Properties

Label 1445.4.a.a
Level $1445$
Weight $4$
Character orbit 1445.a
Self dual yes
Analytic conductor $85.258$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(85.2577599583\)
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

\(f(q)\) \(=\) \( q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} - 6 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} - 6 q^{7} - 23 q^{9} - 20 q^{10} - 32 q^{11} - 16 q^{12} - 38 q^{13} + 24 q^{14} - 10 q^{15} - 64 q^{16} + 92 q^{18} + 100 q^{19} + 40 q^{20} + 12 q^{21} + 128 q^{22} + 78 q^{23} + 25 q^{25} + 152 q^{26} + 100 q^{27} - 48 q^{28} + 50 q^{29} + 40 q^{30} + 108 q^{31} + 256 q^{32} + 64 q^{33} - 30 q^{35} - 184 q^{36} - 266 q^{37} - 400 q^{38} + 76 q^{39} - 22 q^{41} - 48 q^{42} + 442 q^{43} - 256 q^{44} - 115 q^{45} - 312 q^{46} - 514 q^{47} + 128 q^{48} - 307 q^{49} - 100 q^{50} - 304 q^{52} + 2 q^{53} - 400 q^{54} - 160 q^{55} - 200 q^{57} - 200 q^{58} + 500 q^{59} - 80 q^{60} + 518 q^{61} - 432 q^{62} + 138 q^{63} - 512 q^{64} - 190 q^{65} - 256 q^{66} + 126 q^{67} - 156 q^{69} + 120 q^{70} - 412 q^{71} + 878 q^{73} + 1064 q^{74} - 50 q^{75} + 800 q^{76} + 192 q^{77} - 304 q^{78} - 600 q^{79} - 320 q^{80} + 421 q^{81} + 88 q^{82} + 282 q^{83} + 96 q^{84} - 1768 q^{86} - 100 q^{87} - 150 q^{89} + 460 q^{90} + 228 q^{91} + 624 q^{92} - 216 q^{93} + 2056 q^{94} + 500 q^{95} - 512 q^{96} - 386 q^{97} + 1228 q^{98} + 736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) \(-1\)
\(17\) \(1\)

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 1445.4.a.a 1
17.b even 2 1 5.4.a.a 1
51.c odd 2 1 45.4.a.d 1
68.d odd 2 1 80.4.a.d 1
85.c even 2 1 25.4.a.c 1
85.g odd 4 2 25.4.b.a 2
119.d odd 2 1 245.4.a.a 1
119.h odd 6 2 245.4.e.g 2
119.j even 6 2 245.4.e.f 2
136.e odd 2 1 320.4.a.h 1
136.h even 2 1 320.4.a.g 1
153.h even 6 2 405.4.e.l 2
153.i odd 6 2 405.4.e.c 2
187.b odd 2 1 605.4.a.d 1
204.h even 2 1 720.4.a.u 1
221.b even 2 1 845.4.a.b 1
255.h odd 2 1 225.4.a.b 1
255.o even 4 2 225.4.b.c 2
272.k odd 4 2 1280.4.d.l 2
272.r even 4 2 1280.4.d.e 2
323.c odd 2 1 1805.4.a.h 1
340.d odd 2 1 400.4.a.m 1
340.r even 4 2 400.4.c.k 2
357.c even 2 1 2205.4.a.q 1
595.b odd 2 1 1225.4.a.k 1
680.h even 2 1 1600.4.a.bi 1
680.k odd 2 1 1600.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 17.b even 2 1
25.4.a.c 1 85.c even 2 1
25.4.b.a 2 85.g odd 4 2
45.4.a.d 1 51.c odd 2 1
80.4.a.d 1 68.d odd 2 1
225.4.a.b 1 255.h odd 2 1
225.4.b.c 2 255.o even 4 2
245.4.a.a 1 119.d odd 2 1
245.4.e.f 2 119.j even 6 2
245.4.e.g 2 119.h odd 6 2
320.4.a.g 1 136.h even 2 1
320.4.a.h 1 136.e odd 2 1
400.4.a.m 1 340.d odd 2 1
400.4.c.k 2 340.r even 4 2
405.4.e.c 2 153.i odd 6 2
405.4.e.l 2 153.h even 6 2
605.4.a.d 1 187.b odd 2 1
720.4.a.u 1 204.h even 2 1
845.4.a.b 1 221.b even 2 1
1225.4.a.k 1 595.b odd 2 1
1280.4.d.e 2 272.r even 4 2
1280.4.d.l 2 272.k odd 4 2
1445.4.a.a 1 1.a even 1 1 trivial
1600.4.a.s 1 680.k odd 2 1
1600.4.a.bi 1 680.h even 2 1
1805.4.a.h 1 323.c odd 2 1
2205.4.a.q 1 357.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 6 \) Copy content Toggle raw display
$11$ \( T + 32 \) Copy content Toggle raw display
$13$ \( T + 38 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 100 \) Copy content Toggle raw display
$23$ \( T - 78 \) Copy content Toggle raw display
$29$ \( T - 50 \) Copy content Toggle raw display
$31$ \( T - 108 \) Copy content Toggle raw display
$37$ \( T + 266 \) Copy content Toggle raw display
$41$ \( T + 22 \) Copy content Toggle raw display
$43$ \( T - 442 \) Copy content Toggle raw display
$47$ \( T + 514 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 500 \) Copy content Toggle raw display
$61$ \( T - 518 \) Copy content Toggle raw display
$67$ \( T - 126 \) Copy content Toggle raw display
$71$ \( T + 412 \) Copy content Toggle raw display
$73$ \( T - 878 \) Copy content Toggle raw display
$79$ \( T + 600 \) Copy content Toggle raw display
$83$ \( T - 282 \) Copy content Toggle raw display
$89$ \( T + 150 \) Copy content Toggle raw display
$97$ \( T + 386 \) Copy content Toggle raw display
show more
show less