LMFDB - Embedded newform 1080.3.c.c.809.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,3,Mod(809,1080)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1080.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1080.c (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:	$29.4278685509$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.16
Character	$\chi$	$=$	1080.809
Dual form			1080.3.c.c.809.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72017 + 4.69478i) q^{5} -13.6186i q^{7} +1.59573i q^{11} -17.2139i q^{13} -21.0854 q^{17} +11.6883 q^{19} -24.4565 q^{23} +(-19.0820 + 16.1517i) q^{25} +40.5058i q^{29} -38.6629 q^{31} +(63.9362 - 23.4263i) q^{35} +32.5152i q^{37} -5.41815i q^{41} -15.7699i q^{43} +32.8867 q^{47} -136.465 q^{49} -97.5723 q^{53} +(-7.49161 + 2.74493i) q^{55} -88.1744i q^{59} +9.07733 q^{61} +(80.8157 - 29.6110i) q^{65} -26.8943i q^{67} +109.861i q^{71} +29.0319i q^{73} +21.7316 q^{77} -18.3005 q^{79} -23.3872 q^{83} +(-36.2705 - 98.9912i) q^{85} -147.211i q^{89} -234.429 q^{91} +(20.1058 + 54.8738i) q^{95} -100.818i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{25} - 72 q^{31} - 408 q^{49} - 168 q^{55} - 240 q^{61} - 312 q^{79} - 96 q^{85}+O(q^{100}) $ 24 * q - 72 * q^25 - 72 * q^31 - 408 * q^49 - 168 * q^55 - 240 * q^61 - 312 * q^79 - 96 * q^85

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
$5$	1.72017	+	4.69478i	0.344035	+	0.938957i
$5$	1.72017	+	4.69478i	0.344035	+	0.938957i
$6$		0			0
$7$		−	13.6186i		−	1.94551i	−0.231838	−	0.972754i	$-0.574474\pi$
$7$		−	13.6186i		−	1.94551i	0.231838	−	0.972754i	$-0.425526\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.59573i			0.145066i	0.997366	+	0.0725332i	$0.0231083\pi$
$11$			1.59573i			0.145066i	−0.997366	+	0.0725332i	$0.976892\pi$
$12$		0			0
$13$		−	17.2139i		−	1.32415i	−0.749438	−	0.662075i	$-0.769676\pi$
$13$		−	17.2139i		−	1.32415i	0.749438	−	0.662075i	$-0.230324\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21.0854			−1.24031			−0.620157	−	0.784477i	$-0.712931\pi$
$17$	−21.0854			−1.24031			−0.620157	+	0.784477i	$0.712931\pi$
$18$		0			0
$19$	11.6883			0.615171			0.307586	−	0.951520i	$-0.400479\pi$
$19$	11.6883			0.615171			0.307586	+	0.951520i	$0.400479\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−24.4565			−1.06333			−0.531663	−	0.846956i	$-0.678433\pi$
$23$	−24.4565			−1.06333			−0.531663	+	0.846956i	$0.678433\pi$
$24$		0			0
$25$	−19.0820	+	16.1517i	−0.763280	+	0.646068i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			40.5058i			1.39675i	0.715732	+	0.698375i	$0.246093\pi$
$29$			40.5058i			1.39675i	−0.715732	+	0.698375i	$0.753907\pi$
$30$		0			0
$31$	−38.6629			−1.24719			−0.623595	−	0.781748i	$-0.714328\pi$
$31$	−38.6629			−1.24719			−0.623595	+	0.781748i	$0.714328\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	63.9362	−	23.4263i	1.82675	−	0.669323i
$36$		0			0
$37$			32.5152i			0.878789i	0.898294	+	0.439394i	$0.144807\pi$
$37$			32.5152i			0.878789i	−0.898294	+	0.439394i	$0.855193\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.41815i		−	0.132150i	−0.997815	−	0.0660751i	$-0.978952\pi$
$41$		−	5.41815i		−	0.132150i	0.997815	−	0.0660751i	$-0.0210477\pi$
$42$		0			0
$43$		−	15.7699i		−	0.366741i	−0.983044	−	0.183371i	$-0.941299\pi$
$43$		−	15.7699i		−	0.366741i	0.983044	−	0.183371i	$-0.0587008\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	32.8867			0.699717			0.349858	−	0.936803i	$-0.386230\pi$
$47$	32.8867			0.699717			0.349858	+	0.936803i	$0.386230\pi$
$48$		0			0
$49$	−136.465			−2.78501
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−97.5723			−1.84099			−0.920493	−	0.390759i	$-0.872213\pi$
$53$	−97.5723			−1.84099			−0.920493	+	0.390759i	$0.872213\pi$
$54$		0			0
$55$	−7.49161	+	2.74493i	−0.136211	+	0.0499079i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	88.1744i		−	1.49448i	−0.664553	−	0.747241i	$-0.731378\pi$
$59$		−	88.1744i		−	1.49448i	0.664553	−	0.747241i	$-0.268622\pi$
$60$		0			0
$61$	9.07733			0.148809			0.0744044	−	0.997228i	$-0.476294\pi$
$61$	9.07733			0.148809			0.0744044	+	0.997228i	$0.476294\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	80.8157	−	29.6110i	1.24332	−	0.455553i
$66$		0			0
$67$		−	26.8943i		−	0.401408i	−0.979652	−	0.200704i	$-0.935677\pi$
$67$		−	26.8943i		−	0.401408i	0.979652	−	0.200704i	$-0.0643230\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			109.861i			1.54733i	0.633593	+	0.773666i	$0.281579\pi$
$71$			109.861i			1.54733i	−0.633593	+	0.773666i	$0.718421\pi$
$72$		0			0
$73$			29.0319i			0.397697i	0.980030	+	0.198848i	$0.0637202\pi$
$73$			29.0319i			0.397697i	−0.980030	+	0.198848i	$0.936280\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	21.7316			0.282228
$78$		0			0
$79$	−18.3005			−0.231652			−0.115826	−	0.993269i	$-0.536952\pi$
$79$	−18.3005			−0.231652			−0.115826	+	0.993269i	$0.536952\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−23.3872			−0.281773			−0.140887	−	0.990026i	$-0.544995\pi$
$83$	−23.3872			−0.281773			−0.140887	+	0.990026i	$0.544995\pi$
