LMFDB - Embedded newform 231.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,4,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("231.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 231 = 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	231.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:	$13.6294412113$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	231.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.56155 q^{2} -3.00000 q^{3} +4.68466 q^{4} -5.68466 q^{5} -10.6847 q^{6} +7.00000 q^{7} -11.8078 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.56155 q^{2} -3.00000 q^{3} +4.68466 q^{4} -5.68466 q^{5} -10.6847 q^{6} +7.00000 q^{7} -11.8078 q^{8} +9.00000 q^{9} -20.2462 q^{10} -11.0000 q^{11} -14.0540 q^{12} -39.3002 q^{13} +24.9309 q^{14} +17.0540 q^{15} -79.5312 q^{16} -64.2462 q^{17} +32.0540 q^{18} -82.9157 q^{19} -26.6307 q^{20} -21.0000 q^{21} -39.1771 q^{22} -11.5076 q^{23} +35.4233 q^{24} -92.6847 q^{25} -139.970 q^{26} -27.0000 q^{27} +32.7926 q^{28} +128.501 q^{29} +60.7386 q^{30} -101.400 q^{31} -188.793 q^{32} +33.0000 q^{33} -228.816 q^{34} -39.7926 q^{35} +42.1619 q^{36} +188.086 q^{37} -295.309 q^{38} +117.901 q^{39} +67.1231 q^{40} +198.294 q^{41} -74.7926 q^{42} +231.386 q^{43} -51.5312 q^{44} -51.1619 q^{45} -40.9848 q^{46} -285.533 q^{47} +238.594 q^{48} +49.0000 q^{49} -330.101 q^{50} +192.739 q^{51} -184.108 q^{52} +202.570 q^{53} -96.1619 q^{54} +62.5312 q^{55} -82.6543 q^{56} +248.747 q^{57} +457.663 q^{58} +103.762 q^{59} +79.8920 q^{60} -382.924 q^{61} -361.140 q^{62} +63.0000 q^{63} -36.1449 q^{64} +223.408 q^{65} +117.531 q^{66} -259.348 q^{67} -300.972 q^{68} +34.5227 q^{69} -141.723 q^{70} +352.553 q^{71} -106.270 q^{72} +490.012 q^{73} +669.879 q^{74} +278.054 q^{75} -388.432 q^{76} -77.0000 q^{77} +419.909 q^{78} +141.633 q^{79} +452.108 q^{80} +81.0000 q^{81} +706.233 q^{82} -192.847 q^{83} -98.3778 q^{84} +365.218 q^{85} +824.095 q^{86} -385.503 q^{87} +129.885 q^{88} -423.295 q^{89} -182.216 q^{90} -275.101 q^{91} -53.9091 q^{92} +304.199 q^{93} -1016.94 q^{94} +471.348 q^{95} +566.378 q^{96} -1095.89 q^{97} +174.516 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - 6 q^{3} - 3 q^{4} + q^{5} - 9 q^{6} + 14 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 + q^5 - 9 * q^6 + 14 * q^7 - 3 * q^8 + 18 * q^9	\( 2 q + 3 q^{2} - 6 q^{3} - 3 q^{4} + q^{5} - 9 q^{6} + 14 q^{7} - 3 q^{8} + 18 q^{9} - 24 q^{10} - 22 q^{11} + 9 q^{12} - 25 q^{13} + 21 q^{14} - 3 q^{15} - 23 q^{16} - 112 q^{17} + 27 q^{18} - 71 q^{19} - 78 q^{20} - 42 q^{21} - 33 q^{22} - 56 q^{23} + 9 q^{24} - 173 q^{25} - 148 q^{26} - 54 q^{27} - 21 q^{28} - 11 q^{29} + 72 q^{30} - 310 q^{31} - 291 q^{32} + 66 q^{33} - 202 q^{34} + 7 q^{35} - 27 q^{36} - 65 q^{37} - 302 q^{38} + 75 q^{39} + 126 q^{40} + 42 q^{41} - 63 q^{42} - 32 q^{43} + 33 q^{44} + 9 q^{45} - 16 q^{46} + 101 q^{47} + 69 q^{48} + 98 q^{49} - 285 q^{50} + 336 q^{51} - 294 q^{52} + 166 q^{53} - 81 q^{54} - 11 q^{55} - 21 q^{56} + 213 q^{57} + 536 q^{58} - 11 q^{59} + 234 q^{60} - 436 q^{61} - 244 q^{62} + 126 q^{63} - 431 q^{64} + 319 q^{65} + 99 q^{66} - 127 q^{67} + 66 q^{68} + 168 q^{69} - 168 q^{70} + 936 q^{71} - 27 q^{72} - 327 q^{73} + 812 q^{74} + 519 q^{75} - 480 q^{76} - 154 q^{77} + 444 q^{78} - 228 q^{79} + 830 q^{80} + 162 q^{81} + 794 q^{82} - 262 q^{83} + 63 q^{84} + 46 q^{85} + 972 q^{86} + 33 q^{87} + 33 q^{88} + 44 q^{89} - 216 q^{90} - 175 q^{91} + 288 q^{92} + 930 q^{93} - 1234 q^{94} + 551 q^{95} + 873 q^{96} - 2266 q^{97} + 147 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 + q^5 - 9 * q^6 + 14 * q^7 - 3 * q^8 + 18 * q^9 - 24 * q^10 - 22 * q^11 + 9 * q^12 - 25 * q^13 + 21 * q^14 - 3 * q^15 - 23 * q^16 - 112 * q^17 + 27 * q^18 - 71 * q^19 - 78 * q^20 - 42 * q^21 - 33 * q^22 - 56 * q^23 + 9 * q^24 - 173 * q^25 - 148 * q^26 - 54 * q^27 - 21 * q^28 - 11 * q^29 + 72 * q^30 - 310 * q^31 - 291 * q^32 + 66 * q^33 - 202 * q^34 + 7 * q^35 - 27 * q^36 - 65 * q^37 - 302 * q^38 + 75 * q^39 + 126 * q^40 + 42 * q^41 - 63 * q^42 - 32 * q^43 + 33 * q^44 + 9 * q^45 - 16 * q^46 + 101 * q^47 + 69 * q^48 + 98 * q^49 - 285 * q^50 + 336 * q^51 - 294 * q^52 + 166 * q^53 - 81 * q^54 - 11 * q^55 - 21 * q^56 + 213 * q^57 + 536 * q^58 - 11 * q^59 + 234 * q^60 - 436 * q^61 - 244 * q^62 + 126 * q^63 - 431 * q^64 + 319 * q^65 + 99 * q^66 - 127 * q^67 + 66 * q^68 + 168 * q^69 - 168 * q^70 + 936 * q^71 - 27 * q^72 - 327 * q^73 + 812 * q^74 + 519 * q^75 - 480 * q^76 - 154 * q^77 + 444 * q^78 - 228 * q^79 + 830 * q^80 + 162 * q^81 + 794 * q^82 - 262 * q^83 + 63 * q^84 + 46 * q^85 + 972 * q^86 + 33 * q^87 + 33 * q^88 + 44 * q^89 - 216 * q^90 - 175 * q^91 + 288 * q^92 + 930 * q^93 - 1234 * q^94 + 551 * q^95 + 873 * q^96 - 2266 * q^97 + 147 * q^98 - 198 * 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 $

$2$	3.56155	1.25920	0.629600	−	0.776920i	$-0.283219\pi$
$2$	3.56155	1.25920	0.629600	+	0.776920i	$0.283219\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.68466	0.585582
$5$	−5.68466	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$5$	−5.68466	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$6$	−10.6847	−0.726999
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−11.8078	−0.521834
$9$	9.00000	0.333333
$10$	−20.2462	−0.640241
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−14.0540	−0.338086
$13$	−39.3002	−0.838455	−0.419227	−	0.907881i	$-0.637699\pi$
$13$	−39.3002	−0.838455	−0.419227	+	0.907881i	$0.637699\pi$
$14$	24.9309	0.475933
$15$	17.0540	0.293554
$16$	−79.5312	−1.24268
$17$	−64.2462	−0.916588	−0.458294	−	0.888801i	$-0.651539\pi$
$17$	−64.2462	−0.916588	−0.458294	+	0.888801i	$0.651539\pi$
$18$	32.0540	0.419733
$19$	−82.9157	−1.00117	−0.500583	−	0.865688i	$-0.666881\pi$
$19$	−82.9157	−1.00117	−0.500583	+	0.865688i	$0.666881\pi$
$20$	−26.6307	−0.297740
$21$	−21.0000	−0.218218
$22$	−39.1771	−0.379663
$23$	−11.5076	−0.104326	−0.0521630	−	0.998639i	$-0.516612\pi$
$23$	−11.5076	−0.104326	−0.0521630	+	0.998639i	$0.516612\pi$
$24$	35.4233	0.301281
$25$	−92.6847	−0.741477
$26$	−139.970	−1.05578
$27$	−27.0000	−0.192450
$28$	32.7926	0.221329
$29$	128.501	0.822828	0.411414	−	0.911448i	$-0.365035\pi$
