LMFDB - Embedded newform 2700.3.p.a.1601.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,3,Mod(1601,2700)] 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("2700.1601"); 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, 1, 0])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 5x^{2} + 25$ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 3$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.1
Root			$1.93649 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	2700.1601
Dual form			2700.3.p.a.2501.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.87298 + 10.1723i) q^{7} +(-13.1190 - 7.57423i) q^{11} +(-8.87298 - 15.3685i) q^{13} +15.1485i q^{17} -11.2540 q^{19} +(29.2379 - 16.8805i) q^{23} +(-8.23790 - 4.75615i) q^{29} +(-28.1109 - 48.6895i) q^{31} -14.0000 q^{37} +(22.5000 - 12.9904i) q^{41} +(-9.99193 + 17.3065i) q^{43} +(62.6190 + 36.1531i) q^{47} +(-44.4839 - 77.0483i) q^{49} +37.6651i q^{53} +(55.1190 - 31.8229i) q^{59} +(-0.618950 + 1.07205i) q^{61} +(42.9758 + 74.4363i) q^{67} +22.1046i q^{71} +60.2379 q^{73} +(154.095 - 88.9666i) q^{77} +(-51.6190 + 89.4066i) q^{79} +(-78.0000 - 45.0333i) q^{83} +12.0964i q^{89} +208.444 q^{91} +(49.8488 - 86.3406i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 8 q^{7} - 6 q^{11} - 20 q^{13} - 76 q^{19} + 24 q^{23} + 60 q^{29} - 4 q^{31} - 56 q^{37} + 90 q^{41} + 22 q^{43} + 204 q^{47} - 54 q^{49} + 174 q^{59} + 44 q^{61} - 14 q^{67} + 148 q^{73} + 384 q^{77}+ \cdots - 2 q^{97}+O(q^{100})$ 4 * q - 8 * q^7 - 6 * q^11 - 20 * q^13 - 76 * q^19 + 24 * q^23 + 60 * q^29 - 4 * q^31 - 56 * q^37 + 90 * q^41 + 22 * q^43 + 204 * q^47 - 54 * q^49 + 174 * q^59 + 44 * q^61 - 14 * q^67 + 148 * q^73 + 384 * q^77 - 160 * q^79 - 312 * q^83 + 400 * q^91 - 2 * q^97

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
$7$	−5.87298	+	10.1723i	−0.838998	+	1.45319i	0.0517360	+	0.998661i	$0.483525\pi$
$7$	−5.87298	+	10.1723i	−0.838998	+	1.45319i	−0.890734	+	0.454526i	$0.849809\pi$
$8$		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
$13$	−8.87298	−	15.3685i	−0.682537	−	1.18219i	−0.974204	−	0.225669i	$-0.927543\pi$
$13$	−8.87298	−	15.3685i	−0.682537	−	1.18219i	0.291667	−	0.956520i	$-0.405790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			15.1485i			0.891086i	0.895261	+	0.445543i	$0.146989\pi$
$17$			15.1485i			0.891086i	−0.895261	+	0.445543i	$0.853011\pi$
$18$		0			0
$19$	−11.2540			−0.592318			−0.296159	−	0.955139i	$-0.595706\pi$
$19$	−11.2540			−0.592318			−0.296159	+	0.955139i	$0.595706\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	29.2379	−	16.8805i	1.27121	−	0.733935i	0.295997	−	0.955189i	$-0.404348\pi$
$23$	29.2379	−	16.8805i	1.27121	−	0.733935i	0.975216	+	0.221254i	$0.0710149\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.23790	−	4.75615i	−0.284066	−	0.164005i	0.351197	−	0.936302i	$-0.385775\pi$
$29$	−8.23790	−	4.75615i	−0.284066	−	0.164005i	−0.635262	+	0.772296i	$0.719108\pi$
$30$		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
$37$	−14.0000			−0.378378			−0.189189	−	0.981941i	$-0.560586\pi$
$37$	−14.0000			−0.378378			−0.189189	+	0.981941i	$0.560586\pi$
$38$		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
$43$	−9.99193	+	17.3065i	−0.232371	+	0.402478i	−0.958505	−	0.285075i	$-0.907982\pi$
$43$	−9.99193	+	17.3065i	−0.232371	+	0.402478i	0.726135	+	0.687552i	$0.241315\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	62.6190	+	36.1531i	1.33232	+	0.769214i	0.985655	−	0.168775i	$-0.0539812\pi$
$47$	62.6190	+	36.1531i	1.33232	+	0.769214i	0.346664	+	0.937990i	$0.387315\pi$
$48$		0			0
$49$	−44.4839	−	77.0483i	−0.907834	−	1.57241i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			37.6651i			0.710663i	0.934740	+	0.355331i	$0.115632\pi$
$53$			37.6651i			0.710663i	−0.934740	+	0.355331i	$0.884368\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	55.1190	−	31.8229i	0.934219	−	0.539372i	0.0460759	−	0.998938i	$-0.485328\pi$
$59$	55.1190	−	31.8229i	0.934219	−	0.539372i	0.888144	+	0.459566i	$0.151995\pi$
$60$		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
$67$	42.9758	+	74.4363i	0.641430	+	1.11099i	0.985114	+	0.171904i	$0.0549918\pi$
