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Properties

Label 7406.2.a.bt
Level $7406$
Weight $2$
Character orbit 7406.a
Self dual yes
Analytic conductor $59.137$
Analytic rank $0$
Dimension $20$
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] = [7406,2,Mod(1,7406)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7406, 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("7406.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.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: \(59.1372077370\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 46 x^{18} + 45 x^{17} + 870 x^{16} - 815 x^{15} - 8776 x^{14} + 7663 x^{13} + \cdots - 5819 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{12} q^{5} - \beta_1 q^{6} + q^{7} - q^{8} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{12} q^{5} - \beta_1 q^{6} + q^{7} - q^{8} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{19} - \beta_{18} + 3 \beta_{17} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + q^{3} + 20 q^{4} + q^{5} - q^{6} + 20 q^{7} - 20 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + q^{3} + 20 q^{4} + q^{5} - q^{6} + 20 q^{7} - 20 q^{8} + 33 q^{9} - q^{10} + q^{12} + 13 q^{13} - 20 q^{14} + 33 q^{15} + 20 q^{16} - 5 q^{17} - 33 q^{18} - 9 q^{19} + q^{20} + q^{21} - q^{24} + 63 q^{25} - 13 q^{26} - 2 q^{27} + 20 q^{28} + 11 q^{29} - 33 q^{30} + 38 q^{31} - 20 q^{32} - 15 q^{33} + 5 q^{34} + q^{35} + 33 q^{36} + 37 q^{37} + 9 q^{38} + 14 q^{39} - q^{40} + 29 q^{41} - q^{42} + 42 q^{43} - 3 q^{45} - 30 q^{47} + q^{48} + 20 q^{49} - 63 q^{50} - 16 q^{51} + 13 q^{52} - 15 q^{53} + 2 q^{54} + 29 q^{55} - 20 q^{56} + 21 q^{57} - 11 q^{58} - 4 q^{59} + 33 q^{60} - 2 q^{61} - 38 q^{62} + 33 q^{63} + 20 q^{64} - 29 q^{65} + 15 q^{66} + 33 q^{67} - 5 q^{68} - q^{70} - 12 q^{71} - 33 q^{72} + 53 q^{73} - 37 q^{74} + 6 q^{75} - 9 q^{76} - 14 q^{78} + 7 q^{79} + q^{80} + 100 q^{81} - 29 q^{82} + 45 q^{83} + q^{84} + 58 q^{85} - 42 q^{86} - 11 q^{87} + 29 q^{89} + 3 q^{90} + 13 q^{91} + 10 q^{93} + 30 q^{94} - 2 q^{95} - q^{96} - 6 q^{97} - 20 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 46 x^{18} + 45 x^{17} + 870 x^{16} - 815 x^{15} - 8776 x^{14} + 7663 x^{13} + \cdots - 5819 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!92 \nu^{19} + \cdots - 69\!\cdots\!89 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!97 \nu^{19} + \cdots + 88\!\cdots\!72 ) / 21\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!94 \nu^{19} + \cdots - 15\!\cdots\!69 ) / 21\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!20 \nu^{19} + \cdots + 96\!\cdots\!45 ) / 21\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!57 \nu^{19} + \cdots - 34\!\cdots\!63 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!16 \nu^{19} + \cdots + 77\!\cdots\!21 ) / 21\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!24 \nu^{19} + \cdots + 35\!\cdots\!99 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38\!\cdots\!65 \nu^{19} + \cdots - 50\!\cdots\!36 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!05 \nu^{19} + \cdots - 13\!\cdots\!05 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!81 \nu^{19} + \cdots - 19\!\cdots\!59 ) / 21\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72\!\cdots\!51 \nu^{19} + \cdots - 87\!\cdots\!32 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 92\!\cdots\!94 \nu^{19} + \cdots - 59\!\cdots\!72 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!55 \nu^{19} + \cdots - 18\!\cdots\!26 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!49 \nu^{19} + \cdots + 21\!\cdots\!14 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!35 \nu^{19} + \cdots + 20\!