.779901
$732$	0	0
$733$	− 6.14966e15i	− 1.07344i	−0.843759	−	0.536722i	$-0.819662\pi$
$733$	− 6.14966e15i	− 1.07344i	0.843759	−	0.536722i	$-0.180338\pi$
$734$	3.23483e15	0.560432
$735$	0	0
$736$	2.94334e15	0.502357
$737$	6.10732e15i	1.03462i
$738$	0	0
$739$	4.15701e15	0.693804	0.346902	−	0.937901i	$-0.387234\pi$
$739$	4.15701e15	0.693804	0.346902	+	0.937901i	$0.387234\pi$
$740$	0	0
$741$	0	0
$742$	4.15068e15i	0.677482i
$743$	4.90364e15i	0.794474i	0.917716	+	0.397237i	$0.130031\pi$
$743$	4.90364e15i	0.794474i	−0.917716	+	0.397237i	$0.869969\pi$
$744$	0	0
$745$	0	0
$746$	2.46150e15	0.390064
$747$	0	0
$748$	− 3.88226e15i	− 0.606214i
$749$	8.52690e14	0.132173
$750$	0	0
$751$	−2.00580e15	−0.306385	−0.153192	−	0.988196i	$-0.548955\pi$
$751$	−2.00580e15	−0.306385	−0.153192	+	0.988196i	$0.548955\pi$
$752$	2.49837e15i	0.378842i
$753$	0	0
$754$	−3.35873e15	−0.501918
$755$	0	0
$756$	0	0
$757$	− 2.85016e15i	− 0.416718i	−0.978052	−	0.208359i	$-0.933188\pi$
$757$	− 2.85016e15i	− 0.416718i	0.978052	−	0.208359i	$-0.0668122\pi$
$758$	2.32235e15i	0.337091i
$759$	0	0
$760$	0	0
$761$	−8.65962e15	−1.22994	−0.614969	−	0.788551i	$-0.710832\pi$
$761$	−8.65962e15	−1.22994	−0.614969	+	0.788551i	$0.710832\pi$
$762$	0	0
$763$	− 2.49827e15i	− 0.349747i
$764$	−5.78743e15	−0.804400
$765$	0	0
$766$	8.08453e14	0.110763
$767$	− 5.21897e15i	− 0.709922i
$768$	0	0
$769$	7.49727e15	1.00533	0.502665	−	0.864481i	$-0.332353\pi$
$769$	7.49727e15	1.00533	0.502665	+	0.864481i	$0.332353\pi$
$770$	0	0
$771$	0	0
$772$	5.07826e15i	0.666530i
$773$	− 2.07867e15i	− 0.270894i	−0.990785	−	0.135447i	$-0.956753\pi$
$773$	− 2.07867e15i	− 0.270894i	0.990785	−	0.135447i	$-0.0432470\pi$
$774$	0	0
$775$	0	0
$776$	−1.16743e15	−0.148934
$777$	0	0
$778$	− 3.42866e15i	− 0.431258i
$779$	2.62664e15	0.328053
$780$	0	0
$781$	−2.22045e16	−2.73438
$782$	7.13988e14i	0.0873079i
$783$	0	0
$784$	−1.28432e16	−1.54859
$785$	0	0
$786$	0	0
$787$	1.58421e16i	1.87047i	0.354026	+	0.935235i	$0.384812\pi$
$787$	1.58421e16i	1.87047i	−0.354026	+	0.935235i	$0.615188\pi$
$788$	7.07823e15i	0.829909i
$789$	0	0
$790$	0	0
$791$	3.16844e15	0.363810
$792$	0	0
$793$	1.98086e15i	0.224311i
$794$	−7.00826e14	−0.0788129
$795$	0	0
$796$	1.86503e15	0.206854
$797$	− 4.58469e15i	− 0.504997i	−0.967597	−	0.252499i	$-0.918748\pi$
$797$	− 4.58469e15i	− 0.504997i	0.967597	−	0.252499i	$-0.0812523\pi$
$798$	0	0
$799$	−2.26061e15	−0.245594
