LMFDB - Embedded newform 81.10.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,10,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.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:	$41.7179027293$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 $ x^8 - x^7 - 2930x^6 - 1276x^5 + 2487472x^4 + 3423248x^3 - 586568096x^2 - 2825363008x + 965565184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{18}\cdot 17 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$41.7589$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-39.7589 q^{2} +1068.77 q^{4} +846.447 q^{5} -7043.99 q^{7} -22136.7 q^{8} -33653.8 q^{10} +90103.0 q^{11} +68589.0 q^{13} +280061. q^{14} +332920. q^{16} +235749. q^{17} -232999. q^{19} +904660. q^{20} -3.58240e6 q^{22} +266252. q^{23} -1.23665e6 q^{25} -2.72703e6 q^{26} -7.52842e6 q^{28} -3.06183e6 q^{29} +3.59150e6 q^{31} -1.90255e6 q^{32} -9.37312e6 q^{34} -5.96236e6 q^{35} +5.13436e6 q^{37} +9.26378e6 q^{38} -1.87375e7 q^{40} +4.50206e6 q^{41} -3.29912e7 q^{43} +9.62997e7 q^{44} -1.05859e7 q^{46} -1.24308e7 q^{47} +9.26412e6 q^{49} +4.91680e7 q^{50} +7.33061e7 q^{52} -3.42506e7 q^{53} +7.62674e7 q^{55} +1.55931e8 q^{56} +1.21735e8 q^{58} +4.55616e7 q^{59} +6.16460e7 q^{61} -1.42794e8 q^{62} -9.48116e7 q^{64} +5.80570e7 q^{65} +5.84484e6 q^{67} +2.51962e8 q^{68} +2.37057e8 q^{70} +2.53018e8 q^{71} +3.59593e8 q^{73} -2.04137e8 q^{74} -2.49023e8 q^{76} -6.34684e8 q^{77} +2.88923e8 q^{79} +2.81799e8 q^{80} -1.78997e8 q^{82} +4.53300e8 q^{83} +1.99549e8 q^{85} +1.31170e9 q^{86} -1.99458e9 q^{88} +5.93996e8 q^{89} -4.83140e8 q^{91} +2.84563e8 q^{92} +4.94236e8 q^{94} -1.97221e8 q^{95} -2.99011e8 q^{97} -3.68332e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) $ 8 * q + 15 * q^2 + 1793 * q^4 + 453 * q^5 + 343 * q^7 + 7239 * q^8 + 510 * q^10 + 99150 * q^11 - 32435 * q^13 + 394824 * q^14 + 328193 * q^16 + 415539 * q^17 - 85277 * q^19 + 1855164 * q^20 - 529359 * q^22 + 1064559 * q^23 + 2293229 * q^25 - 1218156 * q^26 + 612862 * q^28 - 1309053 * q^29 + 2359819 * q^31 + 5760063 * q^32 - 981801 * q^34 + 15533277 * q^35 + 8195758 * q^37 + 39490203 * q^38 + 16760496 * q^40 + 54747318 * q^41 - 15249608 * q^43 + 166254963 * q^44 + 1195260 * q^46 + 156295545 * q^47 - 15239583 * q^49 + 315590163 * q^50 + 19773358 * q^52 + 262758114 * q^53 - 3789885 * q^55 + 470339790 * q^56 - 55408560 * q^58 + 307774074 * q^59 - 69192125 * q^61 + 457218462 * q^62 - 201794239 * q^64 + 482470359 * q^65 - 14328044 * q^67 + 915409575 * q^68 + 229271934 * q^70 + 619800696 * q^71 + 299306599 * q^73 + 1022736000 * q^74 - 119954093 * q^76 + 717995541 * q^77 - 30257531 * q^79 + 1463913264 * q^80 - 101188011 * q^82 + 1176168291 * q^83 - 4818366 * q^85 + 1426944009 * q^86 - 911312427 * q^88 + 1658520648 * q^89 - 369615061 * q^91 - 76813998 * q^92 + 1954316784 * q^94 - 391400652 * q^95 + 267311278 * q^97 - 2413650159 * q^98

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 $

$2$	−39.7589	−1.75711	−0.878557	−	0.477638i	$-0.841493\pi$
$2$	−39.7589	−1.75711	−0.878557	+	0.477638i	$0.841493\pi$
$3$	0	0
$3$	0	0
$4$	1068.77	2.08745
$5$	846.447	0.605668	0.302834	−	0.953043i	$-0.402067\pi$
$5$	846.447	0.605668	0.302834	+	0.953043i	$0.402067\pi$
$6$	0	0
$7$	−7043.99	−1.10886	−0.554431	−	0.832230i	$-0.687064\pi$
$7$	−7043.99	−1.10886	−0.554431	+	0.832230i	$0.687064\pi$
$8$	−22136.7	−1.91077
$9$	0	0
$10$	−33653.8	−1.06423
$11$	90103.0	1.85555	0.927774	−	0.373142i	$-0.121720\pi$
$11$	90103.0	1.85555	0.927774	+	0.373142i	$0.121720\pi$
$12$	0	0
$13$	68589.0	0.666054	0.333027	−	0.942917i	$-0.391930\pi$
$13$	68589.0	0.666054	0.333027	+	0.942917i	$0.391930\pi$
$14$	280061.	1.94840
$15$	0	0
$16$	332920.	1.26999
$17$	235749.	0.684588	0.342294	−	0.939593i	$-0.388796\pi$
$17$	235749.	0.684588	0.342294	+	0.939593i	$0.388796\pi$
$18$	0	0
$19$	−232999.	−0.410168	−0.205084	−	0.978744i	$-0.565747\pi$
$19$	−232999.	−0.410168	−0.205084	+	0.978744i	$0.565747\pi$
$20$	904660.	1.26430
$21$	0	0
$22$	−3.58240e6	−3.26041
$23$	266252.	0.198389	0.0991944	−	0.995068i	$-0.468373\pi$
$23$	266252.	0.198389	0.0991944	+	0.995068i	$0.468373\pi$
$24$	0	0
$25$	−1.23665e6	−0.633166
$26$	−2.72703e6	−1.17033
$27$	0	0
$28$	−7.52842e6	−2.31469
$29$	−3.06183e6	−0.803877	−0.401938	−	0.915667i	$-0.631663\pi$
$29$	−3.06183e6	−0.803877	−0.401938	+	0.915667i	$0.631663\pi$
$30$	0	0
$31$	3.59150e6	0.698470	0.349235	−	0.937035i	$-0.386441\pi$
$31$	3.59150e6	0.698470	0.349235	+	0.937035i	$0.386441\pi$
$32$	−1.90255e6	−0.320747
$33$	0	0
$34$	−9.37312e6	−1.20290
$35$	−5.96236e6	−0.671602
$36$	0	0
$37$	5.13436e6	0.450380	0.225190	−	0.974315i	$-0.427700\pi$
$37$	5.13436e6	0.450380	0.225190	+	0.974315i	$0.427700\pi$
$38$	9.26378e6	0.720712
$39$	0	0
$40$	−1.87375e7	−1.15729
$41$	4.50206e6	0.248819	0.124410	−	0.992231i	$-0.460296\pi$
$41$	4.50206e6	0.248819	0.124410	+	0.992231i	$0.460296\pi$
$42$	0	0
$43$	−3.29912e7	−1.47160	−0.735801	−	0.677198i	$-0.763194\pi$
$43$	−3.29912e7	−1.47160	−0.735801	+	0.677198i	$0.763194\pi$
$44$	9.62997e7	3.87336
$45$	0	0
$46$	−1.05859e7	−0.348592
$47$	−1.24308e7	−0.371586	−0.185793	−	0.982589i	$-0.559485\pi$
$47$	−1.24308e7	−0.371586	−0.185793	+	0.982589i	$0.559485\pi$
$48$	0	0
$49$	9.26412e6	0.229574
$50$	4.91680e7	1.11254
$51$	0	0
$52$	7.33061e7	1.39035
$53$	−3.42506e7	−0.596248	−0.298124	−	0.954527i	$-0.596361\pi$
$53$	−3.42506e7	−0.596248	−0.298124	+	0.954527i	$0.596361\pi$
$54$	0	0
$55$	7.62674e7	1.12385
