LMFDB - Embedded newform 983.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [983,2,Mod(1,983)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(983, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("983.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 983 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	983.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:	$7.84929451869$
Analytic rank:	$0$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	983.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11673 q^{2} -3.41526 q^{3} +2.48055 q^{4} -2.85207 q^{5} +7.22918 q^{6} +3.49572 q^{7} -1.01719 q^{8} +8.66400 q^{9} +O(q^{10})$	\(q-2.11673 q^{2} -3.41526 q^{3} +2.48055 q^{4} -2.85207 q^{5} +7.22918 q^{6} +3.49572 q^{7} -1.01719 q^{8} +8.66400 q^{9} +6.03706 q^{10} +0.909384 q^{11} -8.47171 q^{12} +5.99798 q^{13} -7.39951 q^{14} +9.74055 q^{15} -2.80798 q^{16} +1.01298 q^{17} -18.3394 q^{18} -6.73710 q^{19} -7.07469 q^{20} -11.9388 q^{21} -1.92492 q^{22} -0.850369 q^{23} +3.47396 q^{24} +3.13429 q^{25} -12.6961 q^{26} -19.3440 q^{27} +8.67131 q^{28} +5.31305 q^{29} -20.6181 q^{30} +4.91598 q^{31} +7.97812 q^{32} -3.10578 q^{33} -2.14421 q^{34} -9.97004 q^{35} +21.4915 q^{36} +7.38510 q^{37} +14.2606 q^{38} -20.4846 q^{39} +2.90109 q^{40} -2.55868 q^{41} +25.2712 q^{42} +10.5240 q^{43} +2.25577 q^{44} -24.7103 q^{45} +1.80000 q^{46} +3.59557 q^{47} +9.58999 q^{48} +5.22009 q^{49} -6.63445 q^{50} -3.45959 q^{51} +14.8783 q^{52} -9.31653 q^{53} +40.9461 q^{54} -2.59362 q^{55} -3.55580 q^{56} +23.0089 q^{57} -11.2463 q^{58} -11.9533 q^{59} +24.1619 q^{60} -9.13376 q^{61} -10.4058 q^{62} +30.2870 q^{63} -11.2716 q^{64} -17.1066 q^{65} +6.57410 q^{66} -1.87854 q^{67} +2.51274 q^{68} +2.90423 q^{69} +21.1039 q^{70} -2.52508 q^{71} -8.81291 q^{72} +9.49510 q^{73} -15.6323 q^{74} -10.7044 q^{75} -16.7117 q^{76} +3.17896 q^{77} +43.3605 q^{78} +7.58030 q^{79} +8.00856 q^{80} +40.0729 q^{81} +5.41602 q^{82} -1.53498 q^{83} -29.6148 q^{84} -2.88909 q^{85} -22.2764 q^{86} -18.1455 q^{87} -0.925013 q^{88} -1.40345 q^{89} +52.3051 q^{90} +20.9673 q^{91} -2.10938 q^{92} -16.7894 q^{93} -7.61086 q^{94} +19.2147 q^{95} -27.2473 q^{96} -4.48053 q^{97} -11.0495 q^{98} +7.87890 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 8 q^{2} + 6 q^{3} + 64 q^{4} + 7 q^{5} + 5 q^{6} + 31 q^{7} + 24 q^{8} + 74 q^{9}+O(q^{10}) $ 54 * q + 8 * q^2 + 6 * q^3 + 64 * q^4 + 7 * q^5 + 5 * q^6 + 31 * q^7 + 24 * q^8 + 74 * q^9	\( 54 q + 8 q^{2} + 6 q^{3} + 64 q^{4} + 7 q^{5} + 5 q^{6} + 31 q^{7} + 24 q^{8} + 74 q^{9} + 15 q^{10} + 8 q^{11} + 10 q^{12} + 34 q^{13} + q^{14} + 3 q^{15} + 80 q^{16} + 32 q^{17} + 28 q^{18} + 15 q^{19} + 4 q^{20} + 11 q^{21} + 25 q^{22} + 15 q^{23} - 7 q^{24} + 125 q^{25} - 14 q^{26} + 12 q^{27} + 78 q^{28} + 8 q^{29} - 32 q^{30} + 16 q^{31} + 36 q^{32} + 38 q^{33} - 8 q^{34} - 2 q^{35} + 65 q^{36} + 80 q^{37} - 14 q^{38} + 16 q^{39} + 36 q^{40} + 30 q^{41} - 10 q^{42} + 53 q^{43} + 6 q^{44} + 3 q^{45} + 24 q^{46} + 22 q^{47} + 7 q^{48} + 111 q^{49} + 14 q^{50} - 10 q^{51} + 45 q^{52} + 10 q^{53} - 8 q^{54} + 12 q^{55} - 30 q^{56} + 106 q^{57} + 61 q^{58} - 4 q^{59} - 61 q^{60} + 24 q^{61} - 12 q^{62} + 73 q^{63} + 86 q^{64} + 32 q^{65} - 43 q^{66} + 54 q^{67} + 33 q^{68} - 30 q^{69} - 21 q^{70} - 6 q^{71} + 60 q^{72} + 172 q^{73} - 32 q^{74} - 36 q^{75} + 7 q^{76} - 12 q^{77} - 3 q^{78} + 28 q^{79} - 66 q^{80} + 86 q^{81} + 4 q^{82} + 14 q^{83} - 58 q^{84} + 99 q^{85} - 31 q^{86} - 27 q^{87} + 5 q^{88} - 5 q^{89} - 45 q^{90} - 11 q^{91} + 20 q^{92} + 27 q^{93} - 39 q^{94} + 5 q^{95} - 87 q^{96} + 127 q^{97} - 29 q^{98} - 21 q^{99}+O(q^{100}) \) 54 * q + 8 * q^2 + 6 * q^3 + 64 * q^4 + 7 * q^5 + 5 * q^6 + 31 * q^7 + 24 * q^8 + 74 * q^9 + 15 * q^10 + 8 * q^11 + 10 * q^12 + 34 * q^13 + q^14 + 3 * q^15 + 80 * q^16 + 32 * q^17 + 28 * q^18 + 15 * q^19 + 4 * q^20 + 11 * q^21 + 25 * q^22 + 15 * q^23 - 7 * q^24 + 125 * q^25 - 14 * q^26 + 12 * q^27 + 78 * q^28 + 8 * q^29 - 32 * q^30 + 16 * q^31 + 36 * q^32 + 38 * q^33 - 8 * q^34 - 2 * q^35 + 65 * q^36 + 80 * q^37 - 14 * q^38 + 16 * q^39 + 36 * q^40 + 30 * q^41 - 10 * q^42 + 53 * q^43 + 6 * q^44 + 3 * q^45 + 24 * q^46 + 22 * q^47 + 7 * q^48 + 111 * q^49 + 14 * q^50 - 10 * q^51 + 45 * q^52 + 10 * q^53 - 8 * q^54 + 12 * q^55 - 30 * q^56 + 106 * q^57 + 61 * q^58 - 4 * q^59 - 61 * q^60 + 24 * q^61 - 12 * q^62 + 73 * q^63 + 86 * q^64 + 32 * q^65 - 43 * q^66 + 54 * q^67 + 33 * q^68 - 30 * q^69 - 21 * q^70 - 6 * q^71 + 60 * q^72 + 172 * q^73 - 32 * q^74 - 36 * q^75 + 7 * q^76 - 12 * q^77 - 3 * q^78 + 28 * q^79 - 66 * q^80 + 86 * q^81 + 4 * q^82 + 14 * q^83 - 58 * q^84 + 99 * q^85 - 31 * q^86 - 27 * q^87 + 5 * q^88 - 5 * q^89 - 45 * q^90 - 11 * q^91 + 20 * q^92 + 27 * q^93 - 39 * q^94 + 5 * q^95 - 87 * q^96 + 127 * q^97 - 29 * q^98 - 21 * 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 $

$2$	−2.11673	−1.49675	−0.748377	−	0.663273i	$-0.769167\pi$
$2$	−2.11673	−1.49675	−0.748377	+	0.663273i	$0.769167\pi$
$3$	−3.41526	−1.97180	−0.985901	−	0.167331i	$-0.946485\pi$
$3$	−3.41526	−1.97180	−0.985901	+	0.167331i	$0.946485\pi$
$4$	2.48055	1.24027
$5$	−2.85207	−1.27548	−0.637742	−	0.770250i	$-0.720131\pi$
$5$	−2.85207	−1.27548	−0.637742	+	0.770250i	$0.720131\pi$
$6$	7.22918	2.95130
$7$	3.49572	1.32126	0.660630	−	0.750712i	$-0.270289\pi$
$7$	3.49572	1.32126	0.660630	+	0.750712i	$0.270289\pi$
$8$	−1.01719	−0.359630
$9$	8.66400	2.88800
$10$	6.03706	1.90909
$11$	0.909384	0.274190	0.137095	−	0.990558i	$-0.456224\pi$
$11$	0.909384	0.274190	0.137095	+	0.990558i	$0.456224\pi$
$12$	−8.47171	−2.44557
$13$	5.99798	1.66354	0.831769	−	0.555121i	$-0.187328\pi$
$13$	5.99798	1.66354	0.831769	+	0.555121i	$0.187328\pi$
$14$	−7.39951	−1.97760
$15$	9.74055	2.51500
$16$	−2.80798	−0.701996
$17$	1.01298	0.245684	0.122842	−	0.992426i	$-0.460799\pi$
$17$	1.01298	0.245684	0.122842	+	0.992426i	$0.460799\pi$
$18$	−18.3394	−4.32263
$19$	−6.73710	−1.54560	−0.772798	−	0.634652i	$-0.781143\pi$
$19$	−6.73710	−1.54560	−0.772798	+	0.634652i	$0.781143\pi$
$20$	−7.07469	−1.58195
$21$	−11.9388	−2.60526
$22$	−1.92492	−0.410394
$23$	−0.850369	−0.177314	−0.0886571	−	0.996062i	$-0.528258\pi$
