LMFDB - Embedded newform 8003.2.a.a.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8003,2,Mod(1,8003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8003 = 53 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8003.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:	$63.9042767376$
Analytic rank:	$1$
Dimension:	$147$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40138 q^{2} +1.08484 q^{3} +3.76663 q^{4} -1.87801 q^{5} -2.60511 q^{6} -2.74065 q^{7} -4.24236 q^{8} -1.82313 q^{9} +O(q^{10})$	\(q-2.40138 q^{2} +1.08484 q^{3} +3.76663 q^{4} -1.87801 q^{5} -2.60511 q^{6} -2.74065 q^{7} -4.24236 q^{8} -1.82313 q^{9} +4.50981 q^{10} +5.80487 q^{11} +4.08618 q^{12} +5.73860 q^{13} +6.58136 q^{14} -2.03733 q^{15} +2.65427 q^{16} -5.60800 q^{17} +4.37803 q^{18} +2.29560 q^{19} -7.07376 q^{20} -2.97316 q^{21} -13.9397 q^{22} -2.06733 q^{23} -4.60227 q^{24} -1.47310 q^{25} -13.7806 q^{26} -5.23231 q^{27} -10.3230 q^{28} -0.0903616 q^{29} +4.89241 q^{30} +1.21394 q^{31} +2.11082 q^{32} +6.29734 q^{33} +13.4669 q^{34} +5.14696 q^{35} -6.86706 q^{36} -0.541111 q^{37} -5.51261 q^{38} +6.22545 q^{39} +7.96718 q^{40} +6.90388 q^{41} +7.13970 q^{42} -3.25966 q^{43} +21.8648 q^{44} +3.42385 q^{45} +4.96445 q^{46} +5.33129 q^{47} +2.87945 q^{48} +0.511184 q^{49} +3.53746 q^{50} -6.08376 q^{51} +21.6152 q^{52} +1.00000 q^{53} +12.5648 q^{54} -10.9016 q^{55} +11.6269 q^{56} +2.49035 q^{57} +0.216993 q^{58} +1.01638 q^{59} -7.67388 q^{60} -12.7832 q^{61} -2.91514 q^{62} +4.99656 q^{63} -10.3774 q^{64} -10.7771 q^{65} -15.1223 q^{66} -0.260079 q^{67} -21.1233 q^{68} -2.24272 q^{69} -12.3598 q^{70} +2.32782 q^{71} +7.73438 q^{72} -4.78757 q^{73} +1.29941 q^{74} -1.59807 q^{75} +8.64668 q^{76} -15.9091 q^{77} -14.9497 q^{78} +9.90330 q^{79} -4.98473 q^{80} -0.206816 q^{81} -16.5789 q^{82} +1.95848 q^{83} -11.1988 q^{84} +10.5319 q^{85} +7.82770 q^{86} -0.0980276 q^{87} -24.6264 q^{88} +10.6960 q^{89} -8.22196 q^{90} -15.7275 q^{91} -7.78688 q^{92} +1.31693 q^{93} -12.8025 q^{94} -4.31115 q^{95} +2.28989 q^{96} -10.3988 q^{97} -1.22755 q^{98} -10.5830 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 147 q - 6 q^{2} - 23 q^{3} + 130 q^{4} - 25 q^{5} - 18 q^{6} - 33 q^{7} - 15 q^{8} + 114 q^{9}+O(q^{10}) $ 147 * q - 6 * q^2 - 23 * q^3 + 130 * q^4 - 25 * q^5 - 18 * q^6 - 33 * q^7 - 15 * q^8 + 114 * q^9	\( 147 q - 6 q^{2} - 23 q^{3} + 130 q^{4} - 25 q^{5} - 18 q^{6} - 33 q^{7} - 15 q^{8} + 114 q^{9} - 24 q^{10} - 13 q^{11} - 52 q^{12} - 119 q^{13} - 6 q^{14} - 22 q^{15} + 100 q^{16} - 42 q^{17} - 6 q^{18} - 31 q^{19} - 50 q^{20} - 44 q^{21} - 38 q^{22} - 16 q^{23} - 30 q^{24} + 80 q^{25} - 18 q^{26} - 68 q^{27} - 72 q^{28} - 40 q^{29} - 3 q^{30} - 44 q^{31} - 41 q^{32} - 71 q^{33} - 55 q^{34} - 24 q^{35} + 108 q^{36} - 145 q^{37} - 39 q^{38} + 28 q^{39} - 81 q^{40} - 28 q^{41} - 54 q^{42} - 47 q^{43} - 33 q^{44} - 89 q^{45} - 43 q^{46} - 65 q^{47} - 82 q^{48} + 52 q^{49} - 43 q^{50} + 7 q^{51} - 215 q^{52} + 147 q^{53} - 88 q^{54} - 48 q^{55} + 19 q^{56} - 66 q^{57} - 110 q^{58} - 66 q^{59} - 118 q^{60} - 80 q^{61} - 41 q^{62} - 52 q^{63} + 33 q^{64} - 17 q^{65} - 86 q^{66} - 114 q^{67} - 77 q^{68} - 49 q^{69} - 56 q^{70} + 10 q^{71} + 5 q^{72} - 143 q^{73} - 30 q^{74} - 97 q^{75} - 104 q^{76} - 116 q^{77} - 33 q^{78} - 31 q^{79} - 83 q^{80} - q^{81} - 72 q^{82} - 70 q^{83} - 64 q^{84} - 85 q^{85} - 43 q^{87} - 149 q^{88} - 130 q^{89} + 42 q^{90} - 35 q^{91} - 31 q^{92} - 149 q^{93} - 94 q^{94} - 9 q^{95} + 6 q^{96} - 211 q^{97} - 18 q^{98} - 19 q^{99}+O(q^{100}) \) 147 * q - 6 * q^2 - 23 * q^3 + 130 * q^4 - 25 * q^5 - 18 * q^6 - 33 * q^7 - 15 * q^8 + 114 * q^9 - 24 * q^10 - 13 * q^11 - 52 * q^12 - 119 * q^13 - 6 * q^14 - 22 * q^15 + 100 * q^16 - 42 * q^17 - 6 * q^18 - 31 * q^19 - 50 * q^20 - 44 * q^21 - 38 * q^22 - 16 * q^23 - 30 * q^24 + 80 * q^25 - 18 * q^26 - 68 * q^27 - 72 * q^28 - 40 * q^29 - 3 * q^30 - 44 * q^31 - 41 * q^32 - 71 * q^33 - 55 * q^34 - 24 * q^35 + 108 * q^36 - 145 * q^37 - 39 * q^38 + 28 * q^39 - 81 * q^40 - 28 * q^41 - 54 * q^42 - 47 * q^43 - 33 * q^44 - 89 * q^45 - 43 * q^46 - 65 * q^47 - 82 * q^48 + 52 * q^49 - 43 * q^50 + 7 * q^51 - 215 * q^52 + 147 * q^53 - 88 * q^54 - 48 * q^55 + 19 * q^56 - 66 * q^57 - 110 * q^58 - 66 * q^59 - 118 * q^60 - 80 * q^61 - 41 * q^62 - 52 * q^63 + 33 * q^64 - 17 * q^65 - 86 * q^66 - 114 * q^67 - 77 * q^68 - 49 * q^69 - 56 * q^70 + 10 * q^71 + 5 * q^72 - 143 * q^73 - 30 * q^74 - 97 * q^75 - 104 * q^76 - 116 * q^77 - 33 * q^78 - 31 * q^79 - 83 * q^80 - q^81 - 72 * q^82 - 70 * q^83 - 64 * q^84 - 85 * q^85 - 43 * q^87 - 149 * q^88 - 130 * q^89 + 42 * q^90 - 35 * q^91 - 31 * q^92 - 149 * q^93 - 94 * q^94 - 9 * q^95 + 6 * q^96 - 211 * q^97 - 18 * q^98 - 19 * 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 $

$2$	−2.40138	−1.69803	−0.849017	−	0.528366i	$-0.822805\pi$
$2$	−2.40138	−1.69803	−0.849017	+	0.528366i	$0.822805\pi$
$3$	1.08484	0.626331	0.313165	−	0.949699i	$-0.398611\pi$
$3$	1.08484	0.626331	0.313165	+	0.949699i	$0.398611\pi$
$4$	3.76663	1.88332
$5$	−1.87801	−0.839870	−0.419935	−	0.907554i	$-0.637947\pi$
$5$	−1.87801	−0.839870	−0.419935	+	0.907554i	$0.637947\pi$
$6$	−2.60511	−1.06353
$7$	−2.74065	−1.03587	−0.517935	−	0.855420i	$-0.673299\pi$
$7$	−2.74065	−1.03587	−0.517935	+	0.855420i	$0.673299\pi$
$8$	−4.24236	−1.49990
$9$	−1.82313	−0.607710
$10$	4.50981	1.42613
$11$	5.80487	1.75023	0.875117	−	0.483911i	$-0.160784\pi$
$11$	5.80487	1.75023	0.875117	+	0.483911i	$0.160784\pi$
$12$	4.08618	1.17958
$13$	5.73860	1.59160	0.795801	−	0.605558i	$-0.207050\pi$
$13$	5.73860	1.59160	0.795801	+	0.605558i	$0.207050\pi$
$14$	6.58136	1.75894
$15$	−2.03733	−0.526036
$16$	2.65427	0.663567
$17$	−5.60800	−1.36014	−0.680070	−	0.733148i	$-0.738050\pi$
$17$	−5.60800	−1.36014	−0.680070	+	0.733148i	$0.738050\pi$
$18$	4.37803	1.03191
$19$	2.29560	0.526646	0.263323	−	0.964708i	$-0.415181\pi$
$19$	2.29560	0.526646	0.263323	+	0.964708i	$0.415181\pi$
$20$	−7.07376	−1.58174
$21$	−2.97316	−0.648797
$22$	−13.9397	−2.97196
$23$	−2.06733	−0.431069	−0.215534	−	0.976496i	$-0.569149\pi$
$23$	−2.06733	−0.431069	−0.215534	+	0.976496i	$0.569149\pi$
$24$	−4.60227	−0.939435
$25$	−1.47310	−0.294619
