LMFDB - Embedded newform 1250.2.b.e.1249.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1250,2,Mod(1249,1250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1250, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1250.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1250 = 2 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1250.b (of order $2$, degree $1$, not 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:	$9.98130025266$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14884000000.15
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 121x^{4} + 304x^{2} + 256 $ x^8 + 19x^6 + 121x^4 + 304*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.8
Root			$2.33275i$ of defining polynomial
Character	$\chi$	$=$	1250.1249
Dual form			1250.2.b.e.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +2.33275i q^{3} -1.00000 q^{4} -2.33275 q^{6} +3.77447i q^{7} -1.00000i q^{8} -2.44172 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +2.33275i q^{3} -1.00000 q^{4} -2.33275 q^{6} +3.77447i q^{7} -1.00000i q^{8} -2.44172 q^{9} -3.77447 q^{11} -2.33275i q^{12} -3.17632i q^{13} -3.77447 q^{14} +1.00000 q^{16} -1.39250i q^{17} -2.44172i q^{18} -3.91385 q^{19} -8.80489 q^{21} -3.77447i q^{22} -0.891031i q^{23} +2.33275 q^{24} +3.17632 q^{26} +1.30233i q^{27} -3.77447i q^{28} -0.0492169 q^{29} -5.58111 q^{31} +1.00000i q^{32} -8.80489i q^{33} +1.39250 q^{34} +2.44172 q^{36} +7.04746i q^{37} -3.91385i q^{38} +7.40955 q^{39} -1.48918 q^{41} -8.80489i q^{42} -2.69767i q^{43} +3.77447 q^{44} +0.891031 q^{46} +3.77447i q^{47} +2.33275i q^{48} -7.24660 q^{49} +3.24836 q^{51} +3.17632i q^{52} -11.6163i q^{53} -1.30233 q^{54} +3.77447 q^{56} -9.13004i q^{57} -0.0492169i q^{58} +0.690074 q^{59} +10.3603 q^{61} -5.58111i q^{62} -9.21619i q^{63} -1.00000 q^{64} +8.80489 q^{66} +15.3234i q^{67} +1.39250i q^{68} +2.07855 q^{69} -6.69767 q^{71} +2.44172i q^{72} -5.16872i q^{73} -7.04746 q^{74} +3.91385 q^{76} -14.2466i q^{77} +7.40955i q^{78} +1.80664 q^{79} -10.3632 q^{81} -1.48918i q^{82} +9.96783i q^{83} +8.80489 q^{84} +2.69767 q^{86} -0.114811i q^{87} +3.77447i q^{88} -14.5103 q^{89} +11.9889 q^{91} +0.891031i q^{92} -13.0193i q^{93} -3.77447 q^{94} -2.33275 q^{96} -0.0901699i q^{97} -7.24660i q^{98} +9.21619 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 + 2 * q^6 - 14 * q^9	$ 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} + 10 q^{19} - 14 q^{21} - 2 q^{24} + 22 q^{26} - 30 q^{29} - 24 q^{31} - 24 q^{34} + 14 q^{36} + 22 q^{39} + 26 q^{41} + 4 q^{44} - 8 q^{46} + 4 q^{49} + 26 q^{51} - 20 q^{54} + 4 q^{56} + 16 q^{61} - 8 q^{64} + 14 q^{66} + 62 q^{69} - 44 q^{71} - 24 q^{74} - 10 q^{76} + 20 q^{79} - 32 q^{81} + 14 q^{84} + 12 q^{86} + 10 q^{89} - 24 q^{91} - 4 q^{94} + 2 q^{96} + 42 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 + 2 * q^6 - 14 * q^9 - 4 * q^11 - 4 * q^14 + 8 * q^16 + 10 * q^19 - 14 * q^21 - 2 * q^24 + 22 * q^26 - 30 * q^29 - 24 * q^31 - 24 * q^34 + 14 * q^36 + 22 * q^39 + 26 * q^41 + 4 * q^44 - 8 * q^46 + 4 * q^49 + 26 * q^51 - 20 * q^54 + 4 * q^56 + 16 * q^61 - 8 * q^64 + 14 * q^66 + 62 * q^69 - 44 * q^71 - 24 * q^74 - 10 * q^76 + 20 * q^79 - 32 * q^81 + 14 * q^84 + 12 * q^86 + 10 * q^89 - 24 * q^91 - 4 * q^94 + 2 * q^96 + 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1250\mathbb{Z}\right)^\times$.

$n$	$627$
$\chi(n)$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	2.33275i	1.34681i	0.739272	+	0.673407i	$0.235170\pi$
$3$	2.33275i	1.34681i	−0.739272	+	0.673407i	$0.764830\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.33275	−0.952341
$7$	3.77447i	1.42661i	0.700851	+	0.713307i	$0.252804\pi$
$7$	3.77447i	1.42661i	−0.700851	+	0.713307i	$0.747196\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−2.44172	−0.813906
$10$	0	0
$11$	−3.77447	−1.13804	−0.569022	−	0.822322i	$-0.692678\pi$
$11$	−3.77447	−1.13804	−0.569022	+	0.822322i	$0.692678\pi$
$12$	− 2.33275i	− 0.673407i
$13$	− 3.17632i	− 0.880951i	−0.897765	−	0.440476i	$-0.854810\pi$
$13$	− 3.17632i	− 0.880951i	0.897765	−	0.440476i	$-0.145190\pi$
$14$	−3.77447	−1.00877
$15$	0	0
$16$	1.00000	0.250000
$17$	− 1.39250i	− 0.337731i	−0.985639	−	0.168866i	$-0.945990\pi$
$17$	− 1.39250i	− 0.337731i	0.985639	−	0.168866i	$-0.0540104\pi$
$18$	− 2.44172i	− 0.575519i
$19$	−3.91385	−0.897900	−0.448950	−	0.893557i	$-0.648202\pi$
$19$	−3.91385	−0.897900	−0.448950	+	0.893557i	$0.648202\pi$
$20$	0	0
$21$	−8.80489	−1.92138
$22$	− 3.77447i	− 0.804719i
$23$	− 0.891031i	− 0.185793i	−0.995676	−	0.0928964i	$-0.970387\pi$
$23$	− 0.891031i	− 0.185793i	0.995676	−	0.0928964i	$-0.0296125\pi$
$24$	2.33275	0.476170
$25$	0	0
$26$	3.17632	0.622927
$27$	1.30233i	0.250634i
$28$	− 3.77447i	− 0.713307i
$29$	−0.0492169	−0.00913936	−0.00456968	−	0.999990i	$-0.501455\pi$
$29$	−0.0492169	−0.00913936	−0.00456968	+	0.999990i	$0.501455\pi$
$30$	0	0
$31$	−5.58111	−1.00240	−0.501198	−	0.865333i	$-0.667107\pi$
$31$	−5.58111	−1.00240	−0.501198	+	0.865333i	$0.667107\pi$
$32$	1.00000i	0.176777i
$33$	− 8.80489i	− 1.53273i
$34$	1.39250	0.238812
$35$	0	0
$36$	2.44172	0.406953
$37$	7.04746i	1.15860i	0.815116	+	0.579298i	$0.196673\pi$
$37$	7.04746i	1.15860i	−0.815116	+	0.579298i	$0.803327\pi$
$38$	− 3.91385i	− 0.634911i
$39$	7.40955	1.18648
$40$	0	0
$41$	−1.48918	−0.232571	−0.116286	−	0.993216i	$-0.537099\pi$
$41$	−1.48918	−0.232571	−0.116286	+	0.993216i	$0.537099\pi$
$42$	− 8.80489i	− 1.35862i
$43$	− 2.69767i	− 0.411391i	−0.978616	−	0.205695i	$-0.934054\pi$
$43$	− 2.69767i	− 0.411391i	0.978616	−	0.205695i	$-0.0659456\pi$
$44$	3.77447	0.569022
$45$	0	0
$46$	0.891031	0.131375
$47$	3.77447i	0.550563i	0.961364	+	0.275281i	$0.0887710\pi$
$47$	3.77447i	0.550563i	−0.961364	+	0.275281i	$0.911229\pi$
$48$	2.33275i	0.336703i
$49$	−7.24660	−1.03523
$50$	0	0
