Properties

Label 605.2.g.l
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{2}) q^{2} + 2 \beta_1 q^{3} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{5} + ( - 4 \beta_{6} - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4) q^{6} + 2 \beta_{4} q^{7} + ( - 3 \beta_{6} + \beta_1) q^{8} + 5 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{2}) q^{2} + 2 \beta_1 q^{3} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{5} + ( - 4 \beta_{6} - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4) q^{6} + 2 \beta_{4} q^{7} + ( - 3 \beta_{6} + \beta_1) q^{8} + 5 \beta_{2} q^{9} + ( - \beta_{5} + 1) q^{10} + (2 \beta_{5} - 8) q^{12} + (2 \beta_{7} + 4 \beta_{2}) q^{13} + ( - 2 \beta_{6} + 2 \beta_1) q^{14} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{15} + ( - 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 3) q^{16} + (4 \beta_{6} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{17} + ( - 5 \beta_{7} - 5 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} - 5 \beta_1) q^{18} + (2 \beta_{7} - \beta_{2}) q^{20} + 4 \beta_{5} q^{21} + 2 \beta_{5} q^{23} + ( - 6 \beta_{7} + 4 \beta_{2}) q^{24} + \beta_{6} q^{25} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{26} + 4 \beta_{3} q^{27} + ( - 2 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 2) q^{28} + (4 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 4 \beta_1) q^{29} + ( - 4 \beta_{6} + 2 \beta_1) q^{30} + (\beta_{5} + 3) q^{32} + ( - 6 \beta_{5} + 8) q^{34} - 2 \beta_{2} q^{35} + (5 \beta_{6} - 10 \beta_1) q^{36} + ( - 4 \beta_{7} - 4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_1) q^{37} + ( - 8 \beta_{6} - 8 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} - 8) q^{39} + (\beta_{7} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1) q^{40} - 6 \beta_{6} q^{41} + ( - 4 \beta_{7} + 8 \beta_{2}) q^{42} + 6 q^{43} - 5 q^{45} + ( - 2 \beta_{7} + 4 \beta_{2}) q^{46} - 2 \beta_1 q^{47} + ( - 6 \beta_{7} - 6 \beta_{5} - 6 \beta_{3} - 6 \beta_1) q^{48} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{49} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{50} + (8 \beta_{7} + 8 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} + 8 \beta_1) q^{51} + ( - 4 \beta_{6} - 6 \beta_1) q^{52} + ( - 4 \beta_{7} + 6 \beta_{2}) q^{53} + ( - 4 \beta_{5} + 8) q^{54} + (2 \beta_{5} - 6) q^{56} + ( - 6 \beta_{6} + 2 \beta_1) q^{58} + (4 \beta_{7} + 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_1) q^{59} + ( - 8 \beta_{6} - 8 \beta_{4} - 2 \beta_{3} - 8 \beta_{2} - 8) q^{60} + (2 \beta_{6} + 2 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 2) q^{61} + 10 \beta_{6} q^{63} + (2 \beta_{7} - 7 \beta_{2}) q^{64} + ( - 2 \beta_{5} - 4) q^{65} + ( - 6 \beta_{5} + 4) q^{67} + (10 \beta_{7} - 12 \beta_{2}) q^{68} + 8 \beta_{6} q^{69} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{70} - 8 \beta_{3} q^{71} + (15 \beta_{6} + 15 \beta_{4} + 5 \beta_{3} + 15 \beta_{2} + 15) q^{72} + ( - 2 \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{73} + (10 \beta_{6} - 6 \beta_1) q^{74} + 2 \beta_{7} q^{75} + 8 q^{78} - 4 \beta_{2} q^{79} - 3 \beta_{6} q^{80} + \beta_{4} q^{81} + ( - 6 \beta_{6} - 6 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} - 6) q^{82} + ( - 6 \beta_{6} - 6 \beta_{4} - 6 \beta_{2} - 6) q^{83} + ( - 4 \beta_{7} - 4 \beta_{5} - 16 \beta_{4} - 4 \beta_{3} - 4 \beta_1) q^{84} + (4 \beta_{6} - 2 \beta_1) q^{85} + (6 \beta_{7} - 6 \beta_{2}) q^{86} + ( - 4 \beta_{5} - 16) q^{87} + (8 \beta_{5} - 2) q^{89} + ( - 5 \beta_{7} + 5 \beta_{2}) q^{90} + (8 \beta_{6} + 4 \beta_1) q^{91} + ( - 2 \beta_{7} - 2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{92} + (4 \beta_{6} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{94} + (4 \beta_{6} + 6 \beta_1) q^{96} + ( - 4 \beta_{7} - 2 \beta_{2}) q^{97} + ( - 3 \beta_{5} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} - 10 q^{9} + 8 q^{10} - 64 q^{12} - 8 q^{13} + 4 q^{14} - 6 q^{16} + 8 q^{17} + 10 q^{18} + 2 q^{20} - 8 q^{24} - 2 q^{25} - 4 q^{28} + 4 q^{29} + 8 q^{30} + 24 q^{32} + 64 q^{34} + 4 q^{35} - 10 q^{36} + 4 q^{37} - 16 q^{39} - 6 q^{40} + 12 q^{41} - 16 q^{42} + 48 q^{43} - 40 q^{45} - 8 q^{46} + 6 q^{49} + 2 q^{50} - 16 q^{51} + 8 q^{52} - 12 q^{53} + 64 q^{54} - 48 q^{56} + 12 q^{58} + 8 q^{59} - 16 q^{60} + 4 q^{61} - 20 q^{63} + 14 q^{64} - 32 q^{65} + 32 q^{67} + 24 q^{68} - 16 q^{69} - 4 q^{70} + 30 q^{72} - 8 q^{73} - 20 q^{74} + 64 q^{78} + 8 q^{79} + 6 q^{80} - 2 q^{81} - 12 q^{82} - 12 q^{83} + 32 q^{84} - 8 q^{85} + 12 q^{86} - 128 q^{87} - 16 q^{89} - 10 q^{90} - 16 q^{91} + 16 q^{92} + 8 q^{94} - 8 q^{96} + 4 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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