$84$		0			0
$85$	−36.2705	−	98.9912i	−0.426712	−	1.16460i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	147.211i		−	1.65406i	−0.562160	−	0.827028i	$-0.690030\pi$
$89$		−	147.211i		−	1.65406i	0.562160	−	0.827028i	$-0.309970\pi$
$90$		0			0
$91$	−234.429			−2.57614
$92$		0			0
$93$		0			0
$94$		0			0
$95$	20.1058	+	54.8738i	0.211640	+	0.577619i
$96$		0			0
$97$		−	100.818i		−	1.03936i	−0.854362	−	0.519678i	$-0.826052\pi$
$97$		−	100.818i		−	1.03936i	0.854362	−	0.519678i	$-0.173948\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	103.221i		−	1.02199i	−0.859582	−	0.510997i	$-0.829276\pi$
$101$		−	103.221i		−	1.02199i	0.859582	−	0.510997i	$-0.170724\pi$
$102$		0			0
$103$		−	42.0848i		−	0.408590i	−0.978909	−	0.204295i	$-0.934510\pi$
$103$		−	42.0848i		−	0.408590i	0.978909	−	0.204295i	$-0.0654902\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	151.752			1.41824			0.709121	−	0.705087i	$-0.249092\pi$
$107$	151.752			1.41824			0.709121	+	0.705087i	$0.249092\pi$
$108$		0			0
$109$	−65.4958			−0.600879			−0.300439	−	0.953801i	$-0.597133\pi$
$109$	−65.4958			−0.600879			−0.300439	+	0.953801i	$0.597133\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−133.768			−1.18379			−0.591893	−	0.806016i	$-0.701619\pi$
$113$	−133.768			−1.18379			−0.591893	+	0.806016i	$0.701619\pi$
$114$		0			0
$115$	−42.0694	−	114.818i	−0.365821	−	0.998417i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			287.152i			2.41304i
$120$		0			0
$121$	118.454			0.978956
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−108.653	−	61.8021i	−0.869225	−	0.494417i
$126$		0			0
$127$		−	20.3722i		−	0.160411i	−0.996778	−	0.0802054i	$-0.974442\pi$
$127$		−	20.3722i		−	0.160411i	0.996778	−	0.0802054i	$-0.0255576\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			109.292i			0.834293i	0.908839	+	0.417146i	$0.136970\pi$
$131$			109.292i			0.834293i	−0.908839	+	0.417146i	$0.863030\pi$
$132$		0			0
$133$		−	159.177i		−	1.19682i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−30.3772			−0.221732			−0.110866	−	0.993835i	$-0.535362\pi$
$137$	−30.3772			−0.221732			−0.110866	+	0.993835i	$0.535362\pi$
$138$		0			0
$139$	85.7921			0.617210			0.308605	−	0.951190i	$-0.400138\pi$
$139$	85.7921			0.617210			0.308605	+	0.951190i	$0.400138\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	27.4688			0.192090
$144$		0			0
$145$	−190.166	+	69.6770i	−1.31149	+	0.480531i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	87.3174i		−	0.586023i	−0.956109	−	0.293011i	$-0.905343\pi$
$149$		−	87.3174i		−	0.586023i	0.956109	−	0.293011i	$-0.0946574\pi$
$150$		0			0
$151$	−218.922			−1.44982			−0.724908	−	0.688846i	$-0.758118\pi$
$151$	−218.922			−1.44982			−0.724908	+	0.688846i	$0.758118\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−66.5069	−	181.514i	−0.429077	−	1.17106i
$156$		0			0
$157$		−	92.4742i		−	0.589008i	−0.955650	−	0.294504i	$-0.904846\pi$
$157$		−	92.4742i		−	0.589008i	0.955650	−	0.294504i	$-0.0951544\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			333.062i			2.06871i
$162$		0			0
$163$			252.317i			1.54796i	0.633211	+	0.773979i	$0.281737\pi$
$163$			252.317i			1.54796i	−0.633211	+	0.773979i	$0.718263\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	91.0894			0.545445			0.272723	−	0.962093i	$-0.412076\pi$
$167$	91.0894			0.545445			0.272723	+	0.962093i	$0.412076\pi$
$168$		0			0
$169$	−127.320			−0.753371
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−185.784			−1.07390			−0.536948	−	0.843616i	$-0.680423\pi$
$173$	−185.784			−1.07390			−0.536948	+	0.843616i	$0.680423\pi$
$174$		0			0
$175$	219.963	+	259.869i	1.25693	+	1.48497i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	150.913i		−	0.843088i	−0.906808	−	0.421544i	$-0.861488\pi$
$179$		−	150.913i		−	0.843088i	0.906808	−	0.421544i	$-0.138512\pi$
$180$		0			0
$181$	16.5251			0.0912990			0.0456495	−	0.998958i	$-0.485464\pi$
$181$	16.5251			0.0912990			0.0456495	+	0.998958i	$0.485464\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−152.652	+	55.9318i	−0.825145	+	0.302334i
$186$		0			0
$187$		−	33.6465i		−	0.179928i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.9230i		−	0.0728953i	−0.999336	−	0.0364476i	$-0.988396\pi$
$191$		−	13.9230i		−	0.0728953i	0.999336	−	0.0364476i	$-0.0116042\pi$
$192$		0			0
$193$			95.7140i			0.495928i	0.968769	+	0.247964i	$0.0797614\pi$
$193$			95.7140i			0.495928i	−0.968769	+	0.247964i	$0.920239\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−254.417			−1.29145			−0.645727	−	0.763568i	$-0.723446\pi$
$197$	−254.417			−1.29145			−0.645727	+	0.763568i	$0.723446\pi$
$198$		0			0
$199$	−325.493			−1.63564			−0.817821	−	0.575472i	$-0.804818\pi$
$199$	−325.493			−1.63564			−0.817821	+	0.575472i	$0.804818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	551.630			2.71739
$204$		0			0
$205$	25.4371	−	9.32017i	0.124083	−	0.0454642i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			18.6513i			0.0892407i
$210$		0			0
$211$	−93.0330			−0.440915			−0.220457	−	0.975397i	$-0.570755\pi$
