LMFDB - Embedded newform 1104.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,2,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1104.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1104.a (trivial)

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:	yes
Analytic conductor:	$8.81548438315$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.41421 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.41421 q^{5} -1.41421 q^{7} +1.00000 q^{9} +5.65685 q^{11} -5.65685 q^{13} -3.41421 q^{15} +6.24264 q^{17} -0.242641 q^{19} +1.41421 q^{21} -1.00000 q^{23} +6.65685 q^{25} -1.00000 q^{27} +3.17157 q^{29} +6.82843 q^{31} -5.65685 q^{33} -4.82843 q^{35} -8.82843 q^{37} +5.65685 q^{39} -2.00000 q^{41} +2.58579 q^{43} +3.41421 q^{45} -6.48528 q^{47} -5.00000 q^{49} -6.24264 q^{51} +7.41421 q^{53} +19.3137 q^{55} +0.242641 q^{57} +3.17157 q^{59} +8.82843 q^{61} -1.41421 q^{63} -19.3137 q^{65} +16.2426 q^{67} +1.00000 q^{69} +11.3137 q^{71} +11.6569 q^{73} -6.65685 q^{75} -8.00000 q^{77} -7.07107 q^{79} +1.00000 q^{81} -1.65685 q^{83} +21.3137 q^{85} -3.17157 q^{87} -1.07107 q^{89} +8.00000 q^{91} -6.82843 q^{93} -0.828427 q^{95} -12.1421 q^{97} +5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} - 4 q^{15} + 4 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} + 8 q^{31} - 4 q^{35} - 12 q^{37} - 4 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} - 10 q^{49} - 4 q^{51} + 12 q^{53} + 16 q^{55} - 8 q^{57} + 12 q^{59} + 12 q^{61} - 16 q^{65} + 24 q^{67} + 2 q^{69} + 12 q^{73} - 2 q^{75} - 16 q^{77} + 2 q^{81} + 8 q^{83} + 20 q^{85} - 12 q^{87} + 12 q^{89} + 16 q^{91} - 8 q^{93} + 4 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 - 4 * q^15 + 4 * q^17 + 8 * q^19 - 2 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^29 + 8 * q^31 - 4 * q^35 - 12 * q^37 - 4 * q^41 + 8 * q^43 + 4 * q^45 + 4 * q^47 - 10 * q^49 - 4 * q^51 + 12 * q^53 + 16 * q^55 - 8 * q^57 + 12 * q^59 + 12 * q^61 - 16 * q^65 + 24 * q^67 + 2 * q^69 + 12 * q^73 - 2 * q^75 - 16 * q^77 + 2 * q^81 + 8 * q^83 + 20 * q^85 - 12 * q^87 + 12 * q^89 + 16 * q^91 - 8 * q^93 + 4 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	6.24264	1.51406	0.757031	−	0.653379i	$-0.226649\pi$
$17$	6.24264	1.51406	0.757031	+	0.653379i	$0.226649\pi$
$18$	0	0
$19$	−0.242641	−0.0556656	−0.0278328	−	0.999613i	$-0.508861\pi$
$19$	−0.242641	−0.0556656	−0.0278328	+	0.999613i	$0.508861\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.17157	0.588946	0.294473	−	0.955660i	$-0.404856\pi$
$29$	3.17157	0.588946	0.294473	+	0.955660i	$0.404856\pi$
$30$	0	0
$31$	6.82843	1.22642	0.613211	−	0.789919i	$-0.289878\pi$
$31$	6.82843	1.22642	0.613211	+	0.789919i	$0.289878\pi$
$32$	0	0
$33$	−5.65685	−0.984732
$34$	0	0
$35$	−4.82843	−0.816153
$36$	0	0
$37$	−8.82843	−1.45138	−0.725692	−	0.688019i	$-0.758480\pi$
$37$	−8.82843	−1.45138	−0.725692	+	0.688019i	$0.758480\pi$
$38$	0	0
$39$	5.65685	0.905822
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	2.58579	0.394329	0.197164	−	0.980370i	$-0.436827\pi$
$43$	2.58579	0.394329	0.197164	+	0.980370i	$0.436827\pi$
$44$	0	0
$45$	3.41421	0.508961
$46$	0	0
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−6.24264	−0.874145
$52$	0	0
$53$	7.41421	1.01842	0.509210	−	0.860642i	$-0.329938\pi$
$53$	7.41421	1.01842	0.509210	+	0.860642i	$0.329938\pi$
$54$	0	0
$55$	19.3137	2.60426
$56$	0	0
$57$	0.242641	0.0321385
$58$	0	0
$59$	3.17157	0.412904	0.206452	−	0.978457i	$-0.433808\pi$
$59$	3.17157	0.412904	0.206452	+	0.978457i	$0.433808\pi$
$60$	0	0
$61$	8.82843	1.13036	0.565182	−	0.824966i	$-0.308806\pi$
$61$	8.82843	1.13036	0.565182	+	0.824966i	$0.308806\pi$
$62$	0	0
$63$	−1.41421	−0.178174
$64$	0	0
$65$	−19.3137	−2.39557
$66$	0	0
$67$	16.2426	1.98435	0.992177	−	0.124838i	$-0.0398410\pi$
$67$	16.2426	1.98435	0.992177	+	0.124838i	$0.0398410\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	11.6569	1.36433	0.682166	−	0.731198i	$-0.261038\pi$
$73$	11.6569	1.36433	0.682166	+	0.731198i	$0.261038\pi$
$74$	0	0
$75$	−6.65685	−0.768667
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−7.07107	−0.795557	−0.397779	−	0.917481i	$-0.630219\pi$
$79$	−7.07107	−0.795557	−0.397779	+	0.917481i	$0.630219\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.65685	−0.181863	−0.0909317	−	0.995857i	$-0.528984\pi$
$83$	−1.65685	−0.181863	−0.0909317	+	0.995857i	$0.528984\pi$
$84$	0	0
$85$	21.3137	2.31180
$86$	0	0
$87$	−3.17157	−0.340028
$88$	0	0
$89$	−1.07107	−0.113533	−0.0567665	−	0.998387i	$-0.518079\pi$
