LMFDB - Embedded newform 2352.4.a.cr.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.145408.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} + 142 $ x^4 - 24*x^2 + 142
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.66254$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -4.11659 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -4.11659 q^{5} +9.00000 q^{9} -17.9344 q^{11} +23.4236 q^{13} -12.3498 q^{15} +76.2280 q^{17} -35.5964 q^{19} +40.7918 q^{23} -108.054 q^{25} +27.0000 q^{27} -178.805 q^{29} -31.6797 q^{31} -53.8032 q^{33} -54.8450 q^{37} +70.2708 q^{39} -190.547 q^{41} +131.964 q^{43} -37.0493 q^{45} -199.573 q^{47} +228.684 q^{51} +321.907 q^{53} +73.8287 q^{55} -106.789 q^{57} -163.392 q^{59} +265.230 q^{61} -96.4254 q^{65} -278.127 q^{67} +122.375 q^{69} +10.5794 q^{71} +584.292 q^{73} -324.161 q^{75} +183.078 q^{79} +81.0000 q^{81} +175.117 q^{83} -313.800 q^{85} -536.416 q^{87} +47.1437 q^{89} -95.0391 q^{93} +146.536 q^{95} -556.805 q^{97} -161.410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} - 152 q^{17} - 224 q^{19} + 8 q^{23} - 28 q^{25} + 108 q^{27} - 144 q^{29} - 400 q^{31} + 120 q^{33} - 304 q^{37} + 144 q^{39} - 152 q^{41} - 160 q^{43} + 72 q^{45} - 544 q^{47} - 456 q^{51} - 1320 q^{53} - 16 q^{55} - 672 q^{57} - 1040 q^{59} + 896 q^{61} - 648 q^{65} + 416 q^{67} + 24 q^{69} - 248 q^{71} - 752 q^{73} - 84 q^{75} - 864 q^{79} + 324 q^{81} - 1456 q^{83} - 1608 q^{85} - 432 q^{87} + 2936 q^{89} - 1200 q^{93} + 80 q^{95} + 144 q^{97} + 360 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 - 152 * q^17 - 224 * q^19 + 8 * q^23 - 28 * q^25 + 108 * q^27 - 144 * q^29 - 400 * q^31 + 120 * q^33 - 304 * q^37 + 144 * q^39 - 152 * q^41 - 160 * q^43 + 72 * q^45 - 544 * q^47 - 456 * q^51 - 1320 * q^53 - 16 * q^55 - 672 * q^57 - 1040 * q^59 + 896 * q^61 - 648 * q^65 + 416 * q^67 + 24 * q^69 - 248 * q^71 - 752 * q^73 - 84 * q^75 - 864 * q^79 + 324 * q^81 - 1456 * q^83 - 1608 * q^85 - 432 * q^87 + 2936 * q^89 - 1200 * q^93 + 80 * q^95 + 144 * q^97 + 360 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−4.11659	−0.368199	−0.184100	−	0.982908i	$-0.558937\pi$
$5$	−4.11659	−0.368199	−0.184100	+	0.982908i	$0.558937\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−17.9344	−0.491585	−0.245792	−	0.969323i	$-0.579048\pi$
$11$	−17.9344	−0.491585	−0.245792	+	0.969323i	$0.579048\pi$
$12$	0	0
$13$	23.4236	0.499733	0.249867	−	0.968280i	$-0.419613\pi$
$13$	23.4236	0.499733	0.249867	+	0.968280i	$0.419613\pi$
$14$	0	0
$15$	−12.3498	−0.212580
$16$	0	0
$17$	76.2280	1.08753	0.543765	−	0.839237i	$-0.316998\pi$
$17$	76.2280	1.08753	0.543765	+	0.839237i	$0.316998\pi$
$18$	0	0
$19$	−35.5964	−0.429809	−0.214905	−	0.976635i	$-0.568944\pi$
$19$	−35.5964	−0.429809	−0.214905	+	0.976635i	$0.568944\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	40.7918	0.369812	0.184906	−	0.982756i	$-0.440802\pi$
$23$	40.7918	0.369812	0.184906	+	0.982756i	$0.440802\pi$
$24$	0	0
$25$	−108.054	−0.864429
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−178.805	−1.14494	−0.572471	−	0.819925i	$-0.694015\pi$
$29$	−178.805	−1.14494	−0.572471	+	0.819925i	$0.694015\pi$
$30$	0	0
$31$	−31.6797	−0.183543	−0.0917717	−	0.995780i	$-0.529253\pi$
$31$	−31.6797	−0.183543	−0.0917717	+	0.995780i	$0.529253\pi$
$32$	0	0
$33$	−53.8032	−0.283816
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−54.8450	−0.243688	−0.121844	−	0.992549i	$-0.538881\pi$
$37$	−54.8450	−0.243688	−0.121844	+	0.992549i	$0.538881\pi$
$38$	0	0
$39$	70.2708	0.288521
$40$	0	0
$41$	−190.547	−0.725815	−0.362907	−	0.931825i	$-0.618216\pi$
$41$	−190.547	−0.725815	−0.362907	+	0.931825i	$0.618216\pi$
$42$	0	0
$43$	131.964	0.468009	0.234004	−	0.972236i	$-0.424817\pi$
$43$	131.964	0.468009	0.234004	+	0.972236i	$0.424817\pi$
$44$	0	0
$45$	−37.0493	−0.122733
$46$	0	0
$47$	−199.573	−0.619377	−0.309689	−	0.950838i	$-0.600225\pi$
$47$	−199.573	−0.619377	−0.309689	+	0.950838i	$0.600225\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	228.684	0.627886
$52$	0	0
$53$	321.907	0.834289	0.417145	−	0.908840i	$-0.363031\pi$
$53$	321.907	0.834289	0.417145	+	0.908840i	$0.363031\pi$
$54$	0	0
$55$	73.8287	0.181001
$56$	0	0
$57$	−106.789	−0.248150
$58$	0	0
$59$	−163.392	−0.360540	−0.180270	−	0.983617i	$-0.557697\pi$
$59$	−163.392	−0.360540	−0.180270	+	0.983617i	$0.557697\pi$
$60$	0	0
$61$	265.230	0.556710	0.278355	−	0.960478i	$-0.410211\pi$
$61$	265.230	0.556710	0.278355	+	0.960478i	$0.410211\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−96.4254	−0.184001
$66$	0	0
$67$	−278.127	−0.507144	−0.253572	−	0.967316i	$-0.581606\pi$
$67$	−278.127	−0.507144	−0.253572	+	0.967316i	$0.581606\pi$
