Properties

Label 325.6.b.d
Level $325$
Weight $6$
Character orbit 325.b
Analytic conductor $52.125$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 85x^{4} + 1668x^{2} + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{5} - 5 \beta_{2}) q^{3} + (\beta_{4} - \beta_{3} + 4) q^{4} + ( - 3 \beta_{4} - 14 \beta_{3} + 11) q^{6} + ( - 6 \beta_{5} + 66 \beta_{2} - 16 \beta_1) q^{7} + ( - 2 \beta_{5} + 57 \beta_{2} + 25 \beta_1) q^{8} + (2 \beta_{4} + 24 \beta_{3} - 55) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{5} - 5 \beta_{2}) q^{3} + (\beta_{4} - \beta_{3} + 4) q^{4} + ( - 3 \beta_{4} - 14 \beta_{3} + 11) q^{6} + ( - 6 \beta_{5} + 66 \beta_{2} - 16 \beta_1) q^{7} + ( - 2 \beta_{5} + 57 \beta_{2} + 25 \beta_1) q^{8} + (2 \beta_{4} + 24 \beta_{3} - 55) q^{9} + ( - 5 \beta_{4} + 24 \beta_{3} - 215) q^{11} + ( - 9 \beta_{5} + 247 \beta_{2} + 10 \beta_1) q^{12} + 169 \beta_{2} q^{13} + ( - 34 \beta_{4} + 44 \beta_{3} + 402) q^{14} + (51 \beta_{4} - 43 \beta_{3} - 592) q^{16} + ( - 60 \beta_{5} + 274 \beta_{2} - 32 \beta_1) q^{17} + ( - 30 \beta_{5} - 715 \beta_{2} - 25 \beta_1) q^{18} + ( - 35 \beta_{4} + 136 \beta_{3} + 1063) q^{19} + ( - 52 \beta_{4} + 368 \beta_{3} - 1404) q^{21} + ( - 9 \beta_{5} - 833 \beta_{2} - 122 \beta_1) q^{22} + (61 \beta_{5} - 2419 \beta_{2} - 120 \beta_1) q^{23} + ( - 113 \beta_{4} - 302 \beta_{3} - 111) q^{24} + (169 \beta_{3} - 169) q^{26} + ( - 146 \beta_{5} - 250 \beta_{2} + 384 \beta_1) q^{27} + ( - 134 \beta_{5} + 1530 \beta_{2} + 284 \beta_1) q^{28} + ( - 2 \beta_{4} + 8 \beta_{3} - 772) q^{29} + ( - 83 \beta_{4} + 896 \beta_{3} - 2453) q^{31} + ( - 174 \beta_{5} + 2087 \beta_{2} - 337 \beta_1) q^{32} + (278 \beta_{5} - 146 \beta_{2} + 216 \beta_1) q^{33} + ( - 212 \beta_{4} - 202 \beta_{3} + 950) q^{34} + ( - 51 \beta_{4} - 167 \beta_{3} - 190) q^{36} + (82 \beta_{5} - 4956 \beta_{2} + 1224 \beta_1) q^{37} + ( - 31 \beta_{5} - 2399 \beta_{2} + 1650 \beta_1) q^{38} + ( - 169 \beta_{4} + 845) q^{39} + (632 \beta_{4} - 352 \beta_{3} + 3090) q^{41} + ( - 212 \beta_{5} - 11028 \beta_{2} - 200 \beta_1) q^{42} + ( - 303 \beta_{5} + 2837 \beta_{2} + 1392 \beta_1) q^{43} + ( - 309 \beta_{4} + 98 \beta_{3} - 2699) q^{44} + (63 \beta_{4} - 1630 \beta_{3} + 5293) q^{46} + (226 \beta_{5} - 2110 \beta_{2} + 1536 \beta_1) q^{47} + (353 \beta_{5} + 16625 \beta_{2} + 622 \beta_1) q^{48} + ( - 440 \beta_{4} - 288 \beta_{3} - 5441) q^{49} + ( - 390 \beta_{4} + 1888 \beta_{3} - 15202) q^{51} + ( - 169 \beta_{5} + 676 \beta_{2} + 169 \beta_1) q^{52} + ( - 1312 \beta_{5} + 4662 \beta_{2} + 3056 \beta_1) q^{53} + ( - 54 \beta_{4} - 2332 \beta_{3} - 9242) q^{54} + ( - 1206 \beta_{4} + 1164 \beta_{3} + 4470) q^{56} + ( - 686 \beta_{5} - 14054 \beta_{2} + 1064 \beta_1) q^{57} + ( - 2 \beta_{5} - 976 \beta_{2} - 738 \beta_1) q^{58} + (1367 \beta_{4} + 3752 \beta_{3} - 2379) q^{59} + ( - 802 \beta_{4} + 6216 \beta_{3} - 31304) q^{61} + ( - 647 \beta_{5} - 26147 \beta_{2} + 86 \beta_1) q^{62} + (838 \beta_{5} + 11070 \beta_{2} + 16 \beta_1) q^{63} + (773 \beta_{4} - 181 \beta_{3} - 10888) q^{64} + (1050 \beta_{4} + 1924 \beta_{3} - 7354) q^{66} + (2796 \beta_{5} + 29128 \beta_{2} - 1440 \beta_1) q^{67} + ( - 1082 \beta_{5} + 16444 \beta_{2} + 1430 \beta_1) q^{68} + (2842 \beta_{4} + 216 \beta_{3} + 3838) q^{69} + (2669 \beta_{4} - 784 \beta_{3} - 39933) q^{71} + ( - 640 \beta_{5} - 18255 \beta_{2} - 865 \beta_1) q^{72} + (502 \beta_{5} - 28584 \beta_{2} - 6616 \beta_1) q^{73} + (1470 \beta_{4} - 6666 \beta_{3} - 28584) q^{74} + (437 \beta_{4} - 1626 \beta_{3} - 7949) q^{76} + (2372 \beta_{5} - 11628 \beta_{2} + 560 \beta_1) q^{77} + (507 \beta_{5} + 1859 \beta_{2} + 2366 \beta_1) q^{78} + ( - 2462 \beta_{4} + 1776 \beta_{3} + 35486) q^{79} + ( - 470 \beta_{4} + 3960 \beta_{3} - 52169) q^{81} + ( - 1544 \beta_{5} + 8802 \beta_{2} - 3302 \beta_1) q^{82} + ( - 366 \beta_{5} - 3082 \beta_{2} + 9296 \beta_1) q^{83} + ( - 2500 \beta_{4} - 760 \beta_{3} - 27228) q^{84} + (483 \beta_{4} - 2674 \beta_{3} - 38603) q^{86} + (794 \beta_{5} + 3362 \beta_{2} + 64 \beta_1) q^{87} + (541 \beta_{5} - 30147 \beta_{2} - 3626 \beta_1) q^{88} + (2376 \beta_{4} + 10576 \beta_{3} + 34670) q^{89} + ( - 1014 \beta_{4} - 2704 \beta_{3} - 