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Properties

Label 351.2.i.b
Level $351$
Weight $2$
Character orbit 351.i
Analytic conductor $2.803$
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] = [351,2,Mod(161,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 184x^{12} + 600x^{10} + 894x^{8} + 600x^{6} + 184x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - 2 \beta_{9} - \beta_{3} - \beta_{2}) q^{4} - \beta_{13} q^{5} + (\beta_{8} + \beta_{6} + \beta_{3}) q^{7} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{5}) q^{8} + ( - \beta_{9} - 3 \beta_{2}) q^{10}+ \cdots + (4 \beta_{11} + \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7} + 16 q^{13} - 8 q^{16} - 32 q^{19} + 8 q^{22} + 40 q^{28} + 16 q^{31} + 24 q^{34} - 32 q^{37} - 72 q^{40} + 48 q^{46} + 48 q^{52} + 32 q^{55} + 56 q^{58} - 64 q^{61} - 32 q^{67} - 40 q^{70}+ \cdots + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 184x^{12} + 600x^{10} + 894x^{8} + 600x^{6} + 184x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 26\nu^{12} + 231\nu^{10} + 944\nu^{8} + 1911\nu^{6} + 1810\nu^{4} + 625\nu^{2} + 52 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{15} + 159\nu^{13} + 1077\nu^{11} + 2661\nu^{9} + 1709\nu^{7} - 1331\nu^{5} - 1001\nu^{3} - 81\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{15} - 24\nu^{13} - 184\nu^{11} - 600\nu^{9} - 894\nu^{7} - 600\nu^{5} - 184\nu^{3} - 23\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{14} + 356\nu^{12} + 2665\nu^{10} + 8288\nu^{8} + 11193\nu^{6} + 6004\nu^{4} + 1151\nu^{2} + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8 \nu^{15} - 31 \nu^{14} + 192 \nu^{13} - 736 \nu^{12} + 1472 \nu^{11} - 5515 \nu^{10} + 4800 \nu^{9} + \cdots - 178 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{14} - 213\nu^{12} - 1585\nu^{10} - 4872\nu^{8} - 6429\nu^{6} - 3297\nu^{4} - 629\nu^{2} - 38 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8 \nu^{15} + 31 \nu^{14} + 192 \nu^{13} + 736 \nu^{12} + 1472 \nu^{11} + 5515 \nu^{10} + 4800 \nu^{9} + \cdots + 178 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{15} - 379\nu^{13} - 2825\nu^{11} - 8704\nu^{9} - 11488\nu^{7} - 5729\nu^{5} - 819\nu^{3} + 16\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -25\nu^{15} - 594\nu^{13} - 4458\nu^{11} - 13943\nu^{9} - 19095\nu^{7} - 10674\nu^{5} - 2330\nu^{3} - 153\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 53 \nu^{15} - 21 \nu^{14} + 1273 \nu^{13} - 498 \nu^{12} + 9773 \nu^{11} - 3721 \nu^{10} + \cdots + 675 \nu ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 71 \nu^{15} + 8 \nu^{14} - 1685 \nu^{13} + 185 \nu^{12} - 12613 \nu^{11} + 1308 \nu^{10} - 39223 \nu^{9} + \cdots - 37 ) / 16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 53 \nu^{15} + 21 \nu^{14} + 1273 \nu^{13} + 498 \nu^{12} + 9773 \nu^{11} + 3721 \nu^{10} + \cdots + 675 \nu ) / 16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 71 \nu^{15} - 8 \nu^{14} - 1685 \nu^{13} - 185 \nu^{12} - 12613 \nu^{11} - 1308 \nu^{10} - 39223 \nu^{9} + \cdots + 37 ) / 16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43 \nu^{15} + \nu^{14} - 1020 \nu^{13} + 23 \nu^{12} - 7628 \nu^{11} + 161 \nu^{10} - 23687 \nu^{9} + \cdots - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 43 \nu^{15} + \nu^{14} + 1020 \nu^{13} + 23 \nu^{12} + 7628 \nu^{11} + 161 \nu^{10} + 23687 \nu^{9} + \cdots - 7 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} - \beta_{14} - \beta_{12} + \beta_{10} + \beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 4 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 8 \beta_{7} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 30 \beta_{15} - 30 \beta_{14} + 26 \beta_{13} + 34 \beta_{12} + 26 \beta_{11} + 34 \beta_{10} + \cdots + 76 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 119 \beta_{15} - 119 \beta_{14} + 72 \beta_{13} - 147 \beta_{12} - 72 \beta_{11} + 147 \beta_{10} + \cdots - 550 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 404 \beta_{15} + 404 \beta_{14} - 320 \beta_{13} - 480 \beta_{12} - 320 \beta_{11} + \cdots - 1104 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 754 \beta_{15} + 754 \beta_{14} - 516 \beta_{13} + 928 \beta_{12} + 516 \beta_{11} - 928 \beta_{10} + \cdots + 3353 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5324 \beta_{15} - 5324 \beta_{14} + 4032 \beta_{13} + 6432 \beta_{12} + 4032 \beta_{11} + \cdots + 14928 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19465 \beta_{15} - 19465 \beta_{14} + 13896 \beta_{13} - 23845 \beta_{12} - 13896 \beta_{11} + \cdots - 85126 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 69670 \beta_{15} + 69670 \beta_{14} - 51722 \beta_{13} - 84674 \beta_{12} - 51722 \beta_{11} + \cdots - 197140 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 126367 \beta_{15} + 126367 \beta_{14} - 91622 \beta_{13} + 154418 \beta_{12} + 91622 \beta_{11} + \cdots + 548873 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 909242 \beta_{15} - 909242 \beta_{14} + 669394 \beta_{13} + 1107490 \beta_{12} + \cdots + 2581468 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3288287 \beta_{15} - 3288287 \beta_{14} + 2398224 \beta_{13} - 4013783 \beta_{12} - 2398224 \beta_{11} + \cdots - 14243558 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 11853384 \beta_{15} + 11853384 \beta_{14} - 8696960 \beta_{13} - 14449856 \beta_{12} + \cdots - 33696160 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(\beta_{9}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.277061i
