LMFDB - Embedded newform 2304.3.e.n.1025.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1025,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("2304.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.e (of order $2$, degree $1$, not 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:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.540942598144.16
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 71x^{4} + 56x^{2} + 9 $ x^8 + 16x^6 + 71x^4 + 56*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.6
Root			$2.69642i$ of defining polynomial
Character	$\chi$	$=$	2304.1025
Dual form			2304.3.e.n.1025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07498i q^{5} -7.21110 q^{7} +O(q^{10})$	$q+1.07498i q^{5} -7.21110 q^{7} +16.3517i q^{11} +21.6045 q^{13} -18.9819i q^{17} -17.0438 q^{19} +1.11567i q^{23} +23.8444 q^{25} +29.4784i q^{29} +5.63331 q^{31} -7.75182i q^{35} -17.0438 q^{37} -27.4671i q^{41} +52.3306 q^{43} +64.5352i q^{47} +3.00000 q^{49} -35.9283i q^{53} -17.5778 q^{55} -56.8069i q^{59} -69.3743 q^{61} +23.2245i q^{65} +69.3743 q^{67} +98.4764i q^{71} -37.6888 q^{73} -117.914i q^{77} -127.322 q^{79} +7.75182i q^{83} +20.4052 q^{85} +76.1475i q^{89} -155.792 q^{91} -18.3218i q^{95} +4.84441 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 40 q^{25} - 128 q^{31} + 24 q^{49} - 256 q^{55} + 160 q^{73} - 384 q^{79} - 192 q^{97}+O(q^{100}) $ 8 * q - 40 * q^25 - 128 * q^31 + 24 * q^49 - 256 * q^55 + 160 * q^73 - 384 * q^79 - 192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.07498i	0.214997i	0.994205	+	0.107498i	$0.0342840\pi$
$5$	1.07498i	0.214997i	−0.994205	+	0.107498i	$0.965716\pi$
$6$	0	0
$7$	−7.21110	−1.03016	−0.515079	−	0.857143i	$-0.672237\pi$
$7$	−7.21110	−1.03016	−0.515079	+	0.857143i	$0.672237\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.3517i	1.48652i	0.669004	+	0.743258i	$0.266721\pi$
$11$	16.3517i	1.48652i	−0.669004	+	0.743258i	$0.733279\pi$
$12$	0	0
$13$	21.6045	1.66189	0.830943	−	0.556357i	$-0.187801\pi$
$13$	21.6045	1.66189	0.830943	+	0.556357i	$0.187801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18.9819i	− 1.11658i	−0.829646	−	0.558290i	$-0.811458\pi$
$17$	− 18.9819i	− 1.11658i	0.829646	−	0.558290i	$-0.188542\pi$
$18$	0	0
$19$	−17.0438	−0.897040	−0.448520	−	0.893773i	$-0.648049\pi$
$19$	−17.0438	−0.897040	−0.448520	+	0.893773i	$0.648049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.11567i	0.0485074i	0.999706	+	0.0242537i	$0.00772094\pi$
$23$	1.11567i	0.0485074i	−0.999706	+	0.0242537i	$0.992279\pi$
$24$	0	0
$25$	23.8444	0.953776
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.4784i	1.01650i	0.861210	+	0.508249i	$0.169707\pi$
$29$	29.4784i	1.01650i	−0.861210	+	0.508249i	$0.830293\pi$
$30$	0	0
$31$	5.63331	0.181720	0.0908598	−	0.995864i	$-0.471038\pi$
$31$	5.63331	0.181720	0.0908598	+	0.995864i	$0.471038\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 7.75182i	− 0.221480i
$36$	0	0
$37$	−17.0438	−0.460642	−0.230321	−	0.973115i	$-0.573978\pi$
$37$	−17.0438	−0.460642	−0.230321	+	0.973115i	$0.573978\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 27.4671i	− 0.669930i	−0.942231	−	0.334965i	$-0.891275\pi$
$41$	− 27.4671i	− 0.669930i	0.942231	−	0.334965i	$-0.108725\pi$
$42$	0	0
$43$	52.3306	1.21699	0.608495	−	0.793558i	$-0.291774\pi$
$43$	52.3306	1.21699	0.608495	+	0.793558i	$0.291774\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	64.5352i	1.37309i	0.727087	+	0.686545i	$0.240874\pi$
$47$	64.5352i	1.37309i	−0.727087	+	0.686545i	$0.759126\pi$
$48$	0	0
$49$	3.00000	0.0612245
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 35.9283i	− 0.677893i	−0.940806	−	0.338946i	$-0.889929\pi$
$53$	− 35.9283i	− 0.677893i	0.940806	−	0.338946i	$-0.110071\pi$
$54$	0	0
$55$	−17.5778	−0.319596
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 56.8069i	− 0.962828i	−0.876493	−	0.481414i	$-0.840123\pi$
$59$	− 56.8069i	− 0.962828i	0.876493	−	0.481414i	$-0.159877\pi$
$60$	0	0
$61$	−69.3743	−1.13728	−0.568642	−	0.822585i	$-0.692531\pi$
$61$	−69.3743	−1.13728	−0.568642	+	0.822585i	$0.692531\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.2245i	0.357300i
$66$	0	0
$67$	69.3743	1.03544	0.517719	−	0.855551i	$-0.326781\pi$
$67$	69.3743	1.03544	0.517719	+	0.855551i	$0.326781\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	98.4764i	1.38699i	0.720461	+	0.693496i	$0.243930\pi$
$71$	98.4764i	1.38699i	−0.720461	+	0.693496i	$0.756070\pi$
$72$	0	0
$73$	−37.6888	−0.516285	−0.258143	−	0.966107i	$-0.583110\pi$
$73$	−37.6888	−0.516285	−0.258143	+	0.966107i	$0.583110\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 117.914i	− 1.53135i
