Properties

Label 688.2.i.i
Level $688$
Weight $2$
Character orbit 688.i
Analytic conductor $5.494$
Analytic rank $0$
Dimension $10$
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] = [688,2,Mod(49,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.i (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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 8x^{8} - 2x^{7} + 50x^{6} - 12x^{5} + 113x^{4} - 50x^{3} + 200x^{2} - 56x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 344)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_1 + 1) q^{3} + ( - \beta_{6} + \beta_1) q^{5} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{8} + \beta_{7} - 2 \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_1 + 1) q^{3} + ( - \beta_{6} + \beta_1) q^{5} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{8} + \beta_{7} - 2 \beta_{3} - \beta_{2}) q^{9} + (\beta_{4} + 1) q^{11} - \beta_{9} q^{13} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{15} + (\beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{2} + \beta_1) q^{17} + (\beta_{8} + \beta_1) q^{19} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{21} + (\beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{4} + \beta_1 - 3) q^{23} + (2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{2} + \beta_1) q^{25} + ( - \beta_{5} - \beta_{3} - 3 \beta_{2} + \beta_1 - 5) q^{27} + (\beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{29} + (\beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + 1) q^{31} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} + \beta_{4}) q^{33} + (\beta_{5} + \beta_{4} + 3 \beta_{3} - 3 \beta_1 + 5) q^{35} + (\beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 2) q^{37} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{39} + ( - \beta_{5} - 2 \beta_{4} - 1) q^{41} + (2 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{43} + (2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} + 2) q^{45} + (\beta_{5} - 6 \beta_{3} + \beta_{2} + 6 \beta_1) q^{47} + (\beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{4} + 3 \beta_1 - 3) q^{49} + ( - 3 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{51} + ( - \beta_{9} + 2 \beta_{6} + \beta_{4} - 2 \beta_1) q^{53} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_1 + 2) q^{55} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_1) q^{57} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{59} + ( - 3 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{61} + ( - 3 \beta_{9} - 4 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{4} + 5 \beta_1 - 2) q^{63} + (\beta_{5} - 4 \beta_{3} + 4 \beta_1 - 2) q^{65} + ( - \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + \beta_{6} + \beta_{4} + 4) q^{67} + (\beta_{9} - \beta_{8} - 6 \beta_{7} + \beta_{6} - \beta_{5} + 6 \beta_{3} + \beta_{2} - \beta_1) q^{69} + (2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 3 \beta_{3} - 2 \beta_{2}) q^{71} + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{73} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} + 4 \beta_1 - 2) q^{75} + ( - \beta_{8} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{3} + \beta_{2} - 3 \beta_1) q^{77} + ( - 7 \beta_{7} - \beta_{3}) q^{79} + ( - \beta_{9} - 3 \beta_{8} - 9 \beta_{7} - 4 \beta_{6} + \beta_{4} + 6 \beta_1 - 9) q^{81} + ( - 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{6} - 3 \beta_1 + 5) q^{83} + (\beta_{5} + 8 \beta_{3} - 2 \beta_{2} - 8 \beta_1 - 4) q^{85} + (\beta_{3} - \beta_1 - 3) q^{87} + ( - 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_1 + 3) q^{89} + (2 \beta_{8} + \beta_{7} - 2 \beta_{6} - 3 \beta_1 + 1) q^{91} + (2 \beta_{8} + 6 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{93} + (\beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + \beta_1) q^{95} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 7) q^{97} + ( - 4 \beta_{8} - \beta_{7} - 5 \beta_{6} + 5 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{3} + 2 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{3} + 2 q^{5} - 4 q^{7} - 6 q^{9} + 8 q^{11} + q^{13} + 6 q^{15} + q^{19} + 2 q^{21} - 15 q^{23} - 9 q^{25} - 52 q^{27} + 8 q^{29} + 10 q^{31} + 2 q^{33} + 44 q^{35} + 7 q^{37} + 6 q^{39} - 2 q^{41} - 11 q^{43} + 24 q^{45} - 2 q^{47} - 13 q^{49} - 5 q^{53} + 14 q^{55} + 12 q^{57} + 20 q^{59} - 21 q^{61} - 11 q^{63} - 24 q^{65} + 20 q^{67} + 32 q^{69} + 11 q^{71} - 2 q^{73} - 14 q^{75} + 7 q^{77} + 35 q^{79} - 41 q^{81} + 19 q^{83} - 48 q^{85} - 30 q^{87} + 12 q^{89} + 11 q^{91} - 26 q^{93} - 10 q^{95} + 62 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 8x^{8} - 2x^{7} + 50x^{6} - 12x^{5} + 113x^{4} - 50x^{3} + 200x^{2} - 56x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 175 \nu^{9} - 4886 \nu^{8} + 1396 \nu^{7} - 30800 \nu^{6} + 16054 \nu^{5} - 245696 \nu^{4} + 50025 \nu^{3} - 552816 \nu^{2} + 156352 \nu - 1288620 ) / 426316 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1209 \nu^{9} - 350 \nu^{8} + 100 \nu^{7} - 374 \nu^{6} + 1150 \nu^{5} - 17600 \nu^{4} + 354775 \nu^{3} - 39600 \nu^{2} + 863832 \nu - 245000 ) / 852632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 523 \nu^{9} - 10339 \nu^{8} + 2954 \nu^{7} - 92048 \nu^{6} + 33971 \nu^{5} - 519904 \nu^{4} + 88601 \nu^{3} - 1169784 \nu^{2} + 330848 \nu - 1055718 ) / 213158 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1209 \nu^{9} + 350 \nu^{8} - 100 \nu^{7} + 374 \nu^{6} - 1150 \nu^{5} + 17600 \nu^{4} - 141617 \nu^{3} + 39600 \nu^{2} - 11200 \nu + 31842 ) / 213158 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11081 \nu^{9} + 2434 \nu^{8} - 122500 \nu^{7} + 31834 \nu^{6} - 556118 \nu^{5} + 244200 \nu^{4} - 1249161 \nu^{3} + 336292 \nu^{2} + 2480008 \nu - 1464 ) / 1705264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30625 \nu^{9} - 2418 \nu^{8} + 244300 \nu^{7} - 61050 \nu^{6} + 1530502 \nu^{5} - 365200 \nu^{4} + 3425425 \nu^{3} - 821700 \nu^{2} + 6045800 \nu - 1692600 ) / 1705264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 91175 \nu^{9} - 12290 \nu^{8} - 727316 \nu^{7} + 59950 \nu^{6} - 4527290 \nu^{5} + 112816 \nu^{4} - 10076175 \nu^{3} + 1959100 \nu^{2} - 17511992 \nu - 76680 ) / 1705264 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13919 \nu^{9} + 2042 \nu^{8} + 110148 \nu^{7} - 27494 \nu^{6} + 685362 \nu^{5} - 163072 \nu^{4} + 1512303 \nu^{3} - 948252 \nu^{2} + 2647576 \nu - 740936 ) / 155024 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{7} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 4\beta_{3} - 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 6\beta_{8} - 13\beta_{7} + \beta_{4} - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + 7\beta_{7} + 8\beta_{6} - 8\beta_{5} - 18\beta_{3} - \beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{5} - 8\beta_{4} + 34\beta_{2} + 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{9} - 10\beta_{8} - 43\beta_{7} - 50\beta_{6} + \beta_{4} + 138\beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50\beta_{9} + 189\beta_{8} + 325\beta_{7} + 12\beta_{6} - 12\beta_{5} - 2\beta_{3} - 189\beta_{2} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 289\beta_{5} - 12\beta_{4} + 452\beta_{3} + 76\beta_{2} - 452\beta _1 + 257 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.14150 1.97713i
0.743304 1.28744i
0.146947 0.254519i
−0.844279 + 1.46233i
−1.18747 + 2.05676i
1.14150 + 1.97713i
0.743304 + 1.28744i
0.146947 + 0.254519i
−0.844279 1.46233i
−1.18747 2.05676i
