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Properties

Label 945.2.bj.k
Level $945$
Weight $2$
Character orbit 945.bj
Analytic conductor $7.546$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(26,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bj (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 28 x^{18} + 322 x^{16} + 1978 x^{14} + 7075 x^{12} + 15064 x^{10} + 18679 x^{8} + 12544 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_1) q^{2} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{4} - \beta_{4} q^{5} + \beta_{12} q^{7} + (\beta_{16} + \beta_{14} - \beta_{13} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_1) q^{2} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{4} - \beta_{4} q^{5} + \beta_{12} q^{7} + (\beta_{16} + \beta_{14} - \beta_{13} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{19} + \beta_{18} + 3 \beta_{16} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 8 q^{4} - 10 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 8 q^{4} - 10 q^{5} + 6 q^{7} + 6 q^{11} + 12 q^{14} - 12 q^{16} - 6 q^{19} - 16 q^{20} + 28 q^{22} + 24 q^{23} - 10 q^{25} - 12 q^{26} + 10 q^{28} + 18 q^{31} - 60 q^{32} - 24 q^{38} + 12 q^{41} - 36 q^{43} + 60 q^{44} + 4 q^{46} - 12 q^{47} - 22 q^{49} - 78 q^{52} + 24 q^{53} + 66 q^{56} + 16 q^{58} - 6 q^{61} - 72 q^{62} - 60 q^{64} - 24 q^{65} - 20 q^{67} + 12 q^{68} + 12 q^{70} + 42 q^{73} + 42 q^{74} - 36 q^{77} + 10 q^{79} - 12 q^{80} + 72 q^{82} + 48 q^{83} - 42 q^{86} + 44 q^{88} + 12 q^{89} - 24 q^{91} + 78 q^{94} + 6 q^{95} - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 28 x^{18} + 322 x^{16} + 1978 x^{14} + 7075 x^{12} + 15064 x^{10} + 18679 x^{8} + 12544 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{19} - 24 \nu^{17} - 219 \nu^{15} - 922 \nu^{13} - 1545 \nu^{11} + 738 \nu^{9} + 5777 \nu^{7} + \cdots - 36 ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{19} + 51 \nu^{17} + 515 \nu^{15} + 2630 \nu^{13} + 7182 \nu^{11} + 10127 \nu^{9} + 6239 \nu^{7} + \cdots + 12 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{19} - 22 \nu^{18} + 5 \nu^{17} - 561 \nu^{16} + 1324 \nu^{15} - 5665 \nu^{14} + 17502 \nu^{13} + \cdots + 1056 ) / 264 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 64 \nu^{19} + 27 \nu^{18} - 1789 \nu^{17} + 780 \nu^{16} - 20503 \nu^{15} + 9303 \nu^{14} + \cdots + 4293 ) / 792 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{19} - 22 \nu^{18} - 5 \nu^{17} - 561 \nu^{16} - 1324 \nu^{15} - 5665 \nu^{14} - 17502 \nu^{13} + \cdots + 1056 ) / 264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{18} - 129 \nu^{16} - 1326 \nu^{14} - 6968 \nu^{12} - 20001 \nu^{10} - 31119 \nu^{8} + \cdots - 18 ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 64 \nu^{19} - 129 \nu^{18} + 1789 \nu^{17} - 3393 \nu^{16} + 20503 \nu^{15} - 35853 \nu^{14} + \cdots - 3384 ) / 792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13 \nu^{19} - 228 \nu^{18} + 400 \nu^{17} - 5835 \nu^{16} + 5116 \nu^{15} - 59250 \nu^{14} + \cdots + 2457 ) / 792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37 \nu^{19} + 48 \nu^{18} - 1031 \nu^{17} + 1262 \nu^{16} - 11783 \nu^{15} + 13334 \nu^{14} + \cdots + 669 ) / 264 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64 \nu^{19} + 129 \nu^{18} + 1789 \nu^{17} + 3393 \nu^{16} + 20503 \nu^{15} + 35853 \nu^{14} + \cdots + 3384 ) / 792 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 91 \nu^{19} - 126 \nu^{18} - 2569 \nu^{17} - 3288 \nu^{16} - 29806 \nu^{15} - 34383 \nu^{14} + \cdots - 2016 ) / 792 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14 \nu^{19} + 294 \nu^{18} - 380 \nu^{17} + 7584 \nu^{16} - 4187 \nu^{15} + 77928 \nu^{14} + \cdots + 1206 ) / 792 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14 \nu^{19} + 294 \nu^{18} + 380 \nu^{17} + 7584 \nu^{16} + 4187 \nu^{15} + 77928 \nu^{14} + \cdots + 1206 ) / 792 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 22 \nu^{19} - 327 \nu^{18} - 385 \nu^{17} - 8409 \nu^{16} - 1144 \nu^{15} - 86013 \nu^{14} + \cdots - 1305 ) / 792 