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Properties

Label 370.2.l.c
Level $370$
Weight $2$
Character orbit 370.l
Analytic conductor $2.954$
Analytic rank $0$
Dimension $12$
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] = [370,2,Mod(11,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.l (of order \(6\), 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.116304318664704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{10} - \beta_{6} + 1) q^{3} + ( - \beta_{6} + 1) q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{9} + \beta_{4}) q^{6} + (\beta_{10} - \beta_{6} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{10} - \beta_{6} + 1) q^{3} + ( - \beta_{6} + 1) q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{9} + \beta_{4}) q^{6} + (\beta_{10} - \beta_{6} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + (3 \beta_{11} - 3 \beta_{10} + \cdots + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 6 q^{4} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 6 q^{4} + 2 q^{7} - 2 q^{9} + 12 q^{10} - 16 q^{11} - 4 q^{12} + 6 q^{13} - 6 q^{16} - 6 q^{17} + 18 q^{19} - 14 q^{21} + 6 q^{22} + 6 q^{25} + 8 q^{26} - 32 q^{27} - 2 q^{28} + 4 q^{30} - 10 q^{33} - 10 q^{34} - 6 q^{35} - 4 q^{36} - 26 q^{37} + 8 q^{38} + 18 q^{39} + 6 q^{40} + 4 q^{41} + 18 q^{42} - 8 q^{44} - 4 q^{46} - 20 q^{47} - 8 q^{48} + 2 q^{49} + 6 q^{52} - 2 q^{53} + 6 q^{55} + 6 q^{56} + 36 q^{57} + 8 q^{58} + 12 q^{59} + 24 q^{61} + 10 q^{62} - 16 q^{63} - 12 q^{64} + 4 q^{65} + 28 q^{67} - 6 q^{69} + 2 q^{70} - 40 q^{71} - 12 q^{73} + 14 q^{74} + 8 q^{75} + 18 q^{76} - 24 q^{77} - 10 q^{78} + 24 q^{79} - 6 q^{81} - 16 q^{83} - 28 q^{84} - 20 q^{85} - 16 q^{86} - 24 q^{87} + 6 q^{89} - 2 q^{90} - 18 q^{91} + 6 q^{92} + 78 q^{93} + 4 q^{95} - 12 q^{98} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + \nu^{9} - 6 \nu^{8} + 7 \nu^{7} + 10 \nu^{6} - 14 \nu^{5} - 12 \nu^{4} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} + \nu^{9} - 6 \nu^{8} + 3 \nu^{7} + 10 \nu^{6} - 12 \nu^{5} - 12 \nu^{4} + \cdots + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{10} - 3 \nu^{9} + 3 \nu^{8} + 7 \nu^{7} - 11 \nu^{6} - 10 \nu^{5} + 24 \nu^{4} + \cdots + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 10 \nu^{10} + 5 \nu^{9} + 14 \nu^{8} - 37 \nu^{7} - 10 \nu^{6} + 78 \nu^{5} - 20 \nu^{4} + \cdots - 256 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 6 \nu^{10} + \nu^{9} - 14 \nu^{8} + 19 \nu^{7} + 18 \nu^{6} - 44 \nu^{5} - 16 \nu^{4} + \cdots + 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} + \nu^{9} - 15 \nu^{8} + 19 \nu^{7} + 19 \nu^{6} - 48 \nu^{5} - 18 \nu^{4} + \cdots + 176 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9 \nu^{11} + 4 \nu^{10} + 19 \nu^{9} - 44 \nu^{8} - 3 \nu^{7} + 88 \nu^{6} - 38 \nu^{5} - 160 \nu^{4} + \cdots + 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 9 \nu^{10} + 3 \nu^{9} + 19 \nu^{8} - 39 \nu^{7} - 15 \nu^{6} + 84 \nu^{5} - 6 \nu^{4} + \cdots - 288 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} + 18 \nu^{10} - 3 \nu^{9} - 46 \nu^{8} + 51 \nu^{7} + 66 \nu^{6} - 130 \nu^{5} + \cdots + 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9 \nu^{11} + 8 \nu^{10} + 19 \nu^{9} - 56 \nu^{8} + 13 \nu^{7} + 100 \nu^{6} - 70 \nu^{5} + \cdots + 256 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17 \nu^{11} + 14 \nu^{10} + 27 \nu^{9} - 82 \nu^{8} + 5 \nu^{7} + 158 \nu^{6} - 78 \nu^{5} + \cdots + 160 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{9} - 3\beta_{5} - 3\beta_{4} + \beta_{3} + 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{11} + 3\beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - 5\beta_{6} + \beta_{5} - \beta_{3} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{10} - \beta_{8} - 3\beta_{7} - 7\beta_{6} - 3\beta_{4} - 3\beta_{2} + 3\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{11} + \beta_{9} + \beta_{5} - 5\beta_{4} - 9\beta_{3} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 9 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} + \cdots - 11 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -11\beta_{10} + 9\beta_{8} + 15\beta_{7} + 5\beta_{6} - 21\beta_{4} - 21\beta_{2} + 7\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3\beta_{11} - 11\beta_{9} + 3\beta_{5} - 13\beta_{4} - \beta_{3} - 3\beta _1 - 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 13 \beta_{11} + 13 \beta_{10} + 15 \beta_{9} + 25 \beta_{8} + 15 \beta_{7} - 19 \beta_{6} + \cdots - 5 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19\beta_{10} + 25\beta_{8} - 29\beta_{7} + 39\beta_{6} - 5\beta_{4} - 5\beta_{2} - 3\beta _1 - 39 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1 - \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.41362 + 0.0408194i
