Properties

Label 688.2.u.e
Level $688$
Weight $2$
Character orbit 688.u
Analytic conductor $5.494$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,2,Mod(97,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.u (of order \(7\), degree \(6\), 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 14 x^{10} - 22 x^{9} + 40 x^{8} + 42 x^{7} + 239 x^{6} + 518 x^{5} + 2208 x^{4} + 3625 x^{3} + 13209 x^{2} + 17931 x + 19321 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 172)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + 1) q^{3} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + \cdots - 1) q^{5}+ \cdots + (\beta_{10} + \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + 1) q^{3} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{3} + 3 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{3} + 3 q^{5} + 12 q^{9} - q^{11} - 4 q^{13} - q^{15} + 17 q^{17} - 13 q^{19} - 7 q^{21} + 14 q^{23} + q^{25} + 22 q^{27} - 6 q^{29} + 23 q^{31} - 2 q^{33} + 16 q^{35} - 24 q^{37} - q^{39} + 8 q^{41} + 7 q^{43} + 3 q^{45} - 22 q^{47} + 24 q^{49} + 13 q^{51} + 27 q^{53} + 29 q^{55} - 19 q^{57} + 17 q^{59} + 8 q^{61} + 7 q^{63} + 29 q^{65} - 29 q^{67} + 17 q^{71} - 10 q^{73} - 5 q^{75} + 15 q^{77} - 14 q^{79} + 36 q^{81} + 6 q^{83} - 70 q^{85} - 12 q^{87} - 16 q^{89} - 46 q^{91} + 46 q^{93} - 67 q^{95} + 10 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 14 x^{10} - 22 x^{9} + 40 x^{8} + 42 x^{7} + 239 x^{6} + 518 x^{5} + 2208 x^{4} + 3625 x^{3} + 13209 x^{2} + 17931 x + 19321 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 230871308369 \nu^{11} + 59758893747878 \nu^{10} - 460711021387736 \nu^{9} + \cdots + 39\!\cdots\!04 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 518046327669711 \nu^{11} + \cdots - 93\!\cdots\!60 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1973168161909 \nu^{11} - 7176206523924 \nu^{10} + 8474865346765 \nu^{9} + 42611497958911 \nu^{8} - 161877463575778 \nu^{7} + \cdots + 64\!\cdots\!91 ) / 86\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 466854998116169 \nu^{11} + \cdots + 52\!\cdots\!09 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!79 \nu^{11} + \cdots + 21\!\cdots\!24 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12234409131736 \nu^{11} + 83393503466864 \nu^{10} - 287048597756255 \nu^{9} + 535701837647792 \nu^{8} + \cdots - 72\!\cdots\!29 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 870218488758043 \nu^{11} + \cdots - 35\!\cdots\!66 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!67 \nu^{11} + \cdots + 28\!\cdots\!73 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!21 \nu^{11} + \cdots + 37\!\cdots\!65 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32664660916021 \nu^{11} - 215793150833575 \nu^{10} + 705667711143615 \nu^{9} + \cdots + 26\!\cdots\!13 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} - 5\beta_{5} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{6} - 7\beta_{5} + 7\beta_{4} + 7\beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 17 \beta_{9} + 2 \beta_{8} + 15 \beta_{7} + 24 \beta_{6} - 21 \beta_{5} + 15 \beta_{4} + 52 \beta_{3} - 3 \beta_{2} + 3 \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{11} - 22 \beta_{10} + 91 \beta_{9} + 82 \beta_{7} + 156 \beta_{6} - 52 \beta_{5} + 39 \beta_{4} + 117 \beta_{3} - 39 \beta_{2} + 17 \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 108 \beta_{11} - 150 \beta_{10} + 498 \beta_{9} - 39 \beta_{8} + 238 \beta_{7} + 730 \beta_{6} - 111 \beta_{5} + 108 \beta_{4} + 260 \beta_{3} - 238 \beta_{2} + 39 \beta _1 - 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 606 \beta_{11} - 544 \beta_{10} + 2012 \beta_{9} - 258 \beta_{8} + 606 \beta_{7} + 2012 \beta_{6} + 258 \beta_{5} + 258 \beta_{4} + 286 \beta_{3} - 968 \beta_{2} - 583 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2618 \beta_{11} - 2346 \beta_{10} + 5882 \beta_{9} - 1150 \beta_{8} + 1150 \beta_{7} + 3814 \beta_{6} + 3150 \beta_{5} - 1150 \beta_{3} - 2618 \beta_{2} - 583 \beta _1 - 3150 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8500 \beta_{11} - 9073 \beta_{10} + 9073 \beta_{9} - 4964 \beta_{8} + 17715 \beta_{5} - 3733 \beta_{4} - 13332 \beta_{3} - 4964 \beta_{2} - 3733 \beta _1 - 13332 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17573 \beta_{11} - 21413 \beta_{10} - 17573 \beta_{9} - 17573 \beta_{8} - 17065 \beta_{7} - 51090 \beta_{6} + 78255 \beta_{5} - 25181 \beta_{4} - 78255 \beta_{3} - 17065 \beta _1 - 51090 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 220187 \beta_{9} - 38986 \beta_{8} - 120501 \beta_{7} - 340150 \beta_{6} + 271995 \beta_{5} - 120501 \beta_{4} - 330148 \beta_{3} + 68155 \beta_{2} - 68155 \beta _1 - 152032 \) Copy content Toggle raw display

