LMFDB - Embedded newform 4800.2.f.k.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 600)
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.k.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -3.00000i q^{13} +6.00000i q^{17} -7.00000 q^{19} -3.00000 q^{21} -6.00000i q^{23} -1.00000i q^{27} -2.00000 q^{29} -5.00000 q^{31} -2.00000i q^{33} -10.0000i q^{37} +3.00000 q^{39} +12.0000 q^{41} +3.00000i q^{43} -10.0000i q^{47} -2.00000 q^{49} -6.00000 q^{51} -7.00000i q^{57} -6.00000 q^{59} +13.0000 q^{61} -3.00000i q^{63} -7.00000i q^{67} +6.00000 q^{69} -4.00000 q^{71} +6.00000i q^{73} -6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000i q^{87} -16.0000 q^{89} +9.00000 q^{91} -5.00000i q^{93} -7.00000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 4 q^{11} - 14 q^{19} - 6 q^{21} - 4 q^{29} - 10 q^{31} + 6 q^{39} + 24 q^{41} - 4 q^{49} - 12 q^{51} - 12 q^{59} + 26 q^{61} + 12 q^{69} - 8 q^{71} + 16 q^{79} + 2 q^{81} - 32 q^{89} + 18 q^{91} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^11 - 14 * q^19 - 6 * q^21 - 4 * q^29 - 10 * q^31 + 6 * q^39 + 24 * q^41 - 4 * q^49 - 12 * q^51 - 12 * q^59 + 26 * q^61 + 12 * q^69 - 8 * q^71 + 16 * q^79 + 2 * q^81 - 32 * q^89 + 18 * q^91 + 4 * q^99

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	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	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
$33$	− 2.00000i	− 0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	3.00000i	0.457496i	0.973486	+	0.228748i	$0.0734631\pi$
$43$	3.00000i	0.457496i	−0.973486	+	0.228748i	$0.926537\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.0000i	− 1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$	− 10.0000i	− 1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 7.00000i	− 0.927173i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.00000i	− 0.214423i
$88$	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
$91$	9.00000	0.943456
$92$	0	0
$93$	− 5.00000i	− 0.518476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.00000i	− 0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$	− 7.00000i	− 0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	12.0000i	1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$	12.0000i	1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$	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
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.00000i	0.277350i
$118$	0	0
$119$	−18.0000	−1.65006
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	12.0000i	1.08200i
$124$	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
$129$	−3.00000	−0.264135
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	− 21.0000i	− 1.82093i
$134$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	0	0
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.00000i	0.718278i	0.933284	+	0.359139i	$0.116930\pi$
$157$	9.00000i	0.718278i	−0.933284	+	0.359139i	$0.883070\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	0	0
$183$	13.0000i	0.960988i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	− 19.0000i	− 1.36765i	−0.729646	−	0.683825i	$-0.760315\pi$
$193$	− 19.0000i	− 1.36765i	0.729646	−	0.683825i	$-0.239685\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000i	0.417029i
$208$	0	0
$209$	14.0000	0.968400
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	0	0
$213$	− 4.00000i	− 0.274075i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 15.0000i	− 1.01827i
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	18.0000	1.21081
$222$	0	0
$223$	− 11.0000i	− 0.736614i	−0.929704	−	0.368307i	$-0.879937\pi$
$223$	− 11.0000i	− 0.736614i	0.929704	−	0.368307i	$-0.120063\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	4.00000i	0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$	4.00000i	0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.0000i	1.33620i
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.00000i	0.249513i	0.992187	+	0.124757i	$0.0398150\pi$
$257$	4.00000i	0.249513i	−0.992187	+	0.124757i	$0.960185\pi$
$258$	0	0
$259$	30.0000	1.86411
$260$	0	0
$261$	2.00000	0.123797
