LMFDB - Embedded newform 2013.4.a.h.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$0$
Dimension:	$39$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32411 q^{2} +3.00000 q^{3} +3.04969 q^{4} -5.05305 q^{5} -9.97232 q^{6} -19.7829 q^{7} +16.4554 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.32411 q^{2} +3.00000 q^{3} +3.04969 q^{4} -5.05305 q^{5} -9.97232 q^{6} -19.7829 q^{7} +16.4554 q^{8} +9.00000 q^{9} +16.7969 q^{10} -11.0000 q^{11} +9.14908 q^{12} +45.4209 q^{13} +65.7604 q^{14} -15.1591 q^{15} -79.0969 q^{16} -100.318 q^{17} -29.9170 q^{18} -118.486 q^{19} -15.4102 q^{20} -59.3486 q^{21} +36.5652 q^{22} -144.840 q^{23} +49.3661 q^{24} -99.4667 q^{25} -150.984 q^{26} +27.0000 q^{27} -60.3317 q^{28} +11.2722 q^{29} +50.3906 q^{30} -214.454 q^{31} +131.284 q^{32} -33.0000 q^{33} +333.466 q^{34} +99.9638 q^{35} +27.4472 q^{36} +354.551 q^{37} +393.859 q^{38} +136.263 q^{39} -83.1497 q^{40} +464.062 q^{41} +197.281 q^{42} +132.501 q^{43} -33.5466 q^{44} -45.4774 q^{45} +481.463 q^{46} -220.759 q^{47} -237.291 q^{48} +48.3619 q^{49} +330.638 q^{50} -300.953 q^{51} +138.520 q^{52} -568.188 q^{53} -89.7509 q^{54} +55.5835 q^{55} -325.534 q^{56} -355.457 q^{57} -37.4699 q^{58} -88.2459 q^{59} -46.2307 q^{60} +61.0000 q^{61} +712.867 q^{62} -178.046 q^{63} +196.374 q^{64} -229.514 q^{65} +109.696 q^{66} +573.794 q^{67} -305.938 q^{68} -434.520 q^{69} -332.290 q^{70} +122.152 q^{71} +148.098 q^{72} -577.597 q^{73} -1178.57 q^{74} -298.400 q^{75} -361.345 q^{76} +217.612 q^{77} -452.952 q^{78} -897.463 q^{79} +399.681 q^{80} +81.0000 q^{81} -1542.59 q^{82} -913.741 q^{83} -180.995 q^{84} +506.909 q^{85} -440.448 q^{86} +33.8165 q^{87} -181.009 q^{88} +277.078 q^{89} +151.172 q^{90} -898.556 q^{91} -441.717 q^{92} -643.361 q^{93} +733.827 q^{94} +598.714 q^{95} +393.851 q^{96} -1766.99 q^{97} -160.760 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + 8 q^{2} + 117 q^{3} + 154 q^{4} + 65 q^{5} + 24 q^{6} + 35 q^{7} + 75 q^{8} + 351 q^{9}+O(q^{10}) $ 39 * q + 8 * q^2 + 117 * q^3 + 154 * q^4 + 65 * q^5 + 24 * q^6 + 35 * q^7 + 75 * q^8 + 351 * q^9	\( 39 q + 8 q^{2} + 117 q^{3} + 154 q^{4} + 65 q^{5} + 24 q^{6} + 35 q^{7} + 75 q^{8} + 351 q^{9} - 21 q^{10} - 429 q^{11} + 462 q^{12} - 27 q^{13} + 164 q^{14} + 195 q^{15} + 686 q^{16} + 170 q^{17} + 72 q^{18} + 139 q^{19} + 1056 q^{20} + 105 q^{21} - 88 q^{22} + 291 q^{23} + 225 q^{24} + 1236 q^{25} + 583 q^{26} + 1053 q^{27} + 976 q^{28} + 374 q^{29} - 63 q^{30} + 232 q^{31} + 933 q^{32} - 1287 q^{33} + 332 q^{34} + 626 q^{35} + 1386 q^{36} + 232 q^{37} + 989 q^{38} - 81 q^{39} - 263 q^{40} + 1014 q^{41} + 492 q^{42} + 515 q^{43} - 1694 q^{44} + 585 q^{45} - 371 q^{46} + 2005 q^{47} + 2058 q^{48} + 2064 q^{49} + 4582 q^{50} + 510 q^{51} + 216 q^{52} + 1485 q^{53} + 216 q^{54} - 715 q^{55} + 2307 q^{56} + 417 q^{57} + 573 q^{58} + 2749 q^{59} + 3168 q^{60} + 2379 q^{61} + 1837 q^{62} + 315 q^{63} + 7295 q^{64} + 3630 q^{65} - 264 q^{66} + 3575 q^{67} + 2630 q^{68} + 873 q^{69} + 4218 q^{70} + 4723 q^{71} + 675 q^{72} + 859 q^{73} + 4232 q^{74} + 3708 q^{75} + 2466 q^{76} - 385 q^{77} + 1749 q^{78} - 1887 q^{79} + 8933 q^{80} + 3159 q^{81} + 6806 q^{82} + 5609 q^{83} + 2928 q^{84} - 565 q^{85} + 5185 q^{86} + 1122 q^{87} - 825 q^{88} + 6725 q^{89} - 189 q^{90} + 2808 q^{91} + 3257 q^{92} + 696 q^{93} + 3184 q^{94} + 3216 q^{95} + 2799 q^{96} + 3512 q^{97} + 4464 q^{98} - 3861 q^{99}+O(q^{100}) \) 39 * q + 8 * q^2 + 117 * q^3 + 154 * q^4 + 65 * q^5 + 24 * q^6 + 35 * q^7 + 75 * q^8 + 351 * q^9 - 21 * q^10 - 429 * q^11 + 462 * q^12 - 27 * q^13 + 164 * q^14 + 195 * q^15 + 686 * q^16 + 170 * q^17 + 72 * q^18 + 139 * q^19 + 1056 * q^20 + 105 * q^21 - 88 * q^22 + 291 * q^23 + 225 * q^24 + 1236 * q^25 + 583 * q^26 + 1053 * q^27 + 976 * q^28 + 374 * q^29 - 63 * q^30 + 232 * q^31 + 933 * q^32 - 1287 * q^33 + 332 * q^34 + 626 * q^35 + 1386 * q^36 + 232 * q^37 + 989 * q^38 - 81 * q^39 - 263 * q^40 + 1014 * q^41 + 492 * q^42 + 515 * q^43 - 1694 * q^44 + 585 * q^45 - 371 * q^46 + 2005 * q^47 + 2058 * q^48 + 2064 * q^49 + 4582 * q^50 + 510 * q^51 + 216 * q^52 + 1485 * q^53 + 216 * q^54 - 715 * q^55 + 2307 * q^56 + 417 * q^57 + 573 * q^58 + 2749 * q^59 + 3168 * q^60 + 2379 * q^61 + 1837 * q^62 + 315 * q^63 + 7295 * q^64 + 3630 * q^65 - 264 * q^66 + 3575 * q^67 + 2630 * q^68 + 873 * q^69 + 4218 * q^70 + 4723 * q^71 + 675 * q^72 + 859 * q^73 + 4232 * q^74 + 3708 * q^75 + 2466 * q^76 - 385 * q^77 + 1749 * q^78 - 1887 * q^79 + 8933 * q^80 + 3159 * q^81 + 6806 * q^82 + 5609 * q^83 + 2928 * q^84 - 565 * q^85 + 5185 * q^86 + 1122 * q^87 - 825 * q^88 + 6725 * q^89 - 189 * q^90 + 2808 * q^91 + 3257 * q^92 + 696 * q^93 + 3184 * q^94 + 3216 * q^95 + 2799 * q^96 + 3512 * q^97 + 4464 * q^98 - 3861 * 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 $

$2$	−3.32411	−1.17525	−0.587625	−	0.809134i	$-0.699937\pi$
$2$	−3.32411	−1.17525	−0.587625	+	0.809134i	$0.699937\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	3.04969	0.381211
$5$	−5.05305	−0.451958	−0.225979	−	0.974132i	$-0.572558\pi$
$5$	−5.05305	−0.451958	−0.225979	+	0.974132i	$0.572558\pi$
$6$	−9.97232	−0.678531
$7$	−19.7829	−1.06817	−0.534087	−	0.845429i	$-0.679345\pi$
$7$	−19.7829	−1.06817	−0.534087	+	0.845429i	$0.679345\pi$
$8$	16.4554	0.727231
$9$	9.00000	0.333333
$10$	16.7969	0.531164
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	9.14908	0.220093
$13$	45.4209	0.969038	0.484519	−	0.874781i	$-0.338995\pi$
$13$	45.4209	0.969038	0.484519	+	0.874781i	$0.338995\pi$
$14$	65.7604	1.25537
$15$	−15.1591	−0.260938
$16$	−79.0969	−1.23589
$17$	−100.318	−1.43121	−0.715605	−	0.698505i	$-0.753849\pi$
$17$	−100.318	−1.43121	−0.715605	+	0.698505i	$0.753849\pi$
$18$	−29.9170	−0.391750
$19$	−118.486	−1.43066	−0.715328	−	0.698789i	$-0.753723\pi$
$19$	−118.486	−1.43066	−0.715328	+	0.698789i	$0.753723\pi$
$20$	−15.4102	−0.172292
$21$	−59.3486	−0.616711
$22$	36.5652	0.354351
$23$	−144.840	−1.31310	−0.656548	−	0.754284i	$-0.727984\pi$
$23$	−144.840	−1.31310	−0.656548	+	0.754284i	$0.727984\pi$
$24$	49.3661	0.419867
$25$	−99.4667	−0.795734
$26$	−150.984	−1.13886
$27$	27.0000	0.192450
$28$	−60.3317	−0.407200
$29$	11.2722	0.0721790	0.0360895	−	0.999349i	$-0.488510\pi$
