LMFDB - Embedded newform 138.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,4,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("138.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	138.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:	$8.14226358079$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +15.3137 q^{5} -6.00000 q^{6} -5.31371 q^{7} +8.00000 q^{8} +9.00000 q^{9} +30.6274 q^{10} -21.2548 q^{11} -12.0000 q^{12} +87.2548 q^{13} -10.6274 q^{14} -45.9411 q^{15} +16.0000 q^{16} +18.0589 q^{17} +18.0000 q^{18} +47.9411 q^{19} +61.2548 q^{20} +15.9411 q^{21} -42.5097 q^{22} +23.0000 q^{23} -24.0000 q^{24} +109.510 q^{25} +174.510 q^{26} -27.0000 q^{27} -21.2548 q^{28} +179.137 q^{29} -91.8823 q^{30} -117.137 q^{31} +32.0000 q^{32} +63.7645 q^{33} +36.1177 q^{34} -81.3726 q^{35} +36.0000 q^{36} -410.902 q^{37} +95.8823 q^{38} -261.765 q^{39} +122.510 q^{40} -205.765 q^{41} +31.8823 q^{42} -149.549 q^{43} -85.0193 q^{44} +137.823 q^{45} +46.0000 q^{46} -299.882 q^{47} -48.0000 q^{48} -314.765 q^{49} +219.019 q^{50} -54.1766 q^{51} +349.019 q^{52} -295.862 q^{53} -54.0000 q^{54} -325.490 q^{55} -42.5097 q^{56} -143.823 q^{57} +358.274 q^{58} +737.450 q^{59} -183.765 q^{60} +550.431 q^{61} -234.274 q^{62} -47.8234 q^{63} +64.0000 q^{64} +1336.20 q^{65} +127.529 q^{66} -527.470 q^{67} +72.2355 q^{68} -69.0000 q^{69} -162.745 q^{70} -297.568 q^{71} +72.0000 q^{72} -714.313 q^{73} -821.803 q^{74} -328.529 q^{75} +191.765 q^{76} +112.942 q^{77} -523.529 q^{78} -834.686 q^{79} +245.019 q^{80} +81.0000 q^{81} -411.529 q^{82} +795.606 q^{83} +63.7645 q^{84} +276.548 q^{85} -299.098 q^{86} -537.411 q^{87} -170.039 q^{88} +46.9605 q^{89} +275.647 q^{90} -463.647 q^{91} +92.0000 q^{92} +351.411 q^{93} -599.765 q^{94} +734.156 q^{95} -96.0000 q^{96} +1037.72 q^{97} -629.529 q^{98} -191.294 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 8 q^{5} - 12 q^{6} + 12 q^{7} + 16 q^{8} + 18 q^{9} + 16 q^{10} + 48 q^{11} - 24 q^{12} + 84 q^{13} + 24 q^{14} - 24 q^{15} + 32 q^{16} + 104 q^{17} + 36 q^{18} + 28 q^{19}+ \cdots + 432 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 8 * q^5 - 12 * q^6 + 12 * q^7 + 16 * q^8 + 18 * q^9 + 16 * q^10 + 48 * q^11 - 24 * q^12 + 84 * q^13 + 24 * q^14 - 24 * q^15 + 32 * q^16 + 104 * q^17 + 36 * q^18 + 28 * q^19 + 32 * q^20 - 36 * q^21 + 96 * q^22 + 46 * q^23 - 48 * q^24 + 38 * q^25 + 168 * q^26 - 54 * q^27 + 48 * q^28 + 132 * q^29 - 48 * q^30 - 8 * q^31 + 64 * q^32 - 144 * q^33 + 208 * q^34 - 208 * q^35 + 72 * q^36 - 324 * q^37 + 56 * q^38 - 252 * q^39 + 64 * q^40 - 140 * q^41 - 72 * q^42 - 548 * q^43 + 192 * q^44 + 72 * q^45 + 92 * q^46 - 464 * q^47 - 96 * q^48 - 358 * q^49 + 76 * q^50 - 312 * q^51 + 336 * q^52 + 336 * q^53 - 108 * q^54 - 832 * q^55 + 96 * q^56 - 84 * q^57 + 264 * q^58 + 72 * q^59 - 96 * q^60 + 60 * q^61 - 16 * q^62 + 108 * q^63 + 128 * q^64 + 1360 * q^65 - 288 * q^66 - 444 * q^67 + 416 * q^68 - 138 * q^69 - 416 * q^70 + 672 * q^71 + 144 * q^72 - 252 * q^73 - 648 * q^74 - 114 * q^75 + 112 * q^76 + 1312 * q^77 - 504 * q^78 - 1692 * q^79 + 128 * q^80 + 162 * q^81 - 280 * q^82 - 400 * q^83 - 144 * q^84 - 352 * q^85 - 1096 * q^86 - 396 * q^87 + 384 * q^88 - 336 * q^89 + 144 * q^90 - 520 * q^91 + 184 * q^92 + 24 * q^93 - 928 * q^94 + 880 * q^95 - 192 * q^96 + 220 * q^97 - 716 * q^98 + 432 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	15.3137	1.36970	0.684850	−	0.728684i	$-0.259868\pi$
$5$	15.3137	1.36970	0.684850	+	0.728684i	$0.259868\pi$
$6$	−6.00000	−0.408248
$7$	−5.31371	−0.286913	−0.143457	−	0.989657i	$-0.545822\pi$
$7$	−5.31371	−0.286913	−0.143457	+	0.989657i	$0.545822\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	30.6274	0.968524
$11$	−21.2548	−0.582598	−0.291299	−	0.956632i	$-0.594087\pi$
$11$	−21.2548	−0.582598	−0.291299	+	0.956632i	$0.594087\pi$
$12$	−12.0000	−0.288675
$13$	87.2548	1.86155	0.930774	−	0.365594i	$-0.119134\pi$
$13$	87.2548	1.86155	0.930774	+	0.365594i	$0.119134\pi$
$14$	−10.6274	−0.202878
$15$	−45.9411	−0.790797
$16$	16.0000	0.250000
$17$	18.0589	0.257642	0.128821	−	0.991668i	$-0.458881\pi$
$17$	18.0589	0.257642	0.128821	+	0.991668i	$0.458881\pi$
$18$	18.0000	0.235702
$19$	47.9411	0.578866	0.289433	−	0.957198i	$-0.406533\pi$
$19$	47.9411	0.578866	0.289433	+	0.957198i	$0.406533\pi$
$20$	61.2548	0.684850
$21$	15.9411	0.165649
$22$	−42.5097	−0.411959
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−24.0000	−0.204124
$25$	109.510	0.876077
$26$	174.510	1.31631
$27$	−27.0000	−0.192450
$28$	−21.2548	−0.143457
$29$	179.137	1.14707	0.573533	−	0.819182i	$-0.305572\pi$
$29$	179.137	1.14707	0.573533	+	0.819182i	$0.305572\pi$
$30$	−91.8823	−0.559178
$31$	−117.137	−0.678659	−0.339330	−	0.940668i	$-0.610200\pi$
$31$	−117.137	−0.678659	−0.339330	+	0.940668i	$0.610200\pi$
$32$	32.0000	0.176777
$33$	63.7645	0.336363
$34$	36.1177	0.182181
$35$	−81.3726	−0.392985
$36$	36.0000	0.166667
$37$	−410.902	−1.82572	−0.912862	−	0.408268i	$-0.866133\pi$
$37$	−410.902	−1.82572	−0.912862	+	0.408268i	$0.866133\pi$
$38$	95.8823	0.409320
$39$	−261.765	−1.07477
$40$	122.510	0.484262
$41$	−205.765	−0.783781	−0.391890	−	0.920012i	$-0.628179\pi$
$41$	−205.765	−0.783781	−0.391890	+	0.920012i	$0.628179\pi$
$42$	31.8823	0.117132
$43$	−149.549	−0.530373	−0.265187	−	0.964197i	$-0.585434\pi$
$43$	−149.549	−0.530373	−0.265187	+	0.964197i	$0.585434\pi$
$44$	−85.0193	−0.291299
$45$	137.823	0.456567
$46$	46.0000	0.147442
$47$	−299.882	−0.930688	−0.465344	−	0.885130i	$-0.654069\pi$
$47$	−299.882	−0.930688	−0.465344	+	0.885130i	$0.654069\pi$
$48$	−48.0000	−0.144338
$49$	−314.765	−0.917681
$50$	219.019	0.619480
$51$	−54.1766	−0.148750
$52$	349.019	0.930774
$53$	−295.862	−0.766788	−0.383394	−	0.923585i	$-0.625245\pi$
$53$	−295.862	−0.766788	−0.383394	+	0.923585i	$0.625245\pi$
$54$	−54.0000	−0.136083
$55$	−325.490	−0.797984
