LMFDB - Embedded newform 1148.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 $ x^15 - 247x^13 - 6x^12 + 23870x^11 + 940x^10 - 1147074x^9 - 8966x^8 + 28798759x^7 - 3139244x^6 - 363586916x^5 + 114207044x^4 + 2060613491x^3 - 892545712x^2 - 4098298668*x + 1720288256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.19235$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.19235 q^{3} -17.0650 q^{5} -7.00000 q^{7} +11.3451 q^{9} +O(q^{10})$	$q-6.19235 q^{3} -17.0650 q^{5} -7.00000 q^{7} +11.3451 q^{9} -60.1420 q^{11} +24.1008 q^{13} +105.672 q^{15} +86.1202 q^{17} -16.6280 q^{19} +43.3464 q^{21} +98.0749 q^{23} +166.213 q^{25} +96.9403 q^{27} +87.6980 q^{29} -303.369 q^{31} +372.420 q^{33} +119.455 q^{35} +380.112 q^{37} -149.240 q^{39} -41.0000 q^{41} -161.387 q^{43} -193.605 q^{45} +362.597 q^{47} +49.0000 q^{49} -533.286 q^{51} -408.448 q^{53} +1026.32 q^{55} +102.966 q^{57} +470.388 q^{59} -335.050 q^{61} -79.4160 q^{63} -411.279 q^{65} -408.546 q^{67} -607.314 q^{69} -286.576 q^{71} +887.853 q^{73} -1029.25 q^{75} +420.994 q^{77} +895.708 q^{79} -906.607 q^{81} +106.276 q^{83} -1469.64 q^{85} -543.056 q^{87} +393.905 q^{89} -168.705 q^{91} +1878.57 q^{93} +283.756 q^{95} +415.346 q^{97} -682.320 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−6.19235	−1.19172	−0.595859	−	0.803089i	$-0.703188\pi$
$3$	−6.19235	−1.19172	−0.595859	+	0.803089i	$0.703188\pi$
$4$	0	0
$5$	−17.0650	−1.52634	−0.763169	−	0.646199i	$-0.776358\pi$
$5$	−17.0650	−1.52634	−0.763169	+	0.646199i	$0.776358\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	11.3451	0.420191
$10$	0	0
$11$	−60.1420	−1.64850	−0.824249	−	0.566227i	$-0.808403\pi$
$11$	−60.1420	−1.64850	−0.824249	+	0.566227i	$0.808403\pi$
$12$	0	0
$13$	24.1008	0.514181	0.257090	−	0.966387i	$-0.417236\pi$
$13$	24.1008	0.514181	0.257090	+	0.966387i	$0.417236\pi$
$14$	0	0
$15$	105.672	1.81896
$16$	0	0
$17$	86.1202	1.22866	0.614330	−	0.789049i	$-0.289426\pi$
$17$	86.1202	1.22866	0.614330	+	0.789049i	$0.289426\pi$
$18$	0	0
$19$	−16.6280	−0.200775	−0.100387	−	0.994948i	$-0.532008\pi$
$19$	−16.6280	−0.200775	−0.100387	+	0.994948i	$0.532008\pi$
$20$	0	0
$21$	43.3464	0.450427
$22$	0	0
$23$	98.0749	0.889132	0.444566	−	0.895746i	$-0.353358\pi$
$23$	98.0749	0.889132	0.444566	+	0.895746i	$0.353358\pi$
$24$	0	0
$25$	166.213	1.32971
$26$	0	0
$27$	96.9403	0.690969
$28$	0	0
$29$	87.6980	0.561555	0.280778	−	0.959773i	$-0.409408\pi$
$29$	87.6980	0.561555	0.280778	+	0.959773i	$0.409408\pi$
$30$	0	0
$31$	−303.369	−1.75764	−0.878818	−	0.477157i	$-0.841667\pi$
$31$	−303.369	−1.75764	−0.878818	+	0.477157i	$0.841667\pi$
$32$	0	0
$33$	372.420	1.96454
$34$	0	0
$35$	119.455	0.576901
$36$	0	0
$37$	380.112	1.68892	0.844459	−	0.535620i	$-0.179922\pi$
$37$	380.112	1.68892	0.844459	+	0.535620i	$0.179922\pi$
$38$	0	0
$39$	−149.240	−0.612758
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	−161.387	−0.572354	−0.286177	−	0.958177i	$-0.592385\pi$
$43$	−161.387	−0.572354	−0.286177	+	0.958177i	$0.592385\pi$
$44$	0	0
$45$	−193.605	−0.641353
$46$	0	0
$47$	362.597	1.12532	0.562662	−	0.826687i	$-0.309777\pi$
$47$	362.597	1.12532	0.562662	+	0.826687i	$0.309777\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−533.286	−1.46422
$52$	0	0
$53$	−408.448	−1.05858	−0.529289	−	0.848442i	$-0.677541\pi$
$53$	−408.448	−1.05858	−0.529289	+	0.848442i	$0.677541\pi$
$54$	0	0
$55$	1026.32	2.51617
$56$	0	0
$57$	102.966	0.239267
$58$	0	0
$59$	470.388	1.03795	0.518977	−	0.854788i	$-0.326313\pi$
$59$	470.388	1.03795	0.518977	+	0.854788i	$0.326313\pi$
$60$	0	0
$61$	−335.050	−0.703259	−0.351630	−	0.936139i	$-0.614372\pi$
$61$	−335.050	−0.703259	−0.351630	+	0.936139i	$0.614372\pi$
$62$	0	0
$63$	−79.4160	−0.158817
$64$	0	0
$65$	−411.279	−0.784813
$66$	0	0
$67$	−408.546	−0.744952	−0.372476	−	0.928042i	$-0.621491\pi$
$67$	−408.546	−0.744952	−0.372476	+	0.928042i	$0.621491\pi$
$68$	0	0
$69$	−607.314	−1.05959
$70$	0	0
$71$	−286.576	−0.479019	−0.239509	−	0.970894i	$-0.576987\pi$
$71$	−286.576	−0.479019	−0.239509	+	0.970894i	$0.576987\pi$
$72$	0	0
$73$	887.853	1.42350	0.711749	−	0.702434i	$-0.247903\pi$
$73$	887.853	1.42350	0.711749	+	0.702434i	$0.247903\pi$
$74$	0	0
$75$	−1029.25	−1.58464
$76$	0	0
$77$	420.994	0.623074
$78$	0	0
$79$	895.708	1.27563	0.637816	−	0.770188i	$-0.279838\pi$
$79$	895.708	1.27563	0.637816	+	0.770188i	$0.279838\pi$
$80$	0	0
$81$	−906.607	−1.24363
$82$	0	0
$83$	106.276	0.140546	0.0702730	−	0.997528i	$-0.477613\pi$
