LMFDB - Embedded newform 2700.3.u.a.2249.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,3,Mod(449,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.u (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2249.4
Root			$-1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	2700.2249
Dual form			2700.3.u.a.449.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.1723 + 5.87298i) q^{7} +(-13.1190 - 7.57423i) q^{11} +(15.3685 - 8.87298i) q^{13} +15.1485 q^{17} +11.2540 q^{19} +(16.8805 + 29.2379i) q^{23} +(8.23790 + 4.75615i) q^{29} +(-28.1109 - 48.6895i) q^{31} +14.0000i q^{37} +(22.5000 - 12.9904i) q^{41} +(-17.3065 - 9.99193i) q^{43} +(36.1531 - 62.6190i) q^{47} +(44.4839 + 77.0483i) q^{49} -37.6651 q^{53} +(-55.1190 + 31.8229i) q^{59} +(-0.618950 + 1.07205i) q^{61} +(74.4363 - 42.9758i) q^{67} +22.1046i q^{71} +60.2379i q^{73} +(-88.9666 - 154.095i) q^{77} +(51.6190 - 89.4066i) q^{79} +(45.0333 - 78.0000i) q^{83} -12.0964i q^{89} +208.444 q^{91} +(-86.3406 - 49.8488i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{11} + 152 q^{19} - 120 q^{29} - 8 q^{31} + 180 q^{41} + 108 q^{49} - 348 q^{59} + 88 q^{61} + 320 q^{79} + 800 q^{91}+O(q^{100}) $ 8 * q - 12 * q^11 + 152 * q^19 - 120 * q^29 - 8 * q^31 + 180 * q^41 + 108 * q^49 - 348 * q^59 + 88 * q^61 + 320 * q^79 + 800 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	10.1723	+	5.87298i	1.45319	+	0.838998i	0.998661	−	0.0517360i	$-0.0164754\pi$
$7$	10.1723	+	5.87298i	1.45319	+	0.838998i	0.454526	+	0.890734i	$0.349809\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.1190	−	7.57423i	−1.19263	−	0.688566i	−0.233729	−	0.972302i	$-0.575093\pi$
$11$	−13.1190	−	7.57423i	−1.19263	−	0.688566i	−0.958903	+	0.283735i	$0.908426\pi$
$12$		0			0
$13$	15.3685	−	8.87298i	1.18219	−	0.682537i	0.225669	−	0.974204i	$-0.427543\pi$
$13$	15.3685	−	8.87298i	1.18219	−	0.682537i	0.956520	+	0.291667i	$0.0942099\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.1485			0.891086			0.445543	−	0.895261i	$-0.353011\pi$
$17$	15.1485			0.891086			0.445543	+	0.895261i	$0.353011\pi$
$18$		0			0
$19$	11.2540			0.592318			0.296159	−	0.955139i	$-0.404294\pi$
$19$	11.2540			0.592318			0.296159	+	0.955139i	$0.404294\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	16.8805	+	29.2379i	0.733935	+	1.27121i	0.955189	+	0.295997i	$0.0956518\pi$
$23$	16.8805	+	29.2379i	0.733935	+	1.27121i	−0.221254	+	0.975216i	$0.571015\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.23790	+	4.75615i	0.284066	+	0.164005i	0.635262	−	0.772296i	$-0.280892\pi$
$29$	8.23790	+	4.75615i	0.284066	+	0.164005i	−0.351197	+	0.936302i	$0.614225\pi$
$30$		0			0
$31$	−28.1109	−	48.6895i	−0.906803	−	1.57063i	−0.818479	−	0.574537i	$-0.805182\pi$
$31$	−28.1109	−	48.6895i	−0.906803	−	1.57063i	−0.0883237	−	0.996092i	$-0.528151\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			14.0000i			0.378378i	0.981941	+	0.189189i	$0.0605859\pi$
$37$			14.0000i			0.378378i	−0.981941	+	0.189189i	$0.939414\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	22.5000	−	12.9904i	0.548780	−	0.316839i	−0.199849	−	0.979827i	$-0.564045\pi$
$41$	22.5000	−	12.9904i	0.548780	−	0.316839i	0.748630	+	0.662988i	$0.230712\pi$
$42$		0			0
$43$	−17.3065	−	9.99193i	−0.402478	−	0.232371i	0.285075	−	0.958505i	$-0.407982\pi$
$43$	−17.3065	−	9.99193i	−0.402478	−	0.232371i	−0.687552	+	0.726135i	$0.741315\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	36.1531	−	62.6190i	0.769214	−	1.33232i	−0.168775	−	0.985655i	$-0.553981\pi$
$47$	36.1531	−	62.6190i	0.769214	−	1.33232i	0.937990	−	0.346664i	$-0.112685\pi$
$48$		0			0
$49$	44.4839	+	77.0483i	0.907834	+	1.57241i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−37.6651			−0.710663			−0.355331	−	0.934740i	$-0.615632\pi$
$53$	−37.6651			−0.710663			−0.355331	+	0.934740i	$0.615632\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−55.1190	+	31.8229i	−0.934219	+	0.539372i	−0.888144	−	0.459566i	$-0.848005\pi$
$59$	−55.1190	+	31.8229i	−0.934219	+	0.539372i	−0.0460759	+	0.998938i	$0.514672\pi$
$60$		0			0
$61$	−0.618950	+	1.07205i	−0.0101467	+	0.0175746i	−0.871054	−	0.491187i	$-0.836563\pi$
$61$	−0.618950	+	1.07205i	−0.0101467	+	0.0175746i	0.860907	+	0.508762i	$0.169897\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	74.4363	−	42.9758i	1.11099	−	0.641430i	0.171904	−	0.985114i	$-0.445008\pi$
$67$	74.4363	−	42.9758i	1.11099	−	0.641430i	0.939085	+	0.343684i	$0.111675\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			22.1046i			0.311332i	0.987810	+	0.155666i	$0.0497524\pi$
$71$			22.1046i			0.311332i	−0.987810	+	0.155666i	$0.950248\pi$
$72$		0			0
$73$			60.2379i			0.825177i	0.910918	+	0.412588i	$0.135375\pi$
$73$			60.2379i			0.825177i	−0.910918	+	0.412588i	$0.864625\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−88.9666	−	154.095i	−1.15541	−	2.00123i
$78$		0			0
$79$	51.6190	−	89.4066i	0.653404	−	1.13173i	−0.328887	−	0.944369i	$-0.606673\pi$
$79$	51.6190	−	89.4066i	0.653404	−	1.13173i	0.982291	−	0.187360i	$-0.0599932\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	45.0333	−	78.0000i	0.542570	−	0.939759i	−0.456185	−	0.889885i	$-0.650785\pi$
$83$	45.0333	−	78.0000i	0.542570	−	0.939759i	0.998756	−	0.0498743i	$-0.0158821\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.0964i		−	0.135915i	−0.997688	−	0.0679574i	$-0.978352\pi$
