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Properties

Label 605.2.j.f
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(9,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} - \beta_{3} q^{3} - \beta_{10} q^{4} + (\beta_{14} + \beta_{10} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{6} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} - \beta_{3} q^{3} - \beta_{10} q^{4} + (\beta_{14} + \beta_{10} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 3 \beta_{11} + 4 \beta_{9}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{6} - 4 q^{9} - 16 q^{10} - 4 q^{15} + 4 q^{16} + 4 q^{19} - 12 q^{20} + 48 q^{21} + 8 q^{24} - 4 q^{25} + 12 q^{26} + 28 q^{31} - 32 q^{34} + 12 q^{35} + 12 q^{36} - 12 q^{39} - 4 q^{40} - 48 q^{45} - 28 q^{46} - 4 q^{49} - 24 q^{50} + 16 q^{51} - 64 q^{54} + 48 q^{56} - 12 q^{59} - 12 q^{60} + 40 q^{61} - 16 q^{64} - 96 q^{65} - 20 q^{69} - 12 q^{70} + 12 q^{71} - 24 q^{74} - 24 q^{75} + 144 q^{76} - 8 q^{79} - 24 q^{80} + 8 q^{81} - 24 q^{84} - 8 q^{85} + 48 q^{86} + 48 q^{89} + 16 q^{90} + 36 q^{91} - 4 q^{94} - 36 q^{95} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 362 ) / 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 780\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 780\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} + 780\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{13} - 2911\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{15} - 2911\nu^{5} ) / 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} - 723\nu^{5} ) / 19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} - 7\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - 11\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{15} + 41\beta_{12} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 209\beta_{2} - 362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 209\beta_{3} - 780\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -780\beta_{6} - 1351\beta_{4} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -780\beta_{7} - 2911\beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2911\beta_{10} + 5042\beta_{8} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -2911\beta_{11} + 7953\beta_{9} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.13551 − 1.56290i
−0.304260 − 0.418778i
0.304260 + 0.418778i
1.13551 + 1.56290i
1.83730 − 0.596975i
0.492303 − 0.159959i
−0.492303 + 0.159959i
−1.83730 + 0.596975i
−1.13551 + 1.56290i
−0.304260 + 0.418778i
0.304260 − 0.418778i
1.13551 − 1.56290i
1.83730 + 0.596975i
0.492303 + 0.159959i
−0.492303 − 0.159959i
−1.83730 − 0.596975i
−1.83730 − 0.596975i 0.304260 + 0.418778i 1.40126 + 1.01807i 0.809764 − 2.08429i −0.309017 − 0.951057i −0.526994 + 0.725345i 0.304260 + 0.418778i 0.844250 − 2.59833i −2.73205 + 3.34607i
9.2 −0.492303 − 0.159959i 1.13551 + 1.56290i −1.40126 − 1.01807i −0.809764 + 2.08429i −0.309017 − 0.951057i 1.96677 − 2.70702i 1.13551 + 1.56290i −0.226216 + 0.696222i 0.732051 − 0.896575i
9.3 0.492303 + 0.159959i −1.13551 − 1.56290i −1.40126 − 1.01807i 1.88023 + 1.21026i −0.309017 − 0.951057i −1.96677 + 2.70702i −1.13551 − 1.56290i −0.226216 + 0.696222i 0.732051 + 0.896575i
9.4 1.83730 + 0.596975i −0.304260 − 0.418778i 1.40126 + 1.01807i −1.88023 − 1.21026i −0.309017 − 0.951057i 0.526994 − 0.725345i −0.304260 − 0.418778i 0.844250 − 2.59833i −2.73205 − 3.34607i
124.1 −1.13551 − 1.56290i −0.492303 + 0.159959i −0.535233 + 1.64728i 2.23251 − 0.126049i 0.809017 + 0.587785i 0.852694 + 0.277057i −0.492303 + 0.159959i −2.21028 + 1.60586i −2.73205 − 3.34607i
124.2 −0.304260 − 0.418778i −1.83730 + 0.596975i 0.535233 − 1.64728i −2.23251 + 0.126049i 0.809017 + 0.587785i −3.18230 − 1.03399i −1.83730 + 0.596975i 0.592242 − 0.430289i 0.732051 + 0.896575i
124.3 0.304260 + 0.418778i 1.83730 − 0.596975i 0.535233 − 1.64728i −0.570005 − 2.16220i 0.809017 + 0.587785i 3.18230 + 1.03399i 1.83730 − 0.596975i 0.592242 − 0.430289i 0.732051 − 0.896575i
124.4 1.13551 + 1.56290i 0.492303 − 0.159959i −0.535233 + 1.64728i 0.570005 + 2.16220i 0.809017 + 0.587785i −0.852694 − 0.277057i 0.492303 − 0.159959i −2.21028 + 1.60586i −2.73205 + 3.34607i
