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Properties

Label 350.4.j.h
Level $350$
Weight $4$
Character orbit 350.j
Analytic conductor $20.651$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [350,4,Mod(149,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,24,0,24,0,0,112,0,52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 134x^{10} + 13467x^{8} - 530084x^{6} + 15364507x^{4} - 160351569x^{2} + 1275989841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - 2 \beta_{5} q^{2} + (\beta_{9} - \beta_{8} - \beta_{5}) q^{3} + 4 \beta_{2} q^{4} + (2 \beta_{4} + 2) q^{6} + ( - \beta_{11} - 7 \beta_{8} + \cdots + \beta_1) q^{7} + 8 \beta_{8} q^{8} + ( - \beta_{7} - 2 \beta_{4} + \cdots + 19) q^{9}+ \cdots + (6 \beta_{6} - 121 \beta_{4} + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{4} + 24 q^{6} + 112 q^{9} + 52 q^{11} + 16 q^{14} - 96 q^{16} - 36 q^{19} + 370 q^{21} + 48 q^{24} + 320 q^{26} + 1616 q^{29} + 1230 q^{31} + 240 q^{34} + 896 q^{36} - 180 q^{39} - 404 q^{41}+ \cdots + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 134x^{10} + 13467x^{8} - 530084x^{6} + 15364507x^{4} - 160351569x^{2} + 1275989841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4489 \nu^{10} - 553898 \nu^{8} + 55666749 \nu^{6} - 1898492369 \nu^{4} + 63510221629 \nu^{2} - 170982638694 ) / 491841341649 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -134\nu^{10} + 13467\nu^{8} - 1661694\nu^{6} + 56671414\nu^{4} - 2927914338\nu^{2} + 5103959364 ) / 7807005423 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -134\nu^{10} + 13467\nu^{8} - 1045173\nu^{6} + 15364507\nu^{4} - 160351569\nu^{2} + 27126706005 ) / 7807005423 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{11} + 402\nu^{9} - 40401\nu^{7} + 458642\nu^{5} - 4786614\nu^{3} - 2286508062\nu ) / 7807005423 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1292 \nu^{10} - 129846 \nu^{8} + 11816481 \nu^{6} - 148141366 \nu^{4} + 1546076322 \nu^{2} + 151889080392 ) / 7807005423 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9649 \nu^{10} + 1483492 \nu^{8} - 149090946 \nu^{6} + 6868672691 \nu^{4} + \cdots + 1775225894022 ) / 54649037961 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4489 \nu^{11} - 553898 \nu^{9} + 55666749 \nu^{7} - 1898492369 \nu^{5} + \cdots - 170982638694 \nu ) / 1387433038383 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4489 \nu^{11} + 553898 \nu^{9} - 55666749 \nu^{7} + 1898492369 \nu^{5} + \cdots + 662823980343 \nu ) / 491841341649 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -314\nu^{11} + 31557\nu^{9} - 2863218\nu^{7} + 36003397\nu^{5} - 375749199\nu^{3} - 44538135093\nu ) / 7807005423 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 534035 \nu^{11} + 79461646 \nu^{9} - 7985895423 \nu^{7} + 353912809666 \nu^{5} + \cdots + 95088056801661 \nu ) / 10328668174629 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 4\beta_{3} + 45\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -67\beta_{9} - 189\beta_{8} + 67\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 67\beta_{7} + 457\beta_{4} + 457\beta_{3} + 3015\beta_{2} - 3015 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -189\beta_{11} - 5245\beta_{9} - 21168\beta_{8} - 21168\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4489\beta_{6} + 43282\beta_{4} - 237726 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 25326\beta_{10} - 1988091\beta_{5} - 437788\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -336484\beta_{7} + 336484\beta_{6} - 3891199\beta_{3} - 19928394\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2545263\beta_{11} + 2545263\beta_{10} + 37512094\beta_{9} + 178132311\beta_{8} - 37512094\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -27331042\beta_{7} - 343452265\beta_{4} - 343452265\beta_{3} - 1710951597\beta_{2} + 1710951597 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 234128097\beta_{11} + 3248746909\beta_{9} + 15701331303\beta_{8} + 15701331303\beta_{5} \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−5.13062 − 2.96216i
−2.95806 − 1.70784i
8.08868 + 4.67000i
−8.08868 − 4.67000i
2.95806 + 1.70784i
5.13062 + 2.96216i
−5.13062 + 2.96216i
−2.95806 + 1.70784i
8.08868 − 4.67000i