$89$	−1.07107	−0.113533	−0.0567665	+	0.998387i	$0.518079\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−6.82843	−0.708075
$94$	0	0
$95$	−0.828427	−0.0849948
$96$	0	0
$97$	−12.1421	−1.23285	−0.616424	−	0.787415i	$-0.711419\pi$
$97$	−12.1421	−1.23285	−0.616424	+	0.787415i	$0.711419\pi$
$98$	0	0
$99$	5.65685	0.568535
$100$	0	0
$101$	17.3137	1.72278	0.861389	−	0.507946i	$-0.169595\pi$
$101$	17.3137	1.72278	0.861389	+	0.507946i	$0.169595\pi$
$102$	0	0
$103$	−12.7279	−1.25412	−0.627060	−	0.778971i	$-0.715742\pi$
$103$	−12.7279	−1.25412	−0.627060	+	0.778971i	$0.715742\pi$
$104$	0	0
$105$	4.82843	0.471206
$106$	0	0
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	8.82843	0.837957
$112$	0	0
$113$	−11.4142	−1.07376	−0.536879	−	0.843659i	$-0.680397\pi$
$113$	−11.4142	−1.07376	−0.536879	+	0.843659i	$0.680397\pi$
$114$	0	0
$115$	−3.41421	−0.318377
$116$	0	0
$117$	−5.65685	−0.522976
$118$	0	0
$119$	−8.82843	−0.809301
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	0.485281	0.0430618	0.0215309	−	0.999768i	$-0.493146\pi$
$127$	0.485281	0.0430618	0.0215309	+	0.999768i	$0.493146\pi$
$128$	0	0
$129$	−2.58579	−0.227666
$130$	0	0
$131$	1.65685	0.144760	0.0723800	−	0.997377i	$-0.476941\pi$
$131$	1.65685	0.144760	0.0723800	+	0.997377i	$0.476941\pi$
$132$	0	0
$133$	0.343146	0.0297545
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	1.07107	0.0915075	0.0457537	−	0.998953i	$-0.485431\pi$
$137$	1.07107	0.0915075	0.0457537	+	0.998953i	$0.485431\pi$
$138$	0	0
$139$	1.65685	0.140533	0.0702663	−	0.997528i	$-0.477615\pi$
$139$	1.65685	0.140533	0.0702663	+	0.997528i	$0.477615\pi$
$140$	0	0
$141$	6.48528	0.546159
$142$	0	0
$143$	−32.0000	−2.67597
$144$	0	0
$145$	10.8284	0.899252
$146$	0	0
$147$	5.00000	0.412393
$148$	0	0
$149$	2.24264	0.183724	0.0918621	−	0.995772i	$-0.470718\pi$
$149$	2.24264	0.183724	0.0918621	+	0.995772i	$0.470718\pi$
$150$	0	0
$151$	−13.6569	−1.11138	−0.555690	−	0.831390i	$-0.687546\pi$
$151$	−13.6569	−1.11138	−0.555690	+	0.831390i	$0.687546\pi$
$152$	0	0
$153$	6.24264	0.504688
$154$	0	0
$155$	23.3137	1.87260
$156$	0	0
$157$	−3.17157	−0.253119	−0.126560	−	0.991959i	$-0.540393\pi$
$157$	−3.17157	−0.253119	−0.126560	+	0.991959i	$0.540393\pi$
$158$	0	0
$159$	−7.41421	−0.587985
$160$	0	0
$161$	1.41421	0.111456
$162$	0	0
$163$	−5.17157	−0.405069	−0.202534	−	0.979275i	$-0.564918\pi$
$163$	−5.17157	−0.405069	−0.202534	+	0.979275i	$0.564918\pi$
$164$	0	0
$165$	−19.3137	−1.50357
$166$	0	0
$167$	−13.6569	−1.05680	−0.528400	−	0.848996i	$-0.677208\pi$
$167$	−13.6569	−1.05680	−0.528400	+	0.848996i	$0.677208\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−0.242641	−0.0185552
$172$	0	0
$173$	5.51472	0.419276	0.209638	−	0.977779i	$-0.432771\pi$
$173$	5.51472	0.419276	0.209638	+	0.977779i	$0.432771\pi$
$174$	0	0
$175$	−9.41421	−0.711648
$176$	0	0
$177$	−3.17157	−0.238390
$178$	0	0
$179$	−14.4853	−1.08268	−0.541340	−	0.840804i	$-0.682083\pi$
$179$	−14.4853	−1.08268	−0.541340	+	0.840804i	$0.682083\pi$
$180$	0	0
$181$	5.31371	0.394965	0.197482	−	0.980306i	$-0.436723\pi$
$181$	5.31371	0.394965	0.197482	+	0.980306i	$0.436723\pi$
$182$	0	0
$183$	−8.82843	−0.652616
$184$	0	0
$185$	−30.1421	−2.21609
$186$	0	0
$187$	35.3137	2.58239
$188$	0	0
$189$	1.41421	0.102869
$190$	0	0
$191$	1.17157	0.0847720	0.0423860	−	0.999101i	$-0.486504\pi$
$191$	1.17157	0.0847720	0.0423860	+	0.999101i	$0.486504\pi$
$192$	0	0
$193$	11.3137	0.814379	0.407189	−	0.913344i	$-0.366509\pi$
$193$	11.3137	0.814379	0.407189	+	0.913344i	$0.366509\pi$
$194$	0	0
$195$	19.3137	1.38308
$196$	0	0
$197$	−8.82843	−0.628999	−0.314500	−	0.949258i	$-0.601837\pi$
$197$	−8.82843	−0.628999	−0.314500	+	0.949258i	$0.601837\pi$
$198$	0	0
$199$	−20.2426	−1.43496	−0.717481	−	0.696578i	$-0.754705\pi$
$199$	−20.2426	−1.43496	−0.717481	+	0.696578i	$0.754705\pi$
$200$	0	0
$201$	−16.2426	−1.14567
$202$	0	0
$203$	−4.48528	−0.314805
$204$	0	0
$205$	−6.82843	−0.476918
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−1.37258	−0.0949436
$210$	0	0
$211$	0.485281	0.0334081	0.0167041	−	0.999860i	$-0.494683\pi$
$211$	0.485281	0.0334081	0.0167041	+	0.999860i	$0.494683\pi$
$212$	0	0
$213$	−11.3137	−0.775203
$214$	0	0
$215$	8.82843	0.602094
$216$	0	0
$217$	−9.65685	−0.655550
$218$	0	0
$219$	−11.6569	−0.787697
$220$	0	0
$221$	−35.3137	−2.37546
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	−10.8284	−0.718708	−0.359354	−	0.933201i	$-0.617003\pi$
$227$	−10.8284	−0.718708	−0.359354	+	0.933201i	$0.617003\pi$
$228$	0	0
