LMFDB - Embedded newform 1150.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 38x^{3} + 38x^{2} + 202x + 101 $ x^5 - x^4 - 38x^3 + 38x^2 + 202*x + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.612813$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -2.73841 q^{3} +4.00000 q^{4} +5.47681 q^{6} -3.11829 q^{7} -8.00000 q^{8} -19.5011 q^{9} +30.7965 q^{11} -10.9536 q^{12} +46.0234 q^{13} +6.23659 q^{14} +16.0000 q^{16} +65.4212 q^{17} +39.0022 q^{18} -145.754 q^{19} +8.53916 q^{21} -61.5931 q^{22} -23.0000 q^{23} +21.9073 q^{24} -92.0468 q^{26} +127.339 q^{27} -12.4732 q^{28} +121.515 q^{29} -112.620 q^{31} -32.0000 q^{32} -84.3335 q^{33} -130.842 q^{34} -78.0045 q^{36} +143.450 q^{37} +291.508 q^{38} -126.031 q^{39} -346.345 q^{41} -17.0783 q^{42} +395.672 q^{43} +123.186 q^{44} +46.0000 q^{46} -373.653 q^{47} -43.8145 q^{48} -333.276 q^{49} -179.150 q^{51} +184.094 q^{52} -573.622 q^{53} -254.678 q^{54} +24.9463 q^{56} +399.133 q^{57} -243.029 q^{58} +241.063 q^{59} +846.540 q^{61} +225.240 q^{62} +60.8102 q^{63} +64.0000 q^{64} +168.667 q^{66} +82.8746 q^{67} +261.685 q^{68} +62.9834 q^{69} -178.779 q^{71} +156.009 q^{72} +1114.99 q^{73} -286.900 q^{74} -583.015 q^{76} -96.0327 q^{77} +252.062 q^{78} -238.255 q^{79} +177.824 q^{81} +692.689 q^{82} -450.052 q^{83} +34.1566 q^{84} -791.344 q^{86} -332.757 q^{87} -246.372 q^{88} -1497.22 q^{89} -143.515 q^{91} -92.0000 q^{92} +308.399 q^{93} +747.306 q^{94} +87.6290 q^{96} +1381.69 q^{97} +666.552 q^{98} -600.567 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−2.73841	−0.527007	−0.263503	−	0.964658i	$-0.584878\pi$
$3$	−2.73841	−0.527007	−0.263503	+	0.964658i	$0.584878\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	5.47681	0.372650
$7$	−3.11829	−0.168372	−0.0841860	−	0.996450i	$-0.526829\pi$
$7$	−3.11829	−0.168372	−0.0841860	+	0.996450i	$0.526829\pi$
$8$	−8.00000	−0.353553
$9$	−19.5011	−0.722264
$10$	0	0
$11$	30.7965	0.844137	0.422069	−	0.906564i	$-0.361304\pi$
$11$	30.7965	0.844137	0.422069	+	0.906564i	$0.361304\pi$
$12$	−10.9536	−0.263503
$13$	46.0234	0.981892	0.490946	−	0.871190i	$-0.336651\pi$
$13$	46.0234	0.981892	0.490946	+	0.871190i	$0.336651\pi$
$14$	6.23659	0.119057
$15$	0	0
$16$	16.0000	0.250000
$17$	65.4212	0.933351	0.466675	−	0.884429i	$-0.345452\pi$
$17$	65.4212	0.933351	0.466675	+	0.884429i	$0.345452\pi$
$18$	39.0022	0.510718
$19$	−145.754	−1.75991	−0.879953	−	0.475061i	$-0.842426\pi$
$19$	−145.754	−1.75991	−0.879953	+	0.475061i	$0.842426\pi$
$20$	0	0
$21$	8.53916	0.0887332
$22$	−61.5931	−0.596895
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	21.9073	0.186325
$25$	0	0
$26$	−92.0468	−0.694303
$27$	127.339	0.907645
$28$	−12.4732	−0.0841860
$29$	121.515	0.778094	0.389047	−	0.921218i	$-0.372804\pi$
$29$	121.515	0.778094	0.389047	+	0.921218i	$0.372804\pi$
$30$	0	0
$31$	−112.620	−0.652488	−0.326244	−	0.945286i	$-0.605783\pi$
$31$	−112.620	−0.652488	−0.326244	+	0.945286i	$0.605783\pi$
$32$	−32.0000	−0.176777
$33$	−84.3335	−0.444866
$34$	−130.842	−0.659979
$35$	0	0
$36$	−78.0045	−0.361132
$37$	143.450	0.637380	0.318690	−	0.947859i	$-0.396757\pi$
$37$	143.450	0.637380	0.318690	+	0.947859i	$0.396757\pi$
$38$	291.508	1.24444
$39$	−126.031	−0.517464
$40$	0	0
$41$	−346.345	−1.31927	−0.659633	−	0.751588i	$-0.729288\pi$
$41$	−346.345	−1.31927	−0.659633	+	0.751588i	$0.729288\pi$
$42$	−17.0783	−0.0627438
$43$	395.672	1.40324	0.701621	−	0.712550i	$-0.252460\pi$
$43$	395.672	1.40324	0.701621	+	0.712550i	$0.252460\pi$
$44$	123.186	0.422069
$45$	0	0
$46$	46.0000	0.147442
$47$	−373.653	−1.15964	−0.579818	−	0.814746i	$-0.696876\pi$
$47$	−373.653	−1.15964	−0.579818	+	0.814746i	$0.696876\pi$
$48$	−43.8145	−0.131752
$49$	−333.276	−0.971651
$50$	0	0
$51$	−179.150	−0.491882
$52$	184.094	0.490946
$53$	−573.622	−1.48666	−0.743330	−	0.668924i	$-0.766755\pi$
$53$	−573.622	−1.48666	−0.743330	+	0.668924i	$0.766755\pi$
$54$	−254.678	−0.641802
$55$	0	0
$56$	24.9463	0.0595285
$57$	399.133	0.927482
$58$	−243.029	−0.550195
$59$	241.063	0.531927	0.265963	−	0.963983i	$-0.414310\pi$
$59$	241.063	0.531927	0.265963	+	0.963983i	$0.414310\pi$
$60$	0	0
$61$	846.540	1.77686	0.888429	−	0.459014i	$-0.151797\pi$
$61$	846.540	1.77686	0.888429	+	0.459014i	$0.151797\pi$
$62$	225.240	0.461379
$63$	60.8102	0.121609
$64$	64.0000	0.125000
$65$	0	0
$66$	168.667	0.314568
$67$	82.8746	0.151116	0.0755578	−	0.997141i	$-0.475926\pi$
$67$	82.8746	0.151116	0.0755578	+	0.997141i	$0.475926\pi$
$68$	261.685	0.466675
$69$	62.9834	0.109889
$70$	0	0
$71$	−178.779	−0.298833	−0.149417	−	0.988774i	$-0.547740\pi$
$71$	−178.779	−0.298833	−0.149417	+	0.988774i	$0.547740\pi$
$72$	156.009	0.255359
$73$	1114.99	1.78767	0.893835	−	0.448396i	$-0.148005\pi$
$73$	1114.99	1.78767	0.893835	+	0.448396i	$0.148005\pi$
$74$	−286.900	−0.450696
$75$	0	0
$76$	−583.015	−0.879953
$77$	−96.0327	−0.142129
$78$	252.062	0.365902
$79$	−238.255	−0.339314	−0.169657	−	0.985503i	$-0.554266\pi$
$79$	−238.255	−0.339314	−0.169657	+	0.985503i	$0.554266\pi$
$80$	0	0
$81$	177.824	0.243929
$82$	692.689	0.932862