$800$	0	0
$801$	0	0
$802$	4.40210e15i	0.468490i
$803$	− 2.59528e16i	− 2.74315i
$804$	0	0
$805$	0	0
$806$	1.78272e15	0.184604
$807$	0	0
$808$	8.90465e15i	0.909608i
$809$	−1.72813e15	−0.175331	−0.0876656	−	0.996150i	$-0.527941\pi$
$809$	−1.72813e15	−0.175331	−0.0876656	+	0.996150i	$0.527941\pi$
$810$	0	0
$811$	4.24638e14	0.0425015	0.0212507	−	0.999774i	$-0.493235\pi$
$811$	4.24638e14	0.0425015	0.0212507	+	0.999774i	$0.493235\pi$
$812$	2.78932e16i	2.77294i
$813$	0	0
$814$	−9.55899e14	−0.0937514
$815$	0	0
$816$	0	0
$817$	− 1.93994e16i	− 1.86452i
$818$	3.76631e15i	0.359561i
$819$	0	0
$820$	0	0
$821$	1.96517e16	1.83871	0.919354	−	0.393431i	$-0.128712\pi$
$821$	1.96517e16	1.83871	0.919354	+	0.393431i	$0.128712\pi$
$822$	0	0
$823$	5.95701e14i	0.0549957i	0.999622	+	0.0274979i	$0.00875394\pi$
$823$	5.95701e14i	0.0549957i	−0.999622	+	0.0274979i	$0.991246\pi$
$824$	2.91398e15	0.267231
$825$	0	0
$826$	5.12743e15	0.463991
$827$	− 1.40975e16i	− 1.26725i	−0.773639	−	0.633626i	$-0.781566\pi$
$827$	− 1.40975e16i	− 1.26725i	0.773639	−	0.633626i	$-0.218434\pi$
$828$	0	0
$829$	7.87187e15	0.698277	0.349139	−	0.937071i	$-0.386474\pi$
$829$	7.87187e15	0.698277	0.349139	+	0.937071i	$0.386474\pi$
$830$	0	0
$831$	0	0
$832$	− 4.32142e15i	− 0.375793i
$833$	− 1.16210e16i	− 1.00391i
$834$	0	0
$835$	0	0
$836$	1.71115e16	1.44929
$837$	0	0
$838$	3.26507e15i	0.272929i
$839$	6.12117e15	0.508327	0.254164	−	0.967161i	$-0.418200\pi$
$839$	6.12117e15	0.508327	0.254164	+	0.967161i	$0.418200\pi$
$840$	0	0
$841$	2.41000e16	1.97533
$842$	1.44092e15i	0.117334i
$843$	0	0
$844$	3.76650e15	0.302729
$845$	0	0
$846$	0	0
$847$	− 2.90043e16i	− 2.28615i
$848$	1.02655e16i	0.803902i
$849$	0	0
$850$	0	0
$851$	−1.48603e15	−0.114134
$852$	0	0
$853$	2.83032e15i	0.214594i	0.994227	+	0.107297i	$0.0342195\pi$
$853$	2.83032e15i	0.214594i	−0.994227	+	0.107297i	$0.965781\pi$
$854$	−1.94611e15	−0.146605
$855$	0	0
$856$	−6.09060e14	−0.0452954
$857$	− 4.83193e14i	− 0.0357048i	−0.999841	−	0.0178524i	$-0.994317\pi$
$857$	− 4.83193e14i	− 0.0357048i	0.999841	−	0.0178524i	$-0.00568289\pi$
$858$	0	0
$859$	−1.23886e16	−0.903775	−0.451888	−	0.892075i	$-0.649249\pi$
$859$	−1.23886e16	−0.903775	−0.451888	+	0.892075i	$0.649249\pi$
$860$	0	0
$861$	0	0
$862$	− 1.63992e15i	− 0.117364i
$863$	2.26458e16i	1.61038i	0.593016	+	0.805191i	$0.297937\pi$
$863$	2.26458e16i	1.61038i	−0.593016	+	0.805191i	$0.702063\pi$
