LMFDB - Embedded newform 8619.2.a.bk.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$68.8230615021$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 17x^{9} + 100x^{7} - 7x^{6} - 241x^{5} + 43x^{4} + 215x^{3} - 48x^{2} - 54x + 12 $ x^11 - 17x^9 + 100x^7 - 7x^6 - 241x^5 + 43x^4 + 215x^3 - 48x^2 - 54x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 663)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.11755$ of defining polynomial
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.11755 q^{2} +1.00000 q^{3} -0.751078 q^{4} +0.988234 q^{5} -1.11755 q^{6} -3.28484 q^{7} +3.07447 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.11755 q^{2} +1.00000 q^{3} -0.751078 q^{4} +0.988234 q^{5} -1.11755 q^{6} -3.28484 q^{7} +3.07447 q^{8} +1.00000 q^{9} -1.10440 q^{10} -3.11937 q^{11} -0.751078 q^{12} +3.67098 q^{14} +0.988234 q^{15} -1.93373 q^{16} -1.00000 q^{17} -1.11755 q^{18} -8.03817 q^{19} -0.742241 q^{20} -3.28484 q^{21} +3.48605 q^{22} -5.92818 q^{23} +3.07447 q^{24} -4.02339 q^{25} +1.00000 q^{27} +2.46717 q^{28} -2.33320 q^{29} -1.10440 q^{30} -3.54004 q^{31} -3.98791 q^{32} -3.11937 q^{33} +1.11755 q^{34} -3.24619 q^{35} -0.751078 q^{36} -0.954614 q^{37} +8.98307 q^{38} +3.03830 q^{40} -8.58809 q^{41} +3.67098 q^{42} -0.108290 q^{43} +2.34289 q^{44} +0.988234 q^{45} +6.62505 q^{46} +3.11492 q^{47} -1.93373 q^{48} +3.79017 q^{49} +4.49635 q^{50} -1.00000 q^{51} +9.23496 q^{53} -1.11755 q^{54} -3.08267 q^{55} -10.0991 q^{56} -8.03817 q^{57} +2.60748 q^{58} -5.61861 q^{59} -0.742241 q^{60} +7.77000 q^{61} +3.95618 q^{62} -3.28484 q^{63} +8.32414 q^{64} +3.48605 q^{66} +12.7075 q^{67} +0.751078 q^{68} -5.92818 q^{69} +3.62779 q^{70} -11.4920 q^{71} +3.07447 q^{72} +6.55037 q^{73} +1.06683 q^{74} -4.02339 q^{75} +6.03729 q^{76} +10.2466 q^{77} -3.05570 q^{79} -1.91097 q^{80} +1.00000 q^{81} +9.59763 q^{82} +6.02478 q^{83} +2.46717 q^{84} -0.988234 q^{85} +0.121020 q^{86} -2.33320 q^{87} -9.59041 q^{88} -7.24068 q^{89} -1.10440 q^{90} +4.45252 q^{92} -3.54004 q^{93} -3.48109 q^{94} -7.94359 q^{95} -3.98791 q^{96} +4.98862 q^{97} -4.23571 q^{98} -3.11937 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{3} + 12 q^{4} - 3 q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^3 + 12 * q^4 - 3 * q^7 + 11 * q^9	$ 11 q + 11 q^{3} + 12 q^{4} - 3 q^{7} + 11 q^{9} - 9 q^{10} - 2 q^{11} + 12 q^{12} + 21 q^{14} + 18 q^{16} - 11 q^{17} + 21 q^{20} - 3 q^{21} + 3 q^{22} + 18 q^{23} + 7 q^{25} + 11 q^{27} - 19 q^{28} + 15 q^{29} - 9 q^{30} + 13 q^{31} - 35 q^{32} - 2 q^{33} + 15 q^{35} + 12 q^{36} + 18 q^{37} + 5 q^{38} + 12 q^{40} + 21 q^{42} - 11 q^{43} - 3 q^{44} + 29 q^{46} - 11 q^{47} + 18 q^{48} + 28 q^{49} + 2 q^{50} - 11 q^{51} + 19 q^{53} + 13 q^{55} + 43 q^{56} - 68 q^{58} + 19 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} + 20 q^{64} + 3 q^{66} - 2 q^{67} - 12 q^{68} + 18 q^{69} + 58 q^{70} - 11 q^{71} - 30 q^{73} + 35 q^{74} + 7 q^{75} + 10 q^{76} + 20 q^{77} - 14 q^{79} + 43 q^{80} + 11 q^{81} - 35 q^{82} - 5 q^{83} - 19 q^{84} + 11 q^{86} + 15 q^{87} - 5 q^{88} - 3 q^{89} - 9 q^{90} + 54 q^{92} + 13 q^{93} + 55 q^{94} + 26 q^{95} - 35 q^{96} - 31 q^{97} + 54 q^{98} - 2 q^{99}+O(q^{100}) $ 11 * q + 11 * q^3 + 12 * q^4 - 3 * q^7 + 11 * q^9 - 9 * q^10 - 2 * q^11 + 12 * q^12 + 21 * q^14 + 18 * q^16 - 11 * q^17 + 21 * q^20 - 3 * q^21 + 3 * q^22 + 18 * q^23 + 7 * q^25 + 11 * q^27 - 19 * q^28 + 15 * q^29 - 9 * q^30 + 13 * q^31 - 35 * q^32 - 2 * q^33 + 15 * q^35 + 12 * q^36 + 18 * q^37 + 5 * q^38 + 12 * q^40 + 21 * q^42 - 11 * q^43 - 3 * q^44 + 29 * q^46 - 11 * q^47 + 18 * q^48 + 28 * q^49 + 2 * q^50 - 11 * q^51 + 19 * q^53 + 13 * q^55 + 43 * q^56 - 68 * q^58 + 19 * q^59 + 21 * q^60 + 18 * q^61 + 6 * q^62 - 3 * q^63 + 20 * q^64 + 3 * q^66 - 2 * q^67 - 12 * q^68 + 18 * q^69 + 58 * q^70 - 11 * q^71 - 30 * q^73 + 35 * q^74 + 7 * q^75 + 10 * q^76 + 20 * q^77 - 14 * q^79 + 43 * q^80 + 11 * q^81 - 35 * q^82 - 5 * q^83 - 19 * q^84 + 11 * q^86 + 15 * q^87 - 5 * q^88 - 3 * q^89 - 9 * q^90 + 54 * q^92 + 13 * q^93 + 55 * q^94 + 26 * q^95 - 35 * q^96 - 31 * q^97 + 54 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.11755	−0.790228	−0.395114	−	0.918632i	$-0.629295\pi$
$2$	−1.11755	−0.790228	−0.395114	+	0.918632i	$0.629295\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.751078	−0.375539
$5$	0.988234	0.441952	0.220976	−	0.975279i	$-0.429076\pi$
$5$	0.988234	0.441952	0.220976	+	0.975279i	$0.429076\pi$
$6$	−1.11755	−0.456239
$7$	−3.28484	−1.24155	−0.620776	−	0.783988i	$-0.713183\pi$
$7$	−3.28484	−1.24155	−0.620776	+	0.783988i	$0.713183\pi$
$8$	3.07447	1.08699
$9$	1.00000	0.333333
$10$	−1.10440	−0.349243
$11$	−3.11937	−0.940525	−0.470262	−	0.882527i	$-0.655841\pi$
$11$	−3.11937	−0.940525	−0.470262	+	0.882527i	$0.655841\pi$
$12$	−0.751078	−0.216818
$13$	0	0
$13$	0	0
$14$	3.67098	0.981110
$15$	0.988234	0.255161
$16$	−1.93373	−0.483431
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.11755	−0.263409
$19$	−8.03817	−1.84408	−0.922041	−	0.387092i	$-0.873480\pi$
$19$	−8.03817	−1.84408	−0.922041	+	0.387092i	$0.873480\pi$
$20$	−0.742241	−0.165970
$21$	−3.28484	−0.716811
$22$	3.48605	0.743229
$23$	−5.92818	−1.23611	−0.618055	−	0.786135i	$-0.712079\pi$
$23$	−5.92818	−1.23611	−0.618055	+	0.786135i	$0.712079\pi$
$24$	3.07447	0.627574
$25$	−4.02339	−0.804679
$26$	0	0
$27$	1.00000	0.192450
$28$	2.46717	0.466252
$29$	−2.33320	−0.433265	−0.216632	−	0.976253i	$-0.569507\pi$
$29$	−2.33320	−0.433265	−0.216632	+	0.976253i	$0.569507\pi$
$30$	−1.10440	−0.201635
$31$	−3.54004	−0.635810	−0.317905	−	0.948123i	$-0.602979\pi$
$31$	−3.54004	−0.635810	−0.317905	+	0.948123i	$0.602979\pi$
$32$	−3.98791	−0.704969
$33$	−3.11937	−0.543012
$34$	1.11755	0.191659
$35$	−3.24619	−0.548706
$36$	−0.751078	−0.125180
$37$	−0.954614	−0.156938	−0.0784688	−	0.996917i	$-0.525003\pi$
$37$	−0.954614	−0.156938	−0.0784688	+	0.996917i	$0.525003\pi$
$38$	8.98307	1.45725
$39$	0	0
