LMFDB - Embedded newform 1080.4.a.a.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.26209$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -10.2621 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -10.2621 q^{7} -5.26209 q^{11} +34.5725 q^{13} +74.8346 q^{17} -50.8830 q^{19} -81.4555 q^{23} +25.0000 q^{25} +152.407 q^{29} +93.9797 q^{31} +51.3104 q^{35} -167.310 q^{37} +18.6209 q^{41} +133.552 q^{43} +46.8346 q^{47} -237.690 q^{49} -105.104 q^{53} +26.3104 q^{55} -77.7099 q^{59} -48.6896 q^{61} -172.863 q^{65} -667.132 q^{67} -344.056 q^{71} -1074.70 q^{73} +54.0000 q^{77} -420.863 q^{79} -296.565 q^{83} -374.173 q^{85} +950.596 q^{89} -354.786 q^{91} +254.415 q^{95} -135.916 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} - 5 q^{7}+O(q^{10}) $ 2 * q - 10 * q^5 - 5 * q^7	$ 2 q - 10 q^{5} - 5 q^{7} + 5 q^{11} - 24 q^{13} + 41 q^{17} + 69 q^{19} + 101 q^{23} + 50 q^{25} + 103 q^{29} - 107 q^{31} + 25 q^{35} - 257 q^{37} - 118 q^{41} - 121 q^{43} - 15 q^{47} - 553 q^{49} + 566 q^{53} - 25 q^{55} - 528 q^{59} - 175 q^{61} + 120 q^{65} - 201 q^{67} + 26 q^{71} - 923 q^{73} + 108 q^{77} - 376 q^{79} - 1152 q^{83} - 205 q^{85} - 148 q^{89} - 663 q^{91} - 345 q^{95} - 1343 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 - 5 * q^7 + 5 * q^11 - 24 * q^13 + 41 * q^17 + 69 * q^19 + 101 * q^23 + 50 * q^25 + 103 * q^29 - 107 * q^31 + 25 * q^35 - 257 * q^37 - 118 * q^41 - 121 * q^43 - 15 * q^47 - 553 * q^49 + 566 * q^53 - 25 * q^55 - 528 * q^59 - 175 * q^61 + 120 * q^65 - 201 * q^67 + 26 * q^71 - 923 * q^73 + 108 * q^77 - 376 * q^79 - 1152 * q^83 - 205 * q^85 - 148 * q^89 - 663 * q^91 - 345 * q^95 - 1343 * 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
$7$	−10.2621	−0.554101	−0.277050	−	0.960855i	$-0.589357\pi$
$7$	−10.2621	−0.554101	−0.277050	+	0.960855i	$0.589357\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.26209	−0.144234	−0.0721172	−	0.997396i	$-0.522976\pi$
$11$	−5.26209	−0.144234	−0.0721172	+	0.997396i	$0.522976\pi$
$12$	0	0
$13$	34.5725	0.737592	0.368796	−	0.929510i	$-0.379770\pi$
$13$	34.5725	0.737592	0.368796	+	0.929510i	$0.379770\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	74.8346	1.06765	0.533825	−	0.845595i	$-0.320754\pi$
$17$	74.8346	1.06765	0.533825	+	0.845595i	$0.320754\pi$
$18$	0	0
$19$	−50.8830	−0.614387	−0.307193	−	0.951647i	$-0.599390\pi$
$19$	−50.8830	−0.614387	−0.307193	+	0.951647i	$0.599390\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−81.4555	−0.738463	−0.369231	−	0.929338i	$-0.620379\pi$
$23$	−81.4555	−0.738463	−0.369231	+	0.929338i	$0.620379\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	152.407	0.975907	0.487953	−	0.872870i	$-0.337744\pi$
$29$	152.407	0.975907	0.487953	+	0.872870i	$0.337744\pi$
$30$	0	0
$31$	93.9797	0.544492	0.272246	−	0.962228i	$-0.412234\pi$
$31$	93.9797	0.544492	0.272246	+	0.962228i	$0.412234\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	51.3104	0.247801
$36$	0	0
$37$	−167.310	−0.743396	−0.371698	−	0.928354i	$-0.621224\pi$
$37$	−167.310	−0.743396	−0.371698	+	0.928354i	$0.621224\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	18.6209	0.0709291	0.0354645	−	0.999371i	$-0.488709\pi$
$41$	18.6209	0.0709291	0.0354645	+	0.999371i	$0.488709\pi$
$42$	0	0
$43$	133.552	0.473640	0.236820	−	0.971554i	$-0.423895\pi$
$43$	133.552	0.473640	0.236820	+	0.971554i	$0.423895\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	46.8346	0.145352	0.0726759	−	0.997356i	$-0.476846\pi$
$47$	46.8346	0.145352	0.0726759	+	0.997356i	$0.476846\pi$
$48$	0	0
$49$	−237.690	−0.692972
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−105.104	−0.272400	−0.136200	−	0.990681i	$-0.543489\pi$
$53$	−105.104	−0.272400	−0.136200	+	0.990681i	$0.543489\pi$
$54$	0	0
$55$	26.3104	0.0645036
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−77.7099	−0.171474	−0.0857370	−	0.996318i	$-0.527324\pi$
$59$	−77.7099	−0.171474	−0.0857370	+	0.996318i	$0.527324\pi$
$60$	0	0
$61$	−48.6896	−0.102198	−0.0510989	−	0.998694i	$-0.516272\pi$
$61$	−48.6896	−0.102198	−0.0510989	+	0.998694i	$0.516272\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−172.863	−0.329861
$66$	0	0
$67$	−667.132	−1.21647	−0.608233	−	0.793759i	$-0.708121\pi$
$67$	−667.132	−1.21647	−0.608233	+	0.793759i	$0.708121\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−344.056	−0.575098	−0.287549	−	0.957766i	$-0.592840\pi$
$71$	−344.056	−0.575098	−0.287549	+	0.957766i	$0.592840\pi$
$72$	0	0
$73$	−1074.70	−1.72308	−0.861539	−	0.507691i	$-0.830499\pi$