$23$	−0.850369	−0.177314	−0.0886571	+	0.996062i	$0.528258\pi$
$24$	3.47396	0.709118
$25$	3.13429	0.626858
$26$	−12.6961	−2.48991
$27$	−19.3440	−3.72276
$28$	8.67131	1.63872
$29$	5.31305	0.986609	0.493305	−	0.869857i	$-0.335789\pi$
$29$	5.31305	0.986609	0.493305	+	0.869857i	$0.335789\pi$
$30$	−20.6181	−3.76434
$31$	4.91598	0.882937	0.441468	−	0.897277i	$-0.354458\pi$
$31$	4.91598	0.882937	0.441468	+	0.897277i	$0.354458\pi$
$32$	7.97812	1.41034
$33$	−3.10578	−0.540647
$34$	−2.14421	−0.367728
$35$	−9.97004	−1.68524
$36$	21.4915	3.58191
$37$	7.38510	1.21410	0.607051	−	0.794663i	$-0.292352\pi$
$37$	7.38510	1.21410	0.607051	+	0.794663i	$0.292352\pi$
$38$	14.2606	2.31338
$39$	−20.4846	−3.28017
$40$	2.90109	0.458702
$41$	−2.55868	−0.399598	−0.199799	−	0.979837i	$-0.564029\pi$
$41$	−2.55868	−0.399598	−0.199799	+	0.979837i	$0.564029\pi$
$42$	25.2712	3.89944
$43$	10.5240	1.60489	0.802446	−	0.596725i	$-0.203532\pi$
$43$	10.5240	1.60489	0.802446	+	0.596725i	$0.203532\pi$
$44$	2.25577	0.340070
$45$	−24.7103	−3.68360
$46$	1.80000	0.265396
$47$	3.59557	0.524468	0.262234	−	0.965004i	$-0.415541\pi$
$47$	3.59557	0.524468	0.262234	+	0.965004i	$0.415541\pi$
$48$	9.58999	1.38420
$49$	5.22009	0.745727
$50$	−6.63445	−0.938253
$51$	−3.45959	−0.484440
$52$	14.8783	2.06324
$53$	−9.31653	−1.27972	−0.639862	−	0.768490i	$-0.721009\pi$
$53$	−9.31653	−1.27972	−0.639862	+	0.768490i	$0.721009\pi$
$54$	40.9461	5.57206
$55$	−2.59362	−0.349724
$56$	−3.55580	−0.475164
$57$	23.0089	3.04761
$58$	−11.2463	−1.47671
$59$	−11.9533	−1.55619	−0.778095	−	0.628146i	$-0.783814\pi$
$59$	−11.9533	−1.55619	−0.778095	+	0.628146i	$0.783814\pi$
$60$	24.1619	3.11929
$61$	−9.13376	−1.16946	−0.584729	−	0.811229i	$-0.698799\pi$
$61$	−9.13376	−1.16946	−0.584729	+	0.811229i	$0.698799\pi$
$62$	−10.4058	−1.32154
$63$	30.2870	3.81580
$64$	−11.2716	−1.40894
$65$	−17.1066	−2.12182
$66$	6.57410	0.809216
$67$	−1.87854	−0.229500	−0.114750	−	0.993394i	$-0.536607\pi$
$67$	−1.87854	−0.229500	−0.114750	+	0.993394i	$0.536607\pi$
$68$	2.51274	0.304715
$69$	2.90423	0.349628
$70$	21.1039	2.52240
$71$	−2.52508	−0.299671	−0.149836	−	0.988711i	$-0.547874\pi$
$71$	−2.52508	−0.299671	−0.149836	+	0.988711i	$0.547874\pi$
$72$	−8.81291	−1.03861
$73$	9.49510	1.11132	0.555659	−	0.831410i	$-0.312466\pi$
$73$	9.49510	1.11132	0.555659	+	0.831410i	$0.312466\pi$
$74$	−15.6323	−1.81721
$75$	−10.7044	−1.23604
$76$	−16.7117	−1.91696
$77$	3.17896	0.362276
$78$	43.3605	4.90961
$79$	7.58030	0.852851	0.426425	−	0.904523i	$-0.359773\pi$
$79$	7.58030	0.852851	0.426425	+	0.904523i	$0.359773\pi$
$80$	8.00856	0.895384
$81$	40.0729	4.45255
$82$	5.41602	0.598100
$83$	−1.53498	−0.168486	−0.0842428	−	0.996445i	$-0.526847\pi$
$83$	−1.53498	−0.168486	−0.0842428	+	0.996445i	$0.526847\pi$
$84$	−29.6148	−3.23124
$85$	−2.88909	−0.313366
$86$	−22.2764	−2.40213
$87$	−18.1455	−1.94540
$88$	−0.925013	−0.0986067
$89$	−1.40345	−0.148765	−0.0743825	−	0.997230i	$-0.523699\pi$
$89$	−1.40345	−0.148765	−0.0743825	+	0.997230i	$0.523699\pi$
$90$	52.3051	5.51344
$91$	20.9673	2.19797
$92$	−2.10938	−0.219918
$93$	−16.7894	−1.74098
$94$	−7.61086	−0.785000
$95$	19.2147	1.97138
$96$	−27.2473	−2.78092
$97$	−4.48053	−0.454929	−0.227465	−	0.973786i	$-0.573044\pi$
$97$	−4.48053	−0.454929	−0.227465	+	0.973786i	$0.573044\pi$
$98$	−11.0495	−1.11617
$99$	7.87890	0.791860
$100$	7.77475	0.777475
$101$	12.0671	1.20072	0.600360	−	0.799730i	$-0.295024\pi$
$101$	12.0671	1.20072	0.600360	+	0.799730i	$0.295024\pi$
$102$	7.32302	0.725087
$103$	−2.11785	−0.208678	−0.104339	−	0.994542i	$-0.533273\pi$
$103$	−2.11785	−0.208678	−0.104339	+	0.994542i	$0.533273\pi$
$104$	−6.10106	−0.598258
$105$	34.0503	3.32297
$106$	19.7206	1.91543
$107$	−4.87354	−0.471143	−0.235571	−	0.971857i	$-0.575696\pi$
$107$	−4.87354	−0.471143	−0.235571	+	0.971857i	$0.575696\pi$
$108$	−47.9838	−4.61724
$109$	−5.59965	−0.536349	−0.268175	−	0.963370i	$-0.586420\pi$
$109$	−5.59965	−0.536349	−0.268175	+	0.963370i	$0.586420\pi$
$110$	5.49000	0.523451
$111$	−25.2220	−2.39397
$112$	−9.81593	−0.927519
$113$	−6.10111	−0.573944	−0.286972	−	0.957939i	$-0.592649\pi$
$113$	−6.10111	−0.573944	−0.286972	+	0.957939i	$0.592649\pi$
$114$	−48.7037	−4.56152
$115$	2.42531	0.226161
$116$	13.1793	1.22366
$117$	51.9665	4.80430
$118$	25.3020	2.32924
$119$	3.54110	0.324612
$120$	−9.90796	−0.904469
$121$	−10.1730	−0.924820
$122$	19.3337	1.75039
$123$	8.73854	0.787928
$124$	12.1943	1.09508
$125$	5.32113	0.475936
$126$	−64.1093	−5.71131
$127$	20.0976	1.78337	0.891687	−	0.452653i	$-0.149522\pi$
$127$	20.0976	1.78337	0.891687	+	0.452653i	$0.149522\pi$
$128$	7.90260	0.698498
$129$	−35.9421	−3.16453
$130$	36.2101	3.17584
$131$	−5.55068	−0.484965	−0.242482	−	0.970156i	$-0.577962\pi$
$131$	−5.55068	−0.484965	−0.242482	+	0.970156i	$0.577962\pi$
$132$	−7.70404	−0.670550
$133$	−23.5510	−2.04213
$134$	3.97637	0.343506
$135$	55.1705	4.74832
$136$	−1.03039	−0.0883552
$137$	14.0494	1.20032	0.600159	−	0.799881i	$-0.295104\pi$
$137$	14.0494	1.20032	0.600159	+	0.799881i	$0.295104\pi$
$138$	−6.14747	−0.523308
$139$	−2.97035	−0.251942	−0.125971	−	0.992034i	$-0.540205\pi$
$139$	−2.97035	−0.251942	−0.125971	+	0.992034i	$0.540205\pi$
$140$	−24.7312	−2.09016
$141$	−12.2798	−1.03415
$142$	5.34490	0.448534
$143$	5.45446	0.456125
$144$	−24.3284	−2.02736
$145$	−15.1532	−1.25840
$146$	−20.0986	−1.66337
$147$	−17.8280	−1.47043
$148$	18.3191	1.50582
$149$	13.1091	1.07394	0.536971	−	0.843601i	$-0.319568\pi$
$149$	13.1091	1.07394	0.536971	+	0.843601i	$0.319568\pi$
$150$	22.6584	1.85005
$151$	−19.3092	−1.57136	−0.785681	−	0.618632i	$-0.787687\pi$
$151$	−19.3092	−1.57136	−0.785681	+	0.618632i	$0.787687\pi$
$152$	6.85288	0.555842
$153$	8.77647	0.709535
$154$	−6.72899	−0.542237
$155$	−14.0207	−1.12617
$156$	−50.8131	−4.06830
$157$	0.676899	0.0540224	0.0270112	−	0.999635i	$-0.491401\pi$
$157$	0.676899	0.0540224	0.0270112	+	0.999635i	$0.491401\pi$
$158$	−16.0455	−1.27651
$159$	31.8184	2.52336
$160$	−22.7541	−1.79887
$161$	−2.97265	−0.234278
$162$	−84.8236	−6.66437
$163$	−10.1331	−0.793687	−0.396843	−	0.917886i	$-0.629894\pi$
$163$	−10.1331	−0.793687	−0.396843	+	0.917886i	$0.629894\pi$
$164$	−6.34691	−0.495611
$165$	8.85790	0.689587
$166$	3.24913	0.252182
