LMFDB - Embedded newform 27.36.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,36,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 36, names="a")

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 36 $
Character orbit:	$[\chi]$	$=$	27.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:	$209.506852711$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 35934374106 x^{10} + 255973111507808 x^{9} + \cdots + 10\!\cdots\!88 $ x^12 - 3x^11 - 35934374106x^10 + 255973111507808x^9 + 466604128527677289312x^8 - 5465816584211724156240384x^7 - 2657959999738216433102994985984x^6 + 35435152384171193876375788221038592x^5 + 6335378590638082582380507769439124504576x^4 - 66540602158451950122945221670684525716504576x^3 - 5281007003757409863831964169846833011293626171392x^2 + 32259129013829557260425287629524671600938175665537024*x + 1073508502520883765460422804350526199942929991981688946688
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	multiple of $ 2^{53}\cdot 3^{96}\cdot 5^{6}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-125967.$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-360417. q^{2} +9.55406e10 q^{4} -1.49977e12 q^{5} -8.36061e14 q^{7} -2.20506e16 q^{8} +O(q^{10})$	\(q-360417. q^{2} +9.55406e10 q^{4} -1.49977e12 q^{5} -8.36061e14 q^{7} -2.20506e16 q^{8} +5.40542e17 q^{10} +1.19996e18 q^{11} -2.90840e19 q^{13} +3.01331e20 q^{14} +4.66467e21 q^{16} +5.08674e21 q^{17} +1.20294e22 q^{19} -1.43289e23 q^{20} -4.32485e23 q^{22} +1.31613e24 q^{23} -6.61074e23 q^{25} +1.04824e25 q^{26} -7.98778e25 q^{28} -1.25051e25 q^{29} -1.02065e26 q^{31} -9.23571e26 q^{32} -1.83335e27 q^{34} +1.25390e27 q^{35} -1.61438e27 q^{37} -4.33561e27 q^{38} +3.30709e28 q^{40} -6.36803e27 q^{41} +6.56885e28 q^{43} +1.14645e29 q^{44} -4.74356e29 q^{46} +2.57962e29 q^{47} +3.20180e29 q^{49} +2.38262e29 q^{50} -2.77871e30 q^{52} +6.46994e29 q^{53} -1.79966e30 q^{55} +1.84357e31 q^{56} +4.50706e30 q^{58} -3.31794e30 q^{59} -3.97906e29 q^{61} +3.67861e31 q^{62} +1.72594e32 q^{64} +4.36194e31 q^{65} -1.06297e32 q^{67} +4.85990e32 q^{68} -4.51927e32 q^{70} -2.09299e32 q^{71} -4.15041e32 q^{73} +5.81851e32 q^{74} +1.14930e33 q^{76} -1.00324e33 q^{77} +1.08282e33 q^{79} -6.99593e33 q^{80} +2.29515e33 q^{82} -1.12823e32 q^{83} -7.62894e33 q^{85} -2.36753e34 q^{86} -2.64598e34 q^{88} -2.33897e34 q^{89} +2.43160e34 q^{91} +1.25744e35 q^{92} -9.29739e34 q^{94} -1.80414e34 q^{95} +7.96728e34 q^{97} -1.15398e35 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 209817 q^{2} + 238170471357 q^{4} + 237590493300 q^{5} - 26315460937956 q^{7} - 11\!\cdots\!85 q^{8}+O(q^{10}) $ 12 * q + 209817 * q^2 + 238170471357 * q^4 + 237590493300 * q^5 - 26315460937956 * q^7 - 1152523679482485 * q^8	$ 12 q + 209817 q^{2} + 238170471357 q^{4} + 237590493300 q^{5} - 26315460937956 q^{7} - 11\!\cdots\!85 q^{8}+ \cdots + 13\!\cdots\!62 q^{98}+O(q^{100}) $ 12 * q + 209817 * q^2 + 238170471357 * q^4 + 237590493300 * q^5 - 26315460937956 * q^7 - 1152523679482485 * q^8 + 53259116553823275 * q^10 + 1496551124826816372 * q^11 - 34328912703023996472 * q^13 + 166669000963239381363 * q^14 + 4860675541765013760705 * q^16 + 7628970615928780710168 * q^17 - 16082766160768990414128 * q^19 - 211198084408179102198645 * q^20 - 52234888231956498211143 * q^22 + 565792395499445750511120 * q^23 + 5555537533169508824637960 * q^25 + 10184460439954596310163124 * q^26 - 53247279099589429012932765 * q^28 - 15537250808393309204019840 * q^29 + 154551329202285014265355884 * q^31 - 569790636301246841664865893 * q^32 - 122663300810295291941317176 * q^34 + 3588084579418444573261981740 * q^35 - 3073092555569253564836155872 * q^37 - 4326541398755059647257334486 * q^38 + 16712219069408859411075107445 * q^40 - 6691483020866136020996784912 * q^41 + 44796529484279073882504569112 * q^43 + 150568080332409818247115161549 * q^44 - 471505816716619581821565726174 * q^46 + 200678144977450668688761056616 * q^47 + 1315326329423501432492384307360 * q^49 + 1819137608178900468376954622580 * q^50 - 1080581934187974962144983076796 * q^52 + 2691222104186670153874638254580 * q^53 + 1500392259136415557290568517820 * q^55 + 27633438158473075328907235551549 * q^56 - 5506073919138421424521005999066 * q^58 + 32766260948981198673864221207520 * q^59 - 11006721411902603259250479295728 * q^61 + 127809302103201321944632426528953 * q^62 + 72419202212575418035703901510825 * q^64 + 190220559832149919172950674352680 * q^65 - 62906959665093330766348132179432 * q^67 + 607941100538476155661581380036520 * q^68 - 23595260156850200715643047044895 * q^70 - 3715605905805552340595540265048 * q^71 - 456868766517955704811323145527276 * q^73 - 496856719188950172720386429591838 * q^74 + 750812250117487485399293980204746 * q^76 - 1491293331618714973103731562151324 * q^77 + 1441181755077379395085116385745496 * q^79 - 6329202616666871203376911405771665 * q^80 - 1025667950479925694157293281726742 * q^82 - 13534009831537273626534025954790652 * q^83 - 507540343243458160211502437320680 * q^85 - 38323435706440776292224238551430650 * q^86 - 3499699986133997560404753135251493 * q^88 - 20560572049351982374812418435974744 * q^89 + 32908658163780767717902805354624520 * q^91 + 22244007710744113308805635019788330 * q^92 - 26483113405825635155782484066278554 * q^94 + 61918684402966765699909525859718000 * q^95 + 9968493320960173817283702572205012 * q^97 + 138741461320002422834263484212313562 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−360417.	−1.94438	−0.972188	−	0.234202i	$-0.924752\pi$
$2$	−360417.	−1.94438	−0.972188	+	0.234202i	$0.924752\pi$
$3$	0	0
$3$	0	0
$4$	9.55406e10	2.78060
$5$	−1.49977e12	−0.879123	−0.439561	−	0.898213i	$-0.644866\pi$
$5$	−1.49977e12	−0.879123	−0.439561	+	0.898213i	$0.644866\pi$
$6$	0	0
$7$	−8.36061e14	−1.35838	−0.679192	−	0.733961i	$-0.737670\pi$
$7$	−8.36061e14	−1.35838	−0.679192	+	0.733961i	$0.737670\pi$
$8$	−2.20506e16	−3.46215
$9$	0	0
$10$	5.40542e17	1.70935
$11$	1.19996e18	0.715804	0.357902	−	0.933759i	$-0.383492\pi$
