LMFDB - Embedded newform 8023.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8023,2,Mod(1,8023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8023, 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("8023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8023 = 71 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8023.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:	$64.0639775417$
Analytic rank:	$0$
Dimension:	$165$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62951 q^{2} +0.0659678 q^{3} +4.91432 q^{4} +1.30562 q^{5} -0.173463 q^{6} -3.45826 q^{7} -7.66322 q^{8} -2.99565 q^{9} +O(q^{10})$	\(q-2.62951 q^{2} +0.0659678 q^{3} +4.91432 q^{4} +1.30562 q^{5} -0.173463 q^{6} -3.45826 q^{7} -7.66322 q^{8} -2.99565 q^{9} -3.43314 q^{10} +0.878711 q^{11} +0.324187 q^{12} +6.68941 q^{13} +9.09353 q^{14} +0.0861289 q^{15} +10.3219 q^{16} +5.11284 q^{17} +7.87708 q^{18} +1.57730 q^{19} +6.41623 q^{20} -0.228134 q^{21} -2.31058 q^{22} +0.769553 q^{23} -0.505526 q^{24} -3.29535 q^{25} -17.5899 q^{26} -0.395520 q^{27} -16.9950 q^{28} +1.88535 q^{29} -0.226477 q^{30} +7.24499 q^{31} -11.8150 q^{32} +0.0579666 q^{33} -13.4443 q^{34} -4.51518 q^{35} -14.7216 q^{36} +6.87948 q^{37} -4.14753 q^{38} +0.441285 q^{39} -10.0053 q^{40} +9.97950 q^{41} +0.599880 q^{42} +0.715171 q^{43} +4.31827 q^{44} -3.91118 q^{45} -2.02355 q^{46} -0.656715 q^{47} +0.680911 q^{48} +4.95957 q^{49} +8.66516 q^{50} +0.337283 q^{51} +32.8739 q^{52} +11.7869 q^{53} +1.04002 q^{54} +1.14726 q^{55} +26.5014 q^{56} +0.104051 q^{57} -4.95754 q^{58} -9.13531 q^{59} +0.423265 q^{60} +9.37437 q^{61} -19.0508 q^{62} +10.3597 q^{63} +10.4239 q^{64} +8.73383 q^{65} -0.152424 q^{66} -1.35028 q^{67} +25.1261 q^{68} +0.0507657 q^{69} +11.8727 q^{70} +1.00000 q^{71} +22.9563 q^{72} -8.79559 q^{73} -18.0897 q^{74} -0.217387 q^{75} +7.75136 q^{76} -3.03881 q^{77} -1.16036 q^{78} -12.4308 q^{79} +13.4765 q^{80} +8.96085 q^{81} -26.2412 q^{82} -7.76578 q^{83} -1.12112 q^{84} +6.67543 q^{85} -1.88055 q^{86} +0.124372 q^{87} -6.73376 q^{88} +8.00646 q^{89} +10.2845 q^{90} -23.1337 q^{91} +3.78183 q^{92} +0.477936 q^{93} +1.72684 q^{94} +2.05936 q^{95} -0.779411 q^{96} +5.71664 q^{97} -13.0412 q^{98} -2.63231 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 165 q + 22 q^{2} + 18 q^{3} + 166 q^{4} + 28 q^{5} + 16 q^{6} + 24 q^{7} + 66 q^{8} + 177 q^{9}+O(q^{10}) $ 165 * q + 22 * q^2 + 18 * q^3 + 166 * q^4 + 28 * q^5 + 16 * q^6 + 24 * q^7 + 66 * q^8 + 177 * q^9	\( 165 q + 22 q^{2} + 18 q^{3} + 166 q^{4} + 28 q^{5} + 16 q^{6} + 24 q^{7} + 66 q^{8} + 177 q^{9} + 14 q^{10} + 18 q^{11} + 54 q^{12} + 44 q^{13} + 26 q^{14} + 24 q^{15} + 168 q^{16} + 143 q^{17} + 57 q^{18} + 20 q^{19} + 49 q^{20} + 39 q^{21} + 25 q^{22} + 52 q^{23} + 27 q^{24} + 175 q^{25} + 48 q^{26} + 69 q^{27} + 28 q^{28} + 58 q^{29} - 11 q^{30} + 28 q^{31} + 114 q^{32} + 110 q^{33} + 55 q^{34} + 67 q^{35} + 202 q^{36} + 44 q^{37} + 35 q^{38} + 27 q^{39} + 53 q^{40} + 141 q^{41} + 40 q^{42} + 29 q^{43} + 52 q^{44} + 54 q^{45} + 29 q^{46} + 87 q^{47} + 53 q^{48} + 143 q^{49} + 16 q^{50} + 37 q^{51} + 105 q^{52} + 101 q^{53} - 36 q^{54} + 72 q^{55} + 57 q^{56} + 82 q^{57} + 4 q^{58} + 103 q^{59} + 53 q^{60} + 16 q^{61} + 54 q^{62} + 126 q^{63} + 136 q^{64} + 159 q^{65} + 53 q^{66} + 60 q^{67} + 220 q^{68} + 81 q^{69} + 16 q^{70} + 165 q^{71} + 176 q^{72} + 124 q^{73} + 29 q^{74} + 44 q^{75} + 18 q^{76} + 127 q^{77} - 91 q^{78} + 14 q^{79} + 158 q^{80} + 213 q^{81} + 20 q^{82} + 116 q^{83} + 67 q^{84} + 59 q^{85} + 30 q^{86} + 28 q^{87} + 79 q^{88} + 195 q^{89} + 16 q^{90} - 26 q^{91} + 173 q^{92} + 116 q^{93} + 53 q^{94} + 26 q^{95} - 36 q^{96} + 88 q^{97} + 150 q^{98} + 12 q^{99}+O(q^{100}) \) 165 * q + 22 * q^2 + 18 * q^3 + 166 * q^4 + 28 * q^5 + 16 * q^6 + 24 * q^7 + 66 * q^8 + 177 * q^9 + 14 * q^10 + 18 * q^11 + 54 * q^12 + 44 * q^13 + 26 * q^14 + 24 * q^15 + 168 * q^16 + 143 * q^17 + 57 * q^18 + 20 * q^19 + 49 * q^20 + 39 * q^21 + 25 * q^22 + 52 * q^23 + 27 * q^24 + 175 * q^25 + 48 * q^26 + 69 * q^27 + 28 * q^28 + 58 * q^29 - 11 * q^30 + 28 * q^31 + 114 * q^32 + 110 * q^33 + 55 * q^34 + 67 * q^35 + 202 * q^36 + 44 * q^37 + 35 * q^38 + 27 * q^39 + 53 * q^40 + 141 * q^41 + 40 * q^42 + 29 * q^43 + 52 * q^44 + 54 * q^45 + 29 * q^46 + 87 * q^47 + 53 * q^48 + 143 * q^49 + 16 * q^50 + 37 * q^51 + 105 * q^52 + 101 * q^53 - 36 * q^54 + 72 * q^55 + 57 * q^56 + 82 * q^57 + 4 * q^58 + 103 * q^59 + 53 * q^60 + 16 * q^61 + 54 * q^62 + 126 * q^63 + 136 * q^64 + 159 * q^65 + 53 * q^66 + 60 * q^67 + 220 * q^68 + 81 * q^69 + 16 * q^70 + 165 * q^71 + 176 * q^72 + 124 * q^73 + 29 * q^74 + 44 * q^75 + 18 * q^76 + 127 * q^77 - 91 * q^78 + 14 * q^79 + 158 * q^80 + 213 * q^81 + 20 * q^82 + 116 * q^83 + 67 * q^84 + 59 * q^85 + 30 * q^86 + 28 * q^87 + 79 * q^88 + 195 * q^89 + 16 * q^90 - 26 * q^91 + 173 * q^92 + 116 * q^93 + 53 * q^94 + 26 * q^95 - 36 * q^96 + 88 * q^97 + 150 * q^98 + 12 * 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 $

$2$	−2.62951	−1.85934	−0.929672	−	0.368389i	$-0.879909\pi$
$2$	−2.62951	−1.85934	−0.929672	+	0.368389i	$0.879909\pi$
$3$	0.0659678	0.0380865	0.0190433	−	0.999819i	$-0.493938\pi$
$3$	0.0659678	0.0380865	0.0190433	+	0.999819i	$0.493938\pi$
$4$	4.91432	2.45716
$5$	1.30562	0.583891	0.291946	−	0.956435i	$-0.405697\pi$
$5$	1.30562	0.583891	0.291946	+	0.956435i	$0.405697\pi$
$6$	−0.173463	−0.0708159
$7$	−3.45826	−1.30710	−0.653550	−	0.756883i	$-0.726721\pi$
$7$	−3.45826	−1.30710	−0.653550	+	0.756883i	$0.726721\pi$
$8$	−7.66322	−2.70936
$9$	−2.99565	−0.998549
$10$	−3.43314	−1.08565
$11$	0.878711	0.264941	0.132471	−	0.991187i	$-0.457709\pi$
$11$	0.878711	0.264941	0.132471	+	0.991187i	$0.457709\pi$
$12$	0.324187	0.0935846
$13$	6.68941	1.85531	0.927654	−	0.373442i	$-0.121822\pi$
$13$	6.68941	1.85531	0.927654	+	0.373442i	$0.121822\pi$
$14$	9.09353	2.43035
$15$	0.0861289	0.0222384
$16$	10.3219	2.58047
$17$	5.11284	1.24005	0.620023	−	0.784584i	$-0.287123\pi$
$17$	5.11284	1.24005	0.620023	+	0.784584i	$0.287123\pi$
$18$	7.87708	1.85665
$19$	1.57730	0.361858	0.180929	−	0.983496i	$-0.442090\pi$
$19$	1.57730	0.361858	0.180929	+	0.983496i	$0.442090\pi$
$20$	6.41623	1.43471
$21$	−0.228134	−0.0497829
$22$	−2.31058	−0.492617