$29$	128.501	0.822828	0.411414	+	0.911448i	$0.365035\pi$
$30$	60.7386	0.369644
$31$	−101.400	−0.587481	−0.293740	−	0.955885i	$-0.594900\pi$
$31$	−101.400	−0.587481	−0.293740	+	0.955885i	$0.594900\pi$
$32$	−188.793	−1.04294
$33$	33.0000	0.174078
$34$	−228.816	−1.15417
$35$	−39.7926	−0.192177
$36$	42.1619	0.195194
$37$	188.086	0.835707	0.417854	−	0.908514i	$-0.362782\pi$
$37$	188.086	0.835707	0.417854	+	0.908514i	$0.362782\pi$
$38$	−295.309	−1.26067
$39$	117.901	0.484082
$40$	67.1231	0.265327
$41$	198.294	0.755323	0.377662	−	0.925944i	$-0.376728\pi$
$41$	198.294	0.755323	0.377662	+	0.925944i	$0.376728\pi$
$42$	−74.7926	−0.274780
$43$	231.386	0.820607	0.410303	−	0.911949i	$-0.365423\pi$
$43$	231.386	0.820607	0.410303	+	0.911949i	$0.365423\pi$
$44$	−51.5312	−0.176560
$45$	−51.1619	−0.169484
$46$	−40.9848	−0.131367
$47$	−285.533	−0.886155	−0.443077	−	0.896483i	$-0.646113\pi$
$47$	−285.533	−0.886155	−0.443077	+	0.896483i	$0.646113\pi$
$48$	238.594	0.717459
$49$	49.0000	0.142857
$50$	−330.101	−0.933667
$51$	192.739	0.529192
$52$	−184.108	−0.490984
$53$	202.570	0.525003	0.262501	−	0.964932i	$-0.415453\pi$
$53$	202.570	0.525003	0.262501	+	0.964932i	$0.415453\pi$
$54$	−96.1619	−0.242333
$55$	62.5312	0.153304
$56$	−82.6543	−0.197235
$57$	248.747	0.578024
$58$	457.663	1.03610
$59$	103.762	0.228961	0.114481	−	0.993425i	$-0.463480\pi$
$59$	103.762	0.228961	0.114481	+	0.993425i	$0.463480\pi$
$60$	79.8920	0.171900
$61$	−382.924	−0.803745	−0.401872	−	0.915696i	$-0.631640\pi$
$61$	−382.924	−0.803745	−0.401872	+	0.915696i	$0.631640\pi$
$62$	−361.140	−0.739756
$63$	63.0000	0.125988
$64$	−36.1449	−0.0705955
$65$	223.408	0.426313
$66$	117.531	0.219198
$67$	−259.348	−0.472901	−0.236450	−	0.971644i	$-0.575984\pi$
$67$	−259.348	−0.472901	−0.236450	+	0.971644i	$0.575984\pi$
$68$	−300.972	−0.536738
$69$	34.5227	0.0602326
$70$	−141.723	−0.241989
$71$	352.553	0.589301	0.294650	−	0.955605i	$-0.404797\pi$
$71$	352.553	0.589301	0.294650	+	0.955605i	$0.404797\pi$
$72$	−106.270	−0.173945
$73$	490.012	0.785638	0.392819	−	0.919616i	$-0.371500\pi$
$73$	490.012	0.785638	0.392819	+	0.919616i	$0.371500\pi$
$74$	669.879	1.05232
$75$	278.054	0.428092
$76$	−388.432	−0.586266
$77$	−77.0000	−0.113961
$78$	419.909	0.609556
$79$	141.633	0.201708	0.100854	−	0.994901i	$-0.467843\pi$
$79$	141.633	0.201708	0.100854	+	0.994901i	$0.467843\pi$
$80$	452.108	0.631840
$81$	81.0000	0.111111
$82$	706.233	0.951102
$83$	−192.847	−0.255032	−0.127516	−	0.991837i	$-0.540700\pi$
$83$	−192.847	−0.255032	−0.127516	+	0.991837i	$0.540700\pi$
$84$	−98.3778	−0.127785
$85$	365.218	0.466040
$86$	824.095	1.03331
$87$	−385.503	−0.475060
$88$	129.885	0.157339
$89$	−423.295	−0.504149	−0.252074	−	0.967708i	$-0.581113\pi$
$89$	−423.295	−0.504149	−0.252074	+	0.967708i	$0.581113\pi$
$90$	−182.216	−0.213414
$91$	−275.101	−0.316906
$92$	−53.9091	−0.0610914
$93$	304.199	0.339182
$94$	−1016.94	−1.11585
$95$	471.348	0.509045
$96$	566.378	0.602143
$97$	−1095.89	−1.14712	−0.573562	−	0.819162i	$-0.694439\pi$
$97$	−1095.89	−1.14712	−0.573562	+	0.819162i	$0.694439\pi$
$98$	174.516	0.179886
$99$	−99.0000	−0.100504
$100$	−434.196	−0.434196
$101$	−1781.30	−1.75491	−0.877455	−	0.479659i	$-0.840760\pi$
$101$	−1781.30	−1.75491	−0.877455	+	0.479659i	$0.840760\pi$
$102$	686.449	0.666358
$103$	10.3239	0.00987611	0.00493806	−	0.999988i	$-0.498428\pi$
$103$	10.3239	0.00987611	0.00493806	+	0.999988i	$0.498428\pi$
$104$	464.047	0.437534
$105$	119.378	0.110953
$106$	721.464	0.661083
$107$	1518.89	1.37230	0.686152	−	0.727458i	$-0.259298\pi$
$107$	1518.89	1.37230	0.686152	+	0.727458i	$0.259298\pi$
$108$	−126.486	−0.112695
$109$	−1648.93	−1.44898	−0.724489	−	0.689286i	$-0.757924\pi$
$109$	−1648.93	−1.44898	−0.724489	+	0.689286i	$0.757924\pi$
$110$	222.708	0.193040
$111$	−564.258	−0.482496
$112$	−556.719	−0.469687
$113$	−188.739	−0.157124	−0.0785621	−	0.996909i	$-0.525033\pi$
$113$	−188.739	−0.157124	−0.0785621	+	0.996909i	$0.525033\pi$
$114$	885.926	0.727847
$115$	65.4166	0.0530446
$116$	601.983	0.481834
$117$	−353.702	−0.279485
$118$	369.555	0.288308
$119$	−449.723	−0.346438
$120$	−201.369	−0.153187
$121$	121.000	0.0909091
$122$	−1363.80	−1.01207
$123$	−594.881	−0.436086
$124$	−475.023	−0.344018
$125$	1237.46	0.885456
$126$	224.378	0.158644
$127$	−2398.92	−1.67614	−0.838069	−	0.545564i	$-0.816316\pi$
$127$	−2398.92	−1.67614	−0.838069	+	0.545564i	$0.816316\pi$
$128$	1381.61	0.954048
$129$	−694.159	−0.473777
$130$	795.680	0.536813
$131$	1560.96	1.04108	0.520540	−	0.853837i	$-0.325731\pi$
$131$	1560.96	1.04108	0.520540	+	0.853837i	$0.325731\pi$
$132$	154.594	0.101937
$133$	−580.410	−0.378405
$134$	−923.680	−0.595476
$135$	153.486	0.0978515
$136$	758.604	0.478307
$137$	821.595	0.512362	0.256181	−	0.966629i	$-0.417536\pi$
$137$	821.595	0.512362	0.256181	+	0.966629i	$0.417536\pi$
$138$	122.955	0.0758448
$139$	−1158.44	−0.706886	−0.353443	−	0.935456i	$-0.614989\pi$
$139$	−1158.44	−0.706886	−0.353443	+	0.935456i	$0.614989\pi$
$140$	−186.415	−0.112535
$141$	856.599	0.511622
$142$	1255.64	0.742047
$143$	432.302	0.252804
$144$	−715.781	−0.414225
$145$	−730.484	−0.418368
$146$	1745.20	0.989275
$147$	−147.000	−0.0824786
$148$	881.119	0.489375
$149$	1234.82	0.678926	0.339463	−	0.940619i	$-0.389755\pi$
$149$	1234.82	0.678926	0.339463	+	0.940619i	$0.389755\pi$
$150$	990.304	0.539053
$151$	−1154.58	−0.622242	−0.311121	−	0.950370i	$-0.600704\pi$
$151$	−1154.58	−0.622242	−0.311121	+	0.950370i	$0.600704\pi$
$152$	979.049	0.522443
$153$	−578.216	−0.305529
$154$	−274.240	−0.143499
$155$	576.422	0.298705
$156$	552.324	0.283470
$157$	−2830.17	−1.43868	−0.719339	−	0.694660i	$-0.755555\pi$
$157$	−2830.17	−1.43868	−0.719339	+	0.694660i	$0.755555\pi$
$158$	504.432	0.253990
$159$	−607.710	−0.303110
$160$	1073.22	0.530285
$161$	−80.5530	−0.0394315
$162$	288.486	0.139911
$163$	3418.64	1.64275	0.821377	−	0.570386i	$-0.193206\pi$
$163$	3418.64	1.64275	0.821377	+	0.570386i	$0.193206\pi$
$164$	928.938	0.442304
$165$	−187.594	−0.0885100
$166$	−686.833	−0.321136
$167$	1575.50	0.730033	0.365017	−	0.931001i	$-0.381063\pi$
$167$	1575.50	0.730033	0.365017	+	0.931001i	$0.381063\pi$
$168$	247.963	0.113874
$169$	−652.495	−0.296994
$170$	1300.74	0.586838
$171$	−746.241	−0.333722
$172$	1083.97	0.480533
