LMFDB - Embedded newform 2304.3.b.q.127.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(127,2304)]

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([1, 1, 0]))

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

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

chi := DirichletCharacter("2304.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.b (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:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.7
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.q.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.29253i q^{5} -2.75787i q^{7} +O(q^{10})$	$q+4.29253i q^{5} -2.75787i q^{7} -13.7980 q^{11} +14.5266i q^{13} -22.8673 q^{17} -16.0399 q^{19} -17.1117i q^{23} +6.57420 q^{25} +21.8667i q^{29} -38.6944i q^{31} +11.8383 q^{35} -66.4204i q^{37} +23.2392 q^{41} +47.9230 q^{43} +14.8512i q^{47} +41.3941 q^{49} +65.5589i q^{53} -59.2281i q^{55} +65.8428 q^{59} -40.1123i q^{61} -62.3559 q^{65} -74.8105 q^{67} -122.681i q^{71} +144.904 q^{73} +38.0530i q^{77} +128.657i q^{79} +22.0417 q^{83} -98.1584i q^{85} -122.075 q^{89} +40.0626 q^{91} -68.8516i q^{95} -88.3072 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{11} - 16 q^{17} + 96 q^{19} + 8 q^{25} + 96 q^{35} + 80 q^{41} + 224 q^{43} - 88 q^{49} + 512 q^{59} + 160 q^{65} + 16 q^{73} + 544 q^{83} - 240 q^{89} - 32 q^{91} + 400 q^{97}+O(q^{100}) $ 8 * q - 32 * q^11 - 16 * q^17 + 96 * q^19 + 8 * q^25 + 96 * q^35 + 80 * q^41 + 224 * q^43 - 88 * q^49 + 512 * q^59 + 160 * q^65 + 16 * q^73 + 544 * q^83 - 240 * q^89 - 32 * q^91 + 400 * 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
$5$	4.29253i	0.858506i	0.903184	+	0.429253i	$0.141223\pi$
$5$	4.29253i	0.858506i	−0.903184	+	0.429253i	$0.858777\pi$
$6$	0	0
$7$	− 2.75787i	− 0.393982i	−0.980405	−	0.196991i	$-0.936883\pi$
$7$	− 2.75787i	− 0.393982i	0.980405	−	0.196991i	$-0.0631170\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−13.7980	−1.25436	−0.627180	−	0.778874i	$-0.715791\pi$
$11$	−13.7980	−1.25436	−0.627180	+	0.778874i	$0.715791\pi$
$12$	0	0
$13$	14.5266i	1.11743i	0.829359	+	0.558716i	$0.188706\pi$
$13$	14.5266i	1.11743i	−0.829359	+	0.558716i	$0.811294\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−22.8673	−1.34513	−0.672567	−	0.740037i	$-0.734808\pi$
$17$	−22.8673	−1.34513	−0.672567	+	0.740037i	$0.734808\pi$
$18$	0	0
$19$	−16.0399	−0.844204	−0.422102	−	0.906548i	$-0.638708\pi$
$19$	−16.0399	−0.844204	−0.422102	+	0.906548i	$0.638708\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 17.1117i	− 0.743986i	−0.928236	−	0.371993i	$-0.878675\pi$
$23$	− 17.1117i	− 0.743986i	0.928236	−	0.371993i	$-0.121325\pi$
$24$	0	0
$25$	6.57420	0.262968
$26$	0	0
$27$	0	0
$28$	0	0
$29$	21.8667i	0.754025i	0.926208	+	0.377013i	$0.123049\pi$
$29$	21.8667i	0.754025i	−0.926208	+	0.377013i	$0.876951\pi$
$30$	0	0
$31$	− 38.6944i	− 1.24821i	−0.781341	−	0.624104i	$-0.785464\pi$
$31$	− 38.6944i	− 1.24821i	0.781341	−	0.624104i	$-0.214536\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11.8383	0.338236
$36$	0	0
$37$	− 66.4204i	− 1.79515i	−0.440866	−	0.897573i	$-0.645329\pi$
$37$	− 66.4204i	− 1.79515i	0.440866	−	0.897573i	$-0.354671\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	23.2392	0.566809	0.283405	−	0.959000i	$-0.408536\pi$
$41$	23.2392	0.566809	0.283405	+	0.959000i	$0.408536\pi$
$42$	0	0
$43$	47.9230	1.11449	0.557244	−	0.830349i	$-0.311859\pi$
$43$	47.9230	1.11449	0.557244	+	0.830349i	$0.311859\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	14.8512i	0.315983i	0.987441	+	0.157991i	$0.0505018\pi$
$47$	14.8512i	0.315983i	−0.987441	+	0.157991i	$0.949498\pi$
$48$	0	0
$49$	41.3941	0.844778
$50$	0	0
$51$	0	0
$52$	0	0
$53$	65.5589i	1.23696i	0.785800	+	0.618480i	$0.212251\pi$
$53$	65.5589i	1.23696i	−0.785800	+	0.618480i	$0.787749\pi$
$54$	0	0
$55$	− 59.2281i	− 1.07688i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	65.8428	1.11598	0.557990	−	0.829848i	$-0.311573\pi$
$59$	65.8428	1.11598	0.557990	+	0.829848i	$0.311573\pi$
$60$	0	0
$61$	− 40.1123i	− 0.657579i	−0.944403	−	0.328790i	$-0.893359\pi$
$61$	− 40.1123i	− 0.657579i	0.944403	−	0.328790i	$-0.106641\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−62.3559	−0.959321
$66$	0	0
$67$	−74.8105	−1.11657	−0.558287	−	0.829648i	$-0.688541\pi$
$67$	−74.8105	−1.11657	−0.558287	+	0.829648i	$0.688541\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 122.681i	− 1.72790i	−0.503578	−	0.863950i	$-0.667983\pi$
$71$	− 122.681i	− 1.72790i	0.503578	−	0.863950i	$-0.332017\pi$
$72$	0	0
$73$	144.904	1.98498	0.992492	−	0.122312i	$-0.0390307\pi$
$73$	144.904	1.98498	0.992492	+	0.122312i	$0.0390307\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	38.0530i	0.494195i
