LMFDB - Embedded newform 1150.4.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(1,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [4,-8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.50148$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} -5.50148 q^{3} +4.00000 q^{4} +11.0030 q^{6} -0.0500526 q^{7} -8.00000 q^{8} +3.26627 q^{9} +35.1757 q^{11} -22.0059 q^{12} -86.1100 q^{13} +0.100105 q^{14} +16.0000 q^{16} -8.82793 q^{17} -6.53253 q^{18} +106.317 q^{19} +0.275364 q^{21} -70.3513 q^{22} -23.0000 q^{23} +44.0118 q^{24} +172.220 q^{26} +130.571 q^{27} -0.200211 q^{28} -280.754 q^{29} -117.077 q^{31} -32.0000 q^{32} -193.518 q^{33} +17.6559 q^{34} +13.0651 q^{36} -93.3088 q^{37} -212.633 q^{38} +473.732 q^{39} -58.0579 q^{41} -0.550727 q^{42} +508.101 q^{43} +140.703 q^{44} +46.0000 q^{46} +407.914 q^{47} -88.0237 q^{48} -342.997 q^{49} +48.5666 q^{51} -344.440 q^{52} -316.251 q^{53} -261.141 q^{54} +0.400421 q^{56} -584.899 q^{57} +561.507 q^{58} +129.500 q^{59} +299.008 q^{61} +234.155 q^{62} -0.163485 q^{63} +64.0000 q^{64} +387.036 q^{66} -596.326 q^{67} -35.3117 q^{68} +126.534 q^{69} -692.108 q^{71} -26.1301 q^{72} -842.098 q^{73} +186.618 q^{74} +425.267 q^{76} -1.76063 q^{77} -947.464 q^{78} -1161.06 q^{79} -806.521 q^{81} +116.116 q^{82} -1121.69 q^{83} +1.10145 q^{84} -1016.20 q^{86} +1544.56 q^{87} -281.405 q^{88} +409.092 q^{89} +4.31003 q^{91} -92.0000 q^{92} +644.098 q^{93} -815.828 q^{94} +176.047 q^{96} +1123.62 q^{97} +685.995 q^{98} +114.893 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} - 26 q^{7} - 32 q^{8} + 64 q^{9} + 93 q^{11} - 16 q^{12} - 32 q^{13} + 52 q^{14} + 64 q^{16} - 108 q^{17} - 128 q^{18} + 185 q^{19} + 302 q^{21} - 186 q^{22}+ \cdots + 2013 q^{99}+O(q^{100}) $ 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 - 26 * q^7 - 32 * q^8 + 64 * q^9 + 93 * q^11 - 16 * q^12 - 32 * q^13 + 52 * q^14 + 64 * q^16 - 108 * q^17 - 128 * q^18 + 185 * q^19 + 302 * q^21 - 186 * q^22 - 92 * q^23 + 32 * q^24 + 64 * q^26 - 259 * q^27 - 104 * q^28 + 294 * q^29 - 211 * q^31 - 128 * q^32 + 237 * q^33 + 216 * q^34 + 256 * q^36 - 5 * q^37 - 370 * q^38 - 421 * q^39 - 369 * q^41 - 604 * q^42 + 100 * q^43 + 372 * q^44 + 184 * q^46 + 363 * q^47 - 64 * q^48 + 600 * q^49 - 315 * q^51 - 128 * q^52 - 21 * q^53 + 518 * q^54 + 208 * q^56 + 160 * q^57 - 588 * q^58 - 33 * q^59 - 307 * q^61 + 422 * q^62 - 167 * q^63 + 256 * q^64 - 474 * q^66 - 725 * q^67 - 432 * q^68 + 92 * q^69 - 1257 * q^71 - 512 * q^72 - 509 * q^73 + 10 * q^74 + 740 * q^76 + 1962 * q^77 + 842 * q^78 + 1202 * q^79 - 992 * q^81 + 738 * q^82 + 1377 * q^83 + 1208 * q^84 - 200 * q^86 + 2829 * q^87 - 744 * q^88 - 984 * q^89 - 995 * q^91 - 368 * q^92 + 1843 * q^93 - 726 * q^94 + 128 * q^96 - 137 * q^97 - 1200 * q^98 + 2013 * 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
$3$	−5.50148	−1.05876	−0.529380	−	0.848385i	$-0.677575\pi$
$3$	−5.50148	−1.05876	−0.529380	+	0.848385i	$0.677575\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	11.0030	0.748656
$7$	−0.0500526	−0.00270259	−0.00135129	−	0.999999i	$-0.500430\pi$
$7$	−0.0500526	−0.00270259	−0.00135129	+	0.999999i	$0.500430\pi$
$8$	−8.00000	−0.353553
$9$	3.26627	0.120973
$10$	0	0
$11$	35.1757	0.964169	0.482085	−	0.876125i	$-0.339880\pi$
$11$	35.1757	0.964169	0.482085	+	0.876125i	$0.339880\pi$
$12$	−22.0059	−0.529380
$13$	−86.1100	−1.83712	−0.918562	−	0.395277i	$-0.870649\pi$
$13$	−86.1100	−1.83712	−0.918562	+	0.395277i	$0.870649\pi$
$14$	0.100105	0.00191102
$15$	0	0
$16$	16.0000	0.250000
$17$	−8.82793	−0.125946	−0.0629731	−	0.998015i	$-0.520058\pi$
$17$	−8.82793	−0.125946	−0.0629731	+	0.998015i	$0.520058\pi$
$18$	−6.53253	−0.0855407
$19$	106.317	1.28372	0.641861	−	0.766821i	$-0.278163\pi$
$19$	106.317	1.28372	0.641861	+	0.766821i	$0.278163\pi$
$20$	0	0
$21$	0.275364	0.00286139
$22$	−70.3513	−0.681771
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	44.0118	0.374328
$25$	0	0
$26$	172.220	1.29904
$27$	130.571	0.930679
$28$	−0.200211	−0.00135129
$29$	−280.754	−1.79775	−0.898873	−	0.438209i	$-0.855613\pi$
$29$	−280.754	−1.79775	−0.898873	+	0.438209i	$0.855613\pi$
$30$	0	0
$31$	−117.077	−0.678313	−0.339156	−	0.940730i	$-0.610142\pi$
$31$	−117.077	−0.678313	−0.339156	+	0.940730i	$0.610142\pi$
$32$	−32.0000	−0.176777
$33$	−193.518	−1.02082
$34$	17.6559	0.0890575
$35$	0	0
$36$	13.0651	0.0604864
$37$	−93.3088	−0.414591	−0.207296	−	0.978278i	$-0.566466\pi$
$37$	−93.3088	−0.414591	−0.207296	+	0.978278i	$0.566466\pi$
$38$	−212.633	−0.907729
$39$	473.732	1.94507
$40$	0	0
$41$	−58.0579	−0.221149	−0.110575	−	0.993868i	$-0.535269\pi$
$41$	−58.0579	−0.221149	−0.110575	+	0.993868i	$0.535269\pi$
$42$	−0.550727	−0.00202331
$43$	508.101	1.80197	0.900985	−	0.433851i	$-0.142845\pi$
$43$	508.101	1.80197	0.900985	+	0.433851i	$0.142845\pi$
$44$	140.703	0.482085
$45$	0	0
$46$	46.0000	0.147442
$47$	407.914	1.26597	0.632983	−	0.774166i	$-0.281830\pi$
$47$	407.914	1.26597	0.632983	+	0.774166i	$0.281830\pi$
$48$	−88.0237	−0.264690
$49$	−342.997	−0.999993
$50$	0	0
$51$	48.5666	0.133347
$52$	−344.440	−0.918562
$53$	−316.251	−0.819629	−0.409815	−	0.912169i	$-0.634407\pi$
$53$	−316.251	−0.819629	−0.409815	+	0.912169i	$0.634407\pi$
$54$	−261.141	−0.658089
$55$	0	0
$56$	0.400421	0.000955509 0
$57$	−584.899	−1.35915
$58$	561.507	1.27120
$59$	129.500	0.285754	0.142877	−	0.989740i	$-0.454365\pi$
$59$	129.500	0.285754	0.142877	+	0.989740i	$0.454365\pi$
$60$	0	0
