LMFDB - Embedded newform 2352.4.a.bp.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.27492$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -9.82475 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -9.82475 q^{5} +9.00000 q^{9} +14.1752 q^{11} +26.1238 q^{13} +29.4743 q^{15} +78.5980 q^{17} +73.1752 q^{19} +96.0000 q^{23} -28.4743 q^{25} -27.0000 q^{27} +173.021 q^{29} +67.2475 q^{31} -42.5257 q^{33} -301.670 q^{37} -78.3713 q^{39} -472.042 q^{41} +463.670 q^{43} -88.4228 q^{45} -91.1960 q^{47} -235.794 q^{51} -163.268 q^{53} -139.268 q^{55} -219.526 q^{57} -600.526 q^{59} -571.691 q^{61} -256.659 q^{65} +539.423 q^{67} -288.000 q^{69} +1064.39 q^{71} +442.680 q^{73} +85.4228 q^{75} +45.7010 q^{79} +81.0000 q^{81} -686.464 q^{83} -772.206 q^{85} -519.062 q^{87} -660.145 q^{89} -201.743 q^{93} -718.929 q^{95} -658.320 q^{97} +127.577 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9} + 51 q^{11} - 61 q^{13} - 9 q^{15} - 24 q^{17} + 169 q^{19} + 192 q^{23} + 11 q^{25} - 54 q^{27} - 39 q^{29} - 92 q^{31} - 153 q^{33} - 173 q^{37} + 183 q^{39} - 174 q^{41} + 497 q^{43} + 27 q^{45} + 180 q^{47} + 72 q^{51} + 285 q^{53} + 333 q^{55} - 507 q^{57} - 1269 q^{59} - 328 q^{61} - 1374 q^{65} + 875 q^{67} - 576 q^{69} + 1404 q^{71} + 1361 q^{73} - 33 q^{75} + 182 q^{79} + 162 q^{81} - 399 q^{83} - 2088 q^{85} + 117 q^{87} - 822 q^{89} + 276 q^{93} + 510 q^{95} - 841 q^{97} + 459 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9 + 51 * q^11 - 61 * q^13 - 9 * q^15 - 24 * q^17 + 169 * q^19 + 192 * q^23 + 11 * q^25 - 54 * q^27 - 39 * q^29 - 92 * q^31 - 153 * q^33 - 173 * q^37 + 183 * q^39 - 174 * q^41 + 497 * q^43 + 27 * q^45 + 180 * q^47 + 72 * q^51 + 285 * q^53 + 333 * q^55 - 507 * q^57 - 1269 * q^59 - 328 * q^61 - 1374 * q^65 + 875 * q^67 - 576 * q^69 + 1404 * q^71 + 1361 * q^73 - 33 * q^75 + 182 * q^79 + 162 * q^81 - 399 * q^83 - 2088 * q^85 + 117 * q^87 - 822 * q^89 + 276 * q^93 + 510 * q^95 - 841 * q^97 + 459 * 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
$5$	−9.82475	−0.878753	−0.439376	−	0.898303i	$-0.644801\pi$
$5$	−9.82475	−0.878753	−0.439376	+	0.898303i	$0.644801\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	14.1752	0.388545	0.194273	−	0.980948i	$-0.437765\pi$
$11$	14.1752	0.388545	0.194273	+	0.980948i	$0.437765\pi$
$12$	0	0
$13$	26.1238	0.557341	0.278670	−	0.960387i	$-0.410106\pi$
$13$	26.1238	0.557341	0.278670	+	0.960387i	$0.410106\pi$
$14$	0	0
$15$	29.4743	0.507348
$16$	0	0
$17$	78.5980	1.12134	0.560671	−	0.828039i	$-0.310543\pi$
$17$	78.5980	1.12134	0.560671	+	0.828039i	$0.310543\pi$
$18$	0	0
$19$	73.1752	0.883555	0.441778	−	0.897125i	$-0.354348\pi$
$19$	73.1752	0.883555	0.441778	+	0.897125i	$0.354348\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	−28.4743	−0.227794
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	173.021	1.10790	0.553951	−	0.832549i	$-0.313120\pi$
$29$	173.021	1.10790	0.553951	+	0.832549i	$0.313120\pi$
$30$	0	0
$31$	67.2475	0.389613	0.194807	−	0.980842i	$-0.437592\pi$
$31$	67.2475	0.389613	0.194807	+	0.980842i	$0.437592\pi$
$32$	0	0
$33$	−42.5257	−0.224327
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−301.670	−1.34039	−0.670193	−	0.742187i	$-0.733789\pi$
$37$	−301.670	−1.34039	−0.670193	+	0.742187i	$0.733789\pi$
$38$	0	0
$39$	−78.3713	−0.321781
$40$	0	0
$41$	−472.042	−1.79806	−0.899031	−	0.437886i	$-0.855727\pi$
$41$	−472.042	−1.79806	−0.899031	+	0.437886i	$0.855727\pi$
$42$	0	0
$43$	463.670	1.64440	0.822198	−	0.569201i	$-0.192747\pi$
$43$	463.670	1.64440	0.822198	+	0.569201i	$0.192747\pi$
$44$	0	0
$45$	−88.4228	−0.292918
$46$	0	0
$47$	−91.1960	−0.283028	−0.141514	−	0.989936i	$-0.545197\pi$
$47$	−91.1960	−0.283028	−0.141514	+	0.989936i	$0.545197\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−235.794	−0.647407
$52$	0	0
$53$	−163.268	−0.423144	−0.211572	−	0.977362i	$-0.567858\pi$
$53$	−163.268	−0.423144	−0.211572	+	0.977362i	$0.567858\pi$
$54$	0	0
$55$	−139.268	−0.341435
$56$	0	0
$57$	−219.526	−0.510121
$58$	0	0
$59$	−600.526	−1.32512	−0.662558	−	0.749011i	$-0.730529\pi$
$59$	−600.526	−1.32512	−0.662558	+	0.749011i	$0.730529\pi$
$60$	0	0
$61$	−571.691	−1.19996	−0.599980	−	0.800015i	$-0.704825\pi$
$61$	−571.691	−1.19996	−0.599980	+	0.800015i	$0.704825\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−256.659	−0.489764
$66$	0	0
$67$	539.423	0.983597	0.491798	−	0.870709i	$-0.336340\pi$
$67$	539.423	0.983597	0.491798	+	0.870709i	$0.336340\pi$
$68$	0	0
$69$	−288.000	−0.502480
$70$	0	0
$71$	1064.39	1.77916	0.889578	−	0.456783i	$-0.150998\pi$
$71$	1064.39	1.77916	0.889578	+	0.456783i	$0.150998\pi$
$72$	0	0
$73$	442.680	0.709751	0.354875	−	0.934914i	$-0.384523\pi$
$73$	442.680	0.709751	0.354875	+	0.934914i	$0.384523\pi$
