LMFDB - Embedded newform 25.16.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,16,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	25.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:	$35.6733762750$
Analytic rank:	$1$
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} - 29397x^{4} + 153469728x^{2} - 65015354624 $ x^6 - 29397x^4 + 153469728x^2 - 65015354624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-150.955$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-301.910 q^{2} +779.290 q^{3} +58381.4 q^{4} -235275. q^{6} +2.08791e6 q^{7} -7.73292e6 q^{8} -1.37416e7 q^{9} +O(q^{10})$	\(q-301.910 q^{2} +779.290 q^{3} +58381.4 q^{4} -235275. q^{6} +2.08791e6 q^{7} -7.73292e6 q^{8} -1.37416e7 q^{9} -7.71305e7 q^{11} +4.54960e7 q^{12} +1.32485e8 q^{13} -6.30361e8 q^{14} +4.21600e8 q^{16} -1.10174e9 q^{17} +4.14872e9 q^{18} +6.23952e9 q^{19} +1.62709e9 q^{21} +2.32864e10 q^{22} +2.49754e10 q^{23} -6.02618e9 q^{24} -3.99986e10 q^{26} -2.18907e10 q^{27} +1.21895e11 q^{28} -1.45836e11 q^{29} +4.01436e10 q^{31} +1.26107e11 q^{32} -6.01070e10 q^{33} +3.32626e11 q^{34} -8.02254e11 q^{36} -5.28137e11 q^{37} -1.88377e12 q^{38} +1.03245e11 q^{39} +6.88546e10 q^{41} -4.91234e11 q^{42} +7.67330e11 q^{43} -4.50298e12 q^{44} -7.54032e12 q^{46} -8.92816e11 q^{47} +3.28549e11 q^{48} -3.88180e11 q^{49} -8.58576e11 q^{51} +7.73468e12 q^{52} +5.86528e11 q^{53} +6.60900e12 q^{54} -1.61457e13 q^{56} +4.86239e12 q^{57} +4.40291e13 q^{58} -3.16543e10 q^{59} +5.85723e12 q^{61} -1.21197e13 q^{62} -2.86913e13 q^{63} -5.18879e13 q^{64} +1.81469e13 q^{66} -3.95005e13 q^{67} -6.43212e13 q^{68} +1.94631e13 q^{69} -7.83244e13 q^{71} +1.06263e14 q^{72} -8.73027e13 q^{73} +1.59450e14 q^{74} +3.64272e14 q^{76} -1.61042e14 q^{77} -3.11705e13 q^{78} +3.01801e13 q^{79} +1.80118e14 q^{81} -2.07879e13 q^{82} +1.08491e14 q^{83} +9.49917e13 q^{84} -2.31664e14 q^{86} -1.13648e14 q^{87} +5.96444e14 q^{88} -5.54023e14 q^{89} +2.76618e14 q^{91} +1.45810e15 q^{92} +3.12835e13 q^{93} +2.69550e14 q^{94} +9.82739e13 q^{96} -1.50497e15 q^{97} +1.17195e14 q^{98} +1.05990e15 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 38568 q^{4} - 515448 q^{6} + 11416662 q^{9}+O(q^{10}) $ 6 * q + 38568 * q^4 - 515448 * q^6 + 11416662 * q^9	$ 6 q + 38568 q^{4} - 515448 q^{6} + 11416662 q^{9} - 108590088 q^{11} - 663751704 q^{14} + 1155522336 q^{16} + 3630995640 q^{19} - 8917537608 q^{21} + 2959765920 q^{24} - 81970953168 q^{26} - 286168468740 q^{29} - 276236748288 q^{31} + 127784939136 q^{34} - 3326879331864 q^{36} - 2186980965936 q^{39} - 6153278882388 q^{41} - 8250173021664 q^{44} - 23334602656488 q^{46} - 11613390856242 q^{49} - 43487373385728 q^{51} - 10162879468560 q^{54} - 59280484297440 q^{56} - 14903258326680 q^{59} - 11352061428588 q^{61} - 73265851251072 q^{64} + 76208211455904 q^{66} + 150489671962824 q^{69} + 131693145807312 q^{71} + 353606797863216 q^{74} + 959127540575520 q^{76} + 26081853939360 q^{79} + 11\!\cdots\!46 q^{81}+ \cdots - 506999099666376 q^{99}+O(q^{100}) $ 6 * q + 38568 * q^4 - 515448 * q^6 + 11416662 * q^9 - 108590088 * q^11 - 663751704 * q^14 + 1155522336 * q^16 + 3630995640 * q^19 - 8917537608 * q^21 + 2959765920 * q^24 - 81970953168 * q^26 - 286168468740 * q^29 - 276236748288 * q^31 + 127784939136 * q^34 - 3326879331864 * q^36 - 2186980965936 * q^39 - 6153278882388 * q^41 - 8250173021664 * q^44 - 23334602656488 * q^46 - 11613390856242 * q^49 - 43487373385728 * q^51 - 10162879468560 * q^54 - 59280484297440 * q^56 - 14903258326680 * q^59 - 11352061428588 * q^61 - 73265851251072 * q^64 + 76208211455904 * q^66 + 150489671962824 * q^69 + 131693145807312 * q^71 + 353606797863216 * q^74 + 959127540575520 * q^76 + 26081853939360 * q^79 + 1148641568615646 * q^81 + 686155736066976 * q^84 + 598387757484072 * q^86 - 59269690709820 * q^89 + 821206462566672 * q^91 + 308164835928456 * q^94 - 1208921682814848 * q^96 - 506999099666376 * q^99

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 $

$2$	−301.910	−1.66783	−0.833915	−	0.551893i	$-0.813906\pi$
$2$	−301.910	−1.66783	−0.833915	+	0.551893i	$0.813906\pi$
$3$	779.290	0.205726	0.102863	−	0.994696i	$-0.467200\pi$
$3$	779.290	0.205726	0.102863	+	0.994696i	$0.467200\pi$
$4$	58381.4	1.78166
$5$	0	0
$5$	0	0
$6$	−235275.	−0.343116
$7$	2.08791e6	0.958246	0.479123	−	0.877748i	$-0.340955\pi$
$7$	2.08791e6	0.958246	0.479123	+	0.877748i	$0.340955\pi$
$8$	−7.73292e6	−1.30367
$9$	−1.37416e7	−0.957677
$10$	0	0
$11$	−7.71305e7	−1.19339	−0.596693	−	0.802469i	$-0.703519\pi$
$11$	−7.71305e7	−1.19339	−0.596693	+	0.802469i	$0.703519\pi$
$12$	4.54960e7	0.366534
$13$	1.32485e8	0.585589	0.292795	−	0.956175i	$-0.405415\pi$
$13$	1.32485e8	0.585589	0.292795	+	0.956175i	$0.405415\pi$
$14$	−6.30361e8	−1.59819
$15$	0	0
$16$	4.21600e8	0.392646
$17$	−1.10174e9	−0.651198	−0.325599	−	0.945508i	$-0.605566\pi$
$17$	−1.10174e9	−0.651198	−0.325599	+	0.945508i	$0.605566\pi$
$18$	4.14872e9	1.59724
$19$	6.23952e9	1.60140	0.800699	−	0.599067i	$-0.204462\pi$
$19$	6.23952e9	1.60140	0.800699	+	0.599067i	$0.204462\pi$
$20$	0	0
$21$	1.62709e9	0.197136
$22$	2.32864e10	1.99037
$23$	2.49754e10	1.52952	0.764759	−	0.644317i	$-0.222858\pi$
$23$	2.49754e10	1.52952	0.764759	+	0.644317i	$0.222858\pi$
$24$	−6.02618e9	−0.268199
$25$	0	0
$26$	−3.99986e10	−0.976663
$27$	−2.18907e10	−0.402745
$28$	1.21895e11	1.70727
$29$	−1.45836e11	−1.56992	−0.784962	−	0.619544i	$-0.787317\pi$
$29$	−1.45836e11	−1.56992	−0.784962	+	0.619544i	$0.787317\pi$
$30$	0	0
$31$	4.01436e10	0.262062	0.131031	−	0.991378i	$-0.458171\pi$
$31$	4.01436e10	0.262062	0.131031	+	0.991378i	$0.458171\pi$
$32$	1.26107e11	0.648805
$33$	−6.01070e10	−0.245511
$34$	3.32626e11	1.08609
$35$	0	0
$36$	−8.02254e11	−1.70625
$37$	−5.28137e11	−0.914605	−0.457302	−	0.889311i	$-0.651184\pi$
$37$	−5.28137e11	−0.914605	−0.457302	+	0.889311i	$0.651184\pi$
$38$	−1.88377e12	−2.67086
$39$	1.03245e11	0.120471
$40$	0	0
$41$	6.88546e10	0.0552146	0.0276073	−	0.999619i	$-0.491211\pi$
$41$	6.88546e10	0.0552146	0.0276073	+	0.999619i	$0.491211\pi$
