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Properties

Label 336.4.bc.c
Level 336336
Weight 44
Character orbit 336.bc
Analytic conductor 19.82519.825
Analytic rank 00
Dimension 1212
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,4,Mod(17,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.17"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 336=24⋅3⋅7 336 = 2^{4} \cdot 3 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 336.bc (of order 66, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 19.824641761919.8246417619
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x12−⋯ )\mathbb{Q}[x]/(x^{12} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x12−14x10−32x9+70x8+224x7−50x6+2016x5+5670x4+⋯+531441 x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 24⋅37 2^{4}\cdot 3^{7}
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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+(β4−β3)q3−β11q5+(−β9−2β6−2β1+2)q7+(−β9−2β7+β6+⋯+1)q9+(β11−β10−2β8+⋯+2)q11+⋯+(−36β11−18β10+⋯−780)q99+O(q100) q + (\beta_{4} - \beta_{3}) q^{3} - \beta_{11} q^{5} + ( - \beta_{9} - 2 \beta_{6} - 2 \beta_1 + 2) q^{7} + ( - \beta_{9} - 2 \beta_{7} + \beta_{6} + \cdots + 1) q^{9} + (\beta_{11} - \beta_{10} - 2 \beta_{8} + \cdots + 2) q^{11}+ \cdots + ( - 36 \beta_{11} - 18 \beta_{10} + \cdots - 780) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+42q7−84q9−132q15−204q19−378q21−444q25−1458q31−108q33+240q37+432q39−342q45−1218q49+300q51+180q57+2148q61−1596q63+⋯−9216q99+O(q100) 12 q + 42 q^{7} - 84 q^{9} - 132 q^{15} - 204 q^{19} - 378 q^{21} - 444 q^{25} - 1458 q^{31} - 108 q^{33} + 240 q^{37} + 432 q^{39} - 342 q^{45} - 1218 q^{49} + 300 q^{51} + 180 q^{57} + 2148 q^{61} - 1596 q^{63}+ \cdots - 9216 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12−14x10−32x9+70x8+224x7−50x6+2016x5+5670x4+⋯+531441 x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441 : Copy content Toggle raw display

β1\beta_{1}== (−36ν11−109ν10+180ν9+1949ν8+2588ν7−1681ν6+⋯+6436341)/2571912 ( - 36 \nu^{11} - 109 \nu^{10} + 180 \nu^{9} + 1949 \nu^{8} + 2588 \nu^{7} - 1681 \nu^{6} + \cdots + 6436341 ) / 2571912 Copy content Toggle raw display
β2\beta_{2}== (871ν11+2952ν10+7489ν9−31292ν8−131909ν7−50236ν6+⋯−192499740)/23147208 ( 871 \nu^{11} + 2952 \nu^{10} + 7489 \nu^{9} - 31292 \nu^{8} - 131909 \nu^{7} - 50236 \nu^{6} + \cdots - 192499740 ) / 23147208 Copy content Toggle raw display
β3\beta_{3}== (109ν11+324ν10−797ν9−5108ν8−6383ν7+9188ν6+⋯−19131876)/2571912 ( 109 \nu^{11} + 324 \nu^{10} - 797 \nu^{9} - 5108 \nu^{8} - 6383 \nu^{7} + 9188 \nu^{6} + \cdots - 19131876 ) / 2571912 Copy content Toggle raw display
β4\beta_{4}== (−109ν11−324ν10+797ν9+5108ν8+6383ν7−9188ν6+⋯+19131876)/2571912 ( - 109 \nu^{11} - 324 \nu^{10} + 797 \nu^{9} + 5108 \nu^{8} + 6383 \nu^{7} - 9188 \nu^{6} + \cdots + 19131876 ) / 2571912 Copy content Toggle raw display
β5\beta_{5}== (−1522ν11−11709ν10+3326ν9+159413ν8+296822ν7+⋯+653377185)/23147208 ( - 1522 \nu^{11} - 11709 \nu^{10} + 3326 \nu^{9} + 159413 \nu^{8} + 296822 \nu^{7} + \cdots + 653377185 ) / 23147208 Copy content Toggle raw display
β6\beta_{6}== (−1643ν11−4941ν10+403ν9+56869ν8+156097ν7+⋯+250426809)/23147208 ( - 1643 \nu^{11} - 4941 \nu^{10} + 403 \nu^{9} + 56869 \nu^{8} + 156097 \nu^{7} + \cdots + 250426809 ) / 23147208 Copy content Toggle raw display
