LMFDB - Embedded newform 2352.3.m.h.1471.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(1471,2352)]

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([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2352.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.m (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.4
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	2352.1471
Dual form			2352.3.m.h.1471.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +4.27492 q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +4.27492 q^{5} -3.00000 q^{9} +3.10302i q^{11} -2.72508 q^{13} +7.40437i q^{15} -5.09967 q^{17} +25.7348i q^{19} -12.1819i q^{23} -6.72508 q^{25} -5.19615i q^{27} +41.0241 q^{29} -0.172632i q^{31} -5.37459 q^{33} -16.9244 q^{37} -4.71998i q^{39} -36.7492 q^{41} +53.3325i q^{43} -12.8248 q^{45} +24.2487i q^{47} -8.83289i q^{51} -103.375 q^{53} +13.2651i q^{55} -44.5739 q^{57} +29.2564i q^{59} +32.9003 q^{61} -11.6495 q^{65} -17.0170i q^{67} +21.0997 q^{69} -76.7857i q^{71} -99.2749 q^{73} -11.6482i q^{75} +87.7275i q^{79} +9.00000 q^{81} +151.452i q^{83} -21.8007 q^{85} +71.0558i q^{87} +68.5498 q^{89} +0.299007 q^{93} +110.014i q^{95} +104.522 q^{97} -9.30906i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^5 - 12 * q^9	$ 4 q + 2 q^{5} - 12 q^{9} - 26 q^{13} + 40 q^{17} - 42 q^{25} - 2 q^{29} + 54 q^{33} + 38 q^{37} + 4 q^{41} - 6 q^{45} - 338 q^{53} + 18 q^{57} + 192 q^{61} + 44 q^{65} + 24 q^{69} - 382 q^{73} + 36 q^{81} - 208 q^{85} + 244 q^{89} - 180 q^{93} - 50 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - 12 * q^9 - 26 * q^13 + 40 * q^17 - 42 * q^25 - 2 * q^29 + 54 * q^33 + 38 * q^37 + 4 * q^41 - 6 * q^45 - 338 * q^53 + 18 * q^57 + 192 * q^61 + 44 * q^65 + 24 * q^69 - 382 * q^73 + 36 * q^81 - 208 * q^85 + 244 * q^89 - 180 * q^93 - 50 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	4.27492	0.854983	0.427492	−	0.904019i	$-0.359397\pi$
$5$	4.27492	0.854983	0.427492	+	0.904019i	$0.359397\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	3.10302i	0.282093i	0.990003	+	0.141046i	$0.0450466\pi$
$11$	3.10302i	0.282093i	−0.990003	+	0.141046i	$0.954953\pi$
$12$	0	0
$13$	−2.72508	−0.209622	−0.104811	−	0.994492i	$-0.533424\pi$
$13$	−2.72508	−0.209622	−0.104811	+	0.994492i	$0.533424\pi$
$14$	0	0
$15$	7.40437i	0.493625i
$16$	0	0
$17$	−5.09967	−0.299981	−0.149990	−	0.988687i	$-0.547924\pi$
$17$	−5.09967	−0.299981	−0.149990	+	0.988687i	$0.547924\pi$
$18$	0	0
$19$	25.7348i	1.35446i	0.735771	+	0.677231i	$0.236820\pi$
$19$	25.7348i	1.35446i	−0.735771	+	0.677231i	$0.763180\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 12.1819i	− 0.529648i	−0.964297	−	0.264824i	$-0.914686\pi$
$23$	− 12.1819i	− 0.529648i	0.964297	−	0.264824i	$-0.0853138\pi$
$24$	0	0
$25$	−6.72508	−0.269003
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	41.0241	1.41462	0.707312	−	0.706902i	$-0.249908\pi$
$29$	41.0241	1.41462	0.707312	+	0.706902i	$0.249908\pi$
$30$	0	0
$31$	− 0.172632i	− 0.00556876i	−0.999996	−	0.00278438i	$-0.999114\pi$
$31$	− 0.172632i	− 0.00556876i	0.999996	−	0.00278438i	$-0.000886297\pi$
$32$	0	0
$33$	−5.37459	−0.162866
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−16.9244	−0.457417	−0.228708	−	0.973495i	$-0.573450\pi$
$37$	−16.9244	−0.457417	−0.228708	+	0.973495i	$0.573450\pi$
$38$	0	0
$39$	− 4.71998i	− 0.121025i
$40$	0	0
$41$	−36.7492	−0.896321	−0.448161	−	0.893953i	$-0.647921\pi$
$41$	−36.7492	−0.896321	−0.448161	+	0.893953i	$0.647921\pi$
$42$	0	0
$43$	53.3325i	1.24029i	0.784487	+	0.620145i	$0.212926\pi$
$43$	53.3325i	1.24029i	−0.784487	+	0.620145i	$0.787074\pi$
$44$	0	0
$45$	−12.8248	−0.284994
$46$	0	0
$47$	24.2487i	0.515930i	0.966154	+	0.257965i	$0.0830519\pi$
$47$	24.2487i	0.515930i	−0.966154	+	0.257965i	$0.916948\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 8.83289i	− 0.173194i
$52$	0	0
$53$	−103.375	−1.95046	−0.975232	−	0.221185i	$-0.929008\pi$
$53$	−103.375	−1.95046	−0.975232	+	0.221185i	$0.929008\pi$
$54$	0	0
$55$	13.2651i	0.241185i
$56$	0	0
$57$	−44.5739	−0.781999
$58$	0	0
$59$	29.2564i	0.495871i	0.968776	+	0.247936i	$0.0797522\pi$
$59$	29.2564i	0.495871i	−0.968776	+	0.247936i	$0.920248\pi$
$60$	0	0
$61$	32.9003	0.539350	0.269675	−	0.962951i	$-0.413084\pi$
$61$	32.9003	0.539350	0.269675	+	0.962951i	$0.413084\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.6495	−0.179223
$66$	0	0
$67$	− 17.0170i	− 0.253985i	−0.991904	−	0.126992i	$-0.959468\pi$
$67$	− 17.0170i	− 0.253985i	0.991904	−	0.126992i	$-0.0405324\pi$
$68$	0	0
$69$	21.0997	0.305792
$70$	0	0
$71$	− 76.7857i	− 1.08149i	−0.841187	−	0.540744i	$-0.818143\pi$
$71$	− 76.7857i	− 1.08149i	0.841187	−	0.540744i	$-0.181857\pi$
$72$	0	0
$73$	−99.2749	−1.35993	−0.679965	−	0.733244i	$-0.738005\pi$
