LMFDB - Embedded newform 2304.2.c.i.2303.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(2303,2304)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2304.2303");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.c (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:	$18.3975326257$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2303.5
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.2303
Dual form			2304.2.c.i.2303.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{5} -3.46410i q^{7} +O(q^{10})$	$q+2.44949i q^{5} -3.46410i q^{7} +2.82843 q^{11} -3.46410 q^{13} +1.41421i q^{17} +4.00000i q^{19} +4.89898 q^{23} -1.00000 q^{25} +2.44949i q^{29} -3.46410i q^{31} +8.48528 q^{35} -1.41421i q^{41} +8.00000i q^{43} +4.89898 q^{47} -5.00000 q^{49} -7.34847i q^{53} +6.92820i q^{55} +11.3137 q^{59} +13.8564 q^{61} -8.48528i q^{65} +4.00000i q^{67} -14.6969 q^{71} +4.00000 q^{73} -9.79796i q^{77} +3.46410i q^{79} +14.1421 q^{83} -3.46410 q^{85} +7.07107i q^{89} +12.0000i q^{91} -9.79796 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} - 40 q^{49} + 32 q^{73} + 64 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 - 40 * q^49 + 32 * q^73 + 64 * 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
$5$	2.44949i	1.09545i	0.836660	+	0.547723i	$0.184505\pi$
$5$	2.44949i	1.09545i	−0.836660	+	0.547723i	$0.815495\pi$
$6$	0	0
$7$	− 3.46410i	− 1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	− 3.46410i	− 1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41421i	0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$	1.41421i	0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.89898	1.02151	0.510754	−	0.859727i	$-0.329366\pi$
$23$	4.89898	1.02151	0.510754	+	0.859727i	$0.329366\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.44949i	0.454859i	0.973795	+	0.227429i	$0.0730321\pi$
$29$	2.44949i	0.454859i	−0.973795	+	0.227429i	$0.926968\pi$
$30$	0	0
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.48528	1.43427
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.41421i	− 0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	− 1.41421i	− 0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.89898	0.714590	0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898	0.714590	0.357295	+	0.933992i	$0.383699\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.34847i	− 1.00939i	−0.863298	−	0.504695i	$-0.831605\pi$
$53$	− 7.34847i	− 1.00939i	0.863298	−	0.504695i	$-0.168395\pi$
$54$	0	0
$55$	6.92820i	0.934199i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	13.8564	1.77413	0.887066	−	0.461644i	$-0.152740\pi$
$61$	13.8564	1.77413	0.887066	+	0.461644i	$0.152740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 8.48528i	− 1.05247i
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.6969	−1.74421	−0.872103	−	0.489323i	$-0.837244\pi$
$71$	−14.6969	−1.74421	−0.872103	+	0.489323i	$0.837244\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.79796i	− 1.11658i
$78$	0	0
$79$	3.46410i	0.389742i	0.980829	+	0.194871i	$0.0624288\pi$
$79$	3.46410i	0.389742i	−0.980829	+	0.194871i	$0.937571\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.07107i	0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$	7.07107i	0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$	0	0
$91$	12.0000i	1.25794i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−9.79796	−1.00525
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.44949i	− 0.243733i	−0.992546	−	0.121867i	$-0.961112\pi$
$101$	− 2.44949i	− 0.243733i	0.992546	−	0.121867i	$-0.0388880\pi$
$102$	0	0
$103$	3.46410i	0.341328i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	3.46410i	0.341328i	−0.985329	+	0.170664i	$0.945409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3137	1.09374	0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137	1.09374	0.546869	+	0.837218i	$0.315820\pi$
$108$	0	0
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.3848i	1.72949i	0.502208	+	0.864747i	$0.332521\pi$
$113$	18.3848i	1.72949i	−0.502208	+	0.864747i	$0.667479\pi$