$73$	−1074.70	−1.72308	−0.861539	+	0.507691i	$0.830499\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	54.0000	0.0799204
$78$	0	0
$79$	−420.863	−0.599377	−0.299688	−	0.954037i	$-0.596883\pi$
$79$	−420.863	−0.599377	−0.299688	+	0.954037i	$0.596883\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−296.565	−0.392195	−0.196098	−	0.980584i	$-0.562827\pi$
$83$	−296.565	−0.392195	−0.196098	+	0.980584i	$0.562827\pi$
$84$	0	0
$85$	−374.173	−0.477468
$86$	0	0
$87$	0	0
$88$	0	0
$89$	950.596	1.13217	0.566084	−	0.824348i	$-0.308458\pi$
$89$	950.596	1.13217	0.566084	+	0.824348i	$0.308458\pi$
$90$	0	0
$91$	−354.786	−0.408700
$92$	0	0
$93$	0	0
$94$	0	0
$95$	254.415	0.274762
$96$	0	0
$97$	−135.916	−0.142270	−0.0711349	−	0.997467i	$-0.522662\pi$
$97$	−135.916	−0.142270	−0.0711349	+	0.997467i	$0.522662\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−392.463	−0.386649	−0.193324	−	0.981135i	$-0.561927\pi$
$101$	−392.463	−0.386649	−0.193324	+	0.981135i	$0.561927\pi$
$102$	0	0
$103$	−799.117	−0.764460	−0.382230	−	0.924067i	$-0.624844\pi$
$103$	−799.117	−0.764460	−0.382230	+	0.924067i	$0.624844\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	374.000	0.337906	0.168953	−	0.985624i	$-0.445961\pi$
$107$	374.000	0.337906	0.168953	+	0.985624i	$0.445961\pi$
$108$	0	0
$109$	428.361	0.376418	0.188209	−	0.982129i	$-0.439732\pi$
$109$	428.361	0.376418	0.188209	+	0.982129i	$0.439732\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	125.649	0.104602	0.0523011	−	0.998631i	$-0.483344\pi$
$113$	125.649	0.104602	0.0523011	+	0.998631i	$0.483344\pi$
$114$	0	0
$115$	407.277	0.330251
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−767.959	−0.591586
$120$	0	0
$121$	−1303.31	−0.979196
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	358.830	0.250716	0.125358	−	0.992112i	$-0.459992\pi$
$127$	358.830	0.250716	0.125358	+	0.992112i	$0.459992\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2084.15	−1.39002	−0.695011	−	0.718999i	$-0.744600\pi$
$131$	−2084.15	−1.39002	−0.695011	+	0.718999i	$0.744600\pi$
$132$	0	0
$133$	522.165	0.340432
$134$	0	0
$135$	0	0
$136$	0	0
$137$	575.517	0.358903	0.179451	−	0.983767i	$-0.442568\pi$
$137$	575.517	0.358903	0.179451	+	0.983767i	$0.442568\pi$
$138$	0	0
$139$	−890.287	−0.543260	−0.271630	−	0.962402i	$-0.587563\pi$
$139$	−890.287	−0.543260	−0.271630	+	0.962402i	$0.587563\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−181.924	−0.106386
$144$	0	0
$145$	−762.036	−0.436439
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1361.89	−0.748794	−0.374397	−	0.927269i	$-0.622150\pi$
$149$	−1361.89	−0.748794	−0.374397	+	0.927269i	$0.622150\pi$
$150$	0	0
$151$	−3154.72	−1.70018	−0.850090	−	0.526638i	$-0.823452\pi$
$151$	−3154.72	−1.70018	−0.850090	+	0.526638i	$0.823452\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−469.898	−0.243504
$156$	0	0
$157$	−1876.99	−0.954139	−0.477070	−	0.878865i	$-0.658301\pi$
$157$	−1876.99	−0.954139	−0.477070	+	0.878865i	$0.658301\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	835.903	0.409183
$162$	0	0
$163$	−1058.19	−0.508488	−0.254244	−	0.967140i	$-0.581827\pi$
$163$	−1058.19	−0.508488	−0.254244	+	0.967140i	$0.581827\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2391.59	1.10819	0.554093	−	0.832455i	$-0.313065\pi$
$167$	2391.59	1.10819	0.554093	+	0.832455i	$0.313065\pi$
$168$	0	0
$169$	−1001.74	−0.455958
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3957.46	1.73919	0.869595	−	0.493766i	$-0.164380\pi$
$173$	3957.46	1.73919	0.869595	+	0.493766i	$0.164380\pi$
$174$	0	0
$175$	−256.552	−0.110820
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1236.91	−0.516486	−0.258243	−	0.966080i	$-0.583144\pi$
$179$	−1236.91	−0.516486	−0.258243	+	0.966080i	$0.583144\pi$
$180$	0	0
$181$	−361.061	−0.148273	−0.0741366	−	0.997248i	$-0.523620\pi$
$181$	−361.061	−0.148273	−0.0741366	+	0.997248i	$0.523620\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	836.552	0.332457
$186$	0	0
$187$	−393.786	−0.153992
$188$	0	0
$189$	0	0
$190$	0	0
$191$	575.135	0.217881	0.108941	−	0.994048i	$-0.465254\pi$
$191$	575.135	0.217881	0.108941	+	0.994048i	$0.465254\pi$
$192$	0	0
$193$	−1755.28	−0.654652	−0.327326	−	0.944912i	$-0.606147\pi$
$193$	−1755.28	−0.654652	−0.327326	+	0.944912i	$0.606147\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2182.56	0.789347	0.394673	−	0.918821i	$-0.370858\pi$
$197$	2182.56	0.789347	0.394673	+	0.918821i	$0.370858\pi$
$198$	0	0
$199$	−2946.07	−1.04945	−0.524727	−	0.851271i	$-0.675833\pi$
$199$	−2946.07	−1.04945	−0.524727	+	0.851271i	$0.675833\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1564.02	−0.540751