$56$	−42.5097	−0.101439
$57$	−143.823	−0.334208
$58$	358.274	0.811098
$59$	737.450	1.62725	0.813625	−	0.581389i	$-0.197491\pi$
$59$	737.450	1.62725	0.813625	+	0.581389i	$0.197491\pi$
$60$	−183.765	−0.395398
$61$	550.431	1.15533	0.577667	−	0.816272i	$-0.303963\pi$
$61$	550.431	1.15533	0.577667	+	0.816272i	$0.303963\pi$
$62$	−234.274	−0.479885
$63$	−47.8234	−0.0956378
$64$	64.0000	0.125000
$65$	1336.20	2.54976
$66$	127.529	0.237844
$67$	−527.470	−0.961802	−0.480901	−	0.876775i	$-0.659690\pi$
$67$	−527.470	−0.961802	−0.480901	+	0.876775i	$0.659690\pi$
$68$	72.2355	0.128821
$69$	−69.0000	−0.120386
$70$	−162.745	−0.277882
$71$	−297.568	−0.497391	−0.248696	−	0.968582i	$-0.580002\pi$
$71$	−297.568	−0.497391	−0.248696	+	0.968582i	$0.580002\pi$
$72$	72.0000	0.117851
$73$	−714.313	−1.14526	−0.572630	−	0.819814i	$-0.694077\pi$
$73$	−714.313	−1.14526	−0.572630	+	0.819814i	$0.694077\pi$
$74$	−821.803	−1.29098
$75$	−328.529	−0.505803
$76$	191.765	0.289433
$77$	112.942	0.167155
$78$	−523.529	−0.759974
$79$	−834.686	−1.18873	−0.594364	−	0.804196i	$-0.702596\pi$
$79$	−834.686	−1.18873	−0.594364	+	0.804196i	$0.702596\pi$
$80$	245.019	0.342425
$81$	81.0000	0.111111
$82$	−411.529	−0.554217
$83$	795.606	1.05216	0.526079	−	0.850436i	$-0.323662\pi$
$83$	795.606	1.05216	0.526079	+	0.850436i	$0.323662\pi$
$84$	63.7645	0.0828247
$85$	276.548	0.352893
$86$	−299.098	−0.375030
$87$	−537.411	−0.662259
$88$	−170.039	−0.205979
$89$	46.9605	0.0559303	0.0279652	−	0.999609i	$-0.491097\pi$
$89$	46.9605	0.0559303	0.0279652	+	0.999609i	$0.491097\pi$
$90$	275.647	0.322841
$91$	−463.647	−0.534103
$92$	92.0000	0.104257
$93$	351.411	0.391824
$94$	−599.765	−0.658096
$95$	734.156	0.792872
$96$	−96.0000	−0.102062
$97$	1037.72	1.08624	0.543118	−	0.839656i	$-0.317244\pi$
$97$	1037.72	1.08624	0.543118	+	0.839656i	$0.317244\pi$
$98$	−629.529	−0.648898
$99$	−191.294	−0.194199
$100$	438.039	0.438039
$101$	−374.548	−0.369000	−0.184500	−	0.982833i	$-0.559066\pi$
$101$	−374.548	−0.369000	−0.184500	+	0.982833i	$0.559066\pi$
$102$	−108.353	−0.105182
$103$	−408.883	−0.391150	−0.195575	−	0.980689i	$-0.562657\pi$
$103$	−408.883	−0.391150	−0.195575	+	0.980689i	$0.562657\pi$
$104$	698.039	0.658157
$105$	244.118	0.226890
$106$	−591.724	−0.542201
$107$	1266.51	1.14428	0.572141	−	0.820155i	$-0.306113\pi$
$107$	1266.51	1.14428	0.572141	+	0.820155i	$0.306113\pi$
$108$	−108.000	−0.0962250
$109$	−1367.57	−1.20174	−0.600868	−	0.799348i	$-0.705178\pi$
$109$	−1367.57	−1.20174	−0.600868	+	0.799348i	$0.705178\pi$
$110$	−650.981	−0.564260
$111$	1232.70	1.05408
$112$	−85.0193	−0.0717283
$113$	94.5703	0.0787294	0.0393647	−	0.999225i	$-0.487467\pi$
$113$	94.5703	0.0787294	0.0393647	+	0.999225i	$0.487467\pi$
$114$	−287.647	−0.236321
$115$	352.215	0.285602
$116$	716.548	0.573533
$117$	785.294	0.620516
$118$	1474.90	1.15064
$119$	−95.9596	−0.0739210
$120$	−367.529	−0.279589
$121$	−879.232	−0.660580
$122$	1100.86	0.816945
$123$	617.294	0.452516
$124$	−468.548	−0.339330
$125$	−237.214	−0.169737
$126$	−95.6468	−0.0676261
$127$	2078.47	1.45224	0.726119	−	0.687569i	$-0.241322\pi$
$127$	2078.47	1.45224	0.726119	+	0.687569i	$0.241322\pi$
$128$	128.000	0.0883883
$129$	448.648	0.306211
$130$	2672.39	1.80295
$131$	749.568	0.499924	0.249962	−	0.968256i	$-0.419582\pi$
$131$	749.568	0.499924	0.249962	+	0.968256i	$0.419582\pi$
$132$	255.058	0.168181
$133$	−254.745	−0.166084
$134$	−1054.94	−0.680097
$135$	−413.470	−0.263599
$136$	144.471	0.0910903
$137$	1443.04	0.899905	0.449953	−	0.893052i	$-0.351441\pi$
$137$	1443.04	0.899905	0.449953	+	0.893052i	$0.351441\pi$
$138$	−138.000	−0.0851257
$139$	−2228.63	−1.35992	−0.679962	−	0.733247i	$-0.738004\pi$
$139$	−2228.63	−1.35992	−0.679962	+	0.733247i	$0.738004\pi$
$140$	−325.490	−0.196493
$141$	899.647	0.537333
$142$	−595.135	−0.351709
$143$	−1854.59	−1.08453
$144$	144.000	0.0833333
$145$	2743.25	1.57114
$146$	−1428.63	−0.809821
$147$	944.294	0.529823
$148$	−1643.61	−0.912862
$149$	1486.29	0.817193	0.408596	−	0.912715i	$-0.366018\pi$
$149$	1486.29	0.817193	0.408596	+	0.912715i	$0.366018\pi$
$150$	−657.058	−0.357657
$151$	348.865	0.188015	0.0940073	−	0.995572i	$-0.470032\pi$
$151$	348.865	0.188015	0.0940073	+	0.995572i	$0.470032\pi$
$152$	383.529	0.204660
$153$	162.530	0.0858808
$154$	225.884	0.118196
$155$	−1793.80	−0.929560
$156$	−1047.06	−0.537383
$157$	1751.14	0.890165	0.445083	−	0.895489i	$-0.353174\pi$
$157$	1751.14	0.890165	0.445083	+	0.895489i	$0.353174\pi$
$158$	−1669.37	−0.840558
$159$	887.586	0.442705
$160$	490.039	0.242131
$161$	−122.215	−0.0598256
$162$	162.000	0.0785674
$163$	420.743	0.202179	0.101089	−	0.994877i	$-0.467767\pi$
$163$	420.743	0.202179	0.101089	+	0.994877i	$0.467767\pi$
$164$	−823.058	−0.391890
$165$	976.471	0.460716
$166$	1591.21	0.743988
$167$	651.606	0.301933	0.150967	−	0.988539i	$-0.451761\pi$
$167$	651.606	0.301933	0.150967	+	0.988539i	$0.451761\pi$
$168$	127.529	0.0585659
$169$	5416.41	2.46536
$170$	553.097	0.249533
$171$	431.470	0.192955
$172$	−598.197	−0.265187
$173$	−1346.35	−0.591684	−0.295842	−	0.955237i	$-0.595600\pi$
$173$	−1346.35	−0.591684	−0.295842	+	0.955237i	$0.595600\pi$
$174$	−1074.82	−0.468288
$175$	−581.902	−0.251358
$176$	−340.077	−0.145649
$177$	−2212.35	−0.939494
$178$	93.9209	0.0395487
$179$	−3424.82	−1.43007	−0.715037	−	0.699086i	$-0.753590\pi$
$179$	−3424.82	−1.43007	−0.715037	+	0.699086i	$0.753590\pi$
$180$	551.294	0.228283
$181$	−3256.98	−1.33751	−0.668755	−	0.743482i	$-0.733173\pi$
$181$	−3256.98	−1.33751	−0.668755	+	0.743482i	$0.733173\pi$
$182$	−927.294	−0.377668
$183$	−1651.29	−0.667033
$184$	184.000	0.0737210
$185$	−6292.43	−2.50069
$186$	702.823	0.277062
$187$	−383.838	−0.150102
$188$	−1199.53	−0.465344
$189$	143.470	0.0552165
$190$	1468.31	0.560645
$191$	−4033.37	−1.52798	−0.763990	−	0.645228i	$-0.776763\pi$
$191$	−4033.37	−1.52798	−0.763990	+	0.645228i	$0.776763\pi$
