LMFDB - Embedded newform 27.11.b.c.26.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,11,Mod(26,27)]

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([1]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	27.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.1546458222$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 188x^{2} + 756 $ x^4 + 188*x^2 + 756
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.2
Root			$-13.5606i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.11.b.c.26.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-23.5425i q^{2} +469.750 q^{4} +4424.52i q^{5} -21568.0 q^{7} -35166.6i q^{8} +O(q^{10})$	\(q-23.5425i q^{2} +469.750 q^{4} +4424.52i q^{5} -21568.0 q^{7} -35166.6i q^{8} +104164. q^{10} -242828. i q^{11} -398104. q^{13} +507765. i q^{14} -346888. q^{16} -2.02619e6i q^{17} -2.77192e6 q^{19} +2.07842e6i q^{20} -5.71678e6 q^{22} +2.47900e6i q^{23} -9.81076e6 q^{25} +9.37237e6i q^{26} -1.01315e7 q^{28} +9.13358e6i q^{29} -1.81463e7 q^{31} -2.78440e7i q^{32} -4.77017e7 q^{34} -9.54280e7i q^{35} +1.14330e7 q^{37} +6.52579e7i q^{38} +1.55595e8 q^{40} +1.37099e8i q^{41} +9.87897e7 q^{43} -1.14068e8i q^{44} +5.83620e7 q^{46} -1.41804e8i q^{47} +1.82702e8 q^{49} +2.30970e8i q^{50} -1.87009e8 q^{52} +1.39084e8i q^{53} +1.07440e9 q^{55} +7.58473e8i q^{56} +2.15027e8 q^{58} -1.06798e9i q^{59} -1.44976e9 q^{61} +4.27210e8i q^{62} -1.01073e9 q^{64} -1.76142e9i q^{65} +1.44613e9 q^{67} -9.51804e8i q^{68} -2.24662e9 q^{70} +1.33807e9i q^{71} -1.46812e9 q^{73} -2.69162e8i q^{74} -1.30211e9 q^{76} +5.23731e9i q^{77} -4.95381e9 q^{79} -1.53481e9i q^{80} +3.22766e9 q^{82} -5.60643e8i q^{83} +8.96494e9 q^{85} -2.32576e9i q^{86} -8.53944e9 q^{88} +4.73263e9i q^{89} +8.58629e9 q^{91} +1.16451e9i q^{92} -3.33843e9 q^{94} -1.22644e10i q^{95} +1.21701e10 q^{97} -4.30128e9i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 548 q^{4} + 20516 q^{7}+O(q^{10}) $ 4 * q - 548 * q^4 + 20516 * q^7	$ 4 q - 548 q^{4} + 20516 q^{7} - 199800 q^{10} - 262420 q^{13} - 3207800 q^{16} - 8233516 q^{19} - 21085704 q^{22} - 47203580 q^{25} - 67604212 q^{28} - 46994920 q^{31} - 78985368 q^{34} + 108086444 q^{37} + 196812720 q^{40} + 78251000 q^{43} + 882645768 q^{46} + 1826240448 q^{49} - 771022300 q^{52} + 1034575200 q^{55} + 1267544592 q^{58} - 3222389524 q^{61} - 1166263952 q^{64} + 1972286516 q^{67} - 17482328280 q^{70} + 4077598940 q^{73} - 603762388 q^{76} - 10402780252 q^{79} - 7945552656 q^{82} + 19825132320 q^{85} - 19783893168 q^{88} + 34160896780 q^{91} + 22345655544 q^{94} + 12969797468 q^{97}+O(q^{100}) $ 4 * q - 548 * q^4 + 20516 * q^7 - 199800 * q^10 - 262420 * q^13 - 3207800 * q^16 - 8233516 * q^19 - 21085704 * q^22 - 47203580 * q^25 - 67604212 * q^28 - 46994920 * q^31 - 78985368 * q^34 + 108086444 * q^37 + 196812720 * q^40 + 78251000 * q^43 + 882645768 * q^46 + 1826240448 * q^49 - 771022300 * q^52 + 1034575200 * q^55 + 1267544592 * q^58 - 3222389524 * q^61 - 1166263952 * q^64 + 1972286516 * q^67 - 17482328280 * q^70 + 4077598940 * q^73 - 603762388 * q^76 - 10402780252 * q^79 - 7945552656 * q^82 + 19825132320 * q^85 - 19783893168 * q^88 + 34160896780 * q^91 + 22345655544 * q^94 + 12969797468 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	− 23.5425i	− 0.735704i	−0.929884	−	0.367852i	$-0.880093\pi$
$2$	− 23.5425i	− 0.735704i	0.929884	−	0.367852i	$-0.119907\pi$
$3$	0	0
$3$	0	0
$4$	469.750	0.458740
$5$	4424.52i	1.41585i	0.706289	+	0.707923i	$0.250368\pi$
$5$	4424.52i	1.41585i	−0.706289	+	0.707923i	$0.749632\pi$
$6$	0	0
$7$	−21568.0	−1.28327	−0.641637	−	0.767009i	$-0.721744\pi$
$7$	−21568.0	−1.28327	−0.641637	+	0.767009i	$0.721744\pi$
$8$	− 35166.6i	− 1.07320i
$9$	0	0
$10$	104164.	1.04164
$11$	− 242828.i	− 1.50777i	−0.657007	−	0.753885i	$-0.728178\pi$
$11$	− 242828.i	− 1.50777i	0.657007	−	0.753885i	$-0.271822\pi$
$12$	0	0
$13$	−398104.	−1.07221	−0.536105	−	0.844152i	$-0.680105\pi$
$13$	−398104.	−1.07221	−0.536105	+	0.844152i	$0.680105\pi$
$14$	507765.i	0.944109i
$15$	0	0
$16$	−346888.	−0.330818
$17$	− 2.02619e6i	− 1.42704i	−0.700634	−	0.713520i	$-0.747100\pi$
$17$	− 2.02619e6i	− 1.42704i	0.700634	−	0.713520i	$-0.252900\pi$
$18$	0	0
$19$	−2.77192e6	−1.11947	−0.559735	−	0.828672i	$-0.689097\pi$
$19$	−2.77192e6	−1.11947	−0.559735	+	0.828672i	$0.689097\pi$
$20$	2.07842e6i	0.649505i
$21$	0	0
$22$	−5.71678e6	−1.10927
$23$	2.47900e6i	0.385157i	0.981282	+	0.192579i	$0.0616850\pi$
$23$	2.47900e6i	0.385157i	−0.981282	+	0.192579i	$0.938315\pi$
$24$	0	0
$25$	−9.81076e6	−1.00462
$26$	9.37237e6i	0.788828i
$27$	0	0
$28$	−1.01315e7	−0.588689
$29$	9.13358e6i	0.445298i	0.974899	+	0.222649i	$0.0714704\pi$
$29$	9.13358e6i	0.445298i	−0.974899	+	0.222649i	$0.928530\pi$
$30$	0	0
$31$	−1.81463e7	−0.633840	−0.316920	−	0.948452i	$-0.602649\pi$
$31$	−1.81463e7	−0.633840	−0.316920	+	0.948452i	$0.602649\pi$
$32$	− 2.78440e7i	− 0.829816i
$33$	0	0
$34$	−4.77017e7	−1.04988
$35$	− 9.54280e7i	− 1.81692i
$36$	0	0
$37$	1.14330e7	0.164874	0.0824369	−	0.996596i	$-0.473730\pi$
$37$	1.14330e7	0.164874	0.0824369	+	0.996596i	$0.473730\pi$
$38$	6.52579e7i	0.823598i
$39$	0	0
$40$	1.55595e8	1.51949
$41$	1.37099e8i	1.18336i	0.806174	+	0.591678i	$0.201534\pi$
$41$	1.37099e8i	1.18336i	−0.806174	+	0.591678i	$0.798466\pi$
$42$	0	0
$43$	9.87897e7	0.672000	0.336000	−	0.941862i	$-0.390926\pi$
$43$	9.87897e7	0.672000	0.336000	+	0.941862i	$0.390926\pi$
$44$	− 1.14068e8i	− 0.691674i
$45$	0	0
$46$	5.83620e7	0.283362
$47$	− 1.41804e8i	− 0.618302i	−0.951013	−	0.309151i	$-0.899955\pi$
$47$	− 1.41804e8i	− 0.618302i	0.951013	−	0.309151i	$-0.100045\pi$
$48$	0	0
$49$	1.82702e8	0.646791
$50$	2.30970e8i	0.739104i
$51$	0	0
$52$	−1.87009e8	−0.491865
$53$	1.39084e8i	0.332582i	0.986077	+	0.166291i	$0.0531792\pi$
