Properties

Label 507.2.j.i
Level 507507
Weight 22
Character orbit 507.j
Analytic conductor 4.0484.048
Analytic rank 00
Dimension 1212
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(316,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 507=3132 507 = 3 \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 507.j (of order 66, degree 22, 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: 4.048415382484.04841538248
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x125x10+19x828x6+31x46x2+1 x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(2β11+β8β1)q2+β7q3+(β93β7β4+3)q4+(β11+3β10++β2)q5+(2β2β1)q6++(2β112β10+2β2)q99+O(q100) q + (2 \beta_{11} + \beta_{8} - \beta_1) q^{2} + \beta_{7} q^{3} + ( - \beta_{9} - 3 \beta_{7} - \beta_{4} + 3) q^{4} + (\beta_{11} + 3 \beta_{10} + \cdots + \beta_{2}) q^{5} + ( - 2 \beta_{2} - \beta_1) q^{6}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+6q3+22q46q92q10+44q1220q1422q162q17+18q2244q2512q27+4q29+2q308q35+22q36+12q4010q4230q43+42q95+O(q100) 12 q + 6 q^{3} + 22 q^{4} - 6 q^{9} - 2 q^{10} + 44 q^{12} - 20 q^{14} - 22 q^{16} - 2 q^{17} + 18 q^{22} - 44 q^{25} - 12 q^{27} + 4 q^{29} + 2 q^{30} - 8 q^{35} + 22 q^{36} + 12 q^{40} - 10 q^{42} - 30 q^{43}+ \cdots - 42 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x125x10+19x828x6+31x46x2+1 x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (25ν11+95ν9361ν7+155ν530ν31563ν)/559 ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 Copy content Toggle raw display
β3\beta_{3}== (25ν1095ν8+361ν6155ν4+30ν2+1004)/559 ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 Copy content Toggle raw display
β4\beta_{4}== (3ν1020ν8+76ν6139ν4+124ν224)/43 ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 Copy content Toggle raw display
β5\beta_{5}== (45ν10171ν8+538ν6279ν4+54ν2+242)/559 ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 Copy content Toggle raw display
β6\beta_{6}== (70ν11266ν9+899ν7434ν5+84ν3+1246ν)/559 ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 Copy content Toggle raw display
β7\beta_{7}== (114ν10545ν8+2071ν62831ν4+3379ν295)/559 ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 Copy content Toggle raw display
β8\beta_{8}== (114ν11545ν9+2071ν72831ν5+3379ν395ν)/559 ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 Copy content Toggle raw display
β9\beta_{9}== (128ν10+710ν82698ν6+4483ν44402ν2+852)/559 ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 Copy content Toggle raw display
β10\beta_{10}== (242ν11+1255ν94769ν7+7314ν57781ν3+1506ν)/559 ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 Copy content Toggle raw display
β11\beta_{11}== (317ν11+1540ν95852ν7+8338ν59548ν3+1848ν)/559 ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β9+β7+β4β3+1 \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 Copy content Toggle raw display
ν3\nu^{3}== β11+3β8+β2 \beta_{11} + 3\beta_{8} + \beta_{2} Copy content Toggle raw display
ν4\nu^{4}== 3β9+2β7+4β42 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 Copy content Toggle raw display
ν5\nu^{5}== 4β11β10+9β89β1 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 5β5+9β314 -5\beta_{5} + 9\beta_{3} - 14 Copy content Toggle raw display
ν7\nu^{7}== 5β614β228β1 -5\beta_{6} - 14\beta_{2} - 28\beta_1 Copy content Toggle raw display
ν8\nu^{8}== 28β914β719β547β4+28β328 -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 Copy content Toggle raw display
ν9\nu^{9}== 47β11+19β1089β819β647β2 -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} Copy content Toggle raw display
ν10\nu^{10}== 89β942β7155β4+42 -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 Copy content Toggle raw display
ν11\nu^{11}== 155β11+66β10286β8+286β1 -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/507Z)×\left(\mathbb{Z}/507\mathbb{Z}\right)^\times.