$29$	11.2722	0.0721790	0.0360895	+	0.999349i	$0.488510\pi$
$30$	50.3906	0.306668
$31$	−214.454	−1.24248	−0.621242	−	0.783619i	$-0.713372\pi$
$31$	−214.454	−1.24248	−0.621242	+	0.783619i	$0.713372\pi$
$32$	131.284	0.725247
$33$	−33.0000	−0.174078
$34$	333.466	1.68203
$35$	99.9638	0.482771
$36$	27.4472	0.127070
$37$	354.551	1.57535	0.787674	−	0.616092i	$-0.211285\pi$
$37$	354.551	1.57535	0.787674	+	0.616092i	$0.211285\pi$
$38$	393.859	1.68138
$39$	136.263	0.559474
$40$	−83.1497	−0.328678
$41$	464.062	1.76767	0.883833	−	0.467802i	$-0.154954\pi$
$41$	464.062	1.76767	0.883833	+	0.467802i	$0.154954\pi$
$42$	197.281	0.724789
$43$	132.501	0.469913	0.234956	−	0.972006i	$-0.424505\pi$
$43$	132.501	0.469913	0.234956	+	0.972006i	$0.424505\pi$
$44$	−33.5466	−0.114940
$45$	−45.4774	−0.150653
$46$	481.463	1.54322
$47$	−220.759	−0.685128	−0.342564	−	0.939495i	$-0.611295\pi$
$47$	−220.759	−0.685128	−0.342564	+	0.939495i	$0.611295\pi$
$48$	−237.291	−0.713541
$49$	48.3619	0.140997
$50$	330.638	0.935185
$51$	−300.953	−0.826310
$52$	138.520	0.369408
$53$	−568.188	−1.47258	−0.736289	−	0.676668i	$-0.763423\pi$
$53$	−568.188	−1.47258	−0.736289	+	0.676668i	$0.763423\pi$
$54$	−89.7509	−0.226177
$55$	55.5835	0.136271
$56$	−325.534	−0.776810
$57$	−355.457	−0.825989
$58$	−37.4699	−0.0848283
$59$	−88.2459	−0.194723	−0.0973613	−	0.995249i	$-0.531040\pi$
$59$	−88.2459	−0.194723	−0.0973613	+	0.995249i	$0.531040\pi$
$60$	−46.2307	−0.0994727
$61$	61.0000	0.128037
$61$	61.0000	0.128037
$62$	712.867	1.46023
$63$	−178.046	−0.356058
$64$	196.374	0.383543
$65$	−229.514	−0.437965
$66$	109.696	0.204585
$67$	573.794	1.04627	0.523135	−	0.852250i	$-0.324762\pi$
$67$	573.794	1.04627	0.523135	+	0.852250i	$0.324762\pi$
$68$	−305.938	−0.545594
$69$	−434.520	−0.758116
$70$	−332.290	−0.567376
$71$	122.152	0.204179	0.102090	−	0.994775i	$-0.467447\pi$
$71$	122.152	0.204179	0.102090	+	0.994775i	$0.467447\pi$
$72$	148.098	0.242410
$73$	−577.597	−0.926063	−0.463032	−	0.886342i	$-0.653238\pi$
$73$	−577.597	−0.926063	−0.463032	+	0.886342i	$0.653238\pi$
$74$	−1178.57	−1.85143
$75$	−298.400	−0.459417
$76$	−361.345	−0.545382
$77$	217.612	0.322067
$78$	−452.952	−0.657522
$79$	−897.463	−1.27813	−0.639066	−	0.769152i	$-0.720679\pi$
$79$	−897.463	−1.27813	−0.639066	+	0.769152i	$0.720679\pi$
$80$	399.681	0.558571
$81$	81.0000	0.111111
$82$	−1542.59	−2.07745
$83$	−913.741	−1.20839	−0.604193	−	0.796838i	$-0.706504\pi$
$83$	−913.741	−1.20839	−0.604193	+	0.796838i	$0.706504\pi$
$84$	−180.995	−0.235097
$85$	506.909	0.646848
$86$	−440.448	−0.552265
$87$	33.8165	0.0416726
$88$	−181.009	−0.219268
$89$	277.078	0.330002	0.165001	−	0.986293i	$-0.447237\pi$
$89$	277.078	0.330002	0.165001	+	0.986293i	$0.447237\pi$
$90$	151.172	0.177055
$91$	−898.556	−1.03510
$92$	−441.717	−0.500567
$93$	−643.361	−0.717349
$94$	733.827	0.805196
$95$	598.714	0.646597
$96$	393.851	0.418722
$97$	−1766.99	−1.84959	−0.924797	−	0.380462i	$-0.875765\pi$
$97$	−1766.99	−1.84959	−0.924797	+	0.380462i	$0.875765\pi$
$98$	−160.760	−0.165707
$99$	−99.0000	−0.100504
$100$	−303.343	−0.303343
$101$	−629.982	−0.620649	−0.310325	−	0.950631i	$-0.600438\pi$
$101$	−629.982	−0.620649	−0.310325	+	0.950631i	$0.600438\pi$
$102$	1000.40	0.971120
$103$	506.154	0.484202	0.242101	−	0.970251i	$-0.422163\pi$
$103$	506.154	0.484202	0.242101	+	0.970251i	$0.422163\pi$
$104$	747.417	0.704714
$105$	299.891	0.278728
$106$	1888.72	1.73065
$107$	1656.14	1.49631	0.748154	−	0.663525i	$-0.230940\pi$
$107$	1656.14	1.49631	0.748154	+	0.663525i	$0.230940\pi$
$108$	82.3417	0.0733642
$109$	−1936.57	−1.70174	−0.850870	−	0.525377i	$-0.823924\pi$
$109$	−1936.57	−1.70174	−0.850870	+	0.525377i	$0.823924\pi$
$110$	−184.766	−0.160152
$111$	1063.65	0.909528
$112$	1564.76	1.32015
$113$	−71.0467	−0.0591461	−0.0295731	−	0.999563i	$-0.509415\pi$
$113$	−71.0467	−0.0591461	−0.0295731	+	0.999563i	$0.509415\pi$
$114$	1181.58	0.970744
$115$	731.883	0.593465
$116$	34.3767	0.0275155
$117$	408.788	0.323013
$118$	293.339	0.228848
$119$	1984.57	1.52878
$120$	−249.449	−0.189762
$121$	121.000	0.0909091
$122$	−202.771	−0.150475
$123$	1392.19	1.02056
$124$	−654.018	−0.473649
$125$	1134.24	0.811597
$126$	591.843	0.418457
$127$	−126.536	−0.0884113	−0.0442056	−	0.999022i	$-0.514076\pi$
$127$	−126.536	−0.0884113	−0.0442056	+	0.999022i	$0.514076\pi$
$128$	−1703.04	−1.17601
$129$	397.504	0.271304
$130$	762.930	0.514718
$131$	−710.910	−0.474141	−0.237071	−	0.971492i	$-0.576187\pi$
$131$	−710.910	−0.474141	−0.237071	+	0.971492i	$0.576187\pi$
$132$	−100.640	−0.0663604
$133$	2343.99	1.52819
$134$	−1907.35	−1.22963
$135$	−136.432	−0.0869795
$136$	−1650.76	−1.04082
$137$	43.1885	0.0269332	0.0134666	−	0.999909i	$-0.495713\pi$
$137$	43.1885	0.0269332	0.0134666	+	0.999909i	$0.495713\pi$
$138$	1444.39	0.890976
$139$	543.397	0.331585	0.165793	−	0.986161i	$-0.446982\pi$
$139$	543.397	0.331585	0.165793	+	0.986161i	$0.446982\pi$
$140$	304.859	0.184038
$141$	−662.277	−0.395559
$142$	−406.045	−0.239961
$143$	−499.630	−0.292176
$144$	−711.872	−0.411963
$145$	−56.9589	−0.0326219
$146$	1919.99	1.08836
$147$	145.086	0.0814046
$148$	1081.27	0.600541
$149$	−2159.54	−1.18736	−0.593679	−	0.804702i	$-0.702325\pi$
$149$	−2159.54	−1.18736	−0.593679	+	0.804702i	$0.702325\pi$
$150$	991.914	0.539930
$151$	−298.113	−0.160663	−0.0803315	−	0.996768i	$-0.525598\pi$
$151$	−298.113	−0.160663	−0.0803315	+	0.996768i	$0.525598\pi$
$152$	−1949.72	−1.04042
$153$	−902.858	−0.477070
$154$	−723.364	−0.378509
$155$	1083.65	0.561551
$156$	415.559	0.213278
$157$	−1334.35	−0.678298	−0.339149	−	0.940733i	$-0.610139\pi$
$157$	−1334.35	−0.678298	−0.339149	+	0.940733i	$0.610139\pi$
$158$	2983.26	1.50212
$159$	−1704.56	−0.850193
$160$	−663.384	−0.327782
$161$	2865.35	1.40262
$162$	−269.253	−0.130583
$163$	1814.58	0.871955	0.435977	−	0.899958i	$-0.356403\pi$
$163$	1814.58	0.871955	0.435977	+	0.899958i	$0.356403\pi$
$164$	1415.25	0.673855
$165$	166.751	0.0786759
$166$	3037.37	1.42016
$167$	−261.808	−0.121313	−0.0606566	−	0.998159i	$-0.519319\pi$
$167$	−261.808	−0.121313	−0.0606566	+	0.998159i	$0.519319\pi$
$168$	−976.603	−0.448491
$169$	−133.940	−0.0609652
$170$	−1685.02	−0.760207
$171$	−1066.37	−0.476885
$172$	404.088	0.179136
$173$	−1290.38	−0.567084	−0.283542	−	0.958960i	$-0.591510\pi$
