LMFDB - Embedded newform 1080.6.a.h.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:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
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			1.3
Root			$-2.47876$ 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)

$f(q)$	$=$	$q+25.0000 q^{5} +99.3349 q^{7} +O(q^{10})$	$q+25.0000 q^{5} +99.3349 q^{7} -377.165 q^{11} -388.245 q^{13} +1158.43 q^{17} +246.899 q^{19} -195.844 q^{23} +625.000 q^{25} -3268.26 q^{29} +160.538 q^{31} +2483.37 q^{35} -9122.21 q^{37} +12382.6 q^{41} +8303.37 q^{43} -11023.0 q^{47} -6939.58 q^{49} +6025.60 q^{53} -9429.11 q^{55} +1557.29 q^{59} -50994.4 q^{61} -9706.11 q^{65} +15506.8 q^{67} -50902.6 q^{71} -25207.0 q^{73} -37465.6 q^{77} +16866.4 q^{79} -108648. q^{83} +28960.7 q^{85} +46103.4 q^{89} -38566.2 q^{91} +6172.49 q^{95} +178039. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	99.3349	0.766226	0.383113	−	0.923702i	$-0.374852\pi$
$7$	99.3349	0.766226	0.383113	+	0.923702i	$0.374852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−377.165	−0.939830	−0.469915	−	0.882712i	$-0.655715\pi$
$11$	−377.165	−0.939830	−0.469915	+	0.882712i	$0.655715\pi$
$12$	0	0
$13$	−388.245	−0.637158	−0.318579	−	0.947896i	$-0.603206\pi$
$13$	−388.245	−0.637158	−0.318579	+	0.947896i	$0.603206\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1158.43	0.972180	0.486090	−	0.873909i	$-0.338423\pi$
$17$	1158.43	0.972180	0.486090	+	0.873909i	$0.338423\pi$
$18$	0	0
$19$	246.899	0.156905	0.0784524	−	0.996918i	$-0.475002\pi$
$19$	246.899	0.156905	0.0784524	+	0.996918i	$0.475002\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−195.844	−0.0771951	−0.0385975	−	0.999255i	$-0.512289\pi$
$23$	−195.844	−0.0771951	−0.0385975	+	0.999255i	$0.512289\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3268.26	−0.721641	−0.360821	−	0.932635i	$-0.617503\pi$
$29$	−3268.26	−0.721641	−0.360821	+	0.932635i	$0.617503\pi$
$30$	0	0
$31$	160.538	0.0300036	0.0150018	−	0.999887i	$-0.495225\pi$
$31$	160.538	0.0300036	0.0150018	+	0.999887i	$0.495225\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2483.37	0.342666
$36$	0	0
$37$	−9122.21	−1.09546	−0.547729	−	0.836656i	$-0.684508\pi$
$37$	−9122.21	−1.09546	−0.547729	+	0.836656i	$0.684508\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12382.6	1.15041	0.575204	−	0.818010i	$-0.304923\pi$
$41$	12382.6	1.15041	0.575204	+	0.818010i	$0.304923\pi$
$42$	0	0
$43$	8303.37	0.684831	0.342416	−	0.939549i	$-0.388755\pi$
$43$	8303.37	0.684831	0.342416	+	0.939549i	$0.388755\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11023.0	−0.727869	−0.363935	−	0.931424i	$-0.618567\pi$
$47$	−11023.0	−0.727869	−0.363935	+	0.931424i	$0.618567\pi$
$48$	0	0
$49$	−6939.58	−0.412898
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6025.60	0.294653	0.147326	−	0.989088i	$-0.452933\pi$
$53$	6025.60	0.294653	0.147326	+	0.989088i	$0.452933\pi$
$54$	0	0
$55$	−9429.11	−0.420305
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1557.29	0.0582423	0.0291212	−	0.999576i	$-0.490729\pi$
$59$	1557.29	0.0582423	0.0291212	+	0.999576i	$0.490729\pi$
$60$	0	0
$61$	−50994.4	−1.75468	−0.877340	−	0.479869i	$-0.840684\pi$
$61$	−50994.4	−1.75468	−0.877340	+	0.479869i	$0.840684\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9706.11	−0.284946
$66$	0	0
$67$	15506.8	0.422021	0.211011	−	0.977484i	$-0.432325\pi$
$67$	15506.8	0.422021	0.211011	+	0.977484i	$0.432325\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−50902.6	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$71$	−50902.6	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$72$	0	0
$73$	−25207.0	−0.553624	−0.276812	−	0.960924i	$-0.589278\pi$
$73$	−25207.0	−0.553624	−0.276812	+	0.960924i	$0.589278\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−37465.6	−0.720122
$78$	0	0
$79$	16866.4	0.304056	0.152028	−	0.988376i	$-0.451420\pi$
$79$	16866.4	0.304056	0.152028	+	0.988376i	$0.451420\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−108648.	−1.73112	−0.865561	−	0.500803i	$-0.833038\pi$
$83$	−108648.	−1.73112	−0.865561	+	0.500803i	$0.833038\pi$
$84$	0	0
$85$	28960.7	0.434772
$86$	0	0
$87$	0	0
$88$	0	0
$89$	46103.4	0.616962	0.308481	−	0.951231i	$-0.400179\pi$
$89$	46103.4	0.616962	0.308481	+	0.951231i	$0.400179\pi$
$90$	0	0
$91$	−38566.2	−0.488207
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6172.49	0.0701699
$96$	0	0
