LMFDB - Embedded newform 4800.2.k.p.2401.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(2401,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4800.2401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$38.3281929702$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2401.8
Root			$0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	4800.2401
Dual form			4800.2.k.p.2401.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.64575 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.64575 q^{7} -1.00000 q^{9} +3.58258i q^{11} -7.02355i q^{13} -7.58258 q^{17} -4.58258i q^{19} +2.64575i q^{21} +2.55040 q^{23} -1.00000i q^{27} +4.37780i q^{29} -4.28245 q^{31} -3.58258 q^{33} +6.92820i q^{37} +7.02355 q^{39} -9.58258 q^{41} +4.58258i q^{43} -5.29150 q^{47} -7.58258i q^{51} +9.66930i q^{53} +4.58258 q^{57} +9.16515i q^{59} -0.0953502i q^{61} -2.64575 q^{63} +4.58258i q^{67} +2.55040i q^{69} -16.5975 q^{71} +2.00000 q^{73} +9.47860i q^{77} -5.29150 q^{79} +1.00000 q^{81} +10.0000i q^{83} -4.37780 q^{87} +3.16515 q^{89} -18.5826i q^{91} -4.28245i q^{93} +0.165151 q^{97} -3.58258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 24 q^{17} + 8 q^{33} - 40 q^{41} + 16 q^{73} + 8 q^{81} - 48 q^{89} - 72 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 24 * q^17 + 8 * q^33 - 40 * q^41 + 16 * q^73 + 8 * q^81 - 48 * q^89 - 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$7$	2.64575	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.58258i	1.08019i	0.841605	+	0.540094i	$0.181611\pi$
$11$	3.58258i	1.08019i	−0.841605	+	0.540094i	$0.818389\pi$
$12$	0	0
$13$	− 7.02355i	− 1.94798i	−0.226584	−	0.973992i	$-0.572756\pi$
$13$	− 7.02355i	− 1.94798i	0.226584	−	0.973992i	$-0.427244\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.58258	−1.83904	−0.919522	−	0.393038i	$-0.871424\pi$
$17$	−7.58258	−1.83904	−0.919522	+	0.393038i	$0.871424\pi$
$18$	0	0
$19$	− 4.58258i	− 1.05131i	−0.850696	−	0.525657i	$-0.823819\pi$
$19$	− 4.58258i	− 1.05131i	0.850696	−	0.525657i	$-0.176181\pi$
$20$	0	0
$21$	2.64575i	0.577350i
$22$	0	0
$23$	2.55040	0.531795	0.265898	−	0.964001i	$-0.414332\pi$
$23$	2.55040	0.531795	0.265898	+	0.964001i	$0.414332\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	4.37780i	0.812937i	0.913665	+	0.406469i	$0.133240\pi$
$29$	4.37780i	0.812937i	−0.913665	+	0.406469i	$0.866760\pi$
$30$	0	0
$31$	−4.28245	−0.769151	−0.384576	−	0.923094i	$-0.625652\pi$
$31$	−4.28245	−0.769151	−0.384576	+	0.923094i	$0.625652\pi$
$32$	0	0
$33$	−3.58258	−0.623646
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.92820i	1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$	6.92820i	1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	7.02355	1.12467
$40$	0	0
$41$	−9.58258	−1.49655	−0.748273	−	0.663390i	$-0.769117\pi$
$41$	−9.58258	−1.49655	−0.748273	+	0.663390i	$0.769117\pi$
$42$	0	0
$43$	4.58258i	0.698836i	0.936967	+	0.349418i	$0.113621\pi$
$43$	4.58258i	0.698836i	−0.936967	+	0.349418i	$0.886379\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.29150	−0.771845	−0.385922	−	0.922531i	$-0.626117\pi$
$47$	−5.29150	−0.771845	−0.385922	+	0.922531i	$0.626117\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 7.58258i	− 1.06177i
$52$	0	0
$53$	9.66930i	1.32818i	0.747652	+	0.664091i	$0.231181\pi$
$53$	9.66930i	1.32818i	−0.747652	+	0.664091i	$0.768819\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.58258	0.606977
$58$	0	0
$59$	9.16515i	1.19320i	0.802538	+	0.596601i	$0.203482\pi$
$59$	9.16515i	1.19320i	−0.802538	+	0.596601i	$0.796518\pi$
$60$	0	0
$61$	− 0.0953502i	− 0.0122083i	−0.999981	−	0.00610417i	$-0.998057\pi$
$61$	− 0.0953502i	− 0.0122083i	0.999981	−	0.00610417i	$-0.00194303\pi$
$62$	0	0
$63$	−2.64575	−0.333333
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.58258i	0.559851i	0.960022	+	0.279925i	$0.0903097\pi$
$67$	4.58258i	0.559851i	−0.960022	+	0.279925i	$0.909690\pi$
$68$	0	0
$69$	2.55040i	0.307032i
$70$	0	0
$71$	−16.5975	−1.96976	−0.984881	−	0.173233i	$-0.944579\pi$
$71$	−16.5975	−1.96976	−0.984881	+	0.173233i	$0.944579\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.47860i	1.08019i
$78$	0	0
$79$	−5.29150	−0.595341	−0.297670	−	0.954669i	$-0.596210\pi$
$79$	−5.29150	−0.595341	−0.297670	+	0.954669i	$0.596210\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0000i	1.09764i	0.835940	+	0.548821i	$0.184923\pi$
$83$	10.0000i	1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.37780	−0.469350
$88$	0	0
$89$	3.16515	0.335505	0.167753	−	0.985829i	$-0.446349\pi$
$89$	3.16515	0.335505	0.167753	+	0.985829i	$0.446349\pi$