$78$	0	0
$79$	−127.322	−1.61167	−0.805836	−	0.592138i	$-0.798284\pi$
$79$	−127.322	−1.61167	−0.805836	+	0.592138i	$0.798284\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.75182i	0.0933954i	0.998909	+	0.0466977i	$0.0148697\pi$
$83$	7.75182i	0.0933954i	−0.998909	+	0.0466977i	$0.985130\pi$
$84$	0	0
$85$	20.4052	0.240061
$86$	0	0
$87$	0	0
$88$	0	0
$89$	76.1475i	0.855590i	0.903876	+	0.427795i	$0.140709\pi$
$89$	76.1475i	0.855590i	−0.903876	+	0.427795i	$0.859291\pi$
$90$	0	0
$91$	−155.792	−1.71200
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 18.3218i	− 0.192861i
$96$	0	0
$97$	4.84441	0.0499424	0.0249712	−	0.999688i	$-0.492051\pi$
$97$	4.84441	0.0499424	0.0249712	+	0.999688i	$0.492051\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	105.635i	1.04589i	0.852366	+	0.522946i	$0.175167\pi$
$101$	105.635i	1.04589i	−0.852366	+	0.522946i	$0.824833\pi$
$102$	0	0
$103$	−104.789	−1.01737	−0.508684	−	0.860953i	$-0.669868\pi$
$103$	−104.789	−1.01737	−0.508684	+	0.860953i	$0.669868\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	65.4067i	0.611278i	0.952147	+	0.305639i	$0.0988701\pi$
$107$	65.4067i	0.611278i	−0.952147	+	0.305639i	$0.901130\pi$
$108$	0	0
$109$	−3.36144	−0.0308389	−0.0154195	−	0.999881i	$-0.504908\pi$
$109$	−3.36144	−0.0308389	−0.0154195	+	0.999881i	$0.504908\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	84.1927i	0.745068i	0.928019	+	0.372534i	$0.121511\pi$
$113$	84.1927i	0.745068i	−0.928019	+	0.372534i	$0.878489\pi$
$114$	0	0
$115$	−1.19933	−0.0104289
$116$	0	0
$117$	0	0
$118$	0	0
$119$	136.880i	1.15025i
$120$	0	0
$121$	−146.378	−1.20973
$122$	0	0
$123$	0	0
$124$	0	0
$125$	52.5069i	0.420056i
$126$	0	0
$127$	−45.5223	−0.358443	−0.179222	−	0.983809i	$-0.557358\pi$
$127$	−45.5223	−0.358443	−0.179222	+	0.983809i	$0.557358\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	129.117i	0.985629i	0.870134	+	0.492814i	$0.164032\pi$
$131$	129.117i	0.985629i	−0.870134	+	0.492814i	$0.835968\pi$
$132$	0	0
$133$	122.904	0.924092
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.9367i	− 0.0798296i	−0.999203	−	0.0399148i	$-0.987291\pi$
$137$	− 10.9367i	− 0.0798296i	0.999203	−	0.0399148i	$-0.0127087\pi$
$138$	0	0
$139$	−1.19933	−0.00862825	−0.00431412	−	0.999991i	$-0.501373\pi$
$139$	−1.19933	−0.00862825	−0.00431412	+	0.999991i	$0.501373\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	353.270i	2.47042i
$144$	0	0
$145$	−31.6888	−0.218544
$146$	0	0
$147$	0	0
$148$	0	0
$149$	29.9323i	0.200888i	0.994943	+	0.100444i	$0.0320263\pi$
$149$	29.9323i	0.200888i	−0.994943	+	0.100444i	$0.967974\pi$
$150$	0	0
$151$	−61.5223	−0.407432	−0.203716	−	0.979030i	$-0.565302\pi$
$151$	−61.5223	−0.407432	−0.203716	+	0.979030i	$0.565302\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.05571i	0.0390691i
$156$	0	0
$157$	137.549	0.876110	0.438055	−	0.898948i	$-0.355667\pi$
$157$	137.549	0.876110	0.438055	+	0.898948i	$0.355667\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.04521i	− 0.0499702i
$162$	0	0
$163$	−191.079	−1.17227	−0.586133	−	0.810215i	$-0.699350\pi$
$163$	−191.079	−1.17227	−0.586133	+	0.810215i	$0.699350\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	233.125i	1.39596i	0.716118	+	0.697980i	$0.245917\pi$
$167$	233.125i	1.39596i	−0.716118	+	0.697980i	$0.754083\pi$
$168$	0	0
$169$	297.755	1.76187
$170$	0	0
$171$	0	0
$172$	0	0
$173$	323.809i	1.87173i	0.352363	+	0.935864i	$0.385378\pi$
$173$	323.809i	1.87173i	−0.352363	+	0.935864i	$0.614622\pi$
$174$	0	0
$175$	−171.944	−0.982540
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 185.924i	− 1.03868i	−0.854567	−	0.519342i	$-0.826177\pi$
$179$	− 185.924i	− 1.03868i	0.854567	−	0.519342i	$-0.173823\pi$
$180$	0	0
$181$	−126.266	−0.697600	−0.348800	−	0.937197i	$-0.613411\pi$
$181$	−126.266	−0.697600	−0.348800	+	0.937197i	$0.613411\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 18.3218i	− 0.0990365i
$186$	0	0
$187$	310.385	1.65982
$188$	0	0
$189$	0	0
$190$	0	0
$191$	233.125i	1.22055i	0.792189	+	0.610275i	$0.208941\pi$
$191$	233.125i	1.22055i	−0.792189	+	0.610275i	$0.791059\pi$
$192$	0	0
$193$	−117.378	−0.608174	−0.304087	−	0.952644i	$-0.598351\pi$
$193$	−117.378	−0.608174	−0.304087	+	0.952644i	$0.598351\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 296.313i	− 1.50413i	−0.659091	−	0.752064i	$-0.729059\pi$
$197$	− 296.313i	− 1.50413i	0.659091	−	0.752064i	$-0.270941\pi$
$198$	0	0
$199$	−233.011	−1.17091	−0.585455	−	0.810705i	$-0.699084\pi$
$199$	−233.011	−1.17091	−0.585455	+	0.810705i	$0.699084\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 212.572i	− 1.04715i
$204$	0	0
$205$	29.5267	0.144033