$73$	−99.2749	−1.35993	−0.679965	+	0.733244i	$0.738005\pi$
$74$	0	0
$75$	− 11.6482i	− 0.155309i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	87.7275i	1.11047i	0.831692	+	0.555237i	$0.187373\pi$
$79$	87.7275i	1.11047i	−0.831692	+	0.555237i	$0.812627\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	151.452i	1.82473i	0.409381	+	0.912363i	$0.365745\pi$
$83$	151.452i	1.82473i	−0.409381	+	0.912363i	$0.634255\pi$
$84$	0	0
$85$	−21.8007	−0.256478
$86$	0	0
$87$	71.0558i	0.816733i
$88$	0	0
$89$	68.5498	0.770223	0.385111	−	0.922870i	$-0.374163\pi$
$89$	68.5498	0.770223	0.385111	+	0.922870i	$0.374163\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.299007	0.00321512
$94$	0	0
$95$	110.014i	1.15804i
$96$	0	0
$97$	104.522	1.07755	0.538775	−	0.842449i	$-0.318887\pi$
$97$	104.522	1.07755	0.538775	+	0.842449i	$0.318887\pi$
$98$	0	0
$99$	− 9.30906i	− 0.0940309i
$100$	0	0
$101$	157.698	1.56136	0.780682	−	0.624929i	$-0.214872\pi$
$101$	157.698	1.56136	0.780682	+	0.624929i	$0.214872\pi$
$102$	0	0
$103$	− 120.448i	− 1.16940i	−0.811250	−	0.584699i	$-0.801213\pi$
$103$	− 120.448i	− 1.16940i	0.811250	−	0.584699i	$-0.198787\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	104.022i	0.972171i	0.873911	+	0.486086i	$0.161576\pi$
$107$	104.022i	0.972171i	−0.873911	+	0.486086i	$0.838424\pi$
$108$	0	0
$109$	−103.625	−0.950692	−0.475346	−	0.879799i	$-0.657677\pi$
$109$	−103.625	−0.950692	−0.475346	+	0.879799i	$0.657677\pi$
$110$	0	0
$111$	− 29.3140i	− 0.264090i
$112$	0	0
$113$	−177.849	−1.57388	−0.786942	−	0.617027i	$-0.788337\pi$
$113$	−177.849	−1.57388	−0.786942	+	0.617027i	$0.788337\pi$
$114$	0	0
$115$	− 52.0766i	− 0.452840i
$116$	0	0
$117$	8.17525	0.0698739
$118$	0	0
$119$	0	0
$120$	0	0
$121$	111.371	0.920424
$122$	0	0
$123$	− 63.6514i	− 0.517491i
$124$	0	0
$125$	−135.622	−1.08498
$126$	0	0
$127$	171.672i	1.35174i	0.737019	+	0.675872i	$0.236233\pi$
$127$	171.672i	1.35174i	−0.737019	+	0.675872i	$0.763767\pi$
$128$	0	0
$129$	−92.3746	−0.716082
$130$	0	0
$131$	− 125.152i	− 0.955360i	−0.878534	−	0.477680i	$-0.841478\pi$
$131$	− 125.152i	− 0.955360i	0.878534	−	0.477680i	$-0.158522\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 22.2131i	− 0.164542i
$136$	0	0
$137$	−180.949	−1.32079	−0.660396	−	0.750918i	$-0.729612\pi$
$137$	−180.949	−1.32079	−0.660396	+	0.750918i	$0.729612\pi$
$138$	0	0
$139$	155.204i	1.11658i	0.829647	+	0.558288i	$0.188542\pi$
$139$	155.204i	1.11658i	−0.829647	+	0.558288i	$0.811458\pi$
$140$	0	0
$141$	−42.0000	−0.297872
$142$	0	0
$143$	− 8.45598i	− 0.0591327i
$144$	0	0
$145$	175.375	1.20948
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−39.1478	−0.262737	−0.131369	−	0.991334i	$-0.541937\pi$
$149$	−39.1478	−0.262737	−0.131369	+	0.991334i	$0.541937\pi$
$150$	0	0
$151$	135.084i	0.894597i	0.894385	+	0.447298i	$0.147614\pi$
$151$	135.084i	0.894597i	−0.894385	+	0.447298i	$0.852386\pi$
$152$	0	0
$153$	15.2990	0.0999935
$154$	0	0
$155$	− 0.737986i	− 0.00476120i
$156$	0	0
$157$	−56.0482	−0.356995	−0.178497	−	0.983940i	$-0.557124\pi$
$157$	−56.0482	−0.356995	−0.178497	+	0.983940i	$0.557124\pi$
$158$	0	0
$159$	− 179.050i	− 1.12610i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 25.5780i	− 0.156920i	−0.996917	−	0.0784600i	$-0.975000\pi$
$163$	− 25.5780i	− 0.156920i	0.996917	−	0.0784600i	$-0.0250003\pi$
$164$	0	0
$165$	−22.9759	−0.139248
$166$	0	0
$167$	− 77.9998i	− 0.467065i	−0.972349	−	0.233532i	$-0.924971\pi$
$167$	− 77.9998i	− 0.467065i	0.972349	−	0.233532i	$-0.0750285\pi$
$168$	0	0
$169$	−161.574	−0.956059
$170$	0	0
$171$	− 77.2043i	− 0.451487i
$172$	0	0
$173$	287.148	1.65981	0.829907	−	0.557902i	$-0.188393\pi$
$173$	287.148	1.65981	0.829907	+	0.557902i	$0.188393\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−50.6736	−0.286291
$178$	0	0
$179$	65.8496i	0.367875i	0.982938	+	0.183937i	$0.0588844\pi$
$179$	65.8496i	0.367875i	−0.982938	+	0.183937i	$0.941116\pi$
$180$	0	0
$181$	−205.323	−1.13438	−0.567191	−	0.823586i	$-0.691970\pi$
$181$	−205.323	−1.13438	−0.567191	+	0.823586i	$0.691970\pi$
$182$	0	0
$183$	56.9850i	0.311394i
$184$	0	0
$185$	−72.3505	−0.391084
$186$	0	0
$187$	− 15.8244i	− 0.0846223i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 98.9830i	− 0.518235i	−0.965846	−	0.259118i	$-0.916568\pi$
$191$	− 98.9830i	− 0.518235i	0.965846	−	0.259118i	$-0.0834318\pi$
$192$	0	0
$193$	−54.5531	−0.282659	−0.141329	−	0.989963i	$-0.545138\pi$
$193$	−54.5531	−0.282659	−0.141329	+	0.989963i	$0.545138\pi$
$194$	0	0
$195$	− 20.1775i	− 0.103475i
$196$	0	0
$197$	−127.450	−0.646955	−0.323478	−	0.946236i	$-0.604852\pi$
$197$	−127.450	−0.646955	−0.323478	+	0.946236i	$0.604852\pi$
$198$	0	0
$199$	− 12.1819i	− 0.0612156i	−0.999531	−	0.0306078i	$-0.990256\pi$
$199$	− 12.1819i	− 0.0612156i	0.999531	−	0.0306078i	$-0.00974428\pi$
