LMFDB - Embedded newform 140.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,6,Mod(1,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("140.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	140.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:	$22.4537347738$
Analytic rank:	$0$
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} - 499x - 210 $ x^3 - 499*x - 210
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.420991$ of defining polynomial
Character	$\chi$	$=$	140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.57901 q^{3} +25.0000 q^{5} +49.0000 q^{7} -240.507 q^{9} +322.771 q^{11} +657.331 q^{13} +39.4752 q^{15} +665.753 q^{17} -2215.14 q^{19} +77.3714 q^{21} +1586.87 q^{23} +625.000 q^{25} -763.461 q^{27} +6386.70 q^{29} +7712.24 q^{31} +509.658 q^{33} +1225.00 q^{35} +3433.65 q^{37} +1037.93 q^{39} -6537.24 q^{41} +16611.0 q^{43} -6012.67 q^{45} +12516.9 q^{47} +2401.00 q^{49} +1051.23 q^{51} +4240.34 q^{53} +8069.27 q^{55} -3497.73 q^{57} +26156.3 q^{59} +32683.8 q^{61} -11784.8 q^{63} +16433.3 q^{65} -6153.18 q^{67} +2505.68 q^{69} -33827.2 q^{71} -82194.4 q^{73} +986.880 q^{75} +15815.8 q^{77} -35205.3 q^{79} +57237.6 q^{81} +17529.6 q^{83} +16643.8 q^{85} +10084.7 q^{87} -72867.8 q^{89} +32209.2 q^{91} +12177.7 q^{93} -55378.5 q^{95} -166047. q^{97} -77628.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33}+ \cdots - 114368 q^{99}+O(q^{100}) $ 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9 - 14 * q^11 - 4 * q^13 + 150 * q^15 + 44 * q^17 + 2328 * q^19 + 294 * q^21 + 3676 * q^23 + 1875 * q^25 + 3726 * q^27 + 4092 * q^29 + 5888 * q^31 + 11318 * q^33 + 3675 * q^35 + 11378 * q^37 + 13702 * q^39 + 11450 * q^41 + 18544 * q^43 + 7025 * q^45 + 21754 * q^47 + 7203 * q^49 + 31762 * q^51 + 7494 * q^53 - 350 * q^55 + 7332 * q^57 + 12388 * q^59 + 27182 * q^61 + 13769 * q^63 - 100 * q^65 - 7676 * q^67 - 62140 * q^69 + 81992 * q^71 + 230 * q^73 + 3750 * q^75 - 686 * q^77 - 15926 * q^79 - 32101 * q^81 - 86100 * q^83 + 1100 * q^85 - 85098 * q^87 - 95710 * q^89 - 196 * q^91 - 229472 * q^93 + 58200 * q^95 - 176188 * q^97 - 114368 * q^99

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
$3$	1.57901	0.101293	0.0506467	−	0.998717i	$-0.483872\pi$
$3$	1.57901	0.101293	0.0506467	+	0.998717i	$0.483872\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−240.507	−0.989740
$10$	0	0
$11$	322.771	0.804290	0.402145	−	0.915576i	$-0.368265\pi$
$11$	322.771	0.804290	0.402145	+	0.915576i	$0.368265\pi$
$12$	0	0
$13$	657.331	1.07876	0.539381	−	0.842062i	$-0.318658\pi$
$13$	657.331	1.07876	0.539381	+	0.842062i	$0.318658\pi$
$14$	0	0
$15$	39.4752	0.0452998
$16$	0	0
$17$	665.753	0.558715	0.279358	−	0.960187i	$-0.409878\pi$
$17$	665.753	0.558715	0.279358	+	0.960187i	$0.409878\pi$
$18$	0	0
$19$	−2215.14	−1.40772	−0.703862	−	0.710337i	$-0.748543\pi$
$19$	−2215.14	−1.40772	−0.703862	+	0.710337i	$0.748543\pi$
$20$	0	0
$21$	77.3714	0.0382853
$22$	0	0
$23$	1586.87	0.625492	0.312746	−	0.949837i	$-0.398751\pi$
$23$	1586.87	0.625492	0.312746	+	0.949837i	$0.398751\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−763.461	−0.201548
$28$	0	0
$29$	6386.70	1.41020	0.705102	−	0.709106i	$-0.250901\pi$
$29$	6386.70	1.41020	0.705102	+	0.709106i	$0.250901\pi$
$30$	0	0
$31$	7712.24	1.44137	0.720686	−	0.693262i	$-0.243827\pi$
$31$	7712.24	1.44137	0.720686	+	0.693262i	$0.243827\pi$
$32$	0	0
$33$	509.658	0.0814693
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	3433.65	0.412337	0.206169	−	0.978517i	$-0.433900\pi$
$37$	3433.65	0.412337	0.206169	+	0.978517i	$0.433900\pi$
$38$	0	0
$39$	1037.93	0.109272
$40$	0	0
$41$	−6537.24	−0.607344	−0.303672	−	0.952777i	$-0.598213\pi$
$41$	−6537.24	−0.607344	−0.303672	+	0.952777i	$0.598213\pi$
$42$	0	0
$43$	16611.0	1.37001	0.685007	−	0.728536i	$-0.259799\pi$
$43$	16611.0	1.37001	0.685007	+	0.728536i	$0.259799\pi$
$44$	0	0
$45$	−6012.67	−0.442625
$46$	0	0
$47$	12516.9	0.826515	0.413257	−	0.910614i	$-0.364391\pi$
$47$	12516.9	0.826515	0.413257	+	0.910614i	$0.364391\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	1051.23	0.0565942
$52$	0	0
$53$	4240.34	0.207353	0.103677	−	0.994611i	$-0.466939\pi$
$53$	4240.34	0.207353	0.103677	+	0.994611i	$0.466939\pi$
$54$	0	0
$55$	8069.27	0.359689
$56$	0	0
$57$	−3497.73	−0.142593
$58$	0	0
$59$	26156.3	0.978243	0.489122	−	0.872216i	$-0.337317\pi$
$59$	26156.3	0.978243	0.489122	+	0.872216i	$0.337317\pi$
$60$	0	0
$61$	32683.8	1.12463	0.562313	−	0.826924i	$-0.309911\pi$
$61$	32683.8	1.12463	0.562313	+	0.826924i	$0.309911\pi$
$62$	0	0
$63$	−11784.8	−0.374086
$64$	0	0
$65$	16433.3	0.482437
$66$	0	0
$67$	−6153.18	−0.167461	−0.0837303	−	0.996488i	$-0.526683\pi$
$67$	−6153.18	−0.167461	−0.0837303	+	0.996488i	$0.526683\pi$
$68$	0	0
$69$	2505.68	0.0633583
$70$	0	0