$83$	−450.052	−0.595177	−0.297588	−	0.954694i	$-0.596182\pi$
$83$	−450.052	−0.595177	−0.297588	+	0.954694i	$0.596182\pi$
$84$	34.1566	0.0443666
$85$	0	0
$86$	−791.344	−0.992242
$87$	−332.757	−0.410061
$88$	−246.372	−0.298447
$89$	−1497.22	−1.78320	−0.891602	−	0.452820i	$-0.850418\pi$
$89$	−1497.22	−1.78320	−0.891602	+	0.452820i	$0.850418\pi$
$90$	0	0
$91$	−143.515	−0.165323
$92$	−92.0000	−0.104257
$93$	308.399	0.343866
$94$	747.306	0.819986
$95$	0	0
$96$	87.6290	0.0931625
$97$	1381.69	1.44629	0.723143	−	0.690699i	$-0.242697\pi$
$97$	1381.69	1.44629	0.723143	+	0.690699i	$0.242697\pi$
$98$	666.552	0.687061
$99$	−600.567	−0.609690
$100$	0	0
$101$	972.455	0.958048	0.479024	−	0.877802i	$-0.340991\pi$
$101$	972.455	0.958048	0.479024	+	0.877802i	$0.340991\pi$
$102$	358.300	0.347813
$103$	−1653.81	−1.58209	−0.791044	−	0.611760i	$-0.790462\pi$
$103$	−1653.81	−1.58209	−0.791044	+	0.611760i	$0.790462\pi$
$104$	−368.187	−0.347151
$105$	0	0
$106$	1147.24	1.05123
$107$	174.048	0.157251	0.0786254	−	0.996904i	$-0.474947\pi$
$107$	174.048	0.157251	0.0786254	+	0.996904i	$0.474947\pi$
$108$	509.356	0.453822
$109$	−1616.11	−1.42014	−0.710070	−	0.704131i	$-0.751337\pi$
$109$	−1616.11	−1.42014	−0.710070	+	0.704131i	$0.751337\pi$
$110$	0	0
$111$	−392.825	−0.335904
$112$	−49.8927	−0.0420930
$113$	1266.16	1.05408	0.527038	−	0.849841i	$-0.323302\pi$
$113$	1266.16	1.05408	0.527038	+	0.849841i	$0.323302\pi$
$114$	−798.266	−0.655829
$115$	0	0
$116$	486.059	0.389047
$117$	−897.508	−0.709185
$118$	−482.125	−0.376129
$119$	−204.002	−0.157150
$120$	0	0
$121$	−382.573	−0.287433
$122$	−1693.08	−1.25643
$123$	948.433	0.695262
$124$	−450.480	−0.326244
$125$	0	0
$126$	−121.620	−0.0859906
$127$	−1077.49	−0.752849	−0.376424	−	0.926447i	$-0.622847\pi$
$127$	−1077.49	−0.752849	−0.376424	+	0.926447i	$0.622847\pi$
$128$	−128.000	−0.0883883
$129$	−1083.51	−0.739518
$130$	0	0
$131$	2734.80	1.82398	0.911988	−	0.410216i	$-0.134547\pi$
$131$	2734.80	1.82398	0.911988	+	0.410216i	$0.134547\pi$
$132$	−337.334	−0.222433
$133$	454.503	0.296319
$134$	−165.749	−0.106855
$135$	0	0
$136$	−523.369	−0.329989
$137$	−657.070	−0.409761	−0.204880	−	0.978787i	$-0.565681\pi$
$137$	−657.070	−0.409761	−0.204880	+	0.978787i	$0.565681\pi$
$138$	−125.967	−0.0777029
$139$	−1923.60	−1.17380	−0.586899	−	0.809660i	$-0.699651\pi$
$139$	−1923.60	−1.17380	−0.586899	+	0.809660i	$0.699651\pi$
$140$	0	0
$141$	1023.21	0.611136
$142$	357.558	0.211307
$143$	1417.36	0.828852
$144$	−312.018	−0.180566
$145$	0	0
$146$	−2229.98	−1.26407
$147$	912.646	0.512067
$148$	573.801	0.318690
$149$	2242.83	1.23315	0.616577	−	0.787295i	$-0.288519\pi$
$149$	2242.83	1.23315	0.616577	+	0.787295i	$0.288519\pi$
$150$	0	0
$151$	2592.34	1.39710	0.698548	−	0.715563i	$-0.253830\pi$
$151$	2592.34	1.39710	0.698548	+	0.715563i	$0.253830\pi$
$152$	1166.03	0.622221
$153$	−1275.79	−0.674125
$154$	192.065	0.100500
$155$	0	0
$156$	−504.124	−0.258732
$157$	2755.08	1.40050	0.700252	−	0.713896i	$-0.253071\pi$
$157$	2755.08	1.40050	0.700252	+	0.713896i	$0.253071\pi$
$158$	476.510	0.239931
$159$	1570.81	0.783480
$160$	0	0
$161$	71.7208	0.0351080
$162$	−355.648	−0.172484
$163$	659.189	0.316758	0.158379	−	0.987378i	$-0.449373\pi$
$163$	659.189	0.316758	0.158379	+	0.987378i	$0.449373\pi$
$164$	−1385.38	−0.659633
$165$	0	0
$166$	900.105	0.420854
$167$	1573.24	0.728987	0.364494	−	0.931206i	$-0.381242\pi$
$167$	1573.24	0.728987	0.364494	+	0.931206i	$0.381242\pi$
$168$	−68.3133	−0.0313719
$169$	−78.8446	−0.0358874
$170$	0	0
$171$	2842.36	1.27112
$172$	1582.69	0.701621
$173$	4377.36	1.92373	0.961863	−	0.273532i	$-0.0881920\pi$
$173$	4377.36	1.92373	0.961863	+	0.273532i	$0.0881920\pi$
$174$	665.514	0.289957
$175$	0	0
$176$	492.745	0.211034
$177$	−660.127	−0.280329
$178$	2994.44	1.26092
$179$	3378.61	1.41078	0.705389	−	0.708821i	$-0.250773\pi$
$179$	3378.61	1.41078	0.705389	+	0.708821i	$0.250773\pi$
$180$	0	0
$181$	−1898.03	−0.779444	−0.389722	−	0.920933i	$-0.627429\pi$
$181$	−1898.03	−0.779444	−0.389722	+	0.920933i	$0.627429\pi$
$182$	287.029	0.116901
$183$	−2318.17	−0.936416
$184$	184.000	0.0737210
$185$	0	0
$186$	−616.799	−0.243150
$187$	2014.75	0.787876
$188$	−1494.61	−0.579818
$189$	−397.080	−0.152822
$190$	0	0
$191$	4638.05	1.75705	0.878527	−	0.477692i	$-0.158526\pi$
$191$	4638.05	1.75705	0.878527	+	0.477692i	$0.158526\pi$
$192$	−175.258	−0.0658758
$193$	−680.705	−0.253877	−0.126938	−	0.991911i	$-0.540515\pi$
$193$	−680.705	−0.253877	−0.126938	+	0.991911i	$0.540515\pi$
$194$	−2763.39	−1.02268
$195$	0	0
$196$	−1333.10	−0.485825
$197$	3629.78	1.31275	0.656373	−	0.754437i	$-0.272090\pi$
$197$	3629.78	1.31275	0.656373	+	0.754437i	$0.272090\pi$
$198$	1201.13	0.431116
$199$	−500.413	−0.178258	−0.0891290	−	0.996020i	$-0.528408\pi$
$199$	−500.413	−0.178258	−0.0891290	+	0.996020i	$0.528408\pi$
$200$	0	0
$201$	−226.944	−0.0796389
$202$	−1944.91	−0.677442
$203$	−378.919	−0.131009
$204$	−716.599	−0.245941
$205$	0	0
$206$	3307.62	1.11870
$207$	448.526	0.150602
$208$	736.375	0.245473
$209$	−4488.71	−1.48560
$210$	0	0
$211$	558.594	0.182252	0.0911261	−	0.995839i	$-0.470953\pi$
