LMFDB - Embedded newform 1080.4.q.b.361.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,4,Mod(361,1080)] 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("1080.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,40,0,-34] 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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 16 x^{14} + 14 x^{13} - 284 x^{12} + 764 x^{11} + 19770 x^{10} + 55106 x^{9} + \cdots + 193472540143 $ x^16 - 6x^15 + 16x^14 + 14x^13 - 284x^12 + 764x^11 + 19770x^10 + 55106x^9 + 266991x^8 + 335444x^7 - 4573302x^6 + 2211506x^5 + 251845642x^4 + 1438820210x^3 + 12022873630x^2 + 75245140836*x + 193472540143
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.6
Root			$-0.887205 + 4.84133i$ of defining polynomial
Character	$\chi$	$=$	1080.361
Dual form			1080.4.q.b.721.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(4.29040 + 7.43119i) q^{7} +(24.3465 + 42.1693i) q^{11} +(-3.15788 + 5.46961i) q^{13} +1.12025 q^{17} -72.2412 q^{19} +(-66.4417 + 115.080i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-43.3334 - 75.0556i) q^{29} +(-49.6174 + 85.9399i) q^{31} +42.9040 q^{35} +168.720 q^{37} +(-72.2997 + 125.227i) q^{41} +(-259.853 - 450.079i) q^{43} +(37.1950 + 64.4236i) q^{47} +(134.685 - 233.281i) q^{49} -399.402 q^{53} +243.465 q^{55} +(-200.086 + 346.559i) q^{59} +(-123.600 - 214.082i) q^{61} +(15.7894 + 27.3481i) q^{65} +(-313.714 + 543.368i) q^{67} -58.8477 q^{71} +9.10248 q^{73} +(-208.912 + 361.846i) q^{77} +(-43.5786 - 75.4803i) q^{79} +(531.267 + 920.182i) q^{83} +(2.80062 - 4.85082i) q^{85} -1055.07 q^{89} -54.1943 q^{91} +(-180.603 + 312.814i) q^{95} +(52.0827 + 90.2099i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) $ 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	4.29040	+	7.43119i	0.231660	+	0.401246i	0.958297	−	0.285775i	$-0.0922511\pi$
$7$	4.29040	+	7.43119i	0.231660	+	0.401246i	−0.726637	+	0.687022i	$0.758918\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	24.3465	+	42.1693i	0.667339	+	1.15587i	0.978645	+	0.205556i	$0.0659002\pi$
$11$	24.3465	+	42.1693i	0.667339	+	1.15587i	−0.311306	+	0.950310i	$0.600766\pi$
$12$		0			0
$13$	−3.15788	+	5.46961i	−0.0673722	+	0.116692i	−0.897744	−	0.440518i	$-0.854795\pi$
$13$	−3.15788	+	5.46961i	−0.0673722	+	0.116692i	0.830372	+	0.557210i	$0.188128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.12025			0.0159824			0.00799118	−	0.999968i	$-0.497456\pi$
$17$	1.12025			0.0159824			0.00799118	+	0.999968i	$0.497456\pi$
$18$		0			0
$19$	−72.2412			−0.872278			−0.436139	−	0.899879i	$-0.643654\pi$
$19$	−72.2412			−0.872278			−0.436139	+	0.899879i	$0.643654\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−66.4417	+	115.080i	−0.602350	+	1.04330i	0.390114	+	0.920767i	$0.372436\pi$
$23$	−66.4417	+	115.080i	−0.602350	+	1.04330i	−0.992464	+	0.122535i	$0.960898\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−43.3334	−	75.0556i	−0.277476	−	0.480602i	0.693281	−	0.720667i	$-0.256165\pi$
$29$	−43.3334	−	75.0556i	−0.277476	−	0.480602i	−0.970757	+	0.240065i	$0.922831\pi$
$30$		0			0
$31$	−49.6174	+	85.9399i	−0.287469	+	0.497911i	−0.973205	−	0.229939i	$-0.926147\pi$
$31$	−49.6174	+	85.9399i	−0.287469	+	0.497911i	0.685736	+	0.727851i	$0.259481\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	42.9040			0.207203
$36$		0			0
$37$	168.720			0.749660			0.374830	−	0.927094i	$-0.377701\pi$
$37$	168.720			0.749660			0.374830	+	0.927094i	$0.377701\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−72.2997	+	125.227i	−0.275398	+	0.477003i	−0.970235	−	0.242163i	$-0.922143\pi$
$41$	−72.2997	+	125.227i	−0.275398	+	0.477003i	0.694837	+	0.719167i	$0.255476\pi$
$42$		0			0
$43$	−259.853	−	450.079i	−0.921563	−	1.59619i	−0.796998	−	0.603982i	$-0.793580\pi$
$43$	−259.853	−	450.079i	−0.921563	−	1.59619i	−0.124565	−	0.992211i	$-0.539754\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	37.1950	+	64.4236i	0.115435	+	0.199939i	0.917954	−	0.396688i	$-0.129840\pi$
$47$	37.1950	+	64.4236i	0.115435	+	0.199939i	−0.802519	+	0.596627i	$0.796507\pi$
$48$		0			0
$49$	134.685	−	233.281i	0.392668	−	0.680120i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−399.402			−1.03513			−0.517567	−	0.855643i	$-0.673162\pi$
$53$	−399.402			−1.03513			−0.517567	+	0.855643i	$0.673162\pi$
$54$		0			0
$55$	243.465			0.596886
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−200.086	+	346.559i	−0.441508	+	0.764714i	−0.997802	−	0.0662717i	$-0.978890\pi$
$59$	−200.086	+	346.559i	−0.441508	+	0.764714i	0.556294	+	0.830986i	$0.312223\pi$
$60$		0			0
$61$	−123.600	−	214.082i	−0.259432	−	0.449350i	0.706658	−	0.707556i	$-0.250202\pi$
$61$	−123.600	−	214.082i	−0.259432	−	0.449350i	−0.966090	+	0.258206i	$0.916869\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	15.7894	+	27.3481i	0.0301298	+	0.0521863i
$66$		0			0
$67$	−313.714	+	543.368i	−0.572033	+	0.990790i	0.424324	+	0.905510i	$0.360512\pi$
$67$	−313.714	+	543.368i	−0.572033	+	0.990790i	−0.996357	+	0.0852799i	$0.972822\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−58.8477			−0.0983653			−0.0491826	−	0.998790i	$-0.515662\pi$
$71$	−58.8477			−0.0983653			−0.0491826	+	0.998790i	$0.515662\pi$
$72$		0			0
$73$	9.10248			0.0145940			0.00729702	−	0.999973i	$-0.497677\pi$
$73$	9.10248			0.0145940			0.00729702	+	0.999973i	$0.497677\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−208.912	+	361.846i	−0.309191	+	0.535535i
$78$		0			0
$79$	−43.5786	−	75.4803i	−0.0620629	−	0.107496i	0.833324	−	0.552784i	$-0.186435\pi$
$79$	−43.5786	−	75.4803i	−0.0620629	−	0.107496i	−0.895387	+	0.445288i	$0.853101\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	531.267	+	920.182i	0.702580	+	1.21690i	0.967558	+	0.252650i	$0.0813022\pi$