$89$		−	12.0964i		−	0.135915i	0.997688	−	0.0679574i	$-0.0216482\pi$
$90$		0			0
$91$	208.444			2.29059
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−86.3406	−	49.8488i	−0.890110	−	0.513905i	−0.0161312	−	0.999870i	$-0.505135\pi$
$97$	−86.3406	−	49.8488i	−0.890110	−	0.513905i	−0.873978	+	0.485965i	$0.838468\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	134.238	+	77.5023i	1.32909	+	0.767349i	0.985159	−	0.171646i	$-0.0549086\pi$
$101$	134.238	+	77.5023i	1.32909	+	0.767349i	0.343929	+	0.938995i	$0.388242\pi$
$102$		0			0
$103$	−134.852	+	77.8569i	−1.30924	+	0.755892i	−0.981970	−	0.189038i	$-0.939463\pi$
$103$	−134.852	+	77.8569i	−1.30924	+	0.755892i	−0.327273	+	0.944930i	$0.606130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	84.8705			0.793182			0.396591	−	0.917995i	$-0.370193\pi$
$107$	84.8705			0.793182			0.396591	+	0.917995i	$0.370193\pi$
$108$		0			0
$109$	38.0323			0.348920			0.174460	−	0.984664i	$-0.444182\pi$
$109$	38.0323			0.348920			0.174460	+	0.984664i	$0.444182\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.46410	+	6.00000i	0.0306558	+	0.0530973i	0.880946	−	0.473216i	$-0.156907\pi$
$113$	3.46410	+	6.00000i	0.0306558	+	0.0530973i	−0.850290	+	0.526314i	$0.823574\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	154.095	+	88.9666i	1.29491	+	0.747619i
$120$		0			0
$121$	54.2379	+	93.9428i	0.448247	+	0.776387i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			145.903i			1.14884i	0.818559	+	0.574422i	$0.194773\pi$
$127$			145.903i			1.14884i	−0.818559	+	0.574422i	$0.805227\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−57.7137	+	33.3210i	−0.440563	+	0.254359i	−0.703836	−	0.710362i	$-0.748531\pi$
$131$	−57.7137	+	33.3210i	−0.440563	+	0.254359i	0.263274	+	0.964721i	$0.415198\pi$
$132$		0			0
$133$	114.479	+	66.0948i	0.860748	+	0.496953i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	36.6070	−	63.4052i	0.267205	−	0.462812i	−0.700934	−	0.713226i	$-0.747233\pi$
$137$	36.6070	−	63.4052i	0.267205	−	0.462812i	0.968139	+	0.250414i	$0.0805667\pi$
$138$		0			0
$139$	−53.3569	−	92.4168i	−0.383862	−	0.664869i	0.607748	−	0.794130i	$-0.292073\pi$
$139$	−53.3569	−	92.4168i	−0.383862	−	0.664869i	−0.991611	+	0.129261i	$0.958740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−268.824			−1.87989
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−144.333	+	83.3305i	−0.968676	+	0.559265i	−0.898832	−	0.438293i	$-0.855583\pi$
$149$	−144.333	+	83.3305i	−0.968676	+	0.559265i	−0.0698433	+	0.997558i	$0.522250\pi$
$150$		0			0
$151$	−1.71370	+	2.96822i	−0.0113490	+	0.0196571i	−0.871644	−	0.490139i	$-0.836946\pi$
$151$	−1.71370	+	2.96822i	−0.0113490	+	0.0196571i	0.860295	+	0.509796i	$0.170279\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−58.8338	+	33.9677i	−0.374738	+	0.216355i	−0.675526	−	0.737336i	$-0.736084\pi$
$157$	−58.8338	+	33.9677i	−0.374738	+	0.216355i	0.300788	+	0.953691i	$0.402750\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			396.556i			2.46308i
$162$		0			0
$163$		−	76.4113i		−	0.468781i	−0.972143	−	0.234390i	$-0.924691\pi$
$163$		−	76.4113i		−	0.468781i	0.972143	−	0.234390i	$-0.0753094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	141.752	+	245.522i	0.848816	+	1.47019i	0.882266	+	0.470752i	$0.156017\pi$
$167$	141.752	+	245.522i	0.848816	+	1.47019i	−0.0334495	+	0.999440i	$0.510649\pi$
$168$		0			0
$169$	72.9597	−	126.370i	0.431714	−	0.747751i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	67.4941	−	116.903i	0.390139	−	0.675741i	−0.602328	−	0.798248i	$-0.705760\pi$
$173$	67.4941	−	116.903i	0.390139	−	0.675741i	0.992468	+	0.122507i	$0.0390935\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	171.445i		−	0.957794i	−0.877871	−	0.478897i	$-0.841037\pi$
$179$		−	171.445i		−	0.957794i	0.877871	−	0.478897i	$-0.158963\pi$
$180$		0			0
$181$	120.794			0.667372			0.333686	−	0.942684i	$-0.391707\pi$
$181$	120.794			0.667372			0.333686	+	0.942684i	$0.391707\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−198.732	−	114.738i	−1.06274	−	0.613572i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	241.808	+	139.608i	1.26601	+	0.730933i	0.974231	−	0.225552i	$-0.0724186\pi$
$191$	241.808	+	139.608i	1.26601	+	0.730933i	0.291782	+	0.956485i	$0.405752\pi$
$192$		0			0
$193$	91.1806	−	52.6431i	0.472438	−	0.272762i	−0.244822	−	0.969568i	$-0.578729\pi$
$193$	91.1806	−	52.6431i	0.472438	−	0.272762i	0.717260	+	0.696806i	$0.245396\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−57.9538			−0.294182			−0.147091	−	0.989123i	$-0.546991\pi$
$197$	−57.9538			−0.294182			−0.147091	+	0.989123i	$0.546991\pi$
$198$		0			0
$199$	25.2702			0.126986			0.0634929	−	0.997982i	$-0.479776\pi$
$199$	25.2702			0.126986			0.0634929	+	0.997982i	$0.479776\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	55.8656	+	96.7621i	0.275200	+	0.476661i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−147.641	−	85.2406i	−0.706417	−	0.407850i
$210$		0			0
$211$	52.0161	+	90.0946i	0.246522	+	0.426989i	0.962558	−	0.271074i	$-0.0873789\pi$
$211$	52.0161	+	90.0946i	0.246522	+	0.426989i	−0.716036	+	0.698063i	$0.754046\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	660.379i		−	3.04322i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	232.808	−	134.412i	1.05343	−	0.608199i
