LMFDB - Embedded newform 1080.4.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(1,1080)]

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, 0]))

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

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

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.a (trivial)

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:	yes
Analytic conductor:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 141x^{2} + 200x + 3500 $ x^4 - x^3 - 141x^2 + 200x + 3500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-10.9094$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +35.5942 q^{7} +O(q^{10})$	$q+5.00000 q^{5} +35.5942 q^{7} -44.7641 q^{11} +77.8220 q^{13} -120.489 q^{17} +121.319 q^{19} +152.855 q^{23} +25.0000 q^{25} +187.926 q^{29} -161.365 q^{31} +177.971 q^{35} +13.3350 q^{37} +188.065 q^{41} -81.5584 q^{43} +48.6219 q^{47} +923.946 q^{49} -707.373 q^{53} -223.821 q^{55} +16.4774 q^{59} -743.868 q^{61} +389.110 q^{65} -59.2676 q^{67} +144.370 q^{71} +657.273 q^{73} -1593.34 q^{77} -454.297 q^{79} -165.513 q^{83} -602.444 q^{85} +535.743 q^{89} +2770.01 q^{91} +606.594 q^{95} -436.761 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{5} + 14 q^{7}+O(q^{10}) $ 4 * q + 20 * q^5 + 14 * q^7	$ 4 q + 20 q^{5} + 14 q^{7} - 4 q^{11} + 30 q^{13} - 28 q^{17} + 78 q^{19} + 182 q^{23} + 100 q^{25} + 202 q^{29} - 76 q^{31} + 70 q^{35} + 302 q^{37} + 380 q^{41} + 178 q^{43} + 114 q^{47} + 958 q^{49} - 256 q^{53} - 20 q^{55} - 204 q^{59} + 766 q^{61} + 150 q^{65} + 330 q^{67} - 1060 q^{71} + 1442 q^{73} + 216 q^{77} + 742 q^{79} - 768 q^{83} - 140 q^{85} - 400 q^{89} + 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) $ 4 * q + 20 * q^5 + 14 * q^7 - 4 * q^11 + 30 * q^13 - 28 * q^17 + 78 * q^19 + 182 * q^23 + 100 * q^25 + 202 * q^29 - 76 * q^31 + 70 * q^35 + 302 * q^37 + 380 * q^41 + 178 * q^43 + 114 * q^47 + 958 * q^49 - 256 * q^53 - 20 * q^55 - 204 * q^59 + 766 * q^61 + 150 * q^65 + 330 * q^67 - 1060 * q^71 + 1442 * q^73 + 216 * q^77 + 742 * q^79 - 768 * q^83 - 140 * q^85 - 400 * q^89 + 3066 * q^91 + 390 * q^95 + 3338 * q^97

Display 100 coefficients 10 coefficients

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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	35.5942	1.92191	0.960953	−	0.276713i	$-0.0892452\pi$
$7$	35.5942	1.92191	0.960953	+	0.276713i	$0.0892452\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−44.7641	−1.22699	−0.613495	−	0.789699i	$-0.710237\pi$
$11$	−44.7641	−1.22699	−0.613495	+	0.789699i	$0.710237\pi$
$12$	0	0
$13$	77.8220	1.66030	0.830152	−	0.557538i	$-0.188254\pi$
$13$	77.8220	1.66030	0.830152	+	0.557538i	$0.188254\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−120.489	−1.71899	−0.859495	−	0.511145i	$-0.829222\pi$
$17$	−120.489	−1.71899	−0.859495	+	0.511145i	$0.829222\pi$
$18$	0	0
$19$	121.319	1.46487	0.732433	−	0.680839i	$-0.238385\pi$
$19$	121.319	1.46487	0.732433	+	0.680839i	$0.238385\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	152.855	1.38576	0.692880	−	0.721053i	$-0.256341\pi$
$23$	152.855	1.38576	0.692880	+	0.721053i	$0.256341\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	187.926	1.20334	0.601672	−	0.798743i	$-0.294501\pi$
$29$	187.926	1.20334	0.601672	+	0.798743i	$0.294501\pi$
$30$	0	0
$31$	−161.365	−0.934903	−0.467452	−	0.884019i	$-0.654828\pi$
$31$	−161.365	−0.934903	−0.467452	+	0.884019i	$0.654828\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	177.971	0.859502
$36$	0	0
$37$	13.3350	0.0592501	0.0296251	−	0.999561i	$-0.490569\pi$
$37$	13.3350	0.0592501	0.0296251	+	0.999561i	$0.490569\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	188.065	0.716360	0.358180	−	0.933653i	$-0.383397\pi$
$41$	188.065	0.716360	0.358180	+	0.933653i	$0.383397\pi$
$42$	0	0
$43$	−81.5584	−0.289245	−0.144623	−	0.989487i	$-0.546197\pi$
$43$	−81.5584	−0.289245	−0.144623	+	0.989487i	$0.546197\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	48.6219	0.150899	0.0754493	−	0.997150i	$-0.475961\pi$
$47$	48.6219	0.150899	0.0754493	+	0.997150i	$0.475961\pi$
$48$	0	0
$49$	923.946	2.69372
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−707.373	−1.83331	−0.916653	−	0.399685i	$-0.869120\pi$
$53$	−707.373	−1.83331	−0.916653	+	0.399685i	$0.869120\pi$
$54$	0	0
$55$	−223.821	−0.548727
$56$	0	0
$57$	0	0
$58$	0	0
$59$	16.4774	0.0363589	0.0181795	−	0.999835i	$-0.494213\pi$
$59$	16.4774	0.0363589	0.0181795	+	0.999835i	$0.494213\pi$
$60$	0	0
$61$	−743.868	−1.56135	−0.780676	−	0.624936i	$-0.785125\pi$
$61$	−743.868	−1.56135	−0.780676	+	0.624936i	$0.785125\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	389.110	0.742510