$211$	−93.0330			−0.440915			−0.220457	+	0.975397i	$0.570755\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	74.0361	−	27.1269i	0.344354	−	0.126172i
$216$		0			0
$217$			526.533i			2.42642i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			362.962i			1.64236i
$222$		0			0
$223$		−	256.345i		−	1.14953i	−0.818318	−	0.574765i	$-0.805093\pi$
$223$		−	256.345i		−	1.14953i	0.818318	−	0.574765i	$-0.194907\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	30.2963			0.133464			0.0667319	−	0.997771i	$-0.478743\pi$
$227$	30.2963			0.133464			0.0667319	+	0.997771i	$0.478743\pi$
$228$		0			0
$229$	265.885			1.16107			0.580536	−	0.814235i	$-0.302843\pi$
$229$	265.885			1.16107			0.580536	+	0.814235i	$0.302843\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	405.182			1.73898			0.869488	−	0.493953i	$-0.164449\pi$
$233$	405.182			1.73898			0.869488	+	0.493953i	$0.164449\pi$
$234$		0			0
$235$	56.5708	+	154.396i	0.240727	+	0.657004i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	369.639i		−	1.54661i	−0.634036	−	0.773303i	$-0.718603\pi$
$239$		−	369.639i		−	1.54661i	0.634036	−	0.773303i	$-0.281397\pi$
$240$		0			0
$241$	340.021			1.41088			0.705439	−	0.708771i	$-0.250750\pi$
$241$	340.021			1.41088			0.705439	+	0.708771i	$0.250750\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−234.744	−	640.675i	−0.958139	−	2.61500i
$246$		0			0
$247$		−	201.201i		−	0.814579i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			103.390i			0.411914i	0.978561	+	0.205957i	$0.0660307\pi$
$251$			103.390i			0.411914i	−0.978561	+	0.205957i	$0.933969\pi$
$252$		0			0
$253$		−	39.0260i		−	0.154253i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−367.987			−1.43185			−0.715927	−	0.698175i	$-0.753996\pi$
$257$	−367.987			−1.43185			−0.715927	+	0.698175i	$0.753996\pi$
$258$		0			0
$259$	442.810			1.70969
$260$		0			0
$261$		0			0
$262$		0			0
$263$	356.509			1.35555			0.677774	−	0.735270i	$-0.262945\pi$
$263$	356.509			1.35555			0.677774	+	0.735270i	$0.262945\pi$
$264$		0			0
$265$	−167.841	−	458.081i	−0.633363	−	1.72861i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	24.4804i		−	0.0910051i	−0.998964	−	0.0455026i	$-0.985511\pi$
$269$		−	24.4804i		−	0.0910051i	0.998964	−	0.0455026i	$-0.0144889\pi$
$270$		0			0
$271$	492.669			1.81797			0.908984	−	0.416831i	$-0.136859\pi$
$271$	492.669			1.81797			0.908984	+	0.416831i	$0.136859\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−25.7738	−	30.4497i	−0.0937227	−	0.110726i
$276$		0			0
$277$		−	54.0702i		−	0.195199i	−0.995226	−	0.0975997i	$-0.968884\pi$
$277$		−	54.0702i		−	0.195199i	0.995226	−	0.0975997i	$-0.0311165\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	84.9003i		−	0.302136i	−0.988523	−	0.151068i	$-0.951729\pi$
$281$		−	84.9003i		−	0.302136i	0.988523	−	0.151068i	$-0.0482713\pi$
$282$		0			0
$283$		−	527.876i		−	1.86529i	−0.360799	−	0.932644i	$-0.617496\pi$
$283$		−	527.876i		−	1.86529i	0.360799	−	0.932644i	$-0.382504\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−73.7875			−0.257099
$288$		0			0
$289$	155.592			0.538381
$290$		0			0
$291$		0			0
$292$		0			0
$293$	296.974			1.01356			0.506781	−	0.862075i	$-0.330835\pi$
$293$	296.974			1.01356			0.506781	+	0.862075i	$0.330835\pi$
$294$		0			0
$295$	413.960	−	151.675i	1.40325	−	0.514154i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			420.993i			1.40800i
$300$		0			0
$301$	−214.763			−0.713498
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.6146	+	42.6161i	0.0511954	+	0.139725i
$306$		0			0
$307$			39.8780i			0.129896i	0.997889	+	0.0649479i	$0.0206881\pi$
$307$			39.8780i			0.129896i	−0.997889	+	0.0649479i	$0.979312\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			321.299i			1.03312i	0.856252	+	0.516558i	$0.172787\pi$
$311$			321.299i			1.03312i	−0.856252	+	0.516558i	$0.827213\pi$
$312$		0			0
$313$		−	79.9726i		−	0.255504i	−0.991806	−	0.127752i	$-0.959224\pi$
$313$		−	79.9726i		−	0.255504i	0.991806	−	0.127752i	$-0.0407761\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−307.837			−0.971095			−0.485547	−	0.874210i	$-0.661380\pi$
$317$	−307.837			−0.971095			−0.485547	+	0.874210i	$0.661380\pi$
$318$		0			0
$319$	−64.6363			−0.202622
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−246.451			−0.763006
$324$		0			0
$325$	278.034	+	328.476i	0.855490	+	1.01070i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	447.870i		−	1.36131i
$330$		0			0
$331$	445.730			1.34662			0.673308	−	0.739362i	$-0.264873\pi$
$331$	445.730			1.34662			0.673308	+	0.739362i	$0.264873\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	126.263	−	46.2630i	0.376905	−	0.138098i
$336$		0			0
$337$			252.981i			0.750685i	0.926886	+	0.375343i	$0.122475\pi$
$337$			252.981i			0.750685i	−0.926886	+	0.375343i	$0.877525\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	61.6955i		−	0.180925i
$342$		0			0
$343$			1191.15i			3.47274i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−240.422			−0.692858			−0.346429	−	0.938076i	$-0.612606\pi$
$347$	−240.422			−0.692858			−0.346429	+	0.938076i	$0.612606\pi$
