LMFDB - Embedded newform 864.2.f.b.431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.170772624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 6x^4 - 12x^3 + 20x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$0.774115 - 1.18353i$ of defining polynomial
Character	$\chi$	$=$	864.431
Dual form			864.2.f.b.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.42703 q^{5} +0.505408i q^{7} +O(q^{10})$	$q-3.42703 q^{5} +0.505408i q^{7} +3.31662i q^{11} -5.43039i q^{13} -1.58457i q^{17} +6.74456 q^{19} +4.30243 q^{23} +6.74456 q^{25} +2.55164 q^{29} -4.92498i q^{31} -1.73205i q^{35} +7.45202i q^{37} -8.51278i q^{41} +4.00000 q^{43} +5.10328 q^{47} +6.74456 q^{49} +0.875393 q^{53} -11.3662i q^{55} -6.63325i q^{59} -10.8608i q^{61} +18.6101i q^{65} +4.00000 q^{67} -9.40571 q^{71} -3.74456 q^{73} -1.67625 q^{77} +6.44121i q^{79} +10.2448i q^{83} +5.43039i q^{85} -3.75906i q^{89} +2.74456 q^{91} -23.1138 q^{95} -1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{19} + 8 q^{25} + 32 q^{43} + 8 q^{49} + 32 q^{67} + 16 q^{73} - 24 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^19 + 8 * q^25 + 32 * q^43 + 8 * q^49 + 32 * q^67 + 16 * q^73 - 24 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	−3.42703	−1.53262	−0.766308	−	0.642473i	$-0.777908\pi$
$5$	−3.42703	−1.53262	−0.766308	+	0.642473i	$0.777908\pi$
$6$	0	0
$7$	0.505408i	0.191026i	0.995428	+	0.0955132i	$0.0304492\pi$
$7$	0.505408i	0.191026i	−0.995428	+	0.0955132i	$0.969551\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.31662i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$11$	3.31662i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$12$	0	0
$13$	− 5.43039i	− 1.50612i	−0.657952	−	0.753059i	$-0.728577\pi$
$13$	− 5.43039i	− 1.50612i	0.657952	−	0.753059i	$-0.271423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.58457i	− 0.384316i	−0.981364	−	0.192158i	$-0.938451\pi$
$17$	− 1.58457i	− 0.384316i	0.981364	−	0.192158i	$-0.0615486\pi$
$18$	0	0
$19$	6.74456	1.54731	0.773654	−	0.633608i	$-0.218427\pi$
$19$	6.74456	1.54731	0.773654	+	0.633608i	$0.218427\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.30243	0.897118	0.448559	−	0.893753i	$-0.351937\pi$
$23$	4.30243	0.897118	0.448559	+	0.893753i	$0.351937\pi$
$24$	0	0
$25$	6.74456	1.34891
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.55164	0.473828	0.236914	−	0.971531i	$-0.423864\pi$
$29$	2.55164	0.473828	0.236914	+	0.971531i	$0.423864\pi$
$30$	0	0
$31$	− 4.92498i	− 0.884553i	−0.896879	−	0.442276i	$-0.854171\pi$
$31$	− 4.92498i	− 0.884553i	0.896879	−	0.442276i	$-0.145829\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.73205i	− 0.292770i
$36$	0	0
$37$	7.45202i	1.22510i	0.790430	+	0.612552i	$0.209857\pi$
$37$	7.45202i	1.22510i	−0.790430	+	0.612552i	$0.790143\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.51278i	− 1.32947i	−0.747078	−	0.664736i	$-0.768544\pi$
$41$	− 8.51278i	− 1.32947i	0.747078	−	0.664736i	$-0.231456\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.10328	0.744390	0.372195	−	0.928154i	$-0.378605\pi$
$47$	5.10328	0.744390	0.372195	+	0.928154i	$0.378605\pi$
$48$	0	0
$49$	6.74456	0.963509
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.875393	0.120244	0.0601222	−	0.998191i	$-0.480851\pi$
$53$	0.875393	0.120244	0.0601222	+	0.998191i	$0.480851\pi$
$54$	0	0
$55$	− 11.3662i	− 1.53262i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.63325i	− 0.863576i	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	− 6.63325i	− 0.863576i	0.901975	−	0.431788i	$-0.142117\pi$
$60$	0	0
$61$	− 10.8608i	− 1.39058i	−0.718729	−	0.695290i	$-0.755276\pi$
$61$	− 10.8608i	− 1.39058i	0.718729	−	0.695290i	$-0.244724\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.6101i	2.30830i
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.40571	−1.11625	−0.558126	−	0.829756i	$-0.688480\pi$
$71$	−9.40571	−1.11625	−0.558126	+	0.829756i	$0.688480\pi$
$72$	0	0
$73$	−3.74456	−0.438268	−0.219134	−	0.975695i	$-0.570323\pi$
$73$	−3.74456	−0.438268	−0.219134	+	0.975695i	$0.570323\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.67625	−0.191026
$78$	0	0
$79$	6.44121i	0.724692i	0.932044	+	0.362346i	$0.118024\pi$
$79$	6.44121i	0.724692i	−0.932044	+	0.362346i	$0.881976\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.2448i	1.12452i	0.826962	+	0.562258i	$0.190067\pi$
$83$	10.2448i	1.12452i	−0.826962	+	0.562258i	$0.809933\pi$
$84$	0	0
$85$	5.43039i	0.589008i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.75906i	− 0.398459i	−0.979953	−	0.199230i	$-0.936156\pi$
