LMFDB - Embedded newform 1875.4.a.l.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,4,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1875.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1875.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:	$110.628581261$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.82486 q^{2} +3.00000 q^{3} +15.2792 q^{4} +14.4746 q^{6} +14.5960 q^{7} +35.1213 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.82486 q^{2} +3.00000 q^{3} +15.2792 q^{4} +14.4746 q^{6} +14.5960 q^{7} +35.1213 q^{8} +9.00000 q^{9} +10.0778 q^{11} +45.8377 q^{12} +5.07513 q^{13} +70.4236 q^{14} +47.2211 q^{16} +97.5413 q^{17} +43.4237 q^{18} -58.7159 q^{19} +43.7880 q^{21} +48.6240 q^{22} +15.8371 q^{23} +105.364 q^{24} +24.4868 q^{26} +27.0000 q^{27} +223.016 q^{28} -220.481 q^{29} +196.868 q^{31} -53.1350 q^{32} +30.2334 q^{33} +470.623 q^{34} +137.513 q^{36} +409.600 q^{37} -283.296 q^{38} +15.2254 q^{39} +200.662 q^{41} +211.271 q^{42} +457.223 q^{43} +153.981 q^{44} +76.4119 q^{46} -280.237 q^{47} +141.663 q^{48} -129.957 q^{49} +292.624 q^{51} +77.5441 q^{52} +586.131 q^{53} +130.271 q^{54} +512.629 q^{56} -176.148 q^{57} -1063.79 q^{58} +192.842 q^{59} +17.2847 q^{61} +949.862 q^{62} +131.364 q^{63} -634.138 q^{64} +145.872 q^{66} -673.397 q^{67} +1490.36 q^{68} +47.5114 q^{69} -384.133 q^{71} +316.091 q^{72} -284.707 q^{73} +1976.26 q^{74} -897.134 q^{76} +147.096 q^{77} +73.4603 q^{78} -774.124 q^{79} +81.0000 q^{81} +968.167 q^{82} -1311.29 q^{83} +669.047 q^{84} +2206.03 q^{86} -661.443 q^{87} +353.945 q^{88} -19.8287 q^{89} +74.0765 q^{91} +241.979 q^{92} +590.605 q^{93} -1352.10 q^{94} -159.405 q^{96} +1590.73 q^{97} -627.024 q^{98} +90.7002 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 72 q^{3} + 133 q^{4} + 3 q^{6} + 62 q^{7} + 27 q^{8} + 216 q^{9}+O(q^{10}) $ 24 * q + q^2 + 72 * q^3 + 133 * q^4 + 3 * q^6 + 62 * q^7 + 27 * q^8 + 216 * q^9	\( 24 q + q^{2} + 72 q^{3} + 133 q^{4} + 3 q^{6} + 62 q^{7} + 27 q^{8} + 216 q^{9} + 96 q^{11} + 399 q^{12} + 156 q^{13} + 92 q^{14} + 845 q^{16} - 46 q^{17} + 9 q^{18} + 182 q^{19} + 186 q^{21} + 158 q^{22} - 286 q^{23} + 81 q^{24} + 478 q^{26} + 648 q^{27} + 701 q^{28} + 1144 q^{29} + 64 q^{31} + 1212 q^{32} + 288 q^{33} + 961 q^{34} + 1197 q^{36} + 762 q^{37} + 474 q^{38} + 468 q^{39} + 1074 q^{41} + 276 q^{42} + 460 q^{43} + 319 q^{44} + 459 q^{46} - 960 q^{47} + 2535 q^{48} + 2680 q^{49} - 138 q^{51} + 2969 q^{52} + 914 q^{53} + 27 q^{54} + 1680 q^{56} + 546 q^{57} + 208 q^{58} + 208 q^{59} + 3520 q^{61} + 334 q^{62} + 558 q^{63} + 5747 q^{64} + 474 q^{66} + 154 q^{67} - 5727 q^{68} - 858 q^{69} - 252 q^{71} + 243 q^{72} + 4414 q^{73} + 5637 q^{74} + 627 q^{76} + 2344 q^{77} + 1434 q^{78} + 1110 q^{79} + 1944 q^{81} + 3714 q^{82} - 1488 q^{83} + 2103 q^{84} + 3036 q^{86} + 3432 q^{87} + 3947 q^{88} + 3402 q^{89} + 3504 q^{91} - 11163 q^{92} + 192 q^{93} + 3408 q^{94} + 3636 q^{96} + 534 q^{97} + 2244 q^{98} + 864 q^{99}+O(q^{100}) \) 24 * q + q^2 + 72 * q^3 + 133 * q^4 + 3 * q^6 + 62 * q^7 + 27 * q^8 + 216 * q^9 + 96 * q^11 + 399 * q^12 + 156 * q^13 + 92 * q^14 + 845 * q^16 - 46 * q^17 + 9 * q^18 + 182 * q^19 + 186 * q^21 + 158 * q^22 - 286 * q^23 + 81 * q^24 + 478 * q^26 + 648 * q^27 + 701 * q^28 + 1144 * q^29 + 64 * q^31 + 1212 * q^32 + 288 * q^33 + 961 * q^34 + 1197 * q^36 + 762 * q^37 + 474 * q^38 + 468 * q^39 + 1074 * q^41 + 276 * q^42 + 460 * q^43 + 319 * q^44 + 459 * q^46 - 960 * q^47 + 2535 * q^48 + 2680 * q^49 - 138 * q^51 + 2969 * q^52 + 914 * q^53 + 27 * q^54 + 1680 * q^56 + 546 * q^57 + 208 * q^58 + 208 * q^59 + 3520 * q^61 + 334 * q^62 + 558 * q^63 + 5747 * q^64 + 474 * q^66 + 154 * q^67 - 5727 * q^68 - 858 * q^69 - 252 * q^71 + 243 * q^72 + 4414 * q^73 + 5637 * q^74 + 627 * q^76 + 2344 * q^77 + 1434 * q^78 + 1110 * q^79 + 1944 * q^81 + 3714 * q^82 - 1488 * q^83 + 2103 * q^84 + 3036 * q^86 + 3432 * q^87 + 3947 * q^88 + 3402 * q^89 + 3504 * q^91 - 11163 * q^92 + 192 * q^93 + 3408 * q^94 + 3636 * q^96 + 534 * q^97 + 2244 * q^98 + 864 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.82486	1.70584	0.852922	−	0.522038i	$-0.174828\pi$
$2$	4.82486	1.70584	0.852922	+	0.522038i	$0.174828\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	15.2792	1.90990
$5$	0	0
$5$	0	0
$6$	14.4746	0.984870
$7$	14.5960	0.788109	0.394055	−	0.919087i	$-0.371072\pi$
$7$	14.5960	0.788109	0.394055	+	0.919087i	$0.371072\pi$
$8$	35.1213	1.55215
$9$	9.00000	0.333333
$10$	0	0
$11$	10.0778	0.276234	0.138117	−	0.990416i	$-0.455895\pi$
$11$	10.0778	0.276234	0.138117	+	0.990416i	$0.455895\pi$
$12$	45.8377	1.10268
$13$	5.07513	0.108276	0.0541380	−	0.998533i	$-0.482759\pi$
$13$	5.07513	0.108276	0.0541380	+	0.998533i	$0.482759\pi$
$14$	70.4236	1.34439
$15$	0	0
$16$	47.2211	0.737830
$17$	97.5413	1.39160	0.695801	−	0.718234i	$-0.255049\pi$
$17$	97.5413	1.39160	0.695801	+	0.718234i	$0.255049\pi$
$18$	43.4237	0.568615
$19$	−58.7159	−0.708966	−0.354483	−	0.935062i	$-0.615343\pi$
$19$	−58.7159	−0.708966	−0.354483	+	0.935062i	$0.615343\pi$
$20$	0	0
$21$	43.7880	0.455015
$22$	48.6240	0.471212
$23$	15.8371	0.143577	0.0717885	−	0.997420i	$-0.477129\pi$
$23$	15.8371	0.143577	0.0717885	+	0.997420i	$0.477129\pi$
$24$	105.364	0.896137
$25$	0	0
$26$	24.4868	0.184702
$27$	27.0000	0.192450
$28$	223.016	1.50521
$29$	−220.481	−1.41180	−0.705902	−	0.708310i	$-0.749458\pi$
$29$	−220.481	−1.41180	−0.705902	+	0.708310i	$0.749458\pi$
$30$	0	0
$31$	196.868	1.14060	0.570300	−	0.821436i	$-0.306827\pi$
$31$	196.868	1.14060	0.570300	+	0.821436i	$0.306827\pi$
$32$	−53.1350	−0.293532
$33$	30.2334	0.159484
$34$	470.623	2.37386
$35$	0	0
$36$	137.513	0.636635
