LMFDB - Embedded newform 275.6.b.a.199.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,6,Mod(199,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.199"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$44.1055504486$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.6.b.a.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} -15.0000i q^{3} +16.0000 q^{4} +60.0000 q^{6} -10.0000i q^{7} +192.000i q^{8} +18.0000 q^{9} -121.000 q^{11} -240.000i q^{12} -1148.00i q^{13} +40.0000 q^{14} -256.000 q^{16} -686.000i q^{17} +72.0000i q^{18} +384.000 q^{19} -150.000 q^{21} -484.000i q^{22} +3709.00i q^{23} +2880.00 q^{24} +4592.00 q^{26} -3915.00i q^{27} -160.000i q^{28} +5424.00 q^{29} -6443.00 q^{31} +5120.00i q^{32} +1815.00i q^{33} +2744.00 q^{34} +288.000 q^{36} -12063.0i q^{37} +1536.00i q^{38} -17220.0 q^{39} -1528.00 q^{41} -600.000i q^{42} -4026.00i q^{43} -1936.00 q^{44} -14836.0 q^{46} -7168.00i q^{47} +3840.00i q^{48} +16707.0 q^{49} -10290.0 q^{51} -18368.0i q^{52} -29862.0i q^{53} +15660.0 q^{54} +1920.00 q^{56} -5760.00i q^{57} +21696.0i q^{58} +6461.00 q^{59} -16980.0 q^{61} -25772.0i q^{62} -180.000i q^{63} -28672.0 q^{64} -7260.00 q^{66} -29999.0i q^{67} -10976.0i q^{68} +55635.0 q^{69} +31023.0 q^{71} +3456.00i q^{72} +1924.00i q^{73} +48252.0 q^{74} +6144.00 q^{76} +1210.00i q^{77} -68880.0i q^{78} -65138.0 q^{79} -54351.0 q^{81} -6112.00i q^{82} -102714. i q^{83} -2400.00 q^{84} +16104.0 q^{86} -81360.0i q^{87} -23232.0i q^{88} -17415.0 q^{89} -11480.0 q^{91} +59344.0i q^{92} +96645.0i q^{93} +28672.0 q^{94} +76800.0 q^{96} -66905.0i q^{97} +66828.0i q^{98} -2178.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{4} + 120 q^{6} + 36 q^{9} - 242 q^{11} + 80 q^{14} - 512 q^{16} + 768 q^{19} - 300 q^{21} + 5760 q^{24} + 9184 q^{26} + 10848 q^{29} - 12886 q^{31} + 5488 q^{34} + 576 q^{36} - 34440 q^{39}+ \cdots - 4356 q^{99}+O(q^{100}) $ 2 * q + 32 * q^4 + 120 * q^6 + 36 * q^9 - 242 * q^11 + 80 * q^14 - 512 * q^16 + 768 * q^19 - 300 * q^21 + 5760 * q^24 + 9184 * q^26 + 10848 * q^29 - 12886 * q^31 + 5488 * q^34 + 576 * q^36 - 34440 * q^39 - 3056 * q^41 - 3872 * q^44 - 29672 * q^46 + 33414 * q^49 - 20580 * q^51 + 31320 * q^54 + 3840 * q^56 + 12922 * q^59 - 33960 * q^61 - 57344 * q^64 - 14520 * q^66 + 111270 * q^69 + 62046 * q^71 + 96504 * q^74 + 12288 * q^76 - 130276 * q^79 - 108702 * q^81 - 4800 * q^84 + 32208 * q^86 - 34830 * q^89 - 22960 * q^91 + 57344 * q^94 + 153600 * q^96 - 4356 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	4.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	− 15.0000i	− 0.962250i	−0.876652	−	0.481125i	$-0.840228\pi$
$3$	− 15.0000i	− 0.962250i	0.876652	−	0.481125i	$-0.159772\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	60.0000	0.680414
$7$	− 10.0000i	− 0.0771356i	−0.999256	−	0.0385678i	$-0.987720\pi$
$7$	− 10.0000i	− 0.0771356i	0.999256	−	0.0385678i	$-0.0122796\pi$
$8$	192.000i	1.06066i
$9$	18.0000	0.0740741
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	− 240.000i	− 0.481125i
$13$	− 1148.00i	− 1.88401i	−0.335597	−	0.942006i	$-0.608938\pi$
$13$	− 1148.00i	− 1.88401i	0.335597	−	0.942006i	$-0.391062\pi$
$14$	40.0000	0.0545431
$15$	0	0
$16$	−256.000	−0.250000
$17$	− 686.000i	− 0.575707i	−0.957674	−	0.287854i	$-0.907058\pi$
$17$	− 686.000i	− 0.575707i	0.957674	−	0.287854i	$-0.0929417\pi$
$18$	72.0000i	0.0523783i
$19$	384.000	0.244032	0.122016	−	0.992528i	$-0.461064\pi$
$19$	384.000	0.244032	0.122016	+	0.992528i	$0.461064\pi$
$20$	0	0
$21$	−150.000	−0.0742238
$22$	− 484.000i	− 0.213201i
$23$	3709.00i	1.46197i	0.682396	+	0.730983i	$0.260938\pi$
$23$	3709.00i	1.46197i	−0.682396	+	0.730983i	$0.739062\pi$
$24$	2880.00	1.02062
$25$	0	0
$26$	4592.00	1.33220
$27$	− 3915.00i	− 1.03353i
$28$	− 160.000i	− 0.0385678i
$29$	5424.00	1.19764	0.598818	−	0.800885i	$-0.295637\pi$
$29$	5424.00	1.19764	0.598818	+	0.800885i	$0.295637\pi$
$30$	0	0
$31$	−6443.00	−1.20416	−0.602080	−	0.798436i	$-0.705661\pi$
$31$	−6443.00	−1.20416	−0.602080	+	0.798436i	$0.705661\pi$
$32$	5120.00i	0.883883i
$33$	1815.00i	0.290129i
$34$	2744.00	0.407087
$35$	0	0
$36$	288.000	0.0370370
$37$	− 12063.0i	− 1.44861i	−0.689481	−	0.724304i	$-0.742161\pi$
$37$	− 12063.0i	− 1.44861i	0.689481	−	0.724304i	$-0.257839\pi$
$38$	1536.00i	0.172557i
$39$	−17220.0	−1.81289
$40$	0	0
$41$	−1528.00	−0.141959	−0.0709796	−	0.997478i	$-0.522613\pi$
$41$	−1528.00	−0.141959	−0.0709796	+	0.997478i	$0.522613\pi$
$42$	− 600.000i	− 0.0524841i
$43$	− 4026.00i	− 0.332049i	−0.986122	−	0.166025i	$-0.946907\pi$
$43$	− 4026.00i	− 0.332049i	0.986122	−	0.166025i	$-0.0530931\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	−14836.0	−1.03377
$47$	− 7168.00i	− 0.473318i	−0.971593	−	0.236659i	$-0.923948\pi$
$47$	− 7168.00i	− 0.473318i	0.971593	−	0.236659i	$-0.0760525\pi$
$48$	3840.00i	0.240563i
$49$	16707.0	0.994050
$50$	0	0
$51$	−10290.0	−0.553975
$52$	− 18368.0i	− 0.942006i
$53$	− 29862.0i	− 1.46026i	−0.683310	−	0.730128i	$-0.739460\pi$
$53$	− 29862.0i	− 1.46026i	0.683310	−	0.730128i	$-0.260540\pi$
$54$	15660.0	0.730815
$55$	0	0
$56$	1920.00	0.0818147
$57$	− 5760.00i	− 0.234820i
$58$	21696.0i	0.846856i
$59$	6461.00	0.241640	0.120820	−	0.992674i	$-0.461448\pi$
$59$	6461.00	0.241640	0.120820	+	0.992674i	$0.461448\pi$
$60$	0	0
$61$	−16980.0	−0.584269	−0.292135	−	0.956377i	$-0.594366\pi$
$61$	−16980.0	−0.584269	−0.292135	+	0.956377i	$0.594366\pi$
$62$	− 25772.0i	− 0.851469i
$63$	− 180.000i	− 0.00571375i
$64$	−28672.0	−0.875000
$65$	0	0
$66$	−7260.00	−0.205152
$67$	− 29999.0i	− 0.816432i	−0.912885	−	0.408216i	$-0.866151\pi$
$67$	− 29999.0i	− 0.816432i	0.912885	−	0.408216i	$-0.133849\pi$
$68$	− 10976.0i	− 0.287854i
$69$	55635.0	1.40678
$70$	0	0
