LMFDB - Embedded newform 2304.3.g.y.1279.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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, 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("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (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:	8.0.3317760000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 16 $ x^8 - 7*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.1
Root			$-0.178197 - 1.40294i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.y.1279.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.89898 q^{5} -7.74597i q^{7} +O(q^{10})$	$q-4.89898 q^{5} -7.74597i q^{7} -12.6491i q^{11} -15.4919 q^{13} -25.2982 q^{17} +8.00000i q^{19} -39.1918i q^{23} -1.00000 q^{25} +24.4949 q^{29} -7.74597i q^{31} +37.9473i q^{35} -46.4758 q^{37} +25.2982 q^{41} +40.0000i q^{43} -39.1918i q^{47} -11.0000 q^{49} -14.6969 q^{53} +61.9677i q^{55} +25.2982i q^{59} +15.4919 q^{61} +75.8947 q^{65} +80.0000i q^{67} +10.0000 q^{73} -97.9796 q^{77} +54.2218i q^{79} -139.140i q^{83} +123.935 q^{85} -50.5964 q^{89} +120.000i q^{91} -39.1918i q^{95} +50.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} - 88 q^{49} + 80 q^{73} + 400 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 - 88 * q^49 + 80 * q^73 + 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
$5$	−4.89898	−0.979796	−0.489898	−	0.871780i	$-0.662966\pi$
$5$	−4.89898	−0.979796	−0.489898	+	0.871780i	$0.662966\pi$
$6$	0	0
$7$	− 7.74597i	− 1.10657i	−0.832993	−	0.553283i	$-0.813375\pi$
$7$	− 7.74597i	− 1.10657i	0.832993	−	0.553283i	$-0.186625\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 12.6491i	− 1.14992i	−0.818182	−	0.574960i	$-0.805018\pi$
$11$	− 12.6491i	− 1.14992i	0.818182	−	0.574960i	$-0.194982\pi$
$12$	0	0
$13$	−15.4919	−1.19169	−0.595844	−	0.803101i	$-0.703182\pi$
$13$	−15.4919	−1.19169	−0.595844	+	0.803101i	$0.703182\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−25.2982	−1.48813	−0.744065	−	0.668107i	$-0.767105\pi$
$17$	−25.2982	−1.48813	−0.744065	+	0.668107i	$0.767105\pi$
$18$	0	0
$19$	8.00000i	0.421053i	0.977588	+	0.210526i	$0.0675178\pi$
$19$	8.00000i	0.421053i	−0.977588	+	0.210526i	$0.932482\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 39.1918i	− 1.70399i	−0.523548	−	0.851996i	$-0.675392\pi$
$23$	− 39.1918i	− 1.70399i	0.523548	−	0.851996i	$-0.324608\pi$
$24$	0	0
$25$	−1.00000	−0.0400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	24.4949	0.844652	0.422326	−	0.906444i	$-0.361214\pi$
$29$	24.4949	0.844652	0.422326	+	0.906444i	$0.361214\pi$
$30$	0	0
$31$	− 7.74597i	− 0.249870i	−0.992165	−	0.124935i	$-0.960128\pi$
$31$	− 7.74597i	− 0.249870i	0.992165	−	0.124935i	$-0.0398722\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	37.9473i	1.08421i
$36$	0	0
$37$	−46.4758	−1.25610	−0.628051	−	0.778172i	$-0.716147\pi$
$37$	−46.4758	−1.25610	−0.628051	+	0.778172i	$0.716147\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	25.2982	0.617030	0.308515	−	0.951220i	$-0.400168\pi$
$41$	25.2982	0.617030	0.308515	+	0.951220i	$0.400168\pi$
$42$	0	0
$43$	40.0000i	0.930233i	0.885250	+	0.465116i	$0.153987\pi$
$43$	40.0000i	0.930233i	−0.885250	+	0.465116i	$0.846013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 39.1918i	− 0.833869i	−0.908937	−	0.416934i	$-0.863104\pi$
$47$	− 39.1918i	− 0.833869i	0.908937	−	0.416934i	$-0.136896\pi$
$48$	0	0
$49$	−11.0000	−0.224490
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−14.6969	−0.277301	−0.138650	−	0.990341i	$-0.544276\pi$
$53$	−14.6969	−0.277301	−0.138650	+	0.990341i	$0.544276\pi$
$54$	0	0
$55$	61.9677i	1.12669i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	25.2982i	0.428783i	0.976748	+	0.214392i	$0.0687769\pi$
$59$	25.2982i	0.428783i	−0.976748	+	0.214392i	$0.931223\pi$
$60$	0	0
$61$	15.4919	0.253966	0.126983	−	0.991905i	$-0.459471\pi$
$61$	15.4919	0.253966	0.126983	+	0.991905i	$0.459471\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	75.8947	1.16761
$66$	0	0
$67$	80.0000i	1.19403i	0.802230	+	0.597015i	$0.203647\pi$
$67$	80.0000i	1.19403i	−0.802230	+	0.597015i	$0.796353\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	10.0000	0.136986	0.0684932	−	0.997652i	$-0.478181\pi$
$73$	10.0000	0.136986	0.0684932	+	0.997652i	$0.478181\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−97.9796	−1.27246
$78$	0	0
$79$	54.2218i	0.686351i	0.939271	+	0.343176i	$0.111503\pi$
$79$	54.2218i	0.686351i	−0.939271	+	0.343176i	$0.888497\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 139.140i	− 1.67639i	−0.545372	−	0.838194i	$-0.683612\pi$
