LMFDB - Embedded newform 1080.6.a.e.1.2

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:	$0$
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			1.2
Root			$6.15876$ 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)

$f(q)$	$=$	$q-25.0000 q^{5} +61.7318 q^{7} +O(q^{10})$	$q-25.0000 q^{5} +61.7318 q^{7} -218.360 q^{11} -864.360 q^{13} -521.206 q^{17} -1456.28 q^{19} -4923.60 q^{23} +625.000 q^{25} +5664.06 q^{29} -2982.68 q^{31} -1543.30 q^{35} -110.503 q^{37} +2417.32 q^{41} +13599.5 q^{43} +15203.5 q^{47} -12996.2 q^{49} -10499.4 q^{53} +5459.00 q^{55} +31719.4 q^{59} -28090.1 q^{61} +21609.0 q^{65} -17898.9 q^{67} +36783.1 q^{71} -49779.7 q^{73} -13479.7 q^{77} +81941.9 q^{79} -97419.7 q^{83} +13030.2 q^{85} +18128.7 q^{89} -53358.5 q^{91} +36407.1 q^{95} +45823.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	61.7318	0.476172	0.238086	−	0.971244i	$-0.423480\pi$
$7$	61.7318	0.476172	0.238086	+	0.971244i	$0.423480\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−218.360	−0.544115	−0.272058	−	0.962281i	$-0.587704\pi$
$11$	−218.360	−0.544115	−0.272058	+	0.962281i	$0.587704\pi$
$12$	0	0
$13$	−864.360	−1.41852	−0.709261	−	0.704946i	$-0.750971\pi$
$13$	−864.360	−1.41852	−0.709261	+	0.704946i	$0.750971\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−521.206	−0.437408	−0.218704	−	0.975791i	$-0.570183\pi$
$17$	−521.206	−0.437408	−0.218704	+	0.975791i	$0.570183\pi$
$18$	0	0
$19$	−1456.28	−0.925469	−0.462735	−	0.886497i	$-0.653132\pi$
$19$	−1456.28	−0.925469	−0.462735	+	0.886497i	$0.653132\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4923.60	−1.94072	−0.970360	−	0.241663i	$-0.922307\pi$
$23$	−4923.60	−1.94072	−0.970360	+	0.241663i	$0.922307\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5664.06	1.25064	0.625321	−	0.780367i	$-0.284968\pi$
$29$	5664.06	1.25064	0.625321	+	0.780367i	$0.284968\pi$
$30$	0	0
$31$	−2982.68	−0.557446	−0.278723	−	0.960372i	$-0.589911\pi$
$31$	−2982.68	−0.557446	−0.278723	+	0.960372i	$0.589911\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1543.30	−0.212951
$36$	0	0
$37$	−110.503	−0.0132700	−0.00663500	−	0.999978i	$-0.502112\pi$
$37$	−110.503	−0.0132700	−0.00663500	+	0.999978i	$0.502112\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2417.32	0.224582	0.112291	−	0.993675i	$-0.464181\pi$
$41$	2417.32	0.224582	0.112291	+	0.993675i	$0.464181\pi$
$42$	0	0
$43$	13599.5	1.12164	0.560818	−	0.827939i	$-0.310487\pi$
$43$	13599.5	1.12164	0.560818	+	0.827939i	$0.310487\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	15203.5	1.00392	0.501959	−	0.864891i	$-0.332613\pi$
$47$	15203.5	1.00392	0.501959	+	0.864891i	$0.332613\pi$
$48$	0	0
$49$	−12996.2	−0.773260
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10499.4	−0.513424	−0.256712	−	0.966488i	$-0.582639\pi$
$53$	−10499.4	−0.513424	−0.256712	+	0.966488i	$0.582639\pi$
$54$	0	0
$55$	5459.00	0.243336
$56$	0	0
$57$	0	0
$58$	0	0
$59$	31719.4	1.18630	0.593150	−	0.805092i	$-0.297884\pi$
$59$	31719.4	1.18630	0.593150	+	0.805092i	$0.297884\pi$
$60$	0	0
$61$	−28090.1	−0.966561	−0.483280	−	0.875466i	$-0.660555\pi$
$61$	−28090.1	−0.966561	−0.483280	+	0.875466i	$0.660555\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21609.0	0.634382
$66$	0	0
$67$	−17898.9	−0.487123	−0.243561	−	0.969886i	$-0.578316\pi$
$67$	−17898.9	−0.487123	−0.243561	+	0.969886i	$0.578316\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	36783.1	0.865969	0.432984	−	0.901401i	$-0.357460\pi$
$71$	36783.1	0.865969	0.432984	+	0.901401i	$0.357460\pi$
$72$	0	0
$73$	−49779.7	−1.09332	−0.546658	−	0.837356i	$-0.684100\pi$
$73$	−49779.7	−1.09332	−0.546658	+	0.837356i	$0.684100\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13479.7	−0.259093
$78$	0	0
$79$	81941.9	1.47720	0.738598	−	0.674146i	$-0.235488\pi$
$79$	81941.9	1.47720	0.738598	+	0.674146i	$0.235488\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−97419.7	−1.55221	−0.776107	−	0.630602i	$-0.782808\pi$
$83$	−97419.7	−1.55221	−0.776107	+	0.630602i	$0.782808\pi$
$84$	0	0
$85$	13030.2	0.195615
$86$	0	0
$87$	0	0
$88$	0	0
$89$	18128.7	0.242601	0.121300	−	0.992616i	$-0.461294\pi$
$89$	18128.7	0.242601	0.121300	+	0.992616i	$0.461294\pi$
$90$	0	0
$91$	−53358.5	−0.675461
$92$	0	0
$93$	0	0
$94$	0	0
$95$	36407.1	0.413882
$96$	0	0
$97$	45823.0	0.494486	0.247243	−	0.968954i	$-0.420475\pi$