$68$	0	0
$69$	122.375	0.213511
$70$	0	0
$71$	10.5794	0.0176836	0.00884182	−	0.999961i	$-0.497186\pi$
$71$	10.5794	0.0176836	0.00884182	+	0.999961i	$0.497186\pi$
$72$	0	0
$73$	584.292	0.936797	0.468398	−	0.883517i	$-0.344831\pi$
$73$	584.292	0.936797	0.468398	+	0.883517i	$0.344831\pi$
$74$	0	0
$75$	−324.161	−0.499079
$76$	0	0
$77$	0	0
$78$	0	0
$79$	183.078	0.260733	0.130366	−	0.991466i	$-0.458385\pi$
$79$	183.078	0.260733	0.130366	+	0.991466i	$0.458385\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	175.117	0.231585	0.115792	−	0.993273i	$-0.463059\pi$
$83$	175.117	0.231585	0.115792	+	0.993273i	$0.463059\pi$
$84$	0	0
$85$	−313.800	−0.400428
$86$	0	0
$87$	−536.416	−0.661033
$88$	0	0
$89$	47.1437	0.0561485	0.0280743	−	0.999606i	$-0.491063\pi$
$89$	47.1437	0.0561485	0.0280743	+	0.999606i	$0.491063\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−95.0391	−0.105969
$94$	0	0
$95$	146.536	0.158255
$96$	0	0
$97$	−556.805	−0.582834	−0.291417	−	0.956596i	$-0.594127\pi$
$97$	−556.805	−0.582834	−0.291417	+	0.956596i	$0.594127\pi$
$98$	0	0
$99$	−161.410	−0.163862
$100$	0	0
$101$	128.804	0.126895	0.0634477	−	0.997985i	$-0.479790\pi$
$101$	128.804	0.126895	0.0634477	+	0.997985i	$0.479790\pi$
$102$	0	0
$103$	−2043.48	−1.95485	−0.977427	−	0.211271i	$-0.932240\pi$
$103$	−2043.48	−1.95485	−0.977427	+	0.211271i	$0.932240\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	268.860	0.242913	0.121456	−	0.992597i	$-0.461244\pi$
$107$	268.860	0.242913	0.121456	+	0.992597i	$0.461244\pi$
$108$	0	0
$109$	1335.64	1.17368	0.586841	−	0.809702i	$-0.300371\pi$
$109$	1335.64	1.17368	0.586841	+	0.809702i	$0.300371\pi$
$110$	0	0
$111$	−164.535	−0.140693
$112$	0	0
$113$	−1675.40	−1.39476	−0.697382	−	0.716700i	$-0.745652\pi$
$113$	−1675.40	−1.39476	−0.697382	+	0.716700i	$0.745652\pi$
$114$	0	0
$115$	−167.923	−0.136164
$116$	0	0
$117$	210.812	0.166578
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1009.36	−0.758345
$122$	0	0
$123$	−571.640	−0.419049
$124$	0	0
$125$	959.387	0.686481
$126$	0	0
$127$	−1626.28	−1.13629	−0.568144	−	0.822929i	$-0.692338\pi$
$127$	−1626.28	−1.13629	−0.568144	+	0.822929i	$0.692338\pi$
$128$	0	0
$129$	395.893	0.270205
$130$	0	0
$131$	−1534.98	−1.02375	−0.511877	−	0.859059i	$-0.671050\pi$
$131$	−1534.98	−1.02375	−0.511877	+	0.859059i	$0.671050\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−111.148	−0.0708600
$136$	0	0
$137$	2466.49	1.53815	0.769076	−	0.639158i	$-0.220717\pi$
$137$	2466.49	1.53815	0.769076	+	0.639158i	$0.220717\pi$
$138$	0	0
$139$	−1322.68	−0.807107	−0.403554	−	0.914956i	$-0.632225\pi$
$139$	−1322.68	−0.807107	−0.403554	+	0.914956i	$0.632225\pi$
$140$	0	0
$141$	−598.719	−0.357598
$142$	0	0
$143$	−420.088	−0.245661
$144$	0	0
$145$	736.069	0.421567
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−648.069	−0.356321	−0.178161	−	0.984001i	$-0.557015\pi$
$149$	−648.069	−0.356321	−0.178161	+	0.984001i	$0.557015\pi$
$150$	0	0
$151$	−2003.70	−1.07986	−0.539930	−	0.841710i	$-0.681549\pi$
$151$	−2003.70	−1.07986	−0.539930	+	0.841710i	$0.681549\pi$
$152$	0	0
$153$	686.052	0.362510
$154$	0	0
$155$	130.412	0.0675805
$156$	0	0
$157$	−773.589	−0.393243	−0.196622	−	0.980479i	$-0.562997\pi$
$157$	−773.589	−0.393243	−0.196622	+	0.980479i	$0.562997\pi$
$158$	0	0
$159$	965.721	0.481677
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−66.4286	−0.0319208	−0.0159604	−	0.999873i	$-0.505081\pi$
$163$	−66.4286	−0.0319208	−0.0159604	+	0.999873i	$0.505081\pi$
$164$	0	0
$165$	221.486	0.104501
$166$	0	0
$167$	−842.616	−0.390441	−0.195220	−	0.980759i	$-0.562542\pi$
$167$	−842.616	−0.390441	−0.195220	+	0.980759i	$0.562542\pi$
$168$	0	0
$169$	−1648.34	−0.750266
$170$	0	0
$171$	−320.368	−0.143270
$172$	0	0
$173$	1570.69	0.690275	0.345138	−	0.938552i	$-0.387832\pi$
$173$	1570.69	0.690275	0.345138	+	0.938552i	$0.387832\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−490.177	−0.208158
$178$	0	0
$179$	−3670.74	−1.53276	−0.766380	−	0.642387i	$-0.777944\pi$
$179$	−3670.74	−1.53276	−0.766380	+	0.642387i	$0.777944\pi$
$180$	0	0
$181$	−802.417	−0.329520	−0.164760	−	0.986334i	$-0.552685\pi$
$181$	−802.417	−0.329520	−0.164760	+	0.986334i	$0.552685\pi$
$182$	0	0
$183$	795.691	0.321416
$184$	0	0
$185$	225.775	0.0897258
$186$	0	0
$187$	−1367.11	−0.534613
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4559.10	1.72714	0.863572	−	0.504225i	$-0.168222\pi$
$191$	4559.10	1.72714	0.863572	+	0.504225i	$0.168222\pi$
$192$	0	0
$193$	3133.71	1.16875	0.584377	−	0.811482i	$-0.301339\pi$
$193$	3133.71	1.16875	0.584377	+	0.811482i	$0.301339\pi$
$194$	0	0
$195$	−289.276	−0.106233
$196$	0	0
$197$	−2805.70	−1.01471	−0.507354	−	0.861738i	$-0.669376\pi$