$222$		0			0
$223$	214.223	+	123.681i	0.960639	+	0.554625i	0.896370	−	0.443307i	$-0.146195\pi$
$223$	214.223	+	123.681i	0.960639	+	0.554625i	0.0642694	+	0.997933i	$0.479528\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	151.114	−	261.738i	0.665702	−	1.15303i	−0.313392	−	0.949624i	$-0.601465\pi$
$227$	151.114	−	261.738i	0.665702	−	1.15303i	0.979094	−	0.203407i	$-0.0652013\pi$
$228$		0			0
$229$	133.111	+	230.555i	0.581270	+	1.00679i	0.995329	+	0.0965395i	$0.0307774\pi$
$229$	133.111	+	230.555i	0.581270	+	1.00679i	−0.414059	+	0.910250i	$0.635889\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	176.969			0.759526			0.379763	−	0.925084i	$-0.376006\pi$
$233$	176.969			0.759526			0.379763	+	0.925084i	$0.376006\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	68.2863	−	39.4251i	0.285717	−	0.164959i	−0.350292	−	0.936641i	$-0.613918\pi$
$239$	68.2863	−	39.4251i	0.285717	−	0.164959i	0.636009	+	0.771682i	$0.280584\pi$
$240$		0			0
$241$	110.246	−	190.952i	0.457452	−	0.792330i	−0.541373	−	0.840782i	$-0.682095\pi$
$241$	110.246	−	190.952i	0.457452	−	0.792330i	0.998826	+	0.0484519i	$0.0154288\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	172.957	−	99.8569i	0.700231	−	0.404279i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	277.373i		−	1.10507i	−0.833490	−	0.552535i	$-0.813661\pi$
$251$		−	277.373i		−	1.10507i	0.833490	−	0.552535i	$-0.186339\pi$
$252$		0			0
$253$		−	511.427i		−	2.02145i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	76.0322	+	131.692i	0.295845	+	0.512418i	0.975181	−	0.221409i	$-0.0710655\pi$
$257$	76.0322	+	131.692i	0.295845	+	0.512418i	−0.679336	+	0.733827i	$0.737732\pi$
$258$		0			0
$259$	−82.2218	+	142.412i	−0.317459	+	0.549854i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.85999	−	4.95365i	0.0108745	−	0.0188352i	−0.860537	−	0.509388i	$-0.829872\pi$
$263$	2.85999	−	4.95365i	0.0108745	−	0.0188352i	0.871411	+	0.490553i	$0.163205\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			361.531i			1.34398i	0.740560	+	0.671990i	$0.234560\pi$
$269$			361.531i			1.34398i	−0.740560	+	0.671990i	$0.765440\pi$
$270$		0			0
$271$	507.427			1.87243			0.936213	−	0.351433i	$-0.114306\pi$
$271$	507.427			1.87243			0.936213	+	0.351433i	$0.114306\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−193.606	−	111.778i	−0.698937	−	0.403532i	0.108014	−	0.994149i	$-0.465551\pi$
$277$	−193.606	−	111.778i	−0.698937	−	0.403532i	−0.806951	+	0.590618i	$0.798884\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	39.8105	+	22.9846i	0.141674	+	0.0817957i	0.569162	−	0.822226i	$-0.307268\pi$
$281$	39.8105	+	22.9846i	0.141674	+	0.0817957i	−0.427487	+	0.904021i	$0.640601\pi$
$282$		0			0
$283$	−124.736	+	72.0161i	−0.440762	+	0.254474i	−0.703921	−	0.710279i	$-0.748569\pi$
$283$	−124.736	+	72.0161i	−0.440762	+	0.254474i	0.263159	+	0.964753i	$0.415236\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	305.169			1.06331
$288$		0			0
$289$	−59.5242			−0.205966
$290$		0			0
$291$		0			0
$292$		0			0
$293$	208.698	+	361.476i	0.712280	+	1.23371i	0.963999	+	0.265905i	$0.0856709\pi$
$293$	208.698	+	361.476i	0.712280	+	1.23371i	−0.251719	+	0.967800i	$0.580996\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	518.855	+	299.561i	1.73530	+	1.00188i
$300$		0			0
$301$	−117.365	−	203.282i	−0.389917	−	0.675355i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	310.048i		−	1.00993i	−0.863140	−	0.504965i	$-0.831505\pi$
$307$		−	310.048i		−	1.00993i	0.863140	−	0.504965i	$-0.168495\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	242.238	−	139.856i	0.778900	−	0.449698i	−0.0571403	−	0.998366i	$-0.518198\pi$
$311$	242.238	−	139.856i	0.778900	−	0.449698i	0.836040	+	0.548668i	$0.184865\pi$
$312$		0			0
$313$	−25.2509	−	14.5786i	−0.0806738	−	0.0465771i	0.459120	−	0.888374i	$-0.348165\pi$
$313$	−25.2509	−	14.5786i	−0.0806738	−	0.0465771i	−0.539794	+	0.841797i	$0.681498\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	62.9859	−	109.095i	0.198694	−	0.344147i	−0.749411	−	0.662105i	$-0.769663\pi$
$317$	62.9859	−	109.095i	0.198694	−	0.344147i	0.948105	+	0.317957i	$0.102997\pi$
$318$		0			0
$319$	−72.0484	−	124.791i	−0.225857	−	0.391196i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	170.481			0.527806
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	735.520	−	424.653i	2.23562	−	1.29074i
$330$		0			0
$331$	232.903	−	403.400i	0.703635	−	1.21873i	−0.263547	−	0.964647i	$-0.584892\pi$
$331$	232.903	−	403.400i	0.703635	−	1.21873i	0.967182	−	0.254085i	$-0.0817743\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	303.259	−	175.087i	0.899878	−	0.519545i	0.0227177	−	0.999742i	$-0.492768\pi$
$337$	303.259	−	175.087i	0.899878	−	0.519545i	0.877161	+	0.480197i	$0.159435\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			851.673i			2.49758i
$342$		0			0
$343$			469.460i			1.36869i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−41.0594	−	71.1169i	−0.118327	−	0.204948i	0.800778	−	0.598961i	$-0.204420\pi$
$347$	−41.0594	−	71.1169i	−0.118327	−	0.204948i	−0.919105	+	0.394013i	$0.871086\pi$
$348$		0			0
$349$	22.7621	−	39.4251i	0.0652209	−	0.112966i	−0.831571	−	0.555418i	$-0.812558\pi$
$349$	22.7621	−	39.4251i	0.0652209	−	0.112966i	0.896792	+	0.442452i	$0.145891\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−262.046	+	453.877i	−0.742340	+	1.28577i	0.209087	+	0.977897i	$0.432951\pi$
