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Properties

Label 161.4.a.d
Level 161161
Weight 44
Character orbit 161.a
Self dual yes
Analytic conductor 9.4999.499
Analytic rank 00
Dimension 1212
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,4,Mod(1,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 161=7⋅23 161 = 7 \cdot 23
Weight: k k == 4 4
Character orbit: [χ][\chi] == 161.a (trivial)

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: yes
Analytic conductor: 9.499307510929.49930751092
Analytic rank: 00
Dimension: 1212
Coefficient field: Q[x]/(x12−⋯ )\mathbb{Q}[x]/(x^{12} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12−4x11−76x10+311x9+1979x8−8466x7−19942x6+95647x5+⋯−70656 x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 24 2^{4}
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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+β1q2−β4q3+(β2+6)q4+(β7+β1+1)q5+(−β6−β5−β4−1)q6+7q7+(β9+β7+β6+⋯−5)q8+⋯+(22β11+31β10+⋯+217)q99+O(q100) q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 6) q^{4} + (\beta_{7} + \beta_1 + 1) q^{5} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 1) q^{6} + 7 q^{7} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots - 5) q^{8}+ \cdots + (22 \beta_{11} + 31 \beta_{10} + \cdots + 217) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+4q2+q3+72q4+16q5−15q6+84q7−21q8+203q9+106q10+50q11−139q12+21q13+28q14+192q15+516q16+26q17+251q18+⋯+2400q99+O(q100) 12 q + 4 q^{2} + q^{3} + 72 q^{4} + 16 q^{5} - 15 q^{6} + 84 q^{7} - 21 q^{8} + 203 q^{9} + 106 q^{10} + 50 q^{11} - 139 q^{12} + 21 q^{13} + 28 q^{14} + 192 q^{15} + 516 q^{16} + 26 q^{17} + 251 q^{18}+ \cdots + 2400 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12−4x11−76x10+311x9+1979x8−8466x7−19942x6+95647x5+⋯−70656 x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2−14 \nu^{2} - 14 Copy content Toggle raw display
β3\beta_{3}== (1057087ν11−736072ν10−81046848ν9+59112285ν8+2160058053ν7+⋯+27581677952)/466611584 ( 1057087 \nu^{11} - 736072 \nu^{10} - 81046848 \nu^{9} + 59112285 \nu^{8} + 2160058053 \nu^{7} + \cdots + 27581677952 ) / 466611584 Copy content Toggle raw display
β4\beta_{4}== (1117105ν11−226867ν10−85952171ν9+21399060ν8+2305747495ν7+⋯+19319250176)/233305792 ( 1117105 \nu^{11} - 226867 \nu^{10} - 85952171 \nu^{9} + 21399060 \nu^{8} + 2305747495 \nu^{7} + \cdots + 19319250176 ) / 233305792 Copy content Toggle raw display
β5\beta_{5}== (1519271ν11−498841ν10−117032797ν9+43441900ν8+3145054901ν7+⋯+27799261696)/233305792 ( 1519271 \nu^{11} - 498841 \nu^{10} - 117032797 \nu^{9} + 43441900 \nu^{8} + 3145054901 \nu^{7} + \cdots + 27799261696 ) / 233305792 Copy content Toggle raw display
β6\beta_{6}== (1605177ν11−326483ν10−123035627ν9+30155740ν8+3282153271ν7+⋯+31578353216)/233305792 ( 1605177 \nu^{11} - 326483 \nu^{10} - 123035627 \nu^{9} + 30155740 \nu^{8} + 3282153271 \nu^{7} + \cdots + 31578353216 ) / 233305792 Copy content Toggle raw display
β7\beta_{7}== (99065ν11−21862ν10−7609238ν9+2030749ν8+203556131ν7+⋯+1798578304)/6964352 ( 99065 \nu^{11} - 21862 \nu^{10} - 7609238 \nu^{9} + 2030749 \nu^{8} + 203556131 \nu^{7} + \cdots + 1798578304 ) / 6964352 Copy content Toggle raw display
β8\beta_{8}== (7097167ν11−2425124ν10−546101432ν9+210879061ν8+⋯+160329330944)/466611584 ( 7097167 \nu^{11} - 2425124 \nu^{10} - 546101432 \nu^{9} + 210879061 \nu^{8} + \cdots + 160329330944 ) / 466611584 Copy content Toggle raw display