$90$	0	0
$91$	− 18.5826i	− 1.94798i
$92$	0	0
$93$	− 4.28245i	− 0.444070i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.165151	0.0167686	0.00838429	−	0.999965i	$-0.497331\pi$
$97$	0.165151	0.0167686	0.00838429	+	0.999965i	$0.497331\pi$
$98$	0	0
$99$	− 3.58258i	− 0.360062i
$100$	0	0
$101$	12.9427i	1.28785i	0.765090	+	0.643924i	$0.222695\pi$
$101$	12.9427i	1.28785i	−0.765090	+	0.643924i	$0.777305\pi$
$102$	0	0
$103$	14.0471	1.38410	0.692051	−	0.721848i	$-0.256707\pi$
$103$	14.0471	1.38410	0.692051	+	0.721848i	$0.256707\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.58258i	− 0.152993i	−0.997070	−	0.0764967i	$-0.975627\pi$
$107$	− 1.58258i	− 0.152993i	0.997070	−	0.0764967i	$-0.0243735\pi$
$108$	0	0
$109$	12.1244i	1.16130i	0.814152	+	0.580651i	$0.197202\pi$
$109$	12.1244i	1.16130i	−0.814152	+	0.580651i	$0.802798\pi$
$110$	0	0
$111$	−6.92820	−0.657596
$112$	0	0
$113$	11.1652	1.05033	0.525165	−	0.851001i	$-0.324004\pi$
$113$	11.1652	1.05033	0.525165	+	0.851001i	$0.324004\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.02355i	0.649328i
$118$	0	0
$119$	−20.0616	−1.83904
$120$	0	0
$121$	−1.83485	−0.166804
$122$	0	0
$123$	− 9.58258i	− 0.864032i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.2197	−1.08432	−0.542162	−	0.840274i	$-0.682394\pi$
$127$	−12.2197	−1.08432	−0.542162	+	0.840274i	$0.682394\pi$
$128$	0	0
$129$	−4.58258	−0.403473
$130$	0	0
$131$	− 5.58258i	− 0.487752i	−0.969806	−	0.243876i	$-0.921581\pi$
$131$	− 5.58258i	− 0.487752i	0.969806	−	0.243876i	$-0.0784190\pi$
$132$	0	0
$133$	− 12.1244i	− 1.05131i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	− 5.29150i	− 0.445625i
$142$	0	0
$143$	25.1624	2.10419
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 5.29150i	− 0.433497i	−0.976228	−	0.216748i	$-0.930455\pi$
$149$	− 5.29150i	− 0.433497i	0.976228	−	0.216748i	$-0.0695451\pi$
$150$	0	0
$151$	−14.6748	−1.19421	−0.597107	−	0.802161i	$-0.703683\pi$
$151$	−14.6748	−1.19421	−0.597107	+	0.802161i	$0.703683\pi$
$152$	0	0
$153$	7.58258	0.613015
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.73205i	0.138233i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	1.73205i	0.138233i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$	0	0
$159$	−9.66930	−0.766826
$160$	0	0
$161$	6.74773	0.531795
$162$	0	0
$163$	− 5.74773i	− 0.450197i	−0.974336	−	0.225098i	$-0.927730\pi$
$163$	− 5.74773i	− 0.450197i	0.974336	−	0.225098i	$-0.0722703\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.0797	1.70858	0.854290	−	0.519797i	$-0.173993\pi$
$167$	22.0797	1.70858	0.854290	+	0.519797i	$0.173993\pi$
$168$	0	0
$169$	−36.3303	−2.79464
$170$	0	0
$171$	4.58258i	0.350438i
$172$	0	0
$173$	− 6.20520i	− 0.471773i	−0.971781	−	0.235886i	$-0.924201\pi$
$173$	− 6.20520i	− 0.471773i	0.971781	−	0.235886i	$-0.0757993\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.16515	−0.688895
$178$	0	0
$179$	− 18.0000i	− 1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$	− 18.0000i	− 1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$	0	0
$181$	19.2433i	1.43034i	0.698950	+	0.715170i	$0.253651\pi$
$181$	19.2433i	1.43034i	−0.698950	+	0.715170i	$0.746349\pi$
$182$	0	0
$183$	0.0953502	0.00704849
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 27.1652i	− 1.98651i
$188$	0	0
$189$	− 2.64575i	− 0.192450i
$190$	0	0
$191$	8.03260	0.581219	0.290609	−	0.956842i	$-0.406142\pi$
$191$	8.03260	0.581219	0.290609	+	0.956842i	$0.406142\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.56490i	0.610224i	0.952316	+	0.305112i	$0.0986939\pi$
$197$	8.56490i	0.610224i	−0.952316	+	0.305112i	$0.901306\pi$
$198$	0	0
$199$	21.9844	1.55843	0.779215	−	0.626757i	$-0.215618\pi$
$199$	21.9844	1.55843	0.779215	+	0.626757i	$0.215618\pi$
$200$	0	0
$201$	−4.58258	−0.323230
$202$	0	0
$203$	11.5826i	0.812937i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.55040	−0.177265
$208$	0	0
$209$	16.4174	1.13562
$210$	0	0
$211$	11.7477i	0.808747i	0.914594	+	0.404373i	$0.132510\pi$
$211$	11.7477i	0.808747i	−0.914594	+	0.404373i	$0.867490\pi$
$212$	0	0
$213$	− 16.5975i	− 1.13724i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.3303	−0.769151
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	53.2566i	3.58243i
$222$	0	0
$223$	−9.76465	−0.653890	−0.326945	−	0.945043i	$-0.606019\pi$
$223$	−9.76465	−0.653890	−0.326945	+	0.945043i	$0.606019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.5826i	− 1.16700i	−0.812114	−	0.583498i	$-0.801683\pi$
$227$	− 17.5826i	− 1.16700i	0.812114	−	0.583498i	$-0.198317\pi$
$228$	0	0
