LMFDB - Embedded newform 4800.2.f.x.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(3649,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.f (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:	$38.3281929702$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4800.3649
Dual form			4800.2.f.x.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -5.00000i q^{13} +5.00000 q^{19} +3.00000 q^{21} -4.00000i q^{23} -1.00000i q^{27} +4.00000 q^{29} -5.00000 q^{31} +10.0000i q^{37} +5.00000 q^{39} -10.0000 q^{41} +1.00000i q^{43} -2.00000i q^{47} -2.00000 q^{49} -10.0000i q^{53} +5.00000i q^{57} -10.0000 q^{59} +5.00000 q^{61} +3.00000i q^{63} +3.00000i q^{67} +4.00000 q^{69} -10.0000 q^{71} +10.0000i q^{73} +1.00000 q^{81} -14.0000i q^{83} +4.00000i q^{87} -16.0000 q^{89} -15.0000 q^{91} -5.00000i q^{93} -5.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 10 q^{19} + 6 q^{21} + 8 q^{29} - 10 q^{31} + 10 q^{39} - 20 q^{41} - 4 q^{49} - 20 q^{59} + 10 q^{61} + 8 q^{69} - 20 q^{71} + 2 q^{81} - 32 q^{89} - 30 q^{91}+O(q^{100}) $ 2 * q - 2 * q^9 + 10 * q^19 + 6 * q^21 + 8 * q^29 - 10 * q^31 + 10 * q^39 - 20 * q^41 - 4 * q^49 - 20 * q^59 + 10 * q^61 + 8 * q^69 - 20 * q^71 + 2 * q^81 - 32 * q^89 - 30 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.00000i	0.662266i
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	3.00000i	0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000i	0.428845i
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−15.0000	−1.57243
$92$	0	0
$93$	− 5.00000i	− 0.518476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.00000i	− 0.507673i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$	− 5.00000i	− 0.507673i	0.967247	−	0.253837i	$-0.0816925\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	20.0000i	1.88144i	0.339182	+	0.940721i	$0.389850\pi$
$113$	20.0000i	1.88144i	−0.339182	+	0.940721i	$0.610150\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.00000i	0.462250i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 10.0000i	− 0.901670i
$124$	0	0
$125$	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	$-0.115500\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	− 15.0000i	− 1.30066i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	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
$141$	2.00000	0.168430
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	20.0000i	1.52057i	0.649589	+	0.760286i	$0.274941\pi$
$173$	20.0000i	1.52057i	−0.649589	+	0.760286i	$0.725059\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 10.0000i	− 0.751646i
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	5.00000i	0.369611i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000i	0.278019i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	0	0
$213$	− 10.0000i	− 0.685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.0000i	1.01827i
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 9.00000i	− 0.602685i	−0.953516	−	0.301342i	$-0.902565\pi$
$223$	− 9.00000i	− 0.602685i	0.953516	−	0.301342i	$-0.0974347\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	30.0000i	1.96537i	0.185296	+	0.982683i	$0.440675\pi$
$233$	30.0000i	1.96537i	−0.185296	+	0.982683i	$0.559325\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 25.0000i	− 1.59071i
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	30.0000	1.86411
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 16.0000i	− 0.979184i
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	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
$273$	− 15.0000i	− 0.907841i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.0000i	− 0.901263i	−0.892710	−	0.450631i	$-0.851199\pi$
$277$	− 15.0000i	− 0.901263i	0.892710	−	0.450631i	$-0.148801\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	− 21.0000i	− 1.24832i	−0.781296	−	0.624160i	$-0.785441\pi$
$283$	− 21.0000i	− 1.24832i	0.781296	−	0.624160i	$-0.214559\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.0000i	1.77084i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	5.00000	0.293105
$292$	0	0
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.0000i	0.741949i	0.928643	+	0.370975i	$0.120976\pi$
$307$	13.0000i	0.741949i	−0.928643	+	0.370975i	$0.879024\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	− 25.0000i	− 1.41308i	−0.707671	−	0.706542i	$-0.750254\pi$