$61$	299.008	0.627607	0.313804	−	0.949488i	$-0.398397\pi$
$61$	299.008	0.627607	0.313804	+	0.949488i	$0.398397\pi$
$62$	234.155	0.479640
$63$	−0.163485	−0.000326940 0
$64$	64.0000	0.125000
$65$	0	0
$66$	387.036	0.721831
$67$	−596.326	−1.08736	−0.543678	−	0.839294i	$-0.682969\pi$
$67$	−596.326	−1.08736	−0.543678	+	0.839294i	$0.682969\pi$
$68$	−35.3117	−0.0629731
$69$	126.534	0.220767
$70$	0	0
$71$	−692.108	−1.15687	−0.578437	−	0.815727i	$-0.696337\pi$
$71$	−692.108	−1.15687	−0.578437	+	0.815727i	$0.696337\pi$
$72$	−26.1301	−0.0427703
$73$	−842.098	−1.35014	−0.675069	−	0.737755i	$-0.735886\pi$
$73$	−842.098	−1.35014	−0.675069	+	0.737755i	$0.735886\pi$
$74$	186.618	0.293160
$75$	0	0
$76$	425.267	0.641861
$77$	−1.76063	−0.00260575
$78$	−947.464	−1.37537
$79$	−1161.06	−1.65354	−0.826769	−	0.562542i	$-0.809823\pi$
$79$	−1161.06	−1.65354	−0.826769	+	0.562542i	$0.809823\pi$
$80$	0	0
$81$	−806.521	−1.10634
$82$	116.116	0.156376
$83$	−1121.69	−1.48340	−0.741699	−	0.670733i	$-0.765980\pi$
$83$	−1121.69	−1.48340	−0.741699	+	0.670733i	$0.765980\pi$
$84$	1.10145	0.00143070
$85$	0	0
$86$	−1016.20	−1.27418
$87$	1544.56	1.90338
$88$	−281.405	−0.340885
$89$	409.092	0.487232	0.243616	−	0.969872i	$-0.421666\pi$
$89$	409.092	0.487232	0.243616	+	0.969872i	$0.421666\pi$
$90$	0	0
$91$	4.31003	0.00496499
$92$	−92.0000	−0.104257
$93$	644.098	0.718170
$94$	−815.828	−0.895173
$95$	0	0
$96$	176.047	0.187164
$97$	1123.62	1.17615	0.588075	−	0.808807i	$-0.299886\pi$
$97$	1123.62	1.17615	0.588075	+	0.808807i	$0.299886\pi$
$98$	685.995	0.707102
$99$	114.893	0.116638
$100$	0	0
$101$	−981.824	−0.967278	−0.483639	−	0.875268i	$-0.660685\pi$
$101$	−981.824	−0.967278	−0.483639	+	0.875268i	$0.660685\pi$
$102$	−97.1333	−0.0942905
$103$	28.9575	0.0277016	0.0138508	−	0.999904i	$-0.495591\pi$
$103$	28.9575	0.0277016	0.0138508	+	0.999904i	$0.495591\pi$
$104$	688.880	0.649521
$105$	0	0
$106$	632.501	0.579565
$107$	187.874	0.169742	0.0848712	−	0.996392i	$-0.472952\pi$
$107$	187.874	0.169742	0.0848712	+	0.996392i	$0.472952\pi$
$108$	522.283	0.465339
$109$	2142.02	1.88228	0.941139	−	0.338018i	$-0.109757\pi$
$109$	2142.02	1.88228	0.941139	+	0.338018i	$0.109757\pi$
$110$	0	0
$111$	513.337	0.438953
$112$	−0.800842	−0.000675647 0
$113$	1668.67	1.38916	0.694582	−	0.719414i	$-0.255589\pi$
$113$	1668.67	1.38916	0.694582	+	0.719414i	$0.255589\pi$
$114$	1169.80	0.961067
$115$	0	0
$116$	−1123.01	−0.898873
$117$	−281.258	−0.222242
$118$	−259.001	−0.202059
$119$	0.441861	0.000340381 0
$120$	0	0
$121$	−93.6729	−0.0703779
$122$	−598.016	−0.443786
$123$	319.404	0.234144
$124$	−468.309	−0.339156
$125$	0	0
$126$	0.326970	0.000231181 0
$127$	354.326	0.247570	0.123785	−	0.992309i	$-0.460497\pi$
$127$	354.326	0.247570	0.123785	+	0.992309i	$0.460497\pi$
$128$	−128.000	−0.0883883
$129$	−2795.31	−1.90785
$130$	0	0
$131$	−922.236	−0.615085	−0.307543	−	0.951534i	$-0.599507\pi$
$131$	−922.236	−0.615085	−0.307543	+	0.951534i	$0.599507\pi$
$132$	−774.073	−0.510412
$133$	−5.32143	−0.00346937
$134$	1192.65	0.768877
$135$	0	0
$136$	70.6234	0.0445287
$137$	1664.96	1.03830	0.519149	−	0.854684i	$-0.326249\pi$
$137$	1664.96	1.03830	0.519149	+	0.854684i	$0.326249\pi$
$138$	−253.068	−0.156106
$139$	2407.80	1.46926	0.734628	−	0.678470i	$-0.237357\pi$
$139$	2407.80	1.46926	0.734628	+	0.678470i	$0.237357\pi$
$140$	0	0
$141$	−2244.13	−1.34035
$142$	1384.22	0.818034
$143$	−3028.97	−1.77130
$144$	52.2603	0.0302432
$145$	0	0
$146$	1684.20	0.954692
$147$	1886.99	1.05875
$148$	−373.235	−0.207296
$149$	1838.74	1.01098	0.505488	−	0.862834i	$-0.331312\pi$
$149$	1838.74	1.01098	0.505488	+	0.862834i	$0.331312\pi$
$150$	0	0
$151$	−1276.86	−0.688139	−0.344070	−	0.938944i	$-0.611806\pi$
$151$	−1276.86	−0.688139	−0.344070	+	0.938944i	$0.611806\pi$
$152$	−850.533	−0.453864
$153$	−28.8344	−0.0152361
$154$	3.52127	0.00184255
$155$	0	0
$156$	1894.93	0.972537
$157$	1288.13	0.654804	0.327402	−	0.944885i	$-0.393827\pi$
$157$	1288.13	0.654804	0.327402	+	0.944885i	$0.393827\pi$
$158$	2322.12	1.16923
$159$	1739.85	0.867791
$160$	0	0
$161$	1.15121	0.000563529 0
$162$	1613.04	0.782299
$163$	995.942	0.478578	0.239289	−	0.970948i	$-0.423086\pi$
$163$	995.942	0.478578	0.239289	+	0.970948i	$0.423086\pi$
$164$	−232.231	−0.110575
$165$	0	0
$166$	2243.39	1.04892
$167$	1568.39	0.726742	0.363371	−	0.931645i	$-0.381626\pi$
$167$	1568.39	0.726742	0.363371	+	0.931645i	$0.381626\pi$
$168$	−2.20291	−0.00101166
$169$	5217.93	2.37502
$170$	0	0
$171$	347.259	0.155295
$172$	2032.40	0.900985
$173$	−2478.94	−1.08942	−0.544711	−	0.838624i	$-0.683361\pi$
$173$	−2478.94	−1.08942	−0.544711	+	0.838624i	$0.683361\pi$
$174$	−3089.12	−1.34589
$175$	0	0
$176$	562.811	0.241042
$177$	−712.444	−0.302545
$178$	−818.183	−0.344525
$179$	3740.99	1.56210	0.781048	−	0.624471i	$-0.214686\pi$
$179$	3740.99	1.56210	0.781048	+	0.624471i	$0.214686\pi$
$180$	0	0
$181$	−3841.87	−1.57770	−0.788851	−	0.614584i	$-0.789324\pi$
$181$	−3841.87	−1.57770	−0.788851	+	0.614584i	$0.789324\pi$
$182$	−8.62006	−0.00351078
$183$	−1644.99	−0.664486
$184$	184.000	0.0737210
$185$	0	0
$186$	−1288.20	−0.507823
$187$	−310.528	−0.121433
$188$	1631.66	0.632983
$189$	−6.53540	−0.00251524
$190$	0	0
$191$	2097.13	0.794465	0.397233	−	0.917718i	$-0.369971\pi$
$191$	2097.13	0.794465	0.397233	+	0.917718i	$0.369971\pi$
$192$	−352.095	−0.132345
$193$	−1264.65	−0.471666	−0.235833	−	0.971794i	$-0.575782\pi$
$193$	−1264.65	−0.471666	−0.235833	+	0.971794i	$0.575782\pi$
$194$	−2247.24	−0.831663