$83$	106.276	0.140546	0.0702730	+	0.997528i	$0.477613\pi$
$84$	0	0
$85$	−1469.64	−1.87535
$86$	0	0
$87$	−543.056	−0.669216
$88$	0	0
$89$	393.905	0.469144	0.234572	−	0.972099i	$-0.424631\pi$
$89$	393.905	0.469144	0.234572	+	0.972099i	$0.424631\pi$
$90$	0	0
$91$	−168.705	−0.194342
$92$	0	0
$93$	1878.57	2.09461
$94$	0	0
$95$	283.756	0.306450
$96$	0	0
$97$	415.346	0.434763	0.217381	−	0.976087i	$-0.430248\pi$
$97$	415.346	0.434763	0.217381	+	0.976087i	$0.430248\pi$
$98$	0	0
$99$	−682.320	−0.692684
$100$	0	0
$101$	247.183	0.243521	0.121760	−	0.992560i	$-0.461146\pi$
$101$	247.183	0.243521	0.121760	+	0.992560i	$0.461146\pi$
$102$	0	0
$103$	−22.7366	−0.0217505	−0.0108753	−	0.999941i	$-0.503462\pi$
$103$	−22.7366	−0.0217505	−0.0108753	+	0.999941i	$0.503462\pi$
$104$	0	0
$105$	−739.706	−0.687504
$106$	0	0
$107$	1747.38	1.57874	0.789371	−	0.613917i	$-0.210407\pi$
$107$	1747.38	1.57874	0.789371	+	0.613917i	$0.210407\pi$
$108$	0	0
$109$	−1299.63	−1.14203	−0.571017	−	0.820938i	$-0.693451\pi$
$109$	−1299.63	−1.14203	−0.571017	+	0.820938i	$0.693451\pi$
$110$	0	0
$111$	−2353.78	−2.01271
$112$	0	0
$113$	−1577.75	−1.31347	−0.656735	−	0.754122i	$-0.728063\pi$
$113$	−1577.75	−1.31347	−0.656735	+	0.754122i	$0.728063\pi$
$114$	0	0
$115$	−1673.65	−1.35712
$116$	0	0
$117$	273.427	0.216054
$118$	0	0
$119$	−602.842	−0.464390
$120$	0	0
$121$	2286.06	1.71755
$122$	0	0
$123$	253.886	0.186115
$124$	0	0
$125$	−703.306	−0.503245
$126$	0	0
$127$	−1564.44	−1.09309	−0.546543	−	0.837431i	$-0.684056\pi$
$127$	−1564.44	−1.09309	−0.546543	+	0.837431i	$0.684056\pi$
$128$	0	0
$129$	999.362	0.682085
$130$	0	0
$131$	−2348.45	−1.56630	−0.783150	−	0.621833i	$-0.786388\pi$
$131$	−2348.45	−1.56630	−0.783150	+	0.621833i	$0.786388\pi$
$132$	0	0
$133$	116.396	0.0758857
$134$	0	0
$135$	−1654.28	−1.05465
$136$	0	0
$137$	982.837	0.612916	0.306458	−	0.951884i	$-0.400856\pi$
$137$	982.837	0.612916	0.306458	+	0.951884i	$0.400856\pi$
$138$	0	0
$139$	−1621.02	−0.989161	−0.494581	−	0.869132i	$-0.664678\pi$
$139$	−1621.02	−0.989161	−0.494581	+	0.869132i	$0.664678\pi$
$140$	0	0
$141$	−2245.33	−1.34107
$142$	0	0
$143$	−1449.47	−0.847626
$144$	0	0
$145$	−1496.56	−0.857123
$146$	0	0
$147$	−303.425	−0.170245
$148$	0	0
$149$	26.5003	0.0145704	0.00728520	−	0.999973i	$-0.497681\pi$
$149$	26.5003	0.0145704	0.00728520	+	0.999973i	$0.497681\pi$
$150$	0	0
$151$	−2326.16	−1.25364	−0.626822	−	0.779162i	$-0.715645\pi$
$151$	−2326.16	−1.25364	−0.626822	+	0.779162i	$0.715645\pi$
$152$	0	0
$153$	977.047	0.516272
$154$	0	0
$155$	5176.99	2.68275
$156$	0	0
$157$	−1201.21	−0.610619	−0.305309	−	0.952253i	$-0.598760\pi$
$157$	−1201.21	−0.610619	−0.305309	+	0.952253i	$0.598760\pi$
$158$	0	0
$159$	2529.25	1.26153
$160$	0	0
$161$	−686.524	−0.336060
$162$	0	0
$163$	−2250.33	−1.08135	−0.540674	−	0.841232i	$-0.681831\pi$
$163$	−2250.33	−1.08135	−0.540674	+	0.841232i	$0.681831\pi$
$164$	0	0
$165$	−6355.34	−2.99856
$166$	0	0
$167$	3392.56	1.57200	0.786000	−	0.618226i	$-0.212148\pi$
$167$	3392.56	1.57200	0.786000	+	0.618226i	$0.212148\pi$
$168$	0	0
$169$	−1616.15	−0.735618
$170$	0	0
$171$	−188.647	−0.0843636
$172$	0	0
$173$	2963.26	1.30227	0.651133	−	0.758963i	$-0.274294\pi$
$173$	2963.26	1.30227	0.651133	+	0.758963i	$0.274294\pi$
$174$	0	0
$175$	−1163.49	−0.502582
$176$	0	0
$177$	−2912.81	−1.23695
$178$	0	0
$179$	−3900.74	−1.62880	−0.814399	−	0.580306i	$-0.802933\pi$
$179$	−3900.74	−1.62880	−0.814399	+	0.580306i	$0.802933\pi$
$180$	0	0
$181$	−2984.86	−1.22576	−0.612881	−	0.790175i	$-0.709990\pi$
$181$	−2984.86	−1.22576	−0.612881	+	0.790175i	$0.709990\pi$
$182$	0	0
$183$	2074.75	0.838086
$184$	0	0
$185$	−6486.60	−2.57786
$186$	0	0
$187$	−5179.44	−2.02544
$188$	0	0
$189$	−678.582	−0.261162
$190$	0	0
$191$	1447.05	0.548192	0.274096	−	0.961702i	$-0.411621\pi$
$191$	1447.05	0.548192	0.274096	+	0.961702i	$0.411621\pi$
$192$	0	0
$193$	317.372	0.118368	0.0591838	−	0.998247i	$-0.481150\pi$
$193$	317.372	0.118368	0.0591838	+	0.998247i	$0.481150\pi$
$194$	0	0
$195$	2546.78	0.935276
$196$	0	0
$197$	4605.32	1.66556	0.832781	−	0.553603i	$-0.186747\pi$
$197$	4605.32	1.66556	0.832781	+	0.553603i	$0.186747\pi$
$198$	0	0
$199$	4559.12	1.62406	0.812029	−	0.583617i	$-0.198363\pi$
$199$	4559.12	1.62406	0.812029	+	0.583617i	$0.198363\pi$
$200$	0	0
$201$	2529.86	0.887773
$202$	0	0
$203$	−613.886	−0.212248
$204$	0	0
$205$	699.664	0.238374
$206$	0	0
$207$	1112.67	0.373605
$208$	0	0
