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Properties

Label 97.4.a.b
Level $97$
Weight $4$
Character orbit 97.a
Self dual yes
Analytic conductor $5.723$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [97,4,Mod(1,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("97.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 97.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13] 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: \(5.72318527056\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 71 x^{11} + 239 x^{10} + 1958 x^{9} - 5334 x^{8} - 26087 x^{7} + 55923 x^{6} + \cdots - 163840 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{8} + \beta_{6} - \beta_{5} + \cdots + 1) q^{5} + ( - \beta_{10} + \beta_{8} - 2 \beta_{6} + \cdots + 2) q^{6}+ \cdots + (7 \beta_{12} - 17 \beta_{11} + \cdots + 42) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 9 q^{2} + 12 q^{3} + 59 q^{4} + 16 q^{5} + 11 q^{6} + 58 q^{7} + 132 q^{8} + 167 q^{9} + 14 q^{10} + 142 q^{11} + 147 q^{12} + 28 q^{13} + 67 q^{14} + 82 q^{15} + 295 q^{16} + 160 q^{17} + 135 q^{18}+ \cdots - 306 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 71 x^{11} + 239 x^{10} + 1958 x^{9} - 5334 x^{8} - 26087 x^{7} + 55923 x^{6} + \cdots - 163840 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 108619 \nu^{12} + 613980 \nu^{11} + 6525453 \nu^{10} - 34136405 \nu^{9} - 149967570 \nu^{8} + \cdots - 7809331200 ) / 840994816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 185559 \nu^{12} - 2493540 \nu^{11} - 3129217 \nu^{10} + 139431665 \nu^{9} - 164522766 \nu^{8} + \cdots - 27375296512 ) / 840994816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 215015 \nu^{12} - 1212356 \nu^{11} - 11803185 \nu^{10} + 62830017 \nu^{9} + 233502386 \nu^{8} + \cdots + 5616975872 ) / 840994816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 253221 \nu^{12} + 1966924 \nu^{11} + 11591651 \nu^{10} - 108366483 \nu^{9} + \cdots + 15790596096 ) / 840994816 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 58026 \nu^{12} - 559979 \nu^{11} - 2054858 \nu^{10} + 30686843 \nu^{9} + 4467439 \nu^{8} + \cdots - 5773678592 ) / 105124352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 109607 \nu^{12} + 657615 \nu^{11} + 5854021 \nu^{10} - 35003118 \nu^{9} - 108732877 \nu^{8} + \cdots + 549824000 ) / 105124352 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 929925 \nu^{12} - 6292108 \nu^{11} - 46108867 \nu^{10} + 336024179 \nu^{9} + 745178998 \nu^{8} + \cdots - 38214287360 ) / 840994816 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1548219 \nu^{12} + 10962220 \nu^{11} + 74413213 \nu^{10} - 591367509 \nu^{9} + \cdots + 63928991744 ) / 840994816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 283378 \nu^{12} - 2048187 \nu^{11} - 13300962 \nu^{10} + 110117587 \nu^{9} + 186071039 \nu^{8} + \cdots - 15170928640 ) / 105124352 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3358641 \nu^{12} + 24946140 \nu^{11} + 156082295 \nu^{10} - 1352237671 \nu^{9} + \cdots + 174041387008 ) / 840994816 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - 2 \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{12} + \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 9 \beta_{6} - 4 \beta_{5} + \cdots + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 37 \beta_{12} + 2 \beta_{11} - 74 \beta_{10} + 27 \beta_{9} - 8 \beta_{8} - 25 \beta_{7} + 61 \beta_{6} + \cdots + 496 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 123 \beta_{12} + 59 \beta_{11} - 234 \beta_{10} - 55 \beta_{9} - 74 \beta_{8} - 13 \beta_{7} + \cdots + 5881 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1252 \beta_{12} + 213 \beta_{11} - 2400 \beta_{10} + 628 \beta_{9} - 434 \beta_{8} - 570 \beta_{7} + \cdots + 19401 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5526 \beta_{12} + 2688 \beta_{11} - 9844 \beta_{10} - 1136 \beta_{9} - 2804 \beta_{8} - 606 \beta_{7} + \cdots + 165580 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 42964 \beta_{12} + 12480 \beta_{11} - 77548 \beta_{10} + 14046 \beta_{9} - 18460 \beta_{8} + \cdots + 698088 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 220885 \beta_{12} + 111798 \beta_{11} - 371410 \beta_{10} - 20265 \beta_{9} - 109232 \beta_{8} + \cdots + 5018752 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1503542 \beta_{12} + 586791 \beta_{11} - 2554392 \beta_{10} + 310072 \beta_{9} - 723834 \beta_{8} + \cdots + 24279431 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8352667 \beta_{12} + 4425444 \beta_{11} - 13399310 \beta_{10} - 316581 \beta_{9} - 4217256 \beta_{8} + \cdots + 159120646 \) 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
5.91646
5.21619
3.55086
3.06021
2.36789
1.95378
0.238679
−1.15266
−1.88201
−3.28029
−3.29102
−4.20378
−4.49430
−4.91646 3.14147 16.1716 16.7985 −15.4449 5.58450 −40.1755 −17.1312 −82.5893
1.2 −4.21619 −3.06734 9.77625 −7.17546 12.9325 −22.7444 −7.48902 −17.5914 30.2531
1.3 −2.55086 9.63927 −1.49313 8.41212 −24.5884 3.81206 24.2156 65.9156 −21.4581
1.4 −2.06021 5.78764 −3.75554 −14.1741 −11.9237 29.7572 24.2189 6.49680 29.2015
1.5 −1.36789 −8.28409 −6.12889 −18.6040 11.3317 −4.25349 19.3267 41.6262 25.4481
1.6 −0.953784 −4.51734 −7.09030 1.75838 4.30857 0.390049 14.3929 −6.59363 −1.67711
1.7 0.761321 4.29446 −7.42039 18.5075 3.26946 9.25870 −11.7399 −8.55758 14.0901
1.8 2.15266 −9.86124 −3.36605 7.32571 −21.2279 33.7493 −24.4673 70.2440 15.7698
1.9 2.88201 9.23116 0.305977 −2.28314 26.6043 5.76345 −22.1742 58.2143 −6.58002
1.10 4.28029 0.999431 10.3209 1.08937 4.27786 31.6904 9.93409 −26.0011 4.66281
1.11 4.29102 4.65748 10.4129 11.0251 19.9854 −19.1952 10.3537 −5.30785 47.3091
1.12 5.20378 −5.47332 19.0793 12.8309 −28.4820 −4.94602 57.6544 2.95721 66.7691
1.13 5.49430 5.45241 22.1873 −19.5110 29.9572 −10.8665 77.9495 2.72875 −107.199
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(97\) \( +1 \)

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 97.4.a.b ✓ 13
3.b odd 2 1 873.4.a.d 13
4.b odd 2 1 1552.4.a.g 13
5.b even 2 1 2425.4.a.b 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.4.a.b ✓ 13 1.a even 1 1 trivial
873.4.a.d 13 3.b odd 2 1
1552.4.a.g 13 4.b odd 2 1
2425.4.a.b 13 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 9 T_{2}^{12} - 41 T_{2}^{11} + 520 T_{2}^{10} + 278 T_{2}^{9} - 10635 T_{2}^{8} + \cdots - 352512 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(97))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 9 T^{12} + \cdots - 352512 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 1092466240 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots + 437583939696 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots + 1407471753368 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 66\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 32\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots - 94\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 69\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 29\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 42\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 24\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 25\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 75\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 59\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 29\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 65\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 34\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T + 97)^{13} \) Copy content Toggle raw display
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