$26$	−13.7806	−2.70259
$27$	−5.23231	−1.00696
$28$	−10.3230	−1.95087
$29$	−0.0903616	−0.0167797	−0.00838986	−	0.999965i	$-0.502671\pi$
$29$	−0.0903616	−0.0167797	−0.00838986	+	0.999965i	$0.502671\pi$
$30$	4.89241	0.893227
$31$	1.21394	0.218031	0.109015	−	0.994040i	$-0.465230\pi$
$31$	1.21394	0.218031	0.109015	+	0.994040i	$0.465230\pi$
$32$	2.11082	0.373144
$33$	6.29734	1.09623
$34$	13.4669	2.30956
$35$	5.14696	0.869996
$36$	−6.86706	−1.14451
$37$	−0.541111	−0.0889581	−0.0444791	−	0.999010i	$-0.514163\pi$
$37$	−0.541111	−0.0889581	−0.0444791	+	0.999010i	$0.514163\pi$
$38$	−5.51261	−0.894263
$39$	6.22545	0.996870
$40$	7.96718	1.25972
$41$	6.90388	1.07821	0.539103	−	0.842240i	$-0.318764\pi$
$41$	6.90388	1.07821	0.539103	+	0.842240i	$0.318764\pi$
$42$	7.13970	1.10168
$43$	−3.25966	−0.497094	−0.248547	−	0.968620i	$-0.579953\pi$
$43$	−3.25966	−0.497094	−0.248547	+	0.968620i	$0.579953\pi$
$44$	21.8648	3.29625
$45$	3.42385	0.510397
$46$	4.96445	0.731969
$47$	5.33129	0.777649	0.388825	−	0.921312i	$-0.372881\pi$
$47$	5.33129	0.777649	0.388825	+	0.921312i	$0.372881\pi$
$48$	2.87945	0.415612
$49$	0.511184	0.0730263
$50$	3.53746	0.500273
$51$	−6.08376	−0.851897
$52$	21.6152	2.99749
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	12.5648	1.70985
$55$	−10.9016	−1.46997
$56$	11.6269	1.55370
$57$	2.49035	0.329855
$58$	0.216993	0.0284925
$59$	1.01638	0.132322	0.0661610	−	0.997809i	$-0.478925\pi$
$59$	1.01638	0.132322	0.0661610	+	0.997809i	$0.478925\pi$
$60$	−7.67388	−0.990693
$61$	−12.7832	−1.63672	−0.818358	−	0.574709i	$-0.805115\pi$
$61$	−12.7832	−1.63672	−0.818358	+	0.574709i	$0.805115\pi$
$62$	−2.91514	−0.370224
$63$	4.99656	0.629508
$64$	−10.3774	−1.29718
$65$	−10.7771	−1.33674
$66$	−15.1223	−1.86143
$67$	−0.260079	−0.0317737	−0.0158869	−	0.999874i	$-0.505057\pi$
$67$	−0.260079	−0.0317737	−0.0158869	+	0.999874i	$0.505057\pi$
$68$	−21.1233	−2.56157
$69$	−2.24272	−0.269992
$70$	−12.3598	−1.47728
$71$	2.32782	0.276262	0.138131	−	0.990414i	$-0.455891\pi$
$71$	2.32782	0.276262	0.138131	+	0.990414i	$0.455891\pi$
$72$	7.73438	0.911505
$73$	−4.78757	−0.560343	−0.280171	−	0.959950i	$-0.590391\pi$
$73$	−4.78757	−0.560343	−0.280171	+	0.959950i	$0.590391\pi$
$74$	1.29941	0.151054
$75$	−1.59807	−0.184529
$76$	8.64668	0.991842
$77$	−15.9091	−1.81301
$78$	−14.9497	−1.69272
$79$	9.90330	1.11421	0.557104	−	0.830443i	$-0.311912\pi$
$79$	9.90330	1.11421	0.557104	+	0.830443i	$0.311912\pi$
$80$	−4.98473	−0.557310
$81$	−0.206816	−0.0229795
$82$	−16.5789	−1.83083
$83$	1.95848	0.214971	0.107485	−	0.994207i	$-0.465720\pi$
$83$	1.95848	0.214971	0.107485	+	0.994207i	$0.465720\pi$
$84$	−11.1988	−1.22189
$85$	10.5319	1.14234
$86$	7.82770	0.844082
$87$	−0.0980276	−0.0105097
$88$	−24.6264	−2.62518
$89$	10.6960	1.13378	0.566889	−	0.823794i	$-0.308147\pi$
$89$	10.6960	1.13378	0.566889	+	0.823794i	$0.308147\pi$
$90$	−8.22196	−0.866671
$91$	−15.7275	−1.64869
$92$	−7.78688	−0.811839
$93$	1.31693	0.136559
$94$	−12.8025	−1.32047
$95$	−4.31115	−0.442314
$96$	2.28989	0.233711
$97$	−10.3988	−1.05584	−0.527920	−	0.849294i	$-0.677028\pi$
$97$	−10.3988	−1.05584	−0.527920	+	0.849294i	$0.677028\pi$
$98$	−1.22755	−0.124001
$99$	−10.5830	−1.06363
$100$	−5.54861	−0.554861
$101$	−0.648803	−0.0645583	−0.0322791	−	0.999479i	$-0.510277\pi$
$101$	−0.648803	−0.0645583	−0.0322791	+	0.999479i	$0.510277\pi$
$102$	14.6094	1.44655
$103$	−7.76817	−0.765421	−0.382710	−	0.923868i	$-0.625009\pi$
$103$	−7.76817	−0.765421	−0.382710	+	0.923868i	$0.625009\pi$
$104$	−24.3453	−2.38725
$105$	5.58362	0.544905
$106$	−2.40138	−0.233243
$107$	−16.4279	−1.58814	−0.794071	−	0.607825i	$-0.792042\pi$
$107$	−16.4279	−1.58814	−0.794071	+	0.607825i	$0.792042\pi$
$108$	−19.7082	−1.89642
$109$	−3.01709	−0.288984	−0.144492	−	0.989506i	$-0.546155\pi$
$109$	−3.01709	−0.288984	−0.144492	+	0.989506i	$0.546155\pi$
$110$	26.1789	2.49606
$111$	−0.587017	−0.0557172
$112$	−7.27443	−0.687369
$113$	1.57045	0.147735	0.0738677	−	0.997268i	$-0.476466\pi$
$113$	1.57045	0.147735	0.0738677	+	0.997268i	$0.476466\pi$
$114$	−5.98028	−0.560105
$115$	3.88246	0.362041
$116$	−0.340359	−0.0316016
$117$	−10.4622	−0.967232
$118$	−2.44073	−0.224687
$119$	15.3696	1.40893
$120$	8.64310	0.789003
$121$	22.6965	2.06332
$122$	30.6972	2.77920
$123$	7.48959	0.675313
$124$	4.57249	0.410621
$125$	12.1565	1.08731
$126$	−11.9987	−1.06893
$127$	4.88981	0.433901	0.216950	−	0.976183i	$-0.430389\pi$
$127$	4.88981	0.433901	0.216950	+	0.976183i	$0.430389\pi$
$128$	20.6985	1.82951
$129$	−3.53620	−0.311345
$130$	25.8800	2.26983
$131$	−1.93485	−0.169048	−0.0845241	−	0.996421i	$-0.526937\pi$
$131$	−1.93485	−0.169048	−0.0845241	+	0.996421i	$0.526937\pi$
$132$	23.7198	2.06454
$133$	−6.29144	−0.545537
$134$	0.624549	0.0539528
$135$	9.82630	0.845714
$136$	23.7912	2.04008
$137$	13.9456	1.19145	0.595726	−	0.803188i	$-0.296864\pi$
$137$	13.9456	1.19145	0.595726	+	0.803188i	$0.296864\pi$
$138$	5.38562	0.458455
$139$	−2.22540	−0.188756	−0.0943780	−	0.995536i	$-0.530086\pi$
$139$	−2.22540	−0.188756	−0.0943780	+	0.995536i	$0.530086\pi$
$140$	19.3867	1.63848
$141$	5.78358	0.487066
$142$	−5.58999	−0.469102
$143$	33.3119	2.78568
$144$	−4.83907	−0.403256
$145$	0.169700	0.0140928
$146$	11.4968	0.951481
$147$	0.554551	0.0457386
$148$	−2.03817	−0.167536
$149$	14.2343	1.16612	0.583061	−	0.812428i	$-0.301855\pi$
$149$	14.2343	1.16612	0.583061	+	0.812428i	$0.301855\pi$
$150$	3.83757	0.313337
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−9.73876	−0.789918
$153$	10.2241	0.826570
$154$	38.2039	3.07856
$155$	−2.27979	−0.183117
$156$	23.4490	1.87742
$157$	6.35541	0.507217	0.253608	−	0.967307i	$-0.418383\pi$
$157$	6.35541	0.507217	0.253608	+	0.967307i	$0.418383\pi$
$158$	−23.7816	−1.89196
$159$	1.08484	0.0860332
$160$	−3.96413	−0.313392
$161$	5.66584	0.446531
$162$	0.496643	0.0390200
$163$	9.47496	0.742136	0.371068	−	0.928606i	$-0.378992\pi$
$163$	9.47496	0.742136	0.371068	+	0.928606i	$0.378992\pi$
$164$	26.0044	2.03060
$165$	−11.8264	−0.920687
$166$	−4.70305	−0.365027
$167$	−16.6213	−1.28620	−0.643098	−	0.765784i	$-0.722351\pi$
$167$	−16.6213	−1.28620	−0.643098	+	0.765784i	$0.722351\pi$
$168$	12.6132	0.973133
$169$	19.9316	1.53320
$170$	−25.2910	−1.93973
$171$	−4.18517	−0.320048
