LMFDB - Embedded newform 1250.4.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1250,4,Mod(1,1250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1250, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1250.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1250 = 2 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1250.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:	$73.7523875072$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} - 125 x^{14} + 990 x^{13} + 6166 x^{12} - 47880 x^{11} - 151199 x^{10} + \cdots - 45086320 $ x^16 - 8x^15 - 125x^14 + 990x^13 + 6166x^12 - 47880x^11 - 151199x^10 + 1143340x^9 + 1899751x^8 - 14035110x^7 - 11070230x^6 + 83435508x^5 + 14900696x^4 - 193725880x^3 + 78433160x^2 + 82757040*x - 45086320
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{15} $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.735805$ of defining polynomial
Character	$\chi$	$=$	1250.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -0.246957 q^{3} +4.00000 q^{4} -0.493914 q^{6} -19.9138 q^{7} +8.00000 q^{8} -26.9390 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -0.246957 q^{3} +4.00000 q^{4} -0.493914 q^{6} -19.9138 q^{7} +8.00000 q^{8} -26.9390 q^{9} -2.21060 q^{11} -0.987829 q^{12} -12.3046 q^{13} -39.8276 q^{14} +16.0000 q^{16} -74.4371 q^{17} -53.8780 q^{18} -50.4660 q^{19} +4.91785 q^{21} -4.42120 q^{22} +199.314 q^{23} -1.97566 q^{24} -24.6092 q^{26} +13.3206 q^{27} -79.6551 q^{28} +241.241 q^{29} -51.2776 q^{31} +32.0000 q^{32} +0.545923 q^{33} -148.874 q^{34} -107.756 q^{36} +278.117 q^{37} -100.932 q^{38} +3.03871 q^{39} +265.583 q^{41} +9.83571 q^{42} -103.714 q^{43} -8.84239 q^{44} +398.627 q^{46} +411.544 q^{47} -3.95132 q^{48} +53.5589 q^{49} +18.3828 q^{51} -49.2183 q^{52} +430.285 q^{53} +26.6413 q^{54} -159.310 q^{56} +12.4629 q^{57} +482.482 q^{58} +550.978 q^{59} -510.914 q^{61} -102.555 q^{62} +536.458 q^{63} +64.0000 q^{64} +1.09185 q^{66} -76.2521 q^{67} -297.748 q^{68} -49.2219 q^{69} -855.645 q^{71} -215.512 q^{72} -516.618 q^{73} +556.233 q^{74} -201.864 q^{76} +44.0214 q^{77} +6.07741 q^{78} +136.253 q^{79} +724.064 q^{81} +531.166 q^{82} +917.533 q^{83} +19.6714 q^{84} -207.427 q^{86} -59.5762 q^{87} -17.6848 q^{88} +1302.75 q^{89} +245.031 q^{91} +797.255 q^{92} +12.6634 q^{93} +823.088 q^{94} -7.90263 q^{96} -443.954 q^{97} +107.118 q^{98} +59.5513 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{2} + 12 q^{3} + 64 q^{4} + 24 q^{6} + 56 q^{7} + 128 q^{8} + 212 q^{9}+O(q^{10}) $ 16 * q + 32 * q^2 + 12 * q^3 + 64 * q^4 + 24 * q^6 + 56 * q^7 + 128 * q^8 + 212 * q^9	\( 16 q + 32 q^{2} + 12 q^{3} + 64 q^{4} + 24 q^{6} + 56 q^{7} + 128 q^{8} + 212 q^{9} + 72 q^{11} + 48 q^{12} + 202 q^{13} + 112 q^{14} + 256 q^{16} + 216 q^{17} + 424 q^{18} + 100 q^{19} + 192 q^{21} + 144 q^{22} + 292 q^{23} + 96 q^{24} + 404 q^{26} + 570 q^{27} + 224 q^{28} + 400 q^{29} + 102 q^{31} + 512 q^{32} + 664 q^{33} + 432 q^{34} + 848 q^{36} + 646 q^{37} + 200 q^{38} + 104 q^{39} + 532 q^{41} + 384 q^{42} + 902 q^{43} + 288 q^{44} + 584 q^{46} + 776 q^{47} + 192 q^{48} + 1038 q^{49} + 442 q^{51} + 808 q^{52} + 632 q^{53} + 1140 q^{54} + 448 q^{56} + 1400 q^{57} + 800 q^{58} + 1000 q^{59} + 662 q^{61} + 204 q^{62} + 932 q^{63} + 1024 q^{64} + 1328 q^{66} + 1326 q^{67} + 864 q^{68} + 1854 q^{69} + 1292 q^{71} + 1696 q^{72} + 2272 q^{73} + 1292 q^{74} + 400 q^{76} + 2582 q^{77} + 208 q^{78} + 320 q^{79} + 2956 q^{81} + 1064 q^{82} + 2842 q^{83} + 768 q^{84} + 1804 q^{86} + 2920 q^{87} + 576 q^{88} + 2780 q^{89} + 812 q^{91} + 1168 q^{92} + 2824 q^{93} + 1552 q^{94} + 384 q^{96} + 3796 q^{97} + 2076 q^{98} + 1054 q^{99}+O(q^{100}) \) 16 * q + 32 * q^2 + 12 * q^3 + 64 * q^4 + 24 * q^6 + 56 * q^7 + 128 * q^8 + 212 * q^9 + 72 * q^11 + 48 * q^12 + 202 * q^13 + 112 * q^14 + 256 * q^16 + 216 * q^17 + 424 * q^18 + 100 * q^19 + 192 * q^21 + 144 * q^22 + 292 * q^23 + 96 * q^24 + 404 * q^26 + 570 * q^27 + 224 * q^28 + 400 * q^29 + 102 * q^31 + 512 * q^32 + 664 * q^33 + 432 * q^34 + 848 * q^36 + 646 * q^37 + 200 * q^38 + 104 * q^39 + 532 * q^41 + 384 * q^42 + 902 * q^43 + 288 * q^44 + 584 * q^46 + 776 * q^47 + 192 * q^48 + 1038 * q^49 + 442 * q^51 + 808 * q^52 + 632 * q^53 + 1140 * q^54 + 448 * q^56 + 1400 * q^57 + 800 * q^58 + 1000 * q^59 + 662 * q^61 + 204 * q^62 + 932 * q^63 + 1024 * q^64 + 1328 * q^66 + 1326 * q^67 + 864 * q^68 + 1854 * q^69 + 1292 * q^71 + 1696 * q^72 + 2272 * q^73 + 1292 * q^74 + 400 * q^76 + 2582 * q^77 + 208 * q^78 + 320 * q^79 + 2956 * q^81 + 1064 * q^82 + 2842 * q^83 + 768 * q^84 + 1804 * q^86 + 2920 * q^87 + 576 * q^88 + 2780 * q^89 + 812 * q^91 + 1168 * q^92 + 2824 * q^93 + 1552 * q^94 + 384 * q^96 + 3796 * q^97 + 2076 * q^98 + 1054 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−0.246957	−0.0475269	−0.0237635	−	0.999718i	$-0.507565\pi$
$3$	−0.246957	−0.0475269	−0.0237635	+	0.999718i	$0.507565\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.493914	−0.0336066
$7$	−19.9138	−1.07524	−0.537622	−	0.843186i	$-0.680677\pi$
$7$	−19.9138	−1.07524	−0.537622	+	0.843186i	$0.680677\pi$
$8$	8.00000	0.353553
$9$	−26.9390	−0.997741
$10$	0	0
$11$	−2.21060	−0.0605928	−0.0302964	−	0.999541i	$-0.509645\pi$
$11$	−2.21060	−0.0605928	−0.0302964	+	0.999541i	$0.509645\pi$
$12$	−0.987829	−0.0237635
$13$	−12.3046	−0.262514	−0.131257	−	0.991348i	$-0.541901\pi$
$13$	−12.3046	−0.262514	−0.131257	+	0.991348i	$0.541901\pi$
$14$	−39.8276	−0.760312
$15$	0	0
$16$	16.0000	0.250000
$17$	−74.4371	−1.06198	−0.530990	−	0.847378i	$-0.678180\pi$
$17$	−74.4371	−1.06198	−0.530990	+	0.847378i	$0.678180\pi$
$18$	−53.8780	−0.705510
$19$	−50.4660	−0.609353	−0.304676	−	0.952456i	$-0.598548\pi$
$19$	−50.4660	−0.609353	−0.304676	+	0.952456i	$0.598548\pi$
$20$	0	0
$21$	4.91785	0.0511030
$22$	−4.42120	−0.0428455
$23$	199.314	1.80695	0.903473	−	0.428644i	$-0.141009\pi$