$42$	−4.91234e11	−0.328790
$43$	7.67330e11	0.430495	0.215248	−	0.976559i	$-0.430944\pi$
$43$	7.67330e11	0.430495	0.215248	+	0.976559i	$0.430944\pi$
$44$	−4.50298e12	−2.12621
$45$	0	0
$46$	−7.54032e12	−2.55098
$47$	−8.92816e11	−0.257056	−0.128528	−	0.991706i	$-0.541025\pi$
$47$	−8.92816e11	−0.257056	−0.128528	+	0.991706i	$0.541025\pi$
$48$	3.28549e11	0.0807776
$49$	−3.88180e11	−0.0817640
$50$	0	0
$51$	−8.58576e11	−0.133968
$52$	7.73468e12	1.04332
$53$	5.86528e11	0.0685836	0.0342918	−	0.999412i	$-0.489082\pi$
$53$	5.86528e11	0.0685836	0.0342918	+	0.999412i	$0.489082\pi$
$54$	6.60900e12	0.671711
$55$	0	0
$56$	−1.61457e13	−1.24924
$57$	4.86239e12	0.329449
$58$	4.40291e13	2.61837
$59$	−3.16543e10	−0.00165593	−0.000827967	−	1.00000i	$-0.500264\pi$
$59$	−3.16543e10	−0.00165593	−0.000827967		1.00000i	$0.500264\pi$
$60$	0	0
$61$	5.85723e12	0.238627	0.119313	−	0.992857i	$-0.461931\pi$
$61$	5.85723e12	0.238627	0.119313	+	0.992857i	$0.461931\pi$
$62$	−1.21197e13	−0.437075
$63$	−2.86913e13	−0.917690
$64$	−5.18879e13	−1.47474
$65$	0	0
$66$	1.81469e13	0.409470
$67$	−3.95005e13	−0.796235	−0.398118	−	0.917334i	$-0.630336\pi$
$67$	−3.95005e13	−0.796235	−0.398118	+	0.917334i	$0.630336\pi$
$68$	−6.43212e13	−1.16021
$69$	1.94631e13	0.314662
$70$	0	0
$71$	−7.83244e13	−1.02202	−0.511010	−	0.859575i	$-0.670729\pi$
$71$	−7.83244e13	−1.02202	−0.511010	+	0.859575i	$0.670729\pi$
$72$	1.06263e14	1.24850
$73$	−8.73027e13	−0.924924	−0.462462	−	0.886639i	$-0.653034\pi$
$73$	−8.73027e13	−0.924924	−0.462462	+	0.886639i	$0.653034\pi$
$74$	1.59450e14	1.52541
$75$	0	0
$76$	3.64272e14	2.85314
$77$	−1.61042e14	−1.14356
$78$	−3.11705e13	−0.200925
$79$	3.01801e13	0.176814	0.0884070	−	0.996084i	$-0.471822\pi$
$79$	3.01801e13	0.176814	0.0884070	+	0.996084i	$0.471822\pi$
$80$	0	0
$81$	1.80118e14	0.874821
$82$	−2.07879e13	−0.0920886
$83$	1.08491e14	0.438841	0.219421	−	0.975630i	$-0.429583\pi$
$83$	1.08491e14	0.438841	0.219421	+	0.975630i	$0.429583\pi$
$84$	9.49917e13	0.351229
$85$	0	0
$86$	−2.31664e14	−0.717993
$87$	−1.13648e14	−0.322974
$88$	5.96444e14	1.55578
$89$	−5.54023e14	−1.32771	−0.663854	−	0.747862i	$-0.731080\pi$
$89$	−5.54023e14	−1.32771	−0.663854	+	0.747862i	$0.731080\pi$
$90$	0	0
$91$	2.76618e14	0.561139
$92$	1.45810e15	2.72508
$93$	3.12835e13	0.0539130
$94$	2.69550e14	0.428726
$95$	0	0
$96$	9.82739e13	0.133476
$97$	−1.50497e15	−1.89121	−0.945606	−	0.325313i	$-0.894530\pi$
$97$	−1.50497e15	−1.89121	−0.945606	+	0.325313i	$0.894530\pi$
$98$	1.17195e14	0.136369
$99$	1.05990e15	1.14288
$100$	0	0
$101$	−1.04693e15	−0.971649	−0.485824	−	0.874056i	$-0.661481\pi$
$101$	−1.04693e15	−0.971649	−0.485824	+	0.874056i	$0.661481\pi$
$102$	2.59212e14	0.223437
$103$	−2.63263e14	−0.210916	−0.105458	−	0.994424i	$-0.533631\pi$
$103$	−2.63263e14	−0.210916	−0.105458	+	0.994424i	$0.533631\pi$
$104$	−1.02450e15	−0.763416
$105$	0	0
$106$	−1.77078e14	−0.114386
$107$	−7.84345e14	−0.472203	−0.236101	−	0.971728i	$-0.575870\pi$
$107$	−7.84345e14	−0.472203	−0.236101	+	0.971728i	$0.575870\pi$
$108$	−1.27801e15	−0.717554
$109$	−8.25405e14	−0.432483	−0.216241	−	0.976340i	$-0.569380\pi$
$109$	−8.25405e14	−0.432483	−0.216241	+	0.976340i	$0.569380\pi$
$110$	0	0
$111$	−4.11572e14	−0.188158
$112$	8.80265e14	0.376252
$113$	−3.56130e15	−1.42403	−0.712016	−	0.702163i	$-0.752218\pi$
$113$	−3.56130e15	−1.42403	−0.712016	+	0.702163i	$0.752218\pi$
$114$	−1.46800e15	−0.549466
$115$	0	0
$116$	−8.51408e15	−2.79707
$117$	−1.82056e15	−0.560805
$118$	9.55674e12	0.00276182
$119$	−2.30034e15	−0.624008
$120$	0	0
$121$	1.77187e15	0.424171
$122$	−1.76835e15	−0.397988
$123$	5.36577e13	0.0113591
$124$	2.34364e15	0.466905
$125$	0	0
$126$	8.66218e15	1.53055
$127$	9.24721e15	1.53987	0.769933	−	0.638125i	$-0.220290\pi$
$127$	9.24721e15	1.53987	0.769933	+	0.638125i	$0.220290\pi$
$128$	1.15332e16	1.81082
$129$	5.97972e14	0.0885642
$130$	0	0
$131$	−3.50922e15	−0.463101	−0.231550	−	0.972823i	$-0.574380\pi$
$131$	−3.50922e15	−0.463101	−0.231550	+	0.972823i	$0.574380\pi$
$132$	−3.50913e15	−0.437416
$133$	1.30276e16	1.53453
$134$	1.19256e16	1.32798
$135$	0	0
$136$	8.51968e15	0.848948
$137$	−7.60397e15	−0.717194	−0.358597	−	0.933493i	$-0.616745\pi$
$137$	−7.60397e15	−0.717194	−0.358597	+	0.933493i	$0.616745\pi$
$138$	−5.87610e15	−0.524802
$139$	−3.33077e15	−0.281795	−0.140898	−	0.990024i	$-0.544999\pi$
$139$	−3.33077e15	−0.281795	−0.140898	+	0.990024i	$0.544999\pi$
$140$	0	0
$141$	−6.95763e14	−0.0528832
$142$	2.36469e16	1.70456
$143$	−1.02187e16	−0.698834
$144$	−5.79347e15	−0.376028
$145$	0	0
$146$	2.63575e16	1.54262
$147$	−3.02505e14	−0.0168210
$148$	−3.08333e16	−1.62951
$149$	−2.20370e16	−1.10728	−0.553638	−	0.832758i	$-0.686761\pi$
$149$	−2.20370e16	−1.10728	−0.553638	+	0.832758i	$0.686761\pi$
$150$	0	0
$151$	−1.16974e16	−0.531815	−0.265907	−	0.963999i	$-0.585672\pi$
$151$	−1.16974e16	−0.531815	−0.265907	+	0.963999i	$0.585672\pi$
$152$	−4.82497e16	−2.08770
$153$	1.51397e16	0.623637
$154$	4.86201e16	1.90726
$155$	0	0
$156$	6.02756e15	0.214638
$157$	1.50192e16	0.509799	0.254899	−	0.966968i	$-0.417958\pi$
$157$	1.50192e16	0.509799	0.254899	+	0.966968i	$0.417958\pi$
$158$	−9.11165e15	−0.294896
$159$	4.57076e14	0.0141094
$160$	0	0
$161$	5.21466e16	1.46565
$162$	−5.43793e16	−1.45905
$163$	−2.37456e16	−0.608382	−0.304191	−	0.952611i	$-0.598386\pi$
$163$	−2.37456e16	−0.608382	−0.304191	+	0.952611i	$0.598386\pi$
$164$	4.01982e15	0.0983735
$165$	0	0
$166$	−3.27544e16	−0.731913
$167$	8.70806e16	1.86015	0.930075	−	0.367370i	$-0.119742\pi$
$167$	8.70806e16	1.86015	0.930075	+	0.367370i	$0.119742\pi$
$168$	−1.25821e16	−0.257001
$169$	−3.36335e16	−0.657085
$170$	0	0
$171$	−8.57411e16	−1.53362
$172$	4.47977e16	0.766995
$173$	4.24844e16	0.696440	0.348220	−	0.937413i	$-0.386786\pi$
$173$	4.24844e16	0.696440	0.348220	+	0.937413i	$0.386786\pi$
$174$	3.43115e16	0.538666
$175$	0	0
$176$	−3.25183e16	−0.468578
$177$	−2.46679e13	−0.000340669 0
$178$	1.67265e17	2.21439
$179$	−6.47281e16	−0.821666	−0.410833	−	0.911711i	$-0.634762\pi$
$179$	−6.47281e16	−0.821666	−0.410833	+	0.911711i	$0.634762\pi$
$180$	0	0