nn 170170 340340
χ(n)\chi(n) 11 1β71 - \beta_{7}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
316.1
1.56052 0.900969i
−1.07992 + 0.623490i
−0.385418 + 0.222521i
0.385418 0.222521i
1.07992 0.623490i
−1.56052 + 0.900969i
1.56052 + 0.900969i
−1.07992 0.623490i
−0.385418 0.222521i
0.385418 + 0.222521i
1.07992 + 0.623490i
−1.56052 0.900969i
−2.33136 1.34601i 0.500000 0.866025i 2.62349 + 4.54402i 1.04892i −2.33136 + 1.34601i −0.480608 + 0.277479i 8.74094i −0.500000 0.866025i −1.41185 + 2.44540i
316.2 −2.04113 1.17845i 0.500000 0.866025i 1.77748 + 3.07868i 3.69202i −2.04113 + 1.17845i 0.694498 0.400969i 3.66487i −0.500000 0.866025i 4.35086 7.53590i
316.3 −1.77441 1.02446i 0.500000 0.866025i 1.09903 + 1.90358i 3.35690i −1.77441 + 1.02446i 1.94594 1.12349i 0.405813i −0.500000 0.866025i −3.43900 + 5.95652i
316.4 1.77441 + 1.02446i 0.500000 0.866025i 1.09903 + 1.90358i 3.35690i 1.77441 1.02446i −1.94594 + 1.12349i 0.405813i −0.500000 0.866025i −3.43900 + 5.95652i
316.5 2.04113 + 1.17845i 0.500000 0.866025i 1.77748 + 3.07868i 3.69202i 2.04113 1.17845i −0.694498 + 0.400969i 3.66487i −0.500000 0.866025i 4.35086 7.53590i
316.6 2.33136 + 1.34601i 0.500000 0.866025i 2.62349 + 4.54402i 1.04892i 2.33136 1.34601i 0.480608 0.277479i 8.74094i −0.500000 0.866025i −1.41185 + 2.44540i
361.1 −2.33136 + 1.34601i 0.500000 + 0.866025i 2.62349 4.54402i 1.04892i −2.33136 1.34601i −0.480608 0.277479i 8.74094i −0.500000 + 0.866025i −1.41185 2.44540i
361.2 −2.04113 + 1.17845i 0.500000 + 0.866025i 1.77748 3.07868i 3.69202i −2.04113 1.17845i 0.694498 + 0.400969i 3.66487i −0.500000 + 0.866025i 4.35086 + 7.53590i
361.3 −1.77441 + 1.02446i 0.500000 + 0.866025i 1.09903 1.90358i 3.35690i −1.77441 1.02446i 1.94594 + 1.12349i 0.405813i −0.500000 + 0.866025i −3.43900 5.95652i
361.4 1.77441 1.02446i 0.500000 + 0.866025i 1.09903 1.90358i 3.35690i 1.77441 + 1.02446i −1.94594 1.12349i 0.405813i −0.500000 + 0.866025i −3.43900 5.95652i
361.5 2.04113 1.17845i 0.500000 + 0.866025i 1.77748 3.07868i 3.69202i 2.04113 + 1.17845i −0.694498 0.400969i 3.66487i −0.500000 + 0.866025i 4.35086 + 7.53590i
361.6 2.33136 1.34601i 0.500000 + 0.866025i 2.62349 4.54402i 1.04892i 2.33136 + 1.34601i 0.480608 + 0.277479i 8.74094i −0.500000 + 0.866025i −1.41185 2.44540i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 316.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.i 12
13.b even 2 1 inner 507.2.j.i 12
13.c even 3 1 507.2.b.f 6
13.c even 3 1 inner 507.2.j.i 12
13.d odd 4 1 507.2.e.i 6
13.d odd 4 1 507.2.e.l 6
13.e even 6 1 507.2.b.f 6
13.e even 6 1 inner 507.2.j.i 12
13.f odd 12 1 507.2.a.i 3
13.f odd 12 1 507.2.a.l yes 3
13.f odd 12 1 507.2.e.i 6
13.f odd 12 1 507.2.e.l 6
39.h odd 6 1 1521.2.b.k 6
39.i odd 6 1 1521.2.b.k 6
39.k even 12 1 1521.2.a.n 3
39.k even 12 1 1521.2.a.s 3
52.l even 12 1 8112.2.a.cg 3
52.l even 12 1 8112.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.f odd 12 1
507.2.a.l yes 3 13.f odd 12 1
507.2.b.f 6 13.c even 3 1
507.2.b.f 6 13.e even 6 1