$23$	199.314	1.80695	0.903473	+	0.428644i	$0.141009\pi$
$24$	−1.97566	−0.0168033
$25$	0	0
$26$	−24.6092	−0.185625
$27$	13.3206	0.0949465
$28$	−79.6551	−0.537622
$29$	241.241	1.54473	0.772367	−	0.635176i	$-0.219072\pi$
$29$	241.241	1.54473	0.772367	+	0.635176i	$0.219072\pi$
$30$	0	0
$31$	−51.2776	−0.297088	−0.148544	−	0.988906i	$-0.547459\pi$
$31$	−51.2776	−0.297088	−0.148544	+	0.988906i	$0.547459\pi$
$32$	32.0000	0.176777
$33$	0.545923	0.00287979
$34$	−148.874	−0.750933
$35$	0	0
$36$	−107.756	−0.498871
$37$	278.117	1.23573	0.617866	−	0.786283i	$-0.287997\pi$
$37$	278.117	1.23573	0.617866	+	0.786283i	$0.287997\pi$
$38$	−100.932	−0.430877
$39$	3.03871	0.0124765
$40$	0	0
$41$	265.583	1.01164	0.505818	−	0.862640i	$-0.331191\pi$
$41$	265.583	1.01164	0.505818	+	0.862640i	$0.331191\pi$
$42$	9.83571	0.0361353
$43$	−103.714	−0.367818	−0.183909	−	0.982943i	$-0.558875\pi$
$43$	−103.714	−0.367818	−0.183909	+	0.982943i	$0.558875\pi$
$44$	−8.84239	−0.0302964
$45$	0	0
$46$	398.627	1.27770
$47$	411.544	1.27723	0.638616	−	0.769526i	$-0.279507\pi$
$47$	411.544	1.27723	0.638616	+	0.769526i	$0.279507\pi$
$48$	−3.95132	−0.0118817
$49$	53.5589	0.156148
$50$	0	0
$51$	18.3828	0.0504726
$52$	−49.2183	−0.131257
$53$	430.285	1.11517	0.557587	−	0.830118i	$-0.311727\pi$
$53$	430.285	1.11517	0.557587	+	0.830118i	$0.311727\pi$
$54$	26.6413	0.0671373
$55$	0	0
$56$	−159.310	−0.380156
$57$	12.4629	0.0289607
$58$	482.482	1.09229
$59$	550.978	1.21578	0.607891	−	0.794020i	$-0.292016\pi$
$59$	550.978	1.21578	0.607891	+	0.794020i	$0.292016\pi$
$60$	0	0
$61$	−510.914	−1.07239	−0.536196	−	0.844094i	$-0.680139\pi$
$61$	−510.914	−1.07239	−0.536196	+	0.844094i	$0.680139\pi$
$62$	−102.555	−0.210073
$63$	536.458	1.07281
$64$	64.0000	0.125000
$65$	0	0
$66$	1.09185	0.00203632
$67$	−76.2521	−0.139040	−0.0695200	−	0.997581i	$-0.522147\pi$
$67$	−76.2521	−0.139040	−0.0695200	+	0.997581i	$0.522147\pi$
$68$	−297.748	−0.530990
$69$	−49.2219	−0.0858786
$70$	0	0
$71$	−855.645	−1.43023	−0.715115	−	0.699007i	$-0.753626\pi$
$71$	−855.645	−1.43023	−0.715115	+	0.699007i	$0.753626\pi$
$72$	−215.512	−0.352755
$73$	−516.618	−0.828295	−0.414147	−	0.910210i	$-0.635920\pi$
$73$	−516.618	−0.828295	−0.414147	+	0.910210i	$0.635920\pi$
$74$	556.233	0.873795
$75$	0	0
$76$	−201.864	−0.304676
$77$	44.0214	0.0651520
$78$	6.07741	0.00882220
$79$	136.253	0.194046	0.0970229	−	0.995282i	$-0.469068\pi$
$79$	136.253	0.194046	0.0970229	+	0.995282i	$0.469068\pi$
$80$	0	0
$81$	724.064	0.993229
$82$	531.166	0.715335
$83$	917.533	1.21340	0.606701	−	0.794930i	$-0.292493\pi$
$83$	917.533	1.21340	0.606701	+	0.794930i	$0.292493\pi$
$84$	19.6714	0.0255515
$85$	0	0
$86$	−207.427	−0.260087
$87$	−59.5762	−0.0734165
$88$	−17.6848	−0.0214228
$89$	1302.75	1.55158	0.775791	−	0.630990i	$-0.217351\pi$
$89$	1302.75	1.55158	0.775791	+	0.630990i	$0.217351\pi$
$90$	0	0
$91$	245.031	0.282266
$92$	797.255	0.903473
$93$	12.6634	0.0141197
$94$	823.088	0.903139
$95$	0	0
$96$	−7.90263	−0.00840165
$97$	−443.954	−0.464708	−0.232354	−	0.972631i	$-0.574643\pi$
$97$	−443.954	−0.464708	−0.232354	+	0.972631i	$0.574643\pi$
$98$	107.118	0.110414
$99$	59.5513	0.0604559
$100$	0	0
$101$	−493.360	−0.486051	−0.243025	−	0.970020i	$-0.578140\pi$
$101$	−493.360	−0.486051	−0.243025	+	0.970020i	$0.578140\pi$
$102$	36.7656	0.0356895
$103$	1721.28	1.64663	0.823313	−	0.567588i	$-0.192123\pi$
$103$	1721.28	1.64663	0.823313	+	0.567588i	$0.192123\pi$
$104$	−98.4367	−0.0928126
$105$	0	0
$106$	860.571	0.788547
$107$	−1241.00	−1.12124	−0.560618	−	0.828074i	$-0.689437\pi$
$107$	−1241.00	−1.12124	−0.560618	+	0.828074i	$0.689437\pi$
$108$	53.2825	0.0474733
$109$	176.719	0.155290	0.0776450	−	0.996981i	$-0.475260\pi$
$109$	176.719	0.155290	0.0776450	+	0.996981i	$0.475260\pi$
$110$	0	0
$111$	−68.6829	−0.0587306
$112$	−318.621	−0.268811
$113$	191.746	0.159628	0.0798138	−	0.996810i	$-0.474567\pi$
$113$	191.746	0.159628	0.0798138	+	0.996810i	$0.474567\pi$
$114$	24.9259	0.0204783
$115$	0	0
$116$	964.964	0.772367
$117$	331.473	0.261921
$118$	1101.96	0.859688
$119$	1482.32	1.14189
$120$	0	0
$121$	−1326.11	−0.996329
$122$	−1021.83	−0.758295
$123$	−65.5877	−0.0480800
$124$	−205.110	−0.148544
$125$	0	0
$126$	1072.92	0.758595
$127$	307.346	0.214744	0.107372	−	0.994219i	$-0.465756\pi$
$127$	307.346	0.214744	0.107372	+	0.994219i	$0.465756\pi$
$128$	128.000	0.0883883
$129$	25.6129	0.0174813
$130$	0	0
$131$	2809.69	1.87392	0.936962	−	0.349432i	$-0.113626\pi$
$131$	2809.69	1.87392	0.936962	+	0.349432i	$0.113626\pi$
$132$	2.18369	0.00143989
$133$	1004.97	0.655202
$134$	−152.504	−0.0983161
$135$	0	0
$136$	−595.497	−0.375466
$137$	−1960.87	−1.22283	−0.611417	−	0.791309i	$-0.709400\pi$
$137$	−1960.87	−1.22283	−0.611417	+	0.791309i	$0.709400\pi$
$138$	−98.4439	−0.0607254
$139$	−1578.55	−0.963244	−0.481622	−	0.876379i	$-0.659952\pi$
$139$	−1578.55	−0.963244	−0.481622	+	0.876379i	$0.659952\pi$
$140$	0	0
$141$	−101.634	−0.0607029
$142$	−1711.29	−1.01133
$143$	27.2005	0.0159064
$144$	−431.024	−0.249435
$145$	0	0
$146$	−1033.24	−0.585693
$147$	−13.2268	−0.00742126
$148$	1112.47	0.617866
$149$	−1448.55	−0.796443	−0.398221	−	0.917289i	$-0.630372\pi$
$149$	−1448.55	−0.796443	−0.398221	+	0.917289i	$0.630372\pi$
$150$	0	0
$151$	−1804.96	−0.972754	−0.486377	−	0.873749i	$-0.661682\pi$
$151$	−1804.96	−0.972754	−0.486377	+	0.873749i	$0.661682\pi$
$152$	−403.728	−0.215439
$153$	2005.26	1.05958
$154$	88.0427	0.0460694
$155$	0	0
$156$	12.1548	0.00623823
$157$	1998.98	1.01615	0.508076	−	0.861312i	$-0.330357\pi$
$157$	1998.98	1.01615	0.508076	+	0.861312i	$0.330357\pi$
$158$	272.505	0.137211
$159$	−106.262	−0.0530008
$160$	0	0
$161$	−3969.09	−1.94291
$162$	1448.13	0.702319
$163$	491.248	0.236058	0.118029	−	0.993010i	$-0.462342\pi$
$163$	491.248	0.236058	0.118029	+	0.993010i	$0.462342\pi$
$164$	1062.33	0.505818
$165$	0	0
$166$	1835.07	0.858005
$167$	340.453	0.157755	0.0788774	−	0.996884i	$-0.474866\pi$
