LMFDB - Embedded newform 2352.4.a.cq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.136768.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 23x^{2} + 18x + 119 $ x^4 - 2x^3 - 23x^2 + 18*x + 119
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 588)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.89590$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} -37.8189 q^{11} -39.9319 q^{13} -24.5082 q^{15} -9.93596 q^{17} -90.4458 q^{19} -118.595 q^{23} -58.2608 q^{25} +27.0000 q^{27} -78.4061 q^{29} +92.0110 q^{31} -113.457 q^{33} +332.435 q^{37} -119.796 q^{39} -71.7451 q^{41} +115.947 q^{43} -73.5246 q^{45} +307.927 q^{47} -29.8079 q^{51} -403.150 q^{53} +308.958 q^{55} -271.337 q^{57} +593.710 q^{59} -333.170 q^{61} +326.220 q^{65} +743.511 q^{67} -355.784 q^{69} +728.272 q^{71} +801.749 q^{73} -174.782 q^{75} -1067.91 q^{79} +81.0000 q^{81} +906.756 q^{83} +81.1709 q^{85} -235.218 q^{87} -1113.27 q^{89} +276.033 q^{93} +738.889 q^{95} -1480.94 q^{97} -340.370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 36 * q^9	$ 4 q + 12 q^{3} + 36 q^{9} - 48 q^{17} + 192 q^{19} - 192 q^{23} + 324 q^{25} + 108 q^{27} + 96 q^{29} + 48 q^{31} + 256 q^{37} - 1008 q^{41} + 112 q^{43} + 864 q^{47} - 144 q^{51} - 648 q^{53} + 2352 q^{55} + 576 q^{57} + 336 q^{59} - 960 q^{61} - 360 q^{65} - 720 q^{67} - 576 q^{69} + 1344 q^{71} - 672 q^{73} + 972 q^{75} + 1984 q^{79} + 324 q^{81} + 3120 q^{83} + 680 q^{85} + 288 q^{87} - 2160 q^{89} + 144 q^{93} + 3744 q^{95} - 2016 q^{97}+O(q^{100}) $ 4 * q + 12 * q^3 + 36 * q^9 - 48 * q^17 + 192 * q^19 - 192 * q^23 + 324 * q^25 + 108 * q^27 + 96 * q^29 + 48 * q^31 + 256 * q^37 - 1008 * q^41 + 112 * q^43 + 864 * q^47 - 144 * q^51 - 648 * q^53 + 2352 * q^55 + 576 * q^57 + 336 * q^59 - 960 * q^61 - 360 * q^65 - 720 * q^67 - 576 * q^69 + 1344 * q^71 - 672 * q^73 + 972 * q^75 + 1984 * q^79 + 324 * q^81 + 3120 * q^83 + 680 * q^85 + 288 * q^87 - 2160 * q^89 + 144 * q^93 + 3744 * q^95 - 2016 * q^97

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−8.16940	−0.730694	−0.365347	−	0.930871i	$-0.619050\pi$
$5$	−8.16940	−0.730694	−0.365347	+	0.930871i	$0.619050\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−37.8189	−1.03662	−0.518310	−	0.855193i	$-0.673439\pi$
$11$	−37.8189	−1.03662	−0.518310	+	0.855193i	$0.673439\pi$
$12$	0	0
$13$	−39.9319	−0.851933	−0.425966	−	0.904739i	$-0.640066\pi$
$13$	−39.9319	−0.851933	−0.425966	+	0.904739i	$0.640066\pi$
$14$	0	0
$15$	−24.5082	−0.421866
$16$	0	0
$17$	−9.93596	−0.141754	−0.0708772	−	0.997485i	$-0.522580\pi$
$17$	−9.93596	−0.141754	−0.0708772	+	0.997485i	$0.522580\pi$
$18$	0	0
$19$	−90.4458	−1.09209	−0.546045	−	0.837756i	$-0.683867\pi$
$19$	−90.4458	−1.09209	−0.546045	+	0.837756i	$0.683867\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−118.595	−1.07516	−0.537580	−	0.843213i	$-0.680661\pi$
$23$	−118.595	−1.07516	−0.537580	+	0.843213i	$0.680661\pi$
$24$	0	0
$25$	−58.2608	−0.466087
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−78.4061	−0.502057	−0.251028	−	0.967980i	$-0.580769\pi$
$29$	−78.4061	−0.502057	−0.251028	+	0.967980i	$0.580769\pi$
$30$	0	0
$31$	92.0110	0.533086	0.266543	−	0.963823i	$-0.414119\pi$
$31$	92.0110	0.533086	0.266543	+	0.963823i	$0.414119\pi$
$32$	0	0
$33$	−113.457	−0.598493
$34$	0	0
$35$	0	0
$36$	0	0
$37$	332.435	1.47708	0.738541	−	0.674209i	$-0.235515\pi$
$37$	332.435	1.47708	0.738541	+	0.674209i	$0.235515\pi$
$38$	0	0
$39$	−119.796	−0.491864
$40$	0	0
$41$	−71.7451	−0.273285	−0.136643	−	0.990620i	$-0.543631\pi$
$41$	−71.7451	−0.273285	−0.136643	+	0.990620i	$0.543631\pi$
$42$	0	0
$43$	115.947	0.411202	0.205601	−	0.978636i	$-0.434085\pi$
$43$	115.947	0.411202	0.205601	+	0.978636i	$0.434085\pi$
$44$	0	0
$45$	−73.5246	−0.243565
$46$	0	0
$47$	307.927	0.955656	0.477828	−	0.878453i	$-0.341424\pi$
$47$	307.927	0.955656	0.477828	+	0.878453i	$0.341424\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−29.8079	−0.0818419
$52$	0	0
$53$	−403.150	−1.04485	−0.522424	−	0.852686i	$-0.674972\pi$
$53$	−403.150	−1.04485	−0.522424	+	0.852686i	$0.674972\pi$
$54$	0	0
$55$	308.958	0.757451
$56$	0	0
$57$	−271.337	−0.630518
$58$	0	0
$59$	593.710	1.31008	0.655038	−	0.755596i	$-0.272653\pi$
$59$	593.710	1.31008	0.655038	+	0.755596i	$0.272653\pi$
$60$	0	0
$61$	−333.170	−0.699313	−0.349657	−	0.936878i	$-0.613702\pi$
$61$	−333.170	−0.699313	−0.349657	+	0.936878i	$0.613702\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	326.220	0.622502
$66$	0	0
$67$	743.511	1.35574	0.677869	−	0.735183i	$-0.262904\pi$
$67$	743.511	1.35574	0.677869	+	0.735183i	$0.262904\pi$
$68$	0	0
$69$	−355.784	−0.620744
$70$	0	0
$71$	728.272	1.21732	0.608662	−	0.793429i	$-0.291706\pi$
