LMFDB - Embedded newform 4800.2.k.r.2401.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(2401,4800)]

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, 1, 0, 0]))

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

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

chi := DirichletCharacter("4800.2401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.k (of order $2$, degree $1$, 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:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 960)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2401.3
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	4800.2401
Dual form			4800.2.k.r.2401.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.913701 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +0.913701 q^{7} -1.00000 q^{9} +3.58258i q^{11} +0.913701i q^{13} -3.58258 q^{17} -4.00000i q^{19} -0.913701i q^{21} +1.00000i q^{27} -7.84190i q^{29} -5.29150 q^{31} +3.58258 q^{33} +7.84190i q^{37} +0.913701 q^{39} +6.00000 q^{41} +7.16515i q^{43} -6.92820 q^{47} -6.16515 q^{49} +3.58258i q^{51} +2.55040i q^{53} -4.00000 q^{57} -7.58258i q^{59} +10.5830i q^{61} -0.913701 q^{63} -15.1652i q^{67} -6.92820 q^{71} -12.0000 q^{73} +3.27340i q^{77} -5.29150 q^{79} +1.00000 q^{81} -11.1652i q^{83} -7.84190 q^{87} +2.00000 q^{89} +0.834849i q^{91} +5.29150i q^{93} +7.16515 q^{97} -3.58258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 8 q^{17} - 8 q^{33} + 48 q^{41} + 24 q^{49} - 32 q^{57} - 96 q^{73} + 8 q^{81} + 16 q^{89} - 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 8 * q^17 - 8 * q^33 + 48 * q^41 + 24 * q^49 - 32 * q^57 - 96 * q^73 + 8 * q^81 + 16 * q^89 - 16 * q^97

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
$7$	0.913701	0.345346	0.172673	−	0.984979i	$-0.444760\pi$
$7$	0.913701	0.345346	0.172673	+	0.984979i	$0.444760\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.58258i	1.08019i	0.841605	+	0.540094i	$0.181611\pi$
$11$	3.58258i	1.08019i	−0.841605	+	0.540094i	$0.818389\pi$
$12$	0	0
$13$	0.913701i	0.253415i	0.991940	+	0.126707i	$0.0404409\pi$
$13$	0.913701i	0.253415i	−0.991940	+	0.126707i	$0.959559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.58258	−0.868902	−0.434451	−	0.900695i	$-0.643058\pi$
$17$	−3.58258	−0.868902	−0.434451	+	0.900695i	$0.643058\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	− 0.913701i	− 0.199386i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 7.84190i	− 1.45620i	−0.685468	−	0.728102i	$-0.740402\pi$
$29$	− 7.84190i	− 1.45620i	0.685468	−	0.728102i	$-0.259598\pi$
$30$	0	0
$31$	−5.29150	−0.950382	−0.475191	−	0.879883i	$-0.657621\pi$
$31$	−5.29150	−0.950382	−0.475191	+	0.879883i	$0.657621\pi$
$32$	0	0
$33$	3.58258	0.623646
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.84190i	1.28920i	0.764520	+	0.644601i	$0.222976\pi$
$37$	7.84190i	1.28920i	−0.764520	+	0.644601i	$0.777024\pi$
$38$	0	0
$39$	0.913701	0.146309
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	7.16515i	1.09268i	0.837565	+	0.546338i	$0.183978\pi$
$43$	7.16515i	1.09268i	−0.837565	+	0.546338i	$0.816022\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−6.16515	−0.880736
$50$	0	0
$51$	3.58258i	0.501661i
$52$	0	0
$53$	2.55040i	0.350325i	0.984540	+	0.175162i	$0.0560450\pi$
$53$	2.55040i	0.350325i	−0.984540	+	0.175162i	$0.943955\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 7.58258i	− 0.987167i	−0.869698	−	0.493584i	$-0.835687\pi$
$59$	− 7.58258i	− 0.987167i	0.869698	−	0.493584i	$-0.164313\pi$
$60$	0	0
$61$	10.5830i	1.35501i	0.735516	+	0.677507i	$0.236940\pi$
$61$	10.5830i	1.35501i	−0.735516	+	0.677507i	$0.763060\pi$
$62$	0	0
$63$	−0.913701	−0.115115
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 15.1652i	− 1.85272i	−0.376642	−	0.926359i	$-0.622921\pi$
$67$	− 15.1652i	− 1.85272i	0.376642	−	0.926359i	$-0.377079\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.27340i	0.373039i
$78$	0	0
$79$	−5.29150	−0.595341	−0.297670	−	0.954669i	$-0.596210\pi$
$79$	−5.29150	−0.595341	−0.297670	+	0.954669i	$0.596210\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 11.1652i	− 1.22553i	−0.790263	−	0.612767i	$-0.790056\pi$
$83$	− 11.1652i	− 1.22553i	0.790263	−	0.612767i	$-0.209944\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.84190	−0.840740
$88$	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
$91$	0.834849i	0.0875159i
$92$	0	0
$93$	5.29150i	0.548703i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.16515	0.727511	0.363755	−	0.931494i	$-0.381494\pi$