$167$	22.7113	1.75746	0.878728	−	0.477323i	$-0.158393\pi$
$167$	22.7113	1.75746	0.878728	+	0.477323i	$0.158393\pi$
$168$	12.1440	0.936930
$169$	22.9757	1.76736
$170$	6.11542	0.469031
$171$	−58.3702	−4.46368
$172$	26.1052	1.99050
$173$	10.5696	0.803592	0.401796	−	0.915729i	$-0.368386\pi$
$173$	10.5696	0.803592	0.401796	+	0.915729i	$0.368386\pi$
$174$	38.4090	2.91178
$175$	10.9566	0.828243
$176$	−2.55353	−0.192480
$177$	40.8237	3.06850
$178$	2.97072	0.222665
$179$	8.30062	0.620417	0.310209	−	0.950668i	$-0.399601\pi$
$179$	8.30062	0.620417	0.310209	+	0.950668i	$0.399601\pi$
$180$	−61.2951	−4.56867
$181$	16.3898	1.21825	0.609123	−	0.793076i	$-0.291522\pi$
$181$	16.3898	1.21825	0.609123	+	0.793076i	$0.291522\pi$
$182$	−44.3820	−3.28982
$183$	31.1942	2.30594
$184$	0.864984	0.0637674
$185$	−21.0628	−1.54857
$186$	35.5385	2.60581
$187$	0.921188	0.0673639
$188$	8.91899	0.650484
$189$	−67.6215	−4.91874
$190$	−40.6722	−2.95067
$191$	13.4562	0.973660	0.486830	−	0.873497i	$-0.338153\pi$
$191$	13.4562	0.973660	0.486830	+	0.873497i	$0.338153\pi$
$192$	38.4953	2.77816
$193$	18.3587	1.32149	0.660745	−	0.750610i	$-0.270240\pi$
$193$	18.3587	1.32149	0.660745	+	0.750610i	$0.270240\pi$
$194$	9.48408	0.680917
$195$	58.4236	4.18380
$196$	12.9487	0.924905
$197$	11.4042	0.812516	0.406258	−	0.913758i	$-0.366833\pi$
$197$	11.4042	0.812516	0.406258	+	0.913758i	$0.366833\pi$
$198$	−16.6775	−1.18522
$199$	−5.89048	−0.417565	−0.208783	−	0.977962i	$-0.566950\pi$
$199$	−5.89048	−0.417565	−0.208783	+	0.977962i	$0.566950\pi$
$200$	−3.18816	−0.225437
$201$	6.41571	0.452529
$202$	−25.5428	−1.79718
$203$	18.5730	1.30357
$204$	−8.58168	−0.600838
$205$	7.29752	0.509681
$206$	4.48293	0.312340
$207$	−7.36760	−0.512083
$208$	−16.8422	−1.16780
$209$	−6.12661	−0.423786
$210$	−72.0753	−4.97367
$211$	−13.5821	−0.935028	−0.467514	−	0.883986i	$-0.654850\pi$
$211$	−13.5821	−0.935028	−0.467514	+	0.883986i	$0.654850\pi$
$212$	−23.1101	−1.58721
$213$	8.62379	0.590892
$214$	10.3160	0.705185
$215$	−30.0151	−2.04701
$216$	19.6765	1.33882
$217$	17.1849	1.16659
$218$	11.8529	0.802783
$219$	−32.4283	−2.19130
$220$	−6.43360	−0.433754
$221$	6.07583	0.408705
$222$	53.3882	3.58318
$223$	−14.7692	−0.989020	−0.494510	−	0.869172i	$-0.664652\pi$
$223$	−14.7692	−0.989020	−0.494510	+	0.869172i	$0.664652\pi$
$224$	27.8893	1.86343
$225$	27.1555	1.81037
$226$	12.9144	0.859053
$227$	−17.4581	−1.15873	−0.579367	−	0.815067i	$-0.696700\pi$
$227$	−17.4581	−1.15873	−0.579367	+	0.815067i	$0.696700\pi$
$228$	57.0747	3.77987
$229$	19.2134	1.26966	0.634828	−	0.772654i	$-0.281071\pi$
$229$	19.2134	1.26966	0.634828	+	0.772654i	$0.281071\pi$
$230$	−5.13372	−0.338508
$231$	−10.8570	−0.714335
$232$	−5.40437	−0.354814
$233$	0.887488	0.0581412	0.0290706	−	0.999577i	$-0.490745\pi$
$233$	0.887488	0.0581412	0.0290706	+	0.999577i	$0.490745\pi$
$234$	−109.999	−7.19086
$235$	−10.2548	−0.668951
$236$	−29.6508	−1.93010
$237$	−25.8887	−1.68165
$238$	−7.49555	−0.485865
$239$	18.5008	1.19672	0.598359	−	0.801228i	$-0.295820\pi$
$239$	18.5008	1.19672	0.598359	+	0.801228i	$0.295820\pi$
$240$	−27.3513	−1.76552
$241$	5.73501	0.369424	0.184712	−	0.982793i	$-0.440865\pi$
$241$	5.73501	0.369424	0.184712	+	0.982793i	$0.440865\pi$
$242$	21.5335	1.38423
$243$	−78.8274	−5.05678
$244$	−22.6567	−1.45045
$245$	−14.8880	−0.951163
$246$	−18.4971	−1.17933
$247$	−40.4089	−2.57116
$248$	−5.00047	−0.317530
$249$	5.24235	0.332220
$250$	−11.2634	−0.712359
$251$	−17.8109	−1.12422	−0.562108	−	0.827064i	$-0.690010\pi$
$251$	−17.8109	−1.12422	−0.562108	+	0.827064i	$0.690010\pi$
$252$	75.1282	4.73263
$253$	−0.773311	−0.0486177
$254$	−42.5412	−2.66927
$255$	9.86699	0.617895
$256$	5.81543	0.363464
$257$	−10.2668	−0.640426	−0.320213	−	0.947346i	$-0.603755\pi$
$257$	−10.2668	−0.640426	−0.320213	+	0.947346i	$0.603755\pi$
$258$	76.0798	4.73652
$259$	25.8163	1.60414
$260$	−42.4338	−2.63163
$261$	46.0323	2.84933
$262$	11.7493	0.725873
$263$	−8.88774	−0.548042	−0.274021	−	0.961724i	$-0.588354\pi$
$263$	−8.88774	−0.548042	−0.274021	+	0.961724i	$0.588354\pi$
$264$	3.15916	0.194433
$265$	26.5714	1.63227
$266$	49.8512	3.05657
$267$	4.79314	0.293335
$268$	−4.65981	−0.284643
$269$	22.2416	1.35609	0.678047	−	0.735018i	$-0.262827\pi$
$269$	22.2416	1.35609	0.678047	+	0.735018i	$0.262827\pi$
$270$	−116.781	−7.10707
$271$	−27.5000	−1.67050	−0.835252	−	0.549867i	$-0.814678\pi$
$271$	−27.5000	−1.67050	−0.835252	+	0.549867i	$0.814678\pi$
$272$	−2.84443	−0.172469
$273$	−71.6087	−4.33395
$274$	−29.7387	−1.79658
$275$	2.85027	0.171878
$276$	7.20408	0.433634
$277$	9.68860	0.582132	0.291066	−	0.956703i	$-0.405990\pi$
$277$	9.68860	0.582132	0.291066	+	0.956703i	$0.405990\pi$
$278$	6.28743	0.377095
$279$	42.5921	2.54992
$280$	10.1414	0.606064
$281$	−19.5829	−1.16822	−0.584109	−	0.811675i	$-0.698556\pi$
$281$	−19.5829	−1.16822	−0.584109	+	0.811675i	$0.698556\pi$
$282$	25.9931	1.54786
$283$	−4.43671	−0.263735	−0.131867	−	0.991267i	$-0.542097\pi$
$283$	−4.43671	−0.263735	−0.131867	+	0.991267i	$0.542097\pi$
$284$	−6.26357	−0.371674
$285$	−65.6231	−3.88717
$286$	−11.5456	−0.682707
$287$	−8.94442	−0.527973
$288$	69.1224	4.07308
$289$	−15.9739	−0.939639
$290$	32.0752	1.88352
$291$	15.3022	0.897030
$292$	23.5530	1.37834
$293$	9.17870	0.536225	0.268113	−	0.963388i	$-0.413600\pi$
$293$	9.17870	0.536225	0.268113	+	0.963388i	$0.413600\pi$
$294$	37.7370	2.20087
$295$	34.0917	1.98490
$296$	−7.51202	−0.436627
$297$	−17.5912	−1.02074
$298$	−27.7485	−1.60743
$299$	−5.10049	−0.294969
$300$	−26.5528	−1.53303
$301$	36.7889	2.12048
$302$	40.8724	2.35194
$303$	−41.2122	−2.36758
$304$	18.9177	1.08500
$305$	26.0501	1.49163
$306$	−18.5774	−1.06200
$307$	−12.0919	−0.690122	−0.345061	−	0.938580i	$-0.612142\pi$
$307$	−12.0919	−0.690122	−0.345061	+	0.938580i	$0.612142\pi$
$308$	7.88554	0.449321
$309$	7.23303	0.411472
$310$	29.6781	1.68560
$311$	30.9053	1.75248	0.876239	−	0.481877i	$-0.160045\pi$
$311$	30.9053	1.75248	0.876239	+	0.481877i	$0.160045\pi$
$312$	20.8367	1.17965
$313$	4.90980	0.277518	0.138759	−	0.990326i	$-0.455689\pi$
$313$	4.90980	0.277518	0.138759	+	0.990326i	$0.455689\pi$
$314$	−1.43281	−0.0808582
$315$	−86.3805	−4.86699
$316$	18.8033	1.05777