$51$	3.24836	0.454861
$52$	3.17632i	0.440476i
$53$	− 11.6163i	− 1.59562i	−0.602910	−	0.797809i	$-0.705992\pi$
$53$	− 11.6163i	− 1.59562i	0.602910	−	0.797809i	$-0.294008\pi$
$54$	−1.30233	−0.177225
$55$	0	0
$56$	3.77447	0.504384
$57$	− 9.13004i	− 1.20930i
$58$	− 0.0492169i	− 0.00646250i
$59$	0.690074	0.0898400	0.0449200	−	0.998991i	$-0.485697\pi$
$59$	0.690074	0.0898400	0.0449200	+	0.998991i	$0.485697\pi$
$60$	0	0
$61$	10.3603	1.32650	0.663252	−	0.748396i	$-0.269176\pi$
$61$	10.3603	1.32650	0.663252	+	0.748396i	$0.269176\pi$
$62$	− 5.58111i	− 0.708801i
$63$	− 9.21619i	− 1.16113i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	8.80489	1.08381
$67$	15.3234i	1.87205i	0.351932	+	0.936026i	$0.385525\pi$
$67$	15.3234i	1.87205i	−0.351932	+	0.936026i	$0.614475\pi$
$68$	1.39250i	0.168866i
$69$	2.07855	0.250228
$70$	0	0
$71$	−6.69767	−0.794867	−0.397434	−	0.917631i	$-0.630099\pi$
$71$	−6.69767	−0.794867	−0.397434	+	0.917631i	$0.630099\pi$
$72$	2.44172i	0.287759i
$73$	− 5.16872i	− 0.604953i	−0.953157	−	0.302477i	$-0.902187\pi$
$73$	− 5.16872i	− 0.604953i	0.953157	−	0.302477i	$-0.0978134\pi$
$74$	−7.04746	−0.819251
$75$	0	0
$76$	3.91385	0.448950
$77$	− 14.2466i	− 1.62355i
$78$	7.40955i	0.838966i
$79$	1.80664	0.203263	0.101631	−	0.994822i	$-0.467594\pi$
$79$	1.80664	0.203263	0.101631	+	0.994822i	$0.467594\pi$
$80$	0	0
$81$	−10.3632	−1.15146
$82$	− 1.48918i	− 0.164453i
$83$	9.96783i	1.09411i	0.837096	+	0.547056i	$0.184251\pi$
$83$	9.96783i	1.09411i	−0.837096	+	0.547056i	$0.815749\pi$
$84$	8.80489	0.960692
$85$	0	0
$86$	2.69767	0.290897
$87$	− 0.114811i	− 0.0123090i
$88$	3.77447i	0.402360i
$89$	−14.5103	−1.53808	−0.769042	−	0.639198i	$-0.779266\pi$
$89$	−14.5103	−1.53808	−0.769042	+	0.639198i	$0.779266\pi$
$90$	0	0
$91$	11.9889	1.25678
$92$	0.891031i	0.0928964i
$93$	− 13.0193i	− 1.35004i
$94$	−3.77447	−0.389307
$95$	0	0
$96$	−2.33275	−0.238085
$97$	− 0.0901699i	− 0.00915537i	−0.999990	−	0.00457769i	$-0.998543\pi$
$97$	− 0.0901699i	− 0.00915537i	0.999990	−	0.00457769i	$-0.00145713\pi$
$98$	− 7.24660i	− 0.732017i
$99$	9.21619	0.926261
$100$	0	0
$101$	16.3785	1.62972	0.814859	−	0.579659i	$-0.196814\pi$
$101$	16.3785	1.62972	0.814859	+	0.579659i	$0.196814\pi$
$102$	3.24836i	0.321635i
$103$	1.21619i	0.119834i	0.998203	+	0.0599172i	$0.0190837\pi$
$103$	1.21619i	0.119834i	−0.998203	+	0.0599172i	$0.980916\pi$
$104$	−3.17632	−0.311463
$105$	0	0
$106$	11.6163	1.12827
$107$	− 10.8125i	− 1.04528i	−0.852553	−	0.522641i	$-0.824947\pi$
$107$	− 10.8125i	− 1.04528i	0.852553	−	0.522641i	$-0.175053\pi$
$108$	− 1.30233i	− 0.125317i
$109$	−11.4153	−1.09339	−0.546695	−	0.837332i	$-0.684114\pi$
$109$	−11.4153	−1.09339	−0.546695	+	0.837332i	$0.684114\pi$
$110$	0	0
$111$	−16.4400	−1.56041
$112$	3.77447i	0.356654i
$113$	10.4229i	0.980506i	0.871580	+	0.490253i	$0.163096\pi$
$113$	10.4229i	0.980506i	−0.871580	+	0.490253i	$0.836904\pi$
$114$	9.13004	0.855107
$115$	0	0
$116$	0.0492169	0.00456968
$117$	7.75567i	0.717012i
$118$	0.690074i	0.0635265i
$119$	5.25595	0.481812
$120$	0	0
$121$	3.24660	0.295146
$122$	10.3603i	0.937980i
$123$	− 3.47389i	− 0.313230i
$124$	5.58111	0.501198
$125$	0	0
$126$	9.21619	0.821043
$127$	− 3.54893i	− 0.314917i	−0.987526	−	0.157459i	$-0.949670\pi$
$127$	− 3.54893i	− 0.314917i	0.987526	−	0.157459i	$-0.0503301\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	6.29298	0.554066
$130$	0	0
$131$	4.05324	0.354133	0.177067	−	0.984199i	$-0.443339\pi$
$131$	4.05324	0.354133	0.177067	+	0.984199i	$0.443339\pi$
$132$	8.80489i	0.766367i
$133$	− 14.7727i	− 1.28096i
$134$	−15.3234	−1.32374
$135$	0	0
$136$	−1.39250	−0.119406
$137$	− 3.56773i	− 0.304812i	−0.988318	−	0.152406i	$-0.951298\pi$
$137$	− 3.56773i	− 0.304812i	0.988318	−	0.152406i	$-0.0487021\pi$
$138$	2.07855i	0.176938i
$139$	9.94602	0.843611	0.421805	−	0.906686i	$-0.361397\pi$
$139$	9.94602	0.843611	0.421805	+	0.906686i	$0.361397\pi$
$140$	0	0
$141$	−8.80489	−0.741505
$142$	− 6.69767i	− 0.562056i
$143$	11.9889i	1.00256i
$144$	−2.44172	−0.203477
$145$	0	0
$146$	5.16872	0.427766
$147$	− 16.9045i	− 1.39426i
$148$	− 7.04746i	− 0.579298i
$149$	19.5103	1.59834	0.799171	−	0.601104i	$-0.205272\pi$
$149$	19.5103	1.59834	0.799171	+	0.601104i	$0.205272\pi$
$150$	0	0
$151$	−18.4324	−1.50001	−0.750003	−	0.661435i	$-0.769948\pi$
$151$	−18.4324	−1.50001	−0.750003	+	0.661435i	$0.769948\pi$
$152$	3.91385i	0.317455i
$153$	3.40010i	0.274881i
$154$	14.2466	1.14802
$155$	0	0
$156$	−7.40955	−0.593239
$157$	10.1564i	0.810572i	0.914190	+	0.405286i	$0.132828\pi$
$157$	10.1564i	0.810572i	−0.914190	+	0.405286i	$0.867172\pi$
$158$	1.80664i	0.143728i
$159$	27.0979	2.14900
$160$	0	0
$161$	3.36317	0.265055
$162$	− 10.3632i	− 0.814207i
$163$	18.6004i	1.45690i	0.685100	+	0.728449i	$0.259758\pi$
$163$	18.6004i	1.45690i	−0.685100	+	0.728449i	$0.740242\pi$
$164$	1.48918	0.116286
$165$	0	0
$166$	−9.96783	−0.773654
$167$	2.17156i	0.168040i	0.996464	+	0.0840201i	$0.0267760\pi$
$167$	2.17156i	0.168040i	−0.996464	+	0.0840201i	$0.973224\pi$
$168$	8.80489i	0.679312i
$169$	2.91102	0.223924
$170$	0	0
$171$	9.55653	0.730806
$172$	2.69767i	0.205695i
$173$	6.39425i	0.486146i	0.970008	+	0.243073i	$0.0781555\pi$
$173$	6.39425i	0.486146i	−0.970008	+	0.243073i	$0.921845\pi$
$174$	0.114811	0.00870378
$175$	0	0
$176$	−3.77447	−0.284511
$177$	1.60977i	0.120998i
$178$	− 14.5103i	− 1.08759i
$179$	−8.01348	−0.598955	−0.299478	−	0.954103i	$-0.596812\pi$
$179$	−8.01348	−0.598955	−0.299478	+	0.954103i	$0.596812\pi$
$180$	0	0
$181$	−10.8629	−0.807432	−0.403716	−	0.914884i	$-0.632282\pi$
$181$	−10.8629	−0.807432	−0.403716	+	0.914884i	$0.632282\pi$
$182$	11.9889i	0.888676i
$183$	24.1681i	1.78655i
$184$	−0.891031	−0.0656877
$185$	0	0
$186$	13.0193	0.954623
$187$	5.25595i	0.384353i
$188$	− 3.77447i	− 0.275281i
$189$	−4.91561	−0.357558
$190$	0	0