$83$	− 139.140i	− 1.67639i	0.545372	−	0.838194i	$-0.316388\pi$
$84$	0	0
$85$	123.935	1.45806
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−50.5964	−0.568499	−0.284250	−	0.958750i	$-0.591744\pi$
$89$	−50.5964	−0.568499	−0.284250	+	0.958750i	$0.591744\pi$
$90$	0	0
$91$	120.000i	1.31868i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 39.1918i	− 0.412546i
$96$	0	0
$97$	50.0000	0.515464	0.257732	−	0.966216i	$-0.417025\pi$
$97$	50.0000	0.515464	0.257732	+	0.966216i	$0.417025\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	122.474	1.21262	0.606309	−	0.795229i	$-0.292649\pi$
$101$	122.474	1.21262	0.606309	+	0.795229i	$0.292649\pi$
$102$	0	0
$103$	− 178.157i	− 1.72968i	−0.502046	−	0.864841i	$-0.667419\pi$
$103$	− 178.157i	− 1.72968i	0.502046	−	0.864841i	$-0.332581\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	25.2982i	0.236432i	0.992988	+	0.118216i	$0.0377175\pi$
$107$	25.2982i	0.236432i	−0.992988	+	0.118216i	$0.962282\pi$
$108$	0	0
$109$	46.4758	0.426383	0.213192	−	0.977010i	$-0.431614\pi$
$109$	46.4758	0.426383	0.213192	+	0.977010i	$0.431614\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	50.5964	0.447756	0.223878	−	0.974617i	$-0.428128\pi$
$113$	50.5964	0.447756	0.223878	+	0.974617i	$0.428128\pi$
$114$	0	0
$115$	192.000i	1.66957i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	195.959i	1.64672i
$120$	0	0
$121$	−39.0000	−0.322314
$122$	0	0
$123$	0	0
$124$	0	0
$125$	127.373	1.01899
$126$	0	0
$127$	− 116.190i	− 0.914878i	−0.889241	−	0.457439i	$-0.848767\pi$
$127$	− 116.190i	− 0.914878i	0.889241	−	0.457439i	$-0.151233\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	50.5964i	0.386232i	0.981176	+	0.193116i	$0.0618594\pi$
$131$	50.5964i	0.386232i	−0.981176	+	0.193116i	$0.938141\pi$
$132$	0	0
$133$	61.9677	0.465923
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−126.491	−0.923293	−0.461646	−	0.887064i	$-0.652741\pi$
$137$	−126.491	−0.923293	−0.461646	+	0.887064i	$0.652741\pi$
$138$	0	0
$139$	208.000i	1.49640i	0.663472	+	0.748201i	$0.269082\pi$
$139$	208.000i	1.49640i	−0.663472	+	0.748201i	$0.730918\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	195.959i	1.37034i
$144$	0	0
$145$	−120.000	−0.827586
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−122.474	−0.821976	−0.410988	−	0.911641i	$-0.634816\pi$
$149$	−122.474	−0.821976	−0.410988	+	0.911641i	$0.634816\pi$
$150$	0	0
$151$	178.157i	1.17985i	0.807458	+	0.589925i	$0.200843\pi$
$151$	178.157i	1.17985i	−0.807458	+	0.589925i	$0.799157\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	37.9473i	0.244821i
$156$	0	0
$157$	77.4597	0.493374	0.246687	−	0.969095i	$-0.420658\pi$
$157$	77.4597	0.493374	0.246687	+	0.969095i	$0.420658\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−303.579	−1.88558
$162$	0	0
$163$	200.000i	1.22699i	0.789697	+	0.613497i	$0.210238\pi$
$163$	200.000i	1.22699i	−0.789697	+	0.613497i	$0.789762\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 39.1918i	− 0.234682i	−0.993092	−	0.117341i	$-0.962563\pi$
$167$	− 39.1918i	− 0.234682i	0.993092	−	0.117341i	$-0.0374370\pi$
$168$	0	0
$169$	71.0000	0.420118
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−181.262	−1.04776	−0.523879	−	0.851793i	$-0.675516\pi$
$173$	−181.262	−1.04776	−0.523879	+	0.851793i	$0.675516\pi$
$174$	0	0
$175$	7.74597i	0.0442627i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	278.280i	1.55464i	0.629106	+	0.777320i	$0.283421\pi$
$179$	278.280i	1.55464i	−0.629106	+	0.777320i	$0.716579\pi$
$180$	0	0
$181$	−325.331	−1.79741	−0.898703	−	0.438557i	$-0.855490\pi$
$181$	−325.331	−1.79741	−0.898703	+	0.438557i	$0.855490\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	227.684	1.23072
$186$	0	0
$187$	320.000i	1.71123i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	195.959i	1.02596i	0.858399	+	0.512982i	$0.171459\pi$
$191$	195.959i	1.02596i	−0.858399	+	0.512982i	$0.828541\pi$
$192$	0	0
$193$	170.000	0.880829	0.440415	−	0.897795i	$-0.354831\pi$
$193$	170.000	0.880829	0.440415	+	0.897795i	$0.354831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−132.272	−0.671434	−0.335717	−	0.941963i	$-0.608979\pi$
$197$	−132.272	−0.671434	−0.335717	+	0.941963i	$0.608979\pi$
$198$	0	0
$199$	69.7137i	0.350320i	0.984540	+	0.175160i	$0.0560443\pi$
$199$	69.7137i	0.350320i	−0.984540	+	0.175160i	$0.943956\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 189.737i	− 0.934663i
$204$	0	0
$205$	−123.935	−0.604563
$206$	0	0
$207$	0	0
$208$	0	0
$209$	101.193	0.484176
$210$	0	0
$211$	272.000i	1.28910i	0.764562	+	0.644550i	$0.222955\pi$