$53$	1.39084e8i	0.332582i	−0.986077	+	0.166291i	$0.946821\pi$
$54$	0	0
$55$	1.07440e9	2.13477
$56$	7.58473e8i	1.37721i
$57$	0	0
$58$	2.15027e8	0.327608
$59$	− 1.06798e9i	− 1.49384i	−0.664916	−	0.746918i	$-0.731533\pi$
$59$	− 1.06798e9i	− 1.49384i	0.664916	−	0.746918i	$-0.268467\pi$
$60$	0	0
$61$	−1.44976e9	−1.71651	−0.858256	−	0.513222i	$-0.828452\pi$
$61$	−1.44976e9	−1.71651	−0.858256	+	0.513222i	$0.828452\pi$
$62$	4.27210e8i	0.466318i
$63$	0	0
$64$	−1.01073e9	−0.941317
$65$	− 1.76142e9i	− 1.51808i
$66$	0	0
$67$	1.44613e9	1.07111	0.535556	−	0.844500i	$-0.320102\pi$
$67$	1.44613e9	1.07111	0.535556	+	0.844500i	$0.320102\pi$
$68$	− 9.51804e8i	− 0.654640i
$69$	0	0
$70$	−2.24662e9	−1.33671
$71$	1.33807e9i	0.741630i	0.928707	+	0.370815i	$0.120922\pi$
$71$	1.33807e9i	0.741630i	−0.928707	+	0.370815i	$0.879078\pi$
$72$	0	0
$73$	−1.46812e9	−0.708187	−0.354094	−	0.935210i	$-0.615211\pi$
$73$	−1.46812e9	−0.708187	−0.354094	+	0.935210i	$0.615211\pi$
$74$	− 2.69162e8i	− 0.121298i
$75$	0	0
$76$	−1.30211e9	−0.513545
$77$	5.23731e9i	1.93488i
$78$	0	0
$79$	−4.95381e9	−1.60992	−0.804959	−	0.593331i	$-0.797813\pi$
$79$	−4.95381e9	−1.60992	−0.804959	+	0.593331i	$0.797813\pi$
$80$	− 1.53481e9i	− 0.468388i
$81$	0	0
$82$	3.22766e9	0.870600
$83$	− 5.60643e8i	− 0.142330i	−0.997465	−	0.0711650i	$-0.977328\pi$
$83$	− 5.60643e8i	− 0.142330i	0.997465	−	0.0711650i	$-0.0226717\pi$
$84$	0	0
$85$	8.96494e9	2.02047
$86$	− 2.32576e9i	− 0.494393i
$87$	0	0
$88$	−8.53944e9	−1.61814
$89$	4.73263e9i	0.847524i	0.905774	+	0.423762i	$0.139291\pi$
$89$	4.73263e9i	0.847524i	−0.905774	+	0.423762i	$0.860709\pi$
$90$	0	0
$91$	8.58629e9	1.37594
$92$	1.16451e9i	0.176687i
$93$	0	0
$94$	−3.33843e9	−0.454887
$95$	− 1.22644e10i	− 1.58500i
$96$	0	0
$97$	1.21701e10	1.41722	0.708609	−	0.705601i	$-0.249323\pi$
$97$	1.21701e10	1.41722	0.708609	+	0.705601i	$0.249323\pi$
$98$	− 4.30128e9i	− 0.475847i
$99$	0	0
$100$	−4.60860e9	−0.460860
$101$	1.19970e10i	1.14148i	0.821132	+	0.570738i	$0.193343\pi$
$101$	1.19970e10i	1.14148i	−0.821132	+	0.570738i	$0.806657\pi$
$102$	0	0
$103$	1.17848e10	1.01657	0.508286	−	0.861188i	$-0.330279\pi$
$103$	1.17848e10	1.01657	0.508286	+	0.861188i	$0.330279\pi$
$104$	1.40000e10i	1.15070i
$105$	0	0
$106$	3.27440e9	0.244682
$107$	− 1.67650e10i	− 1.19532i	−0.801749	−	0.597661i	$-0.796097\pi$
$107$	− 1.67650e10i	− 1.19532i	0.801749	−	0.597661i	$-0.203903\pi$
$108$	0	0
$109$	1.39867e10	0.909040	0.454520	−	0.890736i	$-0.349811\pi$
$109$	1.39867e10	0.909040	0.454520	+	0.890736i	$0.349811\pi$
$110$	− 2.52940e10i	− 1.57056i
$111$	0	0
$112$	7.48167e9	0.424530
$113$	− 1.66178e9i	− 0.0901948i	−0.998983	−	0.0450974i	$-0.985640\pi$
$113$	− 1.66178e9i	− 0.0901948i	0.998983	−	0.0450974i	$-0.0143598\pi$
$114$	0	0
$115$	−1.09684e10	−0.545323
$116$	4.29049e9i	0.204276i
$117$	0	0
$118$	−2.51430e10	−1.09902
$119$	4.37009e10i	1.83128i
$120$	0	0
$121$	−3.30279e10	−1.27337
$122$	3.41310e10i	1.26284i
$123$	0	0
$124$	−8.52421e9	−0.290768
$125$	− 1.99685e8i	− 0.00654329i
$126$	0	0
$127$	2.55071e9	0.0772045	0.0386022	−	0.999255i	$-0.487709\pi$
$127$	2.55071e9	0.0772045	0.0386022	+	0.999255i	$0.487709\pi$
$128$	− 4.71710e9i	− 0.137286i
$129$	0	0
$130$	−4.14682e10	−1.11686
$131$	− 1.62753e10i	− 0.421863i	−0.977501	−	0.210932i	$-0.932350\pi$
$131$	− 1.62753e10i	− 0.421863i	0.977501	−	0.210932i	$-0.0676498\pi$
$132$	0	0
$133$	5.97846e10	1.43659
$134$	− 3.40457e10i	− 0.788021i
$135$	0	0
$136$	−7.12544e10	−1.53150
$137$	− 6.54848e10i	− 1.35687i	−0.734661	−	0.678434i	$-0.762659\pi$
$137$	− 6.54848e10i	− 1.35687i	0.734661	−	0.678434i	$-0.237341\pi$
$138$	0	0
$139$	−3.01191e10	−0.580454	−0.290227	−	0.956958i	$-0.593731\pi$
$139$	−3.01191e10	−0.580454	−0.290227	+	0.956958i	$0.593731\pi$
$140$	− 4.48272e10i	− 0.833493i
$141$	0	0
$142$	3.15015e10	0.545620
$143$	9.66707e10i	1.61664i
$144$	0	0
$145$	−4.04117e10	−0.630474
$146$	3.45633e10i	0.521016i
$147$	0	0
$148$	5.37065e9	0.0756342
$149$	− 9.88459e10i	− 1.34595i	−0.739667	−	0.672973i	$-0.765017\pi$
$149$	− 9.88459e10i	− 1.34595i	0.739667	−	0.672973i	$-0.234983\pi$
$150$	0	0
$151$	8.33061e10	1.06119	0.530594	−	0.847626i	$-0.321969\pi$
$151$	8.33061e10	1.06119	0.530594	+	0.847626i	$0.321969\pi$
$152$	9.74790e10i	1.20141i
$153$	0	0
$154$	1.23299e11	1.42350
$155$	− 8.02887e10i	− 0.897420i
$156$	0	0
$157$	−7.89160e10	−0.827307	−0.413653	−	0.910434i	$-0.635747\pi$
$157$	−7.89160e10	−0.827307	−0.413653	+	0.910434i	$0.635747\pi$
$158$	1.16625e11i	1.18442i
$159$	0	0
$160$	1.23196e11	1.17489
$161$	− 5.34671e10i	− 0.494262i
$162$	0	0
$163$	1.98358e10	0.172390	0.0861951	−	0.996278i	$-0.472529\pi$
$163$	1.98358e10	0.172390	0.0861951	+	0.996278i	$0.472529\pi$
$164$	6.44023e10i	0.542853i
$165$	0	0
$166$	−1.31990e10	−0.104713
$167$	4.31117e10i	0.331905i	0.986134	+	0.165952i	$0.0530698\pi$
$167$	4.31117e10i	0.331905i	−0.986134	+	0.165952i	$0.946930\pi$
$168$	0	0
$169$	2.06281e10	0.149632
$170$	− 2.11057e11i	− 1.48647i
$171$	0	0
$172$	4.64064e10	0.308273
$173$	1.26896e11i	0.818875i	0.912338	+	0.409438i	$0.134275\pi$
$173$	1.26896e11i	0.818875i	−0.912338	+	0.409438i	$0.865725\pi$
$174$	0	0
$175$	2.11598e11	1.28920
$176$	8.42340e10i	0.498797i
$177$	0	0
$178$	1.11418e11	0.623527
$179$	1.71735e11i	0.934531i	0.884117	+	0.467265i	$0.154761\pi$
$179$	1.71735e11i	0.934531i	−0.884117	+	0.467265i	$0.845239\pi$
$180$	0	0
$181$	−9.76637e10	−0.502737	−0.251368	−	0.967892i	$-0.580881\pi$
$181$	−9.76637e10	−0.502737	−0.251368	+	0.967892i	$0.580881\pi$
$182$	− 2.02143e11i	− 1.01228i
$183$	0	0
$184$	8.71782e10	0.413351
$185$	5.05856e10i	0.233436i
$186$	0	0
$187$	−4.92016e11	−2.15165
$188$	− 6.66126e10i	− 0.283640i
$189$	0	0
$190$	−2.88735e11	−1.16609
$191$	− 1.00665e11i	− 0.396015i	−0.980201	−	0.198007i	$-0.936553\pi$