0 −0.641497 + 1.11111i 0 1.88356 3.26243i 0 −0.863964 1.49643i 0 0.676963 + 1.17254i 0
49.2 0 −0.243304 + 0.421416i 0 −0.830511 + 1.43849i 0 −1.67882 2.90780i 0 1.38161 + 2.39301i 0
49.3 0 0.353053 0.611506i 0 −0.0750950 + 0.130068i 0 0.734771 + 1.27266i 0 1.25071 + 2.16629i 0
49.4 0 1.34428 2.32836i 0 1.46988 2.54591i 0 1.88855 + 3.27106i 0 −2.11417 3.66186i 0
49.5 0 1.68747 2.92278i 0 −1.44784 + 2.50773i 0 −2.08054 3.60360i 0 −4.19510 7.26613i 0
337.1 0 −0.641497 1.11111i 0 1.88356 + 3.26243i 0 −0.863964 + 1.49643i 0 0.676963 1.17254i 0
337.2 0 −0.243304 0.421416i 0 −0.830511 1.43849i 0 −1.67882 + 2.90780i 0 1.38161 2.39301i 0
337.3 0 0.353053 + 0.611506i 0 −0.0750950 0.130068i 0 0.734771 1.27266i 0 1.25071 2.16629i 0
337.4 0 1.34428 + 2.32836i 0 1.46988 + 2.54591i 0 1.88855 3.27106i 0 −2.11417 + 3.66186i 0
337.5 0 1.68747 + 2.92278i 0 −1.44784 2.50773i 0 −2.08054 + 3.60360i 0 −4.19510 + 7.26613i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.i.i 10
4.b odd 2 1 344.2.i.d 10
43.c even 3 1 inner 688.2.i.i 10
172.g odd 6 1 344.2.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
344.2.i.d 10 4.b odd 2 1
344.2.i.d 10 172.g odd 6 1
688.2.i.i 10 1.a even 1 1 trivial
688.2.i.i 10 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{10} - 5T_{3}^{9} + 23T_{3}^{8} - 36T_{3}^{7} + 72T_{3}^{6} - 8T_{3}^{5} + 155T_{3}^{4} - 23T_{3}^{3} + 61T_{3}^{2} + 12T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{10} - 2 T_{5}^{9} + 19 T_{5}^{8} - 2 T_{5}^{7} + 201 T_{5}^{6} - 8 T_{5}^{5} + 1112 T_{5}^{4} + 1136 T_{5}^{3} + 3008 T_{5}^{2} + 448 T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 5 T^{9} + 23 T^{8} - 36 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + 19 T^{8} - 2 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{10} + 4 T^{9} + 32 T^{8} + \cdots + 17956 \) Copy content Toggle raw display
$11$ \( (T^{5} - 4 T^{4} - 22 T^{3} + 57 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + 29 T^{8} - 18 T^{7} + \cdots + 25600 \) Copy content Toggle raw display
$17$ \( T^{10} + 77 T^{8} - 94 T^{7} + \cdots + 2430481 \) Copy content Toggle raw display
$19$ \( T^{10} - T^{9} + 16 T^{8} + 29 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{10} + 15 T^{9} + 183 T^{8} + \cdots + 2835856 \) Copy content Toggle raw display
$29$ \( T^{10} - 8 T^{9} + 94 T^{8} - 80 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{10} - 10 T^{9} + 145 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$37$ \( T^{10} - 7 T^{9} + 97 T^{8} - 430 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$41$ \( (T^{5} + T^{4} - 105 T^{3} + 51 T^{2} + \cdots - 4132)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} + \cdots + 147008443 \) Copy content Toggle raw display
$47$ \( (T^{5} + T^{4} - 247 T^{3} - 166 T^{2} + \cdots + 13696)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 5 T^{9} + 85 T^{8} + \cdots + 1607824 \) Copy content Toggle raw display
$59$ \( (T^{5} - 10 T^{4} - 100 T^{3} + 1201 T^{2} + \cdots + 3824)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 21 T^{9} + 397 T^{8} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{10} - 20 T^{9} + \cdots + 264615289 \) Copy content Toggle raw display
$71$ \( T^{10} - 11 T^{9} + 269 T^{8} + \cdots + 1638400 \) Copy content Toggle raw display
$73$ \( T^{10} + 2 T^{9} + 73 T^{8} + \cdots + 22801 \) Copy content Toggle raw display
$79$ \( T^{10} - 35 T^{9} + \cdots + 199261456 \) Copy content Toggle raw display
$83$ \( T^{10} - 19 T^{9} + 349 T^{8} + \cdots + 54051904 \) Copy content Toggle raw display
$89$ \( T^{10} - 12 T^{9} + 219 T^{8} + \cdots + 20566225 \) Copy content Toggle raw display
$97$ \( (T^{5} - 31 T^{4} + 194 T^{3} + \cdots - 13168)^{2} \) Copy content Toggle raw display
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