LMFDB - Embedded newform 1849.4.a.j.1.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.702850 q^{2} -4.82523 q^{3} -7.50600 q^{4} -8.51327 q^{5} -3.39141 q^{6} -25.5957 q^{7} -10.8984 q^{8} -3.71712 q^{9} +O(q^{10})$	\(q+0.702850 q^{2} -4.82523 q^{3} -7.50600 q^{4} -8.51327 q^{5} -3.39141 q^{6} -25.5957 q^{7} -10.8984 q^{8} -3.71712 q^{9} -5.98355 q^{10} -28.9564 q^{11} +36.2182 q^{12} -77.8657 q^{13} -17.9899 q^{14} +41.0785 q^{15} +52.3881 q^{16} +87.6620 q^{17} -2.61258 q^{18} -15.5698 q^{19} +63.9006 q^{20} +123.505 q^{21} -20.3520 q^{22} -9.68121 q^{23} +52.5873 q^{24} -52.5243 q^{25} -54.7279 q^{26} +148.217 q^{27} +192.121 q^{28} +189.097 q^{29} +28.8720 q^{30} +260.919 q^{31} +124.008 q^{32} +139.721 q^{33} +61.6132 q^{34} +217.903 q^{35} +27.9007 q^{36} -57.5937 q^{37} -10.9432 q^{38} +375.720 q^{39} +92.7809 q^{40} -296.291 q^{41} +86.8055 q^{42} +217.347 q^{44} +31.6448 q^{45} -6.80444 q^{46} -360.808 q^{47} -252.785 q^{48} +312.138 q^{49} -36.9167 q^{50} -422.990 q^{51} +584.460 q^{52} +352.316 q^{53} +104.174 q^{54} +246.514 q^{55} +278.952 q^{56} +75.1279 q^{57} +132.907 q^{58} -711.248 q^{59} -308.335 q^{60} -456.857 q^{61} +183.387 q^{62} +95.1422 q^{63} -331.946 q^{64} +662.891 q^{65} +98.2032 q^{66} -488.166 q^{67} -657.991 q^{68} +46.7141 q^{69} +153.153 q^{70} +924.781 q^{71} +40.5106 q^{72} +708.211 q^{73} -40.4797 q^{74} +253.442 q^{75} +116.867 q^{76} +741.159 q^{77} +264.075 q^{78} +191.956 q^{79} -445.994 q^{80} -614.821 q^{81} -208.248 q^{82} +962.592 q^{83} -927.030 q^{84} -746.290 q^{85} -912.436 q^{87} +315.578 q^{88} -158.267 q^{89} +22.2416 q^{90} +1993.02 q^{91} +72.6672 q^{92} -1259.00 q^{93} -253.594 q^{94} +132.550 q^{95} -598.368 q^{96} -1264.08 q^{97} +219.386 q^{98} +107.635 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * 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 $

$2$	0.702850	0.248495	0.124247	−	0.992251i	$-0.460348\pi$
$2$	0.702850	0.248495	0.124247	+	0.992251i	$0.460348\pi$
$3$	−4.82523	−0.928617	−0.464308	−	0.885674i	$-0.653697\pi$
$3$	−4.82523	−0.928617	−0.464308	+	0.885674i	$0.653697\pi$
$4$	−7.50600	−0.938250
$5$	−8.51327	−0.761450	−0.380725	−	0.924688i	$-0.624325\pi$
$5$	−8.51327	−0.761450	−0.380725	+	0.924688i	$0.624325\pi$
$6$	−3.39141	−0.230757
$7$	−25.5957	−1.38204	−0.691018	−	0.722837i	$-0.742838\pi$
$7$	−25.5957	−1.38204	−0.691018	+	0.722837i	$0.742838\pi$
$8$	−10.8984	−0.481645
$9$	−3.71712	−0.137671
$10$	−5.98355	−0.189216
$11$	−28.9564	−0.793699	−0.396850	−	0.917884i	$-0.629897\pi$
$11$	−28.9564	−0.793699	−0.396850	+	0.917884i	$0.629897\pi$
$12$	36.2182	0.871275
$13$	−77.8657	−1.66124	−0.830618	−	0.556843i	$-0.812012\pi$
$13$	−77.8657	−1.66124	−0.830618	+	0.556843i	$0.812012\pi$
$14$	−17.9899	−0.343429
$15$	41.0785	0.707095
$16$	52.3881	0.818564
$17$	87.6620	1.25066	0.625328	−	0.780362i	$-0.284965\pi$
$17$	87.6620	1.25066	0.625328	+	0.780362i	$0.284965\pi$
$18$	−2.61258	−0.0342106
$19$	−15.5698	−0.187998	−0.0939989	−	0.995572i	$-0.529965\pi$
$19$	−15.5698	−0.187998	−0.0939989	+	0.995572i	$0.529965\pi$
$20$	63.9006	0.714430
$21$	123.505	1.28338
$22$	−20.3520	−0.197230
$23$	−9.68121	−0.0877684	−0.0438842	−	0.999037i	$-0.513973\pi$
$23$	−9.68121	−0.0877684	−0.0438842	+	0.999037i	$0.513973\pi$
$24$	52.5873	0.447264
$25$	−52.5243	−0.420194
$26$	−54.7279	−0.412809
$27$	148.217	1.05646
$28$	192.121	1.29670
$29$	189.097	1.21084	0.605421	−	0.795906i	$-0.293005\pi$
$29$	189.097	1.21084	0.605421	+	0.795906i	$0.293005\pi$
$30$	28.8720	0.175709
$31$	260.919	1.51169	0.755847	−	0.654748i	$-0.227225\pi$
$31$	260.919	1.51169	0.755847	+	0.654748i	$0.227225\pi$
$32$	124.008	0.685054
$33$	139.721	0.737042
$34$	61.6132	0.310782
$35$	217.903	1.05235
$36$	27.9007	0.129170
$37$	−57.5937	−0.255901	−0.127951	−	0.991781i	$-0.540840\pi$
$37$	−57.5937	−0.255901	−0.127951	+	0.991781i	$0.540840\pi$
$38$	−10.9432	−0.0467165
$39$	375.720	1.54265
$40$	92.7809	0.366749
$41$	−296.291	−1.12861	−0.564303	−	0.825568i	$-0.690855\pi$
$41$	−296.291	−1.12861	−0.564303	+	0.825568i	$0.690855\pi$
$42$	86.8055	0.318914
$43$	0	0
$43$	0	0
$44$	217.347	0.744688
$45$	31.6448	0.104830
$46$	−6.80444	−0.0218100
$47$	−360.808	−1.11977	−0.559886	−	0.828570i	$-0.689155\pi$
$47$	−360.808	−1.11977	−0.559886	+	0.828570i	$0.689155\pi$
$48$	−252.785	−0.760132
$49$	312.138	0.910025
$50$	−36.9167	−0.104416
$51$	−422.990	−1.16138
$52$	584.460	1.55865
$53$	352.316	0.913101	0.456551	−	0.889697i	$-0.349085\pi$
$53$	352.316	0.913101	0.456551	+	0.889697i	$0.349085\pi$
$54$	104.174	0.262525
$55$	246.514	0.604362
$56$	278.952	0.665651
$57$	75.1279	0.174578
$58$	132.907	0.300888
$59$	−711.248	−1.56943	−0.784717	−	0.619854i	$-0.787192\pi$
$59$	−711.248	−1.56943	−0.784717	+	0.619854i	$0.787192\pi$
$60$	−308.335	−0.663432
$61$	−456.857	−0.958926	−0.479463	−	0.877562i	$-0.659169\pi$
$61$	−456.857	−0.958926	−0.479463	+	0.877562i	$0.659169\pi$
$62$	183.387	0.375648
$63$	95.1422	0.190267
$64$	−331.946	−0.648331
$65$	662.891	1.26495
$66$	98.2032	0.183151
$67$	−488.166	−0.890134	−0.445067	−	0.895497i	$-0.646820\pi$
$67$	−488.166	−0.890134	−0.445067	+	0.895497i	$0.646820\pi$
$68$	−657.991	−1.17343
$69$	46.7141	0.0815032
$70$	153.153	0.261504
$71$	924.781	1.54579	0.772896	−	0.634532i	$-0.218807\pi$
$71$	924.781	1.54579	0.772896	+	0.634532i	$0.218807\pi$
$72$	40.5106	0.0663087
$73$	708.211	1.13548	0.567739	−	0.823209i	$-0.307818\pi$
$73$	708.211	1.13548	0.567739	+	0.823209i	$0.307818\pi$
$74$	−40.4797	−0.0635902