$172$	−12.2780	−0.936186
$173$	−14.1637	−1.07685	−0.538423	−	0.842675i	$-0.680980\pi$
$173$	−14.1637	−1.07685	−0.538423	+	0.842675i	$0.680980\pi$
$174$	0.235402	0.0178458
$175$	4.03725	0.305187
$176$	15.4077	1.16140
$177$	1.10261	0.0828773
$178$	−25.6853	−1.92519
$179$	−6.10607	−0.456389	−0.228195	−	0.973616i	$-0.573282\pi$
$179$	−6.10607	−0.456389	−0.228195	+	0.973616i	$0.573282\pi$
$180$	12.8964	0.961239
$181$	−20.1240	−1.49580	−0.747902	−	0.663809i	$-0.768939\pi$
$181$	−20.1240	−1.49580	−0.747902	+	0.663809i	$0.768939\pi$
$182$	37.7678	2.79954
$183$	−13.8676	−1.02513
$184$	8.77038	0.646561
$185$	1.01621	0.0747132
$186$	−3.16246	−0.231883
$187$	−32.5537	−2.38056
$188$	20.0810	1.46456
$189$	14.3399	1.04308
$190$	10.3527	0.751064
$191$	−22.9328	−1.65936	−0.829679	−	0.558241i	$-0.811477\pi$
$191$	−22.9328	−1.65936	−0.829679	+	0.558241i	$0.811477\pi$
$192$	−11.2578	−0.812462
$193$	11.4125	0.821488	0.410744	−	0.911751i	$-0.365269\pi$
$193$	11.4125	0.821488	0.410744	+	0.911751i	$0.365269\pi$
$194$	24.9716	1.79285
$195$	−11.6914	−0.837241
$196$	1.92544	0.137532
$197$	−8.50738	−0.606126	−0.303063	−	0.952971i	$-0.598009\pi$
$197$	−8.50738	−0.606126	−0.303063	+	0.952971i	$0.598009\pi$
$198$	25.4139	1.80609
$199$	13.2620	0.940120	0.470060	−	0.882634i	$-0.344232\pi$
$199$	13.2620	0.940120	0.470060	+	0.882634i	$0.344232\pi$
$200$	6.24941	0.441900
$201$	−0.282143	−0.0199009
$202$	1.55802	0.109622
$203$	0.247650	0.0173816
$204$	−22.9153	−1.60439
$205$	−12.9655	−0.905552
$206$	18.6543	1.29971
$207$	3.76901	0.261964
$208$	15.2318	1.05613
$209$	13.3257	0.921755
$210$	−13.4084	−0.925267
$211$	12.1061	0.833416	0.416708	−	0.909040i	$-0.363184\pi$
$211$	12.1061	0.833416	0.416708	+	0.909040i	$0.363184\pi$
$212$	3.76663	0.258694
$213$	2.52531	0.173031
$214$	39.4496	2.69672
$215$	6.12167	0.417494
$216$	22.1974	1.51034
$217$	−3.32700	−0.225852
$218$	7.24517	0.490705
$219$	−5.19373	−0.350960
$220$	−41.0623	−2.76842
$221$	−32.1821	−2.16480
$222$	1.40965	0.0946097
$223$	−9.75943	−0.653540	−0.326770	−	0.945104i	$-0.605960\pi$
$223$	−9.75943	−0.653540	−0.326770	+	0.945104i	$0.605960\pi$
$224$	−5.78502	−0.386528
$225$	2.68564	0.179043
$226$	−3.77125	−0.250859
$227$	1.81206	0.120271	0.0601353	−	0.998190i	$-0.480847\pi$
$227$	1.81206	0.120271	0.0601353	+	0.998190i	$0.480847\pi$
$228$	9.38024	0.621221
$229$	7.47603	0.494030	0.247015	−	0.969012i	$-0.420550\pi$
$229$	7.47603	0.494030	0.247015	+	0.969012i	$0.420550\pi$
$230$	−9.32327	−0.614758
$231$	−17.2588	−1.13555
$232$	0.383347	0.0251680
$233$	14.3220	0.938263	0.469131	−	0.883128i	$-0.344567\pi$
$233$	14.3220	0.938263	0.469131	+	0.883128i	$0.344567\pi$
$234$	25.1238	1.64239
$235$	−10.0122	−0.653124
$236$	3.82835	0.249204
$237$	10.7435	0.697863
$238$	−36.9082	−2.39241
$239$	16.3079	1.05487	0.527435	−	0.849595i	$-0.323154\pi$
$239$	16.3079	1.05487	0.527435	+	0.849595i	$0.323154\pi$
$240$	−5.40762	−0.349060
$241$	−13.2025	−0.850446	−0.425223	−	0.905089i	$-0.639804\pi$
$241$	−13.2025	−0.850446	−0.425223	+	0.905089i	$0.639804\pi$
$242$	−54.5030	−3.50359
$243$	15.4726	0.992565
$244$	−48.1495	−3.08246
$245$	−0.960006	−0.0613325
$246$	−17.9854	−1.14670
$247$	13.1735	0.838212
$248$	−5.14999	−0.327025
$249$	2.12463	0.134643
$250$	−29.1924	−1.84629
$251$	16.1583	1.01990	0.509950	−	0.860204i	$-0.329664\pi$
$251$	16.1583	1.01990	0.509950	+	0.860204i	$0.329664\pi$
$252$	18.8202	1.18556
$253$	−12.0006	−0.754471
$254$	−11.7423	−0.736778
$255$	11.4253	0.715483
$256$	−28.9502	−1.80939
$257$	11.8739	0.740671	0.370336	−	0.928898i	$-0.379243\pi$
$257$	11.8739	0.740671	0.370336	+	0.928898i	$0.379243\pi$
$258$	8.49177	0.528675
$259$	1.48300	0.0921490
$260$	−40.5935	−2.51750
$261$	0.164741	0.0101972
$262$	4.64630	0.287049
$263$	5.09804	0.314359	0.157179	−	0.987570i	$-0.449760\pi$
$263$	5.09804	0.314359	0.157179	+	0.987570i	$0.449760\pi$
$264$	−26.7156	−1.64423
$265$	−1.87801	−0.115365
$266$	15.1081	0.926340
$267$	11.6035	0.710120
$268$	−0.979623	−0.0598400
$269$	10.0881	0.615080	0.307540	−	0.951535i	$-0.400494\pi$
$269$	10.0881	0.615080	0.307540	+	0.951535i	$0.400494\pi$
$270$	−23.5967	−1.43605
$271$	−26.7207	−1.62317	−0.811585	−	0.584234i	$-0.801395\pi$
$271$	−26.7207	−1.62317	−0.811585	+	0.584234i	$0.801395\pi$
$272$	−14.8851	−0.902543
$273$	−17.0618	−1.03263
$274$	−33.4887	−2.02313
$275$	−8.55113	−0.515653
$276$	−8.44750	−0.508480
$277$	−18.2620	−1.09726	−0.548629	−	0.836066i	$-0.684850\pi$
$277$	−18.2620	−1.09726	−0.548629	+	0.836066i	$0.684850\pi$
$278$	5.34403	0.320514
$279$	−2.21318	−0.132499
$280$	−21.8353	−1.30491
$281$	2.53899	0.151463	0.0757316	−	0.997128i	$-0.475871\pi$
$281$	2.53899	0.151463	0.0757316	+	0.997128i	$0.475871\pi$
$282$	−13.8886	−0.827054
$283$	11.7724	0.699798	0.349899	−	0.936787i	$-0.386216\pi$
$283$	11.7724	0.699798	0.349899	+	0.936787i	$0.386216\pi$
$284$	8.76806	0.520289
$285$	−4.67689	−0.277035
$286$	−79.9945	−4.73017
$287$	−18.9211	−1.11688
$288$	−3.84829	−0.226763
$289$	14.4496	0.849979
$290$	−0.407513	−0.0239300
$291$	−11.2810	−0.661306
$292$	−18.0330	−1.05530
$293$	−24.5461	−1.43400	−0.717000	−	0.697073i	$-0.754485\pi$
$293$	−24.5461	−1.43400	−0.717000	+	0.697073i	$0.754485\pi$
$294$	−1.33169	−0.0776657
$295$	−1.90878	−0.111133
$296$	2.29559	0.133428
$297$	−30.3729	−1.76241
$298$	−34.1821	−1.98012
$299$	−11.8636	−0.686090
$300$	−6.01934	−0.347527
$301$	8.93361	0.514925
$302$	−2.40138	−0.138184
$303$	−0.703845	−0.0404348
$304$	6.09313	0.349465
$305$	24.0068	1.37463
$306$	−24.5520	−1.40354
$307$	−24.0827	−1.37447	−0.687235	−	0.726435i	$-0.741176\pi$
$307$	−24.0827	−1.37447	−0.687235	+	0.726435i	$0.741176\pi$
$308$	−59.9239	−3.41448
$309$	−8.42720	−0.479407
$310$	5.47466	0.310940
$311$	−3.55799	−0.201755	−0.100878	−	0.994899i	$-0.532165\pi$
$311$	−3.55799	−0.201755	−0.100878	+	0.994899i	$0.532165\pi$
$312$	−26.4106	−1.49521
$313$	−22.9497	−1.29719	−0.648597	−	0.761132i	$-0.724644\pi$
$313$	−22.9497	−1.29719	−0.648597	+	0.761132i	$0.724644\pi$
$314$	−15.2618	−0.861271
$315$	−9.38358	−0.528705
$316$	37.3021	2.09841
$317$	17.9496	1.00815	0.504074	−	0.863660i	$-0.331834\pi$
$317$	17.9496	1.00815	0.504074	+	0.863660i	$0.331834\pi$
$318$	−2.60511	−0.146087
$319$	−0.524537	−0.0293685