$200$	0	0
$201$	29.4743	0.146638
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−157.100	−0.766340
$206$	0	0
$207$	36.5457i	0.176549i
$208$	0	0
$209$	−79.8555	−0.382084
$210$	0	0
$211$	210.243i	0.996411i	0.867059	+	0.498206i	$0.166008\pi$
$211$	210.243i	0.996411i	−0.867059	+	0.498206i	$0.833992\pi$
$212$	0	0
$213$	132.997	0.624398
$214$	0	0
$215$	227.992i	1.06043i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 171.949i	− 0.785156i
$220$	0	0
$221$	13.8970	0.0628824
$222$	0	0
$223$	− 264.323i	− 1.18531i	−0.805458	−	0.592653i	$-0.798080\pi$
$223$	− 264.323i	− 1.18531i	0.805458	−	0.592653i	$-0.201920\pi$
$224$	0	0
$225$	20.1752	0.0896678
$226$	0	0
$227$	351.878i	1.55012i	0.631885	+	0.775062i	$0.282281\pi$
$227$	351.878i	1.55012i	−0.631885	+	0.775062i	$0.717719\pi$
$228$	0	0
$229$	205.371	0.896818	0.448409	−	0.893829i	$-0.351991\pi$
$229$	205.371	0.896818	0.448409	+	0.893829i	$0.351991\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	33.8488	0.145274	0.0726370	−	0.997358i	$-0.476859\pi$
$233$	33.8488	0.145274	0.0726370	+	0.997358i	$0.476859\pi$
$234$	0	0
$235$	103.661i	0.441112i
$236$	0	0
$237$	−151.949	−0.641133
$238$	0	0
$239$	− 78.4285i	− 0.328153i	−0.986448	−	0.164076i	$-0.947536\pi$
$239$	− 78.4285i	− 0.328153i	0.986448	−	0.164076i	$-0.0524644\pi$
$240$	0	0
$241$	−174.474	−0.723960	−0.361980	−	0.932186i	$-0.617899\pi$
$241$	−174.474	−0.723960	−0.361980	+	0.932186i	$0.617899\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 70.1294i	− 0.283925i
$248$	0	0
$249$	−262.323	−1.05351
$250$	0	0
$251$	− 128.124i	− 0.510455i	−0.966881	−	0.255228i	$-0.917850\pi$
$251$	− 128.124i	− 0.510455i	0.966881	−	0.255228i	$-0.0821504\pi$
$252$	0	0
$253$	37.8007	0.149410
$254$	0	0
$255$	− 37.7599i	− 0.148078i
$256$	0	0
$257$	285.601	1.11129	0.555645	−	0.831420i	$-0.312472\pi$
$257$	285.601	1.11129	0.555645	+	0.831420i	$0.312472\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−123.072	−0.471541
$262$	0	0
$263$	510.227i	1.94003i	0.243051	+	0.970013i	$0.421852\pi$
$263$	510.227i	1.94003i	−0.243051	+	0.970013i	$0.578148\pi$
$264$	0	0
$265$	−441.918	−1.66761
$266$	0	0
$267$	118.732i	0.444688i
$268$	0	0
$269$	−233.471	−0.867922	−0.433961	−	0.900932i	$-0.642884\pi$
$269$	−233.471	−0.867922	−0.433961	+	0.900932i	$0.642884\pi$
$270$	0	0
$271$	− 317.582i	− 1.17189i	−0.810350	−	0.585945i	$-0.800723\pi$
$271$	− 317.582i	− 1.17189i	0.810350	−	0.585945i	$-0.199277\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 20.8681i	− 0.0758838i
$276$	0	0
$277$	−522.265	−1.88543	−0.942717	−	0.333595i	$-0.891738\pi$
$277$	−522.265	−1.88543	−0.942717	+	0.333595i	$0.891738\pi$
$278$	0	0
$279$	0.517895i	0.00185625i
$280$	0	0
$281$	−307.752	−1.09520	−0.547602	−	0.836739i	$-0.684459\pi$
$281$	−307.752	−1.09520	−0.547602	+	0.836739i	$0.684459\pi$
$282$	0	0
$283$	− 207.888i	− 0.734586i	−0.930105	−	0.367293i	$-0.880285\pi$
$283$	− 207.888i	− 0.734586i	0.930105	−	0.367293i	$-0.119715\pi$
$284$	0	0
$285$	−190.550	−0.668596
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−262.993	−0.910012
$290$	0	0
$291$	181.038i	0.622124i
$292$	0	0
$293$	341.670	1.16611	0.583055	−	0.812433i	$-0.301857\pi$
$293$	341.670	1.16611	0.583055	+	0.812433i	$0.301857\pi$
$294$	0	0
$295$	125.069i	0.423962i
$296$	0	0
$297$	16.1238	0.0542887
$298$	0	0
$299$	33.1967i	0.111026i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	273.140i	0.901453i
$304$	0	0
$305$	140.646	0.461135
$306$	0	0
$307$	179.913i	0.586036i	0.956107	+	0.293018i	$0.0946597\pi$
$307$	179.913i	0.586036i	−0.956107	+	0.293018i	$0.905340\pi$
$308$	0	0
$309$	208.622	0.675152
$310$	0	0
$311$	262.665i	0.844581i	0.906461	+	0.422290i	$0.138774\pi$
$311$	262.665i	0.844581i	−0.906461	+	0.422290i	$0.861226\pi$
$312$	0	0
$313$	−547.042	−1.74774	−0.873868	−	0.486163i	$-0.838396\pi$
$313$	−547.042	−1.74774	−0.873868	+	0.486163i	$0.838396\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.2815	−0.0765979	−0.0382990	−	0.999266i	$-0.512194\pi$
$317$	−24.2815	−0.0765979	−0.0382990	+	0.999266i	$0.512194\pi$
$318$	0	0
$319$	127.299i	0.399055i
$320$	0	0
$321$	−180.172	−0.561283
$322$	0	0
$323$	− 131.239i	− 0.406312i
$324$	0	0
$325$	18.3264	0.0563889
$326$	0	0
$327$	− 179.484i	− 0.548882i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 306.819i	− 0.926946i	−0.886111	−	0.463473i	$-0.846603\pi$
$331$	− 306.819i	− 0.926946i	0.886111	−	0.463473i	$-0.153397\pi$
$332$	0	0
$333$	50.7733	0.152472
$334$	0	0
$335$	− 72.7461i	− 0.217153i
$336$	0	0
$337$	312.093	0.926092	0.463046	−	0.886334i	$-0.346756\pi$
$337$	312.093	0.926092	0.463046	+	0.886334i	$0.346756\pi$
$338$	0	0
$339$	− 308.043i	− 0.908682i
$340$	0	0
$341$	0.535679	0.00157091
$342$	0	0
$343$	0	0
$344$	0	0
$345$	90.1993	0.261447
$346$	0	0