$167$	340.453	0.157755	0.0788774	+	0.996884i	$0.474866\pi$
$168$	39.3428	0.0180676
$169$	−2045.60	−0.931087
$170$	0	0
$171$	1359.50	0.607976
$172$	−414.855	−0.183909
$173$	2178.16	0.957240	0.478620	−	0.878022i	$-0.341137\pi$
$173$	2178.16	0.957240	0.478620	+	0.878022i	$0.341137\pi$
$174$	−119.152	−0.0519133
$175$	0	0
$176$	−35.3696	−0.0151482
$177$	−136.068	−0.0577824
$178$	2605.49	1.09713
$179$	−3440.06	−1.43644	−0.718219	−	0.695817i	$-0.755042\pi$
$179$	−3440.06	−1.43644	−0.718219	+	0.695817i	$0.755042\pi$
$180$	0	0
$181$	−2022.21	−0.830441	−0.415221	−	0.909721i	$-0.636296\pi$
$181$	−2022.21	−0.830441	−0.415221	+	0.909721i	$0.636296\pi$
$182$	490.062	0.199592
$183$	126.174	0.0509675
$184$	1594.51	0.638852
$185$	0	0
$186$	25.3267	0.00998412
$187$	164.551	0.0643483
$188$	1646.18	0.638616
$189$	−265.264	−0.102091
$190$	0	0
$191$	4826.69	1.82852	0.914260	−	0.405128i	$-0.132773\pi$
$191$	4826.69	1.82852	0.914260	+	0.405128i	$0.132773\pi$
$192$	−15.8053	−0.00594087
$193$	3075.17	1.14692	0.573460	−	0.819233i	$-0.305601\pi$
$193$	3075.17	1.14692	0.573460	+	0.819233i	$0.305601\pi$
$194$	−887.907	−0.328598
$195$	0	0
$196$	214.236	0.0780742
$197$	4234.60	1.53149	0.765743	−	0.643147i	$-0.222372\pi$
$197$	4234.60	1.53149	0.765743	+	0.643147i	$0.222372\pi$
$198$	119.103	0.0427488
$199$	4126.17	1.46983	0.734915	−	0.678159i	$-0.237222\pi$
$199$	4126.17	1.46983	0.734915	+	0.678159i	$0.237222\pi$
$200$	0	0
$201$	18.8310	0.00660814
$202$	−986.720	−0.343690
$203$	−4804.02	−1.66097
$204$	73.5311	0.0252363
$205$	0	0
$206$	3442.55	1.16434
$207$	−5369.31	−1.80287
$208$	−196.873	−0.0656284
$209$	111.560	0.0369224
$210$	0	0
$211$	−5079.84	−1.65739	−0.828697	−	0.559697i	$-0.810918\pi$
$211$	−5079.84	−1.65739	−0.828697	+	0.559697i	$0.810918\pi$
$212$	1721.14	0.557587
$213$	211.308	0.0679744
$214$	−2482.01	−0.792834
$215$	0	0
$216$	106.565	0.0335687
$217$	1021.13	0.319442
$218$	353.438	0.109807
$219$	127.582	0.0393663
$220$	0	0
$221$	915.918	0.278784
$222$	−137.366	−0.0415288
$223$	4935.02	1.48194	0.740972	−	0.671536i	$-0.234365\pi$
$223$	4935.02	1.48194	0.740972	+	0.671536i	$0.234365\pi$
$224$	−637.241	−0.190078
$225$	0	0
$226$	383.492	0.112874
$227$	−698.200	−0.204146	−0.102073	−	0.994777i	$-0.532548\pi$
$227$	−698.200	−0.204146	−0.102073	+	0.994777i	$0.532548\pi$
$228$	49.8518	0.0144803
$229$	6202.52	1.78984	0.894922	−	0.446222i	$-0.147231\pi$
$229$	6202.52	1.78984	0.894922	+	0.446222i	$0.147231\pi$
$230$	0	0
$231$	−10.8714	−0.00309647
$232$	1929.93	0.546146
$233$	2732.97	0.768425	0.384212	−	0.923245i	$-0.374473\pi$
$233$	2732.97	0.768425	0.384212	+	0.923245i	$0.374473\pi$
$234$	662.947	0.185206
$235$	0	0
$236$	2203.91	0.607891
$237$	−33.6486	−0.00922240
$238$	2964.65	0.807436
$239$	−1480.18	−0.400606	−0.200303	−	0.979734i	$-0.564193\pi$
$239$	−1480.18	−0.400606	−0.200303	+	0.979734i	$0.564193\pi$
$240$	0	0
$241$	258.804	0.0691744	0.0345872	−	0.999402i	$-0.488988\pi$
$241$	258.804	0.0691744	0.0345872	+	0.999402i	$0.488988\pi$
$242$	−2652.23	−0.704511
$243$	−538.470	−0.142152
$244$	−2043.66	−0.536196
$245$	0	0
$246$	−131.175	−0.0339977
$247$	620.963	0.159963
$248$	−410.221	−0.105036
$249$	−226.591	−0.0576693
$250$	0	0
$251$	−3432.35	−0.863139	−0.431570	−	0.902080i	$-0.642040\pi$
$251$	−3432.35	−0.863139	−0.431570	+	0.902080i	$0.642040\pi$
$252$	2145.83	0.536407
$253$	−440.602	−0.109488
$254$	614.692	0.151847
$255$	0	0
$256$	256.000	0.0625000
$257$	3106.91	0.754100	0.377050	−	0.926193i	$-0.376939\pi$
$257$	3106.91	0.754100	0.377050	+	0.926193i	$0.376939\pi$
$258$	51.2257	0.0123611
$259$	−5538.36	−1.32871
$260$	0	0
$261$	−6498.79	−1.54125
$262$	5619.38	1.32506
$263$	4956.72	1.16215	0.581073	−	0.813852i	$-0.302633\pi$
$263$	4956.72	1.16215	0.581073	+	0.813852i	$0.302633\pi$
$264$	4.36738	0.00101816
$265$	0	0
$266$	2009.94	0.463298
$267$	−321.722	−0.0737419
$268$	−305.008	−0.0695200
$269$	2396.83	0.543260	0.271630	−	0.962402i	$-0.412437\pi$
$269$	2396.83	0.543260	0.271630	+	0.962402i	$0.412437\pi$
$270$	0	0
$271$	−6490.60	−1.45489	−0.727446	−	0.686165i	$-0.759293\pi$
$271$	−6490.60	−1.45489	−0.727446	+	0.686165i	$0.759293\pi$
$272$	−1190.99	−0.265495
$273$	−60.5121	−0.0134152
$274$	−3921.74	−0.864674
$275$	0	0
$276$	−196.888	−0.0429393
$277$	3803.88	0.825102	0.412551	−	0.910935i	$-0.364638\pi$
$277$	3803.88	0.825102	0.412551	+	0.910935i	$0.364638\pi$
$278$	−3157.10	−0.681117
$279$	1381.37	0.296417
$280$	0	0
$281$	255.549	0.0542519	0.0271259	−	0.999632i	$-0.491364\pi$
$281$	255.549	0.0542519	0.0271259	+	0.999632i	$0.491364\pi$
$282$	−203.268	−0.0429234
$283$	2050.00	0.430600	0.215300	−	0.976548i	$-0.430927\pi$
$283$	2050.00	0.430600	0.215300	+	0.976548i	$0.430927\pi$
$284$	−3422.58	−0.715115
$285$	0	0
$286$	54.4010	0.0112475
$287$	−5288.77	−1.08776
$288$	−862.048	−0.176377
$289$	627.883	0.127800
$290$	0	0
$291$	109.638	0.0220861
$292$	−2066.47	−0.414147
$293$	6743.02	1.34448	0.672238	−	0.740335i	$-0.265333\pi$
$293$	6743.02	1.34448	0.672238	+	0.740335i	$0.265333\pi$
$294$	−26.4535	−0.00524762
$295$	0	0
$296$	2224.93	0.436897
$297$	−29.4466	−0.00575307
$298$	−2897.10	−0.563170
$299$	−2452.47	−0.474348
$300$	0	0
$301$	2065.33	0.395494
$302$	−3609.93	−0.687841
$303$	121.839	0.0231005
$304$	−807.456	−0.152338
$305$	0	0
$306$	4010.52	0.749237
$307$	−4999.71	−0.929474	−0.464737	−	0.885449i	$-0.653851\pi$
$307$	−4999.71	−0.929474	−0.464737	+	0.885449i	$0.653851\pi$
$308$	176.085	0.0325760
$309$	−425.082	−0.0782591
$310$	0	0
$311$	6869.39	1.25250	0.626250	−	0.779622i	$-0.284589\pi$
$311$	6869.39	1.25250	0.626250	+	0.779622i	$0.284589\pi$
$312$	24.3096	0.00441110
$313$	−448.943	−0.0810727	−0.0405363	−	0.999178i	$-0.512907\pi$
$313$	−448.943	−0.0810727	−0.0405363	+	0.999178i	$0.512907\pi$
$314$	3997.96	0.718528
$315$	0	0
$316$	545.010	0.0970229
$317$	6697.35	1.18663	0.593314	−	0.804971i	$-0.297819\pi$
$317$	6697.35	1.18663	0.593314	+	0.804971i	$0.297819\pi$
$318$	−212.524	−0.0374772
