LMFDB - Embedded newform 1728.4.c.j.1727.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1727,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.1727");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 $ x^12 + 3x^10 - 12x^9 + 73x^8 - 12x^7 + 589x^6 + 84x^5 + 2452x^4 + 852x^3 + 6854x^2 - 888x + 9496
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1727.4
Root			$-1.29835 - 1.36719i$ of defining polynomial
Character	$\chi$	$=$	1728.1727
Dual form			1728.4.c.j.1727.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83890i q^{5} +8.83113i q^{7} +O(q^{10})$	$q-5.83890i q^{5} +8.83113i q^{7} +23.6146 q^{11} -54.6531 q^{13} +117.211i q^{17} +109.576i q^{19} +33.5763 q^{23} +90.9072 q^{25} +40.0490i q^{29} -292.510i q^{31} +51.5641 q^{35} -283.265 q^{37} -367.472i q^{41} -323.337i q^{43} -66.2249 q^{47} +265.011 q^{49} +158.506i q^{53} -137.883i q^{55} -848.630 q^{59} +348.716 q^{61} +319.114i q^{65} +194.285i q^{67} -939.761 q^{71} -473.826 q^{73} +208.544i q^{77} +273.221i q^{79} +338.366 q^{83} +684.386 q^{85} +739.884i q^{89} -482.648i q^{91} +639.805 q^{95} -448.629 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) $ 12 * q + 72 * q^13 - 384 * q^25 + 240 * q^37 + 288 * q^49 - 144 * q^61 + 156 * q^73 + 168 * q^85 + 516 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 5.83890i	− 0.522247i	−0.965305	−	0.261124i	$-0.915907\pi$
$5$	− 5.83890i	− 0.522247i	0.965305	−	0.261124i	$-0.0840930\pi$
$6$	0	0
$7$	8.83113i	0.476836i	0.971163	+	0.238418i	$0.0766288\pi$
$7$	8.83113i	0.476836i	−0.971163	+	0.238418i	$0.923371\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	23.6146	0.647279	0.323639	−	0.946180i	$-0.395094\pi$
$11$	23.6146	0.647279	0.323639	+	0.946180i	$0.395094\pi$
$12$	0	0
$13$	−54.6531	−1.16600	−0.583001	−	0.812471i	$-0.698122\pi$
$13$	−54.6531	−1.16600	−0.583001	+	0.812471i	$0.698122\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	117.211i	1.67223i	0.548553	+	0.836116i	$0.315179\pi$
$17$	117.211i	1.67223i	−0.548553	+	0.836116i	$0.684821\pi$
$18$	0	0
$19$	109.576i	1.32308i	0.749910	+	0.661539i	$0.230097\pi$
$19$	109.576i	1.32308i	−0.749910	+	0.661539i	$0.769903\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	33.5763	0.304397	0.152199	−	0.988350i	$-0.451365\pi$
$23$	33.5763	0.304397	0.152199	+	0.988350i	$0.451365\pi$
$24$	0	0
$25$	90.9072	0.727258
$26$	0	0
$27$	0	0
$28$	0	0
$29$	40.0490i	0.256445i	0.991745	+	0.128223i	$0.0409272\pi$
$29$	40.0490i	0.256445i	−0.991745	+	0.128223i	$0.959073\pi$
$30$	0	0
$31$	− 292.510i	− 1.69472i	−0.531020	−	0.847359i	$-0.678191\pi$
$31$	− 292.510i	− 1.69472i	0.531020	−	0.847359i	$-0.321809\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	51.5641	0.249026
$36$	0	0
$37$	−283.265	−1.25861	−0.629304	−	0.777159i	$-0.716660\pi$
$37$	−283.265	−1.25861	−0.629304	+	0.777159i	$0.716660\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 367.472i	− 1.39974i	−0.714269	−	0.699871i	$-0.753241\pi$
$41$	− 367.472i	− 1.39974i	0.714269	−	0.699871i	$-0.246759\pi$
$42$	0	0
$43$	− 323.337i	− 1.14671i	−0.819308	−	0.573354i	$-0.805642\pi$
$43$	− 323.337i	− 1.14671i	0.819308	−	0.573354i	$-0.194358\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−66.2249	−0.205530	−0.102765	−	0.994706i	$-0.532769\pi$
$47$	−66.2249	−0.205530	−0.102765	+	0.994706i	$0.532769\pi$
$48$	0	0
$49$	265.011	0.772627
$50$	0	0
$51$	0	0
$52$	0	0
$53$	158.506i	0.410801i	0.978678	+	0.205401i	$0.0658498\pi$
$53$	158.506i	0.410801i	−0.978678	+	0.205401i	$0.934150\pi$
$54$	0	0
$55$	− 137.883i	− 0.338040i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−848.630	−1.87258	−0.936290	−	0.351228i	$-0.885764\pi$
$59$	−848.630	−1.87258	−0.936290	+	0.351228i	$0.885764\pi$
$60$	0	0
$61$	348.716	0.731943	0.365972	−	0.930626i	$-0.380737\pi$
$61$	348.716	0.731943	0.365972	+	0.930626i	$0.380737\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	319.114i	0.608942i
$66$	0	0
$67$	194.285i	0.354264i	0.984187	+	0.177132i	$0.0566819\pi$
$67$	194.285i	0.354264i	−0.984187	+	0.177132i	$0.943318\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−939.761	−1.57083	−0.785417	−	0.618968i	$-0.787551\pi$
$71$	−939.761	−1.57083	−0.785417	+	0.618968i	$0.787551\pi$
$72$	0	0
$73$	−473.826	−0.759686	−0.379843	−	0.925051i	$-0.624022\pi$
$73$	−473.826	−0.759686	−0.379843	+	0.925051i	$0.624022\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	208.544i	0.308646i
$78$	0	0
$79$	273.221i	0.389112i	0.980891	+	0.194556i	$0.0623265\pi$
$79$	273.221i	0.389112i	−0.980891	+	0.194556i	$0.937673\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	338.366	0.447476	0.223738	−	0.974649i	$-0.428174\pi$
