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Properties

Label 605.2.g.p
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(81,605)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), 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: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 6 x^{10} - 12 x^{9} + 43 x^{8} + 72 x^{7} + 155 x^{6} + 162 x^{5} + 541 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_1 q^{3} + ( - \beta_{11} + 3 \beta_{10} + \cdots - 3) q^{4}+ \cdots + ( - \beta_{9} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_1 q^{3} + ( - \beta_{11} + 3 \beta_{10} + \cdots - 3) q^{4}+ \cdots + ( - 3 \beta_{3} - 3 \beta_{2} - 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - q^{3} - 9 q^{4} + 3 q^{5} + 5 q^{6} - q^{7} - 9 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} - q^{3} - 9 q^{4} + 3 q^{5} + 5 q^{6} - q^{7} - 9 q^{8} - 2 q^{9} + 4 q^{10} + 36 q^{12} + 6 q^{13} - 5 q^{14} + q^{15} - 13 q^{16} + 4 q^{17} + 20 q^{18} + 4 q^{19} + 9 q^{20} + 68 q^{21} - 24 q^{23} + 17 q^{24} - 3 q^{25} - 12 q^{26} - 13 q^{27} - 25 q^{28} + 2 q^{29} - 5 q^{30} - 14 q^{31} + 108 q^{32} - 32 q^{34} + q^{35} - 2 q^{36} - 4 q^{37} + 18 q^{38} - 4 q^{39} + 9 q^{40} + 9 q^{41} + 35 q^{42} - 28 q^{43} - 8 q^{45} + 8 q^{46} + 15 q^{47} - 7 q^{48} - 18 q^{49} + q^{50} - 20 q^{51} + 2 q^{52} + 6 q^{53} - 76 q^{54} - 12 q^{56} - 14 q^{57} - 30 q^{58} - 10 q^{59} + 9 q^{60} + 3 q^{61} + 24 q^{62} - 12 q^{63} - 29 q^{64} + 24 q^{65} + 76 q^{67} + 8 q^{69} + 5 q^{70} - 6 q^{71} + 48 q^{72} - 12 q^{73} - 28 q^{74} - q^{75} - 64 q^{76} - 8 q^{78} + 2 q^{79} + 13 q^{80} - 3 q^{81} + 27 q^{82} + 18 q^{83} - 31 q^{84} - 4 q^{85} + 3 q^{86} - 40 q^{87} + 44 q^{89} - 20 q^{90} - 20 q^{91} + 34 q^{92} - 20 q^{93} - 59 q^{94} - 4 q^{95} - 7 q^{96} + 2 q^{97} - 144 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 6 x^{10} - 12 x^{9} + 43 x^{8} + 72 x^{7} + 155 x^{6} + 162 x^{5} + 541 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\nu^{10} + 2713\nu^{5} + 5190 ) / 24631 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\nu^{10} + 7669\nu^{5} - 67611 ) / 24631 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\nu^{11} + 7669\nu^{6} - 116873\nu ) / 24631 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 141\nu^{11} + 25720\nu^{6} - 320798\nu ) / 24631 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43 \nu^{11} - 258 \nu^{10} + 516 \nu^{9} - 1849 \nu^{8} + 4573 \nu^{7} - 6665 \nu^{6} - 6966 \nu^{5} + \cdots - 43 ) / 24631 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 227 \nu^{11} + 1362 \nu^{10} - 2724 \nu^{9} + 9761 \nu^{8} - 24714 \nu^{7} + 35185 \nu^{6} + \cdots + 227 ) / 24631 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1092 \nu^{11} - 1092 \nu^{10} + 6521 \nu^{9} - 13104 \nu^{8} + 46956 \nu^{7} + 78624 \nu^{6} + \cdots + 5460 ) / 24631 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 638 \nu^{11} + 3828 \nu^{10} - 7656 \nu^{9} + 27434 \nu^{8} - 69569 \nu^{7} + 98890 \nu^{6} + \cdots + 638 ) / 24631 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5460 \nu^{11} + 6552 \nu^{10} - 33852 \nu^{9} + 72041 \nu^{8} - 247884 \nu^{7} - 346164 \nu^{6} + \cdots - 1092 ) / 24631 