$89$	− 3.75906i	− 0.398459i	0.979953	−	0.199230i	$-0.0638439\pi$
$90$	0	0
$91$	2.74456	0.287708
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−23.1138	−2.37143
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.0820	1.10270	0.551348	−	0.834275i	$-0.314114\pi$
$101$	11.0820	1.10270	0.551348	+	0.834275i	$0.314114\pi$
$102$	0	0
$103$	17.3020i	1.70482i	0.522878	+	0.852408i	$0.324858\pi$
$103$	17.3020i	1.70482i	−0.522878	+	0.852408i	$0.675142\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.36530i	− 0.808704i	−0.914603	−	0.404352i	$-0.867497\pi$
$107$	− 8.36530i	− 0.808704i	0.914603	−	0.404352i	$-0.132503\pi$
$108$	0	0
$109$	− 7.45202i	− 0.713774i	−0.934148	−	0.356887i	$-0.883838\pi$
$109$	− 7.45202i	− 0.713774i	0.934148	−	0.356887i	$-0.116162\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.34363i	0.502686i	0.967898	+	0.251343i	$0.0808723\pi$
$113$	5.34363i	0.502686i	−0.967898	+	0.251343i	$0.919128\pi$
$114$	0	0
$115$	−14.7446	−1.37494
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.800857	0.0734144
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.97868	−0.534749
$126$	0	0
$127$	− 1.51622i	− 0.134543i	−0.997735	−	0.0672716i	$-0.978571\pi$
$127$	− 1.51622i	− 0.134543i	0.997735	−	0.0672716i	$-0.0214294\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 15.2935i	− 1.33620i	−0.744072	−	0.668100i	$-0.767108\pi$
$131$	− 15.2935i	− 1.33620i	0.744072	−	0.668100i	$-0.232892\pi$
$132$	0	0
$133$	3.40876i	0.295577i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.75906i	− 0.321158i	−0.987023	−	0.160579i	$-0.948664\pi$
$137$	− 3.75906i	− 0.321158i	0.987023	−	0.160579i	$-0.0513361\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.0106	1.50612
$144$	0	0
$145$	−8.74456	−0.726196
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.48025	0.776652	0.388326	−	0.921522i	$-0.373053\pi$
$149$	9.48025	0.776652	0.388326	+	0.921522i	$0.373053\pi$
$150$	0	0
$151$	− 6.94661i	− 0.565307i	−0.959222	−	0.282654i	$-0.908785\pi$
$151$	− 6.94661i	− 0.565307i	0.959222	−	0.282654i	$-0.0912147\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.8781i	1.35568i
$156$	0	0
$157$	10.8608i	0.866784i	0.901205	+	0.433392i	$0.142684\pi$
$157$	10.8608i	0.866784i	−0.901205	+	0.433392i	$0.857316\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.17448i	0.171373i
$162$	0	0
$163$	−16.2337	−1.27152	−0.635760	−	0.771887i	$-0.719313\pi$
$163$	−16.2337	−1.27152	−0.635760	+	0.771887i	$0.719313\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.7081	1.06077	0.530384	−	0.847758i	$-0.322048\pi$
$167$	13.7081	1.06077	0.530384	+	0.847758i	$0.322048\pi$
$168$	0	0
$169$	−16.4891	−1.26839
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.67625	0.127443	0.0637214	−	0.997968i	$-0.479703\pi$
$173$	1.67625	0.127443	0.0637214	+	0.997968i	$0.479703\pi$
$174$	0	0
$175$	3.40876i	0.257678i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.43710i	− 0.107414i	−0.998557	−	0.0537068i	$-0.982896\pi$
$179$	− 1.43710i	− 0.107414i	0.998557	−	0.0537068i	$-0.0171036\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 25.5383i	− 1.87762i
$186$	0	0
$187$	5.25544	0.384316
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.0106	1.30320	0.651599	−	0.758563i	$-0.274098\pi$
$191$	18.0106	1.30320	0.651599	+	0.758563i	$0.274098\pi$
$192$	0	0
$193$	−21.2337	−1.52843	−0.764217	−	0.644959i	$-0.776874\pi$
$193$	−21.2337	−1.52843	−0.764217	+	0.644959i	$0.776874\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0319	−0.857236	−0.428618	−	0.903486i	$-0.640999\pi$
$197$	−12.0319	−0.857236	−0.428618	+	0.903486i	$0.640999\pi$
$198$	0	0
$199$	22.2270i	1.57563i	0.615913	+	0.787814i	$0.288787\pi$
$199$	22.2270i	1.57563i	−0.615913	+	0.787814i	$0.711213\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.28962i	0.0905136i
$204$	0	0
$205$	29.1736i	2.03757i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.3692i	1.54731i
$210$	0	0
$211$	1.25544	0.0864279	0.0432139	−	0.999066i	$-0.486240\pi$
$211$	1.25544	0.0864279	0.0432139	+	0.999066i	$0.486240\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.7081	−0.934887
$216$	0	0
$217$	2.48913	0.168973
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.60485	−0.578825
$222$	0	0
$223$	− 8.46284i	− 0.566714i	−0.959015	−	0.283357i	$-0.908552\pi$
$223$	− 8.46284i	− 0.566714i	0.959015	−	0.283357i	$-0.0914481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.87419i	0.190767i	0.995441	+	0.0953835i	$0.0304077\pi$