$71$	31023.0	0.730362	0.365181	−	0.930937i	$-0.381007\pi$
$71$	31023.0	0.730362	0.365181	+	0.930937i	$0.381007\pi$
$72$	3456.00i	0.0785674i
$73$	1924.00i	0.0422569i	0.999777	+	0.0211285i	$0.00672590\pi$
$73$	1924.00i	0.0422569i	−0.999777	+	0.0211285i	$0.993274\pi$
$74$	48252.0	1.02432
$75$	0	0
$76$	6144.00	0.122016
$77$	1210.00i	0.0232573i
$78$	− 68880.0i	− 1.28191i
$79$	−65138.0	−1.17427	−0.587133	−	0.809490i	$-0.699744\pi$
$79$	−65138.0	−1.17427	−0.587133	+	0.809490i	$0.699744\pi$
$80$	0	0
$81$	−54351.0	−0.920439
$82$	− 6112.00i	− 0.100380i
$83$	− 102714.i	− 1.63657i	−0.574813	−	0.818285i	$-0.694925\pi$
$83$	− 102714.i	− 1.63657i	0.574813	−	0.818285i	$-0.305075\pi$
$84$	−2400.00	−0.0371119
$85$	0	0
$86$	16104.0	0.234794
$87$	− 81360.0i	− 1.15243i
$88$	− 23232.0i	− 0.319801i
$89$	−17415.0	−0.233050	−0.116525	−	0.993188i	$-0.537175\pi$
$89$	−17415.0	−0.233050	−0.116525	+	0.993188i	$0.537175\pi$
$90$	0	0
$91$	−11480.0	−0.145324
$92$	59344.0i	0.730983i
$93$	96645.0i	1.15870i
$94$	28672.0	0.334687
$95$	0	0
$96$	76800.0	0.850517
$97$	− 66905.0i	− 0.721987i	−0.932568	−	0.360993i	$-0.882438\pi$
$97$	− 66905.0i	− 0.721987i	0.932568	−	0.360993i	$-0.117562\pi$
$98$	66828.0i	0.702900i
$99$	−2178.00	−0.0223342
$100$	0	0
$101$	96730.0	0.943534	0.471767	−	0.881723i	$-0.343616\pi$
$101$	96730.0	0.943534	0.471767	+	0.881723i	$0.343616\pi$
$102$	− 41160.0i	− 0.391719i
$103$	− 95704.0i	− 0.888868i	−0.895812	−	0.444434i	$-0.853405\pi$
$103$	− 95704.0i	− 0.888868i	0.895812	−	0.444434i	$-0.146595\pi$
$104$	220416.	1.99830
$105$	0	0
$106$	119448.	1.03256
$107$	32658.0i	0.275759i	0.990449	+	0.137880i	$0.0440287\pi$
$107$	32658.0i	0.275759i	−0.990449	+	0.137880i	$0.955971\pi$
$108$	− 62640.0i	− 0.516764i
$109$	185438.	1.49497	0.747485	−	0.664279i	$-0.231261\pi$
$109$	185438.	1.49497	0.747485	+	0.664279i	$0.231261\pi$
$110$	0	0
$111$	−180945.	−1.39392
$112$	2560.00i	0.0192839i
$113$	72849.0i	0.536695i	0.963322	+	0.268347i	$0.0864775\pi$
$113$	72849.0i	0.536695i	−0.963322	+	0.268347i	$0.913522\pi$
$114$	23040.0	0.166043
$115$	0	0
$116$	86784.0	0.598818
$117$	− 20664.0i	− 0.139556i
$118$	25844.0i	0.170866i
$119$	−6860.00	−0.0444075
$120$	0	0
$121$	14641.0	0.0909091
$122$	− 67920.0i	− 0.413141i
$123$	22920.0i	0.136600i
$124$	−103088.	−0.602080
$125$	0	0
$126$	720.000	0.00404023
$127$	78184.0i	0.430139i	0.976599	+	0.215069i	$0.0689978\pi$
$127$	78184.0i	0.430139i	−0.976599	+	0.215069i	$0.931002\pi$
$128$	49152.0i	0.265165i
$129$	−60390.0	−0.319515
$130$	0	0
$131$	−462.000	−0.00235214	−0.00117607	−	0.999999i	$-0.500374\pi$
$131$	−462.000	−0.00235214	−0.00117607	+	0.999999i	$0.500374\pi$
$132$	29040.0i	0.145065i
$133$	− 3840.00i	− 0.0188236i
$134$	119996.	0.577304
$135$	0	0
$136$	131712.	0.610630
$137$	− 296233.i	− 1.34844i	−0.738530	−	0.674221i	$-0.764480\pi$
$137$	− 296233.i	− 1.34844i	0.738530	−	0.674221i	$-0.235520\pi$
$138$	222540.i	0.994742i
$139$	399818.	1.75519	0.877597	−	0.479398i	$-0.159145\pi$
$139$	399818.	1.75519	0.877597	+	0.479398i	$0.159145\pi$
$140$	0	0
$141$	−107520.	−0.455451
$142$	124092.i	0.516444i
$143$	138908.i	0.568051i
$144$	−4608.00	−0.0185185
$145$	0	0
$146$	−7696.00	−0.0298802
$147$	− 250605.i	− 0.956525i
$148$	− 193008.i	− 0.724304i
$149$	−72670.0	−0.268157	−0.134079	−	0.990971i	$-0.542807\pi$
$149$	−72670.0	−0.268157	−0.134079	+	0.990971i	$0.542807\pi$
$150$	0	0
$151$	−303082.	−1.08173	−0.540864	−	0.841110i	$-0.681902\pi$
$151$	−303082.	−1.08173	−0.540864	+	0.841110i	$0.681902\pi$
$152$	73728.0i	0.258835i
$153$	− 12348.0i	− 0.0426450i
$154$	−4840.00	−0.0164454
$155$	0	0
$156$	−275520.	−0.906445
$157$	532987.i	1.72571i	0.505453	+	0.862854i	$0.331326\pi$
$157$	532987.i	1.72571i	−0.505453	+	0.862854i	$0.668674\pi$
$158$	− 260552.i	− 0.830332i
$159$	−447930.	−1.40513
$160$	0	0
$161$	37090.0	0.112770
$162$	− 217404.i	− 0.650849i
$163$	282076.i	0.831567i	0.909464	+	0.415783i	$0.136493\pi$
$163$	282076.i	0.831567i	−0.909464	+	0.415783i	$0.863507\pi$
$164$	−24448.0	−0.0709796
$165$	0	0
$166$	410856.	1.15723
$167$	573588.i	1.59151i	0.605620	+	0.795754i	$0.292925\pi$
$167$	573588.i	1.59151i	−0.605620	+	0.795754i	$0.707075\pi$
$168$	− 28800.0i	− 0.0787262i
$169$	−946611.	−2.54950
$170$	0	0
$171$	6912.00	0.0180765
$172$	− 64416.0i	− 0.166025i
$173$	− 386286.i	− 0.981282i	−0.871362	−	0.490641i	$-0.836763\pi$
$173$	− 386286.i	− 0.981282i	0.871362	−	0.490641i	$-0.163237\pi$
$174$	325440.	0.814888
$175$	0	0
$176$	30976.0	0.0753778
$177$	− 96915.0i	− 0.232519i
$178$	− 69660.0i	− 0.164791i
$179$	−545079.	−1.27153	−0.635765	−	0.771882i	$-0.719315\pi$
$179$	−545079.	−1.27153	−0.635765	+	0.771882i	$0.719315\pi$
$180$	0	0
$181$	−279485.	−0.634106	−0.317053	−	0.948408i	$-0.602693\pi$
$181$	−279485.	−0.634106	−0.317053	+	0.948408i	$0.602693\pi$
$182$	− 45920.0i	− 0.102760i
$183$	254700.i	0.562213i
$184$	−712128.	−1.55065
$185$	0	0
$186$	−386580.	−0.819327
$187$	83006.0i	0.173582i
$188$	− 114688.i	− 0.236659i
$189$	−39150.0	−0.0797218
$190$	0	0
$191$	−444437.	−0.881509	−0.440755	−	0.897628i	$-0.645289\pi$
$191$	−444437.	−0.881509	−0.440755	+	0.897628i	$0.645289\pi$
$192$	430080.i	0.841969i
$193$	− 18476.0i	− 0.0357038i	−0.999841	−	0.0178519i	$-0.994317\pi$
$193$	− 18476.0i	− 0.0357038i	0.999841	−	0.0178519i	$-0.00568274\pi$
$194$	267620.	0.510522
$195$	0	0
$196$	267312.	0.497025
$197$	− 270182.i	− 0.496010i	−0.968759	−	0.248005i	$-0.920225\pi$
$197$	− 270182.i	− 0.496010i	0.968759	−	0.248005i	$-0.0797750\pi$
$198$	− 8712.00i	− 0.0157926i
$199$	−43320.0	−0.0775453	−0.0387727	−	0.999248i	$-0.512345\pi$
$199$	−43320.0	−0.0775453	−0.0387727	+	0.999248i	$0.512345\pi$
$200$	0	0
$201$	−449985.	−0.785612
$202$	386920.i	0.667180i
$203$	− 54240.0i	− 0.0923803i
$204$	−164640.	−0.276987
