Properties

Label 7623.2.a.dc
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $0$
Dimension $16$
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] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.a (trivial)

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: yes
Analytic conductor: \(60.8699614608\)
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: \( x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{7} q^{5} + q^{7} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{7} q^{5} + q^{7} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{8} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{10} + \beta_{14} q^{13} + \beta_1 q^{14} + (\beta_{3} + \beta_{2} + 2) q^{16} + (\beta_{15} - \beta_{12} + \beta_{6}) q^{17} + ( - \beta_{13} + \beta_{8} - \beta_{2}) q^{19} + ( - 2 \beta_{10} + \beta_{9} + 2 \beta_{7} - \beta_{6}) q^{20} + ( - \beta_{15} + \beta_{12} + \beta_{10} + \beta_1) q^{23} + ( - \beta_{14} - \beta_{8} + 2 \beta_{5} + \beta_{2} + 4) q^{25} + (\beta_{15} - \beta_{12} - \beta_{11} + \beta_{9} + \beta_{7}) q^{26} + (\beta_{2} + 1) q^{28} + (\beta_{15} + \beta_{11} + 2 \beta_{10} + \beta_{6} - \beta_1) q^{29} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{31} + (\beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{7} + 2 \beta_1) q^{32} + (\beta_{14} + \beta_{8} - 2 \beta_{5} + \beta_{2} - 1) q^{34} + \beta_{7} q^{35} + ( - \beta_{8} + \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{37} + (\beta_{15} + \beta_{12} + 2 \beta_{11} - \beta_{9} + \beta_{7}) q^{38} + (\beta_{14} + \beta_{13} - 5 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 6) q^{40} + ( - 2 \beta_{10} + \beta_{9} + \beta_{6} + 2 \beta_1) q^{41} + ( - \beta_{13} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{43} + ( - \beta_{14} - \beta_{8} + 4 \beta_{5} + 2 \beta_{2} + 7) q^{46} + ( - \beta_{15} - \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{6} - 2 \beta_1) q^{47} + q^{49} + ( - 2 \beta_{15} - 3 \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_1) q^{50} + (\beta_{13} - 2 \beta_{5} - 2 \beta_{3} - 1) q^{52} + ( - \beta_{12} - 2 \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_1) q^{53} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{56} + (\beta_{14} + 2 \beta_{8} + 4 \beta_{5} - \beta_{4} + 2) q^{58} + ( - \beta_{15} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{7}) q^{59} + (\beta_{14} - 2 \beta_{4} - 2) q^{61} + ( - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} + 2 \beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{62}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{14} - 155\nu^{12} + 1002\nu^{10} - 1989\nu^{8} - 2930\nu^{6} + 13717\nu^{4} - 12905\nu^{2} + 3144 ) / 307 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -15\nu^{14} + 329\nu^{12} - 2723\nu^{10} + 10752\nu^{8} - 21906\nu^{6} + 24667\nu^{4} - 14063\nu^{2} + 2087 ) / 614 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47 \nu^{15} + 1563 \nu^{13} - 19625 \nu^{11} + 117562 \nu^{9} - 344202 \nu^{7} + 441351 \nu^{5} - 159537 \nu^{3} - 55925 \nu ) / 6754 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47 \nu^{15} + 1563 \nu^{13} - 19625 \nu^{11} + 117562 \nu^{9} - 344202 \nu^{7} + 441351 \nu^{5} - 166291 \nu^{3} - 22155 \nu ) / 6754 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -63\nu^{14} + 1259\nu^{12} - 8735\nu^{10} + 23300\nu^{8} - 12308\nu^{6} - 26935\nu^{4} + 27141\nu^{2} - 5111 ) / 614 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 76 \nu^{15} - 2240 \nu^{13} + 25483 \nu^{11} - 139086 \nu^{9} + 363590 \nu^{7} - 378848 \nu^{5} + 57151 \nu^{3} + 