Properties

Label 7942.2.a.bx
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 41 x^{9} + 90 x^{8} - 198 x^{7} - 285 x^{6} + 396 x^{5} + 486 x^{4} - 240 x^{3} - 351 x^{2} - 90 x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{11} - \beta_{4} + \beta_{2}) q^{7} + q^{8} + (\beta_{8} + \beta_{7} - \beta_{3} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{11} - \beta_{4} + \beta_{2}) q^{7} + q^{8} + (\beta_{8} + \beta_{7} - \beta_{3} + \beta_1 - 1) q^{9} + ( - \beta_{2} - 1) q^{10} - q^{11} - \beta_1 q^{12} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{13} + ( - \beta_{11} - \beta_{4} + \beta_{2}) q^{14} + ( - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_1) q^{15} + q^{16} + (2 \beta_{11} - \beta_{9} - \beta_{8} - \beta_{2}) q^{17} + (\beta_{8} + \beta_{7} - \beta_{3} + \beta_1 - 1) q^{18} + ( - \beta_{2} - 1) q^{20} + (2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{21} - q^{22} + ( - \beta_{7} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} - \beta_1 q^{24} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} - 2 \beta_{5} + 3 \beta_{2} + \beta_1) q^{25} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{26} + (\beta_{11} + 2 \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{27} + ( - \beta_{11} - \beta_{4} + \beta_{2}) q^{28} + (\beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_1) q^{30} + (\beta_{11} - 3 \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{31} + q^{32} + \beta_1 q^{33} + (2 \beta_{11} - \beta_{9} - \beta_{8} - \beta_{2}) q^{34} + (\beta_{11} + \beta_{10} - \beta_{8} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1) q^{35} + (\beta_{8} + \beta_{7} - \beta_{3} + \beta_1 - 1) q^{36} + (\beta_{9} + 2 \beta_{8} + 2 \beta_{7} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{37} + (\beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3) q^{39} + ( - \beta_{2} - 1) q^{40} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \cdots - 2) q^{41}+ \cdots + ( - \beta_{8} - \beta_{7} + \beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} + 12 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} + 12 q^{8} + 3 q^{9} - 9 q^{10} - 12 q^{11} - 3 q^{12} + 9 q^{15} + 12 q^{16} - 9 q^{17} + 3 q^{18} - 9 q^{20} - 9 q^{21} - 12 q^{22} - 18 q^{23} - 3 q^{24} + 3 q^{25} - 21 q^{27} + 9 q^{29} + 9 q^{30} - 27 q^{31} + 12 q^{32} + 3 q^{33} - 9 q^{34} - 18 q^{35} + 3 q^{36} - 9 q^{37} + 18 q^{39} - 9 q^{40} - 27 q^{41} - 9 q^{42} + 9 q^{43} - 12 q^{44} - 36 q^{45} - 18 q^{46} - 27 q^{47} - 3 q^{48} - 6 q^{49} + 3 q^{50} - 18 q^{51} + 18 q^{53} - 21 q^{54} + 9 q^{55} + 9 q^{58} + 9 q^{60} - 18 q^{61} - 27 q^{62} + 12 q^{64} - 36 q^{65} + 3 q^{66} + 9 q^{67} - 9 q^{68} - 18 q^{69} - 18 q^{70} - 9 q^{71} + 3 q^{72} + 18 q^{73} - 9 q^{74} - 21 q^{75} + 18 q^{78} - 18 q^{79} - 9 q^{80} - 24 q^{81} - 27 q^{82} - 36 q^{83} - 9 q^{84} + 9 q^{85} + 9 q^{86} - 63 