LMFDB - Embedded newform 1872.4.a.bs.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 75x^{3} - 113x^{2} + 1320x + 3267 $ x^5 - 75x^3 - 113x^2 + 1320*x + 3267
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 936)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$7.80718$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.2924 q^{5} +14.9364 q^{7} +55.0472 q^{11} -13.0000 q^{13} -83.5682 q^{17} -64.3057 q^{19} -21.1653 q^{23} +286.780 q^{25} +269.014 q^{29} -159.257 q^{31} +303.094 q^{35} +156.548 q^{37} +472.407 q^{41} -364.827 q^{43} +8.13086 q^{47} -119.905 q^{49} +640.041 q^{53} +1117.04 q^{55} +442.897 q^{59} +271.848 q^{61} -263.801 q^{65} +714.959 q^{67} -1125.27 q^{71} +425.534 q^{73} +822.206 q^{77} -12.3130 q^{79} -475.200 q^{83} -1695.80 q^{85} +302.167 q^{89} -194.173 q^{91} -1304.92 q^{95} -1242.58 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{5} + 18 q^{7} + 28 q^{11} - 65 q^{13} - 48 q^{17} - 90 q^{19} - 52 q^{23} + 155 q^{25} + 200 q^{29} + 162 q^{31} - 308 q^{35} + 54 q^{37} + 898 q^{41} - 8 q^{43} - 480 q^{47} + 601 q^{49} + 676 q^{53}+ \cdots + 126 q^{97}+O(q^{100}) $ 5 * q + 2 * q^5 + 18 * q^7 + 28 * q^11 - 65 * q^13 - 48 * q^17 - 90 * q^19 - 52 * q^23 + 155 * q^25 + 200 * q^29 + 162 * q^31 - 308 * q^35 + 54 * q^37 + 898 * q^41 - 8 * q^43 - 480 * q^47 + 601 * q^49 + 676 * q^53 + 1048 * q^55 - 732 * q^59 + 434 * q^61 - 26 * q^65 + 722 * q^67 - 2184 * q^71 + 998 * q^73 + 2696 * q^77 + 700 * q^79 - 1308 * q^83 + 128 * q^85 + 2718 * q^89 - 234 * q^91 - 3180 * q^95 + 126 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	20.2924	1.81500	0.907502	−	0.420048i	$-0.137987\pi$
$5$	20.2924	1.81500	0.907502	+	0.420048i	$0.137987\pi$
$6$	0	0
$7$	14.9364	0.806488	0.403244	−	0.915092i	$-0.367883\pi$
$7$	14.9364	0.806488	0.403244	+	0.915092i	$0.367883\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	55.0472	1.50885	0.754426	−	0.656385i	$-0.227915\pi$
$11$	55.0472	1.50885	0.754426	+	0.656385i	$0.227915\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−83.5682	−1.19225	−0.596125	−	0.802891i	$-0.703294\pi$
$17$	−83.5682	−1.19225	−0.596125	+	0.802891i	$0.703294\pi$
$18$	0	0
$19$	−64.3057	−0.776460	−0.388230	−	0.921562i	$-0.626913\pi$
$19$	−64.3057	−0.776460	−0.388230	+	0.921562i	$0.626913\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−21.1653	−0.191881	−0.0959407	−	0.995387i	$-0.530586\pi$
$23$	−21.1653	−0.191881	−0.0959407	+	0.995387i	$0.530586\pi$
$24$	0	0
$25$	286.780	2.29424
$26$	0	0
$27$	0	0
$28$	0	0
$29$	269.014	1.72257	0.861287	−	0.508119i	$-0.169659\pi$
$29$	269.014	1.72257	0.861287	+	0.508119i	$0.169659\pi$
$30$	0	0
$31$	−159.257	−0.922691	−0.461346	−	0.887221i	$-0.652633\pi$
$31$	−159.257	−0.922691	−0.461346	+	0.887221i	$0.652633\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	303.094	1.46378
$36$	0	0
$37$	156.548	0.695577	0.347788	−	0.937573i	$-0.386933\pi$
$37$	156.548	0.695577	0.347788	+	0.937573i	$0.386933\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	472.407	1.79945	0.899727	−	0.436453i	$-0.143765\pi$
$41$	472.407	1.79945	0.899727	+	0.436453i	$0.143765\pi$
$42$	0	0
$43$	−364.827	−1.29385	−0.646925	−	0.762554i	$-0.723945\pi$
$43$	−364.827	−1.29385	−0.646925	+	0.762554i	$0.723945\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.13086	0.0252342	0.0126171	−	0.999920i	$-0.495984\pi$
$47$	8.13086	0.0252342	0.0126171	+	0.999920i	$0.495984\pi$
$48$	0	0
$49$	−119.905	−0.349577
$50$	0	0
$51$	0	0
$52$	0	0
$53$	640.041	1.65880	0.829400	−	0.558655i	$-0.188682\pi$
$53$	640.041	1.65880	0.829400	+	0.558655i	$0.188682\pi$
$54$	0	0
$55$	1117.04	2.73857
$56$	0	0
$57$	0	0
$58$	0	0
$59$	442.897	0.977292	0.488646	−	0.872482i	$-0.337491\pi$
$59$	442.897	0.977292	0.488646	+	0.872482i	$0.337491\pi$
$60$	0	0
$61$	271.848	0.570599	0.285299	−	0.958438i	$-0.407907\pi$
$61$	271.848	0.570599	0.285299	+	0.958438i	$0.407907\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−263.801	−0.503391
$66$	0	0
$67$	714.959	1.30367	0.651837	−	0.758359i	$-0.273999\pi$
$67$	714.959	1.30367	0.651837	+	0.758359i	$0.273999\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1125.27	−1.88091	−0.940454	−	0.339920i	$-0.889600\pi$
$71$	−1125.27	−1.88091	−0.940454	+	0.339920i	$0.889600\pi$
$72$	0	0
$73$	425.534	0.682260	0.341130	−	0.940016i	$-0.389190\pi$
$73$	425.534	0.682260	0.341130	+	0.940016i	$0.389190\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	822.206	1.21687
$78$	0	0
$79$	−12.3130	−0.0175357	−0.00876786	−	0.999962i	$-0.502791\pi$