$192$	−192.000	−0.0721688
$193$	298.784	0.111435	0.0557174	−	0.998447i	$-0.482255\pi$
$193$	298.784	0.111435	0.0557174	+	0.998447i	$0.482255\pi$
$194$	2075.45	0.768085
$195$	−4008.59	−1.47211
$196$	−1259.06	−0.458840
$197$	4254.19	1.53857	0.769286	−	0.638905i	$-0.220612\pi$
$197$	4254.19	1.53857	0.769286	+	0.638905i	$0.220612\pi$
$198$	−382.587	−0.137320
$199$	−4731.23	−1.68537	−0.842684	−	0.538409i	$-0.819026\pi$
$199$	−4731.23	−1.68537	−0.842684	+	0.538409i	$0.819026\pi$
$200$	876.077	0.309740
$201$	1582.41	0.555297
$202$	−749.097	−0.260922
$203$	−951.882	−0.329109
$204$	−216.706	−0.0743749
$205$	−3151.02	−1.07354
$206$	−817.766	−0.276585
$207$	207.000	0.0695048
$208$	1396.08	0.465387
$209$	−1018.98	−0.337246
$210$	488.235	0.160435
$211$	5131.40	1.67422	0.837110	−	0.547035i	$-0.184243\pi$
$211$	5131.40	1.67422	0.837110	+	0.547035i	$0.184243\pi$
$212$	−1183.45	−0.383394
$213$	892.703	0.287169
$214$	2533.02	0.809129
$215$	−2290.15	−0.726452
$216$	−216.000	−0.0680414
$217$	622.432	0.194716
$218$	−2735.14	−0.849756
$219$	2142.94	0.661216
$220$	−1301.96	−0.398992
$221$	1575.72	0.479614
$222$	2465.41	0.745349
$223$	2189.41	0.657461	0.328730	−	0.944424i	$-0.393379\pi$
$223$	2189.41	0.657461	0.328730	+	0.944424i	$0.393379\pi$
$224$	−170.039	−0.0507196
$225$	985.587	0.292026
$226$	189.141	0.0556701
$227$	−5853.37	−1.71146	−0.855731	−	0.517421i	$-0.826892\pi$
$227$	−5853.37	−1.71146	−0.855731	+	0.517421i	$0.826892\pi$
$228$	−575.294	−0.167104
$229$	−3427.17	−0.988970	−0.494485	−	0.869186i	$-0.664643\pi$
$229$	−3427.17	−0.988970	−0.494485	+	0.869186i	$0.664643\pi$
$230$	704.431	0.201951
$231$	−338.826	−0.0965070
$232$	1433.10	0.405549
$233$	5723.33	1.60922	0.804609	−	0.593805i	$-0.202375\pi$
$233$	5723.33	1.60922	0.804609	+	0.593805i	$0.202375\pi$
$234$	1570.59	0.438771
$235$	−4592.31	−1.27476
$236$	2949.80	0.813625
$237$	2504.06	0.686313
$238$	−191.919	−0.0522701
$239$	3298.04	0.892604	0.446302	−	0.894882i	$-0.352741\pi$
$239$	3298.04	0.892604	0.446302	+	0.894882i	$0.352741\pi$
$240$	−735.058	−0.197699
$241$	−5391.60	−1.44109	−0.720547	−	0.693406i	$-0.756109\pi$
$241$	−5391.60	−1.44109	−0.720547	+	0.693406i	$0.756109\pi$
$242$	−1758.46	−0.467101
$243$	−243.000	−0.0641500
$244$	2201.72	0.577667
$245$	−4820.21	−1.25695
$246$	1234.59	0.319977
$247$	4183.09	1.07759
$248$	−937.097	−0.239942
$249$	−2386.82	−0.607464
$250$	−474.429	−0.120022
$251$	3592.11	0.903315	0.451657	−	0.892191i	$-0.350833\pi$
$251$	3592.11	0.903315	0.451657	+	0.892191i	$0.350833\pi$
$252$	−191.294	−0.0478189
$253$	−488.861	−0.121480
$254$	4156.94	1.02689
$255$	−829.645	−0.203743
$256$	256.000	0.0625000
$257$	5645.61	1.37029	0.685143	−	0.728409i	$-0.259740\pi$
$257$	5645.61	1.37029	0.685143	+	0.728409i	$0.259740\pi$
$258$	897.295	0.216524
$259$	2183.41	0.523825
$260$	5344.78	1.27488
$261$	1612.23	0.382355
$262$	1499.14	0.353500
$263$	304.427	0.0713756	0.0356878	−	0.999363i	$-0.488638\pi$
$263$	304.427	0.0713756	0.0356878	+	0.999363i	$0.488638\pi$
$264$	510.116	0.118922
$265$	−4530.75	−1.05027
$266$	−509.490	−0.117439
$267$	−140.881	−0.0322914
$268$	−2109.88	−0.480901
$269$	2748.79	0.623035	0.311517	−	0.950240i	$-0.399163\pi$
$269$	2748.79	0.623035	0.311517	+	0.950240i	$0.399163\pi$
$270$	−826.940	−0.186393
$271$	−1324.28	−0.296841	−0.148421	−	0.988924i	$-0.547419\pi$
$271$	−1324.28	−0.296841	−0.148421	+	0.988924i	$0.547419\pi$
$272$	288.942	0.0644106
$273$	1390.94	0.308365
$274$	2886.08	0.636329
$275$	−2327.61	−0.510401
$276$	−276.000	−0.0601929
$277$	5660.74	1.22787	0.613937	−	0.789355i	$-0.289585\pi$
$277$	5660.74	1.22787	0.613937	+	0.789355i	$0.289585\pi$
$278$	−4457.25	−0.961612
$279$	−1054.23	−0.226220
$280$	−650.981	−0.138941
$281$	4981.86	1.05763	0.528813	−	0.848739i	$-0.322637\pi$
$281$	4981.86	1.05763	0.528813	+	0.848739i	$0.322637\pi$
$282$	1799.29	0.379952
$283$	−5093.51	−1.06989	−0.534943	−	0.844888i	$-0.679667\pi$
$283$	−5093.51	−1.06989	−0.534943	+	0.844888i	$0.679667\pi$
$284$	−1190.27	−0.248696
$285$	−2202.47	−0.457765
$286$	−3709.17	−0.766881
$287$	1093.37	0.224877
$288$	288.000	0.0589256
$289$	−4586.88	−0.933620
$290$	5486.51	1.11096
$291$	−3113.17	−0.627139
$292$	−2857.25	−0.572630
$293$	−7943.46	−1.58383	−0.791914	−	0.610632i	$-0.790915\pi$
$293$	−7943.46	−1.58383	−0.791914	+	0.610632i	$0.790915\pi$
$294$	1888.59	0.374642
$295$	11293.1	2.22885
$296$	−3287.21	−0.645491
$297$	573.881	0.112121
$298$	2972.58	0.577842
$299$	2006.86	0.388160
$300$	−1314.12	−0.252902
$301$	794.661	0.152171
$302$	697.729	0.132946
$303$	1123.65	0.213042
$304$	767.058	0.144716
$305$	8429.13	1.58246
$306$	325.060	0.0607269
$307$	7444.59	1.38399	0.691995	−	0.721902i	$-0.256732\pi$
$307$	7444.59	1.38399	0.691995	+	0.721902i	$0.256732\pi$
$308$	451.768	0.0835775
$309$	1226.65	0.225831
$310$	−3587.61	−0.657298
$311$	−6584.66	−1.20058	−0.600292	−	0.799781i	$-0.704949\pi$
$311$	−6584.66	−1.20058	−0.600292	+	0.799781i	$0.704949\pi$
$312$	−2094.12	−0.379987
$313$	2790.86	0.503990	0.251995	−	0.967728i	$-0.418913\pi$
$313$	2790.86	0.503990	0.251995	+	0.967728i	$0.418913\pi$
$314$	3502.27	0.629442
$315$	−732.353	−0.130995
$316$	−3338.75	−0.594364
$317$	−2083.69	−0.369184	−0.184592	−	0.982815i	$-0.559096\pi$
$317$	−2083.69	−0.369184	−0.184592	+	0.982815i	$0.559096\pi$
$318$	1775.17	0.313040
$319$	−3807.53	−0.668278
$320$	980.077	0.171212
$321$	−3799.53	−0.660651
$322$	−244.431	−0.0423031
$323$	865.763	0.149140
$324$	324.000	0.0555556
$325$	9555.25	1.63086
$326$	841.487	0.142962
$327$	4102.70	0.693823
$328$	−1646.12	−0.277108
$329$	1593.49	0.267027
$330$	1952.94	0.325776
$331$	4687.57	0.778405	0.389203	−	0.921152i	$-0.372751\pi$
$331$	4687.57	0.778405	0.389203	+	0.921152i	$0.372751\pi$
$332$	3182.43	0.526079
$333$	−3698.11	−0.608575
$334$	1303.21	0.213499
$335$	−8077.52	−1.31738
$336$	255.058	0.0414124