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 40 \nu^{19} - 81 \nu^{18} - 1026 \nu^{17} - 2120 \nu^{16} - 10459 \nu^{15} - 22255 \nu^{14} + \cdots - 1923 ) / 264 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 59 \nu^{19} + 60 \nu^{18} + 1559 \nu^{17} + 1550 \nu^{16} + 16590 \nu^{15} + 15969 \nu^{14} + \cdots + 531 ) / 264 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 136 \nu^{19} - 291 \nu^{18} - 3682 \nu^{17} - 7545 \nu^{16} - 40504 \nu^{15} - 78108 \nu^{14} + \cdots - 1323 ) / 792 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{9} - \beta_{7} + \beta_{6} - 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} + \beta_{12} - \beta_{9} + \beta_{8} + \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{16} + 9 \beta_{14} - 9 \beta_{13} - 8 \beta_{12} + 2 \beta_{10} - 8 \beta_{9} + 2 \beta_{8} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{19} - 12 \beta_{18} - 10 \beta_{17} + 12 \beta_{16} - \beta_{15} - \beta_{13} - 11 \beta_{12} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{17} - 45 \beta_{16} - \beta_{15} - 68 \beta_{14} + 69 \beta_{13} + 56 \beta_{12} + \beta_{11} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 111 \beta_{19} + 111 \beta_{18} + 79 \beta_{17} - 115 \beta_{16} + 14 \beta_{15} - 3 \beta_{14} + \cdots + 461 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4 \beta_{19} + 4 \beta_{18} - 16 \beta_{17} + 300 \beta_{16} + 14 \beta_{15} + 500 \beta_{14} + \cdots - 203 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 945 \beta_{19} - 945 \beta_{18} - 591 \beta_{17} + 1021 \beta_{16} - 133 \beta_{15} + 62 \beta_{14} + \cdots - 2897 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 80 \beta_{19} - 80 \beta_{18} + 169 \beta_{17} - 2091 \beta_{16} - 121 \beta_{15} - 3696 \beta_{14} + \cdots + 884 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7762 \beta_{19} + 7762 \beta_{18} + 4380 \beta_{17} - 8714 \beta_{16} + 1083 \beta_{15} - 831 \beta_{14} + \cdots + 18783 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1032 \beta_{19} + 1032 \beta_{18} - 1509 \beta_{17} + 15069 \beta_{16} + 799 \beta_{15} + 27626 \beta_{14} + \cdots - 2643 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 62523 \beta_{19} - 62523 \beta_{18} - 32563 \beta_{17} + 72475 \beta_{16} - 8186 \beta_{15} + \cdots - 124649 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10984 \beta_{19} - 10984 \beta_{18} + 12388 \beta_{17} - 111017 \beta_{16} - 4034 \beta_{15} + \cdots - 4634 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 497330 \beta_{19} + 497330 \beta_{18} + 243488 \beta_{17} - 591534 \beta_{16} + 59557 \beta_{15} + \cdots + 843366 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 105188 \beta_{19} + 105188 \beta_{18} - 97165 \beta_{17} + 829361 \beta_{16} + 11029 \beta_{15} + \cdots + 184238 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 3920689 \beta_{19} - 3920689 \beta_{18} - 1830509 \beta_{17} + 4759469 \beta_{16} - 425120 \beta_{15} + \cdots - 5804272 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 943968 \beta_{19} - 943968 \beta_{18} + 743150 \beta_{17} - 6249825 \beta_{16} + 73572 \beta_{15} + \cdots - 2298238 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(\beta_{4}\) \(-1\) \(1\)

Embeddings

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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 2.46131i
− 2.20724i
− 1.46961i
− 1.13389i
0.161835i
0.414428i
0.682882i
1.48084i
1.76719i
2.76486i
2.46131i
2.20724i
1.46961i
1.13389i
− 0.161835i
− 0.414428i
− 0.682882i
− 1.48084i
− 1.76719i
− 2.76486i
−2.13155 − 1.23065i 0 2.02902 + 3.51436i −0.500000 + 0.866025i 0 −1.36864 + 2.26425i − 5.06545i 0 2.13155 − 1.23065i
26.2 −1.91153 − 1.10362i 0 1.43595 + 2.48714i −0.500000 + 0.866025i 0 2.64028 + 0.169989i − 1.92450i 0 1.91153 − 1.10362i
26.3 −1.27272 − 0.734807i 0 0.0798818 + 0.138359i −0.500000 + 0.866025i 0 0.911101 − 2.48393i 2.70444i 0 1.27272 − 0.734807i
26.4 −0.981973 − 0.566943i 0 −0.357152 − 0.618606i −0.500000 + 0.866025i 0 −1.76643 + 1.96971i 3.07771i 0 0.981973 − 0.566943i
