Properties

Label 1386.2.g.a
Level 13861386
Weight 22
Character orbit 1386.g
Analytic conductor 11.06711.067
Analytic rank 00
Dimension 1616
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1386,2,Mod(881,1386)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1386.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1386=232711 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1386.g (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-16,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 11.067265720111.0672657201
Analytic rank: 00
Dimension: 1616
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x164x14+8x12+80x10+1189x82028x6+1800x4+1080x2+324 x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 2934 2^{9}\cdot 3^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2q4β4q5+β7q7β1q8+β3q10β1q11+β11q14+q16+(β11+β10+β1)q17++(β15β13+β10)q98+O(q100) q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} - \beta_1 q^{11} + \beta_{11} q^{14} + q^{16} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{15} - \beta_{13} + \cdots - \beta_{10}) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q16q48q7+16q16+16q22+16q25+8q2816q37+48q4316q46+8q4916q5816q64+16q678q7016q79+112q8516q88+O(q100) 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x164x14+8x12+80x10+1189x82028x6+1800x4+1080x2+324 x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 : Copy content Toggle raw display

β1\beta_{1}== (96575ν14+438755ν12939814ν107435564ν8110537567ν6+51358644)/86730966 ( - 96575 \nu^{14} + 438755 \nu^{12} - 939814 \nu^{10} - 7435564 \nu^{8} - 110537567 \nu^{6} + \cdots - 51358644 ) / 86730966 Copy content Toggle raw display
β2\beta_{2}== (3833ν14+11990ν1216306ν10332152ν84851221ν6+7232976)/1867860 ( - 3833 \nu^{14} + 11990 \nu^{12} - 16306 \nu^{10} - 332152 \nu^{8} - 4851221 \nu^{6} + \cdots - 7232976 ) / 1867860 Copy content Toggle raw display
β3\beta_{3}== (3079634ν15+11597979ν1322413962ν11248772494ν9+5487054570ν)/2601928980 ( - 3079634 \nu^{15} + 11597979 \nu^{13} - 22413962 \nu^{11} - 248772494 \nu^{9} + \cdots - 5487054570 \nu ) / 2601928980 Copy content Toggle raw display
β4\beta_{4}== (12591283ν1557579817ν13+129110888ν11+952830506ν9++3202119162ν)/7805786940 ( 12591283 \nu^{15} - 57579817 \nu^{13} + 129110888 \nu^{11} + 952830506 \nu^{9} + \cdots + 3202119162 \nu ) / 7805786940 Copy content Toggle raw display
β5\beta_{5}== (14545077ν153765719ν1455196585ν13+11685650ν12+9136826568)/5203857960 ( 14545077 \nu^{15} - 3765719 \nu^{14} - 55196585 \nu^{13} + 11685650 \nu^{12} + \cdots - 9136826568 ) / 5203857960 Copy content Toggle raw display
β6\beta_{6}== (11395ν14+45868ν1296938ν10893276ν813578607ν6+6275124)/2256660 ( - 11395 \nu^{14} + 45868 \nu^{12} - 96938 \nu^{10} - 893276 \nu^{8} - 13578607 \nu^{6} + \cdots - 6275124 ) / 2256660 Copy content Toggle raw display
β7\beta_{7}== (14545077ν15+3765719ν1455196585ν1311685650ν12++3932968608)/5203857960 ( 14545077 \nu^{15} + 3765719 \nu^{14} - 55196585 \nu^{13} - 11685650 \nu^{12} + \cdots + 3932968608 ) / 5203857960 Copy content Toggle raw display
