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Properties

Label 930.2.h.d
Level $930$
Weight $2$
Character orbit 930.h
Analytic conductor $7.426$
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] = [930,2,Mod(371,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.371");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.h (of order \(2\), degree \(1\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} + 36x^{10} - 142x^{8} + 324x^{6} + 486x^{4} - 2916x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{12} q^{3} - q^{4} + \beta_{6} q^{5} + \beta_{5} q^{6} - \beta_{9} q^{7} + \beta_{6} q^{8} - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{12} q^{3} - q^{4} + \beta_{6} q^{5} + \beta_{5} q^{6} - \beta_{9} q^{7} + \beta_{6} q^{8} - \beta_{7} q^{9} + q^{10} + (\beta_{12} - \beta_{11} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{14} + \beta_{13} + \cdots - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 4 q^{7} - 8 q^{9} + 16 q^{10} + 16 q^{16} - 8 q^{18} + 20 q^{19} - 16 q^{25} - 4 q^{28} - 8 q^{31} - 24 q^{33} + 8 q^{36} + 8 q^{39} - 16 q^{40} + 8 q^{45} - 28 q^{49} + 16 q^{51} - 4 q^{63} - 16 q^{64} - 44 q^{66} - 24 q^{67} + 32 q^{69} + 4 q^{70} + 8 q^{72} - 20 q^{76} + 40 q^{78} + 8 q^{81} - 48 q^{82} + 16 q^{87} - 8 q^{90} + 12 q^{93} - 40 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 6x^{12} + 36x^{10} - 142x^{8} + 324x^{6} + 486x^{4} - 2916x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{15} - 101\nu^{13} + 111\nu^{11} - 63\nu^{9} - 4355\nu^{7} + 2187\nu^{5} - 22113\nu^{3} - 57591\nu ) / 69984 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{14} - 101\nu^{12} + 111\nu^{10} - 63\nu^{8} - 4355\nu^{6} + 2187\nu^{4} + 1215\nu^{2} - 80919 ) / 23328 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 4\nu^{13} + 6\nu^{11} + 36\nu^{9} - 142\nu^{7} + 324\nu^{5} + 486\nu^{3} - 2916\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\nu^{14} - 65\nu^{12} - 42\nu^{10} + 369\nu^{8} - 1826\nu^{6} + 711\nu^{4} + 5022\nu^{2} - 45927 ) / 11664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} + 4\nu^{12} - 6\nu^{10} - 36\nu^{8} + 142\nu^{6} - 324\nu^{4} - 486\nu^{2} + 2916 ) / 729 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -23\nu^{14} - 61\nu^{12} - 93\nu^{10} - 207\nu^{8} - 1999\nu^{6} - 2493\nu^{4} - 10125\nu^{2} - 57591 ) / 11664 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -19\nu^{14} + 22\nu^{12} + 21\nu^{10} - 684\nu^{8} + 997\nu^{6} + 54\nu^{4} - 14499\nu^{2} + 20412 ) / 5832 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -14\nu^{15} + 65\nu^{13} + 42\nu^{11} - 369\nu^{9} + 1826\nu^{7} - 711\nu^{5} - 5022\nu^{3} + 45927\nu ) / 11664 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -61\nu^{15} - 17\nu^{13} - 51\nu^{11} - 1575\nu^{9} - 5\nu^{7} - 2385\nu^{5} - 39123\nu^{3} - 5103\nu ) / 34992 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7\nu^{15} - 14\nu^{13} - 23\nu^{11} + 210\nu^{9} - 625\nu^{7} + 442\nu^{5} + 4113\nu^{3} - 15390\nu ) / 3888 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 85 \nu^{15} - 168 \nu^{14} - 43 \nu^{13} - 192 \nu^{12} - 273 \nu^{11} + 504 \nu^{10} + \cdots - 117369 \nu ) / 69984 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 179 \nu^{15} - 168 \nu^{14} + 635 \nu^{13} - 192 \nu^{12} + 303 \nu^{11} + 504 \nu^{10} + \cdots + 508113 \nu ) / 69984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 179 \nu^{15} + 168 \nu^{14} + 635 \nu^{13} + 192 \nu^{12} + 303 \nu^{11} - 504 \nu^{10} + \cdots + 508113 \nu ) / 69984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{13} + \beta_{10} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} + 2\beta_{9} + \beta_{8} - 3\beta_{7} - 2\beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + 2\beta_{14} - \beta_{13} + 5\beta_{12} + 2\beta_{11} + 8\beta_{5} - 4\beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{15} + 3\beta_{14} - 4\beta_{8} + \beta_{7} + 8\beta_{6} - 8\beta_{4} + 6\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5\beta_{15} + \beta_{14} + 4\beta_{13} - 8\beta_{12} - 2\beta_{11} - 16\beta_{10} + 7\beta_{5} - 14\beta_{3} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3\beta_{15} + 3\beta_{14} - 12\beta_{9} + 8\beta_{8} - 10\beta_{7} - 16\beta_{6} - 16\beta_{4} - 10\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 29 \beta_{15} - 23 \beta_{14} + 52 \beta_{13} - 32 \beta_{12} + 10 \beta_{11} - 16 \beta_{10} + \cdots + 11 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 39 \beta_{15} + 39 \beta_{14} - 48 \beta_{9} - 52 \beta_{8} + 26 \beta_{7} - 40 \beta_{6} + \cdots - 156 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 38 \beta_{15} - 35 \beta_{14} - 3 \beta_{13} - 152 \beta_{12} - 74 \beta_{11} + 39 \beta_{10} + \cdots - 160 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 112 \beta_{15} - 112 \beta_{14} + 2 \beta_{9} + 111 \beta_{8} - 55 \beta_{7} - 352 \beta_{6} + 162 \beta_{4} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 40 \beta_{15} + 3 \beta_{14} - 43 \beta_{13} + 339 \beta_{12} + 504 \beta_{10} - 168 \beta_{5} + \cdots - 336 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 40 \beta_{15} - 40 \beta_{14} + 80 \beta_{9} - 424 \beta_{8} + 369 \beta_{7} + 544 \beta_{6} + \cdots - 320 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 104 \beta_{15} + 544 \beta_{14} - 648 \beta_{13} + 664 \beta_{12} - 848 \beta_{11} - 400 \beta_{10} + \cdots - 296 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1
