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Properties

Label 350.3.p.a
Level $350$
Weight $3$
Character orbit 350.p
Analytic conductor $9.537$
Analytic rank $0$
Dimension $8$
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] = [350,3,Mod(93,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.93");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.p (of order \(12\), degree \(4\), 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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{2} + (\beta_{6} - \beta_{4} + \beta_{2} + \cdots - 1) q^{3}+ \cdots + ( - 3 \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{2} + (\beta_{6} - \beta_{4} + \beta_{2} + \cdots - 1) q^{3}+ \cdots + (16 \beta_{7} - 16 \beta_{6} + \cdots + 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 2 q^{3} + 8 q^{6} - 18 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 2 q^{3} + 8 q^{6} - 18 q^{7} - 16 q^{8} - 16 q^{11} - 4 q^{12} - 64 q^{13} + 16 q^{16} + 8 q^{17} + 12 q^{18} + 116 q^{21} + 32 q^{22} - 2 q^{23} + 64 q^{26} + 4 q^{27} - 12 q^{28} + 24 q^{31} + 16 q^{32} + 80 q^{33} - 48 q^{36} + 104 q^{37} + 72 q^{38} + 160 q^{41} - 40 q^{42} + 124 q^{43} - 4 q^{46} + 28 q^{47} - 16 q^{48} + 8 q^{51} + 64 q^{52} + 32 q^{53} + 72 q^{56} - 16 q^{57} - 20 q^{58} + 256 q^{61} - 48 q^{62} - 124 q^{63} + 160 q^{66} - 230 q^{67} - 16 q^{68} - 720 q^{71} + 24 q^{72} + 96 q^{73} - 288 q^{76} - 416 q^{77} + 112 q^{78} - 136 q^{81} - 80 q^{82} - 500 q^{83} - 124 q^{86} + 274 q^{87} + 32 q^{88} + 200 q^{91} + 8 q^{92} + 320 q^{93} + 16 q^{96} - 368 q^{97} - 176 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 144\nu + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 32\nu^{4} - 16\nu^{2} + 144\nu - 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 9\nu^{6} + 16\nu^{5} + 80\nu^{3} + 81\nu + 117 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\nu^{7} - 27\nu^{6} + 80\nu^{5} + 256\nu^{3} + 405\nu - 351 ) / 432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - 4\beta_{4} + \beta_{3} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} - 5\beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + 30\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{7} + 8\beta_{6} + 8\beta_{5} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 96\beta_{5} - 13\beta_{3} - 96\beta_{2} - 13\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{4}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−1.26217 + 1.18614i
0.396143 − 1.68614i
1.26217 − 1.18614i
−0.396143 + 1.68614i
1.26217 + 1.18614i
−0.396143 − 1.68614i
−1.26217 − 1.18614i
0.396143 + 1.68614i
0.366025 + 1.36603i −0.423972 + 1.58228i −1.73205 + 1.00000i 0 −2.31662 −6.16228 − 3.32059i −2.00000 − 2.00000i 5.47036 + 3.15831i 0
93.2 0.366025 + 1.36603i 0.789997 − 2.94831i −1.73205 + 1.00000i 0 4.31662 0.796253 + 6.95457i −2.00000 − 2.00000i −0.274205 − 0.158312i 0
107.1 −1.36603 + 0.366025i −2.94831 − 0.789997i 1.73205 − 1.00000i 0 4.31662 −6.95457 + 0.796253i −2.00000 + 2.00000i 0.274205 + 0.158312i 0
107.2 −1.36603 + 0.366025i 1.58228 + 0.423972i 1.73205 − 1.00000i 0 −2.31662 3.32059 − 6.16228i −2.00000 + 2.00000i −5.47036 − 3.15831i 0
193.1 −1.36603 − 0.366025i −2.94831 + 0.789997i 1.73205 + 1.00000i 0 4.31662 −6.95457 − 0.796253i −2.00000 − 2.00000i 0.274205 − 0.158312i 0
193.2 −1.36603 − 0.366025i 1.58228 − 0.423972i 1.73205 + 1.00000i 0 −2.31662 3.32059 + 6.16228i −2.00000 − 2.00000i −5.47036 + 3.15831i 0
207.1 0.366025 − 1.36603i −0.423972 − 1.58228i −1.73205 − 1.00000i 0 −2.31662 −6.16228 + 3.32059i −2.00000 + 2.00000i 5.47036 − 3.15831i 0
207.2 0.366025 − 1.36603i 0.789997 + 2.94831i −1.73205 − 1.00000i 0 4.31662 0.796253 − 6.95457i −2.00000 + 2.00000i −0.274205 + 0.158312i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.p.a ✓ 8
5.b even 2 1 350.3.p.d yes 8
5.c odd 4 1 inner 350.3.p.a ✓ 8
5.c odd 4 1 350.3.p.d yes 8
7.c even 3 1 inner 350.3.p.a ✓ 8
35.j even 6 1 350.3.p.d yes 8
35.l odd 12 1 inner 350.3.p.a ✓ 8
35.l odd 12 1 350.3.p.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.p.a ✓ 8 1.a even 1 1 trivial
350.3.p.a ✓ 8 5.c odd 4 1 inner
350.3.p.a ✓ 8 7.c even 3 1 inner
350.3.p.a ✓ 8 35.l odd 12 1 inner
350.3.p.d yes 8 5.b even 2 1
350.3.p.d yes 8 5.c odd 4 1
350.3.p.d yes 8 35.j even 6 1
350.3.p.d yes 8 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} + 24T_{3}^{5} - T_{3}^{4} - 120T_{3}^{3} + 50T_{3}^{2} - 250T_{3} + 625 \) Copy content Toggle raw display
\( T_{11}^{4} + 8T_{11}^{3} + 224T_{11}^{2} - 1280T_{11} + 25600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 18 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 6146560000 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 5236114321 \) Copy content Toggle raw display
$29$ \( (T^{4} + 3218 T^{2} + 2430481)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + \cdots + 4494400)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 2552632508416 \) Copy content Toggle raw display
$41$ \( (T^{2} - 40 T - 139)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 62 T^{3} + \cdots + 225625)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 19522293907216 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 18431428857856 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11264239538176 \) Copy content Toggle raw display
$61$ \( (T^{4} - 128 T^{3} + \cdots + 14600041)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 458350804538641 \) Copy content Toggle raw display
$71$ \( (T^{2} + 180 T + 5944)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16796160000 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 753919797760000 \) Copy content Toggle raw display
$83$ \( (T^{4} + 250 T^{3} + \cdots + 33953929)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 3941340648961 \) Copy content Toggle raw display
$97$ \( (T^{4} + 184 T^{3} + \cdots + 11833600)^{2} \) Copy content Toggle raw display
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