β8\beta_{8}== (18506049ν151441114ν1472148880ν13+4446220ν12+13615971048)/5203857960 ( 18506049 \nu^{15} - 1441114 \nu^{14} - 72148880 \nu^{13} + 4446220 \nu^{12} + \cdots - 13615971048 ) / 5203857960 Copy content Toggle raw display
β9\beta_{9}== (18506049ν15+1441114ν1472148880ν134446220ν12++18819829008)/5203857960 ( 18506049 \nu^{15} + 1441114 \nu^{14} - 72148880 \nu^{13} - 4446220 \nu^{12} + \cdots + 18819829008 ) / 5203857960 Copy content Toggle raw display
β10\beta_{10}== (60462680ν1534694355ν14274567721ν13+143608152ν12+18889526916)/15611573880 ( 60462680 \nu^{15} - 34694355 \nu^{14} - 274567721 \nu^{13} + 143608152 \nu^{12} + \cdots - 18889526916 ) / 15611573880 Copy content Toggle raw display
β11\beta_{11}== (60462680ν1517310855ν14+274567721ν13+64632252ν12+9644970996)/15611573880 ( - 60462680 \nu^{15} - 17310855 \nu^{14} + 274567721 \nu^{13} + 64632252 \nu^{12} + \cdots - 9644970996 ) / 15611573880 Copy content Toggle raw display
β12\beta_{12}== (81296765ν15+47163030ν14+357230942ν13210264972ν12++25206770376)/15611573880 ( - 81296765 \nu^{15} + 47163030 \nu^{14} + 357230942 \nu^{13} - 210264972 \nu^{12} + \cdots + 25206770376 ) / 15611573880 Copy content Toggle raw display
β13\beta_{13}== (81296765ν1547163030ν14+357230942ν13+210264972ν12+25206770376)/15611573880 ( - 81296765 \nu^{15} - 47163030 \nu^{14} + 357230942 \nu^{13} + 210264972 \nu^{12} + \cdots - 25206770376 ) / 15611573880 Copy content Toggle raw display
β14\beta_{14}== (23298266ν1593544963ν13+176425470ν11+1897812810ν9++37162486386ν)/2601928980 ( 23298266 \nu^{15} - 93544963 \nu^{13} + 176425470 \nu^{11} + 1897812810 \nu^{9} + \cdots + 37162486386 \nu ) / 2601928980 Copy content Toggle raw display
β15\beta_{15}== (15573353ν15+65944679ν13141046900ν111206672370ν9+2860280478ν)/1115112420 ( - 15573353 \nu^{15} + 65944679 \nu^{13} - 141046900 \nu^{11} - 1206672370 \nu^{9} + \cdots - 2860280478 \nu ) / 1115112420 Copy content Toggle raw display
ν\nu== (β15β13β123β3)/6 ( \beta_{15} - \beta_{13} - \beta_{12} - 3\beta_{3} ) / 6 Copy content Toggle raw display
ν2\nu^{2}== (β11+β10+β7β6β5+β2+β1+1)/2 ( \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (β14+β11β10+2β9+2β8β7β5+5β4+β13)/2 ( -\beta_{14} + \beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + \beta _1 - 3 ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (7β13+7β12+2β11+2β10+32β1)/2 ( -7\beta_{13} + 7\beta_{12} + 2\beta_{11} + 2\beta_{10} + 32\beta_1 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (5β14+β13+β127β11+7β10+13β9+13β8+22)/2 ( - 5 \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + \cdots - 22 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (11β13+11β12+45β11+45β10+11β911β8+68)/2 ( - 11 \beta_{13} + 11 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots - 68 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (31β152β14+87β13+87β1267β11+67β10+58)/2 ( - 31 \beta_{15} - 2 \beta_{14} + 87 \beta_{13} + 87 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + \cdots - 58 ) / 2 Copy content Toggle raw display