$78$	0	0
$79$	128.657i	1.62857i	0.580467	+	0.814284i	$0.302870\pi$
$79$	128.657i	1.62857i	−0.580467	+	0.814284i	$0.697130\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	22.0417	0.265563	0.132781	−	0.991145i	$-0.457609\pi$
$83$	22.0417	0.265563	0.132781	+	0.991145i	$0.457609\pi$
$84$	0	0
$85$	− 98.1584i	− 1.15480i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−122.075	−1.37163	−0.685813	−	0.727778i	$-0.740553\pi$
$89$	−122.075	−1.37163	−0.685813	+	0.727778i	$0.740553\pi$
$90$	0	0
$91$	40.0626	0.440248
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 68.8516i	− 0.724754i
$96$	0	0
$97$	−88.3072	−0.910384	−0.455192	−	0.890393i	$-0.650429\pi$
$97$	−88.3072	−0.910384	−0.455192	+	0.890393i	$0.650429\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 104.345i	− 1.03312i	−0.856251	−	0.516560i	$-0.827212\pi$
$101$	− 104.345i	− 1.03312i	0.856251	−	0.516560i	$-0.172788\pi$
$102$	0	0
$103$	− 69.0609i	− 0.670494i	−0.942130	−	0.335247i	$-0.891180\pi$
$103$	− 69.0609i	− 0.670494i	0.942130	−	0.335247i	$-0.108820\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	149.429	1.39653	0.698265	−	0.715839i	$-0.253956\pi$
$107$	149.429	1.39653	0.698265	+	0.715839i	$0.253956\pi$
$108$	0	0
$109$	54.3341i	0.498478i	0.968442	+	0.249239i	$0.0801805\pi$
$109$	54.3341i	0.498478i	−0.968442	+	0.249239i	$0.919820\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.32846	−0.0560041	−0.0280021	−	0.999608i	$-0.508914\pi$
$113$	−6.32846	−0.0560041	−0.0280021	+	0.999608i	$0.508914\pi$
$114$	0	0
$115$	73.4523	0.638716
$116$	0	0
$117$	0	0
$118$	0	0
$119$	63.0651i	0.529958i
$120$	0	0
$121$	69.3837	0.573419
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.533i	1.08427i
$126$	0	0
$127$	16.5228i	0.130101i	0.997882	+	0.0650504i	$0.0207208\pi$
$127$	16.5228i	0.130101i	−0.997882	+	0.0650504i	$0.979279\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−50.9799	−0.389159	−0.194580	−	0.980887i	$-0.562334\pi$
$131$	−50.9799	−0.389159	−0.194580	+	0.980887i	$0.562334\pi$
$132$	0	0
$133$	44.2360i	0.332601i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0157	0.0731070	0.0365535	−	0.999332i	$-0.488362\pi$
$137$	10.0157	0.0731070	0.0365535	+	0.999332i	$0.488362\pi$
$138$	0	0
$139$	65.7088	0.472725	0.236363	−	0.971665i	$-0.424045\pi$
$139$	65.7088	0.472725	0.236363	+	0.971665i	$0.424045\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 200.438i	− 1.40166i
$144$	0	0
$145$	−93.8635	−0.647335
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 94.7716i	− 0.636051i	−0.948082	−	0.318026i	$-0.896980\pi$
$149$	− 94.7716i	− 0.636051i	0.948082	−	0.318026i	$-0.103020\pi$
$150$	0	0
$151$	− 269.769i	− 1.78655i	−0.449508	−	0.893276i	$-0.648401\pi$
$151$	− 269.769i	− 1.78655i	0.449508	−	0.893276i	$-0.351599\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	166.097	1.07159
$156$	0	0
$157$	31.5058i	0.200674i	0.994954	+	0.100337i	$0.0319921\pi$
$157$	31.5058i	0.200674i	−0.994954	+	0.100337i	$0.968008\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−47.1918	−0.293117
$162$	0	0
$163$	201.159	1.23410	0.617051	−	0.786923i	$-0.288327\pi$
$163$	201.159	1.23410	0.617051	+	0.786923i	$0.288327\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 266.393i	− 1.59517i	−0.603208	−	0.797584i	$-0.706111\pi$
$167$	− 266.393i	− 1.59517i	0.603208	−	0.797584i	$-0.293889\pi$
$168$	0	0
$169$	−42.0224	−0.248653
$170$	0	0
$171$	0	0
$172$	0	0
$173$	43.8456i	0.253443i	0.991938	+	0.126721i	$0.0404454\pi$
$173$	43.8456i	0.253443i	−0.991938	+	0.126721i	$0.959555\pi$
$174$	0	0
$175$	− 18.1308i	− 0.103605i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	275.778	1.54066	0.770330	−	0.637646i	$-0.220092\pi$
$179$	275.778	1.54066	0.770330	+	0.637646i	$0.220092\pi$
$180$	0	0
$181$	79.0033i	0.436482i	0.975895	+	0.218241i	$0.0700319\pi$
$181$	79.0033i	0.436482i	−0.975895	+	0.218241i	$0.929968\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	285.111	1.54114
$186$	0	0
$187$	315.522	1.68728
$188$	0	0
$189$	0	0
$190$	0	0
$191$	142.409i	0.745598i	0.927912	+	0.372799i	$0.121602\pi$
$191$	142.409i	0.745598i	−0.927912	+	0.372799i	$0.878398\pi$
$192$	0	0
$193$	−277.818	−1.43947	−0.719736	−	0.694248i	$-0.755737\pi$
$193$	−277.818	−1.43947	−0.719736	+	0.694248i	$0.755737\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.40848i	0.0173019i	0.999963	+	0.00865095i	$0.00275372\pi$
$197$	3.40848i	0.0173019i	−0.999963	+	0.00865095i	$0.997246\pi$
$198$	0	0
$199$	314.323i	1.57951i	0.613421	+	0.789756i	$0.289793\pi$
$199$	314.323i	1.57951i	−0.613421	+	0.789756i	$0.710207\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	60.3057	0.297072
$204$	0	0
