LMFDB - Embedded newform 1080.6.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,6,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, 6, names="a")

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

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
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:	$173.214525398$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.15881.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 29x - 55 $ x^3 - 29*x - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.33626$ 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)

$f(q)$	$=$	$q+25.0000 q^{5} +85.5605 q^{7} +O(q^{10})$	$q+25.0000 q^{5} +85.5605 q^{7} -99.8209 q^{11} -546.179 q^{13} -424.929 q^{17} +2122.20 q^{19} -3450.04 q^{23} +625.000 q^{25} +6876.76 q^{29} +2989.87 q^{31} +2139.01 q^{35} -14200.2 q^{37} +423.869 q^{41} -3569.67 q^{43} -928.769 q^{47} -9486.39 q^{49} -33234.4 q^{53} -2495.52 q^{55} -17830.1 q^{59} +30245.0 q^{61} -13654.5 q^{65} +20790.3 q^{67} +19209.9 q^{71} -83174.8 q^{73} -8540.73 q^{77} +42514.1 q^{79} -93576.0 q^{83} -10623.2 q^{85} +72645.6 q^{89} -46731.4 q^{91} +53055.1 q^{95} +16741.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 75 q^{5} - 30 q^{7}+O(q^{10}) $ 3 * q + 75 * q^5 - 30 * q^7	$ 3 q + 75 q^{5} - 30 q^{7} - 660 q^{11} - 1278 q^{13} + 1731 q^{17} + 1011 q^{19} + 2433 q^{23} + 1875 q^{25} + 3786 q^{29} + 4131 q^{31} - 750 q^{35} - 7122 q^{37} + 17352 q^{41} + 11556 q^{43} - 19548 q^{47} - 7857 q^{49} - 4965 q^{53} - 16500 q^{55} - 32106 q^{59} - 16317 q^{61} - 31950 q^{65} - 51258 q^{67} - 84072 q^{71} - 161892 q^{73} + 142968 q^{77} + 36441 q^{79} - 64581 q^{83} + 43275 q^{85} - 61584 q^{89} - 123588 q^{91} + 25275 q^{95} + 14760 q^{97}+O(q^{100}) $ 3 * q + 75 * q^5 - 30 * q^7 - 660 * q^11 - 1278 * q^13 + 1731 * q^17 + 1011 * q^19 + 2433 * q^23 + 1875 * q^25 + 3786 * q^29 + 4131 * q^31 - 750 * q^35 - 7122 * q^37 + 17352 * q^41 + 11556 * q^43 - 19548 * q^47 - 7857 * q^49 - 4965 * q^53 - 16500 * q^55 - 32106 * q^59 - 16317 * q^61 - 31950 * q^65 - 51258 * q^67 - 84072 * q^71 - 161892 * q^73 + 142968 * q^77 + 36441 * q^79 - 64581 * q^83 + 43275 * q^85 - 61584 * q^89 - 123588 * q^91 + 25275 * q^95 + 14760 * 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
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	85.5605	0.659976	0.329988	−	0.943985i	$-0.392955\pi$
$7$	85.5605	0.659976	0.329988	+	0.943985i	$0.392955\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−99.8209	−0.248737	−0.124368	−	0.992236i	$-0.539690\pi$
$11$	−99.8209	−0.248737	−0.124368	+	0.992236i	$0.539690\pi$
$12$	0	0
$13$	−546.179	−0.896348	−0.448174	−	0.893946i	$-0.647926\pi$
$13$	−546.179	−0.896348	−0.448174	+	0.893946i	$0.647926\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−424.929	−0.356610	−0.178305	−	0.983975i	$-0.557061\pi$
$17$	−424.929	−0.356610	−0.178305	+	0.983975i	$0.557061\pi$
$18$	0	0
$19$	2122.20	1.34866	0.674331	−	0.738429i	$-0.264432\pi$
$19$	2122.20	1.34866	0.674331	+	0.738429i	$0.264432\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3450.04	−1.35989	−0.679946	−	0.733262i	$-0.737997\pi$
$23$	−3450.04	−1.35989	−0.679946	+	0.733262i	$0.737997\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6876.76	1.51841	0.759205	−	0.650852i	$-0.225588\pi$
$29$	6876.76	1.51841	0.759205	+	0.650852i	$0.225588\pi$
$30$	0	0
$31$	2989.87	0.558790	0.279395	−	0.960176i	$-0.409866\pi$
$31$	2989.87	0.558790	0.279395	+	0.960176i	$0.409866\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2139.01	0.295150
$36$	0	0
$37$	−14200.2	−1.70526	−0.852629	−	0.522517i	$-0.824993\pi$
$37$	−14200.2	−1.70526	−0.852629	+	0.522517i	$0.824993\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	423.869	0.0393797	0.0196898	−	0.999806i	$-0.493732\pi$
$41$	423.869	0.0393797	0.0196898	+	0.999806i	$0.493732\pi$
$42$	0	0
$43$	−3569.67	−0.294413	−0.147207	−	0.989106i	$-0.547028\pi$
$43$	−3569.67	−0.294413	−0.147207	+	0.989106i	$0.547028\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−928.769	−0.0613286	−0.0306643	−	0.999530i	$-0.509762\pi$
$47$	−928.769	−0.0613286	−0.0306643	+	0.999530i	$0.509762\pi$
$48$	0	0
$49$	−9486.39	−0.564431
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−33234.4	−1.62517	−0.812584	−	0.582844i	$-0.801940\pi$
$53$	−33234.4	−1.62517	−0.812584	+	0.582844i	$0.801940\pi$
$54$	0	0
$55$	−2495.52	−0.111238
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−17830.1	−0.666845	−0.333423	−	0.942778i	$-0.608204\pi$
$59$	−17830.1	−0.666845	−0.333423	+	0.942778i	$0.608204\pi$
$60$	0	0
$61$	30245.0	1.04071	0.520354	−	0.853951i	$-0.325800\pi$
$61$	30245.0	1.04071	0.520354	+	0.853951i	$0.325800\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13654.5	−0.400859
