LMFDB - Embedded newform 6031.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6031,2,Mod(1,6031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6031, 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("6031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6031 = 37 \cdot 163 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6031.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:	$48.1577774590$
Analytic rank:	$0$
Dimension:	$134$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71257 q^{2} -2.51910 q^{3} +5.35803 q^{4} -0.492593 q^{5} +6.83324 q^{6} -2.96732 q^{7} -9.10890 q^{8} +3.34588 q^{9} +O(q^{10})$	\(q-2.71257 q^{2} -2.51910 q^{3} +5.35803 q^{4} -0.492593 q^{5} +6.83324 q^{6} -2.96732 q^{7} -9.10890 q^{8} +3.34588 q^{9} +1.33619 q^{10} -5.58297 q^{11} -13.4974 q^{12} -0.0975423 q^{13} +8.04906 q^{14} +1.24089 q^{15} +13.9925 q^{16} +7.26207 q^{17} -9.07594 q^{18} -5.27762 q^{19} -2.63933 q^{20} +7.47498 q^{21} +15.1442 q^{22} -2.90332 q^{23} +22.9463 q^{24} -4.75735 q^{25} +0.264590 q^{26} -0.871318 q^{27} -15.8990 q^{28} +7.93439 q^{29} -3.36601 q^{30} +7.95806 q^{31} -19.7377 q^{32} +14.0641 q^{33} -19.6989 q^{34} +1.46168 q^{35} +17.9274 q^{36} +1.00000 q^{37} +14.3159 q^{38} +0.245719 q^{39} +4.48698 q^{40} +5.83515 q^{41} -20.2764 q^{42} -6.83292 q^{43} -29.9138 q^{44} -1.64816 q^{45} +7.87547 q^{46} -7.31684 q^{47} -35.2485 q^{48} +1.80498 q^{49} +12.9046 q^{50} -18.2939 q^{51} -0.522635 q^{52} +0.826458 q^{53} +2.36351 q^{54} +2.75013 q^{55} +27.0290 q^{56} +13.2949 q^{57} -21.5226 q^{58} -10.6432 q^{59} +6.64875 q^{60} -0.618002 q^{61} -21.5868 q^{62} -9.92830 q^{63} +25.5550 q^{64} +0.0480487 q^{65} -38.1498 q^{66} -13.4307 q^{67} +38.9104 q^{68} +7.31377 q^{69} -3.96491 q^{70} +10.9003 q^{71} -30.4773 q^{72} -12.7366 q^{73} -2.71257 q^{74} +11.9843 q^{75} -28.2777 q^{76} +16.5665 q^{77} -0.666531 q^{78} -8.31360 q^{79} -6.89259 q^{80} -7.84271 q^{81} -15.8283 q^{82} -0.0470429 q^{83} +40.0512 q^{84} -3.57725 q^{85} +18.5348 q^{86} -19.9876 q^{87} +50.8547 q^{88} +5.28215 q^{89} +4.47075 q^{90} +0.289439 q^{91} -15.5561 q^{92} -20.0472 q^{93} +19.8474 q^{94} +2.59972 q^{95} +49.7214 q^{96} -3.35851 q^{97} -4.89613 q^{98} -18.6800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9}+O(q^{10}) $ 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9} + 15 q^{10} + 20 q^{11} + 28 q^{12} + 11 q^{13} + 17 q^{14} - 13 q^{15} + 143 q^{16} + 76 q^{17} + 23 q^{18} + 15 q^{19} + 67 q^{20} + 63 q^{21} + 2 q^{22} + 22 q^{23} + 33 q^{24} + 160 q^{25} + 65 q^{26} + 31 q^{27} + 10 q^{28} + 73 q^{29} + 20 q^{30} + 10 q^{31} + 53 q^{32} + 72 q^{33} - 7 q^{34} + 52 q^{35} + 201 q^{36} + 134 q^{37} + 70 q^{38} + 6 q^{39} + 11 q^{40} + 182 q^{41} - 15 q^{42} + 12 q^{43} + 33 q^{44} + 29 q^{45} + 24 q^{46} + 80 q^{47} + 21 q^{48} + 229 q^{49} + 37 q^{50} + 57 q^{51} - 15 q^{52} + 75 q^{53} + 95 q^{54} - 9 q^{55} + 39 q^{56} + 19 q^{57} - 21 q^{58} + 91 q^{59} + 62 q^{60} + 58 q^{61} + 108 q^{62} + 9 q^{63} + 167 q^{64} + 76 q^{65} + 105 q^{66} - 17 q^{67} + 109 q^{68} + 48 q^{69} - 55 q^{70} + 56 q^{71} + 48 q^{72} + 54 q^{73} + 9 q^{74} + 28 q^{75} + 82 q^{76} + 156 q^{77} + 16 q^{78} - 2 q^{79} + 98 q^{80} + 270 q^{81} - 42 q^{82} + 130 q^{83} + 229 q^{84} + 22 q^{85} + 72 q^{86} + 22 q^{87} + 61 q^{88} + 157 q^{89} + 176 q^{90} + 31 q^{91} - 18 q^{92} + 36 q^{93} + 83 q^{94} + 98 q^{95} + 111 q^{96} + 35 q^{97} + 53 q^{98} + 55 q^{99}+O(q^{100}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9 + 15 * q^10 + 20 * q^11 + 28 * q^12 + 11 * q^13 + 17 * q^14 - 13 * q^15 + 143 * q^16 + 76 * q^17 + 23 * q^18 + 15 * q^19 + 67 * q^20 + 63 * q^21 + 2 * q^22 + 22 * q^23 + 33 * q^24 + 160 * q^25 + 65 * q^26 + 31 * q^27 + 10 * q^28 + 73 * q^29 + 20 * q^30 + 10 * q^31 + 53 * q^32 + 72 * q^33 - 7 * q^34 + 52 * q^35 + 201 * q^36 + 134 * q^37 + 70 * q^38 + 6 * q^39 + 11 * q^40 + 182 * q^41 - 15 * q^42 + 12 * q^43 + 33 * q^44 + 29 * q^45 + 24 * q^46 + 80 * q^47 + 21 * q^48 + 229 * q^49 + 37 * q^50 + 57 * q^51 - 15 * q^52 + 75 * q^53 + 95 * q^54 - 9 * q^55 + 39 * q^56 + 19 * q^57 - 21 * q^58 + 91 * q^59 + 62 * q^60 + 58 * q^61 + 108 * q^62 + 9 * q^63 + 167 * q^64 + 76 * q^65 + 105 * q^66 - 17 * q^67 + 109 * q^68 + 48 * q^69 - 55 * q^70 + 56 * q^71 + 48 * q^72 + 54 * q^73 + 9 * q^74 + 28 * q^75 + 82 * q^76 + 156 * q^77 + 16 * q^78 - 2 * q^79 + 98 * q^80 + 270 * q^81 - 42 * q^82 + 130 * q^83 + 229 * q^84 + 22 * q^85 + 72 * q^86 + 22 * q^87 + 61 * q^88 + 157 * q^89 + 176 * q^90 + 31 * q^91 - 18 * q^92 + 36 * q^93 + 83 * q^94 + 98 * q^95 + 111 * q^96 + 35 * q^97 + 53 * q^98 + 55 * 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 $

$2$	−2.71257	−1.91808	−0.959038	−	0.283277i	$-0.908578\pi$
$2$	−2.71257	−1.91808	−0.959038	+	0.283277i	$0.908578\pi$
$3$	−2.51910	−1.45441	−0.727203	−	0.686423i	$-0.759180\pi$
$3$	−2.51910	−1.45441	−0.727203	+	0.686423i	$0.759180\pi$
$4$	5.35803	2.67902
$5$	−0.492593	−0.220294	−0.110147	−	0.993915i	$-0.535132\pi$
$5$	−0.492593	−0.220294	−0.110147	+	0.993915i	$0.535132\pi$
$6$	6.83324	2.78966
$7$	−2.96732	−1.12154	−0.560770	−	0.827971i	$-0.689495\pi$
$7$	−2.96732	−1.12154	−0.560770	+	0.827971i	$0.689495\pi$
$8$	−9.10890	−3.22048
$9$	3.34588	1.11529
$10$	1.33619	0.422541
$11$	−5.58297	−1.68333	−0.841665	−	0.540001i	$-0.818424\pi$
$11$	−5.58297	−1.68333	−0.841665	+	0.540001i	$0.818424\pi$
$12$	−13.4974	−3.89638
$13$	−0.0975423	−0.0270534	−0.0135267	−	0.999909i	$-0.504306\pi$
$13$	−0.0975423	−0.0270534	−0.0135267	+	0.999909i	$0.504306\pi$
$14$	8.04906	2.15120
$15$	1.24089	0.320397
$16$	13.9925	3.49812
$17$	7.26207	1.76131	0.880656	−	0.473757i	$-0.157102\pi$
$17$	7.26207	1.76131	0.880656	+	0.473757i	$0.157102\pi$
$18$	−9.07594	−2.13922
$19$	−5.27762	−1.21077	−0.605384	−	0.795933i	$-0.706981\pi$
$19$	−5.27762	−1.21077	−0.605384	+	0.795933i	$0.706981\pi$
$20$	−2.63933	−0.590172
$21$	7.47498	1.63118
$22$	15.1442	3.22875
$23$	−2.90332	−0.605385	−0.302692	−	0.953088i	$-0.597885\pi$
$23$	−2.90332	−0.605385	−0.302692	+	0.953088i	$0.597885\pi$
$24$	22.9463	4.68389
$25$	−4.75735	−0.951470
$26$	0.264590	0.0518904
$27$	−0.871318	−0.167685
$28$	−15.8990	−3.00463