$211$	558.594	0.182252	0.0911261	+	0.995839i	$0.470953\pi$
$212$	−2294.49	−0.743330
$213$	489.570	0.157487
$214$	−348.095	−0.111193
$215$	0	0
$216$	−1018.71	−0.320901
$217$	351.182	0.109861
$218$	3232.22	1.00419
$219$	−3053.30	−0.942114
$220$	0	0
$221$	3010.91	0.916450
$222$	785.650	0.237520
$223$	4533.20	1.36128	0.680641	−	0.732617i	$-0.261701\pi$
$223$	4533.20	1.36128	0.680641	+	0.732617i	$0.261701\pi$
$224$	99.7854	0.0297643
$225$	0	0
$226$	−2532.33	−0.745345
$227$	4733.75	1.38410	0.692049	−	0.721851i	$-0.256708\pi$
$227$	4733.75	1.38410	0.692049	+	0.721851i	$0.256708\pi$
$228$	1596.53	0.463741
$229$	1812.74	0.523096	0.261548	−	0.965190i	$-0.415767\pi$
$229$	1812.74	0.523096	0.261548	+	0.965190i	$0.415767\pi$
$230$	0	0
$231$	262.977	0.0749030
$232$	−972.118	−0.275098
$233$	579.945	0.163062	0.0815310	−	0.996671i	$-0.474019\pi$
$233$	579.945	0.163062	0.0815310	+	0.996671i	$0.474019\pi$
$234$	1795.02	0.501470
$235$	0	0
$236$	964.250	0.265963
$237$	652.439	0.178821
$238$	408.005	0.111122
$239$	3777.51	1.02237	0.511186	−	0.859470i	$-0.329206\pi$
$239$	3777.51	1.02237	0.511186	+	0.859470i	$0.329206\pi$
$240$	0	0
$241$	6426.57	1.71773	0.858863	−	0.512205i	$-0.171171\pi$
$241$	6426.57	1.71773	0.858863	+	0.512205i	$0.171171\pi$
$242$	765.146	0.203246
$243$	−3925.11	−1.03620
$244$	3386.16	0.888429
$245$	0	0
$246$	−1896.87	−0.491625
$247$	−6708.09	−1.72804
$248$	900.959	0.230689
$249$	1232.43	0.313662
$250$	0	0
$251$	3038.94	0.764209	0.382104	−	0.924119i	$-0.375199\pi$
$251$	3038.94	0.764209	0.382104	+	0.924119i	$0.375199\pi$
$252$	243.241	0.0608045
$253$	−708.320	−0.176015
$254$	2154.98	0.532345
$255$	0	0
$256$	256.000	0.0625000
$257$	−5545.87	−1.34608	−0.673038	−	0.739608i	$-0.735011\pi$
$257$	−5545.87	−1.34608	−0.673038	+	0.739608i	$0.735011\pi$
$258$	2167.02	0.522918
$259$	−447.320	−0.107317
$260$	0	0
$261$	−2369.67	−0.561989
$262$	−5469.61	−1.28975
$263$	3637.09	0.852746	0.426373	−	0.904547i	$-0.359791\pi$
$263$	3637.09	0.852746	0.426373	+	0.904547i	$0.359791\pi$
$264$	674.668	0.157284
$265$	0	0
$266$	−909.006	−0.209529
$267$	4100.00	0.939760
$268$	331.498	0.0755578
$269$	541.024	0.122628	0.0613138	−	0.998119i	$-0.480471\pi$
$269$	541.024	0.122628	0.0613138	+	0.998119i	$0.480471\pi$
$270$	0	0
$271$	−5831.04	−1.30705	−0.653525	−	0.756905i	$-0.726710\pi$
$271$	−5831.04	−1.30705	−0.653525	+	0.756905i	$0.726710\pi$
$272$	1046.74	0.233338
$273$	393.001	0.0871265
$274$	1314.14	0.289745
$275$	0	0
$276$	251.933	0.0549443
$277$	−6150.80	−1.33417	−0.667086	−	0.744981i	$-0.732459\pi$
$277$	−6150.80	−1.33417	−0.667086	+	0.744981i	$0.732459\pi$
$278$	3847.20	0.830000
$279$	2196.22	0.471269
$280$	0	0
$281$	5792.07	1.22963	0.614814	−	0.788672i	$-0.289231\pi$
$281$	5792.07	1.22963	0.614814	+	0.788672i	$0.289231\pi$
$282$	−2046.43	−0.432138
$283$	2060.79	0.432867	0.216434	−	0.976297i	$-0.430558\pi$
$283$	2060.79	0.432867	0.216434	+	0.976297i	$0.430558\pi$
$284$	−715.116	−0.149417
$285$	0	0
$286$	−2834.72	−0.586087
$287$	1080.00	0.222128
$288$	624.036	0.127679
$289$	−633.073	−0.128857
$290$	0	0
$291$	−3783.64	−0.762202
$292$	4459.97	0.893835
$293$	4064.48	0.810408	0.405204	−	0.914226i	$-0.367201\pi$
$293$	4064.48	0.810408	0.405204	+	0.914226i	$0.367201\pi$
$294$	−1825.29	−0.362086
$295$	0	0
$296$	−1147.60	−0.225348
$297$	3921.60	0.766176
$298$	−4485.66	−0.871971
$299$	−1058.54	−0.204739
$300$	0	0
$301$	−1233.82	−0.236267
$302$	−5184.68	−0.987897
$303$	−2662.98	−0.504898
$304$	−2332.06	−0.439976
$305$	0	0
$306$	2551.57	0.476679
$307$	5482.89	1.01930	0.509650	−	0.860382i	$-0.329775\pi$
$307$	5482.89	1.01930	0.509650	+	0.860382i	$0.329775\pi$
$308$	−384.131	−0.0710645
$309$	4528.81	0.833771
$310$	0	0
$311$	−6876.16	−1.25373	−0.626867	−	0.779126i	$-0.715663\pi$
$311$	−6876.16	−1.25373	−0.626867	+	0.779126i	$0.715663\pi$
$312$	1008.25	0.182951
$313$	4839.09	0.873871	0.436936	−	0.899493i	$-0.356064\pi$
$313$	4839.09	0.873871	0.436936	+	0.899493i	$0.356064\pi$
$314$	−5510.16	−0.990306
$315$	0	0
$316$	−953.020	−0.169657
$317$	820.177	0.145318	0.0726589	−	0.997357i	$-0.476852\pi$
$317$	820.177	0.145318	0.0726589	+	0.997357i	$0.476852\pi$
$318$	−3141.62	−0.554004
$319$	3742.23	0.656818
$320$	0	0
$321$	−476.614	−0.0828722
$322$	−143.442	−0.0248251
$323$	−9535.38	−1.64261
$324$	711.297	0.121964
$325$	0	0
$326$	−1318.38	−0.223982
$327$	4425.57	0.748423
$328$	2770.76	0.466431
$329$	1165.16	0.195250
$330$	0	0
$331$	−175.385	−0.0291239	−0.0145620	−	0.999894i	$-0.504635\pi$
$331$	−175.385	−0.0291239	−0.0145620	+	0.999894i	$0.504635\pi$
$332$	−1800.21	−0.297588
$333$	−2797.44	−0.460357
$334$	−3146.48	−0.515472
$335$	0	0
$336$	136.627	0.0221833
$337$	−5814.29	−0.939836	−0.469918	−	0.882710i	$-0.655717\pi$
$337$	−5814.29	−0.939836	−0.469918	+	0.882710i	$0.655717\pi$
$338$	157.689	0.0253762
$339$	−3467.27	−0.555506
$340$	0	0
$341$	−3468.30	−0.550789
$342$	−5684.73	−0.898815
$343$	2108.83	0.331971
$344$	−3165.38	−0.496121
$345$	0	0
$346$	−8754.72	−1.36028
$347$	981.915	0.151908	0.0759538	−	0.997111i	$-0.475800\pi$
$347$	981.915	0.151908	0.0759538	+	0.997111i	$0.475800\pi$