$353$	−262.046	+	453.877i	−0.742340	+	1.28577i	−0.951427	+	0.307873i	$0.900383\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			295.601i			0.823401i	0.911319	+	0.411701i	$0.135065\pi$
$359$			295.601i			0.823401i	−0.911319	+	0.411701i	$0.864935\pi$
$360$		0			0
$361$	−234.347			−0.649160
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	28.1249	+	16.2379i	0.0766345	+	0.0442450i	0.537828	−	0.843055i	$-0.319245\pi$
$367$	28.1249	+	16.2379i	0.0766345	+	0.0442450i	−0.461193	+	0.887300i	$0.652578\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−383.141	−	221.207i	−1.03273	−	0.596244i
$372$		0			0
$373$	214.334	−	123.746i	0.574623	−	0.331759i	−0.184371	−	0.982857i	$-0.559025\pi$
$373$	214.334	−	123.746i	0.574623	−	0.331759i	0.758994	+	0.651098i	$0.225691\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	168.805			0.447759
$378$		0			0
$379$	−300.746			−0.793525			−0.396762	−	0.917921i	$-0.629866\pi$
$379$	−300.746			−0.793525			−0.396762	+	0.917921i	$0.629866\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−106.039	−	183.665i	−0.276865	−	0.479544i	0.693739	−	0.720226i	$-0.255962\pi$
$383$	−106.039	−	183.665i	−0.276865	−	0.479544i	−0.970604	+	0.240683i	$0.922629\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−190.760	−	110.135i	−0.490386	−	0.283124i	0.234349	−	0.972153i	$-0.424704\pi$
$389$	−190.760	−	110.135i	−0.490386	−	0.283124i	−0.724734	+	0.689028i	$0.758038\pi$
$390$		0			0
$391$	255.714	+	442.909i	0.653999	+	1.13276i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	97.4919i		−	0.245572i	−0.992433	−	0.122786i	$-0.960817\pi$
$397$		−	97.4919i		−	0.245572i	0.992433	−	0.122786i	$-0.0391828\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	66.8347	−	38.5870i	0.166670	−	0.0962270i	−0.414345	−	0.910120i	$-0.635989\pi$
$401$	66.8347	−	38.5870i	0.166670	−	0.0962270i	0.581015	+	0.813893i	$0.302656\pi$
$402$		0			0
$403$	−864.042	−	498.855i	−2.14402	−	1.23785i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	106.039	−	183.665i	0.260539	−	0.451266i
$408$		0			0
$409$	−97.4355	−	168.763i	−0.238229	−	0.412624i	0.721978	−	0.691917i	$-0.243233\pi$
$409$	−97.4355	−	168.763i	−0.238229	−	0.412624i	−0.960206	+	0.279293i	$0.909900\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−747.582			−1.81013
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	196.476	−	113.435i	0.468916	−	0.270729i	−0.246870	−	0.969049i	$-0.579402\pi$
$419$	196.476	−	113.435i	0.468916	−	0.270729i	0.715786	+	0.698320i	$0.246069\pi$
$420$		0			0
$421$	150.857	−	261.292i	0.358330	−	0.620645i	−0.629352	−	0.777120i	$-0.716680\pi$
$421$	150.857	−	261.292i	0.358330	−	0.620645i	0.987682	+	0.156475i	$0.0500130\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.5923	+	7.27017i	−0.0294902	+	0.0170262i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			707.661i			1.64191i	0.570996	+	0.820953i	$0.306557\pi$
$431$			707.661i			1.64191i	−0.570996	+	0.820953i	$0.693443\pi$
$432$		0			0
$433$			669.883i			1.54707i	0.633751	+	0.773537i	$0.281514\pi$
$433$			669.883i			1.54707i	−0.633751	+	0.773537i	$0.718486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	189.974	+	329.044i	0.434723	+	0.752962i
$438$		0			0
$439$	−38.3327	+	66.3941i	−0.0873181	+	0.151239i	−0.906377	−	0.422471i	$-0.861163\pi$
$439$	−38.3327	+	66.3941i	−0.0873181	+	0.151239i	0.819059	+	0.573710i	$0.194496\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	340.704	−	590.117i	0.769084	−	1.33209i	−0.168976	−	0.985620i	$-0.554046\pi$
$443$	340.704	−	590.117i	0.769084	−	1.33209i	0.938060	−	0.346472i	$-0.112621\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			630.327i			1.40385i	0.712253	+	0.701923i	$0.247675\pi$
$449$			630.327i			1.40385i	−0.712253	+	0.701923i	$0.752325\pi$
$450$		0			0
$451$	−393.569			−0.872657
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	496.229	+	286.498i	1.08584	+	0.626910i	0.932466	−	0.361258i	$-0.117653\pi$
$457$	496.229	+	286.498i	1.08584	+	0.626910i	0.153374	+	0.988168i	$0.450986\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−615.950	−	355.619i	−1.33612	−	0.771407i	−0.349887	−	0.936792i	$-0.613780\pi$
$461$	−615.950	−	355.619i	−1.33612	−	0.771407i	−0.986229	+	0.165385i	$0.947113\pi$
$462$		0			0
$463$	−556.317	+	321.190i	−1.20155	+	0.693714i	−0.960899	−	0.276899i	$-0.910693\pi$
$463$	−556.317	+	321.190i	−1.20155	+	0.693714i	−0.240648	+	0.970612i	$0.577360\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−506.331			−1.08422			−0.542111	−	0.840307i	$-0.682375\pi$
$467$	−506.331			−1.08422			−0.542111	+	0.840307i	$0.682375\pi$
$468$		0			0
$469$	1009.58			2.15263
$470$		0			0
$471$		0			0
$472$		0			0
$473$	151.362	+	262.167i	0.320005	+	0.554265i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−644.044	−	371.839i	−1.34456	−	0.776282i	−0.357087	−	0.934071i	$-0.616230\pi$
$479$	−644.044	−	371.839i	−1.34456	−	0.776282i	−0.987473	+	0.157789i	$0.949563\pi$
$480$		0			0
$481$	124.222	+	215.158i	0.258257	+	0.447315i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			154.065i			0.316354i	0.987411	+	0.158177i	$0.0505617\pi$
$487$			154.065i			0.316354i	−0.987411	+	0.158177i	$0.949438\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	145.167	−	83.8124i	0.295657	−	0.170697i	−0.344833	−	0.938664i	$-0.612065\pi$
