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Properties

Label 765.2.n.d
Level $765$
Weight $2$
Character orbit 765.n
Analytic conductor $6.109$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [765,2,Mod(188,765)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(765, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("765.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.n (of order \(4\), 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: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 5x^{12} + 4x^{10} - 20x^{8} + 16x^{6} + 80x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + (\beta_{13} - \beta_{7} + \beta_1) q^{4} + ( - \beta_{14} - \beta_{11}) q^{5} + ( - \beta_{13} + \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{14} + \beta_{12} + \cdots + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + (\beta_{13} - \beta_{7} + \beta_1) q^{4} + ( - \beta_{14} - \beta_{11}) q^{5} + ( - \beta_{13} + \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + ( - 5 \beta_{14} - 6 \beta_{12} + \cdots - 6 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} + 44 q^{10} + 4 q^{13} - 24 q^{16} + 8 q^{22} - 16 q^{25} + 112 q^{28} - 24 q^{31} - 8 q^{37} + 40 q^{40} - 4 q^{43} - 40 q^{46} - 72 q^{52} + 48 q^{55} - 8 q^{58} - 120 q^{61} - 32 q^{67} + 40 q^{70} - 32 q^{73} - 128 q^{76} - 104 q^{82} - 8 q^{88} + 136 q^{91} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 5x^{12} + 4x^{10} - 20x^{8} + 16x^{6} + 80x^{4} - 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 4\nu^{12} + 23\nu^{10} - 20\nu^{8} + 264\nu^{4} + 32\nu^{2} - 128 ) / 640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} - \nu^{12} - 13\nu^{10} - 5\nu^{8} + 50\nu^{6} - 4\nu^{4} + 48\nu^{2} + 128 ) / 320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 9\nu^{13} + 7\nu^{11} - 35\nu^{9} - 10\nu^{7} + 36\nu^{5} - 112\nu^{3} + 128\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{14} - 7\nu^{12} - \nu^{10} + 5\nu^{8} - 30\nu^{6} + 92\nu^{4} - 64\nu^{2} - 384 ) / 320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{14} - 2\nu^{12} + 9\nu^{10} - 10\nu^{8} + 20\nu^{6} - 48\nu^{4} - 224\nu^{2} - 64 ) / 320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{15} + 11\nu^{13} - 22\nu^{11} + 15\nu^{9} + 30\nu^{7} - 36\nu^{5} + 32\nu^{3} + 512\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{14} + 9\nu^{12} + 2\nu^{10} - 35\nu^{8} + 30\nu^{6} - 4\nu^{4} - 272\nu^{2} + 448 ) / 320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -13\nu^{15} + 18\nu^{13} - \nu^{11} - 70\nu^{9} + 100\nu^{7} + 272\nu^{5} - 1024\nu^{3} + 896\nu ) / 1280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -13\nu^{15} + 38\nu^{13} - \nu^{11} - 50\nu^{9} + 100\nu^{7} + 112\nu^{5} - 864\nu^{3} + 1536\nu ) / 1280 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7\nu^{15} - 12\nu^{13} - \nu^{11} + 20\nu^{9} - 8\nu^{5} + 576\nu^{3} - 64\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -8\nu^{15} + 23\nu^{13} - 6\nu^{11} - 45\nu^{9} + 170\nu^{7} + 52\nu^{5} - 784\nu^{3} + 896\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -17\nu^{15} + 52\nu^{13} - 9\nu^{11} - 60\nu^{9} + 240\nu^{7} - 152\nu^{5} - 1696\nu^{3} + 3584\nu ) / 1280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\nu^{14} - 54\nu^{12} + 23\nu^{10} + 130\nu^{8} - 260\nu^{6} - 96\nu^{4} + 1952\nu^{2} - 3328 ) / 640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -13\nu^{15} + 38\nu^{13} - 21\nu^{11} - 90\nu^{9} + 160\nu^{7} - 88\nu^{5} - 944\nu^{3} + 2176\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3\nu^{14} - 7\nu^{12} + \nu^{10} + 13\nu^{8} - 34\nu^{6} - 12\nu^{4} + 208\nu^{2} - 384 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} + 2\beta_{13} + 2\beta_{7} - \beta_{5} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - 2\beta_{12} + 2\beta_{10} + 4\beta_{9} - 2\beta_{8} - \beta_{6} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{13} - 2\beta_{7} - 2\beta_{5} + \beta_{4} + 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{14} - \beta_{12} - \beta_{11} + 3\beta_{10} + 4\beta_{9} + 7\beta_{8} + 2\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{15} + 2\beta_{13} - 6\beta_{7} + 5\beta_{5} - 4\beta_{4} + 7\beta_{2} + 4\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{14} - 2\beta_{12} + 12\beta_{11} + 6\beta_{10} - 6\beta_{8} - 3\beta_{6} - 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -8\beta_{15} + \beta_{13} - 30\beta_{7} + 2\beta_{5} - \beta_{4} - 8\beta_{2} - 3\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -14\beta_{14} + 9\beta_{12} + \beta_{11} - 3\beta_{10} + 20\beta_{9} - 7\beta_{8} + 6\beta_{6} - 16\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -5\beta_{15} + 6\beta_{13} - 10\beta_{7} + 19\beta_{5} - 20\beta_{4} - 15\beta_{2} + 20\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23 \beta_{14} + 26 \beta_{12} + 12 \beta_{11} + 