ν8\nu^{8}== (289β9289β8190β7+190β556β21541)/2 ( 289\beta_{9} - 289\beta_{8} - 190\beta_{7} + 190\beta_{5} - 56\beta_{2} - 1541 ) / 2 Copy content Toggle raw display
ν9\nu^{9}== (205β15+28β14+589β13+589β12479β11+479β10++412)/2 ( - 205 \beta_{15} + 28 \beta_{14} + 589 \beta_{13} + 589 \beta_{12} - 479 \beta_{11} + 479 \beta_{10} + \cdots + 412 ) / 2 Copy content Toggle raw display
ν10\nu^{10}== (753β13753β121877β111877β10+753β9753β8+4274)/2 ( 753 \beta_{13} - 753 \beta_{12} - 1877 \beta_{11} - 1877 \beta_{10} + 753 \beta_{9} - 753 \beta_{8} + \cdots - 4274 ) / 2 Copy content Toggle raw display
ν11\nu^{11}== (274β15+1383β14+1027β13+1027β121877β11+1877β10++7396)/2 ( - 274 \beta_{15} + 1383 \beta_{14} + 1027 \beta_{13} + 1027 \beta_{12} - 1877 \beta_{11} + 1877 \beta_{10} + \cdots + 7396 ) / 2 Copy content Toggle raw display
ν12\nu^{12}== (12341β1312341β1211422β1111422β10+4656β653908β1)/2 ( 12341\beta_{13} - 12341\beta_{12} - 11422\beta_{11} - 11422\beta_{10} + 4656\beta_{6} - 53908\beta_1 ) / 2 Copy content Toggle raw display
ν13\nu^{13}== (2328β15+9409β148039β138039β12+12341β1112341β10++51224)/2 ( 2328 \beta_{15} + 9409 \beta_{14} - 8039 \beta_{13} - 8039 \beta_{12} + 12341 \beta_{11} - 12341 \beta_{10} + \cdots + 51224 ) / 2 Copy content Toggle raw display
ν14\nu^{14}== (42169β1342169β1282029β1182029β1042169β9++234634)/2 ( 42169 \beta_{13} - 42169 \beta_{12} - 82029 \beta_{11} - 82029 \beta_{10} - 42169 \beta_{9} + \cdots + 234634 ) / 2 Copy content Toggle raw display
ν15\nu^{15}== (64331β15+18406β14188529β13188529β12+166367β11++142604)/2 ( 64331 \beta_{15} + 18406 \beta_{14} - 188529 \beta_{13} - 188529 \beta_{12} + 166367 \beta_{11} + \cdots + 142604 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1386Z)×\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times.

nn 155155 199199 11351135
χ(n)\chi(n) 1-1 1-1 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
881.1
−0.876932 2.11710i
2.43348 1.00798i
−0.221383 0.534465i
1.12256 0.464978i
−1.12256 + 0.464978i
0.221383 + 0.534465i
−2.43348 + 1.00798i
0.876932 + 2.11710i
−0.876932 + 2.11710i
2.43348 + 1.00798i
−0.221383 + 0.534465i
1.12256 + 0.464978i
−1.12256 0.464978i
0.221383 0.534465i
−2.43348 1.00798i
0.876932 2.11710i
1.00000i 0 −1.00000 −4.23420 0 −2.59178 + 0.531652i 1.00000i 0 4.23420i
881.2 1.00000i 0 −1.00000 −2.01596 0 1.78450 + 1.95335i 1.00000i 0 2.01596i
881.3 1.00000i 0 −1.00000 −1.06893 0 0.884677 2.49346i 1.00000i 0 1.06893i
881.4 1.00000i 0 −1.00000 −0.929956 0 −2.07739 1.63843i 1.00000i 0 0.929956i
881.5 1.00000i 0 −1.00000 0.929956 0 −2.07739 + 1.63843i 1.00000i 0 0.929956i
881.6 1.00000i 0 −1.00000 1.06893 0 0.884677 + 2.49346i 1.00000i 0 1.06893i
881.7 1.00000i 0 −1.00000 2.01596 0 1.78450 1.95335i 1.00000i 0 2.01596i
881.8 1.00000i 0 −1.00000 4.23420 0 −2.59178 0.531652i 1.00000i 0 4.23420i
881.9 1.00000i 0 −1.00000 −4.23420 0 −2.59178 0.531652i 1.00000i 0 4.23420i
881.10 1.00000i 0 −1.00000 −2.01596 0 1.78450 1.95335i 1.00000i 0 2.01596i
881.11 1.00000i 0 −1.00000 −1.06893 0 0.884677 + 2.49346i 1.00000i 0 1.06893i
881.12 1.00000i 0 −1.00000 −0.929956 0 −2.07739 + 1.63843i 1.00000i 0 0.929956i