$71$	728.272	1.21732	0.608662	+	0.793429i	$0.291706\pi$
$72$	0	0
$73$	801.749	1.28545	0.642723	−	0.766098i	$-0.277804\pi$
$73$	801.749	1.28545	0.642723	+	0.766098i	$0.277804\pi$
$74$	0	0
$75$	−174.782	−0.269095
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1067.91	−1.52087	−0.760435	−	0.649414i	$-0.775014\pi$
$79$	−1067.91	−1.52087	−0.760435	+	0.649414i	$0.775014\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	906.756	1.19915	0.599575	−	0.800319i	$-0.295336\pi$
$83$	906.756	1.19915	0.599575	+	0.800319i	$0.295336\pi$
$84$	0	0
$85$	81.1709	0.103579
$86$	0	0
$87$	−235.218	−0.289863
$88$	0	0
$89$	−1113.27	−1.32591	−0.662957	−	0.748658i	$-0.730699\pi$
$89$	−1113.27	−1.32591	−0.662957	+	0.748658i	$0.730699\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	276.033	0.307777
$94$	0	0
$95$	738.889	0.797983
$96$	0	0
$97$	−1480.94	−1.55017	−0.775084	−	0.631858i	$-0.782293\pi$
$97$	−1480.94	−1.55017	−0.775084	+	0.631858i	$0.782293\pi$
$98$	0	0
$99$	−340.370	−0.345540
$100$	0	0
$101$	556.316	0.548075	0.274037	−	0.961719i	$-0.411641\pi$
$101$	556.316	0.548075	0.274037	+	0.961719i	$0.411641\pi$
$102$	0	0
$103$	552.435	0.528476	0.264238	−	0.964458i	$-0.414880\pi$
$103$	552.435	0.528476	0.264238	+	0.964458i	$0.414880\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−533.804	−0.482288	−0.241144	−	0.970489i	$-0.577523\pi$
$107$	−533.804	−0.482288	−0.241144	+	0.970489i	$0.577523\pi$
$108$	0	0
$109$	1094.63	0.961894	0.480947	−	0.876750i	$-0.340293\pi$
$109$	1094.63	0.961894	0.480947	+	0.876750i	$0.340293\pi$
$110$	0	0
$111$	997.306	0.852794
$112$	0	0
$113$	−1425.18	−1.18646	−0.593228	−	0.805035i	$-0.702147\pi$
$113$	−1425.18	−1.18646	−0.593228	+	0.805035i	$0.702147\pi$
$114$	0	0
$115$	968.847	0.785612
$116$	0	0
$117$	−359.387	−0.283978
$118$	0	0
$119$	0	0
$120$	0	0
$121$	99.2659	0.0745800
$122$	0	0
$123$	−215.235	−0.157781
$124$	0	0
$125$	1497.13	1.07126
$126$	0	0
$127$	786.485	0.549522	0.274761	−	0.961513i	$-0.411401\pi$
$127$	786.485	0.549522	0.274761	+	0.961513i	$0.411401\pi$
$128$	0	0
$129$	347.840	0.237407
$130$	0	0
$131$	−27.7303	−0.0184947	−0.00924735	−	0.999957i	$-0.502944\pi$
$131$	−27.7303	−0.0184947	−0.00924735	+	0.999957i	$0.502944\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−220.574	−0.140622
$136$	0	0
$137$	−2362.98	−1.47360	−0.736798	−	0.676113i	$-0.763663\pi$
$137$	−2362.98	−1.47360	−0.736798	+	0.676113i	$0.763663\pi$
$138$	0	0
$139$	2513.28	1.53362	0.766811	−	0.641873i	$-0.221842\pi$
$139$	2513.28	1.53362	0.766811	+	0.641873i	$0.221842\pi$
$140$	0	0
$141$	923.782	0.551748
$142$	0	0
$143$	1510.18	0.883130
$144$	0	0
$145$	640.531	0.366850
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2265.83	1.24580	0.622900	−	0.782302i	$-0.285955\pi$
$149$	2265.83	1.24580	0.622900	+	0.782302i	$0.285955\pi$
$150$	0	0
$151$	−283.146	−0.152597	−0.0762984	−	0.997085i	$-0.524310\pi$
$151$	−283.146	−0.152597	−0.0762984	+	0.997085i	$0.524310\pi$
$152$	0	0
$153$	−89.4237	−0.0472515
$154$	0	0
$155$	−751.675	−0.389522
$156$	0	0
$157$	−192.581	−0.0978956	−0.0489478	−	0.998801i	$-0.515587\pi$
$157$	−192.581	−0.0978956	−0.0489478	+	0.998801i	$0.515587\pi$
$158$	0	0
$159$	−1209.45	−0.603243
$160$	0	0
$161$	0	0
$162$	0	0
$163$	842.833	0.405005	0.202502	−	0.979282i	$-0.435093\pi$
$163$	842.833	0.405005	0.202502	+	0.979282i	$0.435093\pi$
$164$	0	0
$165$	926.873	0.437315
$166$	0	0
$167$	−3859.21	−1.78823	−0.894116	−	0.447836i	$-0.852195\pi$
$167$	−3859.21	−1.78823	−0.894116	+	0.447836i	$0.852195\pi$
$168$	0	0
$169$	−602.441	−0.274211
$170$	0	0
$171$	−814.012	−0.364030
$172$	0	0
$173$	−1510.66	−0.663891	−0.331946	−	0.943298i	$-0.607705\pi$
$173$	−1510.66	−0.663891	−0.331946	+	0.943298i	$0.607705\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1781.13	0.756372
$178$	0	0
$179$	2464.63	1.02913	0.514566	−	0.857451i	$-0.327953\pi$
$179$	2464.63	1.02913	0.514566	+	0.857451i	$0.327953\pi$
$180$	0	0
$181$	3297.36	1.35409	0.677047	−	0.735940i	$-0.263259\pi$
$181$	3297.36	1.35409	0.677047	+	0.735940i	$0.263259\pi$
$182$	0	0
$183$	−999.511	−0.403749
$184$	0	0
$185$	−2715.80	−1.07929
$186$	0	0
$187$	375.767	0.146945
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4729.71	1.79178	0.895889	−	0.444278i	$-0.146540\pi$
$191$	4729.71	1.79178	0.895889	+	0.444278i	$0.146540\pi$
$192$	0	0
$193$	4553.18	1.69816	0.849080	−	0.528264i	$-0.177157\pi$
$193$	4553.18	1.69816	0.849080	+	0.528264i	$0.177157\pi$
$194$	0	0
$195$	978.660	0.359402
$196$	0	0
$197$	−3109.06	−1.12442	−0.562212	−	0.826993i	$-0.690050\pi$
$197$	−3109.06	−1.12442	−0.562212	+	0.826993i	$0.690050\pi$
$198$	0	0
$199$	−443.512	−0.157989	−0.0789943	−	0.996875i	$-0.525171\pi$
$199$	−443.512	−0.157989	−0.0789943	+	0.996875i	$0.525171\pi$
$200$	0	0
