LMFDB - Embedded newform 1024.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,6,Mod(1,1024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1024.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1024.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:	$164.233031488$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 760x^{6} + 148162x^{4} - 2536632x^{2} + 3538161 $ x^8 - 760x^6 + 148162x^4 - 2536632*x^2 + 3538161
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 256)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.23717$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74962 q^{3} -13.3804 q^{5} -217.226 q^{7} -239.939 q^{9} +O(q^{10})$	$q-1.74962 q^{3} -13.3804 q^{5} -217.226 q^{7} -239.939 q^{9} +399.293 q^{11} +480.632 q^{13} +23.4106 q^{15} +473.078 q^{17} -273.135 q^{19} +380.063 q^{21} -1880.50 q^{23} -2945.96 q^{25} +844.960 q^{27} +7073.23 q^{29} +4038.97 q^{31} -698.612 q^{33} +2906.56 q^{35} +2657.13 q^{37} -840.924 q^{39} -15340.6 q^{41} +16703.7 q^{43} +3210.48 q^{45} -4380.31 q^{47} +30379.9 q^{49} -827.708 q^{51} +27729.9 q^{53} -5342.70 q^{55} +477.882 q^{57} -30262.9 q^{59} +42173.1 q^{61} +52120.8 q^{63} -6431.05 q^{65} +15921.9 q^{67} +3290.16 q^{69} +48020.3 q^{71} +8033.35 q^{73} +5154.33 q^{75} -86736.7 q^{77} -74631.9 q^{79} +56826.8 q^{81} -89199.5 q^{83} -6329.98 q^{85} -12375.5 q^{87} +93498.5 q^{89} -104405. q^{91} -7066.68 q^{93} +3654.65 q^{95} -158815. q^{97} -95805.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1096 q^{9}+O(q^{10}) $ 8 * q + 1096 * q^9	$ 8 q + 1096 q^{9} - 128 q^{11} + 192 q^{17} - 4672 q^{19} + 10248 q^{25} - 17856 q^{27} - 39104 q^{33} - 9024 q^{35} - 13952 q^{41} + 59904 q^{43} + 98056 q^{49} - 111360 q^{51} + 103616 q^{57} - 216384 q^{59} - 315312 q^{65} + 111616 q^{67} - 132976 q^{73} + 558400 q^{75} + 506440 q^{81} - 150080 q^{83} + 168592 q^{89} - 405312 q^{91} - 416576 q^{97} + 175936 q^{99}+O(q^{100}) $ 8 * q + 1096 * q^9 - 128 * q^11 + 192 * q^17 - 4672 * q^19 + 10248 * q^25 - 17856 * q^27 - 39104 * q^33 - 9024 * q^35 - 13952 * q^41 + 59904 * q^43 + 98056 * q^49 - 111360 * q^51 + 103616 * q^57 - 216384 * q^59 - 315312 * q^65 + 111616 * q^67 - 132976 * q^73 + 558400 * q^75 + 506440 * q^81 - 150080 * q^83 + 168592 * q^89 - 405312 * q^91 - 416576 * q^97 + 175936 * q^99

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
$3$	−1.74962	−0.112238	−0.0561192	−	0.998424i	$-0.517873\pi$
$3$	−1.74962	−0.112238	−0.0561192	+	0.998424i	$0.517873\pi$
$4$	0	0
$5$	−13.3804	−0.239356	−0.119678	−	0.992813i	$-0.538186\pi$
$5$	−13.3804	−0.239356	−0.119678	+	0.992813i	$0.538186\pi$
$6$	0	0
$7$	−217.226	−1.67558	−0.837791	−	0.545991i	$-0.816153\pi$
$7$	−217.226	−1.67558	−0.837791	+	0.545991i	$0.816153\pi$
$8$	0	0
$9$	−239.939	−0.987403
$10$	0	0
$11$	399.293	0.994970	0.497485	−	0.867473i	$-0.334257\pi$
$11$	399.293	0.994970	0.497485	+	0.867473i	$0.334257\pi$
$12$	0	0
$13$	480.632	0.788777	0.394388	−	0.918944i	$-0.370957\pi$
$13$	480.632	0.788777	0.394388	+	0.918944i	$0.370957\pi$
$14$	0	0
$15$	23.4106	0.0268649
$16$	0	0
$17$	473.078	0.397018	0.198509	−	0.980099i	$-0.436390\pi$
$17$	473.078	0.397018	0.198509	+	0.980099i	$0.436390\pi$
$18$	0	0
$19$	−273.135	−0.173577	−0.0867886	−	0.996227i	$-0.527660\pi$
$19$	−273.135	−0.173577	−0.0867886	+	0.996227i	$0.527660\pi$
$20$	0	0
$21$	380.063	0.188065
$22$	0	0
$23$	−1880.50	−0.741230	−0.370615	−	0.928787i	$-0.620853\pi$
$23$	−1880.50	−0.741230	−0.370615	+	0.928787i	$0.620853\pi$
$24$	0	0
$25$	−2945.96	−0.942709
$26$	0	0
$27$	844.960	0.223063
$28$	0	0
$29$	7073.23	1.56179	0.780895	−	0.624662i	$-0.214763\pi$
$29$	7073.23	1.56179	0.780895	+	0.624662i	$0.214763\pi$
$30$	0	0
$31$	4038.97	0.754861	0.377430	−	0.926038i	$-0.376808\pi$
$31$	4038.97	0.754861	0.377430	+	0.926038i	$0.376808\pi$
$32$	0	0
$33$	−698.612	−0.111674
$34$	0	0
$35$	2906.56	0.401061
$36$	0	0
$37$	2657.13	0.319087	0.159543	−	0.987191i	$-0.448998\pi$
$37$	2657.13	0.319087	0.159543	+	0.987191i	$0.448998\pi$
$38$	0	0
$39$	−840.924	−0.0885310
$40$	0	0
$41$	−15340.6	−1.42522	−0.712612	−	0.701558i	$-0.752488\pi$
$41$	−15340.6	−1.42522	−0.712612	+	0.701558i	$0.752488\pi$
$42$	0	0
$43$	16703.7	1.37766	0.688828	−	0.724925i	$-0.258126\pi$
$43$	16703.7	1.37766	0.688828	+	0.724925i	$0.258126\pi$
$44$	0	0
$45$	3210.48	0.236341
$46$	0	0
$47$	−4380.31	−0.289242	−0.144621	−	0.989487i	$-0.546196\pi$
$47$	−4380.31	−0.289242	−0.144621	+	0.989487i	$0.546196\pi$
$48$	0	0
$49$	30379.9	1.80758
$50$	0	0
$51$	−827.708	−0.0445607
$52$	0	0
$53$	27729.9	1.35599	0.677997	−	0.735064i	$-0.262848\pi$
$53$	27729.9	1.35599	0.677997	+	0.735064i	$0.262848\pi$
$54$	0	0
$55$	−5342.70	−0.238152
$56$	0	0
$57$	477.882	0.0194820
$58$	0	0
$59$	−30262.9	−1.13183	−0.565914	−	0.824464i	$-0.691477\pi$
$59$	−30262.9	−1.13183	−0.565914	+	0.824464i	$0.691477\pi$
$60$	0	0
$61$	42173.1	1.45115	0.725573	−	0.688145i	$-0.241575\pi$
