Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [149,4,Mod(1,149)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("149.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(149, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 149 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 149.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.79128459086\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 4 x^{14} - 65 x^{13} + 245 x^{12} + 1627 x^{11} - 5700 x^{10} - 19836 x^{9} + 63600 x^{8} + \cdots + 1096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{13} - 1) q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{13} - \beta_{10} + \beta_1 - 2) q^{5} + (\beta_{13} + \beta_{10} - \beta_{7} + \cdots - 4) q^{6} + (\beta_{14} + 2 \beta_{13} - \beta_{11} + \cdots - 6) q^{7}+ \cdots + (11 \beta_{14} + 12 \beta_{13} + \cdots - 460) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 4 q^{2} - 22 q^{3} + 26 q^{4} - 16 q^{5} - 41 q^{6} - 46 q^{7} - 45 q^{8} + 73 q^{9} - 70 q^{10} - 142 q^{11} - 137 q^{12} - 199 q^{13} - 180 q^{14} - 270 q^{15} - 106 q^{16} - 29 q^{17} - 150 q^{18}+ \cdots - 6254 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 65 x^{13} + 245 x^{12} + 1627 x^{11} - 5700 x^{10} - 19836 x^{9} + 63600 x^{8} + \cdots + 1096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!65 \nu^{14} + \cdots + 49\!\cdots\!16 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 74\!\cdots\!47 \nu^{14} + \cdots + 37\!\cdots\!60 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86\!\cdots\!31 \nu^{14} + \cdots + 94\!\cdots\!04 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 62\!\cdots\!01 \nu^{14} + \cdots + 93\!\cdots\!12 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!50 \nu^{14} + \cdots - 43\!\cdots\!44 ) / 26\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!63 \nu^{14} + \cdots + 38\!\cdots\!04 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 54\!\cdots\!75 \nu^{14} + \cdots - 64\!\cdots\!68 ) / 70\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 53\!\cdots\!79 \nu^{14} + \cdots - 99\!\cdots\!64 ) / 53\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 36\!\cdots\!97 \nu^{14} + \cdots + 50\!\cdots\!48 ) / 26\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!79 \nu^{14} + \cdots - 23\!\cdots\!44 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!11 \nu^{14} + \cdots - 74\!\cdots\!08 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!01 \nu^{14} + \cdots - 34\!\cdots\!12 ) / 70\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{13} + \beta_{12} - \beta_{11} - 2\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{2} + 16\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} + \beta_{13} + \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \cdots + 166 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{14} - 56 \beta_{13} + 31 \beta_{12} - 32 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} + 5 \beta_{8} + \cdots + 182 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 108 \beta_{14} + 45 \beta_{13} + 9 \beta_{12} - \beta_{11} - 85 \beta_{10} + 66 \beta_{9} - 83 \beta_{8} + \cdots + 3228 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 310 \beta_{14} - 1253 \beta_{13} + 798 \beta_{12} - 863 \beta_{11} - 55 \beta_{10} + 188 \beta_{9} + \cdots + 5330 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3912 \beta_{14} + 1656 \beta_{13} + 406 \beta_{12} - 815 \beta_{11} - 1842 \beta_{10} + 1706 \beta_{9} + \cdots + 68079 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11704 \beta_{14} - 25518 \beta_{13} + 19638 \beta_{12} - 22133 \beta_{11} - 800 \beta_{10} + \cdots + 147315 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 121976 \beta_{14} + 53980 \beta_{13} + 14691 \beta_{12} - 37774 \beta_{11} - 35030 \beta_{10} + \cdots + 1507532 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 390576 \beta_{14} - 482109 \beta_{13} + 476898 \beta_{12} - 558974 \beta_{11} + 3103 \beta_{10} + \cdots + 3962303 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3544456 \beta_{14} + 1639106 \beta_{13} + 490009 \beta_{12} - 1308689 \beta_{11} - 587250 \beta_{10} + \cdots + 34478563 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12191172 \beta_{14} - 8263629 \beta_{13} + 11544741 \beta_{12} - 14091044 \beta_{11} + 751199 \beta_{10} + \cdots + 104792431 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 99204758 \beta_{14} + 47691163 \beta_{13} + 15489332 \beta_{12} - 40416854 \beta_{11} - 7874075 \beta_{10} + \cdots + 807108829 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
5.10403
4.69170
3.54321
3.30680
2.81202
1.58277
0.856691
0.512051
−0.00348498
−1.64621
−2.09933
−2.84471
−3.03644
−4.19870
−4.58040
−5.10403 −0.236516 18.0511 −2.26673 1.20718 −7.07919 −51.3013 −26.9441 11.5695
1.2 −4.69170 5.48672 14.0121 4.61710 −25.7420 −22.2023 −28.2068 3.10405 −21.6620
1.3 −3.54321 −7.93106 4.55437 1.66042 28.1015 −5.89797 12.2086 35.9017 −5.88324
1.4 −3.30680 −8.15242 2.93496 −11.3898 26.9585 31.5144 16.7491 39.4620 37.6638
1.5 −2.81202 3.42868 −0.0925448 −4.34374 −9.64151 8.98787 22.7564 −15.2442 12.2147
1.6 −1.58277 −1.93972 −5.49484 6.60600 3.07013 12.7492 21.3592 −23.2375 −10.4558
1.7 −0.856691 8.46050 −7.26608 −15.7481 −7.24804 3.40644 13.0783 44.5801 13.4913
1.8 −0.512051 −9.58867 −7.73780 21.2386 4.90989 7.68721 8.05856 64.9425 −10.8753
1.9 0.00348498 2.81685 −7.99999 12.7987 0.00981667 −32.3745 −0.0557596 −19.0653 0.0446031
1.10 1.64621 4.34653 −5.29000 −10.5790 7.15530 −12.3764 −21.8781 −8.10765 −17.4152
1.11 2.09933 −3.29168 −3.59281 0.276333 −6.91032 30.4177 −24.3371 −16.1649 0.580114
1.12 2.84471 −3.31458 0.0923837 4.16009 −9.42902 −9.48133 −22.4949 −16.0136 11.8343
1.13 3.03644 0.816212 1.21999 −11.3545 2.47838 −13.9609 −20.5871 −26.3338 −34.4774
1.14 4.19870 −8.30846 9.62908 8.25636 −34.8847 −32.9542 6.84002 42.0305 34.6660
1.15 4.58040 −4.59239 12.9801 −19.9317 −21.0350 −4.43601 22.8109 −5.90997 −91.2953
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(149\) \( -1 \)

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 149.4.a.a ✓ 15
3.b odd 2 1 1341.4.a.b 15
4.b odd 2 1 2384.4.a.f 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
149.4.a.a ✓ 15 1.a even 1 1 trivial
1341.4.a.b 15 3.b odd 2 1
2384.4.a.f 15 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 4 T_{2}^{14} - 65 T_{2}^{13} - 245 T_{2}^{12} + 1627 T_{2}^{11} + 5700 T_{2}^{10} + \cdots - 1096 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(149))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 4 T^{14} + \cdots - 1096 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 188332756 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 552468330972 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 27\!\cdots\!07 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 59\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 58\!\cdots\!11 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 18\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 29\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 68\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 20\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 95\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 61\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 67\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 17\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 30\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 98\!\cdots\!33 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 15\!\cdots\!57 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 31\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 38\!\cdots\!48 \) Copy content Toggle raw display
show more
show less