$491$	145.167	−	83.8124i	0.295657	−	0.170697i	0.640490	+	0.767967i	$0.278731\pi$
$492$		0			0
$493$	124.791	+	72.0484i	0.253127	+	0.146143i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−129.820	+	224.855i	−0.261207	+	0.452424i
$498$		0			0
$499$	247.308	+	428.351i	0.495608	+	0.858418i	0.999987	−	0.00506391i	$-0.00161190\pi$
$499$	247.308	+	428.351i	0.495608	+	0.858418i	−0.504379	+	0.863482i	$0.668279\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−711.349			−1.41421			−0.707106	−	0.707107i	$-0.750000\pi$
$503$	−711.349			−1.41421			−0.707106	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−717.714	+	414.372i	−1.41005	+	0.814091i	−0.995392	−	0.0958882i	$-0.969431\pi$
$509$	−717.714	+	414.372i	−1.41005	+	0.814091i	−0.414654	+	0.909979i	$0.636098\pi$
$510$		0			0
$511$	−353.776	+	612.758i	−0.692321	+	1.19914i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−948.581	+	547.663i	−1.83478	+	1.05931i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			622.686i			1.19518i	0.801804	+	0.597588i	$0.203874\pi$
$521$			622.686i			1.19518i	−0.801804	+	0.597588i	$0.796126\pi$
$522$		0			0
$523$			89.4274i			0.170989i	0.996339	+	0.0854946i	$0.0272471\pi$
$523$			89.4274i			0.170989i	−0.996339	+	0.0854946i	$0.972753\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−425.837	−	737.571i	−0.808039	−	1.39956i
$528$		0			0
$529$	−305.403	+	528.974i	−0.577322	+	0.999951i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	230.527	−	399.284i	0.432508	−	0.749126i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1347.72i		−	2.50042i
$540$		0			0
$541$	834.629			1.54275			0.771376	−	0.636380i	$-0.219569\pi$
$541$	834.629			1.54275			0.771376	+	0.636380i	$0.219569\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−765.189	−	441.782i	−1.39888	−	0.807646i	−0.404608	−	0.914490i	$-0.632592\pi$
$547$	−765.189	−	441.782i	−1.39888	−	0.807646i	−0.994276	+	0.106845i	$0.965925\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	92.7096	+	53.5259i	0.168257	+	0.0971432i
$552$		0			0
$553$	1050.17	−	606.314i	1.89904	−	1.09641i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	256.448			0.460410			0.230205	−	0.973142i	$-0.426060\pi$
$557$	256.448			0.460410			0.230205	+	0.973142i	$0.426060\pi$
$558$		0			0
$559$	−354.633			−0.634406
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−103.937	−	180.024i	−0.184613	−	0.319759i	0.758833	−	0.651285i	$-0.225770\pi$
$563$	−103.937	−	180.024i	−0.184613	−	0.319759i	−0.943446	+	0.331526i	$0.892436\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	430.923	+	248.794i	0.757334	+	0.437247i	0.828338	−	0.560229i	$-0.189287\pi$
$569$	430.923	+	248.794i	0.757334	+	0.437247i	−0.0710034	+	0.997476i	$0.522620\pi$
$570$		0			0
$571$	−453.817	−	786.033i	−0.794775	−	1.37659i	−0.922982	−	0.384843i	$-0.874256\pi$
$571$	−453.817	−	786.033i	−0.794775	−	1.37659i	0.128207	−	0.991747i	$-0.459078\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			758.810i			1.31510i	0.753413	+	0.657548i	$0.228406\pi$
$577$			758.810i			1.31510i	−0.753413	+	0.657548i	$0.771594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	916.185	−	528.960i	1.57691	−	0.910430i
$582$		0			0
$583$	494.127	+	285.284i	0.847559	+	0.489338i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	378.809	−	656.117i	0.645331	−	1.11775i	−0.338894	−	0.940825i	$-0.610053\pi$
$587$	378.809	−	656.117i	0.645331	−	1.11775i	0.984225	−	0.176921i	$-0.0566139\pi$
$588$		0			0
$589$	−316.361	−	547.953i	−0.537115	−	0.930311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	258.648			0.436169			0.218084	−	0.975930i	$-0.430019\pi$
$593$	258.648			0.436169			0.218084	+	0.975930i	$0.430019\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−495.665	+	286.172i	−0.827488	+	0.477750i	−0.852992	−	0.521924i	$-0.825214\pi$
$599$	−495.665	+	286.172i	−0.827488	+	0.477750i	0.0255038	+	0.999675i	$0.491881\pi$
$600$		0			0
$601$	−68.8186	+	119.197i	−0.114507	+	0.198332i	−0.917582	−	0.397545i	$-0.869862\pi$
$601$	−68.8186	+	119.197i	−0.114507	+	0.198332i	0.803076	+	0.595877i	$0.203195\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	16.0005	−	9.23790i	0.0263600	−	0.0152189i	−0.486762	−	0.873535i	$-0.661822\pi$
$607$	16.0005	−	9.23790i	0.0263600	−	0.0152189i	0.513122	+	0.858316i	$0.328489\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1283.14i		−	2.10007i
$612$		0			0
$613$		−	392.569i		−	0.640405i	−0.947349	−	0.320203i	$-0.896249\pi$
$613$		−	392.569i		−	0.640405i	0.947349	−	0.320203i	$-0.103751\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−132.994	−	230.353i	−0.215550	−	0.373343i	0.737893	−	0.674918i	$-0.235821\pi$
$617$	−132.994	−	230.353i	−0.215550	−	0.373343i	−0.953443	+	0.301575i	$0.902488\pi$
$618$		0			0
$619$	416.736	−	721.808i	0.673240	−	1.16609i	−0.303739	−	0.952755i	$-0.598235\pi$
$619$	416.736	−	721.808i	0.673240	−	1.16609i	0.976980	−	0.213332i	$-0.0684315\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	71.0420	−	123.048i	0.114032	−	0.197509i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			212.078i			0.337168i
$630$		0			0
$631$	−351.875			−0.557647			−0.278823	−	0.960342i	$-0.589944\pi$
