LMFDB - Embedded newform 1080.4.f.b.649.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(649,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1080.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.f (of order $2$, degree $1$, 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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 460 x^{16} + 79376 x^{14} + 6573986 x^{12} + 287860456 x^{10} + 6830463040 x^{8} + \cdots + 59049000000 $ x^18 + 460x^16 + 79376x^14 + 6573986x^12 + 287860456x^10 + 6830463040x^8 + 85240081761x^6 + 498602897820x^4 + 981497055600x^2 + 59049000000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{10}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.13
Root			$12.7197i$ of defining polynomial
Character	$\chi$	$=$	1080.649
Dual form			1080.4.f.b.649.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.52596 - 8.26800i) q^{5} -4.10592i q^{7} +O(q^{10})$	$q+(7.52596 - 8.26800i) q^{5} -4.10592i q^{7} +21.9712 q^{11} +32.8844i q^{13} +90.2375i q^{17} +72.9169 q^{19} +148.632i q^{23} +(-11.7197 - 124.449i) q^{25} -207.598 q^{29} -199.244 q^{31} +(-33.9478 - 30.9010i) q^{35} +309.939i q^{37} +213.806 q^{41} +303.041i q^{43} +231.036i q^{47} +326.141 q^{49} -371.685i q^{53} +(165.355 - 181.658i) q^{55} +252.516 q^{59} +609.790 q^{61} +(271.888 + 247.487i) q^{65} -150.842i q^{67} -947.638 q^{71} +255.666i q^{73} -90.2122i q^{77} -474.461 q^{79} -560.004i q^{83} +(746.084 + 679.124i) q^{85} -57.3346 q^{89} +135.021 q^{91} +(548.770 - 602.877i) q^{95} -32.7042i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{5}+O(q^{10}) $ 18 * q - 2 * q^5	$ 18 q - 2 q^{5} - 62 q^{11} + 30 q^{25} + 122 q^{29} + 222 q^{31} + 256 q^{35} - 20 q^{41} - 270 q^{49} - 546 q^{55} + 120 q^{59} + 468 q^{61} + 28 q^{65} - 1268 q^{71} - 822 q^{79} - 1116 q^{85} + 348 q^{89} - 444 q^{91} + 1996 q^{95}+O(q^{100}) $ 18 * q - 2 * q^5 - 62 * q^11 + 30 * q^25 + 122 * q^29 + 222 * q^31 + 256 * q^35 - 20 * q^41 - 270 * q^49 - 546 * q^55 + 120 * q^59 + 468 * q^61 + 28 * q^65 - 1268 * q^71 - 822 * q^79 - 1116 * q^85 + 348 * q^89 - 444 * q^91 + 1996 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	7.52596	−	8.26800i	0.673143	−	0.739513i
$5$	7.52596	−	8.26800i	0.673143	−	0.739513i
$6$		0			0
$7$		−	4.10592i		−	0.221699i	−0.993837	−	0.110849i	$-0.964643\pi$
$7$		−	4.10592i		−	0.221699i	0.993837	−	0.110849i	$-0.0353571\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	21.9712			0.602234			0.301117	−	0.953587i	$-0.402640\pi$
$11$	21.9712			0.602234			0.301117	+	0.953587i	$0.402640\pi$
$12$		0			0
$13$			32.8844i			0.701577i	0.936455	+	0.350788i	$0.114086\pi$
$13$			32.8844i			0.701577i	−0.936455	+	0.350788i	$0.885914\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			90.2375i			1.28740i	0.765278	+	0.643700i	$0.222602\pi$
$17$			90.2375i			1.28740i	−0.765278	+	0.643700i	$0.777398\pi$
$18$		0			0
$19$	72.9169			0.880436			0.440218	−	0.897891i	$-0.354901\pi$
$19$	72.9169			0.880436			0.440218	+	0.897891i	$0.354901\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			148.632i			1.34747i	0.738971	+	0.673737i	$0.235312\pi$
$23$			148.632i			1.34747i	−0.738971	+	0.673737i	$0.764688\pi$
$24$		0			0
$25$	−11.7197	−	124.449i	−0.0937578	−	0.995595i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−207.598			−1.32931			−0.664655	−	0.747151i	$-0.731421\pi$
$29$	−207.598			−1.32931			−0.664655	+	0.747151i	$0.731421\pi$
$30$		0			0
$31$	−199.244			−1.15436			−0.577181	−	0.816616i	$-0.695847\pi$
$31$	−199.244			−1.15436			−0.577181	+	0.816616i	$0.695847\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−33.9478	−	30.9010i	−0.163949	−	0.149235i
$36$		0			0
$37$			309.939i			1.37712i	0.725177	+	0.688562i	$0.241758\pi$
$37$			309.939i			1.37712i	−0.725177	+	0.688562i	$0.758242\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	213.806			0.814412			0.407206	−	0.913336i	$-0.366503\pi$
$41$	213.806			0.814412			0.407206	+	0.913336i	$0.366503\pi$
$42$		0			0
$43$			303.041i			1.07473i	0.843350	+	0.537364i	$0.180580\pi$
$43$			303.041i			1.07473i	−0.843350	+	0.537364i	$0.819420\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			231.036i			0.717023i	0.933525	+	0.358512i	$0.116716\pi$
$47$			231.036i			0.717023i	−0.933525	+	0.358512i	$0.883284\pi$
$48$		0			0
$49$	326.141			0.950850
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	371.685i		−	0.963299i	−0.876364	−	0.481649i	$-0.840038\pi$
$53$		−	371.685i		−	0.963299i	0.876364	−	0.481649i	$-0.159962\pi$
$54$		0			0
$55$	165.355	−	181.658i	0.405390	−	0.445360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	252.516			0.557199			0.278600	−	0.960407i	$-0.410130\pi$
$59$	252.516			0.557199			0.278600	+	0.960407i	$0.410130\pi$
$60$		0			0
$61$	609.790			1.27993			0.639964	−	0.768405i	$-0.278949\pi$
$61$	609.790			1.27993			0.639964	+	0.768405i	$0.278949\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	271.888	+	247.487i	0.518825	+	0.472261i
$66$		0			0
$67$		−	150.842i		−	0.275048i	−0.990498	−	0.137524i	$-0.956086\pi$
$67$		−	150.842i		−	0.275048i	0.990498	−	0.137524i	$-0.0439145\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−947.638			−1.58400			−0.792000	−	0.610522i	$-0.790960\pi$
