LMFDB - Embedded newform 1296.3.g.j.1135.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(1135,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1296.1135");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.g (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:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.856615824.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 $ x^8 + 11x^6 + 36x^4 + 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.1
Root			$-2.06288i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.3.g.j.1135.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.23321 q^{5} -6.15562i q^{7} +O(q^{10})$	$q-9.23321 q^{5} -6.15562i q^{7} -4.27258i q^{11} -1.73847 q^{13} +12.3476 q^{17} -33.9338i q^{19} +3.86865i q^{23} +60.2521 q^{25} -35.6818 q^{29} -44.8326i q^{31} +56.8361i q^{35} -32.7130 q^{37} -43.7130 q^{41} +39.1835i q^{43} +46.0476i q^{47} +11.1083 q^{49} +46.3143 q^{53} +39.4496i q^{55} -26.8710i q^{59} -46.9089 q^{61} +16.0516 q^{65} +65.8160i q^{67} +96.7955i q^{71} -14.0622 q^{73} -26.3004 q^{77} +39.7164i q^{79} +94.3623i q^{83} -114.008 q^{85} -81.8478 q^{89} +10.7014i q^{91} +313.318i q^{95} +15.9806 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5}+O(q^{10}) $ 8 * q - 6 * q^5	$ 8 q - 6 q^{5} - 10 q^{13} + 6 q^{17} + 46 q^{25} - 138 q^{29} - 20 q^{37} - 108 q^{41} - 82 q^{49} - 252 q^{53} - 14 q^{61} - 186 q^{65} + 74 q^{73} - 414 q^{77} + 60 q^{85} - 168 q^{89} + 20 q^{97}+O(q^{100}) $ 8 * q - 6 * q^5 - 10 * q^13 + 6 * q^17 + 46 * q^25 - 138 * q^29 - 20 * q^37 - 108 * q^41 - 82 * q^49 - 252 * q^53 - 14 * q^61 - 186 * q^65 + 74 * q^73 - 414 * q^77 + 60 * q^85 - 168 * q^89 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−9.23321	−1.84664	−0.923321	−	0.384030i	$-0.874536\pi$
$5$	−9.23321	−1.84664	−0.923321	+	0.384030i	$0.874536\pi$
$6$	0	0
$7$	− 6.15562i	− 0.879375i	−0.898151	−	0.439687i	$-0.855089\pi$
$7$	− 6.15562i	− 0.879375i	0.898151	−	0.439687i	$-0.144911\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.27258i	− 0.388416i	−0.980960	−	0.194208i	$-0.937786\pi$
$11$	− 4.27258i	− 0.388416i	0.980960	−	0.194208i	$-0.0622137\pi$
$12$	0	0
$13$	−1.73847	−0.133728	−0.0668642	−	0.997762i	$-0.521299\pi$
$13$	−1.73847	−0.133728	−0.0668642	+	0.997762i	$0.521299\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.3476	0.726329	0.363164	−	0.931725i	$-0.381696\pi$
$17$	12.3476	0.726329	0.363164	+	0.931725i	$0.381696\pi$
$18$	0	0
$19$	− 33.9338i	− 1.78599i	−0.450065	−	0.892996i	$-0.648599\pi$
$19$	− 33.9338i	− 1.78599i	0.450065	−	0.892996i	$-0.351401\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.86865i	0.168202i	0.996457	+	0.0841012i	$0.0268019\pi$
$23$	3.86865i	0.168202i	−0.996457	+	0.0841012i	$0.973198\pi$
$24$	0	0
$25$	60.2521	2.41008
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−35.6818	−1.23041	−0.615204	−	0.788368i	$-0.710926\pi$
$29$	−35.6818	−1.23041	−0.615204	+	0.788368i	$0.710926\pi$
$30$	0	0
$31$	− 44.8326i	− 1.44621i	−0.690736	−	0.723107i	$-0.742713\pi$
$31$	− 44.8326i	− 1.44621i	0.690736	−	0.723107i	$-0.257287\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	56.8361i	1.62389i
$36$	0	0
$37$	−32.7130	−0.884134	−0.442067	−	0.896982i	$-0.645755\pi$
$37$	−32.7130	−0.884134	−0.442067	+	0.896982i	$0.645755\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−43.7130	−1.06617	−0.533085	−	0.846062i	$-0.678967\pi$
$41$	−43.7130	−1.06617	−0.533085	+	0.846062i	$0.678967\pi$
$42$	0	0
$43$	39.1835i	0.911244i	0.890173	+	0.455622i	$0.150583\pi$
$43$	39.1835i	0.911244i	−0.890173	+	0.455622i	$0.849417\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	46.0476i	0.979736i	0.871797	+	0.489868i	$0.162955\pi$
$47$	46.0476i	0.979736i	−0.871797	+	0.489868i	$0.837045\pi$
$48$	0	0
$49$	11.1083	0.226700
$50$	0	0
$51$	0	0
$52$	0	0
$53$	46.3143	0.873854	0.436927	−	0.899497i	$-0.356067\pi$
$53$	46.3143	0.873854	0.436927	+	0.899497i	$0.356067\pi$
$54$	0	0
$55$	39.4496i	0.717265i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 26.8710i	− 0.455441i	−0.973727	−	0.227720i	$-0.926873\pi$
$59$	− 26.8710i	− 0.455441i	0.973727	−	0.227720i	$-0.0731272\pi$
$60$	0	0
$61$	−46.9089	−0.768999	−0.384500	−	0.923125i	$-0.625626\pi$
$61$	−46.9089	−0.768999	−0.384500	+	0.923125i	$0.625626\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.0516	0.246948
$66$	0	0
$67$	65.8160i	0.982329i	0.871067	+	0.491164i	$0.163429\pi$
$67$	65.8160i	0.982329i	−0.871067	+	0.491164i	$0.836571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	96.7955i	1.36332i	0.731671	+	0.681658i	$0.238741\pi$
$71$	96.7955i	1.36332i	−0.731671	+	0.681658i	$0.761259\pi$
$72$	0	0
$73$	−14.0622	−0.192633	−0.0963163	−	0.995351i	$-0.530706\pi$