$56$	1.55931e8	2.11878
$57$	0	0
$58$	1.21735e8	1.41250
$59$	4.55616e7	0.489514	0.244757	−	0.969584i	$-0.421292\pi$
$59$	4.55616e7	0.489514	0.244757	+	0.969584i	$0.421292\pi$
$60$	0	0
$61$	6.16460e7	0.570060	0.285030	−	0.958519i	$-0.407996\pi$
$61$	6.16460e7	0.570060	0.285030	+	0.958519i	$0.407996\pi$
$62$	−1.42794e8	−1.22729
$63$	0	0
$64$	−9.48116e7	−0.706401
$65$	5.80570e7	0.403408
$66$	0	0
$67$	5.84484e6	0.0354353	0.0177176	−	0.999843i	$-0.494360\pi$
$67$	5.84484e6	0.0354353	0.0177176	+	0.999843i	$0.494360\pi$
$68$	2.51962e8	1.42904
$69$	0	0
$70$	2.37057e8	1.18008
$71$	2.53018e8	1.18165	0.590825	−	0.806800i	$-0.298802\pi$
$71$	2.53018e8	1.18165	0.590825	+	0.806800i	$0.298802\pi$
$72$	0	0
$73$	3.59593e8	1.48203	0.741017	−	0.671486i	$-0.234344\pi$
$73$	3.59593e8	1.48203	0.741017	+	0.671486i	$0.234344\pi$
$74$	−2.04137e8	−0.791368
$75$	0	0
$76$	−2.49023e8	−0.856205
$77$	−6.34684e8	−2.05755
$78$	0	0
$79$	2.88923e8	0.834566	0.417283	−	0.908777i	$-0.362982\pi$
$79$	2.88923e8	0.834566	0.417283	+	0.908777i	$0.362982\pi$
$80$	2.81799e8	0.769192
$81$	0	0
$82$	−1.78997e8	−0.437204
$83$	4.53300e8	1.04842	0.524208	−	0.851590i	$-0.324361\pi$
$83$	4.53300e8	1.04842	0.524208	+	0.851590i	$0.324361\pi$
$84$	0	0
$85$	1.99549e8	0.414633
$86$	1.31170e9	2.58577
$87$	0	0
$88$	−1.99458e9	−3.54552
$89$	5.93996e8	1.00353	0.501763	−	0.865005i	$-0.332685\pi$
$89$	5.93996e8	1.00353	0.501763	+	0.865005i	$0.332685\pi$
$90$	0	0
$91$	−4.83140e8	−0.738562
$92$	2.84563e8	0.414126
$93$	0	0
$94$	4.94236e8	0.652919
$95$	−1.97221e8	−0.248426
$96$	0	0
$97$	−2.99011e8	−0.342937	−0.171468	−	0.985190i	$-0.554851\pi$
$97$	−2.99011e8	−0.342937	−0.171468	+	0.985190i	$0.554851\pi$
$98$	−3.68332e8	−0.403387
$99$	0	0
$100$	−1.32170e9	−1.32170
$101$	7.87432e6	0.00752951	0.00376476	−	0.999993i	$-0.498802\pi$
$101$	7.87432e6	0.00752951	0.00376476	+	0.999993i	$0.498802\pi$
$102$	0	0
$103$	−1.82491e9	−1.59762	−0.798809	−	0.601585i	$-0.794536\pi$
$103$	−1.82491e9	−1.59762	−0.798809	+	0.601585i	$0.794536\pi$
$104$	−1.51833e9	−1.27268
$105$	0	0
$106$	1.36177e9	1.04768
$107$	−4.67288e8	−0.344634	−0.172317	−	0.985042i	$-0.555125\pi$
$107$	−4.67288e8	−0.344634	−0.172317	+	0.985042i	$0.555125\pi$
$108$	0	0
$109$	1.80762e9	1.22656	0.613279	−	0.789866i	$-0.289850\pi$
$109$	1.80762e9	1.22656	0.613279	+	0.789866i	$0.289850\pi$
$110$	−3.03231e9	−1.97473
$111$	0	0
$112$	−2.34508e9	−1.40824
$113$	−1.03767e9	−0.598695	−0.299347	−	0.954144i	$-0.596769\pi$
$113$	−1.03767e9	−0.598695	−0.299347	+	0.954144i	$0.596769\pi$
$114$	0	0
$115$	2.25368e8	0.120158
$116$	−3.27240e9	−1.67805
$117$	0	0
$118$	−1.81148e9	−0.860132
$119$	−1.66061e9	−0.759113
$120$	0	0
$121$	5.76061e9	2.44306
$122$	−2.45098e9	−1.00166
$123$	0	0
$124$	3.83850e9	1.45802
$125$	−2.69998e9	−0.989157
$126$	0	0
$127$	5.65026e9	1.92731	0.963655	−	0.267150i	$-0.0860818\pi$
$127$	5.65026e9	1.92731	0.963655	+	0.267150i	$0.0860818\pi$
$128$	4.74371e9	1.56197
$129$	0	0
$130$	−2.30828e9	−0.708833
$131$	6.48482e9	1.92388	0.961939	−	0.273266i	$-0.0881039\pi$
$131$	6.48482e9	1.92388	0.961939	+	0.273266i	$0.0881039\pi$
$132$	0	0
$133$	1.64124e9	0.454820
$134$	−2.32385e8	−0.0622638
$135$	0	0
$136$	−5.21870e9	−1.30809
$137$	2.59430e9	0.629184	0.314592	−	0.949227i	$-0.398132\pi$
$137$	2.59430e9	0.629184	0.314592	+	0.949227i	$0.398132\pi$
$138$	0	0
$139$	4.38155e9	0.995547	0.497773	−	0.867307i	$-0.334151\pi$
$139$	4.38155e9	0.995547	0.497773	+	0.867307i	$0.334151\pi$
$140$	−6.37241e9	−1.40193
$141$	0	0
$142$	−1.00597e10	−2.07629
$143$	6.18008e9	1.23590
$144$	0	0
$145$	−2.59167e9	−0.486883
$146$	−1.42970e10	−2.60410
$147$	0	0
$148$	5.48747e9	0.940144
$149$	3.76602e9	0.625956	0.312978	−	0.949760i	$-0.398673\pi$
$149$	3.76602e9	0.625956	0.312978	+	0.949760i	$0.398673\pi$
$150$	0	0
$151$	9.21201e9	1.44198	0.720988	−	0.692948i	$-0.243688\pi$
$151$	9.21201e9	1.44198	0.720988	+	0.692948i	$0.243688\pi$
$152$	5.15782e9	0.783737
$153$	0	0
$154$	2.52344e10	3.61534
$155$	3.04001e9	0.423041
$156$	0	0
$157$	−3.48252e9	−0.457452	−0.228726	−	0.973491i	$-0.573456\pi$
$157$	−3.48252e9	−0.457452	−0.228726	+	0.973491i	$0.573456\pi$
$158$	−1.14873e10	−1.46643
$159$	0	0
$160$	−1.61041e9	−0.194266
$161$	−1.87547e9	−0.219986
$162$	0	0
$163$	−6.34728e9	−0.704277	−0.352139	−	0.935948i	$-0.614545\pi$
$163$	−6.34728e9	−0.704277	−0.352139	+	0.935948i	$0.614545\pi$
$164$	4.81168e9	0.519397
$165$	0	0
$166$	−1.80227e10	−1.84219
$167$	−6.81454e9	−0.677973	−0.338986	−	0.940791i	$-0.610084\pi$
$167$	−6.81454e9	−0.677973	−0.338986	+	0.940791i	$0.610084\pi$
$168$	0	0
$169$	−5.90005e9	−0.556372
$170$	−7.93385e9	−0.728557
$171$	0	0
$172$	−3.52601e10	−3.07189
$173$	−9.90370e9	−0.840601	−0.420300	−	0.907385i	$-0.638075\pi$
$173$	−9.90370e9	−0.840601	−0.420300	+	0.907385i	$0.638075\pi$
$174$	0	0
$175$	8.71096e9	0.702094
$176$	2.99971e10	2.35653
$177$	0	0
$178$	−2.36167e10	−1.76331
$179$	−1.34679e9	−0.0980532	−0.0490266	−	0.998797i	$-0.515612\pi$
$179$	−1.34679e9	−0.0980532	−0.0490266	+	0.998797i	$0.515612\pi$
$180$	0	0
$181$	−1.53579e10	−1.06360	−0.531800	−	0.846870i	$-0.678484\pi$
$181$	−1.53579e10	−1.06360	−0.531800	+	0.846870i	$0.678484\pi$
$182$	1.92091e10	1.29774
$183$	0	0
$184$	−5.89394e9	−0.379075
$185$	4.34597e9	0.272781
$186$	0	0
$187$	2.12417e10	1.27029
$188$	−1.32857e10	−0.775667
$189$	0	0
$190$	7.84130e9	0.436512
$191$	2.78626e10	1.51486	0.757429	−	0.652918i	$-0.226455\pi$
$191$	2.78626e10	1.51486	0.757429	+	0.652918i	$0.226455\pi$