$229$	13.9518i	0.921958i	0.887411	+	0.460979i	$0.152502\pi$
$229$	13.9518i	0.921958i	−0.887411	+	0.460979i	$0.847498\pi$
$230$	0	0
$231$	−9.47860	−0.623646
$232$	0	0
$233$	−21.5826	−1.41392	−0.706961	−	0.707253i	$-0.749934\pi$
$233$	−21.5826	−1.41392	−0.706961	+	0.707253i	$0.749934\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5.29150i	− 0.343720i
$238$	0	0
$239$	−2.35970	−0.152636	−0.0763182	−	0.997084i	$-0.524316\pi$
$239$	−2.35970	−0.152636	−0.0763182	+	0.997084i	$0.524316\pi$
$240$	0	0
$241$	−28.4955	−1.83555	−0.917777	−	0.397097i	$-0.870018\pi$
$241$	−28.4955	−1.83555	−0.917777	+	0.397097i	$0.870018\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−32.1860	−2.04794
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	− 27.1652i	− 1.71465i	−0.514776	−	0.857325i	$-0.672125\pi$
$251$	− 27.1652i	− 1.71465i	0.514776	−	0.857325i	$-0.327875\pi$
$252$	0	0
$253$	9.13701i	0.574439i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.83485	−0.301590	−0.150795	−	0.988565i	$-0.548183\pi$
$257$	−4.83485	−0.301590	−0.150795	+	0.988565i	$0.548183\pi$
$258$	0	0
$259$	18.3303i	1.13899i
$260$	0	0
$261$	− 4.37780i	− 0.270979i
$262$	0	0
$263$	11.1153	0.685399	0.342700	−	0.939445i	$-0.388659\pi$
$263$	11.1153	0.685399	0.342700	+	0.939445i	$0.388659\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.16515i	0.193704i
$268$	0	0
$269$	− 7.65120i	− 0.466502i	−0.972417	−	0.233251i	$-0.925064\pi$
$269$	− 7.65120i	− 0.466502i	0.972417	−	0.233251i	$-0.0749364\pi$
$270$	0	0
$271$	−27.9035	−1.69502	−0.847509	−	0.530781i	$-0.821899\pi$
$271$	−27.9035	−1.69502	−0.847509	+	0.530781i	$0.821899\pi$
$272$	0	0
$273$	18.5826	1.12467
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.55945i	0.213867i	0.994266	+	0.106933i	$0.0341032\pi$
$277$	3.55945i	0.213867i	−0.994266	+	0.106933i	$0.965897\pi$
$278$	0	0
$279$	4.28245	0.256384
$280$	0	0
$281$	−16.4174	−0.979381	−0.489691	−	0.871896i	$-0.662890\pi$
$281$	−16.4174	−0.979381	−0.489691	+	0.871896i	$0.662890\pi$
$282$	0	0
$283$	− 14.5826i	− 0.866844i	−0.901191	−	0.433422i	$-0.857306\pi$
$283$	− 14.5826i	− 0.866844i	0.901191	−	0.433422i	$-0.142694\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.3531	−1.49655
$288$	0	0
$289$	40.4955	2.38209
$290$	0	0
$291$	0.165151i	0.00968135i
$292$	0	0
$293$	− 8.94630i	− 0.522649i	−0.965251	−	0.261324i	$-0.915841\pi$
$293$	− 8.94630i	− 0.522649i	0.965251	−	0.261324i	$-0.0841592\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.58258	0.207882
$298$	0	0
$299$	− 17.9129i	− 1.03593i
$300$	0	0
$301$	12.1244i	0.698836i
$302$	0	0
$303$	−12.9427	−0.743539
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.5826i	0.832272i	0.909302	+	0.416136i	$0.136616\pi$
$307$	14.5826i	0.832272i	−0.909302	+	0.416136i	$0.863384\pi$
$308$	0	0
$309$	14.0471i	0.799112i
$310$	0	0
$311$	19.1479	1.08578	0.542889	−	0.839804i	$-0.317330\pi$
$311$	19.1479	1.08578	0.542889	+	0.839804i	$0.317330\pi$
$312$	0	0
$313$	−8.16515	−0.461522	−0.230761	−	0.973011i	$-0.574121\pi$
$313$	−8.16515	−0.461522	−0.230761	+	0.973011i	$0.574121\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.9216i	1.68057i	0.542149	+	0.840283i	$0.317611\pi$
$317$	29.9216i	1.68057i	−0.542149	+	0.840283i	$0.682389\pi$
$318$	0	0
$319$	−15.6838	−0.878125
$320$	0	0
$321$	1.58258	0.0883308
$322$	0	0
$323$	34.7477i	1.93342i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.1244	−0.670478
$328$	0	0
$329$	−14.0000	−0.771845
$330$	0	0
$331$	− 7.16515i	− 0.393832i	−0.980420	−	0.196916i	$-0.936907\pi$
$331$	− 7.16515i	− 0.393832i	0.980420	−	0.196916i	$-0.0630927\pi$
$332$	0	0
$333$	− 6.92820i	− 0.379663i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.16515	0.226890	0.113445	−	0.993544i	$-0.463811\pi$
$337$	4.16515	0.226890	0.113445	+	0.993544i	$0.463811\pi$
$338$	0	0
$339$	11.1652i	0.606408i
$340$	0	0
$341$	− 15.3422i	− 0.830827i
$342$	0	0
$343$	−18.5203	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.3303i	1.09139i	0.837985	+	0.545694i	$0.183734\pi$
$347$	20.3303i	1.09139i	−0.837985	+	0.545694i	$0.816266\pi$
$348$	0	0
$349$	− 21.1660i	− 1.13299i	−0.824065	−	0.566495i	$-0.808299\pi$
$349$	− 21.1660i	− 1.13299i	0.824065	−	0.566495i	$-0.191701\pi$
$350$	0	0
$351$	−7.02355	−0.374890
$352$	0	0
$353$	−13.9129	−0.740508	−0.370254	−	0.928931i	$-0.620729\pi$
$353$	−13.9129	−0.740508	−0.370254	+	0.928931i	$0.620729\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 20.0616i	− 1.06177i
$358$	0	0
$359$	−32.2813	−1.70374	−0.851871	−	0.523751i	$-0.824532\pi$
$359$	−32.2813	−1.70374	−0.851871	+	0.523751i	$0.824532\pi$