$83$	338.366	0.447476	0.223738	+	0.974649i	$0.428174\pi$
$84$	0	0
$85$	684.386	0.873318
$86$	0	0
$87$	0	0
$88$	0	0
$89$	739.884i	0.881208i	0.897702	+	0.440604i	$0.145236\pi$
$89$	739.884i	0.881208i	−0.897702	+	0.440604i	$0.854764\pi$
$90$	0	0
$91$	− 482.648i	− 0.555992i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	639.805	0.690974
$96$	0	0
$97$	−448.629	−0.469602	−0.234801	−	0.972044i	$-0.575444\pi$
$97$	−448.629	−0.469602	−0.234801	+	0.972044i	$0.575444\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1152.87i	− 1.13580i	−0.823099	−	0.567898i	$-0.807757\pi$
$101$	− 1152.87i	− 1.13580i	0.823099	−	0.567898i	$-0.192243\pi$
$102$	0	0
$103$	1389.90i	1.32962i	0.747013	+	0.664809i	$0.231487\pi$
$103$	1389.90i	1.32962i	−0.747013	+	0.664809i	$0.768513\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	245.479	0.221788	0.110894	−	0.993832i	$-0.464629\pi$
$107$	245.479	0.221788	0.110894	+	0.993832i	$0.464629\pi$
$108$	0	0
$109$	644.998	0.566785	0.283393	−	0.959004i	$-0.408540\pi$
$109$	644.998	0.566785	0.283393	+	0.959004i	$0.408540\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 825.442i	− 0.687177i	−0.939120	−	0.343589i	$-0.888357\pi$
$113$	− 825.442i	− 0.687177i	0.939120	−	0.343589i	$-0.111643\pi$
$114$	0	0
$115$	− 196.049i	− 0.158971i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1035.11	−0.797380
$120$	0	0
$121$	−773.351	−0.581030
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1260.66i	− 0.902056i
$126$	0	0
$127$	754.649i	0.527278i	0.964621	+	0.263639i	$0.0849228\pi$
$127$	754.649i	0.527278i	−0.964621	+	0.263639i	$0.915077\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2026.43	−1.35153	−0.675765	−	0.737117i	$-0.736186\pi$
$131$	−2026.43	−1.35153	−0.675765	+	0.737117i	$0.736186\pi$
$132$	0	0
$133$	−967.681	−0.630892
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1882.88i	− 1.17420i	−0.809515	−	0.587099i	$-0.800270\pi$
$137$	− 1882.88i	− 1.17420i	0.809515	−	0.587099i	$-0.199730\pi$
$138$	0	0
$139$	− 93.7277i	− 0.0571934i	−0.999591	−	0.0285967i	$-0.990896\pi$
$139$	− 93.7277i	− 0.0571934i	0.999591	−	0.0285967i	$-0.00910385\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1290.61	−0.754729
$144$	0	0
$145$	233.842	0.133928
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2764.69i	1.52008i	0.649874	+	0.760042i	$0.274822\pi$
$149$	2764.69i	1.52008i	−0.649874	+	0.760042i	$0.725178\pi$
$150$	0	0
$151$	− 3694.11i	− 1.99088i	−0.0954051	−	0.995439i	$-0.530415\pi$
$151$	− 3694.11i	− 1.99088i	0.0954051	−	0.995439i	$-0.469585\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1707.94	−0.885062
$156$	0	0
$157$	−812.401	−0.412972	−0.206486	−	0.978450i	$-0.566203\pi$
$157$	−812.401	−0.412972	−0.206486	+	0.978450i	$0.566203\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	296.516i	0.145148i
$162$	0	0
$163$	1034.18i	0.496953i	0.968638	+	0.248477i	$0.0799299\pi$
$163$	1034.18i	0.496953i	−0.968638	+	0.248477i	$0.920070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1453.29	−0.673409	−0.336704	−	0.941610i	$-0.609312\pi$
$167$	−1453.29	−0.673409	−0.336704	+	0.941610i	$0.609312\pi$
$168$	0	0
$169$	789.957	0.359562
$170$	0	0
$171$	0	0
$172$	0	0
$173$	634.902i	0.279022i	0.990221	+	0.139511i	$0.0445530\pi$
$173$	634.902i	0.279022i	−0.990221	+	0.139511i	$0.955447\pi$
$174$	0	0
$175$	802.813i	0.346783i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2909.35	−1.21483	−0.607417	−	0.794383i	$-0.707794\pi$
$179$	−2909.35	−1.21483	−0.607417	+	0.794383i	$0.707794\pi$
$180$	0	0
$181$	−1354.46	−0.556222	−0.278111	−	0.960549i	$-0.589708\pi$
$181$	−1354.46	−0.556222	−0.278111	+	0.960549i	$0.589708\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1653.96i	0.657305i
$186$	0	0
$187$	2767.90i	1.08240i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3488.40	−1.32153	−0.660765	−	0.750593i	$-0.729768\pi$
$191$	−3488.40	−1.32153	−0.660765	+	0.750593i	$0.729768\pi$
$192$	0	0
$193$	2912.99	1.08643	0.543217	−	0.839592i	$-0.317206\pi$
$193$	2912.99	1.08643	0.543217	+	0.839592i	$0.317206\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2432.66i	0.879795i	0.898048	+	0.439897i	$0.144985\pi$
$197$	2432.66i	0.879795i	−0.898048	+	0.439897i	$0.855015\pi$
$198$	0	0
$199$	563.190i	0.200621i	0.994956	+	0.100310i	$0.0319836\pi$
$199$	563.190i	0.200621i	−0.994956	+	0.100310i	$0.968016\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−353.678	−0.122282
$204$	0	0
$205$	−2145.63	−0.731012
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2587.60i	0.856401i
$210$	0	0
