LMFDB - Embedded newform 7200.2.m.e.3599.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7200,2,Mod(3599,7200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7200.3599"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7200.m (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64$ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$2^{16}$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3599.4
Root			$-0.760198 + 1.19252i$ of defining polynomial
Character	$\chi$	$=$	7200.3599
Dual form			7200.2.m.e.3599.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{7} +0.191427i q^{11} -2.63700 q^{13} -6.20522 q^{17} -1.52311 q^{19} -5.25240i q^{23} +0.270718 q^{29} +6.20522i q^{31} +7.61944 q^{37} +9.22508i q^{41} -12.7755i q^{43} -3.79383i q^{47} -5.00000 q^{49} +8.77551i q^{53} -10.4479i q^{59} +0.382853i q^{61} -1.72928i q^{67} +9.72928 q^{71} +5.45856i q^{73} -0.270718i q^{77} +14.3077i q^{79} +15.2389 q^{83} -3.56822i q^{89} +3.72928 q^{91} +7.31695i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 32 q^{19} + 24 q^{29} - 60 q^{49} + 96 q^{71} + 24 q^{91}+O(q^{100})$ 12 * q + 32 * q^19 + 24 * q^29 - 60 * q^49 + 96 * q^71 + 24 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$ .

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.191427i	0.0577173i	0.999584	+	0.0288587i	$0.00918727\pi$
$11$	0.191427i	0.0577173i	−0.999584	+	0.0288587i	$0.990813\pi$
$12$	0	0
$13$	−2.63700	−0.731372	−0.365686	−	0.930738i	$-0.619166\pi$
$13$	−2.63700	−0.731372	−0.365686	+	0.930738i	$0.619166\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.20522	−1.50499	−0.752494	−	0.658599i	$-0.771149\pi$
$17$	−6.20522	−1.50499	−0.752494	+	0.658599i	$0.771149\pi$
$18$	0	0
$19$	−1.52311	−0.349426	−0.174713	−	0.984619i	$-0.555900\pi$
$19$	−1.52311	−0.349426	−0.174713	+	0.984619i	$0.555900\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.25240i	− 1.09520i	−0.836740	−	0.547600i	$-0.815542\pi$
$23$	− 5.25240i	− 1.09520i	0.836740	−	0.547600i	$-0.184458\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.270718	0.0502711	0.0251356	−	0.999684i	$-0.491998\pi$
$29$	0.270718	0.0502711	0.0251356	+	0.999684i	$0.491998\pi$
$30$	0	0
$31$	6.20522i	1.11449i	0.830348	+	0.557245i	$0.188142\pi$
$31$	6.20522i	1.11449i	−0.830348	+	0.557245i	$0.811858\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.61944	1.25263	0.626314	−	0.779571i	$-0.284563\pi$
$37$	7.61944	1.25263	0.626314	+	0.779571i	$0.284563\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.22508i	1.44071i	0.693603	+	0.720357i	$0.256022\pi$
$41$	9.22508i	1.44071i	−0.693603	+	0.720357i	$0.743978\pi$
$42$	0	0
$43$	− 12.7755i	− 1.94825i	−0.226016	−	0.974124i	$-0.572570\pi$
$43$	− 12.7755i	− 1.94825i	0.226016	−	0.974124i	$-0.427430\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.79383i	− 0.553387i	−0.960958	−	0.276694i	$-0.910761\pi$
$47$	− 3.79383i	− 0.553387i	0.960958	−	0.276694i	$-0.0892387\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.77551i	1.20541i	0.797964	+	0.602705i	$0.205910\pi$
$53$	8.77551i	1.20541i	−0.797964	+	0.602705i	$0.794090\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.4479i	− 1.36020i	−0.733121	−	0.680098i	$-0.761937\pi$
$59$	− 10.4479i	− 1.36020i	0.733121	−	0.680098i	$-0.238063\pi$
$60$	0	0
$61$	0.382853i	0.0490194i	0.999700	+	0.0245097i	$0.00780245\pi$
$61$	0.382853i	0.0490194i	−0.999700	+	0.0245097i	$0.992198\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.72928i	− 0.211265i	−0.994405	−	0.105633i	$-0.966313\pi$
$67$	− 1.72928i	− 0.211265i	0.994405	−	0.105633i	$-0.0336868\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.72928	1.15465	0.577327	−	0.816513i	$-0.304096\pi$
$71$	9.72928	1.15465	0.577327	+	0.816513i	$0.304096\pi$
$72$	0	0
$73$	5.45856i	0.638877i	0.947607	+	0.319438i	$0.103494\pi$
$73$	5.45856i	0.638877i	−0.947607	+	0.319438i	$0.896506\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.270718i	− 0.0308512i
$78$	0	0
$79$	14.3077i	1.60974i	0.593454	+	0.804868i	$0.297764\pi$
$79$	14.3077i	1.60974i	−0.593454	+	0.804868i	$0.702236\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.2389	1.67268	0.836342	−	0.548208i	$-0.184690\pi$
$83$	15.2389	1.67268	0.836342	+	0.548208i	$0.184690\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.56822i	− 0.378231i	−0.981955	−	0.189115i	$-0.939438\pi$