$74$	0	0
$75$	85.4228	0.131517
$76$	0	0
$77$	0	0
$78$	0	0
$79$	45.7010	0.0650856	0.0325428	−	0.999470i	$-0.489639\pi$
$79$	45.7010	0.0650856	0.0325428	+	0.999470i	$0.489639\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−686.464	−0.907822	−0.453911	−	0.891047i	$-0.649972\pi$
$83$	−686.464	−0.907822	−0.453911	+	0.891047i	$0.649972\pi$
$84$	0	0
$85$	−772.206	−0.985382
$86$	0	0
$87$	−519.062	−0.639647
$88$	0	0
$89$	−660.145	−0.786238	−0.393119	−	0.919488i	$-0.628604\pi$
$89$	−660.145	−0.786238	−0.393119	+	0.919488i	$0.628604\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−201.743	−0.224943
$94$	0	0
$95$	−718.929	−0.776427
$96$	0	0
$97$	−658.320	−0.689095	−0.344548	−	0.938769i	$-0.611968\pi$
$97$	−658.320	−0.689095	−0.344548	+	0.938769i	$0.611968\pi$
$98$	0	0
$99$	127.577	0.129515
$100$	0	0
$101$	1803.90	1.77717	0.888586	−	0.458709i	$-0.151688\pi$
$101$	1803.90	1.77717	0.888586	+	0.458709i	$0.151688\pi$
$102$	0	0
$103$	−1074.91	−1.02829	−0.514145	−	0.857703i	$-0.671891\pi$
$103$	−1074.91	−1.02829	−0.514145	+	0.857703i	$0.671891\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1659.35	1.49921	0.749605	−	0.661885i	$-0.230243\pi$
$107$	1659.35	1.49921	0.749605	+	0.661885i	$0.230243\pi$
$108$	0	0
$109$	598.928	0.526302	0.263151	−	0.964755i	$-0.415238\pi$
$109$	598.928	0.526302	0.263151	+	0.964755i	$0.415238\pi$
$110$	0	0
$111$	905.011	0.773872
$112$	0	0
$113$	−781.650	−0.650720	−0.325360	−	0.945590i	$-0.605486\pi$
$113$	−781.650	−0.650720	−0.325360	+	0.945590i	$0.605486\pi$
$114$	0	0
$115$	−943.176	−0.764797
$116$	0	0
$117$	235.114	0.185780
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1130.06	−0.849033
$122$	0	0
$123$	1416.12	1.03811
$124$	0	0
$125$	1507.85	1.07893
$126$	0	0
$127$	−25.3722	−0.0177277	−0.00886385	−	0.999961i	$-0.502821\pi$
$127$	−25.3722	−0.0177277	−0.00886385	+	0.999961i	$0.502821\pi$
$128$	0	0
$129$	−1391.01	−0.949393
$130$	0	0
$131$	−475.577	−0.317186	−0.158593	−	0.987344i	$-0.550696\pi$
$131$	−475.577	−0.317186	−0.158593	+	0.987344i	$0.550696\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	265.268	0.169116
$136$	0	0
$137$	1058.54	0.660123	0.330062	−	0.943959i	$-0.392930\pi$
$137$	1058.54	0.660123	0.330062	+	0.943959i	$0.392930\pi$
$138$	0	0
$139$	2580.99	1.57494	0.787470	−	0.616352i	$-0.211390\pi$
$139$	2580.99	1.57494	0.787470	+	0.616352i	$0.211390\pi$
$140$	0	0
$141$	273.588	0.163406
$142$	0	0
$143$	370.311	0.216552
$144$	0	0
$145$	−1699.89	−0.973571
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−101.875	−0.0560131	−0.0280066	−	0.999608i	$-0.508916\pi$
$149$	−101.875	−0.0560131	−0.0280066	+	0.999608i	$0.508916\pi$
$150$	0	0
$151$	1778.42	0.958450	0.479225	−	0.877692i	$-0.340918\pi$
$151$	1778.42	0.958450	0.479225	+	0.877692i	$0.340918\pi$
$152$	0	0
$153$	707.382	0.373781
$154$	0	0
$155$	−660.690	−0.342374
$156$	0	0
$157$	−1399.26	−0.711292	−0.355646	−	0.934621i	$-0.615739\pi$
$157$	−1399.26	−0.711292	−0.355646	+	0.934621i	$0.615739\pi$
$158$	0	0
$159$	489.805	0.244302
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3615.43	−1.73731	−0.868656	−	0.495415i	$-0.835016\pi$
$163$	−3615.43	−1.73731	−0.868656	+	0.495415i	$0.835016\pi$
$164$	0	0
$165$	417.805	0.197128
$166$	0	0
$167$	−1076.72	−0.498917	−0.249459	−	0.968385i	$-0.580253\pi$
$167$	−1076.72	−0.498917	−0.249459	+	0.968385i	$0.580253\pi$
$168$	0	0
$169$	−1514.55	−0.689372
$170$	0	0
$171$	658.577	0.294518
$172$	0	0
$173$	−1873.63	−0.823407	−0.411704	−	0.911318i	$-0.635066\pi$
$173$	−1873.63	−0.823407	−0.411704	+	0.911318i	$0.635066\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1801.58	0.765056
$178$	0	0
$179$	2692.29	1.12420	0.562099	−	0.827070i	$-0.309994\pi$
$179$	2692.29	1.12420	0.562099	+	0.827070i	$0.309994\pi$
$180$	0	0
$181$	−461.670	−0.189589	−0.0947947	−	0.995497i	$-0.530219\pi$
$181$	−461.670	−0.189589	−0.0947947	+	0.995497i	$0.530219\pi$
$182$	0	0
$183$	1715.07	0.692797
$184$	0	0
$185$	2963.84	1.17787
$186$	0	0
$187$	1114.15	0.435692
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3764.83	1.42625	0.713124	−	0.701038i	$-0.247280\pi$
$191$	3764.83	1.42625	0.713124	+	0.701038i	$0.247280\pi$
$192$	0	0
$193$	2926.20	1.09136	0.545680	−	0.837994i	$-0.316272\pi$
$193$	2926.20	1.09136	0.545680	+	0.837994i	$0.316272\pi$
$194$	0	0
$195$	769.978	0.282766
$196$	0	0
$197$	5045.55	1.82477	0.912387	−	0.409330i	$-0.134237\pi$
$197$	5045.55	1.82477	0.912387	+	0.409330i	$0.134237\pi$
$198$	0	0
$199$	4292.25	1.52899	0.764496	−	0.644628i	$-0.222988\pi$
$199$	4292.25	1.52899	0.764496	+	0.644628i	$0.222988\pi$
$200$	0	0
$201$	−1618.27	−0.567880
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4637.69	1.58005
$206$	0	0
$207$	864.000	0.290107
$208$	0	0
$209$	1037.28	0.343301
$210$	0	0