$348$		0			0
$349$	434.750			1.24570			0.622851	−	0.782340i	$-0.285974\pi$
$349$	434.750			1.24570			0.622851	+	0.782340i	$0.285974\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	121.782			0.344992			0.172496	−	0.985010i	$-0.444817\pi$
$353$	121.782			0.344992			0.172496	+	0.985010i	$0.444817\pi$
$354$		0			0
$355$	−515.772	+	188.979i	−1.45288	+	0.532336i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	376.123i		−	1.04770i	−0.851812	−	0.523848i	$-0.824496\pi$
$359$		−	376.123i		−	1.04770i	0.851812	−	0.523848i	$-0.175504\pi$
$360$		0			0
$361$	−224.385			−0.621564
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−136.298	+	49.9399i	−0.373420	+	0.136822i
$366$		0			0
$367$		−	388.055i		−	1.05737i	−0.848818	−	0.528685i	$-0.822685\pi$
$367$		−	388.055i		−	1.05737i	0.848818	−	0.528685i	$-0.177315\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1328.79i			3.58166i
$372$		0			0
$373$			498.003i			1.33513i	0.744552	+	0.667565i	$0.232663\pi$
$373$			498.003i			1.33513i	−0.744552	+	0.667565i	$0.767337\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	697.264			1.84951
$378$		0			0
$379$	381.165			1.00571			0.502856	−	0.864370i	$-0.332283\pi$
$379$	381.165			1.00571			0.502856	+	0.864370i	$0.332283\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	83.2956			0.217482			0.108741	−	0.994070i	$-0.465318\pi$
$383$	83.2956			0.217482			0.108741	+	0.994070i	$0.465318\pi$
$384$		0			0
$385$	37.3821	+	102.025i	0.0970963	+	0.265000i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	322.027i		−	0.827832i	−0.910315	−	0.413916i	$-0.864161\pi$
$389$		−	322.027i		−	0.827832i	0.910315	−	0.413916i	$-0.135839\pi$
$390$		0			0
$391$	515.674			1.31886
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−31.4801	−	85.9171i	−0.0796965	−	0.217512i
$396$		0			0
$397$			683.223i			1.72097i	0.509479	+	0.860483i	$0.329838\pi$
$397$			683.223i			1.72097i	−0.509479	+	0.860483i	$0.670162\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			673.555i			1.67969i	0.542828	+	0.839844i	$0.317354\pi$
$401$			673.555i			1.67969i	−0.542828	+	0.839844i	$0.682646\pi$
$402$		0			0
$403$			665.540i			1.65146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−51.8855			−0.127483
$408$		0			0
$409$	428.725			1.04823			0.524113	−	0.851648i	$-0.324397\pi$
$409$	428.725			1.04823			0.524113	+	0.851648i	$0.324397\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1200.81			−2.90753
$414$		0			0
$415$	−40.2301	−	109.798i	−0.0969399	−	0.264573i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	85.0394i		−	0.202958i	−0.994838	−	0.101479i	$-0.967643\pi$
$419$		−	85.0394i		−	0.202958i	0.994838	−	0.101479i	$-0.0323575\pi$
$420$		0			0
$421$	−99.1071			−0.235409			−0.117704	−	0.993049i	$-0.537554\pi$
$421$	−99.1071			−0.235409			−0.117704	+	0.993049i	$0.537554\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	402.351	−	340.564i	0.946708	−	0.801328i
$426$		0			0
$427$		−	123.620i		−	0.289509i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			133.079i			0.308768i	0.988011	+	0.154384i	$0.0493393\pi$
$431$			133.079i			0.308768i	−0.988011	+	0.154384i	$0.950661\pi$
$432$		0			0
$433$		−	305.104i		−	0.704628i	−0.935882	−	0.352314i	$-0.885395\pi$
$433$		−	305.104i		−	0.704628i	0.935882	−	0.352314i	$-0.114605\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−285.854			−0.654128
$438$		0			0
$439$	−745.820			−1.69891			−0.849453	−	0.527664i	$-0.823068\pi$
$439$	−745.820			−1.69891			−0.849453	+	0.527664i	$0.823068\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−477.301			−1.07743			−0.538715	−	0.842488i	$-0.681090\pi$
$443$	−477.301			−1.07743			−0.538715	+	0.842488i	$0.681090\pi$
$444$		0			0
$445$	691.124	−	253.229i	1.55309	−	0.569053i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			502.795i			1.11981i	0.828557	+	0.559905i	$0.189163\pi$
$449$			502.795i			1.11981i	−0.828557	+	0.559905i	$0.810837\pi$
$450$		0			0
$451$	8.64592			0.0191705
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−403.259	−	1100.59i	−0.886283	−	2.41889i
$456$		0			0
$457$		−	352.233i		−	0.770750i	−0.922760	−	0.385375i	$-0.874072\pi$
$457$		−	352.233i		−	0.770750i	0.922760	−	0.385375i	$-0.125928\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			159.782i			0.346598i	0.984869	+	0.173299i	$0.0554427\pi$
$461$			159.782i			0.346598i	−0.984869	+	0.173299i	$0.944557\pi$
$462$		0			0
$463$		−	486.215i		−	1.05014i	−0.851059	−	0.525070i	$-0.824039\pi$
$463$		−	486.215i		−	1.05014i	0.851059	−	0.525070i	$-0.175961\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	220.960			0.473147			0.236573	−	0.971614i	$-0.423976\pi$
$467$	220.960			0.473147			0.236573	+	0.971614i	$0.423976\pi$
$468$		0			0
$469$	−366.262			−0.780943
$470$		0			0
$471$		0			0
$472$		0			0
$473$	25.1645			0.0532018
$474$		0			0
$475$	−223.035	+	188.785i	−0.469548	+	0.397442i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	343.941i		−	0.718040i	−0.933330	−	0.359020i	$-0.883111\pi$
$479$		−	343.941i		−	0.718040i	0.933330	−	0.359020i	$-0.116889\pi$