$173$	−1290.38	−0.567084	−0.283542	+	0.958960i	$0.591510\pi$
$174$	−112.410	−0.0489756
$175$	1967.74	0.849982
$176$	870.066	0.372635
$177$	−264.738	−0.112423
$178$	−921.037	−0.387835
$179$	−2114.07	−0.882753	−0.441377	−	0.897322i	$-0.645510\pi$
$179$	−2114.07	−0.882753	−0.441377	+	0.897322i	$0.645510\pi$
$180$	−138.692	−0.0574306
$181$	3646.87	1.49762	0.748811	−	0.662783i	$-0.230625\pi$
$181$	3646.87	1.49762	0.748811	+	0.662783i	$0.230625\pi$
$182$	2986.90	1.21650
$183$	183.000	0.0739221
$184$	−2383.39	−0.954924
$185$	−1791.57	−0.711992
$186$	2138.60	0.843064
$187$	1103.49	0.431526
$188$	−673.247	−0.261179
$189$	−534.137	−0.205570
$190$	−1990.19	−0.759913
$191$	4991.14	1.89082	0.945409	−	0.325886i	$-0.105663\pi$
$191$	4991.14	1.89082	0.945409	+	0.325886i	$0.105663\pi$
$192$	589.122	0.221438
$193$	−4153.05	−1.54893	−0.774463	−	0.632619i	$-0.781980\pi$
$193$	−4153.05	−1.54893	−0.774463	+	0.632619i	$0.781980\pi$
$194$	5873.66	2.17373
$195$	−688.542	−0.252859
$196$	147.489	0.0537496
$197$	5160.39	1.86631	0.933154	−	0.359478i	$-0.117045\pi$
$197$	5160.39	1.86631	0.933154	+	0.359478i	$0.117045\pi$
$198$	329.087	0.118117
$199$	4313.37	1.53651	0.768257	−	0.640141i	$-0.221124\pi$
$199$	4313.37	1.53651	0.768257	+	0.640141i	$0.221124\pi$
$200$	−1636.76	−0.578682
$201$	1721.38	0.604064
$202$	2094.13	0.729418
$203$	−222.996	−0.0770997
$204$	−917.813	−0.314999
$205$	−2344.93	−0.798912
$206$	−1682.51	−0.569058
$207$	−1303.56	−0.437699
$208$	−3592.65	−1.19762
$209$	1303.34	0.431359
$210$	−996.871	−0.327575
$211$	2531.23	0.825864	0.412932	−	0.910762i	$-0.364505\pi$
$211$	2531.23	0.825864	0.412932	+	0.910762i	$0.364505\pi$
$212$	−1732.80	−0.561363
$213$	366.455	0.117883
$214$	−5505.18	−1.75854
$215$	−669.535	−0.212381
$216$	444.295	0.139956
$217$	4242.51	1.32719
$218$	6437.36	1.99997
$219$	−1732.79	−0.534663
$220$	169.513	0.0519479
$221$	−4556.51	−1.38690
$222$	−3535.70	−1.06892
$223$	−2986.55	−0.896836	−0.448418	−	0.893824i	$-0.648012\pi$
$223$	−2986.55	−0.896836	−0.448418	+	0.893824i	$0.648012\pi$
$224$	−2597.17	−0.774691
$225$	−895.200	−0.265245
$226$	236.167	0.0695114
$227$	4031.36	1.17873	0.589363	−	0.807869i	$-0.299379\pi$
$227$	4031.36	1.17873	0.589363	+	0.807869i	$0.299379\pi$
$228$	−1084.03	−0.314877
$229$	−1122.34	−0.323869	−0.161935	−	0.986801i	$-0.551773\pi$
$229$	−1122.34	−0.323869	−0.161935	+	0.986801i	$0.551773\pi$
$230$	−2432.86	−0.697469
$231$	652.835	0.185945
$232$	185.488	0.0524908
$233$	−2293.87	−0.644962	−0.322481	−	0.946576i	$-0.604517\pi$
$233$	−2293.87	−0.644962	−0.322481	+	0.946576i	$0.604517\pi$
$234$	−1358.86	−0.379621
$235$	1115.51	0.309649
$236$	−269.123	−0.0742305
$237$	−2692.39	−0.737930
$238$	−6596.92	−1.79670
$239$	4539.84	1.22869	0.614347	−	0.789036i	$-0.289420\pi$
$239$	4539.84	1.22869	0.614347	+	0.789036i	$0.289420\pi$
$240$	1199.04	0.322491
$241$	−3408.63	−0.911075	−0.455538	−	0.890217i	$-0.650553\pi$
$241$	−3408.63	−0.911075	−0.455538	+	0.890217i	$0.650553\pi$
$242$	−402.217	−0.106841
$243$	243.000	0.0641500
$244$	186.031	0.0488091
$245$	−244.375	−0.0637247
$246$	−4627.78	−1.19942
$247$	−5381.72	−1.38636
$248$	−3528.91	−0.903573
$249$	−2741.22	−0.697662
$250$	−3770.34	−0.953829
$251$	3267.55	0.821698	0.410849	−	0.911703i	$-0.365232\pi$
$251$	3267.55	0.821698	0.410849	+	0.911703i	$0.365232\pi$
$252$	−542.985	−0.135733
$253$	1593.24	0.395913
$254$	420.618	0.103905
$255$	1520.73	0.373458
$256$	4090.09	0.998557
$257$	2504.29	0.607835	0.303917	−	0.952698i	$-0.401705\pi$
$257$	2504.29	0.607835	0.303917	+	0.952698i	$0.401705\pi$
$258$	−1321.35	−0.318850
$259$	−7014.05	−1.68275
$260$	−699.947	−0.166957
$261$	101.450	0.0240597
$262$	2363.14	0.557234
$263$	6085.77	1.42686	0.713431	−	0.700726i	$-0.247140\pi$
$263$	6085.77	1.42686	0.713431	+	0.700726i	$0.247140\pi$
$264$	−543.027	−0.126595
$265$	2871.08	0.665544
$266$	−7791.66	−1.79600
$267$	831.234	0.190527
$268$	1749.89	0.398850
$269$	3082.64	0.698706	0.349353	−	0.936991i	$-0.386401\pi$
$269$	3082.64	0.698706	0.349353	+	0.936991i	$0.386401\pi$
$270$	453.516	0.102223
$271$	266.161	0.0596610	0.0298305	−	0.999555i	$-0.490503\pi$
$271$	266.161	0.0596610	0.0298305	+	0.999555i	$0.490503\pi$
$272$	7934.81	1.76882
$273$	−2695.67	−0.597616
$274$	−143.563	−0.0316532
$275$	1094.13	0.239923
$276$	−1325.15	−0.289003
$277$	−8708.61	−1.88899	−0.944494	−	0.328529i	$-0.893447\pi$
$277$	−8708.61	−1.88899	−0.944494	+	0.328529i	$0.893447\pi$
$278$	−1806.31	−0.389695
$279$	−1930.08	−0.414162
$280$	1644.94	0.351086
$281$	1604.71	0.340673	0.170337	−	0.985386i	$-0.445514\pi$
$281$	1604.71	0.340673	0.170337	+	0.985386i	$0.445514\pi$
$282$	2201.48	0.464880
$283$	615.625	0.129311	0.0646557	−	0.997908i	$-0.479405\pi$
$283$	615.625	0.129311	0.0646557	+	0.997908i	$0.479405\pi$
$284$	372.524	0.0778354
$285$	1796.14	0.373313
$286$	1660.82	0.343380
$287$	−9180.48	−1.88818
$288$	1181.55	0.241749
$289$	5150.61	1.04836
$290$	189.337	0.0383389
$291$	−5300.97	−1.06786
$292$	−1761.49	−0.353026
$293$	9574.24	1.90899	0.954493	−	0.298234i	$-0.0963975\pi$
$293$	9574.24	1.90899	0.954493	+	0.298234i	$0.0963975\pi$
$294$	−482.281	−0.0956707
$295$	445.911	0.0880065
$296$	5834.27	1.14564
$297$	−297.000	−0.0580259
$298$	7178.54	1.39544
$299$	−6578.76	−1.27244
$300$	−910.028	−0.175135
$301$	−2621.25	−0.501949
$302$	990.960	0.188819
$303$	−1889.95	−0.358332
$304$	9371.84	1.76813
$305$	−308.236	−0.0578674
$306$	3001.20	0.560676
$307$	−4921.40	−0.914916	−0.457458	−	0.889231i	$-0.651240\pi$
$307$	−4921.40	−0.914916	−0.457458	+	0.889231i	$0.651240\pi$
$308$	663.648	0.122776
$309$	1518.46	0.279554
$310$	−3602.15	−0.659963
$311$	3624.16	0.660796	0.330398	−	0.943842i	$-0.392817\pi$
$311$	3624.16	0.660796	0.330398	+	0.943842i	$0.392817\pi$
$312$	2242.25	0.406867
$313$	−1262.14	−0.227924	−0.113962	−	0.993485i	$-0.536354\pi$
$313$	−1262.14	−0.227924	−0.113962	+	0.993485i	$0.536354\pi$
$314$	4435.52	0.797169
$315$	899.674	0.160924
$316$	−2736.99	−0.487239
$317$	−4168.66	−0.738597	−0.369298	−	0.929311i	$-0.620402\pi$
$317$	−4168.66	−0.738597	−0.369298	+	0.929311i	$0.620402\pi$
$318$	5666.15	0.999189
$319$	−123.994	−0.0217628
$320$	−992.287	−0.173345
$321$	4968.42	0.863894
$322$	−9524.73	−1.64842
$323$	11886.2	2.04757