$71$	−33827.2	−0.796379	−0.398190	−	0.917303i	$-0.630361\pi$
$71$	−33827.2	−0.796379	−0.398190	+	0.917303i	$0.630361\pi$
$72$	0	0
$73$	−82194.4	−1.80524	−0.902621	−	0.430437i	$-0.858360\pi$
$73$	−82194.4	−1.80524	−0.902621	+	0.430437i	$0.858360\pi$
$74$	0	0
$75$	986.880	0.0202587
$76$	0	0
$77$	15815.8	0.303993
$78$	0	0
$79$	−35205.3	−0.634659	−0.317329	−	0.948315i	$-0.602786\pi$
$79$	−35205.3	−0.634659	−0.317329	+	0.948315i	$0.602786\pi$
$80$	0	0
$81$	57237.6	0.969324
$82$	0	0
$83$	17529.6	0.279304	0.139652	−	0.990201i	$-0.455402\pi$
$83$	17529.6	0.279304	0.139652	+	0.990201i	$0.455402\pi$
$84$	0	0
$85$	16643.8	0.249865
$86$	0	0
$87$	10084.7	0.142844
$88$	0	0
$89$	−72867.8	−0.975126	−0.487563	−	0.873088i	$-0.662114\pi$
$89$	−72867.8	−0.975126	−0.487563	+	0.873088i	$0.662114\pi$
$90$	0	0
$91$	32209.2	0.407734
$92$	0	0
$93$	12177.7	0.146002
$94$	0	0
$95$	−55378.5	−0.629553
$96$	0	0
$97$	−166047.	−1.79185	−0.895927	−	0.444202i	$-0.853487\pi$
$97$	−166047.	−1.79185	−0.895927	+	0.444202i	$0.853487\pi$
$98$	0	0
$99$	−77628.6	−0.796038
$100$	0	0
$101$	19577.2	0.190963	0.0954813	−	0.995431i	$-0.469561\pi$
$101$	19577.2	0.190963	0.0954813	+	0.995431i	$0.469561\pi$
$102$	0	0
$103$	83864.5	0.778906	0.389453	−	0.921046i	$-0.372664\pi$
$103$	83864.5	0.778906	0.389453	+	0.921046i	$0.372664\pi$
$104$	0	0
$105$	1934.29	0.0171217
$106$	0	0
$107$	−105704.	−0.892547	−0.446273	−	0.894897i	$-0.647249\pi$
$107$	−105704.	−0.892547	−0.446273	+	0.894897i	$0.647249\pi$
$108$	0	0
$109$	−114386.	−0.922160	−0.461080	−	0.887359i	$-0.652538\pi$
$109$	−114386.	−0.922160	−0.461080	+	0.887359i	$0.652538\pi$
$110$	0	0
$111$	5421.77	0.0417670
$112$	0	0
$113$	−95123.1	−0.700793	−0.350396	−	0.936601i	$-0.613953\pi$
$113$	−95123.1	−0.700793	−0.350396	+	0.936601i	$0.613953\pi$
$114$	0	0
$115$	39671.8	0.279729
$116$	0	0
$117$	−158092.	−1.06769
$118$	0	0
$119$	32621.9	0.211175
$120$	0	0
$121$	−56870.0	−0.353118
$122$	0	0
$123$	−10322.4	−0.0615199
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	97854.5	0.538358	0.269179	−	0.963090i	$-0.413248\pi$
$127$	97854.5	0.538358	0.269179	+	0.963090i	$0.413248\pi$
$128$	0	0
$129$	26228.9	0.138773
$130$	0	0
$131$	−31926.2	−0.162543	−0.0812715	−	0.996692i	$-0.525898\pi$
$131$	−31926.2	−0.162543	−0.0812715	+	0.996692i	$0.525898\pi$
$132$	0	0
$133$	−108542.	−0.532070
$134$	0	0
$135$	−19086.5	−0.0901348
$136$	0	0
$137$	22595.8	0.102855	0.0514276	−	0.998677i	$-0.483623\pi$
$137$	22595.8	0.102855	0.0514276	+	0.998677i	$0.483623\pi$
$138$	0	0
$139$	287232.	1.26094	0.630472	−	0.776212i	$-0.282861\pi$
$139$	287232.	1.26094	0.630472	+	0.776212i	$0.282861\pi$
$140$	0	0
$141$	19764.2	0.0837205
$142$	0	0
$143$	212167.	0.867637
$144$	0	0
$145$	159668.	0.630662
$146$	0	0
$147$	3791.20	0.0144705
$148$	0	0
$149$	−437546.	−1.61458	−0.807288	−	0.590158i	$-0.799065\pi$
$149$	−437546.	−1.61458	−0.807288	+	0.590158i	$0.799065\pi$
$150$	0	0
$151$	505868.	1.80549	0.902744	−	0.430179i	$-0.141550\pi$
$151$	505868.	1.80549	0.902744	+	0.430179i	$0.141550\pi$
$152$	0	0
$153$	−160118.	−0.552983
$154$	0	0
$155$	192806.	0.644601
$156$	0	0
$157$	−214333.	−0.693967	−0.346984	−	0.937871i	$-0.612794\pi$
$157$	−214333.	−0.693967	−0.346984	+	0.937871i	$0.612794\pi$
$158$	0	0
$159$	6695.54	0.0210035
$160$	0	0
$161$	77756.7	0.236414
$162$	0	0
$163$	−587081.	−1.73073	−0.865364	−	0.501143i	$-0.832913\pi$
$163$	−587081.	−1.73073	−0.865364	+	0.501143i	$0.832913\pi$
$164$	0	0
$165$	12741.5	0.0364342
$166$	0	0
$167$	−387234.	−1.07444	−0.537220	−	0.843442i	$-0.680525\pi$
$167$	−387234.	−1.07444	−0.537220	+	0.843442i	$0.680525\pi$
$168$	0	0
$169$	60790.6	0.163727
$170$	0	0
$171$	532757.	1.39328
$172$	0	0
$173$	610438.	1.55069	0.775347	−	0.631535i	$-0.217575\pi$
$173$	610438.	1.55069	0.775347	+	0.631535i	$0.217575\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	41301.1	0.0990897
$178$	0	0
$179$	169505.	0.395412	0.197706	−	0.980261i	$-0.436651\pi$
$179$	169505.	0.395412	0.197706	+	0.980261i	$0.436651\pi$
$180$	0	0
$181$	305167.	0.692374	0.346187	−	0.938166i	$-0.387476\pi$
$181$	305167.	0.692374	0.346187	+	0.938166i	$0.387476\pi$
$182$	0	0
$183$	51608.1	0.113917
$184$	0	0
$185$	85841.4	0.184403
$186$	0	0
$187$	214886.	0.449369
$188$	0	0
$189$	−37409.6	−0.0761778
$190$	0	0
$191$	941471.	1.86734	0.933670	−	0.358134i	$-0.116587\pi$
$191$	941471.	1.86734	0.933670	+	0.358134i	$0.116587\pi$
$192$	0	0
$193$	385057.	0.744101	0.372051	−	0.928212i	$-0.378655\pi$
$193$	385057.	0.744101	0.372051	+	0.928212i	$0.378655\pi$
$194$	0	0
$195$	25948.3	0.0488677
$196$	0	0
$197$	−1.03718e6	−1.90410	−0.952048	−	0.305948i	$-0.901026\pi$
$197$	−1.03718e6	−1.90410	−0.952048	+	0.305948i	$0.901026\pi$
$198$	0	0
$199$	410855.	0.735455	0.367727	−	0.929934i	$-0.380136\pi$