$211$	1447.62i	0.472314i	0.971715	+	0.236157i	$0.0758879\pi$
$211$	1447.62i	0.472314i	−0.971715	+	0.236157i	$0.924112\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1887.93	−0.598865
$216$	0	0
$217$	2583.19	0.808103
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6405.96i	− 1.94983i
$222$	0	0
$223$	5049.28i	1.51625i	0.652107	+	0.758127i	$0.273885\pi$
$223$	5049.28i	1.51625i	−0.652107	+	0.758127i	$0.726115\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5716.54	−1.67145	−0.835727	−	0.549146i	$-0.814953\pi$
$227$	−5716.54	−1.67145	−0.835727	+	0.549146i	$0.814953\pi$
$228$	0	0
$229$	2582.55	0.745240	0.372620	−	0.927984i	$-0.378460\pi$
$229$	2582.55	0.745240	0.372620	+	0.927984i	$0.378460\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	587.204i	0.165103i	0.996587	+	0.0825516i	$0.0263069\pi$
$233$	587.204i	0.165103i	−0.996587	+	0.0825516i	$0.973693\pi$
$234$	0	0
$235$	386.681i	0.107337i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5583.00	−1.51102	−0.755512	−	0.655135i	$-0.772612\pi$
$239$	−5583.00	−1.51102	−0.755512	+	0.655135i	$0.772612\pi$
$240$	0	0
$241$	−2056.95	−0.549791	−0.274895	−	0.961474i	$-0.588643\pi$
$241$	−2056.95	−0.549791	−0.274895	+	0.961474i	$0.588643\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1547.37i	− 0.403503i
$246$	0	0
$247$	− 5988.67i	− 1.54271i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1256.79	−0.316047	−0.158023	−	0.987435i	$-0.550512\pi$
$251$	−1256.79	−0.316047	−0.158023	+	0.987435i	$0.550512\pi$
$252$	0	0
$253$	792.890	0.197030
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 643.773i	− 0.156255i	−0.996943	−	0.0781273i	$-0.975106\pi$
$257$	− 643.773i	− 0.156255i	0.996943	−	0.0781273i	$-0.0248941\pi$
$258$	0	0
$259$	− 2501.55i	− 0.600150i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−917.108	−0.215024	−0.107512	−	0.994204i	$-0.534288\pi$
$263$	−917.108	−0.215024	−0.107512	+	0.994204i	$0.534288\pi$
$264$	0	0
$265$	925.501	0.214540
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6577.00i	1.49073i	0.666656	+	0.745366i	$0.267725\pi$
$269$	6577.00i	1.49073i	−0.666656	+	0.745366i	$0.732275\pi$
$270$	0	0
$271$	− 4656.21i	− 1.04371i	−0.853035	−	0.521854i	$-0.825241\pi$
$271$	− 4656.21i	− 1.04371i	0.853035	−	0.521854i	$-0.174759\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2146.74	0.470739
$276$	0	0
$277$	2881.42	0.625010	0.312505	−	0.949916i	$-0.398832\pi$
$277$	2881.42	0.625010	0.312505	+	0.949916i	$0.398832\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2604.93i	− 0.553015i	−0.961012	−	0.276507i	$-0.910823\pi$
$281$	− 2604.93i	− 0.553015i	0.961012	−	0.276507i	$-0.0891771\pi$
$282$	0	0
$283$	− 5786.02i	− 1.21535i	−0.794187	−	0.607674i	$-0.792103\pi$
$283$	− 5786.02i	− 1.21535i	0.794187	−	0.607674i	$-0.207897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3245.19	0.667448
$288$	0	0
$289$	−8825.50	−1.79636
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3279.55i	− 0.653902i	−0.945041	−	0.326951i	$-0.893979\pi$
$293$	− 3279.55i	− 0.653902i	0.945041	−	0.326951i	$-0.106021\pi$
$294$	0	0
$295$	4955.07i	0.977950i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1835.05	−0.354928
$300$	0	0
$301$	2855.43	0.546792
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2036.12i	− 0.382255i
$306$	0	0
$307$	− 3014.47i	− 0.560407i	−0.959941	−	0.280203i	$-0.909598\pi$
$307$	− 3014.47i	− 0.560407i	0.959941	−	0.280203i	$-0.0904019\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6230.80	−1.13606	−0.568032	−	0.823006i	$-0.692295\pi$
$311$	−6230.80	−1.13606	−0.568032	+	0.823006i	$0.692295\pi$
$312$	0	0
$313$	−3922.49	−0.708346	−0.354173	−	0.935180i	$-0.615238\pi$
$313$	−3922.49	−0.708346	−0.354173	+	0.935180i	$0.615238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6882.19i	1.21938i	0.792641	+	0.609689i	$0.208705\pi$
$317$	6882.19i	1.21938i	−0.792641	+	0.609689i	$0.791295\pi$
$318$	0	0
$319$	945.741i	0.165992i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12843.6	−2.21249
$324$	0	0
$325$	−4968.36	−0.847984
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 584.841i	− 0.0980040i
$330$	0	0
$331$	− 9489.08i	− 1.57573i	−0.615847	−	0.787866i	$-0.711186\pi$
$331$	− 9489.08i	− 1.57573i	0.615847	−	0.787866i	$-0.288814\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1134.41	0.185013
$336$	0	0
$337$	4578.52	0.740083	0.370041	−	0.929015i	$-0.379344\pi$
$337$	4578.52	0.740083	0.370041	+	0.929015i	$0.379344\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6907.50i	− 1.09696i
$342$	0	0
