LMFDB - Embedded newform 1997.4.a.a.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1997,4,Mod(1,1997)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1997, 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("1997.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1997 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1997.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:	$117.826814281$
Analytic rank:	$1$
Dimension:	$239$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	1997.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.88824 q^{2} +6.13395 q^{3} +15.8948 q^{4} +2.37453 q^{5} -29.9842 q^{6} -2.66042 q^{7} -38.5919 q^{8} +10.6254 q^{9} +O(q^{10})$	\(q-4.88824 q^{2} +6.13395 q^{3} +15.8948 q^{4} +2.37453 q^{5} -29.9842 q^{6} -2.66042 q^{7} -38.5919 q^{8} +10.6254 q^{9} -11.6073 q^{10} -21.0960 q^{11} +97.4982 q^{12} +87.1568 q^{13} +13.0048 q^{14} +14.5652 q^{15} +61.4874 q^{16} -48.9331 q^{17} -51.9392 q^{18} +57.2566 q^{19} +37.7428 q^{20} -16.3189 q^{21} +103.122 q^{22} +127.441 q^{23} -236.721 q^{24} -119.362 q^{25} -426.043 q^{26} -100.441 q^{27} -42.2870 q^{28} -141.847 q^{29} -71.1983 q^{30} -270.755 q^{31} +8.17003 q^{32} -129.402 q^{33} +239.196 q^{34} -6.31725 q^{35} +168.888 q^{36} +68.2687 q^{37} -279.884 q^{38} +534.615 q^{39} -91.6375 q^{40} +12.4704 q^{41} +79.7707 q^{42} -289.050 q^{43} -335.318 q^{44} +25.2302 q^{45} -622.962 q^{46} +443.856 q^{47} +377.161 q^{48} -335.922 q^{49} +583.468 q^{50} -300.153 q^{51} +1385.34 q^{52} +21.0211 q^{53} +490.981 q^{54} -50.0931 q^{55} +102.671 q^{56} +351.209 q^{57} +693.383 q^{58} +285.555 q^{59} +231.512 q^{60} -583.337 q^{61} +1323.51 q^{62} -28.2679 q^{63} -531.836 q^{64} +206.956 q^{65} +632.547 q^{66} +248.563 q^{67} -777.784 q^{68} +781.718 q^{69} +30.8802 q^{70} -68.2770 q^{71} -410.052 q^{72} -820.663 q^{73} -333.713 q^{74} -732.158 q^{75} +910.085 q^{76} +56.1243 q^{77} -2613.33 q^{78} +371.618 q^{79} +146.004 q^{80} -902.986 q^{81} -60.9582 q^{82} +569.021 q^{83} -259.387 q^{84} -116.193 q^{85} +1412.95 q^{86} -870.085 q^{87} +814.134 q^{88} -410.437 q^{89} -123.331 q^{90} -231.874 q^{91} +2025.66 q^{92} -1660.80 q^{93} -2169.67 q^{94} +135.957 q^{95} +50.1145 q^{96} +676.884 q^{97} +1642.07 q^{98} -224.153 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9}+O(q^{10}) $ 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9	\( 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9} - 273 q^{10} - 294 q^{11} - 864 q^{12} - 797 q^{13} - 220 q^{14} - 580 q^{15} + 2816 q^{16} - 439 q^{17} - 536 q^{18} - 1704 q^{19} - 933 q^{20} - 596 q^{21} - 1046 q^{22} - 829 q^{23} - 1237 q^{24} + 4364 q^{25} - 818 q^{26} - 3670 q^{27} - 3690 q^{28} - 316 q^{29} - 888 q^{30} - 2595 q^{31} - 1881 q^{32} - 2066 q^{33} - 2605 q^{34} - 2450 q^{35} + 5863 q^{36} - 1912 q^{37} - 1709 q^{38} - 914 q^{39} - 3582 q^{40} - 1064 q^{41} - 3228 q^{42} - 5184 q^{43} - 2656 q^{44} - 3967 q^{45} - 2521 q^{46} - 4909 q^{47} - 7461 q^{48} + 7193 q^{49} - 1906 q^{50} - 3240 q^{51} - 9614 q^{52} - 2722 q^{53} - 3754 q^{54} - 6018 q^{55} - 2347 q^{56} - 2032 q^{57} - 6709 q^{58} - 6318 q^{59} - 5821 q^{60} - 2990 q^{61} - 2117 q^{62} - 8738 q^{63} + 6866 q^{64} - 1738 q^{65} - 3080 q^{66} - 14729 q^{67} - 3897 q^{68} - 2080 q^{69} - 7445 q^{70} - 3240 q^{71} - 8263 q^{72} - 8828 q^{73} - 3103 q^{74} - 12716 q^{75} - 14843 q^{76} - 3818 q^{77} - 8029 q^{78} - 4794 q^{79} - 10336 q^{80} + 11899 q^{81} - 13447 q^{82} - 11434 q^{83} - 7957 q^{84} - 8188 q^{85} - 5196 q^{86} - 11266 q^{87} - 11861 q^{88} - 4845 q^{89} - 7759 q^{90} - 12734 q^{91} - 8644 q^{92} - 10130 q^{93} - 6909 q^{94} - 3686 q^{95} - 11958 q^{96} - 16108 q^{97} - 6845 q^{98} - 12372 q^{99}+O(q^{100}) \) 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9 - 273 * q^10 - 294 * q^11 - 864 * q^12 - 797 * q^13 - 220 * q^14 - 580 * q^15 + 2816 * q^16 - 439 * q^17 - 536 * q^18 - 1704 * q^19 - 933 * q^20 - 596 * q^21 - 1046 * q^22 - 829 * q^23 - 1237 * q^24 + 4364 * q^25 - 818 * q^26 - 3670 * q^27 - 3690 * q^28 - 316 * q^29 - 888 * q^30 - 2595 * q^31 - 1881 * q^32 - 2066 * q^33 - 2605 * q^34 - 2450 * q^35 + 5863 * q^36 - 1912 * q^37 - 1709 * q^38 - 914 * q^39 - 3582 * q^40 - 1064 * q^41 - 3228 * q^42 - 5184 * q^43 - 2656 * q^44 - 3967 * q^45 - 2521 * q^46 - 4909 * q^47 - 7461 * q^48 + 7193 * q^49 - 1906 * q^50 - 3240 * q^51 - 9614 * q^52 - 2722 * q^53 - 3754 * q^54 - 6018 * q^55 - 2347 * q^56 - 2032 * q^57 - 6709 * q^58 - 6318 * q^59 - 5821 * q^60 - 2990 * q^61 - 2117 * q^62 - 8738 * q^63 + 6866 * q^64 - 1738 * q^65 - 3080 * q^66 - 14729 * q^67 - 3897 * q^68 - 2080 * q^69 - 7445 * q^70 - 3240 * q^71 - 8263 * q^72 - 8828 * q^73 - 3103 * q^74 - 12716 * q^75 - 14843 * q^76 - 3818 * q^77 - 8029 * q^78 - 4794 * q^79 - 10336 * q^80 + 11899 * q^81 - 13447 * q^82 - 11434 * q^83 - 7957 * q^84 - 8188 * q^85 - 5196 * q^86 - 11266 * q^87 - 11861 * q^88 - 4845 * q^89 - 7759 * q^90 - 12734 * q^91 - 8644 * q^92 - 10130 * q^93 - 6909 * q^94 - 3686 * q^95 - 11958 * q^96 - 16108 * q^97 - 6845 * q^98 - 12372 * 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 $

$2$	−4.88824	−1.72825	−0.864126	−	0.503275i	$-0.832128\pi$
$2$	−4.88824	−1.72825	−0.864126	+	0.503275i	$0.832128\pi$
$3$	6.13395	1.18048	0.590240	−	0.807228i	$-0.299033\pi$
$3$	6.13395	1.18048	0.590240	+	0.807228i	$0.299033\pi$
$4$	15.8948	1.98686
$5$	2.37453	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$5$	2.37453	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$6$	−29.9842	−2.04017
$7$	−2.66042	−0.143649	−0.0718247	−	0.997417i	$-0.522882\pi$
$7$	−2.66042	−0.143649	−0.0718247	+	0.997417i	$0.522882\pi$
$8$	−38.5919	−1.70554
$9$	10.6254	0.393532
$10$	−11.6073	−0.367054
$11$	−21.0960	−0.578244	−0.289122	−	0.957292i	$-0.593363\pi$
$11$	−21.0960	−0.578244	−0.289122	+	0.957292i	$0.593363\pi$
$12$	97.4982	2.34544
$13$	87.1568	1.85946	0.929729	−	0.368246i	$-0.120041\pi$
$13$	87.1568	1.85946	0.929729	+	0.368246i	$0.120041\pi$
$14$	13.0048	0.248262
$15$	14.5652	0.250715
$16$	61.4874	0.960741
$17$	−48.9331	−0.698119	−0.349059	−	0.937101i	$-0.613499\pi$
$17$	−48.9331	−0.698119	−0.349059	+	0.937101i	$0.613499\pi$
$18$	−51.9392	−0.680122
$19$	57.2566	0.691345	0.345673	−	0.938355i	$-0.387651\pi$
$19$	57.2566	0.691345	0.345673	+	0.938355i	$0.387651\pi$