$66$	0	0
$67$	20790.3	0.565815	0.282908	−	0.959147i	$-0.408701\pi$
$67$	20790.3	0.565815	0.282908	+	0.959147i	$0.408701\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	19209.9	0.452250	0.226125	−	0.974098i	$-0.427394\pi$
$71$	19209.9	0.452250	0.226125	+	0.974098i	$0.427394\pi$
$72$	0	0
$73$	−83174.8	−1.82677	−0.913386	−	0.407094i	$-0.866542\pi$
$73$	−83174.8	−1.82677	−0.913386	+	0.407094i	$0.866542\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8540.73	−0.164160
$78$	0	0
$79$	42514.1	0.766418	0.383209	−	0.923662i	$-0.374819\pi$
$79$	42514.1	0.766418	0.383209	+	0.923662i	$0.374819\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−93576.0	−1.49097	−0.745486	−	0.666522i	$-0.767782\pi$
$83$	−93576.0	−1.49097	−0.745486	+	0.666522i	$0.767782\pi$
$84$	0	0
$85$	−10623.2	−0.159481
$86$	0	0
$87$	0	0
$88$	0	0
$89$	72645.6	0.972152	0.486076	−	0.873917i	$-0.338428\pi$
$89$	72645.6	0.972152	0.486076	+	0.873917i	$0.338428\pi$
$90$	0	0
$91$	−46731.4	−0.591569
$92$	0	0
$93$	0	0
$94$	0	0
$95$	53055.1	0.603140
$96$	0	0
$97$	16741.0	0.180655	0.0903277	−	0.995912i	$-0.471209\pi$
$97$	16741.0	0.180655	0.0903277	+	0.995912i	$0.471209\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5922.83	−0.0577731	−0.0288866	−	0.999583i	$-0.509196\pi$
$101$	−5922.83	−0.0577731	−0.0288866	+	0.999583i	$0.509196\pi$
$102$	0	0
$103$	−142022.	−1.31906	−0.659528	−	0.751680i	$-0.729244\pi$
$103$	−142022.	−1.31906	−0.659528	+	0.751680i	$0.729244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3404.42	−0.0287464	−0.0143732	−	0.999897i	$-0.504575\pi$
$107$	−3404.42	−0.0287464	−0.0143732	+	0.999897i	$0.504575\pi$
$108$	0	0
$109$	132231.	1.06602	0.533012	−	0.846107i	$-0.321060\pi$
$109$	132231.	1.06602	0.533012	+	0.846107i	$0.321060\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	212816.	1.56786	0.783932	−	0.620847i	$-0.213211\pi$
$113$	212816.	1.56786	0.783932	+	0.620847i	$0.213211\pi$
$114$	0	0
$115$	−86250.9	−0.608162
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−36357.1	−0.235354
$120$	0	0
$121$	−151087.	−0.938130
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	304653.	1.67608	0.838042	−	0.545606i	$-0.183700\pi$
$127$	304653.	1.67608	0.838042	+	0.545606i	$0.183700\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−103961.	−0.529290	−0.264645	−	0.964346i	$-0.585255\pi$
$131$	−103961.	−0.529290	−0.264645	+	0.964346i	$0.585255\pi$
$132$	0	0
$133$	181577.	0.890085
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4202.17	0.0191281	0.00956405	−	0.999954i	$-0.496956\pi$
$137$	4202.17	0.0191281	0.00956405	+	0.999954i	$0.496956\pi$
$138$	0	0
$139$	154249.	0.677151	0.338575	−	0.940939i	$-0.390055\pi$
$139$	154249.	0.677151	0.338575	+	0.940939i	$0.390055\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	54520.1	0.222955
$144$	0	0
$145$	171919.	0.679054
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−288038.	−1.06288	−0.531439	−	0.847096i	$-0.678349\pi$
$149$	−288038.	−1.06288	−0.531439	+	0.847096i	$0.678349\pi$
$150$	0	0
$151$	51586.3	0.184116	0.0920581	−	0.995754i	$-0.470655\pi$
$151$	51586.3	0.184116	0.0920581	+	0.995754i	$0.470655\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	74746.9	0.249899
$156$	0	0
$157$	−247275.	−0.800627	−0.400314	−	0.916378i	$-0.631099\pi$
$157$	−247275.	−0.800627	−0.400314	+	0.916378i	$0.631099\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−295187.	−0.897496
$162$	0	0
$163$	−278761.	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$163$	−278761.	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−489181.	−1.35731	−0.678654	−	0.734459i	$-0.737436\pi$
$167$	−489181.	−1.35731	−0.678654	+	0.734459i	$0.737436\pi$
$168$	0	0
$169$	−72981.4	−0.196560
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−577120.	−1.46606	−0.733029	−	0.680198i	$-0.761894\pi$
$173$	−577120.	−1.46606	−0.733029	+	0.680198i	$0.761894\pi$
$174$	0	0
$175$	53475.3	0.131995
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−617817.	−1.44121	−0.720605	−	0.693346i	$-0.756136\pi$
$179$	−617817.	−1.44121	−0.720605	+	0.693346i	$0.756136\pi$
$180$	0	0
$181$	−265079.	−0.601421	−0.300711	−	0.953715i	$-0.597224\pi$
$181$	−265079.	−0.601421	−0.300711	+	0.953715i	$0.597224\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−355005.	−0.762615
$186$	0	0
$187$	42416.8	0.0887021
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−30593.3	−0.0606797	−0.0303398	−	0.999540i	$-0.509659\pi$
$191$	−30593.3	−0.0606797	−0.0303398	+	0.999540i	$0.509659\pi$
$192$	0	0
$193$	463235.	0.895176	0.447588	−	0.894240i	$-0.352283\pi$