$191$	−8.18577	−0.592301	−0.296151	−	0.955141i	$-0.595703\pi$
$191$	−8.18577	−0.592301	−0.296151	+	0.955141i	$0.595703\pi$
$192$	− 2.33275i	− 0.168352i
$193$	17.5576i	1.26382i	0.775040	+	0.631912i	$0.217730\pi$
$193$	17.5576i	1.26382i	−0.775040	+	0.631912i	$0.782270\pi$
$194$	0.0901699	0.00647382
$195$	0	0
$196$	7.24660	0.517615
$197$	4.75742i	0.338952i	0.985534	+	0.169476i	$0.0542076\pi$
$197$	4.75742i	0.338952i	−0.985534	+	0.169476i	$0.945792\pi$
$198$	9.21619i	0.654966i
$199$	−14.5320	−1.03015	−0.515073	−	0.857146i	$-0.672235\pi$
$199$	−14.5320	−1.03015	−0.515073	+	0.857146i	$0.672235\pi$
$200$	0	0
$201$	−35.7457	−2.52130
$202$	16.3785i	1.15238i
$203$	− 0.185768i	− 0.0130383i
$204$	−3.24836	−0.227430
$205$	0	0
$206$	−1.21619	−0.0847357
$207$	2.17565i	0.151218i
$208$	− 3.17632i	− 0.220238i
$209$	14.7727	1.02185
$210$	0	0
$211$	−14.1359	−0.973154	−0.486577	−	0.873638i	$-0.661755\pi$
$211$	−14.1359	−0.973154	−0.486577	+	0.873638i	$0.661755\pi$
$212$	11.6163i	0.797809i
$213$	− 15.6240i	− 1.07054i
$214$	10.8125	0.739126
$215$	0	0
$216$	1.30233	0.0886124
$217$	− 21.0657i	− 1.43003i
$218$	− 11.4153i	− 0.773143i
$219$	12.0573	0.814759
$220$	0	0
$221$	−4.42302	−0.297525
$222$	− 16.4400i	− 1.10338i
$223$	− 4.61226i	− 0.308860i	−0.988004	−	0.154430i	$-0.950646\pi$
$223$	− 4.61226i	− 0.308860i	0.988004	−	0.154430i	$-0.0493540\pi$
$224$	−3.77447	−0.252192
$225$	0	0
$226$	−10.4229	−0.693322
$227$	− 6.28637i	− 0.417241i	−0.977997	−	0.208620i	$-0.933103\pi$
$227$	− 6.28637i	− 0.417241i	0.977997	−	0.208620i	$-0.0668973\pi$
$228$	9.13004i	0.604652i
$229$	−17.7370	−1.17209	−0.586046	−	0.810278i	$-0.699316\pi$
$229$	−17.7370	−1.17209	−0.586046	+	0.810278i	$0.699316\pi$
$230$	0	0
$231$	33.2338	2.18662
$232$	0.0492169i	0.00323125i
$233$	− 2.56406i	− 0.167977i	−0.996467	−	0.0839885i	$-0.973234\pi$
$233$	− 2.56406i	− 0.167977i	0.996467	−	0.0839885i	$-0.0267659\pi$
$234$	−7.75567	−0.507004
$235$	0	0
$236$	−0.690074	−0.0449200
$237$	4.21443i	0.273757i
$238$	5.25595i	0.340693i
$239$	−9.44172	−0.610734	−0.305367	−	0.952235i	$-0.598779\pi$
$239$	−9.44172	−0.610734	−0.305367	+	0.952235i	$0.598779\pi$
$240$	0	0
$241$	−29.4346	−1.89605	−0.948026	−	0.318193i	$-0.896924\pi$
$241$	−29.4346	−1.89605	−0.948026	+	0.318193i	$0.896924\pi$
$242$	3.24660i	0.208700i
$243$	− 20.2677i	− 1.30017i
$244$	−10.3603	−0.663252
$245$	0	0
$246$	3.47389	0.221487
$247$	12.4316i	0.791006i
$248$	5.58111i	0.354401i
$249$	−23.2524	−1.47356
$250$	0	0
$251$	−6.00759	−0.379196	−0.189598	−	0.981862i	$-0.560718\pi$
$251$	−6.00759	−0.379196	−0.189598	+	0.981862i	$0.560718\pi$
$252$	9.21619i	0.580565i
$253$	3.36317i	0.211440i
$254$	3.54893	0.222680
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 13.6286i	− 0.850127i	−0.905164	−	0.425063i	$-0.860252\pi$
$257$	− 13.6286i	− 0.850127i	0.905164	−	0.425063i	$-0.139748\pi$
$258$	6.29298i	0.391784i
$259$	−26.6004	−1.65287
$260$	0	0
$261$	0.120174	0.00743858
$262$	4.05324i	0.250410i
$263$	9.16294i	0.565011i	0.959266	+	0.282506i	$0.0911656\pi$
$263$	9.16294i	0.565011i	−0.959266	+	0.282506i	$0.908834\pi$
$264$	−8.80489	−0.541903
$265$	0	0
$266$	14.7727	0.905773
$267$	− 33.8488i	− 2.07151i
$268$	− 15.3234i	− 0.936026i
$269$	−28.1026	−1.71345	−0.856724	−	0.515776i	$-0.827504\pi$
$269$	−28.1026	−1.71345	−0.856724	+	0.515776i	$0.827504\pi$
$270$	0	0
$271$	−9.09860	−0.552701	−0.276350	−	0.961057i	$-0.589125\pi$
$271$	−9.09860	−0.552701	−0.276350	+	0.961057i	$0.589125\pi$
$272$	− 1.39250i	− 0.0844328i
$273$	27.9671i	1.69265i
$274$	3.56773	0.215535
$275$	0	0
$276$	−2.07855	−0.125114
$277$	18.7890i	1.12892i	0.825459	+	0.564462i	$0.190916\pi$
$277$	18.7890i	1.12892i	−0.825459	+	0.564462i	$0.809084\pi$
$278$	9.94602i	0.596523i
$279$	13.6275	0.815856
$280$	0	0
$281$	7.88158	0.470176	0.235088	−	0.971974i	$-0.424462\pi$
$281$	7.88158	0.470176	0.235088	+	0.971974i	$0.424462\pi$
$282$	− 8.80489i	− 0.524323i
$283$	18.2998i	1.08781i	0.839146	+	0.543906i	$0.183055\pi$
$283$	18.2998i	1.08781i	−0.839146	+	0.543906i	$0.816945\pi$
$284$	6.69767	0.397434
$285$	0	0
$286$	−11.9889	−0.708919
$287$	− 5.62087i	− 0.331789i
$288$	− 2.44172i	− 0.143880i
$289$	15.0609	0.885938
$290$	0	0
$291$	0.210344	0.0123306
$292$	5.16872i	0.302477i
$293$	29.4990i	1.72335i	0.507461	+	0.861675i	$0.330584\pi$
$293$	29.4990i	1.72335i	−0.507461	+	0.861675i	$0.669416\pi$
$294$	16.9045	0.985891
$295$	0	0
$296$	7.04746	0.409626
$297$	− 4.91561i	− 0.285232i
$298$	19.5103i	1.13020i
$299$	−2.83020	−0.163674
$300$	0	0
$301$	10.1823	0.586896
$302$	− 18.4324i	− 1.06066i
$303$	38.2068i	2.19493i
$304$	−3.91385	−0.224475
$305$	0	0
$306$	−3.40010	−0.194371
$307$	13.9131i	0.794064i	0.917805	+	0.397032i	$0.129960\pi$
$307$	13.9131i	0.794064i	−0.917805	+	0.397032i	$0.870040\pi$
$308$	14.2466i	0.811776i
$309$	−2.83706	−0.161394
$310$	0	0
$311$	25.9636	1.47226	0.736130	−	0.676840i	$-0.236651\pi$
$311$	25.9636	1.47226	0.736130	+	0.676840i	$0.236651\pi$
$312$	− 7.40955i	− 0.419483i
$313$	− 9.47389i	− 0.535496i	−0.963489	−	0.267748i	$-0.913721\pi$
$313$	− 9.47389i	− 0.535496i	0.963489	−	0.267748i	$-0.0862794\pi$
$314$	−10.1564	−0.573161
$315$	0	0
$316$	−1.80664	−0.101631
$317$	24.5074i	1.37647i	0.725486	+	0.688237i	$0.241615\pi$
$317$	24.5074i	1.37647i	−0.725486	+	0.688237i	$0.758385\pi$
$318$	27.0979i	1.51957i
$319$	0.185768	0.0104010
$320$	0	0
$321$	25.2228	1.40780
$322$	3.36317i	0.187422i
$323$	5.45005i	0.303249i
$324$	10.3632	0.575731
$325$	0	0
$326$	−18.6004	−1.03018
$327$	− 26.6291i	− 1.47259i
$328$	1.48918i	0.0822263i
$329$	−14.2466	−0.785441
$330$	0	0
$331$	−9.91035	−0.544722	−0.272361	−	0.962195i	$-0.587804\pi$
$331$	−9.91035	−0.544722	−0.272361	+	0.962195i	$0.587804\pi$
$332$	− 9.96783i	− 0.547056i
$333$	− 17.2079i	− 0.942988i
$334$	−2.17156	−0.118822
$335$	0	0
$336$	−8.80489	−0.480346