$20$	37.7428	0.421977
$21$	−16.3189	−0.169575
$22$	103.122	0.999352
$23$	127.441	1.15536	0.577681	−	0.816263i	$-0.303958\pi$
$23$	127.441	1.15536	0.577681	+	0.816263i	$0.303958\pi$
$24$	−236.721	−2.01335
$25$	−119.362	−0.954893
$26$	−426.043	−3.21361
$27$	−100.441	−0.715923
$28$	−42.2870	−0.285411
$29$	−141.847	−0.908290	−0.454145	−	0.890928i	$-0.650055\pi$
$29$	−141.847	−0.908290	−0.454145	+	0.890928i	$0.650055\pi$
$30$	−71.1983	−0.433299
$31$	−270.755	−1.56868	−0.784338	−	0.620333i	$-0.786997\pi$
$31$	−270.755	−1.56868	−0.784338	+	0.620333i	$0.786997\pi$
$32$	8.17003	0.0451334
$33$	−129.402	−0.682605
$34$	239.196	1.20653
$35$	−6.31725	−0.0305089
$36$	168.888	0.781891
$37$	68.2687	0.303333	0.151666	−	0.988432i	$-0.451536\pi$
$37$	68.2687	0.303333	0.151666	+	0.988432i	$0.451536\pi$
$38$	−279.884	−1.19482
$39$	534.615	2.19505
$40$	−91.6375	−0.362229
$41$	12.4704	0.0475011	0.0237506	−	0.999718i	$-0.492439\pi$
$41$	12.4704	0.0475011	0.0237506	+	0.999718i	$0.492439\pi$
$42$	79.7707	0.293069
$43$	−289.050	−1.02511	−0.512555	−	0.858654i	$-0.671301\pi$
$43$	−289.050	−1.02511	−0.512555	+	0.858654i	$0.671301\pi$
$44$	−335.318	−1.14889
$45$	25.2302	0.0835799
$46$	−622.962	−1.99676
$47$	443.856	1.37751	0.688756	−	0.724993i	$-0.258157\pi$
$47$	443.856	1.37751	0.688756	+	0.724993i	$0.258157\pi$
$48$	377.161	1.13413
$49$	−335.922	−0.979365
$50$	583.468	1.65030
$51$	−300.153	−0.824115
$52$	1385.34	3.69447
$53$	21.0211	0.0544806	0.0272403	−	0.999629i	$-0.491328\pi$
$53$	21.0211	0.0544806	0.0272403	+	0.999629i	$0.491328\pi$
$54$	490.981	1.23730
$55$	−50.0931	−0.122810
$56$	102.671	0.244999
$57$	351.209	0.816119
$58$	693.383	1.56975
$59$	285.555	0.630103	0.315051	−	0.949075i	$-0.397978\pi$
$59$	285.555	0.630103	0.315051	+	0.949075i	$0.397978\pi$
$60$	231.512	0.498135
$61$	−583.337	−1.22440	−0.612202	−	0.790701i	$-0.709716\pi$
$61$	−583.337	−1.22440	−0.612202	+	0.790701i	$0.709716\pi$
$62$	1323.51	2.71107
$63$	−28.2679	−0.0565306
$64$	−531.836	−1.03874
$65$	206.956	0.394919
$66$	632.547	1.17971
$67$	248.563	0.453236	0.226618	−	0.973984i	$-0.427233\pi$
$67$	248.563	0.453236	0.226618	+	0.973984i	$0.427233\pi$
$68$	−777.784	−1.38706
$69$	781.718	1.36388
$70$	30.8802	0.0527270
$71$	−68.2770	−0.114127	−0.0570633	−	0.998371i	$-0.518174\pi$
$71$	−68.2770	−0.114127	−0.0570633	+	0.998371i	$0.518174\pi$
$72$	−410.052	−0.671183
$73$	−820.663	−1.31577	−0.657886	−	0.753118i	$-0.728549\pi$
$73$	−820.663	−1.31577	−0.657886	+	0.753118i	$0.728549\pi$
$74$	−333.713	−0.524235
$75$	−732.158	−1.12723
$76$	910.085	1.37360
$77$	56.1243	0.0830644
$78$	−2613.33	−3.79360
$79$	371.618	0.529243	0.264622	−	0.964352i	$-0.414753\pi$
$79$	371.618	0.529243	0.264622	+	0.964352i	$0.414753\pi$
$80$	146.004	0.204046
$81$	−902.986	−1.23866
$82$	−60.9582	−0.0820939
$83$	569.021	0.752509	0.376254	−	0.926516i	$-0.377212\pi$
$83$	569.021	0.752509	0.376254	+	0.926516i	$0.377212\pi$
$84$	−259.387	−0.336921
$85$	−116.193	−0.148269
$86$	1412.95	1.77165
$87$	−870.085	−1.07222
$88$	814.134	0.986216
$89$	−410.437	−0.488834	−0.244417	−	0.969670i	$-0.578597\pi$
$89$	−410.437	−0.488834	−0.244417	+	0.969670i	$0.578597\pi$
$90$	−123.331	−0.144447
$91$	−231.874	−0.267110
$92$	2025.66	2.29554
$93$	−1660.80	−1.85179
$94$	−2169.67	−2.38069
$95$	135.957	0.146831
$96$	50.1145	0.0532791
$97$	676.884	0.708527	0.354264	−	0.935146i	$-0.384732\pi$
$97$	676.884	0.708527	0.354264	+	0.935146i	$0.384732\pi$
$98$	1642.07	1.69259
$99$	−224.153	−0.227557
$100$	−1897.23	−1.89723
$101$	76.1539	0.0750257	0.0375129	−	0.999296i	$-0.488056\pi$
$101$	76.1539	0.0750257	0.0375129	+	0.999296i	$0.488056\pi$
$102$	1467.22	1.42428
$103$	−482.859	−0.461917	−0.230959	−	0.972964i	$-0.574186\pi$
$103$	−482.859	−0.461917	−0.230959	+	0.972964i	$0.574186\pi$
$104$	−3363.54	−3.17137
$105$	−38.7497	−0.0360151
$106$	−102.756	−0.0941562
$107$	1461.70	1.32063	0.660316	−	0.750988i	$-0.270422\pi$
$107$	1461.70	1.32063	0.660316	+	0.750988i	$0.270422\pi$
$108$	−1596.50	−1.42244
$109$	−2018.27	−1.77353	−0.886767	−	0.462217i	$-0.847054\pi$
$109$	−2018.27	−1.77353	−0.886767	+	0.462217i	$0.847054\pi$
$110$	244.867	0.212247
$111$	418.757	0.358078
$112$	−163.583	−0.138010
$113$	−514.968	−0.428709	−0.214354	−	0.976756i	$-0.568765\pi$
$113$	−514.968	−0.428709	−0.214354	+	0.976756i	$0.568765\pi$
$114$	−1716.79	−1.41046
$115$	302.613	0.245381
$116$	−2254.64	−1.80464
$117$	926.072	0.731755
$118$	−1395.86	−1.08898
$119$	130.183	0.100284
$120$	−562.100	−0.427604
$121$	−885.959	−0.665634
$122$	2851.49	2.11608
$123$	76.4927	0.0560741
$124$	−4303.60	−3.11673
$125$	−580.244	−0.415188
$126$	138.180	0.0976991
$127$	−754.321	−0.527049	−0.263524	−	0.964653i	$-0.584885\pi$
$127$	−754.321	−0.527049	−0.263524	+	0.964653i	$0.584885\pi$
$128$	2534.38	1.75008
$129$	−1773.02	−1.21012
$130$	−1011.65	−0.682520
$131$	−552.772	−0.368671	−0.184336	−	0.982863i	$-0.559013\pi$
$131$	−552.772	−0.368671	−0.184336	+	0.982863i	$0.559013\pi$
$132$	−2056.82	−1.35624
$133$	−152.327	−0.0993113
$134$	−1215.04	−0.783307
$135$	−238.501	−0.152051
$136$	1888.42	1.19067
$137$	−1981.45	−1.23567	−0.617835	−	0.786307i	$-0.711990\pi$
$137$	−1981.45	−1.23567	−0.617835	+	0.786307i	$0.711990\pi$
$138$	−3821.22	−2.35713
$139$	−766.803	−0.467909	−0.233955	−	0.972248i	$-0.575167\pi$
$139$	−766.803	−0.467909	−0.233955	+	0.972248i	$0.575167\pi$
$140$	−100.412	−0.0606167
$141$	2722.59	1.62612
$142$	333.754	0.197240
$143$	−1838.66	−1.07522
$144$	653.326	0.378082
$145$	−336.821	−0.192906
$146$	4011.59	2.27399
$147$	−2060.53	−1.15612
$148$	1085.12	0.602678
$149$	−224.491	−0.123430	−0.0617148	−	0.998094i	$-0.519657\pi$
$149$	−224.491	−0.123430	−0.0617148	+	0.998094i	$0.519657\pi$
$150$	3578.96	1.94814
$151$	−2721.48	−1.46669	−0.733346	−	0.679855i	$-0.762043\pi$
$151$	−2721.48	−1.46669	−0.733346	+	0.679855i	$0.762043\pi$
$152$	−2209.64	−1.17911
$153$	−519.932	−0.274732
$154$	−274.349	−0.143556
$155$	−642.915	−0.333162
$156$	8497.63	4.36125
$157$	1625.44	0.826271	0.413135	−	0.910670i	$-0.364434\pi$
$157$	1625.44	0.826271	0.413135	+	0.910670i	$0.364434\pi$
$158$	−1816.55	−0.914666
$159$	128.942	0.0643132
$160$	19.4000	0.00958563
$161$	−339.047	−0.165967
$162$	4414.01	2.14072
$163$	2808.65	1.34964	0.674818	−	0.737984i	$-0.264222\pi$
$163$	2808.65	1.34964	0.674818	+	0.737984i	$0.264222\pi$