$83$	531.267	+	920.182i	0.702580	+	1.21690i	−0.264977	+	0.964255i	$0.585364\pi$
$84$		0			0
$85$	2.80062	−	4.85082i	0.00357376	−	0.00618994i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1055.07			−1.25659			−0.628297	−	0.777974i	$-0.716248\pi$
$89$	−1055.07			−1.25659			−0.628297	+	0.777974i	$0.716248\pi$
$90$		0			0
$91$	−54.1943			−0.0624297
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−180.603	+	312.814i	−0.195047	+	0.337832i
$96$		0			0
$97$	52.0827	+	90.2099i	0.0545175	+	0.0944270i	0.891996	−	0.452043i	$-0.149305\pi$
$97$	52.0827	+	90.2099i	0.0545175	+	0.0944270i	−0.837479	+	0.546470i	$0.815971\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	576.358	+	998.281i	0.567820	+	0.983492i	0.996781	+	0.0801696i	$0.0255462\pi$
$101$	576.358	+	998.281i	0.567820	+	0.983492i	−0.428962	+	0.903323i	$0.641120\pi$
$102$		0			0
$103$	293.341	−	508.081i	0.280619	−	0.486046i	−0.690919	−	0.722933i	$-0.742794\pi$
$103$	293.341	−	508.081i	0.280619	−	0.486046i	0.971537	+	0.236887i	$0.0761270\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1641.72			1.48329			0.741643	−	0.670795i	$-0.234047\pi$
$107$	1641.72			1.48329			0.741643	+	0.670795i	$0.234047\pi$
$108$		0			0
$109$	−1853.84			−1.62904			−0.814520	−	0.580135i	$-0.803000\pi$
$109$	−1853.84			−1.62904			−0.814520	+	0.580135i	$0.803000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−721.143	+	1249.06i	−0.600349	+	1.03984i	0.392419	+	0.919787i	$0.371638\pi$
$113$	−721.143	+	1249.06i	−0.600349	+	1.03984i	−0.992768	+	0.120049i	$0.961695\pi$
$114$		0			0
$115$	332.209	+	575.402i	0.269379	+	0.466578i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.80631	+	8.32477i	0.00370247	+	0.00641286i
$120$		0			0
$121$	−520.000	+	900.666i	−0.390684	+	0.676684i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−1772.04			−1.23814			−0.619068	−	0.785337i	$-0.712489\pi$
$127$	−1772.04			−1.23814			−0.619068	+	0.785337i	$0.712489\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	584.618	−	1012.59i	0.389911	−	0.675346i	−0.602526	−	0.798099i	$-0.705839\pi$
$131$	584.618	−	1012.59i	0.389911	−	0.675346i	0.992437	+	0.122753i	$0.0391724\pi$
$132$		0			0
$133$	−309.944	−	536.838i	−0.202072	−	0.349998i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−566.836	−	981.789i	−0.353490	−	0.612262i	0.633369	−	0.773850i	$-0.281672\pi$
$137$	−566.836	−	981.789i	−0.353490	−	0.612262i	−0.986858	+	0.161588i	$0.948338\pi$
$138$		0			0
$139$	−1083.55	+	1876.77i	−0.661192	+	1.14522i	0.319110	+	0.947718i	$0.396616\pi$
$139$	−1083.55	+	1876.77i	−0.661192	+	1.14522i	−0.980303	+	0.197501i	$0.936717\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−307.533			−0.179841
$144$		0			0
$145$	−433.334			−0.248182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	698.806	−	1210.37i	0.384218	−	0.665485i	−0.607443	−	0.794364i	$-0.707805\pi$
$149$	698.806	−	1210.37i	0.384218	−	0.665485i	0.991660	+	0.128879i	$0.0411379\pi$
$150$		0			0
$151$	−459.152	−	795.275i	−0.247452	−	0.428600i	0.715366	−	0.698750i	$-0.246260\pi$
$151$	−459.152	−	795.275i	−0.247452	−	0.428600i	−0.962818	+	0.270150i	$0.912927\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	248.087	+	429.699i	0.128560	+	0.222673i
$156$		0			0
$157$	−534.904	+	926.481i	−0.271911	+	0.470963i	−0.969351	−	0.245680i	$-0.920989\pi$
$157$	−534.904	+	926.481i	−0.271911	+	0.470963i	0.697440	+	0.716643i	$0.254322\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1140.25			−0.558161
$162$		0			0
$163$	−2475.81			−1.18970			−0.594849	−	0.803837i	$-0.702788\pi$
$163$	−2475.81			−1.18970			−0.594849	+	0.803837i	$0.702788\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1402.36	+	2428.96i	−0.649809	+	1.12550i	0.333360	+	0.942800i	$0.391818\pi$
$167$	−1402.36	+	2428.96i	−0.649809	+	1.12550i	−0.983168	+	0.182702i	$0.941516\pi$
$168$		0			0
$169$	1078.56	+	1868.11i	0.490922	+	0.850302i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	610.586	+	1057.57i	0.268335	+	0.464770i	0.968432	−	0.249278i	$-0.0801932\pi$
$173$	610.586	+	1057.57i	0.268335	+	0.464770i	−0.700097	+	0.714048i	$0.746860\pi$
$174$		0			0
$175$	107.260	−	185.780i	0.0463319	−	0.0802493i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−369.190			−0.154160			−0.0770798	−	0.997025i	$-0.524560\pi$
$179$	−369.190			−0.154160			−0.0770798	+	0.997025i	$0.524560\pi$
$180$		0			0
$181$	−2583.82			−1.06107			−0.530536	−	0.847662i	$-0.678009\pi$
$181$	−2583.82			−1.06107			−0.530536	+	0.847662i	$0.678009\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	421.800	−	730.580i	0.167629	−	0.290342i
$186$		0			0
$187$	27.2741	+	47.2401i	0.0106657	+	0.0184735i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2397.13	+	4151.96i	0.908118	+	1.57291i	0.816676	+	0.577096i	$0.195814\pi$
$191$	2397.13	+	4151.96i	0.908118	+	1.57291i	0.0914413	+	0.995810i	$0.470853\pi$
$192$		0			0
$193$	−1363.49	+	2361.63i	−0.508528	+	0.880796i	0.491423	+	0.870921i	$0.336477\pi$
$193$	−1363.49	+	2361.63i	−0.508528	+	0.880796i	−0.999951	+	0.00987531i	$0.996857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4546.19			−1.64418			−0.822088	−	0.569360i	$-0.807191\pi$
$197$	−4546.19			−1.64418			−0.822088	+	0.569360i	$0.807191\pi$
$198$		0			0
$199$	3523.79			1.25525			0.627625	−	0.778516i	$-0.284027\pi$
$199$	3523.79			1.25525			0.627625	+	0.778516i	$0.284027\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	371.835	−	644.036i	0.128560	−	0.222672i
$204$		0			0
$205$	361.499	+	626.134i	0.123162	+	0.213322i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1758.82	−	3046.36i	−0.582105	−	1.00824i
