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Properties

Label 688.6.a.h
Level $688$
Weight $6$
Character orbit 688.a
Self dual yes
Analytic conductor $110.344$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,6,Mod(1,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

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: yes
Analytic conductor: \(110.344068031\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 3) q^{3} + (\beta_{7} + 14) q^{5} + (\beta_{7} - \beta_{5} - \beta_{3} + \cdots - 6) q^{7}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 134) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 3) q^{3} + (\beta_{7} + 14) q^{5} + (\beta_{7} - \beta_{5} - \beta_{3} + \cdots - 6) q^{7}+ \cdots + ( - 37 \beta_{9} + 258 \beta_{8} + \cdots + 27314) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 8\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1300283 \nu^{9} + 26922273 \nu^{8} + 154132815 \nu^{7} - 5215675021 \nu^{6} + \cdots + 27676046268420 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1397467 \nu^{9} - 1596255 \nu^{8} + 334667631 \nu^{7} + 479070067 \nu^{6} + \cdots + 24355668723972 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 474917 \nu^{9} - 1144977 \nu^{8} + 116809809 \nu^{7} + 421392509 \nu^{6} + \cdots - 5839603776708 ) / 65688726528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2282743 \nu^{9} - 7177205 \nu^{8} - 569012379 \nu^{7} + 1117284305 \nu^{6} + \cdots + 15723616018348 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5726311 \nu^{9} - 8152363 \nu^{8} + 1437405579 \nu^{7} + 3217063567 \nu^{6} + \cdots - 40534043519788 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6235841 \nu^{9} - 47565 \nu^{8} + 1552893885 \nu^{7} + 1540088777 \nu^{6} + \cdots - 19301122595508 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10838435 \nu^{9} + 23722567 \nu^{8} - 2828120727 \nu^{7} - 8018172347 \nu^{6} + \cdots + 81663371814556 ) / 197066179584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17656675 \nu^{9} - 6000871 \nu^{8} + 4377440727 \nu^{7} + 6072341467 \nu^{6} + \cdots - 87327465633820 ) / 197066179584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + \beta_{7} + \beta_{6} - 4\beta_{5} + 9\beta_{4} + \beta_{3} + \beta _1 + 415 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{9} - 4 \beta_{8} + 27 \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 43 \beta_{4} + 3 \beta_{3} + \cdots + 639 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 67 \beta_{9} - 7 \beta_{8} + 53 \beta_{7} + 7 \beta_{6} - 113 \beta_{5} + 291 \beta_{4} + 43 \beta_{3} + \cdots + 9105 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2584 \beta_{9} - 184 \beta_{8} + 4448 \beta_{7} + 176 \beta_{6} - 1480 \beta_{5} + 7504 \beta_{4} + \cdots + 126802 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 33266 \beta_{9} - 4416 \beta_{8} + 36193 \beta_{7} - 5727 \beta_{6} - 48164 \beta_{5} + \cdots + 3762135 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 339754 \beta_{9} - 5108 \beta_{8} + 639047 \beta_{7} - 29553 \beta_{6} - 228008 \beta_{5} + \cdots + 19803219 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1042617 \beta_{9} - 131727 \beta_{8} + 1430958 \beta_{7} - 383568 \beta_{6} - 1322709 \beta_{5} + \cdots + 104680478 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 44351112 \beta_{9} + 56616 \beta_{8} + 88331664 \beta_{7} - 10812960 \beta_{6} - 32440968 \beta_{5} + \cdots + 2824172738 ) / 8 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
11.5305
−4.38824
−9.70631
3.50018
−6.91219
9.86547
−8.57770
2.86024
5.31531
−1.48720
0 −27.4953 0 86.8464 0 19.8137 0 512.989 0
1.2 0 −25.0462 0 −0.456695 0 −166.517 0 384.312 0
1.3 0 −18.3440 0 −72.1865 0 96.4803 0 93.5034 0
1.4 0 −16.8892 0 47.4635 0 −67.4603 0 42.2439 0
1.5 0 −12.8799 0 79.5677 0 172.354 0 −77.1083 0
1.6 0 −1.50169 0 −37.8251 0 124.747 0 −240.745 0
1.7 0 7.84343 0 28.1028 0 −195.604 0 −181.481 0
1.8 0 14.8716 0 −42.2365 0 −202.971 0 −21.8357 0
1.9 0 23.8469 0 −52.5837 0 174.859 0 325.673 0
1.10 0 27.5943 0 101.308 0 −15.7005 0 518.448 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(43\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.6.a.h 10
4.b odd 2 1 43.6.a.b ✓ 10
12.b even 2 1 387.6.a.e 10
20.d odd 2 1 1075.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b ✓ 10 4.b odd 2 1
387.6.a.e 10 12.b even 2 1
688.6.a.h 10 1.a even 1 1 trivial
1075.6.a.b 10 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 28 T_{3}^{9} - 1501 T_{3}^{8} - 45584 T_{3}^{7} + 644833 T_{3}^{6} + 23720286 T_{3}^{5} + \cdots + 316744901280 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 316744901280 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 25\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 50\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 46\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 47\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 49\!\cdots\!66 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 24\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 47\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 22\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( (T + 1849)^{10} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 89\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 95\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 18\!\cdots\!58 \) Copy content Toggle raw display
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