LMFDB - Embedded newform 225.10.b.m.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,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, 10, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
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:	$115.883063137$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 1305x^{4} + 433104x^{2} + 16000000 $ x^6 + 1305x^4 + 433104x^2 + 16000000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$6.48955i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.10.b.m.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.2014i q^{2} +103.903 q^{4} -4010.25i q^{7} +12442.1i q^{8} +O(q^{10})$	$q+20.2014i q^{2} +103.903 q^{4} -4010.25i q^{7} +12442.1i q^{8} +42110.0 q^{11} -123743. i q^{13} +81012.8 q^{14} -198150. q^{16} -319945. i q^{17} -1.08733e6 q^{19} +850682. i q^{22} +1.50672e6i q^{23} +2.49979e6 q^{26} -416676. i q^{28} -2.62160e6 q^{29} +3.27023e6 q^{31} +2.36745e6i q^{32} +6.46335e6 q^{34} -2.51034e6i q^{37} -2.19655e7i q^{38} -2.95349e7 q^{41} +1.42413e7i q^{43} +4.37534e6 q^{44} -3.04378e7 q^{46} -1.35318e6i q^{47} +2.42715e7 q^{49} -1.28573e7i q^{52} +9.73342e7i q^{53} +4.98960e7 q^{56} -5.29599e7i q^{58} -7.48924e6 q^{59} -9.11752e7 q^{61} +6.60633e7i q^{62} -1.49279e8 q^{64} +2.94376e8i q^{67} -3.32432e7i q^{68} -1.56193e8 q^{71} +2.82539e8i q^{73} +5.07124e7 q^{74} -1.12976e8 q^{76} -1.68872e8i q^{77} +5.55294e8 q^{79} -5.96647e8i q^{82} +6.48378e6i q^{83} -2.87694e8 q^{86} +5.23937e8i q^{88} -5.99001e8 q^{89} -4.96242e8 q^{91} +1.56552e8i q^{92} +2.73361e7 q^{94} +9.25317e8i q^{97} +4.90319e8i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 682 q^{4}+O(q^{10}) $ 6 * q - 682 * q^4	$ 6 q - 682 q^{4} + 109398 q^{11} + 545844 q^{14} - 1398494 q^{16} - 1637690 q^{19} + 2560008 q^{26} + 4350960 q^{29} + 8548132 q^{31} + 13677926 q^{34} - 11852622 q^{41} - 43991406 q^{44} + 28446492 q^{46} + 12907858 q^{49} + 339076260 q^{56} + 11341920 q^{59} + 250613852 q^{61} - 335084802 q^{64} - 595101192 q^{71} - 503014716 q^{74} + 178829730 q^{76} + 620050340 q^{79} - 67894392 q^{86} - 2207720070 q^{89} + 2366375312 q^{91} - 3455782264 q^{94}+O(q^{100}) $ 6 * q - 682 * q^4 + 109398 * q^11 + 545844 * q^14 - 1398494 * q^16 - 1637690 * q^19 + 2560008 * q^26 + 4350960 * q^29 + 8548132 * q^31 + 13677926 * q^34 - 11852622 * q^41 - 43991406 * q^44 + 28446492 * q^46 + 12907858 * q^49 + 339076260 * q^56 + 11341920 * q^59 + 250613852 * q^61 - 335084802 * q^64 - 595101192 * q^71 - 503014716 * q^74 + 178829730 * q^76 + 620050340 * q^79 - 67894392 * q^86 - 2207720070 * q^89 + 2366375312 * q^91 - 3455782264 * 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 $

$2$	20.2014i	0.892785i	0.894837	+	0.446393i	$0.147291\pi$
$2$	20.2014i	0.892785i	−0.894837	+	0.446393i	$0.852709\pi$
$3$	0	0
$3$	0	0
$4$	103.903	0.202935
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4010.25i	− 0.631292i	−0.948877	−	0.315646i	$-0.897779\pi$
$7$	− 4010.25i	− 0.631292i	0.948877	−	0.315646i	$-0.102221\pi$
$8$	12442.1i	1.07396i
$9$	0	0
$10$	0	0
$11$	42110.0	0.867198	0.433599	−	0.901106i	$-0.357243\pi$
$11$	42110.0	0.867198	0.433599	+	0.901106i	$0.357243\pi$
$12$	0	0
$13$	− 123743.i	− 1.20165i	−0.799382	−	0.600824i	$-0.794839\pi$
$13$	− 123743.i	− 1.20165i	0.799382	−	0.600824i	$-0.205161\pi$
$14$	81012.8	0.563608
$15$	0	0
$16$	−198150.	−0.755883
$17$	− 319945.i	− 0.929085i	−0.885551	−	0.464543i	$-0.846219\pi$
$17$	− 319945.i	− 0.929085i	0.885551	−	0.464543i	$-0.153781\pi$
$18$	0	0
$19$	−1.08733e6	−1.91412	−0.957059	−	0.289893i	$-0.906380\pi$
$19$	−1.08733e6	−1.91412	−0.957059	+	0.289893i	$0.906380\pi$
$20$	0	0
$21$	0	0
$22$	850682.i	0.774221i
$23$	1.50672e6i	1.12268i	0.827585	+	0.561341i	$0.189714\pi$
$23$	1.50672e6i	1.12268i	−0.827585	+	0.561341i	$0.810286\pi$
$24$	0	0
$25$	0	0
$26$	2.49979e6	1.07281
$27$	0	0
$28$	− 416676.i	− 0.128111i
$29$	−2.62160e6	−0.688295	−0.344148	−	0.938916i	$-0.611832\pi$
$29$	−2.62160e6	−0.688295	−0.344148	+	0.938916i	$0.611832\pi$
$30$	0	0
$31$	3.27023e6	0.635991	0.317996	−	0.948092i	$-0.396990\pi$
$31$	3.27023e6	0.635991	0.317996	+	0.948092i	$0.396990\pi$
$32$	2.36745e6i	0.399122i
$33$	0	0
$34$	6.46335e6	0.829473
$35$	0	0
$36$	0	0
$37$	− 2.51034e6i	− 0.220204i	−0.993920	−	0.110102i	$-0.964882\pi$
$37$	− 2.51034e6i	− 0.220204i	0.993920	−	0.110102i	$-0.0351177\pi$
$38$	− 2.19655e7i	− 1.70890i
$39$	0	0
$40$	0	0
$41$	−2.95349e7	−1.63233	−0.816165	−	0.577819i	$-0.803904\pi$
$41$	−2.95349e7	−1.63233	−0.816165	+	0.577819i	$0.803904\pi$
$42$	0	0
$43$	1.42413e7i	0.635244i	0.948217	+	0.317622i	$0.102884\pi$
$43$	1.42413e7i	0.635244i	−0.948217	+	0.317622i	$0.897116\pi$
$44$	4.37534e6	0.175985
$45$	0	0
$46$	−3.04378e7	−1.00231
$47$	− 1.35318e6i	− 0.0404496i	−0.999795	−	0.0202248i	$-0.993562\pi$
$47$	− 1.35318e6i	− 0.0404496i	0.999795	−	0.0202248i	$-0.00643820\pi$
$48$	0	0
$49$	2.42715e7	0.601470
$50$	0	0
$51$	0	0
$52$	− 1.28573e7i	− 0.243856i
$53$	9.73342e7i	1.69443i	0.531249	+	0.847216i	$0.321723\pi$
$53$	9.73342e7i	1.69443i	−0.531249	+	0.847216i	$0.678277\pi$
$54$	0	0
$55$	0	0
$56$	4.98960e7	0.677984
$57$	0	0
$58$	− 5.29599e7i	− 0.614500i
$59$	−7.48924e6	−0.0804644	−0.0402322	−	0.999190i	$-0.512810\pi$
$59$	−7.48924e6	−0.0804644	−0.0402322	+	0.999190i	$0.512810\pi$
$60$	0	0
$61$	−9.11752e7	−0.843126	−0.421563	−	0.906799i	$-0.638518\pi$