$229$	−26.9706	−1.78226	−0.891132	−	0.453743i	$-0.850088\pi$
$229$	−26.9706	−1.78226	−0.891132	+	0.453743i	$0.850088\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	−15.6569	−1.02571	−0.512857	−	0.858474i	$-0.671413\pi$
$233$	−15.6569	−1.02571	−0.512857	+	0.858474i	$0.671413\pi$
$234$	0	0
$235$	−22.1421	−1.44439
$236$	0	0
$237$	7.07107	0.459315
$238$	0	0
$239$	−10.3431	−0.669042	−0.334521	−	0.942388i	$-0.608575\pi$
$239$	−10.3431	−0.669042	−0.334521	+	0.942388i	$0.608575\pi$
$240$	0	0
$241$	−21.7990	−1.40420	−0.702098	−	0.712080i	$-0.747753\pi$
$241$	−21.7990	−1.40420	−0.702098	+	0.712080i	$0.747753\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−17.0711	−1.09063
$246$	0	0
$247$	1.37258	0.0873354
$248$	0	0
$249$	1.65685	0.104999
$250$	0	0
$251$	−21.4558	−1.35428	−0.677140	−	0.735854i	$-0.736781\pi$
$251$	−21.4558	−1.35428	−0.677140	+	0.735854i	$0.736781\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	−21.3137	−1.33472
$256$	0	0
$257$	−29.3137	−1.82854	−0.914269	−	0.405107i	$-0.867234\pi$
$257$	−29.3137	−1.82854	−0.914269	+	0.405107i	$0.867234\pi$
$258$	0	0
$259$	12.4853	0.775798
$260$	0	0
$261$	3.17157	0.196315
$262$	0	0
$263$	15.7990	0.974207	0.487104	−	0.873344i	$-0.338053\pi$
$263$	15.7990	0.974207	0.487104	+	0.873344i	$0.338053\pi$
$264$	0	0
$265$	25.3137	1.55501
$266$	0	0
$267$	1.07107	0.0655483
$268$	0	0
$269$	23.1716	1.41280	0.706398	−	0.707815i	$-0.250319\pi$
$269$	23.1716	1.41280	0.706398	+	0.707815i	$0.250319\pi$
$270$	0	0
$271$	−18.8284	−1.14375	−0.571873	−	0.820342i	$-0.693783\pi$
$271$	−18.8284	−1.14375	−0.571873	+	0.820342i	$0.693783\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	37.6569	2.27079
$276$	0	0
$277$	12.6274	0.758708	0.379354	−	0.925252i	$-0.376146\pi$
$277$	12.6274	0.758708	0.379354	+	0.925252i	$0.376146\pi$
$278$	0	0
$279$	6.82843	0.408807
$280$	0	0
$281$	25.0711	1.49561	0.747807	−	0.663916i	$-0.231107\pi$
$281$	25.0711	1.49561	0.747807	+	0.663916i	$0.231107\pi$
$282$	0	0
$283$	21.8995	1.30179	0.650895	−	0.759168i	$-0.274394\pi$
$283$	21.8995	1.30179	0.650895	+	0.759168i	$0.274394\pi$
$284$	0	0
$285$	0.828427	0.0490718
$286$	0	0
$287$	2.82843	0.166957
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	12.1421	0.711785
$292$	0	0
$293$	−9.75736	−0.570031	−0.285016	−	0.958523i	$-0.591999\pi$
$293$	−9.75736	−0.570031	−0.285016	+	0.958523i	$0.591999\pi$
$294$	0	0
$295$	10.8284	0.630455
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	−3.65685	−0.210778
$302$	0	0
$303$	−17.3137	−0.994647
$304$	0	0
$305$	30.1421	1.72593
$306$	0	0
$307$	−21.4558	−1.22455	−0.612275	−	0.790645i	$-0.709745\pi$
$307$	−21.4558	−1.22455	−0.612275	+	0.790645i	$0.709745\pi$
$308$	0	0
$309$	12.7279	0.724066
$310$	0	0
$311$	−8.82843	−0.500614	−0.250307	−	0.968166i	$-0.580532\pi$
$311$	−8.82843	−0.500614	−0.250307	+	0.968166i	$0.580532\pi$
$312$	0	0
$313$	−4.34315	−0.245489	−0.122745	−	0.992438i	$-0.539170\pi$
$313$	−4.34315	−0.245489	−0.122745	+	0.992438i	$0.539170\pi$
$314$	0	0
$315$	−4.82843	−0.272051
$316$	0	0
$317$	−8.82843	−0.495854	−0.247927	−	0.968779i	$-0.579749\pi$
$317$	−8.82843	−0.495854	−0.247927	+	0.968779i	$0.579749\pi$
$318$	0	0
$319$	17.9411	1.00451
$320$	0	0
$321$	−5.65685	−0.315735
$322$	0	0
$323$	−1.51472	−0.0842812
$324$	0	0
$325$	−37.6569	−2.08883
$326$	0	0
$327$	7.65685	0.423425
$328$	0	0
$329$	9.17157	0.505645
$330$	0	0
$331$	−2.14214	−0.117742	−0.0588712	−	0.998266i	$-0.518750\pi$
$331$	−2.14214	−0.117742	−0.0588712	+	0.998266i	$0.518750\pi$
$332$	0	0
$333$	−8.82843	−0.483795
$334$	0	0
$335$	55.4558	3.02988
$336$	0	0
$337$	7.17157	0.390660	0.195330	−	0.980738i	$-0.437422\pi$
$337$	7.17157	0.390660	0.195330	+	0.980738i	$0.437422\pi$
$338$	0	0
$339$	11.4142	0.619935
$340$	0	0
$341$	38.6274	2.09179
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	3.41421	0.183815
$346$	0	0
$347$	7.17157	0.384990	0.192495	−	0.981298i	$-0.438342\pi$
$347$	7.17157	0.384990	0.192495	+	0.981298i	$0.438342\pi$
$348$	0	0
$349$	3.31371	0.177379	0.0886894	−	0.996059i	$-0.471732\pi$
$349$	3.31371	0.177379	0.0886894	+	0.996059i	$0.471732\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	16.1421	0.859159	0.429580	−	0.903029i	$-0.358662\pi$
$353$	16.1421	0.859159	0.429580	+	0.903029i	$0.358662\pi$
$354$	0	0
$355$	38.6274	2.05013
$356$	0	0
$357$	8.82843	0.467250
$358$	0	0
$359$	−30.8284	−1.62706	−0.813531	−	0.581521i	$-0.802458\pi$
$359$	−30.8284	−1.62706	−0.813531	+	0.581521i	$0.802458\pi$
$360$	0	0
$361$	−18.9411	−0.996901