18 \beta_{10} + 32 \beta_{9} - 34 \beta_{8} + \cdots + 27 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -16\beta_{15} + 7\beta_{13} - 2\beta_{7} - 42\beta_{5} - 23\beta_{4} - 32\beta_{2} - 5\beta _1 - 81 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 10 \beta_{14} - 33 \beta_{12} + 23 \beta_{11} - 21 \beta_{10} + 108 \beta_{9} - 81 \beta_{8} + \cdots + 24 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 77\beta_{15} - 110\beta_{13} - 86\beta_{7} + 5\beta_{5} - 84\beta_{4} - 25\beta_{2} + 52\beta _1 - 69 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 65 \beta_{14} + 94 \beta_{12} + 28 \beta_{11} + 6 \beta_{10} - 192 \beta_{9} + 58 \beta_{8} + \cdots + 173 \beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/765\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
1.33768 + 0.458926i
1.39770 + 0.215490i
−0.310269 + 1.37976i
1.07740 − 0.916088i
−1.07740 + 0.916088i
0.310269 − 1.37976i
−1.39770 − 0.215490i
−1.33768 − 0.458926i
1.33768 − 0.458926i
1.39770 − 0.215490i
−0.310269 − 1.37976i
1.07740 + 0.916088i
−1.07740 − 0.916088i
0.310269 + 1.37976i
−1.39770 + 0.215490i
−1.33768 + 0.458926i
−1.79661 − 1.79661i 0 4.45559i 0.218058 + 2.22541i 0 3.45559 − 3.45559i 4.41172 − 4.41172i 0 3.60642 − 4.38995i
188.2 −1.61319 − 1.61319i 0 3.20476i −2.15514 + 0.596138i 0 2.20476 − 2.20476i 1.94351 − 1.94351i 0 4.43833 + 2.51496i
188.3 −1.06949 − 1.06949i 0 0.287614i −1.81282 + 1.30908i 0 −0.712386 + 0.712386i −1.83138 + 1.83138i 0 3.33884 + 0.538737i
188.4 −0.161308 − 0.161308i 0 − 1.94796i 0.146727 − 2.23125i 0 −2.94796 + 2.94796i −0.636837 + 0.636837i 0 −0.383586 + 0.336250i
188.5 0.161308 + 0.161308i 0 − 1.94796i −0.146727 + 2.23125i 0 −2.94796 + 2.94796i 0.636837 − 0.636837i 0 −0.383586 + 0.336250i
188.6 1.06949 + 1.06949i 0 0.287614i 1.81282 − 1.30908i 0 −0.712386 + 0.712386i 1.83138 − 1.83138i 0 3.33884 + 0.538737i
188.7 1.61319 + 1.61319i 0 3.20476i 2.15514 − 0.596138i 0 2.20476 − 2.20476i −1.94351 + 1.94351i 0 4.43833 + 2.51496i
188.8 1.79661 + 1.79661i 0 4.45559i −0.218058 − 2.22541i 0 3.45559 − 3.45559i −4.41172 + 4.41172i 0 3.60642 − 4.38995i
647.1 −1.79661 + 1.79661i 0 − 4.45559i 0.218058 − 2.22541i 0 3.45559 + 3.45559i 4.41172 + 4.41172i 0 3.60642 + 4.38995i
647.2 −1.61319 + 1.61319i 0 − 3.20476i −2.15514 − 0.596138i 0 2.20476 + 2.20476i 1.94351 + 1.94351i 0 4.43833 − 2.51496i
647.3 −1.06949 + 1.06949i 0 − 0.287614i −1.81282 − 1.30908i 0 −0.712386 − 0.712386i −1.83138 − 1.83138i 0 3.33884 − 0.538737i
647.4 −0.161308 + 0.161308i 0 1.94796i 0.146727 + 2.23125i 0 −2.94796 − 2.94796i −0.636837 − 0.636837i 0 −0.383586 − 0.336250i
647.5 0.161308 − 0.161308i 0 1.94796i −0.146727 − 2.23125i 0 −2.94796 − 2.94796i 0.636837 + 0.636837i 0 −0.383586 − 0.336250i
647.6 1.06949 − 1.06949i 0 − 0.287614i 1.81282 + 1.30908i 0 −0.712386 − 0.712386i 1.83138 + 1.83138i 0 3.33884 − 0.538737i
647.7 1.61319 − 1.61319i 0 − 3.20476i 2.15514 + 0.596138i 0 2.20476 + 2.20476i −1.94351 − 1.94351i 0 4.43833 − 2.51496i
647.8 1.79661 − 1.79661i 0 − 4.45559i −0.218058 + 2.22541i 0 3.45559 + 3.45559i −4.41172 − 4.41172i 0 3.60642 + 4.38995i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.2.n.d ✓ 16
3.b odd 2 1 inner 765.2.n.d ✓ 16
5.c odd 4 1 inner 765.2.n.d ✓ 16
15.e even 4 1 inner 765.2.n.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
765.2.n.d ✓ 16 1.a even 1 1 trivial
765.2.n.d ✓ 16 3.b odd 2 1 inner
765.2.n.d ✓ 16 5.c odd 4 1 inner
765.2.n.d ✓ 16 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 74T_{2}^{12} + 1489T_{2}^{8} + 5912T_{2}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 74 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 28 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 110 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1065552449536 \) Copy content Toggle raw display
$29$ \( (T^{8} - 80 T^{6} + \cdots + 29584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots + 2840)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 1263376)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 328 T^{6} + \cdots + 198916)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 32319410176 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2897022976 \) Copy content Toggle raw display
$59$ \( (T^{2} - 72)^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 30 T^{3} + \cdots - 800)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 180 T^{6} + \cdots + 322624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 1607824)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 272 T^{6} + \cdots + 1149184)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2897022976 \) Copy content Toggle raw display
$89$ \( (T^{8} - 112 T^{6} + \cdots + 110224)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 16 T^{7} + \cdots + 46566976)^{2} \) Copy content Toggle raw display
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