LMFDB - Embedded newform 2300.2.c.f.1749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2300 = 2^{2} \cdot 5^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2300.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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 92)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.2.c.f.1749.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000i q^{7} +2.00000 q^{9} +1.00000i q^{13} -6.00000i q^{17} -2.00000 q^{19} +2.00000 q^{21} +1.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +5.00000 q^{31} +8.00000i q^{37} +1.00000 q^{39} +3.00000 q^{41} -8.00000i q^{43} +9.00000i q^{47} +3.00000 q^{49} -6.00000 q^{51} -6.00000i q^{53} +2.00000i q^{57} +12.0000 q^{59} +14.0000 q^{61} +4.00000i q^{63} +8.00000i q^{67} +1.00000 q^{69} -15.0000 q^{71} +7.00000i q^{73} +10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -3.00000i q^{87} -2.00000 q^{91} -5.00000i q^{93} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{9} - 4 q^{19} + 4 q^{21} + 6 q^{29} + 10 q^{31} + 2 q^{39} + 6 q^{41} + 6 q^{49} - 12 q^{51} + 24 q^{59} + 28 q^{61} + 2 q^{69} - 30 q^{71} + 20 q^{79} + 2 q^{81} - 4 q^{91}+O(q^{100})$ 2 * q + 4 * q^9 - 4 * q^19 + 4 * q^21 + 6 * q^29 + 10 * q^31 + 2 * q^39 + 6 * q^41 + 6 * q^49 - 12 * q^51 + 24 * q^59 + 28 * q^61 + 2 * q^69 - 30 * q^71 + 20 * q^79 + 2 * q^81 - 4 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$ .

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	4.00000i	0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	− 5.00000i	− 0.518476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 3.00000i	− 0.270501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.0000i	0.976092i	0.872818	+	0.488046i	$0.162290\pi$
$127$	11.0000i	0.976092i	−0.872818	+	0.488046i	$0.837710\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	0	0
$153$	− 12.0000i	− 0.970143i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	− 14.0000i	− 1.03491i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	10.0000	0.727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	7.00000i	0.503871i	0.967744	+	0.251936i	$0.0810671\pi$
$193$	7.00000i	0.503871i	−0.967744	+	0.251936i	$0.918933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.0000i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$	− 15.0000i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000i	0.139010i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	15.0000i	1.02778i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.0000i	0.678844i
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.00000i	0.589610i	0.955557	+	0.294805i	$0.0952546\pi$
$233$	9.00000i	0.589610i	−0.955557	+	0.294805i	$0.904745\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.0000i	1.68421i	0.539311	+	0.842107i	$0.318685\pi$
$257$	27.0000i	1.68421i	−0.539311	+	0.842107i	$0.681315\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	2.00000i	0.121046i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.0000i	− 1.14160i	−0.821089	−	0.570800i	$-0.806633\pi$
$277$	− 19.0000i	− 1.14160i	0.821089	−	0.570800i	$-0.193367\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	4.00000i	0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$	4.00000i	0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 16.0000i	− 0.884802i
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	0	0
$333$	16.0000i	0.876795i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−29.0000	−1.55233	−0.776167	−	0.630527i	$-0.782839\pi$
$349$	−29.0000	−1.55233	−0.776167	+	0.630527i	$0.782839\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	27.0000i	1.43706i	0.695493	+	0.718532i	$0.255186\pi$
$353$	27.0000i	1.43706i	−0.695493	+	0.718532i	$0.744814\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.0000i	− 0.635107i
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	− 20.0000i	− 1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$	− 20.0000i	− 1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.00000i	0.154508i
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	11.0000	0.563547
$382$	0	0
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 16.0000i	− 0.813326i
$388$	0	0
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	− 21.0000i	− 1.05931i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.0000i	0.552074i	0.961147	+	0.276037i	$0.0890213\pi$
