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Properties

Label 945.2.d.d
Level $945$
Weight $2$
Character orbit 945.d
Analytic conductor $7.546$
Analytic rank $0$
Dimension $10$
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(379,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16x^{8} + 92x^{6} + 225x^{4} + 200x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + \beta_{3} q^{5} - \beta_{5} q^{7} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + \beta_{3} q^{5} - \beta_{5} q^{7} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{8} + (\beta_{9} - \beta_{6} + \beta_{5} + \cdots + 1) q^{10}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 12 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 12 q^{4} - 2 q^{5} + 8 q^{10} + 18 q^{11} - 8 q^{16} + 6 q^{19} + 14 q^{20} + 16 q^{25} + 42 q^{26} - 24 q^{29} - 12 q^{31} - 14 q^{34} - 2 q^{35} - 32 q^{40} + 14 q^{41} - 74 q^{44} - 10 q^{46} - 10 q^{49} + 22 q^{50} - 20 q^{55} + 6 q^{56} - 52 q^{59} + 4 q^{61} + 42 q^{64} - 30 q^{65} + 12 q^{70} + 80 q^{71} - 76 q^{74} + 4 q^{76} - 16 q^{79} + 38 q^{80} - 6 q^{85} + 18 q^{86} - 22 q^{89} + 14 q^{91} - 34 q^{94} + 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16x^{8} + 92x^{6} + 225x^{4} + 200x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 2\nu^{8} - 14\nu^{7} - 24\nu^{6} - 64\nu^{5} - 88\nu^{4} - 97\nu^{3} - 90\nu^{2} - 14\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - 14\nu^{7} + 24\nu^{6} - 64\nu^{5} + 88\nu^{4} - 97\nu^{3} + 90\nu^{2} - 14\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 14\nu^{7} + 64\nu^{5} + 101\nu^{3} + 30\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 14\nu^{7} + 68\nu^{5} + 133\nu^{3} + 86\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} - 14\nu^{6} - 62\nu^{4} - 85\nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + 4\nu^{8} + 38\nu^{7} + 52\nu^{6} + 144\nu^{5} + 216\nu^{4} + 123\nu^{3} + 284\nu^{2} - 122\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{9} + 4\nu^{8} - 38\nu^{7} + 52\nu^{6} - 144\nu^{5} + 216\nu^{4} - 123\nu^{3} + 284\nu^{2} + 122\nu + 4 ) / 8 \) 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
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} - 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 9\beta_{5} - 8\beta_{4} - 8\beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{9} - 9\beta_{8} - 10\beta_{7} + 8\beta_{4} - 8\beta_{3} + 34\beta_{2} - 71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} - \beta_{8} - 12\beta_{6} + 66\beta_{5} + 51\beta_{4} + 51\beta_{3} - 89\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 64\beta_{9} + 64\beta_{8} + 76\beta_{7} - 50\beta_{4} + 50\beta_{3} - 189\beta_{2} + 379 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -14\beta_{9} + 14\beta_{8} + 104\beta_{6} - 445\beta_{5} - 303\beta_{4} - 303\beta_{3} + 468\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
− 2.42873i
− 2.20040i
− 1.66057i
− 1.57529i
− 0.143064i
0.143064i
1.57529i
1.66057i
2.20040i
2.42873i
− 2.42873i 0 −3.89875 −1.31877 + 1.80578i 0 − 1.00000i 4.61157i 0 4.38577 + 3.20294i
379.2 − 2.20040i 0 −2.84174 1.72229 + 1.42608i 0 1.00000i 1.85217i 0 3.13795 − 3.78972i
379.3 − 1.66057i 0 −0.757487 −2.17195 − 0.531639i 0 1.00000i − 2.06328i 0 −0.882823 + 3.60667i
379.4 − 1.57529i 0 −0.481550 −1.45725 − 1.69600i 0 − 1.00000i − 2.39200i 0 −2.67170 + 2.29560i
379.5 − 0.143064i 0 1.97953 2.22568 + 0.215336i 0 1.00000i − 0.569328i 0 0.0308068 − 0.318414i
379.6 0.143064i 0 1.97953 2.22568 − 0.215336i 0 − 1.00000i 0.569328i 0 0.0308068 + 0.318414i
379.7 1.57529i 0 −0.481550 −1.45725 + 1.69600i 0 1.00000i 2.39200i 0 −2.67170 − 2.29560i
379.8 1.66057i 0 −0.757487 −2.17195 + 0.531639i 0 − 1.00000i 2.06328i 0 −0.882823 − 3.60667i
379.9 2.20040i 0 −2.84174 1.72229 − 1.42608i 0 − 1.00000i − 1.85217i 0 3.13795 + 3.78972i
379.10 2.42873i 0 −3.89875 −1.31877 − 1.80578i 0 1.00000i − 4.61157i 0 4.38577 − 3.20294i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.d.d ✓ 10
3.b odd 2 1 945.2.d.e yes 10
5.b even 2 1 inner 945.2.d.d ✓ 10
5.c odd 4 1 4725.2.a.ca 5
5.c odd 4 1 4725.2.a.cd 5
15.d odd 2 1 945.2.d.e yes 10
15.e even 4 1 4725.2.a.cb 5
15.e even 4 1 4725.2.a.cc 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.d.d ✓ 10 1.a even 1 1 trivial
945.2.d.d ✓ 10 5.b even 2 1 inner
945.2.d.e yes 10 3.b odd 2 1
945.2.d.e yes 10 15.d odd 2 1
4725.2.a.ca 5 5.c odd 4 1
4725.2.a.cb 5 15.e even 4 1
4725.2.a.cc 5 15.e even 4 1
4725.2.a.cd 5 5.c odd 4 1

Hecke kernels

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

\( T_{2}^{10} + 16T_{2}^{8} + 92T_{2}^{6} + 225T_{2}^{4} + 200T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{5} - 9T_{11}^{4} + 142T_{11}^{2} - 144T_{11} - 384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 16 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( (T^{5} - 9 T^{4} + \cdots - 384)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 73 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{10} + 100 T^{8} + \cdots + 309136 \) Copy content Toggle raw display
$19$ \( (T^{5} - 3 T^{4} - 37 T^{3} + \cdots + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 194 T^{8} + \cdots + 589824 \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots - 4864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 6 T^{4} - 7 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 190 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{5} - 7 T^{4} + \cdots - 8896)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 231 T^{8} + \cdots + 79138816 \) Copy content Toggle raw display
$47$ \( T^{10} + 131 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 593994384 \) Copy content Toggle raw display
$59$ \( (T^{5} + 26 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 2 T^{4} + \cdots + 556)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 241 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$71$ \( (T - 8)^{10} \) Copy content Toggle raw display
$73$ \( T^{10} + 329 T^{8} + \cdots + 589824 \) Copy content Toggle raw display
$79$ \( (T^{5} + 8 T^{4} + \cdots + 4822)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 317 T^{8} + \cdots + 44368921 \) Copy content Toggle raw display
$89$ \( (T^{5} + 11 T^{4} + \cdots + 97536)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 121176064 \) Copy content Toggle raw display
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