LMFDB - Embedded newform 1148.4.a.d.1.13

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:	$0$
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} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 $ x^15 - 3x^14 - 238x^13 + 602x^12 + 21013x^11 - 44923x^10 - 876344x^9 + 1512470x^8 + 18440602x^7 - 23548938x^6 - 192795335x^5 + 157072885x^4 + 930883767x^3 - 339053445x^2 - 1549251522x - 45134496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-5.65179$ 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)

$f(q)$	$=$	$q+6.65179 q^{3} -10.2494 q^{5} -7.00000 q^{7} +17.2463 q^{9} +O(q^{10})$	$q+6.65179 q^{3} -10.2494 q^{5} -7.00000 q^{7} +17.2463 q^{9} +55.0810 q^{11} -34.3146 q^{13} -68.1771 q^{15} -18.8007 q^{17} +18.4580 q^{19} -46.5625 q^{21} +131.854 q^{23} -19.9489 q^{25} -64.8797 q^{27} +54.4063 q^{29} +142.562 q^{31} +366.387 q^{33} +71.7461 q^{35} +296.837 q^{37} -228.253 q^{39} +41.0000 q^{41} -371.274 q^{43} -176.765 q^{45} +94.6413 q^{47} +49.0000 q^{49} -125.058 q^{51} +287.236 q^{53} -564.550 q^{55} +122.779 q^{57} -499.816 q^{59} +294.600 q^{61} -120.724 q^{63} +351.705 q^{65} +1065.34 q^{67} +877.062 q^{69} +754.396 q^{71} -330.173 q^{73} -132.696 q^{75} -385.567 q^{77} +600.533 q^{79} -897.215 q^{81} +1033.77 q^{83} +192.696 q^{85} +361.899 q^{87} +1547.96 q^{89} +240.202 q^{91} +948.293 q^{93} -189.184 q^{95} -1069.02 q^{97} +949.942 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * 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
$3$	6.65179	1.28014	0.640069	−	0.768318i	$-0.278906\pi$
$3$	6.65179	1.28014	0.640069	+	0.768318i	$0.278906\pi$
$4$	0	0
$5$	−10.2494	−0.916738	−0.458369	−	0.888762i	$-0.651566\pi$
$5$	−10.2494	−0.916738	−0.458369	+	0.888762i	$0.651566\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	17.2463	0.638751
$10$	0	0
$11$	55.0810	1.50978	0.754888	−	0.655853i	$-0.227691\pi$
$11$	55.0810	1.50978	0.754888	+	0.655853i	$0.227691\pi$
$12$	0	0
$13$	−34.3146	−0.732088	−0.366044	−	0.930597i	$-0.619288\pi$
$13$	−34.3146	−0.732088	−0.366044	+	0.930597i	$0.619288\pi$
$14$	0	0
$15$	−68.1771	−1.17355
$16$	0	0
$17$	−18.8007	−0.268225	−0.134113	−	0.990966i	$-0.542818\pi$
$17$	−18.8007	−0.268225	−0.134113	+	0.990966i	$0.542818\pi$
$18$	0	0
$19$	18.4580	0.222871	0.111436	−	0.993772i	$-0.464455\pi$
$19$	18.4580	0.222871	0.111436	+	0.993772i	$0.464455\pi$
$20$	0	0
$21$	−46.5625	−0.483846
$22$	0	0
$23$	131.854	1.19536	0.597682	−	0.801733i	$-0.296088\pi$
$23$	131.854	1.19536	0.597682	+	0.801733i	$0.296088\pi$
$24$	0	0
$25$	−19.9489	−0.159591
$26$	0	0
$27$	−64.8797	−0.462448
$28$	0	0
$29$	54.4063	0.348379	0.174189	−	0.984712i	$-0.444269\pi$
$29$	54.4063	0.348379	0.174189	+	0.984712i	$0.444269\pi$
$30$	0	0
$31$	142.562	0.825965	0.412983	−	0.910739i	$-0.364487\pi$
$31$	142.562	0.825965	0.412983	+	0.910739i	$0.364487\pi$
$32$	0	0
$33$	366.387	1.93272
$34$	0	0
$35$	71.7461	0.346494
$36$	0	0
$37$	296.837	1.31891	0.659455	−	0.751744i	$-0.270787\pi$
$37$	296.837	1.31891	0.659455	+	0.751744i	$0.270787\pi$
$38$	0	0
$39$	−228.253	−0.937174
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	−371.274	−1.31672	−0.658358	−	0.752705i	$-0.728749\pi$
$43$	−371.274	−1.31672	−0.658358	+	0.752705i	$0.728749\pi$
$44$	0	0
$45$	−176.765	−0.585567
$46$	0	0
$47$	94.6413	0.293720	0.146860	−	0.989157i	$-0.453083\pi$
$47$	94.6413	0.293720	0.146860	+	0.989157i	$0.453083\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−125.058	−0.343365
$52$	0	0
$53$	287.236	0.744433	0.372217	−	0.928146i	$-0.378598\pi$
$53$	287.236	0.744433	0.372217	+	0.928146i	$0.378598\pi$
$54$	0	0
$55$	−564.550	−1.38407
$56$	0	0
$57$	122.779	0.285306
$58$	0	0
$59$	−499.816	−1.10289	−0.551445	−	0.834211i	$-0.685923\pi$
$59$	−499.816	−1.10289	−0.551445	+	0.834211i	$0.685923\pi$
$60$	0	0
$61$	294.600	0.618355	0.309177	−	0.951004i	$-0.399946\pi$
$61$	294.600	0.618355	0.309177	+	0.951004i	$0.399946\pi$
$62$	0	0
$63$	−120.724	−0.241425
$64$	0	0
$65$	351.705	0.671133
$66$	0	0
$67$	1065.34	1.94256	0.971282	−	0.237932i	$-0.0764694\pi$
$67$	1065.34	1.94256	0.971282	+	0.237932i	$0.0764694\pi$
$68$	0	0
$69$	877.062	1.53023
$70$	0	0
$71$	754.396	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$71$	754.396	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$72$	0	0
$73$	−330.173	−0.529367	−0.264683	−	0.964335i	$-0.585267\pi$
$73$	−330.173	−0.529367	−0.264683	+	0.964335i	$0.585267\pi$
$74$	0	0
$75$	−132.696	−0.204299
$76$	0	0
$77$	−385.567	−0.570642