$205$	0	0
$206$	382816.	0.628524
$207$	66762.0i	0.108294i
$208$	293888.i	0.471003i
$209$	−46464.0	−0.0735785
$210$	0	0
$211$	1.02968e6	1.59220	0.796100	−	0.605165i	$-0.206893\pi$
$211$	1.02968e6	1.59220	0.796100	+	0.605165i	$0.206893\pi$
$212$	− 477792.i	− 0.730128i
$213$	− 465345.i	− 0.702791i
$214$	−130632.	−0.194991
$215$	0	0
$216$	751680.	1.09622
$217$	64430.0i	0.0928835i
$218$	741752.i	1.05710i
$219$	28860.0	0.0406617
$220$	0	0
$221$	−787528.	−1.08464
$222$	− 723780.i	− 0.985653i
$223$	461281.i	0.621160i	0.950547	+	0.310580i	$0.100523\pi$
$223$	461281.i	0.621160i	−0.950547	+	0.310580i	$0.899477\pi$
$224$	51200.0	0.0681789
$225$	0	0
$226$	−291396.	−0.379501
$227$	855570.i	1.10202i	0.834497	+	0.551012i	$0.185758\pi$
$227$	855570.i	1.10202i	−0.834497	+	0.551012i	$0.814242\pi$
$228$	− 92160.0i	− 0.117410i
$229$	665805.	0.838993	0.419497	−	0.907757i	$-0.362207\pi$
$229$	665805.	0.838993	0.419497	+	0.907757i	$0.362207\pi$
$230$	0	0
$231$	18150.0	0.0223793
$232$	1.04141e6i	1.27028i
$233$	1.20934e6i	1.45934i	0.683798	+	0.729671i	$0.260327\pi$
$233$	1.20934e6i	1.45934i	−0.683798	+	0.729671i	$0.739673\pi$
$234$	82656.0	0.0986813
$235$	0	0
$236$	103376.	0.120820
$237$	977070.i	1.12994i
$238$	− 27440.0i	− 0.0314009i
$239$	571482.	0.647154	0.323577	−	0.946202i	$-0.395114\pi$
$239$	571482.	0.647154	0.323577	+	0.946202i	$0.395114\pi$
$240$	0	0
$241$	−267080.	−0.296209	−0.148105	−	0.988972i	$-0.547317\pi$
$241$	−267080.	−0.296209	−0.148105	+	0.988972i	$0.547317\pi$
$242$	58564.0i	0.0642824i
$243$	− 136080.i	− 0.147835i
$244$	−271680.	−0.292135
$245$	0	0
$246$	−91680.0	−0.0965910
$247$	− 440832.i	− 0.459760i
$248$	− 1.23706e6i	− 1.27720i
$249$	−1.54071e6	−1.57479
$250$	0	0
$251$	1.38737e6	1.38998	0.694988	−	0.719022i	$-0.255410\pi$
$251$	1.38737e6	1.38998	0.694988	+	0.719022i	$0.255410\pi$
$252$	− 2880.00i	− 0.00285687i
$253$	− 448789.i	− 0.440799i
$254$	−312736.	−0.304154
$255$	0	0
$256$	−1.11411e6	−1.06250
$257$	885922.i	0.836686i	0.908289	+	0.418343i	$0.137389\pi$
$257$	885922.i	0.836686i	−0.908289	+	0.418343i	$0.862611\pi$
$258$	− 241560.i	− 0.225931i
$259$	−120630.	−0.111739
$260$	0	0
$261$	97632.0	0.0887137
$262$	− 1848.00i	− 0.00166322i
$263$	1.44687e6i	1.28986i	0.764243	+	0.644928i	$0.223113\pi$
$263$	1.44687e6i	1.28986i	−0.764243	+	0.644928i	$0.776887\pi$
$264$	−348480.	−0.307729
$265$	0	0
$266$	15360.0	0.0133103
$267$	261225.i	0.224252i
$268$	− 479984.i	− 0.408216i
$269$	353878.	0.298176	0.149088	−	0.988824i	$-0.452366\pi$
$269$	353878.	0.298176	0.149088	+	0.988824i	$0.452366\pi$
$270$	0	0
$271$	525260.	0.434461	0.217231	−	0.976120i	$-0.430298\pi$
$271$	525260.	0.434461	0.217231	+	0.976120i	$0.430298\pi$
$272$	175616.i	0.143927i
$273$	172200.i	0.139838i
$274$	1.18493e6	0.953492
$275$	0	0
$276$	890160.	0.703389
$277$	595610.i	0.466404i	0.972428	+	0.233202i	$0.0749204\pi$
$277$	595610.i	0.466404i	−0.972428	+	0.233202i	$0.925080\pi$
$278$	1.59927e6i	1.24111i
$279$	−115974.	−0.0891970
$280$	0	0
$281$	732318.	0.553266	0.276633	−	0.960976i	$-0.410781\pi$
$281$	732318.	0.553266	0.276633	+	0.960976i	$0.410781\pi$
$282$	− 430080.i	− 0.322052i
$283$	2.23380e6i	1.65798i	0.559264	+	0.828989i	$0.311084\pi$
$283$	2.23380e6i	1.65798i	−0.559264	+	0.828989i	$0.688916\pi$
$284$	496368.	0.365181
$285$	0	0
$286$	−555632.	−0.401673
$287$	15280.0i	0.0109501i
$288$	92160.0i	0.0654729i
$289$	949261.	0.668561
$290$	0	0
$291$	−1.00358e6	−0.694732
$292$	30784.0i	0.0211285i
$293$	− 1.53108e6i	− 1.04191i	−0.853585	−	0.520953i	$-0.825577\pi$
$293$	− 1.53108e6i	− 1.04191i	0.853585	−	0.520953i	$-0.174423\pi$
$294$	1.00242e6	0.676365
$295$	0	0
$296$	2.31610e6	1.53648
$297$	473715.i	0.311620i
$298$	− 290680.i	− 0.189616i
$299$	4.25793e6	2.75436
$300$	0	0
$301$	−40260.0	−0.0256128
$302$	− 1.21233e6i	− 0.764897i
$303$	− 1.45095e6i	− 0.907916i
$304$	−98304.0	−0.0610081
$305$	0	0
$306$	49392.0	0.0301546
$307$	1.14268e6i	0.691956i	0.938243	+	0.345978i	$0.112453\pi$
$307$	1.14268e6i	0.691956i	−0.938243	+	0.345978i	$0.887547\pi$
$308$	19360.0i	0.0116286i
$309$	−1.43556e6	−0.855313
$310$	0	0
$311$	586956.	0.344116	0.172058	−	0.985087i	$-0.444958\pi$
$311$	586956.	0.344116	0.172058	+	0.985087i	$0.444958\pi$
$312$	− 3.30624e6i	− 1.92286i
$313$	− 233857.i	− 0.134924i	−0.997722	−	0.0674621i	$-0.978510\pi$
$313$	− 233857.i	− 0.134924i	0.997722	−	0.0674621i	$-0.0214902\pi$
$314$	−2.13195e6	−1.22026
$315$	0	0
$316$	−1.04221e6	−0.587133
$317$	935503.i	0.522874i	0.965221	+	0.261437i	$0.0841964\pi$
$317$	935503.i	0.522874i	−0.965221	+	0.261437i	$0.915804\pi$
$318$	− 1.79172e6i	− 0.993579i
$319$	−656304.	−0.361101
$320$	0	0
$321$	489870.	0.265349
$322$	148360.i	0.0797402i
$323$	− 263424.i	− 0.140491i
$324$	−869616.	−0.460219
$325$	0	0
$326$	−1.12830e6	−0.588007
$327$	− 2.78157e6i	− 1.43854i
$328$	− 293376.i	− 0.150571i
$329$	−71680.0	−0.0365097
$330$	0	0
$331$	−1.05823e6	−0.530897	−0.265449	−	0.964125i	$-0.585520\pi$
$331$	−1.05823e6	−0.530897	−0.265449	+	0.964125i	$0.585520\pi$
$332$	− 1.64342e6i	− 0.818285i
$333$	− 217134.i	− 0.107304i
$334$	−2.29435e6	−1.12537
$335$	0	0
$336$	38400.0	0.0185559
$337$	− 506186.i	− 0.242793i	−0.992604	−	0.121396i	$-0.961263\pi$
$337$	− 506186.i	− 0.242793i	0.992604	−	0.121396i	$-0.0387372\pi$
$338$	− 3.78644e6i	− 1.80277i
$339$	1.09274e6	0.516435
$340$	0	0
$341$	779603.	0.363068
$342$	27648.0i	0.0127820i
$343$	− 335140.i	− 0.153812i
$344$	772992.	0.352192
$345$	0	0
$346$	1.54514e6	0.693871
$347$	− 467636.i	− 0.208490i	−0.994552	−	0.104245i	$-0.966757\pi$
$347$	− 467636.i	− 0.208490i	0.994552	−	0.104245i	$-0.0332425\pi$
$348$	− 1.30176e6i	− 0.576213i
$349$	−304470.	−0.133808	−0.0669038	−	0.997759i	$-0.521312\pi$
$349$	−304470.	−0.133808	−0.0669038	+	0.997759i	$0.521312\pi$
