LMFDB - Embedded newform 1925.4.a.p.1.1

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			1.1
Root			$-0.148103$ 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)

$f(q)$	$=$	\(q-4.60395 q^{2} -2.77399 q^{3} +13.1964 q^{4} +12.7713 q^{6} +7.00000 q^{7} -23.9238 q^{8} -19.3050 q^{9} -11.0000 q^{11} -36.6066 q^{12} -24.6401 q^{13} -32.2277 q^{14} +4.57310 q^{16} -17.8800 q^{17} +88.8792 q^{18} +32.1459 q^{19} -19.4179 q^{21} +50.6435 q^{22} -14.1248 q^{23} +66.3643 q^{24} +113.442 q^{26} +128.450 q^{27} +92.3745 q^{28} -41.5471 q^{29} +175.766 q^{31} +170.336 q^{32} +30.5139 q^{33} +82.3187 q^{34} -254.756 q^{36} -292.877 q^{37} -147.998 q^{38} +68.3513 q^{39} +154.296 q^{41} +89.3991 q^{42} +277.144 q^{43} -145.160 q^{44} +65.0300 q^{46} +52.1450 q^{47} -12.6857 q^{48} +49.0000 q^{49} +49.5989 q^{51} -325.160 q^{52} -82.3907 q^{53} -591.375 q^{54} -167.467 q^{56} -89.1723 q^{57} +191.281 q^{58} +712.816 q^{59} -647.078 q^{61} -809.217 q^{62} -135.135 q^{63} -820.804 q^{64} -140.484 q^{66} -260.867 q^{67} -235.951 q^{68} +39.1821 q^{69} +369.025 q^{71} +461.849 q^{72} -1145.77 q^{73} +1348.39 q^{74} +424.209 q^{76} -77.0000 q^{77} -314.686 q^{78} +488.885 q^{79} +164.917 q^{81} -710.372 q^{82} -548.982 q^{83} -256.246 q^{84} -1275.96 q^{86} +115.251 q^{87} +263.162 q^{88} +105.039 q^{89} -172.481 q^{91} -186.396 q^{92} -487.572 q^{93} -240.073 q^{94} -472.510 q^{96} +1361.91 q^{97} -225.594 q^{98} +212.355 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.60395	−1.62774	−0.813871	−	0.581045i	$-0.802644\pi$
$2$	−4.60395	−1.62774	−0.813871	+	0.581045i	$0.802644\pi$
$3$	−2.77399	−0.533854	−0.266927	−	0.963717i	$-0.586008\pi$
$3$	−2.77399	−0.533854	−0.266927	+	0.963717i	$0.586008\pi$
$4$	13.1964	1.64955
$5$	0	0
$5$	0	0
$6$	12.7713	0.868977
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−23.9238	−1.05729
$9$	−19.3050	−0.715000
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−36.6066	−0.880617
$13$	−24.6401	−0.525687	−0.262844	−	0.964838i	$-0.584660\pi$
$13$	−24.6401	−0.525687	−0.262844	+	0.964838i	$0.584660\pi$
$14$	−32.2277	−0.615229
$15$	0	0
$16$	4.57310	0.0714546
$17$	−17.8800	−0.255090	−0.127545	−	0.991833i	$-0.540710\pi$
$17$	−17.8800	−0.255090	−0.127545	+	0.991833i	$0.540710\pi$
$18$	88.8792	1.16384
$19$	32.1459	0.388146	0.194073	−	0.980987i	$-0.437830\pi$
$19$	32.1459	0.388146	0.194073	+	0.980987i	$0.437830\pi$
$20$	0	0
$21$	−19.4179	−0.201778
$22$	50.6435	0.490783
$23$	−14.1248	−0.128054	−0.0640268	−	0.997948i	$-0.520394\pi$
$23$	−14.1248	−0.128054	−0.0640268	+	0.997948i	$0.520394\pi$
$24$	66.3643	0.564440
$25$	0	0
$26$	113.442	0.855683
$27$	128.450	0.915560
$28$	92.3745	0.623470
$29$	−41.5471	−0.266038	−0.133019	−	0.991113i	$-0.542467\pi$
$29$	−41.5471	−0.266038	−0.133019	+	0.991113i	$0.542467\pi$
$30$	0	0
$31$	175.766	1.01834	0.509169	−	0.860667i	$-0.329953\pi$
$31$	175.766	1.01834	0.509169	+	0.860667i	$0.329953\pi$
$32$	170.336	0.940983
$33$	30.5139	0.160963
$34$	82.3187	0.415222
$35$	0	0
$36$	−254.756	−1.17942
$37$	−292.877	−1.30131	−0.650657	−	0.759372i	$-0.725506\pi$
$37$	−292.877	−1.30131	−0.650657	+	0.759372i	$0.725506\pi$
$38$	−147.998	−0.631801
$39$	68.3513	0.280640
$40$	0	0
$41$	154.296	0.587732	0.293866	−	0.955847i	$-0.405058\pi$
$41$	154.296	0.587732	0.293866	+	0.955847i	$0.405058\pi$
$42$	89.3991	0.328443
$43$	277.144	0.982887	0.491443	−	0.870910i	$-0.336470\pi$
$43$	277.144	0.982887	0.491443	+	0.870910i	$0.336470\pi$
$44$	−145.160	−0.497357
$45$	0	0
$46$	65.0300	0.208438
$47$	52.1450	0.161832	0.0809162	−	0.996721i	$-0.474215\pi$
$47$	52.1450	0.161832	0.0809162	+	0.996721i	$0.474215\pi$
$48$	−12.6857	−0.0381464
$49$	49.0000	0.142857
$50$	0	0
$51$	49.5989	0.136181
$52$	−325.160	−0.867145
$53$	−82.3907	−0.213533	−0.106766	−	0.994284i	$-0.534050\pi$
$53$	−82.3907	−0.213533	−0.106766	+	0.994284i	$0.534050\pi$
$54$	−591.375	−1.49030
$55$	0	0
$56$	−167.467	−0.399619
$57$	−89.1723	−0.207213
$58$	191.281	0.433041
$59$	712.816	1.57289	0.786447	−	0.617658i	$-0.211919\pi$
$59$	712.816	1.57289	0.786447	+	0.617658i	$0.211919\pi$
$60$	0	0
$61$	−647.078	−1.35819	−0.679097	−	0.734048i	$-0.737629\pi$
$61$	−647.078	−1.35819	−0.679097	+	0.734048i	$0.737629\pi$
$62$	−809.217	−1.65759
$63$	−135.135	−0.270244
$64$	−820.804	−1.60313
$65$	0	0
$66$	−140.484	−0.262007
$67$	−260.867	−0.475672	−0.237836	−	0.971305i	$-0.576438\pi$
$67$	−260.867	−0.475672	−0.237836	+	0.971305i	$0.576438\pi$
$68$	−235.951	−0.420783
$69$	39.1821	0.0683619
$70$	0	0
$71$	369.025	0.616833	0.308417	−	0.951251i	$-0.400201\pi$
$71$	369.025	0.616833	0.308417	+	0.951251i	$0.400201\pi$
$72$	461.849	0.755964
$73$	−1145.77	−1.83702	−0.918509	−	0.395401i	$-0.870606\pi$
$73$	−1145.77	−1.83702	−0.918509	+	0.395401i	$0.870606\pi$
$74$	1348.39	2.11820
$75$	0	0
$76$	424.209	0.640264
$77$	−77.0000	−0.113961
$78$	−314.686	−0.456810
$79$	488.885	0.696251	0.348125	−	0.937448i	$-0.386818\pi$
$79$	488.885	0.696251	0.348125	+	0.937448i	$0.386818\pi$
$80$	0	0
$81$	164.917	0.226224
$82$	−710.372	−0.956676
$83$	−548.982	−0.726008	−0.363004	−	0.931788i	$-0.618249\pi$