$78$	0	0
$79$	600.533	0.855257	0.427628	−	0.903955i	$-0.359349\pi$
$79$	600.533	0.855257	0.427628	+	0.903955i	$0.359349\pi$
$80$	0	0
$81$	−897.215	−1.23075
$82$	0	0
$83$	1033.77	1.36713	0.683563	−	0.729891i	$-0.260429\pi$
$83$	1033.77	1.36713	0.683563	+	0.729891i	$0.260429\pi$
$84$	0	0
$85$	192.696	0.245892
$86$	0	0
$87$	361.899	0.445973
$88$	0	0
$89$	1547.96	1.84363	0.921814	−	0.387632i	$-0.126707\pi$
$89$	1547.96	1.84363	0.921814	+	0.387632i	$0.126707\pi$
$90$	0	0
$91$	240.202	0.276703
$92$	0	0
$93$	948.293	1.05735
$94$	0	0
$95$	−189.184	−0.204315
$96$	0	0
$97$	−1069.02	−1.11899	−0.559495	−	0.828834i	$-0.689005\pi$
$97$	−1069.02	−1.11899	−0.559495	+	0.828834i	$0.689005\pi$
$98$	0	0
$99$	949.942	0.964371
$100$	0	0
$101$	507.858	0.500334	0.250167	−	0.968203i	$-0.419515\pi$
$101$	507.858	0.500334	0.250167	+	0.968203i	$0.419515\pi$
$102$	0	0
$103$	−710.889	−0.680058	−0.340029	−	0.940415i	$-0.610437\pi$
$103$	−710.889	−0.680058	−0.340029	+	0.940415i	$0.610437\pi$
$104$	0	0
$105$	477.240	0.443560
$106$	0	0
$107$	471.156	0.425685	0.212843	−	0.977086i	$-0.431728\pi$
$107$	471.156	0.425685	0.212843	+	0.977086i	$0.431728\pi$
$108$	0	0
$109$	1600.56	1.40648	0.703238	−	0.710954i	$-0.251737\pi$
$109$	1600.56	1.40648	0.703238	+	0.710954i	$0.251737\pi$
$110$	0	0
$111$	1974.50	1.68839
$112$	0	0
$113$	800.003	0.665999	0.333000	−	0.942927i	$-0.391939\pi$
$113$	800.003	0.665999	0.333000	+	0.942927i	$0.391939\pi$
$114$	0	0
$115$	−1351.43	−1.09584
$116$	0	0
$117$	−591.798	−0.467622
$118$	0	0
$119$	131.605	0.101380
$120$	0	0
$121$	1702.92	1.27943
$122$	0	0
$123$	272.723	0.199924
$124$	0	0
$125$	1485.65	1.06304
$126$	0	0
$127$	1022.17	0.714195	0.357097	−	0.934067i	$-0.383766\pi$
$127$	1022.17	0.714195	0.357097	+	0.934067i	$0.383766\pi$
$128$	0	0
$129$	−2469.64	−1.68558
$130$	0	0
$131$	−1906.29	−1.27140	−0.635701	−	0.771935i	$-0.719289\pi$
$131$	−1906.29	−1.27140	−0.635701	+	0.771935i	$0.719289\pi$
$132$	0	0
$133$	−129.206	−0.0842375
$134$	0	0
$135$	664.981	0.423944
$136$	0	0
$137$	1177.06	0.734040	0.367020	−	0.930213i	$-0.380378\pi$
$137$	1177.06	0.734040	0.367020	+	0.930213i	$0.380378\pi$
$138$	0	0
$139$	902.667	0.550814	0.275407	−	0.961328i	$-0.411187\pi$
$139$	902.667	0.550814	0.275407	+	0.961328i	$0.411187\pi$
$140$	0	0
$141$	629.534	0.376002
$142$	0	0
$143$	−1890.08	−1.10529
$144$	0	0
$145$	−557.634	−0.319372
$146$	0	0
$147$	325.938	0.182877
$148$	0	0
$149$	1718.40	0.944808	0.472404	−	0.881382i	$-0.343386\pi$
$149$	1718.40	0.944808	0.472404	+	0.881382i	$0.343386\pi$
$150$	0	0
$151$	−2103.47	−1.13363	−0.566815	−	0.823845i	$-0.691825\pi$
$151$	−2103.47	−1.13363	−0.566815	+	0.823845i	$0.691825\pi$
$152$	0	0
$153$	−324.241	−0.171329
$154$	0	0
$155$	−1461.18	−0.757194
$156$	0	0
$157$	111.535	0.0566971	0.0283485	−	0.999598i	$-0.490975\pi$
$157$	111.535	0.0566971	0.0283485	+	0.999598i	$0.490975\pi$
$158$	0	0
$159$	1910.64	0.952976
$160$	0	0
$161$	−922.975	−0.451805
$162$	0	0
$163$	−3704.86	−1.78029	−0.890144	−	0.455679i	$-0.849397\pi$
$163$	−3704.86	−1.78029	−0.890144	+	0.455679i	$0.849397\pi$
$164$	0	0
$165$	−3755.26	−1.77180
$166$	0	0
$167$	−2039.81	−0.945180	−0.472590	−	0.881282i	$-0.656681\pi$
$167$	−2039.81	−0.945180	−0.472590	+	0.881282i	$0.656681\pi$
$168$	0	0
$169$	−1019.51	−0.464047
$170$	0	0
$171$	318.332	0.142359
$172$	0	0
$173$	2060.75	0.905639	0.452820	−	0.891602i	$-0.350418\pi$
$173$	2060.75	0.905639	0.452820	+	0.891602i	$0.350418\pi$
$174$	0	0
$175$	139.642	0.0603198
$176$	0	0
$177$	−3324.67	−1.41185
$178$	0	0
$179$	−2558.99	−1.06854	−0.534268	−	0.845315i	$-0.679413\pi$
$179$	−2558.99	−1.06854	−0.534268	+	0.845315i	$0.679413\pi$
$180$	0	0
$181$	−3451.12	−1.41724	−0.708619	−	0.705591i	$-0.750681\pi$
$181$	−3451.12	−1.41724	−0.708619	+	0.705591i	$0.750681\pi$
$182$	0	0
$183$	1959.62	0.791579
$184$	0	0
$185$	−3042.41	−1.20910
$186$	0	0
$187$	−1035.56	−0.404960
$188$	0	0
$189$	454.158	0.174789
$190$	0	0
$191$	−1964.63	−0.744271	−0.372136	−	0.928178i	$-0.621374\pi$
$191$	−1964.63	−0.744271	−0.372136	+	0.928178i	$0.621374\pi$
$192$	0	0
$193$	5218.28	1.94622	0.973109	−	0.230347i	$-0.0739860\pi$
$193$	5218.28	1.94622	0.973109	+	0.230347i	$0.0739860\pi$
$194$	0	0
$195$	2339.47	0.859143
$196$	0	0
$197$	4543.71	1.64328	0.821639	−	0.570008i	$-0.193060\pi$
$197$	4543.71	1.64328	0.821639	+	0.570008i	$0.193060\pi$
$198$	0	0
$199$	2870.58	1.02256	0.511281	−	0.859413i	$-0.329171\pi$
$199$	2870.58	1.02256	0.511281	+	0.859413i	$0.329171\pi$
