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Properties

Label 315.4.j.i
Level 315315
Weight 44
Character orbit 315.j
Analytic conductor 18.58618.586
Analytic rank 00
Dimension 1616
Inner twists 22

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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] = [315,4,Mod(46,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) 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("315.46"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 315=32⋅5⋅7 315 = 3^{2} \cdot 5 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 315.j (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-4,0,-42,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 18.585601651818.5856016518
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x16−⋯ )\mathbb{Q}[x]/(x^{16} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16−4x15+61x14−88x13+1920x12−2032x11+36527x10−1714x9+⋯+13660416 x^{16} - 4 x^{15} + 61 x^{14} - 88 x^{13} + 1920 x^{12} - 2032 x^{11} + 36527 x^{10} - 1714 x^{9} + \cdots + 13660416 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,…,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β6+β1)q2+(β11−6β6−β5+⋯−6)q4−5β6q5+(−β6−β2−β1+1)q7+(−β12+β8+β6+⋯+9)q8+⋯+(6β15+21β14+⋯−96)q98+O(q100) q + (\beta_{6} + \beta_1) q^{2} + (\beta_{11} - 6 \beta_{6} - \beta_{5} + \cdots - 6) q^{4} - 5 \beta_{6} q^{5} + ( - \beta_{6} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{12} + \beta_{8} + \beta_{6} + \cdots + 9) q^{8}+ \cdots + (6 \beta_{15} + 21 \beta_{14} + \cdots - 96) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q−4q2−42q4+40q5+22q7+96q8+20q10−100q11−204q13+178q14−266q16−56q17−420q20−140q22−190q23−200q25−60q26−4q28+⋯−1514q98+O(q100) 16 q - 4 q^{2} - 42 q^{4} + 40 q^{5} + 22 q^{7} + 96 q^{8} + 20 q^{10} - 100 q^{11} - 204 q^{13} + 178 q^{14} - 266 q^{16} - 56 q^{17} - 420 q^{20} - 140 q^{22} - 190 q^{23} - 200 q^{25} - 60 q^{26} - 4 q^{28}+ \cdots - 1514 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−4x15+61x14−88x13+1920x12−2032x11+36527x10−1714x9+⋯+13660416 x^{16} - 4 x^{15} + 61 x^{14} - 88 x^{13} + 1920 x^{12} - 2032 x^{11} + 36527 x^{10} - 1714 x^{9} + \cdots + 13660416 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (60 ⁣⋯ ⁣59ν15+⋯−71 ⁣⋯ ⁣88)/26 ⁣⋯ ⁣44 ( 60\!\cdots\!59 \nu^{15} + \cdots - 71\!\cdots\!88 ) / 26\!\cdots\!44 Copy content Toggle raw display
β3\beta_{3}== (56 ⁣⋯ ⁣51ν15+⋯−20 ⁣⋯ ⁣36)/13 ⁣⋯ ⁣72 ( 56\!\cdots\!51 \nu^{15} + \cdots - 20\!\cdots\!36 ) / 13\!\cdots\!72 Copy content Toggle raw display
