LMFDB - Embedded newform 4600.2.e.b.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.b.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000i q^{13} +6.00000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +3.00000 q^{29} +3.00000 q^{31} +8.00000i q^{37} +3.00000 q^{39} +3.00000 q^{41} -2.00000i q^{43} +11.0000i q^{47} +3.00000 q^{49} -14.0000i q^{53} +8.00000 q^{59} -4.00000 q^{61} -12.0000i q^{63} +4.00000i q^{67} +3.00000 q^{69} +7.00000 q^{71} -9.00000i q^{73} +9.00000 q^{81} +4.00000i q^{83} -9.00000i q^{87} +2.00000 q^{89} -2.00000 q^{91} -9.00000i q^{93} -18.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} + 12 q^{21} + 6 q^{29} + 6 q^{31} + 6 q^{39} + 6 q^{41} + 6 q^{49} + 16 q^{59} - 8 q^{61} + 6 q^{69} + 14 q^{71} + 18 q^{81} + 4 q^{89} - 4 q^{91}+O(q^{100}) $ 2 * q - 12 * q^9 + 12 * q^21 + 6 * q^29 + 6 * q^31 + 6 * q^39 + 6 * q^41 + 6 * q^49 + 16 * q^59 - 8 * q^61 + 6 * q^69 + 14 * q^71 + 18 * q^81 + 4 * q^89 - 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	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
$9$	−6.00000	−2.00000
$10$	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
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	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
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	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
$39$	3.00000	0.480384
$40$	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
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.0000i	1.60451i	0.596978	+	0.802257i	$0.296368\pi$
$47$	11.0000i	1.60451i	−0.596978	+	0.802257i	$0.703632\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	− 12.0000i	− 1.51186i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 9.00000i	− 0.964901i
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	− 9.00000i	− 0.933257i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 18.0000i	− 1.82762i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	− 18.0000i	− 1.82762i	0.406138	−	0.913812i	$-0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 9.00000i	− 0.811503i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.0000i	− 0.976092i	−0.872818	−	0.488046i	$-0.837710\pi$
$127$	− 11.0000i	− 0.976092i	0.872818	−	0.488046i	$-0.162290\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	33.0000	2.77910
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	0	0
$159$	−42.0000	−3.33082
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	7.00000i	0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$	7.00000i	0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 24.0000i	− 1.80395i
$178$	0	0
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	12.0000i	0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−18.0000	−1.30931
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	− 1.00000i	− 0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$	− 1.00000i	− 0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	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
$201$	12.0000	0.846415
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.00000i	− 0.417029i
$208$	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
$213$	− 21.0000i	− 1.43890i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.00000i	0.407307i
$218$	0	0
$219$	−27.0000	−1.82449
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.00000i	− 0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$	− 2.00000i	− 0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0000i	1.37576i	0.725826	+	0.687878i	$0.241458\pi$
$233$	21.0000i	1.37576i	−0.725826	+	0.687878i	$0.758542\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.00000i	− 0.311891i	−0.987766	−	0.155946i	$-0.950158\pi$
$257$	− 5.00000i	− 0.311891i	0.987766	−	0.155946i	$-0.0498425\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	6.00000i	0.363137i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	29.0000i	1.74244i	0.490892	+	0.871221i	$0.336671\pi$
$277$	29.0000i	1.74244i	−0.490892	+	0.871221i	$0.663329\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	10.0000i	0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$	10.0000i	0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	−54.0000	−3.16554
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	− 54.0000i	− 3.10222i
$304$	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