$199$	410855.	0.735455	0.367727	+	0.929934i	$0.380136\pi$
$200$	0	0
$201$	−9715.93	−0.0169627
$202$	0	0
$203$	312949.	0.533007
$204$	0	0
$205$	−163431.	−0.271612
$206$	0	0
$207$	−381653.	−0.619075
$208$	0	0
$209$	−714983.	−1.13222
$210$	0	0
$211$	442714.	0.684568	0.342284	−	0.939597i	$-0.388799\pi$
$211$	442714.	0.684568	0.342284	+	0.939597i	$0.388799\pi$
$212$	0	0
$213$	−53413.4	−0.0806680
$214$	0	0
$215$	415275.	0.612689
$216$	0	0
$217$	377900.	0.544787
$218$	0	0
$219$	−129786.	−0.182859
$220$	0	0
$221$	437620.	0.602721
$222$	0	0
$223$	93933.5	0.126491	0.0632453	−	0.997998i	$-0.479855\pi$
$223$	93933.5	0.126491	0.0632453	+	0.997998i	$0.479855\pi$
$224$	0	0
$225$	−150317.	−0.197948
$226$	0	0
$227$	234538.	0.302099	0.151049	−	0.988526i	$-0.451735\pi$
$227$	234538.	0.302099	0.151049	+	0.988526i	$0.451735\pi$
$228$	0	0
$229$	239032.	0.301208	0.150604	−	0.988594i	$-0.451878\pi$
$229$	239032.	0.301208	0.150604	+	0.988594i	$0.451878\pi$
$230$	0	0
$231$	24973.2	0.0307925
$232$	0	0
$233$	−1.47330e6	−1.77788	−0.888939	−	0.458025i	$-0.848557\pi$
$233$	−1.47330e6	−1.77788	−0.888939	+	0.458025i	$0.848557\pi$
$234$	0	0
$235$	312921.	0.369629
$236$	0	0
$237$	−55589.5	−0.0642868
$238$	0	0
$239$	−1.31478e6	−1.48887	−0.744436	−	0.667694i	$-0.767282\pi$
$239$	−1.31478e6	−1.48887	−0.744436	+	0.667694i	$0.767282\pi$
$240$	0	0
$241$	1.20810e6	1.33986	0.669932	−	0.742423i	$-0.266323\pi$
$241$	1.20810e6	1.33986	0.669932	+	0.742423i	$0.266323\pi$
$242$	0	0
$243$	275900.	0.299734
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−1.45608e6	−1.51860
$248$	0	0
$249$	27679.4	0.0282916
$250$	0	0
$251$	−518940.	−0.519916	−0.259958	−	0.965620i	$-0.583709\pi$
$251$	−518940.	−0.519916	−0.259958	+	0.965620i	$0.583709\pi$
$252$	0	0
$253$	512196.	0.503077
$254$	0	0
$255$	26280.7	0.0253097
$256$	0	0
$257$	−566825.	−0.535324	−0.267662	−	0.963513i	$-0.586251\pi$
$257$	−566825.	−0.535324	−0.267662	+	0.963513i	$0.586251\pi$
$258$	0	0
$259$	168249.	0.155849
$260$	0	0
$261$	−1.53605e6	−1.39573
$262$	0	0
$263$	−1.27389e6	−1.13564	−0.567822	−	0.823151i	$-0.692214\pi$
$263$	−1.27389e6	−1.13564	−0.567822	+	0.823151i	$0.692214\pi$
$264$	0	0
$265$	106009.	0.0927313
$266$	0	0
$267$	−115059.	−0.0987738
$268$	0	0
$269$	−445770.	−0.375604	−0.187802	−	0.982207i	$-0.560136\pi$
$269$	−445770.	−0.375604	−0.187802	+	0.982207i	$0.560136\pi$
$270$	0	0
$271$	1.26776e6	1.04861	0.524306	−	0.851530i	$-0.324325\pi$
$271$	1.26776e6	1.04861	0.524306	+	0.851530i	$0.324325\pi$
$272$	0	0
$273$	50858.6	0.0413007
$274$	0	0
$275$	201732.	0.160858
$276$	0	0
$277$	−309948.	−0.242711	−0.121355	−	0.992609i	$-0.538724\pi$
$277$	−309948.	−0.242711	−0.121355	+	0.992609i	$0.538724\pi$
$278$	0	0
$279$	−1.85484e6	−1.42658
$280$	0	0
$281$	291247.	0.220037	0.110018	−	0.993930i	$-0.464909\pi$
$281$	291247.	0.220037	0.110018	+	0.993930i	$0.464909\pi$
$282$	0	0
$283$	723026.	0.536646	0.268323	−	0.963329i	$-0.413531\pi$
$283$	723026.	0.536646	0.268323	+	0.963329i	$0.413531\pi$
$284$	0	0
$285$	−87443.2	−0.0637696
$286$	0	0
$287$	−320325.	−0.229554
$288$	0	0
$289$	−976630.	−0.687837
$290$	0	0
$291$	−262190.	−0.181503
$292$	0	0
$293$	−1.19632e6	−0.814098	−0.407049	−	0.913406i	$-0.633442\pi$
$293$	−1.19632e6	−0.814098	−0.407049	+	0.913406i	$0.633442\pi$
$294$	0	0
$295$	653908.	0.437484
$296$	0	0
$297$	−246423.	−0.162103
$298$	0	0
$299$	1.04310e6	0.674757
$300$	0	0
$301$	813940.	0.517817
$302$	0	0
$303$	30912.6	0.0193433
$304$	0	0
$305$	817096.	0.502948
$306$	0	0
$307$	−1.95349e6	−1.18295	−0.591475	−	0.806324i	$-0.701454\pi$
$307$	−1.95349e6	−1.18295	−0.591475	+	0.806324i	$0.701454\pi$
$308$	0	0
$309$	132423.	0.0788981
$310$	0	0
$311$	794401.	0.465735	0.232868	−	0.972508i	$-0.425189\pi$
$311$	794401.	0.465735	0.232868	+	0.972508i	$0.425189\pi$
$312$	0	0
$313$	−118598.	−0.0684255	−0.0342127	−	0.999415i	$-0.510892\pi$
$313$	−118598.	−0.0684255	−0.0342127	+	0.999415i	$0.510892\pi$
$314$	0	0
$315$	−294621.	−0.167297
$316$	0	0
$317$	−2.78228e6	−1.55508	−0.777540	−	0.628833i	$-0.783533\pi$
$317$	−2.78228e6	−1.55508	−0.777540	+	0.628833i	$0.783533\pi$
$318$	0	0
$319$	2.06144e6	1.13421
$320$	0	0
$321$	−166907.	−0.0904092
$322$	0	0
$323$	−1.47474e6	−0.786517
$324$	0	0
$325$	410832.	0.215752
$326$	0	0
$327$	−180616.	−0.0934088
$328$	0	0
$329$	613326.	0.312393
$330$	0	0
$331$	−2.38160e6	−1.19481	−0.597404	−	0.801940i	$-0.703801\pi$
$331$	−2.38160e6	−1.19481	−0.597404	+	0.801940i	$0.703801\pi$
$332$	0	0
$333$	−825817.	−0.408106
$334$	0	0
$335$	−153830.	−0.0748907
$336$	0	0
$337$	270123.	0.129565	0.0647823	−	0.997899i	$-0.479365\pi$
$337$	270123.	0.129565	0.0647823	+	0.997899i	$0.479365\pi$
$338$	0	0
$339$	−150200.	−0.0709857
$340$	0	0
$341$	2.48929e6	1.15928
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	62642.1	0.0283347
$346$	0	0