$193$	463235.	0.895176	0.447588	+	0.894240i	$0.352283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	518076.	0.951104	0.475552	−	0.879688i	$-0.342248\pi$
$197$	518076.	0.951104	0.475552	+	0.879688i	$0.342248\pi$
$198$	0	0
$199$	−53461.6	−0.0956993	−0.0478497	−	0.998855i	$-0.515237\pi$
$199$	−53461.6	−0.0956993	−0.0478497	+	0.998855i	$0.515237\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	588380.	1.00211
$204$	0	0
$205$	10596.7	0.0176111
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−211840.	−0.335462
$210$	0	0
$211$	−318155.	−0.491963	−0.245982	−	0.969275i	$-0.579110\pi$
$211$	−318155.	−0.491963	−0.245982	+	0.969275i	$0.579110\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−89241.8	−0.131666
$216$	0	0
$217$	255815.	0.368788
$218$	0	0
$219$	0	0
$220$	0	0
$221$	232087.	0.319647
$222$	0	0
$223$	282126.	0.379911	0.189955	−	0.981793i	$-0.439166\pi$
$223$	282126.	0.379911	0.189955	+	0.981793i	$0.439166\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−599160.	−0.771753	−0.385877	−	0.922550i	$-0.626101\pi$
$227$	−599160.	−0.771753	−0.385877	+	0.922550i	$0.626101\pi$
$228$	0	0
$229$	1.29257e6	1.62879	0.814394	−	0.580313i	$-0.197070\pi$
$229$	1.29257e6	1.62879	0.814394	+	0.580313i	$0.197070\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−611411.	−0.737808	−0.368904	−	0.929467i	$-0.620267\pi$
$233$	−611411.	−0.737808	−0.368904	+	0.929467i	$0.620267\pi$
$234$	0	0
$235$	−23219.2	−0.0274270
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−960138.	−1.08727	−0.543637	−	0.839320i	$-0.682953\pi$
$239$	−960138.	−1.08727	−0.543637	+	0.839320i	$0.682953\pi$
$240$	0	0
$241$	−345926.	−0.383655	−0.191828	−	0.981429i	$-0.561441\pi$
$241$	−345926.	−0.383655	−0.191828	+	0.981429i	$0.561441\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−237160.	−0.252421
$246$	0	0
$247$	−1.15910e6	−1.20887
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−711996.	−0.713335	−0.356667	−	0.934231i	$-0.616087\pi$
$251$	−711996.	−0.713335	−0.356667	+	0.934231i	$0.616087\pi$
$252$	0	0
$253$	344386.	0.338255
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−305560.	−0.288578	−0.144289	−	0.989536i	$-0.546090\pi$
$257$	−305560.	−0.288578	−0.144289	+	0.989536i	$0.546090\pi$
$258$	0	0
$259$	−1.21498e6	−1.12543
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.69549e6	−1.51149	−0.755744	−	0.654867i	$-0.772725\pi$
$263$	−1.69549e6	−1.51149	−0.755744	+	0.654867i	$0.772725\pi$
$264$	0	0
$265$	−830860.	−0.726797
$266$	0	0
$267$	0	0
$268$	0	0
$269$	577207.	0.486352	0.243176	−	0.969982i	$-0.421811\pi$
$269$	577207.	0.486352	0.243176	+	0.969982i	$0.421811\pi$
$270$	0	0
$271$	−1.65995e6	−1.37300	−0.686501	−	0.727129i	$-0.740854\pi$
$271$	−1.65995e6	−1.37300	−0.686501	+	0.727129i	$0.740854\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−62388.1	−0.0497473
$276$	0	0
$277$	1.25880e6	0.985732	0.492866	−	0.870105i	$-0.335949\pi$
$277$	1.25880e6	0.985732	0.492866	+	0.870105i	$0.335949\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.69319e6	−1.27921	−0.639604	−	0.768705i	$-0.720901\pi$
$281$	−1.69319e6	−1.27921	−0.639604	+	0.768705i	$0.720901\pi$
$282$	0	0
$283$	1.97014e6	1.46228	0.731142	−	0.682225i	$-0.238988\pi$
$283$	1.97014e6	1.46228	0.731142	+	0.682225i	$0.238988\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36266.5	0.0259897
$288$	0	0
$289$	−1.23929e6	−0.872829
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.69945e6	−1.83699	−0.918494	−	0.395435i	$-0.870594\pi$
$293$	−2.69945e6	−1.83699	−0.918494	+	0.395435i	$0.870594\pi$
$294$	0	0
$295$	−445754.	−0.298222
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.88434e6	1.21894
$300$	0	0
$301$	−305423.	−0.194306
$302$	0	0
$303$	0	0
$304$	0	0
$305$	756125.	0.465419
$306$	0	0
$307$	1.46875e6	0.889410	0.444705	−	0.895677i	$-0.353308\pi$
$307$	1.46875e6	0.889410	0.444705	+	0.895677i	$0.353308\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−681620.	−0.399614	−0.199807	−	0.979835i	$-0.564032\pi$
$311$	−681620.	−0.399614	−0.199807	+	0.979835i	$0.564032\pi$
$312$	0	0
$313$	−642495.	−0.370688	−0.185344	−	0.982674i	$-0.559340\pi$
$313$	−642495.	−0.370688	−0.185344	+	0.982674i	$0.559340\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.63094e6	−1.47049	−0.735245	−	0.677801i	$-0.762933\pi$
$317$	−2.63094e6	−1.47049	−0.735245	+	0.677801i	$0.762933\pi$
$318$	0	0
$319$	−686445.	−0.377684
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−901786.	−0.480947
$324$	0	0
$325$	−341362.	−0.179270
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−79465.9	−0.0404754
$330$	0	0
$331$	739507.	0.370999	0.185499	−	0.982644i	$-0.440610\pi$
$331$	739507.	0.370999	0.185499	+	0.982644i	$0.440610\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	519759.	0.253040
$336$	0	0