$75$	253.442	0.390200
$76$	116.867	0.176389
$77$	741.159	1.09692
$78$	264.075	0.383341
$79$	191.956	0.273377	0.136688	−	0.990614i	$-0.456354\pi$
$79$	191.956	0.273377	0.136688	+	0.990614i	$0.456354\pi$
$80$	−445.994	−0.623295
$81$	−614.821	−0.843376
$82$	−208.248	−0.280453
$83$	962.592	1.27299	0.636495	−	0.771281i	$-0.280384\pi$
$83$	962.592	1.27299	0.636495	+	0.771281i	$0.280384\pi$
$84$	−927.030	−1.20413
$85$	−746.290	−0.952312
$86$	0	0
$87$	−912.436	−1.12441
$88$	315.578	0.382281
$89$	−158.267	−0.188498	−0.0942489	−	0.995549i	$-0.530045\pi$
$89$	−158.267	−0.188498	−0.0942489	+	0.995549i	$0.530045\pi$
$90$	22.2416	0.0260496
$91$	1993.02	2.29589
$92$	72.6672	0.0823487
$93$	−1259.00	−1.40378
$94$	−253.594	−0.278257
$95$	132.550	0.143151
$96$	−598.368	−0.636153
$97$	−1264.08	−1.32318	−0.661589	−	0.749867i	$-0.730118\pi$
$97$	−1264.08	−1.32318	−0.661589	+	0.749867i	$0.730118\pi$
$98$	219.386	0.226136
$99$	107.635	0.109269
$100$	394.248	0.394248
$101$	347.859	0.342705	0.171353	−	0.985210i	$-0.445186\pi$
$101$	347.859	0.342705	0.171353	+	0.985210i	$0.445186\pi$
$102$	−297.298	−0.288597
$103$	−572.571	−0.547739	−0.273869	−	0.961767i	$-0.588304\pi$
$103$	−572.571	−0.547739	−0.273869	+	0.961767i	$0.588304\pi$
$104$	848.611	0.800126
$105$	−1051.43	−0.977231
$106$	247.625	0.226901
$107$	545.798	0.493124	0.246562	−	0.969127i	$-0.420699\pi$
$107$	545.798	0.493124	0.246562	+	0.969127i	$0.420699\pi$
$108$	−1112.52	−0.991224
$109$	1658.16	1.45709	0.728547	−	0.684996i	$-0.240196\pi$
$109$	1658.16	1.45709	0.728547	+	0.684996i	$0.240196\pi$
$110$	173.262	0.150181
$111$	277.903	0.237634
$112$	−1340.91	−1.13129
$113$	−492.200	−0.409755	−0.204877	−	0.978788i	$-0.565680\pi$
$113$	−492.200	−0.409755	−0.204877	+	0.978788i	$0.565680\pi$
$114$	52.8036	0.0433817
$115$	82.4187	0.0668312
$116$	−1419.36	−1.13607
$117$	289.436	0.228704
$118$	−499.901	−0.389996
$119$	−2243.77	−1.72845
$120$	−447.689	−0.340569
$121$	−492.526	−0.370042
$122$	−321.102	−0.238288
$123$	1429.67	1.04804
$124$	−1958.46	−1.41835
$125$	1511.31	1.08141
$126$	66.8707	0.0472803
$127$	858.473	0.599820	0.299910	−	0.953968i	$-0.403043\pi$
$127$	858.473	0.599820	0.299910	+	0.953968i	$0.403043\pi$
$128$	−1225.37	−0.846161
$129$	0	0
$130$	465.913	0.314333
$131$	67.8806	0.0452729	0.0226365	−	0.999744i	$-0.492794\pi$
$131$	67.8806	0.0452729	0.0226365	+	0.999744i	$0.492794\pi$
$132$	−1048.75	−0.691530
$133$	398.519	0.259820
$134$	−343.107	−0.221194
$135$	−1261.81	−0.804441
$136$	−955.374	−0.602373
$137$	1944.29	1.21249	0.606247	−	0.795276i	$-0.292674\pi$
$137$	1944.29	1.21249	0.606247	+	0.795276i	$0.292674\pi$
$138$	32.8330	0.0202531
$139$	−531.190	−0.324136	−0.162068	−	0.986780i	$-0.551816\pi$
$139$	−531.190	−0.324136	−0.162068	+	0.986780i	$0.551816\pi$
$140$	−1635.58	−0.987369
$141$	1740.98	1.03984
$142$	649.982	0.384122
$143$	2254.71	1.31852
$144$	−194.733	−0.112693
$145$	−1609.83	−0.921995
$146$	497.766	0.282160
$147$	−1506.14	−0.845064
$148$	432.298	0.240099
$149$	110.710	0.0608708	0.0304354	−	0.999537i	$-0.490311\pi$
$149$	110.710	0.0608708	0.0304354	+	0.999537i	$0.490311\pi$
$150$	178.132	0.0969626
$151$	−123.313	−0.0664572	−0.0332286	−	0.999448i	$-0.510579\pi$
$151$	−123.313	−0.0664572	−0.0332286	+	0.999448i	$0.510579\pi$
$152$	169.686	0.0905482
$153$	−325.850	−0.172179
$154$	520.923	0.272579
$155$	−2221.28	−1.15108
$156$	−2820.16	−1.44739
$157$	2208.42	1.12262	0.561308	−	0.827607i	$-0.310298\pi$
$157$	2208.42	1.12262	0.561308	+	0.827607i	$0.310298\pi$
$158$	134.916	0.0679327
$159$	−1700.01	−0.847921
$160$	−1055.71	−0.521634
$161$	247.797	0.121299
$162$	−432.127	−0.209575
$163$	−3433.18	−1.64974	−0.824868	−	0.565325i	$-0.808751\pi$
$163$	−3433.18	−1.64974	−0.824868	+	0.565325i	$0.808751\pi$
$164$	2223.96	1.05891
$165$	−1189.49	−0.561220
$166$	676.558	0.316332
$167$	1177.83	0.545770	0.272885	−	0.962047i	$-0.412022\pi$
$167$	1177.83	0.545770	0.272885	+	0.962047i	$0.412022\pi$
$168$	−1346.01	−0.618135
$169$	3866.07	1.75970
$170$	−524.530	−0.236645
$171$	57.8748	0.0258819
$172$	0	0
$173$	3101.92	1.36321	0.681603	−	0.731722i	$-0.261283\pi$
$173$	3101.92	1.36321	0.681603	+	0.731722i	$0.261283\pi$
$174$	−641.306	−0.279410
$175$	1344.40	0.580724
$176$	−1516.97	−0.649693
$177$	3431.94	1.45740
$178$	−111.238	−0.0468407
$179$	3524.06	1.47151	0.735756	−	0.677246i	$-0.236827\pi$
$179$	3524.06	1.47151	0.735756	+	0.677246i	$0.236827\pi$
$180$	−237.526	−0.0983564
$181$	−4695.34	−1.92819	−0.964093	−	0.265565i	$-0.914442\pi$
$181$	−4695.34	−1.92819	−0.964093	+	0.265565i	$0.914442\pi$
$182$	1400.80	0.570516
$183$	2204.44	0.890475
$184$	105.510	0.0422732
$185$	490.310	0.194856
$186$	−884.886	−0.348833
$187$	−2538.38	−0.992645
$188$	2708.22	1.05063
$189$	−3793.72	−1.46007
$190$	93.1626	0.0355722
$191$	845.793	0.320416	0.160208	−	0.987083i	$-0.448783\pi$
$191$	845.793	0.320416	0.160208	+	0.987083i	$0.448783\pi$
$192$	1601.72	0.602051
$193$	−4042.09	−1.50754	−0.753772	−	0.657136i	$-0.771768\pi$
$193$	−4042.09	−1.50754	−0.753772	+	0.657136i	$0.771768\pi$
$194$	−888.461	−0.328803
$195$	−3198.61	−1.17465
$196$	−2342.91	−0.853831
$197$	−1036.95	−0.375025	−0.187512	−	0.982262i	$-0.560042\pi$
$197$	−1036.95	−0.375025	−0.187512	+	0.982262i	$0.560042\pi$
$198$	75.6509	0.0271529
$199$	−2419.84	−0.862001	−0.431001	−	0.902352i	$-0.641839\pi$
$199$	−2419.84	−0.862001	−0.431001	+	0.902352i	$0.641839\pi$
$200$	572.430	0.202385
$201$	2355.51	0.826593
$202$	244.492	0.0851605
$203$	−4840.06	−1.67343
