Properties

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

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(19.1756207519\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 3) q^{4} + ( - \beta_{3} - 11) q^{6} + (10 \beta_{2} + 11 \beta_1) q^{7} + (4 \beta_{2} + 11 \beta_1) q^{8} + ( - 15 \beta_{3} - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 3) q^{4} + ( - \beta_{3} - 11) q^{6} + (10 \beta_{2} + 11 \beta_1) q^{7} + (4 \beta_{2} + 11 \beta_1) q^{8} + ( - 15 \beta_{3} - 10) q^{9} + ( - 12 \beta_{3} + 46) q^{11} + (28 \beta_{2} + 13 \beta_1) q^{12} - 13 \beta_{2} q^{13} + (\beta_{3} - 45) q^{14} + (15 \beta_{3} - 27) q^{16} + ( - 18 \beta_{2} - 17 \beta_1) q^{17} + ( - 60 \beta_{2} - 10 \beta_1) q^{18} + ( - 32 \beta_{3} + 58) q^{19} + ( - 41 \beta_{3} - 131) q^{21} + ( - 48 \beta_{2} + 46 \beta_1) q^{22} + (104 \beta_{2} + 12 \beta_1) q^{23} + ( - 23 \beta_{3} - 125) q^{24} + (13 \beta_{3} - 13) q^{26} + ( - 172 \beta_{2} - 9 \beta_1) q^{27} + (84 \beta_{2} + 43 \beta_1) q^{28} + (96 \beta_{3} - 26) q^{29} + (34 \beta_{3} - 60) q^{31} + (92 \beta_{2} + 61 \beta_1) q^{32} + ( - 8 \beta_{2} + 90 \beta_1) q^{33} + (\beta_{3} + 67) q^{34} + ( - 70 \beta_{3} - 90) q^{36} + ( - 102 \beta_{2} + 5 \beta_1) q^{37} + ( - 128 \beta_{2} + 58 \beta_1) q^{38} + (39 \beta_{3} + 13) q^{39} + ( - 22 \beta_{3} - 104) q^{41} + ( - 164 \beta_{2} - 131 \beta_1) q^{42} + (72 \beta_{2} - 143 \beta_1) q^{43} + ( - 2 \beta_{3} + 90) q^{44} + ( - 92 \beta_{3} + 44) q^{46} + ( - 278 \beta_{2} - 121 \beta_1) q^{47} + (132 \beta_{2} - 21 \beta_1) q^{48} + ( - 99 \beta_{3} - 142) q^{49} + (71 \beta_{3} + 205) q^{51} + ( - 52 \beta_{2} - 13 \beta_1) q^{52} + ( - 74 \beta_{2} - 30 \beta_1) q^{53} + (163 \beta_{3} - 127) q^{54} + ( - 33 \beta_{3} - 491) q^{56} + ( - 280 \beta_{2} + 46 \beta_1) q^{57} + (384 \beta_{2} - 26 \beta_1) q^{58} + (124 \beta_{3} + 122) q^{59} + (190 \beta_{3} - 624) q^{61} + (136 \beta_{2} - 60 \beta_1) q^{62} + ( - 910 \beta_{2} - 260 \beta_1) q^{63} + (89 \beta_{3} - 429) q^{64} + (98 \beta_{3} - 458) q^{66} + ( - 150 \beta_{2} - 232 \beta_1) q^{67} + ( - 140 \beta_{2} - 69 \beta_1) q^{68} + ( - 324 \beta_{3} - 236) q^{69} + (231 \beta_{3} - 181) q^{71} + ( - 760 \beta_{2} - 170 \beta_1) q^{72} + (98 \beta_{2} - 260 \beta_1) q^{73} + (107 \beta_{3} - 127) q^{74} + ( - 70 \beta_{3} + 46) q^{76} + ( - 188 \beta_{2} + 386 \beta_1) q^{77} + (156 \beta_{2} + 13 \beta_1) q^{78} + (40 \beta_{3} + 484) q^{79} + (120 \beta_{3} + 1) q^{81} + ( - 88 \beta_{2} - 104 \beta_1) q^{82} + (1070 \beta_{2} + 182 \beta_1) q^{83} + ( - 295 \beta_{3} - 557) q^{84} + ( - 215 \beta_{3} + 787) q^{86} + (1432 \beta_{2} + 306 \beta_1) q^{87} + ( - 392 \beta_{2} + 458 \beta_1) q^{88} + ( - 388 \beta_{3} + 554) q^{89} + (143 \beta_{3} - 13) q^{91} + (464 \beta_{2} + 140 \beta_1) q^{92} + (304 \beta_{2} - 44 \beta_1) q^{93} + (157 \beta_{3} + 327) q^{94} + ( - 337 \beta_{3} - 