$210$		0			0
$211$	−478.046	+	828.000i	−0.155972	+	0.270151i	−0.933412	−	0.358805i	$-0.883184\pi$
$211$	−478.046	+	828.000i	−0.155972	+	0.270151i	0.777441	+	0.628956i	$0.216518\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2598.53			−0.824271
$216$		0			0
$217$	−851.514			−0.266380
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.53761	+	6.12732i	−0.00107677	+	0.00186502i
$222$		0			0
$223$	−1572.24	−	2723.20i	−0.472130	−	0.817753i	0.527361	−	0.849641i	$-0.323181\pi$
$223$	−1572.24	−	2723.20i	−0.472130	−	0.817753i	−0.999491	+	0.0318879i	$0.989848\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1935.98	+	3353.22i	0.566060	+	0.980444i	0.996950	+	0.0780401i	$0.0248662\pi$
$227$	1935.98	+	3353.22i	0.566060	+	0.980444i	−0.430890	+	0.902404i	$0.641800\pi$
$228$		0			0
$229$	−776.369	+	1344.71i	−0.224035	+	0.388039i	−0.956029	−	0.293271i	$-0.905256\pi$
$229$	−776.369	+	1344.71i	−0.224035	+	0.388039i	0.731995	+	0.681310i	$0.238590\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3679.79			1.03464			0.517319	−	0.855792i	$-0.326930\pi$
$233$	3679.79			1.03464			0.517319	+	0.855792i	$0.326930\pi$
$234$		0			0
$235$	371.950			0.103248
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1580.63	−	2737.74i	0.427794	−	0.740960i	−0.568883	−	0.822418i	$-0.692624\pi$
$239$	1580.63	−	2737.74i	0.427794	−	0.740960i	0.996677	+	0.0814580i	$0.0259576\pi$
$240$		0			0
$241$	−1741.53	−	3016.42i	−0.465485	−	0.806244i	0.533738	−	0.845650i	$-0.320787\pi$
$241$	−1741.53	−	3016.42i	−0.465485	−	0.806244i	−0.999223	+	0.0394061i	$0.987453\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−673.425	−	1166.41i	−0.175606	−	0.304159i
$246$		0			0
$247$	228.129	−	395.132i	0.0587673	−	0.101788i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4282.14			−1.07684			−0.538419	−	0.842677i	$-0.680978\pi$
$251$	−4282.14			−1.07684			−0.538419	+	0.842677i	$0.680978\pi$
$252$		0			0
$253$	−6470.48			−1.60789
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2271.89	+	3935.03i	−0.551427	+	0.955099i	0.446745	+	0.894661i	$0.352583\pi$
$257$	−2271.89	+	3935.03i	−0.551427	+	0.955099i	−0.998172	+	0.0604376i	$0.980750\pi$
$258$		0			0
$259$	723.877	+	1253.79i	0.173666	+	0.300798i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3277.39	+	5676.61i	0.768413	+	1.33093i	0.938423	+	0.345487i	$0.112286\pi$
$263$	3277.39	+	5676.61i	0.768413	+	1.33093i	−0.170011	+	0.985442i	$0.554380\pi$
$264$		0			0
$265$	−998.506	+	1729.46i	−0.231463	+	0.400906i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	79.0864			0.0179256			0.00896279	−	0.999960i	$-0.497147\pi$
$269$	79.0864			0.0179256			0.00896279	+	0.999960i	$0.497147\pi$
$270$		0			0
$271$	4218.49			0.945590			0.472795	−	0.881172i	$-0.343245\pi$
$271$	4218.49			0.945590			0.472795	+	0.881172i	$0.343245\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	608.661	−	1054.23i	0.133468	−	0.231173i
$276$		0			0
$277$	−1931.93	−	3346.20i	−0.419055	−	0.725825i	0.576789	−	0.816893i	$-0.304305\pi$
$277$	−1931.93	−	3346.20i	−0.419055	−	0.725825i	−0.995845	+	0.0910678i	$0.970972\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2135.70	+	3699.13i	0.453398	+	0.785309i	0.998595	−	0.0529995i	$-0.0168782\pi$
$281$	2135.70	+	3699.13i	0.453398	+	0.785309i	−0.545196	+	0.838309i	$0.683545\pi$
$282$		0			0
$283$	2725.59	−	4720.86i	0.572508	−	0.991612i	−0.423800	−	0.905756i	$-0.639304\pi$
$283$	2725.59	−	4720.86i	0.572508	−	0.991612i	0.996308	−	0.0858564i	$-0.0273626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1240.78			−0.255194
$288$		0			0
$289$	−4911.75			−0.999745
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3101.29	−	5371.59i	0.618359	−	1.07103i	−0.371426	−	0.928462i	$-0.621131\pi$
$293$	3101.29	−	5371.59i	0.618359	−	1.07103i	0.989785	−	0.142567i	$-0.0455355\pi$
$294$		0			0
$295$	1000.43	+	1732.79i	0.197448	+	0.341991i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−419.630	−	726.821i	−0.0811633	−	0.140579i
$300$		0			0
$301$	2229.74	−	3862.03i	0.426978	−	0.739547i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1236.00			−0.232043
$306$		0			0
$307$	7323.74			1.36152			0.680762	−	0.732505i	$-0.261649\pi$
$307$	7323.74			1.36152			0.680762	+	0.732505i	$0.261649\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−953.892	+	1652.19i	−0.173924	+	0.301245i	−0.939788	−	0.341757i	$-0.888978\pi$
$311$	−953.892	+	1652.19i	−0.173924	+	0.301245i	0.765865	+	0.643002i	$0.222311\pi$
$312$		0			0
$313$	268.672	+	465.354i	0.0485184	+	0.0840363i	0.889265	−	0.457393i	$-0.151217\pi$
$313$	268.672	+	465.354i	0.0485184	+	0.0840363i	−0.840746	+	0.541429i	$0.817883\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3163.52	+	5479.38i	0.560508	+	0.970828i	0.997452	+	0.0713396i	$0.0227274\pi$
$317$	3163.52	+	5479.38i	0.560508	+	0.970828i	−0.436944	+	0.899489i	$0.643939\pi$
$318$		0			0
$319$	2110.03	−	3654.67i	0.370341	−	0.641450i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−80.9281			−0.0139411
$324$		0			0
$325$	157.894			0.0269489
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−319.162	+	552.805i	−0.0534832	+	0.0926357i
$330$		0			0
$331$	−317.469	−	549.873i	−0.0527181	−	0.0913104i	0.838462	−	0.544960i	$-0.183455\pi$
$331$	−317.469	−	549.873i	−0.0527181	−	0.0913104i	−0.891180	+	0.453650i	$0.850122\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1568.57	+	2716.84i	0.255821	+	0.443095i
$336$		0			0
$337$	275.760	−	477.630i	0.0445745	−	0.0772053i	−0.842877	−	0.538106i	$-0.819140\pi$
$337$	275.760	−	477.630i	0.0445745	−	0.0772053i	0.887452	+	0.460900i	$0.152473\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4832.03			−0.767358