$83$	−548.982	−0.726008	−0.363004	+	0.931788i	$0.618249\pi$
$84$	−256.246	−0.332842
$85$	0	0
$86$	−1275.96	−1.59989
$87$	115.251	0.142026
$88$	263.162	0.318786
$89$	105.039	0.125102	0.0625510	−	0.998042i	$-0.480076\pi$
$89$	105.039	0.125102	0.0625510	+	0.998042i	$0.480076\pi$
$90$	0	0
$91$	−172.481	−0.198691
$92$	−186.396	−0.211230
$93$	−487.572	−0.543644
$94$	−240.073	−0.263422
$95$	0	0
$96$	−472.510	−0.502348
$97$	1361.91	1.42557	0.712787	−	0.701381i	$-0.247433\pi$
$97$	1361.91	1.42557	0.712787	+	0.701381i	$0.247433\pi$
$98$	−225.594	−0.232535
$99$	212.355	0.215580
$100$	0	0
$101$	−1610.32	−1.58647	−0.793234	−	0.608917i	$-0.791604\pi$
$101$	−1610.32	−1.58647	−0.793234	+	0.608917i	$0.791604\pi$
$102$	−228.351	−0.221668
$103$	123.044	0.117708	0.0588540	−	0.998267i	$-0.481255\pi$
$103$	123.044	0.117708	0.0588540	+	0.998267i	$0.481255\pi$
$104$	589.485	0.555805
$105$	0	0
$106$	379.323	0.347576
$107$	1740.90	1.57289	0.786446	−	0.617660i	$-0.211919\pi$
$107$	1740.90	1.57289	0.786446	+	0.617660i	$0.211919\pi$
$108$	1695.07	1.51026
$109$	248.938	0.218752	0.109376	−	0.994000i	$-0.465115\pi$
$109$	248.938	0.218752	0.109376	+	0.994000i	$0.465115\pi$
$110$	0	0
$111$	812.436	0.694712
$112$	32.0117	0.0270073
$113$	494.465	0.411641	0.205820	−	0.978590i	$-0.434014\pi$
$113$	494.465	0.411641	0.205820	+	0.978590i	$0.434014\pi$
$114$	410.545	0.337290
$115$	0	0
$116$	−548.271	−0.438842
$117$	475.677	0.375866
$118$	−3281.77	−2.56026
$119$	−125.160	−0.0964151
$120$	0	0
$121$	121.000	0.0909091
$122$	2979.12	2.21079
$123$	−428.016	−0.313763
$124$	2319.47	1.67979
$125$	0	0
$126$	622.155	0.439888
$127$	979.104	0.684106	0.342053	−	0.939681i	$-0.388878\pi$
$127$	979.104	0.684106	0.342053	+	0.939681i	$0.388878\pi$
$128$	2416.25	1.66850
$129$	−768.795	−0.524718
$130$	0	0
$131$	−1660.85	−1.10770	−0.553851	−	0.832616i	$-0.686842\pi$
$131$	−1660.85	−1.10770	−0.553851	+	0.832616i	$0.686842\pi$
$132$	402.672	0.265516
$133$	225.021	0.146705
$134$	1201.02	0.774271
$135$	0	0
$136$	427.758	0.269705
$137$	1618.17	1.00912	0.504559	−	0.863377i	$-0.331655\pi$
$137$	1618.17	1.00912	0.504559	+	0.863377i	$0.331655\pi$
$138$	−180.393	−0.111276
$139$	695.736	0.424544	0.212272	−	0.977211i	$-0.431914\pi$
$139$	695.736	0.424544	0.212272	+	0.977211i	$0.431914\pi$
$140$	0	0
$141$	−144.650	−0.0863950
$142$	−1698.97	−1.00405
$143$	271.041	0.158501
$144$	−88.2835	−0.0510900
$145$	0	0
$146$	5275.07	2.99019
$147$	−135.925	−0.0762649
$148$	−3864.90	−2.14658
$149$	−2081.84	−1.14464	−0.572318	−	0.820032i	$-0.693956\pi$
$149$	−2081.84	−1.14464	−0.572318	+	0.820032i	$0.693956\pi$
$150$	0	0
$151$	2679.28	1.44395	0.721975	−	0.691919i	$-0.243234\pi$
$151$	2679.28	1.44395	0.721975	+	0.691919i	$0.243234\pi$
$152$	−769.052	−0.410384
$153$	345.173	0.182390
$154$	354.504	0.185498
$155$	0	0
$156$	901.989	0.462929
$157$	−2410.32	−1.22525	−0.612627	−	0.790372i	$-0.709887\pi$
$157$	−2410.32	−1.22525	−0.612627	+	0.790372i	$0.709887\pi$
$158$	−2250.80	−1.13332
$159$	228.551	0.113995
$160$	0	0
$161$	−98.8738	−0.0483997
$162$	−759.271	−0.368234
$163$	−3629.08	−1.74388	−0.871938	−	0.489616i	$-0.837137\pi$
$163$	−3629.08	−1.74388	−0.871938	+	0.489616i	$0.837137\pi$
$164$	2036.15	0.969491
$165$	0	0
$166$	2527.49	1.18175
$167$	3826.04	1.77286	0.886432	−	0.462859i	$-0.153176\pi$
$167$	3826.04	1.77286	0.886432	+	0.462859i	$0.153176\pi$
$168$	464.550	0.213338
$169$	−1589.87	−0.723653
$170$	0	0
$171$	−620.576	−0.277524
$172$	3657.30	1.62132
$173$	4310.67	1.89442	0.947209	−	0.320617i	$-0.103890\pi$
$173$	4310.67	1.89442	0.947209	+	0.320617i	$0.103890\pi$
$174$	−530.611	−0.231181
$175$	0	0
$176$	−50.3040	−0.0215444
$177$	−1977.34	−0.839696
$178$	−483.593	−0.203634
$179$	2491.12	1.04019	0.520097	−	0.854107i	$-0.325896\pi$
$179$	2491.12	1.04019	0.520097	+	0.854107i	$0.325896\pi$
$180$	0	0
$181$	−4315.49	−1.77220	−0.886098	−	0.463498i	$-0.846594\pi$
$181$	−4315.49	−1.77220	−0.886098	+	0.463498i	$0.846594\pi$
$182$	794.093	0.323418
$183$	1794.99	0.725078
$184$	337.920	0.135390
$185$	0	0
$186$	2244.76	0.884912
$187$	196.680	0.0769127
$188$	688.124	0.266950
$189$	899.147	0.346049
$190$	0	0
$191$	2840.41	1.07605	0.538024	−	0.842930i	$-0.319171\pi$
$191$	2840.41	1.07605	0.538024	+	0.842930i	$0.319171\pi$
$192$	2276.90	0.855839
$193$	1734.68	0.646969	0.323485	−	0.946233i	$-0.395146\pi$
$193$	1734.68	0.646969	0.323485	+	0.946233i	$0.395146\pi$
$194$	−6270.15	−2.32047
$195$	0	0
$196$	646.622	0.235649
$197$	3098.42	1.12057	0.560287	−	0.828298i	$-0.310691\pi$
$197$	3098.42	1.12057	0.560287	+	0.828298i	$0.310691\pi$
$198$	−977.671	−0.350909
$199$	4497.38	1.60206	0.801032	−	0.598622i	$-0.204285\pi$
$199$	4497.38	1.60206	0.801032	+	0.598622i	$0.204285\pi$
$200$	0	0
$201$	723.643	0.253940
$202$	7413.85	2.58236
$203$	−290.830	−0.100553
$204$	654.525	0.224637
$205$	0	0
$206$	−566.491	−0.191598
$207$	272.680	0.0915582
$208$	−112.682	−0.0375628
$209$	−353.605	−0.117030
$210$	0	0
$211$	1262.32	0.411857	0.205929	−	0.978567i	$-0.433979\pi$
$211$	1262.32	0.411857	0.205929	+	0.978567i	$0.433979\pi$
$212$	−1087.26	−0.352232
$213$	−1023.67	−0.329299
$214$	−8015.03	−2.56026
$215$	0	0
$216$	−3073.00	−0.968015