$337$	−5987.61	−0.967851	−0.483926	−	0.875109i	$-0.660789\pi$
$337$	−5987.61	−0.967851	−0.483926	+	0.875109i	$0.660789\pi$
$338$	10832.8	1.74328
$339$	−283.711	−0.0454544
$340$	1106.19	0.176446
$341$	2489.73	0.395385
$342$	862.940	0.136440
$343$	3495.17	0.550208
$344$	−1196.39	−0.187515
$345$	−1056.65	−0.164892
$346$	−2692.71	−0.418384
$347$	−3378.30	−0.522642	−0.261321	−	0.965252i	$-0.584158\pi$
$347$	−3378.30	−0.522642	−0.261321	+	0.965252i	$0.584158\pi$
$348$	−2149.65	−0.331130
$349$	−844.271	−0.129492	−0.0647461	−	0.997902i	$-0.520624\pi$
$349$	−844.271	−0.129492	−0.0647461	+	0.997902i	$0.520624\pi$
$350$	−1163.80	−0.177737
$351$	−2355.88	−0.358255
$352$	−680.155	−0.102990
$353$	3167.13	0.477534	0.238767	−	0.971077i	$-0.423257\pi$
$353$	3167.13	0.477534	0.238767	+	0.971077i	$0.423257\pi$
$354$	−4424.70	−0.664322
$355$	−4556.86	−0.681277
$356$	187.842	0.0279652
$357$	287.879	0.0426783
$358$	−6849.65	−1.01122
$359$	10484.0	1.54130	0.770650	−	0.637259i	$-0.219932\pi$
$359$	10484.0	1.54130	0.770650	+	0.637259i	$0.219932\pi$
$360$	1102.59	0.161421
$361$	−4560.65	−0.664914
$362$	−6513.96	−0.945763
$363$	2637.70	0.381386
$364$	−1854.59	−0.267052
$365$	−10938.8	−1.56866
$366$	−3302.58	−0.471663
$367$	2349.00	0.334106	0.167053	−	0.985948i	$-0.446575\pi$
$367$	2349.00	0.334106	0.167053	+	0.985948i	$0.446575\pi$
$368$	368.000	0.0521286
$369$	−1851.88	−0.261260
$370$	−12584.9	−1.76826
$371$	1572.12	0.220002
$372$	1405.65	0.195912
$373$	−12076.3	−1.67637	−0.838187	−	0.545384i	$-0.816384\pi$
$373$	−12076.3	−1.67637	−0.838187	+	0.545384i	$0.816384\pi$
$374$	−767.677	−0.106138
$375$	711.643	0.0979976
$376$	−2399.06	−0.329048
$377$	15630.6	2.13532
$378$	286.940	0.0390440
$379$	−1987.35	−0.269349	−0.134674	−	0.990890i	$-0.542999\pi$
$379$	−1987.35	−0.269349	−0.134674	+	0.990890i	$0.542999\pi$
$380$	2936.63	0.396436
$381$	−6235.41	−0.838450
$382$	−8066.74	−1.08045
$383$	10474.6	1.39746	0.698731	−	0.715385i	$-0.253749\pi$
$383$	10474.6	1.39746	0.698731	+	0.715385i	$0.253749\pi$
$384$	−384.000	−0.0510310
$385$	1729.56	0.228952
$386$	597.568	0.0787964
$387$	−1345.94	−0.176791
$388$	4150.90	0.543118
$389$	−3978.53	−0.518559	−0.259279	−	0.965802i	$-0.583485\pi$
$389$	−3978.53	−0.518559	−0.259279	+	0.965802i	$0.583485\pi$
$390$	−8017.17	−1.04094
$391$	415.354	0.0537221
$392$	−2518.12	−0.324449
$393$	−2248.70	−0.288631
$394$	8508.38	1.08793
$395$	−12782.1	−1.62820
$396$	−765.174	−0.0970996
$397$	5539.81	0.700340	0.350170	−	0.936686i	$-0.386124\pi$
$397$	5539.81	0.700340	0.350170	+	0.936686i	$0.386124\pi$
$398$	−9462.46	−1.19173
$399$	764.235	0.0958888
$400$	1752.15	0.219019
$401$	2345.20	0.292054	0.146027	−	0.989281i	$-0.453351\pi$
$401$	2345.20	0.292054	0.146027	+	0.989281i	$0.453351\pi$
$402$	3164.82	0.392654
$403$	−10220.8	−1.26336
$404$	−1498.19	−0.184500
$405$	1240.41	0.152189
$406$	−1903.76	−0.232715
$407$	8733.65	1.06366
$408$	−433.413	−0.0525910
$409$	−7263.96	−0.878190	−0.439095	−	0.898441i	$-0.644701\pi$
$409$	−7263.96	−0.878190	−0.439095	+	0.898441i	$0.644701\pi$
$410$	−6302.04	−0.759111
$411$	−4329.11	−0.519561
$412$	−1635.53	−0.195575
$413$	−3918.59	−0.466880
$414$	414.000	0.0491473
$415$	12183.7	1.44114
$416$	2792.15	0.329078
$417$	6685.88	0.785153
$418$	−2037.96	−0.238469
$419$	4094.89	0.477443	0.238721	−	0.971088i	$-0.423272\pi$
$419$	4094.89	0.477443	0.238721	+	0.971088i	$0.423272\pi$
$420$	976.471	0.113445
$421$	4677.67	0.541510	0.270755	−	0.962648i	$-0.412727\pi$
$421$	4677.67	0.541510	0.270755	+	0.962648i	$0.412727\pi$
$422$	10262.8	1.18385
$423$	−2698.94	−0.310229
$424$	−2366.90	−0.271101
$425$	1977.62	0.225715
$426$	1785.41	0.203059
$427$	−2924.83	−0.331481
$428$	5066.04	0.572141
$429$	5563.76	0.626156
$430$	−4580.31	−0.513679
$431$	13219.0	1.47735	0.738676	−	0.674060i	$-0.235451\pi$
$431$	13219.0	1.47735	0.738676	+	0.674060i	$0.235451\pi$
$432$	−432.000	−0.0481125
$433$	3109.77	0.345141	0.172571	−	0.984997i	$-0.444793\pi$
$433$	3109.77	0.345141	0.172571	+	0.984997i	$0.444793\pi$
$434$	1244.86	0.137685
$435$	−8229.76	−0.907096
$436$	−5470.27	−0.600868
$437$	1102.65	0.120702
$438$	4285.88	0.467551
$439$	16425.0	1.78570	0.892851	−	0.450352i	$-0.148702\pi$
$439$	16425.0	1.78570	0.892851	+	0.450352i	$0.148702\pi$
$440$	−2603.92	−0.282130
$441$	−2832.88	−0.305894
$442$	3151.45	0.339138
$443$	−4736.14	−0.507948	−0.253974	−	0.967211i	$-0.581738\pi$
$443$	−4736.14	−0.507948	−0.253974	+	0.967211i	$0.581738\pi$
$444$	4930.82	0.527041
$445$	719.139	0.0766078
$446$	4378.82	0.464895
$447$	−4458.87	−0.471806
$448$	−340.077	−0.0358642
$449$	−12341.8	−1.29721	−0.648605	−	0.761125i	$-0.724647\pi$
$449$	−12341.8	−1.29721	−0.648605	+	0.761125i	$0.724647\pi$
$450$	1971.17	0.206493
$451$	4373.49	0.456629
$452$	378.281	0.0393647
$453$	−1046.59	−0.108550
$454$	−11706.7	−1.21019
$455$	−7100.15	−0.731561
$456$	−1150.59	−0.118160
$457$	5.80491	0.000594184 0	0.000297092	−	1.00000i	$-0.499905\pi$
$457$	5.80491	0.000594184 0	0.000297092		1.00000i	$0.499905\pi$
$458$	−6854.35	−0.699307
$459$	−487.590	−0.0495833
$460$	1408.86	0.142801
$461$	−11502.8	−1.16213	−0.581063	−	0.813859i	$-0.697363\pi$
$461$	−11502.8	−1.16213	−0.581063	+	0.813859i	$0.697363\pi$
$462$	−677.652	−0.0682407
$463$	11739.5	1.17836	0.589182	−	0.808001i	$-0.299450\pi$
$463$	11739.5	1.17836	0.589182	+	0.808001i	$0.299450\pi$
$464$	2866.19	0.286767
$465$	5381.41	0.536681
$466$	11446.7	1.13789
$467$	−12169.8	−1.20590	−0.602948	−	0.797781i	$-0.706007\pi$
$467$	−12169.8	−1.20590	−0.602948	+	0.797781i	$0.706007\pi$
$468$	3141.17	0.310258
$469$	2802.82	0.275954
$470$	−9184.62	−0.901393
$471$	−5253.41	−0.513937
$472$	5899.60	0.575320
$473$	3178.64	0.308994
$474$	5008.12	0.485296
$475$	5250.02	0.507131
$476$	−383.838	−0.0369605
$477$	−2662.76	−0.255596
$478$	6596.08	0.631166
$479$	9096.85	0.867737	0.433868	−	0.900976i	$-0.357148\pi$
$479$	9096.85	0.867737	0.433868	+	0.900976i	$0.357148\pi$