$197$	−2805.70	−1.01471	−0.507354	+	0.861738i	$0.669376\pi$
$198$	0	0
$199$	−5257.19	−1.87272	−0.936362	−	0.351036i	$-0.885829\pi$
$199$	−5257.19	−1.87272	−0.936362	+	0.351036i	$0.885829\pi$
$200$	0	0
$201$	−834.382	−0.292800
$202$	0	0
$203$	0	0
$204$	0	0
$205$	784.403	0.267244
$206$	0	0
$207$	367.126	0.123271
$208$	0	0
$209$	638.400	0.211287
$210$	0	0
$211$	−3230.30	−1.05395	−0.526974	−	0.849882i	$-0.676673\pi$
$211$	−3230.30	−1.05395	−0.526974	+	0.849882i	$0.676673\pi$
$212$	0	0
$213$	31.7381	0.0102097
$214$	0	0
$215$	−543.243	−0.172320
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1752.87	0.540860
$220$	0	0
$221$	1785.53	0.543475
$222$	0	0
$223$	−433.169	−0.130077	−0.0650385	−	0.997883i	$-0.520717\pi$
$223$	−433.169	−0.130077	−0.0650385	+	0.997883i	$0.520717\pi$
$224$	0	0
$225$	−972.483	−0.288143
$226$	0	0
$227$	−4108.60	−1.20131	−0.600655	−	0.799509i	$-0.705093\pi$
$227$	−4108.60	−1.20131	−0.600655	+	0.799509i	$0.705093\pi$
$228$	0	0
$229$	4633.01	1.33693	0.668467	−	0.743742i	$-0.266951\pi$
$229$	4633.01	1.33693	0.668467	+	0.743742i	$0.266951\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	643.222	0.180853	0.0904267	−	0.995903i	$-0.471177\pi$
$233$	643.222	0.180853	0.0904267	+	0.995903i	$0.471177\pi$
$234$	0	0
$235$	821.561	0.228054
$236$	0	0
$237$	549.235	0.150534
$238$	0	0
$239$	−783.612	−0.212082	−0.106041	−	0.994362i	$-0.533818\pi$
$239$	−783.612	−0.212082	−0.106041	+	0.994362i	$0.533818\pi$
$240$	0	0
$241$	−6338.58	−1.69421	−0.847103	−	0.531429i	$-0.821655\pi$
$241$	−6338.58	−1.69421	−0.847103	+	0.531429i	$0.821655\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−833.795	−0.214790
$248$	0	0
$249$	525.350	0.133706
$250$	0	0
$251$	−5863.01	−1.47438	−0.737191	−	0.675685i	$-0.763848\pi$
$251$	−5863.01	−1.47438	−0.737191	+	0.675685i	$0.763848\pi$
$252$	0	0
$253$	−731.577	−0.181794
$254$	0	0
$255$	−941.399	−0.231187
$256$	0	0
$257$	−4213.76	−1.02275	−0.511376	−	0.859357i	$-0.670864\pi$
$257$	−4213.76	−1.02275	−0.511376	+	0.859357i	$0.670864\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1609.25	−0.381648
$262$	0	0
$263$	1955.85	0.458566	0.229283	−	0.973360i	$-0.426362\pi$
$263$	1955.85	0.458566	0.229283	+	0.973360i	$0.426362\pi$
$264$	0	0
$265$	−1325.16	−0.307185
$266$	0	0
$267$	141.431	0.0324174
$268$	0	0
$269$	1527.99	0.346332	0.173166	−	0.984893i	$-0.444600\pi$
$269$	1527.99	0.346332	0.173166	+	0.984893i	$0.444600\pi$
$270$	0	0
$271$	−7673.88	−1.72013	−0.860064	−	0.510185i	$-0.829577\pi$
$271$	−7673.88	−1.72013	−0.860064	+	0.510185i	$0.829577\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1937.88	0.424940
$276$	0	0
$277$	6312.20	1.36918	0.684591	−	0.728927i	$-0.259981\pi$
$277$	6312.20	1.36918	0.684591	+	0.728927i	$0.259981\pi$
$278$	0	0
$279$	−285.117	−0.0611811
$280$	0	0
$281$	−3334.66	−0.707932	−0.353966	−	0.935258i	$-0.615167\pi$
$281$	−3334.66	−0.707932	−0.353966	+	0.935258i	$0.615167\pi$
$282$	0	0
$283$	725.900	0.152474	0.0762372	−	0.997090i	$-0.475709\pi$
$283$	725.900	0.152474	0.0762372	+	0.997090i	$0.475709\pi$
$284$	0	0
$285$	439.607	0.0913688
$286$	0	0
$287$	0	0
$288$	0	0
$289$	897.713	0.182722
$290$	0	0
$291$	−1670.41	−0.336500
$292$	0	0
$293$	−5681.47	−1.13282	−0.566408	−	0.824125i	$-0.691667\pi$
$293$	−5681.47	−1.13282	−0.566408	+	0.824125i	$0.691667\pi$
$294$	0	0
$295$	672.620	0.132751
$296$	0	0
$297$	−484.229	−0.0946055
$298$	0	0
$299$	955.490	0.184807
$300$	0	0
$301$	0	0
$302$	0	0
$303$	386.411	0.0732631
$304$	0	0
$305$	−1091.85	−0.204980
$306$	0	0
$307$	2469.04	0.459008	0.229504	−	0.973308i	$-0.426290\pi$
$307$	2469.04	0.459008	0.229504	+	0.973308i	$0.426290\pi$
$308$	0	0
$309$	−6130.44	−1.12864
$310$	0	0
$311$	−4068.00	−0.741721	−0.370860	−	0.928689i	$-0.620937\pi$
$311$	−4068.00	−0.741721	−0.370860	+	0.928689i	$0.620937\pi$
$312$	0	0
$313$	4912.13	0.887062	0.443531	−	0.896259i	$-0.353726\pi$
$313$	4912.13	0.887062	0.443531	+	0.896259i	$0.353726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2761.17	0.489219	0.244610	−	0.969622i	$-0.421340\pi$
$317$	2761.17	0.489219	0.244610	+	0.969622i	$0.421340\pi$
$318$	0	0
$319$	3206.77	0.562836
$320$	0	0
$321$	806.579	0.140246
$322$	0	0
$323$	−2713.44	−0.467430
$324$	0	0
$325$	−2531.00	−0.431984
$326$	0	0
$327$	4006.93	0.677626
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1191.72	−0.197894	−0.0989470	−	0.995093i	$-0.531547\pi$
$331$	−1191.72	−0.197894	−0.0989470	+	0.995093i	$0.531547\pi$
$332$	0	0
$333$	−493.605	−0.0812294
$334$	0	0
$335$	1144.94	0.186730
$336$	0	0
$337$	4336.34	0.700937	0.350468	−	0.936575i	$-0.386022\pi$
$337$	4336.34	0.700937	0.350468	+	0.936575i	$0.386022\pi$
$338$	0	0