$191$	− 1.00665e11i	− 0.396015i	0.980201	−	0.198007i	$-0.0634470\pi$
$192$	0	0
$193$	−3.28449e11	−1.22654	−0.613270	−	0.789873i	$-0.710146\pi$
$193$	−3.28449e11	−1.22654	−0.613270	+	0.789873i	$0.710146\pi$
$194$	− 2.86516e11i	− 1.04265i
$195$	0	0
$196$	8.58244e10	0.296709
$197$	− 2.22370e11i	− 0.749453i	−0.927135	−	0.374727i	$-0.877737\pi$
$197$	− 2.22370e11i	− 0.749453i	0.927135	−	0.374727i	$-0.122263\pi$
$198$	0	0
$199$	1.76697e11	0.566191	0.283095	−	0.959092i	$-0.408639\pi$
$199$	1.76697e11	0.566191	0.283095	+	0.959092i	$0.408639\pi$
$200$	3.45011e11i	1.07816i
$201$	0	0
$202$	2.82440e11	0.839788
$203$	− 1.96993e11i	− 0.571439i
$204$	0	0
$205$	−6.06598e11	−1.67545
$206$	− 2.77445e11i	− 0.747896i
$207$	0	0
$208$	1.38097e11	0.354706
$209$	6.73098e11i	1.68790i
$210$	0	0
$211$	−2.67258e11	−0.639027	−0.319513	−	0.947582i	$-0.603519\pi$
$211$	−2.67258e11	−0.639027	−0.319513	+	0.947582i	$0.603519\pi$
$212$	6.53349e10i	0.152569i
$213$	0	0
$214$	−3.94691e11	−0.879403
$215$	4.37097e11i	0.951449i
$216$	0	0
$217$	3.91379e11	0.813390
$218$	− 3.29283e11i	− 0.668785i
$219$	0	0
$220$	5.04697e11	0.979304
$221$	8.06635e11i	1.53009i
$222$	0	0
$223$	−5.07380e11	−0.920046	−0.460023	−	0.887907i	$-0.652159\pi$
$223$	−5.07380e11	−0.920046	−0.460023	+	0.887907i	$0.652159\pi$
$224$	6.00539e11i	1.06488i
$225$	0	0
$226$	−3.91225e10	−0.0663566
$227$	− 9.61037e11i	− 1.59445i	−0.603682	−	0.797225i	$-0.706300\pi$
$227$	− 9.61037e11i	− 1.59445i	0.603682	−	0.797225i	$-0.293700\pi$
$228$	0	0
$229$	2.08272e11	0.330715	0.165357	−	0.986234i	$-0.447122\pi$
$229$	2.08272e11	0.330715	0.165357	+	0.986234i	$0.447122\pi$
$230$	2.58224e11i	0.401197i
$231$	0	0
$232$	3.21197e11	0.477894
$233$	− 1.26797e12i	− 1.84642i	−0.384296	−	0.923210i	$-0.625556\pi$
$233$	− 1.26797e12i	− 1.84642i	0.384296	−	0.923210i	$-0.374444\pi$
$234$	0	0
$235$	6.27417e11	0.875420
$236$	− 5.01683e11i	− 0.685282i
$237$	0	0
$238$	1.02883e12	1.34728
$239$	− 9.98539e11i	− 1.28049i	−0.768171	−	0.640244i	$-0.778833\pi$
$239$	− 9.98539e11i	− 1.28049i	0.768171	−	0.640244i	$-0.221167\pi$
$240$	0	0
$241$	5.61004e11	0.690050	0.345025	−	0.938594i	$-0.387870\pi$
$241$	5.61004e11	0.690050	0.345025	+	0.938594i	$0.387870\pi$
$242$	7.77561e11i	0.936823i
$243$	0	0
$244$	−6.81024e11	−0.787432
$245$	8.08371e11i	0.915757i
$246$	0	0
$247$	1.10351e12	1.20031
$248$	6.38144e11i	0.680237i
$249$	0	0
$250$	−4.70110e9	−0.00481392
$251$	1.09342e12i	1.09753i	0.835975	+	0.548767i	$0.184903\pi$
$251$	1.09342e12i	1.09753i	−0.835975	+	0.548767i	$0.815097\pi$
$252$	0	0
$253$	6.01971e11	0.580728
$254$	− 6.00502e10i	− 0.0567996i
$255$	0	0
$256$	−1.14604e12	−1.04232
$257$	− 1.74563e12i	− 1.55699i	−0.627648	−	0.778497i	$-0.715982\pi$
$257$	− 1.74563e12i	− 1.55699i	0.627648	−	0.778497i	$-0.284018\pi$
$258$	0	0
$259$	−2.46587e11	−0.211578
$260$	− 8.27425e11i	− 0.696405i
$261$	0	0
$262$	−3.83161e11	−0.310367
$263$	− 3.82099e11i	− 0.303667i	−0.988406	−	0.151833i	$-0.951482\pi$
$263$	− 3.82099e11i	− 0.303667i	0.988406	−	0.151833i	$-0.0485177\pi$
$264$	0	0
$265$	−6.15382e11	−0.470886
$266$	− 1.40748e12i	− 1.05690i
$267$	0	0
$268$	6.79321e11	0.491361
$269$	1.54623e12i	1.09777i	0.835896	+	0.548887i	$0.184948\pi$
$269$	1.54623e12i	1.09777i	−0.835896	+	0.548887i	$0.815052\pi$
$270$	0	0
$271$	1.85954e12	1.27221	0.636104	−	0.771603i	$-0.280545\pi$
$271$	1.85954e12	1.27221	0.636104	+	0.771603i	$0.280545\pi$
$272$	7.02862e11i	0.472091i
$273$	0	0
$274$	−1.54168e12	−0.998253
$275$	2.38232e12i	1.51474i
$276$	0	0
$277$	−1.96780e12	−1.20665	−0.603326	−	0.797495i	$-0.706158\pi$
$277$	−1.96780e12	−1.20665	−0.603326	+	0.797495i	$0.706158\pi$
$278$	7.09079e11i	0.427042i
$279$	0	0
$280$	−3.35588e12	−1.94992
$281$	− 3.53773e11i	− 0.201926i	−0.994890	−	0.100963i	$-0.967808\pi$
$281$	− 3.53773e11i	− 0.201926i	0.994890	−	0.100963i	$-0.0321924\pi$
$282$	0	0
$283$	−1.55446e12	−0.856341	−0.428170	−	0.903698i	$-0.640842\pi$
$283$	−1.55446e12	−0.856341	−0.428170	+	0.903698i	$0.640842\pi$
$284$	6.28558e11i	0.340215i
$285$	0	0
$286$	2.27587e12	1.18937
$287$	− 2.95695e12i	− 1.51857i
$288$	0	0
$289$	−2.08947e12	−1.03645
$290$	9.51393e11i	0.463842i
$291$	0	0
$292$	−6.89650e11	−0.324874
$293$	3.65602e12i	1.69305i	0.532347	+	0.846526i	$0.321310\pi$
$293$	3.65602e12i	1.69305i	−0.532347	+	0.846526i	$0.678690\pi$
$294$	0	0
$295$	4.72530e12	2.11504
$296$	− 4.02060e11i	− 0.176943i
$297$	0	0
$298$	−2.32708e12	−0.990217
$299$	− 9.86901e11i	− 0.412969i
$300$	0	0
$301$	−2.13069e12	−0.862360
$302$	− 1.96124e12i	− 0.780719i
$303$	0	0
$304$	9.61544e11	0.370341
$305$	− 6.41449e12i	− 2.43032i
$306$	0	0
$307$	4.14571e12	1.52022	0.760111	−	0.649793i	$-0.225144\pi$
$307$	4.14571e12	1.52022	0.760111	+	0.649793i	$0.225144\pi$
$308$	2.46022e12i	0.887607i
$309$	0	0
$310$	−1.89020e12	−0.660235
$311$	− 7.41396e11i	− 0.254829i	−0.991850	−	0.127414i	$-0.959332\pi$
$311$	− 7.41396e11i	− 0.254829i	0.991850	−	0.127414i	$-0.0406678\pi$
$312$	0	0
$313$	−1.19855e12	−0.398966	−0.199483	−	0.979901i	$-0.563926\pi$
$313$	−1.19855e12	−0.398966	−0.199483	+	0.979901i	$0.563926\pi$
$314$	1.85788e12i	0.608653i
$315$	0	0
$316$	−2.32705e12	−0.738533
$317$	− 2.08772e12i	− 0.652191i	−0.945337	−	0.326096i	$-0.894267\pi$
$317$	− 2.08772e12i	− 0.652191i	0.945337	−	0.326096i	$-0.105733\pi$
$318$	0	0
$319$	2.21789e12	0.671407
$320$	− 4.47200e12i	− 1.33276i
$321$	0	0
$322$	−1.25875e12	−0.363631
$323$	5.61644e12i	1.59753i
$324$	0	0
$325$	3.90570e12	1.07716
$326$	− 4.66986e11i	− 0.126828i
$327$	0	0
$328$	4.82132e12	1.26998
$329$	3.05843e12i	0.793450i
$330$	0	0
$331$	5.73492e12	1.44340	0.721701	−	0.692204i	$-0.243360\pi$
$331$	5.73492e12	1.44340	0.721701	+	0.692204i	$0.243360\pi$
$332$	− 2.63362e11i	− 0.0652924i
$333$	0	0