$73$	−14.0622	−0.192633	−0.0963163	+	0.995351i	$0.530706\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−26.3004	−0.341563
$78$	0	0
$79$	39.7164i	0.502739i	0.967891	+	0.251369i	$0.0808809\pi$
$79$	39.7164i	0.502739i	−0.967891	+	0.251369i	$0.919119\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	94.3623i	1.13690i	0.822719	+	0.568448i	$0.192456\pi$
$83$	94.3623i	1.13690i	−0.822719	+	0.568448i	$0.807544\pi$
$84$	0	0
$85$	−114.008	−1.34127
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−81.8478	−0.919639	−0.459819	−	0.888012i	$-0.652086\pi$
$89$	−81.8478	−0.919639	−0.459819	+	0.888012i	$0.652086\pi$
$90$	0	0
$91$	10.7014i	0.117597i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	313.318i	3.29809i
$96$	0	0
$97$	15.9806	0.164748	0.0823741	−	0.996601i	$-0.473750\pi$
$97$	15.9806	0.164748	0.0823741	+	0.996601i	$0.473750\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−101.269	−1.00266	−0.501331	−	0.865256i	$-0.667156\pi$
$101$	−101.269	−1.00266	−0.501331	+	0.865256i	$0.667156\pi$
$102$	0	0
$103$	102.086i	0.991127i	0.868572	+	0.495564i	$0.165038\pi$
$103$	102.086i	0.991127i	−0.868572	+	0.495564i	$0.834962\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 73.1463i	− 0.683610i	−0.939771	−	0.341805i	$-0.888962\pi$
$107$	− 73.1463i	− 0.683610i	0.939771	−	0.341805i	$-0.111038\pi$
$108$	0	0
$109$	−33.9344	−0.311325	−0.155663	−	0.987810i	$-0.549751\pi$
$109$	−33.9344	−0.311325	−0.155663	+	0.987810i	$0.549751\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	27.8585	0.246535	0.123268	−	0.992373i	$-0.460663\pi$
$113$	27.8585	0.246535	0.123268	+	0.992373i	$0.460663\pi$
$114$	0	0
$115$	− 35.7201i	− 0.310609i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 76.0071i	− 0.638715i
$120$	0	0
$121$	102.745	0.849133
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−325.490	−2.60392
$126$	0	0
$127$	117.905i	0.928387i	0.885734	+	0.464193i	$0.153656\pi$
$127$	117.905i	0.928387i	−0.885734	+	0.464193i	$0.846344\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 84.8951i	− 0.648055i	−0.946048	−	0.324027i	$-0.894963\pi$
$131$	− 84.8951i	− 0.648055i	0.946048	−	0.324027i	$-0.105037\pi$
$132$	0	0
$133$	−208.884	−1.57056
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.95951	−0.0580986	−0.0290493	−	0.999578i	$-0.509248\pi$
$137$	−7.95951	−0.0580986	−0.0290493	+	0.999578i	$0.509248\pi$
$138$	0	0
$139$	− 20.6968i	− 0.148898i	−0.997225	−	0.0744488i	$-0.976280\pi$
$139$	− 20.6968i	− 0.148898i	0.997225	−	0.0744488i	$-0.0237197\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.42775i	0.0519423i
$144$	0	0
$145$	329.458	2.27212
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−131.224	−0.880700	−0.440350	−	0.897826i	$-0.645146\pi$
$149$	−131.224	−0.880700	−0.440350	+	0.897826i	$0.645146\pi$
$150$	0	0
$151$	− 236.655i	− 1.56725i	−0.621232	−	0.783627i	$-0.713368\pi$
$151$	− 236.655i	− 1.56725i	0.621232	−	0.783627i	$-0.286632\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	413.949i	2.67064i
$156$	0	0
$157$	149.778	0.954002	0.477001	−	0.878903i	$-0.341724\pi$
$157$	149.778	0.954002	0.477001	+	0.878903i	$0.341724\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.8140	0.147913
$162$	0	0
$163$	− 152.365i	− 0.934756i	−0.884057	−	0.467378i	$-0.845199\pi$
$163$	− 152.365i	− 0.934756i	0.884057	−	0.467378i	$-0.154801\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.8989i	0.149095i	0.997217	+	0.0745476i	$0.0237513\pi$
$167$	24.8989i	0.149095i	−0.997217	+	0.0745476i	$0.976249\pi$
$168$	0	0
$169$	−165.978	−0.982117
$170$	0	0
$171$	0	0
$172$	0	0
$173$	108.090	0.624800	0.312400	−	0.949951i	$-0.398867\pi$
$173$	108.090	0.624800	0.312400	+	0.949951i	$0.398867\pi$
$174$	0	0
$175$	− 370.889i	− 2.11937i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	313.318i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$179$	313.318i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$180$	0	0
$181$	20.5886	0.113749	0.0568746	−	0.998381i	$-0.481886\pi$
$181$	20.5886	0.113749	0.0568746	+	0.998381i	$0.481886\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	302.046	1.63268
$186$	0	0
$187$	− 52.7560i	− 0.282118i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 27.3224i	− 0.143049i	−0.997439	−	0.0715247i	$-0.977214\pi$
$191$	− 27.3224i	− 0.143049i	0.997439	−	0.0715247i	$-0.0227865\pi$
$192$	0	0
$193$	131.445	0.681064	0.340532	−	0.940233i	$-0.389393\pi$
$193$	131.445	0.681064	0.340532	+	0.940233i	$0.389393\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−126.466	−0.641961	−0.320981	−	0.947086i	$-0.604012\pi$
$197$	−126.466	−0.641961	−0.320981	+	0.947086i	$0.604012\pi$
$198$	0	0
$199$	76.0070i	0.381945i	0.981595	+	0.190972i	$0.0611641\pi$
$199$	76.0070i	0.381945i	−0.981595	+	0.190972i	$0.938836\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	219.644i	1.08199i