$339$	−5026.20	−0.805267
$340$	0	0
$341$	568.157	0.0902271
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−503.769	−0.0786146
$346$	0	0
$347$	9450.20	1.46200	0.730999	−	0.682379i	$-0.239054\pi$
$347$	9450.20	1.46200	0.730999	+	0.682379i	$0.239054\pi$
$348$	0	0
$349$	−10654.7	−1.63418	−0.817092	−	0.576507i	$-0.804415\pi$
$349$	−10654.7	−1.63418	−0.817092	+	0.576507i	$0.804415\pi$
$350$	0	0
$351$	632.437	0.0961738
$352$	0	0
$353$	−1702.81	−0.256746	−0.128373	−	0.991726i	$-0.540975\pi$
$353$	−1702.81	−0.256746	−0.128373	+	0.991726i	$0.540975\pi$
$354$	0	0
$355$	−43.5509	−0.00651110
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4108.61	0.604023	0.302011	−	0.953304i	$-0.402342\pi$
$359$	4108.61	0.604023	0.302011	+	0.953304i	$0.402342\pi$
$360$	0	0
$361$	−5591.90	−0.815264
$362$	0	0
$363$	−3028.07	−0.437830
$364$	0	0
$365$	−2405.29	−0.344928
$366$	0	0
$367$	−1179.35	−0.167743	−0.0838715	−	0.996477i	$-0.526729\pi$
$367$	−1179.35	−0.167743	−0.0838715	+	0.996477i	$0.526729\pi$
$368$	0	0
$369$	−1714.92	−0.241938
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7865.22	1.09181	0.545906	−	0.837846i	$-0.316186\pi$
$373$	7865.22	1.09181	0.545906	+	0.837846i	$0.316186\pi$
$374$	0	0
$375$	2878.16	0.396340
$376$	0	0
$377$	−4188.27	−0.572166
$378$	0	0
$379$	−13076.8	−1.77233	−0.886164	−	0.463372i	$-0.846640\pi$
$379$	−13076.8	−1.77233	−0.886164	+	0.463372i	$0.846640\pi$
$380$	0	0
$381$	−4878.83	−0.656036
$382$	0	0
$383$	2410.38	0.321578	0.160789	−	0.986989i	$-0.448596\pi$
$383$	2410.38	0.321578	0.160789	+	0.986989i	$0.448596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1187.68	0.156003
$388$	0	0
$389$	6178.78	0.805339	0.402669	−	0.915346i	$-0.368082\pi$
$389$	6178.78	0.805339	0.402669	+	0.915346i	$0.368082\pi$
$390$	0	0
$391$	3109.48	0.402182
$392$	0	0
$393$	−4604.94	−0.591064
$394$	0	0
$395$	−753.658	−0.0960017
$396$	0	0
$397$	4809.52	0.608017	0.304009	−	0.952669i	$-0.401675\pi$
$397$	4809.52	0.608017	0.304009	+	0.952669i	$0.401675\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6088.29	0.758191	0.379096	−	0.925357i	$-0.376235\pi$
$401$	6088.29	0.758191	0.379096	+	0.925357i	$0.376235\pi$
$402$	0	0
$403$	−742.053	−0.0917228
$404$	0	0
$405$	−333.444	−0.0409110
$406$	0	0
$407$	983.613	0.119793
$408$	0	0
$409$	−7869.74	−0.951426	−0.475713	−	0.879600i	$-0.657810\pi$
$409$	−7869.74	−0.951426	−0.475713	+	0.879600i	$0.657810\pi$
$410$	0	0
$411$	7399.48	0.888052
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−720.884	−0.0852694
$416$	0	0
$417$	−3968.03	−0.465984
$418$	0	0
$419$	−15279.0	−1.78145	−0.890724	−	0.454544i	$-0.849802\pi$
$419$	−15279.0	−1.78145	−0.890724	+	0.454544i	$0.849802\pi$
$420$	0	0
$421$	6909.70	0.799900	0.399950	−	0.916537i	$-0.369027\pi$
$421$	6909.70	0.799900	0.399950	+	0.916537i	$0.369027\pi$
$422$	0	0
$423$	−1796.16	−0.206459
$424$	0	0
$425$	−8236.72	−0.940093
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1260.27	−0.141833
$430$	0	0
$431$	12627.4	1.41124	0.705618	−	0.708592i	$-0.250669\pi$
$431$	12627.4	1.41124	0.705618	+	0.708592i	$0.250669\pi$
$432$	0	0
$433$	−6738.74	−0.747906	−0.373953	−	0.927448i	$-0.621998\pi$
$433$	−6738.74	−0.747906	−0.373953	+	0.927448i	$0.621998\pi$
$434$	0	0
$435$	2208.21	0.243392
$436$	0	0
$437$	−1452.04	−0.158948
$438$	0	0
$439$	1752.54	0.190533	0.0952667	−	0.995452i	$-0.469630\pi$
$439$	1752.54	0.190533	0.0952667	+	0.995452i	$0.469630\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14286.9	1.53226	0.766130	−	0.642686i	$-0.222180\pi$
$443$	14286.9	1.53226	0.766130	+	0.642686i	$0.222180\pi$
$444$	0	0
$445$	−194.071	−0.0206738
$446$	0	0
$447$	−1944.21	−0.205722
$448$	0	0
$449$	−7524.11	−0.790835	−0.395417	−	0.918501i	$-0.629400\pi$
$449$	−7524.11	−0.790835	−0.395417	+	0.918501i	$0.629400\pi$
$450$	0	0
$451$	3417.35	0.356799
$452$	0	0
$453$	−6011.10	−0.623457
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10579.4	1.08290	0.541448	−	0.840734i	$-0.317876\pi$
$457$	10579.4	1.08290	0.541448	+	0.840734i	$0.317876\pi$
$458$	0	0
$459$	2058.16	0.209295
$460$	0	0
$461$	5889.66	0.595030	0.297515	−	0.954717i	$-0.403842\pi$
$461$	5889.66	0.595030	0.297515	+	0.954717i	$0.403842\pi$
$462$	0	0
$463$	−12716.5	−1.27643	−0.638215	−	0.769858i	$-0.720327\pi$
$463$	−12716.5	−1.27643	−0.638215	+	0.769858i	$0.720327\pi$
$464$	0	0
$465$	391.237	0.0390176
$466$	0	0
$467$	8362.46	0.828626	0.414313	−	0.910134i	$-0.364022\pi$
$467$	8362.46	0.828626	0.414313	+	0.910134i	$0.364022\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2320.77	−0.227039
$472$	0	0
$473$	−2366.70	−0.230066
$474$	0	0
$475$	3846.32	0.371540
$476$	0	0
$477$	2897.16	0.278096
$478$	0	0