$61$	42173.1	1.45115	0.725573	+	0.688145i	$0.241575\pi$
$62$	0	0
$63$	52120.8	1.65447
$64$	0	0
$65$	−6431.05	−0.188798
$66$	0	0
$67$	15921.9	0.433320	0.216660	−	0.976247i	$-0.430484\pi$
$67$	15921.9	0.433320	0.216660	+	0.976247i	$0.430484\pi$
$68$	0	0
$69$	3290.16	0.0831944
$70$	0	0
$71$	48020.3	1.13052	0.565261	−	0.824912i	$-0.308776\pi$
$71$	48020.3	1.13052	0.565261	+	0.824912i	$0.308776\pi$
$72$	0	0
$73$	8033.35	0.176437	0.0882184	−	0.996101i	$-0.471883\pi$
$73$	8033.35	0.176437	0.0882184	+	0.996101i	$0.471883\pi$
$74$	0	0
$75$	5154.33	0.105808
$76$	0	0
$77$	−86736.7	−1.66715
$78$	0	0
$79$	−74631.9	−1.34542	−0.672708	−	0.739908i	$-0.734869\pi$
$79$	−74631.9	−1.34542	−0.672708	+	0.739908i	$0.734869\pi$
$80$	0	0
$81$	56826.8	0.962366
$82$	0	0
$83$	−89199.5	−1.42124	−0.710620	−	0.703576i	$-0.751585\pi$
$83$	−89199.5	−1.42124	−0.710620	+	0.703576i	$0.751585\pi$
$84$	0	0
$85$	−6329.98	−0.0950287
$86$	0	0
$87$	−12375.5	−0.175293
$88$	0	0
$89$	93498.5	1.25121	0.625604	−	0.780141i	$-0.284852\pi$
$89$	93498.5	1.25121	0.625604	+	0.780141i	$0.284852\pi$
$90$	0	0
$91$	−104405.	−1.32166
$92$	0	0
$93$	−7066.68	−0.0847243
$94$	0	0
$95$	3654.65	0.0415467
$96$	0	0
$97$	−158815.	−1.71381	−0.856907	−	0.515472i	$-0.827617\pi$
$97$	−158815.	−1.71381	−0.856907	+	0.515472i	$0.827617\pi$
$98$	0	0
$99$	−95805.9	−0.982436
$100$	0	0
$101$	−154717.	−1.50916	−0.754578	−	0.656211i	$-0.772158\pi$
$101$	−154717.	−1.50916	−0.754578	+	0.656211i	$0.772158\pi$
$102$	0	0
$103$	−56607.0	−0.525747	−0.262873	−	0.964830i	$-0.584670\pi$
$103$	−56607.0	−0.525747	−0.262873	+	0.964830i	$0.584670\pi$
$104$	0	0
$105$	−5085.39	−0.0450144
$106$	0	0
$107$	148623.	1.25495	0.627476	−	0.778636i	$-0.284088\pi$
$107$	148623.	1.25495	0.627476	+	0.778636i	$0.284088\pi$
$108$	0	0
$109$	−50447.3	−0.406698	−0.203349	−	0.979106i	$-0.565183\pi$
$109$	−50447.3	−0.406698	−0.203349	+	0.979106i	$0.565183\pi$
$110$	0	0
$111$	−4648.98	−0.0358138
$112$	0	0
$113$	−78832.7	−0.580779	−0.290389	−	0.956909i	$-0.593785\pi$
$113$	−78832.7	−0.580779	−0.290389	+	0.956909i	$0.593785\pi$
$114$	0	0
$115$	25161.8	0.177418
$116$	0	0
$117$	−115322.	−0.778840
$118$	0	0
$119$	−102765.	−0.665237
$120$	0	0
$121$	−1616.01	−0.0100342
$122$	0	0
$123$	26840.3	0.159965
$124$	0	0
$125$	81232.0	0.464999
$126$	0	0
$127$	146680.	0.806977	0.403488	−	0.914985i	$-0.367798\pi$
$127$	146680.	0.806977	0.403488	+	0.914985i	$0.367798\pi$
$128$	0	0
$129$	−29225.1	−0.154626
$130$	0	0
$131$	230556.	1.17381	0.586906	−	0.809655i	$-0.300346\pi$
$131$	230556.	1.17381	0.586906	+	0.809655i	$0.300346\pi$
$132$	0	0
$133$	59331.8	0.290843
$134$	0	0
$135$	−11305.9	−0.0533914
$136$	0	0
$137$	129663.	0.590221	0.295111	−	0.955463i	$-0.404644\pi$
$137$	129663.	0.590221	0.295111	+	0.955463i	$0.404644\pi$
$138$	0	0
$139$	−415331.	−1.82330	−0.911648	−	0.410972i	$-0.865190\pi$
$139$	−415331.	−1.82330	−0.911648	+	0.410972i	$0.865190\pi$
$140$	0	0
$141$	7663.90	0.0324640
$142$	0	0
$143$	191913.	0.784809
$144$	0	0
$145$	−94642.6	−0.373824
$146$	0	0
$147$	−53153.4	−0.202879
$148$	0	0
$149$	−39355.6	−0.145225	−0.0726125	−	0.997360i	$-0.523134\pi$
$149$	−39355.6	−0.145225	−0.0726125	+	0.997360i	$0.523134\pi$
$150$	0	0
$151$	−81358.3	−0.290375	−0.145188	−	0.989404i	$-0.546379\pi$
$151$	−81358.3	−0.290375	−0.145188	+	0.989404i	$0.546379\pi$
$152$	0	0
$153$	−113510.	−0.392017
$154$	0	0
$155$	−54043.1	−0.180680
$156$	0	0
$157$	−286724.	−0.928358	−0.464179	−	0.885741i	$-0.653651\pi$
$157$	−286724.	−0.928358	−0.464179	+	0.885741i	$0.653651\pi$
$158$	0	0
$159$	−48516.8	−0.152195
$160$	0	0
$161$	408492.	1.24199
$162$	0	0
$163$	−541921.	−1.59760	−0.798798	−	0.601599i	$-0.794530\pi$
$163$	−541921.	−1.59760	−0.798798	+	0.601599i	$0.794530\pi$
$164$	0	0
$165$	9347.71	0.0267298
$166$	0	0
$167$	−523932.	−1.45373	−0.726865	−	0.686781i	$-0.759023\pi$
$167$	−523932.	−1.45373	−0.726865	+	0.686781i	$0.759023\pi$
$168$	0	0
$169$	−140286.	−0.377832
$170$	0	0
$171$	65535.6	0.171391
$172$	0	0
$173$	632736.	1.60734	0.803669	−	0.595077i	$-0.202878\pi$
$173$	632736.	1.60734	0.803669	+	0.595077i	$0.202878\pi$
$174$	0	0
$175$	639939.	1.57959
$176$	0	0
$177$	52948.6	0.127035
$178$	0	0
$179$	278563.	0.649816	0.324908	−	0.945746i	$-0.394667\pi$
$179$	278563.	0.649816	0.324908	+	0.945746i	$0.394667\pi$
$180$	0	0
$181$	367651.	0.834140	0.417070	−	0.908874i	$-0.363057\pi$
$181$	367651.	0.834140	0.417070	+	0.908874i	$0.363057\pi$
$182$	0	0
$183$	−73787.1	−0.162874
$184$	0	0
$185$	−35553.5	−0.0763754
$186$	0	0
$187$	188897.	0.395022
$188$	0	0
$189$	−183547.	−0.373760
$190$	0	0
$191$	−187212.	−0.371322	−0.185661	−	0.982614i	$-0.559443\pi$
$191$	−187212.	−0.371322	−0.185661	+	0.982614i	$0.559443\pi$
$192$	0	0
$193$	592251.	1.14449	0.572246	−	0.820082i	$-0.306072\pi$
$193$	592251.	1.14449	0.572246	+	0.820082i	$0.306072\pi$
$194$	0	0
$195$	11251.9	0.0211904
$196$	0	0