$343$	5369.42i	0.845253i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	942.483	0.145807	0.0729037	−	0.997339i	$-0.476773\pi$
$347$	942.483	0.145807	0.0729037	+	0.997339i	$0.476773\pi$
$348$	0	0
$349$	2124.84	0.325903	0.162951	−	0.986634i	$-0.447899\pi$
$349$	2124.84	0.325903	0.162951	+	0.986634i	$0.447899\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3881.60i	− 0.585260i	−0.956226	−	0.292630i	$-0.905470\pi$
$353$	− 3881.60i	− 0.585260i	0.956226	−	0.292630i	$-0.0945304\pi$
$354$	0	0
$355$	5487.18i	0.820363i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8050.44	−1.18353	−0.591763	−	0.806112i	$-0.701568\pi$
$359$	−8050.44	−1.18353	−0.591763	+	0.806112i	$0.701568\pi$
$360$	0	0
$361$	−5147.94	−0.750537
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2766.62i	0.396744i
$366$	0	0
$367$	− 6890.96i	− 0.980123i	−0.871688	−	0.490062i	$-0.836974\pi$
$367$	− 6890.96i	− 0.980123i	0.871688	−	0.490062i	$-0.163026\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1399.79	−0.195885
$372$	0	0
$373$	10512.5	1.45930	0.729649	−	0.683822i	$-0.239684\pi$
$373$	10512.5	1.45930	0.729649	+	0.683822i	$0.239684\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2188.80i	− 0.299016i
$378$	0	0
$379$	3139.69i	0.425528i	0.977104	+	0.212764i	$0.0682466\pi$
$379$	3139.69i	0.425528i	−0.977104	+	0.212764i	$0.931753\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5117.53	−0.682751	−0.341375	−	0.939927i	$-0.610893\pi$
$383$	−5117.53	−0.682751	−0.341375	+	0.939927i	$0.610893\pi$
$384$	0	0
$385$	1217.67	0.161190
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 787.004i	− 0.102578i	−0.998684	−	0.0512888i	$-0.983667\pi$
$389$	− 787.004i	− 0.102578i	0.998684	−	0.0512888i	$-0.0163329\pi$
$390$	0	0
$391$	3935.52i	0.509023i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1595.31	0.203212
$396$	0	0
$397$	−11160.6	−1.41092	−0.705458	−	0.708752i	$-0.749259\pi$
$397$	−11160.6	−1.41092	−0.705458	+	0.708752i	$0.749259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13224.6i	1.64689i	0.567394	+	0.823447i	$0.307952\pi$
$401$	13224.6i	1.64689i	−0.567394	+	0.823447i	$0.692048\pi$
$402$	0	0
$403$	15986.5i	1.97605i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6689.20	−0.814671
$408$	0	0
$409$	8153.09	0.985683	0.492841	−	0.870119i	$-0.335958\pi$
$409$	8153.09	0.985683	0.492841	+	0.870119i	$0.335958\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7494.36i	− 0.892914i
$414$	0	0
$415$	− 1975.69i	− 0.233693i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11682.5	−1.36212	−0.681061	−	0.732226i	$-0.738481\pi$
$419$	−11682.5	−1.36212	−0.681061	+	0.732226i	$0.738481\pi$
$420$	0	0
$421$	−9595.77	−1.11085	−0.555426	−	0.831566i	$-0.687445\pi$
$421$	−9595.77	−1.11085	−0.555426	+	0.831566i	$0.687445\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10655.4i	1.21614i
$426$	0	0
$427$	3079.56i	0.349017i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4294.48	0.479948	0.239974	−	0.970779i	$-0.422861\pi$
$431$	4294.48	0.479948	0.239974	+	0.970779i	$0.422861\pi$
$432$	0	0
$433$	168.392	0.0186892	0.00934460	−	0.999956i	$-0.497025\pi$
$433$	168.392	0.0186892	0.00934460	+	0.999956i	$0.497025\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3679.16i	0.402742i
$438$	0	0
$439$	− 1459.53i	− 0.158677i	−0.996848	−	0.0793387i	$-0.974719\pi$
$439$	− 1459.53i	− 0.158677i	0.996848	−	0.0793387i	$-0.0252809\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	333.411	0.0357581	0.0178790	−	0.999840i	$-0.494309\pi$
$443$	333.411	0.0357581	0.0178790	+	0.999840i	$0.494309\pi$
$444$	0	0
$445$	4320.11	0.460209
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 11454.3i	− 1.20393i	−0.798523	−	0.601964i	$-0.794385\pi$
$449$	− 11454.3i	− 1.20393i	0.798523	−	0.601964i	$-0.205615\pi$
$450$	0	0
$451$	− 8677.70i	− 0.906024i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2818.14	−0.290365
$456$	0	0
$457$	9680.26	0.990861	0.495431	−	0.868648i	$-0.335010\pi$
$457$	9680.26	0.990861	0.495431	+	0.868648i	$0.335010\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 14806.9i	− 1.49593i	−0.663737	−	0.747966i	$-0.731030\pi$
$461$	− 14806.9i	− 1.49593i	0.663737	−	0.747966i	$-0.268970\pi$
$462$	0	0
$463$	3658.08i	0.367183i	0.983003	+	0.183591i	$0.0587723\pi$
$463$	3658.08i	0.367183i	−0.983003	+	0.183591i	$0.941228\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9852.37	0.976260	0.488130	−	0.872771i	$-0.337679\pi$
$467$	9852.37	0.976260	0.488130	+	0.872771i	$0.337679\pi$
$468$	0	0
$469$	−1715.75	−0.168926
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 7635.47i	− 0.742240i
$474$	0	0