$114$	0	0
$115$	12.0000i	1.11901i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.89898	0.449089
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.79796i	0.876356i
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	13.8564	1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.89949i	− 0.845771i	−0.906183	−	0.422885i	$-0.861017\pi$
$137$	− 9.89949i	− 0.845771i	0.906183	−	0.422885i	$-0.138983\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.79796	−0.819346
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.1464i	1.40469i	0.711837	+	0.702345i	$0.247864\pi$
$149$	17.1464i	1.40469i	−0.711837	+	0.702345i	$0.752136\pi$
$150$	0	0
$151$	− 3.46410i	− 0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$	− 3.46410i	− 0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.48528	0.681554
$156$	0	0
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.9706i	− 1.33747i
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.5959	1.51638	0.758189	−	0.652035i	$-0.226085\pi$
$167$	19.5959	1.51638	0.758189	+	0.652035i	$0.226085\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.44949i	0.186231i	0.995655	+	0.0931156i	$0.0296826\pi$
$173$	2.44949i	0.186231i	−0.995655	+	0.0931156i	$0.970317\pi$
$174$	0	0
$175$	3.46410i	0.261861i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	−10.3923	−0.772454	−0.386227	−	0.922404i	$-0.626222\pi$
$181$	−10.3923	−0.772454	−0.386227	+	0.922404i	$0.626222\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.5959	1.41791	0.708955	−	0.705253i	$-0.249167\pi$
$191$	19.5959	1.41791	0.708955	+	0.705253i	$0.249167\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.34847i	0.523557i	0.965128	+	0.261778i	$0.0843089\pi$
$197$	7.34847i	0.523557i	−0.965128	+	0.261778i	$0.915691\pi$
$198$	0	0
$199$	− 10.3923i	− 0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$	− 10.3923i	− 0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.48528	0.595550
$204$	0	0
$205$	3.46410	0.241943
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.3137i	0.782586i
$210$	0	0
$211$	− 20.0000i	− 1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$	− 20.0000i	− 1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.5959	−1.33643
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.89898i	− 0.329541i
$222$	0	0
$223$	− 17.3205i	− 1.15987i	−0.814664	−	0.579934i	$-0.803079\pi$
$223$	− 17.3205i	− 1.15987i	0.814664	−	0.579934i	$-0.196921\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	0	0
$229$	−17.3205	−1.14457	−0.572286	−	0.820054i	$-0.693943\pi$
$229$	−17.3205	−1.14457	−0.572286	+	0.820054i	$0.693943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.41421i	− 0.0926482i	−0.998926	−	0.0463241i	$-0.985249\pi$
$233$	− 1.41421i	− 0.0926482i	0.998926	−	0.0463241i	$-0.0147507\pi$
$234$	0	0
$235$	12.0000i	0.782794i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.79796	0.633777	0.316889	−	0.948463i	$-0.397362\pi$
$239$	9.79796	0.633777	0.316889	+	0.948463i	$0.397362\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12.2474i	− 0.782461i
$246$	0	0
$247$	− 13.8564i	− 0.881662i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.1421	−0.892644	−0.446322	−	0.894873i	$-0.647266\pi$
$251$	−14.1421	−0.892644	−0.446322	+	0.894873i	$0.647266\pi$
$252$	0	0
$253$	13.8564	0.871145
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.0416i	− 1.49968i	−0.661622	−	0.749838i	$-0.730131\pi$
$257$	− 24.0416i	− 1.49968i	0.661622	−	0.749838i	$-0.269869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.79796	0.604168	0.302084	−	0.953281i	$-0.402318\pi$
$263$	9.79796	0.604168	0.302084	+	0.953281i	$0.402318\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.34847i	0.448044i	0.974584	+	0.224022i	$0.0719188\pi$
$269$	7.34847i	0.448044i	−0.974584	+	0.224022i	$0.928081\pi$
$270$	0	0
$271$	− 31.1769i	− 1.89386i	−0.321436	−	0.946931i	$-0.604165\pi$
$271$	− 31.1769i	− 1.89386i	0.321436	−	0.946931i	$-0.395835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	17.3205	1.04069	0.520344	−	0.853957i	$-0.325804\pi$