$480$		0			0
$481$	559.715			1.16365
$482$		0			0
$483$		0			0
$484$		0			0
$485$	473.317	−	173.424i	0.975911	−	0.357575i
$486$		0			0
$487$			38.0395i			0.0781098i	0.999237	+	0.0390549i	$0.0124347\pi$
$487$			38.0395i			0.0781098i	−0.999237	+	0.0390549i	$0.987565\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	203.703i		−	0.414874i	−0.978248	−	0.207437i	$-0.933488\pi$
$491$		−	203.703i		−	0.414874i	0.978248	−	0.207437i	$-0.0665121\pi$
$492$		0			0
$493$		−	854.078i		−	1.73241i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1496.14			3.01035
$498$		0			0
$499$	469.710			0.941302			0.470651	−	0.882319i	$-0.344019\pi$
$499$	469.710			0.941302			0.470651	+	0.882319i	$0.344019\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−213.475			−0.424403			−0.212202	−	0.977226i	$-0.568063\pi$
$503$	−213.475			−0.424403			−0.212202	+	0.977226i	$0.568063\pi$
$504$		0			0
$505$	484.603	−	177.559i	0.959609	−	0.351602i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			535.595i			1.05225i	0.850407	+	0.526125i	$0.176356\pi$
$509$			535.595i			1.05225i	−0.850407	+	0.526125i	$0.823644\pi$
$510$		0			0
$511$	395.372			0.773723
$512$		0			0
$513$		0			0
$514$		0			0
$515$	197.579	−	72.3931i	0.383648	−	0.140569i
$516$		0			0
$517$			52.4783i			0.101505i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			503.822i			0.967029i	0.875336	+	0.483515i	$0.160640\pi$
$521$			503.822i			0.967029i	−0.875336	+	0.483515i	$0.839360\pi$
$522$		0			0
$523$		−	370.237i		−	0.707911i	−0.935262	−	0.353955i	$-0.884836\pi$
$523$		−	370.237i		−	0.707911i	0.935262	−	0.353955i	$-0.115164\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	815.220			1.54691
$528$		0			0
$529$	69.1203			0.130662
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−93.2678			−0.174986
$534$		0			0
$535$	261.040	+	712.443i	0.487925	+	1.33167i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	217.762i		−	0.404011i
$540$		0			0
$541$	−914.913			−1.69115			−0.845576	−	0.533856i	$-0.820743\pi$
$541$	−914.913			−1.69115			−0.845576	+	0.533856i	$0.820743\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−112.664	−	307.489i	−0.206723	−	0.564199i
$546$		0			0
$547$		−	513.322i		−	0.938431i	−0.883084	−	0.469215i	$-0.844537\pi$
$547$		−	513.322i		−	0.938431i	0.883084	−	0.469215i	$-0.155463\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			473.442i			0.859241i
$552$		0			0
$553$			249.227i			0.450682i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−528.051			−0.948028			−0.474014	−	0.880517i	$-0.657195\pi$
$557$	−528.051			−0.948028			−0.474014	+	0.880517i	$0.657195\pi$
$558$		0			0
$559$	−271.461			−0.485620
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−517.680			−0.919503			−0.459752	−	0.888048i	$-0.652062\pi$
$563$	−517.680			−0.919503			−0.459752	+	0.888048i	$0.652062\pi$
$564$		0			0
$565$	−230.104	−	628.011i	−0.407264	−	1.11152i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	977.375i		−	1.71771i	−0.512222	−	0.858853i	$-0.671177\pi$
$569$		−	977.375i		−	1.71771i	0.512222	−	0.858853i	$-0.328823\pi$
$570$		0			0
$571$	−299.724			−0.524911			−0.262456	−	0.964944i	$-0.584532\pi$
$571$	−299.724			−0.524911			−0.262456	+	0.964944i	$0.584532\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	466.679	−	395.014i	0.811615	−	0.686981i
$576$		0			0
$577$		−	117.212i		−	0.203140i	−0.994828	−	0.101570i	$-0.967613\pi$
$577$		−	117.212i		−	0.203140i	0.994828	−	0.101570i	$-0.0323866\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			318.500i			0.548193i
$582$		0			0
$583$		−	155.699i		−	0.267065i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−286.792			−0.488572			−0.244286	−	0.969703i	$-0.578554\pi$
$587$	−286.792			−0.488572			−0.244286	+	0.969703i	$0.578554\pi$
$588$		0			0
$589$	−451.902			−0.767235
$590$		0			0
$591$		0			0
$592$		0			0
$593$	57.1262			0.0963343			0.0481672	−	0.998839i	$-0.484662\pi$
$593$	57.1262			0.0963343			0.0481672	+	0.998839i	$0.484662\pi$
$594$		0			0
$595$	−1348.12	+	493.952i	−2.26574	+	0.830171i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	407.955i		−	0.681060i	−0.940234	−	0.340530i	$-0.889393\pi$
$599$		−	407.955i		−	0.681060i	0.940234	−	0.340530i	$-0.110607\pi$
$600$		0			0
$601$	−442.203			−0.735778			−0.367889	−	0.929870i	$-0.619919\pi$
$601$	−442.203			−0.735778			−0.367889	+	0.929870i	$0.619919\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	203.761	+	556.114i	0.336795	+	0.919197i
$606$		0			0
$607$		−	28.6479i		−	0.0471958i	−0.999722	−	0.0235979i	$-0.992488\pi$
$607$		−	28.6479i		−	0.0471958i	0.999722	−	0.0235979i	$-0.00751215\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	566.110i		−	0.926529i
$612$		0			0
$613$		−	464.159i		−	0.757192i	−0.925562	−	0.378596i	$-0.876407\pi$
$613$		−	464.159i		−	0.757192i	0.925562	−	0.378596i	$-0.123593\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	593.277			0.961551			0.480776	−	0.876844i	$-0.340355\pi$
$617$	593.277			0.961551			0.480776	+	0.876844i	$0.340355\pi$
$618$		0			0