507.2.e.i 6 13.d odd 4 1
507.2.e.i 6 13.f odd 12 1
507.2.e.l 6 13.d odd 4 1
507.2.e.l 6 13.f odd 12 1
507.2.j.i 12 1.a even 1 1 trivial
507.2.j.i 12 13.b even 2 1 inner
507.2.j.i 12 13.c even 3 1 inner
507.2.j.i 12 13.e even 6 1 inner
1521.2.a.n 3 39.k even 12 1
1521.2.a.s 3 39.k even 12 1
1521.2.b.k 6 39.h odd 6 1
1521.2.b.k 6 39.i odd 6 1
8112.2.a.cg 3 52.l even 12 1
8112.2.a.cp 3 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(507,[χ])S_{2}^{\mathrm{new}}(507, [\chi]):

T21217T210+195T281260T26+5963T2415886T22+28561 T_{2}^{12} - 17T_{2}^{10} + 195T_{2}^{8} - 1260T_{2}^{6} + 5963T_{2}^{4} - 15886T_{2}^{2} + 28561 Copy content Toggle raw display
T56+26T54+181T52+169 T_{5}^{6} + 26T_{5}^{4} + 181T_{5}^{2} + 169 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T1217T10++28561 T^{12} - 17 T^{10} + \cdots + 28561 Copy content Toggle raw display
33 (T2T+1)6 (T^{2} - T + 1)^{6} Copy content Toggle raw display
55 (T6+26T4++169)2 (T^{6} + 26 T^{4} + \cdots + 169)^{2} Copy content Toggle raw display
77 T126T10++1 T^{12} - 6 T^{10} + \cdots + 1 Copy content Toggle raw display
1111 T1241T10++2825761 T^{12} - 41 T^{10} + \cdots + 2825761 Copy content Toggle raw display
1313 T12 T^{12} Copy content Toggle raw display
1717 (T6+T5+17T4++169)2 (T^{6} + T^{5} + 17 T^{4} + \cdots + 169)^{2} Copy content Toggle raw display
1919 T1221T10++2401 T^{12} - 21 T^{10} + \cdots + 2401 Copy content Toggle raw display
2323 (T6+49T4++8281)2 (T^{6} + 49 T^{4} + \cdots + 8281)^{2} Copy content Toggle raw display
2929 (T62T5++841)2 (T^{6} - 2 T^{5} + \cdots + 841)^{2} Copy content Toggle raw display
3131 (T6+174T4++38809)2 (T^{6} + 174 T^{4} + \cdots + 38809)^{2} Copy content Toggle raw display
3737 T12++20200652641 T^{12} + \cdots + 20200652641 Copy content Toggle raw display
4141 T1273T10++707281 T^{12} - 73 T^{10} + \cdots + 707281 Copy content Toggle raw display
4343 (T6+15T5++1681)2 (T^{6} + 15 T^{5} + \cdots + 1681)^{2} Copy content Toggle raw display
4747 (T6+21T4++49)2 (T^{6} + 21 T^{4} + \cdots + 49)^{2} Copy content Toggle raw display
5353 (T3+17T2+41)4 (T^{3} + 17 T^{2} + \cdots - 41)^{4} Copy content Toggle raw display
5959 T12++116985856 T^{12} + \cdots + 116985856 Copy content Toggle raw display
6161 (T613T5++27889)2 (T^{6} - 13 T^{5} + \cdots + 27889)^{2} Copy content Toggle raw display
6767 T12213T10++2825761 T^{12} - 213 T^{10} + \cdots + 2825761 Copy content Toggle raw display
7171 T12++1698181681 T^{12} + \cdots + 1698181681 Copy content Toggle raw display
7373 (T6+306T4++851929)2 (T^{6} + 306 T^{4} + \cdots + 851929)^{2} Copy content Toggle raw display
7979 (T33T218T+27)4 (T^{3} - 3 T^{2} - 18 T + 27)^{4} Copy content Toggle raw display
8383 (T6+62T4++1849)2 (T^{6} + 62 T^{4} + \cdots + 1849)^{2} Copy content Toggle raw display
8989 T12++163047361 T^{12} + \cdots + 163047361 Copy content Toggle raw display
9797 T12++7181161893361 T^{12} + \cdots + 7181161893361 Copy content Toggle raw display
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