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Properties

Label 3312.2.e.h
Level $3312$
Weight $2$
Character orbit 3312.e
Analytic conductor $26.446$
Analytic rank $0$
Dimension $16$
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] = [3312,2,Mod(1151,3312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3312.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.e (of order \(2\), degree \(1\), 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: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 10 x^{14} - 24 x^{13} + 80 x^{12} + 192 x^{11} - 120 x^{10} - 984 x^{9} - 28 x^{8} + \cdots + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + \beta_{8} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + \beta_{8} q^{7} - \beta_{10} q^{11} + (\beta_{12} - 1) q^{13} + (\beta_{11} - \beta_{8}) q^{17} + (\beta_{11} + \beta_{3}) q^{19} + q^{23} + ( - \beta_{4} + \beta_1 - 1) q^{25} + ( - \beta_{9} + \beta_{6} + \beta_{2}) q^{29} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_{7}) q^{31}+ \cdots + (\beta_{14} + 2 \beta_{13} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{11} - 8 q^{13} + 16 q^{23} - 8 q^{25} - 32 q^{35} + 64 q^{47} - 24 q^{49} + 32 q^{59} + 16 q^{61} + 48 q^{71} - 16 q^{73} - 8 q^{83} + 24 q^{85} + 80 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 10 x^{14} - 24 x^{13} + 80 x^{12} + 192 x^{11} - 120 x^{10} - 984 x^{9} - 28 x^{8} + \cdots + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10406044374189 \nu^{15} + 41478903847548 \nu^{14} - 132428639982065 \nu^{13} + \cdots + 14\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 226664662260881 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 447148039058761 \nu^{15} + 1360517570500 \nu^{14} + \cdots - 52\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 210707313227593 \nu^{15} + 410266275608855 \nu^{14} + \cdots - 41\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 212067830798093 \nu^{15} - 488370641823105 \nu^{14} + \cdots - 35\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5370562 \nu^{15} + 15328900 \nu^{14} + 24014895 \nu^{13} + 4762113 \nu^{12} + \cdots - 31759274500 ) / 18978194250 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36510266 \nu^{15} - 111024700 \nu^{14} - 158236485 \nu^{13} - 822384 \nu^{12} + \cdots + 229295716000 ) / 111141777000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!93 \nu^{15} + \cdots + 95\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 205444317150524 \nu^{15} + 176344917821988 \nu^{14} + \cdots - 37\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!57 \nu^{15} + \cdots - 92\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!91 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 717342087418176 \nu^{15} - 365649875161655 \nu^{14} + \cdots + 73\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!81 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 53\!\cdots\!19 \nu^{15} + \cdots + 39\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - 3\beta_{12} - 4\beta_{10} - \beta_{4} - 6\beta_{3} + 4\beta _1 - 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 2 \beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + \cdots + 8 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 2 \beta_{10} - 6 \beta_{8} + \cdots + 31 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} + 6 \beta_{14} + 24 \beta_{11} - 12 \beta_{10} - 12 \beta_{7} - 15 \beta_{6} - 2 \beta_{5} + \cdots - 62 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 36 \beta_{15} + 18 \beta_{14} + 5 \beta_{13} + 39 \beta_{12} + 60 \beta_{11} + 20 \beta_{10} + 12 \beta_{9} + \cdots + 89 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7 \beta_{15} + 51 \beta_{14} + 45 \beta_{13} + 48 \beta_{12} + 71 \beta_{11} + 18 \beta_{10} - 2 \beta_{9} + \cdots - 84 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 21 \beta_{15} + 45 \beta_{14} - 10 \beta_{13} + 75 \beta_{12} + 96 \beta_{11} + 2 \beta_{10} + \cdots - 751 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 84 \beta_{15} + 162 \beta_{14} + 162 \beta_{13} + 396 \beta_{12} + 204 \beta_{11} + 324 \beta_{10} + \cdots + 178 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 516 \beta_{15} + 432 \beta_{14} + 518 \beta_{13} + 288 \beta_{12} - 18 \beta_{11} + 176 \beta_{10} + \cdots - 4406 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 508 \beta_{15} - 