$347$	3.58739e6	1.59939	0.799696	−	0.600405i	$-0.204994\pi$
$347$	3.58739e6	1.59939	0.799696	+	0.600405i	$0.204994\pi$
$348$	0	0
$349$	−1.13696e6	−0.499666	−0.249833	−	0.968289i	$-0.580376\pi$
$349$	−1.13696e6	−0.499666	−0.249833	+	0.968289i	$0.580376\pi$
$350$	0	0
$351$	−501847.	−0.217422
$352$	0	0
$353$	3.49344e6	1.49216	0.746081	−	0.665855i	$-0.231933\pi$
$353$	3.49344e6	1.49216	0.746081	+	0.665855i	$0.231933\pi$
$354$	0	0
$355$	−845680.	−0.356152
$356$	0	0
$357$	51510.2	0.0213906
$358$	0	0
$359$	−986571.	−0.404010	−0.202005	−	0.979384i	$-0.564746\pi$
$359$	−986571.	−0.404010	−0.202005	+	0.979384i	$0.564746\pi$
$360$	0	0
$361$	2.43075e6	0.981687
$362$	0	0
$363$	−89798.2	−0.0357685
$364$	0	0
$365$	−2.05486e6	−0.807329
$366$	0	0
$367$	3.28626e6	1.27361	0.636805	−	0.771025i	$-0.280255\pi$
$367$	3.28626e6	1.27361	0.636805	+	0.771025i	$0.280255\pi$
$368$	0	0
$369$	1.57225e6	0.601112
$370$	0	0
$371$	207777.	0.0783722
$372$	0	0
$373$	3.67518e6	1.36775	0.683874	−	0.729600i	$-0.260294\pi$
$373$	3.67518e6	1.36775	0.683874	+	0.729600i	$0.260294\pi$
$374$	0	0
$375$	24672.0	0.00905996
$376$	0	0
$377$	4.19818e6	1.52127
$378$	0	0
$379$	−1.18583e6	−0.424059	−0.212029	−	0.977263i	$-0.568007\pi$
$379$	−1.18583e6	−0.424059	−0.212029	+	0.977263i	$0.568007\pi$
$380$	0	0
$381$	154513.	0.0545322
$382$	0	0
$383$	−2.67224e6	−0.930846	−0.465423	−	0.885088i	$-0.654098\pi$
$383$	−2.67224e6	−0.930846	−0.465423	+	0.885088i	$0.654098\pi$
$384$	0	0
$385$	395394.	0.135950
$386$	0	0
$387$	−3.99506e6	−1.35596
$388$	0	0
$389$	−2.60515e6	−0.872890	−0.436445	−	0.899731i	$-0.643763\pi$
$389$	−2.60515e6	−0.872890	−0.436445	+	0.899731i	$0.643763\pi$
$390$	0	0
$391$	1.05646e6	0.349472
$392$	0	0
$393$	−50411.7	−0.0164646
$394$	0	0
$395$	−880133.	−0.283828
$396$	0	0
$397$	4.59840e6	1.46430	0.732151	−	0.681142i	$-0.238517\pi$
$397$	4.59840e6	1.46430	0.732151	+	0.681142i	$0.238517\pi$
$398$	0	0
$399$	−171389.	−0.0538952
$400$	0	0
$401$	−2.14418e6	−0.665887	−0.332943	−	0.942947i	$-0.608042\pi$
$401$	−2.14418e6	−0.665887	−0.332943	+	0.942947i	$0.608042\pi$
$402$	0	0
$403$	5.06949e6	1.55490
$404$	0	0
$405$	1.43094e6	0.433495
$406$	0	0
$407$	1.10828e6	0.331639
$408$	0	0
$409$	4.01508e6	1.18682	0.593411	−	0.804900i	$-0.297781\pi$
$409$	4.01508e6	1.18682	0.593411	+	0.804900i	$0.297781\pi$
$410$	0	0
$411$	35679.0	0.0104186
$412$	0	0
$413$	1.28166e6	0.369741
$414$	0	0
$415$	438240.	0.124908
$416$	0	0
$417$	453542.	0.127725
$418$	0	0
$419$	2.05677e6	0.572335	0.286168	−	0.958180i	$-0.407619\pi$
$419$	2.05677e6	0.572335	0.286168	+	0.958180i	$0.407619\pi$
$420$	0	0
$421$	−536575.	−0.147545	−0.0737726	−	0.997275i	$-0.523504\pi$
$421$	−536575.	−0.147545	−0.0737726	+	0.997275i	$0.523504\pi$
$422$	0	0
$423$	−3.01039e6	−0.818034
$424$	0	0
$425$	416096.	0.111743
$426$	0	0
$427$	1.60151e6	0.425069
$428$	0	0
$429$	335014.	0.0878860
$430$	0	0
$431$	−5.57365e6	−1.44526	−0.722631	−	0.691234i	$-0.757067\pi$
$431$	−5.57365e6	−1.44526	−0.722631	+	0.691234i	$0.757067\pi$
$432$	0	0
$433$	−600292.	−0.153866	−0.0769331	−	0.997036i	$-0.524513\pi$
$433$	−600292.	−0.153866	−0.0769331	+	0.997036i	$0.524513\pi$
$434$	0	0
$435$	252117.	0.0638820
$436$	0	0
$437$	−3.51514e6	−0.880521
$438$	0	0
$439$	4.08074e6	1.01060	0.505298	−	0.862945i	$-0.331383\pi$
$439$	4.08074e6	1.01060	0.505298	+	0.862945i	$0.331383\pi$
$440$	0	0
$441$	−577457.	−0.141391
$442$	0	0
$443$	−6.45140e6	−1.56187	−0.780935	−	0.624612i	$-0.785257\pi$
$443$	−6.45140e6	−1.56187	−0.780935	+	0.624612i	$0.785257\pi$
$444$	0	0
$445$	−1.82169e6	−0.436089
$446$	0	0
$447$	−690889.	−0.163546
$448$	0	0
$449$	3.62778e6	0.849230	0.424615	−	0.905374i	$-0.360409\pi$
$449$	3.62778e6	0.849230	0.424615	+	0.905374i	$0.360409\pi$
$450$	0	0
$451$	−2.11003e6	−0.488480
$452$	0	0
$453$	798769.	0.182884
$454$	0	0
$455$	805230.	0.182344
$456$	0	0
$457$	5.12392e6	1.14766	0.573828	−	0.818976i	$-0.305458\pi$
$457$	5.12392e6	1.14766	0.573828	+	0.818976i	$0.305458\pi$
$458$	0	0
$459$	−508277.	−0.112608
$460$	0	0
$461$	−3.96610e6	−0.869184	−0.434592	−	0.900627i	$-0.643107\pi$
$461$	−3.96610e6	−0.869184	−0.434592	+	0.900627i	$0.643107\pi$
$462$	0	0
$463$	310011.	0.0672085	0.0336043	−	0.999435i	$-0.489301\pi$
$463$	310011.	0.0672085	0.0336043	+	0.999435i	$0.489301\pi$
$464$	0	0
$465$	304442.	0.0652939
$466$	0	0
$467$	−1.07130e6	−0.227311	−0.113655	−	0.993520i	$-0.536256\pi$
$467$	−1.07130e6	−0.227311	−0.113655	+	0.993520i	$0.536256\pi$
$468$	0	0
$469$	−301506.	−0.0632942
$470$	0	0
$471$	−338433.	−0.0702943
$472$	0	0
$473$	5.36155e6	1.10189
$474$	0	0
$475$	−1.38446e6	−0.281545
$476$	0	0
$477$	−1.01983e6	−0.205226
$478$	0	0
$479$	5.62151e6	1.11947	0.559737	−	0.828670i	$-0.310902\pi$
$479$	5.62151e6	1.11947	0.559737	+	0.828670i	$0.310902\pi$
$480$	0	0
$481$	2.25705e6	0.444813
$482$	0	0