\cdots\!91 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 24\!\cdots\!82 \nu^{19} + \cdots + 26\!\cdots\!32 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 25\!\cdots\!02 \nu^{19} + \cdots - 27\!\cdots\!99 ) / 49\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 34\!\cdots\!40 \nu^{19} + \cdots + 32\!\cdots\!74 ) / 49\!\cdots\!23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} - \beta_{16} + \beta_{14} - \beta_{13} + \beta_{12} + 3 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} + \beta_{16} + 11 \beta_{15} + 10 \beta_{14} - 12 \beta_{13} + 10 \beta_{12} + 14 \beta_{11} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{19} - 14 \beta_{18} - 14 \beta_{17} - 11 \beta_{16} + 13 \beta_{14} - 13 \beta_{13} + 10 \beta_{12} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{19} - 3 \beta_{18} - 4 \beta_{17} + 12 \beta_{16} + 119 \beta_{15} + 104 \beta_{14} + \cdots + 291 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125 \beta_{19} - 167 \beta_{18} - 169 \beta_{17} - 115 \beta_{16} + 13 \beta_{15} + 149 \beta_{14} + \cdots - 80 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 180 \beta_{19} - 54 \beta_{18} - 80 \beta_{17} + 112 \beta_{16} + 1287 \beta_{15} + 1102 \beta_{14} + \cdots + 2940 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1255 \beta_{19} - 1908 \beta_{18} - 1953 \beta_{17} - 1215 \beta_{16} + 319 \beta_{15} + 1659 \beta_{14} + \cdots - 701 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2028 \beta_{19} - 686 \beta_{18} - 1142 \beta_{17} + 942 \beta_{16} + 13917 \beta_{15} + 11761 \beta_{14} + \cdots + 30633 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12451 \beta_{19} - 21393 \beta_{18} - 22109 \beta_{17} - 12937 \beta_{16} + 5388 \beta_{15} + \cdots - 5817 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 22366 \beta_{19} - 7605 \beta_{18} - 14319 \beta_{17} + 7269 \beta_{16} + 150480 \beta_{15} + \cdots + 323749 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 122842 \beta_{19} - 237489 \beta_{18} - 247272 \beta_{17} - 138150 \beta_{16} + 78092 \beta_{15} + \cdots - 43355 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 245131 \beta_{19} - 78688 \beta_{18} - 168283 \beta_{17} + 49499 \beta_{16} + 1627502 \beta_{15} + \cdots + 3447429 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1206160 \beta_{19} - 2621987 \beta_{18} - 2744481 \beta_{17} - 1476397 \beta_{16} + 1044681 \beta_{15} + \cdots - 253939 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2686361 \beta_{19} - 781589 \beta_{18} - 1906611 \beta_{17} + 251861 \beta_{16} + 17613354 \beta_{15} + \cdots + 36880711 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 11774821 \beta_{19} - 28861786 \beta_{18} - 30314055 \beta_{17} - 15782820 \beta_{16} + 13325785 \beta_{15} + \cdots - 416728 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 29507803 \beta_{19} - 7551179 \beta_{18} - 21130039 \beta_{17} - 27559 \beta_{16} + 190802938 \beta_{15} + \cdots + 395854019 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 114100387 \beta_{19} - 317214070 \beta_{18} - 333842595 \beta_{17} - 168804938 \beta_{16} + \cdots + 20485638 \) 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
−3.33557
−3.28499
−3.12783
−2.03438
−1.92685
−1.45828
−1.42441
−1.20820
−0.628238
0.375613
0.445654
0.614189
0.639390
1.26911
1.79166
2.48710
2.56036
2.67447
3.22542
3.34579
−1.00000 −3.33557 1.00000 −3.89939 3.33557 1.00000 −1.00000 8.12605 3.89939
1.2 −1.00000 −3.28499 1.00000 1.96745 3.28499 1.00000 −1.00000 7.79116 −1.96745
1.3 −1.00000 −3.12783 1.00000 −0.721594 3.12783 1.00000 −1.00000 6.78333 0.721594