β4\beta_{4}== (−17 ⁣⋯ ⁣93ν15+⋯−38 ⁣⋯ ⁣72)/38 ⁣⋯ ⁣72 ( - 17\!\cdots\!93 \nu^{15} + \cdots - 38\!\cdots\!72 ) / 38\!\cdots\!72 Copy content Toggle raw display
β5\beta_{5}== (−86 ⁣⋯ ⁣65ν15+⋯−14 ⁣⋯ ⁣92)/12 ⁣⋯ ⁣24 ( - 86\!\cdots\!65 \nu^{15} + \cdots - 14\!\cdots\!92 ) / 12\!\cdots\!24 Copy content Toggle raw display
β6\beta_{6}== (61 ⁣⋯ ⁣99ν15+⋯−61 ⁣⋯ ⁣12)/85 ⁣⋯ ⁣84 ( 61\!\cdots\!99 \nu^{15} + \cdots - 61\!\cdots\!12 ) / 85\!\cdots\!84 Copy content Toggle raw display
β7\beta_{7}== (77 ⁣⋯ ⁣41ν15+⋯+12 ⁣⋯ ⁣80)/33 ⁣⋯ ⁣68 ( 77\!\cdots\!41 \nu^{15} + \cdots + 12\!\cdots\!80 ) / 33\!\cdots\!68 Copy content Toggle raw display
β8\beta_{8}== (−22 ⁣⋯ ⁣55ν15+⋯−13 ⁣⋯ ⁣00)/61 ⁣⋯ ⁣76 ( - 22\!\cdots\!55 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 61\!\cdots\!76 Copy content Toggle raw display
β9\beta_{9}== (10 ⁣⋯ ⁣47ν15+⋯+94 ⁣⋯ ⁣12)/24 ⁣⋯ ⁣04 ( 10\!\cdots\!47 \nu^{15} + \cdots + 94\!\cdots\!12 ) / 24\!\cdots\!04 Copy content Toggle raw display
β10\beta_{10}== (−83 ⁣⋯ ⁣39ν15+⋯+27 ⁣⋯ ⁣48)/13 ⁣⋯ ⁣72 ( - 83\!\cdots\!39 \nu^{15} + \cdots + 27\!\cdots\!48 ) / 13\!\cdots\!72 Copy content Toggle raw display
β11\beta_{11}== (70 ⁣⋯ ⁣51ν15+⋯−75 ⁣⋯ ⁣20)/85 ⁣⋯ ⁣84 ( 70\!\cdots\!51 \nu^{15} + \cdots - 75\!\cdots\!20 ) / 85\!\cdots\!84 Copy content Toggle raw display
β12\beta_{12}== (11 ⁣⋯ ⁣19ν15+⋯−12 ⁣⋯ ⁣76)/13 ⁣⋯ ⁣72 ( 11\!\cdots\!19 \nu^{15} + \cdots - 12\!\cdots\!76 ) / 13\!\cdots\!72 Copy content Toggle raw display
β13\beta_{13}== (60 ⁣⋯ ⁣65ν15+⋯+33 ⁣⋯ ⁣40)/26 ⁣⋯ ⁣44 ( 60\!\cdots\!65 \nu^{15} + \cdots + 33\!\cdots\!40 ) / 26\!\cdots\!44 Copy content Toggle raw display
β14\beta_{14}== (11 ⁣⋯ ⁣44ν15+⋯−13 ⁣⋯ ⁣64)/33 ⁣⋯ ⁣68 ( 11\!\cdots\!44 \nu^{15} + \cdots - 13\!\cdots\!64 ) / 33\!\cdots\!68 Copy content Toggle raw display
β15\beta_{15}== (56 ⁣⋯ ⁣01ν15+⋯+66 ⁣⋯ ⁣04)/13 ⁣⋯ ⁣72 ( 56\!\cdots\!01 \nu^{15} + \cdots + 66\!\cdots\!04 ) / 13\!\cdots\!72 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β11−13β6−β5−β4+β1−13 \beta_{11} - 13\beta_{6} - \beta_{5} - \beta_{4} + \beta _1 - 13 Copy content Toggle raw display
ν3\nu^{3}== −β12+β8+β6−2β5−22β4+β3−15 -\beta_{12} + \beta_{8} + \beta_{6} - 2\beta_{5} - 22\beta_{4} + \beta_{3} - 15 Copy content Toggle raw display
ν4\nu^{4}== 3β14−β13−2β12−32β11+β10+β9+3β8+⋯−50β1 3 \beta_{14} - \beta_{13} - 2 \beta_{12} - 32 \beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{8} + \cdots - 50 \beta_1 Copy content Toggle raw display