$11$	1.19996e18	0.715804	0.357902	+	0.933759i	$0.383492\pi$
$12$	0	0
$13$	−2.90840e19	−0.932494	−0.466247	−	0.884655i	$-0.654394\pi$
$13$	−2.90840e19	−0.932494	−0.466247	+	0.884655i	$0.654394\pi$
$14$	3.01331e20	2.64121
$15$	0	0
$16$	4.66467e21	3.95113
$17$	5.08674e21	1.49136	0.745682	−	0.666302i	$-0.232124\pi$
$17$	5.08674e21	1.49136	0.745682	+	0.666302i	$0.232124\pi$
$18$	0	0
$19$	1.20294e22	0.503567	0.251783	−	0.967784i	$-0.418983\pi$
$19$	1.20294e22	0.503567	0.251783	+	0.967784i	$0.418983\pi$
$20$	−1.43289e23	−2.44449
$21$	0	0
$22$	−4.32485e23	−1.39179
$23$	1.31613e24	1.94564	0.972819	−	0.231567i	$-0.0743851\pi$
$23$	1.31613e24	1.94564	0.972819	+	0.231567i	$0.0743851\pi$
$24$	0	0
$25$	−6.61074e23	−0.227143
$26$	1.04824e25	1.81312
$27$	0	0
$28$	−7.98778e25	−3.77712
$29$	−1.25051e25	−0.319980	−0.159990	−	0.987119i	$-0.551146\pi$
$29$	−1.25051e25	−0.319980	−0.159990	+	0.987119i	$0.551146\pi$
$30$	0	0
$31$	−1.02065e26	−0.812922	−0.406461	−	0.913668i	$-0.633237\pi$
$31$	−1.02065e26	−0.812922	−0.406461	+	0.913668i	$0.633237\pi$
$32$	−9.23571e26	−4.22032
$33$	0	0
$34$	−1.83335e27	−2.89977
$35$	1.25390e27	1.19419
$36$	0	0
$37$	−1.61438e27	−0.581402	−0.290701	−	0.956814i	$-0.593888\pi$
$37$	−1.61438e27	−0.581402	−0.290701	+	0.956814i	$0.593888\pi$
$38$	−4.33561e27	−0.979123
$39$	0	0
$40$	3.30709e28	3.04366
$41$	−6.36803e27	−0.380441	−0.190221	−	0.981741i	$-0.560920\pi$
$41$	−6.36803e27	−0.380441	−0.190221	+	0.981741i	$0.560920\pi$
$42$	0	0
$43$	6.56885e28	1.70526	0.852632	−	0.522512i	$-0.175005\pi$
$43$	6.56885e28	1.70526	0.852632	+	0.522512i	$0.175005\pi$
$44$	1.14645e29	1.99036
$45$	0	0
$46$	−4.74356e29	−3.78305
$47$	2.57962e29	1.41203	0.706014	−	0.708197i	$-0.250491\pi$
$47$	2.57962e29	1.41203	0.706014	+	0.708197i	$0.250491\pi$
$48$	0	0
$49$	3.20180e29	0.845207
$50$	2.38262e29	0.441652
$51$	0	0
$52$	−2.77871e30	−2.59289
$53$	6.46994e29	0.432586	0.216293	−	0.976328i	$-0.430603\pi$
$53$	6.46994e29	0.432586	0.216293	+	0.976328i	$0.430603\pi$
$54$	0	0
$55$	−1.79966e30	−0.629279
$56$	1.84357e31	4.70293
$57$	0	0
$58$	4.50706e30	0.622161
$59$	−3.31794e30	−0.339593	−0.169796	−	0.985479i	$-0.554311\pi$
$59$	−3.31794e30	−0.339593	−0.169796	+	0.985479i	$0.554311\pi$
$60$	0	0
$61$	−3.97906e29	−0.0227252	−0.0113626	−	0.999935i	$-0.503617\pi$
$61$	−3.97906e29	−0.0227252	−0.0113626	+	0.999935i	$0.503617\pi$
$62$	3.67861e31	1.58063
$63$	0	0
$64$	1.72594e32	4.25477
$65$	4.36194e31	0.819777
$66$	0	0
$67$	−1.06297e32	−1.17547	−0.587734	−	0.809054i	$-0.699980\pi$
$67$	−1.06297e32	−1.17547	−0.587734	+	0.809054i	$0.699980\pi$
$68$	4.85990e32	4.14688
$69$	0	0
$70$	−4.51927e32	−2.32195
$71$	−2.09299e32	−0.838967	−0.419484	−	0.907763i	$-0.637789\pi$
$71$	−2.09299e32	−0.838967	−0.419484	+	0.907763i	$0.637789\pi$
$72$	0	0
$73$	−4.15041e32	−1.02315	−0.511577	−	0.859237i	$-0.670939\pi$
$73$	−4.15041e32	−1.02315	−0.511577	+	0.859237i	$0.670939\pi$
$74$	5.81851e32	1.13046
$75$	0	0
$76$	1.14930e33	1.40022
$77$	−1.00324e33	−0.972336
$78$	0	0
$79$	1.08282e33	0.670011	0.335005	−	0.942216i	$-0.391262\pi$
$79$	1.08282e33	0.670011	0.335005	+	0.942216i	$0.391262\pi$
$80$	−6.99593e33	−3.47353
$81$	0	0
$82$	2.29515e33	0.739721
$83$	−1.12823e32	−0.0294125	−0.0147062	−	0.999892i	$-0.504681\pi$
$83$	−1.12823e32	−0.0294125	−0.0147062	+	0.999892i	$0.504681\pi$
$84$	0	0
$85$	−7.62894e33	−1.31109
$86$	−2.36753e34	−3.31567
$87$	0	0
$88$	−2.64598e34	−2.47822
$89$	−2.33897e34	−1.79763	−0.898815	−	0.438328i	$-0.855571\pi$
$89$	−2.33897e34	−1.79763	−0.898815	+	0.438328i	$0.855571\pi$
$90$	0	0
$91$	2.43160e34	1.26668
$92$	1.25744e35	5.41004
$93$	0	0
$94$	−9.29739e34	−2.74552
$95$	−1.80414e34	−0.442697
$96$	0	0
$97$	7.96728e34	1.35770	0.678851	−	0.734276i	$-0.262478\pi$
$97$	7.96728e34	1.35770	0.678851	+	0.734276i	$0.262478\pi$
$98$	−1.15398e35	−1.64340
$99$	0	0
$100$	−6.31594e34	−0.631594
$101$	−1.30975e34	−0.110044	−0.0550219	−	0.998485i	$-0.517523\pi$
$101$	−1.30975e34	−0.110044	−0.0550219	+	0.998485i	$0.517523\pi$
$102$	0	0
$103$	2.34705e35	1.39917	0.699585	−	0.714549i	$-0.253368\pi$
$103$	2.34705e35	1.39917	0.699585	+	0.714549i	$0.253368\pi$
$104$	6.41321e35	3.22844
$105$	0	0
$106$	−2.33188e35	−0.841110
$107$	3.15381e35	0.965205	0.482603	−	0.875839i	$-0.339692\pi$
$107$	3.15381e35	0.965205	0.482603	+	0.875839i	$0.339692\pi$
$108$	0	0
$109$	−8.13991e35	−1.80159	−0.900794	−	0.434246i	$-0.857015\pi$
$109$	−8.13991e35	−1.80159	−0.900794	+	0.434246i	$0.857015\pi$
$110$	6.48628e35	1.22356
$111$	0	0
$112$	−3.89995e36	−5.36715
$113$	−1.01595e36	−1.19674	−0.598368	−	0.801221i	$-0.704184\pi$
$113$	−1.01595e36	−1.19674	−0.598368	+	0.801221i	$0.704184\pi$
$114$	0	0
$115$	−1.97389e36	−1.71045
$116$	−1.19475e36	−0.889736
$117$	0	0
$118$	1.19584e36	0.660296
$119$	−4.25283e36	−2.02584
$120$	0	0
$121$	−1.37035e36	−0.487625
$122$	1.43412e35	0.0441864
$123$	0	0
$124$	−9.75138e36	−2.26041
$125$	5.35636e36	1.07881
$126$	0	0
$127$	6.82512e36	1.04123	0.520613	−	0.853793i	$-0.325703\pi$
$127$	6.82512e36	1.04123	0.520613	+	0.853793i	$0.325703\pi$
$128$	−3.04721e37	−4.05255
$129$	0	0
$130$	−1.57212e37	−1.59395
$131$	−1.68011e36	−0.148967	−0.0744834	−	0.997222i	$-0.523731\pi$
$131$	−1.68011e36	−0.148967	−0.0744834	+	0.997222i	$0.523731\pi$
$132$	0	0
$133$	−1.00573e37	−0.684037
$134$	3.83112e37	2.28555
$135$	0	0
$136$	−1.12166e38	−5.16333
$137$	−2.62087e36	−0.106129	−0.0530647	−	0.998591i	$-0.516899\pi$
$137$	−2.62087e36	−0.106129	−0.0530647	+	0.998591i	$0.516899\pi$
$138$	0	0
$139$	9.00579e36	0.282985	0.141492	−	0.989939i	$-0.454810\pi$
$139$	9.00579e36	0.282985	0.141492	+	0.989939i	$0.454810\pi$
$140$	1.19798e38	3.32055
$141$	0	0
$142$	7.54347e37	1.63127
$143$	−3.48996e37	−0.667483
$144$	0	0
$145$	1.87548e37	0.281302
$146$	1.49588e38	1.98940
$147$	0	0
$148$	−1.54239e38	−1.61664
$149$	2.09149e37	0.194848	0.0974240	−	0.995243i	$-0.468940\pi$
$149$	2.09149e37	0.194848	0.0974240	+	0.995243i	$0.468940\pi$
$150$	0	0
$151$	1.31942e38	0.973389	0.486694	−	0.873572i	$-0.338203\pi$
