LMFDB - Embedded newform 1080.6.a.h.1.1

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:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 460x - 1125 $ x^3 - 460*x - 1125
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-20.1005$ 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)

$f(q)$	$=$	$q+25.0000 q^{5} -33.1079 q^{7} +O(q^{10})$	$q+25.0000 q^{5} -33.1079 q^{7} -137.873 q^{11} -227.972 q^{13} -830.449 q^{17} +270.258 q^{19} +1102.99 q^{23} +625.000 q^{25} +8777.32 q^{29} +5419.24 q^{31} -827.696 q^{35} +4949.02 q^{37} -13972.5 q^{41} -11699.9 q^{43} -10972.8 q^{47} -15710.9 q^{49} -9177.48 q^{53} -3446.84 q^{55} -19034.9 q^{59} +42440.3 q^{61} -5699.30 q^{65} +28649.4 q^{67} +49918.7 q^{71} -54961.7 q^{73} +4564.69 q^{77} -23525.2 q^{79} -30742.0 q^{83} -20761.2 q^{85} +26828.7 q^{89} +7547.66 q^{91} +6756.45 q^{95} -138265. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 75 q^{5} + 140 q^{7}+O(q^{10}) $ 3 * q + 75 * q^5 + 140 * q^7	$ 3 q + 75 q^{5} + 140 q^{7} - 163 q^{11} + 51 q^{13} - 605 q^{17} - 1622 q^{19} + 1645 q^{23} + 1875 q^{25} - 557 q^{29} - 2807 q^{31} + 3500 q^{35} - 6140 q^{37} - 13760 q^{41} - 2369 q^{43} - 14909 q^{47} - 34015 q^{49} + 964 q^{53} - 4075 q^{55} - 48240 q^{59} + 6296 q^{61} + 1275 q^{65} + 4826 q^{67} - 27988 q^{71} - 36328 q^{73} - 6930 q^{77} + 25293 q^{79} - 48246 q^{83} - 15125 q^{85} + 41208 q^{89} + 18204 q^{91} - 40550 q^{95} - 49164 q^{97}+O(q^{100}) $ 3 * q + 75 * q^5 + 140 * q^7 - 163 * q^11 + 51 * q^13 - 605 * q^17 - 1622 * q^19 + 1645 * q^23 + 1875 * q^25 - 557 * q^29 - 2807 * q^31 + 3500 * q^35 - 6140 * q^37 - 13760 * q^41 - 2369 * q^43 - 14909 * q^47 - 34015 * q^49 + 964 * q^53 - 4075 * q^55 - 48240 * q^59 + 6296 * q^61 + 1275 * q^65 + 4826 * q^67 - 27988 * q^71 - 36328 * q^73 - 6930 * q^77 + 25293 * q^79 - 48246 * q^83 - 15125 * q^85 + 41208 * q^89 + 18204 * q^91 - 40550 * q^95 - 49164 * 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
$7$	−33.1079	−0.255379	−0.127690	−	0.991814i	$-0.540756\pi$
$7$	−33.1079	−0.255379	−0.127690	+	0.991814i	$0.540756\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−137.873	−0.343557	−0.171779	−	0.985136i	$-0.554951\pi$
$11$	−137.873	−0.343557	−0.171779	+	0.985136i	$0.554951\pi$
$12$	0	0
$13$	−227.972	−0.374130	−0.187065	−	0.982347i	$-0.559898\pi$
$13$	−227.972	−0.374130	−0.187065	+	0.982347i	$0.559898\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−830.449	−0.696932	−0.348466	−	0.937321i	$-0.613297\pi$
$17$	−830.449	−0.696932	−0.348466	+	0.937321i	$0.613297\pi$
$18$	0	0
$19$	270.258	0.171749	0.0858746	−	0.996306i	$-0.472632\pi$
$19$	270.258	0.171749	0.0858746	+	0.996306i	$0.472632\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1102.99	0.434762	0.217381	−	0.976087i	$-0.430249\pi$
$23$	1102.99	0.434762	0.217381	+	0.976087i	$0.430249\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8777.32	1.93806	0.969030	−	0.246945i	$-0.0794266\pi$
$29$	8777.32	1.93806	0.969030	+	0.246945i	$0.0794266\pi$
$30$	0	0
$31$	5419.24	1.01282	0.506412	−	0.862292i	$-0.330971\pi$
$31$	5419.24	1.01282	0.506412	+	0.862292i	$0.330971\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−827.696	−0.114209
$36$	0	0
$37$	4949.02	0.594312	0.297156	−	0.954829i	$-0.403962\pi$
$37$	4949.02	0.594312	0.297156	+	0.954829i	$0.403962\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−13972.5	−1.29812	−0.649060	−	0.760737i	$-0.724838\pi$
$41$	−13972.5	−1.29812	−0.649060	+	0.760737i	$0.724838\pi$
$42$	0	0
$43$	−11699.9	−0.964968	−0.482484	−	0.875905i	$-0.660265\pi$
$43$	−11699.9	−0.964968	−0.482484	+	0.875905i	$0.660265\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10972.8	−0.724555	−0.362278	−	0.932070i	$-0.618001\pi$
$47$	−10972.8	−0.724555	−0.362278	+	0.932070i	$0.618001\pi$
$48$	0	0
$49$	−15710.9	−0.934781
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9177.48	−0.448780	−0.224390	−	0.974499i	$-0.572039\pi$
$53$	−9177.48	−0.448780	−0.224390	+	0.974499i	$0.572039\pi$
$54$	0	0
$55$	−3446.84	−0.153643
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−19034.9	−0.711904	−0.355952	−	0.934504i	$-0.615843\pi$
$59$	−19034.9	−0.711904	−0.355952	+	0.934504i	$0.615843\pi$
$60$	0	0
$61$	42440.3	1.46034	0.730170	−	0.683266i	$-0.239441\pi$
$61$	42440.3	1.46034	0.730170	+	0.683266i	$0.239441\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5699.30	−0.167316
$66$	0	0
$67$	28649.4	0.779703	0.389851	−	0.920878i	$-0.372526\pi$
$67$	28649.4	0.779703	0.389851	+	0.920878i	$0.372526\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	49918.7	1.17521	0.587607	−	0.809146i	$-0.300070\pi$
$71$	49918.7	1.17521	0.587607	+	0.809146i	$0.300070\pi$