$205$	99.7548i	0.486609i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	221.317	1.05894
$210$	0	0
$211$	1.54933	0.00734281	0.00367140	−	0.999993i	$-0.498831\pi$
$211$	1.54933	0.00734281	0.00367140	+	0.999993i	$0.498831\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	205.711i	0.956794i
$216$	0	0
$217$	−106.714	−0.491772
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 332.184i	− 1.50309i
$222$	0	0
$223$	− 148.964i	− 0.668001i	−0.942573	−	0.334000i	$-0.891601\pi$
$223$	− 148.964i	− 0.668001i	0.942573	−	0.334000i	$-0.108399\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	237.803	1.04759	0.523796	−	0.851844i	$-0.324515\pi$
$227$	237.803	1.04759	0.523796	+	0.851844i	$0.324515\pi$
$228$	0	0
$229$	− 70.0590i	− 0.305935i	−0.988231	−	0.152967i	$-0.951117\pi$
$229$	− 70.0590i	− 0.305935i	0.988231	−	0.152967i	$-0.0488829\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.18107	0.0308200	0.0154100	−	0.999881i	$-0.495095\pi$
$233$	7.18107	0.0308200	0.0154100	+	0.999881i	$0.495095\pi$
$234$	0	0
$235$	−63.7491	−0.271273
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 216.620i	− 0.906359i	−0.891419	−	0.453180i	$-0.850290\pi$
$239$	− 216.620i	− 0.906359i	0.891419	−	0.453180i	$-0.149710\pi$
$240$	0	0
$241$	385.001	1.59751	0.798757	−	0.601653i	$-0.205491\pi$
$241$	385.001	1.59751	0.798757	+	0.601653i	$0.205491\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	177.685i	0.725247i
$246$	0	0
$247$	− 233.005i	− 0.943340i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−85.4882	−0.340590	−0.170295	−	0.985393i	$-0.554472\pi$
$251$	−85.4882	−0.340590	−0.170295	+	0.985393i	$0.554472\pi$
$252$	0	0
$253$	236.106i	0.933226i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	54.2981	0.211277	0.105638	−	0.994405i	$-0.466311\pi$
$257$	54.2981	0.211277	0.105638	+	0.994405i	$0.466311\pi$
$258$	0	0
$259$	−183.179	−0.707255
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 501.827i	− 1.90809i	−0.299670	−	0.954043i	$-0.596877\pi$
$263$	− 501.827i	− 1.90809i	0.299670	−	0.954043i	$-0.403123\pi$
$264$	0	0
$265$	−281.413	−1.06194
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 284.911i	− 1.05915i	−0.848264	−	0.529574i	$-0.822352\pi$
$269$	− 284.911i	− 1.05915i	0.848264	−	0.529574i	$-0.177648\pi$
$270$	0	0
$271$	− 241.190i	− 0.889998i	−0.895531	−	0.444999i	$-0.853204\pi$
$271$	− 241.190i	− 0.889998i	0.895531	−	0.444999i	$-0.146796\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−90.7105	−0.329856
$276$	0	0
$277$	242.877i	0.876813i	0.898777	+	0.438407i	$0.144457\pi$
$277$	242.877i	0.876813i	−0.898777	+	0.438407i	$0.855543\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	105.341	0.374880	0.187440	−	0.982276i	$-0.439981\pi$
$281$	105.341	0.374880	0.187440	+	0.982276i	$0.439981\pi$
$282$	0	0
$283$	−223.867	−0.791049	−0.395525	−	0.918455i	$-0.629437\pi$
$283$	−223.867	−0.791049	−0.395525	+	0.918455i	$0.629437\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 64.0907i	− 0.223313i
$288$	0	0
$289$	233.912	0.809384
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 421.057i	− 1.43705i	−0.695499	−	0.718527i	$-0.744817\pi$
$293$	− 421.057i	− 1.43705i	0.695499	−	0.718527i	$-0.255183\pi$
$294$	0	0
$295$	282.632i	0.958075i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	248.575	0.831353
$300$	0	0
$301$	− 132.166i	− 0.439088i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	172.183	0.564536
$306$	0	0
$307$	−122.865	−0.400211	−0.200106	−	0.979774i	$-0.564129\pi$
$307$	−122.865	−0.400211	−0.200106	+	0.979774i	$0.564129\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 437.535i	− 1.40686i	−0.710762	−	0.703432i	$-0.751650\pi$
$311$	− 437.535i	− 1.40686i	0.710762	−	0.703432i	$-0.248350\pi$
$312$	0	0
$313$	−375.413	−1.19940	−0.599702	−	0.800223i	$-0.704714\pi$
$313$	−375.413	−1.19940	−0.599702	+	0.800223i	$0.704714\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 388.867i	− 1.22671i	−0.789808	−	0.613355i	$-0.789820\pi$
$317$	− 388.867i	− 1.22671i	0.789808	−	0.613355i	$-0.210180\pi$
$318$	0	0
$319$	− 301.716i	− 0.945819i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	366.788	1.13557
$324$	0	0
$325$	95.5008i	0.293849i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	40.9577	0.124491
$330$	0	0
$331$	15.4690	0.0467341	0.0233670	−	0.999727i	$-0.492561\pi$
$331$	15.4690	0.0467341	0.0233670	+	0.999727i	$0.492561\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 321.126i	− 0.958585i
$336$	0	0
$337$	−88.0105	−0.261159	−0.130579	−	0.991438i	$-0.541684\pi$
$337$	−88.0105	−0.261159	−0.130579	+	0.991438i	$0.541684\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	533.904i	1.56570i
$342$	0	0
$343$	− 249.296i	− 0.726810i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−183.589	−0.529076	−0.264538	−	0.964375i	$-0.585219\pi$