$348$	−1331.03	−0.205030
$349$	11132.7	1.70750	0.853752	−	0.520680i	$-0.174321\pi$
$349$	11132.7	1.70750	0.853752	+	0.520680i	$0.174321\pi$
$350$	0	0
$351$	5860.58	0.891209
$352$	−985.489	−0.149224
$353$	3475.84	0.524081	0.262040	−	0.965057i	$-0.415605\pi$
$353$	3475.84	0.524081	0.262040	+	0.965057i	$0.415605\pi$
$354$	1320.25	0.198222
$355$	0	0
$356$	−5988.88	−0.891602
$357$	558.642	0.0828192
$358$	−6757.22	−0.997570
$359$	−1627.67	−0.239290	−0.119645	−	0.992817i	$-0.538176\pi$
$359$	−1627.67	−0.239290	−0.119645	+	0.992817i	$0.538176\pi$
$360$	0	0
$361$	14385.2	2.09727
$362$	3796.06	0.551150
$363$	1047.64	0.151479
$364$	−574.058	−0.0826616
$365$	0	0
$366$	4636.34	0.662146
$367$	−2081.53	−0.296063	−0.148031	−	0.988983i	$-0.547294\pi$
$367$	−2081.53	−0.296063	−0.148031	+	0.988983i	$0.547294\pi$
$368$	−368.000	−0.0521286
$369$	6754.11	0.952859
$370$	0	0
$371$	1788.72	0.250312
$372$	1233.60	0.171933
$373$	−1729.99	−0.240149	−0.120074	−	0.992765i	$-0.538313\pi$
$373$	−1729.99	−0.240149	−0.120074	+	0.992765i	$0.538313\pi$
$374$	−4029.49	−0.557112
$375$	0	0
$376$	2989.22	0.409993
$377$	5592.52	0.764004
$378$	794.161	0.108061
$379$	−10704.8	−1.45083	−0.725417	−	0.688309i	$-0.758353\pi$
$379$	−10704.8	−1.45083	−0.725417	+	0.688309i	$0.758353\pi$
$380$	0	0
$381$	2950.61	0.396756
$382$	−9276.10	−1.24243
$383$	4.17758	0.000557349 0	0.000278674	−	1.00000i	$-0.499911\pi$
$383$	4.17758	0.000557349 0	0.000278674		1.00000i	$0.499911\pi$
$384$	350.516	0.0465813
$385$	0	0
$386$	1361.41	0.179518
$387$	−7716.05	−1.01351
$388$	5526.77	0.723143
$389$	6012.53	0.783669	0.391835	−	0.920036i	$-0.371841\pi$
$389$	6012.53	0.783669	0.391835	+	0.920036i	$0.371841\pi$
$390$	0	0
$391$	−1504.69	−0.194617
$392$	2666.21	0.343530
$393$	−7489.01	−0.961248
$394$	−7259.55	−0.928251
$395$	0	0
$396$	−2402.27	−0.304845
$397$	−1246.96	−0.157640	−0.0788200	−	0.996889i	$-0.525115\pi$
$397$	−1246.96	−0.157640	−0.0788200	+	0.996889i	$0.525115\pi$
$398$	1000.83	0.126047
$399$	−1244.61	−0.156162
$400$	0	0
$401$	−10179.5	−1.26768	−0.633839	−	0.773465i	$-0.718522\pi$
$401$	−10179.5	−1.26768	−0.633839	+	0.773465i	$0.718522\pi$
$402$	453.889	0.0563132
$403$	−5183.15	−0.640673
$404$	3889.82	0.479024
$405$	0	0
$406$	757.837	0.0926375
$407$	4417.77	0.538036
$408$	1433.20	0.173907
$409$	640.201	0.0773983	0.0386992	−	0.999251i	$-0.487679\pi$
$409$	640.201	0.0773983	0.0386992	+	0.999251i	$0.487679\pi$
$410$	0	0
$411$	1799.32	0.215947
$412$	−6615.25	−0.791044
$413$	−751.704	−0.0895615
$414$	−897.052	−0.106492
$415$	0	0
$416$	−1472.75	−0.173576
$417$	5267.61	0.618599
$418$	8977.42	1.05048
$419$	6850.34	0.798714	0.399357	−	0.916795i	$-0.369233\pi$
$419$	6850.34	0.798714	0.399357	+	0.916795i	$0.369233\pi$
$420$	0	0
$421$	−5478.84	−0.634257	−0.317129	−	0.948383i	$-0.602719\pi$
$421$	−5478.84	−0.634257	−0.317129	+	0.948383i	$0.602719\pi$
$422$	−1117.19	−0.128872
$423$	7286.65	0.837563
$424$	4588.97	0.525614
$425$	0	0
$426$	−979.140	−0.111360
$427$	−2639.76	−0.299173
$428$	696.191	0.0786254
$429$	−3881.32	−0.436810
$430$	0	0
$431$	1975.76	0.220810	0.110405	−	0.993887i	$-0.464785\pi$
$431$	1975.76	0.220810	0.110405	+	0.993887i	$0.464785\pi$
$432$	2037.42	0.226911
$433$	−15769.8	−1.75022	−0.875111	−	0.483923i	$-0.839212\pi$
$433$	−15769.8	−1.75022	−0.875111	+	0.483923i	$0.839212\pi$
$434$	−702.364	−0.0776833
$435$	0	0
$436$	−6464.44	−0.710070
$437$	3352.34	0.366966
$438$	6106.60	0.666175
$439$	2307.58	0.250877	0.125438	−	0.992101i	$-0.459966\pi$
$439$	2307.58	0.250877	0.125438	+	0.992101i	$0.459966\pi$
$440$	0	0
$441$	6499.26	0.701788
$442$	−6021.81	−0.648028
$443$	−13876.1	−1.48820	−0.744101	−	0.668068i	$-0.767122\pi$
$443$	−13876.1	−1.48820	−0.744101	+	0.668068i	$0.767122\pi$
$444$	−1571.30	−0.167952
$445$	0	0
$446$	−9066.41	−0.962572
$447$	−6141.79	−0.649880
$448$	−199.571	−0.0210465
$449$	−9252.47	−0.972497	−0.486249	−	0.873821i	$-0.661635\pi$
$449$	−9252.47	−0.972497	−0.486249	+	0.873821i	$0.661635\pi$
$450$	0	0
$451$	−10666.2	−1.11364
$452$	5064.66	0.527038
$453$	−7098.88	−0.736279
$454$	−9467.50	−0.978705
$455$	0	0
$456$	−3193.07	−0.327914
$457$	8941.99	0.915293	0.457646	−	0.889134i	$-0.348693\pi$
$457$	8941.99	0.915293	0.457646	+	0.889134i	$0.348693\pi$
$458$	−3625.47	−0.369885
$459$	8330.67	0.847151
$460$	0	0
$461$	5684.17	0.574269	0.287134	−	0.957890i	$-0.407297\pi$
$461$	5684.17	0.574269	0.287134	+	0.957890i	$0.407297\pi$
$462$	−525.953	−0.0529644
$463$	−12185.0	−1.22308	−0.611538	−	0.791215i	$-0.709449\pi$
$463$	−12185.0	−1.22308	−0.611538	+	0.791215i	$0.709449\pi$
$464$	1944.24	0.194523
$465$	0	0
$466$	−1159.89	−0.115302
$467$	−5783.34	−0.573064	−0.286532	−	0.958071i	$-0.592503\pi$
$467$	−5783.34	−0.573064	−0.286532	+	0.958071i	$0.592503\pi$
$468$	−3590.03	−0.354593
$469$	−258.427	−0.0254436
$470$	0	0
$471$	−7544.53	−0.738075
$472$	−1928.50	−0.188064
$473$	12185.3	1.18453
$474$	−1304.88	−0.126445
$475$	0	0
$476$	−816.009	−0.0785751
$477$	11186.3	1.07376
$478$	−7555.03	−0.722927
$479$	−862.300	−0.0822536	−0.0411268	−	0.999154i	$-0.513095\pi$
