LMFDB - Embedded newform 1250.4.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1250,4,Mod(1,1250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1250, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1250.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1250 = 2 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1250.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:	$73.7523875072$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 145x^{4} + 120x^{3} + 5125x^{2} - 2431x - 1069 $ x^6 - x^5 - 145x^4 + 120x^3 + 5125x^2 - 2431x - 1069
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$8.28587$ of defining polynomial
Character	$\chi$	$=$	1250.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +5.66784 q^{3} +4.00000 q^{4} +11.3357 q^{6} +16.6100 q^{7} +8.00000 q^{8} +5.12438 q^{9} +3.04838 q^{11} +22.6714 q^{12} +15.4682 q^{13} +33.2200 q^{14} +16.0000 q^{16} +43.5583 q^{17} +10.2488 q^{18} -24.2031 q^{19} +94.1429 q^{21} +6.09676 q^{22} +195.996 q^{23} +45.3427 q^{24} +30.9364 q^{26} -123.987 q^{27} +66.4401 q^{28} +10.3063 q^{29} -105.236 q^{31} +32.0000 q^{32} +17.2777 q^{33} +87.1165 q^{34} +20.4975 q^{36} +266.960 q^{37} -48.4063 q^{38} +87.6713 q^{39} +33.7935 q^{41} +188.286 q^{42} -430.248 q^{43} +12.1935 q^{44} +391.992 q^{46} +556.466 q^{47} +90.6854 q^{48} -67.1073 q^{49} +246.881 q^{51} +61.8728 q^{52} +659.826 q^{53} -247.975 q^{54} +132.880 q^{56} -137.179 q^{57} +20.6126 q^{58} -162.048 q^{59} -197.885 q^{61} -210.471 q^{62} +85.1161 q^{63} +64.0000 q^{64} +34.5555 q^{66} +155.284 q^{67} +174.233 q^{68} +1110.87 q^{69} +860.300 q^{71} +40.9951 q^{72} +359.057 q^{73} +533.919 q^{74} -96.8125 q^{76} +50.6337 q^{77} +175.343 q^{78} -1110.76 q^{79} -841.099 q^{81} +67.5871 q^{82} -476.911 q^{83} +376.572 q^{84} -860.496 q^{86} +58.4143 q^{87} +24.3871 q^{88} -1444.77 q^{89} +256.927 q^{91} +783.984 q^{92} -596.459 q^{93} +1112.93 q^{94} +181.371 q^{96} +645.160 q^{97} -134.215 q^{98} +15.6211 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} - 8 q^{3} + 24 q^{4} - 16 q^{6} + 6 q^{7} + 48 q^{8} + 142 q^{9} + 107 q^{11} - 32 q^{12} + 42 q^{13} + 12 q^{14} + 96 q^{16} + 156 q^{17} + 284 q^{18} + 50 q^{19} + 122 q^{21} + 214 q^{22}+ \cdots + 2389 q^{99}+O(q^{100}) $ 6 * q + 12 * q^2 - 8 * q^3 + 24 * q^4 - 16 * q^6 + 6 * q^7 + 48 * q^8 + 142 * q^9 + 107 * q^11 - 32 * q^12 + 42 * q^13 + 12 * q^14 + 96 * q^16 + 156 * q^17 + 284 * q^18 + 50 * q^19 + 122 * q^21 + 214 * q^22 + 2 * q^23 - 64 * q^24 + 84 * q^26 - 680 * q^27 + 24 * q^28 + 100 * q^29 + 742 * q^31 + 192 * q^32 - 601 * q^33 + 312 * q^34 + 568 * q^36 + 71 * q^37 + 100 * q^38 + 259 * q^39 - 513 * q^41 + 244 * q^42 + 92 * q^43 + 428 * q^44 + 4 * q^46 + 311 * q^47 - 128 * q^48 + 348 * q^49 - 1998 * q^51 + 168 * q^52 + 1892 * q^53 - 1360 * q^54 + 48 * q^56 - 1515 * q^57 + 200 * q^58 + 665 * q^59 + 1577 * q^61 + 1484 * q^62 + 747 * q^63 + 384 * q^64 - 1202 * q^66 + 2201 * q^67 + 624 * q^68 + 4044 * q^69 + 1312 * q^71 + 1136 * q^72 - 708 * q^73 + 142 * q^74 + 200 * q^76 - 2318 * q^77 + 518 * q^78 - 1820 * q^79 + 5126 * q^81 - 1026 * q^82 - 1013 * q^83 + 488 * q^84 + 184 * q^86 + 1230 * q^87 + 856 * q^88 - 80 * q^89 + 1402 * q^91 + 8 * q^92 - 396 * q^93 + 622 * q^94 - 256 * q^96 + 996 * q^97 + 696 * q^98 + 2389 * 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
$3$	5.66784	1.09078	0.545388	−	0.838184i	$-0.316382\pi$
$3$	5.66784	1.09078	0.545388	+	0.838184i	$0.316382\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	11.3357	0.771295
$7$	16.6100	0.896857	0.448428	−	0.893819i	$-0.351984\pi$
$7$	16.6100	0.896857	0.448428	+	0.893819i	$0.351984\pi$
$8$	8.00000	0.353553
$9$	5.12438	0.189792
$10$	0	0
$11$	3.04838	0.0835565	0.0417783	−	0.999127i	$-0.486698\pi$
$11$	3.04838	0.0835565	0.0417783	+	0.999127i	$0.486698\pi$
$12$	22.6714	0.545388
$13$	15.4682	0.330008	0.165004	−	0.986293i	$-0.447236\pi$
$13$	15.4682	0.330008	0.165004	+	0.986293i	$0.447236\pi$
$14$	33.2200	0.634173
$15$	0	0
$16$	16.0000	0.250000
$17$	43.5583	0.621437	0.310719	−	0.950502i	$-0.399430\pi$
$17$	43.5583	0.621437	0.310719	+	0.950502i	$0.399430\pi$
$18$	10.2488	0.134203
$19$	−24.2031	−0.292241	−0.146121	−	0.989267i	$-0.546679\pi$
$19$	−24.2031	−0.292241	−0.146121	+	0.989267i	$0.546679\pi$
$20$	0	0
$21$	94.1429	0.978270
$22$	6.09676	0.0590834
$23$	195.996	1.77687	0.888434	−	0.459003i	$-0.151793\pi$
$23$	195.996	1.77687	0.888434	+	0.459003i	$0.151793\pi$
$24$	45.3427	0.385647
$25$	0	0
$26$	30.9364	0.233351
$27$	−123.987	−0.883755
$28$	66.4401	0.448428
$29$	10.3063	0.0659941	0.0329970	−	0.999455i	$-0.489495\pi$
$29$	10.3063	0.0659941	0.0329970	+	0.999455i	$0.489495\pi$
$30$	0	0
$31$	−105.236	−0.609706	−0.304853	−	0.952399i	$-0.598607\pi$
$31$	−105.236	−0.609706	−0.304853	+	0.952399i	$0.598607\pi$
$32$	32.0000	0.176777
$33$	17.2777	0.0911414
$34$	87.1165	0.439422
$35$	0	0
$36$	20.4975	0.0948960
$37$	266.960	1.18616	0.593080	−	0.805144i	$-0.297912\pi$
$37$	266.960	1.18616	0.593080	+	0.805144i	$0.297912\pi$
$38$	−48.4063	−0.206646
$39$	87.6713	0.359965
$40$	0	0
$41$	33.7935	0.128724	0.0643618	−	0.997927i	$-0.479499\pi$
$41$	33.7935	0.128724	0.0643618	+	0.997927i	$0.479499\pi$
$42$	188.286	0.691741
$43$	−430.248	−1.52587	−0.762933	−	0.646478i	$-0.776241\pi$
$43$	−430.248	−1.52587	−0.762933	+	0.646478i	$0.776241\pi$
$44$	12.1935	0.0417783
$45$	0	0
$46$	391.992	1.25644
$47$	556.466	1.72700	0.863498	−	0.504352i	$-0.168269\pi$
