LMFDB - Embedded newform 225.10.a.m.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 652x + 4000 $ x^3 - x^2 - 652*x + 4000
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.48955$ 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)

$f(q)$	$=$	$q+20.2014 q^{2} -103.903 q^{4} -4010.25 q^{7} -12442.1 q^{8} +O(q^{10})$	$q+20.2014 q^{2} -103.903 q^{4} -4010.25 q^{7} -12442.1 q^{8} +42110.0 q^{11} +123743. q^{13} -81012.8 q^{14} -198150. q^{16} -319945. q^{17} +1.08733e6 q^{19} +850682. q^{22} -1.50672e6 q^{23} +2.49979e6 q^{26} +416676. q^{28} +2.62160e6 q^{29} +3.27023e6 q^{31} +2.36745e6 q^{32} -6.46335e6 q^{34} -2.51034e6 q^{37} +2.19655e7 q^{38} -2.95349e7 q^{41} -1.42413e7 q^{43} -4.37534e6 q^{44} -3.04378e7 q^{46} -1.35318e6 q^{47} -2.42715e7 q^{49} -1.28573e7 q^{52} -9.73342e7 q^{53} +4.98960e7 q^{56} +5.29599e7 q^{58} +7.48924e6 q^{59} -9.11752e7 q^{61} +6.60633e7 q^{62} +1.49279e8 q^{64} +2.94376e8 q^{67} +3.32432e7 q^{68} -1.56193e8 q^{71} -2.82539e8 q^{73} -5.07124e7 q^{74} -1.12976e8 q^{76} -1.68872e8 q^{77} -5.55294e8 q^{79} -5.96647e8 q^{82} -6.48378e6 q^{83} -2.87694e8 q^{86} -5.23937e8 q^{88} +5.99001e8 q^{89} -4.96242e8 q^{91} +1.56552e8 q^{92} -2.73361e7 q^{94} +9.25317e8 q^{97} -4.90319e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 33 q^{2} + 341 q^{4} + 5258 q^{7} + 105 q^{8}+O(q^{10}) $ 3 * q - 33 * q^2 + 341 * q^4 + 5258 * q^7 + 105 * q^8	$ 3 q - 33 q^{2} + 341 q^{4} + 5258 q^{7} + 105 q^{8} + 54699 q^{11} + 215884 q^{13} - 272922 q^{14} - 699247 q^{16} - 334983 q^{17} + 818845 q^{19} + 72761 q^{22} - 3526854 q^{23} + 1280004 q^{26} - 428134 q^{28} - 2175480 q^{29} + 4274066 q^{31} + 9464577 q^{32} - 6838963 q^{34} - 10305042 q^{37} + 24180495 q^{38} - 5926311 q^{41} - 24429956 q^{43} + 21995703 q^{44} + 14223246 q^{46} - 66858708 q^{47} - 6453929 q^{49} - 57862932 q^{52} - 132620514 q^{53} + 169538130 q^{56} + 201908320 q^{58} - 5670960 q^{59} + 125306926 q^{61} + 39831174 q^{62} + 167542401 q^{64} + 88829483 q^{67} + 71162559 q^{68} - 297550596 q^{71} + 181321729 q^{73} + 251507358 q^{74} + 89414865 q^{76} - 561214086 q^{77} - 310025170 q^{79} - 1368322979 q^{82} + 731088801 q^{83} - 33947196 q^{86} - 943671285 q^{88} + 1103860035 q^{89} + 1183187656 q^{91} + 190024242 q^{92} + 1727891132 q^{94} - 332236842 q^{97} - 457927581 q^{98}+O(q^{100}) $ 3 * q - 33 * q^2 + 341 * q^4 + 5258 * q^7 + 105 * q^8 + 54699 * q^11 + 215884 * q^13 - 272922 * q^14 - 699247 * q^16 - 334983 * q^17 + 818845 * q^19 + 72761 * q^22 - 3526854 * q^23 + 1280004 * q^26 - 428134 * q^28 - 2175480 * q^29 + 4274066 * q^31 + 9464577 * q^32 - 6838963 * q^34 - 10305042 * q^37 + 24180495 * q^38 - 5926311 * q^41 - 24429956 * q^43 + 21995703 * q^44 + 14223246 * q^46 - 66858708 * q^47 - 6453929 * q^49 - 57862932 * q^52 - 132620514 * q^53 + 169538130 * q^56 + 201908320 * q^58 - 5670960 * q^59 + 125306926 * q^61 + 39831174 * q^62 + 167542401 * q^64 + 88829483 * q^67 + 71162559 * q^68 - 297550596 * q^71 + 181321729 * q^73 + 251507358 * q^74 + 89414865 * q^76 - 561214086 * q^77 - 310025170 * q^79 - 1368322979 * q^82 + 731088801 * q^83 - 33947196 * q^86 - 943671285 * q^88 + 1103860035 * q^89 + 1183187656 * q^91 + 190024242 * q^92 + 1727891132 * q^94 - 332236842 * q^97 - 457927581 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	20.2014	0.892785	0.446393	−	0.894837i	$-0.352709\pi$
$2$	20.2014	0.892785	0.446393	+	0.894837i	$0.352709\pi$
$3$	0	0
$3$	0	0
$4$	−103.903	−0.202935
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4010.25	−0.631292	−0.315646	−	0.948877i	$-0.602221\pi$
$7$	−4010.25	−0.631292	−0.315646	+	0.948877i	$0.602221\pi$
$8$	−12442.1	−1.07396
$9$	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.	1.20165	0.600824	−	0.799382i	$-0.294839\pi$
$13$	123743.	1.20165	0.600824	+	0.799382i	$0.294839\pi$
$14$	−81012.8	−0.563608
$15$	0	0
$16$	−198150.	−0.755883
$17$	−319945.	−0.929085	−0.464543	−	0.885551i	$-0.653781\pi$
$17$	−319945.	−0.929085	−0.464543	+	0.885551i	$0.653781\pi$
$18$	0	0
$19$	1.08733e6	1.91412	0.957059	−	0.289893i	$-0.0936197\pi$
$19$	1.08733e6	1.91412	0.957059	+	0.289893i	$0.0936197\pi$
$20$	0	0
$21$	0	0
$22$	850682.	0.774221
$23$	−1.50672e6	−1.12268	−0.561341	−	0.827585i	$-0.689714\pi$
$23$	−1.50672e6	−1.12268	−0.561341	+	0.827585i	$0.689714\pi$
$24$	0	0
$25$	0	0
$26$	2.49979e6	1.07281
$27$	0	0
$28$	416676.	0.128111
$29$	2.62160e6	0.688295	0.344148	−	0.938916i	$-0.388168\pi$
$29$	2.62160e6	0.688295	0.344148	+	0.938916i	$0.388168\pi$
$30$	0	0
$31$	3.27023e6	0.635991	0.317996	−	0.948092i	$-0.396990\pi$
$31$	3.27023e6	0.635991	0.317996	+	0.948092i	$0.396990\pi$
$32$	2.36745e6	0.399122
$33$	0	0
$34$	−6.46335e6	−0.829473
$35$	0	0
$36$	0	0
$37$	−2.51034e6	−0.220204	−0.110102	−	0.993920i	$-0.535118\pi$
$37$	−2.51034e6	−0.220204	−0.110102	+	0.993920i	$0.535118\pi$
$38$	2.19655e7	1.70890
$39$	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
$43$	−1.42413e7	−0.635244	−0.317622	−	0.948217i	$-0.602884\pi$