$342$		0			0
$343$	5254.62			0.827180
$344$		0			0
$345$		0			0
$346$		0			0
$347$	196.207	−	339.841i	0.0303544	−	0.0525753i	−0.850449	−	0.526057i	$-0.823670\pi$
$347$	196.207	−	339.841i	0.0303544	−	0.0525753i	0.880804	+	0.473482i	$0.157003\pi$
$348$		0			0
$349$	594.722	+	1030.09i	0.0912171	+	0.157993i	0.908024	−	0.418919i	$-0.137591\pi$
$349$	594.722	+	1030.09i	0.0912171	+	0.157993i	−0.816806	+	0.576912i	$0.804258\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3690.32	−	6391.82i	−0.556419	−	0.963746i	−0.997792	−	0.0664223i	$-0.978842\pi$
$353$	−3690.32	−	6391.82i	−0.556419	−	0.963746i	0.441372	−	0.897324i	$-0.354492\pi$
$354$		0			0
$355$	−147.119	+	254.818i	−0.0219951	+	0.0380967i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2784.75			−0.409397			−0.204698	−	0.978825i	$-0.565621\pi$
$359$	−2784.75			−0.409397			−0.204698	+	0.978825i	$0.565621\pi$
$360$		0			0
$361$	−1640.20			−0.239132
$362$		0			0
$363$		0			0
$364$		0			0
$365$	22.7562	−	39.4149i	0.00326333	−	0.00565225i
$366$		0			0
$367$	6068.06	+	10510.2i	0.863080	+	1.49490i	0.868942	+	0.494915i	$0.164801\pi$
$367$	6068.06	+	10510.2i	0.863080	+	1.49490i	−0.00586197	+	0.999983i	$0.501866\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1713.59	−	2968.03i	−0.239799	−	0.415344i
$372$		0			0
$373$	2954.08	−	5116.63i	0.410072	−	0.710265i	−0.584826	−	0.811159i	$-0.698837\pi$
$373$	2954.08	−	5116.63i	0.410072	−	0.710265i	0.994897	+	0.100894i	$0.0321704\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	547.367			0.0747767
$378$		0			0
$379$	7251.83			0.982854			0.491427	−	0.870919i	$-0.336475\pi$
$379$	7251.83			0.982854			0.491427	+	0.870919i	$0.336475\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5119.59	+	8867.39i	−0.683026	+	1.18304i	0.291027	+	0.956715i	$0.406003\pi$
$383$	−5119.59	+	8867.39i	−0.683026	+	1.18304i	−0.974053	+	0.226321i	$0.927330\pi$
$384$		0			0
$385$	1044.56	+	1809.23i	0.138275	+	0.239498i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2592.38	−	4490.13i	−0.337889	−	0.585241i	0.646147	−	0.763213i	$-0.276379\pi$
$389$	−2592.38	−	4490.13i	−0.337889	−	0.585241i	−0.984035	+	0.177973i	$0.943046\pi$
$390$		0			0
$391$	−74.4312	+	128.919i	−0.00962698	+	0.0166744i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−435.786			−0.0555108
$396$		0			0
$397$	8587.28			1.08560			0.542800	−	0.839862i	$-0.317364\pi$
$397$	8587.28			1.08560			0.542800	+	0.839862i	$0.317364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4819.58	−	8347.75i	0.600195	−	1.03957i	−0.392596	−	0.919711i	$-0.628423\pi$
$401$	4819.58	−	8347.75i	0.600195	−	1.03957i	0.992791	−	0.119857i	$-0.0382437\pi$
$402$		0			0
$403$	−313.372	−	542.776i	−0.0387349	−	0.0670908i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4107.74	+	7114.81i	0.500278	+	0.866506i
$408$		0			0
$409$	1168.59	−	2024.06i	0.141279	−	0.244703i	−0.786699	−	0.617336i	$-0.788212\pi$
$409$	1168.59	−	2024.06i	0.141279	−	0.244703i	0.927979	+	0.372634i	$0.121545\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3433.79			−0.409118
$414$		0			0
$415$	5312.67			0.628407
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7907.02	−	13695.4i	0.921917	−	1.59681i	0.125470	−	0.992097i	$-0.459956\pi$
$419$	7907.02	−	13695.4i	0.921917	−	1.59681i	0.796447	−	0.604709i	$-0.206711\pi$
$420$		0			0
$421$	−2482.77	−	4300.28i	−0.287417	−	0.497822i	0.685775	−	0.727814i	$-0.259463\pi$
$421$	−2482.77	−	4300.28i	−0.287417	−	0.497822i	−0.973193	+	0.229992i	$0.926130\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.0031	−	24.2541i	−0.00159824	−	0.00276823i
$426$		0			0
$427$	1060.59	−	1836.99i	0.120200	−	0.208193i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12384.2			1.38405			0.692027	−	0.721872i	$-0.256718\pi$
$431$	12384.2			1.38405			0.692027	+	0.721872i	$0.256718\pi$
$432$		0			0
$433$	−14443.8			−1.60306			−0.801528	−	0.597958i	$-0.795979\pi$
$433$	−14443.8			−1.60306			−0.801528	+	0.597958i	$0.795979\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4799.83	−	8313.55i	0.525417	−	0.910048i
$438$		0			0
$439$	−6745.40	−	11683.4i	−0.733350	−	1.27020i	−0.955444	−	0.295173i	$-0.904623\pi$
$439$	−6745.40	−	11683.4i	−0.733350	−	1.27020i	0.222094	−	0.975025i	$-0.428711\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4639.88	+	8036.50i	0.497623	+	0.861909i	0.999996	−	0.00274202i	$-0.000872812\pi$
$443$	4639.88	+	8036.50i	0.497623	+	0.861909i	−0.502373	+	0.864651i	$0.667539\pi$
$444$		0			0
$445$	−2637.67	+	4568.57i	−0.280983	+	0.486677i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9474.08			−0.995790			−0.497895	−	0.867237i	$-0.665893\pi$
$449$	−9474.08			−0.995790			−0.497895	+	0.867237i	$0.665893\pi$
$450$		0			0
$451$	−7040.97			−0.735136
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−135.486	+	234.668i	−0.0139597	+	0.0241789i
$456$		0			0
$457$	8182.75	+	14172.9i	0.837577	+	1.45073i	0.891915	+	0.452203i	$0.149362\pi$
$457$	8182.75	+	14172.9i	0.837577	+	1.45073i	−0.0543378	+	0.998523i	$0.517305\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2917.71	−	5053.63i	−0.294775	−	0.510566i	0.680157	−	0.733066i	$-0.261911\pi$
$461$	−2917.71	−	5053.63i	−0.294775	−	0.510566i	−0.974933	+	0.222500i	$0.928578\pi$
$462$		0			0
$463$	4309.43	−	7464.16i	0.432562	−	0.749220i	−0.564531	−	0.825412i	$-0.690943\pi$
$463$	4309.43	−	7464.16i	0.432562	−	0.749220i	0.997093	+	0.0761919i	$0.0242762\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−686.868			−0.0680609			−0.0340305	−	0.999421i	$-0.510834\pi$
$467$	−686.868			−0.0680609			−0.0340305	+	0.999421i	$0.510834\pi$