$334$	1.01496e12	0.244184
$335$	6.39845e12i	1.51653i
$336$	0	0
$337$	−3.80990e11	−0.0876524	−0.0438262	−	0.999039i	$-0.513955\pi$
$337$	−3.80990e11	−0.0876524	−0.0438262	+	0.999039i	$0.513955\pi$
$338$	− 4.85638e11i	− 0.110085i
$339$	0	0
$340$	4.21127e12	0.926870
$341$	4.40643e12i	0.955685i
$342$	0	0
$343$	2.15190e12	0.453264
$344$	− 3.47410e12i	− 0.721191i
$345$	0	0
$346$	2.98745e12	0.602450
$347$	− 3.54684e11i	− 0.0705009i	−0.999379	−	0.0352505i	$-0.988777\pi$
$347$	− 3.54684e11i	− 0.0705009i	0.999379	−	0.0352505i	$-0.0112229\pi$
$348$	0	0
$349$	−7.67975e12	−1.48327	−0.741634	−	0.670805i	$-0.765949\pi$
$349$	−7.67975e12	−1.48327	−0.741634	+	0.670805i	$0.765949\pi$
$350$	− 4.98156e12i	− 0.948473i
$351$	0	0
$352$	−6.76130e12	−1.25117
$353$	− 6.58216e12i	− 1.20087i	−0.799674	−	0.600434i	$-0.794995\pi$
$353$	− 6.58216e12i	− 1.20087i	0.799674	−	0.600434i	$-0.205005\pi$
$354$	0	0
$355$	−5.92032e12	−1.05003
$356$	2.22315e12i	0.388793i
$357$	0	0
$358$	4.04307e12	0.687538
$359$	6.68773e12i	1.12152i	0.827979	+	0.560760i	$0.189491\pi$
$359$	6.68773e12i	1.12152i	−0.827979	+	0.560760i	$0.810509\pi$
$360$	0	0
$361$	1.55245e12	0.253211
$362$	2.29925e12i	0.369865i
$363$	0	0
$364$	4.03341e12	0.631197
$365$	− 6.49574e12i	− 1.00268i
$366$	0	0
$367$	−9.31180e12	−1.39863	−0.699316	−	0.714813i	$-0.746512\pi$
$367$	−9.31180e12	−1.39863	−0.699316	+	0.714813i	$0.746512\pi$
$368$	− 8.59936e11i	− 0.127417i
$369$	0	0
$370$	1.19091e12	0.171740
$371$	− 2.99977e12i	− 0.426794i
$372$	0	0
$373$	6.18448e11	0.0856562	0.0428281	−	0.999082i	$-0.486363\pi$
$373$	6.18448e11	0.0856562	0.0428281	+	0.999082i	$0.486363\pi$
$374$	1.15833e13i	1.58298i
$375$	0	0
$376$	−4.98678e12	−0.663562
$377$	− 3.63611e12i	− 0.477453i
$378$	0	0
$379$	6.79881e11	0.0869435	0.0434717	−	0.999055i	$-0.486158\pi$
$379$	6.79881e11	0.0869435	0.0434717	+	0.999055i	$0.486158\pi$
$380$	− 5.76120e12i	− 0.727101i
$381$	0	0
$382$	−2.36991e12	−0.291350
$383$	− 5.62340e12i	− 0.682346i	−0.940000	−	0.341173i	$-0.889176\pi$
$383$	− 5.62340e12i	− 0.682346i	0.940000	−	0.341173i	$-0.110824\pi$
$384$	0	0
$385$	−2.31726e13	−2.73949
$386$	7.73252e12i	0.902370i
$387$	0	0
$388$	5.71692e12	0.650135
$389$	9.64393e12i	1.08270i	0.840799	+	0.541348i	$0.182086\pi$
$389$	9.64393e12i	1.08270i	−0.840799	+	0.541348i	$0.817914\pi$
$390$	0	0
$391$	5.02294e12	0.549635
$392$	− 6.42503e12i	− 0.694137i
$393$	0	0
$394$	−5.23515e12	−0.551376
$395$	− 2.19182e13i	− 2.27940i
$396$	0	0
$397$	−1.48431e11	−0.0150512	−0.00752561	−	0.999972i	$-0.502395\pi$
$397$	−1.48431e11	−0.0150512	−0.00752561	+	0.999972i	$0.502395\pi$
$398$	− 4.15988e12i	− 0.416549i
$399$	0	0
$400$	3.40323e12	0.332347
$401$	− 2.15508e11i	− 0.0207846i	−0.999946	−	0.0103923i	$-0.996692\pi$
$401$	− 2.15508e11i	− 0.0207846i	0.999946	−	0.0103923i	$-0.00330803\pi$
$402$	0	0
$403$	7.22411e12	0.679609
$404$	5.63560e12i	0.523640i
$405$	0	0
$406$	−4.63771e12	−0.420410
$407$	− 2.77625e12i	− 0.248592i
$408$	0	0
$409$	−1.10585e13	−0.966228	−0.483114	−	0.875557i	$-0.660494\pi$
$409$	−1.10585e13	−0.966228	−0.483114	+	0.875557i	$0.660494\pi$
$410$	1.42809e13i	1.23264i
$411$	0	0
$412$	5.53593e12	0.466342
$413$	2.30342e13i	1.91700i
$414$	0	0
$415$	2.48058e12	0.201517
$416$	1.10848e13i	0.889737i
$417$	0	0
$418$	1.58464e13	1.24180
$419$	− 7.75836e12i	− 0.600758i	−0.953820	−	0.300379i	$-0.902887\pi$
$419$	− 7.75836e12i	− 0.600758i	0.953820	−	0.300379i	$-0.0971132\pi$
$420$	0	0
$421$	−9.44013e12	−0.713785	−0.356893	−	0.934145i	$-0.616164\pi$
$421$	−9.44013e12	−0.713785	−0.356893	+	0.934145i	$0.616164\pi$
$422$	6.29194e12i	0.470135i
$423$	0	0
$424$	4.89113e12	0.356928
$425$	1.98785e13i	1.43364i
$426$	0	0
$427$	3.12684e13	2.20275
$428$	− 7.87535e12i	− 0.548342i
$429$	0	0
$430$	1.02904e13	0.699985
$431$	− 1.56255e13i	− 1.05063i	−0.850909	−	0.525313i	$-0.823948\pi$
$431$	− 1.56255e13i	− 1.05063i	0.850909	−	0.525313i	$-0.176052\pi$
$432$	0	0
$433$	−1.68288e13	−1.10564	−0.552820	−	0.833301i	$-0.686448\pi$
$433$	−1.68288e13	−1.10564	−0.552820	+	0.833301i	$0.686448\pi$
$434$	− 9.21405e12i	− 0.598414i
$435$	0	0
$436$	6.57025e12	0.417013
$437$	− 6.87159e12i	− 0.431172i
$438$	0	0
$439$	−6.14121e12	−0.376644	−0.188322	−	0.982107i	$-0.560305\pi$
$439$	−6.14121e12	−0.376644	−0.188322	+	0.982107i	$0.560305\pi$
$440$	− 3.77829e13i	− 2.29104i
$441$	0	0
$442$	1.89902e13	1.12569
$443$	1.37391e12i	0.0805265i	0.999189	+	0.0402633i	$0.0128197\pi$
$443$	1.37391e12i	0.0805265i	−0.999189	+	0.0402633i	$0.987180\pi$
$444$	0	0
$445$	−2.09396e13	−1.19996
$446$	1.19450e13i	0.676881i
$447$	0	0
$448$	2.17994e13	1.20797
$449$	− 1.04566e13i	− 0.573004i	−0.958080	−	0.286502i	$-0.907508\pi$
$449$	− 1.04566e13i	− 0.573004i	0.958080	−	0.286502i	$-0.0924925\pi$
$450$	0	0
$451$	3.32915e13	1.78423
$452$	− 7.80621e11i	− 0.0413759i
$453$	0	0
$454$	−2.26252e13	−1.17304
$455$	3.79902e13i	1.94812i
$456$	0	0
$457$	3.39247e13	1.70190	0.850951	−	0.525245i	$-0.176026\pi$
$457$	3.39247e13	1.70190	0.850951	+	0.525245i	$0.176026\pi$
$458$	− 4.90325e12i	− 0.243308i
$459$	0	0
$460$	−5.15240e12	−0.250162
$461$	− 2.28758e13i	− 1.09868i	−0.835599	−	0.549340i	$-0.814879\pi$
$461$	− 2.28758e13i	− 1.09868i	0.835599	−	0.549340i	$-0.185121\pi$
$462$	0	0
$463$	−3.66690e12	−0.172343	−0.0861716	−	0.996280i	$-0.527463\pi$
$463$	−3.66690e12	−0.172343	−0.0861716	+	0.996280i	$0.527463\pi$
$464$	− 3.16833e12i	− 0.147313i
$465$	0	0
$466$	−2.98513e13	−1.35842
$467$	2.48699e13i	1.11967i	0.828605	+	0.559834i	$0.189135\pi$
$467$	2.48699e13i	1.11967i	−0.828605	+	0.559834i	$0.810865\pi$
$468$	0	0
$469$	−3.11902e13	−1.37453
$470$	− 1.47710e13i	− 0.644050i
$471$	0	0
$472$	−3.75573e13	−1.60319
$473$	− 2.39889e13i	− 1.01322i
$474$	0	0
$475$	2.71946e13	1.12464
$476$	2.05285e13i	0.840083i
$477$	0	0