$71$	−947.638			−1.58400			−0.792000	+	0.610522i	$0.790960\pi$
$72$		0			0
$73$			255.666i			0.409910i	0.978771	+	0.204955i	$0.0657048\pi$
$73$			255.666i			0.409910i	−0.978771	+	0.204955i	$0.934295\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	90.2122i		−	0.133515i
$78$		0			0
$79$	−474.461			−0.675709			−0.337854	−	0.941198i	$-0.609701\pi$
$79$	−474.461			−0.675709			−0.337854	+	0.941198i	$0.609701\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	560.004i		−	0.740583i	−0.928916	−	0.370292i	$-0.879258\pi$
$83$		−	560.004i		−	0.740583i	0.928916	−	0.370292i	$-0.120742\pi$
$84$		0			0
$85$	746.084	+	679.124i	0.952049	+	0.866604i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−57.3346			−0.0682860			−0.0341430	−	0.999417i	$-0.510870\pi$
$89$	−57.3346			−0.0682860			−0.0341430	+	0.999417i	$0.510870\pi$
$90$		0			0
$91$	135.021			0.155539
$92$		0			0
$93$		0			0
$94$		0			0
$95$	548.770	−	602.877i	0.592659	−	0.651094i
$96$		0			0
$97$		−	32.7042i		−	0.0342331i	−0.999854	−	0.0171165i	$-0.994551\pi$
$97$		−	32.7042i		−	0.0342331i	0.999854	−	0.0171165i	$-0.00544863\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	844.504			0.831993			0.415997	−	0.909366i	$-0.363433\pi$
$101$	844.504			0.831993			0.415997	+	0.909366i	$0.363433\pi$
$102$		0			0
$103$			1632.75i			1.56194i	0.624571	+	0.780968i	$0.285274\pi$
$103$			1632.75i			1.56194i	−0.624571	+	0.780968i	$0.714726\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			567.849i			0.513047i	0.966538	+	0.256524i	$0.0825772\pi$
$107$			567.849i			0.513047i	−0.966538	+	0.256524i	$0.917423\pi$
$108$		0			0
$109$	1742.86			1.53152			0.765758	−	0.643129i	$-0.222364\pi$
$109$	1742.86			1.53152			0.765758	+	0.643129i	$0.222364\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	380.738i		−	0.316963i	−0.987362	−	0.158482i	$-0.949340\pi$
$113$		−	380.738i		−	0.316963i	0.987362	−	0.158482i	$-0.0506598\pi$
$114$		0			0
$115$	1228.89	+	1118.60i	0.996474	+	0.907043i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	370.508			0.285415
$120$		0			0
$121$	−848.265			−0.637314
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1117.15	−	839.703i	−0.799367	−	0.600842i
$126$		0			0
$127$			1511.16i			1.05586i	0.849289	+	0.527928i	$0.177031\pi$
$127$			1511.16i			1.05586i	−0.849289	+	0.527928i	$0.822969\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1120.66			−0.747423			−0.373712	−	0.927545i	$-0.621915\pi$
$131$	−1120.66			−0.747423			−0.373712	+	0.927545i	$0.621915\pi$
$132$		0			0
$133$		−	299.391i		−	0.195192i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2443.24i		−	1.52365i	−0.647783	−	0.761825i	$-0.724304\pi$
$137$		−	2443.24i		−	1.52365i	0.647783	−	0.761825i	$-0.275696\pi$
$138$		0			0
$139$	1117.58			0.681954			0.340977	−	0.940072i	$-0.389242\pi$
$139$	1117.58			0.681954			0.340977	+	0.940072i	$0.389242\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			722.511i			0.422514i
$144$		0			0
$145$	−1562.37	+	1716.42i	−0.894815	+	0.983041i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1920.43			1.05589			0.527947	−	0.849277i	$-0.322962\pi$
$149$	1920.43			1.05589			0.527947	+	0.849277i	$0.322962\pi$
$150$		0			0
$151$	3307.71			1.78263			0.891317	−	0.453381i	$-0.149782\pi$
$151$	3307.71			1.78263			0.891317	+	0.453381i	$0.149782\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1499.50	+	1647.35i	−0.777050	+	0.853665i
$156$		0			0
$157$			2861.64i			1.45467i	0.686281	+	0.727337i	$0.259242\pi$
$157$			2861.64i			1.45467i	−0.686281	+	0.727337i	$0.740758\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	610.271			0.298734
$162$		0			0
$163$		−	1776.45i		−	0.853635i	−0.904338	−	0.426817i	$-0.859635\pi$
$163$		−	1776.45i		−	0.853635i	0.904338	−	0.426817i	$-0.140365\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1703.50i			0.789348i	0.918821	+	0.394674i	$0.129142\pi$
$167$			1703.50i			0.789348i	−0.918821	+	0.394674i	$0.870858\pi$
$168$		0			0
$169$	1115.61			0.507790
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1396.16i			0.613575i	0.951778	+	0.306787i	$0.0992540\pi$
$173$			1396.16i			0.613575i	−0.951778	+	0.306787i	$0.900746\pi$
$174$		0			0
$175$	−510.979	+	48.1203i	−0.220722	+	0.0207860i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1952.60			−0.815332			−0.407666	−	0.913131i	$-0.633657\pi$
$179$	−1952.60			−0.815332			−0.407666	+	0.913131i	$0.633657\pi$
$180$		0			0
$181$	2336.31			0.959426			0.479713	−	0.877425i	$-0.340741\pi$
$181$	2336.31			0.959426			0.479713	+	0.877425i	$0.340741\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2562.57	+	2332.59i	1.01840	+	0.927001i
$186$		0			0
$187$			1982.63i			0.775317i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1728.63			−0.654866			−0.327433	−	0.944874i	$-0.606184\pi$
$191$	−1728.63			−0.654866			−0.327433	+	0.944874i	$0.606184\pi$
$192$		0			0
$193$			3074.44i			1.14665i	0.819329	+	0.573324i	$0.194346\pi$
$193$			3074.44i			1.14665i	−0.819329	+	0.573324i	$0.805654\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	73.8115i		−	0.0266947i	−0.999911	−	0.0133473i	$-0.995751\pi$