$40$	3.03830	0.480397
$41$	−8.58809	−1.34123	−0.670617	−	0.741804i	$-0.733971\pi$
$41$	−8.58809	−1.34123	−0.670617	+	0.741804i	$0.733971\pi$
$42$	3.67098	0.566444
$43$	−0.108290	−0.0165141	−0.00825705	−	0.999966i	$-0.502628\pi$
$43$	−0.108290	−0.0165141	−0.00825705	+	0.999966i	$0.502628\pi$
$44$	2.34289	0.353204
$45$	0.988234	0.147317
$46$	6.62505	0.976810
$47$	3.11492	0.454358	0.227179	−	0.973853i	$-0.427050\pi$
$47$	3.11492	0.454358	0.227179	+	0.973853i	$0.427050\pi$
$48$	−1.93373	−0.279109
$49$	3.79017	0.541453
$50$	4.49635	0.635880
$51$	−1.00000	−0.140028
$52$	0	0
$53$	9.23496	1.26852	0.634260	−	0.773120i	$-0.281305\pi$
$53$	9.23496	1.26852	0.634260	+	0.773120i	$0.281305\pi$
$54$	−1.11755	−0.152080
$55$	−3.08267	−0.415667
$56$	−10.0991	−1.34956
$57$	−8.03817	−1.06468
$58$	2.60748	0.342378
$59$	−5.61861	−0.731481	−0.365740	−	0.930717i	$-0.619184\pi$
$59$	−5.61861	−0.731481	−0.365740	+	0.930717i	$0.619184\pi$
$60$	−0.742241	−0.0958229
$61$	7.77000	0.994847	0.497423	−	0.867508i	$-0.334280\pi$
$61$	7.77000	0.994847	0.497423	+	0.867508i	$0.334280\pi$
$62$	3.95618	0.502435
$63$	−3.28484	−0.413851
$64$	8.32414	1.04052
$65$	0	0
$66$	3.48605	0.429104
$67$	12.7075	1.55246	0.776231	−	0.630448i	$-0.217129\pi$
$67$	12.7075	1.55246	0.776231	+	0.630448i	$0.217129\pi$
$68$	0.751078	0.0910816
$69$	−5.92818	−0.713669
$70$	3.62779	0.433603
$71$	−11.4920	−1.36385	−0.681925	−	0.731423i	$-0.738857\pi$
$71$	−11.4920	−1.36385	−0.681925	+	0.731423i	$0.738857\pi$
$72$	3.07447	0.362330
$73$	6.55037	0.766663	0.383331	−	0.923611i	$-0.374777\pi$
$73$	6.55037	0.766663	0.383331	+	0.923611i	$0.374777\pi$
$74$	1.06683	0.124016
$75$	−4.02339	−0.464581
$76$	6.03729	0.692525
$77$	10.2466	1.16771
$78$	0	0
$79$	−3.05570	−0.343793	−0.171896	−	0.985115i	$-0.554989\pi$
$79$	−3.05570	−0.343793	−0.171896	+	0.985115i	$0.554989\pi$
$80$	−1.91097	−0.213653
$81$	1.00000	0.111111
$82$	9.59763	1.05988
$83$	6.02478	0.661305	0.330653	−	0.943753i	$-0.392731\pi$
$83$	6.02478	0.661305	0.330653	+	0.943753i	$0.392731\pi$
$84$	2.46717	0.269190
$85$	−0.988234	−0.107189
$86$	0.121020	0.0130499
$87$	−2.33320	−0.250146
$88$	−9.59041	−1.02234
$89$	−7.24068	−0.767511	−0.383755	−	0.923435i	$-0.625369\pi$
$89$	−7.24068	−0.767511	−0.383755	+	0.923435i	$0.625369\pi$
$90$	−1.10440	−0.116414
$91$	0	0
$92$	4.45252	0.464208
$93$	−3.54004	−0.367085
$94$	−3.48109	−0.359047
$95$	−7.94359	−0.814996
$96$	−3.98791	−0.407014
$97$	4.98862	0.506518	0.253259	−	0.967399i	$-0.418498\pi$
$97$	4.98862	0.506518	0.253259	+	0.967399i	$0.418498\pi$
$98$	−4.23571	−0.427872
$99$	−3.11937	−0.313508
$100$	3.02188	0.302188
$101$	7.20874	0.717297	0.358648	−	0.933473i	$-0.383238\pi$
$101$	7.20874	0.717297	0.358648	+	0.933473i	$0.383238\pi$
$102$	1.11755	0.110654
$103$	9.13714	0.900309	0.450155	−	0.892951i	$-0.351369\pi$
$103$	9.13714	0.900309	0.450155	+	0.892951i	$0.351369\pi$
$104$	0	0
$105$	−3.24619	−0.316796
$106$	−10.3205	−1.00242
$107$	14.7118	1.42225	0.711124	−	0.703067i	$-0.248186\pi$
$107$	14.7118	1.42225	0.711124	+	0.703067i	$0.248186\pi$
$108$	−0.751078	−0.0722725
$109$	17.9903	1.72316	0.861579	−	0.507624i	$-0.169476\pi$
$109$	17.9903	1.72316	0.861579	+	0.507624i	$0.169476\pi$
$110$	3.44504	0.328472
$111$	−0.954614	−0.0906079
$112$	6.35198	0.600206
$113$	15.7297	1.47972	0.739862	−	0.672759i	$-0.234891\pi$
$113$	15.7297	1.47972	0.739862	+	0.672759i	$0.234891\pi$
$114$	8.98307	0.841342
$115$	−5.85843	−0.546301
$116$	1.75242	0.162708
$117$	0	0
$118$	6.27909	0.578037
$119$	3.28484	0.301121
$120$	3.03830	0.277357
$121$	−1.26955	−0.115413
$122$	−8.68338	−0.786156
$123$	−8.58809	−0.774362
$124$	2.65885	0.238772
$125$	−8.91723	−0.797581
$126$	3.67098	0.327037
$127$	−11.1482	−0.989246	−0.494623	−	0.869108i	$-0.664694\pi$
$127$	−11.1482	−0.989246	−0.494623	+	0.869108i	$0.664694\pi$
$128$	−1.32685	−0.117278
$129$	−0.108290	−0.00953442
$130$	0	0
$131$	−16.8113	−1.46881	−0.734405	−	0.678712i	$-0.762539\pi$
$131$	−16.8113	−1.46881	−0.734405	+	0.678712i	$0.762539\pi$
$132$	2.34289	0.203922
$133$	26.4041	2.28953
$134$	−14.2012	−1.22680
$135$	0.988234	0.0850537
$136$	−3.07447	−0.263634
$137$	1.59256	0.136061	0.0680307	−	0.997683i	$-0.478328\pi$
$137$	1.59256	0.136061	0.0680307	+	0.997683i	$0.478328\pi$
$138$	6.62505	0.563961
$139$	−2.00726	−0.170254	−0.0851269	−	0.996370i	$-0.527130\pi$
$139$	−2.00726	−0.170254	−0.0851269	+	0.996370i	$0.527130\pi$
$140$	2.43814	0.206061
$141$	3.11492	0.262324
$142$	12.8429	1.07775
$143$	0	0
$144$	−1.93373	−0.161144
$145$	−2.30575	−0.191482
$146$	−7.32038	−0.605839
$147$	3.79017	0.312608
$148$	0.716989	0.0589362
$149$	−9.93880	−0.814218	−0.407109	−	0.913380i	$-0.633463\pi$
$149$	−9.93880	−0.814218	−0.407109	+	0.913380i	$0.633463\pi$
$150$	4.49635	0.367125
$151$	−13.7397	−1.11812	−0.559059	−	0.829128i	$-0.688838\pi$
$151$	−13.7397	−1.11812	−0.559059	+	0.829128i	$0.688838\pi$
$152$	−24.7131	−2.00450
$153$	−1.00000	−0.0808452
$154$	−11.4511	−0.922758
$155$	−3.49839	−0.280997
$156$	0	0
$157$	−7.43138	−0.593089	−0.296544	−	0.955019i	$-0.595834\pi$
$157$	−7.43138	−0.593089	−0.296544	+	0.955019i	$0.595834\pi$
$158$	3.41490	0.271675
$159$	9.23496	0.732380
$160$	−3.94099	−0.311562
$161$	19.4731	1.53470
$162$	−1.11755	−0.0878032
$163$	1.57873	0.123655	0.0618277	−	0.998087i	$-0.480307\pi$
$163$	1.57873	0.123655	0.0618277	+	0.998087i	$0.480307\pi$
$164$	6.45032	0.503686
$165$	−3.08267	−0.239985
$166$	−6.73300	−0.522582
$167$	−2.34516	−0.181474	−0.0907370	−	0.995875i	$-0.528922\pi$
$167$	−2.34516	−0.181474	−0.0907370	+	0.995875i	$0.528922\pi$
$168$	−10.0991	−0.779166
$169$	0	0
$170$	1.10440	0.0847038
$171$	−8.03817	−0.614694
$172$	0.0813344	0.00620169
$173$	18.6960	1.42143	0.710714	−	0.703481i	$-0.248372\pi$
$173$	18.6960	1.42143	0.710714	+	0.703481i	$0.248372\pi$
$174$	2.60748	0.197672
$175$	13.2162	0.999051
$176$	6.03200	0.454679
$177$	−5.61861	−0.422321
$178$	8.09184	0.606509
$179$	−10.5780	−0.790637	−0.395319	−	0.918544i	$-0.629366\pi$