$200$	0	0
$201$	7086.40	2.48675
$202$	0	0
$203$	−380.844	−0.131675
$204$	0	0
$205$	−420.227	−0.143170
$206$	0	0
$207$	2273.98	0.763540
$208$	0	0
$209$	1016.69	0.336486
$210$	0	0
$211$	−3195.94	−1.04274	−0.521369	−	0.853331i	$-0.674579\pi$
$211$	−3195.94	−1.04274	−0.521369	+	0.853331i	$0.674579\pi$
$212$	0	0
$213$	5018.08	1.61424
$214$	0	0
$215$	3805.35	1.20708
$216$	0	0
$217$	−997.935	−0.312185
$218$	0	0
$219$	−2196.24	−0.677662
$220$	0	0
$221$	645.137	0.196365
$222$	0	0
$223$	2052.95	0.616483	0.308241	−	0.951308i	$-0.400260\pi$
$223$	2052.95	0.616483	0.308241	+	0.951308i	$0.400260\pi$
$224$	0	0
$225$	−344.044	−0.101939
$226$	0	0
$227$	−2223.60	−0.650158	−0.325079	−	0.945687i	$-0.605391\pi$
$227$	−2223.60	−0.650158	−0.325079	+	0.945687i	$0.605391\pi$
$228$	0	0
$229$	−2010.68	−0.580216	−0.290108	−	0.956994i	$-0.593691\pi$
$229$	−2010.68	−0.580216	−0.290108	+	0.956994i	$0.593691\pi$
$230$	0	0
$231$	−2564.71	−0.730500
$232$	0	0
$233$	−3041.76	−0.855245	−0.427623	−	0.903957i	$-0.640649\pi$
$233$	−3041.76	−0.855245	−0.427623	+	0.903957i	$0.640649\pi$
$234$	0	0
$235$	−970.021	−0.269265
$236$	0	0
$237$	3994.62	1.09485
$238$	0	0
$239$	2867.22	0.776004	0.388002	−	0.921659i	$-0.373165\pi$
$239$	2867.22	0.776004	0.388002	+	0.921659i	$0.373165\pi$
$240$	0	0
$241$	−3144.70	−0.840530	−0.420265	−	0.907401i	$-0.638063\pi$
$241$	−3144.70	−0.840530	−0.420265	+	0.907401i	$0.638063\pi$
$242$	0	0
$243$	−4216.33	−1.11308
$244$	0	0
$245$	−502.223	−0.130963
$246$	0	0
$247$	−633.378	−0.163162
$248$	0	0
$249$	6876.45	1.75011
$250$	0	0
$251$	−1716.45	−0.431639	−0.215819	−	0.976433i	$-0.569242\pi$
$251$	−1716.45	−0.431639	−0.215819	+	0.976433i	$0.569242\pi$
$252$	0	0
$253$	7262.63	1.80473
$254$	0	0
$255$	1281.78	0.314776
$256$	0	0
$257$	−3406.40	−0.826792	−0.413396	−	0.910551i	$-0.635657\pi$
$257$	−3406.40	−0.826792	−0.413396	+	0.910551i	$0.635657\pi$
$258$	0	0
$259$	−2077.86	−0.498501
$260$	0	0
$261$	938.305	0.222527
$262$	0	0
$263$	3615.24	0.847623	0.423812	−	0.905750i	$-0.360692\pi$
$263$	3615.24	0.847623	0.423812	+	0.905750i	$0.360692\pi$
$264$	0	0
$265$	−2944.01	−0.682450
$266$	0	0
$267$	10296.7	2.36010
$268$	0	0
$269$	−4149.40	−0.940495	−0.470247	−	0.882535i	$-0.655835\pi$
$269$	−4149.40	−0.940495	−0.470247	+	0.882535i	$0.655835\pi$
$270$	0	0
$271$	6169.28	1.38287	0.691433	−	0.722440i	$-0.256980\pi$
$271$	6169.28	1.38287	0.691433	+	0.722440i	$0.256980\pi$
$272$	0	0
$273$	1597.77	0.354218
$274$	0	0
$275$	−1098.81	−0.240947
$276$	0	0
$277$	−3158.12	−0.685029	−0.342514	−	0.939513i	$-0.611279\pi$
$277$	−3158.12	−0.685029	−0.342514	+	0.939513i	$0.611279\pi$
$278$	0	0
$279$	2458.67	0.527586
$280$	0	0
$281$	3400.58	0.721928	0.360964	−	0.932580i	$-0.382448\pi$
$281$	3400.58	0.721928	0.360964	+	0.932580i	$0.382448\pi$
$282$	0	0
$283$	8767.64	1.84163	0.920816	−	0.389997i	$-0.127524\pi$
$283$	8767.64	1.84163	0.920816	+	0.389997i	$0.127524\pi$
$284$	0	0
$285$	−1258.41	−0.261551
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	−4559.53	−0.928055
$290$	0	0
$291$	−7110.86	−1.43246
$292$	0	0
$293$	662.243	0.132043	0.0660216	−	0.997818i	$-0.478969\pi$
$293$	662.243	0.132043	0.0660216	+	0.997818i	$0.478969\pi$
$294$	0	0
$295$	5122.83	1.01106
$296$	0	0
$297$	−3573.64	−0.698194
$298$	0	0
$299$	−4524.50	−0.875112
$300$	0	0
$301$	2598.92	0.497672
$302$	0	0
$303$	3378.16	0.640496
$304$	0	0
$305$	−3019.48	−0.566869
$306$	0	0
$307$	−1404.52	−0.261109	−0.130554	−	0.991441i	$-0.541676\pi$
$307$	−1404.52	−0.261109	−0.130554	+	0.991441i	$0.541676\pi$
$308$	0	0
$309$	−4728.68	−0.870568
$310$	0	0
$311$	−4282.78	−0.780882	−0.390441	−	0.920628i	$-0.627677\pi$
$311$	−4282.78	−0.780882	−0.390441	+	0.920628i	$0.627677\pi$
$312$	0	0
$313$	10862.3	1.96157	0.980786	−	0.195087i	$-0.0624988\pi$
$313$	10862.3	1.96157	0.980786	+	0.195087i	$0.0624988\pi$
$314$	0	0
$315$	1237.35	0.221324
$316$	0	0
$317$	−2549.11	−0.451647	−0.225824	−	0.974168i	$-0.572507\pi$
$317$	−2549.11	−0.451647	−0.225824	+	0.974168i	$0.572507\pi$
$318$	0	0
$319$	2996.75	0.525974
$320$	0	0
$321$	3134.03	0.544936
$322$	0	0
$323$	−347.023	−0.0597798
$324$	0	0
$325$	684.538	0.116835
$326$	0	0
$327$	10646.6	1.80048
$328$	0	0
$329$	−662.489	−0.111016
$330$	0	0
$331$	−4651.13	−0.772354	−0.386177	−	0.922425i	$-0.626205\pi$
$331$	−4651.13	−0.772354	−0.386177	+	0.922425i	$0.626205\pi$
$332$	0	0
$333$	5119.33	0.842455
$334$	0	0
$335$	−10919.1	−1.78082
$336$	0	0
$337$	3698.97	0.597910	0.298955	−	0.954267i	$-0.403362\pi$