$89$	− 3.56822i	− 0.378231i	0.981955	−	0.189115i	$-0.0605620\pi$
$90$	0	0
$91$	3.72928	0.390935
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.31695i	0.742923i	0.928448	+	0.371462i	$0.121143\pi$
$97$	7.31695i	0.742923i	−0.928448	+	0.371462i	$0.878857\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5510	1.54738	0.773692	−	0.633562i	$-0.218408\pi$
$101$	15.5510	1.54738	0.773692	+	0.633562i	$0.218408\pi$
$102$	0	0
$103$	2.08863	0.205799	0.102899	−	0.994692i	$-0.467188\pi$
$103$	2.08863	0.205799	0.102899	+	0.994692i	$0.467188\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.82843	−0.273434	−0.136717	−	0.990610i	$-0.543655\pi$
$107$	−2.82843	−0.273434	−0.136717	+	0.990610i	$0.543655\pi$
$108$	0	0
$109$	− 5.27400i	− 0.505158i	−0.967576	−	0.252579i	$-0.918721\pi$
$109$	− 5.27400i	− 0.505158i	0.967576	−	0.252579i	$-0.0812787\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.82237	−0.547722	−0.273861	−	0.961769i	$-0.588301\pi$
$113$	−5.82237	−0.547722	−0.273861	+	0.961769i	$0.588301\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.77551	0.804450
$120$	0	0
$121$	10.9634	0.996669
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.5388	1.82252	0.911261	−	0.411829i	$-0.135110\pi$
$127$	20.5388	1.82252	0.911261	+	0.411829i	$0.135110\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.5679i	− 1.18543i	−0.805413	−	0.592715i	$-0.798056\pi$
$131$	− 13.5679i	− 1.18543i	0.805413	−	0.592715i	$-0.201944\pi$
$132$	0	0
$133$	2.15401	0.186776
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.72563	−0.403738	−0.201869	−	0.979413i	$-0.564702\pi$
$137$	−4.72563	−0.403738	−0.201869	+	0.979413i	$0.564702\pi$
$138$	0	0
$139$	15.7572	1.33651	0.668254	−	0.743934i	$-0.267042\pi$
$139$	15.7572	1.33651	0.668254	+	0.743934i	$0.267042\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 0.504792i	− 0.0422129i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.72928	−0.633207	−0.316604	−	0.948558i	$-0.602543\pi$
$149$	−7.72928	−0.633207	−0.316604	+	0.948558i	$0.602543\pi$
$150$	0	0
$151$	22.2098i	1.80741i	0.428158	+	0.903704i	$0.359163\pi$
$151$	22.2098i	1.80741i	−0.428158	+	0.903704i	$0.640837\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.00229	0.638652	0.319326	−	0.947645i	$-0.396543\pi$
$157$	8.00229	0.638652	0.319326	+	0.947645i	$0.396543\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.42801i	0.585409i
$162$	0	0
$163$	− 0.953771i	− 0.0747051i	−0.999302	−	0.0373526i	$-0.988108\pi$
$163$	− 0.953771i	− 0.0747051i	0.999302	−	0.0373526i	$-0.0118925\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.7572i	− 0.909799i	−0.890543	−	0.454899i	$-0.849675\pi$
$167$	− 11.7572i	− 0.909799i	0.890543	−	0.454899i	$-0.150325\pi$
$168$	0	0
$169$	−6.04623	−0.465095
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.22449i	− 0.245153i	−0.992459	−	0.122577i	$-0.960884\pi$
$173$	− 3.22449i	− 0.245153i	0.992459	−	0.122577i	$-0.0391157\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.6019i	0.941908i	0.882158	+	0.470954i	$0.156090\pi$
$179$	12.6019i	0.941908i	−0.882158	+	0.470954i	$0.843910\pi$
$180$	0	0
$181$	24.1070i	1.79186i	0.444195	+	0.895930i	$0.353490\pi$
$181$	24.1070i	1.79186i	−0.444195	+	0.895930i	$0.646510\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.18785i	− 0.0868639i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.9171	−0.789936	−0.394968	−	0.918695i	$-0.629244\pi$
$191$	−10.9171	−0.789936	−0.394968	+	0.918695i	$0.629244\pi$
$192$	0	0
$193$	15.3169i	1.10254i	0.834328	+	0.551269i	$0.185856\pi$
$193$	15.3169i	1.10254i	−0.834328	+	0.551269i	$0.814144\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.0462i	− 1.21449i	−0.794513	−	0.607247i	$-0.792274\pi$
$197$	− 17.0462i	− 1.21449i	0.794513	−	0.607247i	$-0.207726\pi$
$198$	0	0
$199$	− 4.72563i	− 0.334991i	−0.985873	−	0.167496i	$-0.946432\pi$
$199$	− 4.72563i	− 0.334991i	0.985873	−	0.167496i	$-0.0535680\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.382853	−0.0268710
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 0.291565i	− 0.0201680i
$210$	0	0
$211$	−16.8401	−1.15932	−0.579659	−	0.814859i	$-0.696814\pi$
$211$	−16.8401	−1.15932	−0.579659	+	0.814859i	$0.696814\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.77551i	− 0.595720i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.3632	1.10071