$61$	−9.11752e7	−0.843126	−0.421563	+	0.906799i	$0.638518\pi$
$62$	6.60633e7i	0.567803i
$63$	0	0
$64$	−1.49279e8	−1.11221
$65$	0	0
$66$	0	0
$67$	2.94376e8i	1.78470i	0.451344	+	0.892350i	$0.350945\pi$
$67$	2.94376e8i	1.78470i	−0.451344	+	0.892350i	$0.649055\pi$
$68$	− 3.32432e7i	− 0.188544i
$69$	0	0
$70$	0	0
$71$	−1.56193e8	−0.729455	−0.364728	−	0.931114i	$-0.618838\pi$
$71$	−1.56193e8	−0.729455	−0.364728	+	0.931114i	$0.618838\pi$
$72$	0	0
$73$	2.82539e8i	1.16446i	0.813023	+	0.582232i	$0.197820\pi$
$73$	2.82539e8i	1.16446i	−0.813023	+	0.582232i	$0.802180\pi$
$74$	5.07124e7	0.196594
$75$	0	0
$76$	−1.12976e8	−0.388441
$77$	− 1.68872e8i	− 0.547455i
$78$	0	0
$79$	5.55294e8	1.60399	0.801994	−	0.597332i	$-0.203772\pi$
$79$	5.55294e8	1.60399	0.801994	+	0.597332i	$0.203772\pi$
$80$	0	0
$81$	0	0
$82$	− 5.96647e8i	− 1.45732i
$83$	6.48378e6i	0.0149960i	0.999972	+	0.00749802i	$0.00238672\pi$
$83$	6.48378e6i	0.0149960i	−0.999972	+	0.00749802i	$0.997613\pi$
$84$	0	0
$85$	0	0
$86$	−2.87694e8	−0.567137
$87$	0	0
$88$	5.23937e8i	0.931338i
$89$	−5.99001e8	−1.01198	−0.505990	−	0.862539i	$-0.668873\pi$
$89$	−5.99001e8	−1.01198	−0.505990	+	0.862539i	$0.668873\pi$
$90$	0	0
$91$	−4.96242e8	−0.758591
$92$	1.56552e8i	0.227831i
$93$	0	0
$94$	2.73361e7	0.0361128
$95$	0	0
$96$	0	0
$97$	9.25317e8i	1.06125i	0.847606	+	0.530625i	$0.178043\pi$
$97$	9.25317e8i	1.06125i	−0.847606	+	0.530625i	$0.821957\pi$
$98$	4.90319e8i	0.536984i
$99$	0	0
$100$	0	0
$101$	−9.58959e8	−0.916967	−0.458483	−	0.888703i	$-0.651607\pi$
$101$	−9.58959e8	−0.916967	−0.458483	+	0.888703i	$0.651607\pi$
$102$	0	0
$103$	− 1.60441e8i	− 0.140458i	−0.997531	−	0.0702292i	$-0.977627\pi$
$103$	− 1.60441e8i	− 0.140458i	0.997531	−	0.0702292i	$-0.0223731\pi$
$104$	1.53963e9	1.29052
$105$	0	0
$106$	−1.96629e9	−1.51276
$107$	− 9.60457e8i	− 0.708355i	−0.935178	−	0.354178i	$-0.884761\pi$
$107$	− 9.60457e8i	− 0.708355i	0.935178	−	0.354178i	$-0.115239\pi$
$108$	0	0
$109$	−9.98912e8	−0.677810	−0.338905	−	0.940821i	$-0.610057\pi$
$109$	−9.98912e8	−0.677810	−0.338905	+	0.940821i	$0.610057\pi$
$110$	0	0
$111$	0	0
$112$	7.94632e8i	0.477183i
$113$	2.50705e9i	1.44647i	0.690601	+	0.723236i	$0.257346\pi$
$113$	2.50705e9i	1.44647i	−0.690601	+	0.723236i	$0.742654\pi$
$114$	0	0
$115$	0	0
$116$	−2.72391e8	−0.139679
$117$	0	0
$118$	− 1.51293e8i	− 0.0718374i
$119$	−1.28306e9	−0.586524
$120$	0	0
$121$	−5.84695e8	−0.247968
$122$	− 1.84187e9i	− 0.752730i
$123$	0	0
$124$	3.39786e8	0.129065
$125$	0	0
$126$	0	0
$127$	− 2.47541e9i	− 0.844364i	−0.906511	−	0.422182i	$-0.861264\pi$
$127$	− 2.47541e9i	− 0.844364i	0.906511	−	0.422182i	$-0.138736\pi$
$128$	− 1.80351e9i	− 0.593845i
$129$	0	0
$130$	0	0
$131$	−1.92402e9	−0.570808	−0.285404	−	0.958407i	$-0.592128\pi$
$131$	−1.92402e9	−0.570808	−0.285404	+	0.958407i	$0.592128\pi$
$132$	0	0
$133$	4.36045e9i	1.20837i
$134$	−5.94680e9	−1.59335
$135$	0	0
$136$	3.98079e9	0.997802
$137$	− 4.48594e8i	− 0.108796i	−0.998519	−	0.0543978i	$-0.982676\pi$
$137$	− 4.48594e8i	− 0.108796i	0.998519	−	0.0543978i	$-0.0173239\pi$
$138$	0	0
$139$	−4.48415e9	−1.01886	−0.509429	−	0.860513i	$-0.670143\pi$
$139$	−4.48415e9	−1.01886	−0.509429	+	0.860513i	$0.670143\pi$
$140$	0	0
$141$	0	0
$142$	− 3.15532e9i	− 0.651247i
$143$	− 5.21084e9i	− 1.04207i
$144$	0	0
$145$	0	0
$146$	−5.70769e9	−1.03962
$147$	0	0
$148$	− 2.60831e8i	− 0.0446870i
$149$	−2.20480e9	−0.366463	−0.183232	−	0.983070i	$-0.558656\pi$
$149$	−2.20480e9	−0.366463	−0.183232	+	0.983070i	$0.558656\pi$
$150$	0	0
$151$	−3.21248e9	−0.502857	−0.251428	−	0.967876i	$-0.580900\pi$
$151$	−3.21248e9	−0.502857	−0.251428	+	0.967876i	$0.580900\pi$
$152$	− 1.35286e10i	− 2.05569i
$153$	0	0
$154$	3.41145e9	0.488760
$155$	0	0
$156$	0	0
$157$	− 1.08870e10i	− 1.43007i	−0.699088	−	0.715036i	$-0.746410\pi$
$157$	− 1.08870e10i	− 1.43007i	0.699088	−	0.715036i	$-0.253590\pi$
$158$	1.12177e10i	1.43202i
$159$	0	0
$160$	0	0
$161$	6.04232e9	0.708740
$162$	0	0
$163$	− 1.19994e10i	− 1.33142i	−0.746212	−	0.665708i	$-0.768130\pi$
$163$	− 1.19994e10i	− 1.33142i	0.746212	−	0.665708i	$-0.231870\pi$
$164$	−3.06875e9	−0.331257
$165$	0	0
$166$	−1.30982e8	−0.0133882
$167$	− 9.68608e9i	− 0.963660i	−0.876265	−	0.481830i	$-0.839972\pi$
$167$	− 9.68608e9i	− 0.963660i	0.876265	−	0.481830i	$-0.160028\pi$
$168$	0	0
$169$	−4.70793e9	−0.443956
$170$	0	0
$171$	0	0
$172$	1.47971e9i	0.128913i
$173$	7.35665e9i	0.624414i	0.950014	+	0.312207i	$0.101068\pi$
$173$	7.35665e9i	0.624414i	−0.950014	+	0.312207i	$0.898932\pi$
$174$	0	0
$175$	0	0
$176$	−8.34410e9	−0.655500
$177$	0	0
$178$	− 1.21007e10i	− 0.903481i
$179$	−2.00351e9	−0.145866	−0.0729329	−	0.997337i	$-0.523236\pi$
$179$	−2.00351e9	−0.145866	−0.0729329	+	0.997337i	$0.523236\pi$
$180$	0	0
$181$	5.63414e9	0.390188	0.195094	−	0.980785i	$-0.437499\pi$
$181$	5.63414e9	0.390188	0.195094	+	0.980785i	$0.437499\pi$
$182$	− 1.00248e10i	− 0.677258i
$183$	0	0
$184$	−1.87467e10	−1.20572
$185$	0	0
$186$	0	0
$187$	− 1.34729e10i	− 0.805701i
$188$	− 1.40599e8i	− 0.00820864i
$189$	0	0
$190$	0	0
$191$	−9.16925e9	−0.498521	−0.249261	−	0.968436i	$-0.580188\pi$
$191$	−9.16925e9	−0.498521	−0.249261	+	0.968436i	$0.580188\pi$
$192$	0	0
$193$	3.16327e10i	1.64107i	0.571594	+	0.820536i	$0.306325\pi$
$193$	3.16327e10i	1.64107i	−0.571594	+	0.820536i	$0.693675\pi$
$194$	−1.86927e10	−0.947469