$337$	3698.97	0.597910	0.298955	+	0.954267i	$0.403362\pi$
$338$	0	0
$339$	5321.45	0.852571
$340$	0	0
$341$	7852.47	1.24702
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−8989.40	−1.40282
$346$	0	0
$347$	−4331.64	−0.670129	−0.335065	−	0.942195i	$-0.608758\pi$
$347$	−4331.64	−0.670129	−0.335065	+	0.942195i	$0.608758\pi$
$348$	0	0
$349$	2654.80	0.407187	0.203593	−	0.979056i	$-0.434738\pi$
$349$	2654.80	0.407187	0.203593	+	0.979056i	$0.434738\pi$
$350$	0	0
$351$	2226.32	0.338553
$352$	0	0
$353$	−7848.65	−1.18340	−0.591702	−	0.806157i	$-0.701544\pi$
$353$	−7848.65	−1.18340	−0.591702	+	0.806157i	$0.701544\pi$
$354$	0	0
$355$	−7732.14	−1.15600
$356$	0	0
$357$	875.406	0.129780
$358$	0	0
$359$	−5217.45	−0.767038	−0.383519	−	0.923533i	$-0.625288\pi$
$359$	−5217.45	−0.767038	−0.383519	+	0.923533i	$0.625288\pi$
$360$	0	0
$361$	−6518.30	−0.950328
$362$	0	0
$363$	11327.4	1.63784
$364$	0	0
$365$	3384.09	0.485291
$366$	0	0
$367$	4378.44	0.622760	0.311380	−	0.950286i	$-0.399209\pi$
$367$	4378.44	0.622760	0.311380	+	0.950286i	$0.399209\pi$
$368$	0	0
$369$	707.097	0.0997561
$370$	0	0
$371$	−2010.65	−0.281369
$372$	0	0
$373$	3426.00	0.475581	0.237790	−	0.971317i	$-0.423577\pi$
$373$	3426.00	0.475581	0.237790	+	0.971317i	$0.423577\pi$
$374$	0	0
$375$	9882.20	1.36084
$376$	0	0
$377$	−1866.93	−0.255044
$378$	0	0
$379$	−24.7312	−0.00335186	−0.00167593	−	0.999999i	$-0.500533\pi$
$379$	−24.7312	−0.00335186	−0.00167593	+	0.999999i	$0.500533\pi$
$380$	0	0
$381$	6799.25	0.914267
$382$	0	0
$383$	−9221.32	−1.23025	−0.615127	−	0.788428i	$-0.710895\pi$
$383$	−9221.32	−1.23025	−0.615127	+	0.788428i	$0.710895\pi$
$384$	0	0
$385$	3951.85	0.523129
$386$	0	0
$387$	−6403.09	−0.841053
$388$	0	0
$389$	8168.51	1.06468	0.532339	−	0.846531i	$-0.321313\pi$
$389$	8168.51	1.06468	0.532339	+	0.846531i	$0.321313\pi$
$390$	0	0
$391$	−2478.94	−0.320627
$392$	0	0
$393$	−12680.3	−1.62757
$394$	0	0
$395$	−6155.13	−0.784046
$396$	0	0
$397$	4089.85	0.517037	0.258519	−	0.966006i	$-0.416766\pi$
$397$	4089.85	0.517037	0.258519	+	0.966006i	$0.416766\pi$
$398$	0	0
$399$	−859.451	−0.107836
$400$	0	0
$401$	4604.75	0.573442	0.286721	−	0.958014i	$-0.407435\pi$
$401$	4604.75	0.573442	0.286721	+	0.958014i	$0.407435\pi$
$402$	0	0
$403$	−4891.96	−0.604680
$404$	0	0
$405$	9195.96	1.12827
$406$	0	0
$407$	16350.1	1.99126
$408$	0	0
$409$	−3151.27	−0.380978	−0.190489	−	0.981689i	$-0.561007\pi$
$409$	−3151.27	−0.380978	−0.190489	+	0.981689i	$0.561007\pi$
$410$	0	0
$411$	7829.58	0.939671
$412$	0	0
$413$	3498.71	0.416853
$414$	0	0
$415$	−10595.6	−1.25330
$416$	0	0
$417$	6004.35	0.705118
$418$	0	0
$419$	−1014.43	−0.118278	−0.0591388	−	0.998250i	$-0.518835\pi$
$419$	−1014.43	−0.118278	−0.0591388	+	0.998250i	$0.518835\pi$
$420$	0	0
$421$	−12920.1	−1.49569	−0.747847	−	0.663871i	$-0.768912\pi$
$421$	−12920.1	−1.49569	−0.747847	+	0.663871i	$0.768912\pi$
$422$	0	0
$423$	1632.21	0.187614
$424$	0	0
$425$	375.053	0.0428064
$426$	0	0
$427$	−2062.20	−0.233716
$428$	0	0
$429$	−12572.4	−1.41492
$430$	0	0
$431$	887.475	0.0991837	0.0495919	−	0.998770i	$-0.484208\pi$
$431$	887.475	0.0991837	0.0495919	+	0.998770i	$0.484208\pi$
$432$	0	0
$433$	11117.5	1.23388	0.616942	−	0.787009i	$-0.288371\pi$
$433$	11117.5	1.23388	0.616942	+	0.787009i	$0.288371\pi$
$434$	0	0
$435$	−3709.26	−0.408840
$436$	0	0
$437$	2433.75	0.266413
$438$	0	0
$439$	12310.2	1.33835	0.669173	−	0.743106i	$-0.266648\pi$
$439$	12310.2	1.33835	0.669173	+	0.743106i	$0.266648\pi$
$440$	0	0
$441$	845.067	0.0912501
$442$	0	0
$443$	−14930.9	−1.60133	−0.800666	−	0.599111i	$-0.795521\pi$
$443$	−14930.9	−1.60133	−0.800666	+	0.599111i	$0.795521\pi$
$444$	0	0
$445$	−15865.7	−1.69012
$446$	0	0
$447$	11430.4	1.20948
$448$	0	0
$449$	−13428.3	−1.41141	−0.705705	−	0.708506i	$-0.749369\pi$
$449$	−13428.3	−1.41141	−0.705705	+	0.708506i	$0.749369\pi$
$450$	0	0
$451$	2258.32	0.235788
$452$	0	0
$453$	−13991.8	−1.45120
$454$	0	0
$455$	−2461.94	−0.253665
$456$	0	0
$457$	8202.49	0.839598	0.419799	−	0.907617i	$-0.362101\pi$
$457$	8202.49	0.839598	0.419799	+	0.907617i	$0.362101\pi$
$458$	0	0
$459$	1219.78	0.124040
$460$	0	0
$461$	−15440.9	−1.55999	−0.779996	−	0.625785i	$-0.784779\pi$
$461$	−15440.9	−1.55999	−0.779996	+	0.625785i	$0.784779\pi$
$462$	0	0
$463$	15325.6	1.53832	0.769160	−	0.639056i	$-0.220675\pi$
$463$	15325.6	1.53832	0.769160	+	0.639056i	$0.220675\pi$
$464$	0	0
$465$	−9719.48	−0.969312
$466$	0	0
$467$	15060.2	1.49230	0.746148	−	0.665780i	$-0.231901\pi$
