LMFDB - Embedded newform 225.10.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$115.883063137$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4 x^{7} - 3492 x^{6} + 10490 x^{5} + 3539940 x^{4} - 7097368 x^{3} - 848796093 x^{2} + \cdots + 43654030340 $ x^8 - 4x^7 - 3492x^6 + 10490x^5 + 3539940x^4 - 7097368x^3 - 848796093x^2 + 852346526*x + 43654030340
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{8}\cdot 5^{10} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-7.91265$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.2206 q^{2} -362.657 q^{4} +4342.19 q^{7} +10688.8 q^{8} -36750.7 q^{11} +187538. q^{13} -53064.1 q^{14} +55056.9 q^{16} +3942.12 q^{17} -237154. q^{19} +449115. q^{22} +2.19106e6 q^{23} -2.29183e6 q^{26} -1.57473e6 q^{28} +6.47801e6 q^{29} -4.92268e6 q^{31} -6.14550e6 q^{32} -48175.0 q^{34} +6.61220e6 q^{37} +2.89817e6 q^{38} -2.22036e7 q^{41} +1.25053e7 q^{43} +1.33279e7 q^{44} -2.67760e7 q^{46} -3.42169e7 q^{47} -2.14990e7 q^{49} -6.80121e7 q^{52} -3.09549e7 q^{53} +4.64129e7 q^{56} -7.91651e7 q^{58} -7.50725e7 q^{59} +1.30229e7 q^{61} +6.01580e7 q^{62} +4.69125e7 q^{64} +1.37041e8 q^{67} -1.42964e6 q^{68} +2.12149e8 q^{71} +2.65436e8 q^{73} -8.08050e7 q^{74} +8.60058e7 q^{76} -1.59578e8 q^{77} +2.69754e8 q^{79} +2.71341e8 q^{82} -5.22793e8 q^{83} -1.52822e8 q^{86} -3.92822e8 q^{88} -3.29338e8 q^{89} +8.14326e8 q^{91} -7.94603e8 q^{92} +4.18150e8 q^{94} -1.12881e9 q^{97} +2.62731e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 144 q^{4} - 1009088 q^{16} + 2079872 q^{19} + 12077344 q^{31} - 25286480 q^{34} + 3119840 q^{46} + 174619144 q^{49} - 165845744 q^{61} - 91038464 q^{64} + 1551352896 q^{76} + 1098932768 q^{79} - 1465128000 q^{91}+ \cdots + 4117219360 q^{94}+O(q^{100}) $ 8 * q + 144 * q^4 - 1009088 * q^16 + 2079872 * q^19 + 12077344 * q^31 - 25286480 * q^34 + 3119840 * q^46 + 174619144 * q^49 - 165845744 * q^61 - 91038464 * q^64 + 1551352896 * q^76 + 1098932768 * q^79 - 1465128000 * q^91 + 4117219360 * q^94

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 $

$2$	−12.2206	−0.540079	−0.270039	−	0.962849i	$-0.587037\pi$
$2$	−12.2206	−0.540079	−0.270039	+	0.962849i	$0.587037\pi$
$3$	0	0
$3$	0	0
$4$	−362.657	−0.708315
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4342.19	0.683545	0.341773	−	0.939783i	$-0.388973\pi$
$7$	4342.19	0.683545	0.341773	+	0.939783i	$0.388973\pi$
$8$	10688.8	0.922624
$9$	0	0
$10$	0	0
$11$	−36750.7	−0.756830	−0.378415	−	0.925636i	$-0.623531\pi$
$11$	−36750.7	−0.756830	−0.378415	+	0.925636i	$0.623531\pi$
$12$	0	0
$13$	187538.	1.82115	0.910573	−	0.413349i	$-0.135641\pi$
$13$	187538.	1.82115	0.910573	+	0.413349i	$0.135641\pi$
$14$	−53064.1	−0.369168
$15$	0	0
$16$	55056.9	0.210025
$17$	3942.12	0.0114475	0.00572374	−	0.999984i	$-0.498178\pi$
$17$	3942.12	0.0114475	0.00572374	+	0.999984i	$0.498178\pi$
$18$	0	0
$19$	−237154.	−0.417484	−0.208742	−	0.977971i	$-0.566937\pi$
$19$	−237154.	−0.417484	−0.208742	+	0.977971i	$0.566937\pi$
$20$	0	0
$21$	0	0
$22$	449115.	0.408748
$23$	2.19106e6	1.63260	0.816298	−	0.577631i	$-0.196023\pi$
$23$	2.19106e6	1.63260	0.816298	+	0.577631i	$0.196023\pi$
$24$	0	0
$25$	0	0
$26$	−2.29183e6	−0.983562
$27$	0	0
$28$	−1.57473e6	−0.484165
$29$	6.47801e6	1.70079	0.850395	−	0.526145i	$-0.176363\pi$
$29$	6.47801e6	1.70079	0.850395	+	0.526145i	$0.176363\pi$
$30$	0	0
$31$	−4.92268e6	−0.957357	−0.478679	−	0.877990i	$-0.658884\pi$
$31$	−4.92268e6	−0.957357	−0.478679	+	0.877990i	$0.658884\pi$
$32$	−6.14550e6	−1.03605
$33$	0	0
$34$	−48175.0	−0.00618254
$35$	0	0
$36$	0	0
$37$	6.61220e6	0.580014	0.290007	−	0.957025i	$-0.406342\pi$
$37$	6.61220e6	0.580014	0.290007	+	0.957025i	$0.406342\pi$
$38$	2.89817e6	0.225474
$39$	0	0
$40$	0	0
$41$	−2.22036e7	−1.22715	−0.613573	−	0.789638i	$-0.710268\pi$
$41$	−2.22036e7	−1.22715	−0.613573	+	0.789638i	$0.710268\pi$
$42$	0	0
$43$	1.25053e7	0.557808	0.278904	−	0.960319i	$-0.410029\pi$
$43$	1.25053e7	0.557808	0.278904	+	0.960319i	$0.410029\pi$
$44$	1.33279e7	0.536074
$45$	0	0
$46$	−2.67760e7	−0.881730
$47$	−3.42169e7	−1.02282	−0.511411	−	0.859336i	$-0.670877\pi$
$47$	−3.42169e7	−1.02282	−0.511411	+	0.859336i	$0.670877\pi$
$48$	0	0
$49$	−2.14990e7	−0.532766
$50$	0	0
$51$	0	0
$52$	−6.80121e7	−1.28994
$53$	−3.09549e7	−0.538875	−0.269438	−	0.963018i	$-0.586838\pi$
$53$	−3.09549e7	−0.538875	−0.269438	+	0.963018i	$0.586838\pi$
$54$	0	0
$55$	0	0
$56$	4.64129e7	0.630656
$57$	0	0
$58$	−7.91651e7	−0.918560
$59$	−7.50725e7	−0.806579	−0.403290	−	0.915072i	$-0.632133\pi$
$59$	−7.50725e7	−0.806579	−0.403290	+	0.915072i	$0.632133\pi$
$60$	0	0
$61$	1.30229e7	0.120427	0.0602136	−	0.998186i	$-0.480822\pi$
$61$	1.30229e7	0.120427	0.0602136	+	0.998186i	$0.480822\pi$
$62$	6.01580e7	0.517048
$63$	0	0
$64$	4.69125e7	0.349526
$65$	0	0
$66$	0	0
$67$	1.37041e8	0.830836	0.415418	−	0.909631i	$-0.363635\pi$
$67$	1.37041e8	0.830836	0.415418	+	0.909631i	$0.363635\pi$
$68$	−1.42964e6	−0.00810842
$69$	0	0
$70$	0	0
