LMFDB - Embedded newform 1925.4.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,4,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1925.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1925.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:	$113.578676761$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.20317$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.89098 q^{2} -6.57251 q^{3} +15.9217 q^{4} -32.1460 q^{6} +7.00000 q^{7} +38.7449 q^{8} +16.1978 q^{9} -11.0000 q^{11} -104.646 q^{12} -74.3459 q^{13} +34.2369 q^{14} +62.1271 q^{16} +94.0836 q^{17} +79.2234 q^{18} +135.682 q^{19} -46.0075 q^{21} -53.8008 q^{22} -81.1793 q^{23} -254.651 q^{24} -363.625 q^{26} +70.9972 q^{27} +111.452 q^{28} -53.4259 q^{29} -9.50536 q^{31} -6.09673 q^{32} +72.2976 q^{33} +460.161 q^{34} +257.897 q^{36} +9.14224 q^{37} +663.620 q^{38} +488.639 q^{39} -339.461 q^{41} -225.022 q^{42} -433.078 q^{43} -175.139 q^{44} -397.046 q^{46} +54.4784 q^{47} -408.331 q^{48} +49.0000 q^{49} -618.365 q^{51} -1183.71 q^{52} -123.830 q^{53} +347.246 q^{54} +271.215 q^{56} -891.773 q^{57} -261.305 q^{58} -534.396 q^{59} -358.624 q^{61} -46.4905 q^{62} +113.385 q^{63} -526.836 q^{64} +353.606 q^{66} +694.318 q^{67} +1497.97 q^{68} +533.551 q^{69} -278.330 q^{71} +627.585 q^{72} +886.688 q^{73} +44.7145 q^{74} +2160.29 q^{76} -77.0000 q^{77} +2389.93 q^{78} -185.631 q^{79} -903.972 q^{81} -1660.30 q^{82} +122.624 q^{83} -732.519 q^{84} -2118.18 q^{86} +351.142 q^{87} -426.194 q^{88} -847.086 q^{89} -520.422 q^{91} -1292.51 q^{92} +62.4740 q^{93} +266.453 q^{94} +40.0708 q^{96} -1002.49 q^{97} +239.658 q^{98} -178.176 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 14 q^{3} + 26 q^{4} + 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 14 q^{14} + 2 q^{16} - 4 q^{17} + 62 q^{18} + 258 q^{19} - 98 q^{21} - 22 q^{22} - 8 q^{23}+ \cdots - 836 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 14 * q^3 + 26 * q^4 + 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 14 * q^14 + 2 * q^16 - 4 * q^17 + 62 * q^18 + 258 * q^19 - 98 * q^21 - 22 * q^22 - 8 * q^23 - 498 * q^24 - 482 * q^26 - 428 * q^27 + 182 * q^28 - 396 * q^29 - 56 * q^31 - 134 * q^32 + 154 * q^33 + 472 * q^34 - 418 * q^36 - 84 * q^37 + 942 * q^38 - 412 * q^39 + 52 * q^41 + 98 * q^42 - 408 * q^43 - 286 * q^44 + 368 * q^46 - 8 * q^47 - 982 * q^48 + 196 * q^49 - 388 * q^51 - 2030 * q^52 - 624 * q^53 + 92 * q^54 + 126 * q^56 - 48 * q^57 - 864 * q^58 - 238 * q^59 - 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 - 154 * q^66 - 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 3042 * q^76 - 308 * q^77 + 2632 * q^78 - 1324 * q^79 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 490 * q^84 - 4380 * q^86 - 588 * q^87 - 198 * q^88 - 3072 * q^89 - 406 * q^91 - 544 * q^92 + 1264 * q^93 - 1696 * q^94 + 862 * q^96 + 652 * q^97 + 98 * q^98 - 836 * q^99

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 $

$2$	4.89098	1.72922	0.864612	−	0.502441i	$-0.167564\pi$
$2$	4.89098	1.72922	0.864612	+	0.502441i	$0.167564\pi$
$3$	−6.57251	−1.26488	−0.632440	−	0.774610i	$-0.717946\pi$
$3$	−6.57251	−1.26488	−0.632440	+	0.774610i	$0.717946\pi$
$4$	15.9217	1.99021
$5$	0	0
$5$	0	0
$6$	−32.1460	−2.18726
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	38.7449	1.71230
$9$	16.1978	0.599920
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−104.646	−2.51738
$13$	−74.3459	−1.58614	−0.793071	−	0.609129i	$-0.791519\pi$
$13$	−74.3459	−1.58614	−0.793071	+	0.609129i	$0.791519\pi$
$14$	34.2369	0.653585
$15$	0	0
$16$	62.1271	0.970737
$17$	94.0836	1.34227	0.671136	−	0.741334i	$-0.265807\pi$
$17$	94.0836	1.34227	0.671136	+	0.741334i	$0.265807\pi$
$18$	79.2234	1.03740
$19$	135.682	1.63830	0.819149	−	0.573581i	$-0.194446\pi$
$19$	135.682	1.63830	0.819149	+	0.573581i	$0.194446\pi$
$20$	0	0
$21$	−46.0075	−0.478080
$22$	−53.8008	−0.521380
$23$	−81.1793	−0.735959	−0.367979	−	0.929834i	$-0.619950\pi$
$23$	−81.1793	−0.735959	−0.367979	+	0.929834i	$0.619950\pi$
$24$	−254.651	−2.16585
$25$	0	0
$26$	−363.625	−2.74279
$27$	70.9972	0.506053
$28$	111.452	0.752230
$29$	−53.4259	−0.342102	−0.171051	−	0.985262i	$-0.554716\pi$
$29$	−53.4259	−0.342102	−0.171051	+	0.985262i	$0.554716\pi$
$30$	0	0
$31$	−9.50536	−0.0550714	−0.0275357	−	0.999621i	$-0.508766\pi$
$31$	−9.50536	−0.0550714	−0.0275357	+	0.999621i	$0.508766\pi$
$32$	−6.09673	−0.0336800
$33$	72.2976	0.381376
$34$	460.161	2.32109
$35$	0	0
$36$	257.897	1.19397
$37$	9.14224	0.0406209	0.0203105	−	0.999794i	$-0.493535\pi$
$37$	9.14224	0.0406209	0.0203105	+	0.999794i	$0.493535\pi$
$38$	663.620	2.83298
$39$	488.639	2.00628
$40$	0	0
$41$	−339.461	−1.29305	−0.646523	−	0.762894i	$-0.723778\pi$
$41$	−339.461	−1.29305	−0.646523	+	0.762894i	$0.723778\pi$
$42$	−225.022	−0.826706
$43$	−433.078	−1.53590	−0.767950	−	0.640509i	$-0.778723\pi$
$43$	−433.078	−1.53590	−0.767950	+	0.640509i	$0.778723\pi$
$44$	−175.139	−0.600072
$45$	0	0
$46$	−397.046	−1.27264
$47$	54.4784	0.169074	0.0845371	−	0.996420i	$-0.473059\pi$
$47$	54.4784	0.169074	0.0845371	+	0.996420i	$0.473059\pi$
$48$	−408.331	−1.22786
$49$	49.0000	0.142857
$50$	0	0
$51$	−618.365	−1.69781
$52$	−1183.71	−3.15676
$53$	−123.830	−0.320932	−0.160466	−	0.987041i	$-0.551300\pi$
$53$	−123.830	−0.320932	−0.160466	+	0.987041i	$0.551300\pi$
$54$	347.246	0.875078
$55$	0	0
$56$	271.215	0.647189
$57$	−891.773	−2.07225
$58$	−261.305	−0.591570