$209$	1000.04	0.330977
$210$	0	0
$211$	−2121.24	−0.692096	−0.346048	−	0.938217i	$-0.612477\pi$
$211$	−2121.24	−0.692096	−0.346048	+	0.938217i	$0.612477\pi$
$212$	0	0
$213$	1774.58	0.570855
$214$	0	0
$215$	2754.06	0.873606
$216$	0	0
$217$	2123.58	0.664324
$218$	0	0
$219$	−5497.89	−1.69641
$220$	0	0
$221$	2075.56	0.631753
$222$	0	0
$223$	4936.67	1.48244	0.741219	−	0.671263i	$-0.234248\pi$
$223$	4936.67	1.48244	0.741219	+	0.671263i	$0.234248\pi$
$224$	0	0
$225$	1885.72	0.558731
$226$	0	0
$227$	6663.74	1.94840	0.974202	−	0.225676i	$-0.0724590\pi$
$227$	6663.74	1.94840	0.974202	+	0.225676i	$0.0724590\pi$
$228$	0	0
$229$	3651.00	1.05356	0.526779	−	0.850002i	$-0.323400\pi$
$229$	3651.00	1.05356	0.526779	+	0.850002i	$0.323400\pi$
$230$	0	0
$231$	−2606.94	−0.742528
$232$	0	0
$233$	−3798.32	−1.06797	−0.533983	−	0.845495i	$-0.679305\pi$
$233$	−3798.32	−1.06797	−0.533983	+	0.845495i	$0.679305\pi$
$234$	0	0
$235$	−6187.71	−1.71763
$236$	0	0
$237$	−5546.53	−1.52019
$238$	0	0
$239$	−1049.69	−0.284096	−0.142048	−	0.989860i	$-0.545369\pi$
$239$	−1049.69	−0.284096	−0.142048	+	0.989860i	$0.545369\pi$
$240$	0	0
$241$	4378.97	1.17043	0.585217	−	0.810877i	$-0.301009\pi$
$241$	4378.97	1.17043	0.585217	+	0.810877i	$0.301009\pi$
$242$	0	0
$243$	2996.63	0.791087
$244$	0	0
$245$	−836.184	−0.218048
$246$	0	0
$247$	−400.747	−0.103234
$248$	0	0
$249$	−658.098	−0.167491
$250$	0	0
$251$	−6217.89	−1.56362	−0.781812	−	0.623514i	$-0.785704\pi$
$251$	−6217.89	−1.56362	−0.781812	+	0.623514i	$0.785704\pi$
$252$	0	0
$253$	−5898.42	−1.46573
$254$	0	0
$255$	9100.52	2.23489
$256$	0	0
$257$	7939.37	1.92702	0.963510	−	0.267673i	$-0.0862547\pi$
$257$	7939.37	1.92702	0.963510	+	0.267673i	$0.0862547\pi$
$258$	0	0
$259$	−2660.78	−0.638351
$260$	0	0
$261$	994.947	0.235960
$262$	0	0
$263$	−832.809	−0.195259	−0.0976296	−	0.995223i	$-0.531126\pi$
$263$	−832.809	−0.195259	−0.0976296	+	0.995223i	$0.531126\pi$
$264$	0	0
$265$	6970.15	1.61575
$266$	0	0
$267$	−2439.19	−0.559087
$268$	0	0
$269$	−6327.84	−1.43426	−0.717129	−	0.696941i	$-0.754544\pi$
$269$	−6327.84	−1.43426	−0.717129	+	0.696941i	$0.754544\pi$
$270$	0	0
$271$	−2783.29	−0.623885	−0.311942	−	0.950101i	$-0.600980\pi$
$271$	−2783.29	−0.623885	−0.311942	+	0.950101i	$0.600980\pi$
$272$	0	0
$273$	1044.68	0.231601
$274$	0	0
$275$	−9996.40	−2.19202
$276$	0	0
$277$	−4011.72	−0.870184	−0.435092	−	0.900386i	$-0.643284\pi$
$277$	−4011.72	−0.870184	−0.435092	+	0.900386i	$0.643284\pi$
$278$	0	0
$279$	−3441.77	−0.738542
$280$	0	0
$281$	3433.00	0.728811	0.364405	−	0.931240i	$-0.381272\pi$
$281$	3433.00	0.728811	0.364405	+	0.931240i	$0.381272\pi$
$282$	0	0
$283$	−5800.19	−1.21832	−0.609162	−	0.793046i	$-0.708494\pi$
$283$	−5800.19	−1.21832	−0.609162	+	0.793046i	$0.708494\pi$
$284$	0	0
$285$	−1757.11	−0.365202
$286$	0	0
$287$	287.000	0.0590281
$288$	0	0
$289$	2503.69	0.509606
$290$	0	0
$291$	−2571.97	−0.518114
$292$	0	0
$293$	−6221.84	−1.24056	−0.620279	−	0.784381i	$-0.712981\pi$
$293$	−6221.84	−1.24056	−0.620279	+	0.784381i	$0.712981\pi$
$294$	0	0
$295$	−8027.17	−1.58427
$296$	0	0
$297$	−5830.18	−1.13906
$298$	0	0
$299$	2363.68	0.457174
$300$	0	0
$301$	1129.71	0.216330
$302$	0	0
$303$	−1530.64	−0.290208
$304$	0	0
$305$	5717.63	1.07341
$306$	0	0
$307$	−6545.31	−1.21681	−0.608405	−	0.793627i	$-0.708190\pi$
$307$	−6545.31	−1.21681	−0.608405	+	0.793627i	$0.708190\pi$
$308$	0	0
$309$	140.793	0.0259205
$310$	0	0
$311$	1806.01	0.329291	0.164645	−	0.986353i	$-0.447352\pi$
$311$	1806.01	0.329291	0.164645	+	0.986353i	$0.447352\pi$
$312$	0	0
$313$	2609.32	0.471206	0.235603	−	0.971849i	$-0.424293\pi$
$313$	2609.32	0.471206	0.235603	+	0.971849i	$0.424293\pi$
$314$	0	0
$315$	1355.23	0.242409
$316$	0	0
$317$	5343.13	0.946687	0.473344	−	0.880878i	$-0.343047\pi$
$317$	5343.13	0.946687	0.473344	+	0.880878i	$0.343047\pi$
$318$	0	0
$319$	−5274.33	−0.925723
$320$	0	0
$321$	−10820.4	−1.88141
$322$	0	0
$323$	−1432.00	−0.246684
$324$	0	0
$325$	4005.87	0.683710
$326$	0	0
$327$	8047.74	1.36098
$328$	0	0
$329$	−2538.18	−0.425333
$330$	0	0
$331$	−8068.73	−1.33987	−0.669936	−	0.742419i	$-0.733678\pi$
$331$	−8068.73	−1.33987	−0.669936	+	0.742419i	$0.733678\pi$
$332$	0	0
$333$	4312.42	0.709668
$334$	0	0
$335$	6971.82	1.13705
$336$	0	0
$337$	3623.82	0.585763	0.292881	−	0.956149i	$-0.405386\pi$
$337$	3623.82	0.585763	0.292881	+	0.956149i	$0.405386\pi$
$338$	0	0
$339$	9769.96	1.56528
$340$	0	0
$341$	18245.2	2.89746
