Properties

Label 975.6.a.u
Level 975975
Weight 66
Character orbit 975.a
Self dual yes
Analytic conductor 156.374156.374
Analytic rank 11
Dimension 1111
CM no
Inner twists 11

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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] = [975,6,Mod(1,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: N N == 975=35213 975 = 3 \cdot 5^{2} \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11,2,-99,224,0,-18,-55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 156.374224318156.374224318
Analytic rank: 11
Dimension: 1111
Coefficient field: Q[x]/(x11)\mathbb{Q}[x]/(x^{11} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x112x10286x9+442x8+28715x729138x61208172x5+509768x4++55036800 x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 2535311 2^{5}\cdot 3\cdot 5^{3}\cdot 11
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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,,β101,\beta_1,\ldots,\beta_{10} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q29q3+(β2+β1+20)q49β1q6+(β82β15)q7+(β3+22β1+21)q8+81q9+(β9+β82β2+8)q11++(81β9+81β8+648)q99+O(q100) q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + \beta_1 + 20) q^{4} - 9 \beta_1 q^{6} + ( - \beta_{8} - 2 \beta_1 - 5) q^{7} + (\beta_{3} + 22 \beta_1 + 21) q^{8} + 81 q^{9} + (\beta_{9} + \beta_{8} - 2 \beta_{2} + \cdots - 8) q^{11}+ \cdots + (81 \beta_{9} + 81 \beta_{8} + \cdots - 648) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 11q+2q299q3+224q418q655q7+270q8+891q9125q112016q12+1859q131311q14+5756q164507q17+162q18+142q19+495q21+10125q99+O(q100) 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x112x10286x9+442x8+28715x729138x61208172x5+509768x4++55036800 x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν52 \nu^{2} - \nu - 52 Copy content Toggle raw display
β3\beta_{3}== ν386ν21 \nu^{3} - 86\nu - 21 Copy content Toggle raw display
β4\beta_{4}== (1286527ν1090509002ν9346680258ν8+23125223382ν7+10 ⁣ ⁣00)/1971602843264 ( 1286527 \nu^{10} - 90509002 \nu^{9} - 346680258 \nu^{8} + 23125223382 \nu^{7} + \cdots - 10\!\cdots\!00 ) / 1971602843264 Copy content Toggle raw display
β5\beta_{5}== (20385315ν10+30245988ν9+4762321298ν81281153890ν7+17 ⁣ ⁣32)/3943205686528 ( - 20385315 \nu^{10} + 30245988 \nu^{9} + 4762321298 \nu^{8} - 1281153890 \nu^{7} + \cdots - 17\!\cdots\!32 ) / 3943205686528 Copy content Toggle raw display
β6\beta_{6}== (12146105ν10237424970ν91829394126ν8+63617203634ν7++22 ⁣ ⁣84)/1971602843264 ( 12146105 \nu^{10} - 237424970 \nu^{9} - 1829394126 \nu^{8} + 63617203634 \nu^{7} + \cdots + 22\!\cdots\!84 ) / 1971602843264 Copy content Toggle raw display
β7\beta_{7}== (27730735ν10626840400ν97674408554ν8+164251378850ν7++45 ⁣ ⁣68)/3943205686528 ( 27730735 \nu^{10} - 626840400 \nu^{9} - 7674408554 \nu^{8} + 164251378850 \nu^{7} + \cdots + 45\!\cdots\!68 ) / 3943205686528 Copy content Toggle raw display