$479$	5404.57	0.515535	0.257768	−	0.966207i	$-0.417013\pi$
$479$	5404.57	0.515535	0.257768	+	0.966207i	$0.417013\pi$
$480$	0	0
$481$	−1284.67	−0.121779
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2292.14	0.214599
$486$	0	0
$487$	10659.5	0.991842	0.495921	−	0.868368i	$-0.334831\pi$
$487$	10659.5	0.991842	0.495921	+	0.868368i	$0.334831\pi$
$488$	0	0
$489$	−199.286	−0.0184295
$490$	0	0
$491$	277.130	0.0254719	0.0127360	−	0.999919i	$-0.495946\pi$
$491$	277.130	0.0254719	0.0127360	+	0.999919i	$0.495946\pi$
$492$	0	0
$493$	−13630.0	−1.24516
$494$	0	0
$495$	664.458	0.0603337
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18108.3	−1.62453	−0.812266	−	0.583288i	$-0.801766\pi$
$499$	−18108.3	−1.62453	−0.812266	+	0.583288i	$0.801766\pi$
$500$	0	0
$501$	−2527.85	−0.225421
$502$	0	0
$503$	3671.00	0.325411	0.162706	−	0.986675i	$-0.447978\pi$
$503$	3671.00	0.325411	0.162706	+	0.986675i	$0.447978\pi$
$504$	0	0
$505$	−530.232	−0.0467228
$506$	0	0
$507$	−4945.01	−0.433167
$508$	0	0
$509$	−17364.1	−1.51208	−0.756040	−	0.654526i	$-0.772868\pi$
$509$	−17364.1	−1.51208	−0.756040	+	0.654526i	$0.772868\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−961.103	−0.0827168
$514$	0	0
$515$	8412.17	0.719776
$516$	0	0
$517$	3579.23	0.304476
$518$	0	0
$519$	4712.08	0.398531
$520$	0	0
$521$	2870.45	0.241376	0.120688	−	0.992690i	$-0.461490\pi$
$521$	2870.45	0.241376	0.120688	+	0.992690i	$0.461490\pi$
$522$	0	0
$523$	−5735.39	−0.479524	−0.239762	−	0.970832i	$-0.577069\pi$
$523$	−5735.39	−0.479524	−0.239762	+	0.970832i	$0.577069\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2414.88	−0.199609
$528$	0	0
$529$	−10503.0	−0.863239
$530$	0	0
$531$	−1470.53	−0.120180
$532$	0	0
$533$	−4463.29	−0.362714
$534$	0	0
$535$	−1106.79	−0.0894402
$536$	0	0
$537$	−11012.2	−0.884940
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2014.27	0.160074	0.0800372	−	0.996792i	$-0.474496\pi$
$541$	2014.27	0.160074	0.0800372	+	0.996792i	$0.474496\pi$
$542$	0	0
$543$	−2407.25	−0.190249
$544$	0	0
$545$	−5498.30	−0.432149
$546$	0	0
$547$	−18005.2	−1.40740	−0.703699	−	0.710498i	$-0.748470\pi$
$547$	−18005.2	−1.40740	−0.703699	+	0.710498i	$0.748470\pi$
$548$	0	0
$549$	2387.07	0.185570
$550$	0	0
$551$	6364.83	0.492107
$552$	0	0
$553$	0	0
$554$	0	0
$555$	677.324	0.0518032
$556$	0	0
$557$	3093.70	0.235340	0.117670	−	0.993053i	$-0.462458\pi$
$557$	3093.70	0.235340	0.117670	+	0.993053i	$0.462458\pi$
$558$	0	0
$559$	3091.08	0.233880
$560$	0	0
$561$	−4101.32	−0.308659
$562$	0	0
$563$	18245.4	1.36581	0.682907	−	0.730505i	$-0.260715\pi$
$563$	18245.4	1.36581	0.682907	+	0.730505i	$0.260715\pi$
$564$	0	0
$565$	6896.94	0.513551
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17456.7	1.28616	0.643079	−	0.765800i	$-0.277657\pi$
$569$	17456.7	1.28616	0.643079	+	0.765800i	$0.277657\pi$
$570$	0	0
$571$	−10013.7	−0.733902	−0.366951	−	0.930240i	$-0.619598\pi$
$571$	−10013.7	−0.733902	−0.366951	+	0.930240i	$0.619598\pi$
$572$	0	0
$573$	13677.3	0.997167
$574$	0	0
$575$	−4407.70	−0.319676
$576$	0	0
$577$	−26521.5	−1.91353	−0.956764	−	0.290867i	$-0.906056\pi$
$577$	−26521.5	−1.91353	−0.956764	+	0.290867i	$0.906056\pi$
$578$	0	0
$579$	9401.14	0.674781
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5773.21	−0.410124
$584$	0	0
$585$	−867.828	−0.0613338
$586$	0	0
$587$	−6734.74	−0.473547	−0.236774	−	0.971565i	$-0.576090\pi$
$587$	−6734.74	−0.473547	−0.236774	+	0.971565i	$0.576090\pi$
$588$	0	0
$589$	1127.68	0.0788886
$590$	0	0
$591$	−8417.09	−0.585842
$592$	0	0
$593$	−26173.3	−1.81250	−0.906248	−	0.422746i	$-0.861066\pi$
$593$	−26173.3	−1.81250	−0.906248	+	0.422746i	$0.861066\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15771.6	−1.08122
$598$	0	0
$599$	5944.71	0.405499	0.202750	−	0.979231i	$-0.435012\pi$
$599$	5944.71	0.405499	0.202750	+	0.979231i	$0.435012\pi$
$600$	0	0
$601$	−1988.96	−0.134994	−0.0674970	−	0.997719i	$-0.521501\pi$
$601$	−1988.96	−0.134994	−0.0674970	+	0.997719i	$0.521501\pi$
$602$	0	0
$603$	−2503.15	−0.169048
$604$	0	0
$605$	4155.11	0.279222
$606$	0	0
$607$	−1383.89	−0.0925376	−0.0462688	−	0.998929i	$-0.514733\pi$
$607$	−1383.89	−0.0925376	−0.0462688	+	0.998929i	$0.514733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4674.72	−0.309524
$612$	0	0
$613$	1500.35	0.0988558	0.0494279	−	0.998778i	$-0.484260\pi$
$613$	1500.35	0.0988558	0.0494279	+	0.998778i	$0.484260\pi$
$614$	0	0
$615$	2353.21	0.154294
$616$	0	0
$617$	15497.7	1.01121	0.505604	−	0.862766i	$-0.331270\pi$
$617$	15497.7	1.01121	0.505604	+	0.862766i	$0.331270\pi$
$618$	0	0
$619$	−515.002	−0.0334405	−0.0167203	−	0.999860i	$-0.505322\pi$
$619$	−515.002	−0.0334405	−0.0167203	+	0.999860i	$0.505322\pi$