$197$	1.01650e6	1.86613	0.933065	−	0.359707i	$-0.117124\pi$
$197$	1.01650e6	1.86613	0.933065	+	0.359707i	$0.117124\pi$
$198$	0	0
$199$	−285135.	−0.510408	−0.255204	−	0.966887i	$-0.582143\pi$
$199$	−285135.	−0.510408	−0.255204	+	0.966887i	$0.582143\pi$
$200$	0	0
$201$	−27857.4	−0.0486351
$202$	0	0
$203$	−1.53649e6	−2.61691
$204$	0	0
$205$	205264.	0.341136
$206$	0	0
$207$	451204.	0.731893
$208$	0	0
$209$	−109061.	−0.172704
$210$	0	0
$211$	−960777.	−1.48565	−0.742825	−	0.669486i	$-0.766515\pi$
$211$	−960777.	−1.48565	−0.742825	+	0.669486i	$0.766515\pi$
$212$	0	0
$213$	−84017.3	−0.126888
$214$	0	0
$215$	−223502.	−0.329750
$216$	0	0
$217$	−877368.	−1.26483
$218$	0	0
$219$	−14055.3	−0.0198030
$220$	0	0
$221$	227376.	0.313159
$222$	0	0
$223$	−20550.5	−0.0276732	−0.0138366	−	0.999904i	$-0.504404\pi$
$223$	−20550.5	−0.0276732	−0.0138366	+	0.999904i	$0.504404\pi$
$224$	0	0
$225$	706851.	0.930833
$226$	0	0
$227$	−1.32848e6	−1.71116	−0.855579	−	0.517672i	$-0.826799\pi$
$227$	−1.32848e6	−1.71116	−0.855579	+	0.517672i	$0.826799\pi$
$228$	0	0
$229$	−114024.	−0.143684	−0.0718421	−	0.997416i	$-0.522888\pi$
$229$	−114024.	−0.143684	−0.0718421	+	0.997416i	$0.522888\pi$
$230$	0	0
$231$	151756.	0.187119
$232$	0	0
$233$	293436.	0.354098	0.177049	−	0.984202i	$-0.443345\pi$
$233$	293436.	0.354098	0.177049	+	0.984202i	$0.443345\pi$
$234$	0	0
$235$	58610.4	0.0692317
$236$	0	0
$237$	130578.	0.151007
$238$	0	0
$239$	−109538.	−0.124042	−0.0620211	−	0.998075i	$-0.519755\pi$
$239$	−109538.	−0.124042	−0.0620211	+	0.998075i	$0.519755\pi$
$240$	0	0
$241$	−734422.	−0.814522	−0.407261	−	0.913312i	$-0.633516\pi$
$241$	−734422.	−0.814522	−0.407261	+	0.913312i	$0.633516\pi$
$242$	0	0
$243$	−304751.	−0.331077
$244$	0	0
$245$	−406496.	−0.432654
$246$	0	0
$247$	−131277.	−0.136914
$248$	0	0
$249$	156065.	0.159517
$250$	0	0
$251$	−975084.	−0.976917	−0.488458	−	0.872587i	$-0.662441\pi$
$251$	−975084.	−0.976917	−0.488458	+	0.872587i	$0.662441\pi$
$252$	0	0
$253$	−750870.	−0.737502
$254$	0	0
$255$	11075.1	0.0106659
$256$	0	0
$257$	−1.75709e6	−1.65944	−0.829719	−	0.558181i	$-0.811499\pi$
$257$	−1.75709e6	−1.65944	−0.829719	+	0.558181i	$0.811499\pi$
$258$	0	0
$259$	−577197.	−0.534656
$260$	0	0
$261$	−1.69714e6	−1.54212
$262$	0	0
$263$	−1.72852e6	−1.54094	−0.770468	−	0.637478i	$-0.779978\pi$
$263$	−1.72852e6	−1.54094	−0.770468	+	0.637478i	$0.779978\pi$
$264$	0	0
$265$	−371037.	−0.324565
$266$	0	0
$267$	−163587.	−0.140434
$268$	0	0
$269$	281278.	0.237003	0.118502	−	0.992954i	$-0.462191\pi$
$269$	281278.	0.237003	0.118502	+	0.992954i	$0.462191\pi$
$270$	0	0
$271$	1.61535e6	1.33612	0.668059	−	0.744109i	$-0.267126\pi$
$271$	1.61535e6	1.33612	0.668059	+	0.744109i	$0.267126\pi$
$272$	0	0
$273$	182670.	0.148341
$274$	0	0
$275$	−1.17630e6	−0.937967
$276$	0	0
$277$	−591475.	−0.463166	−0.231583	−	0.972815i	$-0.574390\pi$
$277$	−591475.	−0.463166	−0.231583	+	0.972815i	$0.574390\pi$
$278$	0	0
$279$	−969107.	−0.745352
$280$	0	0
$281$	1.87424e6	1.41599	0.707994	−	0.706219i	$-0.249601\pi$
$281$	1.87424e6	1.41599	0.707994	+	0.706219i	$0.249601\pi$
$282$	0	0
$283$	−1.57519e6	−1.16914	−0.584570	−	0.811344i	$-0.698737\pi$
$283$	−1.57519e6	−1.16914	−0.584570	+	0.811344i	$0.698737\pi$
$284$	0	0
$285$	−6394.26	−0.00466314
$286$	0	0
$287$	3.33237e6	2.38808
$288$	0	0
$289$	−1.19605e6	−0.842376
$290$	0	0
$291$	277867.	0.192355
$292$	0	0
$293$	690676.	0.470008	0.235004	−	0.971994i	$-0.424490\pi$
$293$	690676.	0.470008	0.235004	+	0.971994i	$0.424490\pi$
$294$	0	0
$295$	404930.	0.270910
$296$	0	0
$297$	337387.	0.221941
$298$	0	0
$299$	−903826.	−0.584665
$300$	0	0
$301$	−3.62846e6	−2.30838
$302$	0	0
$303$	270696.	0.169385
$304$	0	0
$305$	−564294.	−0.347341
$306$	0	0
$307$	−561176.	−0.339823	−0.169912	−	0.985459i	$-0.554348\pi$
$307$	−561176.	−0.339823	−0.169912	+	0.985459i	$0.554348\pi$
$308$	0	0
$309$	99040.8	0.0590090
$310$	0	0
$311$	−1.80244e6	−1.05672	−0.528361	−	0.849020i	$-0.677193\pi$
$311$	−1.80244e6	−1.05672	−0.528361	+	0.849020i	$0.677193\pi$
$312$	0	0
$313$	−346574.	−0.199956	−0.0999780	−	0.994990i	$-0.531877\pi$
$313$	−346574.	−0.199956	−0.0999780	+	0.994990i	$0.531877\pi$
$314$	0	0
$315$	−697398.	−0.396008
$316$	0	0
$317$	2.04820e6	1.14479	0.572394	−	0.819979i	$-0.306015\pi$
$317$	2.04820e6	1.14479	0.572394	+	0.819979i	$0.306015\pi$
$318$	0	0
$319$	2.82429e6	1.55393
$320$	0	0
$321$	−260034.	−0.140854
$322$	0	0
$323$	−129214.	−0.0689134
$324$	0	0
$325$	−1.41592e6	−0.743587
$326$	0	0
$327$	88263.8	0.0456471
$328$	0	0
$329$	951516.	0.484648
$330$	0	0
$331$	1.24651e6	0.625352	0.312676	−	0.949860i	$-0.398775\pi$
$331$	1.24651e6	0.625352	0.312676	+	0.949860i	$0.398775\pi$
$332$	0	0
$333$	−637549.	−0.315067
$334$	0	0
$335$	−213042.	−0.103718
$336$	0	0
$337$	2.66445e6	1.27801	0.639003	−	0.769205i	$-0.279347\pi$
$337$	2.66445e6	1.27801	0.639003	+	0.769205i	$0.279347\pi$
$338$	0	0
$339$	137928.	0.0651856
$340$	0	0
$341$	1.61273e6	0.751064