$467$	15060.2	1.49230	0.746148	+	0.665780i	$0.231901\pi$
$468$	0	0
$469$	−7457.37	−0.734220
$470$	0	0
$471$	741.905	0.0725800
$472$	0	0
$473$	−20450.1	−1.98795
$474$	0	0
$475$	−368.217	−0.0355683
$476$	0	0
$477$	4953.76	0.475507
$478$	0	0
$479$	−1643.70	−0.156790	−0.0783950	−	0.996922i	$-0.524980\pi$
$479$	−1643.70	−0.156790	−0.0783950	+	0.996922i	$0.524980\pi$
$480$	0	0
$481$	−10185.8	−0.965559
$482$	0	0
$483$	−6139.44	−0.578373
$484$	0	0
$485$	10956.8	1.02582
$486$	0	0
$487$	−19860.0	−1.84793	−0.923967	−	0.382472i	$-0.875073\pi$
$487$	−19860.0	−1.84793	−0.923967	+	0.382472i	$0.875073\pi$
$488$	0	0
$489$	−24643.9	−2.27901
$490$	0	0
$491$	−2785.48	−0.256022	−0.128011	−	0.991773i	$-0.540859\pi$
$491$	−2785.48	−0.256022	−0.128011	+	0.991773i	$0.540859\pi$
$492$	0	0
$493$	−1022.87	−0.0934441
$494$	0	0
$495$	−9736.38	−0.884076
$496$	0	0
$497$	−5280.77	−0.476610
$498$	0	0
$499$	8248.00	0.739942	0.369971	−	0.929043i	$-0.379368\pi$
$499$	8248.00	0.739942	0.369971	+	0.929043i	$0.379368\pi$
$500$	0	0
$501$	−13568.4	−1.20996
$502$	0	0
$503$	−16133.8	−1.43016	−0.715079	−	0.699044i	$-0.753609\pi$
$503$	−16133.8	−1.43016	−0.715079	+	0.699044i	$0.753609\pi$
$504$	0	0
$505$	−5205.26	−0.458675
$506$	0	0
$507$	−6781.57	−0.594043
$508$	0	0
$509$	−5005.95	−0.435923	−0.217961	−	0.975957i	$-0.569941\pi$
$509$	−5005.95	−0.435923	−0.217961	+	0.975957i	$0.569941\pi$
$510$	0	0
$511$	2311.21	0.200082
$512$	0	0
$513$	−1197.55	−0.103067
$514$	0	0
$515$	7286.22	0.623435
$516$	0	0
$517$	5212.94	0.443452
$518$	0	0
$519$	13707.6	1.15934
$520$	0	0
$521$	−10133.1	−0.852090	−0.426045	−	0.904702i	$-0.640094\pi$
$521$	−10133.1	−0.852090	−0.426045	+	0.904702i	$0.640094\pi$
$522$	0	0
$523$	6655.47	0.556450	0.278225	−	0.960516i	$-0.410254\pi$
$523$	6655.47	0.556450	0.278225	+	0.960516i	$0.410254\pi$
$524$	0	0
$525$	928.871	0.0772176
$526$	0	0
$527$	−2680.26	−0.221545
$528$	0	0
$529$	5218.38	0.428896
$530$	0	0
$531$	−8619.96	−0.704472
$532$	0	0
$533$	−1406.90	−0.114333
$534$	0	0
$535$	−4829.08	−0.390242
$536$	0	0
$537$	−17021.9	−1.36787
$538$	0	0
$539$	2698.97	0.215682
$540$	0	0
$541$	−4300.12	−0.341732	−0.170866	−	0.985294i	$-0.554656\pi$
$541$	−4300.12	−0.341732	−0.170866	+	0.985294i	$0.554656\pi$
$542$	0	0
$543$	−22956.1	−1.81426
$544$	0	0
$545$	−16404.9	−1.28937
$546$	0	0
$547$	10607.7	0.829166	0.414583	−	0.910011i	$-0.363927\pi$
$547$	10607.7	0.829166	0.414583	+	0.910011i	$0.363927\pi$
$548$	0	0
$549$	5080.75	0.394975
$550$	0	0
$551$	1004.23	0.0776437
$552$	0	0
$553$	−4203.73	−0.323257
$554$	0	0
$555$	−20237.5	−1.54781
$556$	0	0
$557$	−20024.6	−1.52329	−0.761644	−	0.647996i	$-0.775607\pi$
$557$	−20024.6	−1.52329	−0.761644	+	0.647996i	$0.775607\pi$
$558$	0	0
$559$	12740.1	0.963952
$560$	0	0
$561$	−6888.32	−0.518405
$562$	0	0
$563$	16796.9	1.25738	0.628690	−	0.777656i	$-0.283591\pi$
$563$	16796.9	1.25738	0.628690	+	0.777656i	$0.283591\pi$
$564$	0	0
$565$	−8199.58	−0.610547
$566$	0	0
$567$	6280.51	0.465179
$568$	0	0
$569$	−9672.53	−0.712642	−0.356321	−	0.934364i	$-0.615969\pi$
$569$	−9672.53	−0.712642	−0.356321	+	0.934364i	$0.615969\pi$
$570$	0	0
$571$	−17438.8	−1.27810	−0.639048	−	0.769167i	$-0.720671\pi$
$571$	−17438.8	−1.27810	−0.639048	+	0.769167i	$0.720671\pi$
$572$	0	0
$573$	−13068.3	−0.952770
$574$	0	0
$575$	−2630.34	−0.190770
$576$	0	0
$577$	2820.14	0.203473	0.101736	−	0.994811i	$-0.467560\pi$
$577$	2820.14	0.203473	0.101736	+	0.994811i	$0.467560\pi$
$578$	0	0
$579$	34710.9	2.49142
$580$	0	0
$581$	−7236.42	−0.516725
$582$	0	0
$583$	15821.3	1.12393
$584$	0	0
$585$	6065.61	0.428687
$586$	0	0
$587$	−4130.84	−0.290456	−0.145228	−	0.989398i	$-0.546392\pi$
$587$	−4130.84	−0.290456	−0.145228	+	0.989398i	$0.546392\pi$
$588$	0	0
$589$	2631.41	0.184084
$590$	0	0
$591$	30223.8	2.10362
$592$	0	0
$593$	19525.0	1.35210	0.676052	−	0.736854i	$-0.263690\pi$
$593$	19525.0	1.35210	0.676052	+	0.736854i	$0.263690\pi$
$594$	0	0
$595$	−1348.87	−0.0929386
$596$	0	0
$597$	19094.5	1.30902
$598$	0	0
$599$	−11085.7	−0.756179	−0.378089	−	0.925769i	$-0.623419\pi$
$599$	−11085.7	−0.756179	−0.378089	+	0.925769i	$0.623419\pi$
$600$	0	0
$601$	−2248.79	−0.152629	−0.0763145	−	0.997084i	$-0.524315\pi$
$601$	−2248.79	−0.152629	−0.0763145	+	0.997084i	$0.524315\pi$
$602$	0	0
$603$	18373.1	1.24081
$604$	0	0
$605$	−17453.9	−1.17290
$606$	0	0
$607$	−17537.2	−1.17267	−0.586337	−	0.810067i	$-0.699431\pi$
$607$	−17537.2	−1.17267	−0.586337	+	0.810067i	$0.699431\pi$
$608$	0	0
$609$	−2533.29	−0.168562
$610$	0	0