$350$	0	0
$351$	−4.49442e6	−1.94718
$352$	− 619520.i	− 0.266501i
$353$	2.51868e6i	1.07581i	0.843005	+	0.537906i	$0.180785\pi$
$353$	2.51868e6i	1.07581i	−0.843005	+	0.537906i	$0.819215\pi$
$354$	387660.	0.164416
$355$	0	0
$356$	−278640.	−0.116525
$357$	102900.i	0.0427312i
$358$	− 2.18032e6i	− 0.899108i
$359$	3.01841e6	1.23607	0.618034	−	0.786151i	$-0.287929\pi$
$359$	3.01841e6	1.23607	0.618034	+	0.786151i	$0.287929\pi$
$360$	0	0
$361$	−2.32864e6	−0.940448
$362$	− 1.11794e6i	− 0.448381i
$363$	− 219615.i	− 0.0874773i
$364$	−183680.	−0.0726622
$365$	0	0
$366$	−1.01880e6	−0.397545
$367$	− 994429.i	− 0.385397i	−0.981258	−	0.192699i	$-0.938276\pi$
$367$	− 994429.i	− 0.385397i	0.981258	−	0.192699i	$-0.0617240\pi$
$368$	− 949504.i	− 0.365491i
$369$	−27504.0	−0.0105155
$370$	0	0
$371$	−298620.	−0.112638
$372$	1.54632e6i	0.579351i
$373$	1.72896e6i	0.643446i	0.946834	+	0.321723i	$0.104262\pi$
$373$	1.72896e6i	0.643446i	−0.946834	+	0.321723i	$0.895738\pi$
$374$	−332024.	−0.122741
$375$	0	0
$376$	1.37626e6	0.502030
$377$	− 6.22675e6i	− 2.25636i
$378$	− 156600.i	− 0.0563718i
$379$	−454765.	−0.162626	−0.0813128	−	0.996689i	$-0.525911\pi$
$379$	−454765.	−0.162626	−0.0813128	+	0.996689i	$0.525911\pi$
$380$	0	0
$381$	1.17276e6	0.413901
$382$	− 1.77775e6i	− 0.623321i
$383$	2.27557e6i	0.792673i	0.918105	+	0.396336i	$0.129719\pi$
$383$	2.27557e6i	0.792673i	−0.918105	+	0.396336i	$0.870281\pi$
$384$	737280.	0.255155
$385$	0	0
$386$	73904.0	0.0252464
$387$	− 72468.0i	− 0.0245962i
$388$	− 1.07048e6i	− 0.360993i
$389$	−389781.	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$389$	−389781.	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$390$	0	0
$391$	2.54437e6	0.841665
$392$	3.20774e6i	1.05435i
$393$	6930.00i	0.00226335i
$394$	1.08073e6	0.350732
$395$	0	0
$396$	−34848.0	−0.0111671
$397$	1.61933e6i	0.515655i	0.966191	+	0.257827i	$0.0830066\pi$
$397$	1.61933e6i	0.515655i	−0.966191	+	0.257827i	$0.916993\pi$
$398$	− 173280.i	− 0.0548328i
$399$	−57600.0	−0.0181130
$400$	0	0
$401$	−5.54368e6	−1.72162	−0.860810	−	0.508927i	$-0.830042\pi$
$401$	−5.54368e6	−1.72162	−0.860810	+	0.508927i	$0.830042\pi$
$402$	− 1.79994e6i	− 0.555511i
$403$	7.39656e6i	2.26865i
$404$	1.54768e6	0.471767
$405$	0	0
$406$	216960.	0.0653228
$407$	1.45962e6i	0.436772i
$408$	− 1.97568e6i	− 0.587579i
$409$	2.70493e6	0.799553	0.399776	−	0.916613i	$-0.369088\pi$
$409$	2.70493e6	0.799553	0.399776	+	0.916613i	$0.369088\pi$
$410$	0	0
$411$	−4.44350e6	−1.29754
$412$	− 1.53126e6i	− 0.444434i
$413$	− 64610.0i	− 0.0186391i
$414$	−267048.	−0.0765753
$415$	0	0
$416$	5.87776e6	1.66525
$417$	− 5.99727e6i	− 1.68894i
$418$	− 185856.i	− 0.0520279i
$419$	−3.37337e6	−0.938705	−0.469353	−	0.883011i	$-0.655513\pi$
$419$	−3.37337e6	−0.938705	−0.469353	+	0.883011i	$0.655513\pi$
$420$	0	0
$421$	−4.52551e6	−1.24441	−0.622204	−	0.782855i	$-0.713762\pi$
$421$	−4.52551e6	−1.24441	−0.622204	+	0.782855i	$0.713762\pi$
$422$	4.11874e6i	1.12586i
$423$	− 129024.i	− 0.0350606i
$424$	5.73350e6	1.54884
$425$	0	0
$426$	1.86138e6	0.496948
$427$	169800.i	0.0450680i
$428$	522528.i	0.137880i
$429$	2.08362e6	0.546607
$430$	0	0
$431$	−684534.	−0.177501	−0.0887507	−	0.996054i	$-0.528287\pi$
$431$	−684534.	−0.177501	−0.0887507	+	0.996054i	$0.528287\pi$
$432$	1.00224e6i	0.258382i
$433$	− 4.22591e6i	− 1.08318i	−0.840643	−	0.541589i	$-0.817823\pi$
$433$	− 4.22591e6i	− 1.08318i	0.840643	−	0.541589i	$-0.182177\pi$
$434$	−257720.	−0.0656786
$435$	0	0
$436$	2.96701e6	0.747485
$437$	1.42426e6i	0.356767i
$438$	115440.i	0.0287522i
$439$	2.09185e6	0.518047	0.259023	−	0.965871i	$-0.416599\pi$
$439$	2.09185e6	0.518047	0.259023	+	0.965871i	$0.416599\pi$
$440$	0	0
$441$	300726.	0.0736333
$442$	− 3.15011e6i	− 0.766956i
$443$	1.56284e6i	0.378361i	0.981942	+	0.189180i	$0.0605831\pi$
$443$	1.56284e6i	0.378361i	−0.981942	+	0.189180i	$0.939417\pi$
$444$	−2.89512e6	−0.696962
$445$	0	0
$446$	−1.84512e6	−0.439226
$447$	1.09005e6i	0.258034i
$448$	286720.i	0.0674937i
$449$	3.00449e6	0.703324	0.351662	−	0.936127i	$-0.385617\pi$
$449$	3.00449e6	0.703324	0.351662	+	0.936127i	$0.385617\pi$
$450$	0	0
$451$	184888.	0.0428023
$452$	1.16558e6i	0.268347i
$453$	4.54623e6i	1.04089i
$454$	−3.42228e6	−0.779248
$455$	0	0
$456$	1.10592e6	0.249064
$457$	2.44552e6i	0.547747i	0.961766	+	0.273874i	$0.0883050\pi$
$457$	2.44552e6i	0.547747i	−0.961766	+	0.273874i	$0.911695\pi$
$458$	2.66322e6i	0.593258i
$459$	−2.68569e6	−0.595010
$460$	0	0
$461$	7.79104e6	1.70743	0.853715	−	0.520741i	$-0.174344\pi$
$461$	7.79104e6	1.70743	0.853715	+	0.520741i	$0.174344\pi$
$462$	72600.0i	0.0158246i
$463$	− 1.05196e6i	− 0.228059i	−0.993477	−	0.114029i	$-0.963624\pi$
$463$	− 1.05196e6i	− 0.228059i	0.993477	−	0.114029i	$-0.0363758\pi$
$464$	−1.38854e6	−0.299409
$465$	0	0
$466$	−4.83734e6	−1.03191
$467$	− 3.97003e6i	− 0.842369i	−0.906975	−	0.421184i	$-0.861615\pi$
$467$	− 3.97003e6i	− 0.842369i	0.906975	−	0.421184i	$-0.138385\pi$
$468$	− 330624.i	− 0.0697782i
$469$	−299990.	−0.0629759
$470$	0	0
$471$	7.99480e6	1.66056
$472$	1.24051e6i	0.256298i
$473$	487146.i	0.100117i
$474$	−3.90828e6	−0.798987
$475$	0	0
$476$	−109760.	−0.0222038
$477$	− 537516.i	− 0.108167i
$478$	2.28593e6i	0.457607i
$479$	8.53908e6	1.70048	0.850241	−	0.526393i	$-0.176456\pi$
$479$	8.53908e6	1.70048	0.850241	+	0.526393i	$0.176456\pi$
$480$	0	0
$481$	−1.38483e7	−2.72919
$482$	− 1.06832e6i	− 0.209452i
$483$	− 556350.i	− 0.108513i
$484$	234256.	0.0454545
$485$	0	0
$486$	544320.	0.104535
$487$	1.86487e6i	0.356308i	0.984003	+	0.178154i	$0.0570125\pi$
$487$	1.86487e6i	0.356308i	−0.984003	+	0.178154i	$0.942987\pi$
$488$	− 3.26016e6i	− 0.619711i
$489$	4.23114e6	0.800175
$490$	0	0
$491$	5.15727e6	0.965420	0.482710	−	0.875780i	$-0.339653\pi$