$468$		0			0
$469$	−5383.82			−0.530068
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12653.0	−	21915.6i	1.22999	−	2.13041i
$474$		0			0
$475$	903.016	+	1564.07i	0.0872278	+	0.151083i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8476.47	−	14681.7i	−0.808559	−	1.40047i	−0.913862	−	0.406025i	$-0.866915\pi$
$479$	−8476.47	−	14681.7i	−0.808559	−	1.40047i	0.105303	−	0.994440i	$-0.466419\pi$
$480$		0			0
$481$	−532.798	+	922.834i	−0.0505063	+	0.0874794i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	520.827			0.0487619
$486$		0			0
$487$	1481.87			0.137885			0.0689424	−	0.997621i	$-0.478038\pi$
$487$	1481.87			0.137885			0.0689424	+	0.997621i	$0.478038\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10145.6	−	17572.6i	0.932512	−	1.61516i	0.153500	−	0.988149i	$-0.450945\pi$
$491$	10145.6	−	17572.6i	0.932512	−	1.61516i	0.779012	−	0.627009i	$-0.215721\pi$
$492$		0			0
$493$	−48.5441	−	84.0809i	−0.00443472	−	0.00768116i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−252.480	−	437.308i	−0.0227873	−	0.0394687i
$498$		0			0
$499$	−807.143	+	1398.01i	−0.0724101	+	0.125418i	−0.899957	−	0.435978i	$-0.856402\pi$
$499$	−807.143	+	1398.01i	−0.0724101	+	0.125418i	0.827547	+	0.561396i	$0.189736\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4259.30			−0.377560			−0.188780	−	0.982019i	$-0.560453\pi$
$503$	−4259.30			−0.377560			−0.188780	+	0.982019i	$0.560453\pi$
$504$		0			0
$505$	5763.58			0.507873
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10428.5	−	18062.6i	0.908122	−	1.57291i	0.0914511	−	0.995810i	$-0.470849\pi$
$509$	10428.5	−	18062.6i	0.908122	−	1.57291i	0.816671	−	0.577104i	$-0.195817\pi$
$510$		0			0
$511$	39.0532	+	67.6422i	0.00338085	+	0.00585580i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1466.70	−	2540.41i	−0.125496	−	0.217366i
$516$		0			0
$517$	−1811.13	+	3136.97i	−0.154069	+	0.266855i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10867.2			−0.913822			−0.456911	−	0.889512i	$-0.651044\pi$
$521$	−10867.2			−0.913822			−0.456911	+	0.889512i	$0.651044\pi$
$522$		0			0
$523$	11812.5			0.987622			0.493811	−	0.869569i	$-0.335603\pi$
$523$	11812.5			0.987622			0.493811	+	0.869569i	$0.335603\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−55.5838	+	96.2740i	−0.00459444	+	0.00795780i
$528$		0			0
$529$	−2745.50	−	4755.35i	−0.225651	−	0.390840i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−456.628	−	790.903i	−0.0371084	−	0.0642736i
$534$		0			0
$535$	4104.31	−	7108.88i	0.331673	−	0.574474i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13116.4			1.04817
$540$		0			0
$541$	17737.7			1.40962			0.704808	−	0.709398i	$-0.251033\pi$
$541$	17737.7			1.40962			0.704808	+	0.709398i	$0.251033\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4634.59	+	8027.35i	−0.364265	+	0.630925i
$546$		0			0
$547$	11363.3	+	19681.8i	0.888226	+	1.53845i	0.841971	+	0.539523i	$0.181396\pi$
$547$	11363.3	+	19681.8i	0.888226	+	1.53845i	0.0462556	+	0.998930i	$0.485271\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3130.46	+	5422.11i	0.242036	+	0.419219i
$552$		0			0
$553$	373.939	−	647.681i	0.0287550	−	0.0498050i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5681.80			−0.432218			−0.216109	−	0.976369i	$-0.569337\pi$
$557$	−5681.80			−0.432218			−0.216109	+	0.976369i	$0.569337\pi$
$558$		0			0
$559$	3282.34			0.248351
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2620.03	−	4538.03i	0.196130	−	0.339707i	−0.751140	−	0.660143i	$-0.770496\pi$
$563$	2620.03	−	4538.03i	0.196130	−	0.339707i	0.947270	+	0.320435i	$0.103829\pi$
$564$		0			0
$565$	3605.72	+	6245.28i	0.268484	+	0.465028i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8608.47	+	14910.3i	0.634246	+	1.09855i	0.986674	+	0.162707i	$0.0520226\pi$
$569$	8608.47	+	14910.3i	0.634246	+	1.09855i	−0.352429	+	0.935839i	$0.614644\pi$
$570$		0			0
$571$	−13163.1	+	22799.1i	−0.964724	+	1.67095i	−0.254371	+	0.967107i	$0.581868\pi$
$571$	−13163.1	+	22799.1i	−0.964724	+	1.67095i	−0.710354	+	0.703845i	$0.751465\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3322.09			0.240940
$576$		0			0
$577$	2186.42			0.157750			0.0788752	−	0.996884i	$-0.474867\pi$
$577$	2186.42			0.157750			0.0788752	+	0.996884i	$0.474867\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4558.70	+	7895.89i	−0.325519	+	0.563816i
$582$		0			0
$583$	−9724.03	−	16842.5i	−0.690786	−	1.19648i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1773.66	+	3072.07i	0.124713	+	0.216010i	0.921621	−	0.388092i	$-0.126866\pi$
$587$	1773.66	+	3072.07i	0.124713	+	0.216010i	−0.796908	+	0.604101i	$0.793532\pi$
$588$		0			0
$589$	3584.42	−	6208.40i	0.250753	−	0.434317i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6372.25			−0.441277			−0.220638	−	0.975356i	$-0.570814\pi$
$593$	−6372.25			−0.441277			−0.220638	+	0.975356i	$0.570814\pi$
$594$		0			0
$595$	48.0631			0.00331159
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9418.24	−	16312.9i	0.642435	−	1.11273i	−0.342452	−	0.939535i	$-0.611258\pi$
$599$	9418.24	−	16312.9i	0.642435	−	1.11273i	0.984888	−	0.173195i	$-0.0554091\pi$
$600$		0			0
$601$	4361.60	+	7554.51i	0.296029	+	0.512737i	0.975224	−	0.221221i	$-0.0710042\pi$
$601$	4361.60	+	7554.51i	0.296029	+	0.512737i	−0.679195	+	0.733958i	$0.737671\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2600.00	+	4503.33i	0.174719	+	0.302622i
$606$		0			0
$607$	−474.168	+	821.283i	−0.0317066	+	0.0549174i	−0.881443	−	0.472290i	$-0.843428\pi$
$607$	−474.168	+	821.283i	−0.0317066	+	0.0549174i	0.849737	+	0.527207i	$0.176761\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−469.829			−0.0311084
