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Properties

Label 7803.2.a.cb
Level $7803$
Weight $2$
Character orbit 7803.a
Self dual yes
Analytic conductor $62.307$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7803,2,Mod(1,7803)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7803, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7803.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7803 = 3^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7803.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: \(62.3072686972\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 36 x^{16} + 549 x^{14} - 4621 x^{12} + 23445 x^{10} - 73524 x^{8} + 140584 x^{6} - 155406 x^{4} + \cdots - 18397 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{5} q^{5} + ( - \beta_{17} - \beta_{12} + \beta_{7}) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{5} q^{5} + ( - \beta_{17} - \beta_{12} + \beta_{7}) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - \beta_{17} + \beta_{16} + \beta_{10} + \cdots - 1) q^{10}+ \cdots + ( - 3 \beta_{14} + \beta_{13} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 36 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 36 q^{4} - 6 q^{7} - 12 q^{10} + 30 q^{13} + 36 q^{16} + 18 q^{19} - 12 q^{22} + 72 q^{25} + 18 q^{28} + 6 q^{31} - 72 q^{40} + 30 q^{43} + 24 q^{46} + 12 q^{49} + 102 q^{52} + 90 q^{55} + 12 q^{58} - 12 q^{61} + 78 q^{64} + 96 q^{67} - 60 q^{70} + 48 q^{73} + 96 q^{76} - 24 q^{79} - 24 q^{88} - 36 q^{91} - 42 q^{94} - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 36 x^{16} + 549 x^{14} - 4621 x^{12} + 23445 x^{10} - 73524 x^{8} + 140584 x^{6} - 155406 x^{4} + \cdots - 18397 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 519 \nu^{17} - 22133 \nu^{15} + 380622 \nu^{13} - 3448921 \nu^{11} + 17894650 \nu^{9} + \cdots + 19199183 \nu ) / 128236 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 487 \nu^{17} - 16012 \nu^{15} + 217058 \nu^{13} - 1566176 \nu^{11} + 6478822 \nu^{9} + \cdots + 2526165 \nu ) / 32059 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1051 \nu^{17} + 31725 \nu^{15} - 390888 \nu^{13} + 2534269 \nu^{11} - 9317728 \nu^{9} + \cdots - 4242455 \nu ) / 64118 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3271 \nu^{16} + 99713 \nu^{14} - 1238818 \nu^{12} + 8083537 \nu^{10} - 29806990 \nu^{8} + \cdots - 8115483 ) / 128236 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2601 \nu^{16} + 77750 \nu^{14} - 947964 \nu^{12} + 6086435 \nu^{10} - 22230027 \nu^{8} + \cdots - 7317152 ) / 64118 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2869 \nu^{16} - 92947 \nu^{14} + 1243836 \nu^{12} - 8898581 \nu^{10} + 36808464 \nu^{8} + \cdots + 18773781 ) / 64118 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3575 \nu^{16} + 109774 \nu^{14} - 1382080 \nu^{12} + 9218787 \nu^{10} - 35187671 \nu^{8} + \cdots - 12369482 ) / 64118 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3575 \nu^{17} + 109774 \nu^{15} - 1382080 \nu^{13} + 9218787 \nu^{11} - 35187671 \nu^{9} + \cdots - 12369482 \nu ) / 64118 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1901 \nu^{16} - 60067 \nu^{14} + 784285 \nu^{12} - 5476060 \nu^{10} + 22104134 \nu^{8} + \cdots + 10127274 ) / 32059 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1901 \nu^{17} - 60067 \nu^{15} + 784285 \nu^{13} - 5476060 \nu^{11} + 22104134 \nu^{9} + \cdots + 10127274 \nu ) / 32059 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10421 \nu^{17} + 319261 \nu^{15} - 4002978 \nu^{13} + 26521111 \nu^{11} - 100182332 \nu^{9} + \cdots - 32854447 \nu ) / 128236 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5373 \nu^{17} - 163163 \nu^{15} + 2020594 \nu^{13} - 13152075 \nu^{11} + 48442446 \nu^{9} + \cdots + 12753313 \nu ) / 64118 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 8201 \nu^{16} - 251273 \nu^{14} + 3155048 \nu^{12} - 20971843 \nu^{10} + 79693070 \nu^{8} + \cdots + 28019649 ) / 64118 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 9999 \nu^{16} - 304662 \nu^{14} + 3793562 \nu^{12} - 24905131 \nu^{10} + 92947845 \nu^{8} + \cdots + 29365250 ) / 64118 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - 2\beta_{16} + \beta_{12} - \beta_{8} + 9\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - \beta_{13} - 2\beta_{11} - \beta_{6} + \beta_{5} + 9\beta_{3} + 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{17} - 25\beta_{16} + 11\beta_{12} - 5\beta_{10} - 11\beta_{8} + 2\beta_{7} + 69\beta_{2} + 137 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{15} + 2\beta_{14} - 14\beta_{13} - 30\beta_{11} - 12\beta_{6} + 14\beta_{5} + 69\beta_{3} + 250\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 111 \beta_{17} - 234 \beta_{16} + 94 \beta_{12} - 72 \beta_{10} + \beta_{9} - 95 \beta_{8} + 34 \beta_{7} + \cdots + 903 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 123 \beta_{15} + 33 \beta_{14} - 140 \beta_{13} - 311 \beta_{11} - 111 \beta_{6} + 140 \beta_{5} + \cdots + 1694 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 939 \beta_{17} - 1973 \beta_{16} + 747 \beta_{12} - 720 \beta_{10} + 15 \beta_{9} - 757 \beta_{8} + \cdots + 6153 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1034 \beta_{15} + 373 \beta_{14} - 1226 \beta_{13} - 2789 \beta_{11} - 939 \beta_{6} + \cdots + 11745 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7605 \beta_{17} - 15818 \beta_{16} + 5783 \beta_{12} - 6243 \beta_{10} + 141 \beta_{9} - 5812 \beta_{8} + \cdots + 42891 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8213 \beta_{15} + 3617 \beta_{14} - 10035 \beta_{13} - 23277 \beta_{11} - 7605 \beta_{6} + \cdots + 82878 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 60040 \beta_{17} - 123295 \beta_{16} + 44286 \beta_{12} - 50441 \beta_{10} + 1029 \beta_{9} + \cdots + 303961 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 63255 \beta_{15} + 32407 \beta_{14} - 79009 \beta_{13} - 186653 \beta_{11} - 60040 \beta_{6} + \cdots + 592633 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 466251 \beta_{17} - 944738 \beta_{16} + 337401 \beta_{12} - 391839 \beta_{10} + 5924 \beta_{9} + \cdots + 2180359 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 478487 \beta_{15} + 277220 \beta_{14} - 607337 \beta_{13} - 1461446 \beta_{11} - 466251 \beta_{6} + \cdots + 4279521 \beta_1 \) Copy content Toggle raw display