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	10363.8	1.61730
$346$	0	0
$347$	3678.11	0.569023	0.284512	−	0.958673i	$-0.408169\pi$
$347$	3678.11	0.569023	0.284512	+	0.958673i	$0.408169\pi$
$348$	0	0
$349$	8294.98	1.27226	0.636132	−	0.771580i	$-0.280533\pi$
$349$	8294.98	1.27226	0.636132	+	0.771580i	$0.280533\pi$
$350$	0	0
$351$	2336.33	0.355283
$352$	0	0
$353$	5048.12	0.761145	0.380572	−	0.924751i	$-0.375727\pi$
$353$	5048.12	0.761145	0.380572	+	0.924751i	$0.375727\pi$
$354$	0	0
$355$	4890.41	0.731144
$356$	0	0
$357$	3733.00	0.553422
$358$	0	0
$359$	−3311.32	−0.486811	−0.243405	−	0.969925i	$-0.578264\pi$
$359$	−3311.32	−0.486811	−0.243405	+	0.969925i	$0.578264\pi$
$360$	0	0
$361$	−6582.51	−0.959690
$362$	0	0
$363$	−14156.0	−2.04683
$364$	0	0
$365$	−15151.2	−2.17274
$366$	0	0
$367$	10534.2	1.49831	0.749155	−	0.662395i	$-0.230460\pi$
$367$	10534.2	1.49831	0.749155	+	0.662395i	$0.230460\pi$
$368$	0	0
$369$	−465.151	−0.0656228
$370$	0	0
$371$	2859.13	0.400105
$372$	0	0
$373$	−3825.19	−0.530994	−0.265497	−	0.964112i	$-0.585536\pi$
$373$	−3825.19	−0.530994	−0.265497	+	0.964112i	$0.585536\pi$
$374$	0	0
$375$	4355.11	0.599726
$376$	0	0
$377$	2113.59	0.288741
$378$	0	0
$379$	10972.0	1.48705	0.743525	−	0.668708i	$-0.233152\pi$
$379$	10972.0	1.48705	0.743525	+	0.668708i	$0.233152\pi$
$380$	0	0
$381$	9687.57	1.30265
$382$	0	0
$383$	−13503.3	−1.80154	−0.900768	−	0.434300i	$-0.856996\pi$
$383$	−13503.3	−1.80154	−0.900768	+	0.434300i	$0.856996\pi$
$384$	0	0
$385$	−7184.25	−0.951021
$386$	0	0
$387$	−1830.96	−0.240498
$388$	0	0
$389$	5731.96	0.747100	0.373550	−	0.927610i	$-0.378140\pi$
$389$	5731.96	0.747100	0.373550	+	0.927610i	$0.378140\pi$
$390$	0	0
$391$	8446.23	1.09244
$392$	0	0
$393$	14542.4	1.86659
$394$	0	0
$395$	−15285.2	−1.94705
$396$	0	0
$397$	3097.95	0.391641	0.195821	−	0.980640i	$-0.437263\pi$
$397$	3097.95	0.391641	0.195821	+	0.980640i	$0.437263\pi$
$398$	0	0
$399$	−720.763	−0.0904343
$400$	0	0
$401$	5437.25	0.677116	0.338558	−	0.940946i	$-0.390061\pi$
$401$	5437.25	0.677116	0.338558	+	0.940946i	$0.390061\pi$
$402$	0	0
$403$	−7311.43	−0.903743
$404$	0	0
$405$	15471.2	1.89820
$406$	0	0
$407$	−22860.7	−2.78418
$408$	0	0
$409$	3494.32	0.422453	0.211226	−	0.977437i	$-0.432254\pi$
$409$	3494.32	0.422453	0.211226	+	0.977437i	$0.432254\pi$
$410$	0	0
$411$	−6086.07	−0.730422
$412$	0	0
$413$	−3292.72	−0.392310
$414$	0	0
$415$	−1813.60	−0.214521
$416$	0	0
$417$	10037.9	1.17880
$418$	0	0
$419$	14538.0	1.69505	0.847525	−	0.530755i	$-0.178092\pi$
$419$	14538.0	1.69505	0.847525	+	0.530755i	$0.178092\pi$
$420$	0	0
$421$	−12738.3	−1.47464	−0.737321	−	0.675542i	$-0.763910\pi$
$421$	−12738.3	−1.47464	−0.737321	+	0.675542i	$0.763910\pi$
$422$	0	0
$423$	4113.72	0.472851
$424$	0	0
$425$	14314.3	1.63376
$426$	0	0
$427$	2345.35	0.265807
$428$	0	0
$429$	8975.60	1.01013
$430$	0	0
$431$	−4376.20	−0.489081	−0.244541	−	0.969639i	$-0.578637\pi$
$431$	−4376.20	−0.489081	−0.244541	+	0.969639i	$0.578637\pi$
$432$	0	0
$433$	10805.6	1.19927	0.599634	−	0.800274i	$-0.295313\pi$
$433$	10805.6	1.19927	0.599634	+	0.800274i	$0.295313\pi$
$434$	0	0
$435$	9267.24	1.02145
$436$	0	0
$437$	−1630.79	−0.178515
$438$	0	0
$439$	−217.997	−0.0237003	−0.0118502	−	0.999930i	$-0.503772\pi$
$439$	−217.997	−0.0237003	−0.0118502	+	0.999930i	$0.503772\pi$
$440$	0	0
$441$	555.912	0.0600272
$442$	0	0
$443$	7535.61	0.808189	0.404095	−	0.914717i	$-0.367587\pi$
$443$	7535.61	0.808189	0.404095	+	0.914717i	$0.367587\pi$
$444$	0	0
$445$	−6721.97	−0.716072
$446$	0	0
$447$	−164.099	−0.0173638
$448$	0	0
$449$	6314.61	0.663708	0.331854	−	0.943331i	$-0.392326\pi$
$449$	6314.61	0.663708	0.331854	+	0.943331i	$0.392326\pi$
$450$	0	0
$451$	2465.82	0.257452
$452$	0	0
$453$	14404.4	1.49399
$454$	0	0
$455$	2878.95	0.296632
$456$	0	0
$457$	−1513.76	−0.154946	−0.0774732	−	0.996994i	$-0.524685\pi$
$457$	−1513.76	−0.154946	−0.0774732	+	0.996994i	$0.524685\pi$
$458$	0	0
$459$	8348.52	0.848966
$460$	0	0
$461$	6157.29	0.622069	0.311034	−	0.950399i	$-0.399325\pi$
$461$	6157.29	0.622069	0.311034	+	0.950399i	$0.399325\pi$
$462$	0	0
$463$	−10671.2	−1.07113	−0.535567	−	0.844493i	$-0.679902\pi$
$463$	−10671.2	−1.07113	−0.535567	+	0.844493i	$0.679902\pi$
$464$	0	0
$465$	−32057.7	−3.19708
$466$	0	0
$467$	−14567.2	−1.44345	−0.721723	−	0.692182i	$-0.756649\pi$
$467$	−14567.2	−1.44345	−0.721723	+	0.692182i	$0.756649\pi$
$468$	0	0
$469$	2859.82	0.281566
$470$	0	0
$471$	7438.32	0.727685
$472$	0	0
$473$	9706.11	0.943525