$67$	42.9758	+	74.4363i	0.641430	+	1.11099i	−0.343684	+	0.939085i	$0.611675\pi$
$68$		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
$73$	60.2379			0.825177			0.412588	−	0.910918i	$-0.364625\pi$
$73$	60.2379			0.825177			0.412588	+	0.910918i	$0.364625\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	154.095	−	88.9666i	2.00123	−	1.15541i
$78$		0			0
$79$	−51.6190	+	89.4066i	−0.653404	+	1.13173i	0.328887	+	0.944369i	$0.393327\pi$
$79$	−51.6190	+	89.4066i	−0.653404	+	1.13173i	−0.982291	+	0.187360i	$0.940007\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−78.0000	−	45.0333i	−0.939759	−	0.542570i	−0.0498743	−	0.998756i	$-0.515882\pi$
$83$	−78.0000	−	45.0333i	−0.939759	−	0.542570i	−0.889885	+	0.456185i	$0.849215\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.0964i			0.135915i	0.997688	+	0.0679574i	$0.0216482\pi$
$89$			12.0964i			0.135915i	−0.997688	+	0.0679574i	$0.978352\pi$
$90$		0			0
$91$	208.444			2.29059
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	49.8488	−	86.3406i	0.513905	−	0.890110i	−0.485965	−	0.873978i	$-0.661532\pi$
$97$	49.8488	−	86.3406i	0.513905	−	0.890110i	0.999870	−	0.0161312i	$-0.00513495\pi$
$98$		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
$103$	77.8569	+	134.852i	0.755892	+	1.30924i	0.944930	+	0.327273i	$0.106130\pi$
$103$	77.8569	+	134.852i	0.755892	+	1.30924i	−0.189038	+	0.981970i	$0.560537\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			84.8705i			0.793182i	0.917995	+	0.396591i	$0.129807\pi$
$107$			84.8705i			0.793182i	−0.917995	+	0.396591i	$0.870193\pi$
$108$		0			0
$109$	−38.0323			−0.348920			−0.174460	−	0.984664i	$-0.555818\pi$
$109$	−38.0323			−0.348920			−0.174460	+	0.984664i	$0.555818\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000	−	3.46410i	0.0530973	−	0.0306558i	−0.473216	−	0.880946i	$-0.656907\pi$
$113$	6.00000	−	3.46410i	0.0530973	−	0.0306558i	0.526314	+	0.850290i	$0.323574\pi$
$114$		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
$127$	−145.903			−1.14884			−0.574422	−	0.818559i	$-0.694773\pi$
$127$	−145.903			−1.14884			−0.574422	+	0.818559i	$0.694773\pi$
$128$		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
$133$	66.0948	−	114.479i	0.496953	−	0.860748i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	63.4052	+	36.6070i	0.462812	+	0.267205i	0.713226	−	0.700934i	$-0.247233\pi$
$137$	63.4052	+	36.6070i	0.462812	+	0.267205i	−0.250414	+	0.968139i	$0.580567\pi$
$138$		0			0
$139$	53.3569	+	92.4168i	0.383862	+	0.664869i	0.991611	−	0.129261i	$-0.0412604\pi$
$139$	53.3569	+	92.4168i	0.383862	+	0.664869i	−0.607748	+	0.794130i	$0.707927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			268.824i			1.87989i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	144.333	−	83.3305i	0.968676	−	0.559265i	0.0698433	−	0.997558i	$-0.477750\pi$
$149$	144.333	−	83.3305i	0.968676	−	0.559265i	0.898832	+	0.438293i	$0.144417\pi$
$150$		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
$157$	−33.9677	−	58.8338i	−0.216355	−	0.374738i	0.737336	−	0.675526i	$-0.236084\pi$
$157$	−33.9677	−	58.8338i	−0.216355	−	0.374738i	−0.953691	+	0.300788i	$0.902750\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			396.556i			2.46308i
$162$		0			0
$163$	−76.4113			−0.468781			−0.234390	−	0.972143i	$-0.575309\pi$
$163$	−76.4113			−0.468781			−0.234390	+	0.972143i	$0.575309\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−245.522	+	141.752i	−1.47019	+	0.848816i	−0.999440	−	0.0334495i	$-0.989351\pi$
$167$	−245.522	+	141.752i	−1.47019	+	0.848816i	−0.470752	+	0.882266i	$0.656017\pi$
$168$		0			0
$169$	−72.9597	+	126.370i	−0.431714	+	0.747751i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−116.903	−	67.4941i	−0.675741	−	0.390139i	0.122507	−	0.992468i	$-0.460906\pi$
$173$	−116.903	−	67.4941i	−0.675741	−	0.390139i	−0.798248	+	0.602328i	$0.794240\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			171.445i			0.957794i	0.877871	+	0.478897i	$0.158963\pi$
$179$			171.445i			0.957794i	−0.877871	+	0.478897i	$0.841037\pi$
$180$		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
$187$	114.738	−	198.732i	0.613572	−	1.06274i
$188$		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