$360$	0	0
$361$	−2.00000	−0.105263
$362$	0	0
$363$	− 1.83485i	− 0.0963046i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.83645	0.148062	0.0740308	−	0.997256i	$-0.476414\pi$
$367$	2.83645	0.148062	0.0740308	+	0.997256i	$0.476414\pi$
$368$	0	0
$369$	9.58258	0.498849
$370$	0	0
$371$	25.5826i	1.32818i
$372$	0	0
$373$	− 36.5638i	− 1.89320i	−0.322410	−	0.946600i	$-0.604493\pi$
$373$	− 36.5638i	− 1.89320i	0.322410	−	0.946600i	$-0.395507\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.7477	1.58359
$378$	0	0
$379$	− 15.7477i	− 0.808906i	−0.914559	−	0.404453i	$-0.867462\pi$
$379$	− 15.7477i	− 0.808906i	0.914559	−	0.404453i	$-0.132538\pi$
$380$	0	0
$381$	− 12.2197i	− 0.626034i
$382$	0	0
$383$	33.1950	1.69619	0.848093	−	0.529847i	$-0.177751\pi$
$383$	33.1950	1.69619	0.848093	+	0.529847i	$0.177751\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.58258i	− 0.232945i
$388$	0	0
$389$	− 4.18710i	− 0.212294i	−0.994350	−	0.106147i	$-0.966149\pi$
$389$	− 4.18710i	− 0.212294i	0.994350	−	0.106147i	$-0.0338515\pi$
$390$	0	0
$391$	−19.3386	−0.977996
$392$	0	0
$393$	5.58258	0.281604
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.38685i	− 0.270358i	−0.990821	−	0.135179i	$-0.956839\pi$
$397$	− 5.38685i	− 0.270358i	0.990821	−	0.135179i	$-0.0431610\pi$
$398$	0	0
$399$	12.1244	0.606977
$400$	0	0
$401$	−24.7477	−1.23584	−0.617921	−	0.786240i	$-0.712025\pi$
$401$	−24.7477	−1.23584	−0.617921	+	0.786240i	$0.712025\pi$
$402$	0	0
$403$	30.0780i	1.49829i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.8208	−1.23032
$408$	0	0
$409$	7.33030	0.362460	0.181230	−	0.983441i	$-0.441992\pi$
$409$	7.33030	0.362460	0.181230	+	0.983441i	$0.441992\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	24.2487i	1.19320i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 29.5826i	− 1.44520i	−0.691265	−	0.722602i	$-0.742946\pi$
$419$	− 29.5826i	− 1.44520i	0.691265	−	0.722602i	$-0.257054\pi$
$420$	0	0
$421$	− 10.5830i	− 0.515784i	−0.966174	−	0.257892i	$-0.916972\pi$
$421$	− 10.5830i	− 0.515784i	0.966174	−	0.257892i	$-0.0830279\pi$
$422$	0	0
$423$	5.29150	0.257282
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.252273i	− 0.0122083i
$428$	0	0
$429$	25.1624i	1.21485i
$430$	0	0
$431$	0.723000	0.0348257	0.0174128	−	0.999848i	$-0.494457\pi$
$431$	0.723000	0.0348257	0.0174128	+	0.999848i	$0.494457\pi$
$432$	0	0
$433$	30.1652	1.44964	0.724822	−	0.688936i	$-0.241922\pi$
$433$	30.1652	1.44964	0.724822	+	0.688936i	$0.241922\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.6874i	− 0.559084i
$438$	0	0
$439$	−25.4485	−1.21459	−0.607294	−	0.794477i	$-0.707745\pi$
$439$	−25.4485	−1.21459	−0.607294	+	0.794477i	$0.707745\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.66970i	0.0793297i	0.999213	+	0.0396649i	$0.0126290\pi$
$443$	1.66970i	0.0793297i	−0.999213	+	0.0396649i	$0.987371\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.29150	0.250279
$448$	0	0
$449$	1.66970	0.0787979	0.0393989	−	0.999224i	$-0.487456\pi$
$449$	1.66970	0.0787979	0.0393989	+	0.999224i	$0.487456\pi$
$450$	0	0
$451$	− 34.3303i	− 1.61655i
$452$	0	0
$453$	− 14.6748i	− 0.689480i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.8348	0.506833	0.253416	−	0.967357i	$-0.418446\pi$
$457$	10.8348	0.506833	0.253416	+	0.967357i	$0.418446\pi$
$458$	0	0
$459$	7.58258i	0.353924i
$460$	0	0
$461$	9.28790i	0.432581i	0.976329	+	0.216290i	$0.0693958\pi$
$461$	9.28790i	0.432581i	−0.976329	+	0.216290i	$0.930604\pi$
$462$	0	0
$463$	−13.6657	−0.635099	−0.317550	−	0.948242i	$-0.602860\pi$
$463$	−13.6657	−0.635099	−0.317550	+	0.948242i	$0.602860\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.58258i	− 0.350880i	−0.984490	−	0.175440i	$-0.943865\pi$
$467$	− 7.58258i	− 0.350880i	0.984490	−	0.175440i	$-0.0561348\pi$
$468$	0	0
$469$	12.1244i	0.559851i
$470$	0	0
$471$	−1.73205	−0.0798087
$472$	0	0
$473$	−16.4174	−0.754874
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 9.66930i	− 0.442727i
$478$	0	0
$479$	0.190700	0.00871332	0.00435666	−	0.999991i	$-0.498613\pi$
$479$	0.190700	0.00871332	0.00435666	+	0.999991i	$0.498613\pi$
$480$	0	0
$481$	48.6606	2.21873
$482$	0	0
$483$	6.74773i	0.307032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.74655	0.351030	0.175515	−	0.984477i	$-0.443841\pi$
$487$	7.74655	0.351030	0.175515	+	0.984477i	$0.443841\pi$
$488$	0	0
$489$	5.74773	0.259921
$490$	0	0
$491$	23.1652i	1.04543i	0.852508	+	0.522714i	$0.175081\pi$
$491$	23.1652i	1.04543i	−0.852508	+	0.522714i	$0.824919\pi$
$492$	0	0
$493$	− 33.1950i	− 1.49503i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−43.9129	−1.96976
