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Properties

Label 350.3.i.b
Level $350$
Weight $3$
Character orbit 350.i
Analytic conductor $9.537$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [350,3,Mod(199,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.199"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.11007531417600000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 70)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{4}) q^{2} + (\beta_{12} - \beta_{6}) q^{3} + 2 \beta_{3} q^{4} + ( - \beta_{15} + \beta_{9} + \beta_{8} + \cdots + 1) q^{6} + (2 \beta_{13} - \beta_{12} + \cdots - 2 \beta_1) q^{7}+ \cdots + (4 \beta_{15} - 8 \beta_{14} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 24 q^{9} - 16 q^{11} - 80 q^{14} - 32 q^{16} + 8 q^{21} + 48 q^{24} - 192 q^{26} + 240 q^{29} - 144 q^{31} - 96 q^{36} - 48 q^{39} + 32 q^{44} + 40 q^{46} - 48 q^{49} + 112 q^{51} - 360 q^{54}+ \cdots - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 377\nu^{2} ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 281\nu^{2} ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{12} - 48\nu^{8} + 336\nu^{4} - 1 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} + 17\nu^{13} - 120\nu^{9} + 816\nu^{5} + 305\nu^{3} - 119\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{15} + 56 \nu^{14} + 21 \nu^{13} - 2 \nu^{12} - 384 \nu^{10} - 144 \nu^{9} + 2640 \nu^{6} + \cdots - 322 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{15} - 56 \nu^{14} + 21 \nu^{13} + 2 \nu^{12} + 384 \nu^{10} - 144 \nu^{9} - 2640 \nu^{6} + \cdots + 322 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + 7\nu^{13} - 48\nu^{9} + 336\nu^{5} + 329\nu^{3} - 49\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{14} + 48\nu^{10} - 328\nu^{6} + \nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} + 305\nu^{3} + 119\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91\nu^{15} - \nu^{13} - 624\nu^{11} + 4272\nu^{7} - 13\nu^{3} - 233\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 55 \nu^{15} + \nu^{14} + 56 \nu^{13} - 47 \nu^{12} - 384 \nu^{11} - 384 \nu^{9} + 336 \nu^{8} + \cdots + 329 ) / 144 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55 \nu^{15} + \nu^{14} - 56 \nu^{13} - 47 \nu^{12} + 384 \nu^{11} + 384 \nu^{9} + 336 \nu^{8} + \cdots + 329 ) / 144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -56\nu^{15} + \nu^{13} + 384\nu^{11} - 2640\nu^{7} + 8\nu^{3} + 377\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\nu^{15} + \nu^{13} - 624\nu^{11} + 4272\nu^{7} - 13\nu^{3} + 233\nu ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + 2\beta_{11} - \beta_{6} - \beta_{5} - 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{2} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{10} + 2\beta_{8} - 2\beta_{6} - 2\beta_{5} - 6\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{13} + 3\beta_{12} - 3\beta_{7} - 3\beta_{6} + 3\beta_{5} + 14\beta_{3} - 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{15} + 5\beta_{14} - 5\beta_{13} + 5\beta_{12} + 16\beta_{11} + 3\beta_{10} + 5\beta_{8} - 16\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{9} + 9\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} + 13 \beta_{12} - 16 \beta_{11} - 13 \beta_{6} + \cdots - 26 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\beta_{13} + 21\beta_{12} + 94\beta_{3} - 21\beta _1 - 94 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 21\beta_{10} + 34\beta_{8} + 34\beta_{6} + 34\beta_{5} - 42\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 110\beta_{9} + 123\beta_{7} + 110\beta_{2} - 123\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 55 \beta_{15} - 89 \beta_{14} - 89 \beta_{13} + 89 \beta_{12} - 110 \beta_{11} + 144 \beta_{10} + \cdots + 110 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 36\beta_{7} + 36\beta_{6} - 36\beta_{5} - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 377 \beta_{15} - 233 \beta_{14} + 233 \beta_{13} - 233 \beta_{12} - 754 \beta_{11} + \cdots + 466 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 754\beta_{2} - 843\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 987\beta_{10} - 610\beta_{8} + 610\beta_{6} + 610\beta_{5} + 1974\beta_{4} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 - \beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.159959 + 0.596975i
0.596975 − 0.159959i
−0.418778 − 1.56290i
−1.56290 + 0.418778i
−0.159959 − 0.596975i
−0.596975 + 0.159959i
0.418778 + 1.56290i
1.56290 − 0.418778i
0.159959 − 0.596975i
0.596975 + 0.159959i
−0.418778 + 1.56290i
−1.56290 − 0.418778i
−0.159959 + 0.596975i