48902 \nu ) / 3377 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15 \nu^{15} + 329 \nu^{13} - 2723 \nu^{11} + 10752 \nu^{9} - 21906 \nu^{7} + 24667 \nu^{5} - 14063 \nu^{3} + 2701 \nu ) / 614 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 353 \nu^{15} - 9871 \nu^{13} + 108453 \nu^{11} - 588520 \nu^{9} + 1624528 \nu^{7} - 2124543 \nu^{5} + 1130541 \nu^{3} - 157219 \nu ) / 6754 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 230 \nu^{15} + 6068 \nu^{13} - 63345 \nu^{11} + 331872 \nu^{9} - 913359 \nu^{7} + 1263820 \nu^{5} - 778773 \nu^{3} + 153982 \nu ) / 3377 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 39 \nu^{14} - 1101 \nu^{12} + 12176 \nu^{10} - 66146 \nu^{8} + 180738 \nu^{6} - 227581 \nu^{4} + 108279 \nu^{2} - 11689 ) / 307 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 63 \nu^{14} + 1566 \nu^{12} - 15182 \nu^{10} + 72420 \nu^{8} - 176246 \nu^{6} + 205157 \nu^{4} - 97194 \nu^{2} + 12388 ) / 307 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 799 \nu^{15} + 19817 \nu^{13} - 191791 \nu^{11} + 914537 \nu^{9} - 2231290 \nu^{7} + 2616448 \nu^{5} - 1256642 \nu^{3} + 173816 \nu ) / 3377 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 10\beta_{7} + 8\beta_{6} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{14} - \beta_{13} + \beta_{8} - \beta_{5} + 11\beta_{3} + 46\beta_{2} + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{12} + 14\beta_{11} - 12\beta_{10} - 13\beta_{9} - 80\beta_{7} + 58\beta_{6} + 192\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -13\beta_{14} - 13\beta_{13} + 14\beta_{8} - 14\beta_{5} + 3\beta_{4} + 93\beta_{3} + 307\beta_{2} + 619 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{15} + 123\beta_{12} + 139\beta_{11} - 113\beta_{10} - 119\beta_{9} - 597\beta_{7} + 415\beta_{6} + 1274\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -118\beta_{14} - 119\beta_{13} + 140\beta_{8} - 158\beta_{5} + 54\beta_{4} + 716\beta_{3} + 2088\beta_{2} + 4064 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 22 \beta_{15} + 1028 \beta_{12} + 1210 \beta_{11} - 982 \beta_{10} - 953 \beta_{9} - 4321 \beta_{7} + 2965 \beta_{6} + 8657 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 931 \beta_{14} - 953 \beta_{13} + 1232 \beta_{8} - 1612 \beta_{5} + 654 \beta_{4} + 5274 \beta_{3} + 14405 \beta_{2} + 27164 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 301 \beta_{15} + 8091 \beta_{12} + 9881 \beta_{11} - 8194 \beta_{10} - 7158 \beta_{9} - 30783 \beta_{7} + 21190 \beta_{6} + 59771 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6857 \beta_{14} - 7158 \beta_{13} + 10182 \beta_{8} - 15290 \beta_{5} + 6692 \beta_{4} + 37941 \beta_{3} + 100361 \beta_{2} + 183724 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3325 \beta_{15} + 61672 \beta_{12} + 77902 \beta_{11} - 66615 \beta_{10} - 51956 \beta_{9} - 217373 \beta_{7} + 151508 \beta_{6} + 417076 \beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.68084
−2.60390
−2.10399
−2.08213
−1.31288
−0.899827
−0.759363
−0.400970
0.400970
0.759363
0.899827
1.31288
2.08213
2.10399
2.60390
2.68084
−2.68084 0 5.18691 1.17039 0 1.00000 −8.54361 0 −3.13763
1.2 −2.60390 0 4.78032 4.31860 0 1.00000 −7.23969 0 −11.2452
1.3 −2.10399 0 2.42677 −3.75046 0 1.00000 −0.897910 0 7.89093
1.4 −2.08213 0 2.33525 −1.33222 0 1.00000 −0.698024 0 2.77386
1.5 −1.31288 0 −0.276342 −3.45937 0 1.00000 2.98857 0 4.54174
1.6 −0.899827 0 −1.19031 0.803961 0 1.00000 2.87073 0 −0.723426
1.7 −0.759363 0 −1.42337 2.86525 0 1.00000 2.59958 0 −2.17577
1.8 −0.400970 0 −1.83922 2.30562 0 1.00000 1.53941 0 −0.924485