$337$	− 9.93566i	− 0.541230i	−0.962688	−	0.270615i	$-0.912773\pi$
$337$	− 9.93566i	− 0.541230i	0.962688	−	0.270615i	$-0.0872270\pi$
$338$	2.91102i	0.158339i
$339$	−24.3141	−1.32056
$340$	0	0
$341$	21.0657	1.14077
$342$	9.55653i	0.516758i
$343$	− 0.930796i	− 0.0502583i
$344$	−2.69767	−0.145449
$345$	0	0
$346$	−6.39425	−0.343757
$347$	− 19.6351i	− 1.05407i	−0.849845	−	0.527033i	$-0.823304\pi$
$347$	− 19.6351i	− 1.05407i	0.849845	−	0.527033i	$-0.176696\pi$
$348$	0.114811i	0.00615450i
$349$	−1.48432	−0.0794538	−0.0397269	−	0.999211i	$-0.512649\pi$
$349$	−1.48432	−0.0794538	−0.0397269	+	0.999211i	$0.512649\pi$
$350$	0	0
$351$	4.13662	0.220796
$352$	− 3.77447i	− 0.201180i
$353$	− 10.2788i	− 0.547084i	−0.961860	−	0.273542i	$-0.911805\pi$
$353$	− 10.2788i	− 0.547084i	0.961860	−	0.273542i	$-0.0881952\pi$
$354$	−1.60977	−0.0855583
$355$	0	0
$356$	14.5103	0.769042
$357$	12.2608i	0.648911i
$358$	− 8.01348i	− 0.423525i
$359$	10.3725	0.547440	0.273720	−	0.961809i	$-0.411746\pi$
$359$	10.3725	0.547440	0.273720	+	0.961809i	$0.411746\pi$
$360$	0	0
$361$	−3.68175	−0.193776
$362$	− 10.8629i	− 0.570941i
$363$	7.57351i	0.397506i
$364$	−11.9889	−0.628389
$365$	0	0
$366$	−24.1681	−1.26328
$367$	26.0743i	1.36107i	0.732717	+	0.680534i	$0.238252\pi$
$367$	26.0743i	1.36107i	−0.732717	+	0.680534i	$0.761748\pi$
$368$	− 0.891031i	− 0.0464482i
$369$	3.63616	0.189291
$370$	0	0
$371$	43.8453	2.27633
$372$	13.0193i	0.675020i
$373$	− 9.45593i	− 0.489609i	−0.969572	−	0.244805i	$-0.921276\pi$
$373$	− 9.45593i	− 0.489609i	0.969572	−	0.244805i	$-0.0787238\pi$
$374$	−5.25595	−0.271779
$375$	0	0
$376$	3.77447	0.194653
$377$	0.156329i	0.00805133i
$378$	− 4.91561i	− 0.252832i
$379$	35.3258	1.81456	0.907282	−	0.420523i	$-0.138154\pi$
$379$	35.3258	1.81456	0.907282	+	0.420523i	$0.138154\pi$
$380$	0	0
$381$	8.27877	0.424134
$382$	− 8.18577i	− 0.418820i
$383$	− 29.8173i	− 1.52360i	−0.647815	−	0.761798i	$-0.724317\pi$
$383$	− 29.8173i	− 1.52360i	0.647815	−	0.761798i	$-0.275683\pi$
$384$	2.33275	0.119043
$385$	0	0
$386$	−17.5576	−0.893659
$387$	6.58695i	0.334833i
$388$	0.0901699i	0.00457769i
$389$	21.3767	1.08384	0.541922	−	0.840429i	$-0.317697\pi$
$389$	21.3767	1.08384	0.541922	+	0.840429i	$0.317697\pi$
$390$	0	0
$391$	−1.24076	−0.0627480
$392$	7.24660i	0.366009i
$393$	9.45519i	0.476951i
$394$	−4.75742	−0.239675
$395$	0	0
$396$	−9.21619	−0.463131
$397$	21.0201i	1.05497i	0.849566	+	0.527483i	$0.176864\pi$
$397$	21.0201i	1.05497i	−0.849566	+	0.527483i	$0.823136\pi$
$398$	− 14.5320i	− 0.728423i
$399$	34.4610	1.72521
$400$	0	0
$401$	16.7820	0.838053	0.419026	−	0.907974i	$-0.362371\pi$
$401$	16.7820	0.838053	0.419026	+	0.907974i	$0.362371\pi$
$402$	− 35.7457i	− 1.78283i
$403$	17.7274i	0.883062i
$404$	−16.3785	−0.814859
$405$	0	0
$406$	0.185768	0.00921950
$407$	− 26.6004i	− 1.31853i
$408$	− 3.24836i	− 0.160818i
$409$	−36.3607	−1.79792	−0.898962	−	0.438028i	$-0.855677\pi$
$409$	−36.3607	−1.79792	−0.898962	+	0.438028i	$0.855677\pi$
$410$	0	0
$411$	8.32263	0.410525
$412$	− 1.21619i	− 0.0599172i
$413$	2.60466i	0.128167i
$414$	−2.17565	−0.106927
$415$	0	0
$416$	3.17632	0.155732
$417$	23.2016i	1.13619i
$418$	14.7727i	0.722557i
$419$	34.5389	1.68733	0.843667	−	0.536867i	$-0.180392\pi$
$419$	34.5389	1.68733	0.843667	+	0.536867i	$0.180392\pi$
$420$	0	0
$421$	9.73403	0.474408	0.237204	−	0.971460i	$-0.423769\pi$
$421$	9.73403	0.474408	0.237204	+	0.971460i	$0.423769\pi$
$422$	− 14.1359i	− 0.688124i
$423$	− 9.21619i	− 0.448106i
$424$	−11.6163	−0.564136
$425$	0	0
$426$	15.6240	0.756984
$427$	39.1047i	1.89241i
$428$	10.8125i	0.522641i
$429$	−27.9671	−1.35026
$430$	0	0
$431$	−3.09549	−0.149105	−0.0745523	−	0.997217i	$-0.523753\pi$
$431$	−3.09549	−0.149105	−0.0745523	+	0.997217i	$0.523753\pi$
$432$	1.30233i	0.0626584i
$433$	18.2366i	0.876394i	0.898879	+	0.438197i	$0.144383\pi$
$433$	18.2366i	0.876394i	−0.898879	+	0.438197i	$0.855617\pi$
$434$	21.0657	1.01119
$435$	0	0
$436$	11.4153	0.546695
$437$	3.48736i	0.166823i
$438$	12.0573i	0.576122i
$439$	26.5254	1.26599	0.632994	−	0.774157i	$-0.281826\pi$
$439$	26.5254	1.26599	0.632994	+	0.774157i	$0.281826\pi$
$440$	0	0
$441$	17.6942	0.842579
$442$	− 4.42302i	− 0.210382i
$443$	− 35.5267i	− 1.68793i	−0.536401	−	0.843963i	$-0.680217\pi$
$443$	− 35.5267i	− 1.68793i	0.536401	−	0.843963i	$-0.319783\pi$
$444$	16.4400	0.780206
$445$	0	0
$446$	4.61226	0.218397
$447$	45.5125i	2.15267i
$448$	− 3.77447i	− 0.178327i
$449$	−16.1841	−0.763776	−0.381888	−	0.924209i	$-0.624726\pi$
$449$	−16.1841	−0.763776	−0.381888	+	0.924209i	$0.624726\pi$
$450$	0	0
$451$	5.62087	0.264676
$452$	− 10.4229i	− 0.490253i
$453$	− 42.9981i	− 2.02023i
$454$	6.28637	0.295034
$455$	0	0
$456$	−9.13004	−0.427553
$457$	23.7205i	1.10960i	0.831984	+	0.554799i	$0.187205\pi$
$457$	23.7205i	1.10960i	−0.831984	+	0.554799i	$0.812795\pi$
$458$	− 17.7370i	− 0.828794i
$459$	1.81350	0.0846468
$460$	0	0
$461$	−7.87297	−0.366681	−0.183340	−	0.983050i	$-0.558691\pi$
$461$	−7.87297	−0.366681	−0.183340	+	0.983050i	$0.558691\pi$
$462$	33.2338i	1.54617i
$463$	− 15.4230i	− 0.716769i	−0.933574	−	0.358384i	$-0.883328\pi$
$463$	− 15.4230i	− 0.716769i	0.933574	−	0.358384i	$-0.116672\pi$
$464$	−0.0492169	−0.00228484
$465$	0	0
$466$	2.56406	0.118778
$467$	− 17.0373i	− 0.788391i	−0.919027	−	0.394196i	$-0.871023\pi$
$467$	− 17.0373i	− 0.788391i	0.919027	−	0.394196i	$-0.128977\pi$
$468$	− 7.75567i	− 0.358506i
$469$	−57.8377	−2.67070
$470$	0	0
$471$	−23.6924	−1.09169
$472$	− 0.690074i	− 0.0317632i
$473$	10.1823i	0.468181i
$474$	−4.21443	−0.193575
$475$	0	0
$476$	−5.25595	−0.240906
$477$	28.3637i	1.29868i
$478$	− 9.44172i	− 0.431854i
$479$	25.0971	1.14672	0.573359	−	0.819304i	$-0.305640\pi$
$479$	25.0971	1.14672	0.573359	+	0.819304i	$0.305640\pi$
$480$	0	0
$481$	22.3850	1.02067
$482$	− 29.4346i	− 1.34071i