$317$	15.4344	0.866880	0.433440	−	0.901182i	$-0.357300\pi$
$317$	15.4344	0.866880	0.433440	+	0.901182i	$0.357300\pi$
$318$	−67.3509	−3.77685
$319$	4.83160	0.270518
$320$	32.1472	1.79708
$321$	16.6444	0.929000
$322$	6.29231	0.350657
$323$	−6.82455	−0.379728
$324$	99.4028	5.52238
$325$	18.7994	1.04280
$326$	21.4491	1.18795
$327$	19.1243	1.05757
$328$	2.60265	0.143707
$329$	12.5691	0.692959
$330$	−18.7498	−1.03214
$331$	13.7088	0.753505	0.376753	−	0.926314i	$-0.377041\pi$
$331$	13.7088	0.753505	0.376753	+	0.926314i	$0.377041\pi$
$332$	−3.80758	−0.208968
$333$	63.9845	3.50633
$334$	−48.0738	−2.63048
$335$	5.35773	0.292724
$336$	33.5240	1.82888
$337$	15.9277	0.867640	0.433820	−	0.901000i	$-0.357165\pi$
$337$	15.9277	0.867640	0.433820	+	0.901000i	$0.357165\pi$
$338$	−48.6334	−2.64531
$339$	20.8369	1.13170
$340$	−7.16652	−0.388659
$341$	4.47052	0.242092
$342$	123.554	6.68104
$343$	−6.22208	−0.335961
$344$	−10.7048	−0.577167
$345$	−8.28306	−0.445945
$346$	−22.3730	−1.20278
$347$	21.2333	1.13986	0.569931	−	0.821693i	$-0.306970\pi$
$347$	21.2333	1.13986	0.569931	+	0.821693i	$0.306970\pi$
$348$	−45.0107	−2.41282
$349$	12.4728	0.667653	0.333827	−	0.942634i	$-0.391660\pi$
$349$	12.4728	0.667653	0.333827	+	0.942634i	$0.391660\pi$
$350$	−23.1922	−1.23968
$351$	−116.025	−6.19296
$352$	7.25517	0.386702
$353$	−13.1609	−0.700484	−0.350242	−	0.936659i	$-0.613901\pi$
$353$	−13.1609	−0.700484	−0.350242	+	0.936659i	$0.613901\pi$
$354$	−86.4128	−4.59279
$355$	7.20169	0.382226
$356$	−3.48131	−0.184509
$357$	−12.0938	−0.640071
$358$	−17.5702	−0.928612
$359$	19.5438	1.03148	0.515741	−	0.856745i	$-0.327517\pi$
$359$	19.5438	1.03148	0.515741	+	0.856745i	$0.327517\pi$
$360$	25.1350	1.32473
$361$	26.3885	1.38887
$362$	−34.6928	−1.82341
$363$	34.7435	1.82356
$364$	52.0103	2.72608
$365$	−27.0807	−1.41747
$366$	−66.0297	−3.45143
$367$	−4.01323	−0.209489	−0.104744	−	0.994499i	$-0.533402\pi$
$367$	−4.01323	−0.209489	−0.104744	+	0.994499i	$0.533402\pi$
$368$	2.38782	0.124474
$369$	−22.1684	−1.15404
$370$	44.5843	2.31783
$371$	−32.5680	−1.69085
$372$	−41.6468	−2.15929
$373$	−32.2858	−1.67169	−0.835846	−	0.548963i	$-0.815023\pi$
$373$	−32.2858	−1.67169	−0.835846	+	0.548963i	$0.815023\pi$
$374$	−1.94991	−0.100827
$375$	−18.1730	−0.938452
$376$	−3.65737	−0.188614
$377$	31.8676	1.64126
$378$	143.136	7.36214
$379$	−16.4744	−0.846233	−0.423116	−	0.906075i	$-0.639064\pi$
$379$	−16.4744	−0.846233	−0.423116	+	0.906075i	$0.639064\pi$
$380$	47.6628	2.44505
$381$	−68.6385	−3.51646
$382$	−28.4832	−1.45733
$383$	14.0134	0.716052	0.358026	−	0.933712i	$-0.383450\pi$
$383$	14.0134	0.716052	0.358026	+	0.933712i	$0.383450\pi$
$384$	−26.9894	−1.37730
$385$	−9.06660	−0.462077
$386$	−38.8605	−1.97795
$387$	91.1798	4.63493
$388$	−11.1142	−0.564236
$389$	−1.30914	−0.0663759	−0.0331879	−	0.999449i	$-0.510566\pi$
$389$	−1.30914	−0.0663759	−0.0331879	+	0.999449i	$0.510566\pi$
$390$	−123.667	−6.26212
$391$	−0.861407	−0.0435632
$392$	−5.30980	−0.268186
$393$	18.9570	0.956255
$394$	−24.1396	−1.21614
$395$	−21.6195	−1.08780
$396$	19.5440	0.982122
$397$	23.7915	1.19406	0.597032	−	0.802218i	$-0.296347\pi$
$397$	23.7915	1.19406	0.597032	+	0.802218i	$0.296347\pi$
$398$	12.4686	0.624993
$399$	80.4329	4.02668
$400$	−8.80104	−0.440052
$401$	1.01741	0.0508069	0.0254035	−	0.999677i	$-0.491913\pi$
$401$	1.01741	0.0508069	0.0254035	+	0.999677i	$0.491913\pi$
$402$	−13.5803	−0.677325
$403$	29.4859	1.46880
$404$	29.9330	1.48922
$405$	−114.291	−5.67915
$406$	−39.3140	−1.95112
$407$	6.71589	0.332894
$408$	3.51905	0.174219
$409$	20.5656	1.01690	0.508452	−	0.861090i	$-0.330218\pi$
$409$	20.5656	1.01690	0.508452	+	0.861090i	$0.330218\pi$
$410$	−15.4469	−0.762867
$411$	−47.9822	−2.36679
$412$	−5.25344	−0.258818
$413$	−41.7855	−2.05613
$414$	15.5952	0.766463
$415$	4.37786	0.214901
$416$	47.8525	2.34616
$417$	10.1445	0.496780
$418$	12.9684	0.634304
$419$	23.3443	1.14044	0.570221	−	0.821491i	$-0.306858\pi$
$419$	23.3443	1.14044	0.570221	+	0.821491i	$0.306858\pi$
$420$	84.4633	4.12139
$421$	2.57361	0.125430	0.0627150	−	0.998031i	$-0.480024\pi$
$421$	2.57361	0.125430	0.0627150	+	0.998031i	$0.480024\pi$
$422$	28.7496	1.39951
$423$	31.1521	1.51467
$424$	9.47665	0.460227
$425$	3.17498	0.154009
$426$	−18.2542	−0.884421
$427$	−31.9291	−1.54516
$428$	−12.0890	−0.584346
$429$	−18.6284	−0.899388
$430$	63.5339	3.06387
$431$	5.31830	0.256174	0.128087	−	0.991763i	$-0.459116\pi$
$431$	5.31830	0.256174	0.128087	+	0.991763i	$0.459116\pi$
$432$	54.3177	2.61336
$433$	28.6879	1.37865	0.689326	−	0.724451i	$-0.257907\pi$
$433$	28.6879	1.37865	0.689326	+	0.724451i	$0.257907\pi$
$434$	−36.3758	−1.74610
$435$	51.7521	2.48132
$436$	−13.8902	−0.665220
$437$	5.72902	0.274056
$438$	68.6419	3.27984
$439$	5.45554	0.260379	0.130189	−	0.991489i	$-0.458441\pi$
$439$	5.45554	0.260379	0.130189	+	0.991489i	$0.458441\pi$
$440$	2.63820	0.125771
$441$	45.2269	2.15366
$442$	−12.8609	−0.611730
$443$	21.5363	1.02322	0.511609	−	0.859218i	$-0.329050\pi$
$443$	21.5363	1.02322	0.511609	+	0.859218i	$0.329050\pi$
$444$	−62.5644	−2.96918
$445$	4.00273	0.189747
$446$	31.2624	1.48032
$447$	−44.7711	−2.11760
$448$	−39.4022	−1.86158
$449$	−22.7558	−1.07391	−0.536957	−	0.843610i	$-0.680426\pi$
$449$	−22.7558	−1.07391	−0.536957	+	0.843610i	$0.680426\pi$
$450$	−57.4809	−2.70967
$451$	−2.32682	−0.109566
$452$	−15.1341	−0.711847
$453$	65.9460	3.09841
$454$	36.9541	1.73434
$455$	−59.8001	−2.80347
$456$	−23.4044	−1.09601
$457$	−9.00618	−0.421291	−0.210646	−	0.977563i	$-0.567557\pi$
$457$	−9.00618	−0.421291	−0.210646	+	0.977563i	$0.567557\pi$
$458$	−40.6695	−1.90036
$459$	−19.5951	−0.914623
$460$	6.01609	0.280502
$461$	−11.4985	−0.535537	−0.267768	−	0.963483i	$-0.586286\pi$
$461$	−11.4985	−0.535537	−0.267768	+	0.963483i	$0.586286\pi$
$462$	22.9813	1.06918
$463$	8.21854	0.381948	0.190974	−	0.981595i	$-0.438835\pi$
$463$	8.21854	0.381948	0.190974	+	0.981595i	$0.438835\pi$
$464$	−14.9190	−0.692596
$465$	47.8844	2.22059
$466$	−1.87857	−0.0870232
$467$	−15.2799	−0.707070	−0.353535	−	0.935421i	$-0.615021\pi$
$467$	−15.2799	−0.707070	−0.353535	+	0.935421i	$0.615021\pi$
$468$	128.905	5.95865
$469$	−6.56686	−0.303230
$470$	21.7067	1.00126
$471$	−2.31178	−0.106521
$472$	12.1588	0.559653
