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Properties

Label 98.14.c.n
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: \( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - 64 \beta_1 - 64) q^{2} + (\beta_{4} - 45 \beta_1) q^{3} + 4096 \beta_1 q^{4} + (\beta_{6} + 5 \beta_{4} + \cdots + 16102) q^{5}+ \cdots + (11 \beta_{7} - 5 \beta_{6} + \cdots - 996183) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 64 \beta_1 - 64) q^{2} + (\beta_{4} - 45 \beta_1) q^{3} + 4096 \beta_1 q^{4} + (\beta_{6} + 5 \beta_{4} + \cdots + 16102) q^{5}+ \cdots + (30229148 \beta_{5} + \cdots - 3231197379261) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 256 q^{2} + 182 q^{3} - 16384 q^{4} + 64400 q^{5} - 23296 q^{6} + 2097152 q^{8} - 3983752 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 256 q^{2} + 182 q^{3} - 16384 q^{4} + 64400 q^{5} - 23296 q^{6} + 2097152 q^{8} - 3983752 q^{9} + 4121600 q^{10} - 1008790 q^{11} + 745472 q^{12} - 53807264 q^{13} - 88888916 q^{15} - 67108864 q^{16} + 165333028 q^{17} - 254960128 q^{18} + 423405794 q^{19} - 527564800 q^{20} + 129125120 q^{22} + 286233866 q^{23} + 47710208 q^{24} - 472017432 q^{25} + 1721832448 q^{26} + 10405941644 q^{27} + 18200674816 q^{29} + 2844445312 q^{30} - 1507094246 q^{31} - 4294967296 q^{32} + 16888935028 q^{33} - 21162627584 q^{34} + 32634896384 q^{36} - 18959705336 q^{37} + 27097970816 q^{38} + 35759388756 q^{39} + 16882073600 q^{40} - 113651910624 q^{41} - 91557619424 q^{43} - 4132003840 q^{44} + 119804452768 q^{45} + 18318967424 q^{46} + 81351201078 q^{47} - 6106906624 q^{48} + 60418231296 q^{50} + 14996824142 q^{51} + 110197276672 q^{52} + 87497947440 q^{53} - 332990132608 q^{54} - 701637410348 q^{55} - 1807326928264 q^{57} - 582421594112 q^{58} + 194140265102 q^{59} + 182044499968 q^{60} - 175816313120 q^{61} + 192908063488 q^{62} + 549755813888 q^{64} - 1689866774568 q^{65} + 1080891841792 q^{66} - 243815218758 q^{67} + 677204082688 q^{68} - 6044157952784 q^{69} - 3275394679072 q^{71} - 1044316684288 q^{72} + 3492491920596 q^{73} - 1213421141504 q^{74} + 2370218127424 q^{75} - 3468540264448 q^{76} - 4577201760768 q^{78} - 1016380081246 q^{79} + 1080452710400 q^{80} + 17492694092 q^{81} + 3636861139968 q^{82} - 7027495743296 q^{83} + 569114840528 q^{85} + 2929843821568 q^{86} + 9706955821052 q^{87} - 264448245760 q^{88} + 8034124428036 q^{89} - 15334969954304 q^{90} - 2344827830272 q^{92} - 3388371390552 q^{93} + 5206476868992 q^{94} + 1429435505438 q^{95} + 195421011968 q^{96} - 54351131725632 q^{97} - 25837834576328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27\!\cdots\!26 \nu^{7} + \cdots - 92\!\cdots\!05 ) / 90\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!70 \nu^{7} + \cdots - 42\!\cdots\!05 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!30 \nu^{7} + \cdots - 84\!\cdots\!45 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 99\!\cdots\!39 \nu^{7} + \cdots + 33\!\cdots\!05 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!30 \nu^{7} + \cdots - 85\!\cdots\!05 ) / 71\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!81 \nu^{7} + \cdots + 35\!\cdots\!95 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40\!\cdots\!79 \nu^{7} + \cdots + 29\!\cdots\!50 ) / 11\!