$483$	7.84542i	0.356979i
$484$	−3.24660	−0.147573
$485$	0	0
$486$	20.2677	0.919360
$487$	− 13.6876i	− 0.620244i	−0.950697	−	0.310122i	$-0.899630\pi$
$487$	− 13.6876i	− 0.620244i	0.950697	−	0.310122i	$-0.100370\pi$
$488$	− 10.3603i	− 0.468990i
$489$	−43.3901	−1.96217
$490$	0	0
$491$	−2.58009	−0.116438	−0.0582188	−	0.998304i	$-0.518542\pi$
$491$	−2.58009	−0.116438	−0.0582188	+	0.998304i	$0.518542\pi$
$492$	3.47389i	0.156615i
$493$	0.0685347i	0.00308665i
$494$	−12.4316	−0.559326
$495$	0	0
$496$	−5.58111	−0.250599
$497$	− 25.2801i	− 1.13397i
$498$	− 23.2524i	− 1.04197i
$499$	39.0539	1.74829	0.874146	−	0.485664i	$-0.161422\pi$
$499$	39.0539	1.74829	0.874146	+	0.485664i	$0.161422\pi$
$500$	0	0
$501$	−5.06570	−0.226319
$502$	− 6.00759i	− 0.268132i
$503$	− 11.9536i	− 0.532986i	−0.963837	−	0.266493i	$-0.914135\pi$
$503$	− 11.9536i	− 0.532986i	0.963837	−	0.266493i	$-0.0858648\pi$
$504$	−9.21619	−0.410522
$505$	0	0
$506$	−3.36317	−0.149511
$507$	6.79067i	0.301584i
$508$	3.54893i	0.157459i
$509$	4.87325	0.216003	0.108002	−	0.994151i	$-0.465555\pi$
$509$	4.87325	0.216003	0.108002	+	0.994151i	$0.465555\pi$
$510$	0	0
$511$	19.5092	0.863035
$512$	1.00000i	0.0441942i
$513$	− 5.09713i	− 0.225044i
$514$	13.6286	0.601130
$515$	0	0
$516$	−6.29298	−0.277033
$517$	− 14.2466i	− 0.626565i
$518$	− 26.6004i	− 1.16876i
$519$	−14.9162	−0.654748
$520$	0	0
$521$	−8.74163	−0.382978	−0.191489	−	0.981495i	$-0.561332\pi$
$521$	−8.74163	−0.382978	−0.191489	+	0.981495i	$0.561332\pi$
$522$	0.120174i	0.00525987i
$523$	21.0792i	0.921728i	0.887471	+	0.460864i	$0.152460\pi$
$523$	21.0792i	0.921728i	−0.887471	+	0.460864i	$0.847540\pi$
$524$	−4.05324	−0.177067
$525$	0	0
$526$	−9.16294	−0.399523
$527$	7.77170i	0.338540i
$528$	− 8.80489i	− 0.383183i
$529$	22.2061	0.965481
$530$	0	0
$531$	−1.68497	−0.0731213
$532$	14.7727i	0.640478i
$533$	4.73011i	0.204884i
$534$	33.8488	1.46478
$535$	0	0
$536$	15.3234	0.661870
$537$	− 18.6934i	− 0.806681i
$538$	− 28.1026i	− 1.21159i
$539$	27.3521	1.17814
$540$	0	0
$541$	32.8060	1.41044	0.705220	−	0.708988i	$-0.250848\pi$
$541$	32.8060	1.41044	0.705220	+	0.708988i	$0.250848\pi$
$542$	− 9.09860i	− 0.390818i
$543$	− 25.3404i	− 1.08746i
$544$	1.39250	0.0597030
$545$	0	0
$546$	−27.9671	−1.19688
$547$	− 5.47038i	− 0.233897i	−0.993138	−	0.116948i	$-0.962689\pi$
$547$	− 5.47038i	− 0.233897i	0.993138	−	0.116948i	$-0.0373112\pi$
$548$	3.56773i	0.152406i
$549$	−25.2970	−1.07965
$550$	0	0
$551$	0.192628	0.00820623
$552$	− 2.07855i	− 0.0884690i
$553$	6.81910i	0.289977i
$554$	−18.7890	−0.798269
$555$	0	0
$556$	−9.94602	−0.421805
$557$	− 5.19840i	− 0.220263i	−0.993917	−	0.110132i	$-0.964873\pi$
$557$	− 5.19840i	− 0.220263i	0.993917	−	0.110132i	$-0.0351273\pi$
$558$	13.6275i	0.576897i
$559$	−8.56865	−0.362415
$560$	0	0
$561$	−12.2608	−0.517652
$562$	7.88158i	0.332464i
$563$	5.01523i	0.211367i	0.994400	+	0.105683i	$0.0337030\pi$
$563$	5.01523i	0.211367i	−0.994400	+	0.105683i	$0.966297\pi$
$564$	8.80489	0.370753
$565$	0	0
$566$	−18.2998	−0.769200
$567$	− 39.1154i	− 1.64269i
$568$	6.69767i	0.281028i
$569$	41.8432	1.75416	0.877078	−	0.480347i	$-0.159489\pi$
$569$	41.8432	1.75416	0.877078	+	0.480347i	$0.159489\pi$
$570$	0	0
$571$	−44.5683	−1.86512	−0.932562	−	0.361011i	$-0.882432\pi$
$571$	−44.5683	−1.86512	−0.932562	+	0.361011i	$0.882432\pi$
$572$	− 11.9889i	− 0.501281i
$573$	− 19.0953i	− 0.797719i
$574$	5.62087	0.234611
$575$	0	0
$576$	2.44172	0.101738
$577$	− 25.1417i	− 1.04666i	−0.852129	−	0.523332i	$-0.824689\pi$
$577$	− 25.1417i	− 1.04666i	0.852129	−	0.523332i	$-0.175311\pi$
$578$	15.0609i	0.626453i
$579$	−40.9575	−1.70214
$580$	0	0
$581$	−37.6232	−1.56088
$582$	0.210344i	0.00871903i
$583$	43.8453i	1.81589i
$584$	−5.16872	−0.213883
$585$	0	0
$586$	−29.4990	−1.21859
$587$	21.3556i	0.881439i	0.897645	+	0.440719i	$0.145277\pi$
$587$	21.3556i	0.881439i	−0.897645	+	0.440719i	$0.854723\pi$
$588$	16.9045i	0.697130i
$589$	21.8436	0.900051
$590$	0	0
$591$	−11.0979	−0.456505
$592$	7.04746i	0.289649i
$593$	33.4430i	1.37334i	0.726970	+	0.686669i	$0.240928\pi$
$593$	33.4430i	1.37334i	−0.726970	+	0.686669i	$0.759072\pi$
$594$	4.91561	0.201690
$595$	0	0
$596$	−19.5103	−0.799171
$597$	− 33.8995i	− 1.38741i
$598$	− 2.83020i	− 0.115735i
$599$	−26.3174	−1.07530	−0.537650	−	0.843168i	$-0.680688\pi$
$599$	−26.3174	−1.07530	−0.537650	+	0.843168i	$0.680688\pi$
$600$	0	0
$601$	20.9964	0.856462	0.428231	−	0.903669i	$-0.359137\pi$
$601$	20.9964	0.856462	0.428231	+	0.903669i	$0.359137\pi$
$602$	10.1823i	0.414998i
$603$	− 37.4154i	− 1.52367i
$604$	18.4324	0.750003
$605$	0	0
$606$	−38.2068	−1.55205
$607$	9.32822i	0.378621i	0.981917	+	0.189310i	$0.0606253\pi$
$607$	9.32822i	0.378621i	−0.981917	+	0.189310i	$0.939375\pi$
$608$	− 3.91385i	− 0.158728i
$609$	0.433350	0.0175602
$610$	0	0
$611$	11.9889	0.485019
$612$	− 3.40010i	− 0.137441i
$613$	32.1036i	1.29665i	0.761362	+	0.648327i	$0.224531\pi$
$613$	32.1036i	1.29665i	−0.761362	+	0.648327i	$0.775469\pi$
$614$	−13.9131	−0.561488
$615$	0	0
$616$	−14.2466	−0.574012
$617$	− 31.9256i	− 1.28528i	−0.766169	−	0.642639i	$-0.777840\pi$
$617$	− 31.9256i	− 1.28528i	0.766169	−	0.642639i	$-0.222160\pi$
$618$	− 2.83706i	− 0.114123i
$619$	4.54407	0.182642	0.0913208	−	0.995822i	$-0.470891\pi$
$619$	4.54407	0.182642	0.0913208	+	0.995822i	$0.470891\pi$
$620$	0	0
$621$	1.16042	0.0465659
$622$	25.9636i	1.04105i
$623$	− 54.7685i	− 2.19425i
$624$	7.40955	0.296619
$625$	0	0
$626$	9.47389	0.378653
$627$	34.4610i	1.37624i
$628$	− 10.1564i	− 0.405286i
$629$	9.81360	0.391294
$630$	0	0
$631$	−26.9162	−1.07152	−0.535758	−	0.844371i	$-0.679974\pi$
$631$	−26.9162	−1.07152	−0.535758	+	0.844371i	$0.679974\pi$
$632$	− 1.80664i	− 0.0718642i
$633$	− 32.9755i	− 1.31066i
$634$	−24.5074	−0.973314
$635$	0	0
$636$	−27.0979	−1.07450
$637$	23.0175i	0.911987i
