Properties

Label 98.15.b.c
Level $98$
Weight $15$
Character orbit 98.b
Analytic conductor $121.842$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.97");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 15 \)
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: \(121.842388789\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 65377122 x^{18} + \cdots + 55\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{86}\cdot 3^{12}\cdot 7^{44} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + 8192 q^{4} + ( - \beta_{13} - 4 \beta_1) q^{5} + (\beta_{14} - \beta_{13} + 4 \beta_{11} - 4 \beta_1) q^{6} + 8192 \beta_{2} q^{8} + (\beta_{3} + 3073 \beta_{2} - 1754743) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + 8192 q^{4} + ( - \beta_{13} - 4 \beta_1) q^{5} + (\beta_{14} - \beta_{13} + 4 \beta_{11} - 4 \beta_1) q^{6} + 8192 \beta_{2} q^{8} + (\beta_{3} + 3073 \beta_{2} - 1754743) q^{9} + ( - \beta_{15} - 4 \beta_{14} + 6 \beta_{13} + 6 \beta_{11} + 824 \beta_1) q^{10} + (\beta_{4} + \beta_{3} + 32926 \beta_{2} - 840042) q^{11} + 8192 \beta_1 q^{12} + (\beta_{17} - 102 \beta_{14} - 62 \beta_{13} + 27 \beta_{12} - 2161 \beta_{11} + \cdots - 1577 \beta_1) q^{13}+ \cdots + (102000 \beta_{10} - 387498 \beta_{9} + \cdots + 52834315340337) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 163840 q^{4} - 35094864 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 163840 q^{4} - 35094864 q^{9} - 16800852 q^{11} + 562720524 q^{15} + 1342177280 q^{16} + 503510016 q^{18} + 5394565632 q^{22} + 13810196772 q^{23} - 28330165288 q^{25} + 27884908704 q^{29} - 108207023616 q^{30} - 287497125888 q^{36} + 54052055852 q^{37} + 252250808760 q^{39} + 726682953656 q^{43} - 137632579584 q^{44} + 573329969664 q^{46} + 1161106642944 q^{50} + 4839219891204 q^{51} - 3092542975092 q^{53} + 14789884876092 q^{57} + 4731726081024 q^{58} + 4609806532608 q^{60} + 10995116277760 q^{64} - 15033407865672 q^{65} + 9311641526452 q^{67} + 96606137494152 q^{71} + 4124754051072 q^{72} + 1133064966144 q^{74} + 88663911671808 q^{78} + 121034948165956 q^{79} + 214360529022684 q^{81} + 416326699526124 q^{85} - 4726670349312 q^{86} + 44192281657344 q^{88} + 113133131956224 q^{92} + 188350136380260 q^{93} - 50405029030980 q^{95} + 10\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 65377122 x^{18} + \cdots + 55\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!01 \nu^{18} + \cdots + 28\!\cdots\!67 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39\!\cdots\!73 \nu^{18} + \cdots + 61\!\cdots\!09 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!37 \nu^{18} + \cdots - 17\!\cdots\!21 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51\!\cdots\!07 \nu^{18} + \cdots + 11\!\cdots\!17 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\!\cdots\!13 \nu^{18} + \cdots + 18\!\cdots\!19 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40\!\cdots\!21 \nu^{18} + \cdots + 98\!\cdots\!59 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!78 \nu^{18} + \cdots + 49\!\cdots\!23 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!51 \nu^{18} + \cdots + 33\!\cdots\!37 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 33\!\cdots\!15 \nu^{18} + \cdots - 77\!\cdots\!21 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!55 \nu^{19} + \cdots + 86\!\cdots\!17 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 68\!\cdots\!10 \nu^{19} + \cdots - 79\!\cdots\!39 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!07 \nu^{19} + \cdots + 23\!\cdots\!53 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 58\!\cdots\!09 \nu^{19} + \cdots + 99\!\cdots\!27 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!51 \nu^{19} + \cdots + 92\!\cdots\!49 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!91 \nu^{19} + \cdots + 47\!\cdots\!91 \nu ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 31\!\cdots\!32 \nu^{19} + \cdots + 37\!\cdots\!45 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 36\!\cdots\!61 \nu^{19} + \cdots - 12\!\cdots\!59 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 58\!\cdots\!13 \nu^{19} + \cdots + 18\!\cdots\!35 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3073\beta_{2} - 6537712 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 33 \beta_{19} + 59 \beta_{18} - 17 \beta_{17} - 62 \beta_{16} + 1100 \beta_{15} + 1751 \beta_{14} + 8131 \beta_{13} - 7541 \beta_{12} - 29992 \beta_{11} - 12488022 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 87276 \beta_{10} + 90438 \beta_{9} - 301260 \beta_{8} - 115656 \beta_{7} + 374592 \beta_{6} + 711588 \beta_{5} - 413022 \beta_{4} - 16312944 \beta_{3} - 33245456430 \beta_{2} + \cdots + 81650254405497 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 900201510 \beta_{19} - 888765762 \beta_{18} + 595597182 \beta_{17} + 1237544634 \beta_{16} - 20091000456 \beta_{15} - 12186905796 \beta_{14} + \cdots + 179754487789767 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2198075540904 \beta_{10} - 2373246323658 \beta_{9} + 8027509305816 \beta_{8} + 7710814388304 \beta_{7} - 9429922520568 \beta_{6} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16\!\cdots\!71 \beta_{19} + \cdots - 27\!