$204$	0	0
$205$	403.611	1.96883
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−144.985	−0.693708
$210$	0	0
$211$	106.066i	0.502685i	0.967898	+	0.251342i	$0.0808720\pi$
$211$	106.066i	0.502685i	−0.967898	+	0.251342i	$0.919128\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 361.789i	− 1.68274i
$216$	0	0
$217$	−275.973	−1.27176
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−21.4659	−0.0971308
$222$	0	0
$223$	63.7048i	0.285672i	0.989746	+	0.142836i	$0.0456221\pi$
$223$	63.7048i	0.285672i	−0.989746	+	0.142836i	$0.954378\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 351.771i	− 1.54965i	−0.632174	−	0.774827i	$-0.717837\pi$
$227$	− 351.771i	− 1.54965i	0.632174	−	0.774827i	$-0.282163\pi$
$228$	0	0
$229$	−204.465	−0.892859	−0.446429	−	0.894819i	$-0.647305\pi$
$229$	−204.465	−0.892859	−0.446429	+	0.894819i	$0.647305\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	236.626	1.01556	0.507782	−	0.861486i	$-0.330466\pi$
$233$	236.626	1.01556	0.507782	+	0.861486i	$0.330466\pi$
$234$	0	0
$235$	− 425.167i	− 1.80922i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 18.6575i	− 0.0780647i	−0.999238	−	0.0390324i	$-0.987572\pi$
$239$	− 18.6575i	− 0.0780647i	0.999238	−	0.0390324i	$-0.0124275\pi$
$240$	0	0
$241$	74.4580	0.308955	0.154477	−	0.987996i	$-0.450631\pi$
$241$	74.4580	0.308955	0.154477	+	0.987996i	$0.450631\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−102.565	−0.418633
$246$	0	0
$247$	58.9930i	0.238838i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	206.637i	0.823257i	0.911352	+	0.411628i	$0.135040\pi$
$251$	206.637i	0.823257i	−0.911352	+	0.411628i	$0.864960\pi$
$252$	0	0
$253$	16.5291	0.0653325
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−297.355	−1.15703	−0.578513	−	0.815673i	$-0.696367\pi$
$257$	−297.355	−1.15703	−0.578513	+	0.815673i	$0.696367\pi$
$258$	0	0
$259$	201.369i	0.777485i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	142.871i	0.543237i	0.962405	+	0.271619i	$0.0875589\pi$
$263$	142.871i	0.543237i	−0.962405	+	0.271619i	$0.912441\pi$
$264$	0	0
$265$	−427.629	−1.61369
$266$	0	0
$267$	0	0
$268$	0	0
$269$	370.517	1.37738	0.688692	−	0.725054i	$-0.258185\pi$
$269$	370.517	1.37738	0.688692	+	0.725054i	$0.258185\pi$
$270$	0	0
$271$	368.022i	1.35801i	0.734132	+	0.679007i	$0.237589\pi$
$271$	368.022i	1.35801i	−0.734132	+	0.679007i	$0.762411\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 257.432i	− 0.936115i
$276$	0	0
$277$	30.6468	0.110638	0.0553191	−	0.998469i	$-0.482382\pi$
$277$	30.6468	0.110638	0.0553191	+	0.998469i	$0.482382\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	462.083	1.64442	0.822212	−	0.569182i	$-0.192740\pi$
$281$	462.083	1.64442	0.822212	+	0.569182i	$0.192740\pi$
$282$	0	0
$283$	488.084i	1.72468i	0.506332	+	0.862339i	$0.331001\pi$
$283$	488.084i	1.72468i	−0.506332	+	0.862339i	$0.668999\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	269.081i	0.937563i
$288$	0	0
$289$	−136.537	−0.472447
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−430.059	−1.46778	−0.733890	−	0.679269i	$-0.762297\pi$
$293$	−430.059	−1.46778	−0.733890	+	0.679269i	$0.762297\pi$
$294$	0	0
$295$	248.105i	0.841035i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.72554i	− 0.0224934i
$300$	0	0
$301$	241.199	0.801325
$302$	0	0
$303$	0	0
$304$	0	0
$305$	433.120	1.42007
$306$	0	0
$307$	276.184i	0.899621i	0.893124	+	0.449810i	$0.148508\pi$
$307$	276.184i	0.899621i	−0.893124	+	0.449810i	$0.851492\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 199.761i	− 0.642319i	−0.947025	−	0.321159i	$-0.895927\pi$
$311$	− 199.761i	− 0.642319i	0.947025	−	0.321159i	$-0.104073\pi$
$312$	0	0
$313$	−118.677	−0.379160	−0.189580	−	0.981865i	$-0.560713\pi$
$313$	−118.677	−0.379160	−0.189580	+	0.981865i	$0.560713\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−386.522	−1.21931	−0.609657	−	0.792666i	$-0.708693\pi$
$317$	−386.522	−1.21931	−0.609657	+	0.792666i	$0.708693\pi$
$318$	0	0
$319$	152.453i	0.477910i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 419.001i	− 1.29722i
$324$	0	0
$325$	−104.746	−0.322297
$326$	0	0
$327$	0	0
$328$	0	0
$329$	283.452	0.861555
$330$	0	0
$331$	326.472i	0.986321i	0.869938	+	0.493161i	$0.164159\pi$
$331$	326.472i	0.986321i	−0.869938	+	0.493161i	$0.835841\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 607.693i	− 1.81401i
$336$	0	0
$337$	−114.981	−0.341190	−0.170595	−	0.985341i	$-0.554569\pi$
$337$	−114.981	−0.341190	−0.170595	+	0.985341i	$0.554569\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−191.551	−0.561733
$342$	0	0
$343$	− 370.004i	− 1.07873i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 580.751i	− 1.67363i	−0.547483	−	0.836817i	$-0.684414\pi$
$347$	− 580.751i	− 1.67363i	0.547483	−	0.836817i	$-0.315586\pi$