$262$	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
$267$	− 16.0000i	− 0.979184i
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	9.00000i	0.544705i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.00000i	− 0.0600842i	−0.999549	−	0.0300421i	$-0.990436\pi$
$277$	− 1.00000i	− 0.0600842i	0.999549	−	0.0300421i	$-0.00956413\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	5.00000i	0.297219i	0.988896	+	0.148610i	$0.0474798\pi$
$283$	5.00000i	0.297219i	−0.988896	+	0.148610i	$0.952520\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36.0000i	2.12501i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	7.00000	0.410347
$292$	0	0
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000i	0.116052i
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.00000i	− 0.285365i	−0.989769	−	0.142683i	$-0.954427\pi$
$307$	− 5.00000i	− 0.285365i	0.989769	−	0.142683i	$-0.0455728\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	− 42.0000i	− 2.33694i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.00000i	0.497701i
$328$	0	0
$329$	30.0000	1.65395
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.00000i	− 0.381314i	−0.981657	−	0.190657i	$-0.938938\pi$
$337$	− 7.00000i	− 0.381314i	0.981657	−	0.190657i	$-0.0610619\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	10.0000	0.541530
$342$	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
$349$	−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
$351$	−3.00000	−0.160128
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 18.0000i	− 0.952661i
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 11.0000i	− 0.574195i	−0.957901	−	0.287098i	$-0.907310\pi$
$367$	− 11.0000i	− 0.574195i	0.957901	−	0.287098i	$-0.0926904\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.00000i	0.362446i	0.983442	+	0.181223i	$0.0580056\pi$
$373$	7.00000i	0.362446i	−0.983442	+	0.181223i	$0.941994\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000i	0.309016i
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.00000i	− 0.152499i
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	8.00000i	0.403547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 11.0000i	− 0.552074i	−0.961147	−	0.276037i	$-0.910979\pi$
$397$	− 11.0000i	− 0.552074i	0.961147	−	0.276037i	$-0.0890213\pi$
$398$	0	0
$399$	21.0000	1.05131
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	15.0000i	0.747203i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000i	0.991363i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000i	0.195881i
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	10.0000i	0.486217i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	39.0000i	1.88734i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	− 3.00000i	− 0.144171i	−0.997398	−	0.0720854i	$-0.977035\pi$
$433$	− 3.00000i	− 0.144171i	0.997398	−	0.0720854i	$-0.0229654\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	42.0000i	2.00913i
$438$	0	0
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	− 16.0000i	− 0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	− 16.0000i	− 0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 22.0000i	− 1.04056i
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	1.00000i	0.0469841i
$454$	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
$459$	6.00000	0.280056
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 26.0000i	− 1.20314i	−0.798821	−	0.601568i	$-0.794543\pi$
$467$	− 26.0000i	− 1.20314i	0.798821	−	0.601568i	$-0.205457\pi$
$468$	0	0
$469$	21.0000	0.969690
$470$	0	0
$471$	−9.00000	−0.414698
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.0000	1.73626	0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000	1.73626	0.868132	+	0.496333i	$0.165321\pi$
$480$	0	0
$481$	−30.0000	−1.36788
$482$	0	0
$483$	18.0000i	0.819028i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.0000i	− 0.538274i
$498$	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
$501$	8.00000	0.357414
$502$	0	0
$503$	4.00000i	0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$	4.00000i	0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.00000i	0.177646i