q^{87} - 12 q^{88} - 18 q^{89} - 36 q^{90} + 12 q^{91} - 18 q^{92} + 18 q^{93} - 27 q^{94} - 3 q^{96} + 12 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 15 x^{10} + 41 x^{9} + 90 x^{8} - 198 x^{7} - 285 x^{6} + 396 x^{5} + 486 x^{4} - 240 x^{3} - 351 x^{2} - 90 x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 93 \nu^{11} - 355 \nu^{10} - 215 \nu^{9} + 1497 \nu^{8} - 4066 \nu^{7} + 11828 \nu^{6} + 16444 \nu^{5} - 64398 \nu^{4} - 21577 \nu^{3} + 78629 \nu^{2} + 15183 \nu - 7229 ) / 2069 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 313 \nu^{11} - 1484 \nu^{10} - 3638 \nu^{9} + 21924 \nu^{8} + 13257 \nu^{7} - 115656 \nu^{6} - 16782 \nu^{5} + 260334 \nu^{4} + 17460 \nu^{3} - 222939 \nu^{2} - 35598 \nu + 18207 ) / 6207 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 316 \nu^{11} + 3164 \nu^{10} - 1027 \nu^{9} - 42996 \nu^{8} + 35796 \nu^{7} + 216255 \nu^{6} - 128178 \nu^{5} - 501531 \nu^{4} + 57987 \nu^{3} + 463143 \nu^{2} + \cdots - 12234 ) / 6207 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 487 \nu^{11} + 388 \nu^{10} - 12383 \nu^{9} - 3507 \nu^{8} + 97887 \nu^{7} + 14529 \nu^{6} - 305109 \nu^{5} - 57375 \nu^{4} + 340233 \nu^{3} + 96501 \nu^{2} - 32820 \nu + 744 ) / 6207 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 614 \nu^{11} + 2455 \nu^{10} + 7315 \nu^{9} - 34044 \nu^{8} - 30309 \nu^{7} + 170301 \nu^{6} + 67188 \nu^{5} - 368997 \nu^{4} - 138084 \nu^{3} + 285765 \nu^{2} + \cdots + 26592 ) / 6207 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 679 \nu^{11} + 1613 \nu^{10} + 11759 \nu^{9} - 23010 \nu^{8} - 80616 \nu^{7} + 117495 \nu^{6} + 269892 \nu^{5} - 252774 \nu^{4} - 421362 \nu^{3} + 181131 \nu^{2} + \cdots + 27729 ) / 6207 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 992 \nu^{11} - 3097 \nu^{10} - 15397 \nu^{9} + 44934 \nu^{8} + 93873 \nu^{7} - 233151 \nu^{6} - 286674 \nu^{5} + 513108 \nu^{4} + 438822 \nu^{3} - 397863 \nu^{2} + \cdots - 21936 ) / 6207 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1379 \nu^{11} + 2572 \nu^{10} + 21898 \nu^{9} - 25935 \nu^{8} - 134418 \nu^{7} + 57789 \nu^{6} + 381630 \nu^{5} + 90249 \nu^{4} - 413907 \nu^{3} - 302556 \nu^{2} + \cdots + 35199 ) / 6207 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1642 \nu^{11} - 7091 \nu^{10} - 16388 \nu^{9} + 89769 \nu^{8} + 50727 \nu^{7} - 400275 \nu^{6} - 85401 \nu^{5} + 772281 \nu^{4} + 192930 \nu^{3} - 549474 \nu^{2} + \cdots - 20892 ) / 6207 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 854 \nu^{11} + 2370 \nu^{10} + 12742 \nu^{9} - 31500 \nu^{8} - 72342 \nu^{7} + 144983 \nu^{6} + 195411 \nu^{5} - 269951 \nu^{4} - 238978 \nu^{3} + 165211 \nu^{2} + \cdots - 404 ) / 2069 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{11} - 4 \beta_{10} + 9 \beta_{8} + 10 \beta_{7} - \beta_{6} - 4 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 9 \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{11} - 25 \beta_{10} + 15 \beta_{8} + 17 \beta_{7} - 10 \beta_{6} - 12 \beta_{5} - 15 \beta_{4} - 20 \beta_{3} + 11 \beta_{2} + 36 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 33 \beta_{11} - 61 \beta_{10} - 2 \beta_{9} + 79 \beta_{8} + 94 \beta_{7} - 17 \beta_{6} - 10 \beta_{5} - 57 \beta_{4} - 103 \beta_{3} + 25 \beta_{2} + 85 \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 154 \beta_{11} - 264 \beta_{10} - 3 \beta_{9} + 172 \beta_{8} + 208 \beta_{7} - 93 \beta_{6} - 116 \beta_{5} - 182 \beta_{4} - 262 \beta_{3} + 105 \beta_{2} + 309 \beta _1 - 34 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 410 \beta_{11} - 721 \beta_{10} - 28 \beta_{9} + 718 \beta_{8} + 892 \beta_{7} - 210 \beta_{6} - 181 \beta_{5} - 644 \beta_{4} - 1056 \beta_{3} + 267 \beta_{2} + 838 \beta _1 - 113 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1596 \beta_{11} - 2685 \beta_{10} - 54 \beta_{9} + 1830 \beta_{8} + 2300 \beta_{7} - 885 \beta_{6} - 1091 \beta_{5} - 2013 \beta_{4} - 2987 \beta_{3} + 998 \beta_{2} + 2883 \beta _1 - 647 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4581 \beta_{11} - 7828 \beta_{10} - 295 \beta_{9} + 6767 \beta_{8} + 8631 \beta_{7} - 2335 \beta_{6} - 2323 \beta_{5} - 6798 \beta_{4} - 10749 \beta_{3} + 2766 \beta_{2} + 8408 \beta _1 - 2135 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16254 \beta_{11} - 27080 \beta_{10} - 677 \beta_{9} + 18943 \beta_{8} + 24349 \beta_{7} - 8639 \beta_{6} - 10386 \beta_{5} - 21294 \beta_{4} - 32068 \beta_{3} + 9673 \beta_{2} + 27988 \beta _1 - 8165 \) 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
3.18066
2.35032
2.15246
1.68007
1.60684
−0.0391340
−0.385135
−0.528177
−1.33949
−1.61335
−1.69090
−2.37416
1.00000 −3.18066 1.00000 −3.45219 −3.18066 3.50612 1.00000 7.11660 −3.45219
1.2 1.00000 −2.35032 1.00000 −1.74497 −2.35032 1.14267 1.00000 2.52400 −1.74497
1.3 1.00000 −2.15246 1.00000 −2.91209 −2.15246 −3.24957 1.00000 1.63308 −2.91209
1.4 1.00000 −1.68007 1.00000 2.00084 −1.68007 −2.90736 1.00000 −0.177370 2.00084
1.5 1.00000 −1.60684 1.00000 0.823492 −1.60684 0.0824650 1.00000 −0.418066 0.823492
1.6 1.00000 0.0391340 1.00000 2.72238 0.0391340 0.967809 1.00000 −2.99847 2.72238
1.7 1.00000 0.385135 1.00000 −0.181888 0.385135 4.50277 1.00000 −2.85167 −0.181888
1.8 1.00000 0.528177 1.00000 −3.17061 0.528177 −0.0476453 1.00000 −2.72103 −3.17061
1.9 1.00000 1.33949 1.00000 0.688863 1.33949 −0.213889 1.00000 −1.20577 0.688863
1.10 1.00000 1.61335 1.00000 0.900419 1.61335 −2.89419 1.00000 −0.397104 0.900419
1.11 1.00000 1.69090 1.00000 −3.94474 1.69090 2.32587 1.00000 −0.140861 −3.94474
1.12 1.00000 2.37416 1.00000 −0.729513 2.37416 −3.21505 1.00000 2.63666 −0.729513
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bx 12
19.b odd 2 1 7942.2.a.bt 12
19.f odd 18 2 418.2.j.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.a 24 19.f odd 18 2