$179$	−10.5780	−0.790637	−0.395319	+	0.918544i	$0.629366\pi$
$180$	−0.742241	−0.0553234
$181$	−19.2238	−1.42890	−0.714448	−	0.699689i	$-0.753322\pi$
$181$	−19.2238	−1.42890	−0.714448	+	0.699689i	$0.753322\pi$
$182$	0	0
$183$	7.77000	0.574375
$184$	−18.2260	−1.34364
$185$	−0.943382	−0.0693588
$186$	3.95618	0.290081
$187$	3.11937	0.228111
$188$	−2.33955	−0.170629
$189$	−3.28484	−0.238937
$190$	8.87738	0.644033
$191$	12.8626	0.930708	0.465354	−	0.885125i	$-0.345927\pi$
$191$	12.8626	0.930708	0.465354	+	0.885125i	$0.345927\pi$
$192$	8.32414	0.600743
$193$	−9.66788	−0.695909	−0.347955	−	0.937511i	$-0.613124\pi$
$193$	−9.66788	−0.695909	−0.347955	+	0.937511i	$0.613124\pi$
$194$	−5.57504	−0.400265
$195$	0	0
$196$	−2.84672	−0.203337
$197$	−5.74682	−0.409444	−0.204722	−	0.978820i	$-0.565629\pi$
$197$	−5.74682	−0.409444	−0.204722	+	0.978820i	$0.565629\pi$
$198$	3.48605	0.247743
$199$	2.42578	0.171959	0.0859795	−	0.996297i	$-0.472598\pi$
$199$	2.42578	0.171959	0.0859795	+	0.996297i	$0.472598\pi$
$200$	−12.3698	−0.874678
$201$	12.7075	0.896315
$202$	−8.05615	−0.566828
$203$	7.66420	0.537921
$204$	0.751078	0.0525860
$205$	−8.48704	−0.592761
$206$	−10.2112	−0.711450
$207$	−5.92818	−0.412037
$208$	0	0
$209$	25.0740	1.73440
$210$	3.62779	0.250341
$211$	26.8115	1.84578	0.922889	−	0.385066i	$-0.125821\pi$
$211$	26.8115	1.84578	0.922889	+	0.385066i	$0.125821\pi$
$212$	−6.93618	−0.476379
$213$	−11.4920	−0.787419
$214$	−16.4413	−1.12390
$215$	−0.107016	−0.00729844
$216$	3.07447	0.209191
$217$	11.6285	0.789392
$218$	−20.1051	−1.36169
$219$	6.55037	0.442633
$220$	2.31532	0.156099
$221$	0	0
$222$	1.06683	0.0716010
$223$	22.4757	1.50509	0.752543	−	0.658543i	$-0.228827\pi$
$223$	22.4757	1.50509	0.752543	+	0.658543i	$0.228827\pi$
$224$	13.0996	0.875256
$225$	−4.02339	−0.268226
$226$	−17.5787	−1.16932
$227$	−10.5053	−0.697262	−0.348631	−	0.937260i	$-0.613353\pi$
$227$	−10.5053	−0.697262	−0.348631	+	0.937260i	$0.613353\pi$
$228$	6.03729	0.399829
$229$	0.157413	0.0104021	0.00520106	−	0.999986i	$-0.498344\pi$
$229$	0.157413	0.0104021	0.00520106	+	0.999986i	$0.498344\pi$
$230$	6.54710	0.431703
$231$	10.2466	0.674178
$232$	−7.17337	−0.470955
$233$	8.62405	0.564980	0.282490	−	0.959270i	$-0.408840\pi$
$233$	8.62405	0.564980	0.282490	+	0.959270i	$0.408840\pi$
$234$	0	0
$235$	3.07827	0.200804
$236$	4.22002	0.274700
$237$	−3.05570	−0.198489
$238$	−3.67098	−0.237954
$239$	−22.7765	−1.47329	−0.736645	−	0.676280i	$-0.763591\pi$
$239$	−22.7765	−1.47329	−0.736645	+	0.676280i	$0.763591\pi$
$240$	−1.91097	−0.123353
$241$	11.1067	0.715447	0.357723	−	0.933828i	$-0.383553\pi$
$241$	11.1067	0.715447	0.357723	+	0.933828i	$0.383553\pi$
$242$	1.41878	0.0912029
$243$	1.00000	0.0641500
$244$	−5.83588	−0.373604
$245$	3.74558	0.239296
$246$	9.59763	0.611923
$247$	0	0
$248$	−10.8838	−0.691119
$249$	6.02478	0.381805
$250$	9.96546	0.630271
$251$	22.1887	1.40054	0.700268	−	0.713880i	$-0.253064\pi$
$251$	22.1887	1.40054	0.700268	+	0.713880i	$0.253064\pi$
$252$	2.46717	0.155417
$253$	18.4922	1.16259
$254$	12.4587	0.781731
$255$	−0.988234	−0.0618856
$256$	−15.1655	−0.947841
$257$	−4.84275	−0.302082	−0.151041	−	0.988527i	$-0.548263\pi$
$257$	−4.84275	−0.302082	−0.151041	+	0.988527i	$0.548263\pi$
$258$	0.121020	0.00753437
$259$	3.13575	0.194846
$260$	0	0
$261$	−2.33320	−0.144422
$262$	18.7875	1.16070
$263$	18.4579	1.13816	0.569082	−	0.822281i	$-0.307299\pi$
$263$	18.4579	1.13816	0.569082	+	0.822281i	$0.307299\pi$
$264$	−9.59041	−0.590249
$265$	9.12631	0.560625
$266$	−29.5079	−1.80925
$267$	−7.24068	−0.443123
$268$	−9.54429	−0.583010
$269$	−9.18033	−0.559734	−0.279867	−	0.960039i	$-0.590290\pi$
$269$	−9.18033	−0.559734	−0.279867	+	0.960039i	$0.590290\pi$
$270$	−1.10440	−0.0672118
$271$	−25.6938	−1.56079	−0.780393	−	0.625289i	$-0.784981\pi$
$271$	−25.6938	−1.56079	−0.780393	+	0.625289i	$0.784981\pi$
$272$	1.93373	0.117249
$273$	0	0
$274$	−1.77977	−0.107520
$275$	12.5504	0.756820
$276$	4.45252	0.268010
$277$	−20.2157	−1.21464	−0.607322	−	0.794456i	$-0.707756\pi$
$277$	−20.2157	−1.21464	−0.607322	+	0.794456i	$0.707756\pi$
$278$	2.24322	0.134539
$279$	−3.54004	−0.211937
$280$	−9.98032	−0.596438
$281$	−3.56706	−0.212793	−0.106396	−	0.994324i	$-0.533931\pi$
$281$	−3.56706	−0.212793	−0.106396	+	0.994324i	$0.533931\pi$
$282$	−3.48109	−0.207296
$283$	9.58360	0.569686	0.284843	−	0.958574i	$-0.408059\pi$
$283$	9.58360	0.569686	0.284843	+	0.958574i	$0.408059\pi$
$284$	8.63139	0.512179
$285$	−7.94359	−0.470538
$286$	0	0
$287$	28.2105	1.66521
$288$	−3.98791	−0.234990
$289$	1.00000	0.0588235
$290$	2.57680	0.151315
$291$	4.98862	0.292438
$292$	−4.91984	−0.287912
$293$	4.17655	0.243997	0.121998	−	0.992530i	$-0.461070\pi$
$293$	4.17655	0.243997	0.121998	+	0.992530i	$0.461070\pi$
$294$	−4.23571	−0.247032
$295$	−5.55250	−0.323279
$296$	−2.93493	−0.170590
$297$	−3.11937	−0.181004
$298$	11.1071	0.643418
$299$	0	0
$300$	3.02188	0.174468
$301$	0.355716	0.0205031
$302$	15.3548	0.883569
$303$	7.20874	0.414132
$304$	15.5436	0.891487
$305$	7.67858	0.439674
$306$	1.11755	0.0638862
$307$	−28.5863	−1.63151	−0.815754	−	0.578400i	$-0.803677\pi$
$307$	−28.5863	−1.63151	−0.815754	+	0.578400i	$0.803677\pi$
$308$	−7.69601	−0.438521
$309$	9.13714	0.519794
$310$	3.90963	0.222052
$311$	15.9824	0.906278	0.453139	−	0.891440i	$-0.350304\pi$
$311$	15.9824	0.906278	0.453139	+	0.891440i	$0.350304\pi$
$312$	0	0
$313$	15.2575	0.862405	0.431203	−	0.902255i	$-0.358089\pi$
$313$	15.2575	0.862405	0.431203	+	0.902255i	$0.358089\pi$
$314$	8.30495	0.468676
$315$	−3.24619	−0.182902
$316$	2.29507	0.129108
$317$	−22.5889	−1.26872	−0.634359	−	0.773038i	$-0.718736\pi$
$317$	−22.5889	−1.26872	−0.634359	+	0.773038i	$0.718736\pi$
$318$	−10.3205	−0.578748
$319$	7.27812	0.407496
$320$	8.22620	0.459859
$321$	14.7118	0.821135
$322$	−21.7622	−1.21276
$323$	8.03817	0.447256
$324$	−0.751078	−0.0417266
$325$	0	0
$326$	−1.76431	−0.0977161
$327$	17.9903	0.994865