$29$	7.93439	1.47338	0.736690	−	0.676231i	$-0.236388\pi$
$29$	7.93439	1.47338	0.736690	+	0.676231i	$0.236388\pi$
$30$	−3.36601	−0.614546
$31$	7.95806	1.42931	0.714655	−	0.699477i	$-0.246584\pi$
$31$	7.95806	1.42931	0.714655	+	0.699477i	$0.246584\pi$
$32$	−19.7377	−3.48917
$33$	14.0641	2.44824
$34$	−19.6989	−3.37833
$35$	1.46168	0.247069
$36$	17.9274	2.98789
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	14.3159	2.32235
$39$	0.245719	0.0393466
$40$	4.48698	0.709454
$41$	5.83515	0.911298	0.455649	−	0.890160i	$-0.349407\pi$
$41$	5.83515	0.911298	0.455649	+	0.890160i	$0.349407\pi$
$42$	−20.2764	−3.12872
$43$	−6.83292	−1.04201	−0.521005	−	0.853554i	$-0.674443\pi$
$43$	−6.83292	−1.04201	−0.521005	+	0.853554i	$0.674443\pi$
$44$	−29.9138	−4.50967
$45$	−1.64816	−0.245693
$46$	7.87547	1.16117
$47$	−7.31684	−1.06727	−0.533636	−	0.845714i	$-0.679175\pi$
$47$	−7.31684	−1.06727	−0.533636	+	0.845714i	$0.679175\pi$
$48$	−35.2485	−5.08768
$49$	1.80498	0.257854
$50$	12.9046	1.82499
$51$	−18.2939	−2.56166
$52$	−0.522635	−0.0724765
$53$	0.826458	0.113523	0.0567614	−	0.998388i	$-0.481923\pi$
$53$	0.826458	0.113523	0.0567614	+	0.998388i	$0.481923\pi$
$54$	2.36351	0.321633
$55$	2.75013	0.370828
$56$	27.0290	3.61190
$57$	13.2949	1.76095
$58$	−21.5226	−2.82605
$59$	−10.6432	−1.38563	−0.692815	−	0.721116i	$-0.743630\pi$
$59$	−10.6432	−1.38563	−0.692815	+	0.721116i	$0.743630\pi$
$60$	6.64875	0.858350
$61$	−0.618002	−0.0791271	−0.0395635	−	0.999217i	$-0.512597\pi$
$61$	−0.618002	−0.0791271	−0.0395635	+	0.999217i	$0.512597\pi$
$62$	−21.5868	−2.74153
$63$	−9.92830	−1.25085
$64$	25.5550	3.19438
$65$	0.0480487	0.00595971
$66$	−38.1498	−4.69592
$67$	−13.4307	−1.64082	−0.820410	−	0.571775i	$-0.806255\pi$
$67$	−13.4307	−1.64082	−0.820410	+	0.571775i	$0.806255\pi$
$68$	38.9104	4.71858
$69$	7.31377	0.880475
$70$	−3.96491	−0.473897
$71$	10.9003	1.29363	0.646815	−	0.762647i	$-0.276100\pi$
$71$	10.9003	1.29363	0.646815	+	0.762647i	$0.276100\pi$
$72$	−30.4773	−3.59179
$73$	−12.7366	−1.49070	−0.745350	−	0.666673i	$-0.767718\pi$
$73$	−12.7366	−1.49070	−0.745350	+	0.666673i	$0.767718\pi$
$74$	−2.71257	−0.315330
$75$	11.9843	1.38382
$76$	−28.2777	−3.24367
$77$	16.5665	1.88792
$78$	−0.666531	−0.0754697
$79$	−8.31360	−0.935353	−0.467676	−	0.883900i	$-0.654909\pi$
$79$	−8.31360	−0.935353	−0.467676	+	0.883900i	$0.654909\pi$
$80$	−6.89259	−0.770615
$81$	−7.84271	−0.871412
$82$	−15.8283	−1.74794
$83$	−0.0470429	−0.00516363	−0.00258182	−	0.999997i	$-0.500822\pi$
$83$	−0.0470429	−0.00516363	−0.00258182	+	0.999997i	$0.500822\pi$
$84$	40.0512	4.36995
$85$	−3.57725	−0.388007
$86$	18.5348	1.99865
$87$	−19.9876	−2.14289
$88$	50.8547	5.42113
$89$	5.28215	0.559907	0.279953	−	0.960014i	$-0.409681\pi$
$89$	5.28215	0.559907	0.279953	+	0.960014i	$0.409681\pi$
$90$	4.47075	0.471258
$91$	0.289439	0.0303415
$92$	−15.5561	−1.62184
$93$	−20.0472	−2.07880
$94$	19.8474	2.04711
$95$	2.59972	0.266725
$96$	49.7214	5.07467
$97$	−3.35851	−0.341005	−0.170502	−	0.985357i	$-0.554539\pi$
$97$	−3.35851	−0.341005	−0.170502	+	0.985357i	$0.554539\pi$
$98$	−4.89613	−0.494584
$99$	−18.6800	−1.87741
$100$	−25.4901	−2.54901
$101$	−10.6226	−1.05699	−0.528494	−	0.848937i	$-0.677243\pi$
$101$	−10.6226	−1.05699	−0.528494	+	0.848937i	$0.677243\pi$
$102$	49.6235	4.91346
$103$	−1.30707	−0.128790	−0.0643949	−	0.997924i	$-0.520512\pi$
$103$	−1.30707	−0.128790	−0.0643949	+	0.997924i	$0.520512\pi$
$104$	0.888504	0.0871249
$105$	−3.68212	−0.359339
$106$	−2.24183	−0.217745
$107$	−3.51964	−0.340257	−0.170128	−	0.985422i	$-0.554418\pi$
$107$	−3.51964	−0.340257	−0.170128	+	0.985422i	$0.554418\pi$
$108$	−4.66855	−0.449232
$109$	17.8481	1.70954	0.854769	−	0.519008i	$-0.173698\pi$
$109$	17.8481	1.70954	0.854769	+	0.519008i	$0.173698\pi$
$110$	−7.45993	−0.711276
$111$	−2.51910	−0.239103
$112$	−41.5201	−3.92328
$113$	−0.331085	−0.0311458	−0.0155729	−	0.999879i	$-0.504957\pi$
$113$	−0.331085	−0.0311458	−0.0155729	+	0.999879i	$0.504957\pi$
$114$	−36.0633	−3.37763
$115$	1.43016	0.133363
$116$	42.5127	3.94721
$117$	−0.326365	−0.0301725
$118$	28.8705	2.65774
$119$	−21.5489	−1.97538
$120$	−11.3032	−1.03183
$121$	20.1696	1.83360
$122$	1.67637	0.151772
$123$	−14.6994	−1.32540
$124$	42.6396	3.82915
$125$	4.80640	0.429898
$126$	26.9312	2.39922
$127$	0.579175	0.0513935	0.0256968	−	0.999670i	$-0.491820\pi$
$127$	0.579175	0.0513935	0.0256968	+	0.999670i	$0.491820\pi$
$128$	−29.8443	−2.63789
$129$	17.2128	1.51550
$130$	−0.130335	−0.0114312
$131$	6.50503	0.568347	0.284173	−	0.958773i	$-0.408281\pi$
$131$	6.50503	0.568347	0.284173	+	0.958773i	$0.408281\pi$
$132$	75.3558	6.55888
$133$	15.6604	1.35793
$134$	36.4317	3.14722
$135$	0.429205	0.0369401
$136$	−66.1495	−5.67227
$137$	−18.6412	−1.59263	−0.796313	−	0.604885i	$-0.793219\pi$
$137$	−18.6412	−1.59263	−0.796313	+	0.604885i	$0.793219\pi$
$138$	−19.8391	−1.68882
$139$	−5.10477	−0.432981	−0.216490	−	0.976285i	$-0.569461\pi$
$139$	−5.10477	−0.432981	−0.216490	+	0.976285i	$0.569461\pi$
$140$	7.83173	0.661902
$141$	18.4319	1.55225
$142$	−29.5679	−2.48128
$143$	0.544576	0.0455397
$144$	46.8172	3.90143
$145$	−3.90843	−0.324577
$146$	34.5488	2.85928
$147$	−4.54693	−0.375024
$148$	5.35803	0.440428
$149$	−18.2569	−1.49566	−0.747832	−	0.663889i	$-0.768905\pi$
$149$	−18.2569	−1.49566	−0.747832	+	0.663889i	$0.768905\pi$
$150$	−32.5082	−2.65428
$151$	−11.9474	−0.972267	−0.486133	−	0.873885i	$-0.661593\pi$
$151$	−11.9474	−0.972267	−0.486133	+	0.873885i	$0.661593\pi$
$152$	48.0733	3.89926
$153$	24.2981	1.96438
$154$	−44.9377	−3.62118
$155$	−3.92009	−0.314869
$156$	1.31657	0.105410
$157$	−21.8286	−1.74211	−0.871055	−	0.491186i	$-0.836564\pi$
$157$	−21.8286	−1.74211	−0.871055	+	0.491186i	$0.836564\pi$
$158$	22.5512	1.79408
$159$	−2.08193	−0.165108
$160$	9.72266	0.768644
$161$	8.61508	0.678964
$162$	21.2739	1.67144
$163$	−1.00000	−0.0783260
$163$	−1.00000	−0.0783260
$164$	31.2650	2.44138
$165$	−6.92787	−0.539334
$166$	0.127607	0.00990424
$167$	−2.59706	−0.200967	−0.100483	−	0.994939i	$-0.532039\pi$
$167$	−2.59706	−0.200967	−0.100483	+	0.994939i	$0.532039\pi$
$168$	−68.0889	−5.25317
$169$	−12.9905	−0.999268
$170$	9.70353	0.744227
$171$	−17.6583	−1.35036