$164$	198.215	0.0943779
$165$	−307.268	−0.144975
$166$	−2781.51	−1.30052
$167$	1665.85	0.771899	0.385949	−	0.922520i	$-0.373874\pi$
$167$	1665.85	0.771899	0.385949	+	0.922520i	$0.373874\pi$
$168$	629.777	0.289217
$169$	5399.31	2.45758
$170$	567.979	0.256247
$171$	608.372	0.272066
$172$	−4594.41	−2.03675
$173$	1740.92	0.765085	0.382543	−	0.923938i	$-0.375049\pi$
$173$	1740.92	0.765085	0.382543	+	0.923938i	$0.375049\pi$
$174$	4253.18	1.85306
$175$	317.552	0.137170
$176$	−1297.14	−0.555543
$177$	1751.58	0.743824
$178$	2006.31	0.844829
$179$	377.491	0.157626	0.0788129	−	0.996889i	$-0.474887\pi$
$179$	377.491	0.157626	0.0788129	+	0.996889i	$0.474887\pi$
$180$	401.030	0.166061
$181$	−2412.24	−0.990611	−0.495306	−	0.868719i	$-0.664944\pi$
$181$	−2412.24	−0.990611	−0.495306	+	0.868719i	$0.664944\pi$
$182$	1133.45	0.461633
$183$	−3578.16	−1.44538
$184$	−4918.19	−1.97051
$185$	162.106	0.0644230
$186$	8118.36	3.20036
$187$	1032.29	0.403683
$188$	7055.02	2.73692
$189$	267.216	0.102842
$190$	−664.592	−0.253761
$191$	−1925.47	−0.729435	−0.364717	−	0.931118i	$-0.618834\pi$
$191$	−1925.47	−0.729435	−0.364717	+	0.931118i	$0.618834\pi$
$192$	−3262.26	−1.22621
$193$	−423.393	−0.157909	−0.0789546	−	0.996878i	$-0.525158\pi$
$193$	−423.393	−0.157909	−0.0789546	+	0.996878i	$0.525158\pi$
$194$	−3308.77	−1.22451
$195$	1269.46	0.466194
$196$	−5339.43	−1.94586
$197$	−4603.44	−1.66488	−0.832440	−	0.554115i	$-0.813057\pi$
$197$	−4603.44	−1.66488	−0.832440	+	0.554115i	$0.813057\pi$
$198$	1095.71	0.393276
$199$	−890.171	−0.317098	−0.158549	−	0.987351i	$-0.550682\pi$
$199$	−890.171	−0.317098	−0.158549	+	0.987351i	$0.550682\pi$
$200$	4606.39	1.62860
$201$	1524.67	0.535036
$202$	−372.258	−0.129663
$203$	377.374	0.130475
$204$	−4770.89	−1.63740
$205$	29.6113	0.0100885
$206$	2360.33	0.798309
$207$	1354.11	0.454671
$208$	5359.05	1.78646
$209$	−1207.88	−0.399766
$210$	189.418	0.0622432
$211$	−1843.91	−0.601612	−0.300806	−	0.953685i	$-0.597256\pi$
$211$	−1843.91	−0.601612	−0.300806	+	0.953685i	$0.597256\pi$
$212$	334.127	0.108245
$213$	−418.808	−0.134724
$214$	−7145.12	−2.28238
$215$	−686.358	−0.217717
$216$	3876.22	1.22103
$217$	720.322	0.225339
$218$	9865.78	3.06511
$219$	−5033.91	−1.55324
$220$	−796.221	−0.244006
$221$	−4264.85	−1.29812
$222$	−2046.98	−0.618849
$223$	2853.67	0.856933	0.428466	−	0.903558i	$-0.359054\pi$
$223$	2853.67	0.856933	0.428466	+	0.903558i	$0.359054\pi$
$224$	−21.7357	−0.00648339
$225$	−1268.26	−0.375781
$226$	2517.28	0.740917
$227$	3243.08	0.948242	0.474121	−	0.880460i	$-0.342766\pi$
$227$	3243.08	0.948242	0.474121	+	0.880460i	$0.342766\pi$
$228$	5582.41	1.62151
$229$	1714.98	0.494887	0.247443	−	0.968902i	$-0.420410\pi$
$229$	1714.98	0.494887	0.247443	+	0.968902i	$0.420410\pi$
$230$	−1479.24	−0.424080
$231$	344.264	0.0980558
$232$	5474.16	1.54912
$233$	2609.32	0.733658	0.366829	−	0.930288i	$-0.380443\pi$
$233$	2609.32	0.733658	0.366829	+	0.930288i	$0.380443\pi$
$234$	−4526.86	−1.26466
$235$	1053.95	0.292562
$236$	4538.85	1.25192
$237$	2279.48	0.624761
$238$	−636.364	−0.173317
$239$	2567.85	0.694980	0.347490	−	0.937684i	$-0.387034\pi$
$239$	2567.85	0.694980	0.347490	+	0.937684i	$0.387034\pi$
$240$	895.579	0.240872
$241$	−2150.96	−0.574920	−0.287460	−	0.957793i	$-0.592811\pi$
$241$	−2150.96	−0.574920	−0.287460	+	0.957793i	$0.592811\pi$
$242$	4330.78	1.15038
$243$	−2826.96	−0.746295
$244$	−9272.06	−2.43272
$245$	−797.657	−0.208002
$246$	−373.914	−0.0969102
$247$	4990.30	1.28553
$248$	10448.9	2.67543
$249$	3490.35	0.888321
$250$	2836.37	0.717550
$251$	6112.64	1.53716	0.768579	−	0.639755i	$-0.220964\pi$
$251$	6112.64	1.53716	0.768579	+	0.639755i	$0.220964\pi$
$252$	−449.315	−0.112318
$253$	−2688.50	−0.668081
$254$	3687.30	0.910873
$255$	−712.722	−0.175029
$256$	−8133.96	−1.98583
$257$	426.739	0.103577	0.0517885	−	0.998658i	$-0.483508\pi$
$257$	426.739	0.103577	0.0517885	+	0.998658i	$0.483508\pi$
$258$	8666.94	2.09140
$259$	−181.624	−0.0435735
$260$	3289.54	0.784648
$261$	−1507.18	−0.357441
$262$	2702.08	0.637157
$263$	−4339.34	−1.01740	−0.508698	−	0.860945i	$-0.669873\pi$
$263$	−4339.34	−1.01740	−0.508698	+	0.860945i	$0.669873\pi$
$264$	4993.86	1.16421
$265$	49.9152	0.0115708
$266$	744.609	0.171635
$267$	−2517.60	−0.577059
$268$	3950.88	0.900515
$269$	−672.264	−0.152374	−0.0761871	−	0.997094i	$-0.524275\pi$
$269$	−672.264	−0.152374	−0.0761871	+	0.997094i	$0.524275\pi$
$270$	1165.85	0.262782
$271$	2012.97	0.451214	0.225607	−	0.974218i	$-0.427563\pi$
$271$	2012.97	0.451214	0.225607	+	0.974218i	$0.427563\pi$
$272$	−3008.77	−0.670711
$273$	−1422.30	−0.315318
$274$	9685.81	2.13555
$275$	2518.05	0.552161
$276$	12425.3	2.70983
$277$	2161.87	0.468932	0.234466	−	0.972124i	$-0.424666\pi$
$277$	2161.87	0.468932	0.234466	+	0.972124i	$0.424666\pi$
$278$	3748.31	0.808665
$279$	−2876.86	−0.617324
$280$	243.795	0.0520340
$281$	−939.606	−0.199474	−0.0997370	−	0.995014i	$-0.531800\pi$
$281$	−939.606	−0.199474	−0.0997370	+	0.995014i	$0.531800\pi$
$282$	−13308.7	−2.81035
$283$	−1439.06	−0.302273	−0.151137	−	0.988513i	$-0.548293\pi$
$283$	−1439.06	−0.302273	−0.151137	+	0.988513i	$0.548293\pi$
$284$	−1085.25	−0.226753
$285$	833.956	0.173331
$286$	8987.80	1.85825
$287$	−33.1765	−0.00682351
$288$	86.8094	0.0177614
$289$	−2518.55	−0.512630
$290$	1646.46	0.333391
$291$	4151.97	0.836402
$292$	−13044.3	−2.61425
$293$	−3827.84	−0.763225	−0.381613	−	0.924322i	$-0.624631\pi$
$293$	−3827.84	−0.763225	−0.381613	+	0.924322i	$0.624631\pi$
$294$	10072.4	1.99807
$295$	678.058	0.133824
$296$	−2634.62	−0.517345
$297$	2118.91	0.413978
$298$	1097.36	0.213317
$299$	11107.4	2.14835
$300$	−11637.5	−2.23965
$301$	768.996	0.147256
$302$	13303.2	2.53481
$303$	467.124	0.0885663
$304$	3520.56	0.664204
$305$	−1385.15	−0.260044
$306$	2541.55	0.474806
$307$	−1409.66	−0.262064	−0.131032	−	0.991378i	$-0.541829\pi$
$307$	−1409.66	−0.262064	−0.131032	+	0.991378i	$0.541829\pi$
$308$	892.087	0.165037
$309$	−2961.83	−0.545284
$310$	3142.72	0.575788
$311$	−6817.44	−1.24303	−0.621514	−	0.783403i	$-0.713482\pi$
$311$	−6817.44	−1.24303	−0.621514	+	0.783403i	$0.713482\pi$
$312$	−20631.8	−3.74374
$313$	−1277.73	−0.230739	−0.115370	−	0.993323i	$-0.536805\pi$
$313$	−1277.73	−0.230739	−0.115370	+	0.993323i	$0.536805\pi$
$314$	−7945.55	−1.42800