$173$	3597.33	1.58092	0.790462	−	0.612510i	$-0.209840\pi$
$173$	3597.33	1.58092	0.790462	+	0.612510i	$0.209840\pi$
$174$	−1372.99	−0.598195
$175$	−648.793	−0.280252
$176$	874.844	0.374681
$177$	−311.287	−0.132191
$178$	−1507.59	−0.634823
$179$	1812.71	0.756917	0.378459	−	0.925618i	$-0.376454\pi$
$179$	1812.71	0.756917	0.378459	+	0.925618i	$0.376454\pi$
$180$	−239.676	−0.0992467
$181$	2239.16	0.919535	0.459767	−	0.888039i	$-0.347933\pi$
$181$	2239.16	0.919535	0.459767	+	0.888039i	$0.347933\pi$
$182$	−979.788	−0.399048
$183$	1148.77	0.464042
$184$	135.879	0.0544408
$185$	−1069.21	−0.424917
$186$	1083.42	0.427098
$187$	706.708	0.276362
$188$	−1337.63	−0.518917
$189$	−189.000	−0.0727393
$190$	1678.73	0.640988
$191$	−3960.32	−1.50031	−0.750153	−	0.661264i	$-0.770020\pi$
$191$	−3960.32	−1.50031	−0.750153	+	0.661264i	$0.770020\pi$
$192$	108.435	0.0407583
$193$	−4740.22	−1.76792	−0.883961	−	0.467561i	$-0.845133\pi$
$193$	−4740.22	−1.76792	−0.883961	+	0.467561i	$0.845133\pi$
$194$	−3903.08	−1.44446
$195$	−670.224	−0.246132
$196$	229.548	0.0836546
$197$	−5144.18	−1.86045	−0.930223	−	0.366996i	$-0.880386\pi$
$197$	−5144.18	−1.86045	−0.930223	+	0.366996i	$0.880386\pi$
$198$	−352.594	−0.126554
$199$	−2232.02	−0.795094	−0.397547	−	0.917582i	$-0.630138\pi$
$199$	−2232.02	−0.795094	−0.397547	+	0.917582i	$0.630138\pi$
$200$	1094.40	0.386928
$201$	778.043	0.273029
$202$	−6344.19	−2.20978
$203$	899.507	0.311000
$204$	902.915	0.309886
$205$	−1127.23	−0.384045
$206$	36.7689	0.0124360
$207$	−103.568	−0.0347753
$208$	3125.59	1.04193
$209$	912.073	0.301863
$210$	425.170	0.139712
$211$	−3119.05	−1.01765	−0.508825	−	0.860870i	$-0.669920\pi$
$211$	−3119.05	−1.01765	−0.508825	+	0.860870i	$0.669920\pi$
$212$	948.972	0.307432
$213$	−1057.66	−0.340233
$214$	5409.60	1.72800
$215$	−1315.35	−0.417239
$216$	318.810	0.100427
$217$	−709.797	−0.222047
$218$	−5872.74	−1.82455
$219$	−1470.04	−0.453588
$220$	292.938	0.0897720
$221$	2524.89	0.768517
$222$	−2009.64	−0.607558
$223$	−1138.94	−0.342015	−0.171008	−	0.985270i	$-0.554702\pi$
$223$	−1138.94	−0.342015	−0.171008	+	0.985270i	$0.554702\pi$
$224$	−1321.55	−0.394195
$225$	−834.162	−0.247159
$226$	−672.203	−0.197851
$227$	−1754.97	−0.513133	−0.256567	−	0.966527i	$-0.582591\pi$
$227$	−1754.97	−0.513133	−0.256567	+	0.966527i	$0.582591\pi$
$228$	1165.30	0.338481
$229$	304.271	0.0878025	0.0439013	−	0.999036i	$-0.486021\pi$
$229$	304.271	0.0878025	0.0439013	+	0.999036i	$0.486021\pi$
$230$	232.985	0.0667938
$231$	231.000	0.0657952
$232$	−1517.31	−0.429380
$233$	−3908.74	−1.09901	−0.549506	−	0.835490i	$-0.685184\pi$
$233$	−3908.74	−1.09901	−0.549506	+	0.835490i	$0.685184\pi$
$234$	−1259.73	−0.351927
$235$	1623.16	0.450567
$236$	486.091	0.134076
$237$	−424.898	−0.116456
$238$	−1601.71	−0.436234
$239$	−170.995	−0.0462794	−0.0231397	−	0.999732i	$-0.507366\pi$
$239$	−170.995	−0.0462794	−0.0231397	+	0.999732i	$0.507366\pi$
$240$	−1356.32	−0.364793
$241$	−6654.70	−1.77870	−0.889351	−	0.457226i	$-0.848843\pi$
$241$	−6654.70	−1.77870	−0.889351	+	0.457226i	$0.848843\pi$
$242$	430.948	0.114473
$243$	−243.000	−0.0641500
$244$	−1793.87	−0.470659
$245$	−278.548	−0.0726359
$246$	−2118.70	−0.549119
$247$	3258.60	0.839433
$248$	1197.30	0.306568
$249$	578.540	0.147243
$250$	4407.29	1.11497
$251$	1019.50	0.256376	0.128188	−	0.991750i	$-0.459084\pi$
$251$	1019.50	0.256376	0.128188	+	0.991750i	$0.459084\pi$
$252$	295.133	0.0737764
$253$	126.583	0.0314554
$254$	−8543.87	−2.11059
$255$	−1095.65	−0.269069
$256$	5209.83	1.27193
$257$	−6742.31	−1.63647	−0.818237	−	0.574881i	$-0.805048\pi$
$257$	−6742.31	−1.63647	−0.818237	+	0.574881i	$0.805048\pi$
$258$	−2472.28	−0.596580
$259$	1316.60	0.315868
$260$	1046.59	0.249642
$261$	1156.51	0.274276
$262$	5559.44	1.31093
$263$	1282.62	0.300721	0.150360	−	0.988631i	$-0.451957\pi$
$263$	1282.62	0.300721	0.150360	+	0.988631i	$0.451957\pi$
$264$	−389.656	−0.0908397
$265$	−1151.54	−0.266938
$266$	−2067.16	−0.476488
$267$	1269.89	0.291070
$268$	−1214.95	−0.276922
$269$	−930.364	−0.210875	−0.105437	−	0.994426i	$-0.533624\pi$
$269$	−930.364	−0.210875	−0.105437	+	0.994426i	$0.533624\pi$
$270$	546.648	0.123215
$271$	8335.61	1.86846	0.934230	−	0.356672i	$-0.116089\pi$
$271$	8335.61	1.86846	0.934230	+	0.356672i	$0.116089\pi$
$272$	5109.58	1.13902
$273$	825.304	0.182966
$274$	2926.15	0.645166
$275$	1019.53	0.223564
$276$	161.727	0.0352711
$277$	3493.35	0.757744	0.378872	−	0.925449i	$-0.376312\pi$
$277$	3493.35	0.757744	0.378872	+	0.925449i	$0.376312\pi$
$278$	−4125.83	−0.890111
$279$	−912.597	−0.195827
$280$	469.862	0.100284
$281$	−1031.25	−0.218929	−0.109465	−	0.993991i	$-0.534914\pi$
$281$	−1031.25	−0.218929	−0.109465	+	0.993991i	$0.534914\pi$
$282$	3050.82	0.644234
$283$	−2380.71	−0.500066	−0.250033	−	0.968237i	$-0.580441\pi$
$283$	−2380.71	−0.500066	−0.250033	+	0.968237i	$0.580441\pi$
$284$	1651.59	0.345084
$285$	−1414.04	−0.293897
$286$	1539.67	0.318330
$287$	1388.05	0.285485
$288$	−1699.13	−0.347647
$289$	−785.424	−0.159867
$290$	−2601.66	−0.526809
$291$	3287.68	0.662292
$292$	2295.54	0.460056
$293$	7189.97	1.43359	0.716796	−	0.697283i	$-0.245608\pi$
$293$	7189.97	1.43359	0.716796	+	0.697283i	$0.245608\pi$
$294$	−523.548	−0.103857
$295$	−589.853	−0.116416
$296$	−2220.88	−0.436101
$297$	297.000	0.0580259
$298$	4397.86	0.854903
$299$	452.250	0.0874725
$300$	1302.59	0.250683
$301$	1619.70	0.310160
$302$	−4112.10	−0.783526
$303$	5343.90	1.01320
$304$	6594.39	1.24413
$305$	2176.79	0.408665
$306$	−2059.35	−0.384722
$307$	−805.946	−0.149830	−0.0749150	−	0.997190i	$-0.523869\pi$
$307$	−805.946	−0.149830	−0.0749150	+	0.997190i	$0.523869\pi$
$308$	−360.719	−0.0667333
$309$	−30.9716	−0.00570197
$310$	2052.96	0.376130
$311$	1526.61	0.278348	0.139174	−	0.990268i	$-0.455555\pi$
$311$	1526.61	0.278348	0.139174	+	0.990268i	$0.455555\pi$
$312$	−1392.14	−0.252611
$313$	5698.67	1.02910	0.514550	−	0.857461i	$-0.327959\pi$
$313$	5698.67	1.02910	0.514550	+	0.857461i	$0.327959\pi$
$314$	−10079.8	−1.81158
$315$	−358.133	−0.0640588
$316$	663.500	0.118116
$317$	−4036.30	−0.715146	−0.357573	−	0.933885i	$-0.616396\pi$
$317$	−4036.30	−0.715146	−0.357573	+	0.933885i	$0.616396\pi$
$318$	−2164.39	−0.381676
$319$	−1413.51	−0.248092
$320$	205.471	0.0358944
$321$	−4556.67	−0.792300
