LMFDB - Embedded newform 2160.3.bs.a.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(881,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5, 0]))

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

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

chi := DirichletCharacter("2160.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.bs (of order $6$, degree $2$, 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:	$58.8557371018$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.1
Root			$1.93649 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	2160.881
Dual form			2160.3.bs.a.1601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(-5.87298 - 10.1723i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(-5.87298 - 10.1723i) q^{7} +(13.1190 - 7.57423i) q^{11} +(8.87298 - 15.3685i) q^{13} +15.1485i q^{17} +11.2540 q^{19} +(29.2379 + 16.8805i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-8.23790 + 4.75615i) q^{29} +(28.1109 - 48.6895i) q^{31} +26.2648i q^{35} +14.0000 q^{37} +(22.5000 + 12.9904i) q^{41} +(-9.99193 - 17.3065i) q^{43} +(62.6190 - 36.1531i) q^{47} +(-44.4839 + 77.0483i) q^{49} +37.6651i q^{53} -33.8730 q^{55} +(-55.1190 - 31.8229i) q^{59} +(-0.618950 - 1.07205i) q^{61} +(-34.3649 + 19.8406i) q^{65} +(42.9758 - 74.4363i) q^{67} +22.1046i q^{71} -60.2379 q^{73} +(-154.095 - 88.9666i) q^{77} +(51.6190 + 89.4066i) q^{79} +(-78.0000 + 45.0333i) q^{83} +(16.9365 - 29.3349i) q^{85} -12.0964i q^{89} -208.444 q^{91} +(-21.7933 - 12.5824i) q^{95} +(-49.8488 - 86.3406i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} + 6 q^{11} + 20 q^{13} + 76 q^{19} + 24 q^{23} + 10 q^{25} + 60 q^{29} + 4 q^{31} + 56 q^{37} + 90 q^{41} + 22 q^{43} + 204 q^{47} - 54 q^{49} - 120 q^{55} - 174 q^{59} + 44 q^{61} - 60 q^{65} - 14 q^{67} - 148 q^{73} - 384 q^{77} + 160 q^{79} - 312 q^{83} + 60 q^{85} - 400 q^{91} + 60 q^{95} + 2 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 + 6 * q^11 + 20 * q^13 + 76 * q^19 + 24 * q^23 + 10 * q^25 + 60 * q^29 + 4 * q^31 + 56 * q^37 + 90 * q^41 + 22 * q^43 + 204 * q^47 - 54 * q^49 - 120 * q^55 - 174 * q^59 + 44 * q^61 - 60 * q^65 - 14 * q^67 - 148 * q^73 - 384 * q^77 + 160 * q^79 - 312 * q^83 + 60 * q^85 - 400 * q^91 + 60 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$6$		0			0
$7$	−5.87298	−	10.1723i	−0.838998	−	1.45319i	−0.890734	−	0.454526i	$-0.849809\pi$
$7$	−5.87298	−	10.1723i	−0.838998	−	1.45319i	0.0517360	−	0.998661i	$-0.483525\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	13.1190	−	7.57423i	1.19263	−	0.688566i	0.233729	−	0.972302i	$-0.424907\pi$
$11$	13.1190	−	7.57423i	1.19263	−	0.688566i	0.958903	+	0.283735i	$0.0915737\pi$
$12$		0			0
$13$	8.87298	−	15.3685i	0.682537	−	1.18219i	−0.291667	−	0.956520i	$-0.594210\pi$
$13$	8.87298	−	15.3685i	0.682537	−	1.18219i	0.974204	−	0.225669i	$-0.0724568\pi$
$14$		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	$-0.404294\pi$
$19$	11.2540			0.592318			0.296159	+	0.955139i	$0.404294\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	29.2379	+	16.8805i	1.27121	+	0.733935i	0.975216	−	0.221254i	$-0.0710149\pi$
$23$	29.2379	+	16.8805i	1.27121	+	0.733935i	0.295997	+	0.955189i	$0.404348\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.23790	+	4.75615i	−0.284066	+	0.164005i	−0.635262	−	0.772296i	$-0.719108\pi$
$29$	−8.23790	+	4.75615i	−0.284066	+	0.164005i	0.351197	+	0.936302i	$0.385775\pi$
$30$		0			0
$31$	28.1109	−	48.6895i	0.906803	−	1.57063i	0.0883237	−	0.996092i	$-0.471849\pi$
$31$	28.1109	−	48.6895i	0.906803	−	1.57063i	0.818479	−	0.574537i	$-0.194818\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			26.2648i			0.750422i