$342$	0	0
$343$	−2.94839e6	−1.35316
$344$	0	0
$345$	−44023.7	−0.0199131
$346$	0	0
$347$	−2.01623e6	−0.898910	−0.449455	−	0.893303i	$-0.648382\pi$
$347$	−2.01623e6	−0.898910	−0.449455	+	0.893303i	$0.648382\pi$
$348$	0	0
$349$	−420348.	−0.184733	−0.0923666	−	0.995725i	$-0.529443\pi$
$349$	−420348.	−0.184733	−0.0923666	+	0.995725i	$0.529443\pi$
$350$	0	0
$351$	406115.	0.175947
$352$	0	0
$353$	−1.12097e6	−0.478804	−0.239402	−	0.970921i	$-0.576951\pi$
$353$	−1.12097e6	−0.478804	−0.239402	+	0.970921i	$0.576951\pi$
$354$	0	0
$355$	−642530.	−0.270597
$356$	0	0
$357$	179799.	0.0746651
$358$	0	0
$359$	−3.31336e6	−1.35685	−0.678426	−	0.734669i	$-0.737337\pi$
$359$	−3.31336e6	−1.35685	−0.678426	+	0.734669i	$0.737337\pi$
$360$	0	0
$361$	−2.40150e6	−0.969871
$362$	0	0
$363$	2827.41	0.00112622
$364$	0	0
$365$	−107489.	−0.0422312
$366$	0	0
$367$	−1.22043e6	−0.472986	−0.236493	−	0.971633i	$-0.575998\pi$
$367$	−1.22043e6	−0.472986	−0.236493	+	0.971633i	$0.575998\pi$
$368$	0	0
$369$	3.68081e6	1.40727
$370$	0	0
$371$	−6.02363e6	−2.27208
$372$	0	0
$373$	1.70283e6	0.633722	0.316861	−	0.948472i	$-0.397371\pi$
$373$	1.70283e6	0.633722	0.316861	+	0.948472i	$0.397371\pi$
$374$	0	0
$375$	−142125.	−0.0521907
$376$	0	0
$377$	3.39962e6	1.23190
$378$	0	0
$379$	−3.97957e6	−1.42311	−0.711554	−	0.702631i	$-0.752008\pi$
$379$	−3.97957e6	−1.42311	−0.711554	+	0.702631i	$0.752008\pi$
$380$	0	0
$381$	−256634.	−0.0905737
$382$	0	0
$383$	432846.	0.150777	0.0753887	−	0.997154i	$-0.475980\pi$
$383$	432846.	0.150777	0.0753887	+	0.997154i	$0.475980\pi$
$384$	0	0
$385$	1.16057e6	0.399043
$386$	0	0
$387$	−4.00786e6	−1.36030
$388$	0	0
$389$	2.39317e6	0.801861	0.400931	−	0.916108i	$-0.368687\pi$
$389$	2.39317e6	0.801861	0.400931	+	0.916108i	$0.368687\pi$
$390$	0	0
$391$	−889622.	−0.294282
$392$	0	0
$393$	−403386.	−0.131747
$394$	0	0
$395$	998604.	0.322033
$396$	0	0
$397$	2.90547e6	0.925208	0.462604	−	0.886565i	$-0.346915\pi$
$397$	2.90547e6	0.925208	0.462604	+	0.886565i	$0.346915\pi$
$398$	0	0
$399$	−103808.	−0.0326437
$400$	0	0
$401$	−2.95627e6	−0.918087	−0.459043	−	0.888414i	$-0.651808\pi$
$401$	−2.95627e6	−0.918087	−0.459043	+	0.888414i	$0.651808\pi$
$402$	0	0
$403$	1.94126e6	0.595417
$404$	0	0
$405$	−760365.	−0.230348
$406$	0	0
$407$	1.06098e6	0.317482
$408$	0	0
$409$	−4.47118e6	−1.32164	−0.660821	−	0.750544i	$-0.729792\pi$
$409$	−4.47118e6	−1.32164	−0.660821	+	0.750544i	$0.729792\pi$
$410$	0	0
$411$	−226861.	−0.0662454
$412$	0	0
$413$	6.57388e6	1.89647
$414$	0	0
$415$	1.19353e6	0.340182
$416$	0	0
$417$	726672.	0.204644
$418$	0	0
$419$	3.69637e6	1.02859	0.514293	−	0.857614i	$-0.328054\pi$
$419$	3.69637e6	1.02859	0.514293	+	0.857614i	$0.328054\pi$
$420$	0	0
$421$	−418255.	−0.115010	−0.0575050	−	0.998345i	$-0.518315\pi$
$421$	−418255.	−0.115010	−0.0575050	+	0.998345i	$0.518315\pi$
$422$	0	0
$423$	1.05101e6	0.285598
$424$	0	0
$425$	−1.39367e6	−0.374273
$426$	0	0
$427$	−9.16108e6	−2.43152
$428$	0	0
$429$	−335775.	−0.0880857
$430$	0	0
$431$	4.14971e6	1.07603	0.538016	−	0.842935i	$-0.319174\pi$
$431$	4.14971e6	1.07603	0.538016	+	0.842935i	$0.319174\pi$
$432$	0	0
$433$	1.48115e6	0.379647	0.189823	−	0.981818i	$-0.439208\pi$
$433$	1.48115e6	0.379647	0.189823	+	0.981818i	$0.439208\pi$
$434$	0	0
$435$	165589.	0.0419573
$436$	0	0
$437$	513629.	0.128661
$438$	0	0
$439$	−3.98071e6	−0.985824	−0.492912	−	0.870079i	$-0.664068\pi$
$439$	−3.98071e6	−0.985824	−0.492912	+	0.870079i	$0.664068\pi$
$440$	0	0
$441$	−7.28932e6	−1.78481
$442$	0	0
$443$	1.00955e6	0.244411	0.122205	−	0.992505i	$-0.461003\pi$
$443$	1.00955e6	0.244411	0.122205	+	0.992505i	$0.461003\pi$
$444$	0	0
$445$	−1.25105e6	−0.299484
$446$	0	0
$447$	68857.5	0.0162998
$448$	0	0
$449$	3.43383e6	0.803827	0.401913	−	0.915678i	$-0.368345\pi$
$449$	3.43383e6	0.803827	0.401913	+	0.915678i	$0.368345\pi$
$450$	0	0
$451$	−6.12540e6	−1.41806
$452$	0	0
$453$	142346.	0.0325912
$454$	0	0
$455$	1.39699e6	0.316347
$456$	0	0
$457$	1.19382e6	0.267393	0.133696	−	0.991022i	$-0.457315\pi$
$457$	1.19382e6	0.267393	0.133696	+	0.991022i	$0.457315\pi$
$458$	0	0
$459$	399732.	0.0885600
$460$	0	0
$461$	4.68616e6	1.02699	0.513493	−	0.858094i	$-0.328351\pi$
$461$	4.68616e6	1.02699	0.513493	+	0.858094i	$0.328351\pi$
$462$	0	0
$463$	2.87318e6	0.622888	0.311444	−	0.950265i	$-0.399187\pi$
$463$	2.87318e6	0.622888	0.311444	+	0.950265i	$0.399187\pi$
$464$	0	0
$465$	94555.0	0.0202793
$466$	0	0
$467$	−2.93371e6	−0.622479	−0.311239	−	0.950332i	$-0.600744\pi$
$467$	−2.93371e6	−0.622479	−0.311239	+	0.950332i	$0.600744\pi$
$468$	0	0
$469$	−3.45865e6	−0.726064
$470$	0	0
$471$	501659.	0.104197
$472$	0	0
$473$	6.66966e6	1.37073
$474$	0	0
$475$	804645.	0.163633
$476$	0	0
$477$	−6.65347e6	−1.33891
$478$	0	0
$479$	−5.37108e6	−1.06960	−0.534801	−	0.844978i	$-0.679614\pi$
$479$	−5.37108e6	−1.06960	−0.534801	+	0.844978i	$0.679614\pi$
$480$	0	0
$481$	1.27710e6	0.251688
$482$	0	0
$483$	−714707.	−0.139399