$473$	9.57033	0.440044
$474$	54.7994	2.51702
$475$	−21.1160	−0.968870
$476$	8.78386	0.402608
$477$	−80.7185	−3.69584
$478$	−39.1612	−1.79119
$479$	14.1300	0.645615	0.322807	−	0.946465i	$-0.395373\pi$
$479$	14.1300	0.645615	0.322807	+	0.946465i	$0.395373\pi$
$480$	77.7113	3.54702
$481$	44.2956	2.01971
$482$	−12.1395	−0.552937
$483$	10.1524	0.461950
$484$	−25.2346	−1.14703
$485$	12.7788	0.580255
$486$	166.856	7.56876
$487$	12.2268	0.554050	0.277025	−	0.960863i	$-0.410652\pi$
$487$	12.2268	0.554050	0.277025	+	0.960863i	$0.410652\pi$
$488$	9.29074	0.420572
$489$	34.6072	1.56499
$490$	31.5140	1.42366
$491$	14.0092	0.632228	0.316114	−	0.948721i	$-0.397622\pi$
$491$	14.0092	0.632228	0.316114	+	0.948721i	$0.397622\pi$
$492$	21.6764	0.977246
$493$	5.38202	0.242394
$494$	85.5348	3.84839
$495$	−22.4712	−1.01000
$496$	−13.8040	−0.619818
$497$	−8.82697	−0.395944
$498$	−11.0966	−0.497252
$499$	20.5865	0.921579	0.460790	−	0.887509i	$-0.347566\pi$
$499$	20.5865	0.921579	0.460790	+	0.887509i	$0.347566\pi$
$500$	13.1993	0.590291
$501$	−77.5651	−3.46535
$502$	37.7010	1.68268
$503$	−25.5216	−1.13795	−0.568976	−	0.822354i	$-0.692660\pi$
$503$	−25.5216	−1.13795	−0.568976	+	0.822354i	$0.692660\pi$
$504$	−30.8075	−1.37227
$505$	−34.4161	−1.53150
$506$	1.63689	0.0727687
$507$	−78.4680	−3.48489
$508$	49.8530	2.21187
$509$	18.8883	0.837208	0.418604	−	0.908169i	$-0.362520\pi$
$509$	18.8883	0.837208	0.418604	+	0.908169i	$0.362520\pi$
$510$	−20.8858	−0.924837
$511$	33.1923	1.46834
$512$	−28.1149	−1.24251
$513$	130.323	5.75389
$514$	21.7321	0.958561
$515$	6.04027	0.266166
$516$	−89.1561	−3.92488
$517$	3.26976	0.143804
$518$	−54.6461	−2.40101
$519$	−36.0979	−1.58452
$520$	17.4006	0.763068
$521$	−28.7591	−1.25996	−0.629979	−	0.776612i	$-0.716937\pi$
$521$	−28.7591	−1.25996	−0.629979	+	0.776612i	$0.716937\pi$
$522$	−97.4380	−4.26474
$523$	−11.0610	−0.483665	−0.241833	−	0.970318i	$-0.577749\pi$
$523$	−11.0610	−0.483665	−0.241833	+	0.970318i	$0.577749\pi$
$524$	−13.7687	−0.601489
$525$	−37.4197	−1.63313
$526$	18.8129	0.820283
$527$	4.97980	0.216923
$528$	8.72098	0.379532
$529$	−22.2769	−0.968560
$530$	−56.2444	−2.44310
$531$	−103.564	−4.49428
$532$	−58.4194	−2.53280
$533$	−15.3469	−0.664747
$534$	−10.1458	−0.439051
$535$	13.8997	0.600935
$536$	1.91083	0.0825352
$537$	−28.3488	−1.22334
$538$	−47.0795	−2.02974
$539$	4.74706	0.204471
$540$	136.853	5.88922
$541$	36.7360	1.57940	0.789702	−	0.613491i	$-0.210235\pi$
$541$	36.7360	1.57940	0.789702	+	0.613491i	$0.210235\pi$
$542$	58.2100	2.50033
$543$	−55.9755	−2.40214
$544$	8.08168	0.346499
$545$	15.9706	0.684105
$546$	151.576	6.48686
$547$	−23.3401	−0.997950	−0.498975	−	0.866616i	$-0.666290\pi$
$547$	−23.3401	−0.997950	−0.498975	+	0.866616i	$0.666290\pi$
$548$	34.8501	1.48872
$549$	−79.1349	−3.37740
$550$	−6.03326	−0.257259
$551$	−35.7946	−1.52490
$552$	−2.95414	−0.125737
$553$	26.4986	1.12684
$554$	−20.5082	−0.871308
$555$	71.9349	3.05347
$556$	−7.36810	−0.312477
$557$	−7.74771	−0.328281	−0.164140	−	0.986437i	$-0.552485\pi$
$557$	−7.74771	−0.328281	−0.164140	+	0.986437i	$0.552485\pi$
$558$	−90.1560	−3.81661
$559$	63.1225	2.66980
$560$	27.9957	1.18303
$561$	−3.14610	−0.132828
$562$	41.4517	1.74853
$563$	−2.43750	−0.102728	−0.0513642	−	0.998680i	$-0.516357\pi$
$563$	−2.43750	−0.102728	−0.0513642	+	0.998680i	$0.516357\pi$
$564$	−30.4607	−1.28263
$565$	17.4008	0.732056
$566$	9.39131	0.394746
$567$	140.084	5.88297
$568$	2.56847	0.107771
$569$	−30.8797	−1.29454	−0.647272	−	0.762259i	$-0.724090\pi$
$569$	−30.8797	−1.29454	−0.647272	+	0.762259i	$0.724090\pi$
$570$	138.906	5.81814
$571$	−9.74216	−0.407697	−0.203848	−	0.979002i	$-0.565345\pi$
$571$	−9.74216	−0.407697	−0.203848	+	0.979002i	$0.565345\pi$
$572$	13.5300	0.565720
$573$	−45.9566	−1.91986
$574$	18.9329	0.790245
$575$	−2.66530	−0.111151
$576$	−97.6567	−4.06903
$577$	−17.7111	−0.737323	−0.368662	−	0.929564i	$-0.620184\pi$
$577$	−17.7111	−0.737323	−0.368662	+	0.929564i	$0.620184\pi$
$578$	33.8124	1.40641
$579$	−62.6998	−2.60572
$580$	−37.5882	−1.56076
$581$	−5.36586	−0.222613
$582$	−32.3906	−1.34263
$583$	−8.47230	−0.350887
$584$	−9.65829	−0.399663
$585$	−148.212	−6.12781
$586$	−19.4288	−0.802598
$587$	22.1977	0.916197	0.458099	−	0.888901i	$-0.348531\pi$
$587$	22.1977	0.916197	0.458099	+	0.888901i	$0.348531\pi$
$588$	−44.2231	−1.82373
$589$	−33.1195	−1.36466
$590$	−72.1629	−2.97090
$591$	−38.9483	−1.60212
$592$	−20.7372	−0.852295
$593$	−2.79758	−0.114883	−0.0574415	−	0.998349i	$-0.518294\pi$
$593$	−2.79758	−0.114883	−0.0574415	+	0.998349i	$0.518294\pi$
$594$	37.2357	1.52780
$595$	−10.0995	−0.414037
$596$	32.5178	1.33198
$597$	20.1175	0.823356
$598$	10.7964	0.441496
$599$	36.8186	1.50437	0.752184	−	0.658954i	$-0.229001\pi$
$599$	36.8186	1.50437	0.752184	+	0.658954i	$0.229001\pi$
$600$	10.8884	0.444517
$601$	−12.2447	−0.499473	−0.249737	−	0.968314i	$-0.580344\pi$
$601$	−12.2447	−0.499473	−0.249737	+	0.968314i	$0.580344\pi$
$602$	−77.8722	−3.17383
$603$	−16.2757	−0.662797
$604$	−47.8974	−1.94892
$605$	29.0141	1.17959
$606$	87.2352	3.54369
$607$	7.73482	0.313947	0.156973	−	0.987603i	$-0.449826\pi$
$607$	7.73482	0.313947	0.156973	+	0.987603i	$0.449826\pi$
$608$	−53.7493	−2.17982
$609$	−63.4315	−2.57038
$610$	−55.1411	−2.23260
$611$	21.5662	0.872474
$612$	21.7704	0.880017
$613$	−30.7711	−1.24283	−0.621417	−	0.783480i	$-0.713443\pi$
$613$	−30.7711	−1.24283	−0.621417	+	0.783480i	$0.713443\pi$
$614$	25.5953	1.03294
$615$	−24.9229	−1.00499
$616$	−3.23359	−0.130285
$617$	48.4172	1.94920	0.974602	−	0.223945i	$-0.0718937\pi$
$617$	48.4172	1.94920	0.974602	+	0.223945i	$0.0718937\pi$
$618$	−15.3104	−0.615873
$619$	31.1500	1.25202	0.626011	−	0.779814i	$-0.284686\pi$
$619$	31.1500	1.25202	0.626011	+	0.779814i	$0.284686\pi$
$620$	−34.7790	−1.39676
$621$	16.4496	0.660098
$622$	−65.4181	−2.62303
$623$	−4.90606	−0.196557
$624$	57.5205	2.30266
$625$	−30.8477	−1.23391
$626$	−10.3927	−0.415377
$627$	20.9240	0.835622
$628$	1.67908	0.0670025
$629$	7.48096	0.298285
$630$	182.844	7.28469
$631$	−4.15907	−0.165570	−0.0827850	−	0.996567i	$-0.526381\pi$
$631$	−4.15907	−0.165570	−0.0827850	+	0.996567i	$0.526381\pi$
$632$	−7.71058	−0.306710
$633$	46.3863	1.84369