$79$	−12.3130	−0.0175357	−0.00876786	+	0.999962i	$0.502791\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−475.200	−0.628433	−0.314217	−	0.949351i	$-0.601742\pi$
$83$	−475.200	−0.628433	−0.314217	+	0.949351i	$0.601742\pi$
$84$	0	0
$85$	−1695.80	−2.16394
$86$	0	0
$87$	0	0
$88$	0	0
$89$	302.167	0.359884	0.179942	−	0.983677i	$-0.442409\pi$
$89$	302.167	0.359884	0.179942	+	0.983677i	$0.442409\pi$
$90$	0	0
$91$	−194.173	−0.223680
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1304.92	−1.40928
$96$	0	0
$97$	−1242.58	−1.30066	−0.650332	−	0.759650i	$-0.725370\pi$
$97$	−1242.58	−1.30066	−0.650332	+	0.759650i	$0.725370\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−859.138	−0.846410	−0.423205	−	0.906034i	$-0.639095\pi$
$101$	−859.138	−0.846410	−0.423205	+	0.906034i	$0.639095\pi$
$102$	0	0
$103$	1234.75	1.18120	0.590602	−	0.806963i	$-0.298890\pi$
$103$	1234.75	1.18120	0.590602	+	0.806963i	$0.298890\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	598.360	0.540613	0.270307	−	0.962774i	$-0.412875\pi$
$107$	598.360	0.540613	0.270307	+	0.962774i	$0.412875\pi$
$108$	0	0
$109$	646.709	0.568289	0.284144	−	0.958782i	$-0.408291\pi$
$109$	646.709	0.568289	0.284144	+	0.958782i	$0.408291\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−481.406	−0.400769	−0.200384	−	0.979717i	$-0.564219\pi$
$113$	−481.406	−0.400769	−0.200384	+	0.979717i	$0.564219\pi$
$114$	0	0
$115$	−429.494	−0.348266
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1248.21	−0.961536
$120$	0	0
$121$	1699.20	1.27663
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3282.89	2.34905
$126$	0	0
$127$	−2705.33	−1.89023	−0.945115	−	0.326738i	$-0.894051\pi$
$127$	−2705.33	−1.89023	−0.945115	+	0.326738i	$0.894051\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	906.578	0.604642	0.302321	−	0.953206i	$-0.402239\pi$
$131$	906.578	0.604642	0.302321	+	0.953206i	$0.402239\pi$
$132$	0	0
$133$	−960.494	−0.626206
$134$	0	0
$135$	0	0
$136$	0	0
$137$	424.637	0.264811	0.132406	−	0.991196i	$-0.457730\pi$
$137$	424.637	0.264811	0.132406	+	0.991196i	$0.457730\pi$
$138$	0	0
$139$	2510.62	1.53200	0.766001	−	0.642840i	$-0.222244\pi$
$139$	2510.62	1.53200	0.766001	+	0.642840i	$0.222244\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−715.614	−0.418480
$144$	0	0
$145$	5458.93	3.12648
$146$	0	0
$147$	0	0
$148$	0	0
$149$	280.866	0.154426	0.0772130	−	0.997015i	$-0.475398\pi$
$149$	280.866	0.154426	0.0772130	+	0.997015i	$0.475398\pi$
$150$	0	0
$151$	−2774.83	−1.49545	−0.747724	−	0.664010i	$-0.768853\pi$
$151$	−2774.83	−1.49545	−0.747724	+	0.664010i	$0.768853\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3231.70	−1.67469
$156$	0	0
$157$	15.6740	0.00796763	0.00398382	−	0.999992i	$-0.498732\pi$
$157$	15.6740	0.00796763	0.00398382	+	0.999992i	$0.498732\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−316.133	−0.154750
$162$	0	0
$163$	2600.49	1.24961	0.624805	−	0.780781i	$-0.285178\pi$
$163$	2600.49	1.24961	0.624805	+	0.780781i	$0.285178\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	566.453	0.262476	0.131238	−	0.991351i	$-0.458105\pi$
$167$	566.453	0.262476	0.131238	+	0.991351i	$0.458105\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1683.73	−0.739949	−0.369975	−	0.929042i	$-0.620634\pi$
$173$	−1683.73	−0.739949	−0.369975	+	0.929042i	$0.620634\pi$
$174$	0	0
$175$	4283.45	1.85028
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3508.39	−1.46497	−0.732484	−	0.680784i	$-0.761639\pi$
$179$	−3508.39	−1.46497	−0.732484	+	0.680784i	$0.761639\pi$
$180$	0	0
$181$	2237.07	0.918675	0.459338	−	0.888262i	$-0.348087\pi$
$181$	2237.07	0.918675	0.459338	+	0.888262i	$0.348087\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3176.73	1.26247
$186$	0	0
$187$	−4600.20	−1.79893
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4512.58	−1.70952	−0.854762	−	0.519021i	$-0.826297\pi$
$191$	−4512.58	−1.70952	−0.854762	+	0.519021i	$0.826297\pi$
$192$	0	0
$193$	−1822.56	−0.679743	−0.339871	−	0.940472i	$-0.610384\pi$
$193$	−1822.56	−0.679743	−0.339871	+	0.940472i	$0.610384\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1936.31	0.700287	0.350143	−	0.936696i	$-0.386133\pi$
$197$	1936.31	0.700287	0.350143	+	0.936696i	$0.386133\pi$
$198$	0	0
$199$	2247.90	0.800752	0.400376	−	0.916351i	$-0.368880\pi$
$199$	2247.90	0.800752	0.400376	+	0.916351i	$0.368880\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4018.09	1.38924
$204$	0	0
$205$	9586.26	3.26602