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

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
−2.72847
−2.70719
−2.29936
−2.27787
−2.19456
−1.53624
−1.35946
−1.12403
−0.680530
0.680530
1.12403
1.35946
1.53624
2.19456
2.27787
2.29936
2.70719
2.72847
−2.72847 0 5.44453 0.231509 0 −0.982880 −9.39829 0 −0.631664
1.2 −2.70719 0 5.32888 4.30428 0 2.62880 −9.01192 0 −11.6525
1.3 −2.29936 0 3.28705 −3.30426 0 −4.39277 −2.95940 0 7.59768
1.4 −2.27787 0 3.18870 −2.25364 0 3.59123 −2.70771 0 5.13351
1.5 −2.19456 0 2.81609 3.79807 0 −1.64901 −1.79095 0 −8.33508
1.6 −1.53624 0 0.360028 −3.13036 0 2.45967 2.51939 0 4.80897
1.7 −1.35946 0 −0.151856 2.90602 0 1.28198 2.92537 0 −3.95063
1.8 −1.12403 0 −0.736551 1.20696 0 −2.30021 3.07597 0 −1.35666
1.9 −0.680530 0 −1.53688 −3.50663 0 −3.63681 2.40695 0 2.38636
1.10 0.680530 0 −1.53688 3.50663 0 −3.63681 −2.40695 0 2.38636
1.11 1.12403 0 −0.736551 −1.20696 0 −2.30021 −3.07597 0 −1.35666
1.12 1.35946 0 −0.151856 −2.90602 0 1.28198 −2.92537 0 −3.95063
1.13 1.53624 0 0.360028 3.13036 0 2.45967 −2.51939 0 4.80897
1.14 2.19456 0 2.81609 −3.79807 0 −1.64901 1.79095 0 −8.33508
1.15 2.27787 0 3.18870 2.25364 0 3.59123 2.70771 0 5.13351
1.16 2.29936 0 3.28705 3.30426 0 −4.39277 2.95940 0 7.59768
1.17 2.70719 0 5.32888 −4.30428 0 2.62880 9.01192 0 −11.6525
1.18 2.72847 0 5.44453 −0.231509 0 −0.982880 9.39829 0 −0.631664
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)

