Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} + 3 \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} + 3 \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} - q^{10} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{13} + \zeta_{10} q^{16} + 6 \zeta_{10} q^{17} - 3 \zeta_{10}^{3} q^{18} + 4 \zeta_{10}^{2} q^{19} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{20} + 4 q^{23} + \zeta_{10}^{2} q^{25} - 2 \zeta_{10}^{3} q^{26} + 6 \zeta_{10}^{3} q^{29} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{31} - 5 q^{32} + 6 q^{34} + 3 \zeta_{10}^{2} q^{36} + 2 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{38} - 3 \zeta_{10}^{3} q^{40} - 2 \zeta_{10}^{2} q^{41} - 4 q^{43} - 3 q^{45} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{46} - 12 \zeta_{10}^{2} q^{47} + 7 \zeta_{10} q^{49} + \zeta_{10} q^{50} + 2 \zeta_{10}^{2} q^{52} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{53} + 6 \zeta_{10}^{2} q^{58} - 4 \zeta_{10}^{3} q^{59} - 10 \zeta_{10} q^{61} - 8 \zeta_{10}^{3} q^{62} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{64} - 2 q^{65} - 16 q^{67} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{68} - 8 \zeta_{10} q^{71} + 9 \zeta_{10} q^{72} + 14 \zeta_{10}^{3} q^{73} + 2 \zeta_{10}^{2} q^{74} - 4 q^{76} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{79} - \zeta_{10}^{2} q^{80} - 9 \zeta_{10}^{3} q^{81} - 2 \zeta_{10} q^{82} - 4 \zeta_{10} q^{83} - 6 \zeta_{10}^{2} q^{85} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{86} + 10 q^{89} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{90} + 4 \zeta_{10}^{3} q^{92} - 12 \zeta_{10} q^{94} - 4 \zeta_{10}^{3} q^{95} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 10) q^{97} + 7 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{4} - q^{5} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{4} - q^{5} - 3 q^{8} + 3 q^{9} - 4 q^{10} + 2 q^{13} + q^{16} + 6 q^{17} - 3 q^{18} - 4 q^{19} + q^{20} + 16 q^{23} - q^{25} - 2 q^{26} + 6 q^{29} + 8 q^{31} - 20 q^{32} + 24 q^{34} - 3 q^{36} + 2 q^{37} + 4 q^{38} - 3 q^{40} + 2 q^{41} - 16 q^{43} - 12 q^{45} + 4 q^{46} + 12 q^{47} + 7 q^{49} + q^{50} - 2 q^{52} + 2 q^{53} - 6 q^{58} - 4 q^{59} - 10 q^{61} - 8 q^{62} - 7 q^{64} - 8 q^{65} - 64 q^{67} - 6 q^{68} - 8 q^{71} + 9 q^{72} + 14 q^{73} - 2 q^{74} - 16 q^{76} + 8 q^{79} + q^{80} - 9 q^{81} - 2 q^{82} - 4 q^{83} + 6 q^{85} - 4 q^{86} + 40 q^{89} - 3 q^{90} + 4 q^{92} - 12 q^{94} - 4 q^{95} - 10 q^{97} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.809017 0.587785i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i 0 −0.309017 0.951057i −0.809017 + 0.587785i 0 0 0.927051 2.85317i 2.42705 + 1.76336i −1.00000
251.1 −0.309017 + 0.951057i 0 0.809017 + 0.587785i 0.309017 + 0.951057i 0 0 −2.42705 + 1.76336i −0.927051 + 2.85317i −1.00000
366.1 0.809017 0.587785i 0 −0.309017 + 0.951057i −0.809017 0.587785i 0 0 0.927051 + 2.85317i 2.42705 1.76336i −1.00000
511.1 −0.309017 0.951057i 0 0.809017 0.587785i 0.309017 0.951057i 0 0 −2.42705 1.76336i −0.927051 2.85317i −1.00000
\(n\): e.g. 2-40 or 990-1000
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.c 4
11.b odd 2 1 605.2.g.a 4
11.c even 5 1 605.2.a.b 1
11.c even 5 3 inner 605.2.g.c 4
11.d odd 10 1 55.2.a.a 1
11.d odd 10 3 605.2.g.a 4
33.f even 10 1 495.2.a.a 1
33.h odd 10 1 5445.2.a.i 1
44.g even 10 1 880.2.a.h 1
44.h odd 10 1 9680.2.a.r 1
55.h odd 10 1 275.2.a.a 1
55.j even 10 1 3025.2.a.f 1
55.l even 20 2 275.2.b.b 2
77.l even 10 1 2695.2.a.c 1
88.k even 10 1 3520.2.a.n 1
88.p odd 10 1 3520.2.a.p 1
132.n odd 10 1 7920.2.a.i 1
143.l odd 10 1 9295.2.a.b 1
165.r even 10 1 2475.2.a.i 1
165.u odd 20 2 2475.2.c.f 2
220.o even 10 1 4400.2.a.p 1
220.w odd 20 2 4400.2.b.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 11.d odd 10 1
275.2.a.a 1 55.h odd 10 1
275.2.b.b 2 55.l even 20 2
495.2.a.a 1 33.f even 10 1
605.2.a.b 1 11.c even 5 1
605.2.g.a 4 11.b odd 2 1
605.2.g.a 4 11.d odd 10 3
605.2.g.c 4 1.a even 1 1 trivial
605.2.g.c 4 11.c even 5 3 inner
880.2.a.h 1 44.g even 10 1
2475.2.a.i 1 165.r even 10 1
2475.2.c.f 2 165.u odd 20 2
2695.2.a.c 1 77.l even 10 1
3025.2.a.f 1 55.j even 10 1
3520.2.a.n 1 88.k even 10 1
3520.2.a.p 1 88.p odd 10 1
4400.2.a.p 1 220.o even 10 1
4400.2.b.n 2 220.w odd 20 2
5445.2.a.i 1 33.h odd 10 1
7920.2.a.i 1 132.n odd 10 1
9295.2.a.b 1 143.l odd 10 1
9680.2.a.r 1 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 \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( (T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
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