$204$	0	0
$205$	−93.1044	−0.0317204
$206$	0	0
$207$	0	0
$208$	0	0
$209$	267.751	0.0886158
$210$	0	0
$211$	−2081.63	−0.679171	−0.339585	−	0.940575i	$-0.610287\pi$
$211$	−2081.63	−0.679171	−0.339585	+	0.940575i	$0.610287\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−667.761	−0.211818
$216$	0	0
$217$	−964.427	−0.301703
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2587.22	0.787490
$222$	0	0
$223$	591.288	0.177559	0.0887793	−	0.996051i	$-0.471703\pi$
$223$	591.288	0.177559	0.0887793	+	0.996051i	$0.471703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4369.86	−1.27770	−0.638849	−	0.769332i	$-0.720589\pi$
$227$	−4369.86	−1.27770	−0.638849	+	0.769332i	$0.720589\pi$
$228$	0	0
$229$	−3580.09	−1.03310	−0.516548	−	0.856258i	$-0.672783\pi$
$229$	−3580.09	−1.03310	−0.516548	+	0.856258i	$0.672783\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3853.73	−1.08355	−0.541773	−	0.840525i	$-0.682247\pi$
$233$	−3853.73	−1.08355	−0.541773	+	0.840525i	$0.682247\pi$
$234$	0	0
$235$	−234.173	−0.0650033
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2519.71	−0.681952	−0.340976	−	0.940072i	$-0.610757\pi$
$239$	−2519.71	−0.681952	−0.340976	+	0.940072i	$0.610757\pi$
$240$	0	0
$241$	3498.30	0.935044	0.467522	−	0.883982i	$-0.345147\pi$
$241$	3498.30	0.935044	0.467522	+	0.883982i	$0.345147\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1188.45	0.309907
$246$	0	0
$247$	−1759.15	−0.453167
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1298.83	0.326619	0.163310	−	0.986575i	$-0.447783\pi$
$251$	1298.83	0.326619	0.163310	+	0.986575i	$0.447783\pi$
$252$	0	0
$253$	428.626	0.106512
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4682.87	1.13661	0.568306	−	0.822818i	$-0.307599\pi$
$257$	4682.87	1.13661	0.568306	+	0.822818i	$0.307599\pi$
$258$	0	0
$259$	1716.95	0.411916
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4137.19	−0.970000	−0.485000	−	0.874514i	$-0.661180\pi$
$263$	−4137.19	−0.970000	−0.485000	+	0.874514i	$0.661180\pi$
$264$	0	0
$265$	525.522	0.121821
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−851.771	−0.193061	−0.0965305	−	0.995330i	$-0.530775\pi$
$269$	−851.771	−0.193061	−0.0965305	+	0.995330i	$0.530775\pi$
$270$	0	0
$271$	3045.15	0.682581	0.341291	−	0.939958i	$-0.389136\pi$
$271$	3045.15	0.682581	0.341291	+	0.939958i	$0.389136\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−131.552	−0.0288469
$276$	0	0
$277$	−1512.69	−0.328118	−0.164059	−	0.986451i	$-0.552459\pi$
$277$	−1512.69	−0.328118	−0.164059	+	0.986451i	$0.552459\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1158.06	0.245850	0.122925	−	0.992416i	$-0.460773\pi$
$281$	1158.06	0.245850	0.122925	+	0.992416i	$0.460773\pi$
$282$	0	0
$283$	−8650.56	−1.81704	−0.908520	−	0.417842i	$-0.862787\pi$
$283$	−8650.56	−1.81704	−0.908520	+	0.417842i	$0.862787\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−191.089	−0.0393018
$288$	0	0
$289$	687.219	0.139878
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5920.13	1.18040	0.590201	−	0.807257i	$-0.299049\pi$
$293$	5920.13	1.18040	0.590201	+	0.807257i	$0.299049\pi$
$294$	0	0
$295$	388.550	0.0766855
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2816.12	−0.544684
$300$	0	0
$301$	−1370.52	−0.262444
$302$	0	0
$303$	0	0
$304$	0	0
$305$	243.448	0.0457042
$306$	0	0
$307$	1906.71	0.354469	0.177234	−	0.984169i	$-0.443285\pi$
$307$	1906.71	0.354469	0.177234	+	0.984169i	$0.443285\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6750.08	−1.23075	−0.615373	−	0.788236i	$-0.710994\pi$
$311$	−6750.08	−1.23075	−0.615373	+	0.788236i	$0.710994\pi$
$312$	0	0
$313$	2225.44	0.401883	0.200942	−	0.979603i	$-0.435600\pi$
$313$	2225.44	0.401883	0.200942	+	0.979603i	$0.435600\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7761.66	1.37520	0.687600	−	0.726090i	$-0.258664\pi$
$317$	7761.66	1.37520	0.687600	+	0.726090i	$0.258664\pi$
$318$	0	0
$319$	−801.980	−0.140759
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3807.81	−0.655951
$324$	0	0
$325$	864.313	0.147518
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−480.621	−0.0805395
$330$	0	0
$331$	6513.99	1.08170	0.540848	−	0.841121i	$-0.318104\pi$
$331$	6513.99	1.08170	0.540848	+	0.841121i	$0.318104\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3335.66	0.544020
$336$	0	0
$337$	−3625.89	−0.586097	−0.293049	−	0.956098i	$-0.594670\pi$
$337$	−3625.89	−0.586097	−0.293049	+	0.956098i	$0.594670\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−494.529	−0.0785345
$342$	0	0
$343$	5959.09	0.938077