$59$	−534.396	−1.17919	−0.589596	−	0.807698i	$-0.700713\pi$
$59$	−534.396	−1.17919	−0.589596	+	0.807698i	$0.700713\pi$
$60$	0	0
$61$	−358.624	−0.752740	−0.376370	−	0.926469i	$-0.622828\pi$
$61$	−358.624	−0.752740	−0.376370	+	0.926469i	$0.622828\pi$
$62$	−46.4905	−0.0952307
$63$	113.385	0.226749
$64$	−526.836	−1.02898
$65$	0	0
$66$	353.606	0.659484
$67$	694.318	1.26604	0.633019	−	0.774137i	$-0.281816\pi$
$67$	694.318	1.26604	0.633019	+	0.774137i	$0.281816\pi$
$68$	1497.97	2.67141
$69$	533.551	0.930899
$70$	0	0
$71$	−278.330	−0.465236	−0.232618	−	0.972568i	$-0.574729\pi$
$71$	−278.330	−0.465236	−0.232618	+	0.972568i	$0.574729\pi$
$72$	627.585	1.02724
$73$	886.688	1.42163	0.710815	−	0.703379i	$-0.248326\pi$
$73$	886.688	1.42163	0.710815	+	0.703379i	$0.248326\pi$
$74$	44.7145	0.0702427
$75$	0	0
$76$	2160.29	3.26056
$77$	−77.0000	−0.113961
$78$	2389.93	3.46931
$79$	−185.631	−0.264369	−0.132184	−	0.991225i	$-0.542199\pi$
$79$	−185.631	−0.264369	−0.132184	+	0.991225i	$0.542199\pi$
$80$	0	0
$81$	−903.972	−1.24002
$82$	−1660.30	−2.23597
$83$	122.624	0.162166	0.0810830	−	0.996707i	$-0.474162\pi$
$83$	122.624	0.162166	0.0810830	+	0.996707i	$0.474162\pi$
$84$	−732.519	−0.951480
$85$	0	0
$86$	−2118.18	−2.65592
$87$	351.142	0.432717
$88$	−426.194	−0.516278
$89$	−847.086	−1.00889	−0.504443	−	0.863445i	$-0.668302\pi$
$89$	−847.086	−1.00889	−0.504443	+	0.863445i	$0.668302\pi$
$90$	0	0
$91$	−520.422	−0.599506
$92$	−1292.51	−1.46471
$93$	62.4740	0.0696586
$94$	266.453	0.292367
$95$	0	0
$96$	40.0708	0.0426011
$97$	−1002.49	−1.04935	−0.524676	−	0.851302i	$-0.675814\pi$
$97$	−1002.49	−1.04935	−0.524676	+	0.851302i	$0.675814\pi$
$98$	239.658	0.247032
$99$	−178.176	−0.180883
$100$	0	0
$101$	1124.79	1.10812	0.554062	−	0.832476i	$-0.313077\pi$
$101$	1124.79	1.10812	0.554062	+	0.832476i	$0.313077\pi$
$102$	−3024.41	−2.93590
$103$	−966.118	−0.924218	−0.462109	−	0.886823i	$-0.652907\pi$
$103$	−966.118	−0.924218	−0.462109	+	0.886823i	$0.652907\pi$
$104$	−2880.53	−2.71595
$105$	0	0
$106$	−605.652	−0.554963
$107$	144.202	0.130286	0.0651428	−	0.997876i	$-0.479250\pi$
$107$	144.202	0.130286	0.0651428	+	0.997876i	$0.479250\pi$
$108$	1130.40	1.00715
$109$	−1875.32	−1.64792	−0.823961	−	0.566647i	$-0.808240\pi$
$109$	−1875.32	−1.64792	−0.823961	+	0.566647i	$0.808240\pi$
$110$	0	0
$111$	−60.0874	−0.0513806
$112$	434.890	0.366904
$113$	−1207.46	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$113$	−1207.46	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$114$	−4361.64	−3.58338
$115$	0	0
$116$	−850.632	−0.680855
$117$	−1204.24	−0.951559
$118$	−2613.72	−2.03909
$119$	658.585	0.507331
$120$	0	0
$121$	121.000	0.0909091
$122$	−1754.03	−1.30166
$123$	2231.11	1.63555
$124$	−151.342	−0.109604
$125$	0	0
$126$	554.564	0.392099
$127$	−1143.56	−0.799009	−0.399504	−	0.916731i	$-0.630818\pi$
$127$	−1143.56	−0.799009	−0.399504	+	0.916731i	$0.630818\pi$
$128$	−2527.97	−1.74565
$129$	2846.41	1.94273
$130$	0	0
$131$	2478.40	1.65297	0.826484	−	0.562961i	$-0.190338\pi$
$131$	2478.40	1.65297	0.826484	+	0.562961i	$0.190338\pi$
$132$	1151.10	0.759019
$133$	949.776	0.619218
$134$	3395.90	2.18926
$135$	0	0
$136$	3645.26	2.29837
$137$	835.661	0.521134	0.260567	−	0.965456i	$-0.416091\pi$
$137$	835.661	0.521134	0.260567	+	0.965456i	$0.416091\pi$
$138$	2609.59	1.60973
$139$	−1726.99	−1.05382	−0.526912	−	0.849920i	$-0.676650\pi$
$139$	−1726.99	−1.05382	−0.526912	+	0.849920i	$0.676650\pi$
$140$	0	0
$141$	−358.060	−0.213859
$142$	−1361.31	−0.804497
$143$	817.805	0.478240
$144$	1006.33	0.582365
$145$	0	0
$146$	4336.78	2.45832
$147$	−322.053	−0.180697
$148$	145.560	0.0808443
$149$	−3454.95	−1.89960	−0.949799	−	0.312859i	$-0.898713\pi$
$149$	−3454.95	−1.89960	−0.949799	+	0.312859i	$0.898713\pi$
$150$	0	0
$151$	−2468.17	−1.33018	−0.665089	−	0.746764i	$-0.731606\pi$
$151$	−2468.17	−1.33018	−0.665089	+	0.746764i	$0.731606\pi$
$152$	5257.00	2.80526
$153$	1523.95	0.805256
$154$	−376.606	−0.197063
$155$	0	0
$156$	7779.97	3.99292
$157$	1561.27	0.793649	0.396824	−	0.917895i	$-0.370112\pi$
$157$	1561.27	0.793649	0.396824	+	0.917895i	$0.370112\pi$
$158$	−907.919	−0.457153
$159$	813.875	0.405940
$160$	0	0
$161$	−568.255	−0.278166
$162$	−4421.31	−2.14426
$163$	−3458.59	−1.66195	−0.830973	−	0.556312i	$-0.812216\pi$
$163$	−3458.59	−1.66195	−0.830973	+	0.556312i	$0.812216\pi$
$164$	−5404.80	−2.57344
$165$	0	0
$166$	599.754	0.280421
$167$	972.527	0.450637	0.225319	−	0.974285i	$-0.427658\pi$
$167$	972.527	0.450637	0.225319	+	0.974285i	$0.427658\pi$
$168$	−1782.56	−0.818616
$169$	3330.32	1.51585
$170$	0	0
$171$	2197.76	0.982848
$172$	−6895.34	−3.05677
$173$	1154.18	0.507230	0.253615	−	0.967305i	$-0.418380\pi$
$173$	1154.18	0.507230	0.253615	+	0.967305i	$0.418380\pi$
$174$	1717.43	0.748265
$175$	0	0
$176$	−683.399	−0.292688
$177$	3512.32	1.49154
$178$	−4143.08	−1.74459
$179$	259.234	0.108246	0.0541231	−	0.998534i	$-0.482764\pi$
$179$	259.234	0.108246	0.0541231	+	0.998534i	$0.482764\pi$
$180$	0	0
$181$	−2121.49	−0.871209	−0.435604	−	0.900138i	$-0.643465\pi$
$181$	−2121.49	−0.871209	−0.435604	+	0.900138i	$0.643465\pi$
$182$	−2545.37	−1.03668
$183$	2357.06	0.952126
$184$	−3145.29	−1.26018
$185$	0	0
$186$	305.559	0.120455
$187$	−1034.92	−0.404710
$188$	867.389	0.336494
$189$	496.981	0.191270
$190$	0	0
$191$	2918.27	1.10554	0.552772	−	0.833332i	$-0.313570\pi$
$191$	2918.27	1.10554	0.552772	+	0.833332i	$0.313570\pi$