$97$	178039.	1.92126	0.960632	−	0.277825i	$-0.0896136\pi$
$97$	178039.	1.92126	0.960632	+	0.277825i	$0.0896136\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−22874.9	−0.223129	−0.111565	−	0.993757i	$-0.535586\pi$
$101$	−22874.9	−0.223129	−0.111565	+	0.993757i	$0.535586\pi$
$102$	0	0
$103$	−35513.2	−0.329835	−0.164918	−	0.986307i	$-0.552736\pi$
$103$	−35513.2	−0.329835	−0.164918	+	0.986307i	$0.552736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	181041.	1.52868	0.764342	−	0.644811i	$-0.223064\pi$
$107$	181041.	1.52868	0.764342	+	0.644811i	$0.223064\pi$
$108$	0	0
$109$	−135116.	−1.08928	−0.544641	−	0.838669i	$-0.683334\pi$
$109$	−135116.	−1.08928	−0.544641	+	0.838669i	$0.683334\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	25161.8	0.185373	0.0926863	−	0.995695i	$-0.470455\pi$
$113$	25161.8	0.185373	0.0926863	+	0.995695i	$0.470455\pi$
$114$	0	0
$115$	−4896.09	−0.0345227
$116$	0	0
$117$	0	0
$118$	0	0
$119$	115072.	0.744909
$120$	0	0
$121$	−18797.9	−0.116720
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	199066.	1.09519	0.547594	−	0.836744i	$-0.315544\pi$
$127$	199066.	1.09519	0.547594	+	0.836744i	$0.315544\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−156929.	−0.798962	−0.399481	−	0.916741i	$-0.630810\pi$
$131$	−156929.	−0.798962	−0.399481	+	0.916741i	$0.630810\pi$
$132$	0	0
$133$	24525.7	0.120224
$134$	0	0
$135$	0	0
$136$	0	0
$137$	111075.	0.505611	0.252806	−	0.967517i	$-0.418647\pi$
$137$	111075.	0.505611	0.252806	+	0.967517i	$0.418647\pi$
$138$	0	0
$139$	−38994.0	−0.171183	−0.0855915	−	0.996330i	$-0.527278\pi$
$139$	−38994.0	−0.171183	−0.0855915	+	0.996330i	$0.527278\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	146432.	0.598820
$144$	0	0
$145$	−81706.5	−0.322728
$146$	0	0
$147$	0	0
$148$	0	0
$149$	128256.	0.473272	0.236636	−	0.971598i	$-0.423955\pi$
$149$	128256.	0.473272	0.236636	+	0.971598i	$0.423955\pi$
$150$	0	0
$151$	−40903.4	−0.145988	−0.0729939	−	0.997332i	$-0.523255\pi$
$151$	−40903.4	−0.145988	−0.0729939	+	0.997332i	$0.523255\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4013.45	0.0134180
$156$	0	0
$157$	−278801.	−0.902705	−0.451352	−	0.892346i	$-0.649058\pi$
$157$	−278801.	−0.902705	−0.451352	+	0.892346i	$0.649058\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−19454.1	−0.0591488
$162$	0	0
$163$	−407634.	−1.20171	−0.600857	−	0.799357i	$-0.705174\pi$
$163$	−407634.	−1.20171	−0.600857	+	0.799357i	$0.705174\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−63166.6	−0.175265	−0.0876327	−	0.996153i	$-0.527930\pi$
$167$	−63166.6	−0.175265	−0.0876327	+	0.996153i	$0.527930\pi$
$168$	0	0
$169$	−220559.	−0.594030
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−37294.0	−0.0947380	−0.0473690	−	0.998877i	$-0.515084\pi$
$173$	−37294.0	−0.0947380	−0.0473690	+	0.998877i	$0.515084\pi$
$174$	0	0
$175$	62084.3	0.153245
$176$	0	0
$177$	0	0
$178$	0	0
$179$	88336.1	0.206066	0.103033	−	0.994678i	$-0.467145\pi$
$179$	88336.1	0.206066	0.103033	+	0.994678i	$0.467145\pi$
$180$	0	0
$181$	−205793.	−0.466910	−0.233455	−	0.972368i	$-0.575003\pi$
$181$	−205793.	−0.466910	−0.233455	+	0.972368i	$0.575003\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−228055.	−0.489904
$186$	0	0
$187$	−436918.	−0.913684
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2274.66	−0.00451162	−0.00225581	−	0.999997i	$-0.500718\pi$
$191$	−2274.66	−0.00451162	−0.00225581	+	0.999997i	$0.500718\pi$
$192$	0	0
$193$	−537826.	−1.03932	−0.519659	−	0.854374i	$-0.673941\pi$
$193$	−537826.	−1.03932	−0.519659	+	0.854374i	$0.673941\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−463032.	−0.850052	−0.425026	−	0.905181i	$-0.639735\pi$
$197$	−463032.	−0.850052	−0.425026	+	0.905181i	$0.639735\pi$
$198$	0	0
$199$	−550731.	−0.985841	−0.492920	−	0.870074i	$-0.664071\pi$
$199$	−550731.	−0.985841	−0.492920	+	0.870074i	$0.664071\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−324652.	−0.552940
$204$	0	0
$205$	309565.	0.514478
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−93121.7	−0.147464
$210$	0	0
$211$	−28057.4	−0.0433851	−0.0216926	−	0.999765i	$-0.506905\pi$
$211$	−28057.4	−0.0433851	−0.0216926	+	0.999765i	$0.506905\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	207584.	0.306266
$216$	0	0
$217$	15947.0	0.0229895
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−449753.	−0.619432
$222$	0	0
$223$	52686.4	0.0709473	0.0354737	−	0.999371i	$-0.488706\pi$