$193$	−52.6431	−	91.1806i	−0.272762	−	0.472438i	0.696806	−	0.717260i	$-0.254604\pi$
$193$	−52.6431	−	91.1806i	−0.272762	−	0.472438i	−0.969568	+	0.244822i	$0.921271\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	57.9538i		−	0.294182i	−0.989123	−	0.147091i	$-0.953009\pi$
$197$		−	57.9538i		−	0.294182i	0.989123	−	0.147091i	$-0.0469910\pi$
$198$		0			0
$199$	−25.2702			−0.126986			−0.0634929	−	0.997982i	$-0.520224\pi$
$199$	−25.2702			−0.126986			−0.0634929	+	0.997982i	$0.520224\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	96.7621	−	55.8656i	0.476661	−	0.275200i
$204$		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
$217$	660.379			3.04322
$218$		0			0
$219$		0			0
$220$		0			0
$221$	232.808	−	134.412i	1.05343	−	0.608199i
$222$		0			0
$223$	123.681	−	214.223i	0.554625	−	0.960639i	−0.443307	−	0.896370i	$-0.646195\pi$
$223$	123.681	−	214.223i	0.554625	−	0.960639i	0.997933	−	0.0642694i	$-0.0204717\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	261.738	+	151.114i	1.15303	+	0.665702i	0.949624	−	0.313392i	$-0.101465\pi$
$227$	261.738	+	151.114i	1.15303	+	0.665702i	0.203407	+	0.979094i	$0.434799\pi$
$228$		0			0
$229$	−133.111	−	230.555i	−0.581270	−	1.00679i	−0.995329	−	0.0965395i	$-0.969223\pi$
$229$	−133.111	−	230.555i	−0.581270	−	1.00679i	0.414059	−	0.910250i	$-0.364111\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	176.969i		−	0.759526i	−0.925084	−	0.379763i	$-0.876006\pi$
$233$		−	176.969i		−	0.759526i	0.925084	−	0.379763i	$-0.123994\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−68.2863	+	39.4251i	−0.285717	+	0.164959i	−0.636009	−	0.771682i	$-0.719416\pi$
$239$	−68.2863	+	39.4251i	−0.285717	+	0.164959i	0.350292	+	0.936641i	$0.386082\pi$
$240$		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
$247$	99.8569	+	172.957i	0.404279	+	0.700231i
$248$		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
$253$	−511.427			−2.02145
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−131.692	+	76.0322i	−0.512418	+	0.295845i	−0.733827	−	0.679336i	$-0.762268\pi$
$257$	−131.692	+	76.0322i	−0.512418	+	0.295845i	0.221409	+	0.975181i	$0.428935\pi$
$258$		0			0
$259$	82.2218	−	142.412i	0.317459	−	0.549854i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.95365	−	2.85999i	−0.0188352	−	0.0108745i	0.490553	−	0.871411i	$-0.336795\pi$
$263$	−4.95365	−	2.85999i	−0.0188352	−	0.0108745i	−0.509388	+	0.860537i	$0.670128\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	361.531i		−	1.34398i	−0.740560	−	0.671990i	$-0.765440\pi$
$269$		−	361.531i		−	1.34398i	0.740560	−	0.671990i	$-0.234560\pi$
$270$		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
$277$	111.778	−	193.606i	0.403532	−	0.698937i	−0.590618	−	0.806951i	$-0.701116\pi$
$277$	111.778	−	193.606i	0.403532	−	0.698937i	0.994149	+	0.108014i	$0.0344492\pi$
$278$		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
$283$	72.0161	+	124.736i	0.254474	+	0.440762i	0.964753	−	0.263159i	$-0.0847643\pi$
$283$	72.0161	+	124.736i	0.254474	+	0.440762i	−0.710279	+	0.703921i	$0.751431\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			305.169i			1.06331i
$288$		0			0
$289$	59.5242			0.205966
$290$		0			0
$291$		0			0
$292$		0			0
$293$	361.476	−	208.698i	1.23371	−	0.712280i	0.265905	−	0.963999i	$-0.414329\pi$
$293$	361.476	−	208.698i	1.23371	−	0.712280i	0.967800	+	0.251719i	$0.0809958\pi$
$294$		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
$307$	310.048			1.00993			0.504965	−	0.863140i	$-0.331505\pi$
$307$	310.048			1.00993			0.504965	+	0.863140i	$0.331505\pi$
$308$		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
$313$	−14.5786	+	25.2509i	−0.0465771	+	0.0806738i	−0.888374	−	0.459120i	$-0.848165\pi$
$313$	−14.5786	+	25.2509i	−0.0465771	+	0.0806738i	0.841797	+	0.539794i	$0.181498\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	109.095	+	62.9859i	0.344147	+	0.198694i	0.662105	−	0.749411i	$-0.269663\pi$
$317$	109.095	+	62.9859i	0.344147	+	0.198694i	−0.317957	+	0.948105i	$0.602997\pi$
$318$		0			0
$319$	72.0484	+	124.791i	0.225857	+	0.391196i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	170.481i		−	0.527806i
$324$		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