$320$	19.4888	1.08946
$321$	−17.8216	−0.994702
$322$	−13.6058	−0.758224
$323$	−12.8737	−0.716312
$324$	−0.778999	−0.0432777
$325$	−8.45351	−0.468917
$326$	−22.7530	−1.26017
$327$	−3.27305	−0.181000
$328$	−29.2888	−1.61720
$329$	−14.6112	−0.805543
$330$	28.3998	1.56336
$331$	−4.74079	−0.260577	−0.130289	−	0.991476i	$-0.541590\pi$
$331$	−4.74079	−0.260577	−0.130289	+	0.991476i	$0.541590\pi$
$332$	7.37686	0.404858
$333$	0.986515	0.0540607
$334$	39.9141	2.18400
$335$	0.488430	0.0266858
$336$	−7.89157	−0.430520
$337$	−12.9548	−0.705691	−0.352846	−	0.935682i	$-0.614786\pi$
$337$	−12.9548	−0.705691	−0.352846	+	0.935682i	$0.614786\pi$
$338$	−47.8633	−2.60342
$339$	1.70368	0.0925312
$340$	39.6696	2.15139
$341$	7.04679	0.381605
$342$	10.0502	0.543452
$343$	17.7836	0.960224
$344$	13.8287	0.745592
$345$	4.21184	0.226758
$346$	34.0125	1.82852
$347$	5.81954	0.312410	0.156205	−	0.987725i	$-0.450074\pi$
$347$	5.81954	0.312410	0.156205	+	0.987725i	$0.450074\pi$
$348$	−0.369234	−0.0197930
$349$	−23.3610	−1.25048	−0.625242	−	0.780431i	$-0.715000\pi$
$349$	−23.3610	−1.25048	−0.625242	+	0.780431i	$0.715000\pi$
$350$	−9.69497	−0.518218
$351$	−30.0261	−1.60268
$352$	12.2530	0.653089
$353$	6.56819	0.349590	0.174795	−	0.984605i	$-0.444074\pi$
$353$	6.56819	0.349590	0.174795	+	0.984605i	$0.444074\pi$
$354$	−2.64779	−0.140729
$355$	−4.37166	−0.232024
$356$	40.2881	2.13526
$357$	16.6735	0.882455
$358$	14.6630	0.774964
$359$	5.84036	0.308242	0.154121	−	0.988052i	$-0.450745\pi$
$359$	5.84036	0.308242	0.154121	+	0.988052i	$0.450745\pi$
$360$	−14.5252	−0.765545
$361$	−13.7302	−0.722644
$362$	48.3254	2.53992
$363$	24.6220	1.29232
$364$	−59.2399	−3.10501
$365$	8.99108	0.470615
$366$	33.3015	1.74070
$367$	−5.00156	−0.261079	−0.130540	−	0.991443i	$-0.541671\pi$
$367$	−5.00156	−0.261079	−0.130540	+	0.991443i	$0.541671\pi$
$368$	−5.48725	−0.286043
$369$	−12.5867	−0.655236
$370$	−2.44031	−0.126866
$371$	−2.74065	−0.142288
$372$	4.96040	0.257185
$373$	8.30787	0.430165	0.215082	−	0.976596i	$-0.430998\pi$
$373$	8.30787	0.430165	0.215082	+	0.976596i	$0.430998\pi$
$374$	78.1739	4.04227
$375$	13.1878	0.681017
$376$	−22.6173	−1.16640
$377$	−0.518549	−0.0267067
$378$	−34.4357	−1.77118
$379$	−31.1856	−1.60190	−0.800950	−	0.598732i	$-0.795672\pi$
$379$	−31.1856	−1.60190	−0.800950	+	0.598732i	$0.795672\pi$
$380$	−16.2385	−0.833018
$381$	5.30465	0.271765
$382$	55.0704	2.81765
$383$	−12.8253	−0.655342	−0.327671	−	0.944792i	$-0.606264\pi$
$383$	−12.8253	−0.655342	−0.327671	+	0.944792i	$0.606264\pi$
$384$	22.4545	1.14588
$385$	29.8775	1.52270
$386$	−27.4057	−1.39491
$387$	5.94279	0.302089
$388$	−39.1686	−1.98848
$389$	2.94418	0.149276	0.0746378	−	0.997211i	$-0.476220\pi$
$389$	2.94418	0.149276	0.0746378	+	0.997211i	$0.476220\pi$
$390$	28.0756	1.42166
$391$	11.5936	0.586313
$392$	−2.16863	−0.109532
$393$	−2.09899	−0.105880
$394$	20.4295	1.02922
$395$	−18.5985	−0.935790
$396$	−39.8624	−2.00316
$397$	16.5699	0.831619	0.415809	−	0.909452i	$-0.363498\pi$
$397$	16.5699	0.831619	0.415809	+	0.909452i	$0.363498\pi$
$398$	−31.8472	−1.59636
$399$	−6.82519	−0.341687
$400$	−3.90999	−0.195499
$401$	27.7221	1.38437	0.692187	−	0.721718i	$-0.256647\pi$
$401$	27.7221	1.38437	0.692187	+	0.721718i	$0.256647\pi$
$402$	0.677534	0.0337923
$403$	6.96635	0.347018
$404$	−2.44380	−0.121584
$405$	0.388401	0.0192998
$406$	−0.594702	−0.0295146
$407$	−3.14108	−0.155698
$408$	25.8095	1.27776
$409$	−0.0695549	−0.00343927	−0.00171963	−	0.999999i	$-0.500547\pi$
$409$	−0.0695549	−0.00343927	−0.00171963	+	0.999999i	$0.500547\pi$
$410$	31.1352	1.53766
$411$	15.1287	0.746243
$412$	−29.2599	−1.44153
$413$	−2.78556	−0.137068
$414$	−9.05084	−0.444824
$415$	−3.67803	−0.180547
$416$	12.1132	0.593896
$417$	−2.41420	−0.118224
$418$	−32.0000	−1.56517
$419$	−5.10868	−0.249576	−0.124788	−	0.992183i	$-0.539825\pi$
$419$	−5.10868	−0.249576	−0.124788	+	0.992183i	$0.539825\pi$
$420$	21.0314	1.02623
$421$	30.3982	1.48152	0.740758	−	0.671772i	$-0.234467\pi$
$421$	30.3982	1.48152	0.740758	+	0.671772i	$0.234467\pi$
$422$	−29.0713	−1.41517
$423$	−9.71963	−0.472585
$424$	−4.24236	−0.206027
$425$	8.26112	0.400723
$426$	−6.06423	−0.293813
$427$	35.0342	1.69542
$428$	−61.8778	−2.99097
$429$	36.1379	1.74476
$430$	−14.7005	−0.708919
$431$	−20.4686	−0.985937	−0.492969	−	0.870047i	$-0.664088\pi$
$431$	−20.4686	−0.985937	−0.492969	+	0.870047i	$0.664088\pi$
$432$	−13.8879	−0.668184
$433$	−8.30975	−0.399341	−0.199670	−	0.979863i	$-0.563987\pi$
$433$	−8.30975	−0.399341	−0.199670	+	0.979863i	$0.563987\pi$
$434$	7.98940	0.383504
$435$	0.184096	0.00882675
$436$	−11.3643	−0.544249
$437$	−4.74576	−0.227021
$438$	12.4721	0.595942
$439$	−16.2446	−0.775311	−0.387655	−	0.921804i	$-0.626715\pi$
$439$	−16.2446	−0.775311	−0.387655	+	0.921804i	$0.626715\pi$
$440$	46.2485	2.20481
$441$	−0.931954	−0.0443788
$442$	77.2815	3.67590
$443$	−31.4629	−1.49485	−0.747425	−	0.664346i	$-0.768710\pi$
$443$	−31.4629	−1.49485	−0.747425	+	0.664346i	$0.768710\pi$
$444$	−2.21108	−0.104933
$445$	−20.0872	−0.952226
$446$	23.4361	1.10973
$447$	15.4419	0.730379
$448$	28.4409	1.34371
$449$	0.792162	0.0373844	0.0186922	−	0.999825i	$-0.494050\pi$
$449$	0.792162	0.0373844	0.0186922	+	0.999825i	$0.494050\pi$
$450$	−6.44925	−0.304021
$451$	40.0761	1.88711
$452$	5.91530	0.278232
$453$	1.08484	0.0509701
$454$	−4.35145	−0.204224
$455$	29.5364	1.38469
$456$	−10.5650	−0.494750
$457$	−31.7136	−1.48350	−0.741750	−	0.670676i	$-0.766004\pi$
$457$	−31.7136	−1.48350	−0.741750	+	0.670676i	$0.766004\pi$
$458$	−17.9528	−0.838880
$459$	29.3428	1.36960
$460$	14.6238	0.681839
$461$	−26.2127	−1.22085	−0.610424	−	0.792075i	$-0.709001\pi$
$461$	−26.2127	−1.22085	−0.610424	+	0.792075i	$0.709001\pi$
$462$	41.4450	1.92820
$463$	−18.2190	−0.846711	−0.423355	−	0.905964i	$-0.639148\pi$
$463$	−18.2190	−0.846711	−0.423355	+	0.905964i	$0.639148\pi$
$464$	−0.239844	−0.0111345
$465$	−2.47321	−0.114692
$466$	−34.3925	−1.59320
$467$	−24.7805	−1.14670	−0.573352	−	0.819309i	$-0.694357\pi$
$467$	−24.7805	−1.14670	−0.573352	+	0.819309i	$0.694357\pi$
$468$	−39.4073	−1.82160
$469$	0.712787	0.0329134
$470$	24.0431	1.10903
$471$	6.89458	0.317686
$472$	−4.31187	−0.198470
$473$	−18.9219	−0.870031
$474$	−25.7992	−1.18500