$192$	0	0
$193$	−1.47753e10	−0.766528	−0.383264	−	0.923639i	$-0.625200\pi$
$193$	−1.47753e10	−0.766528	−0.383264	+	0.923639i	$0.625200\pi$
$194$	1.18883e10	0.602578
$195$	0	0
$196$	9.90125e9	0.479223
$197$	2.08977e10	0.988554	0.494277	−	0.869304i	$-0.335433\pi$
$197$	2.08977e10	0.988554	0.494277	+	0.869304i	$0.335433\pi$
$198$	0	0
$199$	2.77242e10	1.25320	0.626600	−	0.779341i	$-0.284446\pi$
$199$	2.77242e10	1.25320	0.626600	+	0.779341i	$0.284446\pi$
$200$	2.73754e10	1.20983
$201$	0	0
$202$	−3.13075e8	−0.0132302
$203$	2.15675e10	0.891388
$204$	0	0
$205$	3.81075e9	0.150702
$206$	7.25563e10	2.80720
$207$	0	0
$208$	2.28347e10	0.845881
$209$	−2.09939e10	−0.761087
$210$	0	0
$211$	−1.09165e10	−0.379152	−0.189576	−	0.981866i	$-0.560711\pi$
$211$	−1.09165e10	−0.379152	−0.189576	+	0.981866i	$0.560711\pi$
$212$	−3.66061e10	−1.24464
$213$	0	0
$214$	1.85789e10	0.605561
$215$	−2.79253e10	−0.891302
$216$	0	0
$217$	−2.52985e10	−0.774507
$218$	−7.18691e10	−2.15520
$219$	0	0
$220$	8.15126e10	2.34597
$221$	1.61698e10	0.455972
$222$	0	0
$223$	−5.60254e10	−1.51709	−0.758547	−	0.651618i	$-0.774091\pi$
$223$	−5.60254e10	−1.51709	−0.758547	+	0.651618i	$0.774091\pi$
$224$	1.34016e10	0.355663
$225$	0	0
$226$	4.12566e10	1.05197
$227$	2.56209e10	0.640439	0.320219	−	0.947343i	$-0.396243\pi$
$227$	2.56209e10	0.640439	0.320219	+	0.947343i	$0.396243\pi$
$228$	0	0
$229$	8.09411e9	0.194495	0.0972477	−	0.995260i	$-0.468996\pi$
$229$	8.09411e9	0.194495	0.0972477	+	0.995260i	$0.468996\pi$
$230$	−8.96039e9	−0.211131
$231$	0	0
$232$	6.77787e10	1.53602
$233$	2.40857e10	0.535374	0.267687	−	0.963506i	$-0.413741\pi$
$233$	2.40857e10	0.535374	0.267687	+	0.963506i	$0.413741\pi$
$234$	0	0
$235$	−1.05220e10	−0.225058
$236$	4.86950e10	1.02183
$237$	0	0
$238$	6.60241e10	1.33385
$239$	−3.66341e10	−0.726264	−0.363132	−	0.931738i	$-0.618293\pi$
$239$	−3.66341e10	−0.726264	−0.363132	+	0.931738i	$0.618293\pi$
$240$	0	0
$241$	−8.26453e10	−1.57813	−0.789063	−	0.614312i	$-0.789434\pi$
$241$	−8.26453e10	−1.57813	−0.789063	+	0.614312i	$0.789434\pi$
$242$	−2.29036e11	−4.29273
$243$	0	0
$244$	6.58855e10	1.18997
$245$	7.84159e9	0.139045
$246$	0	0
$247$	−1.59811e10	−0.273194
$248$	−7.95039e10	−1.33461
$249$	0	0
$250$	1.07348e11	1.73806
$251$	8.40180e10	1.33611	0.668053	−	0.744114i	$-0.267128\pi$
$251$	8.40180e10	1.33611	0.668053	+	0.744114i	$0.267128\pi$
$252$	0	0
$253$	2.39901e10	0.368120
$254$	−2.24648e11	−3.38650
$255$	0	0
$256$	−1.40061e11	−2.03816
$257$	1.23973e11	1.77267	0.886337	−	0.463041i	$-0.153242\pi$
$257$	1.23973e11	1.77267	0.886337	+	0.463041i	$0.153242\pi$
$258$	0	0
$259$	−3.61664e10	−0.499409
$260$	6.20497e10	0.842092
$261$	0	0
$262$	−2.57830e11	−3.38047
$263$	1.07755e11	1.38879	0.694393	−	0.719596i	$-0.255673\pi$
$263$	1.07755e11	1.38879	0.694393	+	0.719596i	$0.255673\pi$
$264$	0	0
$265$	−2.89913e10	−0.361128
$266$	−6.52539e10	−0.799170
$267$	0	0
$268$	6.24681e9	0.0739693
$269$	5.47369e10	0.637375	0.318688	−	0.947860i	$-0.396758\pi$
$269$	5.47369e10	0.637375	0.318688	+	0.947860i	$0.396758\pi$
$270$	0	0
$271$	7.34345e9	0.0827063	0.0413531	−	0.999145i	$-0.486833\pi$
$271$	7.34345e9	0.0827063	0.0413531	+	0.999145i	$0.486833\pi$
$272$	7.84855e10	0.869419
$273$	0	0
$274$	−1.03147e11	−1.10555
$275$	−1.11426e11	−1.17487
$276$	0	0
$277$	7.60255e10	0.775890	0.387945	−	0.921682i	$-0.373185\pi$
$277$	7.60255e10	0.775890	0.387945	+	0.921682i	$0.373185\pi$
$278$	−1.74206e11	−1.74929
$279$	0	0
$280$	1.31987e11	1.28328
$281$	−2.28385e10	−0.218519	−0.109259	−	0.994013i	$-0.534848\pi$
$281$	−2.28385e10	−0.218519	−0.109259	+	0.994013i	$0.534848\pi$
$282$	0	0
$283$	−1.98438e11	−1.83902	−0.919511	−	0.393064i	$-0.871415\pi$
$283$	−1.98438e11	−1.83902	−0.919511	+	0.393064i	$0.871415\pi$
$284$	2.70419e11	2.46663
$285$	0	0
$286$	−2.45713e11	−2.17161
$287$	−3.17124e10	−0.275906
$288$	0	0
$289$	−6.30104e10	−0.531340
$290$	1.03042e11	0.855508
$291$	0	0
$292$	3.84323e11	3.09367
$293$	−4.15371e10	−0.329254	−0.164627	−	0.986356i	$-0.552642\pi$
$293$	−4.15371e10	−0.329254	−0.164627	+	0.986356i	$0.552642\pi$
$294$	0	0
$295$	3.85655e10	0.296483
$296$	−1.13658e11	−0.860571
$297$	0	0
$298$	−1.49733e11	−1.09988
$299$	1.82619e10	0.132138
$300$	0	0
$301$	2.32390e11	1.63180
$302$	−3.66260e11	−2.53371
$303$	0	0
$304$	−7.75699e10	−0.520909
$305$	5.21800e10	0.345267
$306$	0	0
$307$	1.45036e10	0.0931864	0.0465932	−	0.998914i	$-0.485164\pi$
$307$	1.45036e10	0.0931864	0.0465932	+	0.998914i	$0.485164\pi$
$308$	−6.78334e11	−4.29502
$309$	0	0
$310$	−1.20868e11	−0.743331
$311$	2.96131e10	0.179499	0.0897494	−	0.995964i	$-0.471393\pi$
$311$	2.96131e10	0.179499	0.0897494	+	0.995964i	$0.471393\pi$
$312$	0	0
$313$	−9.26854e10	−0.545836	−0.272918	−	0.962037i	$-0.587989\pi$
$313$	−9.26854e10	−0.545836	−0.272918	+	0.962037i	$0.587989\pi$
$314$	1.38461e11	0.803795
$315$	0	0
$316$	3.08793e11	1.74211
$317$	1.36748e11	0.760594	0.380297	−	0.924864i	$-0.375822\pi$
$317$	1.36748e11	0.760594	0.380297	+	0.924864i	$0.375822\pi$
$318$	0	0
$319$	−2.75880e11	−1.49163
$320$	−8.02529e10	−0.427845
$321$	0	0
$322$	7.45668e10	0.386540
$323$	−5.49291e10	−0.280796
$324$	0	0
$325$	−8.48208e10	−0.421723
$326$	2.52361e11	1.23749
$327$	0	0
$328$	−9.96608e10	−0.475436
$329$	8.75626e10	0.412038
$330$	0	0
$331$	3.13607e11	1.43602	0.718009	−	0.696034i	$-0.245054\pi$
$331$	3.13607e11	1.43602	0.718009	+	0.696034i	$0.245054\pi$
$332$	4.84475e11	2.18852
$333$	0	0
$334$	2.70939e11	1.19128
$335$	4.94735e9	0.0214620