$315$	−67.1230	−0.0120062
$316$	5906.80	1.05153
$317$	−2323.84	−0.411734	−0.205867	−	0.978580i	$-0.566001\pi$
$317$	−2323.84	−0.411734	−0.205867	+	0.978580i	$0.566001\pi$
$318$	−630.301	−0.111149
$319$	2992.41	0.525213
$320$	−1262.86	−0.220613
$321$	8965.98	1.55898
$322$	1657.34	0.286833
$323$	−2801.74	−0.482641
$324$	−14352.8	−2.46105
$325$	−10403.2	−1.77558
$326$	−13729.4	−2.33251
$327$	−12380.0	−2.09362
$328$	−481.255	−0.0810149
$329$	−1180.84	−0.197879
$330$	1502.00	0.250553
$331$	−1225.13	−0.203442	−0.101721	−	0.994813i	$-0.532435\pi$
$331$	−1225.13	−0.203442	−0.101721	+	0.994813i	$0.532435\pi$
$332$	9044.51	1.49513
$333$	725.379	0.119371
$334$	−8143.05	−1.33404
$335$	590.220	0.0962603
$336$	−1003.41	−0.162918
$337$	6717.49	1.08583	0.542916	−	0.839787i	$-0.317320\pi$
$337$	6717.49	1.08583	0.542916	+	0.839787i	$0.317320\pi$
$338$	−26393.1	−4.24732
$339$	−3158.79	−0.506082
$340$	−1846.87	−0.294590
$341$	5711.84	0.907078
$342$	−2973.86	−0.470199
$343$	1806.22	0.284334
$344$	11155.0	1.74836
$345$	1856.21	0.289667
$346$	−8510.03	−1.32226
$347$	1392.15	0.215373	0.107687	−	0.994185i	$-0.465656\pi$
$347$	1392.15	0.215373	0.107687	+	0.994185i	$0.465656\pi$
$348$	−13829.9	−2.13034
$349$	−6907.96	−1.05953	−0.529763	−	0.848146i	$-0.677719\pi$
$349$	−6907.96	−1.05953	−0.529763	+	0.848146i	$0.677719\pi$
$350$	−1552.27	−0.237064
$351$	−8754.14	−1.33123
$352$	−172.355	−0.0260981
$353$	−1914.64	−0.288685	−0.144343	−	0.989528i	$-0.546107\pi$
$353$	−1914.64	−0.288685	−0.144343	+	0.989528i	$0.546107\pi$
$354$	−8562.13	−1.28551
$355$	−162.126	−0.0242387
$356$	−6523.83	−0.971243
$357$	798.535	0.118384
$358$	−1845.27	−0.272417
$359$	−11505.7	−1.69150	−0.845752	−	0.533576i	$-0.820848\pi$
$359$	−11505.7	−1.69150	−0.845752	+	0.533576i	$0.820848\pi$
$360$	−973.681	−0.142549
$361$	−3580.68	−0.522042
$362$	11791.6	1.71203
$363$	−5434.43	−0.785767
$364$	−3685.60	−0.530709
$365$	−1948.69	−0.279449
$366$	17490.9	2.49799
$367$	431.549	0.0613806	0.0306903	−	0.999529i	$-0.490229\pi$
$367$	431.549	0.0613806	0.0306903	+	0.999529i	$0.490229\pi$
$368$	7836.03	1.11000
$369$	132.502	0.0186932
$370$	−792.412	−0.111339
$371$	−55.9250	−0.00782610
$372$	−26398.1	−3.67924
$373$	−10260.2	−1.42427	−0.712134	−	0.702044i	$-0.752271\pi$
$373$	−10260.2	−1.42427	−0.712134	+	0.702044i	$0.752271\pi$
$374$	−5046.09	−0.697666
$375$	−3559.19	−0.490121
$376$	−17129.2	−2.34940
$377$	−12363.0	−1.68893
$378$	−1306.22	−0.177737
$379$	−1124.64	−0.152424	−0.0762122	−	0.997092i	$-0.524283\pi$
$379$	−1124.64	−0.152424	−0.0762122	+	0.997092i	$0.524283\pi$
$380$	2161.02	0.291732
$381$	−4626.97	−0.622170
$382$	9412.14	1.26065
$383$	−5696.60	−0.760008	−0.380004	−	0.924985i	$-0.624077\pi$
$383$	−5696.60	−0.760008	−0.380004	+	0.924985i	$0.624077\pi$
$384$	15545.8	2.06593
$385$	133.269	0.0176416
$386$	2069.64	0.272907
$387$	−3071.26	−0.403414
$388$	10759.0	1.40774
$389$	2911.00	0.379418	0.189709	−	0.981840i	$-0.439246\pi$
$389$	2911.00	0.379418	0.189709	+	0.981840i	$0.439246\pi$
$390$	−6205.42	−0.805701
$391$	−6236.09	−0.806579
$392$	12963.9	1.67034
$393$	−3390.68	−0.435209
$394$	22502.7	2.87733
$395$	882.416	0.112403
$396$	−3562.87	−0.452124
$397$	15138.6	1.91382	0.956908	−	0.290391i	$-0.0937855\pi$
$397$	15138.6	1.91382	0.956908	+	0.290391i	$0.0937855\pi$
$398$	4351.37	0.548026
$399$	−934.365	−0.117235
$400$	−7339.24	−0.917405
$401$	−5582.19	−0.695165	−0.347582	−	0.937649i	$-0.612997\pi$
$401$	−5582.19	−0.695165	−0.347582	+	0.937649i	$0.612997\pi$
$402$	−7452.97	−0.924678
$403$	−23598.1	−2.91689
$404$	1210.45	0.149065
$405$	−2144.17	−0.263073
$406$	−1844.69	−0.225494
$407$	−1440.20	−0.175400
$408$	11583.5	1.40556
$409$	1584.61	0.191574	0.0957871	−	0.995402i	$-0.469463\pi$
$409$	1584.61	0.191574	0.0957871	+	0.995402i	$0.469463\pi$
$410$	−144.747	−0.0174355
$411$	−12154.1	−1.45868
$412$	−7674.96	−0.917763
$413$	−759.697	−0.0905139
$414$	−6619.20	−0.785787
$415$	1351.16	0.159821
$416$	712.073	0.0839237
$417$	−4703.53	−0.552357
$418$	5904.43	0.690897
$419$	−7096.10	−0.827368	−0.413684	−	0.910421i	$-0.635758\pi$
$419$	−7096.10	−0.827368	−0.413684	+	0.910421i	$0.635758\pi$
$420$	−615.921	−0.0715568
$421$	−8279.49	−0.958474	−0.479237	−	0.877685i	$-0.659087\pi$
$421$	−8279.49	−0.958474	−0.479237	+	0.877685i	$0.659087\pi$
$422$	9013.48	1.03974
$423$	4716.13	0.542095
$424$	−811.244	−0.0929186
$425$	5840.73	0.666629
$426$	2047.23	0.232837
$427$	1551.92	0.175885
$428$	23233.5	2.62391
$429$	−11278.2	−1.26928
$430$	3355.08	0.376271
$431$	−7314.07	−0.817417	−0.408708	−	0.912665i	$-0.634021\pi$
$431$	−7314.07	−0.817417	−0.408708	+	0.912665i	$0.634021\pi$
$432$	−6175.87	−0.687817
$433$	14736.5	1.63555	0.817773	−	0.575541i	$-0.195209\pi$
$433$	14736.5	1.63555	0.817773	+	0.575541i	$0.195209\pi$
$434$	−3521.10	−0.389443
$435$	−2066.04	−0.227722
$436$	−32080.1	−3.52376
$437$	7296.84	0.798754
$438$	24606.9	2.68439
$439$	−5642.22	−0.613413	−0.306707	−	0.951804i	$-0.599227\pi$
$439$	−5642.22	−0.613413	−0.306707	+	0.951804i	$0.599227\pi$
$440$	1933.19	0.209457
$441$	−3569.29	−0.385411
$442$	20847.6	2.24348
$443$	2207.33	0.236735	0.118367	−	0.992970i	$-0.462234\pi$
$443$	2207.33	0.236735	0.118367	+	0.992970i	$0.462234\pi$
$444$	6656.08	0.711449
$445$	−974.594	−0.103821
$446$	−13949.4	−1.48100
$447$	−1377.02	−0.145706
$448$	1414.91	0.149215
$449$	−1562.78	−0.164259	−0.0821295	−	0.996622i	$-0.526172\pi$
$449$	−1562.78	−0.164259	−0.0821295	+	0.996622i	$0.526172\pi$
$450$	6199.55	0.649444
$451$	−263.075	−0.0274672
$452$	−8185.34	−0.851783
$453$	−16693.4	−1.73140
$454$	−15852.9	−1.63880
$455$	−550.591	−0.0567299
$456$	−13553.8	−1.39192
$457$	6299.02	0.644760	0.322380	−	0.946610i	$-0.395517\pi$
$457$	6299.02	0.644760	0.322380	+	0.946610i	$0.395517\pi$
$458$	−8383.22	−0.855289
$459$	4914.90	0.499800
$460$	4809.98	0.487536
$461$	5614.12	0.567192	0.283596	−	0.958944i	$-0.408473\pi$
$461$	5614.12	0.567192	0.283596	+	0.958944i	$0.408473\pi$
$462$	−1682.84	−0.169465
$463$	−6659.50	−0.668452	−0.334226	−	0.942493i	$-0.608475\pi$
$463$	−6659.50	−0.668452	−0.334226	+	0.942493i	$0.608475\pi$
$464$	−8721.83	−0.872631
$465$	−3943.61	−0.393291
$466$	−12755.0	−1.26795
$467$	−16874.0	−1.67203	−0.836015	−	0.548707i	$-0.815120\pi$
$467$	−16874.0	−1.67203	−0.836015	+	0.548707i	$0.815120\pi$
$468$	14719.8	1.45389
$469$	−661.284	−0.0651071