β7\beta_{7}== (−58ν11−36ν10+569ν9+1631ν8+494ν7+2470ν6+⋯+2834352)/826686 ( - 58 \nu^{11} - 36 \nu^{10} + 569 \nu^{9} + 1631 \nu^{8} + 494 \nu^{7} + 2470 \nu^{6} + \cdots + 2834352 ) / 826686 Copy content Toggle raw display
β8\beta_{8}== (−1931ν11+774ν10−425ν9+5758ν8−83879ν7+542606ν6+⋯−65544390)/23147208 ( - 1931 \nu^{11} + 774 \nu^{10} - 425 \nu^{9} + 5758 \nu^{8} - 83879 \nu^{7} + 542606 \nu^{6} + \cdots - 65544390 ) / 23147208 Copy content Toggle raw display
β9\beta_{9}== (74ν11+207ν10−793ν9−3808ν8−4846ν7+2878ν6+⋯−13581270)/826686 ( 74 \nu^{11} + 207 \nu^{10} - 793 \nu^{9} - 3808 \nu^{8} - 4846 \nu^{7} + 2878 \nu^{6} + \cdots - 13581270 ) / 826686 Copy content Toggle raw display
β10\beta_{10}== (−125ν11−477ν10+454ν9+9949ν8+18097ν7−9712ν6+⋯+40743810)/826686 ( - 125 \nu^{11} - 477 \nu^{10} + 454 \nu^{9} + 9949 \nu^{8} + 18097 \nu^{7} - 9712 \nu^{6} + \cdots + 40743810 ) / 826686 Copy content Toggle raw display
β11\beta_{11}== (−3554ν11−16407ν10+12982ν9+145867ν8+174346ν7+⋯+264480471)/23147208 ( - 3554 \nu^{11} - 16407 \nu^{10} + 12982 \nu^{9} + 145867 \nu^{8} + 174346 \nu^{7} + \cdots + 264480471 ) / 23147208 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β4+β3)/3 ( \beta_{4} + \beta_{3} ) / 3 Copy content Toggle raw display
ν2\nu^{2}== (β9+2β7−β6+β5+β4−β3−β2−12β1−1)/3 ( \beta_{9} + 2\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 12\beta _1 - 1 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== (2β11+β10+β9−2β8+4β7−5β6−2β5+⋯+26)/3 ( 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + \cdots + 26 ) / 3 Copy content Toggle raw display
ν4\nu^{4}== (14β9−5β8+9β7+14β6+16β4+21β3+14β2+56β1+47)/3 ( 14\beta_{9} - 5\beta_{8} + 9\beta_{7} + 14\beta_{6} + 16\beta_{4} + 21\beta_{3} + 14\beta_{2} + 56\beta _1 + 47 ) / 3 Copy content Toggle raw display
ν5\nu^{5}== (14β11+28β10+33β9+21β7−19β6−12β5+78β4+⋯−19)/3 ( 14 \beta_{11} + 28 \beta_{10} + 33 \beta_{9} + 21 \beta_{7} - 19 \beta_{6} - 12 \beta_{5} + 78 \beta_{4} + \cdots - 19 ) / 3 Copy content Toggle raw display
ν6\nu^{6}== (32β11+16β10−110β9+24β8+50β7+60β6+24β5+⋯+721)/3 ( 32 \beta_{11} + 16 \beta_{10} - 110 \beta_{9} + 24 \beta_{8} + 50 \beta_{7} + 60 \beta_{6} + 24 \beta_{5} + \cdots + 721 ) / 3 Copy content Toggle raw display
ν7\nu^{7}== (−70β11+70β10−266β9−164β8+214β7+378β6+⋯−3490)/3 ( - 70 \beta_{11} + 70 \beta_{10} - 266 \beta_{9} - 164 \beta_{8} + 214 \beta_{7} + 378 \beta_{6} + \cdots - 3490 ) / 3 Copy content Toggle raw display
ν8\nu^{8}== (448β11+896β10+1521β9+1566β7−1703β6+45β5+⋯−1703)/3 ( 448 \beta_{11} + 896 \beta_{10} + 1521 \beta_{9} + 1566 \beta_{7} - 1703 \beta_{6} + 45 \beta_{5} + \cdots - 1703 ) / 3 Copy content Toggle raw display
ν9\nu^{9}== (622β11+311β10−3721β9−1966β8+194β7+3527β6+⋯+12070)/3 ( 622 \beta_{11} + 311 \beta_{10} - 3721 \beta_{9} - 1966 \beta_{8} + 194 \beta_{7} + 3527 \beta_{6} + \cdots + 12070 ) / 3 Copy content Toggle raw display
ν10\nu^{10}== (−5376β11+5376β10+6104β9+3415β8+8175β7+4760β6+⋯−48985)/3 ( - 5376 \beta_{11} + 5376 \beta_{10} + 6104 \beta_{9} + 3415 \beta_{8} + 8175 \beta_{7} + 4760 \beta_{6} + \cdots - 48985 ) / 3 Copy content Toggle raw display
ν11\nu^{11}== (10136β11+20272β10−28127β9−25519β7−10121β6+2608β5+⋯−10121)/3 ( 10136 \beta_{11} + 20272 \beta_{10} - 28127 \beta_{9} - 25519 \beta_{7} - 10121 \beta_{6} + 2608 \beta_{5} + \cdots - 10121 ) / 3 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/336Z)×\left(\mathbb{Z}/336\mathbb{Z}\right)^\times.