$47$	556.466	1.72700	0.863498	+	0.504352i	$0.168269\pi$
$48$	90.6854	0.272694
$49$	−67.1073	−0.195648
$50$	0	0
$51$	246.881	0.677849
$52$	61.8728	0.165004
$53$	659.826	1.71008	0.855038	−	0.518565i	$-0.173534\pi$
$53$	659.826	1.71008	0.855038	+	0.518565i	$0.173534\pi$
$54$	−247.975	−0.624909
$55$	0	0
$56$	132.880	0.317087
$57$	−137.179	−0.318769
$58$	20.6126	0.0466649
$59$	−162.048	−0.357575	−0.178787	−	0.983888i	$-0.557217\pi$
$59$	−162.048	−0.357575	−0.178787	+	0.983888i	$0.557217\pi$
$60$	0	0
$61$	−197.885	−0.415353	−0.207677	−	0.978198i	$-0.566590\pi$
$61$	−197.885	−0.415353	−0.207677	+	0.978198i	$0.566590\pi$
$62$	−210.471	−0.431127
$63$	85.1161	0.170216
$64$	64.0000	0.125000
$65$	0	0
$66$	34.5555	0.0644467
$67$	155.284	0.283149	0.141575	−	0.989928i	$-0.454783\pi$
$67$	155.284	0.283149	0.141575	+	0.989928i	$0.454783\pi$
$68$	174.233	0.310719
$69$	1110.87	1.93817
$70$	0	0
$71$	860.300	1.43801	0.719006	−	0.695004i	$-0.244597\pi$
$71$	860.300	1.43801	0.719006	+	0.695004i	$0.244597\pi$
$72$	40.9951	0.0671016
$73$	359.057	0.575678	0.287839	−	0.957679i	$-0.407063\pi$
$73$	359.057	0.575678	0.287839	+	0.957679i	$0.407063\pi$
$74$	533.919	0.838741
$75$	0	0
$76$	−96.8125	−0.146121
$77$	50.6337	0.0749382
$78$	175.343	0.254534
$79$	−1110.76	−1.58190	−0.790951	−	0.611879i	$-0.790414\pi$
$79$	−1110.76	−1.58190	−0.790951	+	0.611879i	$0.790414\pi$
$80$	0	0
$81$	−841.099	−1.15377
$82$	67.5871	0.0910213
$83$	−476.911	−0.630696	−0.315348	−	0.948976i	$-0.602121\pi$
$83$	−476.911	−0.630696	−0.315348	+	0.948976i	$0.602121\pi$
$84$	376.572	0.489135
$85$	0	0
$86$	−860.496	−1.07895
$87$	58.4143	0.0719847
$88$	24.3871	0.0295417
$89$	−1444.77	−1.72074	−0.860368	−	0.509673i	$-0.829766\pi$
$89$	−1444.77	−1.72074	−0.860368	+	0.509673i	$0.829766\pi$
$90$	0	0
$91$	256.927	0.295970
$92$	783.984	0.888434
$93$	−596.459	−0.665053
$94$	1112.93	1.22117
$95$	0	0
$96$	181.371	0.192824
$97$	645.160	0.675321	0.337660	−	0.941268i	$-0.390365\pi$
$97$	645.160	0.675321	0.337660	+	0.941268i	$0.390365\pi$
$98$	−134.215	−0.138344
$99$	15.6211	0.0158584
$100$	0	0
$101$	991.112	0.976429	0.488214	−	0.872724i	$-0.337648\pi$
$101$	991.112	0.976429	0.488214	+	0.872724i	$0.337648\pi$
$102$	493.762	0.479311
$103$	1463.05	1.39960	0.699798	−	0.714341i	$-0.253273\pi$
$103$	1463.05	1.39960	0.699798	+	0.714341i	$0.253273\pi$
$104$	123.746	0.116676
$105$	0	0
$106$	1319.65	1.20921
$107$	−816.480	−0.737684	−0.368842	−	0.929492i	$-0.620246\pi$
$107$	−816.480	−0.737684	−0.368842	+	0.929492i	$0.620246\pi$
$108$	−495.950	−0.441878
$109$	202.993	0.178378	0.0891889	−	0.996015i	$-0.471573\pi$
$109$	202.993	0.178378	0.0891889	+	0.996015i	$0.471573\pi$
$110$	0	0
$111$	1513.08	1.29383
$112$	265.760	0.224214
$113$	1305.27	1.08663	0.543317	−	0.839528i	$-0.317168\pi$
$113$	1305.27	1.08663	0.543317	+	0.839528i	$0.317168\pi$
$114$	−274.359	−0.225404
$115$	0	0
$116$	41.2251	0.0329970
$117$	79.2650	0.0626329
$118$	−324.097	−0.252843
$119$	723.504	0.557340
$120$	0	0
$121$	−1321.71	−0.993018
$122$	−395.770	−0.293699
$123$	191.536	0.140408
$124$	−420.943	−0.304853
$125$	0	0
$126$	170.232	0.120361
$127$	−1453.61	−1.01565	−0.507824	−	0.861461i	$-0.669550\pi$
$127$	−1453.61	−1.01565	−0.507824	+	0.861461i	$0.669550\pi$
$128$	128.000	0.0883883
$129$	−2438.58	−1.66438
$130$	0	0
$131$	278.547	0.185777	0.0928884	−	0.995677i	$-0.470390\pi$
$131$	278.547	0.185777	0.0928884	+	0.995677i	$0.470390\pi$
$132$	69.1109	0.0455707
$133$	−402.015	−0.262098
$134$	310.569	0.200217
$135$	0	0
$136$	348.466	0.219711
$137$	338.479	0.211082	0.105541	−	0.994415i	$-0.466343\pi$
$137$	338.479	0.211082	0.105541	+	0.994415i	$0.466343\pi$
$138$	2221.75	1.37049
$139$	−1295.12	−0.790293	−0.395146	−	0.918618i	$-0.629306\pi$
$139$	−1295.12	−0.790293	−0.395146	+	0.918618i	$0.629306\pi$
$140$	0	0
$141$	3153.96	1.88377
$142$	1720.60	1.01683
$143$	47.1530	0.0275743
$144$	81.9901	0.0474480
$145$	0	0
$146$	718.115	0.407066
$147$	−380.353	−0.213408
$148$	1067.84	0.593080
$149$	−1132.23	−0.622521	−0.311261	−	0.950325i	$-0.600751\pi$
$149$	−1132.23	−0.622521	−0.311261	+	0.950325i	$0.600751\pi$
$150$	0	0
$151$	719.198	0.387599	0.193800	−	0.981041i	$-0.437919\pi$
$151$	719.198	0.387599	0.193800	+	0.981041i	$0.437919\pi$
$152$	−193.625	−0.103323
$153$	223.209	0.117944
$154$	101.267	0.0529893
$155$	0	0
$156$	350.685	0.179983
$157$	−2182.94	−1.10967	−0.554833	−	0.831962i	$-0.687218\pi$
$157$	−2182.94	−1.10967	−0.554833	+	0.831962i	$0.687218\pi$
$158$	−2221.52	−1.11857
$159$	3739.78	1.86531
$160$	0	0
$161$	3255.50	1.59360
$162$	−1682.20	−0.815839
$163$	3031.50	1.45672	0.728359	−	0.685196i	$-0.240283\pi$
$163$	3031.50	1.45672	0.728359	+	0.685196i	$0.240283\pi$
$164$	135.174	0.0643618
$165$	0	0
$166$	−953.822	−0.445969
$167$	−2072.36	−0.960265	−0.480132	−	0.877196i	$-0.659411\pi$
$167$	−2072.36	−0.960265	−0.480132	+	0.877196i	$0.659411\pi$
$168$	753.143	0.345871
$169$	−1957.73	−0.891095
$170$	0	0
$171$	−124.026	−0.0554650
$172$	−1720.99	−0.762933
$173$	737.782	0.324234	0.162117	−	0.986772i	$-0.448168\pi$
$173$	737.782	0.324234	0.162117	+	0.986772i	$0.448168\pi$
$174$	116.829	0.0509009
$175$	0	0
$176$	48.7741	0.0208891
$177$	−918.464	−0.390034
$178$	−2889.54	−1.21674
$179$	−2351.63	−0.981952	−0.490976	−	0.871173i	$-0.663360\pi$
$179$	−2351.63	−0.981952	−0.490976	+	0.871173i	$0.663360\pi$
$180$	0	0
$181$	4327.33	1.77706	0.888530	−	0.458818i	$-0.151727\pi$
$181$	4327.33	1.77706	0.888530	+	0.458818i	$0.151727\pi$
$182$	513.854	0.209282
$183$	−1121.58	−0.453058