$181$	−8.81817e16	−1.02989	−0.514943	−	0.857225i	$-0.672187\pi$
$181$	−8.81817e16	−1.02989	−0.514943	+	0.857225i	$0.672187\pi$
$182$	−8.35137e16	−0.935884
$183$	4.56448e15	0.0490917
$184$	−1.93133e17	−1.99399
$185$	0	0
$186$	−9.44479e15	−0.0899177
$187$	8.49779e16	0.777131
$188$	−5.21238e16	−0.457986
$189$	−4.57058e16	−0.385929
$190$	0	0
$191$	4.71702e16	0.368059	0.184030	−	0.982921i	$-0.441086\pi$
$191$	4.71702e16	0.368059	0.184030	+	0.982921i	$0.441086\pi$
$192$	−4.04357e16	−0.303393
$193$	9.79733e16	0.707014	0.353507	−	0.935432i	$-0.384989\pi$
$193$	9.79733e16	0.707014	0.353507	+	0.935432i	$0.384989\pi$
$194$	4.54365e17	3.15422
$195$	0	0
$196$	−2.26625e16	−0.145676
$197$	−1.82893e17	−1.13162	−0.565810	−	0.824536i	$-0.691437\pi$
$197$	−1.82893e17	−1.13162	−0.565810	+	0.824536i	$0.691437\pi$
$198$	−3.19993e17	−1.90613
$199$	1.95969e16	0.112406	0.0562029	−	0.998419i	$-0.482101\pi$
$199$	1.95969e16	0.112406	0.0562029	+	0.998419i	$0.482101\pi$
$200$	0	0
$201$	−3.07823e16	−0.163806
$202$	3.16080e17	1.62055
$203$	−3.04492e17	−1.50437
$204$	−5.01248e16	−0.238686
$205$	0	0
$206$	7.94815e16	0.351772
$207$	−3.43203e17	−1.46478
$208$	5.58559e16	0.229929
$209$	−4.81257e17	−1.91109
$210$	0	0
$211$	1.71813e17	0.635241	0.317620	−	0.948218i	$-0.397116\pi$
$211$	1.71813e17	0.635241	0.317620	+	0.948218i	$0.397116\pi$
$212$	3.42423e16	0.122192
$213$	−6.10374e16	−0.210256
$214$	2.36801e17	0.787554
$215$	0	0
$216$	1.69279e17	0.525048
$217$	8.38164e16	0.251120
$218$	2.49198e17	0.721307
$219$	−6.80341e16	−0.190281
$220$	0	0
$221$	−1.45965e17	−0.381334
$222$	1.24257e17	0.313816
$223$	5.08815e17	1.24243	0.621217	−	0.783638i	$-0.286638\pi$
$223$	5.08815e17	1.24243	0.621217	+	0.783638i	$0.286638\pi$
$224$	2.63300e17	0.621715
$225$	0	0
$226$	1.07519e18	2.37504
$227$	−5.27724e17	−1.12775	−0.563875	−	0.825860i	$-0.690690\pi$
$227$	−5.27724e17	−1.12775	−0.563875	+	0.825860i	$0.690690\pi$
$228$	2.83873e17	0.586966
$229$	3.25364e17	0.651034	0.325517	−	0.945536i	$-0.394462\pi$
$229$	3.25364e17	0.651034	0.325517	+	0.945536i	$0.394462\pi$
$230$	0	0
$231$	−1.25498e17	−0.235260
$232$	1.12773e18	2.04667
$233$	−5.04233e17	−0.886057	−0.443029	−	0.896507i	$-0.646096\pi$
$233$	−5.04233e17	−0.886057	−0.443029	+	0.896507i	$0.646096\pi$
$234$	5.49646e17	0.935328
$235$	0	0
$236$	−1.84802e15	−0.00295031
$237$	2.35190e16	0.0363753
$238$	6.94495e17	1.04074
$239$	−4.80698e17	−0.698052	−0.349026	−	0.937113i	$-0.613488\pi$
$239$	−4.80698e17	−0.698052	−0.349026	+	0.937113i	$0.613488\pi$
$240$	0	0
$241$	−6.74534e17	−0.920186	−0.460093	−	0.887871i	$-0.652184\pi$
$241$	−6.74534e17	−0.920186	−0.460093	+	0.887871i	$0.652184\pi$
$242$	−5.34944e17	−0.707446
$243$	4.54471e17	0.582719
$244$	3.41953e17	0.425151
$245$	0	0
$246$	−1.61998e16	−0.0189450
$247$	8.26646e17	0.937761
$248$	−3.10427e17	−0.341643
$249$	8.45459e16	0.0902812
$250$	0	0
$251$	−2.82502e17	−0.284098	−0.142049	−	0.989860i	$-0.545369\pi$
$251$	−2.82502e17	−0.284098	−0.142049	+	0.989860i	$0.545369\pi$
$252$	−1.67504e18	−1.63501
$253$	−1.92637e18	−1.82531
$254$	−2.79182e18	−2.56823
$255$	0	0
$256$	−1.78171e18	−1.54539
$257$	−6.91378e17	−0.582394	−0.291197	−	0.956663i	$-0.594054\pi$
$257$	−6.91378e17	−0.582394	−0.291197	+	0.956663i	$0.594054\pi$
$258$	−1.80533e17	−0.147710
$259$	−1.10270e18	−0.876417
$260$	0	0
$261$	2.00402e18	1.50348
$262$	1.05947e18	0.772374
$263$	4.44052e17	0.314605	0.157303	−	0.987550i	$-0.449720\pi$
$263$	4.44052e17	0.314605	0.157303	+	0.987550i	$0.449720\pi$
$264$	4.64803e17	0.320066
$265$	0	0
$266$	−3.93315e18	−2.55934
$267$	−4.31744e17	−0.273144
$268$	−2.30609e18	−1.41862
$269$	3.14482e18	1.88128	0.940641	−	0.339402i	$-0.110225\pi$
$269$	3.14482e18	1.88128	0.940641	+	0.339402i	$0.110225\pi$
$270$	0	0
$271$	4.09117e17	0.231514	0.115757	−	0.993278i	$-0.463071\pi$
$271$	4.09117e17	0.231514	0.115757	+	0.993278i	$0.463071\pi$
$272$	−4.64495e17	−0.255690
$273$	2.15566e17	0.115441
$274$	2.29571e18	1.19616
$275$	0	0
$276$	1.13628e18	0.560620
$277$	2.54882e18	1.22388	0.611942	−	0.790903i	$-0.290389\pi$
$277$	2.54882e18	1.22388	0.611942	+	0.790903i	$0.290389\pi$
$278$	1.00559e18	0.469987
$279$	−5.51638e17	−0.250971
$280$	0	0
$281$	−2.10517e18	−0.907799	−0.453899	−	0.891053i	$-0.649967\pi$
$281$	−2.10517e18	−0.907799	−0.453899	+	0.891053i	$0.649967\pi$
$282$	2.10057e17	0.0882002
$283$	3.14365e18	1.28539	0.642697	−	0.766121i	$-0.277815\pi$
$283$	3.14365e18	1.28539	0.642697	+	0.766121i	$0.277815\pi$
$284$	−4.57268e18	−1.82089
$285$	0	0
$286$	3.08511e18	1.16554
$287$	1.43762e17	0.0529092
$288$	−1.73291e18	−0.621346
$289$	−1.64859e18	−0.575942
$290$	0	0
$291$	−1.17281e18	−0.389072
$292$	−5.09685e18	−1.64790
$293$	4.70542e18	1.48283	0.741415	−	0.671047i	$-0.234155\pi$
$293$	4.70542e18	1.48283	0.741415	+	0.671047i	$0.234155\pi$
$294$	9.13290e16	0.0280546
$295$	0	0
$296$	4.08404e18	1.19234
$297$	1.68844e18	0.480631
$298$	6.65318e18	1.84675
$299$	3.30888e18	0.895669
$300$	0	0
$301$	1.60212e18	0.412521
$302$	3.53154e18	0.886977
$303$	−8.15865e17	−0.199894
$304$	2.63058e18	0.628782
$305$	0	0
$306$	−4.57082e18	−1.04012
$307$	4.22293e18	0.937725	0.468863	−	0.883271i	$-0.344664\pi$
$307$	4.22293e18	0.937725	0.468863	+	0.883271i	$0.344664\pi$
$308$	−9.40184e18	−2.03743
$309$	−2.05158e17	−0.0433910
$310$	0	0
$311$	9.72072e18	1.95882	0.979412	−	0.201872i	$-0.0647024\pi$
$311$	9.72072e18	1.95882	0.979412	+	0.201872i	$0.0647024\pi$
$312$	−7.98382e17	−0.157055
$313$	−6.04916e18	−1.16175	−0.580875	−	0.813993i	$-0.697289\pi$
$313$	−6.04916e18	−1.16175	−0.580875	+	0.813993i	$0.697289\pi$
$314$	−4.53443e18	−0.850258
$315$	0	0
$316$	1.76195e18	0.315022
$317$	9.16096e17	0.159955	0.0799773	−	0.996797i	$-0.474515\pi$
$317$	9.16096e17	0.159955	0.0799773	+	0.996797i	$0.474515\pi$
$318$	−1.37995e17	−0.0235321
$319$	1.12484e19	1.87353
$320$	0	0
$321$	−6.11232e17	−0.0971445
$322$	−1.57435e19	−2.44446
$323$	−6.87434e18	−1.04283
$324$	1.05155e19	1.55863
$325$	0	0
$326$	7.16903e18	1.01468
$327$	−6.43230e17	−0.0889730
$328$	−5.32447e17	−0.0719817
$329$	−1.86412e18	−0.246323
$330$	0	0