$223$	52686.4	0.0709473	0.0354737	+	0.999371i	$0.488706\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	996201.	1.28316	0.641582	−	0.767054i	$-0.278278\pi$
$227$	996201.	1.28316	0.641582	+	0.767054i	$0.278278\pi$
$228$	0	0
$229$	−822292.	−1.03619	−0.518093	−	0.855325i	$-0.673358\pi$
$229$	−822292.	−1.03619	−0.518093	+	0.855325i	$0.673358\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	379890.	0.458424	0.229212	−	0.973376i	$-0.426385\pi$
$233$	379890.	0.458424	0.229212	+	0.973376i	$0.426385\pi$
$234$	0	0
$235$	−275574.	−0.325513
$236$	0	0
$237$	0	0
$238$	0	0
$239$	514548.	0.582682	0.291341	−	0.956619i	$-0.405899\pi$
$239$	514548.	0.582682	0.291341	+	0.956619i	$0.405899\pi$
$240$	0	0
$241$	−1.02435e6	−1.13608	−0.568038	−	0.823003i	$-0.692297\pi$
$241$	−1.02435e6	−1.13608	−0.568038	+	0.823003i	$0.692297\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−173490.	−0.184654
$246$	0	0
$247$	−95857.4	−0.0999731
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−572497.	−0.573574	−0.286787	−	0.957994i	$-0.592587\pi$
$251$	−572497.	−0.573574	−0.286787	+	0.957994i	$0.592587\pi$
$252$	0	0
$253$	73865.2	0.0725502
$254$	0	0
$255$	0	0
$256$	0	0
$257$	687168.	0.648978	0.324489	−	0.945889i	$-0.394808\pi$
$257$	687168.	0.648978	0.324489	+	0.945889i	$0.394808\pi$
$258$	0	0
$259$	−906154.	−0.839368
$260$	0	0
$261$	0	0
$262$	0	0
$263$	185741.	0.165584	0.0827921	−	0.996567i	$-0.473616\pi$
$263$	185741.	0.165584	0.0827921	+	0.996567i	$0.473616\pi$
$264$	0	0
$265$	150640.	0.131773
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−443456.	−0.373654	−0.186827	−	0.982393i	$-0.559820\pi$
$269$	−443456.	−0.373654	−0.186827	+	0.982393i	$0.559820\pi$
$270$	0	0
$271$	−130669.	−0.108081	−0.0540403	−	0.998539i	$-0.517210\pi$
$271$	−130669.	−0.108081	−0.0540403	+	0.998539i	$0.517210\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−235728.	−0.187966
$276$	0	0
$277$	−1.63482e6	−1.28018	−0.640090	−	0.768300i	$-0.721103\pi$
$277$	−1.63482e6	−1.28018	−0.640090	+	0.768300i	$0.721103\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.42734e6	−1.07835	−0.539176	−	0.842193i	$-0.681264\pi$
$281$	−1.42734e6	−1.07835	−0.539176	+	0.842193i	$0.681264\pi$
$282$	0	0
$283$	−1.02228e6	−0.758757	−0.379379	−	0.925242i	$-0.623862\pi$
$283$	−1.02228e6	−0.758757	−0.379379	+	0.925242i	$0.623862\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.23002e6	0.881471
$288$	0	0
$289$	−77902.6	−0.0548665
$290$	0	0
$291$	0	0
$292$	0	0
$293$	505763.	0.344174	0.172087	−	0.985082i	$-0.444949\pi$
$293$	505763.	0.344174	0.172087	+	0.985082i	$0.444949\pi$
$294$	0	0
$295$	38932.2	0.0260468
$296$	0	0
$297$	0	0
$298$	0	0
$299$	76035.2	0.0491854
$300$	0	0
$301$	824815.	0.524735
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.27486e6	−0.784717
$306$	0	0
$307$	−1.88839e6	−1.14353	−0.571763	−	0.820419i	$-0.693740\pi$
$307$	−1.88839e6	−1.14353	−0.571763	+	0.820419i	$0.693740\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	356728.	0.209139	0.104570	−	0.994518i	$-0.466653\pi$
$311$	356728.	0.209139	0.104570	+	0.994518i	$0.466653\pi$
$312$	0	0
$313$	−706711.	−0.407738	−0.203869	−	0.978998i	$-0.565352\pi$
$313$	−706711.	−0.407738	−0.203869	+	0.978998i	$0.565352\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.36781e6	−0.764502	−0.382251	−	0.924058i	$-0.624851\pi$
$317$	−1.36781e6	−0.764502	−0.382251	+	0.924058i	$0.624851\pi$
$318$	0	0
$319$	1.23267e6	0.678220
$320$	0	0
$321$	0	0
$322$	0	0
$323$	286015.	0.152540
$324$	0	0
$325$	−242653.	−0.127432
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.09496e6	−0.557712
$330$	0	0
$331$	−935015.	−0.469082	−0.234541	−	0.972106i	$-0.575359\pi$
$331$	−935015.	−0.469082	−0.234541	+	0.972106i	$0.575359\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	387669.	0.188734
$336$	0	0
$337$	−1.11603e6	−0.535304	−0.267652	−	0.963516i	$-0.586248\pi$
$337$	−1.11603e6	−0.535304	−0.267652	+	0.963516i	$0.586248\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−60549.3	−0.0281983
$342$	0	0
$343$	−2.35886e6	−1.08260
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.64476e6	−1.17913	−0.589566	−	0.807721i	$-0.700701\pi$
$347$	−2.64476e6	−1.17913	−0.589566	+	0.807721i	$0.700701\pi$
$348$	0	0
$349$	−1.93475e6	−0.850279	−0.425139	−	0.905128i	$-0.639775\pi$
$349$	−1.93475e6	−0.850279	−0.425139	+	0.905128i	$0.639775\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	793356.	0.338869	0.169434	−	0.985541i	$-0.445806\pi$