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	39.7990	2.08317
$366$	0	0
$367$	20.2426	1.05666	0.528329	−	0.849040i	$-0.322819\pi$
$367$	20.2426	1.05666	0.528329	+	0.849040i	$0.322819\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−10.4853	−0.544369
$372$	0	0
$373$	−16.8284	−0.871343	−0.435671	−	0.900106i	$-0.643489\pi$
$373$	−16.8284	−0.871343	−0.435671	+	0.900106i	$0.643489\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−17.9411	−0.924015
$378$	0	0
$379$	−16.7279	−0.859256	−0.429628	−	0.903006i	$-0.641355\pi$
$379$	−16.7279	−0.859256	−0.429628	+	0.903006i	$0.641355\pi$
$380$	0	0
$381$	−0.485281	−0.0248617
$382$	0	0
$383$	−11.7990	−0.602900	−0.301450	−	0.953482i	$-0.597471\pi$
$383$	−11.7990	−0.602900	−0.301450	+	0.953482i	$0.597471\pi$
$384$	0	0
$385$	−27.3137	−1.39204
$386$	0	0
$387$	2.58579	0.131443
$388$	0	0
$389$	−10.7279	−0.543927	−0.271964	−	0.962308i	$-0.587673\pi$
$389$	−10.7279	−0.543927	−0.271964	+	0.962308i	$0.587673\pi$
$390$	0	0
$391$	−6.24264	−0.315704
$392$	0	0
$393$	−1.65685	−0.0835772
$394$	0	0
$395$	−24.1421	−1.21472
$396$	0	0
$397$	28.6274	1.43677	0.718384	−	0.695646i	$-0.244882\pi$
$397$	28.6274	1.43677	0.718384	+	0.695646i	$0.244882\pi$
$398$	0	0
$399$	−0.343146	−0.0171788
$400$	0	0
$401$	3.89949	0.194731	0.0973657	−	0.995249i	$-0.468958\pi$
$401$	3.89949	0.194731	0.0973657	+	0.995249i	$0.468958\pi$
$402$	0	0
$403$	−38.6274	−1.92417
$404$	0	0
$405$	3.41421	0.169654
$406$	0	0
$407$	−49.9411	−2.47549
$408$	0	0
$409$	17.6569	0.873075	0.436538	−	0.899686i	$-0.356205\pi$
$409$	17.6569	0.873075	0.436538	+	0.899686i	$0.356205\pi$
$410$	0	0
$411$	−1.07107	−0.0528319
$412$	0	0
$413$	−4.48528	−0.220706
$414$	0	0
$415$	−5.65685	−0.277684
$416$	0	0
$417$	−1.65685	−0.0811365
$418$	0	0
$419$	−8.68629	−0.424353	−0.212177	−	0.977231i	$-0.568055\pi$
$419$	−8.68629	−0.424353	−0.212177	+	0.977231i	$0.568055\pi$
$420$	0	0
$421$	6.68629	0.325870	0.162935	−	0.986637i	$-0.447904\pi$
$421$	6.68629	0.325870	0.162935	+	0.986637i	$0.447904\pi$
$422$	0	0
$423$	−6.48528	−0.315325
$424$	0	0
$425$	41.5563	2.01578
$426$	0	0
$427$	−12.4853	−0.604205
$428$	0	0
$429$	32.0000	1.54497
$430$	0	0
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	−30.9706	−1.48835	−0.744175	−	0.667985i	$-0.767157\pi$
$433$	−30.9706	−1.48835	−0.744175	+	0.667985i	$0.767157\pi$
$434$	0	0
$435$	−10.8284	−0.519183
$436$	0	0
$437$	0.242641	0.0116071
$438$	0	0
$439$	13.6569	0.651806	0.325903	−	0.945403i	$-0.394332\pi$
$439$	13.6569	0.651806	0.325903	+	0.945403i	$0.394332\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	16.2843	0.773689	0.386845	−	0.922145i	$-0.373565\pi$
$443$	16.2843	0.773689	0.386845	+	0.922145i	$0.373565\pi$
$444$	0	0
$445$	−3.65685	−0.173352
$446$	0	0
$447$	−2.24264	−0.106073
$448$	0	0
$449$	28.3431	1.33760	0.668798	−	0.743444i	$-0.266809\pi$
$449$	28.3431	1.33760	0.668798	+	0.743444i	$0.266809\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	0	0
$453$	13.6569	0.641655
$454$	0	0
$455$	27.3137	1.28049
$456$	0	0
$457$	−36.8284	−1.72276	−0.861381	−	0.507960i	$-0.830400\pi$
$457$	−36.8284	−1.72276	−0.861381	+	0.507960i	$0.830400\pi$
$458$	0	0
$459$	−6.24264	−0.291382
$460$	0	0
$461$	−27.4558	−1.27875	−0.639373	−	0.768897i	$-0.720806\pi$
$461$	−27.4558	−1.27875	−0.639373	+	0.768897i	$0.720806\pi$
$462$	0	0
$463$	−28.9706	−1.34638	−0.673188	−	0.739471i	$-0.735076\pi$
$463$	−28.9706	−1.34638	−0.673188	+	0.739471i	$0.735076\pi$
$464$	0	0
$465$	−23.3137	−1.08115
$466$	0	0
$467$	27.1127	1.25463	0.627313	−	0.778767i	$-0.284155\pi$
$467$	27.1127	1.25463	0.627313	+	0.778767i	$0.284155\pi$
$468$	0	0
$469$	−22.9706	−1.06068
$470$	0	0
$471$	3.17157	0.146138
$472$	0	0
$473$	14.6274	0.672569
$474$	0	0
$475$	−1.61522	−0.0741115
$476$	0	0
$477$	7.41421	0.339474
$478$	0	0
$479$	42.6274	1.94770	0.973848	−	0.227200i	$-0.0729570\pi$
$479$	42.6274	1.94770	0.973848	+	0.227200i	$0.0729570\pi$
$480$	0	0
$481$	49.9411	2.27712
$482$	0	0
$483$	−1.41421	−0.0643489
$484$	0	0
$485$	−41.4558	−1.88241
$486$	0	0
$487$	25.6569	1.16262	0.581312	−	0.813681i	$-0.302540\pi$
$487$	25.6569	1.16262	0.581312	+	0.813681i	$0.302540\pi$
$488$	0	0
$489$	5.17157	0.233867
$490$	0	0
$491$	28.1421	1.27004	0.635018	−	0.772497i	$-0.280993\pi$
$491$	28.1421	1.27004	0.635018	+	0.772497i	$0.280993\pi$
$492$	0	0
$493$	19.7990	0.891702
$494$	0	0
$495$	19.3137	0.868087
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	7.51472	0.336405	0.168203	−	0.985752i	$-0.446204\pi$
$499$	7.51472	0.336405	0.168203	+	0.985752i	$0.446204\pi$
