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Properties

Label 650.2.t.e
Level 650650
Weight 22
Character orbit 650.t
Analytic conductor 5.1905.190
Analytic rank 00
Dimension 1212
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] = [650,2,Mod(7,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 11])) 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("650.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 650=2⋅52⋅13 650 = 2 \cdot 5^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 650.t (of order 1212, degree 44, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,6,0,0,0] 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: no
Analytic conductor: 5.190276131385.19027613138
Analytic rank: 00
Dimension: 1212
Relative dimension: 33 over Q(ζ12)\Q(\zeta_{12})
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+24x10+192x8+680x6+1104x4+672x2+4 x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: SU(2)[C12]\mathrm{SU}(2)[C_{12}]

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+β4q2+β8q3+(−β3+1)q4+(β11+β8+⋯+β5)q6+(β5−β1)q7+(β6+β4)q8+(2β10−2β9+β8+⋯−4)q9+⋯+(β10−2β8−β7+⋯+6)q99+O(q100) q + \beta_{4} q^{2} + \beta_{8} q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{11} + \beta_{8} + \cdots + \beta_{5}) q^{6} + (\beta_{5} - \beta_1) q^{7} + (\beta_{6} + \beta_{4}) q^{8} + (2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots - 4) q^{9}+ \cdots + (\beta_{10} - 2 \beta_{8} - \beta_{7} + \cdots + 6) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q+6q4−24q9+6q11+12q13−6q16−12q17−12q18+36q19−24q21−6q22−6q23+6q26+12q27+6q29−24q31+6q33−12q34−24q36+⋯+54q99+O(q100) 12 q + 6 q^{4} - 24 q^{9} + 6 q^{11} + 12 q^{13} - 6 q^{16} - 12 q^{17} - 12 q^{18} + 36 q^{19} - 24 q^{21} - 6 q^{22} - 6 q^{23} + 6 q^{26} + 12 q^{27} + 6 q^{29} - 24 q^{31} + 6 q^{33} - 12 q^{34} - 24 q^{36}+ \cdots + 54 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12+24x10+192x8+680x6+1104x4+672x2+4 x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== (7ν10+136ν8+668ν6+702ν4+176ν2+380ν+1728)/760 ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} + 380\nu + 1728 ) / 760 Copy content Toggle raw display
β2\beta_{2}== (7ν10+136ν8+668ν6+702ν4+176ν2−380ν+1728)/760 ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} - 380\nu + 1728 ) / 760 Copy content Toggle raw display
β3\beta_{3}== (4ν11+87ν9+576ν7+1504ν5+1542ν3+456ν+20)/40 ( 4\nu^{11} + 87\nu^{9} + 576\nu^{7} + 1504\nu^{5} + 1542\nu^{3} + 456\nu + 20 ) / 40 Copy content Toggle raw display
β4\beta_{4}== (−60ν11+36ν10−1315ν9+808ν8−8820ν7+5634ν6−23280ν5+⋯+744)/760 ( - 60 \nu^{11} + 36 \nu^{10} - 1315 \nu^{9} + 808 \nu^{8} - 8820 \nu^{7} + 5634 \nu^{6} - 23280 \nu^{5} + \cdots + 744 ) / 760 Copy content Toggle raw display
β5\beta_{5}== (−11ν11+89ν10−173ν9+1892ν8−154ν7+11886ν6+5194ν5+⋯+1016)/760 ( - 11 \nu^{11} + 89 \nu^{10} - 173 \nu^{9} + 1892 \nu^{8} - 154 \nu^{7} + 11886 \nu^{6} + 5194 \nu^{5} + \cdots + 1016 ) / 760 Copy content Toggle raw display
β6\beta_{6}== (−60ν11−36ν10−1315ν9−808ν8−8820ν7−5634ν6−23280ν5+⋯−744)/760 ( - 60 \nu^{11} - 36 \nu^{10} - 1315 \nu^{9} - 808 \nu^{8} - 8820 \nu^{7} - 5634 \nu^{6} - 23280 \nu^{5} + \cdots - 744 ) / 760 Copy content Toggle raw display