$353$	793356.	0.338869	0.169434	+	0.985541i	$0.445806\pi$
$354$	0	0
$355$	−1.27256e6	−0.535931
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.04612e6	0.837906	0.418953	−	0.908008i	$-0.362397\pi$
$359$	2.04612e6	0.837906	0.418953	+	0.908008i	$0.362397\pi$
$360$	0	0
$361$	−2.41514e6	−0.975381
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−630176.	−0.247588
$366$	0	0
$367$	−3.79742e6	−1.47171	−0.735857	−	0.677137i	$-0.763220\pi$
$367$	−3.79742e6	−1.47171	−0.735857	+	0.677137i	$0.763220\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	598552.	0.225771
$372$	0	0
$373$	2.00590e6	0.746511	0.373256	−	0.927729i	$-0.378241\pi$
$373$	2.00590e6	0.746511	0.373256	+	0.927729i	$0.378241\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.26888e6	0.459800
$378$	0	0
$379$	2.33518e6	0.835070	0.417535	−	0.908661i	$-0.362894\pi$
$379$	2.33518e6	0.835070	0.417535	+	0.908661i	$0.362894\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	109725.	0.0382217	0.0191108	−	0.999817i	$-0.493916\pi$
$383$	109725.	0.0382217	0.0191108	+	0.999817i	$0.493916\pi$
$384$	0	0
$385$	−936640.	−0.322048
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.39896e6	−1.13887	−0.569433	−	0.822038i	$-0.692837\pi$
$389$	−3.39896e6	−1.13887	−0.569433	+	0.822038i	$0.692837\pi$
$390$	0	0
$391$	−226871.	−0.0750475
$392$	0	0
$393$	0	0
$394$	0	0
$395$	421659.	0.135978
$396$	0	0
$397$	4.56111e6	1.45243	0.726213	−	0.687470i	$-0.241279\pi$
$397$	4.56111e6	1.45243	0.726213	+	0.687470i	$0.241279\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.11367e6	0.656412	0.328206	−	0.944606i	$-0.393556\pi$
$401$	2.11367e6	0.656412	0.328206	+	0.944606i	$0.393556\pi$
$402$	0	0
$403$	−62328.0	−0.0191170
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.44058e6	1.02954
$408$	0	0
$409$	−1.71857e6	−0.507995	−0.253997	−	0.967205i	$-0.581745\pi$
$409$	−1.71857e6	−0.507995	−0.253997	+	0.967205i	$0.581745\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	154693.	0.0446268
$414$	0	0
$415$	−2.71621e6	−0.774182
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.57529e6	0.994892	0.497446	−	0.867495i	$-0.334271\pi$
$419$	3.57529e6	0.994892	0.497446	+	0.867495i	$0.334271\pi$
$420$	0	0
$421$	−4.89801e6	−1.34683	−0.673417	−	0.739263i	$-0.735174\pi$
$421$	−4.89801e6	−1.34683	−0.673417	+	0.739263i	$0.735174\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	724017.	0.194436
$426$	0	0
$427$	−5.06552e6	−1.34448
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−557438.	−0.144545	−0.0722726	−	0.997385i	$-0.523025\pi$
$431$	−557438.	−0.144545	−0.0722726	+	0.997385i	$0.523025\pi$
$432$	0	0
$433$	−5.44495e6	−1.39564	−0.697821	−	0.716272i	$-0.745847\pi$
$433$	−5.44495e6	−1.39564	−0.697821	+	0.716272i	$0.745847\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−48353.7	−0.0121123
$438$	0	0
$439$	−658765.	−0.163143	−0.0815716	−	0.996667i	$-0.525994\pi$
$439$	−658765.	−0.163143	−0.0815716	+	0.996667i	$0.525994\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.02447e6	−0.490118	−0.245059	−	0.969508i	$-0.578807\pi$
$443$	−2.02447e6	−0.490118	−0.245059	+	0.969508i	$0.578807\pi$
$444$	0	0
$445$	1.15259e6	0.275914
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.98151e6	−0.932034	−0.466017	−	0.884776i	$-0.654311\pi$
$449$	−3.98151e6	−0.932034	−0.466017	+	0.884776i	$0.654311\pi$
$450$	0	0
$451$	−4.67027e6	−1.08119
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−964156.	−0.218333
$456$	0	0
$457$	−2.69421e6	−0.603449	−0.301725	−	0.953395i	$-0.597562\pi$
$457$	−2.69421e6	−0.603449	−0.301725	+	0.953395i	$0.597562\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.99662e6	0.437565	0.218782	−	0.975774i	$-0.429792\pi$
$461$	1.99662e6	0.437565	0.218782	+	0.975774i	$0.429792\pi$
$462$	0	0
$463$	485244.	0.105198	0.0525990	−	0.998616i	$-0.483249\pi$
$463$	485244.	0.105198	0.0525990	+	0.998616i	$0.483249\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.14247e6	−1.30332	−0.651660	−	0.758511i	$-0.725927\pi$
$467$	−6.14247e6	−1.30332	−0.651660	+	0.758511i	$0.725927\pi$
$468$	0	0
$469$	1.54036e6	0.323364
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.13174e6	−0.643625
$474$	0	0
$475$	154312.	0.0313810
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.22475e6	1.04046	0.520232	−	0.854025i	$-0.325845\pi$
$479$	5.22475e6	1.04046	0.520232	+	0.854025i	$0.325845\pi$
$480$	0	0