nn 8585 113113 127127 241241
χ(n)\chi(n) 11 −1-1 11 −β1-\beta_{1}

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
17.1
−1.67777 − 2.48698i
−2.14770 − 2.09461i
2.99268 + 0.209499i
−2.92030 + 0.686929i
2.88784 + 0.812653i
0.865250 + 2.87252i
−1.67777 + 2.48698i
−2.14770 + 2.09461i
2.99268 − 0.209499i
−2.92030 − 0.686929i
2.88784 − 0.812653i
0.865250 − 2.87252i
0 −4.30758 + 2.90598i 0 −7.41560 − 12.8442i 0 16.6528 + 8.10467i 0 10.1105 − 25.0355i 0
17.2 0 −3.62798 + 3.71992i 0 9.68891 + 16.7817i 0 2.66851 − 18.3270i 0 −0.675594 − 26.9915i 0
17.3 0 0.362863 − 5.18347i 0 7.41560 + 12.8442i 0 16.6528 + 8.10467i 0 −26.7367 − 3.76177i 0
17.4 0 1.18980 + 5.05810i 0 0.619556 + 1.07310i 0 −8.82127 + 16.2845i 0 −24.1688 + 12.0362i 0
17.5 0 1.40756 − 5.00188i 0 −9.68891 − 16.7817i 0 2.66851 − 18.3270i 0 −23.0376 − 14.0809i 0
17.6 0 4.97534 − 1.49866i 0 −0.619556 − 1.07310i 0 −8.82127 + 16.2845i 0 22.5081 − 14.9127i 0
257.1 0 −4.30758 − 2.90598i 0 −7.41560 + 12.8442i 0 16.6528 − 8.10467i 0 10.1105 + 25.0355i 0
257.2 0 −3.62798 − 3.71992i 0 9.68891 − 16.7817i 0 2.66851 + 18.3270i 0 −0.675594 + 26.9915i 0
257.3 0 0.362863 + 5.18347i 0 7.41560 − 12.8442i 0 16.6528 − 8.10467i 0 −26.7367 + 3.76177i 0
257.4 0 1.18980 − 5.05810i 0 0.619556 − 1.07310i 0 −8.82127 − 16.2845i 0 −24.1688 − 12.0362i 0
257.5 0 1.40756 + 5.00188i 0 −9.68891 + 16.7817i 0 2.66851 + 18.3270i 0 −23.0376 + 14.0809i 0
257.6 0 4.97534 + 1.49866i 0 −0.619556 + 1.07310i 0 −8.82127 − 16.2845i 0 22.5081 + 14.9127i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.c 12
3.b odd 2 1 inner 336.4.bc.c 12
4.b odd 2 1 84.4.k.c ✓ 12
7.d odd 6 1 inner 336.4.bc.c 12
12.b even 2 1 84.4.k.c ✓ 12
21.g even 6 1 inner 336.4.bc.c 12
28.d even 2 1 588.4.k.c 12
28.f even 6 1 84.4.k.c ✓ 12
28.f even 6 1 588.4.f.c 12
28.g odd 6 1 588.4.f.c 12
28.g odd 6 1 588.4.k.c 12
84.h odd 2 1 588.4.k.c 12
84.j odd 6 1 84.4.k.c ✓ 12
84.j odd 6 1 588.4.f.c 12
84.n even 6 1 588.4.f.c 12
84.n even 6 1 588.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.c ✓ 12 4.b odd 2 1
84.4.k.c ✓ 12 12.b even 2 1
84.4.k.c ✓ 12 28.f even 6 1
84.4.k.c ✓ 12 84.j odd 6 1
336.4.bc.c 12 1.a even 1 1 trivial
336.4.bc.c 12 3.b odd 2 1 inner
336.4.bc.c 12 7.d odd 6 1 inner
336.4.bc.c 12 21.g even 6 1 inner
588.4.f.c 12 28.f even 6 1
588.4.f.c 12 28.g odd 6 1
588.4.f.c 12 84.j odd 6 1
588.4.f.c 12 84.n even 6 1
588.4.k.c 12 28.d even 2 1
588.4.k.c 12 28.g odd 6 1
588.4.k.c 12 84.h odd 2 1
588.4.k.c 12 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(336,[χ])S_{4}^{\mathrm{new}}(336, [\chi]):