881.13 1.00000i 0 −1.00000 0.929956 0 −2.07739 1.63843i 1.00000i 0 0.929956i
881.14 1.00000i 0 −1.00000 1.06893 0 0.884677 2.49346i 1.00000i 0 1.06893i
881.15 1.00000i 0 −1.00000 2.01596 0 1.78450 + 1.95335i 1.00000i 0 2.01596i
881.16 1.00000i 0 −1.00000 4.23420 0 −2.59178 + 0.531652i 1.00000i 0 4.23420i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.g.a 16
3.b odd 2 1 inner 1386.2.g.a 16
7.b odd 2 1 inner 1386.2.g.a 16
21.c even 2 1 inner 1386.2.g.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.g.a 16 1.a even 1 1 trivial
1386.2.g.a 16 3.b odd 2 1 inner
1386.2.g.a 16 7.b odd 2 1 inner
1386.2.g.a 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5824T56+118T54168T52+72 T_{5}^{8} - 24T_{5}^{6} + 118T_{5}^{4} - 168T_{5}^{2} + 72 acting on S2new(1386,[χ])S_{2}^{\mathrm{new}}(1386, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+1)8 (T^{2} + 1)^{8} Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 (T824T6++72)2 (T^{8} - 24 T^{6} + \cdots + 72)^{2} Copy content Toggle raw display
77 (T8+4T7++2401)2 (T^{8} + 4 T^{7} + \cdots + 2401)^{2} Copy content Toggle raw display
1111 (T2+1)8 (T^{2} + 1)^{8} Copy content Toggle raw display
1313 T16 T^{16} Copy content Toggle raw display
1717 (T884T6++83232)2 (T^{8} - 84 T^{6} + \cdots + 83232)^{2} Copy content Toggle raw display
1919 (T8+104T6++72)2 (T^{8} + 104 T^{6} + \cdots + 72)^{2} Copy content Toggle raw display
2323 (T8+120T6++331776)2 (T^{8} + 120 T^{6} + \cdots + 331776)^{2} Copy content Toggle raw display
2929 (T8+108T6++5184)2 (T^{8} + 108 T^{6} + \cdots + 5184)^{2} Copy content Toggle raw display
3131 (T8+268T6++6139008)2 (T^{8} + 268 T^{6} + \cdots + 6139008)^{2} Copy content Toggle raw display
3737 (T4+4T3++544)4 (T^{4} + 4 T^{3} + \cdots + 544)^{4} Copy content Toggle raw display
4141 (T8132T6++23328)2 (T^{8} - 132 T^{6} + \cdots + 23328)^{2} Copy content Toggle raw display
4343 (T412T3+16)4 (T^{4} - 12 T^{3} + \cdots - 16)^{4} Copy content Toggle raw display
4747 (T8196T6++288)2 (T^{8} - 196 T^{6} + \cdots + 288)^{2} Copy content Toggle raw display
5353 (T8+260T6++8714304)2 (T^{8} + 260 T^{6} + \cdots + 8714304)^{2} Copy content Toggle raw display
5959 (T896T6++18432)2 (T^{8} - 96 T^{6} + \cdots + 18432)^{2} Copy content Toggle raw display
6161 (T8+528T6++294912)2 (T^{8} + 528 T^{6} + \cdots + 294912)^{2} Copy content Toggle raw display
6767 (T44T370T2++16)4 (T^{4} - 4 T^{3} - 70 T^{2} + \cdots + 16)^{4} Copy content Toggle raw display
7171 (T8+312T6++82944)2 (T^{8} + 312 T^{6} + \cdots + 82944)^{2} Copy content Toggle raw display
7373 (T8+196T6++288)2 (T^{8} + 196 T^{6} + \cdots + 288)^{2} Copy content Toggle raw display
7979 (T4+4T3+292)4 (T^{4} + 4 T^{3} + \cdots - 292)^{4} Copy content Toggle raw display
8383 (T8348T6++1492992)2 (T^{8} - 348 T^{6} + \cdots + 1492992)^{2} Copy content Toggle raw display
8989 (T8420T6++1936512)2 (T^{8} - 420 T^{6} + \cdots + 1936512)^{2} Copy content Toggle raw display
9797 (T8+480T6++18874368)2 (T^{8} + 480 T^{6} + \cdots + 18874368)^{2} Copy content Toggle raw display
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