$500$	0	0
$501$	13.6569	0.610143
$502$	0	0
$503$	−23.3137	−1.03951	−0.519753	−	0.854316i	$-0.673976\pi$
$503$	−23.3137	−1.03951	−0.519753	+	0.854316i	$0.673976\pi$
$504$	0	0
$505$	59.1127	2.63048
$506$	0	0
$507$	−19.0000	−0.843820
$508$	0	0
$509$	−5.51472	−0.244436	−0.122218	−	0.992503i	$-0.539001\pi$
$509$	−5.51472	−0.244436	−0.122218	+	0.992503i	$0.539001\pi$
$510$	0	0
$511$	−16.4853	−0.729266
$512$	0	0
$513$	0.242641	0.0107128
$514$	0	0
$515$	−43.4558	−1.91489
$516$	0	0
$517$	−36.6863	−1.61346
$518$	0	0
$519$	−5.51472	−0.242069
$520$	0	0
$521$	2.24264	0.0982519	0.0491259	−	0.998793i	$-0.484356\pi$
$521$	2.24264	0.0982519	0.0491259	+	0.998793i	$0.484356\pi$
$522$	0	0
$523$	41.2132	1.80213	0.901064	−	0.433687i	$-0.142787\pi$
$523$	41.2132	1.80213	0.901064	+	0.433687i	$0.142787\pi$
$524$	0	0
$525$	9.41421	0.410870
$526$	0	0
$527$	42.6274	1.85688
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.17157	0.137635
$532$	0	0
$533$	11.3137	0.490051
$534$	0	0
$535$	19.3137	0.835004
$536$	0	0
$537$	14.4853	0.625086
$538$	0	0
$539$	−28.2843	−1.21829
$540$	0	0
$541$	−32.2843	−1.38801	−0.694005	−	0.719971i	$-0.744155\pi$
$541$	−32.2843	−1.38801	−0.694005	+	0.719971i	$0.744155\pi$
$542$	0	0
$543$	−5.31371	−0.228033
$544$	0	0
$545$	−26.1421	−1.11981
$546$	0	0
$547$	−1.85786	−0.0794365	−0.0397183	−	0.999211i	$-0.512646\pi$
$547$	−1.85786	−0.0794365	−0.0397183	+	0.999211i	$0.512646\pi$
$548$	0	0
$549$	8.82843	0.376788
$550$	0	0
$551$	−0.769553	−0.0327840
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	30.1421	1.27946
$556$	0	0
$557$	30.2426	1.28142	0.640711	−	0.767782i	$-0.278640\pi$
$557$	30.2426	1.28142	0.640711	+	0.767782i	$0.278640\pi$
$558$	0	0
$559$	−14.6274	−0.618674
$560$	0	0
$561$	−35.3137	−1.49095
$562$	0	0
$563$	−21.1716	−0.892275	−0.446138	−	0.894964i	$-0.647201\pi$
$563$	−21.1716	−0.892275	−0.446138	+	0.894964i	$0.647201\pi$
$564$	0	0
$565$	−38.9706	−1.63950
$566$	0	0
$567$	−1.41421	−0.0593914
$568$	0	0
$569$	9.75736	0.409050	0.204525	−	0.978861i	$-0.434435\pi$
$569$	9.75736	0.409050	0.204525	+	0.978861i	$0.434435\pi$
$570$	0	0
$571$	24.7279	1.03483	0.517416	−	0.855734i	$-0.326894\pi$
$571$	24.7279	1.03483	0.517416	+	0.855734i	$0.326894\pi$
$572$	0	0
$573$	−1.17157	−0.0489432
$574$	0	0
$575$	−6.65685	−0.277610
$576$	0	0
$577$	−46.6274	−1.94112	−0.970562	−	0.240850i	$-0.922574\pi$
$577$	−46.6274	−1.94112	−0.970562	+	0.240850i	$0.922574\pi$
$578$	0	0
$579$	−11.3137	−0.470182
$580$	0	0
$581$	2.34315	0.0972101
$582$	0	0
$583$	41.9411	1.73702
$584$	0	0
$585$	−19.3137	−0.798524
$586$	0	0
$587$	33.6569	1.38917	0.694584	−	0.719412i	$-0.255589\pi$
$587$	33.6569	1.38917	0.694584	+	0.719412i	$0.255589\pi$
$588$	0	0
$589$	−1.65685	−0.0682695
$590$	0	0
$591$	8.82843	0.363153
$592$	0	0
$593$	−32.3431	−1.32817	−0.664087	−	0.747655i	$-0.731180\pi$
$593$	−32.3431	−1.32817	−0.664087	+	0.747655i	$0.731180\pi$
$594$	0	0
$595$	−30.1421	−1.23571
$596$	0	0
$597$	20.2426	0.828476
$598$	0	0
$599$	−44.2843	−1.80941	−0.904703	−	0.426043i	$-0.859907\pi$
$599$	−44.2843	−1.80941	−0.904703	+	0.426043i	$0.859907\pi$
$600$	0	0
$601$	35.3137	1.44048	0.720238	−	0.693727i	$-0.244033\pi$
$601$	35.3137	1.44048	0.720238	+	0.693727i	$0.244033\pi$
$602$	0	0
$603$	16.2426	0.661451
$604$	0	0
$605$	71.6985	2.91496
$606$	0	0
$607$	21.6569	0.879025	0.439512	−	0.898237i	$-0.355151\pi$
$607$	21.6569	0.879025	0.439512	+	0.898237i	$0.355151\pi$
$608$	0	0
$609$	4.48528	0.181753
$610$	0	0
$611$	36.6863	1.48417
$612$	0	0
$613$	1.51472	0.0611789	0.0305895	−	0.999532i	$-0.490262\pi$
$613$	1.51472	0.0611789	0.0305895	+	0.999532i	$0.490262\pi$
$614$	0	0
$615$	6.82843	0.275349
$616$	0	0
$617$	7.89949	0.318022	0.159011	−	0.987277i	$-0.449170\pi$
$617$	7.89949	0.318022	0.159011	+	0.987277i	$0.449170\pi$
$618$	0	0
$619$	−8.24264	−0.331300	−0.165650	−	0.986185i	$-0.552972\pi$
$619$	−8.24264	−0.331300	−0.165650	+	0.986185i	$0.552972\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	1.51472	0.0606859
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	1.37258	0.0548157
$628$	0	0
$629$	−55.1127	−2.19749
$630$	0	0
$631$	−4.24264	−0.168897	−0.0844484	−	0.996428i	$-0.526913\pi$
$631$	−4.24264	−0.168897	−0.0844484	+	0.996428i	$0.526913\pi$
$632$	0	0
$633$	−0.485281	−0.0192882
$634$	0	0
$635$	1.65685	0.0657503
$636$	0	0
$637$	28.2843	1.12066
$638$	0	0
$639$	11.3137	0.447563
$640$	0	0
$641$	−19.6985	−0.778043	−0.389022	−	0.921229i	$-0.627187\pi$
$641$	−19.6985	−0.778043	−0.389022	+	0.921229i	$0.627187\pi$