β9\beta_{9}== (−5914582ν11+1189665ν10+454454473ν9−112071589ν8+⋯−101406339072)/233305792 ( - 5914582 \nu^{11} + 1189665 \nu^{10} + 454454473 \nu^{9} - 112071589 \nu^{8} + \cdots - 101406339072 ) / 233305792 Copy content Toggle raw display
β10\beta_{10}== (7229129ν11−1769452ν10−555592220ν9+160203215ν8+⋯+135239697088)/233305792 ( 7229129 \nu^{11} - 1769452 \nu^{10} - 555592220 \nu^{9} + 160203215 \nu^{8} + \cdots + 135239697088 ) / 233305792 Copy content Toggle raw display
β11\beta_{11}== (22127067ν11−4919124ν10−1700971564ν9+457650357ν8+⋯+427632581376)/466611584 ( 22127067 \nu^{11} - 4919124 \nu^{10} - 1700971564 \nu^{9} + 457650357 \nu^{8} + \cdots + 427632581376 ) / 466611584 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}== β2+14 \beta_{2} + 14 Copy content Toggle raw display
ν3\nu^{3}== β9+β7+β6+β5−β3+β2+23β1−5 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 23\beta _1 - 5 Copy content Toggle raw display
ν4\nu^{4}== β11+2β9−2β7+2β6−2β5+6β4+β3+31β2−2β1+315 \beta_{11} + 2\beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 6\beta_{4} + \beta_{3} + 31\beta_{2} - 2\beta _1 + 315 Copy content Toggle raw display
ν5\nu^{5}== −β11−6β10+36β9+40β7+44β6+44β5+14β4+⋯−185 - \beta_{11} - 6 \beta_{10} + 36 \beta_{9} + 40 \beta_{7} + 44 \beta_{6} + 44 \beta_{5} + 14 \beta_{4} + \cdots - 185 Copy content Toggle raw display
ν6\nu^{6}== 27β11−10β10+75β9+16β8−65β7+83β6−109β5+⋯+8130 27 \beta_{11} - 10 \beta_{10} + 75 \beta_{9} + 16 \beta_{8} - 65 \beta_{7} + 83 \beta_{6} - 109 \beta_{5} + \cdots + 8130 Copy content Toggle raw display
ν7\nu^{7}== −58β11−354β10+1026β9+48β8+1242β7+1570β6+⋯−6210 - 58 \beta_{11} - 354 \beta_{10} + 1026 \beta_{9} + 48 \beta_{8} + 1242 \beta_{7} + 1570 \beta_{6} + \cdots - 6210 Copy content Toggle raw display
ν8\nu^{8}== 476β11−748β10+2172β9+1184β8−1532β7+2692β6+⋯+224802 476 \beta_{11} - 748 \beta_{10} + 2172 \beta_{9} + 1184 \beta_{8} - 1532 \beta_{7} + 2692 \beta_{6} + \cdots + 224802 Copy content Toggle raw display
ν9\nu^{9}== −2552β11−15264β10+26977β9+3408β8+36409β7+⋯−202781 - 2552 \beta_{11} - 15264 \beta_{10} + 26977 \beta_{9} + 3408 \beta_{8} + 36409 \beta_{7} + \cdots - 202781 Copy content Toggle raw display
ν10\nu^{10}== 3329β11−36960β10+56722β9+57632β8−29538β7+⋯+6459459 3329 \beta_{11} - 36960 \beta_{10} + 56722 \beta_{9} + 57632 \beta_{8} - 29538 \beta_{7} + \cdots + 6459459 Copy content Toggle raw display
ν11\nu^{11}== −100129β11−579662β10+683620β9+162544β8+1056368β7+⋯−6540689 - 100129 \beta_{11} - 579662 \beta_{10} + 683620 \beta_{9} + 162544 \beta_{8} + 1056368 \beta_{7} + \cdots - 6540689 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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}
1.1
−5.60949
−4.89764
−3.54240
−3.06154
−0.718333
0.285899
1.40034
2.11879
3.76511
3.76781
4.95487
5.53660
−5.60949 −7.98435 23.4664 −5.27021 44.7882 7.00000 −86.7589 36.7498 29.5632
1.2 −4.89764 7.57994 15.9869 −14.0191 −37.1238 7.00000 −39.1171 30.4555 68.6607
1.3 −3.54240 −2.15594 4.54860 6.95559 7.63719 7.00000 12.2262 −22.3519 −24.6395
1.4 −3.06154 5.71887 1.37302 21.1978 −17.5086 7.00000 20.2888 5.70552 −64.8978
1.5 −0.718333 −2.25536 −7.48400 −17.9779 1.62010 7.00000 11.1227 −21.9133 12.9141