$620$	0	0
$621$	1101.38	0.0711703
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9557.30	0.611667
$626$	0	0
$627$	1915.20	0.121987
$628$	0	0
$629$	−4180.73	−0.265018
$630$	0	0
$631$	−22060.1	−1.39176	−0.695880	−	0.718158i	$-0.744985\pi$
$631$	−22060.1	−1.39176	−0.695880	+	0.718158i	$0.744985\pi$
$632$	0	0
$633$	−9690.89	−0.608497
$634$	0	0
$635$	6694.71	0.418380
$636$	0	0
$637$	0	0
$638$	0	0
$639$	95.2142	0.00589455
$640$	0	0
$641$	−17094.1	−1.05332	−0.526660	−	0.850076i	$-0.676556\pi$
$641$	−17094.1	−1.05332	−0.526660	+	0.850076i	$0.676556\pi$
$642$	0	0
$643$	−2999.98	−0.183993	−0.0919965	−	0.995759i	$-0.529325\pi$
$643$	−2999.98	−0.183993	−0.0919965	+	0.995759i	$0.529325\pi$
$644$	0	0
$645$	−1629.73	−0.0994892
$646$	0	0
$647$	8255.72	0.501648	0.250824	−	0.968033i	$-0.419299\pi$
$647$	8255.72	0.501648	0.250824	+	0.968033i	$0.419299\pi$
$648$	0	0
$649$	2930.35	0.177236
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7953.02	−0.476609	−0.238304	−	0.971191i	$-0.576592\pi$
$653$	−7953.02	−0.476609	−0.238304	+	0.971191i	$0.576592\pi$
$654$	0	0
$655$	6318.88	0.376945
$656$	0	0
$657$	5258.62	0.312266
$658$	0	0
$659$	4420.98	0.261331	0.130665	−	0.991427i	$-0.458289\pi$
$659$	4420.98	0.261331	0.130665	+	0.991427i	$0.458289\pi$
$660$	0	0
$661$	5830.81	0.343105	0.171552	−	0.985175i	$-0.445122\pi$
$661$	5830.81	0.343105	0.171552	+	0.985175i	$0.445122\pi$
$662$	0	0
$663$	5356.60	0.313776
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7293.79	−0.423413
$668$	0	0
$669$	−1299.51	−0.0751000
$670$	0	0
$671$	−4756.75	−0.273670
$672$	0	0
$673$	24614.2	1.40982	0.704910	−	0.709297i	$-0.250987\pi$
$673$	24614.2	1.40982	0.704910	+	0.709297i	$0.250987\pi$
$674$	0	0
$675$	−2917.45	−0.166360
$676$	0	0
$677$	130.989	0.00743619	0.00371810	−	0.999993i	$-0.498816\pi$
$677$	130.989	0.00743619	0.00371810	+	0.999993i	$0.498816\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12325.8	−0.693576
$682$	0	0
$683$	4912.86	0.275235	0.137617	−	0.990485i	$-0.456056\pi$
$683$	4912.86	0.275235	0.137617	+	0.990485i	$0.456056\pi$
$684$	0	0
$685$	−10153.5	−0.566346
$686$	0	0
$687$	13899.0	0.771879
$688$	0	0
$689$	7540.22	0.416922
$690$	0	0
$691$	−9639.37	−0.530678	−0.265339	−	0.964155i	$-0.585484\pi$
$691$	−9639.37	−0.530678	−0.265339	+	0.964155i	$0.585484\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5444.92	0.297176
$696$	0	0
$697$	−14525.0	−0.789346
$698$	0	0
$699$	1929.66	0.104416
$700$	0	0
$701$	−18377.9	−0.990188	−0.495094	−	0.868839i	$-0.664866\pi$
$701$	−18377.9	−0.990188	−0.495094	+	0.868839i	$0.664866\pi$
$702$	0	0
$703$	1952.28	0.104739
$704$	0	0
$705$	2464.68	0.131667
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8667.23	−0.459104	−0.229552	−	0.973296i	$-0.573726\pi$
$709$	−8667.23	−0.459104	−0.229552	+	0.973296i	$0.573726\pi$
$710$	0	0
$711$	1647.70	0.0869110
$712$	0	0
$713$	−1292.27	−0.0678765
$714$	0	0
$715$	1729.33	0.0904523
$716$	0	0
$717$	−2350.84	−0.122446
$718$	0	0
$719$	−13439.1	−0.697070	−0.348535	−	0.937296i	$-0.613321\pi$
$719$	−13439.1	−0.697070	−0.348535	+	0.937296i	$0.613321\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19015.7	−0.978150
$724$	0	0
$725$	19320.6	0.989722
$726$	0	0
$727$	25085.9	1.27976	0.639879	−	0.768476i	$-0.278985\pi$
$727$	25085.9	1.27976	0.639879	+	0.768476i	$0.278985\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	10059.4	0.508973
$732$	0	0
$733$	31020.6	1.56313	0.781563	−	0.623826i	$-0.214423\pi$
$733$	31020.6	1.56313	0.781563	+	0.623826i	$0.214423\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4988.05	0.249304
$738$	0	0
$739$	−12992.4	−0.646730	−0.323365	−	0.946274i	$-0.604814\pi$
$739$	−12992.4	−0.646730	−0.323365	+	0.946274i	$0.604814\pi$
$740$	0	0
$741$	−2501.39	−0.124009
$742$	0	0
$743$	23950.7	1.18259	0.591297	−	0.806454i	$-0.298616\pi$
$743$	23950.7	1.18259	0.591297	+	0.806454i	$0.298616\pi$
$744$	0	0
$745$	2667.83	0.131197
$746$	0	0
$747$	1576.05	0.0771950
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−13677.5	−0.664580	−0.332290	−	0.943177i	$-0.607821\pi$
$751$	−13677.5	−0.664580	−0.332290	+	0.943177i	$0.607821\pi$
$752$	0	0
$753$	−17589.0	−0.851235
$754$	0	0
$755$	8248.41	0.397603
$756$	0	0
$757$	−9295.64	−0.446309	−0.223154	−	0.974783i	$-0.571635\pi$
$757$	−9295.64	−0.446309	−0.223154	+	0.974783i	$0.571635\pi$
$758$	0	0
$759$	−2194.73	−0.104959
$760$	0	0
$761$	−19586.0	−0.932970	−0.466485	−	0.884529i	$-0.654480\pi$
$761$	−19586.0	−0.932970	−0.466485	+	0.884529i	$0.654480\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2824.20	−0.133476
$766$	0	0
$767$	−3827.24	−0.180174
$768$	0	0
$769$	5952.29	0.279122	0.139561	−	0.990213i	$-0.455431\pi$