$480$	−1470.12	−0.139794
$481$	−35853.1	−3.39868
$482$	−10783.2	−1.01901
$483$	366.646	0.0345403
$484$	−3516.93	−0.330290
$485$	15891.4	1.48782
$486$	−486.000	−0.0453609
$487$	18741.0	1.74381	0.871906	−	0.489674i	$-0.162884\pi$
$487$	18741.0	1.74381	0.871906	+	0.489674i	$0.162884\pi$
$488$	4403.44	0.408472
$489$	−1262.23	−0.116728
$490$	−9640.42	−0.888796
$491$	−16903.7	−1.55367	−0.776837	−	0.629702i	$-0.783177\pi$
$491$	−16903.7	−1.55367	−0.776837	+	0.629702i	$0.783177\pi$
$492$	2469.17	0.226258
$493$	3235.01	0.295533
$494$	8366.19	0.761969
$495$	−2929.41	−0.265995
$496$	−1874.19	−0.169665
$497$	1581.19	0.142708
$498$	−4773.64	−0.429542
$499$	−2690.01	−0.241326	−0.120663	−	0.992694i	$-0.538502\pi$
$499$	−2690.01	−0.241326	−0.120663	+	0.992694i	$0.538502\pi$
$500$	−948.858	−0.0848684
$501$	−1954.82	−0.174321
$502$	7184.22	0.638740
$503$	2009.56	0.178135	0.0890675	−	0.996026i	$-0.471611\pi$
$503$	2009.56	0.178135	0.0890675	+	0.996026i	$0.471611\pi$
$504$	−382.587	−0.0338131
$505$	−5735.72	−0.505419
$506$	−977.722	−0.0858993
$507$	−16249.2	−1.42338
$508$	8313.88	0.726119
$509$	16797.6	1.46275	0.731377	−	0.681973i	$-0.238878\pi$
$509$	16797.6	1.46275	0.731377	+	0.681973i	$0.238878\pi$
$510$	−1659.29	−0.144068
$511$	3795.65	0.328590
$512$	512.000	0.0441942
$513$	−1294.41	−0.111403
$514$	11291.2	0.968938
$515$	−6261.52	−0.535758
$516$	1794.59	0.153106
$517$	6373.95	0.542216
$518$	4366.82	0.370400
$519$	4039.06	0.341609
$520$	10689.6	0.901477
$521$	3387.35	0.284841	0.142421	−	0.989806i	$-0.454511\pi$
$521$	3387.35	0.284841	0.142421	+	0.989806i	$0.454511\pi$
$522$	3224.47	0.270366
$523$	−4402.98	−0.368124	−0.184062	−	0.982915i	$-0.558925\pi$
$523$	−4402.98	−0.368124	−0.184062	+	0.982915i	$0.558925\pi$
$524$	2998.27	0.249962
$525$	1745.71	0.145122
$526$	608.854	0.0504702
$527$	−2115.36	−0.174851
$528$	1020.23	0.0840907
$529$	529.000	0.0434783
$530$	−9061.49	−0.742653
$531$	6637.05	0.542417
$532$	−1018.98	−0.0830421
$533$	−17953.9	−1.45905
$534$	−281.763	−0.0228335
$535$	19395.0	1.56732
$536$	−4219.76	−0.340048
$537$	10274.5	0.825654
$538$	5497.57	0.440552
$539$	6690.27	0.534639
$540$	−1653.88	−0.131799
$541$	−7658.32	−0.608608	−0.304304	−	0.952575i	$-0.598424\pi$
$541$	−7658.32	−0.608608	−0.304304	+	0.952575i	$0.598424\pi$
$542$	−2648.55	−0.209899
$543$	9770.94	0.772212
$544$	577.884	0.0455452
$545$	−20942.5	−1.64602
$546$	2781.88	0.218047
$547$	3176.92	0.248327	0.124164	−	0.992262i	$-0.460375\pi$
$547$	3176.92	0.248327	0.124164	+	0.992262i	$0.460375\pi$
$548$	5772.15	0.449953
$549$	4953.88	0.385112
$550$	−4655.22	−0.360908
$551$	8588.03	0.663997
$552$	−552.000	−0.0425628
$553$	4435.28	0.341062
$554$	11321.5	0.868238
$555$	18877.3	1.44378
$556$	−8914.50	−0.679962
$557$	−18450.7	−1.40356	−0.701780	−	0.712393i	$-0.747611\pi$
$557$	−18450.7	−1.40356	−0.701780	+	0.712393i	$0.747611\pi$
$558$	−2108.47	−0.159962
$559$	−13048.9	−0.987315
$560$	−1301.96	−0.0982463
$561$	1151.52	0.0866613
$562$	9963.71	0.747854
$563$	−5515.81	−0.412902	−0.206451	−	0.978457i	$-0.566191\pi$
$563$	−5515.81	−0.412902	−0.206451	+	0.978457i	$0.566191\pi$
$564$	3598.59	0.268666
$565$	1448.22	0.107836
$566$	−10187.0	−0.756523
$567$	−430.410	−0.0318793
$568$	−2380.54	−0.175854
$569$	23452.9	1.72794	0.863968	−	0.503546i	$-0.167972\pi$
$569$	23452.9	1.72794	0.863968	+	0.503546i	$0.167972\pi$
$570$	−4404.94	−0.323689
$571$	17731.8	1.29957	0.649784	−	0.760119i	$-0.274859\pi$
$571$	17731.8	1.29957	0.649784	+	0.760119i	$0.274859\pi$
$572$	−7418.35	−0.542267
$573$	12100.1	0.882180
$574$	2186.75	0.159012
$575$	2518.72	0.182675
$576$	576.000	0.0416667
$577$	−15350.1	−1.10751	−0.553756	−	0.832679i	$-0.686806\pi$
$577$	−15350.1	−1.10751	−0.553756	+	0.832679i	$0.686806\pi$
$578$	−9173.75	−0.660169
$579$	−896.352	−0.0643370
$580$	10973.0	0.785568
$581$	−4227.62	−0.301878
$582$	−6226.34	−0.443454
$583$	6288.50	0.446729
$584$	−5714.50	−0.404911
$585$	12025.8	0.849921
$586$	−15886.9	−1.11994
$587$	23291.6	1.63773	0.818866	−	0.573984i	$-0.194603\pi$
$587$	23291.6	1.63773	0.818866	+	0.573984i	$0.194603\pi$
$588$	3777.17	0.264912
$589$	−5615.68	−0.392853
$590$	22586.2	1.57603
$591$	−12762.6	−0.888295
$592$	−6574.43	−0.456431
$593$	−20205.9	−1.39925	−0.699626	−	0.714509i	$-0.746650\pi$
$593$	−20205.9	−1.39925	−0.699626	+	0.714509i	$0.746650\pi$
$594$	1147.76	0.0792815
$595$	−1469.50	−0.101250
$596$	5945.16	0.408596
$597$	14193.7	0.973047
$598$	4013.72	0.274470
$599$	11173.3	0.762153	0.381077	−	0.924543i	$-0.375553\pi$
$599$	11173.3	0.762153	0.381077	+	0.924543i	$0.375553\pi$
$600$	−2628.23	−0.178829
$601$	−16631.6	−1.12881	−0.564405	−	0.825498i	$-0.690894\pi$
$601$	−16631.6	−1.12881	−0.564405	+	0.825498i	$0.690894\pi$
$602$	1589.32	0.107601
$603$	−4747.23	−0.320601
$604$	1395.46	0.0940073
$605$	−13464.3	−0.904796
$606$	2247.29	0.150643
$607$	−23958.5	−1.60205	−0.801025	−	0.598631i	$-0.795712\pi$
$607$	−23958.5	−1.60205	−0.801025	+	0.598631i	$0.795712\pi$
$608$	1534.12	0.102330
$609$	2855.65	0.190011
$610$	16858.3	1.11897
$611$	−26166.2	−1.73252
$612$	650.119	0.0429404
$613$	25615.0	1.68774	0.843868	−	0.536551i	$-0.180273\pi$
$613$	25615.0	1.68774	0.843868	+	0.536551i	$0.180273\pi$
$614$	14889.2	0.978629
$615$	9453.05	0.619811
$616$	903.536	0.0590982
$617$	−11722.2	−0.764859	−0.382429	−	0.923985i	$-0.624912\pi$
$617$	−11722.2	−0.764859	−0.382429	+	0.923985i	$0.624912\pi$
$618$	2453.30	0.159686
$619$	−18949.8	−1.23047	−0.615233	−	0.788346i	$-0.710938\pi$
$619$	−18949.8	−1.23047	−0.615233	+	0.788346i	$0.710938\pi$
$620$	−7175.21	−0.464780
$621$	−621.000	−0.0401286
$622$	−13169.3	−0.848941
$623$	−249.534	−0.0160472
$624$	−4188.23	−0.268691
$625$	−17321.3	−1.10857
$626$	5581.73	0.356375
$627$	3056.94	0.194709
$628$	7004.55	0.445083
$629$	−7420.42	−0.470384
$630$	−1464.71	−0.0926275
$631$	−1559.47	−0.0983859	−0.0491930	−	0.998789i	$-0.515665\pi$