$484$	0	0
$485$	2.12501e6	0.410211
$486$	0	0
$487$	−6.47468e6	−1.23708	−0.618538	−	0.785755i	$-0.712275\pi$
$487$	−6.47468e6	−1.23708	−0.618538	+	0.785755i	$0.712275\pi$
$488$	0	0
$489$	948157.	0.179311
$490$	0	0
$491$	−4.80164e6	−0.898847	−0.449424	−	0.893319i	$-0.648371\pi$
$491$	−4.80164e6	−0.898847	−0.449424	+	0.893319i	$0.648371\pi$
$492$	0	0
$493$	3.34619e6	0.620059
$494$	0	0
$495$	1.28192e6	0.235152
$496$	0	0
$497$	−1.04312e7	−1.89428
$498$	0	0
$499$	3.49344e6	0.628062	0.314031	−	0.949413i	$-0.398320\pi$
$499$	3.49344e6	0.628062	0.314031	+	0.949413i	$0.398320\pi$
$500$	0	0
$501$	916683.	0.163164
$502$	0	0
$503$	9.84544e6	1.73506	0.867531	−	0.497383i	$-0.165706\pi$
$503$	9.84544e6	1.73506	0.867531	+	0.497383i	$0.165706\pi$
$504$	0	0
$505$	2.07017e6	0.361225
$506$	0	0
$507$	245448.	0.0424072
$508$	0	0
$509$	−4.13736e6	−0.707829	−0.353915	−	0.935278i	$-0.615150\pi$
$509$	−4.13736e6	−0.707829	−0.353915	+	0.935278i	$0.615150\pi$
$510$	0	0
$511$	−1.74505e6	−0.295635
$512$	0	0
$513$	−230788.	−0.0387186
$514$	0	0
$515$	757424.	0.125841
$516$	0	0
$517$	−1.74903e6	−0.287787
$518$	0	0
$519$	−1.10705e6	−0.180405
$520$	0	0
$521$	160551.	0.0259130	0.0129565	−	0.999916i	$-0.495876\pi$
$521$	160551.	0.0259130	0.0129565	+	0.999916i	$0.495876\pi$
$522$	0	0
$523$	3.81643e6	0.610103	0.305051	−	0.952336i	$-0.401326\pi$
$523$	3.81643e6	0.610103	0.305051	+	0.952336i	$0.401326\pi$
$524$	0	0
$525$	−1.11965e6	−0.177290
$526$	0	0
$527$	1.91075e6	0.299694
$528$	0	0
$529$	−2.90007e6	−0.450578
$530$	0	0
$531$	7.26125e6	1.11757
$532$	0	0
$533$	−7.37319e6	−1.12418
$534$	0	0
$535$	−1.98864e6	−0.300380
$536$	0	0
$537$	−487380.	−0.0729343
$538$	0	0
$539$	1.21305e7	1.79848
$540$	0	0
$541$	−8.98597e6	−1.31999	−0.659997	−	0.751269i	$-0.729442\pi$
$541$	−8.98597e6	−1.31999	−0.659997	+	0.751269i	$0.729442\pi$
$542$	0	0
$543$	−643250.	−0.0936225
$544$	0	0
$545$	675005.	0.0973455
$546$	0	0
$547$	4.36041e6	0.623101	0.311551	−	0.950230i	$-0.399152\pi$
$547$	4.36041e6	0.623101	0.311551	+	0.950230i	$0.399152\pi$
$548$	0	0
$549$	−1.01190e7	−1.43287
$550$	0	0
$551$	−1.93194e6	−0.271091
$552$	0	0
$553$	1.62119e7	2.25436
$554$	0	0
$555$	62205.2	0.00857224
$556$	0	0
$557$	−5.87571e6	−0.802459	−0.401229	−	0.915978i	$-0.631417\pi$
$557$	−5.87571e6	−0.802459	−0.401229	+	0.915978i	$0.631417\pi$
$558$	0	0
$559$	8.02831e6	1.08666
$560$	0	0
$561$	−330498.	−0.0443365
$562$	0	0
$563$	−2.96184e6	−0.393814	−0.196907	−	0.980422i	$-0.563090\pi$
$563$	−2.96184e6	−0.393814	−0.196907	+	0.980422i	$0.563090\pi$
$564$	0	0
$565$	1.05481e6	0.139013
$566$	0	0
$567$	−1.23442e7	−1.61252
$568$	0	0
$569$	3.46207e6	0.448286	0.224143	−	0.974556i	$-0.428042\pi$
$569$	3.46207e6	0.448286	0.224143	+	0.974556i	$0.428042\pi$
$570$	0	0
$571$	7.48480e6	0.960705	0.480353	−	0.877075i	$-0.340509\pi$
$571$	7.48480e6	0.960705	0.480353	+	0.877075i	$0.340509\pi$
$572$	0	0
$573$	327551.	0.0416766
$574$	0	0
$575$	5.53988e6	0.698764
$576$	0	0
$577$	2.13325e6	0.266749	0.133374	−	0.991066i	$-0.457419\pi$
$577$	2.13325e6	0.266749	0.133374	+	0.991066i	$0.457419\pi$
$578$	0	0
$579$	−1.03622e6	−0.128456
$580$	0	0
$581$	1.93764e7	2.38140
$582$	0	0
$583$	1.10723e7	1.34917
$584$	0	0
$585$	1.54306e6	0.186420
$586$	0	0
$587$	−2.28850e6	−0.274129	−0.137065	−	0.990562i	$-0.543767\pi$
$587$	−2.28850e6	−0.274129	−0.137065	+	0.990562i	$0.543767\pi$
$588$	0	0
$589$	−1.10318e6	−0.131027
$590$	0	0
$591$	−1.77849e6	−0.209451
$592$	0	0
$593$	−1.12281e6	−0.131121	−0.0655603	−	0.997849i	$-0.520883\pi$
$593$	−1.12281e6	−0.131121	−0.0655603	+	0.997849i	$0.520883\pi$
$594$	0	0
$595$	1.37503e6	0.159228
$596$	0	0
$597$	498878.	0.0572873
$598$	0	0
$599$	8.37554e6	0.953774	0.476887	−	0.878965i	$-0.341765\pi$
$599$	8.37554e6	0.953774	0.476887	+	0.878965i	$0.341765\pi$
$600$	0	0
$601$	−3.75444e6	−0.423994	−0.211997	−	0.977270i	$-0.567997\pi$
$601$	−3.75444e6	−0.423994	−0.211997	+	0.977270i	$0.567997\pi$
$602$	0	0
$603$	−3.82029e6	−0.427862
$604$	0	0
$605$	21622.9	0.00240174
$606$	0	0
$607$	−1.04209e7	−1.14798	−0.573988	−	0.818864i	$-0.694604\pi$
$607$	−1.04209e7	−1.14798	−0.573988	+	0.818864i	$0.694604\pi$
$608$	0	0
$609$	2.68827e6	0.293717
$610$	0	0
$611$	−2.10532e6	−0.228147
$612$	0	0
$613$	−1.30282e7	−1.40034	−0.700170	−	0.713976i	$-0.746892\pi$
$613$	−1.30282e7	−1.40034	−0.700170	+	0.713976i	$0.746892\pi$
$614$	0	0
$615$	−359134.	−0.0382885
$616$	0	0
$617$	−1.42219e7	−1.50399	−0.751997	−	0.659167i	$-0.770909\pi$
$617$	−1.42219e7	−1.50399	−0.751997	+	0.659167i	$0.770909\pi$
$618$	0	0
$619$	1.37047e7	1.43762	0.718808	−	0.695209i	$-0.244688\pi$
$619$	1.37047e7	1.43762	0.718808	+	0.695209i	$0.244688\pi$
$620$	0	0
$621$	−1.58895e6	−0.165341
$622$	0	0
$623$	−2.03103e7	−2.09650
$624$	0	0
$625$	8.11922e6	0.831409
$626$	0	0
$627$	190815.	0.0193840
$628$	0	0
$629$	1.25703e6	0.126683
$630$	0	0
$631$	1.49041e7	1.49016	0.745081	−	0.666974i	$-0.232411\pi$