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−29.0000	−1.64444	−0.822220	−	0.569170i	$-0.807264\pi$
$311$	−29.0000	−1.64444	−0.822220	+	0.569170i	$0.807264\pi$
$312$	0	0
$313$	− 20.0000i	− 1.13047i	−0.824931	−	0.565233i	$-0.808786\pi$
$313$	− 20.0000i	− 1.13047i	0.824931	−	0.565233i	$-0.191214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	48.0000	2.67910
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 54.0000i	− 2.98621i
$328$	0	0
$329$	−22.0000	−1.21290
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	− 48.0000i	− 2.63038i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	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	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	0	0
$353$	19.0000i	1.01127i	0.862748	+	0.505634i	$0.168741\pi$
$353$	19.0000i	1.01127i	−0.862748	+	0.505634i	$0.831259\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	33.0000i	1.73205i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	28.0000	1.45369
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.00000i	0.154508i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−33.0000	−1.69064
$382$	0	0
$383$	30.0000i	1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$	30.0000i	1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	27.0000i	1.36197i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 25.0000i	− 1.25471i	−0.778732	−	0.627357i	$-0.784137\pi$
$397$	− 25.0000i	− 1.25471i	0.778732	−	0.627357i	$-0.215863\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	3.00000i	0.149441i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	16.0000i	0.787309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 33.0000i	− 1.61602i
$418$	0	0
$419$	−22.0000	−1.07477	−0.537385	−	0.843337i	$-0.680588\pi$
$419$	−22.0000	−1.07477	−0.537385	+	0.843337i	$0.680588\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	− 66.0000i	− 3.20903i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	33.0000i	1.56788i	0.620838	+	0.783939i	$0.286792\pi$
$443$	33.0000i	1.56788i	−0.620838	+	0.783939i	$0.713208\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 66.0000i	− 3.12169i
$448$	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
$453$	− 21.0000i	− 0.986666i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.00000i	0.187112i	0.995614	+	0.0935561i	$0.0298234\pi$
$457$	4.00000i	0.187112i	−0.995614	+	0.0935561i	$0.970177\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 42.0000i	− 1.94353i	−0.235954	−	0.971764i	$-0.575822\pi$
$467$	− 42.0000i	− 1.94353i	0.235954	−	0.971764i	$-0.424178\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	84.0000i	3.84610i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.0000i	1.13286i	0.824110	+	0.566429i	$0.191675\pi$
$487$	25.0000i	1.13286i	−0.824110	+	0.566429i	$0.808325\pi$
$488$	0	0
$489$	21.0000	0.949653
$490$	0	0
$491$	−31.0000	−1.39901	−0.699505	−	0.714628i	$-0.746596\pi$
$491$	−31.0000	−1.39901	−0.699505	+	0.714628i	$0.746596\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.0000i	0.627986i
$498$	0	0
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	0	0
$503$	− 14.0000i	− 0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$	− 14.0000i	− 0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 36.0000i	− 1.59882i
$508$	0	0
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	42.0000	1.84360
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	42.0000i	1.83653i	0.395964	+	0.918266i	$0.370410\pi$
$523$	42.0000i	1.83653i	−0.395964	+	0.918266i	$0.629590\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−48.0000	−2.08302
$532$	0	0
$533$	3.00000i	0.129944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	63.0000i	2.71865i
$538$	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
$543$	− 36.0000i	− 1.54491i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 35.0000i	− 1.49649i	−0.663421	−	0.748246i	$-0.730896\pi$
$547$	− 35.0000i	− 1.49649i	0.663421	−	0.748246i	$-0.269104\pi$
$548$	0	0
$549$	24.0000	1.02430
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 16.0000i	− 0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$	− 16.0000i	− 0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.0000i	0.755929i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	− 6.00000i	− 0.250654i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.00000i	0.374675i	0.982296	+	0.187337i	$0.0599858\pi$