$631$	−1559.47	−0.0983859	−0.0491930	+	0.998789i	$0.515665\pi$
$632$	−6677.49	−0.420279
$633$	−15394.2	−0.966611
$634$	−4167.37	−0.261053
$635$	31829.1	1.98913
$636$	3550.34	0.221353
$637$	−27464.7	−1.70831
$638$	−7615.06	−0.472544
$639$	−2678.11	−0.165797
$640$	1960.15	0.121065
$641$	−5937.72	−0.365875	−0.182938	−	0.983125i	$-0.558561\pi$
$641$	−5937.72	−0.365875	−0.182938	+	0.983125i	$0.558561\pi$
$642$	−7599.06	−0.467151
$643$	10389.6	0.637209	0.318604	−	0.947888i	$-0.396786\pi$
$643$	10389.6	0.637209	0.318604	+	0.947888i	$0.396786\pi$
$644$	−488.861	−0.0299128
$645$	6870.46	0.419417
$646$	1731.53	0.105458
$647$	23853.1	1.44940	0.724701	−	0.689064i	$-0.241978\pi$
$647$	23853.1	1.44940	0.724701	+	0.689064i	$0.241978\pi$
$648$	648.000	0.0392837
$649$	−15674.4	−0.948032
$650$	19110.5	1.15319
$651$	−1867.30	−0.112420
$652$	1682.97	0.101089
$653$	13748.3	0.823912	0.411956	−	0.911204i	$-0.364846\pi$
$653$	13748.3	0.823912	0.411956	+	0.911204i	$0.364846\pi$
$654$	8205.41	0.490607
$655$	11478.7	0.684746
$656$	−3292.23	−0.195945
$657$	−6428.82	−0.381753
$658$	3186.97	0.188816
$659$	−22821.1	−1.34899	−0.674495	−	0.738279i	$-0.735639\pi$
$659$	−22821.1	−1.34899	−0.674495	+	0.738279i	$0.735639\pi$
$660$	3905.88	0.230358
$661$	−14131.9	−0.831567	−0.415783	−	0.909464i	$-0.636493\pi$
$661$	−14131.9	−0.831567	−0.415783	+	0.909464i	$0.636493\pi$
$662$	9375.14	0.550416
$663$	−4727.17	−0.276905
$664$	6364.85	0.371994
$665$	−3901.09	−0.227486
$666$	−7396.23	−0.430327
$667$	4120.15	0.239180
$668$	2606.43	0.150967
$669$	−6568.23	−0.379585
$670$	−16155.0	−0.931528
$671$	−11699.3	−0.673095
$672$	510.116	0.0292830
$673$	7960.88	0.455972	0.227986	−	0.973664i	$-0.426786\pi$
$673$	7960.88	0.455972	0.227986	+	0.973664i	$0.426786\pi$
$674$	−11975.2	−0.684374
$675$	−2956.76	−0.168601
$676$	21665.6	1.23268
$677$	10547.0	0.598751	0.299376	−	0.954135i	$-0.403222\pi$
$677$	10547.0	0.598751	0.299376	+	0.954135i	$0.403222\pi$
$678$	−567.422	−0.0321411
$679$	−5514.16	−0.311656
$680$	2212.39	0.124766
$681$	17560.1	0.988113
$682$	4979.46	0.279580
$683$	870.274	0.0487557	0.0243778	−	0.999703i	$-0.492240\pi$
$683$	870.274	0.0487557	0.0243778	+	0.999703i	$0.492240\pi$
$684$	1725.88	0.0964776
$685$	22098.3	1.23260
$686$	6990.34	0.389056
$687$	10281.5	0.570982
$688$	−2392.79	−0.132593
$689$	−25815.4	−1.42741
$690$	−2113.29	−0.116597
$691$	19966.0	1.09919	0.549596	−	0.835431i	$-0.314782\pi$
$691$	19966.0	1.09919	0.549596	+	0.835431i	$0.314782\pi$
$692$	−5385.41	−0.295842
$693$	1016.48	0.0557183
$694$	−6756.61	−0.369564
$695$	−34128.5	−1.86269
$696$	−4299.29	−0.234144
$697$	−3715.88	−0.201935
$698$	−1688.54	−0.0915648
$699$	−17170.0	−0.929083
$700$	−2327.61	−0.125679
$701$	−28944.0	−1.55949	−0.779744	−	0.626099i	$-0.784651\pi$
$701$	−28944.0	−1.55949	−0.779744	+	0.626099i	$0.784651\pi$
$702$	−4711.76	−0.253325
$703$	−19699.1	−1.05685
$704$	−1360.31	−0.0728247
$705$	13776.9	0.735985
$706$	6334.26	0.337667
$707$	1990.24	0.105871
$708$	−8849.40	−0.469747
$709$	8061.61	0.427024	0.213512	−	0.976940i	$-0.431510\pi$
$709$	8061.61	0.427024	0.213512	+	0.976940i	$0.431510\pi$
$710$	−9113.73	−0.481735
$711$	−7512.18	−0.396243
$712$	375.684	0.0197744
$713$	−2694.15	−0.141510
$714$	575.758	0.0301781
$715$	−28400.6	−1.48549
$716$	−13699.3	−0.715037
$717$	−9894.12	−0.515345
$718$	20968.1	1.08986
$719$	−21709.2	−1.12603	−0.563015	−	0.826447i	$-0.690359\pi$
$719$	−21709.2	−1.12603	−0.563015	+	0.826447i	$0.690359\pi$
$720$	2205.17	0.114142
$721$	2172.69	0.112226
$722$	−9121.30	−0.470166
$723$	16174.8	0.832016
$724$	−13027.9	−0.668755
$725$	19617.2	1.00492
$726$	5275.39	0.269681
$727$	11612.3	0.592403	0.296202	−	0.955125i	$-0.404280\pi$
$727$	11612.3	0.592403	0.296202	+	0.955125i	$0.404280\pi$
$728$	−3709.17	−0.188834
$729$	729.000	0.0370370
$730$	−21877.6	−1.10921
$731$	−2700.69	−0.136647
$732$	−6605.17	−0.333516
$733$	−25130.2	−1.26631	−0.633155	−	0.774025i	$-0.718240\pi$
$733$	−25130.2	−1.26631	−0.633155	+	0.774025i	$0.718240\pi$
$734$	4698.01	0.236249
$735$	14460.6	0.725699
$736$	736.000	0.0368605
$737$	11211.3	0.560344
$738$	−3703.76	−0.184739
$739$	−14011.9	−0.697478	−0.348739	−	0.937220i	$-0.613390\pi$
$739$	−14011.9	−0.697478	−0.348739	+	0.937220i	$0.613390\pi$
$740$	−25169.7	−1.25035
$741$	−12549.3	−0.622145
$742$	3144.25	0.155565
$743$	−6207.90	−0.306522	−0.153261	−	0.988186i	$-0.548978\pi$
$743$	−6207.90	−0.306522	−0.153261	+	0.988186i	$0.548978\pi$
$744$	2811.29	0.138531
$745$	22760.6	1.11931
$746$	−24152.6	−1.18537
$747$	7160.46	0.350719
$748$	−1535.35	−0.0750509
$749$	−6729.86	−0.328310
$750$	1423.29	0.0692948
$751$	−8787.36	−0.426971	−0.213485	−	0.976946i	$-0.568482\pi$
$751$	−8787.36	−0.426971	−0.213485	+	0.976946i	$0.568482\pi$
$752$	−4798.12	−0.232672
$753$	−10776.3	−0.521529
$754$	31261.2	1.50990
$755$	5342.41	0.257524
$756$	573.881	0.0276082
$757$	35327.3	1.69616	0.848079	−	0.529870i	$-0.177759\pi$
$757$	35327.3	1.69616	0.848079	+	0.529870i	$0.177759\pi$
$758$	−3974.70	−0.190458
$759$	1466.58	0.0701365
$760$	5873.25	0.280323
$761$	20753.0	0.988562	0.494281	−	0.869302i	$-0.335431\pi$
$761$	20753.0	0.988562	0.494281	+	0.869302i	$0.335431\pi$
$762$	−12470.8	−0.592874
$763$	7266.86	0.344794
$764$	−16133.5	−0.763990
$765$	2488.94	0.117631
$766$	20949.2	0.988154
$767$	64346.1	3.02921
$768$	−768.000	−0.0360844
$769$	−3409.43	−0.159879	−0.0799397	−	0.996800i	$-0.525473\pi$
$769$	−3409.43	−0.159879	−0.0799397	+	0.996800i	$0.525473\pi$
$770$	3459.12	0.161894
$771$	−16936.8	−0.791134
$772$	1195.14	0.0557174
$773$	−504.444	−0.0234716	−0.0117358	−	0.999931i	$-0.503736\pi$
$773$	−504.444	−0.0234716	−0.0117358	+	0.999931i	$0.503736\pi$
$774$	−2691.89	−0.125010
$775$	−12827.6	−0.594558
$776$	8301.79	0.384043
$777$	−6550.23	−0.302430
$778$	−7957.05	−0.366676
$779$	−9864.58	−0.453704
$780$	−16034.3	−0.736053
$781$	6324.75	0.289779