$491$	5.15727e6	0.965420	0.482710	+	0.875780i	$0.339653\pi$
$492$	366720.i	0.0683002i
$493$	− 3.72086e6i	− 0.689488i
$494$	1.76333e6	0.325099
$495$	0	0
$496$	1.64941e6	0.301040
$497$	− 310230.i	− 0.0563369i
$498$	− 6.16284e6i	− 1.11354i
$499$	−4.53340e6	−0.815029	−0.407514	−	0.913199i	$-0.633604\pi$
$499$	−4.53340e6	−0.815029	−0.407514	+	0.913199i	$0.633604\pi$
$500$	0	0
$501$	8.60382e6	1.53143
$502$	5.54947e6i	0.982861i
$503$	1.71163e6i	0.301641i	0.988561	+	0.150821i	$0.0481916\pi$
$503$	1.71163e6i	0.301641i	−0.988561	+	0.150821i	$0.951808\pi$
$504$	34560.0	0.00606035
$505$	0	0
$506$	1.79516e6	0.311692
$507$	1.41992e7i	2.45326i
$508$	1.25094e6i	0.215069i
$509$	−9.73822e6	−1.66604	−0.833019	−	0.553244i	$-0.813390\pi$
$509$	−9.73822e6	−1.66604	−0.833019	+	0.553244i	$0.813390\pi$
$510$	0	0
$511$	19240.0	0.00325951
$512$	− 2.88358e6i	− 0.486136i
$513$	− 1.50336e6i	− 0.252214i
$514$	−3.54369e6	−0.591627
$515$	0	0
$516$	−966240.	−0.159757
$517$	867328.i	0.142711i
$518$	− 482520.i	− 0.0790116i
$519$	−5.79429e6	−0.944239
$520$	0	0
$521$	4.30279e6	0.694474	0.347237	−	0.937777i	$-0.387120\pi$
$521$	4.30279e6	0.694474	0.347237	+	0.937777i	$0.387120\pi$
$522$	390528.i	0.0627301i
$523$	2.62280e6i	0.419287i	0.977778	+	0.209643i	$0.0672303\pi$
$523$	2.62280e6i	0.419287i	−0.977778	+	0.209643i	$0.932770\pi$
$524$	−7392.00	−0.00117607
$525$	0	0
$526$	−5.78750e6	−0.912066
$527$	4.41990e6i	0.693243i
$528$	− 464640.i	− 0.0725324i
$529$	−7.32034e6	−1.13734
$530$	0	0
$531$	116298.	0.0178993
$532$	− 61440.0i	− 0.00941179i
$533$	1.75414e6i	0.267453i
$534$	−1.04490e6	−0.158570
$535$	0	0
$536$	5.75981e6	0.865956
$537$	8.17618e6i	1.22353i
$538$	1.41551e6i	0.210842i
$539$	−2.02155e6	−0.299717
$540$	0	0
$541$	−2.49634e6	−0.366700	−0.183350	−	0.983048i	$-0.558694\pi$
$541$	−2.49634e6	−0.366700	−0.183350	+	0.983048i	$0.558694\pi$
$542$	2.10104e6i	0.307211i
$543$	4.19228e6i	0.610169i
$544$	3.51232e6	0.508858
$545$	0	0
$546$	−688800.	−0.0988807
$547$	− 1.14323e7i	− 1.63368i	−0.576868	−	0.816838i	$-0.695725\pi$
$547$	− 1.14323e7i	− 1.63368i	0.576868	−	0.816838i	$-0.304275\pi$
$548$	− 4.73973e6i	− 0.674221i
$549$	−305640.	−0.0432792
$550$	0	0
$551$	2.08282e6	0.292262
$552$	1.06819e7i	1.49211i
$553$	651380.i	0.0905778i
$554$	−2.38244e6	−0.329798
$555$	0	0
$556$	6.39709e6	0.877597
$557$	9.81529e6i	1.34049i	0.742138	+	0.670247i	$0.233812\pi$
$557$	9.81529e6i	1.34049i	−0.742138	+	0.670247i	$0.766188\pi$
$558$	− 463896.i	− 0.0630718i
$559$	−4.62185e6	−0.625585
$560$	0	0
$561$	1.24509e6	0.167030
$562$	2.92927e6i	0.391218i
$563$	− 8.19192e6i	− 1.08922i	−0.838690	−	0.544609i	$-0.816678\pi$
$563$	− 8.19192e6i	− 1.08922i	0.838690	−	0.544609i	$-0.183322\pi$
$564$	−1.72032e6	−0.227725
$565$	0	0
$566$	−8.93522e6	−1.17237
$567$	543510.i	0.0709986i
$568$	5.95642e6i	0.774665i
$569$	7.54286e6	0.976687	0.488344	−	0.872651i	$-0.337601\pi$
$569$	7.54286e6	0.976687	0.488344	+	0.872651i	$0.337601\pi$
$570$	0	0
$571$	−8.69400e6	−1.11591	−0.557956	−	0.829871i	$-0.688414\pi$
$571$	−8.69400e6	−1.11591	−0.557956	+	0.829871i	$0.688414\pi$
$572$	2.22253e6i	0.284025i
$573$	6.66656e6i	0.848233i
$574$	−61120.0	−0.00774290
$575$	0	0
$576$	−516096.	−0.0648148
$577$	− 2.03379e6i	− 0.254312i	−0.991883	−	0.127156i	$-0.959415\pi$
$577$	− 2.03379e6i	− 0.254312i	0.991883	−	0.127156i	$-0.0405849\pi$
$578$	3.79704e6i	0.472744i
$579$	−277140.	−0.0343560
$580$	0	0
$581$	−1.02714e6	−0.126238
$582$	− 4.01430e6i	− 0.491250i
$583$	3.61330e6i	0.440284i
$584$	−369408.	−0.0448202
$585$	0	0
$586$	6.12432e6	0.736739
$587$	− 3.51780e6i	− 0.421381i	−0.977553	−	0.210691i	$-0.932429\pi$
$587$	− 3.51780e6i	− 0.421381i	0.977553	−	0.210691i	$-0.0675713\pi$
$588$	− 4.00968e6i	− 0.478263i
$589$	−2.47411e6	−0.293854
$590$	0	0
$591$	−4.05273e6	−0.477286
$592$	3.08813e6i	0.362152i
$593$	− 8.34535e6i	− 0.974558i	−0.873246	−	0.487279i	$-0.837989\pi$
$593$	− 8.34535e6i	− 0.974558i	0.873246	−	0.487279i	$-0.162011\pi$
$594$	−1.89486e6	−0.220349
$595$	0	0
$596$	−1.16272e6	−0.134079
$597$	649800.i	0.0746180i
$598$	1.70317e7i	1.94763i
$599$	−6.15022e6	−0.700364	−0.350182	−	0.936682i	$-0.613880\pi$
$599$	−6.15022e6	−0.700364	−0.350182	+	0.936682i	$0.613880\pi$
$600$	0	0
$601$	−6.86232e6	−0.774970	−0.387485	−	0.921876i	$-0.626656\pi$
$601$	−6.86232e6	−0.774970	−0.387485	+	0.921876i	$0.626656\pi$
$602$	− 161040.i	− 0.0181110i
$603$	− 539982.i	− 0.0604764i
$604$	−4.84931e6	−0.540864
$605$	0	0
$606$	5.80380e6	0.641994
$607$	9.45536e6i	1.04161i	0.853675	+	0.520807i	$0.174369\pi$
$607$	9.45536e6i	1.04161i	−0.853675	+	0.520807i	$0.825631\pi$
$608$	1.96608e6i	0.215696i
$609$	−813600.	−0.0888930
$610$	0	0
$611$	−8.22886e6	−0.891737
$612$	− 197568.i	− 0.0213225i
$613$	− 4.63658e6i	− 0.498363i	−0.968457	−	0.249182i	$-0.919838\pi$
$613$	− 4.63658e6i	− 0.498363i	0.968457	−	0.249182i	$-0.0801616\pi$
$614$	−4.57072e6	−0.489287
$615$	0	0
$616$	−232320.	−0.0246680
$617$	− 6.05704e6i	− 0.640542i	−0.947326	−	0.320271i	$-0.896226\pi$
$617$	− 6.05704e6i	− 0.640542i	0.947326	−	0.320271i	$-0.103774\pi$
$618$	− 5.74224e6i	− 0.604798i
$619$	5.63994e6	0.591626	0.295813	−	0.955246i	$-0.404409\pi$
$619$	5.63994e6	0.591626	0.295813	+	0.955246i	$0.404409\pi$
$620$	0	0
$621$	1.45207e7	1.51098
$622$	2.34782e6i	0.243327i
$623$	174150.i	0.0179764i
$624$	4.40832e6	0.453223
$625$	0	0
$626$	935428.	0.0954057
$627$	696960.i	0.0708009i
$628$	8.52779e6i	0.862854i
$629$	−8.27522e6	−0.833975
$630$	0	0
$631$	1.12616e6	0.112597	0.0562987	−	0.998414i	$-0.482070\pi$
$631$	1.12616e6	0.112597	0.0562987	+	0.998414i	$0.482070\pi$
$632$	− 1.25065e7i	− 1.24550i
$633$	− 1.54453e7i	− 1.53210i
$634$	−3.74201e6	−0.369728
$635$	0	0
$636$	−7.16688e6	−0.702566
$637$	− 1.91796e7i	− 1.87280i
