Properties

Label 325.4.d.b
Level $325$
Weight $4$
Character orbit 325.d
Analytic conductor $19.176$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 325.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1756207519\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + 3 q^{2} -i q^{3} + q^{4} -3 i q^{6} -15 q^{7} -21 q^{8} + 26 q^{9} +O(q^{10})\) \( q + 3 q^{2} -i q^{3} + q^{4} -3 i q^{6} -15 q^{7} -21 q^{8} + 26 q^{9} + 48 i q^{11} -i q^{12} + ( 39 + 26 i ) q^{13} -45 q^{14} -71 q^{16} + 45 i q^{17} + 78 q^{18} -6 i q^{19} + 15 i q^{21} + 144 i q^{22} + 162 i q^{23} + 21 i q^{24} + ( 117 + 78 i ) q^{26} -53 i q^{27} -15 q^{28} + 144 q^{29} + 264 i q^{31} -45 q^{32} + 48 q^{33} + 135 i q^{34} + 26 q^{36} -303 q^{37} -18 i q^{38} + ( 26 - 39 i ) q^{39} -192 i q^{41} + 45 i q^{42} -97 i q^{43} + 48 i q^{44} + 486 i q^{46} -111 q^{47} + 71 i q^{48} -118 q^{49} + 45 q^{51} + ( 39 + 26 i ) q^{52} -414 i q^{53} -159 i q^{54} + 315 q^{56} -6 q^{57} + 432 q^{58} + 522 i q^{59} + 376 q^{61} + 792 i q^{62} -390 q^{63} + 433 q^{64} + 144 q^{66} -36 q^{67} + 45 i q^{68} + 162 q^{69} + 357 i q^{71} -546 q^{72} -1098 q^{73} -909 q^{74} -6 i q^{76} -720 i q^{77} + ( 78 - 117 i ) q^{78} + 830 q^{79} + 649 q^{81} -576 i q^{82} + 438 q^{83} + 15 i q^{84} -291 i q^{86} -144 i q^{87} -1008 i q^{88} -438 i q^{89} + ( -585 - 390 i ) q^{91} + 162 i q^{92} + 264 q^{93} -333 q^{94} + 45 i q^{96} -852 q^{97} -354 q^{98} + 1248 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{2} + 2 q^{4} - 30 q^{7} - 42 q^{8} + 52 q^{9} + 78 q^{13} - 90 q^{14} - 142 q^{16} + 156 q^{18} + 234 q^{26} - 30 q^{28} + 288 q^{29} - 90 q^{32} + 96 q^{33} + 52 q^{36} - 606 q^{37} + 52 q^{39} - 222 q^{47} - 236 q^{49} + 90 q^{51} + 78 q^{52} + 630 q^{56} - 12 q^{57} + 864 q^{58} + 752 q^{61} - 780 q^{63} + 866 q^{64} + 288 q^{66} - 72 q^{67} + 324 q^{69} - 1092 q^{72} - 2196 q^{73} - 1818 q^{74} + 156 q^{78} + 1660 q^{79} + 1298 q^{81} + 876 q^{83} - 1170 q^{91} + 528 q^{93} - 666 q^{94} - 1704 q^{97} - 708 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.00000i
1.00000i
3.00000 1.00000i 1.00000 0 3.00000i −15.0000 −21.0000 26.0000 0
324.2 3.00000 1.00000i 1.00000 0 3.00000i −15.0000 −21.0000 26.0000 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
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.4.d.b 2
5.b even 2 1 325.4.d.a 2
5.c odd 4 1 13.4.b.a 2
5.c odd 4 1 325.4.c.b 2
13.b even 2 1 325.4.d.a 2
15.e even 4 1 117.4.b.a 2
20.e even 4 1 208.4.f.b 2
40.i odd 4 1 832.4.f.e 2
40.k even 4 1 832.4.f.c 2
65.d even 2 1 inner 325.4.d.b 2
65.f even 4 1 169.4.a.b 1
65.h odd 4 1 13.4.b.a 2
65.h odd 4 1 325.4.c.b 2
65.k even 4 1 169.4.a.c 1
65.o even 12 2 169.4.c.b 2
65.q odd 12 2 169.4.e.d 4
65.r odd 12 2 169.4.e.d 4
65.t even 12 2 169.4.c.c 2
195.j odd 4 1 1521.4.a.d 1
195.s even 4 1 117.4.b.a 2
195.u odd 4 1 1521.4.a.i 1
260.p even 4 1 208.4.f.b 2
520.bc even 4 1 832.4.f.c 2
520.bg odd 4 1 832.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 5.c odd 4 1
13.4.b.a 2 65.h odd 4 1
117.4.b.a 2 15.e even 4 1
117.4.b.a 2 195.s even 4 1
169.4.a.b 1 65.f even 4 1
169.4.a.c 1 65.k even 4 1
169.4.c.b 2 65.o even 12 2
169.4.c.c 2 65.t even 12 2
169.4.e.d 4 65.q odd 12 2
169.4.e.d 4 65.r odd 12 2
208.4.f.b 2 20.e even 4 1
208.4.f.b 2 260.p even 4 1
325.4.c.b 2 5.c odd 4 1
325.4.c.b 2 65.h odd 4 1
325.4.d.a 2 5.b even 2 1
325.4.d.a 2 13.b even 2 1
325.4.d.b 2 1.a even 1 1 trivial
325.4.d.b 2 65.d even 2 1 inner
832.4.f.c 2 40.k even 4 1
832.4.f.c 2 520.bc even 4 1
832.4.f.e 2 40.i odd 4 1
832.4.f.e 2 520.bg odd 4 1
1521.4.a.d 1 195.j odd 4 1
1521.4.a.i 1 195.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(325, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -3 + T )^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 15 + T )^{2} \)
$11$ \( 2304 + T^{2} \)
$13$ \( 2197 - 78 T + T^{2} \)
$17$ \( 2025 + T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 26244 + T^{2} \)
$29$ \( ( -144 + T )^{2} \)
$31$ \( 69696 + T^{2} \)
$37$ \( ( 303 + T )^{2} \)
$41$ \( 36864 + T^{2} \)
$43$ \( 9409 + T^{2} \)
$47$ \( ( 111 + T )^{2} \)
$53$ \( 171396 + T^{2} \)
$59$ \( 272484 + T^{2} \)
$61$ \( ( -376 + T )^{2} \)
$67$ \( ( 36 + T )^{2} \)
$71$ \( 127449 + T^{2} \)
$73$ \( ( 1098 + T )^{2} \)
$79$ \( ( -830 + T )^{2} \)
$83$ \( ( -438 + T )^{2} \)
$89$ \( 191844 + T^{2} \)
$97$ \( ( 852 + T )^{2} \)
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