$43$	−1.42413e7	−0.635244	−0.317622	+	0.948217i	$0.602884\pi$
$44$	−4.37534e6	−0.175985
$45$	0	0
$46$	−3.04378e7	−1.00231
$47$	−1.35318e6	−0.0404496	−0.0202248	−	0.999795i	$-0.506438\pi$
$47$	−1.35318e6	−0.0404496	−0.0202248	+	0.999795i	$0.506438\pi$
$48$	0	0
$49$	−2.42715e7	−0.601470
$50$	0	0
$51$	0	0
$52$	−1.28573e7	−0.243856
$53$	−9.73342e7	−1.69443	−0.847216	−	0.531249i	$-0.821723\pi$
$53$	−9.73342e7	−1.69443	−0.847216	+	0.531249i	$0.821723\pi$
$54$	0	0
$55$	0	0
$56$	4.98960e7	0.677984
$57$	0	0
$58$	5.29599e7	0.614500
$59$	7.48924e6	0.0804644	0.0402322	−	0.999190i	$-0.487190\pi$
$59$	7.48924e6	0.0804644	0.0402322	+	0.999190i	$0.487190\pi$
$60$	0	0
$61$	−9.11752e7	−0.843126	−0.421563	−	0.906799i	$-0.638518\pi$
$61$	−9.11752e7	−0.843126	−0.421563	+	0.906799i	$0.638518\pi$
$62$	6.60633e7	0.567803
$63$	0	0
$64$	1.49279e8	1.11221
$65$	0	0
$66$	0	0
$67$	2.94376e8	1.78470	0.892350	−	0.451344i	$-0.149055\pi$
$67$	2.94376e8	1.78470	0.892350	+	0.451344i	$0.149055\pi$
$68$	3.32432e7	0.188544
$69$	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
$73$	−2.82539e8	−1.16446	−0.582232	−	0.813023i	$-0.697820\pi$
$73$	−2.82539e8	−1.16446	−0.582232	+	0.813023i	$0.697820\pi$
$74$	−5.07124e7	−0.196594
$75$	0	0
$76$	−1.12976e8	−0.388441
$77$	−1.68872e8	−0.547455
$78$	0	0
$79$	−5.55294e8	−1.60399	−0.801994	−	0.597332i	$-0.796228\pi$
$79$	−5.55294e8	−1.60399	−0.801994	+	0.597332i	$0.796228\pi$
$80$	0	0
$81$	0	0
$82$	−5.96647e8	−1.45732
$83$	−6.48378e6	−0.0149960	−0.00749802	−	0.999972i	$-0.502387\pi$
$83$	−6.48378e6	−0.0149960	−0.00749802	+	0.999972i	$0.502387\pi$
$84$	0	0
$85$	0	0
$86$	−2.87694e8	−0.567137
$87$	0	0
$88$	−5.23937e8	−0.931338
$89$	5.99001e8	1.01198	0.505990	−	0.862539i	$-0.331127\pi$
$89$	5.99001e8	1.01198	0.505990	+	0.862539i	$0.331127\pi$
$90$	0	0
$91$	−4.96242e8	−0.758591
$92$	1.56552e8	0.227831
$93$	0	0
$94$	−2.73361e7	−0.0361128
$95$	0	0
$96$	0	0
$97$	9.25317e8	1.06125	0.530625	−	0.847606i	$-0.321957\pi$
$97$	9.25317e8	1.06125	0.530625	+	0.847606i	$0.321957\pi$
$98$	−4.90319e8	−0.536984
$99$	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
$103$	1.60441e8	0.140458	0.0702292	−	0.997531i	$-0.477627\pi$
$103$	1.60441e8	0.140458	0.0702292	+	0.997531i	$0.477627\pi$
$104$	−1.53963e9	−1.29052
$105$	0	0
$106$	−1.96629e9	−1.51276
$107$	−9.60457e8	−0.708355	−0.354178	−	0.935178i	$-0.615239\pi$
$107$	−9.60457e8	−0.708355	−0.354178	+	0.935178i	$0.615239\pi$
$108$	0	0
$109$	9.98912e8	0.677810	0.338905	−	0.940821i	$-0.389943\pi$
$109$	9.98912e8	0.677810	0.338905	+	0.940821i	$0.389943\pi$
$110$	0	0
$111$	0	0
$112$	7.94632e8	0.477183
$113$	−2.50705e9	−1.44647	−0.723236	−	0.690601i	$-0.757346\pi$
$113$	−2.50705e9	−1.44647	−0.723236	+	0.690601i	$0.757346\pi$
$114$	0	0
$115$	0	0
$116$	−2.72391e8	−0.139679
$117$	0	0
$118$	1.51293e8	0.0718374
$119$	1.28306e9	0.586524
$120$	0	0
$121$	−5.84695e8	−0.247968
$122$	−1.84187e9	−0.752730
$123$	0	0
$124$	−3.39786e8	−0.129065
$125$	0	0
$126$	0	0
$127$	−2.47541e9	−0.844364	−0.422182	−	0.906511i	$-0.638736\pi$
$127$	−2.47541e9	−0.844364	−0.422182	+	0.906511i	$0.638736\pi$
$128$	1.80351e9	0.593845
$129$	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
$133$	−4.36045e9	−1.20837
$134$	5.94680e9	1.59335
$135$	0	0
$136$	3.98079e9	0.997802
$137$	−4.48594e8	−0.108796	−0.0543978	−	0.998519i	$-0.517324\pi$
$137$	−4.48594e8	−0.108796	−0.0543978	+	0.998519i	$0.517324\pi$
$138$	0	0
$139$	4.48415e9	1.01886	0.509429	−	0.860513i	$-0.329857\pi$
$139$	4.48415e9	1.01886	0.509429	+	0.860513i	$0.329857\pi$
$140$	0	0
$141$	0	0
$142$	−3.15532e9	−0.651247
$143$	5.21084e9	1.04207
$144$	0	0
$145$	0	0
$146$	−5.70769e9	−1.03962
$147$	0	0
$148$	2.60831e8	0.0446870
$149$	2.20480e9	0.366463	0.183232	−	0.983070i	$-0.441344\pi$
$149$	2.20480e9	0.366463	0.183232	+	0.983070i	$0.441344\pi$
$150$	0	0
$151$	−3.21248e9	−0.502857	−0.251428	−	0.967876i	$-0.580900\pi$
$151$	−3.21248e9	−0.502857	−0.251428	+	0.967876i	$0.580900\pi$
$152$	−1.35286e10	−2.05569
$153$	0	0
$154$	−3.41145e9	−0.488760
$155$	0	0
$156$	0	0
$157$	−1.08870e10	−1.43007	−0.715036	−	0.699088i	$-0.753590\pi$
$157$	−1.08870e10	−1.43007	−0.715036	+	0.699088i	$0.753590\pi$
$158$	−1.12177e10	−1.43202
$159$	0	0
$160$	0	0
$161$	6.04232e9	0.708740
$162$	0	0
$163$	1.19994e10	1.33142	0.665708	−	0.746212i	$-0.268130\pi$
$163$	1.19994e10	1.33142	0.665708	+	0.746212i	$0.268130\pi$
$164$	3.06875e9	0.331257
$165$	0	0
$166$	−1.30982e8	−0.0133882
$167$	−9.68608e9	−0.963660	−0.481830	−	0.876265i	$-0.660028\pi$
$167$	−9.68608e9	−0.963660	−0.481830	+	0.876265i	$0.660028\pi$
$168$	0	0
$169$	4.70793e9	0.443956
$170$	0	0
$171$	0	0
$172$	1.47971e9	0.128913
$173$	−7.35665e9	−0.624414	−0.312207	−	0.950014i	$-0.601068\pi$
$173$	−7.35665e9	−0.624414	−0.312207	+	0.950014i	$0.601068\pi$
$174$	0	0
$175$	0	0
$176$	−8.34410e9	−0.655500
$177$	0	0
$178$	1.21007e10	0.903481
$179$	2.00351e9	0.145866	0.0729329	−	0.997337i	$-0.476764\pi$
$179$	2.00351e9	0.145866	0.0729329	+	0.997337i	$0.476764\pi$
