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Properties

Label 370.2.c.a
Level $370$
Weight $2$
Character orbit 370.c
Analytic conductor $2.954$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(369,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("370.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.c (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: \(2.95446487479\)
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: \( x^{10} + 19x^{8} + 103x^{6} + 210x^{4} + 140x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} - \beta_1 q^{6} + \beta_{2} q^{7} - q^{8} + (\beta_{8} - \beta_{7} - 1) q^{9} + \beta_{3} q^{10} + \beta_{7} q^{11} + \beta_1 q^{12} + (\beta_{8} + \beta_{7} + \cdots - \beta_{4}) q^{13}+ \cdots + (4 \beta_{8} - \beta_{7} + \beta_{6} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 10 q^{8} - 8 q^{9} + 3 q^{10} - 2 q^{13} - 10 q^{15} + 10 q^{16} + 18 q^{17} + 8 q^{18} - 3 q^{20} - 12 q^{21} + 10 q^{23} + 5 q^{25} + 2 q^{26} + 10 q^{30} - 10 q^{32}+ \cdots - 82 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 19x^{8} + 103x^{6} + 210x^{4} + 140x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{9} - 45\nu^{7} - 121\nu^{5} - 26\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 6\nu^{8} + 15\nu^{7} + 94\nu^{6} + 39\nu^{5} + 302\nu^{4} - 10\nu^{3} + 212\nu^{2} - 44\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{9} + 2\nu^{8} + 49\nu^{7} + 34\nu^{6} + 181\nu^{5} + 142\nu^{4} + 186\nu^{3} + 196\nu^{2} + 20\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 6\nu^{8} + 15\nu^{7} - 94\nu^{6} + 39\nu^{5} - 302\nu^{4} - 10\nu^{3} - 212\nu^{2} - 44\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{9} - 2\nu^{8} + 49\nu^{7} - 34\nu^{6} + 181\nu^{5} - 142\nu^{4} + 186\nu^{3} - 196\nu^{2} + 20\nu - 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{8} + 49\nu^{6} + 181\nu^{4} + 186\nu^{2} + 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{8} + 49\nu^{6} + 181\nu^{4} + 190\nu^{2} + 44 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} - 17\nu^{7} - 71\nu^{5} - 100\nu^{3} - 46\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{8} + 10\beta_{7} - 3\beta_{6} - \beta_{5} + 3\beta_{4} + \beta_{3} + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{9} + 14\beta_{6} - 17\beta_{5} + 14\beta_{4} - 17\beta_{3} - 2\beta_{2} + 68\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 170\beta_{8} - 108\beta_{7} + 45\beta_{6} + 16\beta_{5} - 45\beta_{4} - 16\beta_{3} - 330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -170\beta_{9} - 169\beta_{6} + 215\beta_{5} - 169\beta_{4} + 215\beta_{3} + 32\beta_{2} - 748\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1994\beta_{8} + 1224\beta_{7} - 554\beta_{6} - 201\beta_{5} + 554\beta_{4} + 201\beta_{3} + 3698 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1994\beta_{9} + 1979\beta_{6} - 2548\beta_{5} + 1979\beta_{4} - 2548\beta_{3} - 402\beta_{2} + 8542\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
− 3.40359i
− 1.78647i
− 1.76216i
− 0.987983i
− 0.377861i
0.377861i
0.987983i
1.76216i
1.78647i
3.40359i
−1.00000 − 3.40359i 1.00000 −1.28269 − 1.83159i 3.40359i − 2.06225i −1.00000 −8.58443 1.28269 + 1.83159i
369.2 −1.00000 − 1.78647i 1.00000 2.21736 + 0.288618i 1.78647i − 3.14934i −1.00000 −0.191472 −2.21736 − 0.288618i
369.3 −1.00000 − 1.76216i 1.00000 −1.62868 + 1.53213i 1.76216i 1.22131i −1.00000 −0.105209 1.62868 − 1.53213i