$97$	7.16515	0.727511	0.363755	+	0.931494i	$0.381494\pi$
$98$	0	0
$99$	− 3.58258i	− 0.360062i
$100$	0	0
$101$	− 16.5975i	− 1.65151i	−0.564026	−	0.825757i	$-0.690748\pi$
$101$	− 16.5975i	− 1.65151i	0.564026	−	0.825757i	$-0.309252\pi$
$102$	0	0
$103$	−16.5975	−1.63540	−0.817701	−	0.575644i	$-0.804751\pi$
$103$	−16.5975	−1.63540	−0.817701	+	0.575644i	$0.804751\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	− 8.75560i	− 0.838635i	−0.907840	−	0.419317i	$-0.862269\pi$
$109$	− 8.75560i	− 0.838635i	0.907840	−	0.419317i	$-0.137731\pi$
$110$	0	0
$111$	7.84190	0.744321
$112$	0	0
$113$	4.41742	0.415556	0.207778	−	0.978176i	$-0.433377\pi$
$113$	4.41742	0.415556	0.207778	+	0.978176i	$0.433377\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.913701i	− 0.0844716i
$118$	0	0
$119$	−3.27340	−0.300072
$120$	0	0
$121$	−1.83485	−0.166804
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.7701	−1.31064	−0.655318	−	0.755354i	$-0.727465\pi$
$127$	−14.7701	−1.31064	−0.655318	+	0.755354i	$0.727465\pi$
$128$	0	0
$129$	7.16515	0.630856
$130$	0	0
$131$	− 15.5826i	− 1.36146i	−0.732536	−	0.680728i	$-0.761664\pi$
$131$	− 15.5826i	− 1.36146i	0.732536	−	0.680728i	$-0.238336\pi$
$132$	0	0
$133$	− 3.65480i	− 0.316912i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.5826	−1.67305	−0.836526	−	0.547927i	$-0.815417\pi$
$137$	−19.5826	−1.67305	−0.836526	+	0.547927i	$0.815417\pi$
$138$	0	0
$139$	11.1652i	0.947016i	0.880790	+	0.473508i	$0.157012\pi$
$139$	11.1652i	0.947016i	−0.880790	+	0.473508i	$0.842988\pi$
$140$	0	0
$141$	6.92820i	0.583460i
$142$	0	0
$143$	−3.27340	−0.273736
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.16515i	0.508493i
$148$	0	0
$149$	11.4967i	0.941847i	0.882174	+	0.470923i	$0.156079\pi$
$149$	11.4967i	0.941847i	−0.882174	+	0.470923i	$0.843921\pi$
$150$	0	0
$151$	15.8745	1.29185	0.645925	−	0.763401i	$-0.276472\pi$
$151$	15.8745	1.29185	0.645925	+	0.763401i	$0.276472\pi$
$152$	0	0
$153$	3.58258	0.289634
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.6983i	1.73171i	0.500292	+	0.865857i	$0.333226\pi$
$157$	21.6983i	1.73171i	−0.500292	+	0.865857i	$0.666774\pi$
$158$	0	0
$159$	2.55040	0.202260
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.6120	−1.74977	−0.874885	−	0.484331i	$-0.839063\pi$
$167$	−22.6120	−1.74977	−0.874885	+	0.484331i	$0.839063\pi$
$168$	0	0
$169$	12.1652	0.935781
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	11.3060i	0.859580i	0.902929	+	0.429790i	$0.141412\pi$
$173$	11.3060i	0.859580i	−0.902929	+	0.429790i	$0.858588\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.58258	−0.569941
$178$	0	0
$179$	7.58258i	0.566748i	0.959009	+	0.283374i	$0.0914538\pi$
$179$	7.58258i	0.566748i	−0.959009	+	0.283374i	$0.908546\pi$
$180$	0	0
$181$	− 8.75560i	− 0.650799i	−0.945577	−	0.325399i	$-0.894501\pi$
$181$	− 8.75560i	− 0.650799i	0.945577	−	0.325399i	$-0.105499\pi$
$182$	0	0
$183$	10.5830	0.782318
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.8348i	− 0.938577i
$188$	0	0
$189$	0.913701i	0.0664619i
$190$	0	0
$191$	−13.8564	−1.00261	−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564	−1.00261	−0.501307	+	0.865269i	$0.667147\pi$
$192$	0	0
$193$	−3.16515	−0.227833	−0.113916	−	0.993490i	$-0.536340\pi$
$193$	−3.16515	−0.227833	−0.113916	+	0.993490i	$0.536340\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.5438i	1.81992i	0.414695	+	0.909961i	$0.363888\pi$
$197$	25.5438i	1.81992i	−0.414695	+	0.909961i	$0.636112\pi$
$198$	0	0
$199$	1.63670	0.116023	0.0580113	−	0.998316i	$-0.481524\pi$
$199$	1.63670	0.116023	0.0580113	+	0.998316i	$0.481524\pi$
$200$	0	0
$201$	−15.1652	−1.06967
$202$	0	0
$203$	− 7.16515i	− 0.502895i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.3303	0.991248
$210$	0	0
$211$	− 11.1652i	− 0.768641i	−0.923200	−	0.384320i	$-0.874436\pi$
$211$	− 11.1652i	− 0.768641i	0.923200	−	0.384320i	$-0.125564\pi$
$212$	0	0
$213$	6.92820i	0.474713i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.83485	−0.328211
$218$	0	0
$219$	12.0000i	0.810885i
$220$	0	0
$221$	− 3.27340i	− 0.220193i
$222$	0	0
$223$	−0.913701	−0.0611859	−0.0305930	−	0.999532i	$-0.509740\pi$
$223$	−0.913701	−0.0611859	−0.0305930	+	0.999532i	$0.509740\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.16515i	0.210078i	0.994468	+	0.105039i	$0.0334968\pi$
$227$	3.16515i	0.210078i	−0.994468	+	0.105039i	$0.966503\pi$
$228$	0	0
$229$	22.6120i	1.49424i	0.664687	+	0.747122i	$0.268565\pi$
$229$	22.6120i	1.49424i	−0.664687	+	0.747122i	$0.731435\pi$