$180$	0	0
$181$	5.63414e9	0.390188	0.195094	−	0.980785i	$-0.437499\pi$
$181$	5.63414e9	0.390188	0.195094	+	0.980785i	$0.437499\pi$
$182$	−1.00248e10	−0.677258
$183$	0	0
$184$	1.87467e10	1.20572
$185$	0	0
$186$	0	0
$187$	−1.34729e10	−0.805701
$188$	1.40599e8	0.00820864
$189$	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
$193$	−3.16327e10	−1.64107	−0.820536	−	0.571594i	$-0.806325\pi$
$193$	−3.16327e10	−1.64107	−0.820536	+	0.571594i	$0.806325\pi$
$194$	1.86927e10	0.947469
$195$	0	0
$196$	2.52187e9	0.122059
$197$	−2.59858e10	−1.22924	−0.614621	−	0.788822i	$-0.710691\pi$
$197$	−2.59858e10	−1.22924	−0.614621	+	0.788822i	$0.710691\pi$
$198$	0	0
$199$	1.05766e10	0.478088	0.239044	−	0.971009i	$-0.423166\pi$
$199$	1.05766e10	0.478088	0.239044	+	0.971009i	$0.423166\pi$
$200$	0	0
$201$	0	0
$202$	−1.93723e10	−0.818654
$203$	−1.05133e10	−0.434515
$204$	0	0
$205$	0	0
$206$	3.24113e9	0.125399
$207$	0	0
$208$	−2.45198e10	−0.908304
$209$	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	$0.398672\pi$
$212$	1.01133e10	0.343859
$213$	0	0
$214$	−1.94026e10	−0.632409
$215$	0	0
$216$	0	0
$217$	−1.31145e10	−0.401496
$218$	2.01794e10	0.605139
$219$	0	0
$220$	0	0
$221$	−3.95911e10	−1.11643
$222$	0	0
$223$	4.44522e10	1.20371	0.601855	−	0.798606i	$-0.294429\pi$
$223$	4.44522e10	1.20371	0.601855	+	0.798606i	$0.294429\pi$
$224$	−9.49405e9	−0.251962
$225$	0	0
$226$	−5.06460e10	−1.29139
$227$	−5.59677e10	−1.39901	−0.699505	−	0.714627i	$-0.746596\pi$
$227$	−5.59677e10	−1.39901	−0.699505	+	0.714627i	$0.746596\pi$
$228$	0	0
$229$	−1.47705e9	−0.0354923	−0.0177462	−	0.999843i	$-0.505649\pi$
$229$	−1.47705e9	−0.0354923	−0.0177462	+	0.999843i	$0.505649\pi$
$230$	0	0
$231$	0	0
$232$	−3.26182e10	−0.739203
$233$	7.83279e9	0.174106	0.0870532	−	0.996204i	$-0.472255\pi$
$233$	7.83279e9	0.174106	0.0870532	+	0.996204i	$0.472255\pi$
$234$	0	0
$235$	0	0
$236$	−7.78152e8	−0.0163290
$237$	0	0
$238$	2.59197e10	0.523640
$239$	−5.56371e10	−1.10300	−0.551498	−	0.834176i	$-0.685943\pi$
$239$	−5.56371e10	−1.10300	−0.551498	+	0.834176i	$0.685943\pi$
$240$	0	0
$241$	−1.16053e10	−0.221606	−0.110803	−	0.993842i	$-0.535342\pi$
$241$	−1.16053e10	−0.221606	−0.110803	+	0.993842i	$0.535342\pi$
$242$	−1.18117e10	−0.221382
$243$	0	0
$244$	9.47334e9	0.171100
$245$	0	0
$246$	0	0
$247$	1.34549e11	2.30009
$248$	−4.06886e10	−0.683030
$249$	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
$253$	−6.34479e10	−0.973587
$254$	−5.00067e10	−0.753836
$255$	0	0
$256$	−3.99972e10	−0.582036
$257$	3.67735e10	0.525818	0.262909	−	0.964821i	$-0.415318\pi$
$257$	3.67735e10	0.525818	0.262909	+	0.964821i	$0.415318\pi$
$258$	0	0
$259$	1.00671e10	0.139013
$260$	0	0
$261$	0	0
$262$	−3.88680e10	−0.509609
$263$	−1.33758e11	−1.72392	−0.861962	−	0.506974i	$-0.830764\pi$
$263$	−1.33758e11	−1.72392	−0.861962	+	0.506974i	$0.830764\pi$
$264$	0	0
$265$	0	0
$266$	−8.80873e10	−1.07881
$267$	0	0
$268$	−3.05864e10	−0.362178
$269$	1.42461e11	1.65886	0.829429	−	0.558612i	$-0.188666\pi$
$269$	1.42461e11	1.65886	0.829429	+	0.558612i	$0.188666\pi$
$270$	0	0
$271$	−1.11046e11	−1.25067	−0.625333	−	0.780358i	$-0.715037\pi$
$271$	−1.11046e11	−1.25067	−0.625333	+	0.780358i	$0.715037\pi$
$272$	6.33972e10	0.702279
$273$	0	0
$274$	−9.06224e9	−0.0971310
$275$	0	0
$276$	0	0
$277$	2.80726e10	0.286500	0.143250	−	0.989687i	$-0.454245\pi$
$277$	2.80726e10	0.286500	0.143250	+	0.989687i	$0.454245\pi$
$278$	9.05861e10	0.909620
$279$	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
$283$	−1.09950e11	−1.01895	−0.509477	−	0.860484i	$-0.670161\pi$
$283$	−1.09950e11	−1.01895	−0.509477	+	0.860484i	$0.670161\pi$
$284$	1.62288e10	0.148032
$285$	0	0
$286$	1.05266e11	0.930341
$287$	1.18442e11	1.03048
$288$	0	0
$289$	−1.62229e10	−0.136801
$290$	0	0
$291$	0	0
$292$	2.93566e10	0.236310
$293$	2.30453e11	1.82675	0.913373	−	0.407124i	$-0.133468\pi$
$293$	2.30453e11	1.82675	0.913373	+	0.407124i	$0.133468\pi$
$294$	0	0
$295$	0	0
$296$	3.12339e10	0.236490
$297$	0	0
$298$	4.45400e10	0.327173
$299$	−1.86446e11	−1.34907
$300$	0	0
$301$	5.71111e10	0.401025
$302$	−6.48966e10	−0.448943
$303$	0	0
$304$	−2.15454e11	−1.44685
$305$	0	0
$306$	0	0
$307$	6.85036e10	0.440140	0.220070	−	0.975484i	$-0.429371\pi$
$307$	6.85036e10	0.440140	0.220070	+	0.975484i	$0.429371\pi$
$308$	1.75462e10	0.111098
$309$	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
$313$	−6.15202e10	−0.362300	−0.181150	−	0.983455i	$-0.557982\pi$
$313$	−6.15202e10	−0.362300	−0.181150	+	0.983455i	$0.557982\pi$
$314$	−2.19932e11	−1.27675
$315$	0	0
$316$	5.76965e10	0.325505
$317$	2.25932e11	1.25664	0.628320	−	0.777955i	$-0.283743\pi$
$317$	2.25932e11	1.25664	0.628320	+	0.777955i	$0.283743\pi$
$318$	0	0
$319$	1.10395e11	0.596888
$320$	0	0
$321$	0	0
$322$	1.22063e11	0.632753
$323$	−3.47885e11	−1.77838
$324$	0	0
$325$	0	0
$326$	2.42404e11	1.18867
$327$	0	0
$328$	3.67476e11	1.75306
$329$	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	$0.561514\pi$