$336$	0	0
$337$	2.89098e11	1.22099	0.610493	−	0.792022i	$-0.290971\pi$
$337$	2.89098e11	1.22099	0.610493	+	0.792022i	$0.290971\pi$
$338$	2.34580e11	0.977609
$339$	0	0
$340$	2.13272e11	0.865524
$341$	3.23605e11	1.29605
$342$	0	0
$343$	2.18994e11	0.854296
$344$	7.30317e11	2.81189
$345$	0	0
$346$	3.93760e11	1.47703
$347$	2.24860e11	0.832585	0.416293	−	0.909231i	$-0.363329\pi$
$347$	2.24860e11	0.832585	0.416293	+	0.909231i	$0.363329\pi$
$348$	0	0
$349$	−8.79918e10	−0.317488	−0.158744	−	0.987320i	$-0.550744\pi$
$349$	−8.79918e10	−0.317488	−0.158744	+	0.987320i	$0.550744\pi$
$350$	−3.46339e11	−1.23366
$351$	0	0
$352$	−1.71426e11	−0.595161
$353$	−3.82103e11	−1.30977	−0.654884	−	0.755730i	$-0.727282\pi$
$353$	−3.82103e11	−1.30977	−0.654884	+	0.755730i	$0.727282\pi$
$354$	0	0
$355$	2.14166e11	0.715688
$356$	6.34847e11	2.09481
$357$	0	0
$358$	5.35470e10	0.172291
$359$	−4.78348e11	−1.51991	−0.759957	−	0.649973i	$-0.774780\pi$
$359$	−4.78348e11	−1.51991	−0.759957	+	0.649973i	$0.774780\pi$
$360$	0	0
$361$	−2.68399e11	−0.831762
$362$	6.10614e11	1.86887
$363$	0	0
$364$	−5.16367e11	−1.54171
$365$	3.04376e11	0.897621
$366$	0	0
$367$	2.26517e11	0.651783	0.325892	−	0.945407i	$-0.394336\pi$
$367$	2.26517e11	0.651783	0.325892	+	0.945407i	$0.394336\pi$
$368$	8.86405e10	0.251952
$369$	0	0
$370$	−1.72791e11	−0.479307
$371$	2.41261e11	0.661156
$372$	0	0
$373$	4.77159e11	1.27636	0.638180	−	0.769888i	$-0.279688\pi$
$373$	4.77159e11	1.27636	0.638180	+	0.769888i	$0.279688\pi$
$374$	−8.44546e11	−2.23204
$375$	0	0
$376$	2.75178e11	0.710015
$377$	−2.10008e11	−0.535426
$378$	0	0
$379$	−5.69565e10	−0.141797	−0.0708985	−	0.997484i	$-0.522587\pi$
$379$	−5.69565e10	−0.141797	−0.0708985	+	0.997484i	$0.522587\pi$
$380$	−2.10784e11	−0.518576
$381$	0	0
$382$	−1.10779e12	−2.66178
$383$	2.23806e10	0.0531469	0.0265734	−	0.999647i	$-0.491540\pi$
$383$	2.23806e10	0.0531469	0.0265734	+	0.999647i	$0.491540\pi$
$384$	0	0
$385$	−5.37227e11	−1.24619
$386$	5.87449e11	1.34688
$387$	0	0
$388$	−3.19575e11	−0.715862
$389$	−4.22707e11	−0.935980	−0.467990	−	0.883734i	$-0.655022\pi$
$389$	−4.22707e11	−0.935980	−0.467990	+	0.883734i	$0.655022\pi$
$390$	0	0
$391$	6.27685e10	0.135815
$392$	−2.05077e11	−0.438662
$393$	0	0
$394$	−8.30870e11	−1.73700
$395$	2.44558e11	0.505470
$396$	0	0
$397$	5.15461e11	1.04145	0.520725	−	0.853724i	$-0.325662\pi$
$397$	5.15461e11	1.04145	0.520725	+	0.853724i	$0.325662\pi$
$398$	−1.10229e12	−2.20202
$399$	0	0
$400$	−4.11706e11	−0.804114
$401$	−6.41659e9	−0.0123924	−0.00619619	−	0.999981i	$-0.501972\pi$
$401$	−6.41659e9	−0.0123924	−0.00619619	+	0.999981i	$0.501972\pi$
$402$	0	0
$403$	2.46337e11	0.465219
$404$	8.41586e9	0.0157175
$405$	0	0
$406$	−8.57499e11	−1.56627
$407$	4.62622e11	0.835702
$408$	0	0
$409$	−3.48484e11	−0.615783	−0.307891	−	0.951421i	$-0.599623\pi$
$409$	−3.48484e11	−0.615783	−0.307891	+	0.951421i	$0.599623\pi$
$410$	−1.51512e11	−0.264800
$411$	0	0
$412$	−1.95041e12	−3.33494
$413$	−3.20935e11	−0.542803
$414$	0	0
$415$	3.83694e11	0.634993
$416$	−1.30494e11	−0.213635
$417$	0	0
$418$	8.34694e11	1.33732
$419$	−3.79268e11	−0.601150	−0.300575	−	0.953758i	$-0.597179\pi$
$419$	−3.79268e11	−0.601150	−0.300575	+	0.953758i	$0.597179\pi$
$420$	0	0
$421$	1.14436e12	1.77538	0.887690	−	0.460441i	$-0.152309\pi$
$421$	1.14436e12	1.77538	0.887690	+	0.460441i	$0.152309\pi$
$422$	4.34030e11	0.666213
$423$	0	0
$424$	7.58196e11	1.13929
$425$	−2.91539e11	−0.433458
$426$	0	0
$427$	−4.34233e11	−0.632117
$428$	−4.99425e11	−0.719405
$429$	0	0
$430$	1.11028e12	1.56612
$431$	3.08594e11	0.430764	0.215382	−	0.976530i	$-0.430900\pi$
$431$	3.08594e11	0.430764	0.215382	+	0.976530i	$0.430900\pi$
$432$	0	0
$433$	−3.53112e11	−0.482744	−0.241372	−	0.970433i	$-0.577597\pi$
$433$	−3.53112e11	−0.482744	−0.241372	+	0.970433i	$0.577597\pi$
$434$	1.00584e12	1.36090
$435$	0	0
$436$	1.93194e12	2.56038
$437$	−6.20363e10	−0.0813728
$438$	0	0
$439$	−5.94705e11	−0.764207	−0.382104	−	0.924119i	$-0.624800\pi$
$439$	−5.94705e11	−0.764207	−0.382104	+	0.924119i	$0.624800\pi$
$440$	−1.68831e12	−2.14741
$441$	0	0
$442$	−6.42893e11	−0.801195
$443$	1.90700e11	0.235252	0.117626	−	0.993058i	$-0.462472\pi$
$443$	1.90700e11	0.235252	0.117626	+	0.993058i	$0.462472\pi$
$444$	0	0
$445$	5.02786e11	0.607804
$446$	2.22751e12	2.66571
$447$	0	0
$448$	6.67851e11	0.783301
$449$	−2.55836e11	−0.297066	−0.148533	−	0.988907i	$-0.547455\pi$
$449$	−2.55836e11	−0.297066	−0.148533	+	0.988907i	$0.547455\pi$
$450$	0	0
$451$	4.05649e11	0.461696
$452$	−1.10903e12	−1.24974
$453$	0	0
$454$	−1.01866e12	−1.12532
$455$	−4.08952e11	−0.447323
$456$	0	0
$457$	4.04694e11	0.434014	0.217007	−	0.976170i	$-0.430371\pi$
$457$	4.04694e11	0.434014	0.217007	+	0.976170i	$0.430371\pi$
$458$	−3.21813e11	−0.341750
$459$	0	0
$460$	2.40867e11	0.250823
$461$	−5.66036e11	−0.583700	−0.291850	−	0.956464i	$-0.594271\pi$
$461$	−5.66036e11	−0.583700	−0.291850	+	0.956464i	$0.594271\pi$
$462$	0	0
$463$	−4.61864e11	−0.467089	−0.233544	−	0.972346i	$-0.575032\pi$
$463$	−4.61864e11	−0.467089	−0.233544	+	0.972346i	$0.575032\pi$
$464$	−1.01934e12	−1.02091
$465$	0	0
$466$	−9.57621e11	−0.940713
$467$	1.14378e12	1.11280	0.556398	−	0.830916i	$-0.312183\pi$
$467$	1.14378e12	1.11280	0.556398	+	0.830916i	$0.312183\pi$
$468$	0	0
$469$	−4.11710e10	−0.0392928
$470$	4.18345e11	0.395452
$471$	0	0
$472$	−1.00858e12	−0.935348
$473$	−2.97261e12	−2.73063
$474$	0	0
$475$	2.88138e11	0.259705
$476$	−1.77482e12	−1.58461
$477$	0	0