$211$	272.000i	1.28910i	−0.764562	+	0.644550i	$0.777045\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 195.959i	− 0.911438i
$216$	0	0
$217$	−60.0000	−0.276498
$218$	0	0
$219$	0	0
$220$	0	0
$221$	391.918	1.77339
$222$	0	0
$223$	193.649i	0.868382i	0.900821	+	0.434191i	$0.142966\pi$
$223$	193.649i	0.868382i	−0.900821	+	0.434191i	$0.857034\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 215.035i	− 0.947290i	−0.880716	−	0.473645i	$-0.842938\pi$
$227$	− 215.035i	− 0.947290i	0.880716	−	0.473645i	$-0.157062\pi$
$228$	0	0
$229$	108.444	0.473553	0.236776	−	0.971564i	$-0.423909\pi$
$229$	108.444	0.473553	0.236776	+	0.971564i	$0.423909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	252.982	1.08576	0.542880	−	0.839810i	$-0.317334\pi$
$233$	252.982	1.08576	0.542880	+	0.839810i	$0.317334\pi$
$234$	0	0
$235$	192.000i	0.817021i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 195.959i	− 0.819913i	−0.912105	−	0.409956i	$-0.865544\pi$
$239$	− 195.959i	− 0.819913i	0.912105	−	0.409956i	$-0.134456\pi$
$240$	0	0
$241$	−202.000	−0.838174	−0.419087	−	0.907946i	$-0.637650\pi$
$241$	−202.000	−0.838174	−0.419087	+	0.907946i	$0.637650\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	53.8888	0.219954
$246$	0	0
$247$	− 123.935i	− 0.501763i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 316.228i	− 1.25987i	−0.776647	−	0.629936i	$-0.783081\pi$
$251$	− 316.228i	− 1.25987i	0.776647	−	0.629936i	$-0.216919\pi$
$252$	0	0
$253$	−495.742	−1.95945
$254$	0	0
$255$	0	0
$256$	0	0
$257$	202.386	0.787493	0.393747	−	0.919219i	$-0.371179\pi$
$257$	202.386	0.787493	0.393747	+	0.919219i	$0.371179\pi$
$258$	0	0
$259$	360.000i	1.38996i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	156.767i	0.596074i	0.954554	+	0.298037i	$0.0963318\pi$
$263$	156.767i	0.596074i	−0.954554	+	0.298037i	$0.903668\pi$
$264$	0	0
$265$	72.0000	0.271698
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−73.4847	−0.273177	−0.136589	−	0.990628i	$-0.543614\pi$
$269$	−73.4847	−0.273177	−0.136589	+	0.990628i	$0.543614\pi$
$270$	0	0
$271$	69.7137i	0.257246i	0.991694	+	0.128623i	$0.0410557\pi$
$271$	69.7137i	0.257246i	−0.991694	+	0.128623i	$0.958944\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.6491i	0.0459968i
$276$	0	0
$277$	−15.4919	−0.0559276	−0.0279638	−	0.999609i	$-0.508902\pi$
$277$	−15.4919	−0.0559276	−0.0279638	+	0.999609i	$0.508902\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−202.386	−0.720234	−0.360117	−	0.932907i	$-0.617263\pi$
$281$	−202.386	−0.720234	−0.360117	+	0.932907i	$0.617263\pi$
$282$	0	0
$283$	160.000i	0.565371i	0.959213	+	0.282686i	$0.0912253\pi$
$283$	160.000i	0.565371i	−0.959213	+	0.282686i	$0.908775\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 195.959i	− 0.682785i
$288$	0	0
$289$	351.000	1.21453
$290$	0	0
$291$	0	0
$292$	0	0
$293$	259.646	0.886164	0.443082	−	0.896481i	$-0.353885\pi$
$293$	259.646	0.886164	0.443082	+	0.896481i	$0.353885\pi$
$294$	0	0
$295$	− 123.935i	− 0.420120i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	607.157i	2.03063i
$300$	0	0
$301$	309.839	1.02936
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−75.8947	−0.248835
$306$	0	0
$307$	80.0000i	0.260586i	0.991476	+	0.130293i	$0.0415919\pi$
$307$	80.0000i	0.260586i	−0.991476	+	0.130293i	$0.958408\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 391.918i	− 1.26019i	−0.776519	−	0.630094i	$-0.783016\pi$
$311$	− 391.918i	− 1.26019i	0.776519	−	0.630094i	$-0.216984\pi$
$312$	0	0
$313$	−230.000	−0.734824	−0.367412	−	0.930058i	$-0.619756\pi$
$313$	−230.000	−0.734824	−0.367412	+	0.930058i	$0.619756\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−504.595	−1.59178	−0.795891	−	0.605440i	$-0.792997\pi$
$317$	−504.595	−1.59178	−0.795891	+	0.605440i	$0.792997\pi$
$318$	0	0
$319$	− 309.839i	− 0.971281i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 202.386i	− 0.626581i
$324$	0	0
$325$	15.4919	0.0476675
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−303.579	−0.922731
$330$	0	0
$331$	352.000i	1.06344i	0.846919	+	0.531722i	$0.178455\pi$
$331$	352.000i	1.06344i	−0.846919	+	0.531722i	$0.821545\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 391.918i	− 1.16991i
$336$	0	0
$337$	−550.000	−1.63205	−0.816024	−	0.578018i	$-0.803826\pi$
$337$	−550.000	−1.63205	−0.816024	+	0.578018i	$0.803826\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−97.9796	−0.287330
$342$	0	0
$343$	− 294.347i	− 0.858154i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	366.824i	1.05713i	0.848893	+	0.528565i	$0.177270\pi$
$347$	366.824i	1.05713i	−0.848893	+	0.528565i	$0.822730\pi$
$348$	0	0