$172$	−36.6110	−2.79156
$173$	14.9561	1.13709	0.568547	−	0.822650i	$-0.307506\pi$
$173$	14.9561	1.13709	0.568547	+	0.822650i	$0.307506\pi$
$174$	54.2176	4.11023
$175$	14.1166	1.06711
$176$	−78.1195	−5.88848
$177$	26.8114	2.01527
$178$	−14.3282	−1.07394
$179$	16.6223	1.24241	0.621203	−	0.783649i	$-0.286644\pi$
$179$	16.6223	1.24241	0.621203	+	0.783649i	$0.286644\pi$
$180$	−8.83089	−0.658216
$181$	−10.4888	−0.779625	−0.389812	−	0.920894i	$-0.627460\pi$
$181$	−10.4888	−0.779625	−0.389812	+	0.920894i	$0.627460\pi$
$182$	−0.785124	−0.0581973
$183$	1.55681	0.115083
$184$	26.4461	1.94963
$185$	−0.492593	−0.0362162
$186$	54.3794	3.98729
$187$	−40.5439	−2.96487
$188$	−39.2039	−2.85924
$189$	2.58548	0.188066
$190$	−7.05192	−0.511600
$191$	14.4200	1.04340	0.521698	−	0.853130i	$-0.325299\pi$
$191$	14.4200	1.04340	0.521698	+	0.853130i	$0.325299\pi$
$192$	−64.3758	−4.64592
$193$	−11.3389	−0.816194	−0.408097	−	0.912939i	$-0.633807\pi$
$193$	−11.3389	−0.816194	−0.408097	+	0.912939i	$0.633807\pi$
$194$	9.11018	0.654073
$195$	−0.121040	−0.00866783
$196$	9.67114	0.690795
$197$	14.0519	1.00116	0.500580	−	0.865690i	$-0.333120\pi$
$197$	14.0519	1.00116	0.500580	+	0.865690i	$0.333120\pi$
$198$	50.6707	3.60101
$199$	−7.82021	−0.554360	−0.277180	−	0.960818i	$-0.589400\pi$
$199$	−7.82021	−0.554360	−0.277180	+	0.960818i	$0.589400\pi$
$200$	43.3342	3.06419
$201$	33.8333	2.38642
$202$	28.8146	2.02738
$203$	−23.5439	−1.65246
$204$	−98.0194	−6.86273
$205$	−2.87436	−0.200754
$206$	3.54553	0.247029
$207$	−9.71418	−0.675182
$208$	−1.36486	−0.0946358
$209$	29.4648	2.03812
$210$	9.98802	0.689239
$211$	11.3201	0.779308	0.389654	−	0.920961i	$-0.372595\pi$
$211$	11.3201	0.779308	0.389654	+	0.920961i	$0.372595\pi$
$212$	4.42819	0.304129
$213$	−27.4591	−1.88146
$214$	9.54727	0.652638
$215$	3.36585	0.229549
$216$	7.93675	0.540028
$217$	−23.6141	−1.60303
$218$	−48.4142	−3.27903
$219$	32.0847	2.16808
$220$	14.7353	0.993454
$221$	−0.708360	−0.0476494
$222$	6.83324	0.458617
$223$	12.5350	0.839409	0.419704	−	0.907661i	$-0.362134\pi$
$223$	12.5350	0.839409	0.419704	+	0.907661i	$0.362134\pi$
$224$	58.5681	3.91325
$225$	−15.9175	−1.06117
$226$	0.898090	0.0597401
$227$	−29.0863	−1.93053	−0.965264	−	0.261278i	$-0.915856\pi$
$227$	−29.0863	−1.93053	−0.965264	+	0.261278i	$0.915856\pi$
$228$	71.2344	4.71761
$229$	−0.902616	−0.0596465	−0.0298233	−	0.999555i	$-0.509494\pi$
$229$	−0.902616	−0.0596465	−0.0298233	+	0.999555i	$0.509494\pi$
$230$	−3.87940	−0.255800
$231$	−41.7326	−2.74580
$232$	−72.2736	−4.74499
$233$	−20.5420	−1.34575	−0.672875	−	0.739757i	$-0.734941\pi$
$233$	−20.5420	−1.34575	−0.672875	+	0.739757i	$0.734941\pi$
$234$	0.885289	0.0578731
$235$	3.60423	0.235114
$236$	−57.0267	−3.71212
$237$	20.9428	1.36038
$238$	58.4528	3.78893
$239$	−1.60971	−0.104123	−0.0520617	−	0.998644i	$-0.516579\pi$
$239$	−1.60971	−0.104123	−0.0520617	+	0.998644i	$0.516579\pi$
$240$	17.3631	1.12079
$241$	−25.3357	−1.63202	−0.816009	−	0.578039i	$-0.803818\pi$
$241$	−25.3357	−1.63202	−0.816009	+	0.578039i	$0.803818\pi$
$242$	−54.7114	−3.51698
$243$	22.3706	1.43507
$244$	−3.31128	−0.211983
$245$	−0.889120	−0.0568038
$246$	39.8730	2.54221
$247$	0.514791	0.0327554
$248$	−72.4892	−4.60307
$249$	0.118506	0.00751001
$250$	−13.0377	−0.824577
$251$	2.35838	0.148860	0.0744298	−	0.997226i	$-0.476286\pi$
$251$	2.35838	0.148860	0.0744298	+	0.997226i	$0.476286\pi$
$252$	−53.1962	−3.35104
$253$	16.2092	1.01906
$254$	−1.57105	−0.0985767
$255$	9.01146	0.564319
$256$	29.8448	1.86530
$257$	3.96901	0.247580	0.123790	−	0.992308i	$-0.460495\pi$
$257$	3.96901	0.247580	0.123790	+	0.992308i	$0.460495\pi$
$258$	−46.6910	−2.90685
$259$	−2.96732	−0.184380
$260$	0.257446	0.0159662
$261$	26.5476	1.64325
$262$	−17.6453	−1.09013
$263$	8.90023	0.548811	0.274406	−	0.961614i	$-0.411519\pi$
$263$	8.90023	0.548811	0.274406	+	0.961614i	$0.411519\pi$
$264$	−128.108	−7.88452
$265$	−0.407108	−0.0250084
$266$	−42.4799	−2.60461
$267$	−13.3063	−0.814331
$268$	−71.9621	−4.39579
$269$	−11.6458	−0.710058	−0.355029	−	0.934855i	$-0.615529\pi$
$269$	−11.6458	−0.710058	−0.355029	+	0.934855i	$0.615529\pi$
$270$	−1.16425	−0.0708540
$271$	−7.92337	−0.481310	−0.240655	−	0.970611i	$-0.577362\pi$
$271$	−7.92337	−0.481310	−0.240655	+	0.970611i	$0.577362\pi$
$272$	101.614	6.16127
$273$	−0.729127	−0.0441288
$274$	50.5656	3.05478
$275$	26.5602	1.60164
$276$	39.1874	2.35881
$277$	−12.1194	−0.728186	−0.364093	−	0.931363i	$-0.618621\pi$
$277$	−12.1194	−0.728186	−0.364093	+	0.931363i	$0.618621\pi$
$278$	13.8470	0.830490
$279$	26.6268	1.59410
$280$	−13.3143	−0.795682
$281$	−21.8917	−1.30595	−0.652973	−	0.757381i	$-0.726479\pi$
$281$	−21.8917	−1.30595	−0.652973	+	0.757381i	$0.726479\pi$
$282$	−49.9978	−2.97732
$283$	−31.8523	−1.89343	−0.946713	−	0.322080i	$-0.895618\pi$
$283$	−31.8523	−1.89343	−0.946713	+	0.322080i	$0.895618\pi$
$284$	58.4043	3.46566
$285$	−6.54896	−0.387927
$286$	−1.47720	−0.0873487
$287$	−17.3148	−1.02206
$288$	−66.0401	−3.89145
$289$	35.7377	2.10222
$290$	10.6019	0.622564
$291$	8.46043	0.495959
$292$	−68.2429	−3.99361
$293$	−26.0139	−1.51975	−0.759875	−	0.650070i	$-0.774740\pi$
$293$	−26.0139	−1.51975	−0.759875	+	0.650070i	$0.774740\pi$
$294$	12.3339	0.719325
$295$	5.24278	0.305246
$296$	−9.10890	−0.529444
$297$	4.86454	0.282269
$298$	49.5231	2.86880
$299$	0.283197	0.0163777
$300$	64.2121	3.70729
$301$	20.2754	1.16866
$302$	32.4082	1.86488
$303$	26.7594	1.53729
$304$	−73.8469	−4.23541
$305$	0.304424	0.0174312
$306$	−65.9102	−3.76783
$307$	−29.9835	−1.71125	−0.855626	−	0.517595i	$-0.826827\pi$
$307$	−29.9835	−1.71125	−0.855626	+	0.517595i	$0.826827\pi$
$308$	88.7636	5.05778
$309$	3.29266	0.187313
$310$	10.6335	0.603943
$311$	29.5150	1.67364	0.836820	−	0.547478i	$-0.184412\pi$
$311$	29.5150	1.67364	0.836820	+	0.547478i	$0.184412\pi$
$312$	−2.23823	−0.126715
$313$	2.68909	0.151996	0.0759981	−	0.997108i	$-0.475786\pi$
$313$	2.68909	0.151996	0.0759981	+	0.997108i	$0.475786\pi$
$314$	59.2115	3.34150
$315$	4.89061	0.275555
$316$	−44.5445	−2.50583
$317$	−14.9006	−0.836903	−0.418452	−	0.908239i	$-0.637427\pi$
$317$	−14.9006	−0.836903	−0.418452	+	0.908239i	$0.637427\pi$
$318$	5.64739	0.316690
$319$	−44.2975	−2.48018