$151$	1.31942e38	0.973389	0.486694	+	0.873572i	$0.338203\pi$
$152$	−2.65256e38	−1.74342
$153$	0	0
$154$	3.61584e38	1.89059
$155$	1.53075e38	0.714658
$156$	0	0
$157$	−3.65819e38	−1.36465	−0.682327	−	0.731047i	$-0.739032\pi$
$157$	−3.65819e38	−1.36465	−0.682327	+	0.731047i	$0.739032\pi$
$158$	−3.90266e38	−1.30275
$159$	0	0
$160$	1.38514e39	3.71018
$161$	−1.10037e39	−2.64292
$162$	0	0
$163$	−4.02666e38	−0.779223	−0.389612	−	0.920979i	$-0.627391\pi$
$163$	−4.02666e38	−0.779223	−0.389612	+	0.920979i	$0.627391\pi$
$164$	−6.08405e38	−1.05785
$165$	0	0
$166$	4.06634e37	0.0571889
$167$	−2.89665e38	−0.366739	−0.183370	−	0.983044i	$-0.558700\pi$
$167$	−2.89665e38	−0.366739	−0.183370	+	0.983044i	$0.558700\pi$
$168$	0	0
$169$	−1.26905e38	−0.130455
$170$	2.74960e39	2.54925
$171$	0	0
$172$	6.27592e39	4.74165
$173$	8.59380e38	0.586649	0.293325	−	0.956013i	$-0.405238\pi$
$173$	8.59380e38	0.586649	0.293325	+	0.956013i	$0.405238\pi$
$174$	0	0
$175$	5.52698e38	0.308548
$176$	5.59740e39	2.82823
$177$	0	0
$178$	8.43006e39	3.49527
$179$	−9.60805e38	−0.361166	−0.180583	−	0.983560i	$-0.557799\pi$
$179$	−9.60805e38	−0.361166	−0.180583	+	0.983560i	$0.557799\pi$
$180$	0	0
$181$	−9.79162e38	−0.303026	−0.151513	−	0.988455i	$-0.548414\pi$
$181$	−9.79162e38	−0.303026	−0.151513	+	0.988455i	$0.548414\pi$
$182$	−8.76391e39	−2.46291
$183$	0	0
$184$	−2.90215e40	−6.73609
$185$	2.42120e39	0.511123
$186$	0	0
$187$	6.10387e39	1.06752
$188$	2.46459e40	3.92628
$189$	0	0
$190$	6.50242e39	0.860769
$191$	1.42738e40	1.72367	0.861836	−	0.507187i	$-0.169315\pi$
$191$	1.42738e40	1.72367	0.861836	+	0.507187i	$0.169315\pi$
$192$	0	0
$193$	−1.46404e40	−1.47333	−0.736667	−	0.676255i	$-0.763602\pi$
$193$	−1.46404e40	−1.47333	−0.736667	+	0.676255i	$0.763602\pi$
$194$	−2.87154e40	−2.63988
$195$	0	0
$196$	3.05902e40	2.35018
$197$	−3.56337e39	−0.250439	−0.125220	−	0.992129i	$-0.539964\pi$
$197$	−3.56337e39	−0.250439	−0.125220	+	0.992129i	$0.539964\pi$
$198$	0	0
$199$	−1.51074e40	−0.889738	−0.444869	−	0.895596i	$-0.646750\pi$
$199$	−1.51074e40	−0.889738	−0.444869	+	0.895596i	$0.646750\pi$
$200$	1.45771e40	0.786404
$201$	0	0
$202$	4.72057e39	0.213966
$203$	1.04550e40	0.434656
$204$	0	0
$205$	9.55058e39	0.334454
$206$	−8.45916e40	−2.72051
$207$	0	0
$208$	−1.35667e41	−3.68440
$209$	1.44348e40	0.360455
$210$	0	0
$211$	−8.39580e40	−1.77468	−0.887339	−	0.461119i	$-0.847448\pi$
$211$	−8.39580e40	−1.77468	−0.887339	+	0.461119i	$0.847448\pi$
$212$	6.18142e40	1.20285
$213$	0	0
$214$	−1.13669e41	−1.87672
$215$	−9.85177e40	−1.49914
$216$	0	0
$217$	8.53329e40	1.10426
$218$	2.93376e41	3.50297
$219$	0	0
$220$	−1.71941e41	−1.74977
$221$	−1.47943e41	−1.39069
$222$	0	0
$223$	−8.06839e40	−0.647816	−0.323908	−	0.946089i	$-0.604997\pi$
$223$	−8.06839e40	−0.647816	−0.323908	+	0.946089i	$0.604997\pi$
$224$	7.72162e41	5.73282
$225$	0	0
$226$	3.66164e41	2.32690
$227$	8.90100e40	0.523584	0.261792	−	0.965124i	$-0.415687\pi$
$227$	8.90100e40	0.523584	0.261792	+	0.965124i	$0.415687\pi$
$228$	0	0
$229$	2.91956e41	1.47298	0.736488	−	0.676450i	$-0.236483\pi$
$229$	2.91956e41	1.47298	0.736488	+	0.676450i	$0.236483\pi$
$230$	7.11425e41	3.32577
$231$	0	0
$232$	2.75746e41	1.10782
$233$	−1.48690e41	−0.554056	−0.277028	−	0.960862i	$-0.589349\pi$
$233$	−1.48690e41	−0.554056	−0.277028	+	0.960862i	$0.589349\pi$
$234$	0	0
$235$	−3.86884e41	−1.24135
$236$	−3.16998e41	−0.944271
$237$	0	0
$238$	1.53279e42	3.93900
$239$	−5.11721e41	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$239$	−5.11721e41	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$240$	0	0
$241$	1.08856e40	0.0224676	0.0112338	−	0.999937i	$-0.496424\pi$
$241$	1.08856e40	0.0224676	0.0112338	+	0.999937i	$0.496424\pi$
$242$	4.93896e41	0.948127
$243$	0	0
$244$	−3.80162e40	−0.0631898
$245$	−4.80196e41	−0.743040
$246$	0	0
$247$	−3.49864e41	−0.469573
$248$	2.25060e42	2.81446
$249$	0	0
$250$	−1.93052e42	−2.09761
$251$	1.06242e41	0.107648	0.0538242	−	0.998550i	$-0.482859\pi$
$251$	1.06242e41	0.107648	0.0538242	+	0.998550i	$0.482859\pi$
$252$	0	0
$253$	1.57930e42	1.39269
$254$	−2.45989e42	−2.02454
$255$	0	0
$256$	5.05238e42	3.62491
$257$	−7.94004e41	−0.532100	−0.266050	−	0.963959i	$-0.585719\pi$
$257$	−7.94004e41	−0.532100	−0.266050	+	0.963959i	$0.585719\pi$
$258$	0	0
$259$	1.34972e42	0.789767
$260$	4.16742e42	2.27947
$261$	0	0
$262$	6.05539e41	0.289648
$263$	3.00653e42	1.34537	0.672683	−	0.739931i	$-0.265142\pi$
$263$	3.00653e42	1.34537	0.672683	+	0.739931i	$0.265142\pi$
$264$	0	0
$265$	−9.70342e41	−0.380296
$266$	3.62484e42	1.33002
$267$	0	0
$268$	−1.01557e43	−3.26851
$269$	1.34004e42	0.404068	0.202034	−	0.979379i	$-0.435245\pi$
$269$	1.34004e42	0.404068	0.202034	+	0.979379i	$0.435245\pi$
$270$	0	0
$271$	−7.15130e41	−0.189418	−0.0947092	−	0.995505i	$-0.530192\pi$
$271$	−7.15130e41	−0.189418	−0.0947092	+	0.995505i	$0.530192\pi$
$272$	2.37280e43	5.89257
$273$	0	0
$274$	9.44606e41	0.206355
$275$	−7.93260e41	−0.162590
$276$	0	0
$277$	−3.93393e42	−0.710281	−0.355140	−	0.934813i	$-0.615567\pi$
$277$	−3.93393e42	−0.710281	−0.355140	+	0.934813i	$0.615567\pi$
$278$	−3.24584e42	−0.550229
$279$	0	0
$280$	−2.76493e43	−4.13445
$281$	−6.47025e42	−0.908992	−0.454496	−	0.890749i	$-0.650181\pi$
$281$	−6.47025e42	−0.908992	−0.454496	+	0.890749i	$0.650181\pi$
$282$	0	0
$283$	8.77566e42	1.08897	0.544486	−	0.838770i	$-0.316725\pi$
$283$	8.77566e42	1.08897	0.544486	+	0.838770i	$0.316725\pi$
$284$	−1.99965e43	−2.33283
$285$	0	0
$286$	1.25784e43	1.29784
$287$	5.32406e42	0.516785
$288$	0	0
$289$	1.42414e43	1.22416
$290$	−6.75955e42	−0.546956
$291$	0	0
$292$	−3.96533e43	−2.84498
$293$	1.95686e42	0.132244	0.0661222	−	0.997812i	$-0.478937\pi$
$293$	1.95686e42	0.132244	0.0661222	+	0.997812i	$0.478937\pi$
$294$	0	0
$295$	4.97615e42	0.298544
$296$	3.55981e43	2.01290
$297$	0	0
$298$	−7.53808e42	−0.378858
$299$	−3.82784e43	−1.81430
$300$	0	0
$301$	−5.49197e43	−2.31640
$302$	−4.75541e43	−1.89263
$303$	0	0
$304$	5.61133e43	1.98966
$305$	5.96768e41	0.0199783
$306$	0	0