$217$	1230.36	0.384895
$218$	−1146.10	−0.356071
$219$	3178.35	0.980700
$220$	0	0
$221$	440.565	0.134098
$222$	−3740.42	−1.13081
$223$	−2931.38	−0.880268	−0.440134	−	0.897932i	$-0.645069\pi$
$223$	−2931.38	−0.880268	−0.440134	+	0.897932i	$0.645069\pi$
$224$	1192.35	0.355658
$225$	0	0
$226$	−2276.49	−0.670045
$227$	−4298.37	−1.25680	−0.628399	−	0.777891i	$-0.716289\pi$
$227$	−4298.37	−1.25680	−0.628399	+	0.777891i	$0.716289\pi$
$228$	−1176.75	−0.341808
$229$	698.500	0.201564	0.100782	−	0.994909i	$-0.467866\pi$
$229$	698.500	0.201564	0.100782	+	0.994909i	$0.467866\pi$
$230$	0	0
$231$	213.597	0.0608383
$232$	993.965	0.281280
$233$	−1887.78	−0.530783	−0.265391	−	0.964141i	$-0.585501\pi$
$233$	−1887.78	−0.530783	−0.265391	+	0.964141i	$0.585501\pi$
$234$	−2189.99	−0.611813
$235$	0	0
$236$	9406.57	2.59456
$237$	−1356.16	−0.371697
$238$	576.231	0.156939
$239$	−6449.93	−1.74565	−0.872827	−	0.488030i	$-0.837716\pi$
$239$	−6449.93	−1.74565	−0.872827	+	0.488030i	$0.837716\pi$
$240$	0	0
$241$	4636.98	1.23939	0.619697	−	0.784841i	$-0.287256\pi$
$241$	4636.98	1.23939	0.619697	+	0.784841i	$0.287256\pi$
$242$	−557.078	−0.147977
$243$	−3925.62	−1.03633
$244$	−8539.08	−2.24040
$245$	0	0
$246$	1970.56	0.510726
$247$	−792.078	−0.204043
$248$	−4204.98	−1.07668
$249$	1522.87	0.387582
$250$	0	0
$251$	2194.11	0.551758	0.275879	−	0.961192i	$-0.411031\pi$
$251$	2194.11	0.551758	0.275879	+	0.961192i	$0.411031\pi$
$252$	−1783.29	−0.445780
$253$	155.373	0.0386096
$254$	−4507.75	−1.11355
$255$	0	0
$256$	−4557.87	−1.11276
$257$	2966.97	0.720133	0.360067	−	0.932927i	$-0.382754\pi$
$257$	2966.97	0.720133	0.360067	+	0.932927i	$0.382754\pi$
$258$	3539.50	0.854106
$259$	−2050.14	−0.491850
$260$	0	0
$261$	802.066	0.190217
$262$	7646.47	1.80305
$263$	−915.810	−0.214720	−0.107360	−	0.994220i	$-0.534240\pi$
$263$	−915.810	−0.214720	−0.107360	+	0.994220i	$0.534240\pi$
$264$	−730.008	−0.170185
$265$	0	0
$266$	−1035.99	−0.238799
$267$	−291.376	−0.0667863
$268$	−3442.50	−0.784642
$269$	−164.462	−0.0372768	−0.0186384	−	0.999826i	$-0.505933\pi$
$269$	−164.462	−0.0372768	−0.0186384	+	0.999826i	$0.505933\pi$
$270$	0	0
$271$	1502.60	0.336815	0.168407	−	0.985718i	$-0.446138\pi$
$271$	1502.60	0.336815	0.168407	+	0.985718i	$0.446138\pi$
$272$	−81.7670	−0.0182274
$273$	478.459	0.106072
$274$	−7449.96	−1.64259
$275$	0	0
$276$	517.061	0.112766
$277$	7500.11	1.62685	0.813426	−	0.581669i	$-0.197600\pi$
$277$	7500.11	1.62685	0.813426	+	0.581669i	$0.197600\pi$
$278$	−3203.14	−0.691048
$279$	−3393.15	−0.728111
$280$	0	0
$281$	−2945.12	−0.625235	−0.312618	−	0.949879i	$-0.601206\pi$
$281$	−2945.12	−0.625235	−0.312618	+	0.949879i	$0.601206\pi$
$282$	665.959	0.140629
$283$	−5215.29	−1.09547	−0.547733	−	0.836653i	$-0.684509\pi$
$283$	−5215.29	−1.09547	−0.547733	+	0.836653i	$0.684509\pi$
$284$	4869.78	1.01749
$285$	0	0
$286$	−1247.86	−0.257998
$287$	1080.07	0.222142
$288$	−3288.34	−0.672802
$289$	−4593.31	−0.934929
$290$	0	0
$291$	−3777.91	−0.761049
$292$	−15120.0	−3.03024
$293$	−7407.99	−1.47706	−0.738531	−	0.674219i	$-0.764480\pi$
$293$	−7407.99	−1.47706	−0.738531	+	0.674219i	$0.764480\pi$
$294$	625.794	0.124140
$295$	0	0
$296$	7006.72	1.37587
$297$	−1412.94	−0.276052
$298$	9584.68	1.86317
$299$	348.037	0.0673161
$300$	0	0
$301$	1940.01	0.371496
$302$	−12335.3	−2.35038
$303$	4467.02	0.846942
$304$	147.006	0.0277348
$305$	0	0
$306$	−1589.16	−0.296883
$307$	6850.01	1.27345	0.636727	−	0.771089i	$-0.280288\pi$
$307$	6850.01	1.27345	0.636727	+	0.771089i	$0.280288\pi$
$308$	−1016.12	−0.187983
$309$	−341.324	−0.0628390
$310$	0	0
$311$	−5538.62	−1.00986	−0.504929	−	0.863161i	$-0.668481\pi$
$311$	−5538.62	−1.00986	−0.504929	+	0.863161i	$0.668481\pi$
$312$	−1635.22	−0.296719
$313$	−9361.13	−1.69049	−0.845243	−	0.534382i	$-0.820544\pi$
$313$	−9361.13	−1.69049	−0.845243	+	0.534382i	$0.820544\pi$
$314$	11097.0	1.99440
$315$	0	0
$316$	6451.50	1.14850
$317$	−219.221	−0.0388413	−0.0194206	−	0.999811i	$-0.506182\pi$
$317$	−219.221	−0.0388413	−0.0194206	+	0.999811i	$0.506182\pi$
$318$	−1052.24	−0.185555
$319$	457.018	0.0802135
$320$	0	0
$321$	−4829.24	−0.839695
$322$	455.210	0.0787822
$323$	−574.769	−0.0990123
$324$	2176.31	0.373167
$325$	0	0
$326$	16708.1	2.83858
$327$	−690.551	−0.116781
$328$	−3691.35	−0.621405
$329$	365.015	0.0611669
$330$	0	0
$331$	3377.63	0.560880	0.280440	−	0.959872i	$-0.409520\pi$
$331$	3377.63	0.560880	0.280440	+	0.959872i	$0.409520\pi$
$332$	−7244.57	−1.19758
$333$	5653.98	0.930439
$334$	−17614.9	−2.88577
$335$	0	0
$336$	−88.8000	−0.0144180
$337$	−7755.93	−1.25369	−0.626843	−	0.779145i	$-0.715653\pi$
$337$	−7755.93	−1.25369	−0.626843	+	0.779145i	$0.715653\pi$
$338$	7319.66	1.17792
$339$	−1371.64	−0.219756
$340$	0	0
$341$	−1933.42	−0.307040
$342$	2857.10	0.451738
$343$	343.000	0.0539949
$344$	−6630.35	−1.03920
$345$	0	0
$346$	−19846.1	−3.08362
$347$	831.044	0.128567	0.0642835	−	0.997932i	$-0.479524\pi$
$347$	831.044	0.128567	0.0642835	+	0.997932i	$0.479524\pi$
$348$	1520.90	0.234278
$349$	1840.94	0.282359	0.141180	−	0.989984i	$-0.454911\pi$
$349$	1840.94	0.282359	0.141180	+	0.989984i	$0.454911\pi$
$350$	0	0