$201$	2230.53	0.782735
$202$	0	0
$203$	0	0
$204$	0	0
$205$	586.114	0.199688
$206$	0	0
$207$	−1067.35	−0.358387
$208$	0	0
$209$	3420.56	1.13208
$210$	0	0
$211$	−4653.28	−1.51822	−0.759112	−	0.650960i	$-0.774367\pi$
$211$	−4653.28	−1.51822	−0.759112	+	0.650960i	$0.774367\pi$
$212$	0	0
$213$	2184.82	0.702823
$214$	0	0
$215$	−947.214	−0.300463
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2405.25	0.742153
$220$	0	0
$221$	396.762	0.120765
$222$	0	0
$223$	2778.90	0.834481	0.417240	−	0.908796i	$-0.362997\pi$
$223$	2778.90	0.834481	0.417240	+	0.908796i	$0.362997\pi$
$224$	0	0
$225$	−524.347	−0.155362
$226$	0	0
$227$	3217.22	0.940681	0.470340	−	0.882485i	$-0.344131\pi$
$227$	3217.22	0.940681	0.470340	+	0.882485i	$0.344131\pi$
$228$	0	0
$229$	3728.01	1.07578	0.537891	−	0.843014i	$-0.319221\pi$
$229$	3728.01	1.07578	0.537891	+	0.843014i	$0.319221\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3603.51	−1.01319	−0.506596	−	0.862183i	$-0.669097\pi$
$233$	−3603.51	−1.01319	−0.506596	+	0.862183i	$0.669097\pi$
$234$	0	0
$235$	−2515.58	−0.698292
$236$	0	0
$237$	−3203.72	−0.878075
$238$	0	0
$239$	−2348.76	−0.635684	−0.317842	−	0.948144i	$-0.602958\pi$
$239$	−2348.76	−0.635684	−0.317842	+	0.948144i	$0.602958\pi$
$240$	0	0
$241$	5093.39	1.36139	0.680693	−	0.732569i	$-0.261679\pi$
$241$	5093.39	1.36139	0.680693	+	0.732569i	$0.261679\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3611.68	0.930386
$248$	0	0
$249$	2720.27	0.692329
$250$	0	0
$251$	−5939.02	−1.49350	−0.746749	−	0.665106i	$-0.768386\pi$
$251$	−5939.02	−1.49350	−0.746749	+	0.665106i	$0.768386\pi$
$252$	0	0
$253$	4485.11	1.11453
$254$	0	0
$255$	243.513	0.0598014
$256$	0	0
$257$	1515.89	0.367932	0.183966	−	0.982933i	$-0.441106\pi$
$257$	1515.89	0.367932	0.183966	+	0.982933i	$0.441106\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−705.655	−0.167352
$262$	0	0
$263$	6433.92	1.50849	0.754244	−	0.656594i	$-0.228003\pi$
$263$	6433.92	1.50849	0.754244	+	0.656594i	$0.228003\pi$
$264$	0	0
$265$	3293.50	0.763464
$266$	0	0
$267$	−3339.81	−0.765516
$268$	0	0
$269$	6949.79	1.57523	0.787614	−	0.616169i	$-0.211316\pi$
$269$	6949.79	1.57523	0.787614	+	0.616169i	$0.211316\pi$
$270$	0	0
$271$	−961.994	−0.215635	−0.107817	−	0.994171i	$-0.534386\pi$
$271$	−961.994	−0.215635	−0.107817	+	0.994171i	$0.534386\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2203.36	0.483154
$276$	0	0
$277$	760.010	0.164854	0.0824271	−	0.996597i	$-0.473733\pi$
$277$	760.010	0.164854	0.0824271	+	0.996597i	$0.473733\pi$
$278$	0	0
$279$	828.099	0.177695
$280$	0	0
$281$	−4412.07	−0.936662	−0.468331	−	0.883553i	$-0.655145\pi$
$281$	−4412.07	−0.936662	−0.468331	+	0.883553i	$0.655145\pi$
$282$	0	0
$283$	−2602.15	−0.546578	−0.273289	−	0.961932i	$-0.588112\pi$
$283$	−2602.15	−0.546578	−0.273289	+	0.961932i	$0.588112\pi$
$284$	0	0
$285$	2216.67	0.460716
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4814.28	−0.979906
$290$	0	0
$291$	−4442.81	−0.894990
$292$	0	0
$293$	9332.18	1.86072	0.930361	−	0.366644i	$-0.119493\pi$
$293$	9332.18	1.86072	0.930361	+	0.366644i	$0.119493\pi$
$294$	0	0
$295$	−4850.26	−0.957264
$296$	0	0
$297$	−1021.11	−0.199498
$298$	0	0
$299$	4735.71	0.915964
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1668.95	0.316431
$304$	0	0
$305$	2721.80	0.510984
$306$	0	0
$307$	4895.51	0.910103	0.455051	−	0.890465i	$-0.349621\pi$
$307$	4895.51	0.910103	0.455051	+	0.890465i	$0.349621\pi$
$308$	0	0
$309$	1657.30	0.305116
$310$	0	0
$311$	7505.28	1.36844	0.684221	−	0.729275i	$-0.260142\pi$
$311$	7505.28	1.36844	0.684221	+	0.729275i	$0.260142\pi$
$312$	0	0
$313$	6349.27	1.14659	0.573294	−	0.819350i	$-0.305665\pi$
$313$	6349.27	1.14659	0.573294	+	0.819350i	$0.305665\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7999.60	−1.41736	−0.708679	−	0.705531i	$-0.750708\pi$
$317$	−7999.60	−1.41736	−0.708679	+	0.705531i	$0.750708\pi$
$318$	0	0
$319$	2965.23	0.520442
$320$	0	0
$321$	−1601.41	−0.278449
$322$	0	0
$323$	898.666	0.154808
$324$	0	0
$325$	2326.47	0.397074
$326$	0	0
$327$	3283.89	0.555350
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2121.56	0.352300	0.176150	−	0.984363i	$-0.443636\pi$
$331$	2121.56	0.352300	0.176150	+	0.984363i	$0.443636\pi$
$332$	0	0
$333$	2991.92	0.492361
$334$	0	0
$335$	−6074.05	−0.990629
$336$	0	0
$337$	−9114.19	−1.47324	−0.736620	−	0.676307i	$-0.763579\pi$
$337$	−9114.19	−1.47324	−0.736620	+	0.676307i	$0.763579\pi$
$338$	0	0
$339$	−4275.53	−0.685001
$340$	0	0
$341$	−3479.75	−0.552607
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2906.54	0.453574
$346$	0	0
$347$	1673.07	0.258833	0.129416	−	0.991590i	$-0.458690\pi$
$347$	1673.07	0.258833	0.129416	+	0.991590i	$0.458690\pi$
$348$	0	0
