↑

Properties

Label 768.7.g.g
Level $768$
Weight $7$
Character orbit 768.g
Analytic conductor $176.682$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,7,Mod(511,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.511");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), 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: \(176.681536220\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 53x^{6} - 2x^{5} + 2532x^{4} - 772x^{3} - 31349x^{2} - 33880x + 366025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 384)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{3} q^{7} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{5} - \beta_{3} q^{7} - 243 q^{9} + ( - \beta_{4} - 17 \beta_1) q^{11} - \beta_{5} q^{13} + \beta_{7} q^{15} + (\beta_{6} - 818) q^{17} + ( - 5 \beta_{4} + 195 \beta_1) q^{19} + (3 \beta_{5} - 3 \beta_{2}) q^{21} + ( - 4 \beta_{7} - 2 \beta_{3}) q^{23} + (2 \beta_{6} + 7079) q^{25} + 243 \beta_1 q^{27} + ( - 14 \beta_{5} - 47 \beta_{2}) q^{29} + (20 \beta_{7} - 61 \beta_{3}) q^{31} + ( - 3 \beta_{6} - 4212) q^{33} + (33 \beta_{4} + 325 \beta_1) q^{35} + ( - 27 \beta_{5} - 110 \beta_{2}) q^{37} + (\beta_{7} - 81 \beta_{3}) q^{39} + ( - \beta_{6} + 62446) q^{41} + (25 \beta_{4} + 529 \beta_1) q^{43} + 243 \beta_{2} q^{45} + (32 \beta_{7} + 198 \beta_{3}) q^{47} + (12 \beta_{6} - 51839) q^{49} + ( - 81 \beta_{4} + 845 \beta_1) q^{51} + ( - 70 \beta_{5} + 949 \beta_{2}) q^{53} + (76 \beta_{7} + 400 \beta_{3}) q^{55} + ( - 15 \beta_{6} + 46980) q^{57} + ( - 32 \beta_{4} - 10620 \beta_1) q^{59} + (5 \beta_{5} + 462 \beta_{2}) q^{61} + 243 \beta_{3} q^{63} + ( - 31 \beta_{6} - 4512) q^{65} + (140 \beta_{4} + 10568 \beta_1) q^{67} + (6 \beta_{5} - 978 \beta_{2}) q^{69} + (228 \beta_{7} - 94 \beta_{3}) q^{71} + (20 \beta_{6} + 245330) q^{73} + ( - 162 \beta_{4} - 7025 \beta_1) q^{75} + ( - 124 \beta_{5} - 3284 \beta_{2}) q^{77} + ( - 4 \beta_{7} - 1525 \beta_{3}) q^{79} + 59049 q^{81} + ( - 309 \beta_{4} + 13683 \beta_1) q^{83} + (400 \beta_{5} - 4334 \beta_{2}) q^{85} + (61 \beta_{7} - 1134 \beta_{3}) q^{87} + ( - 38 \beta_{6} + 211874) q^{89} + (357 \beta_{4} - 56279 \beta_1) q^{91} + (183 \beta_{5} + 4677 \beta_{2}) q^{93} + (100 \beta_{7} + 2000 \beta_{3}) q^{95} + ( - 10 \beta_{6} + 954094) q^{97} + (243 \beta_{4} + 4131 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1944 q^{9} - 6544 q^{17} + 56632 q^{25} - 33696 q^{33} + 499568 q^{41} - 414712 q^{49} + 375840 q^{57} - 36096 q^{65} + 1962640 q^{73} + 472392 q^{81} + 1694992 q^{89} + 7632752 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 53x^{6} - 2x^{5} + 2532x^{4} - 772x^{3} - 31349x^{2} - 33880x + 366025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 828747 \nu^{7} + 5475258 \nu^{6} - 46400868 \nu^{5} - 318001302 \nu^{4} + 1561303692 \nu^{3} + \cdots - 148555417725 ) / 9239308925 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26514 \nu^{7} + 579378 \nu^{6} + 11306672 \nu^{5} - 31230312 \nu^{4} - 392579888 \nu^{3} + \cdots - 974690090 ) / 263980255 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6425154 \nu^{7} + 3940254 \nu^{6} + 284961216 \nu^{5} + 483802824 \nu^{4} + \cdots + 296213011050 ) / 839937175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19492379 \nu^{7} - 175563642 \nu^{6} + 1247954852 \nu^{5} + 11629186678 \nu^{4} + \cdots + 3833648818285 ) / 1847861785 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22028796 \nu^{7} + 66004572 \nu^{6} + 1655284448 \nu^{5} - 1485058128 \nu^{4} + \cdots + 427089143140 ) / 1847861785 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 