$481$	3.54165e6	0.697980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.45099e6	0.859215
$486$	0	0
$487$	526301.	0.100557	0.0502784	−	0.998735i	$-0.483989\pi$
$487$	526301.	0.100557	0.0502784	+	0.998735i	$0.483989\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.79599e6	1.45938	0.729689	−	0.683779i	$-0.239665\pi$
$491$	7.79599e6	1.45938	0.729689	+	0.683779i	$0.239665\pi$
$492$	0	0
$493$	−3.78604e6	−0.701565
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.05640e6	−0.918227
$498$	0	0
$499$	−2.85501e6	−0.513283	−0.256642	−	0.966507i	$-0.582616\pi$
$499$	−2.85501e6	−0.513283	−0.256642	+	0.966507i	$0.582616\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.58697e6	1.51328	0.756641	−	0.653830i	$-0.226839\pi$
$503$	8.58697e6	1.51328	0.756641	+	0.653830i	$0.226839\pi$
$504$	0	0
$505$	−571874.	−0.0997865
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.78163e6	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$509$	6.78163e6	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$510$	0	0
$511$	−2.50394e6	−0.424201
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−887831.	−0.147507
$516$	0	0
$517$	4.15747e6	0.684073
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.16124e6	−0.994429	−0.497215	−	0.867628i	$-0.665644\pi$
$521$	−6.16124e6	−0.994429	−0.497215	+	0.867628i	$0.665644\pi$
$522$	0	0
$523$	8.56908e6	1.36987	0.684936	−	0.728604i	$-0.259830\pi$
$523$	8.56908e6	1.36987	0.684936	+	0.728604i	$0.259830\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	185972.	0.0291689
$528$	0	0
$529$	−6.39799e6	−0.994041
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.80747e6	−0.732991
$534$	0	0
$535$	4.52603e6	0.683648
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.61737e6	0.388054
$540$	0	0
$541$	−5.18531e6	−0.761695	−0.380848	−	0.924638i	$-0.624368\pi$
$541$	−5.18531e6	−0.761695	−0.380848	+	0.924638i	$0.624368\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.37790e6	−0.487142
$546$	0	0
$547$	4.67763e6	0.668433	0.334216	−	0.942496i	$-0.391528\pi$
$547$	4.67763e6	0.668433	0.334216	+	0.942496i	$0.391528\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−806931.	−0.113229
$552$	0	0
$553$	1.67542e6	0.232976
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.26986e6	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$557$	−8.26986e6	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$558$	0	0
$559$	−3.22374e6	−0.436345
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.98649e6	0.530054	0.265027	−	0.964241i	$-0.414619\pi$
$563$	3.98649e6	0.530054	0.265027	+	0.964241i	$0.414619\pi$
$564$	0	0
$565$	629045.	0.0829012
$566$	0	0
$567$	0	0
$568$	0	0
$569$	225037.	0.0291389	0.0145695	−	0.999894i	$-0.495362\pi$
$569$	225037.	0.0291389	0.0145695	+	0.999894i	$0.495362\pi$
$570$	0	0
$571$	1.22720e7	1.57516	0.787582	−	0.616210i	$-0.211333\pi$
$571$	1.22720e7	1.57516	0.787582	+	0.616210i	$0.211333\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−122402.	−0.0154390
$576$	0	0
$577$	−3.11245e6	−0.389191	−0.194596	−	0.980884i	$-0.562339\pi$
$577$	−3.11245e6	−0.389191	−0.194596	+	0.980884i	$0.562339\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.07926e7	−1.32643
$582$	0	0
$583$	−2.27264e6	−0.276923
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.10819e6	0.611888	0.305944	−	0.952050i	$-0.401028\pi$
$587$	5.10819e6	0.611888	0.305944	+	0.952050i	$0.401028\pi$
$588$	0	0
$589$	39636.8	0.00470771
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.08828e7	−1.27087	−0.635437	−	0.772153i	$-0.719180\pi$
$593$	−1.08828e7	−1.27087	−0.635437	+	0.772153i	$0.719180\pi$
$594$	0	0
$595$	2.87681e6	0.333133
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.72892e6	−0.196883	−0.0984414	−	0.995143i	$-0.531386\pi$
$599$	−1.72892e6	−0.196883	−0.0984414	+	0.995143i	$0.531386\pi$
$600$	0	0
$601$	1.18839e7	1.34206	0.671030	−	0.741430i	$-0.265852\pi$
$601$	1.18839e7	1.34206	0.671030	+	0.741430i	$0.265852\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−469947.	−0.0521988
$606$	0	0
$607$	1.75410e6	0.193234	0.0966170	−	0.995322i	$-0.469198\pi$
$607$	1.75410e6	0.193234	0.0966170	+	0.995322i	$0.469198\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.27960e6	0.463768
$612$	0	0
$613$	−500215.	−0.0537658	−0.0268829	−	0.999639i	$-0.508558\pi$
$613$	−500215.	−0.0537658	−0.0268829	+	0.999639i	$0.508558\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.27374e6	−0.874962	−0.437481	−	0.899228i	$-0.644129\pi$