2.26780i
0.592624i
1.34416i
− 0.743961i
− 1.68741i
− 0.440957i
− 3.60931i
− 0.277061i
− 2.26780i
− 0.592624i
− 1.34416i
0.743961i
1.68741i
0.440957i
3.60931i
−1.94318 − 1.94318i 0 5.55193i −0.426892 − 0.426892i 0 −1.60020 − 1.60020i 6.90206 − 6.90206i 0 1.65906i
161.2 −1.35438 − 1.35438i 0 1.66867i −2.02548 − 2.02548i 0 0.0947876 + 0.0947876i −0.448746 + 0.448746i 0 5.48652i
161.3 −1.14002 − 1.14002i 0 0.599280i 2.84492 + 2.84492i 0 −2.82684 − 2.82684i −1.59685 + 1.59685i 0 − 6.48652i
161.4 −1.04406 − 1.04406i 0 0.180117i 1.27342 + 1.27342i 0 2.33225 + 2.33225i −1.90006 + 1.90006i 0 − 2.65906i
161.5 1.04406 + 1.04406i 0 0.180117i −1.27342 − 1.27342i 0 2.33225 + 2.33225i 1.90006 − 1.90006i 0 − 2.65906i
161.6 1.14002 + 1.14002i 0 0.599280i −2.84492 − 2.84492i 0 −2.82684 − 2.82684i 1.59685 − 1.59685i 0 − 6.48652i
161.7 1.35438 + 1.35438i 0 1.66867i 2.02548 + 2.02548i 0 0.0947876 + 0.0947876i 0.448746 − 0.448746i 0 5.48652i
161.8 1.94318 + 1.94318i 0 5.55193i 0.426892 + 0.426892i 0 −1.60020 − 1.60020i −6.90206 + 6.90206i 0 1.65906i
242.1 −1.94318 + 1.94318i 0 − 5.55193i −0.426892 + 0.426892i 0 −1.60020 + 1.60020i 6.90206 + 6.90206i 0 − 1.65906i
242.2 −1.35438 + 1.35438i 0 − 1.66867i −2.02548 + 2.02548i 0 0.0947876 − 0.0947876i −0.448746 − 0.448746i 0 − 5.48652i
242.3 −1.14002 + 1.14002i 0 − 0.599280i 2.84492 − 2.84492i 0 −2.82684 + 2.82684i −1.59685 − 1.59685i 0 6.48652i
242.4 −1.04406 + 1.04406i 0 − 0.180117i 1.27342 − 1.27342i 0 2.33225 − 2.33225i −1.90006 − 1.90006i 0 2.65906i
242.5 1.04406 − 1.04406i 0 − 0.180117i −1.27342 + 1.27342i 0 2.33225 − 2.33225i 1.90006 + 1.90006i 0 2.65906i
242.6 1.14002 − 1.14002i 0 − 0.599280i −2.84492 + 2.84492i 0 −2.82684 + 2.82684i 1.59685 + 1.59685i 0 6.48652i
242.7 1.35438 − 1.35438i 0 − 1.66867i 2.02548 − 2.02548i 0 0.0947876 − 0.0947876i 0.448746 + 0.448746i 0 − 5.48652i
242.8 1.94318 − 1.94318i 0 − 5.55193i 0.426892 − 0.426892i 0 −1.60020 + 1.60020i −6.90206 − 6.90206i 0 − 1.65906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.2.i.b ✓ 16
3.b odd 2 1 inner 351.2.i.b ✓ 16
13.d odd 4 1 inner 351.2.i.b ✓ 16
39.f even 4 1 inner 351.2.i.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.i.b ✓ 16 1.a even 1 1 trivial
351.2.i.b ✓ 16 3.b odd 2 1 inner
351.2.i.b ✓ 16 13.d odd 4 1 inner
351.2.i.b ✓ 16 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 82T_{2}^{12} + 1611T_{2}^{8} + 11098T_{2}^{4} + 24649 \) acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 82 T^{12} + \cdots + 24649 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 340 T^{12} + \cdots + 24649 \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 688 T^{12} + \cdots + 6310144 \) Copy content Toggle raw display
$13$ \( (T^{8} - 8 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 96 T^{6} + \cdots + 203472)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 16 T^{7} + \cdots + 80656)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 120 T^{6} + \cdots + 22608)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 80 T^{6} + \cdots + 2512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 1336336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 16 T^{7} + \cdots + 824464)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 688 T^{12} + \cdots + 6310144 \) Copy content Toggle raw display
$43$ \( (T^{8} + 84 T^{6} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 67786191498169 \) Copy content Toggle raw display
$53$ \( (T^{8} + 200 T^{6} + \cdots + 40192)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 12628 T^{12} + \cdots + 24649 \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} + \cdots - 311)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots + 327184)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 100990701644569 \) Copy content Toggle raw display
$73$ \( (T^{8} + 28 T^{7} + \cdots + 1763584)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 976)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 530388868409929 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 6897800809 \) Copy content Toggle raw display
$97$ \( (T^{8} - 32 T^{7} + \cdots + 1926544)^{2} \) Copy content Toggle raw display
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