$211$	−3625.76	−1.18297	−0.591487	−	0.806315i	$-0.701459\pi$
$211$	−3625.76	−1.18297	−0.591487	+	0.806315i	$0.701459\pi$
$212$	0	0
$213$	−3193.18	−1.02720
$214$	0	0
$215$	−4555.45	−1.44502
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1328.04	−0.409775
$220$	0	0
$221$	2053.28	0.624969
$222$	0	0
$223$	−3145.00	−0.944417	−0.472208	−	0.881487i	$-0.656543\pi$
$223$	−3145.00	−0.944417	−0.472208	+	0.881487i	$0.656543\pi$
$224$	0	0
$225$	−256.268	−0.0759313
$226$	0	0
$227$	−1545.75	−0.451959	−0.225980	−	0.974132i	$-0.572558\pi$
$227$	−1545.75	−0.451959	−0.225980	+	0.974132i	$0.572558\pi$
$228$	0	0
$229$	4072.15	1.17509	0.587544	−	0.809192i	$-0.300095\pi$
$229$	4072.15	1.17509	0.587544	+	0.809192i	$0.300095\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3015.71	−0.847922	−0.423961	−	0.905680i	$-0.639361\pi$
$233$	−3015.71	−0.847922	−0.423961	+	0.905680i	$0.639361\pi$
$234$	0	0
$235$	895.978	0.248711
$236$	0	0
$237$	−137.103	−0.0375772
$238$	0	0
$239$	−4647.38	−1.25780	−0.628900	−	0.777486i	$-0.716495\pi$
$239$	−4647.38	−1.25780	−0.628900	+	0.777486i	$0.716495\pi$
$240$	0	0
$241$	−1118.28	−0.298898	−0.149449	−	0.988769i	$-0.547750\pi$
$241$	−1118.28	−0.298898	−0.149449	+	0.988769i	$0.547750\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1911.61	0.492441
$248$	0	0
$249$	2059.39	0.524131
$250$	0	0
$251$	883.518	0.222180	0.111090	−	0.993810i	$-0.464566\pi$
$251$	883.518	0.222180	0.111090	+	0.993810i	$0.464566\pi$
$252$	0	0
$253$	1360.82	0.338159
$254$	0	0
$255$	2316.62	0.568911
$256$	0	0
$257$	5868.89	1.42448	0.712240	−	0.701937i	$-0.247681\pi$
$257$	5868.89	1.42448	0.712240	+	0.701937i	$0.247681\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1557.19	0.369301
$262$	0	0
$263$	−4875.61	−1.14313	−0.571565	−	0.820557i	$-0.693664\pi$
$263$	−4875.61	−1.14313	−0.571565	+	0.820557i	$0.693664\pi$
$264$	0	0
$265$	1604.07	0.371839
$266$	0	0
$267$	1980.43	0.453935
$268$	0	0
$269$	5527.10	1.25276	0.626382	−	0.779516i	$-0.284535\pi$
$269$	5527.10	1.25276	0.626382	+	0.779516i	$0.284535\pi$
$270$	0	0
$271$	−3224.73	−0.722836	−0.361418	−	0.932404i	$-0.617707\pi$
$271$	−3224.73	−0.722836	−0.361418	+	0.932404i	$0.617707\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−403.630	−0.0885083
$276$	0	0
$277$	−1958.04	−0.424720	−0.212360	−	0.977191i	$-0.568115\pi$
$277$	−1958.04	−0.424720	−0.212360	+	0.977191i	$0.568115\pi$
$278$	0	0
$279$	605.228	0.129871
$280$	0	0
$281$	8531.16	1.81113	0.905563	−	0.424212i	$-0.139449\pi$
$281$	8531.16	1.81113	0.905563	+	0.424212i	$0.139449\pi$
$282$	0	0
$283$	7909.86	1.66146	0.830728	−	0.556678i	$-0.187924\pi$
$283$	7909.86	1.66146	0.830728	+	0.556678i	$0.187924\pi$
$284$	0	0
$285$	2156.79	0.448270
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1264.65	0.257408
$290$	0	0
$291$	1974.96	0.397849
$292$	0	0
$293$	8321.41	1.65919	0.829594	−	0.558367i	$-0.188572\pi$
$293$	8321.41	1.65919	0.829594	+	0.558367i	$0.188572\pi$
$294$	0	0
$295$	5900.02	1.16445
$296$	0	0
$297$	−382.732	−0.0747756
$298$	0	0
$299$	2507.88	0.485065
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−5411.69	−1.02605
$304$	0	0
$305$	5616.72	1.05447
$306$	0	0
$307$	2541.79	0.472533	0.236267	−	0.971688i	$-0.424076\pi$
$307$	2541.79	0.472533	0.236267	+	0.971688i	$0.424076\pi$
$308$	0	0
$309$	3224.72	0.593683
$310$	0	0
$311$	−607.709	−0.110804	−0.0554020	−	0.998464i	$-0.517644\pi$
$311$	−607.709	−0.110804	−0.0554020	+	0.998464i	$0.517644\pi$
$312$	0	0
$313$	−11027.3	−1.99137	−0.995684	−	0.0928043i	$-0.970417\pi$
$313$	−11027.3	−1.99137	−0.995684	+	0.0928043i	$0.970417\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7261.60	1.28660	0.643300	−	0.765614i	$-0.277565\pi$
$317$	7261.60	1.28660	0.643300	+	0.765614i	$0.277565\pi$
$318$	0	0
$319$	2452.61	0.430470
$320$	0	0
$321$	−4978.05	−0.865570
$322$	0	0
$323$	5751.43	0.990768
$324$	0	0
$325$	−743.855	−0.126959
$326$	0	0
$327$	−1796.78	−0.303860
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6632.73	−1.10141	−0.550706	−	0.834699i	$-0.685642\pi$
$331$	−6632.73	−1.10141	−0.550706	+	0.834699i	$0.685642\pi$
$332$	0	0
$333$	−2715.03	−0.446795
$334$	0	0
$335$	−5299.69	−0.864338
$336$	0	0
$337$	8104.59	1.31005	0.655023	−	0.755609i	$-0.272659\pi$
$337$	8104.59	1.31005	0.655023	+	0.755609i	$0.272659\pi$
$338$	0	0
$339$	2344.95	0.375694
$340$	0	0
$341$	953.250	0.151382
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2829.53	0.441556
$346$	0	0
$347$	5937.28	0.918530	0.459265	−	0.888299i	$-0.348113\pi$
$347$	5937.28	0.918530	0.459265	+	0.888299i	$0.348113\pi$
$348$	0	0
$349$	−268.472	−0.0411775	−0.0205888	−	0.999788i	$-0.506554\pi$
$349$	−268.472	−0.0411775	−0.0205888	+	0.999788i	$0.506554\pi$
$350$	0	0
$351$	−705.341	−0.107260
$352$	0	0