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 278.694i	− 1.33346i
$210$	0	0
$211$	35.2868	0.167236	0.0836181	−	0.996498i	$-0.473352\pi$
$211$	35.2868	0.167236	0.0836181	+	0.996498i	$0.473352\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	56.2545i	0.261649i
$216$	0	0
$217$	−40.6224	−0.187200
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 410.094i	− 1.85563i
$222$	0	0
$223$	−51.8335	−0.232437	−0.116219	−	0.993224i	$-0.537077\pi$
$223$	−51.8335	−0.232437	−0.116219	+	0.993224i	$0.537077\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 334.786i	− 1.47483i	−0.675442	−	0.737413i	$-0.736047\pi$
$227$	− 334.786i	− 1.47483i	0.675442	−	0.737413i	$-0.263953\pi$
$228$	0	0
$229$	−219.407	−0.958108	−0.479054	−	0.877786i	$-0.659020\pi$
$229$	−219.407	−0.958108	−0.479054	+	0.877786i	$0.659020\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 203.867i	− 0.874965i	−0.899227	−	0.437482i	$-0.855870\pi$
$233$	− 203.867i	− 0.874965i	0.899227	−	0.437482i	$-0.144130\pi$
$234$	0	0
$235$	−69.3743	−0.295210
$236$	0	0
$237$	0	0
$238$	0	0
$239$	54.0232i	0.226038i	0.993593	+	0.113019i	$0.0360522\pi$
$239$	54.0232i	0.226038i	−0.993593	+	0.113019i	$0.963948\pi$
$240$	0	0
$241$	327.600	1.35933	0.679667	−	0.733520i	$-0.262124\pi$
$241$	327.600	1.35933	0.679667	+	0.733520i	$0.262124\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.22495i	0.0131631i
$246$	0	0
$247$	−368.222	−1.49078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	362.281i	1.44335i	0.692231	+	0.721676i	$0.256628\pi$
$251$	362.281i	1.44335i	−0.692231	+	0.721676i	$0.743372\pi$
$252$	0	0
$253$	−18.2431	−0.0721070
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2132i	0.0825416i	0.999148	+	0.0412708i	$0.0131406\pi$
$257$	21.2132i	0.0825416i	−0.999148	+	0.0412708i	$0.986859\pi$
$258$	0	0
$259$	122.904	0.474534
$260$	0	0
$261$	0	0
$262$	0	0
$263$	205.878i	0.782806i	0.920219	+	0.391403i	$0.128010\pi$
$263$	205.878i	0.782806i	−0.920219	+	0.391403i	$0.871990\pi$
$264$	0	0
$265$	38.6224	0.145745
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.77109i	− 0.0103014i	−0.999987	−	0.00515072i	$-0.998360\pi$
$269$	− 2.77109i	− 0.0103014i	0.999987	−	0.00515072i	$-0.00163953\pi$
$270$	0	0
$271$	−125.744	−0.464001	−0.232001	−	0.972716i	$-0.574527\pi$
$271$	−125.744	−0.464001	−0.232001	+	0.972716i	$0.574527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	389.896i	1.41780i
$276$	0	0
$277$	−162.752	−0.587552	−0.293776	−	0.955874i	$-0.594912\pi$
$277$	−162.752	−0.587552	−0.293776	+	0.955874i	$0.594912\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	301.227i	1.07198i	0.844223	+	0.535992i	$0.180062\pi$
$281$	301.227i	1.07198i	−0.844223	+	0.535992i	$0.819938\pi$
$282$	0	0
$283$	−31.6888	−0.111975	−0.0559874	−	0.998431i	$-0.517831\pi$
$283$	−31.6888	−0.111975	−0.0559874	+	0.998431i	$0.517831\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	198.068i	0.690134i
$288$	0	0
$289$	−71.3112	−0.246751
$290$	0	0
$291$	0	0
$292$	0	0
$293$	304.913i	1.04066i	0.853966	+	0.520329i	$0.174191\pi$
$293$	304.913i	1.04066i	−0.853966	+	0.520329i	$0.825809\pi$
$294$	0	0
$295$	61.0665	0.207005
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.1035i	0.0806137i
$300$	0	0
$301$	−377.361	−1.25369
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 74.5763i	− 0.244512i
$306$	0	0
$307$	105.860	0.344822	0.172411	−	0.985025i	$-0.444844\pi$
$307$	105.860	0.344822	0.172411	+	0.985025i	$0.444844\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	555.157i	1.78507i	0.450978	+	0.892535i	$0.351075\pi$
$311$	555.157i	1.78507i	−0.450978	+	0.892535i	$0.648925\pi$
$312$	0	0
$313$	−158.000	−0.504792	−0.252396	−	0.967624i	$-0.581219\pi$
$313$	−158.000	−0.504792	−0.252396	+	0.967624i	$0.581219\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	97.0351i	0.306105i	0.988218	+	0.153052i	$0.0489103\pi$
$317$	97.0351i	0.306105i	−0.988218	+	0.153052i	$0.951090\pi$
$318$	0	0
$319$	−482.022	−1.51104
$320$	0	0
$321$	0	0
$322$	0	0
$323$	323.522i	1.00162i
$324$	0	0
$325$	515.147	1.58507
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 465.370i	− 1.41450i
$330$	0	0
$331$	−619.572	−1.87182	−0.935909	−	0.352242i	$-0.885419\pi$
$331$	−619.572	−1.87182	−0.935909	+	0.352242i	$0.885419\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	74.5763i	0.222616i
$336$	0	0
$337$	−170.755	−0.506692	−0.253346	−	0.967376i	$-0.581531\pi$
$337$	−170.755	−0.506692	−0.253346	+	0.967376i	$0.581531\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	92.1141i	0.270129i
$342$	0	0
$343$	331.711	0.967087
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 300.386i	− 0.865666i	−0.901474	−	0.432833i	$-0.857514\pi$