$192$	3462.63	1.30153
$193$	−3757.48	−1.40139	−0.700697	−	0.713459i	$-0.747128\pi$
$193$	−3757.48	−1.40139	−0.700697	+	0.713459i	$0.747128\pi$
$194$	−4903.15	−1.81456
$195$	0	0
$196$	780.164	0.284316
$197$	−1608.39	−0.581689	−0.290845	−	0.956770i	$-0.593936\pi$
$197$	−1608.39	−0.581689	−0.290845	+	0.956770i	$0.593936\pi$
$198$	−871.457	−0.312787
$199$	2865.53	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$199$	2865.53	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$200$	0	0
$201$	−4563.41	−1.60138
$202$	5501.31	1.91619
$203$	−373.981	−0.129302
$204$	−9845.43	−3.37901
$205$	0	0
$206$	−4725.27	−1.59818
$207$	−1314.93	−0.441517
$208$	−4618.90	−1.53973
$209$	−1492.50	−0.493965
$210$	0	0
$211$	821.996	0.268192	0.134096	−	0.990968i	$-0.457187\pi$
$211$	821.996	0.268192	0.134096	+	0.990968i	$0.457187\pi$
$212$	−1971.59	−0.638723
$213$	1829.33	0.588467
$214$	705.290	0.225293
$215$	0	0
$216$	2750.78	0.866514
$217$	−66.5375	−0.0208150
$218$	−9172.18	−2.84963
$219$	−5827.76	−1.79819
$220$	0	0
$221$	−6994.73	−2.12903
$222$	−293.887	−0.0888485
$223$	−109.532	−0.0328916	−0.0164458	−	0.999865i	$-0.505235\pi$
$223$	−109.532	−0.0328916	−0.0164458	+	0.999865i	$0.505235\pi$
$224$	−42.6771	−0.0127298
$225$	0	0
$226$	−5905.67	−1.73823
$227$	−3023.03	−0.883900	−0.441950	−	0.897040i	$-0.645713\pi$
$227$	−3023.03	−0.883900	−0.441950	+	0.897040i	$0.645713\pi$
$228$	−14198.5	−4.12422
$229$	−2278.75	−0.657571	−0.328786	−	0.944405i	$-0.606639\pi$
$229$	−2278.75	−0.657571	−0.328786	+	0.944405i	$0.606639\pi$
$230$	0	0
$231$	506.083	0.144146
$232$	−2069.98	−0.585781
$233$	1864.08	0.524121	0.262061	−	0.965051i	$-0.415598\pi$
$233$	1864.08	0.524121	0.262061	+	0.965051i	$0.415598\pi$
$234$	−5889.94	−1.64546
$235$	0	0
$236$	−8508.49	−2.34685
$237$	1220.06	0.334395
$238$	3221.13	0.877289
$239$	1404.69	0.380174	0.190087	−	0.981767i	$-0.439123\pi$
$239$	1404.69	0.380174	0.190087	+	0.981767i	$0.439123\pi$
$240$	0	0
$241$	3879.32	1.03688	0.518442	−	0.855113i	$-0.326512\pi$
$241$	3879.32	1.03688	0.518442	+	0.855113i	$0.326512\pi$
$242$	591.809	0.157202
$243$	4024.43	1.06242
$244$	−5709.91	−1.49811
$245$	0	0
$246$	10912.3	2.82823
$247$	−10087.4	−2.59857
$248$	−368.284	−0.0942987
$249$	−805.950	−0.205121
$250$	0	0
$251$	−2143.54	−0.539040	−0.269520	−	0.962995i	$-0.586865\pi$
$251$	−2143.54	−0.539040	−0.269520	+	0.962995i	$0.586865\pi$
$252$	1805.28	0.451278
$253$	892.972	0.221900
$254$	−5593.11	−1.38166
$255$	0	0
$256$	−8149.58	−1.98964
$257$	−7288.62	−1.76907	−0.884537	−	0.466471i	$-0.845525\pi$
$257$	−7288.62	−1.76907	−0.884537	+	0.466471i	$0.845525\pi$
$258$	13921.7	3.35941
$259$	63.9957	0.0153533
$260$	0	0
$261$	−865.385	−0.205234
$262$	12121.8	2.85835
$263$	−2670.52	−0.626127	−0.313064	−	0.949732i	$-0.601355\pi$
$263$	−2670.52	−0.626127	−0.313064	+	0.949732i	$0.601355\pi$
$264$	2801.17	0.653030
$265$	0	0
$266$	4645.34	1.07077
$267$	5567.48	1.27612
$268$	11054.7	2.51968
$269$	1073.80	0.243387	0.121693	−	0.992568i	$-0.461168\pi$
$269$	1073.80	0.243387	0.121693	+	0.992568i	$0.461168\pi$
$270$	0	0
$271$	3624.88	0.812530	0.406265	−	0.913755i	$-0.366831\pi$
$271$	3624.88	0.812530	0.406265	+	0.913755i	$0.366831\pi$
$272$	5845.14	1.30299
$273$	3420.47	0.758302
$274$	4087.20	0.901157
$275$	0	0
$276$	8495.05	1.85269
$277$	4905.35	1.06402	0.532010	−	0.846738i	$-0.321437\pi$
$277$	4905.35	1.06402	0.532010	+	0.846738i	$0.321437\pi$
$278$	−8446.68	−1.82230
$279$	−153.966	−0.0330384
$280$	0	0
$281$	−2661.90	−0.565109	−0.282555	−	0.959251i	$-0.591182\pi$
$281$	−2661.90	−0.565109	−0.282555	+	0.959251i	$0.591182\pi$
$282$	−1751.26	−0.369809
$283$	4367.36	0.917359	0.458680	−	0.888602i	$-0.348323\pi$
$283$	4367.36	0.917359	0.458680	+	0.888602i	$0.348323\pi$
$284$	−4431.50	−0.925919
$285$	0	0
$286$	3999.87	0.826984
$287$	−2376.23	−0.488726
$288$	−98.7539	−0.0202053
$289$	3938.72	0.801693
$290$	0	0
$291$	6588.86	1.32730
$292$	14117.6	2.82935
$293$	−1992.30	−0.397240	−0.198620	−	0.980077i	$-0.563646\pi$
$293$	−1992.30	−0.397240	−0.198620	+	0.980077i	$0.563646\pi$
$294$	−1575.15	−0.312466
$295$	0	0
$296$	354.215	0.0695553
$297$	−780.969	−0.152581
$298$	−16898.1	−3.28483
$299$	6035.35	1.16734
$300$	0	0
$301$	−3031.54	−0.580516
$302$	−12071.8	−2.30018
$303$	−7392.67	−1.40164
$304$	8429.55	1.59036
$305$	0	0
$306$	7453.62	1.39247
$307$	−7633.53	−1.41912	−0.709558	−	0.704647i	$-0.751105\pi$
$307$	−7633.53	−1.41912	−0.709558	+	0.704647i	$0.751105\pi$
$308$	−1225.97	−0.226806
$309$	6349.82	1.16902
$310$	0	0
$311$	1453.52	0.265020	0.132510	−	0.991182i	$-0.457696\pi$
$311$	1453.52	0.265020	0.132510	+	0.991182i	$0.457696\pi$
$312$	18932.3	3.43535
$313$	−1936.27	−0.349663	−0.174831	−	0.984598i	$-0.555938\pi$
$313$	−1936.27	−0.349663	−0.174831	+	0.984598i	$0.555938\pi$
$314$	7636.14	1.37240
$315$	0	0
$316$	−2955.56	−0.526150
$317$	1534.36	0.271856	0.135928	−	0.990719i	$-0.456598\pi$
$317$	1534.36	0.271856	0.135928	+	0.990719i	$0.456598\pi$
$318$	3980.65	0.701961
$319$	587.685	0.103148
$320$	0	0
$321$	−947.770	−0.164795
$322$	−2779.32	−0.481012
$323$	12765.5	2.19904
$324$	−14392.8	−2.46790
$325$	0	0
$326$	−16915.9	−2.87388
$327$	12325.6	2.08442
$328$	−13152.4	−2.21408
$329$	381.349	0.0639041
$330$	0	0
$331$	3807.42	0.632251	0.316125	−	0.948717i	$-0.397618\pi$
$331$	3807.42	0.632251	0.316125	+	0.948717i	$0.397618\pi$
$332$	1952.39	0.322745
$333$	148.085	0.0243693
$334$	4756.61	0.779252
$335$	0	0