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3539.85	−1.17156
$210$	0	0
$211$	2001.54	0.653042	0.326521	−	0.945190i	$-0.394124\pi$
$211$	2001.54	0.653042	0.326521	+	0.945190i	$0.394124\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7403.19	−2.34834
$216$	0	0
$217$	−2378.72	−0.744140
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1086.39	0.330671
$222$	0	0
$223$	4774.04	1.43360	0.716801	−	0.697278i	$-0.245606\pi$
$223$	4774.04	1.43360	0.716801	+	0.697278i	$0.245606\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2078.64	0.607773	0.303887	−	0.952708i	$-0.401716\pi$
$227$	2078.64	0.607773	0.303887	+	0.952708i	$0.401716\pi$
$228$	0	0
$229$	−5166.32	−1.49083	−0.745415	−	0.666601i	$-0.767749\pi$
$229$	−5166.32	−1.49083	−0.745415	+	0.666601i	$0.767749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1691.35	−0.475553	−0.237777	−	0.971320i	$-0.576419\pi$
$233$	−1691.35	−0.475553	−0.237777	+	0.971320i	$0.576419\pi$
$234$	0	0
$235$	164.994	0.0458002
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5586.28	−1.51191	−0.755955	−	0.654624i	$-0.772827\pi$
$239$	−5586.28	−1.51191	−0.755955	+	0.654624i	$0.772827\pi$
$240$	0	0
$241$	−2961.10	−0.791458	−0.395729	−	0.918367i	$-0.629508\pi$
$241$	−2961.10	−0.791458	−0.395729	+	0.918367i	$0.629508\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2433.15	−0.634483
$246$	0	0
$247$	835.975	0.215351
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−663.525	−0.166858	−0.0834290	−	0.996514i	$-0.526587\pi$
$251$	−663.525	−0.166858	−0.0834290	+	0.996514i	$0.526587\pi$
$252$	0	0
$253$	−1165.09	−0.289521
$254$	0	0
$255$	0	0
$256$	0	0
$257$	140.799	0.0341743	0.0170871	−	0.999854i	$-0.494561\pi$
$257$	140.799	0.0341743	0.0170871	+	0.999854i	$0.494561\pi$
$258$	0	0
$259$	2338.26	0.560975
$260$	0	0
$261$	0	0
$262$	0	0
$263$	478.308	0.112144	0.0560718	−	0.998427i	$-0.482142\pi$
$263$	478.308	0.112144	0.0560718	+	0.998427i	$0.482142\pi$
$264$	0	0
$265$	12987.9	3.01073
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7177.45	−1.62683	−0.813414	−	0.581686i	$-0.802393\pi$
$269$	−7177.45	−1.62683	−0.813414	+	0.581686i	$0.802393\pi$
$270$	0	0
$271$	2557.12	0.573189	0.286595	−	0.958052i	$-0.407477\pi$
$271$	2557.12	0.573189	0.286595	+	0.958052i	$0.407477\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15786.4	3.46167
$276$	0	0
$277$	5904.26	1.28070	0.640348	−	0.768085i	$-0.278790\pi$
$277$	5904.26	1.28070	0.640348	+	0.768085i	$0.278790\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6528.43	1.38596	0.692978	−	0.720959i	$-0.256298\pi$
$281$	6528.43	1.38596	0.692978	+	0.720959i	$0.256298\pi$
$282$	0	0
$283$	158.324	0.0332558	0.0166279	−	0.999862i	$-0.494707\pi$
$283$	158.324	0.0332558	0.0166279	+	0.999862i	$0.494707\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7056.05	1.45124
$288$	0	0
$289$	2070.64	0.421462
$290$	0	0
$291$	0	0
$292$	0	0
$293$	213.859	0.0426409	0.0213204	−	0.999773i	$-0.493213\pi$
$293$	213.859	0.0426409	0.0213204	+	0.999773i	$0.493213\pi$
$294$	0	0
$295$	8987.42	1.77379
$296$	0	0
$297$	0	0
$298$	0	0
$299$	275.149	0.0532184
$300$	0	0
$301$	−5449.18	−1.04347
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5516.43	1.03564
$306$	0	0
$307$	9044.29	1.68138	0.840692	−	0.541514i	$-0.182149\pi$
$307$	9044.29	1.68138	0.840692	+	0.541514i	$0.182149\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8093.05	−1.47561	−0.737805	−	0.675014i	$-0.764138\pi$
$311$	−8093.05	−1.47561	−0.737805	+	0.675014i	$0.764138\pi$
$312$	0	0
$313$	3866.24	0.698188	0.349094	−	0.937088i	$-0.386489\pi$
$313$	3866.24	0.698188	0.349094	+	0.937088i	$0.386489\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2426.39	0.429903	0.214952	−	0.976625i	$-0.431041\pi$
$317$	2426.39	0.429903	0.214952	+	0.976625i	$0.431041\pi$
$318$	0	0
$319$	14808.5	2.59911
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5373.91	0.925736
$324$	0	0
$325$	−3728.14	−0.636307
$326$	0	0
$327$	0	0
$328$	0	0
$329$	121.446	0.0203511
$330$	0	0
$331$	1848.92	0.307027	0.153514	−	0.988147i	$-0.450941\pi$
$331$	1848.92	0.307027	0.153514	+	0.988147i	$0.450941\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14508.2	2.36617
$336$	0	0
$337$	−5090.31	−0.822810	−0.411405	−	0.911453i	$-0.634962\pi$
$337$	−5090.31	−0.822810	−0.411405	+	0.911453i	$0.634962\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8766.67	−1.39220
$342$	0	0
$343$	−6914.12	−1.08842
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2231.94	0.345294	0.172647	−	0.984984i	$-0.444768\pi$