$631$	1.49041e7	1.49016	0.745081	+	0.666974i	$0.232411\pi$
$632$	0	0
$633$	1.68100e6	0.166747
$634$	0	0
$635$	−1.96263e6	−0.193155
$636$	0	0
$637$	1.46016e7	1.42577
$638$	0	0
$639$	−1.15219e7	−1.11628
$640$	0	0
$641$	−1.23159e7	−1.18391	−0.591956	−	0.805970i	$-0.701644\pi$
$641$	−1.23159e7	−1.18391	−0.591956	+	0.805970i	$0.701644\pi$
$642$	0	0
$643$	6.60309e6	0.629824	0.314912	−	0.949121i	$-0.398025\pi$
$643$	6.60309e6	0.629824	0.314912	+	0.949121i	$0.398025\pi$
$644$	0	0
$645$	391044.	0.0370106
$646$	0	0
$647$	−1.22769e7	−1.15299	−0.576496	−	0.817100i	$-0.695581\pi$
$647$	−1.22769e7	−1.15299	−0.576496	+	0.817100i	$0.695581\pi$
$648$	0	0
$649$	−1.20838e7	−1.12614
$650$	0	0
$651$	1.53506e6	0.141963
$652$	0	0
$653$	7.96661e6	0.731123	0.365562	−	0.930787i	$-0.380877\pi$
$653$	7.96661e6	0.731123	0.365562	+	0.930787i	$0.380877\pi$
$654$	0	0
$655$	−3.08494e6	−0.280959
$656$	0	0
$657$	−1.92751e6	−0.174214
$658$	0	0
$659$	−4.43856e6	−0.398134	−0.199067	−	0.979986i	$-0.563791\pi$
$659$	−4.43856e6	−0.398134	−0.199067	+	0.979986i	$0.563791\pi$
$660$	0	0
$661$	3.18500e6	0.283535	0.141767	−	0.989900i	$-0.454722\pi$
$661$	3.18500e6	0.283535	0.141767	+	0.989900i	$0.454722\pi$
$662$	0	0
$663$	−397823.	−0.0351484
$664$	0	0
$665$	−793884.	−0.0696150
$666$	0	0
$667$	−1.33012e7	−1.15765
$668$	0	0
$669$	35955.6	0.00310600
$670$	0	0
$671$	1.68394e7	1.44385
$672$	0	0
$673$	−5.80254e6	−0.493834	−0.246917	−	0.969037i	$-0.579417\pi$
$673$	−5.80254e6	−0.493834	−0.246917	+	0.969037i	$0.579417\pi$
$674$	0	0
$675$	−2.48922e6	−0.210283
$676$	0	0
$677$	7.59860e6	0.637180	0.318590	−	0.947893i	$-0.396791\pi$
$677$	7.59860e6	0.637180	0.318590	+	0.947893i	$0.396791\pi$
$678$	0	0
$679$	3.44988e7	2.87163
$680$	0	0
$681$	2.32434e6	0.192057
$682$	0	0
$683$	1.12433e7	0.922234	0.461117	−	0.887339i	$-0.347449\pi$
$683$	1.12433e7	0.922234	0.461117	+	0.887339i	$0.347449\pi$
$684$	0	0
$685$	−1.73494e6	−0.141273
$686$	0	0
$687$	199500.	0.0161269
$688$	0	0
$689$	1.33278e7	1.06958
$690$	0	0
$691$	1.55666e7	1.24022	0.620110	−	0.784515i	$-0.287088\pi$
$691$	1.55666e7	1.24022	0.620110	+	0.784515i	$0.287088\pi$
$692$	0	0
$693$	2.08115e7	1.64615
$694$	0	0
$695$	5.55729e6	0.436417
$696$	0	0
$697$	−7.25731e6	−0.565840
$698$	0	0
$699$	−513402.	−0.0397434
$700$	0	0
$701$	1.39060e7	1.06882	0.534412	−	0.845224i	$-0.320533\pi$
$701$	1.39060e7	1.06882	0.534412	+	0.845224i	$0.320533\pi$
$702$	0	0
$703$	−725755.	−0.0553862
$704$	0	0
$705$	−102546.	−0.00777045
$706$	0	0
$707$	3.36084e7	2.52871
$708$	0	0
$709$	−8.54902e6	−0.638706	−0.319353	−	0.947636i	$-0.603465\pi$
$709$	−8.54902e6	−0.638706	−0.319353	+	0.947636i	$0.603465\pi$
$710$	0	0
$711$	1.79071e7	1.32847
$712$	0	0
$713$	−7.59528e6	−0.559526
$714$	0	0
$715$	−2.56787e6	−0.187849
$716$	0	0
$717$	191650.	0.0139223
$718$	0	0
$719$	−2.40812e7	−1.73722	−0.868611	−	0.495495i	$-0.834987\pi$
$719$	−2.40812e7	−1.73722	−0.868611	+	0.495495i	$0.834987\pi$
$720$	0	0
$721$	1.22965e7	0.880932
$722$	0	0
$723$	1.28496e6	0.0914206
$724$	0	0
$725$	−2.08375e7	−1.47231
$726$	0	0
$727$	−5.30046e6	−0.371944	−0.185972	−	0.982555i	$-0.559543\pi$
$727$	−5.30046e6	−0.371944	−0.185972	+	0.982555i	$0.559543\pi$
$728$	0	0
$729$	−1.32757e7	−0.925207
$730$	0	0
$731$	7.90214e6	0.546955
$732$	0	0
$733$	1.62342e7	1.11602	0.558009	−	0.829835i	$-0.311566\pi$
$733$	1.62342e7	1.11602	0.558009	+	0.829835i	$0.311566\pi$
$734$	0	0
$735$	711214.	0.0485604
$736$	0	0
$737$	6.35752e6	0.431141
$738$	0	0
$739$	−1.56254e6	−0.105249	−0.0526247	−	0.998614i	$-0.516759\pi$
$739$	−1.56254e6	−0.105249	−0.0526247	+	0.998614i	$0.516759\pi$
$740$	0	0
$741$	229685.	0.0153670
$742$	0	0
$743$	−2.73631e6	−0.181842	−0.0909209	−	0.995858i	$-0.528981\pi$
$743$	−2.73631e6	−0.181842	−0.0909209	+	0.995858i	$0.528981\pi$
$744$	0	0
$745$	526594.	0.0347605
$746$	0	0
$747$	2.14024e7	1.40334
$748$	0	0
$749$	−3.22847e7	−2.10277
$750$	0	0
$751$	−2.79880e7	−1.81080	−0.905402	−	0.424555i	$-0.860430\pi$
$751$	−2.79880e7	−1.81080	−0.905402	+	0.424555i	$0.860430\pi$
$752$	0	0
$753$	1.70603e6	0.109647
$754$	0	0
$755$	1.08861e6	0.0695030
$756$	0	0
$757$	2.15035e7	1.36386	0.681929	−	0.731419i	$-0.261141\pi$
$757$	2.15035e7	1.36386	0.681929	+	0.731419i	$0.261141\pi$
$758$	0	0
$759$	1.31374e6	0.0827760
$760$	0	0
$761$	1.28357e7	0.803448	0.401724	−	0.915761i	$-0.368411\pi$
$761$	1.28357e7	0.803448	0.401724	+	0.915761i	$0.368411\pi$
$762$	0	0
$763$	1.09584e7	0.681456
$764$	0	0
$765$	1.51881e6	0.0938316
$766$	0	0
$767$	−1.45453e7	−0.892760
$768$	0	0
$769$	479067.	0.0292133	0.0146066	−	0.999893i	$-0.495350\pi$
$769$	479067.	0.0292133	0.0146066	+	0.999893i	$0.495350\pi$
$770$	0	0
$771$	3.07424e6	0.186253
$772$	0	0
$773$	1.02802e7	0.618801	0.309400	−	0.950932i	$-0.399872\pi$
$773$	1.02802e7	0.618801	0.309400	+	0.950932i	$0.399872\pi$
$774$	0	0
$775$	−1.18987e7	−0.711614
$776$	0	0
$777$	1.00988e6	0.0600089
$778$	0	0
$779$	4.19005e6	0.247386