$331$	−9.14544e18	−1.15477	−0.577384	−	0.816473i	$-0.695926\pi$
$331$	−9.14544e18	−1.15477	−0.577384	+	0.816473i	$0.695926\pi$
$332$	6.33385e18	0.781865
$333$	7.25745e18	0.875896
$334$	−2.62905e19	−3.10241
$335$	0	0
$336$	6.85982e17	0.0774048
$337$	−1.21070e19	−1.33601	−0.668006	−	0.744156i	$-0.732852\pi$
$337$	−1.21070e19	−1.33601	−0.668006	+	0.744156i	$0.732852\pi$
$338$	1.01543e19	1.09591
$339$	−2.77528e18	−0.292961
$340$	0	0
$341$	−3.09630e18	−0.312741
$342$	2.58860e19	2.55782
$343$	−1.07230e19	−1.03660
$344$	−5.93369e18	−0.561225
$345$	0	0
$346$	−1.28264e19	−1.16154
$347$	−3.51728e18	−0.311700	−0.155850	−	0.987781i	$-0.549812\pi$
$347$	−3.51728e18	−0.311700	−0.155850	+	0.987781i	$0.549812\pi$
$348$	−6.63493e18	−0.575430
$349$	1.19985e19	1.01844	0.509221	−	0.860636i	$-0.329934\pi$
$349$	1.19985e19	1.01844	0.509221	+	0.860636i	$0.329934\pi$
$350$	0	0
$351$	−2.90019e18	−0.235843
$352$	−9.72670e18	−0.774275
$353$	1.47029e19	1.14576	0.572878	−	0.819641i	$-0.305827\pi$
$353$	1.47029e19	1.14576	0.572878	+	0.819641i	$0.305827\pi$
$354$	7.44747e15	0.000568178 0
$355$	0	0
$356$	−3.23446e19	−2.36552
$357$	−1.79263e18	−0.128375
$358$	1.95420e19	1.37040
$359$	2.86436e18	0.196707	0.0983534	−	0.995152i	$-0.468642\pi$
$359$	2.86436e18	0.196707	0.0983534	+	0.995152i	$0.468642\pi$
$360$	0	0
$361$	2.37505e19	1.56447
$362$	2.66229e19	1.71767
$363$	1.38080e18	0.0872631
$364$	1.61493e19	0.999757
$365$	0	0
$366$	−1.37806e18	−0.0818766
$367$	−2.57378e18	−0.149822	−0.0749110	−	0.997190i	$-0.523867\pi$
$367$	−2.57378e18	−0.149822	−0.0749110	+	0.997190i	$0.523867\pi$
$368$	1.05297e19	0.600559
$369$	−9.46173e17	−0.0528777
$370$	0	0
$371$	1.22462e18	0.0657199
$372$	1.82637e18	0.0960545
$373$	−1.16191e19	−0.598904	−0.299452	−	0.954111i	$-0.596804\pi$
$373$	−1.16191e19	−0.598904	−0.299452	+	0.954111i	$0.596804\pi$
$374$	−2.56556e19	−1.29612
$375$	0	0
$376$	6.90407e18	0.335117
$377$	−1.93211e19	−0.919330
$378$	1.37990e19	0.643664
$379$	−8.35574e18	−0.382112	−0.191056	−	0.981579i	$-0.561191\pi$
$379$	−8.35574e18	−0.382112	−0.191056	+	0.981579i	$0.561191\pi$
$380$	0	0
$381$	7.20626e18	0.316791
$382$	−1.42411e19	−0.613860
$383$	−7.58133e18	−0.320446	−0.160223	−	0.987081i	$-0.551221\pi$
$383$	−7.58133e18	−0.320446	−0.160223	+	0.987081i	$0.551221\pi$
$384$	8.98769e18	0.372532
$385$	0	0
$386$	−2.95791e19	−1.17918
$387$	−1.05443e19	−0.412275
$388$	−8.78623e19	−3.36949
$389$	−3.55842e19	−1.33855	−0.669277	−	0.743013i	$-0.733396\pi$
$389$	−3.55842e19	−1.33855	−0.669277	+	0.743013i	$0.733396\pi$
$390$	0	0
$391$	−2.75165e19	−0.996018
$392$	3.00176e18	0.106593
$393$	−2.73470e18	−0.0952720
$394$	5.52172e19	1.88735
$395$	0	0
$396$	6.18783e19	2.03622
$397$	−4.87189e18	−0.157314	−0.0786572	−	0.996902i	$-0.525063\pi$
$397$	−4.87189e18	−0.157314	−0.0786572	+	0.996902i	$0.525063\pi$
$398$	−5.91648e18	−0.187474
$399$	1.01523e19	0.315694
$400$	0	0
$401$	−8.01043e18	−0.239924	−0.119962	−	0.992779i	$-0.538277\pi$
$401$	−8.01043e18	−0.239924	−0.119962	+	0.992779i	$0.538277\pi$
$402$	9.29347e18	0.273201
$403$	5.31845e18	0.153461
$404$	−6.11215e19	−1.73115
$405$	0	0
$406$	9.19290e19	2.50904
$407$	4.07355e19	1.09148
$408$	6.63930e18	0.174651
$409$	5.16991e19	1.33524	0.667619	−	0.744503i	$-0.267314\pi$
$409$	5.16991e19	1.33524	0.667619	+	0.744503i	$0.267314\pi$
$410$	0	0
$411$	−5.92570e18	−0.147545
$412$	−1.53696e19	−0.375780
$413$	−6.60915e16	−0.00158679
$414$	1.03616e20	2.44301
$415$	0	0
$416$	1.67073e19	0.379933
$417$	−2.59563e18	−0.0579727
$418$	1.45296e20	3.18737
$419$	−4.37426e19	−0.942539	−0.471269	−	0.881989i	$-0.656204\pi$
$419$	−4.37426e19	−0.942539	−0.471269	+	0.881989i	$0.656204\pi$
$420$	0	0
$421$	−1.55963e19	−0.324268	−0.162134	−	0.986769i	$-0.551838\pi$
$421$	−1.55963e19	−0.324268	−0.162134	+	0.986769i	$0.551838\pi$
$422$	−5.18721e19	−1.05947
$423$	1.22687e19	0.246177
$424$	−4.53557e18	−0.0894105
$425$	0	0
$426$	1.84278e19	0.350672
$427$	1.22294e19	0.228663
$428$	−4.57911e19	−0.841304
$429$	−7.96331e18	−0.143769
$430$	0	0
$431$	1.87054e19	0.326127	0.163064	−	0.986616i	$-0.447862\pi$
$431$	1.87054e19	0.326127	0.163064	+	0.986616i	$0.447862\pi$
$432$	−9.22911e18	−0.158136
$433$	−2.79482e19	−0.470647	−0.235323	−	0.971917i	$-0.575615\pi$
$433$	−2.79482e19	−0.470647	−0.235323	+	0.971917i	$0.575615\pi$
$434$	−2.53050e19	−0.418825
$435$	0	0
$436$	−4.81883e19	−0.770536
$437$	1.55835e20	2.44937
$438$	2.05401e19	0.317357
$439$	−7.81127e19	−1.18642	−0.593209	−	0.805049i	$-0.702139\pi$
$439$	−7.81127e19	−1.18642	−0.593209	+	0.805049i	$0.702139\pi$
$440$	0	0
$441$	5.33422e18	0.0783035
$442$	4.40682e19	0.636001
$443$	1.27574e20	1.81023	0.905115	−	0.425167i	$-0.139784\pi$
$443$	1.27574e20	1.81023	0.905115	+	0.425167i	$0.139784\pi$
$444$	−2.40281e19	−0.335233
$445$	0	0
$446$	−1.53616e20	−2.07217
$447$	−1.71732e19	−0.227795
$448$	−1.08337e20	−1.41317
$449$	−5.40721e19	−0.693627	−0.346814	−	0.937934i	$-0.612736\pi$
$449$	−5.40721e19	−0.693627	−0.346814	+	0.937934i	$0.612736\pi$
$450$	0	0
$451$	−5.31079e18	−0.0658924
$452$	−2.07913e20	−2.53714
$453$	−9.11563e18	−0.109408
$454$	1.59325e20	1.88090
$455$	0	0
$456$	−3.76005e19	−0.429494
$457$	3.69928e19	0.415667	0.207833	−	0.978164i	$-0.433359\pi$
$457$	3.69928e19	0.415667	0.207833	+	0.978164i	$0.433359\pi$
$458$	−9.82306e19	−1.08581
$459$	2.41178e19	0.262267
$460$	0	0
$461$	−4.38103e19	−0.461126	−0.230563	−	0.973057i	$-0.574057\pi$
$461$	−4.38103e19	−0.461126	−0.230563	+	0.973057i	$0.574057\pi$
$462$	3.78891e19	0.392373
$463$	−6.17232e19	−0.628913	−0.314457	−	0.949272i	$-0.601822\pi$
$463$	−6.17232e19	−0.628913	−0.314457	+	0.949272i	$0.601822\pi$
$464$	−6.14843e19	−0.616424
$465$	0	0
$466$	1.52233e20	1.47779
$467$	−1.96472e20	−1.87683	−0.938414	−	0.345513i	$-0.887705\pi$
$467$	−1.96472e20	−1.87683	−0.938414	+	0.345513i	$0.887705\pi$
$468$	−1.06287e20	−0.999163
$469$	−8.24736e19	−0.762989
$470$	0	0
$471$	1.17043e19	0.104879
$472$	2.44780e17	0.00215879
$473$	−5.91845e19	−0.513747
$474$	−7.10061e18	−0.0606678
$475$	0	0
$476$	−1.34297e20	−1.11177
$477$	−8.05985e18	−0.0656809
$478$	1.45127e20	1.16423