$72$	0	0
$73$	−54961.7	−1.20713	−0.603564	−	0.797315i	$-0.706253\pi$
$73$	−54961.7	−1.20713	−0.603564	+	0.797315i	$0.706253\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4564.69	0.0877374
$78$	0	0
$79$	−23525.2	−0.424097	−0.212049	−	0.977259i	$-0.568013\pi$
$79$	−23525.2	−0.424097	−0.212049	+	0.977259i	$0.568013\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−30742.0	−0.489821	−0.244910	−	0.969546i	$-0.578759\pi$
$83$	−30742.0	−0.489821	−0.244910	+	0.969546i	$0.578759\pi$
$84$	0	0
$85$	−20761.2	−0.311678
$86$	0	0
$87$	0	0
$88$	0	0
$89$	26828.7	0.359025	0.179513	−	0.983756i	$-0.442548\pi$
$89$	26828.7	0.359025	0.179513	+	0.983756i	$0.442548\pi$
$90$	0	0
$91$	7547.66	0.0955452
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6756.45	0.0768086
$96$	0	0
$97$	−138265.	−1.49205	−0.746024	−	0.665919i	$-0.768040\pi$
$97$	−138265.	−1.49205	−0.746024	+	0.665919i	$0.768040\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−99307.3	−0.968675	−0.484337	−	0.874881i	$-0.660939\pi$
$101$	−99307.3	−0.968675	−0.484337	+	0.874881i	$0.660939\pi$
$102$	0	0
$103$	130782.	1.21466	0.607332	−	0.794448i	$-0.292240\pi$
$103$	130782.	1.21466	0.607332	+	0.794448i	$0.292240\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−53831.1	−0.454542	−0.227271	−	0.973832i	$-0.572980\pi$
$107$	−53831.1	−0.454542	−0.227271	+	0.973832i	$0.572980\pi$
$108$	0	0
$109$	65861.0	0.530960	0.265480	−	0.964116i	$-0.414470\pi$
$109$	65861.0	0.530960	0.265480	+	0.964116i	$0.414470\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−22511.4	−0.165847	−0.0829233	−	0.996556i	$-0.526426\pi$
$113$	−22511.4	−0.165847	−0.0829233	+	0.996556i	$0.526426\pi$
$114$	0	0
$115$	27574.7	0.194432
$116$	0	0
$117$	0	0
$118$	0	0
$119$	27494.4	0.177982
$120$	0	0
$121$	−142042.	−0.881969
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−86796.4	−0.477521	−0.238760	−	0.971078i	$-0.576741\pi$
$127$	−86796.4	−0.477521	−0.238760	+	0.971078i	$0.576741\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	207484.	1.05635	0.528174	−	0.849136i	$-0.322877\pi$
$131$	207484.	1.05635	0.528174	+	0.849136i	$0.322877\pi$
$132$	0	0
$133$	−8947.67	−0.0438612
$134$	0	0
$135$	0	0
$136$	0	0
$137$	163205.	0.742902	0.371451	−	0.928453i	$-0.378860\pi$
$137$	163205.	0.742902	0.371451	+	0.928453i	$0.378860\pi$
$138$	0	0
$139$	−351814.	−1.54446	−0.772229	−	0.635345i	$-0.780858\pi$
$139$	−351814.	−1.54446	−0.772229	+	0.635345i	$0.780858\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	31431.3	0.128535
$144$	0	0
$145$	219433.	0.866726
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−180540.	−0.666205	−0.333102	−	0.942891i	$-0.608095\pi$
$149$	−180540.	−0.666205	−0.333102	+	0.942891i	$0.608095\pi$
$150$	0	0
$151$	−68275.9	−0.243683	−0.121842	−	0.992550i	$-0.538880\pi$
$151$	−68275.9	−0.243683	−0.121842	+	0.992550i	$0.538880\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	135481.	0.452949
$156$	0	0
$157$	69845.1	0.226145	0.113072	−	0.993587i	$-0.463931\pi$
$157$	69845.1	0.226145	0.113072	+	0.993587i	$0.463931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−36517.6	−0.111029
$162$	0	0
$163$	−317301.	−0.935410	−0.467705	−	0.883885i	$-0.654919\pi$
$163$	−317301.	−0.935410	−0.467705	+	0.883885i	$0.654919\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−200657.	−0.556753	−0.278376	−	0.960472i	$-0.589796\pi$
$167$	−200657.	−0.556753	−0.278376	+	0.960472i	$0.589796\pi$
$168$	0	0
$169$	−319322.	−0.860026
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6848.86	0.0173982	0.00869908	−	0.999962i	$-0.497231\pi$
$173$	6848.86	0.0173982	0.00869908	+	0.999962i	$0.497231\pi$
$174$	0	0
$175$	−20692.4	−0.0510759
$176$	0	0
$177$	0	0
$178$	0	0
$179$	481129.	1.12235	0.561176	−	0.827697i	$-0.310349\pi$
$179$	481129.	1.12235	0.561176	+	0.827697i	$0.310349\pi$
$180$	0	0
$181$	−525388.	−1.19202	−0.596011	−	0.802977i	$-0.703248\pi$
$181$	−525388.	−1.19202	−0.596011	+	0.802977i	$0.703248\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	123725.	0.265784
$186$	0	0
$187$	114497.	0.239436
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−278503.	−0.552390	−0.276195	−	0.961102i	$-0.589074\pi$
$191$	−278503.	−0.552390	−0.276195	+	0.961102i	$0.589074\pi$
$192$	0	0
$193$	−79995.0	−0.154586	−0.0772929	−	0.997008i	$-0.524628\pi$
$193$	−79995.0	−0.154586	−0.0772929	+	0.997008i	$0.524628\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	309404.	0.568017	0.284008	−	0.958822i	$-0.408336\pi$
$197$	309404.	0.568017	0.284008	+	0.958822i	$0.408336\pi$
$198$	0	0
$199$	475660.	0.851459	0.425729	−	0.904851i	$-0.360018\pi$