$313$	− 25.0000i	− 1.41308i	0.707671	−	0.706542i	$-0.249746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.0000i	− 1.12331i	−0.827371	−	0.561656i	$-0.810164\pi$
$317$	− 20.0000i	− 1.12331i	0.827371	−	0.561656i	$-0.189836\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.00000i	0.276501i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5.00000i	− 0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$	− 5.00000i	− 0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	− 20.0000i	− 1.06449i	−0.846590	−	0.532246i	$-0.821348\pi$
$353$	− 20.0000i	− 1.06449i	0.846590	−	0.532246i	$-0.178652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−30.0000	−1.55752
$372$	0	0
$373$	5.00000i	0.258890i	0.991587	+	0.129445i	$0.0413196\pi$
$373$	5.00000i	0.258890i	−0.991587	+	0.129445i	$0.958680\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 20.0000i	− 1.03005i
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.00000i	− 0.0508329i
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 10.0000i	− 0.504433i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.0000i	0.752828i	0.926451	+	0.376414i	$0.122843\pi$
$397$	15.0000i	0.752828i	−0.926451	+	0.376414i	$0.877157\pi$
$398$	0	0
$399$	15.0000	0.750939
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	25.0000i	1.24534i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	30.0000i	1.47620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000i	0.979404i
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 15.0000i	− 0.725901i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	− 25.0000i	− 1.20142i	−0.799466	−	0.600712i	$-0.794884\pi$
$433$	− 25.0000i	− 1.20142i	0.799466	−	0.600712i	$-0.205116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.0000i	− 0.956730i
$438$	0	0
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 10.0000i	− 0.472984i
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	5.00000i	0.234920i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	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
$469$	9.00000	0.415581
$470$	0	0
$471$	5.00000	0.230388
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	50.0000	2.27980
$482$	0	0
$483$	− 12.0000i	− 0.546019i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17.0000i	− 0.770344i	−0.922845	−	0.385172i	$-0.874142\pi$
$487$	− 17.0000i	− 0.770344i	0.922845	−	0.385172i	$-0.125858\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	0	0
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.0000i	1.34568i
$498$	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
$501$	18.0000	0.804181
$502$	0	0
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	0	0
$513$	− 5.00000i	− 0.220755i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−20.0000	−0.877903
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	19.0000i	0.830812i	0.909636	+	0.415406i	$0.136360\pi$
$523$	19.0000i	0.830812i	−0.909636	+	0.415406i	$0.863640\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	50.0000i	2.16574i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0000i	0.431532i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−33.0000	−1.41878	−0.709390	−	0.704816i	$-0.751030\pi$
$541$	−33.0000	−1.41878	−0.709390	+	0.704816i	$0.751030\pi$
$542$	0	0
$543$	7.00000i	0.300399i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	0	0
$549$	−5.00000	−0.213395
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.00000i	− 0.125988i
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.0000i	1.04076i	0.853934	+	0.520382i	$0.174210\pi$
$577$	25.0000i	1.04076i	−0.853934	+	0.520382i	$0.825790\pi$
$578$	0	0
$579$	5.00000	0.207793
$580$	0	0
$581$	−42.0000	−1.74245
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	−25.0000	−1.03011
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	10.0000i	0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$	10.0000i	0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 25.0000i	− 1.02318i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	− 3.00000i	− 0.122169i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−10.0000	−0.404557
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	48.0000i	1.92308i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	35.0000	1.39333	0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000	1.39333	0.696664	+	0.717398i	$0.254667\pi$
$632$	0	0