$230$	0	0
$231$	3.27340	0.215374
$232$	0	0
$233$	−19.5826	−1.28290	−0.641449	−	0.767166i	$-0.721666\pi$
$233$	−19.5826	−1.28290	−0.641449	+	0.767166i	$0.721666\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.29150i	0.343720i
$238$	0	0
$239$	−21.1660	−1.36912	−0.684558	−	0.728959i	$-0.740005\pi$
$239$	−21.1660	−1.36912	−0.684558	+	0.728959i	$0.740005\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.65480	0.232549
$248$	0	0
$249$	−11.1652	−0.707563
$250$	0	0
$251$	− 11.5826i	− 0.731086i	−0.930795	−	0.365543i	$-0.880883\pi$
$251$	− 11.5826i	− 0.731086i	0.930795	−	0.365543i	$-0.119117\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.74773	0.171399	0.0856993	−	0.996321i	$-0.472688\pi$
$257$	2.74773	0.171399	0.0856993	+	0.996321i	$0.472688\pi$
$258$	0	0
$259$	7.16515i	0.445221i
$260$	0	0
$261$	7.84190i	0.485402i
$262$	0	0
$263$	8.75560	0.539894	0.269947	−	0.962875i	$-0.412994\pi$
$263$	8.75560	0.539894	0.269947	+	0.962875i	$0.412994\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	0	0
$269$	− 9.66930i	− 0.589548i	−0.955567	−	0.294774i	$-0.904756\pi$
$269$	− 9.66930i	− 0.589548i	0.955567	−	0.294774i	$-0.0952444\pi$
$270$	0	0
$271$	26.0761	1.58401	0.792006	−	0.610514i	$-0.209037\pi$
$271$	26.0761	1.58401	0.792006	+	0.610514i	$0.209037\pi$
$272$	0	0
$273$	0.834849	0.0505273
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.7701i	0.887450i	0.896163	+	0.443725i	$0.146343\pi$
$277$	14.7701i	0.887450i	−0.896163	+	0.443725i	$0.853657\pi$
$278$	0	0
$279$	5.29150	0.316794
$280$	0	0
$281$	−4.33030	−0.258324	−0.129162	−	0.991623i	$-0.541229\pi$
$281$	−4.33030	−0.258324	−0.129162	+	0.991623i	$0.541229\pi$
$282$	0	0
$283$	− 7.16515i	− 0.425924i	−0.977060	−	0.212962i	$-0.931689\pi$
$283$	− 7.16515i	− 0.425924i	0.977060	−	0.212962i	$-0.0683111\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.48220	0.323604
$288$	0	0
$289$	−4.16515	−0.245009
$290$	0	0
$291$	− 7.16515i	− 0.420029i
$292$	0	0
$293$	4.37780i	0.255754i	0.991790	+	0.127877i	$0.0408163\pi$
$293$	4.37780i	0.255754i	−0.991790	+	0.127877i	$0.959184\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.58258	−0.207882
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.54680i	0.377351i
$302$	0	0
$303$	−16.5975	−0.953502
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	16.5975i	0.944199i
$310$	0	0
$311$	−17.5112	−0.992970	−0.496485	−	0.868045i	$-0.665376\pi$
$311$	−17.5112	−0.992970	−0.496485	+	0.868045i	$0.665376\pi$
$312$	0	0
$313$	−19.1652	−1.08328	−0.541639	−	0.840611i	$-0.682196\pi$
$313$	−19.1652	−1.08328	−0.541639	+	0.840611i	$0.682196\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.93180i	− 0.164666i	−0.996605	−	0.0823332i	$-0.973763\pi$
$317$	− 2.93180i	− 0.164666i	0.996605	−	0.0823332i	$-0.0262372\pi$
$318$	0	0
$319$	28.0942	1.57297
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	14.3303i	0.797359i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.75560	−0.484186
$328$	0	0
$329$	−6.33030	−0.349001
$330$	0	0
$331$	26.3303i	1.44724i	0.690196	+	0.723622i	$0.257524\pi$
$331$	26.3303i	1.44724i	−0.690196	+	0.723622i	$0.742476\pi$
$332$	0	0
$333$	− 7.84190i	− 0.429734i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.3303	−0.562727	−0.281364	−	0.959601i	$-0.590787\pi$
$337$	−10.3303	−0.562727	−0.281364	+	0.959601i	$0.590787\pi$
$338$	0	0
$339$	− 4.41742i	− 0.239922i
$340$	0	0
$341$	− 18.9572i	− 1.02659i
$342$	0	0
$343$	−12.0290	−0.649505
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 19.1652i	− 1.02884i	−0.857539	−	0.514420i	$-0.828007\pi$
$347$	− 19.1652i	− 1.02884i	0.857539	−	0.514420i	$-0.171993\pi$
$348$	0	0
$349$	3.27340i	0.175221i	0.996155	+	0.0876106i	$0.0279231\pi$
$349$	3.27340i	0.175221i	−0.996155	+	0.0876106i	$0.972077\pi$
$350$	0	0
$351$	−0.913701	−0.0487697
$352$	0	0
$353$	33.9129	1.80500	0.902500	−	0.430689i	$-0.141730\pi$
$353$	33.9129	1.80500	0.902500	+	0.430689i	$0.141730\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.27340i	0.173247i
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	1.83485i	0.0963046i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−35.9361	−1.87585	−0.937925	−	0.346838i	$-0.887255\pi$
$367$	−35.9361	−1.87585	−0.937925	+	0.346838i	$0.887255\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	2.33030i	0.120983i
$372$	0	0
$373$	6.01450i	0.311419i	0.987803	+	0.155710i	$0.0497664\pi$
$373$	6.01450i	0.311419i	−0.987803	+	0.155710i	$0.950234\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.16515	0.369024