$353$	−2662.81	−0.401493	−0.200747	−	0.979643i	$-0.564337\pi$
$353$	−2662.81	−0.401493	−0.200747	+	0.979643i	$0.564337\pi$
$354$	0	0
$355$	−10457.4	−1.56344
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1563.27	0.229823	0.114911	−	0.993376i	$-0.463342\pi$
$359$	1563.27	0.229823	0.114911	+	0.993376i	$0.463342\pi$
$360$	0	0
$361$	−1504.38	−0.219330
$362$	0	0
$363$	3390.19	0.490189
$364$	0	0
$365$	−4349.22	−0.623695
$366$	0	0
$367$	1210.82	0.172218	0.0861091	−	0.996286i	$-0.472557\pi$
$367$	1210.82	0.172218	0.0861091	+	0.996286i	$0.472557\pi$
$368$	0	0
$369$	−4248.37	−0.599354
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4771.86	−0.662406	−0.331203	−	0.943559i	$-0.607455\pi$
$373$	−4771.86	−0.662406	−0.331203	+	0.943559i	$0.607455\pi$
$374$	0	0
$375$	−4523.54	−0.622919
$376$	0	0
$377$	4519.95	0.617479
$378$	0	0
$379$	4118.66	0.558209	0.279104	−	0.960261i	$-0.409962\pi$
$379$	4118.66	0.558209	0.279104	+	0.960261i	$0.409962\pi$
$380$	0	0
$381$	76.1166	0.0102351
$382$	0	0
$383$	3535.98	0.471749	0.235875	−	0.971783i	$-0.424204\pi$
$383$	3535.98	0.471749	0.235875	+	0.971783i	$0.424204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4173.03	0.548132
$388$	0	0
$389$	9606.46	1.25210	0.626050	−	0.779783i	$-0.284671\pi$
$389$	9606.46	1.25210	0.626050	+	0.779783i	$0.284671\pi$
$390$	0	0
$391$	7545.41	0.975928
$392$	0	0
$393$	1426.73	0.183127
$394$	0	0
$395$	−449.001	−0.0571941
$396$	0	0
$397$	−5367.07	−0.678503	−0.339252	−	0.940696i	$-0.610174\pi$
$397$	−5367.07	−0.678503	−0.339252	+	0.940696i	$0.610174\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14065.8	−1.75166	−0.875828	−	0.482624i	$-0.839684\pi$
$401$	−14065.8	−1.75166	−0.875828	+	0.482624i	$0.839684\pi$
$402$	0	0
$403$	1756.76	0.217147
$404$	0	0
$405$	−795.805	−0.0976392
$406$	0	0
$407$	−4276.25	−0.520801
$408$	0	0
$409$	−3272.44	−0.395628	−0.197814	−	0.980240i	$-0.563384\pi$
$409$	−3272.44	−0.395628	−0.197814	+	0.980240i	$0.563384\pi$
$410$	0	0
$411$	−3175.61	−0.381122
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6744.34	0.797751
$416$	0	0
$417$	−7742.97	−0.909293
$418$	0	0
$419$	8077.07	0.941744	0.470872	−	0.882202i	$-0.343939\pi$
$419$	8077.07	0.941744	0.470872	+	0.882202i	$0.343939\pi$
$420$	0	0
$421$	8051.68	0.932101	0.466051	−	0.884758i	$-0.345676\pi$
$421$	8051.68	0.932101	0.466051	+	0.884758i	$0.345676\pi$
$422$	0	0
$423$	−820.764	−0.0943426
$424$	0	0
$425$	−2238.02	−0.255435
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1110.93	−0.125026
$430$	0	0
$431$	9900.22	1.10644	0.553221	−	0.833034i	$-0.313398\pi$
$431$	9900.22	1.10644	0.553221	+	0.833034i	$0.313398\pi$
$432$	0	0
$433$	511.795	0.0568021	0.0284010	−	0.999597i	$-0.490958\pi$
$433$	511.795	0.0568021	0.0284010	+	0.999597i	$0.490958\pi$
$434$	0	0
$435$	5099.66	0.562092
$436$	0	0
$437$	7024.82	0.768977
$438$	0	0
$439$	−1302.26	−0.141580	−0.0707898	−	0.997491i	$-0.522552\pi$
$439$	−1302.26	−0.141580	−0.0707898	+	0.997491i	$0.522552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8200.32	0.879479	0.439739	−	0.898125i	$-0.355071\pi$
$443$	8200.32	0.879479	0.439739	+	0.898125i	$0.355071\pi$
$444$	0	0
$445$	6485.76	0.690909
$446$	0	0
$447$	305.626	0.0323392
$448$	0	0
$449$	−5788.12	−0.608370	−0.304185	−	0.952613i	$-0.598384\pi$
$449$	−5788.12	−0.608370	−0.304185	+	0.952613i	$0.598384\pi$
$450$	0	0
$451$	−6691.31	−0.698628
$452$	0	0
$453$	−5335.27	−0.553362
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7259.18	0.743041	0.371521	−	0.928425i	$-0.378836\pi$
$457$	7259.18	0.743041	0.371521	+	0.928425i	$0.378836\pi$
$458$	0	0
$459$	−2122.15	−0.215802
$460$	0	0
$461$	5966.92	0.602835	0.301418	−	0.953492i	$-0.402540\pi$
$461$	5966.92	0.602835	0.301418	+	0.953492i	$0.402540\pi$
$462$	0	0
$463$	8884.02	0.891740	0.445870	−	0.895098i	$-0.352894\pi$
$463$	8884.02	0.891740	0.445870	+	0.895098i	$0.352894\pi$
$464$	0	0
$465$	1982.07	0.197669
$466$	0	0
$467$	−4029.36	−0.399265	−0.199632	−	0.979871i	$-0.563975\pi$
$467$	−4029.36	−0.399265	−0.199632	+	0.979871i	$0.563975\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4197.77	0.410665
$472$	0	0
$473$	6572.64	0.638923
$474$	0	0
$475$	−2083.61	−0.201269
$476$	0	0
$477$	−1469.41	−0.141048
$478$	0	0
$479$	−17440.7	−1.66364	−0.831820	−	0.555045i	$-0.812701\pi$
$479$	−17440.7	−1.66364	−0.831820	+	0.555045i	$0.812701\pi$
$480$	0	0
$481$	−7880.76	−0.747052
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6467.83	0.605544
$486$	0	0
$487$	4272.28	0.397526	0.198763	−	0.980048i	$-0.436308\pi$
$487$	4272.28	0.397526	0.198763	+	0.980048i	$0.436308\pi$
$488$	0	0
$489$	10846.3	1.00304
$490$	0	0
$491$	−2013.29	−0.185048	−0.0925240	−	0.995710i	$-0.529493\pi$
$491$	−2013.29	−0.185048	−0.0925240	+	0.995710i	$0.529493\pi$
$492$	0	0
$493$	13599.1	1.24234