$782$	830.708	0.0379873
$783$	−4836.70	−0.220753
$784$	−5036.23	−0.229420
$785$	26816.4	1.21926
$786$	−4497.41	−0.204093
$787$	−8403.22	−0.380613	−0.190306	−	0.981725i	$-0.560948\pi$
$787$	−8403.22	−0.380613	−0.190306	+	0.981725i	$0.560948\pi$
$788$	17016.8	0.769286
$789$	−913.281	−0.0412087
$790$	−25564.3	−1.15131
$791$	−502.519	−0.0225885
$792$	−1530.35	−0.0686598
$793$	48027.7	2.15071
$794$	11079.6	0.495215
$795$	13592.2	0.606373
$796$	−18924.9	−0.842684
$797$	19884.7	0.883754	0.441877	−	0.897076i	$-0.354313\pi$
$797$	19884.7	0.883754	0.441877	+	0.897076i	$0.354313\pi$
$798$	1528.47	0.0678036
$799$	−5415.54	−0.239785
$800$	3504.31	0.154870
$801$	422.644	0.0186434
$802$	4690.40	0.206513
$803$	15182.6	0.667226
$804$	6329.64	0.277648
$805$	−1871.57	−0.0819430
$806$	−20441.6	−0.893329
$807$	−8246.36	−0.359709
$808$	−2996.39	−0.130461
$809$	33037.5	1.43577	0.717883	−	0.696164i	$-0.245111\pi$
$809$	33037.5	1.43577	0.717883	+	0.696164i	$0.245111\pi$
$810$	2480.82	0.107614
$811$	−11788.0	−0.510399	−0.255199	−	0.966888i	$-0.582141\pi$
$811$	−11788.0	−0.510399	−0.255199	+	0.966888i	$0.582141\pi$
$812$	−3807.53	−0.164554
$813$	3972.83	0.171381
$814$	17467.3	0.752123
$815$	6443.14	0.276924
$816$	−866.826	−0.0371875
$817$	−7169.56	−0.307015
$818$	−14527.9	−0.620974
$819$	−4172.82	−0.178034
$820$	−12604.1	−0.536772
$821$	22695.5	0.964772	0.482386	−	0.875959i	$-0.339770\pi$
$821$	22695.5	0.964772	0.482386	+	0.875959i	$0.339770\pi$
$822$	−8658.23	−0.367385
$823$	33804.9	1.43179	0.715896	−	0.698207i	$-0.246018\pi$
$823$	33804.9	1.43179	0.715896	+	0.698207i	$0.246018\pi$
$824$	−3271.06	−0.138292
$825$	6982.83	0.294680
$826$	−7837.19	−0.330134
$827$	25825.6	1.08591	0.542953	−	0.839763i	$-0.317306\pi$
$827$	25825.6	1.08591	0.542953	+	0.839763i	$0.317306\pi$
$828$	828.000	0.0347524
$829$	−11682.4	−0.489442	−0.244721	−	0.969594i	$-0.578696\pi$
$829$	−11682.4	−0.489442	−0.244721	+	0.969594i	$0.578696\pi$
$830$	24367.4	1.01904
$831$	−16982.2	−0.708913
$832$	5584.31	0.232694
$833$	−5684.29	−0.236433
$834$	13371.8	0.555187
$835$	9978.51	0.413558
$836$	−4075.92	−0.168623
$837$	3162.70	0.130608
$838$	8189.78	0.337603
$839$	−20526.4	−0.844635	−0.422317	−	0.906448i	$-0.638783\pi$
$839$	−20526.4	−0.844635	−0.422317	+	0.906448i	$0.638783\pi$
$840$	1952.94	0.0802177
$841$	7701.10	0.315761
$842$	9355.35	0.382906
$843$	−14945.6	−0.610620
$844$	20525.6	0.837110
$845$	82945.3	3.37681
$846$	−5397.88	−0.219365
$847$	4671.98	0.189529
$848$	−4733.79	−0.191697
$849$	15280.5	0.617698
$850$	3955.24	0.159604
$851$	−9450.74	−0.380690
$852$	3570.81	0.143585
$853$	15186.7	0.609593	0.304797	−	0.952417i	$-0.401412\pi$
$853$	15186.7	0.609593	0.304797	+	0.952417i	$0.401412\pi$
$854$	−5849.66	−0.234392
$855$	6607.41	0.264291
$856$	10132.1	0.404565
$857$	37394.9	1.49053	0.745266	−	0.666767i	$-0.232322\pi$
$857$	37394.9	1.49053	0.745266	+	0.666767i	$0.232322\pi$
$858$	11127.5	0.442759
$859$	19572.9	0.777437	0.388719	−	0.921356i	$-0.372918\pi$
$859$	19572.9	0.777437	0.388719	+	0.921356i	$0.372918\pi$
$860$	−9160.61	−0.363226
$861$	−3280.12	−0.129833
$862$	26438.1	1.04465
$863$	16086.0	0.634500	0.317250	−	0.948342i	$-0.397241\pi$
$863$	16086.0	0.634500	0.317250	+	0.948342i	$0.397241\pi$
$864$	−864.000	−0.0340207
$865$	−20617.7	−0.810429
$866$	6219.55	0.244052
$867$	13760.6	0.539026
$868$	2489.73	0.0973582
$869$	17741.1	0.692550
$870$	−16459.5	−0.641414
$871$	−46024.3	−1.79044
$872$	−10940.5	−0.424878
$873$	9339.52	0.362079
$874$	2205.29	0.0853491
$875$	1260.49	0.0486998
$876$	8571.75	0.330608
$877$	1016.11	0.0391239	0.0195620	−	0.999809i	$-0.493773\pi$
$877$	1016.11	0.0391239	0.0195620	+	0.999809i	$0.493773\pi$
$878$	32850.0	1.26268
$879$	23830.4	0.914424
$880$	−5207.85	−0.199496
$881$	7851.35	0.300248	0.150124	−	0.988667i	$-0.452033\pi$
$881$	7851.35	0.300248	0.150124	+	0.988667i	$0.452033\pi$
$882$	−5665.76	−0.216299
$883$	17536.2	0.668336	0.334168	−	0.942513i	$-0.391545\pi$
$883$	17536.2	0.668336	0.334168	+	0.942513i	$0.391545\pi$
$884$	6302.90	0.239807
$885$	−33879.3	−1.28682
$886$	−9472.29	−0.359174
$887$	−32681.8	−1.23714	−0.618572	−	0.785728i	$-0.712289\pi$
$887$	−32681.8	−1.23714	−0.618572	+	0.785728i	$0.712289\pi$
$888$	9861.64	0.372674
$889$	−11044.4	−0.416667
$890$	1438.28	0.0541699
$891$	−1721.64	−0.0647331
$892$	8757.64	0.328730
$893$	−14376.7	−0.538743
$894$	−8917.75	−0.333617
$895$	−52446.8	−1.95877
$896$	−680.155	−0.0253598
$897$	−6020.58	−0.224104
$898$	−24683.7	−0.917266
$899$	−20983.6	−0.778467
$900$	3942.35	0.146013
$901$	−5342.94	−0.197557
$902$	8746.98	0.322885
$903$	−2383.98	−0.0878560
$904$	756.562	0.0278350
$905$	−49876.5	−1.83199
$906$	−2093.19	−0.0767566
$907$	14992.8	0.548874	0.274437	−	0.961605i	$-0.411509\pi$
$907$	14992.8	0.548874	0.274437	+	0.961605i	$0.411509\pi$
$908$	−23413.5	−0.855731
$909$	−3370.94	−0.123000
$910$	−14200.3	−0.517292
$911$	−28483.3	−1.03589	−0.517943	−	0.855415i	$-0.673302\pi$
$911$	−28483.3	−1.03589	−0.517943	+	0.855415i	$0.673302\pi$
$912$	−2301.17	−0.0835521
$913$	−16910.5	−0.612985
$914$	11.6098	0.000420152 0
$915$	−25287.4	−0.913635
$916$	−13708.7	−0.494485
$917$	−3982.98	−0.143435
$918$	−975.179	−0.0350607
$919$	48476.1	1.74002	0.870011	−	0.493032i	$-0.164111\pi$
$919$	48476.1	1.74002	0.870011	+	0.493032i	$0.164111\pi$
$920$	2817.72	0.100976
$921$	−22333.8	−0.799047
$922$	−23005.6	−0.821747
$923$	−25964.2	−0.925918
$924$	−1355.30	−0.0482535
$925$	−44997.7	−1.59948
$926$	23479.1	0.833229
$927$	−3679.95	−0.130383
$928$	5732.39	0.202775
$929$	44980.1	1.58853	0.794267	−	0.607568i	$-0.207855\pi$
$929$	44980.1	1.58853	0.794267	+	0.607568i	$0.207855\pi$
$930$	10762.8	0.379491
$931$	−15090.2	−0.531214
$932$	22893.3	0.804609
$933$	19754.0	0.693158
$934$	−24339.7	−0.852697
$935$	−5877.99	−0.205594
$936$	6282.35	0.219386
$937$	1296.80	0.0452131	0.0226065	−	0.999744i	$-0.492804\pi$