$320$	−12.5882	−0.703703
$321$	8.86634	0.494871
$322$	−23.3690	−1.30230
$323$	−38.3264	−2.13254
$324$	−42.0215	−2.33453
$325$	0.464043	0.0257405
$326$	2.71257	0.150235
$327$	−44.9612	−2.48636
$328$	−53.1518	−2.93482
$329$	21.7114	1.19699
$330$	18.7923	1.03448
$331$	2.53216	0.139180	0.0695899	−	0.997576i	$-0.477831\pi$
$331$	2.53216	0.139180	0.0695899	+	0.997576i	$0.477831\pi$
$332$	−0.252058	−0.0138335
$333$	3.34588	0.183353
$334$	7.04471	0.385469
$335$	6.61587	0.361463
$336$	104.593	5.70604
$337$	18.1688	0.989720	0.494860	−	0.868973i	$-0.335219\pi$
$337$	18.1688	0.989720	0.494860	+	0.868973i	$0.335219\pi$
$338$	35.2376	1.91667
$339$	0.834037	0.0452987
$340$	−19.1670	−1.03948
$341$	−44.4296	−2.40600
$342$	47.8994	2.59010
$343$	15.4153	0.832347
$344$	62.2404	3.35578
$345$	−3.60271	−0.193964
$346$	−40.5696	−2.18103
$347$	−7.60750	−0.408392	−0.204196	−	0.978930i	$-0.565458\pi$
$347$	−7.60750	−0.408392	−0.204196	+	0.978930i	$0.565458\pi$
$348$	−107.094	−5.74084
$349$	−30.3054	−1.62221	−0.811107	−	0.584898i	$-0.801134\pi$
$349$	−30.3054	−1.62221	−0.811107	+	0.584898i	$0.801134\pi$
$350$	−38.2922	−2.04680
$351$	0.0849904	0.00453645
$352$	110.195	5.87342
$353$	23.9513	1.27480	0.637400	−	0.770533i	$-0.280010\pi$
$353$	23.9513	1.27480	0.637400	+	0.770533i	$0.280010\pi$
$354$	−72.7277	−3.86544
$355$	−5.36942	−0.284979
$356$	28.3019	1.50000
$357$	54.2839	2.87301
$358$	−45.0891	−2.38303
$359$	7.52245	0.397020	0.198510	−	0.980099i	$-0.436390\pi$
$359$	7.52245	0.397020	0.198510	+	0.980099i	$0.436390\pi$
$360$	15.0129	0.791250
$361$	8.85326	0.465961
$362$	28.4515	1.49538
$363$	−50.8092	−2.66679
$364$	1.55082	0.0812853
$365$	6.27394	0.328393
$366$	−4.22296	−0.220738
$367$	−21.9753	−1.14710	−0.573551	−	0.819170i	$-0.694434\pi$
$367$	−21.9753	−1.14710	−0.573551	+	0.819170i	$0.694434\pi$
$368$	−40.6246	−2.11771
$369$	19.5237	1.01637
$370$	1.33619	0.0694654
$371$	−2.45236	−0.127320
$372$	−107.414	−5.56913
$373$	4.59229	0.237780	0.118890	−	0.992907i	$-0.462066\pi$
$373$	4.59229	0.237780	0.118890	+	0.992907i	$0.462066\pi$
$374$	109.978	5.68684
$375$	−12.1078	−0.625246
$376$	66.6484	3.43713
$377$	−0.773939	−0.0398599
$378$	−7.01329	−0.360725
$379$	7.78792	0.400039	0.200019	−	0.979792i	$-0.435899\pi$
$379$	7.78792	0.400039	0.200019	+	0.979792i	$0.435899\pi$
$380$	13.9294	0.714562
$381$	−1.45900	−0.0747470
$382$	−39.1153	−2.00131
$383$	6.96546	0.355919	0.177959	−	0.984038i	$-0.443050\pi$
$383$	6.96546	0.355919	0.177959	+	0.984038i	$0.443050\pi$
$384$	75.1810	3.83656
$385$	−8.16052	−0.415899
$386$	30.7576	1.56552
$387$	−22.8621	−1.16215
$388$	−17.9950	−0.913557
$389$	2.33563	0.118421	0.0592106	−	0.998246i	$-0.481142\pi$
$389$	2.33563	0.118421	0.0592106	+	0.998246i	$0.481142\pi$
$390$	0.328328	0.0166256
$391$	−21.0841	−1.06627
$392$	−16.4414	−0.830415
$393$	−16.3868	−0.826607
$394$	−38.1169	−1.92030
$395$	4.09522	0.206053
$396$	−100.088	−5.02961
$397$	25.5088	1.28025	0.640124	−	0.768272i	$-0.278883\pi$
$397$	25.5088	1.28025	0.640124	+	0.768272i	$0.278883\pi$
$398$	21.2129	1.06331
$399$	−39.4501	−1.97498
$400$	−66.5671	−3.32835
$401$	−38.7031	−1.93274	−0.966370	−	0.257154i	$-0.917215\pi$
$401$	−38.7031	−1.93274	−0.966370	+	0.257154i	$0.917215\pi$
$402$	−91.7752	−4.57733
$403$	−0.776248	−0.0386677
$404$	−56.9163	−2.83169
$405$	3.86327	0.191967
$406$	63.8644	3.16954
$407$	−5.58297	−0.276738
$408$	166.637	8.24978
$409$	24.9384	1.23313	0.616563	−	0.787306i	$-0.288525\pi$
$409$	24.9384	1.23313	0.616563	+	0.787306i	$0.288525\pi$
$410$	7.79689	0.385061
$411$	46.9591	2.31632
$412$	−7.00335	−0.345030
$413$	31.5818	1.55404
$414$	26.3504	1.29505
$415$	0.0231730	0.00113752
$416$	1.92526	0.0943938
$417$	12.8594	0.629730
$418$	−79.9253	−3.90927
$419$	−8.05755	−0.393637	−0.196819	−	0.980440i	$-0.563061\pi$
$419$	−8.05755	−0.393637	−0.196819	+	0.980440i	$0.563061\pi$
$420$	−19.7290	−0.962674
$421$	−9.16699	−0.446772	−0.223386	−	0.974730i	$-0.571711\pi$
$421$	−9.16699	−0.446772	−0.223386	+	0.974730i	$0.571711\pi$
$422$	−30.7066	−1.49477
$423$	−24.4813	−1.19032
$424$	−7.52813	−0.365598
$425$	−34.5482	−1.67584
$426$	74.4846	3.60879
$427$	1.83381	0.0887442
$428$	−18.8584	−0.911553
$429$	−1.37184	−0.0662332
$430$	−9.13010	−0.440292
$431$	12.2846	0.591729	0.295865	−	0.955230i	$-0.404392\pi$
$431$	12.2846	0.591729	0.295865	+	0.955230i	$0.404392\pi$
$432$	−12.1919	−0.586582
$433$	−19.2877	−0.926909	−0.463455	−	0.886121i	$-0.653390\pi$
$433$	−19.2877	−0.926909	−0.463455	+	0.886121i	$0.653390\pi$
$434$	64.0549	3.07473
$435$	9.84573	0.472067
$436$	95.6308	4.57988
$437$	15.3226	0.732981
$438$	−87.0320	−4.15855
$439$	31.9453	1.52467	0.762334	−	0.647184i	$-0.224053\pi$
$439$	31.9453	1.52467	0.762334	+	0.647184i	$0.224053\pi$
$440$	−25.0507	−1.19424
$441$	6.03925	0.287583
$442$	1.92147	0.0913952
$443$	−23.9228	−1.13660	−0.568302	−	0.822820i	$-0.692400\pi$
$443$	−23.9228	−1.13660	−0.568302	+	0.822820i	$0.692400\pi$
$444$	−13.4974	−0.640560
$445$	−2.60195	−0.123344
$446$	−34.0022	−1.61005
$447$	45.9910	2.17530
$448$	−75.8299	−3.58263
$449$	7.21491	0.340493	0.170246	−	0.985402i	$-0.445544\pi$
$449$	7.21491	0.340493	0.170246	+	0.985402i	$0.445544\pi$
$450$	43.1775	2.03540
$451$	−32.5775	−1.53401
$452$	−1.77396	−0.0834402
$453$	30.0968	1.41407
$454$	78.8987	3.70290
$455$	−0.142576	−0.00668405
$456$	−121.102	−5.67110
$457$	33.9312	1.58724	0.793618	−	0.608416i	$-0.208195\pi$
$457$	33.9312	1.58724	0.793618	+	0.608416i	$0.208195\pi$
$458$	2.44841	0.114407
$459$	−6.32758	−0.295346
$460$	7.66283	0.357281
$461$	2.69960	0.125733	0.0628664	−	0.998022i	$-0.479976\pi$
$461$	2.69960	0.125733	0.0628664	+	0.998022i	$0.479976\pi$
$462$	113.203	5.26666
$463$	34.9059	1.62222	0.811108	−	0.584896i	$-0.198865\pi$
$463$	34.9059	1.62222	0.811108	+	0.584896i	$0.198865\pi$
$464$	111.022	5.15405
$465$	9.87511	0.457947
$466$	55.7215	2.58125
$467$	−17.1401	−0.793148	−0.396574	−	0.918003i	$-0.629801\pi$
$467$	−17.1401	−0.793148	−0.396574	+	0.918003i	$0.629801\pi$
$468$	−1.74868	−0.0808326
$469$	39.8531	1.84025
$470$	−9.77672	−0.450966
$471$	54.9884	2.53373
$472$	96.9481	4.46240
$473$	38.1480	1.75405
$474$	−56.8089	−2.60932
$475$	25.1075	1.15201
$476$	−115.460	−5.29208