$324$	247.025	0.0423568
$325$	−4517.87	−0.771096
$326$	−6031.85	−1.02476
$327$	−5809.70	−0.982500
$328$	7636.31	1.28550
$329$	4367.25	0.731836
$330$	−554.297	−0.0924638
$331$	2734.54	0.454091	0.227045	−	0.973884i	$-0.427093\pi$
$331$	2734.54	0.454091	0.227045	+	0.973884i	$0.427093\pi$
$332$	−2786.63	−0.460651
$333$	3190.96	0.525116
$334$	870.278	0.142573
$335$	−2899.41	−0.472870
$336$	4694.29	0.762186
$337$	−10588.8	−1.71160	−0.855801	−	0.517305i	$-0.826935\pi$
$337$	−10588.8	−1.71160	−0.855801	+	0.517305i	$0.826935\pi$
$338$	445.232	0.0716493
$339$	−213.140	−0.0341480
$340$	1545.92	0.246586
$341$	2358.99	0.374623
$342$	3544.73	0.560459
$343$	5828.79	0.917565
$344$	2180.36	0.341735
$345$	2195.65	0.342637
$346$	4289.35	0.666465
$347$	1369.21	0.211825	0.105912	−	0.994375i	$-0.466224\pi$
$347$	1369.21	0.211825	0.105912	+	0.994375i	$0.466224\pi$
$348$	103.130	0.0158861
$349$	−7794.72	−1.19553	−0.597767	−	0.801670i	$-0.703945\pi$
$349$	−7794.72	−1.19553	−0.597767	+	0.801670i	$0.703945\pi$
$350$	−6540.97	−0.998941
$351$	1226.36	0.186491
$352$	−1444.12	−0.218670
$353$	5865.97	0.884459	0.442230	−	0.896902i	$-0.354188\pi$
$353$	5865.97	0.884459	0.442230	+	0.896902i	$0.354188\pi$
$354$	880.016	0.132125
$355$	−617.238	−0.0922805
$356$	845.002	0.125801
$357$	5953.71	0.882643
$358$	7027.39	1.03746
$359$	−283.337	−0.0416545	−0.0208273	−	0.999783i	$-0.506630\pi$
$359$	−283.337	−0.0416545	−0.0208273	+	0.999783i	$0.506630\pi$
$360$	−748.348	−0.109559
$361$	7179.84	1.04678
$362$	−12122.6	−1.76008
$363$	363.000	0.0524864
$364$	−2740.32	−0.394593
$365$	2918.63	0.418542
$366$	−608.312	−0.0868769
$367$	12571.1	1.78803	0.894013	−	0.448041i	$-0.147878\pi$
$367$	12571.1	1.78803	0.894013	+	0.448041i	$0.147878\pi$
$368$	11456.4	1.62284
$369$	4176.56	0.589222
$370$	5955.36	0.836769
$371$	11240.4	1.57297
$372$	−1962.05	−0.273462
$373$	6955.48	0.965525	0.482763	−	0.875751i	$-0.339633\pi$
$373$	6955.48	0.965525	0.482763	+	0.875751i	$0.339633\pi$
$374$	−3668.13	−0.507151
$375$	3402.72	0.468576
$376$	−3632.67	−0.498246
$377$	511.993	0.0699442
$378$	1775.53	0.241596
$379$	5246.28	0.711038	0.355519	−	0.934669i	$-0.384304\pi$
$379$	5246.28	0.711038	0.355519	+	0.934669i	$0.384304\pi$
$380$	1825.89	0.246490
$381$	−379.607	−0.0510443
$382$	−16591.1	−2.22218
$383$	−2138.18	−0.285263	−0.142631	−	0.989776i	$-0.545556\pi$
$383$	−2138.18	−0.285263	−0.142631	+	0.989776i	$0.545556\pi$
$384$	−5109.11	−0.678967
$385$	−1099.60	−0.145561
$386$	13805.2	1.82038
$387$	1192.51	0.156638
$388$	−5388.77	−0.705086
$389$	10318.5	1.34490	0.672452	−	0.740141i	$-0.265241\pi$
$389$	10318.5	1.34490	0.672452	+	0.740141i	$0.265241\pi$
$390$	2288.79	0.297173
$391$	14530.0	1.87932
$392$	795.813	0.102537
$393$	−2132.73	−0.273745
$394$	−17153.7	−2.19338
$395$	4534.92	0.577663
$396$	−301.919	−0.0383132
$397$	−13581.2	−1.71693	−0.858465	−	0.512872i	$-0.828582\pi$
$397$	−13581.2	−1.71693	−0.858465	+	0.512872i	$0.828582\pi$
$398$	−14338.1	−1.80579
$399$	7031.96	0.882301
$400$	7867.51	0.983439
$401$	−6869.10	−0.855428	−0.427714	−	0.903914i	$-0.640681\pi$
$401$	−6869.10	−0.855428	−0.427714	+	0.903914i	$0.640681\pi$
$402$	−5722.06	−0.709926
$403$	−9740.68	−1.20401
$404$	−1921.25	−0.236599
$405$	−409.297	−0.0502176
$406$	741.263	0.0906114
$407$	−3900.07	−0.474986
$408$	−4952.28	−0.600918
$409$	7108.10	0.859347	0.429674	−	0.902984i	$-0.358629\pi$
$409$	7108.10	0.859347	0.429674	+	0.902984i	$0.358629\pi$
$410$	7794.79	0.938921
$411$	129.566	0.0155499
$412$	1543.61	0.184583
$413$	1745.76	0.207998
$414$	4333.17	0.514405
$415$	4617.18	0.546141
$416$	5963.03	0.702792
$417$	1630.19	0.191441
$418$	−4332.45	−0.506954
$419$	10424.6	1.21546	0.607729	−	0.794144i	$-0.292081\pi$
$419$	10424.6	1.21546	0.607729	+	0.794144i	$0.292081\pi$
$420$	914.576	0.106254
$421$	−14035.7	−1.62484	−0.812419	−	0.583074i	$-0.801850\pi$
$421$	−14035.7	−1.62484	−0.812419	+	0.583074i	$0.801850\pi$
$422$	−8414.09	−0.970596
$423$	−1986.83	−0.228376
$424$	−9349.73	−1.07090
$425$	9978.25	1.13886
$426$	−1218.13	−0.138542
$427$	−1206.76	−0.136766
$428$	5050.71	0.570410
$429$	−1498.89	−0.168688
$430$	2225.61	0.249601
$431$	11121.8	1.24297	0.621485	−	0.783426i	$-0.286529\pi$
$431$	11121.8	1.24297	0.621485	+	0.783426i	$0.286529\pi$
$432$	−2135.62	−0.237847
$433$	−4427.29	−0.491367	−0.245683	−	0.969350i	$-0.579012\pi$
$433$	−4427.29	−0.491367	−0.245683	+	0.969350i	$0.579012\pi$
$434$	−14102.6	−1.55978
$435$	−170.877	−0.0188343
$436$	−5905.94	−0.648722
$437$	17161.4	1.87859
$438$	5759.98	0.628362
$439$	−789.378	−0.0858199	−0.0429100	−	0.999079i	$-0.513663\pi$
$439$	−789.378	−0.0858199	−0.0429100	+	0.999079i	$0.513663\pi$
$440$	914.647	0.0991002
$441$	435.257	0.0469990
$442$	15146.3	1.62995
$443$	11635.8	1.24793	0.623965	−	0.781452i	$-0.285521\pi$
$443$	11635.8	1.24793	0.623965	+	0.781452i	$0.285521\pi$
$444$	3243.82	0.346723
$445$	−1400.09	−0.149147
$446$	9927.63	1.05401
$447$	−6478.62	−0.685522
$448$	−3884.84	−0.409691
$449$	−12675.4	−1.33227	−0.666136	−	0.745830i	$-0.732053\pi$
$449$	−12675.4	−1.33227	−0.666136	+	0.745830i	$0.732053\pi$
$450$	2975.74	0.311728
$451$	−5104.68	−0.532971
$452$	−216.671	−0.0225472
$453$	−894.340	−0.0927588
$454$	−13400.7	−1.38530
$455$	4540.45	0.467823
$456$	−5849.17	−0.600685
$457$	19486.2	1.99458	0.997291	−	0.0735527i	$-0.0234337\pi$
$457$	19486.2	1.99458	0.997291	+	0.0735527i	$0.0234337\pi$
$458$	3730.77	0.380627
$459$	−2708.57	−0.275437
$460$	2232.02	0.226236
$461$	−4164.73	−0.420761	−0.210380	−	0.977620i	$-0.567470\pi$
$461$	−4164.73	−0.420761	−0.210380	+	0.977620i	$0.567470\pi$
$462$	−2170.09	−0.218532
$463$	4045.72	0.406092	0.203046	−	0.979169i	$-0.434916\pi$
$463$	4045.72	0.406092	0.203046	+	0.979169i	$0.434916\pi$
$464$	−891.594	−0.0892052
$465$	3250.94	0.324212
$466$	7625.06	0.757991
$467$	−5230.53	−0.518287	−0.259143	−	0.965839i	$-0.583440\pi$
$467$	−5230.53	−0.518287	−0.259143	+	0.965839i	$0.583440\pi$
$468$	1246.68	0.123136
$469$	−11351.3	−1.11760
$470$	−3708.06	−0.363915
$471$	−4003.05	−0.391615
$472$	−1452.12	−0.141608
$473$	−1457.51	−0.141684
$474$	8949.79	0.867252
$475$	11785.4	1.13842
$476$	6052.32	0.582789
$477$	−5113.69	−0.490859
$478$	−15090.9	−1.44402
$479$	−4684.28	−0.446827	−0.223414	−	0.974724i	$-0.571720\pi$