$195$	0	0
$196$	2.52187e9	0.122059
$197$	− 2.59858e10i	− 1.22924i	−0.788822	−	0.614621i	$-0.789309\pi$
$197$	− 2.59858e10i	− 1.22924i	0.788822	−	0.614621i	$-0.210691\pi$
$198$	0	0
$199$	−1.05766e10	−0.478088	−0.239044	−	0.971009i	$-0.576834\pi$
$199$	−1.05766e10	−0.478088	−0.239044	+	0.971009i	$0.576834\pi$
$200$	0	0
$201$	0	0
$202$	− 1.93723e10i	− 0.818654i
$203$	1.05133e10i	0.434515i
$204$	0	0
$205$	0	0
$206$	3.24113e9	0.125399
$207$	0	0
$208$	2.45198e10i	0.908304i
$209$	−4.57873e10	−1.65992
$210$	0	0
$211$	1.80228e10	0.625965	0.312983	−	0.949759i	$-0.398672\pi$
$211$	1.80228e10	0.625965	0.312983	+	0.949759i	$0.398672\pi$
$212$	1.01133e10i	0.343859i
$213$	0	0
$214$	1.94026e10	0.632409
$215$	0	0
$216$	0	0
$217$	− 1.31145e10i	− 0.401496i
$218$	− 2.01794e10i	− 0.605139i
$219$	0	0
$220$	0	0
$221$	−3.95911e10	−1.11643
$222$	0	0
$223$	− 4.44522e10i	− 1.20371i	−0.798606	−	0.601855i	$-0.794429\pi$
$223$	− 4.44522e10i	− 1.20371i	0.798606	−	0.601855i	$-0.205571\pi$
$224$	9.49405e9	0.251962
$225$	0	0
$226$	−5.06460e10	−1.29139
$227$	− 5.59677e10i	− 1.39901i	−0.714627	−	0.699505i	$-0.753404\pi$
$227$	− 5.59677e10i	− 1.39901i	0.714627	−	0.699505i	$-0.246596\pi$
$228$	0	0
$229$	1.47705e9	0.0354923	0.0177462	−	0.999843i	$-0.494351\pi$
$229$	1.47705e9	0.0354923	0.0177462	+	0.999843i	$0.494351\pi$
$230$	0	0
$231$	0	0
$232$	− 3.26182e10i	− 0.739203i
$233$	− 7.83279e9i	− 0.174106i	−0.996204	−	0.0870532i	$-0.972255\pi$
$233$	− 7.83279e9i	− 0.174106i	0.996204	−	0.0870532i	$-0.0277450\pi$
$234$	0	0
$235$	0	0
$236$	−7.78152e8	−0.0163290
$237$	0	0
$238$	− 2.59197e10i	− 0.523640i
$239$	5.56371e10	1.10300	0.551498	−	0.834176i	$-0.314057\pi$
$239$	5.56371e10	1.10300	0.551498	+	0.834176i	$0.314057\pi$
$240$	0	0
$241$	−1.16053e10	−0.221606	−0.110803	−	0.993842i	$-0.535342\pi$
$241$	−1.16053e10	−0.221606	−0.110803	+	0.993842i	$0.535342\pi$
$242$	− 1.18117e10i	− 0.221382i
$243$	0	0
$244$	−9.47334e9	−0.171100
$245$	0	0
$246$	0	0
$247$	1.34549e11i	2.30009i
$248$	4.06886e10i	0.683030i
$249$	0	0
$250$	0	0
$251$	3.45974e10	0.550189	0.275094	−	0.961417i	$-0.411291\pi$
$251$	3.45974e10	0.550189	0.275094	+	0.961417i	$0.411291\pi$
$252$	0	0
$253$	6.34479e10i	0.973587i
$254$	5.00067e10	0.753836
$255$	0	0
$256$	−3.99972e10	−0.582036
$257$	3.67735e10i	0.525818i	0.964821	+	0.262909i	$0.0846820\pi$
$257$	3.67735e10i	0.525818i	−0.964821	+	0.262909i	$0.915318\pi$
$258$	0	0
$259$	−1.00671e10	−0.139013
$260$	0	0
$261$	0	0
$262$	− 3.88680e10i	− 0.509609i
$263$	1.33758e11i	1.72392i	0.506974	+	0.861962i	$0.330764\pi$
$263$	1.33758e11i	1.72392i	−0.506974	+	0.861962i	$0.669236\pi$
$264$	0	0
$265$	0	0
$266$	−8.80873e10	−1.07881
$267$	0	0
$268$	3.05864e10i	0.362178i
$269$	−1.42461e11	−1.65886	−0.829429	−	0.558612i	$-0.811334\pi$
$269$	−1.42461e11	−1.65886	−0.829429	+	0.558612i	$0.811334\pi$
$270$	0	0
$271$	−1.11046e11	−1.25067	−0.625333	−	0.780358i	$-0.715037\pi$
$271$	−1.11046e11	−1.25067	−0.625333	+	0.780358i	$0.715037\pi$
$272$	6.33972e10i	0.702279i
$273$	0	0
$274$	9.06224e9	0.0971310
$275$	0	0
$276$	0	0
$277$	2.80726e10i	0.286500i	0.989687	+	0.143250i	$0.0457552\pi$
$277$	2.80726e10i	0.286500i	−0.989687	+	0.143250i	$0.954245\pi$
$278$	− 9.05861e10i	− 0.909620i
$279$	0	0
$280$	0	0
$281$	−5.47143e10	−0.523507	−0.261753	−	0.965135i	$-0.584301\pi$
$281$	−5.47143e10	−0.523507	−0.261753	+	0.965135i	$0.584301\pi$
$282$	0	0
$283$	1.09950e11i	1.01895i	0.860484	+	0.509477i	$0.170161\pi$
$283$	1.09950e11i	1.01895i	−0.860484	+	0.509477i	$0.829839\pi$
$284$	−1.62288e10	−0.148032
$285$	0	0
$286$	1.05266e11	0.930341
$287$	1.18442e11i	1.03048i
$288$	0	0
$289$	1.62229e10	0.136801
$290$	0	0
$291$	0	0
$292$	2.93566e10i	0.236310i
$293$	− 2.30453e11i	− 1.82675i	−0.407124	−	0.913373i	$-0.633468\pi$
$293$	− 2.30453e11i	− 1.82675i	0.407124	−	0.913373i	$-0.366532\pi$
$294$	0	0
$295$	0	0
$296$	3.12339e10	0.236490
$297$	0	0
$298$	− 4.45400e10i	− 0.327173i
$299$	1.86446e11	1.34907
$300$	0	0
$301$	5.71111e10	0.401025
$302$	− 6.48966e10i	− 0.448943i
$303$	0	0
$304$	2.15454e11	1.44685
$305$	0	0
$306$	0	0
$307$	6.85036e10i	0.440140i	0.975484	+	0.220070i	$0.0706286\pi$
$307$	6.85036e10i	0.440140i	−0.975484	+	0.220070i	$0.929371\pi$
$308$	− 1.75462e10i	− 0.111098i
$309$	0	0
$310$	0	0
$311$	1.98797e11	1.20500	0.602502	−	0.798118i	$-0.294171\pi$
$311$	1.98797e11	1.20500	0.602502	+	0.798118i	$0.294171\pi$
$312$	0	0
$313$	6.15202e10i	0.362300i	0.983455	+	0.181150i	$0.0579820\pi$
$313$	6.15202e10i	0.362300i	−0.983455	+	0.181150i	$0.942018\pi$
$314$	2.19932e11	1.27675
$315$	0	0
$316$	5.76965e10	0.325505
$317$	2.25932e11i	1.25664i	0.777955	+	0.628320i	$0.216257\pi$
$317$	2.25932e11i	1.25664i	−0.777955	+	0.628320i	$0.783743\pi$
$318$	0	0
$319$	−1.10395e11	−0.596888
$320$	0	0
$321$	0	0
$322$	1.22063e11i	0.632753i
$323$	3.47885e11i	1.77838i
$324$	0	0
$325$	0	0
$326$	2.42404e11	1.18867
$327$	0	0
$328$	− 3.67476e11i	− 1.75306i
$329$	−5.42658e9	−0.0255355
$330$	0	0
$331$	−8.38825e10	−0.384101	−0.192050	−	0.981385i	$-0.561514\pi$
$331$	−8.38825e10	−0.384101	−0.192050	+	0.981385i	$0.561514\pi$
$332$	6.73682e8i	0.00304322i
$333$	0	0
$334$	1.95673e11	0.860341
$335$	0	0
$336$	0	0
$337$	− 2.19457e11i	− 0.926862i	−0.886133	−	0.463431i	$-0.846618\pi$
$337$	− 2.19457e11i	− 0.926862i	0.886133	−	0.463431i	$-0.153382\pi$
$338$	− 9.51069e10i	− 0.396357i