$336$	−2858.32	−0.464089
$337$	4724.09	0.763614	0.381807	−	0.924242i	$-0.375302\pi$
$337$	4724.09	0.763614	0.381807	+	0.924242i	$0.375302\pi$
$338$	16288.5	2.62124
$339$	7936.05	1.27147
$340$	0	0
$341$	104.559	0.0166046
$342$	10749.2	1.69956
$343$	343.000	0.0539949
$344$	−16779.6	−2.62992
$345$	0	0
$346$	5645.08	0.877113
$347$	−6122.41	−0.947171	−0.473586	−	0.880748i	$-0.657041\pi$
$347$	−6122.41	−0.947171	−0.473586	+	0.880748i	$0.657041\pi$
$348$	5590.79	0.861200
$349$	6778.93	1.03974	0.519868	−	0.854247i	$-0.325981\pi$
$349$	6778.93	1.03974	0.519868	+	0.854247i	$0.325981\pi$
$350$	0	0
$351$	−5278.35	−0.802672
$352$	67.0640	0.0101549
$353$	3486.98	0.525760	0.262880	−	0.964829i	$-0.415328\pi$
$353$	3486.98	0.525760	0.262880	+	0.964829i	$0.415328\pi$
$354$	17178.7	2.57920
$355$	0	0
$356$	−13487.1	−2.00790
$357$	−4328.55	−0.641713
$358$	1267.91	0.187182
$359$	3230.16	0.474878	0.237439	−	0.971402i	$-0.423692\pi$
$359$	3230.16	0.474878	0.237439	+	0.971402i	$0.423692\pi$
$360$	0	0
$361$	11550.7	1.68402
$362$	−10376.2	−1.50651
$363$	−795.273	−0.114989
$364$	−8286.00	−1.19314
$365$	0	0
$366$	11528.3	1.64644
$367$	8190.16	1.16491	0.582456	−	0.812862i	$-0.302092\pi$
$367$	8190.16	1.16491	0.582456	+	0.812862i	$0.302092\pi$
$368$	−5043.44	−0.714422
$369$	−5498.54	−0.775725
$370$	0	0
$371$	−866.812	−0.121301
$372$	994.693	0.138636
$373$	−5008.53	−0.695260	−0.347630	−	0.937632i	$-0.613013\pi$
$373$	−5008.53	−0.695260	−0.347630	+	0.937632i	$0.613013\pi$
$374$	−5061.77	−0.699834
$375$	0	0
$376$	2110.76	0.289506
$377$	3972.00	0.542622
$378$	2430.72	0.330748
$379$	1522.26	0.206314	0.103157	−	0.994665i	$-0.467106\pi$
$379$	1522.26	0.206314	0.103157	+	0.994665i	$0.467106\pi$
$380$	0	0
$381$	7516.02	1.01065
$382$	14273.2	1.91173
$383$	−8520.08	−1.13670	−0.568349	−	0.822787i	$-0.692418\pi$
$383$	−8520.08	−1.13670	−0.568349	+	0.822787i	$0.692418\pi$
$384$	16615.1	2.20804
$385$	0	0
$386$	−18377.8	−2.42332
$387$	−7014.93	−0.921418
$388$	−15961.3	−2.08844
$389$	1588.83	0.207088	0.103544	−	0.994625i	$-0.466982\pi$
$389$	1588.83	0.207088	0.103544	+	0.994625i	$0.466982\pi$
$390$	0	0
$391$	−7637.64	−0.987856
$392$	1898.50	0.244614
$393$	−16289.3	−2.09080
$394$	−7866.59	−1.00587
$395$	0	0
$396$	−2836.87	−0.359995
$397$	1483.46	0.187539	0.0937694	−	0.995594i	$-0.470108\pi$
$397$	1483.46	0.187539	0.0937694	+	0.995594i	$0.470108\pi$
$398$	14015.3	1.76513
$399$	−6242.41	−0.783236
$400$	0	0
$401$	10189.5	1.26892	0.634460	−	0.772956i	$-0.281222\pi$
$401$	10189.5	1.26892	0.634460	+	0.772956i	$0.281222\pi$
$402$	−22319.6	−2.76915
$403$	706.685	0.0873510
$404$	17908.5	2.20540
$405$	0	0
$406$	−1829.14	−0.223592
$407$	−100.565	−0.0122477
$408$	−23958.5	−2.90716
$409$	3465.96	0.419024	0.209512	−	0.977806i	$-0.432812\pi$
$409$	3465.96	0.419024	0.209512	+	0.977806i	$0.432812\pi$
$410$	0	0
$411$	−5492.39	−0.659171
$412$	−15382.2	−1.83939
$413$	−3740.77	−0.445693
$414$	−6431.30	−0.763481
$415$	0	0
$416$	453.267	0.0534213
$417$	11350.7	1.33296
$418$	−7299.81	−0.854176
$419$	2754.83	0.321199	0.160599	−	0.987020i	$-0.448657\pi$
$419$	2754.83	0.321199	0.160599	+	0.987020i	$0.448657\pi$
$420$	0	0
$421$	−3140.19	−0.363524	−0.181762	−	0.983343i	$-0.558180\pi$
$421$	−3140.19	−0.363524	−0.181762	+	0.983343i	$0.558180\pi$
$422$	4020.37	0.463764
$423$	882.433	0.101431
$424$	−4797.80	−0.549532
$425$	0	0
$426$	8947.21	1.01759
$427$	−2510.37	−0.284509
$428$	2295.94	0.259296
$429$	−5375.03	−0.604916
$430$	0	0
$431$	−4476.22	−0.500260	−0.250130	−	0.968212i	$-0.580473\pi$
$431$	−4476.22	−0.500260	−0.250130	+	0.968212i	$0.580473\pi$
$432$	4410.85	0.491244
$433$	−1646.00	−0.182683	−0.0913416	−	0.995820i	$-0.529116\pi$
$433$	−1646.00	−0.182683	−0.0913416	+	0.995820i	$0.529116\pi$
$434$	−325.434	−0.0359938
$435$	0	0
$436$	−29858.4	−3.27972
$437$	−11014.6	−1.20572
$438$	−28503.5	−3.10947
$439$	−7241.05	−0.787236	−0.393618	−	0.919274i	$-0.628777\pi$
$439$	−7241.05	−0.787236	−0.393618	+	0.919274i	$0.628777\pi$
$440$	0	0
$441$	793.695	0.0857029
$442$	−34211.1	−3.68158
$443$	−5887.27	−0.631405	−0.315703	−	0.948858i	$-0.602240\pi$
$443$	−5887.27	−0.631405	−0.315703	+	0.948858i	$0.602240\pi$
$444$	−956.695	−0.102258
$445$	0	0
$446$	−535.721	−0.0568770
$447$	22707.7	2.40276
$448$	−3687.85	−0.388917
$449$	6378.42	0.670415	0.335207	−	0.942144i	$-0.391194\pi$
$449$	6378.42	0.670415	0.335207	+	0.942144i	$0.391194\pi$
$450$	0	0
$451$	3734.07	0.389868
$452$	−19224.8	−2.00058
$453$	16222.1	1.68252
$454$	−14785.6	−1.52846
$455$	0	0
$456$	−34551.7	−3.54831
$457$	18368.0	1.88013	0.940064	−	0.340998i	$-0.110765\pi$
$457$	18368.0	1.88013	0.940064	+	0.340998i	$0.110765\pi$
$458$	−11145.3	−1.13709
$459$	6679.67	0.679260
$460$	0	0
$461$	17510.6	1.76909	0.884546	−	0.466453i	$-0.154468\pi$
$461$	17510.6	1.76909	0.884546	+	0.466453i	$0.154468\pi$
$462$	2475.24	0.249261
$463$	12732.5	1.27803	0.639016	−	0.769193i	$-0.279342\pi$
$463$	12732.5	1.27803	0.639016	+	0.769193i	$0.279342\pi$
$464$	−3319.20	−0.332091
$465$	0	0
$466$	9117.20	0.906323
$467$	4997.23	0.495170	0.247585	−	0.968866i	$-0.420363\pi$
$467$	4997.23	0.495170	0.247585	+	0.968866i	$0.420363\pi$
$468$	−19173.6	−1.89381
$469$	4860.23	0.478517
$470$	0	0
$471$	−10261.5	−1.00387
$472$	−20705.1	−2.01913
$473$	4763.85	0.463091
$474$	5967.30	0.578243
$475$	0	0
$476$	10485.8	1.00970
$477$	−2005.78	−0.192534
$478$	6870.30	0.657406
$479$	−9075.53	−0.865703	−0.432851	−	0.901465i	$-0.642493\pi$