$479$	−862.300	−0.0822536	−0.0411268	+	0.999154i	$0.513095\pi$
$480$	0	0
$481$	6602.07	0.625839
$482$	−12853.1	−1.21462
$483$	−196.401	−0.0185022
$484$	−1530.29	−0.143716
$485$	0	0
$486$	7850.22	0.732702
$487$	9245.42	0.860267	0.430134	−	0.902765i	$-0.358466\pi$
$487$	9245.42	0.860267	0.430134	+	0.902765i	$0.358466\pi$
$488$	−6772.32	−0.628214
$489$	−1805.13	−0.166934
$490$	0	0
$491$	−143.399	−0.0131802	−0.00659011	−	0.999978i	$-0.502098\pi$
$491$	−143.399	−0.0131802	−0.00659011	+	0.999978i	$0.502098\pi$
$492$	3793.73	0.347631
$493$	7949.63	0.726234
$494$	13416.2	1.22191
$495$	0	0
$496$	−1801.92	−0.163122
$497$	557.486	0.0503152
$498$	−2464.85	−0.221793
$499$	−10448.8	−0.937378	−0.468689	−	0.883363i	$-0.655273\pi$
$499$	−10448.8	−0.937378	−0.468689	+	0.883363i	$0.655273\pi$
$500$	0	0
$501$	−4308.17	−0.384181
$502$	−6077.89	−0.540377
$503$	−13208.2	−1.17082	−0.585412	−	0.810736i	$-0.699067\pi$
$503$	−13208.2	−1.17082	−0.585412	+	0.810736i	$0.699067\pi$
$504$	−486.482	−0.0429953
$505$	0	0
$506$	1416.64	0.124461
$507$	215.909	0.0189129
$508$	−4309.96	−0.376424
$509$	18167.0	1.58200	0.791002	−	0.611813i	$-0.209560\pi$
$509$	18167.0	1.58200	0.791002	+	0.611813i	$0.209560\pi$
$510$	0	0
$511$	−3476.87	−0.300994
$512$	−512.000	−0.0441942
$513$	−18560.1	−1.59737
$514$	11091.7	0.951820
$515$	0	0
$516$	−4334.05	−0.369759
$517$	−11507.2	−0.978891
$518$	894.639	0.0758846
$519$	−11987.0	−1.01382
$520$	0	0
$521$	10365.5	0.871631	0.435815	−	0.900036i	$-0.356460\pi$
$521$	10365.5	0.871631	0.435815	+	0.900036i	$0.356460\pi$
$522$	4739.35	0.397386
$523$	−15496.3	−1.29562	−0.647809	−	0.761803i	$-0.724314\pi$
$523$	−15496.3	−1.29562	−0.647809	+	0.761803i	$0.724314\pi$
$524$	10939.2	0.911988
$525$	0	0
$526$	−7274.17	−0.602983
$527$	−7367.73	−0.609000
$528$	−1349.34	−0.111216
$529$	529.000	0.0434783
$530$	0	0
$531$	−4700.99	−0.384191
$532$	1818.01	0.148159
$533$	−15940.0	−1.29538
$534$	−8200.00	−0.664511
$535$	0	0
$536$	−662.997	−0.0534274
$537$	−9252.01	−0.743489
$538$	−1082.05	−0.0867108
$539$	−10263.8	−0.820206
$540$	0	0
$541$	17055.2	1.35538	0.677689	−	0.735349i	$-0.262982\pi$
$541$	17055.2	1.35538	0.677689	+	0.735349i	$0.262982\pi$
$542$	11662.1	0.924223
$543$	5197.58	0.410772
$544$	−2093.48	−0.164995
$545$	0	0
$546$	−786.003	−0.0616077
$547$	−16809.6	−1.31394	−0.656971	−	0.753916i	$-0.728163\pi$
$547$	−16809.6	−1.31394	−0.656971	+	0.753916i	$0.728163\pi$
$548$	−2628.28	−0.204880
$549$	−16508.5	−1.28336
$550$	0	0
$551$	−17711.2	−1.36937
$552$	−503.867	−0.0388515
$553$	742.949	0.0571309
$554$	12301.6	0.943402
$555$	0	0
$556$	−7694.41	−0.586899
$557$	5297.34	0.402972	0.201486	−	0.979491i	$-0.435423\pi$
$557$	5297.34	0.402972	0.201486	+	0.979491i	$0.435423\pi$
$558$	−4392.43	−0.333237
$559$	18210.2	1.37783
$560$	0	0
$561$	−5517.19	−0.415216
$562$	−11584.1	−0.869479
$563$	−21852.9	−1.63586	−0.817932	−	0.575314i	$-0.804880\pi$
$563$	−21852.9	−1.63586	−0.817932	+	0.575314i	$0.804880\pi$
$564$	4092.86	0.305568
$565$	0	0
$566$	−4121.58	−0.306083
$567$	−554.508	−0.0410708
$568$	1430.23	0.105654
$569$	15184.9	1.11877	0.559387	−	0.828906i	$-0.311036\pi$
$569$	15184.9	1.11877	0.559387	+	0.828906i	$0.311036\pi$
$570$	0	0
$571$	25838.9	1.89374	0.946868	−	0.321622i	$-0.104228\pi$
$571$	25838.9	1.89374	0.946868	+	0.321622i	$0.104228\pi$
$572$	5669.45	0.414426
$573$	−12700.9	−0.925980
$574$	−2160.01	−0.157068
$575$	0	0
$576$	−1248.07	−0.0902830
$577$	11113.7	0.801854	0.400927	−	0.916110i	$-0.368688\pi$
$577$	11113.7	0.801854	0.400927	+	0.916110i	$0.368688\pi$
$578$	1266.15	0.0911155
$579$	1864.05	0.133795
$580$	0	0
$581$	1403.40	0.100211
$582$	7567.28	0.538958
$583$	−17665.6	−1.25495
$584$	−8919.93	−0.632037
$585$	0	0
$586$	−8128.96	−0.573045
$587$	20629.9	1.45057	0.725287	−	0.688447i	$-0.241707\pi$
$587$	20629.9	1.45057	0.725287	+	0.688447i	$0.241707\pi$
$588$	3650.58	0.256033
$589$	16414.8	1.14832
$590$	0	0
$591$	−9939.80	−0.691826
$592$	2295.20	0.159345
$593$	22070.7	1.52839	0.764194	−	0.644986i	$-0.223137\pi$
$593$	22070.7	1.52839	0.764194	+	0.644986i	$0.223137\pi$
$594$	−7843.20	−0.541769
$595$	0	0
$596$	8971.32	0.616577
$597$	1370.34	0.0939432
$598$	2117.08	0.144772
$599$	−3299.87	−0.225090	−0.112545	−	0.993647i	$-0.535900\pi$
$599$	−3299.87	−0.225090	−0.112545	+	0.993647i	$0.535900\pi$
$600$	0	0
$601$	−17617.1	−1.19570	−0.597849	−	0.801608i	$-0.703978\pi$
$601$	−17617.1	−1.19570	−0.597849	+	0.801608i	$0.703978\pi$
$602$	2467.64	0.167066
$603$	−1616.15	−0.109145
$604$	10369.4	0.698548
$605$	0	0
$606$	5325.96	0.357017
$607$	74.0280	0.00495009	0.00247504	−	0.999997i	$-0.499212\pi$
$607$	74.0280	0.00495009	0.00247504	+	0.999997i	$0.499212\pi$
$608$	4664.12	0.311110
$609$	1037.63	0.0690428
$610$	0	0
$611$	−17196.8	−1.13864
$612$	−5103.14	−0.337063
$613$	21583.6	1.42211	0.711054	−	0.703138i	$-0.248218\pi$
$613$	21583.6	1.42211	0.711054	+	0.703138i	$0.248218\pi$
$614$	−10965.8	−0.720754
$615$	0	0
$616$	768.261	0.0502502
$617$	10922.4	0.712673	0.356337	−	0.934358i	$-0.384026\pi$
$617$	10922.4	0.712673	0.356337	+	0.934358i	$0.384026\pi$
$618$	−9057.62	−0.589565