$222$	0	0
$223$	−22.3099	−1.49398	−0.746992	−	0.664833i	$-0.768503\pi$
$223$	−22.3099	−1.49398	−0.746992	+	0.664833i	$0.768503\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.86813	0.588599	0.294299	−	0.955713i	$-0.404914\pi$
$227$	8.86813	0.588599	0.294299	+	0.955713i	$0.404914\pi$
$228$	0	0
$229$	21.2786i	1.40613i	0.711126	+	0.703064i	$0.248185\pi$
$229$	21.2786i	1.40613i	−0.711126	+	0.703064i	$0.751815\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.0734	−0.987488	−0.493744	−	0.869607i	$-0.664372\pi$
$233$	−15.0734	−0.987488	−0.493744	+	0.869607i	$0.664372\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.0462	1.23200	0.615999	−	0.787747i	$-0.288753\pi$
$239$	19.0462	1.23200	0.615999	+	0.787747i	$0.288753\pi$
$240$	0	0
$241$	18.5048	1.19200	0.595999	−	0.802985i	$-0.296756\pi$
$241$	18.5048	1.19200	0.595999	+	0.802985i	$0.296756\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.01645	0.255561
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16.8186i	− 1.06158i	−0.847503	−	0.530790i	$-0.821895\pi$
$251$	− 16.8186i	− 1.06158i	0.847503	−	0.530790i	$-0.178105\pi$
$252$	0	0
$253$	1.00545	0.0632120
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.79936	0.611267	0.305634	−	0.952149i	$-0.401132\pi$
$257$	9.79936	0.611267	0.305634	+	0.952149i	$0.401132\pi$
$258$	0	0
$259$	−10.7755	−0.669558
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 5.79383i	− 0.357263i	−0.983916	−	0.178632i	$-0.942833\pi$
$263$	− 5.79383i	− 0.357263i	0.983916	−	0.178632i	$-0.0571670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.1878	0.926019	0.463010	−	0.886353i	$-0.346770\pi$
$269$	15.1878	0.926019	0.463010	+	0.886353i	$0.346770\pi$
$270$	0	0
$271$	10.1304i	0.615377i	0.951487	+	0.307689i	$0.0995555\pi$
$271$	10.1304i	0.615377i	−0.951487	+	0.307689i	$0.900444\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.7559	0.886595	0.443298	−	0.896375i	$-0.353809\pi$
$277$	14.7559	0.886595	0.443298	+	0.896375i	$0.353809\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 21.5442i	− 1.28522i	−0.766193	−	0.642611i	$-0.777851\pi$
$281$	− 21.5442i	− 1.28522i	0.766193	−	0.642611i	$-0.222149\pi$
$282$	0	0
$283$	18.2707i	1.08608i	0.839706	+	0.543041i	$0.182727\pi$
$283$	18.2707i	1.08608i	−0.839706	+	0.543041i	$0.817273\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 13.0462i	− 0.770095i
$288$	0	0
$289$	21.5048	1.26499
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.8506i	0.800999i
$300$	0	0
$301$	18.0673i	1.04138i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.5048i	− 1.28442i	−0.766530	−	0.642208i	$-0.778018\pi$
$307$	− 22.5048i	− 1.28442i	0.766530	−	0.642208i	$-0.221982\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.0462	1.76047	0.880235	−	0.474538i	$-0.157385\pi$
$311$	31.0462	1.76047	0.880235	+	0.474538i	$0.157385\pi$
$312$	0	0
$313$	2.23407i	0.126277i	0.998005	+	0.0631387i	$0.0201110\pi$
$313$	2.23407i	0.126277i	−0.998005	+	0.0631387i	$0.979889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.5048i	− 0.927001i	−0.886097	−	0.463501i	$-0.846593\pi$
$317$	− 16.5048i	− 0.927001i	0.886097	−	0.463501i	$-0.153407\pi$
$318$	0	0
$319$	0.0518227i	0.00290151i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.45126	0.525882
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.36529i	0.295798i
$330$	0	0
$331$	12.9817	0.713538	0.356769	−	0.934193i	$-0.383878\pi$
$331$	12.9817	0.713538	0.356769	+	0.934193i	$0.383878\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 27.0096i	− 1.47131i	−0.677359	−	0.735653i	$-0.736875\pi$
$337$	− 27.0096i	− 1.47131i	0.677359	−	0.735653i	$-0.263125\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.18785	−0.0643254
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.07372	0.272372	0.136186	−	0.990683i	$-0.456516\pi$
$347$	5.07372	0.272372	0.136186	+	0.990683i	$0.456516\pi$
$348$	0	0
$349$	20.1300i	1.07754i	0.842454	+	0.538768i	$0.181110\pi$
$349$	20.1300i	1.07754i	−0.842454	+	0.538768i	$0.818890\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.3077	−0.761519	−0.380760	−	0.924674i	$-0.624337\pi$
$353$	−14.3077	−0.761519	−0.380760	+	0.924674i	$0.624337\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.8217	−1.67949	−0.839744	−	0.542983i	$-0.817295\pi$