$97$	45823.0	0.494486	0.247243	+	0.968954i	$0.420475\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−62187.5	−0.606596	−0.303298	−	0.952896i	$-0.598088\pi$
$101$	−62187.5	−0.606596	−0.303298	+	0.952896i	$0.598088\pi$
$102$	0	0
$103$	−43842.8	−0.407198	−0.203599	−	0.979054i	$-0.565264\pi$
$103$	−43842.8	−0.407198	−0.203599	+	0.979054i	$0.565264\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−59741.4	−0.504447	−0.252224	−	0.967669i	$-0.581162\pi$
$107$	−59741.4	−0.504447	−0.252224	+	0.967669i	$0.581162\pi$
$108$	0	0
$109$	144322.	1.16350	0.581750	−	0.813368i	$-0.302368\pi$
$109$	144322.	1.16350	0.581750	+	0.813368i	$0.302368\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	41891.8	0.308626	0.154313	−	0.988022i	$-0.450684\pi$
$113$	41891.8	0.308626	0.154313	+	0.988022i	$0.450684\pi$
$114$	0	0
$115$	123090.	0.867916
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−32175.0	−0.208282
$120$	0	0
$121$	−113370.	−0.703938
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−267112.	−1.46955	−0.734775	−	0.678311i	$-0.762712\pi$
$127$	−267112.	−1.46955	−0.734775	+	0.678311i	$0.762712\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	149202.	0.759620	0.379810	−	0.925065i	$-0.375989\pi$
$131$	149202.	0.759620	0.379810	+	0.925065i	$0.375989\pi$
$132$	0	0
$133$	−89899.0	−0.440683
$134$	0	0
$135$	0	0
$136$	0	0
$137$	238234.	1.08443	0.542215	−	0.840240i	$-0.317586\pi$
$137$	238234.	1.08443	0.542215	+	0.840240i	$0.317586\pi$
$138$	0	0
$139$	37079.7	0.162779	0.0813896	−	0.996682i	$-0.474064\pi$
$139$	37079.7	0.162779	0.0813896	+	0.996682i	$0.474064\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	188741.	0.771840
$144$	0	0
$145$	−141602.	−0.559304
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−353924.	−1.30600	−0.653002	−	0.757356i	$-0.726491\pi$
$149$	−353924.	−1.30600	−0.653002	+	0.757356i	$0.726491\pi$
$150$	0	0
$151$	−312960.	−1.11698	−0.558491	−	0.829511i	$-0.688619\pi$
$151$	−312960.	−1.11698	−0.558491	+	0.829511i	$0.688619\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	74567.0	0.249297
$156$	0	0
$157$	461545.	1.49439	0.747197	−	0.664603i	$-0.231399\pi$
$157$	461545.	1.49439	0.747197	+	0.664603i	$0.231399\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−303943.	−0.924117
$162$	0	0
$163$	379232.	1.11799	0.558993	−	0.829172i	$-0.311188\pi$
$163$	379232.	1.11799	0.558993	+	0.829172i	$0.311188\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	412408.	1.14429	0.572144	−	0.820153i	$-0.306112\pi$
$167$	412408.	1.14429	0.572144	+	0.820153i	$0.306112\pi$
$168$	0	0
$169$	375825.	1.01221
$170$	0	0
$171$	0	0
$172$	0	0
$173$	730459.	1.85558	0.927792	−	0.373098i	$-0.121704\pi$
$173$	730459.	1.85558	0.927792	+	0.373098i	$0.121704\pi$
$174$	0	0
$175$	38582.4	0.0952344
$176$	0	0
$177$	0	0
$178$	0	0
$179$	217864.	0.508221	0.254110	−	0.967175i	$-0.418217\pi$
$179$	217864.	0.508221	0.254110	+	0.967175i	$0.418217\pi$
$180$	0	0
$181$	541786.	1.22923	0.614613	−	0.788829i	$-0.289312\pi$
$181$	541786.	1.22923	0.614613	+	0.788829i	$0.289312\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2762.58	0.00593453
$186$	0	0
$187$	113810.	0.238001
$188$	0	0
$189$	0	0
$190$	0	0
$191$	809680.	1.60594	0.802972	−	0.596017i	$-0.203251\pi$
$191$	809680.	1.60594	0.802972	+	0.596017i	$0.203251\pi$
$192$	0	0
$193$	−711052.	−1.37407	−0.687033	−	0.726626i	$-0.741087\pi$
$193$	−711052.	−1.37407	−0.687033	+	0.726626i	$0.741087\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	135834.	0.249370	0.124685	−	0.992196i	$-0.460208\pi$
$197$	135834.	0.249370	0.124685	+	0.992196i	$0.460208\pi$
$198$	0	0
$199$	393307.	0.704043	0.352021	−	0.935992i	$-0.385494\pi$
$199$	393307.	0.704043	0.352021	+	0.935992i	$0.385494\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	349653.	0.595521
$204$	0	0
$205$	−60433.1	−0.100436
$206$	0	0
$207$	0	0
$208$	0	0
$209$	317994.	0.503562
$210$	0	0
$211$	−857.535	−0.00132601	−0.000663003	−	1.00000i	$-0.500211\pi$
$211$	−857.535	−0.00132601	−0.000663003		1.00000i	$0.500211\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−339988.	−0.501611
$216$	0	0
$217$	−184126.	−0.265440
$218$	0	0
$219$	0	0
$220$	0	0
$221$	450510.	0.620474
$222$	0	0
$223$	−936796.	−1.26149	−0.630744	−	0.775991i	$-0.717250\pi$
$223$	−936796.	−1.26149	−0.630744	+	0.775991i	$0.717250\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	308948.	0.397943	0.198971	−	0.980005i	$-0.436240\pi$
$227$	308948.	0.397943	0.198971	+	0.980005i	$0.436240\pi$