$349$	3467.56	0.531845	0.265923	−	0.963994i	$-0.414323\pi$
$349$	3467.56	0.531845	0.265923	+	0.963994i	$0.414323\pi$
$350$	0	0
$351$	−1078.16	−0.163955
$352$	0	0
$353$	−3984.24	−0.600736	−0.300368	−	0.953823i	$-0.597109\pi$
$353$	−3984.24	−0.600736	−0.300368	+	0.953823i	$0.597109\pi$
$354$	0	0
$355$	−5949.55	−0.889491
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2170.93	−0.319157	−0.159579	−	0.987185i	$-0.551014\pi$
$359$	−2170.93	−0.319157	−0.159579	+	0.987185i	$0.551014\pi$
$360$	0	0
$361$	1321.45	0.192659
$362$	0	0
$363$	297.798	0.0430588
$364$	0	0
$365$	−6549.81	−0.939268
$366$	0	0
$367$	3592.12	0.510918	0.255459	−	0.966820i	$-0.417773\pi$
$367$	3592.12	0.510918	0.255459	+	0.966820i	$0.417773\pi$
$368$	0	0
$369$	−645.705	−0.0910951
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7439.80	1.03276	0.516378	−	0.856361i	$-0.327280\pi$
$373$	7439.80	1.03276	0.516378	+	0.856361i	$0.327280\pi$
$374$	0	0
$375$	4491.40	0.618492
$376$	0	0
$377$	3130.91	0.427719
$378$	0	0
$379$	−11243.9	−1.52390	−0.761951	−	0.647635i	$-0.775758\pi$
$379$	−11243.9	−1.52390	−0.761951	+	0.647635i	$0.775758\pi$
$380$	0	0
$381$	2359.46	0.317267
$382$	0	0
$383$	3541.25	0.472452	0.236226	−	0.971698i	$-0.424089\pi$
$383$	3541.25	0.472452	0.236226	+	0.971698i	$0.424089\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1043.52	0.137067
$388$	0	0
$389$	−10558.2	−1.37615	−0.688074	−	0.725640i	$-0.741544\pi$
$389$	−10558.2	−1.37615	−0.688074	+	0.725640i	$0.741544\pi$
$390$	0	0
$391$	1178.35	0.152409
$392$	0	0
$393$	−83.1908	−0.0106779
$394$	0	0
$395$	8724.15	1.11129
$396$	0	0
$397$	540.574	0.0683391	0.0341696	−	0.999416i	$-0.489121\pi$
$397$	540.574	0.0683391	0.0341696	+	0.999416i	$0.489121\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2167.03	0.269866	0.134933	−	0.990855i	$-0.456918\pi$
$401$	2167.03	0.269866	0.134933	+	0.990855i	$0.456918\pi$
$402$	0	0
$403$	−3674.17	−0.454153
$404$	0	0
$405$	−661.722	−0.0811882
$406$	0	0
$407$	−12572.3	−1.53117
$408$	0	0
$409$	957.690	0.115782	0.0578908	−	0.998323i	$-0.481562\pi$
$409$	957.690	0.115782	0.0578908	+	0.998323i	$0.481562\pi$
$410$	0	0
$411$	−7088.93	−0.850781
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7407.66	−0.876211
$416$	0	0
$417$	7539.84	0.885437
$418$	0	0
$419$	6464.42	0.753718	0.376859	−	0.926271i	$-0.377004\pi$
$419$	6464.42	0.753718	0.376859	+	0.926271i	$0.377004\pi$
$420$	0	0
$421$	−6201.23	−0.717885	−0.358943	−	0.933360i	$-0.616863\pi$
$421$	−6201.23	−0.717885	−0.358943	+	0.933360i	$0.616863\pi$
$422$	0	0
$423$	2771.35	0.318552
$424$	0	0
$425$	578.877	0.0660698
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4530.54	0.509875
$430$	0	0
$431$	−1415.59	−0.158206	−0.0791029	−	0.996866i	$-0.525206\pi$
$431$	−1415.59	−0.158206	−0.0791029	+	0.996866i	$0.525206\pi$
$432$	0	0
$433$	−8905.73	−0.988411	−0.494206	−	0.869345i	$-0.664541\pi$
$433$	−8905.73	−0.988411	−0.494206	+	0.869345i	$0.664541\pi$
$434$	0	0
$435$	1921.59	0.211801
$436$	0	0
$437$	10726.4	1.17417
$438$	0	0
$439$	10876.5	1.18248	0.591238	−	0.806497i	$-0.298639\pi$
$439$	10876.5	1.18248	0.591238	+	0.806497i	$0.298639\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11403.0	−1.22296	−0.611479	−	0.791260i	$-0.709425\pi$
$443$	−11403.0	−1.22296	−0.611479	+	0.791260i	$0.709425\pi$
$444$	0	0
$445$	9094.74	0.968837
$446$	0	0
$447$	6797.49	0.719262
$448$	0	0
$449$	−12689.0	−1.33370	−0.666852	−	0.745190i	$-0.732359\pi$
$449$	−12689.0	−1.33370	−0.666852	+	0.745190i	$0.732359\pi$
$450$	0	0
$451$	2713.32	0.283293
$452$	0	0
$453$	−849.439	−0.0881019
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2270.05	0.232360	0.116180	−	0.993228i	$-0.462935\pi$
$457$	2270.05	0.232360	0.116180	+	0.993228i	$0.462935\pi$
$458$	0	0
$459$	−268.271	−0.0272806
$460$	0	0
$461$	−10731.2	−1.08417	−0.542085	−	0.840324i	$-0.682365\pi$
$461$	−10731.2	−1.08417	−0.542085	+	0.840324i	$0.682365\pi$
$462$	0	0
$463$	3307.74	0.332017	0.166008	−	0.986124i	$-0.446912\pi$
$463$	3307.74	0.332017	0.166008	+	0.986124i	$0.446912\pi$
$464$	0	0
$465$	−2255.02	−0.224891
$466$	0	0
$467$	−9246.69	−0.916244	−0.458122	−	0.888889i	$-0.651478\pi$
$467$	−9246.69	−0.916244	−0.458122	+	0.888889i	$0.651478\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−577.742	−0.0565201
$472$	0	0
$473$	−4384.96	−0.426260
$474$	0	0
$475$	5269.45	0.509008
$476$	0	0
$477$	−3628.35	−0.348283
$478$	0	0
$479$	15223.3	1.45213	0.726066	−	0.687625i	$-0.241347\pi$
$479$	15223.3	1.45213	0.726066	+	0.687625i	$0.241347\pi$
$480$	0	0
$481$	−13274.8	−1.25837
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12098.4	1.13270
$486$	0	0
$487$	7758.72	0.721933	0.360966	−	0.932579i	$-0.382447\pi$
$487$	7758.72	0.721933	0.360966	+	0.932579i	$0.382447\pi$