$498$	0	0
$499$	− 6.58258i	− 0.294677i	−0.989086	−	0.147338i	$-0.952929\pi$
$499$	− 6.58258i	− 0.294677i	0.989086	−	0.147338i	$-0.0470706\pi$
$500$	0	0
$501$	22.0797i	0.986449i
$502$	0	0
$503$	39.2095	1.74827	0.874133	−	0.485687i	$-0.161430\pi$
$503$	39.2095	1.74827	0.874133	+	0.485687i	$0.161430\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 36.3303i	− 1.61349i
$508$	0	0
$509$	− 22.8027i	− 1.01071i	−0.862911	−	0.505356i	$-0.831361\pi$
$509$	− 22.8027i	− 1.01071i	0.862911	−	0.505356i	$-0.168639\pi$
$510$	0	0
$511$	5.29150	0.234082
$512$	0	0
$513$	−4.58258	−0.202326
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 18.9572i	− 0.833737i
$518$	0	0
$519$	6.20520	0.272378
$520$	0	0
$521$	−22.7477	−0.996596	−0.498298	−	0.867006i	$-0.666041\pi$
$521$	−22.7477	−0.996596	−0.498298	+	0.867006i	$0.666041\pi$
$522$	0	0
$523$	− 35.7477i	− 1.56314i	−0.623818	−	0.781569i	$-0.714419\pi$
$523$	− 35.7477i	− 1.56314i	0.623818	−	0.781569i	$-0.285581\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.4720	1.41450
$528$	0	0
$529$	−16.4955	−0.717194
$530$	0	0
$531$	− 9.16515i	− 0.397734i
$532$	0	0
$533$	67.3037i	2.91525i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 24.1534i	− 1.03843i	−0.854642	−	0.519217i	$-0.826224\pi$
$541$	− 24.1534i	− 1.03843i	0.854642	−	0.519217i	$-0.173776\pi$
$542$	0	0
$543$	−19.2433	−0.825807
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 40.6606i	− 1.73852i	−0.494353	−	0.869261i	$-0.664595\pi$
$547$	− 40.6606i	− 1.73852i	0.494353	−	0.869261i	$-0.335405\pi$
$548$	0	0
$549$	0.0953502i	0.00406945i
$550$	0	0
$551$	20.0616	0.854653
$552$	0	0
$553$	−14.0000	−0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.3923i	− 0.440336i	−0.975462	−	0.220168i	$-0.929339\pi$
$557$	− 10.3923i	− 0.440336i	0.975462	−	0.220168i	$-0.0706606\pi$
$558$	0	0
$559$	32.1860	1.36132
$560$	0	0
$561$	27.1652	1.14691
$562$	0	0
$563$	− 41.0780i	− 1.73123i	−0.500708	−	0.865616i	$-0.666927\pi$
$563$	− 41.0780i	− 1.73123i	0.500708	−	0.865616i	$-0.333073\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.64575	0.111111
$568$	0	0
$569$	−41.0780	−1.72208	−0.861040	−	0.508537i	$-0.830187\pi$
$569$	−41.0780	−1.72208	−0.861040	+	0.508537i	$0.830187\pi$
$570$	0	0
$571$	24.0780i	1.00763i	0.863810	+	0.503817i	$0.168071\pi$
$571$	24.0780i	1.00763i	−0.863810	+	0.503817i	$0.831929\pi$
$572$	0	0
$573$	8.03260i	0.335567i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−44.1652	−1.83862	−0.919310	−	0.393535i	$-0.871252\pi$
$577$	−44.1652	−1.83862	−0.919310	+	0.393535i	$0.871252\pi$
$578$	0	0
$579$	17.0000i	0.706496i
$580$	0	0
$581$	26.4575i	1.09764i
$582$	0	0
$583$	−34.6410	−1.43468
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.4174i	1.42056i	0.703920	+	0.710280i	$0.251431\pi$
$587$	34.4174i	1.42056i	−0.703920	+	0.710280i	$0.748569\pi$
$588$	0	0
$589$	19.6247i	0.808620i
$590$	0	0
$591$	−8.56490	−0.352313
$592$	0	0
$593$	−2.41742	−0.0992717	−0.0496359	−	0.998767i	$-0.515806\pi$
$593$	−2.41742	−0.0992717	−0.0496359	+	0.998767i	$0.515806\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.9844i	0.899760i
$598$	0	0
$599$	24.8208	1.01415	0.507075	−	0.861902i	$-0.330727\pi$
$599$	24.8208	1.01415	0.507075	+	0.861902i	$0.330727\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	0	0
$603$	− 4.58258i	− 0.186617i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.8745	0.644326	0.322163	−	0.946684i	$-0.395590\pi$
$607$	15.8745	0.644326	0.322163	+	0.946684i	$0.395590\pi$
$608$	0	0
$609$	−11.5826	−0.469350
$610$	0	0
$611$	37.1652i	1.50354i
$612$	0	0
$613$	18.9572i	0.765674i	0.923816	+	0.382837i	$0.125053\pi$
$613$	18.9572i	0.765674i	−0.923816	+	0.382837i	$0.874947\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.58258	0.224746	0.112373	−	0.993666i	$-0.464155\pi$
$617$	5.58258	0.224746	0.112373	+	0.993666i	$0.464155\pi$
$618$	0	0
$619$	36.9129i	1.48365i	0.670591	+	0.741827i	$0.266040\pi$
$619$	36.9129i	1.48365i	−0.670591	+	0.741827i	$0.733960\pi$
$620$	0	0
$621$	− 2.55040i	− 0.102344i
$622$	0	0
$623$	8.37420	0.335505
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.4174i	0.655649i
$628$	0	0
$629$	− 52.5336i	− 2.09465i
$630$	0	0
$631$	24.0025	0.955523	0.477761	−	0.878490i	$-0.341448\pi$
$631$	24.0025	0.955523	0.477761	+	0.878490i	$0.341448\pi$
$632$	0	0
$633$	−11.7477	−0.466930
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	16.5975	0.656587
$640$	0	0
$641$	−2.33030	−0.0920414	−0.0460207	−	0.998940i	$-0.514654\pi$
$641$	−2.33030	−0.0920414	−0.0460207	+	0.998940i	$0.514654\pi$
