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Properties

Label 370.2.n.f
Level $370$
Weight $2$
Character orbit 370.n
Analytic conductor $2.954$
Analytic rank $0$
Dimension $12$
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] = [370,2,Mod(269,370)]
 
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([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.n (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: 12.0.89539436150784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{9} - \beta_{7} + \cdots + \beta_{5}) q^{3}+ \cdots + ( - 2 \beta_{10} - 4 \beta_{8} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{9} - \beta_{7} + \cdots + \beta_{5}) q^{3}+ \cdots + ( - 2 \beta_{11} + 8 \beta_{10} + \cdots - 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} + 4 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} + 4 q^{6} + 20 q^{9} + 4 q^{10} - 32 q^{11} + 18 q^{15} - 6 q^{16} + 4 q^{19} + 20 q^{21} + 2 q^{24} - 2 q^{25} - 4 q^{26} - 32 q^{29} - 20 q^{30} + 28 q^{31} - 4 q^{34} + 4 q^{35} + 40 q^{36} - 58 q^{39} + 2 q^{40} - 18 q^{41} - 16 q^{44} + 16 q^{45} - 26 q^{49} - 8 q^{50} + 32 q^{51} - 34 q^{54} - 4 q^{55} + 28 q^{59} + 36 q^{60} + 20 q^{61} - 12 q^{64} - 22 q^{65} + 24 q^{66} + 20 q^{69} - 16 q^{70} - 32 q^{71} - 24 q^{74} + 64 q^{75} - 4 q^{76} - 4 q^{79} - 6 q^{81} + 40 q^{84} + 24 q^{85} - 22 q^{86} - 44 q^{89} - 20 q^{90} - 36 q^{91} + 16 q^{94} + 16 q^{95} - 2 q^{96} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + \cdots + 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} + \cdots + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + \cdots - 12 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + \cdots - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} + \cdots + 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + \cdots - 32 ) / 460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + \cdots - 420 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + \cdots - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} + \cdots + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + \cdots - 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} + \cdots + 612 ) / 230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + \cdots - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\beta_{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.98293 + 0.531325i
−1.16746 − 0.312819i
0.550552 + 0.147520i
−0.147520 + 0.550552i
0.312819 − 1.16746i
−0.531325 + 1.98293i
1.98293 − 0.531325i
−1.16746 + 0.312819i
0.550552 − 0.147520i
−0.147520 − 0.550552i
0.312819 + 1.16746i
−0.531325 − 1.98293i
−0.866025 − 0.500000i −2.18708 + 1.26271i 0.500000 + 0.866025i −1.37659 − 1.76210i 2.52543 −1.91766 + 1.10716i − 1.00000i 1.68889 − 2.92525i 0.311108 + 2.21432i
269.2 −0.866025 − 0.500000i −1.41240 + 0.815449i 0.500000 + 0.866025i −1.60976 + 1.55199i 1.63090 0.466951 − 0.269594i − 1.00000i −0.170086 + 0.294598i 2.17009 − 0.539189i
269.3 −0.866025 − 0.500000i 2.73346 − 1.57816i 0.500000 + 0.866025i 2.12032 + 0.710109i −3.15633 1.45071 − 0.837565i − 1.00000i 3.48119 − 6.02961i −1.48119 − 1.67513i
269.4 0.866025 + 0.500000i −2.73346 + 1.57816i 0.500000 + 0.866025i −0.445186 + 2.19130i −3.15633 −1.45071 + 0.837565i 1.00000i 3.48119 − 6.02961i −1.48119 + 1.67513i
269.5 0.866025 + 0.500000i 1.41240 − 0.815449i 0.500000 + 0.866025i 2.14894 − 0.618092i 1.63090 −0.466951 + 0.269594i 1.00000i −0.170086 + 0.294598i 2.17009 + 0.539189i
269.6 0.866025 + 0.500000i 2.18708 − 1.26271i 0.500000 + 0.866025i −0.837733 − 2.07321i 2.52543 1.91766 − 1.10716i 1.00000i 1.68889 − 2.92525i 0.311108 − 2.21432i
359.1 −0.866025 + 0.500000i −2.18708 − 1.26271i 0.500000 − 0.866025i −1.37659 + 1.76210i 2.52543 −1.91766 − 1.10716i 1.00000i 1.68889 + 2.92525i 0.311108 − 2.21432i
359.2 −0.866025 + 0.500000i −1.41240 − 0.815449i 0.500000 − 0.866025i −1.60976 − 1.55199i 1.63090 0.466951 + 0.269594i 1.00000i −0.170086 − 0.294598i 2.17009 + 0.539189i
359.3 −0.866025 + 0.500000i 2.73346 + 1.57816i 0.500000 − 0.866025i 2.12032 − 0.710109i −3.15633 1.45071 + 0.837565i 1.00000i 3.48119 + 6.02961i −1.48119 + 1.67513i
359.4 0.866025 − 0.500000i −2.73346 − 1.57816i 0.500000 − 0.866025i −0.445186 − 2.19130i −3.15633 −1.45071 − 0.837565i − 1.00000i 3.48119 + 6.02961i −1.48119 − 1.67513i
359.5 0.866025 − 0.500000i 1.41240 + 0.815449i 0.500000 − 0.866025i 2.14894 + 0.618092i 1.63090 −0.466951 − 0.269594i − 1.00000i −0.170086 − 0.294598i 2.17009 − 0.539189i
359.6 0.866025 − 0.500000i 2.18708 + 1.26271i 0.500000 − 0.866025i −0.837733 + 2.07321i 2.52543 1.91766 + 1.10716i − 1.00000i 1.68889 + 2.92525i 0.311108 + 2.21432i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n 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.n.f ✓ 12
5.b even 2 1 inner 370.2.n.f ✓ 12
37.c even 3 1 inner 370.2.n.f ✓ 12
185.n even 6 1 inner 370.2.n.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.n.f ✓ 12 1.a even 1 1 trivial
370.2.n.f ✓ 12 5.b even 2 1 inner
370.2.n.f ✓ 12 37.c even 3 1 inner
370.2.n.f ✓ 12 185.n 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 \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{12} - 19T_{3}^{10} + 254T_{3}^{8} - 1695T_{3}^{6} + 8238T_{3}^{4} - 18083T_{3}^{2} + 28561 \) Copy content Toggle raw display
\( T_{7}^{12} - 8T_{7}^{10} + 48T_{7}^{8} - 120T_{7}^{6} + 224T_{7}^{4} - 64T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - 19 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} + 8 T^{2} + 18 T + 10)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} - 47 T^{10} + \cdots + 390625 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 8 T^{4} + 16 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 8 T^{2} + \cdots - 928)^{4} \) Copy content Toggle raw display
$31$ \( (T^{3} - 7 T^{2} - 7 T + 59)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( (T^{6} + 9 T^{5} + 70 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 51 T^{4} + \cdots + 841)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 64 T^{4} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 35 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$59$ \( (T^{6} - 14 T^{5} + \cdots + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 10 T^{5} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 33556377856 \) Copy content Toggle raw display
$71$ \( (T^{6} + 16 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 256 T^{4} + \cdots + 289444)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 2 T^{5} + \cdots + 67600)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 268435456 \) Copy content Toggle raw display
$89$ \( (T^{6} + 22 T^{5} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 320 T^{4} + \cdots + 952576)^{2} \) Copy content Toggle raw display
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