$638$	0.185768i	0.00735462i
$639$	16.3538	0.646947
$640$	0	0
$641$	35.7599	1.41243	0.706215	−	0.707998i	$-0.250401\pi$
$641$	35.7599	1.41243	0.706215	+	0.707998i	$0.250401\pi$
$642$	25.2228i	0.995465i
$643$	− 28.2255i	− 1.11311i	−0.830812	−	0.556553i	$-0.812124\pi$
$643$	− 28.2255i	− 1.11311i	0.830812	−	0.556553i	$-0.187876\pi$
$644$	−3.36317	−0.132527
$645$	0	0
$646$	−5.45005	−0.214429
$647$	1.27776i	0.0502337i	0.999685	+	0.0251169i	$0.00799579\pi$
$647$	1.27776i	0.0502337i	−0.999685	+	0.0251169i	$0.992004\pi$
$648$	10.3632i	0.407104i
$649$	−2.60466	−0.102242
$650$	0	0
$651$	49.1410	1.92599
$652$	− 18.6004i	− 0.728449i
$653$	27.4108i	1.07267i	0.844006	+	0.536334i	$0.180191\pi$
$653$	27.4108i	1.07267i	−0.844006	+	0.536334i	$0.819809\pi$
$654$	26.6291	1.04128
$655$	0	0
$656$	−1.48918	−0.0581428
$657$	12.6206i	0.492375i
$658$	− 14.2466i	− 0.555390i
$659$	41.2271	1.60598	0.802991	−	0.595992i	$-0.203241\pi$
$659$	41.2271	1.60598	0.802991	+	0.595992i	$0.203241\pi$
$660$	0	0
$661$	−26.5326	−1.03200	−0.515999	−	0.856589i	$-0.672579\pi$
$661$	−26.5326	−1.03200	−0.515999	+	0.856589i	$0.672579\pi$
$662$	− 9.91035i	− 0.385177i
$663$	− 10.3178i	− 0.400710i
$664$	9.96783	0.386827
$665$	0	0
$666$	17.2079	0.666793
$667$	0.0438538i	0.00169803i
$668$	− 2.17156i	− 0.0840201i
$669$	10.7592	0.415976
$670$	0	0
$671$	−39.1047	−1.50962
$672$	− 8.80489i	− 0.339656i
$673$	7.93731i	0.305961i	0.988229	+	0.152980i	$0.0488871\pi$
$673$	7.93731i	0.305961i	−0.988229	+	0.152980i	$0.951113\pi$
$674$	9.93566	0.382707
$675$	0	0
$676$	−2.91102	−0.111962
$677$	29.5008i	1.13381i	0.823783	+	0.566905i	$0.191859\pi$
$677$	29.5008i	1.13381i	−0.823783	+	0.566905i	$0.808141\pi$
$678$	− 24.3141i	− 0.933776i
$679$	0.340344	0.0130612
$680$	0	0
$681$	14.6645	0.561946
$682$	21.0657i	0.806647i
$683$	− 37.7488i	− 1.44442i	−0.691676	−	0.722208i	$-0.743127\pi$
$683$	− 37.7488i	− 1.44442i	0.691676	−	0.722208i	$-0.256873\pi$
$684$	−9.55653	−0.365403
$685$	0	0
$686$	0.930796	0.0355380
$687$	− 41.3759i	− 1.57859i
$688$	− 2.69767i	− 0.102848i
$689$	−36.8970	−1.40566
$690$	0	0
$691$	17.6564	0.671681	0.335840	−	0.941919i	$-0.390980\pi$
$691$	17.6564	0.671681	0.335840	+	0.941919i	$0.390980\pi$
$692$	− 6.39425i	− 0.243073i
$693$	34.7862i	1.32142i
$694$	19.6351	0.745337
$695$	0	0
$696$	−0.114811	−0.00435189
$697$	2.07369i	0.0785465i
$698$	− 1.48432i	− 0.0561823i
$699$	5.98131	0.226234
$700$	0	0
$701$	−16.8372	−0.635931	−0.317965	−	0.948102i	$-0.603000\pi$
$701$	−16.8372	−0.635931	−0.317965	+	0.948102i	$0.603000\pi$
$702$	4.13662i	0.156126i
$703$	− 27.5827i	− 1.04030i
$704$	3.77447	0.142256
$705$	0	0
$706$	10.2788	0.386847
$707$	61.8200i	2.32498i
$708$	− 1.60977i	− 0.0604989i
$709$	28.6292	1.07519	0.537596	−	0.843203i	$-0.319333\pi$
$709$	28.6292	1.07519	0.537596	+	0.843203i	$0.319333\pi$
$710$	0	0
$711$	−4.41130	−0.165437
$712$	14.5103i	0.543795i
$713$	4.97294i	0.186238i
$714$	−12.2608	−0.458849
$715$	0	0
$716$	8.01348	0.299478
$717$	− 22.0252i	− 0.822545i
$718$	10.3725i	0.387099i
$719$	8.24734	0.307574	0.153787	−	0.988104i	$-0.450853\pi$
$719$	8.24734	0.307574	0.153787	+	0.988104i	$0.450853\pi$
$720$	0	0
$721$	−4.59045	−0.170957
$722$	− 3.68175i	− 0.137020i
$723$	− 68.6636i	− 2.55363i
$724$	10.8629	0.403716
$725$	0	0
$726$	−7.57351	−0.281079
$727$	− 2.11383i	− 0.0783977i	−0.999231	−	0.0391988i	$-0.987519\pi$
$727$	− 2.11383i	− 0.0783977i	0.999231	−	0.0391988i	$-0.0124806\pi$
$728$	− 11.9889i	− 0.444338i
$729$	16.1899	0.599626
$730$	0	0
$731$	−3.75651	−0.138939
$732$	− 24.1681i	− 0.893277i
$733$	− 45.1552i	− 1.66785i	−0.551881	−	0.833923i	$-0.686090\pi$
$733$	− 45.1552i	− 1.66785i	0.551881	−	0.833923i	$-0.313910\pi$
$734$	−26.0743	−0.962420
$735$	0	0
$736$	0.891031	0.0328438
$737$	− 57.8377i	− 2.13048i
$738$	3.63616i	0.133849i
$739$	−29.9719	−1.10253	−0.551267	−	0.834329i	$-0.685856\pi$
$739$	−29.9719	−1.10253	−0.551267	+	0.834329i	$0.685856\pi$
$740$	0	0
$741$	−28.9999	−1.06534
$742$	43.8453i	1.60961i
$743$	− 17.6562i	− 0.647741i	−0.946101	−	0.323871i	$-0.895016\pi$
$743$	− 17.6562i	− 0.647741i	0.946101	−	0.323871i	$-0.104984\pi$
$744$	−13.0193	−0.477311
$745$	0	0
$746$	9.45593	0.346206
$747$	− 24.3386i	− 0.890504i
$748$	− 5.25595i	− 0.192177i
$749$	40.8113	1.49121
$750$	0	0
$751$	30.1342	1.09961	0.549806	−	0.835293i	$-0.314702\pi$
$751$	30.1342	1.09961	0.549806	+	0.835293i	$0.314702\pi$
$752$	3.77447i	0.137641i
$753$	− 14.0142i	− 0.510706i
$754$	−0.156329	−0.00569315
$755$	0	0
$756$	4.91561	0.178779
$757$	− 4.90706i	− 0.178350i	−0.996016	−	0.0891750i	$-0.971577\pi$
$757$	− 4.90706i	− 0.178350i	0.996016	−	0.0891750i	$-0.0284231\pi$
$758$	35.3258i	1.28309i
$759$	−7.84542	−0.284771
$760$	0	0
$761$	13.2835	0.481528	0.240764	−	0.970584i	$-0.422602\pi$
$761$	13.2835	0.481528	0.240764	+	0.970584i	$0.422602\pi$
$762$	8.27877i	0.299908i
$763$	− 43.0868i	− 1.55985i
$764$	8.18577	0.296151
$765$	0	0
$766$	29.8173	1.07734
$767$	− 2.19189i	− 0.0791447i
$768$	2.33275i	0.0841758i
$769$	28.5977	1.03126	0.515629	−	0.856812i	$-0.327558\pi$
$769$	28.5977	1.03126	0.515629	+	0.856812i	$0.327558\pi$
$770$	0	0
$771$	31.7920	1.14496
$772$	− 17.5576i	− 0.631912i
$773$	45.0625i	1.62079i	0.585886	+	0.810393i	$0.300747\pi$
$773$	45.0625i	1.62079i	−0.585886	+	0.810393i	$0.699253\pi$
$774$	−6.58695	−0.236763
$775$	0	0
$776$	−0.0901699	−0.00323691
$777$	− 62.0521i	− 2.22611i
$778$	21.3767i	0.766394i
$779$	5.82844	0.208826
$780$	0	0
$781$	25.2801	0.904594
$782$	− 1.24076i	− 0.0443695i
$783$	− 0.0640968i	− 0.00229063i
$784$	−7.24660	−0.258807
$785$	0	0
$786$	−9.45519	−0.337256
$787$	25.7669i	0.918490i	0.888310	+	0.459245i	$0.151880\pi$
$787$	25.7669i	0.918490i	−0.888310	+	0.459245i	$0.848120\pi$
$788$	− 4.75742i	− 0.169476i
$789$	−21.3749	−0.760965
$790$	0	0
$791$	−39.3410	−1.39880
$792$	− 9.21619i	− 0.327483i