$277$	17.3205	1.04069	0.520344	+	0.853957i	$0.325804\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.0416i	1.43420i	0.696969	+	0.717102i	$0.254532\pi$
$281$	24.0416i	1.43420i	−0.696969	+	0.717102i	$0.745468\pi$
$282$	0	0
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.89898	−0.289178
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 26.9444i	− 1.57411i	−0.616884	−	0.787054i	$-0.711605\pi$
$293$	− 26.9444i	− 1.57411i	0.616884	−	0.787054i	$-0.288395\pi$
$294$	0	0
$295$	27.7128i	1.61350i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.9706	−0.981433
$300$	0	0
$301$	27.7128	1.59734
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.9411i	1.94346i
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.79796	−0.555591	−0.277796	−	0.960640i	$-0.589604\pi$
$311$	−9.79796	−0.555591	−0.277796	+	0.960640i	$0.589604\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.2474i	− 0.687885i	−0.938991	−	0.343943i	$-0.888237\pi$
$317$	− 12.2474i	− 0.687885i	0.938991	−	0.343943i	$-0.111763\pi$
$318$	0	0
$319$	6.92820i	0.387905i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	3.46410	0.192154
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 16.9706i	− 0.935617i
$330$	0	0
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.79796	−0.535320
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9.79796i	− 0.530589i
$342$	0	0
$343$	− 6.92820i	− 0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.1421	−0.759190	−0.379595	−	0.925153i	$-0.623937\pi$
$347$	−14.1421	−0.759190	−0.379595	+	0.925153i	$0.623937\pi$
$348$	0	0
$349$	−27.7128	−1.48343	−0.741716	−	0.670714i	$-0.765988\pi$
$349$	−27.7128	−1.48343	−0.741716	+	0.670714i	$0.765988\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.5563i	− 0.827981i	−0.910281	−	0.413990i	$-0.864135\pi$
$353$	− 15.5563i	− 0.827981i	0.910281	−	0.413990i	$-0.135865\pi$
$354$	0	0
$355$	− 36.0000i	− 1.91068i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.6969	0.775675	0.387837	−	0.921728i	$-0.373222\pi$
$359$	14.6969	0.775675	0.387837	+	0.921728i	$0.373222\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.79796i	0.512849i
$366$	0	0
$367$	24.2487i	1.26577i	0.774245	+	0.632886i	$0.218130\pi$
$367$	24.2487i	1.26577i	−0.774245	+	0.632886i	$0.781870\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.4558	−1.32160
$372$	0	0
$373$	13.8564	0.717458	0.358729	−	0.933442i	$-0.383210\pi$
$373$	13.8564	0.717458	0.358729	+	0.933442i	$0.383210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.48528i	− 0.437014i
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.79796	0.500652	0.250326	−	0.968162i	$-0.419462\pi$
$383$	9.79796	0.500652	0.250326	+	0.968162i	$0.419462\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.9444i	1.36613i	0.730355	+	0.683067i	$0.239354\pi$
$389$	26.9444i	1.36613i	−0.730355	+	0.683067i	$0.760646\pi$
$390$	0	0
$391$	6.92820i	0.350374i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.48528	−0.426941
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.41421i	0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$	1.41421i	0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 39.1918i	− 1.92850i
$414$	0	0
$415$	34.6410i	1.70046i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.7990	−0.967244	−0.483622	−	0.875277i	$-0.660679\pi$
$419$	−19.7990	−0.967244	−0.483622	+	0.875277i	$0.660679\pi$
$420$	0	0
$421$	−24.2487	−1.18181	−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487	−1.18181	−0.590905	+	0.806741i	$0.701229\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.41421i	− 0.0685994i
$426$	0	0
$427$	− 48.0000i	− 2.32288i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.6969	−0.707927	−0.353963	−	0.935259i	$-0.615166\pi$
$431$	−14.6969	−0.707927	−0.353963	+	0.935259i	$0.615166\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.5959i	0.937400i
$438$	0	0
$439$	− 3.46410i	− 0.165333i	−0.996577	−	0.0826663i	$-0.973656\pi$
$439$	− 3.46410i	− 0.165333i	0.996577	−	0.0826663i	$-0.0263436\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.82843	0.134383	0.0671913	−	0.997740i	$-0.478596\pi$