$479$	7.54528e19	0.595881	0.297940	−	0.954585i	$-0.403700\pi$
$479$	7.54528e19	0.595881	0.297940	+	0.954585i	$0.403700\pi$
$480$	0	0
$481$	−6.99705e19	−0.535583
$482$	2.03648e20	1.53471
$483$	4.06373e19	0.301523
$484$	1.03444e20	0.755728
$485$	0	0
$486$	−1.37209e20	−0.971876
$487$	1.37432e20	0.958560	0.479280	−	0.877662i	$-0.340898\pi$
$487$	1.37432e20	0.958560	0.479280	+	0.877662i	$0.340898\pi$
$488$	−4.52935e19	−0.311091
$489$	−1.85047e19	−0.125160
$490$	0	0
$491$	1.58161e19	0.103750	0.0518750	−	0.998654i	$-0.483480\pi$
$491$	1.58161e19	0.103750	0.0518750	+	0.998654i	$0.483480\pi$
$492$	3.13261e18	0.0202380
$493$	1.60673e20	1.02233
$494$	−2.49572e20	−1.56403
$495$	0	0
$496$	1.69246e19	0.102898
$497$	−1.63535e20	−0.979347
$498$	−2.55252e19	−0.150574
$499$	−2.37301e20	−1.37894	−0.689470	−	0.724314i	$-0.742157\pi$
$499$	−2.37301e20	−1.37894	−0.689470	+	0.724314i	$0.742157\pi$
$500$	0	0
$501$	6.78610e19	0.382682
$502$	8.52900e19	0.473828
$503$	−1.76122e20	−0.963948	−0.481974	−	0.876185i	$-0.660080\pi$
$503$	−1.76122e20	−0.963948	−0.481974	+	0.876185i	$0.660080\pi$
$504$	2.21867e20	1.19637
$505$	0	0
$506$	5.81589e20	3.04430
$507$	−2.62102e19	−0.135180
$508$	5.39865e20	2.74351
$509$	1.78294e20	0.892796	0.446398	−	0.894834i	$-0.352707\pi$
$509$	1.78294e20	0.892796	0.446398	+	0.894834i	$0.352707\pi$
$510$	0	0
$511$	−1.82280e20	−0.886305
$512$	1.59997e20	0.766632
$513$	−1.36587e20	−0.644955
$514$	2.08734e20	0.971335
$515$	0	0
$516$	3.49104e19	0.157791
$517$	6.88634e19	0.306768
$518$	3.32917e20	1.46171
$519$	3.31076e19	0.143276
$520$	0	0
$521$	2.55797e20	1.07551	0.537753	−	0.843102i	$-0.319273\pi$
$521$	2.55797e20	1.07551	0.537753	+	0.843102i	$0.319273\pi$
$522$	−6.05032e20	−2.50755
$523$	−3.28407e19	−0.134168	−0.0670842	−	0.997747i	$-0.521370\pi$
$523$	−3.28407e19	−0.134168	−0.0670842	+	0.997747i	$0.521370\pi$
$524$	−2.04873e20	−0.825087
$525$	0	0
$526$	−1.34064e20	−0.524708
$527$	−4.42279e19	−0.170654
$528$	−2.53411e19	−0.0963989
$529$	3.57138e20	1.33942
$530$	0	0
$531$	4.34981e17	0.00158585
$532$	7.60568e20	2.73401
$533$	9.12223e18	0.0323331
$534$	1.30348e20	0.455558
$535$	0	0
$536$	3.05454e20	1.03803
$537$	−5.04419e19	−0.169038
$538$	−9.49452e20	−3.13766
$539$	2.99405e19	0.0975761
$540$	0	0
$541$	4.27990e20	1.35661	0.678303	−	0.734782i	$-0.262716\pi$
$541$	4.27990e20	1.35661	0.678303	+	0.734782i	$0.262716\pi$
$542$	−1.23516e20	−0.386127
$543$	−6.87191e19	−0.211874
$544$	−1.38937e20	−0.422500
$545$	0	0
$546$	−6.50813e19	−0.192536
$547$	−4.60215e20	−1.34294	−0.671469	−	0.741032i	$-0.734336\pi$
$547$	−4.60215e20	−1.34294	−0.671469	+	0.741032i	$0.734336\pi$
$548$	−4.43930e20	−1.27779
$549$	−8.04878e19	−0.228527
$550$	0	0
$551$	−9.09944e20	−2.51407
$552$	−1.50507e20	−0.410216
$553$	6.30133e19	0.169431
$554$	−7.69512e20	−2.04123
$555$	0	0
$556$	−1.94455e20	−0.502063
$557$	−2.11016e20	−0.537529	−0.268765	−	0.963206i	$-0.586615\pi$
$557$	−2.11016e20	−0.537529	−0.268765	+	0.963206i	$0.586615\pi$
$558$	1.66545e20	0.418576
$559$	1.01660e20	0.252093
$560$	0	0
$561$	6.62224e19	0.159876
$562$	6.35571e20	1.51405
$563$	2.16952e19	0.0509976	0.0254988	−	0.999675i	$-0.491883\pi$
$563$	2.16952e19	0.0509976	0.0254988	+	0.999675i	$0.491883\pi$
$564$	−4.06196e19	−0.0942198
$565$	0	0
$566$	−9.49098e20	−2.14382
$567$	3.76071e20	0.838294
$568$	6.05676e20	1.33238
$569$	2.23104e19	0.0484356	0.0242178	−	0.999707i	$-0.492290\pi$
$569$	2.23104e19	0.0484356	0.0242178	+	0.999707i	$0.492290\pi$
$570$	0	0
$571$	6.99283e20	1.47871	0.739353	−	0.673317i	$-0.235131\pi$
$571$	6.99283e20	1.47871	0.739353	+	0.673317i	$0.235131\pi$
$572$	−5.96580e20	−1.24508
$573$	3.67593e19	0.0757194
$574$	−4.34032e19	−0.0882435
$575$	0	0
$576$	7.13024e20	1.41233
$577$	−5.07975e20	−0.993171	−0.496585	−	0.867988i	$-0.665413\pi$
$577$	−5.07975e20	−0.993171	−0.496585	+	0.867988i	$0.665413\pi$
$578$	4.97724e20	0.960573
$579$	7.63496e19	0.145451
$580$	0	0
$581$	2.26520e20	0.420518
$582$	3.54082e20	0.648906
$583$	−4.52392e19	−0.0818467
$584$	6.75104e20	1.20580
$585$	0	0
$586$	−1.42061e21	−2.47311
$587$	9.20148e20	1.58151	0.790755	−	0.612133i	$-0.209688\pi$
$587$	9.20148e20	1.58151	0.790755	+	0.612133i	$0.209688\pi$
$588$	−1.76606e19	−0.0299693
$589$	2.50477e20	0.419665
$590$	0	0
$591$	−1.42527e20	−0.232804
$592$	−2.22663e20	−0.359116
$593$	7.45068e20	1.18655	0.593275	−	0.805000i	$-0.297835\pi$
$593$	7.45068e20	1.18655	0.593275	+	0.805000i	$0.297835\pi$
$594$	−5.09755e20	−0.801611
$595$	0	0
$596$	−1.28655e21	−1.97278
$597$	1.52716e19	0.0231248
$598$	−9.98983e20	−1.49382
$599$	3.78607e20	0.559098	0.279549	−	0.960131i	$-0.409815\pi$
$599$	3.78607e20	0.559098	0.279549	+	0.960131i	$0.409815\pi$
$600$	0	0
$601$	5.38635e20	0.775775	0.387887	−	0.921707i	$-0.373205\pi$
$601$	5.38635e20	0.775775	0.387887	+	0.921707i	$0.373205\pi$
$602$	−4.83695e20	−0.688014
$603$	5.42800e20	0.762536
$604$	−6.82907e20	−0.947512
$605$	0	0
$606$	2.46318e20	0.333389
$607$	7.37841e20	0.986388	0.493194	−	0.869919i	$-0.335829\pi$
$607$	7.37841e20	0.986388	0.493194	+	0.869919i	$0.335829\pi$
$608$	7.86847e20	1.03899
$609$	−2.37288e20	−0.309489
$610$	0	0
$611$	−1.18285e20	−0.150529
$612$	8.83877e20	1.11111
$613$	−6.78619e20	−0.842698	−0.421349	−	0.906898i	$-0.638443\pi$
$613$	−6.78619e20	−0.842698	−0.421349	+	0.906898i	$0.638443\pi$
$614$	−1.27494e21	−1.56397
$615$	0	0
$616$	1.24532e21	1.49082
$617$	5.97586e20	0.706743	0.353372	−	0.935483i	$-0.385035\pi$
$617$	5.97586e20	0.706743	0.353372	+	0.935483i	$0.385035\pi$
$618$	6.19391e19	0.0723688
$619$	3.49530e20	0.403463	0.201732	−	0.979441i	$-0.435343\pi$
$619$	3.49530e20	0.403463	0.201732	+	0.979441i	$0.435343\pi$
$620$	0	0
$621$	−5.46729e20	−0.616006
$622$	−2.93478e21	−3.26699
$623$	−1.15675e21	−1.27227
$624$	4.35280e19	0.0473025
$625$	0	0
$626$	1.82630e21	1.93760
$627$	−3.75039e20	−0.393160
$628$	8.76840e20	0.908287
$629$	5.81870e20	0.595589
$630$	0	0
$631$	−1.66495e21	−1.66410	−0.832052	−	0.554697i	$-0.812834\pi$
$631$	−1.66495e21	−1.66410	−0.832052	+	0.554697i	$0.812834\pi$
$632$	−2.33380e20	−0.230508
$633$	1.33892e20	0.130686