$475$	−3.38164	−0.155160
$476$	57.8916	2.65346
$477$	−1.82313	−0.0834753
$478$	−39.1615	−1.79121
$479$	29.6760	1.35593	0.677965	−	0.735094i	$-0.262862\pi$
$479$	29.6760	1.35593	0.677965	+	0.735094i	$0.262862\pi$
$480$	−4.30043	−0.196287
$481$	−3.10522	−0.141586
$482$	31.7042	1.44409
$483$	6.14651	0.279676
$484$	85.4895	3.88589
$485$	19.5291	0.886769
$486$	−37.1555	−1.68541
$487$	36.0876	1.63529	0.817643	−	0.575725i	$-0.195280\pi$
$487$	36.0876	1.63529	0.817643	+	0.575725i	$0.195280\pi$
$488$	54.2308	2.45491
$489$	10.2788	0.464823
$490$	2.30534	0.104145
$491$	−15.6765	−0.707471	−0.353736	−	0.935345i	$-0.615089\pi$
$491$	−15.6765	−0.707471	−0.353736	+	0.935345i	$0.615089\pi$
$492$	28.2105	1.27183
$493$	0.506748	0.0228228
$494$	−31.6347	−1.42331
$495$	19.8750	0.893314
$496$	3.22213	0.144678
$497$	−6.37976	−0.286171
$498$	−5.10204	−0.228628
$499$	−1.75515	−0.0785713	−0.0392856	−	0.999228i	$-0.512508\pi$
$499$	−1.75515	−0.0785713	−0.0392856	+	0.999228i	$0.512508\pi$
$500$	45.7891	2.04775
$501$	−18.0314	−0.805584
$502$	−38.8021	−1.73182
$503$	−13.2635	−0.591391	−0.295695	−	0.955282i	$-0.595551\pi$
$503$	−13.2635	−0.591391	−0.295695	+	0.955282i	$0.595551\pi$
$504$	−21.1972	−0.944201
$505$	1.21845	0.0542205
$506$	28.8180	1.28112
$507$	21.6225	0.960290
$508$	18.4181	0.817173
$509$	−10.4748	−0.464289	−0.232145	−	0.972681i	$-0.574574\pi$
$509$	−10.4748	−0.464289	−0.232145	+	0.972681i	$0.574574\pi$
$510$	−27.4366	−1.21491
$511$	13.1211	0.580442
$512$	28.1234	1.24289
$513$	−12.0113	−0.530311
$514$	−28.5137	−1.25768
$515$	14.5887	0.642854
$516$	−13.3196	−0.586362
$517$	30.9475	1.36107
$518$	−3.56124	−0.156472
$519$	−15.3653	−0.674462
$520$	45.7205	2.00498
$521$	8.90702	0.390223	0.195112	−	0.980781i	$-0.437493\pi$
$521$	8.90702	0.390223	0.195112	+	0.980781i	$0.437493\pi$
$522$	−0.395606	−0.0173152
$523$	−9.76508	−0.426997	−0.213499	−	0.976943i	$-0.568486\pi$
$523$	−9.76508	−0.426997	−0.213499	+	0.976943i	$0.568486\pi$
$524$	−7.28785	−0.318371
$525$	4.37975	0.191148
$526$	−12.2423	−0.533791
$527$	−6.80780	−0.296552
$528$	16.7148	0.727419
$529$	−18.7261	−0.814180
$530$	4.50981	0.195894
$531$	−1.85300	−0.0804133
$532$	−23.6976	−1.02742
$533$	39.6186	1.71607
$534$	−27.8643	−1.20581
$535$	30.8516	1.33383
$536$	1.10335	0.0476575
$537$	−6.62409	−0.285851
$538$	−24.2253	−1.04443
$539$	2.96736	0.127813
$540$	37.0121	1.59275
$541$	10.0459	0.431908	0.215954	−	0.976404i	$-0.430714\pi$
$541$	10.0459	0.431908	0.215954	+	0.976404i	$0.430714\pi$
$542$	64.1667	2.75620
$543$	−21.8312	−0.936868
$544$	−11.8375	−0.507527
$545$	5.66610	0.242709
$546$	40.9719	1.75344
$547$	−16.6768	−0.713048	−0.356524	−	0.934286i	$-0.616038\pi$
$547$	−16.6768	−0.713048	−0.356524	+	0.934286i	$0.616038\pi$
$548$	52.5279	2.24388
$549$	23.3053	0.994648
$550$	20.5345	0.875595
$551$	−0.207434	−0.00883698
$552$	9.51443	0.404961
$553$	−27.1415	−1.15418
$554$	43.8540	1.86318
$555$	1.10242	0.0467952
$556$	−8.38227	−0.355487
$557$	−38.7808	−1.64319	−0.821597	−	0.570069i	$-0.806917\pi$
$557$	−38.7808	−1.64319	−0.821597	+	0.570069i	$0.806917\pi$
$558$	5.31468	0.224988
$559$	−18.7059	−0.791176
$560$	13.6614	0.577300
$561$	−35.3155	−1.49102
$562$	−6.09708	−0.257190
$563$	9.19201	0.387397	0.193699	−	0.981061i	$-0.437952\pi$
$563$	9.19201	0.387397	0.193699	+	0.981061i	$0.437952\pi$
$564$	21.7847	0.917299
$565$	−2.94931	−0.124078
$566$	−28.2701	−1.18828
$567$	0.566810	0.0238038
$568$	−9.87547	−0.414366
$569$	15.0236	0.629821	0.314911	−	0.949121i	$-0.398025\pi$
$569$	15.0236	0.629821	0.314911	+	0.949121i	$0.398025\pi$
$570$	11.2310	0.470415
$571$	13.7694	0.576230	0.288115	−	0.957596i	$-0.406971\pi$
$571$	13.7694	0.576230	0.288115	+	0.957596i	$0.406971\pi$
$572$	125.474	5.24631
$573$	−24.8783	−1.03931
$574$	45.4369	1.89650
$575$	3.04538	0.127001
$576$	18.9194	0.788307
$577$	−35.0023	−1.45716	−0.728582	−	0.684958i	$-0.759820\pi$
$577$	−35.0023	−1.45716	−0.728582	+	0.684958i	$0.759820\pi$
$578$	−34.6991	−1.44329
$579$	12.3807	0.514524
$580$	0.639196	0.0265412
$581$	−5.36750	−0.222682
$582$	27.0901	1.12292
$583$	5.80487	0.240413
$584$	20.3106	0.840460
$585$	19.6481	0.812349
$586$	58.9446	2.43498
$587$	17.4282	0.719341	0.359670	−	0.933079i	$-0.382889\pi$
$587$	17.4282	0.719341	0.359670	+	0.933079i	$0.382889\pi$
$588$	2.08879	0.0861403
$589$	2.78673	0.114825
$590$	4.58370	0.188708
$591$	−9.22912	−0.379635
$592$	−1.43625	−0.0590297
$593$	−21.4909	−0.882526	−0.441263	−	0.897378i	$-0.645469\pi$
$593$	−21.4909	−0.882526	−0.441263	+	0.897378i	$0.645469\pi$
$594$	72.9369	2.99264
$595$	−28.8642	−1.18332
$596$	53.6156	2.19618
$597$	14.3871	0.588826
$598$	28.4890	1.16500
$599$	12.9063	0.527339	0.263669	−	0.964613i	$-0.415067\pi$
$599$	12.9063	0.527339	0.263669	+	0.964613i	$0.415067\pi$
$600$	6.77959	0.276776
$601$	−46.3444	−1.89043	−0.945214	−	0.326452i	$-0.894147\pi$
$601$	−46.3444	−1.89043	−0.945214	+	0.326452i	$0.894147\pi$
$602$	−21.4530	−0.874359
$603$	0.474158	0.0193092
$604$	3.76663	0.153262
$605$	−42.6242	−1.73292
$606$	1.69020	0.0686597
$607$	3.71594	0.150825	0.0754126	−	0.997152i	$-0.475973\pi$
$607$	3.71594	0.150825	0.0754126	+	0.997152i	$0.475973\pi$
$608$	4.84559	0.196515
$609$	0.268660	0.0108866
$610$	−57.6496	−2.33416
$611$	30.5942	1.23771
$612$	38.5105	1.55669
$613$	−9.34301	−0.377361	−0.188680	−	0.982039i	$-0.560421\pi$
$613$	−9.34301	−0.377361	−0.188680	+	0.982039i	$0.560421\pi$
$614$	57.8317	2.33390
$615$	−14.0655	−0.567175
$616$	67.4924	2.71935
$617$	−4.76299	−0.191751	−0.0958755	−	0.995393i	$-0.530565\pi$
$617$	−4.76299	−0.191751	−0.0958755	+	0.995393i	$0.530565\pi$
$618$	20.2369	0.814048
$619$	−28.4305	−1.14272	−0.571358	−	0.820701i	$-0.693583\pi$
$619$	−28.4305	−1.14272	−0.571358	+	0.820701i	$0.693583\pi$
$620$	−8.58715	−0.344868
$621$	10.8169	0.434068
$622$	8.54410	0.342587
$623$	−29.3142	−1.17445
$624$	16.5240	0.661490
$625$	−15.4645	−0.618580
$626$	55.1110	2.20268
$627$	14.4562	0.577323
$628$	23.9385	0.955250
$629$	3.03455	0.120995
$630$	22.5335	0.897758
$631$	−35.8079	−1.42549	−0.712745	−	0.701423i	$-0.752548\pi$
$631$	−35.8079	−1.42549	−0.712745	+	0.701423i	$0.752548\pi$
$632$	−42.0134	−1.67120
$633$	13.1331	0.521994
$634$	−43.1038	−1.71187
$635$	−9.18310	−0.364420