$322$	−286.894	−0.0496521
$323$	5327.02	0.917657
$324$	379.457	0.0650647
$325$	3642.52	0.621695
$326$	12175.7	2.06855
$327$	4946.78	0.836568
$328$	−2341.40	−0.394154
$329$	−1998.73	−0.334935
$330$	−668.125	−0.111452
$331$	2666.48	0.442789	0.221394	−	0.975184i	$-0.428939\pi$
$331$	2666.48	0.442789	0.221394	+	0.975184i	$0.428939\pi$
$332$	−903.420	−0.149342
$333$	1692.78	0.278569
$334$	5611.21	0.919257
$335$	1474.30	0.240447
$336$	1670.16	0.271174
$337$	2057.65	0.332603	0.166302	−	0.986075i	$-0.446817\pi$
$337$	2057.65	0.332603	0.166302	+	0.986075i	$0.446817\pi$
$338$	−2323.90	−0.373974
$339$	566.216	0.0907157
$340$	1710.92	0.272905
$341$	1115.40	0.177132
$342$	−2657.78	−0.420223
$343$	343.000	0.0539949
$344$	−2732.16	−0.428221
$345$	−196.250	−0.0306253
$346$	12812.1	1.99070
$347$	8556.95	1.32381	0.661904	−	0.749589i	$-0.269749\pi$
$347$	8556.95	1.32381	0.661904	+	0.749589i	$0.269749\pi$
$348$	−1805.95	−0.278187
$349$	−9298.02	−1.42611	−0.713054	−	0.701109i	$-0.752689\pi$
$349$	−9298.02	−1.42611	−0.713054	+	0.701109i	$0.752689\pi$
$350$	−2310.71	−0.352893
$351$	1061.11	0.161361
$352$	2076.72	0.314459
$353$	6315.48	0.952236	0.476118	−	0.879381i	$-0.342044\pi$
$353$	6315.48	0.952236	0.476118	+	0.879381i	$0.342044\pi$
$354$	−1108.66	−0.166454
$355$	−2004.14	−0.299631
$356$	−1982.99	−0.295220
$357$	1349.17	0.200016
$358$	6456.06	0.953109
$359$	4366.78	0.641977	0.320988	−	0.947083i	$-0.395985\pi$
$359$	4366.78	0.641977	0.320988	+	0.947083i	$0.395985\pi$
$360$	604.108	0.0884425
$361$	16.0157	0.00233500
$362$	7974.90	1.15788
$363$	−363.000	−0.0524864
$364$	−1288.76	−0.185575
$365$	−2785.55	−0.399459
$366$	4091.41	0.584321
$367$	455.244	0.0647508	0.0323754	−	0.999476i	$-0.489693\pi$
$367$	455.244	0.0647508	0.0323754	+	0.999476i	$0.489693\pi$
$368$	915.212	0.129643
$369$	1784.64	0.251774
$370$	−3808.03	−0.535054
$371$	1417.99	0.198432
$372$	1425.07	0.198619
$373$	−5345.01	−0.741968	−0.370984	−	0.928639i	$-0.620980\pi$
$373$	−5345.01	−0.741968	−0.370984	+	0.928639i	$0.620980\pi$
$374$	2516.98	0.347994
$375$	−3712.39	−0.511218
$376$	3371.51	0.462426
$377$	−5050.11	−0.689904
$378$	−673.133	−0.0915933
$379$	9772.43	1.32448	0.662238	−	0.749294i	$-0.269607\pi$
$379$	9772.43	1.32448	0.662238	+	0.749294i	$0.269607\pi$
$380$	2208.10	0.298087
$381$	7196.76	0.967719
$382$	−14104.9	−1.88918
$383$	821.560	0.109608	0.0548039	−	0.998497i	$-0.482547\pi$
$383$	821.560	0.109608	0.0548039	+	0.998497i	$0.482547\pi$
$384$	−4144.83	−0.550820
$385$	437.719	0.0579434
$386$	−16882.6	−2.22617
$387$	2082.48	0.273536
$388$	−5133.88	−0.671735
$389$	−3632.64	−0.473476	−0.236738	−	0.971574i	$-0.576078\pi$
$389$	−3632.64	−0.473476	−0.236738	+	0.971574i	$0.576078\pi$
$390$	−2387.04	−0.309929
$391$	739.318	0.0956239
$392$	−578.580	−0.0745478
$393$	−4682.87	−0.601068
$394$	−18321.3	−2.34267
$395$	−805.133	−0.102559
$396$	−463.781	−0.0588532
$397$	4985.80	0.630302	0.315151	−	0.949041i	$-0.397945\pi$
$397$	4985.80	0.630302	0.315151	+	0.949041i	$0.397945\pi$
$398$	−7949.45	−1.00118
$399$	1741.23	0.218473
$400$	7371.33	0.921416
$401$	−5585.37	−0.695562	−0.347781	−	0.937576i	$-0.613065\pi$
$401$	−5585.37	−0.695562	−0.347781	+	0.937576i	$0.613065\pi$
$402$	2771.04	0.343798
$403$	3985.02	0.492576
$404$	−8344.78	−1.02764
$405$	−460.457	−0.0564946
$406$	3203.64	0.391611
$407$	−2068.95	−0.251975
$408$	−2275.81	−0.276151
$409$	−5307.38	−0.641645	−0.320823	−	0.947139i	$-0.603959\pi$
$409$	−5307.38	−0.641645	−0.320823	+	0.947139i	$0.603959\pi$
$410$	−4014.69	−0.483589
$411$	−2464.78	−0.295812
$412$	48.3637	0.00578328
$413$	726.336	0.0865391
$414$	−368.864	−0.0437890
$415$	1096.27	0.129671
$416$	7419.58	0.874459
$417$	3475.31	0.408121
$418$	3248.40	0.380106
$419$	12248.3	1.42808	0.714041	−	0.700104i	$-0.246863\pi$
$419$	12248.3	1.42808	0.714041	+	0.700104i	$0.246863\pi$
$420$	559.244	0.0649722
$421$	−6958.41	−0.805539	−0.402770	−	0.915301i	$-0.631952\pi$
$421$	−6958.41	−0.805539	−0.402770	+	0.915301i	$0.631952\pi$
$422$	−11108.7	−1.28142
$423$	−2569.80	−0.295385
$424$	−2391.90	−0.273964
$425$	5954.64	0.679629
$426$	−3766.91	−0.428421
$427$	−2680.47	−0.303787
$428$	7115.48	0.803597
$429$	−1296.91	−0.145956
$430$	−4684.70	−0.525386
$431$	6245.52	0.697996	0.348998	−	0.937123i	$-0.386522\pi$
$431$	6245.52	0.697996	0.348998	+	0.937123i	$0.386522\pi$
$432$	2147.34	0.239153
$433$	8035.72	0.891853	0.445926	−	0.895070i	$-0.352874\pi$
$433$	8035.72	0.891853	0.445926	+	0.895070i	$0.352874\pi$
$434$	−2527.98	−0.279601
$435$	2191.45	0.241545
$436$	−7724.66	−0.848496
$437$	954.159	0.104448
$438$	−5235.61	−0.571158
$439$	10775.4	1.17148	0.585742	−	0.810497i	$-0.300803\pi$
$439$	10775.4	1.17148	0.585742	+	0.810497i	$0.300803\pi$
$440$	−738.354	−0.0799992
$441$	441.000	0.0476190
$442$	8992.52	0.967716
$443$	3395.81	0.364198	0.182099	−	0.983280i	$-0.441711\pi$
$443$	3395.81	0.364198	0.182099	+	0.983280i	$0.441711\pi$
$444$	−2643.36	−0.282541
$445$	2406.29	0.256335
$446$	−4056.41	−0.430665
$447$	−3704.45	−0.391978
$448$	−253.014	−0.0266826
$449$	−10625.6	−1.11682	−0.558409	−	0.829566i	$-0.688588\pi$
$449$	−10625.6	−1.11682	−0.558409	+	0.829566i	$0.688588\pi$
$450$	−2970.91	−0.311222
$451$	−2181.23	−0.227738
$452$	−884.176	−0.0920092
$453$	3463.74	0.359251
$454$	−6250.41	−0.646137
$455$	1563.86	0.161131
$456$	−2937.15	−0.301633
$457$	3189.95	0.326520	0.163260	−	0.986583i	$-0.447799\pi$
$457$	3189.95	0.326520	0.163260	+	0.986583i	$0.447799\pi$
$458$	1083.68	0.110561
$459$	1734.65	0.176397
$460$	306.455	0.0310620
$461$	11839.8	1.19617	0.598084	−	0.801433i	$-0.295929\pi$
$461$	11839.8	1.19617	0.598084	+	0.801433i	$0.295929\pi$
$462$	822.719	0.0828492
$463$	−14847.0	−1.49027	−0.745137	−	0.666911i	$-0.767616\pi$
$463$	−14847.0	−1.49027	−0.745137	+	0.666911i	$0.767616\pi$
$464$	−10219.8	−1.02251
$465$	−1729.27	−0.172458
$466$	−13921.2	−1.38388
$467$	−5678.07	−0.562634	−0.281317	−	0.959615i	$-0.590771\pi$
$467$	−5678.07	−0.562634	−0.281317	+	0.959615i	$0.590771\pi$
$468$	−1656.97	−0.163661
$469$	−1815.43	−0.178740
$470$	5780.96	0.567353
$471$	8490.52	0.830621
$472$	−1225.20	−0.119480
$473$	−2545.25	−0.247422
$474$	−1513.30	−0.146641
$475$	7685.01	0.742342
$476$	−2106.80	−0.202868
$477$	1823.13	0.175001
$478$	−609.009	−0.0582750