$339$	0	0
$340$	0	0
$341$	1.37710e11	0.551530
$342$	0	0
$343$	− 2.59163e11i	− 1.01100i
$344$	−1.77191e11	−0.682228
$345$	0	0
$346$	−1.48615e11	−0.557467
$347$	1.85960e11i	0.688551i	0.938869	+	0.344276i	$0.111875\pi$
$347$	1.85960e11i	0.688551i	−0.938869	+	0.344276i	$0.888125\pi$
$348$	0	0
$349$	−2.73237e11	−0.985881	−0.492940	−	0.870063i	$-0.664078\pi$
$349$	−2.73237e11	−0.985881	−0.492940	+	0.870063i	$0.664078\pi$
$350$	0	0
$351$	0	0
$352$	9.96932e10i	0.346117i
$353$	− 3.02861e11i	− 1.03814i	−0.854731	−	0.519072i	$-0.826278\pi$
$353$	− 3.02861e11i	− 1.03814i	0.854731	−	0.519072i	$-0.173722\pi$
$354$	0	0
$355$	0	0
$356$	−6.22378e10	−0.205366
$357$	0	0
$358$	− 4.04738e10i	− 0.130227i
$359$	2.47660e11	0.786919	0.393460	−	0.919342i	$-0.371278\pi$
$359$	2.47660e11	0.786919	0.393460	+	0.919342i	$0.371278\pi$
$360$	0	0
$361$	8.59591e11	2.66385
$362$	1.13818e11i	0.348354i
$363$	0	0
$364$	−5.15609e10	−0.153944
$365$	0	0
$366$	0	0
$367$	2.66354e11i	0.766412i	0.923663	+	0.383206i	$0.125180\pi$
$367$	2.66354e11i	0.766412i	−0.923663	+	0.383206i	$0.874820\pi$
$368$	− 2.98556e11i	− 0.848616i
$369$	0	0
$370$	0	0
$371$	3.90335e11	1.06968
$372$	0	0
$373$	5.20850e11i	1.39323i	0.717445	+	0.696616i	$0.245312\pi$
$373$	5.20850e11i	1.39323i	−0.717445	+	0.696616i	$0.754688\pi$
$374$	2.72172e11	0.719318
$375$	0	0
$376$	1.68364e10	0.0434414
$377$	3.24405e11i	0.827088i
$378$	0	0
$379$	−3.28308e10	−0.0817344	−0.0408672	−	0.999165i	$-0.513012\pi$
$379$	−3.28308e10	−0.0817344	−0.0408672	+	0.999165i	$0.513012\pi$
$380$	0	0
$381$	0	0
$382$	− 1.85232e11i	− 0.445072i
$383$	− 7.15293e10i	− 0.169859i	−0.996387	−	0.0849297i	$-0.972933\pi$
$383$	− 7.15293e10i	− 0.169859i	0.996387	−	0.0849297i	$-0.0270666\pi$
$384$	0	0
$385$	0	0
$386$	−6.39025e11	−1.46513
$387$	0	0
$388$	9.61429e10i	0.215365i
$389$	−2.02681e11	−0.448787	−0.224394	−	0.974499i	$-0.572040\pi$
$389$	−2.02681e11	−0.448787	−0.224394	+	0.974499i	$0.572040\pi$
$390$	0	0
$391$	4.82067e11	1.04307
$392$	3.01989e11i	0.645956i
$393$	0	0
$394$	5.24949e11	1.09745
$395$	0	0
$396$	0	0
$397$	1.63266e10i	0.0329867i	0.999864	+	0.0164934i	$0.00525024\pi$
$397$	1.63266e10i	0.0329867i	−0.999864	+	0.0164934i	$0.994750\pi$
$398$	− 2.13663e11i	− 0.426830i
$399$	0	0
$400$	0	0
$401$	8.20766e11	1.58515	0.792574	−	0.609776i	$-0.208741\pi$
$401$	8.20766e11	1.58515	0.792574	+	0.609776i	$0.208741\pi$
$402$	0	0
$403$	− 4.04670e11i	− 0.764237i
$404$	−9.96383e10	−0.186085
$405$	0	0
$406$	−2.12383e11	−0.387929
$407$	− 1.05710e11i	− 0.190960i
$408$	0	0
$409$	4.24628e11	0.750332	0.375166	−	0.926958i	$-0.377586\pi$
$409$	4.24628e11	0.750332	0.375166	+	0.926958i	$0.377586\pi$
$410$	0	0
$411$	0	0
$412$	− 1.66702e10i	− 0.0285039i
$413$	3.00337e10i	0.0507966i
$414$	0	0
$415$	0	0
$416$	2.92956e11	0.479603
$417$	0	0
$418$	− 9.24969e11i	− 1.48195i
$419$	−1.26375e10	−0.0200308	−0.0100154	−	0.999950i	$-0.503188\pi$
$419$	−1.26375e10	−0.0200308	−0.0100154	+	0.999950i	$0.503188\pi$
$420$	0	0
$421$	−3.04545e10	−0.0472479	−0.0236240	−	0.999721i	$-0.507520\pi$
$421$	−3.04545e10	−0.0472479	−0.0236240	+	0.999721i	$0.507520\pi$
$422$	3.64085e11i	0.558853i
$423$	0	0
$424$	−1.21104e12	−1.81976
$425$	0	0
$426$	0	0
$427$	3.65635e11i	0.532259i
$428$	− 9.97940e10i	− 0.143750i
$429$	0	0
$430$	0	0
$431$	−4.05830e11	−0.566495	−0.283248	−	0.959047i	$-0.591412\pi$
$431$	−4.05830e11	−0.566495	−0.283248	+	0.959047i	$0.591412\pi$
$432$	0	0
$433$	− 1.36978e11i	− 0.187265i	−0.995607	−	0.0936324i	$-0.970152\pi$
$433$	− 1.36978e11i	− 0.187265i	0.995607	−	0.0936324i	$-0.0298478\pi$
$434$	2.64931e11	0.358450
$435$	0	0
$436$	−1.03790e11	−0.137551
$437$	− 1.63829e12i	− 2.14895i
$438$	0	0
$439$	−7.84981e11	−1.00872	−0.504358	−	0.863495i	$-0.668271\pi$
$439$	−7.84981e11	−1.00872	−0.504358	+	0.863495i	$0.668271\pi$
$440$	0	0
$441$	0	0
$442$	− 7.99797e11i	− 0.996734i
$443$	8.87799e11i	1.09521i	0.836737	+	0.547605i	$0.184460\pi$
$443$	8.87799e11i	1.09521i	−0.836737	+	0.547605i	$0.815540\pi$
$444$	0	0
$445$	0	0
$446$	8.97998e11	1.07465
$447$	0	0
$448$	5.98645e11i	0.702131i
$449$	8.35477e11	0.970122	0.485061	−	0.874480i	$-0.338797\pi$
$449$	8.35477e11	0.970122	0.485061	+	0.874480i	$0.338797\pi$
$450$	0	0
$451$	−1.24371e12	−1.41555
$452$	2.60489e11i	0.293540i
$453$	0	0
$454$	1.13063e12	1.24902
$455$	0	0
$456$	0	0
$457$	6.01172e11i	0.644727i	0.946616	+	0.322364i	$0.104477\pi$
$457$	6.01172e11i	0.644727i	−0.946616	+	0.322364i	$0.895523\pi$
$458$	2.98384e10i	0.0316870i
$459$	0	0
$460$	0	0
$461$	−1.52807e12	−1.57576	−0.787879	−	0.615829i	$-0.788821\pi$
$461$	−1.52807e12	−1.57576	−0.787879	+	0.615829i	$0.788821\pi$
$462$	0	0
$463$	7.80402e11i	0.789231i	0.918846	+	0.394615i	$0.129122\pi$
$463$	7.80402e11i	0.789231i	−0.918846	+	0.394615i	$0.870878\pi$
$464$	5.19469e11	0.520270
$465$	0	0
$466$	1.58234e11	0.155440
$467$	4.04751e11i	0.393788i	0.980425	+	0.196894i	$0.0630855\pi$
$467$	4.04751e11i	0.393788i	−0.980425	+	0.196894i	$0.936915\pi$
$468$	0	0
$469$	1.18052e12	1.12667
$470$	0	0
$471$	0	0
$472$	− 9.31820e10i	− 0.0864158i
$473$	5.99700e11i	0.550883i
$474$	0	0
$475$	0	0
$476$	−1.33313e11	−0.119026
$477$	0	0
$478$	1.12395e12i	0.984738i
$479$	−2.06972e12	−1.79640	−0.898199	−	0.439588i	$-0.855124\pi$
$479$	−2.06972e12	−1.79640	−0.898199	+	0.439588i	$0.855124\pi$
$480$	0	0
$481$	−3.10638e11	−0.264607