$611$	−3247.58	−0.215029
$612$	0	0
$613$	−6465.66	−0.426013	−0.213006	−	0.977051i	$-0.568325\pi$
$613$	−6465.66	−0.426013	−0.213006	+	0.977051i	$0.568325\pi$
$614$	0	0
$615$	−2795.26	−0.183278
$616$	0	0
$617$	1725.07	0.112559	0.0562793	−	0.998415i	$-0.482076\pi$
$617$	1725.07	0.112559	0.0562793	+	0.998415i	$0.482076\pi$
$618$	0	0
$619$	1964.66	0.127571	0.0637853	−	0.997964i	$-0.479683\pi$
$619$	1964.66	0.127571	0.0637853	+	0.997964i	$0.479683\pi$
$620$	0	0
$621$	−8554.62	−0.552794
$622$	0	0
$623$	−10835.7	−0.696826
$624$	0	0
$625$	−12733.4	−0.814939
$626$	0	0
$627$	6762.77	0.430748
$628$	0	0
$629$	−5580.73	−0.353765
$630$	0	0
$631$	−1702.78	−0.107427	−0.0537135	−	0.998556i	$-0.517106\pi$
$631$	−1702.78	−0.107427	−0.0537135	+	0.998556i	$0.517106\pi$
$632$	0	0
$633$	−21258.7	−1.33485
$634$	0	0
$635$	−10476.7	−0.654730
$636$	0	0
$637$	−1681.41	−0.104584
$638$	0	0
$639$	13010.5	0.805459
$640$	0	0
$641$	−10122.2	−0.623715	−0.311857	−	0.950129i	$-0.600951\pi$
$641$	−10122.2	−0.623715	−0.311857	+	0.950129i	$0.600951\pi$
$642$	0	0
$643$	31359.3	1.92331	0.961656	−	0.274257i	$-0.0884318\pi$
$643$	31359.3	1.92331	0.961656	+	0.274257i	$0.0884318\pi$
$644$	0	0
$645$	25312.4	1.54523
$646$	0	0
$647$	−6848.62	−0.416147	−0.208074	−	0.978113i	$-0.566719\pi$
$647$	−6848.62	−0.416147	−0.208074	+	0.978113i	$0.566719\pi$
$648$	0	0
$649$	−27530.3	−1.66512
$650$	0	0
$651$	−6638.05	−0.399640
$652$	0	0
$653$	535.495	0.0320912	0.0160456	−	0.999871i	$-0.494892\pi$
$653$	535.495	0.0320912	0.0160456	+	0.999871i	$0.494892\pi$
$654$	0	0
$655$	19538.5	1.16554
$656$	0	0
$657$	−5694.25	−0.338134
$658$	0	0
$659$	−12352.0	−0.730146	−0.365073	−	0.930979i	$-0.618956\pi$
$659$	−12352.0	−0.730146	−0.365073	+	0.930979i	$0.618956\pi$
$660$	0	0
$661$	−15694.6	−0.923526	−0.461763	−	0.887003i	$-0.652783\pi$
$661$	−15694.6	−0.923526	−0.461763	+	0.887003i	$0.652783\pi$
$662$	0	0
$663$	4291.31	0.251374
$664$	0	0
$665$	1324.29	0.0772237
$666$	0	0
$667$	7173.66	0.416440
$668$	0	0
$669$	13655.8	0.789183
$670$	0	0
$671$	16226.9	0.933578
$672$	0	0
$673$	34213.4	1.95963	0.979813	−	0.199917i	$-0.0640673\pi$
$673$	34213.4	1.95963	0.979813	+	0.199917i	$0.0640673\pi$
$674$	0	0
$675$	1294.28	0.0738027
$676$	0	0
$677$	−31451.2	−1.78547	−0.892737	−	0.450578i	$-0.851218\pi$
$677$	−31451.2	−1.78547	−0.892737	+	0.450578i	$0.851218\pi$
$678$	0	0
$679$	7483.11	0.422939
$680$	0	0
$681$	−14790.9	−0.832291
$682$	0	0
$683$	21568.2	1.20832	0.604160	−	0.796863i	$-0.293509\pi$
$683$	21568.2	1.20832	0.604160	+	0.796863i	$0.293509\pi$
$684$	0	0
$685$	−12064.3	−0.672922
$686$	0	0
$687$	−13374.6	−0.742755
$688$	0	0
$689$	−9856.39	−0.544991
$690$	0	0
$691$	14081.8	0.775250	0.387625	−	0.921817i	$-0.373296\pi$
$691$	14081.8	0.775250	0.387625	+	0.921817i	$0.373296\pi$
$692$	0	0
$693$	−6649.59	−0.364498
$694$	0	0
$695$	−9251.83	−0.504952
$696$	0	0
$697$	−770.827	−0.0418898
$698$	0	0
$699$	−20233.1	−1.09483
$700$	0	0
$701$	26063.9	1.40431	0.702155	−	0.712024i	$-0.252221\pi$
$701$	26063.9	1.40431	0.702155	+	0.712024i	$0.252221\pi$
$702$	0	0
$703$	5479.02	0.293947
$704$	0	0
$705$	−6452.37	−0.344695
$706$	0	0
$707$	−3555.00	−0.189108
$708$	0	0
$709$	4611.50	0.244272	0.122136	−	0.992513i	$-0.461026\pi$
$709$	4611.50	0.244272	0.122136	+	0.992513i	$0.461026\pi$
$710$	0	0
$711$	10357.0	0.546296
$712$	0	0
$713$	18797.3	0.987329
$714$	0	0
$715$	19372.3	1.01326
$716$	0	0
$717$	19072.1	0.993392
$718$	0	0
$719$	−9634.79	−0.499745	−0.249873	−	0.968279i	$-0.580389\pi$
$719$	−9634.79	−0.499745	−0.249873	+	0.968279i	$0.580389\pi$
$720$	0	0
$721$	4976.23	0.257038
$722$	0	0
$723$	−20917.9	−1.07599
$724$	0	0
$725$	−1085.35	−0.0555982
$726$	0	0
$727$	−25023.1	−1.27656	−0.638278	−	0.769806i	$-0.720353\pi$
$727$	−25023.1	−1.27656	−0.638278	+	0.769806i	$0.720353\pi$
$728$	0	0
$729$	−3821.34	−0.194144
$730$	0	0
$731$	6980.20	0.353177
$732$	0	0
$733$	−3995.56	−0.201336	−0.100668	−	0.994920i	$-0.532098\pi$
$733$	−3995.56	−0.201336	−0.100668	+	0.994920i	$0.532098\pi$
$734$	0	0
$735$	−3340.68	−0.167650
$736$	0	0
$737$	58679.9	2.93284
$738$	0	0
$739$	7715.29	0.384048	0.192024	−	0.981390i	$-0.438495\pi$
$739$	7715.29	0.384048	0.192024	+	0.981390i	$0.438495\pi$
$740$	0	0
$741$	−4213.10	−0.208869
$742$	0	0
$743$	16211.5	0.800463	0.400232	−	0.916414i	$-0.368930\pi$
$743$	16211.5	0.800463	0.400232	+	0.916414i	$0.368930\pi$
$744$	0	0
$745$	−17612.6	−0.866142
$746$	0	0
$747$	17828.8	0.873253
$748$	0	0
$749$	−3298.09	−0.160894
$750$	0	0