$617$	−8.27374e6	−0.874962	−0.437481	+	0.899228i	$0.644129\pi$
$618$	0	0
$619$	1.17094e7	1.22831	0.614153	−	0.789187i	$-0.289498\pi$
$619$	1.17094e7	1.22831	0.614153	+	0.789187i	$0.289498\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.57968e6	0.472732
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.05674e7	−1.06498
$630$	0	0
$631$	−367210.	−0.0367148	−0.0183574	−	0.999831i	$-0.505844\pi$
$631$	−367210.	−0.0367148	−0.0183574	+	0.999831i	$0.505844\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.97666e6	0.489783
$636$	0	0
$637$	2.69426e6	0.263081
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.59036e7	1.52880	0.764398	−	0.644745i	$-0.223036\pi$
$641$	1.59036e7	1.52880	0.764398	+	0.644745i	$0.223036\pi$
$642$	0	0
$643$	3.08655e6	0.294405	0.147203	−	0.989106i	$-0.452973\pi$
$643$	3.08655e6	0.294405	0.147203	+	0.989106i	$0.452973\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.94249e6	−0.370263	−0.185131	−	0.982714i	$-0.559271\pi$
$647$	−3.94249e6	−0.370263	−0.185131	+	0.982714i	$0.559271\pi$
$648$	0	0
$649$	−587354.	−0.0547379
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.83440e6	−0.810763	−0.405381	−	0.914148i	$-0.632861\pi$
$653$	−8.83440e6	−0.810763	−0.405381	+	0.914148i	$0.632861\pi$
$654$	0	0
$655$	−3.92324e6	−0.357307
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.93767e6	−0.353204	−0.176602	−	0.984282i	$-0.556511\pi$
$659$	−3.93767e6	−0.353204	−0.176602	+	0.984282i	$0.556511\pi$
$660$	0	0
$661$	9.52284e6	0.847740	0.423870	−	0.905723i	$-0.360671\pi$
$661$	9.52284e6	0.847740	0.423870	+	0.905723i	$0.360671\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	613143.	0.0537660
$666$	0	0
$667$	640067.	0.0557072
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.92333e7	1.64910
$672$	0	0
$673$	−4.47502e6	−0.380853	−0.190426	−	0.981701i	$-0.560987\pi$
$673$	−4.47502e6	−0.380853	−0.190426	+	0.981701i	$0.560987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.60623e7	−1.34690	−0.673452	−	0.739231i	$-0.735189\pi$
$677$	−1.60623e7	−1.34690	−0.673452	+	0.739231i	$0.735189\pi$
$678$	0	0
$679$	1.76855e7	1.47212
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.64450e7	1.34891	0.674456	−	0.738315i	$-0.264378\pi$
$683$	1.64450e7	1.34891	0.674456	+	0.738315i	$0.264378\pi$
$684$	0	0
$685$	2.77689e6	0.226116
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.33941e6	−0.187740
$690$	0	0
$691$	−7.86170e6	−0.626357	−0.313178	−	0.949694i	$-0.601394\pi$
$691$	−7.86170e6	−0.626357	−0.313178	+	0.949694i	$0.601394\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−974849.	−0.0765553
$696$	0	0
$697$	1.43443e7	1.11840
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.57224e7	−1.20844	−0.604219	−	0.796818i	$-0.706515\pi$
$701$	−1.57224e7	−1.20844	−0.604219	+	0.796818i	$0.706515\pi$
$702$	0	0
$703$	−2.25227e6	−0.171883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.27228e6	−0.170967
$708$	0	0
$709$	−1.42345e7	−1.06347	−0.531736	−	0.846910i	$-0.678460\pi$
$709$	−1.42345e7	−1.06347	−0.531736	+	0.846910i	$0.678460\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31440.3	−0.00231613
$714$	0	0
$715$	3.66080e6	0.267800
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.51911e7	1.81729	0.908646	−	0.417568i	$-0.137118\pi$
$719$	2.51911e7	1.81729	0.908646	+	0.417568i	$0.137118\pi$
$720$	0	0
$721$	−3.52770e6	−0.252728
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.04266e6	−0.144328
$726$	0	0
$727$	−5.76666e6	−0.404658	−0.202329	−	0.979318i	$-0.564851\pi$
$727$	−5.76666e6	−0.404658	−0.202329	+	0.979318i	$0.564851\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.61886e6	0.665779
$732$	0	0
$733$	9.99042e6	0.686790	0.343395	−	0.939191i	$-0.388423\pi$
$733$	9.99042e6	0.686790	0.343395	+	0.939191i	$0.388423\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.84861e6	−0.396628
$738$	0	0
$739$	−2.19691e7	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$739$	−2.19691e7	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.21995e7	−0.810721	−0.405361	−	0.914157i	$-0.632854\pi$
$743$	−1.21995e7	−0.810721	−0.405361	+	0.914157i	$0.632854\pi$
$744$	0	0
$745$	3.20639e6	0.211653
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.79837e7	1.17132
$750$	0	0
$751$	9.89445e6	0.640165	0.320082	−	0.947390i	$-0.396289\pi$
$751$	9.89445e6	0.640165	0.320082	+	0.947390i	$0.396289\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.02258e6	−0.0652878