$474$	0	0
$475$	−2763.79	−0.266971
$476$	0	0
$477$	−4633.90	−0.444804
$478$	0	0
$479$	−7994.21	−0.762557	−0.381278	−	0.924460i	$-0.624516\pi$
$479$	−7994.21	−0.762557	−0.381278	+	0.924460i	$0.624516\pi$
$480$	0	0
$481$	9160.98	0.868409
$482$	0	0
$483$	4251.19	0.400489
$484$	0	0
$485$	−7087.87	−0.663595
$486$	0	0
$487$	−14723.8	−1.37002	−0.685009	−	0.728534i	$-0.740202\pi$
$487$	−14723.8	−1.37002	−0.685009	+	0.728534i	$0.740202\pi$
$488$	0	0
$489$	13934.8	1.28866
$490$	0	0
$491$	431.405	0.0396518	0.0198259	−	0.999803i	$-0.493689\pi$
$491$	431.405	0.0396518	0.0198259	+	0.999803i	$0.493689\pi$
$492$	0	0
$493$	7552.57	0.689961
$494$	0	0
$495$	11643.8	1.05727
$496$	0	0
$497$	2006.03	0.181052
$498$	0	0
$499$	10343.3	0.927917	0.463958	−	0.885857i	$-0.346429\pi$
$499$	10343.3	0.927917	0.463958	+	0.885857i	$0.346429\pi$
$500$	0	0
$501$	−21007.9	−1.87338
$502$	0	0
$503$	3949.11	0.350064	0.175032	−	0.984563i	$-0.443997\pi$
$503$	3949.11	0.350064	0.175032	+	0.984563i	$0.443997\pi$
$504$	0	0
$505$	−4218.16	−0.371695
$506$	0	0
$507$	10007.8	0.876649
$508$	0	0
$509$	10152.6	0.884099	0.442050	−	0.896991i	$-0.354251\pi$
$509$	10152.6	0.884099	0.442050	+	0.896991i	$0.354251\pi$
$510$	0	0
$511$	−6214.97	−0.538031
$512$	0	0
$513$	−1611.92	−0.138729
$514$	0	0
$515$	388.000	0.0331987
$516$	0	0
$517$	−21807.3	−1.85510
$518$	0	0
$519$	−18349.5	−1.55193
$520$	0	0
$521$	−11272.9	−0.947937	−0.473968	−	0.880542i	$-0.657179\pi$
$521$	−11272.9	−0.947937	−0.473968	+	0.880542i	$0.657179\pi$
$522$	0	0
$523$	−6350.88	−0.530984	−0.265492	−	0.964113i	$-0.585534\pi$
$523$	−6350.88	−0.530984	−0.265492	+	0.964113i	$0.585534\pi$
$524$	0	0
$525$	7204.76	0.598936
$526$	0	0
$527$	−26126.2	−2.15954
$528$	0	0
$529$	−2548.32	−0.209445
$530$	0	0
$531$	5336.63	0.436139
$532$	0	0
$533$	−988.131	−0.0803015
$534$	0	0
$535$	−29818.9	−2.40969
$536$	0	0
$537$	24154.7	1.94107
$538$	0	0
$539$	−2946.96	−0.235500
$540$	0	0
$541$	3499.13	0.278076	0.139038	−	0.990287i	$-0.455599\pi$
$541$	3499.13	0.278076	0.139038	+	0.990287i	$0.455599\pi$
$542$	0	0
$543$	18483.3	1.46076
$544$	0	0
$545$	22178.1	1.74313
$546$	0	0
$547$	15120.6	1.18192	0.590960	−	0.806701i	$-0.298749\pi$
$547$	15120.6	1.18192	0.590960	+	0.806701i	$0.298749\pi$
$548$	0	0
$549$	−3801.20	−0.295503
$550$	0	0
$551$	−1458.24	−0.112746
$552$	0	0
$553$	−6269.95	−0.482144
$554$	0	0
$555$	40167.3	3.07208
$556$	0	0
$557$	8222.91	0.625522	0.312761	−	0.949832i	$-0.398746\pi$
$557$	8222.91	0.625522	0.312761	+	0.949832i	$0.398746\pi$
$558$	0	0
$559$	−3889.54	−0.294294
$560$	0	0
$561$	32072.9	2.41376
$562$	0	0
$563$	−23992.9	−1.79606	−0.898030	−	0.439935i	$-0.855002\pi$
$563$	−23992.9	−1.79606	−0.898030	+	0.439935i	$0.855002\pi$
$564$	0	0
$565$	26924.2	2.00480
$566$	0	0
$567$	6346.25	0.470048
$568$	0	0
$569$	777.674	0.0572966	0.0286483	−	0.999590i	$-0.490880\pi$
$569$	777.674	0.0572966	0.0286483	+	0.999590i	$0.490880\pi$
$570$	0	0
$571$	−14345.4	−1.05138	−0.525690	−	0.850676i	$-0.676193\pi$
$571$	−14345.4	−1.05138	−0.525690	+	0.850676i	$0.676193\pi$
$572$	0	0
$573$	−8960.61	−0.653289
$574$	0	0
$575$	16301.4	1.18228
$576$	0	0
$577$	841.019	0.0606795	0.0303398	−	0.999540i	$-0.490341\pi$
$577$	841.019	0.0606795	0.0303398	+	0.999540i	$0.490341\pi$
$578$	0	0
$579$	−1965.28	−0.141061
$580$	0	0
$581$	−743.932	−0.0531214
$582$	0	0
$583$	24564.9	1.74506
$584$	0	0
$585$	−4666.02	−0.329771
$586$	0	0
$587$	−19093.1	−1.34252	−0.671259	−	0.741223i	$-0.734246\pi$
$587$	−19093.1	−1.34252	−0.671259	+	0.741223i	$0.734246\pi$
$588$	0	0
$589$	5044.41	0.352889
$590$	0	0
$591$	−28517.8	−1.98488
$592$	0	0
$593$	4196.30	0.290592	0.145296	−	0.989388i	$-0.453586\pi$
$593$	4196.30	0.290592	0.145296	+	0.989388i	$0.453586\pi$
$594$	0	0
$595$	10287.5	0.708816
$596$	0	0
$597$	−28231.7	−1.93542
$598$	0	0
$599$	2091.61	0.142672	0.0713362	−	0.997452i	$-0.477274\pi$
$599$	2091.61	0.142672	0.0713362	+	0.997452i	$0.477274\pi$
$600$	0	0
$601$	5892.10	0.399906	0.199953	−	0.979805i	$-0.435921\pi$
$601$	5892.10	0.399906	0.199953	+	0.979805i	$0.435921\pi$
$602$	0	0
$603$	−4635.01	−0.313022
$604$	0	0
$605$	−39011.5	−2.62156
$606$	0	0
$607$	8429.75	0.563679	0.281839	−	0.959462i	$-0.409055\pi$
$607$	8429.75	0.563679	0.281839	+	0.959462i	$0.409055\pi$
$608$	0	0
$609$	3801.39	0.252940
$610$	0	0
$611$	8738.87	0.578620
$612$	0	0
$613$	−20771.1	−1.36857	−0.684286	−	0.729213i	$-0.739886\pi$
$613$	−20771.1	−1.36857	−0.684286	+	0.729213i	$0.739886\pi$