$482$	− 2.34444e11i	− 0.197846i
$483$	0	0
$484$	−6.07514e10	−0.0503213
$485$	0	0
$486$	0	0
$487$	− 2.41184e11i	− 0.194298i	−0.995270	−	0.0971490i	$-0.969028\pi$
$487$	− 2.41184e11i	− 0.194298i	0.995270	−	0.0971490i	$-0.0309723\pi$
$488$	− 1.13441e12i	− 0.905485i
$489$	0	0
$490$	0	0
$491$	−2.27883e12	−1.76948	−0.884739	−	0.466088i	$-0.845663\pi$
$491$	−2.27883e12	−1.76948	−0.884739	+	0.466088i	$0.845663\pi$
$492$	0	0
$493$	8.38767e11i	0.639485i
$494$	−2.71809e12	−2.05349
$495$	0	0
$496$	−6.47997e11	−0.480735
$497$	6.26373e11i	0.460499i
$498$	0	0
$499$	−9.88752e11	−0.713896	−0.356948	−	0.934124i	$-0.616183\pi$
$499$	−9.88752e11	−0.713896	−0.356948	+	0.934124i	$0.616183\pi$
$500$	0	0
$501$	0	0
$502$	6.98916e11i	0.491200i
$503$	1.22385e12i	0.852455i	0.904616	+	0.426228i	$0.140158\pi$
$503$	1.22385e12i	0.852455i	−0.904616	+	0.426228i	$0.859842\pi$
$504$	0	0
$505$	0	0
$506$	−1.28174e12	−0.869204
$507$	0	0
$508$	− 2.57201e11i	− 0.171351i
$509$	1.58447e12	1.04629	0.523146	−	0.852243i	$-0.324758\pi$
$509$	1.58447e12	1.04629	0.523146	+	0.852243i	$0.324758\pi$
$510$	0	0
$511$	1.13305e12	0.735117
$512$	− 1.73140e12i	− 1.11348i
$513$	0	0
$514$	−7.42877e11	−0.469443
$515$	0	0
$516$	0	0
$517$	− 5.69823e10i	− 0.0350778i
$518$	− 2.03369e11i	− 0.124109i
$519$	0	0
$520$	0	0
$521$	2.71561e12	1.61472	0.807362	−	0.590057i	$-0.200895\pi$
$521$	2.71561e12	1.61472	0.807362	+	0.590057i	$0.200895\pi$
$522$	0	0
$523$	2.16171e12i	1.26340i	0.775214	+	0.631698i	$0.217642\pi$
$523$	2.16171e12i	1.26340i	−0.775214	+	0.631698i	$0.782358\pi$
$524$	−1.99911e11	−0.115837
$525$	0	0
$526$	−2.70210e12	−1.53909
$527$	− 1.04630e12i	− 0.590890i
$528$	0	0
$529$	−4.69046e11	−0.260415
$530$	0	0
$531$	0	0
$532$	4.53062e11i	0.245220i
$533$	3.65475e12i	1.96149i
$534$	0	0
$535$	0	0
$536$	−3.66265e12	−1.91670
$537$	0	0
$538$	− 2.87791e12i	− 1.48100i
$539$	1.02207e12	0.521594
$540$	0	0
$541$	−2.29090e12	−1.14979	−0.574895	−	0.818227i	$-0.694957\pi$
$541$	−2.29090e12	−1.14979	−0.574895	+	0.818227i	$0.694957\pi$
$542$	− 2.24329e12i	− 1.11658i
$543$	0	0
$544$	7.57453e11	0.370818
$545$	0	0
$546$	0	0
$547$	− 3.39447e12i	− 1.62117i	−0.585620	−	0.810586i	$-0.699149\pi$
$547$	− 3.39447e12i	− 1.62117i	0.585620	−	0.810586i	$-0.300851\pi$
$548$	− 4.66101e10i	− 0.0220784i
$549$	0	0
$550$	0	0
$551$	2.85053e12	1.31748
$552$	0	0
$553$	− 2.22687e12i	− 1.01259i
$554$	−5.67107e11	−0.255783
$555$	0	0
$556$	−4.65915e11	−0.206762
$557$	− 1.74706e12i	− 0.769060i	−0.923113	−	0.384530i	$-0.874364\pi$
$557$	− 1.74706e12i	− 0.769060i	0.923113	−	0.384530i	$-0.125636\pi$
$558$	0	0
$559$	1.76226e12	0.763340
$560$	0	0
$561$	0	0
$562$	− 1.10531e12i	− 0.467379i
$563$	− 2.58864e12i	− 1.08588i	−0.839770	−	0.542942i	$-0.817310\pi$
$563$	− 2.58864e12i	− 1.08588i	0.839770	−	0.542942i	$-0.182690\pi$
$564$	0	0
$565$	0	0
$566$	−2.22114e12	−0.909707
$567$	0	0
$568$	− 1.94337e12i	− 0.783407i
$569$	−1.99294e12	−0.797055	−0.398527	−	0.917156i	$-0.630479\pi$
$569$	−1.99294e12	−0.797055	−0.398527	+	0.917156i	$0.630479\pi$
$570$	0	0
$571$	3.50761e10	0.0138086	0.00690428	−	0.999976i	$-0.497802\pi$
$571$	3.50761e10	0.0138086	0.00690428	+	0.999976i	$0.497802\pi$
$572$	− 5.41420e11i	− 0.211471i
$573$	0	0
$574$	−2.39270e12	−0.919995
$575$	0	0
$576$	0	0
$577$	− 7.75900e11i	− 0.291417i	−0.989328	−	0.145708i	$-0.953454\pi$
$577$	− 7.75900e11i	− 0.291417i	0.989328	−	0.145708i	$-0.0465461\pi$
$578$	3.27726e11i	0.122134i
$579$	0	0
$580$	0	0
$581$	2.60016e10	0.00946689
$582$	0	0
$583$	4.09874e12i	1.46941i
$584$	−3.51538e12	−1.25059
$585$	0	0
$586$	4.65548e12	1.63089
$587$	− 1.24477e12i	− 0.432730i	−0.976313	−	0.216365i	$-0.930580\pi$
$587$	− 1.24477e12i	− 0.432730i	0.976313	−	0.216365i	$-0.0694201\pi$
$588$	0	0
$589$	−3.55581e12	−1.21736
$590$	0	0
$591$	0	0
$592$	4.97424e11i	0.166448i
$593$	− 2.57794e11i	− 0.0856103i	−0.999083	−	0.0428052i	$-0.986371\pi$
$593$	− 2.57794e11i	− 0.0856103i	0.999083	−	0.0428052i	$-0.0136295\pi$
$594$	0	0
$595$	0	0
$596$	−2.29084e11	−0.0743681
$597$	0	0
$598$	3.76648e12i	1.20443i
$599$	2.32089e12	0.736604	0.368302	−	0.929706i	$-0.379939\pi$
$599$	2.32089e12	0.736604	0.368302	+	0.929706i	$0.379939\pi$
$600$	0	0
$601$	1.78665e12	0.558605	0.279302	−	0.960203i	$-0.409897\pi$
$601$	1.78665e12	0.558605	0.279302	+	0.960203i	$0.409897\pi$
$602$	1.15373e12i	0.358029i
$603$	0	0
$604$	−3.33785e11	−0.102047
$605$	0	0
$606$	0	0
$607$	− 1.45787e12i	− 0.435883i	−0.975962	−	0.217941i	$-0.930066\pi$
$607$	− 1.45787e12i	− 0.435883i	0.975962	−	0.217941i	$-0.0699342\pi$
$608$	− 2.57419e12i	− 0.763966i
$609$	0	0
$610$	0	0
$611$	−1.67447e11	−0.0486062
$612$	0	0
$613$	− 1.42075e12i	− 0.406394i	−0.979138	−	0.203197i	$-0.934867\pi$
$613$	− 1.42075e12i	− 0.406394i	0.979138	−	0.203197i	$-0.0651331\pi$
$614$	−1.38387e12	−0.392950
$615$	0	0
$616$	2.10112e12	0.587946
$617$	− 1.20441e12i	− 0.334573i	−0.985908	−	0.167286i	$-0.946500\pi$
$617$	− 1.20441e12i	− 0.334573i	0.985908	−	0.167286i	$-0.0535004\pi$
$618$	0	0
$619$	4.91349e12	1.34519	0.672593	−	0.740013i	$-0.265181\pi$
$619$	4.91349e12	1.34519	0.672593	+	0.740013i	$0.265181\pi$
$620$	0	0
$621$	0	0
$622$	4.01598e12i	1.07581i
$623$	2.40214e12i	0.638856i
$624$	0	0
$625$	0	0
$626$	−1.24280e12	−0.323456
$627$	0	0
$628$	− 1.13118e12i	− 0.290211i
$629$	−8.03171e11	−0.204588
$630$	0	0
$631$	4.58663e12	1.15176	0.575879	−	0.817535i	$-0.304660\pi$