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4830.55	−0.747313	−0.373656	−	0.927567i	$-0.621896\pi$
$347$	−4830.55	−0.747313	−0.373656	+	0.927567i	$0.621896\pi$
$348$	0	0
$349$	9300.98	1.42656	0.713281	−	0.700878i	$-0.247208\pi$
$349$	9300.98	1.42656	0.713281	+	0.700878i	$0.247208\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7446.44	−1.12276	−0.561380	−	0.827558i	$-0.689729\pi$
$353$	−7446.44	−1.12276	−0.561380	+	0.827558i	$0.689729\pi$
$354$	0	0
$355$	1720.28	0.257192
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5872.73	−0.863373	−0.431686	−	0.902024i	$-0.642081\pi$
$359$	−5872.73	−0.863373	−0.431686	+	0.902024i	$0.642081\pi$
$360$	0	0
$361$	−4269.92	−0.622529
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5373.52	0.770584
$366$	0	0
$367$	8708.80	1.23868	0.619340	−	0.785123i	$-0.287400\pi$
$367$	8708.80	1.23868	0.619340	+	0.785123i	$0.287400\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1078.59	0.150937
$372$	0	0
$373$	4355.98	0.604676	0.302338	−	0.953201i	$-0.402233\pi$
$373$	4355.98	0.604676	0.302338	+	0.953201i	$0.402233\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5269.10	0.719821
$378$	0	0
$379$	5116.37	0.693431	0.346716	−	0.937970i	$-0.387297\pi$
$379$	5116.37	0.693431	0.346716	+	0.937970i	$0.387297\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2906.74	−0.387801	−0.193900	−	0.981021i	$-0.562114\pi$
$383$	−2906.74	−0.387801	−0.193900	+	0.981021i	$0.562114\pi$
$384$	0	0
$385$	−270.000	−0.0357415
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1446.15	0.188491	0.0942453	−	0.995549i	$-0.469956\pi$
$389$	1446.15	0.188491	0.0942453	+	0.995549i	$0.469956\pi$
$390$	0	0
$391$	−6095.69	−0.788420
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2104.31	0.268049
$396$	0	0
$397$	1377.59	0.174154	0.0870770	−	0.996202i	$-0.472247\pi$
$397$	1377.59	0.174154	0.0870770	+	0.996202i	$0.472247\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6094.96	0.759022	0.379511	−	0.925187i	$-0.376092\pi$
$401$	6094.96	0.759022	0.379511	+	0.925187i	$0.376092\pi$
$402$	0	0
$403$	3249.11	0.401613
$404$	0	0
$405$	0	0
$406$	0	0
$407$	880.402	0.107223
$408$	0	0
$409$	−10329.4	−1.24879	−0.624394	−	0.781109i	$-0.714654\pi$
$409$	−10329.4	−1.24879	−0.624394	+	0.781109i	$0.714654\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	797.466	0.0950139
$414$	0	0
$415$	1482.82	0.175395
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7525.80	−0.877468	−0.438734	−	0.898617i	$-0.644573\pi$
$419$	−7525.80	−0.877468	−0.438734	+	0.898617i	$0.644573\pi$
$420$	0	0
$421$	9532.73	1.10356	0.551778	−	0.833991i	$-0.313950\pi$
$421$	9532.73	1.10356	0.551778	+	0.833991i	$0.313950\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1870.87	0.213530
$426$	0	0
$427$	499.657	0.0566278
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1909.54	0.213409	0.106705	−	0.994291i	$-0.465970\pi$
$431$	1909.54	0.213409	0.106705	+	0.994291i	$0.465970\pi$
$432$	0	0
$433$	7780.10	0.863482	0.431741	−	0.901998i	$-0.357899\pi$
$433$	7780.10	0.863482	0.431741	+	0.901998i	$0.357899\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4144.70	0.453702
$438$	0	0
$439$	11480.5	1.24815	0.624074	−	0.781365i	$-0.285476\pi$
$439$	11480.5	1.24815	0.624074	+	0.781365i	$0.285476\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1562.80	0.167609	0.0838046	−	0.996482i	$-0.473293\pi$
$443$	1562.80	0.167609	0.0838046	+	0.996482i	$0.473293\pi$
$444$	0	0
$445$	−4752.98	−0.506321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7086.91	−0.744882	−0.372441	−	0.928056i	$-0.621479\pi$
$449$	−7086.91	−0.744882	−0.372441	+	0.928056i	$0.621479\pi$
$450$	0	0
$451$	−97.9847	−0.0102304
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1773.93	0.182776
$456$	0	0
$457$	14548.5	1.48917	0.744587	−	0.667526i	$-0.232647\pi$
$457$	14548.5	1.48917	0.744587	+	0.667526i	$0.232647\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2272.13	0.229553	0.114776	−	0.993391i	$-0.463385\pi$
$461$	2272.13	0.229553	0.114776	+	0.993391i	$0.463385\pi$
$462$	0	0
$463$	−2757.88	−0.276824	−0.138412	−	0.990375i	$-0.544200\pi$
$463$	−2757.88	−0.276824	−0.138412	+	0.990375i	$0.544200\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17714.6	1.75532	0.877661	−	0.479282i	$-0.159103\pi$
$467$	17714.6	1.75532	0.877661	+	0.479282i	$0.159103\pi$
$468$	0	0
$469$	6846.17	0.674044
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−702.763	−0.0683152
$474$	0	0
$475$	−1272.07	−0.122877
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9413.68	−0.897959	−0.448979	−	0.893542i	$-0.648212\pi$