$228$	0	0
$229$	−711701.	−0.896827	−0.448413	−	0.893826i	$-0.648011\pi$
$229$	−711701.	−0.896827	−0.448413	+	0.893826i	$0.648011\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12911.2	0.0155804	0.00779018	−	0.999970i	$-0.497520\pi$
$233$	12911.2	0.0155804	0.00779018	+	0.999970i	$0.497520\pi$
$234$	0	0
$235$	−380087.	−0.448966
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−965160.	−1.09296	−0.546481	−	0.837472i	$-0.684033\pi$
$239$	−965160.	−1.09296	−0.546481	+	0.837472i	$0.684033\pi$
$240$	0	0
$241$	−255074.	−0.282894	−0.141447	−	0.989946i	$-0.545175\pi$
$241$	−255074.	−0.282894	−0.141447	+	0.989946i	$0.545175\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	324905.	0.345812
$246$	0	0
$247$	1.25875e6	1.31280
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.09535e6	1.09741	0.548703	−	0.836018i	$-0.315122\pi$
$251$	1.09535e6	1.09741	0.548703	+	0.836018i	$0.315122\pi$
$252$	0	0
$253$	1.07512e6	1.05598
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41430e6	1.33570	0.667852	−	0.744294i	$-0.267214\pi$
$257$	1.41430e6	1.33570	0.667852	+	0.744294i	$0.267214\pi$
$258$	0	0
$259$	−6821.57	−0.00631881
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.61172e6	−1.43681	−0.718407	−	0.695623i	$-0.755128\pi$
$263$	−1.61172e6	−1.43681	−0.718407	+	0.695623i	$0.755128\pi$
$264$	0	0
$265$	262486.	0.229610
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.03815e6	1.71733	0.858667	−	0.512535i	$-0.171293\pi$
$269$	2.03815e6	1.71733	0.858667	+	0.512535i	$0.171293\pi$
$270$	0	0
$271$	−1.13393e6	−0.937914	−0.468957	−	0.883221i	$-0.655370\pi$
$271$	−1.13393e6	−0.937914	−0.468957	+	0.883221i	$0.655370\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−136475.	−0.108823
$276$	0	0
$277$	596450.	0.467062	0.233531	−	0.972349i	$-0.424972\pi$
$277$	596450.	0.467062	0.233531	+	0.972349i	$0.424972\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−867713.	−0.655557	−0.327778	−	0.944755i	$-0.606300\pi$
$281$	−867713.	−0.655557	−0.327778	+	0.944755i	$0.606300\pi$
$282$	0	0
$283$	125401.	0.0930756	0.0465378	−	0.998917i	$-0.485181\pi$
$283$	125401.	0.0930756	0.0465378	+	0.998917i	$0.485181\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	149226.	0.106940
$288$	0	0
$289$	−1.14820e6	−0.808674
$290$	0	0
$291$	0	0
$292$	0	0
$293$	746956.	0.508307	0.254153	−	0.967164i	$-0.418203\pi$
$293$	746956.	0.508307	0.254153	+	0.967164i	$0.418203\pi$
$294$	0	0
$295$	−792984.	−0.530529
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.25576e6	2.75296
$300$	0	0
$301$	839522.	0.534092
$302$	0	0
$303$	0	0
$304$	0	0
$305$	702253.	0.432259
$306$	0	0
$307$	−2.84667e6	−1.72382	−0.861908	−	0.507065i	$-0.830731\pi$
$307$	−2.84667e6	−1.72382	−0.861908	+	0.507065i	$0.830731\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	256851.	0.150585	0.0752924	−	0.997162i	$-0.476011\pi$
$311$	256851.	0.150585	0.0752924	+	0.997162i	$0.476011\pi$
$312$	0	0
$313$	3.28277e6	1.89400	0.946998	−	0.321240i	$-0.104100\pi$
$313$	3.28277e6	1.89400	0.946998	+	0.321240i	$0.104100\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	694224.	0.388018	0.194009	−	0.981000i	$-0.437851\pi$
$317$	694224.	0.388018	0.194009	+	0.981000i	$0.437851\pi$
$318$	0	0
$319$	−1.23680e6	−0.680494
$320$	0	0
$321$	0	0
$322$	0	0
$323$	759024.	0.404808
$324$	0	0
$325$	−540225.	−0.283704
$326$	0	0
$327$	0	0
$328$	0	0
$329$	938538.	0.478038
$330$	0	0
$331$	−645658.	−0.323916	−0.161958	−	0.986798i	$-0.551781\pi$
$331$	−645658.	−0.323916	−0.161958	+	0.986798i	$0.551781\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	447471.	0.217848
$336$	0	0
$337$	19338.3	0.00927562	0.00463781	−	0.999989i	$-0.498524\pi$
$337$	19338.3	0.00927562	0.00463781	+	0.999989i	$0.498524\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	651297.	0.303315
$342$	0	0
$343$	−1.83980e6	−0.844377
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.21004e6	0.539479	0.269739	−	0.962933i	$-0.413062\pi$
$347$	1.21004e6	0.539479	0.269739	+	0.962933i	$0.413062\pi$
$348$	0	0
$349$	−818886.	−0.359882	−0.179941	−	0.983677i	$-0.557591\pi$
$349$	−818886.	−0.359882	−0.179941	+	0.983677i	$0.557591\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.12396e6	1.76148	0.880740	−	0.473599i	$-0.157046\pi$
$353$	4.12396e6	1.76148	0.880740	+	0.473599i	$0.157046\pi$
$354$	0	0
$355$	−919577.	−0.387273