$197$		−	73.8115i		−	0.0266947i	0.999911	−	0.0133473i	$-0.00424872\pi$
$198$		0			0
$199$	3273.72			1.16617			0.583085	−	0.812411i	$-0.301846\pi$
$199$	3273.72			1.16617			0.583085	+	0.812411i	$0.301846\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			852.381i			0.294706i
$204$		0			0
$205$	1609.10	−	1767.75i	0.548215	−	0.602268i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1602.07			0.530229
$210$		0			0
$211$	659.530			0.215184			0.107592	−	0.994195i	$-0.465686\pi$
$211$	659.530			0.215184			0.107592	+	0.994195i	$0.465686\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2505.55	+	2280.68i	0.794776	+	0.723446i
$216$		0			0
$217$			818.079i			0.255921i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2967.41			−0.903210
$222$		0			0
$223$			1498.20i			0.449896i	0.974371	+	0.224948i	$0.0722212\pi$
$223$			1498.20i			0.449896i	−0.974371	+	0.224948i	$0.927779\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2047.67i			0.598716i	0.954141	+	0.299358i	$0.0967725\pi$
$227$			2047.67i			0.598716i	−0.954141	+	0.299358i	$0.903228\pi$
$228$		0			0
$229$	−3493.16			−1.00801			−0.504005	−	0.863701i	$-0.668141\pi$
$229$	−3493.16			−1.00801			−0.504005	+	0.863701i	$0.668141\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1910.97i		−	0.537303i	−0.963237	−	0.268652i	$-0.913422\pi$
$233$		−	1910.97i		−	0.537303i	0.963237	−	0.268652i	$-0.0865781\pi$
$234$		0			0
$235$	1910.21	+	1738.77i	0.530248	+	0.482659i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4514.48			−1.22183			−0.610915	−	0.791696i	$-0.709198\pi$
$239$	−4514.48			−1.22183			−0.610915	+	0.791696i	$0.709198\pi$
$240$		0			0
$241$	3035.42			0.811322			0.405661	−	0.914024i	$-0.367041\pi$
$241$	3035.42			0.811322			0.405661	+	0.914024i	$0.367041\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2454.53	−	2696.54i	0.640057	−	0.703165i
$246$		0			0
$247$			2397.83i			0.617694i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	756.759			0.190304			0.0951518	−	0.995463i	$-0.469666\pi$
$251$	756.759			0.190304			0.0951518	+	0.995463i	$0.469666\pi$
$252$		0			0
$253$			3265.63i			0.811495i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4040.36i		−	0.980665i	−0.871536	−	0.490332i	$-0.836875\pi$
$257$		−	4040.36i		−	0.980665i	0.871536	−	0.490332i	$-0.163125\pi$
$258$		0			0
$259$	1272.58			0.305307
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1544.35i			0.362086i	0.983475	+	0.181043i	$0.0579474\pi$
$263$			1544.35i			0.362086i	−0.983475	+	0.181043i	$0.942053\pi$
$264$		0			0
$265$	−3073.09	−	2797.29i	−0.712372	−	0.648437i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6874.49			1.55816			0.779080	−	0.626925i	$-0.215687\pi$
$269$	6874.49			1.55816			0.779080	+	0.626925i	$0.215687\pi$
$270$		0			0
$271$	−876.663			−0.196507			−0.0982536	−	0.995161i	$-0.531326\pi$
$271$	−876.663			−0.196507			−0.0982536	+	0.995161i	$0.531326\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−257.497	−	2734.31i	−0.0564642	−	0.599581i
$276$		0			0
$277$		−	3131.45i		−	0.679243i	−0.940562	−	0.339622i	$-0.889701\pi$
$277$		−	3131.45i		−	0.679243i	0.940562	−	0.339622i	$-0.110299\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1007.26			−0.213837			−0.106918	−	0.994268i	$-0.534098\pi$
$281$	−1007.26			−0.213837			−0.106918	+	0.994268i	$0.534098\pi$
$282$		0			0
$283$			5473.88i			1.14978i	0.818230	+	0.574892i	$0.194956\pi$
$283$			5473.88i			1.14978i	−0.818230	+	0.574892i	$0.805044\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	877.871i		−	0.180554i
$288$		0			0
$289$	−3229.81			−0.657400
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	5434.77i		−	1.08363i	−0.840499	−	0.541814i	$-0.817738\pi$
$293$		−	5434.77i		−	1.08363i	0.840499	−	0.541814i	$-0.182262\pi$
$294$		0			0
$295$	1900.43	−	2087.80i	0.375075	−	0.412056i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4887.68			−0.945357
$300$		0			0
$301$	1244.26			0.238266
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4589.26	−	5041.74i	0.861574	−	0.946523i
$306$		0			0
$307$		−	6870.26i		−	1.27722i	−0.769531	−	0.638610i	$-0.779510\pi$
$307$		−	6870.26i		−	1.27722i	0.769531	−	0.638610i	$-0.220490\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8964.62			−1.63452			−0.817262	−	0.576267i	$-0.804509\pi$
$311$	−8964.62			−1.63452			−0.817262	+	0.576267i	$0.804509\pi$
$312$		0			0
$313$		−	4518.11i		−	0.815907i	−0.913003	−	0.407954i	$-0.866243\pi$
$313$		−	4518.11i		−	0.815907i	0.913003	−	0.407954i	$-0.133757\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1584.81i			0.280794i	0.990095	+	0.140397i	$0.0448379\pi$
$317$			1584.81i			0.280794i	−0.990095	+	0.140397i	$0.955162\pi$
$318$		0			0
$319$	−4561.18			−0.800556
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6579.84i			1.13347i
$324$		0			0
$325$	4092.45	−	385.396i	0.698486	−	0.0657783i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	948.617			0.158963
$330$		0			0
$331$	−2582.14			−0.428784			−0.214392	−	0.976748i	$-0.568777\pi$
$331$	−2582.14			−0.428784			−0.214392	+	0.976748i	$0.568777\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1247.16	−	1135.23i	−0.203402	−	0.185147i
