↑

Properties

Label 7942.2.a.cb
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, 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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 27 x^{13} + 83 x^{12} + 264 x^{11} - 828 x^{10} - 1171 x^{9} + 3624 x^{8} + 2634 x^{7} + \cdots - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 418)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{12} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{7} + 1) q^{7} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{12} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{7} + 1) q^{7} + q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{12} + 1) q^{10} + q^{11} + \beta_1 q^{12} + (\beta_{11} - \beta_{9} - \beta_{2}) q^{13} + ( - \beta_{7} + 1) q^{14} + ( - \beta_{14} - \beta_{10} + \beta_{9} + \cdots - 1) q^{15}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 12 q^{7} + 15 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 12 q^{7} + 15 q^{8} + 18 q^{9} + 9 q^{10} + 15 q^{11} + 3 q^{12} + 12 q^{14} - 9 q^{15} + 15 q^{16} + 21 q^{17} + 18 q^{18} + 9 q^{20} - 27 q^{21} + 15 q^{22} + 21 q^{23} + 3 q^{24} + 36 q^{25} + 3 q^{27} + 12 q^{28} - 15 q^{29} - 9 q^{30} + 30 q^{31} + 15 q^{32} + 3 q^{33} + 21 q^{34} + 30 q^{35} + 18 q^{36} - 9 q^{37} + 9 q^{40} + 9 q^{41} - 27 q^{42} + 33 q^{43} + 15 q^{44} + 18 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 36 q^{50} + 6 q^{51} - 6 q^{53} + 3 q^{54} + 9 q^{55} + 12 q^{56} - 15 q^{58} - 6 q^{59} - 9 q^{60} + 36 q^{61} + 30 q^{62} + 30 q^{63} + 15 q^{64} + 18 q^{65} + 3 q^{66} + 15 q^{67} + 21 q^{68} + 48 q^{69} + 30 q^{70} - 3 q^{71} + 18 q^{72} + 60 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{77} + 18 q^{79} + 9 q^{80} + 27 q^{81} + 9 q^{82} + 36 q^{83} - 27 q^{84} + 15 q^{85} + 33 q^{86} + 21 q^{87} + 15 q^{88} + 6 q^{89} + 18 q^{90} - 18 q^{91} + 21 q^{92} - 54 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 27 x^{13} + 83 x^{12} + 264 x^{11} - 828 x^{10} - 1171 x^{9} + 3624 x^{8} + 2634 x^{7} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2926695 \nu^{14} + 6601067 \nu^{13} + 79862879 \nu^{12} - 183833163 \nu^{11} - 795872698 \nu^{10} + \cdots + 99971276 ) / 93856416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5533067 \nu^{14} - 19578967 \nu^{13} - 139235931 \nu^{12} + 535039751 \nu^{11} + 1183257770 \nu^{10} + \cdots - 76972908 ) / 93856416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19243227 \nu^{14} + 68795815 \nu^{13} + 480409195 \nu^{12} - 1875659703 \nu^{11} + \cdots + 45920428 ) / 187712832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24992819 \nu^{14} - 80831847 \nu^{13} - 661603979 \nu^{12} + 2234129735 \nu^{11} + \cdots - 827993852 ) / 187712832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12522945 \nu^{14} + 42193829 \nu^{13} + 328768145 \nu^{12} - 1163884245 \nu^{11} + \cdots + 855425636 ) / 93856416 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44100613 \nu^{14} + 152416361 \nu^{13} + 1100399205 \nu^{12} - 4152432697 \nu^{11} + \cdots + 271136628 ) / 187712832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12759611 \nu^{14} - 43798481 \nu^{13} - 323874901 \nu^{12} + 1199166773 \nu^{11} + \cdots - 233877016 ) / 46928208 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53783677 \nu^{14} + 169906697 \nu^{13} + 1416480357 \nu^{12} - 4675557673 \nu^{11} + \cdots + 1319876868 ) / 187712832 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29973621 \nu^{14} - 107855785 \nu^{13} - 745787269 \nu^{12} + 2938807737 \nu^{11} + \cdots - 346029076 ) / 93856416 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39958213 \nu^{14} - 131788201 \nu^{13} - 1036256933 \nu^{12} + 3621335257 \nu^{11} + \cdots - 940348820 ) / 93856416 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 80398797 \nu^{14} - 270783545 \nu^{13} - 2060619989 \nu^{12} + 7423194969 \nu^{11} + \cdots - 1273723076 ) / 187712832 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 54703009 \nu^{14} - 180767221 \nu^{13} - 1420588961 \nu^{12} + 4972841413 \nu^{11} + \cdots - 1893680036 ) / 93856416 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + 3 \beta_{6} - 4 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} + 11 \beta_{13} + \beta_{12} + 2 \beta_{7} - 13 \beta_{6} + 24 \beta_{5} - 14 \beta_{4} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{14} + 10 \beta_{13} - 15 \beta_{12} - 13 \beta_{11} - 2 \beta_{8} - 3 \beta_{7} + \cdots + 236 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 19 \beta_{14} + 103 \beta_{13} + 19 \beta_{12} - 4 \beta_{11} - \beta_{10} + 3 \beta_{9} + \beta_{8} + \cdots - 126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 159 \beta_{14} + 76 \beta_{13} - 178 \beta_{12} - 128 \beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots + 2033 