$227$	2.87419i	0.190767i	−0.995441	+	0.0953835i	$0.969592\pi$
$228$	0	0
$229$	− 12.8824i	− 0.851294i	−0.904889	−	0.425647i	$-0.860047\pi$
$229$	− 12.8824i	− 0.851294i	0.904889	−	0.425647i	$-0.139953\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 27.1229i	− 1.77688i	−0.458992	−	0.888440i	$-0.651789\pi$
$233$	− 27.1229i	− 1.77688i	0.458992	−	0.888440i	$-0.348211\pi$
$234$	0	0
$235$	−17.4891	−1.14086
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.5090	−0.938509	−0.469254	−	0.883063i	$-0.655477\pi$
$239$	−14.5090	−0.938509	−0.469254	+	0.883063i	$0.655477\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−23.1138	−1.47669
$246$	0	0
$247$	− 36.6256i	− 2.33043i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.7306i	1.05603i	0.849236	+	0.528013i	$0.177063\pi$
$251$	16.7306i	1.05603i	−0.849236	+	0.528013i	$0.822937\pi$
$252$	0	0
$253$	14.2695i	0.897118i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.2718i	0.765496i	0.923853	+	0.382748i	$0.125022\pi$
$257$	12.2718i	0.765496i	−0.923853	+	0.382748i	$0.874978\pi$
$258$	0	0
$259$	−3.76631	−0.234027
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.30243	−0.265299	−0.132649	−	0.991163i	$-0.542348\pi$
$263$	−4.30243	−0.265299	−0.132649	+	0.991163i	$0.542348\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.1565	0.680223	0.340112	−	0.940385i	$-0.389535\pi$
$269$	11.1565	0.680223	0.340112	+	0.940385i	$0.389535\pi$
$270$	0	0
$271$	20.8398i	1.26593i	0.774180	+	0.632965i	$0.218162\pi$
$271$	20.8398i	1.26593i	−0.774180	+	0.632965i	$0.781838\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.3692i	1.34891i
$276$	0	0
$277$	− 10.8608i	− 0.652561i	−0.945273	−	0.326280i	$-0.894205\pi$
$277$	− 10.8608i	− 0.652561i	0.945273	−	0.326280i	$-0.105795\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 15.4410i	− 0.921132i	−0.887626	−	0.460566i	$-0.847647\pi$
$281$	− 15.4410i	− 0.921132i	0.887626	−	0.460566i	$-0.152353\pi$
$282$	0	0
$283$	−22.7446	−1.35202	−0.676012	−	0.736891i	$-0.736293\pi$
$283$	−22.7446	−1.35202	−0.676012	+	0.736891i	$0.736293\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.30243	0.253964
$288$	0	0
$289$	14.4891	0.852301
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.8646	−1.45261	−0.726304	−	0.687374i	$-0.758763\pi$
$293$	−24.8646	−1.45261	−0.726304	+	0.687374i	$0.758763\pi$
$294$	0	0
$295$	22.7324i	1.32353i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 23.3639i	− 1.35117i
$300$	0	0
$301$	2.02163i	0.116525i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	37.2203i	2.13123i
$306$	0	0
$307$	18.7446	1.06981	0.534904	−	0.844913i	$-0.320348\pi$
$307$	18.7446	1.06981	0.534904	+	0.844913i	$0.320348\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.1138	1.31067	0.655333	−	0.755340i	$-0.272528\pi$
$311$	23.1138	1.31067	0.655333	+	0.755340i	$0.272528\pi$
$312$	0	0
$313$	−9.23369	−0.521919	−0.260959	−	0.965350i	$-0.584039\pi$
$313$	−9.23369	−0.521919	−0.260959	+	0.965350i	$0.584039\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.5835	0.819093	0.409546	−	0.912289i	$-0.365687\pi$
$317$	14.5835	0.819093	0.409546	+	0.912289i	$0.365687\pi$
$318$	0	0
$319$	8.46284i	0.473828i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 10.6873i	− 0.594655i
$324$	0	0
$325$	− 36.6256i	− 2.03162i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.57924i	0.142198i
$330$	0	0
$331$	33.4891	1.84073	0.920364	−	0.391062i	$-0.127892\pi$
$331$	33.4891	1.84073	0.920364	+	0.391062i	$0.127892\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.7081	−0.748955
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.3343	0.884553
$342$	0	0
$343$	6.94661i	0.375082i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.19082i	− 0.332341i	−0.986097	−	0.166170i	$-0.946860\pi$
$347$	− 6.19082i	− 0.332341i	0.986097	−	0.166170i	$-0.0531401\pi$
$348$	0	0
$349$	− 3.40876i	− 0.182467i	−0.995830	−	0.0912333i	$-0.970919\pi$
$349$	− 3.40876i	− 0.182467i	0.995830	−	0.0912333i	$-0.0290809\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.0202i	− 0.959120i	−0.877509	−	0.479560i	$-0.840796\pi$
$353$	− 18.0202i	− 0.959120i	0.877509	−	0.479560i	$-0.159204\pi$
$354$	0	0
$355$	32.2337	1.71079
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.50157	−0.184806	−0.0924029	−	0.995722i	$-0.529455\pi$
$359$	−3.50157	−0.184806	−0.0924029	+	0.995722i	$0.529455\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.8327	0.671697
$366$	0	0
$367$	− 15.7858i	− 0.824010i	−0.911182	−	0.412005i	$-0.864829\pi$