$347$	−183.589	−0.529076	−0.264538	+	0.964375i	$0.585219\pi$
$348$	0	0
$349$	− 242.556i	− 0.695003i	−0.937679	−	0.347501i	$-0.887030\pi$
$349$	− 242.556i	− 0.695003i	0.937679	−	0.347501i	$-0.112970\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.2850	−0.0291360	−0.0145680	−	0.999894i	$-0.504637\pi$
$353$	−10.2850	−0.0291360	−0.0145680	+	0.999894i	$0.504637\pi$
$354$	0	0
$355$	526.611	1.48341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	137.976i	0.384334i	0.981362	+	0.192167i	$0.0615515\pi$
$359$	137.976i	0.384334i	−0.981362	+	0.192167i	$0.938448\pi$
$360$	0	0
$361$	−103.723	−0.287320
$362$	0	0
$363$	0	0
$364$	0	0
$365$	622.004i	1.70412i
$366$	0	0
$367$	− 400.180i	− 1.09041i	−0.838303	−	0.545205i	$-0.816452\pi$
$367$	− 400.180i	− 1.09041i	0.838303	−	0.545205i	$-0.183548\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	180.803	0.487340
$372$	0	0
$373$	644.881i	1.72890i	0.502717	+	0.864451i	$0.332334\pi$
$373$	644.881i	1.72890i	−0.502717	+	0.864451i	$0.667666\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−317.649	−0.842571
$378$	0	0
$379$	−485.900	−1.28206	−0.641029	−	0.767517i	$-0.721492\pi$
$379$	−485.900	−1.28206	−0.641029	+	0.767517i	$0.721492\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 168.828i	− 0.440804i	−0.975409	−	0.220402i	$-0.929263\pi$
$383$	− 168.828i	− 0.440804i	0.975409	−	0.220402i	$-0.0707370\pi$
$384$	0	0
$385$	−163.344	−0.424270
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 192.592i	− 0.495095i	−0.968876	−	0.247548i	$-0.920375\pi$
$389$	− 192.592i	− 0.495095i	0.968876	−	0.247548i	$-0.0796247\pi$
$390$	0	0
$391$	391.297i	1.00076i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−552.263	−1.39813
$396$	0	0
$397$	− 62.0483i	− 0.156293i	−0.996942	−	0.0781465i	$-0.975100\pi$
$397$	− 62.0483i	− 0.156293i	0.996942	−	0.0781465i	$-0.0249002\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−353.066	−0.880464	−0.440232	−	0.897884i	$-0.645104\pi$
$401$	−353.066	−0.880464	−0.440232	+	0.897884i	$0.645104\pi$
$402$	0	0
$403$	562.099	1.39479
$404$	0	0
$405$	0	0
$406$	0	0
$407$	916.466i	2.25176i
$408$	0	0
$409$	62.9725	0.153967	0.0769835	−	0.997032i	$-0.475471\pi$
$409$	62.9725	0.153967	0.0769835	+	0.997032i	$0.475471\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 181.586i	− 0.439676i
$414$	0	0
$415$	94.6146i	0.227987i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−398.741	−0.951650	−0.475825	−	0.879540i	$-0.657850\pi$
$419$	−398.741	−0.951650	−0.475825	+	0.879540i	$0.657850\pi$
$420$	0	0
$421$	− 533.059i	− 1.26617i	−0.774081	−	0.633086i	$-0.781788\pi$
$421$	− 533.059i	− 1.26617i	0.774081	−	0.633086i	$-0.218212\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−150.334	−0.353727
$426$	0	0
$427$	−110.625	−0.259075
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 284.282i	− 0.659587i	−0.944053	−	0.329793i	$-0.893021\pi$
$431$	− 284.282i	− 0.659587i	0.944053	−	0.329793i	$-0.106979\pi$
$432$	0	0
$433$	−304.599	−0.703462	−0.351731	−	0.936101i	$-0.614407\pi$
$433$	−304.599	−0.703462	−0.351731	+	0.936101i	$0.614407\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	274.469i	0.628075i
$438$	0	0
$439$	− 204.615i	− 0.466094i	−0.972465	−	0.233047i	$-0.925130\pi$
$439$	− 204.615i	− 0.466094i	0.972465	−	0.233047i	$-0.0748696\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	625.828	1.41270	0.706352	−	0.707861i	$-0.250340\pi$
$443$	625.828	1.41270	0.706352	+	0.707861i	$0.250340\pi$
$444$	0	0
$445$	− 524.009i	− 1.17755i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	557.532	1.24172	0.620860	−	0.783921i	$-0.286783\pi$
$449$	557.532	1.24172	0.620860	+	0.783921i	$0.286783\pi$
$450$	0	0
$451$	−320.653	−0.710983
$452$	0	0
$453$	0	0
$454$	0	0
$455$	171.970i	0.377955i
$456$	0	0
$457$	312.142	0.683024	0.341512	−	0.939877i	$-0.389061\pi$
$457$	312.142	0.683024	0.341512	+	0.939877i	$0.389061\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	603.187i	1.30843i	0.756308	+	0.654216i	$0.227001\pi$
$461$	603.187i	1.30843i	−0.756308	+	0.654216i	$0.772999\pi$
$462$	0	0
$463$	707.866i	1.52887i	0.644702	+	0.764434i	$0.276981\pi$
$463$	707.866i	1.52887i	−0.644702	+	0.764434i	$0.723019\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−195.220	−0.418031	−0.209015	−	0.977912i	$-0.567026\pi$
$467$	−195.220	−0.418031	−0.209015	+	0.977912i	$0.567026\pi$
$468$	0	0
$469$	206.318i	0.439910i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−661.239	−1.39797
$474$	0	0
$475$	−105.449	−0.221998
$476$	0	0
$477$	0	0
$478$	0	0
$479$	233.729i	0.487952i	0.969781	+	0.243976i	$0.0784517\pi$
$479$	233.729i	0.487952i	−0.969781	+	0.243976i	$0.921548\pi$
$480$	0	0