$397$	11.0000i	0.552074i	−0.961147	+	0.276037i	$0.910979\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	5.00000i	0.249068i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	24.0000i	1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.0000i	0.538672i
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	18.0000i	0.875190i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.0000i	1.35501i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	16.0000i	0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$	16.0000i	0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	− 21.0000i	− 0.997740i	−0.866677	−	0.498870i	$-0.833748\pi$
$443$	− 21.0000i	− 0.997740i	0.866677	−	0.498870i	$-0.166252\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	7.00000i	0.328889i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	0	0
$459$	−30.0000	−1.40028
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 30.0000i	− 1.38823i	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	− 30.0000i	− 1.38823i	0.719862	−	0.694117i	$-0.244205\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	−42.0000	−1.91903	−0.959514	−	0.281659i	$-0.909115\pi$
$479$	−42.0000	−1.91903	−0.959514	+	0.281659i	$0.909115\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.0000i	1.04223i	0.853487	+	0.521115i	$0.174484\pi$
$487$	23.0000i	1.04223i	−0.853487	+	0.521115i	$0.825516\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 30.0000i	− 1.34568i
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	10.0000i	0.441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	− 38.0000i	− 1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	− 38.0000i	− 1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	3.00000i	0.129944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.00000i	− 0.388379i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	16.0000i	0.686626i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.00000i	− 0.0427569i	−0.999771	−	0.0213785i	$-0.993195\pi$
$547$	− 1.00000i	− 0.0427569i	0.999771	−	0.0213785i	$-0.00680549\pi$
$548$	0	0
$549$	28.0000	1.19501
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	20.0000i	0.850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0000i	1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$	24.0000i	1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 7.00000i	− 0.291414i	−0.989328	−	0.145707i	$-0.953454\pi$
$577$	− 7.00000i	− 0.291414i	0.989328	−	0.145707i	$-0.0465456\pi$
$578$	0	0
$579$	7.00000	0.290910
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0000i	0.866763i	0.901211	+	0.433381i	$0.142680\pi$
$587$	21.0000i	0.866763i	−0.901211	+	0.433381i	$0.857320\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	−15.0000	−0.617018
$592$	0	0
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.00000i	− 0.163709i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	16.0000i	0.651570i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	48.0000	1.91389
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	16.0000i	0.635943i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	0	0
$639$	−30.0000	−1.18678
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	0	0
$653$	− 27.0000i	− 1.05659i	−0.849060	−	0.528296i	$-0.822831\pi$
$653$	− 27.0000i	− 1.05659i	0.849060	−	0.528296i	$-0.177169\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	− 6.00000i	− 0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.00000i	0.116160i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13.0000i	0.501113i	0.968102	+	0.250557i	$0.0806136\pi$
$673$	13.0000i	0.501113i	−0.968102	+	0.250557i	$0.919386\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	− 39.0000i	− 1.49229i	−0.665782	−	0.746147i	$-0.731902\pi$
$683$	− 39.0000i	− 1.49229i	0.665782	−	0.746147i	$-0.268098\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4.00000i	− 0.152610i
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.0000i	− 0.681799i
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	− 16.0000i	− 0.603451i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	5.00000i	0.187251i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.0000i	0.784259i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	4.00000i	0.148762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−48.0000	−1.77534
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	18.0000i	0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000i	0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$	20.0000i	0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	32.0000i	1.15848i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000i	0.433295i