$184$	1567.97	0.628218
$185$	0	0
$186$	−1192.92	−0.470263
$187$	132.782	0.0519251
$188$	2225.86	0.863498
$189$	−2059.43	−0.792602
$190$	0	0
$191$	1764.16	0.668325	0.334163	−	0.942515i	$-0.391547\pi$
$191$	1764.16	0.668325	0.334163	+	0.942515i	$0.391547\pi$
$192$	362.742	0.136347
$193$	−4958.76	−1.84943	−0.924714	−	0.380662i	$-0.875696\pi$
$193$	−4958.76	−1.84943	−0.924714	+	0.380662i	$0.875696\pi$
$194$	1290.32	0.477524
$195$	0	0
$196$	−268.429	−0.0978240
$197$	−3133.00	−1.13308	−0.566540	−	0.824034i	$-0.691718\pi$
$197$	−3133.00	−1.13308	−0.566540	+	0.824034i	$0.691718\pi$
$198$	31.2421	0.0112135
$199$	2883.31	1.02710	0.513549	−	0.858060i	$-0.328330\pi$
$199$	2883.31	1.02710	0.513549	+	0.858060i	$0.328330\pi$
$200$	0	0
$201$	880.127	0.308852
$202$	1982.22	0.690439
$203$	171.187	0.0591872
$204$	987.525	0.338924
$205$	0	0
$206$	2926.10	0.989664
$207$	1004.36	0.337235
$208$	247.491	0.0825021
$209$	−73.7804	−0.0244186
$210$	0	0
$211$	5976.93	1.95009	0.975044	−	0.222011i	$-0.0712622\pi$
$211$	5976.93	1.95009	0.975044	+	0.222011i	$0.0712622\pi$
$212$	2639.30	0.855038
$213$	4876.04	1.56855
$214$	−1632.96	−0.521621
$215$	0	0
$216$	−991.900	−0.312455
$217$	−1747.97	−0.546819
$218$	405.986	0.126132
$219$	2035.08	0.627936
$220$	0	0
$221$	673.768	0.205079
$222$	3026.17	0.914879
$223$	−3951.70	−1.18666	−0.593331	−	0.804959i	$-0.702187\pi$
$223$	−3951.70	−1.18666	−0.593331	+	0.804959i	$0.702187\pi$
$224$	531.521	0.158543
$225$	0	0
$226$	2610.54	0.768366
$227$	−2141.61	−0.626185	−0.313092	−	0.949723i	$-0.601365\pi$
$227$	−2141.61	−0.626185	−0.313092	+	0.949723i	$0.601365\pi$
$228$	−548.718	−0.159385
$229$	2122.54	0.612494	0.306247	−	0.951952i	$-0.400927\pi$
$229$	2122.54	0.612494	0.306247	+	0.951952i	$0.400927\pi$
$230$	0	0
$231$	286.983	0.0817408
$232$	82.4502	0.0233324
$233$	−4051.36	−1.13911	−0.569557	−	0.821952i	$-0.692885\pi$
$233$	−4051.36	−1.13911	−0.569557	+	0.821952i	$0.692885\pi$
$234$	158.530	0.0442882
$235$	0	0
$236$	−648.194	−0.178787
$237$	−6295.61	−1.72550
$238$	1447.01	0.394099
$239$	−4447.49	−1.20370	−0.601850	−	0.798609i	$-0.705569\pi$
$239$	−4447.49	−1.20370	−0.601850	+	0.798609i	$0.705569\pi$
$240$	0	0
$241$	571.376	0.152720	0.0763601	−	0.997080i	$-0.475670\pi$
$241$	571.376	0.152720	0.0763601	+	0.997080i	$0.475670\pi$
$242$	−2643.41	−0.702170
$243$	−1419.55	−0.374750
$244$	−791.540	−0.207677
$245$	0	0
$246$	383.073	0.0992838
$247$	−374.379	−0.0964420
$248$	−841.886	−0.215564
$249$	−2703.05	−0.687948
$250$	0	0
$251$	−3432.31	−0.863129	−0.431565	−	0.902082i	$-0.642038\pi$
$251$	−3432.31	−0.863129	−0.431565	+	0.902082i	$0.642038\pi$
$252$	340.464	0.0851081
$253$	597.471	0.148469
$254$	−2907.22	−0.718171
$255$	0	0
$256$	256.000	0.0625000
$257$	−144.659	−0.0351112	−0.0175556	−	0.999846i	$-0.505588\pi$
$257$	−144.659	−0.0351112	−0.0175556	+	0.999846i	$0.505588\pi$
$258$	−4877.15	−1.17689
$259$	4434.21	1.06382
$260$	0	0
$261$	52.8133	0.0125251
$262$	557.094	0.131364
$263$	−5463.46	−1.28096	−0.640478	−	0.767977i	$-0.721264\pi$
$263$	−5463.46	−1.28096	−0.640478	+	0.767977i	$0.721264\pi$
$264$	138.222	0.0322234
$265$	0	0
$266$	−804.029	−0.185332
$267$	−8188.73	−1.87694
$268$	621.137	0.141575
$269$	2797.61	0.634101	0.317051	−	0.948409i	$-0.397307\pi$
$269$	2797.61	0.634101	0.317051	+	0.948409i	$0.397307\pi$
$270$	0	0
$271$	4284.97	0.960492	0.480246	−	0.877134i	$-0.340547\pi$
$271$	4284.97	0.960492	0.480246	+	0.877134i	$0.340547\pi$
$272$	696.932	0.155359
$273$	1456.22	0.322837
$274$	676.957	0.149257
$275$	0	0
$276$	4443.49	0.969083
$277$	3735.09	0.810179	0.405089	−	0.914277i	$-0.367240\pi$
$277$	3735.09	0.810179	0.405089	+	0.914277i	$0.367240\pi$
$278$	−2590.24	−0.558821
$279$	−539.268	−0.115717
$280$	0	0
$281$	−6712.03	−1.42493	−0.712466	−	0.701707i	$-0.752422\pi$
$281$	−6712.03	−1.42493	−0.712466	+	0.701707i	$0.752422\pi$
$282$	6307.91	1.33202
$283$	−535.045	−0.112386	−0.0561928	−	0.998420i	$-0.517896\pi$
$283$	−535.045	−0.112386	−0.0561928	+	0.998420i	$0.517896\pi$
$284$	3441.20	0.719006
$285$	0	0
$286$	94.3060	0.0194980
$287$	561.311	0.115447
$288$	163.980	0.0335508
$289$	−3015.68	−0.613816
$290$	0	0
$291$	3656.66	0.736623
$292$	1436.23	0.287839
$293$	−9332.20	−1.86073	−0.930363	−	0.366639i	$-0.880509\pi$
$293$	−9332.20	−1.86073	−0.930363	+	0.366639i	$0.880509\pi$
$294$	−760.706	−0.150902
$295$	0	0
$296$	2135.68	0.419371
$297$	−377.961	−0.0738435
$298$	−2264.45	−0.440189
$299$	3031.71	0.586381
$300$	0	0
$301$	−7146.43	−1.36848
$302$	1438.40	0.274074
$303$	5617.46	1.06506
$304$	−387.250	−0.0730603
$305$	0	0
$306$	446.418	0.0833988
$307$	2429.22	0.451606	0.225803	−	0.974173i	$-0.427499\pi$
$307$	2429.22	0.451606	0.225803	+	0.974173i	$0.427499\pi$
$308$	202.535	0.0374691
$309$	8292.32	1.52665
$310$	0	0
$311$	−2332.86	−0.425351	−0.212676	−	0.977123i	$-0.568218\pi$
$311$	−2332.86	−0.425351	−0.212676	+	0.977123i	$0.568218\pi$
$312$	701.370	0.127267
$313$	−1872.59	−0.338163	−0.169081	−	0.985602i	$-0.554080\pi$
$313$	−1872.59	−0.338163	−0.169081	+	0.985602i	$0.554080\pi$
$314$	−4365.88	−0.784652
$315$	0	0
$316$	−4443.04	−0.790951
$317$	4709.54	0.834429	0.417215	−	0.908808i	$-0.363006\pi$
$317$	4709.54	0.834429	0.417215	+	0.908808i	$0.363006\pi$
$318$	7479.57	1.31897
$319$	31.4175	0.00551423
$320$	0	0
$321$	−4627.68	−0.804647
$322$	6510.99	1.12684
$323$	−1054.25	−0.181609
$324$	−3364.40	−0.576885
$325$	0	0
$326$	6062.99	1.03006
$327$	1150.53	0.194570
$328$	270.348	0.0455106
$329$	9242.90	1.54887