$204$	3174.96	1.08967
$205$	2522.40	0.859377
$206$	−402.431	−0.136110
$207$	35.9862	0.0120832
$208$	−4079.23	−1.35983
$209$	450.846	0.149214
$210$	−738.998	−0.242837
$211$	−86.7026	−0.0282884	−0.0141442	−	0.999900i	$-0.504502\pi$
$211$	−86.7026	−0.0282884	−0.0141442	+	0.999900i	$0.504502\pi$
$212$	−2644.49	−0.856718
$213$	−4462.28	−1.43545
$214$	383.614	0.122539
$215$	0	0
$216$	−1615.33	−0.508839
$217$	−6678.41	−2.08922
$218$	1165.44	0.362081
$219$	−3417.29	−1.05442
$220$	−1850.33	−0.567043
$221$	−6825.86	−2.07763
$222$	195.324	0.0590509
$223$	−5215.36	−1.56613	−0.783063	−	0.621942i	$-0.786344\pi$
$223$	−5215.36	−1.56613	−0.783063	+	0.621942i	$0.786344\pi$
$224$	−3174.07	−0.946770
$225$	195.239	0.0578487
$226$	−345.943	−0.101822
$227$	6096.41	1.78252	0.891261	−	0.453490i	$-0.149821\pi$
$227$	6096.41	1.78252	0.891261	+	0.453490i	$0.149821\pi$
$228$	−563.910	−0.163798
$229$	1061.19	0.306225	0.153113	−	0.988209i	$-0.451070\pi$
$229$	1061.19	0.306225	0.153113	+	0.988209i	$0.451070\pi$
$230$	57.9280	0.0166072
$231$	−3576.27	−1.01862
$232$	−2060.85	−0.583196
$233$	−5539.60	−1.55756	−0.778779	−	0.627298i	$-0.784161\pi$
$233$	−5539.60	−1.55756	−0.778779	+	0.627298i	$0.784161\pi$
$234$	203.430	0.0568318
$235$	3071.65	0.852649
$236$	5338.63	1.47252
$237$	−926.233	−0.253862
$238$	−1577.03	−0.429512
$239$	3252.71	0.880337	0.440169	−	0.897915i	$-0.354919\pi$
$239$	3252.71	0.880337	0.440169	+	0.897915i	$0.354919\pi$
$240$	2152.02	0.578802
$241$	6590.62	1.76157	0.880787	−	0.473513i	$-0.157014\pi$
$241$	6590.62	1.76157	0.880787	+	0.473513i	$0.157014\pi$
$242$	−346.172	−0.0919535
$243$	−1035.21	−0.273288
$244$	3429.17	0.899713
$245$	−2657.32	−0.692938
$246$	1004.84	0.260433
$247$	1212.35	0.312308
$248$	−2843.60	−0.728100
$249$	−4644.73	−1.18212
$250$	1062.22	0.268724
$251$	−6086.84	−1.53067	−0.765334	−	0.643633i	$-0.777426\pi$
$251$	−6086.84	−1.53067	−0.765334	+	0.643633i	$0.777426\pi$
$252$	−714.138	−0.178518
$253$	280.333	0.0696617
$254$	603.377	0.149052
$255$	3601.02	0.884332
$256$	1794.31	0.438065
$257$	6729.20	1.63329	0.816646	−	0.577138i	$-0.195831\pi$
$257$	6729.20	1.63329	0.816646	+	0.577138i	$0.195831\pi$
$258$	0	0
$259$	1474.15	0.353665
$260$	−4975.66	−1.18684
$261$	−702.896	−0.166698
$262$	47.7098	0.0112501
$263$	5288.13	1.23985	0.619924	−	0.784662i	$-0.287163\pi$
$263$	5288.13	1.23985	0.619924	+	0.784662i	$0.287163\pi$
$264$	−1522.74	−0.354993
$265$	−2999.36	−0.695281
$266$	280.099	0.0645639
$267$	763.677	0.175042
$268$	3664.17	0.835168
$269$	−7705.95	−1.74662	−0.873309	−	0.487166i	$-0.838031\pi$
$269$	−7705.95	−1.74662	−0.873309	+	0.487166i	$0.838031\pi$
$270$	−886.865	−0.199900
$271$	−474.184	−0.106290	−0.0531450	−	0.998587i	$-0.516925\pi$
$271$	−474.184	−0.106290	−0.0531450	+	0.998587i	$0.516925\pi$
$272$	4592.44	1.02374
$273$	−9616.81	−2.13200
$274$	1366.54	0.301299
$275$	1520.92	0.333508
$276$	−350.636	−0.0764704
$277$	2161.04	0.468752	0.234376	−	0.972146i	$-0.424695\pi$
$277$	2161.04	0.468752	0.234376	+	0.972146i	$0.424695\pi$
$278$	−373.346	−0.0805462
$279$	−969.869	−0.208117
$280$	−2374.79	−0.506860
$281$	−2131.11	−0.452426	−0.226213	−	0.974078i	$-0.572634\pi$
$281$	−2131.11	−0.452426	−0.226213	+	0.974078i	$0.572634\pi$
$282$	1223.65	0.258394
$283$	4185.92	0.879249	0.439624	−	0.898182i	$-0.355112\pi$
$283$	4185.92	0.879249	0.439624	+	0.898182i	$0.355112\pi$
$284$	−6941.41	−1.45034
$285$	−639.584	−0.132932
$286$	1584.72	0.327646
$287$	7583.76	1.55977
$288$	−460.953	−0.0943122
$289$	2771.62	0.564141
$290$	−1131.47	−0.229111
$291$	6099.50	1.22872
$292$	−5315.84	−1.06536
$293$	5909.60	1.17830	0.589151	−	0.808023i	$-0.299462\pi$
$293$	5909.60	1.17830	0.589151	+	0.808023i	$0.299462\pi$
$294$	−1058.59	−0.209994
$295$	6055.04	1.19505
$296$	627.679	0.123254
$297$	−4291.84	−0.838512
$298$	77.8127	0.0151261
$299$	753.834	0.145804
$300$	−1902.34	−0.366105
$301$	0	0
$302$	−86.6702	−0.0165143
$303$	−1678.50	−0.318242
$304$	−815.672	−0.153888
$305$	3889.34	0.730174
$306$	−229.024	−0.0427857
$307$	−4281.17	−0.795893	−0.397946	−	0.917409i	$-0.630277\pi$
$307$	−4281.17	−0.795893	−0.397946	+	0.917409i	$0.630277\pi$
$308$	−5563.14	−1.02919
$309$	2762.79	0.508639
$310$	−1561.22	−0.286037
$311$	−7076.50	−1.29026	−0.645131	−	0.764072i	$-0.723197\pi$
$311$	−7076.50	−1.29026	−0.645131	+	0.764072i	$0.723197\pi$
$312$	−4094.74	−0.743011
$313$	−334.959	−0.0604889	−0.0302444	−	0.999543i	$-0.509629\pi$
$313$	−334.959	−0.0604889	−0.0302444	+	0.999543i	$0.509629\pi$
$314$	1552.18	0.278965
$315$	−809.971	−0.144878
$316$	−1440.82	−0.256496
$317$	3536.49	0.626591	0.313295	−	0.949656i	$-0.398567\pi$
$317$	3536.49	0.626591	0.313295	+	0.949656i	$0.398567\pi$
$318$	−1194.85	−0.210704
$319$	−5475.57	−0.961044
$320$	2825.94	0.493672
$321$	−2633.60	−0.457924
$322$	174.164	0.0301422
$323$	−1364.88	−0.235120
$324$	4614.85	0.791297
$325$	4089.84	0.698042
$326$	−2413.01	−0.409951
$327$	−8001.03	−1.35308
$328$	3229.09	0.543588
$329$	9235.12	1.54756
$330$	−836.030	−0.139460
$331$	−1925.31	−0.319712	−0.159856	−	0.987140i	$-0.551103\pi$
$331$	−1925.31	−0.319712	−0.159856	+	0.987140i	$0.551103\pi$
$332$	−7225.22	−1.19438
$333$	214.083	0.0352302
$334$	827.841	0.135621
$335$	4155.89	0.677792
$336$	6470.20	1.05053
$337$	−2426.42	−0.392212	−0.196106	−	0.980583i	$-0.562830\pi$
$337$	−2426.42	−0.392212	−0.196106	+	0.980583i	$0.562830\pi$
$338$	2717.26	0.437277
$339$	2374.98	0.380505
$340$	5601.65	0.893507
$341$	−7555.29	−1.19983
$342$	40.6773	0.00643151
$343$	789.923	0.124349