$619$	−5974.14	−0.387918	−0.193959	−	0.981010i	$-0.562133\pi$
$619$	−5974.14	−0.387918	−0.193959	+	0.981010i	$0.562133\pi$
$620$	0	0
$621$	−2928.80	−0.189257
$622$	13752.3	0.886524
$623$	4668.77	0.300242
$624$	−2016.49	−0.129366
$625$	0	0
$626$	−9678.19	−0.617920
$627$	12291.9	0.782922
$628$	11020.3	0.700252
$629$	9384.67	0.594899
$630$	0	0
$631$	−11707.3	−0.738603	−0.369301	−	0.929310i	$-0.620403\pi$
$631$	−11707.3	−0.738603	−0.369301	+	0.929310i	$0.620403\pi$
$632$	1906.04	0.119965
$633$	−1529.66	−0.0960481
$634$	−1640.35	−0.102755
$635$	0	0
$636$	6283.24	0.391740
$637$	−15338.5	−0.954057
$638$	−7484.47	−0.464440
$639$	3486.39	0.215837
$640$	0	0
$641$	−7900.35	−0.486810	−0.243405	−	0.969925i	$-0.578264\pi$
$641$	−7900.35	−0.486810	−0.243405	+	0.969925i	$0.578264\pi$
$642$	953.227	0.0585995
$643$	7156.10	0.438894	0.219447	−	0.975624i	$-0.429575\pi$
$643$	7156.10	0.438894	0.219447	+	0.975624i	$0.429575\pi$
$644$	286.883	0.0175540
$645$	0	0
$646$	19070.8	1.16150
$647$	22643.9	1.37592	0.687962	−	0.725747i	$-0.258506\pi$
$647$	22643.9	1.37592	0.687962	+	0.725747i	$0.258506\pi$
$648$	−1422.59	−0.0862419
$649$	7423.89	0.449019
$650$	0	0
$651$	−961.679	−0.0578974
$652$	2636.75	0.158379
$653$	2633.02	0.157792	0.0788958	−	0.996883i	$-0.474861\pi$
$653$	2633.02	0.157792	0.0788958	+	0.996883i	$0.474861\pi$
$654$	−8851.13	−0.529215
$655$	0	0
$656$	−5541.51	−0.329817
$657$	−21743.6	−1.29117
$658$	−2330.32	−0.138063
$659$	28880.3	1.70715	0.853577	−	0.520966i	$-0.174428\pi$
$659$	28880.3	1.70715	0.853577	+	0.520966i	$0.174428\pi$
$660$	0	0
$661$	−9424.84	−0.554590	−0.277295	−	0.960785i	$-0.589438\pi$
$661$	−9424.84	−0.554590	−0.277295	+	0.960785i	$0.589438\pi$
$662$	350.770	0.0205937
$663$	−8245.09	−0.482975
$664$	3600.42	0.210427
$665$	0	0
$666$	5594.88	0.325521
$667$	−2794.84	−0.162244
$668$	6292.96	0.364494
$669$	−12413.8	−0.717405
$670$	0	0
$671$	26070.5	1.49991
$672$	−273.253	−0.0156860
$673$	8544.52	0.489401	0.244701	−	0.969599i	$-0.421310\pi$
$673$	8544.52	0.489401	0.244701	+	0.969599i	$0.421310\pi$
$674$	11628.6	0.664564
$675$	0	0
$676$	−315.378	−0.0179437
$677$	−16025.6	−0.909767	−0.454883	−	0.890551i	$-0.650319\pi$
$677$	−16025.6	−0.909767	−0.454883	+	0.890551i	$0.650319\pi$
$678$	6934.55	0.392802
$679$	−4308.53	−0.243514
$680$	0	0
$681$	−12962.9	−0.729429
$682$	6936.61	0.389467
$683$	22270.1	1.24765	0.623824	−	0.781565i	$-0.285578\pi$
$683$	22270.1	1.24765	0.623824	+	0.781565i	$0.285578\pi$
$684$	11369.5	0.635558
$685$	0	0
$686$	−4217.66	−0.234739
$687$	−4964.01	−0.275675
$688$	6330.75	0.350811
$689$	−26400.0	−1.45974
$690$	0	0
$691$	−27377.9	−1.50724	−0.753621	−	0.657309i	$-0.771695\pi$
$691$	−27377.9	−1.50724	−0.753621	+	0.657309i	$0.771695\pi$
$692$	17509.4	0.961863
$693$	1872.74	0.102655
$694$	−1963.83	−0.107415
$695$	0	0
$696$	2662.06	0.144978
$697$	−22658.3	−1.23134
$698$	−22265.4	−1.20739
$699$	−1588.12	−0.0859348
$700$	0	0
$701$	6907.69	0.372182	0.186091	−	0.982532i	$-0.440418\pi$
$701$	6907.69	0.372182	0.186091	+	0.982532i	$0.440418\pi$
$702$	−11721.2	−0.630180
$703$	−20908.4	−1.12173
$704$	1970.98	0.105517
$705$	0	0
$706$	−6951.69	−0.370581
$707$	−3032.40	−0.161309
$708$	−2640.51	−0.140164
$709$	4227.19	0.223914	0.111957	−	0.993713i	$-0.464288\pi$
$709$	4227.19	0.223914	0.111957	+	0.993713i	$0.464288\pi$
$710$	0	0
$711$	4646.24	0.245074
$712$	11977.8	0.630458
$713$	2590.26	0.136053
$714$	−1117.28	−0.0585620
$715$	0	0
$716$	13514.4	0.705389
$717$	−10344.4	−0.538797
$718$	3255.34	0.169204
$719$	−25555.1	−1.32551	−0.662756	−	0.748835i	$-0.730614\pi$
$719$	−25555.1	−1.32551	−0.662756	+	0.748835i	$0.730614\pi$
$720$	0	0
$721$	5157.07	0.266379
$722$	−28770.3	−1.48299
$723$	−17598.6	−0.905253
$724$	−7592.12	−0.389722
$725$	0	0
$726$	−2095.28	−0.107112
$727$	−13638.7	−0.695778	−0.347889	−	0.937536i	$-0.613101\pi$
$727$	−13638.7	−0.695778	−0.347889	+	0.937536i	$0.613101\pi$
$728$	1148.12	0.0584506
$729$	5947.29	0.302154
$730$	0	0
$731$	25885.3	1.30972
$732$	−9272.69	−0.468208
$733$	788.421	0.0397285	0.0198643	−	0.999803i	$-0.493677\pi$
$733$	788.421	0.0397285	0.0198643	+	0.999803i	$0.493677\pi$
$734$	4163.06	0.209348
$735$	0	0
$736$	736.000	0.0368605
$737$	2552.25	0.127562
$738$	−13508.2	−0.673773
$739$	9564.14	0.476079	0.238040	−	0.971255i	$-0.423495\pi$
$739$	9564.14	0.476079	0.238040	+	0.971255i	$0.423495\pi$
$740$	0	0
$741$	18369.5	0.910688
$742$	−3577.44	−0.176997
$743$	−15210.1	−0.751018	−0.375509	−	0.926819i	$-0.622532\pi$
$743$	−15210.1	−0.751018	−0.375509	+	0.926819i	$0.622532\pi$
$744$	−2467.19	−0.121575
$745$	0	0
$746$	3459.98	0.169811
$747$	8776.53	0.429875
$748$	8058.98	0.393938
$749$	−542.732	−0.0264766
$750$	0	0
$751$	−18877.5	−0.917241	−0.458620	−	0.888632i	$-0.651656\pi$
$751$	−18877.5	−0.917241	−0.458620	+	0.888632i	$0.651656\pi$
$752$	−5978.45	−0.289909
$753$	−8321.86	−0.402743
$754$	−11185.0	−0.540233
$755$	0	0
$756$	−1588.32	−0.0764110
$757$	10995.1	0.527905	0.263953	−	0.964536i	$-0.414974\pi$
$757$	10995.1	0.527905	0.263953	+	0.964536i	$0.414974\pi$
$758$	21409.5	1.02589
$759$	1939.67	0.0927610
$760$	0	0