$66$	0	0
$67$	−59.2676	−0.108070	−0.0540350	−	0.998539i	$-0.517208\pi$
$67$	−59.2676	−0.108070	−0.0540350	+	0.998539i	$0.517208\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	144.370	0.241317	0.120659	−	0.992694i	$-0.461499\pi$
$71$	144.370	0.241317	0.120659	+	0.992694i	$0.461499\pi$
$72$	0	0
$73$	657.273	1.05381	0.526904	−	0.849925i	$-0.323353\pi$
$73$	657.273	1.05381	0.526904	+	0.849925i	$0.323353\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1593.34	−2.35816
$78$	0	0
$79$	−454.297	−0.646992	−0.323496	−	0.946230i	$-0.604858\pi$
$79$	−454.297	−0.646992	−0.323496	+	0.946230i	$0.604858\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−165.513	−0.218885	−0.109442	−	0.993993i	$-0.534907\pi$
$83$	−165.513	−0.218885	−0.109442	+	0.993993i	$0.534907\pi$
$84$	0	0
$85$	−602.444	−0.768755
$86$	0	0
$87$	0	0
$88$	0	0
$89$	535.743	0.638074	0.319037	−	0.947742i	$-0.396641\pi$
$89$	535.743	0.638074	0.319037	+	0.947742i	$0.396641\pi$
$90$	0	0
$91$	2770.01	3.19094
$92$	0	0
$93$	0	0
$94$	0	0
$95$	606.594	0.655108
$96$	0	0
$97$	−436.761	−0.457179	−0.228590	−	0.973523i	$-0.573411\pi$
$97$	−436.761	−0.457179	−0.228590	+	0.973523i	$0.573411\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1376.74	1.35635	0.678174	−	0.734901i	$-0.262771\pi$
$101$	1376.74	1.35635	0.678174	+	0.734901i	$0.262771\pi$
$102$	0	0
$103$	−20.2617	−0.0193830	−0.00969149	−	0.999953i	$-0.503085\pi$
$103$	−20.2617	−0.0193830	−0.00969149	+	0.999953i	$0.503085\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−359.961	−0.325222	−0.162611	−	0.986690i	$-0.551992\pi$
$107$	−359.961	−0.325222	−0.162611	+	0.986690i	$0.551992\pi$
$108$	0	0
$109$	377.931	0.332103	0.166052	−	0.986117i	$-0.446898\pi$
$109$	377.931	0.332103	0.166052	+	0.986117i	$0.446898\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−21.3449	−0.0177696	−0.00888478	−	0.999961i	$-0.502828\pi$
$113$	−21.3449	−0.0177696	−0.00888478	+	0.999961i	$0.502828\pi$
$114$	0	0
$115$	764.276	0.619731
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4288.70	−3.30373
$120$	0	0
$121$	672.826	0.505504
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1756.10	1.22700	0.613499	−	0.789696i	$-0.289762\pi$
$127$	1756.10	1.22700	0.613499	+	0.789696i	$0.289762\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2388.31	1.59288	0.796441	−	0.604716i	$-0.206713\pi$
$131$	2388.31	1.59288	0.796441	+	0.604716i	$0.206713\pi$
$132$	0	0
$133$	4318.24	2.81533
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1124.94	0.701534	0.350767	−	0.936463i	$-0.385921\pi$
$137$	1124.94	0.701534	0.350767	+	0.936463i	$0.385921\pi$
$138$	0	0
$139$	795.105	0.485179	0.242590	−	0.970129i	$-0.422003\pi$
$139$	795.105	0.485179	0.242590	+	0.970129i	$0.422003\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3483.63	−2.03718
$144$	0	0
$145$	939.630	0.538152
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2933.56	1.61293	0.806466	−	0.591281i	$-0.201378\pi$
$149$	2933.56	1.61293	0.806466	+	0.591281i	$0.201378\pi$
$150$	0	0
$151$	1645.26	0.886686	0.443343	−	0.896352i	$-0.353792\pi$
$151$	1645.26	0.886686	0.443343	+	0.896352i	$0.353792\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−806.825	−0.418101
$156$	0	0
$157$	−1001.96	−0.509330	−0.254665	−	0.967029i	$-0.581965\pi$
$157$	−1001.96	−0.509330	−0.254665	+	0.967029i	$0.581965\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5440.75	2.66330
$162$	0	0
$163$	−2135.60	−1.02622	−0.513108	−	0.858324i	$-0.671506\pi$
$163$	−2135.60	−1.02622	−0.513108	+	0.858324i	$0.671506\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3612.36	−1.67385	−0.836925	−	0.547318i	$-0.815649\pi$
$167$	−3612.36	−1.67385	−0.836925	+	0.547318i	$0.815649\pi$
$168$	0	0
$169$	3859.26	1.75661
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2684.80	1.17989	0.589947	−	0.807442i	$-0.299149\pi$
$173$	2684.80	1.17989	0.589947	+	0.807442i	$0.299149\pi$
$174$	0	0
$175$	889.854	0.384381
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2182.63	−0.911382	−0.455691	−	0.890138i	$-0.650608\pi$
$179$	−2182.63	−0.911382	−0.455691	+	0.890138i	$0.650608\pi$
$180$	0	0
$181$	3117.86	1.28038	0.640189	−	0.768217i	$-0.278856\pi$
$181$	3117.86	1.28038	0.640189	+	0.768217i	$0.278856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	66.6748	0.0264975
$186$	0	0
$187$	5393.57	2.10918
$188$	0	0
$189$	0	0
$190$	0	0
$191$	76.0912	0.0288260	0.0144130	−	0.999896i	$-0.495412\pi$
$191$	76.0912	0.0288260	0.0144130	+	0.999896i	$0.495412\pi$
$192$	0	0
$193$	533.717	0.199056	0.0995280	−	0.995035i	$-0.468267\pi$