$638$	− 2.62522e6i	− 0.255337i
$639$	558414.	0.0541009
$640$	0	0
$641$	−1.42020e7	−1.36522	−0.682611	−	0.730782i	$-0.739156\pi$
$641$	−1.42020e7	−1.36522	−0.682611	+	0.730782i	$0.739156\pi$
$642$	1.95948e6i	0.187630i
$643$	1.60794e6i	0.153371i	0.997055	+	0.0766853i	$0.0244337\pi$
$643$	1.60794e6i	0.153371i	−0.997055	+	0.0766853i	$0.975566\pi$
$644$	593440.	0.0563848
$645$	0	0
$646$	1.05370e6	0.0993423
$647$	− 3.10236e6i	− 0.291361i	−0.989332	−	0.145680i	$-0.953463\pi$
$647$	− 3.10236e6i	− 0.291361i	0.989332	−	0.145680i	$-0.0465371\pi$
$648$	− 1.04354e7i	− 0.976273i
$649$	−781781.	−0.0728573
$650$	0	0
$651$	966450.	0.0893772
$652$	4.51322e6i	0.415783i
$653$	6.88852e6i	0.632183i	0.948729	+	0.316091i	$0.102371\pi$
$653$	6.88852e6i	0.632183i	−0.948729	+	0.316091i	$0.897629\pi$
$654$	1.11263e7	1.01720
$655$	0	0
$656$	391168.	0.0354898
$657$	34632.0i	0.00313014i
$658$	− 286720.i	− 0.0258163i
$659$	1.24134e7	1.11347	0.556735	−	0.830690i	$-0.312054\pi$
$659$	1.24134e7	1.11347	0.556735	+	0.830690i	$0.312054\pi$
$660$	0	0
$661$	−8.10994e6	−0.721961	−0.360980	−	0.932573i	$-0.617558\pi$
$661$	−8.10994e6	−0.721961	−0.360980	+	0.932573i	$0.617558\pi$
$662$	− 4.23292e6i	− 0.375401i
$663$	1.18129e7i	1.04369i
$664$	1.97211e7	1.73584
$665$	0	0
$666$	868536.	0.0758756
$667$	2.01176e7i	1.75090i
$668$	9.17741e6i	0.795754i
$669$	6.91922e6	0.597711
$670$	0	0
$671$	2.05458e6	0.176164
$672$	− 768000.i	− 0.0656052i
$673$	1.78063e7i	1.51543i	0.652584	+	0.757717i	$0.273685\pi$
$673$	1.78063e7i	1.51543i	−0.652584	+	0.757717i	$0.726315\pi$
$674$	2.02474e6	0.171680
$675$	0	0
$676$	−1.51458e7	−1.27475
$677$	− 1.55179e7i	− 1.30125i	−0.759398	−	0.650626i	$-0.774507\pi$
$677$	− 1.55179e7i	− 1.30125i	0.759398	−	0.650626i	$-0.225493\pi$
$678$	4.37094e6i	0.365175i
$679$	−669050.	−0.0556909
$680$	0	0
$681$	1.28336e7	1.06042
$682$	3.11841e6i	0.256728i
$683$	− 2.18106e6i	− 0.178902i	−0.995991	−	0.0894510i	$-0.971489\pi$
$683$	− 2.18106e6i	− 0.178902i	0.995991	−	0.0894510i	$-0.0285112\pi$
$684$	110592.	0.00903823
$685$	0	0
$686$	1.34056e6	0.108762
$687$	− 9.98708e6i	− 0.807321i
$688$	1.03066e6i	0.0830123i
$689$	−3.42816e7	−2.75114
$690$	0	0
$691$	2.29892e7	1.83159	0.915795	−	0.401647i	$-0.131562\pi$
$691$	2.29892e7	1.83159	0.915795	+	0.401647i	$0.131562\pi$
$692$	− 6.18058e6i	− 0.490641i
$693$	21780.0i	0.00172276i
$694$	1.87054e6	0.147424
$695$	0	0
$696$	1.56211e7	1.22233
$697$	1.04821e6i	0.0817270i
$698$	− 1.21788e6i	− 0.0946163i
$699$	1.81400e7	1.40425
$700$	0	0
$701$	2.34092e6	0.179925	0.0899626	−	0.995945i	$-0.471325\pi$
$701$	2.34092e6	0.179925	0.0899626	+	0.995945i	$0.471325\pi$
$702$	− 1.79777e7i	− 1.37686i
$703$	− 4.63219e6i	− 0.353507i
$704$	3.46931e6	0.263822
$705$	0	0
$706$	−1.00747e7	−0.760715
$707$	− 967300.i	− 0.0727801i
$708$	− 1.55064e6i	− 0.116259i
$709$	1.92694e7	1.43964	0.719820	−	0.694161i	$-0.244225\pi$
$709$	1.92694e7	1.43964	0.719820	+	0.694161i	$0.244225\pi$
$710$	0	0
$711$	−1.17248e6	−0.0869827
$712$	− 3.34368e6i	− 0.247186i
$713$	− 2.38971e7i	− 1.76044i
$714$	−411600.	−0.0302155
$715$	0	0
$716$	−8.72126e6	−0.635765
$717$	− 8.57223e6i	− 0.622724i
$718$	1.20736e7i	0.874032i
$719$	2.14665e7	1.54860	0.774300	−	0.632819i	$-0.218102\pi$
$719$	2.14665e7	1.54860	0.774300	+	0.632819i	$0.218102\pi$
$720$	0	0
$721$	−957040.	−0.0685633
$722$	− 9.31457e6i	− 0.664997i
$723$	4.00620e6i	0.285028i
$724$	−4.47176e6	−0.317053
$725$	0	0
$726$	878460.	0.0618558
$727$	1.67705e7i	1.17682i	0.808562	+	0.588411i	$0.200246\pi$
$727$	1.67705e7i	1.17682i	−0.808562	+	0.588411i	$0.799754\pi$
$728$	− 2.20416e6i	− 0.154140i
$729$	−1.52485e7	−1.06269
$730$	0	0
$731$	−2.76184e6	−0.191163
$732$	4.07520e6i	0.281107i
$733$	− 1.75373e7i	− 1.20560i	−0.797894	−	0.602798i	$-0.794052\pi$
$733$	− 1.75373e7i	− 1.20560i	0.797894	−	0.602798i	$-0.205948\pi$
$734$	3.97772e6	0.272517
$735$	0	0
$736$	−1.89901e7	−1.29221
$737$	3.62988e6i	0.246163i
$738$	− 110016.i	− 0.00743558i
$739$	−1.47387e7	−0.992766	−0.496383	−	0.868104i	$-0.665339\pi$
$739$	−1.47387e7	−0.992766	−0.496383	+	0.868104i	$0.665339\pi$
$740$	0	0
$741$	−6.61248e6	−0.442404
$742$	− 1.19448e6i	− 0.0796469i
$743$	− 4.80946e6i	− 0.319613i	−0.987148	−	0.159806i	$-0.948913\pi$
$743$	− 4.80946e6i	− 0.319613i	0.987148	−	0.159806i	$-0.0510870\pi$
$744$	−1.85558e7	−1.22899
$745$	0	0
$746$	−6.91583e6	−0.454985
$747$	− 1.84885e6i	− 0.121227i
$748$	1.32810e6i	0.0867912i
$749$	326580.	0.0212709
$750$	0	0
$751$	8.29317e6	0.536563	0.268282	−	0.963341i	$-0.413544\pi$
$751$	8.29317e6	0.536563	0.268282	+	0.963341i	$0.413544\pi$
$752$	1.83501e6i	0.118330i
$753$	− 2.08105e7i	− 1.33750i
$754$	2.49070e7	1.59549
$755$	0	0
$756$	−626400.	−0.0398609
$757$	352294.i	0.0223442i	0.999938	+	0.0111721i	$0.00355627\pi$
$757$	352294.i	0.0223442i	−0.999938	+	0.0111721i	$0.996444\pi$
$758$	− 1.81906e6i	− 0.114994i
$759$	−6.73184e6	−0.424159
$760$	0	0
$761$	1.68985e7	1.05776	0.528878	−	0.848698i	$-0.322613\pi$
$761$	1.68985e7	1.05776	0.528878	+	0.848698i	$0.322613\pi$
$762$	4.69104e6i	0.292672i
$763$	− 1.85438e6i	− 0.115315i
$764$	−7.11099e6	−0.440755
$765$	0	0
$766$	−9.10229e6	−0.560504
$767$	− 7.41723e6i	− 0.455253i
$768$	1.67117e7i	1.02239i
$769$	36652.0	0.00223502	0.00111751	−	0.999999i	$-0.499644\pi$
$769$	36652.0	0.00223502	0.00111751	+	0.999999i	$0.499644\pi$
$770$	0	0
$771$	1.32888e7	0.805102
$772$	− 295616.i	− 0.0178519i
$773$	− 3.17462e7i	− 1.91093i	−0.295113	−	0.955463i	$-0.595357\pi$
$773$	− 3.17462e7i	− 1.91093i	0.295113	−	0.955463i	$-0.404643\pi$
$774$	289872.	0.0173922
$775$	0	0
$776$	1.28458e7	0.765783
$777$	1.80945e6i	0.107521i
$778$	− 1.55912e6i	− 0.0923489i
$779$	−586752.	−0.0346426
$780$	0	0
$781$	−3.75378e6	−0.220212
$782$	1.01775e7i	0.595147i