$642$	0	0
$643$	− 6.33030i	− 0.249643i	−0.992179	−	0.124821i	$-0.960164\pi$
$643$	− 6.33030i	− 0.249643i	0.992179	−	0.124821i	$-0.0398358\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.27340	0.128691	0.0643453	−	0.997928i	$-0.479504\pi$
$647$	3.27340	0.128691	0.0643453	+	0.997928i	$0.479504\pi$
$648$	0	0
$649$	−32.8348	−1.28888
$650$	0	0
$651$	− 11.3303i	− 0.444070i
$652$	0	0
$653$	11.8383i	0.463269i	0.972803	+	0.231634i	$0.0744073\pi$
$653$	11.8383i	0.463269i	−0.972803	+	0.231634i	$0.925593\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	8.74773i	0.340763i	0.985378	+	0.170382i	$0.0545000\pi$
$659$	8.74773i	0.340763i	−0.985378	+	0.170382i	$0.945500\pi$
$660$	0	0
$661$	16.0652i	0.624864i	0.949940	+	0.312432i	$0.101144\pi$
$661$	16.0652i	0.624864i	−0.949940	+	0.312432i	$0.898856\pi$
$662$	0	0
$663$	−53.2566	−2.06832
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.1652i	0.432316i
$668$	0	0
$669$	− 9.76465i	− 0.377523i
$670$	0	0
$671$	0.341599	0.0131873
$672$	0	0
$673$	32.3303	1.24624	0.623121	−	0.782126i	$-0.285864\pi$
$673$	32.3303	1.24624	0.623121	+	0.782126i	$0.285864\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.4720i	1.24800i	0.781424	+	0.624000i	$0.214494\pi$
$677$	32.4720i	1.24800i	−0.781424	+	0.624000i	$0.785506\pi$
$678$	0	0
$679$	0.436950	0.0167686
$680$	0	0
$681$	17.5826	0.673766
$682$	0	0
$683$	− 7.16515i	− 0.274167i	−0.990560	−	0.137083i	$-0.956227\pi$
$683$	− 7.16515i	− 0.274167i	0.990560	−	0.137083i	$-0.0437728\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.9518	−0.532293
$688$	0	0
$689$	67.9129	2.58727
$690$	0	0
$691$	− 19.1652i	− 0.729077i	−0.931188	−	0.364538i	$-0.881227\pi$
$691$	− 19.1652i	− 0.729077i	0.931188	−	0.364538i	$-0.118773\pi$
$692$	0	0
$693$	− 9.47860i	− 0.360062i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	72.6606	2.75222
$698$	0	0
$699$	− 21.5826i	− 0.816328i
$700$	0	0
$701$	− 3.99640i	− 0.150942i	−0.997148	−	0.0754710i	$-0.975954\pi$
$701$	− 3.99640i	− 0.150942i	0.997148	−	0.0754710i	$-0.0240460\pi$
$702$	0	0
$703$	31.7490	1.19744
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.2432i	1.28785i
$708$	0	0
$709$	8.66025i	0.325243i	0.986689	+	0.162621i	$0.0519949\pi$
$709$	8.66025i	0.325243i	−0.986689	+	0.162621i	$0.948005\pi$
$710$	0	0
$711$	5.29150	0.198447
$712$	0	0
$713$	−10.9220	−0.409031
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.35970i	− 0.0881246i
$718$	0	0
$719$	−39.4002	−1.46938	−0.734690	−	0.678403i	$-0.762672\pi$
$719$	−39.4002	−1.46938	−0.734690	+	0.678403i	$0.762672\pi$
$720$	0	0
$721$	37.1652	1.38410
$722$	0	0
$723$	− 28.4955i	− 1.05976i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.1570	0.747580	0.373790	−	0.927513i	$-0.378058\pi$
$727$	20.1570	0.747580	0.373790	+	0.927513i	$0.378058\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 34.7477i	− 1.28519i
$732$	0	0
$733$	− 3.65480i	− 0.134993i	−0.997720	−	0.0674966i	$-0.978499\pi$
$733$	− 3.65480i	− 0.134993i	0.997720	−	0.0674966i	$-0.0215012\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.4174	−0.604744
$738$	0	0
$739$	− 2.33030i	− 0.0857216i	−0.999081	−	0.0428608i	$-0.986353\pi$
$739$	− 2.33030i	− 0.0857216i	0.999081	−	0.0428608i	$-0.0136472\pi$
$740$	0	0
$741$	− 32.1860i	− 1.18238i
$742$	0	0
$743$	12.0290	0.441301	0.220651	−	0.975353i	$-0.429182\pi$
$743$	12.0290	0.441301	0.220651	+	0.975353i	$0.429182\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.0000i	− 0.365881i
$748$	0	0
$749$	− 4.18710i	− 0.152993i
$750$	0	0
$751$	36.6591	1.33771	0.668855	−	0.743393i	$-0.266785\pi$
$751$	36.6591	1.33771	0.668855	+	0.743393i	$0.266785\pi$
$752$	0	0
$753$	27.1652	0.989953
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.9518i	0.507085i	0.967324	+	0.253543i	$0.0815958\pi$
$757$	13.9518i	0.507085i	−0.967324	+	0.253543i	$0.918404\pi$
$758$	0	0
$759$	−9.13701	−0.331652
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	32.0780i	1.16130i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	64.3719	2.32434
$768$	0	0
$769$	31.8348	1.14799	0.573997	−	0.818857i	$-0.305392\pi$
$769$	31.8348	1.14799	0.573997	+	0.818857i	$0.305392\pi$
$770$	0	0
$771$	− 4.83485i	− 0.174123i
$772$	0	0
$773$	21.6983i	0.780434i	0.920723	+	0.390217i	$0.127600\pi$
$773$	21.6983i	0.780434i	−0.920723	+	0.390217i	$0.872400\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−18.3303	−0.657596
$778$	0	0
$779$	43.9129i	1.57334i
$780$	0	0
$781$	− 59.4618i	− 2.12771i
$782$	0	0
$783$	4.37780	0.156450
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.9129i	0.959341i	0.877449	+	0.479670i	$0.159244\pi$