$378$	0	0
$379$	4.83485i	0.248349i	0.992260	+	0.124175i	$0.0396283\pi$
$379$	4.83485i	0.248349i	−0.992260	+	0.124175i	$0.960372\pi$
$380$	0	0
$381$	14.7701i	0.756696i
$382$	0	0
$383$	−22.9934	−1.17491	−0.587454	−	0.809257i	$-0.699870\pi$
$383$	−22.9934	−1.17491	−0.587454	+	0.809257i	$0.699870\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.16515i	− 0.364225i
$388$	0	0
$389$	− 18.4249i	− 0.934180i	−0.884210	−	0.467090i	$-0.845302\pi$
$389$	− 18.4249i	− 0.934180i	0.884210	−	0.467090i	$-0.154698\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.5826	−0.786037
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.84190i	− 0.393574i	−0.980446	−	0.196787i	$-0.936949\pi$
$397$	− 7.84190i	− 0.393574i	0.980446	−	0.196787i	$-0.0630507\pi$
$398$	0	0
$399$	−3.65480	−0.182969
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	− 4.83485i	− 0.240841i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.0942	−1.39258
$408$	0	0
$409$	−18.8348	−0.931323	−0.465662	−	0.884963i	$-0.654184\pi$
$409$	−18.8348	−0.931323	−0.465662	+	0.884963i	$0.654184\pi$
$410$	0	0
$411$	19.5826i	0.965937i
$412$	0	0
$413$	− 6.92820i	− 0.340915i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.1652	0.546760
$418$	0	0
$419$	− 10.7477i	− 0.525061i	−0.964924	−	0.262530i	$-0.915443\pi$
$419$	− 10.7477i	− 0.525061i	0.964924	−	0.262530i	$-0.0845570\pi$
$420$	0	0
$421$	18.9572i	0.923918i	0.886901	+	0.461959i	$0.152853\pi$
$421$	18.9572i	0.923918i	−0.886901	+	0.461959i	$0.847147\pi$
$422$	0	0
$423$	6.92820	0.336861
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.66970i	0.467949i
$428$	0	0
$429$	3.27340i	0.158041i
$430$	0	0
$431$	−17.1298	−0.825114	−0.412557	−	0.910932i	$-0.635364\pi$
$431$	−17.1298	−0.825114	−0.412557	+	0.910932i	$0.635364\pi$
$432$	0	0
$433$	38.3303	1.84204	0.921018	−	0.389519i	$-0.127359\pi$
$433$	38.3303	1.84204	0.921018	+	0.389519i	$0.127359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.01810	−0.0963187	−0.0481594	−	0.998840i	$-0.515336\pi$
$439$	−2.01810	−0.0963187	−0.0481594	+	0.998840i	$0.515336\pi$
$440$	0	0
$441$	6.16515	0.293579
$442$	0	0
$443$	− 12.8348i	− 0.609802i	−0.952384	−	0.304901i	$-0.901377\pi$
$443$	− 12.8348i	− 0.609802i	0.952384	−	0.304901i	$-0.0986234\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.4967	0.543776
$448$	0	0
$449$	−8.33030	−0.393131	−0.196566	−	0.980491i	$-0.562979\pi$
$449$	−8.33030	−0.393131	−0.196566	+	0.980491i	$0.562979\pi$
$450$	0	0
$451$	21.4955i	1.01218i
$452$	0	0
$453$	− 15.8745i	− 0.745849i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.1652	−1.08362	−0.541810	−	0.840501i	$-0.682261\pi$
$457$	−23.1652	−1.08362	−0.541810	+	0.840501i	$0.682261\pi$
$458$	0	0
$459$	− 3.58258i	− 0.167220i
$460$	0	0
$461$	− 25.3531i	− 1.18081i	−0.807106	−	0.590406i	$-0.798968\pi$
$461$	− 25.3531i	− 1.18081i	0.807106	−	0.590406i	$-0.201032\pi$
$462$	0	0
$463$	39.5909	1.83995	0.919973	−	0.391982i	$-0.128210\pi$
$463$	39.5909	1.83995	0.919973	+	0.391982i	$0.128210\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.1652i	− 0.886857i	−0.896310	−	0.443429i	$-0.853762\pi$
$467$	− 19.1652i	− 0.886857i	0.896310	−	0.443429i	$-0.146238\pi$
$468$	0	0
$469$	− 13.8564i	− 0.639829i
$470$	0	0
$471$	21.6983	0.999805
$472$	0	0
$473$	−25.6697	−1.18029
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.55040i	− 0.116775i
$478$	0	0
$479$	31.7490	1.45065	0.725325	−	0.688407i	$-0.241690\pi$
$479$	31.7490	1.45065	0.725325	+	0.688407i	$0.241690\pi$
$480$	0	0
$481$	−7.16515	−0.326703
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.2523	0.917720	0.458860	−	0.888509i	$-0.348258\pi$
$487$	20.2523	0.917720	0.458860	+	0.888509i	$0.348258\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 14.7477i	− 0.665556i	−0.943005	−	0.332778i	$-0.892014\pi$
$491$	− 14.7477i	− 0.665556i	0.943005	−	0.332778i	$-0.107986\pi$
$492$	0	0
$493$	28.0942i	1.26530i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.33030	−0.283953
$498$	0	0
$499$	− 18.3303i	− 0.820577i	−0.911956	−	0.410289i	$-0.865428\pi$
$499$	− 18.3303i	− 0.820577i	0.911956	−	0.410289i	$-0.134572\pi$
$500$	0	0
$501$	22.6120i	1.01023i
$502$	0	0
$503$	−14.2378	−0.634832	−0.317416	−	0.948286i	$-0.602815\pi$
$503$	−14.2378	−0.634832	−0.317416	+	0.948286i	$0.602815\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.1652i	− 0.540273i
$508$	0	0
$509$	8.22330i	0.364492i	0.983253	+	0.182246i	$0.0583367\pi$
$509$	8.22330i	0.364492i	−0.983253	+	0.182246i	$0.941663\pi$
$510$	0	0