$577$	9.00000i	0.374675i	−0.982296	+	0.187337i	$0.940014\pi$
$578$	0	0
$579$	−3.00000	−0.124676
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 33.0000i	− 1.36206i	−0.732257	−	0.681028i	$-0.761533\pi$
$587$	− 33.0000i	− 1.36206i	0.732257	−	0.681028i	$-0.238467\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−9.00000	−0.370211
$592$	0	0
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 12.0000i	− 0.491127i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	0	0
$603$	− 24.0000i	− 0.977356i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	−11.0000	−0.445012
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.0000i	1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$	48.0000i	1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	0	0
$623$	4.00000i	0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	48.0000i	1.90783i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	0	0
$639$	−42.0000	−1.66149
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	39.0000i	1.53325i	0.642096	+	0.766624i	$0.278065\pi$
$647$	39.0000i	1.53325i	−0.642096	+	0.766624i	$0.721935\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	18.0000	0.705476
$652$	0	0
$653$	− 3.00000i	− 0.117399i	−0.998276	−	0.0586995i	$-0.981305\pi$
$653$	− 3.00000i	− 0.117399i	0.998276	−	0.0586995i	$-0.0186954\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	54.0000i	2.10674i
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.00000i	0.116160i
$668$	0	0
$669$	−48.0000	−1.85579
$670$	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
$677$	46.0000i	1.76792i	0.467559	+	0.883962i	$0.345134\pi$
$677$	46.0000i	1.76792i	−0.467559	+	0.883962i	$0.654866\pi$
$678$	0	0
$679$	36.0000	1.38155
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	35.0000i	1.33924i	0.742705	+	0.669619i	$0.233543\pi$
$683$	35.0000i	1.33924i	−0.742705	+	0.669619i	$0.766457\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6.00000i	− 0.228914i
$688$	0	0
$689$	14.0000	0.533358
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	63.0000	2.38288
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.0000i	1.35392i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.00000i	0.112351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.00000i	− 0.112037i
$718$	0	0
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 6.00000i	− 0.223142i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.00000i	0.222528i	0.993791	+	0.111264i	$0.0354899\pi$
$727$	6.00000i	0.222528i	−0.993791	+	0.111264i	$0.964510\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 8.00000i	− 0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$	− 8.00000i	− 0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.0000	−0.625355	−0.312678	−	0.949859i	$-0.601226\pi$
$739$	−17.0000	−0.625355	−0.312678	+	0.949859i	$0.601226\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 24.0000i	− 0.878114i
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	0	0
$753$	− 48.0000i	− 1.74922i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.0000	1.05125	0.525625	−	0.850717i	$-0.323832\pi$
$761$	29.0000	1.05125	0.525625	+	0.850717i	$0.323832\pi$
$762$	0	0
$763$	36.0000i	1.30329i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000i	0.288863i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−15.0000	−0.540212
$772$	0	0
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	48.0000i	1.72199i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	27.0000i	0.964901i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	0	0
$789$	−36.0000	−1.28163
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	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
$801$	−12.0000	−0.423999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	51.0000i	1.79529i
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	0	0
$813$	60.0000i	2.10429i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	33.0000i	1.15031i	0.818045	+	0.575154i	$0.195058\pi$
$823$	33.0000i	1.15031i	−0.818045	+	0.575154i	$0.804942\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	87.0000	3.01800
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.0000i	0.933257i
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	0	0
$849$	30.0000	1.02960
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 41.0000i	− 1.40053i	−0.713881	−	0.700267i	$-0.753064\pi$