$367$	− 15.7858i	− 0.824010i	0.911182	−	0.412005i	$-0.135171\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.442430i	0.0229698i
$372$	0	0
$373$	− 5.43039i	− 0.281175i	−0.990068	−	0.140587i	$-0.955101\pi$
$373$	− 5.43039i	− 0.281175i	0.990068	−	0.140587i	$-0.0448991\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13.8564i	− 0.713641i
$378$	0	0
$379$	−13.4891	−0.692890	−0.346445	−	0.938070i	$-0.612611\pi$
$379$	−13.4891	−0.692890	−0.346445	+	0.938070i	$0.612611\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.6123	1.00214	0.501070	−	0.865407i	$-0.332940\pi$
$383$	19.6123	1.00214	0.501070	+	0.865407i	$0.332940\pi$
$384$	0	0
$385$	5.74456	0.292770
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.8965	1.87073	0.935364	−	0.353687i	$-0.115072\pi$
$389$	36.8965	1.87073	0.935364	+	0.353687i	$0.115072\pi$
$390$	0	0
$391$	− 6.81751i	− 0.344776i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 22.0742i	− 1.11068i
$396$	0	0
$397$	− 1.38712i	− 0.0696178i	−0.999394	−	0.0348089i	$-0.988918\pi$
$397$	− 1.38712i	− 0.0696178i	0.999394	−	0.0348089i	$-0.0110823\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.0256i	0.850216i	0.905143	+	0.425108i	$0.139764\pi$
$401$	17.0256i	0.850216i	−0.905143	+	0.425108i	$0.860236\pi$
$402$	0	0
$403$	−26.7446	−1.33224
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.7156	−1.22510
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.35250	0.164966
$414$	0	0
$415$	− 35.1094i	− 1.72345i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 20.4897i	− 1.00099i	−0.865741	−	0.500493i	$-0.833152\pi$
$419$	− 20.4897i	− 1.00099i	0.865741	−	0.500493i	$-0.166848\pi$
$420$	0	0
$421$	4.04326i	0.197057i	0.995134	+	0.0985283i	$0.0314135\pi$
$421$	4.04326i	0.197057i	−0.995134	+	0.0985283i	$0.968586\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 10.6873i	− 0.518408i
$426$	0	0
$427$	5.48913	0.265637
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.3130	−1.07478	−0.537389	−	0.843334i	$-0.680589\pi$
$431$	−22.3130	−1.07478	−0.537389	+	0.843334i	$0.680589\pi$
$432$	0	0
$433$	4.48913	0.215734	0.107867	−	0.994165i	$-0.465598\pi$
$433$	4.48913	0.215734	0.107867	+	0.994165i	$0.465598\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.0180	1.38812
$438$	0	0
$439$	− 32.0769i	− 1.53095i	−0.643467	−	0.765474i	$-0.722505\pi$
$439$	− 32.0769i	− 1.53095i	0.643467	−	0.765474i	$-0.277495\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.63325i	− 0.315155i	−0.987507	−	0.157578i	$-0.949632\pi$
$443$	− 6.63325i	− 0.315155i	0.987507	−	0.157578i	$-0.0503684\pi$
$444$	0	0
$445$	12.8824i	0.610685i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	28.7075i	1.35479i	0.735620	+	0.677395i	$0.236891\pi$
$449$	28.7075i	1.35479i	−0.735620	+	0.677395i	$0.763109\pi$
$450$	0	0
$451$	28.2337	1.32947
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.40571	−0.440946
$456$	0	0
$457$	27.4674	1.28487	0.642435	−	0.766340i	$-0.277924\pi$
$457$	27.4674	1.28487	0.642435	+	0.766340i	$0.277924\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−41.8507	−1.94918	−0.974591	−	0.223990i	$-0.928092\pi$
$461$	−41.8507	−1.94918	−0.974591	+	0.223990i	$0.928092\pi$
$462$	0	0
$463$	12.7533i	0.592697i	0.955080	+	0.296348i	$0.0957689\pi$
$463$	12.7533i	0.592697i	−0.955080	+	0.296348i	$0.904231\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.9576i	1.75647i	0.478229	+	0.878235i	$0.341279\pi$
$467$	37.9576i	1.75647i	−0.478229	+	0.878235i	$0.658721\pi$
$468$	0	0
$469$	2.02163i	0.0933503i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.2665i	0.609994i
$474$	0	0
$475$	45.4891	2.08718
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.7081	−0.626341	−0.313170	−	0.949697i	$-0.601391\pi$
$479$	−13.7081	−0.626341	−0.313170	+	0.949697i	$0.601391\pi$
$480$	0	0
$481$	40.4674	1.84515
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.42703	0.155614
$486$	0	0
$487$	− 4.41957i	− 0.200270i	−0.994974	−	0.100135i	$-0.968073\pi$
$487$	− 4.41957i	− 0.200270i	0.994974	−	0.100135i	$-0.0319275\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.3624i	1.73127i	0.500675	+	0.865635i	$0.333085\pi$
$491$	38.3624i	1.73127i	−0.500675	+	0.865635i	$0.666915\pi$
$492$	0	0
$493$	− 4.04326i	− 0.182099i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.75372i	− 0.213234i
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.9073	0.575507	0.287754	−	0.957704i	$-0.407092\pi$
$503$	12.9073	0.575507	0.287754	+	0.957704i	$0.407092\pi$
$504$	0	0
$505$	−37.9783	−1.69001
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.0745360	0.00330375	0.00165187	−	0.999999i	$-0.499474\pi$