$636$	4.08618	0.162028
$637$	2.93348	0.116229
$638$	1.25961	0.0498686
$639$	−4.24392	−0.167887
$640$	−38.8719	−1.53655
$641$	3.24790	0.128284	0.0641421	−	0.997941i	$-0.479569\pi$
$641$	3.24790	0.128284	0.0641421	+	0.997941i	$0.479569\pi$
$642$	42.7963	1.68904
$643$	−38.9362	−1.53549	−0.767746	−	0.640754i	$-0.778622\pi$
$643$	−38.9362	−1.53549	−0.767746	+	0.640754i	$0.778622\pi$
$644$	21.3412	0.840959
$645$	6.64101	0.261489
$646$	30.9147	1.21632
$647$	−31.8943	−1.25390	−0.626948	−	0.779061i	$-0.715696\pi$
$647$	−31.8943	−1.25390	−0.626948	+	0.779061i	$0.715696\pi$
$648$	0.877388	0.0344670
$649$	5.89998	0.231594
$650$	20.3001	0.796236
$651$	−3.60925	−0.141458
$652$	35.6887	1.39768
$653$	−35.0831	−1.37291	−0.686454	−	0.727174i	$-0.740833\pi$
$653$	−35.0831	−1.37291	−0.686454	+	0.727174i	$0.740833\pi$
$654$	7.85983	0.307344
$655$	3.63365	0.141978
$656$	18.3247	0.715461
$657$	8.72836	0.340526
$658$	35.0871	1.36784
$659$	−40.1901	−1.56558	−0.782792	−	0.622283i	$-0.786205\pi$
$659$	−40.1901	−1.56558	−0.782792	+	0.622283i	$0.786205\pi$
$660$	−44.5459	−1.73395
$661$	43.0107	1.67292	0.836462	−	0.548026i	$-0.184620\pi$
$661$	43.0107	1.67292	0.836462	+	0.548026i	$0.184620\pi$
$662$	11.3844	0.442469
$663$	−34.9123	−1.35588
$664$	−8.30856	−0.322435
$665$	11.8154	0.458180
$666$	−2.36900	−0.0917969
$667$	0.186807	0.00723321
$668$	−62.6064	−2.42231
$669$	−10.5874	−0.409332
$670$	−1.17291	−0.0453133
$671$	−74.2046	−2.86464
$672$	−6.27581	−0.242095
$673$	38.8325	1.49688	0.748441	−	0.663202i	$-0.230803\pi$
$673$	38.8325	1.49688	0.748441	+	0.663202i	$0.230803\pi$
$674$	31.1093	1.19829
$675$	7.70769	0.296669
$676$	75.0750	2.88750
$677$	−18.2912	−0.702986	−0.351493	−	0.936190i	$-0.614326\pi$
$677$	−18.2912	−0.702986	−0.351493	+	0.936190i	$0.614326\pi$
$678$	−4.09119	−0.157121
$679$	28.4996	1.09371
$680$	−44.6799	−1.71340
$681$	1.96579	0.0753292
$682$	−16.9220	−0.647978
$683$	−16.7152	−0.639591	−0.319795	−	0.947487i	$-0.603614\pi$
$683$	−16.7152	−0.639591	−0.319795	+	0.947487i	$0.603614\pi$
$684$	−15.7640	−0.602752
$685$	−26.1899	−1.00066
$686$	−42.7052	−1.63049
$687$	8.11028	0.309426
$688$	−8.65202	−0.329855
$689$	5.73860	0.218623
$690$	−10.1142	−0.385042
$691$	−25.9513	−0.987235	−0.493618	−	0.869679i	$-0.664326\pi$
$691$	−25.9513	−0.987235	−0.493618	+	0.869679i	$0.664326\pi$
$692$	−53.3495	−2.02804
$693$	29.0044	1.10179
$694$	−13.9749	−0.530482
$695$	4.17931	0.158530
$696$	0.415869	0.0157635
$697$	−38.7169	−1.46651
$698$	56.0986	2.12336
$699$	15.5370	0.587663
$700$	15.2068	0.574764
$701$	24.8084	0.937001	0.468501	−	0.883463i	$-0.344794\pi$
$701$	24.8084	0.937001	0.468501	+	0.883463i	$0.344794\pi$
$702$	72.1042	2.72140
$703$	−1.24217	−0.0468495
$704$	−60.2396	−2.27036
$705$	−10.8616	−0.409072
$706$	−15.7727	−0.593615
$707$	1.77814	0.0668740
$708$	4.15313	0.156084
$709$	7.87408	0.295717	0.147859	−	0.989008i	$-0.452762\pi$
$709$	7.87408	0.295717	0.147859	+	0.989008i	$0.452762\pi$
$710$	10.4980	0.393984
$711$	−18.0550	−0.677115
$712$	−45.3765	−1.70056
$713$	−2.50963	−0.0939862
$714$	−40.0394	−1.49844
$715$	−62.5598	−2.33961
$716$	−22.9993	−0.859525
$717$	17.6914	0.660698
$718$	−14.0249	−0.523406
$719$	−28.7232	−1.07119	−0.535597	−	0.844474i	$-0.679913\pi$
$719$	−28.7232	−1.07119	−0.535597	+	0.844474i	$0.679913\pi$
$720$	9.08780	0.338682
$721$	21.2899	0.792876
$722$	32.9715	1.22707
$723$	−14.3225	−0.532661
$724$	−75.7997	−2.81707
$725$	0.133111	0.00494363
$726$	−59.1269	−2.19440
$727$	−14.3212	−0.531145	−0.265572	−	0.964091i	$-0.585561\pi$
$727$	−14.3212	−0.531145	−0.265572	+	0.964091i	$0.585561\pi$
$728$	66.7219	2.47288
$729$	17.4057	0.644654
$730$	−21.5910	−0.799120
$731$	18.2802	0.676117
$732$	−52.2343	−1.93064
$733$	1.86006	0.0687027	0.0343514	−	0.999410i	$-0.489063\pi$
$733$	1.86006	0.0687027	0.0343514	+	0.999410i	$0.489063\pi$
$734$	12.0106	0.443321
$735$	−1.04145	−0.0384145
$736$	−4.36376	−0.160850
$737$	−1.50973	−0.0556115
$738$	30.2254	1.11261
$739$	1.75067	0.0643993	0.0321997	−	0.999481i	$-0.489749\pi$
$739$	1.75067	0.0643993	0.0321997	+	0.999481i	$0.489749\pi$
$740$	3.82769	0.140709
$741$	14.2911	0.524998
$742$	6.58136	0.241609
$743$	−18.5696	−0.681253	−0.340626	−	0.940199i	$-0.610639\pi$
$743$	−18.5696	−0.681253	−0.340626	+	0.940199i	$0.610639\pi$
$744$	−5.58690	−0.204826
$745$	−26.7322	−0.979391
$746$	−19.9504	−0.730434
$747$	−3.57055	−0.130640
$748$	−122.618	−4.48335
$749$	45.0231	1.64511
$750$	−31.6690	−1.15639
$751$	−15.7533	−0.574846	−0.287423	−	0.957804i	$-0.592799\pi$
$751$	−15.7533	−0.574846	−0.287423	+	0.957804i	$0.592799\pi$
$752$	14.1507	0.516022
$753$	17.5291	0.638795
$754$	1.24524	0.0453488
$755$	−1.87801	−0.0683476
$756$	54.0133	1.96445
$757$	−44.6078	−1.62130	−0.810649	−	0.585532i	$-0.800886\pi$
$757$	−44.6078	−1.62130	−0.810649	+	0.585532i	$0.800886\pi$
$758$	74.8886	2.72008
$759$	−13.0187	−0.472548
$760$	18.2895	0.663428
$761$	36.7232	1.33121	0.665607	−	0.746302i	$-0.268173\pi$
$761$	36.7232	1.33121	0.665607	+	0.746302i	$0.268173\pi$
$762$	−12.7385	−0.461467
$763$	8.26879	0.299350
$764$	−86.3794	−3.12510
$765$	−19.2009	−0.694211
$766$	30.7985	1.11279
$767$	5.83263	0.210604
$768$	−31.4062	−1.13327
$769$	−12.5408	−0.452232	−0.226116	−	0.974100i	$-0.572603\pi$
$769$	−12.5408	−0.452232	−0.226116	+	0.974100i	$0.572603\pi$
$770$	−71.7472	−2.58559
$771$	12.8812	0.463905
$772$	42.9867	1.54712
$773$	44.9553	1.61693	0.808464	−	0.588545i	$-0.200299\pi$
$773$	44.9553	1.61693	0.808464	+	0.588545i	$0.200299\pi$
$774$	−14.2709	−0.512957
$775$	−1.78826	−0.0642361
$776$	44.1156	1.58366
$777$	1.60881	0.0577158
$778$	−7.07009	−0.253475
$779$	15.8485	0.567833
$780$	−44.0373	−1.57679
$781$	13.5127	0.483523
$782$	−27.8406	−0.995579
$783$	0.472800	0.0168965
$784$	1.35682	0.0484578
$785$	−11.9355	−0.425996
$786$	5.04048	0.179788
$787$	51.0563	1.81996	0.909980	−	0.414652i	$-0.136097\pi$
$787$	51.0563	1.81996	0.909980	+	0.414652i	$0.136097\pi$
$788$	−32.0442	−1.14153
$789$	5.53054	0.196893
$790$	44.6620	1.58900
$791$	−4.30405	−0.153035
$792$	44.8971	1.59535
$793$	−73.3575	−2.60500
$794$	−39.7906	−1.41212
$795$	−2.03733	−0.0722566
$796$	49.9532	1.77054
$797$	−13.4666	−0.477011	−0.238505	−	0.971141i	$-0.576657\pi$
