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Properties

Label 72.2.n.b
Level $72$
Weight $2$
Character orbit 72.n
Analytic conductor $0.575$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,2,Mod(13,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.n (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: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{10} - \beta_{3}) q^{3} + ( - \beta_{15} - \beta_{13} + \beta_{8}) q^{4} + (\beta_{13} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{8} + \beta_1 - 1) q^{6} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - \beta_{2} - 1) q^{8} + (\beta_{11} - \beta_{9} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{10} - \beta_{3}) q^{3} + ( - \beta_{15} - \beta_{13} + \beta_{8}) q^{4} + (\beta_{13} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{8} + \beta_1 - 1) q^{6} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - \beta_{2} - 1) q^{8} + (\beta_{11} - \beta_{9} + 1) q^{9} + (\beta_{15} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{7} + \beta_{3} - \beta_1) q^{10} + ( - \beta_{13} - \beta_{12} + \beta_{2}) q^{11} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5}) q^{13} + (\beta_{15} + \beta_{13} - \beta_{8} - \beta_{5} + 2 \beta_{4} + 2) q^{14} + ( - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + \cdots - 1) q^{15}+ \cdots + ( - 4 \beta_{15} - 4 \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - q^{4} - 7 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} - q^{4} - 7 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 16 q^{10} - 16 q^{12} + 16 q^{14} - 10 q^{15} - 9 q^{16} - 28 q^{17} + 4 q^{18} - 8 q^{20} + q^{22} - 10 q^{23} + 7 q^{24} + 2 q^{25} + 28 q^{26} + 4 q^{28} + 22 q^{30} - 10 q^{31} + 11 q^{32} + q^{34} + 27 q^{36} + 23 q^{38} + 2 q^{39} + 6 q^{40} - 8 q^{41} + 8 q^{42} + 18 q^{44} - 20 q^{46} + 6 q^{47} + 39 q^{48} + 18 q^{49} - 23 q^{50} - 8 q^{52} - 29 q^{54} - 4 q^{55} + 10 q^{56} + 10 q^{57} - 14 q^{58} + 6 q^{60} - 52 q^{62} + 2 q^{63} + 26 q^{64} - 14 q^{65} - 72 q^{66} - 39 q^{68} + 72 q^{71} - 77 q^{72} - 44 q^{73} - 38 q^{74} + 5 q^{76} + 10 q^{78} - 30 q^{79} - 96 q^{80} + 10 q^{81} + 38 q^{82} - 28 q^{84} + 7 q^{86} + 42 q^{87} + 31 q^{88} + 64 q^{89} + 64 q^{90} - 30 q^{92} - 12 q^{94} + 44 q^{95} - 26 q^{96} + 66 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 8 \nu^{10} - 16 \nu^{9} - 48 \nu^{8} - 12 \nu^{7} + 16 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} + 256 \nu^{2} + 384 \nu - 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} - \nu^{14} - 3 \nu^{13} - 6 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + 24 \nu^{4} + 80 \nu^{3} + 32 \nu^{2} + 32 \nu + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} - 24 \nu^{6} - 32 \nu^{5} - 72 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 96 \nu - 256 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} + \nu^{14} - \nu^{13} - 2 \nu^{11} + 4 \nu^{10} + 8 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + 32 \nu^{4} - 32 \nu^{3} - 64 \nu + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} - 24 \nu^{8} - 20 \nu^{7} + 64 \nu^{6} + 96 \nu^{5} + 160 \nu^{4} + 256 \nu^{3} + 384 \nu^{2} + 448 \nu + 512 ) / 384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + 4 \nu^{7} + 16 \nu^{6} + 20 \nu^{5} + 64 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 64 \nu + 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + \nu^{14} + 7 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + 20 \nu^{10} + 24 \nu^{9} + 16 \nu^{8} + 20 \nu^{7} - 40 \nu^{6} - 