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Properties

Label 1216.3.d.d
Level $1216$
Weight $3$
Character orbit 1216.d
Analytic conductor $33.134$
Analytic rank $0$
Dimension $14$
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] = [1216,3,Mod(191,1216)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.d (of order \(2\), degree \(1\), not 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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + \cdots + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 76)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{7} q^{5} + \beta_{9} q^{7} + ( - \beta_{11} - \beta_{7} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{7} q^{5} + \beta_{9} q^{7} + ( - \beta_{11} - \beta_{7} - 5) q^{9} - \beta_{8} q^{11} + (\beta_{13} + 2 \beta_{7} - 4) q^{13} + ( - \beta_{9} + \beta_{3} + \cdots - \beta_1) q^{15}+ \cdots + ( - 6 \beta_{9} - 3 \beta_{8} + \cdots + 20 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + \cdots + 16384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} + \cdots + 114688 ) / 20480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{12} + 2 \nu^{11} + 27 \nu^{10} - 142 \nu^{9} + 242 \nu^{8} + 4 \nu^{7} - 1172 \nu^{6} + \cdots - 1024 ) / 1024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21 \nu^{13} - 6 \nu^{12} + 91 \nu^{11} - 246 \nu^{10} + 354 \nu^{9} + 244 \nu^{8} - 2100 \nu^{7} + \cdots - 172032 ) / 20480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{13} + 10 \nu^{12} + 11 \nu^{11} - 102 \nu^{10} + 98 \nu^{9} + 84 \nu^{8} - 948 \nu^{7} + \cdots + 32768 ) / 4096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31 \nu^{13} + 86 \nu^{12} - 281 \nu^{11} + 166 \nu^{10} + 626 \nu^{9} - 1924 \nu^{8} + 3660 \nu^{7} + \cdots + 315392 ) / 20480 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + \cdots + 6144 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + \cdots + 12288 ) / 1024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39 \nu^{13} - 84 \nu^{12} + 269 \nu^{11} - 164 \nu^{10} - 454 \nu^{9} + 2416 \nu^{8} + \cdots - 442368 ) / 10240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39 \nu^{13} + 94 \nu^{12} - 209 \nu^{11} - 146 \nu^{10} + 1314 \nu^{9} - 3156 \nu^{8} + \cdots + 411648 ) / 10240 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9 \nu^{13} - 8 \nu^{12} - 75 \nu^{11} + 232 \nu^{10} - 174 \nu^{9} - 840 \nu^{8} + 3004 \nu^{7} + \cdots - 112640 ) / 2048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11 \nu^{13} + 30 \nu^{12} - 11 \nu^{11} - 114 \nu^{10} + 334 \nu^{9} - 356 \nu^{8} - 620 \nu^{7} + \cdots + 6144 ) / 2048 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25 \nu^{13} - 62 \nu^{12} + \nu^{11} + 370 \nu^{10} - 882 \nu^{9} + 468 \nu^{8} + 3316 \nu^{7} + \cdots - 139264 ) / 4096 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27 \nu^{13} - 62 \nu^{12} - 133 \nu^{11} + 594 \nu^{10} - 782 \nu^{9} - 988 \nu^{8} + 6764 \nu^{7} + \cdots - 262144 ) / 4096 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 4 \beta_{7} - 2 \beta_{5} + \cdots - 23 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 9 \beta_{7} + \beta_{6} + \cdots + 33 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 9 \beta_{10} + 10 \beta_{9} + 14 \beta_{8} - \beta_{7} + \cdots + 37 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9 \beta_{13} - 8 \beta_{12} - 18 \beta_{11} + 9 \beta_{10} - 14 \beta_{9} + 2 \beta_{8} + 7 \beta_{7} + \cdots - 75 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 15 \beta_{13} + 12 \beta_{12} + 26 \beta_{11} - 19 \beta_{10} + 14 \beta_{9} + 42 \beta_{8} + 11 \beta_{7} + \cdots - 15 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 19 \beta_{13} + 8 \beta_{12} + 86 \beta_{11} - 51 \beta_{10} - 22 \beta_{9} - 6 \beta_{8} - 141 \beta_{7} + \cdots - 551 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 83 \beta_{13} + 88 \beta_{12} - 46 \beta_{11} - 39 \beta_{10} + 122 \beta_{9} + 134 \beta_{8} + \cdots + 1237 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 129 \beta_{13} - 120 \beta_{12} - 386 \beta_{11} + 193 \beta_{10} - 126 \beta_{9} + 50 \beta_{8} + \cdots - 915 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 225 \beta_{13} + 384 \beta_{12} - 230 \beta_{11} + 61 \beta_{10} - 86 \beta_{9} + 230 \beta_{8} + \cdots + 7609 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 373 \beta_{13} + 168 \beta_{12} + 1238 \beta_{11} - 107 \beta_{10} + 970 \beta_{9} + 794 \beta_{8} + \cdots - 12911 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 491 \beta_{13} + 1984 \beta_{12} + 66 \beta_{11} - 1023 \beta_{10} + 1362 \beta_{9} - 482 \beta_{8} + \cdots - 23571 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.0607713 + 1.99908i
−1.89728 − 0.632718i
0.711746 − 1.86907i
1.94929 − 0.447510i
1.57398 + 1.23393i
0.645572 − 1.89294i
−1.92254 + 0.551226i
−1.92254 − 0.551226i
0.645572 + 1.89294i
1.57398 − 1.23393i
1.94929 + 0.447510i
0.711746 + 1.86907i
−1.89728 + 0.632718i
−0.0607713 − 1.99908i
0 − 5.37609i 0 5.82257 0 5.45132i 0 −19.9023 0
191.2 0 − 5.34370i 0 −5.79268 0 5.87536i 0 −19.5551 0
191.3 0 − 4.44946i 0 −4.97973 0 − 12.2628i 0 −10.7977 0
191.4 0 − 3.19988i 0 3.90374 0 − 2.64664i 0 −1.23921 0
191.5 0 − 2.90118i 0 −3.66290 0 1.93414i 0 0.583162 0
191.6 0 − 0.820457i 0 2.38184 0 12.3764i 0 8.32685 0
191.7 0 − 0.644704i 0 2.32715 0 8.62924i 0 8.58436 0
191.8 0 0.644704i 0 2.32715 0 − 8.62924i 0 8.58436 0
191.9 0 0.820457i 0 2.38184 0 − 12.3764i 0 8.32685 0
191.10 0 2.90118i 0 −3.66290 0 − 1.93414i 0 0.583162 0
191.11 0 3.19988i 0 3.90374 0 2.64664i 0 −1.23921 0
191.12 0 4.44946i 0 −4.97973 0 12.2628i 0 −10.7977 0
191.13 0 5.34370i 0 −5.79268 0 − 5.87536i 0 −19.5551 0
191.14 0 5.37609i 0 5.82257 0 − 5.45132i 0 −19.9023 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.d.d 14
4.b odd 2 1 inner 1216.3.d.d 14
8.b even 2 1 76.3.b.b ✓ 14
8.d odd 2 1 76.3.b.b ✓ 14
24.f even 2 1 684.3.g.b 14
24.h odd 2 1 684.3.g.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b ✓ 14 8.b even 2 1
76.3.b.b ✓ 14 8.d odd 2 1
684.3.g.b 14 24.f even 2 1
684.3.g.b 14 24.h odd 2 1
1216.3.d.d 14 1.a even 1 1 trivial
1216.3.d.d 14 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 97T_{3}^{12} + 3595T_{3}^{10} + 63443T_{3}^{8} + 539872T_{3}^{6} + 1940896T_{3}^{4} + 1665792T_{3}^{2} + 393984 \) acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 97 T^{12} + \cdots + 393984 \) Copy content Toggle raw display
$5$ \( (T^{7} - 66 T^{5} + \cdots + 13312)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 46106276299 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 94488337600 \) Copy content Toggle raw display
$13$ \( (T^{7} + 27 T^{6} + \cdots + 1265920)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} - 17 T^{6} + \cdots - 69101055)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{7} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 683970816311296 \) Copy content Toggle raw display
$29$ \( (T^{7} + 27 T^{6} + \cdots + 112127472)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + 50 T^{6} + \cdots - 914894720)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} - 112 T^{6} + \cdots + 18882293760)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 64\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{7} + 7 T^{6} + \cdots - 18228603200)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + 14 T^{6} + \cdots - 522558358600)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} - 35 T^{6} + \cdots - 77971926925)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{7} - 23640 T^{5} + \cdots - 88310345728)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 4410892984320)^{2} \) Copy content Toggle raw display
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