$336$		0			0
$337$			5689.97i			0.919741i	0.887986	+	0.459870i	$0.152104\pi$
$337$			5689.97i			0.919741i	−0.887986	+	0.459870i	$0.847896\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4377.63			−0.695196
$342$		0			0
$343$		−	2747.44i		−	0.432501i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			205.403i			0.0317769i	0.999874	+	0.0158885i	$0.00505767\pi$
$347$			205.403i			0.0317769i	−0.999874	+	0.0158885i	$0.994942\pi$
$348$		0			0
$349$	11958.8			1.83421			0.917107	−	0.398641i	$-0.130518\pi$
$349$	11958.8			1.83421			0.917107	+	0.398641i	$0.130518\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6273.97i		−	0.945976i	−0.881069	−	0.472988i	$-0.843175\pi$
$353$		−	6273.97i		−	0.945976i	0.881069	−	0.472988i	$-0.156825\pi$
$354$		0			0
$355$	−7131.89	+	7835.07i	−1.06626	+	1.17139i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9977.93			1.46689			0.733447	−	0.679746i	$-0.237910\pi$
$359$	9977.93			1.46689			0.733447	+	0.679746i	$0.237910\pi$
$360$		0			0
$361$	−1542.12			−0.224832
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2113.85	+	1924.13i	0.303134	+	0.275928i
$366$		0			0
$367$		−	12286.5i		−	1.74755i	−0.486329	−	0.873776i	$-0.661664\pi$
$367$		−	12286.5i		−	1.74755i	0.486329	−	0.873776i	$-0.338336\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1526.11			−0.213562
$372$		0			0
$373$			2908.84i			0.403791i	0.979407	+	0.201895i	$0.0647101\pi$
$373$			2908.84i			0.403791i	−0.979407	+	0.201895i	$0.935290\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6826.74i		−	0.932613i
$378$		0			0
$379$	−5515.20			−0.747485			−0.373743	−	0.927532i	$-0.621926\pi$
$379$	−5515.20			−0.747485			−0.373743	+	0.927532i	$0.621926\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			8051.91i			1.07424i	0.843506	+	0.537119i	$0.180487\pi$
$383$			8051.91i			1.07424i	−0.843506	+	0.537119i	$0.819513\pi$
$384$		0			0
$385$	−745.875	−	678.934i	−0.0987358	−	0.0898744i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5377.05			0.700841			0.350421	−	0.936592i	$-0.386039\pi$
$389$	5377.05			0.700841			0.350421	+	0.936592i	$0.386039\pi$
$390$		0			0
$391$	−13412.2			−1.73474
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3570.77	+	3922.84i	−0.454848	+	0.499695i
$396$		0			0
$397$		−	2894.44i		−	0.365914i	−0.983121	−	0.182957i	$-0.941433\pi$
$397$		−	2894.44i		−	0.365914i	0.983121	−	0.182957i	$-0.0585668\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9854.30			1.22718			0.613592	−	0.789624i	$-0.289724\pi$
$401$	9854.30			1.22718			0.613592	+	0.789624i	$0.289724\pi$
$402$		0			0
$403$		−	6552.01i		−	0.809874i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6809.73i			0.829351i
$408$		0			0
$409$	−12163.3			−1.47051			−0.735255	−	0.677790i	$-0.762938\pi$
$409$	−12163.3			−1.47051			−0.735255	+	0.677790i	$0.762938\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	1036.81i		−	0.123531i
$414$		0			0
$415$	−4630.11	−	4214.57i	−0.547671	−	0.498518i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1642.39			0.191494			0.0957471	−	0.995406i	$-0.469476\pi$
$419$	1642.39			0.191494			0.0957471	+	0.995406i	$0.469476\pi$
$420$		0			0
$421$	−7101.18			−0.822067			−0.411033	−	0.911620i	$-0.634832\pi$
$421$	−7101.18			−0.822067			−0.411033	+	0.911620i	$0.634832\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11230.0	−	1057.56i	1.28173	−	0.120704i
$426$		0			0
$427$		−	2503.75i		−	0.283759i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16557.7			−1.85048			−0.925239	−	0.379384i	$-0.876136\pi$
$431$	−16557.7			−1.85048			−0.925239	+	0.379384i	$0.876136\pi$
$432$		0			0
$433$		−	14254.0i		−	1.58200i	−0.611819	−	0.790998i	$-0.709562\pi$
$433$		−	14254.0i		−	1.58200i	0.611819	−	0.790998i	$-0.290438\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			10837.8i			1.18637i
$438$		0			0
$439$	10455.9			1.13675			0.568373	−	0.822771i	$-0.307573\pi$
$439$	10455.9			1.13675			0.568373	+	0.822771i	$0.307573\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	11181.2i		−	1.19917i	−0.800310	−	0.599586i	$-0.795332\pi$
$443$		−	11181.2i		−	1.19917i	0.800310	−	0.599586i	$-0.204668\pi$
$444$		0			0
$445$	−431.498	+	474.043i	−0.0459662	+	0.0504984i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6315.04			−0.663753			−0.331876	−	0.943323i	$-0.607682\pi$
$449$	−6315.04			−0.663753			−0.331876	+	0.943323i	$0.607682\pi$
$450$		0			0
$451$	4697.58			0.490467
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1016.16	−	1116.35i	0.104700	−	0.115023i
$456$		0			0
$457$		−	2106.93i		−	0.215664i	−0.994169	−	0.107832i	$-0.965609\pi$
$457$		−	2106.93i		−	0.215664i	0.994169	−	0.107832i	$-0.0343908\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16107.3			−1.62731			−0.813656	−	0.581346i	$-0.802526\pi$
$461$	−16107.3			−1.62731			−0.813656	+	0.581346i	$0.802526\pi$
$462$		0			0
$463$		−	7288.38i		−	0.731576i	−0.930698	−	0.365788i	$-0.880800\pi$
$463$		−	7288.38i		−	0.731576i	0.930698	−	0.365788i	$-0.119200\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			11414.2i			1.13102i	0.824742	+	0.565510i	$0.191321\pi$