$642$	0	0
$643$	−16.7279	−0.659685	−0.329842	−	0.944036i	$-0.606996\pi$
$643$	−16.7279	−0.659685	−0.329842	+	0.944036i	$0.606996\pi$
$644$	0	0
$645$	−8.82843	−0.347619
$646$	0	0
$647$	9.79899	0.385238	0.192619	−	0.981274i	$-0.438302\pi$
$647$	9.79899	0.385238	0.192619	+	0.981274i	$0.438302\pi$
$648$	0	0
$649$	17.9411	0.704251
$650$	0	0
$651$	9.65685	0.378482
$652$	0	0
$653$	18.9706	0.742375	0.371188	−	0.928558i	$-0.378951\pi$
$653$	18.9706	0.742375	0.371188	+	0.928558i	$0.378951\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	11.6569	0.454777
$658$	0	0
$659$	0.201010	0.00783024	0.00391512	−	0.999992i	$-0.498754\pi$
$659$	0.201010	0.00783024	0.00391512	+	0.999992i	$0.498754\pi$
$660$	0	0
$661$	8.14214	0.316692	0.158346	−	0.987384i	$-0.449384\pi$
$661$	8.14214	0.316692	0.158346	+	0.987384i	$0.449384\pi$
$662$	0	0
$663$	35.3137	1.37147
$664$	0	0
$665$	1.17157	0.0454316
$666$	0	0
$667$	−3.17157	−0.122804
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	49.9411	1.92796
$672$	0	0
$673$	23.3137	0.898677	0.449339	−	0.893361i	$-0.351660\pi$
$673$	23.3137	0.898677	0.449339	+	0.893361i	$0.351660\pi$
$674$	0	0
$675$	−6.65685	−0.256222
$676$	0	0
$677$	−15.6985	−0.603342	−0.301671	−	0.953412i	$-0.597544\pi$
$677$	−15.6985	−0.603342	−0.301671	+	0.953412i	$0.597544\pi$
$678$	0	0
$679$	17.1716	0.658984
$680$	0	0
$681$	10.8284	0.414946
$682$	0	0
$683$	18.6274	0.712758	0.356379	−	0.934341i	$-0.384011\pi$
$683$	18.6274	0.712758	0.356379	+	0.934341i	$0.384011\pi$
$684$	0	0
$685$	3.65685	0.139721
$686$	0	0
$687$	26.9706	1.02899
$688$	0	0
$689$	−41.9411	−1.59783
$690$	0	0
$691$	25.4558	0.968386	0.484193	−	0.874961i	$-0.339113\pi$
$691$	25.4558	0.968386	0.484193	+	0.874961i	$0.339113\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	−12.4853	−0.472914
$698$	0	0
$699$	15.6569	0.592197
$700$	0	0
$701$	4.78680	0.180795	0.0903974	−	0.995906i	$-0.471186\pi$
$701$	4.78680	0.180795	0.0903974	+	0.995906i	$0.471186\pi$
$702$	0	0
$703$	2.14214	0.0807922
$704$	0	0
$705$	22.1421	0.833921
$706$	0	0
$707$	−24.4853	−0.920864
$708$	0	0
$709$	17.3137	0.650230	0.325115	−	0.945674i	$-0.394597\pi$
$709$	17.3137	0.650230	0.325115	+	0.945674i	$0.394597\pi$
$710$	0	0
$711$	−7.07107	−0.265186
$712$	0	0
$713$	−6.82843	−0.255727
$714$	0	0
$715$	−109.255	−4.08590
$716$	0	0
$717$	10.3431	0.386272
$718$	0	0
$719$	−8.14214	−0.303650	−0.151825	−	0.988407i	$-0.548515\pi$
$719$	−8.14214	−0.303650	−0.151825	+	0.988407i	$0.548515\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	21.7990	0.810713
$724$	0	0
$725$	21.1127	0.784106
$726$	0	0
$727$	24.0416	0.891655	0.445827	−	0.895119i	$-0.352910\pi$
$727$	24.0416	0.891655	0.445827	+	0.895119i	$0.352910\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.1421	0.597038
$732$	0	0
$733$	−22.9706	−0.848437	−0.424219	−	0.905560i	$-0.639451\pi$
$733$	−22.9706	−0.848437	−0.424219	+	0.905560i	$0.639451\pi$
$734$	0	0
$735$	17.0711	0.629676
$736$	0	0
$737$	91.8823	3.38453
$738$	0	0
$739$	−47.3137	−1.74046	−0.870231	−	0.492643i	$-0.836031\pi$
$739$	−47.3137	−1.74046	−0.870231	+	0.492643i	$0.836031\pi$
$740$	0	0
$741$	−1.37258	−0.0504231
$742$	0	0
$743$	−33.4558	−1.22738	−0.613688	−	0.789549i	$-0.710315\pi$
$743$	−33.4558	−1.22738	−0.613688	+	0.789549i	$0.710315\pi$
$744$	0	0
$745$	7.65685	0.280525
$746$	0	0
$747$	−1.65685	−0.0606211
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−53.6985	−1.95949	−0.979743	−	0.200260i	$-0.935821\pi$
$751$	−53.6985	−1.95949	−0.979743	+	0.200260i	$0.935821\pi$
$752$	0	0
$753$	21.4558	0.781894
$754$	0	0
$755$	−46.6274	−1.69695
$756$	0	0
$757$	−1.31371	−0.0477475	−0.0238738	−	0.999715i	$-0.507600\pi$
$757$	−1.31371	−0.0477475	−0.0238738	+	0.999715i	$0.507600\pi$
$758$	0	0
$759$	5.65685	0.205331
$760$	0	0
$761$	−40.8284	−1.48003	−0.740015	−	0.672591i	$-0.765181\pi$
$761$	−40.8284	−1.48003	−0.740015	+	0.672591i	$0.765181\pi$
$762$	0	0
$763$	10.8284	0.392015
$764$	0	0
$765$	21.3137	0.770599
$766$	0	0
$767$	−17.9411	−0.647816
$768$	0	0
$769$	8.34315	0.300862	0.150431	−	0.988621i	$-0.451934\pi$
$769$	8.34315	0.300862	0.150431	+	0.988621i	$0.451934\pi$
$770$	0	0
$771$	29.3137	1.05571
$772$	0	0
$773$	8.87006	0.319034	0.159517	−	0.987195i	$-0.449006\pi$
$773$	8.87006	0.319034	0.159517	+	0.987195i	$0.449006\pi$
$774$	0	0
$775$	45.4558	1.63282
$776$	0	0
$777$	−12.4853	−0.447907
$778$	0	0
$779$	0.485281	0.0173870
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	−3.17157	−0.113343
$784$	0	0
$785$	−10.8284	−0.386483
$786$	0	0