$351$	−3165.01	−0.481298
$352$	−1873.70	−0.283717
$353$	−3409.11	−0.514019	−0.257009	−	0.966409i	$-0.582737\pi$
$353$	−3409.11	−0.514019	−0.257009	+	0.966409i	$0.582737\pi$
$354$	9103.59	1.36681
$355$	0	0
$356$	1386.13	0.206361
$357$	347.193	0.0514716
$358$	−11469.0	−1.69317
$359$	2199.75	0.323394	0.161697	−	0.986840i	$-0.448303\pi$
$359$	2199.75	0.323394	0.161697	+	0.986840i	$0.448303\pi$
$360$	0	0
$361$	−5825.64	−0.849343
$362$	19868.3	2.88468
$363$	−335.653	−0.0485322
$364$	−2276.12	−0.327750
$365$	0	0
$366$	−8264.04	−1.18024
$367$	855.008	0.121611	0.0608053	−	0.998150i	$-0.480633\pi$
$367$	855.008	0.121611	0.0608053	+	0.998150i	$0.480633\pi$
$368$	−64.5942	−0.00915002
$369$	−2978.69	−0.420228
$370$	0	0
$371$	−576.735	−0.0807078
$372$	−6434.18	−0.896765
$373$	−3193.40	−0.443293	−0.221646	−	0.975127i	$-0.571143\pi$
$373$	−3193.40	−0.443293	−0.221646	+	0.975127i	$0.571143\pi$
$374$	−905.505	−0.125194
$375$	0	0
$376$	−1247.51	−0.171104
$377$	1023.72	0.139853
$378$	−4139.63	−0.563279
$379$	−5614.48	−0.760940	−0.380470	−	0.924793i	$-0.624238\pi$
$379$	−5614.48	−0.760940	−0.380470	+	0.924793i	$0.624238\pi$
$380$	0	0
$381$	−2716.02	−0.365213
$382$	−13077.1	−1.75153
$383$	1736.86	0.231721	0.115861	−	0.993265i	$-0.463037\pi$
$383$	1736.86	0.231721	0.115861	+	0.993265i	$0.463037\pi$
$384$	−6702.65	−0.890738
$385$	0	0
$386$	−7986.38	−1.05310
$387$	−5350.27	−0.702763
$388$	17972.2	2.35155
$389$	8710.78	1.13536	0.567679	−	0.823250i	$-0.307841\pi$
$389$	8710.78	1.13536	0.567679	+	0.823250i	$0.307841\pi$
$390$	0	0
$391$	252.552	0.0326652
$392$	−1172.27	−0.151042
$393$	4607.17	0.591352
$394$	−14265.0	−1.82401
$395$	0	0
$396$	2802.31	0.355610
$397$	−11731.6	−1.48311	−0.741553	−	0.670894i	$-0.765911\pi$
$397$	−11731.6	−1.48311	−0.741553	+	0.670894i	$0.765911\pi$
$398$	−20705.7	−2.60775
$399$	−624.206	−0.0783193
$400$	0	0
$401$	−14408.8	−1.79437	−0.897183	−	0.441659i	$-0.854390\pi$
$401$	−14408.8	−1.79437	−0.897183	+	0.441659i	$0.854390\pi$
$402$	−3331.62	−0.413348
$403$	−4330.88	−0.535327
$404$	−21250.4	−2.61695
$405$	0	0
$406$	1338.97	0.163674
$407$	3221.64	0.392361
$408$	−1186.59	−0.143983
$409$	−2155.54	−0.260598	−0.130299	−	0.991475i	$-0.541594\pi$
$409$	−2155.54	−0.260598	−0.130299	+	0.991475i	$0.541594\pi$
$410$	0	0
$411$	−4488.77	−0.538722
$412$	1623.74	0.194165
$413$	4989.71	0.594498
$414$	−1255.40	−0.149033
$415$	0	0
$416$	−4197.10	−0.494663
$417$	−1929.96	−0.226644
$418$	1627.98	0.190495
$419$	8443.41	0.984458	0.492229	−	0.870466i	$-0.336182\pi$
$419$	8443.41	0.984458	0.492229	+	0.870466i	$0.336182\pi$
$420$	0	0
$421$	−3070.28	−0.355430	−0.177715	−	0.984082i	$-0.556871\pi$
$421$	−3070.28	−0.355430	−0.177715	+	0.984082i	$0.556871\pi$
$422$	−5811.67	−0.670398
$423$	−1006.66	−0.115710
$424$	1971.10	0.225767
$425$	0	0
$426$	4712.93	0.536014
$427$	−4529.55	−0.513349
$428$	22973.6	2.59456
$429$	−751.865	−0.0846163
$430$	0	0
$431$	−7004.40	−0.782808	−0.391404	−	0.920219i	$-0.628010\pi$
$431$	−7004.40	−0.782808	−0.391404	+	0.920219i	$0.628010\pi$
$432$	587.412	0.0654210
$433$	−7486.00	−0.830841	−0.415421	−	0.909629i	$-0.636366\pi$
$433$	−7486.00	−0.830841	−0.415421	+	0.909629i	$0.636366\pi$
$434$	−5664.52	−0.626510
$435$	0	0
$436$	3285.07	0.360841
$437$	−454.055	−0.0497034
$438$	−14633.0	−1.59633
$439$	−13046.2	−1.41836	−0.709179	−	0.705029i	$-0.750934\pi$
$439$	−13046.2	−1.41836	−0.709179	+	0.705029i	$0.750934\pi$
$440$	0	0
$441$	−945.944	−0.102143
$442$	−2028.34	−0.218277
$443$	11781.3	1.26354	0.631768	−	0.775157i	$-0.282329\pi$
$443$	11781.3	1.26354	0.631768	+	0.775157i	$0.282329\pi$
$444$	10721.2	1.14596
$445$	0	0
$446$	13495.9	1.43285
$447$	5774.99	0.611069
$448$	−5745.63	−0.605927
$449$	7576.58	0.796349	0.398175	−	0.917310i	$-0.369644\pi$
$449$	7576.58	0.796349	0.398175	+	0.917310i	$0.369644\pi$
$450$	0	0
$451$	−1697.26	−0.177208
$452$	6525.14	0.679020
$453$	−7432.28	−0.770859
$454$	19789.5	2.04574
$455$	0	0
$456$	2133.34	0.219085
$457$	−11793.0	−1.20712	−0.603560	−	0.797318i	$-0.706252\pi$
$457$	−11793.0	−1.20712	−0.603560	+	0.797318i	$0.706252\pi$
$458$	−3215.86	−0.328095
$459$	−2296.68	−0.233551
$460$	0	0
$461$	4228.32	0.427185	0.213592	−	0.976923i	$-0.431484\pi$
$461$	4228.32	0.427185	0.213592	+	0.976923i	$0.431484\pi$
$462$	−983.391	−0.0990292
$463$	−14448.8	−1.45031	−0.725154	−	0.688586i	$-0.758232\pi$
$463$	−14448.8	−1.45031	−0.725154	+	0.688586i	$0.758232\pi$
$464$	−189.999	−0.0190096
$465$	0	0
$466$	8691.23	0.863977
$467$	−16547.5	−1.63967	−0.819836	−	0.572599i	$-0.805935\pi$
$467$	−16547.5	−1.63967	−0.819836	+	0.572599i	$0.805935\pi$
$468$	6277.20	0.620008
$469$	−1826.07	−0.179787
$470$	0	0
$471$	6686.21	0.654107
$472$	−17053.3	−1.66301
$473$	−3048.59	−0.296351
$474$	6243.69	0.605026
$475$	0	0
$476$	−1651.66	−0.159041
$477$	1590.55	0.152676
$478$	29695.2	2.84148
$479$	−2989.34	−0.285149	−0.142575	−	0.989784i	$-0.545538\pi$
$479$	−2989.34	−0.285149	−0.142575	+	0.989784i	$0.545538\pi$
$480$	0	0
$481$	7216.51	0.684084
$482$	−21348.4	−2.01741
$483$	274.275	0.0258384
$484$	1596.76	0.149959
$485$	0	0
$486$	18073.3	1.68688
$487$	5549.61	0.516379	0.258190	−	0.966094i	$-0.416874\pi$