$511$	−10.9644	−0.485037
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 24.8208i	− 1.09162i
$518$	0	0
$519$	11.3060	0.496279
$520$	0	0
$521$	38.6606	1.69375	0.846876	−	0.531791i	$-0.178481\pi$
$521$	38.6606	1.69375	0.846876	+	0.531791i	$0.178481\pi$
$522$	0	0
$523$	− 18.3303i	− 0.801528i	−0.916181	−	0.400764i	$-0.868745\pi$
$523$	− 18.3303i	− 0.801528i	0.916181	−	0.400764i	$-0.131255\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.9572	0.825789
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	7.58258i	0.329056i
$532$	0	0
$533$	5.48220i	0.237461i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.58258	0.327212
$538$	0	0
$539$	− 22.0871i	− 0.951360i
$540$	0	0
$541$	16.0652i	0.690697i	0.938474	+	0.345349i	$0.112239\pi$
$541$	16.0652i	0.690697i	−0.938474	+	0.345349i	$0.887761\pi$
$542$	0	0
$543$	−8.75560	−0.375739
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.4955i	1.26113i	0.776135	+	0.630567i	$0.217178\pi$
$547$	29.4955i	1.26113i	−0.776135	+	0.630567i	$0.782822\pi$
$548$	0	0
$549$	− 10.5830i	− 0.451672i
$550$	0	0
$551$	−31.3676	−1.33631
$552$	0	0
$553$	−4.83485	−0.205599
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.0616i	0.850038i	0.905185	+	0.425019i	$0.139733\pi$
$557$	20.0616i	0.850038i	−0.905185	+	0.425019i	$0.860267\pi$
$558$	0	0
$559$	−6.54680	−0.276900
$560$	0	0
$561$	−12.8348	−0.541888
$562$	0	0
$563$	25.4955i	1.07451i	0.843421	+	0.537253i	$0.180538\pi$
$563$	25.4955i	1.07451i	−0.843421	+	0.537253i	$0.819462\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.913701	0.0383718
$568$	0	0
$569$	0.330303	0.0138470	0.00692351	−	0.999976i	$-0.497796\pi$
$569$	0.330303	0.0138470	0.00692351	+	0.999976i	$0.497796\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	13.8564i	0.578860i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.16515	−0.131767	−0.0658835	−	0.997827i	$-0.520987\pi$
$577$	−3.16515	−0.131767	−0.0658835	+	0.997827i	$0.520987\pi$
$578$	0	0
$579$	3.16515i	0.131539i
$580$	0	0
$581$	− 10.2016i	− 0.423234i
$582$	0	0
$583$	−9.13701	−0.378416
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.3303i	1.74716i	0.486682	+	0.873579i	$0.338207\pi$
$587$	42.3303i	1.74716i	−0.486682	+	0.873579i	$0.661793\pi$
$588$	0	0
$589$	21.1660i	0.872130i
$590$	0	0
$591$	25.5438	1.05073
$592$	0	0
$593$	−9.91288	−0.407073	−0.203537	−	0.979067i	$-0.565244\pi$
$593$	−9.91288	−0.407073	−0.203537	+	0.979067i	$0.565244\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.63670i	− 0.0669857i
$598$	0	0
$599$	−28.0942	−1.14790	−0.573949	−	0.818891i	$-0.694589\pi$
$599$	−28.0942	−1.14790	−0.573949	+	0.818891i	$0.694589\pi$
$600$	0	0
$601$	−5.16515	−0.210691	−0.105345	−	0.994436i	$-0.533595\pi$
$601$	−5.16515	−0.210691	−0.105345	+	0.994436i	$0.533595\pi$
$602$	0	0
$603$	15.1652i	0.617573i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.28790	−0.376984	−0.188492	−	0.982075i	$-0.560360\pi$
$607$	−9.28790	−0.376984	−0.188492	+	0.982075i	$0.560360\pi$
$608$	0	0
$609$	−7.16515	−0.290347
$610$	0	0
$611$	− 6.33030i	− 0.256097i
$612$	0	0
$613$	15.1515i	0.611964i	0.952037	+	0.305982i	$0.0989847\pi$
$613$	15.1515i	0.611964i	−0.952037	+	0.305982i	$0.901015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.41742	0.177839	0.0889194	−	0.996039i	$-0.471659\pi$
$617$	4.41742	0.177839	0.0889194	+	0.996039i	$0.471659\pi$
$618$	0	0
$619$	− 9.49545i	− 0.381655i	−0.981624	−	0.190827i	$-0.938883\pi$
$619$	− 9.49545i	− 0.381655i	0.981624	−	0.190827i	$-0.0611170\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.82740	0.0732133
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 14.3303i	− 0.572297i
$628$	0	0
$629$	− 28.0942i	− 1.12019i
$630$	0	0
$631$	1.63670	0.0651560	0.0325780	−	0.999469i	$-0.489628\pi$
$631$	1.63670	0.0651560	0.0325780	+	0.999469i	$0.489628\pi$
$632$	0	0
$633$	−11.1652	−0.443775
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.63310i	− 0.223192i
$638$	0	0
$639$	6.92820	0.274075
$640$	0	0
$641$	32.3303	1.27697	0.638485	−	0.769634i	$-0.279561\pi$
$641$	32.3303	1.27697	0.638485	+	0.769634i	$0.279561\pi$
$642$	0	0
$643$	− 18.3303i	− 0.722877i	−0.932396	−	0.361438i	$-0.882286\pi$
$643$	− 18.3303i	− 0.722877i	0.932396	−	0.361438i	$-0.117714\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.6054	1.79293	0.896467	−	0.443110i	$-0.146125\pi$
$647$	45.6054	1.79293	0.896467	+	0.443110i	$0.146125\pi$
$648$	0	0
$649$	27.1652	1.06633
$650$	0	0
$651$	4.83485i	0.189493i
$652$	0	0
$653$	− 25.5438i	− 0.999607i	−0.866139	−	0.499803i	$-0.833406\pi$