$494$	0	0
$495$	−1253.41	−0.113812
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4880.46	0.437834	0.218917	−	0.975743i	$-0.429748\pi$
$499$	4880.46	0.437834	0.218917	+	0.975743i	$0.429748\pi$
$500$	0	0
$501$	3230.16	0.288050
$502$	0	0
$503$	12961.9	1.14899	0.574496	−	0.818508i	$-0.305198\pi$
$503$	12961.9	1.14899	0.574496	+	0.818508i	$0.305198\pi$
$504$	0	0
$505$	−17722.8	−1.56170
$506$	0	0
$507$	4543.65	0.398009
$508$	0	0
$509$	−8155.38	−0.710178	−0.355089	−	0.934832i	$-0.615550\pi$
$509$	−8155.38	−0.710178	−0.355089	+	0.934832i	$0.615550\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1975.73	−0.170040
$514$	0	0
$515$	10560.7	0.903612
$516$	0	0
$517$	−1292.73	−0.109969
$518$	0	0
$519$	5620.89	0.475394
$520$	0	0
$521$	332.018	0.0279193	0.0139597	−	0.999903i	$-0.495556\pi$
$521$	332.018	0.0279193	0.0139597	+	0.999903i	$0.495556\pi$
$522$	0	0
$523$	7254.83	0.606561	0.303281	−	0.952901i	$-0.401918\pi$
$523$	7254.83	0.606561	0.303281	+	0.952901i	$0.401918\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5285.52	0.436890
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	−5404.73	−0.441705
$532$	0	0
$533$	−12331.5	−1.00213
$534$	0	0
$535$	−16302.7	−1.31744
$536$	0	0
$537$	−8076.87	−0.649055
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5410.19	0.429949	0.214974	−	0.976620i	$-0.431033\pi$
$541$	5410.19	0.429949	0.214974	+	0.976620i	$0.431033\pi$
$542$	0	0
$543$	1385.01	0.109459
$544$	0	0
$545$	−5884.32	−0.462489
$546$	0	0
$547$	18673.9	1.45967	0.729834	−	0.683625i	$-0.239597\pi$
$547$	18673.9	1.45967	0.729834	+	0.683625i	$0.239597\pi$
$548$	0	0
$549$	−5145.22	−0.399987
$550$	0	0
$551$	12660.8	0.978893
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8891.51	−0.680042
$556$	0	0
$557$	−8209.48	−0.624500	−0.312250	−	0.950000i	$-0.601083\pi$
$557$	−8209.48	−0.624500	−0.312250	+	0.950000i	$0.601083\pi$
$558$	0	0
$559$	12112.8	0.916489
$560$	0	0
$561$	−3342.44	−0.251547
$562$	0	0
$563$	17522.2	1.31167	0.655837	−	0.754902i	$-0.272316\pi$
$563$	17522.2	1.31167	0.655837	+	0.754902i	$0.272316\pi$
$564$	0	0
$565$	7679.51	0.571822
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9056.83	−0.667279	−0.333640	−	0.942701i	$-0.608277\pi$
$569$	−9056.83	−0.667279	−0.333640	+	0.942701i	$0.608277\pi$
$570$	0	0
$571$	−10716.5	−0.785417	−0.392709	−	0.919663i	$-0.628462\pi$
$571$	−10716.5	−0.785417	−0.392709	+	0.919663i	$0.628462\pi$
$572$	0	0
$573$	−11294.5	−0.823444
$574$	0	0
$575$	−2733.53	−0.198254
$576$	0	0
$577$	−3217.23	−0.232123	−0.116062	−	0.993242i	$-0.537027\pi$
$577$	−3217.23	−0.232123	−0.116062	+	0.993242i	$0.537027\pi$
$578$	0	0
$579$	−8778.59	−0.630097
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2314.37	−0.164411
$584$	0	0
$585$	−2309.93	−0.163255
$586$	0	0
$587$	3248.45	0.228412	0.114206	−	0.993457i	$-0.463568\pi$
$587$	3248.45	0.228412	0.114206	+	0.993457i	$0.463568\pi$
$588$	0	0
$589$	4920.85	0.344245
$590$	0	0
$591$	−15136.6	−1.05353
$592$	0	0
$593$	21110.4	1.46189	0.730945	−	0.682437i	$-0.239080\pi$
$593$	21110.4	1.46189	0.730945	+	0.682437i	$0.239080\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12876.7	−0.882764
$598$	0	0
$599$	15738.0	1.07352	0.536758	−	0.843736i	$-0.319649\pi$
$599$	15738.0	1.07352	0.536758	+	0.843736i	$0.319649\pi$
$600$	0	0
$601$	4856.96	0.329650	0.164825	−	0.986323i	$-0.447294\pi$
$601$	4856.96	0.329650	0.164825	+	0.986323i	$0.447294\pi$
$602$	0	0
$603$	4854.80	0.327866
$604$	0	0
$605$	11102.6	0.746089
$606$	0	0
$607$	−17258.6	−1.15404	−0.577022	−	0.816729i	$-0.695785\pi$
$607$	−17258.6	−1.15404	−0.577022	+	0.816729i	$0.695785\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2382.38	−0.157743
$612$	0	0
$613$	16117.1	1.06193	0.530964	−	0.847394i	$-0.321830\pi$
$613$	16117.1	1.06193	0.530964	+	0.847394i	$0.321830\pi$
$614$	0	0
$615$	−13913.1	−0.912243
$616$	0	0
$617$	−17507.3	−1.14233	−0.571164	−	0.820836i	$-0.693508\pi$
$617$	−17507.3	−1.14233	−0.571164	+	0.820836i	$0.693508\pi$
$618$	0	0
$619$	9718.49	0.631048	0.315524	−	0.948918i	$-0.397820\pi$
$619$	9718.49	0.631048	0.315524	+	0.948918i	$0.397820\pi$
$620$	0	0
$621$	−2592.00	−0.167493
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11254.9	−0.720316
$626$	0	0
$627$	−3111.83	−0.198205
$628$	0	0
$629$	−23710.7	−1.50303
$630$	0	0
$631$	−5144.78	−0.324580	−0.162290	−	0.986743i	$-0.551888\pi$
$631$	−5144.78	−0.324580	−0.162290	+	0.986743i	$0.551888\pi$
$632$	0	0
$633$	10877.3	0.682990
$634$	0	0
$635$	249.275	0.0155783
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9579.53	0.593052
$640$	0	0
$641$	21686.1	1.33627	0.668135	−	0.744040i	$-0.267093\pi$
$641$	21686.1	1.33627	0.668135	+	0.744040i	$0.267093\pi$
$642$	0	0
$643$	−14171.1	−0.869132	−0.434566	−	0.900640i	$-0.643098\pi$