$195$	0	0
$196$	−1371.99	−0.499996
$197$	850.129	0.307458	0.153729	−	0.988113i	$-0.450872\pi$
$197$	850.129	0.307458	0.153729	+	0.988113i	$0.450872\pi$
$198$	−229.786	−0.0824757
$199$	−203.089	−0.0723447	−0.0361724	−	0.999346i	$-0.511517\pi$
$199$	−203.089	−0.0723447	−0.0361724	+	0.999346i	$0.511517\pi$
$200$	0	0
$201$	3280.68	1.15125
$202$	1963.65	0.683969
$203$	14.0525	0.00485857
$204$	194.267	0.0666734
$205$	0	0
$206$	−57.9149	−0.0195880
$207$	−75.1241	−0.0252246
$208$	−1377.76	−0.459281
$209$	3739.76	1.23773
$210$	0	0
$211$	5612.25	1.83111	0.915553	−	0.402197i	$-0.131753\pi$
$211$	5612.25	1.83111	0.915553	+	0.402197i	$0.131753\pi$
$212$	−1265.00	−0.409815
$213$	3807.62	1.22485
$214$	−375.747	−0.120026
$215$	0	0
$216$	−1044.57	−0.329045
$217$	5.86003	0.00183320
$218$	−4284.04	−1.33097
$219$	4632.78	1.42947
$220$	0	0
$221$	760.172	0.231379
$222$	−1026.67	−0.310386
$223$	557.909	0.167535	0.0837676	−	0.996485i	$-0.473305\pi$
$223$	557.909	0.167535	0.0837676	+	0.996485i	$0.473305\pi$
$224$	1.60168	0.000477755 0
$225$	0	0
$226$	−3337.35	−0.982287
$227$	684.858	0.200245	0.100123	−	0.994975i	$-0.468077\pi$
$227$	684.858	0.200245	0.100123	+	0.994975i	$0.468077\pi$
$228$	−2339.60	−0.679577
$229$	369.484	0.106621	0.0533104	−	0.998578i	$-0.483023\pi$
$229$	369.484	0.106621	0.0533104	+	0.998578i	$0.483023\pi$
$230$	0	0
$231$	9.68609	0.00275887
$232$	2246.03	0.635599
$233$	5119.84	1.43954	0.719768	−	0.694215i	$-0.244248\pi$
$233$	5119.84	1.43954	0.719768	+	0.694215i	$0.244248\pi$
$234$	562.516	0.157149
$235$	0	0
$236$	518.002	0.142877
$237$	6387.55	1.75070
$238$	−0.883722	−0.000240686 0
$239$	1665.03	0.450635	0.225317	−	0.974285i	$-0.427658\pi$
$239$	1665.03	0.450635	0.225317	+	0.974285i	$0.427658\pi$
$240$	0	0
$241$	5401.56	1.44376	0.721878	−	0.692020i	$-0.243279\pi$
$241$	5401.56	1.44376	0.721878	+	0.692020i	$0.243279\pi$
$242$	187.346	0.0497647
$243$	911.649	0.240668
$244$	1196.03	0.313804
$245$	0	0
$246$	−638.808	−0.165565
$247$	−9154.93	−2.35836
$248$	936.618	0.239820
$249$	6170.98	1.57056
$250$	0	0
$251$	4246.35	1.06784	0.533919	−	0.845535i	$-0.320719\pi$
$251$	4246.35	1.06784	0.533919	+	0.845535i	$0.320719\pi$
$252$	−0.653941	−0.000163470 0
$253$	−809.040	−0.201043
$254$	−708.652	−0.175058
$255$	0	0
$256$	256.000	0.0625000
$257$	5585.56	1.35571	0.677855	−	0.735196i	$-0.262910\pi$
$257$	5585.56	1.35571	0.677855	+	0.735196i	$0.262910\pi$
$258$	5590.61	1.34906
$259$	4.67035	0.00112047
$260$	0	0
$261$	−917.016	−0.217478
$262$	1844.47	0.434931
$263$	2131.01	0.499634	0.249817	−	0.968293i	$-0.419630\pi$
$263$	2131.01	0.499634	0.249817	+	0.968293i	$0.419630\pi$
$264$	1548.15	0.360916
$265$	0	0
$266$	10.6429	0.00245322
$267$	−2250.61	−0.515862
$268$	−2385.31	−0.543678
$269$	−8003.32	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$269$	−8003.32	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$270$	0	0
$271$	1349.33	0.302458	0.151229	−	0.988499i	$-0.451677\pi$
$271$	1349.33	0.302458	0.151229	+	0.988499i	$0.451677\pi$
$272$	−141.247	−0.0314866
$273$	−23.7115	−0.00525673
$274$	−3329.92	−0.734188
$275$	0	0
$276$	506.136	0.110383
$277$	3762.76	0.816181	0.408091	−	0.912941i	$-0.366195\pi$
$277$	3762.76	0.816181	0.408091	+	0.912941i	$0.366195\pi$
$278$	−4815.59	−1.03892
$279$	−382.405	−0.0820574
$280$	0	0
$281$	2885.36	0.612548	0.306274	−	0.951943i	$-0.400918\pi$
$281$	2885.36	0.612548	0.306274	+	0.951943i	$0.400918\pi$
$282$	4488.26	0.947773
$283$	−4809.11	−1.01015	−0.505074	−	0.863076i	$-0.668535\pi$
$283$	−4809.11	−1.01015	−0.505074	+	0.863076i	$0.668535\pi$
$284$	−2768.43	−0.578437
$285$	0	0
$286$	6057.95	1.25250
$287$	2.90595	0.000597675 0
$288$	−104.521	−0.0213852
$289$	−4835.07	−0.984138
$290$	0	0
$291$	−6181.58	−1.24526
$292$	−3368.39	−0.675069
$293$	5362.69	1.06925	0.534627	−	0.845088i	$-0.320452\pi$
$293$	5362.69	1.06925	0.534627	+	0.845088i	$0.320452\pi$
$294$	−3773.99	−0.748651
$295$	0	0
$296$	746.471	0.146580
$297$	4592.91	0.897332
$298$	−3677.48	−0.714868
$299$	1980.53	0.383067
$300$	0	0
$301$	−25.4318	−0.00486998
$302$	2553.71	0.486588
$303$	5401.48	1.02412
$304$	1701.07	0.320931
$305$	0	0
$306$	57.6687	0.0107735
$307$	−6374.74	−1.18510	−0.592549	−	0.805534i	$-0.701879\pi$
$307$	−6374.74	−1.18510	−0.592549	+	0.805534i	$0.701879\pi$
$308$	−7.04254	−0.00130288
$309$	−159.309	−0.0293293
$310$	0	0
$311$	37.2799	0.00679727	0.00339863	−	0.999994i	$-0.498918\pi$
$311$	37.2799	0.00679727	0.00339863	+	0.999994i	$0.498918\pi$
$312$	−3789.86	−0.687687
$313$	3787.66	0.683998	0.341999	−	0.939700i	$-0.388896\pi$
$313$	3787.66	0.683998	0.341999	+	0.939700i	$0.388896\pi$
$314$	−2576.27	−0.463016
$315$	0	0
$316$	−4644.24	−0.826769
$317$	5633.36	0.998110	0.499055	−	0.866570i	$-0.333681\pi$
$317$	5633.36	0.998110	0.499055	+	0.866570i	$0.333681\pi$
$318$	−3479.69	−0.613621
$319$	−9875.70	−1.73333
$320$	0	0
$321$	−1033.58	−0.179716
$322$	−2.30242	−0.000398475 0
$323$	−938.556	−0.161680
$324$	−3226.08	−0.553169
$325$	0	0
$326$	−1991.88	−0.338406
$327$	−11784.3	−1.99288
$328$	464.463	0.0781880
$329$	−20.4172	−0.00342138
$330$	0	0
$331$	4566.91	0.758369	0.379185	−	0.925321i	$-0.376204\pi$
$331$	4566.91	0.758369	0.379185	+	0.925321i	$0.376204\pi$
$332$	−4486.78	−0.741699
$333$	−304.771	−0.0501543
$334$	−3136.79	−0.513884
$335$	0	0
$336$	4.40582	0.000715348 0
$337$	−787.487	−0.127291	−0.0636456	−	0.997973i	$-0.520273\pi$
$337$	−787.487	−0.127291	−0.0636456	+	0.997973i	$0.520273\pi$