$614$	0	0
$615$	−4332.56	−0.284074
$616$	0	0
$617$	−559.840	−0.0365288	−0.0182644	−	0.999833i	$-0.505814\pi$
$617$	−559.840	−0.0365288	−0.0182644	+	0.999833i	$0.505814\pi$
$618$	0	0
$619$	−8164.40	−0.530137	−0.265069	−	0.964230i	$-0.585395\pi$
$619$	−8164.40	−0.530137	−0.265069	+	0.964230i	$0.585395\pi$
$620$	0	0
$621$	9507.40	0.614362
$622$	0	0
$623$	−2757.33	−0.177320
$624$	0	0
$625$	−8774.78	−0.561586
$626$	0	0
$627$	−6192.59	−0.394431
$628$	0	0
$629$	32735.3	2.07511
$630$	0	0
$631$	−2049.77	−0.129319	−0.0646593	−	0.997907i	$-0.520596\pi$
$631$	−2049.77	−0.129319	−0.0646593	+	0.997907i	$0.520596\pi$
$632$	0	0
$633$	13135.5	0.824783
$634$	0	0
$635$	26697.2	1.66842
$636$	0	0
$637$	1180.94	0.0734544
$638$	0	0
$639$	−3251.25	−0.201279
$640$	0	0
$641$	4121.34	0.253952	0.126976	−	0.991906i	$-0.459473\pi$
$641$	4121.34	0.253952	0.126976	+	0.991906i	$0.459473\pi$
$642$	0	0
$643$	18288.4	1.12165	0.560827	−	0.827933i	$-0.310483\pi$
$643$	18288.4	1.12165	0.560827	+	0.827933i	$0.310483\pi$
$644$	0	0
$645$	−17054.1	−1.04109
$646$	0	0
$647$	12852.9	0.780989	0.390494	−	0.920605i	$-0.372304\pi$
$647$	12852.9	0.780989	0.390494	+	0.920605i	$0.372304\pi$
$648$	0	0
$649$	−28290.1	−1.71107
$650$	0	0
$651$	−13150.0	−0.791687
$652$	0	0
$653$	895.057	0.0536391	0.0268195	−	0.999640i	$-0.491462\pi$
$653$	895.057	0.0536391	0.0268195	+	0.999640i	$0.491462\pi$
$654$	0	0
$655$	40076.3	2.39070
$656$	0	0
$657$	10072.8	0.598140
$658$	0	0
$659$	11219.7	0.663211	0.331605	−	0.943418i	$-0.392410\pi$
$659$	11219.7	0.663211	0.331605	+	0.943418i	$0.392410\pi$
$660$	0	0
$661$	92.7766	0.00545929	0.00272965	−	0.999996i	$-0.499131\pi$
$661$	92.7766	0.00545929	0.00272965	+	0.999996i	$0.499131\pi$
$662$	0	0
$663$	−12852.6	−0.752872
$664$	0	0
$665$	−1986.29	−0.115827
$666$	0	0
$667$	8600.97	0.499297
$668$	0	0
$669$	−30569.6	−1.76665
$670$	0	0
$671$	20150.6	1.15932
$672$	0	0
$673$	−2231.66	−0.127822	−0.0639110	−	0.997956i	$-0.520357\pi$
$673$	−2231.66	−0.127822	−0.0639110	+	0.997956i	$0.520357\pi$
$674$	0	0
$675$	16112.8	0.918786
$676$	0	0
$677$	−15857.1	−0.900202	−0.450101	−	0.892978i	$-0.648612\pi$
$677$	−15857.1	−0.900202	−0.450101	+	0.892978i	$0.648612\pi$
$678$	0	0
$679$	−2907.42	−0.164325
$680$	0	0
$681$	−41264.2	−2.32195
$682$	0	0
$683$	390.446	0.0218741	0.0109371	−	0.999940i	$-0.496519\pi$
$683$	390.446	0.0218741	0.0109371	+	0.999940i	$0.496519\pi$
$684$	0	0
$685$	−16772.1	−0.935517
$686$	0	0
$687$	−22608.2	−1.25554
$688$	0	0
$689$	−9843.90	−0.544300
$690$	0	0
$691$	−4885.73	−0.268976	−0.134488	−	0.990915i	$-0.542939\pi$
$691$	−4885.73	−0.268976	−0.134488	+	0.990915i	$0.542939\pi$
$692$	0	0
$693$	4776.24	0.261810
$694$	0	0
$695$	27662.7	1.50979
$696$	0	0
$697$	−3530.93	−0.191884
$698$	0	0
$699$	23520.5	1.27271
$700$	0	0
$701$	−9778.16	−0.526842	−0.263421	−	0.964681i	$-0.584851\pi$
$701$	−9778.16	−0.526842	−0.263421	+	0.964681i	$0.584851\pi$
$702$	0	0
$703$	−6320.49	−0.339092
$704$	0	0
$705$	38316.5	2.04692
$706$	0	0
$707$	−1730.28	−0.0920421
$708$	0	0
$709$	−15546.4	−0.823494	−0.411747	−	0.911298i	$-0.635081\pi$
$709$	−15546.4	−0.823494	−0.411747	+	0.911298i	$0.635081\pi$
$710$	0	0
$711$	10161.9	0.536009
$712$	0	0
$713$	−29752.9	−1.56277
$714$	0	0
$715$	24735.1	1.29376
$716$	0	0
$717$	6500.07	0.338563
$718$	0	0
$719$	9553.32	0.495520	0.247760	−	0.968821i	$-0.420306\pi$
$719$	9553.32	0.495520	0.247760	+	0.968821i	$0.420306\pi$
$720$	0	0
$721$	159.156	0.00822093
$722$	0	0
$723$	−27116.1	−1.39483
$724$	0	0
$725$	14576.6	0.746704
$726$	0	0
$727$	−29930.2	−1.52689	−0.763446	−	0.645871i	$-0.776494\pi$
$727$	−29930.2	−1.52689	−0.763446	+	0.645871i	$0.776494\pi$
$728$	0	0
$729$	5922.18	0.300878
$730$	0	0
$731$	−13898.7	−0.703229
$732$	0	0
$733$	−23658.4	−1.19214	−0.596072	−	0.802931i	$-0.703273\pi$
$733$	−23658.4	−1.19214	−0.596072	+	0.802931i	$0.703273\pi$
$734$	0	0
$735$	5177.94	0.259852
$736$	0	0
$737$	24570.7	1.22805
$738$	0	0
$739$	−33053.1	−1.64530	−0.822651	−	0.568547i	$-0.807506\pi$
$739$	−33053.1	−1.64530	−0.822651	+	0.568547i	$0.807506\pi$
$740$	0	0
$741$	2481.56	0.123026
$742$	0	0
$743$	−11440.6	−0.564891	−0.282446	−	0.959283i	$-0.591146\pi$
$743$	−11440.6	−0.564891	−0.282446	+	0.959283i	$0.591146\pi$
$744$	0	0
$745$	−452.227	−0.0222394
$746$	0	0
$747$	1205.72	0.0590561
$748$	0	0
$749$	−12231.6	−0.596708
$750$	0	0
$751$	−2524.71	−0.122674	−0.0613368	−	0.998117i	$-0.519536\pi$
$751$	−2524.71	−0.122674	−0.0613368	+	0.998117i	$0.519536\pi$
