Properties

Label 325.4.b.b
Level $325$
Weight $4$
Character orbit 325.b
Analytic conductor $19.176$
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: \( k \) \(=\) \( 4 \)
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: \(2\)
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.

\(f(q)\) \(=\) \( q + 5 i q^{2} - 7 i q^{3} - 17 q^{4} + 35 q^{6} + 13 i q^{7} - 45 i q^{8} - 22 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 5 i q^{2} - 7 i q^{3} - 17 q^{4} + 35 q^{6} + 13 i q^{7} - 45 i q^{8} - 22 q^{9} - 26 q^{11} + 119 i q^{12} + 13 i q^{13} - 65 q^{14} + 89 q^{16} - 77 i q^{17} - 110 i q^{18} + 126 q^{19} + 91 q^{21} - 130 i q^{22} - 96 i q^{23} - 315 q^{24} - 65 q^{26} - 35 i q^{27} - 221 i q^{28} + 82 q^{29} + 196 q^{31} + 85 i q^{32} + 182 i q^{33} + 385 q^{34} + 374 q^{36} + 131 i q^{37} + 630 i q^{38} + 91 q^{39} + 336 q^{41} + 455 i q^{42} - 201 i q^{43} + 442 q^{44} + 480 q^{46} + 105 i q^{47} - 623 i q^{48} + 174 q^{49} - 539 q^{51} - 221 i q^{52} - 432 i q^{53} + 175 q^{54} + 585 q^{56} - 882 i q^{57} + 410 i q^{58} + 294 q^{59} - 56 q^{61} + 980 i q^{62} - 286 i q^{63} + 287 q^{64} - 910 q^{66} - 478 i q^{67} + 1309 i q^{68} - 672 q^{69} + 9 q^{71} + 990 i q^{72} + 98 i q^{73} - 655 q^{74} - 2142 q^{76} - 338 i q^{77} + 455 i q^{78} - 1304 q^{79} - 839 q^{81} + 1680 i q^{82} - 308 i q^{83} - 1547 q^{84} + 1005 q^{86} - 574 i q^{87} + 1170 i q^{88} + 1190 q^{89} - 169 q^{91} + 1632 i q^{92} - 1372 i q^{93} - 525 q^{94} + 595 q^{96} - 70 i q^{97} + 870 i q^{98} + 572 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4} + 70 q^{6} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 34 q^{4} + 70 q^{6} - 44 q^{9} - 52 q^{11} - 130 q^{14} + 178 q^{16} + 252 q^{19} + 182 q^{21} - 630 q^{24} - 130 q^{26} + 164 q^{29} + 392 q^{31} + 770 q^{34} + 748 q^{36} + 182 q^{39} + 672 q^{41} + 884 q^{44} + 960 q^{46} + 348 q^{49} - 1078 q^{51} + 350 q^{54} + 1170 q^{56} + 588 q^{59} - 112 q^{61} + 574 q^{64} - 1820 q^{66} - 1344 q^{69} + 18 q^{71} - 1310 q^{74} - 4284 q^{76} - 2608 q^{79} - 1678 q^{81} - 3094 q^{84} + 2010 q^{86} + 2380 q^{89} - 338 q^{91} - 1050 q^{94} + 1190 q^{96} + 1144 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
1.00000i
1.00000i
5.00000i 7.00000i −17.0000 0 35.0000 13.0000i 45.0000i −22.0000 0
274.2 5.00000i 7.00000i −17.0000 0 35.0000 13.0000i 45.0000i −22.0000 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
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.b 2
5.b even 2 1 inner 325.4.b.b 2
5.c odd 4 1 13.4.a.a 1
5.c odd 4 1 325.4.a.d 1
15.e even 4 1 117.4.a.b 1
20.e even 4 1 208.4.a.g 1
35.f even 4 1 637.4.a.a 1
40.i odd 4 1 832.4.a.r 1
40.k even 4 1 832.4.a.a 1
55.e even 4 1 1573.4.a.a 1
60.l odd 4 1 1872.4.a.k 1
65.f even 4 1 169.4.b.a 2
65.h odd 4 1 169.4.a.e 1
65.k even 4 1 169.4.b.a 2
65.o even 12 2 169.4.e.e 4
65.q odd 12 2 169.4.c.e 2
65.r odd 12 2 169.4.c.a 2
65.t even 12 2 169.4.e.e 4
195.s even 4 1 1521.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 5.c odd 4 1
117.4.a.b 1 15.e even 4 1
169.4.a.e 1 65.h odd 4 1
169.4.b.a 2 65.f even 4 1
169.4.b.a 2 65.k even 4 1
169.4.c.a 2 65.r odd 12 2
169.4.c.e 2 65.q odd 12 2
169.4.e.e 4 65.o even 12 2
169.4.e.e 4 65.t even 12 2
208.4.a.g 1 20.e even 4 1
325.4.a.d 1 5.c odd 4 1
325.4.b.b 2 1.a even 1 1 trivial
325.4.b.b 2 5.b even 2 1 inner
637.4.a.a 1 35.f even 4 1
832.4.a.a 1 40.k even 4 1
832.4.a.r 1 40.i odd 4 1
1521.4.a.a 1 195.s even 4 1
1573.4.a.a 1 55.e even 4 1
1872.4.a.k 1 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 \(S_{4}^{\mathrm{new}}(325, [\chi])\):

\( T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 169 \) Copy content Toggle raw display
$11$ \( (T + 26)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{2} + 5929 \) Copy content Toggle raw display
$19$ \( (T - 126)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9216 \) Copy content Toggle raw display
$29$ \( (T - 82)^{2} \) Copy content Toggle raw display
$31$ \( (T - 196)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17161 \) Copy content Toggle raw display
$41$ \( (T - 336)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 40401 \) Copy content Toggle raw display
$47$ \( T^{2} + 11025 \) Copy content Toggle raw display
$53$ \( T^{2} + 186624 \) Copy content Toggle raw display
$59$ \( (T - 294)^{2} \) Copy content Toggle raw display
$61$ \( (T + 56)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 228484 \) Copy content Toggle raw display
$71$ \( (T - 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9604 \) Copy content Toggle raw display
$79$ \( (T + 1304)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 94864 \) Copy content Toggle raw display
$89$ \( (T - 1190)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4900 \) Copy content Toggle raw display
show more
show less