$199$	475660.	0.851459	0.425729	+	0.904851i	$0.360018\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−290598.	−0.494941
$204$	0	0
$205$	−349313.	−0.580537
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−37261.4	−0.0590057
$210$	0	0
$211$	123211.	0.190521	0.0952607	−	0.995452i	$-0.469632\pi$
$211$	123211.	0.190521	0.0952607	+	0.995452i	$0.469632\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−292499.	−0.431547
$216$	0	0
$217$	−179419.	−0.258655
$218$	0	0
$219$	0	0
$220$	0	0
$221$	189319.	0.260744
$222$	0	0
$223$	−976649.	−1.31515	−0.657577	−	0.753388i	$-0.728418\pi$
$223$	−976649.	−1.31515	−0.657577	+	0.753388i	$0.728418\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−291187.	−0.375066	−0.187533	−	0.982258i	$-0.560049\pi$
$227$	−291187.	−0.375066	−0.187533	+	0.982258i	$0.560049\pi$
$228$	0	0
$229$	−579929.	−0.730779	−0.365390	−	0.930855i	$-0.619064\pi$
$229$	−579929.	−0.730779	−0.365390	+	0.930855i	$0.619064\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	211086.	0.254724	0.127362	−	0.991856i	$-0.459349\pi$
$233$	211086.	0.254724	0.127362	+	0.991856i	$0.459349\pi$
$234$	0	0
$235$	−274319.	−0.324031
$236$	0	0
$237$	0	0
$238$	0	0
$239$	442814.	0.501448	0.250724	−	0.968059i	$-0.419331\pi$
$239$	442814.	0.501448	0.250724	+	0.968059i	$0.419331\pi$
$240$	0	0
$241$	−135492.	−0.150269	−0.0751347	−	0.997173i	$-0.523939\pi$
$241$	−135492.	−0.150269	−0.0751347	+	0.997173i	$0.523939\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−392772.	−0.418047
$246$	0	0
$247$	−61611.3	−0.0642566
$248$	0	0
$249$	0	0
$250$	0	0
$251$	537659.	0.538670	0.269335	−	0.963047i	$-0.413196\pi$
$251$	537659.	0.538670	0.269335	+	0.963047i	$0.413196\pi$
$252$	0	0
$253$	−152073.	−0.149366
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.95669e6	−1.84794	−0.923971	−	0.382463i	$-0.875076\pi$
$257$	−1.95669e6	−1.84794	−0.923971	+	0.382463i	$0.875076\pi$
$258$	0	0
$259$	−163851.	−0.151775
$260$	0	0
$261$	0	0
$262$	0	0
$263$	818771.	0.729916	0.364958	−	0.931024i	$-0.381083\pi$
$263$	818771.	0.729916	0.364958	+	0.931024i	$0.381083\pi$
$264$	0	0
$265$	−229437.	−0.200701
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.78705e6	−1.50576	−0.752881	−	0.658157i	$-0.771336\pi$
$269$	−1.78705e6	−1.50576	−0.752881	+	0.658157i	$0.771336\pi$
$270$	0	0
$271$	−1.72414e6	−1.42610	−0.713050	−	0.701113i	$-0.752687\pi$
$271$	−1.72414e6	−1.42610	−0.713050	+	0.701113i	$0.752687\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−86170.9	−0.0687114
$276$	0	0
$277$	−709726.	−0.555765	−0.277882	−	0.960615i	$-0.589633\pi$
$277$	−709726.	−0.555765	−0.277882	+	0.960615i	$0.589633\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.23486e6	0.932935	0.466467	−	0.884538i	$-0.345527\pi$
$281$	1.23486e6	0.932935	0.466467	+	0.884538i	$0.345527\pi$
$282$	0	0
$283$	−1.25698e6	−0.932961	−0.466480	−	0.884532i	$-0.654478\pi$
$283$	−1.25698e6	−0.932961	−0.466480	+	0.884532i	$0.654478\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	462600.	0.331513
$288$	0	0
$289$	−730212.	−0.514285
$290$	0	0
$291$	0	0
$292$	0	0
$293$	594593.	0.404623	0.202311	−	0.979321i	$-0.435155\pi$
$293$	594593.	0.404623	0.202311	+	0.979321i	$0.435155\pi$
$294$	0	0
$295$	−475873.	−0.318373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−251451.	−0.162658
$300$	0	0
$301$	387360.	0.246433
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.06101e6	0.653084
$306$	0	0
$307$	−1.49370e6	−0.904517	−0.452259	−	0.891887i	$-0.649382\pi$
$307$	−1.49370e6	−0.904517	−0.452259	+	0.891887i	$0.649382\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.18519e6	−0.694843	−0.347421	−	0.937709i	$-0.612943\pi$
$311$	−1.18519e6	−0.694843	−0.347421	+	0.937709i	$0.612943\pi$
$312$	0	0
$313$	−1.33430e6	−0.769824	−0.384912	−	0.922953i	$-0.625768\pi$
$313$	−1.33430e6	−0.769824	−0.384912	+	0.922953i	$0.625768\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	370669.	0.207175	0.103588	−	0.994620i	$-0.466968\pi$
$317$	370669.	0.207175	0.103588	+	0.994620i	$0.466968\pi$
$318$	0	0
$319$	−1.21016e6	−0.665834
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−224436.	−0.119698
$324$	0	0
$325$	−142482.	−0.0748261
$326$	0	0
$327$	0	0
$328$	0	0
$329$	363285.	0.185037
$330$	0	0
$331$	−1.03096e6	−0.517214	−0.258607	−	0.965983i	$-0.583264\pi$
$331$	−1.03096e6	−0.517214	−0.258607	+	0.965983i	$0.583264\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	716236.	0.348694
$336$	0	0
$337$	−2.56925e6	−1.23234	−0.616172	−	0.787612i	$-0.711317\pi$
$337$	−2.56925e6	−1.23234	−0.616172	+	0.787612i	$0.711317\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−747169.	−0.347963
$342$	0	0
$343$	1.07660e6	0.494103