0.437016 1.34500i
−0.437016 + 1.34500i
1.14412 0.831254i
−1.14412 + 0.831254i
0.437016 + 1.34500i
−0.437016 1.34500i
1.14412 + 0.831254i
−1.14412 0.831254i
−0.335106 0.243469i 0.874032 2.68999i −0.565015 1.73894i 0.809017 0.587785i −0.947822 + 0.688633i 0.618034 + 1.90211i −0.490035 + 1.50817i −4.04508 2.93893i −0.414214
81.2 1.95314 + 1.41904i −0.874032 + 2.68999i 1.18305 + 3.64105i 0.809017 0.587785i −5.52431 + 4.01365i 0.618034 + 1.90211i −1.36407 + 4.19817i −4.04508 2.93893i 2.41421
251.1 −0.746033 + 2.29605i 2.28825 1.66251i −3.09726 2.25029i −0.309017 0.951057i 2.11010 + 6.49422i −1.61803 1.17557i 3.57117 2.59461i 1.54508 4.75528i 2.41421
251.2 0.127999 0.393941i −2.28825 + 1.66251i 1.47923 + 1.07472i −0.309017 0.951057i 0.362036 + 1.11423i −1.61803 1.17557i 1.28293 0.932102i 1.54508 4.75528i −0.414214
366.1 −0.335106 + 0.243469i 0.874032 + 2.68999i −0.565015 + 1.73894i 0.809017 + 0.587785i −0.947822 0.688633i 0.618034 1.90211i −0.490035 1.50817i −4.04508 + 2.93893i −0.414214
366.2 1.95314 1.41904i −0.874032 2.68999i 1.18305 3.64105i 0.809017 + 0.587785i −5.52431 4.01365i 0.618034 1.90211i −1.36407 4.19817i −4.04508 + 2.93893i 2.41421
511.1 −0.746033 2.29605i 2.28825 + 1.66251i −3.09726 + 2.25029i −0.309017 + 0.951057i 2.11010 6.49422i −1.61803 + 1.17557i 3.57117 + 2.59461i 1.54508 + 4.75528i 2.41421
511.2 0.127999 + 0.393941i −2.28825 1.66251i 1.47923 1.07472i −0.309017 + 0.951057i 0.362036 1.11423i −1.61803 + 1.17557i 1.28293 + 0.932102i 1.54508 + 4.75528i −0.414214
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
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.l 8
11.b odd 2 1 605.2.g.f 8
11.c even 5 1 605.2.a.d 2
11.c even 5 3 inner 605.2.g.l 8
11.d odd 10 1 55.2.a.b 2
11.d odd 10 3 605.2.g.f 8
33.f even 10 1 495.2.a.b 2
33.h odd 10 1 5445.2.a.y 2
44.g even 10 1 880.2.a.m 2
44.h odd 10 1 9680.2.a.bn 2
55.h odd 10 1 275.2.a.c 2
55.j even 10 1 3025.2.a.o 2
55.l even 20 2 275.2.b.d 4
77.l even 10 1 2695.2.a.f 2
88.k even 10 1 3520.2.a.bo 2
88.p odd 10 1 3520.2.a.bn 2
132.n odd 10 1 7920.2.a.ch 2
143.l odd 10 1 9295.2.a.g 2
165.r even 10 1 2475.2.a.x 2
165.u odd 20 2 2475.2.c.l 4
220.o even 10 1 4400.2.a.bn 2
220.w odd 20 2 4400.2.b.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 11.d odd 10 1
275.2.a.c 2 55.h odd 10 1
275.2.b.d 4 55.l even 20 2
495.2.a.b 2 33.f even 10 1
605.2.a.d 2 11.c even 5 1
605.2.g.f 8 11.b odd 2 1
605.2.g.f 8 11.d odd 10 3
605.2.g.l 8 1.a even 1 1 trivial
605.2.g.l 8 11.c even 5 3 inner
880.2.a.m 2 44.g even 10 1
2475.2.a.x 2 165.r even 10 1
2475.2.c.l 4 165.u odd 20 2
2695.2.a.f 2 77.l even 10 1
3025.2.a.o 2 55.j even 10 1
3520.2.a.bn 2 88.p odd 10 1
3520.2.a.bo 2 88.k even 10 1
4400.2.a.bn 2 220.o even 10 1
4400.2.b.q 4 220.w odd 20 2
5445.2.a.y 2 33.h odd 10 1
7920.2.a.ch 2 132.n odd 10 1
9295.2.a.g 2 143.l odd 10 1
9680.2.a.bn 2 44.h odd 10 1

Hecke kernels

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

\( T_{2}^{8} - 2T_{2}^{7} + 5T_{2}^{6} - 12T_{2}^{5} + 29T_{2}^{4} + 12T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 8T_{3}^{6} + 64T_{3}^{4} + 512T_{3}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + 5 T^{6} - 12 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 8 T^{6} + 64 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + 56 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + 56 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + 44 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + 44 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{6} + 64 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + 140 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} + 80 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$61$ \( T^{8} - 4 T^{7} + 140 T^{6} + \cdots + 236421376 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 56)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 128 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + 56 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 6 T^{3} + 36 T^{2} + 216 T + 1296)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 124)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + 44 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
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