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Properties

Label 605.2.j.i
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: 16.0.343361479062744140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 55)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{11} + \cdots + \beta_1) q^{2}+ \cdots + ( - \beta_{15} + \beta_{13} + \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{11} + \cdots + \beta_1) q^{2}+ \cdots + ( - 5 \beta_{14} - 5 \beta_{13} + \cdots + 5 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{4} + 3 q^{5} - 8 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{4} + 3 q^{5} - 8 q^{6} + 2 q^{9} - 40 q^{10} - 12 q^{14} - q^{15} - 14 q^{16} + 16 q^{19} + 12 q^{20} + 48 q^{21} - 4 q^{24} - q^{25} + 12 q^{29} + 6 q^{30} - 2 q^{31} + 64 q^{34} - 18 q^{35} + 30 q^{36} - 28 q^{40} - 12 q^{41} + 72 q^{45} + 8 q^{46} + 20 q^{49} + 18 q^{50} + 28 q^{51} + 80 q^{54} - 240 q^{56} - 18 q^{59} + 18 q^{60} - 20 q^{61} + 2 q^{64} - 14 q^{69} - 24 q^{70} + 6 q^{71} - 12 q^{74} - 15 q^{75} + 96 q^{76} + 28 q^{79} - 6 q^{80} + 8 q^{81} - 48 q^{84} - 2 q^{85} + 12 q^{86} + 24 q^{89} + 28 q^{90} + 44 q^{94} - 12 q^{95} - 36 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5 \nu^{15} + 15 \nu^{14} - 40 \nu^{13} + 40 \nu^{12} - 5709 \nu^{11} + 30 \nu^{10} + \cdots + 32805 ) / 261711 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 160 \nu^{15} + 400 \nu^{14} + 3467 \nu^{12} - 1040 \nu^{11} - 4960 \nu^{10} + 6480 \nu^{9} + \cdots - 524880 ) / 785133 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 3 \nu^{12} - 48 \nu^{11} + 93 \nu^{10} + 81 \nu^{9} + \cdots + 13122 ) / 9693 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{15} - 40 \nu^{14} + 27 \nu^{13} - 72 \nu^{12} + 72 \nu^{11} + 63 \nu^{10} - 664 \nu^{9} + \cdots - 19683 ) / 58158 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\nu^{15} + 718\nu^{10} + 22258\nu^{5} + 146286 ) / 87237 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31 \nu^{15} + 93 \nu^{14} - 248 \nu^{13} + 248 \nu^{12} - 501 \nu^{11} + 186 \nu^{10} + \cdots + 203391 ) / 174474 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 403 \nu^{15} - 806 \nu^{14} + 1053 \nu^{13} - 403 \nu^{12} + 6448 \nu^{11} - 12493 \nu^{10} + \cdots - 1762722 ) / 523422 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 842 \nu^{15} + 1684 \nu^{14} - 324 \nu^{13} + 842 \nu^{12} - 13472 \nu^{11} + 26102 \nu^{10} + \cdots + 3682908 ) / 785133 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1079 \nu^{15} - 2158 \nu^{14} - 6561 \nu^{13} - 1079 \nu^{12} + 17264 \nu^{11} - 33449 \nu^{10} + \cdots - 4719546 ) / 1570266 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 3 \nu^{14} + 8 \nu^{13} - 8 \nu^{12} - 7 \nu^{11} - 6 \nu^{10} - 56 \nu^{9} + 137 \nu^{8} + \cdots - 6561 ) / 2154 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 646 \nu^{15} - 1615 \nu^{14} + 178 \nu^{12} + 4199 \nu^{11} + 20026 \nu^{10} - 26163 \nu^{9} + \cdots + 2119203 ) / 785133 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2 \nu^{15} + 5 \nu^{14} - 15 \nu^{12} - 13 \nu^{11} - 62 \nu^{10} + 81 \nu^{9} - 31 \nu^{8} + \cdots - 6561 ) / 2154 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 599 \nu^{15} - 130 \nu^{14} - 1797 \nu^{13} + 4792 \nu^{12} - 4792 \nu^{11} - 4193 \nu^{10} + \cdots + 1310013 ) / 523422 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2170 \nu^{15} - 5425 \nu^{14} - 8855 \nu^{12} + 14105 \nu^{11} + 67270 \nu^{10} - 87885 \nu^{9} + \cdots + 7118685 ) / 1570266 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 248 \nu^{15} + 744 \nu^{14} - 1984 \nu^{13} + 1984 \nu^{12} - 4008 \nu^{11} + 1488 \nu^{10} + \cdots + 1627128 ) / 261711 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{10} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} - 3\beta_{11} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{8} - 