$319$	−533.287	−0.0935997
$320$	0	0
$321$	306.475	0.0532889
$322$	−7938.18	−1.37384
$323$	3756.54	0.647120
$324$	2896.25	0.496614
$325$	0	0
$326$	982.496	0.166918
$327$	−43.6420	−0.00738046
$328$	2124.66	0.357668
$329$	−8195.40	−1.37333
$330$	0	0
$331$	−421.785	−0.0700405	−0.0350203	−	0.999387i	$-0.511150\pi$
$331$	−421.785	−0.0700405	−0.0350203	+	0.999387i	$0.511150\pi$
$332$	3670.13	0.606701
$333$	−7492.19	−1.23294
$334$	680.906	0.111550
$335$	0	0
$336$	78.6856	0.0127758
$337$	−4566.12	−0.738078	−0.369039	−	0.929414i	$-0.620313\pi$
$337$	−4566.12	−0.738078	−0.369039	+	0.929414i	$0.620313\pi$
$338$	−4091.19	−0.658378
$339$	−47.3530	−0.00758661
$340$	0	0
$341$	113.354	0.0180014
$342$	2719.01	0.429904
$343$	5763.87	0.907346
$344$	−829.710	−0.130043
$345$	0	0
$346$	4356.32	0.676871
$347$	6141.84	0.950177	0.475089	−	0.879938i	$-0.342416\pi$
$347$	6141.84	0.950177	0.475089	+	0.879938i	$0.342416\pi$
$348$	−238.305	−0.0367083
$349$	−7456.55	−1.14367	−0.571834	−	0.820369i	$-0.693768\pi$
$349$	−7456.55	−1.14367	−0.571834	+	0.820369i	$0.693768\pi$
$350$	0	0
$351$	−163.905	−0.0249248
$352$	−70.7391	−0.0107114
$353$	6605.02	0.995891	0.497946	−	0.867208i	$-0.334088\pi$
$353$	6605.02	0.995891	0.497946	+	0.867208i	$0.334088\pi$
$354$	−272.136	−0.0408583
$355$	0	0
$356$	5210.98	0.775791
$357$	−366.071	−0.0542704
$358$	−6880.12	−1.01571
$359$	1064.29	0.156466	0.0782329	−	0.996935i	$-0.475072\pi$
$359$	1064.29	0.156466	0.0782329	+	0.996935i	$0.475072\pi$
$360$	0	0
$361$	−4312.18	−0.628689
$362$	−4044.43	−0.587211
$363$	327.493	0.0473524
$364$	980.123	0.141133
$365$	0	0
$366$	252.348	0.0360395
$367$	−1613.78	−0.229533	−0.114766	−	0.993393i	$-0.536612\pi$
$367$	−1613.78	−0.229533	−0.114766	+	0.993393i	$0.536612\pi$
$368$	3189.02	0.451737
$369$	−7154.55	−1.00935
$370$	0	0
$371$	−8568.61	−1.19908
$372$	50.6535	0.00705984
$373$	−7465.11	−1.03627	−0.518135	−	0.855299i	$-0.673374\pi$
$373$	−7465.11	−1.03627	−0.518135	+	0.855299i	$0.673374\pi$
$374$	329.101	0.0455011
$375$	0	0
$376$	3292.35	0.451569
$377$	−2968.37	−0.405514
$378$	−530.528	−0.0721890
$379$	−7494.14	−1.01569	−0.507847	−	0.861447i	$-0.669558\pi$
$379$	−7494.14	−1.01569	−0.507847	+	0.861447i	$0.669558\pi$
$380$	0	0
$381$	−75.9013	−0.0102061
$382$	9653.39	1.29296
$383$	−4391.97	−0.585951	−0.292975	−	0.956120i	$-0.594645\pi$
$383$	−4391.97	−0.585951	−0.292975	+	0.956120i	$0.594645\pi$
$384$	−31.6105	−0.00420083
$385$	0	0
$386$	6150.34	0.810995
$387$	2793.95	0.366988
$388$	−1775.81	−0.232354
$389$	5764.74	0.751373	0.375687	−	0.926747i	$-0.377407\pi$
$389$	5764.74	0.751373	0.375687	+	0.926747i	$0.377407\pi$
$390$	0	0
$391$	−14836.3	−1.91894
$392$	428.471	0.0552068
$393$	−693.874	−0.0890618
$394$	8469.20	1.08292
$395$	0	0
$396$	238.205	0.0302279
$397$	1597.55	0.201961	0.100981	−	0.994888i	$-0.467802\pi$
$397$	1597.55	0.201961	0.100981	+	0.994888i	$0.467802\pi$
$398$	8252.33	1.03933
$399$	−248.184	−0.0311398
$400$	0	0
$401$	−4722.70	−0.588130	−0.294065	−	0.955785i	$-0.595008\pi$
$401$	−4722.70	−0.588130	−0.294065	+	0.955785i	$0.595008\pi$
$402$	37.6620	0.00467266
$403$	630.949	0.0779896
$404$	−1973.44	−0.243025
$405$	0	0
$406$	−9608.04	−1.17448
$407$	−614.804	−0.0748764
$408$	147.062	0.0178448
$409$	5829.04	0.704712	0.352356	−	0.935866i	$-0.385381\pi$
$409$	5829.04	0.704712	0.352356	+	0.935866i	$0.385381\pi$
$410$	0	0
$411$	484.250	0.0581175
$412$	6885.11	0.823313
$413$	−10972.0	−1.30726
$414$	−10738.6	−1.27482
$415$	0	0
$416$	−393.747	−0.0464063
$417$	389.835	0.0457801
$418$	223.120	0.0261080
$419$	1138.39	0.132731	0.0663653	−	0.997795i	$-0.478860\pi$
$419$	1138.39	0.132731	0.0663653	+	0.997795i	$0.478860\pi$
$420$	0	0
$421$	−1026.17	−0.118794	−0.0593970	−	0.998234i	$-0.518918\pi$
$421$	−1026.17	−0.118794	−0.0593970	+	0.998234i	$0.518918\pi$
$422$	−10159.7	−1.17196
$423$	−11086.6	−1.27435
$424$	3442.28	0.394274
$425$	0	0
$426$	422.615	0.0480652
$427$	10174.2	1.15308
$428$	−4964.01	−0.560618
$429$	−6.71736	−0.000755984 0
$430$	0	0
$431$	−5596.70	−0.625483	−0.312742	−	0.949838i	$-0.601247\pi$
$431$	−5596.70	−0.625483	−0.312742	+	0.949838i	$0.601247\pi$
$432$	213.130	0.0237366
$433$	878.643	0.0975171	0.0487585	−	0.998811i	$-0.484474\pi$
$433$	878.643	0.0975171	0.0487585	+	0.998811i	$0.484474\pi$
$434$	2042.26	0.225879
$435$	0	0
$436$	706.876	0.0776450
$437$	−10058.6	−1.10107
$438$	255.165	0.0278362
$439$	−7004.85	−0.761556	−0.380778	−	0.924666i	$-0.624344\pi$
$439$	−7004.85	−0.761556	−0.380778	+	0.924666i	$0.624344\pi$
$440$	0	0
$441$	−1442.82	−0.155796
$442$	1831.84	0.197130
$443$	−17736.5	−1.90223	−0.951114	−	0.308839i	$-0.900060\pi$
$443$	−17736.5	−1.90223	−0.951114	+	0.308839i	$0.900060\pi$
$444$	−274.732	−0.0293653
$445$	0	0
$446$	9870.04	1.04789
$447$	357.730	0.0378525
$448$	−1274.48	−0.134405
$449$	4292.26	0.451146	0.225573	−	0.974226i	$-0.427575\pi$
$449$	4292.26	0.451146	0.225573	+	0.974226i	$0.427575\pi$
$450$	0	0
$451$	−587.097	−0.0612979
$452$	766.983	0.0798138
$453$	445.749	0.0462320
$454$	−1396.40	−0.144353
$455$	0	0
$456$	99.7036	0.0102391
$457$	−17409.6	−1.78203	−0.891015	−	0.453973i	$-0.850006\pi$
$457$	−17409.6	−1.78203	−0.891015	+	0.453973i	$0.850006\pi$
$458$	12405.0	1.26561
$459$	−991.549	−0.100831
$460$	0	0
$461$	14638.1	1.47889	0.739443	−	0.673219i	$-0.235089\pi$
$461$	14638.1	1.47889	0.739443	+	0.673219i	$0.235089\pi$
$462$	−21.7428	−0.00218954
$463$	7543.68	0.757202	0.378601	−	0.925560i	$-0.376405\pi$
$463$	7543.68	0.757202	0.378601	+	0.925560i	$0.376405\pi$
$464$	3859.85	0.386184
$465$	0	0
$466$	5465.94	0.543358
$467$	−3528.60	−0.349644	−0.174822	−	0.984600i	$-0.555935\pi$
$467$	−3528.60	−0.349644	−0.174822	+	0.984600i	$0.555935\pi$
$468$	1325.89	0.130960
$469$	1518.47	0.149502
$470$	0	0
$471$	−493.662	−0.0482946
$472$	4407.82	0.429844
$473$	229.269	0.0222871
$474$	−67.2971	−0.00652122
$475$	0	0
$476$	5929.30	0.570943
$477$	−11591.5	−1.11266