−8.08868 + 4.67000i
2.95806 − 1.70784i
5.13062 − 2.96216i
−1.73205 − 1.00000i −5.99664 + 3.46216i 2.00000 + 3.46410i 0 13.8487 −12.7473 + 13.4353i − 8.00000i 10.4731 − 18.1400i 0
149.2 −1.73205 − 1.00000i −3.82408 + 2.20784i 2.00000 + 3.46410i 0 8.83134 18.4072 − 2.04309i − 8.00000i −3.75092 + 6.49679i 0
149.3 −1.73205 − 1.00000i 7.22265 − 4.17000i 2.00000 + 3.46410i 0 −16.6800 8.19644 + 16.6078i − 8.00000i 21.2778 − 36.8542i 0
149.4 1.73205 + 1.00000i −7.22265 + 4.17000i 2.00000 + 3.46410i 0 −16.6800 −8.19644 − 16.6078i 8.00000i 21.2778 − 36.8542i 0
149.5 1.73205 + 1.00000i 3.82408 − 2.20784i 2.00000 + 3.46410i 0 8.83134 −18.4072 + 2.04309i 8.00000i −3.75092 + 6.49679i 0
149.6 1.73205 + 1.00000i 5.99664 − 3.46216i 2.00000 + 3.46410i 0 13.8487 12.7473 − 13.4353i 8.00000i 10.4731 − 18.1400i 0
249.1 −1.73205 + 1.00000i −5.99664 − 3.46216i 2.00000 − 3.46410i 0 13.8487 −12.7473 − 13.4353i 8.00000i 10.4731 + 18.1400i 0
249.2 −1.73205 + 1.00000i −3.82408 − 2.20784i 2.00000 − 3.46410i 0 8.83134 18.4072 + 2.04309i 8.00000i −3.75092 − 6.49679i 0
249.3 −1.73205 + 1.00000i 7.22265 + 4.17000i 2.00000 − 3.46410i 0 −16.6800 8.19644 − 16.6078i 8.00000i 21.2778 + 36.8542i 0
249.4 1.73205 − 1.00000i −7.22265 − 4.17000i 2.00000 − 3.46410i 0 −16.6800 −8.19644 + 16.6078i − 8.00000i 21.2778 + 36.8542i 0
249.5 1.73205 − 1.00000i 3.82408 + 2.20784i 2.00000 − 3.46410i 0 8.83134 −18.4072 − 2.04309i − 8.00000i −3.75092 − 6.49679i 0
249.6 1.73205 − 1.00000i 5.99664 + 3.46216i 2.00000 − 3.46410i 0 13.8487 12.7473 + 13.4353i − 8.00000i 10.4731 + 18.1400i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.6
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.h 12
5.b even 2 1 inner 350.4.j.h 12
5.c odd 4 1 350.4.e.i ✓ 6
5.c odd 4 1 350.4.e.j yes 6
7.c even 3 1 inner 350.4.j.h 12
35.j even 6 1 inner 350.4.j.h 12
35.k even 12 1 2450.4.a.cd 3
35.k even 12 1 2450.4.a.ch 3
35.l odd 12 1 350.4.e.i ✓ 6
35.l odd 12 1 350.4.e.j yes 6
35.l odd 12 1 2450.4.a.cc 3
35.l odd 12 1 2450.4.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.4.e.i ✓ 6 5.c odd 4 1
350.4.e.i ✓ 6 35.l odd 12 1
350.4.e.j yes 6 5.c odd 4 1
350.4.e.j yes 6 35.l odd 12 1
350.4.j.h 12 1.a even 1 1 trivial
350.4.j.h 12 5.b even 2 1 inner
350.4.j.h 12 7.c even 3 1 inner
350.4.j.h 12 35.j even 6 1 inner
2450.4.a.cc 3 35.l odd 12 1
2450.4.a.cd 3 35.k even 12 1
2450.4.a.ch 3 35.k even 12 1
2450.4.a.ci 3 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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{12} - 137T_{3}^{10} + 13143T_{3}^{8} - 640712T_{3}^{6} + 22743451T_{3}^{4} - 365830650T_{3}^{2} + 4228250625 \) Copy content Toggle raw display
\( T_{11}^{6} - 26T_{11}^{5} + 2069T_{11}^{4} - 36596T_{11}^{3} + 2887031T_{11}^{2} - 50714951T_{11} + 1325469649 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 4228250625 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{6} - 26 T^{5} + \cdots + 1325469649)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 11630 T^{4} + \cdots + 25832525625)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( (T^{6} + 18 T^{5} + \cdots + 9063992025)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 80\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{3} - 404 T^{2} + \cdots - 1808541)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 57192570880929)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( (T^{3} + 101 T^{2} + \cdots - 7167)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 783985160070225)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 77\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 76\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 86\!\cdots\!25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 647 T^{5} + \cdots + 863302997881)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{3} - 533 T^{2} + \cdots + 74350109)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 30\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 86\!\cdots\!09)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 71\!\cdots\!69)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 760149 T^{4} + \cdots + 779726086441)^{2} \) Copy content Toggle raw display
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