$479$	−9075.53	−0.865703	−0.432851	+	0.901465i	$0.642493\pi$
$480$	0	0
$481$	−679.688	−0.0644306
$482$	18973.7	1.79300
$483$	3734.86	0.351847
$484$	1926.53	0.180929
$485$	0	0
$486$	19683.4	1.83716
$487$	3867.63	0.359875	0.179937	−	0.983678i	$-0.442410\pi$
$487$	3867.63	0.359875	0.179937	+	0.983678i	$0.442410\pi$
$488$	−13894.9	−1.28892
$489$	22731.6	2.10216
$490$	0	0
$491$	7334.24	0.674113	0.337057	−	0.941484i	$-0.390569\pi$
$491$	7334.24	0.674113	0.337057	+	0.941484i	$0.390569\pi$
$492$	35523.1	3.25509
$493$	−5026.50	−0.459193
$494$	−49337.4	−4.49351
$495$	0	0
$496$	−590.541	−0.0534598
$497$	−1948.31	−0.175843
$498$	−3941.89	−0.354699
$499$	−4365.93	−0.391675	−0.195838	−	0.980636i	$-0.562743\pi$
$499$	−4365.93	−0.391675	−0.195838	+	0.980636i	$0.562743\pi$
$500$	0	0
$501$	−6391.94	−0.570002
$502$	−10484.0	−0.932120
$503$	16352.6	1.44956	0.724779	−	0.688981i	$-0.241942\pi$
$503$	16352.6	1.44956	0.724779	+	0.688981i	$0.241942\pi$
$504$	4393.09	0.388262
$505$	0	0
$506$	4367.51	0.383714
$507$	−21888.5	−1.91737
$508$	−18207.4	−1.59020
$509$	−12108.2	−1.05440	−0.527199	−	0.849742i	$-0.676758\pi$
$509$	−12108.2	−1.05440	−0.527199	+	0.849742i	$0.676758\pi$
$510$	0	0
$511$	6206.82	0.537326
$512$	−19635.7	−1.69489
$513$	9633.06	0.829065
$514$	−35648.5	−3.05912
$515$	0	0
$516$	45319.7	3.86645
$517$	−599.262	−0.0509778
$518$	313.002	0.0265492
$519$	−7585.86	−0.641584
$520$	0	0
$521$	−5625.67	−0.473062	−0.236531	−	0.971624i	$-0.576010\pi$
$521$	−5625.67	−0.473062	−0.236531	+	0.971624i	$0.576010\pi$
$522$	−4232.58	−0.354895
$523$	3280.32	0.274261	0.137130	−	0.990553i	$-0.456212\pi$
$523$	3280.32	0.274261	0.137130	+	0.990553i	$0.456212\pi$
$524$	39460.3	3.28976
$525$	0	0
$526$	−13061.5	−1.08271
$527$	−894.298	−0.0739207
$528$	4491.64	0.370215
$529$	−5576.93	−0.458365
$530$	0	0
$531$	−8656.06	−0.707422
$532$	15122.1	1.23238
$533$	25237.6	2.05096
$534$	27230.4	2.20670
$535$	0	0
$536$	26901.3	2.16784
$537$	−1703.82	−0.136918
$538$	5251.96	0.420870
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−7772.10	−0.617650	−0.308825	−	0.951119i	$-0.599936\pi$
$541$	−7772.10	−0.617650	−0.308825	+	0.951119i	$0.599936\pi$
$542$	17729.2	1.40505
$543$	13943.5	1.10197
$544$	−573.602	−0.0452077
$545$	0	0
$546$	16729.5	1.31127
$547$	−10022.2	−0.783398	−0.391699	−	0.920094i	$-0.628112\pi$
$547$	−10022.2	−0.783398	−0.391699	+	0.920094i	$0.628112\pi$
$548$	13305.1	1.03717
$549$	−5808.94	−0.451584
$550$	0	0
$551$	−7248.95	−0.560464
$552$	20672.4	1.59398
$553$	−1299.42	−0.0999220
$554$	23992.0	1.83993
$555$	0	0
$556$	−27496.7	−2.09733
$557$	10849.3	0.825310	0.412655	−	0.910887i	$-0.364601\pi$
$557$	10849.3	0.825310	0.412655	+	0.910887i	$0.364601\pi$
$558$	−753.047	−0.0571308
$559$	32197.6	2.43616
$560$	0	0
$561$	6802.01	0.511910
$562$	−13019.3	−0.977200
$563$	−22019.9	−1.64836	−0.824180	−	0.566328i	$-0.808363\pi$
$563$	−22019.9	−1.64836	−0.824180	+	0.566328i	$0.808363\pi$
$564$	−5700.92	−0.425624
$565$	0	0
$566$	21360.7	1.58632
$567$	−6327.80	−0.468682
$568$	−10783.9	−0.796624
$569$	13742.9	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$569$	13742.9	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$570$	0	0
$571$	19382.4	1.42054	0.710269	−	0.703931i	$-0.248573\pi$
$571$	19382.4	1.42054	0.710269	+	0.703931i	$0.248573\pi$
$572$	13020.9	0.951800
$573$	−19180.4	−1.39838
$574$	−11622.1	−0.845116
$575$	0	0
$576$	−8533.61	−0.617304
$577$	565.348	0.0407899	0.0203949	−	0.999792i	$-0.493508\pi$
$577$	565.348	0.0407899	0.0203949	+	0.999792i	$0.493508\pi$
$578$	19264.2	1.38631
$579$	24696.0	1.77260
$580$	0	0
$581$	858.371	0.0612930
$582$	32226.0	2.29521
$583$	1362.13	0.0967646
$584$	34354.7	2.43426
$585$	0	0
$586$	−9744.30	−0.686917
$587$	20727.4	1.45743	0.728714	−	0.684818i	$-0.240118\pi$
$587$	20727.4	1.45743	0.728714	+	0.684818i	$0.240118\pi$
$588$	−5127.63	−0.359626
$589$	−1289.71	−0.0902233
$590$	0	0
$591$	10571.1	0.735767
$592$	567.981	0.0394322
$593$	2787.10	0.193006	0.0965030	−	0.995333i	$-0.469234\pi$
$593$	2787.10	0.193006	0.0965030	+	0.995333i	$0.469234\pi$
$594$	−3819.71	−0.263846
$595$	0	0
$596$	−55008.7	−3.78061
$597$	−18833.7	−1.29114
$598$	29518.8	2.01858
$599$	18935.5	1.29162	0.645812	−	0.763497i	$-0.276519\pi$
$599$	18935.5	1.29162	0.645812	+	0.763497i	$0.276519\pi$
$600$	0	0
$601$	15821.3	1.07382	0.536908	−	0.843641i	$-0.319592\pi$
$601$	15821.3	1.07382	0.536908	+	0.843641i	$0.319592\pi$
$602$	−14827.2	−1.00384
$603$	11246.5	0.759521
$604$	−39297.5	−2.64734
$605$	0	0
$606$	−36157.4	−2.42375
$607$	2084.63	0.139394	0.0696972	−	0.997568i	$-0.477797\pi$
$607$	2084.63	0.139394	0.0696972	+	0.997568i	$0.477797\pi$
$608$	−827.218	−0.0551778
$609$	2458.00	0.163552
$610$	0	0
$611$	−4050.25	−0.268176
$612$	24263.9	1.60263
$613$	4395.15	0.289590	0.144795	−	0.989462i	$-0.453748\pi$
$613$	4395.15	0.289590	0.144795	+	0.989462i	$0.453748\pi$
$614$	−37335.5	−2.45397
$615$	0	0
$616$	−2983.36	−0.195135
$617$	98.5856	0.00643259	0.00321629	−	0.999995i	$-0.498976\pi$
$617$	98.5856	0.00643259	0.00321629	+	0.999995i	$0.498976\pi$
$618$	31056.8	2.02150
$619$	−16533.9	−1.07359	−0.536796	−	0.843712i	$-0.680366\pi$
$619$	−16533.9	−1.07359	−0.536796	+	0.843712i	$0.680366\pi$
$620$	0	0
$621$	−5763.50	−0.372434
$622$	7109.12	0.458280
$623$	−5929.60	−0.381323
$624$	30357.8	1.94757
$625$	0	0
$626$	−9470.27	−0.604645
$627$	9809.50	0.624806
$628$	24858.1	1.57953