$631$	4.58663e12	1.15176	0.575879	+	0.817535i	$0.304660\pi$
$632$	6.90903e12i	1.72262i
$633$	0	0
$634$	−4.56414e12	−1.12191
$635$	0	0
$636$	0	0
$637$	− 3.00344e12i	− 0.722755i
$638$	− 2.23014e12i	− 0.532893i
$639$	0	0
$640$	0	0
$641$	2.30636e12	0.539593	0.269796	−	0.962917i	$-0.413044\pi$
$641$	2.30636e12	0.539593	0.269796	+	0.962917i	$0.413044\pi$
$642$	0	0
$643$	− 2.59493e12i	− 0.598654i	−0.954151	−	0.299327i	$-0.903238\pi$
$643$	− 2.59493e12i	− 0.598654i	0.954151	−	0.299327i	$-0.0967622\pi$
$644$	6.27813e11	0.143828
$645$	0	0
$646$	−7.02777e12	−1.58771
$647$	− 5.14811e12i	− 1.15499i	−0.816394	−	0.577495i	$-0.804030\pi$
$647$	− 5.14811e12i	− 1.15499i	0.816394	−	0.577495i	$-0.195970\pi$
$648$	0	0
$649$	−3.15372e11	−0.0697786
$650$	0	0
$651$	0	0
$652$	− 1.24676e12i	− 0.270191i
$653$	4.42559e12i	0.952492i	0.879312	+	0.476246i	$0.158003\pi$
$653$	4.42559e12i	0.952492i	−0.879312	+	0.476246i	$0.841997\pi$
$654$	0	0
$655$	0	0
$656$	5.85234e12	1.23385
$657$	0	0
$658$	− 1.09625e11i	− 0.0227977i
$659$	1.09827e12	0.226842	0.113421	−	0.993547i	$-0.463819\pi$
$659$	1.09827e12	0.226842	0.113421	+	0.993547i	$0.463819\pi$
$660$	0	0
$661$	7.99232e11	0.162842	0.0814209	−	0.996680i	$-0.474054\pi$
$661$	7.99232e11	0.162842	0.0814209	+	0.996680i	$0.474054\pi$
$662$	− 1.69455e12i	− 0.342919i
$663$	0	0
$664$	−8.06719e10	−0.0161052
$665$	0	0
$666$	0	0
$667$	− 3.95000e12i	− 0.772736i
$668$	− 1.00641e12i	− 0.195560i
$669$	0	0
$670$	0	0
$671$	−3.83939e12	−0.731157
$672$	0	0
$673$	− 7.68445e12i	− 1.44393i	−0.691931	−	0.721963i	$-0.743240\pi$
$673$	− 7.68445e12i	− 1.44393i	0.691931	−	0.721963i	$-0.256760\pi$
$674$	4.43334e12	0.827488
$675$	0	0
$676$	−4.89167e11	−0.0900942
$677$	− 1.28854e12i	− 0.235749i	−0.993029	−	0.117874i	$-0.962392\pi$
$677$	− 1.28854e12i	− 0.235749i	0.993029	−	0.117874i	$-0.0376080\pi$
$678$	0	0
$679$	3.71076e12	0.669959
$680$	0	0
$681$	0	0
$682$	2.78193e12i	0.492398i
$683$	− 8.11444e11i	− 0.142681i	−0.997452	−	0.0713404i	$-0.977272\pi$
$683$	− 8.11444e11i	− 0.142681i	0.997452	−	0.0713404i	$-0.0227277\pi$
$684$	0	0
$685$	0	0
$686$	5.23546e12	0.902602
$687$	0	0
$688$	− 2.82191e12i	− 0.480170i
$689$	1.20445e13	2.03611
$690$	0	0
$691$	2.26741e12	0.378338	0.189169	−	0.981945i	$-0.439421\pi$
$691$	2.26741e12	0.378338	0.189169	+	0.981945i	$0.439421\pi$
$692$	7.64375e11i	0.126715i
$693$	0	0
$694$	−3.75665e12	−0.614728
$695$	0	0
$696$	0	0
$697$	9.44955e12i	1.51657i
$698$	− 5.51977e12i	− 0.880180i
$699$	0	0
$700$	0	0
$701$	4.92113e12	0.769721	0.384861	−	0.922975i	$-0.374249\pi$
$701$	4.92113e12	0.769721	0.384861	+	0.922975i	$0.374249\pi$
$702$	0	0
$703$	2.72956e12i	0.421496i
$704$	−6.28612e12	−0.964508
$705$	0	0
$706$	6.11822e12	0.926839
$707$	3.84566e12i	0.578874i
$708$	0	0
$709$	6.12354e12	0.910112	0.455056	−	0.890463i	$-0.349619\pi$
$709$	6.12354e12	0.910112	0.455056	+	0.890463i	$0.349619\pi$
$710$	0	0
$711$	0	0
$712$	− 7.45283e12i	− 1.08683i
$713$	4.92732e12i	0.714016i
$714$	0	0
$715$	0	0
$716$	−2.08170e11	−0.0296012
$717$	0	0
$718$	5.00307e12i	0.702550i
$719$	−2.30376e11	−0.0321483	−0.0160741	−	0.999871i	$-0.505117\pi$
$719$	−2.30376e11	−0.0321483	−0.0160741	+	0.999871i	$0.505117\pi$
$720$	0	0
$721$	−6.43408e11	−0.0886702
$722$	1.73650e13i	2.37824i
$723$	0	0
$724$	5.85402e11	0.0791827
$725$	0	0
$726$	0	0
$727$	9.25894e12i	1.22930i	0.788801	+	0.614648i	$0.210702\pi$
$727$	9.25894e12i	1.22930i	−0.788801	+	0.614648i	$0.789298\pi$
$728$	− 6.17430e12i	− 0.814698i
$729$	0	0
$730$	0	0
$731$	4.55643e12	0.590196
$732$	0	0
$733$	7.22531e12i	0.924461i	0.886760	+	0.462231i	$0.152951\pi$
$733$	7.22531e12i	0.924461i	−0.886760	+	0.462231i	$0.847049\pi$
$734$	−5.38074e12	−0.684241
$735$	0	0
$736$	−3.56707e12	−0.448086
$737$	1.23962e13i	1.54769i
$738$	0	0
$739$	1.24725e13	1.53834	0.769170	−	0.639044i	$-0.220670\pi$
$739$	1.24725e13	1.53834	0.769170	+	0.639044i	$0.220670\pi$
$740$	0	0
$741$	0	0
$742$	7.88531e12i	0.954996i
$743$	− 1.15645e13i	− 1.39212i	−0.717986	−	0.696058i	$-0.754936\pi$
$743$	− 1.15645e13i	− 1.39212i	0.717986	−	0.696058i	$-0.245064\pi$
$744$	0	0
$745$	0	0
$746$	−1.05219e13	−1.24386
$747$	0	0
$748$	− 1.39987e12i	− 0.163505i
$749$	−3.85167e12	−0.447179
$750$	0	0
$751$	−3.70732e12	−0.425285	−0.212642	−	0.977130i	$-0.568207\pi$
$751$	−3.70732e12	−0.425285	−0.212642	+	0.977130i	$0.568207\pi$
$752$	2.68132e11i	0.0305752i
$753$	0	0
$754$	−6.55344e12	−0.738412
$755$	0	0
$756$	0	0
$757$	− 4.86635e11i	− 0.0538607i	−0.999637	−	0.0269303i	$-0.991427\pi$
$757$	− 4.86635e11i	− 0.0538607i	0.999637	−	0.0269303i	$-0.00857323\pi$
$758$	− 6.63228e11i	− 0.0729712i
$759$	0	0
$760$	0	0
$761$	4.28395e12	0.463035	0.231518	−	0.972831i	$-0.425631\pi$
$761$	4.28395e12	0.463035	0.231518	+	0.972831i	$0.425631\pi$
$762$	0	0
$763$	4.00589e12i	0.427896i
$764$	−9.52709e11	−0.101167
$765$	0	0
$766$	1.44499e12	0.151648
$767$	9.26745e11i	0.0966899i
$768$	0	0
$769$	4.03625e12	0.416207	0.208103	−	0.978107i	$-0.433271\pi$
$769$	4.03625e12	0.416207	0.208103	+	0.978107i	$0.433271\pi$
$770$	0	0
$771$	0	0
$772$	3.28672e12i	0.333031i
$773$	1.21916e13i	1.22815i	0.789247	+	0.614076i	$0.210471\pi$
$773$	1.21916e13i	1.22815i	−0.789247	+	0.614076i	$0.789529\pi$
$774$	0	0
$775$	0	0
$776$	−1.15129e13	−1.13974
$777$	0	0
$778$	− 4.09445e12i	− 0.400671i
$779$	3.21141e13	3.12447