$349$	573.202	1.64241	0.821206	−	0.570632i	$-0.193302\pi$
$349$	573.202	1.64241	0.821206	+	0.570632i	$0.193302\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	354.175	1.00333	0.501664	−	0.865062i	$-0.332721\pi$
$353$	354.175	1.00333	0.501664	+	0.865062i	$0.332721\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	297.000	0.822715
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−48.9898	−0.134219
$366$	0	0
$367$	− 503.488i	− 1.37190i	−0.727648	−	0.685951i	$-0.759387\pi$
$367$	− 503.488i	− 1.37190i	0.727648	−	0.685951i	$-0.240613\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	113.842i	0.306852i
$372$	0	0
$373$	201.395	0.539933	0.269967	−	0.962870i	$-0.412987\pi$
$373$	201.395	0.539933	0.269967	+	0.962870i	$0.412987\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−379.473	−1.00656
$378$	0	0
$379$	− 152.000i	− 0.401055i	−0.979688	−	0.200528i	$-0.935734\pi$
$379$	− 152.000i	− 0.401055i	0.979688	−	0.200528i	$-0.0642657\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 431.110i	− 1.12561i	−0.826588	−	0.562807i	$-0.809721\pi$
$383$	− 431.110i	− 1.12561i	0.826588	−	0.562807i	$-0.190279\pi$
$384$	0	0
$385$	480.000	1.24675
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.4949	−0.0629689	−0.0314844	−	0.999504i	$-0.510023\pi$
$389$	−24.4949	−0.0629689	−0.0314844	+	0.999504i	$0.510023\pi$
$390$	0	0
$391$	991.484i	2.53576i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 265.631i	− 0.672484i
$396$	0	0
$397$	−46.4758	−0.117068	−0.0585338	−	0.998285i	$-0.518643\pi$
$397$	−46.4758	−0.117068	−0.0585338	+	0.998285i	$0.518643\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−480.666	−1.19867	−0.599334	−	0.800499i	$-0.704568\pi$
$401$	−480.666	−1.19867	−0.599334	+	0.800499i	$0.704568\pi$
$402$	0	0
$403$	120.000i	0.297767i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	587.878i	1.44442i
$408$	0	0
$409$	−62.0000	−0.151589	−0.0757946	−	0.997123i	$-0.524149\pi$
$409$	−62.0000	−0.151589	−0.0757946	+	0.997123i	$0.524149\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	195.959	0.474477
$414$	0	0
$415$	681.645i	1.64252i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 63.2456i	− 0.150944i	−0.997148	−	0.0754720i	$-0.975954\pi$
$419$	− 63.2456i	− 0.150944i	0.997148	−	0.0754720i	$-0.0240464\pi$
$420$	0	0
$421$	−15.4919	−0.0367979	−0.0183990	−	0.999831i	$-0.505857\pi$
$421$	−15.4919	−0.0367979	−0.0183990	+	0.999831i	$0.505857\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25.2982	0.0595252
$426$	0	0
$427$	− 120.000i	− 0.281030i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	587.878i	1.36399i	0.731359	+	0.681993i	$0.238886\pi$
$431$	587.878i	1.36399i	−0.731359	+	0.681993i	$0.761114\pi$
$432$	0	0
$433$	−370.000	−0.854503	−0.427252	−	0.904133i	$-0.640518\pi$
$433$	−370.000	−0.854503	−0.427252	+	0.904133i	$0.640518\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	313.535	0.717471
$438$	0	0
$439$	− 426.028i	− 0.970451i	−0.874389	−	0.485226i	$-0.838737\pi$
$439$	− 426.028i	− 0.970451i	0.874389	−	0.485226i	$-0.161263\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 240.333i	− 0.542513i	−0.962507	−	0.271256i	$-0.912561\pi$
$443$	− 240.333i	− 0.542513i	0.962507	−	0.271256i	$-0.0874391\pi$
$444$	0	0
$445$	247.871	0.557013
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−25.2982	−0.0563435	−0.0281717	−	0.999603i	$-0.508969\pi$
$449$	−25.2982	−0.0563435	−0.0281717	+	0.999603i	$0.508969\pi$
$450$	0	0
$451$	− 320.000i	− 0.709534i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 587.878i	− 1.29204i
$456$	0	0
$457$	130.000	0.284464	0.142232	−	0.989833i	$-0.454572\pi$
$457$	130.000	0.284464	0.142232	+	0.989833i	$0.454572\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	759.342	1.64716	0.823581	−	0.567198i	$-0.191973\pi$
$461$	759.342	1.64716	0.823581	+	0.567198i	$0.191973\pi$
$462$	0	0
$463$	503.488i	1.08745i	0.839265	+	0.543723i	$0.182986\pi$
$463$	503.488i	1.08745i	−0.839265	+	0.543723i	$0.817014\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 366.824i	− 0.785491i	−0.919647	−	0.392745i	$-0.871525\pi$
$467$	− 366.824i	− 0.785491i	0.919647	−	0.392745i	$-0.128475\pi$
$468$	0	0
$469$	619.677	1.32127
$470$	0	0
$471$	0	0
$472$	0	0
$473$	505.964	1.06969
$474$	0	0
$475$	− 8.00000i	− 0.0168421i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	783.837i	1.63640i	0.574932	+	0.818201i	$0.305028\pi$
$479$	783.837i	1.63640i	−0.574932	+	0.818201i	$0.694972\pi$
$480$	0	0
$481$	720.000	1.49688
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−244.949	−0.505049
$486$	0	0
$487$	− 255.617i	− 0.524881i	−0.964948	−	0.262440i	$-0.915473\pi$