$330$	0	0
$331$	7114.17	1.18136	0.590680	−	0.806906i	$-0.298859\pi$
$331$	7114.17	1.18136	0.590680	+	0.806906i	$0.298859\pi$
$332$	−1907.64	−0.315348
$333$	1368.00	0.225124
$334$	−4144.72	−0.679010
$335$	0	0
$336$	1506.29	0.244567
$337$	2327.59	0.376237	0.188119	−	0.982146i	$-0.439761\pi$
$337$	2327.59	0.376237	0.188119	+	0.982146i	$0.439761\pi$
$338$	−3915.47	−0.630099
$339$	7398.07	1.18527
$340$	0	0
$341$	−320.799	−0.0509449
$342$	−248.052	−0.0392197
$343$	−6811.89	−1.07232
$344$	−3441.99	−0.539475
$345$	0	0
$346$	1475.56	0.229268
$347$	−3332.82	−0.515605	−0.257803	−	0.966198i	$-0.582998\pi$
$347$	−3332.82	−0.515605	−0.257803	+	0.966198i	$0.582998\pi$
$348$	233.657	0.0359924
$349$	−2838.44	−0.435353	−0.217677	−	0.976021i	$-0.569848\pi$
$349$	−2838.44	−0.435353	−0.217677	+	0.976021i	$0.569848\pi$
$350$	0	0
$351$	−1917.86	−0.291647
$352$	97.5482	0.0147708
$353$	10997.7	1.65821	0.829107	−	0.559091i	$-0.188850\pi$
$353$	10997.7	1.65821	0.829107	+	0.559091i	$0.188850\pi$
$354$	−1836.93	−0.275796
$355$	0	0
$356$	−5779.09	−0.860368
$357$	4100.70	0.607933
$358$	−4703.27	−0.694345
$359$	995.709	0.146383	0.0731915	−	0.997318i	$-0.476682\pi$
$359$	995.709	0.146383	0.0731915	+	0.997318i	$0.476682\pi$
$360$	0	0
$361$	−6273.21	−0.914595
$362$	8654.66	1.25657
$363$	−7491.22	−1.08316
$364$	1027.71	0.147985
$365$	0	0
$366$	−2243.16	−0.320360
$367$	9187.31	1.30674	0.653370	−	0.757038i	$-0.273354\pi$
$367$	9187.31	1.30674	0.653370	+	0.757038i	$0.273354\pi$
$368$	3135.94	0.444217
$369$	173.171	0.0244307
$370$	0	0
$371$	10959.7	1.53369
$372$	−2385.84	−0.332526
$373$	6058.93	0.841072	0.420536	−	0.907276i	$-0.361842\pi$
$373$	6058.93	0.841072	0.420536	+	0.907276i	$0.361842\pi$
$374$	265.564	0.0367166
$375$	0	0
$376$	4451.72	0.610585
$377$	159.420	0.0217786
$378$	−4118.87	−0.560454
$379$	893.684	0.121123	0.0605613	−	0.998164i	$-0.480711\pi$
$379$	893.684	0.121123	0.0605613	+	0.998164i	$0.480711\pi$
$380$	0	0
$381$	−8238.83	−1.10784
$382$	3528.32	0.472577
$383$	5839.47	0.779068	0.389534	−	0.921012i	$-0.372636\pi$
$383$	5839.47	0.779068	0.389534	+	0.921012i	$0.372636\pi$
$384$	725.483	0.0964119
$385$	0	0
$386$	−9917.53	−1.30774
$387$	−2204.76	−0.289597
$388$	2580.64	0.337660
$389$	−4912.83	−0.640335	−0.320168	−	0.947361i	$-0.603739\pi$
$389$	−4912.83	−0.640335	−0.320168	+	0.947361i	$0.603739\pi$
$390$	0	0
$391$	8537.24	1.10421
$392$	−536.858	−0.0691720
$393$	1578.76	0.202641
$394$	−6266.00	−0.801209
$395$	0	0
$396$	62.4843	0.00792918
$397$	−8419.00	−1.06433	−0.532163	−	0.846642i	$-0.678621\pi$
$397$	−8419.00	−1.06433	−0.532163	+	0.846642i	$0.678621\pi$
$398$	5766.62	0.726268
$399$	−2278.55	−0.285891
$400$	0	0
$401$	−7353.01	−0.915690	−0.457845	−	0.889032i	$-0.651379\pi$
$401$	−7353.01	−0.915690	−0.457845	+	0.889032i	$0.651379\pi$
$402$	1760.25	0.218392
$403$	−1627.81	−0.201208
$404$	3964.45	0.488214
$405$	0	0
$406$	342.375	0.0418517
$407$	813.795	0.0991114
$408$	1975.05	0.239656
$409$	−13980.4	−1.69019	−0.845096	−	0.534615i	$-0.820457\pi$
$409$	−13980.4	−1.69019	−0.845096	+	0.534615i	$0.820457\pi$
$410$	0	0
$411$	1918.44	0.230243
$412$	5852.19	0.699798
$413$	−2691.63	−0.320693
$414$	2008.72	0.238461
$415$	0	0
$416$	494.982	0.0583378
$417$	−7340.53	−0.862032
$418$	−147.561	−0.0172666
$419$	−12778.0	−1.48985	−0.744923	−	0.667150i	$-0.767514\pi$
$419$	−12778.0	−1.48985	−0.744923	+	0.667150i	$0.767514\pi$
$420$	0	0
$421$	1708.71	0.197809	0.0989044	−	0.995097i	$-0.468466\pi$
$421$	1708.71	0.197809	0.0989044	+	0.995097i	$0.468466\pi$
$422$	11953.9	1.37892
$423$	2851.54	0.327770
$424$	5278.60	0.604603
$425$	0	0
$426$	9752.09	1.10913
$427$	−3286.87	−0.372513
$428$	−3265.92	−0.368842
$429$	267.255	0.0300774
$430$	0	0
$431$	−5049.46	−0.564325	−0.282162	−	0.959367i	$-0.591052\pi$
$431$	−5049.46	−0.564325	−0.282162	+	0.959367i	$0.591052\pi$
$432$	−1983.80	−0.220939
$433$	10438.4	1.15852	0.579259	−	0.815143i	$-0.303342\pi$
$433$	10438.4	1.15852	0.579259	+	0.815143i	$0.303342\pi$
$434$	−3495.94	−0.386660
$435$	0	0
$436$	811.971	0.0891889
$437$	−4743.72	−0.519274
$438$	4070.16	0.444018
$439$	15290.1	1.66231	0.831156	−	0.556040i	$-0.187680\pi$
$439$	15290.1	1.66231	0.831156	+	0.556040i	$0.187680\pi$
$440$	0	0
$441$	−343.883	−0.0371324
$442$	1347.54	0.145013
$443$	866.921	0.0929766	0.0464883	−	0.998919i	$-0.485197\pi$
$443$	866.921	0.0929766	0.0464883	+	0.998919i	$0.485197\pi$
$444$	6052.34	0.646917
$445$	0	0
$446$	−7903.41	−0.839097
$447$	−6417.28	−0.679031
$448$	1063.04	0.112107
$449$	400.462	0.0420913	0.0210456	−	0.999779i	$-0.493300\pi$
$449$	400.462	0.0420913	0.0210456	+	0.999779i	$0.493300\pi$
$450$	0	0
$451$	103.016	0.0107557
$452$	5221.09	0.543317
$453$	4076.30	0.422784
$454$	−4283.23	−0.442779
$455$	0	0
$456$	−1097.44	−0.112702
$457$	−11585.7	−1.18590	−0.592949	−	0.805240i	$-0.702036\pi$
$457$	−11585.7	−1.18590	−0.592949	+	0.805240i	$0.702036\pi$
$458$	4245.07	0.433099
$459$	−5400.68	−0.549198
$460$	0	0
$461$	−5722.52	−0.578144	−0.289072	−	0.957307i	$-0.593347\pi$
$461$	−5722.52	−0.578144	−0.289072	+	0.957307i	$0.593347\pi$
$462$	573.967	0.0577995
$463$	−16136.2	−1.61968	−0.809841	−	0.586649i	$-0.800447\pi$
$463$	−16136.2	−1.61968	−0.809841	+	0.586649i	$0.800447\pi$
$464$	164.900	0.0164985
$465$	0	0
$466$	−8102.73	−0.805476
$467$	10508.2	1.04125	0.520623	−	0.853787i	$-0.325700\pi$
$467$	10508.2	1.04125	0.520623	+	0.853787i	$0.325700\pi$
$468$	317.060	0.0313165
$469$	2579.28	0.253944
$470$	0	0
$471$	−12372.5	−1.21040
$472$	−1296.39	−0.126422