$629$	860.135	0.0545243
$630$	0	0
$631$	22032.9	1.39004	0.695022	−	0.718989i	$-0.255395\pi$
$631$	22032.9	1.39004	0.695022	+	0.718989i	$0.255395\pi$
$632$	−7192.27	−0.452679
$633$	−5402.57	−0.339231
$634$	7504.54	0.470100
$635$	0	0
$636$	12958.3	0.807908
$637$	−3642.95	−0.226592
$638$	2874.36	0.178365
$639$	−4508.35	−0.279104
$640$	0	0
$641$	434.281	0.0267598	0.0133799	−	0.999910i	$-0.495741\pi$
$641$	434.281	0.0267598	0.0133799	+	0.999910i	$0.495741\pi$
$642$	−4635.52	−0.284968
$643$	−10963.1	−0.672381	−0.336190	−	0.941794i	$-0.609139\pi$
$643$	−10963.1	−0.672381	−0.336190	+	0.941794i	$0.609139\pi$
$644$	−9047.59	−0.553610
$645$	0	0
$646$	62435.7	3.80263
$647$	452.083	0.0274702	0.0137351	−	0.999906i	$-0.495628\pi$
$647$	452.083	0.0274702	0.0137351	+	0.999906i	$0.495628\pi$
$648$	−35024.3	−2.12328
$649$	5878.35	0.355540
$650$	0	0
$651$	437.318	0.0263285
$652$	−55066.6	−3.30763
$653$	10438.7	0.625572	0.312786	−	0.949824i	$-0.398738\pi$
$653$	10438.7	0.625572	0.312786	+	0.949824i	$0.398738\pi$
$654$	60284.2	3.60443
$655$	0	0
$656$	−21089.8	−1.25521
$657$	14362.4	0.852865
$658$	1865.17	0.110504
$659$	−8738.79	−0.516563	−0.258281	−	0.966070i	$-0.583156\pi$
$659$	−8738.79	−0.516563	−0.258281	+	0.966070i	$0.583156\pi$
$660$	0	0
$661$	12849.4	0.756102	0.378051	−	0.925785i	$-0.376594\pi$
$661$	12849.4	0.756102	0.378051	+	0.925785i	$0.376594\pi$
$662$	18622.0	1.09330
$663$	45972.9	2.69297
$664$	4751.08	0.277677
$665$	0	0
$666$	724.279	0.0421400
$667$	4337.08	0.251773
$668$	15484.3	0.896864
$669$	719.903	0.0416040
$670$	0	0
$671$	3944.87	0.226960
$672$	280.496	0.0161017
$673$	22926.5	1.31315	0.656577	−	0.754259i	$-0.272004\pi$
$673$	22926.5	1.31315	0.656577	+	0.754259i	$0.272004\pi$
$674$	23105.5	1.32046
$675$	0	0
$676$	53024.4	3.01686
$677$	−13633.4	−0.773965	−0.386983	−	0.922087i	$-0.626483\pi$
$677$	−13633.4	−0.773965	−0.386983	+	0.922087i	$0.626483\pi$
$678$	38815.1	2.19865
$679$	−7017.41	−0.396618
$680$	0	0
$681$	19868.9	1.11803
$682$	511.396	0.0287131
$683$	−33307.5	−1.86600	−0.932998	−	0.359882i	$-0.882817\pi$
$683$	−33307.5	−1.86600	−0.932998	+	0.359882i	$0.882817\pi$
$684$	34992.1	1.95608
$685$	0	0
$686$	1677.61	0.0933693
$687$	14977.1	0.831748
$688$	−26905.9	−1.49096
$689$	9206.28	0.509044
$690$	0	0
$691$	−10077.3	−0.554786	−0.277393	−	0.960757i	$-0.589470\pi$
$691$	−10077.3	−0.554786	−0.277393	+	0.960757i	$0.589470\pi$
$692$	18376.5	1.00950
$693$	−1247.23	−0.0683673
$694$	−29944.6	−1.63787
$695$	0	0
$696$	13605.0	0.740942
$697$	−31937.7	−1.73562
$698$	33155.6	1.79794
$699$	−12251.7	−0.662950
$700$	0	0
$701$	−4621.74	−0.249017	−0.124508	−	0.992219i	$-0.539735\pi$
$701$	−4621.74	−0.249017	−0.124508	+	0.992219i	$0.539735\pi$
$702$	−25816.3	−1.38800
$703$	1240.44	0.0665492
$704$	5795.20	0.310248
$705$	0	0
$706$	17054.8	0.909157
$707$	7873.51	0.418831
$708$	55922.1	2.96848
$709$	17746.6	0.940038	0.470019	−	0.882656i	$-0.344247\pi$
$709$	17746.6	0.940038	0.470019	+	0.882656i	$0.344247\pi$
$710$	0	0
$711$	−3006.82	−0.158600
$712$	−32820.3	−1.72752
$713$	771.638	0.0405302
$714$	−21170.9	−1.10966
$715$	0	0
$716$	4127.45	0.215433
$717$	−9232.31	−0.480875
$718$	15798.6	0.821170
$719$	−31652.1	−1.64176	−0.820879	−	0.571103i	$-0.806516\pi$
$719$	−31652.1	−1.64176	−0.820879	+	0.571103i	$0.806516\pi$
$720$	0	0
$721$	−6762.83	−0.349322
$722$	56494.1	2.91204
$723$	−25496.9	−1.31153
$724$	−33777.7	−1.73389
$725$	0	0
$726$	−3889.67	−0.198842
$727$	−21610.8	−1.10248	−0.551239	−	0.834348i	$-0.685845\pi$
$727$	−21610.8	−1.10248	−0.551239	+	0.834348i	$0.685845\pi$
$728$	−20163.7	−1.02653
$729$	−2043.39	−0.103815
$730$	0	0
$731$	−40745.5	−2.06160
$732$	37528.5	1.89493
$733$	7665.43	0.386261	0.193130	−	0.981173i	$-0.438136\pi$
$733$	7665.43	0.386261	0.193130	+	0.981173i	$0.438136\pi$
$734$	40057.9	2.01439
$735$	0	0
$736$	494.928	0.0247871
$737$	−7637.50	−0.381725
$738$	−26893.3	−1.34140
$739$	−13841.5	−0.688996	−0.344498	−	0.938787i	$-0.611951\pi$
$739$	−13841.5	−0.688996	−0.344498	+	0.938787i	$0.611951\pi$
$740$	0	0
$741$	66299.7	3.28688
$742$	−4239.56	−0.209756
$743$	12136.3	0.599243	0.299621	−	0.954058i	$-0.403140\pi$
$743$	12136.3	0.599243	0.299621	+	0.954058i	$0.403140\pi$
$744$	2420.55	0.119277
$745$	0	0
$746$	−24496.6	−1.20226
$747$	1986.25	0.0972867
$748$	−16477.7	−0.805460
$749$	1009.42	0.0492433
$750$	0	0
$751$	28866.7	1.40261	0.701305	−	0.712861i	$-0.252601\pi$
$751$	28866.7	1.40261	0.701305	+	0.712861i	$0.252601\pi$
$752$	3384.59	0.164127
$753$	14088.4	0.681820
$754$	19427.0	0.938314
$755$	0	0
$756$	7912.78	0.380668
$757$	−8370.48	−0.401889	−0.200945	−	0.979603i	$-0.564401\pi$
$757$	−8370.48	−0.401889	−0.200945	+	0.979603i	$0.564401\pi$
$758$	7445.33	0.356763
$759$	−5869.06	−0.280677
$760$	0	0
$761$	30458.1	1.45086	0.725430	−	0.688296i	$-0.241641\pi$
$761$	30458.1	1.45086	0.725430	+	0.688296i	$0.241641\pi$
$762$	36760.7	1.74764
$763$	−13127.3	−0.622856
$764$	46463.9	2.20027
$765$	0	0
$766$	−41671.5	−1.96561
$767$	39730.1	1.87037
$768$	53563.2	2.51666
$769$	−32239.5	−1.51182	−0.755908	−	0.654678i	$-0.772804\pi$
$769$	−32239.5	−1.51182	−0.755908	+	0.654678i	$0.772804\pi$
$770$	0	0
$771$	47904.5	2.23766
$772$	−59825.5	−2.78907
$773$	7524.47	0.350112	0.175056	−	0.984559i	$-0.443989\pi$
$773$	7524.47	0.350112	0.175056	+	0.984559i	$0.443989\pi$
$774$	−34309.9	−1.59334
$775$	0	0
$776$	−38841.3	−1.79681
$777$	−420.612	−0.0194200
$778$	7770.96	0.358101