$479$	224.024	0.0213694	0.0106847	−	0.999943i	$-0.496599\pi$
$479$	224.024	0.0213694	0.0106847	+	0.999943i	$0.496599\pi$
$480$	−3219.66	−0.306160
$481$	−7391.82	−0.700703
$482$	−23701.1	−2.23974
$483$	241.659	0.0227658
$484$	566.844	0.0532348
$485$	6229.77	0.583256
$486$	−865.457	−0.0807777
$487$	−6298.42	−0.586055	−0.293028	−	0.956104i	$-0.594663\pi$
$487$	−6298.42	−0.586055	−0.293028	+	0.956104i	$0.594663\pi$
$488$	4521.48	0.419422
$489$	−10255.9	−0.948444
$490$	−992.064	−0.0914631
$491$	7973.25	0.732847	0.366424	−	0.930448i	$-0.380582\pi$
$491$	7973.25	0.732847	0.366424	+	0.930448i	$0.380582\pi$
$492$	−2786.81	−0.255364
$493$	−8255.70	−0.754195
$494$	11605.7	1.05701
$495$	562.781	0.0511013
$496$	8064.44	0.730048
$497$	2467.87	0.222735
$498$	2060.50	0.185408
$499$	−15518.7	−1.39221	−0.696104	−	0.717941i	$-0.745085\pi$
$499$	−15518.7	−1.39221	−0.696104	+	0.717941i	$0.745085\pi$
$500$	5797.09	0.518508
$501$	−4726.49	−0.421485
$502$	3631.01	0.322829
$503$	8420.95	0.746465	0.373232	−	0.927738i	$-0.378249\pi$
$503$	8420.95	0.746465	0.373232	+	0.927738i	$0.378249\pi$
$504$	−743.889	−0.0657450
$505$	10126.1	0.892286
$506$	450.833	0.0396087
$507$	1957.49	0.171469
$508$	−11238.1	−0.981517
$509$	−22582.1	−1.96647	−0.983234	−	0.182350i	$-0.941630\pi$
$509$	−22582.1	−1.96647	−0.983234	+	0.182350i	$0.941630\pi$
$510$	−3902.23	−0.338811
$511$	3430.09	0.296943
$512$	7502.22	0.647567
$513$	2238.72	0.192675
$514$	−24013.1	−2.06065
$515$	−58.6876	−0.00502152
$516$	−3251.90	−0.277436
$517$	3140.86	0.267186
$518$	4689.15	0.397740
$519$	−10792.0	−0.912747
$520$	−2637.95	−0.222465
$521$	2072.11	0.174243	0.0871215	−	0.996198i	$-0.472233\pi$
$521$	2072.11	0.174243	0.0871215	+	0.996198i	$0.472233\pi$
$522$	4118.97	0.345368
$523$	−10531.2	−0.880491	−0.440245	−	0.897877i	$-0.645109\pi$
$523$	−10531.2	−0.880491	−0.440245	+	0.897877i	$0.645109\pi$
$524$	7312.56	0.609638
$525$	1946.38	0.161804
$526$	4568.11	0.378667
$527$	6514.54	0.538478
$528$	−2624.53	−0.216322
$529$	−12034.6	−0.989116
$530$	−4101.28	−0.336128
$531$	933.861	0.0763203
$532$	−2719.02	−0.221588
$533$	−7792.97	−0.633304
$534$	4522.77	0.366515
$535$	−8634.37	−0.697750
$536$	3062.31	0.246776
$537$	−5438.12	−0.437006
$538$	−3313.54	−0.265533
$539$	−539.000	−0.0430730
$540$	719.028	0.0573001
$541$	17730.0	1.40900	0.704502	−	0.709702i	$-0.251170\pi$
$541$	17730.0	1.40900	0.704502	+	0.709702i	$0.251170\pi$
$542$	29687.7	2.35276
$543$	−6717.49	−0.530894
$544$	12129.2	0.955948
$545$	9373.59	0.736735
$546$	2939.36	0.230390
$547$	−18082.8	−1.41346	−0.706730	−	0.707483i	$-0.749831\pi$
$547$	−18082.8	−1.41346	−0.706730	+	0.707483i	$0.749831\pi$
$548$	3848.89	0.300030
$549$	−3446.32	−0.267915
$550$	3631.11	0.281511
$551$	−10654.7	−0.823789
$552$	−407.636	−0.0314314
$553$	991.428	0.0762383
$554$	12441.8	0.954150
$555$	3207.62	0.245326
$556$	−5426.87	−0.413940
$557$	18748.2	1.42619	0.713095	−	0.701068i	$-0.247293\pi$
$557$	18748.2	1.42619	0.713095	+	0.701068i	$0.247293\pi$
$558$	−3250.26	−0.246585
$559$	−9093.53	−0.688041
$560$	3164.76	0.238813
$561$	−2120.12	−0.159557
$562$	−3672.85	−0.275676
$563$	−9033.83	−0.676253	−0.338127	−	0.941101i	$-0.609793\pi$
$563$	−9033.83	−0.676253	−0.338127	+	0.941101i	$0.609793\pi$
$564$	4012.88	0.299597
$565$	1072.91	0.0798900
$566$	−8479.03	−0.629682
$567$	567.000	0.0419961
$568$	−4162.86	−0.307517
$569$	5852.87	0.431222	0.215611	−	0.976479i	$-0.430826\pi$
$569$	5852.87	0.431222	0.215611	+	0.976479i	$0.430826\pi$
$570$	−5036.19	−0.370075
$571$	6521.25	0.477944	0.238972	−	0.971026i	$-0.423190\pi$
$571$	6521.25	0.477944	0.238972	+	0.971026i	$0.423190\pi$
$572$	2025.19	0.148037
$573$	11880.9	0.866202
$574$	4943.63	0.359483
$575$	1066.58	0.0773553
$576$	−325.304	−0.0235318
$577$	10542.3	0.760627	0.380314	−	0.924858i	$-0.375816\pi$
$577$	10542.3	0.760627	0.380314	+	0.924858i	$0.375816\pi$
$578$	−2797.33	−0.201304
$579$	14220.7	1.02071
$580$	−3422.07	−0.244989
$581$	−1349.93	−0.0963931
$582$	11709.2	0.833958
$583$	−2228.27	−0.158294
$584$	−5785.95	−0.409973
$585$	2010.67	0.142104
$586$	25607.5	1.80518
$587$	20713.3	1.45644	0.728218	−	0.685346i	$-0.240349\pi$
$587$	20713.3	1.45644	0.728218	+	0.685346i	$0.240349\pi$
$588$	−688.645	−0.0482980
$589$	8407.62	0.588166
$590$	−2100.79	−0.146590
$591$	15432.5	1.07413
$592$	−14958.7	−1.03851
$593$	7202.25	0.498754	0.249377	−	0.968406i	$-0.419774\pi$
$593$	7202.25	0.498754	0.249377	+	0.968406i	$0.419774\pi$
$594$	1057.78	0.0730661
$595$	2556.52	0.176147
$596$	5784.69	0.397567
$597$	6696.06	0.459048
$598$	1610.71	0.110145
$599$	−19894.2	−1.35702	−0.678510	−	0.734591i	$-0.737374\pi$
$599$	−19894.2	−1.35702	−0.678510	+	0.734591i	$0.737374\pi$
$600$	−3283.20	−0.223393
$601$	10378.4	0.704397	0.352199	−	0.935925i	$-0.385434\pi$
$601$	10378.4	0.704397	0.352199	+	0.935925i	$0.385434\pi$
$602$	5768.66	0.390553
$603$	−2334.13	−0.157634
$604$	−5408.82	−0.364374
$605$	−687.844	−0.0462228
$606$	19032.6	1.27582
$607$	15224.0	1.01800	0.508998	−	0.860768i	$-0.330016\pi$
$607$	15224.0	1.01800	0.508998	+	0.860768i	$0.330016\pi$
$608$	15653.9	1.04416
$609$	−2698.52	−0.179556
$610$	7752.76	0.514591
$611$	11221.5	0.743001
$612$	−2708.74	−0.178913
$613$	−25967.7	−1.71097	−0.855485	−	0.517828i	$-0.826741\pi$
$613$	−25967.7	−1.71097	−0.855485	+	0.517828i	$0.826741\pi$
$614$	−2870.42	−0.188666
$615$	3381.69	0.221729
$616$	909.198	0.0594685
$617$	−25500.4	−1.66387	−0.831935	−	0.554873i	$-0.812767\pi$
$617$	−25500.4	−1.66387	−0.831935	+	0.554873i	$0.812767\pi$
$618$	−110.307	−0.00717992
$619$	−19794.4	−1.28531	−0.642654	−	0.766157i	$-0.722167\pi$
$619$	−19794.4	−1.28531	−0.642654	+	0.766157i	$0.722167\pi$
$620$	2700.34	0.174917
$621$	310.705	0.0200775
$622$	5437.11	0.350496
$623$	−2963.07	−0.190550
$624$	−9376.78	−0.601557
$625$	4551.03	0.291266
$626$	20296.1	1.29584
$627$	−2736.22	−0.174281
$628$	−13258.4	−0.842464
$629$	−12083.8	−0.765999
$630$	−1275.51	−0.0806628
$631$	19940.5	1.25803	0.629017	−	0.777392i	$-0.283458\pi$
$631$	19940.5	1.25803	0.629017	+	0.777392i	$0.283458\pi$
$632$	−1672.36	−0.105258
$633$	9357.14	0.587540
$634$	−14375.5	−0.900511
$635$	13637.0	0.852235
$636$	−2846.91	−0.177496
$637$	−1925.71	−0.119779
$638$	−5034.29	−0.312397
$639$	3172.98	0.196434