1.33544 − 0.465413i
0.578188 + 1.29062i
0.742163 − 1.20382i
−1.07078 + 0.923815i
0.828615 + 1.14604i
−1.41362 − 0.0408194i
1.33544 + 0.465413i
0.578188 − 1.29062i
0.742163 + 1.20382i
−1.07078 − 0.923815i
0.828615 − 1.14604i
−0.866025 + 0.500000i −0.671462 + 1.16301i 0.500000 − 0.866025i −0.866025 − 0.500000i − 1.34292i −0.461662 + 0.799622i 1.00000i 0.598279 + 1.03625i 1.00000
11.2 −0.866025 + 0.500000i 0.264658 − 0.458402i 0.500000 − 0.866025i −0.866025 − 0.500000i 0.529317i −0.146963 + 0.254547i 1.00000i 1.35991 + 2.35544i 1.00000
11.3 −0.866025 + 0.500000i 1.40680 − 2.43665i 0.500000 − 0.866025i −0.866025 − 0.500000i 2.81361i 1.97465 − 3.42019i 1.00000i −2.45819 − 4.25771i 1.00000
11.4 0.866025 − 0.500000i −0.671462 + 1.16301i 0.500000 − 0.866025i 0.866025 + 0.500000i 1.34292i −1.45444 + 2.51917i − 1.00000i 0.598279 + 1.03625i 1.00000
11.5 0.866025 − 0.500000i 0.264658 − 0.458402i 0.500000 − 0.866025i 0.866025 + 0.500000i − 0.529317i 1.80085 − 3.11916i − 1.00000i 1.35991 + 2.35544i 1.00000
11.6 0.866025 − 0.500000i 1.40680 − 2.43665i 0.500000 − 0.866025i 0.866025 + 0.500000i − 2.81361i −0.712432 + 1.23397i − 1.00000i −2.45819 − 4.25771i 1.00000
101.1 −0.866025 − 0.500000i −0.671462 − 1.16301i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.34292i −0.461662 − 0.799622i − 1.00000i 0.598279 − 1.03625i 1.00000
101.2 −0.866025 − 0.500000i 0.264658 + 0.458402i 0.500000 + 0.866025i −0.866025 + 0.500000i − 0.529317i −0.146963 − 0.254547i − 1.00000i 1.35991 − 2.35544i 1.00000
101.3 −0.866025 − 0.500000i 1.40680 + 2.43665i 0.500000 + 0.866025i −0.866025 + 0.500000i − 2.81361i 1.97465 + 3.42019i − 1.00000i −2.45819 + 4.25771i 1.00000
101.4 0.866025 + 0.500000i −0.671462 − 1.16301i 0.500000 + 0.866025i 0.866025 − 0.500000i − 1.34292i −1.45444 − 2.51917i 1.00000i 0.598279 − 1.03625i 1.00000
101.5 0.866025 + 0.500000i 0.264658 + 0.458402i 0.500000 + 0.866025i 0.866025 − 0.500000i 0.529317i 1.80085 + 3.11916i 1.00000i 1.35991 − 2.35544i 1.00000
101.6 0.866025 + 0.500000i 1.40680 + 2.43665i 0.500000 + 0.866025i 0.866025 − 0.500000i 2.81361i −0.712432 − 1.23397i 1.00000i −2.45819 + 4.25771i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.l.c ✓ 12
37.e even 6 1 inner 370.2.l.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.l.c ✓ 12 1.a even 1 1 trivial
370.2.l.c ✓ 12 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{6} - 2T_{3}^{5} + 7T_{3}^{4} + 2T_{3}^{3} + 13T_{3}^{2} - 6T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{12} - 2 T_{7}^{11} + 22 T_{7}^{10} + 12 T_{7}^{9} + 256 T_{7}^{8} + 232 T_{7}^{7} + 1992 T_{7}^{6} + \cdots + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} + 7 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{5} + 11 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 32761 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{12} - 18 T^{11} + \cdots + 55830784 \) Copy content Toggle raw display
$23$ \( T^{12} + 186 T^{10} + \cdots + 80496784 \) Copy content Toggle raw display
$29$ \( T^{12} + 200 T^{10} + \cdots + 2166784 \) Copy content Toggle raw display
$31$ \( T^{12} + 216 T^{10} + \cdots + 99042304 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} - 4 T^{11} + \cdots + 3139984 \) Copy content Toggle raw display
$43$ \( T^{12} + 276 T^{10} + \cdots + 14348944 \) Copy content Toggle raw display
$47$ \( (T^{6} + 10 T^{5} + \cdots + 436)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 2 T^{11} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 61895468944 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11546791936 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2279489536 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1020972826624 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots - 7328)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 31564496896 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 304986689536 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1867622656 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 3957919744 \) Copy content Toggle raw display
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