160 \nu^{5} - 272 \nu^{4} - 160 \nu^{3} - 128 \nu^{2} - 384 \nu - 640 ) / 384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 2 \nu^{12} - 10 \nu^{11} - 4 \nu^{10} - 16 \nu^{9} + 36 \nu^{7} + 88 \nu^{6} + 64 \nu^{5} + 112 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} - 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} - \nu^{14} - 7 \nu^{13} - 14 \nu^{12} - 22 \nu^{11} - 20 \nu^{10} - 24 \nu^{9} - 16 \nu^{8} - 20 \nu^{7} + 40 \nu^{6} + 160 \nu^{5} + 272 \nu^{4} + 160 \nu^{3} + 512 \nu^{2} + 384 \nu + 640 ) / 384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} - \nu^{14} + 4 \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} + 8 \nu^{7} + 20 \nu^{5} - 52 \nu^{4} - 56 \nu^{3} - 48 \nu^{2} + 16 \nu - 288 ) / 96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{15} + \nu^{14} - 4 \nu^{11} - \nu^{10} - 4 \nu^{9} + 2 \nu^{8} - 8 \nu^{7} - 20 \nu^{5} + 52 \nu^{4} - 40 \nu^{3} + 48 \nu^{2} - 16 \nu + 192 ) / 96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{15} - \nu^{14} + 9 \nu^{13} + 6 \nu^{12} + 26 \nu^{11} + 12 \nu^{10} + 16 \nu^{9} - 16 \nu^{8} + 44 \nu^{7} - 40 \nu^{6} - 32 \nu^{5} - 240 \nu^{4} - 64 \nu^{3} - 256 \nu^{2} - 128 \nu - 896 ) / 384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} - 8 \nu^{8} - 32 \nu^{7} + 12 \nu^{6} + 64 \nu^{5} + 160 \nu^{4} + 96 \nu^{3} + 256 \nu^{2} + 256 \nu + 704 ) / 192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 9 \nu^{15} + \nu^{14} - 9 \nu^{13} - 8 \nu^{12} - 42 \nu^{11} - 12 \nu^{10} - 24 \nu^{9} + 40 \nu^{8} - 52 \nu^{7} + 48 \nu^{6} + 96 \nu^{5} + 384 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} + 64 \nu + 1664 ) / 384 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{11} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} + 2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{15} + \beta_{13} - 2\beta_{10} + \beta_{9} - \beta_{7} + 5\beta_{6} + 5\beta_{5} - \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - 10 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} - \beta_{9} - 7 \beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{3} + \beta_{2} + 7 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} - 3 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} - 11 \beta_{6} + 6 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7 \beta_{15} - \beta_{14} + 12 \beta_{13} + \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} + 13 \beta_{5} + 26 \beta_{4} + 20 \beta_{3} + \beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9 \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 19 \beta_{9} - 27 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 9 \beta_{3} - 19 \beta_{2} + 27 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14 \beta_{15} - 3 \beta_{14} + 45 \beta_{13} + 13 \beta_{12} - 7 \beta_{11} - 11 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} + 29 \beta_{6} + 14 \beta_{4} - 29 \beta_{3} - 6 \beta_{2} + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 47 \beta_{15} + 31 \beta_{14} - 28 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} + 41 \beta_{10} - 42 \beta_{9} + 58 \beta_{8} - 13 \beta_{7} + 53 \beta_{5} - 78 \beta_{4} - 12 \beta_{3} - 31 \beta_{2} - 78 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 55 \beta_{15} + 12 \beta_{14} - 65 \beta_{13} - 22 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + 19 \beta_{9} + 5 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} - 55 \beta_{3} - 19 \beta_{2} - 5 \beta _1 + 46 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.05026 − 0.947078i