$193$	533.717	0.199056	0.0995280	+	0.995035i	$0.468267\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2729.47	0.987142	0.493571	−	0.869706i	$-0.335691\pi$
$197$	2729.47	0.987142	0.493571	+	0.869706i	$0.335691\pi$
$198$	0	0
$199$	−3196.27	−1.13858	−0.569290	−	0.822137i	$-0.692782\pi$
$199$	−3196.27	−1.13858	−0.569290	+	0.822137i	$0.692782\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6689.07	2.31271
$204$	0	0
$205$	940.323	0.320366
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5430.73	−1.79738
$210$	0	0
$211$	−4325.59	−1.41131	−0.705654	−	0.708557i	$-0.749347\pi$
$211$	−4325.59	−1.41131	−0.705654	+	0.708557i	$0.749347\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−407.792	−0.129354
$216$	0	0
$217$	−5743.65	−1.79680
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9376.68	−2.85404
$222$	0	0
$223$	1370.87	0.411660	0.205830	−	0.978588i	$-0.434011\pi$
$223$	1370.87	0.411660	0.205830	+	0.978588i	$0.434011\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2676.95	0.782711	0.391356	−	0.920240i	$-0.372006\pi$
$227$	2676.95	0.782711	0.391356	+	0.920240i	$0.372006\pi$
$228$	0	0
$229$	−2641.40	−0.762222	−0.381111	−	0.924529i	$-0.624459\pi$
$229$	−2641.40	−0.762222	−0.381111	+	0.924529i	$0.624459\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1719.93	−0.483589	−0.241795	−	0.970327i	$-0.577736\pi$
$233$	−1719.93	−0.483589	−0.241795	+	0.970327i	$0.577736\pi$
$234$	0	0
$235$	243.109	0.0674839
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5162.21	−1.39714	−0.698568	−	0.715543i	$-0.746179\pi$
$239$	−5162.21	−1.39714	−0.698568	+	0.715543i	$0.746179\pi$
$240$	0	0
$241$	−717.935	−0.191893	−0.0959465	−	0.995386i	$-0.530588\pi$
$241$	−717.935	−0.191893	−0.0959465	+	0.995386i	$0.530588\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4619.73	1.20467
$246$	0	0
$247$	9441.27	2.43212
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1101.95	−0.277110	−0.138555	−	0.990355i	$-0.544246\pi$
$251$	−1101.95	−0.277110	−0.138555	+	0.990355i	$0.544246\pi$
$252$	0	0
$253$	−6842.43	−1.70031
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2922.73	0.709396	0.354698	−	0.934981i	$-0.384584\pi$
$257$	2922.73	0.709396	0.354698	+	0.934981i	$0.384584\pi$
$258$	0	0
$259$	474.647	0.113873
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1380.33	−0.323631	−0.161815	−	0.986821i	$-0.551735\pi$
$263$	−1380.33	−0.323631	−0.161815	+	0.986821i	$0.551735\pi$
$264$	0	0
$265$	−3536.87	−0.819879
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−85.9555	−0.0194825	−0.00974126	−	0.999953i	$-0.503101\pi$
$269$	−85.9555	−0.0194825	−0.00974126	+	0.999953i	$0.503101\pi$
$270$	0	0
$271$	−2676.24	−0.599889	−0.299944	−	0.953957i	$-0.596968\pi$
$271$	−2676.24	−0.599889	−0.299944	+	0.953957i	$0.596968\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1119.10	−0.245398
$276$	0	0
$277$	6135.26	1.33080	0.665400	−	0.746487i	$-0.268261\pi$
$277$	6135.26	1.33080	0.665400	+	0.746487i	$0.268261\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2899.29	−0.615505	−0.307753	−	0.951466i	$-0.599577\pi$
$281$	−2899.29	−0.615505	−0.307753	+	0.951466i	$0.599577\pi$
$282$	0	0
$283$	−3584.20	−0.752857	−0.376429	−	0.926446i	$-0.622848\pi$
$283$	−3584.20	−0.752857	−0.376429	+	0.926446i	$0.622848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6694.00	1.37678
$288$	0	0
$289$	9604.55	1.95492
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7592.52	1.51386	0.756928	−	0.653498i	$-0.226699\pi$
$293$	7592.52	1.51386	0.756928	+	0.653498i	$0.226699\pi$
$294$	0	0
$295$	82.3871	0.0162602
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11895.5	2.30078
$300$	0	0
$301$	−2903.01	−0.555902
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3719.34	−0.698258
$306$	0	0
$307$	5539.62	1.02985	0.514923	−	0.857237i	$-0.327821\pi$
$307$	5539.62	1.02985	0.514923	+	0.857237i	$0.327821\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8862.85	−1.61597	−0.807984	−	0.589204i	$-0.799441\pi$
$311$	−8862.85	−1.61597	−0.807984	+	0.589204i	$0.799441\pi$
$312$	0	0
$313$	6028.49	1.08866	0.544329	−	0.838872i	$-0.316784\pi$
$313$	6028.49	1.08866	0.544329	+	0.838872i	$0.316784\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10578.3	1.87425	0.937126	−	0.348991i	$-0.113476\pi$
$317$	10578.3	1.87425	0.937126	+	0.348991i	$0.113476\pi$
$318$	0	0
$319$	−8412.34	−1.47649
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14617.6	−2.51809
$324$	0	0
$325$	1945.55	0.332061
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1730.66	0.290013
$330$	0	0
$331$	8051.25	1.33697	0.668484	−	0.743726i	$-0.266943\pi$
$331$	8051.25	1.33697	0.668484	+	0.743726i	$0.266943\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−296.338	−0.0483304