ν5\nu^{5}== 2β15+42β14−42β13−6β12−96β11+14β10+28β9+⋯+722 2 \beta_{15} + 42 \beta_{14} - 42 \beta_{13} - 6 \beta_{12} - 96 \beta_{11} + 14 \beta_{10} + 28 \beta_{9} + \cdots + 722 Copy content Toggle raw display
ν6\nu^{6}== 50β15−74β13+70β12+24β10−172β8+50β7+⋯+7997 50 \beta_{15} - 74 \beta_{13} + 70 \beta_{12} + 24 \beta_{10} - 172 \beta_{8} + 50 \beta_{7} + \cdots + 7997 Copy content Toggle raw display
ν7\nu^{7}== −1563β14+1169β13+1565β12+3820β11−396β10+⋯+17192β1 - 1563 \beta_{14} + 1169 \beta_{13} + 1565 \beta_{12} + 3820 \beta_{11} - 396 \beta_{10} + \cdots + 17192 \beta_1 Copy content Toggle raw display
ν8\nu^{8}== −2013β15−7571β14+6960β13+3469β12+31546β11+⋯−238739 - 2013 \beta_{15} - 7571 \beta_{14} + 6960 \beta_{13} + 3469 \beta_{12} + 31546 \beta_{11} + \cdots - 238739 Copy content Toggle raw display
ν9\nu^{9}== −8760β15+19064β13−38120β12−19016β10+56916β8+⋯−1029984 - 8760 \beta_{15} + 19064 \beta_{13} - 38120 \beta_{12} - 19016 \beta_{10} + 56916 \beta_{8} + \cdots - 1029984 Copy content Toggle raw display
ν10\nu^{10}== 302172β14−149032β13−290552β12−1038829β11+141520β10+⋯−2543549β1 302172 \beta_{14} - 149032 \beta_{13} - 290552 \beta_{12} - 1038829 \beta_{11} + 141520 \beta_{10} + \cdots - 2543549 \beta_1 Copy content Toggle raw display
ν11\nu^{11}== 381240β15+2057257β14−2074373β13−803576β12−5234054β11+⋯+36710499 381240 \beta_{15} + 2057257 \beta_{14} - 2074373 \beta_{13} - 803576 \beta_{12} - 5234054 \beta_{11} + \cdots + 36710499 Copy content Toggle raw display
ν12\nu^{12}== 2814613β15−5445677β13+5869457β12+3894592β10−11488431β8+⋯+246606449 2814613 \beta_{15} - 5445677 \beta_{13} + 5869457 \beta_{12} + 3894592 \beta_{10} - 11488431 \beta_{8} + \cdots + 246606449 Copy content Toggle raw display
ν13\nu^{13}== −74008386β14+43252592β13+74858386β12+188363376β11+⋯+575973043β1 - 74008386 \beta_{14} + 43252592 \beta_{13} + 74858386 \beta_{12} + 188363376 \beta_{11} + \cdots + 575973043 \beta_1 Copy content Toggle raw display
ν14\nu^{14}== −102637418β15−425070592β14+424470416β13+203311002β12+⋯−8285372429 - 102637418 \beta_{15} - 425070592 \beta_{14} + 424470416 \beta_{13} + 203311002 \beta_{12} + \cdots - 8285372429 Copy content Toggle raw display
ν15\nu^{15}== −578940360β15+1195029020β13−1494652213β12−1022446512β10+⋯−45663266503 - 578940360 \beta_{15} + 1195029020 \beta_{13} - 1494652213 \beta_{12} - 1022446512 \beta_{10} + \cdots - 45663266503 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/315Z)×\left(\mathbb{Z}/315\mathbb{Z}\right)^\times.