$479$	−9413.68	−0.897959	−0.448979	+	0.893542i	$0.648212\pi$
$480$	0	0
$481$	−5784.34	−0.548323
$482$	0	0
$483$	0	0
$484$	0	0
$485$	679.580	0.0636250
$486$	0	0
$487$	−4065.97	−0.378330	−0.189165	−	0.981945i	$-0.560578\pi$
$487$	−4065.97	−0.378330	−0.189165	+	0.981945i	$0.560578\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8984.27	−0.825773	−0.412886	−	0.910783i	$-0.635479\pi$
$491$	−8984.27	−0.825773	−0.412886	+	0.910783i	$0.635479\pi$
$492$	0	0
$493$	11405.3	1.04193
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3530.73	0.318662
$498$	0	0
$499$	4151.04	0.372397	0.186199	−	0.982512i	$-0.440383\pi$
$499$	4151.04	0.372397	0.186199	+	0.982512i	$0.440383\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2721.08	−0.241207	−0.120603	−	0.992701i	$-0.538483\pi$
$503$	−2721.08	−0.241207	−0.120603	+	0.992701i	$0.538483\pi$
$504$	0	0
$505$	1962.32	0.172915
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6571.92	0.572289	0.286145	−	0.958186i	$-0.407626\pi$
$509$	6571.92	0.572289	0.286145	+	0.958186i	$0.407626\pi$
$510$	0	0
$511$	11028.7	0.954759
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3995.59	0.341877
$516$	0	0
$517$	−246.448	−0.0209647
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4049.20	−0.340496	−0.170248	−	0.985401i	$-0.554457\pi$
$521$	−4049.20	−0.340496	−0.170248	+	0.985401i	$0.554457\pi$
$522$	0	0
$523$	−3872.83	−0.323799	−0.161900	−	0.986807i	$-0.551762\pi$
$523$	−3872.83	−0.323799	−0.161900	+	0.986807i	$0.551762\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7032.93	0.581327
$528$	0	0
$529$	−5532.00	−0.454673
$530$	0	0
$531$	0	0
$532$	0	0
$533$	643.771	0.0523167
$534$	0	0
$535$	−1870.00	−0.151116
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1250.74	0.0999505
$540$	0	0
$541$	5476.29	0.435202	0.217601	−	0.976038i	$-0.430177\pi$
$541$	5476.29	0.435202	0.217601	+	0.976038i	$0.430177\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2141.81	−0.168339
$546$	0	0
$547$	15867.7	1.24032	0.620158	−	0.784477i	$-0.287068\pi$
$547$	15867.7	1.24032	0.620158	+	0.784477i	$0.287068\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7754.93	−0.599584
$552$	0	0
$553$	4318.93	0.332115
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6121.57	0.465672	0.232836	−	0.972516i	$-0.425200\pi$
$557$	6121.57	0.465672	0.232836	+	0.972516i	$0.425200\pi$
$558$	0	0
$559$	4617.24	0.349353
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7405.33	−0.554347	−0.277174	−	0.960820i	$-0.589398\pi$
$563$	−7405.33	−0.554347	−0.277174	+	0.960820i	$0.589398\pi$
$564$	0	0
$565$	−628.244	−0.0467795
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20196.3	−1.48800	−0.744000	−	0.668180i	$-0.767074\pi$
$569$	−20196.3	−1.48800	−0.744000	+	0.668180i	$0.767074\pi$
$570$	0	0
$571$	5341.83	0.391504	0.195752	−	0.980653i	$-0.437285\pi$
$571$	5341.83	0.391504	0.195752	+	0.980653i	$0.437285\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2036.39	−0.147693
$576$	0	0
$577$	21443.1	1.54712	0.773561	−	0.633722i	$-0.218474\pi$
$577$	21443.1	1.54712	0.773561	+	0.633722i	$0.218474\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3043.37	0.217316
$582$	0	0
$583$	553.068	0.0392895
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18441.6	1.29670	0.648352	−	0.761341i	$-0.275459\pi$
$587$	18441.6	1.29670	0.648352	+	0.761341i	$0.275459\pi$
$588$	0	0
$589$	−4781.96	−0.334529
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6213.39	−0.430276	−0.215138	−	0.976584i	$-0.569020\pi$
$593$	−6213.39	−0.430276	−0.215138	+	0.976584i	$0.569020\pi$
$594$	0	0
$595$	3839.80	0.264565
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22782.0	−1.55400	−0.777000	−	0.629501i	$-0.783259\pi$
$599$	−22782.0	−1.55400	−0.777000	+	0.629501i	$0.783259\pi$
$600$	0	0
$601$	14788.6	1.00373	0.501865	−	0.864946i	$-0.332648\pi$
$601$	14788.6	1.00373	0.501865	+	0.864946i	$0.332648\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6516.55	0.437910
$606$	0	0
$607$	−18559.8	−1.24105	−0.620525	−	0.784186i	$-0.713081\pi$
$607$	−18559.8	−1.24105	−0.620525	+	0.784186i	$0.713081\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1619.19	0.107210
$612$	0	0
$613$	2519.83	0.166028	0.0830138	−	0.996548i	$-0.473545\pi$
$613$	2519.83	0.166028	0.0830138	+	0.996548i	$0.473545\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11012.6	0.718561	0.359281	−	0.933230i	$-0.383022\pi$
$617$	11012.6	0.718561	0.359281	+	0.933230i	$0.383022\pi$
$618$	0	0
$619$	19538.6	1.26870	0.634349	−	0.773047i	$-0.281268\pi$
