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Properties

Label 49.8.c.g
Level $49$
Weight $8$
Character orbit 49.c
Analytic conductor $15.307$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,8,Mod(18,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.18");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(15.3068662487\)
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: \( x^{8} - x^{7} + 103x^{6} - 378x^{5} + 9744x^{4} - 22680x^{3} + 149400x^{2} + 216000x + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{2} + (\beta_{5} + \beta_{3} - 6 \beta_{2} + 6) q^{3} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - 88) q^{4}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + \cdots + 47) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{2} + (\beta_{5} + \beta_{3} - 6 \beta_{2} + 6) q^{3} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - 88) q^{4}+ \cdots + (5385 \beta_{7} + 86599 \beta_{5} + \cdots + 3762037) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 28 q^{3} - 348 q^{4} + 252 q^{5} - 2044 q^{6} - 1968 q^{8} - 2008 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} + 28 q^{3} - 348 q^{4} + 252 q^{5} - 2044 q^{6} - 1968 q^{8} - 2008 q^{9} + 4774 q^{10} + 3972 q^{11} - 5404 q^{12} + 2352 q^{13} - 33112 q^{15} - 57264 q^{16} + 56364 q^{17} + 35908 q^{18} + 41748 q^{19} - 324744 q^{20} - 305908 q^{22} - 131748 q^{23} + 389592 q^{24} + 68016 q^{25} + 113652 q^{26} - 378056 q^{27} + 68112 q^{29} - 605378 q^{30} + 401212 q^{31} + 453408 q^{32} + 24052 q^{33} - 164220 q^{34} - 720176 q^{36} + 5396 q^{37} + 31794 q^{38} - 364840 q^{39} - 443688 q^{40} - 820848 q^{41} + 93088 q^{43} + 1465836 q^{44} - 1209656 q^{45} - 1379202 q^{46} + 1470084 q^{47} + 4037600 q^{48} + 2791056 q^{50} + 1724796 q^{51} - 5269768 q^{52} + 642372 q^{53} - 4567318 q^{54} + 6126680 q^{55} + 6718136 q^{57} + 1972220 q^{58} + 752220 q^{59} - 2123324 q^{60} + 1325772 q^{61} + 6032628 q^{62} - 1731712 q^{64} + 4868808 q^{65} - 6013294 q^{66} + 290916 q^{67} + 2453556 q^{68} - 9830520 q^{69} + 6755520 q^{71} - 4557744 q^{72} + 6706588 q^{73} - 3616878 q^{74} + 7223888 q^{75} - 5846008 q^{76} - 32576264 q^{78} - 3946244 q^{79} + 10695888 q^{80} + 11731292 q^{81} - 17563252 q^{82} - 19084128 q^{83} + 14012136 q^{85} - 23237832 q^{86} + 17836840 q^{87} + 8043336 q^{88} + 16165212 q^{89} - 6831496 q^{90} - 38885256 q^{92} - 12552852 q^{93} + 5720442 q^{94} + 3268452 q^{95} - 8448608 q^{96} - 3066224 q^{97} + 30427952 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 103x^{6} - 378x^{5} + 9744x^{4} - 22680x^{3} + 149400x^{2} + 216000x + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2419 \nu^{7} + 2089856 \nu^{6} + 6596632 \nu^{5} + 191205693 \nu^{4} - 241778964 \nu^{3} + \cdots + 21568329000 ) / 3180326625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51287 \nu^{7} - 221882 \nu^{6} + 4986506 \nu^{5} - 35051121 \nu^{4} + 521375988 \nu^{3} + \cdots + 9362007000 ) / 25442613000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11627 \nu^{7} + 68647 \nu^{6} - 1542751 \nu^{5} + 10026716 \nu^{4} - 161305998 \nu^{3} + \cdots + 4975101000 ) / 605776500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 141661 \nu^{7} + 215659 \nu^{6} + 13007813 \nu^{5} - 17965948 \nu^{4} + 1049560974 \nu^{3} + \cdots + 78343411500 ) / 4240435500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 832833 \nu^{7} + 2650538 \nu^{6} - 59567354 \nu^{5} + 625838339 \nu^{4} + \cdots + 192094254000 ) / 8480871000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5145317 \nu^{7} - 23255012 \nu^{6} + 522625796 \nu^{5} - 4123329861 \nu^{4} + \cdots - 1685376396000 ) / 25442613000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1093321 \nu^{7} - 961699 \nu^{6} - 100392593 \nu^{5} + 138658828 \nu^{4} + \cdots - 692595526500 ) / 4240435500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + 5\beta_{3} - 4\beta_{2} + 4 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{4} - 4\beta_{3} - 104\beta_{2} - \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -36\beta_{7} - 