$470$	−5151.95	−0.505621
$471$	9970.39	0.975395
$472$	−11020.1	−1.07466
$473$	6097.81	0.592764
$474$	−11142.7	−1.07974
$475$	−6834.24	−0.660161
$476$	2069.23	0.199250
$477$	223.357	0.0214398
$478$	−12552.2	−1.20110
$479$	1248.33	0.119076	0.0595382	−	0.998226i	$-0.481037\pi$
$479$	1248.33	0.119076	0.0595382	+	0.998226i	$0.481037\pi$
$480$	118.998	0.0113156
$481$	5950.08	0.564034
$482$	10514.4	0.993608
$483$	−2079.70	−0.195921
$484$	−14082.2	−1.32252
$485$	1607.28	0.150480
$486$	13818.8	1.28979
$487$	−13778.1	−1.28202	−0.641011	−	0.767532i	$-0.721485\pi$
$487$	−13778.1	−1.28202	−0.641011	+	0.767532i	$0.721485\pi$
$488$	22512.1	2.08827
$489$	17228.1	1.59322
$490$	3899.13	0.359479
$491$	−4805.96	−0.441731	−0.220866	−	0.975304i	$-0.570888\pi$
$491$	−4805.96	−0.441731	−0.220866	+	0.975304i	$0.570888\pi$
$492$	1215.84	0.111411
$493$	6941.03	0.634094
$494$	−24393.8	−2.22171
$495$	−532.256	−0.0483296
$496$	−16648.0	−1.50709
$497$	181.646	0.0163942
$498$	−17061.7	−1.53524
$499$	−19232.8	−1.72541	−0.862703	−	0.505711i	$-0.831230\pi$
$499$	−19232.8	−1.72541	−0.862703	+	0.505711i	$0.831230\pi$
$500$	−9222.88	−0.824920
$501$	10218.2	0.911211
$502$	−29880.0	−2.65660
$503$	17304.2	1.53391	0.766955	−	0.641700i	$-0.221771\pi$
$503$	17304.2	1.53391	0.766955	+	0.641700i	$0.221771\pi$
$504$	1090.91	0.0964149
$505$	180.830	0.0159343
$506$	13142.0	1.15461
$507$	33119.1	2.90112
$508$	−11989.8	−1.04717
$509$	−16550.2	−1.44121	−0.720604	−	0.693347i	$-0.756135\pi$
$509$	−16550.2	−1.44121	−0.720604	+	0.693347i	$0.756135\pi$
$510$	3483.95	0.302494
$511$	2183.31	0.189010
$512$	19485.7	1.68194
$513$	−5750.92	−0.494950
$514$	−2086.00	−0.179007
$515$	−1146.56	−0.0981039
$516$	−28181.9	−2.40434
$517$	−9363.59	−0.796538
$518$	887.819	0.0753060
$519$	10678.7	0.903167
$520$	−7986.83	−0.673549
$521$	9810.07	0.824927	0.412464	−	0.910974i	$-0.364668\pi$
$521$	9810.07	0.824927	0.412464	+	0.910974i	$0.364668\pi$
$522$	7367.45	0.617748
$523$	−11093.6	−0.927515	−0.463757	−	0.885962i	$-0.653499\pi$
$523$	−11093.6	−0.927515	−0.463757	+	0.885962i	$0.653499\pi$
$524$	−8786.23	−0.732497
$525$	1947.85	0.161926
$526$	21211.7	1.75832
$527$	13248.9	1.09512
$528$	−7956.59	−0.655807
$529$	4074.24	0.334860
$530$	−243.997	−0.0199973
$531$	3034.12	0.247965
$532$	−2421.21	−0.197317
$533$	1086.88	0.0883263
$534$	12306.6	0.997303
$535$	3470.84	0.280481
$536$	−9592.52	−0.773011
$537$	2315.51	0.186074
$538$	3286.19	0.263341
$539$	7086.61	0.566312
$540$	−3790.93	−0.302103
$541$	465.628	0.0370036	0.0185018	−	0.999829i	$-0.494110\pi$
$541$	465.628	0.0370036	0.0185018	+	0.999829i	$0.494110\pi$
$542$	−9839.86	−0.779812
$543$	−14796.6	−1.16940
$544$	−399.785	−0.0315085
$545$	−4792.44	−0.376671
$546$	6952.55	0.544948
$547$	17820.1	1.39293	0.696463	−	0.717593i	$-0.254756\pi$
$547$	17820.1	1.39293	0.696463	+	0.717593i	$0.254756\pi$
$548$	−31494.9	−2.45510
$549$	−6198.17	−0.481842
$550$	−12308.8	−0.954274
$551$	−8121.70	−0.627942
$552$	−30168.0	−2.32615
$553$	−988.660	−0.0760255
$554$	−10567.7	−0.810432
$555$	994.350	0.0760501
$556$	−12188.2	−0.929668
$557$	8364.68	0.636307	0.318153	−	0.948039i	$-0.396937\pi$
$557$	8364.68	0.636307	0.318153	+	0.948039i	$0.396937\pi$
$558$	14062.8	1.06689
$559$	−25192.7	−1.90615
$560$	−388.431	−0.0293111
$561$	6332.03	0.476539
$562$	4593.01	0.344741
$563$	−13543.8	−1.01386	−0.506930	−	0.861987i	$-0.669220\pi$
$563$	−13543.8	−1.01386	−0.506930	+	0.861987i	$0.669220\pi$
$564$	43275.2	3.23088
$565$	−1222.81	−0.0910510
$566$	7034.48	0.522405
$567$	2402.33	0.177933
$568$	2634.94	0.194647
$569$	22876.3	1.68546	0.842729	−	0.538338i	$-0.180948\pi$
$569$	22876.3	1.68546	0.842729	+	0.538338i	$0.180948\pi$
$570$	−4076.57	−0.299559
$571$	13642.0	0.999825	0.499913	−	0.866076i	$-0.333365\pi$
$571$	13642.0	0.999825	0.499913	+	0.866076i	$0.333365\pi$
$572$	−29225.2	−2.13631
$573$	−11810.7	−0.861082
$574$	162.175	0.0117927
$575$	−15211.6	−1.10325
$576$	−5650.95	−0.408778
$577$	−2880.37	−0.207819	−0.103909	−	0.994587i	$-0.533135\pi$
$577$	−2880.37	−0.207819	−0.103909	+	0.994587i	$0.533135\pi$
$578$	12311.3	0.885954
$579$	−2597.07	−0.186409
$580$	−5353.71	−0.383277
$581$	−1513.84	−0.108097
$582$	−20295.8	−1.44551
$583$	−443.461	−0.0315031
$584$	31670.9	2.24410
$585$	2198.98	0.155413
$586$	18711.4	1.31905
$587$	−16592.9	−1.16672	−0.583359	−	0.812214i	$-0.698262\pi$
$587$	−16592.9	−1.16672	−0.583359	+	0.812214i	$0.698262\pi$
$588$	−32751.8	−2.29704
$589$	−15502.5	−1.08450
$590$	−3314.51	−0.231282
$591$	−28237.3	−1.96536
$592$	4197.67	0.291424
$593$	−8139.89	−0.563685	−0.281843	−	0.959461i	$-0.590946\pi$
$593$	−8139.89	−0.563685	−0.281843	+	0.959461i	$0.590946\pi$
$594$	−10357.7	−0.715459
$595$	309.123	0.0212988
$596$	−3568.25	−0.245237
$597$	−5460.27	−0.374328
$598$	−54295.4	−3.71288
$599$	11148.1	0.760430	0.380215	−	0.924898i	$-0.375850\pi$
$599$	11148.1	0.760430	0.380215	+	0.924898i	$0.375850\pi$
$600$	28255.4	1.92253
$601$	7345.63	0.498560	0.249280	−	0.968431i	$-0.419806\pi$
$601$	7345.63	0.498560	0.249280	+	0.968431i	$0.419806\pi$
$602$	−3759.03	−0.254496
$603$	2641.07	0.178363
$604$	−43257.4	−2.91411
$605$	−2103.73	−0.141370
$606$	−2283.41	−0.153065
$607$	8718.94	0.583017	0.291508	−	0.956568i	$-0.405843\pi$
$607$	8718.94	0.583017	0.291508	+	0.956568i	$0.405843\pi$
$608$	467.788	0.0312028
$609$	2314.79	0.154023
$610$	6770.94	0.449422
$611$	38685.1	2.56142
$612$	−8264.23	−0.545853
$613$	10543.9	0.694724	0.347362	−	0.937731i	$-0.387077\pi$
$613$	10543.9	0.694724	0.347362	+	0.937731i	$0.387077\pi$
$614$	6890.75	0.452912
$615$	181.634	0.0119093
$616$	−2165.94	−0.141669
$617$	−23789.0	−1.55220	−0.776102	−	0.630607i	$-0.782806\pi$
$617$	−23789.0	−1.55220	−0.776102	+	0.630607i	$0.782806\pi$
$618$	14478.1	0.942388
$619$	−7608.36	−0.494032	−0.247016	−	0.969011i	$-0.579450\pi$
$619$	−7608.36	−0.494032	−0.247016	+	0.969011i	$0.579450\pi$
$620$	−10219.0	−0.661945
$621$	−12800.3	−0.827150
$622$	33325.2	2.14826
$623$	1091.94	0.0702207
$624$	32872.1	2.10888
$625$	13542.4	0.866713
$626$	6245.83	0.398776
$627$	−7409.11	−0.471916
$628$	25836.2	1.64168
$629$	−3340.60	−0.211762
$630$	328.113	0.0207498
$631$	11191.0	0.706032	0.353016	−	0.935617i	$-0.385156\pi$
$631$	11191.0	0.706032	0.353016	+	0.935617i	$0.385156\pi$
$632$	−14341.4	−0.902644
$633$	−11310.5	−0.710191