7942.2.a.bt 12 19.b odd 2 1
7942.2.a.bx 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{12} + 3 T_{3}^{11} - 15 T_{3}^{10} - 41 T_{3}^{9} + 90 T_{3}^{8} + 198 T_{3}^{7} - 285 T_{3}^{6} - 396 T_{3}^{5} + 486 T_{3}^{4} + 240 T_{3}^{3} - 351 T_{3}^{2} + 90 T_{3} - 3 \) Copy content Toggle raw display
\( T_{5}^{12} + 9 T_{5}^{11} + 9 T_{5}^{10} - 117 T_{5}^{9} - 261 T_{5}^{8} + 459 T_{5}^{7} + 1287 T_{5}^{6} - 756 T_{5}^{5} - 1998 T_{5}^{4} + 900 T_{5}^{3} + 891 T_{5}^{2} - 324 T_{5} - 81 \) Copy content Toggle raw display
\( T_{13}^{12} - 75 T_{13}^{10} + 26 T_{13}^{9} + 1683 T_{13}^{8} - 1944 T_{13}^{7} - 14682 T_{13}^{6} + 31302 T_{13}^{5} + 27864 T_{13}^{4} - 140461 T_{13}^{3} + 156069 T_{13}^{2} - 71439 T_{13} + 11503 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} - 15 T^{10} - 41 T^{9} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{11} + 9 T^{10} - 117 T^{9} + \cdots - 81 \) Copy content Toggle raw display
$7$ \( T^{12} - 39 T^{10} - 11 T^{9} + 522 T^{8} + \cdots + 3 \) Copy content Toggle raw display
$11$ \( (T + 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 75 T^{10} + 26 T^{9} + \cdots + 11503 \) Copy content Toggle raw display
$17$ \( T^{12} + 9 T^{11} - 81 T^{10} + \cdots + 157797 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 18 T^{11} + 45 T^{10} + \cdots + 10344987 \) Copy content Toggle raw display
$29$ \( T^{12} - 9 T^{11} - 180 T^{10} + \cdots - 6230061 \) Copy content Toggle raw display
$31$ \( T^{12} + 27 T^{11} + 150 T^{10} + \cdots + 23815963 \) Copy content Toggle raw display
$37$ \( T^{12} + 9 T^{11} - 174 T^{10} + \cdots + 133364607 \) Copy content Toggle raw display
$41$ \( T^{12} + 27 T^{11} + 63 T^{10} + \cdots + 19766277 \) Copy content Toggle raw display
$43$ \( T^{12} - 9 T^{11} - 255 T^{10} + \cdots - 266779 \) Copy content Toggle raw display
$47$ \( T^{12} + 27 T^{11} + 45 T^{10} + \cdots + 81815211 \) Copy content Toggle raw display
$53$ \( T^{12} - 18 T^{11} - 36 T^{10} + \cdots + 1796247 \) Copy content Toggle raw display
$59$ \( T^{12} - 333 T^{10} + \cdots + 296333613 \) Copy content Toggle raw display
$61$ \( T^{12} + 18 T^{11} - 3 T^{10} + \cdots + 1638579 \) Copy content Toggle raw display
$67$ \( T^{12} - 9 T^{11} + \cdots - 1220891289 \) Copy content Toggle raw display
$71$ \( T^{12} + 9 T^{11} - 351 T^{10} + \cdots + 198281169 \) Copy content Toggle raw display
$73$ \( T^{12} - 18 T^{11} + \cdots + 516405459 \) Copy content Toggle raw display
$79$ \( T^{12} + 18 T^{11} + \cdots + 487576423 \) Copy content Toggle raw display
$83$ \( T^{12} + 36 T^{11} + \cdots - 1572682851 \) Copy content Toggle raw display
$89$ \( T^{12} + 18 T^{11} + \cdots + 458650431 \) Copy content Toggle raw display
$97$ \( T^{12} - 12 T^{11} + \cdots - 1327182035957 \) Copy content Toggle raw display
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