2\beta_{7} - 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{15} + \beta_{11} - \beta_{8} + \beta_{5} - 10\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{13} + 2\beta_{12} + 2\beta_{7} - 2\beta_{6} + 15\beta_{5} - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{15} - 16\beta_{10} - 16\beta_{6} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -26\beta_{14} + 35\beta_{11} - 35\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 39\beta_{8} + 70\beta_{7} + 39\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68 \beta_{15} - 74 \beta_{13} + 37 \beta_{11} - 37 \beta_{8} + 37 \beta_{5} + 74 \beta_{4} - 68 \beta_{3} + \cdots - 68 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 62\beta_{14} - 62\beta_{10} - 62\beta_{9} - 253\beta_{5} - 62\beta_{4} - 253 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -93\beta_{15} + 506\beta_{6} - 93\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 160\beta_{14} + 160\beta_{12} + 80\beta_{11} + 679\beta_{2} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -1198\beta_{9} - 1079\beta_{8} + 1079\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1797 \beta_{15} + 2158 \beta_{13} - 1797 \beta_{11} + 1797 \beta_{8} - 1797 \beta_{5} + 1797 \beta_{3} + \cdots + 1797 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 718 \beta_{14} - 718 \beta_{13} - 718 \beta_{12} + 718 \beta_{10} + 718 \beta_{9} - 718 \beta_{7} + \cdots + 3596 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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
1.56693 + 0.738055i
−0.144291 − 1.72603i
−0.897801 − 1.48120i
−0.833856 + 1.51812i
−0.217724 − 1.71831i
1.59696 + 0.670602i
1.13127 + 1.31158i
−1.70149 + 0.323920i
1.56693 − 0.738055i
−0.144291 + 1.72603i
−0.897801 + 1.48120i
−0.833856 − 1.51812i
−0.217724 + 1.71831i
1.59696 − 0.670602i
1.13127 − 1.31158i
−1.70149 − 0.323920i
−2.40079 − 0.780063i −0.465695 − 0.640974i 3.53725 + 2.56996i 2.23606 − 0.00508966i 0.618034 + 1.90211i 2.03615 − 2.80252i −3.51992 − 4.84475i 0.733075 − 2.25617i −5.37228 − 1.73205i
9.2 −0.753510 − 0.244830i −1.48377 − 2.04223i −1.11020 − 0.806607i −1.12244 − 1.93394i 0.618034 + 1.90211i −2.03615 + 2.80252i 1.57045 + 2.16154i −1.04209 + 3.20723i 0.372281 + 1.73205i
9.3 0.753510 + 0.244830i 1.48377 + 2.04223i −1.11020 − 0.806607i −0.228670 − 2.22434i 0.618034 + 1.90211i 2.03615 − 2.80252i −1.57045 − 2.16154i −1.04209 + 3.20723i 0.372281 − 1.73205i
9.4 2.40079 + 0.780063i 0.465695 + 0.640974i 3.53725 + 2.56996i −1.81200 + 1.31021i 0.618034 + 1.90211i −2.03615 + 2.80252i 3.51992 + 4.84475i 0.733075 − 2.25617i −5.37228 + 1.73205i
124.1 −1.48377 − 2.04223i 0.753510 − 0.244830i −1.35111 + 4.15829i 0.695822 − 2.12505i −1.61803 − 1.17557i −3.29456 − 1.07047i 5.69534 − 1.85053i −1.91922 + 1.39439i −5.37228 + 1.73205i
124.2 −0.465695 − 0.640974i 2.40079 − 0.780063i 0.424058 − 1.30512i 1.49244 + 1.66512i −1.61803 − 1.17557i 3.29456 + 1.07047i −2.54105 + 0.825636i 2.72823 − 1.98218i 0.372281 − 1.73205i
124.3 0.465695 + 0.640974i −2.40079 + 0.780063i 0.424058 − 1.30512i 2.04481 + 0.904839i −1.61803 − 1.17557i −3.29456 − 1.07047i 2.54105 − 0.825636i 2.72823 − 1.98218i 0.372281 + 1.73205i
124.4 1.48377 + 2.04223i −0.753510 + 0.244830i −1.35111 + 4.15829i −1.80602 + 1.31844i −1.61803 − 1.17557i 3.29456 + 1.07047i −5.69534 + 1.85053i −1.91922 + 1.39439i −5.37228 − 1.73205i
269.1 −2.40079 + 0.780063i −0.465695 + 0.640974i 3.53725 − 2.56996i 2.23606 + 0.00508966i 0.618034 − 1.90211i 2.03615 + 2.80252i −3.51992 + 4.84475i 0.733075 + 2.25617i −5.37228 + 1.73205i
269.2 −0.753510 + 0.244830i −1.48377 + 2.04223i −1.11020 + 0.806607i −1.12244 + 1.93394i 0.618034 − 1.90211i −2.03615 − 2.80252i 1.57045 − 2.16154i −1.04209 − 3.20723i 0.372281 − 1.73205i
269.3 0.753510 − 0.244830i 1.48377 − 2.04223i −1.11020 + 0.806607i −0.228670 + 2.22434i 0.618034 − 1.90211i 2.03615 + 2.80252i −1.57045 + 2.16154i −1.04209 − 3.20723i 0.372281 + 1.73205i