nn 127127 136136 281281
χ(n)\chi(n) 11 β6\beta_{6} 11

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}
46.1
−2.29598 − 3.97675i
−1.71036 − 2.96243i
−1.01204 − 1.75290i
−0.475331 − 0.823298i
0.906849 + 1.57071i
1.15131 + 1.99413i
2.46040 + 4.26153i
2.97515 + 5.15311i
−2.29598 + 3.97675i
−1.71036 + 2.96243i
−1.01204 + 1.75290i
−0.475331 + 0.823298i
0.906849 − 1.57071i
1.15131 − 1.99413i
2.46040 − 4.26153i
2.97515 − 5.15311i
−2.79598 − 4.84278i 0 −11.6350 + 20.1524i 2.50000 + 4.33013i 0 8.61636 + 16.3939i 85.3894 0 13.9799 − 24.2139i
46.2 −2.21036 − 3.82846i 0 −5.77139 + 9.99634i 2.50000 + 4.33013i 0 0.502371 − 18.5134i 15.6616 0 11.0518 − 19.1423i
46.3 −1.51204 − 2.61893i 0 −0.572522 + 0.991637i 2.50000 + 4.33013i 0 −16.1159 + 9.12560i −20.7299 0 7.56019 − 13.0946i
46.4 −0.975331 − 1.68932i 0 2.09746 − 3.63290i 2.50000 + 4.33013i 0 18.2579 + 3.10650i −23.7882 0 4.87666 − 8.44662i
46.5 0.406849 + 0.704684i 0 3.66895 − 6.35480i 2.50000 + 4.33013i 0 0.736536 + 18.5056i 12.4804 0 −2.03425 + 3.52342i
46.6 0.651312 + 1.12811i 0 3.15158 − 5.45870i 2.50000 + 4.33013i 0 −14.0975 − 12.0108i 18.6317 0 −3.25656 + 5.64053i
46.7 1.96040 + 3.39551i 0 −3.68632 + 6.38489i 2.50000 + 4.33013i 0 17.2936 + 6.62818i 2.45977 0 −9.80199 + 16.9775i
46.8 2.47515 + 4.28709i 0 −8.25275 + 14.2942i 2.50000 + 4.33013i 0 −4.19324 − 18.0393i −42.1048 0 −12.3758 + 21.4354i
226.1 −2.79598 + 4.84278i 0 −11.6350 − 20.1524i 2.50000 − 4.33013i 0 8.61636 − 16.3939i 85.3894 0 13.9799 + 24.2139i
226.2 −2.21036 + 3.82846i 0 −5.77139 − 9.99634i 2.50000 − 4.33013i 0 0.502371 + 18.5134i 15.6616 0 11.0518 + 19.1423i
226.3 −1.51204 + 2.61893i 0 −0.572522 − 0.991637i 2.50000 − 4.33013i 0 −16.1159 − 9.12560i −20.7299 0 7.56019 + 13.0946i
226.4 −0.975331 + 1.68932i 0 2.09746 + 3.63290i 2.50000 − 4.33013i 0 18.2579 − 3.10650i −23.7882 0 4.87666 + 8.44662i
226.5 0.406849 − 0.704684i 0 3.66895 + 6.35480i 2.50000 − 4.33013i 0 0.736536 − 18.5056i 12.4804 0 −2.03425 − 3.52342i
226.6 0.651312 − 1.12811i 0 3.15158 + 5.45870i 2.50000 − 4.33013i 0 −14.0975 + 12.0108i 18.6317 0 −3.25656 − 5.64053i
226.7 1.96040 − 3.39551i 0 −3.68632 − 6.38489i 2.50000 − 4.33013i 0 17.2936 − 6.62818i 2.45977 0 −9.80199 − 16.9775i
226.8 2.47515 − 4.28709i 0 −8.25275 − 14.2942i 2.50000 − 4.33013i 0 −4.19324 + 18.0393i −42.1048 0 −12.3758 − 21.4354i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.j.i ✓ 16
3.b odd 2 1 315.4.j.j yes 16
7.c even 3 1 inner 315.4.j.i ✓ 16
7.c even 3 1 2205.4.a.cf 8
7.d odd 6 1 2205.4.a.cg 8
21.g even 6 1 2205.4.a.cb 8
21.h odd 6 1 315.4.j.j yes 16
21.h odd 6 1 2205.4.a.cc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.j.i ✓ 16 1.a even 1 1 trivial
315.4.j.i ✓ 16 7.c even 3 1 inner
315.4.j.j yes 16 3.b odd 2 1
315.4.j.j yes 16 21.h odd 6 1