$467$			11414.2i			1.13102i	−0.824742	+	0.565510i	$0.808679\pi$
$468$		0			0
$469$	−619.344			−0.0609779
$470$		0			0
$471$		0			0
$472$		0			0
$473$			6658.19i			0.647239i
$474$		0			0
$475$	−854.566	−	9074.47i	−0.0825478	−	0.876558i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	71.7507			0.00684420			0.00342210	−	0.999994i	$-0.498911\pi$
$479$	71.7507			0.00684420			0.00342210	+	0.999994i	$0.498911\pi$
$480$		0			0
$481$	−10192.2			−0.966158
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−270.398	−	246.130i	−0.0253158	−	0.0230437i
$486$		0			0
$487$			15353.4i			1.42860i	0.699837	+	0.714302i	$0.253256\pi$
$487$			15353.4i			1.42860i	−0.699837	+	0.714302i	$0.746744\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17693.8			−1.62629			−0.813145	−	0.582061i	$-0.802246\pi$
$491$	−17693.8			−1.62629			−0.813145	+	0.582061i	$0.802246\pi$
$492$		0			0
$493$		−	18733.1i		−	1.71135i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3890.93i			0.351171i
$498$		0			0
$499$	−11595.4			−1.04024			−0.520122	−	0.854092i	$-0.674113\pi$
$499$	−11595.4			−1.04024			−0.520122	+	0.854092i	$0.674113\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			10233.2i			0.907113i	0.891228	+	0.453556i	$0.149845\pi$
$503$			10233.2i			0.907113i	−0.891228	+	0.453556i	$0.850155\pi$
$504$		0			0
$505$	6355.71	−	6982.36i	0.560050	−	0.615270i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12441.9			1.08346			0.541728	−	0.840554i	$-0.317770\pi$
$509$	12441.9			1.08346			0.541728	+	0.840554i	$0.317770\pi$
$510$		0			0
$511$	1049.74			0.0908766
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13499.6	+	12288.0i	1.15507	+	1.05141i
$516$		0			0
$517$			5076.15i			0.431816i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4816.09			−0.404984			−0.202492	−	0.979284i	$-0.564904\pi$
$521$	−4816.09			−0.404984			−0.202492	+	0.979284i	$0.564904\pi$
$522$		0			0
$523$		−	1193.03i		−	0.0997465i	−0.998756	−	0.0498732i	$-0.984118\pi$
$523$		−	1193.03i		−	0.0997465i	0.998756	−	0.0498732i	$-0.0158817\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	17979.3i		−	1.48613i
$528$		0			0
$529$	−9924.47			−0.815688
$530$		0			0
$531$		0			0
$532$		0			0
$533$			7030.89i			0.571372i
$534$		0			0
$535$	4694.98	+	4273.61i	0.379405	+	0.345354i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7165.73			0.572634
$540$		0			0
$541$	−9923.45			−0.788618			−0.394309	−	0.918978i	$-0.629016\pi$
$541$	−9923.45			−0.788618			−0.394309	+	0.918978i	$0.629016\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13116.7	−	14409.9i	1.03093	−	1.13258i
$546$		0			0
$547$		−	17120.3i		−	1.33823i	−0.743158	−	0.669116i	$-0.766673\pi$
$547$		−	17120.3i		−	1.33823i	0.743158	−	0.669116i	$-0.233327\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15137.4			−1.17037
$552$		0			0
$553$			1948.10i			0.149804i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	19051.7i		−	1.44927i	−0.689131	−	0.724637i	$-0.742007\pi$
$557$		−	19051.7i		−	1.44927i	0.689131	−	0.724637i	$-0.257993\pi$
$558$		0			0
$559$	−9965.33			−0.754005
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	5436.04i		−	0.406930i	−0.979082	−	0.203465i	$-0.934780\pi$
$563$		−	5436.04i		−	0.406930i	0.979082	−	0.203465i	$-0.0652203\pi$
$564$		0			0
$565$	−3147.94	−	2865.42i	−0.234398	−	0.213361i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18353.3			1.35222			0.676109	−	0.736801i	$-0.263665\pi$
$569$	18353.3			1.35222			0.676109	+	0.736801i	$0.263665\pi$
$570$		0			0
$571$	−18744.5			−1.37379			−0.686896	−	0.726756i	$-0.741027\pi$
$571$	−18744.5			−1.37379			−0.686896	+	0.726756i	$0.741027\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18497.2	−	1741.93i	1.34154	−	0.126336i
$576$		0			0
$577$		−	11351.1i		−	0.818980i	−0.912315	−	0.409490i	$-0.865707\pi$
$577$		−	11351.1i		−	0.818980i	0.912315	−	0.409490i	$-0.134293\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2299.33			−0.164187
$582$		0			0
$583$		−	8166.37i		−	0.580131i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	11876.6i		−	0.835096i	−0.908655	−	0.417548i	$-0.862890\pi$
$587$		−	11876.6i		−	0.835096i	0.908655	−	0.417548i	$-0.137110\pi$
$588$		0			0
$589$	−14528.2			−1.01634
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	11165.5i		−	0.773206i	−0.922246	−	0.386603i	$-0.873648\pi$
$593$		−	11165.5i		−	0.773206i	0.922246	−	0.386603i	$-0.126352\pi$
$594$		0			0
$595$	2788.43	−	3063.36i	0.192125	−	0.211068i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8260.77			−0.563482			−0.281741	−	0.959490i	$-0.590912\pi$
$599$	−8260.77			−0.563482			−0.281741	+	0.959490i	$0.590912\pi$
$600$		0			0
$601$	25663.7			1.74184			0.870919	−	0.491427i	$-0.163524\pi$
$601$	25663.7			1.74184			0.870919	+	0.491427i	$0.163524\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6384.01	+	7013.46i	−0.429003	+	0.471302i
$606$		0			0
$607$		−	11201.2i		−	0.748999i	−0.927227	−	0.374500i	$-0.877814\pi$
$607$		−	11201.2i		−	0.748999i	0.927227	−	0.374500i	$-0.122186\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7597.49			−0.503047
$612$		0			0
$613$			2870.92i			0.189161i	0.995517	+	0.0945803i	$0.0301509\pi$