$640$	−7853.97	−0.485087
$641$	−22527.9	−1.38814	−0.694070	−	0.719907i	$-0.744184\pi$
$641$	−22527.9	−1.38814	−0.694070	+	0.719907i	$0.744184\pi$
$642$	−16228.8	−0.997664
$643$	−11676.3	−0.716125	−0.358062	−	0.933698i	$-0.616562\pi$
$643$	−11676.3	−0.716125	−0.358062	+	0.933698i	$0.616562\pi$
$644$	−377.363	−0.0230904
$645$	3946.06	0.240893
$646$	18972.5	1.15551
$647$	−9749.45	−0.592412	−0.296206	−	0.955124i	$-0.595721\pi$
$647$	−9749.45	−0.592412	−0.296206	+	0.955124i	$0.595721\pi$
$648$	−956.429	−0.0579816
$649$	−1141.39	−0.0690343
$650$	12973.0	0.782838
$651$	2129.39	0.128199
$652$	16015.2	0.961968
$653$	3439.33	0.206112	0.103056	−	0.994676i	$-0.467138\pi$
$653$	3439.33	0.206112	0.103056	+	0.994676i	$0.467138\pi$
$654$	17618.2	1.05341
$655$	−8873.51	−0.529339
$656$	−15770.5	−0.938622
$657$	4410.11	0.261879
$658$	−7118.59	−0.421750
$659$	16400.6	0.969463	0.484732	−	0.874663i	$-0.338917\pi$
$659$	16400.6	0.969463	0.484732	+	0.874663i	$0.338917\pi$
$660$	−878.813	−0.0518299
$661$	−7571.56	−0.445537	−0.222768	−	0.974871i	$-0.571509\pi$
$661$	−7571.56	−0.445537	−0.222768	+	0.974871i	$0.571509\pi$
$662$	9496.81	0.557559
$663$	−7574.66	−0.443704
$664$	2277.09	0.133085
$665$	3299.43	0.192401
$666$	6028.91	0.350774
$667$	−1478.73	−0.0858423
$668$	7380.66	0.427494
$669$	3416.83	0.197463
$670$	5250.80	0.302771
$671$	4212.17	0.242338
$672$	3964.64	0.227589
$673$	−5876.65	−0.336595	−0.168297	−	0.985736i	$-0.553827\pi$
$673$	−5876.65	−0.336595	−0.168297	+	0.985736i	$0.553827\pi$
$674$	7328.43	0.418814
$675$	2502.49	0.142697
$676$	−3056.72	−0.173914
$677$	−9870.55	−0.560348	−0.280174	−	0.959949i	$-0.590392\pi$
$677$	−9870.55	−0.560348	−0.280174	+	0.959949i	$0.590392\pi$
$678$	2016.61	0.114229
$679$	−7671.24	−0.433572
$680$	−4312.41	−0.243196
$681$	5264.90	0.296258
$682$	3972.54	0.223045
$683$	−9551.67	−0.535116	−0.267558	−	0.963542i	$-0.586217\pi$
$683$	−9551.67	−0.535116	−0.267558	+	0.963542i	$0.586217\pi$
$684$	−3495.89	−0.195422
$685$	−4670.49	−0.260511
$686$	1221.61	0.0679904
$687$	−912.812	−0.0506928
$688$	−18402.4	−1.01975
$689$	−7961.04	−0.440191
$690$	−698.955	−0.0385634
$691$	−10329.4	−0.568665	−0.284333	−	0.958726i	$-0.591772\pi$
$691$	−10329.4	−0.568665	−0.284333	+	0.958726i	$0.591772\pi$
$692$	16852.3	0.925762
$693$	−693.000	−0.0379869
$694$	30476.0	1.66694
$695$	6585.31	0.359417
$696$	4551.93	0.247903
$697$	−12739.6	−0.692320
$698$	−33115.4	−1.79575
$699$	11726.2	0.634515
$700$	−3039.37	−0.164111
$701$	−33570.9	−1.80878	−0.904390	−	0.426708i	$-0.859673\pi$
$701$	−33570.9	−1.80878	−0.904390	+	0.426708i	$0.859673\pi$
$702$	3779.18	0.203185
$703$	−15595.3	−0.836682
$704$	397.594	0.0212853
$705$	−4869.47	−0.260135
$706$	22492.9	1.19905
$707$	−12469.1	−0.663294
$708$	−1458.27	−0.0774085
$709$	22424.4	1.18782	0.593911	−	0.804531i	$-0.297583\pi$
$709$	22424.4	1.18782	0.593911	+	0.804531i	$0.297583\pi$
$710$	−7137.86	−0.377295
$711$	1274.69	0.0672359
$712$	4998.17	0.263082
$713$	1166.86	0.0612895
$714$	4805.14	0.251860
$715$	−2457.49	−0.128538
$716$	8491.92	0.443237
$717$	512.986	0.0267194
$718$	15552.5	0.808377
$719$	−12475.8	−0.647107	−0.323553	−	0.946210i	$-0.604878\pi$
$719$	−12475.8	−0.647107	−0.323553	+	0.946210i	$0.604878\pi$
$720$	4068.97	0.210613
$721$	72.2670	0.00373282
$722$	57.0409	0.00294023
$723$	19964.1	1.02693
$724$	10489.7	0.538463
$725$	−11910.1	−0.610109
$726$	−1292.84	−0.0660908
$727$	35619.7	1.81714	0.908569	−	0.417734i	$-0.137176\pi$
$727$	35619.7	1.81714	0.908569	+	0.417734i	$0.137176\pi$
$728$	3248.33	0.165372
$729$	729.000	0.0370370
$730$	−9920.89	−0.502998
$731$	−14865.7	−0.752158
$732$	5381.61	0.271735
$733$	−23587.0	−1.18855	−0.594274	−	0.804263i	$-0.702560\pi$
$733$	−23587.0	−1.18855	−0.594274	+	0.804263i	$0.702560\pi$
$734$	1621.38	0.0815341
$735$	835.645	0.0419364
$736$	2172.55	0.108806
$737$	2852.82	0.142585
$738$	6356.10	0.317034
$739$	21418.0	1.06613	0.533067	−	0.846073i	$-0.321039\pi$
$739$	21418.0	1.06613	0.533067	+	0.846073i	$0.321039\pi$
$740$	−5008.86	−0.248824
$741$	−9775.81	−0.484647
$742$	5050.25	0.249866
$743$	25868.0	1.27726	0.638630	−	0.769514i	$-0.279501\pi$
$743$	25868.0	1.27726	0.638630	+	0.769514i	$0.279501\pi$
$744$	−3591.91	−0.176997
$745$	−7019.50	−0.345201
$746$	−19036.5	−0.934286
$747$	−1735.62	−0.0850107
$748$	3310.69	0.161832
$749$	10632.2	0.518682
$750$	−13221.9	−0.643726
$751$	−28345.7	−1.37730	−0.688648	−	0.725096i	$-0.741796\pi$
$751$	−28345.7	−1.37730	−0.688648	+	0.725096i	$0.741796\pi$
$752$	22708.8	1.10120
$753$	−3058.51	−0.148019
$754$	−17986.2	−0.868727
$755$	6563.40	0.316380
$756$	−885.400	−0.0425948
$757$	5065.33	0.243200	0.121600	−	0.992579i	$-0.461197\pi$
$757$	5065.33	0.243200	0.121600	+	0.992579i	$0.461197\pi$
$758$	34805.0	1.66778
$759$	−379.750	−0.0181608
$760$	−5565.56	−0.265637
$761$	−24301.0	−1.15757	−0.578784	−	0.815481i	$-0.696473\pi$
$761$	−24301.0	−1.15757	−0.578784	+	0.815481i	$0.696473\pi$
$762$	25631.6	1.21855
$763$	−11542.5	−0.547662
$764$	−18552.7	−0.878553
$765$	3286.96	0.155347
$766$	2926.03	0.138018
$767$	−4077.88	−0.191973
$768$	−15629.5	−0.734350
$769$	−1579.45	−0.0740656	−0.0370328	−	0.999314i	$-0.511791\pi$
$769$	−1579.45	−0.0740656	−0.0370328	+	0.999314i	$0.511791\pi$
$770$	1558.96	0.0729623
$771$	20226.9	0.944818
$772$	−22206.3	−1.03526
$773$	7349.54	0.341973	0.170986	−	0.985273i	$-0.445305\pi$
$773$	7349.54	0.341973	0.170986	+	0.985273i	$0.445305\pi$
$774$	7416.85	0.344436
$775$	9398.19	0.435604
$776$	12940.0	0.598608
$777$	−3949.81	−0.182366
$778$	−12937.8	−0.596200
$779$	−16441.7	−0.756204
$780$	−3139.77	−0.144131
$781$	−3878.08	−0.177681
$782$	2633.12	0.120409
$783$	−3469.53	−0.158353
$784$	−3897.03	−0.177525
$785$	16088.6	0.731497
$786$	−16678.3	−0.756864
$787$	6878.56	0.311555	0.155778	−	0.987792i	$-0.450212\pi$
$787$	6878.56	0.311555	0.155778	+	0.987792i	$0.450212\pi$
$788$	−24098.7	−1.08944
$789$	−3847.85	−0.173621
$790$	−2867.52	−0.129142
$791$	−1321.17	−0.0593874
$792$	1168.97	0.0524463
$793$	15049.0	0.673903
$794$	17757.2	0.793676
$795$	3454.62	0.154117
$796$	−10456.2	−0.465593
$797$	4830.47	0.214685	0.107343	−	0.994222i	$-0.465766\pi$
$797$	4830.47	0.214685	0.107343	+	0.994222i	$0.465766\pi$
$798$	6201.48	0.275100
$799$	18344.4	0.812239