\cdots\!36 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45\!\cdots\!04 \beta_{10} + \cdots + 17\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28\!\cdots\!64 \beta_{19} + \cdots + 41\!\cdots\!73 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 87\!\cdots\!92 \beta_{10} + \cdots - 27\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 46\!\cdots\!49 \beta_{19} + \cdots - 65\!\cdots\!54 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16\!\cdots\!12 \beta_{10} + \cdots + 43\!\cdots\!17 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 72\!\cdots\!74 \beta_{19} + \cdots + 10\!\cdots\!67 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 29\!\cdots\!48 \beta_{10} + \cdots - 68\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11\!\cdots\!47 \beta_{19} + \cdots - 16\!\cdots\!20 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 51\!\cdots\!52 \beta_{10} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 17\!\cdots\!36 \beta_{19} + \cdots + 26\!\cdots\!65 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 91\!\cdots\!92 \beta_{10} + \cdots - 17\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 25\!\cdots\!73 \beta_{19} + \cdots - 43\!\cdots\!46 \beta_1 \) Copy content Toggle raw display

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
3905.91i
3636.42i
1805.34i
1499.57i
302.931i
302.931i
1499.57i
1805.34i
3636.42i
3905.91i
4124.02i
3129.83i
1524.52i
1194.63i
862.049i
862.049i
1194.63i
1524.52i
3129.83i
4124.02i
−90.5097 3905.91i 8192.00 44517.9i 353522.i 0 −741455. −1.04731e7 4.02930e6i
97.2 −90.5097 3636.42i 8192.00 129871.i 329131.i 0 −741455. −8.44055e6 1.17546e7i
97.3 −90.5097 1805.34i 8192.00 29106.1i 163400.i 0 −741455. 1.52373e6 2.63438e6i
97.4 −90.5097 1499.57i 8192.00 77950.8i 135725.i 0 −741455. 2.53426e6 7.05530e6i
97.5 −90.5097 302.931i 8192.00 122619.i 27418.1i 0 −741455. 4.69120e6 1.10982e7i
97.6 −90.5097 302.931i 8192.00 122619.i 27418.1i 0 −741455. 4.69120e6 1.10982e7i
97.7 −90.5097 1499.57i 8192.00 77950.8i 135725.i 0 −741455. 2.53426e6 7.05530e6i
97.8 −90.5097 1805.34i 8192.00 29106.1i 163400.i 0 −741455. 1.52373e6 2.63438e6i
97.9 −90.5097 3636.42i 8192.00 129871.i 329131.i 0 −741455. −8.44055e6 1.17546e7i
97.10 −90.5097 3905.91i 8192.00 44517.9i 353522.i 0 −741455. −1.04731e7 4.02930e6i
97.11 90.5097 4124.02i 8192.00 93595.8i 373263.i 0 741455. −1.22245e7 8.47132e6i
97.12 90.5097 3129.83i 8192.00 11744.1i 283280.i 0 741455. −5.01289e6 1.06295e6i
97.13 90.5097 1524.52i 8192.00 150560.i 137984.i 0 741455. 2.45881e6 1.36272e7i
97.14 90.5097 1194.63i 8192.00 5898.49i 108125.i 0 741455. 3.35583e6 533871.i
97.15 90.5097 862.049i 8192.00 52836.5i 78023.8i 0 741455. 4.03984e6 4.78221e6i
97.16 90.5097 862.049i 8192.00 52836.5i 78023.8i 0 741455. 4.03984e6 4.78221e6i
97.17 90.5097 1194.63i 8192.00 5898.49i 108125.i 0 741455. 3.35583e6 533871.i
97.18 90.5097 1524.52i 8192.00 150560.i 137984.i 0 741455. 2.45881e6 1.36272e7i
97.19 90.5097 3129.83i 8192.00 11744.1i 283280.i 0 741455. −5.01289e6 1.06295e6i
97.20 90.5097 4124.02i 8192.00 93595.8i 373263.i 0 741455. −1.22245e7 8.47132e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.20
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 98.15.b.c 20
7.b odd 2 1 inner 98.15.b.c 20
7.c even 3 1 14.15.d.a 20
7.c even 3 1 98.15.d.b 20
7.d odd 6 1 14.15.d.a 20
7.d odd 6 1 98.15.d.b 20
21.g even 6 1 126.15.n.b 20
21.h odd 6 1 126.15.n.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.15.d.a 20 7.c even 3 1
14.15.d.a 20 7.d odd 6 1
98.15.b.c 20 1.a even 1 1 trivial
98.15.b.c 20 7.b odd 2 1 inner
98.15.d.b 20 7.c even 3 1
98.15.d.b 20 7.d odd 6 1
126.15.n.b 20 21.g even 6 1
126.15.n.b 20 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 65377122 T_{3}^{18} + \cdots + 55\!\cdots\!49 \) acting on \(S_{15}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8192)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + 65377122 T^{18} + \cdots + 55\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + 75200238894 T^{18} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + 8400426 T^{9} + \cdots + 17\!\cdots\!67)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 57\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{10} - 6905098386 T^{9} + \cdots - 57\!\cdots\!93)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 13942454352 T^{9} + \cdots - 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 17\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( (T^{10} - 27026027926 T^{9} + \cdots + 25\!\cdots\!09)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{10} - 363341476828 T^{9} + \cdots + 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 68\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( (T^{10} + 1546271487546 T^{9} + \cdots - 82\!\cdots\!63)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( (T^{10} - 4655820763226 T^{9} + \cdots - 37\!\cdots\!61)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 48303068747076 T^{9} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( (T^{10} - 60517474082978 T^{9} + \cdots + 49\!\cdots\!75)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 70\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
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