$332$	6.73682e8	0.00304322
$333$	0	0
$334$	−1.95673e11	−0.860341
$335$	0	0
$336$	0	0
$337$	−2.19457e11	−0.926862	−0.463431	−	0.886133i	$-0.653382\pi$
$337$	−2.19457e11	−0.926862	−0.463431	+	0.886133i	$0.653382\pi$
$338$	9.51069e10	0.396357
$339$	0	0
$340$	0	0
$341$	1.37710e11	0.551530
$342$	0	0
$343$	2.59163e11	1.01100
$344$	1.77191e11	0.682228
$345$	0	0
$346$	−1.48615e11	−0.557467
$347$	1.85960e11	0.688551	0.344276	−	0.938869i	$-0.388125\pi$
$347$	1.85960e11	0.688551	0.344276	+	0.938869i	$0.388125\pi$
$348$	0	0
$349$	2.73237e11	0.985881	0.492940	−	0.870063i	$-0.335922\pi$
$349$	2.73237e11	0.985881	0.492940	+	0.870063i	$0.335922\pi$
$350$	0	0
$351$	0	0
$352$	9.96932e10	0.346117
$353$	3.02861e11	1.03814	0.519072	−	0.854731i	$-0.326278\pi$
$353$	3.02861e11	1.03814	0.519072	+	0.854731i	$0.326278\pi$
$354$	0	0
$355$	0	0
$356$	−6.22378e10	−0.205366
$357$	0	0
$358$	4.04738e10	0.130227
$359$	−2.47660e11	−0.786919	−0.393460	−	0.919342i	$-0.628722\pi$
$359$	−2.47660e11	−0.786919	−0.393460	+	0.919342i	$0.628722\pi$
$360$	0	0
$361$	8.59591e11	2.66385
$362$	1.13818e11	0.348354
$363$	0	0
$364$	5.15609e10	0.153944
$365$	0	0
$366$	0	0
$367$	2.66354e11	0.766412	0.383206	−	0.923663i	$-0.374820\pi$
$367$	2.66354e11	0.766412	0.383206	+	0.923663i	$0.374820\pi$
$368$	2.98556e11	0.848616
$369$	0	0
$370$	0	0
$371$	3.90335e11	1.06968
$372$	0	0
$373$	−5.20850e11	−1.39323	−0.696616	−	0.717445i	$-0.745312\pi$
$373$	−5.20850e11	−1.39323	−0.696616	+	0.717445i	$0.745312\pi$
$374$	−2.72172e11	−0.719318
$375$	0	0
$376$	1.68364e10	0.0434414
$377$	3.24405e11	0.827088
$378$	0	0
$379$	3.28308e10	0.0817344	0.0408672	−	0.999165i	$-0.486988\pi$
$379$	3.28308e10	0.0817344	0.0408672	+	0.999165i	$0.486988\pi$
$380$	0	0
$381$	0	0
$382$	−1.85232e11	−0.445072
$383$	7.15293e10	0.169859	0.0849297	−	0.996387i	$-0.472933\pi$
$383$	7.15293e10	0.169859	0.0849297	+	0.996387i	$0.472933\pi$
$384$	0	0
$385$	0	0
$386$	−6.39025e11	−1.46513
$387$	0	0
$388$	−9.61429e10	−0.215365
$389$	2.02681e11	0.448787	0.224394	−	0.974499i	$-0.427960\pi$
$389$	2.02681e11	0.448787	0.224394	+	0.974499i	$0.427960\pi$
$390$	0	0
$391$	4.82067e11	1.04307
$392$	3.01989e11	0.645956
$393$	0	0
$394$	−5.24949e11	−1.09745
$395$	0	0
$396$	0	0
$397$	1.63266e10	0.0329867	0.0164934	−	0.999864i	$-0.494750\pi$
$397$	1.63266e10	0.0329867	0.0164934	+	0.999864i	$0.494750\pi$
$398$	2.13663e11	0.426830
$399$	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
$403$	4.04670e11	0.764237
$404$	9.96383e10	0.186085
$405$	0	0
$406$	−2.12383e11	−0.387929
$407$	−1.05710e11	−0.190960
$408$	0	0
$409$	−4.24628e11	−0.750332	−0.375166	−	0.926958i	$-0.622414\pi$
$409$	−4.24628e11	−0.750332	−0.375166	+	0.926958i	$0.622414\pi$
$410$	0	0
$411$	0	0
$412$	−1.66702e10	−0.0285039
$413$	−3.00337e10	−0.0507966
$414$	0	0
$415$	0	0
$416$	2.92956e11	0.479603
$417$	0	0
$418$	9.24969e11	1.48195
$419$	1.26375e10	0.0200308	0.0100154	−	0.999950i	$-0.496812\pi$
$419$	1.26375e10	0.0200308	0.0100154	+	0.999950i	$0.496812\pi$
$420$	0	0
$421$	−3.04545e10	−0.0472479	−0.0236240	−	0.999721i	$-0.507520\pi$
$421$	−3.04545e10	−0.0472479	−0.0236240	+	0.999721i	$0.507520\pi$
$422$	3.64085e11	0.558853
$423$	0	0
$424$	1.21104e12	1.81976
$425$	0	0
$426$	0	0
$427$	3.65635e11	0.532259
$428$	9.97940e10	0.143750
$429$	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
$433$	1.36978e11	0.187265	0.0936324	−	0.995607i	$-0.470152\pi$
$433$	1.36978e11	0.187265	0.0936324	+	0.995607i	$0.470152\pi$
$434$	−2.64931e11	−0.358450
$435$	0	0
$436$	−1.03790e11	−0.137551
$437$	−1.63829e12	−2.14895
$438$	0	0
$439$	7.84981e11	1.00872	0.504358	−	0.863495i	$-0.331729\pi$
$439$	7.84981e11	1.00872	0.504358	+	0.863495i	$0.331729\pi$
$440$	0	0
$441$	0	0
$442$	−7.99797e11	−0.996734
$443$	−8.87799e11	−1.09521	−0.547605	−	0.836737i	$-0.684460\pi$
$443$	−8.87799e11	−1.09521	−0.547605	+	0.836737i	$0.684460\pi$
$444$	0	0
$445$	0	0
$446$	8.97998e11	1.07465
$447$	0	0
$448$	−5.98645e11	−0.702131
$449$	−8.35477e11	−0.970122	−0.485061	−	0.874480i	$-0.661203\pi$
$449$	−8.35477e11	−0.970122	−0.485061	+	0.874480i	$0.661203\pi$
$450$	0	0
$451$	−1.24371e12	−1.41555
$452$	2.60489e11	0.293540
$453$	0	0
$454$	−1.13063e12	−1.24902
$455$	0	0
$456$	0	0
$457$	6.01172e11	0.644727	0.322364	−	0.946616i	$-0.395523\pi$
$457$	6.01172e11	0.644727	0.322364	+	0.946616i	$0.395523\pi$
$458$	−2.98384e10	−0.0316870
$459$	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
$463$	−7.80402e11	−0.789231	−0.394615	−	0.918846i	$-0.629122\pi$
$463$	−7.80402e11	−0.789231	−0.394615	+	0.918846i	$0.629122\pi$
$464$	−5.19469e11	−0.520270
$465$	0	0
$466$	1.58234e11	0.155440
$467$	4.04751e11	0.393788	0.196894	−	0.980425i	$-0.436915\pi$
$467$	4.04751e11	0.393788	0.196894	+	0.980425i	$0.436915\pi$
$468$	0	0
$469$	−1.18052e12	−1.12667
$470$	0	0
$471$	0	0
$472$	−9.31820e10	−0.0864158
$473$	−5.99700e11	−0.550883
$474$	0	0
$475$	0	0
$476$	−1.33313e11	−0.119026
$477$	0	0
$478$	−1.12395e12	−0.984738