$633$	− 25.0000i	− 0.993661i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.0000i	0.396214i
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−15.0000	−0.587896
$652$	0	0
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	9.00000	0.347960
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−15.0000	−0.575647
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.0000i	0.801200i
$688$	0	0
$689$	−50.0000	−1.90485
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−30.0000	−1.13470
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	50.0000i	1.88579i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.0000i	0.749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.0000i	1.12037i
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	5.00000i	0.185952i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.0000i	1.37225i	0.727482	+	0.686127i	$0.240691\pi$
$727$	37.0000i	1.37225i	−0.727482	+	0.686127i	$0.759309\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	10.0000i	0.369358i	0.982799	+	0.184679i	$0.0591246\pi$
$733$	10.0000i	0.369358i	−0.982799	+	0.184679i	$0.940875\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	25.0000	0.918398
$742$	0	0
$743$	− 44.0000i	− 1.61420i	−0.590412	−	0.807102i	$-0.701035\pi$
$743$	− 44.0000i	− 1.61420i	0.590412	−	0.807102i	$-0.298965\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.0000i	0.512233i
$748$	0	0
$749$	−54.0000	−1.97312
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.00000i	0.181728i	0.995863	+	0.0908640i	$0.0289629\pi$
$757$	5.00000i	0.181728i	−0.995863	+	0.0908640i	$0.971037\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	− 15.0000i	− 0.543036i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	50.0000i	1.80540i
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	30.0000i	1.07624i
$778$	0	0
$779$	−50.0000	−1.79144
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 4.00000i	− 0.142948i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.0000i	0.605985i	0.952993	+	0.302992i	$0.0979856\pi$
$787$	17.0000i	0.605985i	−0.952993	+	0.302992i	$0.902014\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	60.0000	2.13335
$792$	0	0
$793$	− 25.0000i	− 0.887776i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	15.0000	0.526721	0.263361	−	0.964697i	$-0.415169\pi$
$811$	15.0000	0.526721	0.263361	+	0.964697i	$0.415169\pi$
$812$	0	0
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.00000i	0.174928i
$818$	0	0
$819$	15.0000	0.524142
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	11.0000i	0.383436i	0.981450	+	0.191718i	$0.0614059\pi$
$823$	11.0000i	0.383436i	−0.981450	+	0.191718i	$0.938594\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.00000i	− 0.0695468i	−0.999395	−	0.0347734i	$-0.988929\pi$
$827$	− 2.00000i	− 0.0695468i	0.999395	−	0.0347734i	$-0.0110710\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	15.0000	0.520344
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.00000i	0.172825i
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	20.0000i	0.688837i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	33.0000i	1.13389i
$848$	0	0
$849$	21.0000	0.720718
$850$	0	0
$851$	40.0000	1.37118
$852$	0	0
$853$	− 15.0000i	− 0.513590i	−0.966466	−	0.256795i	$-0.917333\pi$
$853$	− 15.0000i	− 0.513590i	0.966466	−	0.256795i	$-0.0826666\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−30.0000	−1.02240
$862$	0	0
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.0000i	0.577350i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	15.0000	0.508256
$872$	0	0
$873$	5.00000i	0.169224i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.0000i	0.506514i	0.967399	+	0.253257i	$0.0815018\pi$
$877$	15.0000i	0.506514i	−0.967399	+	0.253257i	$0.918498\pi$
$878$	0	0
$879$	−10.0000	−0.337292
$880$	0	0
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	0	0
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.00000i	0.0671534i	0.999436	+	0.0335767i	$0.0106898\pi$
$887$	2.00000i	0.0671534i	−0.999436	+	0.0335767i	$0.989310\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.0000i	− 0.334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 20.0000i	− 0.667781i
$898$	0	0
$899$	−20.0000	−0.667037
$900$	0	0
$901$	0	0
$902$	0	0
$903$	3.00000i	0.0998337i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.0000i	0.990687i
$918$	0	0
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	−13.0000	−0.428365
