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Properties

Label 2496.2.d.q
Level $2496$
Weight $2$
Character orbit 2496.d
Analytic conductor $19.931$
Analytic rank $0$
Dimension $20$
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] = [2496,2,Mod(1535,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, 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("2496.1535");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.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: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{17} + 18 x^{16} + 8 x^{14} - 8 x^{13} + 241 x^{12} - 44 x^{11} - 112 x^{10} - 132 x^{9} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1248)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{3} + \beta_{12} q^{5} + \beta_{6} q^{7} + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{3} + \beta_{12} q^{5} + \beta_{6} q^{7} + \beta_{8} q^{9} - \beta_{11} q^{11} - q^{13} + ( - \beta_{4} - \beta_1) q^{15} - \beta_{7} q^{17} + (\beta_{14} - \beta_{4} - \beta_1) q^{19} + ( - \beta_{17} + \beta_{12} + \beta_{3} + 1) q^{21} + ( - \beta_{16} + \beta_1) q^{23} + (\beta_{2} - 2) q^{25} + (\beta_{16} - \beta_{14} + \cdots - \beta_1) q^{27}+ \cdots + ( - \beta_{14} + \beta_{11} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{9} - 20 q^{13} + 12 q^{21} - 36 q^{25} - 16 q^{37} - 4 q^{45} - 76 q^{49} - 16 q^{57} + 56 q^{61} - 24 q^{69} + 88 q^{73} + 72 q^{81} - 56 q^{85} - 96 q^{93} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{17} + 18 x^{16} + 8 x^{14} - 8 x^{13} + 241 x^{12} - 44 x^{11} - 112 x^{10} - 132 x^{9} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6401 \nu^{19} + 41394 \nu^{18} - 221679 \nu^{17} + 938150 \nu^{16} + 1283559 \nu^{15} + \cdots + 7664678298 ) / 452236608 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{19} - 960 \nu^{18} + 297 \nu^{17} + 677 \nu^{16} + 2049 \nu^{15} - 10692 \nu^{14} + \cdots + 2263545 ) / 629856 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10505 \nu^{19} + 139536 \nu^{18} + 5103 \nu^{17} + 131066 \nu^{16} - 382977 \nu^{15} + \cdots + 970332534 ) / 226118304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5777 \nu^{19} - 47741 \nu^{18} + 66873 \nu^{17} - 132944 \nu^{16} - 150631 \nu^{15} + \cdots - 1125998820 ) / 75372768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27455 \nu^{19} + 220656 \nu^{18} - 174987 \nu^{17} - 145816 \nu^{16} - 505983 \nu^{15} + \cdots + 490106700 ) / 226118304 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55979 \nu^{19} + 103266 \nu^{18} + 107775 \nu^{17} - 338656 \nu^{16} - 1426923 \nu^{15} + \cdots - 3770869140 ) / 452236608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18533 \nu^{19} + 85695 \nu^{18} - 38367 \nu^{17} + 192284 \nu^{16} - 268053 \nu^{15} + \cdots + 1146810312 ) / 113059152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 42097 \nu^{19} - 122304 \nu^{18} + 20745 \nu^{17} - 170197 \nu^{16} + 533757 \nu^{15} + \cdots - 1076325489 ) / 226118304 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 113183 \nu^{19} - 35001 \nu^{18} + 664803 \nu^{17} + 295249 \nu^{16} + 2233107 \nu^{15} + \cdots - 3932368155 ) / 452236608 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 114941 \nu^{19} + 110322 \nu^{18} - 137673 \nu^{17} + 704800 \nu^{16} + 1970685 \nu^{15} + \cdots + 5700275532 ) / 452236608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 88 \nu^{19} - 360 \nu^{17} + 487 \nu^{16} - 612 \nu^{15} + 1440 \nu^{14} - 4808 \nu^{13} + \cdots + 5924583 ) / 314928 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42375 \nu^{19} + 55610 \nu^{18} + 217515 \nu^{17} + 118776 \nu^{16} - 532967 \nu^{15} + \cdots - 797791356 ) / 150745536 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 131623 \nu^{19} - 36609 \nu^{18} + 30843 \nu^{17} + 345905 \nu^{16} + 2065371 \nu^{15} + \cdots + 2487124197 ) / 452236608 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 66991 \nu^{19} - 135891 \nu^{18} + 179433 \nu^{17} + 409979 \nu^{16} + 1492875 \nu^{15} + \cdots + 1699607367 ) / 226118304 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 37088 \nu^{19} - 85299 \nu^{18} + 6129 \nu^{17} + 234941 \nu^{16} - 220764 \nu^{15} + \cdots + 435289545 ) / 113059152 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 173099 \nu^{19} - 130926 \nu^{18} - 