Character values

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

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-\beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
3.29778 + 1.58813i
−1.67429 0.806295i
1.65762 2.07859i
−0.880144 + 1.10367i
0.571115 2.50222i
−0.472084 + 2.06834i
3.29778 1.58813i
−1.67429 + 0.806295i
0.571115 + 2.50222i
−0.472084 2.06834i
1.65762 + 2.07859i
−0.880144 1.10367i
0 0.777479 0.974928i 0 −2.57526 + 1.24018i 0 2.07401 0 0.321552 + 1.40881i 0
97.2 0 0.777479 0.974928i 0 2.39681 1.15424i 0 −0.381989 0 0.321552 + 1.40881i 0
145.1 0 0.0990311 0.433884i 0 −0.256654 0.321834i 0 −2.98873 0 2.52446 + 1.21572i 0
145.2 0 0.0990311 0.433884i 0 2.28111 + 2.86043i 0 4.34563 0 2.52446 + 1.21572i 0
193.1 0 1.62349 0.781831i 0 −0.694605 3.04326i 0 −4.44744 0 0.153989 0.193096i 0
193.2 0 1.62349 0.781831i 0 0.348594 + 1.52729i 0 1.39852 0 0.153989 0.193096i 0
305.1 0 0.777479 + 0.974928i 0 −2.57526 1.24018i 0 2.07401 0 0.321552 1.40881i 0
305.2 0 0.777479 + 0.974928i 0 2.39681 + 1.15424i 0 −0.381989 0 0.321552 1.40881i 0
385.1 0 1.62349 + 0.781831i 0 −0.694605 + 3.04326i 0 −4.44744 0 0.153989 + 0.193096i 0
385.2 0 1.62349 + 0.781831i 0 0.348594 1.52729i 0 1.39852 0 0.153989 + 0.193096i 0
465.1 0 0.0990311 + 0.433884i 0 −0.256654 + 0.321834i 0 −2.98873 0 2.52446 1.21572i 0
465.2 0 0.0990311 + 0.433884i 0 2.28111 2.86043i 0 4.34563 0 2.52446 1.21572i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.u.e 12
4.b odd 2 1 172.2.i.b 12
43.e even 7 1 inner 688.2.u.e 12
172.j even 14 1 7396.2.a.i 6
172.k odd 14 1 172.2.i.b 12
172.k odd 14 1 7396.2.a.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
172.2.i.b 12 4.b odd 2 1
172.2.i.b 12 172.k odd 14 1
688.2.u.e 12 1.a even 1 1 trivial
688.2.u.e 12 43.e even 7 1 inner
7396.2.a.i 6 172.j even 14 1
7396.2.a.j 6 172.k odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 5T_{3}^{5} + 11T_{3}^{4} - 13T_{3}^{3} + 9T_{3}^{2} - 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 5 T^{5} + 11 T^{4} - 13 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + 9 T^{10} + T^{9} + \cdots + 3136 \) Copy content Toggle raw display
$7$ \( (T^{6} - 27 T^{4} + 7 T^{3} + 152 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + 14 T^{10} + 62 T^{9} + \cdots + 19321 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} - 21 T^{10} + \cdots + 19321 \) Copy content Toggle raw display
$17$ \( T^{12} - 17 T^{11} + 135 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$19$ \( T^{12} + 13 T^{11} + 80 T^{10} + \cdots + 31820881 \) Copy content Toggle raw display
$23$ \( T^{12} - 14 T^{11} + 135 T^{10} + \cdots + 15816529 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + 85 T^{10} + \cdots + 37515625 \) Copy content Toggle raw display
$31$ \( T^{12} - 23 T^{11} + 271 T^{10} + \cdots + 177241 \) Copy content Toggle raw display
$37$ \( (T^{6} + 12 T^{5} + 19 T^{4} - 147 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 8 T^{11} + 63 T^{10} + \cdots + 171688609 \) Copy content Toggle raw display
$43$ \( T^{12} - 7 T^{11} + \cdots + 6321363049 \) Copy content Toggle raw display
$47$ \( T^{12} + 22 T^{11} + 273 T^{10} + \cdots + 5340721 \) Copy content Toggle raw display
$53$ \( T^{12} - 27 T^{11} + \cdots + 3037883689 \) Copy content Toggle raw display
$59$ \( T^{12} - 17 T^{11} + \cdots + 2208906001 \) Copy content Toggle raw display
$61$ \( T^{12} - 8 T^{11} + 105 T^{10} + \cdots + 5139289 \) Copy content Toggle raw display
$67$ \( T^{12} + 29 T^{11} + \cdots + 5116111729 \) Copy content Toggle raw display
$71$ \( T^{12} - 17 T^{11} + 181 T^{10} + \cdots + 11235904 \) Copy content Toggle raw display
$73$ \( T^{12} + 10 T^{11} + \cdots + 24854468409 \) Copy content Toggle raw display
$79$ \( (T^{6} + 7 T^{5} - 269 T^{4} + \cdots + 318016)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 6 T^{11} + 183 T^{10} + \cdots + 292444201 \) Copy content Toggle raw display
$89$ \( T^{12} + 16 T^{11} + 312 T^{10} + \cdots + 82369 \) Copy content Toggle raw display
$97$ \( T^{12} - 10 T^{11} + \cdots + 376889171569 \) Copy content Toggle raw display
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