$347$	2231.94	0.345294	0.172647	+	0.984984i	$0.444768\pi$
$348$	0	0
$349$	−3399.63	−0.521426	−0.260713	−	0.965416i	$-0.583958\pi$
$349$	−3399.63	−0.521426	−0.260713	+	0.965416i	$0.583958\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12189.2	1.83786	0.918929	−	0.394424i	$-0.129056\pi$
$353$	12189.2	1.83786	0.918929	+	0.394424i	$0.129056\pi$
$354$	0	0
$355$	−22834.3	−3.41386
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3323.43	−0.488590	−0.244295	−	0.969701i	$-0.578557\pi$
$359$	−3323.43	−0.488590	−0.244295	+	0.969701i	$0.578557\pi$
$360$	0	0
$361$	−2723.77	−0.397109
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8635.09	1.23830
$366$	0	0
$367$	−12444.2	−1.76998	−0.884992	−	0.465606i	$-0.845836\pi$
$367$	−12444.2	−1.76998	−0.884992	+	0.465606i	$0.845836\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9559.89	1.33780
$372$	0	0
$373$	2984.52	0.414297	0.207148	−	0.978310i	$-0.433582\pi$
$373$	2984.52	0.414297	0.207148	+	0.978310i	$0.433582\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3497.18	−0.477756
$378$	0	0
$379$	−1921.51	−0.260426	−0.130213	−	0.991486i	$-0.541566\pi$
$379$	−1921.51	−0.260426	−0.130213	+	0.991486i	$0.541566\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11537.2	1.53923	0.769613	−	0.638510i	$-0.220449\pi$
$383$	11537.2	1.53923	0.769613	+	0.638510i	$0.220449\pi$
$384$	0	0
$385$	16684.5	2.20863
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12202.9	−1.59052	−0.795261	−	0.606267i	$-0.792666\pi$
$389$	−12202.9	−1.59052	−0.795261	+	0.606267i	$0.792666\pi$
$390$	0	0
$391$	1768.75	0.228771
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−249.860	−0.0318274
$396$	0	0
$397$	10973.3	1.38724	0.693622	−	0.720339i	$-0.256014\pi$
$397$	10973.3	1.38724	0.693622	+	0.720339i	$0.256014\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3165.38	0.394193	0.197096	−	0.980384i	$-0.436849\pi$
$401$	3165.38	0.394193	0.197096	+	0.980384i	$0.436849\pi$
$402$	0	0
$403$	2070.34	0.255908
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8617.54	1.04952
$408$	0	0
$409$	−10619.8	−1.28390	−0.641950	−	0.766746i	$-0.721875\pi$
$409$	−10619.8	−1.28390	−0.641950	+	0.766746i	$0.721875\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6615.27	0.788174
$414$	0	0
$415$	−9642.92	−1.14061
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−646.216	−0.0753454	−0.0376727	−	0.999290i	$-0.511994\pi$
$419$	−646.216	−0.0753454	−0.0376727	+	0.999290i	$0.511994\pi$
$420$	0	0
$421$	17148.5	1.98519	0.992595	−	0.121471i	$-0.0387612\pi$
$421$	17148.5	1.98519	0.992595	+	0.121471i	$0.0387612\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−23965.7	−2.73531
$426$	0	0
$427$	4060.42	0.460181
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5390.71	−0.602463	−0.301231	−	0.953551i	$-0.597398\pi$
$431$	−5390.71	−0.602463	−0.301231	+	0.953551i	$0.597398\pi$
$432$	0	0
$433$	6948.81	0.771220	0.385610	−	0.922662i	$-0.373991\pi$
$433$	6948.81	0.771220	0.385610	+	0.922662i	$0.373991\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1361.05	0.148988
$438$	0	0
$439$	−1963.07	−0.213422	−0.106711	−	0.994290i	$-0.534032\pi$
$439$	−1963.07	−0.213422	−0.106711	+	0.994290i	$0.534032\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9563.45	−1.02567	−0.512837	−	0.858486i	$-0.671405\pi$
$443$	−9563.45	−1.02567	−0.512837	+	0.858486i	$0.671405\pi$
$444$	0	0
$445$	6131.69	0.653191
$446$	0	0
$447$	0	0
$448$	0	0
$449$	701.223	0.0737033	0.0368516	−	0.999321i	$-0.488267\pi$
$449$	701.223	0.0737033	0.0368516	+	0.999321i	$0.488267\pi$
$450$	0	0
$451$	26004.7	2.71511
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3940.22	−0.405979
$456$	0	0
$457$	−3769.75	−0.385868	−0.192934	−	0.981212i	$-0.561800\pi$
$457$	−3769.75	−0.385868	−0.192934	+	0.981212i	$0.561800\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2531.37	0.255743	0.127872	−	0.991791i	$-0.459185\pi$
$461$	2531.37	0.255743	0.127872	+	0.991791i	$0.459185\pi$
$462$	0	0
$463$	5368.85	0.538902	0.269451	−	0.963014i	$-0.413158\pi$
$463$	5368.85	0.538902	0.269451	+	0.963014i	$0.413158\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14709.9	−1.45759	−0.728794	−	0.684733i	$-0.759919\pi$
$467$	−14709.9	−1.45759	−0.728794	+	0.684733i	$0.759919\pi$
$468$	0	0
$469$	10678.9	1.05140
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−20082.7	−1.95223
$474$	0	0
$475$	−18441.6	−1.78139
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6008.04	0.573098	0.286549	−	0.958066i	$-0.407492\pi$
