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Properties

Label 98.9.b.c
Level $98$
Weight $9$
Character orbit 98.b
Analytic conductor $39.923$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,9,Mod(97,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.97");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 98.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(39.9231037860\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{4}\cdot 7^{12} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + 128 q^{4} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{5} + ( - \beta_{6} + \beta_{2}) q^{6} + 128 \beta_1 q^{8} + ( - \beta_{10} - 22 \beta_1 - 3934) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + 128 q^{4} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{5} + ( - \beta_{6} + \beta_{2}) q^{6} + 128 \beta_1 q^{8} + ( - \beta_{10} - 22 \beta_1 - 3934) q^{9} + ( - 2 \beta_{8} - 2 \beta_{6} + \cdots + 18 \beta_{2}) q^{10}+ \cdots + ( - 4956 \beta_{11} + 2661 \beta_{10} + \cdots - 28908858) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 1536 q^{4} - 47208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 1536 q^{4} - 47208 q^{9} - 20604 q^{11} - 255468 q^{15} + 196608 q^{16} - 33792 q^{18} - 941568 q^{22} - 317868 q^{23} - 1677336 q^{25} - 4355256 q^{29} - 1833984 q^{30} - 6042624 q^{36} - 268428 q^{37} - 2670768 q^{39} - 12961896 q^{43} - 2637312 q^{44} - 4690944 q^{46} + 2970624 q^{50} + 5347788 q^{51} + 33081012 q^{53} + 100263780 q^{57} - 18352128 q^{58} - 32699904 q^{60} + 25165824 q^{64} + 93012528 q^{65} + 75500724 q^{67} + 45506424 q^{71} - 4325376 q^{72} - 117674496 q^{74} - 25463808 q^{78} - 119460588 q^{79} + 179354844 q^{81} - 64652220 q^{85} + 30289920 q^{86} - 120520704 q^{88} - 40687104 q^{92} + 22239324 q^{93} + 351836700 q^{95} - 346906296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!84 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!14 \nu^{11} + \cdots + 36\!\cdots\!25 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!12 \nu^{11} + \cdots + 25\!\cdots\!50 ) / 14\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29\!\cdots\!12 \nu^{11} + \cdots - 14\!\cdots\!50 ) / 89\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!46 \nu^{11} + \cdots + 26\!\cdots\!25 ) / 22\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!78 \nu^{11} + \cdots - 10\!\cdots\!25 ) / 74\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!38 \nu^{11} + \cdots - 16\!\cdots\!50 ) / 23\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!24 \nu^{11} + \cdots + 59\!\cdots\!50 ) / 89\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!26 \nu^{11} + \cdots - 69\!\cdots\!00 ) / 53\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!42 \nu^{11} + \cdots + 25\!\cdots\!50 ) / 47\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!58 \nu^{11} + \cdots + 18\!\cdots\!00 ) / 68\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} - 7 \beta_{9} - 21 \beta_{8} + 7 \beta_{7} - 42 \beta_{6} + 12 \beta_{4} + 369 \beta_{3} + \cdots + 784 ) / 4704 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{11} + 29 \beta_{10} + 71 \beta_{9} - 126 \beta_{8} - 224 \beta_{7} - 1155 \beta_{6} + \cdots - 1386896 ) / 4704 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 619\beta_{11} + 1339\beta_{10} + 7569\beta_{9} - 10080\beta_{7} + 2180598\beta _1 - 17391472 ) / 2352 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 553 \beta_{11} + 36877 \beta_{10} + 103349 \beta_{9} + 199416 \beta_{8} - 202594 \beta_{7} + \cdots - 731889480 ) / 2352 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 238371 \beta_{11} - 1457887 \beta_{10} - 5518265 \beta_{9} + 12181134 \beta_{8} + \cdots + 20443188472 ) / 2352 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 280321 \beta_{11} - 9440203 \beta_{10} - 28409699 \beta_{9} + 49068670 \beta_{7} + \cdots + 149321535496 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 251524844 \beta_{11} - 2625038829 \beta_{10} - 8891893808 \beta_{9} - 18800564937 \beta_{8} + \cdots + 37976039366440 ) / 2352 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1421106222 \beta_{11} + 28092390327 \beta_{10} + 87576223540 \beta_{9} - 178451499639 \beta_{8} + \cdots - 419669908306642 ) / 588 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 342028567865 \beta_{11} + 4540877865840 \beta_{10} + 14836252584287 \beta_{9} - 23734836391931 \beta_{7} + \cdots - 66\!\cdots\!24 ) / 1176 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5639840294844 \beta_{11} + 95056384542473 \beta_{10} + 301042962868988 \beta_{9} + \cdots - 14\!\cdots\!84 ) / 1176 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 532018991651105 \beta_{11} + \cdots + 11\!\cdots\!80 ) / 2352 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−10.5605 + 18.2913i
20.6104 + 35.6982i
−9.54988 − 16.5409i
−9.54988 + 16.5409i
20.6104 − 35.6982i
−10.5605 − 18.2913i
9.14374 − 15.8374i
3.92504 + 6.79836i
−12.5688 + 21.7698i
−12.5688 − 21.7698i
3.92504 − 6.79836i
9.14374 + 15.8374i
−11.3137 − 146.013i 128.000 430.555i 1651.95i 0 −1448.15 −14758.8 − 4871.18i
97.2 −11.3137 − 93.7486i 128.000 − 1071.69i 1060.64i 0 −1448.15 −2227.79 12124.8i
97.3 −11.3137 − 25.0948i 128.000 568.295i 283.915i 0 −1448.15 5931.25 − 6429.52i
97.4 −11.3137 25.0948i 128.000 − 568.295i − 283.915i 0 −1448.15 5931.25 6429.52i
97.5 −11.3137 93.7486i 128.000 1071.69i − 1060.64i 0 −1448.15 −2227.79 − 12124.8i
97.6 −11.3137 146.013i 128.000 − 430.555i − 1651.95i 0 −1448.15 −14758.8 4871.18i
97.7 11.3137 − 142.885i 128.000 − 875.684i − 1616.56i 0 1448.15 −13855.0 − 9907.24i
97.8 11.3137 − 104.924i 128.000 − 37.9902i − 1187.08i 0 1448.15 −4448.08 − 429.810i
97.9 11.3137 − 28.4008i 128.000 870.230i − 321.318i 0 1448.15 5754.39 9845.53i
97.10 11.3137 28.4008i 128.000 − 870.230i 321.318i 0 1448.15 5754.39 − 9845.53i
97.11 11.3137 104.924i 128.000 37.9902i 1187.08i 0 1448.15 −4448.08 429.810i
97.12 11.3137 142.885i 128.000 875.684i 1616.56i 0 1448.15 −13855.0 9907.24i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.9.b.c 12
7.b odd 2 1 inner 98.9.b.c 12
7.c even 3 1 14.9.d.a ✓ 12
7.c even 3 1 98.9.d.b 12
7.d odd 6 1 14.9.d.a ✓ 12
7.d odd 6 1 98.9.d.b 12
21.g even 6 1 126.9.n.b 12
21.h odd 6 1 126.9.n.b 12
28.f even 6 1 112.9.s.c 12
28.g odd 6 1 112.9.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.d.a ✓ 12 7.c even 3 1
14.9.d.a ✓ 12 7.d odd 6 1
98.9.b.c 12 1.a even 1 1 trivial
98.9.b.c 12 7.b odd 2 1 inner
98.9.d.b 12 7.c even 3 1
98.9.d.b 12 7.d odd 6 1
112.9.s.c 12 28.f even 6 1
112.9.s.c 12 28.g odd 6 1
126.9.n.b 12 21.g even 6 1
126.9.n.b 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 62970 T_{3}^{10} + 1447192647 T_{3}^{8} + 14637777905772 T_{3}^{6} + \cdots + 21\!\cdots\!81 \) acting on \(S_{9}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 22\!\cdots\!91)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 94\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 15\!\cdots\!79)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 79\!\cdots\!12)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 95\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 45\!\cdots\!89)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 32\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 85\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 10\!\cdots\!29)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 48\!\cdots\!67)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 30\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
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