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 269 \beta_{14} + 930 \beta_{13} + 278 \beta_{12} - 80 \beta_{11} - 15 \beta_{10} + 49 \beta_{9} + \cdots - 1691 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1704 \beta_{14} + 499 \beta_{13} - 1980 \beta_{12} - 1133 \beta_{11} + 17 \beta_{10} - 43 \beta_{9} + \cdots + 18096 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3398 \beta_{14} + 8348 \beta_{13} + 3633 \beta_{12} - 1117 \beta_{11} - 141 \beta_{10} + 558 \beta_{9} + \cdots - 20201 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 17889 \beta_{14} + 2665 \beta_{13} - 21470 \beta_{12} - 9411 \beta_{11} + 208 \beta_{10} + \cdots + 164640 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 40384 \beta_{14} + 75208 \beta_{13} + 44422 \beta_{12} - 13579 \beta_{11} - 989 \beta_{10} + \cdots - 227243 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 186289 \beta_{14} + 6846 \beta_{13} - 229666 \beta_{12} - 74022 \beta_{11} + 2235 \beta_{10} + \cdots + 1523036 \) 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
−3.19779
−2.96776
−2.04918
−1.72125
−0.590584
−0.450488
−0.325486
−0.175562
0.0975736
1.55510
1.81566
2.55482
2.66610
2.87060
2.91824
1.00000 −3.19779 1.00000 −0.357000 −3.19779 4.43696 1.00000 7.22586 −0.357000
1.2 1.00000 −2.96776 1.00000 3.77242 −2.96776 2.89351 1.00000 5.80761 3.77242
1.3 1.00000 −2.04918 1.00000 −1.33678 −2.04918 −2.79903 1.00000 1.19915 −1.33678
1.4 1.00000 −1.72125 1.00000 3.59322 −1.72125 3.50205 1.00000 −0.0372978 3.59322
1.5 1.00000 −0.590584 1.00000 −1.61138 −0.590584 3.93585 1.00000 −2.65121 −1.61138
1.6 1.00000 −0.450488 1.00000 −1.99360 −0.450488 3.08761 1.00000 −2.79706 −1.99360
1.7 1.00000 −0.325486 1.00000 4.31725 −0.325486 2.39653 1.00000 −2.89406 4.31725
1.8 1.00000 −0.175562 1.00000 −0.248996 −0.175562 −2.96694 1.00000 −2.96918 −0.248996
1.9 1.00000 0.0975736 1.00000 1.88847 0.0975736 −3.81151 1.00000 −2.99048 1.88847
1.10 1.00000 1.55510 1.00000 −4.10473 1.55510 0.583172 1.00000 −0.581655 −4.10473
1.11 1.00000 1.81566 1.00000 3.61187 1.81566 1.50482 1.00000 0.296630 3.61187
1.12 1.00000 2.55482 1.00000 −1.03113 2.55482 2.77384 1.00000 3.52711 −1.03113
1.13 1.00000 2.66610 1.00000 3.76828 2.66610 −0.337789 1.00000 4.10809 3.76828
1.14 1.00000 2.87060 1.00000 1.25601 2.87060 1.57964 1.00000 5.24034 1.25601
1.15 1.00000 2.91824 1.00000 −2.52391 2.91824 −4.77871 1.00000 5.51615 −2.52391
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7942.2.a.cb 15
19.b odd 2 1 7942.2.a.bz 15
19.f odd 18 2 418.2.j.c ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.c ✓ 30 19.f odd 18 2
7942.2.a.bz 15 19.b odd 2 1
7942.2.a.cb 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{15} - 3 T_{3}^{14} - 27 T_{3}^{13} + 83 T_{3}^{12} + 264 T_{3}^{11} - 828 T_{3}^{10} - 1171 T_{3}^{9} + \cdots - 8 \) Copy content Toggle raw display
\( T_{5}^{15} - 9 T_{5}^{14} - 15 T_{5}^{13} + 315 T_{5}^{12} - 231 T_{5}^{11} - 3981 T_{5}^{10} + \cdots - 7704 \) Copy content Toggle raw display
\( T_{13}^{15} - 90 T_{13}^{13} - 13 T_{13}^{12} + 2694 T_{13}^{11} + 645 T_{13}^{10} - 31749 T_{13}^{9} + \cdots - 3077 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 3 T^{14} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{15} - 9 T^{14} + \cdots - 7704 \) Copy content Toggle raw display
$7$ \( T^{15} - 12 T^{14} + \cdots + 257256 \) Copy content Toggle raw display
$11$ \( (T - 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} - 90 T^{13} + \cdots - 3077 \) Copy content Toggle raw display
$17$ \( T^{15} - 21 T^{14} + \cdots + 2468808 \) Copy content Toggle raw display
$19$ \( T^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 188961021 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 969316317 \) Copy content Toggle raw display
$31$ \( T^{15} - 30 T^{14} + \cdots - 1472129 \) Copy content Toggle raw display
$37$ \( T^{15} + 9 T^{14} + \cdots - 4953528 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 282351096 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 165703887 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 14520741561 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 1025336531304 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 49488783192 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 56976854223 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 26019018936 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 1421197893 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 26689646904 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 34435413928 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 262500417 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 46147278509361 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 165122431973 \) Copy content Toggle raw display
show more
show less