$479$	6008.04	0.573098	0.286549	+	0.958066i	$0.407492\pi$
$480$	0	0
$481$	−2035.13	−0.192918
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25214.8	−2.36071
$486$	0	0
$487$	18786.3	1.74802	0.874011	−	0.485906i	$-0.161510\pi$
$487$	18786.3	1.74802	0.874011	+	0.485906i	$0.161510\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3175.05	0.291829	0.145914	−	0.989297i	$-0.453388\pi$
$491$	3175.05	0.291829	0.145914	+	0.989297i	$0.453388\pi$
$492$	0	0
$493$	−22481.0	−2.05374
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16807.4	−1.51693
$498$	0	0
$499$	8121.69	0.728611	0.364306	−	0.931279i	$-0.381306\pi$
$499$	8121.69	0.728611	0.364306	+	0.931279i	$0.381306\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11855.0	−1.05087	−0.525436	−	0.850833i	$-0.676098\pi$
$503$	−11855.0	−1.05087	−0.525436	+	0.850833i	$0.676098\pi$
$504$	0	0
$505$	−17433.9	−1.53624
$506$	0	0
$507$	0	0
$508$	0	0
$509$	160.964	0.0140169	0.00700845	−	0.999975i	$-0.497769\pi$
$509$	160.964	0.0140169	0.00700845	+	0.999975i	$0.497769\pi$
$510$	0	0
$511$	6355.93	0.550235
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25056.1	2.14389
$516$	0	0
$517$	447.581	0.0380747
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11369.2	−0.956037	−0.478019	−	0.878350i	$-0.658645\pi$
$521$	−11369.2	−0.956037	−0.478019	+	0.878350i	$0.658645\pi$
$522$	0	0
$523$	−3512.06	−0.293636	−0.146818	−	0.989164i	$-0.546903\pi$
$523$	−3512.06	−0.293636	−0.146818	+	0.989164i	$0.546903\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13308.8	1.10008
$528$	0	0
$529$	−11719.0	−0.963181
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6141.30	−0.499079
$534$	0	0
$535$	12142.1	0.981215
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6600.43	−0.527459
$540$	0	0
$541$	−2671.65	−0.212317	−0.106158	−	0.994349i	$-0.533855\pi$
$541$	−2671.65	−0.212317	−0.106158	+	0.994349i	$0.533855\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13123.2	1.03145
$546$	0	0
$547$	−12080.9	−0.944321	−0.472161	−	0.881513i	$-0.656526\pi$
$547$	−12080.9	−0.944321	−0.472161	+	0.881513i	$0.656526\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17299.1	−1.33751
$552$	0	0
$553$	−183.912	−0.0141423
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2456.10	0.186837	0.0934185	−	0.995627i	$-0.470221\pi$
$557$	2456.10	0.186837	0.0934185	+	0.995627i	$0.470221\pi$
$558$	0	0
$559$	4742.74	0.358849
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4386.95	−0.328397	−0.164199	−	0.986427i	$-0.552504\pi$
$563$	−4386.95	−0.328397	−0.164199	+	0.986427i	$0.552504\pi$
$564$	0	0
$565$	−9768.86	−0.727397
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1607.25	−0.118418	−0.0592088	−	0.998246i	$-0.518858\pi$
$569$	−1607.25	−0.118418	−0.0592088	+	0.998246i	$0.518858\pi$
$570$	0	0
$571$	−14296.0	−1.04776	−0.523879	−	0.851792i	$-0.675516\pi$
$571$	−14296.0	−1.04776	−0.523879	+	0.851792i	$0.675516\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6069.79	−0.440222
$576$	0	0
$577$	−10876.2	−0.784715	−0.392358	−	0.919813i	$-0.628340\pi$
$577$	−10876.2	−0.784715	−0.392358	+	0.919813i	$0.628340\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7097.76	−0.506824
$582$	0	0
$583$	35232.5	2.50288
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4976.96	−0.349951	−0.174975	−	0.984573i	$-0.555985\pi$
$587$	−4976.96	−0.349951	−0.174975	+	0.984573i	$0.555985\pi$
$588$	0	0
$589$	10241.1	0.716433
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10327.2	0.715153	0.357577	−	0.933884i	$-0.383603\pi$
$593$	10327.2	0.715153	0.357577	+	0.933884i	$0.383603\pi$
$594$	0	0
$595$	−25329.0	−1.74519
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16875.2	1.15109	0.575543	−	0.817772i	$-0.304791\pi$
$599$	16875.2	1.15109	0.575543	+	0.817772i	$0.304791\pi$
$600$	0	0
$601$	−19646.1	−1.33341	−0.666706	−	0.745321i	$-0.732296\pi$
$601$	−19646.1	−1.33341	−0.666706	+	0.745321i	$0.732296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	34480.7	2.31709
$606$	0	0
$607$	−25695.1	−1.71817	−0.859087	−	0.511829i	$-0.828968\pi$
$607$	−25695.1	−1.71817	−0.859087	+	0.511829i	$0.828968\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−105.701	−0.00699871
$612$	0	0
$613$	8080.89	0.532437	0.266219	−	0.963913i	$-0.414226\pi$
$613$	8080.89	0.532437	0.266219	+	0.963913i	$0.414226\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11210.3	−0.731458	−0.365729	−	0.930721i	$-0.619180\pi$
$617$	−11210.3	−0.731458	−0.365729	+	0.930721i	$0.619180\pi$
$618$	0	0
