Properties

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

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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: \( x^{6} + 201x^{4} + 10512x^{2} + 65536 \) 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

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.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{5} - 3 \beta_{2} + \beta_1) q^{3} + (4 \beta_{4} + \beta_{3} - 40) q^{4} + (10 \beta_{4} + \beta_{3} - 66) q^{6} + (3 \beta_{5} - 19 \beta_{2} - 3 \beta_1) q^{7} + (28 \beta_{5} - 100 \beta_{2} - 27 \beta_1) q^{8} + (\beta_{4} + 7 \beta_{3} + 66) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{5} - 3 \beta_{2} + \beta_1) q^{3} + (4 \beta_{4} + \beta_{3} - 40) q^{4} + (10 \beta_{4} + \beta_{3} - 66) q^{6} + (3 \beta_{5} - 19 \beta_{2} - 3 \beta_1) q^{7} + (28 \beta_{5} - 100 \beta_{2} - 27 \beta_1) q^{8} + (\beta_{4} + 7 \beta_{3} + 66) q^{9} + ( - 30 \beta_{4} + 26 \beta_{3} + 194) q^{11} + (32 \beta_{5} - 336 \beta_{2} - 83 \beta_1) q^{12} - 169 \beta_{2} q^{13} + ( - 30 \beta_{4} - 31 \beta_{3} + 254) q^{14} + ( - 148 \beta_{4} - 181 \beta_{3} + 868) q^{16} + (93 \beta_{5} + 277 \beta_{2} + 77 \beta_1) q^{17} + (34 \beta_{5} - 348 \beta_{2} + 68 \beta_1) q^{18} + ( - 134 \beta_{4} + 178 \beta_{3} + 10) q^{19} + ( - 31 \beta_{4} - 49 \beta_{3} + 447) q^{21} + ( - 76 \beta_{5} - 1260 \beta_{2} + 370 \beta_1) q^{22} + (152 \beta_{5} - 1232 \beta_{2} + 72 \beta_1) q^{23} + ( - 204 \beta_{4} - 381 \beta_{3} + 4332) q^{24} + ( - 169 \beta_{3} + 338) q^{26} + ( - 257 \beta_{5} - 1527 \beta_{2} + 305 \beta_1) q^{27} + ( - 208 \beta_{5} + 2128 \beta_{2} + 277 \beta_1) q^{28} + ( - 584 \beta_{4} + 56 \beta_{3} + 2938) q^{29} + (268 \beta_{4} + 148 \beta_{3} - 820) q^{31} + ( - 716 \beta_{5} + 11436 \beta_{2} + 563 \beta_1) q^{32} + (134 \beta_{5} + 426 \beta_{2} + 550 \beta_1) q^{33} + ( - 250 \beta_{4} - 265 \beta_{3} - 5418) q^{34} + (100 \beta_{4} - 362 \beta_{3} - 1680) q^{36} + ( - 419 \beta_{5} - 6727 \beta_{2} - 451 \beta_1) q^{37} + ( - 92 \beta_{5} - 11548 \beta_{2} + 858 \beta_1) q^{38} + ( - 169 \beta_{4} - 169 \beta_{3} - 507) q^{39} + (250 \beta_{4} - 858 \beta_{3} - 3952) q^{41} + ( - 382 \beta_{5} + 4350 \beta_{2} + 553 \beta_1) q^{42} + (697 \beta_{5} - 821 \beta_{2} + 431 \beta_1) q^{43} + (976 \beta_{4} - 418 \beta_{3} - 16736) q^{44} + ( - 624 \beta_{4} - 2064 \beta_{3} - 1824) q^{46} + (807 \beta_{5} + 11409 \beta_{2} + 33 \beta_1) q^{47} + ( - 1724 \beta_{5} + 24636 \beta_{2} + 2315 \beta_1) q^{48} + (177 \beta_{4} + 231 \beta_{3} + 14934) q^{49} + (1265 \beta_{4} - 823 \beta_{3} + 4215) q^{51} + ( - 676 \beta_{5} + 6760 \beta_{2} + 169 \beta_1) q^{52} + (2154 \beta_{5} + 4464 \beta_{2} - 822 \beta_1) q^{53} + (2762 \beta_{4} - 547 \beta_{3} - 18714) q^{54} + (1396 \beta_{4} + 1899 \beta_{3} - 15796) q^{56} + (1950 \beta_{5} + 18 \beta_{2} + 1662 \beta_1) q^{57} + ( - 3280 \beta_{5} + 4404 \beta_{2} + 5914 \beta_1) q^{58} + ( - 950 \beta_{4} + 1458 \beta_{3} - 20830) q^{59} + ( - 2330 \beta_{4} - 1910 \beta_{3} - 4888) q^{61} + (2200 \beta_{5} - 12776 \beta_{2} - 2012 \beta_1) q^{62} + (14 \beta_{5} + 546 \beta_{2} + 10 \beta_1) q^{63} + (1812 \beta_{4} + 8661 \beta_{3} - 36244) q^{64} + (1396 \beta_{4} - 794 \beta_{3} - 37716) q^{66} + (1666 \beta_{5} + 18218 \beta_{2} + 1478 \beta_1) q^{67} + (416 \beta_{5} + 17048 \beta_{2} - 1969 \beta_1) q^{68} + ( - 48 \beta_{4} - 2976 \beta_{3} + 5712) q^{69} + ( - 3127 \beta_{4} + 633 \beta_{3} + 26071) q^{71} + (240 \beta_{5} + 9720 \beta_{2} - 366 \beta_1) q^{72} + ( - 2568 \beta_{5} + 13438 \beta_{2} + 2712 \beta_1) q^{73} + (710 \beta_{4} - 4181 \beta_{3} + 42446) q^{74} + ( - 304 \beta_{4} - 6250 \beta_{3} - 35296) q^{76} + (438 \beta_{5} - 6710 \beta_{2} - 922 \beta_1) q^{77} + ( - 1690 \beta_{5} + 11154 \beta_{2} + 169 \beta_1) q^{78} + ( - 3576 \beta_{4} - 2856 \beta_{3} + 19672) q^{79} + (128 \beta_{4} + 2696 \beta_{3} - 35175) q^{81} + ( - 1932 \beta_{5} + 49440 \beta_{2} - 6060 \beta_1) q^{82} + (4840 \beta_{5} - 31428 \beta_{2} - 3808 \beta_1) q^{83} + (3512 \beta_{4} + 4139 \beta_{3} - 33528) q^{84} + ( - 2458 \beta_{4} - 4737 \beta_{3} - 24878) q^{86} + ( - 154 \beta_{5} + 43218 \beta_{2} + 9418 \beta_1) q^{87} + (1752 \beta_{5} - 49272 \beta_{2} - 10194 \beta_1) q^{88} + ( - 432 \beta_{4} - 8864 \beta_{3} - 14186) q^{89} + (507 \beta_{4} + 507 \beta_{3} - 3211) q^{91} + ( - 7136 \beta_{5} + 99776 \beta_{2} + 1536 \beta_1) q^{92} + (932 \beta_{5} - 33924 \beta_{2} - 3620 \beta_1) q^{93} + ( - 4710 \beta_{4} + 7341 \beta_{3} - 21834) q^{94} + (13076 \beta_{4} + 18749 \beta_{3} - 74964) q^{96} + ( - 500 \beta_{5} + 25910 \beta_{2} - 1396 \beta_1) q^{97} + (1986 \beta_{5} + 13452 \beta_{2} + 14280 \beta_1) q^{98} + ( - 1928 \beta_{4} + 4720 \beta_{3} + 21684) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 242 q^{4} - 398 q^{6} + 382 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 242 q^{4} - 398 q^{6} + 382 q^{9} + 1112 q^{11} + 1586 q^{14} + 5570 q^{16} - 296 q^{19} + 2780 q^{21} + 26754 q^{24} + 2366 q^{26} + 17516 q^{29} - 5216 q^{31} - 31978 q^{34} - 9356 q^{36} - 2704 q^{39} - 21996 q^{41} - 99580 q^{44} - 6816 q^{46} + 89142 q^{49} + 26936 q^{51} - 111190 q^{54} - 98574 q^{56} - 127896 q^{59} - 25508 q^{61} - 234786 q^{64} - 224708 q^{66} + 40224 q^{69} + 155160 q^{71} + 263038 q^{74} - 199276 q^{76} + 123744 q^{79} - 216442 q^{81} - 209446 q^{84} - 139794 q^{86} - 67388 q^{89} - 20280 q^{91} - 145686 q^{94} - 487282 q^{96} + 120664 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 201x^{4} + 10512x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 55\nu^{3} + 15344\nu ) / 39936 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 101\nu^{2} + 256 ) / 156 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 153\nu^{2} + 3792 ) / 208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\nu^{5} + 