105363 \nu^{17} + 821116 \nu^{16} + 3459099 \nu^{15} + \cdots + 7085250144 ) / 452236608 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 43795 \nu^{19} + 10281 \nu^{18} + 161559 \nu^{17} + 73552 \nu^{16} - 558003 \nu^{15} + \cdots - 463810212 ) / 113059152 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 99845 \nu^{19} - 330810 \nu^{18} + 105417 \nu^{17} + 185621 \nu^{16} - 1107633 \nu^{15} + \cdots - 844577847 ) / 226118304 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 104660 \nu^{19} + 9075 \nu^{18} - 84510 \nu^{17} + 125545 \nu^{16} + 1419576 \nu^{15} + \cdots + 1248748569 ) / 226118304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{19} - \beta_{18} + 3 \beta_{16} + 2 \beta_{15} - 2 \beta_{13} - 2 \beta_{10} + \beta_{8} + \cdots + 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{19} - 6\beta_{14} + \beta_{13} - 2\beta_{10} + 3\beta_{9} + 6\beta_{4} + 10\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{15} - 3 \beta_{14} + 4 \beta_{13} + \cdots + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{18} + 6\beta_{17} + \beta_{15} - 6\beta_{12} - \beta_{8} - 2\beta_{5} - \beta_{3} - \beta_{2} - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 16 \beta_{19} - \beta_{18} - 3 \beta_{16} - 4 \beta_{15} - 24 \beta_{14} + 4 \beta_{13} + 28 \beta_{10} + \cdots + 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 8 \beta_{16} + 4 \beta_{14} - 11 \beta_{13} + 4 \beta_{11} + 12 \beta_{10} + 3 \beta_{9} + \cdots - 16 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17 \beta_{19} - 29 \beta_{18} - 63 \beta_{16} + 10 \beta_{15} + 48 \beta_{14} + 22 \beta_{13} + \cdots - 272 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 19 \beta_{18} - 30 \beta_{17} - 32 \beta_{15} + 30 \beta_{12} - 4 \beta_{8} + 12 \beta_{7} + 22 \beta_{5} + \cdots - 110 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 23 \beta_{19} + 25 \beta_{18} + 19 \beta_{17} - 45 \beta_{16} - 9 \beta_{15} + 77 \beta_{14} + 88 \beta_{13} + \cdots + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 80 \beta_{19} + 288 \beta_{16} + 450 \beta_{14} - 547 \beta_{13} - 312 \beta_{11} + 110 \beta_{10} + \cdots - 274 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 152 \beta_{19} + 355 \beta_{18} - 768 \beta_{17} + 399 \beta_{16} - 308 \beta_{15} + 360 \beta_{14} + \cdots + 1828 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14 \beta_{18} - 38 \beta_{17} + 70 \beta_{15} + 24 \beta_{12} - 14 \beta_{8} - 94 \beta_{7} + \cdots + 285 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 529 \beta_{19} - 611 \beta_{18} - 768 \beta_{17} + 285 \beta_{16} - 350 \beta_{15} - 120 \beta_{14} + \cdots - 806 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 554 \beta_{19} + 2784 \beta_{16} - 5202 \beta_{14} + 6023 \beta_{13} + 3576 \beta_{11} + \cdots + 1454 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 287 \beta_{19} - 699 \beta_{18} + 2415 \beta_{17} - 47 \beta_{16} + 779 \beta_{15} - 1277 \beta_{14} + \cdots + 4224 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 460 \beta_{18} + 150 \beta_{17} + 1499 \beta_{15} + 234 \beta_{12} + 3379 \beta_{8} + 2892 \beta_{7} + \cdots + 1313 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 1768 \beta_{19} - 875 \beta_{18} + 10752 \beta_{17} + 3075 \beta_{16} - 1328 \beta_{15} - 15312 \beta_{14} + \cdots - 10286 ) / 12 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 2048 \beta_{19} - 17832 \beta_{16} - 868 \beta_{14} + 15283 \beta_{13} - 8364 \beta_{11} + \cdots + 11296 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 9199 \beta_{19} + 10673 \beta_{18} + 10752 \beta_{17} + 15375 \beta_{16} - 12718 \beta_{15} + \cdots - 238684 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1535.1
−1.35086 − 1.08405i
−1.35086 + 1.08405i
0.725959 − 1.57257i
0.725959 + 1.57257i
0.587478 + 1.62938i
0.587478 − 1.62938i
1.50700 − 0.853779i
1.50700 + 0.853779i
−0.933748 − 1.45881i
−0.933748 + 1.45881i
1.45881 + 0.933748i
1.45881 − 0.933748i
−0.853779 + 1.50700i
−0.853779 − 1.50700i
−1.62938 − 0.587478i
−1.62938 + 0.587478i
1.57257 − 0.725959i
1.57257 + 0.725959i
−1.08405 − 1.35086i
−1.08405 + 1.35086i
0 −1.72174 − 0.188664i 0 0.867664i 0 1.51615i 0 2.92881 + 0.649664i 0
1535.2 0 −1.72174 + 0.188664i 0 − 0.867664i 0 − 1.51615i 0 2.92881 − 0.649664i 0