$443$	2.82843	0.134383	0.0671913	+	0.997740i	$0.478596\pi$
$444$	0	0
$445$	−17.3205	−0.821071
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.0416i	− 1.13459i	−0.823513	−	0.567297i	$-0.807989\pi$
$449$	− 24.0416i	− 1.13459i	0.823513	−	0.567297i	$-0.192011\pi$
$450$	0	0
$451$	− 4.00000i	− 0.188353i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−29.3939	−1.37801
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.1464i	0.798589i	0.916823	+	0.399294i	$0.130745\pi$
$461$	17.1464i	0.798589i	−0.916823	+	0.399294i	$0.869255\pi$
$462$	0	0
$463$	17.3205i	0.804952i	0.915430	+	0.402476i	$0.131850\pi$
$463$	17.3205i	0.804952i	−0.915430	+	0.402476i	$0.868150\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.7696	−1.70149	−0.850746	−	0.525577i	$-0.823849\pi$
$467$	−36.7696	−1.70149	−0.850746	+	0.525577i	$0.823849\pi$
$468$	0	0
$469$	13.8564	0.639829
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.6274i	1.04041i
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.4949	−1.11920	−0.559600	−	0.828763i	$-0.689045\pi$
$479$	−24.4949	−1.11920	−0.559600	+	0.828763i	$0.689045\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19.5959i	0.889805i
$486$	0	0
$487$	− 10.3923i	− 0.470920i	−0.971884	−	0.235460i	$-0.924340\pi$
$487$	− 10.3923i	− 0.470920i	0.971884	−	0.235460i	$-0.0756597\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.5980	−1.78703	−0.893516	−	0.449032i	$-0.851769\pi$
$491$	−39.5980	−1.78703	−0.893516	+	0.449032i	$0.851769\pi$
$492$	0	0
$493$	−3.46410	−0.156015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.9117i	2.28370i
$498$	0	0
$499$	− 32.0000i	− 1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	− 32.0000i	− 1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.6969	0.655304	0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969	0.655304	0.327652	+	0.944798i	$0.393743\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.6413i	1.84572i	0.385136	+	0.922860i	$0.374154\pi$
$509$	41.6413i	1.84572i	−0.385136	+	0.922860i	$0.625846\pi$
$510$	0	0
$511$	− 13.8564i	− 0.612971i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.48528	−0.373906
$516$	0	0
$517$	13.8564	0.609404
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 26.8701i	− 1.17720i	−0.808425	−	0.588599i	$-0.799680\pi$
$521$	− 26.8701i	− 1.17720i	0.808425	−	0.588599i	$-0.200320\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.89898	0.213403
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.89898i	0.212198i
$534$	0	0
$535$	27.7128i	1.19813i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14.1421	−0.609145
$540$	0	0
$541$	−10.3923	−0.446800	−0.223400	−	0.974727i	$-0.571716\pi$
$541$	−10.3923	−0.446800	−0.223400	+	0.974727i	$0.571716\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 25.4558i	− 1.09041i
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.79796	−0.417407
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.34847i	0.311365i	0.987807	+	0.155682i	$0.0497576\pi$
$557$	7.34847i	0.311365i	−0.987807	+	0.155682i	$0.950242\pi$
$558$	0	0
$559$	− 27.7128i	− 1.17213i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.1421	0.596020	0.298010	−	0.954563i	$-0.403677\pi$
$563$	14.1421	0.596020	0.298010	+	0.954563i	$0.403677\pi$
$564$	0	0
$565$	−45.0333	−1.89457
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.89949i	− 0.415008i	−0.978234	−	0.207504i	$-0.933466\pi$
$569$	− 9.89949i	− 0.415008i	0.978234	−	0.207504i	$-0.0665341\pi$
$570$	0	0
$571$	− 40.0000i	− 1.67395i	−0.547243	−	0.836974i	$-0.684323\pi$
$571$	− 40.0000i	− 1.67395i	0.547243	−	0.836974i	$-0.315677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.89898	−0.204302
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 48.9898i	− 2.03244i
$582$	0	0
$583$	− 20.7846i	− 0.860811i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.6274	−0.933933	−0.466967	−	0.884275i	$-0.654653\pi$
$587$	−22.6274	−0.933933	−0.466967	+	0.884275i	$0.654653\pi$
$588$	0	0
$589$	13.8564	0.570943