β7\beta_{7}== (11ν11+82ν10+173ν9+1756ν8+154ν7+11218ν6−5194ν5+⋯−712)/760 ( 11 \nu^{11} + 82 \nu^{10} + 173 \nu^{9} + 1756 \nu^{8} + 154 \nu^{7} + 11218 \nu^{6} - 5194 \nu^{5} + \cdots - 712 ) / 760 Copy content Toggle raw display
β8\beta_{8}== (−36ν11+125ν10−808ν9+2700ν8−5634ν7+17520ν6−15336ν5+⋯+240)/760 ( - 36 \nu^{11} + 125 \nu^{10} - 808 \nu^{9} + 2700 \nu^{8} - 5634 \nu^{7} + 17520 \nu^{6} - 15336 \nu^{5} + \cdots + 240 ) / 760 Copy content Toggle raw display
β9\beta_{9}== (3ν11−134ν10+99ν9−2902ν8+1182ν7−18976ν6+6218ν5+⋯−1756)/760 ( 3 \nu^{11} - 134 \nu^{10} + 99 \nu^{9} - 2902 \nu^{8} + 1182 \nu^{7} - 18976 \nu^{6} + 6218 \nu^{5} + \cdots - 1756 ) / 760 Copy content Toggle raw display
β10\beta_{10}== (−105ν11−178ν10−2135ν9−3784ν8−11920ν7−23772ν6+⋯−132)/760 ( - 105 \nu^{11} - 178 \nu^{10} - 2135 \nu^{9} - 3784 \nu^{8} - 11920 \nu^{7} - 23772 \nu^{6} + \cdots - 132 ) / 760 Copy content Toggle raw display
β11\beta_{11}== (−421ν10−9048ν8−58144ν6−140586ν4−112968ν2+380ν−784)/760 ( -421\nu^{10} - 9048\nu^{8} - 58144\nu^{6} - 140586\nu^{4} - 112968\nu^{2} + 380\nu - 784 ) / 760 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== −β2+β1 -\beta_{2} + \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β11+2β7−β6+2β5+β4+β1−4 \beta_{11} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + \beta _1 - 4 Copy content Toggle raw display
ν3\nu^{3}== −2β11−4β8−2β7+5β6−2β5+5β4+6β3+⋯−3 - 2 \beta_{11} - 4 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} + \cdots - 3 Copy content Toggle raw display
ν4\nu^{4}== −13β11+2β10−4β9+2β8−27β7+14β6−29β5+⋯+28 - 13 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 27 \beta_{7} + 14 \beta_{6} - 29 \beta_{5} + \cdots + 28 Copy content Toggle raw display
ν5\nu^{5}== 32β11−5β10+59β8+24β7−77β6+35β5−72β4+⋯+50 32 \beta_{11} - 5 \beta_{10} + 59 \beta_{8} + 24 \beta_{7} - 77 \beta_{6} + 35 \beta_{5} - 72 \beta_{4} + \cdots + 50 Copy content Toggle raw display
ν6\nu^{6}== 168β11−24β10+48β9−24β8+352β7−174β6+⋯−292 168 \beta_{11} - 24 \beta_{10} + 48 \beta_{9} - 24 \beta_{8} + 352 \beta_{7} - 174 \beta_{6} + \cdots - 292 Copy content Toggle raw display
ν7\nu^{7}== −422β11+96β10−748β8−270β7+1008β6−478β5+⋯−680 - 422 \beta_{11} + 96 \beta_{10} - 748 \beta_{8} - 270 \beta_{7} + 1008 \beta_{6} - 478 \beta_{5} + \cdots - 680 Copy content Toggle raw display
ν8\nu^{8}== −2122β11+270β10−540β9+270β8−4460β7+2132β6+⋯+3460 - 2122 \beta_{11} + 270 \beta_{10} - 540 \beta_{9} + 270 \beta_{8} - 4460 \beta_{7} + 2132 \beta_{6} + \cdots + 3460 Copy content Toggle raw display
ν9\nu^{9}== 5316β11−1368β10+9264β8+3156β7−12698β6+6108β5+⋯+8706 5316 \beta_{11} - 1368 \beta_{10} + 9264 \beta_{8} + 3156 \beta_{7} - 12698 \beta_{6} + 6108 \beta_{5} + \cdots + 8706 Copy content Toggle raw display
ν10\nu^{10}== 26474β11−3156β10+6312β9−3156β8+55718β7−26196β6+⋯−42312 26474 \beta_{11} - 3156 \beta_{10} + 6312 \beta_{9} - 3156 \beta_{8} + 55718 \beta_{7} - 26196 \beta_{6} + \cdots - 42312 Copy content Toggle raw display
ν11\nu^{11}== −66116β11+17810β10−114422β8−38016β7+158054β6+⋯−109084 - 66116 \beta_{11} + 17810 \beta_{10} - 114422 \beta_{8} - 38016 \beta_{7} + 158054 \beta_{6} + \cdots - 109084 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/650Z)×\left(\mathbb{Z}/650\mathbb{Z}\right)^\times.