$347$	− 300.386i	− 0.865666i	0.901474	−	0.432833i	$-0.142486\pi$
$348$	0	0
$349$	53.5299	0.153381	0.0766904	−	0.997055i	$-0.475565\pi$
$349$	53.5299	0.153381	0.0766904	+	0.997055i	$0.475565\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	600.475i	1.70106i	0.525925	+	0.850531i	$0.323719\pi$
$353$	600.475i	1.70106i	−0.525925	+	0.850531i	$0.676281\pi$
$354$	0	0
$355$	−105.860	−0.298199
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 148.508i	− 0.413671i	−0.978376	−	0.206836i	$-0.933683\pi$
$359$	− 148.508i	− 0.413671i	0.978376	−	0.206836i	$-0.0663165\pi$
$360$	0	0
$361$	−70.5106	−0.195320
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 40.5149i	− 0.111000i
$366$	0	0
$367$	316.389	0.862094	0.431047	−	0.902329i	$-0.358144\pi$
$367$	316.389	0.862094	0.431047	+	0.902329i	$0.358144\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	259.083i	0.698336i
$372$	0	0
$373$	626.768	1.68034	0.840171	−	0.542321i	$-0.182455\pi$
$373$	626.768	1.68034	0.840171	+	0.542321i	$0.182455\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	636.867i	1.68930i
$378$	0	0
$379$	331.027	0.873423	0.436711	−	0.899602i	$-0.356143\pi$
$379$	331.027	0.873423	0.436711	+	0.899602i	$0.356143\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	407.765i	1.06466i	0.846537	+	0.532330i	$0.178683\pi$
$383$	407.765i	1.06466i	−0.846537	+	0.532330i	$0.821317\pi$
$384$	0	0
$385$	126.755	0.329234
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 53.5819i	− 0.137743i	−0.997626	−	0.0688714i	$-0.978060\pi$
$389$	− 53.5819i	− 0.137743i	0.997626	−	0.0688714i	$-0.0219398\pi$
$390$	0	0
$391$	21.1775	0.0541624
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 136.869i	− 0.346504i
$396$	0	0
$397$	289.744	0.729833	0.364917	−	0.931040i	$-0.381098\pi$
$397$	289.744	0.729833	0.364917	+	0.931040i	$0.381098\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	203.458i	0.507376i	0.967286	+	0.253688i	$0.0816436\pi$
$401$	203.458i	0.507376i	−0.967286	+	0.253688i	$0.918356\pi$
$402$	0	0
$403$	121.705	0.301997
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 278.694i	− 0.684752i
$408$	0	0
$409$	−412.844	−1.00940	−0.504700	−	0.863295i	$-0.668397\pi$
$409$	−412.844	−1.00940	−0.504700	+	0.863295i	$0.668397\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	409.640i	0.991865i
$414$	0	0
$415$	−8.33308	−0.0200797
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 336.482i	− 0.803059i	−0.915846	−	0.401529i	$-0.868479\pi$
$419$	− 336.482i	− 0.803059i	0.915846	−	0.401529i	$-0.131521\pi$
$420$	0	0
$421$	−217.481	−0.516582	−0.258291	−	0.966067i	$-0.583159\pi$
$421$	−217.481	−0.516582	−0.258291	+	0.966067i	$0.583159\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 452.611i	− 1.06497i
$426$	0	0
$427$	500.265	1.17158
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 750.994i	− 1.74245i	−0.490888	−	0.871223i	$-0.663328\pi$
$431$	− 750.994i	− 1.74245i	0.490888	−	0.871223i	$-0.336672\pi$
$432$	0	0
$433$	−176.133	−0.406773	−0.203387	−	0.979098i	$-0.565195\pi$
$433$	−176.133	−0.406773	−0.203387	+	0.979098i	$0.565195\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 19.0152i	− 0.0435130i
$438$	0	0
$439$	−417.788	−0.951681	−0.475841	−	0.879531i	$-0.657856\pi$
$439$	−417.788	−0.951681	−0.475841	+	0.879531i	$0.657856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 61.0471i	− 0.137804i	−0.997623	−	0.0689020i	$-0.978050\pi$
$443$	− 61.0471i	− 0.137804i	0.997623	−	0.0689020i	$-0.0219496\pi$
$444$	0	0
$445$	−81.8573	−0.183949
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 182.905i	− 0.407360i	−0.979038	−	0.203680i	$-0.934710\pi$
$449$	− 182.905i	− 0.407360i	0.979038	−	0.203680i	$-0.0652902\pi$
$450$	0	0
$451$	449.134	0.995863
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 167.474i	− 0.368075i
$456$	0	0
$457$	443.600	0.970678	0.485339	−	0.874326i	$-0.338696\pi$
$457$	443.600	0.970678	0.485339	+	0.874326i	$0.338696\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 728.241i	− 1.57970i	−0.613301	−	0.789850i	$-0.710159\pi$
$461$	− 728.241i	− 1.57970i	0.613301	−	0.789850i	$-0.289841\pi$
$462$	0	0
$463$	−107.722	−0.232662	−0.116331	−	0.993211i	$-0.537113\pi$
$463$	−107.722	−0.232662	−0.116331	+	0.993211i	$0.537113\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 820.248i	− 1.75642i	−0.478276	−	0.878210i	$-0.658738\pi$
$467$	− 820.248i	− 1.75642i	0.478276	−	0.878210i	$-0.341262\pi$
$468$	0	0
$469$	−500.265	−1.06666
$470$	0	0
$471$	0	0
$472$	0	0
$473$	855.693i	1.80908i
$474$	0	0
$475$	−406.398	−0.855575
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 120.790i	− 0.252171i	−0.992019	−	0.126085i	$-0.959759\pi$