$487$	− 255.617i	− 0.524881i	0.964948	−	0.262440i	$-0.0845273\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 657.754i	− 1.33962i	−0.742532	−	0.669810i	$-0.766375\pi$
$491$	− 657.754i	− 1.33962i	0.742532	−	0.669810i	$-0.233625\pi$
$492$	0	0
$493$	−619.677	−1.25695
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 352.000i	− 0.705411i	−0.935734	−	0.352705i	$-0.885262\pi$
$499$	− 352.000i	− 0.705411i	0.935734	−	0.352705i	$-0.114738\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	235.151i	0.467497i	0.972297	+	0.233749i	$0.0750992\pi$
$503$	235.151i	0.467497i	−0.972297	+	0.233749i	$0.924901\pi$
$504$	0	0
$505$	−600.000	−1.18812
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−759.342	−1.49183	−0.745915	−	0.666041i	$-0.767988\pi$
$509$	−759.342	−1.49183	−0.745915	+	0.666041i	$0.767988\pi$
$510$	0	0
$511$	− 77.4597i	− 0.151584i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	872.789i	1.69474i
$516$	0	0
$517$	−495.742	−0.958882
$518$	0	0
$519$	0	0
$520$	0	0
$521$	936.034	1.79661	0.898305	−	0.439372i	$-0.144799\pi$
$521$	936.034	1.79661	0.898305	+	0.439372i	$0.144799\pi$
$522$	0	0
$523$	− 920.000i	− 1.75908i	−0.475823	−	0.879541i	$-0.657850\pi$
$523$	− 920.000i	− 1.75908i	0.475823	−	0.879541i	$-0.342150\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	195.959i	0.371839i
$528$	0	0
$529$	−1007.00	−1.90359
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−391.918	−0.735306
$534$	0	0
$535$	− 123.935i	− 0.231655i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	139.140i	0.258145i
$540$	0	0
$541$	−511.234	−0.944979	−0.472490	−	0.881336i	$-0.656645\pi$
$541$	−511.234	−0.944979	−0.472490	+	0.881336i	$0.656645\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−227.684	−0.417769
$546$	0	0
$547$	− 40.0000i	− 0.0731261i	−0.999331	−	0.0365631i	$-0.988359\pi$
$547$	− 40.0000i	− 0.0731261i	0.999331	−	0.0365631i	$-0.0116410\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	195.959i	0.355643i
$552$	0	0
$553$	420.000	0.759494
$554$	0	0
$555$	0	0
$556$	0	0
$557$	308.636	0.554104	0.277052	−	0.960855i	$-0.410643\pi$
$557$	308.636	0.554104	0.277052	+	0.960855i	$0.410643\pi$
$558$	0	0
$559$	− 619.677i	− 1.10855i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	88.5438i	0.157271i	0.996903	+	0.0786357i	$0.0250564\pi$
$563$	88.5438i	0.157271i	−0.996903	+	0.0786357i	$0.974944\pi$
$564$	0	0
$565$	−247.871	−0.438710
$566$	0	0
$567$	0	0
$568$	0	0
$569$	328.877	0.577991	0.288995	−	0.957330i	$-0.406679\pi$
$569$	328.877	0.577991	0.288995	+	0.957330i	$0.406679\pi$
$570$	0	0
$571$	− 368.000i	− 0.644483i	−0.946657	−	0.322242i	$-0.895564\pi$
$571$	− 368.000i	− 0.644483i	0.946657	−	0.322242i	$-0.104436\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	39.1918i	0.0681597i
$576$	0	0
$577$	710.000	1.23050	0.615251	−	0.788331i	$-0.289055\pi$
$577$	710.000	1.23050	0.615251	+	0.788331i	$0.289055\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1077.78	−1.85504
$582$	0	0
$583$	185.903i	0.318873i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	101.193i	0.172390i	0.996278	+	0.0861950i	$0.0274708\pi$
$587$	101.193i	0.172390i	−0.996278	+	0.0861950i	$0.972529\pi$
$588$	0	0
$589$	61.9677	0.105208
$590$	0	0
$591$	0	0
$592$	0	0
$593$	961.332	1.62113	0.810567	−	0.585646i	$-0.199159\pi$
$593$	961.332	1.62113	0.810567	+	0.585646i	$0.199159\pi$
$594$	0	0
$595$	− 960.000i	− 1.61345i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	195.959i	0.327144i	0.986531	+	0.163572i	$0.0523016\pi$
$599$	195.959i	0.327144i	−0.986531	+	0.163572i	$0.947698\pi$
$600$	0	0
$601$	22.0000	0.0366057	0.0183028	−	0.999832i	$-0.494174\pi$
$601$	22.0000	0.0366057	0.0183028	+	0.999832i	$0.494174\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	191.060	0.315802
$606$	0	0
$607$	− 240.125i	− 0.395593i	−0.980243	−	0.197797i	$-0.936621\pi$
$607$	− 240.125i	− 0.395593i	0.980243	−	0.197797i	$-0.0633785\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	607.157i	0.993711i
$612$	0	0
$613$	−790.089	−1.28889	−0.644444	−	0.764651i	$-0.722911\pi$
$613$	−790.089	−1.28889	−0.644444	+	0.764651i	$0.722911\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1113.12	−1.80409	−0.902044	−	0.431645i	$-0.857933\pi$
$617$	−1113.12	−1.80409	−0.902044	+	0.431645i	$0.857933\pi$
$618$	0	0
$619$	− 32.0000i	− 0.0516963i	−0.999666	−	0.0258481i	$-0.991771\pi$
$619$	− 32.0000i	− 0.0516963i	0.999666	−	0.0258481i	$-0.00822864\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	391.918i	0.629082i
$624$	0	0
$625$	−599.000	−0.958400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1175.76	1.86924