$348$	0	0
$349$	350.927	1.00552	0.502761	−	0.864426i	$-0.332318\pi$
$349$	350.927	1.00552	0.502761	+	0.864426i	$0.332318\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	135.574	0.384063	0.192031	−	0.981389i	$-0.438492\pi$
$353$	135.574	0.384063	0.192031	+	0.981389i	$0.438492\pi$
$354$	0	0
$355$	− 893.732i	− 2.51756i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 108.852i	− 0.303210i	−0.988441	−	0.151605i	$-0.951556\pi$
$359$	− 108.852i	− 0.303210i	0.988441	−	0.151605i	$-0.0484441\pi$
$360$	0	0
$361$	−790.506	−2.18977
$362$	0	0
$363$	0	0
$364$	0	0
$365$	129.839	0.355723
$366$	0	0
$367$	16.9301i	0.0461310i	0.999734	+	0.0230655i	$0.00734262\pi$
$367$	16.9301i	0.0461310i	−0.999734	+	0.0230655i	$0.992657\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 285.093i	− 0.768445i
$372$	0	0
$373$	37.0600	0.0993566	0.0496783	−	0.998765i	$-0.484180\pi$
$373$	37.0600	0.0993566	0.0496783	+	0.998765i	$0.484180\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	62.0318	0.164540
$378$	0	0
$379$	531.193i	1.40156i	0.713375	+	0.700782i	$0.247166\pi$
$379$	531.193i	1.40156i	−0.713375	+	0.700782i	$0.752834\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 374.128i	− 0.976834i	−0.872610	−	0.488417i	$-0.837574\pi$
$383$	− 374.128i	− 0.976834i	0.872610	−	0.488417i	$-0.162426\pi$
$384$	0	0
$385$	242.837	0.630745
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−543.185	−1.39636	−0.698182	−	0.715921i	$-0.746007\pi$
$389$	−543.185	−1.39636	−0.698182	+	0.715921i	$0.746007\pi$
$390$	0	0
$391$	47.7685i	0.122170i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 366.709i	− 0.928378i
$396$	0	0
$397$	606.097	1.52669	0.763346	−	0.645990i	$-0.223555\pi$
$397$	606.097	1.52669	0.763346	+	0.645990i	$0.223555\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	587.059	1.46399	0.731994	−	0.681311i	$-0.238590\pi$
$401$	587.059	1.46399	0.731994	+	0.681311i	$0.238590\pi$
$402$	0	0
$403$	77.9402i	0.193400i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	139.769i	0.343412i
$408$	0	0
$409$	−259.763	−0.635118	−0.317559	−	0.948239i	$-0.602863\pi$
$409$	−259.763	−0.635118	−0.317559	+	0.948239i	$0.602863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−165.408	−0.400503
$414$	0	0
$415$	− 871.267i	− 2.09944i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	344.097i	0.821235i	0.911808	+	0.410617i	$0.134687\pi$
$419$	344.097i	0.821235i	−0.911808	+	0.410617i	$0.865313\pi$
$420$	0	0
$421$	−306.527	−0.728092	−0.364046	−	0.931381i	$-0.618605\pi$
$421$	−306.527	−0.728092	−0.364046	+	0.931381i	$0.618605\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	743.968	1.75051
$426$	0	0
$427$	288.754i	0.676238i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	208.029i	0.482667i	0.970442	+	0.241333i	$0.0775847\pi$
$431$	208.029i	0.482667i	−0.970442	+	0.241333i	$0.922415\pi$
$432$	0	0
$433$	353.874	0.817260	0.408630	−	0.912700i	$-0.366007\pi$
$433$	353.874	0.817260	0.408630	+	0.912700i	$0.366007\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	131.278	0.300408
$438$	0	0
$439$	− 231.351i	− 0.526995i	−0.964660	−	0.263497i	$-0.915124\pi$
$439$	− 231.351i	− 0.526995i	0.964660	−	0.263497i	$-0.0848760\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	255.690i	0.577178i	0.957453	+	0.288589i	$0.0931862\pi$
$443$	255.690i	0.577178i	−0.957453	+	0.288589i	$0.906814\pi$
$444$	0	0
$445$	755.718	1.69824
$446$	0	0
$447$	0	0
$448$	0	0
$449$	323.060	0.719509	0.359755	−	0.933047i	$-0.382860\pi$
$449$	323.060	0.719509	0.359755	+	0.933047i	$0.382860\pi$
$450$	0	0
$451$	186.767i	0.414118i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 98.8079i	− 0.217160i
$456$	0	0
$457$	−554.527	−1.21341	−0.606704	−	0.794928i	$-0.707508\pi$
$457$	−554.527	−1.21341	−0.606704	+	0.794928i	$0.707508\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	737.506	1.59980	0.799898	−	0.600136i	$-0.204887\pi$
$461$	737.506	1.59980	0.799898	+	0.600136i	$0.204887\pi$
$462$	0	0
$463$	134.477i	0.290448i	0.989399	+	0.145224i	$0.0463902\pi$
$463$	134.477i	0.290448i	−0.989399	+	0.145224i	$0.953610\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 595.000i	− 1.27409i	−0.770827	−	0.637045i	$-0.780156\pi$
$467$	− 595.000i	− 1.27409i	0.770827	−	0.637045i	$-0.219844\pi$
$468$	0	0
$469$	405.139	0.863835
$470$	0	0
$471$	0	0
$472$	0	0
$473$	167.415	0.353942
$474$	0	0
$475$	− 2044.58i	− 4.30439i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 449.091i	− 0.937559i	−0.883315	−	0.468779i	$-0.844694\pi$
$479$	− 449.091i	− 0.937559i	0.883315	−	0.468779i	$-0.155306\pi$
$480$	0	0
$481$	56.8705	0.118234
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−147.552	−0.304231
$486$	0	0
$487$	120.044i	0.246497i	0.992376	+	0.123249i	$0.0393312\pi$