$509$	0.0745360	0.00330375	0.00165187	+	0.999999i	$0.499474\pi$
$510$	0	0
$511$	− 1.89253i	− 0.0837207i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 59.2945i	− 2.61283i
$516$	0	0
$517$	16.9257i	0.744390i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.58457i	− 0.0694214i	−0.999397	−	0.0347107i	$-0.988949\pi$
$521$	− 1.58457i	− 0.0694214i	0.999397	−	0.0347107i	$-0.0110510\pi$
$522$	0	0
$523$	0.233688	0.0102185	0.00510923	−	0.999987i	$-0.498374\pi$
$523$	0.233688	0.0102185	0.00510923	+	0.999987i	$0.498374\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.80400	−0.339947
$528$	0	0
$529$	−4.48913	−0.195179
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−46.2277	−2.00234
$534$	0	0
$535$	28.6682i	1.23943i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.3692i	0.963509i
$540$	0	0
$541$	− 17.6783i	− 0.760049i	−0.924976	−	0.380025i	$-0.875916\pi$
$541$	− 17.6783i	− 0.760049i	0.924976	−	0.380025i	$-0.124084\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.5383i	1.09394i
$546$	0	0
$547$	−40.2337	−1.72027	−0.860134	−	0.510068i	$-0.829620\pi$
$547$	−40.2337	−1.72027	−0.860134	+	0.510068i	$0.829620\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.2097	0.733158
$552$	0	0
$553$	−3.25544	−0.138435
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.4376	−0.908340	−0.454170	−	0.890915i	$-0.650064\pi$
$557$	−21.4376	−0.908340	−0.454170	+	0.890915i	$0.650064\pi$
$558$	0	0
$559$	− 21.7216i	− 0.918724i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.6087i	1.41644i	0.705993	+	0.708218i	$0.250501\pi$
$563$	33.6087i	1.41644i	−0.705993	+	0.708218i	$0.749499\pi$
$564$	0	0
$565$	− 18.3128i	− 0.770425i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 34.0511i	− 1.42750i	−0.700402	−	0.713748i	$-0.746996\pi$
$569$	− 34.0511i	− 1.42750i	0.700402	−	0.713748i	$-0.253004\pi$
$570$	0	0
$571$	0.233688	0.00977954	0.00488977	−	0.999988i	$-0.498444\pi$
$571$	0.233688	0.00977954	0.00488977	+	0.999988i	$0.498444\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	29.0180	1.21013
$576$	0	0
$577$	−8.97825	−0.373769	−0.186885	−	0.982382i	$-0.559839\pi$
$577$	−8.97825	−0.373769	−0.186885	+	0.982382i	$0.559839\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.17782	−0.214812
$582$	0	0
$583$	2.90335i	0.120244i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.7143i	− 0.524774i	−0.964963	−	0.262387i	$-0.915490\pi$
$587$	− 12.7143i	− 0.524774i	0.964963	−	0.262387i	$-0.0845097\pi$
$588$	0	0
$589$	− 33.2168i	− 1.36868i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 13.2665i	− 0.544790i	−0.962186	−	0.272395i	$-0.912184\pi$
$593$	− 13.2665i	− 0.544790i	0.962186	−	0.272395i	$-0.0878157\pi$
$594$	0	0
$595$	−2.74456	−0.112516
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−30.9178	−1.26327	−0.631634	−	0.775266i	$-0.717616\pi$
$599$	−30.9178	−1.26327	−0.631634	+	0.775266i	$0.717616\pi$
$600$	0	0
$601$	31.2337	1.27405	0.637024	−	0.770844i	$-0.280165\pi$
$601$	31.2337	1.27405	0.637024	+	0.770844i	$0.280165\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 0.376308i	− 0.0152739i	−0.999971	−	0.00763694i	$-0.997569\pi$
$607$	− 0.376308i	− 0.0152739i	0.999971	−	0.00763694i	$-0.00243094\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 27.7128i	− 1.12114i
$612$	0	0
$613$	41.4215i	1.67300i	0.547969	+	0.836499i	$0.315401\pi$
$613$	41.4215i	1.67300i	−0.547969	+	0.836499i	$0.684599\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0202i	− 0.725467i	−0.931893	−	0.362733i	$-0.881844\pi$
$617$	− 18.0202i	− 0.725467i	0.931893	−	0.362733i	$-0.118156\pi$
$618$	0	0
$619$	7.76631	0.312154	0.156077	−	0.987745i	$-0.450115\pi$
$619$	7.76631	0.312154	0.156077	+	0.987745i	$0.450115\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.89986	0.0761162
$624$	0	0
$625$	−13.2337	−0.529348
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.8083	0.470827
$630$	0	0
$631$	− 25.2594i	− 1.00556i	−0.864414	−	0.502781i	$-0.832310\pi$
$631$	− 25.2594i	− 1.00556i	0.864414	−	0.502781i	$-0.167690\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.19615i	0.206203i
$636$	0	0
$637$	− 36.6256i	− 1.45116i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 8.51278i	− 0.336234i	−0.985767	−	0.168117i	$-0.946231\pi$
$641$	− 8.51278i	− 0.336234i	0.985767	−	0.168117i	$-0.0537687\pi$
$642$	0	0
$643$	−24.4674	−0.964899	−0.482450	−	0.875924i	$-0.660253\pi$
$643$	−24.4674	−0.964899	−0.482450	+	0.875924i	$0.660253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.6260	1.75443	0.877214	−	0.480099i	$-0.159399\pi$