$337$	175.087	+	303.259i	0.519545	+	0.899878i	0.999742	+	0.0227177i	$0.00723189\pi$
$337$	175.087	+	303.259i	0.519545	+	0.899878i	−0.480197	+	0.877161i	$0.659435\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			851.673i			2.49758i
$342$		0			0
$343$	469.460			1.36869
$344$		0			0
$345$		0			0
$346$		0			0
$347$	71.1169	−	41.0594i	0.204948	−	0.118327i	−0.394013	−	0.919105i	$-0.628914\pi$
$347$	71.1169	−	41.0594i	0.204948	−	0.118327i	0.598961	+	0.800778i	$0.295580\pi$
$348$		0			0
$349$	−22.7621	+	39.4251i	−0.0652209	+	0.112966i	−0.896792	−	0.442452i	$-0.854109\pi$
$349$	−22.7621	+	39.4251i	−0.0652209	+	0.112966i	0.831571	+	0.555418i	$0.187442\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	453.877	+	262.046i	1.28577	+	0.742340i	0.977897	−	0.209087i	$-0.0670493\pi$
$353$	453.877	+	262.046i	1.28577	+	0.742340i	0.307873	+	0.951427i	$0.400383\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	295.601i		−	0.823401i	−0.911319	−	0.411701i	$-0.864935\pi$
$359$		−	295.601i		−	0.823401i	0.911319	−	0.411701i	$-0.135065\pi$
$360$		0			0
$361$	−234.347			−0.649160
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−16.2379	+	28.1249i	−0.0442450	+	0.0766345i	−0.887300	−	0.461193i	$-0.847422\pi$
$367$	−16.2379	+	28.1249i	−0.0442450	+	0.0766345i	0.843055	+	0.537828i	$0.180755\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−383.141	−	221.207i	−1.03273	−	0.596244i
$372$		0			0
$373$	−123.746	−	214.334i	−0.331759	−	0.574623i	0.651098	−	0.758994i	$-0.274309\pi$
$373$	−123.746	−	214.334i	−0.331759	−	0.574623i	−0.982857	+	0.184371i	$0.940975\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			168.805i			0.447759i
$378$		0			0
$379$	300.746			0.793525			0.396762	−	0.917921i	$-0.370134\pi$
$379$	300.746			0.793525			0.396762	+	0.917921i	$0.370134\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−183.665	+	106.039i	−0.479544	+	0.276865i	−0.720226	−	0.693739i	$-0.755962\pi$
$383$	−183.665	+	106.039i	−0.479544	+	0.276865i	0.240683	+	0.970604i	$0.422629\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	190.760	+	110.135i	0.490386	+	0.283124i	0.724734	−	0.689028i	$-0.241962\pi$
$389$	190.760	+	110.135i	0.490386	+	0.283124i	−0.234349	+	0.972153i	$0.575296\pi$
$390$		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
$397$	97.4919			0.245572			0.122786	−	0.992433i	$-0.460817\pi$
$397$	97.4919			0.245572			0.122786	+	0.992433i	$0.460817\pi$
$398$		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
$403$	−498.855	+	864.042i	−1.23785	+	2.14402i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	183.665	+	106.039i	0.451266	+	0.260539i
$408$		0			0
$409$	97.4355	+	168.763i	0.238229	+	0.412624i	0.960206	−	0.279293i	$-0.0900999\pi$
$409$	97.4355	+	168.763i	0.238229	+	0.412624i	−0.721978	+	0.691917i	$0.756767\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			747.582i			1.81013i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−196.476	+	113.435i	−0.468916	+	0.270729i	−0.715786	−	0.698320i	$-0.753931\pi$
$419$	−196.476	+	113.435i	−0.468916	+	0.270729i	0.246870	+	0.969049i	$0.420598\pi$
$420$		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
$427$	−7.27017	−	12.5923i	−0.0170262	−	0.0294902i
$428$		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
$433$	669.883			1.54707			0.773537	−	0.633751i	$-0.218486\pi$
$433$	669.883			1.54707			0.773537	+	0.633751i	$0.218486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−329.044	+	189.974i	−0.752962	+	0.434723i
$438$		0			0
$439$	38.3327	−	66.3941i	0.0873181	−	0.151239i	−0.819059	−	0.573710i	$-0.805504\pi$
$439$	38.3327	−	66.3941i	0.0873181	−	0.151239i	0.906377	+	0.422471i	$0.138837\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−590.117	−	340.704i	−1.33209	−	0.769084i	−0.346472	−	0.938060i	$-0.612621\pi$
$443$	−590.117	−	340.704i	−1.33209	−	0.769084i	−0.985620	+	0.168976i	$0.945954\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	630.327i		−	1.40385i	−0.712253	−	0.701923i	$-0.752325\pi$
$449$		−	630.327i		−	1.40385i	0.712253	−	0.701923i	$-0.247675\pi$
$450$		0			0
$451$	−393.569			−0.872657
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−286.498	+	496.229i	−0.626910	+	1.08584i	0.361258	+	0.932466i	$0.382347\pi$
