LMFDB - Embedded newform 5850.2.e.bd.5149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.bd.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{19} +4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{26} +2.00000i q^{28} -8.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +6.00000i q^{37} +6.00000i q^{38} -6.00000 q^{41} +8.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} -1.00000i q^{52} +12.0000i q^{53} -2.00000 q^{56} -8.00000i q^{58} +4.00000 q^{59} +10.0000 q^{61} -2.00000i q^{62} -1.00000 q^{64} -2.00000i q^{67} +16.0000 q^{71} -14.0000i q^{73} -6.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} +4.00000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -8.00000 q^{86} -4.00000i q^{88} -6.00000 q^{89} +2.00000 q^{91} -4.00000i q^{92} +8.00000 q^{94} -10.0000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 8 q^{11} + 4 q^{14} + 2 q^{16} + 12 q^{19} - 2 q^{26} - 16 q^{29} - 4 q^{31} - 12 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{56} + 8 q^{59} + 20 q^{61} - 2 q^{64} + 32 q^{71} - 12 q^{74} - 12 q^{76} + 8 q^{79} - 16 q^{86} - 12 q^{89} + 4 q^{91} + 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^11 + 4 * q^14 + 2 * q^16 + 12 * q^19 - 2 * q^26 - 16 * q^29 - 4 * q^31 - 12 * q^41 - 8 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^56 + 8 * q^59 + 20 * q^61 - 2 * q^64 + 32 * q^71 - 12 * q^74 - 12 * q^76 + 8 * q^79 - 16 * q^86 - 12 * q^89 + 4 * q^91 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	2.00000i	0.377964i
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	6.00000i	0.973329i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	− 8.00000i	− 1.05045i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	− 2.00000i	− 0.254000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−6.00000	−0.688247
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	− 6.00000i	− 0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	8.00000	0.825137
$95$	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	$-0.169494\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−12.0000	−1.16554
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	0	0
$116$	8.00000	0.742781
$117$	0	0
$118$	4.00000i	0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	2.00000	0.172774
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	16.0000i	1.34269i
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	−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
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	− 6.00000i	− 0.486664i
$153$	0	0
$154$	8.00000	0.644658
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 8.00000i	− 0.609994i
$173$	16.0000i	1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$	16.0000i	1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	16.0000i	1.12576i
$203$	16.0000i	1.12298i
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	24.0000	1.66011
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	− 12.0000i	− 0.824163i
$213$	0	0
$214$	−20.0000	−1.36717
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	− 10.0000i	− 0.677285i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 6.00000i	− 0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$	− 6.00000i	− 0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−4.00000	−0.266076
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	30.0000	1.98246	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000	1.98246	0.991228	+	0.132164i	$0.0421925\pi$
$230$	0	0
$231$	0	0
$232$	8.00000i	0.525226i
$233$	20.0000i	1.31024i	0.755523	+	0.655122i	$0.227383\pi$
$233$	20.0000i	1.31024i	−0.755523	+	0.655122i	$0.772617\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	6.00000i	0.381771i
$248$	2.00000i	0.127000i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	− 4.00000i	− 0.247121i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	12.0000	0.735767
$267$	0	0
$268$	2.00000i	0.122169i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	20.0000i	1.19952i
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−4.00000	−0.236525
$287$	12.0000i	0.708338i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	14.0000i	0.819288i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−4.00000	−0.231326
$300$	0	0
