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Properties

Label 7942.2.a.bx
Level 79427942
Weight 22
Character orbit 7942.a
Self dual yes
Analytic conductor 63.41763.417
Analytic rank 11
Dimension 1212
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] = [7942,2,Mod(1,7942)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 7942=2⋅11⋅192 7942 = 2 \cdot 11 \cdot 19^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 7942.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,12,-3,12,-9,-3,0] 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: 63.417189285363.4171892853
Analytic rank: 11
Dimension: 1212
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: x12−3x11−15x10+41x9+90x8−198x7−285x6+396x5+486x4+⋯−3 x^{12} - 3 x^{11} - 15 x^{10} + 41 x^{9} + 90 x^{8} - 198 x^{7} - 285 x^{6} + 396 x^{5} + 486 x^{4} + \cdots - 3 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 418)
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,…,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+q2−β1q3+q4+(−β2−1)q5−β1q6+(−β11−β4+β2)q7+q8+(β8+β7−β3+⋯−1)q9+(−β2−1)q10+⋯+(−β8−β7+β3+⋯+1)q99+O(q100) q + q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{2} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{11} - \beta_{4} + \beta_{2}) q^{7} + q^{8} + (\beta_{8} + \beta_{7} - \beta_{3} + \cdots - 1) q^{9} + ( - \beta_{2} - 1) q^{10}+ \cdots + ( - \beta_{8} - \beta_{7} + \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+12q2−3q3+12q4−9q5−3q6+12q8+3q9−9q10−12q11−3q12+9q15+12q16−9q17+3q18−9q20−9q21−12q22−18q23+⋯−3q99+O(q100) 12 q + 12 q^{2} - 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} + 12 q^{8} + 3 q^{9} - 9 q^{10} - 12 q^{11} - 3 q^{12} + 9 q^{15} + 12 q^{16} - 9 q^{17} + 3 q^{18} - 9 q^{20} - 9 q^{21} - 12 q^{22} - 18 q^{23}+ \cdots - 3 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12−3x11−15x10+41x9+90x8−198x7−285x6+396x5+486x4+⋯−3 x^{12} - 3 x^{11} - 15 x^{10} + 41 x^{9} + 90 x^{8} - 198 x^{7} - 285 x^{6} + 396 x^{5} + 486 x^{4} + \cdots - 3 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (93ν11−355ν10−215ν9+1497ν8−4066ν7+11828ν6+16444ν5+⋯−7229)/2069 ( 93 \nu^{11} - 355 \nu^{10} - 215 \nu^{9} + 1497 \nu^{8} - 4066 \nu^{7} + 11828 \nu^{6} + 16444 \nu^{5} + \cdots - 7229 ) / 2069 Copy content Toggle raw display
β3\beta_{3}== (313ν11−1484ν10−3638ν9+21924ν8+13257ν7−115656ν6+⋯+18207)/6207 ( 313 \nu^{11} - 1484 \nu^{10} - 3638 \nu^{9} + 21924 \nu^{8} + 13257 \nu^{7} - 115656 \nu^{6} + \cdots + 18207 ) / 6207 Copy content Toggle raw display
β4\beta_{4}== (−316ν11+3164ν10−1027ν9−42996ν8+35796ν7+216255ν6+⋯−12234)/6207 ( - 316 \nu^{11} + 3164 \nu^{10} - 1027 \nu^{9} - 42996 \nu^{8} + 35796 \nu^{7} + 216255 \nu^{6} + \cdots - 12234 ) / 6207 Copy content Toggle raw display
β5\beta_{5}== (487ν11+388ν10−12383ν9−3507ν8+97887ν7+14529ν6+⋯+744)/6207 ( 487 \nu^{11} + 388 \nu^{10} - 12383 \nu^{9} - 3507 \nu^{8} + 97887 \nu^{7} + 14529 \nu^{6} + \cdots + 744 ) / 6207 Copy content Toggle raw display
β6\beta_{6}== (−614ν11+2455ν10+7315ν9−34044ν8−30309ν7+170301ν6+⋯+26592)/6207 ( - 614 \nu^{11} + 2455 \nu^{10} + 7315 \nu^{9} - 34044 \nu^{8} - 30309 \nu^{7} + 170301 \nu^{6} + \cdots + 26592 ) / 6207 Copy content Toggle raw display