$36$		0			0
$37$	14.0000			0.378378			0.189189	−	0.981941i	$-0.439414\pi$
$37$	14.0000			0.378378			0.189189	+	0.981941i	$0.439414\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	22.5000	+	12.9904i	0.548780	+	0.316839i	0.748630	−	0.662988i	$-0.230712\pi$
$41$	22.5000	+	12.9904i	0.548780	+	0.316839i	−0.199849	+	0.979827i	$0.564045\pi$
$42$		0			0
$43$	−9.99193	−	17.3065i	−0.232371	−	0.402478i	0.726135	−	0.687552i	$-0.241315\pi$
$43$	−9.99193	−	17.3065i	−0.232371	−	0.402478i	−0.958505	+	0.285075i	$0.907982\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	62.6190	−	36.1531i	1.33232	−	0.769214i	0.346664	−	0.937990i	$-0.387315\pi$
$47$	62.6190	−	36.1531i	1.33232	−	0.769214i	0.985655	+	0.168775i	$0.0539812\pi$
$48$		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
$55$	−33.8730			−0.615872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−55.1190	−	31.8229i	−0.934219	−	0.539372i	−0.0460759	−	0.998938i	$-0.514672\pi$
$59$	−55.1190	−	31.8229i	−0.934219	−	0.539372i	−0.888144	+	0.459566i	$0.848005\pi$
$60$		0			0
$61$	−0.618950	−	1.07205i	−0.0101467	−	0.0175746i	0.860907	−	0.508762i	$-0.169897\pi$
$61$	−0.618950	−	1.07205i	−0.0101467	−	0.0175746i	−0.871054	+	0.491187i	$0.836563\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−34.3649	+	19.8406i	−0.528691	+	0.305240i
$66$		0			0
$67$	42.9758	−	74.4363i	0.641430	−	1.11099i	−0.343684	−	0.939085i	$-0.611675\pi$
$67$	42.9758	−	74.4363i	0.641430	−	1.11099i	0.985114	−	0.171904i	$-0.0549918\pi$
$68$		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	$-0.635375\pi$
$73$	−60.2379			−0.825177			−0.412588	+	0.910918i	$0.635375\pi$
$74$		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	0.982291	+	0.187360i	$0.0599932\pi$
$79$	51.6190	+	89.4066i	0.653404	+	1.13173i	−0.328887	+	0.944369i	$0.606673\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−78.0000	+	45.0333i	−0.939759	+	0.542570i	−0.889885	−	0.456185i	$-0.849215\pi$
$83$	−78.0000	+	45.0333i	−0.939759	+	0.542570i	−0.0498743	+	0.998756i	$0.515882\pi$
$84$		0			0
$85$	16.9365	−	29.3349i	0.199253	−	0.345116i
$86$		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
$95$	−21.7933	−	12.5824i	−0.229404	−	0.132446i
$96$		0			0
$97$	−49.8488	−	86.3406i	−0.513905	−	0.890110i	−0.999870	−	0.0161312i	$-0.994865\pi$
$97$	−49.8488	−	86.3406i	−0.513905	−	0.890110i	0.485965	−	0.873978i	$-0.338468\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	134.238	−	77.5023i	1.32909	−	0.767349i	0.343929	−	0.938995i	$-0.388242\pi$
$101$	134.238	−	77.5023i	1.32909	−	0.767349i	0.985159	+	0.171646i	$0.0549086\pi$
$102$		0			0
$103$	77.8569	−	134.852i	0.755892	−	1.30924i	−0.189038	−	0.981970i	$-0.560537\pi$
$103$	77.8569	−	134.852i	0.755892	−	1.30924i	0.944930	−	0.327273i	$-0.106130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	84.8705i		−	0.793182i	−0.917995	−	0.396591i	$-0.870193\pi$
$107$		−	84.8705i		−	0.793182i	0.917995	−	0.396591i	$-0.129807\pi$
$108$		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	$-0.343093\pi$
$113$	−6.00000	−	3.46410i	−0.0530973	−	0.0306558i	−0.526314	+	0.850290i	$0.676426\pi$
$114$		0			0
$115$	−37.7460	−	65.3779i	−0.328226	−	0.568504i
$116$		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
$125$		−	11.1803i		−	0.0894427i
$126$		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	$-0.251469\pi$
$131$	57.7137	+	33.3210i	0.440563	+	0.254359i	−0.263274	+	0.964721i	$0.584802\pi$
$132$		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	$-0.752767\pi$
$137$	−63.4052	+	36.6070i	−0.462812	+	0.267205i	0.250414	+	0.968139i	$0.419433\pi$