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7.07107i	− 0.290374i	−0.989404	−	0.145187i	$-0.953622\pi$
$593$	− 7.07107i	− 0.290374i	0.989404	−	0.145187i	$-0.0463784\pi$
$594$	0	0
$595$	12.0000i	0.491952i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.89898	0.200167	0.100083	−	0.994979i	$-0.468089\pi$
$599$	4.89898	0.200167	0.100083	+	0.994979i	$0.468089\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7.34847i	− 0.298758i
$606$	0	0
$607$	− 45.0333i	− 1.82785i	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	− 45.0333i	− 1.82785i	0.405887	−	0.913923i	$-0.366962\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.9706	−0.686555
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.0416i	0.967880i	0.875101	+	0.483940i	$0.160795\pi$
$617$	24.0416i	0.967880i	−0.875101	+	0.483940i	$0.839205\pi$
$618$	0	0
$619$	8.00000i	0.321547i	0.986991	+	0.160774i	$0.0513989\pi$
$619$	8.00000i	0.321547i	−0.986991	+	0.160774i	$0.948601\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.4949	0.981367
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	31.1769i	1.24113i	0.784154	+	0.620567i	$0.213097\pi$
$631$	31.1769i	1.24113i	−0.784154	+	0.620567i	$0.786903\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−25.4558	−1.01018
$636$	0	0
$637$	17.3205	0.686264
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 15.5563i	− 0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$	− 15.5563i	− 0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$	0	0
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.6969	0.577796	0.288898	−	0.957360i	$-0.406711\pi$
$647$	14.6969	0.577796	0.288898	+	0.957360i	$0.406711\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 17.1464i	− 0.670992i	−0.942042	−	0.335496i	$-0.891096\pi$
$653$	− 17.1464i	− 0.670992i	0.942042	−	0.335496i	$-0.108904\pi$
$654$	0	0
$655$	13.8564i	0.541415i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11.3137	−0.440720	−0.220360	−	0.975419i	$-0.570723\pi$
$659$	−11.3137	−0.440720	−0.220360	+	0.975419i	$0.570723\pi$
$660$	0	0
$661$	13.8564	0.538952	0.269476	−	0.963007i	$-0.413150\pi$
$661$	13.8564	0.538952	0.269476	+	0.963007i	$0.413150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.9411i	1.31618i
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.1918	1.51298
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.1464i	− 0.658991i	−0.944157	−	0.329495i	$-0.893121\pi$
$677$	− 17.1464i	− 0.658991i	0.944157	−	0.329495i	$-0.106879\pi$
$678$	0	0
$679$	− 27.7128i	− 1.06352i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.7696	1.40695	0.703474	−	0.710721i	$-0.251631\pi$
$683$	36.7696	1.40695	0.703474	+	0.710721i	$0.251631\pi$
$684$	0	0
$685$	24.2487	0.926496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25.4558i	0.969790i
$690$	0	0
$691$	− 32.0000i	− 1.21734i	−0.793424	−	0.608669i	$-0.791704\pi$
$691$	− 32.0000i	− 1.21734i	0.793424	−	0.608669i	$-0.208296\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.79796	0.371658
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 7.34847i	− 0.277548i	−0.990324	−	0.138774i	$-0.955684\pi$
$701$	− 7.34847i	− 0.277548i	0.990324	−	0.138774i	$-0.0443161\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.48528	−0.319122
$708$	0	0
$709$	−3.46410	−0.130097	−0.0650485	−	0.997882i	$-0.520720\pi$
$709$	−3.46410	−0.130097	−0.0650485	+	0.997882i	$0.520720\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 16.9706i	− 0.635553i
$714$	0	0
$715$	− 24.0000i	− 0.897549i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	44.0908	1.64431	0.822155	−	0.569264i	$-0.192772\pi$
$719$	44.0908	1.64431	0.822155	+	0.569264i	$0.192772\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.44949i	− 0.0909718i
$726$	0	0
$727$	− 24.2487i	− 0.899335i	−0.893196	−	0.449667i	$-0.851542\pi$
$727$	− 24.2487i	− 0.899335i	0.893196	−	0.449667i	$-0.148458\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.3137	−0.418453
$732$	0	0
$733$	38.1051	1.40744	0.703722	−	0.710475i	$-0.251520\pi$
$733$	38.1051	1.40744	0.703722	+	0.710475i	$0.251520\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3137i	0.416746i