$478$	−2960.36	−0.283271
$479$	7429.14	0.708655	0.354328	−	0.935121i	$-0.384710\pi$
$479$	7429.14	0.708655	0.354328	+	0.935121i	$0.384710\pi$
$480$	0	0
$481$	−3422.11	−0.324397
$482$	517.608	0.0489137
$483$	980.195	0.0923404
$484$	−5304.45	−0.498164
$485$	0	0
$486$	−1076.94	−0.100516
$487$	14017.6	1.30431	0.652155	−	0.758086i	$-0.273865\pi$
$487$	14017.6	1.30431	0.652155	+	0.758086i	$0.273865\pi$
$488$	−4087.32	−0.379148
$489$	−121.317	−0.0112191
$490$	0	0
$491$	−3602.21	−0.331091	−0.165545	−	0.986202i	$-0.552938\pi$
$491$	−3602.21	−0.331091	−0.165545	+	0.986202i	$0.552938\pi$
$492$	−262.351	−0.0240400
$493$	−17957.3	−1.64048
$494$	1241.93	0.113111
$495$	0	0
$496$	−820.441	−0.0742720
$497$	17039.1	1.53785
$498$	−453.183	−0.0407783
$499$	4905.02	0.440037	0.220019	−	0.975496i	$-0.429388\pi$
$499$	4905.02	0.440037	0.220019	+	0.975496i	$0.429388\pi$
$500$	0	0
$501$	−84.0774	−0.00749760
$502$	−6864.70	−0.610332
$503$	−16243.4	−1.43988	−0.719938	−	0.694039i	$-0.755830\pi$
$503$	−16243.4	−1.43988	−0.719938	+	0.694039i	$0.755830\pi$
$504$	4291.66	0.379297
$505$	0	0
$506$	−881.205	−0.0774196
$507$	505.175	0.0442517
$508$	1229.38	0.107372
$509$	14342.3	1.24894	0.624472	−	0.781047i	$-0.285314\pi$
$509$	14342.3	1.24894	0.624472	+	0.781047i	$0.285314\pi$
$510$	0	0
$511$	10287.8	0.890618
$512$	512.000	0.0441942
$513$	−672.239	−0.0578559
$514$	6213.82	0.533229
$515$	0	0
$516$	102.451	0.00874064
$517$	−909.758	−0.0773910
$518$	−11076.7	−0.939542
$519$	−537.912	−0.0454947
$520$	0	0
$521$	11086.0	0.932217	0.466109	−	0.884727i	$-0.345656\pi$
$521$	11086.0	0.932217	0.466109	+	0.884727i	$0.345656\pi$
$522$	−12997.6	−1.08983
$523$	−16.4805	−0.00137790	−0.000688950	−	1.00000i	$-0.500219\pi$
$523$	−16.4805	−0.00137790	−0.000688950		1.00000i	$0.500219\pi$
$524$	11238.8	0.936962
$525$	0	0
$526$	9913.43	0.821761
$527$	3816.95	0.315501
$528$	8.73477	0.000719947 0
$529$	27558.9	2.26506
$530$	0	0
$531$	−14842.8	−1.21304
$532$	4019.88	0.327601
$533$	−3267.89	−0.265569
$534$	−643.445	−0.0521434
$535$	0	0
$536$	−610.017	−0.0491580
$537$	849.548	0.0682695
$538$	4793.65	0.384143
$539$	−118.397	−0.00946146
$540$	0	0
$541$	−24611.4	−1.95587	−0.977937	−	0.208902i	$-0.933011\pi$
$541$	−24611.4	−1.95587	−0.977937	+	0.208902i	$0.933011\pi$
$542$	−12981.2	−1.02876
$543$	499.400	0.0394683
$544$	−2381.99	−0.187733
$545$	0	0
$546$	−121.024	−0.00948601
$547$	−5316.74	−0.415590	−0.207795	−	0.978172i	$-0.566629\pi$
$547$	−5316.74	−0.415590	−0.207795	+	0.978172i	$0.566629\pi$
$548$	−7843.47	−0.611417
$549$	13763.5	1.06997
$550$	0	0
$551$	−12174.5	−0.941288
$552$	−393.776	−0.0303627
$553$	−2713.31	−0.208646
$554$	7607.77	0.583435
$555$	0	0
$556$	−6314.21	−0.481622
$557$	−9362.34	−0.712199	−0.356100	−	0.934448i	$-0.615894\pi$
$557$	−9362.34	−0.712199	−0.356100	+	0.934448i	$0.615894\pi$
$558$	2762.73	0.209598
$559$	1276.15	0.0965574
$560$	0	0
$561$	−40.6369	−0.00305828
$562$	511.098	0.0383619
$563$	−12142.8	−0.908985	−0.454492	−	0.890751i	$-0.650179\pi$
$563$	−12142.8	−0.908985	−0.454492	+	0.890751i	$0.650179\pi$
$564$	−406.535	−0.0303514
$565$	0	0
$566$	4100.00	0.304480
$567$	−14418.9	−1.06796
$568$	−6845.16	−0.505663
$569$	9331.60	0.687524	0.343762	−	0.939057i	$-0.388299\pi$
$569$	9331.60	0.687524	0.343762	+	0.939057i	$0.388299\pi$
$570$	0	0
$571$	18369.2	1.34628	0.673141	−	0.739514i	$-0.264945\pi$
$571$	18369.2	1.34628	0.673141	+	0.739514i	$0.264945\pi$
$572$	108.802	0.00795321
$573$	−1191.99	−0.0869039
$574$	−10577.5	−0.769160
$575$	0	0
$576$	−1724.10	−0.124718
$577$	288.024	0.0207809	0.0103905	−	0.999946i	$-0.496693\pi$
$577$	288.024	0.0207809	0.0103905	+	0.999946i	$0.496693\pi$
$578$	1255.77	0.0903684
$579$	−759.435	−0.0545096
$580$	0	0
$581$	−18271.6	−1.30470
$582$	219.275	0.0156173
$583$	−951.188	−0.0675715
$584$	−4132.94	−0.292846
$585$	0	0
$586$	13486.0	0.950688
$587$	−15152.9	−1.06546	−0.532732	−	0.846284i	$-0.678835\pi$
$587$	−15152.9	−1.06546	−0.532732	+	0.846284i	$0.678835\pi$
$588$	−52.9070	−0.00371063
$589$	2587.77	0.181031
$590$	0	0
$591$	−1045.76	−0.0727868
$592$	4449.87	0.308933
$593$	−8594.02	−0.595133	−0.297567	−	0.954701i	$-0.596175\pi$
$593$	−8594.02	−0.595133	−0.297567	+	0.954701i	$0.596175\pi$
$594$	−58.8931	−0.00406804
$595$	0	0
$596$	−5794.21	−0.398221
$597$	−1018.99	−0.0698565
$598$	−4904.94	−0.335415
$599$	−12431.0	−0.847942	−0.423971	−	0.905676i	$-0.639364\pi$
$599$	−12431.0	−0.847942	−0.423971	+	0.905676i	$0.639364\pi$
$600$	0	0
$601$	16440.3	1.11583	0.557913	−	0.829899i	$-0.311602\pi$
$601$	16440.3	1.11583	0.557913	+	0.829899i	$0.311602\pi$
$602$	4130.67	0.279657
$603$	2054.16	0.138726
$604$	−7219.85	−0.486377
$605$	0	0
$606$	243.678	0.0163345
$607$	10367.1	0.693228	0.346614	−	0.938008i	$-0.387331\pi$
$607$	10367.1	0.693228	0.346614	+	0.938008i	$0.387331\pi$
$608$	−1614.91	−0.107719
$609$	1186.39	0.0789406
$610$	0	0
$611$	−5063.88	−0.335291
$612$	8021.05	0.529790
$613$	−14487.9	−0.954583	−0.477292	−	0.878745i	$-0.658382\pi$
$613$	−14487.9	−0.954583	−0.477292	+	0.878745i	$0.658382\pi$
$614$	−9999.42	−0.657237
$615$	0	0
$616$	352.171	0.0230347
$617$	−2580.56	−0.168378	−0.0841890	−	0.996450i	$-0.526830\pi$
$617$	−2580.56	−0.168378	−0.0841890	+	0.996450i	$0.526830\pi$
$618$	−850.163	−0.0553375
$619$	5875.93	0.381541	0.190770	−	0.981635i	$-0.438901\pi$
$619$	5875.93	0.381541	0.190770	+	0.981635i	$0.438901\pi$
$620$	0	0
$621$	2654.98	0.171563
$622$	13738.8	0.885651
$623$	−25942.6	−1.66833
$624$	48.6193	0.00311912
$625$	0	0
$626$	−897.886	−0.0573270
$627$	−27.5506	−0.00175481
$628$	7995.92	0.508076
$629$	−20702.2	−1.31232
$630$	0	0
$631$	−4016.85	−0.253420	−0.126710	−	0.991940i	$-0.540442\pi$
$631$	−4016.85	−0.253420	−0.126710	+	0.991940i	$0.540442\pi$
$632$	1090.02	0.0686055
$633$	1254.50	0.0787709
$634$	13394.7	0.839072
$635$	0	0
$636$	−425.048	−0.0265004
$637$	−659.020	−0.0409911
$638$	−1066.57	−0.0661850
$639$	23050.2	1.42700
$640$	0	0