$347$	− 380.004i	− 1.09511i	−0.836769	−	0.547556i	$-0.815558\pi$
$347$	− 380.004i	− 1.09511i	0.836769	−	0.547556i	$-0.184442\pi$
$348$	0	0
$349$	201.395	0.577064	0.288532	−	0.957470i	$-0.406833\pi$
$349$	201.395	0.577064	0.288532	+	0.957470i	$0.406833\pi$
$350$	0	0
$351$	14.1599i	0.0403417i
$352$	0	0
$353$	−319.746	−0.905796	−0.452898	−	0.891562i	$-0.649610\pi$
$353$	−319.746	−0.905796	−0.452898	+	0.891562i	$0.649610\pi$
$354$	0	0
$355$	− 328.252i	− 0.924655i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	547.463i	1.52497i	0.647007	+	0.762484i	$0.276020\pi$
$359$	547.463i	1.52497i	−0.647007	+	0.762484i	$0.723980\pi$
$360$	0	0
$361$	−301.278	−0.834566
$362$	0	0
$363$	192.901i	0.531407i
$364$	0	0
$365$	−424.392	−1.16272
$366$	0	0
$367$	186.250i	0.507494i	0.967271	+	0.253747i	$0.0816630\pi$
$367$	186.250i	0.507494i	−0.967271	+	0.253747i	$0.918337\pi$
$368$	0	0
$369$	110.248	0.298774
$370$	0	0
$371$	0	0
$372$	0	0
$373$	31.8729	0.0854502	0.0427251	−	0.999087i	$-0.486396\pi$
$373$	31.8729	0.0854502	0.0427251	+	0.999087i	$0.486396\pi$
$374$	0	0
$375$	− 234.904i	− 0.626412i
$376$	0	0
$377$	−111.794	−0.296536
$378$	0	0
$379$	− 644.960i	− 1.70174i	−0.525375	−	0.850871i	$-0.676075\pi$
$379$	− 644.960i	− 1.70174i	0.525375	−	0.850871i	$-0.323925\pi$
$380$	0	0
$381$	−297.344	−0.780430
$382$	0	0
$383$	539.678i	1.40908i	0.709664	+	0.704540i	$0.248847\pi$
$383$	539.678i	1.40908i	−0.709664	+	0.704540i	$0.751153\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 159.997i	− 0.413430i
$388$	0	0
$389$	255.704	0.657338	0.328669	−	0.944445i	$-0.393400\pi$
$389$	255.704	0.657338	0.328669	+	0.944445i	$0.393400\pi$
$390$	0	0
$391$	62.1237i	0.158884i
$392$	0	0
$393$	216.770	0.551577
$394$	0	0
$395$	375.028i	0.949438i
$396$	0	0
$397$	193.419	0.487203	0.243601	−	0.969875i	$-0.421671\pi$
$397$	193.419	0.487203	0.243601	+	0.969875i	$0.421671\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.0000	0.0798005	0.0399002	−	0.999204i	$-0.487296\pi$
$401$	32.0000	0.0798005	0.0399002	+	0.999204i	$0.487296\pi$
$402$	0	0
$403$	0.470435i	0.00116733i
$404$	0	0
$405$	38.4743	0.0949982
$406$	0	0
$407$	− 52.5168i	− 0.129034i
$408$	0	0
$409$	370.739	0.906453	0.453226	−	0.891395i	$-0.350273\pi$
$409$	370.739	0.906453	0.453226	+	0.891395i	$0.350273\pi$
$410$	0	0
$411$	− 313.412i	− 0.762560i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	647.446i	1.56011i
$416$	0	0
$417$	−268.821	−0.644656
$418$	0	0
$419$	735.457i	1.75527i	0.479332	+	0.877634i	$0.340879\pi$
$419$	735.457i	1.75527i	−0.479332	+	0.877634i	$0.659121\pi$
$420$	0	0
$421$	−483.021	−1.14732	−0.573659	−	0.819094i	$-0.694476\pi$
$421$	−483.021	−1.14732	−0.573659	+	0.819094i	$0.694476\pi$
$422$	0	0
$423$	− 72.7461i	− 0.171977i
$424$	0	0
$425$	34.2957	0.0806958
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.6462	0.0341403
$430$	0	0
$431$	569.117i	1.32046i	0.751065	+	0.660228i	$0.229540\pi$
$431$	569.117i	1.32046i	−0.751065	+	0.660228i	$0.770460\pi$
$432$	0	0
$433$	−45.3297	−0.104688	−0.0523438	−	0.998629i	$-0.516669\pi$
$433$	−45.3297	−0.104688	−0.0523438	+	0.998629i	$0.516669\pi$
$434$	0	0
$435$	303.758i	0.698294i
$436$	0	0
$437$	313.498	0.717388
$438$	0	0
$439$	− 59.1041i	− 0.134633i	−0.997732	−	0.0673167i	$-0.978556\pi$
$439$	− 59.1041i	− 0.134633i	0.997732	−	0.0673167i	$-0.0214438\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	218.191i	0.492530i	0.969202	+	0.246265i	$0.0792034\pi$
$443$	218.191i	0.492530i	−0.969202	+	0.246265i	$0.920797\pi$
$444$	0	0
$445$	293.045	0.658528
$446$	0	0
$447$	− 67.8061i	− 0.151691i
$448$	0	0
$449$	−369.643	−0.823258	−0.411629	−	0.911351i	$-0.635040\pi$
$449$	−369.643	−0.823258	−0.411629	+	0.911351i	$0.635040\pi$
$450$	0	0
$451$	− 114.033i	− 0.252846i
$452$	0	0
$453$	−233.973	−0.516496
$454$	0	0
$455$	0	0
$456$	0	0
$457$	514.650	1.12615	0.563074	−	0.826407i	$-0.309619\pi$
$457$	514.650	1.12615	0.563074	+	0.826407i	$0.309619\pi$
$458$	0	0
$459$	26.4987i	0.0577313i
$460$	0	0
$461$	656.440	1.42395	0.711974	−	0.702206i	$-0.247801\pi$
$461$	656.440	1.42395	0.711974	+	0.702206i	$0.247801\pi$
$462$	0	0
$463$	− 90.7673i	− 0.196042i	−0.995184	−	0.0980208i	$-0.968749\pi$
$463$	− 90.7673i	− 0.196042i	0.995184	−	0.0980208i	$-0.0312512\pi$
$464$	0	0
$465$	1.27823	0.00274888
$466$	0	0
$467$	341.010i	0.730214i	0.930966	+	0.365107i	$0.118968\pi$
$467$	341.010i	0.730214i	−0.930966	+	0.365107i	$0.881032\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 97.0783i	− 0.206111i
$472$	0	0
$473$	−165.492	−0.349877
$474$	0	0
$475$	− 173.068i	− 0.364355i
$476$	0	0
$477$	310.124	0.650155
$478$	0	0
$479$	− 252.376i	− 0.526881i	−0.964676	−	0.263440i	$-0.915143\pi$
$479$	− 252.376i	− 0.526881i	0.964676	−	0.263440i	$-0.0848573\pi$
$480$	0	0
$481$	46.1204	0.0958845
$482$	0	0
$483$	0	0
$484$	0	0