$800$	17498.2	0.773318
$801$	−3809.66	−0.168050
$802$	−19892.6	−0.875851
$803$	−5390.13	−0.236879
$804$	3644.86	0.159881
$805$	457.917	0.0200490
$806$	14192.9	0.620251
$807$	2791.09	0.121749
$808$	21033.2	0.915772
$809$	−44538.0	−1.93557	−0.967783	−	0.251786i	$-0.918982\pi$
$809$	−44538.0	−1.93557	−0.967783	+	0.251786i	$0.918982\pi$
$810$	−1639.94	−0.0711379
$811$	40738.9	1.76391	0.881957	−	0.471329i	$-0.156226\pi$
$811$	40738.9	1.76391	0.881957	+	0.471329i	$0.156226\pi$
$812$	4213.88	0.182116
$813$	−25006.8	−1.07876
$814$	−7368.67	−0.317287
$815$	−19433.8	−0.835260
$816$	−15328.7	−0.657614
$817$	−19185.6	−0.821564
$818$	−18902.5	−0.807959
$819$	−2475.91	−0.105635
$820$	−5280.69	−0.224890
$821$	29805.1	1.26700	0.633498	−	0.773744i	$-0.281618\pi$
$821$	29805.1	1.26700	0.633498	+	0.773744i	$0.281618\pi$
$822$	−8778.46	−0.372487
$823$	3453.87	0.146287	0.0731435	−	0.997321i	$-0.476697\pi$
$823$	3453.87	0.146287	0.0731435	+	0.997321i	$0.476697\pi$
$824$	−121.902	−0.00515369
$825$	−3058.59	−0.129075
$826$	2586.88	0.108970
$827$	−10258.1	−0.431328	−0.215664	−	0.976468i	$-0.569192\pi$
$827$	−10258.1	−0.431328	−0.215664	+	0.976468i	$0.569192\pi$
$828$	−485.182	−0.0203638
$829$	33599.2	1.40766	0.703830	−	0.710369i	$-0.251472\pi$
$829$	33599.2	1.40766	0.703830	+	0.710369i	$0.251472\pi$
$830$	3904.41	0.163282
$831$	−10480.1	−0.437484
$832$	1420.50	0.0591911
$833$	−3148.06	−0.130941
$834$	12377.5	0.513906
$835$	−8956.16	−0.371186
$836$	4272.75	0.176766
$837$	2737.79	0.113061
$838$	43622.8	1.79824
$839$	−24363.0	−1.00251	−0.501254	−	0.865300i	$-0.667128\pi$
$839$	−24363.0	−1.00251	−0.501254	+	0.865300i	$0.667128\pi$
$840$	−1409.59	−0.0578992
$841$	−7876.51	−0.322953
$842$	−24782.7	−1.01433
$843$	3093.75	0.126399
$844$	−14611.7	−0.595918
$845$	3709.21	0.151007
$846$	−9152.47	−0.371949
$847$	847.000	0.0343604
$848$	−16110.6	−0.652408
$849$	7142.13	0.288713
$850$	21207.8	0.855788
$851$	−2164.42	−0.0871859
$852$	−4954.77	−0.199234
$853$	12928.0	0.518928	0.259464	−	0.965753i	$-0.416454\pi$
$853$	12928.0	0.518928	0.259464	+	0.965753i	$0.416454\pi$
$854$	−9546.63	−0.382528
$855$	4242.13	0.169682
$856$	−17934.7	−0.716116
$857$	−40734.2	−1.62363	−0.811817	−	0.583911i	$-0.801522\pi$
$857$	−40734.2	−1.62363	−0.811817	+	0.583911i	$0.801522\pi$
$858$	−4619.00	−0.183788
$859$	32796.6	1.30269	0.651343	−	0.758784i	$-0.274206\pi$
$859$	32796.6	1.30269	0.651343	+	0.758784i	$0.274206\pi$
$860$	−6161.98	−0.244327
$861$	−4164.16	−0.164825
$862$	22243.8	0.878916
$863$	−879.506	−0.0346915	−0.0173457	−	0.999850i	$-0.505522\pi$
$863$	−879.506	−0.0346915	−0.0173457	+	0.999850i	$0.505522\pi$
$864$	5097.40	0.200714
$865$	−20449.6	−0.803823
$866$	28619.7	1.12302
$867$	2356.27	0.0922990
$868$	−3325.16	−0.130027
$869$	−1557.96	−0.0608172
$870$	7804.97	0.304153
$871$	10192.4	0.396506
$872$	19470.2	0.756127
$873$	−9863.03	−0.382374
$874$	3398.29	0.131520
$875$	8662.24	0.334671
$876$	−6886.62	−0.265613
$877$	17607.7	0.677957	0.338978	−	0.940794i	$-0.389919\pi$
$877$	17607.7	0.677957	0.338978	+	0.940794i	$0.389919\pi$
$878$	38377.2	1.47513
$879$	−21569.9	−0.827685
$880$	−4973.19	−0.190507
$881$	−24818.3	−0.949093	−0.474546	−	0.880230i	$-0.657388\pi$
$881$	−24818.3	−0.949093	−0.474546	+	0.880230i	$0.657388\pi$
$882$	1570.64	0.0599619
$883$	18274.6	0.696477	0.348238	−	0.937406i	$-0.386780\pi$
$883$	18274.6	0.696477	0.348238	+	0.937406i	$0.386780\pi$
$884$	11828.2	0.450030
$885$	1769.56	0.0672125
$886$	12094.4	0.458598
$887$	−21922.4	−0.829856	−0.414928	−	0.909854i	$-0.636193\pi$
$887$	−21922.4	−0.829856	−0.414928	+	0.909854i	$0.636193\pi$
$888$	6662.63	0.251783
$889$	−16792.4	−0.633521
$890$	8570.13	0.322777
$891$	−891.000	−0.0335013
$892$	−5335.57	−0.200278
$893$	23675.2	0.887189
$894$	−13193.6	−0.493579
$895$	−10304.6	−0.384856
$896$	9671.26	0.360596
$897$	−1356.75	−0.0505023
$898$	−37843.5	−1.40630
$899$	−13029.9	−0.483396
$900$	−3907.76	−0.144732
$901$	−13014.4	−0.481211
$902$	−7768.56	−0.286768
$903$	−4859.11	−0.179071
$904$	2228.58	0.0819928
$905$	−12728.9	−0.467539
$906$	12336.3	0.452369
$907$	−9704.52	−0.355274	−0.177637	−	0.984096i	$-0.556845\pi$
$907$	−9704.52	−0.355274	−0.177637	+	0.984096i	$0.556845\pi$
$908$	−8221.42	−0.300482
$909$	−16031.7	−0.584970
$910$	5569.76	0.202896
$911$	40557.0	1.47499	0.737494	−	0.675354i	$-0.236009\pi$
$911$	40557.0	1.47499	0.737494	+	0.675354i	$0.236009\pi$
$912$	−19783.2	−0.718296
$913$	2121.31	0.0768951
$914$	11361.2	0.411153
$915$	−6530.38	−0.235943
$916$	1425.40	0.0514156
$917$	10926.7	0.393491
$918$	6178.04	0.222119
$919$	−10182.1	−0.365482	−0.182741	−	0.983161i	$-0.558497\pi$
$919$	−10182.1	−0.365482	−0.182741	+	0.983161i	$0.558497\pi$
$920$	−772.424	−0.0276805
$921$	2417.84	0.0865044
$922$	42168.0	1.50621
$923$	−13855.4	−0.494102
$924$	1082.16	0.0385285
$925$	−17432.7	−0.619658
$926$	−52878.3	−1.87655
$927$	92.9147	0.00329204
$928$	−24260.0	−0.858162
$929$	35454.0	1.25211	0.626053	−	0.779780i	$-0.284669\pi$
$929$	35454.0	1.25211	0.626053	+	0.779780i	$0.284669\pi$
$930$	−6158.87	−0.217159
$931$	−4062.87	−0.143024
$932$	−18311.1	−0.643562
$933$	−4579.84	−0.160704
$934$	−20222.8	−0.708468
$935$	−4017.40	−0.140516
$936$	4176.43	0.145845
$937$	35786.9	1.24771	0.623856	−	0.781539i	$-0.285565\pi$
$937$	35786.9	1.24771	0.623856	+	0.781539i	$0.285565\pi$
$938$	−6465.76	−0.225069
$939$	−17096.0	−0.594151
$940$	7603.94	0.263844
$941$	−53299.3	−1.84645	−0.923223	−	0.384264i	$-0.874455\pi$
$941$	−53299.3	−1.84645	−0.923223	+	0.384264i	$0.874455\pi$
$942$	30239.4	1.04592
$943$	−2281.88	−0.0787998
$944$	−8252.34	−0.284524
$945$	1074.40	0.0369844
$946$	−9065.04	−0.311554
$947$	−2396.55	−0.0822359	−0.0411180	−	0.999154i	$-0.513092\pi$
$947$	−2396.55	−0.0822359	−0.0411180	+	0.999154i	$0.513092\pi$
$948$	−1990.50	−0.0681946
$949$	−19257.6	−0.658722
$950$	27370.6	0.934757
$951$	12108.9	0.412890
$952$	5310.23	0.180783
$953$	49723.2	1.69013	0.845064	−	0.534665i	$-0.179562\pi$
$953$	49723.2	1.69013	0.845064	+	0.534665i	$0.179562\pi$
$954$	6493.18	0.220361
$955$	22513.0	0.762832
$956$	−801.055	−0.0271004
$957$	4240.53	0.143236
$958$	797.875	0.0269083
$959$	5751.16	0.193655
$960$	−616.414	−0.0207236
$961$	−19509.1	−0.654866
$962$	−26326.4	−0.882324
$963$	13670.0	0.457435