$71$	2.12149e8	0.990784	0.495392	−	0.868670i	$-0.335024\pi$
$71$	2.12149e8	0.990784	0.495392	+	0.868670i	$0.335024\pi$
$72$	0	0
$73$	2.65436e8	1.09397	0.546987	−	0.837141i	$-0.315775\pi$
$73$	2.65436e8	1.09397	0.546987	+	0.837141i	$0.315775\pi$
$74$	−8.08050e7	−0.313253
$75$	0	0
$76$	8.60058e7	0.295710
$77$	−1.59578e8	−0.517328
$78$	0	0
$79$	2.69754e8	0.779194	0.389597	−	0.920985i	$-0.372614\pi$
$79$	2.69754e8	0.779194	0.389597	+	0.920985i	$0.372614\pi$
$80$	0	0
$81$	0	0
$82$	2.71341e8	0.662755
$83$	−5.22793e8	−1.20915	−0.604573	−	0.796550i	$-0.706656\pi$
$83$	−5.22793e8	−1.20915	−0.604573	+	0.796550i	$0.706656\pi$
$84$	0	0
$85$	0	0
$86$	−1.52822e8	−0.301260
$87$	0	0
$88$	−3.92822e8	−0.698270
$89$	−3.29338e8	−0.556400	−0.278200	−	0.960523i	$-0.589738\pi$
$89$	−3.29338e8	−0.556400	−0.278200	+	0.960523i	$0.589738\pi$
$90$	0	0
$91$	8.14326e8	1.24484
$92$	−7.94603e8	−1.15639
$93$	0	0
$94$	4.18150e8	0.552405
$95$	0	0
$96$	0	0
$97$	−1.12881e9	−1.29464	−0.647319	−	0.762219i	$-0.724110\pi$
$97$	−1.12881e9	−1.29464	−0.647319	+	0.762219i	$0.724110\pi$
$98$	2.62731e8	0.287735
$99$	0	0
$100$	0	0
$101$	−5.08135e8	−0.485884	−0.242942	−	0.970041i	$-0.578112\pi$
$101$	−5.08135e8	−0.485884	−0.242942	+	0.970041i	$0.578112\pi$
$102$	0	0
$103$	−1.62711e7	−0.0142446	−0.00712230	−	0.999975i	$-0.502267\pi$
$103$	−1.62711e7	−0.0142446	−0.00712230	+	0.999975i	$0.502267\pi$
$104$	2.00456e9	1.68023
$105$	0	0
$106$	3.78287e8	0.291035
$107$	−1.55057e9	−1.14357	−0.571787	−	0.820402i	$-0.693750\pi$
$107$	−1.55057e9	−1.14357	−0.571787	+	0.820402i	$0.693750\pi$
$108$	0	0
$109$	1.46179e9	0.991893	0.495947	−	0.868353i	$-0.334821\pi$
$109$	1.46179e9	0.991893	0.495947	+	0.868353i	$0.334821\pi$
$110$	0	0
$111$	0	0
$112$	2.39067e8	0.143562
$113$	5.32165e8	0.307039	0.153519	−	0.988146i	$-0.450939\pi$
$113$	5.32165e8	0.307039	0.153519	+	0.988146i	$0.450939\pi$
$114$	0	0
$115$	0	0
$116$	−2.34930e9	−1.20470
$117$	0	0
$118$	9.17430e8	0.435616
$119$	1.71174e7	0.00782487
$120$	0	0
$121$	−1.00733e9	−0.427208
$122$	−1.59148e8	−0.0650401
$123$	0	0
$124$	1.78525e9	0.678110
$125$	0	0
$126$	0	0
$127$	2.55988e9	0.873177	0.436589	−	0.899661i	$-0.356187\pi$
$127$	2.55988e9	0.873177	0.436589	+	0.899661i	$0.356187\pi$
$128$	2.57320e9	0.847283
$129$	0	0
$130$	0	0
$131$	5.60469e9	1.66276	0.831382	−	0.555702i	$-0.187550\pi$
$131$	5.60469e9	1.66276	0.831382	+	0.555702i	$0.187550\pi$
$132$	0	0
$133$	−1.02977e9	−0.285369
$134$	−1.67473e9	−0.448717
$135$	0	0
$136$	4.21367e7	0.0105617
$137$	5.27039e9	1.27820	0.639102	−	0.769122i	$-0.279306\pi$
$137$	5.27039e9	1.27820	0.639102	+	0.769122i	$0.279306\pi$
$138$	0	0
$139$	7.90257e9	1.79557	0.897784	−	0.440436i	$-0.145176\pi$
$139$	7.90257e9	1.79557	0.897784	+	0.440436i	$0.145176\pi$
$140$	0	0
$141$	0	0
$142$	−2.59259e9	−0.535101
$143$	−6.89216e9	−1.37830
$144$	0	0
$145$	0	0
$146$	−3.24379e9	−0.590832
$147$	0	0
$148$	−2.39796e9	−0.410833
$149$	−8.88302e9	−1.47646	−0.738232	−	0.674547i	$-0.764339\pi$
$149$	−8.88302e9	−1.47646	−0.738232	+	0.674547i	$0.764339\pi$
$150$	0	0
$151$	4.35498e9	0.681695	0.340847	−	0.940119i	$-0.389286\pi$
$151$	4.35498e9	0.681695	0.340847	+	0.940119i	$0.389286\pi$
$152$	−2.53490e9	−0.385181
$153$	0	0
$154$	1.95014e9	0.279398
$155$	0	0
$156$	0	0
$157$	2.10491e9	0.276494	0.138247	−	0.990398i	$-0.455853\pi$
$157$	2.10491e9	0.276494	0.138247	+	0.990398i	$0.455853\pi$
$158$	−3.29655e9	−0.420826
$159$	0	0
$160$	0	0
$161$	9.51398e9	1.11595
$162$	0	0
$163$	7.77003e9	0.862142	0.431071	−	0.902318i	$-0.358136\pi$
$163$	7.77003e9	0.862142	0.431071	+	0.902318i	$0.358136\pi$
$164$	8.05230e9	0.869206
$165$	0	0
$166$	6.38884e9	0.653034
$167$	1.81537e10	1.80610	0.903048	−	0.429540i	$-0.141324\pi$
$167$	1.81537e10	1.80610	0.903048	+	0.429540i	$0.141324\pi$
$168$	0	0
$169$	2.45661e10	2.31657
$170$	0	0
$171$	0	0
$172$	−4.53512e9	−0.395104
$173$	8.80586e9	0.747419	0.373710	−	0.927546i	$-0.378086\pi$
$173$	8.80586e9	0.747419	0.373710	+	0.927546i	$0.378086\pi$
$174$	0	0
$175$	0	0
$176$	−2.02338e9	−0.158953
$177$	0	0
$178$	4.02470e9	0.300500
$179$	7.00851e8	0.0510255	0.0255127	−	0.999674i	$-0.491878\pi$
$179$	7.00851e8	0.0510255	0.0255127	+	0.999674i	$0.491878\pi$
$180$	0	0
$181$	1.87883e10	1.30117	0.650586	−	0.759433i	$-0.274523\pi$
$181$	1.87883e10	1.30117	0.650586	+	0.759433i	$0.274523\pi$
$182$	−9.95154e9	−0.672309
$183$	0	0
$184$	2.34198e10	1.50627
$185$	0	0
$186$	0	0
$187$	−1.44876e8	−0.00866380
$188$	1.24090e10	0.724481
$189$	0	0
$190$	0	0
$191$	−2.71653e10	−1.47695	−0.738473	−	0.674283i	$-0.764453\pi$
$191$	−2.71653e10	−1.47695	−0.738473	+	0.674283i	$0.764453\pi$
$192$	0	0
$193$	−3.36849e10	−1.74754	−0.873771	−	0.486338i	$-0.838332\pi$
$193$	−3.36849e10	−1.74754	−0.873771	+	0.486338i	$0.838332\pi$
$194$	1.37947e10	0.699207
$195$	0	0
$196$	7.79678e9	0.377366
$197$	3.52896e8	0.0166936	0.00834678	−	0.999965i	$-0.497343\pi$
$197$	3.52896e8	0.0166936	0.00834678	+	0.999965i	$0.497343\pi$
$198$	0	0
$199$	2.79906e10	1.26524	0.632620	−	0.774463i	$-0.281980\pi$