$477$	2.76523	0.126611
$478$	4.36645	0.199717
$479$	29.4093	1.34374	0.671872	−	0.740667i	$-0.265490\pi$
$479$	29.4093	1.34374	0.671872	+	0.740667i	$0.265490\pi$
$480$	−24.4924	−1.11792
$481$	−0.0975423	−0.00444755
$482$	68.7249	3.13033
$483$	−21.7023	−0.987489
$484$	108.069	4.91224
$485$	1.65438	0.0751214
$486$	−60.6817	−2.75258
$487$	28.8278	1.30631	0.653156	−	0.757224i	$-0.273445\pi$
$487$	28.8278	1.30631	0.653156	+	0.757224i	$0.273445\pi$
$488$	5.62932	0.254827
$489$	2.51910	0.113918
$490$	2.41180	0.108954
$491$	38.7824	1.75023	0.875113	−	0.483918i	$-0.160787\pi$
$491$	38.7824	1.75023	0.875113	+	0.483918i	$0.160787\pi$
$492$	−78.7597	−3.55076
$493$	57.6201	2.59508
$494$	−1.39641	−0.0628273
$495$	9.20163	0.413582
$496$	111.353	4.99989
$497$	−32.3447	−1.45086
$498$	−0.321456	−0.0144048
$499$	−7.21074	−0.322797	−0.161398	−	0.986889i	$-0.551600\pi$
$499$	−7.21074	−0.322797	−0.161398	+	0.986889i	$0.551600\pi$
$500$	25.7529	1.15170
$501$	6.54227	0.292287
$502$	−6.39727	−0.285524
$503$	3.05947	0.136415	0.0682076	−	0.997671i	$-0.478272\pi$
$503$	3.05947	0.136415	0.0682076	+	0.997671i	$0.478272\pi$
$504$	90.4359	4.02834
$505$	5.23262	0.232849
$506$	−43.9685	−1.95464
$507$	32.7244	1.45334
$508$	3.10324	0.137684
$509$	−12.4396	−0.551373	−0.275687	−	0.961248i	$-0.588905\pi$
$509$	−12.4396	−0.551373	−0.275687	+	0.961248i	$0.588905\pi$
$510$	−24.4442	−1.08241
$511$	37.7934	1.67188
$512$	−21.2674	−0.939895
$513$	4.59849	0.203028
$514$	−10.7662	−0.474877
$515$	0.643856	0.0283717
$516$	92.2269	4.06006
$517$	40.8497	1.79657
$518$	8.04906	0.353655
$519$	−37.6761	−1.65380
$520$	−0.437671	−0.0191931
$521$	2.77326	0.121499	0.0607494	−	0.998153i	$-0.480651\pi$
$521$	2.77326	0.121499	0.0607494	+	0.998153i	$0.480651\pi$
$522$	−72.0121	−3.15188
$523$	−5.96841	−0.260980	−0.130490	−	0.991450i	$-0.541655\pi$
$523$	−5.96841	−0.260980	−0.130490	+	0.991450i	$0.541655\pi$
$524$	34.8542	1.52261
$525$	−35.5611	−1.55201
$526$	−24.1425	−1.05266
$527$	57.7920	2.51746
$528$	196.791	8.56424
$529$	−14.5707	−0.633509
$530$	1.10431	0.0479681
$531$	−35.6110	−1.54539
$532$	83.9088	3.63791
$533$	−0.569175	−0.0246537
$534$	36.0942	1.56195
$535$	1.73375	0.0749566
$536$	122.339	5.28423
$537$	−41.8732	−1.80696
$538$	31.5901	1.36195
$539$	−10.0771	−0.434053
$540$	2.29970	0.0989632
$541$	−26.2767	−1.12972	−0.564862	−	0.825185i	$-0.691071\pi$
$541$	−26.2767	−1.12972	−0.564862	+	0.825185i	$0.691071\pi$
$542$	21.4927	0.923190
$543$	26.4223	1.13389
$544$	−143.337	−6.14551
$545$	−8.79185	−0.376602
$546$	1.97781	0.0846424
$547$	−8.99133	−0.384442	−0.192221	−	0.981352i	$-0.561569\pi$
$547$	−8.99133	−0.384442	−0.192221	+	0.981352i	$0.561569\pi$
$548$	−99.8802	−4.26667
$549$	−2.06776	−0.0882500
$550$	−72.0463	−3.07206
$551$	−41.8747	−1.78392
$552$	−66.6204	−2.83555
$553$	24.6691	1.04904
$554$	32.8748	1.39672
$555$	1.24089	0.0526730
$556$	−27.3515	−1.15996
$557$	31.3314	1.32756	0.663778	−	0.747930i	$-0.268952\pi$
$557$	31.3314	1.32756	0.663778	+	0.747930i	$0.268952\pi$
$558$	−72.2269	−3.05761
$559$	0.666499	0.0281899
$560$	20.4525	0.864276
$561$	102.134	4.31212
$562$	59.3826	2.50490
$563$	25.5548	1.07700	0.538502	−	0.842624i	$-0.318990\pi$
$563$	25.5548	1.07700	0.538502	+	0.842624i	$0.318990\pi$
$564$	98.7587	4.15849
$565$	0.163090	0.00686125
$566$	86.4017	3.63173
$567$	23.2718	0.977325
$568$	−99.2900	−4.16612
$569$	−25.9863	−1.08940	−0.544701	−	0.838630i	$-0.683357\pi$
$569$	−25.9863	−1.08940	−0.544701	+	0.838630i	$0.683357\pi$
$570$	17.7645	0.744074
$571$	13.7142	0.573921	0.286960	−	0.957942i	$-0.407355\pi$
$571$	13.7142	0.573921	0.286960	+	0.957942i	$0.407355\pi$
$572$	2.91786	0.122002
$573$	−36.3255	−1.51752
$574$	46.9675	1.96039
$575$	13.8121	0.576006
$576$	85.5041	3.56267
$577$	−36.7244	−1.52886	−0.764429	−	0.644708i	$-0.776979\pi$
$577$	−36.7244	−1.52886	−0.764429	+	0.644708i	$0.776979\pi$
$578$	−96.9410	−4.03221
$579$	28.5639	1.18708
$580$	−20.9415	−0.869548
$581$	0.139591	0.00579122
$582$	−22.9495	−0.951287
$583$	−4.61409	−0.191096
$584$	116.016	4.80078
$585$	0.160765	0.00664683
$586$	70.5646	2.91500
$587$	−12.0380	−0.496861	−0.248430	−	0.968650i	$-0.579915\pi$
$587$	−12.0380	−0.496861	−0.248430	+	0.968650i	$0.579915\pi$
$588$	−24.3626	−1.00470
$589$	−41.9996	−1.73056
$590$	−14.2214	−0.585486
$591$	−35.3983	−1.45609
$592$	13.9925	0.575087
$593$	−16.9755	−0.697102	−0.348551	−	0.937290i	$-0.613326\pi$
$593$	−16.9755	−0.697102	−0.348551	+	0.937290i	$0.613326\pi$
$594$	−13.1954	−0.541414
$595$	10.6148	0.435166
$596$	−97.8211	−4.00691
$597$	19.6999	0.806264
$598$	−0.768191	−0.0314137
$599$	22.2718	0.910001	0.455000	−	0.890491i	$-0.349639\pi$
$599$	22.2718	0.910001	0.455000	+	0.890491i	$0.349639\pi$
$600$	−109.163	−4.45658
$601$	−23.8689	−0.973632	−0.486816	−	0.873505i	$-0.661842\pi$
$601$	−23.8689	−0.973632	−0.486816	+	0.873505i	$0.661842\pi$
$602$	−54.9985	−2.24157
$603$	−44.9375	−1.83000
$604$	−64.0146	−2.60472
$605$	−9.93539	−0.403931
$606$	−72.5869	−2.94864
$607$	22.0262	0.894014	0.447007	−	0.894530i	$-0.352490\pi$
$607$	22.0262	0.894014	0.447007	+	0.894530i	$0.352490\pi$
$608$	104.168	4.22458
$609$	59.3094	2.40334
$610$	−0.825770	−0.0334344
$611$	0.713702	0.0288733
$612$	130.190	5.26261
$613$	−12.3777	−0.499930	−0.249965	−	0.968255i	$-0.580419\pi$
$613$	−12.3777	−0.499930	−0.249965	+	0.968255i	$0.580419\pi$
$614$	81.3325	3.28231
$615$	7.24080	0.291977
$616$	−150.902	−6.08002
$617$	−3.52320	−0.141839	−0.0709194	−	0.997482i	$-0.522593\pi$
$617$	−3.52320	−0.141839	−0.0709194	+	0.997482i	$0.522593\pi$
$618$	−8.93156	−0.359280
$619$	20.0097	0.804256	0.402128	−	0.915583i	$-0.368271\pi$
$619$	20.0097	0.804256	0.402128	+	0.915583i	$0.368271\pi$
$620$	−21.0040	−0.843539
$621$	2.52972	0.101514
$622$	−80.0614	−3.21017
$623$	−15.6738	−0.627958
$624$	3.43822	0.137639
$625$	21.4192	0.856766
$626$	−7.29434	−0.291540
$627$	−74.2249	−2.96426
$628$	−116.958	−4.66714
$629$	7.26207	0.289558
$630$	−13.2661	−0.528535
$631$	−6.77863	−0.269853	−0.134927	−	0.990856i	$-0.543080\pi$
$631$	−6.77863	−0.269853	−0.134927	+	0.990856i	$0.543080\pi$
$632$	75.7277	3.01229
$633$	−28.5165	−1.13343
$634$	40.4190	1.60524
$635$	−0.285298	−0.0113217