$634$	−32.6704	−1.29751
$635$	−57.3197	−2.27466
$636$	78.9270	3.12966
$637$	31.3100	1.24055
$638$	−10.2272	−0.404899
$639$	−21.8773	−0.865451
$640$	−22.5387	−0.890922
$641$	4.18723	0.165386	0.0826928	−	0.996575i	$-0.473648\pi$
$641$	4.18723	0.165386	0.0826928	+	0.996575i	$0.473648\pi$
$642$	−35.2317	−1.39048
$643$	16.4835	0.650044	0.325022	−	0.945706i	$-0.394628\pi$
$643$	16.4835	0.650044	0.325022	+	0.945706i	$0.394628\pi$
$644$	−7.37381	−0.290569
$645$	102.509	4.03630
$646$	14.4457	0.568359
$647$	12.9726	0.510007	0.255003	−	0.966940i	$-0.417923\pi$
$647$	12.9726	0.510007	0.255003	+	0.966940i	$0.417923\pi$
$648$	−40.7617	−1.60127
$649$	−10.8702	−0.426691
$650$	−39.7933	−1.56082
$651$	−58.6910	−2.30028
$652$	−25.1357	−0.984388
$653$	48.1515	1.88432	0.942158	−	0.335170i	$-0.108794\pi$
$653$	48.1515	1.88432	0.942158	+	0.335170i	$0.108794\pi$
$654$	−40.4809	−1.58293
$655$	15.8309	0.618565
$656$	7.18472	0.280516
$657$	82.2656	3.20949
$658$	−26.6055	−1.03719
$659$	−40.2150	−1.56655	−0.783277	−	0.621673i	$-0.786453\pi$
$659$	−40.2150	−1.56655	−0.783277	+	0.621673i	$0.786453\pi$
$660$	21.9724	0.855276
$661$	11.3484	0.441403	0.220701	−	0.975341i	$-0.429165\pi$
$661$	11.3484	0.441403	0.220701	+	0.975341i	$0.429165\pi$
$662$	−29.0179	−1.12781
$663$	−20.7505	−0.805884
$664$	1.56136	0.0605925
$665$	67.1691	2.60471
$666$	−135.438	−5.24811
$667$	−4.51805	−0.174940
$668$	56.3365	2.17973
$669$	50.4407	1.95015
$670$	−11.3409	−0.438136
$671$	−8.30610	−0.320653
$672$	−95.2492	−3.67432
$673$	45.9299	1.77047	0.885233	−	0.465147i	$-0.153999\pi$
$673$	45.9299	1.77047	0.885233	+	0.465147i	$0.153999\pi$
$674$	−33.7147	−1.29864
$675$	−60.6299	−2.33364
$676$	56.9923	2.19201
$677$	−18.5887	−0.714423	−0.357212	−	0.934023i	$-0.616273\pi$
$677$	−18.5887	−0.714423	−0.357212	+	0.934023i	$0.616273\pi$
$678$	−44.1060	−1.69388
$679$	−15.6627	−0.601080
$680$	2.93874	0.112696
$681$	59.6239	2.28479
$682$	−9.46287	−0.362352
$683$	−13.0007	−0.497458	−0.248729	−	0.968573i	$-0.580013\pi$
$683$	−13.0007	−0.497458	−0.248729	+	0.968573i	$0.580013\pi$
$684$	−144.790	−5.53619
$685$	−40.0697	−1.53099
$686$	13.1705	0.502850
$687$	−65.6187	−2.50351
$688$	−29.5511	−1.12663
$689$	−55.8803	−2.12887
$690$	17.5330	0.667470
$691$	−43.5167	−1.65545	−0.827727	−	0.561130i	$-0.810367\pi$
$691$	−43.5167	−1.65545	−0.827727	+	0.561130i	$0.810367\pi$
$692$	26.2184	0.996673
$693$	27.5425	1.04625
$694$	−44.9451	−1.70609
$695$	8.47165	0.321348
$696$	18.4573	0.699623
$697$	−2.59189	−0.0981748
$698$	−26.4015	−0.999313
$699$	−3.03100	−0.114643
$700$	27.1784	1.02725
$701$	17.5184	0.661662	0.330831	−	0.943690i	$-0.392671\pi$
$701$	17.5184	0.661662	0.330831	+	0.943690i	$0.392671\pi$
$702$	245.594	9.26934
$703$	−49.7541	−1.87651
$704$	−10.2502	−0.386318
$705$	35.0229	1.31904
$706$	27.8581	1.04845
$707$	42.1832	1.58646
$708$	101.265	3.80578
$709$	−19.9077	−0.747648	−0.373824	−	0.927500i	$-0.621954\pi$
$709$	−19.9077	−0.747648	−0.373824	+	0.927500i	$0.621954\pi$
$710$	−15.2440	−0.572098
$711$	65.6758	2.46303
$712$	1.42757	0.0535003
$713$	−4.18040	−0.156557
$714$	25.5993	0.958029
$715$	−15.5565	−0.581780
$716$	20.5901	0.769487
$717$	−63.1850	−2.35969
$718$	−41.3689	−1.54387
$719$	37.8404	1.41121	0.705604	−	0.708606i	$-0.250676\pi$
$719$	37.8404	1.41121	0.705604	+	0.708606i	$0.250676\pi$
$720$	69.3862	2.58587
$721$	−7.40344	−0.275718
$722$	−55.8573	−2.07879
$723$	−19.5865	−0.728431
$724$	40.6557	1.51096
$725$	16.6527	0.618464
$726$	−73.5426	−2.72942
$727$	−32.1739	−1.19326	−0.596632	−	0.802515i	$-0.703495\pi$
$727$	−32.1739	−1.19326	−0.596632	+	0.802515i	$0.703495\pi$
$728$	−21.3276	−0.790454
$729$	148.997	5.51842
$730$	57.3225	2.12160
$731$	10.6606	0.394296
$732$	77.3786	2.86000
$733$	−5.46941	−0.202017	−0.101009	−	0.994886i	$-0.532207\pi$
$733$	−5.46941	−0.202017	−0.101009	+	0.994886i	$0.532207\pi$
$734$	8.49492	0.313553
$735$	50.8466	1.87550
$736$	−6.78434	−0.250074
$737$	−1.70832	−0.0629266
$738$	46.9245	1.72731
$739$	3.56734	0.131227	0.0656134	−	0.997845i	$-0.479100\pi$
$739$	3.56734	0.131227	0.0656134	+	0.997845i	$0.479100\pi$
$740$	−52.2472	−1.92065
$741$	138.007	5.06982
$742$	68.9377	2.53078
$743$	20.3163	0.745333	0.372667	−	0.927965i	$-0.378444\pi$
$743$	20.3163	0.745333	0.372667	+	0.927965i	$0.378444\pi$
$744$	17.0779	0.626107
$745$	−37.3882	−1.36980
$746$	68.3402	2.50211
$747$	−13.2990	−0.486587
$748$	2.28505	0.0835497
$749$	−17.0365	−0.622502
$750$	38.4674	1.40463
$751$	−7.00647	−0.255670	−0.127835	−	0.991795i	$-0.540803\pi$
$751$	−7.00647	−0.255670	−0.127835	+	0.991795i	$0.540803\pi$
$752$	−10.0963	−0.368175
$753$	60.8290	2.21673
$754$	−67.4550	−2.45657
$755$	55.0712	2.00425
$756$	−167.738	−6.10058
$757$	26.8219	0.974859	0.487430	−	0.873162i	$-0.337935\pi$
$757$	26.8219	0.974859	0.487430	+	0.873162i	$0.337935\pi$
$758$	34.8718	1.26660
$759$	2.64106	0.0958644
$760$	−19.5449	−0.708968
$761$	23.9639	0.868692	0.434346	−	0.900746i	$-0.356980\pi$
$761$	23.9639	0.868692	0.434346	+	0.900746i	$0.356980\pi$
$762$	145.289	5.26327
$763$	−19.5748	−0.708657
$764$	33.3788	1.20760
$765$	−25.0311	−0.905001
$766$	−29.6626	−1.07175
$767$	−71.6958	−2.58878
$768$	−19.8612	−0.716680
$769$	−28.6002	−1.03135	−0.515675	−	0.856784i	$-0.672459\pi$
$769$	−28.6002	−1.03135	−0.515675	+	0.856784i	$0.672459\pi$
$770$	19.1915	0.691615
$771$	35.0638	1.26279
$772$	45.5397	1.63901
$773$	40.1320	1.44345	0.721724	−	0.692181i	$-0.243350\pi$
$773$	40.1320	1.44345	0.721724	+	0.692181i	$0.243350\pi$
$774$	−193.003	−6.93735
$775$	15.4081	0.553476
$776$	4.55754	0.163606
$777$	−88.1693	−3.16305
$778$	2.77109	0.0993484
$779$	17.2380	0.617617
$780$	144.922	5.18906
$781$	−2.29626	−0.0821667
$782$	1.82337	0.0652034
$783$	−102.776	−3.67291
$784$	−14.6579	−0.523497
$785$	−1.93056	−0.0689047
$786$	−40.1269	−1.43128
$787$	46.8617	1.67044	0.835220	−	0.549916i	$-0.185340\pi$
$787$	46.8617	1.67044	0.835220	+	0.549916i	$0.185340\pi$
$788$	28.2887	1.00774
$789$	30.3539	1.08063
$790$	45.7627	1.62816
$791$	−21.3278	−0.758329
$792$	−8.01432	−0.284776
$793$	−54.7841	−1.94544
$794$	−50.3603	−1.78722
$795$	−90.7482	−3.21851
$796$	−14.6116	−0.517895
$797$	−3.41471	−0.120955	−0.0604776	−	0.998170i	$-0.519262\pi$