$475$	9961.26i	0.962219i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11107.9	1.05957	0.529784	−	0.848133i	$-0.322273\pi$
$479$	11107.9	1.05957	0.529784	+	0.848133i	$0.322273\pi$
$480$	0	0
$481$	15481.3	1.46754
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2619.50i	0.245248i
$486$	0	0
$487$	− 11703.7i	− 1.08900i	−0.838761	−	0.544500i	$-0.816719\pi$
$487$	− 11703.7i	− 1.08900i	0.838761	−	0.544500i	$-0.183281\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2851.25	0.262067	0.131034	−	0.991378i	$-0.458170\pi$
$491$	2851.25	0.262067	0.131034	+	0.991378i	$0.458170\pi$
$492$	0	0
$493$	−4694.20	−0.428836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8299.15i	− 0.749030i
$498$	0	0
$499$	− 14936.0i	− 1.33994i	−0.742389	−	0.669969i	$-0.766307\pi$
$499$	− 14936.0i	− 1.33994i	0.742389	−	0.669969i	$-0.233693\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9047.91	−0.802041	−0.401020	−	0.916069i	$-0.631344\pi$
$503$	−9047.91	−0.802041	−0.401020	+	0.916069i	$0.631344\pi$
$504$	0	0
$505$	−6731.52	−0.593166
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15173.5i	1.32132i	0.750683	+	0.660662i	$0.229724\pi$
$509$	15173.5i	1.32132i	−0.750683	+	0.660662i	$0.770276\pi$
$510$	0	0
$511$	− 4184.41i	− 0.362246i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8115.47	0.694389
$516$	0	0
$517$	−1563.87	−0.133035
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7375.96i	0.620243i	0.950697	+	0.310121i	$0.100370\pi$
$521$	7375.96i	0.620243i	−0.950697	+	0.310121i	$0.899630\pi$
$522$	0	0
$523$	8514.96i	0.711918i	0.934502	+	0.355959i	$0.115846\pi$
$523$	8514.96i	0.711918i	−0.934502	+	0.355959i	$0.884154\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34285.4	2.83396
$528$	0	0
$529$	−11039.6	−0.907342
$530$	0	0
$531$	0	0
$532$	0	0
$533$	20083.5i	1.63210i
$534$	0	0
$535$	− 1433.33i	− 0.115828i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6258.13	0.500105
$540$	0	0
$541$	2214.98	0.176025	0.0880125	−	0.996119i	$-0.471948\pi$
$541$	2214.98	0.176025	0.0880125	+	0.996119i	$0.471948\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3766.08i	− 0.296002i
$546$	0	0
$547$	− 3906.55i	− 0.305360i	−0.988276	−	0.152680i	$-0.951210\pi$
$547$	− 3906.55i	− 0.305360i	0.988276	−	0.152680i	$-0.0487904\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4388.41	−0.339297
$552$	0	0
$553$	−2412.85	−0.185542
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2978.14i	0.226549i	0.993564	+	0.113275i	$0.0361340\pi$
$557$	2978.14i	0.226549i	−0.993564	+	0.113275i	$0.963866\pi$
$558$	0	0
$559$	17671.4i	1.33706i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8786.44	−0.657734	−0.328867	−	0.944376i	$-0.606667\pi$
$563$	−8786.44	−0.657734	−0.328867	+	0.944376i	$0.606667\pi$
$564$	0	0
$565$	−4819.67	−0.358876
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19271.0i	1.41983i	0.704288	+	0.709914i	$0.251266\pi$
$569$	19271.0i	1.41983i	−0.704288	+	0.709914i	$0.748734\pi$
$570$	0	0
$571$	15535.7i	1.13861i	0.822125	+	0.569307i	$0.192788\pi$
$571$	15535.7i	1.13861i	−0.822125	+	0.569307i	$0.807212\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3052.33	0.221375
$576$	0	0
$577$	3783.58	0.272985	0.136493	−	0.990641i	$-0.456417\pi$
$577$	3783.58	0.272985	0.136493	+	0.990641i	$0.456417\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2988.15i	0.213373i
$582$	0	0
$583$	3743.06i	0.265903i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15515.4	1.09095	0.545477	−	0.838126i	$-0.316349\pi$
$587$	15515.4	1.09095	0.545477	+	0.838126i	$0.316349\pi$
$588$	0	0
$589$	32052.1	2.24225
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14098.1i	0.976292i	0.872762	+	0.488146i	$0.162327\pi$
$593$	14098.1i	0.976292i	−0.872762	+	0.488146i	$0.837673\pi$
$594$	0	0
$595$	6043.90i	0.416430i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−572.539	−0.0390539	−0.0195270	−	0.999809i	$-0.506216\pi$
$599$	−572.539	−0.0390539	−0.0195270	+	0.999809i	$0.506216\pi$
$600$	0	0
$601$	−23463.4	−1.59250	−0.796249	−	0.604969i	$-0.793185\pi$
$601$	−23463.4	−1.59250	−0.796249	+	0.604969i	$0.793185\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4515.52i	0.303441i
$606$	0	0
$607$	− 1980.51i	− 0.132433i	−0.997805	−	0.0662163i	$-0.978907\pi$
$607$	− 1980.51i	− 0.132433i	0.997805	−	0.0662163i	$-0.0210927\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3619.39	0.239648
$612$	0	0
$613$	−2479.01	−0.163338	−0.0816691	−	0.996660i	$-0.526025\pi$
$613$	−2479.01	−0.163338	−0.0816691	+	0.996660i	$0.526025\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20002.8i	− 1.30516i	−0.757722	−	0.652578i	$-0.773688\pi$
