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Properties

Label 98.14.c.o
Level $98$
Weight $14$
Character orbit 98.c
Analytic conductor $105.086$
Analytic rank $0$
Dimension $8$
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] = [98,14,Mod(67,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not 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: \(105.086310373\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + \cdots + 75\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 64 \beta_1 + 64) q^{2} + ( - \beta_{2} - 46 \beta_1) q^{3} - 4096 \beta_1 q^{4} + (\beta_{7} - \beta_{5} - 12 \beta_{3} + \cdots - 442) q^{5}+ \cdots + ( - 27 \beta_{7} + 27 \beta_{5} + \cdots + 149791) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 64 \beta_1 + 64) q^{2} + ( - \beta_{2} - 46 \beta_1) q^{3} - 4096 \beta_1 q^{4} + (\beta_{7} - \beta_{5} - 12 \beta_{3} + \cdots - 442) q^{5}+ \cdots + (127291050 \beta_{6} + \cdots + 3091428017331) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 256 q^{2} - 182 q^{3} - 16384 q^{4} - 1792 q^{5} - 23296 q^{6} - 2097152 q^{8} + 599840 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 256 q^{2} - 182 q^{3} - 16384 q^{4} - 1792 q^{5} - 23296 q^{6} - 2097152 q^{8} + 599840 q^{9} + 114688 q^{10} - 8726914 q^{11} - 745472 q^{12} - 1438416 q^{13} - 136873212 q^{15} - 67108864 q^{16} + 7943068 q^{17} - 38389760 q^{18} + 215706806 q^{19} + 14680064 q^{20} - 1117044992 q^{22} - 61927978 q^{23} + 47710208 q^{24} - 1327844792 q^{25} - 46029312 q^{26} + 3268643812 q^{27} - 6325846064 q^{29} - 4379942784 q^{30} - 6113775570 q^{31} + 4294967296 q^{32} - 25235960652 q^{33} + 1016712704 q^{34} - 4913889280 q^{36} - 3945652880 q^{37} - 13805235584 q^{38} - 23545599116 q^{39} + 469762048 q^{40} - 86378579952 q^{41} - 109074124256 q^{43} - 35745439744 q^{44} + 104964468168 q^{45} + 3963390592 q^{46} + 3141202722 q^{47} + 6106906624 q^{48} - 169964133376 q^{50} - 241278267462 q^{51} + 2945875968 q^{52} + 149625680376 q^{53} + 104596601984 q^{54} - 174912009748 q^{55} - 128207489960 q^{57} - 202427074048 q^{58} + 866297313938 q^{59} + 280316338176 q^{60} + 477908594184 q^{61} - 782563272960 q^{62} + 549755813888 q^{64} - 1099748343120 q^{65} + 1615101481728 q^{66} + 1895501016278 q^{67} + 32534806528 q^{68} - 3503301895632 q^{69} + 638832672128 q^{71} - 157244456960 q^{72} + 2966596192756 q^{73} + 252521784320 q^{74} + 1331079867376 q^{75} - 1767070154752 q^{76} - 3013836686848 q^{78} - 6505959677634 q^{79} - 30064771072 q^{80} + 2449216493684 q^{81} - 2764114558464 q^{82} + 3379817135968 q^{83} - 16684556982464 q^{85} - 3490371976192 q^{86} + 5273311164492 q^{87} + 2287708143616 q^{88} - 9586601667468 q^{89} + 13435451925504 q^{90} + 507313995776 q^{92} - 14195747226896 q^{93} - 201036974208 q^{94} + 14384410136978 q^{95} + 195421011968 q^{96} + 44560735311568 q^{97} + 24706419985464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + \cdots + 75\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 36\!\cdots\!43 \nu^{7} + \cdots + 22\!\cdots\!25 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\!\cdots\!19 \nu^{7} + \cdots - 20\!\cdots\!25 ) / 65\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!46 \nu^{7} + \cdots + 17\!\cdots\!19 ) / 12\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 93\!\cdots\!78 \nu^{7} + \cdots - 32\!\cdots\!30 ) / 15\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44\!\cdots\!81 \nu^{7} + \cdots - 11\!\cdots\!43 ) / 36\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!91 \nu^{7} + \cdots + 58\!\cdots\!67 ) / 51\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\!\cdots\!27 \nu^{7} + \cdots - 22\!