$613$			2870.92i			0.189161i	−0.995517	+	0.0945803i	$0.969849\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2561.41i			0.167129i	0.996502	+	0.0835643i	$0.0266304\pi$
$617$			2561.41i			0.167129i	−0.996502	+	0.0835643i	$0.973370\pi$
$618$		0			0
$619$	7053.16			0.457981			0.228991	−	0.973429i	$-0.426458\pi$
$619$	7053.16			0.457981			0.228991	+	0.973429i	$0.426458\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			235.411i			0.0151389i
$624$		0			0
$625$	−15350.3	+	2917.03i	−0.982419	+	0.186690i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27968.1			−1.77291
$630$		0			0
$631$	5383.88			0.339665			0.169833	−	0.985473i	$-0.445677\pi$
$631$	5383.88			0.339665			0.169833	+	0.985473i	$0.445677\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12494.3	+	11372.9i	0.780818	+	0.710741i
$636$		0			0
$637$			10725.0i			0.667094i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4395.19			−0.270826			−0.135413	−	0.990789i	$-0.543236\pi$
$641$	−4395.19			−0.270826			−0.135413	+	0.990789i	$0.543236\pi$
$642$		0			0
$643$			31325.2i			1.92122i	0.277901	+	0.960610i	$0.410361\pi$
$643$			31325.2i			1.92122i	−0.277901	+	0.960610i	$0.589639\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4822.81i			0.293051i	0.989207	+	0.146526i	$0.0468091\pi$
$647$			4822.81i			0.293051i	−0.989207	+	0.146526i	$0.953191\pi$
$648$		0			0
$649$	5548.09			0.335565
$650$		0			0
$651$		0			0
$652$		0			0
$653$			7087.81i			0.424759i	0.977187	+	0.212379i	$0.0681213\pi$
$653$			7087.81i			0.424759i	−0.977187	+	0.212379i	$0.931879\pi$
$654$		0			0
$655$	−8434.04	+	9265.61i	−0.503122	+	0.552729i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3876.12			0.229123			0.114562	−	0.993416i	$-0.463454\pi$
$659$	3876.12			0.229123			0.114562	+	0.993416i	$0.463454\pi$
$660$		0			0
$661$	−21686.2			−1.27609			−0.638046	−	0.769999i	$-0.720257\pi$
$661$	−21686.2			−1.27609			−0.638046	+	0.769999i	$0.720257\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2475.37	−	2253.21i	−0.144347	−	0.131392i
$666$		0			0
$667$		−	30855.7i		−	1.79121i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13397.8			0.770816
$672$		0			0
$673$			5878.83i			0.336720i	0.985726	+	0.168360i	$0.0538471\pi$
$673$			5878.83i			0.336720i	−0.985726	+	0.168360i	$0.946153\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1102.13i		−	0.0625678i	−0.999511	−	0.0312839i	$-0.990040\pi$
$677$		−	1102.13i		−	0.0625678i	0.999511	−	0.0312839i	$-0.00995960\pi$
$678$		0			0
$679$	−134.281			−0.00758943
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	283.197i		−	0.0158656i	−0.999969	−	0.00793282i	$-0.997475\pi$
$683$		−	283.197i		−	0.0158656i	0.999969	−	0.00793282i	$-0.00252512\pi$
$684$		0			0
$685$	−20200.7	−	18387.7i	−1.12676	−	1.02563i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12222.6			0.675828
$690$		0			0
$691$	34007.1			1.87220			0.936100	−	0.351735i	$-0.114408\pi$
$691$	34007.1			1.87220			0.936100	+	0.351735i	$0.114408\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8410.84	−	9240.12i	0.459052	−	0.504314i
$696$		0			0
$697$			19293.3i			1.04847i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−33379.2			−1.79845			−0.899227	−	0.437482i	$-0.855870\pi$
$701$	−33379.2			−1.79845			−0.899227	+	0.437482i	$0.855870\pi$
$702$		0			0
$703$			22599.8i			1.21247i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3467.47i		−	0.184452i
$708$		0			0
$709$	5254.37			0.278325			0.139162	−	0.990270i	$-0.455559\pi$
$709$	5254.37			0.278325			0.139162	+	0.990270i	$0.455559\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	29614.0i		−	1.55547i
$714$		0			0
$715$	5973.73	+	5437.60i	0.312454	+	0.284412i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12283.1			0.637108			0.318554	−	0.947905i	$-0.396803\pi$
$719$	12283.1			0.637108			0.318554	+	0.947905i	$0.396803\pi$
$720$		0			0
$721$	6703.93			0.346279
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2432.99	+	25835.4i	0.124633	+	1.32345i
$726$		0			0
$727$		−	26631.2i		−	1.35859i	−0.733863	−	0.679297i	$-0.762285\pi$
$727$		−	26631.2i		−	1.35859i	0.733863	−	0.679297i	$-0.237715\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−27345.7			−1.38361
$732$		0			0
$733$		−	13519.5i		−	0.681249i	−0.940199	−	0.340625i	$-0.889362\pi$
$733$		−	13519.5i		−	0.681249i	0.940199	−	0.340625i	$-0.110638\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3314.18i		−	0.165644i
$738$		0			0
$739$	14287.8			0.711213			0.355607	−	0.934636i	$-0.384274\pi$
$739$	14287.8			0.711213			0.355607	+	0.934636i	$0.384274\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	32914.7i		−	1.62520i	−0.582820	−	0.812601i	$-0.698051\pi$
$743$		−	32914.7i		−	1.62520i	0.582820	−	0.812601i	$-0.301949\pi$
$744$		0			0
$745$	14453.1	−	15878.2i	0.710767	−	0.780847i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2331.54			0.113742
$750$		0			0
$751$	11121.4			0.540381			0.270191	−	0.962807i	$-0.412913\pi$
$751$	11121.4			0.540381			0.270191	+	0.962807i	$0.412913\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24893.7	−	27348.2i	1.19997	−	1.31828i
$756$		0			0