$37$	409.600	1.81994	0.909970	−	0.414674i	$-0.136105\pi$
$37$	409.600	1.81994	0.909970	+	0.414674i	$0.136105\pi$
$38$	−283.296	−1.20939
$39$	15.2254	0.0625131
$40$	0	0
$41$	200.662	0.764346	0.382173	−	0.924091i	$-0.375176\pi$
$41$	200.662	0.764346	0.382173	+	0.924091i	$0.375176\pi$
$42$	211.271	0.776185
$43$	457.223	1.62153	0.810766	−	0.585371i	$-0.199051\pi$
$43$	457.223	1.62153	0.810766	+	0.585371i	$0.199051\pi$
$44$	153.981	0.527580
$45$	0	0
$46$	76.4119	0.244920
$47$	−280.237	−0.869717	−0.434859	−	0.900499i	$-0.643202\pi$
$47$	−280.237	−0.869717	−0.434859	+	0.900499i	$0.643202\pi$
$48$	141.663	0.425986
$49$	−129.957	−0.378884
$50$	0	0
$51$	292.624	0.803442
$52$	77.5441	0.206797
$53$	586.131	1.51908	0.759541	−	0.650460i	$-0.225424\pi$
$53$	586.131	1.51908	0.759541	+	0.650460i	$0.225424\pi$
$54$	130.271	0.328290
$55$	0	0
$56$	512.629	1.22327
$57$	−176.148	−0.409322
$58$	−1063.79	−2.40832
$59$	192.842	0.425524	0.212762	−	0.977104i	$-0.431754\pi$
$59$	192.842	0.425524	0.212762	+	0.977104i	$0.431754\pi$
$60$	0	0
$61$	17.2847	0.0362801	0.0181400	−	0.999835i	$-0.494226\pi$
$61$	17.2847	0.0362801	0.0181400	+	0.999835i	$0.494226\pi$
$62$	949.862	1.94569
$63$	131.364	0.262703
$64$	−634.138	−1.23855
$65$	0	0
$66$	145.872	0.272054
$67$	−673.397	−1.22789	−0.613944	−	0.789349i	$-0.710418\pi$
$67$	−673.397	−1.22789	−0.613944	+	0.789349i	$0.710418\pi$
$68$	1490.36	2.65783
$69$	47.5114	0.0828942
$70$	0	0
$71$	−384.133	−0.642088	−0.321044	−	0.947064i	$-0.604034\pi$
$71$	−384.133	−0.642088	−0.321044	+	0.947064i	$0.604034\pi$
$72$	316.091	0.517385
$73$	−284.707	−0.456472	−0.228236	−	0.973606i	$-0.573296\pi$
$73$	−284.707	−0.456472	−0.228236	+	0.973606i	$0.573296\pi$
$74$	1976.26	3.10453
$75$	0	0
$76$	−897.134	−1.35406
$77$	147.096	0.217703
$78$	73.4603	0.106638
$79$	−774.124	−1.10248	−0.551239	−	0.834347i	$-0.685845\pi$
$79$	−774.124	−1.10248	−0.551239	+	0.834347i	$0.685845\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	968.167	1.30386
$83$	−1311.29	−1.73413	−0.867063	−	0.498199i	$-0.833995\pi$
$83$	−1311.29	−1.73413	−0.867063	+	0.498199i	$0.833995\pi$
$84$	669.047	0.869035
$85$	0	0
$86$	2206.03	2.76608
$87$	−661.443	−0.815105
$88$	353.945	0.428758
$89$	−19.8287	−0.0236161	−0.0118081	−	0.999930i	$-0.503759\pi$
$89$	−19.8287	−0.0236161	−0.0118081	+	0.999930i	$0.503759\pi$
$90$	0	0
$91$	74.0765	0.0853333
$92$	241.979	0.274218
$93$	590.605	0.658526
$94$	−1352.10	−1.48360
$95$	0	0
$96$	−159.405	−0.169471
$97$	1590.73	1.66510	0.832550	−	0.553950i	$-0.186880\pi$
$97$	1590.73	1.66510	0.832550	+	0.553950i	$0.186880\pi$
$98$	−627.024	−0.646316
$99$	90.7002	0.0920779
$100$	0	0
$101$	354.725	0.349470	0.174735	−	0.984615i	$-0.444093\pi$
$101$	354.725	0.349470	0.174735	+	0.984615i	$0.444093\pi$
$102$	1411.87	1.37055
$103$	−264.934	−0.253444	−0.126722	−	0.991938i	$-0.540446\pi$
$103$	−264.934	−0.253444	−0.126722	+	0.991938i	$0.540446\pi$
$104$	178.245	0.168061
$105$	0	0
$106$	2828.00	2.59132
$107$	1360.19	1.22892	0.614461	−	0.788947i	$-0.289374\pi$
$107$	1360.19	1.22892	0.614461	+	0.788947i	$0.289374\pi$
$108$	412.539	0.367561
$109$	654.826	0.575422	0.287711	−	0.957717i	$-0.407106\pi$
$109$	654.826	0.575422	0.287711	+	0.957717i	$0.407106\pi$
$110$	0	0
$111$	1228.80	1.05074
$112$	689.239	0.581491
$113$	−1833.84	−1.52666	−0.763332	−	0.646007i	$-0.776438\pi$
$113$	−1833.84	−1.52666	−0.763332	+	0.646007i	$0.776438\pi$
$114$	−849.888	−0.698239
$115$	0	0
$116$	−3368.78	−2.69641
$117$	45.6762	0.0360920
$118$	930.436	0.725878
$119$	1423.71	1.09674
$120$	0	0
$121$	−1229.44	−0.923695
$122$	83.3964	0.0618882
$123$	601.987	0.441296
$124$	3008.00	2.17844
$125$	0	0
$126$	633.812	0.448131
$127$	315.848	0.220685	0.110342	−	0.993894i	$-0.464805\pi$
$127$	315.848	0.220685	0.110342	+	0.993894i	$0.464805\pi$
$128$	−2634.54	−1.81924
$129$	1371.67	0.936191
$130$	0	0
$131$	327.323	0.218308	0.109154	−	0.994025i	$-0.465186\pi$
$131$	327.323	0.218308	0.109154	+	0.994025i	$0.465186\pi$
$132$	461.943	0.304599
$133$	−857.017	−0.558743
$134$	−3249.04	−2.09459
$135$	0	0
$136$	3425.77	2.15998
$137$	2914.04	1.81725	0.908624	−	0.417614i	$-0.137134\pi$
$137$	2914.04	1.81725	0.908624	+	0.417614i	$0.137134\pi$
$138$	229.236	0.141405
$139$	−931.043	−0.568130	−0.284065	−	0.958805i	$-0.591683\pi$
$139$	−931.043	−0.568130	−0.284065	+	0.958805i	$0.591683\pi$
$140$	0	0
$141$	−840.710	−0.502131
$142$	−1853.39	−1.09530
$143$	51.1461	0.0299095
$144$	424.990	0.245943
$145$	0	0
$146$	−1373.67	−0.778669
$147$	−389.871	−0.218748
$148$	6258.37	3.47591
$149$	−1864.01	−1.02487	−0.512434	−	0.858727i	$-0.671256\pi$
$149$	−1864.01	−1.02487	−0.512434	+	0.858727i	$0.671256\pi$
$150$	0	0
$151$	−917.653	−0.494553	−0.247277	−	0.968945i	$-0.579536\pi$
$151$	−917.653	−0.494553	−0.247277	+	0.968945i	$0.579536\pi$
$152$	−2062.18	−1.10043
$153$	877.872	0.463868
$154$	709.715	0.371367
$155$	0	0
$156$	232.632	0.119394
$157$	706.778	0.359280	0.179640	−	0.983732i	$-0.442507\pi$
$157$	706.778	0.359280	0.179640	+	0.983732i	$0.442507\pi$
$158$	−3735.04	−1.88066
$159$	1758.39	0.877042
$160$	0	0
$161$	231.159	0.113154
$162$	390.813	0.189538
$163$	3185.13	1.53054	0.765272	−	0.643707i	$-0.222604\pi$
$163$	3185.13	1.53054	0.765272	+	0.643707i	$0.222604\pi$
$164$	3065.97	1.45983
$165$	0	0
$166$	−6326.77	−2.95815
$167$	2332.85	1.08097	0.540484	−	0.841354i	$-0.318241\pi$
$167$	2332.85	1.08097	0.540484	+	0.841354i	$0.318241\pi$
$168$	1537.89	0.706254
$169$	−2171.24	−0.988276
$170$	0	0
$171$	−528.443	−0.236322
$172$	6986.02	3.09697
$173$	−62.4472	−0.0274438	−0.0137219	−	0.999906i	$-0.504368\pi$
$173$	−62.4472	−0.0274438	−0.0137219	+	0.999906i	$0.504368\pi$
$174$	−3191.37	−1.39044
$175$	0	0
$176$	475.885	0.203814