$922$	0	0
$923$	50.0000i	1.64577i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.0000i	0.525509i
$928$	0	0
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	0	0
$933$	− 30.0000i	− 0.982156i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.0000i	1.47009i	0.678021	+	0.735043i	$0.262838\pi$
$937$	45.0000i	1.47009i	−0.678021	+	0.735043i	$0.737162\pi$
$938$	0	0
$939$	25.0000	0.815844
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	40.0000i	1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	0	0
$949$	50.0000	1.62307
$950$	0	0
$951$	20.0000	0.648544
$952$	0	0
$953$	40.0000i	1.29573i	0.761756	+	0.647864i	$0.224337\pi$
$953$	40.0000i	1.29573i	−0.761756	+	0.647864i	$0.775663\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−30.0000	−0.968751
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	18.0000i	0.580042i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.0000	0.320915	0.160458	−	0.987043i	$-0.448703\pi$
$971$	10.0000	0.320915	0.160458	+	0.987043i	$0.448703\pi$
$972$	0	0
$973$	− 60.0000i	− 1.92351i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.0000i	0.319928i	0.987123	+	0.159964i	$0.0511379\pi$
$977$	10.0000i	0.319928i	−0.987123	+	0.159964i	$0.948862\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5.00000	−0.159638
$982$	0	0
$983$	4.00000i	0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$	4.00000i	0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	35.0000	1.11181	0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000	1.11181	0.555906	+	0.831245i	$0.312372\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 30.0000i	− 0.950110i	−0.879956	−	0.475055i	$-0.842428\pi$
$997$	− 30.0000i	− 0.950110i	0.879956	−	0.475055i	$-0.157572\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.f.x.3649.2		2
4.3	odd	2		4800.2.f.m.3649.1		2
5.2	odd	4		4800.2.a.cl.1.1		1
5.3	odd	4		4800.2.a.i.1.1		1
5.4	even	2	inner	4800.2.f.x.3649.1		2
8.3	odd	2		2400.2.f.m.1249.2		2
8.5	even	2		2400.2.f.f.1249.1		2
20.3	even	4		4800.2.a.cm.1.1		1
20.7	even	4		4800.2.a.h.1.1		1
20.19	odd	2		4800.2.f.m.3649.2		2
24.5	odd	2		7200.2.f.j.6049.1		2
24.11	even	2		7200.2.f.t.6049.2		2
40.3	even	4		2400.2.a.l.1.1	✓	1
40.13	odd	4		2400.2.a.v.1.1	yes	1
40.19	odd	2		2400.2.f.m.1249.1		2
40.27	even	4		2400.2.a.w.1.1	yes	1
40.29	even	2		2400.2.f.f.1249.2		2
40.37	odd	4		2400.2.a.m.1.1	yes	1
120.29	odd	2		7200.2.f.j.6049.2		2
120.53	even	4		7200.2.a.g.1.1		1
120.59	even	2		7200.2.f.t.6049.1		2
120.77	even	4		7200.2.a.bu.1.1		1
120.83	odd	4		7200.2.a.bt.1.1		1
120.107	odd	4		7200.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.a.l.1.1	✓	1	40.3	even	4
2400.2.a.m.1.1	yes	1	40.37	odd	4
2400.2.a.v.1.1	yes	1	40.13	odd	4
2400.2.a.w.1.1	yes	1	40.27	even	4
2400.2.f.f.1249.1		2	8.5	even	2
2400.2.f.f.1249.2		2	40.29	even	2
2400.2.f.m.1249.1		2	40.19	odd	2
2400.2.f.m.1249.2		2	8.3	odd	2
4800.2.a.h.1.1		1	20.7	even	4
4800.2.a.i.1.1		1	5.3	odd	4
4800.2.a.cl.1.1		1	5.2	odd	4
4800.2.a.cm.1.1		1	20.3	even	4
4800.2.f.m.3649.1		2	4.3	odd	2
4800.2.f.m.3649.2		2	20.19	odd	2
4800.2.f.x.3649.1		2	5.4	even	2	inner
4800.2.f.x.3649.2		2	1.1	even	1	trivial
7200.2.a.g.1.1		1	120.53	even	4
7200.2.a.h.1.1		1	120.107	odd	4
7200.2.a.bt.1.1		1	120.83	odd	4
7200.2.a.bu.1.1		1	120.77	even	4
7200.2.f.j.6049.1		2	24.5	odd	2
7200.2.f.j.6049.2		2	120.29	odd	2
7200.2.f.t.6049.1		2	120.59	even	2
7200.2.f.t.6049.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -5.00000i q^{13} +5.00000 q^{19} +3.00000 q^{21} -4.00000i q^{23} -1.00000i q^{27} +4.00000 q^{29} -5.00000 q^{31} +10.0000i q^{37} +5.00000 q^{39} -10.0000 q^{41} +1.00000i q^{43} -2.00000i q^{47} -2.00000 q^{49} -10.0000i q^{53} +5.00000i q^{57} -10.0000 q^{59} +5.00000 q^{61} +3.00000i q^{63} +3.00000i q^{67} +4.00000 q^{69} -10.0000 q^{71} +10.0000i q^{73} +1.00000 q^{81} -14.0000i q^{83} +4.00000i q^{87} -16.0000 q^{89} -15.0000 q^{91} -5.00000i q^{93} -5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 10 q^{19} + 6 q^{21} + 8 q^{29} - 10 q^{31} + 10 q^{39} - 20 q^{41} - 4 q^{49} - 20 q^{59} + 10 q^{61} + 8 q^{69} - 20 q^{71} + 2 q^{81} - 32 q^{89} - 30 q^{91}+O(q^{100}) \) 2 * q - 2 * q^9 + 10 * q^19 + 6 * q^21 + 8 * q^29 - 10 * q^31 + 10 * q^39 - 20 * q^41 - 4 * q^49 - 20 * q^59 + 10 * q^61 + 8 * q^69 - 20 * q^71 + 2 * q^81 - 32 * q^89 - 30 * q^91

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