$199$	2.79906e10	1.26524	0.632620	+	0.774463i	$0.281980\pi$
$200$	0	0
$201$	0	0
$202$	6.20970e9	0.262416
$203$	2.81287e10	1.16257
$204$	0	0
$205$	0	0
$206$	1.98843e8	0.00769320
$207$	0	0
$208$	1.03253e10	0.382487
$209$	8.71559e9	0.315965
$210$	0	0
$211$	1.91333e10	0.664536	0.332268	−	0.943185i	$-0.392186\pi$
$211$	1.91333e10	0.664536	0.332268	+	0.943185i	$0.392186\pi$
$212$	1.12260e10	0.381694
$213$	0	0
$214$	1.89489e10	0.617620
$215$	0	0
$216$	0	0
$217$	−2.13752e10	−0.654397
$218$	−1.78639e10	−0.535700
$219$	0	0
$220$	0	0
$221$	7.39299e8	0.0208475
$222$	0	0
$223$	5.66446e10	1.53386	0.766932	−	0.641728i	$-0.221782\pi$
$223$	5.66446e10	1.53386	0.766932	+	0.641728i	$0.221782\pi$
$224$	−2.66849e10	−0.708190
$225$	0	0
$226$	−6.50336e9	−0.165825
$227$	−3.31522e10	−0.828698	−0.414349	−	0.910118i	$-0.635991\pi$
$227$	−3.31522e10	−0.828698	−0.414349	+	0.910118i	$0.635991\pi$
$228$	0	0
$229$	1.68693e10	0.405356	0.202678	−	0.979245i	$-0.435036\pi$
$229$	1.68693e10	0.405356	0.202678	+	0.979245i	$0.435036\pi$
$230$	0	0
$231$	0	0
$232$	6.92423e10	1.56919
$233$	−5.99796e10	−1.33322	−0.666610	−	0.745407i	$-0.732255\pi$
$233$	−5.99796e10	−1.33322	−0.666610	+	0.745407i	$0.732255\pi$
$234$	0	0
$235$	0	0
$236$	2.72256e10	0.571312
$237$	0	0
$238$	−2.09185e8	−0.00422605
$239$	1.06603e10	0.211339	0.105669	−	0.994401i	$-0.466301\pi$
$239$	1.06603e10	0.211339	0.105669	+	0.994401i	$0.466301\pi$
$240$	0	0
$241$	−8.69990e9	−0.166126	−0.0830630	−	0.996544i	$-0.526470\pi$
$241$	−8.69990e9	−0.166126	−0.0830630	+	0.996544i	$0.526470\pi$
$242$	1.23102e10	0.230726
$243$	0	0
$244$	−4.72286e9	−0.0853004
$245$	0	0
$246$	0	0
$247$	−4.44755e10	−0.760300
$248$	−5.26177e10	−0.883281
$249$	0	0
$250$	0	0
$251$	6.50269e10	1.03410	0.517049	−	0.855956i	$-0.327031\pi$
$251$	6.50269e10	1.03410	0.517049	+	0.855956i	$0.327031\pi$
$252$	0	0
$253$	−8.05229e10	−1.23560
$254$	−3.12832e10	−0.471584
$255$	0	0
$256$	−5.54652e10	−0.807125
$257$	−6.68013e9	−0.0955182	−0.0477591	−	0.998859i	$-0.515208\pi$
$257$	−6.68013e9	−0.0955182	−0.0477591	+	0.998859i	$0.515208\pi$
$258$	0	0
$259$	2.87114e10	0.396466
$260$	0	0
$261$	0	0
$262$	−6.84925e10	−0.898023
$263$	9.64903e10	1.24361	0.621803	−	0.783174i	$-0.286401\pi$
$263$	9.64903e10	1.24361	0.621803	+	0.783174i	$0.286401\pi$
$264$	0	0
$265$	0	0
$266$	1.25844e10	0.154122
$267$	0	0
$268$	−4.96991e10	−0.588494
$269$	−7.10118e10	−0.826886	−0.413443	−	0.910530i	$-0.635674\pi$
$269$	−7.10118e10	−0.826886	−0.413443	+	0.910530i	$0.635674\pi$
$270$	0	0
$271$	−4.09389e10	−0.461078	−0.230539	−	0.973063i	$-0.574049\pi$
$271$	−4.09389e10	−0.461078	−0.230539	+	0.973063i	$0.574049\pi$
$272$	2.17041e8	0.00240426
$273$	0	0
$274$	−6.44072e10	−0.690331
$275$	0	0
$276$	0	0
$277$	7.44596e10	0.759909	0.379954	−	0.925005i	$-0.375940\pi$
$277$	7.44596e10	0.759909	0.379954	+	0.925005i	$0.375940\pi$
$278$	−9.65740e10	−0.969748
$279$	0	0
$280$	0	0
$281$	−1.38658e11	−1.32668	−0.663340	−	0.748318i	$-0.730862\pi$
$281$	−1.38658e11	−1.32668	−0.663340	+	0.748318i	$0.730862\pi$
$282$	0	0
$283$	5.71956e10	0.530059	0.265029	−	0.964240i	$-0.414618\pi$
$283$	5.71956e10	0.530059	0.265029	+	0.964240i	$0.414618\pi$
$284$	−7.69375e10	−0.701787
$285$	0	0
$286$	8.42262e10	0.744389
$287$	−9.64122e10	−0.838810
$288$	0	0
$289$	−1.18572e11	−0.999869
$290$	0	0
$291$	0	0
$292$	−9.62624e10	−0.774879
$293$	1.09734e11	0.869832	0.434916	−	0.900471i	$-0.356778\pi$
$293$	1.09734e11	0.869832	0.434916	+	0.900471i	$0.356778\pi$
$294$	0	0
$295$	0	0
$296$	7.06767e10	0.535135
$297$	0	0
$298$	1.08556e11	0.797406
$299$	4.10907e11	2.97319
$300$	0	0
$301$	5.43001e10	0.381287
$302$	−5.32204e10	−0.368169
$303$	0	0
$304$	−1.30570e10	−0.0876823
$305$	0	0
$306$	0	0
$307$	1.89369e11	1.21670	0.608352	−	0.793667i	$-0.291831\pi$
$307$	1.89369e11	1.21670	0.608352	+	0.793667i	$0.291831\pi$
$308$	5.78722e10	0.366431
$309$	0	0
$310$	0	0
$311$	−1.40583e11	−0.852139	−0.426070	−	0.904690i	$-0.640102\pi$
$311$	−1.40583e11	−0.852139	−0.426070	+	0.904690i	$0.640102\pi$
$312$	0	0
$313$	1.43754e11	0.846585	0.423292	−	0.905993i	$-0.360874\pi$
$313$	1.43754e11	0.846585	0.423292	+	0.905993i	$0.360874\pi$
$314$	−2.57232e10	−0.149328
$315$	0	0
$316$	−9.78282e10	−0.551915
$317$	−1.67042e11	−0.929091	−0.464545	−	0.885549i	$-0.653782\pi$
$317$	−1.67042e11	−0.929091	−0.464545	+	0.885549i	$0.653782\pi$
$318$	0	0
$319$	−2.38071e11	−1.28721
$320$	0	0
$321$	0	0
$322$	−1.16266e11	−0.602702
$323$	−9.34892e8	−0.00477914
$324$	0	0
$325$	0	0
$326$	−9.49544e10	−0.465624
$327$	0	0
$328$	−2.37331e11	−1.13220
$329$	−1.48576e11	−0.699145
$330$	0	0
$331$	−1.98423e10	−0.0908587	−0.0454294	−	0.998968i	$-0.514466\pi$
$331$	−1.98423e10	−0.0908587	−0.0454294	+	0.998968i	$0.514466\pi$
$332$	1.89595e11	0.856456
$333$	0	0
$334$	−2.21849e11	−0.975433
$335$	0	0
$336$	0	0
$337$	6.71805e10	0.283732	0.141866	−	0.989886i	$-0.454690\pi$
$337$	6.71805e10	0.283732	0.141866	+	0.989886i	$0.454690\pi$
$338$	−3.00212e11	−1.25113
$339$	0	0
$340$	0	0
$341$	1.80912e11	0.724557
$342$	0	0