$797$	−3.41471	−0.120955	−0.0604776	+	0.998170i	$0.519262\pi$
$798$	−170.255	−6.02695
$799$	3.64225	0.128853
$800$	25.0057	0.884086
$801$	−12.1595	−0.429634
$802$	−2.15358	−0.0760455
$803$	8.63469	0.304712
$804$	15.9145	0.561260
$805$	8.47821	0.298818
$806$	−62.4138	−2.19843
$807$	−75.9609	−2.67395
$808$	−12.2745	−0.431814
$809$	4.96564	0.174583	0.0872913	−	0.996183i	$-0.472179\pi$
$809$	4.96564	0.174583	0.0872913	+	0.996183i	$0.472179\pi$
$810$	241.923	8.50030
$811$	2.16230	0.0759285	0.0379642	−	0.999279i	$-0.487913\pi$
$811$	2.16230	0.0759285	0.0379642	+	0.999279i	$0.487913\pi$
$812$	46.0711	1.61678
$813$	93.9195	3.29390
$814$	−14.2157	−0.498261
$815$	28.9003	1.01233
$816$	9.71448	0.340075
$817$	−70.9010	−2.48051
$818$	−43.5318	−1.52205
$819$	181.660	6.34773
$820$	18.1018	0.632143
$821$	−28.6304	−0.999207	−0.499603	−	0.866254i	$-0.666521\pi$
$821$	−28.6304	−0.999207	−0.499603	+	0.866254i	$0.666521\pi$
$822$	101.565	3.54250
$823$	−8.77235	−0.305785	−0.152893	−	0.988243i	$-0.548859\pi$
$823$	−8.77235	−0.305785	−0.152893	+	0.988243i	$0.548859\pi$
$824$	2.15425	0.0750470
$825$	−9.73443	−0.338909
$826$	88.4487	3.07752
$827$	13.2036	0.459136	0.229568	−	0.973293i	$-0.426269\pi$
$827$	13.2036	0.459136	0.229568	+	0.973293i	$0.426269\pi$
$828$	−18.2757	−0.635123
$829$	56.4850	1.96181	0.980903	−	0.194497i	$-0.0623075\pi$
$829$	56.4850	1.96181	0.980903	+	0.194497i	$0.0623075\pi$
$830$	−9.26675	−0.321653
$831$	−33.0891	−1.14785
$832$	−67.6065	−2.34383
$833$	5.28785	0.183213
$834$	−21.4732	−0.743557
$835$	−64.7743	−2.24161
$836$	−15.1973	−0.525611
$837$	−95.0950	−3.28696
$838$	−49.4135	−1.70696
$839$	−32.2278	−1.11263	−0.556314	−	0.830972i	$-0.687785\pi$
$839$	−32.2278	−1.11263	−0.556314	+	0.830972i	$0.687785\pi$
$840$	−34.6355	−1.19504
$841$	−0.771464	−0.0266022
$842$	−5.44763	−0.187738
$843$	66.8807	2.30349
$844$	−33.6909	−1.15969
$845$	−65.5283	−2.25424
$846$	−65.9405	−2.26708
$847$	−35.5621	−1.22193
$848$	26.1607	0.898361
$849$	15.1525	0.520033
$850$	−6.72057	−0.230514
$851$	−6.28005	−0.215277
$852$	21.3917	0.732868
$853$	2.66825	0.0913591	0.0456795	−	0.998956i	$-0.485455\pi$
$853$	2.66825	0.0913591	0.0456795	+	0.998956i	$0.485455\pi$
$854$	67.5853	2.31272
$855$	166.476	5.69335
$856$	4.95730	0.169437
$857$	−17.2309	−0.588595	−0.294298	−	0.955714i	$-0.595086\pi$
$857$	−17.2309	−0.588595	−0.294298	+	0.955714i	$0.595086\pi$
$858$	39.4313	1.34616
$859$	23.9443	0.816967	0.408484	−	0.912766i	$-0.366058\pi$
$859$	23.9443	0.816967	0.408484	+	0.912766i	$0.366058\pi$
$860$	−74.4538	−2.53885
$861$	30.5475	1.04106
$862$	−11.2574	−0.383429
$863$	−0.955599	−0.0325290	−0.0162645	−	0.999868i	$-0.505177\pi$
$863$	−0.955599	−0.0325290	−0.0162645	+	0.999868i	$0.505177\pi$
$864$	−154.329	−5.25038
$865$	−30.1452	−1.02497
$866$	−60.7245	−2.06350
$867$	54.5549	1.85278
$868$	42.6280	1.44689
$869$	6.89340	0.233843
$870$	−109.545	−3.71393
$871$	−11.2674	−0.381783
$872$	5.69589	0.192887
$873$	−38.8194	−1.31384
$874$	−12.1268	−0.410194
$875$	18.6012	0.628835
$876$	−80.4398	−2.71781
$877$	−25.9737	−0.877069	−0.438534	−	0.898714i	$-0.644502\pi$
$877$	−25.9737	−0.877069	−0.438534	+	0.898714i	$0.644502\pi$
$878$	−11.5479	−0.389723
$879$	−31.3477	−1.05733
$880$	7.28285	0.245505
$881$	32.1184	1.08210	0.541049	−	0.840991i	$-0.318027\pi$
$881$	32.1184	1.08210	0.541049	+	0.840991i	$0.318027\pi$
$882$	−95.7331	−3.22350
$883$	50.2340	1.69051	0.845255	−	0.534363i	$-0.179449\pi$
$883$	50.2340	1.69051	0.845255	+	0.534363i	$0.179449\pi$
$884$	15.0714	0.506905
$885$	−116.432	−3.91382
$886$	−45.5864	−1.53151
$887$	−32.5233	−1.09202	−0.546012	−	0.837777i	$-0.683855\pi$
$887$	−32.5233	−1.09202	−0.546012	+	0.837777i	$0.683855\pi$
$888$	25.6555	0.860942
$889$	70.2557	2.35630
$890$	−8.47269	−0.284005
$891$	36.4417	1.22084
$892$	−36.6357	−1.22665
$893$	−24.2237	−0.810616
$894$	94.7684	3.16953
$895$	−23.6739	−0.791332
$896$	27.6253	0.922897
$897$	17.4195	0.581620
$898$	48.1679	1.60739
$899$	26.1189	0.871114
$900$	67.3605	2.24535
$901$	−9.43746	−0.314408
$902$	4.92524	0.163993
$903$	−125.644	−4.18116
$904$	6.20597	0.206407
$905$	−46.7449	−1.55385
$906$	−139.590	−4.63756
$907$	0.0846461	0.00281063	0.00140531	−	0.999999i	$-0.499553\pi$
$907$	0.0846461	0.00281063	0.00140531	+	0.999999i	$0.499553\pi$
$908$	−43.3056	−1.43715
$909$	104.549	3.46768
$910$	126.581	4.19611
$911$	15.5822	0.516262	0.258131	−	0.966110i	$-0.416893\pi$
$911$	15.5822	0.516262	0.258131	+	0.966110i	$0.416893\pi$
$912$	−64.6087	−2.13941
$913$	−1.39588	−0.0461970
$914$	19.0636	0.630569
$915$	−88.9679	−2.94119
$916$	47.6597	1.57472
$917$	−19.4036	−0.640765
$918$	41.4776	1.36897
$919$	48.6275	1.60407	0.802036	−	0.597276i	$-0.203750\pi$
$919$	48.6275	1.60407	0.802036	+	0.597276i	$0.203750\pi$
$920$	−2.46699	−0.0813343
$921$	41.2970	1.36078
$922$	24.3391	0.801567
$923$	−15.1453	−0.498515
$924$	−26.9312	−0.885971
$925$	23.1470	0.761070
$926$	−17.3964	−0.571682
$927$	−18.3491	−0.602664
$928$	42.3882	1.39146
$929$	8.57588	0.281366	0.140683	−	0.990055i	$-0.455070\pi$
$929$	8.57588	0.281366	0.140683	+	0.990055i	$0.455070\pi$
$930$	−101.358	−3.32367
$931$	−35.1682	−1.15259
$932$	2.20145	0.0721110
$933$	−105.550	−3.45554
$934$	32.3435	1.05831
$935$	−2.62729	−0.0859216
$936$	−52.8596	−1.72777
$937$	−5.42426	−0.177203	−0.0886015	−	0.996067i	$-0.528240\pi$
$937$	−5.42426	−0.177203	−0.0886015	+	0.996067i	$0.528240\pi$
$938$	13.9003	0.453860
$939$	−16.7682	−0.547211
$940$	−25.4376	−0.829682
$941$	−23.7907	−0.775554	−0.387777	−	0.921753i	$-0.626757\pi$
$941$	−23.7907	−0.775554	−0.387777	+	0.921753i	$0.626757\pi$
$942$	4.89342	0.159436
$943$	2.17582	0.0708544
$944$	33.5647	1.09244
$945$	192.861	6.27377
$946$	−20.2578	−0.658638
$947$	21.3398	0.693450	0.346725	−	0.937967i	$-0.387294\pi$
$947$	21.3398	0.693450	0.346725	+	0.937967i	$0.387294\pi$
$948$	−64.2181	−2.08571
$949$	56.9514	1.84872
$950$	44.6969	1.45016
$951$	−52.7124	−1.70932
$952$	−3.60196	−0.116740
$953$	−40.9605	−1.32684	−0.663420	−	0.748247i	$-0.730896\pi$
$953$	−40.9605	−1.32684	−0.663420	+	0.748247i	$0.730896\pi$
$954$	170.859	5.53177
$955$	−38.3781	−1.24189
$956$	45.8921	1.48426
$957$	−16.5012	−0.533408
$958$	−29.9093	−0.966327
$959$	49.1127	1.58593
$960$	−109.791	−3.54349