$301$	16.0000	0.922225
$302$	18.0000i	1.03578i
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	8.00000i	0.455842i
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	−32.0000	−1.79166
$320$	0	0
$321$	0	0
$322$	8.00000i	0.445823i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−10.0000	−0.553849
$327$	0	0
$328$	6.00000i	0.331295i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	8.00000	0.431331
$345$	0	0
$346$	−16.0000	−0.860165
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 18.0000i	− 0.946059i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	−8.00000	−0.412568
$377$	− 8.00000i	− 0.412021i
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	4.00000i	0.204658i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	10.0000i	0.507673i
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	− 8.00000i	− 0.401004i
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	−16.0000	−0.796030
$405$	0	0
$406$	−16.0000	−0.794067
$407$	24.0000i	1.18964i
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	24.0000i	1.17388i
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	8.00000i	0.389434i
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	− 20.0000i	− 0.966736i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	10.0000	0.478913
$437$	24.0000i	1.14808i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	6.00000	0.284108
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	− 4.00000i	− 0.188144i
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	30.0000i	1.40181i
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−20.0000	−0.926482
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	− 4.00000i	− 0.184115i
$473$	32.0000i	1.47136i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	− 24.0000i	− 1.09773i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	6.00000i	0.273293i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	− 10.0000i	− 0.452679i
$489$	0	0
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	0	0
$493$	0	0
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	− 32.0000i	− 1.43540i
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	32.0000i	1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$	32.0000i	1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$	0	0
$505$	0	0
$506$	−16.0000	−0.711287
$507$	0	0
$508$	8.00000i	0.354943i
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	12.0000i	0.520266i
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	22.0000i	0.944981i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	− 18.0000i	− 0.768922i
$549$	0	0
$550$	0	0
$551$	−48.0000	−2.04487
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	6.00000	0.254916
$555$	0	0
$556$	−20.0000	−0.848189
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	10.0000i	0.421825i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	− 16.0000i	− 0.671345i
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	−12.0000	−0.500870
$575$	0	0
$576$	0	0
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	17.0000i	0.707107i
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	48.0000i	1.98796i
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	6.00000i	0.246598i
$593$	− 38.0000i	− 1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$593$	− 38.0000i	− 1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	− 4.00000i	− 0.163572i
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000i	0.652111i
$603$	0	0
$604$	−18.0000	−0.732410
$605$	0	0
$606$	0	0
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	6.00000i	0.243332i
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−8.00000	−0.322329
$617$	− 14.0000i	− 0.563619i	−0.959470	−	0.281809i	$-0.909065\pi$
$617$	− 14.0000i	− 0.563619i	0.959470	−	0.281809i	$-0.0909346\pi$
$618$	0	0
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	0	0
$622$	12.0000i	0.481156i
$623$	12.0000i	0.480770i
$624$	0	0
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	0	0
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	− 32.0000i	− 1.26689i
$639$	0	0
$640$	0	0
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	0	0
$643$	10.0000i	0.394362i	0.980367	+	0.197181i	$0.0631786\pi$