$756$	0	0
$757$	1.59336e7	1.01059	0.505293	−	0.862948i	$-0.331385\pi$
$757$	1.59336e7	1.01059	0.505293	+	0.862948i	$0.331385\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.00477e6	0.501057	0.250528	−	0.968109i	$-0.419396\pi$
$761$	8.00477e6	0.501057	0.250528	+	0.968109i	$0.419396\pi$
$762$	0	0
$763$	−1.34217e7	−0.834636
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−604608.	−0.0371096
$768$	0	0
$769$	1.46154e7	0.891238	0.445619	−	0.895223i	$-0.352984\pi$
$769$	1.46154e7	0.891238	0.445619	+	0.895223i	$0.352984\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.74241e7	1.65076	0.825379	−	0.564579i	$-0.190961\pi$
$773$	2.74241e7	1.65076	0.825379	+	0.564579i	$0.190961\pi$
$774$	0	0
$775$	100336.	0.00600073
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.05725e6	0.180504
$780$	0	0
$781$	1.91986e7	1.12627
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.97003e6	−0.403702
$786$	0	0
$787$	2.22194e7	1.27878	0.639388	−	0.768884i	$-0.279188\pi$
$787$	2.22194e7	1.27878	0.639388	+	0.768884i	$0.279188\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.49944e6	0.142037
$792$	0	0
$793$	1.97983e7	1.11801
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.54647e6	−0.0862376	−0.0431188	−	0.999070i	$-0.513729\pi$
$797$	−1.54647e6	−0.0862376	−0.0431188	+	0.999070i	$0.513729\pi$
$798$	0	0
$799$	−1.27693e7	−0.707620
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.50720e6	0.520312
$804$	0	0
$805$	−486352.	−0.0264522
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.22034e7	0.655555	0.327778	−	0.944755i	$-0.393700\pi$
$809$	1.22034e7	0.655555	0.327778	+	0.944755i	$0.393700\pi$
$810$	0	0
$811$	5.99193e6	0.319900	0.159950	−	0.987125i	$-0.448867\pi$
$811$	5.99193e6	0.319900	0.159950	+	0.987125i	$0.448867\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.01908e7	−0.537423
$816$	0	0
$817$	2.05010e6	0.107453
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.38572e7	−1.23527	−0.617635	−	0.786465i	$-0.711909\pi$
$821$	−2.38572e7	−1.23527	−0.617635	+	0.786465i	$0.711909\pi$
$822$	0	0
$823$	2.39131e7	1.23065	0.615327	−	0.788272i	$-0.289024\pi$
$823$	2.39131e7	1.23065	0.615327	+	0.788272i	$0.289024\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.76663e6	0.0898216	0.0449108	−	0.998991i	$-0.485700\pi$
$827$	1.76663e6	0.0898216	0.0449108	+	0.998991i	$0.485700\pi$
$828$	0	0
$829$	1.68441e7	0.851258	0.425629	−	0.904898i	$-0.360053\pi$
$829$	1.68441e7	0.851258	0.425629	+	0.904898i	$0.360053\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.03901e6	−0.401412
$834$	0	0
$835$	−1.57917e6	−0.0783811
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.52232e7	−1.23707	−0.618537	−	0.785756i	$-0.712274\pi$
$839$	−2.52232e7	−1.23707	−0.618537	+	0.785756i	$0.712274\pi$
$840$	0	0
$841$	−9.82963e6	−0.479234
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.51398e6	−0.265658
$846$	0	0
$847$	−1.86728e6	−0.0894338
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.78653e6	0.0845640
$852$	0	0
$853$	−1.96723e7	−0.925727	−0.462864	−	0.886430i	$-0.653178\pi$
$853$	−1.96723e7	−0.925727	−0.462864	+	0.886430i	$0.653178\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.31831e7	−1.07825	−0.539126	−	0.842225i	$-0.681245\pi$
$857$	−2.31831e7	−1.07825	−0.539126	+	0.842225i	$0.681245\pi$
$858$	0	0
$859$	4.11763e7	1.90399	0.951995	−	0.306112i	$-0.0990284\pi$
$859$	4.11763e7	1.90399	0.951995	+	0.306112i	$0.0990284\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.85382e7	0.847307	0.423653	−	0.905824i	$-0.360747\pi$
$863$	1.85382e7	0.847307	0.423653	+	0.905824i	$0.360747\pi$
$864$	0	0
$865$	−932351.	−0.0423681
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.36140e6	−0.285761
$870$	0	0
$871$	−6.02042e6	−0.268894
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.55211e6	0.0685333
$876$	0	0
$877$	2.32018e7	1.01865	0.509323	−	0.860576i	$-0.329896\pi$
$877$	2.32018e7	1.01865	0.509323	+	0.860576i	$0.329896\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.30268e6	−0.0565453	−0.0282727	−	0.999600i	$-0.509001\pi$
$881$	−1.30268e6	−0.0565453	−0.0282727	+	0.999600i	$0.509001\pi$
$882$	0	0
$883$	3.39367e7	1.46476	0.732382	−	0.680894i	$-0.238409\pi$
$883$	3.39367e7	1.46476	0.732382	+	0.680894i	$0.238409\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.47651e7	0.630124	0.315062	−	0.949071i	$-0.397975\pi$
$887$	1.47651e7	0.630124	0.315062	+	0.949071i	$0.397975\pi$