$617$	− 20002.8i	− 1.30516i	0.757722	−	0.652578i	$-0.226312\pi$
$618$	0	0
$619$	− 22292.4i	− 1.44751i	−0.690058	−	0.723754i	$-0.742415\pi$
$619$	− 22292.4i	− 1.44751i	0.690058	−	0.723754i	$-0.257585\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6534.01	−0.420192
$624$	0	0
$625$	4002.52	0.256161
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 33201.9i	− 2.10469i
$630$	0	0
$631$	24203.8i	1.52700i	0.645807	+	0.763501i	$0.276521\pi$
$631$	24203.8i	1.52700i	−0.645807	+	0.763501i	$0.723479\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4406.32	0.275370
$636$	0	0
$637$	−14483.7	−0.900885
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 4423.87i	− 0.272593i	−0.990668	−	0.136297i	$-0.956480\pi$
$641$	− 4423.87i	− 0.272593i	0.990668	−	0.136297i	$-0.0435200\pi$
$642$	0	0
$643$	− 11961.5i	− 0.733619i	−0.930296	−	0.366810i	$-0.880450\pi$
$643$	− 11961.5i	− 0.733619i	0.930296	−	0.366810i	$-0.119550\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6287.16	0.382030	0.191015	−	0.981587i	$-0.438822\pi$
$647$	6287.16	0.382030	0.191015	+	0.981587i	$0.438822\pi$
$648$	0	0
$649$	−20040.1	−1.21208
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18793.2i	− 1.12624i	−0.826376	−	0.563119i	$-0.809601\pi$
$653$	− 18793.2i	− 1.12624i	0.826376	−	0.563119i	$-0.190399\pi$
$654$	0	0
$655$	11832.2i	0.705833i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20827.2	−1.23113	−0.615563	−	0.788088i	$-0.711071\pi$
$659$	−20827.2	−1.23113	−0.615563	+	0.788088i	$0.711071\pi$
$660$	0	0
$661$	6038.16	0.355306	0.177653	−	0.984093i	$-0.443150\pi$
$661$	6038.16	0.355306	0.177653	+	0.984093i	$0.443150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5650.20i	0.329482i
$666$	0	0
$667$	1344.70i	0.0780612i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8234.79	0.473771
$672$	0	0
$673$	−19811.5	−1.13474	−0.567368	−	0.823464i	$-0.692038\pi$
$673$	−19811.5	−1.13474	−0.567368	+	0.823464i	$0.692038\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22962.5i	− 1.30357i	−0.758402	−	0.651787i	$-0.774020\pi$
$677$	− 22962.5i	− 1.30357i	0.758402	−	0.651787i	$-0.225980\pi$
$678$	0	0
$679$	− 3961.90i	− 0.223923i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33097.3	1.85422	0.927109	−	0.374791i	$-0.122286\pi$
$683$	33097.3	1.85422	0.927109	+	0.374791i	$0.122286\pi$
$684$	0	0
$685$	−10993.9	−0.613222
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 8662.84i	− 0.478995i
$690$	0	0
$691$	− 21151.0i	− 1.16443i	−0.813034	−	0.582216i	$-0.802186\pi$
$691$	− 21151.0i	− 1.16443i	0.813034	−	0.582216i	$-0.197814\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−547.267	−0.0298691
$696$	0	0
$697$	43071.9	2.34069
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18752.3i	1.01036i	0.863013	+	0.505182i	$0.168575\pi$
$701$	18752.3i	1.01036i	−0.863013	+	0.505182i	$0.831425\pi$
$702$	0	0
$703$	− 31039.1i	− 1.66524i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10181.2	0.541588
$708$	0	0
$709$	25964.1	1.37532	0.687660	−	0.726033i	$-0.258638\pi$
$709$	25964.1	1.37532	0.687660	+	0.726033i	$0.258638\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 9821.38i	− 0.515868i
$714$	0	0
$715$	7535.75i	0.394155i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11741.1	0.608998	0.304499	−	0.952513i	$-0.401511\pi$
$719$	11741.1	0.608998	0.304499	+	0.952513i	$0.401511\pi$
$720$	0	0
$721$	−12274.4	−0.634010
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3640.74i	0.186502i
$726$	0	0
$727$	− 5637.87i	− 0.287616i	−0.989606	−	0.143808i	$-0.954065\pi$
$727$	− 5637.87i	− 0.287616i	0.989606	−	0.143808i	$-0.0459348\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	37898.8	1.91756
$732$	0	0
$733$	23807.3	1.19965	0.599825	−	0.800131i	$-0.295237\pi$
$733$	23807.3	1.19965	0.599825	+	0.800131i	$0.295237\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4587.96i	0.229307i
$738$	0	0
$739$	− 3556.31i	− 0.177024i	−0.996075	−	0.0885122i	$-0.971789\pi$
$739$	− 3556.31i	− 0.177024i	0.996075	−	0.0885122i	$-0.0282112\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24054.6	1.18772	0.593860	−	0.804568i	$-0.297603\pi$
$743$	24054.6	1.18772	0.593860	+	0.804568i	$0.297603\pi$
$744$	0	0
$745$	16142.8	0.793860
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2167.86i	0.105757i
$750$	0	0
$751$	− 15373.0i	− 0.746960i	−0.927638	−	0.373480i	$-0.878164\pi$
$751$	− 15373.0i	− 0.746960i	0.927638	−	0.373480i	$-0.121836\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21569.6	−1.03973
$756$	0	0
$757$	−26000.8	−1.24837	−0.624184	−	0.781277i	$-0.714569\pi$