$307$	4.72877e43	1.41197	0.705987	−	0.708224i	$-0.250503\pi$
$307$	4.72877e43	1.41197	0.705987	+	0.708224i	$0.250503\pi$
$308$	−9.58500e43	−2.70368
$309$	0	0
$310$	−5.51706e43	−1.38956
$311$	6.37771e43	1.51830	0.759151	−	0.650915i	$-0.225614\pi$
$311$	6.37771e43	1.51830	0.759151	+	0.650915i	$0.225614\pi$
$312$	0	0
$313$	−5.24974e43	−1.11715	−0.558576	−	0.829453i	$-0.688652\pi$
$313$	−5.24974e43	−1.11715	−0.558576	+	0.829453i	$0.688652\pi$
$314$	1.31847e44	2.65340
$315$	0	0
$316$	1.03453e44	1.86303
$317$	4.65845e43	0.793788	0.396894	−	0.917864i	$-0.370088\pi$
$317$	4.65845e43	0.793788	0.396894	+	0.917864i	$0.370088\pi$
$318$	0	0
$319$	−1.50056e43	−0.229043
$320$	−2.58851e44	−3.74046
$321$	0	0
$322$	3.96591e44	5.13884
$323$	6.11906e43	0.751001
$324$	0	0
$325$	1.92267e43	0.211810
$326$	1.45128e44	1.51510
$327$	0	0
$328$	1.40419e44	1.31715
$329$	−2.15672e44	−1.91808
$330$	0	0
$331$	2.56542e43	0.205196	0.102598	−	0.994723i	$-0.467284\pi$
$331$	2.56542e43	0.205196	0.102598	+	0.994723i	$0.467284\pi$
$332$	−1.07792e43	−0.0817842
$333$	0	0
$334$	1.04400e44	0.713079
$335$	1.59421e44	1.03338
$336$	0	0
$337$	9.58331e43	0.559748	0.279874	−	0.960037i	$-0.409707\pi$
$337$	9.58331e43	0.559748	0.279874	+	0.960037i	$0.409707\pi$
$338$	4.57386e43	0.253653
$339$	0	0
$340$	−7.28874e44	−3.64562
$341$	−1.22474e44	−0.581892
$342$	0	0
$343$	4.90255e43	0.210269
$344$	−1.44847e45	−5.90388
$345$	0	0
$346$	−3.09735e44	−1.14067
$347$	−9.98693e43	−0.349676	−0.174838	−	0.984597i	$-0.555940\pi$
$347$	−9.98693e43	−0.349676	−0.174838	+	0.984597i	$0.555940\pi$
$348$	0	0
$349$	−7.75785e42	−0.0245638	−0.0122819	−	0.999925i	$-0.503910\pi$
$349$	−7.75785e42	−0.0245638	−0.0122819	+	0.999925i	$0.503910\pi$
$350$	−1.99202e44	−0.599933
$351$	0	0
$352$	−1.10825e45	−3.02092
$353$	3.18432e44	0.825961	0.412981	−	0.910740i	$-0.364488\pi$
$353$	3.18432e44	0.825961	0.412981	+	0.910740i	$0.364488\pi$
$354$	0	0
$355$	3.13900e44	0.737555
$356$	−2.23467e45	−4.99849
$357$	0	0
$358$	3.46290e44	0.702243
$359$	1.13107e44	0.218442	0.109221	−	0.994017i	$-0.465164\pi$
$359$	1.13107e44	0.218442	0.109221	+	0.994017i	$0.465164\pi$
$360$	0	0
$361$	−4.25951e44	−0.746421
$362$	3.52906e44	0.589196
$363$	0	0
$364$	2.32317e45	3.52214
$365$	6.22466e44	0.899478
$366$	0	0
$367$	−6.78333e44	−0.890814	−0.445407	−	0.895328i	$-0.646941\pi$
$367$	−6.78333e44	−0.890814	−0.445407	+	0.895328i	$0.646941\pi$
$368$	6.13931e45	7.68746
$369$	0	0
$370$	−8.72642e44	−0.993816
$371$	−5.40927e44	−0.587618
$372$	0	0
$373$	3.43130e44	0.339277	0.169639	−	0.985506i	$-0.445740\pi$
$373$	3.43130e44	0.339277	0.169639	+	0.985506i	$0.445740\pi$
$374$	−2.19994e45	−2.07567
$375$	0	0
$376$	−5.68823e45	−4.88866
$377$	3.63699e44	0.298379
$378$	0	0
$379$	−6.72380e44	−0.502839	−0.251419	−	0.967878i	$-0.580897\pi$
$379$	−6.72380e44	−0.502839	−0.251419	+	0.967878i	$0.580897\pi$
$380$	−1.72368e45	−1.23096
$381$	0	0
$382$	−5.14451e45	−3.35147
$383$	9.00722e44	0.560547	0.280274	−	0.959920i	$-0.409575\pi$
$383$	9.00722e44	0.560547	0.280274	+	0.959920i	$0.409575\pi$
$384$	0	0
$385$	1.50463e45	0.854803
$386$	5.27666e45	2.86472
$387$	0	0
$388$	7.61199e45	3.77522
$389$	−6.39727e44	−0.303303	−0.151652	−	0.988434i	$-0.548459\pi$
$389$	−6.39727e44	−0.303303	−0.151652	+	0.988434i	$0.548459\pi$
$390$	0	0
$391$	6.69482e45	2.90165
$392$	−7.06017e45	−2.92623
$393$	0	0
$394$	1.28430e45	0.486948
$395$	−1.62398e45	−0.589022
$396$	0	0
$397$	1.89046e45	0.627674	0.313837	−	0.949477i	$-0.398385\pi$
$397$	1.89046e45	0.627674	0.313837	+	0.949477i	$0.398385\pi$
$398$	5.44498e45	1.72999
$399$	0	0
$400$	−3.08369e45	−0.897471
$401$	2.08571e44	0.0581068	0.0290534	−	0.999578i	$-0.490751\pi$
$401$	2.08571e44	0.0581068	0.0290534	+	0.999578i	$0.490751\pi$
$402$	0	0
$403$	2.96847e45	0.758044
$404$	−1.25135e45	−0.305987
$405$	0	0
$406$	−3.76818e45	−0.845134
$407$	−1.93719e45	−0.416169
$408$	0	0
$409$	3.58803e45	0.707454	0.353727	−	0.935349i	$-0.384914\pi$
$409$	3.58803e45	0.707454	0.353727	+	0.935349i	$0.384914\pi$
$410$	−3.44219e45	−0.650305
$411$	0	0
$412$	2.24239e46	3.89053
$413$	2.77400e45	0.461297
$414$	0	0
$415$	1.69209e44	0.0258572
$416$	2.68612e46	3.93543
$417$	0	0
$418$	−5.20255e45	−0.700860
$419$	−1.42475e46	−1.84074	−0.920370	−	0.391049i	$-0.872112\pi$
$419$	−1.42475e46	−1.84074	−0.920370	+	0.391049i	$0.872112\pi$
$420$	0	0
$421$	−6.91183e45	−0.821592	−0.410796	−	0.911727i	$-0.634749\pi$
$421$	−6.91183e45	−0.821592	−0.410796	+	0.911727i	$0.634749\pi$
$422$	3.02599e46	3.45064
$423$	0	0
$424$	−1.42666e46	−1.49768
$425$	−3.36271e45	−0.338753
$426$	0	0
$427$	3.32674e44	0.0308696
$428$	3.01317e46	2.68385
$429$	0	0
$430$	3.55074e46	2.91489
$431$	−9.61854e44	−0.0758153	−0.0379077	−	0.999281i	$-0.512069\pi$
$431$	−9.61854e44	−0.0758153	−0.0379077	+	0.999281i	$0.512069\pi$
$432$	0	0
$433$	3.14548e45	0.228638	0.114319	−	0.993444i	$-0.463531\pi$
$433$	3.14548e45	0.228638	0.114319	+	0.993444i	$0.463531\pi$
$434$	−3.07554e46	−2.14710
$435$	0	0
$436$	−7.77692e46	−5.00949
$437$	1.58323e46	0.979759
$438$	0	0
$439$	1.17185e46	0.669491	0.334745	−	0.942309i	$-0.391350\pi$
$439$	1.17185e46	0.669491	0.334745	+	0.942309i	$0.391350\pi$
$440$	3.96836e46	2.17866
$441$	0	0
$442$	5.33211e46	2.70402
$443$	−6.43225e45	−0.313544	−0.156772	−	0.987635i	$-0.550109\pi$
$443$	−6.43225e45	−0.313544	−0.156772	+	0.987635i	$0.550109\pi$
$444$	0	0
$445$	3.50792e46	1.58034
$446$	2.90799e46	1.25960
$447$	0	0
$448$	−1.44299e47	−5.77961
$449$	−3.90374e45	−0.150373	−0.0751865	−	0.997169i	$-0.523955\pi$
$449$	−3.90374e45	−0.150373	−0.0751865	+	0.997169i	$0.523955\pi$
$450$	0	0
$451$	−7.64136e45	−0.272321
$452$	−9.70642e46	−3.32764
$453$	0	0
$454$	−3.20807e46	−1.01804
$455$	−3.64685e46	−1.11357
$456$	0	0
$457$	1.38449e46	0.391523	0.195761	−	0.980652i	$-0.437282\pi$
$457$	1.38449e46	0.391523	0.195761	+	0.980652i	$0.437282\pi$
$458$	−1.05226e47	−2.86402
$459$	0	0
$460$	−1.88587e47	−4.75609
$461$	2.24190e46	0.544316	0.272158	−	0.962253i	$-0.412263\pi$
$461$	2.24190e46	0.544316	0.272158	+	0.962253i	$0.412263\pi$
