Properties

Label 5796.2.k.b
Level $5796$
Weight $2$
Character orbit 5796.k
Analytic conductor $46.281$
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] = [5796,2,Mod(5473,5796)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5796, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5796.5473");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5796.k (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: \(46.2812930115\)
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: \( x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 644)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + \beta_{8} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + \beta_{8} q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} - \beta_{8} - \beta_1) q^{11} + \beta_{5} q^{13} + \beta_{9} q^{17} + ( - \beta_{11} - \beta_{9}) q^{19} + (\beta_{12} + \beta_{6} + 1) q^{23} + (\beta_{6} - \beta_{4}) q^{25} + ( - \beta_{13} + \beta_{10} - \beta_{4}) q^{29} + ( - \beta_{5} - \beta_{3}) q^{31} + (\beta_{10} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4}) q^{35} + ( - \beta_{15} + \beta_{12}) q^{37} + ( - \beta_{13} - \beta_{10} - \beta_{5} + 2 \beta_{3}) q^{41} + 2 \beta_{14} q^{43} + ( - \beta_{13} - \beta_{10} + \beta_{5} - \beta_{3}) q^{47} + ( - \beta_{13} - \beta_{7} - \beta_{4} - \beta_{3}) q^{49} + (\beta_{15} + 2 \beta_{14} + \beta_{12} + \beta_{8} + \beta_1) q^{53} + ( - \beta_{13} - \beta_{10} + \beta_{7} + \beta_{5} + 3 \beta_{3}) q^{55} + (\beta_{13} + \beta_{10} + \beta_{7} - \beta_{5}) q^{59} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{2} + \beta_1) q^{61} + (\beta_{15} - 2 \beta_{14} - \beta_{12} - 2 \beta_{8} - 2 \beta_1) q^{65} + (2 \beta_{15} - \beta_{14} - 2 \beta_{12}) q^{67} + (\beta_{13} - \beta_{10} - \beta_{6} + 2 \beta_{4} + 1) q^{71} + ( - \beta_{5} + 2 \beta_{3}) q^{73} + ( - \beta_{13} + \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{3} + 2) q^{77} + (4 \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{8} - \beta_1) q^{79} + (2 \beta_{11} - \beta_{8} + \beta_1) q^{83} + ( - \beta_{13} + \beta_{10} + \beta_{6} - \beta_{4} + 1) q^{85} + (2 \beta_{11} + 3 \beta_{9} - \beta_{8} + 2 \beta_{2} + \beta_1) q^{89} + (\beta_{15} - 2 \beta_{14} + 2 \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{2} - 2 \beta_1) q^{91} + (\beta_{13} - \beta_{10} - 2 \beta_{6} - 4) q^{95} + ( - 2 \beta_{11} + 2 \beta_{8} - \beta_{2} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{23} + 8 q^{25} + 4 q^{35} + 4 q^{49} + 8 q^{71} + 28 q^{77} + 16 q^{85} - 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 849 \nu^{13} - 4069 \nu^{11} - 62787 \nu^{9} - 201333 \nu^{7} - 3304819 \nu^{5} - 7782327 \nu^{3} - 107279081 \nu ) / 45177216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22 \nu^{14} - 495 \nu^{12} + 4384 \nu^{10} + 72405 \nu^{8} + 242430 \nu^{6} + 3396043 \nu^{4} + 31981320 \nu^{2} + 197532671 ) / 56471520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{14} + 60 \nu^{12} + 3497 \nu^{10} + 988 \nu^{8} + 135161 \nu^{6} + 331828 \nu^{4} + 5001283 \nu^{2} - 470596 ) / 7529536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1087 \nu^{14} + 4575 \nu^{12} - 37901 \nu^{10} + 249285 \nu^{8} - 950805 \nu^{6} + 22222333 \nu^{4} + 10336305 \nu^{2} + 846955151 ) / 225886080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{14} - 452 \nu^{12} - 2713 \nu^{10} - 27644 \nu^{8} - 117129 \nu^{6} - 1331036 \nu^{4} - 6000099 \nu^{2} - 35294700 ) / 3764768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2889 \nu^{14} - 935 \nu^{12} - 89387 \nu^{10} + 312675 \nu^{8} - 4683075 \nu^{6} - 318549 \nu^{4} - 26567065 \nu^{2} + 1410964457 ) / 451772160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} - 2\nu^{13} + 