$343$	−2.68576e11	−1.04771
$344$	1.33666e11	0.514647
$345$	0	0
$346$	−1.07613e11	−0.403665
$347$	2.47932e11	0.918014	0.459007	−	0.888433i	$-0.348205\pi$
$347$	2.47932e11	0.918014	0.459007	+	0.888433i	$0.348205\pi$
$348$	0	0
$349$	−1.62415e11	−0.586018	−0.293009	−	0.956110i	$-0.594657\pi$
$349$	−1.62415e11	−0.586018	−0.293009	+	0.956110i	$0.594657\pi$
$350$	0	0
$351$	0	0
$352$	2.25852e11	0.784117
$353$	2.36285e11	0.809935	0.404968	−	0.914331i	$-0.367283\pi$
$353$	2.36285e11	0.809935	0.404968	+	0.914331i	$0.367283\pi$
$354$	0	0
$355$	0	0
$356$	1.19437e11	0.394106
$357$	0	0
$358$	−8.56480e9	−0.0275578
$359$	−4.86202e11	−1.54487	−0.772434	−	0.635095i	$-0.780961\pi$
$359$	−4.86202e11	−1.54487	−0.772434	+	0.635095i	$0.780961\pi$
$360$	0	0
$361$	−2.66445e11	−0.825707
$362$	−2.29604e11	−0.702735
$363$	0	0
$364$	−2.95321e11	−0.881736
$365$	0	0
$366$	0	0
$367$	−6.91679e11	−1.99025	−0.995124	−	0.0986269i	$-0.968555\pi$
$367$	−6.91679e11	−1.99025	−0.995124	+	0.0986269i	$0.968555\pi$
$368$	1.20633e11	0.342886
$369$	0	0
$370$	0	0
$371$	−1.34412e11	−0.368346
$372$	0	0
$373$	5.41625e11	1.44880	0.724401	−	0.689379i	$-0.242116\pi$
$373$	5.41625e11	1.44880	0.724401	+	0.689379i	$0.242116\pi$
$374$	1.77047e9	0.00467913
$375$	0	0
$376$	−3.65738e11	−0.943681
$377$	1.21487e12	3.09739
$378$	0	0
$379$	−7.17554e11	−1.78640	−0.893199	−	0.449662i	$-0.851544\pi$
$379$	−7.17554e11	−1.78640	−0.893199	+	0.449662i	$0.851544\pi$
$380$	0	0
$381$	0	0
$382$	3.31976e11	0.797667
$383$	1.81450e11	0.430885	0.215443	−	0.976516i	$-0.430881\pi$
$383$	1.81450e11	0.430885	0.215443	+	0.976516i	$0.430881\pi$
$384$	0	0
$385$	0	0
$386$	4.11649e11	0.943810
$387$	0	0
$388$	4.09372e11	0.917012
$389$	−1.56142e11	−0.345737	−0.172868	−	0.984945i	$-0.555304\pi$
$389$	−1.56142e11	−0.345737	−0.172868	+	0.984945i	$0.555304\pi$
$390$	0	0
$391$	8.63742e9	0.0186891
$392$	−2.29799e11	−0.491543
$393$	0	0
$394$	−4.31260e9	−0.00901583
$395$	0	0
$396$	0	0
$397$	6.12411e9	0.0123733	0.00618665	−	0.999981i	$-0.498031\pi$
$397$	6.12411e9	0.0123733	0.00618665	+	0.999981i	$0.498031\pi$
$398$	−3.42061e11	−0.683329
$399$	0	0
$400$	0	0
$401$	7.63256e11	1.47408	0.737039	−	0.675851i	$-0.236224\pi$
$401$	7.63256e11	1.47408	0.737039	+	0.675851i	$0.236224\pi$
$402$	0	0
$403$	−9.23190e11	−1.74349
$404$	1.84279e11	0.344159
$405$	0	0
$406$	−3.43749e11	−0.627877
$407$	−2.43003e11	−0.438972
$408$	0	0
$409$	−2.15366e11	−0.380559	−0.190279	−	0.981730i	$-0.560939\pi$
$409$	−2.15366e11	−0.380559	−0.190279	+	0.981730i	$0.560939\pi$
$410$	0	0
$411$	0	0
$412$	5.90084e9	0.0100897
$413$	−3.25979e11	−0.551333
$414$	0	0
$415$	0	0
$416$	−1.15252e12	−1.88681
$417$	0	0
$418$	−1.06510e11	−0.170646
$419$	5.13491e11	0.813898	0.406949	−	0.913451i	$-0.366593\pi$
$419$	5.13491e11	0.813898	0.406949	+	0.913451i	$0.366593\pi$
$420$	0	0
$421$	4.25812e11	0.660615	0.330308	−	0.943873i	$-0.392848\pi$
$421$	4.25812e11	0.660615	0.330308	+	0.943873i	$0.392848\pi$
$422$	−2.33820e11	−0.358902
$423$	0	0
$424$	−3.30872e11	−0.497180
$425$	0	0
$426$	0	0
$427$	5.65480e10	0.0823174
$428$	5.62325e11	0.810010
$429$	0	0
$430$	0	0
$431$	−6.11767e11	−0.853962	−0.426981	−	0.904261i	$-0.640423\pi$
$431$	−6.11767e11	−0.853962	−0.426981	+	0.904261i	$0.640423\pi$
$432$	0	0
$433$	9.75502e10	0.133362	0.0666811	−	0.997774i	$-0.478759\pi$
$433$	9.75502e10	0.133362	0.0666811	+	0.997774i	$0.478759\pi$
$434$	2.61217e11	0.353426
$435$	0	0
$436$	−5.30128e11	−0.702573
$437$	−5.19619e11	−0.681583
$438$	0	0
$439$	1.10090e12	1.41467	0.707337	−	0.706877i	$-0.249896\pi$
$439$	1.10090e12	1.41467	0.707337	+	0.706877i	$0.249896\pi$
$440$	0	0
$441$	0	0
$442$	−9.03466e9	−0.0112593
$443$	3.39709e11	0.419073	0.209537	−	0.977801i	$-0.432804\pi$
$443$	3.39709e11	0.419073	0.209537	+	0.977801i	$0.432804\pi$
$444$	0	0
$445$	0	0
$446$	−6.92230e11	−0.828407
$447$	0	0
$448$	2.03703e11	0.238917
$449$	−3.57373e11	−0.414966	−0.207483	−	0.978239i	$-0.566527\pi$
$449$	−3.57373e11	−0.414966	−0.207483	+	0.978239i	$0.566527\pi$
$450$	0	0
$451$	8.15998e11	0.928741
$452$	−1.92993e11	−0.217480
$453$	0	0
$454$	4.05140e11	0.447562
$455$	0	0
$456$	0	0
$457$	1.48293e12	1.59037	0.795184	−	0.606369i	$-0.207374\pi$
$457$	1.48293e12	1.59037	0.795184	+	0.606369i	$0.207374\pi$
$458$	−2.06152e11	−0.218924
$459$	0	0
$460$	0	0
$461$	1.01555e12	1.04724	0.523622	−	0.851951i	$-0.324580\pi$
$461$	1.01555e12	1.04724	0.523622	+	0.851951i	$0.324580\pi$
$462$	0	0
$463$	6.56647e11	0.664076	0.332038	−	0.943266i	$-0.392264\pi$
$463$	6.56647e11	0.664076	0.332038	+	0.943266i	$0.392264\pi$
$464$	3.56659e11	0.357209
$465$	0	0
$466$	7.32986e11	0.720044
$467$	−9.82330e11	−0.955721	−0.477861	−	0.878436i	$-0.658588\pi$
$467$	−9.82330e11	−0.955721	−0.477861	+	0.878436i	$0.658588\pi$
$468$	0	0
$469$	5.95059e11	0.567914
$470$	0	0
$471$	0	0
$472$	−8.02437e11	−0.744170
$473$	−4.59577e11	−0.422166
$474$	0	0
$475$	0	0
$476$	−6.20776e9	−0.00554248
$477$	0	0
$478$	−1.30275e11	−0.114140
$479$	1.43057e11	0.124165	0.0620826	−	0.998071i	$-0.480226\pi$