$487$	5549.61	0.516379	0.258190	+	0.966094i	$0.416874\pi$
$488$	15480.6	1.43601
$489$	10067.0	0.930976
$490$	0	0
$491$	−7751.20	−0.712438	−0.356219	−	0.934403i	$-0.615934\pi$
$491$	−7751.20	−0.712438	−0.356219	+	0.934403i	$0.615934\pi$
$492$	−5648.25	−0.517567
$493$	742.862	0.0678638
$494$	3646.69	0.332130
$495$	0	0
$496$	803.793	0.0727649
$497$	2583.17	0.233141
$498$	−7011.22	−0.630884
$499$	−6841.83	−0.613792	−0.306896	−	0.951743i	$-0.599290\pi$
$499$	−6841.83	−0.613792	−0.306896	+	0.951743i	$0.599290\pi$
$500$	0	0
$501$	−10613.4	−0.946451
$502$	−10101.6	−0.898120
$503$	11518.5	1.02105	0.510523	−	0.859864i	$-0.329452\pi$
$503$	11518.5	1.02105	0.510523	+	0.859864i	$0.329452\pi$
$504$	3232.94	0.285727
$505$	0	0
$506$	−715.330	−0.0628465
$507$	4410.27	0.386325
$508$	12920.6	1.12846
$509$	8292.40	0.722110	0.361055	−	0.932545i	$-0.382417\pi$
$509$	8292.40	0.722110	0.361055	+	0.932545i	$0.382417\pi$
$510$	0	0
$511$	−8020.39	−0.694327
$512$	1654.21	0.142786
$513$	4129.12	0.355371
$514$	−13659.8	−1.17219
$515$	0	0
$516$	−10145.3	−0.865547
$517$	−573.595	−0.0487943
$518$	9438.72	0.800606
$519$	−11957.8	−1.01134
$520$	0	0
$521$	10891.6	0.915868	0.457934	−	0.888986i	$-0.348590\pi$
$521$	10891.6	0.915868	0.457934	+	0.888986i	$0.348590\pi$
$522$	−3692.67	−0.309624
$523$	−14662.0	−1.22586	−0.612928	−	0.790139i	$-0.710008\pi$
$523$	−14662.0	−1.22586	−0.612928	+	0.790139i	$0.710008\pi$
$524$	−21917.2	−1.82721
$525$	0	0
$526$	4216.34	0.349508
$527$	−3142.69	−0.259768
$528$	139.543	0.0115016
$529$	−11967.5	−0.983602
$530$	0	0
$531$	−13760.9	−1.12462
$532$	2969.46	0.241997
$533$	−3801.87	−0.308963
$534$	1341.48	0.108711
$535$	0	0
$536$	6240.94	0.502924
$537$	−6910.33	−0.555313
$538$	757.177	0.0606770
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−19825.8	−1.57556	−0.787778	−	0.615960i	$-0.788768\pi$
$541$	−19825.8	−1.57556	−0.787778	+	0.615960i	$0.788768\pi$
$542$	−6917.92	−0.548247
$543$	11971.1	0.946094
$544$	−3045.61	−0.240036
$545$	0	0
$546$	−2202.80	−0.172658
$547$	−12706.1	−0.993187	−0.496593	−	0.867983i	$-0.665416\pi$
$547$	−12706.1	−0.993187	−0.496593	+	0.867983i	$0.665416\pi$
$548$	21353.9	1.66459
$549$	12491.8	0.971109
$550$	0	0
$551$	−1335.57	−0.103262
$552$	−937.385	−0.0722786
$553$	3422.19	0.263158
$554$	−34530.1	−2.64809
$555$	0	0
$556$	9181.19	0.700304
$557$	12599.1	0.958421	0.479211	−	0.877700i	$-0.340923\pi$
$557$	12599.1	0.958421	0.479211	+	0.877700i	$0.340923\pi$
$558$	15621.9	1.18518
$559$	−6828.86	−0.516691
$560$	0	0
$561$	−545.588	−0.0410602
$562$	13559.2	1.01772
$563$	−6004.47	−0.449482	−0.224741	−	0.974419i	$-0.572154\pi$
$563$	−6004.47	−0.449482	−0.224741	+	0.974419i	$0.572154\pi$
$564$	−1908.85	−0.142512
$565$	0	0
$566$	24010.9	1.78314
$567$	1154.42	0.0855046
$568$	−8828.47	−0.652173
$569$	−3145.89	−0.231779	−0.115890	−	0.993262i	$-0.536972\pi$
$569$	−3145.89	−0.231779	−0.115890	+	0.993262i	$0.536972\pi$
$570$	0	0
$571$	23549.1	1.72592	0.862960	−	0.505273i	$-0.168608\pi$
$571$	23549.1	1.72592	0.862960	+	0.505273i	$0.168608\pi$
$572$	3576.76	0.261454
$573$	−7879.27	−0.574453
$574$	−4972.60	−0.361590
$575$	0	0
$576$	15845.6	1.14624
$577$	−327.335	−0.0236172	−0.0118086	−	0.999930i	$-0.503759\pi$
$577$	−327.335	−0.0236172	−0.0118086	+	0.999930i	$0.503759\pi$
$578$	21147.4	1.52182
$579$	−4811.98	−0.345387
$580$	0	0
$581$	−3842.88	−0.274405
$582$	17393.3	1.23879
$583$	906.298	0.0643825
$584$	27411.2	1.94226
$585$	0	0
$586$	34106.0	2.40428
$587$	13270.8	0.933123	0.466562	−	0.884489i	$-0.345493\pi$
$587$	13270.8	0.933123	0.466562	+	0.884489i	$0.345493\pi$
$588$	−1793.72	−0.125802
$589$	5650.14	0.395263
$590$	0	0
$591$	−8594.98	−0.598224
$592$	−1339.35	−0.0929849
$593$	−5098.92	−0.353099	−0.176549	−	0.984292i	$-0.556494\pi$
$593$	−5098.92	−0.353099	−0.176549	+	0.984292i	$0.556494\pi$
$594$	6505.13	0.449341
$595$	0	0
$596$	−27472.7	−1.88813
$597$	−12475.7	−0.855268
$598$	−1602.35	−0.109573
$599$	19358.2	1.32046	0.660230	−	0.751064i	$-0.270459\pi$
$599$	19358.2	1.32046	0.660230	+	0.751064i	$0.270459\pi$
$600$	0	0
$601$	1238.87	0.0840841	0.0420420	−	0.999116i	$-0.486614\pi$
$601$	1238.87	0.0840841	0.0420420	+	0.999116i	$0.486614\pi$
$602$	−8931.71	−0.604700
$603$	5036.04	0.340105
$604$	35356.7	2.38186
$605$	0	0
$606$	−20565.9	−1.37860
$607$	14175.6	0.947888	0.473944	−	0.880555i	$-0.342830\pi$
$607$	14175.6	0.947888	0.473944	+	0.880555i	$0.342830\pi$
$608$	5475.60	0.365239
$609$	806.758	0.0536806
$610$	0	0
$611$	−1284.86	−0.0850732
$612$	4555.03	0.300860
$613$	−6906.67	−0.455070	−0.227535	−	0.973770i	$-0.573067\pi$
$613$	−6906.67	−0.455070	−0.227535	+	0.973770i	$0.573067\pi$
$614$	−31537.1	−2.07286
$615$	0	0
$616$	1842.13	0.120490
$617$	−12104.3	−0.789789	−0.394894	−	0.918727i	$-0.629219\pi$
$617$	−12104.3	−0.789789	−0.394894	+	0.918727i	$0.629219\pi$
$618$	1571.44	0.102286
$619$	14945.2	0.970437	0.485218	−	0.874393i	$-0.338740\pi$
$619$	14945.2	0.970437	0.485218	+	0.874393i	$0.338740\pi$
$620$	0	0
$621$	−1814.33	−0.117241
$622$	25499.5	1.64379
$623$	735.271	0.0472841
$624$	312.577	0.0200530
$625$	0	0
$626$	43098.2	2.75168
$627$	980.895	0.0624772