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.59734e6	1.06363	0.531817	−	0.846859i	$-0.321509\pi$
$359$	2.59734e6	1.06363	0.531817	+	0.846859i	$0.321509\pi$
$360$	0	0
$361$	−355336.	−0.143506
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.24449e6	0.488946
$366$	0	0
$367$	1.37067e6	0.531212	0.265606	−	0.964082i	$-0.414428\pi$
$367$	1.37067e6	0.531212	0.265606	+	0.964082i	$0.414428\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−648149.	−0.244478
$372$	0	0
$373$	4.95671e6	1.84468	0.922340	−	0.386379i	$-0.126274\pi$
$373$	4.95671e6	1.84468	0.922340	+	0.386379i	$0.126274\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.89579e6	−1.77406
$378$	0	0
$379$	1.37710e6	0.492456	0.246228	−	0.969212i	$-0.420809\pi$
$379$	1.37710e6	0.492456	0.246228	+	0.969212i	$0.420809\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.87817e6	0.654241	0.327121	−	0.944983i	$-0.393922\pi$
$383$	1.87817e6	0.654241	0.327121	+	0.944983i	$0.393922\pi$
$384$	0	0
$385$	336994.	0.115870
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.59135e6	−1.20333	−0.601663	−	0.798750i	$-0.705495\pi$
$389$	−3.59135e6	−1.20333	−0.601663	+	0.798750i	$0.705495\pi$
$390$	0	0
$391$	2.56621e6	0.848887
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.04855e6	−0.660622
$396$	0	0
$397$	−128220.	−0.0408300	−0.0204150	−	0.999792i	$-0.506499\pi$
$397$	−128220.	−0.0408300	−0.0204150	+	0.999792i	$0.506499\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.60952e6	0.810400	0.405200	−	0.914228i	$-0.367202\pi$
$401$	2.60952e6	0.810400	0.405200	+	0.914228i	$0.367202\pi$
$402$	0	0
$403$	2.57811e6	0.790749
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24129.5	0.00722041
$408$	0	0
$409$	4.00485e6	1.18380	0.591900	−	0.806012i	$-0.298378\pi$
$409$	4.00485e6	1.18380	0.591900	+	0.806012i	$0.298378\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.95809e6	0.564883
$414$	0	0
$415$	2.43549e6	0.694171
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−950749.	−0.264564	−0.132282	−	0.991212i	$-0.542230\pi$
$419$	−950749.	−0.264564	−0.132282	+	0.991212i	$0.542230\pi$
$420$	0	0
$421$	−1.46256e6	−0.402170	−0.201085	−	0.979574i	$-0.564447\pi$
$421$	−1.46256e6	−0.402170	−0.201085	+	0.979574i	$0.564447\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−325754.	−0.0874817
$426$	0	0
$427$	−1.73405e6	−0.460249
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.57304e6	1.44510	0.722551	−	0.691317i	$-0.242969\pi$
$431$	5.57304e6	1.44510	0.722551	+	0.691317i	$0.242969\pi$
$432$	0	0
$433$	−2.12055e6	−0.543535	−0.271768	−	0.962363i	$-0.587608\pi$
$433$	−2.12055e6	−0.543535	−0.271768	+	0.962363i	$0.587608\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.17016e6	1.79608
$438$	0	0
$439$	−5.88072e6	−1.45636	−0.728180	−	0.685386i	$-0.759634\pi$
$439$	−5.88072e6	−1.45636	−0.728180	+	0.685386i	$0.759634\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.10594e6	0.267747	0.133873	−	0.990998i	$-0.457258\pi$
$443$	1.10594e6	0.267747	0.133873	+	0.990998i	$0.457258\pi$
$444$	0	0
$445$	−453218.	−0.108494
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.18891e6	0.746494	0.373247	−	0.927732i	$-0.378244\pi$
$449$	3.18891e6	0.746494	0.373247	+	0.927732i	$0.378244\pi$
$450$	0	0
$451$	−527846.	−0.122199
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.33396e6	0.302075
$456$	0	0
$457$	−107720.	−0.0241272	−0.0120636	−	0.999927i	$-0.503840\pi$
$457$	−107720.	−0.0241272	−0.0120636	+	0.999927i	$0.503840\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.77462e6	1.04637	0.523186	−	0.852219i	$-0.324743\pi$
$461$	4.77462e6	1.04637	0.523186	+	0.852219i	$0.324743\pi$
$462$	0	0
$463$	−5.90712e6	−1.28063	−0.640315	−	0.768113i	$-0.721196\pi$
$463$	−5.90712e6	−1.28063	−0.640315	+	0.768113i	$0.721196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.51853e6	−0.322204	−0.161102	−	0.986938i	$-0.551505\pi$
$467$	−1.51853e6	−0.322204	−0.161102	+	0.986938i	$0.551505\pi$
$468$	0	0
$469$	−1.10493e6	−0.231954
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.96958e6	−0.610299
$474$	0	0
$475$	−910177.	−0.185094
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.09026e6	−0.217116	−0.108558	−	0.994090i	$-0.534623\pi$
$479$	−1.09026e6	−0.217116	−0.108558	+	0.994090i	$0.534623\pi$
$480$	0	0
$481$	95514.6	0.0188238
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.14557e6	−0.221141
$486$	0	0
$487$	1.09380e6	0.208986	0.104493	−	0.994526i	$-0.466678\pi$