$488$	0	0
$489$	2528.50	0.233830
$490$	0	0
$491$	8342.30	0.766767	0.383384	−	0.923589i	$-0.374759\pi$
$491$	8342.30	0.766767	0.383384	+	0.923589i	$0.374759\pi$
$492$	0	0
$493$	779.040	0.0711688
$494$	0	0
$495$	2780.62	0.252484
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2586.95	0.232079	0.116040	−	0.993245i	$-0.462980\pi$
$499$	2586.95	0.232079	0.116040	+	0.993245i	$0.462980\pi$
$500$	0	0
$501$	−11577.6	−1.03244
$502$	0	0
$503$	409.682	0.0363157	0.0181578	−	0.999835i	$-0.494220\pi$
$503$	409.682	0.0363157	0.0181578	+	0.999835i	$0.494220\pi$
$504$	0	0
$505$	−4544.77	−0.400475
$506$	0	0
$507$	−1807.32	−0.158316
$508$	0	0
$509$	−4390.51	−0.382330	−0.191165	−	0.981558i	$-0.561226\pi$
$509$	−4390.51	−0.382330	−0.191165	+	0.981558i	$0.561226\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2442.04	−0.210173
$514$	0	0
$515$	−4513.06	−0.386154
$516$	0	0
$517$	−11645.5	−0.990652
$518$	0	0
$519$	−4531.98	−0.383298
$520$	0	0
$521$	19429.5	1.63382	0.816912	−	0.576762i	$-0.195684\pi$
$521$	19429.5	1.63382	0.816912	+	0.576762i	$0.195684\pi$
$522$	0	0
$523$	17948.7	1.50065	0.750326	−	0.661067i	$-0.229896\pi$
$523$	17948.7	1.50065	0.750326	+	0.661067i	$0.229896\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−914.217	−0.0755672
$528$	0	0
$529$	1897.67	0.155968
$530$	0	0
$531$	5343.39	0.436692
$532$	0	0
$533$	2864.92	0.232821
$534$	0	0
$535$	4360.86	0.352405
$536$	0	0
$537$	7393.88	0.594170
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23094.0	1.83528	0.917641	−	0.397411i	$-0.130091\pi$
$541$	23094.0	1.83528	0.917641	+	0.397411i	$0.130091\pi$
$542$	0	0
$543$	9892.09	0.781787
$544$	0	0
$545$	−8942.47	−0.702850
$546$	0	0
$547$	2266.68	0.177178	0.0885889	−	0.996068i	$-0.471764\pi$
$547$	2266.68	0.177178	0.0885889	+	0.996068i	$0.471764\pi$
$548$	0	0
$549$	−2998.53	−0.233104
$550$	0	0
$551$	7091.50	0.548291
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8147.40	−0.623131
$556$	0	0
$557$	−11038.5	−0.839709	−0.419854	−	0.907592i	$-0.637919\pi$
$557$	−11038.5	−0.839709	−0.419854	+	0.907592i	$0.637919\pi$
$558$	0	0
$559$	−4629.97	−0.350316
$560$	0	0
$561$	1127.30	0.0848390
$562$	0	0
$563$	−7059.36	−0.528448	−0.264224	−	0.964461i	$-0.585116\pi$
$563$	−7059.36	−0.528448	−0.264224	+	0.964461i	$0.585116\pi$
$564$	0	0
$565$	11642.9	0.866936
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1352.81	−0.0996706	−0.0498353	−	0.998757i	$-0.515870\pi$
$569$	−1352.81	−0.0996706	−0.0498353	+	0.998757i	$0.515870\pi$
$570$	0	0
$571$	20839.2	1.52731	0.763655	−	0.645625i	$-0.223403\pi$
$571$	20839.2	1.52731	0.763655	+	0.645625i	$0.223403\pi$
$572$	0	0
$573$	14189.1	1.03448
$574$	0	0
$575$	6909.42	0.501117
$576$	0	0
$577$	−10014.6	−0.722556	−0.361278	−	0.932458i	$-0.617659\pi$
$577$	−10014.6	−0.722556	−0.361278	+	0.932458i	$0.617659\pi$
$578$	0	0
$579$	13659.5	0.980433
$580$	0	0
$581$	0	0
$582$	0	0
$583$	15246.7	1.08311
$584$	0	0
$585$	2935.98	0.207501
$586$	0	0
$587$	20864.8	1.46709	0.733546	−	0.679640i	$-0.237864\pi$
$587$	20864.8	1.46709	0.733546	+	0.679640i	$0.237864\pi$
$588$	0	0
$589$	−8322.01	−0.582177
$590$	0	0
$591$	−9327.19	−0.649187
$592$	0	0
$593$	−17547.7	−1.21517	−0.607586	−	0.794254i	$-0.707862\pi$
$593$	−17547.7	−1.21517	−0.607586	+	0.794254i	$0.707862\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1330.54	−0.0912147
$598$	0	0
$599$	130.682	0.00891405	0.00445703	−	0.999990i	$-0.498581\pi$
$599$	130.682	0.00891405	0.00445703	+	0.999990i	$0.498581\pi$
$600$	0	0
$601$	5964.47	0.404818	0.202409	−	0.979301i	$-0.435123\pi$
$601$	5964.47	0.404818	0.202409	+	0.979301i	$0.435123\pi$
$602$	0	0
$603$	6691.60	0.451912
$604$	0	0
$605$	−810.944	−0.0544951
$606$	0	0
$607$	−9911.63	−0.662769	−0.331384	−	0.943496i	$-0.607516\pi$
$607$	−9911.63	−0.662769	−0.331384	+	0.943496i	$0.607516\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12296.1	−0.814155
$612$	0	0
$613$	−6482.53	−0.427124	−0.213562	−	0.976930i	$-0.568506\pi$
$613$	−6482.53	−0.427124	−0.213562	+	0.976930i	$0.568506\pi$
$614$	0	0
$615$	1758.34	0.115290
$616$	0	0
$617$	7189.36	0.469097	0.234548	−	0.972104i	$-0.424639\pi$
$617$	7189.36	0.469097	0.234548	+	0.972104i	$0.424639\pi$
$618$	0	0
$619$	1861.39	0.120865	0.0604327	−	0.998172i	$-0.480752\pi$
$619$	1861.39	0.120865	0.0604327	+	0.998172i	$0.480752\pi$
$620$	0	0
$621$	−3202.05	−0.206915
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4948.07	−0.316677
$626$	0	0
$627$	10261.7	0.653607
$628$	0	0
$629$	−3303.06	−0.209383
$630$	0	0
$631$	29032.0	1.83161	0.915803	−	0.401627i	$-0.131555\pi$
$631$	29032.0	1.83161	0.915803	+	0.401627i	$0.131555\pi$
$632$	0	0
$633$	−13959.9	−0.876547
$634$	0	0
$635$	−6425.12	−0.401532
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6554.45	0.405775