−0.179748 − 1.40274i
1.41411 − 0.0174668i
−1.12494 − 0.857038i
0.820200 + 1.15207i
0.587625 + 1.28635i
−1.34532 + 0.436011i
−0.722180 + 1.21592i
1.05026 + 0.947078i
−0.179748 + 1.40274i
1.41411 + 0.0174668i
−1.12494 + 0.857038i
0.820200 − 1.15207i
0.587625 − 1.28635i
−1.34532 − 0.436011i
−0.722180 − 1.21592i
−1.34532 + 0.436011i 1.52768 + 0.816201i 1.61979 − 1.17315i −0.602794 − 0.348023i −2.41110 − 0.431967i 0.795065 + 1.37709i −1.66763 + 2.28452i 1.66763 + 2.49379i 0.962695 + 0.205379i
13.2 −1.12494 − 0.857038i −0.986088 + 1.42395i 0.530970 + 1.92823i 1.19115 + 0.687709i 2.32967 − 0.756738i 1.80469 + 3.12581i 1.05526 − 2.62420i −1.05526 − 2.80828i −0.750573 − 1.79449i
13.3 −0.722180 + 1.21592i 0.294546 − 1.70682i −0.956913 − 1.75622i 3.17262 + 1.83171i 1.86264 + 1.59078i −0.191926 − 0.332426i 2.82649 + 0.104780i −2.82649 − 1.00547i −4.51841 + 2.53482i
13.4 −0.179748 − 1.40274i 0.986088 − 1.42395i −1.93538 + 0.504281i −1.19115 − 0.687709i −2.17468 − 1.12728i 1.80469 + 3.12581i 1.05526 + 2.62420i −1.05526 − 2.80828i −0.750573 + 1.79449i
13.5 0.587625 + 1.28635i 1.69028 − 0.378078i −1.30939 + 1.51178i −1.97542 − 1.14051i 1.47959 + 1.95213i −0.907824 − 1.57240i −2.71411 − 0.795980i 2.71411 − 1.27812i 0.306290 − 3.21128i
13.6 0.820200 + 1.15207i −1.69028 + 0.378078i −0.654545 + 1.88986i 1.97542 + 1.14051i −1.82194 − 1.63723i −0.907824 − 1.57240i −2.71411 + 0.795980i 2.71411 − 1.27812i 0.306290 + 3.21128i
13.7 1.05026 − 0.947078i −1.52768 − 0.816201i 0.206086 − 1.98935i 0.602794 + 0.348023i −2.37747 + 0.589613i 0.795065 + 1.37709i −1.66763 − 2.28452i 1.66763 + 2.49379i 0.962695 − 0.205379i
13.8 1.41411 − 0.0174668i −0.294546 + 1.70682i 1.99939 − 0.0493999i −3.17262 − 1.83171i −0.386706 + 2.41877i −0.191926 − 0.332426i 2.82649 − 0.104780i −2.82649 − 1.00547i −4.51841 − 2.53482i
61.1 −1.34532 − 0.436011i 1.52768 − 0.816201i 1.61979 + 1.17315i −0.602794 + 0.348023i −2.41110 + 0.431967i 0.795065 − 1.37709i −1.66763 − 2.28452i 1.66763 − 2.49379i 0.962695 − 0.205379i
61.2 −1.12494 + 0.857038i −0.986088 − 1.42395i 0.530970 − 1.92823i 1.19115 − 0.687709i 2.32967 + 0.756738i 1.80469 − 3.12581i 1.05526 + 2.62420i −1.05526 + 2.80828i −0.750573 + 1.79449i
61.3 −0.722180 − 1.21592i 0.294546 + 1.70682i −0.956913 + 1.75622i 3.17262 − 1.83171i 1.86264 − 1.59078i −0.191926 + 0.332426i 2.82649 − 0.104780i −2.82649 + 1.00547i −4.51841 − 2.53482i
61.4 −0.179748 + 1.40274i 0.986088 + 1.42395i −1.93538 − 0.504281i −1.19115 + 0.687709i −2.17468 + 1.12728i 1.80469 − 3.12581i 1.05526 − 2.62420i −1.05526 + 2.80828i −0.750573 − 1.79449i
61.5 0.587625 − 1.28635i 1.69028 + 0.378078i −1.30939 − 1.51178i −1.97542 + 1.14051i 1.47959 − 1.95213i −0.907824 + 1.57240i −2.71411 + 0.795980i 2.71411 + 1.27812i 0.306290 + 3.21128i
61.6 0.820200 − 1.15207i −1.69028 − 0.378078i −0.654545 − 1.88986i 1.97542 − 1.14051i −1.82194 + 1.63723i −0.907824 + 1.57240i −2.71411 − 0.795980i 2.71411 + 1.27812i 0.306290 − 3.21128i
61.7 1.05026 + 0.947078i −1.52768 + 0.816201i 0.206086 + 1.98935i 0.602794 − 0.348023i −2.37747 − 0.589613i 0.795065 − 1.37709i −1.66763 + 2.28452i 1.66763 − 2.49379i 0.962695 + 0.205379i
61.8 1.41411 + 0.0174668i −0.294546 − 1.70682i 1.99939 + 0.0493999i −3.17262 + 1.83171i −0.386706 − 2.41877i −0.191926 + 0.332426i 2.82649 + 0.104780i −2.82649 + 1.00547i −4.51841 + 2.53482i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.n.b ✓ 16
3.b odd 2 1 216.2.n.b 16
4.b odd 2 1 288.2.r.b 16
8.b even 2 1 inner 72.2.n.b ✓ 16
8.d odd 2 1 288.2.r.b 16
9.c even 3 1 inner 72.2.n.b ✓ 16
9.c even 3 1 648.2.d.j 8