$487$	1.09380e6	0.208986	0.104493	+	0.994526i	$0.466678\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−403601.	−0.0755524	−0.0377762	−	0.999286i	$-0.512027\pi$
$491$	−403601.	−0.0755524	−0.0377762	+	0.999286i	$0.512027\pi$
$492$	0	0
$493$	−2.95214e6	−0.547042
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.27069e6	0.412350
$498$	0	0
$499$	1.05199e7	1.89130	0.945650	−	0.325187i	$-0.105427\pi$
$499$	1.05199e7	1.89130	0.945650	+	0.325187i	$0.105427\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.40828e6	1.65802	0.829011	−	0.559232i	$-0.188904\pi$
$503$	9.40828e6	1.65802	0.829011	+	0.559232i	$0.188904\pi$
$504$	0	0
$505$	1.55469e6	0.271278
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.79366e6	−0.306863	−0.153432	−	0.988159i	$-0.549033\pi$
$509$	−1.79366e6	−0.306863	−0.153432	+	0.988159i	$0.549033\pi$
$510$	0	0
$511$	−3.07299e6	−0.520606
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.09607e6	0.182104
$516$	0	0
$517$	−3.31983e6	−0.546247
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.62374e6	0.907676	0.453838	−	0.891084i	$-0.350054\pi$
$521$	5.62374e6	0.907676	0.453838	+	0.891084i	$0.350054\pi$
$522$	0	0
$523$	−8.41420e6	−1.34511	−0.672556	−	0.740046i	$-0.734804\pi$
$523$	−8.41420e6	−1.34511	−0.672556	+	0.740046i	$0.734804\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.55459e6	0.243831
$528$	0	0
$529$	1.78055e7	2.76640
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.08944e6	−0.318575
$534$	0	0
$535$	1.49353e6	0.225596
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.83784e6	0.420743
$540$	0	0
$541$	9.46117e6	1.38980	0.694899	−	0.719107i	$-0.255449\pi$
$541$	9.46117e6	1.38980	0.694899	+	0.719107i	$0.255449\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.60805e6	−0.520333
$546$	0	0
$547$	−1.14111e7	−1.63065	−0.815325	−	0.579004i	$-0.803442\pi$
$547$	−1.14111e7	−1.63065	−0.815325	+	0.579004i	$0.803442\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.24849e6	−1.15743
$552$	0	0
$553$	5.05842e6	0.703400
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.49930e6	0.751051	0.375526	−	0.926812i	$-0.377462\pi$
$557$	5.49930e6	0.751051	0.375526	+	0.926812i	$0.377462\pi$
$558$	0	0
$559$	−1.17549e7	−1.59107
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.82754e6	−0.375957	−0.187978	−	0.982173i	$-0.560193\pi$
$563$	−2.82754e6	−0.375957	−0.187978	+	0.982173i	$0.560193\pi$
$564$	0	0
$565$	−1.04730e6	−0.138022
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.89752e6	−1.15210	−0.576048	−	0.817416i	$-0.695406\pi$
$569$	−8.89752e6	−1.15210	−0.576048	+	0.817416i	$0.695406\pi$
$570$	0	0
$571$	−352454.	−0.0452390	−0.0226195	−	0.999744i	$-0.507201\pi$
$571$	−352454.	−0.0452390	−0.0226195	+	0.999744i	$0.507201\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.07725e6	−0.388144
$576$	0	0
$577$	4.69637e6	0.587250	0.293625	−	0.955921i	$-0.405138\pi$
$577$	4.69637e6	0.587250	0.293625	+	0.955921i	$0.405138\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.01389e6	−0.739121
$582$	0	0
$583$	2.29265e6	0.279362
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.87559e6	−0.464240	−0.232120	−	0.972687i	$-0.574566\pi$
$587$	−3.87559e6	−0.464240	−0.232120	+	0.972687i	$0.574566\pi$
$588$	0	0
$589$	4.34363e6	0.515899
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.43492e7	−1.67567	−0.837837	−	0.545921i	$-0.816180\pi$
$593$	−1.43492e7	−1.67567	−0.837837	+	0.545921i	$0.816180\pi$
$594$	0	0
$595$	804375.	0.0931464
$596$	0	0
$597$	0	0
$598$	0	0
$599$	614144.	0.0699364	0.0349682	−	0.999388i	$-0.488867\pi$
$599$	614144.	0.0699364	0.0349682	+	0.999388i	$0.488867\pi$
$600$	0	0
$601$	7.48024e6	0.844752	0.422376	−	0.906421i	$-0.361196\pi$
$601$	7.48024e6	0.844752	0.422376	+	0.906421i	$0.361196\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.83425e6	0.314811
$606$	0	0
$607$	−8.16923e6	−0.899932	−0.449966	−	0.893046i	$-0.648564\pi$
$607$	−8.16923e6	−0.899932	−0.449966	+	0.893046i	$0.648564\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.31413e7	−1.42408
$612$	0	0
$613$	−6.59061e6	−0.708393	−0.354197	−	0.935171i	$-0.615246\pi$
$613$	−6.59061e6	−0.708393	−0.354197	+	0.935171i	$0.615246\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.08486e7	−1.14726	−0.573631	−	0.819114i	$-0.694465\pi$
$617$	−1.08486e7	−1.14726	−0.573631	+	0.819114i	$0.694465\pi$
$618$	0	0
$619$	−2.77057e6	−0.290631	−0.145316	−	0.989385i	$-0.546420\pi$