$634$	11359.5	0.711581
$635$	−1791.16	−0.111937
$636$	2049.52	0.127781
$637$	−29277.9	−1.82109
$638$	−14627.6	−0.907701
$639$	−725.467	−0.0449124
$640$	6017.96	0.371689
$641$	23069.3	1.42150	0.710752	−	0.703442i	$-0.248355\pi$
$641$	23069.3	1.42150	0.710752	+	0.703442i	$0.248355\pi$
$642$	−43827.8	−2.69431
$643$	−5744.09	−0.352294	−0.176147	−	0.984364i	$-0.556363\pi$
$643$	−5744.09	−0.352294	−0.176147	+	0.984364i	$0.556363\pi$
$644$	−5389.11	−0.329752
$645$	−4210.09	−0.257011
$646$	13695.6	0.834126
$647$	−3984.08	−0.242087	−0.121043	−	0.992647i	$-0.538624\pi$
$647$	−3984.08	−0.242087	−0.121043	+	0.992647i	$0.538624\pi$
$648$	34847.9	2.11259
$649$	−6024.07	−0.364353
$650$	50853.2	3.06865
$651$	4418.42	0.266009
$652$	44643.1	2.68153
$653$	17596.2	1.05451	0.527253	−	0.849708i	$-0.323222\pi$
$653$	17596.2	1.05451	0.527253	+	0.849708i	$0.323222\pi$
$654$	60516.2	3.61830
$655$	−1312.57	−0.0783000
$656$	766.772	0.0456363
$657$	−8719.84	−0.517798
$658$	5772.25	0.341984
$659$	12046.4	0.712080	0.356040	−	0.934471i	$-0.384127\pi$
$659$	12046.4	0.712080	0.356040	+	0.934471i	$0.384127\pi$
$660$	−4883.98	−0.288044
$661$	−30282.9	−1.78195	−0.890974	−	0.454055i	$-0.849977\pi$
$661$	−30282.9	−1.78195	−0.890974	+	0.454055i	$0.849977\pi$
$662$	5988.72	0.351599
$663$	−26160.4	−1.53241
$664$	−21959.6	−1.28343
$665$	−361.704	−0.0210922
$666$	−3545.82	−0.206303
$667$	−18077.2	−1.04940
$668$	26478.4	1.53365
$669$	17504.3	1.01159
$670$	−2885.14	−0.166362
$671$	12306.1	0.708005
$672$	−133.326	−0.00765351
$673$	2363.93	0.135398	0.0676990	−	0.997706i	$-0.478434\pi$
$673$	2363.93	0.135398	0.0676990	+	0.997706i	$0.478434\pi$
$674$	−32836.7	−1.87659
$675$	11988.8	0.683630
$676$	85821.1	4.88286
$677$	−11586.9	−0.657787	−0.328893	−	0.944367i	$-0.606676\pi$
$677$	−11586.9	−0.657787	−0.328893	+	0.944367i	$0.606676\pi$
$678$	15440.9	0.874638
$679$	−1800.80	−0.101779
$680$	4484.11	0.252879
$681$	19892.9	1.11938
$682$	−27920.8	−1.56766
$683$	−8367.65	−0.468784	−0.234392	−	0.972142i	$-0.575310\pi$
$683$	−8367.65	−0.468784	−0.234392	+	0.972142i	$0.575310\pi$
$684$	9669.97	0.540557
$685$	−4705.01	−0.262437
$686$	−8829.23	−0.491402
$687$	10519.6	0.584203
$688$	−17773.0	−0.984866
$689$	1832.13	0.101304
$690$	−9073.60	−0.500617
$691$	−17082.4	−0.940441	−0.470221	−	0.882549i	$-0.655826\pi$
$691$	−17082.4	−0.940441	−0.470221	+	0.882549i	$0.655826\pi$
$692$	27671.7	1.52011
$693$	596.341	0.0326885
$694$	−6805.16	−0.372220
$695$	−1820.79	−0.0993765
$696$	33578.2	1.82871
$697$	−610.214	−0.0331614
$698$	33767.7	1.83113
$699$	16005.4	0.866068
$700$	5047.45	0.272537
$701$	18397.3	0.991235	0.495617	−	0.868541i	$-0.334942\pi$
$701$	18397.3	0.991235	0.495617	+	0.868541i	$0.334942\pi$
$702$	42792.3	2.30070
$703$	3908.83	0.209707
$704$	11219.6	0.600647
$705$	6464.87	0.345363
$706$	9359.21	0.498921
$707$	−202.602	−0.0107774
$708$	27841.1	1.47787
$709$	−15804.2	−0.837152	−0.418576	−	0.908182i	$-0.637471\pi$
$709$	−15804.2	−0.837152	−0.418576	+	0.908182i	$0.637471\pi$
$710$	792.509	0.0418906
$711$	3948.57	0.208274
$712$	15839.5	0.833724
$713$	−34505.3	−1.81239
$714$	−3903.43	−0.204597
$715$	−4365.95	−0.228360
$716$	6000.16	0.313180
$717$	15751.1	0.820410
$718$	56242.8	2.92335
$719$	−30787.5	−1.59691	−0.798455	−	0.602054i	$-0.794349\pi$
$719$	−30787.5	−1.59691	−0.798455	+	0.602054i	$0.794349\pi$
$720$	1551.34	0.0802987
$721$	1284.61	0.0663541
$722$	17503.2	0.902220
$723$	−13193.9	−0.678682
$724$	−38342.3	−1.96820
$725$	16931.1	0.867319
$726$	26564.8	1.35800
$727$	−23847.6	−1.21659	−0.608294	−	0.793711i	$-0.708146\pi$
$727$	−23847.6	−1.21659	−0.608294	+	0.793711i	$0.708146\pi$
$728$	8948.45	0.455566
$729$	7040.20	0.357679
$730$	9525.64	0.482959
$731$	14144.1	0.715649
$732$	−56874.3	−2.87177
$733$	−181.931	−0.00916750	−0.00458375	−	0.999989i	$-0.501459\pi$
$733$	−181.931	−0.00916750	−0.00458375	+	0.999989i	$0.501459\pi$
$734$	−2109.51	−0.106081
$735$	−4892.79	−0.245542
$736$	1041.20	0.0521454
$737$	−5243.69	−0.262081
$738$	−647.702	−0.0323066
$739$	20127.6	1.00190	0.500952	−	0.865475i	$-0.332983\pi$
$739$	20127.6	1.00190	0.500952	+	0.865475i	$0.332983\pi$
$740$	2576.65	0.127999
$741$	30610.3	1.51754
$742$	273.375	0.0135255
$743$	1786.69	0.0882199	0.0441100	−	0.999027i	$-0.485955\pi$
$743$	1786.69	0.0882199	0.0441100	+	0.999027i	$0.485955\pi$
$744$	64093.2	3.15830
$745$	−533.060	−0.0262145
$746$	50154.2	2.46149
$747$	6046.06	0.296136
$748$	16408.1	0.802060
$749$	−3888.73	−0.189708
$750$	17398.1	0.847054
$751$	−38846.1	−1.88750	−0.943750	−	0.330659i	$-0.892729\pi$
$751$	−38846.1	−1.88750	−0.943750	+	0.330659i	$0.892729\pi$
$752$	27291.6	1.32343
$753$	37494.6	1.81458
$754$	60433.1	2.91889
$755$	−6462.22	−0.311502
$756$	4247.36	0.204332
$757$	−30739.6	−1.47589	−0.737946	−	0.674860i	$-0.764204\pi$
$757$	−30739.6	−1.47589	−0.737946	+	0.674860i	$0.764204\pi$
$758$	5497.50	0.263428
$759$	−16491.1	−0.788656
$760$	−5246.85	−0.250425
$761$	−14415.3	−0.686670	−0.343335	−	0.939213i	$-0.611557\pi$
$761$	−14415.3	−0.686670	−0.343335	+	0.939213i	$0.611557\pi$
$762$	22617.7	1.07527
$763$	5369.45	0.254767
$764$	−30605.0	−1.44928
$765$	−1234.59	−0.0583487
$766$	27846.3	1.31348
$767$	24888.0	1.17165
$768$	−49893.3	−2.34423
$769$	−4044.91	−0.189679	−0.0948394	−	0.995493i	$-0.530234\pi$
$769$	−4044.91	−0.189679	−0.0948394	+	0.995493i	$0.530234\pi$
$770$	−651.449	−0.0304891
$771$	2617.60	0.122270
$772$	−6729.77	−0.313743
$773$	19396.9	0.902535	0.451267	−	0.892389i	$-0.350972\pi$
$773$	19396.9	0.902535	0.451267	+	0.892389i	$0.350972\pi$
$774$	15013.1	0.697200
$775$	32317.7	1.49792
$776$	−26122.2	−1.20842
$777$	−1114.07	−0.0514376
$778$	−14229.7	−0.655730
$779$	714.011	0.0328397
$780$	20177.9	0.926261
$781$	1440.37	0.0659930
$782$	30483.5	1.39397
$783$	14247.3	0.650266
$784$	−20655.0	−0.940916
$785$	3859.66	0.175487
$786$	16574.4	0.752151
$787$	2447.43	0.110853	0.0554266	−	0.998463i	$-0.482348\pi$
$787$	2447.43	0.110853	0.0554266	+	0.998463i	$0.482348\pi$
$788$	−73171.0	−3.30788
$789$	−26617.3	−1.20102
$790$	−4313.46	−0.194261
$791$	1370.03	0.0615838
$792$	8650.47	0.388107
$793$	−50841.8	−2.27673
$794$	−74001.1	−3.30756
$795$	306.177	0.0136591
$796$	−14149.1	−0.630029
$797$	10252.6	0.455665	0.227832	−	0.973700i	$-0.426836\pi$
$797$	10252.6	0.455665	0.227832	+	0.973700i	$0.426836\pi$