$478$	−2.35081e13	−0.942060
$479$	− 4.40756e12i	− 0.174792i	−0.996174	−	0.0873959i	$-0.972145\pi$
$479$	− 4.40756e12i	− 0.174792i	0.996174	−	0.0873959i	$-0.0278545\pi$
$480$	0	0
$481$	−4.55152e12	−0.176779
$482$	− 1.32074e13i	− 0.507672i
$483$	0	0
$484$	−1.55149e13	−0.584145
$485$	5.38470e13i	2.00656i
$486$	0	0
$487$	−9.42862e12	−0.344194	−0.172097	−	0.985080i	$-0.555054\pi$
$487$	−9.42862e12	−0.344194	−0.172097	+	0.985080i	$0.555054\pi$
$488$	5.09831e13i	1.84216i
$489$	0	0
$490$	1.90311e13	0.673726
$491$	− 1.40743e13i	− 0.493195i	−0.969118	−	0.246597i	$-0.920688\pi$
$491$	− 1.40743e13i	− 0.493195i	0.969118	−	0.246597i	$-0.0793125\pi$
$492$	0	0
$493$	1.85064e13	0.635459
$494$	− 2.59794e13i	− 0.883069i
$495$	0	0
$496$	6.29473e12	0.209686
$497$	− 2.88595e13i	− 0.951714i
$498$	0	0
$499$	−3.37708e13	−1.09154	−0.545769	−	0.837935i	$-0.683763\pi$
$499$	−3.37708e13	−1.09154	−0.545769	+	0.837935i	$0.683763\pi$
$500$	− 9.38021e10i	− 0.00300167i
$501$	0	0
$502$	2.57418e13	0.807460
$503$	− 4.24460e12i	− 0.131825i	−0.997825	−	0.0659124i	$-0.979004\pi$
$503$	− 4.24460e12i	− 0.131825i	0.997825	−	0.0659124i	$-0.0209958\pi$
$504$	0	0
$505$	−5.30811e13	−1.61615
$506$	− 1.41719e13i	− 0.427244i
$507$	0	0
$508$	1.19819e12	0.0354168
$509$	− 2.01121e13i	− 0.588664i	−0.955703	−	0.294332i	$-0.904903\pi$
$509$	− 2.01121e13i	− 0.588664i	0.955703	−	0.294332i	$-0.0950972\pi$
$510$	0	0
$511$	3.16644e13	0.908798
$512$	2.21504e13i	0.629552i
$513$	0	0
$514$	−4.10966e13	−1.14549
$515$	5.21423e13i	1.43931i
$516$	0	0
$517$	−3.44341e13	−0.932257
$518$	5.80527e12i	0.155659i
$519$	0	0
$520$	−6.19432e13	−1.62921
$521$	− 1.35570e12i	− 0.0353163i	−0.999844	−	0.0176581i	$-0.994379\pi$
$521$	− 1.35570e12i	− 0.0353163i	0.999844	−	0.0176581i	$-0.00562105\pi$
$522$	0	0
$523$	5.72383e13	1.46278	0.731389	−	0.681961i	$-0.238872\pi$
$523$	5.72383e13	1.46278	0.731389	+	0.681961i	$0.238872\pi$
$524$	− 7.64530e12i	− 0.193526i
$525$	0	0
$526$	−8.99557e12	−0.223409
$527$	3.67679e13i	0.904515i
$528$	0	0
$529$	3.52811e13	0.851654
$530$	1.44877e13i	0.346432i
$531$	0	0
$532$	2.80838e13	0.659019
$533$	− 5.45797e13i	− 1.26881i
$534$	0	0
$535$	7.41771e13	1.69239
$536$	− 5.08557e13i	− 1.14952i
$537$	0	0
$538$	3.64022e13	0.807637
$539$	− 4.43653e13i	− 0.975212i
$540$	0	0
$541$	2.34478e13	0.505961	0.252980	−	0.967471i	$-0.418589\pi$
$541$	2.34478e13	0.505961	0.252980	+	0.967471i	$0.418589\pi$
$542$	− 4.37782e13i	− 0.935969i
$543$	0	0
$544$	−5.64174e13	−1.18418
$545$	6.18845e13i	1.28706i
$546$	0	0
$547$	−1.36836e13	−0.279424	−0.139712	−	0.990192i	$-0.544618\pi$
$547$	−1.36836e13	−0.279424	−0.139712	+	0.990192i	$0.544618\pi$
$548$	− 3.07615e13i	− 0.622449i
$549$	0	0
$550$	5.60859e13	1.11440
$551$	− 2.53175e13i	− 0.498498i
$552$	0	0
$553$	1.06844e14	2.06597
$554$	4.63269e13i	0.887738i
$555$	0	0
$556$	−1.41484e13	−0.266277
$557$	− 6.54044e13i	− 1.21992i	−0.792432	−	0.609960i	$-0.791186\pi$
$557$	− 6.54044e13i	− 1.21992i	0.792432	−	0.609960i	$-0.208814\pi$
$558$	0	0
$559$	−3.93285e13	−0.720525
$560$	3.31028e13i	0.601069i
$561$	0	0
$562$	−8.32871e12	−0.148558
$563$	− 3.14419e13i	− 0.555862i	−0.960601	−	0.277931i	$-0.910351\pi$
$563$	− 3.14419e13i	− 0.555862i	0.960601	−	0.277931i	$-0.0896486\pi$
$564$	0	0
$565$	7.35258e12	0.127702
$566$	3.65959e13i	0.630013i
$567$	0	0
$568$	4.70554e13	0.795917
$569$	1.01433e14i	1.70066i	0.526247	+	0.850332i	$0.323599\pi$
$569$	1.01433e14i	1.70066i	−0.526247	+	0.850332i	$0.676401\pi$
$570$	0	0
$571$	2.86156e13	0.471435	0.235718	−	0.971822i	$-0.424256\pi$
$571$	2.86156e13	0.471435	0.235718	+	0.971822i	$0.424256\pi$
$572$	4.54110e13i	0.741619i
$573$	0	0
$574$	−6.96141e13	−1.11722
$575$	− 2.43209e13i	− 0.386937i
$576$	0	0
$577$	−1.60161e13	−0.250425	−0.125212	−	0.992130i	$-0.539961\pi$
$577$	−1.60161e13	−0.250425	−0.125212	+	0.992130i	$0.539961\pi$
$578$	4.91913e13i	0.762517i
$579$	0	0
$580$	−1.89834e13	−0.289223
$581$	1.20919e13i	0.182648i
$582$	0	0
$583$	3.37736e13	0.501458
$584$	5.16289e13i	0.760027i
$585$	0	0
$586$	8.60719e13	1.24559
$587$	1.31766e14i	1.89066i	0.326115	+	0.945330i	$0.394260\pi$
$587$	1.31766e14i	1.89066i	−0.326115	+	0.945330i	$0.605740\pi$
$588$	0	0
$589$	5.03000e13	0.709564
$590$	− 1.11246e14i	− 1.55605i
$591$	0	0
$592$	−3.96597e12	−0.0545432
$593$	− 4.28136e13i	− 0.583859i	−0.956440	−	0.291930i	$-0.905703\pi$
$593$	− 4.28136e13i	− 0.583859i	0.956440	−	0.291930i	$-0.0942973\pi$
$594$	0	0
$595$	−1.93356e14	−2.59282
$596$	− 4.64328e13i	− 0.617439i
$597$	0	0
$598$	−2.32341e13	−0.303823
$599$	− 1.26813e14i	− 1.64448i	−0.569141	−	0.822240i	$-0.692724\pi$
$599$	− 1.26813e14i	− 1.64448i	0.569141	−	0.822240i	$-0.307276\pi$
$600$	0	0
$601$	−1.25899e14	−1.60565	−0.802823	−	0.596217i	$-0.796670\pi$
$601$	−1.25899e14	−1.60565	−0.802823	+	0.596217i	$0.796670\pi$
$602$	5.01619e13i	0.634442i
$603$	0	0
$604$	3.91330e13	0.486809
$605$	− 1.46133e14i	− 1.80290i
$606$	0	0
$607$	−5.15751e13	−0.625889	−0.312944	−	0.949771i	$-0.601315\pi$
$607$	−5.15751e13	−0.625889	−0.312944	+	0.949771i	$0.601315\pi$
$608$	7.71813e13i	0.928954i
$609$	0	0
$610$	−1.51013e14	−1.78799
$611$	5.64529e13i	0.662949i
$612$	0	0
$613$	6.86500e13	0.793119	0.396560	−	0.918009i	$-0.370204\pi$
$613$	6.86500e13	0.793119	0.396560	+	0.918009i	$0.370204\pi$
$614$	− 9.76005e13i	− 1.11843i
$615$	0	0
$616$	1.84178e14	2.07652
$617$	9.31549e13i	1.04179i	0.853621	+	0.520895i	$0.174402\pi$
$617$	9.31549e13i	1.04179i	−0.853621	+	0.520895i	$0.825598\pi$
$618$	0	0
$619$	−9.30635e13	−1.02406	−0.512031	−	0.858967i	$-0.671107\pi$
$619$	−9.30635e13	−1.02406	−0.512031	+	0.858967i	$0.671107\pi$
$620$	− 3.77156e13i	− 0.411682i
$621$	0	0
$622$	−1.74543e13	−0.187478
$623$	− 1.02073e14i	− 1.08761i
$624$	0	0
$625$	−9.49247e13	−0.995357