$483$	122778.	0.0239472
$484$	0	0
$485$	−4.15118e6	−0.801341
$486$	0	0
$487$	−4.04558e6	−0.772962	−0.386481	−	0.922297i	$-0.626309\pi$
$487$	−4.04558e6	−0.772962	−0.386481	+	0.922297i	$0.626309\pi$
$488$	0	0
$489$	−927006.	−0.175312
$490$	0	0
$491$	−4.15461e6	−0.777727	−0.388863	−	0.921295i	$-0.627132\pi$
$491$	−4.15461e6	−0.777727	−0.388863	+	0.921295i	$0.627132\pi$
$492$	0	0
$493$	4.25197e6	0.787903
$494$	0	0
$495$	−1.94071e6	−0.355999
$496$	0	0
$497$	−1.65753e6	−0.301003
$498$	0	0
$499$	−2.79365e6	−0.502251	−0.251125	−	0.967955i	$-0.580801\pi$
$499$	−2.79365e6	−0.502251	−0.251125	+	0.967955i	$0.580801\pi$
$500$	0	0
$501$	−611445.	−0.108834
$502$	0	0
$503$	−6.95171e6	−1.22510	−0.612551	−	0.790431i	$-0.709856\pi$
$503$	−6.95171e6	−1.22510	−0.612551	+	0.790431i	$0.709856\pi$
$504$	0	0
$505$	489431.	0.0854010
$506$	0	0
$507$	95988.9	0.0165845
$508$	0	0
$509$	−7.84503e6	−1.34215	−0.671073	−	0.741391i	$-0.734166\pi$
$509$	−7.84503e6	−1.34215	−0.671073	+	0.741391i	$0.734166\pi$
$510$	0	0
$511$	−4.02753e6	−0.682317
$512$	0	0
$513$	1.69118e6	0.283723
$514$	0	0
$515$	2.09661e6	0.348337
$516$	0	0
$517$	4.04008e6	0.664758
$518$	0	0
$519$	963887.	0.157075
$520$	0	0
$521$	−6.03460e6	−0.973989	−0.486995	−	0.873405i	$-0.661907\pi$
$521$	−6.03460e6	−0.973989	−0.486995	+	0.873405i	$0.661907\pi$
$522$	0	0
$523$	−1.59256e6	−0.254590	−0.127295	−	0.991865i	$-0.540629\pi$
$523$	−1.59256e6	−0.254590	−0.127295	+	0.991865i	$0.540629\pi$
$524$	0	0
$525$	48357.1	0.00765707
$526$	0	0
$527$	5.13444e6	0.805317
$528$	0	0
$529$	−3.91818e6	−0.608759
$530$	0	0
$531$	−6.29078e6	−0.968206
$532$	0	0
$533$	−4.29713e6	−0.655179
$534$	0	0
$535$	−2.64259e6	−0.399159
$536$	0	0
$537$	267650.	0.0400526
$538$	0	0
$539$	774973.	0.114899
$540$	0	0
$541$	−1.73741e6	−0.255217	−0.127608	−	0.991825i	$-0.540730\pi$
$541$	−1.73741e6	−0.255217	−0.127608	+	0.991825i	$0.540730\pi$
$542$	0	0
$543$	481861.	0.0701330
$544$	0	0
$545$	−2.85965e6	−0.412402
$546$	0	0
$547$	1.14309e7	1.63348	0.816738	−	0.577009i	$-0.195780\pi$
$547$	1.14309e7	1.63348	0.816738	+	0.577009i	$0.195780\pi$
$548$	0	0
$549$	−7.86068e6	−1.11309
$550$	0	0
$551$	−1.41475e7	−1.98518
$552$	0	0
$553$	−1.72506e6	−0.239879
$554$	0	0
$555$	135544.	0.0186788
$556$	0	0
$557$	1.08461e7	1.48127	0.740635	−	0.671908i	$-0.234525\pi$
$557$	1.08461e7	1.48127	0.740635	+	0.671908i	$0.234525\pi$
$558$	0	0
$559$	1.09189e7	1.47792
$560$	0	0
$561$	339306.	0.0455182
$562$	0	0
$563$	203287.	0.0270295	0.0135148	−	0.999909i	$-0.495698\pi$
$563$	203287.	0.0270295	0.0135148	+	0.999909i	$0.495698\pi$
$564$	0	0
$565$	−2.37808e6	−0.313404
$566$	0	0
$567$	2.80464e6	0.366370
$568$	0	0
$569$	1.04184e7	1.34903	0.674514	−	0.738262i	$-0.264353\pi$
$569$	1.04184e7	1.34903	0.674514	+	0.738262i	$0.264353\pi$
$570$	0	0
$571$	−4.44964e6	−0.571129	−0.285565	−	0.958359i	$-0.592181\pi$
$571$	−4.44964e6	−0.571129	−0.285565	+	0.958359i	$0.592181\pi$
$572$	0	0
$573$	1.48659e6	0.189149
$574$	0	0
$575$	991794.	0.125098
$576$	0	0
$577$	−4.10826e6	−0.513711	−0.256855	−	0.966450i	$-0.582686\pi$
$577$	−4.10826e6	−0.513711	−0.256855	+	0.966450i	$0.582686\pi$
$578$	0	0
$579$	608009.	0.0753726
$580$	0	0
$581$	858950.	0.105567
$582$	0	0
$583$	1.36866e6	0.166772
$584$	0	0
$585$	−3.95231e6	−0.477487
$586$	0	0
$587$	−1.10539e7	−1.32409	−0.662047	−	0.749463i	$-0.730312\pi$
$587$	−1.10539e7	−1.32409	−0.662047	+	0.749463i	$0.730312\pi$
$588$	0	0
$589$	−1.70837e7	−2.02905
$590$	0	0
$591$	−1.63772e6	−0.192872
$592$	0	0
$593$	1.01113e7	1.18078	0.590390	−	0.807118i	$-0.298974\pi$
$593$	1.01113e7	1.18078	0.590390	+	0.807118i	$0.298974\pi$
$594$	0	0
$595$	815547.	0.0944402
$596$	0	0
$597$	648744.	0.0744968
$598$	0	0
$599$	−1.10092e6	−0.125369	−0.0626845	−	0.998033i	$-0.519966\pi$
$599$	−1.10092e6	−0.125369	−0.0626845	+	0.998033i	$0.519966\pi$
$600$	0	0
$601$	2.03055e6	0.229312	0.114656	−	0.993405i	$-0.463423\pi$
$601$	2.03055e6	0.229312	0.114656	+	0.993405i	$0.463423\pi$
$602$	0	0
$603$	1.47988e6	0.165742
$604$	0	0
$605$	−1.42175e6	−0.157919
$606$	0	0
$607$	4.20572e6	0.463307	0.231654	−	0.972798i	$-0.425586\pi$
$607$	4.20572e6	0.463307	0.231654	+	0.972798i	$0.425586\pi$
$608$	0	0
$609$	494148.	0.0539901
$610$	0	0
$611$	8.22771e6	0.891612
$612$	0	0
$613$	−1.34913e7	−1.45012	−0.725060	−	0.688686i	$-0.758188\pi$
$613$	−1.34913e7	−1.45012	−0.725060	+	0.688686i	$0.758188\pi$
$614$	0	0
$615$	−258059.	−0.0275126
$616$	0	0
$617$	9.26037e6	0.979299	0.489650	−	0.871919i	$-0.337125\pi$
$617$	9.26037e6	0.979299	0.489650	+	0.871919i	$0.337125\pi$
$618$	0	0
$619$	1.11560e7	1.17026	0.585131	−	0.810939i	$-0.301043\pi$
$619$	1.11560e7	1.17026	0.585131	+	0.810939i	$0.301043\pi$
$620$	0	0
$621$	−1.21151e6	−0.126067
$622$	0	0
$623$	−3.57052e6	−0.368563
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.12896e6	−0.114686