$23$	0.769553	0.160463	0.0802315	−	0.996776i	$-0.474434\pi$
$23$	0.769553	0.160463	0.0802315	+	0.996776i	$0.474434\pi$
$24$	−0.505526	−0.103190
$25$	−3.29535	−0.659071
$26$	−17.5899	−3.44965
$27$	−0.395520	−0.0761178
$28$	−16.9950	−3.21175
$29$	1.88535	0.350101	0.175050	−	0.984559i	$-0.443991\pi$
$29$	1.88535	0.350101	0.175050	+	0.984559i	$0.443991\pi$
$30$	−0.226477	−0.0413488
$31$	7.24499	1.30124	0.650619	−	0.759404i	$-0.274509\pi$
$31$	7.24499	1.30124	0.650619	+	0.759404i	$0.274509\pi$
$32$	−11.8150	−2.08862
$33$	0.0579666	0.0100907
$34$	−13.4443	−2.30567
$35$	−4.51518	−0.763204
$36$	−14.7216	−2.45359
$37$	6.87948	1.13098	0.565490	−	0.824755i	$-0.308687\pi$
$37$	6.87948	1.13098	0.565490	+	0.824755i	$0.308687\pi$
$38$	−4.14753	−0.672818
$39$	0.441285	0.0706622
$40$	−10.0053	−1.58197
$41$	9.97950	1.55854	0.779268	−	0.626691i	$-0.215591\pi$
$41$	9.97950	1.55854	0.779268	+	0.626691i	$0.215591\pi$
$42$	0.599880	0.0925635
$43$	0.715171	0.109063	0.0545313	−	0.998512i	$-0.482634\pi$
$43$	0.715171	0.109063	0.0545313	+	0.998512i	$0.482634\pi$
$44$	4.31827	0.651003
$45$	−3.91118	−0.583044
$46$	−2.02355	−0.298356
$47$	−0.656715	−0.0957918	−0.0478959	−	0.998852i	$-0.515252\pi$
$47$	−0.656715	−0.0957918	−0.0478959	+	0.998852i	$0.515252\pi$
$48$	0.680911	0.0982811
$49$	4.95957	0.708510
$50$	8.66516	1.22544
$51$	0.337283	0.0472290
$52$	32.8739	4.55878
$53$	11.7869	1.61906	0.809530	−	0.587078i	$-0.199722\pi$
$53$	11.7869	1.61906	0.809530	+	0.587078i	$0.199722\pi$
$54$	1.04002	0.141529
$55$	1.14726	0.154697
$56$	26.5014	3.54140
$57$	0.104051	0.0137819
$58$	−4.95754	−0.650957
$59$	−9.13531	−1.18932	−0.594658	−	0.803979i	$-0.702713\pi$
$59$	−9.13531	−1.18932	−0.594658	+	0.803979i	$0.702713\pi$
$60$	0.423265	0.0546433
$61$	9.37437	1.20027	0.600133	−	0.799900i	$-0.295114\pi$
$61$	9.37437	1.20027	0.600133	+	0.799900i	$0.295114\pi$
$62$	−19.0508	−2.41945
$63$	10.3597	1.30520
$64$	10.4239	1.30299
$65$	8.73383	1.08330
$66$	−0.152424	−0.0187621
$67$	−1.35028	−0.164963	−0.0824814	−	0.996593i	$-0.526285\pi$
$67$	−1.35028	−0.164963	−0.0824814	+	0.996593i	$0.526285\pi$
$68$	25.1261	3.04699
$69$	0.0507657	0.00611148
$70$	11.8727	1.41906
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	22.9563	2.70543
$73$	−8.79559	−1.02945	−0.514723	−	0.857356i	$-0.672105\pi$
$73$	−8.79559	−1.02945	−0.514723	+	0.857356i	$0.672105\pi$
$74$	−18.0897	−2.10288
$75$	−0.217387	−0.0251017
$76$	7.75136	0.889142
$77$	−3.03881	−0.346305
$78$	−1.16036	−0.131385
$79$	−12.4308	−1.39857	−0.699286	−	0.714842i	$-0.746499\pi$
$79$	−12.4308	−1.39857	−0.699286	+	0.714842i	$0.746499\pi$
$80$	13.4765	1.50671
$81$	8.96085	0.995650
$82$	−26.2412	−2.89785
$83$	−7.76578	−0.852405	−0.426202	−	0.904628i	$-0.640149\pi$
$83$	−7.76578	−0.852405	−0.426202	+	0.904628i	$0.640149\pi$
$84$	−1.12112	−0.122324
$85$	6.67543	0.724052
$86$	−1.88055	−0.202785
$87$	0.124372	0.0133341
$88$	−6.73376	−0.717821
$89$	8.00646	0.848683	0.424342	−	0.905502i	$-0.360506\pi$
$89$	8.00646	0.848683	0.424342	+	0.905502i	$0.360506\pi$
$90$	10.2845	1.08408
$91$	−23.1337	−2.42507
$92$	3.78183	0.394283
$93$	0.477936	0.0495597
$94$	1.72684	0.178110
$95$	2.05936	0.211286
$96$	−0.779411	−0.0795483
$97$	5.71664	0.580437	0.290218	−	0.956960i	$-0.406272\pi$
$97$	5.71664	0.580437	0.290218	+	0.956960i	$0.406272\pi$
$98$	−13.0412	−1.31736
$99$	−2.63231	−0.264557
$100$	−16.1944	−1.61944
$101$	−14.9607	−1.48865	−0.744324	−	0.667819i	$-0.767228\pi$
$101$	−14.9607	−1.48865	−0.744324	+	0.667819i	$0.767228\pi$
$102$	−0.886888	−0.0878150
$103$	−3.53944	−0.348751	−0.174376	−	0.984679i	$-0.555791\pi$
$103$	−3.53944	−0.348751	−0.174376	+	0.984679i	$0.555791\pi$
$104$	−51.2624	−5.02669
$105$	−0.297856	−0.0290678
$106$	−30.9939	−3.01039
$107$	−2.92018	−0.282304	−0.141152	−	0.989988i	$-0.545081\pi$
$107$	−2.92018	−0.282304	−0.141152	+	0.989988i	$0.545081\pi$
$108$	−1.94371	−0.187034
$109$	4.12598	0.395197	0.197599	−	0.980283i	$-0.436686\pi$
$109$	4.12598	0.395197	0.197599	+	0.980283i	$0.436686\pi$
$110$	−3.01674	−0.287635
$111$	0.453824	0.0430751
$112$	−35.6957	−3.37293
$113$	−1.00000	−0.0940721
$113$	−1.00000	−0.0940721
$114$	−0.273603	−0.0256253
$115$	1.00474	0.0936929
$116$	9.26521	0.860253
$117$	−20.0391	−1.85262
$118$	24.0214	2.21135
$119$	−17.6815	−1.62086
$120$	−0.660025	−0.0602518
$121$	−10.2279	−0.929806
$122$	−24.6500	−2.23171
$123$	0.658326	0.0593592
$124$	35.6042	3.19735
$125$	−10.8306	−0.968717
$126$	−27.2410	−2.42682
$127$	−20.8877	−1.85349	−0.926743	−	0.375695i	$-0.877404\pi$
$127$	−20.8877	−1.85349	−0.926743	+	0.375695i	$0.877404\pi$
$128$	−3.77982	−0.334092
$129$	0.0471783	0.00415381
$130$	−22.9657	−2.01422
$131$	9.76504	0.853175	0.426588	−	0.904446i	$-0.359716\pi$
$131$	9.76504	0.853175	0.426588	+	0.904446i	$0.359716\pi$
$132$	0.284866	0.0247944
$133$	−5.45472	−0.472985
$134$	3.55057	0.306723
$135$	−0.516399	−0.0444445
$136$	−39.1808	−3.35973
$137$	14.5921	1.24669	0.623344	−	0.781948i	$-0.285774\pi$
$137$	14.5921	1.24669	0.623344	+	0.781948i	$0.285774\pi$
$138$	−0.133489	−0.0113633
$139$	11.9092	1.01013	0.505064	−	0.863082i	$-0.331469\pi$
$139$	11.9092	1.01013	0.505064	+	0.863082i	$0.331469\pi$
$140$	−22.1890	−1.87531
$141$	−0.0433221	−0.00364838
$142$	−2.62951	−0.220663
$143$	5.87806	0.491548
$144$	−30.9207	−2.57673
$145$	2.46155	0.204421
$146$	23.1281	1.91409
$147$	0.327172	0.0269847
$148$	33.8080	2.77900
$149$	−1.57994	−0.129434	−0.0647170	−	0.997904i	$-0.520614\pi$
$149$	−1.57994	−0.129434	−0.0647170	+	0.997904i	$0.520614\pi$
$150$	0.571622	0.0466727
$151$	−8.16772	−0.664679	−0.332340	−	0.943160i	$-0.607838\pi$
$151$	−8.16772	−0.664679	−0.332340	+	0.943160i	$0.607838\pi$
$152$	−12.0872	−0.980403
$153$	−15.3163	−1.23825
$154$	7.99059	0.643900
$155$	9.45921	0.759782
$156$	2.16862	0.173628
$157$	9.58662	0.765095	0.382548	−	0.923936i	$-0.375047\pi$
$157$	9.58662	0.765095	0.382548	+	0.923936i	$0.375047\pi$
$158$	32.6869	2.60043
$159$	0.777558	0.0616644
$160$	−15.4259	−1.21953
$161$	−2.66132	−0.209741
$162$	−23.5626	−1.85126
$163$	−8.49773	−0.665594	−0.332797	−	0.942999i	$-0.607992\pi$
$163$	−8.49773	−0.665594	−0.332797	+	0.942999i	$0.607992\pi$
$164$	49.0424	3.82957
$165$	0.0756825	0.00589187
$166$	20.4202	1.58491
$167$	20.4642	1.58357	0.791784	−	0.610801i	$-0.209152\pi$