$653$	− 25.5438i	− 0.999607i	0.866139	−	0.499803i	$-0.166594\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	16.4174i	0.639532i	0.947497	+	0.319766i	$0.103604\pi$
$659$	16.4174i	0.639532i	−0.947497	+	0.319766i	$0.896396\pi$
$660$	0	0
$661$	45.6054i	1.77385i	0.461918	+	0.886923i	$0.347161\pi$
$661$	45.6054i	1.77385i	−0.461918	+	0.886923i	$0.652839\pi$
$662$	0	0
$663$	−3.27340	−0.127128
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0.913701i	0.0353257i
$670$	0	0
$671$	−37.9144	−1.46367
$672$	0	0
$673$	40.0000	1.54189	0.770943	−	0.636904i	$-0.219785\pi$
$673$	40.0000	1.54189	0.770943	+	0.636904i	$0.219785\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 23.3350i	− 0.896837i	−0.893824	−	0.448419i	$-0.851987\pi$
$677$	− 23.3350i	− 0.896837i	0.893824	−	0.448419i	$-0.148013\pi$
$678$	0	0
$679$	6.54680	0.251243
$680$	0	0
$681$	3.16515	0.121289
$682$	0	0
$683$	49.4955i	1.89389i	0.321394	+	0.946945i	$0.395849\pi$
$683$	49.4955i	1.89389i	−0.321394	+	0.946945i	$0.604151\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.6120	0.862702
$688$	0	0
$689$	−2.33030	−0.0887775
$690$	0	0
$691$	− 32.6606i	− 1.24247i	−0.783625	−	0.621234i	$-0.786632\pi$
$691$	− 32.6606i	− 1.24247i	0.783625	−	0.621234i	$-0.213368\pi$
$692$	0	0
$693$	− 3.27340i	− 0.124346i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−21.4955	−0.814198
$698$	0	0
$699$	19.5826i	0.740681i
$700$	0	0
$701$	− 40.6555i	− 1.53554i	−0.640727	−	0.767769i	$-0.721367\pi$
$701$	− 40.6555i	− 1.53554i	0.640727	−	0.767769i	$-0.278633\pi$
$702$	0	0
$703$	31.3676	1.18305
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.1652i	− 0.570344i
$708$	0	0
$709$	− 5.10080i	− 0.191565i	−0.995402	−	0.0957823i	$-0.969465\pi$
$709$	− 5.10080i	− 0.191565i	0.995402	−	0.0957823i	$-0.0305353\pi$
$710$	0	0
$711$	5.29150	0.198447
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.1660i	0.790459i
$718$	0	0
$719$	−52.1522	−1.94495	−0.972475	−	0.233008i	$-0.925143\pi$
$719$	−52.1522	−1.94495	−0.972475	+	0.233008i	$0.925143\pi$
$720$	0	0
$721$	−15.1652	−0.564780
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.1443	0.858375	0.429187	−	0.903215i	$-0.358800\pi$
$727$	23.1443	0.858375	0.429187	+	0.903215i	$0.358800\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 25.6697i	− 0.949428i
$732$	0	0
$733$	− 39.5909i	− 1.46232i	−0.682204	−	0.731162i	$-0.738978\pi$
$733$	− 39.5909i	− 1.46232i	0.682204	−	0.731162i	$-0.261022\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.3303	2.00128
$738$	0	0
$739$	− 20.0000i	− 0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$	− 20.0000i	− 0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	0	0
$741$	− 3.65480i	− 0.134263i
$742$	0	0
$743$	26.6482	0.977628	0.488814	−	0.872388i	$-0.337430\pi$
$743$	26.6482	0.977628	0.488814	+	0.872388i	$0.337430\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.1652i	0.408512i
$748$	0	0
$749$	− 3.65480i	− 0.133544i
$750$	0	0
$751$	22.4213	0.818165	0.409083	−	0.912497i	$-0.365849\pi$
$751$	22.4213	0.818165	0.409083	+	0.912497i	$0.365849\pi$
$752$	0	0
$753$	−11.5826	−0.422093
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12.9427i	− 0.470411i	−0.971946	−	0.235205i	$-0.924424\pi$
$757$	− 12.9427i	− 0.470411i	0.971946	−	0.235205i	$-0.0755763\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.92820	0.250163
$768$	0	0
$769$	31.4955	1.13576	0.567878	−	0.823113i	$-0.307765\pi$
$769$	31.4955	1.13576	0.567878	+	0.823113i	$0.307765\pi$
$770$	0	0
$771$	− 2.74773i	− 0.0989570i
$772$	0	0
$773$	− 14.5794i	− 0.524385i	−0.965016	−	0.262192i	$-0.915554\pi$
$773$	− 14.5794i	− 0.524385i	0.965016	−	0.262192i	$-0.0844455\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.16515	0.257048
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	− 24.8208i	− 0.888159i
$782$	0	0
$783$	7.84190	0.280247
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	0	0
$789$	− 8.75560i	− 0.311708i
$790$	0	0
$791$	4.03620	0.143511
$792$	0	0
$793$	−9.66970	−0.343381
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.6084i	− 0.942518i	−0.881995	−	0.471259i	$-0.843800\pi$
$797$	− 26.6084i	− 0.942518i	0.881995	−	0.471259i	$-0.156200\pi$
$798$	0	0
$799$	24.8208	0.878097
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	− 42.9909i	− 1.51712i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.66930	−0.340376
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	− 19.1652i	− 0.672979i	−0.941687	−	0.336490i	$-0.890760\pi$