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.57999e6	1.15025	0.575127	−	0.818064i	$-0.304953\pi$
$347$	2.57999e6	1.15025	0.575127	+	0.818064i	$0.304953\pi$
$348$	0	0
$349$	−868978.	−0.381896	−0.190948	−	0.981600i	$-0.561156\pi$
$349$	−868978.	−0.381896	−0.190948	+	0.981600i	$0.561156\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.08631e6	−1.74540	−0.872698	−	0.488260i	$-0.837632\pi$
$353$	−4.08631e6	−1.74540	−0.872698	+	0.488260i	$0.837632\pi$
$354$	0	0
$355$	1.24797e6	0.525572
$356$	0	0
$357$	0	0
$358$	0	0
$359$	99760.1	0.0408527	0.0204264	−	0.999791i	$-0.493498\pi$
$359$	99760.1	0.0408527	0.0204264	+	0.999791i	$0.493498\pi$
$360$	0	0
$361$	−2.40306e6	−0.970502
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.37404e6	−0.539844
$366$	0	0
$367$	−1.32250e6	−0.512544	−0.256272	−	0.966605i	$-0.582494\pi$
$367$	−1.32250e6	−0.512544	−0.256272	+	0.966605i	$0.582494\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	303847.	0.114609
$372$	0	0
$373$	1.36755e6	0.508945	0.254473	−	0.967080i	$-0.418098\pi$
$373$	1.36755e6	0.508945	0.254473	+	0.967080i	$0.418098\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00098e6	−0.725087
$378$	0	0
$379$	−5.39902e6	−1.93071	−0.965354	−	0.260944i	$-0.915966\pi$
$379$	−5.39902e6	−1.93071	−0.965354	+	0.260944i	$0.915966\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.17266e6	0.756825	0.378412	−	0.925637i	$-0.376470\pi$
$383$	2.17266e6	0.756825	0.378412	+	0.925637i	$0.376470\pi$
$384$	0	0
$385$	114117.	0.0392374
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.16671e6	−0.390922	−0.195461	−	0.980711i	$-0.562620\pi$
$389$	−1.16671e6	−0.390922	−0.195461	+	0.980711i	$0.562620\pi$
$390$	0	0
$391$	−915976.	−0.303000
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−588130.	−0.189662
$396$	0	0
$397$	−216888.	−0.0690653	−0.0345327	−	0.999404i	$-0.510994\pi$
$397$	−216888.	−0.0690653	−0.0345327	+	0.999404i	$0.510994\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−919793.	−0.285647	−0.142823	−	0.989748i	$-0.545618\pi$
$401$	−919793.	−0.285647	−0.142823	+	0.989748i	$0.545618\pi$
$402$	0	0
$403$	−1.23543e6	−0.378928
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−682338.	−0.204180
$408$	0	0
$409$	3.18388e6	0.941128	0.470564	−	0.882366i	$-0.344050\pi$
$409$	3.18388e6	0.941128	0.470564	+	0.882366i	$0.344050\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	630206.	0.181806
$414$	0	0
$415$	−768551.	−0.219055
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.69517e6	0.749981	0.374991	−	0.927029i	$-0.377646\pi$
$419$	2.69517e6	0.749981	0.374991	+	0.927029i	$0.377646\pi$
$420$	0	0
$421$	−2.03839e6	−0.560509	−0.280255	−	0.959926i	$-0.590419\pi$
$421$	−2.03839e6	−0.560509	−0.280255	+	0.959926i	$0.590419\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−519031.	−0.139386
$426$	0	0
$427$	−1.40511e6	−0.372941
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.39661e6	1.39935	0.699677	−	0.714459i	$-0.253327\pi$
$431$	5.39661e6	1.39935	0.699677	+	0.714459i	$0.253327\pi$
$432$	0	0
$433$	347462.	0.0890611	0.0445305	−	0.999008i	$-0.485821\pi$
$433$	347462.	0.0890611	0.0445305	+	0.999008i	$0.485821\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	298092.	0.0746701
$438$	0	0
$439$	−343865.	−0.0851583	−0.0425792	−	0.999093i	$-0.513557\pi$
$439$	−343865.	−0.0851583	−0.0425792	+	0.999093i	$0.513557\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.57697e6	0.865976	0.432988	−	0.901400i	$-0.357459\pi$
$443$	3.57697e6	0.865976	0.432988	+	0.901400i	$0.357459\pi$
$444$	0	0
$445$	670718.	0.160561
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.97834e6	0.931293	0.465647	−	0.884971i	$-0.345822\pi$
$449$	3.97834e6	0.931293	0.465647	+	0.884971i	$0.345822\pi$
$450$	0	0
$451$	1.92644e6	0.445978
$452$	0	0
$453$	0	0
$454$	0	0
$455$	188692.	0.0427291
$456$	0	0
$457$	−4.41560e6	−0.989006	−0.494503	−	0.869176i	$-0.664650\pi$
$457$	−4.41560e6	−0.989006	−0.494503	+	0.869176i	$0.664650\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.37721e6	−0.301820	−0.150910	−	0.988547i	$-0.548220\pi$
$461$	−1.37721e6	−0.301820	−0.150910	+	0.988547i	$0.548220\pi$
$462$	0	0
$463$	−3.70214e6	−0.802602	−0.401301	−	0.915946i	$-0.631442\pi$
$463$	−3.70214e6	−0.802602	−0.401301	+	0.915946i	$0.631442\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	143413.	0.0304297	0.0152148	−	0.999884i	$-0.495157\pi$
$467$	143413.	0.0304297	0.0152148	+	0.999884i	$0.495157\pi$
$468$	0	0
$469$	−948522.	−0.199120
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.61311e6	0.331522
$474$	0	0
$475$	168911.	0.0343498
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.08333e6	0.414877	0.207438	−	0.978248i	$-0.433487\pi$
