Properties

Label 325.6.b.a
Level $325$
Weight $6$
Character orbit 325.b
Analytic conductor $52.125$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(274,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + 5 i q^{2} - 6 i q^{3} + 7 q^{4} + 30 q^{6} - 244 i q^{7} + 195 i q^{8} + 207 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 5 i q^{2} - 6 i q^{3} + 7 q^{4} + 30 q^{6} - 244 i q^{7} + 195 i q^{8} + 207 q^{9} + 794 q^{11} - 42 i q^{12} + 169 i q^{13} + 1220 q^{14} - 751 q^{16} - 1534 i q^{17} + 1035 i q^{18} - 2706 q^{19} - 1464 q^{21} + 3970 i q^{22} + 702 i q^{23} + 1170 q^{24} - 845 q^{26} - 2700 i q^{27} - 1708 i q^{28} + 5038 q^{29} - 3634 q^{31} + 2485 i q^{32} - 4764 i q^{33} + 7670 q^{34} + 1449 q^{36} - 7058 i q^{37} - 13530 i q^{38} + 1014 q^{39} - 294 q^{41} - 7320 i q^{42} - 7618 i q^{43} + 5558 q^{44} - 3510 q^{46} - 3020 i q^{47} + 4506 i q^{48} - 42729 q^{49} - 9204 q^{51} + 1183 i q^{52} - 626 i q^{53} + 13500 q^{54} + 47580 q^{56} + 16236 i q^{57} + 25190 i q^{58} + 30066 q^{59} - 5806 q^{61} - 18170 i q^{62} - 50508 i q^{63} - 36457 q^{64} + 23820 q^{66} - 12436 i q^{67} - 10738 i q^{68} + 4212 q^{69} + 4734 q^{71} + 40365 i q^{72} + 14694 i q^{73} + 35290 q^{74} - 18942 q^{76} - 193736 i q^{77} + 5070 i q^{78} + 39804 q^{79} + 34101 q^{81} - 1470 i q^{82} + 41776 i q^{83} - 10248 q^{84} + 38090 q^{86} - 30228 i q^{87} + 154830 i q^{88} - 7970 q^{89} + 41236 q^{91} + 4914 i q^{92} + 21804 i q^{93} + 15100 q^{94} + 14910 q^{96} - 78050 i q^{97} - 213645 i q^{98} + 164358 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 60 q^{6} + 414 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} + 60 q^{6} + 414 q^{9} + 1588 q^{11} + 2440 q^{14} - 1502 q^{16} - 5412 q^{19} - 2928 q^{21} + 2340 q^{24} - 1690 q^{26} + 10076 q^{29} - 7268 q^{31} + 15340 q^{34} + 2898 q^{36} + 2028 q^{39} - 588 q^{41} + 11116 q^{44} - 7020 q^{46} - 85458 q^{49} - 18408 q^{51} + 27000 q^{54} + 95160 q^{56} + 60132 q^{59} - 11612 q^{61} - 72914 q^{64} + 47640 q^{66} + 8424 q^{69} + 9468 q^{71} + 70580 q^{74} - 37884 q^{76} + 79608 q^{79} + 68202 q^{81} - 20496 q^{84} + 76180 q^{86} - 15940 q^{89} + 82472 q^{91} + 30200 q^{94} + 29820 q^{96} + 328716 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 6.00000i 7.00000 0 30.0000 244.000i 195.000i 207.000 0
274.2 5.00000i 6.00000i 7.00000 0 30.0000 244.000i 195.000i 207.000 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 325.6.b.a 2
5.b even 2 1 inner 325.6.b.a 2
5.c odd 4 1 65.6.a.a 1
5.c odd 4 1 325.6.a.a 1
15.e even 4 1 585.6.a.a 1
20.e even 4 1 1040.6.a.a 1
65.h odd 4 1 845.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.a 1 5.c odd 4 1
325.6.a.a 1 5.c odd 4 1
325.6.b.a 2 1.a even 1 1 trivial
325.6.b.a 2 5.b even 2 1 inner
585.6.a.a 1 15.e even 4 1
845.6.a.a 1 65.h odd 4 1
1040.6.a.a 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 25 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{2} + 36 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 59536 \) Copy content Toggle raw display
$11$ \( (T - 794)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 28561 \) Copy content Toggle raw display
$17$ \( T^{2} + 2353156 \) Copy content Toggle raw display
$19$ \( (T + 2706)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 492804 \) Copy content Toggle raw display
$29$ \( (T - 5038)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3634)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49815364 \) Copy content Toggle raw display
$41$ \( (T + 294)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 58033924 \) Copy content Toggle raw display
$47$ \( T^{2} + 9120400 \) Copy content Toggle raw display
$53$ \( T^{2} + 391876 \) Copy content Toggle raw display
$59$ \( (T - 30066)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5806)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 154654096 \) Copy content Toggle raw display
$71$ \( (T - 4734)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 215913636 \) Copy content Toggle raw display
$79$ \( (T - 39804)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1745234176 \) Copy content Toggle raw display
$89$ \( (T + 7970)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6091802500 \) Copy content Toggle raw display
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