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Properties

Label 684.7.y.c
Level $684$
Weight $7$
Character orbit 684.y
Analytic conductor $157.357$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,7,Mod(145,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), 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: \(157.356993196\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 56 x^{19} + 93542 x^{18} - 1625808 x^{17} + 5494347739 x^{16} - 30640959724 x^{15} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{9}\cdot 19^{4} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 6 \beta_1 + 6) q^{5} + (\beta_{10} + 23) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 6 \beta_1 + 6) q^{5} + (\beta_{10} + 23) q^{7} + ( - \beta_{14} + 2 \beta_{7} + \cdots + 183) q^{11}+ \cdots + ( - 6 \beta_{19} - 33 \beta_{18} + \cdots - 5011) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 56 q^{5} + 464 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 56 q^{5} + 464 q^{7} + 3644 q^{11} - 7140 q^{13} - 1132 q^{17} + 2110 q^{19} - 832 q^{23} - 27698 q^{25} + 10920 q^{29} - 4172 q^{35} - 109206 q^{41} + 110740 q^{43} - 107080 q^{47} + 136092 q^{49} - 254796 q^{53} + 354840 q^{55} + 610638 q^{59} + 47864 q^{61} - 839562 q^{67} - 366660 q^{71} + 854482 q^{73} - 763088 q^{77} + 1718592 q^{79} - 439612 q^{83} - 400236 q^{85} - 478032 q^{89} + 599856 q^{91} + 1055660 q^{95} - 191286 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 56 x^{19} + 93542 x^{18} - 1625808 x^{17} + 5494347739 x^{16} - 30640959724 x^{15} + \cdots + 80\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 58\!\cdots\!92 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\!\cdots\!92 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40\!\cdots\!94 \nu^{19} + \cdots - 23\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29\!\cdots\!59 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!63 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!86 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 96\!\cdots\!17 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55\!\cdots\!29 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!21 \nu^{19} + \cdots + 74\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!74 \nu^{19} + \cdots + 45\!\cdots\!00 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!56 \nu^{19} + \cdots + 84\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!93 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!67 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!46 \nu^{19} + \cdots - 40\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25\!\cdots\!56 \nu^{19} + \cdots + 77\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30\!\cdots\!47 \nu^{19} + \cdots - 20\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 19\!\cdots\!27 \nu^{19} + \cdots + 20\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 28\!\cdots\!31 \nu^{19} + \cdots - 40\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 25\!\cdots\!98 \nu^{19} + \cdots - 38\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{18} + \beta_{17} + \beta_{16} + 2 \beta_{15} - 4 \beta_{13} - 2 \beta_{12} + \cdots + 18401 \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{19} + 86 \beta_{18} + 43 \beta_{17} - 43 \beta_{16} + 97 \beta_{15} + 8 \beta_{14} + \cdots - 544213 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 21165 \beta_{19} - 30904 \beta_{18} - 61808 \beta_{17} - 9739 \beta_{16} - 142648 \beta_{15} + \cdots - 528188352 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1194401 \beta_{18} + 1194401 \beta_{17} + 3926796 \beta_{16} + 3799137 \beta_{15} + \cdots - 25198820335 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 498860163 \beta_{19} + 2053514520 \beta_{18} + 1026757260 \beta_{17} - 1026757260 \beta_{16} + \cdots + 17536357843062 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 106158338018 \beta_{19} - 22905882597 \beta_{18} - 45811765194 \beta_{17} - 129064220615 \beta_{16} + \cdots + 12\!\cdots\!17 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 37049180696714 \beta_{18} + 37049180696714 \beta_{17} + 58601755264255 \beta_{16} + \cdots + 64\!\cdots\!20 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 54\!\cdots\!84 \beta_{19} - 115726765258042 \beta_{18} - 57863382629021 \beta_{17} + \cdots - 62\!\cdots\!63 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 62\!\cdots\!79 \beta_{19} + \cdots - 25\!\cdots\!10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 39\!\cdots\!15 \beta_{18} + \cdots - 30\!\cdots\!87 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28\!\cdots\!49 \beta_{19} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11\!\cdots\!60 \beta_{19} + \cdots + 14\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 22\!\cdots\!44 \beta_{18} + \cdots + 45\!\cdots\!02 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 48\!\cdots\!10 \beta_{19} + \cdots - 70\!\cdots\!33 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 66\!\cdots\!17 \beta_{19} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 97\!\cdots\!45 \beta_{18} + \cdots - 32\!\cdots\!93 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 31\!\cdots\!11 \beta_{19} + \cdots + 86\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 92\!\cdots\!58 \beta_{19} + \cdots + 15\!\cdots\!69 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−93.3237 − 161.641i
−72.3330 − 125.284i
−71.3406 − 123.566i
−20.1615 − 34.9207i
−16.0954 − 27.8780i
−15.3686 − 26.6192i
67.5013 + 116.916i
70.2011 + 121.592i
72.6742 + 125.875i
106.246 + 184.024i
−93.3237 + 161.641i
−72.3330 + 125.284i
−71.3406 + 123.566i
−20.1615 + 34.9207i
−16.0954 + 27.8780i
−15.3686 + 26.6192i
67.5013 − 116.916i
70.2011 − 121.592i
72.6742 − 125.875i
106.246 − 184.024i
0 0 0 −93.3237 − 161.641i 0 −434.577 0 0 0
145.2 0 0 0 −72.3330 − 125.284i 0 479.542 0 0 0
145.3 0 0 0 −71.3406 − 123.566i 0 421.828 0 0 0
145.4 0 0 0 −20.1615 − 34.9207i 0 53.8286 0 0 0
145.5 0 0 0 −16.0954 − 27.8780i 0 −416.808 0 0 0
145.6 0 0 0 −15.3686 − 26.6192i 0 −77.2273 0 0 0
145.7 0 0 0 67.5013 + 116.916i 0 −157.715 0 0 0
145.8 0 0 0 70.2011 + 121.592i 0 −229.488 0 0 0
145.9 0 0 0 72.6742 + 125.875i 0 621.925 0 0 0
145.10 0 0 0 106.246 + 184.024i 0 −29.3098 0 0 0
217.1 0 0 0 −93.3237 + 161.641i 0 −434.577 0 0 0
217.2 0 0 0 −72.3330 + 125.284i 0 479.542 0 0 0
217.3 0 0 0 −71.3406 + 123.566i 0 421.828 0 0 0
217.4 0 0 0 −20.1615 + 34.9207i 0 53.8286 0 0 0
217.5 0 0 0 −16.0954 + 27.8780i 0 −416.808 0 0 0
217.6 0 0 0 −15.3686 + 26.6192i 0 −77.2273 0 0 0
217.7 0 0 0 67.5013 − 116.916i 0 −157.715 0 0 0
217.8 0 0 0 70.2011 − 121.592i 0 −229.488 0 0 0
217.9 0 0 0 72.6742 − 125.875i 0 621.925 0 0 0
217.10 0 0 0 106.246 − 184.024i 0 −29.3098 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.7.y.c 20
3.b odd 2 1 76.7.h.a ✓ 20
19.d odd 6 1 inner 684.7.y.c 20
57.f even 6 1 76.7.h.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.h.a ✓ 20 3.b odd 2 1
76.7.h.a ✓ 20 57.f even 6 1
684.7.y.c 20 1.a even 1 1 trivial
684.7.y.c 20 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 56 T_{5}^{19} + 93542 T_{5}^{18} - 1625808 T_{5}^{17} + 5494347739 T_{5}^{16} + \cdots + 80\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 39\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 82\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
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