\cdots\!25 ) / 17\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - \beta_{6} - \beta_{4} + 52\beta_{2} + 26\beta_1 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 890\beta_{7} - 890\beta_{5} - 455\beta_{4} - 628\beta_{3} - 628\beta_{2} + 35124622\beta _1 - 35124622 ) / 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7549\beta_{6} + 30461\beta_{5} + 367933\beta_{3} + 375601993 ) / 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -233797382\beta_{7} + 71270609\beta_{6} + 71270609\beta_{4} + 1371990508\beta_{2} - 6422569264258\beta_1 ) / 336 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 117654471422 \beta_{7} - 117654471422 \beta_{5} + 11657352703 \beta_{4} - 1103337193564 \beta_{3} + \cdots - 20\!\cdots\!58 ) / 336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -96184064357\beta_{6} + 518498001245\beta_{5} + 3799909018822\beta_{3} + 12410398903428382 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 28\!\cdots\!66 \beta_{7} - 405888753844057 \beta_{6} - 405888753844057 \beta_{4} + \cdots - 57\!\cdots\!94 \beta_1 ) / 336 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
114.279 − 197.937i
169.399 − 293.408i
−248.013 + 429.572i
−35.6648 + 61.7733i
114.279 + 197.937i
169.399 + 293.408i
−248.013 − 429.572i
−35.6648 − 61.7733i
32.0000 + 55.4256i −888.214 + 1538.43i −2048.00 + 3547.24i 13071.0 + 22639.6i −113691. 0 −262144. −780686. − 1.35219e6i −836543. + 1.44893e6i
67.2 32.0000 + 55.4256i −194.343 + 336.611i −2048.00 + 3547.24i 12951.1 + 22432.0i −24875.9 0 −262144. 721623. + 1.24989e6i −828871. + 1.43565e6i
67.3 32.0000 + 55.4256i 244.702 − 423.837i −2048.00 + 3547.24i −34096.3 − 59056.5i 31321.9 0 −262144. 677403. + 1.17330e6i 2.18216e6 − 3.77962e6i
67.4 32.0000 + 55.4256i 746.854 − 1293.59i −2048.00 + 3547.24i 7178.20 + 12433.0i 95597.3 0 −262144. −318420. − 551520.i −459405. + 795712.i
79.1 32.0000 − 55.4256i −888.214 − 1538.43i −2048.00 − 3547.24i 13071.0 − 22639.6i −113691. 0 −262144. −780686. + 1.35219e6i −836543. − 1.44893e6i
79.2 32.0000 − 55.4256i −194.343 − 336.611i −2048.00 − 3547.24i 12951.1 − 22432.0i −24875.9 0 −262144. 721623. − 1.24989e6i −828871. − 1.43565e6i
79.3 32.0000 − 55.4256i 244.702 + 423.837i −2048.00 − 3547.24i −34096.3 + 59056.5i 31321.9 0 −262144. 677403. − 1.17330e6i 2.18216e6 + 3.77962e6i
79.4 32.0000 − 55.4256i 746.854 + 1293.59i −2048.00 − 3547.24i 7178.20 − 12433.0i 95597.3 0 −262144. −318420. + 551520.i −459405. − 795712.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.14.c.o 8
7.b odd 2 1 14.14.c.b ✓ 8
7.c even 3 1 98.14.a.j 4
7.c even 3 1 inner 98.14.c.o 8
7.d odd 6 1 14.14.c.b ✓ 8
7.d odd 6 1 98.14.a.h 4
21.c even 2 1 126.14.g.b 8
21.g even 6 1 126.14.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.b ✓ 8 7.b odd 2 1
14.14.c.b ✓ 8 7.d odd 6 1
98.14.a.h 4 7.d odd 6 1
98.14.a.j 4 7.c even 3 1
98.14.c.o 8 1.a even 1 1 trivial
98.14.c.o 8 7.c even 3 1 inner
126.14.g.b 8 21.c even 2 1
126.14.g.b 8 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 182 T_{3}^{7} + 2905288 T_{3}^{6} - 949684092 T_{3}^{5} + 7705719718857 T_{3}^{4} + \cdots + 25\!\cdots\!25 \) acting on \(S_{14}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 64 T + 4096)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 61\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 12\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 94\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 82\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 96\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 91\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 90\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 52\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 76\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 25\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 20\!\cdots\!80)^{2} \) Copy content Toggle raw display
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