$787$	20.5269	0.731705	0.365853	−	0.930673i	$-0.380777\pi$
$787$	20.5269	0.731705	0.365853	+	0.930673i	$0.380777\pi$
$788$	0	0
$789$	−15.7990	−0.562459
$790$	0	0
$791$	16.1421	0.573948
$792$	0	0
$793$	−49.9411	−1.77346
$794$	0	0
$795$	−25.3137	−0.897785
$796$	0	0
$797$	−19.8995	−0.704876	−0.352438	−	0.935835i	$-0.614647\pi$
$797$	−19.8995	−0.704876	−0.352438	+	0.935835i	$0.614647\pi$
$798$	0	0
$799$	−40.4853	−1.43227
$800$	0	0
$801$	−1.07107	−0.0378443
$802$	0	0
$803$	65.9411	2.32701
$804$	0	0
$805$	4.82843	0.170180
$806$	0	0
$807$	−23.1716	−0.815678
$808$	0	0
$809$	43.1716	1.51783	0.758916	−	0.651189i	$-0.225729\pi$
$809$	43.1716	1.51783	0.758916	+	0.651189i	$0.225729\pi$
$810$	0	0
$811$	5.17157	0.181598	0.0907992	−	0.995869i	$-0.471058\pi$
$811$	5.17157	0.181598	0.0907992	+	0.995869i	$0.471058\pi$
$812$	0	0
$813$	18.8284	0.660342
$814$	0	0
$815$	−17.6569	−0.618493
$816$	0	0
$817$	−0.627417	−0.0219505
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	2.68629	0.0937522	0.0468761	−	0.998901i	$-0.485073\pi$
$821$	2.68629	0.0937522	0.0468761	+	0.998901i	$0.485073\pi$
$822$	0	0
$823$	−43.7990	−1.52674	−0.763368	−	0.645963i	$-0.776456\pi$
$823$	−43.7990	−1.52674	−0.763368	+	0.645963i	$0.776456\pi$
$824$	0	0
$825$	−37.6569	−1.31104
$826$	0	0
$827$	48.7696	1.69588	0.847942	−	0.530089i	$-0.177842\pi$
$827$	48.7696	1.69588	0.847942	+	0.530089i	$0.177842\pi$
$828$	0	0
$829$	−52.9706	−1.83974	−0.919872	−	0.392219i	$-0.871708\pi$
$829$	−52.9706	−1.83974	−0.919872	+	0.392219i	$0.871708\pi$
$830$	0	0
$831$	−12.6274	−0.438040
$832$	0	0
$833$	−31.2132	−1.08147
$834$	0	0
$835$	−46.6274	−1.61361
$836$	0	0
$837$	−6.82843	−0.236025
$838$	0	0
$839$	6.82843	0.235743	0.117872	−	0.993029i	$-0.462393\pi$
$839$	6.82843	0.235743	0.117872	+	0.993029i	$0.462393\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	0	0
$843$	−25.0711	−0.863493
$844$	0	0
$845$	64.8701	2.23160
$846$	0	0
$847$	−29.6985	−1.02045
$848$	0	0
$849$	−21.8995	−0.751589
$850$	0	0
$851$	8.82843	0.302635
$852$	0	0
$853$	19.9411	0.682771	0.341386	−	0.939923i	$-0.389104\pi$
$853$	19.9411	0.682771	0.341386	+	0.939923i	$0.389104\pi$
$854$	0	0
$855$	−0.828427	−0.0283316
$856$	0	0
$857$	−26.9706	−0.921297	−0.460648	−	0.887583i	$-0.652383\pi$
$857$	−26.9706	−0.921297	−0.460648	+	0.887583i	$0.652383\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−2.82843	−0.0963925
$862$	0	0
$863$	9.51472	0.323885	0.161942	−	0.986800i	$-0.448224\pi$
$863$	9.51472	0.323885	0.161942	+	0.986800i	$0.448224\pi$
$864$	0	0
$865$	18.8284	0.640186
$866$	0	0
$867$	−21.9706	−0.746159
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−91.8823	−3.11331
$872$	0	0
$873$	−12.1421	−0.410949
$874$	0	0
$875$	−8.00000	−0.270449
$876$	0	0
$877$	−30.9706	−1.04580	−0.522901	−	0.852394i	$-0.675150\pi$
$877$	−30.9706	−1.04580	−0.522901	+	0.852394i	$0.675150\pi$
$878$	0	0
$879$	9.75736	0.329108
$880$	0	0
$881$	−50.2426	−1.69272	−0.846359	−	0.532613i	$-0.821210\pi$
$881$	−50.2426	−1.69272	−0.846359	+	0.532613i	$0.821210\pi$
$882$	0	0
$883$	7.79899	0.262457	0.131228	−	0.991352i	$-0.458108\pi$
$883$	7.79899	0.262457	0.131228	+	0.991352i	$0.458108\pi$
$884$	0	0
$885$	−10.8284	−0.363994
$886$	0	0
$887$	−29.7990	−1.00055	−0.500276	−	0.865866i	$-0.666768\pi$
$887$	−29.7990	−1.00055	−0.500276	+	0.865866i	$0.666768\pi$
$888$	0	0
$889$	−0.686292	−0.0230175
$890$	0	0
$891$	5.65685	0.189512
$892$	0	0
$893$	1.57359	0.0526583
$894$	0	0
$895$	−49.4558	−1.65313
$896$	0	0
$897$	−5.65685	−0.188877
$898$	0	0
$899$	21.6569	0.722297
$900$	0	0
$901$	46.2843	1.54195
$902$	0	0
$903$	3.65685	0.121692
$904$	0	0
$905$	18.1421	0.603065
$906$	0	0
$907$	−3.07107	−0.101973	−0.0509866	−	0.998699i	$-0.516237\pi$
$907$	−3.07107	−0.101973	−0.0509866	+	0.998699i	$0.516237\pi$
$908$	0	0
$909$	17.3137	0.574259
$910$	0	0
$911$	21.9411	0.726942	0.363471	−	0.931606i	$-0.381592\pi$
$911$	21.9411	0.726942	0.363471	+	0.931606i	$0.381592\pi$
$912$	0	0
$913$	−9.37258	−0.310187
$914$	0	0
$915$	−30.1421	−0.996468
$916$	0	0
$917$	−2.34315	−0.0773775
$918$	0	0
$919$	24.0416	0.793060	0.396530	−	0.918022i	$-0.370214\pi$
$919$	24.0416	0.793060	0.396530	+	0.918022i	$0.370214\pi$
$920$	0	0
$921$	21.4558	0.706994
$922$	0	0
$923$	−64.0000	−2.10659
$924$	0	0
$925$	−58.7696	−1.93233
$926$	0	0
$927$	−12.7279	−0.418040
$928$	0	0
$929$	30.9706	1.01611	0.508056	−	0.861324i	$-0.330364\pi$
$929$	30.9706	1.01611	0.508056	+	0.861324i	$0.330364\pi$
$930$	0	0
$931$	1.21320	0.0397611
$932$	0	0
$933$	8.82843	0.289030
$934$	0	0