$780$	0	0
$781$	−6.57728e12	−0.632582
$782$	9.73844e12i	0.931235i
$783$	0	0
$784$	−4.80940e12	−0.454641
$785$	0	0
$786$	0	0
$787$	− 1.20299e12i	− 0.111783i	−0.998437	−	0.0558913i	$-0.982200\pi$
$787$	− 1.20299e12i	− 0.111783i	0.998437	−	0.0558913i	$-0.0178000\pi$
$788$	− 2.69999e12i	− 0.249456i
$789$	0	0
$790$	0	0
$791$	1.00539e13	0.913147
$792$	0	0
$793$	1.12823e13i	1.01314i
$794$	−3.29821e11	−0.0294501
$795$	0	0
$796$	−1.09894e12	−0.0970208
$797$	− 1.72611e13i	− 1.51533i	−0.652646	−	0.757663i	$-0.726341\pi$
$797$	− 1.72611e13i	− 1.51533i	0.652646	−	0.757663i	$-0.273659\pi$
$798$	0	0
$799$	−4.32943e11	−0.0375811
$800$	0	0
$801$	0	0
$802$	1.65806e13i	1.41520i
$803$	1.18977e13i	1.00982i
$804$	0	0
$805$	0	0
$806$	8.17490e12	0.682299
$807$	0	0
$808$	− 1.19315e13i	− 0.984788i
$809$	7.44408e12	0.611002	0.305501	−	0.952192i	$-0.401176\pi$
$809$	7.44408e12	0.611002	0.305501	+	0.952192i	$0.401176\pi$
$810$	0	0
$811$	−9.59955e12	−0.779214	−0.389607	−	0.920981i	$-0.627389\pi$
$811$	−9.59955e12	−0.779214	−0.389607	+	0.920981i	$0.627389\pi$
$812$	1.09235e12i	0.0881783i
$813$	0	0
$814$	2.13550e12	0.170486
$815$	0	0
$816$	0	0
$817$	− 1.54849e13i	− 1.21593i
$818$	8.57808e12i	0.669885i
$819$	0	0
$820$	0	0
$821$	5.04043e12	0.387189	0.193595	−	0.981082i	$-0.437985\pi$
$821$	5.04043e12	0.387189	0.193595	+	0.981082i	$0.437985\pi$
$822$	0	0
$823$	− 1.62323e13i	− 1.23333i	−0.787225	−	0.616666i	$-0.788483\pi$
$823$	− 1.62323e13i	− 1.23333i	0.787225	−	0.616666i	$-0.211517\pi$
$824$	1.99622e12	0.150847
$825$	0	0
$826$	−6.06724e11	−0.0453504
$827$	1.74898e13i	1.30020i	0.759850	+	0.650099i	$0.225273\pi$
$827$	1.74898e13i	1.30020i	−0.759850	+	0.650099i	$0.774727\pi$
$828$	0	0
$829$	1.04102e13	0.765535	0.382768	−	0.923845i	$-0.374971\pi$
$829$	1.04102e13	0.765535	0.382768	+	0.923845i	$0.374971\pi$
$830$	0	0
$831$	0	0
$832$	1.84722e13i	1.33649i
$833$	− 7.76555e12i	− 0.558817i
$834$	0	0
$835$	0	0
$836$	−4.75742e12	−0.336855
$837$	0	0
$838$	− 2.55296e11i	− 0.0178832i
$839$	−2.66219e13	−1.85485	−0.927426	−	0.374006i	$-0.877984\pi$
$839$	−2.66219e13	−1.85485	−0.927426	+	0.374006i	$0.877984\pi$
$840$	0	0
$841$	−7.63439e12	−0.526250
$842$	− 6.15225e11i	− 0.0421822i
$843$	0	0
$844$	1.87261e12	0.127030
$845$	0	0
$846$	0	0
$847$	2.34477e12i	0.156540i
$848$	− 1.92868e13i	− 1.28079i
$849$	0	0
$850$	0	0
$851$	3.78237e12	0.247219
$852$	0	0
$853$	7.30064e12i	0.472161i	0.971734	+	0.236080i	$0.0758629\pi$
$853$	7.30064e12i	0.472161i	−0.971734	+	0.236080i	$0.924137\pi$
$854$	−7.38635e12	−0.475192
$855$	0	0
$856$	1.19501e13	0.760747
$857$	1.18598e13i	0.751041i	0.926814	+	0.375520i	$0.122536\pi$
$857$	1.18598e13i	0.751041i	−0.926814	+	0.375520i	$0.877464\pi$
$858$	0	0
$859$	−1.67692e13	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$859$	−1.67692e13	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$860$	0	0
$861$	0	0
$862$	− 8.19834e12i	− 0.505759i
$863$	6.71596e12i	0.412154i	0.978536	+	0.206077i	$0.0660698\pi$
$863$	6.71596e12i	0.412154i	−0.978536	+	0.206077i	$0.933930\pi$
$864$	0	0
$865$	0	0
$866$	2.76716e12	0.167187
$867$	0	0
$868$	− 1.36263e12i	− 0.0814776i
$869$	2.33834e13	1.39098
$870$	0	0
$871$	3.64270e13	2.14458
$872$	− 1.24286e13i	− 0.727943i
$873$	0	0
$874$	3.30959e13	1.91855
$875$	0	0
$876$	0	0
$877$	− 2.79757e13i	− 1.59692i	−0.602051	−	0.798458i	$-0.705649\pi$
$877$	− 2.79757e13i	− 1.59692i	0.602051	−	0.798458i	$-0.294351\pi$
$878$	− 1.58577e13i	− 0.900566i
$879$	0	0
$880$	0	0
$881$	−2.06225e13	−1.15332	−0.576660	−	0.816984i	$-0.695644\pi$
$881$	−2.06225e13	−1.15332	−0.576660	+	0.816984i	$0.695644\pi$
$882$	0	0
$883$	2.02048e13i	1.11849i	0.829003	+	0.559244i	$0.188909\pi$
$883$	2.02048e13i	1.11849i	−0.829003	+	0.559244i	$0.811091\pi$
$884$	−4.11362e12	−0.226563
$885$	0	0
$886$	−1.79348e13	−0.977788
$887$	− 3.19954e13i	− 1.73553i	−0.496978	−	0.867763i	$-0.665557\pi$
$887$	− 3.19954e13i	− 1.73553i	0.496978	−	0.867763i	$-0.334443\pi$
$888$	0	0
$889$	−9.92701e12	−0.533041
$890$	0	0
$891$	0	0
$892$	− 4.61870e12i	− 0.244275i
$893$	1.47135e12i	0.0774254i
$894$	0	0
$895$	0	0
$896$	−7.23252e12	−0.374890
$897$	0	0
$898$	1.68778e13i	0.866110i
$899$	−8.57323e12	−0.437750
$900$	0	0
$901$	3.11416e13	1.57427
$902$	− 2.51248e13i	− 1.26378i
$903$	0	0
$904$	−3.11930e13	−1.55346
$905$	0	0
$906$	0	0
$907$	2.40537e12i	0.118018i	0.998257	+	0.0590091i	$0.0187941\pi$
$907$	2.40537e12i	0.118018i	−0.998257	+	0.0590091i	$0.981206\pi$
$908$	− 5.81519e12i	− 0.283908i
$909$	0	0
$910$	0	0
$911$	3.11773e13	1.49970	0.749852	−	0.661606i	$-0.230125\pi$
$911$	3.11773e13	1.49970	0.749852	+	0.661606i	$0.230125\pi$
$912$	0	0
$913$	2.73032e11i	0.0130045i
$914$	−1.21445e13	−0.575603
$915$	0	0
$916$	1.53469e11	0.00720263
$917$	7.71582e12i	0.360346i
$918$	0	0
$919$	−3.42677e13	−1.58477	−0.792383	−	0.610024i	$-0.791160\pi$
$919$	−3.42677e13	−1.58477	−0.792383	+	0.610024i	$0.791160\pi$
$920$	0	0
$921$	0	0
$922$	− 3.08692e13i	− 1.40681i
$923$	1.93278e13i	0.876548i
$924$	0	0
$925$	0	0
$926$	−1.57652e13	−0.704614
$927$	0	0
$928$	− 6.20648e12i	− 0.274713i
$929$	−2.52782e13	−1.11346	−0.556730	−	0.830693i	$-0.687944\pi$
$929$	−2.52782e13	−1.11346	−0.556730	+	0.830693i	$0.687944\pi$
$930$	0	0
$931$	−2.63910e13	−1.15129
$932$	− 8.13848e11i	− 0.0353323i
$933$	0	0
$934$	−8.17655e12	−0.351568
$935$	0	0
$936$	0	0