$473$	−1311.56	−0.127496
$474$	−12591.2	−1.22011
$475$	0	0
$476$	2894.01	0.278670
$477$	3381.20	0.324559
$478$	−8894.98	−0.851144
$479$	−12405.2	−1.18331	−0.591656	−	0.806191i	$-0.701526\pi$
$479$	−12405.2	−1.18331	−0.591656	+	0.806191i	$0.701526\pi$
$480$	0	0
$481$	4129.39	0.391442
$482$	1142.75	0.107989
$483$	18451.6	1.73826
$484$	−5286.83	−0.496509
$485$	0	0
$486$	−2839.10	−0.264988
$487$	−6720.06	−0.625288	−0.312644	−	0.949870i	$-0.601215\pi$
$487$	−6720.06	−0.625288	−0.312644	+	0.949870i	$0.601215\pi$
$488$	−1583.08	−0.146850
$489$	17182.0	1.58895
$490$	0	0
$491$	3549.65	0.326260	0.163130	−	0.986605i	$-0.447841\pi$
$491$	3549.65	0.326260	0.163130	+	0.986605i	$0.447841\pi$
$492$	766.145	0.0702042
$493$	448.924	0.0410112
$494$	−748.758	−0.0681948
$495$	0	0
$496$	−1683.77	−0.152427
$497$	14289.6	1.28969
$498$	−5406.11	−0.486453
$499$	15925.4	1.42869	0.714346	−	0.699793i	$-0.246724\pi$
$499$	15925.4	1.42869	0.714346	+	0.699793i	$0.246724\pi$
$500$	0	0
$501$	−11745.8	−1.04743
$502$	−6864.62	−0.610324
$503$	−799.366	−0.0708588	−0.0354294	−	0.999372i	$-0.511280\pi$
$503$	−799.366	−0.0708588	−0.0354294	+	0.999372i	$0.511280\pi$
$504$	680.929	0.0601805
$505$	0	0
$506$	1194.94	0.104983
$507$	−11096.1	−0.971984
$508$	−5814.45	−0.507824
$509$	−17678.2	−1.53943	−0.769717	−	0.638386i	$-0.779602\pi$
$509$	−17678.2	−1.53943	−0.769717	+	0.638386i	$0.779602\pi$
$510$	0	0
$511$	5963.95	0.516301
$512$	512.000	0.0441942
$513$	3000.88	0.258270
$514$	−289.318	−0.0248273
$515$	0	0
$516$	−9754.31	−0.832189
$517$	1696.32	0.144302
$518$	8868.41	0.752231
$519$	4181.63	0.353667
$520$	0	0
$521$	−5119.04	−0.430459	−0.215230	−	0.976563i	$-0.569050\pi$
$521$	−5119.04	−0.430459	−0.215230	+	0.976563i	$0.569050\pi$
$522$	105.627	0.00885661
$523$	−17639.5	−1.47480	−0.737402	−	0.675455i	$-0.763947\pi$
$523$	−17639.5	−1.47480	−0.737402	+	0.675455i	$0.763947\pi$
$524$	1114.19	0.0928884
$525$	0	0
$526$	−10926.9	−0.905772
$527$	−4583.89	−0.378894
$528$	276.444	0.0227854
$529$	26247.4	2.15726
$530$	0	0
$531$	−830.398	−0.0678648
$532$	−1608.06	−0.131049
$533$	522.725	0.0424798
$534$	−16377.5	−1.32720
$535$	0	0
$536$	1242.27	0.100108
$537$	−13328.7	−1.07109
$538$	5595.22	0.448377
$539$	−204.569	−0.0163477
$540$	0	0
$541$	2856.92	0.227040	0.113520	−	0.993536i	$-0.463787\pi$
$541$	2856.92	0.227040	0.113520	+	0.993536i	$0.463787\pi$
$542$	8569.93	0.679170
$543$	24526.6	1.93837
$544$	1393.86	0.109856
$545$	0	0
$546$	2912.44	0.228280
$547$	−14400.0	−1.12559	−0.562796	−	0.826596i	$-0.690274\pi$
$547$	−14400.0	−1.12559	−0.562796	+	0.826596i	$0.690274\pi$
$548$	1353.91	0.105541
$549$	−1014.04	−0.0788308
$550$	0	0
$551$	−249.444	−0.0192862
$552$	8886.99	0.685245
$553$	−18449.7	−1.41874
$554$	7470.17	0.572883
$555$	0	0
$556$	−5180.48	−0.395146
$557$	−9685.69	−0.736797	−0.368398	−	0.929668i	$-0.620094\pi$
$557$	−9685.69	−0.736797	−0.368398	+	0.929668i	$0.620094\pi$
$558$	−1078.54	−0.0818245
$559$	−6655.17	−0.503548
$560$	0	0
$561$	752.588	0.0566387
$562$	−13424.1	−1.00758
$563$	−20951.5	−1.56838	−0.784192	−	0.620518i	$-0.786922\pi$
$563$	−20951.5	−1.56838	−0.784192	+	0.620518i	$0.786922\pi$
$564$	12615.8	0.941883
$565$	0	0
$566$	−1070.09	−0.0794686
$567$	−13970.7	−1.03477
$568$	6882.40	0.508414
$569$	6768.07	0.498651	0.249325	−	0.968420i	$-0.419791\pi$
$569$	6768.07	0.498651	0.249325	+	0.968420i	$0.419791\pi$
$570$	0	0
$571$	−17740.7	−1.30022	−0.650109	−	0.759841i	$-0.725277\pi$
$571$	−17740.7	−1.30022	−0.650109	+	0.759841i	$0.725277\pi$
$572$	188.612	0.0137872
$573$	9998.97	0.728993
$574$	1122.62	0.0816330
$575$	0	0
$576$	327.960	0.0237240
$577$	−18908.1	−1.36422	−0.682110	−	0.731249i	$-0.738938\pi$
$577$	−18908.1	−1.36422	−0.682110	+	0.731249i	$0.738938\pi$
$578$	−6031.36	−0.434033
$579$	−28105.5	−2.01731
$580$	0	0
$581$	−7921.50	−0.565644
$582$	7313.33	0.520871
$583$	2011.40	0.142888
$584$	2872.46	0.203533
$585$	0	0
$586$	−18664.4	−1.31573
$587$	15584.1	1.09579	0.547893	−	0.836548i	$-0.315430\pi$
$587$	15584.1	1.09579	0.547893	+	0.836548i	$0.315430\pi$
$588$	−1521.41	−0.106704
$589$	2547.03	0.178181
$590$	0	0
$591$	−17757.3	−1.23594
$592$	4271.36	0.296540
$593$	5757.33	0.398694	0.199347	−	0.979929i	$-0.436118\pi$
$593$	5757.33	0.398694	0.199347	+	0.979929i	$0.436118\pi$
$594$	−755.922	−0.0522152
$595$	0	0
$596$	−4528.91	−0.311261
$597$	16342.1	1.12033
$598$	6063.41	0.414634
$599$	−6004.95	−0.409608	−0.204804	−	0.978803i	$-0.565656\pi$
$599$	−6004.95	−0.409608	−0.204804	+	0.978803i	$0.565656\pi$
$600$	0	0
$601$	−26366.0	−1.78951	−0.894753	−	0.446562i	$-0.852648\pi$
$601$	−26366.0	−1.78951	−0.894753	+	0.446562i	$0.852648\pi$
$602$	−14292.9	−0.967664
$603$	795.737	0.0537395
$604$	2876.79	0.193800
$605$	0	0
$606$	11234.9	0.753114
$607$	−3986.79	−0.266588	−0.133294	−	0.991077i	$-0.542555\pi$
$607$	−3986.79	−0.266588	−0.133294	+	0.991077i	$0.542555\pi$
$608$	−774.500	−0.0516614
$609$	970.263	0.0645600
$610$	0	0
$611$	8607.52	0.569923
$612$	892.837	0.0589719
$613$	16845.0	1.10989	0.554947	−	0.831886i	$-0.312739\pi$
$613$	16845.0	1.10989	0.554947	+	0.831886i	$0.312739\pi$
$614$	4858.45	0.319334
$615$	0	0
$616$	405.069	0.0264947
$617$	1679.95	0.109615	0.0548074	−	0.998497i	$-0.482546\pi$
$617$	1679.95	0.109615	0.0548074	+	0.998497i	$0.482546\pi$
$618$	16584.6	1.07950
$619$	3608.29	0.234296	0.117148	−	0.993114i	$-0.462625\pi$
$619$	3608.29	0.234296	0.117148	+	0.993114i	$0.462625\pi$
$620$	0	0
$621$	−24301.0	−1.57032
$622$	−4665.72	−0.300769