$628$	0	0
$629$	2.28597e6	0.230379
$630$	0	0
$631$	8.72868e6	0.872721	0.436360	−	0.899772i	$-0.356267\pi$
$631$	8.72868e6	0.872721	0.436360	+	0.899772i	$0.356267\pi$
$632$	0	0
$633$	699049.	0.0693423
$634$	0	0
$635$	2.44636e6	0.240761
$636$	0	0
$637$	1.57825e6	0.154109
$638$	0	0
$639$	8.13566e6	0.788208
$640$	0	0
$641$	646402.	0.0621380	0.0310690	−	0.999517i	$-0.490109\pi$
$641$	646402.	0.0621380	0.0310690	+	0.999517i	$0.490109\pi$
$642$	0	0
$643$	1.36895e7	1.30575	0.652873	−	0.757468i	$-0.273564\pi$
$643$	1.36895e7	1.30575	0.652873	+	0.757468i	$0.273564\pi$
$644$	0	0
$645$	655723.	0.0620614
$646$	0	0
$647$	−1.12743e7	−1.05884	−0.529418	−	0.848361i	$-0.677590\pi$
$647$	−1.12743e7	−1.05884	−0.529418	+	0.848361i	$0.677590\pi$
$648$	0	0
$649$	8.44250e6	0.786791
$650$	0	0
$651$	596707.	0.0551834
$652$	0	0
$653$	1.23867e7	1.13677	0.568384	−	0.822763i	$-0.307569\pi$
$653$	1.23867e7	1.13677	0.568384	+	0.822763i	$0.307569\pi$
$654$	0	0
$655$	−798154.	−0.0726915
$656$	0	0
$657$	1.97683e7	1.78672
$658$	0	0
$659$	9.93495e6	0.891153	0.445576	−	0.895244i	$-0.352999\pi$
$659$	9.93495e6	0.891153	0.445576	+	0.895244i	$0.352999\pi$
$660$	0	0
$661$	−4.30434e6	−0.383181	−0.191590	−	0.981475i	$-0.561364\pi$
$661$	−4.30434e6	−0.383181	−0.191590	+	0.981475i	$0.561364\pi$
$662$	0	0
$663$	691005.	0.0610517
$664$	0	0
$665$	−2.71355e6	−0.237949
$666$	0	0
$667$	1.01349e7	0.882072
$668$	0	0
$669$	148322.	0.0128127
$670$	0	0
$671$	1.05494e7	0.904526
$672$	0	0
$673$	−252232.	−0.0214665	−0.0107333	−	0.999942i	$-0.503417\pi$
$673$	−252232.	−0.0214665	−0.0107333	+	0.999942i	$0.503417\pi$
$674$	0	0
$675$	−477163.	−0.0403095
$676$	0	0
$677$	−1.06012e7	−0.888960	−0.444480	−	0.895789i	$-0.646612\pi$
$677$	−1.06012e7	−0.888960	−0.444480	+	0.895789i	$0.646612\pi$
$678$	0	0
$679$	−8.13632e6	−0.677257
$680$	0	0
$681$	370338.	0.0306006
$682$	0	0
$683$	−1.30105e7	−1.06719	−0.533594	−	0.845741i	$-0.679159\pi$
$683$	−1.30105e7	−1.06719	−0.533594	+	0.845741i	$0.679159\pi$
$684$	0	0
$685$	564895.	0.0459983
$686$	0	0
$687$	377433.	0.0305104
$688$	0	0
$689$	2.78731e6	0.223685
$690$	0	0
$691$	−9.67711e6	−0.770993	−0.385496	−	0.922709i	$-0.625970\pi$
$691$	−9.67711e6	−0.770993	−0.385496	+	0.922709i	$0.625970\pi$
$692$	0	0
$693$	−3.80380e6	−0.300874
$694$	0	0
$695$	7.18080e6	0.563911
$696$	0	0
$697$	−4.35218e6	−0.339332
$698$	0	0
$699$	−2.32636e6	−0.180087
$700$	0	0
$701$	−1.63711e7	−1.25829	−0.629147	−	0.777286i	$-0.716596\pi$
$701$	−1.63711e7	−1.25829	−0.629147	+	0.777286i	$0.716596\pi$
$702$	0	0
$703$	−7.60603e6	−0.580457
$704$	0	0
$705$	494106.	0.0374410
$706$	0	0
$707$	959285.	0.0721771
$708$	0	0
$709$	7.10343e6	0.530704	0.265352	−	0.964152i	$-0.414512\pi$
$709$	7.10343e6	0.530704	0.265352	+	0.964152i	$0.414512\pi$
$710$	0	0
$711$	8.46711e6	0.628147
$712$	0	0
$713$	1.22383e7	0.901567
$714$	0	0
$715$	5.30418e6	0.388019
$716$	0	0
$717$	−2.07604e6	−0.150813
$718$	0	0
$719$	−6.87623e6	−0.496053	−0.248027	−	0.968753i	$-0.579782\pi$
$719$	−6.87623e6	−0.496053	−0.248027	+	0.968753i	$0.579782\pi$
$720$	0	0
$721$	4.10936e6	0.294399
$722$	0	0
$723$	1.90760e6	0.135719
$724$	0	0
$725$	3.99169e6	0.282041
$726$	0	0
$727$	9.71330e6	0.681602	0.340801	−	0.940136i	$-0.389302\pi$
$727$	9.71330e6	0.681602	0.340801	+	0.940136i	$0.389302\pi$
$728$	0	0
$729$	−1.34731e7	−0.938963
$730$	0	0
$731$	1.10588e7	0.765448
$732$	0	0
$733$	−1.14783e7	−0.789070	−0.394535	−	0.918881i	$-0.629094\pi$
$733$	−1.14783e7	−0.789070	−0.394535	+	0.918881i	$0.629094\pi$
$734$	0	0
$735$	94780.0	0.00647140
$736$	0	0
$737$	−1.98607e6	−0.134687
$738$	0	0
$739$	1.55073e7	1.04454	0.522270	−	0.852780i	$-0.325085\pi$
$739$	1.55073e7	1.04454	0.522270	+	0.852780i	$0.325085\pi$
$740$	0	0
$741$	−2.29916e6	−0.153824
$742$	0	0
$743$	2.49648e7	1.65904	0.829520	−	0.558477i	$-0.188614\pi$
$743$	2.49648e7	1.65904	0.829520	+	0.558477i	$0.188614\pi$
$744$	0	0
$745$	−1.09387e7	−0.722060
$746$	0	0
$747$	−4.21599e6	−0.276438
$748$	0	0
$749$	−5.17949e6	−0.337351
$750$	0	0
$751$	1.59690e6	0.103319	0.0516593	−	0.998665i	$-0.483549\pi$
$751$	1.59690e6	0.103319	0.0516593	+	0.998665i	$0.483549\pi$
$752$	0	0
$753$	−819411.	−0.0526641
$754$	0	0
$755$	1.26467e7	0.807438
$756$	0	0
$757$	−8.30133e6	−0.526512	−0.263256	−	0.964726i	$-0.584796\pi$
$757$	−8.30133e6	−0.526512	−0.263256	+	0.964726i	$0.584796\pi$
$758$	0	0
$759$	808762.	0.0509584
$760$	0	0
$761$	911464.	0.0570529	0.0285265	−	0.999593i	$-0.490919\pi$
$761$	911464.	0.0570529	0.0285265	+	0.999593i	$0.490919\pi$
$762$	0	0
$763$	−5.60491e6	−0.348544
$764$	0	0
$765$	−4.00295e6	−0.247301
$766$	0	0
$767$	1.71934e7	1.05529
$768$	0	0
$769$	−2.51616e7	−1.53434	−0.767171	−	0.641442i	$-0.778336\pi$
$769$	−2.51616e7	−1.53434	−0.767171	+	0.641442i	$0.778336\pi$
$770$	0	0