$485$	446.825	0.921288
$486$	0	0
$487$	− 586.411i	− 1.20413i	−0.798447	−	0.602065i	$-0.794345\pi$
$487$	− 586.411i	− 1.20413i	0.798447	−	0.602065i	$-0.205655\pi$
$488$	0	0
$489$	44.3023	0.0905978
$490$	0	0
$491$	− 462.384i	− 0.941719i	−0.882208	−	0.470860i	$-0.843944\pi$
$491$	− 462.384i	− 0.941719i	0.882208	−	0.470860i	$-0.156056\pi$
$492$	0	0
$493$	−209.209	−0.424360
$494$	0	0
$495$	− 39.7954i	− 0.0803948i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 882.142i	− 1.76782i	−0.467658	−	0.883910i	$-0.654902\pi$
$499$	− 882.142i	− 1.76782i	0.467658	−	0.883910i	$-0.345098\pi$
$500$	0	0
$501$	135.100	0.269660
$502$	0	0
$503$	19.1303i	0.0380323i	0.999819	+	0.0190162i	$0.00605340\pi$
$503$	19.1303i	0.0380323i	−0.999819	+	0.0190162i	$0.993947\pi$
$504$	0	0
$505$	674.145	1.33494
$506$	0	0
$507$	− 279.854i	− 0.551981i
$508$	0	0
$509$	427.725	0.840324	0.420162	−	0.907449i	$-0.361973\pi$
$509$	427.725	0.840324	0.420162	+	0.907449i	$0.361973\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	133.722	0.260666
$514$	0	0
$515$	− 514.905i	− 0.999816i
$516$	0	0
$517$	−75.2442	−0.145540
$518$	0	0
$519$	497.355i	0.958294i
$520$	0	0
$521$	787.244	1.51103	0.755513	−	0.655134i	$-0.227388\pi$
$521$	787.244	1.51103	0.755513	+	0.655134i	$0.227388\pi$
$522$	0	0
$523$	− 603.935i	− 1.15475i	−0.816479	−	0.577376i	$-0.804077\pi$
$523$	− 603.935i	− 1.15475i	0.816479	−	0.577376i	$-0.195923\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.880364i	0.00167052i
$528$	0	0
$529$	380.601	0.719473
$530$	0	0
$531$	− 87.7692i	− 0.165290i
$532$	0	0
$533$	100.145	0.187888
$534$	0	0
$535$	444.687i	0.831190i
$536$	0	0
$537$	−114.055	−0.212393
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−851.914	−1.57470	−0.787352	−	0.616504i	$-0.788548\pi$
$541$	−851.914	−1.57470	−0.787352	+	0.616504i	$0.788548\pi$
$542$	0	0
$543$	− 355.630i	− 0.654936i
$544$	0	0
$545$	−442.990	−0.812826
$546$	0	0
$547$	887.898i	1.62321i	0.584204	+	0.811607i	$0.301407\pi$
$547$	887.898i	1.62321i	−0.584204	+	0.811607i	$0.698593\pi$
$548$	0	0
$549$	−98.7010	−0.179783
$550$	0	0
$551$	1055.75i	1.91605i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 125.315i	− 0.225792i
$556$	0	0
$557$	592.908	1.06447	0.532233	−	0.846598i	$-0.321353\pi$
$557$	592.908	1.06447	0.532233	+	0.846598i	$0.321353\pi$
$558$	0	0
$559$	− 145.335i	− 0.259992i
$560$	0	0
$561$	27.4086	0.0488567
$562$	0	0
$563$	− 153.849i	− 0.273266i	−0.990622	−	0.136633i	$-0.956372\pi$
$563$	− 153.849i	− 0.273266i	0.990622	−	0.136633i	$-0.0436282\pi$
$564$	0	0
$565$	−760.289	−1.34564
$566$	0	0
$567$	0	0
$568$	0	0
$569$	289.450	0.508700	0.254350	−	0.967112i	$-0.418139\pi$
$569$	289.450	0.508700	0.254350	+	0.967112i	$0.418139\pi$
$570$	0	0
$571$	− 640.702i	− 1.12207i	−0.827792	−	0.561035i	$-0.810403\pi$
$571$	− 640.702i	− 1.12207i	0.827792	−	0.561035i	$-0.189597\pi$
$572$	0	0
$573$	171.444	0.299203
$574$	0	0
$575$	81.9243i	0.142477i
$576$	0	0
$577$	190.209	0.329652	0.164826	−	0.986323i	$-0.447294\pi$
$577$	190.209	0.329652	0.164826	+	0.986323i	$0.447294\pi$
$578$	0	0
$579$	− 94.4888i	− 0.163193i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 320.773i	− 0.550211i
$584$	0	0
$585$	34.9485	0.0597410
$586$	0	0
$587$	− 338.241i	− 0.576219i	−0.957597	−	0.288109i	$-0.906973\pi$
$587$	− 338.241i	− 0.576219i	0.957597	−	0.288109i	$-0.0930267\pi$
$588$	0	0
$589$	4.44263	0.00754267
$590$	0	0
$591$	− 220.750i	− 0.373520i
$592$	0	0
$593$	736.736	1.24239	0.621194	−	0.783657i	$-0.286648\pi$
$593$	736.736	1.24239	0.621194	+	0.783657i	$0.286648\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.0997	0.0353428
$598$	0	0
$599$	371.266i	0.619810i	0.950768	+	0.309905i	$0.100297\pi$
$599$	371.266i	0.619810i	−0.950768	+	0.309905i	$0.899703\pi$
$600$	0	0
$601$	965.894	1.60714	0.803572	−	0.595207i	$-0.202930\pi$
$601$	965.894	1.60714	0.803572	+	0.595207i	$0.202930\pi$
$602$	0	0
$603$	51.0509i	0.0846615i
$604$	0	0
$605$	476.103	0.786947
$606$	0	0
$607$	304.887i	0.502285i	0.967950	+	0.251143i	$0.0808063\pi$
$607$	304.887i	0.502285i	−0.967950	+	0.251143i	$0.919194\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 66.0797i	− 0.108150i
$612$	0	0
$613$	425.341	0.693867	0.346934	−	0.937890i	$-0.387223\pi$
$613$	425.341	0.693867	0.346934	+	0.937890i	$0.387223\pi$
$614$	0	0
$615$	− 272.105i	− 0.442447i
$616$	0	0
$617$	821.189	1.33094	0.665470	−	0.746425i	$-0.268231\pi$
$617$	821.189	1.33094	0.665470	+	0.746425i	$0.268231\pi$
$618$	0	0
$619$	− 897.086i	− 1.44925i	−0.689143	−	0.724625i	$-0.742013\pi$
$619$	− 897.086i	− 1.44925i	0.689143	−	0.724625i	$-0.257987\pi$
$620$	0	0
$621$	−63.2990	−0.101931
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−411.646	−0.658634
$626$	0	0
$627$	− 138.314i	− 0.220596i
$628$	0	0
$629$	86.3089	0.137216
$630$	0	0