$138$		0			0
$139$	−53.3569	+	92.4168i	−0.383862	+	0.664869i	−0.991611	−	0.129261i	$-0.958740\pi$
$139$	−53.3569	+	92.4168i	−0.383862	+	0.664869i	0.607748	+	0.794130i	$0.292073\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	268.824i		−	1.87989i
$144$		0			0
$145$	21.2702			0.146691
$146$		0			0
$147$		0			0
$148$		0			0
$149$	144.333	+	83.3305i	0.968676	+	0.559265i	0.898832	−	0.438293i	$-0.144417\pi$
$149$	144.333	+	83.3305i	0.968676	+	0.559265i	0.0698433	+	0.997558i	$0.477750\pi$
$150$		0			0
$151$	1.71370	+	2.96822i	0.0113490	+	0.0196571i	0.871644	−	0.490139i	$-0.163054\pi$
$151$	1.71370	+	2.96822i	0.0113490	+	0.0196571i	−0.860295	+	0.509796i	$0.829721\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−108.873	+	62.8578i	−0.702406	+	0.405534i
$156$		0			0
$157$	33.9677	−	58.8338i	0.216355	−	0.374738i	−0.737336	−	0.675526i	$-0.763916\pi$
$157$	33.9677	−	58.8338i	0.216355	−	0.374738i	0.953691	+	0.300788i	$0.0972498\pi$
$158$		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	−0.470752	−	0.882266i	$-0.656017\pi$
$167$	−245.522	−	141.752i	−1.47019	−	0.848816i	−0.999440	+	0.0334495i	$0.989351\pi$
$168$		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	$-0.539094\pi$
$173$	116.903	−	67.4941i	0.675741	−	0.390139i	0.798248	+	0.602328i	$0.205760\pi$
$174$		0			0
$175$	29.3649	−	50.8615i	0.167800	−	0.290637i
$176$		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
$185$	−27.1109	−	15.6525i	−0.146545	−	0.0846080i
$186$		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	$-0.927581\pi$
$191$	−241.808	+	139.608i	−1.26601	+	0.730933i	−0.291782	+	0.956485i	$0.594248\pi$
$192$		0			0
$193$	52.6431	−	91.1806i	0.272762	−	0.472438i	−0.696806	−	0.717260i	$-0.745396\pi$
$193$	52.6431	−	91.1806i	0.272762	−	0.472438i	0.969568	+	0.244822i	$0.0787294\pi$
$194$		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	$-0.479776\pi$
$199$	25.2702			0.126986			0.0634929	+	0.997982i	$0.479776\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	96.7621	+	55.8656i	0.476661	+	0.275200i
$204$		0			0
$205$	−29.0474	−	50.3115i	−0.141695	−	0.245422i
$206$		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	$-0.912621\pi$
$211$	−52.0161	+	90.0946i	−0.246522	+	0.426989i	0.716036	+	0.698063i	$0.245954\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			44.6853i			0.207839i
$216$		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	0.997933	+	0.0642694i	$0.0204717\pi$
$223$	123.681	+	214.223i	0.554625	+	0.960639i	−0.443307	+	0.896370i	$0.646195\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	261.738	−	151.114i	1.15303	−	0.665702i	0.203407	−	0.979094i	$-0.434799\pi$
$227$	261.738	−	151.114i	1.15303	−	0.665702i	0.949624	+	0.313392i	$0.101465\pi$
$228$		0			0
$229$	−133.111	+	230.555i	−0.581270	+	1.00679i	0.414059	+	0.910250i	$0.364111\pi$
$229$	−133.111	+	230.555i	−0.581270	+	1.00679i	−0.995329	+	0.0965395i	$0.969223\pi$
$230$		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
$235$	−161.681			−0.688006
$236$		0			0
$237$		0			0
$238$		0			0
$239$	68.2863	+	39.4251i	0.285717	+	0.164959i	0.636009	−	0.771682i	$-0.280584\pi$
$239$	68.2863	+	39.4251i	0.285717	+	0.164959i	−0.350292	+	0.936641i	$0.613918\pi$
$240$		0			0
$241$	110.246	+	190.952i	0.457452	+	0.792330i	0.998826	−	0.0484519i	$-0.0154288\pi$
$241$	110.246	+	190.952i	0.457452	+	0.792330i	−0.541373	+	0.840782i	$0.682095\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	172.285	−	99.4689i	0.703205	−	0.405996i
$246$		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	$-0.237732\pi$