$619$	−6332.66	−0.411197	−0.205599	−	0.978636i	$-0.565914\pi$
$619$	−6332.66	−0.411197	−0.205599	+	0.978636i	$0.565914\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4513.29	0.290242
$624$	0	0
$625$	30770.2	1.96929
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13082.4	−0.829302
$630$	0	0
$631$	19230.9	1.21327	0.606633	−	0.794982i	$-0.292520\pi$
$631$	19230.9	1.21327	0.606633	+	0.794982i	$0.292520\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−54897.5	−3.43077
$636$	0	0
$637$	1558.76	0.0969551
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17566.7	1.08244	0.541220	−	0.840881i	$-0.317963\pi$
$641$	17566.7	1.08244	0.541220	+	0.840881i	$0.317963\pi$
$642$	0	0
$643$	−12050.3	−0.739061	−0.369530	−	0.929219i	$-0.620481\pi$
$643$	−12050.3	−0.739061	−0.369530	+	0.929219i	$0.620481\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31097.4	1.88959	0.944797	−	0.327657i	$-0.106259\pi$
$647$	31097.4	1.88959	0.944797	+	0.327657i	$0.106259\pi$
$648$	0	0
$649$	24380.2	1.47459
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24981.3	1.49708	0.748540	−	0.663090i	$-0.230755\pi$
$653$	24981.3	1.49708	0.748540	+	0.663090i	$0.230755\pi$
$654$	0	0
$655$	18396.6	1.09743
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26069.8	−1.54103	−0.770513	−	0.637424i	$-0.780000\pi$
$659$	−26069.8	−1.54103	−0.770513	+	0.637424i	$0.780000\pi$
$660$	0	0
$661$	−1597.47	−0.0940008	−0.0470004	−	0.998895i	$-0.514966\pi$
$661$	−1597.47	−0.0940008	−0.0470004	+	0.998895i	$0.514966\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19490.7	−1.13657
$666$	0	0
$667$	−5693.77	−0.330530
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14964.5	0.860949
$672$	0	0
$673$	−12156.0	−0.696256	−0.348128	−	0.937447i	$-0.613183\pi$
$673$	−12156.0	−0.696256	−0.348128	+	0.937447i	$0.613183\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20729.1	−1.17679	−0.588393	−	0.808575i	$-0.700239\pi$
$677$	−20729.1	−1.17679	−0.588393	+	0.808575i	$0.700239\pi$
$678$	0	0
$679$	−18559.6	−1.04897
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17293.6	−0.968846	−0.484423	−	0.874834i	$-0.660971\pi$
$683$	−17293.6	−0.968846	−0.484423	+	0.874834i	$0.660971\pi$
$684$	0	0
$685$	8616.88	0.480634
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8320.53	−0.460068
$690$	0	0
$691$	−13226.1	−0.728137	−0.364069	−	0.931372i	$-0.618613\pi$
$691$	−13226.1	−0.728137	−0.364069	+	0.931372i	$0.618613\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	50946.4	2.78059
$696$	0	0
$697$	−39478.2	−2.14540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5761.09	0.310404	0.155202	−	0.987883i	$-0.450397\pi$
$701$	5761.09	0.310404	0.155202	+	0.987883i	$0.450397\pi$
$702$	0	0
$703$	−10066.9	−0.540088
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12832.4	−0.682620
$708$	0	0
$709$	−4024.76	−0.213192	−0.106596	−	0.994302i	$-0.533995\pi$
$709$	−4024.76	−0.213192	−0.106596	+	0.994302i	$0.533995\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3370.73	0.177047
$714$	0	0
$715$	−14521.5	−0.759543
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19265.3	−0.999271	−0.499635	−	0.866236i	$-0.666533\pi$
$719$	−19265.3	−0.999271	−0.499635	+	0.866236i	$0.666533\pi$
$720$	0	0
$721$	18442.8	0.952627
$722$	0	0
$723$	0	0
$724$	0	0
$725$	77147.8	3.95199
$726$	0	0
$727$	−2098.78	−0.107069	−0.0535347	−	0.998566i	$-0.517049\pi$
$727$	−2098.78	−0.107069	−0.0535347	+	0.998566i	$0.517049\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30487.9	1.54259
$732$	0	0
$733$	−3121.61	−0.157298	−0.0786489	−	0.996902i	$-0.525061\pi$
$733$	−3121.61	−0.157298	−0.0786489	+	0.996902i	$0.525061\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39356.5	1.96705
$738$	0	0
$739$	−30904.1	−1.53833	−0.769166	−	0.639049i	$-0.779328\pi$
$739$	−30904.1	−1.53833	−0.769166	+	0.639049i	$0.779328\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12460.7	0.615259	0.307629	−	0.951506i	$-0.400464\pi$
$743$	12460.7	0.615259	0.307629	+	0.951506i	$0.400464\pi$
$744$	0	0
$745$	5699.44	0.280284
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8937.32	0.435998
$750$	0	0
$751$	7333.71	0.356339	0.178170	−	0.984000i	$-0.442982\pi$
$751$	7333.71	0.356339	0.178170	+	0.984000i	$0.442982\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−56307.9	−2.71424
$756$	0	0
$757$	−10643.5	−0.511025	−0.255513	−	0.966806i	$-0.582244\pi$
$757$	−10643.5	−0.511025	−0.255513	+	0.966806i	$0.582244\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36572.1	1.74210	0.871050	−	0.491195i	$-0.163440\pi$