269.1 −1.83730 + 0.596975i 0.304260 − 0.418778i 1.40126 − 1.01807i 0.809764 + 2.08429i −0.309017 + 0.951057i −0.526994 − 0.725345i 0.304260 − 0.418778i 0.844250 + 2.59833i −2.73205 − 3.34607i
269.2 −0.492303 + 0.159959i 1.13551 − 1.56290i −1.40126 + 1.01807i −0.809764 − 2.08429i −0.309017 + 0.951057i 1.96677 + 2.70702i 1.13551 − 1.56290i −0.226216 − 0.696222i 0.732051 + 0.896575i
269.3 0.492303 − 0.159959i −1.13551 + 1.56290i −1.40126 + 1.01807i 1.88023 − 1.21026i −0.309017 + 0.951057i −1.96677 − 2.70702i −1.13551 + 1.56290i −0.226216 − 0.696222i 0.732051 − 0.896575i
269.4 1.83730 − 0.596975i −0.304260 + 0.418778i 1.40126 − 1.01807i −1.88023 + 1.21026i −0.309017 + 0.951057i 0.526994 + 0.725345i −0.304260 + 0.418778i 0.844250 + 2.59833i −2.73205 + 3.34607i
444.1 −1.13551 + 1.56290i −0.492303 − 0.159959i −0.535233 − 1.64728i 2.23251 + 0.126049i 0.809017 − 0.587785i 0.852694 − 0.277057i −0.492303 − 0.159959i −2.21028 − 1.60586i −2.73205 + 3.34607i
444.2 −0.304260 + 0.418778i −1.83730 − 0.596975i 0.535233 + 1.64728i −2.23251 − 0.126049i 0.809017 − 0.587785i −3.18230 + 1.03399i −1.83730 − 0.596975i 0.592242 + 0.430289i 0.732051 − 0.896575i
444.3 0.304260 − 0.418778i 1.83730 + 0.596975i 0.535233 + 1.64728i −0.570005 + 2.16220i 0.809017 − 0.587785i 3.18230 − 1.03399i 1.83730 + 0.596975i 0.592242 + 0.430289i 0.732051 + 0.896575i
444.4 1.13551 − 1.56290i 0.492303 + 0.159959i −0.535233 − 1.64728i 0.570005 − 2.16220i 0.809017 − 0.587785i −0.852694 + 0.277057i 0.492303 + 0.159959i −2.21028 − 1.60586i −2.73205 − 3.34607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 3 inner
55.j even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.f 16
5.b even 2 1 inner 605.2.j.f 16
11.b odd 2 1 605.2.j.e 16
11.c even 5 1 605.2.b.d ✓ 4
11.c even 5 3 inner 605.2.j.f 16
11.d odd 10 1 605.2.b.e yes 4
11.d odd 10 3 605.2.j.e 16
55.d odd 2 1 605.2.j.e 16
55.h odd 10 1 605.2.b.e yes 4
55.h odd 10 3 605.2.j.e 16
55.j even 10 1 605.2.b.d ✓ 4
55.j even 10 3 inner 605.2.j.f 16
55.k odd 20 2 3025.2.a.y 4
55.l even 20 2 3025.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d ✓ 4 11.c even 5 1
605.2.b.d ✓ 4 55.j even 10 1
605.2.b.e yes 4 11.d odd 10 1
605.2.b.e yes 4 55.h odd 10 1
605.2.j.e 16 11.b odd 2 1
605.2.j.e 16 11.d odd 10 3
605.2.j.e 16 55.d odd 2 1
605.2.j.e 16 55.h odd 10 3
605.2.j.f 16 1.a even 1 1 trivial
605.2.j.f 16 5.b even 2 1 inner
605.2.j.f 16 11.c even 5 3 inner
605.2.j.f 16 55.j even 10 3 inner
3025.2.a.y 4 55.k odd 20 2
3025.2.a.z 4 55.l even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{16} - 4T_{2}^{14} + 15T_{2}^{12} - 56T_{2}^{10} + 209T_{2}^{8} - 56T_{2}^{6} + 15T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{8} - 2T_{19}^{7} + 30T_{19}^{6} - 112T_{19}^{5} + 1004T_{19}^{4} + 2912T_{19}^{3} + 20280T_{19}^{2} + 35152T_{19} + 456976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 18 T^{6} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 16 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{8} - 2 T^{7} + \cdots + 456976)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 52 T^{2} + 484)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 48 T^{6} + \cdots + 5308416)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 4477456)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$41$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 156 T^{2} + 4761)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23811286661761 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 208827064576 \) Copy content Toggle raw display
$59$ \( (T^{8} + 6 T^{7} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 20 T^{7} + \cdots + 88529281)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 228 T^{2} + 1089)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 6 T^{7} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 24 T^{6} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 4 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 98 T^{6} + \cdots + 92236816)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 3)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} - 84 T^{14} + \cdots + 1679616 \) Copy content Toggle raw display
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