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 7803.2.a.cb ✓ 18
3.b odd 2 1 inner 7803.2.a.cb ✓ 18
17.b even 2 1 7803.2.a.cc yes 18
51.c odd 2 1 7803.2.a.cc yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7803.2.a.cb ✓ 18 1.a even 1 1 trivial
7803.2.a.cb ✓ 18 3.b odd 2 1 inner
7803.2.a.cc yes 18 17.b even 2 1
7803.2.a.cc yes 18 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):

\( T_{2}^{18} - 36 T_{2}^{16} + 549 T_{2}^{14} - 4621 T_{2}^{12} + 23445 T_{2}^{10} - 73524 T_{2}^{8} + \cdots - 18397 \) Copy content Toggle raw display
\( T_{5}^{18} - 81 T_{5}^{16} + 2772 T_{5}^{14} - 52118 T_{5}^{12} + 584847 T_{5}^{10} - 3968439 T_{5}^{8} + \cdots - 1177408 \) Copy content Toggle raw display
\( T_{7}^{9} + 3T_{7}^{8} - 30T_{7}^{7} - 76T_{7}^{6} + 303T_{7}^{5} + 621T_{7}^{4} - 1158T_{7}^{3} - 1953T_{7}^{2} + 1269T_{7} + 1773 \) Copy content Toggle raw display
\( T_{11}^{18} - 138 T_{11}^{16} + 8013 T_{11}^{14} - 253884 T_{11}^{12} + 4747908 T_{11}^{10} + \cdots - 10596672 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 36 T^{16} + \cdots - 18397 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 81 T^{16} + \cdots - 1177408 \) Copy content Toggle raw display
$7$ \( (T^{9} + 3 T^{8} + \cdots + 1773)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} - 138 T^{16} + \cdots - 10596672 \) Copy content Toggle raw display
$13$ \( (T^{9} - 15 T^{8} + \cdots + 155557)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} \) Copy content Toggle raw display
$19$ \( (T^{9} - 9 T^{8} + \cdots + 575919)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 465830878528 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 1091891679552 \) Copy content Toggle raw display
$31$ \( (T^{9} - 3 T^{8} + \cdots - 110771)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} - 192 T^{7} + \cdots + 23509)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 63450847971648 \) Copy content Toggle raw display
$43$ \( (T^{9} - 15 T^{8} + \cdots + 215937)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 212964379622208 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 480760411968 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 27561943872 \) Copy content Toggle raw display
$61$ \( (T^{9} + 6 T^{8} + \cdots + 31463)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} - 48 T^{8} + \cdots + 12004469)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 172298836055872 \) Copy content Toggle raw display
$73$ \( (T^{9} - 24 T^{8} + \cdots - 4012737)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + 12 T^{8} + \cdots + 3917376)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 340270912 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 2206652154688 \) Copy content Toggle raw display
$97$ \( (T^{9} + 15 T^{8} + \cdots - 4520179)^{2} \) Copy content Toggle raw display
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