$479$	2.06972e12	1.79640	0.898199	−	0.439588i	$-0.144876\pi$
$479$	2.06972e12	1.79640	0.898199	+	0.439588i	$0.144876\pi$
$480$	0	0
$481$	−3.10638e11	−0.264607
$482$	−2.34444e11	−0.197846
$483$	0	0
$484$	6.07514e10	0.0503213
$485$	0	0
$486$	0	0
$487$	−2.41184e11	−0.194298	−0.0971490	−	0.995270i	$-0.530972\pi$
$487$	−2.41184e11	−0.194298	−0.0971490	+	0.995270i	$0.530972\pi$
$488$	1.13441e12	0.905485
$489$	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
$493$	−8.38767e11	−0.639485
$494$	2.71809e12	2.05349
$495$	0	0
$496$	−6.47997e11	−0.480735
$497$	6.26373e11	0.460499
$498$	0	0
$499$	9.88752e11	0.713896	0.356948	−	0.934124i	$-0.383817\pi$
$499$	9.88752e11	0.713896	0.356948	+	0.934124i	$0.383817\pi$
$500$	0	0
$501$	0	0
$502$	6.98916e11	0.491200
$503$	−1.22385e12	−0.852455	−0.426228	−	0.904616i	$-0.640158\pi$
$503$	−1.22385e12	−0.852455	−0.426228	+	0.904616i	$0.640158\pi$
$504$	0	0
$505$	0	0
$506$	−1.28174e12	−0.869204
$507$	0	0
$508$	2.57201e11	0.171351
$509$	−1.58447e12	−1.04629	−0.523146	−	0.852243i	$-0.675242\pi$
$509$	−1.58447e12	−1.04629	−0.523146	+	0.852243i	$0.675242\pi$
$510$	0	0
$511$	1.13305e12	0.735117
$512$	−1.73140e12	−1.11348
$513$	0	0
$514$	7.42877e11	0.469443
$515$	0	0
$516$	0	0
$517$	−5.69823e10	−0.0350778
$518$	2.03369e11	0.124109
$519$	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
$523$	−2.16171e12	−1.26340	−0.631698	−	0.775214i	$-0.717642\pi$
$523$	−2.16171e12	−1.26340	−0.631698	+	0.775214i	$0.717642\pi$
$524$	1.99911e11	0.115837
$525$	0	0
$526$	−2.70210e12	−1.53909
$527$	−1.04630e12	−0.590890
$528$	0	0
$529$	4.69046e11	0.260415
$530$	0	0
$531$	0	0
$532$	4.53062e11	0.245220
$533$	−3.65475e12	−1.96149
$534$	0	0
$535$	0	0
$536$	−3.66265e12	−1.91670
$537$	0	0
$538$	2.87791e12	1.48100
$539$	−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	$0.694957\pi$
$542$	−2.24329e12	−1.11658
$543$	0	0
$544$	−7.57453e11	−0.370818
$545$	0	0
$546$	0	0
$547$	−3.39447e12	−1.62117	−0.810586	−	0.585620i	$-0.800851\pi$
$547$	−3.39447e12	−1.62117	−0.810586	+	0.585620i	$0.800851\pi$
$548$	4.66101e10	0.0220784
$549$	0	0
$550$	0	0
$551$	2.85053e12	1.31748
$552$	0	0
$553$	2.22687e12	1.01259
$554$	5.67107e11	0.255783
$555$	0	0
$556$	−4.65915e11	−0.206762
$557$	−1.74706e12	−0.769060	−0.384530	−	0.923113i	$-0.625636\pi$
$557$	−1.74706e12	−0.769060	−0.384530	+	0.923113i	$0.625636\pi$
$558$	0	0
$559$	−1.76226e12	−0.763340
$560$	0	0
$561$	0	0
$562$	−1.10531e12	−0.467379
$563$	2.58864e12	1.08588	0.542942	−	0.839770i	$-0.317310\pi$
$563$	2.58864e12	1.08588	0.542942	+	0.839770i	$0.317310\pi$
$564$	0	0
$565$	0	0
$566$	−2.22114e12	−0.909707
$567$	0	0
$568$	1.94337e12	0.783407
$569$	1.99294e12	0.797055	0.398527	−	0.917156i	$-0.369521\pi$
$569$	1.99294e12	0.797055	0.398527	+	0.917156i	$0.369521\pi$
$570$	0	0
$571$	3.50761e10	0.0138086	0.00690428	−	0.999976i	$-0.497802\pi$
$571$	3.50761e10	0.0138086	0.00690428	+	0.999976i	$0.497802\pi$
$572$	−5.41420e11	−0.211471
$573$	0	0
$574$	2.39270e12	0.919995
$575$	0	0
$576$	0	0
$577$	−7.75900e11	−0.291417	−0.145708	−	0.989328i	$-0.546546\pi$
$577$	−7.75900e11	−0.291417	−0.145708	+	0.989328i	$0.546546\pi$
$578$	−3.27726e11	−0.122134
$579$	0	0
$580$	0	0
$581$	2.60016e10	0.00946689
$582$	0	0
$583$	−4.09874e12	−1.46941
$584$	3.51538e12	1.25059
$585$	0	0
$586$	4.65548e12	1.63089
$587$	−1.24477e12	−0.432730	−0.216365	−	0.976313i	$-0.569420\pi$
$587$	−1.24477e12	−0.432730	−0.216365	+	0.976313i	$0.569420\pi$
$588$	0	0
$589$	3.55581e12	1.21736
$590$	0	0
$591$	0	0
$592$	4.97424e11	0.166448
$593$	2.57794e11	0.0856103	0.0428052	−	0.999083i	$-0.486371\pi$
$593$	2.57794e11	0.0856103	0.0428052	+	0.999083i	$0.486371\pi$
$594$	0	0
$595$	0	0
$596$	−2.29084e11	−0.0743681
$597$	0	0
$598$	−3.76648e12	−1.20443
$599$	−2.32089e12	−0.736604	−0.368302	−	0.929706i	$-0.620061\pi$
$599$	−2.32089e12	−0.736604	−0.368302	+	0.929706i	$0.620061\pi$
$600$	0	0
$601$	1.78665e12	0.558605	0.279302	−	0.960203i	$-0.409897\pi$
$601$	1.78665e12	0.558605	0.279302	+	0.960203i	$0.409897\pi$
$602$	1.15373e12	0.358029
$603$	0	0
$604$	3.33785e11	0.102047
$605$	0	0
$606$	0	0
$607$	−1.45787e12	−0.435883	−0.217941	−	0.975962i	$-0.569934\pi$
$607$	−1.45787e12	−0.435883	−0.217941	+	0.975962i	$0.569934\pi$
$608$	2.57419e12	0.763966
$609$	0	0
$610$	0	0
$611$	−1.67447e11	−0.0486062
$612$	0	0
$613$	1.42075e12	0.406394	0.203197	−	0.979138i	$-0.434867\pi$
$613$	1.42075e12	0.406394	0.203197	+	0.979138i	$0.434867\pi$
$614$	1.38387e12	0.392950
$615$	0	0
$616$	2.10112e12	0.587946
$617$	−1.20441e12	−0.334573	−0.167286	−	0.985908i	$-0.553500\pi$
$617$	−1.20441e12	−0.334573	−0.167286	+	0.985908i	$0.553500\pi$
$618$	0	0
$619$	−4.91349e12	−1.34519	−0.672593	−	0.740013i	$-0.734819\pi$
$619$	−4.91349e12	−1.34519	−0.672593	+	0.740013i	$0.734819\pi$
$620$	0	0
$621$	0	0
$622$	4.01598e12	1.07581
$623$	−2.40214e12	−0.638856
$624$	0	0
$625$	0	0
$626$	−1.24280e12	−0.323456
$627$	0	0
$628$	1.13118e12	0.290211
$629$	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	$0.304660\pi$