$636$	−11.1551	−0.442327
$637$	−0.176062	−0.00697582
$638$	120.160	4.75718
$639$	36.4712	1.44278
$640$	14.7011	0.581112
$641$	−41.4826	−1.63846	−0.819231	−	0.573463i	$-0.805600\pi$
$641$	−41.4826	−1.63846	−0.819231	+	0.573463i	$0.805600\pi$
$642$	−24.0506	−0.949200
$643$	21.2140	0.836599	0.418299	−	0.908309i	$-0.362626\pi$
$643$	21.2140	0.836599	0.418299	+	0.908309i	$0.362626\pi$
$644$	46.1599	1.81896
$645$	−8.47892	−0.333857
$646$	103.963	4.09038
$647$	−5.75611	−0.226296	−0.113148	−	0.993578i	$-0.536093\pi$
$647$	−5.75611	−0.226296	−0.113148	+	0.993578i	$0.536093\pi$
$648$	71.4385	2.80637
$649$	59.4208	2.33247
$650$	−1.25875	−0.0493722
$651$	59.4864	2.33146
$652$	−5.35803	−0.209837
$653$	12.9849	0.508137	0.254068	−	0.967186i	$-0.418231\pi$
$653$	12.9849	0.508137	0.254068	+	0.967186i	$0.418231\pi$
$654$	121.960	4.76903
$655$	−3.20433	−0.125204
$656$	81.6482	3.18783
$657$	−42.6150	−1.66257
$658$	−58.8937	−2.29592
$659$	−40.8136	−1.58987	−0.794936	−	0.606694i	$-0.792496\pi$
$659$	−40.8136	−1.58987	−0.794936	+	0.606694i	$0.792496\pi$
$660$	−37.1198	−1.44488
$661$	−35.4459	−1.37869	−0.689343	−	0.724435i	$-0.742101\pi$
$661$	−35.4459	−1.37869	−0.689343	+	0.724435i	$0.742101\pi$
$662$	−6.86865	−0.266958
$663$	1.78443	0.0693016
$664$	0.428509	0.0166294
$665$	−7.71419	−0.299144
$666$	−9.07594	−0.351686
$667$	−23.0361	−0.891962
$668$	−13.9151	−0.538393
$669$	−31.5771	−1.22084
$670$	−17.9460	−0.693314
$671$	3.45029	0.133197
$672$	−147.539	−5.69145
$673$	14.8248	0.571453	0.285727	−	0.958311i	$-0.407765\pi$
$673$	14.8248	0.571453	0.285727	+	0.958311i	$0.407765\pi$
$674$	−49.2842	−1.89836
$675$	4.14517	0.159548
$676$	−69.6035	−2.67706
$677$	34.2099	1.31479	0.657397	−	0.753544i	$-0.271657\pi$
$677$	34.2099	1.31479	0.657397	+	0.753544i	$0.271657\pi$
$678$	−2.26238	−0.0868863
$679$	9.96576	0.382451
$680$	32.5848	1.24957
$681$	73.2715	2.80777
$682$	120.518	4.61489
$683$	−3.65986	−0.140041	−0.0700204	−	0.997546i	$-0.522306\pi$
$683$	−3.65986	−0.140041	−0.0700204	+	0.997546i	$0.522306\pi$
$684$	−94.6138	−3.61765
$685$	9.18253	0.350846
$686$	−41.8150	−1.59651
$687$	2.27378	0.0867502
$688$	−95.6093	−3.64507
$689$	−0.0806147	−0.00307117
$690$	9.77261	0.372037
$691$	−11.2409	−0.427624	−0.213812	−	0.976875i	$-0.568588\pi$
$691$	−11.2409	−0.427624	−0.213812	+	0.976875i	$0.568588\pi$
$692$	80.1355	3.04630
$693$	55.4294	2.10559
$694$	20.6359	0.783327
$695$	2.51457	0.0953832
$696$	182.065	6.90114
$697$	42.3753	1.60508
$698$	82.2056	3.11153
$699$	51.7473	1.95726
$700$	75.6371	2.85881
$701$	10.0777	0.380631	0.190316	−	0.981723i	$-0.439049\pi$
$701$	10.0777	0.380631	0.190316	+	0.981723i	$0.439049\pi$
$702$	−0.230542	−0.00870126
$703$	−5.27762	−0.199049
$704$	−142.673	−5.37719
$705$	−9.07942	−0.341951
$706$	−64.9696	−2.44516
$707$	31.5206	1.18546
$708$	143.656	5.39893
$709$	−1.42986	−0.0536997	−0.0268498	−	0.999639i	$-0.508548\pi$
$709$	−1.42986	−0.0536997	−0.0268498	+	0.999639i	$0.508548\pi$
$710$	14.5649	0.546612
$711$	−27.8163	−1.04319
$712$	−48.1146	−1.80317
$713$	−23.1048	−0.865283
$714$	−147.249	−5.51065
$715$	−0.268254	−0.0100321
$716$	89.0627	3.32843
$717$	4.05503	0.151438
$718$	−20.4052	−0.761514
$719$	1.53987	0.0574273	0.0287136	−	0.999588i	$-0.490859\pi$
$719$	1.53987	0.0574273	0.0287136	+	0.999588i	$0.490859\pi$
$720$	−23.0618	−0.859463
$721$	3.87850	0.144443
$722$	−24.0151	−0.893749
$723$	63.8233	2.37362
$724$	−56.1992	−2.08863
$725$	−37.7467	−1.40188
$726$	137.824	5.11511
$727$	15.9197	0.590428	0.295214	−	0.955431i	$-0.404609\pi$
$727$	15.9197	0.590428	0.295214	+	0.955431i	$0.404609\pi$
$728$	−2.63647	−0.0977142
$729$	−32.8256	−1.21576
$730$	−17.0185	−0.629883
$731$	−49.6211	−1.83530
$732$	8.34145	0.308309
$733$	14.4224	0.532703	0.266352	−	0.963876i	$-0.414182\pi$
$733$	14.4224	0.532703	0.266352	+	0.963876i	$0.414182\pi$
$734$	59.6095	2.20023
$735$	2.23979	0.0826157
$736$	57.3050	2.11229
$737$	74.9832	2.76204
$738$	−52.9595	−1.94947
$739$	−14.2991	−0.525999	−0.263000	−	0.964796i	$-0.584712\pi$
$739$	−14.2991	−0.525999	−0.263000	+	0.964796i	$0.584712\pi$
$740$	−2.63933	−0.0970237
$741$	−1.29681	−0.0476396
$742$	6.65221	0.244210
$743$	2.57983	0.0946447	0.0473223	−	0.998880i	$-0.484931\pi$
$743$	2.57983	0.0946447	0.0473223	+	0.998880i	$0.484931\pi$
$744$	182.608	6.69473
$745$	8.99322	0.329486
$746$	−12.4569	−0.456080
$747$	−0.157400	−0.00575897
$748$	−217.236	−7.94293
$749$	10.4439	0.381612
$750$	32.8433	1.19927
$751$	−13.3872	−0.488507	−0.244254	−	0.969711i	$-0.578543\pi$
$751$	−13.3872	−0.488507	−0.244254	+	0.969711i	$0.578543\pi$
$752$	−102.381	−3.73344
$753$	−5.94101	−0.216502
$754$	2.09936	0.0764543
$755$	5.88521	0.214185
$756$	13.8531	0.503832
$757$	32.0023	1.16314	0.581571	−	0.813496i	$-0.302438\pi$
$757$	32.0023	1.16314	0.581571	+	0.813496i	$0.302438\pi$
$758$	−21.1253	−0.767305
$759$	−40.8326	−1.48213
$760$	−23.6806	−0.858985
$761$	42.9623	1.55738	0.778692	−	0.627407i	$-0.215884\pi$
$761$	42.9623	1.55738	0.778692	+	0.627407i	$0.215884\pi$
$762$	3.95765	0.143370
$763$	−52.9610	−1.91732
$764$	77.2629	2.79527
$765$	−11.9691	−0.432742
$766$	−18.8943	−0.682679
$767$	1.03816	0.0374860
$768$	−75.1821	−2.71290
$769$	20.0487	0.722976	0.361488	−	0.932377i	$-0.382269\pi$
$769$	20.0487	0.722976	0.361488	+	0.932377i	$0.382269\pi$
$770$	22.1360	0.797725
$771$	−9.99834	−0.360081
$772$	−60.7544	−2.18660
$773$	1.23677	0.0444837	0.0222418	−	0.999753i	$-0.492920\pi$
$773$	1.23677	0.0444837	0.0222418	+	0.999753i	$0.492920\pi$
$774$	62.0152	2.22909
$775$	−37.8593	−1.35995
$776$	30.5923	1.09820
$777$	7.47498	0.268164
$778$	−6.33556	−0.227141
$779$	−30.7957	−1.10337
$780$	−0.648534	−0.0232213
$781$	−60.8562	−2.17761
$782$	57.1922	2.04519
$783$	−6.91338	−0.247064
$784$	25.2561	0.902003
$785$	10.7526	0.383777
$786$	44.4504	1.58549
$787$	0.612912	0.0218480	0.0109240	−	0.999940i	$-0.496523\pi$
$787$	0.612912	0.0218480	0.0109240	+	0.999940i	$0.496523\pi$
$788$	75.2908	2.68212
$789$	−22.4206	−0.798194
$790$	−11.1086	−0.395225
$791$	0.982434	0.0349313
$792$	170.154	6.04616
$793$	0.0602814	0.00214065
$794$	−69.1943	−2.45561
$795$	1.02555	0.0363724
$796$	−41.9010	−1.48514
$797$	−26.0481	−0.922670	−0.461335	−	0.887226i	$-0.652629\pi$