$487$	120.044i	0.246497i	−0.992376	+	0.123249i	$0.960669\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	429.897i	0.875554i	0.899084	+	0.437777i	$0.144234\pi$
$491$	429.897i	0.875554i	−0.899084	+	0.437777i	$0.855766\pi$
$492$	0	0
$493$	−440.584	−0.893680
$494$	0	0
$495$	0	0
$496$	0	0
$497$	595.837	1.19887
$498$	0	0
$499$	− 737.989i	− 1.47893i	−0.673192	−	0.739467i	$-0.735077\pi$
$499$	− 737.989i	− 1.47893i	0.673192	−	0.739467i	$-0.264923\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 951.782i	− 1.89221i	−0.323859	−	0.946105i	$-0.604980\pi$
$503$	− 951.782i	− 1.89221i	0.323859	−	0.946105i	$-0.395020\pi$
$504$	0	0
$505$	935.036	1.85156
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−75.7797	−0.148880	−0.0744398	−	0.997226i	$-0.523717\pi$
$509$	−75.7797	−0.148880	−0.0744398	+	0.997226i	$0.523717\pi$
$510$	0	0
$511$	86.5615i	0.169396i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 942.582i	− 1.83026i
$516$	0	0
$517$	196.742	0.380545
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.6152	0.0472461	0.0236230	−	0.999721i	$-0.492480\pi$
$521$	24.6152	0.0472461	0.0236230	+	0.999721i	$0.492480\pi$
$522$	0	0
$523$	− 165.798i	− 0.317013i	−0.987358	−	0.158506i	$-0.949332\pi$
$523$	− 165.798i	− 0.317013i	0.987358	−	0.158506i	$-0.0506678\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 553.575i	− 1.05043i
$528$	0	0
$529$	514.034	0.971708
$530$	0	0
$531$	0	0
$532$	0	0
$533$	75.9937	0.142577
$534$	0	0
$535$	675.375i	1.26238i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 47.4611i	− 0.0880539i
$540$	0	0
$541$	−184.323	−0.340708	−0.170354	−	0.985383i	$-0.554491\pi$
$541$	−184.323	−0.340708	−0.170354	+	0.985383i	$0.554491\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	313.324	0.574906
$546$	0	0
$547$	927.633i	1.69586i	0.530111	+	0.847928i	$0.322150\pi$
$547$	927.633i	1.69586i	−0.530111	+	0.847928i	$0.677850\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1210.82i	2.19750i
$552$	0	0
$553$	244.479	0.442096
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−492.087	−0.883459	−0.441730	−	0.897148i	$-0.645635\pi$
$557$	−492.087	−0.883459	−0.441730	+	0.897148i	$0.645635\pi$
$558$	0	0
$559$	− 68.1193i	− 0.121859i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	723.366i	1.28484i	0.766352	+	0.642421i	$0.222070\pi$
$563$	723.366i	1.28484i	−0.766352	+	0.642421i	$0.777930\pi$
$564$	0	0
$565$	−257.223	−0.455262
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−871.632	−1.53187	−0.765933	−	0.642920i	$-0.777723\pi$
$569$	−871.632	−1.53187	−0.765933	+	0.642920i	$0.777723\pi$
$570$	0	0
$571$	243.236i	0.425983i	0.977054	+	0.212991i	$0.0683206\pi$
$571$	243.236i	0.425983i	−0.977054	+	0.212991i	$0.931679\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	233.094i	0.405382i
$576$	0	0
$577$	−201.625	−0.349436	−0.174718	−	0.984619i	$-0.555901\pi$
$577$	−201.625	−0.349436	−0.174718	+	0.984619i	$0.555901\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	580.859	0.999757
$582$	0	0
$583$	− 197.881i	− 0.339419i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 795.569i	− 1.35531i	−0.735379	−	0.677656i	$-0.762996\pi$
$587$	− 795.569i	− 1.35531i	0.735379	−	0.677656i	$-0.237004\pi$
$588$	0	0
$589$	−1521.34	−2.58293
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1078.05	−1.81796	−0.908980	−	0.416839i	$-0.863138\pi$
$593$	−1078.05	−1.81796	−0.908980	+	0.416839i	$0.863138\pi$
$594$	0	0
$595$	701.789i	1.17948i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 242.140i	− 0.404240i	−0.979361	−	0.202120i	$-0.935217\pi$
$599$	− 242.140i	− 0.404240i	0.979361	−	0.202120i	$-0.0647832\pi$
$600$	0	0
$601$	−270.811	−0.450601	−0.225300	−	0.974289i	$-0.572336\pi$
$601$	−270.811	−0.450601	−0.225300	+	0.974289i	$0.572336\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−948.666	−1.56804
$606$	0	0
$607$	− 387.583i	− 0.638523i	−0.947667	−	0.319261i	$-0.896565\pi$
$607$	− 387.583i	− 0.638523i	0.947667	−	0.319261i	$-0.103435\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 80.0524i	− 0.131019i
$612$	0	0
$613$	−1120.09	−1.82722	−0.913610	−	0.406591i	$-0.866718\pi$
$613$	−1120.09	−1.82722	−0.913610	+	0.406591i	$0.866718\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	532.578	0.863174	0.431587	−	0.902071i	$-0.357954\pi$
$617$	532.578	0.863174	0.431587	+	0.902071i	$0.357954\pi$
$618$	0	0
$619$	879.667i	1.42111i	0.703642	+	0.710555i	$0.251556\pi$
$619$	879.667i	1.42111i	−0.703642	+	0.710555i	$0.748444\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	503.825i	0.808707i
$624$	0	0
$625$	1499.01	2.39842
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−403.926	−0.642172
$630$	0	0
$631$	310.499i	0.492075i	0.969260	+	0.246037i	$0.0791286\pi$