$757$	−26000.8	−1.24837	−0.624184	+	0.781277i	$0.714569\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9461.72i	0.450706i	0.974277	+	0.225353i	$0.0723535\pi$
$761$	9461.72i	0.450706i	−0.974277	+	0.225353i	$0.927646\pi$
$762$	0	0
$763$	5696.06i	0.270264i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46380.2	2.18343
$768$	0	0
$769$	11196.2	0.525028	0.262514	−	0.964928i	$-0.415448\pi$
$769$	11196.2	0.525028	0.262514	+	0.964928i	$0.415448\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39575.0i	1.84141i	0.390255	+	0.920707i	$0.372387\pi$
$773$	39575.0i	1.84141i	−0.390255	+	0.920707i	$0.627613\pi$
$774$	0	0
$775$	− 26591.2i	− 1.23250i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40266.2	1.85197
$780$	0	0
$781$	−22192.1	−1.01677
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4743.53i	0.215674i
$786$	0	0
$787$	25069.2i	1.13548i	0.823209	+	0.567739i	$0.192182\pi$
$787$	25069.2i	1.13548i	−0.823209	+	0.567739i	$0.807818\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7289.58	0.327671
$792$	0	0
$793$	−19058.4	−0.853448
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5639.39i	− 0.250637i	−0.992117	−	0.125318i	$-0.960005\pi$
$797$	− 5639.39i	− 0.250637i	0.992117	−	0.125318i	$-0.0399952\pi$
$798$	0	0
$799$	− 7762.31i	− 0.343693i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11189.2	−0.491729
$804$	0	0
$805$	1731.33	0.0758030
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 34032.0i	− 1.47899i	−0.673163	−	0.739494i	$-0.735065\pi$
$809$	− 34032.0i	− 1.47899i	0.673163	−	0.739494i	$-0.264935\pi$
$810$	0	0
$811$	− 20119.5i	− 0.871137i	−0.900156	−	0.435569i	$-0.856547\pi$
$811$	− 20119.5i	− 0.871137i	0.900156	−	0.435569i	$-0.143453\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6038.49	0.259532
$816$	0	0
$817$	35430.0	1.51718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9057.81i	0.385042i	0.981293	+	0.192521i	$0.0616664\pi$
$821$	9057.81i	0.385042i	−0.981293	+	0.192521i	$0.938334\pi$
$822$	0	0
$823$	26277.7i	1.11298i	0.830853	+	0.556491i	$0.187853\pi$
$823$	26277.7i	1.11298i	−0.830853	+	0.556491i	$0.812147\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21737.7	0.914018	0.457009	−	0.889462i	$-0.348921\pi$
$827$	21737.7	0.914018	0.457009	+	0.889462i	$0.348921\pi$
$828$	0	0
$829$	−27552.6	−1.15433	−0.577167	−	0.816626i	$-0.695842\pi$
$829$	−27552.6	−1.15433	−0.577167	+	0.816626i	$0.695842\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31062.3i	1.29201i
$834$	0	0
$835$	8485.65i	0.351686i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12231.7	−0.503320	−0.251660	−	0.967816i	$-0.580977\pi$
$839$	−12231.7	−0.503320	−0.251660	+	0.967816i	$0.580977\pi$
$840$	0	0
$841$	22785.1	0.934236
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4612.48i	− 0.187780i
$846$	0	0
$847$	− 6829.56i	− 0.277056i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9511.00	−0.383117
$852$	0	0
$853$	2257.34	0.0906093	0.0453046	−	0.998973i	$-0.485574\pi$
$853$	2257.34	0.0906093	0.0453046	+	0.998973i	$0.485574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24908.2i	− 0.992820i	−0.868088	−	0.496410i	$-0.834651\pi$
$857$	− 24908.2i	− 0.992820i	0.868088	−	0.496410i	$-0.165349\pi$
$858$	0	0
$859$	− 11850.7i	− 0.470710i	−0.971909	−	0.235355i	$-0.924375\pi$
$859$	− 11850.7i	− 0.470710i	0.971909	−	0.235355i	$-0.0756253\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43145.0	1.70182	0.850910	−	0.525311i	$-0.176051\pi$
$863$	43145.0	1.70182	0.850910	+	0.525311i	$0.176051\pi$
$864$	0	0
$865$	3707.13	0.145718
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6452.01i	0.251864i
$870$	0	0
$871$	− 10618.3i	− 0.413072i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11133.1	0.430133
$876$	0	0
$877$	14333.4	0.551886	0.275943	−	0.961174i	$-0.411010\pi$
$877$	14333.4	0.551886	0.275943	+	0.961174i	$0.411010\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 5066.51i	− 0.193752i	−0.995296	−	0.0968758i	$-0.969115\pi$
$881$	− 5066.51i	− 0.193752i	0.995296	−	0.0968758i	$-0.0308850\pi$
$882$	0	0
$883$	7046.42i	0.268551i	0.990944	+	0.134276i	$0.0428708\pi$
$883$	7046.42i	0.268551i	−0.990944	+	0.134276i	$0.957129\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52122.7	−1.97307	−0.986534	−	0.163559i	$-0.947703\pi$
$887$	−52122.7	−1.97307	−0.986534	+	0.163559i	$0.947703\pi$
$888$	0	0
$889$	−6664.41	−0.251425
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7256.67i	− 0.271932i
$894$	0	0
$895$	16987.4i	0.634444i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11714.7	0.434602
$900$	0	0
$901$	−18578.7	−0.686955