$177$	578.527	0.245676
$178$	−95.6705	−0.0402854
$179$	1051.75	0.439171	0.219586	−	0.975593i	$-0.429529\pi$
$179$	1051.75	0.439171	0.219586	+	0.975593i	$0.429529\pi$
$180$	0	0
$181$	−2328.28	−0.956130	−0.478065	−	0.878324i	$-0.658662\pi$
$181$	−2328.28	−0.956130	−0.478065	+	0.878324i	$0.658662\pi$
$182$	357.409	0.145565
$183$	51.8542	0.0209463
$184$	556.220	0.222854
$185$	0	0
$186$	2849.59	1.12334
$187$	983.002	0.384408
$188$	−4281.80	−1.66108
$189$	394.092	0.151672
$190$	0	0
$191$	−1216.96	−0.461027	−0.230513	−	0.973069i	$-0.574041\pi$
$191$	−1216.96	−0.461027	−0.230513	+	0.973069i	$0.574041\pi$
$192$	−1902.41	−0.715077
$193$	−613.403	−0.228776	−0.114388	−	0.993436i	$-0.536491\pi$
$193$	−613.403	−0.228776	−0.114388	+	0.993436i	$0.536491\pi$
$194$	7675.07	2.84040
$195$	0	0
$196$	−1985.64	−0.723631
$197$	−2923.42	−1.05728	−0.528642	−	0.848845i	$-0.677299\pi$
$197$	−2923.42	−1.05728	−0.528642	+	0.848845i	$0.677299\pi$
$198$	437.616	0.157071
$199$	−3733.69	−1.33002	−0.665010	−	0.746834i	$-0.731573\pi$
$199$	−3733.69	−1.33002	−0.665010	+	0.746834i	$0.731573\pi$
$200$	0	0
$201$	−2020.19	−0.708922
$202$	1711.50	0.596141
$203$	−3218.14	−1.11266
$204$	4471.07	1.53450
$205$	0	0
$206$	−1278.27	−0.432336
$207$	142.534	0.0478590
$208$	239.653	0.0798892
$209$	−591.728	−0.195840
$210$	0	0
$211$	−1388.03	−0.452870	−0.226435	−	0.974026i	$-0.572707\pi$
$211$	−1388.03	−0.452870	−0.226435	+	0.974026i	$0.572707\pi$
$212$	8955.64	2.90130
$213$	−1152.40	−0.370710
$214$	6562.72	2.09635
$215$	0	0
$216$	948.274	0.298712
$217$	2873.49	0.898918
$218$	3159.44	0.981580
$219$	−854.121	−0.263544
$220$	0	0
$221$	495.035	0.150677
$222$	5928.78	1.79240
$223$	4947.16	1.48559	0.742795	−	0.669519i	$-0.233500\pi$
$223$	4947.16	1.48559	0.742795	+	0.669519i	$0.233500\pi$
$224$	−775.558	−0.231335
$225$	0	0
$226$	−8848.00	−2.60425
$227$	−4665.54	−1.36415	−0.682077	−	0.731281i	$-0.738923\pi$
$227$	−4665.54	−1.36415	−0.682077	+	0.731281i	$0.738923\pi$
$228$	−2691.40	−0.781765
$229$	3249.34	0.937652	0.468826	−	0.883290i	$-0.344677\pi$
$229$	3249.34	0.937652	0.468826	+	0.883290i	$0.344677\pi$
$230$	0	0
$231$	441.287	0.125691
$232$	−7743.57	−2.19134
$233$	231.304	0.0650353	0.0325177	−	0.999471i	$-0.489647\pi$
$233$	231.304	0.0650353	0.0325177	+	0.999471i	$0.489647\pi$
$234$	220.381	0.0615673
$235$	0	0
$236$	2946.48	0.812710
$237$	−2322.37	−0.636516
$238$	6869.21	1.87086
$239$	2344.91	0.634644	0.317322	−	0.948318i	$-0.397216\pi$
$239$	2344.91	0.634644	0.317322	+	0.948318i	$0.397216\pi$
$240$	0	0
$241$	−972.058	−0.259817	−0.129908	−	0.991526i	$-0.541468\pi$
$241$	−972.058	−0.259817	−0.129908	+	0.991526i	$0.541468\pi$
$242$	−5931.86	−1.57568
$243$	243.000	0.0641500
$244$	264.098	0.0692915
$245$	0	0
$246$	2904.50	0.752781
$247$	−297.991	−0.0767640
$248$	6914.27	1.77039
$249$	−3933.86	−1.00120
$250$	0	0
$251$	−5817.22	−1.46287	−0.731433	−	0.681913i	$-0.761148\pi$
$251$	−5817.22	−1.46287	−0.731433	+	0.681913i	$0.761148\pi$
$252$	2007.14	0.501738
$253$	159.603	0.0396608
$254$	1523.92	0.376454
$255$	0	0
$256$	−7638.19	−1.86479
$257$	1995.51	0.484344	0.242172	−	0.970233i	$-0.422140\pi$
$257$	1995.51	0.484344	0.242172	+	0.970233i	$0.422140\pi$
$258$	6618.10	1.59700
$259$	5978.51	1.43431
$260$	0	0
$261$	−1984.33	−0.470601
$262$	1579.28	0.372399
$263$	−4131.47	−0.968660	−0.484330	−	0.874885i	$-0.660937\pi$
$263$	−4131.47	−0.968660	−0.484330	+	0.874885i	$0.660937\pi$
$264$	1061.84	0.247543
$265$	0	0
$266$	−4134.98	−0.953128
$267$	−59.4860	−0.0136348
$268$	−10289.0	−2.34515
$269$	4596.52	1.04184	0.520920	−	0.853605i	$-0.325589\pi$
$269$	4596.52	1.04184	0.520920	+	0.853605i	$0.325589\pi$
$270$	0	0
$271$	−1607.50	−0.360328	−0.180164	−	0.983637i	$-0.557663\pi$
$271$	−1607.50	−0.360328	−0.180164	+	0.983637i	$0.557663\pi$
$272$	4606.01	1.02677
$273$	222.230	0.0492672
$274$	14059.8	3.09994
$275$	0	0
$276$	725.938	0.158320
$277$	6529.93	1.41641	0.708205	−	0.706007i	$-0.249505\pi$
$277$	6529.93	1.41641	0.708205	+	0.706007i	$0.249505\pi$
$278$	−4492.15	−0.969141
$279$	1771.82	0.380200
$280$	0	0
$281$	6003.53	1.27452	0.637261	−	0.770648i	$-0.280067\pi$
$281$	6003.53	1.27452	0.637261	+	0.770648i	$0.280067\pi$
$282$	−4056.30	−0.856558
$283$	−8778.34	−1.84388	−0.921940	−	0.387333i	$-0.873396\pi$
$283$	−8778.34	−1.84388	−0.921940	+	0.387333i	$0.873396\pi$
$284$	−5869.26	−1.22633
$285$	0	0
$286$	246.773	0.0510209
$287$	2928.87	0.602389
$288$	−478.215	−0.0978440
$289$	4601.31	0.936559
$290$	0	0
$291$	4772.20	0.961346
$292$	−4350.10	−0.871817
$293$	−9390.14	−1.87228	−0.936139	−	0.351629i	$-0.885628\pi$
$293$	−9390.14	−1.87228	−0.936139	+	0.351629i	$0.885628\pi$
$294$	−1881.07	−0.373151
$295$	0	0
$296$	14385.7	2.82483
$297$	272.101	0.0531612
$298$	−8993.56	−1.74827
$299$	80.3755	0.0155459
$300$	0	0
$301$	6673.62	1.27794
$302$	−4427.55	−0.843631
$303$	1064.18	0.201767
$304$	−2772.63	−0.523096
$305$	0	0
$306$	4235.61	0.791286
$307$	4708.16	0.875274	0.437637	−	0.899152i	$-0.355815\pi$
$307$	4708.16	0.875274	0.437637	+	0.899152i	$0.355815\pi$
$308$	2247.51	0.415791
$309$	−794.802	−0.146326
$310$	0	0
$311$	−417.593	−0.0761400	−0.0380700	−	0.999275i	$-0.512121\pi$
$311$	−417.593	−0.0761400	−0.0380700	+	0.999275i	$0.512121\pi$
$312$	534.735	0.0970301
$313$	475.968	0.0859530	0.0429765	−	0.999076i	$-0.486316\pi$
$313$	475.968	0.0859530	0.0429765	+	0.999076i	$0.486316\pi$
$314$	3410.10	0.612876
$315$	0	0
$316$	−11828.0	−2.10563
$317$	−10492.5	−1.85904	−0.929520	−	0.368772i	$-0.879778\pi$
$317$	−10492.5	−1.85904	−0.929520	+	0.368772i	$0.879778\pi$
$318$	8484.00	1.49610
$319$	−2221.97	−0.389988
$320$	0	0
$321$	4080.57	0.709518
$322$	1115.31	0.193024
$323$	−5727.23	−0.986599