$336$	0	0
$337$	−3573.37	−0.577608	−0.288804	−	0.957388i	$-0.593258\pi$
$337$	−3573.37	−0.577608	−0.288804	+	0.957388i	$0.593258\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7223.36	1.14712
$342$	0	0
$343$	20678.3	3.25517
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3268.28	−0.505621	−0.252811	−	0.967516i	$-0.581355\pi$
$347$	−3268.28	−0.505621	−0.252811	+	0.967516i	$0.581355\pi$
$348$	0	0
$349$	5831.57	0.894432	0.447216	−	0.894426i	$-0.352415\pi$
$349$	5831.57	0.894432	0.447216	+	0.894426i	$0.352415\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12021.4	−1.81257	−0.906285	−	0.422667i	$-0.861094\pi$
$353$	−12021.4	−1.81257	−0.906285	+	0.422667i	$0.861094\pi$
$354$	0	0
$355$	721.849	0.107920
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1598.32	−0.234975	−0.117487	−	0.993074i	$-0.537484\pi$
$359$	−1598.32	−0.234975	−0.117487	+	0.993074i	$0.537484\pi$
$360$	0	0
$361$	7859.26	1.14583
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3286.37	0.471277
$366$	0	0
$367$	−12200.7	−1.73534	−0.867671	−	0.497138i	$-0.834384\pi$
$367$	−12200.7	−1.73534	−0.867671	+	0.497138i	$0.834384\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25178.4	−3.52344
$372$	0	0
$373$	−7613.13	−1.05682	−0.528409	−	0.848990i	$-0.677211\pi$
$373$	−7613.13	−1.05682	−0.528409	+	0.848990i	$0.677211\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14624.8	1.99792
$378$	0	0
$379$	−7463.85	−1.01159	−0.505794	−	0.862654i	$-0.668800\pi$
$379$	−7463.85	−1.01159	−0.505794	+	0.862654i	$0.668800\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4064.01	−0.542196	−0.271098	−	0.962552i	$-0.587387\pi$
$383$	−4064.01	−0.542196	−0.271098	+	0.962552i	$0.587387\pi$
$384$	0	0
$385$	−7966.71	−1.05460
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1258.30	0.164006	0.0820028	−	0.996632i	$-0.473868\pi$
$389$	1258.30	0.164006	0.0820028	+	0.996632i	$0.473868\pi$
$390$	0	0
$391$	−18417.3	−2.38211
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2271.48	−0.289344
$396$	0	0
$397$	−8486.66	−1.07288	−0.536440	−	0.843938i	$-0.680231\pi$
$397$	−8486.66	−1.07288	−0.536440	+	0.843938i	$0.680231\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−829.986	−0.103360	−0.0516802	−	0.998664i	$-0.516458\pi$
$401$	−829.986	−0.103360	−0.0516802	+	0.998664i	$0.516458\pi$
$402$	0	0
$403$	−12557.7	−1.55222
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−596.928	−0.0726993
$408$	0	0
$409$	−8600.00	−1.03971	−0.519857	−	0.854254i	$-0.674015\pi$
$409$	−8600.00	−1.03971	−0.519857	+	0.854254i	$0.674015\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	586.500	0.0698784
$414$	0	0
$415$	−827.566	−0.0978882
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2926.84	−0.341254	−0.170627	−	0.985336i	$-0.554579\pi$
$419$	−2926.84	−0.341254	−0.170627	+	0.985336i	$0.554579\pi$
$420$	0	0
$421$	−707.503	−0.0819041	−0.0409520	−	0.999161i	$-0.513039\pi$
$421$	−707.503	−0.0819041	−0.0409520	+	0.999161i	$0.513039\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3012.22	−0.343798
$426$	0	0
$427$	−26477.4	−3.00077
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4693.17	0.524506	0.262253	−	0.964999i	$-0.415535\pi$
$431$	4693.17	0.524506	0.262253	+	0.964999i	$0.415535\pi$
$432$	0	0
$433$	−279.149	−0.0309816	−0.0154908	−	0.999880i	$-0.504931\pi$
$433$	−279.149	−0.0309816	−0.0154908	+	0.999880i	$0.504931\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18544.2	2.02995
$438$	0	0
$439$	−2651.13	−0.288226	−0.144113	−	0.989561i	$-0.546033\pi$
$439$	−2651.13	−0.288226	−0.144113	+	0.989561i	$0.546033\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1899.98	0.203771	0.101886	−	0.994796i	$-0.467512\pi$
$443$	1899.98	0.203771	0.101886	+	0.994796i	$0.467512\pi$
$444$	0	0
$445$	2678.71	0.285355
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6033.52	0.634164	0.317082	−	0.948398i	$-0.397297\pi$
$449$	6033.52	0.634164	0.317082	+	0.948398i	$0.397297\pi$
$450$	0	0
$451$	−8418.54	−0.878966
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13850.1	1.42703
$456$	0	0
$457$	−7920.25	−0.810708	−0.405354	−	0.914160i	$-0.632852\pi$
$457$	−7920.25	−0.810708	−0.405354	+	0.914160i	$0.632852\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3754.65	0.379331	0.189666	−	0.981849i	$-0.439260\pi$
$461$	3754.65	0.379331	0.189666	+	0.981849i	$0.439260\pi$
$462$	0	0
$463$	−3094.28	−0.310591	−0.155295	−	0.987868i	$-0.549633\pi$
$463$	−3094.28	−0.310591	−0.155295	+	0.987868i	$0.549633\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13827.8	1.37018	0.685092	−	0.728457i	$-0.259762\pi$