$752$	0	0
$753$	38503.3	1.86340
$754$	0	0
$755$	39695.9	1.91349
$756$	0	0
$757$	32527.7	1.56174	0.780870	−	0.624693i	$-0.214776\pi$
$757$	32527.7	1.56174	0.780870	+	0.624693i	$0.214776\pi$
$758$	0	0
$759$	36525.0	1.74674
$760$	0	0
$761$	−24229.7	−1.15417	−0.577087	−	0.816682i	$-0.695811\pi$
$761$	−24229.7	−1.15417	−0.577087	+	0.816682i	$0.695811\pi$
$762$	0	0
$763$	9097.38	0.431648
$764$	0	0
$765$	−16673.3	−0.788005
$766$	0	0
$767$	11336.7	0.533696
$768$	0	0
$769$	4404.73	0.206552	0.103276	−	0.994653i	$-0.467067\pi$
$769$	4404.73	0.206552	0.103276	+	0.994653i	$0.467067\pi$
$770$	0	0
$771$	−49163.3	−2.29646
$772$	0	0
$773$	−7921.00	−0.368562	−0.184281	−	0.982874i	$-0.558996\pi$
$773$	−7921.00	−0.368562	−0.184281	+	0.982874i	$0.558996\pi$
$774$	0	0
$775$	−50424.0	−2.33714
$776$	0	0
$777$	16476.5	0.760734
$778$	0	0
$779$	681.747	0.0313557
$780$	0	0
$781$	17235.3	0.789662
$782$	0	0
$783$	8501.47	0.388017
$784$	0	0
$785$	20498.7	0.932011
$786$	0	0
$787$	−21186.7	−0.959625	−0.479813	−	0.877371i	$-0.659295\pi$
$787$	−21186.7	−0.959625	−0.479813	+	0.877371i	$0.659295\pi$
$788$	0	0
$789$	5157.04	0.232694
$790$	0	0
$791$	11044.2	0.496445
$792$	0	0
$793$	−8074.97	−0.361602
$794$	0	0
$795$	−43161.6	−1.92551
$796$	0	0
$797$	−10871.0	−0.483148	−0.241574	−	0.970382i	$-0.577664\pi$
$797$	−10871.0	−0.483148	−0.241574	+	0.970382i	$0.577664\pi$
$798$	0	0
$799$	31227.0	1.38264
$800$	0	0
$801$	4468.91	0.197130
$802$	0	0
$803$	−53397.2	−2.34663
$804$	0	0
$805$	11715.5	0.512941
$806$	0	0
$807$	39184.2	1.70923
$808$	0	0
$809$	−25632.6	−1.11396	−0.556981	−	0.830525i	$-0.688040\pi$
$809$	−25632.6	−1.11396	−0.556981	+	0.830525i	$0.688040\pi$
$810$	0	0
$811$	23535.1	1.01903	0.509513	−	0.860463i	$-0.329826\pi$
$811$	23535.1	1.01903	0.509513	+	0.860463i	$0.329826\pi$
$812$	0	0
$813$	17235.1	0.743494
$814$	0	0
$815$	38401.9	1.65050
$816$	0	0
$817$	2683.53	0.114914
$818$	0	0
$819$	−1913.99	−0.0816607
$820$	0	0
$821$	−5599.77	−0.238043	−0.119021	−	0.992892i	$-0.537976\pi$
$821$	−5599.77	−0.238043	−0.119021	+	0.992892i	$0.537976\pi$
$822$	0	0
$823$	−9460.66	−0.400702	−0.200351	−	0.979724i	$-0.564208\pi$
$823$	−9460.66	−0.400702	−0.200351	+	0.979724i	$0.564208\pi$
$824$	0	0
$825$	61901.2	2.61227
$826$	0	0
$827$	11792.6	0.495851	0.247926	−	0.968779i	$-0.420251\pi$
$827$	11792.6	0.495851	0.247926	+	0.968779i	$0.420251\pi$
$828$	0	0
$829$	−22853.8	−0.957472	−0.478736	−	0.877959i	$-0.658905\pi$
$829$	−22853.8	−0.957472	−0.478736	+	0.877959i	$0.658905\pi$
$830$	0	0
$831$	24842.0	1.03701
$832$	0	0
$833$	4219.89	0.175523
$834$	0	0
$835$	−57893.9	−2.39940
$836$	0	0
$837$	−29408.7	−1.21447
$838$	0	0
$839$	−9762.97	−0.401735	−0.200867	−	0.979618i	$-0.564376\pi$
$839$	−9762.97	−0.401735	−0.200867	+	0.979618i	$0.564376\pi$
$840$	0	0
$841$	−16698.1	−0.684655
$842$	0	0
$843$	−21258.3	−0.868537
$844$	0	0
$845$	27579.6	1.12280
$846$	0	0
$847$	−16002.4	−0.649172
$848$	0	0
$849$	35916.8	1.45190
$850$	0	0
$851$	37279.4	1.50167
$852$	0	0
$853$	−4450.64	−0.178648	−0.0893242	−	0.996003i	$-0.528471\pi$
$853$	−4450.64	−0.178648	−0.0893242	+	0.996003i	$0.528471\pi$
$854$	0	0
$855$	3219.25	0.128767
$856$	0	0
$857$	−26937.5	−1.07371	−0.536854	−	0.843675i	$-0.680387\pi$
$857$	−26937.5	−1.07371	−0.536854	+	0.843675i	$0.680387\pi$
$858$	0	0
$859$	23535.8	0.934843	0.467421	−	0.884035i	$-0.345183\pi$
$859$	23535.8	0.934843	0.467421	+	0.884035i	$0.345183\pi$
$860$	0	0
$861$	−1777.20	−0.0703449
$862$	0	0
$863$	80.0533	0.00315764	0.00157882	−	0.999999i	$-0.499497\pi$
$863$	80.0533	0.00315764	0.00157882	+	0.999999i	$0.499497\pi$
$864$	0	0
$865$	−50567.9	−1.98770
$866$	0	0
$867$	−15503.7	−0.607306
$868$	0	0
$869$	−53869.6	−2.10288
$870$	0	0
$871$	−9846.27	−0.383040
$872$	0	0
$873$	4712.16	0.182683
$874$	0	0
$875$	4923.14	0.190209
$876$	0	0
$877$	11472.4	0.441729	0.220865	−	0.975304i	$-0.429112\pi$
$877$	11472.4	0.441729	0.220865	+	0.975304i	$0.429112\pi$
$878$	0	0
$879$	38527.8	1.47840
$880$	0	0
$881$	−9543.15	−0.364945	−0.182473	−	0.983211i	$-0.558410\pi$
$881$	−9543.15	−0.364945	−0.182473	+	0.983211i	$0.558410\pi$
$882$	0	0
$883$	−25854.4	−0.985358	−0.492679	−	0.870211i	$-0.663982\pi$
$883$	−25854.4	−0.985358	−0.492679	+	0.870211i	$0.663982\pi$
$884$	0	0
$885$	49707.0	1.88800
$886$	0	0
$887$	−13246.0	−0.501419	−0.250710	−	0.968062i	$-0.580664\pi$
$887$	−13246.0	−0.501419	−0.250710	+	0.968062i	$0.580664\pi$
$888$	0	0
$889$	10951.1	0.413148
$890$	0	0