$857$	− 41.0000i	− 1.40053i	0.713881	−	0.700267i	$-0.246936\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	− 17.0000i	− 0.578687i	−0.957225	−	0.289343i	$-0.906563\pi$
$863$	− 17.0000i	− 0.578687i	0.957225	−	0.289343i	$-0.0934369\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 51.0000i	− 1.73205i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	108.000i	3.65525i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	−72.0000	−2.42850
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.0000i	0.503651i	0.967773	+	0.251825i	$0.0810309\pi$
$887$	15.0000i	0.503651i	−0.967773	+	0.251825i	$0.918969\pi$
$888$	0	0
$889$	22.0000	0.737856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.00000i	0.100167i
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 12.0000i	− 0.399335i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	0	0
$909$	−108.000	−3.58213
$910$	0	0
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.0000i	− 0.594412i
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	60.0000	1.97707
$922$	0	0
$923$	7.00000i	0.230408i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	24.0000i	0.788263i
$928$	0	0
$929$	−19.0000	−0.623370	−0.311685	−	0.950186i	$-0.600893\pi$
$929$	−19.0000	−0.623370	−0.311685	+	0.950186i	$0.600893\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	87.0000i	2.84825i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.0000i	1.43742i	0.695311	+	0.718709i	$0.255266\pi$
$937$	44.0000i	1.43742i	−0.695311	+	0.718709i	$0.744734\pi$
$938$	0	0
$939$	−60.0000	−1.95803
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	3.00000i	0.0976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.0000i	1.52729i	0.645634	+	0.763647i	$0.276593\pi$
$947$	47.0000i	1.52729i	−0.645634	+	0.763647i	$0.723407\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	0	0
$951$	42.0000	1.36194
$952$	0	0
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	− 96.0000i	− 3.09356i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43.0000i	− 1.38279i	−0.722478	−	0.691393i	$-0.756997\pi$
$967$	− 43.0000i	− 1.38279i	0.722478	−	0.691393i	$-0.243003\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	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	$-0.721338\pi$
$977$	− 48.0000i	− 1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−108.000	−3.44817
$982$	0	0
$983$	− 14.0000i	− 0.446531i	−0.974758	−	0.223265i	$-0.928328\pi$
$983$	− 14.0000i	− 0.446531i	0.974758	−	0.223265i	$-0.0716716\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	66.0000i	2.10080i
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	21.0000i	0.666415i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 58.0000i	− 1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$	− 58.0000i	− 1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$	0	0
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.b.4049.1		2
5.2	odd	4		920.2.a.a.1.1	✓	1
5.3	odd	4		4600.2.a.p.1.1		1
5.4	even	2	inner	4600.2.e.b.4049.2		2
15.2	even	4		8280.2.a.d.1.1		1
20.3	even	4		9200.2.a.c.1.1		1
20.7	even	4		1840.2.a.i.1.1		1
40.27	even	4		7360.2.a.a.1.1		1
40.37	odd	4		7360.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.a.1.1	✓	1	5.2	odd	4
1840.2.a.i.1.1		1	20.7	even	4
4600.2.a.p.1.1		1	5.3	odd	4
4600.2.e.b.4049.1		2	1.1	even	1	trivial
4600.2.e.b.4049.2		2	5.4	even	2	inner
7360.2.a.a.1.1		1	40.27	even	4
7360.2.a.ba.1.1		1	40.37	odd	4
8280.2.a.d.1.1		1	15.2	even	4
9200.2.a.c.1.1		1	20.3	even	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000i q^{13} +6.00000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +3.00000 q^{29} +3.00000 q^{31} +8.00000i q^{37} +3.00000 q^{39} +3.00000 q^{41} -2.00000i q^{43} +11.0000i q^{47} +3.00000 q^{49} -14.0000i q^{53} +8.00000 q^{59} -4.00000 q^{61} -12.0000i q^{63} +4.00000i q^{67} +3.00000 q^{69} +7.00000 q^{71} -9.00000i q^{73} +9.00000 q^{81} +4.00000i q^{83} -9.00000i q^{87} +2.00000 q^{89} -2.00000 q^{91} -9.00000i q^{93} -18.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} + 12 q^{21} + 6 q^{29} + 6 q^{31} + 6 q^{39} + 6 q^{41} + 6 q^{49} + 16 q^{59} - 8 q^{61} + 6 q^{69} + 14 q^{71} + 18 q^{81} + 4 q^{89} - 4 q^{91}+O(q^{100}) \) 2 * q - 12 * q^9 + 12 * q^21 + 6 * q^29 + 6 * q^31 + 6 * q^39 + 6 * q^41 + 6 * q^49 + 16 * q^59 - 8 * q^61 + 6 * q^69 + 14 * q^71 + 18 * q^81 + 4 * q^89 - 4 * q^91

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