$457$	−286.498	+	496.229i	−0.626910	+	1.08584i	−0.988168	+	0.153374i	$0.950986\pi$
$458$		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
$463$	321.190	+	556.317i	0.693714	+	1.20155i	0.970612	+	0.240648i	$0.0773601\pi$
$463$	321.190	+	556.317i	0.693714	+	1.20155i	−0.276899	+	0.960899i	$0.589307\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	506.331i		−	1.08422i	−0.840307	−	0.542111i	$-0.817625\pi$
$467$		−	506.331i		−	1.08422i	0.840307	−	0.542111i	$-0.182375\pi$
$468$		0			0
$469$	−1009.58			−2.15263
$470$		0			0
$471$		0			0
$472$		0			0
$473$	262.167	−	151.362i	0.554265	−	0.320005i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	644.044	+	371.839i	1.34456	+	0.776282i	0.987473	−	0.157789i	$-0.0504365\pi$
$479$	644.044	+	371.839i	1.34456	+	0.776282i	0.357087	+	0.934071i	$0.383770\pi$
$480$		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
$487$	−154.065			−0.316354			−0.158177	−	0.987411i	$-0.550562\pi$
$487$	−154.065			−0.316354			−0.158177	+	0.987411i	$0.550562\pi$
$488$		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
$493$	72.0484	−	124.791i	0.146143	−	0.253127i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−224.855	−	129.820i	−0.452424	−	0.261207i
$498$		0			0
$499$	−247.308	−	428.351i	−0.495608	−	0.858418i	0.504379	−	0.863482i	$-0.331721\pi$
$499$	−247.308	−	428.351i	−0.495608	−	0.858418i	−0.999987	+	0.00506391i	$0.998388\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			711.349i			1.41421i	0.707107	+	0.707106i	$0.250000\pi$
$503$			711.349i			1.41421i	−0.707107	+	0.707106i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	717.714	−	414.372i	1.41005	−	0.814091i	0.414654	−	0.909979i	$-0.363902\pi$
$509$	717.714	−	414.372i	1.41005	−	0.814091i	0.995392	+	0.0958882i	$0.0305691\pi$
$510$		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
$517$	−547.663	−	948.581i	−1.05931	−	1.83478i
$518$		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
$523$	89.4274			0.170989			0.0854946	−	0.996339i	$-0.472753\pi$
$523$	89.4274			0.170989			0.0854946	+	0.996339i	$0.472753\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	737.571	−	425.837i	1.39956	−	0.808039i
$528$		0			0
$529$	305.403	−	528.974i	0.577322	−	0.999951i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−399.284	−	230.527i	−0.749126	−	0.432508i
$534$		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
$547$	441.782	−	765.189i	0.807646	−	1.39888i	−0.106845	−	0.994276i	$-0.534075\pi$
$547$	441.782	−	765.189i	0.807646	−	1.39888i	0.914490	−	0.404608i	$-0.132592\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	92.7096	+	53.5259i	0.168257	+	0.0971432i
$552$		0			0
$553$	−606.314	−	1050.17i	−1.09641	−	1.89904i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			256.448i			0.460410i	0.973142	+	0.230205i	$0.0739396\pi$
$557$			256.448i			0.460410i	−0.973142	+	0.230205i	$0.926060\pi$
$558$		0			0
$559$	354.633			0.634406
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−180.024	+	103.937i	−0.319759	+	0.184613i	−0.651285	−	0.758833i	$-0.725770\pi$
$563$	−180.024	+	103.937i	−0.319759	+	0.184613i	0.331526	+	0.943446i	$0.392436\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−430.923	−	248.794i	−0.757334	−	0.437247i	0.0710034	−	0.997476i	$-0.477380\pi$
$569$	−430.923	−	248.794i	−0.757334	−	0.437247i	−0.828338	+	0.560229i	$0.810713\pi$
$570$		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
$577$	−758.810			−1.31510			−0.657548	−	0.753413i	$-0.728406\pi$
$577$	−758.810			−1.31510			−0.657548	+	0.753413i	$0.728406\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	916.185	−	528.960i	1.57691	−	0.910430i
$582$		0			0
$583$	285.284	−	494.127i	0.489338	−	0.847559i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	656.117	+	378.809i	1.11775	+	0.645331i	0.940825	−	0.338894i	$-0.110053\pi$
$587$	656.117	+	378.809i	1.11775	+	0.645331i	0.176921	+	0.984225i	$0.443386\pi$
$588$		0			0
$589$	316.361	+	547.953i	0.537115	+	0.930311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	258.648i		−	0.436169i	−0.975930	−	0.218084i	$-0.930019\pi$