$619$	−2.77057e6	−0.290631	−0.145316	+	0.989385i	$0.546420\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.11912e6	0.115520
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	57595.0	0.00580441
$630$	0	0
$631$	−1.80075e7	−1.80045	−0.900225	−	0.435426i	$-0.856598\pi$
$631$	−1.80075e7	−1.80045	−0.900225	+	0.435426i	$0.856598\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.67780e6	0.657202
$636$	0	0
$637$	1.12334e7	1.09689
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.37062e6	−0.324015	−0.162008	−	0.986790i	$-0.551797\pi$
$641$	−3.37062e6	−0.324015	−0.162008	+	0.986790i	$0.551797\pi$
$642$	0	0
$643$	4.08376e6	0.389523	0.194762	−	0.980851i	$-0.437607\pi$
$643$	4.08376e6	0.389523	0.194762	+	0.980851i	$0.437607\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11635.6	0.00109277	0.000546386	−	1.00000i	$-0.499826\pi$
$647$	11635.6	0.00109277	0.000546386		1.00000i	$0.499826\pi$
$648$	0	0
$649$	−6.92623e6	−0.645484
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.15712e7	−1.06193	−0.530966	−	0.847393i	$-0.678171\pi$
$653$	−1.15712e7	−1.06193	−0.530966	+	0.847393i	$0.678171\pi$
$654$	0	0
$655$	−3.73005e6	−0.339712
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.48159e6	0.581390	0.290695	−	0.956816i	$-0.406113\pi$
$659$	6.48159e6	0.581390	0.290695	+	0.956816i	$0.406113\pi$
$660$	0	0
$661$	1.80346e7	1.60547	0.802734	−	0.596337i	$-0.203378\pi$
$661$	1.80346e7	1.60547	0.802734	+	0.596337i	$0.203378\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.24748e6	0.197079
$666$	0	0
$667$	−2.78876e7	−2.42715
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.13376e6	0.525921
$672$	0	0
$673$	−1.23135e7	−1.04795	−0.523977	−	0.851732i	$-0.675552\pi$
$673$	−1.23135e7	−1.04795	−0.523977	+	0.851732i	$0.675552\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.41788e6	0.789735	0.394868	−	0.918738i	$-0.370790\pi$
$677$	9.41788e6	0.789735	0.394868	+	0.918738i	$0.370790\pi$
$678$	0	0
$679$	2.82873e6	0.235460
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.20155e7	−0.985575	−0.492788	−	0.870150i	$-0.664022\pi$
$683$	−1.20155e7	−0.985575	−0.492788	+	0.870150i	$0.664022\pi$
$684$	0	0
$685$	−5.95584e6	−0.484972
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.07528e6	0.728303
$690$	0	0
$691$	9.13020e6	0.727420	0.363710	−	0.931512i	$-0.381510\pi$
$691$	9.13020e6	0.727420	0.363710	+	0.931512i	$0.381510\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−926992.	−0.0727971
$696$	0	0
$697$	−1.25992e6	−0.0982340
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.86299e6	−0.373773	−0.186887	−	0.982382i	$-0.559840\pi$
$701$	−4.86299e6	−0.373773	−0.186887	+	0.982382i	$0.559840\pi$
$702$	0	0
$703$	160924.	0.0122810
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.83894e6	−0.288844
$708$	0	0
$709$	2.40699e7	1.79829	0.899143	−	0.437655i	$-0.144191\pi$
$709$	2.40699e7	1.79829	0.899143	+	0.437655i	$0.144191\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.46855e7	1.08185
$714$	0	0
$715$	−4.71854e6	−0.345177
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.27412e7	−0.919155	−0.459578	−	0.888138i	$-0.651999\pi$
$719$	−1.27412e7	−0.919155	−0.459578	+	0.888138i	$0.651999\pi$
$720$	0	0
$721$	−2.70650e6	−0.193896
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.54004e6	0.250129
$726$	0	0
$727$	8.22701e6	0.577306	0.288653	−	0.957434i	$-0.406793\pi$
$727$	8.22701e6	0.577306	0.288653	+	0.957434i	$0.406793\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.08814e6	−0.490613
$732$	0	0
$733$	2.05392e7	1.41196	0.705981	−	0.708230i	$-0.250506\pi$
$733$	2.05392e7	1.41196	0.705981	+	0.708230i	$0.250506\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.90839e6	0.265051
$738$	0	0
$739$	−669477.	−0.0450946	−0.0225473	−	0.999746i	$-0.507178\pi$
$739$	−669477.	−0.0450946	−0.0225473	+	0.999746i	$0.507178\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.32841e6	−0.0882793	−0.0441397	−	0.999025i	$-0.514055\pi$
$743$	−1.32841e6	−0.0882793	−0.0441397	+	0.999025i	$0.514055\pi$
$744$	0	0
$745$	8.84811e6	0.584063
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.68794e6	−0.240204
$750$	0	0
$751$	−8.73731e6	−0.565298	−0.282649	−	0.959223i	$-0.591213\pi$
$751$	−8.73731e6	−0.565298	−0.282649	+	0.959223i	$0.591213\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.82399e6	0.499529
$756$	0	0
$757$	−2.14412e7	−1.35991	−0.679953	−	0.733255i	$-0.738000\pi$