$769$	5952.29	0.279122	0.139561	+	0.990213i	$0.455431\pi$
$770$	0	0
$771$	−12641.3	−0.590486
$772$	0	0
$773$	34379.6	1.59968	0.799838	−	0.600216i	$-0.204919\pi$
$773$	34379.6	1.59968	0.799838	+	0.600216i	$0.204919\pi$
$774$	0	0
$775$	3423.11	0.158660
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6782.78	0.311962
$780$	0	0
$781$	−189.735	−0.00869300
$782$	0	0
$783$	−4827.75	−0.220344
$784$	0	0
$785$	3184.55	0.144792
$786$	0	0
$787$	19167.6	0.868172	0.434086	−	0.900872i	$-0.357071\pi$
$787$	19167.6	0.868172	0.434086	+	0.900872i	$0.357071\pi$
$788$	0	0
$789$	5867.55	0.264753
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6212.65	0.278206
$794$	0	0
$795$	−3975.48	−0.177353
$796$	0	0
$797$	2679.24	0.119076	0.0595380	−	0.998226i	$-0.481037\pi$
$797$	2679.24	0.119076	0.0595380	+	0.998226i	$0.481037\pi$
$798$	0	0
$799$	−15213.1	−0.673592
$800$	0	0
$801$	424.293	0.0187162
$802$	0	0
$803$	−10478.9	−0.460515
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4583.98	0.199955
$808$	0	0
$809$	8075.98	0.350972	0.175486	−	0.984482i	$-0.443850\pi$
$809$	8075.98	0.350972	0.175486	+	0.984482i	$0.443850\pi$
$810$	0	0
$811$	31378.3	1.35862	0.679311	−	0.733851i	$-0.262279\pi$
$811$	31378.3	1.35862	0.679311	+	0.733851i	$0.262279\pi$
$812$	0	0
$813$	−23021.6	−0.993117
$814$	0	0
$815$	273.459	0.0117532
$816$	0	0
$817$	−4697.45	−0.201154
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3099.20	0.131745	0.0658726	−	0.997828i	$-0.479017\pi$
$821$	3099.20	0.131745	0.0658726	+	0.997828i	$0.479017\pi$
$822$	0	0
$823$	965.740	0.0409035	0.0204517	−	0.999791i	$-0.493490\pi$
$823$	965.740	0.0409035	0.0204517	+	0.999791i	$0.493490\pi$
$824$	0	0
$825$	5813.64	0.245339
$826$	0	0
$827$	7538.11	0.316960	0.158480	−	0.987362i	$-0.449341\pi$
$827$	7538.11	0.316960	0.158480	+	0.987362i	$0.449341\pi$
$828$	0	0
$829$	12841.2	0.537991	0.268995	−	0.963141i	$-0.413308\pi$
$829$	12841.2	0.537991	0.268995	+	0.963141i	$0.413308\pi$
$830$	0	0
$831$	18936.6	0.790498
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3468.71	0.143760
$836$	0	0
$837$	−855.352	−0.0353229
$838$	0	0
$839$	−13909.3	−0.572351	−0.286176	−	0.958177i	$-0.592384\pi$
$839$	−13909.3	−0.572351	−0.286176	+	0.958177i	$0.592384\pi$
$840$	0	0
$841$	7582.39	0.310894
$842$	0	0
$843$	−10004.0	−0.408725
$844$	0	0
$845$	6785.52	0.276248
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2177.70	0.0880311
$850$	0	0
$851$	−2237.22	−0.0901188
$852$	0	0
$853$	33036.0	1.32606	0.663030	−	0.748593i	$-0.269270\pi$
$853$	33036.0	1.32606	0.663030	+	0.748593i	$0.269270\pi$
$854$	0	0
$855$	1318.82	0.0527518
$856$	0	0
$857$	15827.5	0.630873	0.315437	−	0.948947i	$-0.397849\pi$
$857$	15827.5	0.630873	0.315437	+	0.948947i	$0.397849\pi$
$858$	0	0
$859$	−6933.95	−0.275417	−0.137709	−	0.990473i	$-0.543974\pi$
$859$	−6933.95	−0.275417	−0.137709	+	0.990473i	$0.543974\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25595.4	1.00959	0.504795	−	0.863239i	$-0.331568\pi$
$863$	25595.4	1.00959	0.504795	+	0.863239i	$0.331568\pi$
$864$	0	0
$865$	−6465.91	−0.254159
$866$	0	0
$867$	2693.14	0.105495
$868$	0	0
$869$	−3283.40	−0.128172
$870$	0	0
$871$	−6514.74	−0.253437
$872$	0	0
$873$	−5011.24	−0.194278
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44274.9	−1.70474	−0.852369	−	0.522940i	$-0.824835\pi$
$877$	−44274.9	−1.70474	−0.852369	+	0.522940i	$0.824835\pi$
$878$	0	0
$879$	−17044.4	−0.654032
$880$	0	0
$881$	5773.97	0.220806	0.110403	−	0.993887i	$-0.464786\pi$
$881$	5773.97	0.220806	0.110403	+	0.993887i	$0.464786\pi$
$882$	0	0
$883$	48171.4	1.83590	0.917948	−	0.396701i	$-0.129845\pi$
$883$	48171.4	1.83590	0.917948	+	0.396701i	$0.129845\pi$
$884$	0	0
$885$	2017.86	0.0766437
$886$	0	0
$887$	−48168.5	−1.82338	−0.911692	−	0.410875i	$-0.865223\pi$
$887$	−48168.5	−1.82338	−0.911692	+	0.410875i	$0.865223\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1452.69	−0.0546205
$892$	0	0
$893$	7104.08	0.266214
$894$	0	0
$895$	15110.9	0.564361
$896$	0	0
$897$	2866.47	0.106699
$898$	0	0
$899$	5664.51	0.210147
$900$	0	0
$901$	24538.3	0.907315
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3303.22	0.121329
$906$	0	0
$907$	−31992.5	−1.17122	−0.585608	−	0.810595i	$-0.699144\pi$
$907$	−31992.5	−1.17122	−0.585608	+	0.810595i	$0.699144\pi$
$908$	0	0
$909$	1159.23	0.0422985
$910$	0	0
$911$	21184.7	0.770450	0.385225	−	0.922823i	$-0.374124\pi$
$911$	21184.7	0.770450	0.385225	+	0.922823i	$0.374124\pi$
$912$	0	0
$913$	−3140.61	−0.113844
$914$	0	0
$915$	−3275.54	−0.118345
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−21647.9	−0.777040	−0.388520	−	0.921440i	$-0.627014\pi$
$919$	−21647.9	−0.777040	−0.388520	+	0.921440i	$0.627014\pi$