$359$	−31.8217	−1.67949	−0.839744	+	0.542983i	$0.817295\pi$
$360$	0	0
$361$	−16.6801	−0.877901
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.9844	1.19977	0.599887	−	0.800085i	$-0.295212\pi$
$367$	22.9844	1.19977	0.599887	+	0.800085i	$0.295212\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 12.4104i	− 0.644318i
$372$	0	0
$373$	14.2423	0.737437	0.368718	−	0.929541i	$-0.379797\pi$
$373$	14.2423	0.737437	0.368718	+	0.929541i	$0.379797\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.713884	−0.0367669
$378$	0	0
$379$	−8.71096	−0.447452	−0.223726	−	0.974652i	$-0.571822\pi$
$379$	−8.71096	−0.447452	−0.223726	+	0.974652i	$0.571822\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.7110i	0.956085i	0.878337	+	0.478043i	$0.158654\pi$
$383$	18.7110i	0.956085i	−0.878337	+	0.478043i	$0.841346\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.0096	1.36944	0.684720	−	0.728806i	$-0.259924\pi$
$389$	27.0096	1.36944	0.684720	+	0.728806i	$0.259924\pi$
$390$	0	0
$391$	32.5923i	1.64826i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	25.3952	1.27455	0.637274	−	0.770638i	$-0.280062\pi$
$397$	25.3952	1.27455	0.637274	+	0.770638i	$0.280062\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6.01380i	− 0.300315i	−0.988662	−	0.150157i	$-0.952022\pi$
$401$	− 6.01380i	− 0.300315i	0.988662	−	0.150157i	$-0.0479780\pi$
$402$	0	0
$403$	− 16.3632i	− 0.815108i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.45856i	0.0722983i
$408$	0	0
$409$	19.5510	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$409$	19.5510	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.7755i	0.727055i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.96258i	− 0.0958784i	−0.998850	−	0.0479392i	$-0.984735\pi$
$419$	− 1.96258i	− 0.0958784i	0.998850	−	0.0479392i	$-0.0152654\pi$
$420$	0	0
$421$	4.30802i	0.209960i	0.994474	+	0.104980i	$0.0334778\pi$
$421$	4.30802i	0.209960i	−0.994474	+	0.104980i	$0.966522\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.541436i	− 0.0262019i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.8217	0.569433	0.284717	−	0.958612i	$-0.408101\pi$
$431$	11.8217	0.569433	0.284717	+	0.958612i	$0.408101\pi$
$432$	0	0
$433$	− 9.45856i	− 0.454550i	−0.973831	−	0.227275i	$-0.927018\pi$
$433$	− 9.45856i	− 0.454550i	0.973831	−	0.227275i	$-0.0729816\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	− 16.0393i	− 0.765516i	−0.923849	−	0.382758i	$-0.874974\pi$
$439$	− 16.0393i	− 0.765516i	0.923849	−	0.382758i	$-0.125026\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.0730	−1.19125	−0.595627	−	0.803261i	$-0.703096\pi$
$443$	−25.0730	−1.19125	−0.595627	+	0.803261i	$0.703096\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.24264i	0.200223i	0.994976	+	0.100111i	$0.0319199\pi$
$449$	4.24264i	0.200223i	−0.994976	+	0.100111i	$0.968080\pi$
$450$	0	0
$451$	−1.76593	−0.0831542
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.8680i	1.35039i	0.737641	+	0.675193i	$0.235940\pi$
$457$	28.8680i	1.35039i	−0.737641	+	0.675193i	$0.764060\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	36.7389	1.71110	0.855550	−	0.517721i	$-0.173219\pi$
$461$	36.7389	1.71110	0.855550	+	0.517721i	$0.173219\pi$
$462$	0	0
$463$	−16.6136	−0.772100	−0.386050	−	0.922478i	$-0.626161\pi$
$463$	−16.6136	−0.772100	−0.386050	+	0.922478i	$0.626161\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.47959	0.0684673	0.0342336	−	0.999414i	$-0.489101\pi$
$467$	1.47959	0.0684673	0.0342336	+	0.999414i	$0.489101\pi$
$468$	0	0
$469$	2.44557i	0.112926i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.44557	0.112448
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.81215	0.128491	0.0642453	−	0.997934i	$-0.479536\pi$
$479$	2.81215	0.128491	0.0642453	+	0.997934i	$0.479536\pi$
$480$	0	0
$481$	−20.0925	−0.916137
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.9447	−0.767835	−0.383918	−	0.923367i	$-0.625425\pi$
$487$	−16.9447	−0.767835	−0.383918	+	0.923367i	$0.625425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.0696i	1.17650i	0.808678	+	0.588252i	$0.200184\pi$
$491$	26.0696i	1.17650i	−0.808678	+	0.588252i	$0.799816\pi$
$492$	0	0
$493$	−1.67987	−0.0756574
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.7593	−0.617188
$498$	0	0
$499$	22.5327	1.00870	0.504351	−	0.863499i	$-0.331732\pi$