$647$	44.6260	1.75443	0.877214	+	0.480099i	$0.159399\pi$
$648$	0	0
$649$	22.0000	0.863576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.6942	1.20116	0.600579	−	0.799565i	$-0.294937\pi$
$653$	30.6942	1.20116	0.600579	+	0.799565i	$0.294937\pi$
$654$	0	0
$655$	52.4114i	2.04788i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.2757i	1.02356i	0.859118	+	0.511778i	$0.171013\pi$
$659$	26.2757i	1.02356i	−0.859118	+	0.511778i	$0.828987\pi$
$660$	0	0
$661$	3.40876i	0.132585i	0.997800	+	0.0662926i	$0.0211171\pi$
$661$	3.40876i	0.132585i	−0.997800	+	0.0662926i	$0.978883\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.6819i	− 0.453006i
$666$	0	0
$667$	10.9783	0.425080
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36.0211	1.39058
$672$	0	0
$673$	−30.4891	−1.17527	−0.587635	−	0.809126i	$-0.699941\pi$
$673$	−30.4891	−1.17527	−0.587635	+	0.809126i	$0.699941\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.55478	0.367220	0.183610	−	0.982999i	$-0.441222\pi$
$677$	9.55478	0.367220	0.183610	+	0.982999i	$0.441222\pi$
$678$	0	0
$679$	− 0.505408i	− 0.0193958i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.7306i	0.640179i	0.947387	+	0.320089i	$0.103713\pi$
$683$	16.7306i	0.640179i	−0.947387	+	0.320089i	$0.896287\pi$
$684$	0	0
$685$	12.8824i	0.492212i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.75372i	− 0.181102i
$690$	0	0
$691$	−26.5109	−1.00852	−0.504261	−	0.863552i	$-0.668235\pi$
$691$	−26.5109	−1.00852	−0.504261	+	0.863552i	$0.668235\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.4163	1.03996
$696$	0	0
$697$	−13.4891	−0.510937
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.53032	−0.322186	−0.161093	−	0.986939i	$-0.551502\pi$
$701$	−8.53032	−0.322186	−0.161093	+	0.986939i	$0.551502\pi$
$702$	0	0
$703$	50.2606i	1.89562i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.60091i	0.210644i
$708$	0	0
$709$	36.6256i	1.37550i	0.725946	+	0.687752i	$0.241402\pi$
$709$	36.6256i	1.37550i	−0.725946	+	0.687752i	$0.758598\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 21.1894i	− 0.793548i
$714$	0	0
$715$	−61.7228	−2.30830
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.8114	−0.701548	−0.350774	−	0.936460i	$-0.614081\pi$
$719$	−18.8114	−0.701548	−0.350774	+	0.936460i	$0.614081\pi$
$720$	0	0
$721$	−8.74456	−0.325665
$722$	0	0
$723$	0	0
$724$	0	0
$725$	17.2097	0.639152
$726$	0	0
$727$	47.9918i	1.77992i	0.456042	+	0.889958i	$0.349267\pi$
$727$	47.9918i	1.77992i	−0.456042	+	0.889958i	$0.650733\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6.33830i	− 0.234430i
$732$	0	0
$733$	− 11.4953i	− 0.424588i	−0.977206	−	0.212294i	$-0.931907\pi$
$733$	− 11.4953i	− 0.424588i	0.977206	−	0.212294i	$-0.0680935\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.2665i	0.488678i
$738$	0	0
$739$	18.7446	0.689530	0.344765	−	0.938689i	$-0.387959\pi$
$739$	18.7446	0.689530	0.344765	+	0.938689i	$0.387959\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.800857	0.0293806	0.0146903	−	0.999892i	$-0.495324\pi$
$743$	0.800857	0.0293806	0.0146903	+	0.999892i	$0.495324\pi$
$744$	0	0
$745$	−32.4891	−1.19031
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.22789	0.154484
$750$	0	0
$751$	− 52.4114i	− 1.91252i	−0.292522	−	0.956259i	$-0.594494\pi$
$751$	− 52.4114i	− 1.91252i	0.292522	−	0.956259i	$-0.405506\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23.8063i	0.866399i
$756$	0	0
$757$	41.4215i	1.50549i	0.658313	+	0.752745i	$0.271271\pi$
$757$	41.4215i	1.50549i	−0.658313	+	0.752745i	$0.728729\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.3894i	1.46411i	0.681243	+	0.732057i	$0.261440\pi$
$761$	40.3894i	1.46411i	−0.681243	+	0.732057i	$0.738560\pi$
$762$	0	0
$763$	3.76631	0.136350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.0211	−1.30065
$768$	0	0
$769$	−10.2554	−0.369821	−0.184910	−	0.982755i	$-0.559199\pi$
$769$	−10.2554	−0.369821	−0.184910	+	0.982755i	$0.559199\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−45.2778	−1.62853	−0.814264	−	0.580495i	$-0.802859\pi$
$773$	−45.2778	−1.62853	−0.814264	+	0.580495i	$0.802859\pi$
$774$	0	0
$775$	− 33.2168i	− 1.19318i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 57.4150i	− 2.05710i
$780$	0	0
$781$	− 31.1952i	− 1.11625i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 37.2203i	− 1.32845i
$786$	0	0
$787$	21.4891	0.766005	0.383002	−	0.923747i	$-0.374890\pi$