$632$	6.90903e12	1.72262
$633$	0	0
$634$	4.56414e12	1.12191
$635$	0	0
$636$	0	0
$637$	−3.00344e12	−0.722755
$638$	2.23014e12	0.532893
$639$	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
$643$	2.59493e12	0.598654	0.299327	−	0.954151i	$-0.403238\pi$
$643$	2.59493e12	0.598654	0.299327	+	0.954151i	$0.403238\pi$
$644$	−6.27813e11	−0.143828
$645$	0	0
$646$	−7.02777e12	−1.58771
$647$	−5.14811e12	−1.15499	−0.577495	−	0.816394i	$-0.695970\pi$
$647$	−5.14811e12	−1.15499	−0.577495	+	0.816394i	$0.695970\pi$
$648$	0	0
$649$	3.15372e11	0.0697786
$650$	0	0
$651$	0	0
$652$	−1.24676e12	−0.270191
$653$	−4.42559e12	−0.952492	−0.476246	−	0.879312i	$-0.658003\pi$
$653$	−4.42559e12	−0.952492	−0.476246	+	0.879312i	$0.658003\pi$
$654$	0	0
$655$	0	0
$656$	5.85234e12	1.23385
$657$	0	0
$658$	1.09625e11	0.0227977
$659$	−1.09827e12	−0.226842	−0.113421	−	0.993547i	$-0.536181\pi$
$659$	−1.09827e12	−0.226842	−0.113421	+	0.993547i	$0.536181\pi$
$660$	0	0
$661$	7.99232e11	0.162842	0.0814209	−	0.996680i	$-0.474054\pi$
$661$	7.99232e11	0.162842	0.0814209	+	0.996680i	$0.474054\pi$
$662$	−1.69455e12	−0.342919
$663$	0	0
$664$	8.06719e10	0.0161052
$665$	0	0
$666$	0	0
$667$	−3.95000e12	−0.772736
$668$	1.00641e12	0.195560
$669$	0	0
$670$	0	0
$671$	−3.83939e12	−0.731157
$672$	0	0
$673$	7.68445e12	1.44393	0.721963	−	0.691931i	$-0.243240\pi$
$673$	7.68445e12	1.44393	0.721963	+	0.691931i	$0.243240\pi$
$674$	−4.43334e12	−0.827488
$675$	0	0
$676$	−4.89167e11	−0.0900942
$677$	−1.28854e12	−0.235749	−0.117874	−	0.993029i	$-0.537608\pi$
$677$	−1.28854e12	−0.235749	−0.117874	+	0.993029i	$0.537608\pi$
$678$	0	0
$679$	−3.71076e12	−0.669959
$680$	0	0
$681$	0	0
$682$	2.78193e12	0.492398
$683$	8.11444e11	0.142681	0.0713404	−	0.997452i	$-0.477272\pi$
$683$	8.11444e11	0.142681	0.0713404	+	0.997452i	$0.477272\pi$
$684$	0	0
$685$	0	0
$686$	5.23546e12	0.902602
$687$	0	0
$688$	2.82191e12	0.480170
$689$	−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	$0.439421\pi$
$692$	7.64375e11	0.126715
$693$	0	0
$694$	3.75665e12	0.614728
$695$	0	0
$696$	0	0
$697$	9.44955e12	1.51657
$698$	5.51977e12	0.880180
$699$	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
$703$	−2.72956e12	−0.421496
$704$	6.28612e12	0.964508
$705$	0	0
$706$	6.11822e12	0.926839
$707$	3.84566e12	0.578874
$708$	0	0
$709$	−6.12354e12	−0.910112	−0.455056	−	0.890463i	$-0.650381\pi$
$709$	−6.12354e12	−0.910112	−0.455056	+	0.890463i	$0.650381\pi$
$710$	0	0
$711$	0	0
$712$	−7.45283e12	−1.08683
$713$	−4.92732e12	−0.714016
$714$	0	0
$715$	0	0
$716$	−2.08170e11	−0.0296012
$717$	0	0
$718$	−5.00307e12	−0.702550
$719$	2.30376e11	0.0321483	0.0160741	−	0.999871i	$-0.494883\pi$
$719$	2.30376e11	0.0321483	0.0160741	+	0.999871i	$0.494883\pi$
$720$	0	0
$721$	−6.43408e11	−0.0886702
$722$	1.73650e13	2.37824
$723$	0	0
$724$	−5.85402e11	−0.0791827
$725$	0	0
$726$	0	0
$727$	9.25894e12	1.22930	0.614648	−	0.788801i	$-0.289298\pi$
$727$	9.25894e12	1.22930	0.614648	+	0.788801i	$0.289298\pi$
$728$	6.17430e12	0.814698
$729$	0	0
$730$	0	0
$731$	4.55643e12	0.590196
$732$	0	0
$733$	−7.22531e12	−0.924461	−0.462231	−	0.886760i	$-0.652951\pi$
$733$	−7.22531e12	−0.924461	−0.462231	+	0.886760i	$0.652951\pi$
$734$	5.38074e12	0.684241
$735$	0	0
$736$	−3.56707e12	−0.448086
$737$	1.23962e13	1.54769
$738$	0	0
$739$	−1.24725e13	−1.53834	−0.769170	−	0.639044i	$-0.779330\pi$
$739$	−1.24725e13	−1.53834	−0.769170	+	0.639044i	$0.779330\pi$
$740$	0	0
$741$	0	0
$742$	7.88531e12	0.954996
$743$	1.15645e13	1.39212	0.696058	−	0.717986i	$-0.254936\pi$
$743$	1.15645e13	1.39212	0.696058	+	0.717986i	$0.254936\pi$
$744$	0	0
$745$	0	0
$746$	−1.05219e13	−1.24386
$747$	0	0
$748$	1.39987e12	0.163505
$749$	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	$0.568207\pi$
$752$	2.68132e11	0.0305752
$753$	0	0
$754$	6.55344e12	0.738412
$755$	0	0
$756$	0	0
$757$	−4.86635e11	−0.0538607	−0.0269303	−	0.999637i	$-0.508573\pi$
$757$	−4.86635e11	−0.0538607	−0.0269303	+	0.999637i	$0.508573\pi$
$758$	6.63228e11	0.0729712
$759$	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
$763$	−4.00589e12	−0.427896
$764$	9.52709e11	0.101167
$765$	0	0
$766$	1.44499e12	0.151648
$767$	9.26745e11	0.0966899
$768$	0	0
$769$	−4.03625e12	−0.416207	−0.208103	−	0.978107i	$-0.566729\pi$
$769$	−4.03625e12	−0.416207	−0.208103	+	0.978107i	$0.566729\pi$
$770$	0	0
$771$	0	0
$772$	3.28672e12	0.333031
$773$	−1.21916e13	−1.22815	−0.614076	−	0.789247i	$-0.710471\pi$
$773$	−1.21916e13	−1.22815	−0.614076	+	0.789247i	$0.710471\pi$
$774$	0	0
$775$	0	0
$776$	−1.15129e13	−1.13974
$777$	0	0
$778$	4.09445e12	0.400671
$779$	−3.21141e13	−3.12447
$780$	0	0
$781$	−6.57728e12	−0.632582
$782$	9.73844e12	0.931235
$783$	0	0
$784$	4.80940e12	0.454641
$785$	0	0
$786$	0	0
$787$	−1.20299e12	−0.111783	−0.0558913	−	0.998437i	$-0.517800\pi$
$787$	−1.20299e12	−0.111783	−0.0558913	+	0.998437i	$0.517800\pi$
$788$	2.69999e12	0.249456
$789$	0	0
$790$	0	0
$791$	1.00539e13	0.913147
$792$	0	0
$793$	−1.12823e13	−1.01314