$479$	1.43057e11	0.124165	0.0620826	+	0.998071i	$0.480226\pi$
$480$	0	0
$481$	1.24004e12	1.05629
$482$	1.06318e11	0.0897211
$483$	0	0
$484$	3.65317e11	0.302598
$485$	0	0
$486$	0	0
$487$	−1.86986e12	−1.50636	−0.753181	−	0.657813i	$-0.771482\pi$
$487$	−1.86986e12	−1.50636	−0.753181	+	0.657813i	$0.771482\pi$
$488$	1.39200e11	0.111109
$489$	0	0
$490$	0	0
$491$	−2.39101e11	−0.185658	−0.0928292	−	0.995682i	$-0.529591\pi$
$491$	−2.39101e11	−0.185658	−0.0928292	+	0.995682i	$0.529591\pi$
$492$	0	0
$493$	2.55371e10	0.0194698
$494$	5.43517e11	0.410622
$495$	0	0
$496$	−2.71027e11	−0.201069
$497$	9.21191e11	0.677246
$498$	0	0
$499$	2.13098e12	1.53861	0.769304	−	0.638883i	$-0.220603\pi$
$499$	2.13098e12	1.53861	0.769304	+	0.638883i	$0.220603\pi$
$500$	0	0
$501$	0	0
$502$	−7.94667e11	−0.558494
$503$	9.25247e11	0.644469	0.322234	−	0.946660i	$-0.395566\pi$
$503$	9.25247e11	0.644469	0.322234	+	0.946660i	$0.395566\pi$
$504$	0	0
$505$	0	0
$506$	9.84037e11	0.667320
$507$	0	0
$508$	−9.28358e11	−0.618485
$509$	−1.48004e12	−0.977336	−0.488668	−	0.872470i	$-0.662517\pi$
$509$	−1.48004e12	−0.977336	−0.488668	+	0.872470i	$0.662517\pi$
$510$	0	0
$511$	1.15257e12	0.747781
$512$	−6.39661e11	−0.411372
$513$	0	0
$514$	8.16351e10	0.0515873
$515$	0	0
$516$	0	0
$517$	1.25749e12	0.774103
$518$	−3.50870e11	−0.214123
$519$	0	0
$520$	0	0
$521$	8.54955e11	0.508363	0.254181	−	0.967157i	$-0.418194\pi$
$521$	8.54955e11	0.508363	0.254181	+	0.967157i	$0.418194\pi$
$522$	0	0
$523$	−1.34309e12	−0.784960	−0.392480	−	0.919761i	$-0.628383\pi$
$523$	−1.34309e12	−0.784960	−0.392480	+	0.919761i	$0.628383\pi$
$524$	−2.03258e12	−1.17776
$525$	0	0
$526$	−1.17917e12	−0.671645
$527$	−1.94058e10	−0.0109593
$528$	0	0
$529$	2.99958e12	1.66537
$530$	0	0
$531$	0	0
$532$	3.73453e11	0.202131
$533$	−4.16403e12	−2.23481
$534$	0	0
$535$	0	0
$536$	1.46481e12	0.766549
$537$	0	0
$538$	8.67806e11	0.446583
$539$	7.90104e11	0.403213
$540$	0	0
$541$	8.64727e11	0.434002	0.217001	−	0.976171i	$-0.430373\pi$
$541$	8.64727e11	0.434002	0.217001	+	0.976171i	$0.430373\pi$
$542$	5.00298e11	0.249018
$543$	0	0
$544$	−2.42263e10	−0.0118602
$545$	0	0
$546$	0	0
$547$	1.90338e12	0.909041	0.454521	−	0.890736i	$-0.349811\pi$
$547$	1.90338e12	0.909041	0.454521	+	0.890736i	$0.349811\pi$
$548$	−1.91135e12	−0.905371
$549$	0	0
$550$	0	0
$551$	−1.53629e12	−0.710053
$552$	0	0
$553$	1.17132e12	0.532615
$554$	−9.09939e11	−0.410411
$555$	0	0
$556$	−2.86593e12	−1.27183
$557$	−1.55326e12	−0.683747	−0.341874	−	0.939746i	$-0.611061\pi$
$557$	−1.55326e12	−0.683747	−0.341874	+	0.939746i	$0.611061\pi$
$558$	0	0
$559$	2.34521e12	1.01585
$560$	0	0
$561$	0	0
$562$	1.69448e12	0.716512
$563$	3.94106e12	1.65320	0.826601	−	0.562789i	$-0.190272\pi$
$563$	3.94106e12	1.65320	0.826601	+	0.562789i	$0.190272\pi$
$564$	0	0
$565$	0	0
$566$	−6.98964e11	−0.286273
$567$	0	0
$568$	2.26763e12	0.914121
$569$	1.03063e11	0.0412189	0.0206094	−	0.999788i	$-0.493439\pi$
$569$	1.03063e11	0.0412189	0.0206094	+	0.999788i	$0.493439\pi$
$570$	0	0
$571$	2.92191e12	1.15028	0.575141	−	0.818054i	$-0.304947\pi$
$571$	2.92191e12	1.15028	0.575141	+	0.818054i	$0.304947\pi$
$572$	2.49949e12	0.976269
$573$	0	0
$574$	1.17821e12	0.453023
$575$	0	0
$576$	0	0
$577$	−2.67123e12	−1.00327	−0.501637	−	0.865078i	$-0.667269\pi$
$577$	−2.67123e12	−1.00327	−0.501637	+	0.865078i	$0.667269\pi$
$578$	1.44902e12	0.540008
$579$	0	0
$580$	0	0
$581$	−2.27007e12	−0.826506
$582$	0	0
$583$	1.13761e12	0.407837
$584$	2.83720e12	1.00933
$585$	0	0
$586$	−1.34101e12	−0.469777
$587$	−2.14358e12	−0.745191	−0.372596	−	0.927994i	$-0.621532\pi$
$587$	−2.14358e12	−0.745191	−0.372596	+	0.927994i	$0.621532\pi$
$588$	0	0
$589$	1.16744e12	0.399681
$590$	0	0
$591$	0	0
$592$	3.64047e11	0.121818
$593$	2.79629e12	0.928616	0.464308	−	0.885674i	$-0.346303\pi$
$593$	2.79629e12	0.928616	0.464308	+	0.885674i	$0.346303\pi$
$594$	0	0
$595$	0	0
$596$	3.22149e12	1.04580
$597$	0	0
$598$	−5.02152e12	−1.60576
$599$	5.67415e12	1.80086	0.900430	−	0.435002i	$-0.143252\pi$
$599$	5.67415e12	1.80086	0.900430	+	0.435002i	$0.143252\pi$
$600$	0	0
$601$	3.70560e12	1.15857	0.579287	−	0.815124i	$-0.303331\pi$
$601$	3.70560e12	1.15857	0.579287	+	0.815124i	$0.303331\pi$
$602$	−6.63579e11	−0.205925
$603$	0	0
$604$	−1.57937e12	−0.482855
$605$	0	0
$606$	0	0
$607$	−3.05665e12	−0.913894	−0.456947	−	0.889494i	$-0.651057\pi$
$607$	−3.05665e12	−0.913894	−0.456947	+	0.889494i	$0.651057\pi$
$608$	1.45743e12	0.432537
$609$	0	0
$610$	0	0
$611$	−6.41698e12	−1.86271
$612$	0	0
$613$	−1.97493e12	−0.564911	−0.282456	−	0.959280i	$-0.591149\pi$
$613$	−1.97493e12	−0.564911	−0.282456	+	0.959280i	$0.591149\pi$
$614$	−2.31419e12	−0.657116
$615$	0	0
$616$	−1.70570e12	−0.477299
$617$	3.84845e12	1.06906	0.534531	−	0.845149i	$-0.320488\pi$
$617$	3.84845e12	1.06906	0.534531	+	0.845149i	$0.320488\pi$
$618$	0	0
$619$	−4.09595e11	−0.112136	−0.0560682	−	0.998427i	$-0.517856\pi$
$619$	−4.09595e11	−0.112136	−0.0560682	+	0.998427i	$0.517856\pi$
$620$	0	0
$621$	0	0