$780$	0	0
$781$	1.91742e7	1.12483
$782$	0	0
$783$	5.97660e6	0.348377
$784$	0	0
$785$	3.83649e6	0.222208
$786$	0	0
$787$	−1.30703e6	−0.0752228	−0.0376114	−	0.999292i	$-0.511975\pi$
$787$	−1.30703e6	−0.0752228	−0.0376114	+	0.999292i	$0.511975\pi$
$788$	0	0
$789$	3.02425e6	0.172952
$790$	0	0
$791$	1.71245e7	0.973142
$792$	0	0
$793$	2.02697e7	1.14463
$794$	0	0
$795$	649174.	0.0364287
$796$	0	0
$797$	−6.17423e6	−0.344300	−0.172150	−	0.985071i	$-0.555071\pi$
$797$	−6.17423e6	−0.344300	−0.172150	+	0.985071i	$0.555071\pi$
$798$	0	0
$799$	−2.07223e6	−0.114834
$800$	0	0
$801$	−2.24339e7	−1.23545
$802$	0	0
$803$	3.20766e6	0.175549
$804$	0	0
$805$	−5.46579e6	−0.297278
$806$	0	0
$807$	−492129.	−0.0266008
$808$	0	0
$809$	−1.80829e7	−0.971396	−0.485698	−	0.874127i	$-0.661435\pi$
$809$	−1.80829e7	−0.971396	−0.485698	+	0.874127i	$0.661435\pi$
$810$	0	0
$811$	3.24108e7	1.73036	0.865182	−	0.501459i	$-0.167203\pi$
$811$	3.24108e7	1.73036	0.865182	+	0.501459i	$0.167203\pi$
$812$	0	0
$813$	−2.82626e6	−0.149964
$814$	0	0
$815$	7.25112e6	0.382394
$816$	0	0
$817$	−4.56235e6	−0.239130
$818$	0	0
$819$	2.50509e7	1.30501
$820$	0	0
$821$	−5.84855e6	−0.302824	−0.151412	−	0.988471i	$-0.548382\pi$
$821$	−5.84855e6	−0.302824	−0.151412	+	0.988471i	$0.548382\pi$
$822$	0	0
$823$	−1.35656e7	−0.698133	−0.349067	−	0.937098i	$-0.613501\pi$
$823$	−1.35656e7	−0.698133	−0.349067	+	0.937098i	$0.613501\pi$
$824$	0	0
$825$	2.05809e6	0.105276
$826$	0	0
$827$	2.91211e6	0.148062	0.0740311	−	0.997256i	$-0.476414\pi$
$827$	2.91211e6	0.148062	0.0740311	+	0.997256i	$0.476414\pi$
$828$	0	0
$829$	−3.41915e7	−1.72795	−0.863975	−	0.503534i	$-0.832033\pi$
$829$	−3.41915e7	−1.72795	−0.863975	+	0.503534i	$0.832033\pi$
$830$	0	0
$831$	1.03486e6	0.0519850
$832$	0	0
$833$	1.43721e7	0.717641
$834$	0	0
$835$	7.01042e6	0.347959
$836$	0	0
$837$	3.41277e6	0.168381
$838$	0	0
$839$	9.89108e6	0.485108	0.242554	−	0.970138i	$-0.422015\pi$
$839$	9.89108e6	0.485108	0.242554	+	0.970138i	$0.422015\pi$
$840$	0	0
$841$	2.95194e7	1.43919
$842$	0	0
$843$	−3.27921e6	−0.158928
$844$	0	0
$845$	1.87709e6	0.0904362
$846$	0	0
$847$	351039.	0.0168131
$848$	0	0
$849$	2.75598e6	0.131222
$850$	0	0
$851$	−4.99673e6	−0.236517
$852$	0	0
$853$	−1.56361e7	−0.735793	−0.367897	−	0.929867i	$-0.619922\pi$
$853$	−1.56361e7	−0.735793	−0.367897	+	0.929867i	$0.619922\pi$
$854$	0	0
$855$	−876893.	−0.0410234
$856$	0	0
$857$	1.52495e7	0.709257	0.354628	−	0.935007i	$-0.384607\pi$
$857$	1.52495e7	0.709257	0.354628	+	0.935007i	$0.384607\pi$
$858$	0	0
$859$	1.70283e7	0.787390	0.393695	−	0.919241i	$-0.371197\pi$
$859$	1.70283e7	0.787390	0.393695	+	0.919241i	$0.371197\pi$
$860$	0	0
$861$	−5.83040e6	−0.268034
$862$	0	0
$863$	−3.08994e7	−1.41229	−0.706145	−	0.708067i	$-0.749567\pi$
$863$	−3.08994e7	−1.41229	−0.706145	+	0.708067i	$0.749567\pi$
$864$	0	0
$865$	−8.46626e6	−0.384726
$866$	0	0
$867$	2.09264e6	0.0945469
$868$	0	0
$869$	−2.98000e7	−1.33865
$870$	0	0
$871$	7.65259e6	0.341793
$872$	0	0
$873$	3.81060e7	1.69222
$874$	0	0
$875$	−1.76457e7	−0.779144
$876$	0	0
$877$	−3.29965e7	−1.44867	−0.724334	−	0.689449i	$-0.757853\pi$
$877$	−3.29965e7	−1.44867	−0.724334	+	0.689449i	$0.757853\pi$
$878$	0	0
$879$	−1.20842e6	−0.0527529
$880$	0	0
$881$	−3.57739e7	−1.55284	−0.776419	−	0.630218i	$-0.782966\pi$
$881$	−3.57739e7	−1.55284	−0.776419	+	0.630218i	$0.782966\pi$
$882$	0	0
$883$	3.53653e7	1.52643	0.763213	−	0.646147i	$-0.223621\pi$
$883$	3.53653e7	1.52643	0.763213	+	0.646147i	$0.223621\pi$
$884$	0	0
$885$	−708474.	−0.0304065
$886$	0	0
$887$	3.34384e7	1.42704	0.713521	−	0.700633i	$-0.247099\pi$
$887$	3.34384e7	1.42704	0.713521	+	0.700633i	$0.247099\pi$
$888$	0	0
$889$	−3.18626e7	−1.35216
$890$	0	0
$891$	2.26905e7	0.957526
$892$	0	0
$893$	1.19642e6	0.0502057
$894$	0	0
$895$	−3.72728e6	−0.155537
$896$	0	0
$897$	1.58135e6	0.0656218
$898$	0	0
$899$	2.85686e7	1.17893
$900$	0	0
$901$	1.31184e7	0.538355
$902$	0	0
$903$	6.34844e6	0.259088
$904$	0	0
$905$	−4.91931e6	−0.199656
$906$	0	0
$907$	−2.16989e7	−0.875829	−0.437914	−	0.899017i	$-0.644283\pi$
$907$	−2.16989e7	−0.875829	−0.437914	+	0.899017i	$0.644283\pi$
$908$	0	0
$909$	3.71226e7	1.49014
$910$	0	0
$911$	−479747.	−0.0191521	−0.00957604	−	0.999954i	$-0.503048\pi$
$911$	−479747.	−0.0191521	−0.00957604	+	0.999954i	$0.503048\pi$
$912$	0	0
$913$	−3.56167e7	−1.41409
$914$	0	0
$915$	987300.	0.0389849
$916$	0	0
$917$	−5.00827e7	−1.96682
$918$	0	0
$919$	9.26403e6	0.361835	0.180918	−	0.983498i	$-0.442093\pi$
$919$	9.26403e6	0.361835	0.180918	+	0.983498i	$0.442093\pi$
$920$	0	0
$921$	981845.	0.0381412
$922$	0	0
$923$	2.30801e7	0.891729
$924$	0	0
$925$	−7.82782e6	−0.300806
$926$	0	0
$927$	1.35822e7	0.519124
$928$	0	0
$929$	3.65567e7	1.38972	0.694861	−	0.719144i	$-0.255466\pi$
$929$	3.65567e7	1.38972	0.694861	+	0.719144i	$0.255466\pi$
$930$	0	0
$931$	−8.29781e6	−0.313754
$932$	0	0
$933$	3.15359e6	0.118605
$934$	0	0