$631$	310.499i	0.492075i	−0.969260	+	0.246037i	$0.920871\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1088.64i	− 1.71440i
$636$	0	0
$637$	−19.3114	−0.0303162
$638$	0	0
$639$	0	0
$640$	0	0
$641$	551.220	0.859938	0.429969	−	0.902844i	$-0.358524\pi$
$641$	551.220	0.859938	0.429969	+	0.902844i	$0.358524\pi$
$642$	0	0
$643$	406.506i	0.632202i	0.948726	+	0.316101i	$0.102374\pi$
$643$	406.506i	0.632202i	−0.948726	+	0.316101i	$0.897626\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	652.891i	1.00910i	0.863381	+	0.504552i	$0.168342\pi$
$647$	652.891i	1.00910i	−0.863381	+	0.504552i	$0.831658\pi$
$648$	0	0
$649$	−114.808	−0.176900
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1027.53	−1.57356	−0.786779	−	0.617234i	$-0.788253\pi$
$653$	−1027.53	−1.57356	−0.786779	+	0.617234i	$0.788253\pi$
$654$	0	0
$655$	783.854i	1.19672i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23.7614i	0.0360568i	0.999837	+	0.0180284i	$0.00573893\pi$
$659$	23.7614i	0.0360568i	−0.999837	+	0.0180284i	$0.994261\pi$
$660$	0	0
$661$	−656.327	−0.992931	−0.496465	−	0.868057i	$-0.665369\pi$
$661$	−656.327	−0.992931	−0.496465	+	0.868057i	$0.665369\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1928.67	2.90025
$666$	0	0
$667$	− 138.041i	− 0.206957i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	200.422i	0.298692i
$672$	0	0
$673$	−810.337	−1.20407	−0.602034	−	0.798471i	$-0.705643\pi$
$673$	−810.337	−1.20407	−0.602034	+	0.798471i	$0.705643\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	399.160	0.589602	0.294801	−	0.955559i	$-0.404747\pi$
$677$	399.160	0.589602	0.294801	+	0.955559i	$0.404747\pi$
$678$	0	0
$679$	− 98.3704i	− 0.144875i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 203.612i	− 0.298114i	−0.988829	−	0.149057i	$-0.952376\pi$
$683$	− 203.612i	− 0.298114i	0.988829	−	0.149057i	$-0.0476238\pi$
$684$	0	0
$685$	73.4918	0.107287
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−80.5159	−0.116859
$690$	0	0
$691$	− 295.194i	− 0.427199i	−0.976921	−	0.213599i	$-0.931481\pi$
$691$	− 295.194i	− 0.427199i	0.976921	−	0.213599i	$-0.0685187\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	191.097i	0.274960i
$696$	0	0
$697$	−539.750	−0.774390
$698$	0	0
$699$	0	0
$700$	0	0
$701$	283.069	0.403808	0.201904	−	0.979405i	$-0.435287\pi$
$701$	283.069	0.403808	0.201904	+	0.979405i	$0.435287\pi$
$702$	0	0
$703$	1110.08i	1.57906i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	623.373i	0.881715i
$708$	0	0
$709$	418.798	0.590688	0.295344	−	0.955391i	$-0.404566\pi$
$709$	418.798	0.590688	0.295344	+	0.955391i	$0.404566\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	173.442	0.243257
$714$	0	0
$715$	− 68.5819i	− 0.0959188i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	454.879i	0.632655i	0.948650	+	0.316328i	$0.102450\pi$
$719$	454.879i	0.632655i	−0.948650	+	0.316328i	$0.897550\pi$
$720$	0	0
$721$	628.404	0.871572
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2149.90	−2.96538
$726$	0	0
$727$	− 239.456i	− 0.329375i	−0.986346	−	0.164688i	$-0.947338\pi$
$727$	− 239.456i	− 0.329375i	0.986346	−	0.164688i	$-0.0526616\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	483.822i	0.661863i
$732$	0	0
$733$	421.946	0.575643	0.287822	−	0.957684i	$-0.407069\pi$
$733$	421.946	0.575643	0.287822	+	0.957684i	$0.407069\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	281.204	0.381552
$738$	0	0
$739$	150.203i	0.203251i	0.994823	+	0.101626i	$0.0324044\pi$
$739$	150.203i	0.203251i	−0.994823	+	0.101626i	$0.967596\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 839.185i	− 1.12946i	−0.825277	−	0.564728i	$-0.808981\pi$
$743$	− 839.185i	− 1.12946i	0.825277	−	0.564728i	$-0.191019\pi$
$744$	0	0
$745$	1211.62	1.62634
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−450.261	−0.601150
$750$	0	0
$751$	732.644i	0.975557i	0.872967	+	0.487779i	$0.162193\pi$
$751$	732.644i	0.975557i	−0.872967	+	0.487779i	$0.837807\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2185.09i	2.89415i
$756$	0	0
$757$	−1455.19	−1.92231	−0.961153	−	0.276015i	$-0.910986\pi$
$757$	−1455.19	−1.92231	−0.961153	+	0.276015i	$0.910986\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−730.459	−0.959867	−0.479934	−	0.877305i	$-0.659339\pi$
$761$	−730.459	−0.959867	−0.479934	+	0.877305i	$0.659339\pi$
$762$	0	0
$763$	208.888i	0.273772i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46.7144i	0.0609054i
$768$	0	0
$769$	−861.492	−1.12028	−0.560138	−	0.828399i	$-0.689252\pi$
$769$	−861.492	−1.12028	−0.560138	+	0.828399i	$0.689252\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−981.517	−1.26975	−0.634875	−	0.772615i	$-0.718948\pi$
$773$	−981.517	−1.26975	−0.634875	+	0.772615i	$0.718948\pi$
$774$	0	0
$775$	− 2701.26i	− 3.48550i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1483.35i	1.90417i