$797$	−13.4666	−0.477011	−0.238505	+	0.971141i	$0.576657\pi$
$798$	16.3899	0.580195
$799$	−29.8979	−1.05771
$800$	−3.10944	−0.109935
$801$	−19.5003	−0.689008
$802$	−66.5713	−2.35071
$803$	−27.7912	−0.980731
$804$	−1.06273	−0.0374796
$805$	−10.6405	−0.375028
$806$	−16.7289	−0.589249
$807$	10.9439	0.385243
$808$	2.75246	0.0968311
$809$	12.4881	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$809$	12.4881	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$810$	−0.932699	−0.0327717
$811$	−3.74673	−0.131565	−0.0657827	−	0.997834i	$-0.520954\pi$
$811$	−3.74673	−0.131565	−0.0657827	+	0.997834i	$0.520954\pi$
$812$	0.932807	0.0327351
$813$	−28.9877	−1.01664
$814$	7.54293	0.264380
$815$	−17.7940	−0.623298
$816$	−16.1479	−0.565291
$817$	−7.48288	−0.261793
$818$	0.167028	0.00583999
$819$	28.6733	1.00193
$820$	−48.8364	−1.70544
$821$	50.1255	1.74939	0.874697	−	0.484671i	$-0.161061\pi$
$821$	50.1255	1.74939	0.874697	+	0.484671i	$0.161061\pi$
$822$	−36.3298	−1.26715
$823$	−48.6217	−1.69485	−0.847423	−	0.530918i	$-0.821847\pi$
$823$	−48.6217	−1.69485	−0.847423	+	0.530918i	$0.821847\pi$
$824$	32.9554	1.14806
$825$	−9.27658	−0.322969
$826$	6.68919	0.232747
$827$	4.57364	0.159041	0.0795206	−	0.996833i	$-0.474661\pi$
$827$	4.57364	0.159041	0.0795206	+	0.996833i	$0.474661\pi$
$828$	14.1965	0.493362
$829$	15.6758	0.544445	0.272222	−	0.962234i	$-0.412241\pi$
$829$	15.6758	0.544445	0.272222	+	0.962234i	$0.412241\pi$
$830$	8.83235	0.306575
$831$	−19.8113	−0.687246
$832$	−59.5519	−2.06459
$833$	−2.86672	−0.0993259
$834$	5.79741	0.200748
$835$	31.2149	1.08024
$836$	50.1929	1.73596
$837$	−6.35173	−0.219548
$838$	12.2679	0.423788
$839$	23.9978	0.828497	0.414248	−	0.910164i	$-0.364044\pi$
$839$	23.9978	0.828497	0.414248	+	0.910164i	$0.364044\pi$
$840$	−23.6877	−0.817304
$841$	−28.9918	−0.999718
$842$	−72.9976	−2.51566
$843$	2.75439	0.0948661
$844$	45.5991	1.56959
$845$	−37.4316	−1.28769
$846$	23.3406	0.802465
$847$	−62.2033	−2.13733
$848$	2.65427	0.0911479
$849$	12.7712	0.438305
$850$	−19.8381	−0.680441
$851$	1.11866	0.0383470
$852$	9.51191	0.325873
$853$	1.38225	0.0473274	0.0236637	−	0.999720i	$-0.492467\pi$
$853$	1.38225	0.0473274	0.0236637	+	0.999720i	$0.492467\pi$
$854$	−84.1305	−2.87889
$855$	7.85977	0.268799
$856$	69.6930	2.38206
$857$	12.4568	0.425515	0.212757	−	0.977105i	$-0.431756\pi$
$857$	12.4568	0.425515	0.212757	+	0.977105i	$0.431756\pi$
$858$	−86.7810	−2.96265
$859$	9.40158	0.320778	0.160389	−	0.987054i	$-0.448725\pi$
$859$	9.40158	0.320778	0.160389	+	0.987054i	$0.448725\pi$
$860$	23.0581	0.786274
$861$	−20.5264	−0.699537
$862$	49.1529	1.67415
$863$	16.4882	0.561266	0.280633	−	0.959815i	$-0.409456\pi$
$863$	16.4882	0.561266	0.280633	+	0.959815i	$0.409456\pi$
$864$	−11.0445	−0.375740
$865$	26.5995	0.904411
$866$	19.9549	0.678094
$867$	15.6755	0.532368
$868$	−12.5316	−0.425350
$869$	57.4874	1.95013
$870$	−0.442086	−0.0149881
$871$	−1.49249	−0.0505711
$872$	12.7996	0.433448
$873$	18.9584	0.641645
$874$	11.3964	0.385489
$875$	−33.3168	−1.12631
$876$	−19.5629	−0.660969
$877$	−48.9077	−1.65149	−0.825747	−	0.564040i	$-0.809246\pi$
$877$	−48.9077	−1.65149	−0.825747	+	0.564040i	$0.809246\pi$
$878$	39.0094	1.31650
$879$	−26.6285	−0.898158
$880$	−28.9357	−0.975423
$881$	−20.5911	−0.693732	−0.346866	−	0.937915i	$-0.612754\pi$
$881$	−20.5911	−0.693732	−0.346866	+	0.937915i	$0.612754\pi$
$882$	2.23798	0.0753566
$883$	−3.12697	−0.105231	−0.0526155	−	0.998615i	$-0.516756\pi$
$883$	−3.12697	−0.105231	−0.0526155	+	0.998615i	$0.516756\pi$
$884$	−121.218	−4.07701
$885$	−2.07071	−0.0696062
$886$	75.5545	2.53830
$887$	−10.4187	−0.349826	−0.174913	−	0.984584i	$-0.555964\pi$
$887$	−10.4187	−0.349826	−0.174913	+	0.984584i	$0.555964\pi$
$888$	2.49034	0.0835704
$889$	−13.4013	−0.449465
$890$	48.2371	1.61691
$891$	−1.20054	−0.0402196
$892$	−36.7602	−1.23082
$893$	12.2385	0.409546
$894$	−37.0820	−1.24021
$895$	11.4672	0.383307
$896$	−56.7274	−1.89513
$897$	−12.8701	−0.429719
$898$	−1.90228	−0.0634800
$899$	−0.109694	−0.00365850
$900$	10.1158	0.337195
$901$	−5.60800	−0.186830
$902$	−96.2381	−3.20438
$903$	9.69151	0.322513
$904$	−6.66241	−0.221589
$905$	37.7929	1.25628
$906$	−2.60511	−0.0865489
$907$	8.11358	0.269407	0.134703	−	0.990886i	$-0.456992\pi$
$907$	8.11358	0.269407	0.134703	+	0.990886i	$0.456992\pi$
$908$	6.82537	0.226508
$909$	1.18285	0.0392327
$910$	−70.9281	−2.35124
$911$	−37.3576	−1.23771	−0.618857	−	0.785504i	$-0.712404\pi$
$911$	−37.3576	−1.23771	−0.618857	+	0.785504i	$0.712404\pi$
$912$	6.61005	0.218881
$913$	11.3687	0.376249
$914$	76.1565	2.51903
$915$	26.0435	0.860972
$916$	28.1595	0.930416
$917$	5.30274	0.175112
$918$	−70.4632	−2.32563
$919$	5.25797	0.173444	0.0867221	−	0.996233i	$-0.472361\pi$
$919$	5.25797	0.173444	0.0867221	+	0.996233i	$0.472361\pi$
$920$	−16.4708	−0.543027
$921$	−26.1258	−0.860873
$922$	62.9467	2.07304
$923$	13.3585	0.439699
$924$	−65.0077	−2.13860
$925$	0.797108	0.0262088
$926$	43.7509	1.43774
$927$	14.1624	0.465153
$928$	−0.190737	−0.00626125
$929$	−46.6981	−1.53212	−0.766058	−	0.642772i	$-0.777784\pi$
$929$	−46.6981	−1.53212	−0.766058	+	0.642772i	$0.777784\pi$
$930$	5.93911	0.194751
$931$	1.17347	0.0384590
$932$	53.9456	1.76705
$933$	−3.85984	−0.126366
$934$	59.5074	1.94714
$935$	61.1360	1.99936
$936$	44.3845	1.45075
$937$	42.0509	1.37374	0.686872	−	0.726778i	$-0.258983\pi$
$937$	42.0509	1.37374	0.686872	+	0.726778i	$0.258983\pi$
$938$	−1.71167	−0.0558881
$939$	−24.8967	−0.812473
$940$	−37.7123	−1.23004
$941$	4.77335	0.155607	0.0778034	−	0.996969i	$-0.475209\pi$
$941$	4.77335	0.155607	0.0778034	+	0.996969i	$0.475209\pi$
$942$	−16.5565	−0.539441
$943$	−14.2726	−0.464780
$944$	2.69776	0.0878045
$945$	−26.9305	−0.876049
$946$	45.4388	1.47734
$947$	−17.6884	−0.574795	−0.287397	−	0.957811i	$-0.592790\pi$
$947$	−17.6884	−0.574795	−0.287397	+	0.957811i	$0.592790\pi$
$948$	40.4667	1.31430
$949$	−27.4740	−0.891843
$950$	8.12060	0.263467
$951$	19.4724	0.631435
$952$	−65.2034	−2.11325
$953$	−14.5963	−0.472822	−0.236411	−	0.971653i	$-0.575971\pi$
$953$	−14.5963	−0.472822	−0.236411	+	0.971653i	$0.575971\pi$
$954$	4.37803	0.141744
$955$	43.0679	1.39364
$956$	61.4259	1.98666
$957$	−0.569038	−0.0183944
$958$	−71.2634	−2.30242
$959$	−38.2200	−1.23419
