Properties

Label 325.4.a.f
Level $325$
Weight $4$
Character orbit 325.a
Self dual yes
Analytic conductor $19.176$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1756207519\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + (\beta - 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + (\beta - 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} + (12 \beta + 34) q^{11} + ( - 13 \beta + 28) q^{12} + 13 q^{13} + (\beta + 44) q^{14} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + ( - 10 \beta + 60) q^{18} + ( - 32 \beta - 26) q^{19} + (41 \beta - 172) q^{21} + ( - 46 \beta - 48) q^{22} + (12 \beta - 104) q^{23} + ( - 23 \beta + 148) q^{24} - 13 \beta q^{26} + (9 \beta - 172) q^{27} + (43 \beta - 84) q^{28} + (96 \beta - 70) q^{29} + ( - 34 \beta - 26) q^{31} + ( - 61 \beta + 92) q^{32} + (90 \beta + 8) q^{33} + (\beta - 68) q^{34} + (70 \beta - 160) q^{36} + ( - 5 \beta - 102) q^{37} + (58 \beta + 128) q^{38} + (39 \beta - 52) q^{39} + (22 \beta - 126) q^{41} + (131 \beta - 164) q^{42} + ( - 143 \beta - 72) q^{43} + ( - 2 \beta - 88) q^{44} + (92 \beta - 48) q^{46} + (121 \beta - 278) q^{47} + ( - 21 \beta - 132) q^{48} + ( - 99 \beta + 241) q^{49} + ( - 71 \beta + 276) q^{51} + (13 \beta - 52) q^{52} + ( - 30 \beta + 74) q^{53} + (163 \beta - 36) q^{54} + (33 \beta - 524) q^{56} + ( - 46 \beta - 280) q^{57} + ( - 26 \beta - 384) q^{58} + (124 \beta - 246) q^{59} + ( - 190 \beta - 434) q^{61} + (60 \beta + 136) q^{62} + ( - 260 \beta + 910) q^{63} + (89 \beta + 340) q^{64} + ( - 98 \beta - 360) q^{66} + (232 \beta - 150) q^{67} + ( - 69 \beta + 140) q^{68} + ( - 324 \beta + 560) q^{69} + ( - 231 \beta + 50) q^{71} + (170 \beta - 760) q^{72} + ( - 260 \beta - 98) q^{73} + (107 \beta + 20) q^{74} + (70 \beta - 24) q^{76} + ( - 386 \beta - 188) q^{77} + (13 \beta - 156) q^{78} + (40 \beta - 524) q^{79} + ( - 120 \beta + 121) q^{81} + (104 \beta - 88) q^{82} + (182 \beta - 1070) q^{83} + ( - 295 \beta + 852) q^{84} + (215 \beta + 572) q^{86} + ( - 306 \beta + 1432) q^{87} + (458 \beta + 392) q^{88} + ( - 388 \beta - 166) q^{89} + ( - 143 \beta + 130) q^{91} + ( - 140 \beta + 464) q^{92} + ( - 44 \beta - 304) q^{93} + (157 \beta - 484) q^{94} + (337 \beta - 1100) q^{96} + ( - 508 \beta + 718) q^{97} + ( - 142 \beta + 396) q^{98} + ( - 390 \beta + 130) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 5 q^{3} - 7 q^{4} - 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 5 q^{3} - 7 q^{4} - 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 80 q^{11} + 43 q^{12} + 26 q^{13} + 89 q^{14} - 39 q^{16} - 19 q^{17} + 110 q^{18} - 84 q^{19} - 303 q^{21} - 142 q^{22} - 196 q^{23} + 273 q^{24} - 13 q^{26} - 335 q^{27} - 125 q^{28} - 44 q^{29} - 86 q^{31} + 123 q^{32} + 106 q^{33} - 135 q^{34} - 250 q^{36} - 209 q^{37} + 314 q^{38} - 65 q^{39} - 230 q^{41} - 197 q^{42} - 287 q^{43} - 178 q^{44} - 4 q^{46} - 435 q^{47} - 285 q^{48} + 383 q^{49} + 481 q^{51} - 91 q^{52} + 118 q^{53} + 91 q^{54} - 1015 q^{56} - 606 q^{57} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} + 211 q^{68} + 796 q^{69} - 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} + 22 q^{76} - 762 q^{77} - 299 q^{78} - 1008 q^{79} + 122 q^{81} - 72 q^{82} - 1958 q^{83} + 1409 q^{84} + 1359 q^{86} + 2558 q^{87} + 1242 q^{88} - 720 q^{89} + 117 q^{91} + 788 q^{92} - 652 q^{93} - 811 q^{94} - 1863 q^{96} + 928 q^{97} + 650 q^{98} - 130 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 3.68466 −1.43845 0 −9.43845 −18.1771 24.1771 −13.4233 0
1.2 1.56155 −8.68466 −5.56155 0 −13.5616 27.1771 −21.1771 48.4233 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(13\) \(-1\)

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 325.4.a.f 2
5.b even 2 1 13.4.a.b 2
5.c odd 4 2 325.4.b.e 4
15.d odd 2 1 117.4.a.d 2
20.d odd 2 1 208.4.a.h 2
35.c odd 2 1 637.4.a.b 2
40.e odd 2 1 832.4.a.z 2
40.f even 2 1 832.4.a.s 2
55.d odd 2 1 1573.4.a.b 2
60.h even 2 1 1872.4.a.bb 2
65.d even 2 1 169.4.a.g 2
65.g odd 4 2 169.4.b.f 4
65.l even 6 2 169.4.c.j 4
65.n even 6 2 169.4.c.g 4
65.s odd 12 4 169.4.e.f 8
195.e odd 2 1 1521.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 5.b even 2 1
117.4.a.d 2 15.d odd 2 1
169.4.a.g 2 65.d even 2 1
169.4.b.f 4 65.g odd 4 2
169.4.c.g 4 65.n even 6 2
169.4.c.j 4 65.l even 6 2
169.4.e.f 8 65.s odd 12 4
208.4.a.h 2 20.d odd 2 1
325.4.a.f 2 1.a even 1 1 trivial
325.4.b.e 4 5.c odd 4 2
637.4.a.b 2 35.c odd 2 1
832.4.a.s 2 40.f even 2 1
832.4.a.z 2 40.e odd 2 1
1521.4.a.r 2 195.e odd 2 1
1573.4.a.b 2 55.d odd 2 1
1872.4.a.bb 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(325))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{3}^{2} + 5T_{3} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 9T - 494 \) Copy content Toggle raw display
$11$ \( T^{2} - 80T + 988 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} + 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} + 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} + 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} + 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} + 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} + 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} + 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} + 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} - 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} + 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} + 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} + 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} + 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} + 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} + 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} + 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} + 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} - 928T - 881476 \) Copy content Toggle raw display
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