$864$	0	0
$865$	0	0
$866$	5.03459e15	0.351249
$867$	0	0
$868$	− 1.48050e16i	− 1.01988i
$869$	−1.95751e15	−0.133997
$870$	0	0
$871$	−9.08565e15	−0.614124
$872$	1.78446e15i	0.119858i
$873$	0	0
$874$	−3.14698e15	−0.208728
$875$	0	0
$876$	0	0
$877$	8.89622e15i	0.579039i	0.957172	+	0.289519i	$0.0934955\pi$
$877$	8.89622e15i	0.579039i	−0.957172	+	0.289519i	$0.906505\pi$
$878$	− 1.78023e15i	− 0.115148i
$879$	0	0
$880$	0	0
$881$	1.82720e16	1.15989	0.579946	−	0.814655i	$-0.303073\pi$
$881$	1.82720e16	1.15989	0.579946	+	0.814655i	$0.303073\pi$
$882$	0	0
$883$	8.75758e15i	0.549035i	0.961582	+	0.274518i	$0.0885182\pi$
$883$	8.75758e15i	0.549035i	−0.961582	+	0.274518i	$0.911482\pi$
$884$	5.77551e15	0.359834
$885$	0	0
$886$	−1.87398e15	−0.115314
$887$	8.52159e15i	0.521124i	0.965457	+	0.260562i	$0.0839078\pi$
$887$	8.52159e15i	0.521124i	−0.965457	+	0.260562i	$0.916092\pi$
$888$	0	0
$889$	−2.10613e15	−0.127212
$890$	0	0
$891$	0	0
$892$	1.83132e16i	1.08582i
$893$	− 9.96390e15i	− 0.587146i
$894$	0	0
$895$	0	0
$896$	3.04042e16	1.75889
$897$	0	0
$898$	− 2.38209e15i	− 0.136125i
$899$	−1.92673e16	−1.09432
$900$	0	0
$901$	−9.28858e15	−0.521150
$902$	2.68181e15i	0.149552i
$903$	0	0
$904$	−2.26316e15	−0.124677
$905$	0	0
$906$	0	0
$907$	− 2.10550e16i	− 1.13898i	−0.821998	−	0.569490i	$-0.807141\pi$
$907$	− 2.10550e16i	− 1.13898i	0.821998	−	0.569490i	$-0.192859\pi$
$908$	− 1.24005e16i	− 0.666756i
$909$	0	0
$910$	0	0
$911$	−3.04479e16	−1.60771	−0.803854	−	0.594827i	$-0.797220\pi$
$911$	−3.04479e16	−1.60771	−0.803854	+	0.594827i	$0.797220\pi$
$912$	0	0
$913$	9.71950e15i	0.507055i
$914$	−5.13939e15	−0.266506
$915$	0	0
$916$	−1.86674e16	−0.956443
$917$	− 1.75805e16i	− 0.895363i
$918$	0	0
$919$	4.47695e15	0.225293	0.112646	−	0.993635i	$-0.464067\pi$
$919$	4.47695e15	0.225293	0.112646	+	0.993635i	$0.464067\pi$
$920$	0	0
$921$	0	0
$922$	6.61799e15i	0.327119i
$923$	− 3.30328e16i	− 1.62307i
$924$	0	0
$925$	0	0
$926$	−6.19818e14	−0.0299160
$927$	0	0
$928$	− 3.04417e16i	− 1.45196i
$929$	2.63310e16	1.24848	0.624238	−	0.781234i	$-0.285409\pi$
$929$	2.63310e16	1.24848	0.624238	+	0.781234i	$0.285409\pi$
$930$	0	0
$931$	5.12208e16	2.40007
$932$	− 1.23515e16i	− 0.575348i
$933$	0	0
$934$	1.16481e15	0.0536224
$935$	0	0
$936$	0	0
$937$	− 1.18259e16i	− 0.534894i	−0.963573	−	0.267447i	$-0.913820\pi$
$937$	− 1.18259e16i	− 0.534894i	0.963573	−	0.267447i	$-0.0861800\pi$