$794$	3.29821e11	0.0294501
$795$	0	0
$796$	−1.09894e12	−0.0970208
$797$	−1.72611e13	−1.51533	−0.757663	−	0.652646i	$-0.773659\pi$
$797$	−1.72611e13	−1.51533	−0.757663	+	0.652646i	$0.773659\pi$
$798$	0	0
$799$	4.32943e11	0.0375811
$800$	0	0
$801$	0	0
$802$	1.65806e13	1.41520
$803$	−1.18977e13	−1.00982
$804$	0	0
$805$	0	0
$806$	8.17490e12	0.682299
$807$	0	0
$808$	1.19315e13	0.984788
$809$	−7.44408e12	−0.611002	−0.305501	−	0.952192i	$-0.598824\pi$
$809$	−7.44408e12	−0.611002	−0.305501	+	0.952192i	$0.598824\pi$
$810$	0	0
$811$	−9.59955e12	−0.779214	−0.389607	−	0.920981i	$-0.627389\pi$
$811$	−9.59955e12	−0.779214	−0.389607	+	0.920981i	$0.627389\pi$
$812$	1.09235e12	0.0881783
$813$	0	0
$814$	−2.13550e12	−0.170486
$815$	0	0
$816$	0	0
$817$	−1.54849e13	−1.21593
$818$	−8.57808e12	−0.669885
$819$	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
$823$	1.62323e13	1.23333	0.616666	−	0.787225i	$-0.288483\pi$
$823$	1.62323e13	1.23333	0.616666	+	0.787225i	$0.288483\pi$
$824$	−1.99622e12	−0.150847
$825$	0	0
$826$	−6.06724e11	−0.0453504
$827$	1.74898e13	1.30020	0.650099	−	0.759850i	$-0.274727\pi$
$827$	1.74898e13	1.30020	0.650099	+	0.759850i	$0.274727\pi$
$828$	0	0
$829$	−1.04102e13	−0.765535	−0.382768	−	0.923845i	$-0.625029\pi$
$829$	−1.04102e13	−0.765535	−0.382768	+	0.923845i	$0.625029\pi$
$830$	0	0
$831$	0	0
$832$	1.84722e13	1.33649
$833$	7.76555e12	0.558817
$834$	0	0
$835$	0	0
$836$	−4.75742e12	−0.336855
$837$	0	0
$838$	2.55296e11	0.0178832
$839$	2.66219e13	1.85485	0.927426	−	0.374006i	$-0.122016\pi$
$839$	2.66219e13	1.85485	0.927426	+	0.374006i	$0.122016\pi$
$840$	0	0
$841$	−7.63439e12	−0.526250
$842$	−6.15225e11	−0.0421822
$843$	0	0
$844$	−1.87261e12	−0.127030
$845$	0	0
$846$	0	0
$847$	2.34477e12	0.156540
$848$	1.92868e13	1.28079
$849$	0	0
$850$	0	0
$851$	3.78237e12	0.247219
$852$	0	0
$853$	−7.30064e12	−0.472161	−0.236080	−	0.971734i	$-0.575863\pi$
$853$	−7.30064e12	−0.472161	−0.236080	+	0.971734i	$0.575863\pi$
$854$	7.38635e12	0.475192
$855$	0	0
$856$	1.19501e13	0.760747
$857$	1.18598e13	0.751041	0.375520	−	0.926814i	$-0.377464\pi$
$857$	1.18598e13	0.751041	0.375520	+	0.926814i	$0.377464\pi$
$858$	0	0
$859$	1.67692e13	1.05085	0.525427	−	0.850839i	$-0.323906\pi$
$859$	1.67692e13	1.05085	0.525427	+	0.850839i	$0.323906\pi$
$860$	0	0
$861$	0	0
$862$	−8.19834e12	−0.505759
$863$	−6.71596e12	−0.412154	−0.206077	−	0.978536i	$-0.566070\pi$
$863$	−6.71596e12	−0.412154	−0.206077	+	0.978536i	$0.566070\pi$
$864$	0	0
$865$	0	0
$866$	2.76716e12	0.167187
$867$	0	0
$868$	1.36263e12	0.0814776
$869$	−2.33834e13	−1.39098
$870$	0	0
$871$	3.64270e13	2.14458
$872$	−1.24286e13	−0.727943
$873$	0	0
$874$	−3.30959e13	−1.91855
$875$	0	0
$876$	0	0
$877$	−2.79757e13	−1.59692	−0.798458	−	0.602051i	$-0.794351\pi$
$877$	−2.79757e13	−1.59692	−0.798458	+	0.602051i	$0.794351\pi$
$878$	1.58577e13	0.900566
$879$	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
$883$	−2.02048e13	−1.11849	−0.559244	−	0.829003i	$-0.688909\pi$
$883$	−2.02048e13	−1.11849	−0.559244	+	0.829003i	$0.688909\pi$
$884$	4.11362e12	0.226563
$885$	0	0
$886$	−1.79348e13	−0.977788
$887$	−3.19954e13	−1.73553	−0.867763	−	0.496978i	$-0.834443\pi$
$887$	−3.19954e13	−1.73553	−0.867763	+	0.496978i	$0.834443\pi$
$888$	0	0
$889$	9.92701e12	0.533041
$890$	0	0
$891$	0	0
$892$	−4.61870e12	−0.244275
$893$	−1.47135e12	−0.0774254
$894$	0	0
$895$	0	0
$896$	−7.23252e12	−0.374890
$897$	0	0
$898$	−1.68778e13	−0.866110
$899$	8.57323e12	0.437750
$900$	0	0
$901$	3.11416e13	1.57427
$902$	−2.51248e13	−1.26378
$903$	0	0
$904$	3.11930e13	1.55346
$905$	0	0
$906$	0	0
$907$	2.40537e12	0.118018	0.0590091	−	0.998257i	$-0.481206\pi$
$907$	2.40537e12	0.118018	0.0590091	+	0.998257i	$0.481206\pi$
$908$	5.81519e12	0.283908
$909$	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
$913$	−2.73032e11	−0.0130045
$914$	1.21445e13	0.575603
$915$	0	0
$916$	1.53469e11	0.00720263
$917$	7.71582e12	0.360346
$918$	0	0
$919$	3.42677e13	1.58477	0.792383	−	0.610024i	$-0.208840\pi$
$919$	3.42677e13	1.58477	0.792383	+	0.610024i	$0.208840\pi$
$920$	0	0
$921$	0	0
$922$	−3.08692e13	−1.40681
$923$	−1.93278e13	−0.876548
$924$	0	0
$925$	0	0
$926$	−1.57652e13	−0.704614
$927$	0	0
$928$	6.20648e12	0.274713
$929$	2.52782e13	1.11346	0.556730	−	0.830693i	$-0.312056\pi$
$929$	2.52782e13	1.11346	0.556730	+	0.830693i	$0.312056\pi$
$930$	0	0
$931$	−2.63910e13	−1.15129
$932$	−8.13848e11	−0.0353323
$933$	0	0
$934$	8.17655e12	0.351568
$935$	0	0
$936$	0	0
$937$	2.72636e13	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$937$	2.72636e13	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$938$	−2.38482e13	−1.00587
$939$	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
$943$	4.45008e13	1.83259
$944$	−1.48399e12	−0.0608217
$945$	0	0
$946$	−1.21148e13	−0.491820
$947$	9.71128e12	0.392375	0.196188	−	0.980566i	$-0.437144\pi$
$947$	9.71128e12	0.392375	0.196188	+	0.980566i	$0.437144\pi$
$948$	0	0
$949$	−3.49624e13	−1.39927
$950$	0	0
$951$	0	0
$952$	−1.59640e13	−0.629905