$751$	22564.5	1.09639	0.548195	−	0.836350i	$-0.315315\pi$
$751$	22564.5	1.09639	0.548195	+	0.836350i	$0.315315\pi$
$752$	0	0
$753$	−11417.5	−0.552557
$754$	0	0
$755$	21559.4	1.03924
$756$	0	0
$757$	−29305.8	−1.40705	−0.703525	−	0.710671i	$-0.748391\pi$
$757$	−29305.8	−1.40705	−0.703525	+	0.710671i	$0.748391\pi$
$758$	0	0
$759$	48309.5	2.31031
$760$	0	0
$761$	21230.3	1.01130	0.505649	−	0.862739i	$-0.331253\pi$
$761$	21230.3	1.01130	0.505649	+	0.862739i	$0.331253\pi$
$762$	0	0
$763$	−11203.9	−0.531598
$764$	0	0
$765$	3323.29	0.157064
$766$	0	0
$767$	17151.0	0.807413
$768$	0	0
$769$	−37875.3	−1.77609	−0.888047	−	0.459752i	$-0.847938\pi$
$769$	−37875.3	−1.77609	−0.888047	+	0.459752i	$0.847938\pi$
$770$	0	0
$771$	−22658.7	−1.05841
$772$	0	0
$773$	−13213.6	−0.614825	−0.307413	−	0.951576i	$-0.599463\pi$
$773$	−13213.6	−0.614825	−0.307413	+	0.951576i	$0.599463\pi$
$774$	0	0
$775$	−2843.96	−0.131817
$776$	0	0
$777$	−13821.5	−0.638150
$778$	0	0
$779$	756.778	0.0348067
$780$	0	0
$781$	41552.9	1.90381
$782$	0	0
$783$	−3529.86	−0.161107
$784$	0	0
$785$	−1143.17	−0.0519764
$786$	0	0
$787$	−7800.96	−0.353334	−0.176667	−	0.984271i	$-0.556532\pi$
$787$	−7800.96	−0.353334	−0.176667	+	0.984271i	$0.556532\pi$
$788$	0	0
$789$	24047.8	1.08507
$790$	0	0
$791$	−5600.02	−0.251724
$792$	0	0
$793$	−10109.1	−0.452690
$794$	0	0
$795$	−19583.0	−0.873630
$796$	0	0
$797$	590.385	0.0262390	0.0131195	−	0.999914i	$-0.495824\pi$
$797$	590.385	0.0262390	0.0131195	+	0.999914i	$0.495824\pi$
$798$	0	0
$799$	−1779.32	−0.0787832
$800$	0	0
$801$	26696.5	1.17762
$802$	0	0
$803$	−18186.2	−0.799226
$804$	0	0
$805$	9459.98	0.414187
$806$	0	0
$807$	−27600.9	−1.20396
$808$	0	0
$809$	38125.9	1.65690	0.828451	−	0.560061i	$-0.189222\pi$
$809$	38125.9	1.65690	0.828451	+	0.560061i	$0.189222\pi$
$810$	0	0
$811$	14007.4	0.606494	0.303247	−	0.952912i	$-0.401929\pi$
$811$	14007.4	0.606494	0.303247	+	0.952912i	$0.401929\pi$
$812$	0	0
$813$	41036.7	1.77026
$814$	0	0
$815$	37972.8	1.63206
$816$	0	0
$817$	−6852.98	−0.293458
$818$	0	0
$819$	4142.59	0.176745
$820$	0	0
$821$	−29308.9	−1.24591	−0.622953	−	0.782259i	$-0.714067\pi$
$821$	−29308.9	−1.24591	−0.622953	+	0.782259i	$0.714067\pi$
$822$	0	0
$823$	40727.5	1.72499	0.862497	−	0.506061i	$-0.168899\pi$
$823$	40727.5	1.72499	0.862497	+	0.506061i	$0.168899\pi$
$824$	0	0
$825$	−7309.02	−0.308445
$826$	0	0
$827$	−5721.10	−0.240559	−0.120279	−	0.992740i	$-0.538379\pi$
$827$	−5721.10	−0.240559	−0.120279	+	0.992740i	$0.538379\pi$
$828$	0	0
$829$	26736.7	1.12015	0.560074	−	0.828443i	$-0.310773\pi$
$829$	26736.7	1.12015	0.560074	+	0.828443i	$0.310773\pi$
$830$	0	0
$831$	−21007.1	−0.876931
$832$	0	0
$833$	−921.233	−0.0383179
$834$	0	0
$835$	20906.9	0.866483
$836$	0	0
$837$	−9249.39	−0.381966
$838$	0	0
$839$	−25617.5	−1.05413	−0.527064	−	0.849825i	$-0.676707\pi$
$839$	−25617.5	−1.05413	−0.527064	+	0.849825i	$0.676707\pi$
$840$	0	0
$841$	−21429.0	−0.878632
$842$	0	0
$843$	22620.0	0.924167
$844$	0	0
$845$	10449.4	0.425409
$846$	0	0
$847$	−11920.4	−0.483577
$848$	0	0
$849$	58320.5	2.35754
$850$	0	0
$851$	39139.0	1.57658
$852$	0	0
$853$	−14738.5	−0.591604	−0.295802	−	0.955249i	$-0.595587\pi$
$853$	−14738.5	−0.591604	−0.295802	+	0.955249i	$0.595587\pi$
$854$	0	0
$855$	−3262.72	−0.130506
$856$	0	0
$857$	8667.64	0.345486	0.172743	−	0.984967i	$-0.444737\pi$
$857$	8667.64	0.345486	0.172743	+	0.984967i	$0.444737\pi$
$858$	0	0
$859$	−6617.36	−0.262842	−0.131421	−	0.991327i	$-0.541954\pi$
$859$	−6617.36	−0.262842	−0.131421	+	0.991327i	$0.541954\pi$
$860$	0	0
$861$	−1909.06	−0.0755641
$862$	0	0
$863$	−33929.2	−1.33831	−0.669156	−	0.743122i	$-0.733344\pi$
$863$	−33929.2	−1.33831	−0.669156	+	0.743122i	$0.733344\pi$
$864$	0	0
$865$	−21121.5	−0.830234
$866$	0	0
$867$	−30329.1	−1.18804
$868$	0	0
$869$	33078.0	1.29125
$870$	0	0
$871$	−36556.6	−1.42213
$872$	0	0
$873$	−18436.5	−0.714756
$874$	0	0
$875$	−10399.5	−0.401792
$876$	0	0
$877$	−3160.79	−0.121702	−0.0608508	−	0.998147i	$-0.519381\pi$
$877$	−3160.79	−0.121702	−0.0608508	+	0.998147i	$0.519381\pi$
$878$	0	0
$879$	4405.10	0.169033
$880$	0	0
$881$	12998.2	0.497071	0.248536	−	0.968623i	$-0.420051\pi$
$881$	12998.2	0.497071	0.248536	+	0.968623i	$0.420051\pi$
$882$	0	0
$883$	−7662.62	−0.292036	−0.146018	−	0.989282i	$-0.546646\pi$
$883$	−7662.62	−0.292036	−0.146018	+	0.989282i	$0.546646\pi$
$884$	0	0
$885$	34076.0	1.29430
$886$	0	0
$887$	10069.3	0.381166	0.190583	−	0.981671i	$-0.438962\pi$