11154) q^{91} + (3393 \beta_{5} - 28483 \beta_{2} - 2374 \beta_1) q^{92} + (4494 \beta_{5} - 5018 \beta_{2} + 10552 \beta_1) q^{93} + (2214 \beta_{4} - 3148 \beta_{3} - 40718) q^{94} + ( - 1935 \beta_{4} + 8894 \beta_{3} - 39089) q^{96} + (3304 \beta_{5} - 59658 \beta_{2} - 6864 \beta_1) q^{97} + (1608 \beta_{5} + 4975 \beta_{2} - 2057 \beta_1) q^{98} + ( - 667 \beta_{4} - 3864 \beta_{3} + 24215) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 20 q^{4} + 44 q^{6} - 286 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 20 q^{4} + 44 q^{6} - 286 q^{9} - 1232 q^{11} + 2568 q^{14} - 3740 q^{16} + 6720 q^{19} - 7584 q^{21} - 1044 q^{24} - 676 q^{26} - 4612 q^{29} - 12760 q^{31} + 5720 q^{34} - 1372 q^{36} + 5408 q^{39} + 16572 q^{41} - 15380 q^{44} + 28372 q^{46} - 32342 q^{49} - 86656 q^{51} - 60008 q^{54} + 31560 q^{56} - 9504 q^{59} - 173788 q^{61} - 67236 q^{64} - 42376 q^{66} + 17776 q^{69} - 246504 q^{71} - 187776 q^{74} - 51820 q^{76} + 221392 q^{79} - 304154 q^{81} - 159888 q^{84} - 237932 q^{86} + 224420 q^{89} - 70304 q^{91} - 255032 q^{94} - 212876 q^{96} + 138896 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 85x^{4} + 1668x^{2} + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 133\nu^{3} + 3732\nu ) / 4320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 43\nu^{2} - 48 ) / 90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 73\nu^{2} + 762 ) / 30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{5} - 377\nu^{3} - 5700\nu ) / 288 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 3\beta_{3} - 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 75\beta_{2} - 45\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -43\beta_{4} + 219\beta_{3} + 1209 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -133\beta_{5} - 5655\beta_{2} + 2253\beta_1 \) 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.

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
7.47640i
5.25457i
1.22183i
1.22183i
5.25457i
7.47640i
6.47640i 11.4674i −9.94380 0 −74.2675 146.818i 142.845i 111.499 0
274.2 6.25457i 21.3742i −7.11967 0 133.687 116.319i 155.616i −213.858 0
274.3 2.22183i 16.8416i 27.0635 0 −37.4193 177.501i 131.229i −40.6407 0
274.4 2.22183i 16.8416i 27.0635 0 −37.4193 177.501i 131.229i −40.6407 0
274.5 6.25457i 21.3742i −7.11967 0 133.687 116.319i 155.616i −213.858 0
274.6 6.47640i 11.4674i −9.94380 0 −74.2675 146.818i 142.845i 111.499 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.6
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.d 6
5.b even 2 1 inner 325.6.b.d 6
5.c odd 4 1 65.6.a.b 3
5.c odd 4 1 325.6.a.d 3
15.e even 4 1 585.6.a.c 3
20.e even 4 1 1040.6.a.k 3
65.h odd 4 1 845.6.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.b 3 5.c odd 4 1
325.6.a.d 3 5.c odd 4 1
325.6.b.d 6 1.a even 1 1 trivial
325.6.b.d 6 5.b even 2 1 inner
585.6.a.c 3 15.e even 4 1
845.6.a.c 3 65.h odd 4 1
1040.6.a.k 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 86T_{2}^{4} + 2041T_{2}^{2} + 8100 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 86 T^{4} + \cdots + 8100 \) Copy content Toggle raw display
$3$ \( T^{6} + 872 T^{4} + \cdots + 17040384 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 9188755439616 \) Copy content Toggle raw display
$11$ \( (T^{3} + 616 T^{2} + \cdots + 295968)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 3360 T^{2} + \cdots - 281302528)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 81\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2306 T^{2} + \cdots + 450963192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6380 T^{2} + \cdots - 148752304496)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8286 T^{2} + \cdots - 340059823848)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 56\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots + 17139260691072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 58378672604984)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 96871211913744)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 87960642878336)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 433282552140840)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
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