$630$	0	0
$631$	116.190i	0.184135i	0.995753	+	0.0920677i	$0.0293476\pi$
$631$	116.190i	0.184135i	−0.995753	+	0.0920677i	$0.970652\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	569.210i	0.896394i
$636$	0	0
$637$	170.411	0.267522
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1239.61	−1.93387	−0.966937	−	0.255017i	$-0.917919\pi$
$641$	−1239.61	−1.93387	−0.966937	+	0.255017i	$0.917919\pi$
$642$	0	0
$643$	− 1000.00i	− 1.55521i	−0.628753	−	0.777605i	$-0.716434\pi$
$643$	− 1000.00i	− 1.55521i	0.628753	−	0.777605i	$-0.283566\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	823.029i	1.27207i	0.771661	+	0.636034i	$0.219426\pi$
$647$	823.029i	1.27207i	−0.771661	+	0.636034i	$0.780574\pi$
$648$	0	0
$649$	320.000	0.493066
$650$	0	0
$651$	0	0
$652$	0	0
$653$	328.232	0.502652	0.251326	−	0.967903i	$-0.419133\pi$
$653$	328.232	0.502652	0.251326	+	0.967903i	$0.419133\pi$
$654$	0	0
$655$	− 247.871i	− 0.378429i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 708.350i	− 1.07489i	−0.843300	−	0.537443i	$-0.819390\pi$
$659$	− 708.350i	− 1.07489i	0.843300	−	0.537443i	$-0.180610\pi$
$660$	0	0
$661$	−728.121	−1.10154	−0.550772	−	0.834656i	$-0.685667\pi$
$661$	−728.121	−1.10154	−0.550772	+	0.834656i	$0.685667\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−303.579	−0.456509
$666$	0	0
$667$	− 960.000i	− 1.43928i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 195.959i	− 0.292041i
$672$	0	0
$673$	1070.00	1.58990	0.794948	−	0.606678i	$-0.207498\pi$
$673$	1070.00	1.58990	0.794948	+	0.606678i	$0.207498\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1092.47	1.61370	0.806848	−	0.590759i	$-0.201172\pi$
$677$	1092.47	1.61370	0.806848	+	0.590759i	$0.201172\pi$
$678$	0	0
$679$	− 387.298i	− 0.570395i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	594.508i	0.870437i	0.900325	+	0.435218i	$0.143329\pi$
$683$	594.508i	0.870437i	−0.900325	+	0.435218i	$0.856671\pi$
$684$	0	0
$685$	619.677	0.904638
$686$	0	0
$687$	0	0
$688$	0	0
$689$	227.684	0.330456
$690$	0	0
$691$	8.00000i	0.0115774i	0.999983	+	0.00578871i	$0.00184261\pi$
$691$	8.00000i	0.0115774i	−0.999983	+	0.00578871i	$0.998157\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1018.99i	− 1.46617i
$696$	0	0
$697$	−640.000	−0.918221
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1249.24	−1.78208	−0.891041	−	0.453922i	$-0.850024\pi$
$701$	−1249.24	−1.78208	−0.891041	+	0.453922i	$0.850024\pi$
$702$	0	0
$703$	− 371.806i	− 0.528885i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 948.683i	− 1.34184i
$708$	0	0
$709$	914.024	1.28917	0.644587	−	0.764531i	$-0.277029\pi$
$709$	914.024	1.28917	0.644587	+	0.764531i	$0.277029\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−303.579	−0.425777
$714$	0	0
$715$	− 960.000i	− 1.34266i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1175.76i	− 1.63526i	−0.575741	−	0.817632i	$-0.695286\pi$
$719$	− 1175.76i	− 1.63526i	0.575741	−	0.817632i	$-0.304714\pi$
$720$	0	0
$721$	−1380.00	−1.91401
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.4949	−0.0337861
$726$	0	0
$727$	549.964i	0.756484i	0.925707	+	0.378242i	$0.123471\pi$
$727$	549.964i	0.756484i	−0.925707	+	0.378242i	$0.876529\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1011.93i	− 1.38431i
$732$	0	0
$733$	−945.008	−1.28923	−0.644617	−	0.764506i	$-0.722983\pi$
$733$	−945.008	−1.28923	−0.644617	+	0.764506i	$0.722983\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1011.93	1.37304
$738$	0	0
$739$	− 1072.00i	− 1.45061i	−0.688428	−	0.725304i	$-0.741699\pi$
$739$	− 1072.00i	− 1.45061i	0.688428	−	0.725304i	$-0.258301\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	431.110i	0.580229i	0.956992	+	0.290115i	$0.0936934\pi$
$743$	431.110i	0.580229i	−0.956992	+	0.290115i	$0.906307\pi$
$744$	0	0
$745$	600.000	0.805369
$746$	0	0
$747$	0	0
$748$	0	0
$749$	195.959	0.261628
$750$	0	0
$751$	1293.58i	1.72247i	0.508205	+	0.861236i	$0.330309\pi$
$751$	1293.58i	1.72247i	−0.508205	+	0.861236i	$0.669691\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 872.789i	− 1.15601i
$756$	0	0
$757$	790.089	1.04371	0.521855	−	0.853034i	$-0.325240\pi$
$757$	790.089	1.04371	0.521855	+	0.853034i	$0.325240\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−733.648	−0.964058	−0.482029	−	0.876155i	$-0.660100\pi$
$761$	−733.648	−0.964058	−0.482029	+	0.876155i	$0.660100\pi$
$762$	0	0
$763$	− 360.000i	− 0.471822i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 391.918i	− 0.510976i
$768$	0	0
$769$	938.000	1.21977	0.609883	−	0.792491i	$-0.291216\pi$