$757$	−2.14412e7	−1.35991	−0.679953	+	0.733255i	$0.738000\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.59944e7	−1.00117	−0.500583	−	0.865689i	$-0.666881\pi$
$761$	−1.59944e7	−1.00117	−0.500583	+	0.865689i	$0.666881\pi$
$762$	0	0
$763$	8.90925e6	0.554026
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.74169e7	−1.68279
$768$	0	0
$769$	3.22642e6	0.196746	0.0983729	−	0.995150i	$-0.468636\pi$
$769$	3.22642e6	0.196746	0.0983729	+	0.995150i	$0.468636\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.35896e7	−0.818010	−0.409005	−	0.912532i	$-0.634124\pi$
$773$	−1.35896e7	−0.818010	−0.409005	+	0.912532i	$0.634124\pi$
$774$	0	0
$775$	−1.86417e6	−0.111489
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.52031e6	−0.207844
$780$	0	0
$781$	−8.03195e6	−0.471187
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.15386e7	−0.668313
$786$	0	0
$787$	6.13774e6	0.353242	0.176621	−	0.984279i	$-0.443483\pi$
$787$	6.13774e6	0.353242	0.176621	+	0.984279i	$0.443483\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.58606e6	0.146959
$792$	0	0
$793$	2.42800e7	1.37109
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.67400e6	−0.539461	−0.269731	−	0.962936i	$-0.586935\pi$
$797$	−9.67400e6	−0.539461	−0.269731	+	0.962936i	$0.586935\pi$
$798$	0	0
$799$	−7.92414e6	−0.439122
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.08699e7	0.594890
$804$	0	0
$805$	7.59856e6	0.413278
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.21470e7	−1.18972	−0.594859	−	0.803830i	$-0.702792\pi$
$809$	−2.21470e7	−1.18972	−0.594859	+	0.803830i	$0.702792\pi$
$810$	0	0
$811$	−1.11479e7	−0.595168	−0.297584	−	0.954696i	$-0.596181\pi$
$811$	−1.11479e7	−0.595168	−0.297584	+	0.954696i	$0.596181\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.48081e6	−0.499979
$816$	0	0
$817$	−1.98047e7	−1.03804
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.04925e7	−0.543278	−0.271639	−	0.962399i	$-0.587566\pi$
$821$	−1.04925e7	−0.543278	−0.271639	+	0.962399i	$0.587566\pi$
$822$	0	0
$823$	2.60842e7	1.34239	0.671195	−	0.741281i	$-0.265781\pi$
$823$	2.60842e7	1.34239	0.671195	+	0.741281i	$0.265781\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.64602e6	0.337908	0.168954	−	0.985624i	$-0.445961\pi$
$827$	6.64602e6	0.337908	0.168954	+	0.985624i	$0.445961\pi$
$828$	0	0
$829$	−7.03700e6	−0.355632	−0.177816	−	0.984064i	$-0.556903\pi$
$829$	−7.03700e6	−0.355632	−0.177816	+	0.984064i	$0.556903\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.77369e6	0.338231
$834$	0	0
$835$	−1.03102e7	−0.511741
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.15192e7	−1.54586	−0.772931	−	0.634490i	$-0.781210\pi$
$839$	−3.15192e7	−1.54586	−0.772931	+	0.634490i	$0.781210\pi$
$840$	0	0
$841$	1.15705e7	0.564107
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.39562e6	−0.452672
$846$	0	0
$847$	−6.99854e6	−0.335196
$848$	0	0
$849$	0	0
$850$	0	0
$851$	544074.	0.0257534
$852$	0	0
$853$	2.10469e7	0.990411	0.495206	−	0.868776i	$-0.335093\pi$
$853$	2.10469e7	0.990411	0.495206	+	0.868776i	$0.335093\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.83597e6	−0.410963	−0.205481	−	0.978661i	$-0.565876\pi$
$857$	−8.83597e6	−0.410963	−0.205481	+	0.978661i	$0.565876\pi$
$858$	0	0
$859$	−1.34325e7	−0.621117	−0.310558	−	0.950554i	$-0.600516\pi$
$859$	−1.34325e7	−0.621117	−0.310558	+	0.950554i	$0.600516\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.38060e7	−0.631018	−0.315509	−	0.948923i	$-0.602175\pi$
$863$	−1.38060e7	−0.631018	−0.315509	+	0.948923i	$0.602175\pi$
$864$	0	0
$865$	−1.82615e7	−0.829842
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.78928e7	−0.803765
$870$	0	0
$871$	1.54711e7	0.690994
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−964560.	−0.0425901
$876$	0	0
$877$	3.58782e7	1.57519	0.787593	−	0.616196i	$-0.211327\pi$
$877$	3.58782e7	1.57519	0.787593	+	0.616196i	$0.211327\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.42379e6	0.192024	0.0960118	−	0.995380i	$-0.469391\pi$
$881$	4.42379e6	0.192024	0.0960118	+	0.995380i	$0.469391\pi$
$882$	0	0
$883$	−1.69958e7	−0.733566	−0.366783	−	0.930307i	$-0.619541\pi$
$883$	−1.69958e7	−0.733566	−0.366783	+	0.930307i	$0.619541\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.04106e7	−0.444291	−0.222146	−	0.975013i	$-0.571306\pi$
$887$	−1.04106e7	−0.444291	−0.222146	+	0.975013i	$0.571306\pi$
$888$	0	0
$889$	−1.64893e7	−0.699758