$798$	4567.40	0.202612
$799$	−21719.2	−0.961667
$800$	−975.187	−0.0430976
$801$	−4361.04	−0.192372
$802$	27287.0	1.20142
$803$	17312.7	0.760837
$804$	24234.5	1.06304
$805$	−805.078	−0.0352488
$806$	115353.	5.04112
$807$	−4123.64	−0.179875
$808$	−2938.92	−0.127959
$809$	2381.11	0.103480	0.0517401	−	0.998661i	$-0.483523\pi$
$809$	2381.11	0.103480	0.0517401	+	0.998661i	$0.483523\pi$
$810$	10481.2	0.454656
$811$	13144.1	0.569112	0.284556	−	0.958659i	$-0.408154\pi$
$811$	13144.1	0.569112	0.284556	+	0.958659i	$0.408154\pi$
$812$	5998.30	0.259235
$813$	12347.4	0.532649
$814$	7040.02	0.303136
$815$	6669.23	0.286642
$816$	−18455.6	−0.791761
$817$	−16550.0	−0.708705
$818$	−7745.94	−0.331089
$819$	−2463.74	−0.105116
$820$	470.667	0.0200444
$821$	−19118.2	−0.812705	−0.406353	−	0.913716i	$-0.633200\pi$
$821$	−19118.2	−0.812705	−0.406353	+	0.913716i	$0.633200\pi$
$822$	59412.3	2.52097
$823$	−31990.2	−1.35493	−0.677467	−	0.735554i	$-0.736922\pi$
$823$	−31990.2	−1.35493	−0.677467	+	0.735554i	$0.736922\pi$
$824$	18634.4	0.787816
$825$	15445.6	0.651815
$826$	3713.58	0.156431
$827$	−27895.0	−1.17292	−0.586459	−	0.809979i	$-0.699479\pi$
$827$	−27895.0	−1.17292	−0.586459	+	0.809979i	$0.699479\pi$
$828$	21523.3	0.903367
$829$	43170.5	1.80865	0.904327	−	0.426839i	$-0.140373\pi$
$829$	43170.5	1.80865	0.904327	+	0.426839i	$0.140373\pi$
$830$	−6604.78	−0.276211
$831$	13260.8	0.553564
$832$	−46353.2	−1.93150
$833$	16437.7	0.683713
$834$	22992.0	0.954612
$835$	3955.60	0.163939
$836$	−19199.1	−0.794278
$837$	27194.9	1.12305
$838$	34687.4	1.42990
$839$	−38059.1	−1.56609	−0.783044	−	0.621967i	$-0.786334\pi$
$839$	−38059.1	−1.56609	−0.783044	+	0.621967i	$0.786334\pi$
$840$	1495.42	0.0614250
$841$	−4268.32	−0.175010
$842$	40472.1	1.65649
$843$	−5763.50	−0.235475
$844$	−29308.7	−1.19532
$845$	12820.8	0.521952
$846$	−23053.5	−0.936876
$847$	2357.03	0.0956179
$848$	1292.53	0.0523417
$849$	−8827.14	−0.356828
$850$	−28550.9	−1.15210
$851$	8700.24	0.350459
$852$	−6656.89	−0.267677
$853$	20192.7	0.810535	0.405268	−	0.914198i	$-0.367178\pi$
$853$	20192.7	0.810535	0.405268	+	0.914198i	$0.367178\pi$
$854$	−7586.17	−0.303974
$855$	1444.60	0.0577826
$856$	−56409.6	−2.25239
$857$	−23004.2	−0.916929	−0.458465	−	0.888713i	$-0.651600\pi$
$857$	−23004.2	−0.916929	−0.458465	+	0.888713i	$0.651600\pi$
$858$	55130.7	2.19363
$859$	−16197.1	−0.643350	−0.321675	−	0.946850i	$-0.604246\pi$
$859$	−16197.1	−0.643350	−0.321675	+	0.946850i	$0.604246\pi$
$860$	−10909.6	−0.432573
$861$	−203.503	−0.00805501
$862$	35752.9	1.41270
$863$	−934.946	−0.0368782	−0.0184391	−	0.999830i	$-0.505870\pi$
$863$	−934.946	−0.0368782	−0.0184391	+	0.999830i	$0.505870\pi$
$864$	−820.608	−0.0323121
$865$	4133.87	0.162492
$866$	−72035.5	−2.82663
$867$	−15448.7	−0.605149
$868$	11449.4	0.447717
$869$	−7839.64	−0.306032
$870$	10099.3	0.393561
$871$	21664.0	0.842774
$872$	77888.8	3.02483
$873$	7192.13	0.278828
$874$	−35668.7	−1.38045
$875$	1543.69	0.0596416
$876$	−80013.2	−3.08607
$877$	48374.7	1.86260	0.931298	−	0.364259i	$-0.118678\pi$
$877$	48374.7	1.86260	0.931298	+	0.364259i	$0.118678\pi$
$878$	27580.5	1.06013
$879$	−23479.8	−0.900972
$880$	−3080.09	−0.117989
$881$	−39734.7	−1.51952	−0.759760	−	0.650204i	$-0.774684\pi$
$881$	−39734.7	−1.51952	−0.759760	+	0.650204i	$0.774684\pi$
$882$	17447.5	0.666088
$883$	−2902.47	−0.110618	−0.0553091	−	0.998469i	$-0.517614\pi$
$883$	−2902.47	−0.110618	−0.0553091	+	0.998469i	$0.517614\pi$
$884$	−67789.2	−2.57918
$885$	4159.18	0.157976
$886$	−10790.0	−0.409137
$887$	41913.2	1.58659	0.793295	−	0.608837i	$-0.208364\pi$
$887$	41913.2	1.58659	0.793295	+	0.608837i	$0.208364\pi$
$888$	−16160.6	−0.610715
$889$	2006.81	0.0757102
$890$	4764.05	0.179428
$891$	19049.4	0.716250
$892$	45358.7	1.70260
$893$	25413.7	0.952336
$894$	6731.17	0.251817
$895$	896.363	0.0334772
$896$	−6742.53	−0.251397
$897$	68132.0	2.53608
$898$	7639.26	0.283881
$899$	38405.8	1.42481
$900$	−20158.8	−0.746622
$901$	−1028.63	−0.0380339
$902$	1285.97	0.0474703
$903$	4716.98	0.173833
$904$	19873.6	0.731179
$905$	−5727.94	−0.210390
$906$	81601.3	2.99230
$907$	26607.4	0.974074	0.487037	−	0.873381i	$-0.338078\pi$
$907$	26607.4	0.974074	0.487037	+	0.873381i	$0.338078\pi$
$908$	51548.3	1.88402
$909$	809.162	0.0295250
$910$	2691.42	0.0980436
$911$	−10440.5	−0.379702	−0.189851	−	0.981813i	$-0.560800\pi$
$911$	−10440.5	−0.379702	−0.189851	+	0.981813i	$0.560800\pi$
$912$	21594.9	0.784079
$913$	−12004.1	−0.435134
$914$	−30791.1	−1.11431
$915$	−8496.45	−0.306977
$916$	27259.3	0.983268
$917$	1470.61	0.0529594
$918$	−24025.2	−0.863780
$919$	4557.23	0.163579	0.0817894	−	0.996650i	$-0.473936\pi$
$919$	4557.23	0.163579	0.0817894	+	0.996650i	$0.473936\pi$
$920$	−11678.4	−0.418506
$921$	−8646.79	−0.309361
$922$	−27443.1	−0.980251
$923$	−5950.80	−0.212214
$924$	5472.02	0.194823
$925$	−8148.66	−0.289650
$926$	32553.2	1.15525
$927$	−5130.54	−0.181779
$928$	−1158.90	−0.0409942
$929$	21954.0	0.775336	0.387668	−	0.921799i	$-0.373281\pi$
$929$	21954.0	0.775336	0.387668	+	0.921799i	$0.373281\pi$
$930$	19277.3	0.679706
$931$	−19233.8	−0.677079
$932$	41474.8	1.45767
$933$	−41817.8	−1.46737
$934$	82484.3	2.88969
$935$	2451.21	0.0857359
$936$	−35738.9	−1.24804
$937$	−39162.9	−1.36542	−0.682709	−	0.730691i	$-0.739198\pi$
$937$	−39162.9	−1.36542	−0.682709	+	0.730691i	$0.739198\pi$
$938$	3232.51	0.112522
$939$	−7837.52	−0.272383
$940$	16752.4	0.581278
$941$	−6862.21	−0.237728	−0.118864	−	0.992911i	$-0.537925\pi$
$941$	−6862.21	−0.237728	−0.118864	+	0.992911i	$0.537925\pi$
$942$	−48737.6	−1.68573
$943$	1589.24	0.0548810
$944$	17558.0	0.605366
$945$	634.513	0.0218420
$946$	−29807.5	−1.02445
$947$	34573.1	1.18635	0.593175	−	0.805073i	$-0.297874\pi$
$947$	34573.1	1.18635	0.593175	+	0.805073i	$0.297874\pi$
$948$	36232.0	1.24131
$949$	−71526.3	−2.44662
$950$	33407.4	1.14092
$951$	−14254.3	−0.486044
$952$	−5024.00	−0.171039
$953$	17455.5	0.593326	0.296663	−	0.954982i	$-0.404126\pi$
$953$	17455.5	0.593326	0.296663	+	0.954982i	$0.404126\pi$
$954$	−1091.82	−0.0370534
$955$	−4572.08	−0.154920
$956$	40815.5	1.38083
$957$	18355.3	0.620003
$958$	−6102.12	−0.205794
$959$	5271.50	0.177503
$960$	−7746.32	−0.260429
$961$	43517.1	1.46075
$962$	−29085.4	−0.974793
$963$	15531.1	0.519710
$964$	−34189.3	−1.14228
$965$	−1005.36	−0.0335374