$793$	− 32.9077i	− 1.16859i
$794$	−21.0201	−0.745974
$795$	0	0
$796$	14.5320	0.515073
$797$	50.4446i	1.78684i	0.449222	+	0.893420i	$0.351701\pi$
$797$	50.4446i	1.78684i	−0.449222	+	0.893420i	$0.648299\pi$
$798$	34.4610i	1.21991i
$799$	5.25595	0.185942
$800$	0	0
$801$	35.4299	1.25186
$802$	16.7820i	0.592593i
$803$	19.5092i	0.688464i
$804$	35.7457	1.26065
$805$	0	0
$806$	−17.7274	−0.624419
$807$	− 65.5564i	− 2.30769i
$808$	− 16.3785i	− 0.576192i
$809$	3.61706	0.127169	0.0635844	−	0.997976i	$-0.479747\pi$
$809$	3.61706	0.127169	0.0635844	+	0.997976i	$0.479747\pi$
$810$	0	0
$811$	50.4347	1.77100	0.885502	−	0.464636i	$-0.153815\pi$
$811$	50.4347	1.77100	0.885502	+	0.464636i	$0.153815\pi$
$812$	0.185768i	0.00651917i
$813$	− 21.2248i	− 0.744385i
$814$	26.6004	0.932344
$815$	0	0
$816$	3.24836	0.113715
$817$	10.5583i	0.369388i
$818$	− 36.3607i	− 1.27132i
$819$	−29.2735	−1.02290
$820$	0	0
$821$	−11.2846	−0.393836	−0.196918	−	0.980420i	$-0.563093\pi$
$821$	−11.2846	−0.393836	−0.196918	+	0.980420i	$0.563093\pi$
$822$	8.32263i	0.290285i
$823$	− 1.07680i	− 0.0375348i	−0.999824	−	0.0187674i	$-0.994026\pi$
$823$	− 1.07680i	− 0.0375348i	0.999824	−	0.0187674i	$-0.00597421\pi$
$824$	1.21619	0.0423678
$825$	0	0
$826$	−2.60466	−0.0906278
$827$	− 29.3800i	− 1.02164i	−0.859687	−	0.510821i	$-0.829341\pi$
$827$	− 29.3800i	− 1.02164i	0.859687	−	0.510821i	$-0.170659\pi$
$828$	− 2.17565i	− 0.0756089i
$829$	−21.0798	−0.732132	−0.366066	−	0.930589i	$-0.619296\pi$
$829$	−21.0798	−0.732132	−0.366066	+	0.930589i	$0.619296\pi$
$830$	0	0
$831$	−43.8301	−1.52045
$832$	3.17632i	0.110119i
$833$	10.0909i	0.349629i
$834$	−23.2016	−0.803405
$835$	0	0
$836$	−14.7727	−0.510925
$837$	− 7.26845i	− 0.251234i
$838$	34.5389i	1.19312i
$839$	3.41379	0.117857	0.0589285	−	0.998262i	$-0.481232\pi$
$839$	3.41379	0.117857	0.0589285	+	0.998262i	$0.481232\pi$
$840$	0	0
$841$	−28.9976	−0.999916
$842$	9.73403i	0.335457i
$843$	18.3857i	0.633239i
$844$	14.1359	0.486577
$845$	0	0
$846$	9.21619	0.316859
$847$	12.2542i	0.421059i
$848$	− 11.6163i	− 0.398905i
$849$	−42.6889	−1.46508
$850$	0	0
$851$	6.27951	0.215259
$852$	15.6240i	0.535269i
$853$	40.5963i	1.38999i	0.719014	+	0.694995i	$0.244594\pi$
$853$	40.5963i	1.38999i	−0.719014	+	0.694995i	$0.755406\pi$
$854$	−39.1047	−1.33814
$855$	0	0
$856$	−10.8125	−0.369563
$857$	− 43.0463i	− 1.47043i	−0.677833	−	0.735216i	$-0.737081\pi$
$857$	− 43.0463i	− 1.47043i	0.677833	−	0.735216i	$-0.262919\pi$
$858$	− 27.9671i	− 0.954781i
$859$	−7.63508	−0.260506	−0.130253	−	0.991481i	$-0.541579\pi$
$859$	−7.63508	−0.260506	−0.130253	+	0.991481i	$0.541579\pi$
$860$	0	0
$861$	13.1121	0.446858
$862$	− 3.09549i	− 0.105433i
$863$	− 6.47451i	− 0.220395i	−0.993910	−	0.110197i	$-0.964852\pi$
$863$	− 6.47451i	− 0.220395i	0.993910	−	0.110197i	$-0.0351483\pi$
$864$	−1.30233	−0.0443062
$865$	0	0
$866$	−18.2366	−0.619704
$867$	35.1334i	1.19319i
$868$	21.0657i	0.715016i
$869$	−6.81910	−0.231322
$870$	0	0
$871$	48.6720	1.64919
$872$	11.4153i	0.386572i
$873$	0.220170i	0.00745161i
$874$	−3.48736	−0.117962
$875$	0	0
$876$	−12.0573	−0.407379
$877$	− 7.87108i	− 0.265788i	−0.991130	−	0.132894i	$-0.957573\pi$
$877$	− 7.87108i	− 0.265788i	0.991130	−	0.132894i	$-0.0424269\pi$
$878$	26.5254i	0.895188i
$879$	−68.8137	−2.32103
$880$	0	0
$881$	19.2652	0.649061	0.324530	−	0.945875i	$-0.394794\pi$
$881$	19.2652	0.649061	0.324530	+	0.945875i	$0.394794\pi$
$882$	17.6942i	0.595793i
$883$	− 6.27395i	− 0.211135i	−0.994412	−	0.105568i	$-0.966334\pi$
$883$	− 6.27395i	− 0.211135i	0.994412	−	0.105568i	$-0.0336659\pi$
$884$	4.42302	0.148762
$885$	0	0
$886$	35.5267	1.19354
$887$	23.2728i	0.781424i	0.920513	+	0.390712i	$0.127771\pi$
$887$	23.2728i	0.781424i	−0.920513	+	0.390712i	$0.872229\pi$
$888$	16.4400i	0.551689i
$889$	13.3953	0.449265
$890$	0	0
$891$	39.1154	1.31042
$892$	4.61226i	0.154430i
$893$	− 14.7727i	− 0.494350i
$894$	−45.5125	−1.52217
$895$	0	0
$896$	3.77447	0.126096
$897$	− 6.60214i	− 0.220439i
$898$	− 16.1841i	− 0.540071i
$899$	0.274685	0.00916126
$900$	0	0
$901$	−16.1757	−0.538890
$902$	5.62087i	0.187155i
$903$	23.7527i	0.790439i
$904$	10.4229	0.346661
$905$	0	0
$906$	42.9981	1.42852
$907$	− 17.6544i	− 0.586206i	−0.956081	−	0.293103i	$-0.905312\pi$
$907$	− 17.6544i	− 0.586206i	0.956081	−	0.293103i	$-0.0946879\pi$
$908$	6.28637i	0.208620i
$909$	−39.9916	−1.32644
$910$	0	0
$911$	−34.9519	−1.15801	−0.579003	−	0.815325i	$-0.696558\pi$
$911$	−34.9519	−1.15801	−0.579003	+	0.815325i	$0.696558\pi$
$912$	− 9.13004i	− 0.302326i
$913$	− 37.6232i	− 1.24515i
$914$	−23.7205	−0.784604
$915$	0	0
$916$	17.7370	0.586046
$917$	15.2988i	0.505212i
$918$	1.81350i	0.0598544i
$919$	−36.3226	−1.19817	−0.599086	−	0.800684i	$-0.704469\pi$
$919$	−36.3226	−1.19817	−0.599086	+	0.800684i	$0.704469\pi$
$920$	0	0
$921$	−32.4558	−1.06946
$922$	− 7.87297i	− 0.259282i
$923$	21.2739i	0.700239i
$924$	−33.2338	−1.09331
$925$	0	0
$926$	15.4230	0.506832
$927$	− 2.96958i	− 0.0975339i
$928$	− 0.0492169i	− 0.00161563i
$929$	−19.0543	−0.625152	−0.312576	−	0.949893i	$-0.601192\pi$
$929$	−19.0543	−0.625152	−0.312576	+	0.949893i	$0.601192\pi$
$930$	0	0
$931$	28.3621	0.929532
$932$	2.56406i	0.0839885i
$933$	60.5665i	1.98286i
$934$	17.0373	0.557477
$935$	0	0
$936$	7.75567	0.253502
$937$	59.5279i	1.94469i	0.233548	+	0.972345i	$0.424966\pi$
$937$	59.5279i	1.94469i	−0.233548	+	0.972345i	$0.575034\pi$
$938$	− 57.8377i	− 1.88847i
$939$	22.1002	0.721213
$940$	0	0
$941$	15.1539	0.494005	0.247002	−	0.969015i	$-0.420554\pi$
$941$	15.1539	0.494005	0.247002	+	0.969015i	$0.420554\pi$
$942$	− 23.6924i	− 0.771940i
$943$	1.32691i	0.0432101i
$944$	0.690074	0.0224600
$945$	0	0
$946$	−10.1823	−0.331054
$947$	2.52089i	0.0819180i	0.999161	+	0.0409590i	$0.0130413\pi$
$947$	2.52089i	0.0819180i	−0.999161	+	0.0409590i	$0.986959\pi$
$948$	− 4.21443i	− 0.136878i