$619$	19538.6	1.26870	0.634349	+	0.773047i	$0.281268\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9755.09	−0.627335
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12520.6	−0.793688
$630$	0	0
$631$	8676.78	0.547412	0.273706	−	0.961813i	$-0.411750\pi$
$631$	8676.78	0.547412	0.273706	+	0.961813i	$0.411750\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1794.15	−0.112124
$636$	0	0
$637$	−8217.53	−0.511131
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7988.18	0.492222	0.246111	−	0.969242i	$-0.420847\pi$
$641$	7988.18	0.492222	0.246111	+	0.969242i	$0.420847\pi$
$642$	0	0
$643$	−2200.43	−0.134956	−0.0674779	−	0.997721i	$-0.521495\pi$
$643$	−2200.43	−0.134956	−0.0674779	+	0.997721i	$0.521495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31423.1	1.90938	0.954691	−	0.297597i	$-0.0961853\pi$
$647$	31423.1	1.90938	0.954691	+	0.297597i	$0.0961853\pi$
$648$	0	0
$649$	408.916	0.0247325
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8769.09	0.525515	0.262757	−	0.964862i	$-0.415368\pi$
$653$	8769.09	0.525515	0.262757	+	0.964862i	$0.415368\pi$
$654$	0	0
$655$	10420.7	0.621637
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28986.3	−1.71342	−0.856712	−	0.515795i	$-0.827497\pi$
$659$	−28986.3	−1.71342	−0.856712	+	0.515795i	$0.827497\pi$
$660$	0	0
$661$	3755.04	0.220959	0.110480	−	0.993878i	$-0.464761\pi$
$661$	3755.04	0.220959	0.110480	+	0.993878i	$0.464761\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2610.83	−0.152246
$666$	0	0
$667$	−12414.4	−0.720671
$668$	0	0
$669$	0	0
$670$	0	0
$671$	256.209	0.0147404
$672$	0	0
$673$	3404.61	0.195004	0.0975022	−	0.995235i	$-0.468915\pi$
$673$	3404.61	0.195004	0.0975022	+	0.995235i	$0.468915\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21691.9	1.23144	0.615721	−	0.787964i	$-0.288865\pi$
$677$	21691.9	1.23144	0.615721	+	0.787964i	$0.288865\pi$
$678$	0	0
$679$	1394.78	0.0788318
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6003.49	−0.336335	−0.168168	−	0.985758i	$-0.553785\pi$
$683$	−6003.49	−0.336335	−0.168168	+	0.985758i	$0.553785\pi$
$684$	0	0
$685$	−2877.58	−0.160506
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3633.72	−0.200920
$690$	0	0
$691$	−22090.1	−1.21613	−0.608065	−	0.793887i	$-0.708054\pi$
$691$	−22090.1	−1.21613	−0.608065	+	0.793887i	$0.708054\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4451.44	0.242953
$696$	0	0
$697$	1393.49	0.0757275
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24940.0	−1.34375	−0.671876	−	0.740663i	$-0.734511\pi$
$701$	−24940.0	−1.34375	−0.671876	+	0.740663i	$0.734511\pi$
$702$	0	0
$703$	8513.25	0.456733
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4027.49	0.214242
$708$	0	0
$709$	−20007.0	−1.05977	−0.529886	−	0.848069i	$-0.677765\pi$
$709$	−20007.0	−1.05977	−0.529886	+	0.848069i	$0.677765\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7655.16	−0.402087
$714$	0	0
$715$	909.618	0.0475773
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16188.8	0.839692	0.419846	−	0.907595i	$-0.362084\pi$
$719$	16188.8	0.839692	0.419846	+	0.907595i	$0.362084\pi$
$720$	0	0
$721$	8200.61	0.423588
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3810.18	0.195181
$726$	0	0
$727$	−7264.04	−0.370575	−0.185288	−	0.982684i	$-0.559322\pi$
$727$	−7264.04	−0.370575	−0.185288	+	0.982684i	$0.559322\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9994.33	0.505682
$732$	0	0
$733$	−27522.4	−1.38685	−0.693426	−	0.720528i	$-0.743900\pi$
$733$	−27522.4	−1.38685	−0.693426	+	0.720528i	$0.743900\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3510.51	0.175456
$738$	0	0
$739$	−2936.48	−0.146171	−0.0730853	−	0.997326i	$-0.523285\pi$
$739$	−2936.48	−0.146171	−0.0730853	+	0.997326i	$0.523285\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8395.29	−0.414527	−0.207263	−	0.978285i	$-0.566456\pi$
$743$	−8395.29	−0.414527	−0.207263	+	0.978285i	$0.566456\pi$
$744$	0	0
$745$	6809.44	0.334871
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3838.02	−0.187234
$750$	0	0
$751$	−35222.9	−1.71145	−0.855727	−	0.517428i	$-0.826890\pi$
$751$	−35222.9	−1.71145	−0.855727	+	0.517428i	$0.826890\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15773.6	0.760343
$756$	0	0
$757$	−1328.71	−0.0637952	−0.0318976	−	0.999491i	$-0.510155\pi$
$757$	−1328.71	−0.0637952	−0.0318976	+	0.999491i	$0.510155\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39056.4	−1.86044	−0.930220	−	0.367003i	$-0.880384\pi$
$761$	−39056.4	−1.86044	−0.930220	+	0.367003i	$0.880384\pi$
$762$	0	0
$763$	−4395.88	−0.208574
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2686.63	−0.126478