$640$	0	0
$641$	24457.7	1.50705	0.753527	−	0.657417i	$-0.228351\pi$
$641$	24457.7	1.50705	0.753527	+	0.657417i	$0.228351\pi$
$642$	0	0
$643$	10968.2	0.672695	0.336348	−	0.941738i	$-0.390808\pi$
$643$	10968.2	0.672695	0.336348	+	0.941738i	$0.390808\pi$
$644$	0	0
$645$	−2841.64	−0.173472
$646$	0	0
$647$	5484.74	0.333272	0.166636	−	0.986018i	$-0.446709\pi$
$647$	5484.74	0.333272	0.166636	+	0.986018i	$0.446709\pi$
$648$	0	0
$649$	−22453.4	−1.35805
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10330.0	−0.619055	−0.309527	−	0.950891i	$-0.600171\pi$
$653$	−10330.0	−0.619055	−0.309527	+	0.950891i	$0.600171\pi$
$654$	0	0
$655$	226.540	0.0135140
$656$	0	0
$657$	7215.74	0.428482
$658$	0	0
$659$	−26822.2	−1.58550	−0.792751	−	0.609546i	$-0.791352\pi$
$659$	−26822.2	−1.58550	−0.792751	+	0.609546i	$0.791352\pi$
$660$	0	0
$661$	−2498.76	−0.147035	−0.0735177	−	0.997294i	$-0.523423\pi$
$661$	−2498.76	−0.147035	−0.0735177	+	0.997294i	$0.523423\pi$
$662$	0	0
$663$	1190.29	0.0697238
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9298.54	0.539791
$668$	0	0
$669$	8336.71	0.481788
$670$	0	0
$671$	12600.1	0.724922
$672$	0	0
$673$	10092.1	0.578044	0.289022	−	0.957322i	$-0.406670\pi$
$673$	10092.1	0.578044	0.289022	+	0.957322i	$0.406670\pi$
$674$	0	0
$675$	−1573.04	−0.0896984
$676$	0	0
$677$	26366.4	1.49681	0.748406	−	0.663241i	$-0.230820\pi$
$677$	26366.4	1.49681	0.748406	+	0.663241i	$0.230820\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9651.67	0.543102
$682$	0	0
$683$	−22570.8	−1.26449	−0.632245	−	0.774769i	$-0.717866\pi$
$683$	−22570.8	−1.26449	−0.632245	+	0.774769i	$0.717866\pi$
$684$	0	0
$685$	19304.1	1.07675
$686$	0	0
$687$	11184.0	0.621103
$688$	0	0
$689$	16098.6	0.890140
$690$	0	0
$691$	−11931.5	−0.656867	−0.328434	−	0.944527i	$-0.606521\pi$
$691$	−11931.5	−0.656867	−0.328434	+	0.944527i	$0.606521\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20532.0	−1.12061
$696$	0	0
$697$	712.856	0.0387394
$698$	0	0
$699$	−10810.5	−0.584967
$700$	0	0
$701$	16592.6	0.893997	0.446999	−	0.894535i	$-0.352493\pi$
$701$	16592.6	0.893997	0.446999	+	0.894535i	$0.352493\pi$
$702$	0	0
$703$	−30067.4	−1.61311
$704$	0	0
$705$	−7546.75	−0.403159
$706$	0	0
$707$	0	0
$708$	0	0
$709$	18714.0	0.991283	0.495641	−	0.868527i	$-0.334933\pi$
$709$	18714.0	0.991283	0.495641	+	0.868527i	$0.334933\pi$
$710$	0	0
$711$	−9611.15	−0.506957
$712$	0	0
$713$	−10912.0	−0.573152
$714$	0	0
$715$	−12337.3	−0.645298
$716$	0	0
$717$	−7046.27	−0.367012
$718$	0	0
$719$	−24854.8	−1.28919	−0.644594	−	0.764525i	$-0.722974\pi$
$719$	−24854.8	−1.28919	−0.644594	+	0.764525i	$0.722974\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	15280.2	0.785997
$724$	0	0
$725$	4568.00	0.234002
$726$	0	0
$727$	−22506.0	−1.14814	−0.574071	−	0.818805i	$-0.694637\pi$
$727$	−22506.0	−1.14814	−0.574071	+	0.818805i	$0.694637\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1152.04	−0.0582897
$732$	0	0
$733$	23792.0	1.19888	0.599440	−	0.800420i	$-0.295390\pi$
$733$	23792.0	1.19888	0.599440	+	0.800420i	$0.295390\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28118.8	−1.40538
$738$	0	0
$739$	−4803.24	−0.239093	−0.119547	−	0.992829i	$-0.538144\pi$
$739$	−4803.24	−0.239093	−0.119547	+	0.992829i	$0.538144\pi$
$740$	0	0
$741$	10835.0	0.537159
$742$	0	0
$743$	−5076.10	−0.250638	−0.125319	−	0.992116i	$-0.539995\pi$
$743$	−5076.10	−0.250638	−0.125319	+	0.992116i	$0.539995\pi$
$744$	0	0
$745$	−18510.5	−0.910298
$746$	0	0
$747$	8160.80	0.399716
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9158.72	0.445015	0.222508	−	0.974931i	$-0.428576\pi$
$751$	9158.72	0.445015	0.222508	+	0.974931i	$0.428576\pi$
$752$	0	0
$753$	−17817.1	−0.862271
$754$	0	0
$755$	2313.14	0.111502
$756$	0	0
$757$	33682.2	1.61717	0.808587	−	0.588377i	$-0.200233\pi$
$757$	33682.2	1.61717	0.808587	+	0.588377i	$0.200233\pi$
$758$	0	0
$759$	13455.3	0.643475
$760$	0	0
$761$	−13346.3	−0.635745	−0.317873	−	0.948133i	$-0.602968\pi$
$761$	−13346.3	−0.635745	−0.317873	+	0.948133i	$0.602968\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	730.538	0.0345264
$766$	0	0
$767$	−23708.0	−1.11610
$768$	0	0
$769$	15530.6	0.728281	0.364140	−	0.931344i	$-0.381363\pi$
$769$	15530.6	0.728281	0.364140	+	0.931344i	$0.381363\pi$
$770$	0	0
$771$	4547.66	0.212425
$772$	0	0
$773$	11015.2	0.512536	0.256268	−	0.966606i	$-0.417507\pi$
$773$	11015.2	0.512536	0.256268	+	0.966606i	$0.417507\pi$
$774$	0	0
$775$	−5360.63	−0.248464
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6489.04	0.298452
$780$	0	0
$781$	−27542.4	−1.26190
$782$	0	0
$783$	−2116.96	−0.0966209
$784$	0	0
$785$	1573.27	0.0715317
$786$	0	0
$787$	21192.6	0.959892	0.479946	−	0.877298i	$-0.340656\pi$