$966$	10166.1	0.338600
$967$	−7102.55	−0.236197	−0.118099	−	0.993002i	$-0.537680\pi$
$967$	−7102.55	−0.236197	−0.118099	+	0.993002i	$0.537680\pi$
$968$	34190.8	1.13526
$969$	−17185.7	−0.569748
$970$	−7856.76	−0.260067
$971$	18176.5	0.600734	0.300367	−	0.953824i	$-0.402891\pi$
$971$	18176.5	0.600734	0.300367	+	0.953824i	$0.402891\pi$
$972$	−44934.1	−1.48278
$973$	2040.02	0.0672148
$974$	67350.5	2.21566
$975$	−63812.6	−2.09604
$976$	−35867.9	−1.17634
$977$	−21224.3	−0.695011	−0.347505	−	0.937678i	$-0.612971\pi$
$977$	−21224.3	−0.695011	−0.347505	+	0.937678i	$0.612971\pi$
$978$	−84215.2	−2.75348
$979$	8658.58	0.282665
$980$	−12678.6	−0.413269
$981$	−21444.8	−0.697942
$982$	23492.7	0.763423
$983$	31229.7	1.01330	0.506650	−	0.862152i	$-0.330884\pi$
$983$	31229.7	1.01330	0.506650	+	0.862152i	$0.330884\pi$
$984$	−2952.00	−0.0956364
$985$	−10931.0	−0.353594
$986$	−33929.4	−1.09587
$987$	−7243.24	−0.233592
$988$	79320.1	2.55416
$989$	−36836.9	−1.18437
$990$	2601.80	0.0835257
$991$	−20862.7	−0.668743	−0.334371	−	0.942441i	$-0.608524\pi$
$991$	−20862.7	−0.668743	−0.334371	+	0.942441i	$0.608524\pi$
$992$	−2212.07	−0.0707998
$993$	−7514.89	−0.240159
$994$	−887.927	−0.0283333
$995$	−2113.74	−0.0673467
$996$	55478.6	1.76497
$997$	21427.4	0.680656	0.340328	−	0.940307i	$-0.389462\pi$
$997$	21427.4	0.680656	0.340328	+	0.940307i	$0.389462\pi$
$998$	94014.4	2.98194
$999$	−6856.99	−0.217163

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1997.4.a.a.1.18	✓	239

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1997.4.a.a.1.18	✓	239	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:	\( N \)	\(=\)	\( 1997 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1997.a (trivial)

\(f(q)\)	\(=\)	\(q-4.88824 q^{2} +6.13395 q^{3} +15.8948 q^{4} +2.37453 q^{5} -29.9842 q^{6} -2.66042 q^{7} -38.5919 q^{8} +10.6254 q^{9} +O(q^{10})\)	\(q-4.88824 q^{2} +6.13395 q^{3} +15.8948 q^{4} +2.37453 q^{5} -29.9842 q^{6} -2.66042 q^{7} -38.5919 q^{8} +10.6254 q^{9} -11.6073 q^{10} -21.0960 q^{11} +97.4982 q^{12} +87.1568 q^{13} +13.0048 q^{14} +14.5652 q^{15} +61.4874 q^{16} -48.9331 q^{17} -51.9392 q^{18} +57.2566 q^{19} +37.7428 q^{20} -16.3189 q^{21} +103.122 q^{22} +127.441 q^{23} -236.721 q^{24} -119.362 q^{25} -426.043 q^{26} -100.441 q^{27} -42.2870 q^{28} -141.847 q^{29} -71.1983 q^{30} -270.755 q^{31} +8.17003 q^{32} -129.402 q^{33} +239.196 q^{34} -6.31725 q^{35} +168.888 q^{36} +68.2687 q^{37} -279.884 q^{38} +534.615 q^{39} -91.6375 q^{40} +12.4704 q^{41} +79.7707 q^{42} -289.050 q^{43} -335.318 q^{44} +25.2302 q^{45} -622.962 q^{46} +443.856 q^{47} +377.161 q^{48} -335.922 q^{49} +583.468 q^{50} -300.153 q^{51} +1385.34 q^{52} +21.0211 q^{53} +490.981 q^{54} -50.0931 q^{55} +102.671 q^{56} +351.209 q^{57} +693.383 q^{58} +285.555 q^{59} +231.512 q^{60} -583.337 q^{61} +1323.51 q^{62} -28.2679 q^{63} -531.836 q^{64} +206.956 q^{65} +632.547 q^{66} +248.563 q^{67} -777.784 q^{68} +781.718 q^{69} +30.8802 q^{70} -68.2770 q^{71} -410.052 q^{72} -820.663 q^{73} -333.713 q^{74} -732.158 q^{75} +910.085 q^{76} +56.1243 q^{77} -2613.33 q^{78} +371.618 q^{79} +146.004 q^{80} -902.986 q^{81} -60.9582 q^{82} +569.021 q^{83} -259.387 q^{84} -116.193 q^{85} +1412.95 q^{86} -870.085 q^{87} +814.134 q^{88} -410.437 q^{89} -123.331 q^{90} -231.874 q^{91} +2025.66 q^{92} -1660.80 q^{93} -2169.67 q^{94} +135.957 q^{95} +50.1145 q^{96} +676.884 q^{97} +1642.07 q^{98} -224.153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9}+O(q^{10}) \) 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9	\( 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9} - 273 q^{10} - 294 q^{11} - 864 q^{12} - 797 q^{13} - 220 q^{14} - 580 q^{15} + 2816 q^{16} - 439 q^{17} - 536 q^{18} - 1704 q^{19} - 933 q^{20} - 596 q^{21} - 1046 q^{22} - 829 q^{23} - 1237 q^{24} + 4364 q^{25} - 818 q^{26} - 3670 q^{27} - 3690 q^{28} - 316 q^{29} - 888 q^{30} - 2595 q^{31} - 1881 q^{32} - 2066 q^{33} - 2605 q^{34} - 2450 q^{35} + 5863 q^{36} - 1912 q^{37} - 1709 q^{38} - 914 q^{39} - 3582 q^{40} - 1064 q^{41} - 3228 q^{42} - 5184 q^{43} - 2656 q^{44} - 3967 q^{45} - 2521 q^{46} - 4909 q^{47} - 7461 q^{48} + 7193 q^{49} - 1906 q^{50} - 3240 q^{51} - 9614 q^{52} - 2722 q^{53} - 3754 q^{54} - 6018 q^{55} - 2347 q^{56} - 2032 q^{57} - 6709 q^{58} - 6318 q^{59} - 5821 q^{60} - 2990 q^{61} - 2117 q^{62} - 8738 q^{63} + 6866 q^{64} - 1738 q^{65} - 3080 q^{66} - 14729 q^{67} - 3897 q^{68} - 2080 q^{69} - 7445 q^{70} - 3240 q^{71} - 8263 q^{72} - 8828 q^{73} - 3103 q^{74} - 12716 q^{75} - 14843 q^{76} - 3818 q^{77} - 8029 q^{78} - 4794 q^{79} - 10336 q^{80} + 11899 q^{81} - 13447 q^{82} - 11434 q^{83} - 7957 q^{84} - 8188 q^{85} - 5196 q^{86} - 11266 q^{87} - 11861 q^{88} - 4845 q^{89} - 7759 q^{90} - 12734 q^{91} - 8644 q^{92} - 10130 q^{93} - 6909 q^{94} - 3686 q^{95} - 11958 q^{96} - 16108 q^{97} - 6845 q^{98} - 12372 q^{99}+O(q^{100}) \) 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9 - 273 * q^10 - 294 * q^11 - 864 * q^12 - 797 * q^13 - 220 * q^14 - 580 * q^15 + 2816 * q^16 - 439 * q^17 - 536 * q^18 - 1704 * q^19 - 933 * q^20 - 596 * q^21 - 1046 * q^22 - 829 * q^23 - 1237 * q^24 + 4364 * q^25 - 818 * q^26 - 3670 * q^27 - 3690 * q^28 - 316 * q^29 - 888 * q^30 - 2595 * q^31 - 1881 * q^32 - 2066 * q^33 - 2605 * q^34 - 2450 * q^35 + 5863 * q^36 - 1912 * q^37 - 1709 * q^38 - 914 * q^39 - 3582 * q^40 - 1064 * q^41 - 3228 * q^42 - 5184 * q^43 - 2656 * q^44 - 3967 * q^45 - 2521 * q^46 - 4909 * q^47 - 7461 * q^48 + 7193 * q^49 - 1906 * q^50 - 3240 * q^51 - 9614 * q^52 - 2722 * q^53 - 3754 * q^54 - 6018 * q^55 - 2347 * q^56 - 2032 * q^57 - 6709 * q^58 - 6318 * q^59 - 5821 * q^60 - 2990 * q^61 - 2117 * q^62 - 8738 * q^63 + 6866 * q^64 - 1738 * q^65 - 3080 * q^66 - 14729 * q^67 - 3897 * q^68 - 2080 * q^69 - 7445 * q^70 - 3240 * q^71 - 8263 * q^72 - 8828 * q^73 - 3103 * q^74 - 12716 * q^75 - 14843 * q^76 - 3818 * q^77 - 8029 * q^78 - 4794 * q^79 - 10336 * q^80 + 11899 * q^81 - 13447 * q^82 - 11434 * q^83 - 7957 * q^84 - 8188 * q^85 - 5196 * q^86 - 11266 * q^87 - 11861 * q^88 - 4845 * q^89 - 7759 * q^90 - 12734 * q^91 - 8644 * q^92 - 10130 * q^93 - 6909 * q^94 - 3686 * q^95 - 11958 * q^96 - 16108 * q^97 - 6845 * q^98 - 12372 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more