$631$	−351.875			−0.557647			−0.278823	+	0.960342i	$0.589944\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1367.30	+	789.409i	2.14646	+	1.23926i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−355.645	−	205.332i	−0.554829	−	0.320331i	0.196239	−	0.980556i	$-0.437127\pi$
$641$	−355.645	−	205.332i	−0.554829	−	0.320331i	−0.751067	+	0.660226i	$0.770461\pi$
$642$		0			0
$643$	191.552	−	110.593i	0.297904	−	0.171995i	−0.343597	−	0.939117i	$-0.611645\pi$
$643$	191.552	−	110.593i	0.297904	−	0.171995i	0.641501	+	0.767122i	$0.278312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1095.26			−1.69282			−0.846412	−	0.532529i	$-0.821242\pi$
$647$	−1095.26			−1.69282			−0.846412	+	0.532529i	$0.821242\pi$
$648$		0			0
$649$	964.137			1.48557
$650$		0			0
$651$		0			0
$652$		0			0
$653$	516.179	+	894.048i	0.790473	+	1.36914i	0.925674	+	0.378322i	$0.123499\pi$
$653$	516.179	+	894.048i	0.790473	+	1.36914i	−0.135201	+	0.990818i	$0.543168\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−988.851	−	570.913i	−1.50053	−	0.866333i	−1.00000		0.000614829i	$-0.999804\pi$
$659$	−988.851	−	570.913i	−1.50053	−	0.866333i	−0.500532	−	0.865718i	$-0.666862\pi$
$660$		0			0
$661$	513.282	+	889.031i	0.776524	+	1.34498i	0.933934	+	0.357445i	$0.116352\pi$
$661$	513.282	+	889.031i	0.776524	+	1.34498i	−0.157410	+	0.987533i	$0.550315\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			321.145i			0.481477i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.2399	−	9.37614i	0.0242026	−	0.0139734i
$672$		0			0
$673$	−242.403	−	139.952i	−0.360183	−	0.207952i	0.308978	−	0.951069i	$-0.400013\pi$
$673$	−242.403	−	139.952i	−0.360183	−	0.207952i	−0.669161	+	0.743117i	$0.733346\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−144.060	+	249.520i	−0.212792	+	0.368567i	−0.952587	−	0.304265i	$-0.901589\pi$
$677$	−144.060	+	249.520i	−0.212792	+	0.368567i	0.739795	+	0.672832i	$0.234922\pi$
$678$		0			0
$679$	−585.522	−	1014.15i	−0.862330	−	1.49360i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	283.805			0.415527			0.207763	−	0.978179i	$-0.433382\pi$
$683$	283.805			0.415527			0.207763	+	0.978179i	$0.433382\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−578.855	+	334.202i	−0.840138	+	0.485054i
$690$		0			0
$691$	−440.331	+	762.675i	−0.637237	+	1.10373i	0.348800	+	0.937197i	$0.386589\pi$
$691$	−440.331	+	762.675i	−0.637237	+	1.10373i	−0.986037	+	0.166529i	$0.946744\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	340.840	−	196.784i	0.489011	−	0.282330i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	680.500i		−	0.970757i	−0.874304	−	0.485378i	$-0.838682\pi$
$701$		−	680.500i		−	0.970757i	0.874304	−	0.485378i	$-0.161318\pi$
$702$		0			0
$703$			157.556i			0.224120i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	910.339	+	1576.75i	1.28761	+	2.23020i
$708$		0			0
$709$	−289.696	+	501.767i	−0.408597	+	0.707711i	−0.994733	−	0.102502i	$-0.967315\pi$
$709$	−289.696	+	501.767i	−0.408597	+	0.707711i	0.586135	+	0.810213i	$0.300649\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	949.052	−	1643.81i	1.33107	−	2.30548i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	597.306i		−	0.830746i	−0.909651	−	0.415373i	$-0.863651\pi$
$719$		−	597.306i		−	0.830746i	0.909651	−	0.415373i	$-0.136349\pi$
$720$		0			0
$721$	−1829.01			−2.53677
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	583.537	+	336.905i	0.802664	+	0.463419i	0.844402	−	0.535710i	$-0.179956\pi$
$727$	583.537	+	336.905i	0.802664	+	0.463419i	−0.0417376	+	0.999129i	$0.513289\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−262.167	−	151.362i	−0.358642	−	0.207062i
$732$		0			0
$733$	676.816	−	390.760i	0.923351	−	0.533097i	0.0386484	−	0.999253i	$-0.487695\pi$
$733$	676.816	−	390.760i	0.923351	−	0.533097i	0.884703	+	0.466156i	$0.154361\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1302.03			−1.76667
$738$		0			0
$739$	−232.335			−0.314391			−0.157195	−	0.987568i	$-0.550245\pi$
$739$	−232.335			−0.314391			−0.157195	+	0.987568i	$0.550245\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	251.036	+	434.806i	0.337868	+	0.585204i	0.984031	−	0.177995i	$-0.0569611\pi$
$743$	251.036	+	434.806i	0.337868	+	0.585204i	−0.646164	+	0.763199i	$0.723628\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	863.329	+	498.443i	1.15264	+	0.665478i
$750$		0			0
$751$	−477.190	−	826.516i	−0.635405	−	1.10055i	−0.986429	−	0.164188i	$-0.947500\pi$
$751$	−477.190	−	826.516i	−0.635405	−	1.10055i	0.351024	−	0.936367i	$-0.385834\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	266.379i		−	0.351888i	−0.984400	−	0.175944i	$-0.943702\pi$
$757$		−	266.379i		−	0.351888i	0.984400	−	0.175944i	$-0.0562977\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−294.665	+	170.125i	−0.387208	+	0.223555i	−0.680950	−	0.732330i	$-0.738433\pi$
$761$	−294.665	+	170.125i	−0.387208	+	0.223555i	0.293742	+	0.955885i	$0.405099\pi$
$762$		0			0
$763$	386.876	+	223.363i	0.507046	+	0.292743i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−564.729	+	978.139i	−0.736283	+	1.27528i
$768$		0			0
$769$	644.552	+	1116.40i	0.838170	+	1.45175i	0.891423	+	0.453171i	$0.149707\pi$
$769$	644.552	+	1116.40i	0.838170	+	1.45175i	−0.0532540	+	0.998581i	$0.516959\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−919.244			−1.18919			−0.594595	−	0.804025i	$-0.702688\pi$