$631$	347.398i	0.550552i	0.961365	+	0.275276i	$0.0887693\pi$
$631$	347.398i	0.550552i	−0.961365	+	0.275276i	$0.911231\pi$
$632$	0	0
$633$	−364.151	−0.575278
$634$	0	0
$635$	733.882i	1.15572i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	230.357i	0.360496i
$640$	0	0
$641$	405.855	0.633160	0.316580	−	0.948566i	$-0.397465\pi$
$641$	405.855	0.633160	0.316580	+	0.948566i	$0.397465\pi$
$642$	0	0
$643$	− 52.5787i	− 0.0817709i	−0.999164	−	0.0408854i	$-0.986982\pi$
$643$	− 52.5787i	− 0.0817709i	0.999164	−	0.0408854i	$-0.0130179\pi$
$644$	0	0
$645$	−394.894	−0.612238
$646$	0	0
$647$	1121.44i	1.73329i	0.498928	+	0.866644i	$0.333727\pi$
$647$	1121.44i	1.73329i	−0.498928	+	0.866644i	$0.666273\pi$
$648$	0	0
$649$	−90.7832	−0.139882
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−41.9660	−0.0642664	−0.0321332	−	0.999484i	$-0.510230\pi$
$653$	−41.9660	−0.0642664	−0.0321332	+	0.999484i	$0.510230\pi$
$654$	0	0
$655$	− 535.015i	− 0.816817i
$656$	0	0
$657$	297.825	0.453310
$658$	0	0
$659$	− 1224.43i	− 1.85801i	−0.370069	−	0.929004i	$-0.620666\pi$
$659$	− 1224.43i	− 1.85801i	0.370069	−	0.929004i	$-0.379334\pi$
$660$	0	0
$661$	−27.1171	−0.0410244	−0.0205122	−	0.999790i	$-0.506530\pi$
$661$	−27.1171	−0.0410244	−0.0205122	+	0.999790i	$0.506530\pi$
$662$	0	0
$663$	24.0703i	0.0363052i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 499.751i	− 0.749252i
$668$	0	0
$669$	457.821	0.684337
$670$	0	0
$671$	102.090i	0.152147i
$672$	0	0
$673$	−418.238	−0.621453	−0.310726	−	0.950499i	$-0.600572\pi$
$673$	−418.238	−0.621453	−0.310726	+	0.950499i	$0.600572\pi$
$674$	0	0
$675$	34.9446i	0.0517697i
$676$	0	0
$677$	233.410	0.344770	0.172385	−	0.985030i	$-0.444853\pi$
$677$	233.410	0.344770	0.172385	+	0.985030i	$0.444853\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−609.471	−0.894965
$682$	0	0
$683$	414.221i	0.606472i	0.952915	+	0.303236i	$0.0980671\pi$
$683$	414.221i	0.606472i	−0.952915	+	0.303236i	$0.901933\pi$
$684$	0	0
$685$	−773.540	−1.12926
$686$	0	0
$687$	355.713i	0.517778i
$688$	0	0
$689$	281.704	0.408860
$690$	0	0
$691$	471.232i	0.681956i	0.940071	+	0.340978i	$0.110758\pi$
$691$	471.232i	0.681956i	−0.940071	+	0.340978i	$0.889242\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	663.485i	0.954654i
$696$	0	0
$697$	187.409	0.268879
$698$	0	0
$699$	58.6279i	0.0838740i
$700$	0	0
$701$	1150.00	1.64051	0.820257	−	0.571995i	$-0.193830\pi$
$701$	1150.00	1.64051	0.820257	+	0.571995i	$0.193830\pi$
$702$	0	0
$703$	− 435.546i	− 0.619553i
$704$	0	0
$705$	−179.547	−0.254676
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1388.97	1.95906	0.979528	−	0.201310i	$-0.0645198\pi$
$709$	1388.97	1.95906	0.979528	+	0.201310i	$0.0645198\pi$
$710$	0	0
$711$	− 263.183i	− 0.370158i
$712$	0	0
$713$	−2.10298	−0.00294948
$714$	0	0
$715$	− 36.1486i	− 0.0505575i
$716$	0	0
$717$	135.842	0.189459
$718$	0	0
$719$	1179.12i	1.63995i	0.572400	+	0.819975i	$0.306013\pi$
$719$	1179.12i	1.63995i	−0.572400	+	0.819975i	$0.693987\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 302.198i	− 0.417978i
$724$	0	0
$725$	−275.890	−0.380538
$726$	0	0
$727$	− 1168.99i	− 1.60796i	−0.594656	−	0.803980i	$-0.702712\pi$
$727$	− 1168.99i	− 1.60796i	0.594656	−	0.803980i	$-0.297288\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 271.978i	− 0.372063i
$732$	0	0
$733$	−768.375	−1.04826	−0.524130	−	0.851638i	$-0.675609\pi$
$733$	−768.375	−1.04826	−0.524130	+	0.851638i	$0.675609\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.8040	0.0716472
$738$	0	0
$739$	− 378.288i	− 0.511891i	−0.966691	−	0.255946i	$-0.917613\pi$
$739$	− 378.288i	− 0.511891i	0.966691	−	0.255946i	$-0.0823868\pi$
$740$	0	0
$741$	121.468	0.163924
$742$	0	0
$743$	− 1461.21i	− 1.96664i	−0.181889	−	0.983319i	$-0.558221\pi$
$743$	− 1461.21i	− 1.96664i	0.181889	−	0.983319i	$-0.441779\pi$
$744$	0	0
$745$	−167.354	−0.224636
$746$	0	0
$747$	− 454.357i	− 0.608242i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1109.38i	1.47721i	0.674139	+	0.738605i	$0.264515\pi$
$751$	1109.38i	1.47721i	−0.674139	+	0.738605i	$0.735485\pi$
$752$	0	0
$753$	221.918	0.294712
$754$	0	0
$755$	577.474i	0.764866i
$756$	0	0
$757$	925.286	1.22231	0.611153	−	0.791512i	$-0.290706\pi$
$757$	925.286	1.22231	0.611153	+	0.791512i	$0.290706\pi$
$758$	0	0
$759$	65.4727i	0.0862617i
$760$	0	0
$761$	−1373.79	−1.80524	−0.902620	−	0.430439i	$-0.858359\pi$
$761$	−1373.79	−1.80524	−0.902620	+	0.430439i	$0.858359\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	65.4020	0.0854928
$766$	0	0
$767$	− 79.7261i	− 0.103945i
$768$	0	0
$769$	1212.84	1.57716	0.788580	−	0.614933i	$-0.210817\pi$
$769$	1212.84	1.57716	0.788580	+	0.614933i	$0.210817\pi$
$770$	0	0
$771$	494.676i	0.641603i
$772$	0	0
$773$	796.392	1.03026	0.515131	−	0.857112i	$-0.327743\pi$