$167$	20.4642	1.58357	0.791784	+	0.610801i	$0.209152\pi$
$168$	1.74824	0.134880
$169$	31.7481	2.44216
$170$	−17.5531	−1.34626
$171$	−4.72504	−0.361333
$172$	3.51458	0.267984
$173$	−12.3402	−0.938211	−0.469106	−	0.883142i	$-0.655424\pi$
$173$	−12.3402	−0.938211	−0.469106	+	0.883142i	$0.655424\pi$
$174$	−0.327038	−0.0247927
$175$	11.3962	0.861471
$176$	9.06995	0.683673
$177$	−0.602636	−0.0452969
$178$	−21.0531	−1.57799
$179$	−14.0593	−1.05084	−0.525420	−	0.850843i	$-0.676092\pi$
$179$	−14.0593	−1.05084	−0.525420	+	0.850843i	$0.676092\pi$
$180$	−19.2208	−1.43263
$181$	17.1618	1.27562	0.637812	−	0.770192i	$-0.279840\pi$
$181$	17.1618	1.27562	0.637812	+	0.770192i	$0.279840\pi$
$182$	60.8303	4.50904
$183$	0.618407	0.0457139
$184$	−5.89726	−0.434752
$185$	8.98200	0.660370
$186$	−1.25674	−0.0921484
$187$	4.49271	0.328540
$188$	−3.22731	−0.235376
$189$	1.36781	0.0994936
$190$	−5.41510	−0.392853
$191$	−7.36068	−0.532600	−0.266300	−	0.963890i	$-0.585801\pi$
$191$	−7.36068	−0.532600	−0.266300	+	0.963890i	$0.585801\pi$
$192$	0.687645	0.0496265
$193$	−14.9756	−1.07797	−0.538983	−	0.842316i	$-0.681191\pi$
$193$	−14.9756	−1.07797	−0.538983	+	0.842316i	$0.681191\pi$
$194$	−15.0319	−1.07923
$195$	0.576151	0.0412591
$196$	24.3729	1.74092
$197$	−11.2705	−0.802987	−0.401494	−	0.915862i	$-0.631509\pi$
$197$	−11.2705	−0.802987	−0.401494	+	0.915862i	$0.631509\pi$
$198$	6.92168	0.491903
$199$	19.5823	1.38815	0.694077	−	0.719901i	$-0.255813\pi$
$199$	19.5823	1.38815	0.694077	+	0.719901i	$0.255813\pi$
$200$	25.2530	1.78566
$201$	−0.0890750	−0.00628286
$202$	39.3394	2.76791
$203$	−6.52003	−0.457616
$204$	1.65751	0.116049
$205$	13.0294	0.910016
$206$	9.30699	0.648449
$207$	−2.30531	−0.160230
$208$	69.0472	4.78756
$209$	1.38599	0.0958712
$210$	0.783216	0.0540470
$211$	−18.3658	−1.26435	−0.632176	−	0.774825i	$-0.717838\pi$
$211$	−18.3658	−1.26435	−0.632176	+	0.774825i	$0.717838\pi$
$212$	57.9247	3.97829
$213$	0.0659678	0.00452004
$214$	7.67863	0.524900
$215$	0.933742	0.0636807
$216$	3.03096	0.206230
$217$	−25.0551	−1.70085
$218$	−10.8493	−0.734808
$219$	−0.580226	−0.0392080
$220$	5.63802	0.380115
$221$	34.2019	2.30067
$222$	−1.19334	−0.0800914
$223$	−23.9503	−1.60383	−0.801916	−	0.597436i	$-0.796186\pi$
$223$	−23.9503	−1.60383	−0.801916	+	0.597436i	$0.796186\pi$
$224$	40.8594	2.73003
$225$	9.87172	0.658115
$226$	2.62951	0.174912
$227$	3.73445	0.247864	0.123932	−	0.992291i	$-0.460450\pi$
$227$	3.73445	0.247864	0.123932	+	0.992291i	$0.460450\pi$
$228$	0.511340	0.0338643
$229$	−14.2355	−0.940706	−0.470353	−	0.882478i	$-0.655873\pi$
$229$	−14.2355	−0.940706	−0.470353	+	0.882478i	$0.655873\pi$
$230$	−2.64199	−0.174207
$231$	−0.200464	−0.0131896
$232$	−14.4479	−0.948548
$233$	8.04370	0.526960	0.263480	−	0.964665i	$-0.415130\pi$
$233$	8.04370	0.526960	0.263480	+	0.964665i	$0.415130\pi$
$234$	52.6930	3.44465
$235$	−0.857421	−0.0559320
$236$	−44.8938	−2.92234
$237$	−0.820031	−0.0532668
$238$	46.4938	3.01374
$239$	8.74182	0.565461	0.282731	−	0.959199i	$-0.408760\pi$
$239$	8.74182	0.565461	0.282731	+	0.959199i	$0.408760\pi$
$240$	0.889012	0.0573855
$241$	−13.9679	−0.899754	−0.449877	−	0.893090i	$-0.648532\pi$
$241$	−13.9679	−0.899754	−0.449877	+	0.893090i	$0.648532\pi$
$242$	26.8943	1.72883
$243$	1.77769	0.114039
$244$	46.0686	2.94924
$245$	6.47532	0.413693
$246$	−1.73107	−0.110369
$247$	10.5512	0.671358
$248$	−55.5200	−3.52552
$249$	−0.512291	−0.0324651
$250$	28.4791	1.80118
$251$	−4.96135	−0.313158	−0.156579	−	0.987665i	$-0.550047\pi$
$251$	−4.96135	−0.313158	−0.156579	+	0.987665i	$0.550047\pi$
$252$	50.9110	3.20709
$253$	0.676215	0.0425133
$254$	54.9245	3.44627
$255$	0.440363	0.0275766
$256$	−10.9088	−0.681801
$257$	26.8086	1.67227	0.836136	−	0.548521i	$-0.184809\pi$
$257$	26.8086	1.67227	0.836136	+	0.548521i	$0.184809\pi$
$258$	−0.124056	−0.00772337
$259$	−23.7911	−1.47830
$260$	42.9208	2.66183
$261$	−5.64784	−0.349593
$262$	−25.6773	−1.58635
$263$	27.7664	1.71215	0.856074	−	0.516853i	$-0.172897\pi$
$263$	27.7664	1.71215	0.856074	+	0.516853i	$0.172897\pi$
$264$	−0.444211	−0.0273393
$265$	15.3893	0.945355
$266$	14.3432	0.879441
$267$	0.528169	0.0323234
$268$	−6.63570	−0.405340
$269$	−11.5765	−0.705831	−0.352915	−	0.935655i	$-0.614810\pi$
$269$	−11.5765	−0.705831	−0.352915	+	0.935655i	$0.614810\pi$
$270$	1.35788	0.0826376
$271$	14.9334	0.907139	0.453570	−	0.891221i	$-0.350150\pi$
$271$	14.9334	0.907139	0.453570	+	0.891221i	$0.350150\pi$
$272$	52.7741	3.19990
$273$	−1.52608	−0.0923626
$274$	−38.3701	−2.31802
$275$	−2.89567	−0.174615
$276$	0.249479	0.0150169
$277$	24.1988	1.45396	0.726982	−	0.686656i	$-0.240922\pi$
$277$	24.1988	1.45396	0.726982	+	0.686656i	$0.240922\pi$
$278$	−31.3154	−1.87817
$279$	−21.7034	−1.29935
$280$	34.6008	2.06779
$281$	−1.20513	−0.0718921	−0.0359460	−	0.999354i	$-0.511444\pi$
$281$	−1.20513	−0.0718921	−0.0359460	+	0.999354i	$0.511444\pi$
$282$	0.113916	0.00678358
$283$	20.6614	1.22819	0.614096	−	0.789231i	$-0.289521\pi$
$283$	20.6614	1.22819	0.614096	+	0.789231i	$0.289521\pi$
$284$	4.91432	0.291611
$285$	0.135851	0.00804714
$286$	−15.4564	−0.913956
$287$	−34.5117	−2.03716
$288$	35.3936	2.08559
$289$	9.14113	0.537714
$290$	−6.47267	−0.380088
$291$	0.377114	0.0221068
$292$	−43.2243	−2.52951
$293$	23.4170	1.36804	0.684019	−	0.729464i	$-0.260231\pi$
$293$	23.4170	1.36804	0.684019	+	0.729464i	$0.260231\pi$
$294$	−0.860301	−0.0501738
$295$	−11.9273	−0.694432
$296$	−52.7190	−3.06423
$297$	−0.347548	−0.0201668
$298$	4.15448	0.240662
$299$	5.14785	0.297708
$300$	−1.06831	−0.0616789
$301$	−2.47325	−0.142556
$302$	21.4771	1.23587
$303$	−0.986926	−0.0566974
$304$	16.2807	0.933763
$305$	12.2394	0.700825
$306$	40.2743	2.30233
$307$	−4.52383	−0.258189	−0.129094	−	0.991632i	$-0.541207\pi$
$307$	−4.52383	−0.258189	−0.129094	+	0.991632i	$0.541207\pi$
$308$	−14.9337	−0.850926
$309$	−0.233489	−0.0132827
$310$	−24.8731	−1.41270
$311$	26.3271	1.49288	0.746438	−	0.665455i	$-0.231763\pi$
$311$	26.3271	1.49288	0.746438	+	0.665455i	$0.231763\pi$
$312$	−3.38167	−0.191449
$313$	33.0277	1.86683	0.933417	−	0.358793i	$-0.116811\pi$
$313$	33.0277	1.86683	0.933417	+	0.358793i	$0.116811\pi$
$314$	−25.2081	−1.42258
$315$	13.5259	0.762097
$316$	−61.0888	−3.43651
$317$	15.7906	0.886886	0.443443	−	0.896303i	$-0.353757\pi$
$317$	15.7906	0.886886	0.443443	+	0.896303i	$0.353757\pi$