$641$	4056.35	0.249947	0.124974	−	0.992160i	$-0.460115\pi$
$641$	4056.35	0.249947	0.124974	+	0.992160i	$0.460115\pi$
$642$	612.949	0.0376810
$643$	−3520.39	−0.215911	−0.107955	−	0.994156i	$-0.534430\pi$
$643$	−3520.39	−0.215911	−0.107955	+	0.994156i	$0.534430\pi$
$644$	−15876.4	−0.971454
$645$	0	0
$646$	7513.09	0.457583
$647$	21301.9	1.29438	0.647192	−	0.762327i	$-0.275943\pi$
$647$	21301.9	1.29438	0.647192	+	0.762327i	$0.275943\pi$
$648$	5792.51	0.351159
$649$	−1217.99	−0.0736676
$650$	0	0
$651$	−252.176	−0.0151821
$652$	1964.99	0.118029
$653$	8898.59	0.533275	0.266637	−	0.963797i	$-0.414087\pi$
$653$	8898.59	0.533275	0.266637	+	0.963797i	$0.414087\pi$
$654$	−87.2840	−0.00521877
$655$	0	0
$656$	4249.33	0.252909
$657$	13917.2	0.826424
$658$	−16390.8	−0.971094
$659$	−22224.2	−1.31371	−0.656853	−	0.754019i	$-0.728113\pi$
$659$	−22224.2	−1.31371	−0.656853	+	0.754019i	$0.728113\pi$
$660$	0	0
$661$	28002.3	1.64775	0.823874	−	0.566773i	$-0.191808\pi$
$661$	28002.3	1.64775	0.823874	+	0.566773i	$0.191808\pi$
$662$	−843.570	−0.0495261
$663$	−226.192	−0.0132498
$664$	7340.27	0.429002
$665$	0	0
$666$	−14984.4	−0.871821
$667$	48082.6	2.79125
$668$	1361.81	0.0788774
$669$	−1218.74	−0.0704322
$670$	0	0
$671$	1129.43	0.0649792
$672$	157.371	0.00903382
$673$	21072.1	1.20694	0.603471	−	0.797385i	$-0.293784\pi$
$673$	21072.1	1.20694	0.603471	+	0.797385i	$0.293784\pi$
$674$	−9132.24	−0.521900
$675$	0	0
$676$	−8182.39	−0.465543
$677$	12438.6	0.706137	0.353068	−	0.935597i	$-0.385138\pi$
$677$	12438.6	0.706137	0.353068	+	0.935597i	$0.385138\pi$
$678$	−94.7060	−0.00536455
$679$	8840.80	0.499674
$680$	0	0
$681$	172.425	0.00970243
$682$	226.708	0.0127289
$683$	−5849.01	−0.327681	−0.163840	−	0.986487i	$-0.552388\pi$
$683$	−5849.01	−0.327681	−0.163840	+	0.986487i	$0.552388\pi$
$684$	5438.02	0.303988
$685$	0	0
$686$	11527.7	0.641590
$687$	−1531.76	−0.0850658
$688$	−1659.42	−0.0919546
$689$	−5294.48	−0.292749
$690$	0	0
$691$	−7322.31	−0.403117	−0.201558	−	0.979477i	$-0.564601\pi$
$691$	−7322.31	−0.403117	−0.201558	+	0.979477i	$0.564601\pi$
$692$	8712.64	0.478620
$693$	−1185.89	−0.0650048
$694$	12283.7	0.671877
$695$	0	0
$696$	−476.609	−0.0259567
$697$	−19769.2	−1.07434
$698$	−14913.1	−0.808695
$699$	−674.927	−0.0365209
$700$	0	0
$701$	15238.8	0.821060	0.410530	−	0.911847i	$-0.365344\pi$
$701$	15238.8	0.821060	0.410530	+	0.911847i	$0.365344\pi$
$702$	−327.810	−0.0176245
$703$	−14035.4	−0.752997
$704$	−141.478	−0.00757409
$705$	0	0
$706$	13210.0	0.704201
$707$	9824.66	0.522623
$708$	−544.271	−0.0288912
$709$	−11126.4	−0.589366	−0.294683	−	0.955595i	$-0.595214\pi$
$709$	−11126.4	−0.589366	−0.294683	+	0.955595i	$0.595214\pi$
$710$	0	0
$711$	−3670.51	−0.193607
$712$	10422.0	0.548567
$713$	−10220.3	−0.536822
$714$	−732.141	−0.0383749
$715$	0	0
$716$	−13760.2	−0.718219
$717$	365.541	0.0190396
$718$	2128.59	0.110638
$719$	−26127.0	−1.35518	−0.677588	−	0.735442i	$-0.736974\pi$
$719$	−26127.0	−1.35518	−0.677588	+	0.735442i	$0.736974\pi$
$720$	0	0
$721$	−34277.1	−1.77052
$722$	−8624.36	−0.444551
$723$	−63.9135	−0.00328765
$724$	−8088.85	−0.415221
$725$	0	0
$726$	654.986	0.0334832
$727$	−3909.48	−0.199443	−0.0997213	−	0.995015i	$-0.531795\pi$
$727$	−3909.48	−0.199443	−0.0997213	+	0.995015i	$0.531795\pi$
$728$	1960.25	0.0997961
$729$	−19416.7	−0.986473
$730$	0	0
$731$	7720.15	0.390616
$732$	504.696	0.0254837
$733$	−4662.67	−0.234952	−0.117476	−	0.993076i	$-0.537480\pi$
$733$	−4662.67	−0.234952	−0.117476	+	0.993076i	$0.537480\pi$
$734$	−3227.55	−0.162304
$735$	0	0
$736$	6378.04	0.319426
$737$	168.563	0.00842481
$738$	−14309.1	−0.713720
$739$	−17612.3	−0.876695	−0.438348	−	0.898806i	$-0.644436\pi$
$739$	−17612.3	−0.876695	−0.438348	+	0.898806i	$0.644436\pi$
$740$	0	0
$741$	−153.351	−0.00760257
$742$	−17137.2	−0.847880
$743$	26680.2	1.31737	0.658683	−	0.752421i	$-0.271114\pi$
$743$	26680.2	1.31737	0.658683	+	0.752421i	$0.271114\pi$
$744$	101.307	0.00499206
$745$	0	0
$746$	−14930.2	−0.732754
$747$	−24717.4	−1.21066
$748$	658.202	0.0321741
$749$	24713.1	1.20560
$750$	0	0
$751$	8017.53	0.389566	0.194783	−	0.980846i	$-0.437600\pi$
$751$	8017.53	0.389566	0.194783	+	0.980846i	$0.437600\pi$
$752$	6584.70	0.319308
$753$	847.643	0.0410224
$754$	−5936.74	−0.286742
$755$	0	0
$756$	−1061.06	−0.0510453
$757$	−4716.28	−0.226441	−0.113221	−	0.993570i	$-0.536117\pi$
$757$	−4716.28	−0.226441	−0.113221	+	0.993570i	$0.536117\pi$
$758$	−14988.3	−0.718204
$759$	108.810	0.00520362
$760$	0	0
$761$	−5308.13	−0.252851	−0.126425	−	0.991976i	$-0.540350\pi$
$761$	−5308.13	−0.252851	−0.126425	+	0.991976i	$0.540350\pi$
$762$	−151.803	−0.00721683
$763$	−3519.14	−0.166975
$764$	19306.8	0.914260
$765$	0	0
$766$	−8783.94	−0.414330
$767$	−6779.55	−0.319160
$768$	−63.2210	−0.00297043
$769$	41443.4	1.94342	0.971708	−	0.236187i	$-0.0758979\pi$
$769$	41443.4	1.94342	0.971708	+	0.236187i	$0.0758979\pi$
$770$	0	0
$771$	−767.273	−0.0358400
$772$	12300.7	0.573460
$773$	8343.51	0.388222	0.194111	−	0.980980i	$-0.437818\pi$
$773$	8343.51	0.388222	0.194111	+	0.980980i	$0.437818\pi$
$774$	5587.89	0.259499
$775$	0	0
$776$	−3551.63	−0.164299
$777$	1367.74	0.0631497
$778$	11529.5	0.531301
$779$	−13402.9	−0.616444
$780$	0	0
$781$	1891.49	0.0866616
$782$	−29672.7	−1.35690
$783$	3213.48	0.146667
$784$	856.943	0.0390371
$785$	0	0
$786$	−1387.75	−0.0629762
$787$	−8095.65	−0.366682	−0.183341	−	0.983049i	$-0.558691\pi$
$787$	−8095.65	−0.366682	−0.183341	+	0.983049i	$0.558691\pi$
$788$	16938.4	0.765743
$789$	−1224.10	−0.0552332
$790$	0	0
$791$	−3818.38	−0.171639
$792$	476.411	0.0213744
$793$	6286.59	0.281517
$794$	3195.09	0.142808
$795$	0	0
$796$	16504.7	0.734915
$797$	−20038.9	−0.890606	−0.445303	−	0.895380i	$-0.646904\pi$
$797$	−20038.9	−0.890606	−0.445303	+	0.895380i	$0.646904\pi$
$798$	−496.369	−0.0220191
$799$	−30634.1	−1.35639
$800$	0	0
$801$	−35094.7	−1.54808
$802$	−9445.39	−0.415871
$803$	1142.03	0.0501887