$935$	−2.52752e6	−0.0945507
$936$	0	0
$937$	−8.48280e6	−0.315639	−0.157819	−	0.987468i	$-0.550446\pi$
$937$	−8.48280e6	−0.315639	−0.157819	+	0.987468i	$0.550446\pi$
$938$	0	0
$939$	606373.	0.0224427
$940$	0	0
$941$	−1.93368e7	−0.711885	−0.355943	−	0.934508i	$-0.615840\pi$
$941$	−1.93368e7	−0.711885	−0.355943	+	0.934508i	$0.615840\pi$
$942$	0	0
$943$	2.88480e7	1.05642
$944$	0	0
$945$	2.45593e6	0.0894617
$946$	0	0
$947$	−3.71714e7	−1.34690	−0.673448	−	0.739235i	$-0.735187\pi$
$947$	−3.71714e7	−1.34690	−0.673448	+	0.739235i	$0.735187\pi$
$948$	0	0
$949$	3.86108e6	0.139169
$950$	0	0
$951$	−3.58358e6	−0.128489
$952$	0	0
$953$	2.85225e6	0.101732	0.0508658	−	0.998705i	$-0.483802\pi$
$953$	2.85225e6	0.101732	0.0508658	+	0.998705i	$0.483802\pi$
$954$	0	0
$955$	2.50497e6	0.0888781
$956$	0	0
$957$	−4.94144e6	−0.174411
$958$	0	0
$959$	−2.81661e7	−0.988964
$960$	0	0
$961$	−1.23158e7	−0.430185
$962$	0	0
$963$	−3.56605e7	−1.23914
$964$	0	0
$965$	−7.92456e6	−0.273941
$966$	0	0
$967$	2.35407e6	0.0809568	0.0404784	−	0.999180i	$-0.487112\pi$
$967$	2.35407e6	0.0809568	0.0404784	+	0.999180i	$0.487112\pi$
$968$	0	0
$969$	226076.	0.00773472
$970$	0	0
$971$	4.43206e6	0.150854	0.0754271	−	0.997151i	$-0.475968\pi$
$971$	4.43206e6	0.150854	0.0754271	+	0.997151i	$0.475968\pi$
$972$	0	0
$973$	9.02205e7	3.05508
$974$	0	0
$975$	2.47733e6	0.0834589
$976$	0	0
$977$	−574910.	−0.0192692	−0.00963460	−	0.999954i	$-0.503067\pi$
$977$	−574910.	−0.0192692	−0.00963460	+	0.999954i	$0.503067\pi$
$978$	0	0
$979$	3.73333e7	1.24492
$980$	0	0
$981$	1.21043e7	0.401574
$982$	0	0
$983$	2.93611e7	0.969145	0.484572	−	0.874751i	$-0.338975\pi$
$983$	2.93611e7	0.969145	0.484572	+	0.874751i	$0.338975\pi$
$984$	0	0
$985$	−1.36012e7	−0.446669
$986$	0	0
$987$	−1.66479e6	−0.0543961
$988$	0	0
$989$	−3.14112e7	−1.02116
$990$	0	0
$991$	−5.12370e7	−1.65730	−0.828648	−	0.559770i	$-0.810889\pi$
$991$	−5.12370e7	−1.65730	−0.828648	+	0.559770i	$0.810889\pi$
$992$	0	0
$993$	−2.18092e6	−0.0701885
$994$	0	0
$995$	3.81521e6	0.122169
$996$	0	0
$997$	2.61915e7	0.834493	0.417247	−	0.908793i	$-0.362995\pi$
$997$	2.61915e7	0.834493	0.417247	+	0.908793i	$0.362995\pi$
$998$	0	0
$999$	2.24517e6	0.0711764

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.6.a.d.1.3		8
4.3	odd	2		1024.6.a.e.1.5		8
8.3	odd	2	inner	1024.6.a.d.1.4		8
8.5	even	2		1024.6.a.e.1.6		8
32.3	odd	8		256.6.e.b.65.5	yes	16
32.5	even	8		256.6.e.a.193.5	yes	16
32.11	odd	8		256.6.e.b.193.5	yes	16
32.13	even	8		256.6.e.a.65.5	yes	16
32.19	odd	8		256.6.e.a.65.4	✓	16
32.21	even	8		256.6.e.b.193.4	yes	16
32.27	odd	8		256.6.e.a.193.4	yes	16
32.29	even	8		256.6.e.b.65.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.6.e.a.65.4	✓	16	32.19	odd	8
256.6.e.a.65.5	yes	16	32.13	even	8
256.6.e.a.193.4	yes	16	32.27	odd	8
256.6.e.a.193.5	yes	16	32.5	even	8
256.6.e.b.65.4	yes	16	32.29	even	8
256.6.e.b.65.5	yes	16	32.3	odd	8
256.6.e.b.193.4	yes	16	32.21	even	8
256.6.e.b.193.5	yes	16	32.11	odd	8
1024.6.a.d.1.3		8	1.1	even	1	trivial
1024.6.a.d.1.4		8	8.3	odd	2	inner
1024.6.a.e.1.5		8	4.3	odd	2
1024.6.a.e.1.6		8	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74962 q^{3} -13.3804 q^{5} -217.226 q^{7} -239.939 q^{9} +O(q^{10})\)	\(q-1.74962 q^{3} -13.3804 q^{5} -217.226 q^{7} -239.939 q^{9} +399.293 q^{11} +480.632 q^{13} +23.4106 q^{15} +473.078 q^{17} -273.135 q^{19} +380.063 q^{21} -1880.50 q^{23} -2945.96 q^{25} +844.960 q^{27} +7073.23 q^{29} +4038.97 q^{31} -698.612 q^{33} +2906.56 q^{35} +2657.13 q^{37} -840.924 q^{39} -15340.6 q^{41} +16703.7 q^{43} +3210.48 q^{45} -4380.31 q^{47} +30379.9 q^{49} -827.708 q^{51} +27729.9 q^{53} -5342.70 q^{55} +477.882 q^{57} -30262.9 q^{59} +42173.1 q^{61} +52120.8 q^{63} -6431.05 q^{65} +15921.9 q^{67} +3290.16 q^{69} +48020.3 q^{71} +8033.35 q^{73} +5154.33 q^{75} -86736.7 q^{77} -74631.9 q^{79} +56826.8 q^{81} -89199.5 q^{83} -6329.98 q^{85} -12375.5 q^{87} +93498.5 q^{89} -104405. q^{91} -7066.68 q^{93} +3654.65 q^{95} -158815. q^{97} -95805.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1096 q^{9}+O(q^{10}) \) 8 * q + 1096 * q^9	\( 8 q + 1096 q^{9} - 128 q^{11} + 192 q^{17} - 4672 q^{19} + 10248 q^{25} - 17856 q^{27} - 39104 q^{33} - 9024 q^{35} - 13952 q^{41} + 59904 q^{43} + 98056 q^{49} - 111360 q^{51} + 103616 q^{57} - 216384 q^{59} - 315312 q^{65} + 111616 q^{67} - 132976 q^{73} + 558400 q^{75} + 506440 q^{81} - 150080 q^{83} + 168592 q^{89} - 405312 q^{91} - 416576 q^{97} + 175936 q^{99}+O(q^{100}) \) 8 * q + 1096 * q^9 - 128 * q^11 + 192 * q^17 - 4672 * q^19 + 10248 * q^25 - 17856 * q^27 - 39104 * q^33 - 9024 * q^35 - 13952 * q^41 + 59904 * q^43 + 98056 * q^49 - 111360 * q^51 + 103616 * q^57 - 216384 * q^59 - 315312 * q^65 + 111616 * q^67 - 132976 * q^73 + 558400 * q^75 + 506440 * q^81 - 150080 * q^83 + 168592 * q^89 - 405312 * q^91 - 416576 * q^97 + 175936 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more