$344$	0	0
$345$	−397.690	−0.0620605
$346$	2180.18	0.338750
$347$	−588.164	−0.0909922	−0.0454961	−	0.998965i	$-0.514487\pi$
$347$	−588.164	−0.0909922	−0.0454961	+	0.998965i	$0.514487\pi$
$348$	6848.75	1.05498
$349$	8121.04	1.24558	0.622792	−	0.782387i	$-0.285998\pi$
$349$	8121.04	1.24558	0.622792	+	0.782387i	$0.285998\pi$
$350$	944.908	0.144307
$351$	−11541.0	−1.75503
$352$	−3590.83	−0.543727
$353$	7819.91	1.17907	0.589535	−	0.807743i	$-0.299311\pi$
$353$	7819.91	1.17907	0.589535	+	0.807743i	$0.299311\pi$
$354$	2412.14	0.362157
$355$	−7872.90	−1.17704
$356$	1187.95	0.176858
$357$	10826.7	1.60507
$358$	2476.89	0.365663
$359$	6276.39	0.922716	0.461358	−	0.887214i	$-0.347362\pi$
$359$	6276.39	0.922716	0.461358	+	0.887214i	$0.347362\pi$
$360$	−344.878	−0.0504907
$361$	−6616.58	−0.964657
$362$	−3300.12	−0.479144
$363$	2376.55	0.343627
$364$	−14959.6	−2.15412
$365$	−6029.19	−0.864609
$366$	1549.39	0.221278
$367$	817.330	0.116251	0.0581257	−	0.998309i	$-0.481488\pi$
$367$	817.330	0.116251	0.0581257	+	0.998309i	$0.481488\pi$
$368$	−507.180	−0.0718440
$369$	1101.35	0.155376
$370$	344.615	0.0484207
$371$	−9017.77	−1.26194
$372$	9450.04	1.31710
$373$	8113.99	1.12635	0.563173	−	0.826339i	$-0.309581\pi$
$373$	8113.99	1.12635	0.563173	+	0.826339i	$0.309581\pi$
$374$	−1784.10	−0.246667
$375$	−7292.43	−1.00421
$376$	3932.23	0.539333
$377$	−14724.2	−2.01149
$378$	−2666.42	−0.362819
$379$	95.6236	0.0129600	0.00648002	−	0.999979i	$-0.497937\pi$
$379$	95.6236	0.0129600	0.00648002	+	0.999979i	$0.497937\pi$
$380$	−994.919	−0.134311
$381$	−4142.33	−0.557003
$382$	594.466	0.0796218
$383$	−7564.64	−1.00923	−0.504615	−	0.863345i	$-0.668365\pi$
$383$	−7564.64	−1.00923	−0.504615	+	0.863345i	$0.668365\pi$
$384$	5912.71	0.785760
$385$	−6309.68	−0.835250
$386$	−2840.98	−0.374617
$387$	0	0
$388$	9488.21	1.24147
$389$	−5243.67	−0.683457	−0.341729	−	0.939799i	$-0.611012\pi$
$389$	−5243.67	−0.683457	−0.341729	+	0.939799i	$0.611012\pi$
$390$	−2248.14	−0.291895
$391$	−848.674	−0.109768
$392$	−3401.81	−0.438309
$393$	−327.540	−0.0420412
$394$	−728.823	−0.0931918
$395$	−1634.17	−0.208163
$396$	−807.905	−0.102522
$397$	−5525.23	−0.698497	−0.349249	−	0.937030i	$-0.613563\pi$
$397$	−5525.23	−0.698497	−0.349249	+	0.937030i	$0.613563\pi$
$398$	−1700.79	−0.214203
$399$	−1922.95	−0.241273
$400$	−2751.65	−0.343956
$401$	5427.82	0.675941	0.337971	−	0.941157i	$-0.390260\pi$
$401$	5427.82	0.675941	0.337971	+	0.941157i	$0.390260\pi$
$402$	1655.57	0.205404
$403$	−20316.7	−2.51128
$404$	−2611.03	−0.321543
$405$	5234.13	0.642188
$406$	−3401.84	−0.415838
$407$	1667.71	0.203109
$408$	4609.90	0.559373
$409$	4847.71	0.586073	0.293037	−	0.956101i	$-0.405334\pi$
$409$	4847.71	0.586073	0.293037	+	0.956101i	$0.405334\pi$
$410$	1772.87	0.213551
$411$	−9381.65	−1.12594
$412$	4297.72	0.513916
$413$	18204.9	2.16902
$414$	25.2929	0.00300261
$415$	−8194.80	−0.969318
$416$	−9655.98	−1.13804
$417$	2563.11	0.300998
$418$	316.877	0.0370788
$419$	6156.12	0.717771	0.358886	−	0.933382i	$-0.383157\pi$
$419$	6156.12	0.717771	0.358886	+	0.933382i	$0.383157\pi$
$420$	7892.05	0.916887
$421$	5676.26	0.657111	0.328556	−	0.944485i	$-0.393438\pi$
$421$	5676.26	0.657111	0.328556	+	0.944485i	$0.393438\pi$
$422$	−60.9389	−0.00702952
$423$	1341.17	0.154160
$424$	−3839.68	−0.439791
$425$	−4604.39	−0.525519
$426$	−3136.31	−0.356702
$427$	11693.6	1.32527
$428$	−4096.76	−0.462674
$429$	−10879.5	−1.22440
$430$	0	0
$431$	−10888.1	−1.21685	−0.608425	−	0.793612i	$-0.708198\pi$
$431$	−10888.1	−1.21685	−0.608425	+	0.793612i	$0.708198\pi$
$432$	7764.82	0.864780
$433$	−2719.36	−0.301811	−0.150905	−	0.988548i	$-0.548219\pi$
$433$	−2719.36	−0.301811	−0.150905	+	0.988548i	$0.548219\pi$
$434$	−4693.92	−0.519160
$435$	7767.81	0.856180
$436$	−12446.2	−1.36712
$437$	150.735	0.0165003
$438$	−2401.84	−0.262019
$439$	857.486	0.0932245	0.0466123	−	0.998913i	$-0.485157\pi$
$439$	857.486	0.0932245	0.0466123	+	0.998913i	$0.485157\pi$
$440$	−2686.60	−0.291088
$441$	−1160.26	−0.125284
$442$	−4797.55	−0.516281
$443$	12384.2	1.32819	0.664097	−	0.747646i	$-0.268816\pi$
$443$	12384.2	1.32819	0.664097	+	0.747646i	$0.268816\pi$
$444$	−2085.94	−0.222960
$445$	1347.37	0.143532
$446$	−3665.61	−0.389174
$447$	−534.203	−0.0565256
$448$	8496.37	0.896018
$449$	8314.20	0.873878	0.436939	−	0.899491i	$-0.356063\pi$
$449$	8314.20	0.873878	0.436939	+	0.899491i	$0.356063\pi$
$450$	137.224	0.0143751
$451$	8579.52	0.895773
$452$	3694.45	0.384452
$453$	595.012	0.0617132
$454$	4284.86	0.442948
$455$	−16967.1	−1.74820
$456$	−818.773	−0.0840846
$457$	4900.06	0.501565	0.250782	−	0.968043i	$-0.419312\pi$
$457$	4900.06	0.501565	0.250782	+	0.968043i	$0.419312\pi$
$458$	745.859	0.0760954
$459$	12993.0	1.32127
$460$	−618.635	−0.0627044
$461$	10050.6	1.01541	0.507706	−	0.861530i	$-0.330494\pi$
$461$	10050.6	1.01541	0.507706	+	0.861530i	$0.330494\pi$
$462$	−2513.58	−0.253122
$463$	−4096.77	−0.411217	−0.205608	−	0.978634i	$-0.565917\pi$
$463$	−4096.77	−0.411217	−0.205608	+	0.978634i	$0.565917\pi$
$464$	9906.42	0.991151
$465$	10718.2	1.06891
$466$	−3893.50	−0.387045
$467$	8517.70	0.844009	0.422005	−	0.906594i	$-0.361327\pi$
$467$	8517.70	0.844009	0.422005	+	0.906594i	$0.361327\pi$
$468$	−2172.51	−0.214582
$469$	12494.9	1.23020
$470$	2158.91	0.211879
$471$	−10656.1	−1.04248
$472$	7751.46	0.755911
$473$	0	0
$474$	−651.003	−0.0630834
$475$	817.793	0.0789956
$476$	16841.7	1.62172
$477$	−1309.60	−0.125708
$478$	2286.17	0.218759