$761$	36572.1	1.74210	0.871050	+	0.491195i	$0.163440\pi$
$762$	0	0
$763$	9659.48	0.458318
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5757.65	−0.271052
$768$	0	0
$769$	−10738.5	−0.503563	−0.251782	−	0.967784i	$-0.581016\pi$
$769$	−10738.5	−0.503563	−0.251782	+	0.967784i	$0.581016\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21069.6	−0.980363	−0.490182	−	0.871620i	$-0.663070\pi$
$773$	−21069.6	−0.980363	−0.490182	+	0.871620i	$0.663070\pi$
$774$	0	0
$775$	−45671.7	−2.11687
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30378.5	−1.39721
$780$	0	0
$781$	−61942.8	−2.83801
$782$	0	0
$783$	0	0
$784$	0	0
$785$	318.062	0.0144613
$786$	0	0
$787$	−30557.9	−1.38408	−0.692040	−	0.721859i	$-0.743288\pi$
$787$	−30557.9	−1.38408	−0.692040	+	0.721859i	$0.743288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7190.46	−0.323215
$792$	0	0
$793$	−3534.02	−0.158256
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23807.1	−1.05808	−0.529040	−	0.848597i	$-0.677448\pi$
$797$	−23807.1	−1.05808	−0.529040	+	0.848597i	$0.677448\pi$
$798$	0	0
$799$	−679.481	−0.0300855
$800$	0	0
$801$	0	0
$802$	0	0
$803$	23424.5	1.02943
$804$	0	0
$805$	−6415.09	−0.280872
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11562.3	−0.502482	−0.251241	−	0.967925i	$-0.580839\pi$
$809$	−11562.3	−0.502482	−0.251241	+	0.967925i	$0.580839\pi$
$810$	0	0
$811$	−40267.5	−1.74351	−0.871754	−	0.489944i	$-0.837017\pi$
$811$	−40267.5	−1.74351	−0.871754	+	0.489944i	$0.837017\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	52770.2	2.26805
$816$	0	0
$817$	23460.4	1.00462
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27437.3	1.16635	0.583173	−	0.812348i	$-0.301811\pi$
$821$	27437.3	1.16635	0.583173	+	0.812348i	$0.301811\pi$
$822$	0	0
$823$	30067.9	1.27351	0.636756	−	0.771065i	$-0.280276\pi$
$823$	30067.9	1.27351	0.636756	+	0.771065i	$0.280276\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33130.2	1.39305	0.696523	−	0.717534i	$-0.254729\pi$
$827$	33130.2	1.39305	0.696523	+	0.717534i	$0.254729\pi$
$828$	0	0
$829$	−36696.9	−1.53744	−0.768719	−	0.639587i	$-0.779105\pi$
$829$	−36696.9	−1.53744	−0.768719	+	0.639587i	$0.779105\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10020.2	0.416783
$834$	0	0
$835$	11494.7	0.476395
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12906.1	−0.531072	−0.265536	−	0.964101i	$-0.585549\pi$
$839$	−12906.1	−0.531072	−0.265536	+	0.964101i	$0.585549\pi$
$840$	0	0
$841$	47979.5	1.96726
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3429.41	0.139616
$846$	0	0
$847$	25379.9	1.02959
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3313.39	−0.133468
$852$	0	0
$853$	22827.3	0.916286	0.458143	−	0.888879i	$-0.348515\pi$
$853$	22827.3	0.916286	0.458143	+	0.888879i	$0.348515\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24504.7	−0.976739	−0.488369	−	0.872637i	$-0.662408\pi$
$857$	−24504.7	−0.976739	−0.488369	+	0.872637i	$0.662408\pi$
$858$	0	0
$859$	4905.41	0.194843	0.0974217	−	0.995243i	$-0.468940\pi$
$859$	4905.41	0.194843	0.0974217	+	0.995243i	$0.468940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42275.8	−1.66754	−0.833770	−	0.552112i	$-0.813822\pi$
$863$	−42275.8	−1.66754	−0.833770	+	0.552112i	$0.813822\pi$
$864$	0	0
$865$	−34166.8	−1.34301
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−677.797	−0.0264588
$870$	0	0
$871$	−9294.46	−0.361574
$872$	0	0
$873$	0	0
$874$	0	0
$875$	49034.5	1.89448
$876$	0	0
$877$	−27485.3	−1.05828	−0.529140	−	0.848534i	$-0.677485\pi$
$877$	−27485.3	−1.05828	−0.529140	+	0.848534i	$0.677485\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34620.6	1.32395	0.661973	−	0.749528i	$-0.269719\pi$
$881$	34620.6	1.32395	0.661973	+	0.749528i	$0.269719\pi$
$882$	0	0
$883$	24023.5	0.915578	0.457789	−	0.889061i	$-0.348642\pi$
$883$	24023.5	0.915578	0.457789	+	0.889061i	$0.348642\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8593.11	−0.325286	−0.162643	−	0.986685i	$-0.552002\pi$
$887$	−8593.11	−0.325286	−0.162643	+	0.986685i	$0.552002\pi$
$888$	0	0
$889$	−40407.8	−1.52445
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−522.861	−0.0195934
$894$	0	0
$895$	−71193.5	−2.65892
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−42842.4	−1.58940
$900$	0	0
$901$	−53487.1	−1.97771
$902$	0	0
$903$	0	0
$904$	0	0
$905$	45395.5	1.66740
$906$	0	0
$907$	36918.7	1.35156	0.675781	−	0.737102i	$-0.263806\pi$