$783$	− 2.12350e7i	− 1.23779i
$784$	−4.27699e6	−0.248513
$785$	0	0
$786$	−27720.0	−0.00160043
$787$	− 2.01985e7i	− 1.16247i	−0.813735	−	0.581236i	$-0.802569\pi$
$787$	− 2.01985e7i	− 1.16247i	0.813735	−	0.581236i	$-0.197431\pi$
$788$	− 4.32291e6i	− 0.248005i
$789$	2.17031e7	1.24116
$790$	0	0
$791$	728490.	0.0413983
$792$	− 418176.i	− 0.0236890i
$793$	1.94930e7i	1.10077i
$794$	−6.47732e6	−0.364623
$795$	0	0
$796$	−693120.	−0.0387727
$797$	− 1.55660e7i	− 0.868023i	−0.900907	−	0.434011i	$-0.857098\pi$
$797$	− 1.55660e7i	− 0.868023i	0.900907	−	0.434011i	$-0.142902\pi$
$798$	− 230400.i	− 0.0128078i
$799$	−4.91725e6	−0.272493
$800$	0	0
$801$	−313470.	−0.0172629
$802$	− 2.21747e7i	− 1.21737i
$803$	− 232804.i	− 0.0127409i
$804$	−7.19976e6	−0.392806
$805$	0	0
$806$	−2.95863e7	−1.60418
$807$	− 5.30817e6i	− 0.286920i
$808$	1.85722e7i	1.00077i
$809$	2.91667e7	1.56681	0.783404	−	0.621513i	$-0.213482\pi$
$809$	2.91667e7	1.56681	0.783404	+	0.621513i	$0.213482\pi$
$810$	0	0
$811$	−1.65215e7	−0.882057	−0.441029	−	0.897493i	$-0.645386\pi$
$811$	−1.65215e7	−0.882057	−0.441029	+	0.897493i	$0.645386\pi$
$812$	− 867840.i	− 0.0461902i
$813$	− 7.87890e6i	− 0.418061i
$814$	−5.83849e6	−0.308844
$815$	0	0
$816$	2.63424e6	0.138494
$817$	− 1.54598e6i	− 0.0810307i
$818$	1.08197e7i	0.565369i
$819$	−206640.	−0.0107648
$820$	0	0
$821$	5.56614e6	0.288202	0.144101	−	0.989563i	$-0.453971\pi$
$821$	5.56614e6	0.288202	0.144101	+	0.989563i	$0.453971\pi$
$822$	− 1.77740e7i	− 0.917498i
$823$	1.18801e7i	0.611391i	0.952129	+	0.305696i	$0.0988890\pi$
$823$	1.18801e7i	0.611391i	−0.952129	+	0.305696i	$0.901111\pi$
$824$	1.83752e7	0.942786
$825$	0	0
$826$	258440.	0.0131798
$827$	1.32856e7i	0.675489i	0.941238	+	0.337745i	$0.109664\pi$
$827$	1.32856e7i	0.675489i	−0.941238	+	0.337745i	$0.890336\pi$
$828$	1.06819e6i	0.0541469i
$829$	653987.	0.0330509	0.0165254	−	0.999863i	$-0.494740\pi$
$829$	653987.	0.0330509	0.0165254	+	0.999863i	$0.494740\pi$
$830$	0	0
$831$	8.93415e6	0.448798
$832$	3.29155e7i	1.64851i
$833$	− 1.14610e7i	− 0.572282i
$834$	2.39891e7	1.19426
$835$	0	0
$836$	−743424.	−0.0367892
$837$	2.52243e7i	1.24453i
$838$	− 1.34935e7i	− 0.663765i
$839$	−2.47747e7	−1.21508	−0.607538	−	0.794290i	$-0.707843\pi$
$839$	−2.47747e7	−1.21508	−0.607538	+	0.794290i	$0.707843\pi$
$840$	0	0
$841$	8.90863e6	0.434331
$842$	− 1.81021e7i	− 0.879929i
$843$	− 1.09848e7i	− 0.532380i
$844$	1.64749e7	0.796100
$845$	0	0
$846$	516096.	0.0247916
$847$	− 146410.i	− 0.00701233i
$848$	7.64467e6i	0.365064i
$849$	3.35071e7	1.59539
$850$	0	0
$851$	4.47417e7	2.11782
$852$	− 7.44552e6i	− 0.351395i
$853$	2.71291e7i	1.27662i	0.769779	+	0.638311i	$0.220367\pi$
$853$	2.71291e7i	1.27662i	−0.769779	+	0.638311i	$0.779633\pi$
$854$	−679200.	−0.0318679
$855$	0	0
$856$	−6.27034e6	−0.292487
$857$	2.84232e7i	1.32197i	0.750400	+	0.660984i	$0.229861\pi$
$857$	2.84232e7i	1.32197i	−0.750400	+	0.660984i	$0.770139\pi$
$858$	8.33448e6i	0.386510i
$859$	−2.65922e7	−1.22962	−0.614810	−	0.788675i	$-0.710767\pi$
$859$	−2.65922e7	−1.22962	−0.614810	+	0.788675i	$0.710767\pi$
$860$	0	0
$861$	229200.	0.0105368
$862$	− 2.73814e6i	− 0.125512i
$863$	− 2.22500e7i	− 1.01696i	−0.861074	−	0.508479i	$-0.830208\pi$
$863$	− 2.22500e7i	− 1.01696i	0.861074	−	0.508479i	$-0.169792\pi$
$864$	2.00448e7	0.913519
$865$	0	0
$866$	1.69036e7	0.765923
$867$	− 1.42389e7i	− 0.643323i
$868$	1.03088e6i	0.0464418i
$869$	7.88170e6	0.354055
$870$	0	0
$871$	−3.44389e7	−1.53817
$872$	3.56041e7i	1.58566i
$873$	− 1.20429e6i	− 0.0534805i
$874$	−5.69702e6	−0.252272
$875$	0	0
$876$	461760.	0.0203309
$877$	− 2.83428e7i	− 1.24435i	−0.782877	−	0.622176i	$-0.786249\pi$
$877$	− 2.83428e7i	− 1.24435i	0.782877	−	0.622176i	$-0.213751\pi$
$878$	8.36739e6i	0.366314i
$879$	−2.29662e7	−1.00258
$880$	0	0
$881$	3.66445e7	1.59063	0.795315	−	0.606196i	$-0.207305\pi$
$881$	3.66445e7	1.59063	0.795315	+	0.606196i	$0.207305\pi$
$882$	1.20290e6i	0.0520666i
$883$	1.68772e7i	0.728447i	0.931312	+	0.364223i	$0.118666\pi$
$883$	1.68772e7i	0.728447i	−0.931312	+	0.364223i	$0.881334\pi$
$884$	−1.26004e7	−0.542320
$885$	0	0
$886$	−6.25137e6	−0.267541
$887$	− 2.73941e6i	− 0.116909i	−0.998290	−	0.0584544i	$-0.981383\pi$
$887$	− 2.73941e6i	− 0.116909i	0.998290	−	0.0584544i	$-0.0186172\pi$
$888$	− 3.47414e7i	− 1.47848i
$889$	781840.	0.0331790
$890$	0	0
$891$	6.57647e6	0.277523
$892$	7.38050e6i	0.310580i
$893$	− 2.75251e6i	− 0.115505i
$894$	−4.36020e6	−0.182458
$895$	0	0
$896$	491520.	0.0204537
$897$	− 6.38690e7i	− 2.65038i
$898$	1.20180e7i	0.497325i
$899$	−3.49468e7	−1.44214
$900$	0	0
$901$	−2.04853e7	−0.840681
$902$	739552.i	0.0302658i
$903$	603900.i	0.0246460i
$904$	−1.39870e7	−0.569251
$905$	0	0
$906$	−1.81849e7	−0.736022
$907$	3.13286e7i	1.26451i	0.774760	+	0.632255i	$0.217871\pi$
$907$	3.13286e7i	1.26451i	−0.774760	+	0.632255i	$0.782129\pi$
$908$	1.36891e7i	0.551012i
$909$	1.74114e6	0.0698914
$910$	0	0
$911$	−2.49762e7	−0.997081	−0.498541	−	0.866866i	$-0.666131\pi$
$911$	−2.49762e7	−0.997081	−0.498541	+	0.866866i	$0.666131\pi$
$912$	1.47456e6i	0.0587050i
$913$	1.24284e7i	0.493444i
$914$	−9.78206e6	−0.387316
$915$	0	0
$916$	1.06529e7	0.419497
$917$	4620.00i	0 0.000181434i
$918$	− 1.07428e7i	− 0.420736i
$919$	1.10613e7	0.432032	0.216016	−	0.976390i	$-0.430694\pi$
$919$	1.10613e7	0.432032	0.216016	+	0.976390i	$0.430694\pi$
$920$	0	0
$921$	1.71402e7	0.665835
$922$	3.11641e7i	1.20734i
$923$	− 3.56144e7i	− 1.37601i
$924$	290400.	0.0111897
$925$	0	0
$926$	4.20784e6	0.161262
$927$	− 1.72267e6i	− 0.0658420i
$928$	2.77709e7i	1.05857i
$929$	2.01739e7	0.766919	0.383460	−	0.923558i	$-0.374733\pi$
$929$	2.01739e7	0.766919	0.383460	+	0.923558i	$0.374733\pi$
$930$	0	0
$931$	6.41549e6	0.242580
$932$	1.93494e7i	0.729671i