$938$	− 8.92628e15i	− 0.401379i
$939$	0	0
$940$	0	0
$941$	−6.96003e15	−0.307517	−0.153758	−	0.988108i	$-0.549138\pi$
$941$	−6.96003e15	−0.307517	−0.153758	+	0.988108i	$0.549138\pi$
$942$	0	0
$943$	4.16911e15i	0.182066i
$944$	1.26812e16	0.550573
$945$	0	0
$946$	1.98069e16	0.849993
$947$	− 3.38501e16i	− 1.44423i	−0.691775	−	0.722113i	$-0.743171\pi$
$947$	− 3.38501e16i	− 1.44423i	0.691775	−	0.722113i	$-0.256829\pi$
$948$	0	0
$949$	3.86091e16	1.62827
$950$	0	0
$951$	0	0
$952$	1.20197e16i	0.498183i
$953$	− 4.50062e16i	− 1.85465i	−0.374262	−	0.927323i	$-0.622104\pi$
$953$	− 4.50062e16i	− 1.85465i	0.374262	−	0.927323i	$-0.377896\pi$
$954$	0	0
$955$	0	0
$956$	−3.23235e16	−1.30918
$957$	0	0
$958$	− 1.41644e16i	− 0.567139i
$959$	−4.42907e16	−1.76323
$960$	0	0
$961$	−1.51819e16	−0.597513
$962$	− 1.42206e15i	− 0.0556486i
$963$	0	0
$964$	−2.52107e16	−0.975353
$965$	0	0
$966$	0	0
$967$	1.48359e16i	0.564246i	0.959378	+	0.282123i	$0.0910386\pi$
$967$	1.48359e16i	0.564246i	−0.959378	+	0.282123i	$0.908961\pi$
$968$	2.07172e16i	0.783461i
$969$	0	0
$970$	0	0
$971$	−1.14414e16	−0.425378	−0.212689	−	0.977120i	$-0.568222\pi$
$971$	−1.14414e16	−0.425378	−0.212689	+	0.977120i	$0.568222\pi$
$972$	0	0
$973$	− 4.93669e16i	− 1.81474i
$974$	6.72835e15	0.245943
$975$	0	0
$976$	−4.81314e15	−0.173962
$977$	− 2.64437e16i	− 0.950390i	−0.879881	−	0.475195i	$-0.842378\pi$
$977$	− 2.64437e16i	− 0.950390i	0.879881	−	0.475195i	$-0.157622\pi$
$978$	0	0
$979$	−3.57508e15	−0.127052
$980$	0	0
$981$	0	0
$982$	− 6.47820e15i	− 0.226382i
$983$	− 3.63656e16i	− 1.26371i	−0.775088	−	0.631854i	$-0.782294\pi$
$983$	− 3.63656e16i	− 1.26371i	0.775088	−	0.631854i	$-0.217706\pi$
$984$	0	0
$985$	0	0
$986$	7.38447e15	0.252346
$987$	0	0
$988$	2.54562e16i	0.860262i
$989$	3.07915e16	1.03479
$990$	0	0
$991$	5.09067e16	1.69188	0.845941	−	0.533276i	$-0.179039\pi$
$991$	5.09067e16	1.69188	0.845941	+	0.533276i	$0.179039\pi$
$992$	1.61576e16i	0.534027i
$993$	0	0
$994$	3.24534e16	1.06080
$995$	0	0
$996$	0	0
$997$	4.28291e16i	1.37694i	0.725264	+	0.688471i	$0.241718\pi$
$997$	4.28291e16i	1.37694i	−0.725264	+	0.688471i	$0.758282\pi$
$998$	− 9.36259e15i	− 0.299349i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.b.i.199.3		4
3.2	odd	2		75.12.b.d.49.2		4
5.2	odd	4		45.12.a.c.1.1		2
5.3	odd	4		225.12.a.i.1.2		2
5.4	even	2	inner	225.12.b.i.199.2		4
15.2	even	4		15.12.a.c.1.2	✓	2
