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Properties

Label 805.2.d.e
Level $805$
Weight $2$
Character orbit 805.d
Analytic conductor $6.428$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [805,2,Mod(804,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.804");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.d (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: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2x^{10} - 5x^{8} - 28x^{6} - 125x^{4} + 1250x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + q^{3} + ( - \beta_{2} - 2) q^{4} - \beta_{11} q^{5} + \beta_{3} q^{6} + (\beta_{11} - \beta_{8}) q^{7} + ( - \beta_{5} - 2 \beta_{3}) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + q^{3} + ( - \beta_{2} - 2) q^{4} - \beta_{11} q^{5} + \beta_{3} q^{6} + (\beta_{11} - \beta_{8}) q^{7} + ( - \beta_{5} - 2 \beta_{3}) q^{8} - 2 q^{9} + ( - \beta_{10} - \beta_{9} + \cdots - \beta_1) q^{10}+ \cdots + 2 \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 28 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 28 q^{4} - 24 q^{9} - 28 q^{12} + 4 q^{13} + 36 q^{16} - 4 q^{25} - 60 q^{27} - 60 q^{29} + 24 q^{35} + 56 q^{36} + 4 q^{39} - 24 q^{46} + 44 q^{47} + 36 q^{48} + 20 q^{49} - 12 q^{50} + 140 q^{52} - 28 q^{55} + 36 q^{62} - 16 q^{64} - 52 q^{70} - 48 q^{71} - 4 q^{75} + 28 q^{77} + 12 q^{81} - 92 q^{82} + 36 q^{85} - 60 q^{87} + 8 q^{95} + 116 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2x^{10} - 5x^{8} - 28x^{6} - 125x^{4} + 1250x^{2} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} - 2\nu^{6} + 30\nu^{4} + 78\nu^{2} + 175 ) / 400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{10} + 21\nu^{8} + 60\nu^{6} + 556\nu^{4} - 450\nu^{2} - 5625 ) / 10000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{10} - 3 \nu^{9} - 23 \nu^{8} + 110 \nu^{7} + 570 \nu^{6} - 378 \nu^{5} + 722 \nu^{4} + \cdots - 4375 ) / 20000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} + 2\nu^{8} - 5\nu^{6} - 28\nu^{4} + 500\nu^{2} + 1250 ) / 1250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{10} + 9\nu^{8} + 90\nu^{6} - 526\nu^{4} + 4050\nu^{2} - 4375 ) / 10000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{11} + 33\nu^{9} + 30\nu^{7} - 3362\nu^{5} + 5050\nu^{3} + 8125\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 3\nu^{9} + 15\nu^{7} + 3\nu^{5} - 640\nu^{3} ) / 5000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\nu^{11} - 78\nu^{9} + 370\nu^{7} + 1442\nu^{5} - 4825\nu^{3} - 10000\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{11} + 19\nu^{9} + 65\nu^{7} - 41\nu^{5} + 2800\nu^{3} - 10000\nu ) / 12500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} - 2\nu^{9} + 5\nu^{7} + 28\nu^{5} + 125\nu^{3} - 1250\nu ) / 3125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{10} - 4\beta_{8} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 8\beta_{3} + 6\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{11} + 6\beta_{10} - 4\beta_{8} - 16\beta_{7} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{9} - 16\beta_{8} - 16\beta_{7} + 9\beta_{6} + 7\beta_{5} + 32\beta_{4} - 7\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{11} + 48\beta_{10} + 80\beta_{9} + 48\beta_{8} + 32\beta_{7} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32\beta_{9} + 32\beta_{8} + 32\beta_{7} + 64\beta_{5} - 64\beta_{4} + 240\beta_{3} - 128\beta_{2} + 113 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -496\beta_{11} + 240\beta_{10} - 160\beta_{9} + 272\beta_{8} - 144\beta_{7} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 144 \beta_{9} - 144 \beta_{8} - 144 \beta_{7} - 511 \beta_{6} + 657 \beta_{5} + 288 \beta_{4} + \cdots - 913 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -2191\beta_{11} + 178\beta_{10} + 720\beta_{9} - 916\beta_{8} - 590\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/805\mathbb{Z}\right)^\times\).