$628$	−31807.5	−2.02111
$629$	5236.63	0.331953
$630$	0	0
$631$	−17711.9	−1.11743	−0.558717	−	0.829358i	$-0.688706\pi$
$631$	−17711.9	−1.11743	−0.558717	+	0.829358i	$0.688706\pi$
$632$	−11696.0	−0.736141
$633$	−3501.67	−0.219872
$634$	1009.28	0.0632236
$635$	0	0
$636$	3016.04	0.188041
$637$	−1207.36	−0.0750982
$638$	−2104.09	−0.130567
$639$	−7124.02	−0.441036
$640$	0	0
$641$	−17994.4	−1.10879	−0.554396	−	0.832253i	$-0.687051\pi$
$641$	−17994.4	−1.10879	−0.554396	+	0.832253i	$0.687051\pi$
$642$	22233.6	1.36681
$643$	−27947.8	−1.71408	−0.857039	−	0.515251i	$-0.827699\pi$
$643$	−27947.8	−1.71408	−0.857039	+	0.515251i	$0.827699\pi$
$644$	−1304.77	−0.0798375
$645$	0	0
$646$	2646.21	0.161167
$647$	14336.2	0.871122	0.435561	−	0.900159i	$-0.356550\pi$
$647$	14336.2	0.871122	0.435561	+	0.900159i	$0.356550\pi$
$648$	−3945.45	−0.239185
$649$	−7840.97	−0.474245
$650$	0	0
$651$	−3413.00	−0.205478
$652$	−47890.7	−2.87660
$653$	−4315.79	−0.258637	−0.129318	−	0.991603i	$-0.541279\pi$
$653$	−4315.79	−0.258637	−0.129318	+	0.991603i	$0.541279\pi$
$654$	3179.26	0.190090
$655$	0	0
$656$	705.611	0.0419962
$657$	22119.1	1.31347
$658$	−1680.51	−0.0995640
$659$	4002.23	0.236578	0.118289	−	0.992979i	$-0.462259\pi$
$659$	4002.23	0.236578	0.118289	+	0.992979i	$0.462259\pi$
$660$	0	0
$661$	−12223.4	−0.719268	−0.359634	−	0.933094i	$-0.617098\pi$
$661$	−12223.4	−0.719268	−0.359634	+	0.933094i	$0.617098\pi$
$662$	−15550.4	−0.912968
$663$	−1222.12	−0.0715887
$664$	13133.7	0.767603
$665$	0	0
$666$	−26030.6	−1.51451
$667$	586.846	0.0340671
$668$	50489.9	2.92442
$669$	8131.61	0.469935
$670$	0	0
$671$	7117.86	0.409511
$672$	−3307.57	−0.189870
$673$	6121.53	0.350621	0.175310	−	0.984513i	$-0.443907\pi$
$673$	6121.53	0.350621	0.175310	+	0.984513i	$0.443907\pi$
$674$	35707.9	2.04068
$675$	0	0
$676$	−20980.4	−1.19370
$677$	−9626.46	−0.546492	−0.273246	−	0.961944i	$-0.588097\pi$
$677$	−9626.46	−0.546492	−0.273246	+	0.961944i	$0.588097\pi$
$678$	6314.97	0.357706
$679$	9533.34	0.538816
$680$	0	0
$681$	11923.6	0.670947
$682$	8901.38	0.499782
$683$	−11554.4	−0.647315	−0.323658	−	0.946174i	$-0.604913\pi$
$683$	−11554.4	−0.647315	−0.323658	+	0.946174i	$0.604913\pi$
$684$	−8189.34	−0.457789
$685$	0	0
$686$	−1579.16	−0.0878898
$687$	−1937.63	−0.107606
$688$	1267.41	0.0702318
$689$	2030.12	0.112251
$690$	0	0
$691$	2991.45	0.164689	0.0823444	−	0.996604i	$-0.473759\pi$
$691$	2991.45	0.164689	0.0823444	+	0.996604i	$0.473759\pi$
$692$	56885.2	3.12493
$693$	1486.48	0.0814818
$694$	−3826.08	−0.209274
$695$	0	0
$696$	−2757.25	−0.150163
$697$	−2758.82	−0.149925
$698$	−8475.60	−0.459608
$699$	5236.67	0.283361
$700$	0	0
$701$	30772.8	1.65802	0.829010	−	0.559234i	$-0.188905\pi$
$701$	30772.8	1.65802	0.829010	+	0.559234i	$0.188905\pi$
$702$	14571.5	0.783429
$703$	−9414.77	−0.505099
$704$	9028.84	0.483363
$705$	0	0
$706$	15695.4	0.836690
$707$	−11272.3	−0.599628
$708$	−26093.7	−1.38512
$709$	−21698.3	−1.14936	−0.574681	−	0.818377i	$-0.694874\pi$
$709$	−21698.3	−1.14936	−0.574681	+	0.818377i	$0.694874\pi$
$710$	0	0
$711$	−9437.91	−0.497819
$712$	−2512.93	−0.132269
$713$	−2482.66	−0.130402
$714$	−1598.46	−0.0837826
$715$	0	0
$716$	32873.7	1.71585
$717$	17892.0	0.931925
$718$	−10127.5	−0.526402
$719$	−12364.2	−0.641318	−0.320659	−	0.947195i	$-0.603904\pi$
$719$	−12364.2	−0.641318	−0.320659	+	0.947195i	$0.603904\pi$
$720$	0	0
$721$	861.311	0.0444895
$722$	26821.0	1.38251
$723$	−12862.9	−0.661656
$724$	−56948.7	−2.92332
$725$	0	0
$726$	1545.33	0.0789979
$727$	16017.2	0.817119	0.408559	−	0.912732i	$-0.366031\pi$
$727$	16017.2	0.817119	0.408559	+	0.912732i	$0.366031\pi$
$728$	4126.39	0.210075
$729$	6436.85	0.327026
$730$	0	0
$731$	−4955.34	−0.250725
$732$	23687.3	1.19605
$733$	−10564.4	−0.532340	−0.266170	−	0.963926i	$-0.585758\pi$
$733$	−10564.4	−0.532340	−0.266170	+	0.963926i	$0.585758\pi$
$734$	−3936.42	−0.197951
$735$	0	0
$736$	−2405.97	−0.120496
$737$	2869.54	0.143420
$738$	13713.7	0.684023
$739$	1760.32	0.0876242	0.0438121	−	0.999040i	$-0.486050\pi$
$739$	1760.32	0.0876242	0.0438121	+	0.999040i	$0.486050\pi$
$740$	0	0
$741$	2197.21	0.108929
$742$	2655.26	0.131371
$743$	−11383.4	−0.562067	−0.281034	−	0.959698i	$-0.590677\pi$
$743$	−11383.4	−0.562067	−0.281034	+	0.959698i	$0.590677\pi$
$744$	11664.6	0.574790
$745$	0	0
$746$	14702.3	0.721566
$747$	10598.1	0.519095
$748$	2595.46	0.126871
$749$	12186.3	0.594497
$750$	0	0
$751$	−15058.3	−0.731670	−0.365835	−	0.930680i	$-0.619217\pi$
$751$	−15058.3	−0.731670	−0.365835	+	0.930680i	$0.619217\pi$
$752$	238.464	0.0115637
$753$	−6086.45	−0.294558
$754$	−4713.18	−0.227644
$755$	0	0
$756$	11865.5	0.570824
$757$	−38073.4	−1.82801	−0.914004	−	0.405706i	$-0.867026\pi$
$757$	−38073.4	−1.82801	−0.914004	+	0.405706i	$0.867026\pi$
$758$	25848.8	1.23861
$759$	−431.003	−0.0206119
$760$	0	0
$761$	15232.1	0.725577	0.362788	−	0.931872i	$-0.381825\pi$
$761$	15232.1	0.725577	0.362788	+	0.931872i	$0.381825\pi$
$762$	12504.4	0.594473
$763$	1742.57	0.0826803
$764$	37483.1	1.77499
$765$	0	0
$766$	−7996.40	−0.377182
$767$	−17563.8	−0.826850
$768$	12643.5	0.594053
$769$	12013.1	0.563332	0.281666	−	0.959513i	$-0.409113\pi$