$935$	120.569	3.94301
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	4.34315	0.141733
$940$	0	0
$941$	18.0416	0.588140	0.294070	−	0.955784i	$-0.404990\pi$
$941$	18.0416	0.588140	0.294070	+	0.955784i	$0.404990\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	4.82843	0.157069
$946$	0	0
$947$	−9.79899	−0.318424	−0.159212	−	0.987244i	$-0.550895\pi$
$947$	−9.79899	−0.318424	−0.159212	+	0.987244i	$0.550895\pi$
$948$	0	0
$949$	−65.9411	−2.14054
$950$	0	0
$951$	8.82843	0.286281
$952$	0	0
$953$	−3.89949	−0.126317	−0.0631585	−	0.998004i	$-0.520117\pi$
$953$	−3.89949	−0.126317	−0.0631585	+	0.998004i	$0.520117\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	−17.9411	−0.579954
$958$	0	0
$959$	−1.51472	−0.0489128
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	5.65685	0.182290
$964$	0	0
$965$	38.6274	1.24346
$966$	0	0
$967$	5.45584	0.175448	0.0877241	−	0.996145i	$-0.472041\pi$
$967$	5.45584	0.175448	0.0877241	+	0.996145i	$0.472041\pi$
$968$	0	0
$969$	1.51472	0.0486598
$970$	0	0
$971$	−41.4558	−1.33038	−0.665191	−	0.746674i	$-0.731650\pi$
$971$	−41.4558	−1.33038	−0.665191	+	0.746674i	$0.731650\pi$
$972$	0	0
$973$	−2.34315	−0.0751178
$974$	0	0
$975$	37.6569	1.20598
$976$	0	0
$977$	43.2132	1.38251	0.691256	−	0.722610i	$-0.257058\pi$
$977$	43.2132	1.38251	0.691256	+	0.722610i	$0.257058\pi$
$978$	0	0
$979$	−6.05887	−0.193642
$980$	0	0
$981$	−7.65685	−0.244465
$982$	0	0
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	0	0
$985$	−30.1421	−0.960408
$986$	0	0
$987$	−9.17157	−0.291934
$988$	0	0
$989$	−2.58579	−0.0822232
$990$	0	0
$991$	12.7696	0.405638	0.202819	−	0.979216i	$-0.434990\pi$
$991$	12.7696	0.405638	0.202819	+	0.979216i	$0.434990\pi$
$992$	0	0
$993$	2.14214	0.0679786
$994$	0	0
$995$	−69.1127	−2.19102
$996$	0	0
$997$	−4.68629	−0.148416	−0.0742082	−	0.997243i	$-0.523643\pi$
$997$	−4.68629	−0.148416	−0.0742082	+	0.997243i	$0.523643\pi$
$998$	0	0
$999$	8.82843	0.279319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.a.l.1.2		2
3.2	odd	2		3312.2.a.s.1.1		2
4.3	odd	2		276.2.a.b.1.2	✓	2
8.3	odd	2		4416.2.a.bc.1.1		2
8.5	even	2		4416.2.a.bi.1.1		2
12.11	even	2		828.2.a.e.1.1		2
20.3	even	4		6900.2.f.l.6349.3		4
20.7	even	4		6900.2.f.l.6349.2		4
20.19	odd	2		6900.2.a.m.1.1		2
92.91	even	2		6348.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.a.b.1.2	✓	2	4.3	odd	2
828.2.a.e.1.1		2	12.11	even	2
1104.2.a.l.1.2		2	1.1	even	1	trivial
3312.2.a.s.1.1		2	3.2	odd	2
4416.2.a.bc.1.1		2	8.3	odd	2
4416.2.a.bi.1.1		2	8.5	even	2
6348.2.a.h.1.1		2	92.91	even	2
6900.2.a.m.1.1		2	20.19	odd	2
6900.2.f.l.6349.2		4	20.7	even	4
6900.2.f.l.6349.3		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.41421 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.41421 q^{5} -1.41421 q^{7} +1.00000 q^{9} +5.65685 q^{11} -5.65685 q^{13} -3.41421 q^{15} +6.24264 q^{17} -0.242641 q^{19} +1.41421 q^{21} -1.00000 q^{23} +6.65685 q^{25} -1.00000 q^{27} +3.17157 q^{29} +6.82843 q^{31} -5.65685 q^{33} -4.82843 q^{35} -8.82843 q^{37} +5.65685 q^{39} -2.00000 q^{41} +2.58579 q^{43} +3.41421 q^{45} -6.48528 q^{47} -5.00000 q^{49} -6.24264 q^{51} +7.41421 q^{53} +19.3137 q^{55} +0.242641 q^{57} +3.17157 q^{59} +8.82843 q^{61} -1.41421 q^{63} -19.3137 q^{65} +16.2426 q^{67} +1.00000 q^{69} +11.3137 q^{71} +11.6569 q^{73} -6.65685 q^{75} -8.00000 q^{77} -7.07107 q^{79} +1.00000 q^{81} -1.65685 q^{83} +21.3137 q^{85} -3.17157 q^{87} -1.07107 q^{89} +8.00000 q^{91} -6.82843 q^{93} -0.828427 q^{95} -12.1421 q^{97} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} - 4 q^{15} + 4 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} + 8 q^{31} - 4 q^{35} - 12 q^{37} - 4 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} - 10 q^{49} - 4 q^{51} + 12 q^{53} + 16 q^{55} - 8 q^{57} + 12 q^{59} + 12 q^{61} - 16 q^{65} + 24 q^{67} + 2 q^{69} + 12 q^{73} - 2 q^{75} - 16 q^{77} + 2 q^{81} + 8 q^{83} + 20 q^{85} - 12 q^{87} + 12 q^{89} + 16 q^{91} - 8 q^{93} + 4 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 - 4 * q^15 + 4 * q^17 + 8 * q^19 - 2 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^29 + 8 * q^31 - 4 * q^35 - 12 * q^37 - 4 * q^41 + 8 * q^43 + 4 * q^45 + 4 * q^47 - 10 * q^49 - 4 * q^51 + 12 * q^53 + 16 * q^55 - 8 * q^57 + 12 * q^59 + 12 * q^61 - 16 * q^65 + 24 * q^67 + 2 * q^69 + 12 * q^73 - 2 * q^75 - 16 * q^77 + 2 * q^81 + 8 * q^83 + 20 * q^85 - 12 * q^87 + 12 * q^89 + 16 * q^91 - 8 * q^93 + 4 * q^95 + 4 * q^97

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