$479$	2.08333e6	0.414877	0.207438	+	0.978248i	$0.433487\pi$
$480$	0	0
$481$	−1.12824e6	−0.222350
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.45662e6	−0.667264
$486$	0	0
$487$	−419216.	−0.0800969	−0.0400484	−	0.999198i	$-0.512751\pi$
$487$	−419216.	−0.0800969	−0.0400484	+	0.999198i	$0.512751\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.99525e6	0.373503	0.186751	−	0.982407i	$-0.440204\pi$
$491$	1.99525e6	0.373503	0.186751	+	0.982407i	$0.440204\pi$
$492$	0	0
$493$	−7.28912e6	−1.35070
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.65270e6	−0.300126
$498$	0	0
$499$	6.17451e6	1.11007	0.555036	−	0.831826i	$-0.312704\pi$
$499$	6.17451e6	1.11007	0.555036	+	0.831826i	$0.312704\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.75817e6	−0.486072	−0.243036	−	0.970017i	$-0.578143\pi$
$503$	−2.75817e6	−0.486072	−0.243036	+	0.970017i	$0.578143\pi$
$504$	0	0
$505$	−2.48268e6	−0.433204
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.45815e6	−0.762712	−0.381356	−	0.924428i	$-0.624543\pi$
$509$	−4.45815e6	−0.762712	−0.381356	+	0.924428i	$0.624543\pi$
$510$	0	0
$511$	1.81966e6	0.308276
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.26956e6	0.543214
$516$	0	0
$517$	1.51285e6	0.248926
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.14709e7	−1.85141	−0.925705	−	0.378247i	$-0.876527\pi$
$521$	−1.14709e7	−1.85141	−0.925705	+	0.378247i	$0.876527\pi$
$522$	0	0
$523$	−2.02101e6	−0.323083	−0.161541	−	0.986866i	$-0.551647\pi$
$523$	−2.02101e6	−0.323083	−0.161541	+	0.986866i	$0.551647\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.50040e6	−0.705870
$528$	0	0
$529$	−5.21976e6	−0.810982
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.18534e6	0.485666
$534$	0	0
$535$	−1.34578e6	−0.203277
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.16611e6	0.321151
$540$	0	0
$541$	6.26733e6	0.920639	0.460319	−	0.887753i	$-0.347735\pi$
$541$	6.26733e6	0.920639	0.460319	+	0.887753i	$0.347735\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.64653e6	0.237453
$546$	0	0
$547$	−1.01427e6	−0.144938	−0.0724692	−	0.997371i	$-0.523088\pi$
$547$	−1.01427e6	−0.144938	−0.0724692	+	0.997371i	$0.523088\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.37214e6	0.332860
$552$	0	0
$553$	778868.	0.108306
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.94075e6	−0.811341	−0.405671	−	0.914019i	$-0.632962\pi$
$557$	−5.94075e6	−0.811341	−0.405671	+	0.914019i	$0.632962\pi$
$558$	0	0
$559$	2.66726e6	0.361024
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−330820.	−0.0439866	−0.0219933	−	0.999758i	$-0.507001\pi$
$563$	−330820.	−0.0439866	−0.0219933	+	0.999758i	$0.507001\pi$
$564$	0	0
$565$	−562785.	−0.0741689
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.02192e6	0.391293	0.195646	−	0.980675i	$-0.437320\pi$
$569$	3.02192e6	0.391293	0.195646	+	0.980675i	$0.437320\pi$
$570$	0	0
$571$	4.88380e6	0.626856	0.313428	−	0.949612i	$-0.398523\pi$
$571$	4.88380e6	0.626856	0.313428	+	0.949612i	$0.398523\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	689368.	0.0869524
$576$	0	0
$577$	2.56920e6	0.321261	0.160630	−	0.987015i	$-0.448647\pi$
$577$	2.56920e6	0.321261	0.160630	+	0.987015i	$0.448647\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.01780e6	0.125090
$582$	0	0
$583$	1.26533e6	0.154182
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.98845e6	0.597545	0.298772	−	0.954324i	$-0.403423\pi$
$587$	4.98845e6	0.597545	0.298772	+	0.954324i	$0.403423\pi$
$588$	0	0
$589$	1.46459e6	0.173952
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.04060e6	−0.938970	−0.469485	−	0.882940i	$-0.655560\pi$
$593$	−8.04060e6	−0.938970	−0.469485	+	0.882940i	$0.655560\pi$
$594$	0	0
$595$	687360.	0.0795961
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.15088e7	1.31058	0.655290	−	0.755377i	$-0.272546\pi$
$599$	1.15088e7	1.31058	0.655290	+	0.755377i	$0.272546\pi$
$600$	0	0
$601$	−4.61127e6	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$601$	−4.61127e6	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.55105e6	−0.394428
$606$	0	0
$607$	1.30684e7	1.43963	0.719815	−	0.694166i	$-0.244227\pi$
$607$	1.30684e7	1.43963	0.719815	+	0.694166i	$0.244227\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.50148e6	0.271078
$612$	0	0
$613$	−2.68983e6	−0.289117	−0.144558	−	0.989496i	$-0.546176\pi$
$613$	−2.68983e6	−0.289117	−0.144558	+	0.989496i	$0.546176\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.71781e6	−0.710420	−0.355210	−	0.934787i	$-0.615591\pi$
$617$	−6.71781e6	−0.710420	−0.355210	+	0.934787i	$0.615591\pi$
$618$	0	0
$619$	−1.26128e7	−1.32308	−0.661538	−	0.749912i	$-0.730096\pi$