9.d odd 6 1 216.2.n.b 16
9.d odd 6 1 648.2.d.k 8
12.b even 2 1 864.2.r.b 16
24.f even 2 1 864.2.r.b 16
24.h odd 2 1 216.2.n.b 16
36.f odd 6 1 288.2.r.b 16
36.f odd 6 1 2592.2.d.j 8
36.h even 6 1 864.2.r.b 16
36.h even 6 1 2592.2.d.k 8
72.j odd 6 1 216.2.n.b 16
72.j odd 6 1 648.2.d.k 8
72.l even 6 1 864.2.r.b 16
72.l even 6 1 2592.2.d.k 8
72.n even 6 1 inner 72.2.n.b ✓ 16
72.n even 6 1 648.2.d.j 8
72.p odd 6 1 288.2.r.b 16
72.p odd 6 1 2592.2.d.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b ✓ 16 1.a even 1 1 trivial
72.2.n.b ✓ 16 8.b even 2 1 inner
72.2.n.b ✓ 16 9.c even 3 1 inner
72.2.n.b ✓ 16 72.n even 6 1 inner
216.2.n.b 16 3.b odd 2 1
216.2.n.b 16 9.d odd 6 1
216.2.n.b 16 24.h odd 2 1
216.2.n.b 16 72.j odd 6 1
288.2.r.b 16 4.b odd 2 1
288.2.r.b 16 8.d odd 2 1
288.2.r.b 16 36.f odd 6 1
288.2.r.b 16 72.p odd 6 1
648.2.d.j 8 9.c even 3 1
648.2.d.j 8 72.n even 6 1
648.2.d.k 8 9.d odd 6 1
648.2.d.k 8 72.j odd 6 1
864.2.r.b 16 12.b even 2 1
864.2.r.b 16 24.f even 2 1
864.2.r.b 16 36.h even 6 1
864.2.r.b 16 72.l even 6 1
2592.2.d.j 8 36.f odd 6 1
2592.2.d.j 8 72.p odd 6 1
2592.2.d.k 8 36.h even 6 1
2592.2.d.k 8 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 21T_{5}^{14} + 326T_{5}^{12} - 2049T_{5}^{10} + 9318T_{5}^{8} - 18357T_{5}^{6} + 26129T_{5}^{4} - 11712T_{5}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{15} + T^{14} + 2 T^{12} - 4 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{14} - 2 T^{12} + 9 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} - 21 T^{14} + 326 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{8} - 3 T^{7} + 14 T^{6} - 3 T^{5} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 40 T^{14} + 1134 T^{12} + \cdots + 10556001 \) Copy content Toggle raw display
$13$ \( T^{16} - 53 T^{14} + 2094 T^{12} + \cdots + 20736 \) Copy content Toggle raw display
$17$ \( (T^{4} + 7 T^{3} - 2 T^{2} - 48 T + 36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 83 T^{6} + 1660 T^{4} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 5 T^{7} + 60 T^{6} + 275 T^{5} + \cdots + 90000)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 109 T^{14} + \cdots + 4032758016 \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{7} + 76 T^{6} + 155 T^{5} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 152 T^{6} + 8080 T^{4} + \cdots + 1327104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + 90 T^{6} + 616 T^{5} + \cdots + 335241)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 20 T^{14} + 294 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( (T^{8} - 3 T^{7} + 90 T^{6} + \cdots + 2178576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 160 T^{6} + 7744 T^{4} + \cdots + 451584)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 108 T^{14} + \cdots + 131079601 \) Copy content Toggle raw display
$61$ \( T^{16} - 165 T^{14} + \cdots + 176319369216 \) Copy content Toggle raw display
$67$ \( T^{16} - 240 T^{14} + \cdots + 9881774573841 \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + 48 T^{2} + 252 T - 864)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11 T^{3} - 14 T^{2} - 84 T + 36)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 15 T^{7} + 236 T^{6} + 1065 T^{5} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 105 T^{14} + \cdots + 31713911056 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + 52 T^{2} + 156 T - 504)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 134 T^{6} + 1560 T^{5} + \cdots + 1324801)^{2} \) Copy content Toggle raw display
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