$738$	0	0
$739$	40.0000i	1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$	40.0000i	1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.1918	−1.43781	−0.718905	−	0.695109i	$-0.755356\pi$
$743$	−39.1918	−1.43781	−0.718905	+	0.695109i	$0.755356\pi$
$744$	0	0
$745$	−42.0000	−1.53876
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 39.1918i	− 1.43204i
$750$	0	0
$751$	− 3.46410i	− 0.126407i	−0.998001	−	0.0632034i	$-0.979868\pi$
$751$	− 3.46410i	− 0.126407i	0.998001	−	0.0632034i	$-0.0201317\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.48528	0.308811
$756$	0	0
$757$	10.3923	0.377715	0.188857	−	0.982005i	$-0.439522\pi$
$757$	10.3923	0.377715	0.188857	+	0.982005i	$0.439522\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.3848i	− 0.666448i	−0.942848	−	0.333224i	$-0.891864\pi$
$761$	− 18.3848i	− 0.666448i	0.942848	−	0.333224i	$-0.108136\pi$
$762$	0	0
$763$	36.0000i	1.30329i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−39.1918	−1.41514
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0454i	0.792918i	0.918052	+	0.396459i	$0.129761\pi$
$773$	22.0454i	0.792918i	−0.918052	+	0.396459i	$0.870239\pi$
$774$	0	0
$775$	3.46410i	0.124434i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.65685	0.202678
$780$	0	0
$781$	−41.5692	−1.48746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 33.9411i	− 1.21141i
$786$	0	0
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	63.6867	2.26444
$792$	0	0
$793$	−48.0000	−1.70453
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.2474i	0.433827i	0.976191	+	0.216913i	$0.0695989\pi$
$797$	12.2474i	0.433827i	−0.976191	+	0.216913i	$0.930401\pi$
$798$	0	0
$799$	6.92820i	0.245102i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3137	0.399252
$804$	0	0
$805$	41.5692	1.46512
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1.41421i	− 0.0497211i	−0.999691	−	0.0248606i	$-0.992086\pi$
$809$	− 1.41421i	− 0.0497211i	0.999691	−	0.0248606i	$-0.00791417\pi$
$810$	0	0
$811$	− 4.00000i	− 0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$	− 4.00000i	− 0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−39.1918	−1.37283
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.44949i	− 0.0854878i	−0.999086	−	0.0427439i	$-0.986390\pi$
$821$	− 2.44949i	− 0.0854878i	0.999086	−	0.0427439i	$-0.0136099\pi$
$822$	0	0
$823$	− 17.3205i	− 0.603755i	−0.953347	−	0.301877i	$-0.902387\pi$
$823$	− 17.3205i	− 0.603755i	0.953347	−	0.301877i	$-0.0976134\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.65685	−0.196708	−0.0983540	−	0.995151i	$-0.531358\pi$
$827$	−5.65685	−0.196708	−0.0983540	+	0.995151i	$0.531358\pi$
$828$	0	0
$829$	−10.3923	−0.360940	−0.180470	−	0.983581i	$-0.557762\pi$
$829$	−10.3923	−0.360940	−0.180470	+	0.983581i	$0.557762\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 7.07107i	− 0.244998i
$834$	0	0
$835$	48.0000i	1.66111i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.89898	−0.169132	−0.0845658	−	0.996418i	$-0.526950\pi$
$839$	−4.89898	−0.169132	−0.0845658	+	0.996418i	$0.526950\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.44949i	− 0.0842650i
$846$	0	0
$847$	10.3923i	0.357084i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−13.8564	−0.474434	−0.237217	−	0.971457i	$-0.576235\pi$
$853$	−13.8564	−0.474434	−0.237217	+	0.971457i	$0.576235\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.41421i	− 0.0483086i	−0.999708	−	0.0241543i	$-0.992311\pi$
$857$	− 1.41421i	− 0.0483086i	0.999708	−	0.0241543i	$-0.00768930\pi$
$858$	0	0
$859$	− 40.0000i	− 1.36478i	−0.730987	−	0.682391i	$-0.760940\pi$
$859$	− 40.0000i	− 1.36478i	0.730987	−	0.682391i	$-0.239060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.79796i	0.332373i
$870$	0	0
$871$	− 13.8564i	− 0.469506i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33.9411	1.14742
$876$	0	0
$877$	−55.4256	−1.87159	−0.935795	−	0.352544i	$-0.885317\pi$
$877$	−55.4256	−1.87159	−0.935795	+	0.352544i	$0.885317\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43.8406i	1.47703i	0.674238	+	0.738514i	$0.264472\pi$