68\nu^{11} - 46\nu^{9} + 4950\nu^{7} - 2254\nu^{5} + 163268\nu^{3} - 235298\nu ) / 823543 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 769 \nu^{15} + 20697 \nu^{13} + 137485 \nu^{11} + 1705539 \nu^{9} + 3410709 \nu^{7} + 75371947 \nu^{5} + 364911183 \nu^{3} + 3093815753 \nu ) / 632481024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 841 \nu^{14} + 1699 \nu^{12} + 52827 \nu^{10} - 166527 \nu^{8} + 2170659 \nu^{6} - 1388023 \nu^{4} + 104513129 \nu^{2} - 603892317 ) / 90354432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1207 \nu^{15} - 11871 \nu^{13} + 2549 \nu^{11} - 619365 \nu^{9} + 774525 \nu^{7} - 20886397 \nu^{5} - 116076345 \nu^{3} - 1242255791 \nu ) / 632481024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1559 \nu^{15} - 6535 \nu^{13} - 113117 \nu^{11} + 291675 \nu^{9} - 4951245 \nu^{7} - 7641109 \nu^{5} - 153507935 \nu^{3} + 1086253217 \nu ) / 790601280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1153 \nu^{14} - 1277 \nu^{12} + 21123 \nu^{10} - 225567 \nu^{8} + 2728395 \nu^{6} - 11082967 \nu^{4} + 19517729 \nu^{2} - 773306877 ) / 90354432 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1251 \nu^{15} + 885 \nu^{13} - 79433 \nu^{11} + 295735 \nu^{9} - 4190065 \nu^{7} + 14587839 \nu^{5} - 65775395 \nu^{3} + 738482773 \nu ) / 527067520 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4507 \nu^{15} + 14555 \nu^{13} + 218521 \nu^{11} + 182865 \nu^{9} + 11979225 \nu^{7} + 24501617 \nu^{5} + 348325075 \nu^{3} - 1142254141 \nu ) / 790601280 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 4\beta_{14} - \beta_{12} - \beta_{11} + \beta_{9} + 4\beta_{8} + 4\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{13} - \beta_{10} - 16\beta_{7} + 2\beta_{6} + 10\beta_{5} + 2\beta_{4} + 4\beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{15} + 28\beta_{14} - 29\beta_{12} + 17\beta_{11} - \beta_{9} + 13\beta_{8} - 12\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51\beta_{13} - 32\beta_{10} + 43\beta_{7} + 44\beta_{5} + 51\beta_{4} - 13\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -68\beta_{15} - 72\beta_{14} + 8\beta_{12} - 100\beta_{11} - 140\beta_{9} + 213\beta_{8} - 248\beta_{2} - 48\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 72\beta_{13} - 184\beta_{10} + 296\beta_{7} - 136\beta_{6} - 468\beta_{5} - 80\beta_{4} + 128\beta_{3} - 1511 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 484 \beta_{15} - 856 \beta_{14} + 1928 \beta_{12} + 12 \beta_{11} + 324 \beta_{9} - 512 \beta_{8} + 168 \beta_{2} - 2059 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1600 \beta_{13} - 395 \beta_{10} - 1011 \beta_{7} + 16 \beta_{6} - 2292 \beta_{5} + 2443 \beta_{4} - 1323 \beta_{3} + 348 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2767 \beta_{15} - 8468 \beta_{14} - 2333 \beta_{12} + 6039 \beta_{11} + 6201 \beta_{9} - 10700 \beta_{8} + 3708 \beta_{2} - 1651 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1897 \beta_{13} + 10251 \beta_{10} + 18064 \beta_{7} - 5422 \beta_{6} + 3774 \beta_{5} - 9086 \beta_{4} - 24692 \beta_{3} + 49153 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12137 \beta_{15} - 24396 \beta_{14} - 10913 \beta_{12} - 62935 \beta_{11} - 25673 \beta_{9} - 48927 \beta_{8} - 14228 \beta_{2} + 20916 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 25657 \beta_{13} + 31776 \beta_{10} - 176041 \beta_{7} - 13680 \beta_{6} - 53384 \beta_{5} - 155001 \beta_{4} + 74215 \beta_{3} + 345224 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 392248 \beta_{15} + 254096 \beta_{14} + 283808 \beta_{12} + 160616 \beta_{11} + 69368 \beta_{9} - 248383 \beta_{8} + 274608 \beta_{2} + 378032 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5796\mathbb{Z}\right)^\times\).