$467$	13827.8	1.37018	0.685092	+	0.728457i	$0.259762\pi$
$468$	0	0
$469$	−2109.58	−0.207700
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3650.89	0.354901
$474$	0	0
$475$	3032.97	0.292973
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1776.55	−0.169463	−0.0847313	−	0.996404i	$-0.527003\pi$
$479$	−1776.55	−0.169463	−0.0847313	+	0.996404i	$0.527003\pi$
$480$	0	0
$481$	1037.75	0.0983732
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2183.81	−0.204457
$486$	0	0
$487$	−3223.84	−0.299971	−0.149986	−	0.988688i	$-0.547923\pi$
$487$	−3223.84	−0.299971	−0.149986	+	0.988688i	$0.547923\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8699.20	−0.799571	−0.399786	−	0.916609i	$-0.630915\pi$
$491$	−8699.20	−0.799571	−0.399786	+	0.916609i	$0.630915\pi$
$492$	0	0
$493$	−22643.0	−2.06854
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5138.72	0.463789
$498$	0	0
$499$	11833.7	1.06163	0.530813	−	0.847489i	$-0.321887\pi$
$499$	11833.7	1.06163	0.530813	+	0.847489i	$0.321887\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15882.5	−1.40789	−0.703943	−	0.710256i	$-0.748579\pi$
$503$	−15882.5	−1.40789	−0.703943	+	0.710256i	$0.748579\pi$
$504$	0	0
$505$	6883.72	0.606577
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20662.8	−1.79934	−0.899670	−	0.436570i	$-0.856193\pi$
$509$	−20662.8	−1.79934	−0.899670	+	0.436570i	$0.856193\pi$
$510$	0	0
$511$	23395.1	2.02532
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−101.309	−0.00866834
$516$	0	0
$517$	−2176.52	−0.185151
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13333.8	1.12124	0.560618	−	0.828075i	$-0.310564\pi$
$521$	13333.8	1.12124	0.560618	+	0.828075i	$0.310564\pi$
$522$	0	0
$523$	−8822.73	−0.737650	−0.368825	−	0.929499i	$-0.620240\pi$
$523$	−8822.73	−0.737650	−0.368825	+	0.929499i	$0.620240\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19442.7	1.60709
$528$	0	0
$529$	11197.7	0.920333
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14635.6	1.18937
$534$	0	0
$535$	−1799.80	−0.145444
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−41359.6	−3.30517
$540$	0	0
$541$	−16975.6	−1.34905	−0.674527	−	0.738250i	$-0.735652\pi$
$541$	−16975.6	−1.34905	−0.674527	+	0.738250i	$0.735652\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1889.66	0.148521
$546$	0	0
$547$	9986.07	0.780573	0.390287	−	0.920693i	$-0.372376\pi$
$547$	9986.07	0.780573	0.390287	+	0.920693i	$0.372376\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22799.0	1.76274
$552$	0	0
$553$	−16170.3	−1.24346
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15740.3	−1.19737	−0.598686	−	0.800984i	$-0.704310\pi$
$557$	−15740.3	−1.19737	−0.598686	+	0.800984i	$0.704310\pi$
$558$	0	0
$559$	−6347.04	−0.480235
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10829.5	−0.810674	−0.405337	−	0.914167i	$-0.632846\pi$
$563$	−10829.5	−0.810674	−0.405337	+	0.914167i	$0.632846\pi$
$564$	0	0
$565$	−106.725	−0.00794679
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11858.7	−0.873711	−0.436855	−	0.899532i	$-0.643908\pi$
$569$	−11858.7	−0.873711	−0.436855	+	0.899532i	$0.643908\pi$
$570$	0	0
$571$	16839.3	1.23415	0.617077	−	0.786903i	$-0.288317\pi$
$571$	16839.3	1.23415	0.617077	+	0.786903i	$0.288317\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3821.38	0.277152
$576$	0	0
$577$	2182.04	0.157434	0.0787169	−	0.996897i	$-0.474918\pi$
$577$	2182.04	0.157434	0.0787169	+	0.996897i	$0.474918\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5891.31	−0.420676
$582$	0	0
$583$	31664.9	2.24945
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2121.16	0.149148	0.0745739	−	0.997215i	$-0.476240\pi$
$587$	2121.16	0.149148	0.0745739	+	0.997215i	$0.476240\pi$
$588$	0	0
$589$	−19576.6	−1.36951
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1602.22	0.110953	0.0554766	−	0.998460i	$-0.482332\pi$
$593$	1602.22	0.110953	0.0554766	+	0.998460i	$0.482332\pi$
$594$	0	0
$595$	−21443.5	−1.47748
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18383.4	−1.25396	−0.626982	−	0.779033i	$-0.715710\pi$
$599$	−18383.4	−1.25396	−0.626982	+	0.779033i	$0.715710\pi$
$600$	0	0
$601$	−25606.2	−1.73793	−0.868967	−	0.494870i	$-0.835216\pi$
$601$	−25606.2	−1.73793	−0.868967	+	0.494870i	$0.835216\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3364.13	0.226068
$606$	0	0
$607$	−28967.2	−1.93697	−0.968487	−	0.249065i	$-0.919877\pi$
$607$	−28967.2	−1.93697	−0.968487	+	0.249065i	$0.919877\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3783.85	0.250537
$612$	0	0
$613$	9101.21	0.599665	0.299832	−	0.953992i	$-0.403069\pi$
$613$	9101.21	0.599665	0.299832	+	0.953992i	$0.403069\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13945.3	−0.909913	−0.454957	−	0.890514i	$-0.650345\pi$