$324$	1237.62	0.212212
$325$	0	0
$326$	15367.8	2.61087
$327$	1964.48	0.332220
$328$	7047.51	1.18638
$329$	−4090.33	−0.685432
$330$	0	0
$331$	−11008.0	−1.82797	−0.913983	−	0.405752i	$-0.867010\pi$
$331$	−11008.0	−1.82797	−0.913983	+	0.405752i	$0.867010\pi$
$332$	−20035.5	−3.31201
$333$	3686.40	0.606647
$334$	11255.7	1.84396
$335$	0	0
$336$	2067.72	0.335724
$337$	−2462.50	−0.398044	−0.199022	−	0.979995i	$-0.563777\pi$
$337$	−2462.50	−0.398044	−0.199022	+	0.979995i	$0.563777\pi$
$338$	−10475.9	−1.68585
$339$	−5501.51	−0.881419
$340$	0	0
$341$	1984.00	0.315073
$342$	−2549.66	−0.403129
$343$	−6903.28	−1.08671
$344$	16058.2	2.51687
$345$	0	0
$346$	−301.299	−0.0468148
$347$	190.027	0.0293982	0.0146991	−	0.999892i	$-0.495321\pi$
$347$	190.027	0.0293982	0.0146991	+	0.999892i	$0.495321\pi$
$348$	−10106.3	−1.55677
$349$	6611.29	1.01402	0.507012	−	0.861939i	$-0.330750\pi$
$349$	6611.29	1.01402	0.507012	+	0.861939i	$0.330750\pi$
$350$	0	0
$351$	137.028	0.0208377
$352$	−535.484	−0.0810835
$353$	3543.48	0.534279	0.267140	−	0.963658i	$-0.413921\pi$
$353$	3543.48	0.534279	0.267140	+	0.963658i	$0.413921\pi$
$354$	2791.31	0.419086
$355$	0	0
$356$	−302.967	−0.0451046
$357$	4271.14	0.633200
$358$	5074.55	0.749158
$359$	1843.74	0.271056	0.135528	−	0.990774i	$-0.456727\pi$
$359$	1843.74	0.271056	0.135528	+	0.990774i	$0.456727\pi$
$360$	0	0
$361$	−3411.44	−0.497367
$362$	−11233.6	−1.63101
$363$	−3688.31	−0.533295
$364$	1131.83	0.162978
$365$	0	0
$366$	250.189	0.0357311
$367$	−10827.3	−1.54000	−0.770000	−	0.638044i	$-0.779744\pi$
$367$	−10827.3	−1.54000	−0.770000	+	0.638044i	$0.779744\pi$
$368$	747.847	0.105935
$369$	1805.96	0.254782
$370$	0	0
$371$	8555.17	1.19720
$372$	9024.00	1.25772
$373$	−10889.6	−1.51164	−0.755819	−	0.654780i	$-0.772761\pi$
$373$	−10889.6	−1.51164	−0.755819	+	0.654780i	$0.772761\pi$
$374$	4742.85	0.655740
$375$	0	0
$376$	−9842.26	−1.34994
$377$	−1118.97	−0.152864
$378$	1901.44	0.258728
$379$	−6458.61	−0.875347	−0.437674	−	0.899134i	$-0.644197\pi$
$379$	−6458.61	−0.875347	−0.437674	+	0.899134i	$0.644197\pi$
$380$	0	0
$381$	947.544	0.127412
$382$	−5871.65	−0.786440
$383$	−1241.89	−0.165685	−0.0828426	−	0.996563i	$-0.526400\pi$
$383$	−1241.89	−0.165685	−0.0828426	+	0.996563i	$0.526400\pi$
$384$	−7903.63	−1.05034
$385$	0	0
$386$	−2959.58	−0.390256
$387$	4115.01	0.540510
$388$	24305.2	3.18018
$389$	−12947.2	−1.68752	−0.843762	−	0.536717i	$-0.819664\pi$
$389$	−12947.2	−1.68752	−0.843762	+	0.536717i	$0.819664\pi$
$390$	0	0
$391$	1544.77	0.199802
$392$	−4564.25	−0.588086
$393$	981.968	0.126040
$394$	−14105.1	−1.80356
$395$	0	0
$396$	1385.83	0.175860
$397$	−10817.0	−1.36748	−0.683739	−	0.729727i	$-0.739647\pi$
$397$	−10817.0	−1.36748	−0.683739	+	0.729727i	$0.739647\pi$
$398$	−18014.5	−2.26881
$399$	−2571.05	−0.322590
$400$	0	0
$401$	3473.67	0.432585	0.216293	−	0.976329i	$-0.430603\pi$
$401$	3473.67	0.432585	0.216293	+	0.976329i	$0.430603\pi$
$402$	−9747.13	−1.20931
$403$	999.133	0.123500
$404$	5419.93	0.667454
$405$	0	0
$406$	−15527.1	−1.89802
$407$	4127.87	0.502729
$408$	10277.3	1.24707
$409$	15262.8	1.84523	0.922613	−	0.385727i	$-0.126049\pi$
$409$	15262.8	1.84523	0.922613	+	0.385727i	$0.126049\pi$
$410$	0	0
$411$	8742.12	1.04919
$412$	−4047.99	−0.484053
$413$	2814.72	0.335359
$414$	687.707	0.0816400
$415$	0	0
$416$	−269.667	−0.0317825
$417$	−2793.13	−0.328010
$418$	−2855.00	−0.334073
$419$	−11209.0	−1.30692	−0.653458	−	0.756963i	$-0.726682\pi$
$419$	−11209.0	−1.30692	−0.653458	+	0.756963i	$0.726682\pi$
$420$	0	0
$421$	12600.7	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$421$	12600.7	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$422$	−6697.02	−0.772526
$423$	−2522.13	−0.289906
$424$	20585.7	2.35785
$425$	0	0
$426$	−5560.16	−0.632373
$427$	252.288	0.0285927
$428$	20782.7	2.34712
$429$	153.438	0.0172682
$430$	0	0
$431$	−1480.60	−0.165471	−0.0827354	−	0.996572i	$-0.526366\pi$
$431$	−1480.60	−0.165471	−0.0827354	+	0.996572i	$0.526366\pi$
$432$	1274.97	0.141995
$433$	−1374.85	−0.152589	−0.0762944	−	0.997085i	$-0.524309\pi$
$433$	−1374.85	−0.152589	−0.0762944	+	0.997085i	$0.524309\pi$
$434$	13864.2	1.53341
$435$	0	0
$436$	10005.2	1.09900
$437$	−929.892	−0.101791
$438$	−4121.01	−0.449565
$439$	6523.14	0.709186	0.354593	−	0.935021i	$-0.384619\pi$
$439$	6523.14	0.709186	0.354593	+	0.935021i	$0.384619\pi$
$440$	0	0
$441$	−1169.61	−0.126295
$442$	2388.47	0.257032
$443$	11119.1	1.19251	0.596257	−	0.802794i	$-0.296654\pi$
$443$	11119.1	1.19251	0.596257	+	0.802794i	$0.296654\pi$
$444$	18775.1	2.00682
$445$	0	0
$446$	23869.3	2.53418
$447$	−5592.02	−0.591708
$448$	−9255.87	−0.976113
$449$	−17054.7	−1.79256	−0.896280	−	0.443489i	$-0.853740\pi$
$449$	−17054.7	−1.79256	−0.896280	+	0.443489i	$0.853740\pi$
$450$	0	0
$451$	2022.24	0.211138
$452$	−28019.6	−2.91578
$453$	−2752.96	−0.285531
$454$	−22510.6	−2.32703
$455$	0	0
$456$	−6186.53	−0.635331
$457$	4405.39	0.450931	0.225466	−	0.974251i	$-0.427610\pi$
$457$	4405.39	0.450931	0.225466	+	0.974251i	$0.427610\pi$
$458$	15677.6	1.59949
$459$	2633.62	0.267814
$460$	0	0
$461$	3509.59	0.354572	0.177286	−	0.984159i	$-0.443268\pi$
$461$	3509.59	0.354572	0.177286	+	0.984159i	$0.443268\pi$
$462$	2129.14	0.214409
$463$	−2816.30	−0.282688	−0.141344	−	0.989961i	$-0.545142\pi$
$463$	−2816.30	−0.282688	−0.141344	+	0.989961i	$0.545142\pi$
$464$	−10411.4	−1.04167
$465$	0	0
$466$	1116.01	0.110940
$467$	−3440.14	−0.340880	−0.170440	−	0.985368i	$-0.554519\pi$
$467$	−3440.14	−0.340880	−0.170440	+	0.985368i	$0.554519\pi$
$468$	697.897	0.0689322
$469$	−9828.90	−0.967711
$470$	0	0
$471$	2120.33	0.207431
$472$	6772.86	0.660479
$473$	4607.80	0.447922
$474$	−11205.1	−1.08580
$475$	0	0