26.5 0.140153 + 0.0809175i 0 −0.986905 − 1.70937i −0.500000 + 0.866025i 0 2.13574 + 1.56161i − 0.643102i 0 −0.140153 + 0.0809175i
26.6 0.358905 + 0.207214i 0 −0.914125 − 1.58331i −0.500000 + 0.866025i 0 −1.21855 − 2.34843i − 1.58653i 0 −0.358905 + 0.207214i
26.7 0.591393 + 0.341441i 0 −0.766836 − 1.32820i −0.500000 + 0.866025i 0 2.37102 − 1.17399i − 2.41308i 0 −0.591393 + 0.341441i
26.8 1.28245 + 0.740422i 0 0.0964494 + 0.167055i −0.500000 + 0.866025i 0 −0.498659 + 2.59833i − 2.67603i 0 −1.28245 + 0.740422i
26.9 1.53044 + 0.883597i 0 0.561489 + 0.972527i −0.500000 + 0.866025i 0 −1.64688 − 2.07070i − 1.54987i 0 −1.53044 + 0.883597i
26.10 2.39444 + 1.38243i 0 2.82223 + 4.88825i −0.500000 + 0.866025i 0 1.44101 − 2.21890i 10.0764i 0 −2.39444 + 1.38243i
836.1 −2.13155 + 1.23065i 0 2.02902 − 3.51436i −0.500000 − 0.866025i 0 −1.36864 − 2.26425i 5.06545i 0 2.13155 + 1.23065i
836.2 −1.91153 + 1.10362i 0 1.43595 − 2.48714i −0.500000 − 0.866025i 0 2.64028 − 0.169989i 1.92450i 0 1.91153 + 1.10362i
836.3 −1.27272 + 0.734807i 0 0.0798818 − 0.138359i −0.500000 − 0.866025i 0 0.911101 + 2.48393i − 2.70444i 0 1.27272 + 0.734807i
836.4 −0.981973 + 0.566943i 0 −0.357152 + 0.618606i −0.500000 − 0.866025i 0 −1.76643 − 1.96971i − 3.07771i 0 0.981973 + 0.566943i
836.5 0.140153 − 0.0809175i 0 −0.986905 + 1.70937i −0.500000 − 0.866025i 0 2.13574 − 1.56161i 0.643102i 0 −0.140153 − 0.0809175i
836.6 0.358905 − 0.207214i 0 −0.914125 + 1.58331i −0.500000 − 0.866025i 0 −1.21855 + 2.34843i 1.58653i 0 −0.358905 − 0.207214i
836.7 0.591393 − 0.341441i 0 −0.766836 + 1.32820i −0.500000 − 0.866025i 0 2.37102 + 1.17399i 2.41308i 0 −0.591393 − 0.341441i
836.8 1.28245 − 0.740422i 0 0.0964494 − 0.167055i −0.500000 − 0.866025i 0 −0.498659 − 2.59833i 2.67603i 0 −1.28245 − 0.740422i
836.9 1.53044 − 0.883597i 0 0.561489 − 0.972527i −0.500000 − 0.866025i 0 −1.64688 + 2.07070i 1.54987i 0 −1.53044 − 0.883597i
836.10 2.39444 − 1.38243i 0 2.82223 − 4.88825i −0.500000 − 0.866025i 0 1.44101 + 2.21890i − 10.0764i 0 −2.39444 − 1.38243i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bj.k ✓ 20
3.b odd 2 1 945.2.bj.l yes 20
7.d odd 6 1 945.2.bj.l yes 20
21.g even 6 1 inner 945.2.bj.k ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bj.k ✓ 20 1.a even 1 1 trivial
945.2.bj.k ✓ 20 21.g even 6 1 inner
945.2.bj.l yes 20 3.b odd 2 1
945.2.bj.l yes 20 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{20} - 14 T_{2}^{18} + 133 T_{2}^{16} + 12 T_{2}^{15} - 668 T_{2}^{14} - 48 T_{2}^{13} + 2416 T_{2}^{12} + \cdots + 9 \) Copy content Toggle raw display
\( T_{11}^{20} - 6 T_{11}^{19} - 29 T_{11}^{18} + 246 T_{11}^{17} + 823 T_{11}^{16} - 5430 T_{11}^{15} + \cdots + 9566649 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 14 T^{18} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} - 6 T^{19} + \cdots + 9566649 \) Copy content Toggle raw display
$13$ \( T^{20} + 144 T^{18} + \cdots + 42341049 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 5046255369 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 1039586199201 \) Copy content Toggle raw display
$23$ \( T^{20} - 24 T^{19} + \cdots + 1879641 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 2182926465729 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 817094868489 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 36877065877201 \) Copy content Toggle raw display
$41$ \( (T^{10} - 6 T^{9} + \cdots + 10961973)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 18 T^{9} + \cdots + 182547013)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 5541038231481 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 839702820609 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 7057295094249 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 18870136176529 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( (T^{10} - 24 T^{9} + \cdots + 29133513)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 97\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10583329759209 \) Copy content Toggle raw display
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