β8\beta_{8}== (29180789ν10+5549056ν98355371630ν82320337258ν7+11 ⁣ ⁣92)/3943205686528 ( 29180789 \nu^{10} + 5549056 \nu^{9} - 8355371630 \nu^{8} - 2320337258 \nu^{7} + \cdots - 11\!\cdots\!92 ) / 3943205686528 Copy content Toggle raw display
β9\beta_{9}== (60411079ν10+315478660ν9+15624013090ν876119492426ν7+784025003900992)/1971602843264 ( - 60411079 \nu^{10} + 315478660 \nu^{9} + 15624013090 \nu^{8} - 76119492426 \nu^{7} + \cdots - 784025003900992 ) / 1971602843264 Copy content Toggle raw display
β10\beta_{10}== (105564843ν10+501938874ν9+27968787146ν8113547750174ν7++34 ⁣ ⁣28)/1971602843264 ( - 105564843 \nu^{10} + 501938874 \nu^{9} + 27968787146 \nu^{8} - 113547750174 \nu^{7} + \cdots + 34\!\cdots\!28 ) / 1971602843264 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+52 \beta_{2} + \beta _1 + 52 Copy content Toggle raw display
ν3\nu^{3}== β3+86β1+21 \beta_{3} + 86\beta _1 + 21 Copy content Toggle raw display
ν4\nu^{4}== β102β9β8β6β5+β4β3+118β2+125β1+4480 \beta_{10} - 2\beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 118\beta_{2} + 125\beta _1 + 4480 Copy content Toggle raw display
ν5\nu^{5}== 3β102β9+7β8β69β5+11β4+140β3++2799 3 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} - \beta_{6} - 9 \beta_{5} + 11 \beta_{4} + 140 \beta_{3} + \cdots + 2799 Copy content Toggle raw display
ν6\nu^{6}== 176β10352β9160β818β7172β6166β5++442426 176 \beta_{10} - 352 \beta_{9} - 160 \beta_{8} - 18 \beta_{7} - 172 \beta_{6} - 166 \beta_{5} + \cdots + 442426 Copy content Toggle raw display
ν7\nu^{7}== 732β10400β9+2144β8+24β740β61848β5++347191 732 \beta_{10} - 400 \beta_{9} + 2144 \beta_{8} + 24 \beta_{7} - 40 \beta_{6} - 1848 \beta_{5} + \cdots + 347191 Copy content Toggle raw display
ν8\nu^{8}== 24839β1049910β922131β85076β722923β6++46147118 24839 \beta_{10} - 49910 \beta_{9} - 22131 \beta_{8} - 5076 \beta_{7} - 22923 \beta_{6} + \cdots + 46147118 Copy content Toggle raw display
ν9\nu^{9}== 126407β1064186β9+406779β8+844β7+10251β6++47264721 126407 \beta_{10} - 64186 \beta_{9} + 406779 \beta_{8} + 844 \beta_{7} + 10251 \beta_{6} + \cdots + 47264721 Copy content Toggle raw display
ν10\nu^{10}== 3254970β106569668β92776370β8940778β72811870β6++4960185176 3254970 \beta_{10} - 6569668 \beta_{9} - 2776370 \beta_{8} - 940778 \beta_{7} - 2811870 \beta_{6} + \cdots + 4960185176 Copy content Toggle raw display

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}
1.1
−10.5321
−9.62146
−5.81158
−4.77830
−3.50720
0.592612
2.29989
4.89893
8.18506
9.28847
10.9857
−10.5321 −9.00000 78.9248 0 94.7888 88.0004 −494.216 81.0000 0
1.2 −9.62146 −9.00000 60.5725 0 86.5931 −139.353 −274.909 81.0000 0
1.3 −5.81158 −9.00000 1.77443 0 52.3042 186.151 175.658 81.0000 0
1.4 −4.77830 −9.00000 −9.16786 0 43.0047 26.6297 196.712 81.0000 0
1.5 −3.50720 −9.00000 −19.6996 0 31.5648 −147.807 181.321 81.0000 0
1.6 0.592612 −9.00000 −31.6488 0 −5.33351 209.544 −37.7190 81.0000 0
1.7 2.29989 −9.00000 −26.7105 0 −20.6990 −228.917 −135.028 81.0000 0
1.8 4.89893 −9.00000 −8.00051 0 −44.0903 56.8551 −195.960 81.0000 0
1.9 8.18506 −9.00000 34.9953 0 −73.6656 −143.020 24.5164 81.0000 0