763) q^{96} + (718 \beta_{2} + 508 \beta_1) q^{97} + ( - 396 \beta_{2} - 142 \beta_1) q^{98} + ( - 390 \beta_{3} + 260) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{4} - 46 q^{6} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{4} - 46 q^{6} - 70 q^{9} + 160 q^{11} - 178 q^{14} - 78 q^{16} + 168 q^{19} - 606 q^{21} - 546 q^{24} - 26 q^{26} + 88 q^{29} - 172 q^{31} + 270 q^{34} - 500 q^{36} + 130 q^{39} - 460 q^{41} + 356 q^{44} - 8 q^{46} - 766 q^{49} + 962 q^{51} - 182 q^{54} - 2030 q^{56} + 736 q^{59} - 2116 q^{61} - 1538 q^{64} - 1636 q^{66} - 1592 q^{69} - 262 q^{71} - 294 q^{74} + 44 q^{76} + 2016 q^{79} + 244 q^{81} - 2818 q^{84} + 2718 q^{86} + 1440 q^{89} + 234 q^{91} + 1622 q^{94} - 3726 q^{96} + 260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 3.68466i 1.43845 0 −9.43845 18.1771i 24.1771i 13.4233 0
274.2 1.56155i 8.68466i 5.56155 0 −13.5616 27.1771i 21.1771i −48.4233 0
274.3 1.56155i 8.68466i 5.56155 0 −13.5616 27.1771i 21.1771i −48.4233 0
274.4 2.56155i 3.68466i 1.43845 0 −9.43845 18.1771i 24.1771i 13.4233 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
5.b even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{4} + 89T_{3}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 89T^{2} + 1024 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1069 T^{2} + 244036 \) Copy content Toggle raw display
$11$ \( (T^{2} - 80 T + 988)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2637 T^{2} + \cdots + 1295044 \) Copy content Toggle raw display
$19$ \( (T^{2} - 84 T - 2588)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 20432 T^{2} + \cdots + 80856064 \) Copy content Toggle raw display
$29$ \( (T^{2} - 44 T - 38684)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 86 T - 3064)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 22053 T^{2} + \cdots + 116942596 \) Copy content Toggle raw display
$41$ \( (T^{2} + 230 T + 11168)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 215001 T^{2} + \cdots + 4397811856 \) Copy content Toggle raw display
$47$ \( T^{4} + 219061 T^{2} + \cdots + 222546724 \) Copy content Toggle raw display
$53$ \( T^{4} + 14612 T^{2} + \cdots + 118336 \) Copy content Toggle raw display
$59$ \( (T^{2} - 368 T - 31492)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1058 T + 126416)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 459816 T^{2} + \cdots + 51799939216 \) Copy content Toggle raw display
$71$ \( (T^{2} + 131 T - 222494)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 678568 T^{2} + \cdots + 55373619856 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1008 T + 247216)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 2198436 T^{2} + \cdots + 668574416896 \) Copy content Toggle raw display
$89$ \( (T^{2} - 720 T - 510212)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 2624136 T^{2} + \cdots + 776999938576 \) Copy content Toggle raw display
show more
show less