$887$	10069.3	0.381166	0.190583	+	0.981671i	$0.438962\pi$
$888$	0	0
$889$	−7155.18	−0.269940
$890$	0	0
$891$	−49419.5	−1.85815
$892$	0	0
$893$	1746.89	0.0654618
$894$	0	0
$895$	26228.2	0.979568
$896$	0	0
$897$	−30096.0	−1.12026
$898$	0	0
$899$	7756.27	0.287749
$900$	0	0
$901$	−5400.24	−0.199676
$902$	0	0
$903$	17287.5	0.637088
$904$	0	0
$905$	35372.1	1.29924
$906$	0	0
$907$	−3119.93	−0.114218	−0.0571089	−	0.998368i	$-0.518188\pi$
$907$	−3119.93	−0.114218	−0.0571089	+	0.998368i	$0.518188\pi$
$908$	0	0
$909$	8758.65	0.319589
$910$	0	0
$911$	39256.6	1.42770	0.713848	−	0.700301i	$-0.246951\pi$
$911$	39256.6	1.42770	0.713848	+	0.700301i	$0.246951\pi$
$912$	0	0
$913$	56941.3	2.06406
$914$	0	0
$915$	−20085.0	−0.725671
$916$	0	0
$917$	13344.1	0.480545
$918$	0	0
$919$	17574.1	0.630812	0.315406	−	0.948957i	$-0.397859\pi$
$919$	17574.1	0.630812	0.315406	+	0.948957i	$0.397859\pi$
$920$	0	0
$921$	−9342.59	−0.334255
$922$	0	0
$923$	−25886.8	−0.923156
$924$	0	0
$925$	−5921.57	−0.210486
$926$	0	0
$927$	−12260.2	−0.434388
$928$	0	0
$929$	−4612.53	−0.162898	−0.0814489	−	0.996678i	$-0.525955\pi$
$929$	−4612.53	−0.162898	−0.0814489	+	0.996678i	$0.525955\pi$
$930$	0	0
$931$	904.442	0.0318388
$932$	0	0
$933$	−28488.2	−0.999636
$934$	0	0
$935$	10613.9	0.371243
$936$	0	0
$937$	−7791.39	−0.271647	−0.135824	−	0.990733i	$-0.543368\pi$
$937$	−7791.39	−0.271647	−0.135824	+	0.990733i	$0.543368\pi$
$938$	0	0
$939$	72253.5	2.51108
$940$	0	0
$941$	14235.7	0.493166	0.246583	−	0.969122i	$-0.420692\pi$
$941$	14235.7	0.493166	0.246583	+	0.969122i	$0.420692\pi$
$942$	0	0
$943$	5406.00	0.186685
$944$	0	0
$945$	−4654.87	−0.160236
$946$	0	0
$947$	17872.6	0.613284	0.306642	−	0.951825i	$-0.400795\pi$
$947$	17872.6	0.613284	0.306642	+	0.951825i	$0.400795\pi$
$948$	0	0
$949$	11329.7	0.387543
$950$	0	0
$951$	−16956.1	−0.578170
$952$	0	0
$953$	37945.6	1.28980	0.644899	−	0.764268i	$-0.276899\pi$
$953$	37945.6	1.28980	0.644899	+	0.764268i	$0.276899\pi$
$954$	0	0
$955$	20136.4	0.682302
$956$	0	0
$957$	19933.8	0.673319
$958$	0	0
$959$	−8239.45	−0.277441
$960$	0	0
$961$	−9467.03	−0.317782
$962$	0	0
$963$	8125.68	0.271907
$964$	0	0
$965$	−53484.5	−1.78417
$966$	0	0
$967$	−30535.4	−1.01546	−0.507731	−	0.861515i	$-0.669516\pi$
$967$	−30535.4	−1.01546	−0.507731	+	0.861515i	$0.669516\pi$
$968$	0	0
$969$	−2308.32	−0.0765263
$970$	0	0
$971$	−39967.0	−1.32091	−0.660455	−	0.750866i	$-0.729636\pi$
$971$	−39967.0	−1.32091	−0.660455	+	0.750866i	$0.729636\pi$
$972$	0	0
$973$	−6318.67	−0.208188
$974$	0	0
$975$	4553.40	0.149565
$976$	0	0
$977$	−7515.67	−0.246108	−0.123054	−	0.992400i	$-0.539269\pi$
$977$	−7515.67	−0.246108	−0.123054	+	0.992400i	$0.539269\pi$
$978$	0	0
$979$	85262.9	2.78347
$980$	0	0
$981$	27603.7	0.898388
$982$	0	0
$983$	18509.8	0.600581	0.300290	−	0.953848i	$-0.402916\pi$
$983$	18509.8	0.600581	0.300290	+	0.953848i	$0.402916\pi$
$984$	0	0
$985$	−46570.5	−1.50646
$986$	0	0
$987$	−4406.74	−0.142115
$988$	0	0
$989$	−48953.8	−1.57395
$990$	0	0
$991$	−15614.3	−0.500509	−0.250254	−	0.968180i	$-0.580514\pi$
$991$	−15614.3	−0.500509	−0.250254	+	0.968180i	$0.580514\pi$
$992$	0	0
$993$	−30938.3	−0.988719
$994$	0	0
$995$	−29421.8	−0.937422
$996$	0	0
$997$	966.198	0.0306919	0.0153459	−	0.999882i	$-0.495115\pi$
$997$	966.198	0.0306919	0.0153459	+	0.999882i	$0.495115\pi$
$998$	0	0
$999$	−19258.7	−0.609928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.d.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.d.1.13	✓	15	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+6.65179 q^{3} -10.2494 q^{5} -7.00000 q^{7} +17.2463 q^{9} +O(q^{10})\)	\(q+6.65179 q^{3} -10.2494 q^{5} -7.00000 q^{7} +17.2463 q^{9} +55.0810 q^{11} -34.3146 q^{13} -68.1771 q^{15} -18.8007 q^{17} +18.4580 q^{19} -46.5625 q^{21} +131.854 q^{23} -19.9489 q^{25} -64.8797 q^{27} +54.4063 q^{29} +142.562 q^{31} +366.387 q^{33} +71.7461 q^{35} +296.837 q^{37} -228.253 q^{39} +41.0000 q^{41} -371.274 q^{43} -176.765 q^{45} +94.6413 q^{47} +49.0000 q^{49} -125.058 q^{51} +287.236 q^{53} -564.550 q^{55} +122.779 q^{57} -499.816 q^{59} +294.600 q^{61} -120.724 q^{63} +351.705 q^{65} +1065.34 q^{67} +877.062 q^{69} +754.396 q^{71} -330.173 q^{73} -132.696 q^{75} -385.567 q^{77} +600.533 q^{79} -897.215 q^{81} +1033.77 q^{83} +192.696 q^{85} +361.899 q^{87} +1547.96 q^{89} +240.202 q^{91} +948.293 q^{93} -189.184 q^{95} -1069.02 q^{97} +949.942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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