$476$	21753.2	2.09466
$477$	5275.18	0.506361
$478$	11313.9	1.08260
$479$	−1767.47	−0.168597	−0.0842984	−	0.996441i	$-0.526865\pi$
$479$	−1767.47	−0.168597	−0.0842984	+	0.996441i	$0.526865\pi$
$480$	0	0
$481$	2078.77	0.197056
$482$	−4690.04	−0.443206
$483$	693.476	0.0653297
$484$	−18784.9	−1.76417
$485$	0	0
$486$	1172.44	0.109430
$487$	4290.09	0.399184	0.199592	−	0.979879i	$-0.436038\pi$
$487$	4290.09	0.399184	0.199592	+	0.979879i	$0.436038\pi$
$488$	607.062	0.0563123
$489$	9555.39	0.883660
$490$	0	0
$491$	13036.8	1.19825	0.599126	−	0.800655i	$-0.295515\pi$
$491$	13036.8	1.19825	0.599126	+	0.800655i	$0.295515\pi$
$492$	9197.90	0.842832
$493$	−21506.0	−1.96467
$494$	−1437.76	−0.130947
$495$	0	0
$496$	9296.35	0.841569
$497$	−5606.81	−0.506036
$498$	−18980.3	−1.70789
$499$	6222.24	0.558208	0.279104	−	0.960261i	$-0.409963\pi$
$499$	6222.24	0.558208	0.279104	+	0.960261i	$0.409963\pi$
$500$	0	0
$501$	6998.56	0.624097
$502$	−28067.2	−2.49542
$503$	−7982.98	−0.707641	−0.353820	−	0.935313i	$-0.615118\pi$
$503$	−7982.98	−0.707641	−0.353820	+	0.935313i	$0.615118\pi$
$504$	4613.67	0.407756
$505$	0	0
$506$	770.064	0.0676552
$507$	−6513.73	−0.570582
$508$	4825.91	0.421487
$509$	10660.0	0.928286	0.464143	−	0.885760i	$-0.346362\pi$
$509$	10660.0	0.928286	0.464143	+	0.885760i	$0.346362\pi$
$510$	0	0
$511$	−4155.58	−0.359750
$512$	−15776.8	−1.36180
$513$	−1585.33	−0.136441
$514$	9628.04	0.826216
$515$	0	0
$516$	20958.0	1.78804
$517$	−2824.17	−0.240245
$518$	28845.5	2.44671
$519$	−187.342	−0.0158447
$520$	0	0
$521$	13119.7	1.10323	0.551617	−	0.834097i	$-0.314011\pi$
$521$	13119.7	1.10323	0.551617	+	0.834097i	$0.314011\pi$
$522$	−9574.11	−0.802772
$523$	15980.9	1.33613	0.668067	−	0.744101i	$-0.267122\pi$
$523$	15980.9	1.33613	0.668067	+	0.744101i	$0.267122\pi$
$524$	5001.24	0.416947
$525$	0	0
$526$	−19933.8	−1.65238
$527$	19202.8	1.58726
$528$	1427.66	0.117672
$529$	−11916.2	−0.979386
$530$	0	0
$531$	1735.58	0.141841
$532$	−13094.6	−1.06715
$533$	1018.39	0.0827603
$534$	−287.012	−0.0232588
$535$	0	0
$536$	−23650.6	−1.90587
$537$	3155.26	0.253556
$538$	22177.6	1.77722
$539$	−1309.68	−0.104660
$540$	0	0
$541$	12125.9	0.963650	0.481825	−	0.876268i	$-0.339974\pi$
$541$	12125.9	0.963650	0.481825	+	0.876268i	$0.339974\pi$
$542$	−7755.98	−0.614664
$543$	−6984.83	−0.552022
$544$	−5182.86	−0.408480
$545$	0	0
$546$	1072.23	0.0840422
$547$	−2614.22	−0.204344	−0.102172	−	0.994767i	$-0.532579\pi$
$547$	−2614.22	−0.204344	−0.102172	+	0.994767i	$0.532579\pi$
$548$	44524.3	3.47077
$549$	155.563	0.0120934
$550$	0	0
$551$	12945.8	1.00092
$552$	1668.66	0.128665
$553$	−11299.1	−0.868873
$554$	31506.0	2.41617
$555$	0	0
$556$	−14225.6	−1.08507
$557$	4706.55	0.358030	0.179015	−	0.983846i	$-0.442709\pi$
$557$	4706.55	0.358030	0.179015	+	0.983846i	$0.442709\pi$
$558$	8548.76	0.648562
$559$	2320.46	0.175573
$560$	0	0
$561$	2949.01	0.221938
$562$	28966.2	2.17414
$563$	−16894.8	−1.26471	−0.632353	−	0.774681i	$-0.717911\pi$
$563$	−16894.8	−1.26471	−0.632353	+	0.774681i	$0.717911\pi$
$564$	−12845.4	−0.959023
$565$	0	0
$566$	−42354.2	−3.14537
$567$	1182.28	0.0875677
$568$	−13491.2	−0.996620
$569$	−18190.0	−1.34019	−0.670093	−	0.742277i	$-0.733746\pi$
$569$	−18190.0	−1.34019	−0.670093	+	0.742277i	$0.733746\pi$
$570$	0	0
$571$	−23938.9	−1.75449	−0.877245	−	0.480043i	$-0.840621\pi$
$571$	−23938.9	−1.75449	−0.877245	+	0.480043i	$0.840621\pi$
$572$	781.474	0.0571242
$573$	−3650.88	−0.266174
$574$	14131.4	1.02758
$575$	0	0
$576$	−5707.24	−0.412850
$577$	−19835.7	−1.43114	−0.715571	−	0.698540i	$-0.753833\pi$
$577$	−19835.7	−1.43114	−0.715571	+	0.698540i	$0.753833\pi$
$578$	22200.7	1.59762
$579$	−1840.21	−0.132084
$580$	0	0
$581$	−19139.5	−1.36668
$582$	23025.2	1.63991
$583$	5906.92	0.419622
$584$	−9999.26	−0.708514
$585$	0	0
$586$	−45306.1	−3.19382
$587$	−3309.98	−0.232739	−0.116369	−	0.993206i	$-0.537126\pi$
$587$	−3309.98	−0.232739	−0.116369	+	0.993206i	$0.537126\pi$
$588$	−5956.93	−0.417789
$589$	−11559.3	−0.808647
$590$	0	0
$591$	−8770.26	−0.610424
$592$	19341.7	1.34281
$593$	−22674.2	−1.57018	−0.785091	−	0.619381i	$-0.787384\pi$
$593$	−22674.2	−1.57018	−0.785091	+	0.619381i	$0.787384\pi$
$594$	1312.85	0.0906848
$595$	0	0
$596$	−28480.6	−1.95740
$597$	−11201.1	−0.767888
$598$	387.800	0.0265189
$599$	−26557.9	−1.81157	−0.905783	−	0.423742i	$-0.860716\pi$
$599$	−26557.9	−1.81157	−0.905783	+	0.423742i	$0.860716\pi$
$600$	0	0
$601$	−10465.7	−0.710326	−0.355163	−	0.934804i	$-0.615575\pi$
$601$	−10465.7	−0.710326	−0.355163	+	0.934804i	$0.615575\pi$
$602$	32199.3	2.17997
$603$	−6060.57	−0.409296
$604$	−14021.0	−0.944550
$605$	0	0
$606$	5134.49	0.344182
$607$	26709.9	1.78603	0.893015	−	0.450027i	$-0.148586\pi$
$607$	26709.9	1.78603	0.893015	+	0.450027i	$0.148586\pi$
$608$	3119.87	0.208104
$609$	−9654.42	−0.642392
$610$	0	0
$611$	−1422.24	−0.0941695
$612$	13413.2	0.885943
$613$	2826.13	0.186209	0.0931046	−	0.995656i	$-0.470321\pi$
$613$	2826.13	0.186209	0.0931046	+	0.995656i	$0.470321\pi$
$614$	22716.2	1.49308
$615$	0	0
$616$	5166.18	0.337908
$617$	−9037.79	−0.589705	−0.294852	−	0.955543i	$-0.595270\pi$
$617$	−9037.79	−0.589705	−0.294852	+	0.955543i	$0.595270\pi$
$618$	−3834.80	−0.249609
$619$	17609.1	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$619$	17609.1	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$620$	0	0
$621$	427.602	0.0276314
$622$	−2014.83	−0.129883
$623$	−289.419	−0.0186121
$624$	718.959	0.0461241
$625$	0	0
$626$	2296.48	0.146622
$627$	−1775.18	−0.113069
$628$	10799.0	0.686191
$629$	39952.9	2.53263
$630$	0	0
$631$	21956.7	1.38524	0.692618	−	0.721305i	$-0.256457\pi$
$631$	21956.7	1.38524	0.692618	+	0.721305i	$0.256457\pi$
$632$	−27188.2	−1.71122
