↑

Properties

Label 7935.2.a.bq
Level $7935$
Weight $2$
Character orbit 7935.a
Self dual yes
Analytic conductor $63.361$
Analytic rank $1$
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] = [7935,2,Mod(1,7935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7935, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7935.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.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: \(63.3612940039\)
Analytic rank: \(1\)
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: \( x^{15} - 21x^{13} + 172x^{11} - 696x^{9} + 1466x^{7} - 1583x^{5} + 803x^{3} - 11x^{2} - 143x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{12} - \beta_{9} + \cdots + \beta_{2}) q^{4}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{12} - \beta_{9} + \cdots + \beta_{2}) q^{4}+ \cdots + (\beta_{14} + \beta_{13} + \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{3} + 12 q^{4} + 15 q^{5} + 5 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{3} + 12 q^{4} + 15 q^{5} + 5 q^{7} + 15 q^{9} - 13 q^{11} - 12 q^{12} - 24 q^{13} - 15 q^{14} - 15 q^{15} + 2 q^{16} - 2 q^{17} - 13 q^{19} + 12 q^{20} - 5 q^{21} + 9 q^{22} + 15 q^{25} - 9 q^{26} - 15 q^{27} - q^{28} - 3 q^{29} - 22 q^{31} + 13 q^{33} - q^{34} + 5 q^{35} + 12 q^{36} + 6 q^{37} + 17 q^{38} + 24 q^{39} - 20 q^{41} + 15 q^{42} + 4 q^{43} + 18 q^{44} + 15 q^{45} + 7 q^{47} - 2 q^{48} - 10 q^{49} + 2 q^{51} - 55 q^{52} - 4 q^{53} - 13 q^{55} - 25 q^{56} + 13 q^{57} + 2 q^{58} + 9 q^{59} - 12 q^{60} - 13 q^{61} - 7 q^{62} + 5 q^{63} - 26 q^{64} - 24 q^{65} - 9 q^{66} + 32 q^{67} - 27 q^{68} - 15 q^{70} + 14 q^{71} - 43 q^{73} + 11 q^{74} - 15 q^{75} - 88 q^{76} - 21 q^{77} + 9 q^{78} - 33 q^{79} + 2 q^{80} + 15 q^{81} - 35 q^{82} + 12 q^{83} + q^{84} - 2 q^{85} - 77 q^{86} + 3 q^{87} + 37 q^{88} - 29 q^{89} - 20 q^{91} + 22 q^{93} - 20 q^{94} - 13 q^{95} - 18 q^{97} - 19 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 21x^{13} + 172x^{11} - 696x^{9} + 1466x^{7} - 1583x^{5} + 803x^{3} - 11x^{2} - 143x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 254730 \nu^{14} - 914678 \nu^{13} + 5362553 \nu^{12} + 17241718 \nu^{11} - 43440600 \nu^{10} + \cdots + 37594449 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 522683 \nu^{14} - 137164 \nu^{13} - 10352146 \nu^{12} + 3375373 \nu^{11} + 80365547 \nu^{10} + \cdots - 65127440 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 535066 \nu^{14} + 1512080 \nu^{13} + 10077123 \nu^{12} - 30736529 \nu^{11} - 72418300 \nu^{10} + \cdots - 15124076 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 669354 \nu^{14} + 1415769 \nu^{13} - 13119967 \nu^{12} - 28775618 \nu^{11} + 96921391 \nu^{10} + \cdots - 58230526 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 914678 \nu^{14} + 13223 \nu^{13} + 17241718 \nu^{12} + 372960 \nu^{11} - 119465807 \nu^{10} + \cdots + 22837429 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1362533 \nu^{14} + 839850 \nu^{13} - 27636179 \nu^{12} - 17284033 \nu^{11} + 213696270 \nu^{10} + \cdots - 40044505 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1374916 \nu^{14} - 535066 \nu^{13} - 27361156 \nu^{12} + 10077123 \nu^{11} + 205749023 \nu^{10} + \cdots + 40207011 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1450839 \nu^{14} - 3727601 \nu^{13} - 29695598 \nu^{12} + 74458883 \nu^{11} + 234536640 \nu^{10} + \cdots + 37458936 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1683331 \nu^{14} - 1652724 \nu^{13} - 33977760 \nu^{12} + 33706163 \nu^{11} + 262688872 \nu^{10} + \cdots + 33982744 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2287008 \nu^{14} - 4707646 \nu^{13} + 45906150 \nu^{12} + 94597938 \nu^{11} - 353150424 \nu^{10} + \cdots + 84535715 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2835161 \nu^{14} - 1918997 \nu^{13} - 58297556 \nu^{12} + 38891630 \nu^{11} + 461181847 \nu^{10} + \cdots - 35392397 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2866608 \nu^{14} - 2791134 \nu^{13} - 58471216 \nu^{12} + 56251386 \nu^{11} + 459392728 \nu^{10} + \cdots + 50131441 ) / 20035399 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3514963 \nu^{14} + 915758 \nu^{13} - 68863680 \nu^{12} - 17115799 \nu^{11} + 506071864 \nu^{10} + \cdots + 71752559 ) / 20035399 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} + 6 \beta_{12} - \beta_{11} + \beta_{10} - 5 \beta_{9} - 6 \beta_{8} - \beta_{7} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 2 \beta_{10} - 7 \beta_{8} + 7 \beta_{7} - \beta_{6} - \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{13} + 33 \beta_{12} - 9 \beta_{11} + 8 \beta_{10} - 21 \beta_{9} - 33 \beta_{8} - 8 \beta_{7} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{14} + 12 \beta_{13} - 24 \beta_{10} - 45 \beta_{8} + 44 \beta_{7} - 12 \beta_{6} - 14 \beta_{5} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 92 \beta_{13} + 181 \beta_{12} - 66 \beta_{11} + 53 \beta_{10} - 78 \beta_{9} - 177 \beta_{8} + \cdots + 187 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15 \beta_{14} + 104 \beta_{13} + \beta_{12} - 208 \beta_{10} - 289 \beta_{8} + 271 \beta_{7} - 103 \beta_{6} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - \beta_{14} - 687 \beta_{13} + 1000 \beta_{12} - 454 \beta_{11} + 334 \beta_{10} - 224 \beta_{9} + \cdots + 1076 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 153 \beta_{14} + 791 \beta_{13} + 20 \beta_{12} + \beta_{11} - 1583 \beta_{10} - 3 \beta_{9} + \cdots + 621 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 15 \beta_{14} - 4827 \beta_{13} + 5565 \beta_{12} - 3027 \beta_{11} + 2066 \beta_{10} - 100 \beta_{9} + \cdots + 6313 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1324 \beta_{14} + 5617 \beta_{13} + 248 \beta_{12} + 17 \beta_{11} - 11255 \beta_{10} - 62 \beta_{9} + \cdots + 4113 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 146 \beta_{14} - 32681 \beta_{13} + 31164 \beta_{12} - 19799 \beta_{11} + 12668 \beta_{10} + \cdots + 37431 \) 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
−2.46438
−2.38025
−2.12733
−1.52166
−1.05174
−0.910856
−0.710736
0.0791955
0.523893
0.977162
1.15836
1.43887
2.17565
2.33043
2.48339
−2.46438 −1.00000 4.07318 1.00000 2.46438 1.15044 −5.10911 1.00000 −2.46438
1.2 −2.38025 −1.00000 3.66560 1.00000 2.38025 3.58597 −3.96455 1.00000 −2.38025
1.3 −2.12733 −1.00000 2.52552 1.00000 2.12733 −2.46529 −1.11794 1.00000 −2.12733
1.4 −1.52166 −1.00000 0.315440 1.00000 1.52166 2.43525 2.56332 1.00000 −1.52166
1.5 −1.05174 −1.00000 −0.893844 1.00000 1.05174 0.00828334 3.04357 1.00000 −1.05174
1.6 −0.910856 −1.00000 −1.17034 1.00000 0.910856 −2.43067 2.88772 1.00000 −0.910856
1.7 −0.710736 −1.00000 −1.49485 1.00000 0.710736 4.35754 2.48392 1.00000 −0.710736
1.8 0.0791955 −1.00000 −1.99373 1.00000 −0.0791955 −1.92769 −0.316285 1.00000 0.0791955
1.9 0.523893 −1.00000 −1.72554 1.00000 −0.523893 3.44277 −1.95178 1.00000 0.523893
1.10 0.977162 −1.00000 −1.04515 1.00000 −0.977162 −2.04484 −2.97561 1.00000 0.977162
1.11 1.15836 −1.00000 −0.658198 1.00000 −1.15836 −1.58372 −3.07915 1.00000 1.15836
1.12 1.43887 −1.00000 0.0703351 1.00000 −1.43887 3.25243 −2.77653 1.00000 1.43887
1.13 2.17565 −1.00000 2.73345 1.00000 −2.17565 −1.00697 1.59572 1.00000 2.17565
1.14 2.33043 −1.00000 3.43091 1.00000 −2.33043 1.17356 3.33465 1.00000 2.33043
1.15 2.48339 −1.00000 4.16722 1.00000 −2.48339 −2.94705 5.38206 1.00000 2.48339
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 7935.2.a.bq 15
23.b odd 2 1 7935.2.a.bp 15
23.d odd 22 2 345.2.m.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.a ✓ 30 23.d odd 22 2
7935.2.a.bp 15 23.b odd 2 1
7935.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):