$626$	2.82170e13i	0.293521i
$627$	0	0
$628$	−3.70707e13	−0.379519
$629$	− 2.31655e13i	− 0.235282i
$630$	0	0
$631$	8.95590e13	0.895288	0.447644	−	0.894212i	$-0.352263\pi$
$631$	8.95590e13	0.895288	0.447644	+	0.894212i	$0.352263\pi$
$632$	1.74209e14i	1.72776i
$633$	0	0
$634$	−4.91501e13	−0.479820
$635$	1.12857e13i	0.109310i
$636$	0	0
$637$	−7.27345e13	−0.693495
$638$	− 5.22147e13i	− 0.493957i
$639$	0	0
$640$	2.08709e13	0.194375
$641$	1.07669e14i	0.994946i	0.867480	+	0.497473i	$0.165739\pi$
$641$	1.07669e14i	0.994946i	−0.867480	+	0.497473i	$0.834261\pi$
$642$	0	0
$643$	4.75748e13	0.432835	0.216417	−	0.976301i	$-0.430563\pi$
$643$	4.75748e13	0.432835	0.216417	+	0.976301i	$0.430563\pi$
$644$	− 2.51161e13i	− 0.226738i
$645$	0	0
$646$	1.32225e14	1.17531
$647$	− 1.19789e13i	− 0.105656i	−0.998604	−	0.0528280i	$-0.983176\pi$
$647$	− 1.19789e13i	− 0.105656i	0.998604	−	0.0528280i	$-0.0168235\pi$
$648$	0	0
$649$	−2.59335e14	−2.25236
$650$	− 9.19500e13i	− 0.792474i
$651$	0	0
$652$	9.31788e12	0.0790823
$653$	− 1.60289e13i	− 0.135001i	−0.997719	−	0.0675007i	$-0.978498\pi$
$653$	− 1.60289e13i	− 0.135001i	0.997719	−	0.0675007i	$-0.0215025\pi$
$654$	0	0
$655$	7.20103e13	0.597294
$656$	− 4.75580e13i	− 0.391476i
$657$	0	0
$658$	7.20033e13	0.583744
$659$	− 2.23451e14i	− 1.79786i	−0.438093	−	0.898930i	$-0.644346\pi$
$659$	− 2.23451e14i	− 1.79786i	0.438093	−	0.898930i	$-0.355654\pi$
$660$	0	0
$661$	1.81957e13	0.144198	0.0720992	−	0.997397i	$-0.477030\pi$
$661$	1.81957e13	0.144198	0.0720992	+	0.997397i	$0.477030\pi$
$662$	− 1.35015e14i	− 1.06192i
$663$	0	0
$664$	−1.97159e13	−0.152749
$665$	2.64518e14i	2.03398i
$666$	0	0
$667$	−2.26422e13	−0.171510
$668$	2.02517e13i	0.152258i
$669$	0	0
$670$	1.50636e14	1.11572
$671$	3.52042e14i	2.58810i
$672$	0	0
$673$	2.39243e14	1.73286	0.866432	−	0.499294i	$-0.166407\pi$
$673$	2.39243e14	1.73286	0.866432	+	0.499294i	$0.166407\pi$
$674$	8.96946e12i	0.0644862i
$675$	0	0
$676$	9.69004e12	0.0686423
$677$	2.31720e14i	1.62937i	0.579902	+	0.814686i	$0.303091\pi$
$677$	2.31720e14i	1.62937i	−0.579902	+	0.814686i	$0.696909\pi$
$678$	0	0
$679$	−2.62485e14	−1.81868
$680$	− 3.15267e14i	− 2.16837i
$681$	0	0
$682$	1.03738e14	0.703101
$683$	1.67561e14i	1.12738i	0.825986	+	0.563690i	$0.190619\pi$
$683$	1.67561e14i	1.12738i	−0.825986	+	0.563690i	$0.809381\pi$
$684$	0	0
$685$	2.89739e14	1.92112
$686$	− 5.06611e13i	− 0.333468i
$687$	0	0
$688$	−3.42689e13	−0.222310
$689$	− 5.53701e13i	− 0.356598i
$690$	0	0
$691$	1.92427e14	1.22145	0.610725	−	0.791843i	$-0.290878\pi$
$691$	1.92427e14	1.22145	0.610725	+	0.791843i	$0.290878\pi$
$692$	5.96094e13i	0.375651i
$693$	0	0
$694$	−8.35017e12	−0.0518678
$695$	− 1.33262e14i	− 0.821833i
$696$	0	0
$697$	2.77790e14	1.68870
$698$	1.80801e14i	1.09125i
$699$	0	0
$700$	9.93982e13	0.591409
$701$	− 3.26565e14i	− 1.92921i	−0.263702	−	0.964604i	$-0.584943\pi$
$701$	− 3.26565e14i	− 1.92921i	0.263702	−	0.964604i	$-0.415057\pi$
$702$	0	0
$703$	−3.16913e13	−0.184571
$704$	2.45434e14i	1.41929i
$705$	0	0
$706$	−1.54961e14	−0.883483
$707$	− 2.58752e14i	− 1.46483i
$708$	0	0
$709$	−3.21622e14	−1.79521	−0.897604	−	0.440804i	$-0.854694\pi$
$709$	−3.21622e14	−1.79521	−0.897604	+	0.440804i	$0.854694\pi$
$710$	1.39379e14i	0.772514i
$711$	0	0
$712$	1.66431e14	0.909563
$713$	− 4.49847e13i	− 0.244128i
$714$	0	0
$715$	−4.27721e14	−2.28892
$716$	8.06724e13i	0.428706i
$717$	0	0
$718$	1.57446e14	0.825106
$719$	− 2.74758e13i	− 0.142990i	−0.997441	−	0.0714950i	$-0.977223\pi$
$719$	− 2.74758e13i	− 0.142990i	0.997441	−	0.0714950i	$-0.0227770\pi$
$720$	0	0
$721$	−2.54175e14	−1.30454
$722$	− 3.65487e13i	− 0.186288i
$723$	0	0
$724$	−4.58775e13	−0.230625
$725$	− 8.96073e13i	− 0.447356i
$726$	0	0
$727$	8.42042e13	0.414631	0.207316	−	0.978274i	$-0.433527\pi$
$727$	8.42042e13	0.414631	0.207316	+	0.978274i	$0.433527\pi$
$728$	− 3.01951e14i	− 1.47666i
$729$	0	0
$730$	−1.52926e14	−0.737679
$731$	− 2.00167e14i	− 0.958971i
$732$	0	0
$733$	3.62014e14	1.71082	0.855412	−	0.517947i	$-0.173304\pi$
$733$	3.62014e14	1.71082	0.855412	+	0.517947i	$0.173304\pi$
$734$	2.19223e14i	1.02898i
$735$	0	0
$736$	6.90254e13	0.319610
$737$	− 3.51162e14i	− 1.61499i
$738$	0	0
$739$	−2.44206e14	−1.10799	−0.553993	−	0.832522i	$-0.686896\pi$
$739$	−2.44206e14	−1.10799	−0.553993	+	0.832522i	$0.686896\pi$
$740$	2.37625e13i	0.107086i
$741$	0	0
$742$	−7.06222e13	−0.313994
$743$	6.89118e13i	0.304333i	0.988355	+	0.152167i	$0.0486250\pi$
$743$	6.89118e13i	0.304333i	−0.988355	+	0.152167i	$0.951375\pi$
$744$	0	0
$745$	4.37346e14	1.90565
$746$	− 1.45598e13i	− 0.0630176i
$747$	0	0
$748$	−2.31124e14	−0.987047
$749$	3.61587e14i	1.53392i
$750$	0	0
$751$	−2.44541e14	−1.02365	−0.511825	−	0.859090i	$-0.671030\pi$
$751$	−2.44541e14	−1.02365	−0.511825	+	0.859090i	$0.671030\pi$
$752$	4.91902e13i	0.204545i
$753$	0	0
$754$	−8.56032e13	−0.351264
$755$	3.68589e14i	1.50248i
$756$	0	0
$757$	−3.72001e13	−0.149646	−0.0748230	−	0.997197i	$-0.523839\pi$
$757$	−3.72001e13	−0.149646	−0.0748230	+	0.997197i	$0.523839\pi$
$758$	− 1.60061e13i	− 0.0639646i
$759$	0	0
$760$	−4.31298e14	−1.70102
$761$	5.57377e13i	0.218386i	0.994021	+	0.109193i	$0.0348267\pi$
$761$	5.57377e13i	0.218386i	−0.994021	+	0.109193i	$0.965173\pi$
$762$	0	0
$763$	−3.01665e14	−1.16655
$764$	− 4.72873e13i	− 0.181668i
$765$	0	0
$766$	−1.32389e14	−0.502005
$767$	4.25167e14i	1.60171i
$768$	0	0
$769$	4.18923e13	0.155777	0.0778883	−	0.996962i	$-0.475182\pi$
$769$	4.18923e13	0.155777	0.0778883	+	0.996962i	$0.475182\pi$
$770$	5.45541e14i	2.01546i
$771$	0	0
$772$	−1.54289e14	−0.562663
$773$	− 4.44998e13i	− 0.161236i	−0.996745	−	0.0806178i	$-0.974311\pi$
$773$	− 4.44998e13i	− 0.161236i	0.996745	−	0.0806178i	$-0.0256893\pi$
$774$	0	0
$775$	1.78029e14	0.636769
$776$	− 4.27983e14i	− 1.52096i