$761$	20823.5	0.991919	0.495959	−	0.868346i	$-0.334817\pi$
$761$	20823.5	0.991919	0.495959	+	0.868346i	$0.334817\pi$
$762$	−5901.22	−0.280549
$763$	5039.50	0.239112
$764$	18552.2	0.878527
$765$	0	0
$766$	−8.35516	−0.000394105 0
$767$	11094.5	0.522295
$768$	−701.032	−0.0329379
$769$	−6039.20	−0.283198	−0.141599	−	0.989924i	$-0.545224\pi$
$769$	−6039.20	−0.283198	−0.141599	+	0.989924i	$0.545224\pi$
$770$	0	0
$771$	15186.8	0.709391
$772$	−2722.82	−0.126938
$773$	2376.05	0.110557	0.0552784	−	0.998471i	$-0.482395\pi$
$773$	2376.05	0.110557	0.0552784	+	0.998471i	$0.482395\pi$
$774$	15432.1	0.716661
$775$	0	0
$776$	−11053.5	−0.511339
$777$	1224.94	0.0565568
$778$	−12025.1	−0.554138
$779$	50481.0	2.32178
$780$	0	0
$781$	−5505.78	−0.252256
$782$	3009.37	0.137615
$783$	15473.6	0.706233
$784$	−5332.42	−0.242913
$785$	0	0
$786$	14978.0	0.679705
$787$	1845.74	0.0836004	0.0418002	−	0.999126i	$-0.486691\pi$
$787$	1845.74	0.0836004	0.0418002	+	0.999126i	$0.486691\pi$
$788$	14519.1	0.656373
$789$	−9959.82	−0.449403
$790$	0	0
$791$	−3948.27	−0.177477
$792$	4804.54	0.215558
$793$	38960.7	1.74468
$794$	2493.92	0.111468
$795$	0	0
$796$	−2001.65	−0.0891290
$797$	4811.23	0.213830	0.106915	−	0.994268i	$-0.465903\pi$
$797$	4811.23	0.213830	0.106915	+	0.994268i	$0.465903\pi$
$798$	2489.23	0.110423
$799$	−24444.8	−1.08235
$800$	0	0
$801$	29197.5	1.28794
$802$	20359.0	0.896384
$803$	34337.9	1.50904
$804$	−907.777	−0.0398195
$805$	0	0
$806$	10366.3	0.453024
$807$	−1481.54	−0.0646255
$808$	−7779.64	−0.338721
$809$	−8654.37	−0.376108	−0.188054	−	0.982159i	$-0.560218\pi$
$809$	−8654.37	−0.376108	−0.188054	+	0.982159i	$0.560218\pi$
$810$	0	0
$811$	−17826.3	−0.771846	−0.385923	−	0.922531i	$-0.626117\pi$
$811$	−17826.3	−0.771846	−0.385923	+	0.922531i	$0.626117\pi$
$812$	−1515.67	−0.0655046
$813$	15967.8	0.688824
$814$	−8835.54	−0.380449
$815$	0	0
$816$	−2866.40	−0.122971
$817$	−57670.7	−2.46957
$818$	−1280.40	−0.0547289
$819$	2798.70	0.119407
$820$	0	0
$821$	−8505.57	−0.361567	−0.180783	−	0.983523i	$-0.557863\pi$
$821$	−8505.57	−0.361567	−0.180783	+	0.983523i	$0.557863\pi$
$822$	−3598.65	−0.152697
$823$	−5281.84	−0.223710	−0.111855	−	0.993725i	$-0.535679\pi$
$823$	−5281.84	−0.223710	−0.111855	+	0.993725i	$0.535679\pi$
$824$	13230.5	0.559352
$825$	0	0
$826$	1503.41	0.0633296
$827$	−18254.5	−0.767560	−0.383780	−	0.923425i	$-0.625378\pi$
$827$	−18254.5	−0.767560	−0.383780	+	0.923425i	$0.625378\pi$
$828$	1794.10	0.0753012
$829$	17262.0	0.723201	0.361601	−	0.932333i	$-0.382230\pi$
$829$	17262.0	0.723201	0.361601	+	0.932333i	$0.382230\pi$
$830$	0	0
$831$	16843.4	0.703118
$832$	2945.50	0.122737
$833$	−21803.3	−0.906891
$834$	−10535.2	−0.437416
$835$	0	0
$836$	−17954.8	−0.742801
$837$	−14340.9	−0.592228
$838$	−13700.7	−0.564776
$839$	13744.9	0.565588	0.282794	−	0.959181i	$-0.408739\pi$
$839$	13744.9	0.565588	0.282794	+	0.959181i	$0.408739\pi$
$840$	0	0
$841$	−9623.17	−0.394570
$842$	10957.7	0.448488
$843$	−15861.0	−0.648023
$844$	2234.38	0.0911261
$845$	0	0
$846$	−14573.3	−0.592246
$847$	1192.97	0.0483956
$848$	−9177.95	−0.371665
$849$	−5643.29	−0.228124
$850$	0	0
$851$	−3299.35	−0.132903
$852$	1958.28	0.0787436
$853$	48141.5	1.93240	0.966198	−	0.257801i	$-0.0829980\pi$
$853$	48141.5	1.93240	0.966198	+	0.257801i	$0.0829980\pi$
$854$	5279.52	0.211547
$855$	0	0
$856$	−1392.38	−0.0555965
$857$	2897.47	0.115491	0.0577454	−	0.998331i	$-0.481609\pi$
$857$	2897.47	0.115491	0.0577454	+	0.998331i	$0.481609\pi$
$858$	7762.63	0.308872
$859$	−34787.8	−1.38177	−0.690887	−	0.722962i	$-0.742780\pi$
$859$	−34787.8	−1.38177	−0.690887	+	0.722962i	$0.742780\pi$
$860$	0	0
$861$	−2957.49	−0.117063
$862$	−3951.53	−0.156136
$863$	28071.4	1.10725	0.553627	−	0.832765i	$-0.313243\pi$
$863$	28071.4	1.10725	0.553627	+	0.832765i	$0.313243\pi$
$864$	−4074.85	−0.160450
$865$	0	0
$866$	31539.5	1.23759
$867$	1733.61	0.0679084
$868$	1404.73	0.0549304
$869$	−7337.43	−0.286427
$870$	0	0
$871$	3814.17	0.148379
$872$	12928.9	0.502095
$873$	−26944.6	−1.04460
$874$	−6704.67	−0.259484
$875$	0	0
$876$	−12213.2	−0.471057
$877$	−6108.99	−0.235218	−0.117609	−	0.993060i	$-0.537523\pi$
$877$	−6108.99	−0.235218	−0.117609	+	0.993060i	$0.537523\pi$
$878$	−4615.16	−0.177397
$879$	−11130.2	−0.427090
$880$	0	0
$881$	−15515.1	−0.593322	−0.296661	−	0.954983i	$-0.595873\pi$
$881$	−15515.1	−0.593322	−0.296661	+	0.954983i	$0.595873\pi$
$882$	−12998.5	−0.496239
$883$	50045.4	1.90732	0.953658	−	0.300892i	$-0.0972843\pi$
$883$	50045.4	1.90732	0.953658	+	0.300892i	$0.0972843\pi$
$884$	12043.6	0.458225
$885$	0	0
$886$	27752.2	1.05232
$887$	−25049.6	−0.948232	−0.474116	−	0.880462i	$-0.657232\pi$
$887$	−25049.6	−0.948232	−0.474116	+	0.880462i	$0.657232\pi$
$888$	3142.60	0.118760
$889$	3359.93	0.126759
$890$	0	0
$891$	5476.37	0.205909
$892$	18132.8	0.680641
$893$	54461.3	2.04085
$894$	12283.6	0.459535
$895$	0	0
$896$	399.142	0.0148821
$897$	2898.71	0.107899
$898$	18504.9	0.687659
$899$	−13685.0	−0.507697
$900$	0	0
$901$	−37527.0	−1.38758
$902$	21332.4	0.787464
$903$	3378.71	0.124514
$904$	−10129.3	−0.372672
$905$	0	0
$906$	14197.8	0.520628