$622$	1.71800e12	0.460222
$623$	−1.43005e12	−0.380324
$624$	0	0
$625$	0	0
$626$	−1.75676e12	−0.457222
$627$	0	0
$628$	−7.63361e11	−0.195845
$629$	2.60661e10	0.00663970
$630$	0	0
$631$	2.20647e12	0.554072	0.277036	−	0.960860i	$-0.410648\pi$
$631$	2.20647e12	0.554072	0.277036	+	0.960860i	$0.410648\pi$
$632$	2.88335e12	0.718904
$633$	0	0
$634$	2.04135e12	0.501782
$635$	0	0
$636$	0	0
$637$	−4.03189e12	−0.970244
$638$	2.90937e12	0.695194
$639$	0	0
$640$	0	0
$641$	7.59163e12	1.77613	0.888064	−	0.459720i	$-0.152050\pi$
$641$	7.59163e12	1.77613	0.888064	+	0.459720i	$0.152050\pi$
$642$	0	0
$643$	6.12938e12	1.41406	0.707029	−	0.707184i	$-0.250035\pi$
$643$	6.12938e12	1.41406	0.707029	+	0.707184i	$0.250035\pi$
$644$	−3.45032e12	−0.790446
$645$	0	0
$646$	1.14249e10	0.00258111
$647$	−1.67991e12	−0.376893	−0.188446	−	0.982084i	$-0.560345\pi$
$647$	−1.67991e12	−0.376893	−0.188446	+	0.982084i	$0.560345\pi$
$648$	0	0
$649$	2.75897e12	0.610443
$650$	0	0
$651$	0	0
$652$	−2.81786e12	−0.610668
$653$	2.09466e12	0.450821	0.225410	−	0.974264i	$-0.427628\pi$
$653$	2.09466e12	0.450821	0.225410	+	0.974264i	$0.427628\pi$
$654$	0	0
$655$	0	0
$656$	−1.22246e12	−0.257732
$657$	0	0
$658$	1.81569e12	0.377594
$659$	−7.17038e11	−0.148101	−0.0740505	−	0.997254i	$-0.523593\pi$
$659$	−7.17038e11	−0.148101	−0.0740505	+	0.997254i	$0.523593\pi$
$660$	0	0
$661$	4.83658e12	0.985444	0.492722	−	0.870187i	$-0.336002\pi$
$661$	4.83658e12	0.985444	0.492722	+	0.870187i	$0.336002\pi$
$662$	2.42485e11	0.0490708
$663$	0	0
$664$	−5.58804e12	−1.11559
$665$	0	0
$666$	0	0
$667$	1.41937e13	2.77670
$668$	−6.58357e12	−1.27928
$669$	0	0
$670$	0	0
$671$	−4.78602e11	−0.0911429
$672$	0	0
$673$	−4.63436e12	−0.870808	−0.435404	−	0.900235i	$-0.643394\pi$
$673$	−4.63436e12	−0.870808	−0.435404	+	0.900235i	$0.643394\pi$
$674$	−8.20985e11	−0.153238
$675$	0	0
$676$	−8.90907e12	−1.64086
$677$	6.94095e12	1.26990	0.634950	−	0.772553i	$-0.281021\pi$
$677$	6.94095e12	1.26990	0.634950	+	0.772553i	$0.281021\pi$
$678$	0	0
$679$	−4.90151e12	−0.884944
$680$	0	0
$681$	0	0
$682$	−2.21085e12	−0.391318
$683$	−6.91635e12	−1.21614	−0.608070	−	0.793883i	$-0.708056\pi$
$683$	−6.91635e12	−1.21614	−0.608070	+	0.793883i	$0.708056\pi$
$684$	0	0
$685$	0	0
$686$	3.28215e12	0.565848
$687$	0	0
$688$	6.88500e11	0.117154
$689$	−5.80523e12	−0.981371
$690$	0	0
$691$	1.45798e12	0.243277	0.121638	−	0.992574i	$-0.461185\pi$
$691$	1.45798e12	0.243277	0.121638	+	0.992574i	$0.461185\pi$
$692$	−3.19351e12	−0.529408
$693$	0	0
$694$	−3.02987e12	−0.495800
$695$	0	0
$696$	0	0
$697$	−8.75294e10	−0.0140477
$698$	1.98480e12	0.316496
$699$	0	0
$700$	0	0
$701$	5.87637e12	0.919132	0.459566	−	0.888144i	$-0.348005\pi$
$701$	5.87637e12	0.919132	0.459566	+	0.888144i	$0.348005\pi$
$702$	0	0
$703$	−1.56811e12	−0.242147
$704$	−1.72407e12	−0.264531
$705$	0	0
$706$	−2.88754e12	−0.437429
$707$	−2.20642e12	−0.332124
$708$	0	0
$709$	−7.62910e12	−1.13388	−0.566938	−	0.823761i	$-0.691872\pi$
$709$	−7.62910e12	−1.13388	−0.566938	+	0.823761i	$0.691872\pi$
$710$	0	0
$711$	0	0
$712$	−3.52024e12	−0.513348
$713$	−1.07859e13	−1.56298
$714$	0	0
$715$	0	0
$716$	−2.54169e11	−0.0361421
$717$	0	0
$718$	5.94167e12	0.834350
$719$	2.38732e12	0.333142	0.166571	−	0.986029i	$-0.446730\pi$
$719$	2.38732e12	0.333142	0.166571	+	0.986029i	$0.446730\pi$
$720$	0	0
$721$	−7.06523e10	−0.00973682
$722$	3.25612e12	0.445947
$723$	0	0
$724$	−6.81373e12	−0.921640
$725$	0	0
$726$	0	0
$727$	−5.38271e12	−0.714654	−0.357327	−	0.933979i	$-0.616312\pi$
$727$	−5.38271e12	−0.714654	−0.357327	+	0.933979i	$0.616312\pi$
$728$	8.70419e12	1.14852
$729$	0	0
$730$	0	0
$731$	4.92972e10	0.00638549
$732$	0	0
$733$	−7.93743e12	−1.01557	−0.507787	−	0.861483i	$-0.669536\pi$
$733$	−7.93743e12	−1.01557	−0.507787	+	0.861483i	$0.669536\pi$
$734$	8.45272e12	1.07489
$735$	0	0
$736$	−1.34652e13	−1.69146
$737$	−5.03637e12	−0.628801
$738$	0	0
$739$	−2.88800e12	−0.356203	−0.178101	−	0.984012i	$-0.556996\pi$
$739$	−2.88800e12	−0.356203	−0.178101	+	0.984012i	$0.556996\pi$
$740$	0	0
$741$	0	0
$742$	1.64259e12	0.198936
$743$	9.68056e12	1.16533	0.582667	−	0.812711i	$-0.302009\pi$
$743$	9.68056e12	1.16533	0.582667	+	0.812711i	$0.302009\pi$
$744$	0	0
$745$	0	0
$746$	−6.61898e12	−0.782467
$747$	0	0
$748$	5.25402e10	0.00613670
$749$	−6.73286e12	−0.781684
$750$	0	0
$751$	1.55121e12	0.177947	0.0889735	−	0.996034i	$-0.471641\pi$
$751$	1.55121e12	0.177947	0.0889735	+	0.996034i	$0.471641\pi$
$752$	−1.88388e12	−0.214819
$753$	0	0
$754$	−1.48465e13	−1.67283
$755$	0	0
$756$	0	0
$757$	−2.05679e12	−0.227646	−0.113823	−	0.993501i	$-0.536310\pi$
$757$	−2.05679e12	−0.227646	−0.113823	+	0.993501i	$0.536310\pi$
$758$	8.76893e12	0.964795
$759$	0	0
$760$	0	0
$761$	−4.01826e12	−0.434317	−0.217159	−	0.976136i	$-0.569679\pi$
$761$	−4.01826e12	−0.434317	−0.217159	+	0.976136i	$0.569679\pi$
$762$	0	0
$763$	6.34735e12	0.678004
$764$	9.85171e12	1.04614
$765$	0	0
$766$	−2.21742e12	−0.232712
$767$	−1.40790e13	−1.46890
$768$	0	0
