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Properties

Label 605.6.a.k
Level $605$
Weight $6$
Character orbit 605.a
Self dual yes
Analytic conductor $97.032$
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] = [605,6,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

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: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 296x^{8} + 30555x^{6} - 1272708x^{4} + 17770320x^{2} - 446400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 4) q^{3} + (\beta_{4} + \beta_{2} + 27) q^{4} + 25 q^{5} + ( - \beta_{3} - \beta_1) q^{6} + ( - \beta_{7} - 5 \beta_1) q^{7} + (\beta_{5} + \beta_{3} + 26 \beta_1) q^{8} + ( - \beta_{6} + 2 \beta_{4} + \cdots + 121) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 4) q^{3} + (\beta_{4} + \beta_{2} + 27) q^{4} + 25 q^{5} + ( - \beta_{3} - \beta_1) q^{6} + ( - \beta_{7} - 5 \beta_1) q^{7} + (\beta_{5} + \beta_{3} + 26 \beta_1) q^{8} + ( - \beta_{6} + 2 \beta_{4} + \cdots + 121) q^{9}+ \cdots + ( - 42 \beta_{9} + 204 \beta_{7} + \cdots + 25304 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 36 q^{3} + 272 q^{4} + 250 q^{5} + 1182 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 36 q^{3} + 272 q^{4} + 250 q^{5} + 1182 q^{9} - 1534 q^{12} - 3014 q^{14} + 900 q^{15} + 6420 q^{16} + 6800 q^{20} + 10348 q^{23} + 6250 q^{25} - 5632 q^{26} + 16248 q^{27} + 16648 q^{31} + 21142 q^{34} + 56914 q^{36} + 13440 q^{37} + 39614 q^{38} + 91226 q^{42} + 29550 q^{45} + 95652 q^{47} - 120110 q^{48} + 135270 q^{49} + 59392 q^{53} - 86762 q^{56} + 114890 q^{58} - 7984 q^{59} - 38350 q^{60} + 66888 q^{64} + 177300 q^{67} + 25160 q^{69} - 75350 q^{70} - 79760 q^{71} + 22500 q^{75} + 600640 q^{78} + 160500 q^{80} + 174018 q^{81} - 540444 q^{82} - 56124 q^{86} + 12192 q^{89} + 117632 q^{91} + 807588 q^{92} - 284556 q^{93} + 155580 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 296x^{8} + 30555x^{6} - 1272708x^{4} + 17770320x^{2} - 446400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{8} + 988\nu^{6} - 93185\nu^{4} + 2432184\nu^{2} - 4709520 ) / 565440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{9} + 988\nu^{7} - 93185\nu^{5} + 2432184\nu^{3} - 7536720\nu ) / 565440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{8} - 988\nu^{6} + 93185\nu^{4} - 1866744\nu^{2} - 28651440 ) / 565440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} - 988\nu^{7} + 93185\nu^{5} - 1866744\nu^{3} - 43352880\nu ) / 565440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -67\nu^{8} + 14212\nu^{6} - 887425\nu^{4} + 14337456\nu^{2} + 43248720 ) / 282720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{9} + 988\nu^{7} - 112033\nu^{5} + 5052056\nu^{3} - 75125648\nu ) / 75392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{8} - 2641\nu^{6} + 210135\nu^{4} - 5917053\nu^{2} + 23452740 ) / 35340 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -101\nu^{9} + 29336\nu^{7} - 2964455\nu^{5} + 120557268\nu^{3} - 1642127040\nu ) / 282720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} + 59 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + 90\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} + 2\beta_{6} + 113\beta_{4} + 141\beta_{2} + 5267 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} + 139\beta_{5} + 169\beta_{3} + 8924\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 304\beta_{8} + 268\beta_{6} + 12085\beta_{4} + 17949\beta_{2} + 519115 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -72\beta_{9} - 176\beta_{7} + 15965\beta_{5} + 22133\beta_{3} + 925486\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 37994\beta_{8} + 26138\beta_{6} + 1280753\beta_{4} + 2153757\beta_{2} + 53623187 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -23712\beta_{9} + 66284\beta_{7} + 1750963\beta_{5} + 2661961\beta_{3} + 98052356\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.5846
−10.0568
−7.32039
−5.40487
−0.158638
0.158638
5.40487
7.32039
10.0568
10.5846
−10.5846 −22.2800 80.0342 25.0000 235.825 247.400 −508.423 253.397 −264.615
1.2 −10.0568 10.4999 69.1399 25.0000 −105.596 −205.739 −373.510 −132.751 −251.421
1.3 −7.32039 29.9438 21.5882 25.0000 −219.200 151.442 76.2187 653.629 −183.010
1.4 −5.40487 −12.3845 −2.78739 25.0000 66.9368 −23.3592 188.021 −89.6230 −135.122
1.5 −0.158638 12.2208 −31.9748 25.0000 −1.93868 −157.017 10.1488 −93.6520 −3.96594
1.6 0.158638 12.2208 −31.9748 25.0000 1.93868 157.017 −10.1488 −93.6520 3.96594
1.7 5.40487 −12.3845 −2.78739 25.0000 −66.9368 23.3592 −188.021 −89.6230 135.122
1.8 7.32039 29.9438 21.5882 25.0000 219.200 −151.442 −76.2187 653.629 183.010
1.9 10.0568 10.4999 69.1399 25.0000 105.596 205.739 373.510 −132.751 251.421
1.10 10.5846 −22.2800 80.0342 25.0000 −235.825 −247.400 508.423 253.397 264.615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.6.a.k ✓ 10
11.b odd 2 1 inner 605.6.a.k ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.6.a.k ✓ 10 1.a even 1 1 trivial
605.6.a.k ✓ 10 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 296T_{2}^{8} + 30555T_{2}^{6} - 1272708T_{2}^{4} + 17770320T_{2}^{2} - 446400 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 296 T^{8} + \cdots - 446400 \) Copy content Toggle raw display
$3$ \( (T^{5} - 18 T^{4} + \cdots - 1060200)^{2} \) Copy content Toggle raw display
$5$ \( (T - 25)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 93\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 49\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
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