$811$	− 19.1652i	− 0.672979i	0.941687	−	0.336490i	$-0.109240\pi$
$812$	0	0
$813$	− 26.0761i	− 0.914529i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.6606	1.00271
$818$	0	0
$819$	− 0.834849i	− 0.0291720i
$820$	0	0
$821$	− 9.66930i	− 0.337461i	−0.985662	−	0.168731i	$-0.946033\pi$
$821$	− 9.66930i	− 0.337461i	0.985662	−	0.168731i	$-0.0539668\pi$
$822$	0	0
$823$	−14.7701	−0.514854	−0.257427	−	0.966298i	$-0.582875\pi$
$823$	−14.7701	−0.514854	−0.257427	+	0.966298i	$0.582875\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	16.0652i	0.557968i	0.960296	+	0.278984i	$0.0899976\pi$
$829$	16.0652i	0.557968i	−0.960296	+	0.278984i	$0.910002\pi$
$830$	0	0
$831$	14.7701	0.512369
$832$	0	0
$833$	22.0871	0.765273
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.29150i	− 0.182901i
$838$	0	0
$839$	−13.4750	−0.465209	−0.232604	−	0.972571i	$-0.574725\pi$
$839$	−13.4750	−0.465209	−0.232604	+	0.972571i	$0.574725\pi$
$840$	0	0
$841$	−32.4955	−1.12053
$842$	0	0
$843$	4.33030i	0.149144i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.67650	−0.0576053
$848$	0	0
$849$	−7.16515	−0.245907
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 23.5257i	− 0.805505i	−0.915309	−	0.402753i	$-0.868054\pi$
$853$	− 23.5257i	− 0.805505i	0.915309	−	0.402753i	$-0.131946\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.4174	−0.697446	−0.348723	−	0.937226i	$-0.613385\pi$
$857$	−20.4174	−0.697446	−0.348723	+	0.937226i	$0.613385\pi$
$858$	0	0
$859$	43.1652i	1.47278i	0.676559	+	0.736388i	$0.263470\pi$
$859$	43.1652i	1.47278i	−0.676559	+	0.736388i	$0.736530\pi$
$860$	0	0
$861$	− 5.48220i	− 0.186833i
$862$	0	0
$863$	−2.20880	−0.0751885	−0.0375942	−	0.999293i	$-0.511969\pi$
$863$	−2.20880	−0.0751885	−0.0375942	+	0.999293i	$0.511969\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.16515i	0.141456i
$868$	0	0
$869$	− 18.9572i	− 0.643079i
$870$	0	0
$871$	13.8564	0.469506
$872$	0	0
$873$	−7.16515	−0.242504
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 54.8933i	− 1.85362i	−0.375536	−	0.926808i	$-0.622541\pi$
$877$	− 54.8933i	− 1.85362i	0.375536	−	0.926808i	$-0.377459\pi$
$878$	0	0
$879$	4.37780	0.147660
$880$	0	0
$881$	−0.330303	−0.0111282	−0.00556409	−	0.999985i	$-0.501771\pi$
$881$	−0.330303	−0.0111282	−0.00556409	+	0.999985i	$0.501771\pi$
$882$	0	0
$883$	− 2.33030i	− 0.0784209i	−0.999231	−	0.0392105i	$-0.987516\pi$
$883$	− 2.33030i	− 0.0784209i	0.999231	−	0.0392105i	$-0.0124843\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.3024	0.513805	0.256902	−	0.966437i	$-0.417298\pi$
$887$	15.3024	0.513805	0.256902	+	0.966437i	$0.417298\pi$
$888$	0	0
$889$	−13.4955	−0.452623
$890$	0	0
$891$	3.58258i	0.120021i
$892$	0	0
$893$	27.7128i	0.927374i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.4955i	1.38395i
$900$	0	0
$901$	− 9.13701i	− 0.304398i
$902$	0	0
$903$	6.54680	0.217864
$904$	0	0
$905$	0	0
$906$	0	0
$907$	37.4955i	1.24502i	0.782613	+	0.622508i	$0.213886\pi$
$907$	37.4955i	1.24502i	−0.782613	+	0.622508i	$0.786114\pi$
$908$	0	0
$909$	16.5975i	0.550505i
$910$	0	0
$911$	35.0224	1.16034	0.580172	−	0.814494i	$-0.302985\pi$
$911$	35.0224	1.16034	0.580172	+	0.814494i	$0.302985\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.2378i	− 0.470174i
$918$	0	0
$919$	−43.5873	−1.43781	−0.718907	−	0.695107i	$-0.755357\pi$
$919$	−43.5873	−1.43781	−0.718907	+	0.695107i	$0.755357\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	− 6.33030i	− 0.208364i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.5975	0.545134
$928$	0	0
$929$	42.6606	1.39965	0.699825	−	0.714315i	$-0.253262\pi$
$929$	42.6606	1.39965	0.699825	+	0.714315i	$0.253262\pi$
$930$	0	0
$931$	24.6606i	0.808219i
$932$	0	0
$933$	17.5112i	0.573291i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.3303	0.729499	0.364750	−	0.931106i	$-0.381155\pi$
$937$	22.3303	0.729499	0.364750	+	0.931106i	$0.381155\pi$
$938$	0	0
$939$	19.1652i	0.625431i
$940$	0	0
$941$	− 44.3103i	− 1.44448i	−0.691645	−	0.722238i	$-0.743114\pi$
$941$	− 44.3103i	− 1.44448i	0.691645	−	0.722238i	$-0.256886\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	− 10.9644i	− 0.355920i
$950$	0	0
$951$	−2.93180	−0.0950702
$952$	0	0
$953$	−56.2432	−1.82190	−0.910948	−	0.412522i	$-0.864648\pi$
$953$	−56.2432	−1.82190	−0.910948	+	0.412522i	$0.864648\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 28.0942i	− 0.908157i
$958$	0	0
$959$	−17.8926	−0.577782
$960$	0	0
$961$	−3.00000	−0.0967742
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	11.1153	0.357444	0.178722	−	0.983900i	$-0.442804\pi$