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15030 \nu^{11} - 18036 \nu^{10} + 93186 \nu^{9} - 198446 \nu^{8} + 682362 \nu^{7} + 952902 \nu^{6} + \cdots + 3006 ) / 24631 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + 3\beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 4\beta_{10} + 6\beta_{8} + 6\beta_{6} - 6\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{11} + 19\beta_{10} + 6\beta_{9} + 12\beta_{8} - 19\beta_{7} - 6\beta_{5} + 19\beta_{4} - 6\beta_{3} - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{3} + 43\beta_{2} - 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 43\beta_{5} - 141\beta_{4} - 109\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 109\beta_{9} - 370\beta_{7} - 336\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -336\beta_{11} - 1117\beta_{10} - 924\beta_{8} - 924\beta_{6} + 924\beta_{2} + 924\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 924 \beta_{11} - 3108 \beta_{10} - 924 \beta_{9} - 2713 \beta_{8} + 3108 \beta_{7} + 924 \beta_{5} + \cdots + 3108 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2713\beta_{3} - 7669\beta_{2} + 9063 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -7669\beta_{5} + 25720\beta_{4} + 22158\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4} - \beta_{7} + \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.0651271 − 0.200441i
0.511560 − 1.57442i
−0.885704 + 2.72592i
2.31880 − 1.68471i
−1.33928 + 0.973045i
−0.170505 + 0.123879i
0.0651271 + 0.200441i
0.511560 + 1.57442i
−0.885704 − 2.72592i
2.31880 + 1.68471i
−1.33928 − 0.973045i
−0.170505 − 0.123879i
−2.22061 − 1.61337i −0.0651271 + 0.200441i 1.71012 + 5.26321i 0.809017 − 0.587785i 0.468007 − 0.340027i 0.717944 + 2.20960i 2.99759 − 9.22564i 2.39112 + 1.73725i −2.74483
81.2 1.12933 + 0.820508i −0.511560 + 1.57442i −0.0158755 − 0.0488598i 0.809017 − 0.587785i −1.86955 + 1.35830i −1.45449 − 4.47645i 0.884894 − 2.72343i 0.209948 + 0.152536i 1.39593
81.3 1.90030 + 1.38065i 0.885704 − 2.72592i 1.08691 + 3.34515i 0.809017 − 0.587785i 5.44662 − 3.95720i 1.04556 + 3.21790i −1.10133 + 3.38955i −4.21910 − 3.06535i 2.34889
251.1 −0.725848 + 2.23393i −2.31880 + 1.68471i −2.84556 − 2.06742i −0.309017 − 0.951057i −2.08042 − 6.40289i −2.73731 − 1.98877i 2.88333 − 2.09486i 1.61155 − 4.95985i 2.34889
251.2 −0.431367 + 1.32761i 1.33928 − 0.973045i 0.0415626 + 0.0301970i −0.309017 − 0.951057i 0.714103 + 2.19778i 3.80789 + 2.76660i −2.31668 + 1.68317i −0.0801932 + 0.246809i 1.39593
251.3 0.848198 − 2.61048i 0.170505 − 0.123879i −4.47716 − 3.25284i −0.309017 − 0.951057i −0.178763 − 0.550175i −1.87960 − 1.36561i −7.84779 + 5.70176i −0.913325 + 2.81093i −2.74483
366.1 −2.22061 + 1.61337i −0.0651271 − 0.200441i 1.71012 − 5.26321i 0.809017 + 0.587785i 0.468007 + 0.340027i 0.717944 − 2.20960i 2.99759 + 9.22564i 2.39112 − 1.73725i −2.74483
366.2 1.12933 − 0.820508i −0.511560 − 1.57442i −0.0158755 + 0.0488598i 0.809017 + 0.587785i −1.86955 − 1.35830i −1.45449 + 4.47645i 0.884894 + 2.72343i 0.209948 − 0.152536i 1.39593
366.3 1.90030 − 1.38065i 0.885704 + 2.72592i 1.08691 − 3.34515i 0.809017 + 0.587785i 5.44662 + 3.95720i 1.04556 − 3.21790i −1.10133 − 3.38955i −4.21910 + 3.06535i 2.34889