$257$	131.692	+	76.0322i	0.512418	+	0.295845i	−0.221409	+	0.975181i	$0.571065\pi$
$258$		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	−0.509388	−	0.860537i	$-0.670128\pi$
$263$	−4.95365	+	2.85999i	−0.0188352	+	0.0108745i	0.490553	+	0.871411i	$0.336795\pi$
$264$		0			0
$265$	42.1109	−	72.9382i	0.158909	−	0.275238i
$266$		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	$-0.885694\pi$
$271$	−507.427			−1.87243			−0.936213	+	0.351433i	$0.885694\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	65.5948	+	37.8711i	0.238526	+	0.137713i
$276$		0			0
$277$	−111.778	−	193.606i	−0.403532	−	0.698937i	0.590618	−	0.806951i	$-0.298884\pi$
$277$	−111.778	−	193.606i	−0.403532	−	0.698937i	−0.994149	+	0.108014i	$0.965551\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	39.8105	−	22.9846i	0.141674	−	0.0817957i	−0.427487	−	0.904021i	$-0.640601\pi$
$281$	39.8105	−	22.9846i	0.141674	−	0.0817957i	0.569162	+	0.822226i	$0.307268\pi$
$282$		0			0
$283$	72.0161	−	124.736i	0.254474	−	0.440762i	−0.710279	−	0.703921i	$-0.751431\pi$
$283$	72.0161	−	124.736i	0.254474	−	0.440762i	0.964753	+	0.263159i	$0.0847643\pi$
$284$		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	$-0.585671\pi$
$293$	−361.476	−	208.698i	−1.23371	−	0.712280i	−0.967800	+	0.251719i	$0.919004\pi$
$294$		0			0
$295$	71.1583	+	123.250i	0.241214	+	0.417796i
$296$		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
$305$			2.76803i			0.00907550i
$306$		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	$-0.481802\pi$
$311$	−242.238	−	139.856i	−0.778900	−	0.449698i	−0.836040	+	0.548668i	$0.815135\pi$
$312$		0			0
$313$	14.5786	+	25.2509i	0.0465771	+	0.0806738i	0.888374	−	0.459120i	$-0.151835\pi$
$313$	14.5786	+	25.2509i	0.0465771	+	0.0806738i	−0.841797	+	0.539794i	$0.818502\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−109.095	+	62.9859i	−0.344147	+	0.198694i	−0.662105	−	0.749411i	$-0.730337\pi$
$317$	−109.095	+	62.9859i	−0.344147	+	0.198694i	0.317957	+	0.948105i	$0.397003\pi$
$318$		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
$325$	88.7298			0.273015
$326$		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	−0.967182	−	0.254085i	$-0.918226\pi$
$331$	−232.903	−	403.400i	−0.703635	−	1.21873i	0.263547	−	0.964647i	$-0.415108\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−166.445	+	96.0968i	−0.496849	+	0.286856i
$336$		0			0
$337$	−175.087	+	303.259i	−0.519545	+	0.899878i	0.480197	+	0.877161i	$0.340565\pi$
$337$	−175.087	+	303.259i	−0.519545	+	0.899878i	−0.999742	+	0.0227177i	$0.992768\pi$
$338$		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	0.598961	−	0.800778i	$-0.295580\pi$
$347$	71.1169	+	41.0594i	0.204948	+	0.118327i	−0.394013	+	0.919105i	$0.628914\pi$
$348$		0			0
$349$	−22.7621	−	39.4251i	−0.0652209	−	0.112966i	0.831571	−	0.555418i	$-0.187442\pi$
$349$	−22.7621	−	39.4251i	−0.0652209	−	0.112966i	−0.896792	+	0.442452i	$0.854109\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−453.877	+	262.046i	−1.28577	+	0.742340i	−0.977897	−	0.209087i	$-0.932951\pi$
$353$	−453.877	+	262.046i	−1.28577	+	0.742340i	−0.307873	+	0.951427i	$0.599617\pi$
$354$		0			0
$355$	24.7137	−	42.8054i	0.0696161	−	0.120579i
$356$		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
$365$	116.650	+	67.3480i	0.319590	+	0.184515i
$366$		0			0
$367$	−16.2379	−	28.1249i	−0.0442450	−	0.0766345i	0.843055	−	0.537828i	$-0.180755\pi$
$367$	−16.2379	−	28.1249i	−0.0442450	−	0.0766345i	−0.887300	+	0.461193i	$0.847422\pi$
$368$		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	$-0.725691\pi$
$373$	123.746	−	214.334i	0.331759	−	0.574623i	0.982857	+	0.184371i	$0.0590247\pi$