1410 \beta_{14} - 1470 \beta_{13} + 60 \beta_{12} - 986 \beta_{11} + 636 \beta_{10} + \cdots - 6388 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1800 \beta_{15} - 864 \beta_{14} + 2362 \beta_{13} - 366 \beta_{12} - 2430 \beta_{11} + 3244 \beta_{10} + \cdots + 2698 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4116 \beta_{15} - 4124 \beta_{14} - 2028 \beta_{13} - 4936 \beta_{12} - 4524 \beta_{11} + \cdots - 15288 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1404 \beta_{15} - 56976 \beta_{14} - 39890 \beta_{13} - 39414 \beta_{12} - 23568 \beta_{11} + \cdots + 44898 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 29480 \beta_{15} - 64932 \beta_{14} + 20916 \beta_{13} - 118320 \beta_{12} - 50812 \beta_{11} + \cdots + 110176 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 76092 \beta_{15} - 403812 \beta_{14} - 300352 \beta_{13} - 378468 \beta_{12} - 139212 \beta_{11} + \cdots - 7556 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(2305\) \(2485\) \(2945\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−1.62819 − 1.94897i
−1.93285 − 1.64730i
0.265208 − 1.21662i
2.32315 − 1.02589i
0.746855 − 0.996352i
−1.01412 − 0.722538i
2.46263 − 0.620455i
−1.22268 − 0.235704i
−1.22268 + 0.235704i
2.46263 + 0.620455i
−1.01412 + 0.722538i
0.746855 + 0.996352i
2.32315 + 1.02589i
0.265208 + 1.21662i
−1.93285 + 1.64730i
−1.62819 + 1.94897i
0 0 0 − 3.89795i 0 − 4.76908i 0 0 0
1151.2 0 0 0 − 3.29459i 0 1.70049i 0 0 0
1151.3 0 0 0 − 2.43325i 0 − 1.67598i 0 0 0
1151.4 0 0 0 − 2.05177i 0 2.99794i 0 0 0
1151.5 0 0 0 − 1.99270i 0 0.297239i 0 0 0
1151.6 0 0 0 − 1.44508i 0 1.03398i 0 0 0
1151.7 0 0 0 − 1.24091i 0 − 4.88585i 0 0 0
1151.8 0 0 0 − 0.471408i 0 − 2.35345i 0 0 0
1151.9 0 0 0 0.471408i 0 2.35345i 0 0 0
1151.10 0 0 0 1.24091i 0 4.88585i 0 0 0
1151.11 0 0 0 1.44508i 0 − 1.03398i 0 0 0
1151.12 0 0 0 1.99270i 0 − 0.297239i 0 0 0
1151.13 0 0 0 2.05177i 0 − 2.99794i 0 0 0
1151.14 0 0 0 2.43325i 0 1.67598i 0 0 0
1151.15 0 0 0 3.29459i 0 − 1.70049i 0 0 0
1151.16 0 0 0 3.89795i 0 4.76908i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3312.2.e.h yes 16
3.b odd 2 1 3312.2.e.g ✓ 16
4.b odd 2 1 3312.2.e.g ✓ 16
12.b even 2 1 inner 3312.2.e.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3312.2.e.g ✓ 16 3.b odd 2 1
3312.2.e.g ✓ 16 4.b odd 2 1
3312.2.e.h yes 16 1.a even 1 1 trivial
3312.2.e.h yes 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3312, [\chi])\):

\( T_{5}^{16} + 44 T_{5}^{14} + 756 T_{5}^{12} + 6584 T_{5}^{10} + 31624 T_{5}^{8} + 84624 T_{5}^{6} + \cdots + 11664 \) Copy content Toggle raw display
\( T_{11}^{8} - 4T_{11}^{7} - 40T_{11}^{6} + 200T_{11}^{5} + 110T_{11}^{4} - 1408T_{11}^{3} + 928T_{11}^{2} + 1536T_{11} - 1152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 44 T^{14} + \cdots + 11664 \) Copy content Toggle raw display
$7$ \( T^{16} + 68 T^{14} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots + 640)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1412557056 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 6566833296 \) Copy content Toggle raw display
$23$ \( (T - 1)^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 248 T^{14} + \cdots + 1327104 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 6046617600 \) Copy content Toggle raw display
$37$ \( (T^{8} - 188 T^{6} + \cdots - 18000)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 2473901162496 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 806787216 \) Copy content Toggle raw display
$47$ \( (T^{8} - 32 T^{7} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 35255319696 \) Copy content Toggle raw display
$59$ \( (T^{8} - 16 T^{7} + \cdots + 1272384)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 8 T^{7} + \cdots + 47664)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1199997557136 \) Copy content Toggle raw display
$71$ \( (T^{8} - 24 T^{7} + \cdots + 9679104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 8 T^{7} + \cdots + 519744)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 61946470289664 \) Copy content Toggle raw display
$83$ \( (T^{8} + 4 T^{7} + \cdots + 22991040)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 699496704 \) Copy content Toggle raw display
$97$ \( (T^{8} - 4 T^{7} + \cdots + 10496896)^{2} \) Copy content Toggle raw display
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