$462$	0	0
$463$	4.89333e46	1.10138	0.550690	−	0.834710i	$-0.314365\pi$
$463$	4.89333e46	1.10138	0.550690	+	0.834710i	$0.314365\pi$
$464$	−5.83322e46	−1.26428
$465$	0	0
$466$	5.35905e46	1.07729
$467$	5.14443e46	0.996075	0.498037	−	0.867156i	$-0.334054\pi$
$467$	5.14443e46	0.996075	0.498037	+	0.867156i	$0.334054\pi$
$468$	0	0
$469$	8.88707e46	1.59674
$470$	1.39440e47	2.41364
$471$	0	0
$472$	7.31627e46	1.17572
$473$	7.88235e46	1.22063
$474$	0	0
$475$	−7.95234e45	−0.114382
$476$	−4.06318e47	−5.63306
$477$	0	0
$478$	1.84433e47	2.37602
$479$	1.28910e47	1.60109	0.800543	−	0.599275i	$-0.204544\pi$
$479$	1.28910e47	1.60109	0.800543	+	0.599275i	$0.204544\pi$
$480$	0	0
$481$	4.69528e46	0.542154
$482$	−3.92337e45	−0.0436854
$483$	0	0
$484$	−1.30924e47	−1.35589
$485$	−1.19491e47	−1.19359
$486$	0	0
$487$	1.69049e47	1.57129	0.785645	−	0.618678i	$-0.212331\pi$
$487$	1.69049e47	1.57129	0.785645	+	0.618678i	$0.212331\pi$
$488$	8.77408e45	0.0786782
$489$	0	0
$490$	1.73071e47	1.44475
$491$	−8.79868e46	−0.708747	−0.354374	−	0.935104i	$-0.615306\pi$
$491$	−8.79868e46	−0.708747	−0.354374	+	0.935104i	$0.615306\pi$
$492$	0	0
$493$	−6.36103e46	−0.477206
$494$	1.26097e47	0.913026
$495$	0	0
$496$	−4.76101e47	−3.21196
$497$	1.74986e47	1.13964
$498$	0	0
$499$	1.00911e47	0.612605	0.306303	−	0.951934i	$-0.400908\pi$
$499$	1.00911e47	0.612605	0.306303	+	0.951934i	$0.400908\pi$
$500$	5.11750e47	2.99974
$501$	0	0
$502$	−3.82915e46	−0.209309
$503$	−6.02067e46	−0.317838	−0.158919	−	0.987292i	$-0.550801\pi$
$503$	−6.02067e46	−0.317838	−0.158919	+	0.987292i	$0.550801\pi$
$504$	0	0
$505$	1.96433e46	0.0967420
$506$	−5.69207e47	−2.70792
$507$	0	0
$508$	6.52076e47	2.89523
$509$	2.56312e47	1.09953	0.549767	−	0.835318i	$-0.314717\pi$
$509$	2.56312e47	1.09953	0.549767	+	0.835318i	$0.314717\pi$
$510$	0	0
$511$	3.47000e47	1.38984
$512$	−7.73950e47	−2.99563
$513$	0	0
$514$	2.86172e47	1.03460
$515$	−3.52003e47	−1.23004
$516$	0	0
$517$	3.09544e47	1.01074
$518$	−4.86463e47	−1.53560
$519$	0	0
$520$	−9.61834e47	−2.83819
$521$	−6.53782e47	−1.86540	−0.932701	−	0.360650i	$-0.882555\pi$
$521$	−6.53782e47	−1.86540	−0.932701	+	0.360650i	$0.882555\pi$
$522$	0	0
$523$	5.27229e47	1.40676	0.703380	−	0.710814i	$-0.251673\pi$
$523$	5.27229e47	1.40676	0.703380	+	0.710814i	$0.251673\pi$
$524$	−1.60519e47	−0.414217
$525$	0	0
$526$	−1.08360e48	−2.61590
$527$	−5.19180e47	−1.21236
$528$	0	0
$529$	1.27461e48	2.78551
$530$	3.49728e47	0.739439
$531$	0	0
$532$	−9.60885e47	−1.90203
$533$	1.85208e47	0.354759
$534$	0	0
$535$	−4.72999e47	−0.848534
$536$	2.34391e48	4.06965
$537$	0	0
$538$	−4.82974e47	−0.785659
$539$	3.84203e47	0.605002
$540$	0	0
$541$	7.55787e47	1.11544	0.557721	−	0.830028i	$-0.311676\pi$
$541$	7.55787e47	1.11544	0.557721	+	0.830028i	$0.311676\pi$
$542$	2.57745e47	0.368301
$543$	0	0
$544$	−4.69797e48	−6.29404
$545$	1.22080e48	1.58382
$546$	0	0
$547$	−2.31185e47	−0.281308	−0.140654	−	0.990059i	$-0.544921\pi$
$547$	−2.31185e47	−0.281308	−0.140654	+	0.990059i	$0.544921\pi$
$548$	−2.50400e47	−0.295103
$549$	0	0
$550$	2.85904e47	0.316136
$551$	−1.50429e47	−0.161131
$552$	0	0
$553$	−9.05303e47	−0.910132
$554$	1.41785e48	1.38105
$555$	0	0
$556$	8.60419e47	0.786867
$557$	1.07886e48	0.956088	0.478044	−	0.878336i	$-0.341346\pi$
$557$	1.07886e48	0.956088	0.478044	+	0.878336i	$0.341346\pi$
$558$	0	0
$559$	−1.91049e48	−1.59015
$560$	5.84902e48	4.71838
$561$	0	0
$562$	2.33199e48	1.76742
$563$	−9.30654e47	−0.683740	−0.341870	−	0.939747i	$-0.611060\pi$
$563$	−9.30654e47	−0.683740	−0.341870	+	0.939747i	$0.611060\pi$
$564$	0	0
$565$	1.52369e48	1.05208
$566$	−3.16290e48	−2.11737
$567$	0	0
$568$	4.61516e48	2.90463
$569$	1.17111e48	0.714713	0.357356	−	0.933968i	$-0.383678\pi$
$569$	1.17111e48	0.714713	0.357356	+	0.933968i	$0.383678\pi$
$570$	0	0
$571$	3.01053e48	1.72787	0.863934	−	0.503604i	$-0.167993\pi$
$571$	3.01053e48	1.72787	0.863934	+	0.503604i	$0.167993\pi$
$572$	−3.33433e48	−1.85600
$573$	0	0
$574$	−1.91888e48	−1.00482
$575$	−8.70060e47	−0.441938
$576$	0	0
$577$	−1.98741e48	−0.949970	−0.474985	−	0.879994i	$-0.657546\pi$
$577$	−1.98741e48	−0.949970	−0.474985	+	0.879994i	$0.657546\pi$
$578$	−5.13283e48	−2.38024
$579$	0	0
$580$	1.79184e48	0.782187
$581$	9.43271e46	0.0399534
$582$	0	0
$583$	7.76366e47	0.309647
$584$	9.15192e48	3.54232
$585$	0	0
$586$	−7.05287e47	−0.257133
$587$	−3.07714e48	−1.08888	−0.544441	−	0.838799i	$-0.683258\pi$
$587$	−3.07714e48	−1.08888	−0.544441	+	0.838799i	$0.683258\pi$
$588$	0	0
$589$	−1.22779e48	−0.409360
$590$	−1.79349e48	−0.580481
$591$	0	0
$592$	−7.53056e48	−2.29719
$593$	1.35475e48	0.401239	0.200620	−	0.979669i	$-0.435704\pi$
$593$	1.35475e48	0.401239	0.200620	+	0.979669i	$0.435704\pi$
$594$	0	0
$595$	6.37826e48	1.78097
$596$	1.99822e48	0.541794
$597$	0	0
$598$	1.37962e49	3.52767
$599$	−6.09713e48	−1.51411	−0.757053	−	0.653354i	$-0.773361\pi$
$599$	−6.09713e48	−1.51411	−0.757053	+	0.653354i	$0.773361\pi$
$600$	0	0
$601$	−4.65101e48	−1.08954	−0.544771	−	0.838585i	$-0.683383\pi$
$601$	−4.65101e48	−1.08954	−0.544771	+	0.838585i	$0.683383\pi$
$602$	1.97940e49	4.50396
$603$	0	0
$604$	1.26058e49	2.70660
$605$	2.05520e48	0.428682
$606$	0	0
$607$	−3.90009e48	−0.767844	−0.383922	−	0.923366i	$-0.625427\pi$
$607$	−3.90009e48	−0.767844	−0.383922	+	0.923366i	$0.625427\pi$
$608$	−1.11100e49	−2.12521
$609$	0	0
$610$	−2.15085e47	−0.0388453
$611$	−7.50258e48	−1.31671
$612$	0	0
$613$	−6.89040e48	−1.14205	−0.571026	−	0.820932i	$-0.693455\pi$
$613$	−6.89040e48	−1.14205	−0.571026	+	0.820932i	$0.693455\pi$
$614$	−1.70433e49	−2.74541
$615$	0	0
$616$	2.21220e49	3.36638
$617$	5.19940e48	0.769064	0.384532	−	0.923112i	$-0.374363\pi$
$617$	5.19940e48	0.769064	0.384532	+	0.923112i	$0.374363\pi$
$618$	0	0
$619$	6.27382e48	0.876891	0.438445	−	0.898758i	$-0.355529\pi$
$619$	6.27382e48	0.876891	0.438445	+	0.898758i	$0.355529\pi$
$620$	1.46248e49	1.98718
$621$	0	0
$622$	−2.29864e49	−2.95215
$623$	1.95553e49	2.44187
$624$	0	0
$625$	−6.10933e48	−0.721263
$626$	1.89210e49	2.17216