$374$		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	$-0.629866\pi$
$379$	−300.746			−0.793525			−0.396762	+	0.917921i	$0.629866\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−183.665	−	106.039i	−0.479544	−	0.276865i	0.240683	−	0.970604i	$-0.422629\pi$
$383$	−183.665	−	106.039i	−0.479544	−	0.276865i	−0.720226	+	0.693739i	$0.755962\pi$
$384$		0			0
$385$	198.935	+	344.566i	0.516715	+	0.894977i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	190.760	−	110.135i	0.490386	−	0.283124i	−0.234349	−	0.972153i	$-0.575296\pi$
$389$	190.760	−	110.135i	0.490386	−	0.283124i	0.724734	+	0.689028i	$0.241962\pi$
$390$		0			0
$391$	−255.714	+	442.909i	−0.653999	+	1.13276i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	230.847i		−	0.584423i
$396$		0			0
$397$	−97.4919			−0.245572			−0.122786	−	0.992433i	$-0.539183\pi$
$397$	−97.4919			−0.245572			−0.122786	+	0.992433i	$0.539183\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	66.8347	+	38.5870i	0.166670	+	0.0962270i	0.581015	−	0.813893i	$-0.302656\pi$
$401$	66.8347	+	38.5870i	0.166670	+	0.0962270i	−0.414345	+	0.910120i	$0.635989\pi$
$402$		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	−0.721978	−	0.691917i	$-0.756767\pi$
$409$	97.4355	−	168.763i	0.238229	−	0.412624i	0.960206	+	0.279293i	$0.0900999\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			747.582i			1.81013i
$414$		0			0
$415$	201.395			0.485289
$416$		0			0
$417$		0			0
$418$		0			0
$419$	196.476	+	113.435i	0.468916	+	0.270729i	0.715786	−	0.698320i	$-0.246069\pi$
$419$	196.476	+	113.435i	0.468916	+	0.270729i	−0.246870	+	0.969049i	$0.579402\pi$
$420$		0			0
$421$	150.857	+	261.292i	0.358330	+	0.620645i	0.987682	−	0.156475i	$-0.0500130\pi$
$421$	150.857	+	261.292i	0.358330	+	0.620645i	−0.629352	+	0.777120i	$0.716680\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−65.5948	+	37.8711i	−0.154341	+	0.0891086i
$426$		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	$-0.781514\pi$
$433$	−669.883			−1.54707			−0.773537	+	0.633751i	$0.781514\pi$
$434$		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	$-0.194496\pi$
$439$	−38.3327	−	66.3941i	−0.0873181	−	0.151239i	−0.906377	+	0.422471i	$0.861163\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−590.117	+	340.704i	−1.33209	+	0.769084i	−0.985620	−	0.168976i	$-0.945954\pi$
$443$	−590.117	+	340.704i	−1.33209	+	0.769084i	−0.346472	+	0.938060i	$0.612621\pi$
$444$		0			0
$445$	−13.5242	+	23.4246i	−0.0303915	+	0.0526396i
$446$		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
$455$	403.649	+	233.047i	0.887141	+	0.512191i
$456$		0			0
$457$	286.498	+	496.229i	0.626910	+	1.08584i	0.988168	+	0.153374i	$0.0490140\pi$
$457$	286.498	+	496.229i	0.626910	+	1.08584i	−0.361258	+	0.932466i	$0.617653\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−615.950	+	355.619i	−1.33612	+	0.771407i	−0.986229	−	0.165385i	$-0.947113\pi$
$461$	−615.950	+	355.619i	−1.33612	+	0.771407i	−0.349887	+	0.936792i	$0.613780\pi$
$462$		0			0
$463$	321.190	−	556.317i	0.693714	−	1.20155i	−0.276899	−	0.960899i	$-0.589307\pi$
$463$	321.190	−	556.317i	0.693714	−	1.20155i	0.970612	−	0.240648i	$-0.0773601\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			506.331i			1.08422i	0.840307	+	0.542111i	$0.182375\pi$
$467$			506.331i			1.08422i	−0.840307	+	0.542111i	$0.817625\pi$
$468$		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
$475$	28.1351	+	48.7314i	0.0592318	+	0.102592i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−644.044	+	371.839i	−1.34456	+	0.776282i	−0.987473	−	0.157789i	$-0.949563\pi$
