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Properties

Label 350.4.j.g
Level $350$
Weight $4$
Character orbit 350.j
Analytic conductor $20.651$
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] = [350,4,Mod(149,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.149");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\), degree \(2\), not 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: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1485512441856.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 529x^{4} + 279841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 70)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + 4 \beta_{2} q^{4} + ( - \beta_{7} - \beta_{6} + 2) q^{6} + (\beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{7} - 4 \beta_{5} q^{8} + ( - \beta_{7} - 20 \beta_{2} + 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + 4 \beta_{2} q^{4} + ( - \beta_{7} - \beta_{6} + 2) q^{6} + (\beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{7} - 4 \beta_{5} q^{8} + ( - \beta_{7} - 20 \beta_{2} + 20) q^{9} + ( - \beta_{6} + 34 \beta_{2}) q^{11} + ( - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3}) q^{12} + ( - 11 \beta_{5} - 4 \beta_{3}) q^{13} + (3 \beta_{7} + \beta_{6} + 8 \beta_{2} + 2) q^{14} + (16 \beta_{2} - 16) q^{16} + (37 \beta_{5} - 8 \beta_{4} + 37 \beta_1) q^{17} + (18 \beta_{5} - 4 \beta_{4} + 18 \beta_1) q^{18} + ( - 2 \beta_{7} + 116 \beta_{2} - 116) q^{19} + ( - 2 \beta_{7} - 3 \beta_{6} + \cdots - 41) q^{21}+ \cdots + ( - 54 \beta_{7} - 54 \beta_{6} + 864) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + 16 q^{6} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 16 q^{6} + 80 q^{9} + 136 q^{11} + 48 q^{14} - 64 q^{16} - 464 q^{19} + 220 q^{21} + 32 q^{24} - 208 q^{26} + 72 q^{29} - 392 q^{31} + 1312 q^{34} + 640 q^{36} - 632 q^{39} + 248 q^{41} - 544 q^{44} - 792 q^{46} - 1892 q^{49} + 1800 q^{51} + 680 q^{54} - 96 q^{56} - 160 q^{59} + 348 q^{61} - 512 q^{64} + 1008 q^{66} + 4728 q^{69} - 3328 q^{71} - 640 q^{74} - 3712 q^{76} - 1152 q^{79} + 2740 q^{81} - 2848 q^{84} + 792 q^{86} - 364 q^{89} + 3368 q^{91} - 656 q^{94} - 128 q^{96} + 6912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 529x^{4} + 279841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} ) / 529 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 23\nu^{5} - 529\nu^{3} + 12167\nu ) / 12167 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 529\nu^{2} + 12167\nu ) / 12167 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} ) / 12167 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + 24334\nu ) / 12167 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 46\nu^{5} + 1058\nu^{3} ) / 12167 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + 2\beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 23\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 23\beta_{7} + 23\beta_{6} + 23\beta_{5} - 46\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 529\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -529\beta_{7} + 1058\beta_{5} + 1058\beta_{4} - 1058\beta_{3} + 529\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -12167\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12167\beta_{6} - 12167\beta_{5} - 24334\beta_{4} - 12167\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.24125 − 4.63242i
−1.24125 + 4.63242i
4.63242 + 1.24125i
−4.63242 − 1.24125i
1.24125 + 4.63242i
−1.24125 − 4.63242i
4.63242 − 1.24125i
−4.63242 + 1.24125i
−1.73205 − 1.00000i −6.73970 + 3.89116i 2.00000 + 3.46410i 0 15.5647 1.54354 − 18.4558i − 8.00000i 16.7823 − 29.0678i 0
149.2 −1.73205 − 1.00000i 5.00764 − 2.89116i 2.00000 + 3.46410i 0 −11.5647 −10.2038 + 15.4558i − 8.00000i 3.21767 − 5.57317i 0