$318$	−2.04460	−0.114655
$319$	1.65668	0.0927562
$320$	13.6097	0.760807
$321$	−0.192638	−0.0107520
$322$	6.99795	0.389981
$323$	8.06450	0.448721
$324$	44.0365	2.44647
$325$	−22.0440	−1.22278
$326$	22.3449	1.23757
$327$	0.272182	0.0150517
$328$	−76.4751	−4.22263
$329$	2.27109	0.125209
$330$	−0.199008	−0.0109550
$331$	−2.72430	−0.149741	−0.0748705	−	0.997193i	$-0.523854\pi$
$331$	−2.72430	−0.149741	−0.0748705	+	0.997193i	$0.523854\pi$
$332$	−38.1635	−2.09449
$333$	−20.6085	−1.12934
$334$	−53.8109	−2.94440
$335$	−1.76295	−0.0963204
$336$	−2.35477	−0.128463
$337$	5.25428	0.286219	0.143109	−	0.989707i	$-0.454290\pi$
$337$	5.25428	0.286219	0.143109	+	0.989707i	$0.454290\pi$
$338$	−83.4820	−4.54082
$339$	−0.0659678	−0.00358288
$340$	32.8052	1.77911
$341$	6.36625	0.344752
$342$	12.4245	0.671842
$343$	7.05634	0.381006
$344$	−5.48051	−0.295490
$345$	0.0662808	0.00356844
$346$	32.4488	1.74446
$347$	25.8660	1.38856	0.694279	−	0.719706i	$-0.255724\pi$
$347$	25.8660	1.38856	0.694279	+	0.719706i	$0.255724\pi$
$348$	0.611205	0.0327640
$349$	−29.9926	−1.60547	−0.802734	−	0.596338i	$-0.796622\pi$
$349$	−29.9926	−1.60547	−0.802734	+	0.596338i	$0.796622\pi$
$350$	−29.9664	−1.60177
$351$	−2.64579	−0.141222
$352$	−10.3820	−0.553362
$353$	−5.20960	−0.277279	−0.138640	−	0.990343i	$-0.544273\pi$
$353$	−5.20960	−0.277279	−0.138640	+	0.990343i	$0.544273\pi$
$354$	1.58464	0.0842226
$355$	1.30562	0.0692952
$356$	39.3463	2.08535
$357$	−1.16641	−0.0617331
$358$	36.9690	1.95387
$359$	−18.7814	−0.991245	−0.495623	−	0.868538i	$-0.665060\pi$
$359$	−18.7814	−0.991245	−0.495623	+	0.868538i	$0.665060\pi$
$360$	29.9722	1.57968
$361$	−16.5121	−0.869059
$362$	−45.1270	−2.37182
$363$	−0.674710	−0.0354131
$364$	−113.686	−5.95879
$365$	−11.4837	−0.601085
$366$	−1.62611	−0.0849979
$367$	−12.2766	−0.640832	−0.320416	−	0.947277i	$-0.603823\pi$
$367$	−12.2766	−0.640832	−0.320416	+	0.947277i	$0.603823\pi$
$368$	7.94323	0.414070
$369$	−29.8951	−1.55628
$370$	−23.6182	−1.22785
$371$	−40.7623	−2.11627
$372$	2.34873	0.121776
$373$	7.30961	0.378477	0.189239	−	0.981931i	$-0.439398\pi$
$373$	7.30961	0.378477	0.189239	+	0.981931i	$0.439398\pi$
$374$	−11.8136	−0.610868
$375$	−0.714470	−0.0368951
$376$	5.03255	0.259534
$377$	12.6119	0.649544
$378$	−3.59667	−0.184993
$379$	−2.87335	−0.147594	−0.0737969	−	0.997273i	$-0.523512\pi$
$379$	−2.87335	−0.147594	−0.0737969	+	0.997273i	$0.523512\pi$
$380$	10.1203	0.519163
$381$	−1.37792	−0.0705929
$382$	19.3550	0.990287
$383$	20.6315	1.05422	0.527110	−	0.849797i	$-0.323276\pi$
$383$	20.6315	1.05422	0.527110	+	0.849797i	$0.323276\pi$
$384$	−0.249347	−0.0127244
$385$	−3.96754	−0.202204
$386$	39.3784	2.00431
$387$	−2.14240	−0.108904
$388$	28.0934	1.42622
$389$	−0.567625	−0.0287797	−0.0143899	−	0.999896i	$-0.504581\pi$
$389$	−0.567625	−0.0287797	−0.0143899	+	0.999896i	$0.504581\pi$
$390$	−1.51499	−0.0767147
$391$	3.93460	0.198981
$392$	−38.0063	−1.91961
$393$	0.644178	0.0324945
$394$	29.6358	1.49303
$395$	−16.2299	−0.816614
$396$	−12.9360	−0.650059
$397$	−22.6754	−1.13804	−0.569022	−	0.822322i	$-0.692678\pi$
$397$	−22.6754	−1.13804	−0.569022	+	0.822322i	$0.692678\pi$
$398$	−51.4919	−2.58105
$399$	−0.359836	−0.0180143
$400$	−34.0142	−1.70071
$401$	−21.2573	−1.06154	−0.530769	−	0.847517i	$-0.678097\pi$
$401$	−21.2573	−1.06154	−0.530769	+	0.847517i	$0.678097\pi$
$402$	0.234223	0.0116820
$403$	48.4647	2.41420
$404$	−73.5217	−3.65784
$405$	11.6995	0.581352
$406$	17.1445	0.850866
$407$	6.04508	0.299643
$408$	−2.58467	−0.127960
$409$	−2.29684	−0.113571	−0.0567857	−	0.998386i	$-0.518085\pi$
$409$	−2.29684	−0.113571	−0.0567857	+	0.998386i	$0.518085\pi$
$410$	−34.2610	−1.69203
$411$	0.962609	0.0474820
$412$	−17.3939	−0.856938
$413$	31.5923	1.55456
$414$	6.06184	0.297923
$415$	−10.1392	−0.497712
$416$	−79.0355	−3.87503
$417$	0.785625	0.0384722
$418$	−3.64448	−0.178257
$419$	6.85635	0.334954	0.167477	−	0.985876i	$-0.446438\pi$
$419$	6.85635	0.334954	0.167477	+	0.985876i	$0.446438\pi$
$420$	−1.46376	−0.0714242
$421$	−33.8172	−1.64815	−0.824074	−	0.566483i	$-0.808304\pi$
$421$	−33.8172	−1.64815	−0.824074	+	0.566483i	$0.808304\pi$
$422$	48.2930	2.35086
$423$	1.96729	0.0956528
$424$	−90.3259	−4.38661
$425$	−16.8486	−0.817278
$426$	−0.173463	−0.00840431
$427$	−32.4190	−1.56887
$428$	−14.3507	−0.693666
$429$	0.387762	0.0187213
$430$	−2.45528	−0.118404
$431$	21.2467	1.02342	0.511709	−	0.859159i	$-0.329013\pi$
$431$	21.2467	1.02342	0.511709	+	0.859159i	$0.329013\pi$
$432$	−4.08251	−0.196420
$433$	5.22810	0.251246	0.125623	−	0.992078i	$-0.459907\pi$
$433$	5.22810	0.251246	0.125623	+	0.992078i	$0.459907\pi$
$434$	65.8825	3.16246
$435$	0.162383	0.00778568
$436$	20.2764	0.971063
$437$	1.21382	0.0580648
$438$	1.52571	0.0729012
$439$	3.61876	0.172714	0.0863569	−	0.996264i	$-0.472477\pi$
$439$	3.61876	0.172714	0.0863569	+	0.996264i	$0.472477\pi$
$440$	−8.79174	−0.419130
$441$	−14.8571	−0.707482
$442$	−89.9341	−4.27773
$443$	−23.2763	−1.10589	−0.552945	−	0.833218i	$-0.686496\pi$
$443$	−23.2763	−1.10589	−0.552945	+	0.833218i	$0.686496\pi$
$444$	2.23024	0.105842
$445$	10.4534	0.495539
$446$	62.9776	2.98208
$447$	−0.104225	−0.00492969
$448$	−36.0487	−1.70314
$449$	−32.1771	−1.51853	−0.759265	−	0.650781i	$-0.774441\pi$
$449$	−32.1771	−1.51853	−0.759265	+	0.650781i	$0.774441\pi$
$450$	−25.9578	−1.22366
$451$	8.76910	0.412921
$452$	−4.91432	−0.231150
$453$	−0.538806	−0.0253153
$454$	−9.81976	−0.460864
$455$	−30.2039	−1.41598
$456$	−0.797367	−0.0373401
$457$	−12.0767	−0.564926	−0.282463	−	0.959278i	$-0.591151\pi$
$457$	−12.0767	−0.564926	−0.282463	+	0.959278i	$0.591151\pi$
$458$	37.4323	1.74910
$459$	−2.02223	−0.0943896
$460$	4.93763	0.230218
$461$	21.9605	1.02280	0.511400	−	0.859343i	$-0.329127\pi$
$461$	21.9605	1.02280	0.511400	+	0.859343i	$0.329127\pi$
$462$	0.527121	0.0245239
$463$	7.48845	0.348018	0.174009	−	0.984744i	$-0.444328\pi$
$463$	7.48845	0.348018	0.174009	+	0.984744i	$0.444328\pi$
$464$	19.4603	0.903424
$465$	0.624003	0.0289375
$466$	−21.1510	−0.979800
$467$	7.77097	0.359597	0.179799	−	0.983703i	$-0.442455\pi$
$467$	7.77097	0.359597	0.179799	+	0.983703i	$0.442455\pi$
$468$	−98.4785	−4.55217
$469$	4.66962	0.215623
$470$	2.25460	0.103997
$471$	0.632408	0.0291398
$472$	70.0059	3.22228