$634$	−2.76578e20	−0.266777
$635$	0	0
$636$	2.66847e19	0.0251382
$637$	−5.14282e19	−0.0478801
$638$	−3.39599e21	−3.12472
$639$	1.07630e21	0.978765
$640$	0	0
$641$	1.45703e21	1.29430	0.647150	−	0.762363i	$-0.275961\pi$
$641$	1.45703e21	1.29430	0.647150	+	0.762363i	$0.275961\pi$
$642$	1.84537e20	0.162021
$643$	1.13525e21	0.985166	0.492583	−	0.870266i	$-0.336053\pi$
$643$	1.13525e21	0.985166	0.492583	+	0.870266i	$0.336053\pi$
$644$	3.04439e21	2.61129
$645$	0	0
$646$	2.07543e21	1.73926
$647$	1.77938e21	1.47397	0.736983	−	0.675912i	$-0.236250\pi$
$647$	1.77938e21	1.47397	0.736983	+	0.675912i	$0.236250\pi$
$648$	−1.39284e21	−1.14048
$649$	2.44151e18	0.00197617
$650$	0	0
$651$	6.53173e19	0.0516619
$652$	−1.38630e21	−1.08393
$653$	−5.03322e20	−0.389043	−0.194521	−	0.980898i	$-0.562315\pi$
$653$	−5.03322e20	−0.389043	−0.194521	+	0.980898i	$0.562315\pi$
$654$	1.94197e20	0.148392
$655$	0	0
$656$	2.90291e19	0.0216798
$657$	1.19968e21	0.885779
$658$	5.62797e20	0.410825
$659$	−2.35681e21	−1.70092	−0.850461	−	0.526038i	$-0.823677\pi$
$659$	−2.35681e21	−1.70092	−0.850461	+	0.526038i	$0.823677\pi$
$660$	0	0
$661$	1.75909e21	1.24102	0.620508	−	0.784200i	$-0.286926\pi$
$661$	1.75909e21	1.24102	0.620508	+	0.784200i	$0.286926\pi$
$662$	2.76110e21	1.92596
$663$	−1.13749e20	−0.0784505
$664$	−8.38951e20	−0.572105
$665$	0	0
$666$	−2.19109e21	−1.46085
$667$	−3.64231e21	−2.40123
$668$	5.08388e21	3.31415
$669$	3.96514e20	0.255601
$670$	0	0
$671$	−4.51771e20	−0.284774
$672$	2.05187e20	0.127903
$673$	−1.14413e21	−0.705281	−0.352641	−	0.935759i	$-0.614716\pi$
$673$	−1.14413e21	−0.705281	−0.352641	+	0.935759i	$0.614716\pi$
$674$	3.65520e21	2.22824
$675$	0	0
$676$	−1.96357e21	−1.17070
$677$	−1.38202e21	−0.814889	−0.407444	−	0.913230i	$-0.633580\pi$
$677$	−1.38202e21	−0.814889	−0.407444	+	0.913230i	$0.633580\pi$
$678$	8.37884e20	0.488608
$679$	−3.14225e21	−1.81225
$680$	0	0
$681$	−4.11250e20	−0.232008
$682$	9.34802e20	0.521599
$683$	2.51269e21	1.38670	0.693351	−	0.720600i	$-0.256134\pi$
$683$	2.51269e21	1.38670	0.693351	+	0.720600i	$0.256134\pi$
$684$	−5.00568e21	−2.73239
$685$	0	0
$686$	3.23737e21	1.72887
$687$	2.53553e20	0.133935
$688$	3.23507e20	0.169032
$689$	7.77065e19	0.0401618
$690$	0	0
$691$	−3.22434e21	−1.63063	−0.815314	−	0.579019i	$-0.803436\pi$
$691$	−3.22434e21	−1.63063	−0.815314	+	0.579019i	$0.803436\pi$
$692$	2.48030e21	1.24082
$693$	2.21297e21	1.09516
$694$	1.06190e21	0.519862
$695$	0	0
$696$	8.78832e20	0.421053
$697$	−7.58600e19	−0.0359556
$698$	−3.62246e21	−1.69859
$699$	−3.92944e20	−0.182285
$700$	0	0
$701$	2.11598e21	0.960787	0.480393	−	0.877053i	$-0.340494\pi$
$701$	2.11598e21	0.960787	0.480393	+	0.877053i	$0.340494\pi$
$702$	8.75596e20	0.393347
$703$	−3.29532e21	−1.46465
$704$	4.00214e21	1.75994
$705$	0	0
$706$	−4.43894e21	−1.91093
$707$	−2.18591e21	−0.931079
$708$	−1.44014e18	−0.000606955 0
$709$	2.68880e21	1.12127	0.560637	−	0.828062i	$-0.310556\pi$
$709$	2.68880e21	1.12127	0.560637	+	0.828062i	$0.310556\pi$
$710$	0	0
$711$	−4.14723e20	−0.169331
$712$	4.28421e21	1.73090
$713$	1.00261e21	0.400828
$714$	5.41213e20	0.214107
$715$	0	0
$716$	−3.77891e21	−1.46393
$717$	−3.74603e20	−0.143608
$718$	−8.64778e20	−0.328073
$719$	1.17866e21	0.442510	0.221255	−	0.975216i	$-0.428985\pi$
$719$	1.17866e21	0.442510	0.221255	+	0.975216i	$0.428985\pi$
$720$	0	0
$721$	−5.49669e20	−0.202110
$722$	−7.17050e21	−2.60928
$723$	−5.25657e20	−0.189306
$724$	−5.14817e21	−1.83490
$725$	0	0
$726$	−4.16876e20	−0.145540
$727$	−1.95489e21	−0.675484	−0.337742	−	0.941239i	$-0.609663\pi$
$727$	−1.95489e21	−0.675484	−0.337742	+	0.941239i	$0.609663\pi$
$728$	−2.13906e21	−0.731541
$729$	−2.23033e21	−0.754941
$730$	0	0
$731$	−8.45399e20	−0.280338
$732$	2.66481e20	0.0874646
$733$	−5.70177e21	−1.85238	−0.926189	−	0.377059i	$-0.876935\pi$
$733$	−5.70177e21	−1.85238	−0.926189	+	0.377059i	$0.876935\pi$
$734$	7.77048e20	0.249878
$735$	0	0
$736$	3.14958e21	0.992359
$737$	3.04669e21	0.950216
$738$	2.85659e20	0.0881911
$739$	−5.49336e21	−1.67882	−0.839411	−	0.543497i	$-0.817100\pi$
$739$	−5.49336e21	−1.67882	−0.839411	+	0.543497i	$0.817100\pi$
$740$	0	0
$741$	6.44197e20	0.192922
$742$	−3.69725e20	−0.109610
$743$	6.28661e20	0.184502	0.0922509	−	0.995736i	$-0.470594\pi$
$743$	6.28661e20	0.184502	0.0922509	+	0.995736i	$0.470594\pi$
$744$	−2.41913e20	−0.0702849
$745$	0	0
$746$	3.50792e21	0.998870
$747$	−1.49084e21	−0.420268
$748$	4.96113e21	1.38458
$749$	−1.63764e21	−0.452487
$750$	0	0
$751$	−2.29542e21	−0.621673	−0.310837	−	0.950463i	$-0.600609\pi$
$751$	−2.29542e21	−0.621673	−0.310837	+	0.950463i	$0.600609\pi$
$752$	−3.76412e20	−0.100932
$753$	−2.20151e20	−0.0584465
$754$	5.83322e21	1.53329
$755$	0	0
$756$	−2.66837e21	−0.687594
$757$	2.60470e21	0.664567	0.332284	−	0.943180i	$-0.392181\pi$
$757$	2.60470e21	0.664567	0.332284	+	0.943180i	$0.392181\pi$
$758$	2.52268e21	0.637298
$759$	−1.50120e21	−0.375513
$760$	0	0
$761$	−4.31540e20	−0.105837	−0.0529183	−	0.998599i	$-0.516852\pi$
$761$	−4.31540e20	−0.105837	−0.0529183	+	0.998599i	$0.516852\pi$
$762$	−2.17564e21	−0.528353
$763$	−1.72337e21	−0.414425
$764$	2.75386e21	0.655756
$765$	0	0
$766$	2.28887e21	0.534449
$767$	−4.19374e18	−0.000969697 0
$768$	−1.38847e21	−0.317927
$769$	4.79403e21	1.08706	0.543530	−	0.839390i	$-0.317088\pi$
$769$	4.79403e21	1.08706	0.543530	+	0.839390i	$0.317088\pi$
$770$	0	0
$771$	−5.38784e20	−0.119814
$772$	5.71981e21	1.25966
$773$	4.26764e21	0.930767	0.465384	−	0.885109i	$-0.345916\pi$
$773$	4.26764e21	0.930767	0.465384	+	0.885109i	$0.345916\pi$
$774$	3.18344e21	0.687605
$775$	0	0
$776$	1.16378e22	2.46552
$777$	−8.59326e20	−0.180302
$778$	1.07432e22	2.23248
$779$	4.29620e20	0.0884205
$780$	0	0
$781$	6.04120e21	1.21967
$782$	8.30749e21	1.66119
$783$	3.19244e21	0.632279
$784$	−1.63657e20	−0.0321043
$785$	0	0
$786$	8.25631e20	0.158897
$787$	5.95930e21	1.13602	0.568008	−	0.823023i	$-0.307714\pi$
$787$	5.95930e21	1.13602	0.568008	+	0.823023i	$0.307714\pi$
$788$	−1.06776e22	−2.01616
$789$	3.46045e20	0.0647225
$790$	0	0
$791$	−7.43568e21	−1.36457