$643$	−14171.1	−0.869132	−0.434566	+	0.900640i	$0.643098\pi$
$644$	0	0
$645$	13666.3	0.834281
$646$	0	0
$647$	20053.3	1.21851	0.609254	−	0.792975i	$-0.291469\pi$
$647$	20053.3	1.21851	0.609254	+	0.792975i	$0.291469\pi$
$648$	0	0
$649$	−8512.60	−0.514867
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23641.2	1.41677	0.708385	−	0.705827i	$-0.249424\pi$
$653$	23641.2	1.41677	0.708385	+	0.705827i	$0.249424\pi$
$654$	0	0
$655$	4672.43	0.278728
$656$	0	0
$657$	3984.12	0.236584
$658$	0	0
$659$	−2367.34	−0.139937	−0.0699685	−	0.997549i	$-0.522290\pi$
$659$	−2367.34	−0.139937	−0.0699685	+	0.997549i	$0.522290\pi$
$660$	0	0
$661$	389.981	0.0229478	0.0114739	−	0.999934i	$-0.496348\pi$
$661$	389.981	0.0229478	0.0114739	+	0.999934i	$0.496348\pi$
$662$	0	0
$663$	−6159.83	−0.360826
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16610.0	0.964230
$668$	0	0
$669$	9435.01	0.545259
$670$	0	0
$671$	−8103.86	−0.466239
$672$	0	0
$673$	28764.5	1.64753	0.823766	−	0.566930i	$-0.191869\pi$
$673$	28764.5	1.64753	0.823766	+	0.566930i	$0.191869\pi$
$674$	0	0
$675$	768.805	0.0438390
$676$	0	0
$677$	1762.12	0.100035	0.0500176	−	0.998748i	$-0.484072\pi$
$677$	1762.12	0.100035	0.0500176	+	0.998748i	$0.484072\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4637.24	0.260939
$682$	0	0
$683$	2228.87	0.124869	0.0624344	−	0.998049i	$-0.480114\pi$
$683$	2228.87	0.124869	0.0624344	+	0.998049i	$0.480114\pi$
$684$	0	0
$685$	−10399.9	−0.580085
$686$	0	0
$687$	−12216.4	−0.678437
$688$	0	0
$689$	−4265.18	−0.235835
$690$	0	0
$691$	17344.3	0.954859	0.477429	−	0.878670i	$-0.341569\pi$
$691$	17344.3	0.954859	0.477429	+	0.878670i	$0.341569\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25357.6	−1.38398
$696$	0	0
$697$	−37101.5	−2.01624
$698$	0	0
$699$	9047.14	0.489548
$700$	0	0
$701$	−23697.7	−1.27682	−0.638409	−	0.769697i	$-0.720407\pi$
$701$	−23697.7	−1.27682	−0.638409	+	0.769697i	$0.720407\pi$
$702$	0	0
$703$	−22074.8	−1.18431
$704$	0	0
$705$	−2687.93	−0.143594
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−22791.0	−1.20724	−0.603620	−	0.797272i	$-0.706275\pi$
$709$	−22791.0	−1.20724	−0.603620	+	0.797272i	$0.706275\pi$
$710$	0	0
$711$	411.309	0.0216952
$712$	0	0
$713$	6455.76	0.339089
$714$	0	0
$715$	−3638.21	−0.190296
$716$	0	0
$717$	13942.2	0.726191
$718$	0	0
$719$	31844.8	1.65175	0.825876	−	0.563852i	$-0.190681\pi$
$719$	31844.8	1.65175	0.825876	+	0.563852i	$0.190681\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3354.83	0.172569
$724$	0	0
$725$	−4926.64	−0.252373
$726$	0	0
$727$	−22500.0	−1.14784	−0.573920	−	0.818912i	$-0.694578\pi$
$727$	−22500.0	−1.14784	−0.573920	+	0.818912i	$0.694578\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	36443.6	1.84393
$732$	0	0
$733$	3222.75	0.162394	0.0811972	−	0.996698i	$-0.474126\pi$
$733$	3222.75	0.162394	0.0811972	+	0.996698i	$0.474126\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7646.45	0.382172
$738$	0	0
$739$	−18359.5	−0.913893	−0.456946	−	0.889494i	$-0.651057\pi$
$739$	−18359.5	−0.913893	−0.456946	+	0.889494i	$0.651057\pi$
$740$	0	0
$741$	−5734.84	−0.284311
$742$	0	0
$743$	−12890.4	−0.636478	−0.318239	−	0.948010i	$-0.603091\pi$
$743$	−12890.4	−0.636478	−0.318239	+	0.948010i	$0.603091\pi$
$744$	0	0
$745$	1000.90	0.0492217
$746$	0	0
$747$	−6178.18	−0.302607
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15390.2	−0.747798	−0.373899	−	0.927469i	$-0.621979\pi$
$751$	−15390.2	−0.747798	−0.373899	+	0.927469i	$0.621979\pi$
$752$	0	0
$753$	−2650.55	−0.128276
$754$	0	0
$755$	−17472.6	−0.842241
$756$	0	0
$757$	13974.6	0.670957	0.335479	−	0.942048i	$-0.391102\pi$
$757$	13974.6	0.670957	0.335479	+	0.942048i	$0.391102\pi$
$758$	0	0
$759$	−4082.47	−0.195236
$760$	0	0
$761$	3375.14	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$761$	3375.14	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6949.85	−0.328461
$766$	0	0
$767$	−15688.0	−0.738540
$768$	0	0
$769$	31253.9	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$769$	31253.9	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$770$	0	0
$771$	−17606.7	−0.822423
$772$	0	0
$773$	−23163.5	−1.07779	−0.538897	−	0.842372i	$-0.681159\pi$
$773$	−23163.5	−1.07779	−0.538897	+	0.842372i	$0.681159\pi$
$774$	0	0
$775$	−1914.82	−0.0887516
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−34541.8	−1.58869
$780$	0	0
$781$	15088.0	0.691283
$782$	0	0
$783$	−4671.56	−0.213216
$784$	0	0
$785$	13747.4	0.625050
$786$	0	0
$787$	−9902.95	−0.448541	−0.224271	−	0.974527i	$-0.572000\pi$
$787$	−9902.95	−0.448541	−0.224271	+	0.974527i	$0.572000\pi$
$788$	0	0
$789$	14626.8	0.659986
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14934.7	−0.668786
$794$	0	0
$795$	−4812.21	−0.214681
$796$	0	0