$479$	8478.48	0.808751	0.404375	−	0.914593i	$-0.367489\pi$
$479$	8478.48	0.808751	0.404375	+	0.914593i	$0.367489\pi$
$480$	5094.06	0.484398
$481$	4484.57	0.425112
$482$	4632.22	0.437742
$483$	−1195.68	−0.112640
$484$	3696.90	0.347192
$485$	10761.5	1.00753
$486$	−727.599	−0.0679106
$487$	−13015.1	−1.21102	−0.605512	−	0.795836i	$-0.707032\pi$
$487$	−13015.1	−1.21102	−0.605512	+	0.795836i	$0.707032\pi$
$488$	4979.00	0.461862
$489$	16565.9	1.53197
$490$	−1867.69	−0.172192
$491$	−13090.1	−1.20315	−0.601577	−	0.798815i	$-0.705461\pi$
$491$	−13090.1	−1.20315	−0.601577	+	0.798815i	$0.705461\pi$
$492$	−10731.1	−0.983326
$493$	16576.6	1.51435
$494$	852.102	0.0776071
$495$	−916.321	−0.0832032
$496$	13669.1	1.23742
$497$	−23670.4	−2.13634
$498$	−3264.55	−0.293751
$499$	−15411.4	−1.38258	−0.691290	−	0.722578i	$-0.742957\pi$
$499$	−15411.4	−1.38258	−0.691290	+	0.722578i	$0.742957\pi$
$500$	−11343.9	−1.01463
$501$	−5683.33	−0.506811
$502$	−4278.13	−0.380363
$503$	19402.6	1.71991	0.859957	−	0.510366i	$-0.170490\pi$
$503$	19402.6	1.71991	0.859957	+	0.510366i	$0.170490\pi$
$504$	−1036.90	−0.0916410
$505$	−2961.41	−0.260953
$506$	197.032	0.0173106
$507$	−18654.7	−1.63409
$508$	−6443.70	−0.562781
$509$	−12.0380	−0.00104828	−0.000524140	−	1.00000i	$-0.500167\pi$
$509$	−12.0380	−0.00104828	−0.000524140		1.00000i	$0.500167\pi$
$510$	2530.98	0.219752
$511$	−18127.1	−1.56927
$512$	11064.1	0.955018
$513$	−2307.71	−0.198612
$514$	4729.62	0.405865
$515$	4874.45	0.417076
$516$	0	0
$517$	10447.7	0.888761
$518$	1036.11	0.0878839
$519$	−14967.5	−1.26590
$520$	−7224.45	−0.609256
$521$	1360.35	0.114391	0.0571956	−	0.998363i	$-0.481784\pi$
$521$	1360.35	0.114391	0.0571956	+	0.998363i	$0.481784\pi$
$522$	−494.030	−0.0414236
$523$	−6659.15	−0.556757	−0.278379	−	0.960471i	$-0.589797\pi$
$523$	−6659.15	−0.556757	−0.278379	+	0.960471i	$0.589797\pi$
$524$	−509.512	−0.0424773
$525$	−6487.02	−0.539270
$526$	3716.76	0.308096
$527$	22872.7	1.89061
$528$	7319.74	0.603316
$529$	−12073.3	−0.992297
$530$	−2108.10	−0.172774
$531$	2643.80	0.216066
$532$	−2991.29	−0.243776
$533$	23070.9	1.87488
$534$	536.750	0.0434971
$535$	−4646.53	−0.375489
$536$	5320.22	0.428729
$537$	−17004.4	−1.36647
$538$	−5416.13	−0.434026
$539$	−9038.41	−0.722286
$540$	9471.17	0.754767
$541$	19303.1	1.53402	0.767009	−	0.641636i	$-0.221744\pi$
$541$	19303.1	1.53402	0.767009	+	0.641636i	$0.221744\pi$
$542$	−333.280	−0.0264125
$543$	22656.1	1.79055
$544$	10870.8	0.856767
$545$	−14116.4	−1.10950
$546$	−6759.17	−0.529791
$547$	16113.9	1.25956	0.629782	−	0.776772i	$-0.283144\pi$
$547$	16113.9	1.25956	0.629782	+	0.776772i	$0.283144\pi$
$548$	−14593.8	−1.13762
$549$	1698.19	0.132016
$550$	1068.98	0.0828750
$551$	−2944.20	−0.227635
$552$	−509.109	−0.0392556
$553$	−4913.25	−0.377816
$554$	1518.89	0.116483
$555$	−2365.86	−0.180946
$556$	3987.11	0.304121
$557$	7897.57	0.600773	0.300387	−	0.953818i	$-0.402884\pi$
$557$	7897.57	0.600773	0.300387	+	0.953818i	$0.402884\pi$
$558$	−681.672	−0.0517159
$559$	0	0
$560$	11415.5	0.861417
$561$	12248.3	0.921786
$562$	−1497.85	−0.112425
$563$	−11672.7	−0.873790	−0.436895	−	0.899513i	$-0.643922\pi$
$563$	−11672.7	−0.873790	−0.436895	+	0.899513i	$0.643922\pi$
$564$	−13067.8	−0.975628
$565$	4190.23	0.312008
$566$	2942.08	0.218489
$567$	15736.8	1.16558
$568$	−10078.6	−0.744524
$569$	13214.7	0.973622	0.486811	−	0.873507i	$-0.338160\pi$
$569$	13214.7	0.973622	0.486811	+	0.873507i	$0.338160\pi$
$570$	−449.531	−0.0330330
$571$	6180.52	0.452972	0.226486	−	0.974014i	$-0.427276\pi$
$571$	6180.52	0.452972	0.226486	+	0.974014i	$0.427276\pi$
$572$	−16923.9	−1.23710
$573$	−4081.15	−0.297544
$574$	5330.24	0.387596
$575$	508.499	0.0368798
$576$	1233.88	0.0892565
$577$	2583.84	0.186424	0.0932120	−	0.995646i	$-0.470287\pi$
$577$	2583.84	0.186424	0.0932120	+	0.995646i	$0.470287\pi$
$578$	1948.03	0.140186
$579$	19504.0	1.39993
$580$	12083.4	0.865062
$581$	−24638.2	−1.75932
$582$	4287.03	0.305332
$583$	−10201.8	−0.724728
$584$	−7718.36	−0.546898
$585$	−2464.05	−0.174147
$586$	4153.56	0.292802
$587$	−4777.54	−0.335929	−0.167964	−	0.985793i	$-0.553719\pi$
$587$	−4777.54	−0.335929	−0.167964	+	0.985793i	$0.553719\pi$
$588$	11305.1	0.792881
$589$	−4062.46	−0.284195
$590$	4255.79	0.296963
$591$	5003.54	0.348254
$592$	−3017.22	−0.209472
$593$	−6495.84	−0.449835	−0.224918	−	0.974378i	$-0.572211\pi$
$593$	−6495.84	−0.449835	−0.224918	+	0.974378i	$0.572211\pi$
$594$	−3016.52	−0.208366
$595$	19101.8	1.31613
$596$	−830.992	−0.0571120
$597$	11676.3	0.800469
$598$	529.832	0.0362315
$599$	3799.23	0.259152	0.129576	−	0.991569i	$-0.458638\pi$
$599$	3799.23	0.259152	0.129576	+	0.991569i	$0.458638\pi$
$600$	−2762.11	−0.187938
$601$	−7204.53	−0.488983	−0.244492	−	0.969651i	$-0.578621\pi$
$601$	−7204.53	−0.488983	−0.244492	+	0.969651i	$0.578621\pi$
$602$	0	0
$603$	1814.57	0.122546
$604$	925.584	0.0623535
$605$	4193.00	0.281768
$606$	−1179.73	−0.0790814
$607$	−16521.2	−1.10474	−0.552368	−	0.833601i	$-0.686276\pi$
$607$	−16521.2	−1.10474	−0.552368	+	0.833601i	$0.686276\pi$
$608$	−1930.78	−0.128789
$609$	23354.4	1.55397
$610$	2733.62	0.181445
$611$	28094.6	1.86020
$612$	2445.83	0.161547
$613$	12480.3	0.822307	0.411153	−	0.911566i	$-0.365126\pi$
$613$	12480.3	0.822307	0.411153	+	0.911566i	$0.365126\pi$
$614$	−3009.02	−0.197775
$615$	−12171.2	−0.798031
$616$	−8077.44	−0.528327
$617$	−10157.2	−0.662745	−0.331372	−	0.943500i	$-0.607512\pi$
$617$	−10157.2	−0.662745	−0.331372	+	0.943500i	$0.607512\pi$
$618$	1941.83	0.126394