$787$	26.9129i	0.959341i	−0.877449	+	0.479670i	$0.840756\pi$
$788$	0	0
$789$	11.1153i	0.395715i
$790$	0	0
$791$	29.5402	1.05033
$792$	0	0
$793$	−0.669697	−0.0237816
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 0.381401i	− 0.0135099i	−0.999977	−	0.00675495i	$-0.997850\pi$
$797$	− 0.381401i	− 0.0135099i	0.999977	−	0.00675495i	$-0.00215019\pi$
$798$	0	0
$799$	40.1232	1.41946
$800$	0	0
$801$	−3.16515	−0.111835
$802$	0	0
$803$	7.16515i	0.252853i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.65120	0.269335
$808$	0	0
$809$	−3.25227	−0.114344	−0.0571719	−	0.998364i	$-0.518208\pi$
$809$	−3.25227	−0.114344	−0.0571719	+	0.998364i	$0.518208\pi$
$810$	0	0
$811$	41.4174i	1.45436i	0.686446	+	0.727181i	$0.259170\pi$
$811$	41.4174i	1.45436i	−0.686446	+	0.727181i	$0.740830\pi$
$812$	0	0
$813$	− 27.9035i	− 0.978619i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	21.0000	0.734697
$818$	0	0
$819$	18.5826i	0.649328i
$820$	0	0
$821$	− 36.6591i	− 1.27941i	−0.768620	−	0.639706i	$-0.779056\pi$
$821$	− 36.6591i	− 1.27941i	0.768620	−	0.639706i	$-0.220944\pi$
$822$	0	0
$823$	8.12795	0.283323	0.141661	−	0.989915i	$-0.454756\pi$
$823$	8.12795	0.283323	0.141661	+	0.989915i	$0.454756\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.2523i	0.391280i	0.980676	+	0.195640i	$0.0626784\pi$
$827$	11.2523i	0.391280i	−0.980676	+	0.195640i	$0.937322\pi$
$828$	0	0
$829$	50.7062i	1.76110i	0.473953	+	0.880550i	$0.342827\pi$
$829$	50.7062i	1.76110i	−0.473953	+	0.880550i	$0.657173\pi$
$830$	0	0
$831$	−3.55945	−0.123476
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.28245i	0.148023i
$838$	0	0
$839$	−9.86001	−0.340405	−0.170203	−	0.985409i	$-0.554442\pi$
$839$	−9.86001	−0.340405	−0.170203	+	0.985409i	$0.554442\pi$
$840$	0	0
$841$	9.83485	0.339133
$842$	0	0
$843$	− 16.4174i	− 0.565446i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.85455	−0.166804
$848$	0	0
$849$	14.5826	0.500473
$850$	0	0
$851$	17.6697i	0.605709i
$852$	0	0
$853$	24.5348i	0.840054i	0.907512	+	0.420027i	$0.137979\pi$
$853$	24.5348i	0.840054i	−0.907512	+	0.420027i	$0.862021\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.8348	0.848342	0.424171	−	0.905582i	$-0.360565\pi$
$857$	24.8348	0.848342	0.424171	+	0.905582i	$0.360565\pi$
$858$	0	0
$859$	3.16515i	0.107994i	0.998541	+	0.0539968i	$0.0171961\pi$
$859$	3.16515i	0.107994i	−0.998541	+	0.0539968i	$0.982804\pi$
$860$	0	0
$861$	− 25.3531i	− 0.864032i
$862$	0	0
$863$	−26.2668	−0.894133	−0.447066	−	0.894501i	$-0.647531\pi$
$863$	−26.2668	−0.894133	−0.447066	+	0.894501i	$0.647531\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	40.4955i	1.37530i
$868$	0	0
$869$	− 18.9572i	− 0.643079i
$870$	0	0
$871$	32.1860	1.09058
$872$	0	0
$873$	−0.165151	−0.00558953
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 34.7364i	− 1.17296i	−0.809962	−	0.586482i	$-0.800513\pi$
$877$	− 34.7364i	− 1.17296i	0.809962	−	0.586482i	$-0.199487\pi$
$878$	0	0
$879$	8.94630	0.301751
$880$	0	0
$881$	34.3303	1.15662	0.578309	−	0.815818i	$-0.303713\pi$
$881$	34.3303	1.15662	0.578309	+	0.815818i	$0.303713\pi$
$882$	0	0
$883$	− 7.08712i	− 0.238501i	−0.992864	−	0.119250i	$-0.961951\pi$
$883$	− 7.08712i	− 0.238501i	0.992864	−	0.119250i	$-0.0380491\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.3205	0.581566	0.290783	−	0.956789i	$-0.406084\pi$
$887$	17.3205	0.581566	0.290783	+	0.956789i	$0.406084\pi$
$888$	0	0
$889$	−32.3303	−1.08432
$890$	0	0
$891$	3.58258i	0.120021i
$892$	0	0
$893$	24.2487i	0.811452i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.9129	0.598094
$898$	0	0
$899$	− 18.7477i	− 0.625272i
$900$	0	0
$901$	− 73.3182i	− 2.44258i
$902$	0	0
$903$	−12.1244	−0.403473
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.6697i	0.453895i	0.973907	+	0.226947i	$0.0728746\pi$
$907$	13.6697i	0.453895i	−0.973907	+	0.226947i	$0.927125\pi$
$908$	0	0
$909$	− 12.9427i	− 0.429282i
$910$	0	0
$911$	38.1051	1.26248	0.631239	−	0.775588i	$-0.282547\pi$
$911$	38.1051	1.26248	0.631239	+	0.775588i	$0.282547\pi$
$912$	0	0
$913$	−35.8258	−1.18566
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.7701i	− 0.487752i
$918$	0	0
$919$	33.8227	1.11571	0.557853	−	0.829940i	$-0.311625\pi$
$919$	33.8227	1.11571	0.557853	+	0.829940i	$0.311625\pi$
$920$	0	0
$921$	−14.5826	−0.480512
$922$	0	0
$923$	116.573i	3.83706i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.0471	−0.461368
$928$	0	0
$929$	48.2432	1.58281	0.791404	−	0.611294i	$-0.209351\pi$