$337$	2.86873e6	1.37599	0.687996	−	0.725715i	$-0.258491\pi$
$337$	2.86873e6	1.37599	0.687996	+	0.725715i	$0.258491\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−298452.	−0.138992
$342$	0	0
$343$	−2.24968e6	−1.03249
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−358901.	−0.160011	−0.0800057	−	0.996794i	$-0.525494\pi$
$347$	−358901.	−0.160011	−0.0800057	+	0.996794i	$0.525494\pi$
$348$	0	0
$349$	2.96201e6	1.30174	0.650869	−	0.759190i	$-0.274405\pi$
$349$	2.96201e6	1.30174	0.650869	+	0.759190i	$0.274405\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.02769e6	0.866095	0.433048	−	0.901371i	$-0.357438\pi$
$353$	2.02769e6	0.866095	0.433048	+	0.901371i	$0.357438\pi$
$354$	0	0
$355$	480246.	0.202252
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.52540e6	−1.85319	−0.926596	−	0.376058i	$-0.877279\pi$
$359$	−4.52540e6	−1.85319	−0.926596	+	0.376058i	$0.877279\pi$
$360$	0	0
$361$	2.02765e6	0.818890
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.07937e6	−0.816958
$366$	0	0
$367$	−3.58235e6	−1.38836	−0.694182	−	0.719800i	$-0.744234\pi$
$367$	−3.58235e6	−1.38836	−0.694182	+	0.719800i	$0.744234\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.84355e6	−1.07257
$372$	0	0
$373$	−1.87995e6	−0.699638	−0.349819	−	0.936817i	$-0.613757\pi$
$373$	−1.87995e6	−0.699638	−0.349819	+	0.936817i	$0.613757\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.75594e6	−1.36102
$378$	0	0
$379$	−1.50759e6	−0.539121	−0.269560	−	0.962983i	$-0.586878\pi$
$379$	−1.50759e6	−0.539121	−0.269560	+	0.962983i	$0.586878\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.72039e6	1.29596	0.647980	−	0.761658i	$-0.275614\pi$
$383$	3.72039e6	1.29596	0.647980	+	0.761658i	$0.275614\pi$
$384$	0	0
$385$	−213518.	−0.0734148
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.42840e6	−0.478603	−0.239302	−	0.970945i	$-0.576919\pi$
$389$	−1.42840e6	−0.478603	−0.239302	+	0.970945i	$0.576919\pi$
$390$	0	0
$391$	1.46602e6	0.484951
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.06285e6	0.342752
$396$	0	0
$397$	−2.52884e6	−0.805276	−0.402638	−	0.915359i	$-0.631907\pi$
$397$	−2.52884e6	−0.805276	−0.402638	+	0.915359i	$0.631907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	730346.	0.226813	0.113406	−	0.993549i	$-0.463824\pi$
$401$	730346.	0.226813	0.113406	+	0.993549i	$0.463824\pi$
$402$	0	0
$403$	−1.63301e6	−0.500871
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.41748e6	0.424160
$408$	0	0
$409$	−1.51732e6	−0.448506	−0.224253	−	0.974531i	$-0.571994\pi$
$409$	−1.51732e6	−0.448506	−0.224253	+	0.974531i	$0.571994\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.52556e6	−0.440102
$414$	0	0
$415$	−2.33940e6	−0.666783
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.36661e6	−0.380286	−0.190143	−	0.981756i	$-0.560895\pi$
$419$	−1.36661e6	−0.380286	−0.190143	+	0.981756i	$0.560895\pi$
$420$	0	0
$421$	3.20736e6	0.881946	0.440973	−	0.897520i	$-0.354633\pi$
$421$	3.20736e6	0.881946	0.440973	+	0.897520i	$0.354633\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−265581.	−0.0713221
$426$	0	0
$427$	2.58778e6	0.686843
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.51920e6	−1.17184	−0.585920	−	0.810369i	$-0.699267\pi$
$431$	−4.51920e6	−1.17184	−0.585920	+	0.810369i	$0.699267\pi$
$432$	0	0
$433$	5.08506e6	1.30340	0.651698	−	0.758478i	$-0.274057\pi$
$433$	5.08506e6	1.30340	0.651698	+	0.758478i	$0.274057\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.32169e6	−1.83403
$438$	0	0
$439$	1.67597e6	0.415054	0.207527	−	0.978229i	$-0.433459\pi$
$439$	1.67597e6	0.415054	0.207527	+	0.978229i	$0.433459\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.33137e6	−1.77491	−0.887454	−	0.460897i	$-0.847528\pi$
$443$	−7.33137e6	−1.77491	−0.887454	+	0.460897i	$0.847528\pi$
$444$	0	0
$445$	1.81614e6	0.434759
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.68385e6	−0.862354	−0.431177	−	0.902267i	$-0.641902\pi$
$449$	−3.68385e6	−0.862354	−0.431177	+	0.902267i	$0.641902\pi$
$450$	0	0
$451$	−42311.0	−0.00979518
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.16828e6	−0.264557
$456$	0	0
$457$	−810253.	−0.181480	−0.0907402	−	0.995875i	$-0.528923\pi$
$457$	−810253.	−0.181480	−0.0907402	+	0.995875i	$0.528923\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.76965e6	1.04528	0.522642	−	0.852552i	$-0.324947\pi$
$461$	4.76965e6	1.04528	0.522642	+	0.852552i	$0.324947\pi$
$462$	0	0
$463$	3.70653e6	0.803554	0.401777	−	0.915738i	$-0.368393\pi$
$463$	3.70653e6	0.803554	0.401777	+	0.915738i	$0.368393\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.33319e6	0.707241	0.353621	−	0.935389i	$-0.384950\pi$
$467$	3.33319e6	0.707241	0.353621	+	0.935389i	$0.384950\pi$