$964$	−31175.0	−1.04158
$965$	26946.6	0.898902
$966$	860.682	0.0286666
$967$	−34465.6	−1.14616	−0.573081	−	0.819499i	$-0.694252\pi$
$967$	−34465.6	−1.14616	−0.573081	+	0.819499i	$0.694252\pi$
$968$	−1428.74	−0.0474395
$969$	−15981.1	−0.529810
$970$	22187.7	0.734436
$971$	−29569.6	−0.977276	−0.488638	−	0.872487i	$-0.662506\pi$
$971$	−29569.6	−0.977276	−0.488638	+	0.872487i	$0.662506\pi$
$972$	−1138.37	−0.0375651
$973$	−8109.05	−0.267178
$974$	−22432.2	−0.737960
$975$	−10927.6	−0.358936
$976$	30454.4	0.998794
$977$	49863.2	1.63282	0.816411	−	0.577472i	$-0.195961\pi$
$977$	49863.2	1.63282	0.816411	+	0.577472i	$0.195961\pi$
$978$	−36527.0	−1.19428
$979$	4656.25	0.152007
$980$	−1304.90	−0.0425343
$981$	−14840.4	−0.482993
$982$	28397.2	0.922800
$983$	45383.3	1.47254	0.736268	−	0.676690i	$-0.236586\pi$
$983$	45383.3	1.47254	0.736268	+	0.676690i	$0.236586\pi$
$984$	7024.21	0.227565
$985$	29242.9	0.945946
$986$	−29403.1	−0.949681
$987$	5996.20	0.193375
$988$	15265.4	0.491557
$989$	−2662.70	−0.0856105
$990$	2004.37	0.0643467
$991$	−54822.7	−1.75732	−0.878658	−	0.477451i	$-0.841561\pi$
$991$	−54822.7	−1.75732	−0.878658	+	0.477451i	$0.841561\pi$
$992$	19143.5	0.612708
$993$	−7999.44	−0.255644
$994$	8789.45	0.280467
$995$	12688.3	0.404266
$996$	2710.26	0.0862228
$997$	29649.9	0.941849	0.470924	−	0.882174i	$-0.343920\pi$
$997$	29649.9	0.941849	0.470924	+	0.882174i	$0.343920\pi$
$998$	−55270.6	−1.75307
$999$	−5078.33	−0.160832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.4.a.h.1.2	✓	2
3.2	odd	2		693.4.a.g.1.1		2
7.6	odd	2		1617.4.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.h.1.2	✓	2	1.1	even	1	trivial
693.4.a.g.1.1		2	3.2	odd	2
1617.4.a.m.1.2		2	7.6	odd	2

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:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

\(f(q)\)	\(=\)	\(q+3.56155 q^{2} -3.00000 q^{3} +4.68466 q^{4} -5.68466 q^{5} -10.6847 q^{6} +7.00000 q^{7} -11.8078 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.56155 q^{2} -3.00000 q^{3} +4.68466 q^{4} -5.68466 q^{5} -10.6847 q^{6} +7.00000 q^{7} -11.8078 q^{8} +9.00000 q^{9} -20.2462 q^{10} -11.0000 q^{11} -14.0540 q^{12} -39.3002 q^{13} +24.9309 q^{14} +17.0540 q^{15} -79.5312 q^{16} -64.2462 q^{17} +32.0540 q^{18} -82.9157 q^{19} -26.6307 q^{20} -21.0000 q^{21} -39.1771 q^{22} -11.5076 q^{23} +35.4233 q^{24} -92.6847 q^{25} -139.970 q^{26} -27.0000 q^{27} +32.7926 q^{28} +128.501 q^{29} +60.7386 q^{30} -101.400 q^{31} -188.793 q^{32} +33.0000 q^{33} -228.816 q^{34} -39.7926 q^{35} +42.1619 q^{36} +188.086 q^{37} -295.309 q^{38} +117.901 q^{39} +67.1231 q^{40} +198.294 q^{41} -74.7926 q^{42} +231.386 q^{43} -51.5312 q^{44} -51.1619 q^{45} -40.9848 q^{46} -285.533 q^{47} +238.594 q^{48} +49.0000 q^{49} -330.101 q^{50} +192.739 q^{51} -184.108 q^{52} +202.570 q^{53} -96.1619 q^{54} +62.5312 q^{55} -82.6543 q^{56} +248.747 q^{57} +457.663 q^{58} +103.762 q^{59} +79.8920 q^{60} -382.924 q^{61} -361.140 q^{62} +63.0000 q^{63} -36.1449 q^{64} +223.408 q^{65} +117.531 q^{66} -259.348 q^{67} -300.972 q^{68} +34.5227 q^{69} -141.723 q^{70} +352.553 q^{71} -106.270 q^{72} +490.012 q^{73} +669.879 q^{74} +278.054 q^{75} -388.432 q^{76} -77.0000 q^{77} +419.909 q^{78} +141.633 q^{79} +452.108 q^{80} +81.0000 q^{81} +706.233 q^{82} -192.847 q^{83} -98.3778 q^{84} +365.218 q^{85} +824.095 q^{86} -385.503 q^{87} +129.885 q^{88} -423.295 q^{89} -182.216 q^{90} -275.101 q^{91} -53.9091 q^{92} +304.199 q^{93} -1016.94 q^{94} +471.348 q^{95} +566.378 q^{96} -1095.89 q^{97} +174.516 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 6 q^{3} - 3 q^{4} + q^{5} - 9 q^{6} + 14 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 + q^5 - 9 * q^6 + 14 * q^7 - 3 * q^8 + 18 * q^9	\( 2 q + 3 q^{2} - 6 q^{3} - 3 q^{4} + q^{5} - 9 q^{6} + 14 q^{7} - 3 q^{8} + 18 q^{9} - 24 q^{10} - 22 q^{11} + 9 q^{12} - 25 q^{13} + 21 q^{14} - 3 q^{15} - 23 q^{16} - 112 q^{17} + 27 q^{18} - 71 q^{19} - 78 q^{20} - 42 q^{21} - 33 q^{22} - 56 q^{23} + 9 q^{24} - 173 q^{25} - 148 q^{26} - 54 q^{27} - 21 q^{28} - 11 q^{29} + 72 q^{30} - 310 q^{31} - 291 q^{32} + 66 q^{33} - 202 q^{34} + 7 q^{35} - 27 q^{36} - 65 q^{37} - 302 q^{38} + 75 q^{39} + 126 q^{40} + 42 q^{41} - 63 q^{42} - 32 q^{43} + 33 q^{44} + 9 q^{45} - 16 q^{46} + 101 q^{47} + 69 q^{48} + 98 q^{49} - 285 q^{50} + 336 q^{51} - 294 q^{52} + 166 q^{53} - 81 q^{54} - 11 q^{55} - 21 q^{56} + 213 q^{57} + 536 q^{58} - 11 q^{59} + 234 q^{60} - 436 q^{61} - 244 q^{62} + 126 q^{63} - 431 q^{64} + 319 q^{65} + 99 q^{66} - 127 q^{67} + 66 q^{68} + 168 q^{69} - 168 q^{70} + 936 q^{71} - 27 q^{72} - 327 q^{73} + 812 q^{74} + 519 q^{75} - 480 q^{76} - 154 q^{77} + 444 q^{78} - 228 q^{79} + 830 q^{80} + 162 q^{81} + 794 q^{82} - 262 q^{83} + 63 q^{84} + 46 q^{85} + 972 q^{86} + 33 q^{87} + 33 q^{88} + 44 q^{89} - 216 q^{90} - 175 q^{91} + 288 q^{92} + 930 q^{93} - 1234 q^{94} + 551 q^{95} + 873 q^{96} - 2266 q^{97} + 147 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 + q^5 - 9 * q^6 + 14 * q^7 - 3 * q^8 + 18 * q^9 - 24 * q^10 - 22 * q^11 + 9 * q^12 - 25 * q^13 + 21 * q^14 - 3 * q^15 - 23 * q^16 - 112 * q^17 + 27 * q^18 - 71 * q^19 - 78 * q^20 - 42 * q^21 - 33 * q^22 - 56 * q^23 + 9 * q^24 - 173 * q^25 - 148 * q^26 - 54 * q^27 - 21 * q^28 - 11 * q^29 + 72 * q^30 - 310 * q^31 - 291 * q^32 + 66 * q^33 - 202 * q^34 + 7 * q^35 - 27 * q^36 - 65 * q^37 - 302 * q^38 + 75 * q^39 + 126 * q^40 + 42 * q^41 - 63 * q^42 - 32 * q^43 + 33 * q^44 + 9 * q^45 - 16 * q^46 + 101 * q^47 + 69 * q^48 + 98 * q^49 - 285 * q^50 + 336 * q^51 - 294 * q^52 + 166 * q^53 - 81 * q^54 - 11 * q^55 - 21 * q^56 + 213 * q^57 + 536 * q^58 - 11 * q^59 + 234 * q^60 - 436 * q^61 - 244 * q^62 + 126 * q^63 - 431 * q^64 + 319 * q^65 + 99 * q^66 - 127 * q^67 + 66 * q^68 + 168 * q^69 - 168 * q^70 + 936 * q^71 - 27 * q^72 - 327 * q^73 + 812 * q^74 + 519 * q^75 - 480 * q^76 - 154 * q^77 + 444 * q^78 - 228 * q^79 + 830 * q^80 + 162 * q^81 + 794 * q^82 - 262 * q^83 + 63 * q^84 + 46 * q^85 + 972 * q^86 + 33 * q^87 + 33 * q^88 + 44 * q^89 - 216 * q^90 - 175 * q^91 + 288 * q^92 + 930 * q^93 - 1234 * q^94 + 551 * q^95 + 873 * q^96 - 2266 * q^97 + 147 * q^98 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more