$929$	48.2432	1.58281	0.791404	+	0.611294i	$0.209351\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	19.1479i	0.626874i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.8348	−0.451965	−0.225982	−	0.974131i	$-0.572559\pi$
$937$	−13.8348	−0.451965	−0.225982	+	0.974131i	$0.572559\pi$
$938$	0	0
$939$	− 8.16515i	− 0.266460i
$940$	0	0
$941$	− 36.6591i	− 1.19505i	−0.801849	−	0.597527i	$-0.796150\pi$
$941$	− 36.6591i	− 1.19505i	0.801849	−	0.597527i	$-0.203850\pi$
$942$	0	0
$943$	−24.4394	−0.795857
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.0780i	− 1.33486i	−0.744674	−	0.667428i	$-0.767395\pi$
$947$	− 41.0780i	− 1.33486i	0.744674	−	0.667428i	$-0.232605\pi$
$948$	0	0
$949$	− 14.0471i	− 0.455988i
$950$	0	0
$951$	−29.9216	−0.970275
$952$	0	0
$953$	−48.6606	−1.57627	−0.788136	−	0.615501i	$-0.788954\pi$
$953$	−48.6606	−1.57627	−0.788136	+	0.615501i	$0.788954\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 15.6838i	− 0.506985i
$958$	0	0
$959$	−15.8745	−0.512615
$960$	0	0
$961$	−12.6606	−0.408407
$962$	0	0
$963$	1.58258i	0.0509978i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.0405	−1.19114	−0.595571	−	0.803302i	$-0.703074\pi$
$967$	−37.0405	−1.19114	−0.595571	+	0.803302i	$0.703074\pi$
$968$	0	0
$969$	−34.7477	−1.11626
$970$	0	0
$971$	18.0000i	0.577647i	0.957382	+	0.288824i	$0.0932642\pi$
$971$	18.0000i	0.577647i	−0.957382	+	0.288824i	$0.906736\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.4955	0.495743	0.247872	−	0.968793i	$-0.420269\pi$
$977$	15.4955	0.495743	0.247872	+	0.968793i	$0.420269\pi$
$978$	0	0
$979$	11.3394i	0.362409i
$980$	0	0
$981$	− 12.1244i	− 0.387101i
$982$	0	0
$983$	12.4104	0.395830	0.197915	−	0.980219i	$-0.436583\pi$
$983$	12.4104	0.395830	0.197915	+	0.980219i	$0.436583\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 14.0000i	− 0.445625i
$988$	0	0
$989$	11.6874i	0.371638i
$990$	0	0
$991$	−51.7153	−1.64279	−0.821395	−	0.570360i	$-0.806804\pi$
$991$	−51.7153	−1.64279	−0.821395	+	0.570360i	$0.806804\pi$
$992$	0	0
$993$	7.16515	0.227379
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.6772i	− 1.22492i	−0.790502	−	0.612460i	$-0.790180\pi$
$997$	− 38.6772i	− 1.22492i	0.790502	−	0.612460i	$-0.209820\pi$
$998$	0	0
$999$	6.92820	0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.k.p.2401.8	yes	8
4.3	odd	2	inner	4800.2.k.p.2401.1	✓	8
5.2	odd	4		4800.2.d.t.1249.7		8
5.3	odd	4		4800.2.d.s.1249.3		8
5.4	even	2		4800.2.k.s.2401.2	yes	8
8.3	odd	2	inner	4800.2.k.p.2401.6	yes	8
8.5	even	2	inner	4800.2.k.p.2401.3	yes	8
20.3	even	4		4800.2.d.t.1249.6		8
20.7	even	4		4800.2.d.s.1249.2		8
20.19	odd	2		4800.2.k.s.2401.7	yes	8
40.3	even	4		4800.2.d.s.1249.7		8
40.13	odd	4		4800.2.d.t.1249.2		8
40.19	odd	2		4800.2.k.s.2401.4	yes	8
40.27	even	4		4800.2.d.t.1249.3		8
40.29	even	2		4800.2.k.s.2401.5	yes	8
40.37	odd	4		4800.2.d.s.1249.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4800.2.d.s.1249.2		8	20.7	even	4
4800.2.d.s.1249.3		8	5.3	odd	4
4800.2.d.s.1249.6		8	40.37	odd	4
4800.2.d.s.1249.7		8	40.3	even	4
4800.2.d.t.1249.2		8	40.13	odd	4
4800.2.d.t.1249.3		8	40.27	even	4
4800.2.d.t.1249.6		8	20.3	even	4
4800.2.d.t.1249.7		8	5.2	odd	4
4800.2.k.p.2401.1	✓	8	4.3	odd	2	inner
4800.2.k.p.2401.3	yes	8	8.5	even	2	inner
4800.2.k.p.2401.6	yes	8	8.3	odd	2	inner
4800.2.k.p.2401.8	yes	8	1.1	even	1	trivial
4800.2.k.s.2401.2	yes	8	5.4	even	2
4800.2.k.s.2401.4	yes	8	40.19	odd	2
4800.2.k.s.2401.5	yes	8	40.29	even	2
4800.2.k.s.2401.7	yes	8	20.19	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:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.64575 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.64575 q^{7} -1.00000 q^{9} +3.58258i q^{11} -7.02355i q^{13} -7.58258 q^{17} -4.58258i q^{19} +2.64575i q^{21} +2.55040 q^{23} -1.00000i q^{27} +4.37780i q^{29} -4.28245 q^{31} -3.58258 q^{33} +6.92820i q^{37} +7.02355 q^{39} -9.58258 q^{41} +4.58258i q^{43} -5.29150 q^{47} -7.58258i q^{51} +9.66930i q^{53} +4.58258 q^{57} +9.16515i q^{59} -0.0953502i q^{61} -2.64575 q^{63} +4.58258i q^{67} +2.55040i q^{69} -16.5975 q^{71} +2.00000 q^{73} +9.47860i q^{77} -5.29150 q^{79} +1.00000 q^{81} +10.0000i q^{83} -4.37780 q^{87} +3.16515 q^{89} -18.5826i q^{91} -4.28245i q^{93} +0.165151 q^{97} -3.58258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 24 q^{17} + 8 q^{33} - 40 q^{41} + 16 q^{73} + 8 q^{81} - 48 q^{89} - 72 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 24 * q^17 + 8 * q^33 - 40 * q^41 + 16 * q^73 + 8 * q^81 - 48 * q^89 - 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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