$508$	0	0
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	0	0
$513$	7.00000i	0.309058i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	20.0000i	0.879599i
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	29.0000i	1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$	29.0000i	1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.0000i	0.776757i
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	0	0
$543$	19.0000i	0.815368i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−13.0000	−0.554826
$550$	0	0
$551$	14.0000	0.596420
$552$	0	0
$553$	24.0000i	1.02058i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 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
$559$	9.00000	0.380659
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	− 26.0000i	− 1.09577i	−0.836554	−	0.547885i	$-0.815433\pi$
$563$	− 26.0000i	− 1.09577i	0.836554	−	0.547885i	$-0.184567\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.00000i	0.125988i
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	39.0000	1.63210	0.816050	−	0.577982i	$-0.196160\pi$
$571$	39.0000	1.63210	0.816050	+	0.577982i	$0.196160\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	0	0
$579$	19.0000	0.789613
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0000i	0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$	16.0000i	0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$	0	0
$589$	35.0000	1.44215
$590$	0	0
$591$	14.0000	0.575883
$592$	0	0
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.00000i	0.122782i
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	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
$603$	7.00000i	0.285062i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−30.0000	−1.21367
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 40.0000i	− 1.61034i	−0.593045	−	0.805170i	$-0.702074\pi$
$617$	− 40.0000i	− 1.61034i	0.593045	−	0.805170i	$-0.297926\pi$
$618$	0	0
$619$	5.00000	0.200967	0.100483	−	0.994939i	$-0.467961\pi$
$619$	5.00000	0.200967	0.100483	+	0.994939i	$0.467961\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	− 48.0000i	− 1.92308i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.0000i	0.559106i
$628$	0	0
$629$	60.0000	2.39236
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	− 9.00000i	− 0.357718i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	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	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	15.0000	0.587896
$652$	0	0
$653$	26.0000i	1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$	26.0000i	1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	18.0000i	0.699062i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	11.0000	0.425285
$670$	0	0
$671$	−26.0000	−1.00372
$672$	0	0
$673$	50.0000i	1.92736i	0.267063	+	0.963679i	$0.413947\pi$
$673$	50.0000i	1.92736i	−0.267063	+	0.963679i	$0.586053\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.0000i	0.384331i	0.981363	+	0.192166i	$0.0615511\pi$
$677$	10.0000i	0.384331i	−0.981363	+	0.192166i	$0.938449\pi$
$678$	0	0
$679$	21.0000	0.805906
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	40.0000i	1.53056i	0.643699	+	0.765279i	$0.277399\pi$
$683$	40.0000i	1.53056i	−0.643699	+	0.765279i	$0.722601\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 19.0000i	− 0.724895i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	6.00000i	0.227921i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	72.0000i	2.72719i
$698$	0	0
$699$	−4.00000	−0.151294
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	70.0000i	2.64010i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	30.0000i	1.12351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	−36.0000	−1.34071
$722$	0	0
$723$	25.0000i	0.929760i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 5.00000i	− 0.185440i	−0.995692	−	0.0927199i	$-0.970444\pi$
$727$	− 5.00000i	− 0.185440i	0.995692	−	0.0927199i	$-0.0295561\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−18.0000	−0.665754
$732$	0	0
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.0000i	0.515697i
$738$	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