$949$	−16.4175	−0.532934
$950$	0	0
$951$	−57.1697	−1.85385
$952$	− 5.25595i	− 0.170346i
$953$	− 31.3684i	− 1.01612i	−0.861321	−	0.508061i	$-0.830363\pi$
$953$	− 31.3684i	− 1.01612i	0.861321	−	0.508061i	$-0.169637\pi$
$954$	−28.3637	−0.918308
$955$	0	0
$956$	9.44172	0.305367
$957$	0.433350i	0.0140082i
$958$	25.0971i	0.810852i
$959$	13.4663	0.434849
$960$	0	0
$961$	0.148734	0.00479787
$962$	22.3850i	0.721720i
$963$	26.4010i	0.850761i
$964$	29.4346	0.948026
$965$	0	0
$966$	−7.84542	−0.252422
$967$	49.1061i	1.57915i	0.613656	+	0.789573i	$0.289698\pi$
$967$	49.1061i	1.57915i	−0.613656	+	0.789573i	$0.710302\pi$
$968$	− 3.24660i	− 0.104350i
$969$	−12.7136	−0.408419
$970$	0	0
$971$	24.9840	0.801776	0.400888	−	0.916127i	$-0.368702\pi$
$971$	24.9840	0.801776	0.400888	+	0.916127i	$0.368702\pi$
$972$	20.2677i	0.650086i
$973$	37.5409i	1.20351i
$974$	13.6876	0.438579
$975$	0	0
$976$	10.3603	0.331626
$977$	15.5861i	0.498645i	0.968421	+	0.249323i	$0.0802079\pi$
$977$	15.5861i	0.498645i	−0.968421	+	0.249323i	$0.919792\pi$
$978$	− 43.3901i	− 1.38746i
$979$	54.7685	1.75041
$980$	0	0
$981$	27.8730	0.889917
$982$	− 2.58009i	− 0.0823339i
$983$	13.5589i	0.432462i	0.976342	+	0.216231i	$0.0693764\pi$
$983$	13.5589i	0.432462i	−0.976342	+	0.216231i	$0.930624\pi$
$984$	−3.47389	−0.110744
$985$	0	0
$986$	−0.0685347	−0.00218259
$987$	− 33.2338i	− 1.05784i
$988$	− 12.4316i	− 0.395503i
$989$	−2.40371	−0.0764334
$990$	0	0
$991$	31.0401	0.986023	0.493011	−	0.870023i	$-0.335896\pi$
$991$	31.0401	0.986023	0.493011	+	0.870023i	$0.335896\pi$
$992$	− 5.58111i	− 0.177200i
$993$	− 23.1184i	− 0.733639i
$994$	25.2801	0.801837
$995$	0	0
$996$	23.2524	0.736782
$997$	1.54918i	0.0490630i	0.999699	+	0.0245315i	$0.00780940\pi$
$997$	1.54918i	0.0490630i	−0.999699	+	0.0245315i	$0.992191\pi$
$998$	39.0539i	1.23623i
$999$	−9.17813	−0.290383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1250.2.b.e.1249.8		8
5.2	odd	4		1250.2.a.f.1.4		4
5.3	odd	4		1250.2.a.l.1.1		4
5.4	even	2	inner	1250.2.b.e.1249.1		8
20.3	even	4		10000.2.a.t.1.4		4
20.7	even	4		10000.2.a.x.1.1		4
25.3	odd	20		50.2.d.b.41.1	yes	8
25.4	even	10		250.2.e.c.49.2		16
25.6	even	5		250.2.e.c.199.2		16
25.8	odd	20		50.2.d.b.11.1	✓	8
25.17	odd	20		250.2.d.d.51.2		8
25.19	even	10		250.2.e.c.199.3		16
25.21	even	5		250.2.e.c.49.3		16
25.22	odd	20		250.2.d.d.201.2		8
75.8	even	20		450.2.h.e.361.2		8
75.53	even	20		450.2.h.e.91.2		8
100.3	even	20		400.2.u.d.241.2		8
100.83	even	20		400.2.u.d.161.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.b.11.1	✓	8	25.8	odd	20
50.2.d.b.41.1	yes	8	25.3	odd	20
250.2.d.d.51.2		8	25.17	odd	20
250.2.d.d.201.2		8	25.22	odd	20
250.2.e.c.49.2		16	25.4	even	10
250.2.e.c.49.3		16	25.21	even	5
250.2.e.c.199.2		16	25.6	even	5
250.2.e.c.199.3		16	25.19	even	10
400.2.u.d.161.2		8	100.83	even	20
400.2.u.d.241.2		8	100.3	even	20
450.2.h.e.91.2		8	75.53	even	20
450.2.h.e.361.2		8	75.8	even	20
1250.2.a.f.1.4		4	5.2	odd	4
1250.2.a.l.1.1		4	5.3	odd	4
1250.2.b.e.1249.1		8	5.4	even	2	inner
1250.2.b.e.1249.8		8	1.1	even	1	trivial
10000.2.a.t.1.4		4	20.3	even	4
10000.2.a.x.1.1		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1250.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.33275i q^{3} -1.00000 q^{4} -2.33275 q^{6} +3.77447i q^{7} -1.00000i q^{8} -2.44172 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +2.33275i q^{3} -1.00000 q^{4} -2.33275 q^{6} +3.77447i q^{7} -1.00000i q^{8} -2.44172 q^{9} -3.77447 q^{11} -2.33275i q^{12} -3.17632i q^{13} -3.77447 q^{14} +1.00000 q^{16} -1.39250i q^{17} -2.44172i q^{18} -3.91385 q^{19} -8.80489 q^{21} -3.77447i q^{22} -0.891031i q^{23} +2.33275 q^{24} +3.17632 q^{26} +1.30233i q^{27} -3.77447i q^{28} -0.0492169 q^{29} -5.58111 q^{31} +1.00000i q^{32} -8.80489i q^{33} +1.39250 q^{34} +2.44172 q^{36} +7.04746i q^{37} -3.91385i q^{38} +7.40955 q^{39} -1.48918 q^{41} -8.80489i q^{42} -2.69767i q^{43} +3.77447 q^{44} +0.891031 q^{46} +3.77447i q^{47} +2.33275i q^{48} -7.24660 q^{49} +3.24836 q^{51} +3.17632i q^{52} -11.6163i q^{53} -1.30233 q^{54} +3.77447 q^{56} -9.13004i q^{57} -0.0492169i q^{58} +0.690074 q^{59} +10.3603 q^{61} -5.58111i q^{62} -9.21619i q^{63} -1.00000 q^{64} +8.80489 q^{66} +15.3234i q^{67} +1.39250i q^{68} +2.07855 q^{69} -6.69767 q^{71} +2.44172i q^{72} -5.16872i q^{73} -7.04746 q^{74} +3.91385 q^{76} -14.2466i q^{77} +7.40955i q^{78} +1.80664 q^{79} -10.3632 q^{81} -1.48918i q^{82} +9.96783i q^{83} +8.80489 q^{84} +2.69767 q^{86} -0.114811i q^{87} +3.77447i q^{88} -14.5103 q^{89} +11.9889 q^{91} +0.891031i q^{92} -13.0193i q^{93} -3.77447 q^{94} -2.33275 q^{96} -0.0901699i q^{97} -7.24660i q^{98} +9.21619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9}+O(q^{10}) \) 8 * q - 8 * q^4 + 2 * q^6 - 14 * q^9	\( 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} + 10 q^{19} - 14 q^{21} - 2 q^{24} + 22 q^{26} - 30 q^{29} - 24 q^{31} - 24 q^{34} + 14 q^{36} + 22 q^{39} + 26 q^{41} + 4 q^{44} - 8 q^{46} + 4 q^{49} + 26 q^{51} - 20 q^{54} + 4 q^{56} + 16 q^{61} - 8 q^{64} + 14 q^{66} + 62 q^{69} - 44 q^{71} - 24 q^{74} - 10 q^{76} + 20 q^{79} - 32 q^{81} + 14 q^{84} + 12 q^{86} + 10 q^{89} - 24 q^{91} - 4 q^{94} + 2 q^{96} + 42 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 + 2 * q^6 - 14 * q^9 - 4 * q^11 - 4 * q^14 + 8 * q^16 + 10 * q^19 - 14 * q^21 - 2 * q^24 + 22 * q^26 - 30 * q^29 - 24 * q^31 - 24 * q^34 + 14 * q^36 + 22 * q^39 + 26 * q^41 + 4 * q^44 - 8 * q^46 + 4 * q^49 + 26 * q^51 - 20 * q^54 + 4 * q^56 + 16 * q^61 - 8 * q^64 + 14 * q^66 + 62 * q^69 - 44 * q^71 - 24 * q^74 - 10 * q^76 + 20 * q^79 - 32 * q^81 + 14 * q^84 + 12 * q^86 + 10 * q^89 - 24 * q^91 - 4 * q^94 + 2 * q^96 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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