$328$	−26.4038	−1.45791
$329$	−10.2320	−0.564110
$330$	3.44504	0.189643
$331$	−24.2519	−1.33300	−0.666502	−	0.745503i	$-0.732209\pi$
$331$	−24.2519	−1.33300	−0.666502	+	0.745503i	$0.732209\pi$
$332$	−4.52508	−0.248346
$333$	−0.954614	−0.0523125
$334$	2.62084	0.143406
$335$	12.5579	0.686114
$336$	6.35198	0.346529
$337$	21.7640	1.18556	0.592781	−	0.805364i	$-0.298030\pi$
$337$	21.7640	1.18556	0.592781	+	0.805364i	$0.298030\pi$
$338$	0	0
$339$	15.7297	0.854319
$340$	0.742241	0.0402537
$341$	11.0427	0.597995
$342$	8.98307	0.485749
$343$	10.5438	0.569310
$344$	−0.332935	−0.0179507
$345$	−5.85843	−0.315407
$346$	−20.8937	−1.12325
$347$	−11.0428	−0.592810	−0.296405	−	0.955062i	$-0.595788\pi$
$347$	−11.0428	−0.592810	−0.296405	+	0.955062i	$0.595788\pi$
$348$	1.75242	0.0939395
$349$	−8.62677	−0.461780	−0.230890	−	0.972980i	$-0.574164\pi$
$349$	−8.62677	−0.461780	−0.230890	+	0.972980i	$0.574164\pi$
$350$	−14.7698	−0.789478
$351$	0	0
$352$	12.4397	0.663041
$353$	9.70868	0.516741	0.258370	−	0.966046i	$-0.416815\pi$
$353$	9.70868	0.516741	0.258370	+	0.966046i	$0.416815\pi$
$354$	6.27909	0.333730
$355$	−11.3568	−0.602756
$356$	5.43832	0.288230
$357$	3.28484	0.173852
$358$	11.8215	0.624784
$359$	21.9613	1.15907	0.579537	−	0.814946i	$-0.303234\pi$
$359$	21.9613	1.15907	0.579537	+	0.814946i	$0.303234\pi$
$360$	3.03830	0.160132
$361$	45.6122	2.40064
$362$	21.4836	1.12915
$363$	−1.26955	−0.0666339
$364$	0	0
$365$	6.47330	0.338828
$366$	−8.68338	−0.453887
$367$	6.54228	0.341504	0.170752	−	0.985314i	$-0.445380\pi$
$367$	6.54228	0.341504	0.170752	+	0.985314i	$0.445380\pi$
$368$	11.4635	0.597575
$369$	−8.58809	−0.447078
$370$	1.05428	0.0548093
$371$	−30.3354	−1.57493
$372$	2.65885	0.137855
$373$	−14.5583	−0.753802	−0.376901	−	0.926253i	$-0.623010\pi$
$373$	−14.5583	−0.753802	−0.376901	+	0.926253i	$0.623010\pi$
$374$	−3.48605	−0.180260
$375$	−8.91723	−0.460484
$376$	9.57674	0.493883
$377$	0	0
$378$	3.67098	0.188815
$379$	26.1395	1.34270	0.671349	−	0.741141i	$-0.265715\pi$
$379$	26.1395	1.34270	0.671349	+	0.741141i	$0.265715\pi$
$380$	5.96626	0.306063
$381$	−11.1482	−0.571142
$382$	−14.3747	−0.735472
$383$	−7.19270	−0.367530	−0.183765	−	0.982970i	$-0.558829\pi$
$383$	−7.19270	−0.367530	−0.183765	+	0.982970i	$0.558829\pi$
$384$	−1.32685	−0.0677104
$385$	10.1261	0.516072
$386$	10.8044	0.549927
$387$	−0.108290	−0.00550470
$388$	−3.74685	−0.190217
$389$	18.4423	0.935064	0.467532	−	0.883976i	$-0.345143\pi$
$389$	18.4423	0.935064	0.467532	+	0.883976i	$0.345143\pi$
$390$	0	0
$391$	5.92818	0.299801
$392$	11.6528	0.588554
$393$	−16.8113	−0.848018
$394$	6.42237	0.323554
$395$	−3.01975	−0.151940
$396$	2.34289	0.117735
$397$	27.4835	1.37936	0.689678	−	0.724116i	$-0.257752\pi$
$397$	27.4835	1.37936	0.689678	+	0.724116i	$0.257752\pi$
$398$	−2.71093	−0.135887
$399$	26.4041	1.32186
$400$	7.78014	0.389007
$401$	0.978598	0.0488688	0.0244344	−	0.999701i	$-0.492222\pi$
$401$	0.978598	0.0488688	0.0244344	+	0.999701i	$0.492222\pi$
$402$	−14.2012	−0.708294
$403$	0	0
$404$	−5.41433	−0.269373
$405$	0.988234	0.0491058
$406$	−8.56514	−0.425081
$407$	2.97779	0.147604
$408$	−3.07447	−0.152209
$409$	0.448686	0.0221861	0.0110930	−	0.999938i	$-0.496469\pi$
$409$	0.448686	0.0221861	0.0110930	+	0.999938i	$0.496469\pi$
$410$	9.48471	0.468416
$411$	1.59256	0.0785551
$412$	−6.86271	−0.338101
$413$	18.4562	0.908172
$414$	6.62505	0.325603
$415$	5.95389	0.292265
$416$	0	0
$417$	−2.00726	−0.0982960
$418$	−28.0215	−1.37058
$419$	−19.4195	−0.948707	−0.474354	−	0.880334i	$-0.657318\pi$
$419$	−19.4195	−0.948707	−0.474354	+	0.880334i	$0.657318\pi$
$420$	2.43814	0.118969
$421$	3.92642	0.191362	0.0956811	−	0.995412i	$-0.469497\pi$
$421$	3.92642	0.191362	0.0956811	+	0.995412i	$0.469497\pi$
$422$	−29.9632	−1.45859
$423$	3.11492	0.151453
$424$	28.3926	1.37887
$425$	4.02339	0.195163
$426$	12.8429	0.622241
$427$	−25.5232	−1.23515
$428$	−11.0497	−0.534110
$429$	0	0
$430$	0.119596	0.00576743
$431$	−25.0813	−1.20812	−0.604062	−	0.796938i	$-0.706452\pi$
$431$	−25.0813	−1.20812	−0.604062	+	0.796938i	$0.706452\pi$
$432$	−1.93373	−0.0930364
$433$	−30.8904	−1.48450	−0.742250	−	0.670123i	$-0.766241\pi$
$433$	−30.8904	−1.48450	−0.742250	+	0.670123i	$0.766241\pi$
$434$	−12.9954	−0.623800
$435$	−2.30575	−0.110552
$436$	−13.5121	−0.647113
$437$	47.6517	2.27949
$438$	−7.32038	−0.349781
$439$	3.66912	0.175118	0.0875589	−	0.996159i	$-0.472093\pi$
$439$	3.66912	0.175118	0.0875589	+	0.996159i	$0.472093\pi$
$440$	−9.47757	−0.451825
$441$	3.79017	0.180484
$442$	0	0
$443$	33.3024	1.58225	0.791123	−	0.611657i	$-0.209497\pi$
$443$	33.3024	1.58225	0.791123	+	0.611657i	$0.209497\pi$
$444$	0.716989	0.0340268
$445$	−7.15549	−0.339203
$446$	−25.1178	−1.18936
$447$	−9.93880	−0.470089
$448$	−27.3435	−1.29186
$449$	−7.71269	−0.363984	−0.181992	−	0.983300i	$-0.558255\pi$
$449$	−7.71269	−0.363984	−0.181992	+	0.983300i	$0.558255\pi$
$450$	4.49635	0.211960
$451$	26.7894	1.26146
$452$	−11.8142	−0.555694
$453$	−13.7397	−0.645546
$454$	11.7402	0.550996
$455$	0	0
$456$	−24.7131	−1.15730
$457$	42.6886	1.99689	0.998444	−	0.0557695i	$-0.0177612\pi$
$457$	42.6886	1.99689	0.998444	+	0.0557695i	$0.0177612\pi$
$458$	−0.175917	−0.00822005
$459$	−1.00000	−0.0466760
$460$	4.40014	0.205157
$461$	4.05294	0.188764	0.0943821	−	0.995536i	$-0.469912\pi$
$461$	4.05294	0.188764	0.0943821	+	0.995536i	$0.469912\pi$
$462$	−11.4511	−0.532755
$463$	−36.4057	−1.69192	−0.845958	−	0.533249i	$-0.820971\pi$
$463$	−36.4057	−1.69192	−0.845958	+	0.533249i	$0.820971\pi$
$464$	4.51177	0.209454
$465$	−3.49839	−0.162234
$466$	−9.63782	−0.446463
$467$	−37.5210	−1.73627	−0.868133	−	0.496332i	$-0.834680\pi$
$467$	−37.5210	−1.73627	−0.868133	+	0.496332i	$0.834680\pi$
$468$	0	0
$469$	−41.7420	−1.92746
$470$	−3.44013	−0.158681
$471$	−7.43138	−0.342420
$472$	−17.2743	−0.795112
$473$	0.337797	0.0155319
$474$	3.41490	0.156852
$475$	32.3407	1.48389
$476$	−2.46717	−0.113083
$477$	9.23496	0.422840
$478$	25.4539	1.16424