$617$	−13945.3	−0.909913	−0.454957	+	0.890514i	$0.650345\pi$
$618$	0	0
$619$	−4276.76	−0.277702	−0.138851	−	0.990313i	$-0.544341\pi$
$619$	−4276.76	−0.277702	−0.138851	+	0.990313i	$0.544341\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	19069.3	1.22632
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1606.71	−0.101850
$630$	0	0
$631$	−18459.0	−1.16457	−0.582283	−	0.812987i	$-0.697840\pi$
$631$	−18459.0	−1.16457	−0.582283	+	0.812987i	$0.697840\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8780.50	0.548730
$636$	0	0
$637$	71903.3	4.47239
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25102.1	−1.54676	−0.773381	−	0.633941i	$-0.781436\pi$
$641$	−25102.1	−1.54676	−0.773381	+	0.633941i	$0.781436\pi$
$642$	0	0
$643$	−14935.1	−0.915991	−0.457996	−	0.888954i	$-0.651432\pi$
$643$	−14935.1	−0.915991	−0.457996	+	0.888954i	$0.651432\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18764.3	−1.14019	−0.570093	−	0.821580i	$-0.693093\pi$
$647$	−18764.3	−1.14019	−0.570093	+	0.821580i	$0.693093\pi$
$648$	0	0
$649$	−737.597	−0.0446121
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10200.0	0.611264	0.305632	−	0.952150i	$-0.401132\pi$
$653$	10200.0	0.611264	0.305632	+	0.952150i	$0.401132\pi$
$654$	0	0
$655$	11941.5	0.712359
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6908.98	−0.408400	−0.204200	−	0.978929i	$-0.565459\pi$
$659$	−6908.98	−0.408400	−0.204200	+	0.978929i	$0.565459\pi$
$660$	0	0
$661$	−15449.8	−0.909118	−0.454559	−	0.890717i	$-0.650203\pi$
$661$	−15449.8	−0.909118	−0.454559	+	0.890717i	$0.650203\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21591.2	1.25906
$666$	0	0
$667$	28725.5	1.66755
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33298.6	1.91576
$672$	0	0
$673$	15782.7	0.903977	0.451989	−	0.892024i	$-0.350715\pi$
$673$	15782.7	0.903977	0.451989	+	0.892024i	$0.350715\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6982.32	0.396384	0.198192	−	0.980163i	$-0.436493\pi$
$677$	6982.32	0.396384	0.198192	+	0.980163i	$0.436493\pi$
$678$	0	0
$679$	−15546.2	−0.878655
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1267.31	−0.0709988	−0.0354994	−	0.999370i	$-0.511302\pi$
$683$	−1267.31	−0.0709988	−0.0354994	+	0.999370i	$0.511302\pi$
$684$	0	0
$685$	5624.71	0.313736
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−55049.2	−3.04384
$690$	0	0
$691$	24220.7	1.33343	0.666713	−	0.745315i	$-0.267701\pi$
$691$	24220.7	1.33343	0.666713	+	0.745315i	$0.267701\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3975.52	0.216979
$696$	0	0
$697$	−22659.7	−1.23141
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11822.1	0.636967	0.318483	−	0.947928i	$-0.396826\pi$
$701$	11822.1	0.636967	0.318483	+	0.947928i	$0.396826\pi$
$702$	0	0
$703$	1617.78	0.0867935
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49004.1	2.60677
$708$	0	0
$709$	−15140.6	−0.801997	−0.400998	−	0.916079i	$-0.631337\pi$
$709$	−15140.6	−0.801997	−0.400998	+	0.916079i	$0.631337\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24665.5	−1.29555
$714$	0	0
$715$	−17418.2	−0.911052
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1628.91	−0.0844895	−0.0422448	−	0.999107i	$-0.513451\pi$
$719$	−1628.91	−0.0844895	−0.0422448	+	0.999107i	$0.513451\pi$
$720$	0	0
$721$	−721.200	−0.0372523
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4698.15	0.240669
$726$	0	0
$727$	−30179.1	−1.53959	−0.769793	−	0.638293i	$-0.779641\pi$
$727$	−30179.1	−1.53959	−0.769793	+	0.638293i	$0.779641\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9826.87	0.497209
$732$	0	0
$733$	9233.18	0.465260	0.232630	−	0.972565i	$-0.425267\pi$
$733$	9233.18	0.465260	0.232630	+	0.972565i	$0.425267\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2653.06	0.132601
$738$	0	0
$739$	21276.8	1.05911	0.529554	−	0.848276i	$-0.322359\pi$
$739$	21276.8	1.05911	0.529554	+	0.848276i	$0.322359\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9647.90	0.476375	0.238188	−	0.971219i	$-0.423447\pi$
$743$	9647.90	0.476375	0.238188	+	0.971219i	$0.423447\pi$
$744$	0	0
$745$	14667.8	0.721325
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12812.5	−0.625045
$750$	0	0
$751$	−19151.6	−0.930560	−0.465280	−	0.885163i	$-0.654046\pi$
$751$	−19151.6	−0.930560	−0.465280	+	0.885163i	$0.654046\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8226.31	0.396538
$756$	0	0
$757$	6689.40	0.321176	0.160588	−	0.987022i	$-0.448661\pi$
$757$	6689.40	0.321176	0.160588	+	0.987022i	$0.448661\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24594.6	1.17156	0.585779	−	0.810471i	$-0.300789\pi$