$797$	−16844.5	−0.748638	−0.374319	−	0.927300i	$-0.622123\pi$
$797$	−16844.5	−0.748638	−0.374319	+	0.927300i	$0.622123\pi$
$798$	0	0
$799$	−7167.83	−0.317371
$800$	0	0
$801$	−5941.30	−0.262079
$802$	0	0
$803$	6275.10	0.275770
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16581.3	−0.723284
$808$	0	0
$809$	−34430.7	−1.49631	−0.748157	−	0.663522i	$-0.769061\pi$
$809$	−34430.7	−1.49631	−0.748157	+	0.663522i	$0.769061\pi$
$810$	0	0
$811$	−62.4498	−0.00270396	−0.00135198	−	0.999999i	$-0.500430\pi$
$811$	−62.4498	−0.00270396	−0.00135198	+	0.999999i	$0.500430\pi$
$812$	0	0
$813$	9674.20	0.417329
$814$	0	0
$815$	35520.7	1.52667
$816$	0	0
$817$	33929.2	1.45292
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38951.7	1.65581	0.827906	−	0.560866i	$-0.189532\pi$
$821$	38951.7	1.65581	0.827906	+	0.560866i	$0.189532\pi$
$822$	0	0
$823$	35482.3	1.50284	0.751420	−	0.659825i	$-0.229369\pi$
$823$	35482.3	1.50284	0.751420	+	0.659825i	$0.229369\pi$
$824$	0	0
$825$	1210.89	0.0511003
$826$	0	0
$827$	−7945.87	−0.334105	−0.167053	−	0.985948i	$-0.553425\pi$
$827$	−7945.87	−0.334105	−0.167053	+	0.985948i	$0.553425\pi$
$828$	0	0
$829$	−40636.4	−1.70249	−0.851243	−	0.524772i	$-0.824151\pi$
$829$	−40636.4	−1.70249	−0.851243	+	0.524772i	$0.824151\pi$
$830$	0	0
$831$	5874.13	0.245212
$832$	0	0
$833$	0	0
$834$	0	0
$835$	10578.5	0.438425
$836$	0	0
$837$	−1815.68	−0.0749811
$838$	0	0
$839$	27723.5	1.14079	0.570394	−	0.821371i	$-0.306791\pi$
$839$	27723.5	1.14079	0.570394	+	0.821371i	$0.306791\pi$
$840$	0	0
$841$	5547.19	0.227446
$842$	0	0
$843$	−25593.5	−1.04565
$844$	0	0
$845$	14880.1	0.605787
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−23729.6	−0.959243
$850$	0	0
$851$	−28960.3	−1.16657
$852$	0	0
$853$	−37599.1	−1.50922	−0.754612	−	0.656171i	$-0.772175\pi$
$853$	−37599.1	−1.50922	−0.754612	+	0.656171i	$0.772175\pi$
$854$	0	0
$855$	−6470.36	−0.258809
$856$	0	0
$857$	−25610.0	−1.02079	−0.510397	−	0.859939i	$-0.670502\pi$
$857$	−25610.0	−1.02079	−0.510397	+	0.859939i	$0.670502\pi$
$858$	0	0
$859$	−3979.39	−0.158062	−0.0790309	−	0.996872i	$-0.525183\pi$
$859$	−3979.39	−0.158062	−0.0790309	+	0.996872i	$0.525183\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24820.3	0.979020	0.489510	−	0.871998i	$-0.337176\pi$
$863$	24820.3	0.979020	0.489510	+	0.871998i	$0.337176\pi$
$864$	0	0
$865$	18407.9	0.723571
$866$	0	0
$867$	−3793.94	−0.148615
$868$	0	0
$869$	647.823	0.0252887
$870$	0	0
$871$	14091.7	0.548198
$872$	0	0
$873$	−5924.88	−0.229698
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50019.4	1.92593	0.962963	−	0.269635i	$-0.0869030\pi$
$877$	50019.4	1.92593	0.962963	+	0.269635i	$0.0869030\pi$
$878$	0	0
$879$	−24964.2	−0.957933
$880$	0	0
$881$	10386.4	0.397193	0.198596	−	0.980081i	$-0.436362\pi$
$881$	10386.4	0.397193	0.198596	+	0.980081i	$0.436362\pi$
$882$	0	0
$883$	−36366.4	−1.38599	−0.692994	−	0.720944i	$-0.743709\pi$
$883$	−36366.4	−1.38599	−0.692994	+	0.720944i	$0.743709\pi$
$884$	0	0
$885$	−17700.0	−0.672295
$886$	0	0
$887$	−7607.31	−0.287969	−0.143984	−	0.989580i	$-0.545992\pi$
$887$	−7607.31	−0.287969	−0.143984	+	0.989580i	$0.545992\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1148.20	0.0431717
$892$	0	0
$893$	−6673.29	−0.250071
$894$	0	0
$895$	−26451.1	−0.987891
$896$	0	0
$897$	−7523.64	−0.280053
$898$	0	0
$899$	11635.2	0.431653
$900$	0	0
$901$	−12832.6	−0.474489
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4535.80	0.166602
$906$	0	0
$907$	−25803.1	−0.944628	−0.472314	−	0.881430i	$-0.656581\pi$
$907$	−25803.1	−0.944628	−0.472314	+	0.881430i	$0.656581\pi$
$908$	0	0
$909$	16235.1	0.592391
$910$	0	0
$911$	−8165.21	−0.296954	−0.148477	−	0.988916i	$-0.547437\pi$
$911$	−8165.21	−0.296954	−0.148477	+	0.988916i	$0.547437\pi$
$912$	0	0
$913$	−9730.80	−0.352730
$914$	0	0
$915$	−16850.2	−0.608797
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−593.607	−0.0213072	−0.0106536	−	0.999943i	$-0.503391\pi$
$919$	−593.607	−0.0213072	−0.0106536	+	0.999943i	$0.503391\pi$
$920$	0	0
$921$	−7625.37	−0.272817
$922$	0	0
$923$	27805.9	0.991596
$924$	0	0
$925$	8589.84	0.305332
$926$	0	0
$927$	−9674.17	−0.342763
$928$	0	0
$929$	33147.4	1.17065	0.585323	−	0.810800i	$-0.300968\pi$
$929$	33147.4	1.17065	0.585323	+	0.810800i	$0.300968\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1823.13	0.0639727
$934$	0	0
$935$	−10946.2	−0.382866
$936$	0	0
$937$	24883.8	0.867575	0.433788	−	0.901015i	$-0.357177\pi$
$937$	24883.8	0.867575	0.433788	+	0.901015i	$0.357177\pi$
$938$	0	0
$939$	33081.8	1.14972
$940$	0	0
$941$	26130.3	0.905232	0.452616	−	0.891706i	$-0.350491\pi$
$941$	26130.3	0.905232	0.452616	+	0.891706i	$0.350491\pi$
$942$	0	0
$943$	−45316.0	−1.56489
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7968.69	−0.273440	−0.136720	−	0.990610i	$-0.543656\pi$
