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Properties

Label 1216.3.g.d
Level $1216$
Weight $3$
Character orbit 1216.g
Analytic conductor $33.134$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,3,Mod(417,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.417");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.g (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: \(33.1336001462\)
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: \( x^{16} - 34x^{14} + 509x^{12} - 4794x^{10} + 30356x^{8} - 106386x^{6} + 288389x^{4} - 166634x^{2} + 28561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{6} + \beta_{5}) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{6} + \beta_{5}) q^{7} + (\beta_{2} - 1) q^{9} + (4 \beta_{13} + 5 \beta_{11}) q^{11} - \beta_{10} q^{13} + \beta_{7} q^{15} + (6 \beta_{2} - 5) q^{17} + ( - 3 \beta_{13} + \beta_{11} + \cdots - \beta_1) q^{19}+ \cdots + (10 \beta_{13} + 3 \beta_{11}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{9} - 32 q^{17} - 40 q^{25} - 224 q^{49} + 136 q^{57} - 416 q^{73} - 1152 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 34x^{14} + 509x^{12} - 4794x^{10} + 30356x^{8} - 106386x^{6} + 288389x^{4} - 166634x^{2} + 28561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 169102461 \nu^{14} - 3612486279 \nu^{12} + 34361326496 \nu^{10} - 323940314258 \nu^{8} + \cdots + 461251684944973 ) / 141914985852224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 153627 \nu^{14} - 4809003 \nu^{12} + 63338848 \nu^{10} - 510002526 \nu^{8} + \cdots + 56437596273 ) / 22098253792 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1571505795 \nu^{14} + 49698854845 \nu^{12} - 661252294720 \nu^{10} + \cdots - 357850444010527 ) / 70957492926112 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 178040787 \nu^{14} - 5959747101 \nu^{12} + 87642326560 \nu^{10} - 811891078862 \nu^{8} + \cdots - 14056758927425 ) / 839733644096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 51775128969 \nu^{15} + 1834569911071 \nu^{13} - 28950860955936 \nu^{11} + \cdots + 17\!\cdots\!47 \nu ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41875148377 \nu^{15} + 1429086947329 \nu^{13} - 21459376777248 \nu^{11} + \cdots + 11\!\cdots\!41 \nu ) / 922447408039456 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 324648095 \nu^{14} - 10864164143 \nu^{12} + 159484111472 \nu^{10} - 1475269888454 \nu^{8} + \cdots - 25417508935195 ) / 466825611356 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 643206577 \nu^{14} - 21659315191 \nu^{12} + 320359006240 \nu^{10} - 2978518882906 \nu^{8} + \cdots - 50324586311651 ) / 839733644096 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 88511626345 \nu^{15} + 3121299198191 \nu^{13} - 48726562704896 \nu^{11} + \cdots + 28\!\cdots\!43 \nu ) / 922447408039456 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 262843054661 \nu^{15} + 9014742285199 \nu^{13} - 136262585471360 \nu^{11} + \cdots + 72\!\cdots\!51 \nu ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 70644768241 \nu^{15} + 2373845182500 \nu^{13} - 35035440894392 \nu^{11} + \cdots + 54\!\cdots\!92 \nu ) / 461223704019728 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3338997653 \nu^{14} - 112817837749 \nu^{12} + 1674203108144 \nu^{10} + \cdots - 261960285444273 ) / 1867302445424 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 133095395 \nu^{15} + 4481361397 \nu^{13} - 66247940416 \nu^{11} + \cdots + 10069465083977 \nu ) / 667956124576 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 459913846149 \nu^{15} + 15466599382227 \nu^{13} - 228428497840512 \nu^{11} + \cdots + 35\!\cdots\!27 \nu ) / 922447408039456 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 298352393633 \nu^{15} + 10041957431973 \nu^{13} - 148386925874736 \nu^{11} + \cdots + 22\!\cdots\!13 \nu ) / 461223704019728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{13} - \beta_{9} + 2\beta_{6} + 2\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - \beta_{7} + 7\beta_{4} + 3\beta_{3} + 8\beta_{2} - 10\beta _1 + 30 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{15} - 2\beta_{14} - 8\beta_{13} - 2\beta_{10} - \beta_{9} + 6\beta_{6} + 4\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{12} + 11\beta_{8} - 15\beta_{7} + 43\beta_{4} + 59\beta_{3} + 208\beta_{2} - 82\beta _1 + 34 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 129 \beta_{15} + 89 \beta_{14} - 411 \beta_{13} - 300 \beta_{11} - 108 \beta_{10} - 21 \beta_{9} + \cdots + 126 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{12} + 52\beta_{8} - 16\beta_{7} + 80\beta_{3} + 260\beta_{2} - 108\beta _1 + 91 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1483 \beta_{15} + 2337 \beta_{14} - 5459 \beta_{13} - 6776 \beta_{11} - 888 \beta_{10} + \cdots + 1726 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2044 \beta_{12} + 6525 \beta_{8} - 1919 \beta_{7} + 13 \beta_{4} + 3941 \beta_{3} + 12960 \beta_{2} + \cdots + 24962 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6788 \beta_{15} + 7150 \beta_{14} - 22888 \beta_{13} - 22176 \beta_{11} - 1374 \beta_{10} + \cdots + 5548 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 25344 \beta_{12} + 82137 \beta_{8} - 37881 \beta_{7} + 41553 \beta_{4} + 16245 \beta_{3} + \cdots + 299118 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 446291 \beta_{15} + 307295 \beta_{14} - 1434025 \beta_{13} - 1019612 \beta_{11} + \cdots + 164626 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 39104 \beta_{12} + 128824 \beta_{8} - 72384 \beta_{7} + 101952 \beta_{4} + 9152 \beta_{3} + \cdots + 157993 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 5931225 \beta_{15} + 3852575 \beta_{14} - 18986345 \beta_{13} - 12890800 \beta_{11} + \cdots - 48318 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 4073752 \beta_{12} + 13347671 \beta_{8} - 7025665 \beta_{7} + 9206607 \beta_{4} - 880301 \beta_{3} + \cdots - 15539234 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17231668 \beta_{15} + 13085766 \beta_{14} - 55868328 \beta_{13} - 42696000 \beta_{11} + \cdots - 5585436 \beta_{5} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
−0.592744 − 0.0718888i
0.592744 + 0.0718888i
−0.592744 + 0.0718888i
0.592744 − 0.0718888i
1.69895 + 1.19682i
−1.69895 − 1.19682i
1.69895 − 1.19682i
−1.69895 + 1.19682i
−2.40873 − 1.69682i
2.40873 + 1.69682i
−2.40873 + 1.69682i
2.40873 − 1.69682i
−3.52990 − 0.428111i
3.52990 + 0.428111i
−3.52990 + 0.428111i
3.52990 − 0.428111i
0 −3.24985 0 − 3.61513i 0 −8.24529 0 1.56155 0
417.2 0 −3.24985 0 − 3.61513i 0 8.24529 0 1.56155 0
417.3 0 −3.24985 0 3.61513i 0 −8.24529 0 1.56155 0
417.4 0 −3.24985 0 3.61513i 0 8.24529 0 1.56155 0
417.5 0 −2.53741 0 − 6.47540i 0 −1.41956 0 −2.56155 0
417.6 0 −2.53741 0 − 6.47540i 0 1.41956 0 −2.56155 0
417.7 0 −2.53741 0 6.47540i 0 −1.41956 0 −2.56155 0
417.8 0 −2.53741 0 6.47540i 0 1.41956 0 −2.56155 0
417.9 0 2.53741 0 − 6.47540i 0 −1.41956 0 −2.56155 0
417.10 0 2.53741 0 − 6.47540i 0 1.41956 0 −2.56155 0
417.11 0 2.53741 0 6.47540i 0 −1.41956 0 −2.56155 0
417.12 0 2.53741 0 6.47540i 0 1.41956 0 −2.56155 0
417.13 0 3.24985 0 − 3.61513i 0 −8.24529 0 1.56155 0
417.14 0 3.24985 0 − 3.61513i 0 8.24529 0 1.56155 0
417.15 0 3.24985 0 3.61513i 0 −8.24529 0 1.56155 0
417.16 0 3.24985 0 3.61513i 0 8.24529 0 1.56155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner
152.g odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.g.d ✓ 16
4.b odd 2 1 inner 1216.3.g.d ✓ 16
8.b even 2 1 inner 1216.3.g.d ✓ 16
8.d odd 2 1 inner 1216.3.g.d ✓ 16
19.b odd 2 1 inner 1216.3.g.d ✓ 16
76.d even 2 1 inner 1216.3.g.d ✓ 16
152.b even 2 1 inner 1216.3.g.d ✓ 16
152.g odd 2 1 inner 1216.3.g.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.3.g.d ✓ 16 1.a even 1 1 trivial
1216.3.g.d ✓ 16 4.b odd 2 1 inner
1216.3.g.d ✓ 16 8.b even 2 1 inner
1216.3.g.d ✓ 16 8.d odd 2 1 inner
1216.3.g.d ✓ 16 19.b odd 2 1 inner
1216.3.g.d ✓ 16 76.d even 2 1 inner
1216.3.g.d ✓ 16 152.b even 2 1 inner
1216.3.g.d ✓ 16 152.g odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{4} - 17T_{3}^{2} + 68 \) Copy content Toggle raw display
\( T_{5}^{4} + 55T_{5}^{2} + 548 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} - 17 T^{2} + 68)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 55 T^{2} + 548)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 70 T^{2} + 137)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 273 T^{2} + 5776)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 391 T^{2} + 37264)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 149)^{8} \) Copy content Toggle raw display
$19$ \( (T^{8} - 1072 T^{6} + \cdots + 16983563041)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2075 T^{2} + 791312)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 867 T^{2} + 9316)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1836 T^{2} + 149056)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4828 T^{2} + 2384896)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3400 T^{2} + 170000)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3757 T^{2} + 334084)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1611 T^{2} + 177552)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 10931 T^{2} + 17225284)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4709 T^{2} + 183872)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8363 T^{2} + 15100688)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 15793 T^{2} + 693668)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 14892 T^{2} + 25190464)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 52 T - 1381)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24344 T^{2} + 82316176)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 12936 T^{2} + 4999696)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3944 T^{2} + 946832)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 18836 T^{2} + 2941952)^{4} \) Copy content Toggle raw display
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