$478$	1.45653e12	1.27613
$479$	1.43929e12	1.24922	0.624610	−	0.780936i	$-0.285258\pi$
$479$	1.43929e12	1.24922	0.624610	+	0.780936i	$0.285258\pi$
$480$	0	0
$481$	3.52161e11	0.299977
$482$	3.28589e12	2.77295
$483$	0	0
$484$	6.15678e12	5.09976
$485$	−2.53097e11	−0.207706
$486$	0	0
$487$	3.07201e11	0.247481	0.123740	−	0.992315i	$-0.460511\pi$
$487$	3.07201e11	0.247481	0.123740	+	0.992315i	$0.460511\pi$
$488$	−1.36464e12	−1.08925
$489$	0	0
$490$	−3.11773e11	−0.244319
$491$	−1.32999e12	−1.03272	−0.516358	−	0.856373i	$-0.672713\pi$
$491$	−1.32999e12	−1.03272	−0.516358	+	0.856373i	$0.672713\pi$
$492$	0	0
$493$	−7.21822e11	−0.550324
$494$	6.35393e11	0.480033
$495$	0	0
$496$	1.19568e12	0.887050
$497$	−1.78226e12	−1.31029
$498$	0	0
$499$	−2.71582e11	−0.196087	−0.0980433	−	0.995182i	$-0.531258\pi$
$499$	−2.71582e11	−0.196087	−0.0980433	+	0.995182i	$0.531258\pi$
$500$	−2.88566e12	−2.06481
$501$	0	0
$502$	−3.34047e12	−2.34769
$503$	−2.43504e12	−1.69610	−0.848049	−	0.529917i	$-0.822223\pi$
$503$	−2.43504e12	−1.69610	−0.848049	+	0.529917i	$0.822223\pi$
$504$	0	0
$505$	6.66519e9	0.00456039
$506$	−9.53820e11	−0.646829
$507$	0	0
$508$	6.03884e12	4.02316
$509$	2.49952e12	1.65054	0.825272	−	0.564736i	$-0.191022\pi$
$509$	2.49952e12	1.65054	0.825272	+	0.564736i	$0.191022\pi$
$510$	0	0
$511$	−2.53297e12	−1.64337
$512$	3.13991e12	2.01931
$513$	0	0
$514$	−4.92904e12	−3.11479
$515$	−1.54469e12	−0.967626
$516$	0	0
$517$	−1.12006e12	−0.689496
$518$	1.43794e12	0.877518
$519$	0	0
$520$	−1.28519e12	−0.770819
$521$	−1.97181e11	−0.117245	−0.0586225	−	0.998280i	$-0.518671\pi$
$521$	−1.97181e11	−0.117245	−0.0586225	+	0.998280i	$0.518671\pi$
$522$	0	0
$523$	−1.84314e12	−1.07721	−0.538606	−	0.842558i	$-0.681049\pi$
$523$	−1.84314e12	−1.07721	−0.538606	+	0.842558i	$0.681049\pi$
$524$	6.93080e12	4.01599
$525$	0	0
$526$	−4.28421e12	−2.44025
$527$	8.46691e11	0.478164
$528$	0	0
$529$	−1.73026e12	−0.960642
$530$	1.15266e12	0.634543
$531$	0	0
$532$	1.75411e12	0.949412
$533$	3.08792e11	0.165727
$534$	0	0
$535$	−3.95535e11	−0.208734
$536$	−1.29385e11	−0.0677086
$537$	0	0
$538$	−2.17628e12	−1.11994
$539$	8.34726e11	0.425985
$540$	0	0
$541$	1.54291e12	0.774378	0.387189	−	0.922000i	$-0.373446\pi$
$541$	1.54291e12	0.774378	0.387189	+	0.922000i	$0.373446\pi$
$542$	−2.91968e11	−0.145324
$543$	0	0
$544$	−4.48524e11	−0.219579
$545$	1.53006e12	0.742887
$546$	0	0
$547$	1.05810e12	0.505342	0.252671	−	0.967552i	$-0.418691\pi$
$547$	1.05810e12	0.505342	0.252671	+	0.967552i	$0.418691\pi$
$548$	2.77272e12	1.31339
$549$	0	0
$550$	4.43019e12	2.06438
$551$	7.13401e11	0.329725
$552$	0	0
$553$	−2.03517e12	−0.925418
$554$	−3.02269e12	−1.36333
$555$	0	0
$556$	4.68289e12	2.07815
$557$	−3.41690e11	−0.150413	−0.0752063	−	0.997168i	$-0.523962\pi$
$557$	−3.41690e11	−0.150413	−0.0752063	+	0.997168i	$0.523962\pi$
$558$	0	0
$559$	−2.26283e12	−0.980166
$560$	−1.98499e12	−0.852927
$561$	0	0
$562$	9.08034e11	0.383962
$563$	2.39315e12	1.00388	0.501941	−	0.864902i	$-0.332619\pi$
$563$	2.39315e12	1.00388	0.501941	+	0.864902i	$0.332619\pi$
$564$	0	0
$565$	−8.78331e11	−0.362610
$566$	7.88970e12	3.23137
$567$	0	0
$568$	−5.60099e12	−2.25786
$569$	1.33349e12	0.533317	0.266659	−	0.963791i	$-0.414080\pi$
$569$	1.33349e12	0.533317	0.266659	+	0.963791i	$0.414080\pi$
$570$	0	0
$571$	1.56871e12	0.617561	0.308781	−	0.951133i	$-0.400079\pi$
$571$	1.56871e12	0.617561	0.308781	+	0.951133i	$0.400079\pi$
$572$	6.60510e12	2.57987
$573$	0	0
$574$	1.26085e12	0.484798
$575$	−3.29261e11	−0.125613
$576$	0	0
$577$	−1.18747e12	−0.445996	−0.222998	−	0.974819i	$-0.571584\pi$
$577$	−1.18747e12	−0.445996	−0.222998	+	0.974819i	$0.571584\pi$
$578$	2.50523e12	0.933624
$579$	0	0
$580$	−2.76991e12	−1.01634
$581$	−3.19304e12	−1.16255
$582$	0	0
$583$	−3.08608e12	−1.10637
$584$	−7.96020e12	−2.83182
$585$	0	0
$586$	1.65147e12	0.578537
$587$	−5.57957e12	−1.93968	−0.969839	−	0.243746i	$-0.921624\pi$
$587$	−5.57957e12	−1.93968	−0.969839	+	0.243746i	$0.921624\pi$
$588$	0	0
$589$	−8.36814e11	−0.286490
$590$	−1.53332e12	−0.520954
$591$	0	0
$592$	1.70933e12	0.571977
$593$	−3.44549e12	−1.14421	−0.572105	−	0.820181i	$-0.693873\pi$
$593$	−3.44549e12	−1.14421	−0.572105	+	0.820181i	$0.693873\pi$
$594$	0	0
$595$	−1.40562e12	−0.459771
$596$	4.02502e12	1.30665
$597$	0	0
$598$	−7.26075e11	−0.232181
$599$	−1.84366e12	−0.585142	−0.292571	−	0.956244i	$-0.594511\pi$
$599$	−1.84366e12	−0.585142	−0.292571	+	0.956244i	$0.594511\pi$
$600$	0	0
$601$	5.04168e12	1.57630	0.788152	−	0.615481i	$-0.211038\pi$
$601$	5.04168e12	1.57630	0.788152	+	0.615481i	$0.211038\pi$
$602$	−9.23956e12	−2.86726
$603$	0	0
$604$	9.84554e12	3.01005
$605$	4.87605e12	1.47968
$606$	0	0
$607$	9.82998e11	0.293903	0.146951	−	0.989144i	$-0.453054\pi$
$607$	9.82998e11	0.293903	0.146951	+	0.989144i	$0.453054\pi$
$608$	4.43292e11	0.131560
$609$	0	0
$610$	−2.07462e12	−0.606673
$611$	−8.52618e11	−0.247497
$612$	0	0
$613$	−5.27728e12	−1.50952	−0.754758	−	0.656003i	$-0.772246\pi$
$613$	−5.27728e12	−1.50952	−0.754758	+	0.656003i	$0.772246\pi$
$614$	−5.76647e11	−0.163739
$615$	0	0
$616$	1.40498e13	3.93149
$617$	−5.00458e12	−1.39022	−0.695111	−	0.718902i	$-0.744645\pi$
$617$	−5.00458e12	−1.39022	−0.695111	+	0.718902i	$0.744645\pi$
$618$	0	0
$619$	5.24307e12	1.43542	0.717708	−	0.696344i	$-0.245191\pi$
$619$	5.24307e12	1.43542	0.717708	+	0.696344i	$0.245191\pi$
$620$	3.24908e12	0.883076
$621$	0	0
$622$	−1.17738e12	−0.315400
$623$	−4.18410e12	−1.11277
$624$	0	0