$787$	21192.6	0.959892	0.479946	+	0.877298i	$0.340656\pi$
$788$	0	0
$789$	19301.8	0.870927
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13304.1	0.595768
$794$	0	0
$795$	9880.49	0.440786
$796$	0	0
$797$	−18189.9	−0.808432	−0.404216	−	0.914663i	$-0.632456\pi$
$797$	−18189.9	−0.808432	−0.404216	+	0.914663i	$0.632456\pi$
$798$	0	0
$799$	−3059.56	−0.135468
$800$	0	0
$801$	−10019.4	−0.441971
$802$	0	0
$803$	−30321.2	−1.33252
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20849.4	0.909458
$808$	0	0
$809$	45516.5	1.97809	0.989045	−	0.147611i	$-0.0471584\pi$
$809$	45516.5	1.97809	0.989045	+	0.147611i	$0.0471584\pi$
$810$	0	0
$811$	−42099.9	−1.82285	−0.911423	−	0.411472i	$-0.865015\pi$
$811$	−42099.9	−1.82285	−0.911423	+	0.411472i	$0.865015\pi$
$812$	0	0
$813$	−2885.98	−0.124497
$814$	0	0
$815$	−6885.45	−0.295935
$816$	0	0
$817$	−10486.9	−0.449069
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26555.9	−1.12888	−0.564439	−	0.825475i	$-0.690907\pi$
$821$	−26555.9	−1.12888	−0.564439	+	0.825475i	$0.690907\pi$
$822$	0	0
$823$	1554.25	0.0658295	0.0329147	−	0.999458i	$-0.489521\pi$
$823$	1554.25	0.0658295	0.0329147	+	0.999458i	$0.489521\pi$
$824$	0	0
$825$	6610.07	0.278949
$826$	0	0
$827$	10548.3	0.443531	0.221766	−	0.975100i	$-0.428818\pi$
$827$	10548.3	0.443531	0.221766	+	0.975100i	$0.428818\pi$
$828$	0	0
$829$	−4331.26	−0.181461	−0.0907303	−	0.995875i	$-0.528920\pi$
$829$	−4331.26	−0.181461	−0.0907303	+	0.995875i	$0.528920\pi$
$830$	0	0
$831$	2280.03	0.0951786
$832$	0	0
$833$	0	0
$834$	0	0
$835$	31527.5	1.30665
$836$	0	0
$837$	2484.30	0.102592
$838$	0	0
$839$	−23557.7	−0.969369	−0.484685	−	0.874689i	$-0.661066\pi$
$839$	−23557.7	−0.969369	−0.484685	+	0.874689i	$0.661066\pi$
$840$	0	0
$841$	−18241.5	−0.747939
$842$	0	0
$843$	−13236.2	−0.540782
$844$	0	0
$845$	4921.59	0.200364
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7806.45	−0.315567
$850$	0	0
$851$	−39425.0	−1.58810
$852$	0	0
$853$	11493.6	0.461350	0.230675	−	0.973031i	$-0.425907\pi$
$853$	11493.6	0.461350	0.230675	+	0.973031i	$0.425907\pi$
$854$	0	0
$855$	6650.00	0.265994
$856$	0	0
$857$	2534.23	0.101012	0.0505062	−	0.998724i	$-0.483917\pi$
$857$	2534.23	0.101012	0.0505062	+	0.998724i	$0.483917\pi$
$858$	0	0
$859$	3127.09	0.124208	0.0621041	−	0.998070i	$-0.480219\pi$
$859$	3127.09	0.124208	0.0621041	+	0.998070i	$0.480219\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15309.9	−0.603887	−0.301943	−	0.953326i	$-0.597635\pi$
$863$	−15309.9	−0.603887	−0.301943	+	0.953326i	$0.597635\pi$
$864$	0	0
$865$	12341.2	0.485101
$866$	0	0
$867$	−14442.8	−0.565749
$868$	0	0
$869$	40387.0	1.57656
$870$	0	0
$871$	−29689.8	−1.15500
$872$	0	0
$873$	−13328.4	−0.516723
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42420.7	−1.63335	−0.816673	−	0.577100i	$-0.804184\pi$
$877$	−42420.7	−1.63335	−0.816673	+	0.577100i	$0.804184\pi$
$878$	0	0
$879$	27996.5	1.07429
$880$	0	0
$881$	−10060.8	−0.384740	−0.192370	−	0.981322i	$-0.561617\pi$
$881$	−10060.8	−0.384740	−0.192370	+	0.981322i	$0.561617\pi$
$882$	0	0
$883$	−17437.8	−0.664585	−0.332293	−	0.943176i	$-0.607822\pi$
$883$	−17437.8	−0.664585	−0.332293	+	0.943176i	$0.607822\pi$
$884$	0	0
$885$	−14550.8	−0.552677
$886$	0	0
$887$	−16016.7	−0.606299	−0.303149	−	0.952943i	$-0.598038\pi$
$887$	−16016.7	−0.606299	−0.303149	+	0.952943i	$0.598038\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3063.33	−0.115180
$892$	0	0
$893$	−27850.8	−1.04366
$894$	0	0
$895$	−20134.5	−0.751981
$896$	0	0
$897$	14207.1	0.528832
$898$	0	0
$899$	−7214.22	−0.267639
$900$	0	0
$901$	4005.69	0.148112
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−26937.5	−0.989428
$906$	0	0
$907$	10131.0	0.370887	0.185443	−	0.982655i	$-0.440628\pi$
$907$	10131.0	0.370887	0.185443	+	0.982655i	$0.440628\pi$
$908$	0	0
$909$	5006.85	0.182692
$910$	0	0
$911$	−1320.58	−0.0480273	−0.0240137	−	0.999712i	$-0.507645\pi$
$911$	−1320.58	−0.0480273	−0.0240137	+	0.999712i	$0.507645\pi$
$912$	0	0
$913$	−34292.5	−1.24306
$914$	0	0
$915$	8165.41	0.295017
$916$	0	0
$917$	0	0
$918$	0	0
$919$	7285.76	0.261518	0.130759	−	0.991414i	$-0.458259\pi$
$919$	7285.76	0.261518	0.130759	+	0.991414i	$0.458259\pi$
$920$	0	0
$921$	14686.5	0.525448
$922$	0	0
$923$	−29081.3	−1.03708
$924$	0	0
$925$	−19368.0	−0.688448
$926$	0	0
$927$	4971.91	0.176159
$928$	0	0
$929$	−23708.0	−0.837280	−0.418640	−	0.908152i	$-0.637493\pi$
$929$	−23708.0	−0.837280	−0.418640	+	0.908152i	$0.637493\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	22515.8	0.790070
$934$	0	0
$935$	−3069.79	−0.107372
$936$	0	0
$937$	8853.60	0.308682	0.154341	−	0.988018i	$-0.450675\pi$
$937$	8853.60	0.308682	0.154341	+	0.988018i	$0.450675\pi$
$938$	0	0
$939$	19047.8	0.661983