β7\beta_{7}== (−679ν11+1613ν10+11759ν9−23010ν8−80616ν7+117495ν6+⋯+27729)/6207 ( - 679 \nu^{11} + 1613 \nu^{10} + 11759 \nu^{9} - 23010 \nu^{8} - 80616 \nu^{7} + 117495 \nu^{6} + \cdots + 27729 ) / 6207 Copy content Toggle raw display
β8\beta_{8}== (992ν11−3097ν10−15397ν9+44934ν8+93873ν7−233151ν6+⋯−21936)/6207 ( 992 \nu^{11} - 3097 \nu^{10} - 15397 \nu^{9} + 44934 \nu^{8} + 93873 \nu^{7} - 233151 \nu^{6} + \cdots - 21936 ) / 6207 Copy content Toggle raw display
β9\beta_{9}== (−1379ν11+2572ν10+21898ν9−25935ν8−134418ν7+57789ν6+⋯+35199)/6207 ( - 1379 \nu^{11} + 2572 \nu^{10} + 21898 \nu^{9} - 25935 \nu^{8} - 134418 \nu^{7} + 57789 \nu^{6} + \cdots + 35199 ) / 6207 Copy content Toggle raw display
β10\beta_{10}== (1642ν11−7091ν10−16388ν9+89769ν8+50727ν7−400275ν6+⋯−20892)/6207 ( 1642 \nu^{11} - 7091 \nu^{10} - 16388 \nu^{9} + 89769 \nu^{8} + 50727 \nu^{7} - 400275 \nu^{6} + \cdots - 20892 ) / 6207 Copy content Toggle raw display
β11\beta_{11}== (−854ν11+2370ν10+12742ν9−31500ν8−72342ν7+144983ν6+⋯−404)/2069 ( - 854 \nu^{11} + 2370 \nu^{10} + 12742 \nu^{9} - 31500 \nu^{8} - 72342 \nu^{7} + 144983 \nu^{6} + \cdots - 404 ) / 2069 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}== β8+β7−β3+β1+2 \beta_{8} + \beta_{7} - \beta_{3} + \beta _1 + 2 Copy content Toggle raw display
ν3\nu^{3}== −β11−2β10+β8+β7−β6−β5−β4−β3+β2+5β1+1 -\beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== −2β11−4β10+9β8+10β7−β6−4β4−10β3+⋯+6 - 2 \beta_{11} - 4 \beta_{10} + 9 \beta_{8} + 10 \beta_{7} - \beta_{6} - 4 \beta_{4} - 10 \beta_{3} + \cdots + 6 Copy content Toggle raw display
ν5\nu^{5}== −14β11−25β10+15β8+17β7−10β6−12β5−15β4+⋯+2 - 14 \beta_{11} - 25 \beta_{10} + 15 \beta_{8} + 17 \beta_{7} - 10 \beta_{6} - 12 \beta_{5} - 15 \beta_{4} + \cdots + 2 Copy content Toggle raw display
ν6\nu^{6}== −33β11−61β10−2β9+79β8+94β7−17β6−10β5+⋯+10 - 33 \beta_{11} - 61 \beta_{10} - 2 \beta_{9} + 79 \beta_{8} + 94 \beta_{7} - 17 \beta_{6} - 10 \beta_{5} + \cdots + 10 Copy content Toggle raw display
ν7\nu^{7}== −154β11−264β10−3β9+172β8+208β7−93β6+⋯−34 - 154 \beta_{11} - 264 \beta_{10} - 3 \beta_{9} + 172 \beta_{8} + 208 \beta_{7} - 93 \beta_{6} + \cdots - 34 Copy content Toggle raw display
ν8\nu^{8}== −410β11−721β10−28β9+718β8+892β7−210β6+⋯−113 - 410 \beta_{11} - 721 \beta_{10} - 28 \beta_{9} + 718 \beta_{8} + 892 \beta_{7} - 210 \beta_{6} + \cdots - 113 Copy content Toggle raw display
ν9\nu^{9}== −1596β11−2685β10−54β9+1830β8+2300β7−885β6+⋯−647 - 1596 \beta_{11} - 2685 \beta_{10} - 54 \beta_{9} + 1830 \beta_{8} + 2300 \beta_{7} - 885 \beta_{6} + \cdots - 647 Copy content Toggle raw display
ν10\nu^{10}== −4581β11−7828β10−295β9+6767β8+8631β7−2335β6+⋯−2135 - 4581 \beta_{11} - 7828 \beta_{10} - 295 \beta_{9} + 6767 \beta_{8} + 8631 \beta_{7} - 2335 \beta_{6} + \cdots - 2135 Copy content Toggle raw display
ν11\nu^{11}== −16254β11−27080β10−677β9+18943β8+24349β7−8639β6+⋯−8165 - 16254 \beta_{11} - 27080 \beta_{10} - 677 \beta_{9} + 18943 \beta_{8} + 24349 \beta_{7} - 8639 \beta_{6} + \cdots - 8165 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.