$741$	−21.0000	−0.771454
$742$	0	0
$743$	− 20.0000i	− 0.733729i	−0.930274	−	0.366864i	$-0.880431\pi$
$743$	− 20.0000i	− 0.733729i	0.930274	−	0.366864i	$-0.119569\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	− 28.0000i	− 1.02038i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 13.0000i	− 0.472493i	−0.971693	−	0.236247i	$-0.924083\pi$
$757$	− 13.0000i	− 0.472493i	0.971693	−	0.236247i	$-0.0759173\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−52.0000	−1.88500	−0.942499	−	0.334208i	$-0.891531\pi$
$761$	−52.0000	−1.88500	−0.942499	+	0.334208i	$0.891531\pi$
$762$	0	0
$763$	27.0000i	0.977466i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.0000i	0.649942i
$768$	0	0
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	0	0
$773$	− 12.0000i	− 0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$	− 12.0000i	− 0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	30.0000i	1.07624i
$778$	0	0
$779$	−84.0000	−3.00961
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	2.00000i	0.0714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 37.0000i	− 1.31891i	−0.751745	−	0.659454i	$-0.770788\pi$
$787$	− 37.0000i	− 1.31891i	0.751745	−	0.659454i	$-0.229212\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	36.0000	1.28001
$792$	0	0
$793$	− 39.0000i	− 1.38493i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.0000i	0.708436i	0.935163	+	0.354218i	$0.115253\pi$
$797$	20.0000i	0.708436i	−0.935163	+	0.354218i	$0.884747\pi$
$798$	0	0
$799$	60.0000	2.12265
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.00000i	− 0.211210i
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 21.0000i	− 0.734697i
$818$	0	0
$819$	−9.00000	−0.314485
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	9.00000i	0.313720i	0.987621	+	0.156860i	$0.0501372\pi$
$823$	9.00000i	0.313720i	−0.987621	+	0.156860i	$0.949863\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24.0000i	− 0.834562i	−0.908778	−	0.417281i	$-0.862983\pi$
$827$	− 24.0000i	− 0.834562i	0.908778	−	0.417281i	$-0.137017\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	1.00000	0.0346896
$832$	0	0
$833$	− 12.0000i	− 0.415775i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.00000i	0.172825i
$838$	0	0
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	2.00000i	0.0688837i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 21.0000i	− 0.721569i
$848$	0	0
$849$	−5.00000	−0.171600
$850$	0	0
$851$	−60.0000	−2.05677
$852$	0	0
$853$	− 25.0000i	− 0.855984i	−0.903783	−	0.427992i	$-0.859221\pi$
$853$	− 25.0000i	− 0.855984i	0.903783	−	0.427992i	$-0.140779\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.00000i	− 0.136637i	−0.997664	−	0.0683187i	$-0.978237\pi$
$857$	− 4.00000i	− 0.136637i	0.997664	−	0.0683187i	$-0.0217635\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−21.0000	−0.711558
$872$	0	0
$873$	7.00000i	0.236914i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 31.0000i	− 1.04680i	−0.852088	−	0.523398i	$-0.824664\pi$
$877$	− 31.0000i	− 1.04680i	0.852088	−	0.523398i	$-0.175336\pi$
$878$	0	0
$879$	2.00000	0.0674583
$880$	0	0
$881$	8.00000	0.269527	0.134763	−	0.990878i	$-0.456973\pi$
$881$	8.00000	0.269527	0.134763	+	0.990878i	$0.456973\pi$
$882$	0	0
$883$	− 53.0000i	− 1.78359i	−0.452438	−	0.891796i	$-0.649446\pi$
$883$	− 53.0000i	− 1.78359i	0.452438	−	0.891796i	$-0.350554\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.0000i	1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$	42.0000i	1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	70.0000i	2.34246i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 18.0000i	− 0.601003i
$898$	0	0
$899$	10.0000	0.333519
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 9.00000i	− 0.299501i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24.0000i	0.792550i
$918$	0	0
$919$	−41.0000	−1.35247	−0.676233	−	0.736688i	$-0.736389\pi$
$919$	−41.0000	−1.35247	−0.676233	+	0.736688i	$0.736389\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.0000i	− 0.394132i
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	14.0000	0.458831
$932$	0	0