$891$	54525.1	2.05012
$892$	0	0
$893$	−6029.26	−0.225937
$894$	0	0
$895$	66566.0	2.48610
$896$	0	0
$897$	−14636.7	−0.544823
$898$	0	0
$899$	−26604.9	−0.987010
$900$	0	0
$901$	−35175.6	−1.30063
$902$	0	0
$903$	−6995.54	−0.257804
$904$	0	0
$905$	50936.6	1.87093
$906$	0	0
$907$	3877.57	0.141954	0.0709771	−	0.997478i	$-0.477388\pi$
$907$	3877.57	0.141954	0.0709771	+	0.997478i	$0.477388\pi$
$908$	0	0
$909$	2804.32	0.102325
$910$	0	0
$911$	19792.5	0.719818	0.359909	−	0.932987i	$-0.382808\pi$
$911$	19792.5	0.719818	0.359909	+	0.932987i	$0.382808\pi$
$912$	0	0
$913$	−6391.65	−0.231690
$914$	0	0
$915$	−35405.5	−1.27920
$916$	0	0
$917$	16439.2	0.592005
$918$	0	0
$919$	−33398.8	−1.19883	−0.599415	−	0.800438i	$-0.704600\pi$
$919$	−33398.8	−1.19883	−0.599415	+	0.800438i	$0.704600\pi$
$920$	0	0
$921$	40530.8	1.45009
$922$	0	0
$923$	−6906.70	−0.246302
$924$	0	0
$925$	63179.7	2.24577
$926$	0	0
$927$	−257.950	−0.00913937
$928$	0	0
$929$	36473.0	1.28809	0.644047	−	0.764986i	$-0.277254\pi$
$929$	36473.0	1.28809	0.644047	+	0.764986i	$0.277254\pi$
$930$	0	0
$931$	−814.771	−0.0286821
$932$	0	0
$933$	−11183.4	−0.392422
$934$	0	0
$935$	88387.0	3.09151
$936$	0	0
$937$	−29908.9	−1.04277	−0.521387	−	0.853320i	$-0.674585\pi$
$937$	−29908.9	−1.04277	−0.521387	+	0.853320i	$0.674585\pi$
$938$	0	0
$939$	−16157.8	−0.561545
$940$	0	0
$941$	26334.9	0.912319	0.456159	−	0.889898i	$-0.349225\pi$
$941$	26334.9	0.912319	0.456159	+	0.889898i	$0.349225\pi$
$942$	0	0
$943$	−4021.07	−0.138859
$944$	0	0
$945$	11580.0	0.398621
$946$	0	0
$947$	−38743.9	−1.32947	−0.664734	−	0.747080i	$-0.731455\pi$
$947$	−38743.9	−1.32947	−0.664734	+	0.747080i	$0.731455\pi$
$948$	0	0
$949$	21397.9	0.731935
$950$	0	0
$951$	−33086.5	−1.12818
$952$	0	0
$953$	148.778	0.00505706	0.00252853	−	0.999997i	$-0.499195\pi$
$953$	148.778	0.00505706	0.00252853	+	0.999997i	$0.499195\pi$
$954$	0	0
$955$	−24693.8	−0.836726
$956$	0	0
$957$	32660.5	1.10320
$958$	0	0
$959$	−6879.86	−0.231660
$960$	0	0
$961$	62241.9	2.08928
$962$	0	0
$963$	19824.3	0.663372
$964$	0	0
$965$	−5415.95	−0.180669
$966$	0	0
$967$	−39330.8	−1.30796	−0.653978	−	0.756513i	$-0.726901\pi$
$967$	−39330.8	−1.30796	−0.653978	+	0.756513i	$0.726901\pi$
$968$	0	0
$969$	8867.47	0.293977
$970$	0	0
$971$	−2910.65	−0.0961968	−0.0480984	−	0.998843i	$-0.515316\pi$
$971$	−2910.65	−0.0961968	−0.0480984	+	0.998843i	$0.515316\pi$
$972$	0	0
$973$	11347.2	0.373868
$974$	0	0
$975$	−24805.7	−0.814789
$976$	0	0
$977$	−57474.3	−1.88205	−0.941026	−	0.338333i	$-0.890137\pi$
$977$	−57474.3	−1.88205	−0.941026	+	0.338333i	$0.890137\pi$
$978$	0	0
$979$	−23690.2	−0.773383
$980$	0	0
$981$	−14744.5	−0.479872
$982$	0	0
$983$	−32910.0	−1.06782	−0.533910	−	0.845541i	$-0.679278\pi$
$983$	−32910.0	−1.06782	−0.533910	+	0.845541i	$0.679278\pi$
$984$	0	0
$985$	−78589.7	−2.54221
$986$	0	0
$987$	15717.3	0.506876
$988$	0	0
$989$	−15828.0	−0.508898
$990$	0	0
$991$	11029.6	0.353550	0.176775	−	0.984251i	$-0.443434\pi$
$991$	11029.6	0.353550	0.176775	+	0.984251i	$0.443434\pi$
$992$	0	0
$993$	49964.4	1.59675
$994$	0	0
$995$	−77801.3	−2.47886
$996$	0	0
$997$	−54163.0	−1.72052	−0.860261	−	0.509854i	$-0.829700\pi$
$997$	−54163.0	−1.72052	−0.860261	+	0.509854i	$0.829700\pi$
$998$	0	0
$999$	36848.1	1.16699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.b.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.b.1.3	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-6.19235 q^{3} -17.0650 q^{5} -7.00000 q^{7} +11.3451 q^{9} +O(q^{10})\)	\(q-6.19235 q^{3} -17.0650 q^{5} -7.00000 q^{7} +11.3451 q^{9} -60.1420 q^{11} +24.1008 q^{13} +105.672 q^{15} +86.1202 q^{17} -16.6280 q^{19} +43.3464 q^{21} +98.0749 q^{23} +166.213 q^{25} +96.9403 q^{27} +87.6980 q^{29} -303.369 q^{31} +372.420 q^{33} +119.455 q^{35} +380.112 q^{37} -149.240 q^{39} -41.0000 q^{41} -161.387 q^{43} -193.605 q^{45} +362.597 q^{47} +49.0000 q^{49} -533.286 q^{51} -408.448 q^{53} +1026.32 q^{55} +102.966 q^{57} +470.388 q^{59} -335.050 q^{61} -79.4160 q^{63} -411.279 q^{65} -408.546 q^{67} -607.314 q^{69} -286.576 q^{71} +887.853 q^{73} -1029.25 q^{75} +420.994 q^{77} +895.708 q^{79} -906.607 q^{81} +106.276 q^{83} -1469.64 q^{85} -543.056 q^{87} +393.905 q^{89} -168.705 q^{91} +1878.57 q^{93} +283.756 q^{95} +415.346 q^{97} -682.320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Introduction

L-functions

Modular forms

Varieties

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