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7908.56i	0.290486i
$906$	0	0
$907$	41205.1i	1.50848i	0.656599	+	0.754240i	$0.271994\pi$
$907$	41205.1i	1.50848i	−0.656599	+	0.754240i	$0.728006\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34305.8	1.24764	0.623821	−	0.781567i	$-0.285580\pi$
$911$	34305.8	1.24764	0.623821	+	0.781567i	$0.285580\pi$
$912$	0	0
$913$	7990.37	0.289642
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 17895.7i	− 0.644458i
$918$	0	0
$919$	43537.1i	1.56274i	0.624068	+	0.781370i	$0.285479\pi$
$919$	43537.1i	1.56274i	−0.624068	+	0.781370i	$0.714521\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	51360.8	1.83160
$924$	0	0
$925$	−25750.9	−0.915333
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8764.21i	0.309520i	0.987952	+	0.154760i	$0.0494605\pi$
$929$	8764.21i	0.309520i	−0.987952	+	0.154760i	$0.950539\pi$
$930$	0	0
$931$	29038.9i	1.02225i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16161.5	0.565281
$936$	0	0
$937$	−22745.0	−0.793006	−0.396503	−	0.918033i	$-0.629776\pi$
$937$	−22745.0	−0.793006	−0.396503	+	0.918033i	$0.629776\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 24740.5i	− 0.857084i	−0.903522	−	0.428542i	$-0.859027\pi$
$941$	− 24740.5i	− 0.857084i	0.903522	−	0.428542i	$-0.140973\pi$
$942$	0	0
$943$	− 12338.3i	− 0.426078i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1886.69	−0.0647405	−0.0323703	−	0.999476i	$-0.510306\pi$
$947$	−1886.69	−0.0647405	−0.0323703	+	0.999476i	$0.510306\pi$
$948$	0	0
$949$	25896.0	0.885796
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36292.1i	1.23360i	0.787122	+	0.616798i	$0.211570\pi$
$953$	36292.1i	1.23360i	−0.787122	+	0.616798i	$0.788430\pi$
$954$	0	0
$955$	20368.5i	0.690165i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16627.9	0.559900
$960$	0	0
$961$	−55770.8	−1.87207
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 17008.7i	− 0.567387i
$966$	0	0
$967$	− 12599.4i	− 0.418996i	−0.977809	−	0.209498i	$-0.932817\pi$
$967$	− 12599.4i	− 0.418996i	0.977809	−	0.209498i	$-0.0671830\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41974.3	−1.38725	−0.693625	−	0.720336i	$-0.743988\pi$
$971$	−41974.3	−1.38725	−0.693625	+	0.720336i	$0.743988\pi$
$972$	0	0
$973$	827.721	0.0272719
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33488.8i	1.09662i	0.836274	+	0.548312i	$0.184729\pi$
$977$	33488.8i	1.09662i	−0.836274	+	0.548312i	$0.815271\pi$
$978$	0	0
$979$	17472.1i	0.570387i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3461.54	−0.112315	−0.0561576	−	0.998422i	$-0.517885\pi$
$983$	−3461.54	−0.112315	−0.0561576	+	0.998422i	$0.517885\pi$
$984$	0	0
$985$	14204.0	0.459470
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10856.5i	− 0.349055i
$990$	0	0
$991$	20067.6i	0.643256i	0.946866	+	0.321628i	$0.104230\pi$
$991$	20067.6i	0.643256i	−0.946866	+	0.321628i	$0.895770\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3288.41	0.104774
$996$	0	0
$997$	−821.726	−0.0261026	−0.0130513	−	0.999915i	$-0.504154\pi$
$997$	−821.726	−0.0261026	−0.0130513	+	0.999915i	$0.504154\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.c.j.1727.4		12
3.2	odd	2	inner	1728.4.c.j.1727.10		12
4.3	odd	2	inner	1728.4.c.j.1727.3		12
8.3	odd	2		108.4.b.b.107.4	yes	12
8.5	even	2		108.4.b.b.107.10	yes	12
12.11	even	2	inner	1728.4.c.j.1727.9		12
24.5	odd	2		108.4.b.b.107.3	✓	12
24.11	even	2		108.4.b.b.107.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.b.b.107.3	✓	12	24.5	odd	2
108.4.b.b.107.4	yes	12	8.3	odd	2
108.4.b.b.107.9	yes	12	24.11	even	2
108.4.b.b.107.10	yes	12	8.5	even	2
1728.4.c.j.1727.3		12	4.3	odd	2	inner
1728.4.c.j.1727.4		12	1.1	even	1	trivial
1728.4.c.j.1727.9		12	12.11	even	2	inner
1728.4.c.j.1727.10		12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.83890i q^{5} +8.83113i q^{7} +O(q^{10})\)	\(q-5.83890i q^{5} +8.83113i q^{7} +23.6146 q^{11} -54.6531 q^{13} +117.211i q^{17} +109.576i q^{19} +33.5763 q^{23} +90.9072 q^{25} +40.0490i q^{29} -292.510i q^{31} +51.5641 q^{35} -283.265 q^{37} -367.472i q^{41} -323.337i q^{43} -66.2249 q^{47} +265.011 q^{49} +158.506i q^{53} -137.883i q^{55} -848.630 q^{59} +348.716 q^{61} +319.114i q^{65} +194.285i q^{67} -939.761 q^{71} -473.826 q^{73} +208.544i q^{77} +273.221i q^{79} +338.366 q^{83} +684.386 q^{85} +739.884i q^{89} -482.648i q^{91} +639.805 q^{95} -448.629 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) \) 12 * q + 72 * q^13 - 384 * q^25 + 240 * q^37 + 288 * q^49 - 144 * q^61 + 156 * q^73 + 168 * q^85 + 516 * q^97

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