1.9 0.400970 0 −1.83922 −2.30562 0 1.00000 −1.53941 0 −0.924485
1.10 0.759363 0 −1.42337 −2.86525 0 1.00000 −2.59958 0 −2.17577
1.11 0.899827 0 −1.19031 −0.803961 0 1.00000 −2.87073 0 −0.723426
1.12 1.31288 0 −0.276342 3.45937 0 1.00000 −2.98857 0 4.54174
1.13 2.08213 0 2.33525 1.33222 0 1.00000 0.698024 0 2.77386
1.14 2.10399 0 2.42677 3.75046 0 1.00000 0.897910 0 7.89093
1.15 2.60390 0 4.78032 −4.31860 0 1.00000 7.23969 0 −11.2452
1.16 2.68084 0 5.18691 −1.17039 0 1.00000 8.54361 0 −3.13763
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.dc 16
3.b odd 2 1 inner 7623.2.a.dc 16
11.b odd 2 1 7623.2.a.db 16
11.c even 5 2 693.2.m.k 32
33.d even 2 1 7623.2.a.db 16
33.h odd 10 2 693.2.m.k 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.k 32 11.c even 5 2
693.2.m.k 32 33.h odd 10 2
7623.2.a.db 16 11.b odd 2 1
7623.2.a.db 16 33.d even 2 1
7623.2.a.dc 16 1.a even 1 1 trivial
7623.2.a.dc 16 3.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{16} - 26T_{2}^{14} + 268T_{2}^{12} - 1395T_{2}^{10} + 3876T_{2}^{8} - 5635T_{2}^{6} + 4042T_{2}^{4} - 1272T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{5}^{16} - 62 T_{5}^{14} + 1527 T_{5}^{12} - 19130 T_{5}^{10} + 129721 T_{5}^{8} - 470400 T_{5}^{6} + 858208 T_{5}^{4} - 723136 T_{5}^{2} + 215296 \) Copy content Toggle raw display
\( T_{13}^{8} - 81T_{13}^{6} + 60T_{13}^{5} + 2016T_{13}^{4} - 2280T_{13}^{3} - 16016T_{13}^{2} + 12640T_{13} + 42176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 26 T^{14} + 268 T^{12} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 62 T^{14} + 1527 T^{12} + \cdots + 215296 \) Copy content Toggle raw display
$7$ \( (T - 1)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 81 T^{6} + 60 T^{5} + 2016 T^{4} + \cdots + 42176)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 189 T^{14} + \cdots + 1249198336 \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} - 99 T^{6} + 280 T^{5} + \cdots + 6336)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 227 T^{14} + 19672 T^{12} + \cdots + 929296 \) Copy content Toggle raw display
$29$ \( T^{16} - 294 T^{14} + 30086 T^{12} + \cdots + 3920400 \) Copy content Toggle raw display
$31$ \( (T^{8} - 15 T^{7} - 34 T^{6} + \cdots + 306416)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 19 T^{7} - 2 T^{6} + 1665 T^{5} + \cdots + 84016)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 372 T^{14} + \cdots + 20759046400 \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} - 162 T^{6} + 1694 T^{5} + \cdots - 43244)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 563 T^{14} + \cdots + 658640896 \) Copy content Toggle raw display
$53$ \( T^{16} - 497 T^{14} + \cdots + 98834784400 \) Copy content Toggle raw display
$59$ \( T^{16} - 401 T^{14} + \cdots + 1358954496 \) Copy content Toggle raw display
$61$ \( (T^{8} + 14 T^{7} - 119 T^{6} + \cdots - 281920)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 26 T^{7} + 53 T^{6} + \cdots + 706816)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - 302 T^{14} + \cdots + 3891264400 \) Copy content Toggle raw display
$73$ \( (T^{8} + 7 T^{7} - 283 T^{6} + \cdots + 5996736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 27 T^{7} - 16 T^{6} + 4467 T^{5} + \cdots + 1980)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 636 T^{14} + \cdots + 74848798310400 \) Copy content Toggle raw display
$89$ \( T^{16} - 768 T^{14} + \cdots + 39082502560000 \) Copy content Toggle raw display
$97$ \( (T^{8} - 54 T^{7} + 1053 T^{6} + \cdots + 10823936)^{2} \) Copy content Toggle raw display
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