$633$	−4164.08	−0.261465
$634$	−50624.6	−3.17123
$635$	0	0
$636$	26866.9	1.67507
$637$	−659.549	−0.0410240
$638$	−10720.7	−0.665259
$639$	−3457.20	−0.214029
$640$	0	0
$641$	11686.0	0.720075	0.360037	−	0.932938i	$-0.382764\pi$
$641$	11686.0	0.720075	0.360037	+	0.932938i	$0.382764\pi$
$642$	19688.2	1.21033
$643$	−21869.7	−1.34130	−0.670651	−	0.741773i	$-0.733985\pi$
$643$	−21869.7	−1.34130	−0.670651	+	0.741773i	$0.733985\pi$
$644$	3531.93	0.216114
$645$	0	0
$646$	−27633.1	−1.68298
$647$	5865.88	0.356432	0.178216	−	0.983991i	$-0.442967\pi$
$647$	5865.88	0.356432	0.178216	+	0.983991i	$0.442967\pi$
$648$	2844.82	0.172462
$649$	1943.43	0.117544
$650$	0	0
$651$	8620.47	0.518991
$652$	48666.4	2.92319
$653$	−18388.0	−1.10196	−0.550980	−	0.834519i	$-0.685746\pi$
$653$	−18388.0	−1.10196	−0.550980	+	0.834519i	$0.685746\pi$
$654$	9478.33	0.566715
$655$	0	0
$656$	9475.50	0.563957
$657$	−2562.36	−0.152157
$658$	−19735.3	−1.16924
$659$	25899.5	1.53096	0.765478	−	0.643462i	$-0.222503\pi$
$659$	25899.5	1.53096	0.765478	+	0.643462i	$0.222503\pi$
$660$	0	0
$661$	9947.37	0.585337	0.292669	−	0.956214i	$-0.405457\pi$
$661$	9947.37	0.585337	0.292669	+	0.956214i	$0.405457\pi$
$662$	−53112.2	−3.11823
$663$	1485.10	0.0869935
$664$	−46054.0	−2.69163
$665$	0	0
$666$	17786.3	1.03484
$667$	−3491.79	−0.202702
$668$	35644.2	2.06455
$669$	14841.5	0.857706
$670$	0	0
$671$	174.192	0.0100218
$672$	−2326.67	−0.133562
$673$	−8891.28	−0.509262	−0.254631	−	0.967038i	$-0.581954\pi$
$673$	−8891.28	−0.509262	−0.254631	+	0.967038i	$0.581954\pi$
$674$	−11881.2	−0.679001
$675$	0	0
$676$	−33174.9	−1.88751
$677$	19321.6	1.09688	0.548441	−	0.836190i	$-0.315222\pi$
$677$	19321.6	1.09688	0.548441	+	0.836190i	$0.315222\pi$
$678$	−26544.0	−1.50356
$679$	23218.3	1.31228
$680$	0	0
$681$	−13996.6	−0.787594
$682$	9572.52	0.537465
$683$	−7010.63	−0.392759	−0.196380	−	0.980528i	$-0.562918\pi$
$683$	−7010.63	−0.392759	−0.196380	+	0.980528i	$0.562918\pi$
$684$	−8074.21	−0.451352
$685$	0	0
$686$	−33307.3	−1.85376
$687$	9748.02	0.541354
$688$	21590.6	1.19641
$689$	2974.69	0.164480
$690$	0	0
$691$	−4857.64	−0.267429	−0.133714	−	0.991020i	$-0.542690\pi$
$691$	−4857.64	−0.267429	−0.133714	+	0.991020i	$0.542690\pi$
$692$	−954.145	−0.0524149
$693$	1323.86	0.0725675
$694$	916.853	0.0501488
$695$	0	0
$696$	−23230.7	−1.26517
$697$	19572.9	1.06367
$698$	31898.5	1.72977
$699$	693.912	0.0375481
$700$	0	0
$701$	−14721.0	−0.793156	−0.396578	−	0.918001i	$-0.629802\pi$
$701$	−14721.0	−0.793156	−0.396578	+	0.918001i	$0.629802\pi$
$702$	661.143	0.0355459
$703$	−24050.0	−1.29028
$704$	−6390.71	−0.342129
$705$	0	0
$706$	17096.8	0.911397
$707$	5177.57	0.275421
$708$	8839.44	0.469218
$709$	12470.9	0.660584	0.330292	−	0.943879i	$-0.392853\pi$
$709$	12470.9	0.660584	0.330292	+	0.943879i	$0.392853\pi$
$710$	0	0
$711$	−6967.12	−0.367493
$712$	−696.408	−0.0366559
$713$	3117.83	0.163764
$714$	20607.6	1.08014
$715$	0	0
$716$	16070.0	0.838775
$717$	7034.74	0.366412
$718$	8895.79	0.462379
$719$	−12044.6	−0.624738	−0.312369	−	0.949961i	$-0.601123\pi$
$719$	−12044.6	−0.624738	−0.312369	+	0.949961i	$0.601123\pi$
$720$	0	0
$721$	−3866.97	−0.199741
$722$	−16459.7	−0.848430
$723$	−2916.17	−0.150005
$724$	−35574.3	−1.82612
$725$	0	0
$726$	−17795.6	−0.909719
$727$	−7141.34	−0.364316	−0.182158	−	0.983269i	$-0.558308\pi$
$727$	−7141.34	−0.364316	−0.182158	+	0.983269i	$0.558308\pi$
$728$	2601.66	0.132450
$729$	729.000	0.0370370
$730$	0	0
$731$	44598.1	2.25653
$732$	792.293	0.0400055
$733$	−34632.8	−1.74514	−0.872572	−	0.488485i	$-0.837550\pi$
$733$	−34632.8	−1.74514	−0.872572	+	0.488485i	$0.837550\pi$
$734$	−52240.1	−2.62700
$735$	0	0
$736$	−841.506	−0.0421444
$737$	−6786.36	−0.339184
$738$	8713.50	0.434619
$739$	23868.4	1.18811	0.594056	−	0.804424i	$-0.297526\pi$
$739$	23868.4	1.18811	0.594056	+	0.804424i	$0.297526\pi$
$740$	0	0
$741$	−893.973	−0.0443197
$742$	41277.5	2.04224
$743$	−5629.99	−0.277987	−0.138993	−	0.990293i	$-0.544387\pi$
$743$	−5629.99	−0.277987	−0.138993	+	0.990293i	$0.544387\pi$
$744$	20742.8	1.02213
$745$	0	0
$746$	−52540.7	−2.57862
$747$	−11801.6	−0.578042
$748$	15019.5	0.734182
$749$	19853.3	0.968525
$750$	0	0
$751$	−13201.9	−0.641471	−0.320736	−	0.947169i	$-0.603930\pi$
$751$	−13201.9	−0.641471	−0.320736	+	0.947169i	$0.603930\pi$
$752$	−13233.1	−0.641703
$753$	−17451.7	−0.844587
$754$	−5398.87	−0.260763
$755$	0	0
$756$	6021.42	0.289678
$757$	20592.0	0.988680	0.494340	−	0.869269i	$-0.335410\pi$
$757$	20592.0	0.988680	0.494340	+	0.869269i	$0.335410\pi$
$758$	−31161.9	−1.49321
$759$	478.810	0.0228982
$760$	0	0
$761$	527.855	0.0251442	0.0125721	−	0.999921i	$-0.495998\pi$
$761$	527.855	0.0251442	0.0125721	+	0.999921i	$0.495998\pi$
$762$	4571.76	0.217346
$763$	9557.84	0.453495
$764$	−18594.2	−0.880517
$765$	0	0
$766$	−5991.93	−0.282633
$767$	978.699	0.0460740
$768$	−22914.6	−1.07664
$769$	8694.99	0.407737	0.203868	−	0.978998i	$-0.434649\pi$
$769$	8694.99	0.407737	0.203868	+	0.978998i	$0.434649\pi$
$770$	0	0
$771$	5986.53	0.279636
$772$	−9372.32	−0.436939
$773$	39872.6	1.85526	0.927632	−	0.373495i	$-0.121841\pi$
$773$	39872.6	1.85526	0.927632	+	0.373495i	$0.121841\pi$
$774$	19854.3	0.922027
$775$	0	0
$776$	55868.6	2.58449
$777$	17935.5	0.828100
$778$	−62468.2	−2.87865
$779$	−11782.1	−0.541896
$780$	0	0
$781$	−3871.22	−0.177366
$782$	7453.31	0.340831
$783$	−5952.99	−0.271702
$784$	−6136.71	−0.279552
$785$	0	0
$786$	4737.85	0.215005
$787$	12956.4	0.586845	0.293423	−	0.955983i	$-0.405206\pi$
$787$	12956.4	0.586845	0.293423	+	0.955983i	$0.405206\pi$
$788$	−44667.6	−2.01931
$789$	−12394.4	−0.559256
$790$	0	0
$791$	−26766.7	−1.20318
$792$	3185.51	0.142919
$793$	87.7223	0.00392826