$792$	−8.19610e21	−1.48994
$793$	7.75998e20	0.139737
$794$	1.47087e21	0.262374
$795$	0	0
$796$	1.14409e21	0.200269
$797$	−8.33436e21	−1.44522	−0.722611	−	0.691255i	$-0.757058\pi$
$797$	−8.33436e21	−1.44522	−0.722611	+	0.691255i	$0.757058\pi$
$798$	−3.06506e21	−0.526523
$799$	9.83653e20	0.167394
$800$	0	0
$801$	7.61317e21	1.27152
$802$	2.41842e21	0.400152
$803$	6.73370e21	1.10379
$804$	−1.79711e21	−0.291847
$805$	0	0
$806$	−1.60569e21	−0.255946
$807$	2.45073e21	0.387029
$808$	8.09586e21	1.26671
$809$	4.00498e20	0.0620850	0.0310425	−	0.999518i	$-0.490117\pi$
$809$	4.00498e20	0.0620850	0.0310425	+	0.999518i	$0.490117\pi$
$810$	0	0
$811$	−4.61659e21	−0.702529	−0.351265	−	0.936276i	$-0.614248\pi$
$811$	−4.61659e21	−0.702529	−0.351265	+	0.936276i	$0.614248\pi$
$812$	−1.77767e22	−2.68028
$813$	3.18821e20	0.0476286
$814$	−1.22984e22	−1.82040
$815$	0	0
$816$	−3.61976e20	−0.0526022
$817$	4.78777e21	0.689394
$818$	−1.56085e22	−2.22695
$819$	−3.80118e21	−0.537390
$820$	0	0
$821$	8.29377e20	0.115127	0.0575636	−	0.998342i	$-0.481667\pi$
$821$	8.29377e20	0.115127	0.0575636	+	0.998342i	$0.481667\pi$
$822$	1.78902e21	0.246081
$823$	−7.90016e21	−1.07681	−0.538403	−	0.842688i	$-0.680972\pi$
$823$	−7.90016e21	−1.07681	−0.538403	+	0.842688i	$0.680972\pi$
$824$	2.03579e21	0.274966
$825$	0	0
$826$	1.99536e19	0.00264650
$827$	9.62655e21	1.26526	0.632629	−	0.774455i	$-0.281976\pi$
$827$	9.62655e21	1.26526	0.632629	+	0.774455i	$0.281976\pi$
$828$	−2.00367e22	−2.60974
$829$	−3.97587e21	−0.513185	−0.256592	−	0.966520i	$-0.582600\pi$
$829$	−3.97587e21	−0.513185	−0.256592	+	0.966520i	$0.582600\pi$
$830$	0	0
$831$	1.98627e21	0.251785
$832$	−6.87439e21	−0.863594
$833$	4.27674e20	0.0532446
$834$	7.83647e20	0.0966885
$835$	0	0
$836$	−2.80965e22	−3.40490
$837$	−8.78770e20	−0.105544
$838$	1.32063e22	1.57199
$839$	5.99428e21	0.707168	0.353584	−	0.935403i	$-0.384963\pi$
$839$	5.99428e21	0.707168	0.353584	+	0.935403i	$0.384963\pi$
$840$	0	0
$841$	1.26388e22	1.46466
$842$	4.70866e21	0.540825
$843$	−1.64054e21	−0.186758
$844$	1.00307e22	1.13178
$845$	0	0
$846$	−3.70405e21	−0.410581
$847$	3.69951e21	0.406461
$848$	2.47281e20	0.0269291
$849$	2.44982e21	0.264439
$850$	0	0
$851$	−1.31905e22	−1.39890
$852$	−3.56345e21	−0.374605
$853$	−5.27177e21	−0.549337	−0.274668	−	0.961539i	$-0.588568\pi$
$853$	−5.27177e21	−0.549337	−0.274668	+	0.961539i	$0.588568\pi$
$854$	−3.69217e21	−0.381371
$855$	0	0
$856$	6.06527e21	0.615598
$857$	−1.15549e21	−0.116254	−0.0581271	−	0.998309i	$-0.518513\pi$
$857$	−1.15549e21	−0.116254	−0.0581271	+	0.998309i	$0.518513\pi$
$858$	2.40420e21	0.239781
$859$	1.29668e22	1.28199	0.640995	−	0.767545i	$-0.278522\pi$
$859$	1.29668e22	1.28199	0.640995	+	0.767545i	$0.278522\pi$
$860$	0	0
$861$	1.12033e20	0.0108848
$862$	−5.64734e21	−0.543925
$863$	8.97402e21	0.856852	0.428426	−	0.903577i	$-0.359068\pi$
$863$	8.97402e21	0.856852	0.428426	+	0.903577i	$0.359068\pi$
$864$	−2.76056e21	−0.261303
$865$	0	0
$866$	8.43784e21	0.784959
$867$	−1.28473e21	−0.118486
$868$	4.89332e21	0.447410
$869$	−2.32780e21	−0.211008
$870$	0	0
$871$	−5.23324e21	−0.466267
$872$	6.38279e21	0.563815
$873$	2.06807e22	1.81117
$874$	−4.70480e22	−4.08513
$875$	0	0
$876$	−3.97192e21	−0.339016
$877$	3.50814e21	0.296879	0.148440	−	0.988921i	$-0.452575\pi$
$877$	3.50814e21	0.296879	0.148440	+	0.988921i	$0.452575\pi$
$878$	2.35830e22	1.97874
$879$	3.66689e21	0.305057
$880$	0	0
$881$	1.73060e22	1.41540	0.707699	−	0.706514i	$-0.249733\pi$
$881$	1.73060e22	1.41540	0.707699	+	0.706514i	$0.249733\pi$
$882$	−1.61045e21	−0.130597
$883$	−1.95732e22	−1.57383	−0.786914	−	0.617062i	$-0.788323\pi$
$883$	−1.95732e22	−1.57383	−0.786914	+	0.617062i	$0.788323\pi$
$884$	−8.52162e21	−0.679407
$885$	0	0
$886$	−3.85158e22	−3.01916
$887$	−6.58506e21	−0.511838	−0.255919	−	0.966698i	$-0.582378\pi$
$887$	−6.58506e21	−0.511838	−0.255919	+	0.966698i	$0.582378\pi$
$888$	3.18265e21	0.245297
$889$	1.93074e22	1.47557
$890$	0	0
$891$	−1.38926e22	−1.04400
$892$	2.97053e22	2.21359
$893$	−5.57075e21	−0.411649
$894$	5.18476e21	0.379924
$895$	0	0
$896$	2.40803e22	1.73521
$897$	2.57858e21	0.184263
$898$	1.63249e22	1.15685
$899$	−5.85437e21	−0.411417
$900$	0	0
$901$	−6.46203e20	−0.0446615
$902$	1.60338e21	0.109897
$903$	1.24851e21	0.0848663
$904$	2.75392e22	1.85647
$905$	0	0
$906$	2.75209e21	0.182474
$907$	−2.54107e22	−1.67095	−0.835473	−	0.549531i	$-0.814806\pi$
$907$	−2.54107e22	−1.67095	−0.835473	+	0.549531i	$0.814806\pi$
$908$	−3.08092e22	−2.00926
$909$	1.43866e22	0.930525
$910$	0	0
$911$	1.32653e22	0.843975	0.421987	−	0.906602i	$-0.361333\pi$
$911$	1.32653e22	0.843975	0.421987	+	0.906602i	$0.361333\pi$
$912$	2.04999e21	0.129357
$913$	−8.36796e21	−0.523707
$914$	−1.11685e22	−0.693262
$915$	0	0
$916$	1.89952e22	1.15992
$917$	−7.32694e21	−0.443765
$918$	−7.28141e21	−0.437417
$919$	−2.44915e22	−1.45932	−0.729659	−	0.683812i	$-0.760321\pi$
$919$	−2.44915e22	−1.45932	−0.729659	+	0.683812i	$0.760321\pi$
$920$	0	0
$921$	3.29088e21	0.192915
$922$	1.32268e22	0.769080
$923$	−1.03768e22	−0.598484
$924$	−7.32676e21	−0.419152
$925$	0	0
$926$	1.86348e22	1.04892
$927$	3.61765e21	0.201990
$928$	−1.83909e22	−1.01857
$929$	1.27699e22	0.701570	0.350785	−	0.936456i	$-0.385915\pi$
$929$	1.27699e22	0.701570	0.350785	+	0.936456i	$0.385915\pi$
$930$	0	0
$931$	−2.42206e21	−0.130937
$932$	−2.94378e22	−1.57865
$933$	7.57526e21	0.402981
$934$	5.93168e22	3.13023
$935$	0	0
$936$	1.40783e22	0.731106
$937$	−3.23835e22	−1.66831	−0.834156	−	0.551528i	$-0.814045\pi$
$937$	−3.23835e22	−1.66831	−0.834156	+	0.551528i	$0.814045\pi$
$938$	2.48996e22	1.27254
$939$	−4.71405e21	−0.239002
$940$	0	0
$941$	1.02130e22	0.509600	0.254800	−	0.966994i	$-0.417990\pi$
$941$	1.02130e22	0.509600	0.254800	+	0.966994i	$0.417990\pi$
$942$	−3.53364e21	−0.174920
$943$	1.71967e21	0.0844517
$944$	−1.33455e19	−0.000650196 0
$945$	0	0
$946$	1.78684e22	0.856843
$947$	3.01106e22	1.43250	0.716249	−	0.697844i	$-0.245857\pi$
$947$	3.01106e22	1.43250	0.716249	+	0.697844i	$0.245857\pi$
$948$	1.37307e21	0.0648083
$949$	−1.15663e22	−0.541626
$950$	0	0