$625$	1.29950e11	0.0340655
$626$	3.68507e12	0.959095
$627$	0	0
$628$	−3.72203e12	−0.954907
$629$	1.21042e12	0.308324
$630$	0	0
$631$	4.84088e12	1.21560	0.607802	−	0.794089i	$-0.292052\pi$
$631$	4.84088e12	1.21560	0.607802	+	0.794089i	$0.292052\pi$
$632$	−6.39581e12	−1.59466
$633$	0	0
$634$	−5.43694e12	−1.33645
$635$	4.78264e12	1.16731
$636$	0	0
$637$	6.35417e11	0.152908
$638$	1.09687e13	2.62097
$639$	0	0
$640$	4.01530e12	0.946037
$641$	−2.44216e12	−0.571364	−0.285682	−	0.958325i	$-0.592220\pi$
$641$	−2.44216e12	−0.571364	−0.285682	+	0.958325i	$0.592220\pi$
$642$	0	0
$643$	5.59949e12	1.29181	0.645906	−	0.763417i	$-0.276480\pi$
$643$	5.59949e12	1.29181	0.645906	+	0.763417i	$0.276480\pi$
$644$	−2.00446e12	−0.459209
$645$	0	0
$646$	2.18392e12	0.493391
$647$	7.09599e12	1.59200	0.796002	−	0.605295i	$-0.206945\pi$
$647$	7.09599e12	1.59200	0.796002	+	0.605295i	$0.206945\pi$
$648$	0	0
$649$	4.10524e12	0.908317
$650$	3.37238e12	0.741015
$651$	0	0
$652$	−6.78380e12	−1.47014
$653$	1.97712e11	0.0425523	0.0212762	−	0.999774i	$-0.493227\pi$
$653$	1.97712e11	0.0425523	0.0212762	+	0.999774i	$0.493227\pi$
$654$	0	0
$655$	5.48906e12	1.16523
$656$	1.49883e12	0.315998
$657$	0	0
$658$	−3.48139e12	−0.723997
$659$	4.95568e12	1.02357	0.511786	−	0.859113i	$-0.328984\pi$
$659$	4.95568e12	1.02357	0.511786	+	0.859113i	$0.328984\pi$
$660$	0	0
$661$	−3.62405e12	−0.738393	−0.369196	−	0.929351i	$-0.620367\pi$
$661$	−3.62405e12	−0.738393	−0.369196	+	0.929351i	$0.620367\pi$
$662$	−1.24687e13	−2.52325
$663$	0	0
$664$	−1.00346e13	−2.00328
$665$	1.38922e12	0.275470
$666$	0	0
$667$	−8.15216e11	−0.159480
$668$	−7.28319e12	−1.41523
$669$	0	0
$670$	−1.96701e11	−0.0377112
$671$	5.55449e12	1.05777
$672$	0	0
$673$	−3.33531e12	−0.626713	−0.313356	−	0.949636i	$-0.601453\pi$
$673$	−3.33531e12	−0.626713	−0.313356	+	0.949636i	$0.601453\pi$
$674$	−1.14942e13	−2.14541
$675$	0	0
$676$	−6.30581e12	−1.16140
$677$	4.18668e12	0.765986	0.382993	−	0.923751i	$-0.374893\pi$
$677$	4.18668e12	0.765986	0.382993	+	0.923751i	$0.374893\pi$
$678$	0	0
$679$	2.10623e12	0.380269
$680$	−4.41735e12	−0.792268
$681$	0	0
$682$	−1.28662e13	−2.27730
$683$	−3.28484e12	−0.577591	−0.288796	−	0.957391i	$-0.593255\pi$
$683$	−3.28484e12	−0.577591	−0.288796	+	0.957391i	$0.593255\pi$
$684$	0	0
$685$	2.19594e12	0.381077
$686$	−8.70696e12	−1.50110
$687$	0	0
$688$	−1.09834e13	−1.86892
$689$	−2.34922e12	−0.397133
$690$	0	0
$691$	−2.69533e12	−0.449738	−0.224869	−	0.974389i	$-0.572195\pi$
$691$	−2.69533e12	−0.449738	−0.224869	+	0.974389i	$0.572195\pi$
$692$	−1.05848e13	−1.75471
$693$	0	0
$694$	−8.94018e12	−1.46295
$695$	3.70875e12	0.602971
$696$	0	0
$697$	1.06135e12	0.170339
$698$	3.49846e12	0.557863
$699$	0	0
$700$	9.31004e12	1.46558
$701$	−8.26238e12	−1.29233	−0.646166	−	0.763197i	$-0.723628\pi$
$701$	−8.26238e12	−1.29233	−0.646166	+	0.763197i	$0.723628\pi$
$702$	0	0
$703$	−1.19630e12	−0.184732
$704$	−8.54281e12	−1.31076
$705$	0	0
$706$	1.51920e13	2.30141
$707$	−5.54666e10	−0.00834919
$708$	0	0
$709$	−7.89394e12	−1.17324	−0.586619	−	0.809863i	$-0.699541\pi$
$709$	−7.89394e12	−1.17324	−0.586619	+	0.809863i	$0.699541\pi$
$710$	−8.51503e12	−1.25754
$711$	0	0
$712$	−1.31491e13	−1.91751
$713$	9.56242e11	0.138569
$714$	0	0
$715$	5.23111e12	0.748543
$716$	−1.43942e12	−0.204681
$717$	0	0
$718$	1.90186e13	2.67066
$719$	9.37950e12	1.30888	0.654440	−	0.756114i	$-0.272904\pi$
$719$	9.37950e12	1.30888	0.654440	+	0.756114i	$0.272904\pi$
$720$	0	0
$721$	1.28546e13	1.77154
$722$	1.06713e13	1.46150
$723$	0	0
$724$	−1.64141e13	−2.22021
$725$	3.78642e12	0.508988
$726$	0	0
$727$	3.79110e11	0.0503339	0.0251670	−	0.999683i	$-0.491988\pi$
$727$	3.79110e11	0.0503339	0.0251670	+	0.999683i	$0.491988\pi$
$728$	1.06951e13	1.41122
$729$	0	0
$730$	−1.21017e13	−1.57722
$731$	−7.77763e12	−1.00744
$732$	0	0
$733$	−8.09870e12	−1.03621	−0.518104	−	0.855317i	$-0.673362\pi$
$733$	−8.09870e12	−1.03621	−0.518104	+	0.855317i	$0.673362\pi$
$734$	−9.00607e12	−1.14526
$735$	0	0
$736$	−5.06558e11	−0.0636325
$737$	5.26638e11	0.0657519
$738$	0	0
$739$	5.44252e12	0.671274	0.335637	−	0.941991i	$-0.391048\pi$
$739$	5.44252e12	0.671274	0.335637	+	0.941991i	$0.391048\pi$
$740$	4.64485e12	0.569415
$741$	0	0
$742$	−9.59227e12	−1.16173
$743$	1.03123e13	1.24138	0.620691	−	0.784056i	$-0.286852\pi$
$743$	1.03123e13	1.24138	0.620691	+	0.784056i	$0.286852\pi$
$744$	0	0
$745$	3.18773e12	0.379122
$746$	−1.89713e13	−2.24271
$747$	0	0
$748$	2.27025e13	2.65165
$749$	3.29157e12	0.382151
$750$	0	0
$751$	−2.63156e12	−0.301880	−0.150940	−	0.988543i	$-0.548230\pi$
$751$	−2.63156e12	−0.301880	−0.150940	+	0.988543i	$0.548230\pi$
$752$	−4.13847e12	−0.471910
$753$	0	0
$754$	8.34968e12	0.940803
$755$	7.79747e12	0.873359
$756$	0	0
$757$	−4.56786e12	−0.505570	−0.252785	−	0.967523i	$-0.581346\pi$
$757$	−4.56786e12	−0.505570	−0.252785	+	0.967523i	$0.581346\pi$
$758$	2.26453e12	0.249153
$759$	0	0
$760$	4.36582e12	0.474684
$761$	−5.48565e12	−0.592921	−0.296461	−	0.955045i	$-0.595806\pi$
$761$	−5.48565e12	−0.592921	−0.296461	+	0.955045i	$0.595806\pi$
$762$	0	0
$763$	−1.27329e13	−1.36008
$764$	2.97788e13	3.16219
$765$	0	0
$766$	−8.89830e11	−0.0933851
$767$	3.12503e12	0.326043
$768$	0	0
$769$	9.09600e12	0.937955	0.468977	−	0.883210i	$-0.344623\pi$
$769$	9.09600e12	0.937955	0.468977	+	0.883210i	$0.344623\pi$
$770$	2.13596e13	2.18970
$771$	0	0
$772$	−1.57914e13	−1.60009
$773$	4.56540e12	0.459908	0.229954	−	0.973202i	$-0.426142\pi$