$794$	−52190.4	−2.33270
$795$	0	0
$796$	−57047.9	−2.54021
$797$	−21323.8	−0.947713	−0.473857	−	0.880602i	$-0.657139\pi$
$797$	−21323.8	−0.947713	−0.473857	+	0.880602i	$0.657139\pi$
$798$	−12405.0	−0.550289
$799$	−27334.7	−1.21030
$800$	0	0
$801$	−178.458	−0.00787204
$802$	16760.0	0.737923
$803$	−2869.22	−0.126093
$804$	−30867.0	−1.35397
$805$	0	0
$806$	4820.67	0.210671
$807$	13789.6	0.601507
$808$	12458.4	0.542432
$809$	1640.13	0.0712781	0.0356391	−	0.999365i	$-0.488653\pi$
$809$	1640.13	0.0712781	0.0356391	+	0.999365i	$0.488653\pi$
$810$	0	0
$811$	15959.7	0.691025	0.345512	−	0.938414i	$-0.387705\pi$
$811$	15959.7	0.691025	0.345512	+	0.938414i	$0.387705\pi$
$812$	−49170.7	−2.12507
$813$	−4822.51	−0.208036
$814$	19916.4	0.857577
$815$	0	0
$816$	13818.0	0.592804
$817$	−26846.3	−1.14961
$818$	73640.9	3.14767
$819$	666.689	0.0284444
$820$	0	0
$821$	5803.88	0.246720	0.123360	−	0.992362i	$-0.460633\pi$
$821$	5803.88	0.246720	0.123360	+	0.992362i	$0.460633\pi$
$822$	42179.4	1.78975
$823$	26562.8	1.12506	0.562528	−	0.826779i	$-0.309829\pi$
$823$	26562.8	1.12506	0.562528	+	0.826779i	$0.309829\pi$
$824$	−9304.81	−0.393384
$825$	0	0
$826$	13580.6	0.572071
$827$	−21278.3	−0.894703	−0.447352	−	0.894358i	$-0.647633\pi$
$827$	−21278.3	−0.894703	−0.447352	+	0.894358i	$0.647633\pi$
$828$	2177.81	0.0914061
$829$	−4084.90	−0.171139	−0.0855697	−	0.996332i	$-0.527271\pi$
$829$	−4084.90	−0.171139	−0.0855697	+	0.996332i	$0.527271\pi$
$830$	0	0
$831$	19589.8	0.817764
$832$	−3218.33	−0.134105
$833$	−12676.2	−0.527255
$834$	−13476.4	−0.559534
$835$	0	0
$836$	−9041.14	−0.374037
$837$	5315.45	0.219509
$838$	−54082.0	−2.22939
$839$	−28121.6	−1.15717	−0.578584	−	0.815623i	$-0.696395\pi$
$839$	−28121.6	−1.15717	−0.578584	+	0.815623i	$0.696395\pi$
$840$	0	0
$841$	24222.9	0.993190
$842$	60796.6	2.48835
$843$	18010.6	0.735846
$844$	−21208.0	−0.864939
$845$	0	0
$846$	−12168.9	−0.494534
$847$	−17944.9	−0.727973
$848$	27677.8	1.12082
$849$	−26335.0	−1.06456
$850$	0	0
$851$	6486.88	0.261301
$852$	−17607.8	−0.708020
$853$	29669.7	1.19094	0.595471	−	0.803377i	$-0.296966\pi$
$853$	29669.7	1.19094	0.595471	+	0.803377i	$0.296966\pi$
$854$	1217.25	0.0487746
$855$	0	0
$856$	47771.6	1.90748
$857$	−4544.19	−0.181128	−0.0905639	−	0.995891i	$-0.528867\pi$
$857$	−4544.19	−0.181128	−0.0905639	+	0.995891i	$0.528867\pi$
$858$	740.318	0.0294569
$859$	30746.4	1.22125	0.610626	−	0.791919i	$-0.290918\pi$
$859$	30746.4	1.22125	0.610626	+	0.791919i	$0.290918\pi$
$860$	0	0
$861$	8786.60	0.347789
$862$	−7143.68	−0.282268
$863$	29828.1	1.17655	0.588275	−	0.808661i	$-0.299807\pi$
$863$	29828.1	1.17655	0.588275	+	0.808661i	$0.299807\pi$
$864$	−1434.64	−0.0564903
$865$	0	0
$866$	−6633.44	−0.260293
$867$	13803.9	0.540722
$868$	43904.7	1.71685
$869$	−7801.47	−0.304542
$870$	0	0
$871$	−3417.58	−0.132951
$872$	22998.3	0.893144
$873$	14316.6	0.555033
$874$	−4486.59	−0.173640
$875$	0	0
$876$	−13050.3	−0.503344
$877$	−10612.5	−0.408617	−0.204309	−	0.978907i	$-0.565495\pi$
$877$	−10612.5	−0.408617	−0.204309	+	0.978907i	$0.565495\pi$
$878$	31473.2	1.20976
$879$	−28170.4	−1.08096
$880$	0	0
$881$	15062.2	0.576003	0.288002	−	0.957630i	$-0.407009\pi$
$881$	15062.2	0.576003	0.288002	+	0.957630i	$0.407009\pi$
$882$	−5643.22	−0.215439
$883$	19380.8	0.738637	0.369318	−	0.929303i	$-0.379591\pi$
$883$	19380.8	0.738637	0.369318	+	0.929303i	$0.379591\pi$
$884$	7563.75	0.287779
$885$	0	0
$886$	53648.0	2.03424
$887$	30106.6	1.13966	0.569832	−	0.821761i	$-0.307008\pi$
$887$	30106.6	1.13966	0.569832	+	0.821761i	$0.307008\pi$
$888$	43157.0	1.63092
$889$	4610.11	0.173924
$890$	0	0
$891$	816.302	0.0306926
$892$	75588.9	2.83733
$893$	16454.4	0.616600
$894$	−26980.7	−1.00936
$895$	0	0
$896$	−38453.8	−1.43376
$897$	241.126	0.00897545
$898$	−82286.3	−3.05783
$899$	−43405.8	−1.61030
$900$	0	0
$901$	57172.0	2.11396
$902$	9757.00	0.360169
$903$	20020.9	0.737821
$904$	−64406.7	−2.36962
$905$	0	0
$906$	−13282.6	−0.487071
$907$	26364.7	0.965189	0.482594	−	0.875844i	$-0.339695\pi$
$907$	26364.7	0.965189	0.482594	+	0.875844i	$0.339695\pi$
$908$	−71285.9	−2.60540
$909$	3192.53	0.116490
$910$	0	0
$911$	32653.6	1.18756	0.593778	−	0.804629i	$-0.297636\pi$
$911$	32653.6	1.18756	0.593778	+	0.804629i	$0.297636\pi$
$912$	−8317.89	−0.302010
$913$	−13214.9	−0.479024
$914$	21255.4	0.769219
$915$	0	0
$916$	49647.4	1.79083
$917$	4777.60	0.172050
$918$	12706.8	0.456849
$919$	5291.84	0.189947	0.0949737	−	0.995480i	$-0.469723\pi$
$919$	5291.84	0.189947	0.0949737	+	0.995480i	$0.469723\pi$
$920$	0	0
$921$	14124.5	0.505340
$922$	16933.3	0.604845
$923$	−1949.53	−0.0695227
$924$	6742.52	0.240057
$925$	0	0
$926$	−13588.2	−0.482221
$927$	−2384.40	−0.0844813
$928$	11715.3	0.414410
$929$	−52969.5	−1.87069	−0.935346	−	0.353735i	$-0.884911\pi$
$929$	−52969.5	−1.87069	−0.935346	+	0.353735i	$0.884911\pi$
$930$	0	0
$931$	7630.55	0.268616
$932$	3534.15	0.124211
$933$	−1252.78	−0.0439595
$934$	−16598.2	−0.581487
$935$	0	0
$936$	1604.20	0.0560203
$937$	−29788.2	−1.03857	−0.519284	−	0.854602i	$-0.673801\pi$
$937$	−29788.2	−1.03857	−0.519284	+	0.854602i	$0.673801\pi$
$938$	−47423.0	−1.65076
$939$	1427.90	0.0496250
$940$	0	0
$941$	−35950.4	−1.24543	−0.622714	−	0.782449i	$-0.713970\pi$
$941$	−35950.4	−1.24543	−0.622714	+	0.782449i	$0.713970\pi$
$942$	10230.3	0.353844
$943$	3177.92	0.109742
$944$	9106.22	0.313964
$945$	0	0
$946$	22232.0	0.764085
$947$	29852.4	1.02436	0.512182	−	0.858877i	$-0.328837\pi$
$947$	29852.4	1.02436	0.512182	+	0.858877i	$0.328837\pi$
$948$	−35484.1	−1.21568
$949$	−1444.92	−0.0494249
$950$	0	0
$951$	−31477.4	−1.07332
$952$	50002.6	1.70230
$953$	−10092.7	−0.343058	−0.171529	−	0.985179i	$-0.554871\pi$