$777$	0	0
$778$	2.27043e14	0.796543
$779$	− 3.80027e14i	− 1.32473i
$780$	0	0
$781$	3.24921e14	1.11821
$782$	− 1.18253e14i	− 0.404369i
$783$	0	0
$784$	−6.33773e13	−0.213970
$785$	− 3.49165e14i	− 1.17134i
$786$	0	0
$787$	−5.29172e12	−0.0175276	−0.00876381	−	0.999962i	$-0.502790\pi$
$787$	−5.29172e12	−0.0175276	−0.00876381	+	0.999962i	$0.502790\pi$
$788$	− 1.04458e14i	− 0.343804i
$789$	0	0
$790$	−5.16010e14	−1.67696
$791$	3.58412e13i	0.115745i
$792$	0	0
$793$	5.77155e14	1.84046
$794$	3.49444e12i	0.0110732i
$795$	0	0
$796$	8.30031e13	0.259734
$797$	− 2.29589e14i	− 0.713936i	−0.934117	−	0.356968i	$-0.883810\pi$
$797$	− 2.29589e14i	− 0.713936i	0.934117	−	0.356968i	$-0.116190\pi$
$798$	0	0
$799$	−2.87323e14	−0.882342
$800$	2.73171e14i	0.833651i
$801$	0	0
$802$	−5.07360e12	−0.0152913
$803$	3.56501e14i	1.06778i
$804$	0	0
$805$	2.36566e14	0.699799
$806$	− 1.70074e14i	− 0.499991i
$807$	0	0
$808$	4.21895e14	1.22503
$809$	2.21639e14i	0.639594i	0.947486	+	0.319797i	$0.103615\pi$
$809$	2.21639e14i	0.639594i	−0.947486	+	0.319797i	$0.896385\pi$
$810$	0	0
$811$	2.70111e14	0.769907	0.384953	−	0.922936i	$-0.374218\pi$
$811$	2.70111e14	0.769907	0.384953	+	0.922936i	$0.374218\pi$
$812$	− 9.25373e13i	− 0.262142i
$813$	0	0
$814$	−6.53600e13	−0.182890
$815$	8.77641e13i	0.244078i
$816$	0	0
$817$	−2.73837e14	−0.752283
$818$	2.60345e14i	0.710858i
$819$	0	0
$820$	−2.84949e14	−0.768596
$821$	− 1.46064e14i	− 0.391586i	−0.980645	−	0.195793i	$-0.937272\pi$
$821$	− 1.46064e14i	− 0.391586i	0.980645	−	0.195793i	$-0.0627281\pi$
$822$	0	0
$823$	−3.82425e13	−0.101286	−0.0506428	−	0.998717i	$-0.516127\pi$
$823$	−3.82425e13	−0.101286	−0.0506428	+	0.998717i	$0.516127\pi$
$824$	− 4.14434e14i	− 1.09099i
$825$	0	0
$826$	5.42283e14	1.41035
$827$	− 3.49746e13i	− 0.0904119i	−0.998978	−	0.0452060i	$-0.985606\pi$
$827$	− 3.49746e13i	− 0.0904119i	0.998978	−	0.0452060i	$-0.0143944\pi$
$828$	0	0
$829$	6.43554e14	1.64366	0.821831	−	0.569732i	$-0.192953\pi$
$829$	6.43554e14	1.64366	0.821831	+	0.569732i	$0.192953\pi$
$830$	− 5.83991e13i	− 0.148257i
$831$	0	0
$832$	4.02376e14	1.00929
$833$	− 3.70191e14i	− 0.922997i
$834$	0	0
$835$	−1.90749e14	−0.469926
$836$	3.16188e14i	0.774308i
$837$	0	0
$838$	−1.82651e14	−0.441980
$839$	− 7.76705e13i	− 0.186830i	−0.995627	−	0.0934149i	$-0.970222\pi$
$839$	− 7.76705e13i	− 0.186830i	0.995627	−	0.0934149i	$-0.0297783\pi$
$840$	0	0
$841$	3.37285e14	0.801710
$842$	2.22245e14i	0.525135i
$843$	0	0
$844$	−1.25545e14	−0.293147
$845$	9.12695e13i	0.211857i
$846$	0	0
$847$	7.12346e14	1.63408
$848$	− 4.82467e13i	− 0.110024i
$849$	0	0
$850$	4.67990e14	1.05473
$851$	2.83425e13i	0.0635023i
$852$	0	0
$853$	−3.53687e14	−0.783202	−0.391601	−	0.920135i	$-0.628079\pi$
$853$	−3.53687e14	−0.783202	−0.391601	+	0.920135i	$0.628079\pi$
$854$	− 7.36136e14i	− 1.62057i
$855$	0	0
$856$	−5.89569e14	−1.28282
$857$	− 5.53847e14i	− 1.19808i	−0.800719	−	0.599040i	$-0.795549\pi$
$857$	− 5.53847e14i	− 1.19808i	0.800719	−	0.599040i	$-0.204451\pi$
$858$	0	0
$859$	4.82091e14	1.03077	0.515387	−	0.856958i	$-0.327648\pi$
$859$	4.82091e14	1.03077	0.515387	+	0.856958i	$0.327648\pi$
$860$	2.05326e14i	0.436467i
$861$	0	0
$862$	−3.67864e14	−0.772950
$863$	− 2.02122e14i	− 0.422240i	−0.977460	−	0.211120i	$-0.932289\pi$
$863$	− 2.02122e14i	− 0.422240i	0.977460	−	0.211120i	$-0.0677111\pi$
$864$	0	0
$865$	−5.61454e14	−1.15940
$866$	3.96193e14i	0.813424i
$867$	0	0
$868$	1.83850e14	0.373134
$869$	1.20292e15i	2.42739i
$870$	0	0
$871$	−5.75712e14	−1.14846
$872$	− 4.91866e14i	− 0.975583i
$873$	0	0
$874$	−1.61775e14	−0.317215
$875$	4.30681e12i	0.00839683i
$876$	0	0
$877$	−2.83664e14	−0.546771	−0.273386	−	0.961905i	$-0.588143\pi$
$877$	−2.83664e14	−0.546771	−0.273386	+	0.961905i	$0.588143\pi$
$878$	1.44580e14i	0.277099i
$879$	0	0
$880$	−3.72695e14	−0.706221
$881$	7.92783e14i	1.49374i	0.664970	+	0.746870i	$0.268444\pi$
$881$	7.92783e14i	1.49374i	−0.664970	+	0.746870i	$0.731556\pi$
$882$	0	0
$883$	5.58732e14	1.04088	0.520439	−	0.853899i	$-0.325768\pi$
$883$	5.58732e14	1.04088	0.520439	+	0.853899i	$0.325768\pi$
$884$	3.78917e14i	0.701911i
$885$	0	0
$886$	3.23453e13	0.0592437
$887$	− 5.60792e13i	− 0.102137i	−0.998695	−	0.0510685i	$-0.983737\pi$
$887$	− 5.60792e13i	− 0.102137i	0.998695	−	0.0510685i	$-0.0162627\pi$
$888$	0	0
$889$	−5.50137e13	−0.0990745
$890$	4.92971e14i	0.882818i
$891$	0	0
$892$	−2.38342e14	−0.422062
$893$	3.93070e14i	0.692170i
$894$	0	0
$895$	−7.59845e14	−1.32315
$896$	1.01738e14i	0.176175i
$897$	0	0
$898$	−2.46174e14	−0.421561
$899$	− 1.65741e14i	− 0.282248i
$900$	0	0
$901$	2.81812e14	0.474609
$902$	− 7.83766e14i	− 1.31266i
$903$	0	0
$904$	−5.84392e13	−0.0967971
$905$	− 4.32115e14i	− 0.711798i
$906$	0	0
$907$	−1.02359e14	−0.166759	−0.0833795	−	0.996518i	$-0.526571\pi$
$907$	−1.02359e14	−0.166759	−0.0833795	+	0.996518i	$0.526571\pi$
$908$	− 4.51447e14i	− 0.731438i
$909$	0	0
$910$	8.94386e14	1.43324
$911$	− 2.03072e14i	− 0.323637i	−0.986821	−	0.161818i	$-0.948264\pi$
$911$	− 2.03072e14i	− 0.323637i	0.986821	−	0.161818i	$-0.0517358\pi$
$912$	0	0
$913$	−1.36140e14	−0.214601
$914$	− 7.98672e14i	− 1.25210i
$915$	0	0
$916$	9.78357e13	0.151712
$917$	3.51025e14i	0.541366i
$918$	0	0
$919$	−1.21378e15	−1.85167	−0.925835	−	0.377929i	$-0.876636\pi$
$919$	−1.21378e15	−1.85167	−0.925835	+	0.377929i	$0.876636\pi$
$920$	3.85722e14i	0.585241i
$921$	0	0
$922$	−5.38554e14	−0.808304
$923$	− 5.32691e14i	− 0.795182i
$924$	0	0
$925$	−1.12166e14	−0.165636
$926$	8.63281e13i	0.126794i
$927$	0	0
$928$	2.54315e14	0.369516
$929$	7.35885e14i	1.06348i	0.846906	+	0.531742i	$0.178463\pi$
$929$	7.35885e14i	1.06348i	−0.846906	+	0.531742i	$0.821537\pi$
$930$	0	0
$931$	−5.06436e14	−0.724063
$932$	− 5.95630e14i	− 0.847026i
$933$	0	0
$934$	5.85499e14	0.823744