$338$	−10435.9	−1.67940
$339$	−9180.17	−1.47079
$340$	0	0
$341$	−4118.27	−0.654008
$342$	−694.517	−0.109810
$343$	34.3360	0.00540516
$344$	−4064.81	−0.637092
$345$	0	0
$346$	4957.87	0.770338
$347$	3738.52	0.578370	0.289185	−	0.957273i	$-0.406616\pi$
$347$	3738.52	0.578370	0.289185	+	0.957273i	$0.406616\pi$
$348$	6178.24	0.951691
$349$	425.210	0.0652177	0.0326088	−	0.999468i	$-0.489618\pi$
$349$	425.210	0.0652177	0.0326088	+	0.999468i	$0.489618\pi$
$350$	0	0
$351$	−11243.4	−1.70977
$352$	−1125.62	−0.170443
$353$	−355.990	−0.0536755	−0.0268377	−	0.999640i	$-0.508544\pi$
$353$	−355.990	−0.0536755	−0.0268377	+	0.999640i	$0.508544\pi$
$354$	1424.89	0.213932
$355$	0	0
$356$	1636.37	0.243616
$357$	−2.43089	−0.000360382 0
$358$	−7481.99	−1.10457
$359$	−3713.17	−0.545888	−0.272944	−	0.962030i	$-0.587997\pi$
$359$	−3713.17	−0.545888	−0.272944	+	0.962030i	$0.587997\pi$
$360$	0	0
$361$	4444.24	0.647942
$362$	7683.75	1.11560
$363$	515.340	0.0745133
$364$	17.2401	0.00248249
$365$	0	0
$366$	3289.97	0.469862
$367$	−3908.40	−0.555904	−0.277952	−	0.960595i	$-0.589656\pi$
$367$	−3908.40	−0.555904	−0.277952	+	0.960595i	$0.589656\pi$
$368$	−368.000	−0.0521286
$369$	−189.632	−0.0267530
$370$	0	0
$371$	15.8292	0.00221512
$372$	2576.39	0.359085
$373$	−13956.5	−1.93737	−0.968685	−	0.248293i	$-0.920130\pi$
$373$	−13956.5	−1.93737	−0.968685	+	0.248293i	$0.920130\pi$
$374$	621.056	0.0858664
$375$	0	0
$376$	−3263.31	−0.447586
$377$	24175.7	3.30268
$378$	13.0708	0.00177854
$379$	−343.162	−0.0465093	−0.0232547	−	0.999730i	$-0.507403\pi$
$379$	−343.162	−0.0465093	−0.0232547	+	0.999730i	$0.507403\pi$
$380$	0	0
$381$	−1949.32	−0.262117
$382$	−4194.26	−0.561772
$383$	13870.9	1.85057	0.925285	−	0.379273i	$-0.123826\pi$
$383$	13870.9	1.85057	0.925285	+	0.379273i	$0.123826\pi$
$384$	704.189	0.0935821
$385$	0	0
$386$	2529.30	0.333518
$387$	1659.59	0.217989
$388$	4494.49	0.588075
$389$	80.0920	0.0104391	0.00521957	−	0.999986i	$-0.498339\pi$
$389$	80.0920	0.0104391	0.00521957	+	0.999986i	$0.498339\pi$
$390$	0	0
$391$	203.042	0.0262616
$392$	2743.98	0.353551
$393$	5073.66	0.651228
$394$	−1700.26	−0.217406
$395$	0	0
$396$	459.572	0.0583191
$397$	5077.38	0.641880	0.320940	−	0.947099i	$-0.396001\pi$
$397$	5077.38	0.641880	0.320940	+	0.947099i	$0.396001\pi$
$398$	406.178	0.0511554
$399$	29.2757	0.00367323
$400$	0	0
$401$	1588.41	0.197809	0.0989045	−	0.995097i	$-0.468466\pi$
$401$	1588.41	0.197809	0.0989045	+	0.995097i	$0.468466\pi$
$402$	−6561.35	−0.814056
$403$	10081.5	1.24614
$404$	−3927.29	−0.483639
$405$	0	0
$406$	−28.1049	−0.00343553
$407$	−3282.20	−0.399736
$408$	−388.533	−0.0471452
$409$	4704.53	0.568763	0.284381	−	0.958711i	$-0.408212\pi$
$409$	4704.53	0.568763	0.284381	+	0.958711i	$0.408212\pi$
$410$	0	0
$411$	−9159.73	−1.09931
$412$	115.830	0.0138508
$413$	−6.48184	−0.000772277 0
$414$	150.248	0.0178365
$415$	0	0
$416$	2755.52	0.324761
$417$	−13246.4	−1.55559
$418$	−7479.52	−0.875204
$419$	8774.62	1.02307	0.511537	−	0.859261i	$-0.329076\pi$
$419$	8774.62	1.02307	0.511537	+	0.859261i	$0.329076\pi$
$420$	0	0
$421$	−12089.9	−1.39959	−0.699794	−	0.714345i	$-0.746725\pi$
$421$	−12089.9	−1.39959	−0.699794	+	0.714345i	$0.746725\pi$
$422$	−11224.5	−1.29479
$423$	1332.36	0.153147
$424$	2530.00	0.289783
$425$	0	0
$426$	−7615.23	−0.866102
$427$	−14.9661	−0.00169616
$428$	751.495	0.0848712
$429$	16663.8	1.87538
$430$	0	0
$431$	−901.917	−0.100798	−0.0503989	−	0.998729i	$-0.516049\pi$
$431$	−901.917	−0.100798	−0.0503989	+	0.998729i	$0.516049\pi$
$432$	2089.13	0.232670
$433$	−11009.9	−1.22194	−0.610972	−	0.791652i	$-0.709221\pi$
$433$	−11009.9	−1.22194	−0.610972	+	0.791652i	$0.709221\pi$
$434$	−11.7201	−0.00129627
$435$	0	0
$436$	8568.08	0.941139
$437$	−2445.28	−0.267675
$438$	−9265.56	−1.01079
$439$	10645.7	1.15738	0.578690	−	0.815548i	$-0.303564\pi$
$439$	10645.7	1.15738	0.578690	+	0.815548i	$0.303564\pi$
$440$	0	0
$441$	−1120.32	−0.120972
$442$	−1520.34	−0.163610
$443$	−3031.48	−0.325124	−0.162562	−	0.986698i	$-0.551976\pi$
$443$	−3031.48	−0.325124	−0.162562	+	0.986698i	$0.551976\pi$
$444$	2053.35	0.219476
$445$	0	0
$446$	−1115.82	−0.118465
$447$	−10115.8	−1.07038
$448$	−3.20337	−0.000337824 0
$449$	−2924.54	−0.307388	−0.153694	−	0.988118i	$-0.549117\pi$
$449$	−2924.54	−0.307388	−0.153694	+	0.988118i	$0.549117\pi$
$450$	0	0
$451$	−2042.22	−0.213225
$452$	6674.69	0.694582
$453$	7024.59	0.728574
$454$	−1369.72	−0.141595
$455$	0	0
$456$	4679.19	0.480533
$457$	−16680.1	−1.70736	−0.853681	−	0.520797i	$-0.825635\pi$
$457$	−16680.1	−1.70736	−0.853681	+	0.520797i	$0.825635\pi$
$458$	−738.968	−0.0753924
$459$	−1152.67	−0.117216
$460$	0	0
$461$	10948.6	1.10613	0.553065	−	0.833138i	$-0.313458\pi$
$461$	10948.6	1.10613	0.553065	+	0.833138i	$0.313458\pi$
$462$	−19.3722	−0.00195081
$463$	−9101.09	−0.913528	−0.456764	−	0.889588i	$-0.650992\pi$
$463$	−9101.09	−0.913528	−0.456764	+	0.889588i	$0.650992\pi$
$464$	−4492.06	−0.449437
$465$	0	0
$466$	−10239.7	−1.01791
$467$	−12728.2	−1.26122	−0.630609	−	0.776101i	$-0.717195\pi$
$467$	−12728.2	−1.26122	−0.630609	+	0.776101i	$0.717195\pi$
$468$	−1125.03	−0.111121
$469$	29.8477	0.00293868
$470$	0	0
$471$	−7086.63	−0.693280
$472$	−1036.00	−0.101029
$473$	17872.8	1.73740
$474$	−12775.1	−1.23793
$475$	0	0
$476$	1.76744	0.000170190 0
$477$	−1032.96	−0.0991529
$478$	−3330.05	−0.318647
$479$	−858.175	−0.0818602	−0.0409301	−	0.999162i	$-0.513032\pi$
$479$	−858.175	−0.0818602	−0.0409301	+	0.999162i	$0.513032\pi$
$480$	0	0