$769$	−7.34878e12	−0.757786	−0.378893	−	0.925441i	$-0.623695\pi$
$769$	−7.34878e12	−0.757786	−0.378893	+	0.925441i	$0.623695\pi$
$770$	0	0
$771$	0	0
$772$	1.22161e13	1.23781
$773$	−7.03720e12	−0.708912	−0.354456	−	0.935073i	$-0.615334\pi$
$773$	−7.03720e12	−0.708912	−0.354456	+	0.935073i	$0.615334\pi$
$774$	0	0
$775$	0	0
$776$	−1.20657e13	−1.19447
$777$	0	0
$778$	1.90814e12	0.186725
$779$	5.26569e12	0.512314
$780$	0	0
$781$	−7.79663e12	−0.749855
$782$	−1.05554e11	−0.0100936
$783$	0	0
$784$	−1.18367e12	−0.111894
$785$	0	0
$786$	0	0
$787$	−3.89415e12	−0.361848	−0.180924	−	0.983497i	$-0.557909\pi$
$787$	−3.89415e12	−0.361848	−0.180924	+	0.983497i	$0.557909\pi$
$788$	−1.27980e11	−0.0118243
$789$	0	0
$790$	0	0
$791$	2.31076e12	0.209875
$792$	0	0
$793$	2.44230e12	0.219315
$794$	−7.48402e10	−0.00668256
$795$	0	0
$796$	−1.01510e13	−0.896188
$797$	−6.27475e12	−0.550851	−0.275425	−	0.961322i	$-0.588819\pi$
$797$	−6.27475e12	−0.550851	−0.275425	+	0.961322i	$0.588819\pi$
$798$	0	0
$799$	−1.34887e11	−0.0117087
$800$	0	0
$801$	0	0
$802$	−9.32743e12	−0.796118
$803$	−9.75496e12	−0.827953
$804$	0	0
$805$	0	0
$806$	1.12819e13	0.941620
$807$	0	0
$808$	−5.43136e12	−0.448289
$809$	−8.39256e12	−0.688852	−0.344426	−	0.938813i	$-0.611926\pi$
$809$	−8.39256e12	−0.688852	−0.344426	+	0.938813i	$0.611926\pi$
$810$	0	0
$811$	4.06885e12	0.330277	0.165138	−	0.986270i	$-0.447193\pi$
$811$	4.06885e12	0.330277	0.165138	+	0.986270i	$0.447193\pi$
$812$	−1.02011e13	−0.823464
$813$	0	0
$814$	2.96964e12	0.237079
$815$	0	0
$816$	0	0
$817$	−2.96568e12	−0.232876
$818$	2.63189e12	0.205532
$819$	0	0
$820$	0	0
$821$	−1.00927e13	−0.775287	−0.387643	−	0.921809i	$-0.626711\pi$
$821$	−1.00927e13	−0.775287	−0.387643	+	0.921809i	$0.626711\pi$
$822$	0	0
$823$	−1.21960e13	−0.926654	−0.463327	−	0.886187i	$-0.653344\pi$
$823$	−1.21960e13	−0.926654	−0.463327	+	0.886187i	$0.653344\pi$
$824$	−1.73919e11	−0.0131424
$825$	0	0
$826$	3.98365e12	0.297763
$827$	1.65725e13	1.23201	0.616003	−	0.787743i	$-0.288751\pi$
$827$	1.65725e13	1.23201	0.616003	+	0.787743i	$0.288751\pi$
$828$	0	0
$829$	−7.37572e12	−0.542387	−0.271193	−	0.962525i	$-0.587418\pi$
$829$	−7.37572e12	−0.542387	−0.271193	+	0.962525i	$0.587418\pi$
$830$	0	0
$831$	0	0
$832$	8.79789e12	0.636537
$833$	−8.47518e10	−0.00609883
$834$	0	0
$835$	0	0
$836$	−3.16077e12	−0.223803
$837$	0	0
$838$	−6.27516e12	−0.439569
$839$	7.96014e12	0.554615	0.277308	−	0.960781i	$-0.410558\pi$
$839$	7.96014e12	0.554615	0.277308	+	0.960781i	$0.410558\pi$
$840$	0	0
$841$	2.74575e13	1.89269
$842$	−5.20367e12	−0.356784
$843$	0	0
$844$	−6.93883e12	−0.470701
$845$	0	0
$846$	0	0
$847$	−4.37404e12	−0.292016
$848$	−1.70428e12	−0.113178
$849$	0	0
$850$	0	0
$851$	1.44877e13	0.946928
$852$	0	0
$853$	7.49408e12	0.484672	0.242336	−	0.970192i	$-0.422086\pi$
$853$	7.49408e12	0.484672	0.242336	+	0.970192i	$0.422086\pi$
$854$	−6.91049e11	−0.0444579
$855$	0	0
$856$	−1.65738e13	−1.05509
$857$	−2.06124e12	−0.130531	−0.0652656	−	0.997868i	$-0.520789\pi$
$857$	−2.06124e12	−0.130531	−0.0652656	+	0.997868i	$0.520789\pi$
$858$	0	0
$859$	1.61145e13	1.00983	0.504914	−	0.863169i	$-0.331524\pi$
$859$	1.61145e13	1.00983	0.504914	+	0.863169i	$0.331524\pi$
$860$	0	0
$861$	0	0
$862$	7.47615e12	0.461207
$863$	−1.14800e13	−0.704521	−0.352261	−	0.935902i	$-0.614587\pi$
$863$	−1.14800e13	−0.704521	−0.352261	+	0.935902i	$0.614587\pi$
$864$	0	0
$865$	0	0
$866$	−1.19212e12	−0.0720260
$867$	0	0
$868$	7.75187e12	0.463519
$869$	−9.91364e12	−0.589718
$870$	0	0
$871$	2.57005e13	1.51307
$872$	1.56248e13	0.915145
$873$	0	0
$874$	6.35005e12	0.368108
$875$	0	0
$876$	0	0
$877$	1.14652e13	0.654460	0.327230	−	0.944945i	$-0.393885\pi$
$877$	1.14652e13	0.654460	0.327230	+	0.944945i	$0.393885\pi$
$878$	−1.34536e13	−0.764035
$879$	0	0
$880$	0	0
$881$	2.36703e13	1.32377	0.661885	−	0.749605i	$-0.269757\pi$
$881$	2.36703e13	1.32377	0.661885	+	0.749605i	$0.269757\pi$
$882$	0	0
$883$	2.01686e13	1.11648	0.558242	−	0.829678i	$-0.311476\pi$
$883$	2.01686e13	1.11648	0.558242	+	0.829678i	$0.311476\pi$
$884$	−2.68112e11	−0.0147666
$885$	0	0
$886$	−4.15144e12	−0.226332
$887$	2.58905e13	1.40438	0.702188	−	0.711992i	$-0.252207\pi$
$887$	2.58905e13	1.40438	0.702188	+	0.711992i	$0.252207\pi$
$888$	0	0
$889$	1.11155e13	0.596856
$890$	0	0
$891$	0	0
$892$	−2.05426e13	−1.08646
$893$	8.11469e12	0.427012
$894$	0	0
$895$	0	0
$896$	1.11733e13	0.579157
$897$	0	0
$898$	4.36730e12	0.224115
$899$	−3.18892e13	−1.62826
$900$	0	0
$901$	−1.22028e11	−0.00616877
$902$	−9.97197e12	−0.501593
$903$	0	0
$904$	5.68821e12	0.283281
$905$	0	0
$906$	0	0
$907$	2.09156e13	1.02621	0.513107	−	0.858325i	$-0.328494\pi$
$907$	2.09156e13	1.02621	0.513107	+	0.858325i	$0.328494\pi$
$908$	1.20229e13	0.586979
$909$	0	0
$910$	0	0
$911$	−4.47158e12	−0.215094	−0.107547	−	0.994200i	$-0.534300\pi$
$911$	−4.47158e12	−0.215094	−0.107547	+	0.994200i	$0.534300\pi$
$912$	0	0
$913$	1.92130e13	0.915118
$914$	−1.81223e13	−0.858923
$915$	0	0
$916$	−6.11776e12	−0.287120
$917$	2.43366e13	1.13657
$918$	0	0