$499$	22.5327	1.00870	0.504351	+	0.863499i	$0.331732\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.80342i	0.392525i	0.980551	+	0.196262i	$0.0628805\pi$
$503$	8.80342i	0.392525i	−0.980551	+	0.196262i	$0.937120\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.6339	0.914581	0.457291	−	0.889317i	$-0.348820\pi$
$509$	20.6339	0.914581	0.457291	+	0.889317i	$0.348820\pi$
$510$	0	0
$511$	− 7.71957i	− 0.341494i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.726241	0.0319400
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.55066i	0.374611i	0.982302	+	0.187306i	$0.0599755\pi$
$521$	8.55066i	0.374611i	−0.982302	+	0.187306i	$0.940025\pi$
$522$	0	0
$523$	33.0096i	1.44341i	0.692202	+	0.721704i	$0.256641\pi$
$523$	33.0096i	1.44341i	−0.692202	+	0.721704i	$0.743359\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 38.5048i	− 1.67730i
$528$	0	0
$529$	−4.58767	−0.199464
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 24.3265i	− 1.05370i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 0.957133i	− 0.0412267i
$540$	0	0
$541$	27.6493i	1.18874i	0.804193	+	0.594369i	$0.202598\pi$
$541$	27.6493i	1.18874i	−0.804193	+	0.594369i	$0.797402\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.85838i	0.421514i	0.977538	+	0.210757i	$0.0675929\pi$
$547$	9.85838i	0.421514i	−0.977538	+	0.210757i	$0.932407\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.412335	−0.0175661
$552$	0	0
$553$	− 20.2341i	− 0.860440i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.68305i	0.113685i	0.998383	+	0.0568423i	$0.0181032\pi$
$557$	2.68305i	0.113685i	−0.998383	+	0.0568423i	$0.981897\pi$
$558$	0	0
$559$	33.6890i	1.42489i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0794	0.509087	0.254543	−	0.967061i	$-0.418075\pi$
$563$	12.0794	0.509087	0.254543	+	0.967061i	$0.418075\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 15.8479i	− 0.664379i	−0.943213	−	0.332190i	$-0.892213\pi$
$569$	− 15.8479i	− 0.664379i	0.943213	−	0.332190i	$-0.107787\pi$
$570$	0	0
$571$	−1.11078	−0.0464847	−0.0232423	−	0.999730i	$-0.507399\pi$
$571$	−1.11078	−0.0464847	−0.0232423	+	0.999730i	$0.507399\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.7755i	0.615113i	0.951530	+	0.307556i	$0.0995112\pi$
$577$	14.7755i	0.615113i	−0.951530	+	0.307556i	$0.900489\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−21.5510	−0.894087
$582$	0	0
$583$	−1.67987	−0.0695730
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.5590	−0.559640	−0.279820	−	0.960052i	$-0.590275\pi$
$587$	−13.5590	−0.559640	−0.279820	+	0.960052i	$0.590275\pi$
$588$	0	0
$589$	− 9.45126i	− 0.389433i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.9921	1.31376	0.656879	−	0.753996i	$-0.271876\pi$
$593$	31.9921	1.31376	0.656879	+	0.753996i	$0.271876\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.82174	0.156152	0.0780760	−	0.996947i	$-0.475122\pi$
$599$	3.82174	0.156152	0.0780760	+	0.996947i	$0.475122\pi$
$600$	0	0
$601$	31.6435	1.29076	0.645382	−	0.763860i	$-0.276698\pi$
$601$	31.6435	1.29076	0.645382	+	0.763860i	$0.276698\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.7204	1.40926	0.704629	−	0.709576i	$-0.251114\pi$
$607$	34.7204	1.40926	0.704629	+	0.709576i	$0.251114\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.0043i	0.404732i
$612$	0	0
$613$	−26.3612	−1.06472	−0.532359	−	0.846519i	$-0.678694\pi$
$613$	−26.3612	−1.06472	−0.532359	+	0.846519i	$0.678694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.88509	0.317442	0.158721	−	0.987323i	$-0.449263\pi$
$617$	7.88509	0.317442	0.158721	+	0.987323i	$0.449263\pi$
$618$	0	0
$619$	−38.9325	−1.56483	−0.782415	−	0.622757i	$-0.786012\pi$
$619$	−38.9325	−1.56483	−0.782415	+	0.622757i	$0.786012\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.04623i	0.202173i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−47.2803	−1.88519
$630$	0	0
$631$	0.800468i	0.0318661i	0.999873	+	0.0159331i	$0.00507186\pi$
$631$	0.800468i	0.0318661i	−0.999873	+	0.0159331i	$0.994928\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.1850	0.522409
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 22.8931i	− 0.904222i	−0.891962	−	0.452111i	$-0.850671\pi$
$641$	− 22.8931i	− 0.904222i	0.891962	−	0.452111i	$-0.149329\pi$
$642$	0	0