$768$	0	0
$769$	−10509.2	−0.492813	−0.246406	−	0.969167i	$-0.579250\pi$
$769$	−10509.2	−0.492813	−0.246406	+	0.969167i	$0.579250\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20180.8	−0.939009	−0.469505	−	0.882930i	$-0.655567\pi$
$773$	−20180.8	−0.939009	−0.469505	+	0.882930i	$0.655567\pi$
$774$	0	0
$775$	2349.49	0.108898
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−947.485	−0.0435779
$780$	0	0
$781$	1810.45	0.0829489
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9384.94	0.426704
$786$	0	0
$787$	−19486.2	−0.882604	−0.441302	−	0.897359i	$-0.645483\pi$
$787$	−19486.2	−0.882604	−0.441302	+	0.897359i	$0.645483\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1289.42	−0.0579602
$792$	0	0
$793$	−1683.32	−0.0753802
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38955.5	1.73134	0.865668	−	0.500619i	$-0.166894\pi$
$797$	38955.5	1.73134	0.865668	+	0.500619i	$0.166894\pi$
$798$	0	0
$799$	3504.85	0.155185
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5655.19	0.248527
$804$	0	0
$805$	−4179.52	−0.182992
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−41073.8	−1.78501	−0.892507	−	0.451033i	$-0.851056\pi$
$809$	−41073.8	−1.78501	−0.892507	+	0.451033i	$0.851056\pi$
$810$	0	0
$811$	7397.33	0.320290	0.160145	−	0.987093i	$-0.448804\pi$
$811$	7397.33	0.320290	0.160145	+	0.987093i	$0.448804\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5290.93	0.227403
$816$	0	0
$817$	−6795.53	−0.290998
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20477.0	0.870465	0.435233	−	0.900318i	$-0.356666\pi$
$821$	20477.0	0.870465	0.435233	+	0.900318i	$0.356666\pi$
$822$	0	0
$823$	−12109.6	−0.512896	−0.256448	−	0.966558i	$-0.582552\pi$
$823$	−12109.6	−0.512896	−0.256448	+	0.966558i	$0.582552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34564.6	1.45336	0.726680	−	0.686976i	$-0.241062\pi$
$827$	34564.6	1.45336	0.726680	+	0.686976i	$0.241062\pi$
$828$	0	0
$829$	−34571.8	−1.44841	−0.724203	−	0.689587i	$-0.757792\pi$
$829$	−34571.8	−1.44841	−0.724203	+	0.689587i	$0.757792\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−17787.4	−0.739852
$834$	0	0
$835$	−11958.0	−0.495596
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13192.9	0.542870	0.271435	−	0.962457i	$-0.412502\pi$
$839$	13192.9	0.542870	0.271435	+	0.962457i	$0.412502\pi$
$840$	0	0
$841$	−1161.07	−0.0476061
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5008.70	0.203911
$846$	0	0
$847$	13374.7	0.542573
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13628.4	0.548971
$852$	0	0
$853$	25796.2	1.03546	0.517729	−	0.855545i	$-0.326778\pi$
$853$	25796.2	1.03546	0.517729	+	0.855545i	$0.326778\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26349.2	−1.05026	−0.525130	−	0.851022i	$-0.675983\pi$
$857$	−26349.2	−1.05026	−0.525130	+	0.851022i	$0.675983\pi$
$858$	0	0
$859$	13466.3	0.534884	0.267442	−	0.963574i	$-0.413822\pi$
$859$	13466.3	0.534884	0.267442	+	0.963574i	$0.413822\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44430.9	1.75254	0.876272	−	0.481816i	$-0.160023\pi$
$863$	44430.9	1.75254	0.876272	+	0.481816i	$0.160023\pi$
$864$	0	0
$865$	−19787.3	−0.777789
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2214.62	0.0864507
$870$	0	0
$871$	−23064.5	−0.897255
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1282.76	0.0495603
$876$	0	0
$877$	13993.4	0.538797	0.269398	−	0.963029i	$-0.413175\pi$
$877$	13993.4	0.538797	0.269398	+	0.963029i	$0.413175\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−49227.2	−1.88253	−0.941264	−	0.337672i	$-0.890361\pi$
$881$	−49227.2	−1.88253	−0.941264	+	0.337672i	$0.890361\pi$
$882$	0	0
$883$	−44724.5	−1.70453	−0.852264	−	0.523112i	$-0.824771\pi$
$883$	−44724.5	−1.70453	−0.852264	+	0.523112i	$0.824771\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20387.9	0.771769	0.385885	−	0.922547i	$-0.373896\pi$
$887$	20387.9	0.771769	0.385885	+	0.922547i	$0.373896\pi$
$888$	0	0
$889$	−3682.34	−0.138922
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2383.08	−0.0893022
$894$	0	0
$895$	6184.55	0.230980
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14323.2	0.531373
$900$	0	0
$901$	−7865.44	−0.290828
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1805.31	0.0663098
$906$	0	0
$907$	19788.7	0.724446	0.362223	−	0.932091i	$-0.382018\pi$
$907$	19788.7	0.724446	0.362223	+	0.932091i	$0.382018\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11895.2	0.432607	0.216304	−	0.976326i	$-0.430600\pi$
$911$	11895.2	0.432607	0.216304	+	0.976326i	$0.430600\pi$
$912$	0	0
$913$	1560.55	0.0565681