$481$	8034.82	0.761656
$482$	−10803.1	−1.02089
$483$	−6.33336	−0.000596642 0
$484$	−374.692	−0.0351889
$485$	0	0
$486$	−1823.30	−0.170178
$487$	−1777.12	−0.165357	−0.0826784	−	0.996576i	$-0.526347\pi$
$487$	−1777.12	−0.165357	−0.0826784	+	0.996576i	$0.526347\pi$
$488$	−2392.06	−0.221893
$489$	−5479.15	−0.506699
$490$	0	0
$491$	13458.2	1.23699	0.618495	−	0.785789i	$-0.287743\pi$
$491$	13458.2	1.23699	0.618495	+	0.785789i	$0.287743\pi$
$492$	1277.62	0.117072
$493$	2478.47	0.226419
$494$	18309.9	1.66761
$495$	0	0
$496$	−1873.24	−0.169578
$497$	34.6418	0.00312656
$498$	−12342.0	−1.11055
$499$	8470.02	0.759860	0.379930	−	0.925015i	$-0.375948\pi$
$499$	8470.02	0.759860	0.379930	+	0.925015i	$0.375948\pi$
$500$	0	0
$501$	−8628.48	−0.769445
$502$	−8492.71	−0.755076
$503$	3558.11	0.315404	0.157702	−	0.987487i	$-0.449591\pi$
$503$	3558.11	0.315404	0.157702	+	0.987487i	$0.449591\pi$
$504$	1.30788	0.000115591 0
$505$	0	0
$506$	1618.08	0.142159
$507$	−28706.3	−2.51458
$508$	1417.30	0.123785
$509$	−11039.9	−0.961370	−0.480685	−	0.876893i	$-0.659612\pi$
$509$	−11039.9	−0.961370	−0.480685	+	0.876893i	$0.659612\pi$
$510$	0	0
$511$	42.1492	0.00364887
$512$	−512.000	−0.0441942
$513$	13881.8	1.19473
$514$	−11171.1	−0.958632
$515$	0	0
$516$	−11181.2	−0.953927
$517$	14348.6	1.22060
$518$	−9.34071	−0.000792292 0
$519$	13637.8	1.15344
$520$	0	0
$521$	5846.79	0.491656	0.245828	−	0.969314i	$-0.420940\pi$
$521$	5846.79	0.491656	0.245828	+	0.969314i	$0.420940\pi$
$522$	1834.03	0.153780
$523$	−17121.8	−1.43152	−0.715759	−	0.698348i	$-0.753919\pi$
$523$	−17121.8	−1.43152	−0.715759	+	0.698348i	$0.753919\pi$
$524$	−3688.95	−0.307543
$525$	0	0
$526$	−4262.02	−0.353294
$527$	1033.55	0.0854310
$528$	−3096.29	−0.255206
$529$	529.000	0.0434783
$530$	0	0
$531$	422.983	0.0345685
$532$	−21.2857	−0.00173469
$533$	4999.36	0.406278
$534$	4501.22	0.364769
$535$	0	0
$536$	4770.61	0.384438
$537$	−20581.0	−1.65388
$538$	16006.6	1.28270
$539$	−12065.2	−0.964162
$540$	0	0
$541$	12373.9	0.983352	0.491676	−	0.870778i	$-0.336384\pi$
$541$	12373.9	0.983352	0.491676	+	0.870778i	$0.336384\pi$
$542$	−2698.66	−0.213870
$543$	21136.0	1.67041
$544$	282.494	0.0222644
$545$	0	0
$546$	47.4231	0.00371707
$547$	16142.0	1.26176	0.630881	−	0.775880i	$-0.282694\pi$
$547$	16142.0	1.26176	0.630881	+	0.775880i	$0.282694\pi$
$548$	6659.83	0.519149
$549$	976.640	0.0759234
$550$	0	0
$551$	−29848.8	−2.30781
$552$	−1012.27	−0.0780528
$553$	58.1141	0.00446883
$554$	−7525.52	−0.577127
$555$	0	0
$556$	9631.19	0.734628
$557$	−8855.85	−0.673671	−0.336835	−	0.941564i	$-0.609357\pi$
$557$	−8855.85	−0.673671	−0.336835	+	0.941564i	$0.609357\pi$
$558$	764.811	0.0580233
$559$	−43752.6	−3.31044
$560$	0	0
$561$	1708.36	0.128569
$562$	−5770.72	−0.433137
$563$	25526.5	1.91086	0.955428	−	0.295223i	$-0.0953939\pi$
$563$	25526.5	1.91086	0.955428	+	0.295223i	$0.0953939\pi$
$564$	−8976.52	−0.670177
$565$	0	0
$566$	9618.22	0.714283
$567$	40.3685	0.00298998
$568$	5536.86	0.409017
$569$	5758.98	0.424304	0.212152	−	0.977237i	$-0.431953\pi$
$569$	5758.98	0.424304	0.212152	+	0.977237i	$0.431953\pi$
$570$	0	0
$571$	−14978.9	−1.09781	−0.548903	−	0.835886i	$-0.684955\pi$
$571$	−14978.9	−1.09781	−0.548903	+	0.835886i	$0.684955\pi$
$572$	−12115.9	−0.885649
$573$	−11537.3	−0.841148
$574$	−5.81190	−0.000422620 0
$575$	0	0
$576$	209.041	0.0151216
$577$	−19218.9	−1.38664	−0.693322	−	0.720627i	$-0.743854\pi$
$577$	−19218.9	−1.38664	−0.693322	+	0.720627i	$0.743854\pi$
$578$	9670.14	0.695890
$579$	6957.45	0.499381
$580$	0	0
$581$	56.1438	0.00400901
$582$	12363.2	0.880532
$583$	−11124.3	−0.790261
$584$	6736.78	0.477346
$585$	0	0
$586$	−10725.4	−0.756077
$587$	4648.60	0.326863	0.163431	−	0.986555i	$-0.447744\pi$
$587$	4648.60	0.326863	0.163431	+	0.986555i	$0.447744\pi$
$588$	7547.97	0.529376
$589$	−12447.3	−0.870765
$590$	0	0
$591$	−4676.97	−0.325524
$592$	−1492.94	−0.103648
$593$	22294.2	1.54387	0.771933	−	0.635703i	$-0.219290\pi$
$593$	22294.2	1.54387	0.771933	+	0.635703i	$0.219290\pi$
$594$	−9185.82	−0.634509
$595$	0	0
$596$	7354.96	0.505488
$597$	1117.29	0.0765957
$598$	−3961.06	−0.270869
$599$	14695.1	1.00238	0.501191	−	0.865336i	$-0.332895\pi$
$599$	14695.1	1.00238	0.501191	+	0.865336i	$0.332895\pi$
$600$	0	0
$601$	−4381.79	−0.297399	−0.148699	−	0.988882i	$-0.547509\pi$
$601$	−4381.79	−0.297399	−0.148699	+	0.988882i	$0.547509\pi$
$602$	50.8636	0.00344360
$603$	−1947.76	−0.131541
$604$	−5107.42	−0.344070
$605$	0	0
$606$	−10803.0	−0.724159
$607$	−14325.9	−0.957943	−0.478971	−	0.877831i	$-0.658990\pi$
$607$	−14325.9	−0.957943	−0.478971	+	0.877831i	$0.658990\pi$
$608$	−3402.13	−0.226932
$609$	−77.3093	−0.00514406
$610$	0	0
$611$	−35125.5	−2.32574
$612$	−115.337	−0.00761804
$613$	−12551.9	−0.827028	−0.413514	−	0.910498i	$-0.635699\pi$
$613$	−12551.9	−0.827028	−0.413514	+	0.910498i	$0.635699\pi$
$614$	12749.5	0.837991
$615$	0	0
$616$	14.0851	0.000921273 0
$617$	−5556.60	−0.362561	−0.181281	−	0.983431i	$-0.558024\pi$
$617$	−5556.60	−0.362561	−0.181281	+	0.983431i	$0.558024\pi$
$618$	318.618	0.0207390
$619$	16504.3	1.07167	0.535834	−	0.844323i	$-0.319997\pi$
$619$	16504.3	1.07167	0.535834	+	0.844323i	$0.319997\pi$
$620$	0	0
$621$	−3003.12	−0.194060
$622$	−74.5598	−0.00480639
$623$	−20.4761	−0.00131679
$624$	7579.71	0.486268
$625$	0	0
$626$	−7575.33	−0.483660
$627$	−20574.2	−1.31045
$628$	5152.53	0.327402
$629$	823.723	0.0522162
$630$	0	0
$631$	20737.5	1.30832	0.654158	−	0.756358i	$-0.273023\pi$
