Properties

Label 325.2.c.e
Level $325$
Weight $2$
Character orbit 325.c
Analytic conductor $2.595$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + 2 q^{3} + q^{4} + 2 i q^{6} + 3 i q^{8} + q^{9} +O(q^{10})\) \( q + i q^{2} + 2 q^{3} + q^{4} + 2 i q^{6} + 3 i q^{8} + q^{9} -2 i q^{11} + 2 q^{12} + ( -2 + 3 i ) q^{13} - q^{16} + i q^{18} -6 i q^{19} + 2 q^{22} + 6 q^{23} + 6 i q^{24} + ( -3 - 2 i ) q^{26} -4 q^{27} -6 q^{29} -6 i q^{31} + 5 i q^{32} -4 i q^{33} + q^{36} -6 i q^{37} + 6 q^{38} + ( -4 + 6 i ) q^{39} + 8 i q^{41} -6 q^{43} -2 i q^{44} + 6 i q^{46} + 8 i q^{47} -2 q^{48} + 7 q^{49} + ( -2 + 3 i ) q^{52} -12 q^{53} -4 i q^{54} -12 i q^{57} -6 i q^{58} + 2 i q^{59} + 6 q^{61} + 6 q^{62} -7 q^{64} + 4 q^{66} -12 i q^{67} + 12 q^{69} + 2 i q^{71} + 3 i q^{72} -6 i q^{73} + 6 q^{74} -6 i q^{76} + ( -6 - 4 i ) q^{78} -11 q^{81} -8 q^{82} -4 i q^{83} -6 i q^{86} -12 q^{87} + 6 q^{88} -8 i q^{89} + 6 q^{92} -12 i q^{93} -8 q^{94} + 10 i q^{96} + 6 i q^{97} + 7 i q^{98} -2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} + 4 q^{12} - 4 q^{13} - 2 q^{16} + 4 q^{22} + 12 q^{23} - 6 q^{26} - 8 q^{27} - 12 q^{29} + 2 q^{36} + 12 q^{38} - 8 q^{39} - 12 q^{43} - 4 q^{48} + 14 q^{49} - 4 q^{52} - 24 q^{53} + 12 q^{61} + 12 q^{62} - 14 q^{64} + 8 q^{66} + 24 q^{69} + 12 q^{74} - 12 q^{78} - 22 q^{81} - 16 q^{82} - 24 q^{87} + 12 q^{88} + 12 q^{92} - 16 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.00000i
1.00000i
1.00000i 2.00000 1.00000 0 2.00000i 0 3.00000i 1.00000 0
51.2 1.00000i 2.00000 1.00000 0 2.00000i 0 3.00000i 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.c.e 2
5.b even 2 1 325.2.c.b 2
5.c odd 4 1 65.2.d.a 2
5.c odd 4 1 65.2.d.b yes 2
13.b even 2 1 inner 325.2.c.e 2
13.d odd 4 1 4225.2.a.h 1
13.d odd 4 1 4225.2.a.m 1
15.e even 4 1 585.2.h.b 2
15.e even 4 1 585.2.h.c 2
20.e even 4 1 1040.2.f.a 2
20.e even 4 1 1040.2.f.b 2
65.d even 2 1 325.2.c.b 2
65.f even 4 1 845.2.b.a 2
65.f even 4 1 845.2.b.b 2
65.g odd 4 1 4225.2.a.e 1
65.g odd 4 1 4225.2.a.k 1
65.h odd 4 1 65.2.d.a 2
65.h odd 4 1 65.2.d.b yes 2
65.k even 4 1 845.2.b.a 2
65.k even 4 1 845.2.b.b 2
65.o even 12 2 845.2.n.a 4
65.o even 12 2 845.2.n.b 4
65.q odd 12 2 845.2.l.a 4
65.q odd 12 2 845.2.l.b 4
65.r odd 12 2 845.2.l.a 4
65.r odd 12 2 845.2.l.b 4
65.t even 12 2 845.2.n.a 4
65.t even 12 2 845.2.n.b 4
195.s even 4 1 585.2.h.b 2
195.s even 4 1 585.2.h.c 2
260.p even 4 1 1040.2.f.a 2
260.p even 4 1 1040.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 5.c odd 4 1
65.2.d.a 2 65.h odd 4 1
65.2.d.b yes 2 5.c odd 4 1
65.2.d.b yes 2 65.h odd 4 1
325.2.c.b 2 5.b even 2 1
325.2.c.b 2 65.d even 2 1
325.2.c.e 2 1.a even 1 1 trivial
325.2.c.e 2 13.b even 2 1 inner
585.2.h.b 2 15.e even 4 1
585.2.h.b 2 195.s even 4 1
585.2.h.c 2 15.e even 4 1
585.2.h.c 2 195.s even 4 1
845.2.b.a 2 65.f even 4 1
845.2.b.a 2 65.k even 4 1
845.2.b.b 2 65.f even 4 1
845.2.b.b 2 65.k even 4 1
845.2.l.a 4 65.q odd 12 2
845.2.l.a 4 65.r odd 12 2
845.2.l.b 4 65.q odd 12 2
845.2.l.b 4 65.r odd 12 2
845.2.n.a 4 65.o even 12 2
845.2.n.a 4 65.t even 12 2
845.2.n.b 4 65.o even 12 2
845.2.n.b 4 65.t even 12 2
1040.2.f.a 2 20.e even 4 1
1040.2.f.a 2 260.p even 4 1
1040.2.f.b 2 20.e even 4 1
1040.2.f.b 2 260.p even 4 1
4225.2.a.e 1 65.g odd 4 1
4225.2.a.h 1 13.d odd 4 1
4225.2.a.k 1 65.g odd 4 1
4225.2.a.m 1 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{3} - 2 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 13 + 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( 4 + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 4 + T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( 64 + T^{2} \)
$97$ \( 36 + T^{2} \)
show more
show less