$947$	−7968.69	−0.273440	−0.136720	+	0.990610i	$0.543656\pi$
$948$	0	0
$949$	11564.5	0.395573
$950$	0	0
$951$	−21784.8	−0.742819
$952$	0	0
$953$	−6742.36	−0.229178	−0.114589	−	0.993413i	$-0.536555\pi$
$953$	−6742.36	−0.229178	−0.114589	+	0.993413i	$0.536555\pi$
$954$	0	0
$955$	−36988.5	−1.25332
$956$	0	0
$957$	−7357.84	−0.248532
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25268.8	−0.848202
$962$	0	0
$963$	14934.2	0.499737
$964$	0	0
$965$	−28749.2	−0.959035
$966$	0	0
$967$	7202.50	0.239521	0.119760	−	0.992803i	$-0.461787\pi$
$967$	7202.50	0.239521	0.119760	+	0.992803i	$0.461787\pi$
$968$	0	0
$969$	−17254.3	−0.572020
$970$	0	0
$971$	13705.7	0.452973	0.226487	−	0.974014i	$-0.427276\pi$
$971$	13705.7	0.452973	0.226487	+	0.974014i	$0.427276\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2231.56	0.0732997
$976$	0	0
$977$	−56548.0	−1.85172	−0.925860	−	0.377868i	$-0.876657\pi$
$977$	−56548.0	−1.85172	−0.925860	+	0.377868i	$0.876657\pi$
$978$	0	0
$979$	−9357.71	−0.305489
$980$	0	0
$981$	5390.35	0.175434
$982$	0	0
$983$	17363.7	0.563395	0.281698	−	0.959503i	$-0.409103\pi$
$983$	17363.7	0.563395	0.281698	+	0.959503i	$0.409103\pi$
$984$	0	0
$985$	−49571.2	−1.60352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	44512.3	1.43115
$990$	0	0
$991$	−49039.4	−1.57194	−0.785968	−	0.618267i	$-0.787835\pi$
$991$	−49039.4	−1.57194	−0.785968	+	0.618267i	$0.787835\pi$
$992$	0	0
$993$	19898.2	0.635901
$994$	0	0
$995$	−42170.3	−1.34361
$996$	0	0
$997$	9116.60	0.289595	0.144797	−	0.989461i	$-0.453747\pi$
$997$	9116.60	0.289595	0.144797	+	0.989461i	$0.453747\pi$
$998$	0	0
$999$	8145.10	0.257957

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bp.1.1		2
4.3	odd	2		588.4.a.h.1.1		2
7.3	odd	6		336.4.q.h.289.1		4
7.5	odd	6		336.4.q.h.193.1		4
7.6	odd	2		2352.4.a.cb.1.2		2
12.11	even	2		1764.4.a.p.1.2		2
28.3	even	6		84.4.i.b.37.1	yes	4
28.11	odd	6		588.4.i.i.373.2		4
28.19	even	6		84.4.i.b.25.1	✓	4
28.23	odd	6		588.4.i.i.361.2		4
28.27	even	2		588.4.a.g.1.2		2
84.11	even	6		1764.4.k.z.1549.1		4
84.23	even	6		1764.4.k.z.361.1		4
84.47	odd	6		252.4.k.d.109.2		4
84.59	odd	6		252.4.k.d.37.2		4
84.83	odd	2		1764.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.i.b.25.1	✓	4	28.19	even	6
84.4.i.b.37.1	yes	4	28.3	even	6
252.4.k.d.37.2		4	84.59	odd	6
252.4.k.d.109.2		4	84.47	odd	6
336.4.q.h.193.1		4	7.5	odd	6
336.4.q.h.289.1		4	7.3	odd	6
588.4.a.g.1.2		2	28.27	even	2
588.4.a.h.1.1		2	4.3	odd	2
588.4.i.i.361.2		4	28.23	odd	6
588.4.i.i.373.2		4	28.11	odd	6
1764.4.a.p.1.2		2	12.11	even	2
1764.4.a.x.1.1		2	84.83	odd	2
1764.4.k.z.361.1		4	84.23	even	6
1764.4.k.z.1549.1		4	84.11	even	6
2352.4.a.bp.1.1		2	1.1	even	1	trivial
2352.4.a.cb.1.2		2	7.6	odd	2

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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -9.82475 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -9.82475 q^{5} +9.00000 q^{9} +14.1752 q^{11} +26.1238 q^{13} +29.4743 q^{15} +78.5980 q^{17} +73.1752 q^{19} +96.0000 q^{23} -28.4743 q^{25} -27.0000 q^{27} +173.021 q^{29} +67.2475 q^{31} -42.5257 q^{33} -301.670 q^{37} -78.3713 q^{39} -472.042 q^{41} +463.670 q^{43} -88.4228 q^{45} -91.1960 q^{47} -235.794 q^{51} -163.268 q^{53} -139.268 q^{55} -219.526 q^{57} -600.526 q^{59} -571.691 q^{61} -256.659 q^{65} +539.423 q^{67} -288.000 q^{69} +1064.39 q^{71} +442.680 q^{73} +85.4228 q^{75} +45.7010 q^{79} +81.0000 q^{81} -686.464 q^{83} -772.206 q^{85} -519.062 q^{87} -660.145 q^{89} -201.743 q^{93} -718.929 q^{95} -658.320 q^{97} +127.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9} + 51 q^{11} - 61 q^{13} - 9 q^{15} - 24 q^{17} + 169 q^{19} + 192 q^{23} + 11 q^{25} - 54 q^{27} - 39 q^{29} - 92 q^{31} - 153 q^{33} - 173 q^{37} + 183 q^{39} - 174 q^{41} + 497 q^{43} + 27 q^{45} + 180 q^{47} + 72 q^{51} + 285 q^{53} + 333 q^{55} - 507 q^{57} - 1269 q^{59} - 328 q^{61} - 1374 q^{65} + 875 q^{67} - 576 q^{69} + 1404 q^{71} + 1361 q^{73} - 33 q^{75} + 182 q^{79} + 162 q^{81} - 399 q^{83} - 2088 q^{85} + 117 q^{87} - 822 q^{89} + 276 q^{93} + 510 q^{95} - 841 q^{97} + 459 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9 + 51 * q^11 - 61 * q^13 - 9 * q^15 - 24 * q^17 + 169 * q^19 + 192 * q^23 + 11 * q^25 - 54 * q^27 - 39 * q^29 - 92 * q^31 - 153 * q^33 - 173 * q^37 + 183 * q^39 - 174 * q^41 + 497 * q^43 + 27 * q^45 + 180 * q^47 + 72 * q^51 + 285 * q^53 + 333 * q^55 - 507 * q^57 - 1269 * q^59 - 328 * q^61 - 1374 * q^65 + 875 * q^67 - 576 * q^69 + 1404 * q^71 + 1361 * q^73 - 33 * q^75 + 182 * q^79 + 162 * q^81 - 399 * q^83 - 2088 * q^85 + 117 * q^87 - 822 * q^89 + 276 * q^93 + 510 * q^95 - 841 * q^97 + 459 * q^99

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