$643$	− 12.9538i	− 0.510847i	−0.966829	−	0.255423i	$-0.917785\pi$
$643$	− 12.9538i	− 0.510847i	0.966829	−	0.255423i	$-0.0822149\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6647i	0.615844i	0.951412	+	0.307922i	$0.0996336\pi$
$647$	15.6647i	0.615844i	−0.951412	+	0.307922i	$0.900366\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 41.5144i	− 1.62458i	−0.583252	−	0.812292i	$-0.698220\pi$
$653$	− 41.5144i	− 1.62458i	0.583252	−	0.812292i	$-0.301780\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 34.8859i	− 1.35896i	−0.733693	−	0.679481i	$-0.762205\pi$
$659$	− 34.8859i	− 1.35896i	0.733693	−	0.679481i	$-0.237795\pi$
$660$	0	0
$661$	19.7990i	0.770091i	0.922897	+	0.385046i	$0.125814\pi$
$661$	19.7990i	0.770091i	−0.922897	+	0.385046i	$0.874186\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.42192i	− 0.0550569i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.0732884	−0.00282927
$672$	0	0
$673$	− 34.1849i	− 1.31773i	−0.752260	−	0.658866i	$-0.771036\pi$
$673$	− 34.1849i	− 1.31773i	0.752260	−	0.658866i	$-0.228964\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.0462i	− 0.655140i	−0.944827	−	0.327570i	$-0.893770\pi$
$677$	− 17.0462i	− 0.655140i	0.944827	−	0.327570i	$-0.106230\pi$
$678$	0	0
$679$	− 10.3477i	− 0.397109i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.6671	−1.09692	−0.548459	−	0.836178i	$-0.684785\pi$
$683$	−28.6671	−1.09692	−0.548459	+	0.836178i	$0.684785\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 23.1410i	− 0.881603i
$690$	0	0
$691$	27.0741	1.02995	0.514974	−	0.857206i	$-0.327801\pi$
$691$	27.0741	1.02995	0.514974	+	0.857206i	$0.327801\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 57.2437i	− 2.16826i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23.5510	−0.889510	−0.444755	−	0.895652i	$-0.646709\pi$
$701$	−23.5510	−0.889510	−0.444755	+	0.895652i	$0.646709\pi$
$702$	0	0
$703$	−11.6053	−0.437701
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.9925	−0.827112
$708$	0	0
$709$	− 7.85033i	− 0.294825i	−0.989075	−	0.147413i	$-0.952905\pi$
$709$	− 7.85033i	− 0.294825i	0.989075	−	0.147413i	$-0.0470945\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.5923	1.22059
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.1416	0.378218	0.189109	−	0.981956i	$-0.439440\pi$
$719$	10.1416	0.378218	0.189109	+	0.981956i	$0.439440\pi$
$720$	0	0
$721$	−2.95377	−0.110004
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.86524	0.180442	0.0902208	−	0.995922i	$-0.471243\pi$
$727$	4.86524	0.180442	0.0902208	+	0.995922i	$0.471243\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	79.2749i	2.93209i
$732$	0	0
$733$	−9.09903	−0.336080	−0.168040	−	0.985780i	$-0.553744\pi$
$733$	−9.09903	−0.336080	−0.168040	+	0.985780i	$0.553744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.331031	0.0121937
$738$	0	0
$739$	12.5693	0.462371	0.231185	−	0.972910i	$-0.425740\pi$
$739$	12.5693	0.462371	0.231185	+	0.972910i	$0.425740\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23.6647i	− 0.868175i	−0.900871	−	0.434087i	$-0.857071\pi$
$743$	− 23.6647i	− 0.868175i	0.900871	−	0.434087i	$-0.142929\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	− 31.2782i	− 1.14136i	−0.821173	−	0.570679i	$-0.806680\pi$
$751$	− 31.2782i	− 1.14136i	0.821173	−	0.570679i	$-0.193320\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39.1150	−1.42166	−0.710829	−	0.703365i	$-0.751680\pi$
$757$	−39.1150	−1.42166	−0.710829	+	0.703365i	$0.751680\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.76305i	0.100160i	0.998745	+	0.0500802i	$0.0159477\pi$
$761$	2.76305i	0.100160i	−0.998745	+	0.0500802i	$0.984052\pi$
$762$	0	0
$763$	7.45856i	0.270018i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.5510i	0.994810i
$768$	0	0
$769$	−46.5606	−1.67902	−0.839509	−	0.543345i	$-0.817157\pi$
$769$	−46.5606	−1.67902	−0.839509	+	0.543345i	$0.817157\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.5877i	1.06419i	0.846683	+	0.532097i	$0.178596\pi$
$773$	29.5877i	1.06419i	−0.846683	+	0.532097i	$0.821404\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 14.0508i	− 0.503424i
$780$	0	0
$781$	1.86244i	0.0666435i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.9267i	1.28065i	0.768105	+	0.640324i	$0.221200\pi$
$787$	35.9267i	1.28065i	−0.768105	+	0.640324i	$0.778800\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.23407	0.292770