T512+597T510+272898T58+49602429T56+6898376178T54+10590781509T52+16083058761 T_{5}^{12} + 597T_{5}^{10} + 272898T_{5}^{8} + 49602429T_{5}^{6} + 6898376178T_{5}^{4} + 10590781509T_{5}^{2} + 16083058761 Copy content Toggle raw display
T136+11724T134+42650496T132+49422707712 T_{13}^{6} + 11724T_{13}^{4} + 42650496T_{13}^{2} + 49422707712 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12 T^{12} Copy content Toggle raw display
33 T12+⋯+387420489 T^{12} + \cdots + 387420489 Copy content Toggle raw display
55 T12+⋯+16083058761 T^{12} + \cdots + 16083058761 Copy content Toggle raw display
77 (T6−21T5+⋯+40353607)2 (T^{6} - 21 T^{5} + \cdots + 40353607)^{2} Copy content Toggle raw display
1111 T12+⋯+11724549836769 T^{12} + \cdots + 11724549836769 Copy content Toggle raw display
1313 (T6+11724T4+⋯+49422707712)2 (T^{6} + 11724 T^{4} + \cdots + 49422707712)^{2} Copy content Toggle raw display
1717 T12+⋯+12 ⁣⋯ ⁣96 T^{12} + \cdots + 12\!\cdots\!96 Copy content Toggle raw display
1919 (T6+102T5+⋯+33716904588)2 (T^{6} + 102 T^{5} + \cdots + 33716904588)^{2} Copy content Toggle raw display
2323 T12+⋯+45 ⁣⋯ ⁣04 T^{12} + \cdots + 45\!\cdots\!04 Copy content Toggle raw display
2929 (T6+55377T4+⋯+42952073472)2 (T^{6} + 55377 T^{4} + \cdots + 42952073472)^{2} Copy content Toggle raw display
3131 (T6+⋯+2248345829547)2 (T^{6} + \cdots + 2248345829547)^{2} Copy content Toggle raw display
3737 (T6−120T5+⋯+236777613604)2 (T^{6} - 120 T^{5} + \cdots + 236777613604)^{2} Copy content Toggle raw display
4141 (T6+⋯−18213269969664)2 (T^{6} + \cdots - 18213269969664)^{2} Copy content Toggle raw display
4343 (T3−100848T−12209024)4 (T^{3} - 100848 T - 12209024)^{4} Copy content Toggle raw display
4747 T12+⋯+18 ⁣⋯ ⁣36 T^{12} + \cdots + 18\!\cdots\!36 Copy content Toggle raw display
5353 T12+⋯+17 ⁣⋯ ⁣89 T^{12} + \cdots + 17\!\cdots\!89 Copy content Toggle raw display
5959 T12+⋯+36 ⁣⋯ ⁣01 T^{12} + \cdots + 36\!\cdots\!01 Copy content Toggle raw display
6161 (T6+⋯+687780167594688)2 (T^{6} + \cdots + 687780167594688)^{2} Copy content Toggle raw display
6767 (T6+⋯+11 ⁣⋯ ⁣44)2 (T^{6} + \cdots + 11\!\cdots\!44)^{2} Copy content Toggle raw display
7171 (T6+⋯+15 ⁣⋯ ⁣72)2 (T^{6} + \cdots + 15\!\cdots\!72)^{2} Copy content Toggle raw display
7373 (T6+⋯+13 ⁣⋯ ⁣72)2 (T^{6} + \cdots + 13\!\cdots\!72)^{2} Copy content Toggle raw display
7979 (T6+⋯+23 ⁣⋯ ⁣01)2 (T^{6} + \cdots + 23\!\cdots\!01)^{2} Copy content Toggle raw display
8383 (T6+⋯−18 ⁣⋯ ⁣04)2 (T^{6} + \cdots - 18\!\cdots\!04)^{2} Copy content Toggle raw display
8989 T12+⋯+14 ⁣⋯ ⁣76 T^{12} + \cdots + 14\!\cdots\!76 Copy content Toggle raw display
9797 (T6+⋯+59 ⁣⋯ ⁣52)2 (T^{6} + \cdots + 59\!\cdots\!52)^{2} Copy content Toggle raw display
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