$761$	24594.6	1.17156	0.585779	+	0.810471i	$0.300789\pi$
$762$	0	0
$763$	13452.2	0.638271
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1282.31	0.0603669
$768$	0	0
$769$	−2631.04	−0.123378	−0.0616890	−	0.998095i	$-0.519649\pi$
$769$	−2631.04	−0.123378	−0.0616890	+	0.998095i	$0.519649\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15347.0	−0.714091	−0.357045	−	0.934087i	$-0.616216\pi$
$773$	−15347.0	−0.714091	−0.357045	+	0.934087i	$0.616216\pi$
$774$	0	0
$775$	−4034.12	−0.186981
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22815.8	1.04937
$780$	0	0
$781$	−6462.58	−0.296094
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5009.78	−0.227779
$786$	0	0
$787$	42630.0	1.93087	0.965436	−	0.260639i	$-0.0839331\pi$
$787$	42630.0	1.93087	0.965436	+	0.260639i	$0.0839331\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−759.755	−0.0341514
$792$	0	0
$793$	−57889.3	−2.59232
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10002.6	−0.444554	−0.222277	−	0.974984i	$-0.571349\pi$
$797$	−10002.6	−0.444554	−0.222277	+	0.974984i	$0.571349\pi$
$798$	0	0
$799$	−5858.39	−0.259393
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−29422.3	−1.29301
$804$	0	0
$805$	27203.8	1.19106
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13642.8	0.592898	0.296449	−	0.955049i	$-0.404198\pi$
$809$	13642.8	0.592898	0.296449	+	0.955049i	$0.404198\pi$
$810$	0	0
$811$	−3786.57	−0.163951	−0.0819756	−	0.996634i	$-0.526123\pi$
$811$	−3786.57	−0.163951	−0.0819756	+	0.996634i	$0.526123\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10678.0	−0.458938
$816$	0	0
$817$	−9894.57	−0.423705
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31609.5	1.34370	0.671851	−	0.740686i	$-0.265499\pi$
$821$	31609.5	1.34370	0.671851	+	0.740686i	$0.265499\pi$
$822$	0	0
$823$	20737.3	0.878320	0.439160	−	0.898409i	$-0.355276\pi$
$823$	20737.3	0.878320	0.439160	+	0.898409i	$0.355276\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10152.0	0.426866	0.213433	−	0.976958i	$-0.431535\pi$
$827$	10152.0	0.426866	0.213433	+	0.976958i	$0.431535\pi$
$828$	0	0
$829$	−2857.26	−0.119706	−0.0598532	−	0.998207i	$-0.519063\pi$
$829$	−2857.26	−0.119706	−0.0598532	+	0.998207i	$0.519063\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−111325.	−4.63047
$834$	0	0
$835$	−18061.8	−0.748568
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24638.0	−1.01382	−0.506911	−	0.861998i	$-0.669213\pi$
$839$	−24638.0	−1.01382	−0.506911	+	0.861998i	$0.669213\pi$
$840$	0	0
$841$	10927.2	0.448037
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19296.3	0.785578
$846$	0	0
$847$	23948.7	0.971531
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2038.32	0.0821065
$852$	0	0
$853$	1.13034	4.53719e−5 0	2.26860e−5	−	1.00000i	$-0.499993\pi$
$853$	1.13034	4.53719e−5 0	2.26860e−5		1.00000i	$0.499993\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37803.4	1.50681	0.753406	−	0.657555i	$-0.228409\pi$
$857$	37803.4	1.50681	0.753406	+	0.657555i	$0.228409\pi$
$858$	0	0
$859$	−23555.7	−0.935635	−0.467817	−	0.883825i	$-0.654959\pi$
$859$	−23555.7	−0.935635	−0.467817	+	0.883825i	$0.654959\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9572.78	−0.377591	−0.188796	−	0.982016i	$-0.560458\pi$
$863$	−9572.78	−0.377591	−0.188796	+	0.982016i	$0.560458\pi$
$864$	0	0
$865$	13424.0	0.527665
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20336.2	0.793853
$870$	0	0
$871$	−4612.32	−0.179429
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4449.27	0.171900
$876$	0	0
$877$	1559.23	0.0600358	0.0300179	−	0.999549i	$-0.490444\pi$
$877$	1559.23	0.0600358	0.0300179	+	0.999549i	$0.490444\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36729.5	−1.40459	−0.702297	−	0.711884i	$-0.747842\pi$
$881$	−36729.5	−1.40459	−0.702297	+	0.711884i	$0.747842\pi$
$882$	0	0
$883$	6625.31	0.252502	0.126251	−	0.991998i	$-0.459706\pi$
$883$	6625.31	0.252502	0.126251	+	0.991998i	$0.459706\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32562.3	1.23262	0.616311	−	0.787503i	$-0.288626\pi$
$887$	32562.3	1.23262	0.616311	+	0.787503i	$0.288626\pi$
$888$	0	0
$889$	62506.9	2.35817
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5898.75	0.221046
$894$	0	0
$895$	−10913.1	−0.407582
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30324.7	−1.12501
$900$	0	0
$901$	85230.5	3.15143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15589.3	0.572603
$906$	0	0
$907$	−37343.5	−1.36711	−0.683556	−	0.729898i	$-0.739568\pi$
$907$	−37343.5	−1.36711	−0.683556	+	0.729898i	$0.739568\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13207.3	−0.480328	−0.240164	−	0.970732i	$-0.577201\pi$