$593$		−	258.648i		−	0.436169i	0.975930	−	0.218084i	$-0.0699808\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	495.665	−	286.172i	0.827488	−	0.477750i	−0.0255038	−	0.999675i	$-0.508119\pi$
$599$	495.665	−	286.172i	0.827488	−	0.477750i	0.852992	+	0.521924i	$0.174786\pi$
$600$		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
$607$	9.23790	+	16.0005i	0.0152189	+	0.0263600i	0.873535	−	0.486762i	$-0.161822\pi$
$607$	9.23790	+	16.0005i	0.0152189	+	0.0263600i	−0.858316	+	0.513122i	$0.828489\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1283.14i		−	2.10007i
$612$		0			0
$613$	−392.569			−0.640405			−0.320203	−	0.947349i	$-0.603751\pi$
$613$	−392.569			−0.640405			−0.320203	+	0.947349i	$0.603751\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	230.353	−	132.994i	0.373343	−	0.215550i	−0.301575	−	0.953443i	$-0.597512\pi$
$617$	230.353	−	132.994i	0.373343	−	0.215550i	0.674918	+	0.737893i	$0.264179\pi$
$618$		0			0
$619$	−416.736	+	721.808i	−0.673240	+	1.16609i	0.303739	+	0.952755i	$0.401765\pi$
$619$	−416.736	+	721.808i	−0.673240	+	1.16609i	−0.976980	+	0.213332i	$0.931569\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−123.048	−	71.0420i	−0.197509	−	0.114032i
$624$		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
$637$	−789.409	+	1367.30i	−1.23926	+	2.14646i
$638$		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
$643$	−110.593	−	191.552i	−0.171995	−	0.297904i	0.767122	−	0.641501i	$-0.221688\pi$
$643$	−110.593	−	191.552i	−0.171995	−	0.297904i	−0.939117	+	0.343597i	$0.888355\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1095.26i		−	1.69282i	−0.532529	−	0.846412i	$-0.678758\pi$
$647$		−	1095.26i		−	1.69282i	0.532529	−	0.846412i	$-0.321242\pi$
$648$		0			0
$649$	−964.137			−1.48557
$650$		0			0
$651$		0			0
$652$		0			0
$653$	894.048	−	516.179i	1.36914	−	0.790473i	0.378322	−	0.925674i	$-0.376501\pi$
$653$	894.048	−	516.179i	1.36914	−	0.790473i	0.990818	+	0.135201i	$0.0431680\pi$
$654$		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	$0.000195706\pi$
$659$	988.851	+	570.913i	1.50053	+	0.866333i	0.500532	+	0.865718i	$0.333138\pi$
$660$		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
$667$	−321.145			−0.481477
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.2399	−	9.37614i	0.0242026	−	0.0139734i
$672$		0			0
$673$	−139.952	+	242.403i	−0.207952	+	0.360183i	−0.951069	−	0.308978i	$-0.900013\pi$
$673$	−139.952	+	242.403i	−0.207952	+	0.360183i	0.743117	+	0.669161i	$0.233346\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−249.520	−	144.060i	−0.368567	−	0.212792i	0.304265	−	0.952587i	$-0.401589\pi$
$677$	−249.520	−	144.060i	−0.368567	−	0.212792i	−0.672832	+	0.739795i	$0.734922\pi$
$678$		0			0
$679$	585.522	+	1014.15i	0.862330	+	1.49360i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	283.805i		−	0.415527i	−0.978179	−	0.207763i	$-0.933382\pi$
$683$		−	283.805i		−	0.415527i	0.978179	−	0.207763i	$-0.0666184\pi$
$684$		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
$697$	196.784	+	340.840i	0.282330	+	0.489011i
$698$		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
$703$	157.556			0.224120
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1576.75	+	910.339i	−2.23020	+	1.28761i
$708$		0			0
$709$	289.696	−	501.767i	0.408597	−	0.707711i	−0.586135	−	0.810213i	$-0.699351\pi$
$709$	289.696	−	501.767i	0.408597	−	0.707711i	0.994733	+	0.102502i	$0.0326847\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1643.81	−	949.052i	−2.30548	−	1.33107i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			597.306i			0.830746i	0.909651	+	0.415373i	$0.136349\pi$
$719$			597.306i			0.830746i	−0.909651	+	0.415373i	$0.863651\pi$
$720$		0			0
$721$	−1829.01			−2.53677
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−336.905	+	583.537i	−0.463419	+	0.802664i	−0.999129	−	0.0417376i	$-0.986711\pi$
$727$	−336.905	+	583.537i	−0.463419	+	0.802664i	0.535710	+	0.844402i	$0.320044\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−262.167	−	151.362i	−0.358642	−	0.207062i
$732$		0			0
$733$	−390.760	−	676.816i	−0.533097	−	0.923351i	−0.999253	−	0.0386484i	$-0.987695\pi$
$733$	−390.760	−	676.816i	−0.533097	−	0.923351i	0.466156	−	0.884703i	$-0.345639\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1302.03i		−	1.76667i