$612$		0			0
$613$	16627.7			1.09557			0.547787	−	0.836618i	$-0.315471\pi$
$613$	16627.7			1.09557			0.547787	+	0.836618i	$0.315471\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7503.77	+	12996.9i	−0.489612	+	0.848033i	−0.999929	−	0.0119539i	$-0.996195\pi$
$617$	−7503.77	+	12996.9i	−0.489612	+	0.848033i	0.510317	+	0.859987i	$0.329528\pi$
$618$		0			0
$619$	−5561.22	−	9632.32i	−0.361105	−	0.625453i	0.627038	−	0.778989i	$-0.284267\pi$
$619$	−5561.22	−	9632.32i	−0.361105	−	0.625453i	−0.988143	+	0.153536i	$0.950934\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4526.66	−	7840.40i	−0.291102	−	0.504204i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	189.008			0.0119813
$630$		0			0
$631$	22734.4			1.43430			0.717151	−	0.696918i	$-0.245446\pi$
$631$	22734.4			1.43430			0.717151	+	0.696918i	$0.245446\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4430.10	+	7673.17i	−0.276856	+	0.479528i
$636$		0			0
$637$	850.639	+	1473.35i	0.0529098	+	0.0916424i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5742.48	−	9946.28i	−0.353845	−	0.612877i	0.633075	−	0.774091i	$-0.281793\pi$
$641$	−5742.48	−	9946.28i	−0.353845	−	0.612877i	−0.986920	+	0.161213i	$0.948459\pi$
$642$		0			0
$643$	−6891.02	+	11935.6i	−0.422636	+	0.732028i	−0.996196	−	0.0871358i	$-0.972229\pi$
$643$	−6891.02	+	11935.6i	−0.422636	+	0.732028i	0.573560	+	0.819164i	$0.305562\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18878.3			−1.14712			−0.573558	−	0.819165i	$-0.694437\pi$
$647$	−18878.3			−1.14712			−0.573558	+	0.819165i	$0.694437\pi$
$648$		0			0
$649$	−19485.5			−1.17854
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13052.5	−	22607.6i	0.782210	−	1.35483i	−0.148442	−	0.988921i	$-0.547426\pi$
$653$	13052.5	−	22607.6i	0.782210	−	1.35483i	0.930652	−	0.365906i	$-0.119241\pi$
$654$		0			0
$655$	−2923.09	−	5062.94i	−0.174374	−	0.302024i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4472.29	+	7746.23i	0.264364	+	0.457891i	0.967397	−	0.253266i	$-0.0815048\pi$
$659$	4472.29	+	7746.23i	0.264364	+	0.457891i	−0.703033	+	0.711157i	$0.748171\pi$
$660$		0			0
$661$	10765.3	−	18646.0i	0.633464	−	1.09719i	−0.353374	−	0.935482i	$-0.614966\pi$
$661$	10765.3	−	18646.0i	0.633464	−	1.09719i	0.986838	−	0.161710i	$-0.0517010\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3099.44			−0.180738
$666$		0			0
$667$	11516.6			0.668551
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6018.45	−	10424.3i	0.346259	−	0.599738i
$672$		0			0
$673$	12769.5	+	22117.5i	0.731395	+	1.26681i	0.956287	+	0.292429i	$0.0944636\pi$
$673$	12769.5	+	22117.5i	0.731395	+	1.26681i	−0.224892	+	0.974384i	$0.572203\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15596.9	+	27014.6i	0.885432	+	1.53361i	0.845218	+	0.534422i	$0.179471\pi$
$677$	15596.9	+	27014.6i	0.885432	+	1.53361i	0.0402138	+	0.999191i	$0.487196\pi$
$678$		0			0
$679$	−446.911	+	774.072i	−0.0252590	+	0.0437499i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8565.42			−0.479863			−0.239932	−	0.970790i	$-0.577125\pi$
$683$	−8565.42			−0.479863			−0.239932	+	0.970790i	$0.577125\pi$
$684$		0			0
$685$	−5668.36			−0.316171
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1261.27	−	2184.58i	0.0697393	−	0.120792i
$690$		0			0
$691$	−9503.88	−	16461.2i	−0.523219	−	0.906243i	−0.999635	−	0.0270223i	$-0.991397\pi$
$691$	−9503.88	−	16461.2i	−0.523219	−	0.906243i	0.476415	−	0.879220i	$-0.341936\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5417.76	+	9383.84i	0.295694	+	0.512157i
$696$		0			0
$697$	−80.9936	+	140.285i	−0.00440151	+	0.00762364i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26916.1			1.45022			0.725112	−	0.688631i	$-0.241788\pi$
$701$	26916.1			1.45022			0.725112	+	0.688631i	$0.241788\pi$
$702$		0			0
$703$	−12188.6			−0.653912
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4945.61	+	8566.05i	−0.263082	+	0.455671i
$708$		0			0
$709$	−11715.1	−	20291.2i	−0.620551	−	1.07483i	−0.989383	−	0.145330i	$-0.953576\pi$
$709$	−11715.1	−	20291.2i	−0.620551	−	1.07483i	0.368832	−	0.929496i	$-0.379758\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6593.33	−	11420.0i	−0.346314	−	0.599834i
$714$		0			0
$715$	−768.832	+	1331.66i	−0.0402136	+	0.0696519i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5636.54			0.292361			0.146180	−	0.989258i	$-0.453302\pi$
$719$	5636.54			0.292361			0.146180	+	0.989258i	$0.453302\pi$
$720$		0			0
$721$	5034.19			0.260032
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1083.33	+	1876.39i	−0.0554952	+	0.0961205i
$726$		0			0
$727$	−9359.59	−	16211.3i	−0.477480	−	0.827020i	0.522187	−	0.852831i	$-0.325116\pi$
$727$	−9359.59	−	16211.3i	−0.477480	−	0.827020i	−0.999667	+	0.0258116i	$0.991783\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−291.100	−	504.200i	−0.0147287	−	0.0255109i
$732$		0			0
$733$	13273.1	−	22989.7i	0.668831	−	1.15845i	−0.309400	−	0.950932i	$-0.600128\pi$
$733$	13273.1	−	22989.7i	0.668831	−	1.15845i	0.978231	−	0.207518i	$-0.0665385\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−30551.3			−1.52696
$738$		0			0
$739$	−17564.7			−0.874326			−0.437163	−	0.899382i	$-0.644017\pi$
$739$	−17564.7			−0.874326			−0.437163	+	0.899382i	$0.644017\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−906.746	+	1570.53i	−0.0447716	+	0.0775467i	−0.887543	−	0.460725i	$-0.847589\pi$
$743$	−906.746	+	1570.53i	−0.0447716	+	0.0775467i	0.842771	+	0.538272i	$0.180923\pi$
$744$		0			0
$745$	−3494.03	−	6051.84i	−0.171827	−	0.297614i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7043.65	+	12200.0i	0.343617	+	0.595163i
$750$		0			0