$479$	−644.044	+	371.839i	−1.34456	+	0.776282i	−0.357087	+	0.934071i	$0.616230\pi$
$480$		0			0
$481$	124.222	−	215.158i	0.258257	−	0.447315i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			222.931i			0.459651i
$486$		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	$-0.387935\pi$
$491$	−145.167	−	83.8124i	−0.295657	−	0.170697i	−0.640490	+	0.767967i	$0.721269\pi$
$492$		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	$-0.668279\pi$
$499$	247.308	−	428.351i	0.495608	−	0.858418i	0.999987	+	0.00506391i	$0.00161190\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	711.349i		−	1.41421i	−0.707107	−	0.707106i	$-0.750000\pi$
$503$		−	711.349i		−	1.41421i	0.707107	−	0.707106i	$-0.250000\pi$
$504$		0			0
$505$	−346.601			−0.686338
$506$		0			0
$507$		0			0
$508$		0			0
$509$	717.714	+	414.372i	1.41005	+	0.814091i	0.995392	−	0.0958882i	$-0.0305691\pi$
$509$	717.714	+	414.372i	1.41005	+	0.814091i	0.414654	+	0.909979i	$0.363902\pi$
$510$		0			0
$511$	353.776	+	612.758i	0.692321	+	1.19914i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−301.538	+	174.093i	−0.585511	+	0.338045i
$516$		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	$-0.796126\pi$
$521$		−	622.686i		−	1.19518i	0.801804	−	0.597588i	$-0.203874\pi$
$522$		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
$535$	−94.8881	+	164.351i	−0.177361	+	0.307198i
$536$		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
$545$	73.6492	+	42.5214i	0.135136	+	0.0780209i
$546$		0			0
$547$	441.782	+	765.189i	0.807646	+	1.39888i	0.914490	+	0.404608i	$0.132592\pi$
$547$	441.782	+	765.189i	0.807646	+	1.39888i	−0.106845	+	0.994276i	$0.534075\pi$
$548$		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	0.331526	−	0.943446i	$-0.392436\pi$
$563$	−180.024	−	103.937i	−0.319759	−	0.184613i	−0.651285	+	0.758833i	$0.725770\pi$
$564$		0			0
$565$	7.74597	+	13.4164i	0.0137097	+	0.0237459i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−430.923	+	248.794i	−0.757334	+	0.437247i	−0.828338	−	0.560229i	$-0.810713\pi$
$569$	−430.923	+	248.794i	−0.757334	+	0.437247i	0.0710034	+	0.997476i	$0.477380\pi$
$570$		0			0
$571$	453.817	−	786.033i	0.794775	−	1.37659i	−0.128207	−	0.991747i	$-0.540922\pi$
$571$	453.817	−	786.033i	0.794775	−	1.37659i	0.922982	−	0.384843i	$-0.125744\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			168.805i			0.293574i
$576$		0			0
$577$	758.810			1.31510			0.657548	−	0.753413i	$-0.271594\pi$
$577$	758.810			1.31510			0.657548	+	0.753413i	$0.271594\pi$
$578$		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	0.176921	−	0.984225i	$-0.443386\pi$
$587$	656.117	−	378.809i	1.11775	−	0.645331i	0.940825	+	0.338894i	$0.110053\pi$
$588$		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
$595$	−397.871			−0.668691
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−495.665	−	286.172i	−0.827488	−	0.477750i	0.0255038	−	0.999675i	$-0.491881\pi$
$599$	−495.665	−	286.172i	−0.827488	−	0.477750i	−0.852992	+	0.521924i	$0.825214\pi$
$600$		0			0
$601$	−68.8186	−	119.197i	−0.114507	−	0.198332i	0.803076	−	0.595877i	$-0.203195\pi$
$601$	−68.8186	−	119.197i	−0.114507	−	0.198332i	−0.917582	+	0.397545i	$0.869862\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−210.062	+	121.280i	−0.347211	+	0.200462i
$606$		0			0
$607$	9.23790	−	16.0005i	0.0152189	−	0.0263600i	−0.858316	−	0.513122i	$-0.828489\pi$
$607$	9.23790	−	16.0005i	0.0152189	−	0.0263600i	0.873535	+	0.486762i	$0.161822\pi$
$608$		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	$-0.396249\pi$