$479$	35.7575	1.63380	0.816902	−	0.576777i	$-0.195690\pi$
$479$	35.7575	1.63380	0.816902	+	0.576777i	$0.195690\pi$
$480$	−3.94099	−0.179881
$481$	0	0
$482$	−12.4123	−0.565367
$483$	19.4731	0.886057
$484$	0.953529	0.0433422
$485$	4.92993	0.223856
$486$	−1.11755	−0.0506932
$487$	14.3949	0.652297	0.326148	−	0.945319i	$-0.394249\pi$
$487$	14.3949	0.652297	0.326148	+	0.945319i	$0.394249\pi$
$488$	23.8887	1.08139
$489$	1.57873	0.0713925
$490$	−4.18588	−0.189099
$491$	15.3451	0.692514	0.346257	−	0.938140i	$-0.387452\pi$
$491$	15.3451	0.692514	0.346257	+	0.938140i	$0.387452\pi$
$492$	6.45032	0.290803
$493$	2.33320	0.105082
$494$	0	0
$495$	−3.08267	−0.138556
$496$	6.84547	0.307371
$497$	37.7494	1.69329
$498$	−6.73300	−0.301713
$499$	−39.3910	−1.76338	−0.881692	−	0.471825i	$-0.843595\pi$
$499$	−39.3910	−1.76338	−0.881692	+	0.471825i	$0.843595\pi$
$500$	6.69753	0.299523
$501$	−2.34516	−0.104774
$502$	−24.7970	−1.10674
$503$	−0.197790	−0.00881901	−0.00440951	−	0.999990i	$-0.501404\pi$
$503$	−0.197790	−0.00881901	−0.00440951	+	0.999990i	$0.501404\pi$
$504$	−10.0991	−0.449852
$505$	7.12393	0.317011
$506$	−20.6660	−0.918714
$507$	0	0
$508$	8.37320	0.371501
$509$	5.42573	0.240491	0.120246	−	0.992744i	$-0.461632\pi$
$509$	5.42573	0.240491	0.120246	+	0.992744i	$0.461632\pi$
$510$	1.10440	0.0489038
$511$	−21.5169	−0.951852
$512$	19.6019	0.866289
$513$	−8.03817	−0.354894
$514$	5.41202	0.238714
$515$	9.02964	0.397893
$516$	0.0813344	0.00358055
$517$	−9.71659	−0.427335
$518$	−3.50437	−0.153973
$519$	18.6960	0.820662
$520$	0	0
$521$	16.4376	0.720145	0.360073	−	0.932924i	$-0.382752\pi$
$521$	16.4376	0.720145	0.360073	+	0.932924i	$0.382752\pi$
$522$	2.60748	0.114126
$523$	−40.0834	−1.75272	−0.876361	−	0.481654i	$-0.840036\pi$
$523$	−40.0834	−1.75272	−0.876361	+	0.481654i	$0.840036\pi$
$524$	12.6266	0.551595
$525$	13.2162	0.576802
$526$	−20.6277	−0.899409
$527$	3.54004	0.154207
$528$	6.03200	0.262509
$529$	12.1433	0.527969
$530$	−10.1991	−0.443021
$531$	−5.61861	−0.243827
$532$	−19.8315	−0.859806
$533$	0	0
$534$	8.09184	0.350168
$535$	14.5388	0.628565
$536$	39.0687	1.68751
$537$	−10.5780	−0.456475
$538$	10.2595	0.442318
$539$	−11.8229	−0.509250
$540$	−0.742241	−0.0319410
$541$	13.8150	0.593954	0.296977	−	0.954885i	$-0.404021\pi$
$541$	13.8150	0.593954	0.296977	+	0.954885i	$0.404021\pi$
$542$	28.7141	1.23338
$543$	−19.2238	−0.824973
$544$	3.98791	0.170980
$545$	17.7786	0.761553
$546$	0	0
$547$	−6.62133	−0.283108	−0.141554	−	0.989931i	$-0.545210\pi$
$547$	−6.62133	−0.283108	−0.141554	+	0.989931i	$0.545210\pi$
$548$	−1.19614	−0.0510964
$549$	7.77000	0.331616
$550$	−14.0258	−0.598061
$551$	18.7547	0.798976
$552$	−18.2260	−0.775751
$553$	10.0375	0.426837
$554$	22.5921	0.959845
$555$	−0.943382	−0.0400443
$556$	1.50761	0.0639369
$557$	−0.269020	−0.0113988	−0.00569938	−	0.999984i	$-0.501814\pi$
$557$	−0.269020	−0.0113988	−0.00569938	+	0.999984i	$0.501814\pi$
$558$	3.95618	0.167478
$559$	0	0
$560$	6.27724	0.265262
$561$	3.11937	0.131700
$562$	3.98637	0.168155
$563$	33.4879	1.41135	0.705673	−	0.708538i	$-0.250645\pi$
$563$	33.4879	1.41135	0.705673	+	0.708538i	$0.250645\pi$
$564$	−2.33955	−0.0985128
$565$	15.5446	0.653967
$566$	−10.7102	−0.450182
$567$	−3.28484	−0.137950
$568$	−35.3318	−1.48249
$569$	6.38032	0.267477	0.133739	−	0.991017i	$-0.457302\pi$
$569$	6.38032	0.267477	0.133739	+	0.991017i	$0.457302\pi$
$570$	8.87738	0.371832
$571$	−32.0302	−1.34042	−0.670212	−	0.742170i	$-0.733797\pi$
$571$	−32.0302	−1.34042	−0.670212	+	0.742170i	$0.733797\pi$
$572$	0	0
$573$	12.8626	0.537345
$574$	−31.5267	−1.31590
$575$	23.8514	0.994672
$576$	8.32414	0.346839
$577$	40.9685	1.70554	0.852771	−	0.522285i	$-0.174920\pi$
$577$	40.9685	1.70554	0.852771	+	0.522285i	$0.174920\pi$
$578$	−1.11755	−0.0464840
$579$	−9.66788	−0.401783
$580$	1.73180	0.0719091
$581$	−19.7904	−0.821045
$582$	−5.57504	−0.231093
$583$	−28.8072	−1.19307
$584$	20.1389	0.833355
$585$	0	0
$586$	−4.66752	−0.192813
$587$	25.4056	1.04860	0.524302	−	0.851533i	$-0.324326\pi$
$587$	25.4056	1.04860	0.524302	+	0.851533i	$0.324326\pi$
$588$	−2.84672	−0.117397
$589$	28.4554	1.17249
$590$	6.20521	0.255465
$591$	−5.74682	−0.236392
$592$	1.84596	0.0758685
$593$	12.7651	0.524200	0.262100	−	0.965041i	$-0.415585\pi$
$593$	12.7651	0.524200	0.262100	+	0.965041i	$0.415585\pi$
$594$	3.48605	0.143035
$595$	3.24619	0.133081
$596$	7.46482	0.305771
$597$	2.42578	0.0992805
$598$	0	0
$599$	−28.3629	−1.15888	−0.579438	−	0.815016i	$-0.696728\pi$
$599$	−28.3629	−1.15888	−0.579438	+	0.815016i	$0.696728\pi$
$600$	−12.3698	−0.504995
$601$	−31.7035	−1.29321	−0.646607	−	0.762823i	$-0.723813\pi$
$601$	−31.7035	−1.29321	−0.646607	+	0.762823i	$0.723813\pi$
$602$	−0.397531	−0.0162022
$603$	12.7075	0.517488
$604$	10.3196	0.419897
$605$	−1.25461	−0.0510071
$606$	−8.05615	−0.327259
$607$	−1.19574	−0.0485337	−0.0242669	−	0.999706i	$-0.507725\pi$
$607$	−1.19574	−0.0485337	−0.0242669	+	0.999706i	$0.507725\pi$
$608$	32.0555	1.30002
$609$	7.66420	0.310569
$610$	−8.58121	−0.347443
$611$	0	0
$612$	0.751078	0.0303605
$613$	−41.2856	−1.66751	−0.833754	−	0.552135i	$-0.813813\pi$
$613$	−41.2856	−1.66751	−0.833754	+	0.552135i	$0.813813\pi$
$614$	31.9467	1.28926
$615$	−8.48704	−0.342231
$616$	31.5030	1.26929
$617$	41.2475	1.66056	0.830281	−	0.557345i	$-0.188180\pi$
$617$	41.2475	1.66056	0.830281	+	0.557345i	$0.188180\pi$
$618$	−10.2112	−0.410756
$619$	38.7581	1.55782	0.778910	−	0.627135i	$-0.215773\pi$
$619$	38.7581	1.55782	0.778910	+	0.627135i	$0.215773\pi$
$620$	2.62756	0.105526
$621$	−5.92818	−0.237890
$622$	−17.8611	−0.716167
$623$	23.7845	0.952905
$624$	0	0
$625$	11.3047	0.452186
$626$	−17.0510	−0.681497
$627$	25.0740	1.00136
$628$	5.58155	0.222728
$629$	0.954614	0.0380629
$630$	3.62779	0.144534
$631$	−34.4370	−1.37092	−0.685458	−	0.728112i	$-0.740398\pi$
$631$	−34.4370	−1.37092	−0.685458	+	0.728112i	$0.740398\pi$
$632$	−9.39466	−0.373699
$633$	26.8115	1.06566
$634$	25.2443	1.00258