$773$	796.392	1.03026	0.515131	+	0.857112i	$0.327743\pi$
$774$	0	0
$775$	1.16096i	0.00149801i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 945.731i	− 1.21403i
$780$	0	0
$781$	238.267	0.305080
$782$	0	0
$783$	− 213.167i	− 0.272244i
$784$	0	0
$785$	−239.601	−0.305225
$786$	0	0
$787$	169.301i	0.215122i	0.994199	+	0.107561i	$0.0343040\pi$
$787$	169.301i	0.215122i	−0.994199	+	0.107561i	$0.965696\pi$
$788$	0	0
$789$	−883.739	−1.12008
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−89.6561	−0.113059
$794$	0	0
$795$	− 765.424i	− 0.962798i
$796$	0	0
$797$	181.725	0.228011	0.114006	−	0.993480i	$-0.463632\pi$
$797$	181.725	0.228011	0.114006	+	0.993480i	$0.463632\pi$
$798$	0	0
$799$	− 123.660i	− 0.154769i
$800$	0	0
$801$	−205.650	−0.256741
$802$	0	0
$803$	− 308.052i	− 0.383626i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 404.384i	− 0.501095i
$808$	0	0
$809$	1284.05	1.58720	0.793602	−	0.608437i	$-0.208203\pi$
$809$	1284.05	1.58720	0.793602	+	0.608437i	$0.208203\pi$
$810$	0	0
$811$	627.609i	0.773871i	0.922107	+	0.386936i	$0.126466\pi$
$811$	627.609i	0.773871i	−0.922107	+	0.386936i	$0.873534\pi$
$812$	0	0
$813$	550.069	0.676592
$814$	0	0
$815$	− 109.344i	− 0.134164i
$816$	0	0
$817$	−1372.50	−1.67993
$818$	0	0
$819$	0	0
$820$	0	0
$821$	644.577	0.785112	0.392556	−	0.919728i	$-0.371591\pi$
$821$	644.577	0.785112	0.392556	+	0.919728i	$0.371591\pi$
$822$	0	0
$823$	698.701i	0.848969i	0.905435	+	0.424484i	$0.139545\pi$
$823$	698.701i	0.848969i	−0.905435	+	0.424484i	$0.860455\pi$
$824$	0	0
$825$	36.1445	0.0438116
$826$	0	0
$827$	− 736.603i	− 0.890693i	−0.895358	−	0.445347i	$-0.853080\pi$
$827$	− 736.603i	− 0.890693i	0.895358	−	0.445347i	$-0.146920\pi$
$828$	0	0
$829$	−854.107	−1.03029	−0.515143	−	0.857104i	$-0.672261\pi$
$829$	−854.107	−1.03029	−0.515143	+	0.857104i	$0.672261\pi$
$830$	0	0
$831$	− 904.589i	− 1.08856i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 333.443i	− 0.399333i
$836$	0	0
$837$	−0.897020	−0.00107171
$838$	0	0
$839$	261.597i	0.311796i	0.987773	+	0.155898i	$0.0498272\pi$
$839$	261.597i	0.311796i	−0.987773	+	0.155898i	$0.950173\pi$
$840$	0	0
$841$	841.976	1.00116
$842$	0	0
$843$	− 533.043i	− 0.632317i
$844$	0	0
$845$	−690.715	−0.817414
$846$	0	0
$847$	0	0
$848$	0	0
$849$	360.072	0.424113
$850$	0	0
$851$	206.172i	0.242270i
$852$	0	0
$853$	300.278	0.352026	0.176013	−	0.984388i	$-0.443680\pi$
$853$	300.278	0.352026	0.176013	+	0.984388i	$0.443680\pi$
$854$	0	0
$855$	− 330.042i	− 0.386014i
$856$	0	0
$857$	256.344	0.299118	0.149559	−	0.988753i	$-0.452215\pi$
$857$	256.344	0.299118	0.149559	+	0.988753i	$0.452215\pi$
$858$	0	0
$859$	− 887.342i	− 1.03299i	−0.856289	−	0.516497i	$-0.827236\pi$
$859$	− 887.342i	− 1.03299i	0.856289	−	0.516497i	$-0.172764\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1543.01i	1.78796i	0.448106	+	0.893980i	$0.352099\pi$
$863$	1543.01i	1.78796i	−0.448106	+	0.893980i	$0.647901\pi$
$864$	0	0
$865$	1227.53	1.41911
$866$	0	0
$867$	− 455.518i	− 0.525395i
$868$	0	0
$869$	−272.220	−0.313257
$870$	0	0
$871$	46.3726i	0.0532407i
$872$	0	0
$873$	−313.567	−0.359184
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−574.028	−0.654536	−0.327268	−	0.944932i	$-0.606128\pi$
$877$	−574.028	−0.654536	−0.327268	+	0.944932i	$0.606128\pi$
$878$	0	0
$879$	591.790i	0.673254i
$880$	0	0
$881$	−201.842	−0.229106	−0.114553	−	0.993417i	$-0.536544\pi$
$881$	−201.842	−0.229106	−0.114553	+	0.993417i	$0.536544\pi$
$882$	0	0
$883$	− 661.078i	− 0.748673i	−0.927293	−	0.374336i	$-0.877871\pi$
$883$	− 661.078i	− 0.748673i	0.927293	−	0.374336i	$-0.122129\pi$
$884$	0	0
$885$	−216.625	−0.244774
$886$	0	0
$887$	− 430.783i	− 0.485663i	−0.970069	−	0.242831i	$-0.921924\pi$
$887$	− 430.783i	− 0.485663i	0.970069	−	0.242831i	$-0.0780762\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	27.9272i	0.0313436i
$892$	0	0
$893$	−624.035	−0.698807
$894$	0	0
$895$	281.501i	0.314527i
$896$	0	0
$897$	−57.4983	−0.0641007
$898$	0	0
$899$	− 7.08205i	− 0.00787770i
$900$	0	0
$901$	527.176	0.585101
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−877.739	−0.969878
$906$	0	0
$907$	− 1012.97i	− 1.11684i	−0.829559	−	0.558419i	$-0.811408\pi$
$907$	− 1012.97i	− 1.11684i	0.829559	−	0.558419i	$-0.188592\pi$
$908$	0	0
$909$	−473.093	−0.520454
$910$	0	0
$911$	− 507.799i	− 0.557408i	−0.960377	−	0.278704i	$-0.910095\pi$
$911$	− 507.799i	− 0.557408i	0.960377	−	0.278704i	$-0.0899049\pi$
$912$	0	0
$913$	−469.959	−0.514742
$914$	0	0
$915$	243.606i	0.266236i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	179.536i	0.195360i	0.995218	+	0.0976802i	$0.0311422\pi$
$919$	179.536i	0.195360i	−0.995218	+	0.0976802i	$0.968858\pi$
$920$	0	0
$921$	−311.619	−0.338348
$922$	0	0
$923$	209.247i	0.226703i
$924$	0	0
$925$	113.818	0.123047
$926$	0	0