$613$	392.569			0.640405			0.320203	+	0.947349i	$0.396249\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−230.353	−	132.994i	−0.373343	−	0.215550i	0.301575	−	0.953443i	$-0.402488\pi$
$617$	−230.353	−	132.994i	−0.373343	−	0.215550i	−0.674918	+	0.737893i	$0.735821\pi$
$618$		0			0
$619$	416.736	+	721.808i	0.673240	+	1.16609i	0.976980	+	0.213332i	$0.0684315\pi$
$619$	416.736	+	721.808i	0.673240	+	1.16609i	−0.303739	+	0.952755i	$0.598235\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−123.048	+	71.0420i	−0.197509	+	0.114032i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		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	$-0.410056\pi$
$631$	351.875			0.557647			0.278823	+	0.960342i	$0.410056\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	282.540	+	163.125i	0.444945	+	0.256889i
$636$		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	−0.751067	−	0.660226i	$-0.770461\pi$
$641$	−355.645	+	205.332i	−0.554829	+	0.320331i	0.196239	+	0.980556i	$0.437127\pi$
$642$		0			0
$643$	−110.593	+	191.552i	−0.171995	+	0.297904i	−0.939117	−	0.343597i	$-0.888355\pi$
$643$	−110.593	+	191.552i	−0.171995	+	0.297904i	0.767122	+	0.641501i	$0.221688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1095.26i			1.69282i	0.532529	+	0.846412i	$0.321242\pi$
$647$			1095.26i			1.69282i	−0.532529	+	0.846412i	$0.678758\pi$
$648$		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	$-0.623499\pi$
$653$	−894.048	−	516.179i	−1.36914	−	0.790473i	−0.990818	+	0.135201i	$0.956832\pi$
$654$		0			0
$655$	−74.5081	−	129.052i	−0.113753	−	0.197026i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−988.851	+	570.913i	−1.50053	+	0.866333i	−0.500532	+	0.865718i	$0.666862\pi$
$659$	−988.851	+	570.913i	−1.50053	+	0.866333i	−1.00000		0.000614829i	$0.999804\pi$
$660$		0			0
$661$	513.282	−	889.031i	0.776524	−	1.34498i	−0.157410	−	0.987533i	$-0.550315\pi$
$661$	513.282	−	889.031i	0.776524	−	1.34498i	0.933934	−	0.357445i	$-0.116352\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			295.585i			0.444488i
$666$		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	$-0.0999869\pi$
$673$	139.952	+	242.403i	0.207952	+	0.360183i	−0.743117	+	0.669161i	$0.766654\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	249.520	−	144.060i	0.368567	−	0.212792i	−0.304265	−	0.952587i	$-0.598411\pi$
$677$	249.520	−	144.060i	0.368567	−	0.212792i	0.672832	+	0.739795i	$0.265078\pi$
$678$		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	$0.0666184\pi$
$683$			283.805i			0.415527i	−0.978179	+	0.207763i	$0.933382\pi$
$684$		0			0
$685$	163.712			0.238995
$686$		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	0.986037	+	0.166529i	$0.0532560\pi$
$691$	440.331	+	762.675i	0.637237	+	1.10373i	−0.348800	+	0.937197i	$0.613411\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	206.650	−	119.310i	0.297338	−	0.171668i
$696$		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	$0.161318\pi$
$701$			680.500i			0.970757i	−0.874304	+	0.485378i	$0.838682\pi$
$702$		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	0.994733	−	0.102502i	$-0.0326847\pi$
$709$	289.696	+	501.767i	0.408597	+	0.707711i	−0.586135	+	0.810213i	$0.699351\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1643.81	−	949.052i	2.30548	−	1.33107i
$714$		0			0
$715$	−300.554	+	520.576i	−0.420356	+	0.728078i
$716$		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
$725$	−41.1895	−	23.7808i	−0.0568131	−	0.0328011i
$726$		0			0
$727$	−336.905	−	583.537i	−0.463419	−	0.802664i	0.535710	−	0.844402i	$-0.320044\pi$
$727$	−336.905	−	583.537i	−0.463419	−	0.802664i	−0.999129	+	0.0417376i	$0.986711\pi$