369.4 −1.00000 − 0.987983i 1.00000 −1.85396 − 1.25013i 0.987983i 4.78937i −1.00000 2.02389 1.85396 + 1.25013i
369.5 −1.00000 − 0.377861i 1.00000 1.04797 − 1.97529i 0.377861i − 0.631751i −1.00000 2.85722 −1.04797 + 1.97529i
369.6 −1.00000 0.377861i 1.00000 1.04797 + 1.97529i − 0.377861i 0.631751i −1.00000 2.85722 −1.04797 − 1.97529i
369.7 −1.00000 0.987983i 1.00000 −1.85396 + 1.25013i − 0.987983i − 4.78937i −1.00000 2.02389 1.85396 − 1.25013i
369.8 −1.00000 1.76216i 1.00000 −1.62868 − 1.53213i − 1.76216i − 1.22131i −1.00000 −0.105209 1.62868 + 1.53213i
369.9 −1.00000 1.78647i 1.00000 2.21736 − 0.288618i − 1.78647i 3.14934i −1.00000 −0.191472 −2.21736 + 0.288618i
369.10 −1.00000 3.40359i 1.00000 −1.28269 + 1.83159i − 3.40359i 2.06225i −1.00000 −8.58443 1.28269 − 1.83159i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.c.a ✓ 10
3.b odd 2 1 3330.2.e.d 10
5.b even 2 1 370.2.c.b yes 10
5.c odd 4 2 1850.2.d.i 20
15.d odd 2 1 3330.2.e.c 10
37.b even 2 1 370.2.c.b yes 10
111.d odd 2 1 3330.2.e.c 10
185.d even 2 1 inner 370.2.c.a ✓ 10
185.h odd 4 2 1850.2.d.i 20
555.b odd 2 1 3330.2.e.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a ✓ 10 1.a even 1 1 trivial
370.2.c.a ✓ 10 185.d even 2 1 inner
370.2.c.b yes 10 5.b even 2 1
370.2.c.b yes 10 37.b even 2 1
1850.2.d.i 20 5.c odd 4 2
1850.2.d.i 20 185.h odd 4 2
3330.2.e.c 10 15.d odd 2 1
3330.2.e.c 10 111.d odd 2 1
3330.2.e.d 10 3.b odd 2 1
3330.2.e.d 10 555.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{5} + T_{13}^{4} - 39T_{13}^{3} - 100T_{13}^{2} + 160T_{13} + 488 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 19 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 39 T^{8} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( (T^{5} - 28 T^{3} + \cdots - 48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + T^{4} - 39 T^{3} + \cdots + 488)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 9 T^{4} + \cdots - 144)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 118 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
$23$ \( (T^{5} - 5 T^{4} + \cdots - 768)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 182 T^{8} + \cdots + 2262016 \) Copy content Toggle raw display
$31$ \( T^{10} + 116 T^{8} + \cdots + 60516 \) Copy content Toggle raw display
$37$ \( T^{10} + 8 T^{9} + \cdots + 69343957 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} - 44 T^{3} + \cdots - 36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 5 T^{4} + \cdots - 2624)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 310 T^{8} + \cdots + 4596736 \) Copy content Toggle raw display
$53$ \( T^{10} + 257 T^{8} + \cdots + 39337984 \) Copy content Toggle raw display
$59$ \( T^{10} + 238 T^{8} + \cdots + 21827584 \) Copy content Toggle raw display
$61$ \( T^{10} + 150 T^{8} + \cdots + 82944 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 559417104 \) Copy content Toggle raw display
$71$ \( (T^{5} + 10 T^{4} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 189 T^{8} + \cdots + 589824 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 186486336 \) Copy content Toggle raw display
$83$ \( T^{10} + 466 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1230045184 \) Copy content Toggle raw display
$97$ \( (T^{5} - T^{4} - 178 T^{3} + \cdots + 1168)^{2} \) Copy content Toggle raw display
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