$797$	−26.0481	−0.922670	−0.461335	+	0.887226i	$0.652629\pi$
$798$	107.011	3.78815
$799$	−53.1355	−1.87980
$800$	93.8993	3.31984
$801$	17.6735	0.624461
$802$	104.985	3.70714
$803$	71.1078	2.50934
$804$	181.280	6.39325
$805$	−4.24373	−0.149572
$806$	2.10563	0.0741676
$807$	29.3370	1.03271
$808$	96.7602	3.40401
$809$	−24.0827	−0.846702	−0.423351	−	0.905966i	$-0.639146\pi$
$809$	−24.0827	−0.846702	−0.423351	+	0.905966i	$0.639146\pi$
$810$	−10.4794	−0.368208
$811$	1.14723	0.0402846	0.0201423	−	0.999797i	$-0.493588\pi$
$811$	1.14723	0.0402846	0.0201423	+	0.999797i	$0.493588\pi$
$812$	−126.149	−4.42696
$813$	19.9598	0.700020
$814$	15.1442	0.530804
$815$	0.492593	0.0172548
$816$	−255.977	−8.96098
$817$	36.0615	1.26163
$818$	−67.6472	−2.36523
$819$	0.968430	0.0338397
$820$	−15.4009	−0.537823
$821$	1.09208	0.0381140	0.0190570	−	0.999818i	$-0.493934\pi$
$821$	1.09208	0.0381140	0.0190570	+	0.999818i	$0.493934\pi$
$822$	−127.380	−4.44288
$823$	−30.2009	−1.05274	−0.526369	−	0.850256i	$-0.676447\pi$
$823$	−30.2009	−1.05274	−0.526369	+	0.850256i	$0.676447\pi$
$824$	11.9060	0.414765
$825$	−66.9078	−2.32943
$826$	−85.6679	−2.98077
$827$	23.3256	0.811109	0.405555	−	0.914071i	$-0.367078\pi$
$827$	23.3256	0.811109	0.405555	+	0.914071i	$0.367078\pi$
$828$	−52.0489	−1.80883
$829$	−29.6973	−1.03143	−0.515715	−	0.856760i	$-0.672473\pi$
$829$	−29.6973	−1.03143	−0.515715	+	0.856760i	$0.672473\pi$
$830$	−0.0628584	−0.00218185
$831$	30.5301	1.05908
$832$	−2.49270	−0.0864187
$833$	13.1079	0.454161
$834$	−34.8821	−1.20787
$835$	1.27929	0.0442718
$836$	157.873	5.46016
$837$	−6.93401	−0.239674
$838$	21.8567	0.755026
$839$	−41.0980	−1.41886	−0.709431	−	0.704775i	$-0.751048\pi$
$839$	−41.0980	−1.41886	−0.709431	+	0.704775i	$0.751048\pi$
$840$	33.5401	1.15724
$841$	33.9546	1.17085
$842$	24.8661	0.856942
$843$	55.1473	1.89938
$844$	60.6535	2.08778
$845$	6.39902	0.220133
$846$	66.4073	2.28313
$847$	−59.8495	−2.05645
$848$	11.5642	0.397116
$849$	80.2394	2.75381
$850$	93.7145	3.21438
$851$	−2.90332	−0.0995246
$852$	−147.127	−5.04047
$853$	−14.4301	−0.494078	−0.247039	−	0.969006i	$-0.579458\pi$
$853$	−14.4301	−0.494078	−0.247039	+	0.969006i	$0.579458\pi$
$854$	−4.97433	−0.170218
$855$	8.69836	0.297478
$856$	32.0601	1.09579
$857$	9.54993	0.326219	0.163110	−	0.986608i	$-0.447848\pi$
$857$	9.54993	0.326219	0.163110	+	0.986608i	$0.447848\pi$
$858$	3.72122	0.127040
$859$	−37.4090	−1.27638	−0.638189	−	0.769879i	$-0.720316\pi$
$859$	−37.4090	−1.27638	−0.638189	+	0.769879i	$0.720316\pi$
$860$	18.0343	0.614965
$861$	43.6177	1.48649
$862$	−33.3229	−1.13498
$863$	46.6941	1.58949	0.794743	−	0.606946i	$-0.207605\pi$
$863$	46.6941	1.58949	0.794743	+	0.606946i	$0.207605\pi$
$864$	17.1978	0.585082
$865$	−7.36729	−0.250496
$866$	52.3193	1.77788
$867$	−90.0270	−3.05748
$868$	−126.525	−4.29454
$869$	46.4146	1.57451
$870$	−26.7072	−0.905460
$871$	1.31006	0.0443897
$872$	−162.577	−5.50554
$873$	−11.2372	−0.380321
$874$	−41.5637	−1.40591
$875$	−14.2621	−0.482148
$876$	171.911	5.80833
$877$	19.2459	0.649886	0.324943	−	0.945734i	$-0.394655\pi$
$877$	19.2459	0.649886	0.324943	+	0.945734i	$0.394655\pi$
$878$	−86.6540	−2.92443
$879$	65.5318	2.21033
$880$	38.4811	1.29720
$881$	−19.3170	−0.650806	−0.325403	−	0.945575i	$-0.605500\pi$
$881$	−19.3170	−0.650806	−0.325403	+	0.945575i	$0.605500\pi$
$882$	−16.3819	−0.551607
$883$	48.4432	1.63024	0.815122	−	0.579289i	$-0.196670\pi$
$883$	48.4432	1.63024	0.815122	+	0.579289i	$0.196670\pi$
$884$	−3.79541	−0.127654
$885$	−13.2071	−0.443952
$886$	64.8922	2.18009
$887$	−11.9980	−0.402852	−0.201426	−	0.979504i	$-0.564558\pi$
$887$	−11.9980	−0.402852	−0.201426	+	0.979504i	$0.564558\pi$
$888$	22.9463	0.770026
$889$	−1.71860	−0.0576399
$890$	7.05797	0.236584
$891$	43.7856	1.46687
$892$	67.1632	2.24879
$893$	38.6155	1.29222
$894$	−124.754	−4.17239
$895$	−8.18802	−0.273695
$896$	88.5576	2.95850
$897$	−0.713403	−0.0238198
$898$	−19.5709	−0.653091
$899$	63.1424	2.10592
$900$	−85.2868	−2.84289
$901$	6.00180	0.199949
$902$	88.3687	2.94236
$903$	−51.0759	−1.69970
$904$	3.01582	0.100305
$905$	5.16670	0.171747
$906$	−81.6396	−2.71229
$907$	19.7675	0.656368	0.328184	−	0.944614i	$-0.393563\pi$
$907$	19.7675	0.656368	0.328184	+	0.944614i	$0.393563\pi$
$908$	−155.846	−5.17191
$909$	−35.5420	−1.17885
$910$	0.386747	0.0128205
$911$	−0.711932	−0.0235874	−0.0117937	−	0.999930i	$-0.503754\pi$
$911$	−0.711932	−0.0235874	−0.0117937	+	0.999930i	$0.503754\pi$
$912$	186.028	6.16000
$913$	0.262639	0.00869209
$914$	−92.0408	−3.04444
$915$	−0.766874	−0.0253521
$916$	−4.83624	−0.159794
$917$	−19.3025	−0.637424
$918$	17.1640	0.566496
$919$	−43.6763	−1.44075	−0.720375	−	0.693585i	$-0.756030\pi$
$919$	−43.6763	−1.44075	−0.720375	+	0.693585i	$0.756030\pi$
$920$	−13.0272	−0.429493
$921$	75.5317	2.48885
$922$	−7.32285	−0.241165
$923$	−1.06324	−0.0349971
$924$	−223.605	−7.35606
$925$	−4.75735	−0.156421
$926$	−94.6848	−3.11153
$927$	−4.37332	−0.143639
$928$	−156.607	−5.14087
$929$	34.2217	1.12278	0.561389	−	0.827552i	$-0.310267\pi$
$929$	34.2217	1.12278	0.561389	+	0.827552i	$0.310267\pi$
$930$	−26.7869	−0.878377
$931$	−9.52599	−0.312202
$932$	−110.065	−3.60528
$933$	−74.3512	−2.43415
$934$	46.4936	1.52132
$935$	19.9717	0.653143
$936$	2.97283	0.0971700
$937$	−14.3689	−0.469410	−0.234705	−	0.972067i	$-0.575412\pi$
$937$	−14.3689	−0.469410	−0.234705	+	0.972067i	$0.575412\pi$
$938$	−108.104	−3.52973
$939$	−6.77409	−0.221064
$940$	19.3116	0.629874
$941$	3.96841	0.129366	0.0646832	−	0.997906i	$-0.479396\pi$
$941$	3.96841	0.129366	0.0646832	+	0.997906i	$0.479396\pi$
$942$	−149.160	−4.85989
$943$	−16.9413	−0.551686
$944$	−148.925	−4.84709
$945$	−1.27359	−0.0414298
$946$	−103.479	−3.36439
$947$	−52.2821	−1.69894	−0.849470	−	0.527638i	$-0.823078\pi$
$947$	−52.2821	−1.69894	−0.849470	+	0.527638i	$0.823078\pi$
$948$	112.212	3.64449
$949$	1.24235	0.0403285
$950$	−68.1058	−2.20964
$951$	37.5363	1.21720
$952$	196.287	6.36169
$953$	40.9150	1.32537	0.662684	−	0.748899i	$-0.269417\pi$
$953$	40.9150	1.32537	0.662684	+	0.748899i	$0.269417\pi$
$954$	−7.50089	−0.242850
$955$	−7.10320	−0.229854
$956$	−8.62488	−0.278949
$957$	111.590	3.60719
$958$	−79.7747	−2.57740