$619$	303.903	0.0197333	0.00986664	−	0.999951i	$-0.496859\pi$
$619$	303.903	0.0197333	0.00986664	+	0.999951i	$0.496859\pi$
$620$	16672.9	1.08000
$621$	−1434.92	−0.0927238
$622$	−4973.72	−0.320623
$623$	4050.96	0.260511
$624$	19683.3	1.26276
$625$	−6300.66	−0.403242
$626$	−235.426	−0.0150312
$627$	−2175.44	−0.138562
$628$	−16576.4	−1.05330
$629$	−5048.78	−0.320044
$630$	−569.288	−0.0360015
$631$	−24763.4	−1.56231	−0.781153	−	0.624340i	$-0.785368\pi$
$631$	−24763.4	−1.56231	−0.781153	+	0.624340i	$0.785368\pi$
$632$	−2092.01	−0.131671
$633$	418.360	0.0262691
$634$	2485.62	0.155705
$635$	−7308.41	−0.456733
$636$	12760.3	0.795562
$637$	−24304.9	−1.51176
$638$	−3848.50	−0.238814
$639$	−3437.52	−0.212811
$640$	10431.9	0.644309
$641$	−17677.2	−1.08925	−0.544624	−	0.838680i	$-0.683328\pi$
$641$	−17677.2	−1.08925	−0.544624	+	0.838680i	$0.683328\pi$
$642$	−1851.03	−0.113792
$643$	1817.63	0.111478	0.0557390	−	0.998445i	$-0.482249\pi$
$643$	1817.63	0.111478	0.0557390	+	0.998445i	$0.482249\pi$
$644$	−1859.97	−0.113809
$645$	0	0
$646$	−959.305	−0.0584262
$647$	−7234.22	−0.439578	−0.219789	−	0.975547i	$-0.570537\pi$
$647$	−7234.22	−0.439578	−0.219789	+	0.975547i	$0.570537\pi$
$648$	6700.56	0.406208
$649$	20595.2	1.24566
$650$	2874.54	0.173460
$651$	32224.9	1.94008
$652$	25769.4	1.54787
$653$	−13094.2	−0.784710	−0.392355	−	0.919814i	$-0.628339\pi$
$653$	−13094.2	−0.784710	−0.392355	+	0.919814i	$0.628339\pi$
$654$	−5623.52	−0.336234
$655$	−577.885	−0.0344730
$656$	−15522.1	−0.923836
$657$	−2632.51	−0.156323
$658$	6490.90	0.384562
$659$	23832.6	1.40878	0.704391	−	0.709813i	$-0.251220\pi$
$659$	23832.6	1.40878	0.704391	+	0.709813i	$0.251220\pi$
$660$	8928.29	0.526565
$661$	−6199.16	−0.364780	−0.182390	−	0.983226i	$-0.558383\pi$
$661$	−6199.16	−0.364780	−0.182390	+	0.983226i	$0.558383\pi$
$662$	−1353.20	−0.0794468
$663$	32936.4	1.92933
$664$	−10490.7	−0.613130
$665$	−3392.70	−0.197840
$666$	150.468	0.00875453
$667$	−1830.69	−0.106274
$668$	−8840.83	−0.512069
$669$	25165.3	1.45433
$670$	2920.96	0.168428
$671$	13228.9	0.761099
$672$	15315.6	0.879186
$673$	−14970.2	−0.857442	−0.428721	−	0.903437i	$-0.641036\pi$
$673$	−14970.2	−0.857442	−0.428721	+	0.903437i	$0.641036\pi$
$674$	−1705.41	−0.0974626
$675$	−7785.01	−0.443919
$676$	−29018.7	−1.65104
$677$	14646.4	0.831470	0.415735	−	0.909486i	$-0.363524\pi$
$677$	14646.4	0.831470	0.415735	+	0.909486i	$0.363524\pi$
$678$	1669.25	0.0945536
$679$	32355.1	1.82868
$680$	8133.36	0.458676
$681$	−29416.6	−1.65528
$682$	−5310.24	−0.298152
$683$	−18578.7	−1.04084	−0.520420	−	0.853910i	$-0.674225\pi$
$683$	−18578.7	−1.04084	−0.520420	+	0.853910i	$0.674225\pi$
$684$	−434.408	−0.0242837
$685$	−16552.2	−0.923254
$686$	555.197	0.0309002
$687$	−5120.50	−0.284366
$688$	0	0
$689$	−27433.4	−1.51688
$690$	−279.516	−0.0154217
$691$	20405.3	1.12338	0.561690	−	0.827348i	$-0.310151\pi$
$691$	20405.3	1.12338	0.561690	+	0.827348i	$0.310151\pi$
$692$	−23283.0	−1.27903
$693$	−2754.98	−0.151014
$694$	−413.391	−0.0226111
$695$	4522.16	0.246813
$696$	9944.09	0.541566
$697$	−25973.4	−1.41150
$698$	5707.87	0.309522
$699$	26729.8	1.44637
$700$	−10091.0	−0.544865
$701$	−9377.79	−0.505270	−0.252635	−	0.967562i	$-0.581297\pi$
$701$	−9377.79	−0.505270	−0.252635	+	0.967562i	$0.581297\pi$
$702$	−8111.62	−0.436116
$703$	896.722	0.0481088
$704$	9611.96	0.514580
$705$	−14821.4	−0.791784
$706$	5496.22	0.292993
$707$	−8903.67	−0.473631
$708$	−25760.1	−1.36741
$709$	−23415.3	−1.24031	−0.620155	−	0.784479i	$-0.712930\pi$
$709$	−23415.3	−1.24031	−0.620155	+	0.784479i	$0.712930\pi$
$710$	−5533.47	−0.292489
$711$	−713.524	−0.0376361
$712$	1724.86	0.0907891
$713$	−2526.02	−0.132679
$714$	7609.54	0.398852
$715$	−19195.0	−1.00399
$716$	−26451.6	−1.38065
$717$	−15695.1	−0.817496
$718$	4411.36	0.229290
$719$	17381.0	0.901531	0.450765	−	0.892642i	$-0.351151\pi$
$719$	17381.0	0.901531	0.450765	+	0.892642i	$0.351151\pi$
$720$	1657.81	0.0858098
$721$	14655.3	0.756995
$722$	−4650.46	−0.239712
$723$	−31801.3	−1.63583
$724$	35243.2	1.80912
$725$	−9932.18	−0.508789
$726$	1670.36	0.0853896
$727$	−10089.7	−0.514728	−0.257364	−	0.966315i	$-0.582854\pi$
$727$	−10089.7	−0.514728	−0.257364	+	0.966315i	$0.582854\pi$
$728$	−21720.8	−1.10580
$729$	21595.3	1.09716
$730$	−4237.62	−0.214851
$731$	0	0
$732$	−16546.5	−0.835488
$733$	18021.1	0.908081	0.454041	−	0.890981i	$-0.349982\pi$
$733$	18021.1	0.908081	0.454041	+	0.890981i	$0.349982\pi$
$734$	574.460	0.0288879
$735$	12822.2	0.643474
$736$	−1200.55	−0.0601261
$737$	14135.5	0.706498
$738$	774.083	0.0386103
$739$	−10773.6	−0.536284	−0.268142	−	0.963379i	$-0.586410\pi$
$739$	−10773.6	−0.536284	−0.268142	+	0.963379i	$0.586410\pi$
$740$	−3680.27	−0.182824
$741$	−5849.89	−0.290015
$742$	−6338.14	−0.313585
$743$	−14689.6	−0.725316	−0.362658	−	0.931922i	$-0.618131\pi$
$743$	−14689.6	−0.725316	−0.362658	+	0.931922i	$0.618131\pi$
$744$	13721.0	0.676126
$745$	−942.507	−0.0463500
$746$	5702.92	0.279891
$747$	−3578.07	−0.175254
$748$	19053.1	0.931349
$749$	−13970.1	−0.681516
$750$	−5125.48	−0.249542
$751$	−24097.2	−1.17086	−0.585432	−	0.810721i	$-0.699075\pi$
$751$	−24097.2	−1.17086	−0.585432	+	0.810721i	$0.699075\pi$
$752$	−18902.0	−0.916604
$753$	29370.4	1.42140
$754$	−10348.9	−0.499846
$755$	1049.79	0.0506038
$756$	28475.7	1.36991
$757$	21261.7	1.02083	0.510415	−	0.859928i	$-0.329492\pi$
$757$	21261.7	1.02083	0.510415	+	0.859928i	$0.329492\pi$
$758$	67.2090	0.00322050
$759$	−1352.67	−0.0646890
$760$	−1444.58	−0.0689479