$468$	0	0
$469$	1.77883e6	0.373425
$470$	0	0
$471$	0	0
$472$	0	0
$473$	356328.	0.0732314
$474$	0	0
$475$	1.32638e6	0.269732
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.99488e6	−0.596404	−0.298202	−	0.954503i	$-0.596387\pi$
$479$	−2.99488e6	−0.596404	−0.298202	+	0.954503i	$0.596387\pi$
$480$	0	0
$481$	7.75585e6	1.52850
$482$	0	0
$483$	0	0
$484$	0	0
$485$	418524.	0.0807916
$486$	0	0
$487$	−7.71593e6	−1.47423	−0.737116	−	0.675766i	$-0.763813\pi$
$487$	−7.71593e6	−1.47423	−0.737116	+	0.675766i	$0.763813\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.65091e6	−0.496239	−0.248120	−	0.968729i	$-0.579813\pi$
$491$	−2.65091e6	−0.496239	−0.248120	+	0.968729i	$0.579813\pi$
$492$	0	0
$493$	−2.92214e6	−0.541481
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.64361e6	0.298474
$498$	0	0
$499$	3.29296e6	0.592019	0.296010	−	0.955185i	$-0.404344\pi$
$499$	3.29296e6	0.592019	0.296010	+	0.955185i	$0.404344\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.22340e6	0.215600	0.107800	−	0.994173i	$-0.465619\pi$
$503$	1.22340e6	0.215600	0.107800	+	0.994173i	$0.465619\pi$
$504$	0	0
$505$	−148071.	−0.0258369
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.10782e6	0.873858	0.436929	−	0.899496i	$-0.356066\pi$
$509$	5.10782e6	0.873858	0.436929	+	0.899496i	$0.356066\pi$
$510$	0	0
$511$	−7.11648e6	−1.20563
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.55055e6	−0.589899
$516$	0	0
$517$	92710.6	0.0152547
$518$	0	0
$519$	0	0
$520$	0	0
$521$	115812.	0.0186922	0.00934611	−	0.999956i	$-0.497025\pi$
$521$	115812.	0.0186922	0.00934611	+	0.999956i	$0.497025\pi$
$522$	0	0
$523$	−6.72323e6	−1.07479	−0.537395	−	0.843330i	$-0.680592\pi$
$523$	−6.72323e6	−1.07479	−0.537395	+	0.843330i	$0.680592\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.27048e6	−0.199270
$528$	0	0
$529$	5.46642e6	0.849305
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−231509.	−0.0352979
$534$	0	0
$535$	−85110.6	−0.0128558
$536$	0	0
$537$	0	0
$538$	0	0
$539$	946941.	0.140395
$540$	0	0
$541$	−9.33904e6	−1.37186	−0.685929	−	0.727669i	$-0.740604\pi$
$541$	−9.33904e6	−1.37186	−0.685929	+	0.727669i	$0.740604\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.30578e6	0.476741
$546$	0	0
$547$	−1.29689e6	−0.185325	−0.0926626	−	0.995698i	$-0.529538\pi$
$547$	−1.29689e6	−0.185325	−0.0926626	+	0.995698i	$0.529538\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.45939e7	2.04782
$552$	0	0
$553$	3.63753e6	0.505818
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.88461e6	0.393958	0.196979	−	0.980408i	$-0.436887\pi$
$557$	2.88461e6	0.393958	0.196979	+	0.980408i	$0.436887\pi$
$558$	0	0
$559$	1.94968e6	0.263897
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.96177e6	−1.05862	−0.529308	−	0.848430i	$-0.677548\pi$
$563$	−7.96177e6	−1.05862	−0.529308	+	0.848430i	$0.677548\pi$
$564$	0	0
$565$	5.32040e6	0.701170
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.36853e6	−0.436174	−0.218087	−	0.975929i	$-0.569982\pi$
$569$	−3.36853e6	−0.436174	−0.218087	+	0.975929i	$0.569982\pi$
$570$	0	0
$571$	−1.43608e7	−1.84327	−0.921633	−	0.388063i	$-0.873144\pi$
$571$	−1.43608e7	−1.84327	−0.921633	+	0.388063i	$0.873144\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.15627e6	−0.271978
$576$	0	0
$577$	−5.67748e6	−0.709930	−0.354965	−	0.934880i	$-0.615507\pi$
$577$	−5.67748e6	−0.709930	−0.354965	+	0.934880i	$0.615507\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.00641e6	−0.984006
$582$	0	0
$583$	3.31749e6	0.404239
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.30937e6	0.516201	0.258101	−	0.966118i	$-0.416903\pi$
$587$	4.30937e6	0.516201	0.258101	+	0.966118i	$0.416903\pi$
$588$	0	0
$589$	6.34513e6	0.753619
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.30041e6	0.618975	0.309487	−	0.950904i	$-0.399843\pi$
$593$	5.30041e6	0.618975	0.309487	+	0.950904i	$0.399843\pi$
$594$	0	0
$595$	−908929.	−0.105254
$596$	0	0
$597$	0	0
$598$	0	0
$599$	479201.	0.0545696	0.0272848	−	0.999628i	$-0.491314\pi$
$599$	479201.	0.0545696	0.0272848	+	0.999628i	$0.491314\pi$
$600$	0	0
$601$	6.07294e6	0.685825	0.342912	−	0.939367i	$-0.388587\pi$
$601$	6.07294e6	0.685825	0.342912	+	0.939367i	$0.388587\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.77717e6	−0.419545
$606$	0	0
$607$	−7.45020e6	−0.820722	−0.410361	−	0.911923i	$-0.634597\pi$
$607$	−7.45020e6	−0.820722	−0.410361	+	0.911923i	$0.634597\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	507274.	0.0549718
$612$	0	0
$613$	1.15156e7	1.23776	0.618878	−	0.785487i	$-0.287587\pi$
$613$	1.15156e7	1.23776	0.618878	+	0.785487i	$0.287587\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.00826e7	1.06625	0.533126	−	0.846036i	$-0.321017\pi$