$643$	10.0000i	0.394362i	−0.980367	+	0.197181i	$0.936821\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 10.0000i	− 0.391630i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	− 16.0000i	− 0.623745i
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	− 10.0000i	− 0.388661i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	− 32.0000i	− 1.23904i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	34.0000	1.30963
$675$	0	0
$676$	1.00000	0.0384615
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	8.00000i	0.304997i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	− 16.0000i	− 0.608229i
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 22.0000i	− 0.832712i
$699$	0	0
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	36.0000i	1.35777i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	− 32.0000i	− 1.20348i
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	17.0000i	0.632674i
$723$	0	0
$724$	18.0000	0.668965
$725$	0	0
$726$	0	0
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	54.0000	1.98642	0.993211	−	0.116326i	$-0.0371118\pi$
$739$	54.0000	1.98642	0.993211	+	0.116326i	$0.0371118\pi$
$740$	0	0
$741$	0	0
$742$	24.0000i	0.881068i
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	0	0
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	− 18.0000i	− 0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	− 18.0000i	− 0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$	− 10.0000i	− 0.363216i
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	−4.00000	−0.144715
$765$	0	0
$766$	0	0
$767$	4.00000i	0.144432i
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	− 12.0000i	− 0.430221i
$779$	−36.0000	−1.28983
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	10.0000i	0.355110i
$794$	10.0000	0.354887
$795$	0	0
$796$	8.00000	0.283552
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	6.00000i	0.211867i
$803$	− 56.0000i	− 1.97620i
$804$	0	0
$805$	0	0
$806$	2.00000	0.0704470
$807$	0	0
$808$	− 16.0000i	− 0.562878i
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	50.0000	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$812$	− 16.0000i	− 0.561490i
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	48.0000i	1.67931i
$818$	26.0000i	0.909069i
$819$	0	0
$820$	0	0
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	0	0
$823$	12.0000i	0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$	12.0000i	0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	8.00000	0.278356
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	36.0000i	1.24360i
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	10.0000i	0.344623i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	12.0000i	0.412082i
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	20.0000	0.683586
$857$	28.0000i	0.956462i	0.878234	+	0.478231i	$0.158722\pi$
$857$	28.0000i	0.956462i	−0.878234	+	0.478231i	$0.841278\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.00000i	0.272323i	0.990687	+	0.136162i	$0.0434766\pi$
$863$	8.00000i	0.272323i	−0.990687	+	0.136162i	$0.956523\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	− 4.00000i	− 0.135769i
$869$	16.0000	0.542763
$870$	0	0
$871$	2.00000	0.0677674
$872$	10.0000i	0.338643i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	4.00000i	0.134993i
$879$	0	0
$880$	0	0
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	− 44.0000i	− 1.47738i	−0.674048	−	0.738688i	$-0.735446\pi$
$887$	− 44.0000i	− 1.47738i	0.674048	−	0.738688i	$-0.264554\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	6.00000i	0.200895i
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	26.0000i	0.867631i
$899$	16.0000	0.533630
$900$	0	0
$901$	0	0
$902$	− 24.0000i	− 0.799113i
$903$	0	0
$904$	4.00000	0.133038
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	− 48.0000i	− 1.58857i
$914$	2.00000	0.0661541
$915$	0	0
$916$	−30.0000	−0.991228
$917$	8.00000i	0.264183i
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	0	0
$922$	− 30.0000i	− 0.987997i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	22.0000	0.722965
$927$	0	0
$928$	− 8.00000i	− 0.262613i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	− 20.0000i	− 0.655122i
$933$	0	0
$934$	−8.00000	−0.261768
$935$	0	0
$936$	0	0
$937$	30.0000i	0.980057i	0.871706	+	0.490029i	$0.163014\pi$