$937$	2.72636e13i	1.15546i	0.816227	+	0.577731i	$0.196062\pi$
$937$	2.72636e13i	1.15546i	−0.816227	+	0.577731i	$0.803938\pi$
$938$	2.38482e13i	1.00587i
$939$	0	0
$940$	0	0
$941$	−1.57383e13	−0.654340	−0.327170	−	0.944966i	$-0.606095\pi$
$941$	−1.57383e13	−0.654340	−0.327170	+	0.944966i	$0.606095\pi$
$942$	0	0
$943$	− 4.45008e13i	− 1.83259i
$944$	1.48399e12	0.0608217
$945$	0	0
$946$	−1.21148e13	−0.491820
$947$	9.71128e12i	0.392375i	0.980566	+	0.196188i	$0.0628562\pi$
$947$	9.71128e12i	0.392375i	−0.980566	+	0.196188i	$0.937144\pi$
$948$	0	0
$949$	3.49624e13	1.39927
$950$	0	0
$951$	0	0
$952$	− 1.59640e13i	− 0.629905i
$953$	4.15385e12i	0.163130i	0.996668	+	0.0815648i	$0.0259917\pi$
$953$	4.15385e12i	0.163130i	−0.996668	+	0.0815648i	$0.974008\pi$
$954$	0	0
$955$	0	0
$956$	5.78084e12	0.223836
$957$	0	0
$958$	− 4.18114e13i	− 1.60380i
$959$	−1.79897e12	−0.0686818
$960$	0	0
$961$	−1.57452e13	−0.595515
$962$	− 6.27532e12i	− 0.236237i
$963$	0	0
$964$	−1.20582e12	−0.0449715
$965$	0	0
$966$	0	0
$967$	1.84544e13i	0.678706i	0.940659	+	0.339353i	$0.110208\pi$
$967$	1.84544e13i	0.678706i	−0.940659	+	0.339353i	$0.889792\pi$
$968$	− 7.27484e12i	− 0.266308i
$969$	0	0
$970$	0	0
$971$	2.95973e12	0.106848	0.0534238	−	0.998572i	$-0.482987\pi$
$971$	2.95973e12	0.106848	0.0534238	+	0.998572i	$0.482987\pi$
$972$	0	0
$973$	1.79826e13i	0.643197i
$974$	4.87226e12	0.173466
$975$	0	0
$976$	1.80664e13	0.637304
$977$	− 4.44178e13i	− 1.55967i	−0.625988	−	0.779833i	$-0.715304\pi$
$977$	− 4.44178e13i	− 1.55967i	0.625988	−	0.779833i	$-0.284696\pi$
$978$	0	0
$979$	−2.52239e13	−0.877588
$980$	0	0
$981$	0	0
$982$	− 4.60356e13i	− 1.57976i
$983$	4.57558e13i	1.56299i	0.623914	+	0.781493i	$0.285541\pi$
$983$	4.57558e13i	1.56299i	−0.623914	+	0.781493i	$0.714459\pi$
$984$	0	0
$985$	0	0
$986$	−1.69443e13	−0.570922
$987$	0	0
$988$	1.39800e13i	0.466769i
$989$	−2.14576e13	−0.713177
$990$	0	0
$991$	1.30469e13	0.429710	0.214855	−	0.976646i	$-0.431072\pi$
$991$	1.30469e13	0.429710	0.214855	+	0.976646i	$0.431072\pi$
$992$	7.74210e12i	0.253838i
$993$	0	0
$994$	−1.26536e13	−0.411127
$995$	0	0
$996$	0	0
$997$	− 5.07578e12i	− 0.162695i	−0.996686	−	0.0813476i	$-0.974078\pi$
$997$	− 5.07578e12i	− 0.162695i	0.996686	−	0.0813476i	$-0.0259224\pi$
$998$	− 1.99742e13i	− 0.637356i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.b.m.199.4		6
3.2	odd	2		25.10.b.c.24.3		6
5.2	odd	4		225.10.a.p.1.1		3
5.3	odd	4		225.10.a.m.1.3		3
5.4	even	2	inner	225.10.b.m.199.3		6
12.11	even	2		400.10.c.q.49.4		6
15.2	even	4		25.10.a.c.1.3	✓	3
15.8	even	4		25.10.a.d.1.1	yes	3
15.14	odd	2		25.10.b.c.24.4		6
60.23	odd	4		400.10.a.u.1.2		3
60.47	odd	4		400.10.a.y.1.2		3
60.59	even	2		400.10.c.q.49.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.3	✓	3	15.2	even	4
25.10.a.d.1.1	yes	3	15.8	even	4
25.10.b.c.24.3		6	3.2	odd	2
25.10.b.c.24.4		6	15.14	odd	2
225.10.a.m.1.3		3	5.3	odd	4
225.10.a.p.1.1		3	5.2	odd	4
225.10.b.m.199.3		6	5.4	even	2	inner
225.10.b.m.199.4		6	1.1	even	1	trivial
400.10.a.u.1.2		3	60.23	odd	4
400.10.a.y.1.2		3	60.47	odd	4
400.10.c.q.49.3		6	60.59	even	2
400.10.c.q.49.4		6	12.11	even	2

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+20.2014i q^{2} +103.903 q^{4} -4010.25i q^{7} +12442.1i q^{8} +O(q^{10})\)	\(q+20.2014i q^{2} +103.903 q^{4} -4010.25i q^{7} +12442.1i q^{8} +42110.0 q^{11} -123743. i q^{13} +81012.8 q^{14} -198150. q^{16} -319945. i q^{17} -1.08733e6 q^{19} +850682. i q^{22} +1.50672e6i q^{23} +2.49979e6 q^{26} -416676. i q^{28} -2.62160e6 q^{29} +3.27023e6 q^{31} +2.36745e6i q^{32} +6.46335e6 q^{34} -2.51034e6i q^{37} -2.19655e7i q^{38} -2.95349e7 q^{41} +1.42413e7i q^{43} +4.37534e6 q^{44} -3.04378e7 q^{46} -1.35318e6i q^{47} +2.42715e7 q^{49} -1.28573e7i q^{52} +9.73342e7i q^{53} +4.98960e7 q^{56} -5.29599e7i q^{58} -7.48924e6 q^{59} -9.11752e7 q^{61} +6.60633e7i q^{62} -1.49279e8 q^{64} +2.94376e8i q^{67} -3.32432e7i q^{68} -1.56193e8 q^{71} +2.82539e8i q^{73} +5.07124e7 q^{74} -1.12976e8 q^{76} -1.68872e8i q^{77} +5.55294e8 q^{79} -5.96647e8i q^{82} +6.48378e6i q^{83} -2.87694e8 q^{86} +5.23937e8i q^{88} -5.99001e8 q^{89} -4.96242e8 q^{91} +1.56552e8i q^{92} +2.73361e7 q^{94} +9.25317e8i q^{97} +4.90319e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 682 q^{4}+O(q^{10}) \) 6 * q - 682 * q^4	\( 6 q - 682 q^{4} + 109398 q^{11} + 545844 q^{14} - 1398494 q^{16} - 1637690 q^{19} + 2560008 q^{26} + 4350960 q^{29} + 8548132 q^{31} + 13677926 q^{34} - 11852622 q^{41} - 43991406 q^{44} + 28446492 q^{46} + 12907858 q^{49} + 339076260 q^{56} + 11341920 q^{59} + 250613852 q^{61} - 335084802 q^{64} - 595101192 q^{71} - 503014716 q^{74} + 178829730 q^{76} + 620050340 q^{79} - 67894392 q^{86} - 2207720070 q^{89} + 2366375312 q^{91} - 3455782264 q^{94}+O(q^{100}) \) 6 * q - 682 * q^4 + 109398 * q^11 + 545844 * q^14 - 1398494 * q^16 - 1637690 * q^19 + 2560008 * q^26 + 4350960 * q^29 + 8548132 * q^31 + 13677926 * q^34 - 11852622 * q^41 - 43991406 * q^44 + 28446492 * q^46 + 12907858 * q^49 + 339076260 * q^56 + 11341920 * q^59 + 250613852 * q^61 - 335084802 * q^64 - 595101192 * q^71 - 503014716 * q^74 + 178829730 * q^76 + 620050340 * q^79 - 67894392 * q^86 - 2207720070 * q^89 + 2366375312 * q^91 - 3455782264 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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