$773$	−919.244			−1.18919			−0.594595	+	0.804025i	$0.702688\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	253.216	−	146.194i	0.325052	−	0.187669i
$780$		0			0
$781$	167.425	−	289.989i	0.214373	−	0.371305i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1094.40	+	631.855i	−1.39060	+	0.802865i	−0.993382	−	0.114858i	$-0.963359\pi$
$787$	−1094.40	+	631.855i	−1.39060	+	0.802865i	−0.397221	+	0.917723i	$0.630025\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			81.3784i			0.102880i
$792$		0			0
$793$			21.9677i			0.0277021i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	306.461	+	530.806i	0.384518	+	0.666006i	0.991702	−	0.128556i	$-0.0410342\pi$
$797$	306.461	+	530.806i	0.384518	+	0.666006i	−0.607184	+	0.794561i	$0.707701\pi$
$798$		0			0
$799$	547.663	−	948.581i	0.685436	−	1.18721i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	456.256	−	790.258i	0.568189	−	0.984132i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	273.189i		−	0.337687i	−0.985643	−	0.168844i	$-0.945997\pi$
$809$		−	273.189i		−	0.337687i	0.985643	−	0.168844i	$-0.0540033\pi$
$810$		0			0
$811$	−1446.49			−1.78359			−0.891793	−	0.452444i	$-0.850552\pi$
$811$	−1446.49			−1.78359			−0.891793	+	0.452444i	$0.850552\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−194.768	−	112.450i	−0.238395	−	0.137637i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−221.813	−	128.064i	−0.270174	−	0.155985i	0.358793	−	0.933417i	$-0.383188\pi$
$821$	−221.813	−	128.064i	−0.270174	−	0.155985i	−0.628967	+	0.777432i	$0.716522\pi$
$822$		0			0
$823$	−1065.59	+	615.218i	−1.29476	+	0.747531i	−0.979494	−	0.201473i	$-0.935427\pi$
$823$	−1065.59	+	615.218i	−1.29476	+	0.747531i	−0.315267	+	0.949003i	$0.602094\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	726.246			0.878169			0.439085	−	0.898446i	$-0.355303\pi$
$827$	726.246			0.878169			0.439085	+	0.898446i	$0.355303\pi$
$828$		0			0
$829$	−1439.23			−1.73610			−0.868052	−	0.496474i	$-0.834628\pi$
$829$	−1439.23			−1.73610			−0.868052	+	0.496474i	$0.834628\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	673.862	+	1167.16i	0.808958	+	1.40116i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	968.135	+	558.953i	1.15392	+	0.666213i	0.949838	−	0.312742i	$-0.101247\pi$
$839$	968.135	+	558.953i	1.15392	+	0.666213i	0.204077	+	0.978955i	$0.434581\pi$
$840$		0			0
$841$	−375.258	−	649.966i	−0.446205	−	0.772849i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			1274.15i			1.50431i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−409.331	+	236.327i	−0.481000	+	0.277705i
$852$		0			0
$853$	−602.834	−	348.046i	−0.706722	−	0.408026i	0.103124	−	0.994669i	$-0.467116\pi$
$853$	−602.834	−	348.046i	−0.706722	−	0.408026i	−0.809846	+	0.586642i	$0.800449\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	430.149	−	745.040i	0.501924	−	0.869358i	−0.498073	−	0.867135i	$-0.665959\pi$
$857$	430.149	−	745.040i	0.501924	−	0.869358i	0.999998	−	0.00222347i	$-0.000707752\pi$
$858$		0			0
$859$	−321.216	−	556.362i	−0.373942	−	0.647686i	0.616226	−	0.787569i	$-0.288661\pi$
$859$	−321.216	−	556.362i	−0.373942	−	0.647686i	−0.990168	+	0.139883i	$0.955327\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1466.58			−1.69940			−0.849698	−	0.527269i	$-0.823216\pi$
$863$	−1466.58			−1.69940			−0.849698	+	0.527269i	$0.823216\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1354.37	+	781.948i	−1.55854	+	0.899825i
$870$		0			0
$871$	762.647	−	1320.94i	0.875599	−	1.51658i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−167.461	+	96.6835i	−0.190947	+	0.110243i	−0.592426	−	0.805625i	$-0.701830\pi$
$877$	−167.461	+	96.6835i	−0.190947	+	0.110243i	0.401479	+	0.915868i	$0.368496\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	554.368i		−	0.629249i	−0.949216	−	0.314624i	$-0.898121\pi$
$881$		−	554.368i		−	0.629249i	0.949216	−	0.314624i	$-0.101879\pi$
$882$		0			0
$883$		−	352.460i		−	0.399162i	−0.979881	−	0.199581i	$-0.936042\pi$
$883$		−	352.460i		−	0.399162i	0.979881	−	0.199581i	$-0.0639580\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−538.140	−	932.087i	−0.606697	−	1.05083i	−0.991781	−	0.127949i	$-0.959161\pi$
$887$	−538.140	−	932.087i	−0.606697	−	1.05083i	0.385084	−	0.922882i	$-0.374173\pi$
$888$		0			0
$889$	−856.887	+	1484.17i	−0.963877	+	1.66948i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	406.868	−	704.716i	0.455619	−	0.789155i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	534.799i		−	0.594882i
$900$		0			0
$901$	−570.569			−0.633261
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	191.385	+	110.496i	0.211008	+	0.121826i	0.601780	−	0.798662i	$-0.294458\pi$
$907$	191.385	+	110.496i	0.211008	+	0.121826i	−0.390772	+	0.920488i	$0.627792\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1535.28	−	886.394i	−1.68527	−	0.972991i	−0.958057	−	0.286578i	$-0.907482\pi$
$911$	−1535.28	−	886.394i	−1.68527	−	0.972991i	−0.727212	−	0.686413i	$-0.759184\pi$
$912$		0			0
$913$	−1181.58	+	682.185i	−1.29417	+	0.747191i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−782.775			−0.853626
$918$		0			0
$919$	−1006.85			−1.09560			−0.547799	−	0.836610i	$-0.684534\pi$
$919$	−1006.85			−1.09560			−0.547799	+	0.836610i	$0.684534\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	196.134	+	339.714i	0.212496	+	0.368054i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−974.758	−	562.777i	−1.04926	−	0.605788i	−0.126814	−	0.991926i	$-0.540475\pi$