$773$	4.56540e12	0.459908	0.229954	+	0.973202i	$0.426142\pi$
$774$	0	0
$775$	−4.44143e12	−0.442248
$776$	6.61911e12	0.655272
$777$	0	0
$778$	1.68064e13	1.64462
$779$	−1.04897e12	−0.102058
$780$	0	0
$781$	2.27977e13	2.19261
$782$	−2.49561e12	−0.238642
$783$	0	0
$784$	3.08421e12	0.291556
$785$	−2.94777e12	−0.277064
$786$	0	0
$787$	2.93316e12	0.272552	0.136276	−	0.990671i	$-0.456487\pi$
$787$	2.93316e12	0.272552	0.136276	+	0.990671i	$0.456487\pi$
$788$	2.23349e13	2.06356
$789$	0	0
$790$	−9.72337e12	−0.888168
$791$	7.30932e12	0.663870
$792$	0	0
$793$	4.22824e12	0.379691
$794$	−2.04942e13	−1.82995
$795$	0	0
$796$	2.96309e13	2.61599
$797$	−1.70674e13	−1.49832	−0.749162	−	0.662387i	$-0.769543\pi$
$797$	−1.70674e13	−1.49832	−0.749162	+	0.662387i	$0.769543\pi$
$798$	0	0
$799$	−2.93055e12	−0.254383
$800$	2.35280e12	0.203086
$801$	0	0
$802$	2.55117e11	0.0217748
$803$	3.24004e13	2.74999
$804$	0	0
$805$	−1.58749e12	−0.133238
$806$	−9.79411e12	−0.817443
$807$	0	0
$808$	−1.74311e11	−0.0143872
$809$	−1.70485e13	−1.39932	−0.699662	−	0.714474i	$-0.746666\pi$
$809$	−1.70485e13	−1.39932	−0.699662	+	0.714474i	$0.746666\pi$
$810$	0	0
$811$	1.00699e13	0.817390	0.408695	−	0.912671i	$-0.365984\pi$
$811$	1.00699e13	0.817390	0.408695	+	0.912671i	$0.365984\pi$
$812$	2.30507e13	1.86073
$813$	0	0
$814$	−1.83933e13	−1.46842
$815$	−5.37264e12	−0.426558
$816$	0	0
$817$	7.68691e12	0.603604
$818$	1.38553e13	1.08200
$819$	0	0
$820$	4.07283e12	0.314582
$821$	−1.27721e13	−0.981110	−0.490555	−	0.871410i	$-0.663206\pi$
$821$	−1.27721e13	−0.981110	−0.490555	+	0.871410i	$0.663206\pi$
$822$	0	0
$823$	8.68293e12	0.659731	0.329865	−	0.944028i	$-0.392997\pi$
$823$	8.68293e12	0.659731	0.329865	+	0.944028i	$0.392997\pi$
$824$	4.03974e13	3.05268
$825$	0	0
$826$	1.27601e13	0.953767
$827$	1.72868e13	1.28511	0.642554	−	0.766240i	$-0.277875\pi$
$827$	1.72868e13	1.28511	0.642554	+	0.766240i	$0.277875\pi$
$828$	0	0
$829$	1.88685e12	0.138753	0.0693763	−	0.997591i	$-0.477899\pi$
$829$	1.88685e12	0.138753	0.0693763	+	0.997591i	$0.477899\pi$
$830$	−1.52553e13	−1.11575
$831$	0	0
$832$	−6.50303e12	−0.470501
$833$	2.18401e12	0.157163
$834$	0	0
$835$	−5.76814e12	−0.410626
$836$	−2.24377e13	−1.58873
$837$	0	0
$838$	1.50793e13	1.05629
$839$	−9.65939e12	−0.673009	−0.336504	−	0.941682i	$-0.609245\pi$
$839$	−9.65939e12	−0.673009	−0.336504	+	0.941682i	$0.609245\pi$
$840$	0	0
$841$	−5.13237e12	−0.353782
$842$	−4.54984e13	−3.11955
$843$	0	0
$844$	−1.16673e13	−0.791460
$845$	−4.99408e12	−0.336977
$846$	0	0
$847$	−4.05776e13	−2.70902
$848$	−1.14027e13	−0.757228
$849$	0	0
$850$	1.15913e13	0.761634
$851$	1.36703e12	0.0893503
$852$	0	0
$853$	1.26577e13	0.818621	0.409311	−	0.912395i	$-0.365769\pi$
$853$	1.26577e13	0.818621	0.409311	+	0.912395i	$0.365769\pi$
$854$	1.72647e13	1.11070
$855$	0	0
$856$	1.03442e13	0.658515
$857$	−1.63024e13	−1.03238	−0.516189	−	0.856475i	$-0.672650\pi$
$857$	−1.63024e13	−1.03238	−0.516189	+	0.856475i	$0.672650\pi$
$858$	0	0
$859$	−2.65609e13	−1.66446	−0.832231	−	0.554429i	$-0.812937\pi$
$859$	−2.65609e13	−1.66446	−0.832231	+	0.554429i	$0.812937\pi$
$860$	−2.98458e13	−1.86055
$861$	0	0
$862$	−1.22694e13	−0.756901
$863$	−1.50602e13	−0.924234	−0.462117	−	0.886819i	$-0.652910\pi$
$863$	−1.50602e13	−0.924234	−0.462117	+	0.886819i	$0.652910\pi$
$864$	0	0
$865$	−8.38295e12	−0.509125
$866$	1.40394e13	0.848236
$867$	0	0
$868$	−2.70383e13	−1.61674
$869$	2.60329e13	1.54858
$870$	0	0
$871$	4.00892e11	0.0236018
$872$	−4.00148e13	−2.34367
$873$	0	0
$874$	2.46650e12	0.142981
$875$	1.90186e13	1.09684
$876$	0	0
$877$	−2.66464e13	−1.52104	−0.760520	−	0.649315i	$-0.775056\pi$
$877$	−2.66464e13	−1.52104	−0.760520	+	0.649315i	$0.775056\pi$
$878$	2.36448e13	1.34280
$879$	0	0
$880$	2.53910e13	1.42727
$881$	3.46233e13	1.93632	0.968159	−	0.250336i	$-0.0805412\pi$
$881$	3.46233e13	1.93632	0.968159	+	0.250336i	$0.0805412\pi$
$882$	0	0
$883$	5.22763e12	0.289389	0.144694	−	0.989476i	$-0.453780\pi$
$883$	5.22763e12	0.289389	0.144694	+	0.989476i	$0.453780\pi$
$884$	1.72818e13	0.951819
$885$	0	0
$886$	−7.58203e12	−0.413365
$887$	3.41986e13	1.85504	0.927518	−	0.373779i	$-0.121938\pi$
$887$	3.41986e13	1.85504	0.927518	+	0.373779i	$0.121938\pi$
$888$	0	0
$889$	−3.98003e13	−2.13712
$890$	−1.99902e13	−1.06798
$891$	0	0
$892$	−5.98784e13	−3.16686
$893$	2.89636e12	0.152413
$894$	0	0
$895$	−1.13999e12	−0.0593877
$896$	−3.34147e13	−1.73201
$897$	0	0
$898$	1.01718e13	0.521979
$899$	−1.09965e13	−0.561484
$900$	0	0
$901$	−8.07454e12	−0.408184
$902$	−1.61282e13	−0.811253
$903$	0	0
$904$	2.29706e13	1.14397
$905$	−1.29996e13	−0.644189
$906$	0	0
$907$	−7.35881e12	−0.361056	−0.180528	−	0.983570i	$-0.557781\pi$
$907$	−7.35881e12	−0.361056	−0.180528	+	0.983570i	$0.557781\pi$
$908$	2.73829e13	1.33688
$909$	0	0
$910$	1.62595e13	0.785998
$911$	1.30433e13	0.627415	0.313708	−	0.949520i	$-0.398429\pi$
$911$	1.30433e13	0.627415	0.313708	+	0.949520i	$0.398429\pi$
$912$	0	0
$913$	4.08437e13	1.94539
$914$	−1.60902e13	−0.762611
$915$	0	0
$916$	8.65076e12	0.405999
$917$	−4.56790e13	−2.13331
$918$	0	0
$919$	−6.75575e12	−0.312431	−0.156215	−	0.987723i	$-0.549929\pi$
$919$	−6.75575e12	−0.312431	−0.156215	+	0.987723i	$0.549929\pi$
$920$	−4.98890e12	−0.229594
$921$	0	0
$922$	2.25050e13	1.02563
$923$	1.73543e13	0.787043
$924$	0	0
$925$	−6.34942e12	−0.285165
$926$	1.83632e13	0.820727
$927$	0	0
$928$	5.82529e12	0.257841
$929$	−1.73927e13	−0.766120	−0.383060	−	0.923723i	$-0.625130\pi$
$929$	−1.73927e13	−0.766120	−0.383060	+	0.923723i	$0.625130\pi$