$792$	0	0
$793$	− 1.00958i	− 0.0358514i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 9.31695i	− 0.330023i	−0.986292	−	0.165012i	$-0.947234\pi$
$797$	− 9.31695i	− 0.330023i	0.986292	−	0.165012i	$-0.0527662\pi$
$798$	0	0
$799$	23.5416i	0.832841i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.04491	−0.0368742
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	34.9207i	1.22775i	0.789405	+	0.613873i	$0.210389\pi$
$809$	34.9207i	1.22775i	−0.789405	+	0.613873i	$0.789611\pi$
$810$	0	0
$811$	25.7938	0.905744	0.452872	−	0.891576i	$-0.350399\pi$
$811$	25.7938	0.905744	0.452872	+	0.891576i	$0.350399\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.4586i	0.680769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1878	1.08846	0.544232	−	0.838935i	$-0.316821\pi$
$821$	31.1878	1.08846	0.544232	+	0.838935i	$0.316821\pi$
$822$	0	0
$823$	33.5718	1.17024	0.585120	−	0.810947i	$-0.301047\pi$
$823$	33.5718	1.17024	0.585120	+	0.810947i	$0.301047\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.4362	−1.85816	−0.929079	−	0.369881i	$-0.879399\pi$
$827$	−53.4362	−1.85816	−0.929079	+	0.369881i	$0.879399\pi$
$828$	0	0
$829$	0.634952i	0.0220528i	0.999939	+	0.0110264i	$0.00350988\pi$
$829$	0.634952i	0.0220528i	−0.999939	+	0.0110264i	$0.996490\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.0261	1.07499
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.9046	0.859803	0.429901	−	0.902876i	$-0.358548\pi$
$839$	24.9046	0.859803	0.429901	+	0.902876i	$0.358548\pi$
$840$	0	0
$841$	−28.9267	−0.997473
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.5045	−0.532742
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 40.0203i	− 1.37188i
$852$	0	0
$853$	17.5448	0.600724	0.300362	−	0.953825i	$-0.402893\pi$
$853$	17.5448	0.600724	0.300362	+	0.953825i	$0.402893\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.30196	−0.249430	−0.124715	−	0.992193i	$-0.539802\pi$
$857$	−7.30196	−0.249430	−0.124715	+	0.992193i	$0.539802\pi$
$858$	0	0
$859$	15.2890	0.521655	0.260828	−	0.965385i	$-0.416005\pi$
$859$	15.2890	0.521655	0.260828	+	0.965385i	$0.416005\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.9787i	1.42897i	0.699649	+	0.714487i	$0.253340\pi$
$863$	41.9787i	1.42897i	−0.699649	+	0.714487i	$0.746660\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.73887	−0.0929097
$870$	0	0
$871$	4.56012i	0.154514i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.4800	−1.29938	−0.649689	−	0.760200i	$-0.725101\pi$
$877$	−38.4800	−1.29938	−0.649689	+	0.760200i	$0.725101\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.11053i	− 0.239560i	−0.992800	−	0.119780i	$-0.961781\pi$
$881$	− 7.11053i	− 0.239560i	0.992800	−	0.119780i	$-0.0382189\pi$
$882$	0	0
$883$	− 26.5048i	− 0.891957i	−0.895044	−	0.445979i	$-0.852856\pi$
$883$	− 26.5048i	− 0.891957i	0.895044	−	0.445979i	$-0.147144\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.3449i	1.32107i	0.750795	+	0.660535i	$0.229671\pi$
$887$	39.3449i	1.32107i	−0.750795	+	0.660535i	$0.770329\pi$
$888$	0	0
$889$	−29.0462	−0.974179
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.77844i	0.193368i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.67987i	0.0560267i
$900$	0	0
$901$	− 54.4540i	− 1.81413i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	22.1974i	0.737054i	0.929617	+	0.368527i	$0.120138\pi$
$907$	22.1974i	0.737054i	−0.929617	+	0.368527i	$0.879862\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.9538	0.694229	0.347115	−	0.937823i	$-0.387161\pi$
$911$	20.9538	0.694229	0.347115	+	0.937823i	$0.387161\pi$
$912$	0	0
$913$	2.91713i	0.0965428i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.1878i	0.633638i
$918$	0	0
$919$	7.88509i	0.260105i	0.991507	+	0.130053i	$0.0415146\pi$
$919$	7.88509i	0.260105i	−0.991507	+	0.130053i	$0.958485\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25.6561	−0.844481
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.35718i	0.208572i	0.994547	+	0.104286i	$0.0332558\pi$
$929$	6.35718i	0.208572i	−0.994547	+	0.104286i	$0.966744\pi$
$930$	0	0
$931$	7.61557	0.249590
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.0096i	1.14371i	0.820354	+	0.571857i	$0.193777\pi$