$481$	964.863	2.00595
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 379.061i	− 0.781569i
$486$	0	0
$487$	224.058i	0.460078i	0.973181	+	0.230039i	$0.0738854\pi$
$487$	224.058i	0.460078i	−0.973181	+	0.230039i	$0.926115\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	194.772	0.396684	0.198342	−	0.980133i	$-0.436444\pi$
$491$	194.772	0.396684	0.198342	+	0.980133i	$0.436444\pi$
$492$	0	0
$493$	− 500.032i	− 1.01426i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−338.339	−0.680762
$498$	0	0
$499$	99.7462	0.199892	0.0999461	−	0.994993i	$-0.468133\pi$
$499$	99.7462	0.199892	0.0999461	+	0.994993i	$0.468133\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	772.172i	1.53513i	0.640969	+	0.767567i	$0.278533\pi$
$503$	772.172i	1.53513i	−0.640969	+	0.767567i	$0.721467\pi$
$504$	0	0
$505$	447.904	0.886939
$506$	0	0
$507$	0	0
$508$	0	0
$509$	292.306i	0.574276i	0.957889	+	0.287138i	$0.0927038\pi$
$509$	292.306i	0.574276i	−0.957889	+	0.287138i	$0.907296\pi$
$510$	0	0
$511$	− 399.627i	− 0.782048i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	296.446	0.575623
$516$	0	0
$517$	− 204.916i	− 0.396356i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−412.035	−0.790853	−0.395427	−	0.918498i	$-0.629403\pi$
$521$	−412.035	−0.790853	−0.395427	+	0.918498i	$0.629403\pi$
$522$	0	0
$523$	124.693	0.238420	0.119210	−	0.992869i	$-0.461964\pi$
$523$	124.693	0.238420	0.119210	+	0.992869i	$0.461964\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	884.836i	1.67901i
$528$	0	0
$529$	236.191	0.446486
$530$	0	0
$531$	0	0
$532$	0	0
$533$	337.586i	0.633370i
$534$	0	0
$535$	641.427i	1.19893i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−571.154	−1.05966
$540$	0	0
$541$	− 572.988i	− 1.05913i	−0.848270	−	0.529564i	$-0.822356\pi$
$541$	− 572.988i	− 1.05913i	0.848270	−	0.529564i	$-0.177644\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−233.231	−0.427946
$546$	0	0
$547$	682.433	1.24759	0.623796	−	0.781587i	$-0.285590\pi$
$547$	682.433	1.24759	0.623796	+	0.781587i	$0.285590\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 350.739i	− 0.636551i
$552$	0	0
$553$	354.820	0.641627
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 856.125i	− 1.53703i	−0.639832	−	0.768515i	$-0.720996\pi$
$557$	− 856.125i	− 1.53703i	0.639832	−	0.768515i	$-0.279004\pi$
$558$	0	0
$559$	696.158i	1.24536i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−680.745	−1.20914	−0.604570	−	0.796552i	$-0.706655\pi$
$563$	−680.745	−1.20914	−0.604570	+	0.796552i	$0.706655\pi$
$564$	0	0
$565$	− 27.1651i	− 0.0480798i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	343.401	0.603516	0.301758	−	0.953384i	$-0.402426\pi$
$569$	343.401	0.603516	0.301758	+	0.953384i	$0.402426\pi$
$570$	0	0
$571$	329.137	0.576422	0.288211	−	0.957567i	$-0.406940\pi$
$571$	329.137	0.576422	0.288211	+	0.957567i	$0.406940\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 112.495i	− 0.195644i
$576$	0	0
$577$	734.738	1.27338	0.636688	−	0.771122i	$-0.280304\pi$
$577$	734.738	1.27338	0.636688	+	0.771122i	$0.280304\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 60.7883i	− 0.104627i
$582$	0	0
$583$	− 904.579i	− 1.55159i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−433.815	−0.739037	−0.369519	−	0.929223i	$-0.620477\pi$
$587$	−433.815	−0.739037	−0.369519	+	0.929223i	$0.620477\pi$
$588$	0	0
$589$	620.654i	1.05374i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	425.075	0.716822	0.358411	−	0.933564i	$-0.383319\pi$
$593$	425.075	0.716822	0.358411	+	0.933564i	$0.383319\pi$
$594$	0	0
$595$	−270.709	−0.454972
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1048.73i	− 1.75081i	−0.483394	−	0.875403i	$-0.660596\pi$
$599$	− 1048.73i	− 1.75081i	0.483394	−	0.875403i	$-0.339404\pi$
$600$	0	0
$601$	478.816	0.796699	0.398349	−	0.917234i	$-0.369583\pi$
$601$	478.816	0.796699	0.398349	+	0.917234i	$0.369583\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	297.831i	0.492283i
$606$	0	0
$607$	18.7009i	0.0308088i	0.999881	+	0.0154044i	$0.00490356\pi$
$607$	18.7009i	0.0308088i	−0.999881	+	0.0154044i	$0.995096\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−215.737	−0.353089
$612$	0	0
$613$	− 314.964i	− 0.513807i	−0.966437	−	0.256904i	$-0.917298\pi$
$613$	− 314.964i	− 0.513807i	0.966437	−	0.256904i	$-0.0827023\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	786.590	1.27486	0.637431	−	0.770507i	$-0.279997\pi$
$617$	786.590	1.27486	0.637431	+	0.770507i	$0.279997\pi$
$618$	0	0
$619$	−505.431	−0.816528	−0.408264	−	0.912864i	$-0.633866\pi$
$619$	−505.431	−0.816528	−0.408264	+	0.912864i	$0.633866\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	336.667i	0.540396i
$624$	0	0
$625$	−417.425	−0.667880