$728$		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	−0.466156	−	0.884703i	$-0.654361\pi$
$733$	390.760	−	676.816i	0.533097	−	0.923351i	0.999253	−	0.0386484i	$-0.0123052\pi$
$734$		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	$-0.550245\pi$
$739$	−232.335			−0.314391			−0.157195	+	0.987568i	$0.550245\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	434.806	+	251.036i	0.585204	+	0.337868i	0.763199	−	0.646164i	$-0.223628\pi$
$743$	434.806	+	251.036i	0.585204	+	0.337868i	−0.177995	+	0.984031i	$0.556961\pi$
$744$		0			0
$745$	−186.333	−	322.738i	−0.250111	−	0.433205i
$746$		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	−0.351024	−	0.936367i	$-0.614166\pi$
$751$	477.190	−	826.516i	0.635405	−	1.10055i	0.986429	−	0.164188i	$-0.0525004\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	7.66390i		−	0.0101509i
$756$		0			0
$757$	−266.379			−0.351888			−0.175944	−	0.984400i	$-0.556298\pi$
$757$	−266.379			−0.351888			−0.175944	+	0.984400i	$0.556298\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−294.665	−	170.125i	−0.387208	−	0.223555i	0.293742	−	0.955885i	$-0.405099\pi$
$761$	−294.665	−	170.125i	−0.387208	−	0.223555i	−0.680950	+	0.732330i	$0.738433\pi$
$762$		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	0.0532540	+	0.998581i	$0.483041\pi$
$769$	−644.552	+	1116.40i	−0.838170	+	1.45175i	−0.891423	+	0.453171i	$0.850293\pi$
$770$		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
$775$	281.109			0.362721
$776$		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
$785$	−131.556	+	75.9542i	−0.167588	+	0.0967569i
$786$		0			0
$787$	−631.855	+	1094.40i	−0.802865	+	1.39060i	0.114858	+	0.993382i	$0.463359\pi$
$787$	−631.855	+	1094.40i	−0.802865	+	1.39060i	−0.917723	+	0.397221i	$0.869975\pi$
$788$		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

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.bs (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-5.87298 - 10.1723i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-5.87298 - 10.1723i) q^{7} +(13.1190 - 7.57423i) q^{11} +(8.87298 - 15.3685i) q^{13} +15.1485i q^{17} +11.2540 q^{19} +(29.2379 + 16.8805i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-8.23790 + 4.75615i) q^{29} +(28.1109 - 48.6895i) q^{31} +26.2648i q^{35} +14.0000 q^{37} +(22.5000 + 12.9904i) q^{41} +(-9.99193 - 17.3065i) q^{43} +(62.6190 - 36.1531i) q^{47} +(-44.4839 + 77.0483i) q^{49} +37.6651i q^{53} -33.8730 q^{55} +(-55.1190 - 31.8229i) q^{59} +(-0.618950 - 1.07205i) q^{61} +(-34.3649 + 19.8406i) q^{65} +(42.9758 - 74.4363i) q^{67} +22.1046i q^{71} -60.2379 q^{73} +(-154.095 - 88.9666i) q^{77} +(51.6190 + 89.4066i) q^{79} +(-78.0000 + 45.0333i) q^{83} +(16.9365 - 29.3349i) q^{85} -12.0964i q^{89} -208.444 q^{91} +(-21.7933 - 12.5824i) q^{95} +(-49.8488 - 86.3406i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} + 6 q^{11} + 20 q^{13} + 76 q^{19} + 24 q^{23} + 10 q^{25} + 60 q^{29} + 4 q^{31} + 56 q^{37} + 90 q^{41} + 22 q^{43} + 204 q^{47} - 54 q^{49} - 120 q^{55} - 174 q^{59} + 44 q^{61} - 60 q^{65} - 14 q^{67} - 148 q^{73} - 384 q^{77} + 160 q^{79} - 312 q^{83} + 60 q^{85} - 400 q^{91} + 60 q^{95} + 2 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 + 6 * q^11 + 20 * q^13 + 76 * q^19 + 24 * q^23 + 10 * q^25 + 60 * q^29 + 4 * q^31 + 56 * q^37 + 90 * q^41 + 22 * q^43 + 204 * q^47 - 54 * q^49 - 120 * q^55 - 174 * q^59 + 44 * q^61 - 60 * q^65 - 14 * q^67 - 148 * q^73 - 384 * q^77 + 160 * q^79 - 312 * q^83 + 60 * q^85 - 400 * q^91 + 60 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

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