\( T_{2}^{15} - 21T_{2}^{13} + 172T_{2}^{11} - 696T_{2}^{9} + 1466T_{2}^{7} - 1583T_{2}^{5} + 803T_{2}^{3} - 11T_{2}^{2} - 143T_{2} + 11 \) Copy content Toggle raw display
\( T_{7}^{15} - 5 T_{7}^{14} - 35 T_{7}^{13} + 168 T_{7}^{12} + 553 T_{7}^{11} - 2246 T_{7}^{10} + \cdots + 529 \) Copy content Toggle raw display
\( T_{11}^{15} + 13 T_{11}^{14} + 26 T_{11}^{13} - 339 T_{11}^{12} - 1714 T_{11}^{11} + 867 T_{11}^{10} + \cdots - 1979 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 21 T^{13} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( (T + 1)^{15} \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 5 T^{14} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{15} + 13 T^{14} + \cdots - 1979 \) Copy content Toggle raw display
$13$ \( T^{15} + 24 T^{14} + \cdots - 2189 \) Copy content Toggle raw display
$17$ \( T^{15} + 2 T^{14} + \cdots - 13003 \) Copy content Toggle raw display
$19$ \( T^{15} + 13 T^{14} + \cdots - 79607 \) Copy content Toggle raw display
$23$ \( T^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + 3 T^{14} + \cdots + 552661 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 234131371 \) Copy content Toggle raw display
$37$ \( T^{15} - 6 T^{14} + \cdots - 1199243 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 835582177 \) Copy content Toggle raw display
$43$ \( T^{15} - 4 T^{14} + \cdots - 28295719 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 1536259429 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 3355744919 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 393816853 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 54452259443 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 250050916103 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 73909198837 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 420345253151 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 494402124919 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 302656025783 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 870101506889 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 21016263293651 \) Copy content Toggle raw display
show more
show less