$888$	0	0
$889$	1.97742e7	0.839161
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.72156e6	−0.114206
$894$	0	0
$895$	2.20840e6	0.0921554
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−524680.	−0.0216519
$900$	0	0
$901$	6.98022e6	0.286456
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.14482e6	−0.208809
$906$	0	0
$907$	−1.33294e7	−0.538011	−0.269006	−	0.963139i	$-0.586695\pi$
$907$	−1.33294e7	−0.538011	−0.269006	+	0.963139i	$0.586695\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.33053e7	−0.930375	−0.465187	−	0.885212i	$-0.654013\pi$
$911$	−2.33053e7	−0.930375	−0.465187	+	0.885212i	$0.654013\pi$
$912$	0	0
$913$	4.09783e7	1.62696
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.55886e7	−0.612185
$918$	0	0
$919$	4.50929e7	1.76124	0.880622	−	0.473819i	$-0.157125\pi$
$919$	4.50929e7	1.76124	0.880622	+	0.473819i	$0.157125\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.97626e7	0.763556
$924$	0	0
$925$	−5.70138e6	−0.219092
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.55017e7	1.34962	0.674808	−	0.737993i	$-0.264226\pi$
$929$	3.55017e7	1.34962	0.674808	+	0.737993i	$0.264226\pi$
$930$	0	0
$931$	−1.71338e6	−0.0647857
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.09229e7	−0.408612
$936$	0	0
$937$	−2.53866e7	−0.944616	−0.472308	−	0.881434i	$-0.656579\pi$
$937$	−2.53866e7	−0.944616	−0.472308	+	0.881434i	$0.656579\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.52084e6	−0.129620	−0.0648100	−	0.997898i	$-0.520644\pi$
$941$	−3.52084e6	−0.129620	−0.0648100	+	0.997898i	$0.520644\pi$
$942$	0	0
$943$	−2.42505e6	−0.0888058
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.87280e6	−0.140330	−0.0701650	−	0.997535i	$-0.522353\pi$
$947$	−3.87280e6	−0.140330	−0.0701650	+	0.997535i	$0.522353\pi$
$948$	0	0
$949$	9.78650e6	0.352746
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.66754e7	−0.951435	−0.475718	−	0.879598i	$-0.657812\pi$
$953$	−2.66754e7	−0.951435	−0.475718	+	0.879598i	$0.657812\pi$
$954$	0	0
$955$	−56866.4	−0.00201766
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.10337e7	0.387412
$960$	0	0
$961$	−2.86034e7	−0.999100
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.34456e7	−0.464797
$966$	0	0
$967$	3.00067e7	1.03193	0.515967	−	0.856608i	$-0.327433\pi$
$967$	3.00067e7	1.03193	0.515967	+	0.856608i	$0.327433\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.42664e7	0.825956	0.412978	−	0.910741i	$-0.364489\pi$
$971$	2.42664e7	0.825956	0.412978	+	0.910741i	$0.364489\pi$
$972$	0	0
$973$	−3.87346e6	−0.131165
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.67601e7	0.896915	0.448458	−	0.893804i	$-0.351974\pi$
$977$	2.67601e7	0.896915	0.448458	+	0.893804i	$0.351974\pi$
$978$	0	0
$979$	−1.73886e7	−0.579839
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.48680e7	0.820839	0.410419	−	0.911897i	$-0.365382\pi$
$983$	2.48680e7	0.820839	0.410419	+	0.911897i	$0.365382\pi$
$984$	0	0
$985$	−1.15758e7	−0.380155
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.62616e6	−0.0528656
$990$	0	0
$991$	4.05276e7	1.31089	0.655446	−	0.755242i	$-0.272481\pi$
$991$	4.05276e7	1.31089	0.655446	+	0.755242i	$0.272481\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.37683e7	−0.440881
$996$	0	0
$997$	−4.07721e7	−1.29905	−0.649524	−	0.760341i	$-0.725032\pi$
$997$	−4.07721e7	−1.29905	−0.649524	+	0.760341i	$0.725032\pi$
$998$	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.1.3	yes	3
3.2	odd	2		1080.6.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.f.1.3	✓	3	3.2	odd	2
1080.6.a.h.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)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} +99.3349 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} +99.3349 q^{7} -377.165 q^{11} -388.245 q^{13} +1158.43 q^{17} +246.899 q^{19} -195.844 q^{23} +625.000 q^{25} -3268.26 q^{29} +160.538 q^{31} +2483.37 q^{35} -9122.21 q^{37} +12382.6 q^{41} +8303.37 q^{43} -11023.0 q^{47} -6939.58 q^{49} +6025.60 q^{53} -9429.11 q^{55} +1557.29 q^{59} -50994.4 q^{61} -9706.11 q^{65} +15506.8 q^{67} -50902.6 q^{71} -25207.0 q^{73} -37465.6 q^{77} +16866.4 q^{79} -108648. q^{83} +28960.7 q^{85} +46103.4 q^{89} -38566.2 q^{91} +6172.49 q^{95} +178039. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 75 q^{5} + 140 q^{7}+O(q^{10}) \) 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

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