49248 \nu^{7} - 678816 \nu^{6} + 2184192 \nu^{5} + 3708288 \nu^{4} - 4436352 \nu^{3} + \cdots - 24850917216 ) / 3054317 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1068066 \nu^{7} - 986094 \nu^{6} - 47369664 \nu^{5} - 80423496 \nu^{4} + 1635589152 \nu^{3} + \cdots - 39175581258 ) / 33597487 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 3\beta_{5} - 9\beta_{3} - 30\beta_{2} - 48\beta _1 + 432 ) / 1728 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} + 6\beta_{5} + 27\beta_{4} + 18\beta_{3} - 60\beta_{2} + 5271\beta _1 + 47520 ) / 3456 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -212\beta_{7} + 9\beta_{6} - 900\beta_{3} + 143424 ) / 3456 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 107 \beta_{7} - 81 \beta_{6} + 153 \beta_{5} + 729 \beta_{4} - 459 \beta_{3} - 3042 \beta_{2} + \cdots - 784080 ) / 1728 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9032 \beta_{7} + 825 \beta_{6} - 9288 \beta_{5} + 7425 \beta_{4} - 27864 \beta_{3} + \cdots + 8080128 ) / 6912 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7042\beta_{7} - 3423\beta_{6} - 22050\beta_{3} - 28285632 ) / 864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 88549 \beta_{7} - 12705 \beta_{6} - 80847 \beta_{5} + 114345 \beta_{4} + 242541 \beta_{3} + \cdots - 106111728 ) / 1728 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
−5.30399 − 3.63961i
3.53301 + 1.46244i
−3.03301 − 2.32846i
5.80399 + 2.77359i
−5.30399 + 3.63961i
3.53301 − 1.46244i
−3.03301 + 2.32846i
5.80399 − 2.77359i
0 − 15.5885i 0 −195.235 0 − 277.510i 0 −243.000 0
511.2 0 − 15.5885i 0 −85.3891 0 511.824i 0 −243.000 0
511.3 0 − 15.5885i 0 85.3891 0 − 511.824i 0 −243.000 0
511.4 0 − 15.5885i 0 195.235 0 277.510i 0 −243.000 0
511.5 0 15.5885i 0 −195.235 0 277.510i 0 −243.000 0
511.6 0 15.5885i 0 −85.3891 0 − 511.824i 0 −243.000 0
511.7 0 15.5885i 0 85.3891 0 511.824i 0 −243.000 0
511.8 0 15.5885i 0 195.235 0 − 277.510i 0 −243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.7.g.g 8
4.b odd 2 1 inner 768.7.g.g 8
8.b even 2 1 inner 768.7.g.g 8
8.d odd 2 1 inner 768.7.g.g 8
16.e even 4 2 384.7.b.c ✓ 8
16.f odd 4 2 384.7.b.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.c ✓ 8 16.e even 4 2
384.7.b.c ✓ 8 16.f odd 4 2
768.7.g.g 8 1.a even 1 1 trivial
768.7.g.g 8 4.b odd 2 1 inner
768.7.g.g 8 8.b even 2 1 inner
768.7.g.g 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 45408T_{5}^{2} + 277920000 \) acting on \(S_{7}^{\mathrm{new}}(768, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 45408 T^{2} + 277920000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 338976 T^{2} + 20174323968)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 4522189383936)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 11711515004928)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1636 T - 58718780)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 29\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 64\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 124892 T + 3840115012)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 80\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 24\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 55\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 490660 T + 36431647300)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 27\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 423748 T - 40865541500)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1908188 T + 904356570436)^{4} \) Copy content Toggle raw display
show more
show less