$619$	−1.26128e7	−1.32308	−0.661538	+	0.749912i	$0.730096\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−888241.	−0.0916876
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.10990e6	−0.414195
$630$	0	0
$631$	−1.60794e7	−1.60767	−0.803834	−	0.594854i	$-0.797210\pi$
$631$	−1.60794e7	−1.60767	−0.803834	+	0.594854i	$0.797210\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.16991e6	−0.213554
$636$	0	0
$637$	3.58164e6	0.349730
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.09952e7	−1.05696	−0.528482	−	0.848945i	$-0.677239\pi$
$641$	−1.09952e7	−1.05696	−0.528482	+	0.848945i	$0.677239\pi$
$642$	0	0
$643$	1.69080e7	1.61274	0.806369	−	0.591412i	$-0.201429\pi$
$643$	1.69080e7	1.61274	0.806369	+	0.591412i	$0.201429\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.13123e7	1.06241	0.531203	−	0.847245i	$-0.321740\pi$
$647$	1.13123e7	1.06241	0.531203	+	0.847245i	$0.321740\pi$
$648$	0	0
$649$	2.62441e6	0.244580
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.39483e6	−0.678649	−0.339324	−	0.940669i	$-0.610198\pi$
$653$	−7.39483e6	−0.678649	−0.339324	+	0.940669i	$0.610198\pi$
$654$	0	0
$655$	5.18710e6	0.472413
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.15556e6	−0.193351	−0.0966755	−	0.995316i	$-0.530821\pi$
$659$	−2.15556e6	−0.193351	−0.0966755	+	0.995316i	$0.530821\pi$
$660$	0	0
$661$	1.08558e7	0.966407	0.483203	−	0.875508i	$-0.339473\pi$
$661$	1.08558e7	0.966407	0.483203	+	0.875508i	$0.339473\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−223692.	−0.0196153
$666$	0	0
$667$	9.68129e6	0.842595
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.85139e6	−0.501710
$672$	0	0
$673$	1.05830e7	0.900684	0.450342	−	0.892856i	$-0.351302\pi$
$673$	1.05830e7	0.900684	0.450342	+	0.892856i	$0.351302\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.33266e7	−1.11750	−0.558751	−	0.829336i	$-0.688719\pi$
$677$	−1.33266e7	−1.11750	−0.558751	+	0.829336i	$0.688719\pi$
$678$	0	0
$679$	4.57766e6	0.381039
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.58506e7	1.30015	0.650076	−	0.759870i	$-0.274737\pi$
$683$	1.58506e7	1.30015	0.650076	+	0.759870i	$0.274737\pi$
$684$	0	0
$685$	4.08012e6	0.332236
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.09221e6	0.167902
$690$	0	0
$691$	−1.08677e7	−0.865854	−0.432927	−	0.901429i	$-0.642519\pi$
$691$	−1.08677e7	−0.865854	−0.432927	+	0.901429i	$0.642519\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.79535e6	−0.690702
$696$	0	0
$697$	1.16035e7	0.904702
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.32082e7	1.01519	0.507596	−	0.861595i	$-0.330534\pi$
$701$	1.32082e7	1.01519	0.507596	+	0.861595i	$0.330534\pi$
$702$	0	0
$703$	1.33751e6	0.102073
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.28785e6	0.247380
$708$	0	0
$709$	3.88652e6	0.290366	0.145183	−	0.989405i	$-0.453623\pi$
$709$	3.88652e6	0.290366	0.145183	+	0.989405i	$0.453623\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.97736e6	0.440338
$714$	0	0
$715$	785782.	0.0574827
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.12534e6	0.586164	0.293082	−	0.956087i	$-0.405319\pi$
$719$	8.12534e6	0.586164	0.293082	+	0.956087i	$0.405319\pi$
$720$	0	0
$721$	−4.32993e6	−0.310200
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.48583e6	0.387612
$726$	0	0
$727$	−7.16766e6	−0.502969	−0.251484	−	0.967861i	$-0.580919\pi$
$727$	−7.16766e6	−0.502969	−0.251484	+	0.967861i	$0.580919\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.71621e6	0.672517
$732$	0	0
$733$	2.55569e7	1.75691	0.878454	−	0.477828i	$-0.158576\pi$
$733$	2.55569e7	1.75691	0.878454	+	0.477828i	$0.158576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.95000e6	−0.267872
$738$	0	0
$739$	2.15168e7	1.44933	0.724664	−	0.689102i	$-0.241995\pi$
$739$	2.15168e7	1.44933	0.724664	+	0.689102i	$0.241995\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.89110e7	1.25673	0.628364	−	0.777919i	$-0.283725\pi$
$743$	1.89110e7	1.25673	0.628364	+	0.777919i	$0.283725\pi$
$744$	0	0
$745$	−4.51350e6	−0.297936
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.78223e6	0.116081
$750$	0	0
$751$	1.93031e7	1.24890	0.624451	−	0.781064i	$-0.285323\pi$
$751$	1.93031e7	1.24890	0.624451	+	0.781064i	$0.285323\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.70690e6	−0.108978
$756$	0	0
$757$	−3.11575e6	−0.197617	−0.0988083	−	0.995106i	$-0.531503\pi$
$757$	−3.11575e6	−0.197617	−0.0988083	+	0.995106i	$0.531503\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−150251.	−0.00940494	−0.00470247	−	0.999989i	$-0.501497\pi$
$761$	−150251.	−0.00940494	−0.00470247	+	0.999989i	$0.501497\pi$
$762$	0	0
$763$	−2.18052e6	−0.135596