$635$	−11.0171	−0.437199
$636$	−6.93618	−0.275037
$637$	0	0
$638$	−8.13367	−0.322015
$639$	−11.4920	−0.454616
$640$	−1.31124	−0.0518312
$641$	−12.4094	−0.490141	−0.245070	−	0.969505i	$-0.578811\pi$
$641$	−12.4094	−0.490141	−0.245070	+	0.969505i	$0.578811\pi$
$642$	−16.4413	−0.648884
$643$	−43.8118	−1.72777	−0.863885	−	0.503689i	$-0.831976\pi$
$643$	−43.8118	−1.72777	−0.863885	+	0.503689i	$0.831976\pi$
$644$	−14.6258	−0.576338
$645$	−0.107016	−0.00421376
$646$	−8.98307	−0.353434
$647$	−6.25593	−0.245946	−0.122973	−	0.992410i	$-0.539243\pi$
$647$	−6.25593	−0.245946	−0.122973	+	0.992410i	$0.539243\pi$
$648$	3.07447	0.120777
$649$	17.5265	0.687976
$650$	0	0
$651$	11.6285	0.455756
$652$	−1.18575	−0.0464375
$653$	−12.1346	−0.474863	−0.237432	−	0.971404i	$-0.576306\pi$
$653$	−12.1346	−0.474863	−0.237432	+	0.971404i	$0.576306\pi$
$654$	−20.1051	−0.786171
$655$	−16.6135	−0.649143
$656$	16.6070	0.648395
$657$	6.55037	0.255554
$658$	11.4348	0.445775
$659$	−30.9090	−1.20404	−0.602022	−	0.798480i	$-0.705638\pi$
$659$	−30.9090	−1.20404	−0.602022	+	0.798480i	$0.705638\pi$
$660$	2.31532	0.0901238
$661$	−21.1832	−0.823933	−0.411966	−	0.911199i	$-0.635158\pi$
$661$	−21.1832	−0.823933	−0.411966	+	0.911199i	$0.635158\pi$
$662$	27.1028	1.05338
$663$	0	0
$664$	18.5230	0.718832
$665$	26.0934	1.01186
$666$	1.06683	0.0413388
$667$	13.8316	0.535563
$668$	1.76140	0.0681506
$669$	22.4757	0.868962
$670$	−14.0342	−0.542187
$671$	−24.2375	−0.935678
$672$	13.0996	0.505329
$673$	−13.7877	−0.531475	−0.265737	−	0.964045i	$-0.585615\pi$
$673$	−13.7877	−0.531475	−0.265737	+	0.964045i	$0.585615\pi$
$674$	−24.3224	−0.936865
$675$	−4.02339	−0.154860
$676$	0	0
$677$	−12.5458	−0.482175	−0.241088	−	0.970503i	$-0.577504\pi$
$677$	−12.5458	−0.482175	−0.241088	+	0.970503i	$0.577504\pi$
$678$	−17.5787	−0.675107
$679$	−16.3868	−0.628869
$680$	−3.03830	−0.116513
$681$	−10.5053	−0.402564
$682$	−12.3408	−0.472553
$683$	7.86134	0.300806	0.150403	−	0.988625i	$-0.451943\pi$
$683$	7.86134	0.300806	0.150403	+	0.988625i	$0.451943\pi$
$684$	6.03729	0.230842
$685$	1.57382	0.0601326
$686$	−11.7832	−0.449885
$687$	0.157413	0.00600567
$688$	0.209404	0.00798344
$689$	0	0
$690$	6.54710	0.249244
$691$	−24.2167	−0.921248	−0.460624	−	0.887595i	$-0.652374\pi$
$691$	−24.2167	−0.921248	−0.460624	+	0.887595i	$0.652374\pi$
$692$	−14.0421	−0.533802
$693$	10.2466	0.389237
$694$	12.3409	0.468455
$695$	−1.98365	−0.0752440
$696$	−7.17337	−0.271906
$697$	8.58809	0.325297
$698$	9.64086	0.364912
$699$	8.62405	0.326191
$700$	−9.92640	−0.375183
$701$	−30.9599	−1.16934	−0.584669	−	0.811272i	$-0.698776\pi$
$701$	−30.9599	−1.16934	−0.584669	+	0.811272i	$0.698776\pi$
$702$	0	0
$703$	7.67335	0.289406
$704$	−25.9661	−0.978633
$705$	3.07827	0.115934
$706$	−10.8499	−0.408343
$707$	−23.6796	−0.890562
$708$	4.22002	0.158598
$709$	38.4119	1.44259	0.721294	−	0.692629i	$-0.243548\pi$
$709$	38.4119	1.44259	0.721294	+	0.692629i	$0.243548\pi$
$710$	12.6918	0.476315
$711$	−3.05570	−0.114598
$712$	−22.2613	−0.834277
$713$	20.9860	0.785932
$714$	−3.67098	−0.137383
$715$	0	0
$716$	7.94491	0.296915
$717$	−22.7765	−0.850604
$718$	−24.5429	−0.915932
$719$	20.5344	0.765805	0.382903	−	0.923789i	$-0.374924\pi$
$719$	20.5344	0.765805	0.382903	+	0.923789i	$0.374924\pi$
$720$	−1.91097	−0.0712178
$721$	−30.0141	−1.11778
$722$	−50.9739	−1.89705
$723$	11.1067	0.413063
$724$	14.4386	0.536606
$725$	9.38739	0.348639
$726$	1.41878	0.0526560
$727$	−13.8723	−0.514494	−0.257247	−	0.966346i	$-0.582815\pi$
$727$	−13.8723	−0.514494	−0.257247	+	0.966346i	$0.582815\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−7.23425	−0.267752
$731$	0.108290	0.00400526
$732$	−5.83588	−0.215700
$733$	−33.7997	−1.24842	−0.624210	−	0.781257i	$-0.714579\pi$
$733$	−33.7997	−1.24842	−0.624210	+	0.781257i	$0.714579\pi$
$734$	−7.31134	−0.269866
$735$	3.74558	0.138158
$736$	23.6410	0.871419
$737$	−39.6392	−1.46013
$738$	9.59763	0.353294
$739$	−28.1242	−1.03456	−0.517282	−	0.855815i	$-0.673056\pi$
$739$	−28.1242	−1.03456	−0.517282	+	0.855815i	$0.673056\pi$
$740$	0.708554	0.0260469
$741$	0	0
$742$	33.9013	1.24456
$743$	−9.46574	−0.347264	−0.173632	−	0.984811i	$-0.555550\pi$
$743$	−9.46574	−0.347264	−0.173632	+	0.984811i	$0.555550\pi$
$744$	−10.8838	−0.399018
$745$	−9.82186	−0.359845
$746$	16.2697	0.595676
$747$	6.02478	0.220435
$748$	−2.34289	−0.0856645
$749$	−48.3261	−1.76580
$750$	9.96546	0.363887
$751$	42.1688	1.53876	0.769380	−	0.638791i	$-0.220565\pi$
$751$	42.1688	1.53876	0.769380	+	0.638791i	$0.220565\pi$
$752$	−6.02340	−0.219651
$753$	22.1887	0.808600
$754$	0	0
$755$	−13.5780	−0.494154
$756$	2.46717	0.0897301
$757$	−33.1406	−1.20451	−0.602257	−	0.798302i	$-0.705732\pi$
$757$	−33.1406	−1.20451	−0.602257	+	0.798302i	$0.705732\pi$
$758$	−29.2123	−1.06104
$759$	18.4922	0.671223
$760$	−24.4224	−0.885892
$761$	43.0643	1.56108	0.780540	−	0.625106i	$-0.214944\pi$
$761$	43.0643	1.56108	0.780540	+	0.625106i	$0.214944\pi$
$762$	12.4587	0.451332
$763$	−59.0952	−2.13939
$764$	−9.66085	−0.349517
$765$	−0.988234	−0.0357297
$766$	8.03822	0.290433
$767$	0	0
$768$	−15.1655	−0.547237
$769$	44.3242	1.59837	0.799186	−	0.601084i	$-0.205264\pi$
$769$	44.3242	1.59837	0.799186	+	0.601084i	$0.205264\pi$
$770$	−11.3164	−0.407815
$771$	−4.84275	−0.174407
$772$	7.26133	0.261341
$773$	20.5733	0.739972	0.369986	−	0.929037i	$-0.379363\pi$
$773$	20.5733	0.739972	0.369986	+	0.929037i	$0.379363\pi$
$774$	0.121020	0.00434997
$775$	14.2430	0.511623
$776$	15.3374	0.550580
$777$	3.13575	0.112495
$778$	−20.6103	−0.738914
$779$	69.0325	2.47335
$780$	0	0
$781$	35.8478	1.28273
$782$	−6.62505	−0.236911
$783$	−2.33320	−0.0833819
$784$	−7.32915	−0.261755
$785$	−7.34395	−0.262117
$786$	18.7875	0.670128
$787$	11.7307	0.418155	0.209078	−	0.977899i	$-0.432954\pi$
$787$	11.7307	0.418155	0.209078	+	0.977899i	$0.432954\pi$
$788$	4.31631	0.153762
$789$	18.4579	0.657119
$790$	3.37472	0.120067
$791$	−51.6695	−1.83716
$792$	−9.59041	−0.340780