$937$	1296.80	0.0452131	0.0226065	+	0.999744i	$0.492804\pi$
$938$	5605.65	0.195129
$939$	−8372.59	−0.290979
$940$	−18369.2	−0.637381
$941$	42721.5	1.48000	0.740000	−	0.672607i	$-0.234825\pi$
$941$	42721.5	1.48000	0.740000	+	0.672607i	$0.234825\pi$
$942$	−10506.8	−0.363408
$943$	−4732.58	−0.163430
$944$	11799.2	0.406813
$945$	2197.06	0.0756300
$946$	6357.29	0.218492
$947$	−13367.3	−0.458688	−0.229344	−	0.973345i	$-0.573658\pi$
$947$	−13367.3	−0.458688	−0.229344	+	0.973345i	$0.573658\pi$
$948$	10016.2	0.343156
$949$	−62327.2	−2.13196
$950$	10500.0	0.358596
$951$	6251.06	0.213149
$952$	−767.677	−0.0261350
$953$	7309.26	0.248447	0.124224	−	0.992254i	$-0.460356\pi$
$953$	7309.26	0.248447	0.124224	+	0.992254i	$0.460356\pi$
$954$	−5325.52	−0.180734
$955$	−61765.8	−2.09287
$956$	13192.2	0.446302
$957$	11422.6	0.385831
$958$	18193.7	0.613583
$959$	−7667.88	−0.258195
$960$	−2940.23	−0.0988496
$961$	−16069.9	−0.539421
$962$	−71706.3	−2.40323
$963$	11398.6	0.381427
$964$	−21566.4	−0.720547
$965$	4575.49	0.152632
$966$	733.292	0.0244237
$967$	41664.5	1.38556	0.692782	−	0.721147i	$-0.256385\pi$
$967$	41664.5	1.38556	0.692782	+	0.721147i	$0.256385\pi$
$968$	−7033.86	−0.233550
$969$	−2597.29	−0.0861062
$970$	31782.8	1.05205
$971$	54110.9	1.78836	0.894182	−	0.447705i	$-0.147758\pi$
$971$	54110.9	1.78836	0.894182	+	0.447705i	$0.147758\pi$
$972$	−972.000	−0.0320750
$973$	11842.3	0.390181
$974$	37482.0	1.23306
$975$	−28665.7	−0.941578
$976$	8806.89	0.288834
$977$	5877.98	0.192480	0.0962401	−	0.995358i	$-0.469318\pi$
$977$	5877.98	0.192480	0.0962401	+	0.995358i	$0.469318\pi$
$978$	−2524.46	−0.0825392
$979$	−998.137	−0.0325849
$980$	−19280.8	−0.628474
$981$	−12308.1	−0.400579
$982$	−33807.4	−1.09861
$983$	−14598.5	−0.473674	−0.236837	−	0.971549i	$-0.576111\pi$
$983$	−14598.5	−0.473674	−0.236837	+	0.971549i	$0.576111\pi$
$984$	4938.35	0.159989
$985$	65147.5	2.10738
$986$	6470.03	0.208973
$987$	−4780.46	−0.154168
$988$	16732.4	0.538793
$989$	−3439.63	−0.110590
$990$	−5858.83	−0.188087
$991$	36254.6	1.16212	0.581062	−	0.813859i	$-0.302637\pi$
$991$	36254.6	1.16212	0.581062	+	0.813859i	$0.302637\pi$
$992$	−3748.39	−0.119971
$993$	−14062.7	−0.449412
$994$	3162.38	0.100910
$995$	−72452.7	−2.30845
$996$	−9547.28	−0.303732
$997$	8640.87	0.274483	0.137241	−	0.990538i	$-0.456176\pi$
$997$	8640.87	0.274483	0.137241	+	0.990538i	$0.456176\pi$
$998$	−5380.02	−0.170643
$999$	11094.3	0.351361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.a.e.1.2	✓	2
3.2	odd	2		414.4.a.g.1.1		2
4.3	odd	2		1104.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.a.e.1.2	✓	2	1.1	even	1	trivial
414.4.a.g.1.1		2	3.2	odd	2
1104.4.a.p.1.2		2	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +15.3137 q^{5} -6.00000 q^{6} -5.31371 q^{7} +8.00000 q^{8} +9.00000 q^{9} +30.6274 q^{10} -21.2548 q^{11} -12.0000 q^{12} +87.2548 q^{13} -10.6274 q^{14} -45.9411 q^{15} +16.0000 q^{16} +18.0589 q^{17} +18.0000 q^{18} +47.9411 q^{19} +61.2548 q^{20} +15.9411 q^{21} -42.5097 q^{22} +23.0000 q^{23} -24.0000 q^{24} +109.510 q^{25} +174.510 q^{26} -27.0000 q^{27} -21.2548 q^{28} +179.137 q^{29} -91.8823 q^{30} -117.137 q^{31} +32.0000 q^{32} +63.7645 q^{33} +36.1177 q^{34} -81.3726 q^{35} +36.0000 q^{36} -410.902 q^{37} +95.8823 q^{38} -261.765 q^{39} +122.510 q^{40} -205.765 q^{41} +31.8823 q^{42} -149.549 q^{43} -85.0193 q^{44} +137.823 q^{45} +46.0000 q^{46} -299.882 q^{47} -48.0000 q^{48} -314.765 q^{49} +219.019 q^{50} -54.1766 q^{51} +349.019 q^{52} -295.862 q^{53} -54.0000 q^{54} -325.490 q^{55} -42.5097 q^{56} -143.823 q^{57} +358.274 q^{58} +737.450 q^{59} -183.765 q^{60} +550.431 q^{61} -234.274 q^{62} -47.8234 q^{63} +64.0000 q^{64} +1336.20 q^{65} +127.529 q^{66} -527.470 q^{67} +72.2355 q^{68} -69.0000 q^{69} -162.745 q^{70} -297.568 q^{71} +72.0000 q^{72} -714.313 q^{73} -821.803 q^{74} -328.529 q^{75} +191.765 q^{76} +112.942 q^{77} -523.529 q^{78} -834.686 q^{79} +245.019 q^{80} +81.0000 q^{81} -411.529 q^{82} +795.606 q^{83} +63.7645 q^{84} +276.548 q^{85} -299.098 q^{86} -537.411 q^{87} -170.039 q^{88} +46.9605 q^{89} +275.647 q^{90} -463.647 q^{91} +92.0000 q^{92} +351.411 q^{93} -599.765 q^{94} +734.156 q^{95} -96.0000 q^{96} +1037.72 q^{97} -629.529 q^{98} -191.294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 8 q^{5} - 12 q^{6} + 12 q^{7} + 16 q^{8} + 18 q^{9} + 16 q^{10} + 48 q^{11} - 24 q^{12} + 84 q^{13} + 24 q^{14} - 24 q^{15} + 32 q^{16} + 104 q^{17} + 36 q^{18} + 28 q^{19}+ \cdots + 432 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 8 * q^5 - 12 * q^6 + 12 * q^7 + 16 * q^8 + 18 * q^9 + 16 * q^10 + 48 * q^11 - 24 * q^12 + 84 * q^13 + 24 * q^14 - 24 * q^15 + 32 * q^16 + 104 * q^17 + 36 * q^18 + 28 * q^19 + 32 * q^20 - 36 * q^21 + 96 * q^22 + 46 * q^23 - 48 * q^24 + 38 * q^25 + 168 * q^26 - 54 * q^27 + 48 * q^28 + 132 * q^29 - 48 * q^30 - 8 * q^31 + 64 * q^32 - 144 * q^33 + 208 * q^34 - 208 * q^35 + 72 * q^36 - 324 * q^37 + 56 * q^38 - 252 * q^39 + 64 * q^40 - 140 * q^41 - 72 * q^42 - 548 * q^43 + 192 * q^44 + 72 * q^45 + 92 * q^46 - 464 * q^47 - 96 * q^48 - 358 * q^49 + 76 * q^50 - 312 * q^51 + 336 * q^52 + 336 * q^53 - 108 * q^54 - 832 * q^55 + 96 * q^56 - 84 * q^57 + 264 * q^58 + 72 * q^59 - 96 * q^60 + 60 * q^61 - 16 * q^62 + 108 * q^63 + 128 * q^64 + 1360 * q^65 - 288 * q^66 - 444 * q^67 + 416 * q^68 - 138 * q^69 - 416 * q^70 + 672 * q^71 + 144 * q^72 - 252 * q^73 - 648 * q^74 - 114 * q^75 + 112 * q^76 + 1312 * q^77 - 504 * q^78 - 1692 * q^79 + 128 * q^80 + 162 * q^81 - 280 * q^82 - 400 * q^83 - 144 * q^84 - 352 * q^85 - 1096 * q^86 - 396 * q^87 + 384 * q^88 - 336 * q^89 + 144 * q^90 - 520 * q^91 + 184 * q^92 + 24 * q^93 - 928 * q^94 + 880 * q^95 - 192 * q^96 + 220 * q^97 - 716 * q^98 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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