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Properties

Label 350.4.e.h
Level $350$
Weight $4$
Character orbit 350.e
Analytic conductor $20.651$
Analytic rank $0$
Dimension $4$
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] = [350,4,Mod(51,350)]
 
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([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.e (of order \(3\), 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: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{46})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 46x^{2} + 2116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - 4 \beta_{2} - 4) q^{4} + (2 \beta_{3} + 2) q^{6} + ( - 2 \beta_{3} + 5 \beta_{2} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{3} + 20 \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - 4 \beta_{2} - 4) q^{4} + (2 \beta_{3} + 2) q^{6} + ( - 2 \beta_{3} + 5 \beta_{2} + \cdots + 1) q^{7}+ \cdots + ( - 108 \beta_{3} - 864) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 8 q^{6} - 6 q^{7} + 32 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 8 q^{6} - 6 q^{7} + 32 q^{8} - 40 q^{9} + 68 q^{11} - 8 q^{12} + 104 q^{13} - 24 q^{14} - 32 q^{16} + 164 q^{17} - 80 q^{18} + 232 q^{19} + 110 q^{21} - 272 q^{22} - 198 q^{23} - 16 q^{24} - 104 q^{26} + 340 q^{27} + 72 q^{28} - 36 q^{29} - 196 q^{31} - 64 q^{32} + 252 q^{33} - 656 q^{34} + 320 q^{36} + 160 q^{37} + 464 q^{38} + 316 q^{39} + 124 q^{41} - 932 q^{42} - 396 q^{43} + 272 q^{44} - 396 q^{46} + 164 q^{47} + 64 q^{48} + 946 q^{49} + 900 q^{51} - 208 q^{52} + 40 q^{53} - 340 q^{54} - 48 q^{56} + 272 q^{57} + 36 q^{58} + 80 q^{59} + 174 q^{61} + 784 q^{62} + 496 q^{63} + 256 q^{64} + 504 q^{66} - 1054 q^{67} + 656 q^{68} - 2364 q^{69} - 1664 q^{71} - 320 q^{72} + 820 q^{73} + 320 q^{74} - 1856 q^{76} - 796 q^{77} - 1264 q^{78} + 576 q^{79} + 1370 q^{81} - 124 q^{82} + 596 q^{83} + 1424 q^{84} + 396 q^{86} + 1674 q^{87} + 544 q^{88} + 182 q^{89} + 1684 q^{91} + 1584 q^{92} - 2956 q^{93} + 328 q^{94} - 64 q^{96} - 1784 q^{97} - 2236 q^{98} - 3456 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 46x^{2} + 2116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 46\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 46\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
−3.39116 − 5.87367i
3.39116 + 5.87367i
−3.39116 + 5.87367i
3.39116 − 5.87367i
−1.00000 + 1.73205i −3.89116 − 6.73970i −2.00000 − 3.46410i 0 15.5647 −18.4558 − 1.54354i 8.00000 −16.7823 + 29.0678i 0
51.2 −1.00000 + 1.73205i 2.89116 + 5.00764i −2.00000 − 3.46410i 0 −11.5647 15.4558 + 10.2038i 8.00000 −3.21767 + 5.57317i 0
151.1 −1.00000 − 1.73205i −3.89116 + 6.73970i −2.00000 + 3.46410i 0 15.5647 −18.4558 + 1.54354i 8.00000 −16.7823 − 29.0678i 0
151.2 −1.00000 − 1.73205i 2.89116 − 5.00764i −2.00000 + 3.46410i 0 −11.5647 15.4558 − 10.2038i 8.00000 −3.21767 − 5.57317i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.e.h 4
5.b even 2 1 70.4.e.d ✓ 4
5.c odd 4 2 350.4.j.g 8
7.c even 3 1 inner 350.4.e.h 4
7.c even 3 1 2450.4.a.bz 2
7.d odd 6 1 2450.4.a.bv 2
15.d odd 2 1 630.4.k.l 4
20.d odd 2 1 560.4.q.j 4
35.c odd 2 1 490.4.e.u 4
35.i odd 6 1 490.4.a.t 2
35.i odd 6 1 490.4.e.u 4
35.j even 6 1 70.4.e.d ✓ 4
35.j even 6 1 490.4.a.r 2
35.l odd 12 2 350.4.j.g 8
105.o odd 6 1 630.4.k.l 4
140.p odd 6 1 560.4.q.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.d ✓ 4 5.b even 2 1
70.4.e.d ✓ 4 35.j even 6 1
350.4.e.h 4 1.a even 1 1 trivial
350.4.e.h 4 7.c even 3 1 inner
350.4.j.g 8 5.c odd 4 2
350.4.j.g 8 35.l odd 12 2
490.4.a.r 2 35.j even 6 1
490.4.a.t 2 35.i odd 6 1
490.4.e.u 4 35.c odd 2 1
490.4.e.u 4 35.i odd 6 1
560.4.q.j 4 20.d odd 2 1
560.4.q.j 4 140.p odd 6 1
630.4.k.l 4 15.d odd 2 1
630.4.k.l 4 105.o odd 6 1
2450.4.a.bv 2 7.d odd 6 1
2450.4.a.bz 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 49T_{3}^{2} - 90T_{3} + 2025 \) 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

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 68 T^{3} + \cdots + 944784 \) Copy content Toggle raw display
$13$ \( (T^{2} - 52 T - 60)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 164 T^{3} + \cdots + 14288400 \) Copy content Toggle raw display
$19$ \( T^{4} - 232 T^{3} + \cdots + 161798400 \) Copy content Toggle raw display
$23$ \( T^{4} + 198 T^{3} + \cdots + 301401 \) Copy content Toggle raw display
$29$ \( (T^{2} + 18 T - 14823)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1010985616 \) Copy content Toggle raw display
$37$ \( T^{4} - 160 T^{3} + \cdots + 50176 \) Copy content Toggle raw display
$41$ \( (T^{2} - 62 T - 65463)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 198 T - 18949)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 164 T^{3} + \cdots + 14288400 \) Copy content Toggle raw display
$53$ \( T^{4} - 40 T^{3} + \cdots + 6471936 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 245952516096 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 121560309025 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 5074425225 \) Copy content Toggle raw display
$71$ \( (T^{2} + 832 T + 99456)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 3858397456 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 11124342784 \) Copy content Toggle raw display
$83$ \( (T^{2} - 298 T - 719733)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 66262482225 \) Copy content Toggle raw display
$97$ \( (T^{2} + 892 T + 92932)^{2} \) Copy content Toggle raw display
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