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1518.85i	2.41471i
$630$	0	0
$631$	682.834i	1.08215i	0.840976	+	0.541073i	$0.181982\pi$
$631$	682.834i	1.08215i	−0.840976	+	0.541073i	$0.818018\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−70.9246	−0.111692
$636$	0	0
$637$	601.316i	0.943982i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−816.719	−1.27413	−0.637066	−	0.770809i	$-0.719852\pi$
$641$	−816.719	−1.27413	−0.637066	+	0.770809i	$0.719852\pi$
$642$	0	0
$643$	−1164.05	−1.81034	−0.905172	−	0.425045i	$-0.860258\pi$
$643$	−1164.05	−1.81034	−0.905172	+	0.425045i	$0.860258\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	471.519i	0.728778i	0.931247	+	0.364389i	$0.118722\pi$
$647$	471.519i	0.728778i	−0.931247	+	0.364389i	$0.881278\pi$
$648$	0	0
$649$	−908.496	−1.39984
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1187.25i	1.81814i	0.416642	+	0.909071i	$0.363207\pi$
$653$	1187.25i	1.81814i	−0.416642	+	0.909071i	$0.636793\pi$
$654$	0	0
$655$	− 218.833i	− 0.334096i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−245.640	−0.372747	−0.186373	−	0.982479i	$-0.559673\pi$
$659$	−245.640	−0.372747	−0.186373	+	0.982479i	$0.559673\pi$
$660$	0	0
$661$	− 1244.70i	− 1.88305i	−0.336936	−	0.941527i	$-0.609391\pi$
$661$	− 1244.70i	− 1.88305i	0.336936	−	0.941527i	$-0.390609\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−189.884	−0.285540
$666$	0	0
$667$	374.176	0.560984
$668$	0	0
$669$	0	0
$670$	0	0
$671$	553.468i	0.824841i
$672$	0	0
$673$	−852.278	−1.26639	−0.633193	−	0.773994i	$-0.718256\pi$
$673$	−852.278	−1.26639	−0.633193	+	0.773994i	$0.718256\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 555.765i	− 0.820923i	−0.911878	−	0.410461i	$-0.865368\pi$
$677$	− 555.765i	− 0.820923i	0.911878	−	0.410461i	$-0.134632\pi$
$678$	0	0
$679$	243.540i	0.358675i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	335.241	0.490836	0.245418	−	0.969417i	$-0.421075\pi$
$683$	335.241	0.490836	0.245418	+	0.969417i	$0.421075\pi$
$684$	0	0
$685$	42.9925i	0.0627628i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−952.349	−1.38222
$690$	0	0
$691$	−1274.20	−1.84400	−0.921999	−	0.387192i	$-0.873445\pi$
$691$	−1274.20	−1.84400	−0.921999	+	0.387192i	$0.873445\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	282.057i	0.405837i
$696$	0	0
$697$	−531.416	−0.762434
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 676.056i	− 0.964416i	−0.876057	−	0.482208i	$-0.839835\pi$
$701$	− 676.056i	− 0.964416i	0.876057	−	0.482208i	$-0.160165\pi$
$702$	0	0
$703$	1065.37i	1.51547i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−287.771	−0.407031
$708$	0	0
$709$	62.3418i	0.0879292i	0.999033	+	0.0439646i	$0.0139989\pi$
$709$	62.3418i	0.0879292i	−0.999033	+	0.0439646i	$0.986001\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−662.127	−0.928649
$714$	0	0
$715$	860.384	1.20333
$716$	0	0
$717$	0	0
$718$	0	0
$719$	322.341i	0.448318i	0.974553	+	0.224159i	$0.0719636\pi$
$719$	322.341i	0.448318i	−0.974553	+	0.224159i	$0.928036\pi$
$720$	0	0
$721$	−190.461	−0.264163
$722$	0	0
$723$	0	0
$724$	0	0
$725$	143.756i	0.198284i
$726$	0	0
$727$	368.070i	0.506287i	0.967429	+	0.253143i	$0.0814644\pi$
$727$	368.070i	0.506287i	−0.967429	+	0.253143i	$0.918536\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1095.87	−1.49913
$732$	0	0
$733$	430.587i	0.587431i	0.955893	+	0.293715i	$0.0948918\pi$
$733$	430.587i	0.587431i	−0.955893	+	0.293715i	$0.905108\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1032.23	1.40059
$738$	0	0
$739$	−818.050	−1.10697	−0.553485	−	0.832859i	$-0.686702\pi$
$739$	−818.050	−1.10697	−0.553485	+	0.832859i	$0.686702\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	685.438i	0.922528i	0.887263	+	0.461264i	$0.152604\pi$
$743$	685.438i	0.922528i	−0.887263	+	0.461264i	$0.847396\pi$
$744$	0	0
$745$	406.810	0.546053
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 412.106i	− 0.550208i
$750$	0	0
$751$	104.336i	0.138929i	0.997584	+	0.0694647i	$0.0221291\pi$
$751$	104.336i	0.138929i	−0.997584	+	0.0694647i	$0.977871\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1157.99	1.53377
$756$	0	0
$757$	539.949i	0.713274i	0.934243	+	0.356637i	$0.116077\pi$
$757$	539.949i	0.713274i	−0.934243	+	0.356637i	$0.883923\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1192.21	1.56663	0.783317	−	0.621623i	$-0.213526\pi$
$761$	1192.21	1.56663	0.783317	+	0.621623i	$0.213526\pi$
$762$	0	0
$763$	149.847	0.196391
$764$	0	0
$765$	0	0
$766$	0	0
$767$	956.473i	1.24703i
$768$	0	0
$769$	−321.506	−0.418083	−0.209042	−	0.977907i	$-0.567034\pi$
$769$	−321.506	−0.418083	−0.209042	+	0.977907i	$0.567034\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1453.29i	− 1.88006i	−0.341094	−	0.940029i	$-0.610798\pi$