$967$	11.1153	0.357444	0.178722	+	0.983900i	$0.442804\pi$
$968$	0	0
$969$	14.3303	0.460356
$970$	0	0
$971$	− 49.9129i	− 1.60178i	−0.598811	−	0.800890i	$-0.704360\pi$
$971$	− 49.9129i	− 1.60178i	0.598811	−	0.800890i	$-0.295640\pi$
$972$	0	0
$973$	10.2016i	0.327048i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.7477	1.11168	0.555839	−	0.831290i	$-0.312397\pi$
$977$	34.7477	1.11168	0.555839	+	0.831290i	$0.312397\pi$
$978$	0	0
$979$	7.16515i	0.228999i
$980$	0	0
$981$	8.75560i	0.279545i
$982$	0	0
$983$	50.7062	1.61728	0.808639	−	0.588306i	$-0.200205\pi$
$983$	50.7062	1.61728	0.808639	+	0.588306i	$0.200205\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.33030i	0.201496i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−43.5873	−1.38460	−0.692298	−	0.721611i	$-0.743402\pi$
$991$	−43.5873	−1.38460	−0.692298	+	0.721611i	$0.743402\pi$
$992$	0	0
$993$	26.3303	0.835567
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 39.2095i	− 1.24178i	−0.783899	−	0.620889i	$-0.786772\pi$
$997$	− 39.2095i	− 1.24178i	0.783899	−	0.620889i	$-0.213228\pi$
$998$	0	0
$999$	−7.84190	−0.248107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.k.r.2401.3		8
4.3	odd	2	inner	4800.2.k.r.2401.6		8
5.2	odd	4		960.2.d.e.289.8	yes	8
5.3	odd	4		960.2.d.f.289.2	yes	8
5.4	even	2		4800.2.k.q.2401.6		8
8.3	odd	2	inner	4800.2.k.r.2401.2		8
8.5	even	2	inner	4800.2.k.r.2401.7		8
15.2	even	4		2880.2.d.j.289.1		8
15.8	even	4		2880.2.d.i.289.7		8
20.3	even	4		960.2.d.e.289.2	yes	8
20.7	even	4		960.2.d.f.289.8	yes	8
20.19	odd	2		4800.2.k.q.2401.3		8
40.3	even	4		960.2.d.f.289.7	yes	8
40.13	odd	4		960.2.d.e.289.7	yes	8
40.19	odd	2		4800.2.k.q.2401.7		8
40.27	even	4		960.2.d.e.289.1	✓	8
40.29	even	2		4800.2.k.q.2401.2		8
40.37	odd	4		960.2.d.f.289.1	yes	8
60.23	odd	4		2880.2.d.j.289.7		8
60.47	odd	4		2880.2.d.i.289.1		8
80.3	even	4		3840.2.f.k.769.3		8
80.13	odd	4		3840.2.f.i.769.7		8
80.27	even	4		3840.2.f.i.769.2		8
80.37	odd	4		3840.2.f.k.769.6		8
80.43	even	4		3840.2.f.i.769.6		8
80.53	odd	4		3840.2.f.k.769.2		8
80.67	even	4		3840.2.f.k.769.7		8
80.77	odd	4		3840.2.f.i.769.3		8
120.53	even	4		2880.2.d.j.289.2		8
120.77	even	4		2880.2.d.i.289.8		8
120.83	odd	4		2880.2.d.i.289.2		8
120.107	odd	4		2880.2.d.j.289.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.d.e.289.1	✓	8	40.27	even	4
960.2.d.e.289.2	yes	8	20.3	even	4
960.2.d.e.289.7	yes	8	40.13	odd	4
960.2.d.e.289.8	yes	8	5.2	odd	4
960.2.d.f.289.1	yes	8	40.37	odd	4
960.2.d.f.289.2	yes	8	5.3	odd	4
960.2.d.f.289.7	yes	8	40.3	even	4
960.2.d.f.289.8	yes	8	20.7	even	4
2880.2.d.i.289.1		8	60.47	odd	4
2880.2.d.i.289.2		8	120.83	odd	4
2880.2.d.i.289.7		8	15.8	even	4
2880.2.d.i.289.8		8	120.77	even	4
2880.2.d.j.289.1		8	15.2	even	4
2880.2.d.j.289.2		8	120.53	even	4
2880.2.d.j.289.7		8	60.23	odd	4
2880.2.d.j.289.8		8	120.107	odd	4
3840.2.f.i.769.2		8	80.27	even	4
3840.2.f.i.769.3		8	80.77	odd	4
3840.2.f.i.769.6		8	80.43	even	4
3840.2.f.i.769.7		8	80.13	odd	4
3840.2.f.k.769.2		8	80.53	odd	4
3840.2.f.k.769.3		8	80.3	even	4
3840.2.f.k.769.6		8	80.37	odd	4
3840.2.f.k.769.7		8	80.67	even	4
4800.2.k.q.2401.2		8	40.29	even	2
4800.2.k.q.2401.3		8	20.19	odd	2
4800.2.k.q.2401.6		8	5.4	even	2
4800.2.k.q.2401.7		8	40.19	odd	2
4800.2.k.r.2401.2		8	8.3	odd	2	inner
4800.2.k.r.2401.3		8	1.1	even	1	trivial
4800.2.k.r.2401.6		8	4.3	odd	2	inner
4800.2.k.r.2401.7		8	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.913701 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +0.913701 q^{7} -1.00000 q^{9} +3.58258i q^{11} +0.913701i q^{13} -3.58258 q^{17} -4.00000i q^{19} -0.913701i q^{21} +1.00000i q^{27} -7.84190i q^{29} -5.29150 q^{31} +3.58258 q^{33} +7.84190i q^{37} +0.913701 q^{39} +6.00000 q^{41} +7.16515i q^{43} -6.92820 q^{47} -6.16515 q^{49} +3.58258i q^{51} +2.55040i q^{53} -4.00000 q^{57} -7.58258i q^{59} +10.5830i q^{61} -0.913701 q^{63} -15.1652i q^{67} -6.92820 q^{71} -12.0000 q^{73} +3.27340i q^{77} -5.29150 q^{79} +1.00000 q^{81} -11.1652i q^{83} -7.84190 q^{87} +2.00000 q^{89} +0.834849i q^{91} +5.29150i q^{93} +7.16515 q^{97} -3.58258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 8 q^{17} - 8 q^{33} + 48 q^{41} + 24 q^{49} - 32 q^{57} - 96 q^{73} + 8 q^{81} + 16 q^{89} - 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 8 * q^17 - 8 * q^33 + 48 * q^41 + 24 * q^49 - 32 * q^57 - 96 * q^73 + 8 * q^81 + 16 * q^89 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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