$479$	− 120.790i	− 0.252171i	0.992019	−	0.126085i	$-0.0402413\pi$
$480$	0	0
$481$	−368.222	−0.765534
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.20766i	0.0107374i
$486$	0	0
$487$	128.234	0.263314	0.131657	−	0.991295i	$-0.457970\pi$
$487$	128.234	0.263314	0.131657	+	0.991295i	$0.457970\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	382.144i	0.778298i	0.921175	+	0.389149i	$0.127231\pi$
$491$	382.144i	0.778298i	−0.921175	+	0.389149i	$0.872769\pi$
$492$	0	0
$493$	559.555	1.13500
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 710.123i	− 1.42882i
$498$	0	0
$499$	−554.995	−1.11221	−0.556107	−	0.831111i	$-0.687705\pi$
$499$	−554.995	−1.11221	−0.556107	+	0.831111i	$0.687705\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	78.3943i	0.155854i	0.996959	+	0.0779268i	$0.0248300\pi$
$503$	78.3943i	0.155854i	−0.996959	+	0.0779268i	$0.975170\pi$
$504$	0	0
$505$	−113.556	−0.224863
$506$	0	0
$507$	0	0
$508$	0	0
$509$	514.941i	1.01167i	0.862630	+	0.505836i	$0.168816\pi$
$509$	514.941i	1.01167i	−0.862630	+	0.505836i	$0.831184\pi$
$510$	0	0
$511$	271.778	0.531855
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 112.646i	− 0.218731i
$516$	0	0
$517$	−1055.26	−2.04112
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 489.695i	− 0.939914i	−0.882689	−	0.469957i	$-0.844270\pi$
$521$	− 489.695i	− 0.939914i	0.882689	−	0.469957i	$-0.155730\pi$
$522$	0	0
$523$	774.165	1.48024	0.740119	−	0.672476i	$-0.234769\pi$
$523$	774.165	1.48024	0.740119	+	0.672476i	$0.234769\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 106.931i	− 0.202905i
$528$	0	0
$529$	527.755	0.997647
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 593.415i	− 1.11335i
$534$	0	0
$535$	−70.3112	−0.131423
$536$	0	0
$537$	0	0
$538$	0	0
$539$	49.0551i	0.0910112i
$540$	0	0
$541$	−590.045	−1.09066	−0.545328	−	0.838223i	$-0.683595\pi$
$541$	−590.045	−1.09066	−0.545328	+	0.838223i	$0.683595\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.61350i	− 0.00663027i
$546$	0	0
$547$	−707.189	−1.29285	−0.646425	−	0.762977i	$-0.723737\pi$
$547$	−707.189	−1.29285	−0.646425	+	0.762977i	$0.723737\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 502.423i	− 0.911838i
$552$	0	0
$553$	918.133	1.66028
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 75.0816i	− 0.134796i	−0.997726	−	0.0673982i	$-0.978530\pi$
$557$	− 75.0816i	− 0.134796i	0.997726	−	0.0673982i	$-0.0214698\pi$
$558$	0	0
$559$	1130.58	2.02250
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 114.462i	− 0.203307i	−0.994820	−	0.101653i	$-0.967587\pi$
$563$	− 114.462i	− 0.203307i	0.994820	−	0.101653i	$-0.0324133\pi$
$564$	0	0
$565$	−90.5058	−0.160187
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 45.8199i	− 0.0805270i	−0.999189	−	0.0402635i	$-0.987180\pi$
$569$	− 45.8199i	− 0.0805270i	0.999189	−	0.0402635i	$-0.0128197\pi$
$570$	0	0
$571$	−830.093	−1.45375	−0.726877	−	0.686768i	$-0.759029\pi$
$571$	−830.093	−1.45375	−0.726877	+	0.686768i	$0.759029\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	26.6025i	0.0462652i
$576$	0	0
$577$	−69.3776	−0.120239	−0.0601193	−	0.998191i	$-0.519148\pi$
$577$	−69.3776	−0.120239	−0.0601193	+	0.998191i	$0.519148\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 55.8991i	− 0.0962120i
$582$	0	0
$583$	587.489	1.00770
$584$	0	0
$585$	0	0
$586$	0	0
$587$	428.655i	0.730248i	0.930959	+	0.365124i	$0.118973\pi$
$587$	428.655i	0.730248i	−0.930959	+	0.365124i	$0.881027\pi$
$588$	0	0
$589$	−96.0127	−0.163010
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 77.0276i	− 0.129895i	−0.997889	−	0.0649474i	$-0.979312\pi$
$593$	− 77.0276i	− 0.129895i	0.997889	−	0.0649474i	$-0.0206880\pi$
$594$	0	0
$595$	−147.144	−0.247301
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 963.566i	− 1.60862i	−0.594207	−	0.804312i	$-0.702534\pi$
$599$	− 963.566i	− 1.60862i	0.594207	−	0.804312i	$-0.297466\pi$
$600$	0	0
$601$	−840.133	−1.39789	−0.698946	−	0.715175i	$-0.746347\pi$
$601$	−840.133	−1.39789	−0.698946	+	0.715175i	$0.746347\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 157.354i	− 0.260089i
$606$	0	0
$607$	156.611	0.258008	0.129004	−	0.991644i	$-0.458822\pi$
$607$	156.611	0.258008	0.129004	+	0.991644i	$0.458822\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1394.25i	2.28192i
$612$	0	0
$613$	426.094	0.695096	0.347548	−	0.937662i	$-0.387014\pi$
$613$	426.094	0.695096	0.347548	+	0.937662i	$0.387014\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 963.520i	− 1.56162i	−0.624769	−	0.780810i	$-0.714807\pi$
$617$	− 963.520i	− 1.56162i	0.624769	−	0.780810i	$-0.285193\pi$
$618$	0	0
$619$	175.235	0.283093	0.141547	−	0.989932i	$-0.454792\pi$