$959$	55.3144	1.78619
$960$	31.7110	1.02347
$961$	32.3308	1.04293
$962$	0.264590	0.00853074
$963$	−11.7763	−0.379486
$964$	−135.750	−4.37220
$965$	5.58548	0.179803
$966$	58.8690	1.89408
$967$	23.1223	0.743562	0.371781	−	0.928320i	$-0.378747\pi$
$967$	23.1223	0.743562	0.371781	+	0.928320i	$0.378747\pi$
$968$	−183.723	−5.90507
$969$	96.5483	3.10158
$970$	−4.48761	−0.144089
$971$	35.6593	1.14436	0.572180	−	0.820128i	$-0.306098\pi$
$971$	35.6593	1.14436	0.572180	+	0.820128i	$0.306098\pi$
$972$	119.862	3.84458
$973$	15.1475	0.485606
$974$	−78.1974	−2.50561
$975$	−1.16897	−0.0374371
$976$	−8.64737	−0.276796
$977$	−51.9325	−1.66147	−0.830735	−	0.556669i	$-0.812079\pi$
$977$	−51.9325	−1.66147	−0.830735	+	0.556669i	$0.812079\pi$
$978$	−6.83324	−0.218503
$979$	−29.4901	−0.942507
$980$	−4.76393	−0.152178
$981$	59.7177	1.90664
$982$	−105.200	−3.35707
$983$	53.1163	1.69415	0.847074	−	0.531475i	$-0.178362\pi$
$983$	53.1163	1.69415	0.847074	+	0.531475i	$0.178362\pi$
$984$	133.895	4.26842
$985$	−6.92189	−0.220550
$986$	−156.299	−4.97756
$987$	−54.6933	−1.74091
$988$	2.75827	0.0877522
$989$	19.8382	0.630817
$990$	−24.9601	−0.793283
$991$	8.36725	0.265794	0.132897	−	0.991130i	$-0.457572\pi$
$991$	8.36725	0.265794	0.132897	+	0.991130i	$0.457572\pi$
$992$	−157.074	−4.98711
$993$	−6.37876	−0.202424
$994$	87.7374	2.78286
$995$	3.85218	0.122122
$996$	0.634959	0.0201195
$997$	−15.2551	−0.483134	−0.241567	−	0.970384i	$-0.577661\pi$
$997$	−15.2551	−0.483134	−0.241567	+	0.970384i	$0.577661\pi$
$998$	19.5596	0.619149
$999$	−0.871318	−0.0275673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6031.2.a.e.1.3	✓	134

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6031.2.a.e.1.3	✓	134	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:	\( N \)	\(=\)	\( 6031 = 37 \cdot 163 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71257 q^{2} -2.51910 q^{3} +5.35803 q^{4} -0.492593 q^{5} +6.83324 q^{6} -2.96732 q^{7} -9.10890 q^{8} +3.34588 q^{9} +O(q^{10})\)	\(q-2.71257 q^{2} -2.51910 q^{3} +5.35803 q^{4} -0.492593 q^{5} +6.83324 q^{6} -2.96732 q^{7} -9.10890 q^{8} +3.34588 q^{9} +1.33619 q^{10} -5.58297 q^{11} -13.4974 q^{12} -0.0975423 q^{13} +8.04906 q^{14} +1.24089 q^{15} +13.9925 q^{16} +7.26207 q^{17} -9.07594 q^{18} -5.27762 q^{19} -2.63933 q^{20} +7.47498 q^{21} +15.1442 q^{22} -2.90332 q^{23} +22.9463 q^{24} -4.75735 q^{25} +0.264590 q^{26} -0.871318 q^{27} -15.8990 q^{28} +7.93439 q^{29} -3.36601 q^{30} +7.95806 q^{31} -19.7377 q^{32} +14.0641 q^{33} -19.6989 q^{34} +1.46168 q^{35} +17.9274 q^{36} +1.00000 q^{37} +14.3159 q^{38} +0.245719 q^{39} +4.48698 q^{40} +5.83515 q^{41} -20.2764 q^{42} -6.83292 q^{43} -29.9138 q^{44} -1.64816 q^{45} +7.87547 q^{46} -7.31684 q^{47} -35.2485 q^{48} +1.80498 q^{49} +12.9046 q^{50} -18.2939 q^{51} -0.522635 q^{52} +0.826458 q^{53} +2.36351 q^{54} +2.75013 q^{55} +27.0290 q^{56} +13.2949 q^{57} -21.5226 q^{58} -10.6432 q^{59} +6.64875 q^{60} -0.618002 q^{61} -21.5868 q^{62} -9.92830 q^{63} +25.5550 q^{64} +0.0480487 q^{65} -38.1498 q^{66} -13.4307 q^{67} +38.9104 q^{68} +7.31377 q^{69} -3.96491 q^{70} +10.9003 q^{71} -30.4773 q^{72} -12.7366 q^{73} -2.71257 q^{74} +11.9843 q^{75} -28.2777 q^{76} +16.5665 q^{77} -0.666531 q^{78} -8.31360 q^{79} -6.89259 q^{80} -7.84271 q^{81} -15.8283 q^{82} -0.0470429 q^{83} +40.0512 q^{84} -3.57725 q^{85} +18.5348 q^{86} -19.9876 q^{87} +50.8547 q^{88} +5.28215 q^{89} +4.47075 q^{90} +0.289439 q^{91} -15.5561 q^{92} -20.0472 q^{93} +19.8474 q^{94} +2.59972 q^{95} +49.7214 q^{96} -3.35851 q^{97} -4.89613 q^{98} -18.6800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9}+O(q^{10}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9	\( 134 q + 9 q^{2} + 7 q^{3} + 149 q^{4} + 22 q^{5} + 20 q^{6} + 11 q^{7} + 27 q^{8} + 181 q^{9} + 15 q^{10} + 20 q^{11} + 28 q^{12} + 11 q^{13} + 17 q^{14} - 13 q^{15} + 143 q^{16} + 76 q^{17} + 23 q^{18} + 15 q^{19} + 67 q^{20} + 63 q^{21} + 2 q^{22} + 22 q^{23} + 33 q^{24} + 160 q^{25} + 65 q^{26} + 31 q^{27} + 10 q^{28} + 73 q^{29} + 20 q^{30} + 10 q^{31} + 53 q^{32} + 72 q^{33} - 7 q^{34} + 52 q^{35} + 201 q^{36} + 134 q^{37} + 70 q^{38} + 6 q^{39} + 11 q^{40} + 182 q^{41} - 15 q^{42} + 12 q^{43} + 33 q^{44} + 29 q^{45} + 24 q^{46} + 80 q^{47} + 21 q^{48} + 229 q^{49} + 37 q^{50} + 57 q^{51} - 15 q^{52} + 75 q^{53} + 95 q^{54} - 9 q^{55} + 39 q^{56} + 19 q^{57} - 21 q^{58} + 91 q^{59} + 62 q^{60} + 58 q^{61} + 108 q^{62} + 9 q^{63} + 167 q^{64} + 76 q^{65} + 105 q^{66} - 17 q^{67} + 109 q^{68} + 48 q^{69} - 55 q^{70} + 56 q^{71} + 48 q^{72} + 54 q^{73} + 9 q^{74} + 28 q^{75} + 82 q^{76} + 156 q^{77} + 16 q^{78} - 2 q^{79} + 98 q^{80} + 270 q^{81} - 42 q^{82} + 130 q^{83} + 229 q^{84} + 22 q^{85} + 72 q^{86} + 22 q^{87} + 61 q^{88} + 157 q^{89} + 176 q^{90} + 31 q^{91} - 18 q^{92} + 36 q^{93} + 83 q^{94} + 98 q^{95} + 111 q^{96} + 35 q^{97} + 53 q^{98} + 55 q^{99}+O(q^{100}) \) 134 * q + 9 * q^2 + 7 * q^3 + 149 * q^4 + 22 * q^5 + 20 * q^6 + 11 * q^7 + 27 * q^8 + 181 * q^9 + 15 * q^10 + 20 * q^11 + 28 * q^12 + 11 * q^13 + 17 * q^14 - 13 * q^15 + 143 * q^16 + 76 * q^17 + 23 * q^18 + 15 * q^19 + 67 * q^20 + 63 * q^21 + 2 * q^22 + 22 * q^23 + 33 * q^24 + 160 * q^25 + 65 * q^26 + 31 * q^27 + 10 * q^28 + 73 * q^29 + 20 * q^30 + 10 * q^31 + 53 * q^32 + 72 * q^33 - 7 * q^34 + 52 * q^35 + 201 * q^36 + 134 * q^37 + 70 * q^38 + 6 * q^39 + 11 * q^40 + 182 * q^41 - 15 * q^42 + 12 * q^43 + 33 * q^44 + 29 * q^45 + 24 * q^46 + 80 * q^47 + 21 * q^48 + 229 * q^49 + 37 * q^50 + 57 * q^51 - 15 * q^52 + 75 * q^53 + 95 * q^54 - 9 * q^55 + 39 * q^56 + 19 * q^57 - 21 * q^58 + 91 * q^59 + 62 * q^60 + 58 * q^61 + 108 * q^62 + 9 * q^63 + 167 * q^64 + 76 * q^65 + 105 * q^66 - 17 * q^67 + 109 * q^68 + 48 * q^69 - 55 * q^70 + 56 * q^71 + 48 * q^72 + 54 * q^73 + 9 * q^74 + 28 * q^75 + 82 * q^76 + 156 * q^77 + 16 * q^78 - 2 * q^79 + 98 * q^80 + 270 * q^81 - 42 * q^82 + 130 * q^83 + 229 * q^84 + 22 * q^85 + 72 * q^86 + 22 * q^87 + 61 * q^88 + 157 * q^89 + 176 * q^90 + 31 * q^91 - 18 * q^92 + 36 * q^93 + 83 * q^94 + 98 * q^95 + 111 * q^96 + 35 * q^97 + 53 * q^98 + 55 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more