$933$	2.00000i	0.0654771i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.0000i	1.27407i	0.770833	+	0.637037i	$0.219840\pi$
$937$	39.0000i	1.27407i	−0.770833	+	0.637037i	$0.780160\pi$
$938$	0	0
$939$	19.0000	0.620042
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	− 72.0000i	− 2.34464i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.00000i	− 0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$	− 6.00000i	− 0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$	0	0
$949$	18.0000	0.584305
$950$	0	0
$951$	32.0000	1.03767
$952$	0	0
$953$	8.00000i	0.259145i	0.991570	+	0.129573i	$0.0413606\pi$
$953$	8.00000i	0.259145i	−0.991570	+	0.129573i	$0.958639\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.00000i	0.129302i
$958$	0	0
$959$	30.0000	0.968751
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	− 16.0000i	− 0.515593i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	0	0
$969$	42.0000	1.34923
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.0000i	1.85558i	0.373097	+	0.927792i	$0.378296\pi$
$977$	58.0000i	1.85558i	−0.373097	+	0.927792i	$0.621704\pi$
$978$	0	0
$979$	32.0000	1.02272
$980$	0	0
$981$	−9.00000	−0.287348
$982$	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
$987$	30.0000i	0.954911i
$988$	0	0
$989$	18.0000	0.572367
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	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.k.3649.2		2
4.3	odd	2		4800.2.f.z.3649.1		2
5.2	odd	4		4800.2.a.bp.1.1		1
5.3	odd	4		4800.2.a.bc.1.1		1
5.4	even	2	inner	4800.2.f.k.3649.1		2
8.3	odd	2		1200.2.f.c.49.2		2
8.5	even	2		600.2.f.d.49.1		2
20.3	even	4		4800.2.a.bs.1.1		1
20.7	even	4		4800.2.a.bd.1.1		1
20.19	odd	2		4800.2.f.z.3649.2		2
24.5	odd	2		1800.2.f.e.649.2		2
24.11	even	2		3600.2.f.o.2449.1		2
40.3	even	4		1200.2.a.b.1.1		1
40.13	odd	4		600.2.a.i.1.1	yes	1
40.19	odd	2		1200.2.f.c.49.1		2
40.27	even	4		1200.2.a.q.1.1		1
40.29	even	2		600.2.f.d.49.2		2
40.37	odd	4		600.2.a.b.1.1	✓	1
120.29	odd	2		1800.2.f.e.649.1		2
120.53	even	4		1800.2.a.t.1.1		1
120.59	even	2		3600.2.f.o.2449.2		2
120.77	even	4		1800.2.a.e.1.1		1
120.83	odd	4		3600.2.a.i.1.1		1
120.107	odd	4		3600.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.a.b.1.1	✓	1	40.37	odd	4
600.2.a.i.1.1	yes	1	40.13	odd	4
600.2.f.d.49.1		2	8.5	even	2
600.2.f.d.49.2		2	40.29	even	2
1200.2.a.b.1.1		1	40.3	even	4
1200.2.a.q.1.1		1	40.27	even	4
1200.2.f.c.49.1		2	40.19	odd	2
1200.2.f.c.49.2		2	8.3	odd	2
1800.2.a.e.1.1		1	120.77	even	4
1800.2.a.t.1.1		1	120.53	even	4
1800.2.f.e.649.1		2	120.29	odd	2
1800.2.f.e.649.2		2	24.5	odd	2
3600.2.a.i.1.1		1	120.83	odd	4
3600.2.a.bl.1.1		1	120.107	odd	4
3600.2.f.o.2449.1		2	24.11	even	2
3600.2.f.o.2449.2		2	120.59	even	2
4800.2.a.bc.1.1		1	5.3	odd	4
4800.2.a.bd.1.1		1	20.7	even	4
4800.2.a.bp.1.1		1	5.2	odd	4
4800.2.a.bs.1.1		1	20.3	even	4
4800.2.f.k.3649.1		2	5.4	even	2	inner
4800.2.f.k.3649.2		2	1.1	even	1	trivial
4800.2.f.z.3649.1		2	4.3	odd	2
4800.2.f.z.3649.2		2	20.19	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -3.00000i q^{13} +6.00000i q^{17} -7.00000 q^{19} -3.00000 q^{21} -6.00000i q^{23} -1.00000i q^{27} -2.00000 q^{29} -5.00000 q^{31} -2.00000i q^{33} -10.0000i q^{37} +3.00000 q^{39} +12.0000 q^{41} +3.00000i q^{43} -10.0000i q^{47} -2.00000 q^{49} -6.00000 q^{51} -7.00000i q^{57} -6.00000 q^{59} +13.0000 q^{61} -3.00000i q^{63} -7.00000i q^{67} +6.00000 q^{69} -4.00000 q^{71} +6.00000i q^{73} -6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000i q^{87} -16.0000 q^{89} +9.00000 q^{91} -5.00000i q^{93} -7.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 4 q^{11} - 14 q^{19} - 6 q^{21} - 4 q^{29} - 10 q^{31} + 6 q^{39} + 24 q^{41} - 4 q^{49} - 12 q^{51} - 12 q^{59} + 26 q^{61} + 12 q^{69} - 8 q^{71} + 16 q^{79} + 2 q^{81} - 32 q^{89} + 18 q^{91} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^11 - 14 * q^19 - 6 * q^21 - 4 * q^29 - 10 * q^31 + 6 * q^39 + 24 * q^41 - 4 * q^49 - 12 * q^51 - 12 * q^59 + 26 * q^61 + 12 * q^69 - 8 * q^71 + 16 * q^79 + 2 * q^81 - 32 * q^89 + 18 * q^91 + 4 * q^99

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