−0.596975 − 0.159959i
0.418778 − 1.56290i
1.56290 + 0.418778i
−1.22474 + 0.707107i −2.16226 + 3.74514i 1.00000 − 1.73205i 0 − 6.11578i 6.91361 + 1.09634i 2.82843i −4.85071 − 8.40167i 0
199.2 −1.22474 + 0.707107i 1.15093 − 1.99347i 1.00000 − 1.73205i 0 3.25533i −3.02596 + 6.31218i 2.82843i 1.85071 + 3.20552i 0
199.3 −1.22474 + 0.707107i 1.65495 − 2.86646i 1.00000 − 1.73205i 0 4.68091i 4.40626 + 5.43919i 2.82843i −0.977722 − 1.69346i 0
199.4 −1.22474 + 0.707107i 1.80586 − 3.12785i 1.00000 − 1.73205i 0 5.10775i 3.95353 − 5.77664i 2.82843i −2.02228 − 3.50269i 0
199.5 1.22474 − 0.707107i −1.80586 + 3.12785i 1.00000 − 1.73205i 0 5.10775i −3.95353 + 5.77664i − 2.82843i −2.02228 − 3.50269i 0
199.6 1.22474 − 0.707107i −1.65495 + 2.86646i 1.00000 − 1.73205i 0 4.68091i −4.40626 − 5.43919i − 2.82843i −0.977722 − 1.69346i 0
199.7 1.22474 − 0.707107i −1.15093 + 1.99347i 1.00000 − 1.73205i 0 3.25533i 3.02596 − 6.31218i − 2.82843i 1.85071 + 3.20552i 0
199.8 1.22474 − 0.707107i 2.16226 − 3.74514i 1.00000 − 1.73205i 0 − 6.11578i −6.91361 − 1.09634i − 2.82843i −4.85071 − 8.40167i 0
299.1 −1.22474 − 0.707107i −2.16226 − 3.74514i 1.00000 + 1.73205i 0 6.11578i 6.91361 − 1.09634i − 2.82843i −4.85071 + 8.40167i 0
299.2 −1.22474 − 0.707107i 1.15093 + 1.99347i 1.00000 + 1.73205i 0 − 3.25533i −3.02596 − 6.31218i − 2.82843i 1.85071 − 3.20552i 0
299.3 −1.22474 − 0.707107i 1.65495 + 2.86646i 1.00000 + 1.73205i 0 − 4.68091i 4.40626 − 5.43919i − 2.82843i −0.977722 + 1.69346i 0
299.4 −1.22474 − 0.707107i 1.80586 + 3.12785i 1.00000 + 1.73205i 0 − 5.10775i 3.95353 + 5.77664i − 2.82843i −2.02228 + 3.50269i 0
299.5 1.22474 + 0.707107i −1.80586 − 3.12785i 1.00000 + 1.73205i 0 − 5.10775i −3.95353 − 5.77664i 2.82843i −2.02228 + 3.50269i 0
299.6 1.22474 + 0.707107i −1.65495 − 2.86646i 1.00000 + 1.73205i 0 − 4.68091i −4.40626 + 5.43919i 2.82843i −0.977722 + 1.69346i 0
299.7 1.22474 + 0.707107i −1.15093 − 1.99347i 1.00000 + 1.73205i 0 − 3.25533i 3.02596 + 6.31218i 2.82843i 1.85071 − 3.20552i 0
299.8 1.22474 + 0.707107i 2.16226 + 3.74514i 1.00000 + 1.73205i 0 6.11578i −6.91361 + 1.09634i 2.82843i −4.85071 + 8.40167i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.i.b 16
5.b even 2 1 inner 350.3.i.b 16
5.c odd 4 1 70.3.j.a ✓ 8
5.c odd 4 1 350.3.k.b 8
7.d odd 6 1 inner 350.3.i.b 16
15.e even 4 1 630.3.v.a 8
20.e even 4 1 560.3.bx.a 8
35.f even 4 1 490.3.j.a 8
35.i odd 6 1 inner 350.3.i.b 16
35.k even 12 1 70.3.j.a ✓ 8
35.k even 12 1 350.3.k.b 8
35.k even 12 1 490.3.b.b 8
35.l odd 12 1 490.3.b.b 8
35.l odd 12 1 490.3.j.a 8
105.w odd 12 1 630.3.v.a 8
140.x odd 12 1 560.3.bx.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.j.a ✓ 8 5.c odd 4 1
70.3.j.a ✓ 8 35.k even 12 1
350.3.i.b 16 1.a even 1 1 trivial
350.3.i.b 16 5.b even 2 1 inner
350.3.i.b 16 7.d odd 6 1 inner
350.3.i.b 16 35.i odd 6 1 inner
350.3.k.b 8 5.c odd 4 1
350.3.k.b 8 35.k even 12 1
490.3.b.b 8 35.k even 12 1
490.3.b.b 8 35.l odd 12 1
490.3.j.a 8 35.f even 4 1
490.3.j.a 8 35.l odd 12 1
560.3.bx.a 8 20.e even 4 1
560.3.bx.a 8 140.x odd 12 1
630.3.v.a 8 15.e even 4 1
630.3.v.a 8 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 48 T_{3}^{14} + 1486 T_{3}^{12} + 27648 T_{3}^{10} + 376179 T_{3}^{8} + 3391488 T_{3}^{6} + \cdots + 200533921 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 200533921 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} + 8 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 1056 T^{6} + \cdots + 2206744576)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 96717311574016 \) Copy content Toggle raw display
$19$ \( (T^{8} - 960 T^{6} + \cdots + 2265760000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{4} - 60 T^{3} + \cdots + 5569)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 72 T^{7} + \cdots + 46214680576)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 4105968689761)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 10368 T^{6} + \cdots + 551218638481)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 41420242452736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 72 T^{7} + \cdots + 216605537281)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 60\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{4} - 32 T^{3} + \cdots - 4477424)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 102942614692096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 2396523821041)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 132 T^{7} + \cdots + 35539413361)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 511556553523456)^{2} \) Copy content Toggle raw display
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