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Properties

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

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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: \(0\)
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: \( x^{15} - 3 x^{14} - 33 x^{13} + 101 x^{12} + 408 x^{11} - 1314 x^{10} - 2271 x^{9} + 8292 x^{8} + \cdots - 3592 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{8} + 1) q^{5} + \beta_1 q^{6} - \beta_{10} q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{8} + 1) q^{5} + \beta_1 q^{6} - \beta_{10} q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{3} + 2) q^{9} + (\beta_{8} + 1) q^{10} - q^{11} + \beta_1 q^{12} + (\beta_{13} - \beta_{12} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - \beta_{9} + \beta_{7} + \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} + 21 q^{15} + 15 q^{16} + 21 q^{17} + 30 q^{18} + 9 q^{20} - 9 q^{21} - 15 q^{22} + 21 q^{23} + 3 q^{24} + 24 q^{25} + 3 q^{27} - 9 q^{29} + 21 q^{30} + 18 q^{31} + 15 q^{32} - 3 q^{33} + 21 q^{34} + 18 q^{35} + 30 q^{36} - 9 q^{37} + 9 q^{40} + 15 q^{41} - 9 q^{42} + 3 q^{43} - 15 q^{44} + 54 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 24 q^{50} + 30 q^{51} + 18 q^{53} + 3 q^{54} - 9 q^{55} - 9 q^{58} + 6 q^{59} + 21 q^{60} - 30 q^{61} + 18 q^{62} + 24 q^{63} + 15 q^{64} + 6 q^{65} - 3 q^{66} + 9 q^{67} + 21 q^{68} - 42 q^{69} + 18 q^{70} + 9 q^{71} + 30 q^{72} + 12 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{79} + 9 q^{80} + 63 q^{81} + 15 q^{82} + 30 q^{83} - 9 q^{84} - 3 q^{85} + 3 q^{86} + 9 q^{87} - 15 q^{88} - 6 q^{89} + 54 q^{90} + 96 q^{91} + 21 q^{92} + 102 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 33 x^{13} + 101 x^{12} + 408 x^{11} - 1314 x^{10} - 2271 x^{9} + 8292 x^{8} + \cdots - 3592 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4041344225 \nu^{14} + 6626698403 \nu^{13} - 127869811855 \nu^{12} - 221689950789 \nu^{11} + \cdots - 2168207233588 ) / 222397362672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1432127461 \nu^{14} + 1940271789 \nu^{13} + 47458954663 \nu^{12} - 64858458563 \nu^{11} + \cdots - 4539845929460 ) / 74132454224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3028825681 \nu^{14} + 1764428881 \nu^{13} - 100570930577 \nu^{12} - 58806932067 \nu^{11} + \cdots + 3110313788056 ) / 111198681336 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3948428071 \nu^{14} + 17764928303 \nu^{13} + 147623274605 \nu^{12} - 587058086001 \nu^{11} + \cdots - 25984188002824 ) / 55599340668 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17390490625 \nu^{14} - 32764674113 \nu^{13} - 609748705379 \nu^{12} + 1073418798447 \nu^{11} + \cdots + 54077807515300 ) / 222397362672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1465855903 \nu^{14} + 677560785 \nu^{13} + 50032860231 \nu^{12} - 20981990719 \nu^{11} + \cdots - 1848964958544 ) / 18533113556 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3511525285 \nu^{14} + 8575536681 \nu^{13} + 124573127299 \nu^{12} - 281142512903 \nu^{11} + \cdots - 13058692892900 ) / 37066227112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7295551073 \nu^{14} + 4650514929 \nu^{13} + 247590395587 \nu^{12} - 148786421439 \nu^{11} + \cdots - 12306368034756 ) / 74132454224 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13059257297 \nu^{14} + 39244134781 \nu^{13} + 468148786135 \nu^{12} + \cdots - 60336934112828 ) / 111198681336 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16495511999 \nu^{14} - 31098804457 \nu^{13} - 578678421463 \nu^{12} + 1019939889939 \nu^{11} + \cdots + 53128770758240 ) / 111198681336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60223605067 \nu^{14} - 97888867343 \nu^{13} - 2103104210453 \nu^{12} + 3209609309289 \nu^{11} + \cdots + 178802193292228 ) / 222397362672 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 61955642611 \nu^{14} + 93675792959 \nu^{13} + 2164154956325 \nu^{12} + \cdots - 168300233929828 ) / 222397362672 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 77912907985 \nu^{14} - 147891891665 \nu^{13} - 2736053340203 \nu^{12} + \cdots + 251431984059796 ) / 111198681336 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{7} - \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{10} - \beta_{5} - 2\beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{11} - \beta_{10} + 10 \beta_{9} - \beta_{8} - 12 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 13 \beta_{13} - 15 \beta_{12} - 7 \beta_{11} + 11 \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 7 \beta_{13} - 7 \beta_{12} - 73 \beta_{11} - 15 \beta_{10} + 97 \beta_{9} - 16 \beta_{8} + \cdots + 350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26 \beta_{14} - 147 \beta_{13} - 195 \beta_{12} - 147 \beta_{11} + 114 \beta_{10} + 38 \beta_{9} + \cdots + 78 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 31 \beta_{14} - 168 \beta_{13} - 175 \beta_{12} - 1027 \beta_{11} - 178 \beta_{10} + 967 \beta_{9} + \cdots + 3314 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 458 \beta_{14} - 1650 \beta_{13} - 2423 \beta_{12} - 2299 \beta_{11} + 1188 \beta_{10} + 756 \beta_{9} + \cdots + 1237 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 672 \beta_{14} - 2804 \beta_{13} - 3103 \beta_{12} - 13221 \beta_{11} - 1911 \beta_{10} + 9988 \beta_{9} + \cdots + 32357 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6881 \beta_{14} - 18807 \beta_{13} - 29446 \beta_{12} - 32185 \beta_{11} + 12422 \beta_{10} + \cdots + 18046 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12025 \beta_{14} - 40528 \beta_{13} - 47740 \beta_{12} - 163466 \beta_{11} - 19159 \beta_{10} + \cdots + 323393 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 95188 \beta_{14} - 217735 \beta_{13} - 353449 \beta_{12} - 426141 \beta_{11} + 130015 \beta_{10} + \cdots + 249016 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 191039 \beta_{14} - 544842 \beta_{13} - 679460 \beta_{12} - 1979865 \beta_{11} - 181069 \beta_{10} + \cdots + 3297325 \) 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
−3.11106
−3.05702
−2.83505
−1.75198
−1.52164
−1.35740
0.337420
0.841073
1.11596
1.13626
1.37516
2.62401
2.62560
3.14157
3.43709
1.00000 −3.11106 1.00000 2.93101 −3.11106 4.25846 1.00000 6.67869 2.93101
1.2 1.00000 −3.05702 1.00000 −2.49410 −3.05702 −0.608525 1.00000 6.34538 −2.49410
1.3 1.00000 −2.83505 1.00000 −0.662259 −2.83505 −1.47776 1.00000 5.03749 −0.662259
1.4 1.00000 −1.75198 1.00000 2.75002 −1.75198 3.83483 1.00000 0.0694508 2.75002
1.5 1.00000 −1.52164 1.00000 −2.93560 −1.52164 −0.464814 1.00000 −0.684609 −2.93560
1.6 1.00000 −1.35740 1.00000 2.80170 −1.35740 −4.57307 1.00000 −1.15746 2.80170
1.7 1.00000 0.337420 1.00000 −2.62790 0.337420 −0.195559 1.00000 −2.88615 −2.62790
1.8 1.00000 0.841073 1.00000 −1.34881 0.841073 −0.547221 1.00000 −2.29260 −1.34881
1.9 1.00000 1.11596 1.00000 1.31280 1.11596 4.27450 1.00000 −1.75463 1.31280
1.10 1.00000 1.13626 1.00000 3.34560 1.13626 0.960360 1.00000 −1.70892 3.34560
1.11 1.00000 1.37516 1.00000 −2.56135 1.37516 −5.05243 1.00000 −1.10892 −2.56135
1.12 1.00000 2.62401 1.00000 −0.0573255 2.62401 3.60292 1.00000 3.88544 −0.0573255
1.13 1.00000 2.62560 1.00000 3.59746 2.62560 −4.57964 1.00000 3.89377 3.59746
1.14 1.00000 3.14157 1.00000 4.25314 3.14157 −0.789032 1.00000 6.86947 4.25314
1.15 1.00000 3.43709 1.00000 0.695614 3.43709 1.35698 1.00000 8.81359 0.695614
\(n\): 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.ca 15
19.b odd 2 1 7942.2.a.by 15
19.e even 9 2 418.2.j.d ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.d ✓ 30 19.e even 9 2
7942.2.a.by 15 19.b odd 2 1
7942.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{15} - 3 T_{3}^{14} - 33 T_{3}^{13} + 101 T_{3}^{12} + 408 T_{3}^{11} - 1314 T_{3}^{10} + \cdots - 3592 \) Copy content Toggle raw display
\( T_{5}^{15} - 9 T_{5}^{14} - 9 T_{5}^{13} + 273 T_{5}^{12} - 303 T_{5}^{11} - 3225 T_{5}^{10} + \cdots + 2664 \) Copy content Toggle raw display
\( T_{13}^{15} - 138 T_{13}^{13} + 37 T_{13}^{12} + 7392 T_{13}^{11} - 3429 T_{13}^{10} - 194979 T_{13}^{9} + \cdots + 17665057 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 3 T^{14} + \cdots - 3592 \) Copy content Toggle raw display
$5$ \( T^{15} - 9 T^{14} + \cdots + 2664 \) Copy content Toggle raw display
$7$ \( T^{15} - 69 T^{13} + \cdots + 1224 \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} - 138 T^{13} + \cdots + 17665057 \) Copy content Toggle raw display
$17$ \( T^{15} - 21 T^{14} + \cdots + 6578568 \) Copy content Toggle raw display
$19$ \( T^{15} \) Copy content Toggle raw display
$23$ \( T^{15} - 21 T^{14} + \cdots - 4112523 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 1094326677 \) Copy content Toggle raw display
$31$ \( T^{15} - 18 T^{14} + \cdots + 3776833 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 4518962856 \) Copy content Toggle raw display
$41$ \( T^{15} - 15 T^{14} + \cdots + 15839064 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 12044752147 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 949957443 \) Copy content Toggle raw display
$53$ \( T^{15} - 18 T^{14} + \cdots + 10857816 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 5414485176 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 56455635693 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 2705369544 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 278304249061161 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 255146083360152 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 2234956168 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 2418781653 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 110977169319 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 7094932136549 \) Copy content Toggle raw display
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