Properties

Label 605.4.a.d
Level $605$
Weight $4$
Character orbit 605.a
Self dual yes
Analytic conductor $35.696$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.6961555535\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} + 8 q^{6} - 6 q^{7} - 23 q^{9} + O(q^{10}) \) \( q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} + 8 q^{6} - 6 q^{7} - 23 q^{9} - 20 q^{10} + 16 q^{12} + 38 q^{13} - 24 q^{14} - 10 q^{15} - 64 q^{16} - 26 q^{17} - 92 q^{18} - 100 q^{19} - 40 q^{20} - 12 q^{21} - 78 q^{23} + 25 q^{25} + 152 q^{26} - 100 q^{27} - 48 q^{28} + 50 q^{29} - 40 q^{30} - 108 q^{31} - 256 q^{32} - 104 q^{34} + 30 q^{35} - 184 q^{36} + 266 q^{37} - 400 q^{38} + 76 q^{39} - 22 q^{41} - 48 q^{42} - 442 q^{43} + 115 q^{45} - 312 q^{46} - 514 q^{47} - 128 q^{48} - 307 q^{49} + 100 q^{50} - 52 q^{51} + 304 q^{52} + 2 q^{53} - 400 q^{54} - 200 q^{57} + 200 q^{58} + 500 q^{59} - 80 q^{60} + 518 q^{61} - 432 q^{62} + 138 q^{63} - 512 q^{64} - 190 q^{65} + 126 q^{67} - 208 q^{68} - 156 q^{69} + 120 q^{70} + 412 q^{71} + 878 q^{73} + 1064 q^{74} + 50 q^{75} - 800 q^{76} + 304 q^{78} - 600 q^{79} + 320 q^{80} + 421 q^{81} - 88 q^{82} - 282 q^{83} - 96 q^{84} + 130 q^{85} - 1768 q^{86} + 100 q^{87} - 150 q^{89} + 460 q^{90} - 228 q^{91} - 624 q^{92} - 216 q^{93} - 2056 q^{94} + 500 q^{95} - 512 q^{96} + 386 q^{97} - 1228 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
4.00000 2.00000 8.00000 −5.00000 8.00000 −6.00000 0 −23.0000 −20.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.4.a.d 1
11.b odd 2 1 5.4.a.a 1
33.d even 2 1 45.4.a.d 1
44.c even 2 1 80.4.a.d 1
55.d odd 2 1 25.4.a.c 1
55.e even 4 2 25.4.b.a 2
77.b even 2 1 245.4.a.a 1
77.h odd 6 2 245.4.e.f 2
77.i even 6 2 245.4.e.g 2
88.b odd 2 1 320.4.a.g 1
88.g even 2 1 320.4.a.h 1
99.g even 6 2 405.4.e.c 2
99.h odd 6 2 405.4.e.l 2
132.d odd 2 1 720.4.a.u 1
143.d odd 2 1 845.4.a.b 1
165.d even 2 1 225.4.a.b 1
165.l odd 4 2 225.4.b.c 2
176.i even 4 2 1280.4.d.l 2
176.l odd 4 2 1280.4.d.e 2
187.b odd 2 1 1445.4.a.a 1
209.d even 2 1 1805.4.a.h 1
220.g even 2 1 400.4.a.m 1
220.i odd 4 2 400.4.c.k 2
231.h odd 2 1 2205.4.a.q 1
385.h even 2 1 1225.4.a.k 1
440.c even 2 1 1600.4.a.s 1
440.o odd 2 1 1600.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 11.b odd 2 1
25.4.a.c 1 55.d odd 2 1
25.4.b.a 2 55.e even 4 2
45.4.a.d 1 33.d even 2 1
80.4.a.d 1 44.c even 2 1
225.4.a.b 1 165.d even 2 1
225.4.b.c 2 165.l odd 4 2
245.4.a.a 1 77.b even 2 1
245.4.e.f 2 77.h odd 6 2
245.4.e.g 2 77.i even 6 2
320.4.a.g 1 88.b odd 2 1
320.4.a.h 1 88.g even 2 1
400.4.a.m 1 220.g even 2 1
400.4.c.k 2 220.i odd 4 2
405.4.e.c 2 99.g even 6 2
405.4.e.l 2 99.h odd 6 2
605.4.a.d 1 1.a even 1 1 trivial
720.4.a.u 1 132.d odd 2 1
845.4.a.b 1 143.d odd 2 1
1225.4.a.k 1 385.h even 2 1
1280.4.d.e 2 176.l odd 4 2
1280.4.d.l 2 176.i even 4 2
1445.4.a.a 1 187.b odd 2 1
1600.4.a.s 1 440.c even 2 1
1600.4.a.bi 1 440.o odd 2 1
1805.4.a.h 1 209.d even 2 1
2205.4.a.q 1 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2} - 4 \)
\( T_{3} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -2 + T \)
$5$ \( 5 + T \)
$7$ \( 6 + T \)
$11$ \( T \)
$13$ \( -38 + T \)
$17$ \( 26 + T \)
$19$ \( 100 + T \)
$23$ \( 78 + T \)
$29$ \( -50 + T \)
$31$ \( 108 + T \)
$37$ \( -266 + T \)
$41$ \( 22 + T \)
$43$ \( 442 + T \)
$47$ \( 514 + T \)
$53$ \( -2 + T \)
$59$ \( -500 + T \)
$61$ \( -518 + T \)
$67$ \( -126 + T \)
$71$ \( -412 + T \)
$73$ \( -878 + T \)
$79$ \( 600 + T \)
$83$ \( 282 + T \)
$89$ \( 150 + T \)
$97$ \( -386 + T \)
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