269.4 2.40079 − 0.780063i 0.465695 − 0.640974i 3.53725 − 2.56996i −1.81200 − 1.31021i 0.618034 − 1.90211i −2.03615 − 2.80252i 3.51992 − 4.84475i 0.733075 + 2.25617i −5.37228 − 1.73205i
444.1 −1.48377 + 2.04223i 0.753510 + 0.244830i −1.35111 − 4.15829i 0.695822 + 2.12505i −1.61803 + 1.17557i −3.29456 + 1.07047i 5.69534 + 1.85053i −1.91922 − 1.39439i −5.37228 − 1.73205i
444.2 −0.465695 + 0.640974i 2.40079 + 0.780063i 0.424058 + 1.30512i 1.49244 − 1.66512i −1.61803 + 1.17557i 3.29456 − 1.07047i −2.54105 − 0.825636i 2.72823 + 1.98218i 0.372281 + 1.73205i
444.3 0.465695 − 0.640974i −2.40079 − 0.780063i 0.424058 + 1.30512i 2.04481 − 0.904839i −1.61803 + 1.17557i −3.29456 + 1.07047i 2.54105 + 0.825636i 2.72823 + 1.98218i 0.372281 − 1.73205i
444.4 1.48377 − 2.04223i −0.753510 − 0.244830i −1.35111 − 4.15829i −1.80602 − 1.31844i −1.61803 + 1.17557i 3.29456 − 1.07047i −5.69534 − 1.85053i −1.91922 − 1.39439i −5.37228 + 1.73205i
\(n\): 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.i 16
5.b even 2 1 inner 605.2.j.i 16
11.b odd 2 1 605.2.j.j 16
11.c even 5 1 55.2.b.a ✓ 4
11.c even 5 3 inner 605.2.j.i 16
11.d odd 10 1 605.2.b.c 4
11.d odd 10 3 605.2.j.j 16
33.h odd 10 1 495.2.c.a 4
44.h odd 10 1 880.2.b.h 4
55.d odd 2 1 605.2.j.j 16
55.h odd 10 1 605.2.b.c 4
55.h odd 10 3 605.2.j.j 16
55.j even 10 1 55.2.b.a ✓ 4
55.j even 10 3 inner 605.2.j.i 16
55.k odd 20 2 275.2.a.h 4
55.l even 20 2 3025.2.a.ba 4
165.o odd 10 1 495.2.c.a 4
165.v even 20 2 2475.2.a.bi 4
220.n odd 10 1 880.2.b.h 4
220.v even 20 2 4400.2.a.cc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a ✓ 4 11.c even 5 1
55.2.b.a ✓ 4 55.j even 10 1
275.2.a.h 4 55.k odd 20 2
495.2.c.a 4 33.h odd 10 1
495.2.c.a 4 165.o odd 10 1
605.2.b.c 4 11.d odd 10 1
605.2.b.c 4 55.h odd 10 1
605.2.j.i 16 1.a even 1 1 trivial
605.2.j.i 16 5.b even 2 1 inner
605.2.j.i 16 11.c even 5 3 inner
605.2.j.i 16 55.j even 10 3 inner
605.2.j.j 16 11.b odd 2 1
605.2.j.j 16 11.d odd 10 3
605.2.j.j 16 55.d odd 2 1
605.2.j.j 16 55.h odd 10 3
880.2.b.h 4 44.h odd 10 1
880.2.b.h 4 220.n odd 10 1
2475.2.a.bi 4 165.v even 20 2
3025.2.a.ba 4 55.l even 20 2
4400.2.a.cc 4 220.v even 20 2

Hecke kernels

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

\( T_{2}^{16} - 7T_{2}^{14} + 45T_{2}^{12} - 287T_{2}^{10} + 1829T_{2}^{8} - 1148T_{2}^{6} + 720T_{2}^{4} - 448T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{19}^{4} - 4T_{19}^{3} + 16T_{19}^{2} - 64T_{19} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 7 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} - 3 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} - 12 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} - 28 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 7 T^{2} + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} - 6 T^{7} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + T^{7} + 9 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$41$ \( (T^{8} + 6 T^{7} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} - 44 T^{6} + \cdots + 3748096)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1099511627776 \) Copy content Toggle raw display
$59$ \( (T^{8} + 9 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 10 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 87 T^{2} + 36)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 3 T^{7} + \cdots + 26873856)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 48 T^{6} + \cdots + 5308416)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 14 T^{7} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 44 T^{6} + \cdots + 3748096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T - 6)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 110075314176 \) Copy content Toggle raw display
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