$619$	175.235	0.283093	0.141547	+	0.989932i	$0.454792\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 549.107i	− 0.881392i
$624$	0	0
$625$	539.666	0.863466
$626$	0	0
$627$	0	0
$628$	0	0
$629$	323.522i	0.514344i
$630$	0	0
$631$	1140.72	1.80780	0.903900	−	0.427744i	$-0.140692\pi$
$631$	1140.72	1.80780	0.903900	+	0.427744i	$0.140692\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 48.9357i	− 0.0770641i
$636$	0	0
$637$	64.8136	0.101748
$638$	0	0
$639$	0	0
$640$	0	0
$641$	351.730i	0.548721i	0.961627	+	0.274360i	$0.0884661\pi$
$641$	351.730i	0.548721i	−0.961627	+	0.274360i	$0.911534\pi$
$642$	0	0
$643$	86.4181	0.134398	0.0671991	−	0.997740i	$-0.478594\pi$
$643$	86.4181	0.134398	0.0671991	+	0.997740i	$0.478594\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 430.723i	− 0.665723i	−0.942976	−	0.332861i	$-0.891986\pi$
$647$	− 430.723i	− 0.665723i	0.942976	−	0.332861i	$-0.108014\pi$
$648$	0	0
$649$	928.888	1.43126
$650$	0	0
$651$	0	0
$652$	0	0
$653$	886.670i	1.35784i	0.734212	+	0.678920i	$0.237552\pi$
$653$	886.670i	1.35784i	−0.734212	+	0.678920i	$0.762448\pi$
$654$	0	0
$655$	−138.799	−0.211907
$656$	0	0
$657$	0	0
$658$	0	0
$659$	352.833i	0.535407i	0.963501	+	0.267704i	$0.0862648\pi$
$659$	352.833i	0.535407i	−0.963501	+	0.267704i	$0.913735\pi$
$660$	0	0
$661$	1206.26	1.82489	0.912447	−	0.409195i	$-0.134190\pi$
$661$	1206.26	1.82489	0.912447	+	0.409195i	$0.134190\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	132.120i	0.198677i
$666$	0	0
$667$	−32.8882	−0.0493076
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1134.39i	− 1.69059i
$672$	0	0
$673$	−956.133	−1.42070	−0.710351	−	0.703847i	$-0.751464\pi$
$673$	−956.133	−1.42070	−0.710351	+	0.703847i	$0.751464\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 420.557i	− 0.621207i	−0.950540	−	0.310604i	$-0.899469\pi$
$677$	− 420.557i	− 0.621207i	0.950540	−	0.310604i	$-0.100531\pi$
$678$	0	0
$679$	−34.9335	−0.0514485
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 539.725i	− 0.790227i	−0.918632	−	0.395113i	$-0.870705\pi$
$683$	− 539.725i	− 0.790227i	0.918632	−	0.395113i	$-0.129295\pi$
$684$	0	0
$685$	11.7567	0.0171631
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 776.214i	− 1.12658i
$690$	0	0
$691$	432.090	0.625312	0.312656	−	0.949866i	$-0.398781\pi$
$691$	432.090	0.625312	0.312656	+	0.949866i	$0.398781\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.28926i	− 0.00185504i
$696$	0	0
$697$	−521.378	−0.748031
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 880.339i	− 1.25583i	−0.778280	−	0.627917i	$-0.783908\pi$
$701$	− 880.339i	− 1.25583i	0.778280	−	0.627917i	$-0.216092\pi$
$702$	0	0
$703$	290.489	0.413214
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 761.745i	− 1.07743i
$708$	0	0
$709$	799.367	1.12746	0.563729	−	0.825960i	$-0.309366\pi$
$709$	799.367	1.12746	0.563729	+	0.825960i	$0.309366\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.28491i	0.00881474i
$714$	0	0
$715$	−379.760	−0.531133
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 258.786i	− 0.359924i	−0.983674	−	0.179962i	$-0.942402\pi$
$719$	− 258.786i	− 0.359924i	0.983674	−	0.179962i	$-0.0575975\pi$
$720$	0	0
$721$	755.643	1.04805
$722$	0	0
$723$	0	0
$724$	0	0
$725$	702.896i	0.969511i
$726$	0	0
$727$	663.211	0.912257	0.456129	−	0.889914i	$-0.349236\pi$
$727$	663.211	0.912257	0.456129	+	0.889914i	$0.349236\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 993.332i	− 1.35887i
$732$	0	0
$733$	908.826	1.23987	0.619936	−	0.784653i	$-0.287159\pi$
$733$	908.826	1.23987	0.619936	+	0.784653i	$0.287159\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1134.39i	1.53920i
$738$	0	0
$739$	1112.39	1.50526	0.752631	−	0.658443i	$-0.228785\pi$
$739$	1112.39	1.50526	0.752631	+	0.658443i	$0.228785\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 541.298i	− 0.728530i	−0.931295	−	0.364265i	$-0.881320\pi$
$743$	− 541.298i	− 0.728530i	0.931295	−	0.364265i	$-0.118680\pi$
$744$	0	0
$745$	−32.1767	−0.0431902
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 471.655i	− 0.629713i
$750$	0	0
$751$	−763.699	−1.01691	−0.508455	−	0.861089i	$-0.669783\pi$
$751$	−763.699	−1.01691	−0.508455	+	0.861089i	$0.669783\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 66.1354i	− 0.0875966i
$756$	0	0
$757$	−419.134	−0.553678	−0.276839	−	0.960916i	$-0.589287\pi$
$757$	−419.134	−0.553678	−0.276839	+	0.960916i	$0.589287\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1330.24i	1.74802i	0.485912	+	0.874008i	$0.338488\pi$
$761$	1330.24i	1.74802i	−0.485912	+	0.874008i	$0.661512\pi$
$762$	0	0
$763$	24.2397	0.0317690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1227.29i	− 1.60011i