$627$	0	0
$628$	−3.49506e49	−3.79455
$629$	−8.21195e48	−0.867081
$630$	0	0
$631$	−1.30211e49	−1.30057	−0.650283	−	0.759692i	$-0.725350\pi$
$631$	−1.30211e49	−1.30057	−0.650283	+	0.759692i	$0.725350\pi$
$632$	−2.38768e49	−2.31968
$633$	0	0
$634$	−1.67899e49	−1.54342
$635$	−1.02361e49	−0.915366
$636$	0	0
$637$	−9.31213e48	−0.788150
$638$	5.40828e48	0.445345
$639$	0	0
$640$	4.57012e49	3.56269
$641$	1.28689e49	0.976171	0.488086	−	0.872796i	$-0.337695\pi$
$641$	1.28689e49	0.976171	0.488086	+	0.872796i	$0.337695\pi$
$642$	0	0
$643$	1.27395e49	0.915086	0.457543	−	0.889187i	$-0.348730\pi$
$643$	1.27395e49	0.915086	0.457543	+	0.889187i	$0.348730\pi$
$644$	−1.05130e50	−7.34891
$645$	0	0
$646$	−2.20541e49	−1.46023
$647$	−6.67807e48	−0.430354	−0.215177	−	0.976575i	$-0.569033\pi$
$647$	−6.67807e48	−0.430354	−0.215177	+	0.976575i	$0.569033\pi$
$648$	0	0
$649$	−3.98139e48	−0.243082
$650$	−6.92962e48	−0.411838
$651$	0	0
$652$	−3.84710e49	−2.16671
$653$	2.43017e49	1.33247	0.666233	−	0.745744i	$-0.267906\pi$
$653$	2.43017e49	1.33247	0.666233	+	0.745744i	$0.267906\pi$
$654$	0	0
$655$	2.51978e48	0.130960
$656$	−2.97047e49	−1.50317
$657$	0	0
$658$	7.77319e49	3.72946
$659$	5.67449e48	0.265114	0.132557	−	0.991175i	$-0.457681\pi$
$659$	5.67449e48	0.265114	0.132557	+	0.991175i	$0.457681\pi$
$660$	0	0
$661$	−2.40890e49	−1.06732	−0.533658	−	0.845700i	$-0.679183\pi$
$661$	−2.40890e49	−1.06732	−0.533658	+	0.845700i	$0.679183\pi$
$662$	−9.24621e48	−0.398978
$663$	0	0
$664$	2.48782e48	0.101830
$665$	1.50837e49	0.601352
$666$	0	0
$667$	−1.64584e49	−0.622565
$668$	−2.76747e49	−1.01975
$669$	0	0
$670$	−5.74579e49	−2.00928
$671$	−4.77470e47	−0.0162668
$672$	0	0
$673$	1.44608e49	0.467659	0.233830	−	0.972278i	$-0.424874\pi$
$673$	1.44608e49	0.467659	0.233830	+	0.972278i	$0.424874\pi$
$674$	−3.45399e49	−1.08836
$675$	0	0
$676$	−1.21245e49	−0.362742
$677$	−4.48710e49	−1.30817	−0.654084	−	0.756422i	$-0.726946\pi$
$677$	−4.48710e49	−1.30817	−0.654084	+	0.756422i	$0.726946\pi$
$678$	0	0
$679$	−6.66113e49	−1.84428
$680$	1.68223e50	4.53920
$681$	0	0
$682$	4.41417e49	1.13142
$683$	4.74827e49	1.18624	0.593121	−	0.805114i	$-0.297896\pi$
$683$	4.74827e49	1.18624	0.593121	+	0.805114i	$0.297896\pi$
$684$	0	0
$685$	3.93070e48	0.0933007
$686$	−1.76696e49	−0.408841
$687$	0	0
$688$	3.06415e50	6.73771
$689$	−1.88172e49	−0.403384
$690$	0	0
$691$	−4.21383e49	−0.858640	−0.429320	−	0.903152i	$-0.641247\pi$
$691$	−4.21383e49	−0.858640	−0.429320	+	0.903152i	$0.641247\pi$
$692$	8.21057e49	1.63124
$693$	0	0
$694$	3.59946e49	0.679902
$695$	−1.35066e49	−0.248778
$696$	0	0
$697$	−3.23925e49	−0.567376
$698$	2.79606e48	0.0477613
$699$	0	0
$700$	5.28051e49	0.857947
$701$	5.95016e49	0.942895	0.471447	−	0.881894i	$-0.343732\pi$
$701$	5.95016e49	0.942895	0.471447	+	0.881894i	$0.343732\pi$
$702$	0	0
$703$	−1.94201e49	−0.292775
$704$	2.07105e50	3.04558
$705$	0	0
$706$	−1.14768e50	−1.60598
$707$	1.09503e49	0.149482
$708$	0	0
$709$	1.73340e48	0.0225211	0.0112605	−	0.999937i	$-0.496416\pi$
$709$	1.73340e48	0.0225211	0.0112605	+	0.999937i	$0.496416\pi$
$710$	−1.13135e50	−1.43408
$711$	0	0
$712$	5.15758e50	6.22367
$713$	−1.34331e50	−1.58165
$714$	0	0
$715$	5.23414e49	0.586799
$716$	−9.17959e49	−1.00426
$717$	0	0
$718$	−4.07656e49	−0.424733
$719$	−8.52595e49	−0.866936	−0.433468	−	0.901169i	$-0.642710\pi$
$719$	−8.52595e49	−0.866936	−0.433468	+	0.901169i	$0.642710\pi$
$720$	0	0
$721$	−1.96228e50	−1.90061
$722$	1.53520e50	1.45132
$723$	0	0
$724$	−9.35497e49	−0.842592
$725$	8.26680e48	0.0726813
$726$	0	0
$727$	−2.18593e50	−1.83140	−0.915702	−	0.401857i	$-0.868365\pi$
$727$	−2.18593e50	−1.83140	−0.915702	+	0.401857i	$0.868365\pi$
$728$	−5.36184e50	−4.38546
$729$	0	0
$730$	−2.24347e50	−1.74892
$731$	3.34141e50	2.54317
$732$	0	0
$733$	1.07844e50	0.782487	0.391244	−	0.920287i	$-0.372045\pi$
$733$	1.07844e50	0.782487	0.391244	+	0.920287i	$0.372045\pi$
$734$	2.44483e50	1.73208
$735$	0	0
$736$	−1.21554e51	−8.21122
$737$	−1.27552e50	−0.841405
$738$	0	0
$739$	1.35081e50	0.849799	0.424899	−	0.905241i	$-0.360310\pi$
$739$	1.35081e50	0.849799	0.424899	+	0.905241i	$0.360310\pi$
$740$	2.31323e50	1.42123
$741$	0	0
$742$	1.94959e50	1.14255
$743$	1.39790e50	0.800149	0.400074	−	0.916483i	$-0.368984\pi$
$743$	1.39790e50	0.800149	0.400074	+	0.916483i	$0.368984\pi$
$744$	0	0
$745$	−3.13675e49	−0.171295
$746$	−1.23670e50	−0.659682
$747$	0	0
$748$	5.83168e50	2.96835
$749$	−2.63678e50	−1.31112
$750$	0	0
$751$	2.50376e50	1.18821	0.594106	−	0.804387i	$-0.297506\pi$
$751$	2.50376e50	1.18821	0.594106	+	0.804387i	$0.297506\pi$
$752$	1.20331e51	5.57911
$753$	0	0
$754$	−1.31083e50	−0.580162
$755$	−1.97883e50	−0.855728
$756$	0	0
$757$	−3.41209e50	−1.40878	−0.704390	−	0.709813i	$-0.748779\pi$
$757$	−3.41209e50	−1.40878	−0.704390	+	0.709813i	$0.748779\pi$
$758$	2.42337e50	0.977707
$759$	0	0
$760$	3.97824e50	1.53268
$761$	−3.05095e48	−0.0114869	−0.00574347	−	0.999984i	$-0.501828\pi$
$761$	−3.05095e48	−0.0114869	−0.00574347	+	0.999984i	$0.501828\pi$
$762$	0	0
$763$	6.80546e50	2.44725
$764$	1.36372e51	4.79284
$765$	0	0
$766$	−3.24636e50	−1.08991
$767$	9.64992e49	0.316668
$768$	0	0
$769$	−3.86757e50	−1.21263	−0.606314	−	0.795225i	$-0.707353\pi$
$769$	−3.86757e50	−1.21263	−0.606314	+	0.795225i	$0.707353\pi$
$770$	−5.42293e50	−1.66206
$771$	0	0
$772$	−1.39876e51	−4.09675
$773$	3.08884e50	0.884413	0.442206	−	0.896913i	$-0.354196\pi$
$773$	3.08884e50	0.884413	0.442206	+	0.896913i	$0.354196\pi$
$774$	0	0
$775$	6.74727e49	0.184650
$776$	−1.75683e51	−4.70057
$777$	0	0
$778$	2.30569e50	0.589735
$779$	−7.66038e49	−0.191578
$780$	0	0
$781$	−2.51149e50	−0.600536
$782$	−2.41293e51	−5.64191
$783$	0	0
$784$	1.49353e51	3.33952
$785$	5.48644e50	1.19970
$786$	0	0
$787$	2.58325e49	0.0540267	0.0270134	−	0.999635i	$-0.491400\pi$
$787$	2.58325e49	0.0540267	0.0270134	+	0.999635i	$0.491400\pi$
$788$	−3.40447e50	−0.696370
$789$	0	0
$790$	5.85310e50	1.14528
$791$	8.49394e50	1.62563
$792$	0	0
$793$	1.15727e49	0.0211912
$794$	−6.81355e50	−1.22043