$780$	0	0
$781$	413.566	0.529534
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1382.93	−1.76170
$786$	0	0
$787$	− 709.323i	− 0.901300i	−0.892701	−	0.450650i	$-0.851192\pi$
$787$	− 709.323i	− 0.901300i	0.892701	−	0.450650i	$-0.148808\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 171.486i	− 0.216797i
$792$	0	0
$793$	81.5498	0.102837
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1183.71	1.48521	0.742605	−	0.669730i	$-0.233590\pi$
$797$	1183.71	1.48521	0.742605	+	0.669730i	$0.233590\pi$
$798$	0	0
$799$	568.577i	0.711610i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	60.0817i	0.0748216i
$804$	0	0
$805$	−219.879	−0.273142
$806$	0	0
$807$	0	0
$808$	0	0
$809$	522.491	0.645847	0.322924	−	0.946425i	$-0.395334\pi$
$809$	522.491	0.645847	0.322924	+	0.946425i	$0.395334\pi$
$810$	0	0
$811$	− 115.368i	− 0.142254i	−0.997467	−	0.0711271i	$-0.977340\pi$
$811$	− 115.368i	− 0.142254i	0.997467	−	0.0711271i	$-0.0226596\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1406.82i	1.72616i
$816$	0	0
$817$	1329.65	1.62747
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−835.946	−1.01820	−0.509102	−	0.860706i	$-0.670023\pi$
$821$	−835.946	−1.01820	−0.509102	+	0.860706i	$0.670023\pi$
$822$	0	0
$823$	727.340i	0.883766i	0.897073	+	0.441883i	$0.145689\pi$
$823$	727.340i	0.883766i	−0.897073	+	0.441883i	$0.854311\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1271.70i	− 1.53773i	−0.639413	−	0.768864i	$-0.720822\pi$
$827$	− 1271.70i	− 1.53773i	0.639413	−	0.768864i	$-0.279178\pi$
$828$	0	0
$829$	577.896	0.697100	0.348550	−	0.937290i	$-0.386674\pi$
$829$	577.896	0.697100	0.348550	+	0.937290i	$0.386674\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	137.161	0.164659
$834$	0	0
$835$	− 229.897i	− 0.275325i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 716.237i	− 0.853679i	−0.904327	−	0.426839i	$-0.859627\pi$
$839$	− 716.237i	− 0.853679i	0.904327	−	0.426839i	$-0.140373\pi$
$840$	0	0
$841$	432.192	0.513903
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1532.51	1.81362
$846$	0	0
$847$	− 632.460i	− 0.746706i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 126.555i	− 0.148713i
$852$	0	0
$853$	1045.28	1.22541	0.612705	−	0.790311i	$-0.290081\pi$
$853$	1045.28	1.22541	0.612705	+	0.790311i	$0.290081\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	822.736	0.960018	0.480009	−	0.877263i	$-0.340633\pi$
$857$	822.736	0.960018	0.480009	+	0.877263i	$0.340633\pi$
$858$	0	0
$859$	− 62.0794i	− 0.0722694i	−0.999347	−	0.0361347i	$-0.988495\pi$
$859$	− 62.0794i	− 0.0722694i	0.999347	−	0.0361347i	$-0.0115045\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1250.20i	1.44867i	0.689448	+	0.724335i	$0.257853\pi$
$863$	1250.20i	1.44867i	−0.689448	+	0.724335i	$0.742147\pi$
$864$	0	0
$865$	−998.022	−1.15378
$866$	0	0
$867$	0	0
$868$	0	0
$869$	169.691	0.195272
$870$	0	0
$871$	− 114.419i	− 0.131365i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2003.59i	2.28982i
$876$	0	0
$877$	210.604	0.240141	0.120071	−	0.992765i	$-0.461688\pi$
$877$	210.604	0.240141	0.120071	+	0.992765i	$0.461688\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−366.361	−0.415846	−0.207923	−	0.978145i	$-0.566670\pi$
$881$	−366.361	−0.415846	−0.207923	+	0.978145i	$0.566670\pi$
$882$	0	0
$883$	− 266.329i	− 0.301619i	−0.988563	−	0.150809i	$-0.951812\pi$
$883$	− 266.329i	− 0.301619i	0.988563	−	0.150809i	$-0.0481879\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	202.410i	0.228196i	0.993470	+	0.114098i	$0.0363977\pi$
$887$	202.410i	0.228196i	−0.993470	+	0.114098i	$0.963602\pi$
$888$	0	0
$889$	725.780	0.816400
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1562.57	1.74980
$894$	0	0
$895$	− 2892.93i	− 3.23233i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1599.71i	1.77943i
$900$	0	0
$901$	571.869	0.634705
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−190.099	−0.210054
$906$	0	0
$907$	− 1211.15i	− 1.33534i	−0.744458	−	0.667669i	$-0.767292\pi$
$907$	− 1211.15i	− 1.33534i	0.744458	−	0.667669i	$-0.232708\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 739.923i	− 0.812210i	−0.913826	−	0.406105i	$-0.866887\pi$
$911$	− 739.923i	− 0.812210i	0.913826	−	0.406105i	$-0.133113\pi$
$912$	0	0
$913$	403.171	0.441589
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−522.583	−0.569883
$918$	0	0
$919$	1080.71i	1.17596i	0.808874	+	0.587982i	$0.200078\pi$
$919$	1080.71i	1.17596i	−0.808874	+	0.587982i	$0.799922\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 168.276i	− 0.182314i
$924$	0	0
$925$	−1971.02	−2.13084
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1189.06	−1.27994	−0.639968	−	0.768402i	$-0.721052\pi$
$929$	−1189.06	−1.27994	−0.639968	+	0.768402i	$0.721052\pi$
$930$	0	0