$919$	1.06549e13	0.492753	0.246377	−	0.969174i	$-0.420760\pi$
$919$	1.06549e13	0.492753	0.246377	+	0.969174i	$0.420760\pi$
$920$	0	0
$921$	0	0
$922$	−1.24106e13	−0.565594
$923$	3.97861e13	1.80436
$924$	0	0
$925$	0	0
$926$	−8.02461e12	−0.358653
$927$	0	0
$928$	−3.98106e13	−1.76211
$929$	3.87854e13	1.70843	0.854217	−	0.519917i	$-0.174037\pi$
$929$	3.87854e13	1.70843	0.854217	+	0.519917i	$0.174037\pi$
$930$	0	0
$931$	5.09859e12	0.222421
$932$	2.17520e13	0.944340
$933$	0	0
$934$	1.20046e13	0.516165
$935$	0	0
$936$	0	0
$937$	6.15535e11	0.0260870	0.0130435	−	0.999915i	$-0.495848\pi$
$937$	6.15535e11	0.0260870	0.0130435	+	0.999915i	$0.495848\pi$
$938$	−7.27197e12	−0.306718
$939$	0	0
$940$	0	0
$941$	−3.81077e13	−1.58438	−0.792190	−	0.610275i	$-0.791059\pi$
$941$	−3.81077e13	−1.58438	−0.792190	+	0.610275i	$0.791059\pi$
$942$	0	0
$943$	−4.86494e13	−2.00343
$944$	−4.13326e12	−0.169402
$945$	0	0
$946$	5.61629e12	0.228003
$947$	−2.37836e13	−0.960955	−0.480478	−	0.877007i	$-0.659537\pi$
$947$	−2.37836e13	−0.960955	−0.480478	+	0.877007i	$0.659537\pi$
$948$	0	0
$949$	4.97794e13	1.99229
$950$	0	0
$951$	0	0
$952$	1.82965e11	0.00721942
$953$	−4.19004e12	−0.164551	−0.0822755	−	0.996610i	$-0.526219\pi$
$953$	−4.19004e12	−0.164551	−0.0822755	+	0.996610i	$0.526219\pi$
$954$	0	0
$955$	0	0
$956$	−3.86604e12	−0.149694
$957$	0	0
$958$	−1.74824e12	−0.0670589
$959$	2.28850e13	0.873710
$960$	0	0
$961$	−2.20685e12	−0.0834675
$962$	−1.51540e13	−0.570480
$963$	0	0
$964$	3.15508e12	0.117670
$965$	0	0
$966$	0	0
$967$	−2.08426e13	−0.766537	−0.383268	−	0.923637i	$-0.625202\pi$
$967$	−2.08426e13	−0.766537	−0.383268	+	0.923637i	$0.625202\pi$
$968$	−1.07672e13	−0.394153
$969$	0	0
$970$	0	0
$971$	1.13186e13	0.408606	0.204303	−	0.978908i	$-0.434507\pi$
$971$	1.13186e13	0.408606	0.204303	+	0.978908i	$0.434507\pi$
$972$	0	0
$973$	3.43144e13	1.22735
$974$	2.28508e13	0.813554
$975$	0	0
$976$	7.17002e11	0.0252928
$977$	−3.51597e13	−1.23458	−0.617291	−	0.786735i	$-0.711770\pi$
$977$	−3.51597e13	−1.23458	−0.617291	+	0.786735i	$0.711770\pi$
$978$	0	0
$979$	1.21034e13	0.421100
$980$	0	0
$981$	0	0
$982$	2.92195e12	0.100270
$983$	2.63427e12	0.0899850	0.0449925	−	0.998987i	$-0.485674\pi$
$983$	2.63427e12	0.0899850	0.0449925	+	0.998987i	$0.485674\pi$
$984$	0	0
$985$	0	0
$986$	−3.12078e11	−0.0105152
$987$	0	0
$988$	1.61294e13	0.538532
$989$	2.73997e13	0.910674
$990$	0	0
$991$	−9.08136e12	−0.299102	−0.149551	−	0.988754i	$-0.547783\pi$
$991$	−9.08136e12	−0.299102	−0.149551	+	0.988754i	$0.547783\pi$
$992$	3.02524e13	0.991874
$993$	0	0
$994$	−1.12575e13	−0.365766
$995$	0	0
$996$	0	0
$997$	−3.23962e13	−1.03840	−0.519201	−	0.854652i	$-0.673771\pi$
$997$	−3.23962e13	−1.03840	−0.519201	+	0.854652i	$0.673771\pi$
$998$	−2.60419e13	−0.830969
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.x.1.4		8
3.2	odd	2	inner	225.10.a.x.1.6		8
5.2	odd	4		45.10.b.d.19.3	✓	8
5.3	odd	4		45.10.b.d.19.5	yes	8
5.4	even	2	inner	225.10.a.x.1.5		8
15.2	even	4		45.10.b.d.19.6	yes	8
15.8	even	4		45.10.b.d.19.4	yes	8
15.14	odd	2	inner	225.10.a.x.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.10.b.d.19.3	✓	8	5.2	odd	4
45.10.b.d.19.4	yes	8	15.8	even	4
45.10.b.d.19.5	yes	8	5.3	odd	4
45.10.b.d.19.6	yes	8	15.2	even	4
225.10.a.x.1.3		8	15.14	odd	2	inner
225.10.a.x.1.4		8	1.1	even	1	trivial
225.10.a.x.1.5		8	5.4	even	2	inner
225.10.a.x.1.6		8	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-12.2206 q^{2} -362.657 q^{4} +4342.19 q^{7} +10688.8 q^{8} -36750.7 q^{11} +187538. q^{13} -53064.1 q^{14} +55056.9 q^{16} +3942.12 q^{17} -237154. q^{19} +449115. q^{22} +2.19106e6 q^{23} -2.29183e6 q^{26} -1.57473e6 q^{28} +6.47801e6 q^{29} -4.92268e6 q^{31} -6.14550e6 q^{32} -48175.0 q^{34} +6.61220e6 q^{37} +2.89817e6 q^{38} -2.22036e7 q^{41} +1.25053e7 q^{43} +1.33279e7 q^{44} -2.67760e7 q^{46} -3.42169e7 q^{47} -2.14990e7 q^{49} -6.80121e7 q^{52} -3.09549e7 q^{53} +4.64129e7 q^{56} -7.91651e7 q^{58} -7.50725e7 q^{59} +1.30229e7 q^{61} +6.01580e7 q^{62} +4.69125e7 q^{64} +1.37041e8 q^{67} -1.42964e6 q^{68} +2.12149e8 q^{71} +2.65436e8 q^{73} -8.08050e7 q^{74} +8.60058e7 q^{76} -1.59578e8 q^{77} +2.69754e8 q^{79} +2.71341e8 q^{82} -5.22793e8 q^{83} -1.52822e8 q^{86} -3.92822e8 q^{88} -3.29338e8 q^{89} +8.14326e8 q^{91} -7.94603e8 q^{92} +4.18150e8 q^{94} -1.12881e9 q^{97} +2.62731e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 144 q^{4} - 1009088 q^{16} + 2079872 q^{19} + 12077344 q^{31} - 25286480 q^{34} + 3119840 q^{46} + 174619144 q^{49} - 165845744 q^{61} - 91038464 q^{64} + 1551352896 q^{76} + 1098932768 q^{79} - 1465128000 q^{91}+ \cdots + 4117219360 q^{94}+O(q^{100}) \) 8 * q + 144 * q^4 - 1009088 * q^16 + 2079872 * q^19 + 12077344 * q^31 - 25286480 * q^34 + 3119840 * q^46 + 174619144 * q^49 - 165845744 * q^61 - 91038464 * q^64 + 1551352896 * q^76 + 1098932768 * q^79 - 1465128000 * q^91 + 4117219360 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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