\cdots\!35 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 4\beta_{6} + 8\beta_{4} - 4\beta_{3} + 8\beta_{2} + 170\beta _1 + 174 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 193\beta_{7} - 106\beta_{6} - 193\beta_{5} + 5558\beta_{4} + 29070590\beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -813235\beta_{5} + 2158324\beta_{3} - 17462624\beta_{2} - 56301080754 ) / 336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 47369614 \beta_{7} + 74108644 \beta_{6} - 1074838820 \beta_{4} + 74108644 \beta_{3} + \cdots - 5071555289829 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 737261146321 \beta_{7} + 1596388283524 \beta_{6} + 737261146321 \beta_{5} + \cdots - 64\!\cdots\!10 \beta_1 ) / 336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 176934179918233\beta_{5} - 325903905472666\beta_{3} + 3961750843826918\beta_{2} + 17241586909054883976 ) / 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 68\!\cdots\!55 \beta_{7} + \cdots + 63\!\cdots\!34 ) / 336 \) 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
−217.937 − 377.477i
481.229 + 833.513i
−258.759 − 448.183i
−3.53342 − 6.12007i
−217.937 + 377.477i
481.229 − 833.513i
−258.759 + 448.183i
−3.53342 + 6.12007i
−32.0000 − 55.4256i −1142.35 + 1978.62i −2048.00 + 3547.24i 18256.5 + 31621.1i 146221. 0 262144. −1.81279e6 − 3.13984e6i 1.16841e6 − 2.02375e6i
67.2 −32.0000 − 55.4256i −301.970 + 523.027i −2048.00 + 3547.24i −2521.96 − 4368.16i 38652.2 0 262144. 614790. + 1.06485e6i −161406. + 279563.i
67.3 −32.0000 − 55.4256i 679.656 − 1177.20i −2048.00 + 3547.24i 29011.3 + 50249.0i −86996.0 0 262144. −126703. − 219456.i 1.85672e6 − 3.21593e6i
67.4 −32.0000 − 55.4256i 855.669 − 1482.06i −2048.00 + 3547.24i −12545.8 − 21729.9i −109526. 0 262144. −667176. − 1.15558e6i −802928. + 1.39071e6i
79.1 −32.0000 + 55.4256i −1142.35 − 1978.62i −2048.00 − 3547.24i 18256.5 − 31621.1i 146221. 0 262144. −1.81279e6 + 3.13984e6i 1.16841e6 + 2.02375e6i
79.2 −32.0000 + 55.4256i −301.970 − 523.027i −2048.00 − 3547.24i −2521.96 + 4368.16i 38652.2 0 262144. 614790. − 1.06485e6i −161406. − 279563.i
79.3 −32.0000 + 55.4256i 679.656 + 1177.20i −2048.00 − 3547.24i 29011.3 − 50249.0i −86996.0 0 262144. −126703. + 219456.i 1.85672e6 + 3.21593e6i
79.4 −32.0000 + 55.4256i 855.669 + 1482.06i −2048.00 − 3547.24i −12545.8 + 21729.9i −109526. 0 262144. −667176. + 1.15558e6i −802928. − 1.39071e6i
\(n\): 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.n 8
7.b odd 2 1 14.14.c.a ✓ 8
7.c even 3 1 98.14.a.k 4
7.c even 3 1 inner 98.14.c.n 8
7.d odd 6 1 14.14.c.a ✓ 8
7.d odd 6 1 98.14.a.l 4
21.c even 2 1 126.14.g.d 8
21.g even 6 1 126.14.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.a ✓ 8 7.b odd 2 1
14.14.c.a ✓ 8 7.d odd 6 1
98.14.a.k 4 7.c even 3 1
98.14.a.l 4 7.d odd 6 1
98.14.c.n 8 1.a even 1 1 trivial
98.14.c.n 8 7.c even 3 1 inner
126.14.g.d 8 21.c even 2 1
126.14.g.d 8 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 182 T_{3}^{7} + 5197084 T_{3}^{6} - 4025617932 T_{3}^{5} + 23908528667637 T_{3}^{4} + \cdots + 10\!\cdots\!25 \) 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
$3$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 71\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 94\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 47\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 55\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 79\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
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