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Properties

Label 3696.2.q.e
Level $3696$
Weight $2$
Character orbit 3696.q
Analytic conductor $29.513$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3696,2,Mod(769,3696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3696.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.q (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: \(29.5127085871\)
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: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} - \beta_{3} q^{5} + (\beta_{14} + \beta_{7} - \beta_{6}) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} - \beta_{3} q^{5} + (\beta_{14} + \beta_{7} - \beta_{6}) q^{7} - q^{9} + (\beta_{7} + \beta_{4} - 1) q^{11} + ( - \beta_{13} + \beta_{12} - \beta_{11}) q^{13} + ( - \beta_{2} + 1) q^{15} + (\beta_{15} + 2 \beta_{14} + \cdots - \beta_{6}) q^{17}+ \cdots + ( - \beta_{7} - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 16 q^{11} + 8 q^{15} - 24 q^{23} - 24 q^{25} + 8 q^{37} + 24 q^{53} + 32 q^{67} + 48 q^{71} + 16 q^{81} - 16 q^{91} - 40 q^{93} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 52907 \nu^{15} + 3769822 \nu^{13} + 7571087 \nu^{11} - 23218303 \nu^{9} - 220485308 \nu^{7} + \cdots + 5720425875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 75631 \nu^{14} + 26336 \nu^{12} - 423219 \nu^{10} - 3371514 \nu^{8} + 2776621 \nu^{6} + \cdots + 47672250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 739864 \nu^{15} - 254886 \nu^{13} - 6858931 \nu^{11} - 26926811 \nu^{9} + 104639904 \nu^{7} + \cdots - 1014860875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3523301 \nu^{14} + 408316 \nu^{12} - 11381739 \nu^{10} - 167934984 \nu^{8} + 176298151 \nu^{6} + \cdots + 2332929625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64321 \nu^{14} - 44136 \nu^{12} + 365249 \nu^{10} + 2994144 \nu^{8} - 2128291 \nu^{6} + \cdots - 30499350 ) / 3123275 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3653383 \nu^{14} + 2120282 \nu^{12} + 14044947 \nu^{10} + 157272982 \nu^{8} + \cdots - 948936625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 752571 \nu^{14} - 450738 \nu^{12} - 2572243 \nu^{10} - 31461918 \nu^{8} + 59211447 \nu^{6} + \cdots + 338358275 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + \cdots + 955700 \nu ) / 633875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9990424 \nu^{14} - 4060336 \nu^{12} - 33805356 \nu^{10} - 441784386 \nu^{8} + 731294504 \nu^{6} + \cdots + 4406749000 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 276139 \nu^{15} - 1073735 \nu^{14} + 41490 \nu^{13} + 529596 \nu^{12} - 1383000 \nu^{11} + \cdots - 434987125 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 246140 \nu^{15} + 289514 \nu^{13} + 711399 \nu^{11} + 9979979 \nu^{9} - 25675526 \nu^{7} + \cdots - 72914275 \nu ) / 15616375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16157471 \nu^{15} - 26843375 \nu^{14} - 2214014 \nu^{13} + 13239900 \nu^{12} + \cdots - 10874678125 ) / 858900625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16157471 \nu^{15} - 26843375 \nu^{14} + 2214014 \nu^{13} + 13239900 \nu^{12} + \cdots - 10874678125 ) / 858900625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 23106644 \nu^{15} - 18540595 \nu^{14} + 6020806 \nu^{13} + 10934930 \nu^{12} + \cdots - 6601820000 ) / 858900625 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2615982 \nu^{15} - 26528 \nu^{13} - 9744263 \nu^{11} - 121007403 \nu^{9} + 150031092 \nu^{7} + \cdots + 1357510875 \nu ) / 78081875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{14} - \beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + \beta_{12} - \beta_{9} + \beta_{7} - 3\beta_{6} + \beta_{5} + 2\beta_{4} + 3\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14 \beta_{14} - 11 \beta_{13} + 9 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 14 \beta_{8} + \cdots + 4 \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{13} + 4\beta_{12} + 9\beta_{9} - 5\beta_{7} + 3\beta_{6} + \beta_{5} - 6\beta_{4} + 9\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{15} + 14 \beta_{14} - 6 \beta_{13} - 14 \beta_{12} - 6 \beta_{11} + 20 \beta_{10} + \cdots - 7 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 47\beta_{13} + 47\beta_{12} + 32\beta_{9} - \beta_{7} - 57\beta_{6} - 4\beta_{5} - 2\beta_{4} - 32\beta_{2} + 150 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 46 \beta_{15} + 38 \beta_{14} + 11 \beta_{13} - 33 \beta_{12} - 24 \beta_{11} + 22 \beta_{10} + \cdots + 46 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 103 \beta_{13} + 103 \beta_{12} + 15 \beta_{9} + 129 \beta_{7} - 91 \beta_{6} + 15 \beta_{5} + \cdots - 130 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 304 \beta_{15} + 2 \beta_{14} - 495 \beta_{13} + 113 \beta_{12} + 78 \beta_{11} + 382 \beta_{10} + \cdots + 76 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 276\beta_{13} + 276\beta_{12} + 496\beta_{9} - 67\beta_{7} + 237\beta_{6} - 248\beta_{4} - 398 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 194 \beta_{15} - 1482 \beta_{14} + 247 \beta_{13} - 1961 \beta_{12} + 300 \beta_{11} + 1714 \beta_{10} + \cdots + 38 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1947 \beta_{13} + 1947 \beta_{12} + 988 \beta_{9} + 1337 \beta_{7} - 1491 \beta_{6} - 988 \beta_{5} + \cdots + 3586 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1870 \beta_{15} - 7486 \beta_{14} + 2939 \beta_{13} - 2601 \beta_{12} + 2208 \beta_{11} + \cdots + 1870 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5003 \beta_{13} + 5003 \beta_{12} - 1462 \beta_{9} + 13555 \beta_{7} - 1749 \beta_{6} - 338 \beta_{5} + \cdots - 25310 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 6184 \beta_{15} - 6918 \beta_{14} - 2953 \beta_{13} + 873 \beta_{12} + 3092 \beta_{11} + \cdots + 3459 \beta_{6} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3696\mathbb{Z}\right)^\times\).