$773$	− 1453.29i	− 1.88006i	0.341094	−	0.940029i	$-0.389202\pi$
$774$	0	0
$775$	− 254.385i	− 0.328239i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−372.753	−0.478502
$780$	0	0
$781$	1692.75i	2.16741i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−135.240	−0.172280
$786$	0	0
$787$	−231.223	−0.293802	−0.146901	−	0.989151i	$-0.546930\pi$
$787$	−231.223	−0.293802	−0.146901	+	0.989151i	$0.546930\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17.4531i	0.0220646i
$792$	0	0
$793$	582.696	0.734800
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 51.5068i	− 0.0646259i	−0.999478	−	0.0323129i	$-0.989713\pi$
$797$	− 51.5068i	− 0.0646259i	0.999478	−	0.0323129i	$-0.0102873\pi$
$798$	0	0
$799$	− 339.606i	− 0.425039i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1999.38	−2.48988
$804$	0	0
$805$	− 202.572i	− 0.251643i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	580.914	0.718065	0.359032	−	0.933325i	$-0.383107\pi$
$809$	580.914	0.718065	0.359032	+	0.933325i	$0.383107\pi$
$810$	0	0
$811$	−1351.98	−1.66706	−0.833529	−	0.552475i	$-0.813683\pi$
$811$	−1351.98	−1.66706	−0.833529	+	0.552475i	$0.813683\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	863.480i	1.05948i
$816$	0	0
$817$	−768.678	−0.940855
$818$	0	0
$819$	0	0
$820$	0	0
$821$	179.511i	0.218650i	0.994006	+	0.109325i	$0.0348688\pi$
$821$	179.511i	0.218650i	−0.994006	+	0.109325i	$0.965131\pi$
$822$	0	0
$823$	− 665.446i	− 0.808561i	−0.914635	−	0.404281i	$-0.867522\pi$
$823$	− 665.446i	− 0.808561i	0.914635	−	0.404281i	$-0.132478\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−901.166	−1.08968	−0.544840	−	0.838540i	$-0.683410\pi$
$827$	−901.166	−1.08968	−0.544840	+	0.838540i	$0.683410\pi$
$828$	0	0
$829$	− 183.151i	− 0.220930i	−0.993880	−	0.110465i	$-0.964766\pi$
$829$	− 183.151i	− 0.220930i	0.993880	−	0.110465i	$-0.0352340\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−946.571	−1.13634
$834$	0	0
$835$	1143.50	1.36946
$836$	0	0
$837$	0	0
$838$	0	0
$839$	846.082i	1.00844i	0.863575	+	0.504220i	$0.168220\pi$
$839$	846.082i	1.00844i	−0.863575	+	0.504220i	$0.831780\pi$
$840$	0	0
$841$	362.846	0.431446
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 180.382i	− 0.213470i
$846$	0	0
$847$	− 191.351i	− 0.225917i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1136.56	−1.33556
$852$	0	0
$853$	540.635i	0.633804i	0.948458	+	0.316902i	$0.102643\pi$
$853$	540.635i	0.633804i	−0.948458	+	0.316902i	$0.897357\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1333.41	1.55590	0.777952	−	0.628323i	$-0.216258\pi$
$857$	1333.41	1.55590	0.777952	+	0.628323i	$0.216258\pi$
$858$	0	0
$859$	−439.034	−0.511099	−0.255549	−	0.966796i	$-0.582256\pi$
$859$	−439.034	−0.511099	−0.255549	+	0.966796i	$0.582256\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1164.41i	− 1.34926i	−0.738155	−	0.674631i	$-0.764303\pi$
$863$	− 1164.41i	− 1.34926i	0.738155	−	0.674631i	$-0.235697\pi$
$864$	0	0
$865$	−188.208	−0.217582
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1775.20i	− 2.04281i
$870$	0	0
$871$	− 1086.74i	− 1.24769i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	373.783	0.427181
$876$	0	0
$877$	404.456i	0.461181i	0.973051	+	0.230591i	$0.0740658\pi$
$877$	404.456i	0.461181i	−0.973051	+	0.230591i	$0.925934\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	227.084	0.257757	0.128879	−	0.991660i	$-0.458862\pi$
$881$	227.084	0.257757	0.128879	+	0.991660i	$0.458862\pi$
$882$	0	0
$883$	−673.730	−0.763001	−0.381501	−	0.924369i	$-0.624593\pi$
$883$	−673.730	−0.763001	−0.381501	+	0.924369i	$0.624593\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1370.95i	− 1.54560i	−0.634648	−	0.772801i	$-0.718855\pi$
$887$	− 1370.95i	− 1.54560i	0.634648	−	0.772801i	$-0.281145\pi$
$888$	0	0
$889$	45.5678	0.0512574
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 238.211i	− 0.266754i
$894$	0	0
$895$	1183.79i	1.32267i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	846.121	0.941180
$900$	0	0
$901$	− 1499.15i	− 1.66388i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−339.124	−0.374723
$906$	0	0
$907$	1469.46	1.62013	0.810063	−	0.586342i	$-0.199433\pi$
$907$	1469.46	1.62013	0.810063	+	0.586342i	$0.199433\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	871.203i	0.956315i	0.878274	+	0.478157i	$0.158695\pi$
$911$	871.203i	0.956315i	−0.878274	+	0.478157i	$0.841305\pi$
$912$	0	0
$913$	−304.131	−0.333111
$914$	0	0
$915$	0	0
$916$	0	0
$917$	140.596i	0.153322i
$918$	0	0
$919$	256.361i	0.278957i	0.990225	+	0.139478i	$0.0445426\pi$
$919$	256.361i	0.278957i	−0.990225	+	0.139478i	$0.955457\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1782.14	1.93081