$751$	−5479.66	+	9491.05i	−0.266252	+	0.461163i	−0.967891	−	0.251370i	$-0.919119\pi$
$751$	−5479.66	+	9491.05i	−0.266252	+	0.461163i	0.701639	+	0.712533i	$0.252452\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4591.52			−0.221328
$756$		0			0
$757$	9955.99			0.478014			0.239007	−	0.971018i	$-0.423178\pi$
$757$	9955.99			0.478014			0.239007	+	0.971018i	$0.423178\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12171.3	+	21081.3i	−0.579775	+	1.00420i	0.415730	+	0.909488i	$0.363526\pi$
$761$	−12171.3	+	21081.3i	−0.579775	+	1.00420i	−0.995505	+	0.0947118i	$0.969807\pi$
$762$		0			0
$763$	−7953.70	−	13776.2i	−0.377383	−	0.653647i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1263.70	−	2188.78i	−0.0594907	−	0.103041i
$768$		0			0
$769$	13430.2	−	23261.8i	0.629787	−	1.09082i	−0.357807	−	0.933795i	$-0.616476\pi$
$769$	13430.2	−	23261.8i	0.629787	−	1.09082i	0.987594	−	0.157028i	$-0.0501912\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13695.6			0.637254			0.318627	−	0.947880i	$-0.396778\pi$
$773$	13695.6			0.637254			0.318627	+	0.947880i	$0.396778\pi$
$774$		0			0
$775$	2480.87			0.114988
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5223.02	−	9046.54i	0.240224	−	0.416079i
$780$		0			0
$781$	−1432.73	−	2481.57i	−0.0656430	−	0.113697i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2674.52	+	4632.41i	0.121602	+	0.210621i
$786$		0			0
$787$	−10871.1	+	18829.3i	−0.492393	+	0.852850i	−0.999962	−	0.00876125i	$-0.997211\pi$
$787$	−10871.1	+	18829.3i	−0.492393	+	0.852850i	0.507568	+	0.861612i	$0.330545\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12376.0			−0.556307
$792$		0			0
$793$	1561.26			0.0699141
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13253.1	+	22955.1i	−0.589021	+	1.02021i	0.405340	+	0.914166i	$0.367153\pi$
$797$	−13253.1	+	22955.1i	−0.589021	+	1.02021i	−0.994361	+	0.106048i	$0.966180\pi$
$798$		0			0
$799$	41.6676	+	72.1704i	0.00184492	+	0.00319550i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	221.613	+	383.845i	0.00973917	+	0.0168687i
$804$		0			0
$805$	−2850.61	+	4937.41i	−0.124809	+	0.216175i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18438.3			−0.801304			−0.400652	−	0.916230i	$-0.631216\pi$
$809$	−18438.3			−0.801304			−0.400652	+	0.916230i	$0.631216\pi$
$810$		0			0
$811$	−29237.5			−1.26593			−0.632965	−	0.774181i	$-0.718162\pi$
$811$	−29237.5			−1.26593			−0.632965	+	0.774181i	$0.718162\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6189.54	+	10720.6i	−0.266025	+	0.460768i
$816$		0			0
$817$	18772.1	+	32514.2i	0.803859	+	1.39232i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	650.252	+	1126.27i	0.0276419	+	0.0478771i	0.879515	−	0.475870i	$-0.157867\pi$
$821$	650.252	+	1126.27i	0.0276419	+	0.0478771i	−0.851874	+	0.523747i	$0.824534\pi$
$822$		0			0
$823$	−18327.9	+	31744.8i	−0.776268	+	1.34454i	0.157811	+	0.987469i	$0.449556\pi$
$823$	−18327.9	+	31744.8i	−0.776268	+	1.34454i	−0.934079	+	0.357067i	$0.883777\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24331.0			−1.02306			−0.511531	−	0.859265i	$-0.670922\pi$
$827$	−24331.0			−1.02306			−0.511531	+	0.859265i	$0.670922\pi$
$828$		0			0
$829$	−45642.9			−1.91224			−0.956119	−	0.292980i	$-0.905353\pi$
$829$	−45642.9			−1.91224			−0.956119	+	0.292980i	$0.905353\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	150.881	−	261.333i	0.00627575	−	0.0108699i
$834$		0			0
$835$	7011.81	+	12144.8i	0.290603	+	0.503340i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2402.81	+	4161.79i	0.0988728	+	0.171253i	0.911218	−	0.411924i	$-0.135143\pi$
$839$	2402.81	+	4161.79i	0.0988728	+	0.171253i	−0.812346	+	0.583176i	$0.801810\pi$
$840$		0			0
$841$	8438.94	−	14616.7i	0.346014	−	0.599314i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10785.6			0.439094
$846$		0			0
$847$	−8924.02			−0.362023
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11210.1	+	19416.4i	−0.451558	+	0.782121i
$852$		0			0
$853$	12122.3	+	20996.4i	0.486587	+	0.842793i	0.999881	−	0.0154196i	$-0.00490841\pi$
$853$	12122.3	+	20996.4i	0.486587	+	0.842793i	−0.513294	+	0.858213i	$0.671575\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14991.0	+	25965.2i	0.597529	+	1.03495i	0.993185	+	0.116552i	$0.0371842\pi$
$857$	14991.0	+	25965.2i	0.597529	+	1.03495i	−0.395655	+	0.918399i	$0.629482\pi$
$858$		0			0
$859$	23640.6	−	40946.8i	0.939008	−	1.62641i	0.171681	−	0.985153i	$-0.445080\pi$
$859$	23640.6	−	40946.8i	0.939008	−	1.62641i	0.767327	−	0.641256i	$-0.221586\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13655.1			−0.538614			−0.269307	−	0.963054i	$-0.586795\pi$
$863$	−13655.1			−0.538614			−0.269307	+	0.963054i	$0.586795\pi$
$864$		0			0
$865$	6105.86			0.240006
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2121.97	−	3675.36i	0.0828341	−	0.143473i
$870$		0			0
$871$	−1981.34	−	3431.78i	−0.0770783	−	0.133504i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−536.300	−	928.898i	−0.0207203	−	0.0358886i
$876$		0			0
$877$	14193.0	−	24582.9i	0.546479	−	0.946529i	−0.452033	−	0.892001i	$-0.649301\pi$
$877$	14193.0	−	24582.9i	0.546479	−	0.946529i	0.998512	−	0.0545280i	$-0.0173654\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−13719.4			−0.524651			−0.262326	−	0.964979i	$-0.584489\pi$
$881$	−13719.4			−0.524651			−0.262326	+	0.964979i	$0.584489\pi$
$882$		0			0
$883$	−18580.8			−0.708145			−0.354073	−	0.935218i	$-0.615203\pi$
$883$	−18580.8			−0.708145			−0.354073	+	0.935218i	$0.615203\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11633.2	+	20149.2i	−0.440364	+	0.762733i	−0.997716	−	0.0675431i	$-0.978484\pi$
$887$	−11633.2	+	20149.2i	−0.440364	+	0.762733i	0.557352	+	0.830276i	$0.311817\pi$