$757$			1574.67i			0.0756044i	0.999285	+	0.0378022i	$0.0120357\pi$
$757$			1574.67i			0.0756044i	−0.999285	+	0.0378022i	$0.987964\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21697.4			−1.03355			−0.516773	−	0.856122i	$-0.672867\pi$
$761$	−21697.4			−1.03355			−0.516773	+	0.856122i	$0.672867\pi$
$762$		0			0
$763$		−	7156.03i		−	0.339536i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			8303.84i			0.390918i
$768$		0			0
$769$	40007.6			1.87609			0.938043	−	0.346519i	$-0.112636\pi$
$769$	40007.6			1.87609			0.938043	+	0.346519i	$0.112636\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			21599.8i			1.00504i	0.864567	+	0.502518i	$0.167593\pi$
$773$			21599.8i			1.00504i	−0.864567	+	0.502518i	$0.832407\pi$
$774$		0			0
$775$	2335.08	+	24795.8i	0.108230	+	1.14928i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15590.1			0.717038
$780$		0			0
$781$	−20820.8			−0.953939
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23660.1	+	21536.6i	1.07575	+	0.979203i
$786$		0			0
$787$			32020.2i			1.45031i	0.688584	+	0.725157i	$0.258233\pi$
$787$			32020.2i			1.45031i	−0.688584	+	0.725157i	$0.741767\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1563.28			−0.0702704
$792$		0			0
$793$			20052.6i			0.897967i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			31929.3i			1.41906i	0.704673	+	0.709532i	$0.251094\pi$
$797$			31929.3i			1.41906i	−0.704673	+	0.709532i	$0.748906\pi$
$798$		0			0
$799$	−20848.1			−0.923096
$800$		0			0
$801$		0			0
$802$		0			0
$803$			5617.30i			0.246862i
$804$		0			0
$805$	4592.88	−	5045.73i	0.201090	−	0.220917i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39944.6			1.73594			0.867971	−	0.496614i	$-0.165424\pi$
$809$	39944.6			1.73594			0.867971	+	0.496614i	$0.165424\pi$
$810$		0			0
$811$	−8433.88			−0.365171			−0.182585	−	0.983190i	$-0.558447\pi$
$811$	−8433.88			−0.365171			−0.182585	+	0.983190i	$0.558447\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14687.7	−	13369.5i	−0.631274	−	0.574618i
$816$		0			0
$817$			22096.8i			0.946230i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37670.5			−1.60135			−0.800676	−	0.599098i	$-0.795526\pi$
$821$	−37670.5			−1.60135			−0.800676	+	0.599098i	$0.795526\pi$
$822$		0			0
$823$		−	18397.7i		−	0.779225i	−0.920979	−	0.389612i	$-0.872609\pi$
$823$		−	18397.7i		−	0.779225i	0.920979	−	0.389612i	$-0.127391\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			29310.7i			1.23244i	0.787572	+	0.616222i	$0.211338\pi$
$827$			29310.7i			1.23244i	−0.787572	+	0.616222i	$0.788662\pi$
$828$		0			0
$829$	3689.81			0.154587			0.0772934	−	0.997008i	$-0.475372\pi$
$829$	3689.81			0.154587			0.0772934	+	0.997008i	$0.475372\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			29430.2i			1.22412i
$834$		0			0
$835$	14084.6	+	12820.5i	0.583733	+	0.531344i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16130.2			0.663737			0.331869	−	0.943326i	$-0.392321\pi$
$839$	16130.2			0.663737			0.331869	+	0.943326i	$0.392321\pi$
$840$		0			0
$841$	18707.9			0.767063
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8396.08	−	9223.91i	0.341815	−	0.375517i
$846$		0			0
$847$			3482.91i			0.141292i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−46066.8			−1.85564
$852$		0			0
$853$			23772.2i			0.954214i	0.878845	+	0.477107i	$0.158315\pi$
$853$			23772.2i			0.954214i	−0.878845	+	0.477107i	$0.841685\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			11421.8i			0.455265i	0.973747	+	0.227633i	$0.0730985\pi$
$857$			11421.8i			0.455265i	−0.973747	+	0.227633i	$0.926902\pi$
$858$		0			0
$859$	−23966.9			−0.951966			−0.475983	−	0.879455i	$-0.657908\pi$
$859$	−23966.9			−0.951966			−0.475983	+	0.879455i	$0.657908\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			44038.7i			1.73707i	0.495626	+	0.868536i	$0.334939\pi$
$863$			44038.7i			1.73707i	−0.495626	+	0.868536i	$0.665061\pi$
$864$		0			0
$865$	11543.5	+	10507.5i	0.453746	+	0.413023i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10424.5			−0.406935
$870$		0			0
$871$	4960.34			0.192968
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3447.75	+	4586.93i	−0.133206	+	0.177219i
$876$		0			0
$877$			31644.7i			1.21843i	0.793005	+	0.609216i	$0.208516\pi$
$877$			31644.7i			1.21843i	−0.793005	+	0.609216i	$0.791484\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−42975.3			−1.64345			−0.821723	−	0.569888i	$-0.806987\pi$
$881$	−42975.3			−1.64345			−0.821723	+	0.569888i	$0.806987\pi$
$882$		0			0
$883$		−	21560.7i		−	0.821717i	−0.911699	−	0.410858i	$-0.865229\pi$
$883$		−	21560.7i		−	0.821717i	0.911699	−	0.410858i	$-0.134771\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	10130.8i		−	0.383492i	−0.981445	−	0.191746i	$-0.938585\pi$
$887$		−	10130.8i		−	0.383492i	0.981445	−	0.191746i	$-0.0614150\pi$
$888$		0			0
$889$	6204.70			0.234082
$890$		0			0
$891$		0			0
$892$		0			0
$893$			16846.4i			0.631293i
$894$		0			0
$895$	−14695.2	+	16144.1i	−0.548835	+	0.602948i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	41362.6			1.53450
$900$		0			0
$901$	33539.9			1.24015
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17582.9	−	19316.6i	0.645831	−	0.709508i