$937$	35.0096i	1.14371i	−0.820354	+	0.571857i	$0.806223\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.5606	−0.931049	−0.465525	−	0.885035i	$-0.654134\pi$
$941$	−28.5606	−0.931049	−0.465525	+	0.885035i	$0.654134\pi$
$942$	0	0
$943$	48.4538	1.57787
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.7650	0.674771	0.337385	−	0.941367i	$-0.390457\pi$
$947$	20.7650	0.674771	0.337385	+	0.941367i	$0.390457\pi$
$948$	0	0
$949$	− 14.3942i	− 0.467257i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.4147	1.24437	0.622186	−	0.782869i	$-0.286245\pi$
$953$	38.4147	1.24437	0.622186	+	0.782869i	$0.286245\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.68305	0.215807
$960$	0	0
$961$	−7.50479	−0.242090
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.6013	0.533861	0.266930	−	0.963716i	$-0.413991\pi$
$967$	16.6013	0.533861	0.266930	+	0.963716i	$0.413991\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.3171i	0.331092i	0.986202	+	0.165546i	$0.0529386\pi$
$971$	10.3171i	0.331092i	−0.986202	+	0.165546i	$0.947061\pi$
$972$	0	0
$973$	−22.2840	−0.714393
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.0439	1.02518	0.512588	−	0.858635i	$-0.328687\pi$
$977$	32.0439	1.02518	0.512588	+	0.858635i	$0.328687\pi$
$978$	0	0
$979$	0.683053	0.0218305
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 24.8034i	− 0.791106i	−0.918443	−	0.395553i	$-0.870553\pi$
$983$	− 24.8034i	− 0.791106i	0.918443	−	0.395553i	$-0.129447\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−67.1020	−2.13372
$990$	0	0
$991$	− 15.7872i	− 0.501498i	−0.968052	−	0.250749i	$-0.919323\pi$
$991$	− 15.7872i	− 0.501498i	0.968052	−	0.250749i	$-0.0806769\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51.3823	1.62729	0.813647	−	0.581359i	$-0.197479\pi$
$997$	51.3823	1.62729	0.813647	+	0.581359i	$0.197479\pi$
$998$	0	0
$999$	0	0

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	7200.2.m.e.3599.4		12
3.2	odd	2		7200.2.m.d.3599.3		12
4.3	odd	2		1800.2.m.e.899.6		12
5.2	odd	4		7200.2.b.e.4751.2		6
5.3	odd	4		1440.2.b.d.431.5		6
5.4	even	2	inner	7200.2.m.e.3599.10		12
8.3	odd	2		7200.2.m.d.3599.10		12
8.5	even	2		1800.2.m.d.899.5		12
12.11	even	2		1800.2.m.d.899.7		12
15.2	even	4		7200.2.b.d.4751.2		6
15.8	even	4		1440.2.b.c.431.5		6
15.14	odd	2		7200.2.m.d.3599.9		12
20.3	even	4		360.2.b.d.251.6	yes	6
20.7	even	4		1800.2.b.d.251.1		6
20.19	odd	2		1800.2.m.e.899.7		12
24.5	odd	2		1800.2.m.e.899.8		12
24.11	even	2	inner	7200.2.m.e.3599.9		12
40.3	even	4		1440.2.b.c.431.2		6
40.13	odd	4		360.2.b.c.251.2	yes	6
40.19	odd	2		7200.2.m.d.3599.4		12
40.27	even	4		7200.2.b.d.4751.5		6
40.29	even	2		1800.2.m.d.899.8		12
40.37	odd	4		1800.2.b.e.251.5		6
60.23	odd	4		360.2.b.c.251.1	✓	6
60.47	odd	4		1800.2.b.e.251.6		6
60.59	even	2		1800.2.m.d.899.6		12
120.29	odd	2		1800.2.m.e.899.5		12
120.53	even	4		360.2.b.d.251.5	yes	6
120.59	even	2	inner	7200.2.m.e.3599.3		12
120.77	even	4		1800.2.b.d.251.2		6
120.83	odd	4		1440.2.b.d.431.2		6
120.107	odd	4		7200.2.b.e.4751.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.b.c.251.1	✓	6	60.23	odd	4
360.2.b.c.251.2	yes	6	40.13	odd	4
360.2.b.d.251.5	yes	6	120.53	even	4
360.2.b.d.251.6	yes	6	20.3	even	4
1440.2.b.c.431.2		6	40.3	even	4
1440.2.b.c.431.5		6	15.8	even	4
1440.2.b.d.431.2		6	120.83	odd	4
1440.2.b.d.431.5		6	5.3	odd	4
1800.2.b.d.251.1		6	20.7	even	4
1800.2.b.d.251.2		6	120.77	even	4
1800.2.b.e.251.5		6	40.37	odd	4
1800.2.b.e.251.6		6	60.47	odd	4
1800.2.m.d.899.5		12	8.5	even	2
1800.2.m.d.899.6		12	60.59	even	2
1800.2.m.d.899.7		12	12.11	even	2
1800.2.m.d.899.8		12	40.29	even	2
1800.2.m.e.899.5		12	120.29	odd	2
1800.2.m.e.899.6		12	4.3	odd	2
1800.2.m.e.899.7		12	20.19	odd	2
1800.2.m.e.899.8		12	24.5	odd	2
7200.2.b.d.4751.2		6	15.2	even	4
7200.2.b.d.4751.5		6	40.27	even	4
7200.2.b.e.4751.2		6	5.2	odd	4
7200.2.b.e.4751.5		6	120.107	odd	4
7200.2.m.d.3599.3		12	3.2	odd	2
7200.2.m.d.3599.4		12	40.19	odd	2
7200.2.m.d.3599.9		12	15.14	odd	2
7200.2.m.d.3599.10		12	8.3	odd	2
7200.2.m.e.3599.3		12	120.59	even	2	inner
7200.2.m.e.3599.4		12	1.1	even	1	trivial
7200.2.m.e.3599.9		12	24.11	even	2	inner
7200.2.m.e.3599.10		12	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	7200.2.m.e.3599.4

Level	$7200$
Weight	$2$
Character	7200.3599
Analytic conductor	$57.492$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$