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 $1$

Related objects

Downloads

Learn more about

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

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

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

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} \)
$3$ \( 1 - 2 T + 27 T^{2} \)
$5$ \( 1 + 5 T \)
$7$ \( 1 + 6 T + 343 T^{2} \)
$11$ 1
$13$ \( 1 - 38 T + 2197 T^{2} \)
$17$ \( 1 + 26 T + 4913 T^{2} \)
$19$ \( 1 + 100 T + 6859 T^{2} \)
$23$ \( 1 + 78 T + 12167 T^{2} \)
$29$ \( 1 - 50 T + 24389 T^{2} \)
$31$ \( 1 + 108 T + 29791 T^{2} \)
$37$ \( 1 - 266 T + 50653 T^{2} \)
$41$ \( 1 + 22 T + 68921 T^{2} \)
$43$ \( 1 + 442 T + 79507 T^{2} \)
$47$ \( 1 + 514 T + 103823 T^{2} \)
$53$ \( 1 - 2 T + 148877 T^{2} \)
$59$ \( 1 - 500 T + 205379 T^{2} \)
$61$ \( 1 - 518 T + 226981 T^{2} \)
$67$ \( 1 - 126 T + 300763 T^{2} \)
$71$ \( 1 - 412 T + 357911 T^{2} \)
$73$ \( 1 - 878 T + 389017 T^{2} \)
$79$ \( 1 + 600 T + 493039 T^{2} \)
$83$ \( 1 + 282 T + 571787 T^{2} \)
$89$ \( 1 + 150 T + 704969 T^{2} \)
$97$ \( 1 - 386 T + 912673 T^{2} \)
show more
show less