$930$	0	0
$931$	−2.15853e12	−0.0941638
$932$	2.57421e13	1.11757
$933$	0	0
$934$	−4.54754e13	−1.95531
$935$	1.79799e13	0.769372
$936$	0	0
$937$	−6.24202e12	−0.264544	−0.132272	−	0.991213i	$-0.542227\pi$
$937$	−6.24202e12	−0.264544	−0.132272	+	0.991213i	$0.542227\pi$
$938$	1.63691e12	0.0690419
$939$	0	0
$940$	−1.12457e13	−0.469797
$941$	−7.12554e12	−0.296254	−0.148127	−	0.988968i	$-0.547324\pi$
$941$	−7.12554e12	−0.296254	−0.148127	+	0.988968i	$0.547324\pi$
$942$	0	0
$943$	1.19868e12	0.0493629
$944$	1.51684e13	0.621678
$945$	0	0
$946$	1.18188e14	4.79802
$947$	−1.84126e13	−0.743946	−0.371973	−	0.928244i	$-0.621319\pi$
$947$	−1.84126e13	−0.743946	−0.371973	+	0.928244i	$0.621319\pi$
$948$	0	0
$949$	2.46641e13	0.987115
$950$	−1.14561e13	−0.456331
$951$	0	0
$952$	3.67604e13	1.45049
$953$	−8.34892e12	−0.327878	−0.163939	−	0.986470i	$-0.552420\pi$
$953$	−8.34892e12	−0.327878	−0.163939	+	0.986470i	$0.552420\pi$
$954$	0	0
$955$	2.35842e13	0.917501
$956$	−3.91535e13	−1.51604
$957$	0	0
$958$	−5.72247e13	−2.19502
$959$	−1.82742e13	−0.697678
$960$	0	0
$961$	−1.35408e13	−0.512139
$962$	−1.40015e13	−0.527094
$963$	0	0
$964$	−8.83291e13	−3.29426
$965$	−1.25065e13	−0.464261
$966$	0	0
$967$	−3.00761e13	−1.10612	−0.553059	−	0.833142i	$-0.686540\pi$
$967$	−3.00761e13	−1.10612	−0.553059	+	0.833142i	$0.686540\pi$
$968$	−1.27521e14	−4.66812
$969$	0	0
$970$	1.00629e13	0.364963
$971$	4.68313e12	0.169064	0.0845318	−	0.996421i	$-0.473061\pi$
$971$	4.68313e12	0.169064	0.0845318	+	0.996421i	$0.473061\pi$
$972$	0	0
$973$	−3.08636e13	−1.10392
$974$	−1.22140e13	−0.434852
$975$	0	0
$976$	2.05232e13	0.723970
$977$	−1.56361e13	−0.549037	−0.274519	−	0.961582i	$-0.588518\pi$
$977$	−1.56361e13	−0.549037	−0.274519	+	0.961582i	$0.588518\pi$
$978$	0	0
$979$	5.35209e13	1.86209
$980$	8.38088e12	0.290250
$981$	0	0
$982$	5.28789e13	1.81460
$983$	4.63668e12	0.158386	0.0791929	−	0.996859i	$-0.474766\pi$
$983$	4.63668e12	0.158386	0.0791929	+	0.996859i	$0.474766\pi$
$984$	0	0
$985$	1.76888e13	0.598736
$986$	2.86989e13	0.966982
$987$	0	0
$988$	−1.70802e13	−0.570279
$989$	−8.78397e12	−0.291949
$990$	0	0
$991$	2.04024e13	0.671970	0.335985	−	0.941867i	$-0.390931\pi$
$991$	2.04024e13	0.671970	0.335985	+	0.941867i	$0.390931\pi$
$992$	−6.83302e12	−0.224032
$993$	0	0
$994$	7.08606e13	2.30232
$995$	2.34671e13	0.759024
$996$	0	0
$997$	−2.11715e13	−0.678616	−0.339308	−	0.940675i	$-0.610193\pi$
$997$	−2.11715e13	−0.678616	−0.339308	+	0.940675i	$0.610193\pi$
$998$	1.07978e13	0.344547
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.d.1.1		8
3.2	odd	2		81.10.a.c.1.8		8
9.2	odd	6		9.10.c.a.4.1	✓	16
9.4	even	3		27.10.c.a.19.8		16
9.5	odd	6		9.10.c.a.7.1	yes	16
9.7	even	3		27.10.c.a.10.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.1	✓	16	9.2	odd	6
9.10.c.a.7.1	yes	16	9.5	odd	6
27.10.c.a.10.8		16	9.7	even	3
27.10.c.a.19.8		16	9.4	even	3
81.10.a.c.1.8		8	3.2	odd	2
81.10.a.d.1.1		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-39.7589 q^{2} +1068.77 q^{4} +846.447 q^{5} -7043.99 q^{7} -22136.7 q^{8} -33653.8 q^{10} +90103.0 q^{11} +68589.0 q^{13} +280061. q^{14} +332920. q^{16} +235749. q^{17} -232999. q^{19} +904660. q^{20} -3.58240e6 q^{22} +266252. q^{23} -1.23665e6 q^{25} -2.72703e6 q^{26} -7.52842e6 q^{28} -3.06183e6 q^{29} +3.59150e6 q^{31} -1.90255e6 q^{32} -9.37312e6 q^{34} -5.96236e6 q^{35} +5.13436e6 q^{37} +9.26378e6 q^{38} -1.87375e7 q^{40} +4.50206e6 q^{41} -3.29912e7 q^{43} +9.62997e7 q^{44} -1.05859e7 q^{46} -1.24308e7 q^{47} +9.26412e6 q^{49} +4.91680e7 q^{50} +7.33061e7 q^{52} -3.42506e7 q^{53} +7.62674e7 q^{55} +1.55931e8 q^{56} +1.21735e8 q^{58} +4.55616e7 q^{59} +6.16460e7 q^{61} -1.42794e8 q^{62} -9.48116e7 q^{64} +5.80570e7 q^{65} +5.84484e6 q^{67} +2.51962e8 q^{68} +2.37057e8 q^{70} +2.53018e8 q^{71} +3.59593e8 q^{73} -2.04137e8 q^{74} -2.49023e8 q^{76} -6.34684e8 q^{77} +2.88923e8 q^{79} +2.81799e8 q^{80} -1.78997e8 q^{82} +4.53300e8 q^{83} +1.99549e8 q^{85} +1.31170e9 q^{86} -1.99458e9 q^{88} +5.93996e8 q^{89} -4.83140e8 q^{91} +2.84563e8 q^{92} +4.94236e8 q^{94} -1.97221e8 q^{95} -2.99011e8 q^{97} -3.68332e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) \) 8 * q + 15 * q^2 + 1793 * q^4 + 453 * q^5 + 343 * q^7 + 7239 * q^8 + 510 * q^10 + 99150 * q^11 - 32435 * q^13 + 394824 * q^14 + 328193 * q^16 + 415539 * q^17 - 85277 * q^19 + 1855164 * q^20 - 529359 * q^22 + 1064559 * q^23 + 2293229 * q^25 - 1218156 * q^26 + 612862 * q^28 - 1309053 * q^29 + 2359819 * q^31 + 5760063 * q^32 - 981801 * q^34 + 15533277 * q^35 + 8195758 * q^37 + 39490203 * q^38 + 16760496 * q^40 + 54747318 * q^41 - 15249608 * q^43 + 166254963 * q^44 + 1195260 * q^46 + 156295545 * q^47 - 15239583 * q^49 + 315590163 * q^50 + 19773358 * q^52 + 262758114 * q^53 - 3789885 * q^55 + 470339790 * q^56 - 55408560 * q^58 + 307774074 * q^59 - 69192125 * q^61 + 457218462 * q^62 - 201794239 * q^64 + 482470359 * q^65 - 14328044 * q^67 + 915409575 * q^68 + 229271934 * q^70 + 619800696 * q^71 + 299306599 * q^73 + 1022736000 * q^74 - 119954093 * q^76 + 717995541 * q^77 - 30257531 * q^79 + 1463913264 * q^80 - 101188011 * q^82 + 1176168291 * q^83 - 4818366 * q^85 + 1426944009 * q^86 - 911312427 * q^88 + 1658520648 * q^89 - 369615061 * q^91 - 76813998 * q^92 + 1954316784 * q^94 - 391400652 * q^95 + 267311278 * q^97 - 2413650159 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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