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.33943e6	0.266345
$768$	0	0
$769$	−6.18870e6	−0.377384	−0.188692	−	0.982036i	$-0.560425\pi$
$769$	−6.18870e6	−0.377384	−0.188692	+	0.982036i	$0.560425\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.53085e7	−1.52341	−0.761705	−	0.647924i	$-0.775638\pi$
$773$	−2.53085e7	−1.52341	−0.761705	+	0.647924i	$0.775638\pi$
$774$	0	0
$775$	3.38702e6	0.202565
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.77619e6	−0.222951
$780$	0	0
$781$	−6.88246e6	−0.403753
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.74613e6	0.101135
$786$	0	0
$787$	−3.78088e6	−0.217599	−0.108799	−	0.994064i	$-0.534701\pi$
$787$	−3.78088e6	−0.217599	−0.108799	+	0.994064i	$0.534701\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	745305.	0.0423538
$792$	0	0
$793$	−9.67520e6	−0.546358
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.72025e7	−0.959279	−0.479640	−	0.877466i	$-0.659233\pi$
$797$	−1.72025e7	−0.959279	−0.479640	+	0.877466i	$0.659233\pi$
$798$	0	0
$799$	9.11232e6	0.504966
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.57776e6	0.414717
$804$	0	0
$805$	−912940.	−0.0496538
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.08310e6	−0.111902	−0.0559511	−	0.998434i	$-0.517819\pi$
$809$	−2.08310e6	−0.111902	−0.0559511	+	0.998434i	$0.517819\pi$
$810$	0	0
$811$	2.73921e7	1.46242	0.731212	−	0.682151i	$-0.238955\pi$
$811$	2.73921e7	1.46242	0.731212	+	0.682151i	$0.238955\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.93251e6	−0.418328
$816$	0	0
$817$	−3.16201e6	−0.165732
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.14075e7	−0.590654	−0.295327	−	0.955396i	$-0.595429\pi$
$821$	−1.14075e7	−0.590654	−0.295327	+	0.955396i	$0.595429\pi$
$822$	0	0
$823$	7.66158e6	0.394293	0.197146	−	0.980374i	$-0.436833\pi$
$823$	7.66158e6	0.394293	0.197146	+	0.980374i	$0.436833\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.90683e7	−0.969501	−0.484751	−	0.874652i	$-0.661090\pi$
$827$	−1.90683e7	−0.969501	−0.484751	+	0.874652i	$0.661090\pi$
$828$	0	0
$829$	1.87753e7	0.948856	0.474428	−	0.880294i	$-0.342655\pi$
$829$	1.87753e7	0.948856	0.474428	+	0.880294i	$0.342655\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.30471e7	0.651479
$834$	0	0
$835$	−5.01641e6	−0.248987
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.54970e7	1.25050	0.625251	−	0.780424i	$-0.284997\pi$
$839$	2.54970e7	1.25050	0.625251	+	0.780424i	$0.284997\pi$
$840$	0	0
$841$	5.65302e7	2.75607
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.98304e6	−0.384616
$846$	0	0
$847$	4.70270e6	0.225237
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.45871e6	0.258384
$852$	0	0
$853$	582788.	0.0274245	0.0137122	−	0.999906i	$-0.495635\pi$
$853$	582788.	0.0274245	0.0137122	+	0.999906i	$0.495635\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.12119e7	−0.521469	−0.260734	−	0.965411i	$-0.583965\pi$
$857$	−1.12119e7	−0.521469	−0.260734	+	0.965411i	$0.583965\pi$
$858$	0	0
$859$	2.42006e7	1.11904	0.559518	−	0.828818i	$-0.310986\pi$
$859$	2.42006e7	1.11904	0.559518	+	0.828818i	$0.310986\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.26534e7	1.49246	0.746228	−	0.665691i	$-0.231863\pi$
$863$	3.26534e7	1.49246	0.746228	+	0.665691i	$0.231863\pi$
$864$	0	0
$865$	171222.	0.00778069
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.24350e6	0.145702
$870$	0	0
$871$	−6.53127e6	−0.291711
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−517310.	−0.0228418
$876$	0	0
$877$	3.37517e7	1.48182	0.740912	−	0.671602i	$-0.234393\pi$
$877$	3.37517e7	1.48182	0.740912	+	0.671602i	$0.234393\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.32496e6	0.187734	0.0938670	−	0.995585i	$-0.470077\pi$
$881$	4.32496e6	0.187734	0.0938670	+	0.995585i	$0.470077\pi$
$882$	0	0
$883$	−1.78610e7	−0.770912	−0.385456	−	0.922726i	$-0.625956\pi$
$883$	−1.78610e7	−0.770912	−0.385456	+	0.922726i	$0.625956\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.11624e7	−0.903144	−0.451572	−	0.892235i	$-0.649137\pi$
$887$	−2.11624e7	−0.903144	−0.451572	+	0.892235i	$0.649137\pi$
$888$	0	0
$889$	2.87364e6	0.121949
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.96548e6	−0.124442
$894$	0	0
$895$	1.20282e7	0.501931
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.75664e7	1.96291
$900$	0	0
$901$	7.62143e6	0.312770
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.31347e7	−0.533088
$906$	0	0
$907$	−2.36325e6	−0.0953876	−0.0476938	−	0.998862i	$-0.515187\pi$
$907$	−2.36325e6	−0.0953876	−0.0476938	+	0.998862i	$0.515187\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.28155e7	1.31003	0.655017	−	0.755614i	$-0.272661\pi$