$960$	21.1422	0.682362
$961$	−29.5263	−0.952463
$962$	7.45682	0.240418
$963$	29.9501	0.965129
$964$	−49.7289	−1.60166
$965$	−21.4327	−0.689943
$966$	−14.7601	−0.474899
$967$	4.00885	0.128916	0.0644580	−	0.997920i	$-0.479468\pi$
$967$	4.00885	0.128916	0.0644580	+	0.997920i	$0.479468\pi$
$968$	−96.2869	−3.09478
$969$	−13.9659	−0.448649
$970$	−46.8967	−1.50576
$971$	39.3665	1.26333	0.631665	−	0.775241i	$-0.282372\pi$
$971$	39.3665	1.26333	0.631665	+	0.775241i	$0.282372\pi$
$972$	58.2795	1.86932
$973$	6.09905	0.195527
$974$	−86.6602	−2.77677
$975$	−9.17068	−0.293697
$976$	−33.9299	−1.08607
$977$	11.8823	0.380147	0.190074	−	0.981770i	$-0.439127\pi$
$977$	11.8823	0.380147	0.190074	+	0.981770i	$0.439127\pi$
$978$	−24.6833	−0.789285
$979$	62.0892	1.98438
$980$	−3.61599	−0.115509
$981$	5.50054	0.175619
$982$	37.6453	1.20131
$983$	23.9569	0.764106	0.382053	−	0.924141i	$-0.375217\pi$
$983$	23.9569	0.764106	0.382053	+	0.924141i	$0.375217\pi$
$984$	−31.7736	−1.01290
$985$	15.9769	0.509067
$986$	−1.21689	−0.0387538
$987$	−15.8508	−0.504537
$988$	49.6199	1.57862
$989$	6.73881	0.214282
$990$	−47.7274	−1.51688
$991$	−22.1544	−0.703759	−0.351880	−	0.936045i	$-0.614457\pi$
$991$	−22.1544	−0.703759	−0.351880	+	0.936045i	$0.614457\pi$
$992$	2.56242	0.0813568
$993$	−5.14298	−0.163208
$994$	15.3202	0.485928
$995$	−24.9062	−0.789578
$996$	8.00269	0.253575
$997$	−19.6431	−0.622103	−0.311051	−	0.950393i	$-0.600681\pi$
$997$	−19.6431	−0.622103	−0.311051	+	0.950393i	$0.600681\pi$
$998$	4.21478	0.133417
$999$	2.83126	0.0895771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.a.1.14	✓	147

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.a.1.14	✓	147	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8003 = 53 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40138 q^{2} +1.08484 q^{3} +3.76663 q^{4} -1.87801 q^{5} -2.60511 q^{6} -2.74065 q^{7} -4.24236 q^{8} -1.82313 q^{9} +O(q^{10})\)	\(q-2.40138 q^{2} +1.08484 q^{3} +3.76663 q^{4} -1.87801 q^{5} -2.60511 q^{6} -2.74065 q^{7} -4.24236 q^{8} -1.82313 q^{9} +4.50981 q^{10} +5.80487 q^{11} +4.08618 q^{12} +5.73860 q^{13} +6.58136 q^{14} -2.03733 q^{15} +2.65427 q^{16} -5.60800 q^{17} +4.37803 q^{18} +2.29560 q^{19} -7.07376 q^{20} -2.97316 q^{21} -13.9397 q^{22} -2.06733 q^{23} -4.60227 q^{24} -1.47310 q^{25} -13.7806 q^{26} -5.23231 q^{27} -10.3230 q^{28} -0.0903616 q^{29} +4.89241 q^{30} +1.21394 q^{31} +2.11082 q^{32} +6.29734 q^{33} +13.4669 q^{34} +5.14696 q^{35} -6.86706 q^{36} -0.541111 q^{37} -5.51261 q^{38} +6.22545 q^{39} +7.96718 q^{40} +6.90388 q^{41} +7.13970 q^{42} -3.25966 q^{43} +21.8648 q^{44} +3.42385 q^{45} +4.96445 q^{46} +5.33129 q^{47} +2.87945 q^{48} +0.511184 q^{49} +3.53746 q^{50} -6.08376 q^{51} +21.6152 q^{52} +1.00000 q^{53} +12.5648 q^{54} -10.9016 q^{55} +11.6269 q^{56} +2.49035 q^{57} +0.216993 q^{58} +1.01638 q^{59} -7.67388 q^{60} -12.7832 q^{61} -2.91514 q^{62} +4.99656 q^{63} -10.3774 q^{64} -10.7771 q^{65} -15.1223 q^{66} -0.260079 q^{67} -21.1233 q^{68} -2.24272 q^{69} -12.3598 q^{70} +2.32782 q^{71} +7.73438 q^{72} -4.78757 q^{73} +1.29941 q^{74} -1.59807 q^{75} +8.64668 q^{76} -15.9091 q^{77} -14.9497 q^{78} +9.90330 q^{79} -4.98473 q^{80} -0.206816 q^{81} -16.5789 q^{82} +1.95848 q^{83} -11.1988 q^{84} +10.5319 q^{85} +7.82770 q^{86} -0.0980276 q^{87} -24.6264 q^{88} +10.6960 q^{89} -8.22196 q^{90} -15.7275 q^{91} -7.78688 q^{92} +1.31693 q^{93} -12.8025 q^{94} -4.31115 q^{95} +2.28989 q^{96} -10.3988 q^{97} -1.22755 q^{98} -10.5830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 147 q - 6 q^{2} - 23 q^{3} + 130 q^{4} - 25 q^{5} - 18 q^{6} - 33 q^{7} - 15 q^{8} + 114 q^{9}+O(q^{10}) \) 147 * q - 6 * q^2 - 23 * q^3 + 130 * q^4 - 25 * q^5 - 18 * q^6 - 33 * q^7 - 15 * q^8 + 114 * q^9	\( 147 q - 6 q^{2} - 23 q^{3} + 130 q^{4} - 25 q^{5} - 18 q^{6} - 33 q^{7} - 15 q^{8} + 114 q^{9} - 24 q^{10} - 13 q^{11} - 52 q^{12} - 119 q^{13} - 6 q^{14} - 22 q^{15} + 100 q^{16} - 42 q^{17} - 6 q^{18} - 31 q^{19} - 50 q^{20} - 44 q^{21} - 38 q^{22} - 16 q^{23} - 30 q^{24} + 80 q^{25} - 18 q^{26} - 68 q^{27} - 72 q^{28} - 40 q^{29} - 3 q^{30} - 44 q^{31} - 41 q^{32} - 71 q^{33} - 55 q^{34} - 24 q^{35} + 108 q^{36} - 145 q^{37} - 39 q^{38} + 28 q^{39} - 81 q^{40} - 28 q^{41} - 54 q^{42} - 47 q^{43} - 33 q^{44} - 89 q^{45} - 43 q^{46} - 65 q^{47} - 82 q^{48} + 52 q^{49} - 43 q^{50} + 7 q^{51} - 215 q^{52} + 147 q^{53} - 88 q^{54} - 48 q^{55} + 19 q^{56} - 66 q^{57} - 110 q^{58} - 66 q^{59} - 118 q^{60} - 80 q^{61} - 41 q^{62} - 52 q^{63} + 33 q^{64} - 17 q^{65} - 86 q^{66} - 114 q^{67} - 77 q^{68} - 49 q^{69} - 56 q^{70} + 10 q^{71} + 5 q^{72} - 143 q^{73} - 30 q^{74} - 97 q^{75} - 104 q^{76} - 116 q^{77} - 33 q^{78} - 31 q^{79} - 83 q^{80} - q^{81} - 72 q^{82} - 70 q^{83} - 64 q^{84} - 85 q^{85} - 43 q^{87} - 149 q^{88} - 130 q^{89} + 42 q^{90} - 35 q^{91} - 31 q^{92} - 149 q^{93} - 94 q^{94} - 9 q^{95} + 6 q^{96} - 211 q^{97} - 18 q^{98} - 19 q^{99}+O(q^{100}) \) 147 * q - 6 * q^2 - 23 * q^3 + 130 * q^4 - 25 * q^5 - 18 * q^6 - 33 * q^7 - 15 * q^8 + 114 * q^9 - 24 * q^10 - 13 * q^11 - 52 * q^12 - 119 * q^13 - 6 * q^14 - 22 * q^15 + 100 * q^16 - 42 * q^17 - 6 * q^18 - 31 * q^19 - 50 * q^20 - 44 * q^21 - 38 * q^22 - 16 * q^23 - 30 * q^24 + 80 * q^25 - 18 * q^26 - 68 * q^27 - 72 * q^28 - 40 * q^29 - 3 * q^30 - 44 * q^31 - 41 * q^32 - 71 * q^33 - 55 * q^34 - 24 * q^35 + 108 * q^36 - 145 * q^37 - 39 * q^38 + 28 * q^39 - 81 * q^40 - 28 * q^41 - 54 * q^42 - 47 * q^43 - 33 * q^44 - 89 * q^45 - 43 * q^46 - 65 * q^47 - 82 * q^48 + 52 * q^49 - 43 * q^50 + 7 * q^51 - 215 * q^52 + 147 * q^53 - 88 * q^54 - 48 * q^55 + 19 * q^56 - 66 * q^57 - 110 * q^58 - 66 * q^59 - 118 * q^60 - 80 * q^61 - 41 * q^62 - 52 * q^63 + 33 * q^64 - 17 * q^65 - 86 * q^66 - 114 * q^67 - 77 * q^68 - 49 * q^69 - 56 * q^70 + 10 * q^71 + 5 * q^72 - 143 * q^73 - 30 * q^74 - 97 * q^75 - 104 * q^76 - 116 * q^77 - 33 * q^78 - 31 * q^79 - 83 * q^80 - q^81 - 72 * q^82 - 70 * q^83 - 64 * q^84 - 85 * q^85 - 43 * q^87 - 149 * q^88 - 130 * q^89 + 42 * q^90 - 35 * q^91 - 31 * q^92 - 149 * q^93 - 94 * q^94 - 9 * q^95 + 6 * q^96 - 211 * q^97 - 18 * q^98 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more