\(n\) \(463\) \(673\) \(1585\) \(2465\) \(2773\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
−1.86824 + 0.357358i
1.86824 + 0.357358i
−0.0566033 − 1.17421i
0.0566033 − 1.17421i
−0.644389 + 0.983224i
0.644389 + 0.983224i
0.917186 − 1.66637i
−0.917186 − 1.66637i
0.917186 + 1.66637i
−0.917186 + 1.66637i
−0.644389 − 0.983224i
0.644389 − 0.983224i
−0.0566033 + 1.17421i
0.0566033 + 1.17421i
−1.86824 − 0.357358i
1.86824 − 0.357358i
0 − 1.00000i 0 − 2.77447i 0 −2.60278 − 0.474903i 0 −1.00000 0
769.2 0 − 1.00000i 0 − 2.77447i 0 2.60278 + 0.474903i 0 −1.00000 0
769.3 0 − 1.00000i 0 − 0.833366i 0 −2.19849 + 1.47195i 0 −1.00000 0
769.4 0 − 1.00000i 0 − 0.833366i 0 2.19849 − 1.47195i 0 −1.00000 0
769.5 0 − 1.00000i 0 1.83337i 0 −1.47195 + 2.19849i 0 −1.00000 0
769.6 0 − 1.00000i 0 1.83337i 0 1.47195 − 2.19849i 0 −1.00000 0
769.7 0 − 1.00000i 0 3.77447i 0 −0.474903 − 2.60278i 0 −1.00000 0
769.8 0 − 1.00000i 0 3.77447i 0 0.474903 + 2.60278i 0 −1.00000 0
769.9 0 1.00000i 0 − 3.77447i 0 −0.474903 + 2.60278i 0 −1.00000 0
769.10 0 1.00000i 0 − 3.77447i 0 0.474903 − 2.60278i 0 −1.00000 0
769.11 0 1.00000i 0 − 1.83337i 0 −1.47195 − 2.19849i 0 −1.00000 0
769.12 0 1.00000i 0 − 1.83337i 0 1.47195 + 2.19849i 0 −1.00000 0
769.13 0 1.00000i 0 0.833366i 0 −2.19849 − 1.47195i 0 −1.00000 0
769.14 0 1.00000i 0 0.833366i 0 2.19849 + 1.47195i 0 −1.00000 0
769.15 0 1.00000i 0 2.77447i 0 −2.60278 + 0.474903i 0 −1.00000 0
769.16 0 1.00000i 0 2.77447i 0 2.60278 − 0.474903i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3696.2.q.e 16
4.b odd 2 1 231.2.c.a ✓ 16
7.b odd 2 1 inner 3696.2.q.e 16
11.b odd 2 1 inner 3696.2.q.e 16
12.b even 2 1 693.2.c.e 16
28.d even 2 1 231.2.c.a ✓ 16
44.c even 2 1 231.2.c.a ✓ 16
77.b even 2 1 inner 3696.2.q.e 16
84.h odd 2 1 693.2.c.e 16
132.d odd 2 1 693.2.c.e 16
308.g odd 2 1 231.2.c.a ✓ 16
924.n even 2 1 693.2.c.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.c.a ✓ 16 4.b odd 2 1
231.2.c.a ✓ 16 28.d even 2 1
231.2.c.a ✓ 16 44.c even 2 1
231.2.c.a ✓ 16 308.g odd 2 1
693.2.c.e 16 12.b even 2 1
693.2.c.e 16 84.h odd 2 1
693.2.c.e 16 132.d odd 2 1
693.2.c.e 16 924.n even 2 1
3696.2.q.e 16 1.a even 1 1 trivial
3696.2.q.e 16 7.b odd 2 1 inner
3696.2.q.e 16 11.b odd 2 1 inner
3696.2.q.e 16 77.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 \(S_{2}^{\mathrm{new}}(3696, [\chi])\):

\( T_{5}^{8} + 26T_{5}^{6} + 201T_{5}^{4} + 496T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{8} - 34T_{13}^{6} + 301T_{13}^{4} - 544T_{13}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 26 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 34 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 116 T^{6} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 50 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 44 T^{2} + \cdots - 64)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 74 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 140 T^{6} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 136 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 204 T^{6} + \cdots + 4946176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 146 T^{6} + \cdots + 1597696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 6 T^{3} + \cdots + 176)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 194 T^{6} + \cdots + 215296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 176 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots + 176)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 1984)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 594 T^{6} + \cdots + 426670336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 464 T^{6} + \cdots + 38738176)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 484 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 184 T^{6} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 716 T^{6} + \cdots + 126877696)^{2} \) Copy content Toggle raw display
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