$793$	0	0
$794$	−30.7142	−1.09001
$795$	9.12631	0.323677
$796$	−1.82195	−0.0645773
$797$	54.8493	1.94286	0.971431	−	0.237323i	$-0.0762701\pi$
$797$	54.8493	1.94286	0.971431	+	0.237323i	$0.0762701\pi$
$798$	−29.5079	−1.04457
$799$	−3.11492	−0.110198
$800$	16.0449	0.567273
$801$	−7.24068	−0.255837
$802$	−1.09363	−0.0386175
$803$	−20.4330	−0.721065
$804$	−9.54429	−0.336601
$805$	19.2440	0.678262
$806$	0	0
$807$	−9.18033	−0.323163
$808$	22.1631	0.779695
$809$	5.83287	0.205073	0.102536	−	0.994729i	$-0.467304\pi$
$809$	5.83287	0.205073	0.102536	+	0.994729i	$0.467304\pi$
$810$	−1.10440	−0.0388048
$811$	4.54553	0.159615	0.0798076	−	0.996810i	$-0.474569\pi$
$811$	4.54553	0.159615	0.0798076	+	0.996810i	$0.474569\pi$
$812$	−5.75641	−0.202010
$813$	−25.6938	−0.901121
$814$	−3.32784	−0.116641
$815$	1.56015	0.0546498
$816$	1.93373	0.0676939
$817$	0.870455	0.0304534
$818$	−0.501429	−0.0175321
$819$	0	0
$820$	6.37443	0.222605
$821$	50.5967	1.76584	0.882919	−	0.469526i	$-0.155575\pi$
$821$	50.5967	1.76584	0.882919	+	0.469526i	$0.155575\pi$
$822$	−1.77977	−0.0620765
$823$	39.8476	1.38900	0.694500	−	0.719492i	$-0.255625\pi$
$823$	39.8476	1.38900	0.694500	+	0.719492i	$0.255625\pi$
$824$	28.0919	0.978627
$825$	12.5504	0.436950
$826$	−20.6258	−0.717663
$827$	51.3028	1.78397	0.891987	−	0.452061i	$-0.149311\pi$
$827$	51.3028	1.78397	0.891987	+	0.452061i	$0.149311\pi$
$828$	4.45252	0.154736
$829$	−15.1959	−0.527777	−0.263888	−	0.964553i	$-0.585005\pi$
$829$	−15.1959	−0.527777	−0.263888	+	0.964553i	$0.585005\pi$
$830$	−6.65378	−0.230956
$831$	−20.2157	−0.701274
$832$	0	0
$833$	−3.79017	−0.131322
$834$	2.24322	0.0776763
$835$	−2.31757	−0.0802028
$836$	−18.8325	−0.651337
$837$	−3.54004	−0.122362
$838$	21.7024	0.749695
$839$	−14.0377	−0.484635	−0.242317	−	0.970197i	$-0.577908\pi$
$839$	−14.0377	−0.484635	−0.242317	+	0.970197i	$0.577908\pi$
$840$	−9.98032	−0.344354
$841$	−23.5562	−0.812281
$842$	−4.38798	−0.151220
$843$	−3.56706	−0.122856
$844$	−20.1375	−0.693162
$845$	0	0
$846$	−3.48109	−0.119682
$847$	4.17026	0.143292
$848$	−17.8579	−0.613242
$849$	9.58360	0.328908
$850$	−4.49635	−0.154224
$851$	5.65912	0.193992
$852$	8.63139	0.295706
$853$	43.0108	1.47266	0.736330	−	0.676622i	$-0.236557\pi$
$853$	43.0108	1.47266	0.736330	+	0.676622i	$0.236557\pi$
$854$	28.5235	0.976054
$855$	−7.94359	−0.271665
$856$	45.2312	1.54597
$857$	−13.5706	−0.463563	−0.231781	−	0.972768i	$-0.574455\pi$
$857$	−13.5706	−0.463563	−0.231781	+	0.972768i	$0.574455\pi$
$858$	0	0
$859$	−21.3176	−0.727347	−0.363674	−	0.931526i	$-0.618478\pi$
$859$	−21.3176	−0.727347	−0.363674	+	0.931526i	$0.618478\pi$
$860$	0.0803775	0.00274085
$861$	28.2105	0.961411
$862$	28.0296	0.954693
$863$	21.2984	0.725006	0.362503	−	0.931983i	$-0.381922\pi$
$863$	21.2984	0.725006	0.362503	+	0.931983i	$0.381922\pi$
$864$	−3.98791	−0.135671
$865$	18.4760	0.628202
$866$	34.5217	1.17309
$867$	1.00000	0.0339618
$868$	−8.73389	−0.296447
$869$	9.53184	0.323346
$870$	2.57680	0.0873616
$871$	0	0
$872$	55.3107	1.87306
$873$	4.98862	0.168839
$874$	−53.2532	−1.80132
$875$	29.2917	0.990239
$876$	−4.91984	−0.166226
$877$	−11.8563	−0.400358	−0.200179	−	0.979759i	$-0.564152\pi$
$877$	−11.8563	−0.400358	−0.200179	+	0.979759i	$0.564152\pi$
$878$	−4.10044	−0.138383
$879$	4.17655	0.140872
$880$	5.96103	0.200946
$881$	−55.3017	−1.86316	−0.931581	−	0.363533i	$-0.881570\pi$
$881$	−55.3017	−1.86316	−0.931581	+	0.363533i	$0.881570\pi$
$882$	−4.23571	−0.142624
$883$	−20.3031	−0.683252	−0.341626	−	0.939836i	$-0.610978\pi$
$883$	−20.3031	−0.683252	−0.341626	+	0.939836i	$0.610978\pi$
$884$	0	0
$885$	−5.55250	−0.186645
$886$	−37.2172	−1.25034
$887$	−37.1669	−1.24794	−0.623972	−	0.781447i	$-0.714482\pi$
$887$	−37.1669	−1.24794	−0.623972	+	0.781447i	$0.714482\pi$
$888$	−2.93493	−0.0984899
$889$	36.6202	1.22820
$890$	7.99663	0.268048
$891$	−3.11937	−0.104503
$892$	−16.8810	−0.565219
$893$	−25.0383	−0.837874
$894$	11.1071	0.371478
$895$	−10.4535	−0.349424
$896$	4.35848	0.145607
$897$	0	0
$898$	8.61933	0.287631
$899$	8.25963	0.275474
$900$	3.02188	0.100729
$901$	−9.23496	−0.307661
$902$	−29.9385	−0.996844
$903$	0.355716	0.0118375
$904$	48.3605	1.60845
$905$	−18.9976	−0.631503
$906$	15.3548	0.510129
$907$	3.65237	0.121275	0.0606374	−	0.998160i	$-0.480687\pi$
$907$	3.65237	0.121275	0.0606374	+	0.998160i	$0.480687\pi$
$908$	7.89030	0.261849
$909$	7.20874	0.239099
$910$	0	0
$911$	49.0501	1.62510	0.812551	−	0.582890i	$-0.198078\pi$
$911$	49.0501	1.62510	0.812551	+	0.582890i	$0.198078\pi$
$912$	15.5436	0.514700
$913$	−18.7935	−0.621974
$914$	−47.7067	−1.57800
$915$	7.67858	0.253846
$916$	−0.118229	−0.00390640
$917$	55.2224	1.82360
$918$	1.11755	0.0368847
$919$	13.4967	0.445214	0.222607	−	0.974908i	$-0.428543\pi$
$919$	13.4967	0.445214	0.222607	+	0.974908i	$0.428543\pi$
$920$	−18.0116	−0.593824
$921$	−28.5863	−0.941951
$922$	−4.52937	−0.149167
$923$	0	0
$924$	−7.69601	−0.253180
$925$	3.84079	0.126284
$926$	40.6852	1.33700
$927$	9.13714	0.300103
$928$	9.30460	0.305438
$929$	36.6838	1.20356	0.601779	−	0.798663i	$-0.294459\pi$
$929$	36.6838	1.20356	0.601779	+	0.798663i	$0.294459\pi$
$930$	3.90963	0.128202
$931$	−30.4660	−0.998484
$932$	−6.47733	−0.212172
$933$	15.9824	0.523240
$934$	41.9317	1.37205
$935$	3.08267	0.100814
$936$	0	0
$937$	−23.2732	−0.760302	−0.380151	−	0.924924i	$-0.624128\pi$
$937$	−23.2732	−0.760302	−0.380151	+	0.924924i	$0.624128\pi$
$938$	46.6488	1.52314
$939$	15.2575	0.497910
$940$	−2.31202	−0.0754099
$941$	−13.2488	−0.431900	−0.215950	−	0.976404i	$-0.569285\pi$
$941$	−13.2488	−0.431900	−0.215950	+	0.976404i	$0.569285\pi$
$942$	8.30495	0.270590
$943$	50.9117	1.65791
$944$	10.8649	0.353621
$945$	−3.24619	−0.105599
$946$	−0.377506	−0.0122738
$947$	50.1700	1.63031	0.815154	−	0.579245i	$-0.196652\pi$
$947$	50.1700	1.63031	0.815154	+	0.579245i	$0.196652\pi$
$948$	2.29507	0.0745403
$949$	0	0
$950$	−36.1424	−1.17261
$951$	−22.5889	−0.732495
$952$	10.0991	0.327315