2205.4.a.cb 8 21.g even 6 1
2205.4.a.cc 8 21.h odd 6 1
2205.4.a.cf 8 7.c even 3 1
2205.4.a.cg 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(315,[χ])S_{4}^{\mathrm{new}}(315, [\chi]):

T216+4T215+61T214+148T213+2115T212+4642T211+⋯+9000000 T_{2}^{16} + 4 T_{2}^{15} + 61 T_{2}^{14} + 148 T_{2}^{13} + 2115 T_{2}^{12} + 4642 T_{2}^{11} + \cdots + 9000000 Copy content Toggle raw display
T138+102T137−6020T136−866738T135−9517198T134+⋯−1519211972487 T_{13}^{8} + 102 T_{13}^{7} - 6020 T_{13}^{6} - 866738 T_{13}^{5} - 9517198 T_{13}^{4} + \cdots - 1519211972487 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16+4T15+⋯+9000000 T^{16} + 4 T^{15} + \cdots + 9000000 Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 (T2−5T+25)8 (T^{2} - 5 T + 25)^{8} Copy content Toggle raw display
77 T16+⋯+19 ⁣⋯ ⁣01 T^{16} + \cdots + 19\!\cdots\!01 Copy content Toggle raw display
1111 T16+⋯+39 ⁣⋯ ⁣00 T^{16} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
1313 (T8+⋯−1519211972487)2 (T^{8} + \cdots - 1519211972487)^{2} Copy content Toggle raw display
1717 T16+⋯+17 ⁣⋯ ⁣00 T^{16} + \cdots + 17\!\cdots\!00 Copy content Toggle raw display
1919 T16+⋯+13 ⁣⋯ ⁣21 T^{16} + \cdots + 13\!\cdots\!21 Copy content Toggle raw display
2323 T16+⋯+13 ⁣⋯ ⁣00 T^{16} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
2929 (T8+⋯−32 ⁣⋯ ⁣40)2 (T^{8} + \cdots - 32\!\cdots\!40)^{2} Copy content Toggle raw display
3131 T16+⋯+10 ⁣⋯ ⁣00 T^{16} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
3737 T16+⋯+18 ⁣⋯ ⁣41 T^{16} + \cdots + 18\!\cdots\!41 Copy content Toggle raw display
4141 (T8+⋯+54 ⁣⋯ ⁣80)2 (T^{8} + \cdots + 54\!\cdots\!80)^{2} Copy content Toggle raw display
4343 (T8+⋯+53 ⁣⋯ ⁣00)2 (T^{8} + \cdots + 53\!\cdots\!00)^{2} Copy content Toggle raw display
4747 T16+⋯+19 ⁣⋯ ⁣00 T^{16} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
5353 T16+⋯+14 ⁣⋯ ⁣00 T^{16} + \cdots + 14\!\cdots\!00 Copy content Toggle raw display
5959 T16+⋯+68 ⁣⋯ ⁣00 T^{16} + \cdots + 68\!\cdots\!00 Copy content Toggle raw display
6161 T16+⋯+19 ⁣⋯ ⁣84 T^{16} + \cdots + 19\!\cdots\!84 Copy content Toggle raw display
6767 T16+⋯+47 ⁣⋯ ⁣96 T^{16} + \cdots + 47\!\cdots\!96 Copy content Toggle raw display
7171 (T8+⋯+24 ⁣⋯ ⁣80)2 (T^{8} + \cdots + 24\!\cdots\!80)^{2} Copy content Toggle raw display
7373 T16+⋯+10 ⁣⋯ ⁣56 T^{16} + \cdots + 10\!\cdots\!56 Copy content Toggle raw display
7979 T16+⋯+90 ⁣⋯ ⁣96 T^{16} + \cdots + 90\!\cdots\!96 Copy content Toggle raw display
8383 (T8+⋯+31 ⁣⋯ ⁣60)2 (T^{8} + \cdots + 31\!\cdots\!60)^{2} Copy content Toggle raw display
8989 T16+⋯+39 ⁣⋯ ⁣00 T^{16} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
9797 (T8+⋯−69 ⁣⋯ ⁣88)2 (T^{8} + \cdots - 69\!\cdots\!88)^{2} Copy content Toggle raw display
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