1.6 0.285899 8.01264 −7.91826 1.18183 2.29081 7.00000 −4.55102 37.2024 0.337884
1.7 1.40034 −4.22464 −6.03905 13.7756 −5.91594 7.00000 −19.6594 −9.15238 19.2904
1.8 2.11879 −9.53944 −3.51074 −13.8523 −20.2120 7.00000 −24.3888 64.0009 −29.3501
1.9 3.76511 9.31158 6.17602 1.25334 35.0591 7.00000 −6.86747 59.7056 4.71897
1.10 3.76781 2.58468 6.19640 21.4417 9.73858 7.00000 −6.79562 −20.3194 80.7883
1.11 4.95487 3.24611 16.5508 −2.30377 16.0841 7.00000 42.3679 −16.4628 −11.4149
1.12 5.53660 −9.29409 22.6539 3.61751 −51.4577 7.00000 81.1328 59.3802 20.0287
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
77 −1 -1
2323 −1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.a.d ✓ 12
3.b odd 2 1 1449.4.a.o 12
7.b odd 2 1 1127.4.a.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.a.d ✓ 12 1.a even 1 1 trivial
1127.4.a.h 12 7.b odd 2 1
1449.4.a.o 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T212−4T211−76T210+311T29+1979T28−8466T27+⋯−70656 T_{2}^{12} - 4 T_{2}^{11} - 76 T_{2}^{10} + 311 T_{2}^{9} + 1979 T_{2}^{8} - 8466 T_{2}^{7} + \cdots - 70656 acting on S4new(Γ0(161))S_{4}^{\mathrm{new}}(\Gamma_0(161)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12−4T11+⋯−70656 T^{12} - 4 T^{11} + \cdots - 70656 Copy content Toggle raw display
33 T12+⋯+394600512 T^{12} + \cdots + 394600512 Copy content Toggle raw display
55 T12+⋯−9891773184 T^{12} + \cdots - 9891773184 Copy content Toggle raw display
77 (T−7)12 (T - 7)^{12} Copy content Toggle raw display
1111 T12+⋯−23 ⁣⋯ ⁣04 T^{12} + \cdots - 23\!\cdots\!04 Copy content Toggle raw display
1313 T12+⋯+82 ⁣⋯ ⁣04 T^{12} + \cdots + 82\!\cdots\!04 Copy content Toggle raw display
1717 T12+⋯−14 ⁣⋯ ⁣72 T^{12} + \cdots - 14\!\cdots\!72 Copy content Toggle raw display
1919 T12+⋯+63 ⁣⋯ ⁣52 T^{12} + \cdots + 63\!\cdots\!52 Copy content Toggle raw display
2323 (T−23)12 (T - 23)^{12} Copy content Toggle raw display
2929 T12+⋯+52 ⁣⋯ ⁣36 T^{12} + \cdots + 52\!\cdots\!36 Copy content Toggle raw display
3131 T12+⋯−33 ⁣⋯ ⁣64 T^{12} + \cdots - 33\!\cdots\!64 Copy content Toggle raw display
3737 T12+⋯−28 ⁣⋯ ⁣52 T^{12} + \cdots - 28\!\cdots\!52 Copy content Toggle raw display
4141 T12+⋯+32 ⁣⋯ ⁣52 T^{12} + \cdots + 32\!\cdots\!52 Copy content Toggle raw display
4343 T12+⋯+75 ⁣⋯ ⁣08 T^{12} + \cdots + 75\!\cdots\!08 Copy content Toggle raw display
4747 T12+⋯+36 ⁣⋯ ⁣48 T^{12} + \cdots + 36\!\cdots\!48 Copy content Toggle raw display
5353 T12+⋯+39 ⁣⋯ ⁣48 T^{12} + \cdots + 39\!\cdots\!48 Copy content Toggle raw display
5959 T12+⋯−41 ⁣⋯ ⁣32 T^{12} + \cdots - 41\!\cdots\!32 Copy content Toggle raw display
6161 T12+⋯−60 ⁣⋯ ⁣16 T^{12} + \cdots - 60\!\cdots\!16 Copy content Toggle raw display
6767 T12+⋯−10 ⁣⋯ ⁣64 T^{12} + \cdots - 10\!\cdots\!64 Copy content Toggle raw display
7171 T12+⋯+32 ⁣⋯ ⁣88 T^{12} + \cdots + 32\!\cdots\!88 Copy content Toggle raw display
7373 T12+⋯−13 ⁣⋯ ⁣24 T^{12} + \cdots - 13\!\cdots\!24 Copy content Toggle raw display
7979 T12+⋯−26 ⁣⋯ ⁣72 T^{12} + \cdots - 26\!\cdots\!72 Copy content Toggle raw display
8383 T12+⋯+76 ⁣⋯ ⁣04 T^{12} + \cdots + 76\!\cdots\!04 Copy content Toggle raw display
8989 T12+⋯−81 ⁣⋯ ⁣12 T^{12} + \cdots - 81\!\cdots\!12 Copy content Toggle raw display
9797 T12+⋯+22 ⁣⋯ ⁣52 T^{12} + \cdots + 22\!\cdots\!52 Copy content Toggle raw display
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