$623$	−23997.7	−1.54325
$624$	1402.74	0.0899913
$625$	0	0
$626$	−3745.18	−0.239117
$627$	−418.175	−0.0266353
$628$	−8731.76	−0.554833
$629$	11628.3	0.737124
$630$	0	0
$631$	−2066.53	−0.130376	−0.0651881	−	0.997873i	$-0.520765\pi$
$631$	−2066.53	−0.130376	−0.0651881	+	0.997873i	$0.520765\pi$
$632$	−8886.08	−0.559287
$633$	33876.2	2.12711
$634$	9419.08	0.590031
$635$	0	0
$636$	14959.1	0.932655
$637$	−1038.03	−0.0645655
$638$	62.8349	0.00389915
$639$	4408.51	0.272923
$640$	0	0
$641$	−27937.8	−1.72149	−0.860745	−	0.509037i	$-0.830002\pi$
$641$	−27937.8	−1.72149	−0.860745	+	0.509037i	$0.830002\pi$
$642$	−9255.36	−0.568972
$643$	27067.1	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$643$	27067.1	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$644$	13022.0	0.796798
$645$	0	0
$646$	−2108.49	−0.128417
$647$	−2460.23	−0.149493	−0.0747463	−	0.997203i	$-0.523815\pi$
$647$	−2460.23	−0.149493	−0.0747463	+	0.997203i	$0.523815\pi$
$648$	−6728.79	−0.407920
$649$	−493.985	−0.0298777
$650$	0	0
$651$	−9907.20	−0.596457
$652$	12126.0	0.728359
$653$	7190.60	0.430919	0.215459	−	0.976513i	$-0.430875\pi$
$653$	7190.60	0.430919	0.215459	+	0.976513i	$0.430875\pi$
$654$	2301.06	0.137582
$655$	0	0
$656$	540.697	0.0321809
$657$	1839.95	0.109259
$658$	18485.8	1.09522
$659$	−26124.2	−1.54424	−0.772121	−	0.635475i	$-0.780804\pi$
$659$	−26124.2	−1.54424	−0.772121	+	0.635475i	$0.780804\pi$
$660$	0	0
$661$	13907.5	0.818362	0.409181	−	0.912453i	$-0.365814\pi$
$661$	13907.5	0.818362	0.409181	+	0.912453i	$0.365814\pi$
$662$	14228.3	0.835348
$663$	3818.81	0.223696
$664$	−3815.29	−0.222985
$665$	0	0
$666$	2736.01	0.159186
$667$	2019.99	0.117263
$668$	−8289.45	−0.480132
$669$	−22397.6	−1.29438
$670$	0	0
$671$	−603.229	−0.0347055
$672$	3012.57	0.172935
$673$	16079.5	0.920982	0.460491	−	0.887664i	$-0.347673\pi$
$673$	16079.5	0.920982	0.460491	+	0.887664i	$0.347673\pi$
$674$	4655.18	0.266040
$675$	0	0
$676$	−7830.94	−0.445547
$677$	−15882.2	−0.901626	−0.450813	−	0.892618i	$-0.648866\pi$
$677$	−15882.2	−0.901626	−0.450813	+	0.892618i	$0.648866\pi$
$678$	14796.1	0.838115
$679$	10716.1	0.605666
$680$	0	0
$681$	−12138.3	−0.683027
$682$	−641.597	−0.0360235
$683$	−308.314	−0.0172728	−0.00863640	−	0.999963i	$-0.502749\pi$
$683$	−308.314	−0.0172728	−0.00863640	+	0.999963i	$0.502749\pi$
$684$	−496.105	−0.0277325
$685$	0	0
$686$	−13623.8	−0.758248
$687$	12030.2	0.668093
$688$	−6883.97	−0.381466
$689$	10206.3	0.564339
$690$	0	0
$691$	11388.7	0.626988	0.313494	−	0.949590i	$-0.398501\pi$
$691$	11388.7	0.626988	0.313494	+	0.949590i	$0.398501\pi$
$692$	2951.13	0.162117
$693$	259.466	0.0142227
$694$	−6665.63	−0.364588
$695$	0	0
$696$	467.315	0.0254504
$697$	1471.99	0.0799936
$698$	−5676.89	−0.307841
$699$	−22962.5	−1.24252
$700$	0	0
$701$	30888.8	1.66427	0.832135	−	0.554573i	$-0.187118\pi$
$701$	30888.8	1.66427	0.832135	+	0.554573i	$0.187118\pi$
$702$	−3835.73	−0.206225
$703$	−6461.26	−0.346645
$704$	195.096	0.0104446
$705$	0	0
$706$	21995.4	1.17253
$707$	16462.4	0.875717
$708$	−3673.86	−0.195017
$709$	1748.32	0.0926089	0.0463044	−	0.998927i	$-0.485256\pi$
$709$	1748.32	0.0926089	0.0463044	+	0.998927i	$0.485256\pi$
$710$	0	0
$711$	−5691.96	−0.300232
$712$	−11558.2	−0.608372
$713$	−20625.8	−1.08337
$714$	8201.40	0.429874
$715$	0	0
$716$	−9406.53	−0.490976
$717$	−25207.6	−1.31297
$718$	1991.42	0.103508
$719$	−9493.76	−0.492431	−0.246215	−	0.969215i	$-0.579187\pi$
$719$	−9493.76	−0.492431	−0.246215	+	0.969215i	$0.579187\pi$
$720$	0	0
$721$	24301.3	1.25524
$722$	−12546.4	−0.646716
$723$	3238.47	0.166584
$724$	17309.3	0.888530
$725$	0	0
$726$	−14982.4	−0.765910
$727$	−19255.5	−0.982321	−0.491161	−	0.871069i	$-0.663427\pi$
$727$	−19255.5	−0.982321	−0.491161	+	0.871069i	$0.663427\pi$
$728$	2055.42	0.104641
$729$	14663.9	0.745003
$730$	0	0
$731$	−18740.9	−0.948230
$732$	−4486.32	−0.226529
$733$	−23675.5	−1.19301	−0.596503	−	0.802611i	$-0.703444\pi$
$733$	−23675.5	−1.19301	−0.596503	+	0.802611i	$0.703444\pi$
$734$	18374.6	0.924005
$735$	0	0
$736$	6271.87	0.314109
$737$	473.366	0.0236590
$738$	346.342	0.0172751
$739$	14116.4	0.702680	0.351340	−	0.936248i	$-0.385726\pi$
$739$	14116.4	0.702680	0.351340	+	0.936248i	$0.385726\pi$
$740$	0	0
$741$	−2121.92	−0.105197
$742$	21919.4	1.08448
$743$	−22922.9	−1.13184	−0.565921	−	0.824459i	$-0.691479\pi$
$743$	−22922.9	−1.13184	−0.565921	+	0.824459i	$0.691479\pi$
$744$	−4771.67	−0.235132
$745$	0	0
$746$	12117.9	0.594727
$747$	−2443.87	−0.119701
$748$	531.129	0.0259626
$749$	−13561.8	−0.661596
$750$	0	0
$751$	3611.62	0.175486	0.0877429	−	0.996143i	$-0.472035\pi$
$751$	3611.62	0.175486	0.0877429	+	0.996143i	$0.472035\pi$
$752$	8903.45	0.431749
$753$	−19453.8	−0.941480
$754$	318.839	0.0153998
$755$	0	0
$756$	−8237.74	−0.396301
$757$	−22716.3	−1.09067	−0.545334	−	0.838219i	$-0.683597\pi$
$757$	−22716.3	−1.09067	−0.545334	+	0.838219i	$0.683597\pi$
$758$	1787.37	0.0856466
$759$	3386.37	0.161946
$760$	0	0
$761$	−7913.07	−0.376936	−0.188468	−	0.982079i	$-0.560352\pi$
$761$	−7913.07	−0.376936	−0.188468	+	0.982079i	$0.560352\pi$
$762$	−16477.7	−0.783363
$763$	3371.71	0.159979
$764$	7056.64	0.334163
$765$	0	0
$766$	11678.9	0.550884
$767$	−2506.60	−0.118003
$768$	1450.97	0.0681735
$769$	−6790.99	−0.318452	−0.159226	−	0.987242i	$-0.550900\pi$
$769$	−6790.99	−0.318452	−0.159226	+	0.987242i	$0.550900\pi$
$770$	0	0
$771$	−819.903	−0.0382984
$772$	−19835.1	−0.924714
$773$	6088.83	0.283312	0.141656	−	0.989916i	$-0.454757\pi$