$738$		0			0
$739$	232.335			0.314391			0.157195	−	0.987568i	$-0.449755\pi$
$739$	232.335			0.314391			0.157195	+	0.987568i	$0.449755\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	434.806	−	251.036i	0.585204	−	0.337868i	−0.177995	−	0.984031i	$-0.556961\pi$
$743$	434.806	−	251.036i	0.585204	−	0.337868i	0.763199	+	0.646164i	$0.223628\pi$
$744$		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
$757$	266.379			0.351888			0.175944	−	0.984400i	$-0.443702\pi$
$757$	266.379			0.351888			0.175944	+	0.984400i	$0.443702\pi$
$758$		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
$763$	223.363	−	386.876i	0.292743	−	0.507046i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−978.139	−	564.729i	−1.27528	−	0.736283i
$768$		0			0
$769$	−644.552	−	1116.40i	−0.838170	−	1.45175i	−0.891423	−	0.453171i	$-0.850293\pi$
$769$	−644.552	−	1116.40i	−0.838170	−	1.45175i	0.0532540	−	0.998581i	$-0.483041\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			919.244i			1.18919i	0.804025	+	0.594595i	$0.202688\pi$
$773$			919.244i			1.18919i	−0.804025	+	0.594595i	$0.797312\pi$
$774$		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
$787$	−631.855	−	1094.40i	−0.802865	−	1.39060i	−0.917723	−	0.397221i	$-0.869975\pi$
$787$	−631.855	−	1094.40i	−0.802865	−	1.39060i	0.114858	−	0.993382i	$-0.463359\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			81.3784i			0.102880i
$792$		0			0
$793$	21.9677			0.0277021
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−530.806	+	306.461i	−0.666006	+	0.384518i	−0.794561	−	0.607184i	$-0.792299\pi$
$797$	−530.806	+	306.461i	−0.666006	+	0.384518i	0.128556	+	0.991702i	$0.458966\pi$
$798$		0			0
$799$	−547.663	+	948.581i	−0.685436	+	1.18721i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−790.258	−	456.256i	−0.984132	−	0.568189i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			273.189i			0.337687i	0.985643	+	0.168844i	$0.0540033\pi$
$809$			273.189i			0.337687i	−0.985643	+	0.168844i	$0.945997\pi$
$810$		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
$817$	112.450	−	194.768i	0.137637	−	0.238395i
$818$		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
$823$	615.218	+	1065.59i	0.747531	+	1.29476i	0.949003	+	0.315267i	$0.102094\pi$
$823$	615.218	+	1065.59i	0.747531	+	1.29476i	−0.201473	+	0.979494i	$0.564573\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			726.246i			0.878169i	0.898446	+	0.439085i	$0.144697\pi$
$827$			726.246i			0.878169i	−0.898446	+	0.439085i	$0.855303\pi$
$828$		0			0
$829$	1439.23			1.73610			0.868052	−	0.496474i	$-0.165372\pi$
$829$	1439.23			1.73610			0.868052	+	0.496474i	$0.165372\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1167.16	−	673.862i	1.40116	−	0.808958i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−968.135	−	558.953i	−1.15392	−	0.666213i	−0.204077	−	0.978955i	$-0.565419\pi$
$839$	−968.135	−	558.953i	−1.15392	−	0.666213i	−0.949838	+	0.312742i	$0.898753\pi$
$840$		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
$847$	−1274.15			−1.50431
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−409.331	+	236.327i	−0.481000	+	0.277705i
$852$		0			0
$853$	−348.046	+	602.834i	−0.408026	+	0.706722i	−0.994669	−	0.103124i	$-0.967116\pi$
$853$	−348.046	+	602.834i	−0.408026	+	0.706722i	0.586642	+	0.809846i	$0.300449\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	745.040	+	430.149i	0.869358	+	0.501924i	0.867135	−	0.498073i	$-0.165959\pi$
$857$	745.040	+	430.149i	0.869358	+	0.501924i	0.00222347	+	0.999998i	$0.499292\pi$
$858$		0			0
$859$	321.216	+	556.362i	0.373942	+	0.647686i	0.990168	−	0.139883i	$-0.0446727\pi$
$859$	321.216	+	556.362i	0.373942	+	0.647686i	−0.616226	+	0.787569i	$0.711339\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1466.58i			1.69940i	0.527269	+	0.849698i	$0.323216\pi$
$863$			1466.58i			1.69940i	−0.527269	+	0.849698i	$0.676784\pi$
$864$		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
$877$	−96.6835	−	167.461i	−0.110243	−	0.190947i	0.805625	−	0.592426i	$-0.201830\pi$
$877$	−96.6835	−	167.461i	−0.110243	−	0.190947i	−0.915868	+	0.401479i	$0.868496\pi$
$878$		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