$619$	794.099			1.28287			0.641437	−	0.767176i	$-0.278338\pi$
$619$	794.099			1.28287			0.641437	+	0.767176i	$0.278338\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2004.80			−3.21798
$624$		0			0
$625$	103.245	−	616.413i	0.165193	−	0.986261i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	685.594i		−	1.08998i
$630$		0			0
$631$	58.8087			0.0931991			0.0465996	−	0.998914i	$-0.485162\pi$
$631$	58.8087			0.0931991			0.0465996	+	0.998914i	$0.485162\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	95.6429	−	35.0437i	0.150619	−	0.0551869i
$636$		0			0
$637$			2349.10i			3.68776i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			916.323i			1.42952i	0.699369	+	0.714760i	$0.253464\pi$
$641$			916.323i			1.42952i	−0.699369	+	0.714760i	$0.746536\pi$
$642$		0			0
$643$			94.0391i			0.146251i	0.997323	+	0.0731253i	$0.0232973\pi$
$643$			94.0391i			0.146251i	−0.997323	+	0.0731253i	$0.976703\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−250.055			−0.386484			−0.193242	−	0.981151i	$-0.561900\pi$
$647$	−250.055			−0.386484			−0.193242	+	0.981151i	$0.561900\pi$
$648$		0			0
$649$	140.703			0.216799
$650$		0			0
$651$		0			0
$652$		0			0
$653$	716.952			1.09794			0.548968	−	0.835843i	$-0.315021\pi$
$653$	716.952			1.09794			0.548968	+	0.835843i	$0.315021\pi$
$654$		0			0
$655$	−513.104	+	188.002i	−0.783365	+	0.287026i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	421.359i		−	0.639391i	−0.947520	−	0.319696i	$-0.896419\pi$
$659$		−	421.359i		−	0.639391i	0.947520	−	0.319696i	$-0.103581\pi$
$660$		0			0
$661$	−426.618			−0.645413			−0.322707	−	0.946499i	$-0.604593\pi$
$661$	−426.618			−0.645413			−0.322707	+	0.946499i	$0.604593\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	747.303	−	273.813i	1.12376	−	0.411748i
$666$		0			0
$667$		−	990.629i		−	1.48520i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			14.4850i			0.0215871i
$672$		0			0
$673$			8.65133i			0.0128549i	0.999979	+	0.00642744i	$0.00204593\pi$
$673$			8.65133i			0.0128549i	−0.999979	+	0.00642744i	$0.997954\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−69.4880			−0.102641			−0.0513206	−	0.998682i	$-0.516343\pi$
$677$	−69.4880			−0.102641			−0.0513206	+	0.998682i	$0.516343\pi$
$678$		0			0
$679$	−1372.99			−2.02208
$680$		0			0
$681$		0			0
$682$		0			0
$683$	217.382			0.318276			0.159138	−	0.987256i	$-0.449129\pi$
$683$	217.382			0.318276			0.159138	+	0.987256i	$0.449129\pi$
$684$		0			0
$685$	−52.2541	−	142.615i	−0.0762834	−	0.208196i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1679.60i			2.43774i
$690$		0			0
$691$	−1063.94			−1.53971			−0.769853	−	0.638221i	$-0.779671\pi$
$691$	−1063.94			−1.53971			−0.769853	+	0.638221i	$0.779671\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	147.577	+	402.776i	0.212342	+	0.579533i
$696$		0			0
$697$			114.244i			0.163908i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1095.79i			1.56318i	0.623795	+	0.781588i	$0.285590\pi$
$701$			1095.79i			1.56318i	−0.623795	+	0.781588i	$0.714410\pi$
$702$		0			0
$703$			380.046i			0.540606i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1405.73			−1.98830
$708$		0			0
$709$	−784.926			−1.10709			−0.553545	−	0.832820i	$-0.686725\pi$
$709$	−784.926			−1.10709			−0.553545	+	0.832820i	$0.686725\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	945.558			1.32617
$714$		0			0
$715$	47.2511	+	128.960i	0.0660855	+	0.180364i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1086.25i			1.51078i	0.655274	+	0.755391i	$0.272553\pi$
$719$			1086.25i			1.51078i	−0.655274	+	0.755391i	$0.727447\pi$
$720$		0			0
$721$	−573.134			−0.794915
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−654.237	−	772.931i	−0.902396	−	1.06611i
$726$		0			0
$727$		−	790.496i		−	1.08734i	−0.839299	−	0.543670i	$-0.817034\pi$
$727$		−	790.496i		−	1.08734i	0.839299	−	0.543670i	$-0.182966\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			332.513i			0.454874i
$732$		0			0
$733$		−	82.0567i		−	0.111946i	−0.998432	−	0.0559732i	$-0.982174\pi$
$733$		−	82.0567i		−	0.111946i	0.998432	−	0.0559732i	$-0.0178261\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	42.9161			0.0582309
$738$		0			0
$739$	119.154			0.161237			0.0806186	−	0.996745i	$-0.474310\pi$
$739$	119.154			0.161237			0.0806186	+	0.996745i	$0.474310\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1233.25			1.65982			0.829912	−	0.557894i	$-0.188391\pi$
$743$	1233.25			1.65982			0.829912	+	0.557894i	$0.188391\pi$
$744$		0			0
$745$	409.936	−	150.201i	0.550250	−	0.201612i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	2066.64i		−	2.75920i
$750$		0			0
$751$	−39.0008			−0.0519318			−0.0259659	−	0.999663i	$-0.508266\pi$
$751$	−39.0008			−0.0519318			−0.0259659	+	0.999663i	$0.508266\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−376.584	−	1027.79i	−0.498787	−	1.36131i
$756$		0			0
$757$		−	300.471i		−	0.396923i	−0.980109	−	0.198461i	$-0.936406\pi$
$757$		−	300.471i		−	0.396923i	0.980109	−	0.198461i	$-0.0635944\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			618.814i			0.813159i	0.913615	+	0.406579i	$0.133279\pi$