$924$	0	0
$925$	− 436.661i	− 0.472066i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1399.93	1.50692	0.753462	−	0.657491i	$-0.228382\pi$
$929$	1399.93	1.50692	0.753462	+	0.657491i	$0.228382\pi$
$930$	0	0
$931$	−663.956	−0.713165
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1354.39i	1.44854i
$936$	0	0
$937$	−365.610	−0.390192	−0.195096	−	0.980784i	$-0.562502\pi$
$937$	−365.610	−0.390192	−0.195096	+	0.980784i	$0.562502\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	407.498i	0.433048i	0.976277	+	0.216524i	$0.0694720\pi$
$941$	407.498i	0.433048i	−0.976277	+	0.216524i	$0.930528\pi$
$942$	0	0
$943$	− 397.661i	− 0.421698i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1150.64	−1.21503	−0.607516	−	0.794307i	$-0.707834\pi$
$947$	−1150.64	−1.21503	−0.607516	+	0.794307i	$0.707834\pi$
$948$	0	0
$949$	2104.96i	2.21808i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1341.53	−1.40769	−0.703844	−	0.710355i	$-0.748534\pi$
$953$	−1341.53	−1.40769	−0.703844	+	0.710355i	$0.748534\pi$
$954$	0	0
$955$	−611.295	−0.640100
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 27.6219i	− 0.0288029i
$960$	0	0
$961$	−536.260	−0.558023
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1192.54i	− 1.23579i
$966$	0	0
$967$	− 1227.25i	− 1.26914i	−0.772867	−	0.634568i	$-0.781178\pi$
$967$	− 1227.25i	− 1.26914i	0.772867	−	0.634568i	$-0.218822\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1864.31	−1.91999	−0.959994	−	0.280020i	$-0.909659\pi$
$971$	−1864.31	−1.91999	−0.959994	+	0.280020i	$0.909659\pi$
$972$	0	0
$973$	− 181.217i	− 0.186245i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1209.56	1.23803	0.619017	−	0.785378i	$-0.287531\pi$
$977$	1209.56	1.23803	0.619017	+	0.785378i	$0.287531\pi$
$978$	0	0
$979$	1684.38	1.72051
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1018.26i	− 1.03587i	−0.855420	−	0.517935i	$-0.826701\pi$
$983$	− 1018.26i	− 1.03587i	0.855420	−	0.517935i	$-0.173299\pi$
$984$	0	0
$985$	−14.6310	−0.0148538
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 820.042i	− 0.829163i
$990$	0	0
$991$	1627.52i	1.64230i	0.570712	+	0.821150i	$0.306667\pi$
$991$	1627.52i	1.64230i	−0.570712	+	0.821150i	$0.693333\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1349.24	−1.35602
$996$	0	0
$997$	− 540.192i	− 0.541817i	−0.962605	−	0.270909i	$-0.912676\pi$
$997$	− 540.192i	− 0.541817i	0.962605	−	0.270909i	$-0.0873241\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.q.127.7		8
3.2	odd	2		768.3.b.f.127.5		8
4.3	odd	2		2304.3.b.t.127.7		8
8.3	odd	2	inner	2304.3.b.q.127.2		8
8.5	even	2		2304.3.b.t.127.2		8
12.11	even	2		768.3.b.e.127.1		8
16.3	odd	4		1152.3.g.c.127.7		8
16.5	even	4		1152.3.g.f.127.2		8
16.11	odd	4		1152.3.g.f.127.1		8
16.13	even	4		1152.3.g.c.127.8		8
24.5	odd	2		768.3.b.e.127.4		8
24.11	even	2		768.3.b.f.127.8		8
48.5	odd	4		384.3.g.a.127.4	✓	8
48.11	even	4		384.3.g.a.127.8	yes	8
48.29	odd	4		384.3.g.b.127.5	yes	8
48.35	even	4		384.3.g.b.127.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.4	✓	8	48.5	odd	4
384.3.g.a.127.8	yes	8	48.11	even	4
384.3.g.b.127.1	yes	8	48.35	even	4
384.3.g.b.127.5	yes	8	48.29	odd	4
768.3.b.e.127.1		8	12.11	even	2
768.3.b.e.127.4		8	24.5	odd	2
768.3.b.f.127.5		8	3.2	odd	2
768.3.b.f.127.8		8	24.11	even	2
1152.3.g.c.127.7		8	16.3	odd	4
1152.3.g.c.127.8		8	16.13	even	4
1152.3.g.f.127.1		8	16.11	odd	4
1152.3.g.f.127.2		8	16.5	even	4
2304.3.b.q.127.2		8	8.3	odd	2	inner
2304.3.b.q.127.7		8	1.1	even	1	trivial
2304.3.b.t.127.2		8	8.5	even	2
2304.3.b.t.127.7		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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.29253i q^{5} -2.75787i q^{7} +O(q^{10})\)	\(q+4.29253i q^{5} -2.75787i q^{7} -13.7980 q^{11} +14.5266i q^{13} -22.8673 q^{17} -16.0399 q^{19} -17.1117i q^{23} +6.57420 q^{25} +21.8667i q^{29} -38.6944i q^{31} +11.8383 q^{35} -66.4204i q^{37} +23.2392 q^{41} +47.9230 q^{43} +14.8512i q^{47} +41.3941 q^{49} +65.5589i q^{53} -59.2281i q^{55} +65.8428 q^{59} -40.1123i q^{61} -62.3559 q^{65} -74.8105 q^{67} -122.681i q^{71} +144.904 q^{73} +38.0530i q^{77} +128.657i q^{79} +22.0417 q^{83} -98.1584i q^{85} -122.075 q^{89} +40.0626 q^{91} -68.8516i q^{95} -88.3072 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{11} - 16 q^{17} + 96 q^{19} + 8 q^{25} + 96 q^{35} + 80 q^{41} + 224 q^{43} - 88 q^{49} + 512 q^{59} + 160 q^{65} + 16 q^{73} + 544 q^{83} - 240 q^{89} - 32 q^{91} + 400 q^{97}+O(q^{100}) \) 8 * q - 32 * q^11 - 16 * q^17 + 96 * q^19 + 8 * q^25 + 96 * q^35 + 80 * q^41 + 224 * q^43 - 88 * q^49 + 512 * q^59 + 160 * q^65 + 16 * q^73 + 544 * q^83 - 240 * q^89 - 32 * q^91 + 400 * q^97

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