$907$	−36780.7	−1.34651	−0.673254	−	0.739412i	$-0.735104\pi$
$907$	−36780.7	−1.34651	−0.673254	+	0.739412i	$0.735104\pi$
$908$	18935.0	0.692049
$909$	−18964.0	−0.691964
$910$	0	0
$911$	−16915.8	−0.615199	−0.307600	−	0.951516i	$-0.599526\pi$
$911$	−16915.8	−0.615199	−0.307600	+	0.951516i	$0.599526\pi$
$912$	6386.13	0.231871
$913$	−13860.1	−0.502411
$914$	−17884.0	−0.647210
$915$	0	0
$916$	7250.94	0.261548
$917$	−8527.92	−0.307107
$918$	−16661.3	−0.599026
$919$	−36523.8	−1.31100	−0.655500	−	0.755195i	$-0.727542\pi$
$919$	−36523.8	−1.31100	−0.655500	+	0.755195i	$0.727542\pi$
$920$	0	0
$921$	−15014.4	−0.537178
$922$	−11368.3	−0.406069
$923$	−8228.03	−0.293422
$924$	1051.91	0.0374515
$925$	0	0
$926$	24370.0	0.864846
$927$	32251.2	1.14268
$928$	−3888.47	−0.137549
$929$	14928.7	0.527226	0.263613	−	0.964628i	$-0.415086\pi$
$929$	14928.7	0.527226	0.263613	+	0.964628i	$0.415086\pi$
$930$	0	0
$931$	48576.3	1.71001
$932$	2319.78	0.0815310
$933$	18829.7	0.660727
$934$	11566.7	0.405218
$935$	0	0
$936$	7180.07	0.250735
$937$	16078.4	0.560574	0.280287	−	0.959916i	$-0.409570\pi$
$937$	16078.4	0.560574	0.280287	+	0.959916i	$0.409570\pi$
$938$	516.855	0.0179914
$939$	−13251.4	−0.460536
$940$	0	0
$941$	14396.0	0.498720	0.249360	−	0.968411i	$-0.419780\pi$
$941$	14396.0	0.498720	0.249360	+	0.968411i	$0.419780\pi$
$942$	15089.1	0.521898
$943$	7965.92	0.275086
$944$	3857.00	0.132982
$945$	0	0
$946$	−24370.7	−0.837588
$947$	18355.4	0.629854	0.314927	−	0.949116i	$-0.398020\pi$
$947$	18355.4	0.629854	0.314927	+	0.949116i	$0.398020\pi$
$948$	2609.76	0.0894103
$949$	51315.7	1.75530
$950$	0	0
$951$	−2245.98	−0.0765835
$952$	1632.02	0.0555610
$953$	−8601.43	−0.292369	−0.146184	−	0.989257i	$-0.546699\pi$
$953$	−8601.43	−0.292369	−0.146184	+	0.989257i	$0.546699\pi$
$954$	−22372.5	−0.759264
$955$	0	0
$956$	15110.1	0.511186
$957$	−10247.8	−0.346147
$958$	1724.60	0.0581621
$959$	2048.94	0.0689923
$960$	0	0
$961$	−17107.8	−0.574259
$962$	−13204.1	−0.442535
$963$	−3394.13	−0.113577
$964$	25706.3	0.858863
$965$	0	0
$966$	392.801	0.0130830
$967$	11749.0	0.390716	0.195358	−	0.980732i	$-0.437413\pi$
$967$	11749.0	0.390716	0.195358	+	0.980732i	$0.437413\pi$
$968$	3060.58	0.101623
$969$	26111.8	0.865666
$970$	0	0
$971$	−9950.94	−0.328878	−0.164439	−	0.986387i	$-0.552581\pi$
$971$	−9950.94	−0.328878	−0.164439	+	0.986387i	$0.552581\pi$
$972$	−15700.4	−0.518098
$973$	5998.36	0.197635
$974$	−18490.8	−0.608301
$975$	0	0
$976$	13544.6	0.444215
$977$	36998.6	1.21156	0.605778	−	0.795634i	$-0.292862\pi$
$977$	36998.6	1.21156	0.605778	+	0.795634i	$0.292862\pi$
$978$	3610.25	0.118040
$979$	−46109.2	−1.50527
$980$	0	0
$981$	31515.9	1.02572
$982$	286.797	0.00931982
$983$	16860.1	0.547053	0.273527	−	0.961864i	$-0.411810\pi$
$983$	16860.1	0.547053	0.273527	+	0.961864i	$0.411810\pi$
$984$	−7587.46	−0.245812
$985$	0	0
$986$	−15899.3	−0.513525
$987$	−3190.68	−0.102898
$988$	−26832.4	−0.864019
$989$	−9100.46	−0.292596
$990$	0	0
$991$	−36029.8	−1.15492	−0.577460	−	0.816419i	$-0.695956\pi$
$991$	−36029.8	−1.15492	−0.577460	+	0.816419i	$0.695956\pi$
$992$	3603.84	0.115345
$993$	480.275	0.0153485
$994$	−1114.97	−0.0355782
$995$	0	0
$996$	4929.71	0.156831
$997$	−41319.5	−1.31254	−0.656269	−	0.754527i	$-0.727866\pi$
$997$	−41319.5	−1.31254	−0.656269	+	0.754527i	$0.727866\pi$
$998$	20897.6	0.662826
$999$	18266.8	0.578515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.s.1.1	✓	5
5.2	odd	4		1150.4.b.p.599.5		10
5.3	odd	4		1150.4.b.p.599.6		10
5.4	even	2		1150.4.a.t.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.s.1.1	✓	5	1.1	even	1	trivial
1150.4.a.t.1.5	yes	5	5.4	even	2
1150.4.b.p.599.5		10	5.2	odd	4
1150.4.b.p.599.6		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -2.73841 q^{3} +4.00000 q^{4} +5.47681 q^{6} -3.11829 q^{7} -8.00000 q^{8} -19.5011 q^{9} +30.7965 q^{11} -10.9536 q^{12} +46.0234 q^{13} +6.23659 q^{14} +16.0000 q^{16} +65.4212 q^{17} +39.0022 q^{18} -145.754 q^{19} +8.53916 q^{21} -61.5931 q^{22} -23.0000 q^{23} +21.9073 q^{24} -92.0468 q^{26} +127.339 q^{27} -12.4732 q^{28} +121.515 q^{29} -112.620 q^{31} -32.0000 q^{32} -84.3335 q^{33} -130.842 q^{34} -78.0045 q^{36} +143.450 q^{37} +291.508 q^{38} -126.031 q^{39} -346.345 q^{41} -17.0783 q^{42} +395.672 q^{43} +123.186 q^{44} +46.0000 q^{46} -373.653 q^{47} -43.8145 q^{48} -333.276 q^{49} -179.150 q^{51} +184.094 q^{52} -573.622 q^{53} -254.678 q^{54} +24.9463 q^{56} +399.133 q^{57} -243.029 q^{58} +241.063 q^{59} +846.540 q^{61} +225.240 q^{62} +60.8102 q^{63} +64.0000 q^{64} +168.667 q^{66} +82.8746 q^{67} +261.685 q^{68} +62.9834 q^{69} -178.779 q^{71} +156.009 q^{72} +1114.99 q^{73} -286.900 q^{74} -583.015 q^{76} -96.0327 q^{77} +252.062 q^{78} -238.255 q^{79} +177.824 q^{81} +692.689 q^{82} -450.052 q^{83} +34.1566 q^{84} -791.344 q^{86} -332.757 q^{87} -246.372 q^{88} -1497.22 q^{89} -143.515 q^{91} -92.0000 q^{92} +308.399 q^{93} +747.306 q^{94} +87.6290 q^{96} +1381.69 q^{97} +666.552 q^{98} -600.567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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