$779$	−46058.8	−2.11839
$780$	0	0
$781$	3061.63	0.140274
$782$	−37355.5	−1.70822
$783$	−3793.09	−0.173121
$784$	3044.23	0.138677
$785$	0	0
$786$	−79670.7	−3.61547
$787$	−33196.6	−1.50360	−0.751798	−	0.659394i	$-0.770813\pi$
$787$	−33196.6	−1.50360	−0.751798	+	0.659394i	$0.770813\pi$
$788$	−25608.3	−1.15769
$789$	17552.0	0.791976
$790$	0	0
$791$	−8452.23	−0.379932
$792$	−6903.43	−0.309726
$793$	26662.3	1.19395
$794$	7255.59	0.324296
$795$	0	0
$796$	45624.1	2.03154
$797$	−8817.06	−0.391865	−0.195932	−	0.980617i	$-0.562773\pi$
$797$	−8817.06	−0.391865	−0.195932	+	0.980617i	$0.562773\pi$
$798$	−30531.5	−1.35439
$799$	5125.52	0.226944
$800$	0	0
$801$	−13721.0	−0.605252
$802$	49836.4	2.19425
$803$	−9753.57	−0.428638
$804$	−72657.3	−3.18710
$805$	0	0
$806$	3456.38	0.151049
$807$	−7057.59	−0.307855
$808$	43579.8	1.89744
$809$	−28930.3	−1.25727	−0.628637	−	0.777699i	$-0.716387\pi$
$809$	−28930.3	−1.25727	−0.628637	+	0.777699i	$0.716387\pi$
$810$	0	0
$811$	−45923.3	−1.98839	−0.994195	−	0.107594i	$-0.965685\pi$
$811$	−45923.3	−1.98839	−0.994195	+	0.107594i	$0.965685\pi$
$812$	−5954.42	−0.257339
$813$	−23824.5	−1.02775
$814$	−491.860	−0.0211790
$815$	0	0
$816$	−38417.3	−1.64813
$817$	−58761.0	−2.51626
$818$	16952.0	0.724586
$819$	−8429.71	−0.359656
$820$	0	0
$821$	16382.8	0.696423	0.348211	−	0.937416i	$-0.386789\pi$
$821$	16382.8	0.696423	0.348211	+	0.937416i	$0.386789\pi$
$822$	−26863.2	−1.13985
$823$	32395.3	1.37209	0.686045	−	0.727559i	$-0.259345\pi$
$823$	32395.3	1.37209	0.686045	+	0.727559i	$0.259345\pi$
$824$	−37432.2	−1.58254
$825$	0	0
$826$	−18296.0	−0.770703
$827$	14329.2	0.602508	0.301254	−	0.953544i	$-0.402595\pi$
$827$	14329.2	0.602508	0.301254	+	0.953544i	$0.402595\pi$
$828$	−20935.9	−0.878712
$829$	16445.8	0.689005	0.344503	−	0.938785i	$-0.388048\pi$
$829$	16445.8	0.689005	0.344503	+	0.938785i	$0.388048\pi$
$830$	0	0
$831$	−32240.4	−1.34586
$832$	39168.1	1.63210
$833$	4610.10	0.191753
$834$	55515.9	2.30499
$835$	0	0
$836$	−23763.2	−0.983096
$837$	−674.854	−0.0278690
$838$	13473.8	0.555424
$839$	40514.7	1.66713	0.833565	−	0.552422i	$-0.186296\pi$
$839$	40514.7	1.66713	0.833565	+	0.552422i	$0.186296\pi$
$840$	0	0
$841$	−21534.7	−0.882967
$842$	−15358.6	−0.628614
$843$	17495.4	0.714795
$844$	13087.6	0.533760
$845$	0	0
$846$	4315.96	0.175397
$847$	847.000	0.0343604
$848$	−7693.22	−0.311540
$849$	−28704.5	−1.16035
$850$	0	0
$851$	−742.160	−0.0298953
$852$	29126.0	1.17118
$853$	27962.7	1.12242	0.561211	−	0.827673i	$-0.310336\pi$
$853$	27962.7	1.12242	0.561211	+	0.827673i	$0.310336\pi$
$854$	−12278.2	−0.491980
$855$	0	0
$856$	5587.10	0.223088
$857$	4639.52	0.184928	0.0924639	−	0.995716i	$-0.470526\pi$
$857$	4639.52	0.184928	0.0924639	+	0.995716i	$0.470526\pi$
$858$	−26289.2	−1.04603
$859$	18521.2	0.735663	0.367831	−	0.929892i	$-0.380100\pi$
$859$	18521.2	0.735663	0.367831	+	0.929892i	$0.380100\pi$
$860$	0	0
$861$	15617.8	0.618179
$862$	−21893.1	−0.865061
$863$	−40491.4	−1.59715	−0.798577	−	0.601893i	$-0.794413\pi$
$863$	−40491.4	−1.59715	−0.798577	+	0.601893i	$0.794413\pi$
$864$	−432.851	−0.0170439
$865$	0	0
$866$	−8050.57	−0.315900
$867$	−25887.3	−1.01405
$868$	−1059.39	−0.0414263
$869$	2041.94	0.0797102
$870$	0	0
$871$	−51619.8	−2.00812
$872$	−72659.3	−2.82174
$873$	−16238.1	−0.629528
$874$	−53872.1	−2.08496
$875$	0	0
$876$	−92788.0	−3.57878
$877$	48530.3	1.86859	0.934295	−	0.356502i	$-0.116031\pi$
$877$	48530.3	1.86859	0.934295	+	0.356502i	$0.116031\pi$
$878$	−35415.9	−1.36131
$879$	13094.4	0.502461
$880$	0	0
$881$	−11590.0	−0.443219	−0.221609	−	0.975135i	$-0.571131\pi$
$881$	−11590.0	−0.443219	−0.221609	+	0.975135i	$0.571131\pi$
$882$	3881.95	0.148199
$883$	−41900.7	−1.59691	−0.798455	−	0.602054i	$-0.794349\pi$
$883$	−41900.7	−1.59691	−0.798455	+	0.602054i	$0.794349\pi$
$884$	−111368.	−4.23723
$885$	0	0
$886$	−28794.5	−1.09184
$887$	17136.8	0.648700	0.324350	−	0.945937i	$-0.394854\pi$
$887$	17136.8	0.648700	0.324350	+	0.945937i	$0.394854\pi$
$888$	−2328.08	−0.0879790
$889$	−8004.89	−0.301997
$890$	0	0
$891$	9943.69	0.373879
$892$	−1743.94	−0.0654614
$893$	7391.75	0.276994
$894$	111063.	4.15492
$895$	0	0
$896$	−17695.8	−0.659794
$897$	−39667.4	−1.47654
$898$	31196.7	1.15930
$899$	507.832	0.0188400
$900$	0	0
$901$	−11650.4	−0.430778
$902$	18263.3	0.674169
$903$	19924.8	0.734283
$904$	−46783.0	−1.72122
$905$	0	0
$906$	79341.9	2.90944
$907$	−40350.7	−1.47720	−0.738601	−	0.674143i	$-0.764513\pi$
$907$	−40350.7	−1.47720	−0.738601	+	0.674143i	$0.764513\pi$
$908$	−48131.8	−1.75915
$909$	18219.1	0.664786
$910$	0	0
$911$	−32322.7	−1.17552	−0.587760	−	0.809035i	$-0.699990\pi$
$911$	−32322.7	−1.17552	−0.587760	+	0.809035i	$0.699990\pi$
$912$	−55403.3	−2.01161
$913$	−1348.87	−0.0488949
$914$	89837.5	3.25116
$915$	0	0
$916$	−36281.5	−1.30871
$917$	17348.8	0.624763
$918$	32670.2	1.17459
$919$	2431.28	0.0872694	0.0436347	−	0.999048i	$-0.486106\pi$
$919$	2431.28	0.0872694	0.0436347	+	0.999048i	$0.486106\pi$
$920$	0	0
$921$	50171.4	1.79501
$922$	85644.2	3.05916
$923$	20692.7	0.737930
$924$	8057.71	0.286882
$925$	0	0
$926$	62274.4	2.21000
$927$	−15649.0	−0.554457
$928$	325.724	0.0115220
$929$	40221.0	1.42046	0.710230	−	0.703970i	$-0.248591\pi$
$929$	40221.0	1.42046	0.710230	+	0.703970i	$0.248591\pi$
$930$	0	0
$931$	6648.43	0.234042
$932$	29679.4	1.04311
$933$	−9553.25	−0.335219
$934$	24441.4	0.856260
$935$	0	0
$936$	−46658.4	−1.62936
$937$	−41503.4	−1.44702	−0.723510	−	0.690314i	$-0.757472\pi$