$881$	43.8406i	1.47703i	−0.674238	+	0.738514i	$0.735528\pi$
$882$	0	0
$883$	40.0000i	1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$	40.0000i	1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.79796	−0.328983	−0.164492	−	0.986378i	$-0.552598\pi$
$887$	−9.79796	−0.328983	−0.164492	+	0.986378i	$0.552598\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.5959i	0.655752i
$894$	0	0
$895$	13.8564i	0.463169i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.48528	0.283000
$900$	0	0
$901$	10.3923	0.346218
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 25.4558i	− 0.846181i
$906$	0	0
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.1918	−1.29848	−0.649242	−	0.760582i	$-0.724914\pi$
$911$	−39.1918	−1.29848	−0.649242	+	0.760582i	$0.724914\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 19.5959i	− 0.647114i
$918$	0	0
$919$	10.3923i	0.342811i	0.985201	+	0.171405i	$0.0548307\pi$
$919$	10.3923i	0.342811i	−0.985201	+	0.171405i	$0.945169\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	50.9117	1.67578
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.41421i	0.0463988i	0.999731	+	0.0231994i	$0.00738527\pi$
$929$	1.41421i	0.0463988i	−0.999731	+	0.0231994i	$0.992615\pi$
$930$	0	0
$931$	− 20.0000i	− 0.655474i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.79796	−0.320428
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 2.44949i	− 0.0798511i	−0.999203	−	0.0399255i	$-0.987288\pi$
$941$	− 2.44949i	− 0.0798511i	0.999203	−	0.0399255i	$-0.0127121\pi$
$942$	0	0
$943$	− 6.92820i	− 0.225613i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.6274	0.735292	0.367646	−	0.929966i	$-0.380164\pi$
$947$	22.6274	0.735292	0.367646	+	0.929966i	$0.380164\pi$
$948$	0	0
$949$	−13.8564	−0.449798
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 26.8701i	− 0.870407i	−0.900332	−	0.435203i	$-0.856677\pi$
$953$	− 26.8701i	− 0.870407i	0.900332	−	0.435203i	$-0.143323\pi$
$954$	0	0
$955$	48.0000i	1.55324i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.2929	−1.10737
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	34.2929i	1.10393i
$966$	0	0
$967$	45.0333i	1.44817i	0.689709	+	0.724087i	$0.257739\pi$
$967$	45.0333i	1.44817i	−0.689709	+	0.724087i	$0.742261\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.1127	−0.998454	−0.499227	−	0.866471i	$-0.666383\pi$
$971$	−31.1127	−0.998454	−0.499227	+	0.866471i	$0.666383\pi$
$972$	0	0
$973$	−13.8564	−0.444216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.0416i	− 0.769160i	−0.923092	−	0.384580i	$-0.874346\pi$
$977$	− 24.0416i	− 0.769160i	0.923092	−	0.384580i	$-0.125654\pi$
$978$	0	0
$979$	20.0000i	0.639203i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.9898	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$983$	−48.9898	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.1918i	1.24623i
$990$	0	0
$991$	51.9615i	1.65061i	0.564686	+	0.825306i	$0.308997\pi$
$991$	51.9615i	1.65061i	−0.564686	+	0.825306i	$0.691003\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.4558	0.807005
$996$	0	0
$997$	−13.8564	−0.438837	−0.219418	−	0.975631i	$-0.570416\pi$
$997$	−13.8564	−0.438837	−0.219418	+	0.975631i	$0.570416\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.c.i.2303.5		8
3.2	odd	2	inner	2304.2.c.i.2303.1		8
4.3	odd	2	inner	2304.2.c.i.2303.7		8
8.3	odd	2	inner	2304.2.c.i.2303.4		8
8.5	even	2	inner	2304.2.c.i.2303.2		8
12.11	even	2	inner	2304.2.c.i.2303.3		8
16.3	odd	4		288.2.f.a.143.3		4
16.5	even	4		288.2.f.a.143.2		4
16.11	odd	4		72.2.f.a.35.4	yes	4
16.13	even	4		72.2.f.a.35.2	yes	4
24.5	odd	2	inner	2304.2.c.i.2303.6		8
24.11	even	2	inner	2304.2.c.i.2303.8		8
48.5	odd	4		288.2.f.a.143.4		4
48.11	even	4		72.2.f.a.35.1	✓	4
48.29	odd	4		72.2.f.a.35.3	yes	4
48.35	even	4		288.2.f.a.143.1		4
80.3	even	4		7200.2.m.c.3599.4		8
80.13	odd	4		1800.2.m.c.899.8		8
80.19	odd	4		7200.2.b.c.4751.4		4
80.27	even	4		1800.2.m.c.899.4		8
80.29	even	4		1800.2.b.c.251.3		4
80.37	odd	4		7200.2.m.c.3599.3		8