$617$	1.00826e7	1.06625	0.533126	+	0.846036i	$0.321017\pi$
$618$	0	0
$619$	−1.39730e7	−1.46576	−0.732882	−	0.680355i	$-0.761825\pi$
$619$	−1.39730e7	−1.46576	−0.732882	+	0.680355i	$0.761825\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.21559e6	0.641597
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.03407e6	0.608113
$630$	0	0
$631$	6.09736e6	0.609633	0.304817	−	0.952411i	$-0.401405\pi$
$631$	6.09736e6	0.609633	0.304817	+	0.952411i	$0.401405\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.61632e6	0.749568
$636$	0	0
$637$	5.18127e6	0.505927
$638$	0	0
$639$	0	0
$640$	0	0
$641$	548849.	0.0527604	0.0263802	−	0.999652i	$-0.491602\pi$
$641$	548849.	0.0527604	0.0263802	+	0.999652i	$0.491602\pi$
$642$	0	0
$643$	−1.29095e7	−1.23135	−0.615677	−	0.787999i	$-0.711117\pi$
$643$	−1.29095e7	−1.23135	−0.615677	+	0.787999i	$0.711117\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.71727e6	−0.161279	−0.0806397	−	0.996743i	$-0.525696\pi$
$647$	−1.71727e6	−0.161279	−0.0806397	+	0.996743i	$0.525696\pi$
$648$	0	0
$649$	1.77982e6	0.165869
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.15426e7	1.05931	0.529653	−	0.848214i	$-0.322322\pi$
$653$	1.15426e7	1.05931	0.529653	+	0.848214i	$0.322322\pi$
$654$	0	0
$655$	−2.59904e6	−0.236706
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.09831e7	0.985169	0.492585	−	0.870265i	$-0.336052\pi$
$659$	1.09831e7	0.985169	0.492585	+	0.870265i	$0.336052\pi$
$660$	0	0
$661$	1.79023e7	1.59370	0.796849	−	0.604178i	$-0.206499\pi$
$661$	1.79023e7	1.59370	0.796849	+	0.604178i	$0.206499\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.53942e6	0.398058
$666$	0	0
$667$	−2.37251e7	−2.06487
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.01908e6	−0.258862
$672$	0	0
$673$	1.51352e7	1.28810	0.644050	−	0.764983i	$-0.277253\pi$
$673$	1.51352e7	1.28810	0.644050	+	0.764983i	$0.277253\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.74374e6	0.146221	0.0731105	−	0.997324i	$-0.476707\pi$
$677$	1.74374e6	0.146221	0.0731105	+	0.997324i	$0.476707\pi$
$678$	0	0
$679$	1.43237e6	0.119228
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.91354e6	0.321010	0.160505	−	0.987035i	$-0.448688\pi$
$683$	3.91354e6	0.321010	0.160505	+	0.987035i	$0.448688\pi$
$684$	0	0
$685$	105054.	0.00855434
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.81519e7	1.45672
$690$	0	0
$691$	2.09646e7	1.67029	0.835143	−	0.550032i	$-0.185385\pi$
$691$	2.09646e7	1.67029	0.835143	+	0.550032i	$0.185385\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.85622e6	0.302831
$696$	0	0
$697$	−180114.	−0.0140432
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.37397e7	−1.05604	−0.528022	−	0.849231i	$-0.677066\pi$
$701$	−1.37397e7	−1.05604	−0.528022	+	0.849231i	$0.677066\pi$
$702$	0	0
$703$	−3.01357e7	−2.29982
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−506761.	−0.0381289
$708$	0	0
$709$	8.09118e6	0.604500	0.302250	−	0.953229i	$-0.402262\pi$
$709$	8.09118e6	0.604500	0.302250	+	0.953229i	$0.402262\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.03152e7	−0.759894
$714$	0	0
$715$	1.36300e6	0.0997084
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.50696e6	−0.613694	−0.306847	−	0.951759i	$-0.599274\pi$
$719$	−8.50696e6	−0.613694	−0.306847	+	0.951759i	$0.599274\pi$
$720$	0	0
$721$	−1.21515e7	−0.870545
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.29798e6	0.303682
$726$	0	0
$727$	−1.35919e7	−0.953771	−0.476885	−	0.878966i	$-0.658234\pi$
$727$	−1.35919e7	−0.953771	−0.476885	+	0.878966i	$0.658234\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.51686e6	0.104991
$732$	0	0
$733$	1.21454e7	0.834932	0.417466	−	0.908692i	$-0.362918\pi$
$733$	1.21454e7	0.834932	0.417466	+	0.908692i	$0.362918\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.07531e6	−0.140739
$738$	0	0
$739$	−4.50729e6	−0.303602	−0.151801	−	0.988411i	$-0.548507\pi$
$739$	−4.50729e6	−0.303602	−0.151801	+	0.988411i	$0.548507\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.82273e6	−0.519860	−0.259930	−	0.965628i	$-0.583699\pi$
$743$	−7.82273e6	−0.519860	−0.259930	+	0.965628i	$0.583699\pi$
$744$	0	0
$745$	−7.20094e6	−0.475334
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−291284.	−0.0189720
$750$	0	0
$751$	1.62764e7	1.05307	0.526537	−	0.850152i	$-0.323490\pi$
$751$	1.62764e7	1.05307	0.526537	+	0.850152i	$0.323490\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.28966e6	0.0823393
$756$	0	0
$757$	5.05461e6	0.320588	0.160294	−	0.987069i	$-0.448756\pi$
$757$	5.05461e6	0.320588	0.160294	+	0.987069i	$0.448756\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	786485.	0.0492299	0.0246149	−	0.999697i	$-0.492164\pi$
$761$	786485.	0.0492299	0.0246149	+	0.999697i	$0.492164\pi$