$769$	938.000	1.21977	0.609883	+	0.792491i	$0.291216\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−426.211	−0.551373	−0.275686	−	0.961248i	$-0.588905\pi$
$773$	−426.211	−0.551373	−0.275686	+	0.961248i	$0.588905\pi$
$774$	0	0
$775$	7.74597i	0.00999480i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	202.386i	0.259802i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−379.473	−0.483406
$786$	0	0
$787$	− 520.000i	− 0.660737i	−0.943852	−	0.330368i	$-0.892827\pi$
$787$	− 520.000i	− 0.660737i	0.943852	−	0.330368i	$-0.107173\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 391.918i	− 0.495472i
$792$	0	0
$793$	−240.000	−0.302648
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−83.2827	−0.104495	−0.0522476	−	0.998634i	$-0.516638\pi$
$797$	−83.2827	−0.104495	−0.0522476	+	0.998634i	$0.516638\pi$
$798$	0	0
$799$	991.484i	1.24091i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 126.491i	− 0.157523i
$804$	0	0
$805$	1487.23	1.84749
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−430.070	−0.531607	−0.265803	−	0.964027i	$-0.585637\pi$
$809$	−430.070	−0.531607	−0.265803	+	0.964027i	$0.585637\pi$
$810$	0	0
$811$	− 872.000i	− 1.07522i	−0.843195	−	0.537608i	$-0.819328\pi$
$811$	− 872.000i	− 1.07522i	0.843195	−	0.537608i	$-0.180672\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 979.796i	− 1.20220i
$816$	0	0
$817$	−320.000	−0.391677
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1053.28	−1.28292	−0.641462	−	0.767155i	$-0.721672\pi$
$821$	−1053.28	−1.28292	−0.641462	+	0.767155i	$0.721672\pi$
$822$	0	0
$823$	240.125i	0.291768i	0.989302	+	0.145884i	$0.0466026\pi$
$823$	240.125i	0.291768i	−0.989302	+	0.145884i	$0.953397\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1087.82i	1.31539i	0.753286	+	0.657693i	$0.228467\pi$
$827$	1087.82i	1.31539i	−0.753286	+	0.657693i	$0.771533\pi$
$828$	0	0
$829$	1533.70	1.85006	0.925031	−	0.379892i	$-0.124039\pi$
$829$	1533.70	1.85006	0.925031	+	0.379892i	$0.124039\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	278.280	0.334070
$834$	0	0
$835$	192.000i	0.229940i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 783.837i	− 0.934251i	−0.884191	−	0.467126i	$-0.845290\pi$
$839$	− 783.837i	− 0.934251i	0.884191	−	0.467126i	$-0.154710\pi$
$840$	0	0
$841$	−241.000	−0.286564
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−347.828	−0.411630
$846$	0	0
$847$	302.093i	0.356662i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1821.47i	2.14039i
$852$	0	0
$853$	−1037.96	−1.21683	−0.608417	−	0.793617i	$-0.708195\pi$
$853$	−1037.96	−1.21683	−0.608417	+	0.793617i	$0.708195\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	936.034	1.09222	0.546111	−	0.837713i	$-0.316108\pi$
$857$	936.034	1.09222	0.546111	+	0.837713i	$0.316108\pi$
$858$	0	0
$859$	− 872.000i	− 1.01513i	−0.861612	−	0.507567i	$-0.830545\pi$
$859$	− 872.000i	− 1.01513i	0.861612	−	0.507567i	$-0.169455\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 352.727i	− 0.408721i	−0.978896	−	0.204361i	$-0.934488\pi$
$863$	− 352.727i	− 0.408721i	0.978896	−	0.204361i	$-0.0655115\pi$
$864$	0	0
$865$	888.000	1.02659
$866$	0	0
$867$	0	0
$868$	0	0
$869$	685.857	0.789249
$870$	0	0
$871$	− 1239.35i	− 1.42291i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 986.631i	− 1.12758i
$876$	0	0
$877$	−480.250	−0.547605	−0.273803	−	0.961786i	$-0.588282\pi$
$877$	−480.250	−0.547605	−0.273803	+	0.961786i	$0.588282\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−404.772	−0.459446	−0.229723	−	0.973256i	$-0.573782\pi$
$881$	−404.772	−0.459446	−0.229723	+	0.973256i	$0.573782\pi$
$882$	0	0
$883$	− 1720.00i	− 1.94790i	−0.226752	−	0.973952i	$-0.572811\pi$
$883$	− 1720.00i	− 1.94790i	0.226752	−	0.973952i	$-0.427189\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	548.686i	0.618586i	0.950967	+	0.309293i	$0.100092\pi$
$887$	548.686i	0.618586i	−0.950967	+	0.309293i	$0.899908\pi$
$888$	0	0
$889$	−900.000	−1.01237
$890$	0	0
$891$	0	0
$892$	0	0
$893$	313.535	0.351103
$894$	0	0
$895$	− 1363.29i	− 1.52323i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 189.737i	− 0.211053i
$900$	0	0
$901$	371.806	0.412660
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1593.79	1.76109
$906$	0	0
$907$	760.000i	0.837927i	0.908003	+	0.418964i	$0.137607\pi$
$907$	760.000i	0.837927i	−0.908003	+	0.418964i	$0.862393\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1371.71i	1.50572i	0.658179	+	0.752862i	$0.271327\pi$
$911$	1371.71i	1.50572i	−0.658179	+	0.752862i	$0.728673\pi$
$912$	0	0
$913$	−1760.00	−1.92771
$914$	0	0
$915$	0	0
$916$	0	0
$917$	391.918	0.427392
$918$	0	0