$953$	−10092.7	−0.343058	−0.171529	+	0.985179i	$0.554871\pi$
$954$	25452.0	0.863772
$955$	0	0
$956$	35828.5	1.21211
$957$	−6665.90	−0.225160
$958$	−8527.80	−0.287600
$959$	42533.3	1.43219
$960$	0	0
$961$	8966.21	0.300970
$962$	10029.8	0.336146
$963$	12241.7	0.409641
$964$	−14852.3	−0.496225
$965$	0	0
$966$	3345.92	0.111442
$967$	32797.3	1.09068	0.545341	−	0.838214i	$-0.316400\pi$
$967$	32797.3	1.09068	0.545341	+	0.838214i	$0.316400\pi$
$968$	−43179.4	−1.43372
$969$	−17181.7	−0.569613
$970$	0	0
$971$	8700.78	0.287561	0.143780	−	0.989610i	$-0.454074\pi$
$971$	8700.78	0.287561	0.143780	+	0.989610i	$0.454074\pi$
$972$	3712.85	0.122520
$973$	−13589.5	−0.447749
$974$	20699.1	0.680945
$975$	0	0
$976$	816.205	0.0267685
$977$	1703.74	0.0557908	0.0278954	−	0.999611i	$-0.491119\pi$
$977$	1703.74	0.0557908	0.0278954	+	0.999611i	$0.491119\pi$
$978$	46103.4	1.50739
$979$	−199.830	−0.00652358
$980$	0	0
$981$	5893.44	0.191807
$982$	62900.6	2.04403
$983$	−28395.4	−0.921335	−0.460667	−	0.887573i	$-0.652390\pi$
$983$	−28395.4	−0.921335	−0.460667	+	0.887573i	$0.652390\pi$
$984$	21142.5	0.684959
$985$	0	0
$986$	−103763.	−3.35142
$987$	−12271.0	−0.395735
$988$	−4553.07	−0.146612
$989$	7241.10	0.232814
$990$	0	0
$991$	35097.9	1.12505	0.562523	−	0.826782i	$-0.309831\pi$
$991$	35097.9	1.12505	0.562523	+	0.826782i	$0.309831\pi$
$992$	−10460.6	−0.334803
$993$	−33024.1	−1.05538
$994$	−27052.0	−0.863218
$995$	0	0
$996$	−60106.4	−1.91219
$997$	−36374.9	−1.15547	−0.577736	−	0.816224i	$-0.696064\pi$
$997$	−36374.9	−1.15547	−0.577736	+	0.816224i	$0.696064\pi$
$998$	30021.4	0.952215
$999$	11059.2	0.350248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.l.1.21	yes	24
5.4	even	2		1875.4.a.k.1.4	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.4.a.k.1.4	✓	24	5.4	even	2
1875.4.a.l.1.21	yes	24	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+4.82486 q^{2} +3.00000 q^{3} +15.2792 q^{4} +14.4746 q^{6} +14.5960 q^{7} +35.1213 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.82486 q^{2} +3.00000 q^{3} +15.2792 q^{4} +14.4746 q^{6} +14.5960 q^{7} +35.1213 q^{8} +9.00000 q^{9} +10.0778 q^{11} +45.8377 q^{12} +5.07513 q^{13} +70.4236 q^{14} +47.2211 q^{16} +97.5413 q^{17} +43.4237 q^{18} -58.7159 q^{19} +43.7880 q^{21} +48.6240 q^{22} +15.8371 q^{23} +105.364 q^{24} +24.4868 q^{26} +27.0000 q^{27} +223.016 q^{28} -220.481 q^{29} +196.868 q^{31} -53.1350 q^{32} +30.2334 q^{33} +470.623 q^{34} +137.513 q^{36} +409.600 q^{37} -283.296 q^{38} +15.2254 q^{39} +200.662 q^{41} +211.271 q^{42} +457.223 q^{43} +153.981 q^{44} +76.4119 q^{46} -280.237 q^{47} +141.663 q^{48} -129.957 q^{49} +292.624 q^{51} +77.5441 q^{52} +586.131 q^{53} +130.271 q^{54} +512.629 q^{56} -176.148 q^{57} -1063.79 q^{58} +192.842 q^{59} +17.2847 q^{61} +949.862 q^{62} +131.364 q^{63} -634.138 q^{64} +145.872 q^{66} -673.397 q^{67} +1490.36 q^{68} +47.5114 q^{69} -384.133 q^{71} +316.091 q^{72} -284.707 q^{73} +1976.26 q^{74} -897.134 q^{76} +147.096 q^{77} +73.4603 q^{78} -774.124 q^{79} +81.0000 q^{81} +968.167 q^{82} -1311.29 q^{83} +669.047 q^{84} +2206.03 q^{86} -661.443 q^{87} +353.945 q^{88} -19.8287 q^{89} +74.0765 q^{91} +241.979 q^{92} +590.605 q^{93} -1352.10 q^{94} -159.405 q^{96} +1590.73 q^{97} -627.024 q^{98} +90.7002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 72 q^{3} + 133 q^{4} + 3 q^{6} + 62 q^{7} + 27 q^{8} + 216 q^{9}+O(q^{10}) \) 24 * q + q^2 + 72 * q^3 + 133 * q^4 + 3 * q^6 + 62 * q^7 + 27 * q^8 + 216 * q^9	\( 24 q + q^{2} + 72 q^{3} + 133 q^{4} + 3 q^{6} + 62 q^{7} + 27 q^{8} + 216 q^{9} + 96 q^{11} + 399 q^{12} + 156 q^{13} + 92 q^{14} + 845 q^{16} - 46 q^{17} + 9 q^{18} + 182 q^{19} + 186 q^{21} + 158 q^{22} - 286 q^{23} + 81 q^{24} + 478 q^{26} + 648 q^{27} + 701 q^{28} + 1144 q^{29} + 64 q^{31} + 1212 q^{32} + 288 q^{33} + 961 q^{34} + 1197 q^{36} + 762 q^{37} + 474 q^{38} + 468 q^{39} + 1074 q^{41} + 276 q^{42} + 460 q^{43} + 319 q^{44} + 459 q^{46} - 960 q^{47} + 2535 q^{48} + 2680 q^{49} - 138 q^{51} + 2969 q^{52} + 914 q^{53} + 27 q^{54} + 1680 q^{56} + 546 q^{57} + 208 q^{58} + 208 q^{59} + 3520 q^{61} + 334 q^{62} + 558 q^{63} + 5747 q^{64} + 474 q^{66} + 154 q^{67} - 5727 q^{68} - 858 q^{69} - 252 q^{71} + 243 q^{72} + 4414 q^{73} + 5637 q^{74} + 627 q^{76} + 2344 q^{77} + 1434 q^{78} + 1110 q^{79} + 1944 q^{81} + 3714 q^{82} - 1488 q^{83} + 2103 q^{84} + 3036 q^{86} + 3432 q^{87} + 3947 q^{88} + 3402 q^{89} + 3504 q^{91} - 11163 q^{92} + 192 q^{93} + 3408 q^{94} + 3636 q^{96} + 534 q^{97} + 2244 q^{98} + 864 q^{99}+O(q^{100}) \) 24 * q + q^2 + 72 * q^3 + 133 * q^4 + 3 * q^6 + 62 * q^7 + 27 * q^8 + 216 * q^9 + 96 * q^11 + 399 * q^12 + 156 * q^13 + 92 * q^14 + 845 * q^16 - 46 * q^17 + 9 * q^18 + 182 * q^19 + 186 * q^21 + 158 * q^22 - 286 * q^23 + 81 * q^24 + 478 * q^26 + 648 * q^27 + 701 * q^28 + 1144 * q^29 + 64 * q^31 + 1212 * q^32 + 288 * q^33 + 961 * q^34 + 1197 * q^36 + 762 * q^37 + 474 * q^38 + 468 * q^39 + 1074 * q^41 + 276 * q^42 + 460 * q^43 + 319 * q^44 + 459 * q^46 - 960 * q^47 + 2535 * q^48 + 2680 * q^49 - 138 * q^51 + 2969 * q^52 + 914 * q^53 + 27 * q^54 + 1680 * q^56 + 546 * q^57 + 208 * q^58 + 208 * q^59 + 3520 * q^61 + 334 * q^62 + 558 * q^63 + 5747 * q^64 + 474 * q^66 + 154 * q^67 - 5727 * q^68 - 858 * q^69 - 252 * q^71 + 243 * q^72 + 4414 * q^73 + 5637 * q^74 + 627 * q^76 + 2344 * q^77 + 1434 * q^78 + 1110 * q^79 + 1944 * q^81 + 3714 * q^82 - 1488 * q^83 + 2103 * q^84 + 3036 * q^86 + 3432 * q^87 + 3947 * q^88 + 3402 * q^89 + 3504 * q^91 - 11163 * q^92 + 192 * q^93 + 3408 * q^94 + 3636 * q^96 + 534 * q^97 + 2244 * q^98 + 864 * q^99

Introduction

L-functions

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