$804$	75.3240	0.00330407
$805$	0	0
$806$	1261.90	0.0551470
$807$	−591.913	−0.0258195
$808$	−3946.88	−0.171845
$809$	4433.03	0.192654	0.0963270	−	0.995350i	$-0.469291\pi$
$809$	4433.03	0.192654	0.0963270	+	0.995350i	$0.469291\pi$
$810$	0	0
$811$	−5413.40	−0.234390	−0.117195	−	0.993109i	$-0.537390\pi$
$811$	−5413.40	−0.234390	−0.117195	+	0.993109i	$0.537390\pi$
$812$	−19216.1	−0.830483
$813$	1602.90	0.0691465
$814$	−1229.61	−0.0529456
$815$	0	0
$816$	294.124	0.0126182
$817$	5234.02	0.224131
$818$	11658.1	0.498307
$819$	−6600.89	−0.281629
$820$	0	0
$821$	21736.8	0.924017	0.462009	−	0.886875i	$-0.347129\pi$
$821$	21736.8	0.924017	0.462009	+	0.886875i	$0.347129\pi$
$822$	968.501	0.0410953
$823$	−25738.9	−1.09016	−0.545080	−	0.838384i	$-0.683501\pi$
$823$	−25738.9	−1.09016	−0.545080	+	0.838384i	$0.683501\pi$
$824$	13770.2	0.582170
$825$	0	0
$826$	−21944.1	−0.924374
$827$	−3145.96	−0.132280	−0.0661401	−	0.997810i	$-0.521068\pi$
$827$	−3145.96	−0.132280	−0.0661401	+	0.997810i	$0.521068\pi$
$828$	−21477.3	−0.901433
$829$	12097.6	0.506837	0.253419	−	0.967357i	$-0.418445\pi$
$829$	12097.6	0.506837	0.253419	+	0.967357i	$0.418445\pi$
$830$	0	0
$831$	−939.396	−0.0392146
$832$	−787.493	−0.0328142
$833$	−3986.77	−0.165826
$834$	779.669	0.0323714
$835$	0	0
$836$	446.240	0.0184612
$837$	−683.049	−0.0282075
$838$	2276.78	0.0938547
$839$	−1481.68	−0.0609693	−0.0304846	−	0.999535i	$-0.509705\pi$
$839$	−1481.68	−0.0609693	−0.0304846	+	0.999535i	$0.509705\pi$
$840$	0	0
$841$	33808.2	1.38621
$842$	−2052.33	−0.0840000
$843$	−63.1097	−0.00257842
$844$	−20319.3	−0.828697
$845$	0	0
$846$	−22173.2	−0.901099
$847$	26407.9	1.07130
$848$	6884.57	0.278794
$849$	−506.262	−0.0204651
$850$	0	0
$851$	55432.5	2.23290
$852$	845.230	0.0339872
$853$	−17049.6	−0.684368	−0.342184	−	0.939633i	$-0.611167\pi$
$853$	−17049.6	−0.684368	−0.342184	+	0.939633i	$0.611167\pi$
$854$	20348.5	0.815352
$855$	0	0
$856$	−9928.03	−0.396417
$857$	−5794.04	−0.230946	−0.115473	−	0.993311i	$-0.536838\pi$
$857$	−5794.04	−0.230946	−0.115473	+	0.993311i	$0.536838\pi$
$858$	−13.4347	−0.000534561 0
$859$	−19989.5	−0.793983	−0.396992	−	0.917822i	$-0.629946\pi$
$859$	−19989.5	−0.793983	−0.396992	+	0.917822i	$0.629946\pi$
$860$	0	0
$861$	1306.10	0.0516977
$862$	−11193.4	−0.442284
$863$	48891.0	1.92847	0.964234	−	0.265052i	$-0.0853890\pi$
$863$	48891.0	1.92847	0.964234	+	0.265052i	$0.0853890\pi$
$864$	426.260	0.0167843
$865$	0	0
$866$	1757.29	0.0689550
$867$	−155.060	−0.00607395
$868$	4084.52	0.159721
$869$	−301.200	−0.0117578
$870$	0	0
$871$	938.250	0.0364999
$872$	1413.75	0.0549033
$873$	11959.7	0.463658
$874$	−20117.1	−0.778572
$875$	0	0
$876$	510.330	0.0196832
$877$	−6336.93	−0.243994	−0.121997	−	0.992530i	$-0.538930\pi$
$877$	−6336.93	−0.243994	−0.121997	+	0.992530i	$0.538930\pi$
$878$	−14009.7	−0.538502
$879$	−1665.24	−0.0638988
$880$	0	0
$881$	19575.1	0.748583	0.374292	−	0.927311i	$-0.377886\pi$
$881$	19575.1	0.748583	0.374292	+	0.927311i	$0.377886\pi$
$882$	−2885.65	−0.110164
$883$	46818.0	1.78432	0.892158	−	0.451724i	$-0.149191\pi$
$883$	46818.0	1.78432	0.892158	+	0.451724i	$0.149191\pi$
$884$	3663.67	0.139392
$885$	0	0
$886$	−35473.0	−1.34508
$887$	3230.51	0.122288	0.0611442	−	0.998129i	$-0.480525\pi$
$887$	3230.51	0.122288	0.0611442	+	0.998129i	$0.480525\pi$
$888$	−549.463	−0.0207644
$889$	−6120.42	−0.230903
$890$	0	0
$891$	−1600.61	−0.0601825
$892$	19740.1	0.740972
$893$	−20769.0	−0.778284
$894$	715.461	0.0267658
$895$	0	0
$896$	−2548.96	−0.0950390
$897$	605.656	0.0225443
$898$	8584.52	0.319008
$899$	−12370.2	−0.458922
$900$	0	0
$901$	−32029.2	−1.18429
$902$	−1174.19	−0.0433441
$903$	−510.049	−0.0187966
$904$	1533.97	0.0564369
$905$	0	0
$906$	891.497	0.0326910
$907$	24727.7	0.905258	0.452629	−	0.891699i	$-0.350486\pi$
$907$	24727.7	0.905258	0.452629	+	0.891699i	$0.350486\pi$
$908$	−2792.80	−0.102073
$909$	13290.6	0.484953
$910$	0	0
$911$	−12612.5	−0.458695	−0.229347	−	0.973345i	$-0.573659\pi$
$911$	−12612.5	−0.458695	−0.229347	+	0.973345i	$0.573659\pi$
$912$	199.407	0.00724016
$913$	−2028.30	−0.0735234
$914$	−34819.3	−1.26009
$915$	0	0
$916$	24810.1	0.894922
$917$	−55951.6	−2.01492
$918$	−1983.10	−0.0712985
$919$	−29945.0	−1.07486	−0.537429	−	0.843309i	$-0.680604\pi$
$919$	−29945.0	−1.07486	−0.537429	+	0.843309i	$0.680604\pi$
$920$	0	0
$921$	1234.71	0.0441750
$922$	29276.3	1.04573
$923$	10528.4	0.375455
$924$	−43.4856	−0.00154824
$925$	0	0
$926$	15087.4	0.535423
$927$	−46369.5	−1.64291
$928$	7719.71	0.273073
$929$	10963.4	0.387187	0.193594	−	0.981082i	$-0.437986\pi$
$929$	10963.4	0.387187	0.193594	+	0.981082i	$0.437986\pi$
$930$	0	0
$931$	−2702.90	−0.0951494
$932$	10931.9	0.384212
$933$	−1696.45	−0.0595275
$934$	−7057.19	−0.247236
$935$	0	0
$936$	2651.79	0.0926029
$937$	48684.1	1.69737	0.848687	−	0.528895i	$-0.177394\pi$
$937$	48684.1	1.69737	0.848687	+	0.528895i	$0.177394\pi$
$938$	3036.94	0.105714
$939$	110.870	0.00385314
$940$	0	0
$941$	12525.2	0.433910	0.216955	−	0.976182i	$-0.430387\pi$
$941$	12525.2	0.433910	0.216955	+	0.976182i	$0.430387\pi$
$942$	−987.325	−0.0341494
$943$	52934.3	1.82797
$944$	8815.64	0.303946
$945$	0	0
$946$	458.539	0.0157594
$947$	−33428.9	−1.14709	−0.573545	−	0.819174i	$-0.694432\pi$
$947$	−33428.9	−1.14709	−0.573545	+	0.819174i	$0.694432\pi$
$948$	−134.594	−0.00461120
$949$	6356.76	0.217439
$950$	0	0
$951$	−1653.96	−0.0563968
$952$	11858.6	0.403718
$953$	35396.9	1.20317	0.601583	−	0.798811i	$-0.294537\pi$
$953$	35396.9	1.20317	0.601583	+	0.798811i	$0.294537\pi$
$954$	−23182.9	−0.786766
$955$	0	0
$956$	−5920.72	−0.200303
$957$	131.699	0.00444851
$958$	14858.3	0.501095
$959$	39048.3	1.31484
$960$	0	0
$961$	−27161.6	−0.911739
$962$	−6844.22	−0.229383
$963$	33431.4	1.11870
$964$	1035.22	0.0345872
$965$	0	0
$966$	1960.39	0.0652946
$967$	−22468.9	−0.747208	−0.373604	−	0.927588i	$-0.621878\pi$
$967$	−22468.9	−0.747208	−0.373604	+	0.927588i	$0.621878\pi$