$911$	3.28155e7	1.31003	0.655017	+	0.755614i	$0.272661\pi$
$912$	0	0
$913$	4.23851e6	0.168281
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.86936e6	−0.269769
$918$	0	0
$919$	−4.30080e7	−1.67981	−0.839906	−	0.542733i	$-0.817390\pi$
$919$	−4.30080e7	−1.67981	−0.839906	+	0.542733i	$0.817390\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.13801e7	−0.439683
$924$	0	0
$925$	3.09313e6	0.118862
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.83568e6	−0.259862	−0.129931	−	0.991523i	$-0.541476\pi$
$929$	−6.83568e6	−0.259862	−0.129931	+	0.991523i	$0.541476\pi$
$930$	0	0
$931$	−4.24599e6	−0.160548
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.86242e6	0.107079
$936$	0	0
$937$	−1.51219e7	−0.562673	−0.281337	−	0.959609i	$-0.590778\pi$
$937$	−1.51219e7	−0.562673	−0.281337	+	0.959609i	$0.590778\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.36326e6	0.160634	0.0803169	−	0.996769i	$-0.474407\pi$
$941$	4.36326e6	0.160634	0.0803169	+	0.996769i	$0.474407\pi$
$942$	0	0
$943$	−1.54115e7	−0.564374
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.24772e7	−1.53915	−0.769575	−	0.638557i	$-0.779532\pi$
$947$	−4.24772e7	−1.53915	−0.769575	+	0.638557i	$0.779532\pi$
$948$	0	0
$949$	1.25297e7	0.451623
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.30802e7	0.466533	0.233266	−	0.972413i	$-0.425059\pi$
$953$	1.30802e7	0.466533	0.233266	+	0.972413i	$0.425059\pi$
$954$	0	0
$955$	−6.96257e6	−0.247036
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.40336e6	−0.189722
$960$	0	0
$961$	739000.	0.0258129
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.99987e6	−0.0691329
$966$	0	0
$967$	3.54258e7	1.21830	0.609149	−	0.793056i	$-0.291511\pi$
$967$	3.54258e7	1.21830	0.609149	+	0.793056i	$0.291511\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.86993e6	0.267869	0.133935	−	0.990990i	$-0.457239\pi$
$971$	7.86993e6	0.267869	0.133935	+	0.990990i	$0.457239\pi$
$972$	0	0
$973$	1.16478e7	0.394423
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.49639e6	0.117188	0.0585941	−	0.998282i	$-0.481338\pi$
$977$	3.49639e6	0.117188	0.0585941	+	0.998282i	$0.481338\pi$
$978$	0	0
$979$	−3.69897e6	−0.123346
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.75550e7	−0.909531	−0.454765	−	0.890611i	$-0.650277\pi$
$983$	−2.75550e7	−0.909531	−0.454765	+	0.890611i	$0.650277\pi$
$984$	0	0
$985$	7.73511e6	0.254025
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.29049e7	−0.419531
$990$	0	0
$991$	3.04424e7	0.984680	0.492340	−	0.870403i	$-0.336142\pi$
$991$	3.04424e7	0.984680	0.492340	+	0.870403i	$0.336142\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.18915e7	0.380784
$996$	0	0
$997$	−8.46733e6	−0.269779	−0.134890	−	0.990861i	$-0.543068\pi$
$997$	−8.46733e6	−0.269779	−0.134890	+	0.990861i	$0.543068\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.f.1.1	✓	3	3.2	odd	2
1080.6.a.h.1.1	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)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} -33.1079 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} -33.1079 q^{7} -137.873 q^{11} -227.972 q^{13} -830.449 q^{17} +270.258 q^{19} +1102.99 q^{23} +625.000 q^{25} +8777.32 q^{29} +5419.24 q^{31} -827.696 q^{35} +4949.02 q^{37} -13972.5 q^{41} -11699.9 q^{43} -10972.8 q^{47} -15710.9 q^{49} -9177.48 q^{53} -3446.84 q^{55} -19034.9 q^{59} +42440.3 q^{61} -5699.30 q^{65} +28649.4 q^{67} +49918.7 q^{71} -54961.7 q^{73} +4564.69 q^{77} -23525.2 q^{79} -30742.0 q^{83} -20761.2 q^{85} +26828.7 q^{89} +7547.66 q^{91} +6756.45 q^{95} -138265. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 75 q^{5} + 140 q^{7}+O(q^{10}) \) 3 * q + 75 * q^5 + 140 * q^7	\( 3 q + 75 q^{5} + 140 q^{7} - 163 q^{11} + 51 q^{13} - 605 q^{17} - 1622 q^{19} + 1645 q^{23} + 1875 q^{25} - 557 q^{29} - 2807 q^{31} + 3500 q^{35} - 6140 q^{37} - 13760 q^{41} - 2369 q^{43} - 14909 q^{47} - 34015 q^{49} + 964 q^{53} - 4075 q^{55} - 48240 q^{59} + 6296 q^{61} + 1275 q^{65} + 4826 q^{67} - 27988 q^{71} - 36328 q^{73} - 6930 q^{77} + 25293 q^{79} - 48246 q^{83} - 15125 q^{85} + 41208 q^{89} + 18204 q^{91} - 40550 q^{95} - 49164 q^{97}+O(q^{100}) \) 3 * q + 75 * q^5 + 140 * q^7 - 163 * q^11 + 51 * q^13 - 605 * q^17 - 1622 * q^19 + 1645 * q^23 + 1875 * q^25 - 557 * q^29 - 2807 * q^31 + 3500 * q^35 - 6140 * q^37 - 13760 * q^41 - 2369 * q^43 - 14909 * q^47 - 34015 * q^49 + 964 * q^53 - 4075 * q^55 - 48240 * q^59 + 6296 * q^61 + 1275 * q^65 + 4826 * q^67 - 27988 * q^71 - 36328 * q^73 - 6930 * q^77 + 25293 * q^79 - 48246 * q^83 - 15125 * q^85 + 41208 * q^89 + 18204 * q^91 - 40550 * q^95 - 49164 * q^97

Introduction

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