$795$	0	0
$796$	−1.44337e51	−2.47400
$797$	−1.30460e50	−0.218754	−0.109377	−	0.994000i	$-0.534886\pi$
$797$	−1.30460e50	−0.218754	−0.109377	+	0.994000i	$0.534886\pi$
$798$	0	0
$799$	1.31219e51	2.10585
$800$	6.10549e50	0.958618
$801$	0	0
$802$	−7.51724e49	−0.112981
$803$	−4.98032e50	−0.732378
$804$	0	0
$805$	1.65030e51	2.32345
$806$	−1.06989e51	−1.47392
$807$	0	0
$808$	2.88809e50	0.380988
$809$	−4.28086e50	−0.552627	−0.276314	−	0.961068i	$-0.589113\pi$
$809$	−4.28086e50	−0.552627	−0.276314	+	0.961068i	$0.589113\pi$
$810$	0	0
$811$	9.34908e50	1.15586	0.577928	−	0.816087i	$-0.303861\pi$
$811$	9.34908e50	1.15586	0.577928	+	0.816087i	$0.303861\pi$
$812$	9.98882e50	1.20860
$813$	0	0
$814$	6.98196e50	0.809190
$815$	6.03906e50	0.685033
$816$	0	0
$817$	7.90196e50	0.858714
$818$	−1.29319e51	−1.37556
$819$	0	0
$820$	9.12468e50	0.929983
$821$	1.64827e50	0.164446	0.0822228	−	0.996614i	$-0.473798\pi$
$821$	1.64827e50	0.164446	0.0822228	+	0.996614i	$0.473798\pi$
$822$	0	0
$823$	−6.46789e50	−0.618394	−0.309197	−	0.950998i	$-0.600060\pi$
$823$	−6.46789e50	−0.618394	−0.309197	+	0.950998i	$0.600060\pi$
$824$	−5.17539e51	−4.84414
$825$	0	0
$826$	−9.99798e50	−0.896936
$827$	2.06690e51	1.81540	0.907700	−	0.419619i	$-0.137836\pi$
$827$	2.06690e51	1.81540	0.907700	+	0.419619i	$0.137836\pi$
$828$	0	0
$829$	−9.47612e49	−0.0797859	−0.0398929	−	0.999204i	$-0.512702\pi$
$829$	−9.47612e49	−0.0797859	−0.0398929	+	0.999204i	$0.512702\pi$
$830$	−6.09857e49	−0.0502761
$831$	0	0
$832$	−5.01973e51	−3.96755
$833$	1.62867e51	1.26051
$834$	0	0
$835$	4.34430e50	0.322409
$836$	1.37911e51	1.00228
$837$	0	0
$838$	5.13503e51	3.57909
$839$	1.55356e51	1.06046	0.530231	−	0.847854i	$-0.322105\pi$
$839$	1.55356e51	1.06046	0.530231	+	0.847854i	$0.322105\pi$
$840$	0	0
$841$	−1.37094e51	−0.897613
$842$	2.49114e51	1.59748
$843$	0	0
$844$	−8.02140e51	−4.93466
$845$	1.90328e50	0.114686
$846$	0	0
$847$	1.14569e51	0.662382
$848$	3.01801e51	1.70920
$849$	0	0
$850$	1.21198e51	0.658663
$851$	−2.12474e51	−1.13120
$852$	0	0
$853$	2.07482e51	1.06016	0.530082	−	0.847946i	$-0.322161\pi$
$853$	2.07482e51	1.06016	0.530082	+	0.847946i	$0.322161\pi$
$854$	−1.19901e50	−0.0600221
$855$	0	0
$856$	−6.95435e51	−3.34169
$857$	−2.71728e51	−1.27929	−0.639647	−	0.768669i	$-0.720920\pi$
$857$	−2.71728e51	−1.27929	−0.639647	+	0.768669i	$0.720920\pi$
$858$	0	0
$859$	−9.40895e50	−0.425266	−0.212633	−	0.977132i	$-0.568204\pi$
$859$	−9.40895e50	−0.425266	−0.212633	+	0.977132i	$0.568204\pi$
$860$	−9.41244e51	−4.16850
$861$	0	0
$862$	3.46669e50	0.147414
$863$	6.30271e50	0.262626	0.131313	−	0.991341i	$-0.458081\pi$
$863$	6.30271e50	0.262626	0.131313	+	0.991341i	$0.458081\pi$
$864$	0	0
$865$	−1.28887e51	−0.515737
$866$	−1.13368e51	−0.444558
$867$	0	0
$868$	8.15276e51	3.07050
$869$	1.29934e51	0.479596
$870$	0	0
$871$	3.09154e51	1.09612
$872$	1.79490e52	6.23737
$873$	0	0
$874$	−5.70623e51	−1.90502
$875$	−4.47825e51	−1.46544
$876$	0	0
$877$	−8.26190e50	−0.259769	−0.129884	−	0.991529i	$-0.541461\pi$
$877$	−8.26190e50	−0.259769	−0.129884	+	0.991529i	$0.541461\pi$
$878$	−4.22356e51	−1.30174
$879$	0	0
$880$	−8.39482e51	−2.48636
$881$	2.55409e51	0.741579	0.370789	−	0.928717i	$-0.379087\pi$
$881$	2.55409e51	0.741579	0.370789	+	0.928717i	$0.379087\pi$
$882$	0	0
$883$	1.64868e51	0.460070	0.230035	−	0.973182i	$-0.426116\pi$
$883$	1.64868e51	0.460070	0.230035	+	0.973182i	$0.426116\pi$
$884$	−1.41346e52	−3.86694
$885$	0	0
$886$	2.31829e51	0.609646
$887$	1.38488e51	0.357067	0.178534	−	0.983934i	$-0.442865\pi$
$887$	1.38488e51	0.357067	0.178534	+	0.983934i	$0.442865\pi$
$888$	0	0
$889$	−5.70622e51	−1.41438
$890$	−1.26431e52	−3.07277
$891$	0	0
$892$	−7.70859e51	−1.80132
$893$	3.10314e51	0.711051
$894$	0	0
$895$	1.44099e51	0.317510
$896$	2.54766e52	5.50492
$897$	0	0
$898$	1.40697e51	0.292381
$899$	1.27634e51	0.260119
$900$	0	0
$901$	3.29109e51	0.645143
$902$	2.75408e51	0.529495
$903$	0	0
$904$	2.24023e52	4.14328
$905$	1.46852e51	0.266397
$906$	0	0
$907$	−5.89961e51	−1.02966	−0.514832	−	0.857291i	$-0.672146\pi$
$907$	−5.89961e51	−1.02966	−0.514832	+	0.857291i	$0.672146\pi$
$908$	8.50407e51	1.45588
$909$	0	0
$910$	1.31439e52	2.16520
$911$	8.58981e51	1.38807	0.694036	−	0.719941i	$-0.255831\pi$
$911$	8.58981e51	1.38807	0.694036	+	0.719941i	$0.255831\pi$
$912$	0	0
$913$	−1.35383e50	−0.0210536
$914$	−4.98995e51	−0.761268
$915$	0	0
$916$	2.78936e52	4.09575
$917$	1.40467e51	0.202354
$918$	0	0
$919$	1.24801e52	1.73060	0.865298	−	0.501258i	$-0.167129\pi$
$919$	1.24801e52	1.73060	0.865298	+	0.501258i	$0.167129\pi$
$920$	4.35256e52	5.92185
$921$	0	0
$922$	−8.08020e51	−1.05835
$923$	6.08725e51	0.782332
$924$	0	0
$925$	1.06723e51	0.132061
$926$	−1.76364e52	−2.14150
$927$	0	0
$928$	1.15494e52	1.35042
$929$	1.09687e52	1.25857	0.629287	−	0.777173i	$-0.283347\pi$
$929$	1.09687e52	1.25857	0.629287	+	0.777173i	$0.283347\pi$
$930$	0	0
$931$	3.85158e51	0.425618
$932$	−1.42060e52	−1.54061
$933$	0	0
$934$	−1.85414e52	−1.93674
$935$	−9.15440e51	−0.938484
$936$	0	0
$937$	−1.82928e52	−1.80650	−0.903248	−	0.429120i	$-0.858824\pi$
$937$	−1.82928e52	−1.80650	−0.903248	+	0.429120i	$0.858824\pi$
$938$	−3.20305e52	−3.10466
$939$	0	0
$940$	−3.69631e52	−3.45169
$941$	2.90698e51	0.266455	0.133227	−	0.991086i	$-0.457466\pi$
$941$	2.90698e51	0.266455	0.133227	+	0.991086i	$0.457466\pi$
$942$	0	0
$943$	−8.38116e51	−0.740201
$944$	−1.54771e52	−1.34177
$945$	0	0
$946$	−2.84093e52	−2.37337
$947$	1.62485e52	1.33256	0.666282	−	0.745700i	$-0.267885\pi$
$947$	1.62485e52	1.33256	0.666282	+	0.745700i	$0.267885\pi$
$948$	0	0
$949$	1.20711e52	0.954085
$950$	2.86616e51	0.222401
$951$	0	0
$952$	9.37775e52	7.01378
$953$	−2.15176e51	−0.158004	−0.0790019	−	0.996874i	$-0.525173\pi$
$953$	−2.15176e51	−0.158004	−0.0790019	+	0.996874i	$0.525173\pi$
$954$	0	0
$955$	−2.14074e52	−1.51532
$956$	−4.88902e52	−3.39788
$957$	0	0
$958$	−4.64613e52	−3.11311
$959$	2.19121e51	0.144164
$960$	0	0
$961$	−5.34641e51	−0.339159
$962$	−1.69226e52	−1.05415
$963$	0	0
$964$	1.04002e51	0.0624732