$769$	12013.1	0.563332	0.281666	+	0.959513i	$0.409113\pi$
$770$	0	0
$771$	−8230.33	−0.384446
$772$	22891.5	1.06720
$773$	14258.5	0.663443	0.331721	−	0.943377i	$-0.392371\pi$
$773$	14258.5	0.663443	0.331721	+	0.943377i	$0.392371\pi$
$774$	24632.4	1.14392
$775$	0	0
$776$	−32582.0	−1.50725
$777$	5687.05	0.262576
$778$	−40104.0	−1.84807
$779$	4959.99	0.228126
$780$	0	0
$781$	−4059.27	−0.185982
$782$	−1162.74	−0.0531706
$783$	−5336.70	−0.243574
$784$	224.082	0.0102078
$785$	0	0
$786$	−21211.2	−0.962568
$787$	14377.9	0.651227	0.325613	−	0.945503i	$-0.394429\pi$
$787$	14377.9	0.651227	0.325613	+	0.945503i	$0.394429\pi$
$788$	40887.9	1.84844
$789$	2540.45	0.114629
$790$	0	0
$791$	3461.26	0.155585
$792$	−5080.34	−0.227932
$793$	15944.1	0.713986
$794$	54011.8	2.41412
$795$	0	0
$796$	59349.0	2.64268
$797$	−38515.9	−1.71180	−0.855899	−	0.517144i	$-0.826995\pi$
$797$	−38515.9	−1.71180	−0.855899	+	0.517144i	$0.826995\pi$
$798$	2873.81	0.127484
$799$	−932.352	−0.0412819
$800$	0	0
$801$	−2027.77	−0.0894479
$802$	66337.4	2.92076
$803$	12603.5	0.553882
$804$	9549.46	0.418885
$805$	0	0
$806$	19939.2	0.871374
$807$	456.217	0.0199004
$808$	38525.1	1.67736
$809$	11438.6	0.497106	0.248553	−	0.968618i	$-0.420045\pi$
$809$	11438.6	0.497106	0.248553	+	0.968618i	$0.420045\pi$
$810$	0	0
$811$	15872.9	0.687265	0.343632	−	0.939104i	$-0.388343\pi$
$811$	15872.9	0.687265	0.343632	+	0.939104i	$0.388343\pi$
$812$	−3837.89	−0.165867
$813$	−4168.21	−0.179810
$814$	−14832.3	−0.638662
$815$	0	0
$816$	226.821	0.00973077
$817$	8909.05	0.381503
$818$	9924.00	0.424186
$819$	3329.74	0.142064
$820$	0	0
$821$	−11988.8	−0.509638	−0.254819	−	0.966989i	$-0.582016\pi$
$821$	−11988.8	−0.509638	−0.254819	+	0.966989i	$0.582016\pi$
$822$	20666.1	0.876901
$823$	222.224	0.00941219	0.00470610	−	0.999989i	$-0.498502\pi$
$823$	222.224	0.00941219	0.00470610	+	0.999989i	$0.498502\pi$
$824$	−2943.69	−0.124452
$825$	0	0
$826$	−22972.4	−0.967689
$827$	−7524.36	−0.316382	−0.158191	−	0.987409i	$-0.550566\pi$
$827$	−7524.36	−0.316382	−0.158191	+	0.987409i	$0.550566\pi$
$828$	3598.38	0.151029
$829$	−11807.1	−0.494663	−0.247332	−	0.968931i	$-0.579554\pi$
$829$	−11807.1	−0.494663	−0.247332	+	0.968931i	$0.579554\pi$
$830$	0	0
$831$	−20805.2	−0.868502
$832$	20224.7	0.842746
$833$	−876.120	−0.0364415
$834$	8885.46	0.368919
$835$	0	0
$836$	−4666.30	−0.193047
$837$	22577.0	0.932349
$838$	−38873.1	−1.60244
$839$	−29112.9	−1.19796	−0.598980	−	0.800764i	$-0.704427\pi$
$839$	−29112.9	−1.19796	−0.598980	+	0.800764i	$0.704427\pi$
$840$	0	0
$841$	−22662.8	−0.929224
$842$	14135.4	0.578549
$843$	8169.73	0.333785
$844$	16658.1	0.679377
$845$	0	0
$846$	4634.60	0.188346
$847$	847.000	0.0343604
$848$	−376.781	−0.0152579
$849$	14467.2	0.584819
$850$	0	0
$851$	4136.83	0.166638
$852$	−13508.7	−0.543194
$853$	24892.1	0.999169	0.499584	−	0.866265i	$-0.333486\pi$
$853$	24892.1	0.999169	0.499584	+	0.866265i	$0.333486\pi$
$854$	20853.8	0.835601
$855$	0	0
$856$	−41649.0	−1.66301
$857$	27299.2	1.08812	0.544062	−	0.839045i	$-0.316886\pi$
$857$	27299.2	1.08812	0.544062	+	0.839045i	$0.316886\pi$
$858$	3461.55	0.137733
$859$	−6975.13	−0.277053	−0.138526	−	0.990359i	$-0.544237\pi$
$859$	−6975.13	−0.277053	−0.138526	+	0.990359i	$0.544237\pi$
$860$	0	0
$861$	−2996.11	−0.118591
$862$	32247.9	1.27421
$863$	4481.56	0.176772	0.0883858	−	0.996086i	$-0.471829\pi$
$863$	4481.56	0.176772	0.0883858	+	0.996086i	$0.471829\pi$
$864$	21879.6	0.861526
$865$	0	0
$866$	34465.2	1.35240
$867$	12741.8	0.499116
$868$	16236.3	0.634902
$869$	−5377.73	−0.209928
$870$	0	0
$871$	6427.80	0.250055
$872$	−5955.54	−0.231284
$873$	−26291.6	−1.01928
$874$	2090.45	0.0809044
$875$	0	0
$876$	41942.7	1.61771
$877$	7207.96	0.277532	0.138766	−	0.990325i	$-0.455686\pi$
$877$	7207.96	0.277532	0.138766	+	0.990325i	$0.455686\pi$
$878$	60063.9	2.30872
$879$	20549.7	0.788536
$880$	0	0
$881$	−38413.7	−1.46900	−0.734502	−	0.678607i	$-0.762584\pi$
$881$	−38413.7	−1.46900	−0.734502	+	0.678607i	$0.762584\pi$
$882$	4355.08	0.166262
$883$	8705.04	0.331764	0.165882	−	0.986146i	$-0.446953\pi$
$883$	8705.04	0.331764	0.165882	+	0.986146i	$0.446953\pi$
$884$	5813.86	0.221200
$885$	0	0
$886$	−54240.6	−2.05671
$887$	44100.0	1.66937	0.834687	−	0.550725i	$-0.185649\pi$
$887$	44100.0	1.66937	0.834687	+	0.550725i	$0.185649\pi$
$888$	−19436.6	−0.734514
$889$	6853.73	0.258568
$890$	0	0
$891$	−1814.09	−0.0682091
$892$	−38683.6	−1.45204
$893$	1676.25	0.0628146
$894$	−26587.8	−0.994663
$895$	0	0
$896$	16913.8	0.630635
$897$	−965.451	−0.0359370
$898$	−34882.2	−1.29625
$899$	−7302.55	−0.270916
$900$	0	0
$901$	1473.15	0.0544702
$902$	7814.09	0.288449
$903$	−5381.57	−0.198325
$904$	−11829.5	−0.435224
$905$	0	0
$906$	34217.9	1.25476
$907$	−19921.2	−0.729296	−0.364648	−	0.931145i	$-0.618811\pi$
$907$	−19921.2	−0.729296	−0.364648	+	0.931145i	$0.618811\pi$
$908$	−56722.9	−2.07314
$909$	31087.3	1.13432
$910$	0	0
$911$	−14446.7	−0.525402	−0.262701	−	0.964877i	$-0.584613\pi$
$911$	−14446.7	−0.525402	−0.262701	+	0.964877i	$0.584613\pi$
$912$	−407.793	−0.0148063
$913$	6038.81	0.218900
$914$	54294.4	1.96488
$915$	0	0