511.1 −0.725848 − 2.23393i −2.31880 − 1.68471i −2.84556 + 2.06742i −0.309017 + 0.951057i −2.08042 + 6.40289i −2.73731 + 1.98877i 2.88333 + 2.09486i 1.61155 + 4.95985i 2.34889
511.2 −0.431367 − 1.32761i 1.33928 + 0.973045i 0.0415626 − 0.0301970i −0.309017 + 0.951057i 0.714103 − 2.19778i 3.80789 − 2.76660i −2.31668 − 1.68317i −0.0801932 − 0.246809i 1.39593
511.3 0.848198 + 2.61048i 0.170505 + 0.123879i −4.47716 + 3.25284i −0.309017 + 0.951057i −0.178763 + 0.550175i −1.87960 + 1.36561i −7.84779 − 5.70176i −0.913325 − 2.81093i −2.74483
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.p 12
11.b odd 2 1 605.2.g.o 12
11.c even 5 1 605.2.a.g ✓ 3
11.c even 5 3 inner 605.2.g.p 12
11.d odd 10 1 605.2.a.h yes 3
11.d odd 10 3 605.2.g.o 12
33.f even 10 1 5445.2.a.bb 3
33.h odd 10 1 5445.2.a.bd 3
44.g even 10 1 9680.2.a.cb 3
44.h odd 10 1 9680.2.a.bz 3
55.h odd 10 1 3025.2.a.p 3
55.j even 10 1 3025.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.g ✓ 3 11.c even 5 1
605.2.a.h yes 3 11.d odd 10 1
605.2.g.o 12 11.b odd 2 1
605.2.g.o 12 11.d odd 10 3
605.2.g.p 12 1.a even 1 1 trivial
605.2.g.p 12 11.c even 5 3 inner
3025.2.a.p 3 55.h odd 10 1
3025.2.a.u 3 55.j even 10 1
5445.2.a.bb 3 33.f even 10 1
5445.2.a.bd 3 33.h odd 10 1
9680.2.a.bz 3 44.h odd 10 1
9680.2.a.cb 3 44.g even 10 1

Hecke kernels

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

\( T_{2}^{12} - T_{2}^{11} + 8 T_{2}^{10} - 6 T_{2}^{9} + 53 T_{2}^{8} - 102 T_{2}^{7} + 419 T_{2}^{6} + \cdots + 6561 \) Copy content Toggle raw display
\( T_{3}^{12} + T_{3}^{11} + 6 T_{3}^{10} + 12 T_{3}^{9} + 43 T_{3}^{8} - 72 T_{3}^{7} + 155 T_{3}^{6} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 6561 \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 1874161 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} + \cdots + 5308416 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots + 1679616 \) Copy content Toggle raw display
$23$ \( (T^{3} + 6 T^{2} + 4 T - 12)^{4} \) Copy content Toggle raw display
$29$ \( T^{12} - 2 T^{11} + \cdots + 26873856 \) Copy content Toggle raw display
$31$ \( T^{12} + 14 T^{11} + \cdots + 1679616 \) Copy content Toggle raw display
$37$ \( T^{12} + 4 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 7780827681 \) Copy content Toggle raw display
$43$ \( (T^{3} + 7 T^{2} - 3 T - 63)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 411651843201 \) Copy content Toggle raw display
$53$ \( T^{12} - 6 T^{11} + \cdots + 20736 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14666178816 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 713283282721 \) Copy content Toggle raw display
$67$ \( (T^{3} - 19 T^{2} + \cdots - 59)^{4} \) Copy content Toggle raw display
$71$ \( T^{12} + 6 T^{11} + \cdots + 20736 \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots + 1048576 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 7676563456 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 11019960576 \) Copy content Toggle raw display
$89$ \( (T^{3} - 11 T^{2} + \cdots + 1719)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 754507653376 \) Copy content Toggle raw display
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