$787$	21.4891	0.766005	0.383002	+	0.923747i	$0.374890\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.70071	−0.0960263
$792$	0	0
$793$	−58.9783	−2.09438
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.7400	−0.911758	−0.455879	−	0.890042i	$-0.650675\pi$
$797$	−25.7400	−0.911758	−0.455879	+	0.890042i	$0.650675\pi$
$798$	0	0
$799$	− 8.08653i	− 0.286081i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 12.4193i	− 0.438268i
$804$	0	0
$805$	− 7.45202i	− 0.262649i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.9686i	1.51070i	0.655323	+	0.755349i	$0.272532\pi$
$809$	42.9686i	1.51070i	−0.655323	+	0.755349i	$0.727468\pi$
$810$	0	0
$811$	6.74456	0.236834	0.118417	−	0.992964i	$-0.462218\pi$
$811$	6.74456	0.236834	0.118417	+	0.992964i	$0.462218\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	55.6334	1.94875
$816$	0	0
$817$	26.9783	0.943850
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0743	−1.46840	−0.734202	−	0.678931i	$-0.762444\pi$
$821$	−42.0743	−1.46840	−0.734202	+	0.678931i	$0.762444\pi$
$822$	0	0
$823$	21.5925i	0.752666i	0.926485	+	0.376333i	$0.122815\pi$
$823$	21.5925i	0.752666i	−0.926485	+	0.376333i	$0.877185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.63325i	− 0.230661i	−0.993327	−	0.115330i	$-0.963207\pi$
$827$	− 6.63325i	− 0.230661i	0.993327	−	0.115330i	$-0.0367927\pi$
$828$	0	0
$829$	23.7432i	0.824635i	0.911040	+	0.412318i	$0.135281\pi$
$829$	23.7432i	0.824635i	−0.911040	+	0.412318i	$0.864719\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 10.6873i	− 0.370292i
$834$	0	0
$835$	−46.9783	−1.62575
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.4268	−1.56831	−0.784154	−	0.620566i	$-0.786903\pi$
$839$	−45.4268	−1.56831	−0.784154	+	0.620566i	$0.786903\pi$
$840$	0	0
$841$	−22.4891	−0.775487
$842$	0	0
$843$	0	0
$844$	0	0
$845$	56.5088	1.94396
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.0618i	1.09906i
$852$	0	0
$853$	45.4647i	1.55668i	0.627841	+	0.778342i	$0.283939\pi$
$853$	45.4647i	1.55668i	−0.627841	+	0.778342i	$0.716061\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.6357i	1.21729i	0.793442	+	0.608646i	$0.208287\pi$
$857$	35.6357i	1.21729i	−0.793442	+	0.608646i	$0.791713\pi$
$858$	0	0
$859$	−48.4674	−1.65369	−0.826843	−	0.562433i	$-0.809865\pi$
$859$	−48.4674	−1.65369	−0.826843	+	0.562433i	$0.809865\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−56.4343	−1.92104	−0.960522	−	0.278203i	$-0.910261\pi$
$863$	−56.4343	−1.92104	−0.960522	+	0.278203i	$0.910261\pi$
$864$	0	0
$865$	−5.74456	−0.195321
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21.3631	−0.724692
$870$	0	0
$871$	− 21.7216i	− 0.736007i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.02167i	− 0.102151i
$876$	0	0
$877$	2.02163i	0.0682657i	0.999417	+	0.0341328i	$0.0108669\pi$
$877$	2.02163i	0.0682657i	−0.999417	+	0.0341328i	$0.989133\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 38.8048i	− 1.30737i	−0.756768	−	0.653684i	$-0.773223\pi$
$881$	− 38.8048i	− 1.30737i	0.756768	−	0.653684i	$-0.226777\pi$
$882$	0	0
$883$	−33.7228	−1.13486	−0.567432	−	0.823421i	$-0.692063\pi$
$883$	−33.7228	−1.13486	−0.567432	+	0.823421i	$0.692063\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−55.6334	−1.86799	−0.933993	−	0.357290i	$-0.883701\pi$
$887$	−55.6334	−1.86799	−0.933993	+	0.357290i	$0.883701\pi$
$888$	0	0
$889$	0.766312	0.0257013
$890$	0	0
$891$	0	0
$892$	0	0
$893$	34.4194	1.15180
$894$	0	0
$895$	4.92498i	0.164624i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 12.5668i	− 0.419126i
$900$	0	0
$901$	− 1.38712i	− 0.0462118i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.7663	0.656330	0.328165	−	0.944620i	$-0.393570\pi$
$907$	19.7663	0.656330	0.328165	+	0.944620i	$0.393570\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.10328	0.169079	0.0845397	−	0.996420i	$-0.473058\pi$
$911$	5.10328	0.169079	0.0845397	+	0.996420i	$0.473058\pi$
$912$	0	0
$913$	−33.9783	−1.12452
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.72946	0.255249
$918$	0	0
$919$	44.5830i	1.47066i	0.677710	+	0.735329i	$0.262972\pi$
$919$	44.5830i	1.47066i	−0.677710	+	0.735329i	$0.737028\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	51.0767i	1.68121i
$924$	0	0
$925$	50.2606i	1.65256i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	49.8968i	1.63706i	0.574462	+	0.818531i	$0.305211\pi$
$929$	49.8968i	1.63706i	−0.574462	+	0.818531i	$0.694789\pi$
$930$	0	0
$931$	45.4891	1.49085
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.0106	−0.589008
$936$	0	0