1.4 −1.00000 −2.03438 1.00000 1.14511 2.03438 1.00000 −1.00000 1.13872 −1.14511
1.5 −1.00000 −1.92685 1.00000 −3.92034 1.92685 1.00000 −1.00000 0.712762 3.92034
1.6 −1.00000 −1.45828 1.00000 −4.21795 1.45828 1.00000 −1.00000 −0.873406 4.21795
1.7 −1.00000 −1.42441 1.00000 3.24532 1.42441 1.00000 −1.00000 −0.971069 −3.24532
1.8 −1.00000 −1.20820 1.00000 −2.97956 1.20820 1.00000 −1.00000 −1.54025 2.97956
1.9 −1.00000 −0.628238 1.00000 2.50357 0.628238 1.00000 −1.00000 −2.60532 −2.50357
1.10 −1.00000 0.375613 1.00000 −1.82566 −0.375613 1.00000 −1.00000 −2.85891 1.82566
1.11 −1.00000 0.445654 1.00000 −0.0356934 −0.445654 1.00000 −1.00000 −2.80139 0.0356934
1.12 −1.00000 0.614189 1.00000 3.99841 −0.614189 1.00000 −1.00000 −2.62277 −3.99841
1.13 −1.00000 0.639390 1.00000 0.904622 −0.639390 1.00000 −1.00000 −2.59118 −0.904622
1.14 −1.00000 1.26911 1.00000 3.71932 −1.26911 1.00000 −1.00000 −1.38936 −3.71932
1.15 −1.00000 1.79166 1.00000 −1.79345 −1.79166 1.00000 −1.00000 0.210032 1.79345
1.16 −1.00000 2.48710 1.00000 −1.83557 −2.48710 1.00000 −1.00000 3.18566 1.83557
1.17 −1.00000 2.56036 1.00000 −3.88639 −2.56036 1.00000 −1.00000 3.55544 3.88639
1.18 −1.00000 2.67447 1.00000 3.86429 −2.67447 1.00000 −1.00000 4.15280 −3.86429
1.19 −1.00000 3.22542 1.00000 1.57945 −3.22542 1.00000 −1.00000 7.40335 −1.57945
1.20 −1.00000 3.34579 1.00000 3.18805 −3.34579 1.00000 −1.00000 8.19434 −3.18805
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 7406.2.a.bt 20
23.b odd 2 1 7406.2.a.bs 20
23.c even 11 2 322.2.i.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.e ✓ 40 23.c even 11 2
7406.2.a.bs 20 23.b odd 2 1
7406.2.a.bt 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):

\( T_{3}^{20} - T_{3}^{19} - 46 T_{3}^{18} + 45 T_{3}^{17} + 870 T_{3}^{16} - 815 T_{3}^{15} - 8776 T_{3}^{14} + \cdots - 5819 \) Copy content Toggle raw display
\( T_{5}^{20} - T_{5}^{19} - 81 T_{5}^{18} + 91 T_{5}^{17} + 2753 T_{5}^{16} - 3389 T_{5}^{15} + \cdots + 553817 \) Copy content Toggle raw display
\( T_{11}^{20} - 168 T_{11}^{18} + 66 T_{11}^{17} + 12064 T_{11}^{16} - 8998 T_{11}^{15} + \cdots + 4807238656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots - 5819 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{19} + \cdots + 553817 \) Copy content Toggle raw display
$7$ \( (T - 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 4807238656 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 141080929 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 14970487808 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 136853729057 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 3015161936896 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 1740477887488 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 601954804736 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 118827630593024 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 2851436094464 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 12329761989632 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 14565426045952 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 3563392522349 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 1301397154816 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 17717010281333 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 15\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 185615574947 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 87\!\cdots\!77 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 75\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 250156029438976 \) Copy content Toggle raw display
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