$762$	0	0
$763$	1.13138e7	0.703551
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.73845e6	0.597725
$768$	0	0
$769$	1.44258e7	0.879679	0.439840	−	0.898076i	$-0.355035\pi$
$769$	1.44258e7	0.879679	0.439840	+	0.898076i	$0.355035\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.81060e7	1.08987	0.544935	−	0.838478i	$-0.316554\pi$
$773$	1.81060e7	1.08987	0.544935	+	0.838478i	$0.316554\pi$
$774$	0	0
$775$	1.86867e6	0.111758
$776$	0	0
$777$	0	0
$778$	0	0
$779$	899537.	0.0531099
$780$	0	0
$781$	−1.91755e6	−0.112491
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.18187e6	−0.358051
$786$	0	0
$787$	1.86788e7	1.07501	0.537504	−	0.843261i	$-0.319367\pi$
$787$	1.86788e7	1.07501	0.537504	+	0.843261i	$0.319367\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.82087e7	1.03475
$792$	0	0
$793$	−1.65192e7	−0.932836
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.99446e6	−0.334275	−0.167138	−	0.985934i	$-0.553452\pi$
$797$	−5.99446e6	−0.334275	−0.167138	+	0.985934i	$0.553452\pi$
$798$	0	0
$799$	394661.	0.0218704
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.30259e6	0.454386
$804$	0	0
$805$	−7.37968e6	−0.401373
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.85790e7	0.998047	0.499023	−	0.866589i	$-0.333692\pi$
$809$	1.85790e7	0.998047	0.499023	+	0.866589i	$0.333692\pi$
$810$	0	0
$811$	−2.31663e7	−1.23682	−0.618408	−	0.785857i	$-0.712222\pi$
$811$	−2.31663e7	−1.23682	−0.618408	+	0.785857i	$0.712222\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.96902e6	−0.367517
$816$	0	0
$817$	−7.57558e6	−0.397064
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.02639e7	0.531440	0.265720	−	0.964050i	$-0.414390\pi$
$821$	1.02639e7	0.531440	0.265720	+	0.964050i	$0.414390\pi$
$822$	0	0
$823$	3.46900e7	1.78527	0.892637	−	0.450776i	$-0.148853\pi$
$823$	3.46900e7	1.78527	0.892637	+	0.450776i	$0.148853\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.50380e7	0.764585	0.382293	−	0.924041i	$-0.375135\pi$
$827$	1.50380e7	0.764585	0.382293	+	0.924041i	$0.375135\pi$
$828$	0	0
$829$	−1.38937e7	−0.702155	−0.351078	−	0.936346i	$-0.614185\pi$
$829$	−1.38937e7	−0.702155	−0.351078	+	0.936346i	$0.614185\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.03104e6	0.201282
$834$	0	0
$835$	−1.22295e7	−0.607006
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.79014e7	−0.877973	−0.438986	−	0.898494i	$-0.644662\pi$
$839$	−1.79014e7	−0.877973	−0.438986	+	0.898494i	$0.644662\pi$
$840$	0	0
$841$	2.67787e7	1.30557
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.82454e6	−0.0879044
$846$	0	0
$847$	−1.29271e7	−0.619144
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.89912e7	2.31897
$852$	0	0
$853$	1.07670e7	0.506664	0.253332	−	0.967379i	$-0.418473\pi$
$853$	1.07670e7	0.506664	0.253332	+	0.967379i	$0.418473\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.76117e7	0.819122	0.409561	−	0.912283i	$-0.365682\pi$
$857$	1.76117e7	0.819122	0.409561	+	0.912283i	$0.365682\pi$
$858$	0	0
$859$	1.95268e7	0.902917	0.451458	−	0.892292i	$-0.350904\pi$
$859$	1.95268e7	0.902917	0.451458	+	0.892292i	$0.350904\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.78308e7	1.72910	0.864548	−	0.502550i	$-0.167605\pi$
$863$	3.78308e7	1.72910	0.864548	+	0.502550i	$0.167605\pi$
$864$	0	0
$865$	−1.44280e7	−0.655641
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.24380e6	−0.190636
$870$	0	0
$871$	−1.13553e7	−0.507168
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.33688e6	0.0590301
$876$	0	0
$877$	4.51942e6	0.198419	0.0992096	−	0.995067i	$-0.468369\pi$
$877$	4.51942e6	0.198419	0.0992096	+	0.995067i	$0.468369\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.48399e7	1.07823	0.539114	−	0.842233i	$-0.318759\pi$
$881$	2.48399e7	1.07823	0.539114	+	0.842233i	$0.318759\pi$
$882$	0	0
$883$	−4.40390e6	−0.190080	−0.0950399	−	0.995473i	$-0.530298\pi$
$883$	−4.40390e6	−0.190080	−0.0950399	+	0.995473i	$0.530298\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.67390e7	1.56790	0.783951	−	0.620823i	$-0.213201\pi$
$887$	3.67390e7	1.56790	0.783951	+	0.620823i	$0.213201\pi$
$888$	0	0
$889$	2.60663e7	1.10618
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.97104e6	−0.0827116
$894$	0	0
$895$	−1.54454e7	−0.644528
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.05607e7	0.848473
$900$	0	0
$901$	1.41223e7	0.579552
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.62697e6	−0.268964
$906$	0	0
$907$	−3.61944e6	−0.146091	−0.0730454	−	0.997329i	$-0.523272\pi$
$907$	−3.61944e6	−0.146091	−0.0730454	+	0.997329i	$0.523272\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.06846e7	0.426543	0.213271	−	0.976993i	$-0.431588\pi$