$479$	−4684.28	−0.446827	−0.223414	+	0.974724i	$0.571720\pi$
$480$	−1990.15	−0.189245
$481$	16104.1	1.52657
$482$	11330.7	1.07074
$483$	8596.05	0.809801
$484$	369.013	0.0346556
$485$	8928.68	0.835939
$486$	−807.758	−0.0753923
$487$	10104.2	0.940177	0.470088	−	0.882619i	$-0.344222\pi$
$487$	10104.2	0.940177	0.470088	+	0.882619i	$0.344222\pi$
$488$	1003.78	0.0931124
$489$	5443.73	0.503423
$490$	812.330	0.0748925
$491$	11569.3	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$491$	11569.3	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$492$	4245.74	0.389050
$493$	−1130.80	−0.103303
$494$	17889.4	1.62932
$495$	500.252	0.0454235
$496$	16962.6	1.53557
$497$	−2416.51	−0.218099
$498$	9112.12	0.819927
$499$	−13929.9	−1.24968	−0.624838	−	0.780755i	$-0.714835\pi$
$499$	−13929.9	−1.24968	−0.624838	+	0.780755i	$0.714835\pi$
$500$	3459.09	0.309390
$501$	−785.424	−0.0700403
$502$	−10861.7	−0.965700
$503$	−7477.70	−0.662851	−0.331426	−	0.943481i	$-0.607530\pi$
$503$	−7477.70	−0.662851	−0.331426	+	0.943481i	$0.607530\pi$
$504$	−2929.81	−0.258937
$505$	3183.33	0.280508
$506$	−5296.10	−0.465297
$507$	−401.821	−0.0351982
$508$	−385.895	−0.0337034
$509$	19790.6	1.72338	0.861692	−	0.507432i	$-0.169405\pi$
$509$	19790.6	1.72338	0.861692	+	0.507432i	$0.169405\pi$
$510$	−5055.07	−0.438906
$511$	11426.5	0.989197
$512$	28.4011	0.00245149
$513$	−3199.11	−0.275330
$514$	−8324.54	−0.714358
$515$	−2557.62	−0.218839
$516$	1212.26	0.103424
$517$	2428.35	0.206574
$518$	23315.4	1.97765
$519$	−3871.13	−0.327406
$520$	−3776.74	−0.318502
$521$	−2965.02	−0.249328	−0.124664	−	0.992199i	$-0.539785\pi$
$521$	−2965.02	−0.249328	−0.124664	+	0.992199i	$0.539785\pi$
$522$	−337.229	−0.0282761
$523$	5477.62	0.457972	0.228986	−	0.973430i	$-0.426459\pi$
$523$	5477.62	0.457972	0.228986	+	0.973430i	$0.426459\pi$
$524$	−2168.06	−0.180748
$525$	5903.21	0.490738
$526$	−20229.8	−1.67692
$527$	21513.5	1.77826
$528$	2610.20	0.215141
$529$	8811.60	0.724222
$530$	−9543.78	−0.782180
$531$	−794.213	−0.0649075
$532$	7148.43	0.582564
$533$	21078.1	1.71294
$534$	−2763.11	−0.223917
$535$	−8368.55	−0.676269
$536$	9441.98	0.760880
$537$	−6342.20	−0.509658
$538$	−10247.0	−0.821154
$539$	−531.981	−0.0425122
$540$	−416.077	−0.0331576
$541$	−14414.8	−1.14555	−0.572775	−	0.819713i	$-0.694133\pi$
$541$	−14414.8	−1.14555	−0.572775	+	0.819713i	$0.694133\pi$
$542$	−884.748	−0.0701166
$543$	10940.6	0.864653
$544$	−13170.1	−1.03798
$545$	9785.57	0.769115
$546$	8960.69	0.702348
$547$	6894.79	0.538939	0.269470	−	0.963009i	$-0.413152\pi$
$547$	6894.79	0.538939	0.269470	+	0.963009i	$0.413152\pi$
$548$	131.712	0.0102672
$549$	549.000	0.0426790
$550$	−3637.02	−0.281969
$551$	−1335.59	−0.103263
$552$	−7150.18	−0.551326
$553$	17754.4	1.36527
$554$	28948.4	2.22003
$555$	−5374.70	−0.411069
$556$	1657.19	0.126404
$557$	−10742.4	−0.817181	−0.408590	−	0.912718i	$-0.633980\pi$
$557$	−10742.4	−0.817181	−0.408590	+	0.912718i	$0.633980\pi$
$558$	6415.80	0.486743
$559$	6018.33	0.455363
$560$	−7906.83	−0.596651
$561$	3310.48	0.249142
$562$	−5334.24	−0.400376
$563$	−3345.15	−0.250411	−0.125205	−	0.992131i	$-0.539959\pi$
$563$	−3345.15	−0.250411	−0.125205	+	0.992131i	$0.539959\pi$
$564$	−2019.74	−0.150791
$565$	359.002	0.0267316
$566$	−2046.40	−0.151973
$567$	−1602.41	−0.118686
$568$	2010.05	0.148485
$569$	−15002.0	−1.10530	−0.552650	−	0.833414i	$-0.686383\pi$
$569$	−15002.0	−1.10530	−0.552650	+	0.833414i	$0.686383\pi$
$570$	−5970.57	−0.438736
$571$	−22557.0	−1.65321	−0.826605	−	0.562782i	$-0.809731\pi$
$571$	−22557.0	−1.65321	−0.826605	+	0.562782i	$0.809731\pi$
$572$	−1523.72	−0.111381
$573$	14973.4	1.09166
$574$	30516.9	2.21908
$575$	14406.7	1.04487
$576$	1767.36	0.127848
$577$	174.745	0.0126079	0.00630394	−	0.999980i	$-0.497993\pi$
$577$	174.745	0.0126079	0.00630394	+	0.999980i	$0.497993\pi$
$578$	−17121.2	−1.23209
$579$	−12459.1	−0.894273
$580$	−173.707	−0.0124358
$581$	18076.4	1.29077
$582$	17621.0	1.25501
$583$	6250.06	0.443999
$584$	−9504.57	−0.673462
$585$	−2065.63	−0.145988
$586$	−31825.8	−2.24353
$587$	8048.82	0.565946	0.282973	−	0.959128i	$-0.408679\pi$
$587$	8048.82	0.565946	0.282973	+	0.959128i	$0.408679\pi$
$588$	442.467	0.0310324
$589$	25409.7	1.77757
$590$	−1482.26	−0.103430
$591$	15481.2	1.07751
$592$	−28043.9	−1.94696
$593$	19208.4	1.33018	0.665089	−	0.746764i	$-0.268394\pi$
$593$	19208.4	1.33018	0.665089	+	0.746764i	$0.268394\pi$
$594$	987.260	0.0681949
$595$	−10028.1	−0.690946
$596$	−6585.93	−0.452635
$597$	12940.1	0.887107
$598$	21868.5	1.49543
$599$	−8554.65	−0.583528	−0.291764	−	0.956490i	$-0.594242\pi$
$599$	−8554.65	−0.583528	−0.291764	+	0.956490i	$0.594242\pi$
$600$	−4910.28	−0.334102
$601$	5798.38	0.393545	0.196773	−	0.980449i	$-0.436954\pi$
$601$	5798.38	0.393545	0.196773	+	0.980449i	$0.436954\pi$
$602$	8713.33	0.589915
$603$	5164.14	0.348756
$604$	−909.153	−0.0612466
$605$	−611.419	−0.0410871
$606$	6282.39	0.421130
$607$	9838.53	0.657881	0.328941	−	0.944351i	$-0.393308\pi$
$607$	9838.53	0.657881	0.328941	+	0.944351i	$0.393308\pi$
$608$	−15555.2	−1.03758
$609$	−668.988	−0.0445136
$610$	1024.61	0.0680086
$611$	−10027.1	−0.663915
$612$	−2753.44	−0.181865
$613$	−22032.6	−1.45170	−0.725848	−	0.687855i	$-0.758553\pi$
$613$	−22032.6	−1.45170	−0.725848	+	0.687855i	$0.758553\pi$
$614$	16359.3	1.07525
$615$	−7034.79	−0.461252
$616$	3580.88	0.234217
$617$	−11876.8	−0.774944	−0.387472	−	0.921881i	$-0.626652\pi$
$617$	−11876.8	−0.774944	−0.387472	+	0.921881i	$0.626652\pi$
$618$	−5047.53	−0.328546
$619$	27997.8	1.81798	0.908988	−	0.416822i	$-0.136856\pi$
$619$	27997.8	1.81798	0.908988	+	0.416822i	$0.136856\pi$
$620$	3304.78	0.214070
$621$	−3910.68	−0.252705
$622$	−12047.1	−0.776600
$623$	−5481.40	−0.352500
$624$	−10778.0	−0.691448
$625$	6701.96	0.428925
$626$	4195.48	0.267868
$627$	3910.02	0.249045
$628$	−4069.36	−0.258575
$629$	−35567.7	−2.25466
$630$	−2990.61	−0.189125
$631$	−9589.80	−0.605014	−0.302507	−	0.953147i	$-0.597824\pi$
$631$	−9589.80	−0.605014	−0.302507	+	0.953147i	$0.597824\pi$
$632$	−14768.1	−0.929498
$633$	7593.70	0.476813
$634$	13857.1	0.868036
$635$	639.391	0.0399582
$636$	−5198.39	−0.324103
$637$	2196.64	0.136631
$638$	412.169	0.0255767
$639$	1099.36	0.0680597
$640$	8605.54	0.531506