149.3 1.73205 + 1.00000i −5.00764 + 2.89116i 2.00000 + 3.46410i 0 −11.5647 10.2038 − 15.4558i 8.00000i 3.21767 − 5.57317i 0
149.4 1.73205 + 1.00000i 6.73970 − 3.89116i 2.00000 + 3.46410i 0 15.5647 −1.54354 + 18.4558i 8.00000i 16.7823 − 29.0678i 0
249.1 −1.73205 + 1.00000i −6.73970 − 3.89116i 2.00000 − 3.46410i 0 15.5647 1.54354 + 18.4558i 8.00000i 16.7823 + 29.0678i 0
249.2 −1.73205 + 1.00000i 5.00764 + 2.89116i 2.00000 − 3.46410i 0 −11.5647 −10.2038 − 15.4558i 8.00000i 3.21767 + 5.57317i 0
249.3 1.73205 − 1.00000i −5.00764 − 2.89116i 2.00000 − 3.46410i 0 −11.5647 10.2038 + 15.4558i − 8.00000i 3.21767 + 5.57317i 0
249.4 1.73205 − 1.00000i 6.73970 + 3.89116i 2.00000 − 3.46410i 0 15.5647 −1.54354 − 18.4558i − 8.00000i 16.7823 + 29.0678i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.j.g 8
5.b even 2 1 inner 350.4.j.g 8
5.c odd 4 1 70.4.e.d ✓ 4
5.c odd 4 1 350.4.e.h 4
7.c even 3 1 inner 350.4.j.g 8
15.e even 4 1 630.4.k.l 4
20.e even 4 1 560.4.q.j 4
35.f even 4 1 490.4.e.u 4
35.j even 6 1 inner 350.4.j.g 8
35.k even 12 1 490.4.a.t 2
35.k even 12 1 490.4.e.u 4
35.k even 12 1 2450.4.a.bv 2
35.l odd 12 1 70.4.e.d ✓ 4
35.l odd 12 1 350.4.e.h 4
35.l odd 12 1 490.4.a.r 2
35.l odd 12 1 2450.4.a.bz 2
105.x even 12 1 630.4.k.l 4
140.w even 12 1 560.4.q.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.d ✓ 4 5.c odd 4 1
70.4.e.d ✓ 4 35.l odd 12 1
350.4.e.h 4 5.c odd 4 1
350.4.e.h 4 35.l odd 12 1
350.4.j.g 8 1.a even 1 1 trivial
350.4.j.g 8 5.b even 2 1 inner
350.4.j.g 8 7.c even 3 1 inner
350.4.j.g 8 35.j even 6 1 inner
490.4.a.r 2 35.l odd 12 1
490.4.a.t 2 35.k even 12 1
490.4.e.u 4 35.f even 4 1
490.4.e.u 4 35.k even 12 1
560.4.q.j 4 20.e even 4 1
560.4.q.j 4 140.w even 12 1
630.4.k.l 4 15.e even 4 1
630.4.k.l 4 105.x even 12 1
2450.4.a.bv 2 35.k even 12 1
2450.4.a.bz 2 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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{8} - 94T_{3}^{6} + 6811T_{3}^{4} - 190350T_{3}^{2} + 4100625 \) Copy content Toggle raw display
\( T_{11}^{4} - 68T_{11}^{3} + 3652T_{11}^{2} - 66096T_{11} + 944784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 94 T^{6} + \cdots + 4100625 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{4} - 68 T^{3} + \cdots + 944784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2824 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 204158374560000 \) Copy content Toggle raw display
$19$ \( (T^{4} + 232 T^{3} + \cdots + 161798400)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 90842562801 \) Copy content Toggle raw display
$29$ \( (T^{2} - 18 T - 14823)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 196 T^{3} + \cdots + 1010985616)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 2517630976 \) Copy content Toggle raw display
$41$ \( (T^{2} - 62 T - 65463)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 77102 T^{2} + 359064601)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 204158374560000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41885955588096 \) Copy content Toggle raw display
$59$ \( (T^{4} + 80 T^{3} + \cdots + 245952516096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 174 T^{3} + \cdots + 121560309025)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{2} + 832 T + 99456)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} + 576 T^{3} + \cdots + 11124342784)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1528270 T^{2} + 518015591289)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 182 T^{3} + \cdots + 66262482225)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 609800 T^{2} + 8636356624)^{2} \) Copy content Toggle raw display
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