$473$	0.628429	0.0288952
$474$	2.15628	0.0990412
$475$	−5.19777	−0.238490
$476$	−86.8927	−3.98272
$477$	−35.3095	−1.61671
$478$	−22.9867	−1.05139
$479$	13.7222	0.626985	0.313493	−	0.949591i	$-0.398501\pi$
$479$	13.7222	0.626985	0.313493	+	0.949591i	$0.398501\pi$
$480$	−1.01761	−0.0464476
$481$	46.0197	2.09832
$482$	36.7288	1.67295
$483$	−0.175561	−0.00798831
$484$	−50.2630	−2.28468
$485$	7.46376	0.338912
$486$	−4.67444	−0.212037
$487$	22.7976	1.03306	0.516530	−	0.856269i	$-0.327223\pi$
$487$	22.7976	1.03306	0.516530	+	0.856269i	$0.327223\pi$
$488$	−71.8379	−3.25195
$489$	−0.560577	−0.0253502
$490$	−17.0269	−0.769197
$491$	27.0894	1.22253	0.611263	−	0.791427i	$-0.290662\pi$
$491$	27.0894	1.22253	0.611263	+	0.791427i	$0.290662\pi$
$492$	3.23522	0.145855
$493$	9.63949	0.434141
$494$	−27.7445	−1.24828
$495$	−3.43680	−0.154473
$496$	74.7819	3.35781
$497$	−3.45826	−0.155124
$498$	1.34707	0.0603638
$499$	30.3203	1.35732	0.678660	−	0.734452i	$-0.262561\pi$
$499$	30.3203	1.35732	0.678660	+	0.734452i	$0.262561\pi$
$500$	−53.2249	−2.38029
$501$	1.34998	0.0603126
$502$	13.0459	0.582268
$503$	34.9028	1.55624	0.778119	−	0.628117i	$-0.216174\pi$
$503$	34.9028	1.55624	0.778119	+	0.628117i	$0.216174\pi$
$504$	−79.3889	−3.53626
$505$	−19.5330	−0.869208
$506$	−1.77811	−0.0790468
$507$	2.09436	0.0930136
$508$	−102.649	−4.55431
$509$	4.58934	0.203419	0.101710	−	0.994814i	$-0.467569\pi$
$509$	4.58934	0.203419	0.101710	+	0.994814i	$0.467569\pi$
$510$	−1.15794	−0.0512744
$511$	30.4175	1.34559
$512$	36.2445	1.60179
$513$	−0.623854	−0.0275438
$514$	−70.4934	−3.10933
$515$	−4.62117	−0.203633
$516$	0.231849	0.0102066
$517$	−0.577063	−0.0253792
$518$	62.5588	2.74868
$519$	−0.814058	−0.0357332
$520$	−66.9292	−2.93504
$521$	−28.9044	−1.26632	−0.633162	−	0.774019i	$-0.718243\pi$
$521$	−28.9044	−1.26632	−0.633162	+	0.774019i	$0.718243\pi$
$522$	14.8511	0.650013
$523$	16.7070	0.730547	0.365274	−	0.930900i	$-0.380975\pi$
$523$	16.7070	0.730547	0.365274	+	0.930900i	$0.380975\pi$
$524$	47.9885	2.09639
$525$	0.751782	0.0328105
$526$	−73.0120	−3.18347
$527$	37.0425	1.61360
$528$	0.598325	0.0260387
$529$	−22.4078	−0.974252
$530$	−40.4662	−1.75774
$531$	27.3662	1.18759
$532$	−26.8062	−1.16220
$533$	66.7569	2.89156
$534$	−1.38882	−0.0601003
$535$	−3.81264	−0.164835
$536$	10.3475	0.446944
$537$	−0.927459	−0.0400228
$538$	30.4405	1.31238
$539$	4.35803	0.187714
$540$	−2.53775	−0.109207
$541$	18.2844	0.786108	0.393054	−	0.919515i	$-0.371419\pi$
$541$	18.2844	0.786108	0.393054	+	0.919515i	$0.371419\pi$
$542$	−39.2675	−1.68668
$543$	1.13212	0.0485841
$544$	−60.4083	−2.58998
$545$	5.38697	0.230752
$546$	4.01284	0.171734
$547$	0.363248	0.0155314	0.00776569	−	0.999970i	$-0.497528\pi$
$547$	0.363248	0.0155314	0.00776569	+	0.999970i	$0.497528\pi$
$548$	71.7102	3.06331
$549$	−28.0823	−1.19852
$550$	7.61418	0.324670
$551$	2.97377	0.126687
$552$	−0.389029	−0.0165582
$553$	42.9889	1.82807
$554$	−63.6309	−2.70342
$555$	0.592523	0.0251512
$556$	58.5257	2.48204
$557$	41.9754	1.77856	0.889278	−	0.457367i	$-0.151207\pi$
$557$	41.9754	1.77856	0.889278	+	0.457367i	$0.151207\pi$
$558$	57.0694	2.41594
$559$	4.78407	0.202345
$560$	−46.6051	−1.96942
$561$	0.296374	0.0125129
$562$	3.16890	0.133672
$563$	−41.1543	−1.73445	−0.867223	−	0.497919i	$-0.834098\pi$
$563$	−41.1543	−1.73445	−0.867223	+	0.497919i	$0.834098\pi$
$564$	−0.212898	−0.00896464
$565$	−1.30562	−0.0549279
$566$	−54.3293	−2.28363
$567$	−30.9890	−1.30141
$568$	−7.66322	−0.321542
$569$	−23.4390	−0.982613	−0.491306	−	0.870987i	$-0.663480\pi$
$569$	−23.4390	−0.982613	−0.491306	+	0.870987i	$0.663480\pi$
$570$	−0.357222	−0.0149624
$571$	−37.7352	−1.57917	−0.789584	−	0.613643i	$-0.789703\pi$
$571$	−37.7352	−1.57917	−0.789584	+	0.613643i	$0.789703\pi$
$572$	28.8866	1.20781
$573$	−0.485568	−0.0202849
$574$	90.7489	3.78778
$575$	−2.53595	−0.105756
$576$	−31.2265	−1.30110
$577$	−9.26295	−0.385622	−0.192811	−	0.981236i	$-0.561760\pi$
$577$	−9.26295	−0.385622	−0.192811	+	0.981236i	$0.561760\pi$
$578$	−24.0367	−0.999795
$579$	−0.987907	−0.0410560
$580$	12.0968	0.502294
$581$	26.8561	1.11418
$582$	−0.991625	−0.0411042
$583$	10.3573	0.428956
$584$	67.4026	2.78914
$585$	−26.1635	−1.08173
$586$	−61.5753	−2.54365
$587$	−16.1623	−0.667088	−0.333544	−	0.942735i	$-0.608245\pi$
$587$	−16.1623	−0.667088	−0.333544	+	0.942735i	$0.608245\pi$
$588$	1.60783	0.0663056
$589$	11.4275	0.470864
$590$	31.3628	1.29119
$591$	−0.743488	−0.0305830
$592$	71.0092	2.91846
$593$	31.6043	1.29783	0.648917	−	0.760859i	$-0.275222\pi$
$593$	31.6043	1.29783	0.648917	+	0.760859i	$0.275222\pi$
$594$	0.913880	0.0374969
$595$	−23.0854	−0.946408
$596$	−7.76435	−0.318040
$597$	1.29180	0.0528699
$598$	−13.5363	−0.553542
$599$	−21.9322	−0.896126	−0.448063	−	0.894002i	$-0.647886\pi$
$599$	−21.9322	−0.896126	−0.448063	+	0.894002i	$0.647886\pi$
$600$	1.66589	0.0680095
$601$	−11.3106	−0.461371	−0.230685	−	0.973028i	$-0.574097\pi$
$601$	−11.3106	−0.461371	−0.230685	+	0.973028i	$0.574097\pi$
$602$	6.50343	0.265060
$603$	4.04496	0.164724
$604$	−40.1387	−1.63322
$605$	−13.3537	−0.542906
$606$	2.59513	0.105420
$607$	1.32011	0.0535817	0.0267909	−	0.999641i	$-0.491471\pi$
$607$	1.32011	0.0535817	0.0267909	+	0.999641i	$0.491471\pi$
$608$	−18.6359	−0.755784
$609$	−0.430112	−0.0174290
$610$	−32.1835	−1.30307
$611$	−4.39303	−0.177723
$612$	−75.2690	−3.04257
$613$	13.5415	0.546935	0.273468	−	0.961881i	$-0.411829\pi$
$613$	13.5415	0.546935	0.273468	+	0.961881i	$0.411829\pi$
$614$	11.8955	0.480062
$615$	0.859524	0.0346593
$616$	23.2871	0.938264
$617$	−17.1276	−0.689532	−0.344766	−	0.938689i	$-0.612042\pi$
$617$	−17.1276	−0.689532	−0.344766	+	0.938689i	$0.612042\pi$
$618$	0.613962	0.0246972
$619$	8.44209	0.339316	0.169658	−	0.985503i	$-0.445734\pi$
$619$	8.44209	0.339316	0.169658	+	0.985503i	$0.445734\pi$
$620$	46.4855	1.86690
$621$	−0.304374	−0.0122141
$622$	−69.2274	−2.77577
$623$	−27.6884	−1.10931
$624$	4.55489	0.182342
$625$	2.33613	0.0934453
$626$	−86.8466	−3.47109
$627$	0.0914309	0.00365140
$628$	47.1117	1.87996
$629$	35.1737	1.40247
$630$	−35.5664	−1.41700
$631$	1.12696	0.0448637	0.0224318	−	0.999748i	$-0.492859\pi$
$631$	1.12696	0.0448637	0.0224318	+	0.999748i	$0.492859\pi$
$632$	95.2598	3.78923
$633$	−1.21155	−0.0481548
$634$	−41.5214	−1.64903