$951$	7.13904e20	0.0329068
$952$	1.77883e22	0.813502
$953$	−1.54106e22	−0.699233	−0.349616	−	0.936893i	$-0.613688\pi$
$953$	−1.54106e22	−0.699233	−0.349616	+	0.936893i	$0.613688\pi$
$954$	2.43334e21	0.109545
$955$	0	0
$956$	−2.80638e22	−1.24369
$957$	8.76574e21	0.385433
$958$	−2.27799e22	−0.993828
$959$	−1.58764e22	−0.687248
$960$	0	0
$961$	−2.18538e22	−0.931324
$962$	2.11247e22	0.893261
$963$	1.07782e22	0.452218
$964$	−3.93802e22	−1.63946
$965$	0	0
$966$	−1.22688e22	−0.502890
$967$	−4.24903e22	−1.72819	−0.864094	−	0.503330i	$-0.832108\pi$
$967$	−4.24903e22	−1.72819	−0.864094	+	0.503330i	$0.832108\pi$
$968$	−1.37017e22	−0.552980
$969$	−5.35710e21	−0.214537
$970$	0	0
$971$	1.25803e22	0.496073	0.248037	−	0.968751i	$-0.420215\pi$
$971$	1.25803e22	0.496073	0.248037	+	0.968751i	$0.420215\pi$
$972$	2.65326e22	1.03821
$973$	−6.95436e21	−0.270029
$974$	−4.14919e22	−1.59872
$975$	0	0
$976$	2.46941e21	0.0936958
$977$	−2.40977e21	−0.0907331	−0.0453665	−	0.998970i	$-0.514446\pi$
$977$	−2.40977e21	−0.0907331	−0.0453665	+	0.998970i	$0.514446\pi$
$978$	5.58675e21	0.208746
$979$	4.27321e22	1.58447
$980$	0	0
$981$	1.13424e22	0.414178
$982$	−4.77503e21	−0.173037
$983$	4.16509e22	1.49787	0.748934	−	0.662645i	$-0.230566\pi$
$983$	4.16509e22	1.49787	0.748934	+	0.662645i	$0.230566\pi$
$984$	−4.14930e20	−0.0148085
$985$	0	0
$986$	−4.85087e22	−1.70507
$987$	−1.45269e21	−0.0506751
$988$	4.82607e22	1.67077
$989$	1.91644e22	0.658450
$990$	0	0
$991$	−2.71460e21	−0.0918656	−0.0459328	−	0.998945i	$-0.514626\pi$
$991$	−2.71460e21	−0.0918656	−0.0459328	+	0.998945i	$0.514626\pi$
$992$	5.06239e21	0.170027
$993$	−7.12695e21	−0.237566
$994$	4.93726e22	1.63338
$995$	0	0
$996$	4.93590e21	0.160850
$997$	4.95508e22	1.60264	0.801321	−	0.598234i	$-0.204131\pi$
$997$	4.95508e22	1.60264	0.801321	+	0.598234i	$0.204131\pi$
$998$	7.16434e22	2.29984
$999$	1.15613e22	0.368353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.16.a.f.1.1		6
5.2	odd	4		5.16.b.a.4.1	✓	6
5.3	odd	4		5.16.b.a.4.6	yes	6
5.4	even	2	inner	25.16.a.f.1.6		6
15.2	even	4		45.16.b.b.19.6		6
15.8	even	4		45.16.b.b.19.1		6
20.3	even	4		80.16.c.a.49.3		6
20.7	even	4		80.16.c.a.49.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.16.b.a.4.1	✓	6	5.2	odd	4
5.16.b.a.4.6	yes	6	5.3	odd	4
25.16.a.f.1.1		6	1.1	even	1	trivial
25.16.a.f.1.6		6	5.4	even	2	inner
45.16.b.b.19.1		6	15.8	even	4
45.16.b.b.19.6		6	15.2	even	4
80.16.c.a.49.3		6	20.3	even	4
80.16.c.a.49.4		6	20.7	even	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:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q-301.910 q^{2} +779.290 q^{3} +58381.4 q^{4} -235275. q^{6} +2.08791e6 q^{7} -7.73292e6 q^{8} -1.37416e7 q^{9} +O(q^{10})\)	\(q-301.910 q^{2} +779.290 q^{3} +58381.4 q^{4} -235275. q^{6} +2.08791e6 q^{7} -7.73292e6 q^{8} -1.37416e7 q^{9} -7.71305e7 q^{11} +4.54960e7 q^{12} +1.32485e8 q^{13} -6.30361e8 q^{14} +4.21600e8 q^{16} -1.10174e9 q^{17} +4.14872e9 q^{18} +6.23952e9 q^{19} +1.62709e9 q^{21} +2.32864e10 q^{22} +2.49754e10 q^{23} -6.02618e9 q^{24} -3.99986e10 q^{26} -2.18907e10 q^{27} +1.21895e11 q^{28} -1.45836e11 q^{29} +4.01436e10 q^{31} +1.26107e11 q^{32} -6.01070e10 q^{33} +3.32626e11 q^{34} -8.02254e11 q^{36} -5.28137e11 q^{37} -1.88377e12 q^{38} +1.03245e11 q^{39} +6.88546e10 q^{41} -4.91234e11 q^{42} +7.67330e11 q^{43} -4.50298e12 q^{44} -7.54032e12 q^{46} -8.92816e11 q^{47} +3.28549e11 q^{48} -3.88180e11 q^{49} -8.58576e11 q^{51} +7.73468e12 q^{52} +5.86528e11 q^{53} +6.60900e12 q^{54} -1.61457e13 q^{56} +4.86239e12 q^{57} +4.40291e13 q^{58} -3.16543e10 q^{59} +5.85723e12 q^{61} -1.21197e13 q^{62} -2.86913e13 q^{63} -5.18879e13 q^{64} +1.81469e13 q^{66} -3.95005e13 q^{67} -6.43212e13 q^{68} +1.94631e13 q^{69} -7.83244e13 q^{71} +1.06263e14 q^{72} -8.73027e13 q^{73} +1.59450e14 q^{74} +3.64272e14 q^{76} -1.61042e14 q^{77} -3.11705e13 q^{78} +3.01801e13 q^{79} +1.80118e14 q^{81} -2.07879e13 q^{82} +1.08491e14 q^{83} +9.49917e13 q^{84} -2.31664e14 q^{86} -1.13648e14 q^{87} +5.96444e14 q^{88} -5.54023e14 q^{89} +2.76618e14 q^{91} +1.45810e15 q^{92} +3.12835e13 q^{93} +2.69550e14 q^{94} +9.82739e13 q^{96} -1.50497e15 q^{97} +1.17195e14 q^{98} +1.05990e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 38568 q^{4} - 515448 q^{6} + 11416662 q^{9}+O(q^{10}) \) 6 * q + 38568 * q^4 - 515448 * q^6 + 11416662 * q^9	\( 6 q + 38568 q^{4} - 515448 q^{6} + 11416662 q^{9} - 108590088 q^{11} - 663751704 q^{14} + 1155522336 q^{16} + 3630995640 q^{19} - 8917537608 q^{21} + 2959765920 q^{24} - 81970953168 q^{26} - 286168468740 q^{29} - 276236748288 q^{31} + 127784939136 q^{34} - 3326879331864 q^{36} - 2186980965936 q^{39} - 6153278882388 q^{41} - 8250173021664 q^{44} - 23334602656488 q^{46} - 11613390856242 q^{49} - 43487373385728 q^{51} - 10162879468560 q^{54} - 59280484297440 q^{56} - 14903258326680 q^{59} - 11352061428588 q^{61} - 73265851251072 q^{64} + 76208211455904 q^{66} + 150489671962824 q^{69} + 131693145807312 q^{71} + 353606797863216 q^{74} + 959127540575520 q^{76} + 26081853939360 q^{79} + 11\!\cdots\!46 q^{81}+ \cdots - 506999099666376 q^{99}+O(q^{100}) \) 6 * q + 38568 * q^4 - 515448 * q^6 + 11416662 * q^9 - 108590088 * q^11 - 663751704 * q^14 + 1155522336 * q^16 + 3630995640 * q^19 - 8917537608 * q^21 + 2959765920 * q^24 - 81970953168 * q^26 - 286168468740 * q^29 - 276236748288 * q^31 + 127784939136 * q^34 - 3326879331864 * q^36 - 2186980965936 * q^39 - 6153278882388 * q^41 - 8250173021664 * q^44 - 23334602656488 * q^46 - 11613390856242 * q^49 - 43487373385728 * q^51 - 10162879468560 * q^54 - 59280484297440 * q^56 - 14903258326680 * q^59 - 11352061428588 * q^61 - 73265851251072 * q^64 + 76208211455904 * q^66 + 150489671962824 * q^69 + 131693145807312 * q^71 + 353606797863216 * q^74 + 959127540575520 * q^76 + 26081853939360 * q^79 + 1148641568615646 * q^81 + 686155736066976 * q^84 + 598387757484072 * q^86 - 59269690709820 * q^89 + 821206462566672 * q^91 + 308164835928456 * q^94 - 1208921682814848 * q^96 - 506999099666376 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more