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
3.18066
2.35032
2.15246
1.68007
1.60684
−0.0391340
−0.385135
−0.528177
−1.33949
−1.61335
−1.69090
−2.37416
1.00000 −3.18066 1.00000 −3.45219 −3.18066 3.50612 1.00000 7.11660 −3.45219
1.2 1.00000 −2.35032 1.00000 −1.74497 −2.35032 1.14267 1.00000 2.52400 −1.74497
1.3 1.00000 −2.15246 1.00000 −2.91209 −2.15246 −3.24957 1.00000 1.63308 −2.91209
1.4 1.00000 −1.68007 1.00000 2.00084 −1.68007 −2.90736 1.00000 −0.177370 2.00084
1.5 1.00000 −1.60684 1.00000 0.823492 −1.60684 0.0824650 1.00000 −0.418066 0.823492
1.6 1.00000 0.0391340 1.00000 2.72238 0.0391340 0.967809 1.00000 −2.99847 2.72238
1.7 1.00000 0.385135 1.00000 −0.181888 0.385135 4.50277 1.00000 −2.85167 −0.181888
1.8 1.00000 0.528177 1.00000 −3.17061 0.528177 −0.0476453 1.00000 −2.72103 −3.17061
1.9 1.00000 1.33949 1.00000 0.688863 1.33949 −0.213889 1.00000 −1.20577 0.688863
1.10 1.00000 1.61335 1.00000 0.900419 1.61335 −2.89419 1.00000 −0.397104 0.900419
1.11 1.00000 1.69090 1.00000 −3.94474 1.69090 2.32587 1.00000 −0.140861 −3.94474
1.12 1.00000 2.37416 1.00000 −0.729513 2.37416 −3.21505 1.00000 2.63666 −0.729513
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
22 −1 -1
1111 +1 +1
1919 −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 7942.2.a.bx 12
19.b odd 2 1 7942.2.a.bt 12
19.f odd 18 2 418.2.j.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.a ✓ 24 19.f odd 18 2
7942.2.a.bt 12 19.b odd 2 1
7942.2.a.bx 12 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 S2new(Γ0(7942))S_{2}^{\mathrm{new}}(\Gamma_0(7942)):

T312+3T311−15T310−41T39+90T38+198T37−285T36+⋯−3 T_{3}^{12} + 3 T_{3}^{11} - 15 T_{3}^{10} - 41 T_{3}^{9} + 90 T_{3}^{8} + 198 T_{3}^{7} - 285 T_{3}^{6} + \cdots - 3 Copy content Toggle raw display
T512+9T511+9T510−117T59−261T58+459T57+1287T56+⋯−81 T_{5}^{12} + 9 T_{5}^{11} + 9 T_{5}^{10} - 117 T_{5}^{9} - 261 T_{5}^{8} + 459 T_{5}^{7} + 1287 T_{5}^{6} + \cdots - 81 Copy content Toggle raw display
T1312−75T1310+26T139+1683T138−1944T137−14682T136+⋯+11503 T_{13}^{12} - 75 T_{13}^{10} + 26 T_{13}^{9} + 1683 T_{13}^{8} - 1944 T_{13}^{7} - 14682 T_{13}^{6} + \cdots + 11503 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T−1)12 (T - 1)^{12} Copy content Toggle raw display
33 T12+3T11+⋯−3 T^{12} + 3 T^{11} + \cdots - 3 Copy content Toggle raw display
55 T12+9T11+⋯−81 T^{12} + 9 T^{11} + \cdots - 81 Copy content Toggle raw display
77 T12−39T10+⋯+3 T^{12} - 39 T^{10} + \cdots + 3 Copy content Toggle raw display
1111 (T+1)12 (T + 1)^{12} Copy content Toggle raw display
1313 T12−75T10+⋯+11503 T^{12} - 75 T^{10} + \cdots + 11503 Copy content Toggle raw display
1717 T12+9T11+⋯+157797 T^{12} + 9 T^{11} + \cdots + 157797 Copy content Toggle raw display
1919 T12 T^{12} Copy content Toggle raw display
2323 T12+18T11+⋯+10344987 T^{12} + 18 T^{11} + \cdots + 10344987 Copy content Toggle raw display
2929 T12−9T11+⋯−6230061 T^{12} - 9 T^{11} + \cdots - 6230061 Copy content Toggle raw display
3131 T12+27T11+⋯+23815963 T^{12} + 27 T^{11} + \cdots + 23815963 Copy content Toggle raw display
3737 T12+⋯+133364607 T^{12} + \cdots + 133364607 Copy content Toggle raw display
4141 T12+27T11+⋯+19766277 T^{12} + 27 T^{11} + \cdots + 19766277 Copy content Toggle raw display
4343 T12−9T11+⋯−266779 T^{12} - 9 T^{11} + \cdots - 266779 Copy content Toggle raw display
4747 T12+27T11+⋯+81815211 T^{12} + 27 T^{11} + \cdots + 81815211 Copy content Toggle raw display
5353 T12−18T11+⋯+1796247 T^{12} - 18 T^{11} + \cdots + 1796247 Copy content Toggle raw display
5959 T12+⋯+296333613 T^{12} + \cdots + 296333613 Copy content Toggle raw display
6161 T12+18T11+⋯+1638579 T^{12} + 18 T^{11} + \cdots + 1638579 Copy content Toggle raw display
6767 T12+⋯−1220891289 T^{12} + \cdots - 1220891289 Copy content Toggle raw display
7171 T12+⋯+198281169 T^{12} + \cdots + 198281169 Copy content Toggle raw display
7373 T12+⋯+516405459 T^{12} + \cdots + 516405459 Copy content Toggle raw display
7979 T12+⋯+487576423 T^{12} + \cdots + 487576423 Copy content Toggle raw display
8383 T12+⋯−1572682851 T^{12} + \cdots - 1572682851 Copy content Toggle raw display
8989 T12+⋯+458650431 T^{12} + \cdots + 458650431 Copy content Toggle raw display
9797 T12+⋯−1327182035957 T^{12} + \cdots - 1327182035957 Copy content Toggle raw display
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