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21387.7	0.770212
$918$	0	0
$919$	21020.2	0.754507	0.377253	−	0.926110i	$-0.376869\pi$
$919$	21020.2	0.754507	0.377253	+	0.926110i	$0.376869\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11894.9	−0.424187
$924$	0	0
$925$	−4182.76	−0.148679
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50942.1	−1.79909	−0.899546	−	0.436827i	$-0.856102\pi$
$929$	−50942.1	−1.79909	−0.899546	+	0.436827i	$0.856102\pi$
$930$	0	0
$931$	12094.3	0.425753
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1968.93	0.0688673
$936$	0	0
$937$	−29785.2	−1.03846	−0.519232	−	0.854633i	$-0.673782\pi$
$937$	−29785.2	−1.03846	−0.519232	+	0.854633i	$0.673782\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32940.7	1.14116	0.570582	−	0.821240i	$-0.306718\pi$
$941$	32940.7	1.14116	0.570582	+	0.821240i	$0.306718\pi$
$942$	0	0
$943$	−1516.77	−0.0523785
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14141.4	0.485253	0.242626	−	0.970120i	$-0.421991\pi$
$947$	14141.4	0.485253	0.242626	+	0.970120i	$0.421991\pi$
$948$	0	0
$949$	−37155.3	−1.27093
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−35528.4	−1.20764	−0.603819	−	0.797122i	$-0.706355\pi$
$953$	−35528.4	−1.20764	−0.603819	+	0.797122i	$0.706355\pi$
$954$	0	0
$955$	−2875.68	−0.0974394
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5906.00	−0.198868
$960$	0	0
$961$	−20958.8	−0.703529
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8776.40	0.292769
$966$	0	0
$967$	2325.32	0.0773290	0.0386645	−	0.999252i	$-0.487690\pi$
$967$	2325.32	0.0773290	0.0386645	+	0.999252i	$0.487690\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35146.5	−1.16159	−0.580796	−	0.814049i	$-0.697259\pi$
$971$	−35146.5	−1.16159	−0.580796	+	0.814049i	$0.697259\pi$
$972$	0	0
$973$	9136.21	0.301021
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35451.4	1.16089	0.580445	−	0.814299i	$-0.302878\pi$
$977$	35451.4	1.16089	0.580445	+	0.814299i	$0.302878\pi$
$978$	0	0
$979$	−5002.12	−0.163298
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29728.4	−0.964586	−0.482293	−	0.876010i	$-0.660196\pi$
$983$	−29728.4	−0.964586	−0.482293	+	0.876010i	$0.660196\pi$
$984$	0	0
$985$	−10912.8	−0.353007
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10878.6	−0.349765
$990$	0	0
$991$	−37667.8	−1.20742	−0.603712	−	0.797202i	$-0.706312\pi$
$991$	−37667.8	−1.20742	−0.603712	+	0.797202i	$0.706312\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14730.3	0.469330
$996$	0	0
$997$	−37460.1	−1.18994	−0.594972	−	0.803746i	$-0.702837\pi$
$997$	−37460.1	−1.18994	−0.594972	+	0.803746i	$0.702837\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.a.1.1	✓	2
3.2	odd	2		1080.4.a.b.1.1	yes	2
4.3	odd	2		2160.4.a.y.1.2		2
12.11	even	2		2160.4.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.a.1.1	✓	2	1.1	even	1	trivial
1080.4.a.b.1.1	yes	2	3.2	odd	2
2160.4.a.y.1.2		2	4.3	odd	2
2160.4.a.bd.1.2		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -10.2621 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -10.2621 q^{7} -5.26209 q^{11} +34.5725 q^{13} +74.8346 q^{17} -50.8830 q^{19} -81.4555 q^{23} +25.0000 q^{25} +152.407 q^{29} +93.9797 q^{31} +51.3104 q^{35} -167.310 q^{37} +18.6209 q^{41} +133.552 q^{43} +46.8346 q^{47} -237.690 q^{49} -105.104 q^{53} +26.3104 q^{55} -77.7099 q^{59} -48.6896 q^{61} -172.863 q^{65} -667.132 q^{67} -344.056 q^{71} -1074.70 q^{73} +54.0000 q^{77} -420.863 q^{79} -296.565 q^{83} -374.173 q^{85} +950.596 q^{89} -354.786 q^{91} +254.415 q^{95} -135.916 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} - 5 q^{7}+O(q^{10}) \) 2 * q - 10 * q^5 - 5 * q^7	\( 2 q - 10 q^{5} - 5 q^{7} + 5 q^{11} - 24 q^{13} + 41 q^{17} + 69 q^{19} + 101 q^{23} + 50 q^{25} + 103 q^{29} - 107 q^{31} + 25 q^{35} - 257 q^{37} - 118 q^{41} - 121 q^{43} - 15 q^{47} - 553 q^{49} + 566 q^{53} - 25 q^{55} - 528 q^{59} - 175 q^{61} + 120 q^{65} - 201 q^{67} + 26 q^{71} - 923 q^{73} + 108 q^{77} - 376 q^{79} - 1152 q^{83} - 205 q^{85} - 148 q^{89} - 663 q^{91} - 345 q^{95} - 1343 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 - 5 * q^7 + 5 * q^11 - 24 * q^13 + 41 * q^17 + 69 * q^19 + 101 * q^23 + 50 * q^25 + 103 * q^29 - 107 * q^31 + 25 * q^35 - 257 * q^37 - 118 * q^41 - 121 * q^43 - 15 * q^47 - 553 * q^49 + 566 * q^53 - 25 * q^55 - 528 * q^59 - 175 * q^61 + 120 * q^65 - 201 * q^67 + 26 * q^71 - 923 * q^73 + 108 * q^77 - 376 * q^79 - 1152 * q^83 - 205 * q^85 - 148 * q^89 - 663 * q^91 - 345 * q^95 - 1343 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more