1535.3 0 −1.62531 − 0.598646i 0 − 3.43429i 0 5.05054i 0 2.28325 + 1.94597i 0
1535.4 0 −1.62531 + 0.598646i 0 3.43429i 0 − 5.05054i 0 2.28325 − 1.94597i 0
1535.5 0 −0.736734 − 1.56755i 0 2.65543i 0 − 4.30836i 0 −1.91445 + 2.30974i 0
1535.6 0 −0.736734 + 1.56755i 0 − 2.65543i 0 4.30836i 0 −1.91445 − 2.30974i 0
1535.7 0 −0.461900 − 1.66933i 0 − 1.99193i 0 1.52766i 0 −2.57330 + 1.54212i 0
1535.8 0 −0.461900 + 1.66933i 0 1.99193i 0 − 1.52766i 0 −2.57330 − 1.54212i 0
1535.9 0 −0.371272 − 1.69179i 0 3.23012i 0 2.30164i 0 −2.72431 + 1.25623i 0
1535.10 0 −0.371272 + 1.69179i 0 − 3.23012i 0 − 2.30164i 0 −2.72431 − 1.25623i 0
1535.11 0 0.371272 − 1.69179i 0 − 3.23012i 0 2.30164i 0 −2.72431 − 1.25623i 0
1535.12 0 0.371272 + 1.69179i 0 3.23012i 0 − 2.30164i 0 −2.72431 + 1.25623i 0
1535.13 0 0.461900 − 1.66933i 0 1.99193i 0 1.52766i 0 −2.57330 − 1.54212i 0
1535.14 0 0.461900 + 1.66933i 0 − 1.99193i 0 − 1.52766i 0 −2.57330 + 1.54212i 0
1535.15 0 0.736734 − 1.56755i 0 − 2.65543i 0 − 4.30836i 0 −1.91445 − 2.30974i 0
1535.16 0 0.736734 + 1.56755i 0 2.65543i 0 4.30836i 0 −1.91445 + 2.30974i 0
1535.17 0 1.62531 − 0.598646i 0 3.43429i 0 5.05054i 0 2.28325 − 1.94597i 0
1535.18 0 1.62531 + 0.598646i 0 − 3.43429i 0 − 5.05054i 0 2.28325 + 1.94597i 0
1535.19 0 1.72174 − 0.188664i 0 − 0.867664i 0 1.51615i 0 2.92881 − 0.649664i 0
1535.20 0 1.72174 + 0.188664i 0 0.867664i 0 − 1.51615i 0 2.92881 + 0.649664i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1535.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
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 2496.2.d.q 20
3.b odd 2 1 inner 2496.2.d.q 20
4.b odd 2 1 inner 2496.2.d.q 20
8.b even 2 1 1248.2.d.d ✓ 20
8.d odd 2 1 1248.2.d.d ✓ 20
12.b even 2 1 inner 2496.2.d.q 20
24.f even 2 1 1248.2.d.d ✓ 20
24.h odd 2 1 1248.2.d.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.d.d ✓ 20 8.b even 2 1
1248.2.d.d ✓ 20 8.d odd 2 1
1248.2.d.d ✓ 20 24.f even 2 1
1248.2.d.d ✓ 20 24.h odd 2 1
2496.2.d.q 20 1.a even 1 1 trivial
2496.2.d.q 20 3.b odd 2 1 inner
2496.2.d.q 20 4.b odd 2 1 inner
2496.2.d.q 20 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 \(S_{2}^{\mathrm{new}}(2496, [\chi])\):

\( T_{5}^{10} + 34T_{5}^{8} + 421T_{5}^{6} + 2276T_{5}^{4} + 4932T_{5}^{2} + 2592 \) Copy content Toggle raw display
\( T_{7}^{10} + 54T_{7}^{8} + 941T_{7}^{6} + 6048T_{7}^{4} + 15412T_{7}^{2} + 13456 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{23}^{10} - 96T_{23}^{8} + 2528T_{23}^{6} - 18432T_{23}^{4} + 24832T_{23}^{2} - 8192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( (T^{10} + 34 T^{8} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 54 T^{8} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{10} \) Copy content Toggle raw display
$13$ \( (T + 1)^{20} \) Copy content Toggle raw display
$17$ \( (T^{10} + 102 T^{8} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 80 T^{8} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 96 T^{8} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{10} \) Copy content Toggle raw display
$31$ \( (T^{10} + 192 T^{8} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 4 T^{4} + \cdots - 6824)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + 236 T^{8} + \cdots + 158277632)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 290 T^{8} + \cdots + 32993536)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 218 T^{8} + \cdots - 16222208)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 336 T^{8} + \cdots + 2654208)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8)^{10} \) Copy content Toggle raw display
$61$ \( (T^{5} - 14 T^{4} + \cdots - 91136)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + 304 T^{8} + \cdots + 28047616)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 650 T^{8} + \cdots - 9590571008)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 22 T^{4} + \cdots + 35136)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + 500 T^{8} + \cdots + 1230045184)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 472 T^{8} + \cdots - 10616832)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 572 T^{8} + \cdots + 4806705152)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 18 T^{4} + \cdots + 12608)^{4} \) Copy content Toggle raw display
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