$641$	26352.6	1.62382	0.811909	−	0.583784i	$-0.198429\pi$
$641$	26352.6	1.62382	0.811909	+	0.583784i	$0.198429\pi$
$642$	−16515.6	−1.01529
$643$	10967.3	0.672641	0.336321	−	0.941748i	$-0.390817\pi$
$643$	10967.3	0.672641	0.336321	+	0.941748i	$0.390817\pi$
$644$	8738.43	0.534693
$645$	−2008.61	−0.122618
$646$	−39511.0	−2.40640
$647$	−8452.64	−0.513613	−0.256806	−	0.966463i	$-0.582670\pi$
$647$	−8452.64	−0.513613	−0.256806	+	0.966463i	$0.582670\pi$
$648$	1332.88	0.0808034
$649$	970.705	0.0587111
$650$	15017.9	0.906230
$651$	12727.5	0.766254
$652$	5533.90	0.332399
$653$	7193.50	0.431092	0.215546	−	0.976494i	$-0.430847\pi$
$653$	7193.50	0.431092	0.215546	+	0.976494i	$0.430847\pi$
$654$	19312.1	1.15468
$655$	3592.26	0.214292
$656$	−36705.9	−2.18464
$657$	−5198.37	−0.308688
$658$	−14517.2	−0.860090
$659$	−5945.39	−0.351441	−0.175720	−	0.984440i	$-0.556225\pi$
$659$	−5945.39	−0.351441	−0.175720	+	0.984440i	$0.556225\pi$
$660$	508.538	0.0299921
$661$	30.3464	0.00178568	0.000892842	−	1.00000i	$-0.499716\pi$
$661$	30.3464	0.00178568	0.000892842		1.00000i	$0.499716\pi$
$662$	−9089.92	−0.533670
$663$	−13669.5	−0.800726
$664$	−15035.9	−0.878776
$665$	−11844.3	−0.690678
$666$	−10607.1	−0.617143
$667$	−1632.66	−0.0947779
$668$	−798.434	−0.0462460
$669$	−8959.66	−0.517789
$670$	9637.94	0.555741
$671$	−671.000	−0.0386046
$672$	−7791.51	−0.447268
$673$	−15880.4	−0.909579	−0.454789	−	0.890599i	$-0.650285\pi$
$673$	−15880.4	−0.909579	−0.454789	+	0.890599i	$0.650285\pi$
$674$	35198.4	2.01156
$675$	−2685.60	−0.153139
$676$	−408.477	−0.0232406
$677$	−24012.2	−1.36317	−0.681583	−	0.731741i	$-0.738708\pi$
$677$	−24012.2	−1.36317	−0.681583	+	0.731741i	$0.738708\pi$
$678$	708.501	0.0401325
$679$	34956.1	1.97569
$680$	8341.38	0.470408
$681$	12094.1	0.680537
$682$	−7841.54	−0.440276
$683$	1302.87	0.0729911	0.0364955	−	0.999334i	$-0.488381\pi$
$683$	1302.87	0.0729911	0.0364955	+	0.999334i	$0.488381\pi$
$684$	−3252.10	−0.181794
$685$	−218.234	−0.0121727
$686$	−19375.5	−1.07837
$687$	−3367.01	−0.186986
$688$	−10480.4	−0.580760
$689$	−25807.6	−1.42698
$690$	−7298.58	−0.402684
$691$	20041.8	1.10337	0.551684	−	0.834054i	$-0.313986\pi$
$691$	20041.8	1.10337	0.551684	+	0.834054i	$0.313986\pi$
$692$	−3935.25	−0.216179
$693$	1958.50	0.107356
$694$	−4551.41	−0.248947
$695$	−2745.81	−0.149863
$696$	556.463	0.0303056
$697$	−46553.6	−2.52990
$698$	25910.5	1.40505
$699$	−6881.60	−0.372369
$700$	6000.99	0.324023
$701$	4889.19	0.263427	0.131713	−	0.991288i	$-0.457952\pi$
$701$	4889.19	0.263427	0.131713	+	0.991288i	$0.457952\pi$
$702$	−4076.57	−0.219174
$703$	−42009.2	−2.25378
$704$	−2160.11	−0.115642
$705$	3346.52	0.178776
$706$	−19499.1	−1.03946
$707$	12462.9	0.662962
$708$	−807.368	−0.0428570
$709$	−23454.2	−1.24237	−0.621186	−	0.783663i	$-0.713349\pi$
$709$	−23454.2	−1.24237	−0.621186	+	0.783663i	$0.713349\pi$
$710$	2051.76	0.108453
$711$	−8077.17	−0.426044
$712$	4559.42	0.239988
$713$	31061.5	1.63150
$714$	−19790.8	−1.03733
$715$	2524.66	0.132051
$716$	−6447.25	−0.336516
$717$	13619.5	0.709386
$718$	941.844	0.0489544
$719$	−23310.0	−1.20906	−0.604531	−	0.796581i	$-0.706640\pi$
$719$	−23310.0	−1.20906	−0.604531	+	0.796581i	$0.706640\pi$
$720$	3597.13	0.186190
$721$	−10013.2	−0.517212
$722$	−23866.5	−1.23022
$723$	−10225.9	−0.526009
$724$	11121.8	0.570911
$725$	−1121.21	−0.0574352
$726$	−1206.65	−0.0616846
$727$	10702.7	0.545997	0.272998	−	0.962014i	$-0.411985\pi$
$727$	10702.7	0.545997	0.272998	+	0.962014i	$0.411985\pi$
$728$	−14786.1	−0.752758
$729$	729.000	0.0370370
$730$	−9701.83	−0.491891
$731$	−13292.2	−0.672544
$732$	558.094	0.0281800
$733$	−6274.70	−0.316182	−0.158091	−	0.987425i	$-0.550534\pi$
$733$	−6274.70	−0.316182	−0.158091	+	0.987425i	$0.550534\pi$
$734$	−41787.6	−2.10138
$735$	−733.126	−0.0367915
$736$	−19015.1	−0.952320
$737$	−6311.73	−0.315462
$738$	−13883.3	−0.692483
$739$	−5899.14	−0.293645	−0.146822	−	0.989163i	$-0.546905\pi$
$739$	−5899.14	−0.293645	−0.146822	+	0.989163i	$0.546905\pi$
$740$	−5463.72	−0.271420
$741$	−16145.2	−0.800415
$742$	−37364.2	−1.84863
$743$	32753.9	1.61726	0.808630	−	0.588318i	$-0.200210\pi$
$743$	32753.9	1.61726	0.808630	+	0.588318i	$0.200210\pi$
$744$	−10586.7	−0.521678
$745$	10912.3	0.536637
$746$	−23120.8	−1.13473
$747$	−8223.67	−0.402795
$748$	3365.31	0.164503
$749$	−32763.2	−1.59832
$750$	−11311.0	−0.550693
$751$	−729.611	−0.0354512	−0.0177256	−	0.999843i	$-0.505643\pi$
$751$	−729.611	−0.0354512	−0.0177256	+	0.999843i	$0.505643\pi$
$752$	17461.4	0.846742
$753$	9802.66	0.474408
$754$	−1701.92	−0.0822019
$755$	1506.38	0.0726130
$756$	−1628.95	−0.0783658
$757$	5723.76	0.274813	0.137407	−	0.990515i	$-0.456123\pi$
$757$	5723.76	0.274813	0.137407	+	0.990515i	$0.456123\pi$
$758$	−17439.2	−0.835647
$759$	4779.72	0.228581
$760$	9852.05	0.470225
$761$	31903.4	1.51971	0.759854	−	0.650093i	$-0.225270\pi$
$761$	31903.4	1.51971	0.759854	+	0.650093i	$0.225270\pi$
$762$	1261.86	0.0599898
$763$	38310.9	1.81775
$764$	15221.4	0.720802
$765$	4562.19	0.215616
$766$	7107.52	0.335255
$767$	−4008.21	−0.188694
$768$	12270.3	0.576517
$769$	28681.9	1.34499	0.672494	−	0.740103i	$-0.265223\pi$
$769$	28681.9	1.34499	0.672494	+	0.740103i	$0.265223\pi$
$770$	3655.20	0.171070
$771$	7512.88	0.350934
$772$	−12665.5	−0.590469
$773$	−39769.2	−1.85045	−0.925225	−	0.379420i	$-0.876124\pi$
$773$	−39769.2	−1.85045	−0.925225	+	0.379420i	$0.876124\pi$
$774$	−3964.04	−0.184088
$775$	21331.0	0.988687
$776$	−29076.4	−1.34508
$777$	−21042.1	−0.971535
$778$	−34299.7	−1.58060
$779$	−54984.7	−2.52892
$780$	−2099.84	−0.0963928
$781$	−1343.67	−0.0615623
$782$	−48299.2	−2.20867
$783$	304.349	0.0138909
$784$	−3825.28	−0.174257
$785$	6742.54	0.306562
$786$	7089.42	0.321719
$787$	−16563.3	−0.750215	−0.375107	−	0.926981i	$-0.622394\pi$
$787$	−16563.3	−0.750215	−0.375107	+	0.926981i	$0.622394\pi$
$788$	15737.6	0.711458
$789$	18257.3	0.823799
$790$	−15074.6	−0.678898
$791$	1405.51	0.0631784
$792$	−1629.08	−0.0730895
$793$	2770.68	0.124073
$794$	45145.4	2.01782
$795$	8613.24	0.384252
$796$	13154.4	0.585737
$797$	16416.3	0.729606	0.364803	−	0.931085i	$-0.381136\pi$
$797$	16416.3	0.729606	0.364803	+	0.931085i	$0.381136\pi$
$798$	−23375.0	−1.03692
$799$	22146.0	0.980562
$800$	−13058.4	−0.577104
$801$	2493.70	0.110001