\(n\) \(162\) \(281\) \(346\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
804.1
2.16661 − 0.553012i
−2.16661 + 0.553012i
0.450080 + 2.19030i
−0.450080 − 2.19030i
1.45026 + 1.70198i
−1.45026 − 1.70198i
1.45026 − 1.70198i
−1.45026 + 1.70198i
0.450080 − 2.19030i
−0.450080 + 2.19030i
2.16661 + 0.553012i
−2.16661 − 0.553012i
− 2.62897i 1.00000 −4.91150 −2.16661 − 0.553012i − 2.62897i 2.58731 + 0.553012i 7.65427i −2.00000 −1.45385 + 5.69595i
804.2 − 2.62897i 1.00000 −4.91150 2.16661 + 0.553012i − 2.62897i −2.58731 − 0.553012i 7.65427i −2.00000 1.45385 − 5.69595i
804.3 − 2.26483i 1.00000 −3.12946 −0.450080 + 2.19030i − 2.26483i −1.48411 − 2.19030i 2.55804i −2.00000 4.96067 + 1.01936i
804.4 − 2.26483i 1.00000 −3.12946 0.450080 − 2.19030i − 2.26483i 1.48411 + 2.19030i 2.55804i −2.00000 −4.96067 − 1.01936i
804.5 − 0.979304i 1.00000 1.04096 −1.45026 + 1.70198i − 0.979304i −2.02565 − 1.70198i − 2.97803i −2.00000 1.66676 + 1.42024i
804.6 − 0.979304i 1.00000 1.04096 1.45026 − 1.70198i − 0.979304i 2.02565 + 1.70198i − 2.97803i −2.00000 −1.66676 − 1.42024i
804.7 0.979304i 1.00000 1.04096 −1.45026 − 1.70198i 0.979304i −2.02565 + 1.70198i 2.97803i −2.00000 1.66676 − 1.42024i
804.8 0.979304i 1.00000 1.04096 1.45026 + 1.70198i 0.979304i 2.02565 − 1.70198i 2.97803i −2.00000 −1.66676 + 1.42024i
804.9 2.26483i 1.00000 −3.12946 −0.450080 − 2.19030i 2.26483i −1.48411 + 2.19030i − 2.55804i −2.00000 4.96067 − 1.01936i
804.10 2.26483i 1.00000 −3.12946 0.450080 + 2.19030i 2.26483i 1.48411 − 2.19030i − 2.55804i −2.00000 −4.96067 + 1.01936i
804.11 2.62897i 1.00000 −4.91150 −2.16661 + 0.553012i 2.62897i 2.58731 − 0.553012i − 7.65427i −2.00000 −1.45385 − 5.69595i
804.12 2.62897i 1.00000 −4.91150 2.16661 − 0.553012i 2.62897i −2.58731 + 0.553012i − 7.65427i −2.00000 1.45385 + 5.69595i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 804.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner
35.c odd 2 1 inner
805.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.d.e yes 12
5.b even 2 1 805.2.d.d ✓ 12
7.b odd 2 1 805.2.d.d ✓ 12
23.b odd 2 1 inner 805.2.d.e yes 12
35.c odd 2 1 inner 805.2.d.e yes 12
115.c odd 2 1 805.2.d.d ✓ 12
161.c even 2 1 805.2.d.d ✓ 12
805.d even 2 1 inner 805.2.d.e yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.d.d ✓ 12 5.b even 2 1
805.2.d.d ✓ 12 7.b odd 2 1
805.2.d.d ✓ 12 115.c odd 2 1
805.2.d.d ✓ 12 161.c even 2 1
805.2.d.e yes 12 1.a even 1 1 trivial
805.2.d.e yes 12 23.b odd 2 1 inner
805.2.d.e yes 12 35.c odd 2 1 inner
805.2.d.e yes 12 805.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\):

\( T_{2}^{6} + 13T_{2}^{4} + 47T_{2}^{2} + 34 \) Copy content Toggle raw display
\( T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 13 T^{4} + \cdots + 34)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} - 10 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 49 T^{4} + \cdots + 1700)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - T^{2} - 37 T - 43)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 53 T^{4} + \cdots + 1088)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 50 T^{4} + \cdots - 3200)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( (T + 5)^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} + 172 T^{4} + \cdots + 102850)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 90 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 74 T^{4} + \cdots + 3400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 28 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 11 T^{2} + \cdots + 323)^{4} \) Copy content Toggle raw display
$53$ \( (T^{6} - 198 T^{4} + \cdots - 8)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 170 T^{4} + \cdots + 157216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 70 T^{4} + 396 T^{2} - 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 112 T^{4} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} + \cdots - 68)^{4} \) Copy content Toggle raw display
$73$ \( (T^{3} - 37 T + 16)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 253 T^{4} + \cdots + 392768)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 396 T^{4} + \cdots + 1257728)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 294 T^{4} + \cdots - 24200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 429 T^{4} + \cdots + 2106368)^{2} \) Copy content Toggle raw display
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