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.21406e7	−0.929095
$894$	0	0
$895$	−5.44659e6	−0.227283
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.68941e7	−0.697165
$900$	0	0
$901$	5.47236e6	0.224576
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.35447e7	−0.549726
$906$	0	0
$907$	−2.49530e7	−1.00718	−0.503588	−	0.863944i	$-0.667987\pi$
$907$	−2.49530e7	−1.00718	−0.503588	+	0.863944i	$0.667987\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.66692e6	−0.106467	−0.0532333	−	0.998582i	$-0.516953\pi$
$911$	−2.66692e6	−0.106467	−0.0532333	+	0.998582i	$0.516953\pi$
$912$	0	0
$913$	2.12725e7	0.844583
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.21051e6	0.361710
$918$	0	0
$919$	1.85809e7	0.725735	0.362867	−	0.931841i	$-0.381798\pi$
$919$	1.85809e7	0.725735	0.362867	+	0.931841i	$0.381798\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.17938e7	−1.22840
$924$	0	0
$925$	−69064.6	−0.00265400
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.28293e7	1.24802	0.624011	−	0.781415i	$-0.285502\pi$
$929$	3.28293e7	1.24802	0.624011	+	0.781415i	$0.285502\pi$
$930$	0	0
$931$	1.89261e7	0.715629
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.84526e6	−0.106437
$936$	0	0
$937$	−2.77296e7	−1.03180	−0.515898	−	0.856650i	$-0.672542\pi$
$937$	−2.77296e7	−1.03180	−0.515898	+	0.856650i	$0.672542\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.72967e7	1.37308	0.686542	−	0.727090i	$-0.259128\pi$
$941$	3.72967e7	1.37308	0.686542	+	0.727090i	$0.259128\pi$
$942$	0	0
$943$	−1.19019e7	−0.435851
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.05019e7	−0.742882	−0.371441	−	0.928457i	$-0.621136\pi$
$947$	−2.05019e7	−0.742882	−0.371441	+	0.928457i	$0.621136\pi$
$948$	0	0
$949$	4.30276e7	1.55089
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.05113e7	0.374907	0.187454	−	0.982273i	$-0.439977\pi$
$953$	1.05113e7	0.374907	0.187454	+	0.982273i	$0.439977\pi$
$954$	0	0
$955$	−2.02420e7	−0.718200
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.47066e7	0.516375
$960$	0	0
$961$	−1.97328e7	−0.689255
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.77763e7	0.614501
$966$	0	0
$967$	2.13682e7	0.734854	0.367427	−	0.930052i	$-0.380239\pi$
$967$	2.13682e7	0.734854	0.367427	+	0.930052i	$0.380239\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−356908.	−0.0121481	−0.00607405	−	0.999982i	$-0.501933\pi$
$971$	−356908.	−0.0121481	−0.00607405	+	0.999982i	$0.501933\pi$
$972$	0	0
$973$	2.28900e6	0.0775110
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.13784e7	−1.72204	−0.861021	−	0.508569i	$-0.830175\pi$
$977$	−5.13784e7	−1.72204	−0.861021	+	0.508569i	$0.830175\pi$
$978$	0	0
$979$	−3.95858e6	−0.132003
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.49482e6	−0.115356	−0.0576782	−	0.998335i	$-0.518370\pi$
$983$	−3.49482e6	−0.115356	−0.0576782	+	0.998335i	$0.518370\pi$
$984$	0	0
$985$	−3.39585e6	−0.111521
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.69585e7	−2.17678
$990$	0	0
$991$	−1.95776e7	−0.633249	−0.316624	−	0.948551i	$-0.602549\pi$
$991$	−1.95776e7	−0.633249	−0.316624	+	0.948551i	$0.602549\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.83267e6	−0.314857
$996$	0	0
$997$	1.77382e6	0.0565161	0.0282580	−	0.999601i	$-0.491004\pi$
$997$	1.77382e6	0.0565161	0.0282580	+	0.999601i	$0.491004\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.6.a.e.1.2	✓	3
3.2	odd	2		1080.6.a.g.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.e.1.2	✓	3	1.1	even	1	trivial
1080.6.a.g.1.2	yes	3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} +61.7318 q^{7} +O(q^{10})\)	\(q-25.0000 q^{5} +61.7318 q^{7} -218.360 q^{11} -864.360 q^{13} -521.206 q^{17} -1456.28 q^{19} -4923.60 q^{23} +625.000 q^{25} +5664.06 q^{29} -2982.68 q^{31} -1543.30 q^{35} -110.503 q^{37} +2417.32 q^{41} +13599.5 q^{43} +15203.5 q^{47} -12996.2 q^{49} -10499.4 q^{53} +5459.00 q^{55} +31719.4 q^{59} -28090.1 q^{61} +21609.0 q^{65} -17898.9 q^{67} +36783.1 q^{71} -49779.7 q^{73} -13479.7 q^{77} +81941.9 q^{79} -97419.7 q^{83} +13030.2 q^{85} +18128.7 q^{89} -53358.5 q^{91} +36407.1 q^{95} +45823.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 75 q^{5} - 30 q^{7}+O(q^{10}) \) 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

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Modular forms

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