$961$	−6.83311	−0.220423
$962$	−93.7619	−3.02300
$963$	−42.2243	−1.36066
$964$	14.2260	0.458187
$965$	−52.3603	−1.68554
$966$	−21.4899	−0.691425
$967$	2.51405	0.0808462	0.0404231	−	0.999183i	$-0.487129\pi$
$967$	2.51405	0.0808462	0.0404231	+	0.999183i	$0.487129\pi$
$968$	10.3479	0.332593
$969$	23.3076	0.748748
$970$	−27.0492	−0.868499
$971$	−44.4034	−1.42497	−0.712486	−	0.701687i	$-0.752431\pi$
$971$	−44.4034	−1.42497	−0.712486	+	0.701687i	$0.752431\pi$
$972$	−195.535	−6.27179
$973$	−10.3835	−0.332881
$974$	−25.8809	−0.829277
$975$	−64.2048	−2.05620
$976$	25.6474	0.820955
$977$	−22.9880	−0.735450	−0.367725	−	0.929935i	$-0.619863\pi$
$977$	−22.9880	−0.735450	−0.367725	+	0.929935i	$0.619863\pi$
$978$	−73.2541	−2.34241
$979$	−1.27627	−0.0407898
$980$	−36.9305	−1.17970
$981$	−48.5154	−1.54898
$982$	−29.6538	−0.946289
$983$	1.00000	0.0318950
$983$	1.00000	0.0318950
$984$	−8.88873	−0.283362
$985$	−32.5256	−1.03635
$986$	−11.3923	−0.362804
$987$	−42.9269	−1.36638
$988$	−100.236	−3.18894
$989$	−8.94926	−0.284570
$990$	47.5654	1.51173
$991$	−35.7494	−1.13562	−0.567809	−	0.823161i	$-0.692209\pi$
$991$	−35.7494	−1.13562	−0.567809	+	0.823161i	$0.692209\pi$
$992$	39.2203	1.24525
$993$	−46.8192	−1.48576
$994$	18.6843	0.592630
$995$	16.8001	0.532598
$996$	13.0039	0.412044
$997$	−24.4569	−0.774558	−0.387279	−	0.921962i	$-0.626585\pi$
$997$	−24.4569	−0.774558	−0.387279	+	0.921962i	$0.626585\pi$
$998$	−43.5761	−1.37938
$999$	−142.858	−4.51982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.2.a.b.1.8	✓	54
3.2	odd	2		8847.2.a.g.1.47		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.2.a.b.1.8	✓	54	1.1	even	1	trivial
8847.2.a.g.1.47		54	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 983 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	983.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11673 q^{2} -3.41526 q^{3} +2.48055 q^{4} -2.85207 q^{5} +7.22918 q^{6} +3.49572 q^{7} -1.01719 q^{8} +8.66400 q^{9} +O(q^{10})\)	\(q-2.11673 q^{2} -3.41526 q^{3} +2.48055 q^{4} -2.85207 q^{5} +7.22918 q^{6} +3.49572 q^{7} -1.01719 q^{8} +8.66400 q^{9} +6.03706 q^{10} +0.909384 q^{11} -8.47171 q^{12} +5.99798 q^{13} -7.39951 q^{14} +9.74055 q^{15} -2.80798 q^{16} +1.01298 q^{17} -18.3394 q^{18} -6.73710 q^{19} -7.07469 q^{20} -11.9388 q^{21} -1.92492 q^{22} -0.850369 q^{23} +3.47396 q^{24} +3.13429 q^{25} -12.6961 q^{26} -19.3440 q^{27} +8.67131 q^{28} +5.31305 q^{29} -20.6181 q^{30} +4.91598 q^{31} +7.97812 q^{32} -3.10578 q^{33} -2.14421 q^{34} -9.97004 q^{35} +21.4915 q^{36} +7.38510 q^{37} +14.2606 q^{38} -20.4846 q^{39} +2.90109 q^{40} -2.55868 q^{41} +25.2712 q^{42} +10.5240 q^{43} +2.25577 q^{44} -24.7103 q^{45} +1.80000 q^{46} +3.59557 q^{47} +9.58999 q^{48} +5.22009 q^{49} -6.63445 q^{50} -3.45959 q^{51} +14.8783 q^{52} -9.31653 q^{53} +40.9461 q^{54} -2.59362 q^{55} -3.55580 q^{56} +23.0089 q^{57} -11.2463 q^{58} -11.9533 q^{59} +24.1619 q^{60} -9.13376 q^{61} -10.4058 q^{62} +30.2870 q^{63} -11.2716 q^{64} -17.1066 q^{65} +6.57410 q^{66} -1.87854 q^{67} +2.51274 q^{68} +2.90423 q^{69} +21.1039 q^{70} -2.52508 q^{71} -8.81291 q^{72} +9.49510 q^{73} -15.6323 q^{74} -10.7044 q^{75} -16.7117 q^{76} +3.17896 q^{77} +43.3605 q^{78} +7.58030 q^{79} +8.00856 q^{80} +40.0729 q^{81} +5.41602 q^{82} -1.53498 q^{83} -29.6148 q^{84} -2.88909 q^{85} -22.2764 q^{86} -18.1455 q^{87} -0.925013 q^{88} -1.40345 q^{89} +52.3051 q^{90} +20.9673 q^{91} -2.10938 q^{92} -16.7894 q^{93} -7.61086 q^{94} +19.2147 q^{95} -27.2473 q^{96} -4.48053 q^{97} -11.0495 q^{98} +7.87890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 8 q^{2} + 6 q^{3} + 64 q^{4} + 7 q^{5} + 5 q^{6} + 31 q^{7} + 24 q^{8} + 74 q^{9}+O(q^{10}) \) 54 * q + 8 * q^2 + 6 * q^3 + 64 * q^4 + 7 * q^5 + 5 * q^6 + 31 * q^7 + 24 * q^8 + 74 * q^9	\( 54 q + 8 q^{2} + 6 q^{3} + 64 q^{4} + 7 q^{5} + 5 q^{6} + 31 q^{7} + 24 q^{8} + 74 q^{9} + 15 q^{10} + 8 q^{11} + 10 q^{12} + 34 q^{13} + q^{14} + 3 q^{15} + 80 q^{16} + 32 q^{17} + 28 q^{18} + 15 q^{19} + 4 q^{20} + 11 q^{21} + 25 q^{22} + 15 q^{23} - 7 q^{24} + 125 q^{25} - 14 q^{26} + 12 q^{27} + 78 q^{28} + 8 q^{29} - 32 q^{30} + 16 q^{31} + 36 q^{32} + 38 q^{33} - 8 q^{34} - 2 q^{35} + 65 q^{36} + 80 q^{37} - 14 q^{38} + 16 q^{39} + 36 q^{40} + 30 q^{41} - 10 q^{42} + 53 q^{43} + 6 q^{44} + 3 q^{45} + 24 q^{46} + 22 q^{47} + 7 q^{48} + 111 q^{49} + 14 q^{50} - 10 q^{51} + 45 q^{52} + 10 q^{53} - 8 q^{54} + 12 q^{55} - 30 q^{56} + 106 q^{57} + 61 q^{58} - 4 q^{59} - 61 q^{60} + 24 q^{61} - 12 q^{62} + 73 q^{63} + 86 q^{64} + 32 q^{65} - 43 q^{66} + 54 q^{67} + 33 q^{68} - 30 q^{69} - 21 q^{70} - 6 q^{71} + 60 q^{72} + 172 q^{73} - 32 q^{74} - 36 q^{75} + 7 q^{76} - 12 q^{77} - 3 q^{78} + 28 q^{79} - 66 q^{80} + 86 q^{81} + 4 q^{82} + 14 q^{83} - 58 q^{84} + 99 q^{85} - 31 q^{86} - 27 q^{87} + 5 q^{88} - 5 q^{89} - 45 q^{90} - 11 q^{91} + 20 q^{92} + 27 q^{93} - 39 q^{94} + 5 q^{95} - 87 q^{96} + 127 q^{97} - 29 q^{98} - 21 q^{99}+O(q^{100}) \) 54 * q + 8 * q^2 + 6 * q^3 + 64 * q^4 + 7 * q^5 + 5 * q^6 + 31 * q^7 + 24 * q^8 + 74 * q^9 + 15 * q^10 + 8 * q^11 + 10 * q^12 + 34 * q^13 + q^14 + 3 * q^15 + 80 * q^16 + 32 * q^17 + 28 * q^18 + 15 * q^19 + 4 * q^20 + 11 * q^21 + 25 * q^22 + 15 * q^23 - 7 * q^24 + 125 * q^25 - 14 * q^26 + 12 * q^27 + 78 * q^28 + 8 * q^29 - 32 * q^30 + 16 * q^31 + 36 * q^32 + 38 * q^33 - 8 * q^34 - 2 * q^35 + 65 * q^36 + 80 * q^37 - 14 * q^38 + 16 * q^39 + 36 * q^40 + 30 * q^41 - 10 * q^42 + 53 * q^43 + 6 * q^44 + 3 * q^45 + 24 * q^46 + 22 * q^47 + 7 * q^48 + 111 * q^49 + 14 * q^50 - 10 * q^51 + 45 * q^52 + 10 * q^53 - 8 * q^54 + 12 * q^55 - 30 * q^56 + 106 * q^57 + 61 * q^58 - 4 * q^59 - 61 * q^60 + 24 * q^61 - 12 * q^62 + 73 * q^63 + 86 * q^64 + 32 * q^65 - 43 * q^66 + 54 * q^67 + 33 * q^68 - 30 * q^69 - 21 * q^70 - 6 * q^71 + 60 * q^72 + 172 * q^73 - 32 * q^74 - 36 * q^75 + 7 * q^76 - 12 * q^77 - 3 * q^78 + 28 * q^79 - 66 * q^80 + 86 * q^81 + 4 * q^82 + 14 * q^83 - 58 * q^84 + 99 * q^85 - 31 * q^86 - 27 * q^87 + 5 * q^88 - 5 * q^89 - 45 * q^90 - 11 * q^91 + 20 * q^92 + 27 * q^93 - 39 * q^94 + 5 * q^95 - 87 * q^96 + 127 * q^97 - 29 * q^98 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more