1.10 9.28847 −9.00000 54.2758 0 −83.5963 85.6782 206.908 81.0000 0
1.11 10.9857 −9.00000 88.6845 0 −98.8709 −48.7613 622.716 81.0000 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
55 1 -1
1313 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 975.6.a.u yes 11
5.b even 2 1 975.6.a.r 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.6.a.r 11 5.b even 2 1
975.6.a.u yes 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S6new(Γ0(975))S_{6}^{\mathrm{new}}(\Gamma_0(975)):

T2112T210286T29+442T28+28715T2729138T26++55036800 T_{2}^{11} - 2 T_{2}^{10} - 286 T_{2}^{9} + 442 T_{2}^{8} + 28715 T_{2}^{7} - 29138 T_{2}^{6} + \cdots + 55036800 Copy content Toggle raw display
T711+55T710105532T794590338T78+3818034718T77++14 ⁣ ⁣24 T_{7}^{11} + 55 T_{7}^{10} - 105532 T_{7}^{9} - 4590338 T_{7}^{8} + 3818034718 T_{7}^{7} + \cdots + 14\!\cdots\!24 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T112T10++55036800 T^{11} - 2 T^{10} + \cdots + 55036800 Copy content Toggle raw display
33 (T+9)11 (T + 9)^{11} Copy content Toggle raw display
55 T11 T^{11} Copy content Toggle raw display
77 T11++14 ⁣ ⁣24 T^{11} + \cdots + 14\!\cdots\!24 Copy content Toggle raw display
1111 T11++53 ⁣ ⁣80 T^{11} + \cdots + 53\!\cdots\!80 Copy content Toggle raw display
1313 (T169)11 (T - 169)^{11} Copy content Toggle raw display
1717 T11++48 ⁣ ⁣56 T^{11} + \cdots + 48\!\cdots\!56 Copy content Toggle raw display
1919 T11++27 ⁣ ⁣40 T^{11} + \cdots + 27\!\cdots\!40 Copy content Toggle raw display
2323 T11++94 ⁣ ⁣60 T^{11} + \cdots + 94\!\cdots\!60 Copy content Toggle raw display
2929 T11+10 ⁣ ⁣95 T^{11} + \cdots - 10\!\cdots\!95 Copy content Toggle raw display
3131 T11+11 ⁣ ⁣76 T^{11} + \cdots - 11\!\cdots\!76 Copy content Toggle raw display
3737 T11++10 ⁣ ⁣76 T^{11} + \cdots + 10\!\cdots\!76 Copy content Toggle raw display
4141 T11++13 ⁣ ⁣04 T^{11} + \cdots + 13\!\cdots\!04 Copy content Toggle raw display
4343 T11++83 ⁣ ⁣88 T^{11} + \cdots + 83\!\cdots\!88 Copy content Toggle raw display
4747 T11+19 ⁣ ⁣75 T^{11} + \cdots - 19\!\cdots\!75 Copy content Toggle raw display
5353 T11+72 ⁣ ⁣05 T^{11} + \cdots - 72\!\cdots\!05 Copy content Toggle raw display
5959 T11++27 ⁣ ⁣00 T^{11} + \cdots + 27\!\cdots\!00 Copy content Toggle raw display
6161 T11+70 ⁣ ⁣40 T^{11} + \cdots - 70\!\cdots\!40 Copy content Toggle raw display
6767 T11++40 ⁣ ⁣89 T^{11} + \cdots + 40\!\cdots\!89 Copy content Toggle raw display
7171 T11+43 ⁣ ⁣60 T^{11} + \cdots - 43\!\cdots\!60 Copy content Toggle raw display
7373 T11++12 ⁣ ⁣68 T^{11} + \cdots + 12\!\cdots\!68 Copy content Toggle raw display
7979 T11++10 ⁣ ⁣64 T^{11} + \cdots + 10\!\cdots\!64 Copy content Toggle raw display
8383 T11+17 ⁣ ⁣52 T^{11} + \cdots - 17\!\cdots\!52 Copy content Toggle raw display
8989 T11++22 ⁣ ⁣00 T^{11} + \cdots + 22\!\cdots\!00 Copy content Toggle raw display
9797 T11++13 ⁣ ⁣00 T^{11} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
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