$931$	− 376.947i	− 0.404884i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	487.107i	0.520970i
$936$	0	0
$937$	1165.30	1.24364	0.621822	−	0.783158i	$-0.286393\pi$
$937$	1165.30	1.24364	0.621822	+	0.783158i	$0.286393\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−863.485	−0.917625	−0.458812	−	0.888533i	$-0.651725\pi$
$941$	−863.485	−0.917625	−0.458812	+	0.888533i	$0.651725\pi$
$942$	0	0
$943$	− 169.110i	− 0.179332i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1148.53i	1.21281i	0.795156	+	0.606404i	$0.207389\pi$
$947$	1148.53i	1.21281i	−0.795156	+	0.606404i	$0.792611\pi$
$948$	0	0
$949$	24.4467	0.0257604
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1447.80	−1.51920	−0.759600	−	0.650391i	$-0.774605\pi$
$953$	−1447.80	−1.51920	−0.759600	+	0.650391i	$0.774605\pi$
$954$	0	0
$955$	252.274i	0.264161i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	48.9957i	0.0510905i
$960$	0	0
$961$	−1048.97	−1.09154
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1213.66	−1.25768
$966$	0	0
$967$	1522.82i	1.57478i	0.616453	+	0.787392i	$0.288569\pi$
$967$	1522.82i	1.57478i	−0.616453	+	0.787392i	$0.711431\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 266.782i	− 0.274750i	−0.990519	−	0.137375i	$-0.956133\pi$
$971$	− 266.782i	− 0.274750i	0.990519	−	0.137375i	$-0.0438666\pi$
$972$	0	0
$973$	−127.401	−0.130937
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1749.83	−1.79102	−0.895512	−	0.445037i	$-0.853190\pi$
$977$	−1749.83	−1.79102	−0.895512	+	0.445037i	$0.853190\pi$
$978$	0	0
$979$	349.701i	0.357203i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1301.42i	1.32393i	0.749535	+	0.661964i	$0.230277\pi$
$983$	1301.42i	1.32393i	−0.749535	+	0.661964i	$0.769723\pi$
$984$	0	0
$985$	1167.69	1.18547
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−151.587	−0.153273
$990$	0	0
$991$	− 136.009i	− 0.137245i	−0.997643	−	0.0686223i	$-0.978140\pi$
$991$	− 136.009i	− 0.137245i	0.997643	−	0.0686223i	$-0.0218603\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 701.788i	− 0.705315i
$996$	0	0
$997$	−1274.70	−1.27853	−0.639266	−	0.768986i	$-0.720762\pi$
$997$	−1274.70	−1.27853	−0.639266	+	0.768986i	$0.720762\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.g.j.1135.1		8
3.2	odd	2		1296.3.g.k.1135.7		8
4.3	odd	2	inner	1296.3.g.j.1135.2		8
9.2	odd	6		432.3.o.b.415.1		8
9.4	even	3		144.3.o.c.79.2	yes	8
9.5	odd	6		432.3.o.a.127.1		8
9.7	even	3		144.3.o.a.31.3	✓	8
12.11	even	2		1296.3.g.k.1135.8		8
36.7	odd	6		144.3.o.c.31.2	yes	8
36.11	even	6		432.3.o.a.415.1		8
36.23	even	6		432.3.o.b.127.1		8
36.31	odd	6		144.3.o.a.79.3	yes	8
72.5	odd	6		1728.3.o.e.127.4		8
72.11	even	6		1728.3.o.e.1279.4		8
72.13	even	6		576.3.o.d.511.3		8
72.29	odd	6		1728.3.o.f.1279.4		8
72.43	odd	6		576.3.o.d.319.3		8
72.59	even	6		1728.3.o.f.127.4		8
72.61	even	6		576.3.o.f.319.2		8
72.67	odd	6		576.3.o.f.511.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.a.31.3	✓	8	9.7	even	3
144.3.o.a.79.3	yes	8	36.31	odd	6
144.3.o.c.31.2	yes	8	36.7	odd	6
144.3.o.c.79.2	yes	8	9.4	even	3
432.3.o.a.127.1		8	9.5	odd	6
432.3.o.a.415.1		8	36.11	even	6
432.3.o.b.127.1		8	36.23	even	6
432.3.o.b.415.1		8	9.2	odd	6
576.3.o.d.319.3		8	72.43	odd	6
576.3.o.d.511.3		8	72.13	even	6
576.3.o.f.319.2		8	72.61	even	6
576.3.o.f.511.2		8	72.67	odd	6
1296.3.g.j.1135.1		8	1.1	even	1	trivial
1296.3.g.j.1135.2		8	4.3	odd	2	inner
1296.3.g.k.1135.7		8	3.2	odd	2
1296.3.g.k.1135.8		8	12.11	even	2
1728.3.o.e.127.4		8	72.5	odd	6
1728.3.o.e.1279.4		8	72.11	even	6
1728.3.o.f.127.4		8	72.59	even	6
1728.3.o.f.1279.4		8	72.29	odd	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.23321 q^{5} -6.15562i q^{7} +O(q^{10})\)	\(q-9.23321 q^{5} -6.15562i q^{7} -4.27258i q^{11} -1.73847 q^{13} +12.3476 q^{17} -33.9338i q^{19} +3.86865i q^{23} +60.2521 q^{25} -35.6818 q^{29} -44.8326i q^{31} +56.8361i q^{35} -32.7130 q^{37} -43.7130 q^{41} +39.1835i q^{43} +46.0476i q^{47} +11.1083 q^{49} +46.3143 q^{53} +39.4496i q^{55} -26.8710i q^{59} -46.9089 q^{61} +16.0516 q^{65} +65.8160i q^{67} +96.7955i q^{71} -14.0622 q^{73} -26.3004 q^{77} +39.7164i q^{79} +94.3623i q^{83} -114.008 q^{85} -81.8478 q^{89} +10.7014i q^{91} +313.318i q^{95} +15.9806 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5}+O(q^{10}) \) 8 * q - 6 * q^5	\( 8 q - 6 q^{5} - 10 q^{13} + 6 q^{17} + 46 q^{25} - 138 q^{29} - 20 q^{37} - 108 q^{41} - 82 q^{49} - 252 q^{53} - 14 q^{61} - 186 q^{65} + 74 q^{73} - 414 q^{77} + 60 q^{85} - 168 q^{89} + 20 q^{97}+O(q^{100}) \) 8 * q - 6 * q^5 - 10 * q^13 + 6 * q^17 + 46 * q^25 - 138 * q^29 - 20 * q^37 - 108 * q^41 - 82 * q^49 - 252 * q^53 - 14 * q^61 - 186 * q^65 + 74 * q^73 - 414 * q^77 + 60 * q^85 - 168 * q^89 + 20 * q^97

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