$771$	−895022.	−0.0542248
$772$	0	0
$773$	−4.10309e6	−0.246980	−0.123490	−	0.992346i	$-0.539409\pi$
$773$	−4.10309e6	−0.246980	−0.123490	+	0.992346i	$0.539409\pi$
$774$	0	0
$775$	4.82015e6	0.288274
$776$	0	0
$777$	265667.	0.0157865
$778$	0	0
$779$	1.44809e7	0.854972
$780$	0	0
$781$	−1.09184e7	−0.640520
$782$	0	0
$783$	−4.87600e6	−0.284223
$784$	0	0
$785$	−5.35831e6	−0.310352
$786$	0	0
$787$	−2.32406e7	−1.33755	−0.668777	−	0.743463i	$-0.733182\pi$
$787$	−2.32406e7	−1.33755	−0.668777	+	0.743463i	$0.733182\pi$
$788$	0	0
$789$	−2.01148e6	−0.115033
$790$	0	0
$791$	−4.66103e6	−0.264875
$792$	0	0
$793$	2.14841e7	1.21320
$794$	0	0
$795$	167388.	0.00939307
$796$	0	0
$797$	−9.23259e6	−0.514846	−0.257423	−	0.966299i	$-0.582873\pi$
$797$	−9.23259e6	−0.514846	−0.257423	+	0.966299i	$0.582873\pi$
$798$	0	0
$799$	8.33313e6	0.461787
$800$	0	0
$801$	1.75252e7	0.965120
$802$	0	0
$803$	−2.65300e7	−1.45194
$804$	0	0
$805$	1.94392e6	0.105728
$806$	0	0
$807$	−703874.	−0.0380462
$808$	0	0
$809$	−3.27151e7	−1.75743	−0.878714	−	0.477349i	$-0.841598\pi$
$809$	−3.27151e7	−1.75743	−0.878714	+	0.477349i	$0.841598\pi$
$810$	0	0
$811$	−1.01835e7	−0.543681	−0.271840	−	0.962342i	$-0.587632\pi$
$811$	−1.01835e7	−0.543681	−0.271840	+	0.962342i	$0.587632\pi$
$812$	0	0
$813$	2.00181e6	0.106218
$814$	0	0
$815$	−1.46770e7	−0.774005
$816$	0	0
$817$	−3.67958e7	−1.92860
$818$	0	0
$819$	−7.74653e6	−0.403550
$820$	0	0
$821$	3.37256e7	1.74623	0.873116	−	0.487512i	$-0.162096\pi$
$821$	3.37256e7	1.74623	0.873116	+	0.487512i	$0.162096\pi$
$822$	0	0
$823$	−1.33753e7	−0.688343	−0.344171	−	0.938907i	$-0.611840\pi$
$823$	−1.33753e7	−0.688343	−0.344171	+	0.938907i	$0.611840\pi$
$824$	0	0
$825$	318536.	0.0162939
$826$	0	0
$827$	1.05590e7	0.536856	0.268428	−	0.963300i	$-0.413496\pi$
$827$	1.05590e7	0.536856	0.268428	+	0.963300i	$0.413496\pi$
$828$	0	0
$829$	−3.01166e7	−1.52202	−0.761009	−	0.648741i	$-0.775296\pi$
$829$	−3.01166e7	−1.52202	−0.761009	+	0.648741i	$0.775296\pi$
$830$	0	0
$831$	−489410.	−0.0245850
$832$	0	0
$833$	1.59847e6	0.0798165
$834$	0	0
$835$	−9.68084e6	−0.480504
$836$	0	0
$837$	−5.88799e6	−0.290505
$838$	0	0
$839$	2.47935e6	0.121600	0.0607998	−	0.998150i	$-0.480635\pi$
$839$	2.47935e6	0.121600	0.0607998	+	0.998150i	$0.480635\pi$
$840$	0	0
$841$	2.02788e7	0.988674
$842$	0	0
$843$	459881.	0.0222883
$844$	0	0
$845$	1.51977e6	0.0732208
$846$	0	0
$847$	−2.78663e6	−0.133466
$848$	0	0
$849$	1.14166e6	0.0543587
$850$	0	0
$851$	5.44877e6	0.257914
$852$	0	0
$853$	−1.23084e6	−0.0579199	−0.0289600	−	0.999581i	$-0.509220\pi$
$853$	−1.23084e6	−0.0579199	−0.0289600	+	0.999581i	$0.509220\pi$
$854$	0	0
$855$	1.33189e7	0.623094
$856$	0	0
$857$	−2.75099e7	−1.27949	−0.639746	−	0.768586i	$-0.720961\pi$
$857$	−2.75099e7	−1.27949	−0.639746	+	0.768586i	$0.720961\pi$
$858$	0	0
$859$	−1.28833e7	−0.595722	−0.297861	−	0.954609i	$-0.596273\pi$
$859$	−1.28833e7	−0.595722	−0.297861	+	0.954609i	$0.596273\pi$
$860$	0	0
$861$	−505795.	−0.0232524
$862$	0	0
$863$	−1.57981e7	−0.722067	−0.361033	−	0.932553i	$-0.617576\pi$
$863$	−1.57981e7	−0.722067	−0.361033	+	0.932553i	$0.617576\pi$
$864$	0	0
$865$	1.52610e7	0.693492
$866$	0	0
$867$	−1.54211e6	−0.0696734
$868$	0	0
$869$	−1.13632e7	−0.510450
$870$	0	0
$871$	−4.04468e6	−0.180650
$872$	0	0
$873$	3.99355e7	1.77347
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−3.99471e7	−1.75383	−0.876913	−	0.480649i	$-0.840401\pi$
$877$	−3.99471e7	−1.75383	−0.876913	+	0.480649i	$0.840401\pi$
$878$	0	0
$879$	−1.88899e6	−0.0824628
$880$	0	0
$881$	4.20714e7	1.82620	0.913098	−	0.407740i	$-0.133683\pi$
$881$	4.20714e7	1.82620	0.913098	+	0.407740i	$0.133683\pi$
$882$	0	0
$883$	−2.82226e7	−1.21813	−0.609067	−	0.793118i	$-0.708456\pi$
$883$	−2.82226e7	−1.21813	−0.609067	+	0.793118i	$0.708456\pi$
$884$	0	0
$885$	1.03253e6	0.0443142
$886$	0	0
$887$	−1.26687e7	−0.540658	−0.270329	−	0.962768i	$-0.587132\pi$
$887$	−1.26687e7	−0.540658	−0.270329	+	0.962768i	$0.587132\pi$
$888$	0	0
$889$	4.79487e6	0.203480
$890$	0	0
$891$	1.84746e7	0.779618
$892$	0	0
$893$	−2.77266e7	−1.16350
$894$	0	0
$895$	4.23762e6	0.176833
$896$	0	0
$897$	1.64706e6	0.0683485
$898$	0	0
$899$	4.92558e7	2.03263
$900$	0	0
$901$	2.82302e6	0.115852
$902$	0	0
$903$	1.28522e6	0.0524514
$904$	0	0
$905$	7.62917e6	0.309639
$906$	0	0
$907$	−1.50368e7	−0.606928	−0.303464	−	0.952843i	$-0.598143\pi$
$907$	−1.50368e7	−0.606928	−0.303464	+	0.952843i	$0.598143\pi$
$908$	0	0
$909$	−4.70846e6	−0.189003
$910$	0	0
$911$	−1.13882e7	−0.454630	−0.227315	−	0.973821i	$-0.572995\pi$
$911$	−1.13882e7	−0.454630	−0.227315	+	0.973821i	$0.572995\pi$
$912$	0	0
$913$	5.65804e6	0.224641
$914$	0	0
$915$	1.29020e6	0.0509454
$916$	0	0
$917$	−1.56438e6	−0.0614355
$918$	0	0
$919$	1.66749e7	0.651292	0.325646	−	0.945492i	$-0.394418\pi$