$888$		0			0
$889$	−7602.76	−	13168.4i	−0.286826	−	0.496797i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2687.01	−	4654.04i	−0.100691	−	0.174402i
$894$		0			0
$895$	−922.975	+	1598.64i	−0.0344711	+	0.0597057i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8600.35			0.319063
$900$		0			0
$901$	−447.430			−0.0165439
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6459.56	+	11188.3i	−0.237263	+	0.410952i
$906$		0			0
$907$	17839.7	+	30899.2i	0.653095	+	1.13119i	0.982368	+	0.186959i	$0.0598630\pi$
$907$	17839.7	+	30899.2i	0.653095	+	1.13119i	−0.329273	+	0.944235i	$0.606804\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1533.53	−	2656.15i	−0.0557717	−	0.0965995i	0.836792	−	0.547521i	$-0.184429\pi$
$911$	−1533.53	−	2656.15i	−0.0557717	−	0.0965995i	−0.892563	+	0.450922i	$0.851095\pi$
$912$		0			0
$913$	−25869.0	+	44806.3i	−0.937719	+	1.62418i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10033.0			0.361307
$918$		0			0
$919$	−8770.06			−0.314796			−0.157398	−	0.987535i	$-0.550311\pi$
$919$	−8770.06			−0.314796			−0.157398	+	0.987535i	$0.550311\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	185.834	−	321.874i	0.00662709	−	0.0114785i
$924$		0			0
$925$	−2109.00	−	3652.90i	−0.0749660	−	0.129845i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5330.92	+	9233.43i	0.188269	+	0.326091i	0.944673	−	0.328013i	$-0.106379\pi$
$929$	5330.92	+	9233.43i	0.188269	+	0.326091i	−0.756404	+	0.654104i	$0.773046\pi$
$930$		0			0
$931$	−9729.81	+	16852.5i	−0.342515	+	0.593254i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	272.741			0.00953965
$936$		0			0
$937$	−18445.1			−0.643089			−0.321545	−	0.946894i	$-0.604202\pi$
$937$	−18445.1			−0.643089			−0.321545	+	0.946894i	$0.604202\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1233.19	−	2135.95i	0.0427214	−	0.0739956i	−0.843874	−	0.536541i	$-0.819731\pi$
$941$	1233.19	−	2135.95i	0.0427214	−	0.0739956i	0.886595	+	0.462546i	$0.153064\pi$
$942$		0			0
$943$	−9607.43	−	16640.6i	−0.331772	−	0.574646i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11889.2	+	20592.6i	0.407968	+	0.706622i	0.994662	−	0.103187i	$-0.0329041\pi$
$947$	11889.2	+	20592.6i	0.407968	+	0.706622i	−0.586694	+	0.809809i	$0.699571\pi$
$948$		0			0
$949$	−28.7446	+	49.7870i	−0.000983233	+	0.00170301i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4728.75			−0.160734			−0.0803668	−	0.996765i	$-0.525609\pi$
$953$	−4728.75			−0.160734			−0.0803668	+	0.996765i	$0.525609\pi$
$954$		0			0
$955$	23971.3			0.812245
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4863.90	−	8424.53i	0.163779	−	0.283673i
$960$		0			0
$961$	9971.73	+	17271.5i	0.334723	+	0.579757i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6817.43	+	11808.1i	0.227421	+	0.393904i
$966$		0			0
$967$	−8729.62	+	15120.1i	−0.290306	+	0.502824i	−0.973882	−	0.227055i	$-0.927090\pi$
$967$	−8729.62	+	15120.1i	−0.290306	+	0.502824i	0.683576	+	0.729879i	$0.260424\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−35241.4			−1.16473			−0.582364	−	0.812928i	$-0.697872\pi$
$971$	−35241.4			−1.16473			−0.582364	+	0.812928i	$0.697872\pi$
$972$		0			0
$973$	−18595.5			−0.612686
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17091.9	+	29604.0i	−0.559691	+	0.969413i	0.437831	+	0.899057i	$0.355747\pi$
$977$	−17091.9	+	29604.0i	−0.559691	+	0.969413i	−0.997522	+	0.0703555i	$0.977587\pi$
$978$		0			0
$979$	−25687.1	−	44491.4i	−0.838575	−	1.45245i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17521.3	−	30347.9i	−0.568509	−	0.984686i	−0.996714	−	0.0810043i	$-0.974187\pi$
$983$	−17521.3	−	30347.9i	−0.568509	−	0.984686i	0.428205	−	0.903682i	$-0.359146\pi$
$984$		0			0
$985$	−11365.5	+	19685.6i	−0.367649	+	0.636787i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	69060.3			2.22041
$990$		0			0
$991$	−553.013			−0.0177266			−0.00886329	−	0.999961i	$-0.502821\pi$
$991$	−553.013			−0.0177266			−0.00886329	+	0.999961i	$0.502821\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8809.48	−	15258.5i	0.280683	−	0.486156i
$996$		0			0
$997$	28962.3	+	50164.1i	0.920004	+	1.59349i	0.799405	+	0.600792i	$0.205148\pi$
$997$	28962.3	+	50164.1i	0.920004	+	1.59349i	0.120599	+	0.992701i	$0.461519\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.q.b.361.6		16
3.2	odd	2		360.4.q.b.121.4	✓	16
9.2	odd	6		360.4.q.b.241.4	yes	16
9.7	even	3	inner	1080.4.q.b.721.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.q.b.121.4	✓	16	3.2	odd	2
360.4.q.b.241.4	yes	16	9.2	odd	6
1080.4.q.b.361.6		16	1.1	even	1	trivial
1080.4.q.b.721.6		16	9.7	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 4.33013i) q^{5} +(4.29040 + 7.43119i) q^{7} +(24.3465 + 42.1693i) q^{11} +(-3.15788 + 5.46961i) q^{13} +1.12025 q^{17} -72.2412 q^{19} +(-66.4417 + 115.080i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-43.3334 - 75.0556i) q^{29} +(-49.6174 + 85.9399i) q^{31} +42.9040 q^{35} +168.720 q^{37} +(-72.2997 + 125.227i) q^{41} +(-259.853 - 450.079i) q^{43} +(37.1950 + 64.4236i) q^{47} +(134.685 - 233.281i) q^{49} -399.402 q^{53} +243.465 q^{55} +(-200.086 + 346.559i) q^{59} +(-123.600 - 214.082i) q^{61} +(15.7894 + 27.3481i) q^{65} +(-313.714 + 543.368i) q^{67} -58.8477 q^{71} +9.10248 q^{73} +(-208.912 + 361.846i) q^{77} +(-43.5786 - 75.4803i) q^{79} +(531.267 + 920.182i) q^{83} +(2.80062 - 4.85082i) q^{85} -1055.07 q^{89} -54.1943 q^{91} +(-180.603 + 312.814i) q^{95} +(52.0827 + 90.2099i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) \) 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

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