$802$	22833.6	1.00534
$803$	6353.57	0.279219
$804$	5249.68	0.230276
$805$	−14478.8	−0.633924
$806$	32379.1	1.41502
$807$	9247.93	0.403398
$808$	−10366.6	−0.451355
$809$	21993.0	0.955789	0.477894	−	0.878417i	$-0.341400\pi$
$809$	21993.0	0.955789	0.477894	+	0.878417i	$0.341400\pi$
$810$	1360.55	0.0590182
$811$	4615.92	0.199861	0.0999303	−	0.994994i	$-0.468138\pi$
$811$	4615.92	0.199861	0.0999303	+	0.994994i	$0.468138\pi$
$812$	−680.069	−0.0293913
$813$	798.483	0.0344453
$814$	12964.2	0.558227
$815$	−9169.15	−0.394087
$816$	23804.4	1.02123
$817$	−15699.5	−0.672283
$818$	−23628.1	−1.00995
$819$	−8087.00	−0.345034
$820$	−7151.31	−0.304554
$821$	−25832.5	−1.09813	−0.549063	−	0.835781i	$-0.685015\pi$
$821$	−25832.5	−1.09813	−0.549063	+	0.835781i	$0.685015\pi$
$822$	−430.690	−0.0182750
$823$	41660.5	1.76451	0.882255	−	0.470771i	$-0.156024\pi$
$823$	41660.5	1.76451	0.882255	+	0.470771i	$0.156024\pi$
$824$	8328.94	0.352127
$825$	3282.40	0.138519
$826$	−5803.08	−0.244449
$827$	−9239.55	−0.388501	−0.194251	−	0.980952i	$-0.562228\pi$
$827$	−9239.55	−0.388501	−0.194251	+	0.980952i	$0.562228\pi$
$828$	−3975.45	−0.166856
$829$	−19699.0	−0.825302	−0.412651	−	0.910889i	$-0.635397\pi$
$829$	−19699.0	−0.825302	−0.412651	+	0.910889i	$0.635397\pi$
$830$	−15348.0	−0.641851
$831$	−26125.8	−1.09061
$832$	8919.48	0.371667
$833$	−4851.55	−0.201796
$834$	−5418.93	−0.224991
$835$	1322.93	0.0548286
$836$	3974.79	0.164439
$837$	−5790.25	−0.239116
$838$	−34652.6	−1.42847
$839$	27557.6	1.13396	0.566981	−	0.823731i	$-0.308111\pi$
$839$	27557.6	1.13396	0.566981	+	0.823731i	$0.308111\pi$
$840$	4934.82	0.202699
$841$	−24261.9	−0.994790
$842$	46656.1	1.90959
$843$	4814.14	0.196688
$844$	7719.48	0.314829
$845$	676.808	0.0275537
$846$	6604.44	0.268399
$847$	−2393.73	−0.0971068
$848$	44941.9	1.81994
$849$	1846.87	0.0746579
$850$	−33168.8	−1.33845
$851$	−51353.2	−2.06858
$852$	1117.57	0.0449383
$853$	1839.84	0.0738511	0.0369256	−	0.999318i	$-0.488244\pi$
$853$	1839.84	0.0738511	0.0369256	+	0.999318i	$0.488244\pi$
$854$	4011.38	0.160734
$855$	5388.42	0.215532
$856$	27252.4	1.08816
$857$	−27637.9	−1.10163	−0.550813	−	0.834628i	$-0.685682\pi$
$857$	−27637.9	−1.10163	−0.550813	+	0.834628i	$0.685682\pi$
$858$	4982.47	0.198250
$859$	27989.2	1.11173	0.555867	−	0.831271i	$-0.312386\pi$
$859$	27989.2	1.11173	0.555867	+	0.831271i	$0.312386\pi$
$860$	−2041.88	−0.0809621
$861$	−27541.4	−1.09014
$862$	−36970.2	−1.46080
$863$	22628.0	0.892543	0.446272	−	0.894898i	$-0.352752\pi$
$863$	22628.0	0.892543	0.446272	+	0.894898i	$0.352752\pi$
$864$	3544.66	0.139574
$865$	6520.34	0.256298
$866$	14716.8	0.577479
$867$	15451.8	0.605273
$868$	12938.3	0.505940
$869$	9872.09	0.385371
$870$	568.012	0.0221350
$871$	26062.2	1.01387
$872$	−31866.9	−1.23756
$873$	−15902.9	−0.616531
$874$	−57046.5	−2.20781
$875$	−22438.5	−0.866927
$876$	−5284.48	−0.203820
$877$	−23197.8	−0.893199	−0.446600	−	0.894734i	$-0.647365\pi$
$877$	−23197.8	−0.893199	−0.446600	+	0.894734i	$0.647365\pi$
$878$	2623.98	0.100860
$879$	28722.7	1.10215
$880$	−4396.49	−0.168415
$881$	−11633.6	−0.444886	−0.222443	−	0.974946i	$-0.571403\pi$
$881$	−11633.6	−0.444886	−0.222443	+	0.974946i	$0.571403\pi$
$882$	−1446.84	−0.0552355
$883$	−802.962	−0.0306023	−0.0153012	−	0.999883i	$-0.504871\pi$
$883$	−802.962	−0.0306023	−0.0153012	+	0.999883i	$0.504871\pi$
$884$	−13896.0	−0.528701
$885$	1337.73	0.0508106
$886$	−38678.6	−1.46663
$887$	−43234.7	−1.63662	−0.818309	−	0.574779i	$-0.805088\pi$
$887$	−43234.7	−1.63662	−0.818309	+	0.574779i	$0.805088\pi$
$888$	17502.8	0.661437
$889$	2503.24	0.0944387
$890$	4654.04	0.175285
$891$	−891.000	−0.0335013
$892$	−9108.07	−0.341884
$893$	26156.8	0.980182
$894$	21535.6	0.805659
$895$	10682.5	0.398968
$896$	33691.0	1.25618
$897$	−19736.3	−0.734644
$898$	42134.5	1.56575
$899$	−2417.36	−0.0896813
$900$	−2730.08	−0.101114
$901$	56999.2	2.10757
$902$	16968.5	0.626374
$903$	−7863.76	−0.289800
$904$	−1169.10	−0.0430129
$905$	−18427.8	−0.676863
$906$	2972.88	0.109015
$907$	33617.1	1.23069	0.615345	−	0.788258i	$-0.289017\pi$
$907$	33617.1	1.23069	0.615345	+	0.788258i	$0.289017\pi$
$908$	12294.4	0.449344
$909$	−5669.84	−0.206883
$910$	−15092.9	−0.549809
$911$	−9550.60	−0.347339	−0.173669	−	0.984804i	$-0.555562\pi$
$911$	−9550.60	−0.347339	−0.173669	+	0.984804i	$0.555562\pi$
$912$	28115.5	1.02083
$913$	10051.1	0.364342
$914$	−64774.1	−2.34413
$915$	−924.708	−0.0334097
$916$	−3422.78	−0.123463
$917$	14063.8	0.506465
$918$	9003.59	0.323707
$919$	8830.93	0.316981	0.158490	−	0.987361i	$-0.449337\pi$
$919$	8830.93	0.316981	0.158490	+	0.987361i	$0.449337\pi$
$920$	12043.4	0.431586
$921$	−14764.2	−0.528227
$922$	13844.0	0.494499
$923$	5548.23	0.197857
$924$	1990.94	0.0708845
$925$	−35266.1	−1.25356
$926$	−13448.4	−0.477259
$927$	4555.38	0.161401
$928$	1479.85	0.0523476
$929$	−15460.9	−0.546024	−0.273012	−	0.962011i	$-0.588020\pi$
$929$	−15460.9	−0.546024	−0.273012	+	0.962011i	$0.588020\pi$
$930$	−10806.5	−0.381030
$931$	−5730.19	−0.201718
$932$	−6995.58	−0.245867
$933$	10872.5	0.381511
$934$	17386.8	0.609117
$935$	−5576.00	−0.195032
$936$	6726.76	0.234905
$937$	5635.49	0.196482	0.0982409	−	0.995163i	$-0.468678\pi$
$937$	5635.49	0.196482	0.0982409	+	0.995163i	$0.468678\pi$
$938$	37732.9	1.31346
$939$	−3786.41	−0.131592
$940$	3401.95	0.118042
$941$	17826.1	0.617549	0.308775	−	0.951135i	$-0.400081\pi$
$941$	17826.1	0.617549	0.308775	+	0.951135i	$0.400081\pi$
$942$	13306.6	0.460246
$943$	−67214.7	−2.32112
$944$	6979.98	0.240656
$945$	2699.02	0.0929092
$946$	4844.93	0.166514
$947$	46881.4	1.60870	0.804352	−	0.594153i	$-0.202513\pi$
$947$	46881.4	1.60870	0.804352	+	0.594153i	$0.202513\pi$
$948$	−8210.96	−0.281307
$949$	−26235.0	−0.897391
$950$	−39175.8	−1.33793
$951$	−12506.0	−0.426429
$952$	32656.8	1.11178
$953$	−31061.9	−1.05582	−0.527909	−	0.849301i	$-0.677024\pi$
$953$	−31061.9	−1.05582	−0.527909	+	0.849301i	$0.677024\pi$
$954$	16998.5	0.576882
$955$	−25220.5	−0.854571
$956$	13845.1	0.468392
$957$	−371.982	−0.0125647
$958$	15571.0	0.525133
$959$	−854.393	−0.0287693
$960$	−2976.86	−0.100081
$961$	16199.4	0.543768
$962$	−53531.6	−1.79410
$963$	14905.2	0.498769
$964$	−10395.3	−0.347312
$965$	20985.6	0.700051
$966$	−28574.2	−0.951718