$916$	9217.66	0.332489
$917$	−11625.9	−0.418672
$918$	10573.8	0.380160
$919$	44381.1	1.59303	0.796516	−	0.604617i	$-0.206674\pi$
$919$	44381.1	1.59303	0.796516	+	0.604617i	$0.206674\pi$
$920$	0	0
$921$	−19001.8	−0.679839
$922$	−19467.0	−0.695347
$923$	−9092.80	−0.324261
$924$	2818.71	0.100356
$925$	0	0
$926$	66521.6	2.36073
$927$	−2375.37	−0.0841612
$928$	−7076.97	−0.250337
$929$	−3455.30	−0.122029	−0.0610145	−	0.998137i	$-0.519434\pi$
$929$	−3455.30	−0.122029	−0.0610145	+	0.998137i	$0.519434\pi$
$930$	0	0
$931$	1575.15	0.0554494
$932$	−24911.8	−0.875550
$933$	15364.1	0.539117
$934$	76183.8	2.66896
$935$	0	0
$936$	−11380.0	−0.397400
$937$	14962.8	0.521678	0.260839	−	0.965382i	$-0.416001\pi$
$937$	14962.8	0.521678	0.260839	+	0.965382i	$0.416001\pi$
$938$	8407.14	0.292647
$939$	25967.7	0.902474
$940$	0	0
$941$	−6116.95	−0.211909	−0.105955	−	0.994371i	$-0.533790\pi$
$941$	−6116.95	−0.211909	−0.105955	+	0.994371i	$0.533790\pi$
$942$	−30783.0	−1.06472
$943$	−2179.41	−0.0752611
$944$	3259.77	0.112390
$945$	0	0
$946$	14035.6	0.482384
$947$	−53343.7	−1.83045	−0.915225	−	0.402943i	$-0.867987\pi$
$947$	−53343.7	−1.83045	−0.915225	+	0.402943i	$0.867987\pi$
$948$	−17896.4	−0.613130
$949$	28231.9	0.965696
$950$	0	0
$951$	608.117	0.0207356
$952$	2994.30	0.101939
$953$	−1979.62	−0.0672887	−0.0336443	−	0.999434i	$-0.510711\pi$
$953$	−1979.62	−0.0672887	−0.0336443	+	0.999434i	$0.510711\pi$
$954$	−7322.82	−0.248517
$955$	0	0
$956$	−85115.6	−2.87954
$957$	−1267.76	−0.0428223
$958$	13762.8	0.464149
$959$	11327.2	0.381411
$960$	0	0
$961$	1102.58	0.0370104
$962$	−33224.4	−1.11351
$963$	−33608.1	−1.12462
$964$	61191.2	2.04444
$965$	0	0
$966$	−1262.75	−0.0420582
$967$	38892.0	1.29336	0.646681	−	0.762761i	$-0.276157\pi$
$967$	38892.0	1.29336	0.646681	+	0.762761i	$0.276157\pi$
$968$	−2894.78	−0.0961175
$969$	1594.40	0.0528582
$970$	0	0
$971$	−47826.1	−1.58065	−0.790325	−	0.612688i	$-0.790088\pi$
$971$	−47826.1	−1.58065	−0.790325	+	0.612688i	$0.790088\pi$
$972$	−51803.8	−1.70947
$973$	4870.15	0.160462
$974$	−25550.1	−0.840533
$975$	0	0
$976$	−2959.15	−0.0970493
$977$	−20840.5	−0.682442	−0.341221	−	0.939983i	$-0.610840\pi$
$977$	−20840.5	−0.682442	−0.341221	+	0.939983i	$0.610840\pi$
$978$	−46348.1	−1.51539
$979$	−1155.43	−0.0377197
$980$	0	0
$981$	−4805.74	−0.156407
$982$	35686.2	1.15966
$983$	54430.5	1.76609	0.883044	−	0.469290i	$-0.155490\pi$
$983$	54430.5	1.76609	0.883044	+	0.469290i	$0.155490\pi$
$984$	10239.8	0.331740
$985$	0	0
$986$	−3420.10	−0.110465
$987$	−1012.55	−0.0326542
$988$	−10452.5	−0.336579
$989$	−3914.62	−0.125862
$990$	0	0
$991$	−35161.9	−1.12710	−0.563550	−	0.826082i	$-0.690565\pi$
$991$	−35161.9	−1.12710	−0.563550	+	0.826082i	$0.690565\pi$
$992$	29939.2	0.958238
$993$	−9369.50	−0.299428
$994$	−11892.8	−0.379494
$995$	0	0
$996$	20096.4	0.639335
$997$	−26961.6	−0.856451	−0.428225	−	0.903672i	$-0.640861\pi$
$997$	−26961.6	−0.856451	−0.428225	+	0.903672i	$0.640861\pi$
$998$	31499.4	0.999096
$999$	−37619.8	−1.19143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.4.a.p.1.1		4
5.4	even	2		77.4.a.d.1.4	✓	4
15.14	odd	2		693.4.a.l.1.1		4
20.19	odd	2		1232.4.a.s.1.3		4
35.34	odd	2		539.4.a.g.1.4		4
55.54	odd	2		847.4.a.d.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.4	✓	4	5.4	even	2
539.4.a.g.1.4		4	35.34	odd	2
693.4.a.l.1.1		4	15.14	odd	2
847.4.a.d.1.1		4	55.54	odd	2
1232.4.a.s.1.3		4	20.19	odd	2
1925.4.a.p.1.1		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)

\(f(q)\)	\(=\)	\(q-4.60395 q^{2} -2.77399 q^{3} +13.1964 q^{4} +12.7713 q^{6} +7.00000 q^{7} -23.9238 q^{8} -19.3050 q^{9} -11.0000 q^{11} -36.6066 q^{12} -24.6401 q^{13} -32.2277 q^{14} +4.57310 q^{16} -17.8800 q^{17} +88.8792 q^{18} +32.1459 q^{19} -19.4179 q^{21} +50.6435 q^{22} -14.1248 q^{23} +66.3643 q^{24} +113.442 q^{26} +128.450 q^{27} +92.3745 q^{28} -41.5471 q^{29} +175.766 q^{31} +170.336 q^{32} +30.5139 q^{33} +82.3187 q^{34} -254.756 q^{36} -292.877 q^{37} -147.998 q^{38} +68.3513 q^{39} +154.296 q^{41} +89.3991 q^{42} +277.144 q^{43} -145.160 q^{44} +65.0300 q^{46} +52.1450 q^{47} -12.6857 q^{48} +49.0000 q^{49} +49.5989 q^{51} -325.160 q^{52} -82.3907 q^{53} -591.375 q^{54} -167.467 q^{56} -89.1723 q^{57} +191.281 q^{58} +712.816 q^{59} -647.078 q^{61} -809.217 q^{62} -135.135 q^{63} -820.804 q^{64} -140.484 q^{66} -260.867 q^{67} -235.951 q^{68} +39.1821 q^{69} +369.025 q^{71} +461.849 q^{72} -1145.77 q^{73} +1348.39 q^{74} +424.209 q^{76} -77.0000 q^{77} -314.686 q^{78} +488.885 q^{79} +164.917 q^{81} -710.372 q^{82} -548.982 q^{83} -256.246 q^{84} -1275.96 q^{86} +115.251 q^{87} +263.162 q^{88} +105.039 q^{89} -172.481 q^{91} -186.396 q^{92} -487.572 q^{93} -240.073 q^{94} -472.510 q^{96} +1361.91 q^{97} -225.594 q^{98} +212.355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 14 q^{3} + 26 q^{4} + 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 14 q^{14} + 2 q^{16} - 4 q^{17} + 62 q^{18} + 258 q^{19} - 98 q^{21} - 22 q^{22} - 8 q^{23}+ \cdots - 836 q^{99}+O(q^{100}) \) 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

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