$635$	−27.2715	−1.08223
$636$	3.82117	0.151519
$637$	33.1766	1.31450
$638$	−4.35625	−0.172466
$639$	−2.99565	−0.118506
$640$	−4.93501	−0.195074
$641$	33.0692	1.30615	0.653077	−	0.757291i	$-0.273478\pi$
$641$	33.0692	1.30615	0.653077	+	0.757291i	$0.273478\pi$
$642$	0.506542	0.0199916
$643$	4.14698	0.163541	0.0817705	−	0.996651i	$-0.473943\pi$
$643$	4.14698	0.163541	0.0817705	+	0.996651i	$0.473943\pi$
$644$	−13.0786	−0.515367
$645$	0.0615969	0.00242538
$646$	−21.2057	−0.834326
$647$	33.0265	1.29840	0.649202	−	0.760616i	$-0.275103\pi$
$647$	33.0265	1.29840	0.649202	+	0.760616i	$0.275103\pi$
$648$	−68.6690	−2.69757
$649$	−8.02730	−0.315099
$650$	57.9648	2.27357
$651$	−1.65283	−0.0647794
$652$	−41.7605	−1.63547
$653$	−14.0822	−0.551080	−0.275540	−	0.961290i	$-0.588857\pi$
$653$	−14.0822	−0.551080	−0.275540	+	0.961290i	$0.588857\pi$
$654$	−0.715705	−0.0279863
$655$	12.7494	0.498162
$656$	103.007	4.02175
$657$	26.3485	1.02795
$658$	−5.97186	−0.232807
$659$	31.7444	1.23658	0.618292	−	0.785948i	$-0.287825\pi$
$659$	31.7444	1.23658	0.618292	+	0.785948i	$0.287825\pi$
$660$	0.371928	0.0144773
$661$	43.4108	1.68848	0.844242	−	0.535962i	$-0.180051\pi$
$661$	43.4108	1.68848	0.844242	+	0.535962i	$0.180051\pi$
$662$	7.16356	0.278420
$663$	2.25622	0.0876244
$664$	59.5109	2.30947
$665$	−7.12180	−0.276172
$666$	54.1903	2.09983
$667$	1.45088	0.0561782
$668$	100.568	3.89108
$669$	−1.57995	−0.0610844
$670$	4.63570	0.179093
$671$	8.23737	0.318000
$672$	2.69541	0.103978
$673$	−16.1016	−0.620671	−0.310335	−	0.950627i	$-0.600441\pi$
$673$	−16.1016	−0.620671	−0.310335	+	0.950627i	$0.600441\pi$
$674$	−13.8162	−0.532179
$675$	1.30338	0.0501670
$676$	156.020	6.00079
$677$	5.37049	0.206405	0.103202	−	0.994660i	$-0.467091\pi$
$677$	5.37049	0.206405	0.103202	+	0.994660i	$0.467091\pi$
$678$	0.173463	0.00666180
$679$	−19.7696	−0.758689
$680$	−51.1553	−1.96172
$681$	0.246353	0.00944027
$682$	−16.7401	−0.641012
$683$	8.67725	0.332026	0.166013	−	0.986124i	$-0.446911\pi$
$683$	8.67725	0.332026	0.166013	+	0.986124i	$0.446911\pi$
$684$	−23.2204	−0.887853
$685$	19.0518	0.727930
$686$	−18.5547	−0.708422
$687$	−0.939082	−0.0358282
$688$	7.38191	0.281433
$689$	78.8476	3.00385
$690$	−0.174286	−0.00663495
$691$	−21.7086	−0.825835	−0.412917	−	0.910769i	$-0.635490\pi$
$691$	−21.7086	−0.825835	−0.412917	+	0.910769i	$0.635490\pi$
$692$	−60.6438	−2.30533
$693$	9.10321	0.345803
$694$	−68.0148	−2.58181
$695$	15.5489	0.589805
$696$	−0.953093	−0.0361269
$697$	51.0236	1.93266
$698$	78.8658	2.98511
$699$	0.530625	0.0200701
$700$	56.0045	2.11677
$701$	−20.7541	−0.783872	−0.391936	−	0.919992i	$-0.628195\pi$
$701$	−20.7541	−0.783872	−0.391936	+	0.919992i	$0.628195\pi$
$702$	6.95713	0.262580
$703$	10.8510	0.409254
$704$	9.15964	0.345217
$705$	−0.0565622	−0.00213026
$706$	13.6987	0.515557
$707$	51.7381	1.94581
$708$	−2.96155	−0.111302
$709$	−17.3376	−0.651129	−0.325564	−	0.945520i	$-0.605554\pi$
$709$	−17.3376	−0.651129	−0.325564	+	0.945520i	$0.605554\pi$
$710$	−3.43314	−0.128843
$711$	37.2383	1.39654
$712$	−61.3553	−2.29939
$713$	5.57541	0.208801
$714$	3.06709	0.114783
$715$	7.67451	0.287010
$716$	−69.0917	−2.58208
$717$	0.576679	0.0215365
$718$	49.3859	1.84307
$719$	−39.0416	−1.45601	−0.728003	−	0.685573i	$-0.759552\pi$
$719$	−39.0416	−1.45601	−0.728003	+	0.685573i	$0.759552\pi$
$720$	−40.3707	−1.50453
$721$	12.2403	0.455853
$722$	43.4188	1.61588
$723$	−0.921434	−0.0342685
$724$	84.3384	3.13441
$725$	−6.21290	−0.230741
$726$	1.77416	0.0658451
$727$	−20.5575	−0.762436	−0.381218	−	0.924485i	$-0.624495\pi$
$727$	−20.5575	−0.762436	−0.381218	+	0.924485i	$0.624495\pi$
$728$	177.279	6.57039
$729$	−26.7653	−0.991307
$730$	30.1965	1.11762
$731$	3.65656	0.135243
$732$	3.03905	0.112326
$733$	18.3980	0.679546	0.339773	−	0.940508i	$-0.389650\pi$
$733$	18.3980	0.679546	0.339773	+	0.940508i	$0.389650\pi$
$734$	32.2814	1.19153
$735$	0.427162	0.0157561
$736$	−9.09229	−0.335146
$737$	−1.18651	−0.0437055
$738$	78.6094	2.89365
$739$	−24.1084	−0.886841	−0.443421	−	0.896314i	$-0.646235\pi$
$739$	−24.1084	−0.886841	−0.443421	+	0.896314i	$0.646235\pi$
$740$	44.1404	1.62263
$741$	0.696040	0.0255697
$742$	107.185	3.93488
$743$	4.65022	0.170600	0.0853000	−	0.996355i	$-0.472815\pi$
$743$	4.65022	0.170600	0.0853000	+	0.996355i	$0.472815\pi$
$744$	−3.66253	−0.134275
$745$	−2.06281	−0.0755754
$746$	−19.2207	−0.703719
$747$	23.2635	0.851168
$748$	22.0786	0.807274
$749$	10.0987	0.369000
$750$	1.87871	0.0686006
$751$	5.56026	0.202897	0.101448	−	0.994841i	$-0.467652\pi$
$751$	5.56026	0.202897	0.101448	+	0.994841i	$0.467652\pi$
$752$	−6.77853	−0.247188
$753$	−0.327289	−0.0119271
$754$	−33.1630	−1.20773
$755$	−10.6639	−0.388101
$756$	6.72185	0.244471
$757$	−9.05627	−0.329156	−0.164578	−	0.986364i	$-0.552626\pi$
$757$	−9.05627	−0.329156	−0.164578	+	0.986364i	$0.552626\pi$
$758$	7.55549	0.274428
$759$	0.0446084	0.00161918
$760$	−15.7813	−0.572449
$761$	−5.74102	−0.208112	−0.104056	−	0.994571i	$-0.533182\pi$
$761$	−5.74102	−0.208112	−0.104056	+	0.994571i	$0.533182\pi$
$762$	3.62325	0.131256
$763$	−14.2687	−0.516562
$764$	−36.1727	−1.30868
$765$	−19.9972	−0.723002
$766$	−54.2507	−1.96016
$767$	−61.1098	−2.20655
$768$	−0.719631	−0.0259674
$769$	−53.7969	−1.93996	−0.969982	−	0.243177i	$-0.921811\pi$
$769$	−53.7969	−1.93996	−0.969982	+	0.243177i	$0.921811\pi$
$770$	10.4327	0.375967
$771$	1.76850	0.0636911
$772$	−73.5948	−2.64873
$773$	31.5138	1.13347	0.566737	−	0.823899i	$-0.308206\pi$
$773$	31.5138	1.13347	0.566737	+	0.823899i	$0.308206\pi$
$774$	5.63346	0.202491
$775$	−23.8748	−0.857608
$776$	−43.8079	−1.57261
$777$	−1.56944	−0.0563035
$778$	1.49258	0.0535114
$779$	15.7407	0.563969
$780$	2.83139	0.101380
$781$	0.878711	0.0314428
$782$	−10.3461	−0.369975
$783$	−0.745693	−0.0266489
$784$	51.1921	1.82829
$785$	12.5165	0.446733
$786$	−1.69387	−0.0604184
$787$	−13.5063	−0.481446	−0.240723	−	0.970594i	$-0.577385\pi$
$787$	−13.5063	−0.481446	−0.240723	+	0.970594i	$0.577385\pi$
$788$	−55.3866	−1.97307
$789$	1.83169	0.0652098
$790$	42.6766	1.51837
$791$	3.45826	0.122962
$792$	20.1720	0.716780
$793$	62.7090	2.22686
$794$	59.6251	2.11601
$795$	1.01520	0.0360053
$796$	96.2337	3.41091
$797$	14.4748	0.512722	0.256361	−	0.966581i	$-0.417476\pi$
$797$	14.4748	0.512722	0.256361	+	0.966581i	$0.417476\pi$