$937$	30.0000i	0.980057i	−0.871706	+	0.490029i	$0.836986\pi$
$938$	− 4.00000i	− 0.130605i
$939$	0	0
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	4.00000	0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28.0000i	0.907009i	0.891254	+	0.453504i	$0.149826\pi$
$953$	28.0000i	0.907009i	−0.891254	+	0.453504i	$0.850174\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 6.00000i	− 0.193448i
$963$	0	0
$964$	−6.00000	−0.193247
$965$	0	0
$966$	0	0
$967$	− 58.0000i	− 1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	− 58.0000i	− 1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	− 40.0000i	− 1.28234i
$974$	−38.0000	−1.21760
$975$	0	0
$976$	10.0000	0.320092
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	16.0000i	0.510581i
$983$	− 32.0000i	− 1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$	− 32.0000i	− 1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 6.00000i	− 0.190885i
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	0	0
$994$	32.0000	1.01498
$995$	0	0
$996$	0	0
$997$	38.0000i	1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$	38.0000i	1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$	− 22.0000i	− 0.696398i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.bd.5149.2		2
3.2	odd	2		5850.2.e.d.5149.1		2
5.2	odd	4		5850.2.a.v.1.1		1
5.3	odd	4		234.2.a.d.1.1	yes	1
5.4	even	2	inner	5850.2.e.bd.5149.1		2
15.2	even	4		5850.2.a.bv.1.1		1
15.8	even	4		234.2.a.a.1.1	✓	1
15.14	odd	2		5850.2.e.d.5149.2		2
20.3	even	4		1872.2.a.p.1.1		1
40.3	even	4		7488.2.a.s.1.1		1
40.13	odd	4		7488.2.a.j.1.1		1
45.13	odd	12		2106.2.e.e.703.1		2
45.23	even	12		2106.2.e.z.703.1		2
45.38	even	12		2106.2.e.z.1405.1		2
45.43	odd	12		2106.2.e.e.1405.1		2
60.23	odd	4		1872.2.a.g.1.1		1
65.8	even	4		3042.2.b.b.1351.2		2
65.18	even	4		3042.2.b.b.1351.1		2
65.38	odd	4		3042.2.a.b.1.1		1
120.53	even	4		7488.2.a.bp.1.1		1
120.83	odd	4		7488.2.a.bu.1.1		1
195.8	odd	4		3042.2.b.c.1351.1		2
195.38	even	4		3042.2.a.o.1.1		1
195.83	odd	4		3042.2.b.c.1351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.a.a.1.1	✓	1	15.8	even	4
234.2.a.d.1.1	yes	1	5.3	odd	4
1872.2.a.g.1.1		1	60.23	odd	4
1872.2.a.p.1.1		1	20.3	even	4
2106.2.e.e.703.1		2	45.13	odd	12
2106.2.e.e.1405.1		2	45.43	odd	12
2106.2.e.z.703.1		2	45.23	even	12
2106.2.e.z.1405.1		2	45.38	even	12
3042.2.a.b.1.1		1	65.38	odd	4
3042.2.a.o.1.1		1	195.38	even	4
3042.2.b.b.1351.1		2	65.18	even	4
3042.2.b.b.1351.2		2	65.8	even	4
3042.2.b.c.1351.1		2	195.8	odd	4
3042.2.b.c.1351.2		2	195.83	odd	4
5850.2.a.v.1.1		1	5.2	odd	4
5850.2.a.bv.1.1		1	15.2	even	4
5850.2.e.d.5149.1		2	3.2	odd	2
5850.2.e.d.5149.2		2	15.14	odd	2
5850.2.e.bd.5149.1		2	5.4	even	2	inner
5850.2.e.bd.5149.2		2	1.1	even	1	trivial
7488.2.a.j.1.1		1	40.13	odd	4
7488.2.a.s.1.1		1	40.3	even	4
7488.2.a.bp.1.1		1	120.53	even	4
7488.2.a.bu.1.1		1	120.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{19} +4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{26} +2.00000i q^{28} -8.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +6.00000i q^{37} +6.00000i q^{38} -6.00000 q^{41} +8.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} -1.00000i q^{52} +12.0000i q^{53} -2.00000 q^{56} -8.00000i q^{58} +4.00000 q^{59} +10.0000 q^{61} -2.00000i q^{62} -1.00000 q^{64} -2.00000i q^{67} +16.0000 q^{71} -14.0000i q^{73} -6.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} +4.00000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -8.00000 q^{86} -4.00000i q^{88} -6.00000 q^{89} +2.00000 q^{91} -4.00000i q^{92} +8.00000 q^{94} -10.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 8 q^{11} + 4 q^{14} + 2 q^{16} + 12 q^{19} - 2 q^{26} - 16 q^{29} - 4 q^{31} - 12 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{56} + 8 q^{59} + 20 q^{61} - 2 q^{64} + 32 q^{71} - 12 q^{74} - 12 q^{76} + 8 q^{79} - 16 q^{86} - 12 q^{89} + 4 q^{91} + 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^11 + 4 * q^14 + 2 * q^16 + 12 * q^19 - 2 * q^26 - 16 * q^29 - 4 * q^31 - 12 * q^41 - 8 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^56 + 8 * q^59 + 20 * q^61 - 2 * q^64 + 32 * q^71 - 12 * q^74 - 12 * q^76 + 8 * q^79 - 16 * q^86 - 12 * q^89 + 4 * q^91 + 16 * q^94

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