$907$	36918.7	1.35156	0.675781	+	0.737102i	$0.263806\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37525.8	−1.36475	−0.682373	−	0.731004i	$-0.739052\pi$
$911$	−37525.8	−1.36475	−0.682373	+	0.731004i	$0.739052\pi$
$912$	0	0
$913$	−26158.4	−0.948212
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13541.0	0.487636
$918$	0	0
$919$	33444.0	1.20045	0.600227	−	0.799830i	$-0.295077\pi$
$919$	33444.0	1.20045	0.600227	+	0.799830i	$0.295077\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14628.5	0.521670
$924$	0	0
$925$	44894.8	1.59582
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8887.12	−0.313861	−0.156930	−	0.987610i	$-0.550160\pi$
$929$	−8887.12	−0.313861	−0.156930	+	0.987610i	$0.550160\pi$
$930$	0	0
$931$	7710.57	0.271433
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−93348.9	−3.26506
$936$	0	0
$937$	−31175.0	−1.08692	−0.543459	−	0.839436i	$-0.682886\pi$
$937$	−31175.0	−1.08692	−0.543459	+	0.839436i	$0.682886\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6429.51	−0.222738	−0.111369	−	0.993779i	$-0.535523\pi$
$941$	−6429.51	−0.222738	−0.111369	+	0.993779i	$0.535523\pi$
$942$	0	0
$943$	−9998.65	−0.345282
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16495.2	−0.566019	−0.283010	−	0.959117i	$-0.591333\pi$
$947$	−16495.2	−0.566019	−0.283010	+	0.959117i	$0.591333\pi$
$948$	0	0
$949$	−5531.94	−0.189225
$950$	0	0
$951$	0	0
$952$	0	0
$953$	145.122	0.00493282	0.00246641	−	0.999997i	$-0.499215\pi$
$953$	145.122	0.00493282	0.00246641	+	0.999997i	$0.499215\pi$
$954$	0	0
$955$	−91570.9	−3.10279
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6342.53	0.213567
$960$	0	0
$961$	−4428.16	−0.148641
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−36983.9	−1.23374
$966$	0	0
$967$	17874.3	0.594415	0.297207	−	0.954813i	$-0.403945\pi$
$967$	17874.3	0.594415	0.297207	+	0.954813i	$0.403945\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−661.941	−0.0218771	−0.0109386	−	0.999940i	$-0.503482\pi$
$971$	−661.941	−0.0218771	−0.0109386	+	0.999940i	$0.503482\pi$
$972$	0	0
$973$	37499.6	1.23554
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29539.7	−0.967306	−0.483653	−	0.875260i	$-0.660690\pi$
$977$	−29539.7	−0.967306	−0.483653	+	0.875260i	$0.660690\pi$
$978$	0	0
$979$	16633.5	0.543012
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5390.86	0.174915	0.0874577	−	0.996168i	$-0.472126\pi$
$983$	5390.86	0.174915	0.0874577	+	0.996168i	$0.472126\pi$
$984$	0	0
$985$	39292.3	1.27102
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7721.67	0.248266
$990$	0	0
$991$	50580.0	1.62132	0.810660	−	0.585517i	$-0.199109\pi$
$991$	50580.0	1.62132	0.810660	+	0.585517i	$0.199109\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	45615.2	1.45337
$996$	0	0
$997$	−5946.26	−0.188887	−0.0944434	−	0.995530i	$-0.530107\pi$
$997$	−5946.26	−0.188887	−0.0944434	+	0.995530i	$0.530107\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bs.1.5		5
3.2	odd	2		1872.4.a.br.1.1		5
4.3	odd	2		936.4.a.q.1.5	yes	5
12.11	even	2		936.4.a.p.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.4.a.p.1.1	✓	5	12.11	even	2
936.4.a.q.1.5	yes	5	4.3	odd	2
1872.4.a.br.1.1		5	3.2	odd	2
1872.4.a.bs.1.5		5	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+20.2924 q^{5} +14.9364 q^{7} +55.0472 q^{11} -13.0000 q^{13} -83.5682 q^{17} -64.3057 q^{19} -21.1653 q^{23} +286.780 q^{25} +269.014 q^{29} -159.257 q^{31} +303.094 q^{35} +156.548 q^{37} +472.407 q^{41} -364.827 q^{43} +8.13086 q^{47} -119.905 q^{49} +640.041 q^{53} +1117.04 q^{55} +442.897 q^{59} +271.848 q^{61} -263.801 q^{65} +714.959 q^{67} -1125.27 q^{71} +425.534 q^{73} +822.206 q^{77} -12.3130 q^{79} -475.200 q^{83} -1695.80 q^{85} +302.167 q^{89} -194.173 q^{91} -1304.92 q^{95} -1242.58 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{5} + 18 q^{7} + 28 q^{11} - 65 q^{13} - 48 q^{17} - 90 q^{19} - 52 q^{23} + 155 q^{25} + 200 q^{29} + 162 q^{31} - 308 q^{35} + 54 q^{37} + 898 q^{41} - 8 q^{43} - 480 q^{47} + 601 q^{49} + 676 q^{53}+ \cdots + 126 q^{97}+O(q^{100}) \) 5 * q + 2 * q^5 + 18 * q^7 + 28 * q^11 - 65 * q^13 - 48 * q^17 - 90 * q^19 - 52 * q^23 + 155 * q^25 + 200 * q^29 + 162 * q^31 - 308 * q^35 + 54 * q^37 + 898 * q^41 - 8 * q^43 - 480 * q^47 + 601 * q^49 + 676 * q^53 + 1048 * q^55 - 732 * q^59 + 434 * q^61 - 26 * q^65 + 722 * q^67 - 2184 * q^71 + 998 * q^73 + 2696 * q^77 + 700 * q^79 - 1308 * q^83 + 128 * q^85 + 2718 * q^89 - 234 * q^91 - 3180 * q^95 + 126 * q^97

Introduction

L-functions

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