$631$	20737.5	1.30832	0.654158	+	0.756358i	$0.273023\pi$
$632$	9288.48	0.584614
$633$	−30875.7	−1.93870
$634$	−11266.7	−0.705770
$635$	0	0
$636$	6959.38	0.433895
$637$	29535.5	1.83711
$638$	19751.4	1.22565
$639$	−2260.61	−0.139950
$640$	0	0
$641$	−5138.54	−0.316630	−0.158315	−	0.987389i	$-0.550606\pi$
$641$	−5138.54	−0.316630	−0.158315	+	0.987389i	$0.550606\pi$
$642$	2067.17	0.127079
$643$	20288.1	1.24430	0.622149	−	0.782899i	$-0.286260\pi$
$643$	20288.1	1.24430	0.622149	+	0.782899i	$0.286260\pi$
$644$	4.60484	0.000281764 0
$645$	0	0
$646$	1877.11	0.114325
$647$	22823.9	1.38686	0.693432	−	0.720522i	$-0.256098\pi$
$647$	22823.9	1.38686	0.693432	+	0.720522i	$0.256098\pi$
$648$	6452.17	0.391150
$649$	4555.26	0.275516
$650$	0	0
$651$	−32.2388	−0.00194092
$652$	3983.77	0.239289
$653$	−27010.0	−1.61866	−0.809329	−	0.587356i	$-0.800169\pi$
$653$	−27010.0	−1.61866	−0.809329	+	0.587356i	$0.800169\pi$
$654$	23568.6	1.40918
$655$	0	0
$656$	−928.926	−0.0552873
$657$	−2750.52	−0.163330
$658$	40.8344	0.00241928
$659$	−4630.23	−0.273700	−0.136850	−	0.990592i	$-0.543698\pi$
$659$	−4630.23	−0.273700	−0.136850	+	0.990592i	$0.543698\pi$
$660$	0	0
$661$	−12644.2	−0.744028	−0.372014	−	0.928227i	$-0.621333\pi$
$661$	−12644.2	−0.744028	−0.372014	+	0.928227i	$0.621333\pi$
$662$	−9133.83	−0.536248
$663$	−4182.07	−0.244975
$664$	8973.56	0.524460
$665$	0	0
$666$	609.543	0.0354644
$667$	6457.33	0.374856
$668$	6273.57	0.363371
$669$	−3069.32	−0.177380
$670$	0	0
$671$	10517.8	0.605120
$672$	−8.81163	−0.000505828 0
$673$	10099.8	0.578482	0.289241	−	0.957256i	$-0.406597\pi$
$673$	10099.8	0.578482	0.289241	+	0.957256i	$0.406597\pi$
$674$	1574.97	0.0900085
$675$	0	0
$676$	20871.7	1.18751
$677$	10253.3	0.582075	0.291037	−	0.956712i	$-0.406000\pi$
$677$	10253.3	0.582075	0.291037	+	0.956712i	$0.406000\pi$
$678$	18360.3	1.04001
$679$	−56.2402	−0.00317865
$680$	0	0
$681$	−3767.73	−0.212012
$682$	8236.54	0.462454
$683$	5525.56	0.309560	0.154780	−	0.987949i	$-0.450533\pi$
$683$	5525.56	0.309560	0.154780	+	0.987949i	$0.450533\pi$
$684$	1389.03	0.0776477
$685$	0	0
$686$	−68.6720	−0.00382202
$687$	−2032.71	−0.112886
$688$	8129.62	0.450492
$689$	27232.3	1.50576
$690$	0	0
$691$	15754.4	0.867332	0.433666	−	0.901074i	$-0.357220\pi$
$691$	15754.4	0.867332	0.433666	+	0.901074i	$0.357220\pi$
$692$	−9915.74	−0.544711
$693$	−5.75070	−0.000315225 0
$694$	−7477.05	−0.408969
$695$	0	0
$696$	−12356.5	−0.672947
$697$	512.530	0.0278529
$698$	−850.420	−0.0461158
$699$	−28166.7	−1.52412
$700$	0	0
$701$	10304.0	0.555176	0.277588	−	0.960700i	$-0.410465\pi$
$701$	10304.0	0.555176	0.277588	+	0.960700i	$0.410465\pi$
$702$	22486.9	1.20899
$703$	−9920.29	−0.532220
$704$	2251.24	0.120521
$705$	0	0
$706$	711.980	0.0379543
$707$	49.1429	0.00261415
$708$	−2849.77	−0.151273
$709$	−4916.67	−0.260436	−0.130218	−	0.991485i	$-0.541568\pi$
$709$	−4916.67	−0.260436	−0.130218	+	0.991485i	$0.541568\pi$
$710$	0	0
$711$	−3792.33	−0.200033
$712$	−3272.73	−0.172262
$713$	2692.78	0.141438
$714$	4.86178	0.000254828 0
$715$	0	0
$716$	14964.0	0.781048
$717$	−9160.11	−0.477114
$718$	7426.35	0.386001
$719$	−3323.08	−0.172364	−0.0861821	−	0.996279i	$-0.527467\pi$
$719$	−3323.08	−0.172364	−0.0861821	+	0.996279i	$0.527467\pi$
$720$	0	0
$721$	−1.44940	−7.48660e−5 0
$722$	−8888.47	−0.458164
$723$	−29716.6	−1.52859
$724$	−15367.5	−0.788851
$725$	0	0
$726$	−1030.68	−0.0526888
$727$	29941.1	1.52745	0.763724	−	0.645543i	$-0.223369\pi$
$727$	29941.1	1.52745	0.763724	+	0.645543i	$0.223369\pi$
$728$	−34.4802	−0.00175539
$729$	16760.6	0.851529
$730$	0	0
$731$	−4485.48	−0.226951
$732$	−6579.95	−0.332243
$733$	16395.5	0.826170	0.413085	−	0.910693i	$-0.364451\pi$
$733$	16395.5	0.826170	0.413085	+	0.910693i	$0.364451\pi$
$734$	7816.79	0.393083
$735$	0	0
$736$	736.000	0.0368605
$737$	−20976.2	−1.04840
$738$	379.265	0.0189172
$739$	−24881.9	−1.23856	−0.619279	−	0.785171i	$-0.712575\pi$
$739$	−24881.9	−1.23856	−0.619279	+	0.785171i	$0.712575\pi$
$740$	0	0
$741$	50365.6	2.49693
$742$	−31.6583	−0.00156633
$743$	8845.30	0.436746	0.218373	−	0.975865i	$-0.429925\pi$
$743$	8845.30	0.436746	0.218373	+	0.975865i	$0.429925\pi$
$744$	−5152.78	−0.253912
$745$	0	0
$746$	27913.0	1.36993
$747$	−3663.75	−0.179451
$748$	−1242.11	−0.0607167
$749$	−9.40358	−0.000458744 0
$750$	0	0
$751$	33958.9	1.65004	0.825020	−	0.565104i	$-0.191164\pi$
$751$	33958.9	1.65004	0.825020	+	0.565104i	$0.191164\pi$
$752$	6526.63	0.316491
$753$	−23361.2	−1.13058
$754$	−48351.4	−2.33535
$755$	0	0
$756$	−26.1416	−0.00125762
$757$	22718.8	1.09079	0.545395	−	0.838179i	$-0.316379\pi$
$757$	22718.8	1.09079	0.545395	+	0.838179i	$0.316379\pi$
$758$	686.323	0.0328871
$759$	4450.92	0.212856
$760$	0	0
$761$	21813.2	1.03906	0.519532	−	0.854451i	$-0.326106\pi$
$761$	21813.2	1.03906	0.519532	+	0.854451i	$0.326106\pi$
$762$	3898.63	0.185345
$763$	−107.214	−0.00508703
$764$	8388.51	0.397233
$765$	0	0
$766$	−27741.7	−1.30855
$767$	−11151.3	−0.524966
$768$	−1408.38	−0.0661725
$769$	−15098.4	−0.708011	−0.354006	−	0.935243i	$-0.615181\pi$
$769$	−15098.4	−0.708011	−0.354006	+	0.935243i	$0.615181\pi$
$770$	0	0
$771$	−30728.8	−1.43537
$772$	−5058.60	−0.235833
$773$	−4343.28	−0.202092	−0.101046	−	0.994882i	$-0.532219\pi$
$773$	−4343.28	−0.202092	−0.101046	+	0.994882i	$0.532219\pi$
$774$	−3319.19	−0.154142
$775$	0	0
$776$	−8988.97	−0.415832
$777$	−25.6938	−0.00118631
$778$	−160.184	−0.00738159
$779$	−6172.52	−0.283894
$780$	0	0
$781$	−24345.4	−1.11542