$953$	−4.15385e12	−0.163130	−0.0815648	−	0.996668i	$-0.525992\pi$
$953$	−4.15385e12	−0.163130	−0.0815648	+	0.996668i	$0.525992\pi$
$954$	0	0
$955$	0	0
$956$	5.78084e12	0.223836
$957$	0	0
$958$	4.18114e13	1.60380
$959$	1.79897e12	0.0686818
$960$	0	0
$961$	−1.57452e13	−0.595515
$962$	−6.27532e12	−0.236237
$963$	0	0
$964$	1.20582e12	0.0449715
$965$	0	0
$966$	0	0
$967$	1.84544e13	0.678706	0.339353	−	0.940659i	$-0.389792\pi$
$967$	1.84544e13	0.678706	0.339353	+	0.940659i	$0.389792\pi$
$968$	7.27484e12	0.266308
$969$	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
$973$	−1.79826e13	−0.643197
$974$	−4.87226e12	−0.173466
$975$	0	0
$976$	1.80664e13	0.637304
$977$	−4.44178e13	−1.55967	−0.779833	−	0.625988i	$-0.784696\pi$
$977$	−4.44178e13	−1.55967	−0.779833	+	0.625988i	$0.784696\pi$
$978$	0	0
$979$	2.52239e13	0.877588
$980$	0	0
$981$	0	0
$982$	−4.60356e13	−1.57976
$983$	−4.57558e13	−1.56299	−0.781493	−	0.623914i	$-0.785541\pi$
$983$	−4.57558e13	−1.56299	−0.781493	+	0.623914i	$0.785541\pi$
$984$	0	0
$985$	0	0
$986$	−1.69443e13	−0.570922
$987$	0	0
$988$	−1.39800e13	−0.466769
$989$	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	$0.431072\pi$
$992$	7.74210e12	0.253838
$993$	0	0
$994$	1.26536e13	0.411127
$995$	0	0
$996$	0	0
$997$	−5.07578e12	−0.162695	−0.0813476	−	0.996686i	$-0.525922\pi$
$997$	−5.07578e12	−0.162695	−0.0813476	+	0.996686i	$0.525922\pi$
$998$	1.99742e13	0.637356
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.3	✓	3	15.14	odd	2
25.10.a.d.1.1	yes	3	3.2	odd	2
25.10.b.c.24.3		6	15.2	even	4
25.10.b.c.24.4		6	15.8	even	4
225.10.a.m.1.3		3	1.1	even	1	trivial
225.10.a.p.1.1		3	5.4	even	2
225.10.b.m.199.3		6	5.3	odd	4
225.10.b.m.199.4		6	5.2	odd	4
400.10.a.u.1.2		3	12.11	even	2
400.10.a.y.1.2		3	60.59	even	2
400.10.c.q.49.3		6	60.23	odd	4
400.10.c.q.49.4		6	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+20.2014 q^{2} -103.903 q^{4} -4010.25 q^{7} -12442.1 q^{8} +O(q^{10})\)	\(q+20.2014 q^{2} -103.903 q^{4} -4010.25 q^{7} -12442.1 q^{8} +42110.0 q^{11} +123743. q^{13} -81012.8 q^{14} -198150. q^{16} -319945. q^{17} +1.08733e6 q^{19} +850682. q^{22} -1.50672e6 q^{23} +2.49979e6 q^{26} +416676. q^{28} +2.62160e6 q^{29} +3.27023e6 q^{31} +2.36745e6 q^{32} -6.46335e6 q^{34} -2.51034e6 q^{37} +2.19655e7 q^{38} -2.95349e7 q^{41} -1.42413e7 q^{43} -4.37534e6 q^{44} -3.04378e7 q^{46} -1.35318e6 q^{47} -2.42715e7 q^{49} -1.28573e7 q^{52} -9.73342e7 q^{53} +4.98960e7 q^{56} +5.29599e7 q^{58} +7.48924e6 q^{59} -9.11752e7 q^{61} +6.60633e7 q^{62} +1.49279e8 q^{64} +2.94376e8 q^{67} +3.32432e7 q^{68} -1.56193e8 q^{71} -2.82539e8 q^{73} -5.07124e7 q^{74} -1.12976e8 q^{76} -1.68872e8 q^{77} -5.55294e8 q^{79} -5.96647e8 q^{82} -6.48378e6 q^{83} -2.87694e8 q^{86} -5.23937e8 q^{88} +5.99001e8 q^{89} -4.96242e8 q^{91} +1.56552e8 q^{92} -2.73361e7 q^{94} +9.25317e8 q^{97} -4.90319e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 33 q^{2} + 341 q^{4} + 5258 q^{7} + 105 q^{8}+O(q^{10}) \) 3 * q - 33 * q^2 + 341 * q^4 + 5258 * q^7 + 105 * q^8	\( 3 q - 33 q^{2} + 341 q^{4} + 5258 q^{7} + 105 q^{8} + 54699 q^{11} + 215884 q^{13} - 272922 q^{14} - 699247 q^{16} - 334983 q^{17} + 818845 q^{19} + 72761 q^{22} - 3526854 q^{23} + 1280004 q^{26} - 428134 q^{28} - 2175480 q^{29} + 4274066 q^{31} + 9464577 q^{32} - 6838963 q^{34} - 10305042 q^{37} + 24180495 q^{38} - 5926311 q^{41} - 24429956 q^{43} + 21995703 q^{44} + 14223246 q^{46} - 66858708 q^{47} - 6453929 q^{49} - 57862932 q^{52} - 132620514 q^{53} + 169538130 q^{56} + 201908320 q^{58} - 5670960 q^{59} + 125306926 q^{61} + 39831174 q^{62} + 167542401 q^{64} + 88829483 q^{67} + 71162559 q^{68} - 297550596 q^{71} + 181321729 q^{73} + 251507358 q^{74} + 89414865 q^{76} - 561214086 q^{77} - 310025170 q^{79} - 1368322979 q^{82} + 731088801 q^{83} - 33947196 q^{86} - 943671285 q^{88} + 1103860035 q^{89} + 1183187656 q^{91} + 190024242 q^{92} + 1727891132 q^{94} - 332236842 q^{97} - 457927581 q^{98}+O(q^{100}) \) 3 * q - 33 * q^2 + 341 * q^4 + 5258 * q^7 + 105 * q^8 + 54699 * q^11 + 215884 * q^13 - 272922 * q^14 - 699247 * q^16 - 334983 * q^17 + 818845 * q^19 + 72761 * q^22 - 3526854 * q^23 + 1280004 * q^26 - 428134 * q^28 - 2175480 * q^29 + 4274066 * q^31 + 9464577 * q^32 - 6838963 * q^34 - 10305042 * q^37 + 24180495 * q^38 - 5926311 * q^41 - 24429956 * q^43 + 21995703 * q^44 + 14223246 * q^46 - 66858708 * q^47 - 6453929 * q^49 - 57862932 * q^52 - 132620514 * q^53 + 169538130 * q^56 + 201908320 * q^58 - 5670960 * q^59 + 125306926 * q^61 + 39831174 * q^62 + 167542401 * q^64 + 88829483 * q^67 + 71162559 * q^68 - 297550596 * q^71 + 181321729 * q^73 + 251507358 * q^74 + 89414865 * q^76 - 561214086 * q^77 - 310025170 * q^79 - 1368322979 * q^82 + 731088801 * q^83 - 33947196 * q^86 - 943671285 * q^88 + 1103860035 * q^89 + 1183187656 * q^91 + 190024242 * q^92 + 1727891132 * q^94 - 332236842 * q^97 - 457927581 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more