$937$	−41503.4	−1.44702	−0.723510	+	0.690314i	$0.757472\pi$
$938$	23771.3	0.827463
$939$	12726.2	0.442282
$940$	0	0
$941$	−9032.78	−0.312923	−0.156461	−	0.987684i	$-0.550009\pi$
$941$	−9032.78	−0.312923	−0.156461	+	0.987684i	$0.550009\pi$
$942$	−50188.6	−1.73592
$943$	27557.2	0.951629
$944$	−33200.5	−1.14469
$945$	0	0
$946$	23299.9	0.800789
$947$	−5565.96	−0.190992	−0.0954960	−	0.995430i	$-0.530444\pi$
$947$	−5565.96	−0.190992	−0.0954960	+	0.995430i	$0.530444\pi$
$948$	19425.5	0.665517
$949$	−65921.7	−2.25491
$950$	0	0
$951$	−10084.6	−0.343865
$952$	25516.8	0.868703
$953$	−32657.4	−1.11005	−0.555024	−	0.831834i	$-0.687291\pi$
$953$	−32657.4	−1.11005	−0.555024	+	0.831834i	$0.687291\pi$
$954$	−9810.25	−0.332934
$955$	0	0
$956$	22365.0	0.756628
$957$	−3862.56	−0.130469
$958$	−44388.3	−1.49699
$959$	5849.63	0.196970
$960$	0	0
$961$	−29700.6	−0.996967
$962$	−3324.34	−0.111415
$963$	2335.76	0.0781609
$964$	61765.4	2.06362
$965$	0	0
$966$	18267.1	0.608422
$967$	610.079	0.0202883	0.0101442	−	0.999949i	$-0.496771\pi$
$967$	610.079	0.0202883	0.0101442	+	0.999949i	$0.496771\pi$
$968$	4688.14	0.155664
$969$	−83901.2	−2.78152
$970$	0	0
$971$	−45371.4	−1.49952	−0.749762	−	0.661708i	$-0.769832\pi$
$971$	−45371.4	−1.49952	−0.749762	+	0.661708i	$0.769832\pi$
$972$	64075.9	2.11444
$973$	−12088.9	−0.398308
$974$	18916.5	0.622304
$975$	0	0
$976$	−22280.3	−0.730713
$977$	−27229.0	−0.891639	−0.445820	−	0.895123i	$-0.647088\pi$
$977$	−27229.0	−0.891639	−0.445820	+	0.895123i	$0.647088\pi$
$978$	111180.	3.63511
$979$	9317.94	0.304191
$980$	0	0
$981$	−30376.2	−0.988622
$982$	35871.6	1.16569
$983$	−7458.49	−0.242003	−0.121002	−	0.992652i	$-0.538611\pi$
$983$	−7458.49	−0.242003	−0.121002	+	0.992652i	$0.538611\pi$
$984$	86444.2	2.80055
$985$	0	0
$986$	−24584.5	−0.794048
$987$	−2506.42	−0.0808310
$988$	−160609.	−5.17171
$989$	35156.9	1.13036
$990$	0	0
$991$	−24357.8	−0.780778	−0.390389	−	0.920650i	$-0.627660\pi$
$991$	−24357.8	−0.780778	−0.390389	+	0.920650i	$0.627660\pi$
$992$	57.9516	0.00185480
$993$	−25024.3	−0.799721
$994$	−9529.17	−0.304071
$995$	0	0
$996$	−12832.1	−0.408234
$997$	−19954.3	−0.633862	−0.316931	−	0.948449i	$-0.602652\pi$
$997$	−19954.3	−0.633862	−0.316931	+	0.948449i	$0.602652\pi$
$998$	−21353.7	−0.677294
$999$	649.074	0.0205563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.4.a.p.1.4		4
5.4	even	2		77.4.a.d.1.1	✓	4
15.14	odd	2		693.4.a.l.1.4		4
20.19	odd	2		1232.4.a.s.1.2		4
35.34	odd	2		539.4.a.g.1.1		4
55.54	odd	2		847.4.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.1	✓	4	5.4	even	2
539.4.a.g.1.1		4	35.34	odd	2
693.4.a.l.1.4		4	15.14	odd	2
847.4.a.d.1.4		4	55.54	odd	2
1232.4.a.s.1.2		4	20.19	odd	2
1925.4.a.p.1.4		4	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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q+4.89098 q^{2} -6.57251 q^{3} +15.9217 q^{4} -32.1460 q^{6} +7.00000 q^{7} +38.7449 q^{8} +16.1978 q^{9} -11.0000 q^{11} -104.646 q^{12} -74.3459 q^{13} +34.2369 q^{14} +62.1271 q^{16} +94.0836 q^{17} +79.2234 q^{18} +135.682 q^{19} -46.0075 q^{21} -53.8008 q^{22} -81.1793 q^{23} -254.651 q^{24} -363.625 q^{26} +70.9972 q^{27} +111.452 q^{28} -53.4259 q^{29} -9.50536 q^{31} -6.09673 q^{32} +72.2976 q^{33} +460.161 q^{34} +257.897 q^{36} +9.14224 q^{37} +663.620 q^{38} +488.639 q^{39} -339.461 q^{41} -225.022 q^{42} -433.078 q^{43} -175.139 q^{44} -397.046 q^{46} +54.4784 q^{47} -408.331 q^{48} +49.0000 q^{49} -618.365 q^{51} -1183.71 q^{52} -123.830 q^{53} +347.246 q^{54} +271.215 q^{56} -891.773 q^{57} -261.305 q^{58} -534.396 q^{59} -358.624 q^{61} -46.4905 q^{62} +113.385 q^{63} -526.836 q^{64} +353.606 q^{66} +694.318 q^{67} +1497.97 q^{68} +533.551 q^{69} -278.330 q^{71} +627.585 q^{72} +886.688 q^{73} +44.7145 q^{74} +2160.29 q^{76} -77.0000 q^{77} +2389.93 q^{78} -185.631 q^{79} -903.972 q^{81} -1660.30 q^{82} +122.624 q^{83} -732.519 q^{84} -2118.18 q^{86} +351.142 q^{87} -426.194 q^{88} -847.086 q^{89} -520.422 q^{91} -1292.51 q^{92} +62.4740 q^{93} +266.453 q^{94} +40.0708 q^{96} -1002.49 q^{97} +239.658 q^{98} -178.176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 14 q^{3} + 26 q^{4} + 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 14 q^{14} + 2 q^{16} - 4 q^{17} + 62 q^{18} + 258 q^{19} - 98 q^{21} - 22 q^{22} - 8 q^{23}+ \cdots - 836 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 14 * q^3 + 26 * q^4 + 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 14 * q^14 + 2 * q^16 - 4 * q^17 + 62 * q^18 + 258 * q^19 - 98 * q^21 - 22 * q^22 - 8 * q^23 - 498 * q^24 - 482 * q^26 - 428 * q^27 + 182 * q^28 - 396 * q^29 - 56 * q^31 - 134 * q^32 + 154 * q^33 + 472 * q^34 - 418 * q^36 - 84 * q^37 + 942 * q^38 - 412 * q^39 + 52 * q^41 + 98 * q^42 - 408 * q^43 - 286 * q^44 + 368 * q^46 - 8 * q^47 - 982 * q^48 + 196 * q^49 - 388 * q^51 - 2030 * q^52 - 624 * q^53 + 92 * q^54 + 126 * q^56 - 48 * q^57 - 864 * q^58 - 238 * q^59 - 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 - 154 * q^66 - 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 3042 * q^76 - 308 * q^77 + 2632 * q^78 - 1324 * q^79 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 490 * q^84 - 4380 * q^86 - 588 * q^87 - 198 * q^88 - 3072 * q^89 - 406 * q^91 - 544 * q^92 + 1264 * q^93 - 1696 * q^94 + 862 * q^96 + 652 * q^97 + 98 * q^98 - 836 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more