$967$	−21356.2	−0.710207	−0.355103	−	0.934827i	$-0.615554\pi$
$967$	−21356.2	−0.710207	−0.355103	+	0.934827i	$0.615554\pi$
$968$	1991.10	0.0661119
$969$	35658.5	1.18216
$970$	−29679.9	−0.982437
$971$	−37886.0	−1.25213	−0.626065	−	0.779770i	$-0.715336\pi$
$971$	−37886.0	−1.25213	−0.626065	+	0.779770i	$0.715336\pi$
$972$	741.075	0.0244547
$973$	−10749.9	−0.354191
$974$	−33587.5	−1.10494
$975$	−13553.6	−0.445193
$976$	−4824.91	−0.158239
$977$	35506.5	1.16270	0.581349	−	0.813655i	$-0.302525\pi$
$977$	35506.5	1.16270	0.581349	+	0.813655i	$0.302525\pi$
$978$	−18095.5	−0.591648
$979$	−3047.86	−0.0994994
$980$	−745.269	−0.0242926
$981$	−17429.1	−0.567246
$982$	−38457.6	−1.24973
$983$	40077.7	1.30039	0.650194	−	0.759768i	$-0.274688\pi$
$983$	40077.7	1.30039	0.650194	+	0.759768i	$0.274688\pi$
$984$	22908.9	0.742185
$985$	−26075.7	−0.843493
$986$	3758.89	0.121407
$987$	13101.7	0.422526
$988$	−16412.6	−0.528496
$989$	−19191.5	−0.617041
$990$	−1662.89	−0.0533840
$991$	12804.9	0.410455	0.205228	−	0.978714i	$-0.434206\pi$
$991$	12804.9	0.410455	0.205228	+	0.978714i	$0.434206\pi$
$992$	−28154.3	−0.901109
$993$	8203.63	0.262170
$994$	8032.73	0.256321
$995$	−21795.7	−0.694441
$996$	−8359.88	−0.265957
$997$	−48944.1	−1.55474	−0.777369	−	0.629045i	$-0.783446\pi$
$997$	−48944.1	−1.55474	−0.777369	+	0.629045i	$0.783446\pi$
$998$	46304.5	1.46868
$999$	9572.89	0.303176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.h.1.9	✓	39

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.h.1.9	✓	39	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-3.32411 q^{2} +3.00000 q^{3} +3.04969 q^{4} -5.05305 q^{5} -9.97232 q^{6} -19.7829 q^{7} +16.4554 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.32411 q^{2} +3.00000 q^{3} +3.04969 q^{4} -5.05305 q^{5} -9.97232 q^{6} -19.7829 q^{7} +16.4554 q^{8} +9.00000 q^{9} +16.7969 q^{10} -11.0000 q^{11} +9.14908 q^{12} +45.4209 q^{13} +65.7604 q^{14} -15.1591 q^{15} -79.0969 q^{16} -100.318 q^{17} -29.9170 q^{18} -118.486 q^{19} -15.4102 q^{20} -59.3486 q^{21} +36.5652 q^{22} -144.840 q^{23} +49.3661 q^{24} -99.4667 q^{25} -150.984 q^{26} +27.0000 q^{27} -60.3317 q^{28} +11.2722 q^{29} +50.3906 q^{30} -214.454 q^{31} +131.284 q^{32} -33.0000 q^{33} +333.466 q^{34} +99.9638 q^{35} +27.4472 q^{36} +354.551 q^{37} +393.859 q^{38} +136.263 q^{39} -83.1497 q^{40} +464.062 q^{41} +197.281 q^{42} +132.501 q^{43} -33.5466 q^{44} -45.4774 q^{45} +481.463 q^{46} -220.759 q^{47} -237.291 q^{48} +48.3619 q^{49} +330.638 q^{50} -300.953 q^{51} +138.520 q^{52} -568.188 q^{53} -89.7509 q^{54} +55.5835 q^{55} -325.534 q^{56} -355.457 q^{57} -37.4699 q^{58} -88.2459 q^{59} -46.2307 q^{60} +61.0000 q^{61} +712.867 q^{62} -178.046 q^{63} +196.374 q^{64} -229.514 q^{65} +109.696 q^{66} +573.794 q^{67} -305.938 q^{68} -434.520 q^{69} -332.290 q^{70} +122.152 q^{71} +148.098 q^{72} -577.597 q^{73} -1178.57 q^{74} -298.400 q^{75} -361.345 q^{76} +217.612 q^{77} -452.952 q^{78} -897.463 q^{79} +399.681 q^{80} +81.0000 q^{81} -1542.59 q^{82} -913.741 q^{83} -180.995 q^{84} +506.909 q^{85} -440.448 q^{86} +33.8165 q^{87} -181.009 q^{88} +277.078 q^{89} +151.172 q^{90} -898.556 q^{91} -441.717 q^{92} -643.361 q^{93} +733.827 q^{94} +598.714 q^{95} +393.851 q^{96} -1766.99 q^{97} -160.760 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + 8 q^{2} + 117 q^{3} + 154 q^{4} + 65 q^{5} + 24 q^{6} + 35 q^{7} + 75 q^{8} + 351 q^{9}+O(q^{10}) \) 39 * q + 8 * q^2 + 117 * q^3 + 154 * q^4 + 65 * q^5 + 24 * q^6 + 35 * q^7 + 75 * q^8 + 351 * q^9	\( 39 q + 8 q^{2} + 117 q^{3} + 154 q^{4} + 65 q^{5} + 24 q^{6} + 35 q^{7} + 75 q^{8} + 351 q^{9} - 21 q^{10} - 429 q^{11} + 462 q^{12} - 27 q^{13} + 164 q^{14} + 195 q^{15} + 686 q^{16} + 170 q^{17} + 72 q^{18} + 139 q^{19} + 1056 q^{20} + 105 q^{21} - 88 q^{22} + 291 q^{23} + 225 q^{24} + 1236 q^{25} + 583 q^{26} + 1053 q^{27} + 976 q^{28} + 374 q^{29} - 63 q^{30} + 232 q^{31} + 933 q^{32} - 1287 q^{33} + 332 q^{34} + 626 q^{35} + 1386 q^{36} + 232 q^{37} + 989 q^{38} - 81 q^{39} - 263 q^{40} + 1014 q^{41} + 492 q^{42} + 515 q^{43} - 1694 q^{44} + 585 q^{45} - 371 q^{46} + 2005 q^{47} + 2058 q^{48} + 2064 q^{49} + 4582 q^{50} + 510 q^{51} + 216 q^{52} + 1485 q^{53} + 216 q^{54} - 715 q^{55} + 2307 q^{56} + 417 q^{57} + 573 q^{58} + 2749 q^{59} + 3168 q^{60} + 2379 q^{61} + 1837 q^{62} + 315 q^{63} + 7295 q^{64} + 3630 q^{65} - 264 q^{66} + 3575 q^{67} + 2630 q^{68} + 873 q^{69} + 4218 q^{70} + 4723 q^{71} + 675 q^{72} + 859 q^{73} + 4232 q^{74} + 3708 q^{75} + 2466 q^{76} - 385 q^{77} + 1749 q^{78} - 1887 q^{79} + 8933 q^{80} + 3159 q^{81} + 6806 q^{82} + 5609 q^{83} + 2928 q^{84} - 565 q^{85} + 5185 q^{86} + 1122 q^{87} - 825 q^{88} + 6725 q^{89} - 189 q^{90} + 2808 q^{91} + 3257 q^{92} + 696 q^{93} + 3184 q^{94} + 3216 q^{95} + 2799 q^{96} + 3512 q^{97} + 4464 q^{98} - 3861 q^{99}+O(q^{100}) \) 39 * q + 8 * q^2 + 117 * q^3 + 154 * q^4 + 65 * q^5 + 24 * q^6 + 35 * q^7 + 75 * q^8 + 351 * q^9 - 21 * q^10 - 429 * q^11 + 462 * q^12 - 27 * q^13 + 164 * q^14 + 195 * q^15 + 686 * q^16 + 170 * q^17 + 72 * q^18 + 139 * q^19 + 1056 * q^20 + 105 * q^21 - 88 * q^22 + 291 * q^23 + 225 * q^24 + 1236 * q^25 + 583 * q^26 + 1053 * q^27 + 976 * q^28 + 374 * q^29 - 63 * q^30 + 232 * q^31 + 933 * q^32 - 1287 * q^33 + 332 * q^34 + 626 * q^35 + 1386 * q^36 + 232 * q^37 + 989 * q^38 - 81 * q^39 - 263 * q^40 + 1014 * q^41 + 492 * q^42 + 515 * q^43 - 1694 * q^44 + 585 * q^45 - 371 * q^46 + 2005 * q^47 + 2058 * q^48 + 2064 * q^49 + 4582 * q^50 + 510 * q^51 + 216 * q^52 + 1485 * q^53 + 216 * q^54 - 715 * q^55 + 2307 * q^56 + 417 * q^57 + 573 * q^58 + 2749 * q^59 + 3168 * q^60 + 2379 * q^61 + 1837 * q^62 + 315 * q^63 + 7295 * q^64 + 3630 * q^65 - 264 * q^66 + 3575 * q^67 + 2630 * q^68 + 873 * q^69 + 4218 * q^70 + 4723 * q^71 + 675 * q^72 + 859 * q^73 + 4232 * q^74 + 3708 * q^75 + 2466 * q^76 - 385 * q^77 + 1749 * q^78 - 1887 * q^79 + 8933 * q^80 + 3159 * q^81 + 6806 * q^82 + 5609 * q^83 + 2928 * q^84 - 565 * q^85 + 5185 * q^86 + 1122 * q^87 - 825 * q^88 + 6725 * q^89 - 189 * q^90 + 2808 * q^91 + 3257 * q^92 + 696 * q^93 + 3184 * q^94 + 3216 * q^95 + 2799 * q^96 + 3512 * q^97 + 4464 * q^98 - 3861 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more