$968$	−10608.9	−0.352255
$969$	−927.706	−0.0307556
$970$	0	0
$971$	57130.6	1.88817	0.944083	−	0.329709i	$-0.106951\pi$
$971$	57130.6	1.88817	0.944083	+	0.329709i	$0.106951\pi$
$972$	−2153.88	−0.0710758
$973$	31434.9	1.03572
$974$	28035.2	0.922287
$975$	0	0
$976$	−8174.63	−0.268098
$977$	32672.6	1.06990	0.534949	−	0.844884i	$-0.320331\pi$
$977$	32672.6	1.06990	0.534949	+	0.844884i	$0.320331\pi$
$978$	−242.634	−0.00793312
$979$	−2879.85	−0.0940146
$980$	0	0
$981$	−4760.63	−0.154939
$982$	−7204.42	−0.234116
$983$	−51049.8	−1.65640	−0.828198	−	0.560436i	$-0.810634\pi$
$983$	−51049.8	−1.65640	−0.828198	+	0.560436i	$0.810634\pi$
$984$	−524.701	−0.0169988
$985$	0	0
$986$	−35914.5	−1.15999
$987$	2023.91	0.0652704
$988$	2483.85	0.0799817
$989$	−20671.6	−0.664628
$990$	0	0
$991$	55604.9	1.78239	0.891194	−	0.453622i	$-0.149868\pi$
$991$	55604.9	1.78239	0.891194	+	0.453622i	$0.149868\pi$
$992$	−1640.88	−0.0525182
$993$	104.163	0.00332881
$994$	34078.2	1.08742
$995$	0	0
$996$	−906.366	−0.0288346
$997$	10014.6	0.318121	0.159060	−	0.987269i	$-0.449154\pi$
$997$	10014.6	0.318121	0.159060	+	0.987269i	$0.449154\pi$
$998$	9810.03	0.311153
$999$	3704.69	0.117329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1250.4.a.n.1.8		16
5.4	even	2		1250.4.a.m.1.9		16
25.3	odd	20		50.4.e.a.9.3	✓	32
25.4	even	10		250.4.d.d.201.5		32
25.6	even	5		250.4.d.c.51.4		32
25.8	odd	20		250.4.e.b.199.6		32
25.17	odd	20		50.4.e.a.39.3	yes	32
25.19	even	10		250.4.d.d.51.5		32
25.21	even	5		250.4.d.c.201.4		32
25.22	odd	20		250.4.e.b.49.6		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.e.a.9.3	✓	32	25.3	odd	20
50.4.e.a.39.3	yes	32	25.17	odd	20
250.4.d.c.51.4		32	25.6	even	5
250.4.d.c.201.4		32	25.21	even	5
250.4.d.d.51.5		32	25.19	even	10
250.4.d.d.201.5		32	25.4	even	10
250.4.e.b.49.6		32	25.22	odd	20
250.4.e.b.199.6		32	25.8	odd	20
1250.4.a.m.1.9		16	5.4	even	2
1250.4.a.n.1.8		16	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:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1250.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -0.246957 q^{3} +4.00000 q^{4} -0.493914 q^{6} -19.9138 q^{7} +8.00000 q^{8} -26.9390 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -0.246957 q^{3} +4.00000 q^{4} -0.493914 q^{6} -19.9138 q^{7} +8.00000 q^{8} -26.9390 q^{9} -2.21060 q^{11} -0.987829 q^{12} -12.3046 q^{13} -39.8276 q^{14} +16.0000 q^{16} -74.4371 q^{17} -53.8780 q^{18} -50.4660 q^{19} +4.91785 q^{21} -4.42120 q^{22} +199.314 q^{23} -1.97566 q^{24} -24.6092 q^{26} +13.3206 q^{27} -79.6551 q^{28} +241.241 q^{29} -51.2776 q^{31} +32.0000 q^{32} +0.545923 q^{33} -148.874 q^{34} -107.756 q^{36} +278.117 q^{37} -100.932 q^{38} +3.03871 q^{39} +265.583 q^{41} +9.83571 q^{42} -103.714 q^{43} -8.84239 q^{44} +398.627 q^{46} +411.544 q^{47} -3.95132 q^{48} +53.5589 q^{49} +18.3828 q^{51} -49.2183 q^{52} +430.285 q^{53} +26.6413 q^{54} -159.310 q^{56} +12.4629 q^{57} +482.482 q^{58} +550.978 q^{59} -510.914 q^{61} -102.555 q^{62} +536.458 q^{63} +64.0000 q^{64} +1.09185 q^{66} -76.2521 q^{67} -297.748 q^{68} -49.2219 q^{69} -855.645 q^{71} -215.512 q^{72} -516.618 q^{73} +556.233 q^{74} -201.864 q^{76} +44.0214 q^{77} +6.07741 q^{78} +136.253 q^{79} +724.064 q^{81} +531.166 q^{82} +917.533 q^{83} +19.6714 q^{84} -207.427 q^{86} -59.5762 q^{87} -17.6848 q^{88} +1302.75 q^{89} +245.031 q^{91} +797.255 q^{92} +12.6634 q^{93} +823.088 q^{94} -7.90263 q^{96} -443.954 q^{97} +107.118 q^{98} +59.5513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{2} + 12 q^{3} + 64 q^{4} + 24 q^{6} + 56 q^{7} + 128 q^{8} + 212 q^{9}+O(q^{10}) \) 16 * q + 32 * q^2 + 12 * q^3 + 64 * q^4 + 24 * q^6 + 56 * q^7 + 128 * q^8 + 212 * q^9	\( 16 q + 32 q^{2} + 12 q^{3} + 64 q^{4} + 24 q^{6} + 56 q^{7} + 128 q^{8} + 212 q^{9} + 72 q^{11} + 48 q^{12} + 202 q^{13} + 112 q^{14} + 256 q^{16} + 216 q^{17} + 424 q^{18} + 100 q^{19} + 192 q^{21} + 144 q^{22} + 292 q^{23} + 96 q^{24} + 404 q^{26} + 570 q^{27} + 224 q^{28} + 400 q^{29} + 102 q^{31} + 512 q^{32} + 664 q^{33} + 432 q^{34} + 848 q^{36} + 646 q^{37} + 200 q^{38} + 104 q^{39} + 532 q^{41} + 384 q^{42} + 902 q^{43} + 288 q^{44} + 584 q^{46} + 776 q^{47} + 192 q^{48} + 1038 q^{49} + 442 q^{51} + 808 q^{52} + 632 q^{53} + 1140 q^{54} + 448 q^{56} + 1400 q^{57} + 800 q^{58} + 1000 q^{59} + 662 q^{61} + 204 q^{62} + 932 q^{63} + 1024 q^{64} + 1328 q^{66} + 1326 q^{67} + 864 q^{68} + 1854 q^{69} + 1292 q^{71} + 1696 q^{72} + 2272 q^{73} + 1292 q^{74} + 400 q^{76} + 2582 q^{77} + 208 q^{78} + 320 q^{79} + 2956 q^{81} + 1064 q^{82} + 2842 q^{83} + 768 q^{84} + 1804 q^{86} + 2920 q^{87} + 576 q^{88} + 2780 q^{89} + 812 q^{91} + 1168 q^{92} + 2824 q^{93} + 1552 q^{94} + 384 q^{96} + 3796 q^{97} + 2076 q^{98} + 1054 q^{99}+O(q^{100}) \) 16 * q + 32 * q^2 + 12 * q^3 + 64 * q^4 + 24 * q^6 + 56 * q^7 + 128 * q^8 + 212 * q^9 + 72 * q^11 + 48 * q^12 + 202 * q^13 + 112 * q^14 + 256 * q^16 + 216 * q^17 + 424 * q^18 + 100 * q^19 + 192 * q^21 + 144 * q^22 + 292 * q^23 + 96 * q^24 + 404 * q^26 + 570 * q^27 + 224 * q^28 + 400 * q^29 + 102 * q^31 + 512 * q^32 + 664 * q^33 + 432 * q^34 + 848 * q^36 + 646 * q^37 + 200 * q^38 + 104 * q^39 + 532 * q^41 + 384 * q^42 + 902 * q^43 + 288 * q^44 + 584 * q^46 + 776 * q^47 + 192 * q^48 + 1038 * q^49 + 442 * q^51 + 808 * q^52 + 632 * q^53 + 1140 * q^54 + 448 * q^56 + 1400 * q^57 + 800 * q^58 + 1000 * q^59 + 662 * q^61 + 204 * q^62 + 932 * q^63 + 1024 * q^64 + 1328 * q^66 + 1326 * q^67 + 864 * q^68 + 1854 * q^69 + 1292 * q^71 + 1696 * q^72 + 2272 * q^73 + 1292 * q^74 + 400 * q^76 + 2582 * q^77 + 208 * q^78 + 320 * q^79 + 2956 * q^81 + 1064 * q^82 + 2842 * q^83 + 768 * q^84 + 1804 * q^86 + 2920 * q^87 + 576 * q^88 + 2780 * q^89 + 812 * q^91 + 1168 * q^92 + 2824 * q^93 + 1552 * q^94 + 384 * q^96 + 3796 * q^97 + 2076 * q^98 + 1054 * q^99

Introduction

L-functions

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