\(n\) \(829\) \(1289\) \(2899\) \(4789\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5473.1
2.38903 1.13690i
2.38903 + 1.13690i
−1.54356 2.14882i
−1.54356 + 2.14882i
−0.986926 2.45479i
−0.986926 + 2.45479i
−2.33151 1.25062i
−2.33151 + 1.25062i
2.33151 1.25062i
2.33151 + 1.25062i
0.986926 2.45479i
0.986926 + 2.45479i
1.54356 2.14882i
1.54356 + 2.14882i
−2.38903 1.13690i
−2.38903 + 1.13690i
0 0 0 −3.38856 0 −2.38903 1.13690i 0 0 0
5473.2 0 0 0 −3.38856 0 −2.38903 + 1.13690i 0 0 0
5473.3 0 0 0 −2.66041 0 1.54356 2.14882i 0 0 0
5473.4 0 0 0 −2.66041 0 1.54356 + 2.14882i 0 0 0
5473.5 0 0 0 −1.77749 0 0.986926 2.45479i 0 0 0
5473.6 0 0 0 −1.77749 0 0.986926 + 2.45479i 0 0 0
5473.7 0 0 0 −0.529536 0 2.33151 1.25062i 0 0 0
5473.8 0 0 0 −0.529536 0 2.33151 + 1.25062i 0 0 0
5473.9 0 0 0 0.529536 0 −2.33151 1.25062i 0 0 0
5473.10 0 0 0 0.529536 0 −2.33151 + 1.25062i 0 0 0
5473.11 0 0 0 1.77749 0 −0.986926 2.45479i 0 0 0
5473.12 0 0 0 1.77749 0 −0.986926 + 2.45479i 0 0 0
5473.13 0 0 0 2.66041 0 −1.54356 2.14882i 0 0 0
5473.14 0 0 0 2.66041 0 −1.54356 + 2.14882i 0 0 0
5473.15 0 0 0 3.38856 0 2.38903 1.13690i 0 0 0
5473.16 0 0 0 3.38856 0 2.38903 + 1.13690i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5473.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5796.2.k.b 16
3.b odd 2 1 644.2.d.a 16
7.b odd 2 1 inner 5796.2.k.b 16
12.b even 2 1 2576.2.f.g 16
21.c even 2 1 644.2.d.a 16
23.b odd 2 1 inner 5796.2.k.b 16
69.c even 2 1 644.2.d.a 16
84.h odd 2 1 2576.2.f.g 16
161.c even 2 1 inner 5796.2.k.b 16
276.h odd 2 1 2576.2.f.g 16
483.c odd 2 1 644.2.d.a 16
1932.b even 2 1 2576.2.f.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.d.a 16 3.b odd 2 1
644.2.d.a 16 21.c even 2 1
644.2.d.a 16 69.c even 2 1
644.2.d.a 16 483.c odd 2 1
2576.2.f.g 16 12.b even 2 1
2576.2.f.g 16 84.h odd 2 1
2576.2.f.g 16 276.h odd 2 1
2576.2.f.g 16 1932.b even 2 1
5796.2.k.b 16 1.a even 1 1 trivial
5796.2.k.b 16 7.b odd 2 1 inner
5796.2.k.b 16 23.b odd 2 1 inner
5796.2.k.b 16 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 22T_{5}^{6} + 146T_{5}^{4} - 296T_{5}^{2} + 72 \) acting on \(S_{2}^{\mathrm{new}}(5796, [\chi])\). 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
$5$ \( (T^{8} - 22 T^{6} + 146 T^{4} - 296 T^{2} + \cdots + 72)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{14} + 68 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 54 T^{6} + 568 T^{4} + 1624 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 40 T^{6} + 507 T^{4} + 2152 T^{2} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 60 T^{6} + 966 T^{4} - 4724 T^{2} + \cdots + 648)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 78 T^{6} + 1980 T^{4} + \cdots + 28800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 8 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 93 T^{2} + 34 T + 1854)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 114 T^{6} + 2425 T^{4} + \cdots + 450)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 84 T^{6} + 1252 T^{4} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 264 T^{6} + 24587 T^{4} + \cdots + 12781568)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 176 T^{6} + 7296 T^{4} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 258 T^{6} + 17585 T^{4} + \cdots + 55778)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 310 T^{6} + 35716 T^{4} + \cdots + 34105600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 154 T^{6} + 8494 T^{4} + \cdots + 1620000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 102 T^{6} + 3702 T^{4} + \cdots + 307328)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 412 T^{6} + 51368 T^{4} + \cdots + 36144144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 2 T^{3} - 123 T^{2} + 468 T + 1080)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 240 T^{6} + 11659 T^{4} + \cdots + 968832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 502 T^{6} + 86040 T^{4} + \cdots + 114318864)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 250 T^{6} + 6716 T^{4} + \cdots + 28800)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 482 T^{6} + 74038 T^{4} + \cdots + 165888)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 342 T^{6} + 26466 T^{4} + \cdots + 19208)^{2} \) Copy content Toggle raw display
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