$768$	0	0
$769$	671.511	0.873226	0.436613	−	0.899649i	$-0.356178\pi$
$769$	671.511	0.873226	0.436613	+	0.899649i	$0.356178\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1421.13i	1.83846i	0.393724	+	0.919229i	$0.371187\pi$
$773$	1421.13i	1.83846i	−0.393724	+	0.919229i	$0.628813\pi$
$774$	0	0
$775$	134.323	0.173320
$776$	0	0
$777$	0	0
$778$	0	0
$779$	468.143i	0.600954i
$780$	0	0
$781$	−1610.25	−2.06179
$782$	0	0
$783$	0	0
$784$	0	0
$785$	147.863i	0.188361i
$786$	0	0
$787$	649.808	0.825677	0.412839	−	0.910804i	$-0.364537\pi$
$787$	649.808	0.825677	0.412839	+	0.910804i	$0.364537\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 607.122i	− 0.767538i
$792$	0	0
$793$	−1498.80	−1.89004
$794$	0	0
$795$	0	0
$796$	0	0
$797$	422.946i	0.530672i	0.964156	+	0.265336i	$0.0854830\pi$
$797$	422.946i	0.530672i	−0.964156	+	0.265336i	$0.914517\pi$
$798$	0	0
$799$	1225.00	1.53317
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 616.276i	− 0.767467i
$804$	0	0
$805$	8.64847	0.0107434
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1353.87i	− 1.67351i	−0.547574	−	0.836757i	$-0.684448\pi$
$809$	− 1353.87i	− 1.67351i	0.547574	−	0.836757i	$-0.315552\pi$
$810$	0	0
$811$	−842.340	−1.03864	−0.519322	−	0.854579i	$-0.673815\pi$
$811$	−842.340	−1.03864	−0.519322	+	0.854579i	$0.673815\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 205.407i	− 0.252033i
$816$	0	0
$817$	−891.909	−1.09169
$818$	0	0
$819$	0	0
$820$	0	0
$821$	616.897i	0.751397i	0.926742	+	0.375698i	$0.122597\pi$
$821$	616.897i	0.751397i	−0.926742	+	0.375698i	$0.877403\pi$
$822$	0	0
$823$	−980.500	−1.19137	−0.595686	−	0.803217i	$-0.703120\pi$
$823$	−980.500	−1.19137	−0.595686	+	0.803217i	$0.703120\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	70.4951i	0.0852419i	0.999091	+	0.0426210i	$0.0135708\pi$
$827$	70.4951i	0.0852419i	−0.999091	+	0.0426210i	$0.986429\pi$
$828$	0	0
$829$	515.147	0.621408	0.310704	−	0.950507i	$-0.399435\pi$
$829$	515.147	0.621408	0.310704	+	0.950507i	$0.399435\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 56.9456i	− 0.0683621i
$834$	0	0
$835$	−250.606	−0.300127
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.1556i	0.0562045i	0.999605	+	0.0281022i	$0.00894640\pi$
$839$	47.1556i	0.0562045i	−0.999605	+	0.0281022i	$0.991054\pi$
$840$	0	0
$841$	−27.9773	−0.0332667
$842$	0	0
$843$	0	0
$844$	0	0
$845$	320.082i	0.378795i
$846$	0	0
$847$	1055.54	1.24622
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 19.0152i	− 0.0223445i
$852$	0	0
$853$	−763.118	−0.894628	−0.447314	−	0.894377i	$-0.647619\pi$
$853$	−763.118	−0.894628	−0.447314	+	0.894377i	$0.647619\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1641.05i	1.91488i	0.288631	+	0.957440i	$0.406800\pi$
$857$	1641.05i	1.91488i	−0.288631	+	0.957440i	$0.593200\pi$
$858$	0	0
$859$	1366.84	1.59120	0.795602	−	0.605819i	$-0.207155\pi$
$859$	1366.84	1.59120	0.795602	+	0.605819i	$0.207155\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 836.082i	− 0.968809i	−0.874844	−	0.484405i	$-0.839036\pi$
$863$	− 836.082i	− 0.968809i	0.874844	−	0.484405i	$-0.160964\pi$
$864$	0	0
$865$	−348.089	−0.402415
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2081.93i	− 2.39578i
$870$	0	0
$871$	1498.80	1.72078
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 378.633i	− 0.432723i
$876$	0	0
$877$	158.191	0.180377	0.0901887	−	0.995925i	$-0.471253\pi$
$877$	158.191	0.180377	0.0901887	+	0.995925i	$0.471253\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	237.430i	0.269500i	0.990880	+	0.134750i	$0.0430232\pi$
$881$	237.430i	0.269500i	−0.990880	+	0.134750i	$0.956977\pi$
$882$	0	0
$883$	−592.933	−0.671499	−0.335749	−	0.941951i	$-0.608989\pi$
$883$	−592.933	−0.671499	−0.335749	+	0.941951i	$0.608989\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1420.72i	− 1.60171i	−0.598858	−	0.800855i	$-0.704379\pi$
$887$	− 1420.72i	− 1.60171i	0.598858	−	0.800855i	$-0.295621\pi$
$888$	0	0
$889$	328.266	0.369253
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1099.92i	− 1.23172i
$894$	0	0
$895$	199.866	0.223313
$896$	0	0
$897$	0	0
$898$	0	0
$899$	166.061i	0.184717i
$900$	0	0
$901$	−681.987	−0.756922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 135.734i	− 0.149982i
$906$	0	0
$907$	816.648	0.900383	0.450192	−	0.892932i	$-0.351356\pi$
$907$	816.648	0.900383	0.450192	+	0.892932i	$0.351356\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	469.895i	0.515801i	0.966171	+	0.257901i	$0.0830307\pi$
$911$	469.895i	0.515801i	−0.966171	+	0.257901i	$0.916969\pi$
$912$	0	0
$913$	−126.755	−0.138834
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 931.079i	− 1.01535i