0.206738 + 1.71967i
−1.29725 + 1.14767i
1.60627 + 0.647994i
−1.64143 + 0.552909i
1.64143 − 0.552909i
−1.60627 − 0.647994i
1.29725 − 1.14767i
−0.206738 − 1.71967i
0.206738 − 1.71967i
−1.29725 − 1.14767i
1.60627 − 0.647994i
−1.64143 − 0.552909i
1.64143 + 0.552909i
−1.60627 + 0.647994i
1.29725 + 1.14767i
−0.206738 + 1.71967i
− 1.00000i −1.71967 − 0.206738i −1.00000 1.00000i −0.206738 + 1.71967i −1.18164 1.00000i 2.91452 + 0.711040i 1.00000
371.2 − 1.00000i −1.14767 + 1.29725i −1.00000 1.00000i 1.29725 + 1.14767i 3.69457 1.00000i −0.365730 − 2.97762i 1.00000
371.3 − 1.00000i −0.647994 − 1.60627i −1.00000 1.00000i −1.60627 + 0.647994i 0.794255 1.00000i −2.16021 + 2.08171i 1.00000
371.4 − 1.00000i −0.552909 + 1.64143i −1.00000 1.00000i 1.64143 + 0.552909i −2.30718 1.00000i −2.38858 − 1.81512i 1.00000
371.5 − 1.00000i 0.552909 − 1.64143i −1.00000 1.00000i −1.64143 − 0.552909i −2.30718 1.00000i −2.38858 − 1.81512i 1.00000
371.6 − 1.00000i 0.647994 + 1.60627i −1.00000 1.00000i 1.60627 − 0.647994i 0.794255 1.00000i −2.16021 + 2.08171i 1.00000
371.7 − 1.00000i 1.14767 − 1.29725i −1.00000 1.00000i −1.29725 − 1.14767i 3.69457 1.00000i −0.365730 − 2.97762i 1.00000
371.8 − 1.00000i 1.71967 + 0.206738i −1.00000 1.00000i 0.206738 − 1.71967i −1.18164 1.00000i 2.91452 + 0.711040i 1.00000
371.9 1.00000i −1.71967 + 0.206738i −1.00000 − 1.00000i −0.206738 − 1.71967i −1.18164 − 1.00000i 2.91452 − 0.711040i 1.00000
371.10 1.00000i −1.14767 − 1.29725i −1.00000 − 1.00000i 1.29725 − 1.14767i 3.69457 − 1.00000i −0.365730 + 2.97762i 1.00000
371.11 1.00000i −0.647994 + 1.60627i −1.00000 − 1.00000i −1.60627 − 0.647994i 0.794255 − 1.00000i −2.16021 − 2.08171i 1.00000
371.12 1.00000i −0.552909 − 1.64143i −1.00000 − 1.00000i 1.64143 − 0.552909i −2.30718 − 1.00000i −2.38858 + 1.81512i 1.00000
371.13 1.00000i 0.552909 + 1.64143i −1.00000 − 1.00000i −1.64143 + 0.552909i −2.30718 − 1.00000i −2.38858 + 1.81512i 1.00000
371.14 1.00000i 0.647994 − 1.60627i −1.00000 − 1.00000i 1.60627 + 0.647994i 0.794255 − 1.00000i −2.16021 − 2.08171i 1.00000
371.15 1.00000i 1.14767 + 1.29725i −1.00000 − 1.00000i −1.29725 + 1.14767i 3.69457 − 1.00000i −0.365730 + 2.97762i 1.00000
371.16 1.00000i 1.71967 − 0.206738i −1.00000 − 1.00000i 0.206738 + 1.71967i −1.18164 − 1.00000i 2.91452 − 0.711040i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 371.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.d ✓ 16
3.b odd 2 1 inner 930.2.h.d ✓ 16
31.b odd 2 1 inner 930.2.h.d ✓ 16
93.c even 2 1 inner 930.2.h.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.d ✓ 16 1.a even 1 1 trivial
930.2.h.d ✓ 16 3.b odd 2 1 inner
930.2.h.d ✓ 16 31.b odd 2 1 inner
930.2.h.d ✓ 16 93.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - T_{7}^{3} - 10T_{7}^{2} - 2T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} - 10 T^{2} + \cdots + 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 43 T^{6} + 134 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 20 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 60 T^{6} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} + \cdots + 256)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 27 T^{6} + \cdots + 512)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 152 T^{6} + \cdots + 61952)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 264 T^{6} + \cdots + 3463712)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 100 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 135 T^{6} + \cdots + 209952)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 176 T^{6} + \cdots + 135424)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 279 T^{6} + \cdots + 4892192)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 272 T^{6} + \cdots + 4734976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 192 T^{6} + \cdots + 247808)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} - 128 T^{2} + \cdots - 64)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 357 T^{6} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 287 T^{6} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 419 T^{6} + \cdots + 59710592)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 420 T^{6} + \cdots + 31490048)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 179 T^{6} + \cdots + 1131008)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2 T^{3} + \cdots + 592)^{4} \) Copy content Toggle raw display
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