29\beta_{5} - 362\beta_{4} - 29\beta_{2} + 29\beta _1 + 1279 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{6} + 115\beta_{5} + 442\beta_{3} + 8607\beta_{2} - 8607 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3138\beta_{7} - 3138\beta_{6} + 38300\beta_{4} - 38300\beta_{3} - 281950\beta_{2} - 3383\beta _1 + 38300 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2010\beta_{7} - 11341\beta_{5} - 48424\beta_{4} - 11341\beta_{2} + 11341\beta _1 + 750491 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 293766\beta_{6} + 432779\beta_{5} + 3984392\beta_{3} + 36494421\beta_{2} - 36494421 ) / 14 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−1.02959 + 1.78331i
4.11776 − 7.13217i
2.57145 − 4.45388i
−5.15962 + 8.93672i
−1.02959 − 1.78331i
4.11776 + 7.13217i
2.57145 + 4.45388i
−5.15962 − 8.93672i
−9.50966 − 16.4712i 19.7308 − 34.1747i −116.867 + 202.420i 175.002 + 303.113i −750.531 0 2011.00 314.894 + 545.412i 3328.43 − 5765.00i
18.2 −1.59276 − 2.75874i −20.1021 + 34.8178i 58.9262 − 102.063i −0.294487 − 0.510067i 128.071 0 −783.169 285.314 + 494.179i −0.938097 + 1.62483i
18.3 3.69038 + 6.39193i 37.1217 − 64.2967i 36.7621 − 63.6739i −145.409 − 251.857i 547.974 0 1487.40 −1662.54 − 2879.61i 1073.23 − 1858.89i
18.4 10.4120 + 18.0342i −22.7504 + 39.4049i −152.821 + 264.694i 96.7016 + 167.492i −947.513 0 −3699.23 58.3362 + 101.041i −2013.72 + 3487.87i
30.1 −9.50966 + 16.4712i 19.7308 + 34.1747i −116.867 − 202.420i 175.002 − 303.113i −750.531 0 2011.00 314.894 − 545.412i 3328.43 + 5765.00i
30.2 −1.59276 + 2.75874i −20.1021 − 34.8178i 58.9262 + 102.063i −0.294487 + 0.510067i 128.071 0 −783.169 285.314 − 494.179i −0.938097 − 1.62483i
30.3 3.69038 − 6.39193i 37.1217 + 64.2967i 36.7621 + 63.6739i −145.409 + 251.857i 547.974 0 1487.40 −1662.54 + 2879.61i 1073.23 + 1858.89i
30.4 10.4120 − 18.0342i −22.7504 − 39.4049i −152.821 − 264.694i 96.7016 − 167.492i −947.513 0 −3699.23 58.3362 − 101.041i −2013.72 − 3487.87i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.4
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 49.8.c.g 8
7.b odd 2 1 7.8.c.a ✓ 8
7.c even 3 1 49.8.a.e 4
7.c even 3 1 inner 49.8.c.g 8
7.d odd 6 1 7.8.c.a ✓ 8
7.d odd 6 1 49.8.a.f 4
21.c even 2 1 63.8.e.b 8
21.g even 6 1 63.8.e.b 8
21.g even 6 1 441.8.a.s 4
21.h odd 6 1 441.8.a.t 4
28.d even 2 1 112.8.i.c 8
28.f even 6 1 112.8.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.8.c.a ✓ 8 7.b odd 2 1
7.8.c.a ✓ 8 7.d odd 6 1
49.8.a.e 4 7.c even 3 1
49.8.a.f 4 7.d odd 6 1
49.8.c.g 8 1.a even 1 1 trivial
49.8.c.g 8 7.c even 3 1 inner
63.8.e.b 8 21.c even 2 1
63.8.e.b 8 21.g even 6 1
112.8.i.c 8 28.d even 2 1
112.8.i.c 8 28.f even 6 1
441.8.a.s 4 21.g even 6 1
441.8.a.t 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{8} - 6 T_{2}^{7} + 448 T_{2}^{6} - 936 T_{2}^{5} + 170656 T_{2}^{4} - 590304 T_{2}^{3} + \cdots + 86713344 \) Copy content Toggle raw display
\( T_{3}^{8} - 28 T_{3}^{7} + 5770 T_{3}^{6} + 53424 T_{3}^{5} + 20707299 T_{3}^{4} + 85273776 T_{3}^{3} + \cdots + 28723950837729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} + \cdots + 86713344 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 28723950837729 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 134435328890625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 27\!\cdots\!09 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 59\!\cdots\!69 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 15\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 32\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 32\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 39\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 80\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 31\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 62\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 61\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 42\!\cdots\!36)^{2} \) Copy content Toggle raw display
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