$953$	55.6743	1.80347	0.901734	−	0.432292i	$-0.142295\pi$
$953$	55.6743	1.80347	0.901734	+	0.432292i	$0.142295\pi$
$954$	−10.3205	−0.334140
$955$	12.7113	0.411328
$956$	17.1069	0.553278
$957$	7.27812	0.235268
$958$	−39.9609	−1.29108
$959$	−5.23130	−0.168927
$960$	8.22620	0.265500
$961$	−18.4681	−0.595746
$962$	0	0
$963$	14.7118	0.474083
$964$	−8.34202	−0.268678
$965$	−9.55413	−0.307558
$966$	−21.7622	−0.700188
$967$	−38.5524	−1.23976	−0.619880	−	0.784696i	$-0.712819\pi$
$967$	−38.5524	−1.23976	−0.619880	+	0.784696i	$0.712819\pi$
$968$	−3.90319	−0.125453
$969$	8.03817	0.258223
$970$	−5.50945	−0.176898
$971$	−29.1352	−0.934994	−0.467497	−	0.883995i	$-0.654844\pi$
$971$	−29.1352	−0.934994	−0.467497	+	0.883995i	$0.654844\pi$
$972$	−0.751078	−0.0240908
$973$	6.59353	0.211379
$974$	−16.0871	−0.515464
$975$	0	0
$976$	−15.0250	−0.480940
$977$	19.2759	0.616692	0.308346	−	0.951274i	$-0.400225\pi$
$977$	19.2759	0.616692	0.308346	+	0.951274i	$0.400225\pi$
$978$	−1.76431	−0.0564164
$979$	22.5864	0.721863
$980$	−2.81322	−0.0898651
$981$	17.9903	0.574386
$982$	−17.1489	−0.547244
$983$	61.8016	1.97116	0.985582	−	0.169196i	$-0.0541170\pi$
$983$	61.8016	1.97116	0.985582	+	0.169196i	$0.0541170\pi$
$984$	−26.4038	−0.841724
$985$	−5.67920	−0.180954
$986$	−2.60748	−0.0830389
$987$	−10.2320	−0.325689
$988$	0	0
$989$	0.641964	0.0204133
$990$	3.44504	0.109491
$991$	−22.8915	−0.727172	−0.363586	−	0.931561i	$-0.618448\pi$
$991$	−22.8915	−0.727172	−0.363586	+	0.931561i	$0.618448\pi$
$992$	14.1174	0.448226
$993$	−24.2519	−0.769611
$994$	−42.1869	−1.33809
$995$	2.39724	0.0759975
$996$	−4.52508	−0.143383
$997$	18.2793	0.578910	0.289455	−	0.957192i	$-0.406526\pi$
$997$	18.2793	0.578910	0.289455	+	0.957192i	$0.406526\pi$
$998$	44.0215	1.39348
$999$	−0.954614	−0.0302026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.bk.1.4		11
13.4	even	6		663.2.i.h.562.4	yes	22
13.10	even	6		663.2.i.h.256.4	✓	22
13.12	even	2		8619.2.a.bl.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.2.i.h.256.4	✓	22	13.10	even	6
663.2.i.h.562.4	yes	22	13.4	even	6
8619.2.a.bk.1.4		11	1.1	even	1	trivial
8619.2.a.bl.1.8		11	13.12	even	2

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:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

\(f(q)\)	\(=\)	\(q-1.11755 q^{2} +1.00000 q^{3} -0.751078 q^{4} +0.988234 q^{5} -1.11755 q^{6} -3.28484 q^{7} +3.07447 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.11755 q^{2} +1.00000 q^{3} -0.751078 q^{4} +0.988234 q^{5} -1.11755 q^{6} -3.28484 q^{7} +3.07447 q^{8} +1.00000 q^{9} -1.10440 q^{10} -3.11937 q^{11} -0.751078 q^{12} +3.67098 q^{14} +0.988234 q^{15} -1.93373 q^{16} -1.00000 q^{17} -1.11755 q^{18} -8.03817 q^{19} -0.742241 q^{20} -3.28484 q^{21} +3.48605 q^{22} -5.92818 q^{23} +3.07447 q^{24} -4.02339 q^{25} +1.00000 q^{27} +2.46717 q^{28} -2.33320 q^{29} -1.10440 q^{30} -3.54004 q^{31} -3.98791 q^{32} -3.11937 q^{33} +1.11755 q^{34} -3.24619 q^{35} -0.751078 q^{36} -0.954614 q^{37} +8.98307 q^{38} +3.03830 q^{40} -8.58809 q^{41} +3.67098 q^{42} -0.108290 q^{43} +2.34289 q^{44} +0.988234 q^{45} +6.62505 q^{46} +3.11492 q^{47} -1.93373 q^{48} +3.79017 q^{49} +4.49635 q^{50} -1.00000 q^{51} +9.23496 q^{53} -1.11755 q^{54} -3.08267 q^{55} -10.0991 q^{56} -8.03817 q^{57} +2.60748 q^{58} -5.61861 q^{59} -0.742241 q^{60} +7.77000 q^{61} +3.95618 q^{62} -3.28484 q^{63} +8.32414 q^{64} +3.48605 q^{66} +12.7075 q^{67} +0.751078 q^{68} -5.92818 q^{69} +3.62779 q^{70} -11.4920 q^{71} +3.07447 q^{72} +6.55037 q^{73} +1.06683 q^{74} -4.02339 q^{75} +6.03729 q^{76} +10.2466 q^{77} -3.05570 q^{79} -1.91097 q^{80} +1.00000 q^{81} +9.59763 q^{82} +6.02478 q^{83} +2.46717 q^{84} -0.988234 q^{85} +0.121020 q^{86} -2.33320 q^{87} -9.59041 q^{88} -7.24068 q^{89} -1.10440 q^{90} +4.45252 q^{92} -3.54004 q^{93} -3.48109 q^{94} -7.94359 q^{95} -3.98791 q^{96} +4.98862 q^{97} -4.23571 q^{98} -3.11937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{3} + 12 q^{4} - 3 q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^3 + 12 * q^4 - 3 * q^7 + 11 * q^9	\( 11 q + 11 q^{3} + 12 q^{4} - 3 q^{7} + 11 q^{9} - 9 q^{10} - 2 q^{11} + 12 q^{12} + 21 q^{14} + 18 q^{16} - 11 q^{17} + 21 q^{20} - 3 q^{21} + 3 q^{22} + 18 q^{23} + 7 q^{25} + 11 q^{27} - 19 q^{28} + 15 q^{29} - 9 q^{30} + 13 q^{31} - 35 q^{32} - 2 q^{33} + 15 q^{35} + 12 q^{36} + 18 q^{37} + 5 q^{38} + 12 q^{40} + 21 q^{42} - 11 q^{43} - 3 q^{44} + 29 q^{46} - 11 q^{47} + 18 q^{48} + 28 q^{49} + 2 q^{50} - 11 q^{51} + 19 q^{53} + 13 q^{55} + 43 q^{56} - 68 q^{58} + 19 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} + 20 q^{64} + 3 q^{66} - 2 q^{67} - 12 q^{68} + 18 q^{69} + 58 q^{70} - 11 q^{71} - 30 q^{73} + 35 q^{74} + 7 q^{75} + 10 q^{76} + 20 q^{77} - 14 q^{79} + 43 q^{80} + 11 q^{81} - 35 q^{82} - 5 q^{83} - 19 q^{84} + 11 q^{86} + 15 q^{87} - 5 q^{88} - 3 q^{89} - 9 q^{90} + 54 q^{92} + 13 q^{93} + 55 q^{94} + 26 q^{95} - 35 q^{96} - 31 q^{97} + 54 q^{98} - 2 q^{99}+O(q^{100}) \) 11 * q + 11 * q^3 + 12 * q^4 - 3 * q^7 + 11 * q^9 - 9 * q^10 - 2 * q^11 + 12 * q^12 + 21 * q^14 + 18 * q^16 - 11 * q^17 + 21 * q^20 - 3 * q^21 + 3 * q^22 + 18 * q^23 + 7 * q^25 + 11 * q^27 - 19 * q^28 + 15 * q^29 - 9 * q^30 + 13 * q^31 - 35 * q^32 - 2 * q^33 + 15 * q^35 + 12 * q^36 + 18 * q^37 + 5 * q^38 + 12 * q^40 + 21 * q^42 - 11 * q^43 - 3 * q^44 + 29 * q^46 - 11 * q^47 + 18 * q^48 + 28 * q^49 + 2 * q^50 - 11 * q^51 + 19 * q^53 + 13 * q^55 + 43 * q^56 - 68 * q^58 + 19 * q^59 + 21 * q^60 + 18 * q^61 + 6 * q^62 - 3 * q^63 + 20 * q^64 + 3 * q^66 - 2 * q^67 - 12 * q^68 + 18 * q^69 + 58 * q^70 - 11 * q^71 - 30 * q^73 + 35 * q^74 + 7 * q^75 + 10 * q^76 + 20 * q^77 - 14 * q^79 + 43 * q^80 + 11 * q^81 - 35 * q^82 - 5 * q^83 - 19 * q^84 + 11 * q^86 + 15 * q^87 - 5 * q^88 - 3 * q^89 - 9 * q^90 + 54 * q^92 + 13 * q^93 + 55 * q^94 + 26 * q^95 - 35 * q^96 - 31 * q^97 + 54 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more