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.8327	−0.418335	−0.209168	−	0.977880i	$-0.567075\pi$
$941$	−12.8327	−0.418335	−0.209168	+	0.977880i	$0.567075\pi$
$942$	0	0
$943$	− 36.6256i	− 1.19269i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.43710i	− 0.0466994i	−0.999727	−	0.0233497i	$-0.992567\pi$
$947$	− 1.43710i	− 0.0466994i	0.999727	−	0.0233497i	$-0.00743311\pi$
$948$	0	0
$949$	20.3344i	0.660084i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.7384i	1.44922i	0.689160	+	0.724609i	$0.257980\pi$
$953$	44.7384i	1.44922i	−0.689160	+	0.724609i	$0.742020\pi$
$954$	0	0
$955$	−61.7228	−1.99730
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.89986	0.0613496
$960$	0	0
$961$	6.74456	0.217567
$962$	0	0
$963$	0	0
$964$	0	0
$965$	72.7686	2.34250
$966$	0	0
$967$	3.91416i	0.125871i	0.998018	+	0.0629355i	$0.0200462\pi$
$967$	3.91416i	0.125871i	−0.998018	+	0.0629355i	$0.979954\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 47.7600i	− 1.53269i	−0.642428	−	0.766346i	$-0.722073\pi$
$971$	− 47.7600i	− 1.53269i	0.642428	−	0.766346i	$-0.277927\pi$
$972$	0	0
$973$	− 4.04326i	− 0.129621i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.74839i	0.183907i	0.995763	+	0.0919536i	$0.0293112\pi$
$977$	5.74839i	0.183907i	−0.995763	+	0.0919536i	$0.970689\pi$
$978$	0	0
$979$	12.4674	0.398459
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.7261	1.36275	0.681376	−	0.731934i	$-0.261382\pi$
$983$	42.7261	1.36275	0.681376	+	0.731934i	$0.261382\pi$
$984$	0	0
$985$	41.2337	1.31381
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.2097	0.547237
$990$	0	0
$991$	13.3878i	0.425278i	0.977131	+	0.212639i	$0.0682058\pi$
$991$	13.3878i	0.425278i	−0.977131	+	0.212639i	$0.931794\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 76.1726i	− 2.41483i
$996$	0	0
$997$	− 56.9600i	− 1.80394i	−0.431796	−	0.901971i	$-0.642120\pi$
$997$	− 56.9600i	− 1.80394i	0.431796	−	0.901971i	$-0.357880\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.f.b.431.2		8
3.2	odd	2	inner	864.2.f.b.431.8		8
4.3	odd	2		216.2.f.b.107.4	yes	8
8.3	odd	2	inner	864.2.f.b.431.7		8
8.5	even	2		216.2.f.b.107.6	yes	8
9.2	odd	6		2592.2.p.d.2159.1		8
9.4	even	3		2592.2.p.d.431.4		8
9.5	odd	6		2592.2.p.e.431.1		8
9.7	even	3		2592.2.p.e.2159.4		8
12.11	even	2		216.2.f.b.107.5	yes	8
24.5	odd	2		216.2.f.b.107.3	✓	8
24.11	even	2	inner	864.2.f.b.431.1		8
36.7	odd	6		648.2.l.e.539.4		8
36.11	even	6		648.2.l.d.539.1		8
36.23	even	6		648.2.l.e.107.3		8
36.31	odd	6		648.2.l.d.107.2		8
72.5	odd	6		648.2.l.e.107.4		8
72.11	even	6		2592.2.p.d.2159.4		8
72.13	even	6		648.2.l.d.107.1		8
72.29	odd	6		648.2.l.d.539.2		8
72.43	odd	6		2592.2.p.e.2159.1		8
72.59	even	6		2592.2.p.e.431.4		8
72.61	even	6		648.2.l.e.539.3		8
72.67	odd	6		2592.2.p.d.431.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.f.b.107.3	✓	8	24.5	odd	2
216.2.f.b.107.4	yes	8	4.3	odd	2
216.2.f.b.107.5	yes	8	12.11	even	2
216.2.f.b.107.6	yes	8	8.5	even	2
648.2.l.d.107.1		8	72.13	even	6
648.2.l.d.107.2		8	36.31	odd	6
648.2.l.d.539.1		8	36.11	even	6
648.2.l.d.539.2		8	72.29	odd	6
648.2.l.e.107.3		8	36.23	even	6
648.2.l.e.107.4		8	72.5	odd	6
648.2.l.e.539.3		8	72.61	even	6
648.2.l.e.539.4		8	36.7	odd	6
864.2.f.b.431.1		8	24.11	even	2	inner
864.2.f.b.431.2		8	1.1	even	1	trivial
864.2.f.b.431.7		8	8.3	odd	2	inner
864.2.f.b.431.8		8	3.2	odd	2	inner
2592.2.p.d.431.1		8	72.67	odd	6
2592.2.p.d.431.4		8	9.4	even	3
2592.2.p.d.2159.1		8	9.2	odd	6
2592.2.p.d.2159.4		8	72.11	even	6
2592.2.p.e.431.1		8	9.5	odd	6
2592.2.p.e.431.4		8	72.59	even	6
2592.2.p.e.2159.1		8	72.43	odd	6
2592.2.p.e.2159.4		8	9.7	even	3

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.42703 q^{5} +0.505408i q^{7} +O(q^{10})\)	\(q-3.42703 q^{5} +0.505408i q^{7} +3.31662i q^{11} -5.43039i q^{13} -1.58457i q^{17} +6.74456 q^{19} +4.30243 q^{23} +6.74456 q^{25} +2.55164 q^{29} -4.92498i q^{31} -1.73205i q^{35} +7.45202i q^{37} -8.51278i q^{41} +4.00000 q^{43} +5.10328 q^{47} +6.74456 q^{49} +0.875393 q^{53} -11.3662i q^{55} -6.63325i q^{59} -10.8608i q^{61} +18.6101i q^{65} +4.00000 q^{67} -9.40571 q^{71} -3.74456 q^{73} -1.67625 q^{77} +6.44121i q^{79} +10.2448i q^{83} +5.43039i q^{85} -3.75906i q^{89} +2.74456 q^{91} -23.1138 q^{95} -1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{19} + 8 q^{25} + 32 q^{43} + 8 q^{49} + 32 q^{67} + 16 q^{73} - 24 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^19 + 8 * q^25 + 32 * q^43 + 8 * q^49 + 32 * q^67 + 16 * q^73 - 24 * q^91 - 8 * q^97

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