$883$	−352.460			−0.399162			−0.199581	−	0.979881i	$-0.563958\pi$
$883$	−352.460			−0.399162			−0.199581	+	0.979881i	$0.563958\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	932.087	−	538.140i	1.05083	−	0.606697i	0.127949	−	0.991781i	$-0.459161\pi$
$887$	932.087	−	538.140i	1.05083	−	0.606697i	0.922882	+	0.385084i	$0.125827\pi$
$888$		0			0
$889$	856.887	−	1484.17i	0.963877	−	1.66948i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−704.716	−	406.868i	−0.789155	−	0.455619i
$894$		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
$907$	−110.496	+	191.385i	−0.121826	+	0.211008i	−0.920488	−	0.390772i	$-0.872208\pi$
$907$	−110.496	+	191.385i	−0.121826	+	0.211008i	0.798662	+	0.601780i	$0.205542\pi$
$908$		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
$913$	682.185	+	1181.58i	0.747191	+	1.29417i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	782.775i		−	0.853626i
$918$		0			0
$919$	1006.85			1.09560			0.547799	−	0.836610i	$-0.315466\pi$
$919$	1006.85			1.09560			0.547799	+	0.836610i	$0.315466\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	339.714	−	196.134i	0.368054	−	0.212496i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	974.758	+	562.777i	1.04926	+	0.605788i	0.922441	−	0.386139i	$-0.126191\pi$
$929$	974.758	+	562.777i	1.04926	+	0.605788i	0.126814	+	0.991926i	$0.459525\pi$
$930$		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
$937$	−1343.20			−1.43351			−0.716756	−	0.697324i	$-0.754374\pi$
$937$	−1343.20			−1.43351			−0.716756	+	0.697324i	$0.754374\pi$
$938$		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
$943$	438.569	−	759.623i	0.465078	−	0.805539i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−620.456	−	358.220i	−0.655180	−	0.378268i	0.135258	−	0.990810i	$-0.456814\pi$
$947$	−620.456	−	358.220i	−0.655180	−	0.378268i	−0.790438	+	0.612542i	$0.790147\pi$
$948$		0			0
$949$	−534.490	−	925.764i	−0.563214	−	0.975515i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			793.300i			0.832424i	0.909268	+	0.416212i	$0.136643\pi$
$953$			793.300i			0.832424i	−0.909268	+	0.416212i	$0.863357\pi$
$954$		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
$967$	914.331	+	1583.67i	0.945533	+	1.63771i	0.754680	+	0.656093i	$0.227792\pi$
$967$	914.331	+	1583.67i	0.945533	+	1.63771i	0.190853	+	0.981619i	$0.438875\pi$
$968$		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
$973$	−1253.46			−1.28824
$974$		0			0
$975$		0			0
$976$		0			0
$977$	289.591	−	167.195i	0.296408	−	0.171131i	−0.344420	−	0.938816i	$-0.611924\pi$
$977$	289.591	−	167.195i	0.296408	−	0.171131i	0.640828	+	0.767684i	$0.278591\pi$
$978$		0			0
$979$	91.6210	−	158.692i	0.0935863	−	0.162096i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1604.71	+	926.480i	1.63246	+	0.942502i	0.983331	+	0.181825i	$0.0582004\pi$
$983$	1604.71	+	926.480i	1.63246	+	0.942502i	0.649131	+	0.760677i	$0.275133\pi$
$984$		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
$997$	596.808	−	1033.70i	0.598604	−	1.03681i	−0.394423	−	0.918929i	$-0.629056\pi$
$997$	596.808	−	1033.70i	0.598604	−	1.03681i	0.993027	−	0.117884i	$-0.0376111\pi$
$998$		0			0
$999$		0			0

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

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.o.a.41.2	✓	4	45.4	even	6
180.3.o.a.101.2	yes	4	15.14	odd	2
540.3.o.a.341.1		4	45.14	odd	6
540.3.o.a.521.1		4	5.4	even	2
720.3.bs.a.401.2		4	180.139	odd	6
720.3.bs.a.641.2		4	60.59	even	2
900.3.p.b.101.1		4	3.2	odd	2
900.3.p.b.401.1		4	9.4	even	3
900.3.u.b.149.1		8	45.13	odd	12
900.3.u.b.149.4		8	45.22	odd	12
900.3.u.b.749.1		8	15.2	even	4
900.3.u.b.749.4		8	15.8	even	4
1620.3.g.a.161.1		4	45.34	even	6
1620.3.g.a.161.3		4	45.29	odd	6
2160.3.bs.a.881.1		4	180.59	even	6
2160.3.bs.a.1601.1		4	20.19	odd	2
2700.3.p.a.1601.1		4	1.1	even	1	trivial
2700.3.p.a.2501.1		4	9.5	odd	6	inner
2700.3.u.a.449.1		8	45.23	even	12
2700.3.u.a.449.4		8	45.32	even	12
2700.3.u.a.2249.1		8	5.2	odd	4
2700.3.u.a.2249.4		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2700.3.p.a.1601.1

Level	$2700$
Weight	$3$
Character	2700.1601
Analytic conductor	$73.570$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$