$761$	12987.5	0.618657	0.309328	−	0.950955i	$-0.399896\pi$
$761$	12987.5	0.618657	0.309328	+	0.950955i	$0.399896\pi$
$762$	−2911.44	−0.138412
$763$	−42441.8	−2.01376
$764$	−6348.53	−0.300630
$765$	2774.05	0.131106
$766$	−5316.80	−0.250788
$767$	55381.8	2.60720
$768$	−8657.98	−0.406794
$769$	19917.6	0.934001	0.467000	−	0.884257i	$-0.345335\pi$
$769$	19917.6	0.934001	0.467000	+	0.884257i	$0.345335\pi$
$770$	−4434.76	−0.207555
$771$	−32470.0	−1.51670
$772$	30339.9	1.41445
$773$	28483.0	1.32531	0.662653	−	0.748927i	$-0.269431\pi$
$773$	28483.0	1.32531	0.662653	+	0.748927i	$0.269431\pi$
$774$	0	0
$775$	−13704.6	−0.635205
$776$	13776.5	0.637302
$777$	−7113.12	−0.328419
$778$	−3685.51	−0.169836
$779$	4613.19	0.212175
$780$	24008.7	1.10212
$781$	−26778.3	−1.22689
$782$	−596.490	−0.0272768
$783$	28027.4	1.27921
$784$	16352.3	0.744913
$785$	−18800.8	−0.854816
$786$	−230.211	−0.0104470
$787$	14243.4	0.645135	0.322567	−	0.946547i	$-0.395454\pi$
$787$	14243.4	0.645135	0.322567	+	0.946547i	$0.395454\pi$
$788$	7783.38	0.351867
$789$	−25516.5	−1.15134
$790$	−1148.58	−0.0517273
$791$	12598.2	0.566296
$792$	−1173.04	−0.0526291
$793$	35573.5	1.59300
$794$	−3883.41	−0.173573
$795$	14472.6	0.645649
$796$	18163.4	0.808773
$797$	−5179.01	−0.230176	−0.115088	−	0.993355i	$-0.536715\pi$
$797$	−5179.01	−0.230176	−0.115088	+	0.993355i	$0.536715\pi$
$798$	−1351.54	−0.0599551
$799$	−31629.1	−1.40045
$800$	−6513.44	−0.287856
$801$	588.299	0.0259507
$802$	3814.94	0.167968
$803$	−20507.3	−0.901228
$804$	−17680.5	−0.775551
$805$	−2109.56	−0.0923631
$806$	−14279.6	−0.624040
$807$	37183.0	1.62194
$808$	−3791.10	−0.165062
$809$	−25454.2	−1.10621	−0.553103	−	0.833113i	$-0.686556\pi$
$809$	−25454.2	−1.10621	−0.553103	+	0.833113i	$0.686556\pi$
$810$	3678.81	0.159580
$811$	−654.805	−0.0283518	−0.0141759	−	0.999900i	$-0.504512\pi$
$811$	−654.805	−0.0283518	−0.0141759	+	0.999900i	$0.504512\pi$
$812$	36329.5	1.57009
$813$	2288.05	0.0987027
$814$	1172.15	0.0504714
$815$	29227.5	1.25619
$816$	−22159.6	−0.950664
$817$	0	0
$818$	3407.21	0.145636
$819$	−7408.31	−0.316077
$820$	−18933.2	−0.806310
$821$	39663.1	1.68606	0.843029	−	0.537869i	$-0.180770\pi$
$821$	39663.1	1.68606	0.843029	+	0.537869i	$0.180770\pi$
$822$	−6593.89	−0.279791
$823$	−23291.6	−0.986507	−0.493254	−	0.869886i	$-0.664193\pi$
$823$	−23291.6	−0.986507	−0.493254	+	0.869886i	$0.664193\pi$
$824$	6240.10	0.263816
$825$	−7338.78	−0.309701
$826$	12795.3	0.538989
$827$	−17656.3	−0.742404	−0.371202	−	0.928552i	$-0.621054\pi$
$827$	−17656.3	−0.742404	−0.371202	+	0.928552i	$0.621054\pi$
$828$	−270.113	−0.0113370
$829$	−36113.7	−1.51300	−0.756502	−	0.653991i	$-0.773093\pi$
$829$	−36113.7	−1.51300	−0.756502	+	0.653991i	$0.773093\pi$
$830$	−5759.71	−0.240871
$831$	−10427.5	−0.435291
$832$	25847.2	1.07703
$833$	27362.7	1.13813
$834$	1801.48	0.0747965
$835$	−10027.2	−0.415576
$836$	−3384.05	−0.140000
$837$	38672.8	1.59704
$838$	4326.83	0.178363
$839$	−23151.9	−0.952674	−0.476337	−	0.879263i	$-0.658036\pi$
$839$	−23151.9	−0.952674	−0.476337	+	0.879263i	$0.658036\pi$
$840$	11458.9	0.470679
$841$	11368.6	0.466137
$842$	3989.56	0.163289
$843$	10283.1	0.420130
$844$	650.790	0.0265416
$845$	−32912.9	−1.33992
$846$	942.639	0.0383080
$847$	12606.5	0.511411
$848$	18457.2	0.747432
$849$	−20198.1	−0.816485
$850$	−3236.19	−0.130589
$851$	557.577	0.0224600
$852$	33493.9	1.34681
$853$	−14236.2	−0.571441	−0.285721	−	0.958313i	$-0.592233\pi$
$853$	−14236.2	−0.571441	−0.285721	+	0.958313i	$0.592233\pi$
$854$	8218.81	0.329323
$855$	−492.704	−0.0197077
$856$	−5948.32	−0.237511
$857$	−12778.3	−0.509332	−0.254666	−	0.967029i	$-0.581965\pi$
$857$	−12778.3	−0.509332	−0.254666	+	0.967029i	$0.581965\pi$
$858$	−7646.66	−0.304257
$859$	8165.70	0.324342	0.162171	−	0.986763i	$-0.448150\pi$
$859$	8165.70	0.324342	0.162171	+	0.986763i	$0.448150\pi$
$860$	0	0
$861$	−36593.4	−1.44843
$862$	−7652.71	−0.302381
$863$	−8527.22	−0.336350	−0.168175	−	0.985757i	$-0.553787\pi$
$863$	−8527.22	−0.336350	−0.168175	+	0.985757i	$0.553787\pi$
$864$	18380.1	0.723733
$865$	−26407.5	−1.03801
$866$	−1911.30	−0.0749984
$867$	−13373.7	−0.523870
$868$	50128.1	1.96021
$869$	−5558.36	−0.216979
$870$	5459.61	0.212756
$871$	38011.4	1.47872
$872$	−18071.3	−0.701803
$873$	4698.75	0.182163
$874$	105.944	0.00410023
$875$	−38683.0	−1.49454
$876$	25650.2	0.989313
$877$	6640.78	0.255694	0.127847	−	0.991794i	$-0.459193\pi$
$877$	6640.78	0.255694	0.127847	+	0.991794i	$0.459193\pi$
$878$	602.684	0.0231658
$879$	−28515.2	−1.09419
$880$	12914.4	0.494709
$881$	442.425	0.0169190	0.00845952	−	0.999964i	$-0.497307\pi$
$881$	442.425	0.0169190	0.00845952	+	0.999964i	$0.497307\pi$
$882$	−815.486	−0.0311325
$883$	−12877.2	−0.490773	−0.245386	−	0.969425i	$-0.578915\pi$
$883$	−12877.2	−0.490773	−0.245386	+	0.969425i	$0.578915\pi$
$884$	51234.9	1.94934
$885$	−29217.0	−1.10974
$886$	8704.22	0.330050
$887$	41065.2	1.55449	0.777246	−	0.629196i	$-0.216616\pi$
$887$	41065.2	1.55449	0.777246	+	0.629196i	$0.216616\pi$
$888$	−3028.70	−0.114455
$889$	−21973.2	−0.828973
$890$	947.000	0.0356669
$891$	17803.0	0.669386
$892$	39146.5	1.46942
$893$	5617.71	0.210514
$894$	−375.465	−0.0140463
$895$	−30001.3	−1.12048
$896$	31364.2	1.16943
$897$	−3637.43	−0.135396
$898$	5843.63	0.217154
$899$	49339.0	1.83042
$900$	−1465.47	−0.0542765
$901$	30884.7	1.14198
$902$	6030.11	0.222595
$903$	0	0
$904$	5364.19	0.197356
$905$	39972.7	1.46822
$906$	418.204	0.0153354
$907$	10096.2	0.369614	0.184807	−	0.982775i	$-0.440834\pi$