$798$	0.946192	0.0334948
$799$	−3.35768	−0.118786
$800$	38.9347	1.37655
$801$	−23.9845	−0.847452
$802$	55.8962	1.97376
$803$	−7.72879	−0.272743
$804$	−0.437743	−0.0154380
$805$	−3.47467	−0.122466
$806$	−127.438	−4.48882
$807$	−0.763675	−0.0268826
$808$	114.647	4.03328
$809$	19.2892	0.678173	0.339086	−	0.940755i	$-0.389882\pi$
$809$	19.2892	0.678173	0.339086	+	0.940755i	$0.389882\pi$
$810$	−30.7639	−1.08093
$811$	−46.9589	−1.64895	−0.824475	−	0.565898i	$-0.808530\pi$
$811$	−46.9589	−1.64895	−0.824475	+	0.565898i	$0.808530\pi$
$812$	−32.0415	−1.12444
$813$	0.985123	0.0345498
$814$	−15.8956	−0.557140
$815$	−11.0948	−0.388634
$816$	3.48139	0.121873
$817$	1.12804	0.0394652
$818$	6.03956	0.211168
$819$	69.3005	2.42155
$820$	64.0308	2.23605
$821$	35.1040	1.22514	0.612569	−	0.790417i	$-0.290136\pi$
$821$	35.1040	1.22514	0.612569	+	0.790417i	$0.290136\pi$
$822$	−2.53119	−0.0882854
$823$	44.2758	1.54336	0.771679	−	0.636012i	$-0.219417\pi$
$823$	44.2758	1.54336	0.771679	+	0.636012i	$0.219417\pi$
$824$	27.1235	0.944893
$825$	−0.191021	−0.00665049
$826$	−83.0722	−2.89045
$827$	20.1241	0.699784	0.349892	−	0.936790i	$-0.386218\pi$
$827$	20.1241	0.699784	0.349892	+	0.936790i	$0.386218\pi$
$828$	−11.3290	−0.393711
$829$	−29.3781	−1.02034	−0.510172	−	0.860072i	$-0.670418\pi$
$829$	−29.3781	−1.02034	−0.510172	+	0.860072i	$0.670418\pi$
$830$	26.6610	0.925417
$831$	1.59634	0.0553764
$832$	69.7300	2.41745
$833$	25.3575	0.878585
$834$	−2.06581	−0.0715331
$835$	26.7185	0.924632
$836$	6.81121	0.235571
$837$	−2.86554	−0.0990474
$838$	−18.0288	−0.622795
$839$	35.6940	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$839$	35.6940	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$840$	2.28254	0.0787551
$841$	−25.4455	−0.877430
$842$	88.9225	3.06447
$843$	−0.0794998	−0.00273812
$844$	−90.2552	−3.10671
$845$	41.4510	1.42596
$846$	−5.17300	−0.177851
$847$	35.3706	1.21535
$848$	121.663	4.17793
$849$	1.36299	0.0467776
$850$	44.3036	1.51960
$851$	5.29413	0.181480
$852$	0.324187	0.0111065
$853$	9.89194	0.338694	0.169347	−	0.985557i	$-0.445834\pi$
$853$	9.89194	0.338694	0.169347	+	0.985557i	$0.445834\pi$
$854$	85.2461	2.91706
$855$	−6.16911	−0.210979
$856$	22.3780	0.764863
$857$	41.8572	1.42981	0.714907	−	0.699219i	$-0.246469\pi$
$857$	41.8572	1.42981	0.714907	+	0.699219i	$0.246469\pi$
$858$	−1.01962	−0.0348094
$859$	−34.1834	−1.16632	−0.583161	−	0.812357i	$-0.698184\pi$
$859$	−34.1834	−1.16632	−0.583161	+	0.812357i	$0.698184\pi$
$860$	4.58870	0.156474
$861$	−2.27666	−0.0775884
$862$	−55.8684	−1.90289
$863$	16.2831	0.554282	0.277141	−	0.960829i	$-0.410613\pi$
$863$	16.2831	0.554282	0.277141	+	0.960829i	$0.410613\pi$
$864$	4.67307	0.158981
$865$	−16.1117	−0.547813
$866$	−13.7473	−0.467153
$867$	0.603021	0.0204797
$868$	−123.129	−4.17925
$869$	−10.9231	−0.370540
$870$	−0.426988	−0.0144762
$871$	−9.03257	−0.306057
$872$	−31.6183	−1.07073
$873$	−17.1250	−0.579595
$874$	−3.19175	−0.107962
$875$	37.4550	1.26621
$876$	−2.85141	−0.0963404
$877$	22.8163	0.770451	0.385226	−	0.922822i	$-0.374124\pi$
$877$	22.8163	0.770451	0.385226	+	0.922822i	$0.374124\pi$
$878$	−9.51555	−0.321134
$879$	1.54477	0.0521038
$880$	11.8419	0.399191
$881$	10.3085	0.347301	0.173650	−	0.984807i	$-0.444444\pi$
$881$	10.3085	0.347301	0.173650	+	0.984807i	$0.444444\pi$
$882$	39.0669	1.31545
$883$	−51.2114	−1.72340	−0.861700	−	0.507419i	$-0.830600\pi$
$883$	−51.2114	−1.72340	−0.861700	+	0.507419i	$0.830600\pi$
$884$	168.079	5.65310
$885$	−0.786815	−0.0264485
$886$	61.2052	2.05623
$887$	−35.5100	−1.19231	−0.596155	−	0.802870i	$-0.703306\pi$
$887$	−35.5100	−1.19231	−0.596155	+	0.802870i	$0.703306\pi$
$888$	−3.47776	−0.116706
$889$	72.2352	2.42269
$890$	−27.4873	−0.921377
$891$	7.87400	0.263789
$892$	−117.700	−3.94087
$893$	−1.03584	−0.0346630
$894$	0.274062	0.00916599
$895$	−18.3561	−0.613576
$896$	13.0716	0.436692
$897$	0.339593	0.0113387
$898$	84.6099	2.82347
$899$	13.6593	0.455564
$900$	48.5128	1.61709
$901$	60.2647	2.00771
$902$	−23.0584	−0.767762
$903$	−0.163155	−0.00542945
$904$	7.66322	0.254875
$905$	22.4068	0.744826
$906$	1.41680	0.0470699
$907$	−12.9870	−0.431226	−0.215613	−	0.976479i	$-0.569175\pi$
$907$	−12.9870	−0.431226	−0.215613	+	0.976479i	$0.569175\pi$
$908$	18.3522	0.609041
$909$	44.8171	1.48649
$910$	79.4213	2.63279
$911$	−24.8421	−0.823056	−0.411528	−	0.911397i	$-0.635005\pi$
$911$	−24.8421	−0.823056	−0.411528	+	0.911397i	$0.635005\pi$
$912$	1.07400	0.0355638
$913$	−6.82387	−0.225837
$914$	31.7559	1.05039
$915$	0.807405	0.0266920
$916$	−69.9576	−2.31146
$917$	−33.7701	−1.11519
$918$	5.31747	0.175503
$919$	29.2503	0.964877	0.482439	−	0.875930i	$-0.339751\pi$
$919$	29.2503	0.964877	0.482439	+	0.875930i	$0.339751\pi$
$920$	−7.69958	−0.253848
$921$	−0.298427	−0.00983352
$922$	−57.7452	−1.90174
$923$	6.68941	0.220184
$924$	−0.985143	−0.0324088
$925$	−22.6703	−0.745396
$926$	−19.6909	−0.647085
$927$	10.6029	0.348246
$928$	−22.2754	−0.731227
$929$	4.12712	0.135406	0.0677032	−	0.997706i	$-0.478433\pi$
$929$	4.12712	0.135406	0.0677032	+	0.997706i	$0.478433\pi$
$930$	−1.64082	−0.0538047
$931$	7.82274	0.256380
$932$	39.5293	1.29482
$933$	1.73674	0.0568584
$934$	−20.4338	−0.668615
$935$	5.86578	0.191831
$936$	153.564	5.01940
$937$	−13.1621	−0.429986	−0.214993	−	0.976616i	$-0.568973\pi$
$937$	−13.1621	−0.429986	−0.214993	+	0.976616i	$0.568973\pi$
$938$	−12.2788	−0.400917
$939$	2.17876	0.0711012
$940$	−4.21364	−0.137434
$941$	54.7027	1.78326	0.891628	−	0.452769i	$-0.149564\pi$
$941$	54.7027	1.78326	0.891628	+	0.452769i	$0.149564\pi$
$942$	−1.66292	−0.0541809
$943$	7.67976	0.250087
$944$	−94.2936	−3.06899
$945$	1.78584	0.0580934
$946$	−1.65246	−0.0537261
$947$	−10.4149	−0.338439	−0.169219	−	0.985578i	$-0.554125\pi$
$947$	−10.4149	−0.338439	−0.169219	+	0.985578i	$0.554125\pi$
$948$	−4.02989	−0.130885
$949$	−58.8373	−1.90994
$950$	13.6676	0.443435
$951$	1.04167	0.0337784
$952$	135.498	4.39150
$953$	−2.81965	−0.0913375	−0.0456687	−	0.998957i	$-0.514542\pi$
$953$	−2.81965	−0.0913375	−0.0456687	+	0.998957i	$0.514542\pi$
$954$	92.8467	3.00602
$955$	−9.61026	−0.310981
$956$	42.9601	1.38943
$957$	0.109287	0.00353276
$958$	−36.0828	−1.16578
$959$	−50.4633	−1.62955
$960$	0.897803	0.0289765
$961$	21.4899	0.693222
$962$	−121.009	−3.90149