$933$	− 8.80434e6i	− 0.331126i
$934$	1.58801e7	0.595645
$935$	0	0
$936$	3.96749e6	0.148022
$937$	− 9.10734e6i	− 0.338877i	−0.985541	−	0.169439i	$-0.945805\pi$
$937$	− 9.10734e6i	− 0.338877i	0.985541	−	0.169439i	$-0.0541954\pi$
$938$	− 1.19996e6i	− 0.0445307i
$939$	−3.50785e6	−0.129831
$940$	0	0
$941$	−3.67709e7	−1.35372	−0.676861	−	0.736110i	$-0.736660\pi$
$941$	−3.67709e7	−1.35372	−0.676861	+	0.736110i	$0.736660\pi$
$942$	3.19792e7i	1.17420i
$943$	− 5.66735e6i	− 0.207540i
$944$	−1.65402e6	−0.0604101
$945$	0	0
$946$	−1.94858e6	−0.0707932
$947$	4.95743e7i	1.79631i	0.439677	+	0.898156i	$0.355093\pi$
$947$	4.95743e7i	1.79631i	−0.439677	+	0.898156i	$0.644907\pi$
$948$	1.56331e7i	0.564969i
$949$	2.20875e6	0.0796125
$950$	0	0
$951$	1.40325e7	0.503136
$952$	− 1.31712e6i	− 0.0471013i
$953$	3.53787e7i	1.26186i	0.775841	+	0.630928i	$0.217326\pi$
$953$	3.53787e7i	1.26186i	−0.775841	+	0.630928i	$0.782674\pi$
$954$	2.15006e6	0.0764857
$955$	0	0
$956$	9.14371e6	0.323577
$957$	9.84456e6i	0.347469i
$958$	3.41563e7i	1.20242i
$959$	−2.96233e6	−0.104013
$960$	0	0
$961$	1.28831e7	0.449999
$962$	− 5.53933e7i	− 1.92983i
$963$	587844.i	0.0204266i
$964$	−4.27328e6	−0.148105
$965$	0	0
$966$	2.22540e6	0.0767300
$967$	− 2.78059e7i	− 0.956248i	−0.878292	−	0.478124i	$-0.841317\pi$
$967$	− 2.78059e7i	− 0.956248i	0.878292	−	0.478124i	$-0.158683\pi$
$968$	2.81107e6i	0.0964237i
$969$	−3.95136e6	−0.135188
$970$	0	0
$971$	1.56835e7	0.533821	0.266910	−	0.963721i	$-0.413997\pi$
$971$	1.56835e7	0.533821	0.266910	+	0.963721i	$0.413997\pi$
$972$	− 2.17728e6i	− 0.0739177i
$973$	− 3.99818e6i	− 0.135388i
$974$	−7.45947e6	−0.251948
$975$	0	0
$976$	4.34688e6	0.146067
$977$	− 2.01140e7i	− 0.674157i	−0.941476	−	0.337079i	$-0.890561\pi$
$977$	− 2.01140e7i	− 0.674157i	0.941476	−	0.337079i	$-0.109439\pi$
$978$	1.69246e7i	0.565810i
$979$	2.10722e6	0.0702671
$980$	0	0
$981$	3.33788e6	0.110739
$982$	2.06291e7i	0.682655i
$983$	− 2.09269e7i	− 0.690750i	−0.938465	−	0.345375i	$-0.887752\pi$
$983$	− 2.09269e7i	− 0.690750i	0.938465	−	0.345375i	$-0.112248\pi$
$984$	−4.40064e6	−0.144887
$985$	0	0
$986$	1.48835e7	0.487541
$987$	1.07520e6i	0.0351315i
$988$	− 7.05331e6i	− 0.229880i
$989$	1.49324e7	0.485445
$990$	0	0
$991$	−3.18663e7	−1.03074	−0.515368	−	0.856969i	$-0.672345\pi$
$991$	−3.18663e7	−1.03074	−0.515368	+	0.856969i	$0.672345\pi$
$992$	− 3.29882e7i	− 1.06434i
$993$	1.58735e7i	0.510856i
$994$	1.24092e6	0.0398362
$995$	0	0
$996$	−2.46514e7	−0.787395
$997$	− 1.38913e6i	− 0.0442595i	−0.999755	−	0.0221297i	$-0.992955\pi$
$997$	− 1.38913e6i	− 0.0442595i	0.999755	−	0.0221297i	$-0.00704469\pi$
$998$	− 1.81336e7i	− 0.576313i
$999$	−4.72266e7	−1.49718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.b.a.199.2		2
5.2	odd	4		11.6.a.a.1.1	✓	1
5.3	odd	4		275.6.a.a.1.1		1
5.4	even	2	inner	275.6.b.a.199.1		2
15.2	even	4		99.6.a.c.1.1		1
20.7	even	4		176.6.a.c.1.1		1
35.27	even	4		539.6.a.c.1.1		1
40.27	even	4		704.6.a.c.1.1		1
40.37	odd	4		704.6.a.h.1.1		1
55.32	even	4		121.6.a.b.1.1		1
165.32	odd	4		1089.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.a.a.1.1	✓	1	5.2	odd	4
99.6.a.c.1.1		1	15.2	even	4
121.6.a.b.1.1		1	55.32	even	4
176.6.a.c.1.1		1	20.7	even	4
275.6.a.a.1.1		1	5.3	odd	4
275.6.b.a.199.1		2	5.4	even	2	inner
275.6.b.a.199.2		2	1.1	even	1	trivial
539.6.a.c.1.1		1	35.27	even	4
704.6.a.c.1.1		1	40.27	even	4
704.6.a.h.1.1		1	40.37	odd	4
1089.6.a.c.1.1		1	165.32	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -15.0000i q^{3} +16.0000 q^{4} +60.0000 q^{6} -10.0000i q^{7} +192.000i q^{8} +18.0000 q^{9} -121.000 q^{11} -240.000i q^{12} -1148.00i q^{13} +40.0000 q^{14} -256.000 q^{16} -686.000i q^{17} +72.0000i q^{18} +384.000 q^{19} -150.000 q^{21} -484.000i q^{22} +3709.00i q^{23} +2880.00 q^{24} +4592.00 q^{26} -3915.00i q^{27} -160.000i q^{28} +5424.00 q^{29} -6443.00 q^{31} +5120.00i q^{32} +1815.00i q^{33} +2744.00 q^{34} +288.000 q^{36} -12063.0i q^{37} +1536.00i q^{38} -17220.0 q^{39} -1528.00 q^{41} -600.000i q^{42} -4026.00i q^{43} -1936.00 q^{44} -14836.0 q^{46} -7168.00i q^{47} +3840.00i q^{48} +16707.0 q^{49} -10290.0 q^{51} -18368.0i q^{52} -29862.0i q^{53} +15660.0 q^{54} +1920.00 q^{56} -5760.00i q^{57} +21696.0i q^{58} +6461.00 q^{59} -16980.0 q^{61} -25772.0i q^{62} -180.000i q^{63} -28672.0 q^{64} -7260.00 q^{66} -29999.0i q^{67} -10976.0i q^{68} +55635.0 q^{69} +31023.0 q^{71} +3456.00i q^{72} +1924.00i q^{73} +48252.0 q^{74} +6144.00 q^{76} +1210.00i q^{77} -68880.0i q^{78} -65138.0 q^{79} -54351.0 q^{81} -6112.00i q^{82} -102714. i q^{83} -2400.00 q^{84} +16104.0 q^{86} -81360.0i q^{87} -23232.0i q^{88} -17415.0 q^{89} -11480.0 q^{91} +59344.0i q^{92} +96645.0i q^{93} +28672.0 q^{94} +76800.0 q^{96} -66905.0i q^{97} +66828.0i q^{98} -2178.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{4} + 120 q^{6} + 36 q^{9} - 242 q^{11} + 80 q^{14} - 512 q^{16} + 768 q^{19} - 300 q^{21} + 5760 q^{24} + 9184 q^{26} + 10848 q^{29} - 12886 q^{31} + 5488 q^{34} + 576 q^{36} - 34440 q^{39}+ \cdots - 4356 q^{99}+O(q^{100}) \) 2 * q + 32 * q^4 + 120 * q^6 + 36 * q^9 - 242 * q^11 + 80 * q^14 - 512 * q^16 + 768 * q^19 - 300 * q^21 + 5760 * q^24 + 9184 * q^26 + 10848 * q^29 - 12886 * q^31 + 5488 * q^34 + 576 * q^36 - 34440 * q^39 - 3056 * q^41 - 3872 * q^44 - 29672 * q^46 + 33414 * q^49 - 20580 * q^51 + 31320 * q^54 + 3840 * q^56 + 12922 * q^59 - 33960 * q^61 - 57344 * q^64 - 14520 * q^66 + 111270 * q^69 + 62046 * q^71 + 96504 * q^74 + 12288 * q^76 - 130276 * q^79 - 108702 * q^81 - 4800 * q^84 + 32208 * q^86 - 34830 * q^89 - 22960 * q^91 + 57344 * q^94 + 153600 * q^96 - 4356 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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