$918$	0	0
$919$	1648.21	1.79348	0.896741	−	0.442555i	$-0.145928\pi$
$919$	1648.21	1.79348	0.896741	+	0.442555i	$0.145928\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2127.53i	2.30502i
$924$	0	0
$925$	−406.398	−0.439349
$926$	0	0
$927$	0	0
$928$	0	0
$929$	421.311i	0.453510i	0.973952	+	0.226755i	$0.0728116\pi$
$929$	421.311i	0.453510i	−0.973952	+	0.226755i	$0.927188\pi$
$930$	0	0
$931$	−51.1313	−0.0549208
$932$	0	0
$933$	0	0
$934$	0	0
$935$	333.659i	0.356855i
$936$	0	0
$937$	1434.27	1.53070	0.765350	−	0.643614i	$-0.222566\pi$
$937$	1434.27	1.53070	0.765350	+	0.643614i	$0.222566\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	732.876i	0.778827i	0.921063	+	0.389413i	$0.127322\pi$
$941$	732.876i	0.778827i	−0.921063	+	0.389413i	$0.872678\pi$
$942$	0	0
$943$	30.6443	0.0324966
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1385.53i	1.46308i	0.681800	+	0.731538i	$0.261197\pi$
$947$	1385.53i	1.46308i	−0.681800	+	0.731538i	$0.738803\pi$
$948$	0	0
$949$	−814.249	−0.858007
$950$	0	0
$951$	0	0
$952$	0	0
$953$	957.328i	1.00454i	0.864711	+	0.502270i	$0.167502\pi$
$953$	957.328i	1.00454i	−0.864711	+	0.502270i	$0.832498\pi$
$954$	0	0
$955$	−250.606	−0.262414
$956$	0	0
$957$	0	0
$958$	0	0
$959$	78.8654i	0.0822371i
$960$	0	0
$961$	−929.266	−0.966978
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 126.179i	− 0.130755i
$966$	0	0
$967$	−1225.90	−1.26773	−0.633867	−	0.773442i	$-0.718533\pi$
$967$	−1225.90	−1.26773	−0.633867	+	0.773442i	$0.718533\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 752.906i	− 0.775393i	−0.921787	−	0.387696i	$-0.873271\pi$
$971$	− 752.906i	− 0.775393i	0.921787	−	0.387696i	$-0.126729\pi$
$972$	0	0
$973$	8.64847	0.00888845
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 642.932i	− 0.658068i	−0.944318	−	0.329034i	$-0.893277\pi$
$977$	− 642.932i	− 0.658068i	0.944318	−	0.329034i	$-0.106723\pi$
$978$	0	0
$979$	−1245.14	−1.27185
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 161.251i	− 0.164040i	−0.996631	−	0.0820200i	$-0.973863\pi$
$983$	− 161.251i	− 0.164040i	0.996631	−	0.0820200i	$-0.0261371\pi$
$984$	0	0
$985$	318.532	0.323382
$986$	0	0
$987$	0	0
$988$	0	0
$989$	58.3836i	0.0590330i
$990$	0	0
$991$	811.500	0.818870	0.409435	−	0.912339i	$-0.365726\pi$
$991$	811.500	0.818870	0.409435	+	0.912339i	$0.365726\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 250.483i	− 0.251742i
$996$	0	0
$997$	390.554	0.391729	0.195864	−	0.980631i	$-0.437249\pi$
$997$	390.554	0.391729	0.195864	+	0.980631i	$0.437249\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.e.n.1025.6		8
3.2	odd	2	inner	2304.3.e.n.1025.4		8
4.3	odd	2		2304.3.e.o.1025.6		8
8.3	odd	2		2304.3.e.o.1025.3		8
8.5	even	2	inner	2304.3.e.n.1025.3		8
12.11	even	2		2304.3.e.o.1025.4		8
16.3	odd	4		288.3.h.a.17.6		8
16.5	even	4		72.3.h.a.53.8	yes	8
16.11	odd	4		288.3.h.a.17.3		8
16.13	even	4		72.3.h.a.53.2	yes	8
24.5	odd	2	inner	2304.3.e.n.1025.5		8
24.11	even	2		2304.3.e.o.1025.5		8
48.5	odd	4		72.3.h.a.53.1	✓	8
48.11	even	4		288.3.h.a.17.5		8
48.29	odd	4		72.3.h.a.53.7	yes	8
48.35	even	4		288.3.h.a.17.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.h.a.53.1	✓	8	48.5	odd	4
72.3.h.a.53.2	yes	8	16.13	even	4
72.3.h.a.53.7	yes	8	48.29	odd	4
72.3.h.a.53.8	yes	8	16.5	even	4
288.3.h.a.17.3		8	16.11	odd	4
288.3.h.a.17.4		8	48.35	even	4
288.3.h.a.17.5		8	48.11	even	4
288.3.h.a.17.6		8	16.3	odd	4
2304.3.e.n.1025.3		8	8.5	even	2	inner
2304.3.e.n.1025.4		8	3.2	odd	2	inner
2304.3.e.n.1025.5		8	24.5	odd	2	inner
2304.3.e.n.1025.6		8	1.1	even	1	trivial
2304.3.e.o.1025.3		8	8.3	odd	2
2304.3.e.o.1025.4		8	12.11	even	2
2304.3.e.o.1025.5		8	24.11	even	2
2304.3.e.o.1025.6		8	4.3	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.07498i q^{5} -7.21110 q^{7} +O(q^{10})\)	\(q+1.07498i q^{5} -7.21110 q^{7} +16.3517i q^{11} +21.6045 q^{13} -18.9819i q^{17} -17.0438 q^{19} +1.11567i q^{23} +23.8444 q^{25} +29.4784i q^{29} +5.63331 q^{31} -7.75182i q^{35} -17.0438 q^{37} -27.4671i q^{41} +52.3306 q^{43} +64.5352i q^{47} +3.00000 q^{49} -35.9283i q^{53} -17.5778 q^{55} -56.8069i q^{59} -69.3743 q^{61} +23.2245i q^{65} +69.3743 q^{67} +98.4764i q^{71} -37.6888 q^{73} -117.914i q^{77} -127.322 q^{79} +7.75182i q^{83} +20.4052 q^{85} +76.1475i q^{89} -155.792 q^{91} -18.3218i q^{95} +4.84441 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 40 q^{25} - 128 q^{31} + 24 q^{49} - 256 q^{55} + 160 q^{73} - 384 q^{79} - 192 q^{97}+O(q^{100}) \) 8 * q - 40 * q^25 - 128 * q^31 + 24 * q^49 - 256 * q^55 + 160 * q^73 - 384 * q^79 - 192 * q^97

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