$963$	8.74782	0.281895
$964$	−68.6429	−2.21084
$965$	−19.5524	−0.629415
$966$	0.461640	0.0148530
$967$	−27.6818	−0.890185	−0.445093	−	0.895485i	$-0.646829\pi$
$967$	−27.6818	−0.890185	−0.445093	+	0.895485i	$0.646829\pi$
$968$	78.3784	2.51918
$969$	0.531997	0.0170902
$970$	−19.6260	−0.630154
$971$	−54.3199	−1.74321	−0.871604	−	0.490211i	$-0.836920\pi$
$971$	−54.3199	−1.74321	−0.871604	+	0.490211i	$0.836920\pi$
$972$	8.73612	0.280211
$973$	−41.1852	−1.32034
$974$	−59.9466	−1.92081
$975$	−1.45419	−0.0465714
$976$	96.7611	3.09725
$977$	−27.9432	−0.893982	−0.446991	−	0.894538i	$-0.647504\pi$
$977$	−27.9432	−0.893982	−0.446991	+	0.894538i	$0.647504\pi$
$978$	1.47404	0.0471346
$979$	7.03537	0.224851
$980$	31.8218	1.01651
$981$	−12.3600	−0.394624
$982$	−71.2317	−2.27310
$983$	−12.5545	−0.400427	−0.200214	−	0.979752i	$-0.564164\pi$
$983$	−12.5545	−0.400427	−0.200214	+	0.979752i	$0.564164\pi$
$984$	−5.04490	−0.160825
$985$	−14.7149	−0.468857
$986$	−25.3471	−0.807217
$987$	0.149819	0.00476879
$988$	51.8520	1.64963
$989$	0.550362	0.0175005
$990$	9.03709	0.287218
$991$	−15.5705	−0.494612	−0.247306	−	0.968937i	$-0.579545\pi$
$991$	−15.5705	−0.494612	−0.247306	+	0.968937i	$0.579545\pi$
$992$	−85.5997	−2.71779
$993$	−0.179716	−0.00570311
$994$	9.09353	0.288429
$995$	25.5671	0.810531
$996$	−2.51756	−0.0797720
$997$	25.4794	0.806940	0.403470	−	0.914993i	$-0.367804\pi$
$997$	25.4794	0.806940	0.403470	+	0.914993i	$0.367804\pi$
$998$	−79.7274	−2.52373
$999$	−2.72097	−0.0860877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8023.2.a.d.1.5	✓	165

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8023.2.a.d.1.5	✓	165	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:	\( N \)	\(=\)	\( 8023 = 71 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62951 q^{2} +0.0659678 q^{3} +4.91432 q^{4} +1.30562 q^{5} -0.173463 q^{6} -3.45826 q^{7} -7.66322 q^{8} -2.99565 q^{9} +O(q^{10})\)	\(q-2.62951 q^{2} +0.0659678 q^{3} +4.91432 q^{4} +1.30562 q^{5} -0.173463 q^{6} -3.45826 q^{7} -7.66322 q^{8} -2.99565 q^{9} -3.43314 q^{10} +0.878711 q^{11} +0.324187 q^{12} +6.68941 q^{13} +9.09353 q^{14} +0.0861289 q^{15} +10.3219 q^{16} +5.11284 q^{17} +7.87708 q^{18} +1.57730 q^{19} +6.41623 q^{20} -0.228134 q^{21} -2.31058 q^{22} +0.769553 q^{23} -0.505526 q^{24} -3.29535 q^{25} -17.5899 q^{26} -0.395520 q^{27} -16.9950 q^{28} +1.88535 q^{29} -0.226477 q^{30} +7.24499 q^{31} -11.8150 q^{32} +0.0579666 q^{33} -13.4443 q^{34} -4.51518 q^{35} -14.7216 q^{36} +6.87948 q^{37} -4.14753 q^{38} +0.441285 q^{39} -10.0053 q^{40} +9.97950 q^{41} +0.599880 q^{42} +0.715171 q^{43} +4.31827 q^{44} -3.91118 q^{45} -2.02355 q^{46} -0.656715 q^{47} +0.680911 q^{48} +4.95957 q^{49} +8.66516 q^{50} +0.337283 q^{51} +32.8739 q^{52} +11.7869 q^{53} +1.04002 q^{54} +1.14726 q^{55} +26.5014 q^{56} +0.104051 q^{57} -4.95754 q^{58} -9.13531 q^{59} +0.423265 q^{60} +9.37437 q^{61} -19.0508 q^{62} +10.3597 q^{63} +10.4239 q^{64} +8.73383 q^{65} -0.152424 q^{66} -1.35028 q^{67} +25.1261 q^{68} +0.0507657 q^{69} +11.8727 q^{70} +1.00000 q^{71} +22.9563 q^{72} -8.79559 q^{73} -18.0897 q^{74} -0.217387 q^{75} +7.75136 q^{76} -3.03881 q^{77} -1.16036 q^{78} -12.4308 q^{79} +13.4765 q^{80} +8.96085 q^{81} -26.2412 q^{82} -7.76578 q^{83} -1.12112 q^{84} +6.67543 q^{85} -1.88055 q^{86} +0.124372 q^{87} -6.73376 q^{88} +8.00646 q^{89} +10.2845 q^{90} -23.1337 q^{91} +3.78183 q^{92} +0.477936 q^{93} +1.72684 q^{94} +2.05936 q^{95} -0.779411 q^{96} +5.71664 q^{97} -13.0412 q^{98} -2.63231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 165 q + 22 q^{2} + 18 q^{3} + 166 q^{4} + 28 q^{5} + 16 q^{6} + 24 q^{7} + 66 q^{8} + 177 q^{9}+O(q^{10}) \) 165 * q + 22 * q^2 + 18 * q^3 + 166 * q^4 + 28 * q^5 + 16 * q^6 + 24 * q^7 + 66 * q^8 + 177 * q^9	\( 165 q + 22 q^{2} + 18 q^{3} + 166 q^{4} + 28 q^{5} + 16 q^{6} + 24 q^{7} + 66 q^{8} + 177 q^{9} + 14 q^{10} + 18 q^{11} + 54 q^{12} + 44 q^{13} + 26 q^{14} + 24 q^{15} + 168 q^{16} + 143 q^{17} + 57 q^{18} + 20 q^{19} + 49 q^{20} + 39 q^{21} + 25 q^{22} + 52 q^{23} + 27 q^{24} + 175 q^{25} + 48 q^{26} + 69 q^{27} + 28 q^{28} + 58 q^{29} - 11 q^{30} + 28 q^{31} + 114 q^{32} + 110 q^{33} + 55 q^{34} + 67 q^{35} + 202 q^{36} + 44 q^{37} + 35 q^{38} + 27 q^{39} + 53 q^{40} + 141 q^{41} + 40 q^{42} + 29 q^{43} + 52 q^{44} + 54 q^{45} + 29 q^{46} + 87 q^{47} + 53 q^{48} + 143 q^{49} + 16 q^{50} + 37 q^{51} + 105 q^{52} + 101 q^{53} - 36 q^{54} + 72 q^{55} + 57 q^{56} + 82 q^{57} + 4 q^{58} + 103 q^{59} + 53 q^{60} + 16 q^{61} + 54 q^{62} + 126 q^{63} + 136 q^{64} + 159 q^{65} + 53 q^{66} + 60 q^{67} + 220 q^{68} + 81 q^{69} + 16 q^{70} + 165 q^{71} + 176 q^{72} + 124 q^{73} + 29 q^{74} + 44 q^{75} + 18 q^{76} + 127 q^{77} - 91 q^{78} + 14 q^{79} + 158 q^{80} + 213 q^{81} + 20 q^{82} + 116 q^{83} + 67 q^{84} + 59 q^{85} + 30 q^{86} + 28 q^{87} + 79 q^{88} + 195 q^{89} + 16 q^{90} - 26 q^{91} + 173 q^{92} + 116 q^{93} + 53 q^{94} + 26 q^{95} - 36 q^{96} + 88 q^{97} + 150 q^{98} + 12 q^{99}+O(q^{100}) \) 165 * q + 22 * q^2 + 18 * q^3 + 166 * q^4 + 28 * q^5 + 16 * q^6 + 24 * q^7 + 66 * q^8 + 177 * q^9 + 14 * q^10 + 18 * q^11 + 54 * q^12 + 44 * q^13 + 26 * q^14 + 24 * q^15 + 168 * q^16 + 143 * q^17 + 57 * q^18 + 20 * q^19 + 49 * q^20 + 39 * q^21 + 25 * q^22 + 52 * q^23 + 27 * q^24 + 175 * q^25 + 48 * q^26 + 69 * q^27 + 28 * q^28 + 58 * q^29 - 11 * q^30 + 28 * q^31 + 114 * q^32 + 110 * q^33 + 55 * q^34 + 67 * q^35 + 202 * q^36 + 44 * q^37 + 35 * q^38 + 27 * q^39 + 53 * q^40 + 141 * q^41 + 40 * q^42 + 29 * q^43 + 52 * q^44 + 54 * q^45 + 29 * q^46 + 87 * q^47 + 53 * q^48 + 143 * q^49 + 16 * q^50 + 37 * q^51 + 105 * q^52 + 101 * q^53 - 36 * q^54 + 72 * q^55 + 57 * q^56 + 82 * q^57 + 4 * q^58 + 103 * q^59 + 53 * q^60 + 16 * q^61 + 54 * q^62 + 126 * q^63 + 136 * q^64 + 159 * q^65 + 53 * q^66 + 60 * q^67 + 220 * q^68 + 81 * q^69 + 16 * q^70 + 165 * q^71 + 176 * q^72 + 124 * q^73 + 29 * q^74 + 44 * q^75 + 18 * q^76 + 127 * q^77 - 91 * q^78 + 14 * q^79 + 158 * q^80 + 213 * q^81 + 20 * q^82 + 116 * q^83 + 67 * q^84 + 59 * q^85 + 30 * q^86 + 28 * q^87 + 79 * q^88 + 195 * q^89 + 16 * q^90 - 26 * q^91 + 173 * q^92 + 116 * q^93 + 53 * q^94 + 26 * q^95 - 36 * q^96 + 88 * q^97 + 150 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more