$965$	2.19573e52	1.29524
$966$	0	0
$967$	−2.65879e52	−1.51259	−0.756294	−	0.654232i	$-0.772992\pi$
$967$	−2.65879e52	−1.51259	−0.756294	+	0.654232i	$0.772992\pi$
$968$	3.02170e52	1.68823
$969$	0	0
$970$	4.30665e52	2.32078
$971$	−2.03247e52	−1.07569	−0.537844	−	0.843044i	$-0.680761\pi$
$971$	−2.03247e52	−1.07569	−0.537844	+	0.843044i	$0.680761\pi$
$972$	0	0
$973$	−7.52940e51	−0.384402
$974$	−6.09282e52	−3.05518
$975$	0	0
$976$	−1.85610e51	−0.0897903
$977$	−7.24552e51	−0.344282	−0.172141	−	0.985072i	$-0.555069\pi$
$977$	−7.24552e51	−0.344282	−0.172141	+	0.985072i	$0.555069\pi$
$978$	0	0
$979$	−2.80667e52	−1.28675
$980$	−4.58783e52	−2.06610
$981$	0	0
$982$	3.17119e52	1.37807
$983$	−3.52420e52	−1.50444	−0.752219	−	0.658913i	$-0.771017\pi$
$983$	−3.52420e52	−1.50444	−0.752219	+	0.658913i	$0.771017\pi$
$984$	0	0
$985$	5.34424e51	0.220167
$986$	2.29262e52	0.927869
$987$	0	0
$988$	−3.34263e52	−1.30569
$989$	8.64547e52	3.31783
$990$	0	0
$991$	1.33571e52	0.494792	0.247396	−	0.968915i	$-0.420425\pi$
$991$	1.33571e52	0.494792	0.247396	+	0.968915i	$0.420425\pi$
$992$	9.42646e52	3.43079
$993$	0	0
$994$	−6.30681e52	−2.21589
$995$	2.26577e52	0.782189
$996$	0	0
$997$	1.26912e52	0.422996	0.211498	−	0.977378i	$-0.432166\pi$
$997$	1.26912e52	0.422996	0.211498	+	0.977378i	$0.432166\pi$
$998$	−3.63702e52	−1.19114
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.36.a.e.1.1	yes	12
3.2	odd	2		27.36.a.c.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.36.a.c.1.12	✓	12	3.2	odd	2
27.36.a.e.1.1	yes	12	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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-360417. q^{2} +9.55406e10 q^{4} -1.49977e12 q^{5} -8.36061e14 q^{7} -2.20506e16 q^{8} +O(q^{10})\)	\(q-360417. q^{2} +9.55406e10 q^{4} -1.49977e12 q^{5} -8.36061e14 q^{7} -2.20506e16 q^{8} +5.40542e17 q^{10} +1.19996e18 q^{11} -2.90840e19 q^{13} +3.01331e20 q^{14} +4.66467e21 q^{16} +5.08674e21 q^{17} +1.20294e22 q^{19} -1.43289e23 q^{20} -4.32485e23 q^{22} +1.31613e24 q^{23} -6.61074e23 q^{25} +1.04824e25 q^{26} -7.98778e25 q^{28} -1.25051e25 q^{29} -1.02065e26 q^{31} -9.23571e26 q^{32} -1.83335e27 q^{34} +1.25390e27 q^{35} -1.61438e27 q^{37} -4.33561e27 q^{38} +3.30709e28 q^{40} -6.36803e27 q^{41} +6.56885e28 q^{43} +1.14645e29 q^{44} -4.74356e29 q^{46} +2.57962e29 q^{47} +3.20180e29 q^{49} +2.38262e29 q^{50} -2.77871e30 q^{52} +6.46994e29 q^{53} -1.79966e30 q^{55} +1.84357e31 q^{56} +4.50706e30 q^{58} -3.31794e30 q^{59} -3.97906e29 q^{61} +3.67861e31 q^{62} +1.72594e32 q^{64} +4.36194e31 q^{65} -1.06297e32 q^{67} +4.85990e32 q^{68} -4.51927e32 q^{70} -2.09299e32 q^{71} -4.15041e32 q^{73} +5.81851e32 q^{74} +1.14930e33 q^{76} -1.00324e33 q^{77} +1.08282e33 q^{79} -6.99593e33 q^{80} +2.29515e33 q^{82} -1.12823e32 q^{83} -7.62894e33 q^{85} -2.36753e34 q^{86} -2.64598e34 q^{88} -2.33897e34 q^{89} +2.43160e34 q^{91} +1.25744e35 q^{92} -9.29739e34 q^{94} -1.80414e34 q^{95} +7.96728e34 q^{97} -1.15398e35 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 209817 q^{2} + 238170471357 q^{4} + 237590493300 q^{5} - 26315460937956 q^{7} - 11\!\cdots\!85 q^{8}+O(q^{10}) \) 12 * q + 209817 * q^2 + 238170471357 * q^4 + 237590493300 * q^5 - 26315460937956 * q^7 - 1152523679482485 * q^8	\( 12 q + 209817 q^{2} + 238170471357 q^{4} + 237590493300 q^{5} - 26315460937956 q^{7} - 11\!\cdots\!85 q^{8}+ \cdots + 13\!\cdots\!62 q^{98}+O(q^{100}) \) 12 * q + 209817 * q^2 + 238170471357 * q^4 + 237590493300 * q^5 - 26315460937956 * q^7 - 1152523679482485 * q^8 + 53259116553823275 * q^10 + 1496551124826816372 * q^11 - 34328912703023996472 * q^13 + 166669000963239381363 * q^14 + 4860675541765013760705 * q^16 + 7628970615928780710168 * q^17 - 16082766160768990414128 * q^19 - 211198084408179102198645 * q^20 - 52234888231956498211143 * q^22 + 565792395499445750511120 * q^23 + 5555537533169508824637960 * q^25 + 10184460439954596310163124 * q^26 - 53247279099589429012932765 * q^28 - 15537250808393309204019840 * q^29 + 154551329202285014265355884 * q^31 - 569790636301246841664865893 * q^32 - 122663300810295291941317176 * q^34 + 3588084579418444573261981740 * q^35 - 3073092555569253564836155872 * q^37 - 4326541398755059647257334486 * q^38 + 16712219069408859411075107445 * q^40 - 6691483020866136020996784912 * q^41 + 44796529484279073882504569112 * q^43 + 150568080332409818247115161549 * q^44 - 471505816716619581821565726174 * q^46 + 200678144977450668688761056616 * q^47 + 1315326329423501432492384307360 * q^49 + 1819137608178900468376954622580 * q^50 - 1080581934187974962144983076796 * q^52 + 2691222104186670153874638254580 * q^53 + 1500392259136415557290568517820 * q^55 + 27633438158473075328907235551549 * q^56 - 5506073919138421424521005999066 * q^58 + 32766260948981198673864221207520 * q^59 - 11006721411902603259250479295728 * q^61 + 127809302103201321944632426528953 * q^62 + 72419202212575418035703901510825 * q^64 + 190220559832149919172950674352680 * q^65 - 62906959665093330766348132179432 * q^67 + 607941100538476155661581380036520 * q^68 - 23595260156850200715643047044895 * q^70 - 3715605905805552340595540265048 * q^71 - 456868766517955704811323145527276 * q^73 - 496856719188950172720386429591838 * q^74 + 750812250117487485399293980204746 * q^76 - 1491293331618714973103731562151324 * q^77 + 1441181755077379395085116385745496 * q^79 - 6329202616666871203376911405771665 * q^80 - 1025667950479925694157293281726742 * q^82 - 13534009831537273626534025954790652 * q^83 - 507540343243458160211502437320680 * q^85 - 38323435706440776292224238551430650 * q^86 - 3499699986133997560404753135251493 * q^88 - 20560572049351982374812418435974744 * q^89 + 32908658163780767717902805354624520 * q^91 + 22244007710744113308805635019788330 * q^92 - 26483113405825635155782484066278554 * q^94 + 61918684402966765699909525859718000 * q^95 + 9968493320960173817283702572205012 * q^97 + 138741461320002422834263484212313562 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more