$911$	1.06846e7	0.426543	0.213271	+	0.976993i	$0.431588\pi$
$912$	0	0
$913$	9.34084e6	0.370859
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.89499e6	−0.349319
$918$	0	0
$919$	423307.	0.0165336	0.00826679	−	0.999966i	$-0.497369\pi$
$919$	423307.	0.0165336	0.00826679	+	0.999966i	$0.497369\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.04920e7	−0.405373
$924$	0	0
$925$	−8.87512e6	−0.341052
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.51666e6	−0.0956720	−0.0478360	−	0.998855i	$-0.515232\pi$
$929$	−2.51666e6	−0.0956720	−0.0478360	+	0.998855i	$0.515232\pi$
$930$	0	0
$931$	−2.01321e7	−0.761227
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.06042e6	0.0396688
$936$	0	0
$937$	−1.39054e7	−0.517408	−0.258704	−	0.965957i	$-0.583295\pi$
$937$	−1.39054e7	−0.517408	−0.258704	+	0.965957i	$0.583295\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.15658e7	0.425795	0.212897	−	0.977075i	$-0.431710\pi$
$941$	1.15658e7	0.425795	0.212897	+	0.977075i	$0.431710\pi$
$942$	0	0
$943$	−1.46237e6	−0.0535521
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.37017e7	−0.858826	−0.429413	−	0.903108i	$-0.641279\pi$
$947$	−2.37017e7	−0.858826	−0.429413	+	0.903108i	$0.641279\pi$
$948$	0	0
$949$	4.54283e7	1.63742
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.67313e6	0.0596757	0.0298378	−	0.999555i	$-0.490501\pi$
$953$	1.67313e6	0.0596757	0.0298378	+	0.999555i	$0.490501\pi$
$954$	0	0
$955$	−764833.	−0.0271368
$956$	0	0
$957$	0	0
$958$	0	0
$959$	359540.	0.0126241
$960$	0	0
$961$	−1.96898e7	−0.687753
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.15809e7	0.400335
$966$	0	0
$967$	−2.69820e7	−0.927915	−0.463957	−	0.885858i	$-0.653571\pi$
$967$	−2.69820e7	−0.927915	−0.463957	+	0.885858i	$0.653571\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.59915e7	−1.22504	−0.612522	−	0.790454i	$-0.709845\pi$
$971$	−3.59915e7	−1.22504	−0.612522	+	0.790454i	$0.709845\pi$
$972$	0	0
$973$	1.31976e7	0.446903
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.65489e7	1.56017	0.780087	−	0.625671i	$-0.215175\pi$
$977$	4.65489e7	1.56017	0.780087	+	0.625671i	$0.215175\pi$
$978$	0	0
$979$	−7.25155e6	−0.241810
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.41850e7	1.12837	0.564186	−	0.825648i	$-0.309190\pi$
$983$	3.41850e7	1.12837	0.564186	+	0.825648i	$0.309190\pi$
$984$	0	0
$985$	1.29519e7	0.425346
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.23155e7	0.400370
$990$	0	0
$991$	−4.37070e7	−1.41373	−0.706866	−	0.707347i	$-0.749892\pi$
$991$	−4.37070e7	−1.41373	−0.706866	+	0.707347i	$0.749892\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.33654e6	−0.0427980
$996$	0	0
$997$	−1.73468e7	−0.552689	−0.276344	−	0.961059i	$-0.589123\pi$
$997$	−1.73468e7	−0.552689	−0.276344	+	0.961059i	$0.589123\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.6.a.g.1.3	yes	3
3.2	odd	2		1080.6.a.e.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.e.1.3	✓	3	3.2	odd	2
1080.6.a.g.1.3	yes	3	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} +85.5605 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} +85.5605 q^{7} -99.8209 q^{11} -546.179 q^{13} -424.929 q^{17} +2122.20 q^{19} -3450.04 q^{23} +625.000 q^{25} +6876.76 q^{29} +2989.87 q^{31} +2139.01 q^{35} -14200.2 q^{37} +423.869 q^{41} -3569.67 q^{43} -928.769 q^{47} -9486.39 q^{49} -33234.4 q^{53} -2495.52 q^{55} -17830.1 q^{59} +30245.0 q^{61} -13654.5 q^{65} +20790.3 q^{67} +19209.9 q^{71} -83174.8 q^{73} -8540.73 q^{77} +42514.1 q^{79} -93576.0 q^{83} -10623.2 q^{85} +72645.6 q^{89} -46731.4 q^{91} +53055.1 q^{95} +16741.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 75 q^{5} - 30 q^{7}+O(q^{10}) \) 3 * q + 75 * q^5 - 30 * q^7	\( 3 q + 75 q^{5} - 30 q^{7} - 660 q^{11} - 1278 q^{13} + 1731 q^{17} + 1011 q^{19} + 2433 q^{23} + 1875 q^{25} + 3786 q^{29} + 4131 q^{31} - 750 q^{35} - 7122 q^{37} + 17352 q^{41} + 11556 q^{43} - 19548 q^{47} - 7857 q^{49} - 4965 q^{53} - 16500 q^{55} - 32106 q^{59} - 16317 q^{61} - 31950 q^{65} - 51258 q^{67} - 84072 q^{71} - 161892 q^{73} + 142968 q^{77} + 36441 q^{79} - 64581 q^{83} + 43275 q^{85} - 61584 q^{89} - 123588 q^{91} + 25275 q^{95} + 14760 q^{97}+O(q^{100}) \) 3 * q + 75 * q^5 - 30 * q^7 - 660 * q^11 - 1278 * q^13 + 1731 * q^17 + 1011 * q^19 + 2433 * q^23 + 1875 * q^25 + 3786 * q^29 + 4131 * q^31 - 750 * q^35 - 7122 * q^37 + 17352 * q^41 + 11556 * q^43 - 19548 * q^47 - 7857 * q^49 - 4965 * q^53 - 16500 * q^55 - 32106 * q^59 - 16317 * q^61 - 31950 * q^65 - 51258 * q^67 - 84072 * q^71 - 161892 * q^73 + 142968 * q^77 + 36441 * q^79 - 64581 * q^83 + 43275 * q^85 - 61584 * q^89 - 123588 * q^91 + 25275 * q^95 + 14760 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more