$761$			618.814i			0.813159i	−0.913615	+	0.406579i	$0.866721\pi$
$762$		0			0
$763$			891.959i			1.16902i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1517.83			−1.97892
$768$		0			0
$769$	−1333.56			−1.73414			−0.867072	−	0.498183i	$-0.834001\pi$
$769$	−1333.56			−1.73414			−0.867072	+	0.498183i	$0.834001\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−835.665			−1.08107			−0.540534	−	0.841322i	$-0.681778\pi$
$773$	−835.665			−1.08107			−0.540534	+	0.841322i	$0.681778\pi$
$774$		0			0
$775$	737.765	−	624.471i	0.951955	−	0.805769i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	63.3288i		−	0.0812950i
$780$		0			0
$781$	−175.308			−0.224466
$782$		0			0
$783$		0			0
$784$		0			0
$785$	434.147	−	159.072i	0.553053	−	0.202639i
$786$		0			0
$787$		−	646.301i		−	0.821221i	−0.911811	−	0.410610i	$-0.865316\pi$
$787$		−	646.301i		−	0.821221i	0.911811	−	0.410610i	$-0.134684\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1821.73i			2.30307i
$792$		0			0
$793$		−	156.257i		−	0.197045i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−747.026			−0.937297			−0.468649	−	0.883385i	$-0.655259\pi$
$797$	−747.026			−0.937297			−0.468649	+	0.883385i	$0.655259\pi$
$798$		0			0
$799$	−693.428			−0.867869
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−46.3271			−0.0576925
$804$		0			0
$805$	−1563.66	+	572.925i	−1.94243	+	0.711708i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			963.606i			1.19111i	0.803315	+	0.595554i	$0.203067\pi$
$809$			963.606i			1.19111i	−0.803315	+	0.595554i	$0.796933\pi$
$810$		0			0
$811$	862.849			1.06393			0.531966	−	0.846766i	$-0.321453\pi$
$811$	862.849			1.06393			0.531966	+	0.846766i	$0.321453\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1184.58	+	434.030i	−1.45347	+	0.532552i
$816$		0			0
$817$		−	184.322i		−	0.225609i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1050.18i		−	1.27915i	−0.768729	−	0.639575i	$-0.779111\pi$
$821$		−	1050.18i		−	1.27915i	0.768729	−	0.639575i	$-0.220889\pi$
$822$		0			0
$823$		−	216.916i		−	0.263567i	−0.991278	−	0.131784i	$-0.957930\pi$
$823$		−	216.916i		−	0.263567i	0.991278	−	0.131784i	$-0.0420704\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1051.87			1.27191			0.635955	−	0.771726i	$-0.280607\pi$
$827$	1051.87			1.27191			0.635955	+	0.771726i	$0.280607\pi$
$828$		0			0
$829$	647.285			0.780802			0.390401	−	0.920645i	$-0.372336\pi$
$829$	647.285			0.780802			0.390401	+	0.920645i	$0.372336\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2877.42			3.45428
$834$		0			0
$835$	156.690	+	427.645i	0.187652	+	0.512150i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	480.093i		−	0.572221i	−0.958197	−	0.286110i	$-0.907638\pi$
$839$		−	480.093i		−	0.572221i	0.958197	−	0.286110i	$-0.0923624\pi$
$840$		0			0
$841$	−799.718			−0.950913
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−219.012	−	597.738i	−0.259186	−	0.707383i
$846$		0			0
$847$		−	1613.17i		−	1.90457i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	795.208i		−	0.934439i
$852$		0			0
$853$			109.664i			0.128563i	0.997932	+	0.0642815i	$0.0204756\pi$
$853$			109.664i			0.128563i	−0.997932	+	0.0642815i	$0.979524\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−434.234			−0.506690			−0.253345	−	0.967376i	$-0.581531\pi$
$857$	−434.234			−0.506690			−0.253345	+	0.967376i	$0.581531\pi$
$858$		0			0
$859$	63.0244			0.0733695			0.0366848	−	0.999327i	$-0.488320\pi$
$859$	63.0244			0.0733695			0.0366848	+	0.999327i	$0.488320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−717.730			−0.831669			−0.415834	−	0.909440i	$-0.636510\pi$
$863$	−717.730			−0.831669			−0.415834	+	0.909440i	$0.636510\pi$
$864$		0			0
$865$	−319.581	−	872.215i	−0.369457	−	1.00834i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	29.2027i		−	0.0336050i
$870$		0			0
$871$	−462.958			−0.531524
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−841.656	+	1479.70i	−0.961893	+	1.69108i
$876$		0			0
$877$		−	874.660i		−	0.997332i	−0.866794	−	0.498666i	$-0.833824\pi$
$877$		−	874.660i		−	0.997332i	0.866794	−	0.498666i	$-0.166176\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	338.427i		−	0.384140i	−0.981381	−	0.192070i	$-0.938480\pi$
$881$		−	338.427i		−	0.384140i	0.981381	−	0.192070i	$-0.0615200\pi$
$882$		0			0
$883$		−	1147.73i		−	1.29981i	−0.760014	−	0.649906i	$-0.774808\pi$
$883$		−	1147.73i		−	1.29981i	0.760014	−	0.649906i	$-0.225192\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1090.15			−1.22903			−0.614516	−	0.788904i	$-0.710649\pi$
$887$	−1090.15			−1.22903			−0.614516	+	0.788904i	$0.710649\pi$
$888$		0			0
$889$	−277.440			−0.312081
$890$		0			0
$891$		0			0
$892$		0			0
$893$	384.388			0.430446
$894$		0			0
$895$	708.502	−	259.596i	0.791623	−	0.290052i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1566.07i		−	1.74201i
$900$		0			0
$901$	2057.35			2.28340
$902$		0			0
$903$		0			0
$904$		0			0
$905$	28.4261	+	77.5819i	0.0314100	+	0.0857258i
$906$		0			0