$940$	0	0
$941$	53971.6	1.86974	0.934869	−	0.354994i	$-0.115517\pi$
$941$	53971.6	1.86974	0.934869	+	0.354994i	$0.115517\pi$
$942$	0	0
$943$	8508.57	0.293825
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8019.80	−0.275194	−0.137597	−	0.990488i	$-0.543938\pi$
$947$	−8019.80	−0.275194	−0.137597	+	0.990488i	$0.543938\pi$
$948$	0	0
$949$	−32015.4	−1.09511
$950$	0	0
$951$	−23998.8	−0.818312
$952$	0	0
$953$	42628.0	1.44896	0.724479	−	0.689297i	$-0.242080\pi$
$953$	42628.0	1.44896	0.724479	+	0.689297i	$0.242080\pi$
$954$	0	0
$955$	−38638.9	−1.30924
$956$	0	0
$957$	8895.69	0.300477
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21325.0	−0.715820
$962$	0	0
$963$	−4804.24	−0.160763
$964$	0	0
$965$	−37196.8	−1.24084
$966$	0	0
$967$	43161.7	1.43535	0.717676	−	0.696377i	$-0.245206\pi$
$967$	43161.7	1.43535	0.717676	+	0.696377i	$0.245206\pi$
$968$	0	0
$969$	2696.00	0.0893787
$970$	0	0
$971$	−39775.4	−1.31458	−0.657288	−	0.753640i	$-0.728296\pi$
$971$	−39775.4	−1.31458	−0.657288	+	0.753640i	$0.728296\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6979.40	0.229251
$976$	0	0
$977$	−26658.1	−0.872945	−0.436472	−	0.899718i	$-0.643772\pi$
$977$	−26658.1	−0.872945	−0.436472	+	0.899718i	$0.643772\pi$
$978$	0	0
$979$	42102.6	1.37447
$980$	0	0
$981$	9851.66	0.320631
$982$	0	0
$983$	−2942.43	−0.0954719	−0.0477359	−	0.998860i	$-0.515201\pi$
$983$	−2942.43	−0.0954719	−0.0477359	+	0.998860i	$0.515201\pi$
$984$	0	0
$985$	25399.2	0.821610
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13750.6	−0.442108
$990$	0	0
$991$	−44969.8	−1.44149	−0.720743	−	0.693203i	$-0.756199\pi$
$991$	−44969.8	−1.44149	−0.720743	+	0.693203i	$0.756199\pi$
$992$	0	0
$993$	6364.68	0.203401
$994$	0	0
$995$	3623.23	0.115441
$996$	0	0
$997$	−27116.4	−0.861370	−0.430685	−	0.902502i	$-0.641728\pi$
$997$	−27116.4	−0.861370	−0.430685	+	0.902502i	$0.641728\pi$
$998$	0	0
$999$	8975.75	0.284265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cq.1.2		4
4.3	odd	2		588.4.a.j.1.2	✓	4
7.6	odd	2		2352.4.a.cl.1.3		4
12.11	even	2		1764.4.a.bc.1.3		4
28.3	even	6		588.4.i.k.373.2		8
28.11	odd	6		588.4.i.l.373.3		8
28.19	even	6		588.4.i.k.361.2		8
28.23	odd	6		588.4.i.l.361.3		8
28.27	even	2		588.4.a.k.1.3	yes	4
84.11	even	6		1764.4.k.bb.1549.2		8
84.23	even	6		1764.4.k.bb.361.2		8
84.47	odd	6		1764.4.k.bd.361.3		8
84.59	odd	6		1764.4.k.bd.1549.3		8
84.83	odd	2		1764.4.a.ba.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.2	✓	4	4.3	odd	2
588.4.a.k.1.3	yes	4	28.27	even	2
588.4.i.k.361.2		8	28.19	even	6
588.4.i.k.373.2		8	28.3	even	6
588.4.i.l.361.3		8	28.23	odd	6
588.4.i.l.373.3		8	28.11	odd	6
1764.4.a.ba.1.2		4	84.83	odd	2
1764.4.a.bc.1.3		4	12.11	even	2
1764.4.k.bb.361.2		8	84.23	even	6
1764.4.k.bb.1549.2		8	84.11	even	6
1764.4.k.bd.361.3		8	84.47	odd	6
1764.4.k.bd.1549.3		8	84.59	odd	6
2352.4.a.cl.1.3		4	7.6	odd	2
2352.4.a.cq.1.2		4	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} -37.8189 q^{11} -39.9319 q^{13} -24.5082 q^{15} -9.93596 q^{17} -90.4458 q^{19} -118.595 q^{23} -58.2608 q^{25} +27.0000 q^{27} -78.4061 q^{29} +92.0110 q^{31} -113.457 q^{33} +332.435 q^{37} -119.796 q^{39} -71.7451 q^{41} +115.947 q^{43} -73.5246 q^{45} +307.927 q^{47} -29.8079 q^{51} -403.150 q^{53} +308.958 q^{55} -271.337 q^{57} +593.710 q^{59} -333.170 q^{61} +326.220 q^{65} +743.511 q^{67} -355.784 q^{69} +728.272 q^{71} +801.749 q^{73} -174.782 q^{75} -1067.91 q^{79} +81.0000 q^{81} +906.756 q^{83} +81.1709 q^{85} -235.218 q^{87} -1113.27 q^{89} +276.033 q^{93} +738.889 q^{95} -1480.94 q^{97} -340.370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 36 * q^9	\( 4 q + 12 q^{3} + 36 q^{9} - 48 q^{17} + 192 q^{19} - 192 q^{23} + 324 q^{25} + 108 q^{27} + 96 q^{29} + 48 q^{31} + 256 q^{37} - 1008 q^{41} + 112 q^{43} + 864 q^{47} - 144 q^{51} - 648 q^{53} + 2352 q^{55} + 576 q^{57} + 336 q^{59} - 960 q^{61} - 360 q^{65} - 720 q^{67} - 576 q^{69} + 1344 q^{71} - 672 q^{73} + 972 q^{75} + 1984 q^{79} + 324 q^{81} + 3120 q^{83} + 680 q^{85} + 288 q^{87} - 2160 q^{89} + 144 q^{93} + 3744 q^{95} - 2016 q^{97}+O(q^{100}) \) 4 * q + 12 * q^3 + 36 * q^9 - 48 * q^17 + 192 * q^19 - 192 * q^23 + 324 * q^25 + 108 * q^27 + 96 * q^29 + 48 * q^31 + 256 * q^37 - 1008 * q^41 + 112 * q^43 + 864 * q^47 - 144 * q^51 - 648 * q^53 + 2352 * q^55 + 576 * q^57 + 336 * q^59 - 960 * q^61 - 360 * q^65 - 720 * q^67 - 576 * q^69 + 1344 * q^71 - 672 * q^73 + 972 * q^75 + 1984 * q^79 + 324 * q^81 + 3120 * q^83 + 680 * q^85 + 288 * q^87 - 2160 * q^89 + 144 * q^93 + 3744 * q^95 - 2016 * q^97

Introduction

L-functions

Modular forms

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Database

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