$911$	−13207.3	−0.480328	−0.240164	+	0.970732i	$0.577201\pi$
$912$	0	0
$913$	7409.05	0.268569
$914$	0	0
$915$	0	0
$916$	0	0
$917$	85009.9	3.06137
$918$	0	0
$919$	51007.6	1.83089	0.915444	−	0.402445i	$-0.131839\pi$
$919$	51007.6	1.83089	0.915444	+	0.402445i	$0.131839\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11235.1	0.400660
$924$	0	0
$925$	333.374	0.0118500
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25322.8	−0.894311	−0.447155	−	0.894456i	$-0.647563\pi$
$929$	−25322.8	−0.894311	−0.447155	+	0.894456i	$0.647563\pi$
$930$	0	0
$931$	112092.	3.94594
$932$	0	0
$933$	0	0
$934$	0	0
$935$	26967.9	0.943255
$936$	0	0
$937$	12667.9	0.441668	0.220834	−	0.975311i	$-0.429122\pi$
$937$	12667.9	0.441668	0.220834	+	0.975311i	$0.429122\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25676.4	−0.889507	−0.444753	−	0.895653i	$-0.646709\pi$
$941$	−25676.4	−0.889507	−0.444753	+	0.895653i	$0.646709\pi$
$942$	0	0
$943$	28746.6	0.992703
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50265.9	1.72484	0.862420	−	0.506193i	$-0.168948\pi$
$947$	50265.9	1.72484	0.862420	+	0.506193i	$0.168948\pi$
$948$	0	0
$949$	51150.3	1.74964
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39606.9	1.34627	0.673134	−	0.739520i	$-0.264948\pi$
$953$	39606.9	1.34627	0.673134	+	0.739520i	$0.264948\pi$
$954$	0	0
$955$	380.456	0.0128914
$956$	0	0
$957$	0	0
$958$	0	0
$959$	40041.4	1.34828
$960$	0	0
$961$	−3752.35	−0.125956
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2668.59	0.0890205
$966$	0	0
$967$	13997.6	0.465493	0.232746	−	0.972537i	$-0.425229\pi$
$967$	13997.6	0.465493	0.232746	+	0.972537i	$0.425229\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29067.3	−0.960675	−0.480337	−	0.877084i	$-0.659486\pi$
$971$	−29067.3	−0.960675	−0.480337	+	0.877084i	$0.659486\pi$
$972$	0	0
$973$	28301.1	0.932468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32811.7	−1.07445	−0.537226	−	0.843438i	$-0.680528\pi$
$977$	−32811.7	−1.07445	−0.537226	+	0.843438i	$0.680528\pi$
$978$	0	0
$979$	−23982.0	−0.782911
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2957.58	−0.0959635	−0.0479818	−	0.998848i	$-0.515279\pi$
$983$	−2957.58	−0.0959635	−0.0479818	+	0.998848i	$0.515279\pi$
$984$	0	0
$985$	13647.4	0.441463
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12466.6	−0.400825
$990$	0	0
$991$	−8950.60	−0.286907	−0.143454	−	0.989657i	$-0.545821\pi$
$991$	−8950.60	−0.286907	−0.143454	+	0.989657i	$0.545821\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15981.3	−0.509188
$996$	0	0
$997$	−53211.5	−1.69029	−0.845147	−	0.534533i	$-0.820487\pi$
$997$	−53211.5	−1.69029	−0.845147	+	0.534533i	$0.820487\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.p.1.4	yes	4
3.2	odd	2		1080.4.a.o.1.4	✓	4
4.3	odd	2		2160.4.a.bv.1.1		4
12.11	even	2		2160.4.a.bu.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.o.1.4	✓	4	3.2	odd	2
1080.4.a.p.1.4	yes	4	1.1	even	1	trivial
2160.4.a.bu.1.1		4	12.11	even	2
2160.4.a.bv.1.1		4	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +35.5942 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} +35.5942 q^{7} -44.7641 q^{11} +77.8220 q^{13} -120.489 q^{17} +121.319 q^{19} +152.855 q^{23} +25.0000 q^{25} +187.926 q^{29} -161.365 q^{31} +177.971 q^{35} +13.3350 q^{37} +188.065 q^{41} -81.5584 q^{43} +48.6219 q^{47} +923.946 q^{49} -707.373 q^{53} -223.821 q^{55} +16.4774 q^{59} -743.868 q^{61} +389.110 q^{65} -59.2676 q^{67} +144.370 q^{71} +657.273 q^{73} -1593.34 q^{77} -454.297 q^{79} -165.513 q^{83} -602.444 q^{85} +535.743 q^{89} +2770.01 q^{91} +606.594 q^{95} -436.761 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{5} + 14 q^{7}+O(q^{10}) \) 4 * q + 20 * q^5 + 14 * q^7	\( 4 q + 20 q^{5} + 14 q^{7} - 4 q^{11} + 30 q^{13} - 28 q^{17} + 78 q^{19} + 182 q^{23} + 100 q^{25} + 202 q^{29} - 76 q^{31} + 70 q^{35} + 302 q^{37} + 380 q^{41} + 178 q^{43} + 114 q^{47} + 958 q^{49} - 256 q^{53} - 20 q^{55} - 204 q^{59} + 766 q^{61} + 150 q^{65} + 330 q^{67} - 1060 q^{71} + 1442 q^{73} + 216 q^{77} + 742 q^{79} - 768 q^{83} - 140 q^{85} - 400 q^{89} + 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) \) 4 * q + 20 * q^5 + 14 * q^7 - 4 * q^11 + 30 * q^13 - 28 * q^17 + 78 * q^19 + 182 * q^23 + 100 * q^25 + 202 * q^29 - 76 * q^31 + 70 * q^35 + 302 * q^37 + 380 * q^41 + 178 * q^43 + 114 * q^47 + 958 * q^49 - 256 * q^53 - 20 * q^55 - 204 * q^59 + 766 * q^61 + 150 * q^65 + 330 * q^67 - 1060 * q^71 + 1442 * q^73 + 216 * q^77 + 742 * q^79 - 768 * q^83 - 140 * q^85 - 400 * q^89 + 3066 * q^91 + 390 * q^95 + 3338 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more