$927$	361.344i	0.389799i
$928$	0	0
$929$	−298.578	−0.321397	−0.160699	−	0.987004i	$-0.551375\pi$
$929$	−298.578	−0.321397	−0.160699	+	0.987004i	$0.551375\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−454.949	−0.487619
$934$	0	0
$935$	− 67.6479i	− 0.0723507i
$936$	0	0
$937$	−568.691	−0.606927	−0.303464	−	0.952843i	$-0.598143\pi$
$937$	−568.691	−0.606927	−0.303464	+	0.952843i	$0.598143\pi$
$938$	0	0
$939$	− 947.504i	− 1.00906i
$940$	0	0
$941$	570.715	0.606499	0.303249	−	0.952911i	$-0.401929\pi$
$941$	570.715	0.606499	0.303249	+	0.952911i	$0.401929\pi$
$942$	0	0
$943$	447.675i	0.474735i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 435.346i	− 0.459711i	−0.973225	−	0.229855i	$-0.926175\pi$
$947$	− 435.346i	− 0.459711i	0.973225	−	0.229855i	$-0.0738253\pi$
$948$	0	0
$949$	270.532	0.285071
$950$	0	0
$951$	− 42.0569i	− 0.0442238i
$952$	0	0
$953$	−1282.02	−1.34525	−0.672624	−	0.739984i	$-0.734833\pi$
$953$	−1282.02	−1.34525	−0.672624	+	0.739984i	$0.734833\pi$
$954$	0	0
$955$	− 423.144i	− 0.443083i
$956$	0	0
$957$	−220.488	−0.230394
$958$	0	0
$959$	0	0
$960$	0	0
$961$	960.970	0.999969
$962$	0	0
$963$	− 312.067i	− 0.324057i
$964$	0	0
$965$	−233.210	−0.241669
$966$	0	0
$967$	625.993i	0.647355i	0.946167	+	0.323678i	$0.104919\pi$
$967$	625.993i	0.647355i	−0.946167	+	0.323678i	$0.895081\pi$
$968$	0	0
$969$	227.312	0.234584
$970$	0	0
$971$	1260.98i	1.29864i	0.760516	+	0.649319i	$0.224946\pi$
$971$	1260.98i	1.29864i	−0.760516	+	0.649319i	$0.775054\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	31.7423i	0.0325562i
$976$	0	0
$977$	1575.49	1.61257	0.806287	−	0.591524i	$-0.201474\pi$
$977$	1575.49	1.61257	0.806287	+	0.591524i	$0.201474\pi$
$978$	0	0
$979$	212.711i	0.217274i
$980$	0	0
$981$	310.876	0.316897
$982$	0	0
$983$	1047.12i	1.06523i	0.846358	+	0.532614i	$0.178790\pi$
$983$	1047.12i	1.06523i	−0.846358	+	0.532614i	$0.821210\pi$
$984$	0	0
$985$	−544.839	−0.553136
$986$	0	0
$987$	0	0
$988$	0	0
$989$	649.691	0.656917
$990$	0	0
$991$	781.385i	0.788481i	0.919007	+	0.394241i	$0.128992\pi$
$991$	781.385i	0.788481i	−0.919007	+	0.394241i	$0.871008\pi$
$992$	0	0
$993$	531.426	0.535172
$994$	0	0
$995$	− 52.0766i	− 0.0523383i
$996$	0	0
$997$	700.155	0.702262	0.351131	−	0.936326i	$-0.385797\pi$
$997$	700.155	0.702262	0.351131	+	0.936326i	$0.385797\pi$
$998$	0	0
$999$	87.9419i	0.0880299i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.m.h.1471.4		4
4.3	odd	2	inner	2352.3.m.h.1471.2		4
7.3	odd	6		336.3.be.a.79.2	✓	4
7.5	odd	6		336.3.be.b.319.2	yes	4
7.6	odd	2		2352.3.m.g.1471.1		4
21.5	even	6		1008.3.cd.f.991.1		4
21.17	even	6		1008.3.cd.g.415.1		4
28.3	even	6		336.3.be.b.79.2	yes	4
28.19	even	6		336.3.be.a.319.2	yes	4
28.27	even	2		2352.3.m.g.1471.3		4
84.47	odd	6		1008.3.cd.g.991.1		4
84.59	odd	6		1008.3.cd.f.415.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.be.a.79.2	✓	4	7.3	odd	6
336.3.be.a.319.2	yes	4	28.19	even	6
336.3.be.b.79.2	yes	4	28.3	even	6
336.3.be.b.319.2	yes	4	7.5	odd	6
1008.3.cd.f.415.1		4	84.59	odd	6
1008.3.cd.f.991.1		4	21.5	even	6
1008.3.cd.g.415.1		4	21.17	even	6
1008.3.cd.g.991.1		4	84.47	odd	6
2352.3.m.g.1471.1		4	7.6	odd	2
2352.3.m.g.1471.3		4	28.27	even	2
2352.3.m.h.1471.2		4	4.3	odd	2	inner
2352.3.m.h.1471.4		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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +4.27492 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +4.27492 q^{5} -3.00000 q^{9} +3.10302i q^{11} -2.72508 q^{13} +7.40437i q^{15} -5.09967 q^{17} +25.7348i q^{19} -12.1819i q^{23} -6.72508 q^{25} -5.19615i q^{27} +41.0241 q^{29} -0.172632i q^{31} -5.37459 q^{33} -16.9244 q^{37} -4.71998i q^{39} -36.7492 q^{41} +53.3325i q^{43} -12.8248 q^{45} +24.2487i q^{47} -8.83289i q^{51} -103.375 q^{53} +13.2651i q^{55} -44.5739 q^{57} +29.2564i q^{59} +32.9003 q^{61} -11.6495 q^{65} -17.0170i q^{67} +21.0997 q^{69} -76.7857i q^{71} -99.2749 q^{73} -11.6482i q^{75} +87.7275i q^{79} +9.00000 q^{81} +151.452i q^{83} -21.8007 q^{85} +71.0558i q^{87} +68.5498 q^{89} +0.299007 q^{93} +110.014i q^{95} +104.522 q^{97} -9.30906i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^5 - 12 * q^9	\( 4 q + 2 q^{5} - 12 q^{9} - 26 q^{13} + 40 q^{17} - 42 q^{25} - 2 q^{29} + 54 q^{33} + 38 q^{37} + 4 q^{41} - 6 q^{45} - 338 q^{53} + 18 q^{57} + 192 q^{61} + 44 q^{65} + 24 q^{69} - 382 q^{73} + 36 q^{81} - 208 q^{85} + 244 q^{89} - 180 q^{93} - 50 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - 12 * q^9 - 26 * q^13 + 40 * q^17 - 42 * q^25 - 2 * q^29 + 54 * q^33 + 38 * q^37 + 4 * q^41 - 6 * q^45 - 338 * q^53 + 18 * q^57 + 192 * q^61 + 44 * q^65 + 24 * q^69 - 382 * q^73 + 36 * q^81 - 208 * q^85 + 244 * q^89 - 180 * q^93 - 50 * q^97

Introduction

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