$782$	−406.085	−0.0185698
$783$	−36658.2	−1.67312
$784$	−5487.96	−0.249998
$785$	0	0
$786$	−10147.3	−0.460487
$787$	−4399.04	−0.199249	−0.0996243	−	0.995025i	$-0.531764\pi$
$787$	−4399.04	−0.199249	−0.0996243	+	0.995025i	$0.531764\pi$
$788$	3400.52	0.153729
$789$	−11723.7	−0.528992
$790$	0	0
$791$	−83.5215	−0.00375434
$792$	−919.145	−0.0412379
$793$	−25747.6	−1.15299
$794$	−10154.8	−0.453878
$795$	0	0
$796$	−812.356	−0.0361724
$797$	−16911.6	−0.751619	−0.375809	−	0.926697i	$-0.622635\pi$
$797$	−16911.6	−0.751619	−0.375809	+	0.926697i	$0.622635\pi$
$798$	−58.5515	−0.00259737
$799$	−3601.03	−0.159444
$800$	0	0
$801$	1336.20	0.0589418
$802$	−3176.82	−0.139872
$803$	−29621.3	−1.30176
$804$	13122.7	0.575625
$805$	0	0
$806$	−20163.0	−0.881157
$807$	44030.1	1.92061
$808$	7854.59	0.341984
$809$	−21310.5	−0.926128	−0.463064	−	0.886325i	$-0.653250\pi$
$809$	−21310.5	−0.926128	−0.463064	+	0.886325i	$0.653250\pi$
$810$	0	0
$811$	−5999.06	−0.259748	−0.129874	−	0.991531i	$-0.541457\pi$
$811$	−5999.06	−0.259748	−0.129874	+	0.991531i	$0.541457\pi$
$812$	56.2098	0.00242928
$813$	−7423.32	−0.320230
$814$	6564.40	0.282656
$815$	0	0
$816$	777.066	0.0333367
$817$	54019.6	2.31323
$818$	−9409.06	−0.402176
$819$	14.0777	0.000600629 0
$820$	0	0
$821$	−34257.7	−1.45628	−0.728138	−	0.685431i	$-0.759614\pi$
$821$	−34257.7	−1.45628	−0.728138	+	0.685431i	$0.759614\pi$
$822$	18319.5	0.777329
$823$	46224.1	1.95780	0.978902	−	0.204331i	$-0.0655020\pi$
$823$	46224.1	1.95780	0.978902	+	0.204331i	$0.0655020\pi$
$824$	−231.660	−0.00979399
$825$	0	0
$826$	12.9637	0.000546082 0
$827$	−37548.2	−1.57881	−0.789407	−	0.613870i	$-0.789612\pi$
$827$	−37548.2	−1.57881	−0.789407	+	0.613870i	$0.789612\pi$
$828$	−300.496	−0.0126123
$829$	−30920.8	−1.29544	−0.647721	−	0.761877i	$-0.724278\pi$
$829$	−30920.8	−1.29544	−0.647721	+	0.761877i	$0.724278\pi$
$830$	0	0
$831$	−20700.7	−0.864140
$832$	−5511.04	−0.229640
$833$	3027.96	0.125945
$834$	26492.9	1.09997
$835$	0	0
$836$	14959.0	0.618863
$837$	−15286.9	−0.631291
$838$	−17549.2	−0.723423
$839$	4014.99	0.165212	0.0826060	−	0.996582i	$-0.473676\pi$
$839$	4014.99	0.165212	0.0826060	+	0.996582i	$0.473676\pi$
$840$	0	0
$841$	54433.6	2.23189
$842$	24179.8	0.989658
$843$	−15873.7	−0.648542
$844$	22449.0	0.915553
$845$	0	0
$846$	−2664.71	−0.108292
$847$	4.68858	0.000190202 0
$848$	−5060.01	−0.204907
$849$	26457.2	1.06950
$850$	0	0
$851$	2146.10	0.0864483
$852$	15230.5	0.612426
$853$	4140.42	0.166196	0.0830981	−	0.996541i	$-0.473519\pi$
$853$	4140.42	0.166196	0.0830981	+	0.996541i	$0.473519\pi$
$854$	29.9323	0.00119937
$855$	0	0
$856$	−1502.99	−0.0600130
$857$	21402.7	0.853093	0.426547	−	0.904466i	$-0.359730\pi$
$857$	21402.7	0.853093	0.426547	+	0.904466i	$0.359730\pi$
$858$	−33327.7	−1.32609
$859$	32876.0	1.30584	0.652919	−	0.757427i	$-0.273544\pi$
$859$	32876.0	1.30584	0.652919	+	0.757427i	$0.273544\pi$
$860$	0	0
$861$	−15.9870	−0.000632794 0
$862$	1803.83	0.0712747
$863$	−29257.0	−1.15402	−0.577010	−	0.816737i	$-0.695781\pi$
$863$	−29257.0	−1.15402	−0.577010	+	0.816737i	$0.695781\pi$
$864$	−4178.26	−0.164522
$865$	0	0
$866$	22019.8	0.864046
$867$	26600.0	1.04197
$868$	23.4401	0.000916600 0
$869$	−40841.0	−1.59429
$870$	0	0
$871$	51349.6	1.99761
$872$	−17136.2	−0.665486
$873$	3670.05	0.142282
$874$	4890.57	0.189274
$875$	0	0
$876$	18531.1	0.714736
$877$	7545.57	0.290531	0.145266	−	0.989393i	$-0.453596\pi$
$877$	7545.57	0.290531	0.145266	+	0.989393i	$0.453596\pi$
$878$	−21291.3	−0.818391
$879$	−29502.7	−1.13208
$880$	0	0
$881$	19544.5	0.747414	0.373707	−	0.927547i	$-0.378087\pi$
$881$	19544.5	0.747414	0.373707	+	0.927547i	$0.378087\pi$
$882$	2240.64	0.0855401
$883$	−28251.9	−1.07673	−0.538365	−	0.842712i	$-0.680958\pi$
$883$	−28251.9	−1.07673	−0.538365	+	0.842712i	$0.680958\pi$
$884$	3040.69	0.115689
$885$	0	0
$886$	6062.96	0.229897
$887$	−8880.90	−0.336180	−0.168090	−	0.985772i	$-0.553760\pi$
$887$	−8880.90	−0.336180	−0.168090	+	0.985772i	$0.553760\pi$
$888$	−4106.69	−0.155193
$889$	−17.7349	−0.000669079 0
$890$	0	0
$891$	−28369.9	−1.06670
$892$	2231.64	0.0837676
$893$	43368.1	1.62515
$894$	20231.6	0.756874
$895$	0	0
$896$	6.40674	0.000238877 0
$897$	−10895.8	−0.405576
$898$	5849.07	0.217356
$899$	32869.9	1.21943
$900$	0	0
$901$	2791.84	0.103229
$902$	4084.45	0.150773
$903$	139.913	0.00515614
$904$	−13349.4	−0.491144
$905$	0	0
$906$	−14049.2	−0.515180
$907$	−15437.3	−0.565147	−0.282574	−	0.959246i	$-0.591188\pi$
$907$	−15437.3	−0.565147	−0.282574	+	0.959246i	$0.591188\pi$
$908$	2739.43	0.100123
$909$	−3206.90	−0.117014
$910$	0	0
$911$	33155.3	1.20580	0.602900	−	0.797817i	$-0.294012\pi$
$911$	33155.3	1.20580	0.602900	+	0.797817i	$0.294012\pi$
$912$	−9358.38	−0.339788
$913$	−39456.3	−1.43025
$914$	33360.3	1.20729
$915$	0	0
$916$	1477.94	0.0533104
$917$	46.1604	0.00166232
$918$	2305.34	0.0828839
$919$	−12375.1	−0.444196	−0.222098	−	0.975024i	$-0.571291\pi$
$919$	−12375.1	−0.444196	−0.222098	+	0.975024i	$0.571291\pi$
$920$	0	0
$921$	35070.5	1.25474
$922$	−21897.1	−0.782152
$923$	59597.4	2.12532
$924$	38.7444	0.00137943
$925$	0	0
$926$	18202.2	0.645962
$927$	94.5828	0.00335114
$928$	8984.12	0.317800
$929$	49693.8	1.75501	0.877504	−	0.479570i	$-0.159207\pi$
$929$	49693.8	1.75501	0.877504	+	0.479570i	$0.159207\pi$
$930$	0	0
$931$	−36466.4	−1.28371
$932$	20479.4	0.719768
$933$	−205.095	−0.00719668
$934$	25456.3	0.891816
$935$	0	0
$936$	2250.06	0.0785744
$937$	42717.9	1.48936	0.744682	−	0.667420i	$-0.232601\pi$