15.8	even	4		75.12.a.c.1.1		2
15.14	odd	2		75.12.b.d.49.3		4
60.47	odd	4		240.12.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.2	✓	2	15.2	even	4
45.12.a.c.1.1		2	5.2	odd	4
75.12.a.c.1.1		2	15.8	even	4
75.12.b.d.49.2		4	3.2	odd	2
75.12.b.d.49.3		4	15.14	odd	2
225.12.a.i.1.2		2	5.3	odd	4
225.12.b.i.199.2		4	5.4	even	2	inner
225.12.b.i.199.3		4	1.1	even	1	trivial
240.12.a.m.1.1		2	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+14.7191i q^{2} +1831.35 q^{4} -79941.2i q^{7} +57100.5i q^{8} -805067. q^{11} -1.19767e6i q^{13} +1.17666e6 q^{14} +2.91013e6 q^{16} +2.63319e6i q^{17} -1.16061e7 q^{19} -1.18499e7i q^{22} -1.84216e7i q^{23} +1.76286e7 q^{26} -1.46400e8i q^{28} -1.90527e8 q^{29} +1.01127e8 q^{31} +1.59776e8i q^{32} -3.87581e7 q^{34} -8.06675e7i q^{37} -1.70831e8i q^{38} -2.26316e8 q^{41} +1.67149e9i q^{43} -1.47436e9 q^{44} +2.71150e8 q^{46} +8.58507e8i q^{47} -4.41327e9 q^{49} -2.19335e9i q^{52} +3.52750e9i q^{53} +4.56468e9 q^{56} -2.80438e9i q^{58} +4.35760e9 q^{59} -1.65393e9 q^{61} +1.48849e9i q^{62} +3.60819e9 q^{64} -7.58610e9i q^{67} +4.82228e9i q^{68} +2.75809e10 q^{71} +3.22368e10i q^{73} +1.18735e9 q^{74} -2.12548e10 q^{76} +6.43580e10i q^{77} +2.43149e9 q^{79} -3.33116e9i q^{82} -1.20729e10i q^{83} -2.46028e10 q^{86} -4.59697e10i q^{88} +4.44073e9 q^{89} -9.57433e10 q^{91} -3.37364e10i q^{92} -1.26364e10 q^{94} +2.04453e10i q^{97} -6.49594e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6222 q^{4} - 591136 q^{11} + 6353592 q^{14} + 5948130 q^{16} - 35255952 q^{19} + 138104132 q^{26} - 403763896 q^{29} - 142114016 q^{31} + 252161404 q^{34} + 655310296 q^{41} - 1644753136 q^{44} + 3217895664 q^{46}+ \cdots + 40951225552 q^{94}+O(q^{100}) \) 4 * q + 6222 * q^4 - 591136 * q^11 + 6353592 * q^14 + 5948130 * q^16 - 35255952 * q^19 + 138104132 * q^26 - 403763896 * q^29 - 142114016 * q^31 + 252161404 * q^34 + 655310296 * q^41 - 1644753136 * q^44 + 3217895664 * q^46 - 15285229540 * q^49 + 22440862920 * q^56 - 2957541152 * q^59 - 16529782920 * q^61 - 3092633362 * q^64 + 40437776512 * q^71 - 32284748068 * q^74 - 57921409528 * q^76 - 44649990880 * q^79 + 35119396184 * q^86 + 158419366152 * q^89 - 459213424096 * q^91 + 40951225552 * q^94

Introduction

L-functions

Modular forms

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Database

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