7399\nu^{3} + 247280\nu ) / 39936 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4\beta_{4} - 3\beta_{3} - 68 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} + 188\beta_{2} - 97\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -404\beta_{4} + 459\beta_{3} + 6612 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 220\beta_{5} - 29596\beta_{2} + 10009\beta_1 \) 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
8.96778i
10.6486i
2.68079i
2.68079i
10.6486i
8.96778i
10.9678i 15.7989i −88.2923 0 −173.279 75.3967i 617.402i −6.60562 0
274.2 8.64858i 10.2870i −42.7979 0 −88.9676 2.86088i 93.3863i 137.178 0
274.3 4.68079i 13.5120i 10.0902 0 63.2466 12.5359i 197.015i 60.4272 0
274.4 4.68079i 13.5120i 10.0902 0 63.2466 12.5359i 197.015i 60.4272 0
274.5 8.64858i 10.2870i −42.7979 0 −88.9676 2.86088i 93.3863i 137.178 0
274.6 10.9678i 15.7989i −88.2923 0 −173.279 75.3967i 617.402i −6.60562 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.c 6
5.b even 2 1 inner 325.6.b.c 6
5.c odd 4 1 13.6.a.b 3
5.c odd 4 1 325.6.a.c 3
15.e even 4 1 117.6.a.d 3
20.e even 4 1 208.6.a.j 3
35.f even 4 1 637.6.a.b 3
40.i odd 4 1 832.6.a.s 3
40.k even 4 1 832.6.a.t 3
65.f even 4 1 169.6.b.b 6
65.h odd 4 1 169.6.a.b 3
65.k even 4 1 169.6.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.a.b 3 5.c odd 4 1
117.6.a.d 3 15.e even 4 1
169.6.a.b 3 65.h odd 4 1
169.6.b.b 6 65.f even 4 1
169.6.b.b 6 65.k even 4 1
208.6.a.j 3 20.e even 4 1
325.6.a.c 3 5.c odd 4 1
325.6.b.c 6 1.a even 1 1 trivial
325.6.b.c 6 5.b even 2 1 inner
637.6.a.b 3 35.f even 4 1
832.6.a.s 3 40.i odd 4 1
832.6.a.t 3 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 217T_{2}^{4} + 13272T_{2}^{2} + 197136 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 217 T^{4} + \cdots + 197136 \) Copy content Toggle raw display
$3$ \( T^{6} + 538 T^{4} + \cdots + 4822416 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 5850 T^{4} + \cdots + 7311616 \) Copy content Toggle raw display
$11$ \( (T^{3} - 556 T^{2} + \cdots + 39698256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 60\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{3} + 148 T^{2} + \cdots - 1415854512)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{3} - 8758 T^{2} + \cdots + 221025174456)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 2608 T^{2} + \cdots - 1607044480)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{3} + 10998 T^{2} + \cdots + 29456898048)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots + 1932677407728)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 18650455523968)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 37395101110464)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 2044988893184)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 28887869991912)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
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