$906$		0			0
$907$			31077.5i			1.13772i	0.822434	+	0.568860i	$0.192616\pi$
$907$			31077.5i			1.13772i	−0.822434	+	0.568860i	$0.807384\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	32659.7			1.18778			0.593888	−	0.804548i	$-0.297592\pi$
$911$	32659.7			1.18778			0.593888	+	0.804548i	$0.297592\pi$
$912$		0			0
$913$		−	12304.0i		−	0.446005i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			4601.34i			0.165703i
$918$		0			0
$919$	31126.9			1.11728			0.558641	−	0.829410i	$-0.311323\pi$
$919$	31126.9			1.11728			0.558641	+	0.829410i	$0.311323\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	31162.5i		−	1.11130i
$924$		0			0
$925$	38571.7	−	3632.40i	1.37106	−	0.129116i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−22180.2			−0.783325			−0.391662	−	0.920109i	$-0.628100\pi$
$929$	−22180.2			−0.783325			−0.391662	+	0.920109i	$0.628100\pi$
$930$		0			0
$931$	23781.2			0.837162
$932$		0			0
$933$		0			0
$934$		0			0
$935$	16392.4	+	14921.2i	0.573357	+	0.521899i
$936$		0			0
$937$			52733.8i			1.83857i	0.393596	+	0.919284i	$0.371231\pi$
$937$			52733.8i			1.83857i	−0.393596	+	0.919284i	$0.628769\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−43188.1			−1.49617			−0.748083	−	0.663605i	$-0.769025\pi$
$941$	−43188.1			−1.49617			−0.748083	+	0.663605i	$0.769025\pi$
$942$		0			0
$943$			31778.4i			1.09740i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	42403.6i		−	1.45505i	−0.686081	−	0.727525i	$-0.740670\pi$
$947$		−	42403.6i		−	1.45505i	0.686081	−	0.727525i	$-0.259330\pi$
$948$		0			0
$949$	−8407.43			−0.287583
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	20346.1i		−	0.691578i	−0.938312	−	0.345789i	$-0.887611\pi$
$953$		−	20346.1i		−	0.691578i	0.938312	−	0.345789i	$-0.112389\pi$
$954$		0			0
$955$	−13009.6	+	14292.3i	−0.440819	+	0.484282i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10031.7			−0.337791
$960$		0			0
$961$	9907.05			0.332552
$962$		0			0
$963$		0			0
$964$		0			0
$965$	25419.5	+	23138.1i	0.847960	+	0.771857i
$966$		0			0
$967$		−	22451.1i		−	0.746617i	−0.927707	−	0.373309i	$-0.878223\pi$
$967$		−	22451.1i		−	0.746617i	0.927707	−	0.373309i	$-0.121777\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	53962.8			1.78347			0.891734	−	0.452560i	$-0.149489\pi$
$971$	53962.8			1.78347			0.891734	+	0.452560i	$0.149489\pi$
$972$		0			0
$973$		−	4588.68i		−	0.151188i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	15492.3i		−	0.507310i	−0.967295	−	0.253655i	$-0.918367\pi$
$977$		−	15492.3i		−	0.507310i	0.967295	−	0.253655i	$-0.0816329\pi$
$978$		0			0
$979$	−1259.71			−0.0411242
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	51359.3i		−	1.66644i	−0.552945	−	0.833218i	$-0.686496\pi$
$983$		−	51359.3i		−	1.66644i	0.552945	−	0.833218i	$-0.313504\pi$
$984$		0			0
$985$	−610.274	−	555.503i	−0.0197411	−	0.0179693i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−45041.6			−1.44817
$990$		0			0
$991$	−12375.6			−0.396695			−0.198348	−	0.980132i	$-0.563557\pi$
$991$	−12375.6			−0.396695			−0.198348	+	0.980132i	$0.563557\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24637.9	−	27067.1i	0.784998	−	0.862397i
$996$		0			0
$997$			6415.54i			0.203794i	0.994795	+	0.101897i	$0.0324911\pi$
$997$			6415.54i			0.203794i	−0.994795	+	0.101897i	$0.967509\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.f.b.649.13	✓	18
3.2	odd	2		1080.4.f.c.649.6	yes	18
5.4	even	2	inner	1080.4.f.b.649.14	yes	18
15.14	odd	2		1080.4.f.c.649.5	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.f.b.649.13	✓	18	1.1	even	1	trivial
1080.4.f.b.649.14	yes	18	5.4	even	2	inner
1080.4.f.c.649.5	yes	18	15.14	odd	2
1080.4.f.c.649.6	yes	18	3.2	odd	2

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(7.52596 - 8.26800i) q^{5} -4.10592i q^{7} +O(q^{10})\)	\(q+(7.52596 - 8.26800i) q^{5} -4.10592i q^{7} +21.9712 q^{11} +32.8844i q^{13} +90.2375i q^{17} +72.9169 q^{19} +148.632i q^{23} +(-11.7197 - 124.449i) q^{25} -207.598 q^{29} -199.244 q^{31} +(-33.9478 - 30.9010i) q^{35} +309.939i q^{37} +213.806 q^{41} +303.041i q^{43} +231.036i q^{47} +326.141 q^{49} -371.685i q^{53} +(165.355 - 181.658i) q^{55} +252.516 q^{59} +609.790 q^{61} +(271.888 + 247.487i) q^{65} -150.842i q^{67} -947.638 q^{71} +255.666i q^{73} -90.2122i q^{77} -474.461 q^{79} -560.004i q^{83} +(746.084 + 679.124i) q^{85} -57.3346 q^{89} +135.021 q^{91} +(548.770 - 602.877i) q^{95} -32.7042i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{5}+O(q^{10}) \) 18 * q - 2 * q^5	\( 18 q - 2 q^{5} - 62 q^{11} + 30 q^{25} + 122 q^{29} + 222 q^{31} + 256 q^{35} - 20 q^{41} - 270 q^{49} - 546 q^{55} + 120 q^{59} + 468 q^{61} + 28 q^{65} - 1268 q^{71} - 822 q^{79} - 1116 q^{85} + 348 q^{89} - 444 q^{91} + 1996 q^{95}+O(q^{100}) \) 18 * q - 2 * q^5 - 62 * q^11 + 30 * q^25 + 122 * q^29 + 222 * q^31 + 256 * q^35 - 20 * q^41 - 270 * q^49 - 546 * q^55 + 120 * q^59 + 468 * q^61 + 28 * q^65 - 1268 * q^71 - 822 * q^79 - 1116 * q^85 + 348 * q^89 - 444 * q^91 + 1996 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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