$935$	− 2.17694e15i	− 3.04640i
$936$	0	0
$937$	−9.68431e14	−1.34082	−0.670410	−	0.741991i	$-0.733882\pi$
$937$	−9.68431e14	−1.34082	−0.670410	+	0.741991i	$0.733882\pi$
$938$	7.34296e14i	1.01125i
$939$	0	0
$940$	2.94729e14	0.401590
$941$	5.71203e14i	0.774180i	0.922042	+	0.387090i	$0.126520\pi$
$941$	5.71203e14i	0.774180i	−0.922042	+	0.387090i	$0.873480\pi$
$942$	0	0
$943$	−3.39869e14	−0.455778
$944$	3.70469e14i	0.494188i
$945$	0	0
$946$	−5.64759e14	−0.745431
$947$	5.06129e14i	0.664525i	0.943187	+	0.332263i	$0.107812\pi$
$947$	5.06129e14i	0.664525i	−0.943187	+	0.332263i	$0.892188\pi$
$948$	0	0
$949$	5.84465e14	0.759325
$950$	− 6.40229e14i	− 0.827404i
$951$	0	0
$952$	1.53681e15	1.96533
$953$	1.56038e14i	0.198503i	0.995062	+	0.0992515i	$0.0316448\pi$
$953$	1.56038e14i	0.198503i	−0.995062	+	0.0992515i	$0.968355\pi$
$954$	0	0
$955$	4.45394e14	0.560696
$956$	− 4.69063e14i	− 0.587411i
$957$	0	0
$958$	−1.03765e14	−0.128595
$959$	1.41237e15i	1.74123i
$960$	0	0
$961$	−4.90340e14	−0.598247
$962$	1.07154e14i	0.130057i
$963$	0	0
$964$	2.63531e14	0.316553
$965$	− 1.45323e15i	− 1.73659i
$966$	0	0
$967$	−8.20480e14	−0.970366	−0.485183	−	0.874413i	$-0.661247\pi$
$967$	−8.20480e14	−0.970366	−0.485183	+	0.874413i	$0.661247\pi$
$968$	1.16148e15i	1.36658i
$969$	0	0
$970$	1.26769e15	1.47624
$971$	3.49738e14i	0.405179i	0.979264	+	0.202590i	$0.0649357\pi$
$971$	3.49738e14i	0.405179i	−0.979264	+	0.202590i	$0.935064\pi$
$972$	0	0
$973$	6.49608e14	0.744881
$974$	2.21974e14i	0.253225i
$975$	0	0
$976$	5.02904e14	0.567853
$977$	− 4.03993e13i	− 0.0453838i	−0.999743	−	0.0226919i	$-0.992776\pi$
$977$	− 4.03993e13i	− 0.0453838i	0.999743	−	0.0226919i	$-0.00722368\pi$
$978$	0	0
$979$	1.14921e15	1.27787
$980$	3.79732e14i	0.420094i
$981$	0	0
$982$	−3.31344e14	−0.362845
$983$	− 3.08888e14i	− 0.336537i	−0.985741	−	0.168269i	$-0.946182\pi$
$983$	− 3.08888e14i	− 0.336537i	0.985741	−	0.168269i	$-0.0538176\pi$
$984$	0	0
$985$	9.83880e14	1.06111
$986$	− 4.35687e14i	− 0.467509i
$987$	0	0
$988$	5.18373e14	0.550628
$989$	2.44900e14i	0.258826i
$990$	0	0
$991$	1.06886e15	1.11828	0.559140	−	0.829073i	$-0.311131\pi$
$991$	1.06886e15	1.11828	0.559140	+	0.829073i	$0.311131\pi$
$992$	5.05266e14i	0.525971i
$993$	0	0
$994$	−6.79425e14	−0.700179
$995$	7.81798e14i	0.801639i
$996$	0	0
$997$	9.25262e14	0.939267	0.469633	−	0.882862i	$-0.344386\pi$
$997$	9.25262e14	0.939267	0.469633	+	0.882862i	$0.344386\pi$
$998$	7.95051e14i	0.803049i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.11.b.c.26.2	✓	4
3.2	odd	2	inner	27.11.b.c.26.3	yes	4
9.2	odd	6		81.11.d.e.53.3		8
9.4	even	3		81.11.d.e.26.3		8
9.5	odd	6		81.11.d.e.26.2		8
9.7	even	3		81.11.d.e.53.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.11.b.c.26.2	✓	4	1.1	even	1	trivial
27.11.b.c.26.3	yes	4	3.2	odd	2	inner
81.11.d.e.26.2		8	9.5	odd	6
81.11.d.e.26.3		8	9.4	even	3
81.11.d.e.53.2		8	9.7	even	3
81.11.d.e.53.3		8	9.2	odd	6

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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-23.5425i q^{2} +469.750 q^{4} +4424.52i q^{5} -21568.0 q^{7} -35166.6i q^{8} +O(q^{10})\)	\(q-23.5425i q^{2} +469.750 q^{4} +4424.52i q^{5} -21568.0 q^{7} -35166.6i q^{8} +104164. q^{10} -242828. i q^{11} -398104. q^{13} +507765. i q^{14} -346888. q^{16} -2.02619e6i q^{17} -2.77192e6 q^{19} +2.07842e6i q^{20} -5.71678e6 q^{22} +2.47900e6i q^{23} -9.81076e6 q^{25} +9.37237e6i q^{26} -1.01315e7 q^{28} +9.13358e6i q^{29} -1.81463e7 q^{31} -2.78440e7i q^{32} -4.77017e7 q^{34} -9.54280e7i q^{35} +1.14330e7 q^{37} +6.52579e7i q^{38} +1.55595e8 q^{40} +1.37099e8i q^{41} +9.87897e7 q^{43} -1.14068e8i q^{44} +5.83620e7 q^{46} -1.41804e8i q^{47} +1.82702e8 q^{49} +2.30970e8i q^{50} -1.87009e8 q^{52} +1.39084e8i q^{53} +1.07440e9 q^{55} +7.58473e8i q^{56} +2.15027e8 q^{58} -1.06798e9i q^{59} -1.44976e9 q^{61} +4.27210e8i q^{62} -1.01073e9 q^{64} -1.76142e9i q^{65} +1.44613e9 q^{67} -9.51804e8i q^{68} -2.24662e9 q^{70} +1.33807e9i q^{71} -1.46812e9 q^{73} -2.69162e8i q^{74} -1.30211e9 q^{76} +5.23731e9i q^{77} -4.95381e9 q^{79} -1.53481e9i q^{80} +3.22766e9 q^{82} -5.60643e8i q^{83} +8.96494e9 q^{85} -2.32576e9i q^{86} -8.53944e9 q^{88} +4.73263e9i q^{89} +8.58629e9 q^{91} +1.16451e9i q^{92} -3.33843e9 q^{94} -1.22644e10i q^{95} +1.21701e10 q^{97} -4.30128e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 548 q^{4} + 20516 q^{7}+O(q^{10}) \) 4 * q - 548 * q^4 + 20516 * q^7	\( 4 q - 548 q^{4} + 20516 q^{7} - 199800 q^{10} - 262420 q^{13} - 3207800 q^{16} - 8233516 q^{19} - 21085704 q^{22} - 47203580 q^{25} - 67604212 q^{28} - 46994920 q^{31} - 78985368 q^{34} + 108086444 q^{37} + 196812720 q^{40} + 78251000 q^{43} + 882645768 q^{46} + 1826240448 q^{49} - 771022300 q^{52} + 1034575200 q^{55} + 1267544592 q^{58} - 3222389524 q^{61} - 1166263952 q^{64} + 1972286516 q^{67} - 17482328280 q^{70} + 4077598940 q^{73} - 603762388 q^{76} - 10402780252 q^{79} - 7945552656 q^{82} + 19825132320 q^{85} - 19783893168 q^{88} + 34160896780 q^{91} + 22345655544 q^{94} + 12969797468 q^{97}+O(q^{100}) \) 4 * q - 548 * q^4 + 20516 * q^7 - 199800 * q^10 - 262420 * q^13 - 3207800 * q^16 - 8233516 * q^19 - 21085704 * q^22 - 47203580 * q^25 - 67604212 * q^28 - 46994920 * q^31 - 78985368 * q^34 + 108086444 * q^37 + 196812720 * q^40 + 78251000 * q^43 + 882645768 * q^46 + 1826240448 * q^49 - 771022300 * q^52 + 1034575200 * q^55 + 1267544592 * q^58 - 3222389524 * q^61 - 1166263952 * q^64 + 1972286516 * q^67 - 17482328280 * q^70 + 4077598940 * q^73 - 603762388 * q^76 - 10402780252 * q^79 - 7945552656 * q^82 + 19825132320 * q^85 - 19783893168 * q^88 + 34160896780 * q^91 + 22345655544 * q^94 + 12969797468 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more