$907$	10096.2	0.369614	0.184807	+	0.982775i	$0.440834\pi$
$908$	−45759.6	−1.67245
$909$	−1293.03	−0.0471806
$910$	−11925.4	−0.434419
$911$	−33124.2	−1.20467	−0.602334	−	0.798244i	$-0.705762\pi$
$911$	−33124.2	−1.20467	−0.602334	+	0.798244i	$0.705762\pi$
$912$	3935.81	0.142903
$913$	−27873.2	−1.01037
$914$	3444.01	0.124636
$915$	−18767.0	−0.678052
$916$	−7965.32	−0.287316
$917$	−1737.45	−0.0625688
$918$	9132.14	0.328329
$919$	41284.6	1.48189	0.740944	−	0.671567i	$-0.234379\pi$
$919$	41284.6	1.48189	0.740944	+	0.671567i	$0.234379\pi$
$920$	−898.231	−0.0321889
$921$	20657.6	0.739079
$922$	7064.09	0.252325
$923$	−72008.7	−2.56793
$924$	26843.5	0.955720
$925$	3025.07	0.107528
$926$	−2879.42	−0.102185
$927$	2128.32	0.0754078
$928$	23449.5	0.829492
$929$	24029.9	0.848650	0.424325	−	0.905510i	$-0.360511\pi$
$929$	24029.9	0.848650	0.424325	+	0.905510i	$0.360511\pi$
$930$	7533.27	0.265619
$931$	−4859.93	−0.171083
$932$	41580.2	1.46138
$933$	34145.8	1.19816
$934$	5986.67	0.209732
$935$	21609.9	0.755849
$936$	−3154.39	−0.110154
$937$	3930.36	0.137032	0.0685162	−	0.997650i	$-0.478174\pi$
$937$	3930.36	0.137032	0.0685162	+	0.997650i	$0.478174\pi$
$938$	8782.06	0.305698
$939$	1616.26	0.0561710
$940$	−23055.8	−0.799998
$941$	20402.1	0.706790	0.353395	−	0.935474i	$-0.385027\pi$
$941$	20402.1	0.706790	0.353395	+	0.935474i	$0.385027\pi$
$942$	−7489.65	−0.259051
$943$	2868.45	0.0990559
$944$	−37260.9	−1.28468
$945$	32297.0	1.11177
$946$	0	0
$947$	31210.9	1.07098	0.535491	−	0.844541i	$-0.320127\pi$
$947$	31210.9	1.07098	0.535491	+	0.844541i	$0.320127\pi$
$948$	6952.31	0.238186
$949$	−55145.4	−1.88630
$950$	574.786	0.0196300
$951$	−17064.4	−0.581863
$952$	24453.5	0.832501
$953$	30067.7	1.02202	0.511011	−	0.859574i	$-0.329271\pi$
$953$	30067.7	1.02202	0.511011	+	0.859574i	$0.329271\pi$
$954$	−920.454	−0.0312377
$955$	−7200.46	−0.243981
$956$	−24414.9	−0.825977
$957$	26420.9	0.892441
$958$	5959.10	0.200970
$959$	−49765.4	−1.67571
$960$	−13635.8	−0.458432
$961$	38287.9	1.28522
$962$	3151.98	0.105638
$963$	−2028.80	−0.0678890
$964$	−49469.2	−1.65280
$965$	34411.4	1.14792
$966$	−840.383	−0.0279906
$967$	13292.5	0.442046	0.221023	−	0.975269i	$-0.429060\pi$
$967$	13292.5	0.442046	0.221023	+	0.975269i	$0.429060\pi$
$968$	5367.74	0.178229
$969$	6585.86	0.218337
$970$	7563.70	0.250367
$971$	8526.63	0.281805	0.140902	−	0.990023i	$-0.455000\pi$
$971$	8526.63	0.281805	0.140902	+	0.990023i	$0.455000\pi$
$972$	7770.31	0.256412
$973$	13596.2	0.447968
$974$	−9147.63	−0.300933
$975$	−19734.4	−0.648213
$976$	−23933.8	−0.784942
$977$	18477.0	0.605049	0.302524	−	0.953142i	$-0.402171\pi$
$977$	18477.0	0.605049	0.302524	+	0.953142i	$0.402171\pi$
$978$	11643.3	0.380687
$979$	4582.86	0.149611
$980$	19945.8	0.650149
$981$	−6163.60	−0.200600
$982$	−9200.38	−0.298978
$983$	−9172.54	−0.297618	−0.148809	−	0.988866i	$-0.547544\pi$
$983$	−9172.54	−0.297618	−0.148809	+	0.988866i	$0.547544\pi$
$984$	−15581.1	−0.504785
$985$	8827.86	0.285562
$986$	11650.9	0.376307
$987$	−44561.6	−1.43709
$988$	−9099.92	−0.293023
$989$	0	0
$990$	−644.036	−0.0206756
$991$	−48061.3	−1.54058	−0.770292	−	0.637691i	$-0.779890\pi$
$991$	−48061.3	−1.54058	−0.770292	+	0.637691i	$0.779890\pi$
$992$	32356.1	1.03559
$993$	9290.08	0.296890
$994$	−16636.7	−0.530870
$995$	20600.8	0.656371
$996$	34863.4	1.10912
$997$	−49132.9	−1.56074	−0.780369	−	0.625320i	$-0.784969\pi$
$997$	−49132.9	−1.56074	−0.780369	+	0.625320i	$0.784969\pi$
$998$	−10831.9	−0.343564
$999$	−8536.38	−0.270350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.26	yes	50
43.42	odd	2		1849.4.a.i.1.25	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.25	✓	50	43.42	odd	2
1849.4.a.j.1.26	yes	50	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+0.702850 q^{2} -4.82523 q^{3} -7.50600 q^{4} -8.51327 q^{5} -3.39141 q^{6} -25.5957 q^{7} -10.8984 q^{8} -3.71712 q^{9} +O(q^{10})\)	\(q+0.702850 q^{2} -4.82523 q^{3} -7.50600 q^{4} -8.51327 q^{5} -3.39141 q^{6} -25.5957 q^{7} -10.8984 q^{8} -3.71712 q^{9} -5.98355 q^{10} -28.9564 q^{11} +36.2182 q^{12} -77.8657 q^{13} -17.9899 q^{14} +41.0785 q^{15} +52.3881 q^{16} +87.6620 q^{17} -2.61258 q^{18} -15.5698 q^{19} +63.9006 q^{20} +123.505 q^{21} -20.3520 q^{22} -9.68121 q^{23} +52.5873 q^{24} -52.5243 q^{25} -54.7279 q^{26} +148.217 q^{27} +192.121 q^{28} +189.097 q^{29} +28.8720 q^{30} +260.919 q^{31} +124.008 q^{32} +139.721 q^{33} +61.6132 q^{34} +217.903 q^{35} +27.9007 q^{36} -57.5937 q^{37} -10.9432 q^{38} +375.720 q^{39} +92.7809 q^{40} -296.291 q^{41} +86.8055 q^{42} +217.347 q^{44} +31.6448 q^{45} -6.80444 q^{46} -360.808 q^{47} -252.785 q^{48} +312.138 q^{49} -36.9167 q^{50} -422.990 q^{51} +584.460 q^{52} +352.316 q^{53} +104.174 q^{54} +246.514 q^{55} +278.952 q^{56} +75.1279 q^{57} +132.907 q^{58} -711.248 q^{59} -308.335 q^{60} -456.857 q^{61} +183.387 q^{62} +95.1422 q^{63} -331.946 q^{64} +662.891 q^{65} +98.2032 q^{66} -488.166 q^{67} -657.991 q^{68} +46.7141 q^{69} +153.153 q^{70} +924.781 q^{71} +40.5106 q^{72} +708.211 q^{73} -40.4797 q^{74} +253.442 q^{75} +116.867 q^{76} +741.159 q^{77} +264.075 q^{78} +191.956 q^{79} -445.994 q^{80} -614.821 q^{81} -208.248 q^{82} +962.592 q^{83} -927.030 q^{84} -746.290 q^{85} -912.436 q^{87} +315.578 q^{88} -158.267 q^{89} +22.2416 q^{90} +1993.02 q^{91} +72.6672 q^{92} -1259.00 q^{93} -253.594 q^{94} +132.550 q^{95} -598.368 q^{96} -1264.08 q^{97} +219.386 q^{98} +107.635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more