80.43	even	4		1800.2.m.c.899.5		8
80.53	odd	4		7200.2.m.c.3599.8		8
80.59	odd	4		1800.2.b.c.251.1		4
80.67	even	4		7200.2.m.c.3599.7		8
80.69	even	4		7200.2.b.c.4751.2		4
80.77	odd	4		1800.2.m.c.899.1		8
144.5	odd	12		2592.2.p.a.431.1		4
144.11	even	12		648.2.l.a.539.2		4
144.13	even	12		648.2.l.a.107.1		4
144.29	odd	12		648.2.l.c.539.1		4
144.43	odd	12		648.2.l.a.539.1		4
144.59	even	12		648.2.l.c.107.2		4
144.61	even	12		648.2.l.c.539.2		4
144.67	odd	12		2592.2.p.c.431.1		4
144.77	odd	12		648.2.l.a.107.2		4
144.83	even	12		2592.2.p.a.2159.2		4
144.85	even	12		2592.2.p.a.431.2		4
144.101	odd	12		2592.2.p.c.2159.1		4
144.115	odd	12		2592.2.p.a.2159.1		4
144.131	even	12		2592.2.p.c.431.2		4
144.133	even	12		2592.2.p.c.2159.2		4
144.139	odd	12		648.2.l.c.107.1		4
240.29	odd	4		1800.2.b.c.251.2		4
240.53	even	4		7200.2.m.c.3599.5		8
240.59	even	4		1800.2.b.c.251.4		4
240.77	even	4		1800.2.m.c.899.7		8
240.83	odd	4		7200.2.m.c.3599.1		8
240.107	odd	4		1800.2.m.c.899.6		8
240.149	odd	4		7200.2.b.c.4751.1		4
240.173	even	4		1800.2.m.c.899.2		8
240.179	even	4		7200.2.b.c.4751.3		4
240.197	even	4		7200.2.m.c.3599.2		8
240.203	odd	4		1800.2.m.c.899.3		8
240.227	odd	4		7200.2.m.c.3599.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.f.a.35.1	✓	4	48.11	even	4
72.2.f.a.35.2	yes	4	16.13	even	4
72.2.f.a.35.3	yes	4	48.29	odd	4
72.2.f.a.35.4	yes	4	16.11	odd	4
288.2.f.a.143.1		4	48.35	even	4
288.2.f.a.143.2		4	16.5	even	4
288.2.f.a.143.3		4	16.3	odd	4
288.2.f.a.143.4		4	48.5	odd	4
648.2.l.a.107.1		4	144.13	even	12
648.2.l.a.107.2		4	144.77	odd	12
648.2.l.a.539.1		4	144.43	odd	12
648.2.l.a.539.2		4	144.11	even	12
648.2.l.c.107.1		4	144.139	odd	12
648.2.l.c.107.2		4	144.59	even	12
648.2.l.c.539.1		4	144.29	odd	12
648.2.l.c.539.2		4	144.61	even	12
1800.2.b.c.251.1		4	80.59	odd	4
1800.2.b.c.251.2		4	240.29	odd	4
1800.2.b.c.251.3		4	80.29	even	4
1800.2.b.c.251.4		4	240.59	even	4
1800.2.m.c.899.1		8	80.77	odd	4
1800.2.m.c.899.2		8	240.173	even	4
1800.2.m.c.899.3		8	240.203	odd	4
1800.2.m.c.899.4		8	80.27	even	4
1800.2.m.c.899.5		8	80.43	even	4
1800.2.m.c.899.6		8	240.107	odd	4
1800.2.m.c.899.7		8	240.77	even	4
1800.2.m.c.899.8		8	80.13	odd	4
2304.2.c.i.2303.1		8	3.2	odd	2	inner
2304.2.c.i.2303.2		8	8.5	even	2	inner
2304.2.c.i.2303.3		8	12.11	even	2	inner
2304.2.c.i.2303.4		8	8.3	odd	2	inner
2304.2.c.i.2303.5		8	1.1	even	1	trivial
2304.2.c.i.2303.6		8	24.5	odd	2	inner
2304.2.c.i.2303.7		8	4.3	odd	2	inner
2304.2.c.i.2303.8		8	24.11	even	2	inner
2592.2.p.a.431.1		4	144.5	odd	12
2592.2.p.a.431.2		4	144.85	even	12
2592.2.p.a.2159.1		4	144.115	odd	12
2592.2.p.a.2159.2		4	144.83	even	12
2592.2.p.c.431.1		4	144.67	odd	12
2592.2.p.c.431.2		4	144.131	even	12
2592.2.p.c.2159.1		4	144.101	odd	12
2592.2.p.c.2159.2		4	144.133	even	12
7200.2.b.c.4751.1		4	240.149	odd	4
7200.2.b.c.4751.2		4	80.69	even	4
7200.2.b.c.4751.3		4	240.179	even	4
7200.2.b.c.4751.4		4	80.19	odd	4
7200.2.m.c.3599.1		8	240.83	odd	4
7200.2.m.c.3599.2		8	240.197	even	4
7200.2.m.c.3599.3		8	80.37	odd	4
7200.2.m.c.3599.4		8	80.3	even	4
7200.2.m.c.3599.5		8	240.53	even	4
7200.2.m.c.3599.6		8	240.227	odd	4
7200.2.m.c.3599.7		8	80.67	even	4
7200.2.m.c.3599.8		8	80.53	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{5} -3.46410i q^{7} +O(q^{10})\)	\(q+2.44949i q^{5} -3.46410i q^{7} +2.82843 q^{11} -3.46410 q^{13} +1.41421i q^{17} +4.00000i q^{19} +4.89898 q^{23} -1.00000 q^{25} +2.44949i q^{29} -3.46410i q^{31} +8.48528 q^{35} -1.41421i q^{41} +8.00000i q^{43} +4.89898 q^{47} -5.00000 q^{49} -7.34847i q^{53} +6.92820i q^{55} +11.3137 q^{59} +13.8564 q^{61} -8.48528i q^{65} +4.00000i q^{67} -14.6969 q^{71} +4.00000 q^{73} -9.79796i q^{77} +3.46410i q^{79} +14.1421 q^{83} -3.46410 q^{85} +7.07107i q^{89} +12.0000i q^{91} -9.79796 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} - 40 q^{49} + 32 q^{73} + 64 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 - 40 * q^49 + 32 * q^73 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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