$773$	6088.83	0.283312	0.141656	+	0.989916i	$0.454757\pi$
$774$	−4409.51	−0.204776
$775$	0	0
$776$	5161.28	0.238762
$777$	25132.4	1.16038
$778$	−9825.66	−0.452785
$779$	−817.910	−0.0376183
$780$	0	0
$781$	2622.52	0.120155
$782$	17074.5	0.780796
$783$	−1277.85	−0.0583226
$784$	−1073.72	−0.0489120
$785$	0	0
$786$	3157.52	0.143289
$787$	3296.47	0.149309	0.0746547	−	0.997209i	$-0.476215\pi$
$787$	3296.47	0.149309	0.0746547	+	0.997209i	$0.476215\pi$
$788$	−12532.0	−0.566540
$789$	−30966.0	−1.39724
$790$	0	0
$791$	21680.6	0.974555
$792$	124.969	0.00560677
$793$	−3060.92	−0.137070
$794$	−16838.0	−0.752593
$795$	0	0
$796$	11533.2	0.513549
$797$	40635.4	1.80600	0.902998	−	0.429645i	$-0.141361\pi$
$797$	40635.4	1.80600	0.902998	+	0.429645i	$0.141361\pi$
$798$	−4557.11	−0.202155
$799$	24238.7	1.07322
$800$	0	0
$801$	−7403.57	−0.326582
$802$	−14706.0	−0.647491
$803$	1094.54	0.0481016
$804$	3520.51	0.154426
$805$	0	0
$806$	−3255.62	−0.142276
$807$	15856.4	0.691663
$808$	7928.89	0.345220
$809$	−3564.32	−0.154901	−0.0774505	−	0.996996i	$-0.524678\pi$
$809$	−3564.32	−0.154901	−0.0774505	+	0.996996i	$0.524678\pi$
$810$	0	0
$811$	−41160.2	−1.78216	−0.891079	−	0.453847i	$-0.850051\pi$
$811$	−41160.2	−1.78216	−0.891079	+	0.453847i	$0.850051\pi$
$812$	684.750	0.0295936
$813$	24286.5	1.04768
$814$	1627.59	0.0700823
$815$	0	0
$816$	3950.10	0.169462
$817$	10413.4	0.445921
$818$	−27960.9	−1.19515
$819$	1316.59	0.0561728
$820$	0	0
$821$	−1498.30	−0.0636919	−0.0318460	−	0.999493i	$-0.510139\pi$
$821$	−1498.30	−0.0636919	−0.0318460	+	0.999493i	$0.510139\pi$
$822$	3836.88	0.162806
$823$	4166.11	0.176454	0.0882268	−	0.996100i	$-0.471880\pi$
$823$	4166.11	0.176454	0.0882268	+	0.996100i	$0.471880\pi$
$824$	11704.4	0.494832
$825$	0	0
$826$	−5383.25	−0.226764
$827$	28706.4	1.20704	0.603519	−	0.797349i	$-0.293765\pi$
$827$	28706.4	1.20704	0.603519	+	0.797349i	$0.293765\pi$
$828$	4017.43	0.168618
$829$	12315.4	0.515961	0.257980	−	0.966150i	$-0.416943\pi$
$829$	12315.4	0.515961	0.257980	+	0.966150i	$0.416943\pi$
$830$	0	0
$831$	21169.9	0.883723
$832$	989.965	0.0412510
$833$	−2923.08	−0.121583
$834$	−14681.1	−0.609549
$835$	0	0
$836$	−295.122	−0.0122093
$837$	13047.9	0.538831
$838$	−25556.0	−1.05348
$839$	7143.26	0.293937	0.146968	−	0.989141i	$-0.453048\pi$
$839$	7143.26	0.293937	0.146968	+	0.989141i	$0.453048\pi$
$840$	0	0
$841$	−24282.8	−0.995645
$842$	3417.42	0.139872
$843$	−38042.7	−1.55428
$844$	23907.7	0.975044
$845$	0	0
$846$	5703.08	0.231768
$847$	−21953.6	−0.890595
$848$	10557.2	0.427519
$849$	−3032.55	−0.122587
$850$	0	0
$851$	52323.0	2.10765
$852$	19504.2	0.784275
$853$	25259.1	1.01390	0.506949	−	0.861976i	$-0.330773\pi$
$853$	25259.1	1.01390	0.506949	+	0.861976i	$0.330773\pi$
$854$	−6573.74	−0.263406
$855$	0	0
$856$	−6531.84	−0.260811
$857$	32491.3	1.29508	0.647539	−	0.762032i	$-0.275798\pi$
$857$	32491.3	1.29508	0.647539	+	0.762032i	$0.275798\pi$
$858$	534.511	0.0212680
$859$	19552.4	0.776624	0.388312	−	0.921528i	$-0.373058\pi$
$859$	19552.4	0.776624	0.388312	+	0.921528i	$0.373058\pi$
$860$	0	0
$861$	3181.42	0.125926
$862$	−10098.9	−0.399038
$863$	−27438.7	−1.08230	−0.541150	−	0.840926i	$-0.682011\pi$
$863$	−27438.7	−1.08230	−0.541150	+	0.840926i	$0.682011\pi$
$864$	−3967.60	−0.156227
$865$	0	0
$866$	20876.8	0.819196
$867$	−17092.4	−0.669536
$868$	−6991.87	−0.273410
$869$	−3386.02	−0.132178
$870$	0	0
$871$	2401.97	0.0934416
$872$	1623.94	0.0630661
$873$	3306.05	0.128170
$874$	−9487.43	−0.367182
$875$	0	0
$876$	8140.32	0.313968
$877$	47003.2	1.80979	0.904895	−	0.425635i	$-0.139949\pi$
$877$	47003.2	1.80979	0.904895	+	0.425635i	$0.139949\pi$
$878$	30580.1	1.17543
$879$	−52893.4	−2.02964
$880$	0	0
$881$	−24523.1	−0.937804	−0.468902	−	0.883250i	$-0.655350\pi$
$881$	−24523.1	−0.937804	−0.468902	+	0.883250i	$0.655350\pi$
$882$	−687.767	−0.0262566
$883$	−30961.2	−1.17999	−0.589993	−	0.807408i	$-0.700870\pi$
$883$	−30961.2	−1.17999	−0.589993	+	0.807408i	$0.700870\pi$
$884$	2695.07	0.102540
$885$	0	0
$886$	1733.84	0.0657444
$887$	21566.0	0.816363	0.408182	−	0.912901i	$-0.366163\pi$
$887$	21566.0	0.816363	0.408182	+	0.912901i	$0.366163\pi$
$888$	12104.7	0.457439
$889$	−24144.5	−0.910890
$890$	0	0
$891$	−2563.99	−0.0964051
$892$	−15806.8	−0.593331
$893$	−13468.2	−0.504699
$894$	−12834.6	−0.480147
$895$	0	0
$896$	2126.08	0.0792717
$897$	17183.2	0.639611
$898$	800.924	0.0297630
$899$	−1084.59	−0.0402370
$900$	0	0
$901$	28740.9	1.06270
$902$	206.031	0.00760542
$903$	−40504.8	−1.49271
$904$	10442.2	0.384183
$905$	0	0
$906$	8152.60	0.298954
$907$	40311.8	1.47578	0.737889	−	0.674923i	$-0.235823\pi$
$907$	40311.8	1.47578	0.737889	+	0.674923i	$0.235823\pi$
$908$	−8566.46	−0.313092
$909$	5078.84	0.185318
$910$	0	0
$911$	21257.1	0.773084	0.386542	−	0.922272i	$-0.373669\pi$
$911$	21257.1	0.773084	0.386542	+	0.922272i	$0.373669\pi$
$912$	−2194.87	−0.0796924
$913$	−1453.81	−0.0526988
$914$	−23171.4	−0.838556
$915$	0	0
$916$	8490.14	0.306247
$917$	4626.67	0.166615
$918$	−10801.4	−0.388342
$919$	−5398.84	−0.193788	−0.0968940	−	0.995295i	$-0.530891\pi$
$919$	−5398.84	−0.193788	−0.0968940	+	0.995295i	$0.530891\pi$
$920$	0	0
$921$	13768.5	0.492601
$922$	−11445.0	−0.408810
$923$	13307.3	0.474556
$924$	1147.93	0.0408704
$925$	0	0
$926$	−32272.4	−1.14529
$927$	7497.22	0.265632
$928$	329.801	0.0116662
$929$	28181.3	0.995261	0.497631	−	0.867389i	$-0.334204\pi$
$929$	28181.3	0.995261	0.497631	+	0.867389i	$0.334204\pi$
$930$	0	0
$931$	1624.21	0.0571764