$768$	0	0
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	0	0
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.0000i	0.573997i
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 15.0000i	− 0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	36.0000	1.28001
$792$	0	0
$793$	14.0000i	0.497155i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	54.0000	1.91038
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 15.0000i	− 0.528025i
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	− 41.0000i	− 1.42917i	−0.699549	−	0.714585i	$-0.746616\pi$
$823$	− 41.0000i	− 1.42917i	0.699549	−	0.714585i	$-0.253384\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	0	0
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 25.0000i	− 0.864126i
$838$	0	0
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	− 30.0000i	− 1.03325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	− 2.00000i	− 0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$	− 2.00000i	− 0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 33.0000i	− 1.12726i	−0.826028	−	0.563629i	$-0.809405\pi$
$857$	− 33.0000i	− 1.12726i	0.826028	−	0.563629i	$-0.190595\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	9.00000i	0.306364i	0.988198	+	0.153182i	$0.0489520\pi$
$863$	9.00000i	0.306364i	−0.988198	+	0.153182i	$0.951048\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 51.0000i	− 1.71241i	−0.516634	−	0.856206i	$-0.672815\pi$
$887$	− 51.0000i	− 1.71241i	0.516634	−	0.856206i	$-0.327185\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.00000i	0.0333890i
$898$	0	0
$899$	15.0000	0.500278
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50.0000i	1.66022i	0.557598	+	0.830111i	$0.311723\pi$
$907$	50.0000i	1.66022i	−0.557598	+	0.830111i	$0.688277\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42.0000i	1.38696i
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	− 15.0000i	− 0.493731i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	15.0000i	0.491078i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28.0000i	− 0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$	− 28.0000i	− 0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$	0	0
$939$	4.00000	0.130535
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	3.00000i	0.0976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.00000i	− 0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$	− 3.00000i	− 0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47.0000i	1.51142i	0.654907	+	0.755709i	$0.272708\pi$
$967$	47.0000i	1.51142i	−0.654907	+	0.755709i	$0.727292\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	− 22.0000i	− 0.705288i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.0000i	1.53566i	0.640656	+	0.767828i	$0.278662\pi$
$977$	48.0000i	1.53566i	−0.640656	+	0.767828i	$0.721338\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	32.0000	1.02168
$982$	0	0
$983$	18.0000i	0.574111i	0.957914	+	0.287055i	$0.0926764\pi$
$983$	18.0000i	0.574111i	−0.957914	+	0.287055i	$0.907324\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	18.0000i	0.572946i
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	25.0000i	0.793351i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	62.0000i	1.96356i	0.190022	+	0.981780i	$0.439144\pi$
$997$	62.0000i	1.96356i	−0.190022	+	0.981780i	$0.560856\pi$
$998$	0	0
$999$	40.0000	1.26554

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.c.f.1749.1		2
5.2	odd	4		2300.2.a.c.1.1		1
5.3	odd	4		92.2.a.b.1.1	✓	1
5.4	even	2	inner	2300.2.c.f.1749.2		2
15.8	even	4		828.2.a.b.1.1		1
20.3	even	4		368.2.a.b.1.1		1
20.7	even	4		9200.2.a.ba.1.1		1
35.13	even	4		4508.2.a.a.1.1		1
40.3	even	4		1472.2.a.j.1.1		1
40.13	odd	4		1472.2.a.c.1.1		1
60.23	odd	4		3312.2.a.g.1.1		1
115.68	even	4		2116.2.a.d.1.1		1
460.183	odd	4		8464.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.2.a.b.1.1	✓	1	5.3	odd	4
368.2.a.b.1.1		1	20.3	even	4
828.2.a.b.1.1		1	15.8	even	4
1472.2.a.c.1.1		1	40.13	odd	4
1472.2.a.j.1.1		1	40.3	even	4
2116.2.a.d.1.1		1	115.68	even	4
2300.2.a.c.1.1		1	5.2	odd	4
2300.2.c.f.1749.1		2	1.1	even	1	trivial
2300.2.c.f.1749.2		2	5.4	even	2	inner
3312.2.a.g.1.1		1	60.23	odd	4
4508.2.a.a.1.1		1	35.13	even	4
8464.2.a.f.1.1		1	460.183	odd	4
9200.2.a.ba.1.1		1	20.7	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2300.2.c.f.1749.1

Level	$2300$
Weight	$2$
Character	2300.1749
Analytic conductor	$18.366$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$