$920$	0	0
$921$	7407.11	0.265008
$922$	0	0
$923$	247.807	0.00883711
$924$	0	0
$925$	5926.20	0.210651
$926$	0	0
$927$	−18391.3	−0.651618
$928$	0	0
$929$	9480.99	0.334834	0.167417	−	0.985886i	$-0.446457\pi$
$929$	9480.99	0.334834	0.167417	+	0.985886i	$0.446457\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12204.0	−0.428233
$934$	0	0
$935$	5627.82	0.196844
$936$	0	0
$937$	−2941.86	−0.102568	−0.0512841	−	0.998684i	$-0.516331\pi$
$937$	−2941.86	−0.102568	−0.0512841	+	0.998684i	$0.516331\pi$
$938$	0	0
$939$	14736.4	0.512145
$940$	0	0
$941$	52683.8	1.82513	0.912563	−	0.408937i	$-0.134100\pi$
$941$	52683.8	1.82513	0.912563	+	0.408937i	$0.134100\pi$
$942$	0	0
$943$	−7772.74	−0.268415
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45003.1	1.54425	0.772125	−	0.635471i	$-0.219194\pi$
$947$	45003.1	1.54425	0.772125	+	0.635471i	$0.219194\pi$
$948$	0	0
$949$	13686.2	0.468149
$950$	0	0
$951$	8283.50	0.282451
$952$	0	0
$953$	40137.2	1.36429	0.682146	−	0.731216i	$-0.261047\pi$
$953$	40137.2	1.36429	0.682146	+	0.731216i	$0.261047\pi$
$954$	0	0
$955$	−18767.9	−0.635933
$956$	0	0
$957$	9620.31	0.324954
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28787.4	−0.966312
$962$	0	0
$963$	2419.74	0.0809709
$964$	0	0
$965$	−12900.2	−0.430335
$966$	0	0
$967$	42055.0	1.39855	0.699276	−	0.714852i	$-0.253506\pi$
$967$	42055.0	1.39855	0.699276	+	0.714852i	$0.253506\pi$
$968$	0	0
$969$	−8140.33	−0.269871
$970$	0	0
$971$	9772.69	0.322987	0.161494	−	0.986874i	$-0.448369\pi$
$971$	9772.69	0.322987	0.161494	+	0.986874i	$0.448369\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7593.01	−0.249406
$976$	0	0
$977$	−11978.1	−0.392236	−0.196118	−	0.980580i	$-0.562834\pi$
$977$	−11978.1	−0.392236	−0.196118	+	0.980580i	$0.562834\pi$
$978$	0	0
$979$	−845.494	−0.0276018
$980$	0	0
$981$	12020.8	0.391228
$982$	0	0
$983$	56474.0	1.83239	0.916197	−	0.400729i	$-0.131243\pi$
$983$	56474.0	1.83239	0.916197	+	0.400729i	$0.131243\pi$
$984$	0	0
$985$	11549.9	0.373615
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5383.06	0.173075
$990$	0	0
$991$	−48427.6	−1.55233	−0.776163	−	0.630533i	$-0.782837\pi$
$991$	−48427.6	−1.55233	−0.776163	+	0.630533i	$0.782837\pi$
$992$	0	0
$993$	−3575.16	−0.114254
$994$	0	0
$995$	21641.7	0.689535
$996$	0	0
$997$	29268.6	0.929734	0.464867	−	0.885381i	$-0.346102\pi$
$997$	29268.6	0.929734	0.464867	+	0.885381i	$0.346102\pi$
$998$	0	0
$999$	−1480.81	−0.0468978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cr.1.2		4
4.3	odd	2		1176.4.a.bb.1.2	✓	4
7.6	odd	2		2352.4.a.ck.1.3		4
28.27	even	2		1176.4.a.bc.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.bb.1.2	✓	4	4.3	odd	2
1176.4.a.bc.1.3	yes	4	28.27	even	2
2352.4.a.ck.1.3		4	7.6	odd	2
2352.4.a.cr.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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -4.11659 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -4.11659 q^{5} +9.00000 q^{9} -17.9344 q^{11} +23.4236 q^{13} -12.3498 q^{15} +76.2280 q^{17} -35.5964 q^{19} +40.7918 q^{23} -108.054 q^{25} +27.0000 q^{27} -178.805 q^{29} -31.6797 q^{31} -53.8032 q^{33} -54.8450 q^{37} +70.2708 q^{39} -190.547 q^{41} +131.964 q^{43} -37.0493 q^{45} -199.573 q^{47} +228.684 q^{51} +321.907 q^{53} +73.8287 q^{55} -106.789 q^{57} -163.392 q^{59} +265.230 q^{61} -96.4254 q^{65} -278.127 q^{67} +122.375 q^{69} +10.5794 q^{71} +584.292 q^{73} -324.161 q^{75} +183.078 q^{79} +81.0000 q^{81} +175.117 q^{83} -313.800 q^{85} -536.416 q^{87} +47.1437 q^{89} -95.0391 q^{93} +146.536 q^{95} -556.805 q^{97} -161.410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} - 152 q^{17} - 224 q^{19} + 8 q^{23} - 28 q^{25} + 108 q^{27} - 144 q^{29} - 400 q^{31} + 120 q^{33} - 304 q^{37} + 144 q^{39} - 152 q^{41} - 160 q^{43} + 72 q^{45} - 544 q^{47} - 456 q^{51} - 1320 q^{53} - 16 q^{55} - 672 q^{57} - 1040 q^{59} + 896 q^{61} - 648 q^{65} + 416 q^{67} + 24 q^{69} - 248 q^{71} - 752 q^{73} - 84 q^{75} - 864 q^{79} + 324 q^{81} - 1456 q^{83} - 1608 q^{85} - 432 q^{87} + 2936 q^{89} - 1200 q^{93} + 80 q^{95} + 144 q^{97} + 360 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 - 152 * q^17 - 224 * q^19 + 8 * q^23 - 28 * q^25 + 108 * q^27 - 144 * q^29 - 400 * q^31 + 120 * q^33 - 304 * q^37 + 144 * q^39 - 152 * q^41 - 160 * q^43 + 72 * q^45 - 544 * q^47 - 456 * q^51 - 1320 * q^53 - 16 * q^55 - 672 * q^57 - 1040 * q^59 + 896 * q^61 - 648 * q^65 + 416 * q^67 + 24 * q^69 - 248 * q^71 - 752 * q^73 - 84 * q^75 - 864 * q^79 + 324 * q^81 - 1456 * q^83 - 1608 * q^85 - 432 * q^87 + 2936 * q^89 - 1200 * q^93 + 80 * q^95 + 144 * q^97 + 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more