$932$	−16205.5	−0.569557
$933$	−13222.3	−0.463963
$934$	21016.4	0.736272
$935$	0	0
$936$	634.120	0.0221441
$937$	−44657.5	−1.55699	−0.778494	−	0.627652i	$-0.784016\pi$
$937$	−44657.5	−1.55699	−0.778494	+	0.627652i	$0.784016\pi$
$938$	5158.55	0.179566
$939$	−10613.5	−0.368860
$940$	0	0
$941$	30715.7	1.06408	0.532042	−	0.846718i	$-0.321425\pi$
$941$	30715.7	1.06408	0.532042	+	0.846718i	$0.321425\pi$
$942$	−24745.1	−0.855880
$943$	6623.40	0.228725
$944$	−2592.77	−0.0893937
$945$	0	0
$946$	−2623.12	−0.0901533
$947$	7384.38	0.253390	0.126695	−	0.991942i	$-0.459563\pi$
$947$	7384.38	0.253390	0.126695	+	0.991942i	$0.459563\pi$
$948$	−25182.4	−0.862750
$949$	5553.97	0.189979
$950$	0	0
$951$	26692.9	0.910175
$952$	5788.03	0.197049
$953$	45543.4	1.54806	0.774028	−	0.633152i	$-0.218239\pi$
$953$	45543.4	1.54806	0.774028	+	0.633152i	$0.218239\pi$
$954$	6762.40	0.229498
$955$	0	0
$956$	−17790.0	−0.601850
$957$	178.069	0.00601479
$958$	−24810.3	−0.836728
$959$	5622.13	0.189310
$960$	0	0
$961$	−18716.4	−0.628258
$962$	8258.77	0.276792
$963$	−4183.96	−0.140006
$964$	2285.50	0.0763601
$965$	0	0
$966$	36903.3	1.22913
$967$	26437.0	0.879169	0.439584	−	0.898201i	$-0.355126\pi$
$967$	26437.0	0.879169	0.439584	+	0.898201i	$0.355126\pi$
$968$	−10573.7	−0.351085
$969$	−5975.30	−0.198095
$970$	0	0
$971$	47997.0	1.58630	0.793150	−	0.609026i	$-0.208440\pi$
$971$	47997.0	1.58630	0.793150	+	0.609026i	$0.208440\pi$
$972$	−5678.21	−0.187375
$973$	−21512.0	−0.708779
$974$	−13440.1	−0.442145
$975$	0	0
$976$	−3166.16	−0.103838
$977$	8794.09	0.287971	0.143986	−	0.989580i	$-0.454008\pi$
$977$	8794.09	0.287971	0.143986	+	0.989580i	$0.454008\pi$
$978$	34364.1	1.12356
$979$	−4404.22	−0.143779
$980$	0	0
$981$	1040.21	0.0338547
$982$	7099.30	0.230700
$983$	−46163.7	−1.49786	−0.748928	−	0.662651i	$-0.769431\pi$
$983$	−46163.7	−1.49786	−0.748928	+	0.662651i	$0.769431\pi$
$984$	1532.29	0.0496419
$985$	0	0
$986$	897.847	0.0289993
$987$	52387.3	1.68947
$988$	−1497.52	−0.0482210
$989$	−84326.9	−2.71126
$990$	0	0
$991$	−1898.54	−0.0608568	−0.0304284	−	0.999537i	$-0.509687\pi$
$991$	−1898.54	−0.0608568	−0.0304284	+	0.999537i	$0.509687\pi$
$992$	−3367.54	−0.107782
$993$	40322.0	1.28860
$994$	28579.2	0.911949
$995$	0	0
$996$	−10812.2	−0.343974
$997$	−44422.1	−1.41109	−0.705547	−	0.708663i	$-0.749299\pi$
$997$	−44422.1	−1.41109	−0.705547	+	0.708663i	$0.749299\pi$
$998$	31850.7	1.01024
$999$	−33099.7	−1.04827

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1250.4.a.f.1.5	yes	6
5.4	even	2		1250.4.a.c.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.4.a.c.1.2	✓	6	5.4	even	2
1250.4.a.f.1.5	yes	6	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:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1250.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +5.66784 q^{3} +4.00000 q^{4} +11.3357 q^{6} +16.6100 q^{7} +8.00000 q^{8} +5.12438 q^{9} +3.04838 q^{11} +22.6714 q^{12} +15.4682 q^{13} +33.2200 q^{14} +16.0000 q^{16} +43.5583 q^{17} +10.2488 q^{18} -24.2031 q^{19} +94.1429 q^{21} +6.09676 q^{22} +195.996 q^{23} +45.3427 q^{24} +30.9364 q^{26} -123.987 q^{27} +66.4401 q^{28} +10.3063 q^{29} -105.236 q^{31} +32.0000 q^{32} +17.2777 q^{33} +87.1165 q^{34} +20.4975 q^{36} +266.960 q^{37} -48.4063 q^{38} +87.6713 q^{39} +33.7935 q^{41} +188.286 q^{42} -430.248 q^{43} +12.1935 q^{44} +391.992 q^{46} +556.466 q^{47} +90.6854 q^{48} -67.1073 q^{49} +246.881 q^{51} +61.8728 q^{52} +659.826 q^{53} -247.975 q^{54} +132.880 q^{56} -137.179 q^{57} +20.6126 q^{58} -162.048 q^{59} -197.885 q^{61} -210.471 q^{62} +85.1161 q^{63} +64.0000 q^{64} +34.5555 q^{66} +155.284 q^{67} +174.233 q^{68} +1110.87 q^{69} +860.300 q^{71} +40.9951 q^{72} +359.057 q^{73} +533.919 q^{74} -96.8125 q^{76} +50.6337 q^{77} +175.343 q^{78} -1110.76 q^{79} -841.099 q^{81} +67.5871 q^{82} -476.911 q^{83} +376.572 q^{84} -860.496 q^{86} +58.4143 q^{87} +24.3871 q^{88} -1444.77 q^{89} +256.927 q^{91} +783.984 q^{92} -596.459 q^{93} +1112.93 q^{94} +181.371 q^{96} +645.160 q^{97} -134.215 q^{98} +15.6211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 8 q^{3} + 24 q^{4} - 16 q^{6} + 6 q^{7} + 48 q^{8} + 142 q^{9} + 107 q^{11} - 32 q^{12} + 42 q^{13} + 12 q^{14} + 96 q^{16} + 156 q^{17} + 284 q^{18} + 50 q^{19} + 122 q^{21} + 214 q^{22}+ \cdots + 2389 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 8 * q^3 + 24 * q^4 - 16 * q^6 + 6 * q^7 + 48 * q^8 + 142 * q^9 + 107 * q^11 - 32 * q^12 + 42 * q^13 + 12 * q^14 + 96 * q^16 + 156 * q^17 + 284 * q^18 + 50 * q^19 + 122 * q^21 + 214 * q^22 + 2 * q^23 - 64 * q^24 + 84 * q^26 - 680 * q^27 + 24 * q^28 + 100 * q^29 + 742 * q^31 + 192 * q^32 - 601 * q^33 + 312 * q^34 + 568 * q^36 + 71 * q^37 + 100 * q^38 + 259 * q^39 - 513 * q^41 + 244 * q^42 + 92 * q^43 + 428 * q^44 + 4 * q^46 + 311 * q^47 - 128 * q^48 + 348 * q^49 - 1998 * q^51 + 168 * q^52 + 1892 * q^53 - 1360 * q^54 + 48 * q^56 - 1515 * q^57 + 200 * q^58 + 665 * q^59 + 1577 * q^61 + 1484 * q^62 + 747 * q^63 + 384 * q^64 - 1202 * q^66 + 2201 * q^67 + 624 * q^68 + 4044 * q^69 + 1312 * q^71 + 1136 * q^72 - 708 * q^73 + 142 * q^74 + 200 * q^76 - 2318 * q^77 + 518 * q^78 - 1820 * q^79 + 5126 * q^81 - 1026 * q^82 - 1013 * q^83 + 488 * q^84 + 184 * q^86 + 1230 * q^87 + 856 * q^88 - 80 * q^89 + 1402 * q^91 + 8 * q^92 - 396 * q^93 + 622 * q^94 - 256 * q^96 + 996 * q^97 + 696 * q^98 + 2389 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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