$937$	42717.9	1.48936	0.744682	+	0.667420i	$0.232601\pi$
$938$	−59.6954	−0.00207796
$939$	−20837.8	−0.724190
$940$	0	0
$941$	20273.7	0.702344	0.351172	−	0.936311i	$-0.385783\pi$
$941$	20273.7	0.702344	0.351172	+	0.936311i	$0.385783\pi$
$942$	14173.3	0.490223
$943$	1335.33	0.0461128
$944$	2072.01	0.0714386
$945$	0	0
$946$	−35745.6	−1.22853
$947$	−20905.2	−0.717347	−0.358674	−	0.933463i	$-0.616771\pi$
$947$	−20905.2	−0.717347	−0.358674	+	0.933463i	$0.616771\pi$
$948$	25550.2	0.875350
$949$	72513.0	2.48037
$950$	0	0
$951$	−30991.8	−1.05676
$952$	−3.53489	−0.000120343 0
$953$	−39537.9	−1.34392	−0.671960	−	0.740587i	$-0.734548\pi$
$953$	−39537.9	−1.34392	−0.671960	+	0.740587i	$0.734548\pi$
$954$	2065.92	0.0701117
$955$	0	0
$956$	6660.11	0.225317
$957$	54330.9	1.83518
$958$	1716.35	0.0578839
$959$	−83.3355	−0.00280609
$960$	0	0
$961$	−16083.9	−0.539892
$962$	−16069.6	−0.538572
$963$	613.646	0.0205342
$964$	21606.3	0.721878
$965$	0	0
$966$	12.6667	0.000421889 0
$967$	−26356.2	−0.876482	−0.438241	−	0.898858i	$-0.644398\pi$
$967$	−26356.2	−0.876482	−0.438241	+	0.898858i	$0.644398\pi$
$968$	749.383	0.0248823
$969$	5163.44	0.171180
$970$	0	0
$971$	11885.8	0.392827	0.196413	−	0.980521i	$-0.437071\pi$
$971$	11885.8	0.392827	0.196413	+	0.980521i	$0.437071\pi$
$972$	3646.60	0.120334
$973$	−120.517	−0.00397080
$974$	3554.23	0.116925
$975$	0	0
$976$	4784.13	0.156902
$977$	11798.1	0.386341	0.193171	−	0.981165i	$-0.438123\pi$
$977$	11798.1	0.386341	0.193171	+	0.981165i	$0.438123\pi$
$978$	10958.3	0.358290
$979$	14390.1	0.469774
$980$	0	0
$981$	6996.41	0.227705
$982$	−26916.5	−0.874684
$983$	5991.38	0.194400	0.0972001	−	0.995265i	$-0.469011\pi$
$983$	5991.38	0.194400	0.0972001	+	0.995265i	$0.469011\pi$
$984$	−2555.23	−0.0827823
$985$	0	0
$986$	−4956.94	−0.160103
$987$	112.325	0.00362242
$988$	−36619.7	−1.17918
$989$	−11686.3	−0.375737
$990$	0	0
$991$	−18597.3	−0.596129	−0.298065	−	0.954546i	$-0.596341\pi$
$991$	−18597.3	−0.596129	−0.298065	+	0.954546i	$0.596341\pi$
$992$	3746.47	0.119910
$993$	−25124.8	−0.802931
$994$	−69.2837	−0.00221081
$995$	0	0
$996$	24683.9	0.785281
$997$	−21172.0	−0.672541	−0.336271	−	0.941765i	$-0.609166\pi$
$997$	−21172.0	−0.672541	−0.336271	+	0.941765i	$0.609166\pi$
$998$	−16940.0	−0.537302
$999$	−12183.4	−0.385851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.o.1.2		4
5.2	odd	4		1150.4.b.m.599.3		8
5.3	odd	4		1150.4.b.m.599.6		8
5.4	even	2		230.4.a.i.1.3	✓	4
15.14	odd	2		2070.4.a.bi.1.2		4
20.19	odd	2		1840.4.a.l.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.3	✓	4	5.4	even	2
1150.4.a.o.1.2		4	1.1	even	1	trivial
1150.4.b.m.599.3		8	5.2	odd	4
1150.4.b.m.599.6		8	5.3	odd	4
1840.4.a.l.1.2		4	20.19	odd	2
2070.4.a.bi.1.2		4	15.14	odd	2

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -5.50148 q^{3} +4.00000 q^{4} +11.0030 q^{6} -0.0500526 q^{7} -8.00000 q^{8} +3.26627 q^{9} +35.1757 q^{11} -22.0059 q^{12} -86.1100 q^{13} +0.100105 q^{14} +16.0000 q^{16} -8.82793 q^{17} -6.53253 q^{18} +106.317 q^{19} +0.275364 q^{21} -70.3513 q^{22} -23.0000 q^{23} +44.0118 q^{24} +172.220 q^{26} +130.571 q^{27} -0.200211 q^{28} -280.754 q^{29} -117.077 q^{31} -32.0000 q^{32} -193.518 q^{33} +17.6559 q^{34} +13.0651 q^{36} -93.3088 q^{37} -212.633 q^{38} +473.732 q^{39} -58.0579 q^{41} -0.550727 q^{42} +508.101 q^{43} +140.703 q^{44} +46.0000 q^{46} +407.914 q^{47} -88.0237 q^{48} -342.997 q^{49} +48.5666 q^{51} -344.440 q^{52} -316.251 q^{53} -261.141 q^{54} +0.400421 q^{56} -584.899 q^{57} +561.507 q^{58} +129.500 q^{59} +299.008 q^{61} +234.155 q^{62} -0.163485 q^{63} +64.0000 q^{64} +387.036 q^{66} -596.326 q^{67} -35.3117 q^{68} +126.534 q^{69} -692.108 q^{71} -26.1301 q^{72} -842.098 q^{73} +186.618 q^{74} +425.267 q^{76} -1.76063 q^{77} -947.464 q^{78} -1161.06 q^{79} -806.521 q^{81} +116.116 q^{82} -1121.69 q^{83} +1.10145 q^{84} -1016.20 q^{86} +1544.56 q^{87} -281.405 q^{88} +409.092 q^{89} +4.31003 q^{91} -92.0000 q^{92} +644.098 q^{93} -815.828 q^{94} +176.047 q^{96} +1123.62 q^{97} +685.995 q^{98} +114.893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} + 8 q^{6} - 26 q^{7} - 32 q^{8} + 64 q^{9} + 93 q^{11} - 16 q^{12} - 32 q^{13} + 52 q^{14} + 64 q^{16} - 108 q^{17} - 128 q^{18} + 185 q^{19} + 302 q^{21} - 186 q^{22}+ \cdots + 2013 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 4 * q^3 + 16 * q^4 + 8 * q^6 - 26 * q^7 - 32 * q^8 + 64 * q^9 + 93 * q^11 - 16 * q^12 - 32 * q^13 + 52 * q^14 + 64 * q^16 - 108 * q^17 - 128 * q^18 + 185 * q^19 + 302 * q^21 - 186 * q^22 - 92 * q^23 + 32 * q^24 + 64 * q^26 - 259 * q^27 - 104 * q^28 + 294 * q^29 - 211 * q^31 - 128 * q^32 + 237 * q^33 + 216 * q^34 + 256 * q^36 - 5 * q^37 - 370 * q^38 - 421 * q^39 - 369 * q^41 - 604 * q^42 + 100 * q^43 + 372 * q^44 + 184 * q^46 + 363 * q^47 - 64 * q^48 + 600 * q^49 - 315 * q^51 - 128 * q^52 - 21 * q^53 + 518 * q^54 + 208 * q^56 + 160 * q^57 - 588 * q^58 - 33 * q^59 - 307 * q^61 + 422 * q^62 - 167 * q^63 + 256 * q^64 - 474 * q^66 - 725 * q^67 - 432 * q^68 + 92 * q^69 - 1257 * q^71 - 512 * q^72 - 509 * q^73 + 10 * q^74 + 740 * q^76 + 1962 * q^77 + 842 * q^78 + 1202 * q^79 - 992 * q^81 + 738 * q^82 + 1377 * q^83 + 1208 * q^84 - 200 * q^86 + 2829 * q^87 - 744 * q^88 - 984 * q^89 - 995 * q^91 - 368 * q^92 + 1843 * q^93 - 726 * q^94 + 128 * q^96 - 137 * q^97 - 1200 * q^98 + 2013 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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