$919$	1231.61i	1.34016i	0.742288	+	0.670081i	$0.233741\pi$
$919$	1231.61i	1.34016i	−0.742288	+	0.670081i	$0.766259\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	46.4758	0.0502441
$926$	0	0
$927$	0	0
$928$	0	0
$929$	430.070	0.462938	0.231469	−	0.972842i	$-0.425647\pi$
$929$	430.070	0.462938	0.231469	+	0.972842i	$0.425647\pi$
$930$	0	0
$931$	− 88.0000i	− 0.0945220i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1567.67i	− 1.67666i
$936$	0	0
$937$	1330.00	1.41942	0.709712	−	0.704492i	$-0.248825\pi$
$937$	1330.00	1.41942	0.709712	+	0.704492i	$0.248825\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	563.383	0.598706	0.299353	−	0.954142i	$-0.403229\pi$
$941$	563.383	0.598706	0.299353	+	0.954142i	$0.403229\pi$
$942$	0	0
$943$	− 991.484i	− 1.05141i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1037.23i	1.09528i	0.836715	+	0.547638i	$0.184473\pi$
$947$	1037.23i	1.09528i	−0.836715	+	0.547638i	$0.815527\pi$
$948$	0	0
$949$	−154.919	−0.163245
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.2982	0.0265459	0.0132729	−	0.999912i	$-0.495775\pi$
$953$	25.2982	0.0265459	0.0132729	+	0.999912i	$0.495775\pi$
$954$	0	0
$955$	− 960.000i	− 1.00524i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	979.796i	1.02168i
$960$	0	0
$961$	901.000	0.937565
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−832.827	−0.863033
$966$	0	0
$967$	− 1107.67i	− 1.14547i	−0.819739	−	0.572737i	$-0.805882\pi$
$967$	− 1107.67i	− 1.14547i	0.819739	−	0.572737i	$-0.194118\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	746.298i	0.768587i	0.923211	+	0.384293i	$0.125555\pi$
$971$	746.298i	0.768587i	−0.923211	+	0.384293i	$0.874445\pi$
$972$	0	0
$973$	1611.16	1.65587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−480.666	−0.491982	−0.245991	−	0.969272i	$-0.579113\pi$
$977$	−480.666	−0.491982	−0.245991	+	0.969272i	$0.579113\pi$
$978$	0	0
$979$	640.000i	0.653728i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 548.686i	− 0.558175i	−0.960266	−	0.279087i	$-0.909968\pi$
$983$	− 548.686i	− 0.558175i	0.960266	−	0.279087i	$-0.0900319\pi$
$984$	0	0
$985$	648.000	0.657868
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1567.67	1.58511
$990$	0	0
$991$	− 1185.13i	− 1.19590i	−0.801535	−	0.597948i	$-0.795983\pi$
$991$	− 1185.13i	− 1.19590i	0.801535	−	0.597948i	$-0.204017\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 341.526i	− 0.343242i
$996$	0	0
$997$	−666.153	−0.668158	−0.334079	−	0.942545i	$-0.608425\pi$
$997$	−666.153	−0.668158	−0.334079	+	0.942545i	$0.608425\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.y.1279.1		8
3.2	odd	2	inner	2304.3.g.y.1279.5		8
4.3	odd	2	inner	2304.3.g.y.1279.3		8
8.3	odd	2	inner	2304.3.g.y.1279.8		8
8.5	even	2	inner	2304.3.g.y.1279.6		8
12.11	even	2	inner	2304.3.g.y.1279.7		8
16.3	odd	4		72.3.b.c.19.4	yes	4
16.5	even	4		72.3.b.c.19.3	yes	4
16.11	odd	4		288.3.b.c.271.1		4
16.13	even	4		288.3.b.c.271.4		4
24.5	odd	2	inner	2304.3.g.y.1279.2		8
24.11	even	2	inner	2304.3.g.y.1279.4		8
48.5	odd	4		72.3.b.c.19.2	yes	4
48.11	even	4		288.3.b.c.271.3		4
48.29	odd	4		288.3.b.c.271.2		4
48.35	even	4		72.3.b.c.19.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.b.c.19.1	✓	4	48.35	even	4
72.3.b.c.19.2	yes	4	48.5	odd	4
72.3.b.c.19.3	yes	4	16.5	even	4
72.3.b.c.19.4	yes	4	16.3	odd	4
288.3.b.c.271.1		4	16.11	odd	4
288.3.b.c.271.2		4	48.29	odd	4
288.3.b.c.271.3		4	48.11	even	4
288.3.b.c.271.4		4	16.13	even	4
2304.3.g.y.1279.1		8	1.1	even	1	trivial
2304.3.g.y.1279.2		8	24.5	odd	2	inner
2304.3.g.y.1279.3		8	4.3	odd	2	inner
2304.3.g.y.1279.4		8	24.11	even	2	inner
2304.3.g.y.1279.5		8	3.2	odd	2	inner
2304.3.g.y.1279.6		8	8.5	even	2	inner
2304.3.g.y.1279.7		8	12.11	even	2	inner
2304.3.g.y.1279.8		8	8.3	odd	2	inner

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.89898 q^{5} -7.74597i q^{7} +O(q^{10})\)	\(q-4.89898 q^{5} -7.74597i q^{7} -12.6491i q^{11} -15.4919 q^{13} -25.2982 q^{17} +8.00000i q^{19} -39.1918i q^{23} -1.00000 q^{25} +24.4949 q^{29} -7.74597i q^{31} +37.9473i q^{35} -46.4758 q^{37} +25.2982 q^{41} +40.0000i q^{43} -39.1918i q^{47} -11.0000 q^{49} -14.6969 q^{53} +61.9677i q^{55} +25.2982i q^{59} +15.4919 q^{61} +75.8947 q^{65} +80.0000i q^{67} +10.0000 q^{73} -97.9796 q^{77} +54.2218i q^{79} -139.140i q^{83} +123.935 q^{85} -50.5964 q^{89} +120.000i q^{91} -39.1918i q^{95} +50.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} - 88 q^{49} + 80 q^{73} + 400 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 - 88 * q^49 + 80 * q^73 + 400 * q^97

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