Properties

Label 325.4.d.a
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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.a 2
5.b even 2 1 325.4.d.b 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.b 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.a 2
65.f even 4 1 169.4.a.c 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.b 1
65.o even 12 2 169.4.c.c 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.b 2
195.j odd 4 1 1521.4.a.i 1
195.s even 4 1 117.4.b.a 2
195.u odd 4 1 1521.4.a.d 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.k even 4 1
169.4.a.c 1 65.f even 4 1
169.4.c.b 2 65.t even 12 2
169.4.c.c 2 65.o 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 1.a even 1 1 trivial
325.4.d.a 2 65.d even 2 1 inner
325.4.d.b 2 5.b even 2 1
325.4.d.b 2 13.b even 2 1
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.u odd 4 1
1521.4.a.i 1 195.j odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

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