$919$	1.66749e7	0.651292	0.325646	+	0.945492i	$0.394418\pi$
$920$	0	0
$921$	−3.08458e6	−0.119825
$922$	0	0
$923$	−2.22356e7	−0.859103
$924$	0	0
$925$	2.14603e6	0.0824674
$926$	0	0
$927$	−2.01700e7	−0.770914
$928$	0	0
$929$	3.13311e7	1.19107	0.595534	−	0.803330i	$-0.296941\pi$
$929$	3.13311e7	1.19107	0.595534	+	0.803330i	$0.296941\pi$
$930$	0	0
$931$	−5.31856e6	−0.201103
$932$	0	0
$933$	1.25437e6	0.0471759
$934$	0	0
$935$	5.37214e6	0.200964
$936$	0	0
$937$	4.22607e7	1.57249	0.786245	−	0.617914i	$-0.212022\pi$
$937$	4.22607e7	1.57249	0.786245	+	0.617914i	$0.212022\pi$
$938$	0	0
$939$	−187268.	−0.00693105
$940$	0	0
$941$	−2.46633e7	−0.907980	−0.453990	−	0.891007i	$-0.650000\pi$
$941$	−2.46633e7	−0.907980	−0.453990	+	0.891007i	$0.650000\pi$
$942$	0	0
$943$	−1.03738e7	−0.379889
$944$	0	0
$945$	−935240.	−0.0340678
$946$	0	0
$947$	−1.33679e7	−0.484383	−0.242191	−	0.970229i	$-0.577866\pi$
$947$	−1.33679e7	−0.484383	−0.242191	+	0.970229i	$0.577866\pi$
$948$	0	0
$949$	−5.40289e7	−1.94743
$950$	0	0
$951$	−4.39325e6	−0.157519
$952$	0	0
$953$	−2.17678e7	−0.776395	−0.388198	−	0.921576i	$-0.626902\pi$
$953$	−2.17678e7	−0.776395	−0.388198	+	0.921576i	$0.626902\pi$
$954$	0	0
$955$	2.35368e7	0.835100
$956$	0	0
$957$	3.25504e6	0.114888
$958$	0	0
$959$	1.10719e6	0.0388756
$960$	0	0
$961$	3.08494e7	1.07755
$962$	0	0
$963$	2.54225e7	0.883389
$964$	0	0
$965$	9.62643e6	0.332772
$966$	0	0
$967$	2.58647e7	0.889491	0.444746	−	0.895657i	$-0.353294\pi$
$967$	2.58647e7	0.889491	0.444746	+	0.895657i	$0.353294\pi$
$968$	0	0
$969$	−2.32862e6	−0.0796691
$970$	0	0
$971$	3.76225e7	1.28056	0.640280	−	0.768142i	$-0.278819\pi$
$971$	3.76225e7	1.28056	0.640280	+	0.768142i	$0.278819\pi$
$972$	0	0
$973$	1.40744e7	0.476592
$974$	0	0
$975$	648707.	0.0218543
$976$	0	0
$977$	4.22400e7	1.41575	0.707876	−	0.706336i	$-0.249653\pi$
$977$	4.22400e7	1.41575	0.707876	+	0.706336i	$0.249653\pi$
$978$	0	0
$979$	−2.35196e7	−0.784284
$980$	0	0
$981$	2.75106e7	0.912698
$982$	0	0
$983$	2.18464e7	0.721100	0.360550	−	0.932740i	$-0.382589\pi$
$983$	2.18464e7	0.721100	0.360550	+	0.932740i	$0.382589\pi$
$984$	0	0
$985$	−2.59295e7	−0.851538
$986$	0	0
$987$	968447.	0.0316434
$988$	0	0
$989$	2.63595e7	0.856933
$990$	0	0
$991$	4.62904e6	0.149729	0.0748646	−	0.997194i	$-0.476148\pi$
$991$	4.62904e6	0.149729	0.0748646	+	0.997194i	$0.476148\pi$
$992$	0	0
$993$	−3.76056e6	−0.121026
$994$	0	0
$995$	1.02714e7	0.328905
$996$	0	0
$997$	−4.99110e7	−1.59023	−0.795113	−	0.606462i	$-0.792588\pi$
$997$	−4.99110e7	−1.59023	−0.795113	+	0.606462i	$0.792588\pi$
$998$	0	0
$999$	−2.62146e6	−0.0831055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.6.a.d.1.2	✓	3
4.3	odd	2		560.6.a.t.1.2		3
5.2	odd	4		700.6.e.g.449.3		6
5.3	odd	4		700.6.e.g.449.4		6
5.4	even	2		700.6.a.i.1.2		3
7.6	odd	2		980.6.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.d.1.2	✓	3	1.1	even	1	trivial
560.6.a.t.1.2		3	4.3	odd	2
700.6.a.i.1.2		3	5.4	even	2
700.6.e.g.449.3		6	5.2	odd	4
700.6.e.g.449.4		6	5.3	odd	4
980.6.a.h.1.2		3	7.6	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:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	140.a (trivial)

\(f(q)\)	\(=\)	\(q+1.57901 q^{3} +25.0000 q^{5} +49.0000 q^{7} -240.507 q^{9} +322.771 q^{11} +657.331 q^{13} +39.4752 q^{15} +665.753 q^{17} -2215.14 q^{19} +77.3714 q^{21} +1586.87 q^{23} +625.000 q^{25} -763.461 q^{27} +6386.70 q^{29} +7712.24 q^{31} +509.658 q^{33} +1225.00 q^{35} +3433.65 q^{37} +1037.93 q^{39} -6537.24 q^{41} +16611.0 q^{43} -6012.67 q^{45} +12516.9 q^{47} +2401.00 q^{49} +1051.23 q^{51} +4240.34 q^{53} +8069.27 q^{55} -3497.73 q^{57} +26156.3 q^{59} +32683.8 q^{61} -11784.8 q^{63} +16433.3 q^{65} -6153.18 q^{67} +2505.68 q^{69} -33827.2 q^{71} -82194.4 q^{73} +986.880 q^{75} +15815.8 q^{77} -35205.3 q^{79} +57237.6 q^{81} +17529.6 q^{83} +16643.8 q^{85} +10084.7 q^{87} -72867.8 q^{89} +32209.2 q^{91} +12177.7 q^{93} -55378.5 q^{95} -166047. q^{97} -77628.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33}+ \cdots - 114368 q^{99}+O(q^{100}) \) 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9 - 14 * q^11 - 4 * q^13 + 150 * q^15 + 44 * q^17 + 2328 * q^19 + 294 * q^21 + 3676 * q^23 + 1875 * q^25 + 3726 * q^27 + 4092 * q^29 + 5888 * q^31 + 11318 * q^33 + 3675 * q^35 + 11378 * q^37 + 13702 * q^39 + 11450 * q^41 + 18544 * q^43 + 7025 * q^45 + 21754 * q^47 + 7203 * q^49 + 31762 * q^51 + 7494 * q^53 - 350 * q^55 + 7332 * q^57 + 12388 * q^59 + 27182 * q^61 + 13769 * q^63 - 100 * q^65 - 7676 * q^67 - 62140 * q^69 + 81992 * q^71 + 230 * q^73 + 3750 * q^75 - 686 * q^77 - 15926 * q^79 - 32101 * q^81 - 86100 * q^83 + 1100 * q^85 - 85098 * q^87 - 95710 * q^89 - 196 * q^91 - 229472 * q^93 + 58200 * q^95 - 176188 * q^97 - 114368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more