$929$	−974.758	−	562.777i	−1.04926	−	0.605788i	−0.922441	+	0.386139i	$0.873809\pi$
$930$		0			0
$931$	500.623	+	867.104i	0.537726	+	0.931369i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1343.20i			1.43351i	0.697324	+	0.716756i	$0.254374\pi$
$937$			1343.20i			1.43351i	−0.697324	+	0.716756i	$0.745626\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−618.375	+	357.019i	−0.657147	+	0.379404i	−0.791189	−	0.611572i	$-0.790538\pi$
$941$	−618.375	+	357.019i	−0.657147	+	0.379404i	0.134042	+	0.990976i	$0.457204\pi$
$942$		0			0
$943$	759.623	+	438.569i	0.805539	+	0.465078i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−358.220	+	620.456i	−0.378268	+	0.655180i	−0.990810	−	0.135258i	$-0.956814\pi$
$947$	−358.220	+	620.456i	−0.378268	+	0.655180i	0.612542	+	0.790438i	$0.290147\pi$
$948$		0			0
$949$	534.490	+	925.764i	0.563214	+	0.975515i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−793.300			−0.832424			−0.416212	−	0.909268i	$-0.636643\pi$
$953$	−793.300			−0.832424			−0.416212	+	0.909268i	$0.636643\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	744.756	−	429.985i	0.776596	−	0.448368i
$960$		0			0
$961$	−1099.94	+	1905.16i	−1.14458	+	1.98247i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1583.67	−	914.331i	1.63771	−	0.945533i	0.656093	−	0.754680i	$-0.272208\pi$
$967$	1583.67	−	914.331i	1.63771	−	0.945533i	0.981619	−	0.190853i	$-0.0611254\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1843.33i			1.89839i	0.314691	+	0.949194i	$0.398099\pi$
$971$			1843.33i			1.89839i	−0.314691	+	0.949194i	$0.601901\pi$
$972$		0			0
$973$		−	1253.46i		−	1.28824i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−167.195	−	289.591i	−0.171131	−	0.296408i	0.767684	−	0.640828i	$-0.221409\pi$
$977$	−167.195	−	289.591i	−0.171131	−	0.296408i	−0.938816	+	0.344420i	$0.888076\pi$
$978$		0			0
$979$	−91.6210	+	158.692i	−0.0935863	+	0.162096i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−926.480	+	1604.71i	−0.942502	+	1.63246i	−0.181825	+	0.983331i	$0.558200\pi$
$983$	−926.480	+	1604.71i	−0.942502	+	1.63246i	−0.760677	+	0.649131i	$0.775133\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	674.676i		−	0.682180i
$990$		0			0
$991$	415.419			0.419192			0.209596	−	0.977788i	$-0.432785\pi$
$991$	415.419			0.419192			0.209596	+	0.977788i	$0.432785\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1033.70	−	596.808i	−1.03681	−	0.598604i	−0.117884	−	0.993027i	$-0.537611\pi$
$997$	−1033.70	−	596.808i	−1.03681	−	0.598604i	−0.918929	+	0.394423i	$0.870944\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.u.a.2249.4		8
3.2	odd	2		900.3.u.b.749.4		8
5.2	odd	4		2700.3.p.a.1601.1		4
5.3	odd	4		540.3.o.a.521.1		4
5.4	even	2	inner	2700.3.u.a.2249.1		8
9.4	even	3		900.3.u.b.149.1		8
9.5	odd	6	inner	2700.3.u.a.449.1		8
15.2	even	4		900.3.p.b.101.1		4
15.8	even	4		180.3.o.a.101.2	yes	4
15.14	odd	2		900.3.u.b.749.1		8
20.3	even	4		2160.3.bs.a.1601.1		4
45.4	even	6		900.3.u.b.149.4		8
45.13	odd	12		180.3.o.a.41.2	✓	4
45.14	odd	6	inner	2700.3.u.a.449.4		8
45.22	odd	12		900.3.p.b.401.1		4
45.23	even	12		540.3.o.a.341.1		4
45.32	even	12		2700.3.p.a.2501.1		4
45.38	even	12		1620.3.g.a.161.3		4
45.43	odd	12		1620.3.g.a.161.1		4
60.23	odd	4		720.3.bs.a.641.2		4
180.23	odd	12		2160.3.bs.a.881.1		4
180.103	even	12		720.3.bs.a.401.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.o.a.41.2	✓	4	45.13	odd	12
180.3.o.a.101.2	yes	4	15.8	even	4
540.3.o.a.341.1		4	45.23	even	12
540.3.o.a.521.1		4	5.3	odd	4
720.3.bs.a.401.2		4	180.103	even	12
720.3.bs.a.641.2		4	60.23	odd	4
900.3.p.b.101.1		4	15.2	even	4
900.3.p.b.401.1		4	45.22	odd	12
900.3.u.b.149.1		8	9.4	even	3
900.3.u.b.149.4		8	45.4	even	6
900.3.u.b.749.1		8	15.14	odd	2
900.3.u.b.749.4		8	3.2	odd	2
1620.3.g.a.161.1		4	45.43	odd	12
1620.3.g.a.161.3		4	45.38	even	12
2160.3.bs.a.881.1		4	180.23	odd	12
2160.3.bs.a.1601.1		4	20.3	even	4
2700.3.p.a.1601.1		4	5.2	odd	4
2700.3.p.a.2501.1		4	45.32	even	12
2700.3.u.a.449.1		8	9.5	odd	6	inner
2700.3.u.a.449.4		8	45.14	odd	6	inner
2700.3.u.a.2249.1		8	5.4	even	2	inner
2700.3.u.a.2249.4		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.u (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(10.1723 + 5.87298i) q^{7} +(-13.1190 - 7.57423i) q^{11} +(15.3685 - 8.87298i) q^{13} +15.1485 q^{17} +11.2540 q^{19} +(16.8805 + 29.2379i) q^{23} +(8.23790 + 4.75615i) q^{29} +(-28.1109 - 48.6895i) q^{31} +14.0000i q^{37} +(22.5000 - 12.9904i) q^{41} +(-17.3065 - 9.99193i) q^{43} +(36.1531 - 62.6190i) q^{47} +(44.4839 + 77.0483i) q^{49} -37.6651 q^{53} +(-55.1190 + 31.8229i) q^{59} +(-0.618950 + 1.07205i) q^{61} +(74.4363 - 42.9758i) q^{67} +22.1046i q^{71} +60.2379i q^{73} +(-88.9666 - 154.095i) q^{77} +(51.6190 - 89.4066i) q^{79} +(45.0333 - 78.0000i) q^{83} -12.0964i q^{89} +208.444 q^{91} +(-86.3406 - 49.8488i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{11} + 152 q^{19} - 120 q^{29} - 8 q^{31} + 180 q^{41} + 108 q^{49} - 348 q^{59} + 88 q^{61} + 320 q^{79} + 800 q^{91}+O(q^{100}) \) 8 * q - 12 * q^11 + 152 * q^19 - 120 * q^29 - 8 * q^31 + 180 * q^41 + 108 * q^49 - 348 * q^59 + 88 * q^61 + 320 * q^79 + 800 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more