$907$		−	809.177i		−	0.892147i	−0.894996	−	0.446073i	$-0.852822\pi$
$907$		−	809.177i		−	0.892147i	0.894996	−	0.446073i	$-0.147178\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			80.7549i			0.0886442i	0.999017	+	0.0443221i	$0.0141128\pi$
$911$			80.7549i			0.0886442i	−0.999017	+	0.0443221i	$0.985887\pi$
$912$		0			0
$913$		−	37.3197i		−	0.0408759i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1488.40			1.62312
$918$		0			0
$919$	−473.589			−0.515331			−0.257665	−	0.966234i	$-0.582953\pi$
$919$	−473.589			−0.515331			−0.257665	+	0.966234i	$0.582953\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1891.13			2.04890
$924$		0			0
$925$	−525.175	−	620.455i	−0.567757	−	0.670762i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			166.653i			0.179390i	0.995969	+	0.0896949i	$0.0285892\pi$
$929$			166.653i			0.179390i	−0.995969	+	0.0896949i	$0.971411\pi$
$930$		0			0
$931$	−1595.04			−1.71326
$932$		0			0
$933$		0			0
$934$		0			0
$935$	157.963	−	57.8779i	0.168945	−	0.0619015i
$936$		0			0
$937$			1128.54i			1.20442i	0.798338	+	0.602210i	$0.205713\pi$
$937$			1128.54i			1.20442i	−0.798338	+	0.602210i	$0.794287\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			6.17302i			0.00656007i	0.999995	+	0.00328003i	$0.00104407\pi$
$941$			6.17302i			0.00656007i	−0.999995	+	0.00328003i	$0.998956\pi$
$942$		0			0
$943$			132.509i			0.140519i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1078.59			1.13895			0.569477	−	0.822007i	$-0.307146\pi$
$947$	1078.59			1.13895			0.569477	+	0.822007i	$0.307146\pi$
$948$		0			0
$949$	499.753			0.526610
$950$		0			0
$951$		0			0
$952$		0			0
$953$	230.467			0.241833			0.120917	−	0.992663i	$-0.461417\pi$
$953$	230.467			0.241833			0.120917	+	0.992663i	$0.461417\pi$
$954$		0			0
$955$	65.3655	−	23.9500i	0.0684455	−	0.0250785i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			413.694i			0.431381i
$960$		0			0
$961$	533.817			0.555481
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−449.357	+	164.645i	−0.465655	+	0.170616i
$966$		0			0
$967$			1452.71i			1.50229i	0.660140	+	0.751143i	$0.270497\pi$
$967$			1452.71i			1.50229i	−0.660140	+	0.751143i	$0.729503\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1192.60i		−	1.22822i	−0.789222	−	0.614108i	$-0.789516\pi$
$971$		−	1192.60i		−	1.22822i	0.789222	−	0.614108i	$-0.210484\pi$
$972$		0			0
$973$		−	1168.37i		−	1.20079i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	540.885			0.553619			0.276809	−	0.960925i	$-0.410723\pi$
$977$	540.885			0.553619			0.276809	+	0.960925i	$0.410723\pi$
$978$		0			0
$979$	234.909			0.239948
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1508.68			1.53477			0.767386	−	0.641186i	$-0.221557\pi$
$983$	1508.68			1.53477			0.767386	+	0.641186i	$0.221557\pi$
$984$		0			0
$985$	−437.641	−	1194.43i	−0.444305	−	1.21262i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			385.676i			0.389965i
$990$		0			0
$991$	−483.637			−0.488030			−0.244015	−	0.969771i	$-0.578465\pi$
$991$	−483.637			−0.488030			−0.244015	+	0.969771i	$0.578465\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−559.905	−	1528.12i	−0.562718	−	1.53580i
$996$		0			0
$997$			1777.55i			1.78290i	0.453120	+	0.891450i	$0.350311\pi$
$997$			1777.55i			1.78290i	−0.453120	+	0.891450i	$0.649689\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.3.c.c.809.16	yes	24
3.2	odd	2	inner	1080.3.c.c.809.9	✓	24
4.3	odd	2		2160.3.c.q.1889.16		24
5.4	even	2	inner	1080.3.c.c.809.10	yes	24
12.11	even	2		2160.3.c.q.1889.9		24
15.14	odd	2	inner	1080.3.c.c.809.15	yes	24
20.19	odd	2		2160.3.c.q.1889.10		24
60.59	even	2		2160.3.c.q.1889.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.3.c.c.809.9	✓	24	3.2	odd	2	inner
1080.3.c.c.809.10	yes	24	5.4	even	2	inner
1080.3.c.c.809.15	yes	24	15.14	odd	2	inner
1080.3.c.c.809.16	yes	24	1.1	even	1	trivial
2160.3.c.q.1889.9		24	12.11	even	2
2160.3.c.q.1889.10		24	20.19	odd	2
2160.3.c.q.1889.15		24	60.59	even	2
2160.3.c.q.1889.16		24	4.3	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1080.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.72017 + 4.69478i) q^{5} -13.6186i q^{7} +1.59573i q^{11} -17.2139i q^{13} -21.0854 q^{17} +11.6883 q^{19} -24.4565 q^{23} +(-19.0820 + 16.1517i) q^{25} +40.5058i q^{29} -38.6629 q^{31} +(63.9362 - 23.4263i) q^{35} +32.5152i q^{37} -5.41815i q^{41} -15.7699i q^{43} +32.8867 q^{47} -136.465 q^{49} -97.5723 q^{53} +(-7.49161 + 2.74493i) q^{55} -88.1744i q^{59} +9.07733 q^{61} +(80.8157 - 29.6110i) q^{65} -26.8943i q^{67} +109.861i q^{71} +29.0319i q^{73} +21.7316 q^{77} -18.3005 q^{79} -23.3872 q^{83} +(-36.2705 - 98.9912i) q^{85} -147.211i q^{89} -234.429 q^{91} +(20.1058 + 54.8738i) q^{95} -100.818i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{25} - 72 q^{31} - 408 q^{49} - 168 q^{55} - 240 q^{61} - 312 q^{79} - 96 q^{85}+O(q^{100}) \) 24 * q - 72 * q^25 - 72 * q^31 - 408 * q^49 - 168 * q^55 - 240 * q^61 - 312 * q^79 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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