nn 2727 301301
χ(n)\chi(n) β4+β6\beta_{4} + \beta_{6} −β6-\beta_{6}

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}
7.1
1.62980i
− 0.0775341i
− 1.55227i
− 1.62980i
0.0775341i
1.55227i
2.19540i
1.32083i
− 3.51623i
− 2.19540i
− 1.32083i
3.51623i
0.866025 + 0.500000i −0.814901 − 3.04125i 0.500000 + 0.866025i 0 0.814901 − 3.04125i −0.402397 − 0.696972i 1.00000i −5.98707 + 3.45663i 0
7.2 0.866025 + 0.500000i 0.0387670 + 0.144681i 0.500000 + 0.866025i 0 −0.0387670 + 0.144681i −1.10259 − 1.90974i 1.00000i 2.57865 − 1.48878i 0
7.3 0.866025 + 0.500000i 0.776134 + 2.89657i 0.500000 + 0.866025i 0 −0.776134 + 2.89657i 0.638961 + 1.10671i 1.00000i −5.18966 + 2.99625i 0
93.1 0.866025 − 0.500000i −0.814901 + 3.04125i 0.500000 − 0.866025i 0 0.814901 + 3.04125i −0.402397 + 0.696972i − 1.00000i −5.98707 − 3.45663i 0
93.2 0.866025 − 0.500000i 0.0387670 − 0.144681i 0.500000 − 0.866025i 0 −0.0387670 − 0.144681i −1.10259 + 1.90974i − 1.00000i 2.57865 + 1.48878i 0
93.3 0.866025 − 0.500000i 0.776134 − 2.89657i 0.500000 − 0.866025i 0 −0.776134 − 2.89657i 0.638961 − 1.10671i − 1.00000i −5.18966 − 2.99625i 0
557.1 −0.866025 + 0.500000i −1.09770 − 0.294128i 0.500000 − 0.866025i 0 1.09770 − 0.294128i 2.54283 − 4.40431i 1.00000i −1.47964 − 0.854273i 0
557.2 −0.866025 + 0.500000i −0.660414 − 0.176957i 0.500000 − 0.866025i 0 0.660414 − 0.176957i −0.189447 + 0.328132i 1.00000i −2.19324 − 1.26627i 0
557.3 −0.866025 + 0.500000i 1.75811 + 0.471085i 0.500000 − 0.866025i 0 −1.75811 + 0.471085i −1.48736 + 2.57618i 1.00000i 0.270964 + 0.156441i 0
643.1 −0.866025 − 0.500000i −1.09770 + 0.294128i 0.500000 + 0.866025i 0 1.09770 + 0.294128i 2.54283 + 4.40431i − 1.00000i −1.47964 + 0.854273i 0
643.2 −0.866025 − 0.500000i −0.660414 + 0.176957i 0.500000 + 0.866025i 0 0.660414 + 0.176957i −0.189447 − 0.328132i − 1.00000i −2.19324 + 1.26627i 0
643.3 −0.866025 − 0.500000i 1.75811 − 0.471085i 0.500000 + 0.866025i 0 −1.75811 − 0.471085i −1.48736 − 2.57618i − 1.00000i 0.270964 − 0.156441i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.t.e 12
5.b even 2 1 130.2.p.a ✓ 12
5.c odd 4 1 130.2.s.a yes 12
5.c odd 4 1 650.2.w.e 12
13.f odd 12 1 650.2.w.e 12
65.o even 12 1 130.2.p.a ✓ 12
65.s odd 12 1 130.2.s.a yes 12
65.t even 12 1 inner 650.2.t.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.p.a ✓ 12 5.b even 2 1
130.2.p.a ✓ 12 65.o even 12 1
130.2.s.a yes 12 5.c odd 4 1
130.2.s.a yes 12 65.s odd 12 1
650.2.t.e 12 1.a even 1 1 trivial
650.2.t.e 12 65.t even 12 1 inner
650.2.w.e 12 5.c odd 4 1
650.2.w.e 12 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T312+12T310−4T39+24T38−24T37−280T36−48T35+⋯+4 T_{3}^{12} + 12 T_{3}^{10} - 4 T_{3}^{9} + 24 T_{3}^{8} - 24 T_{3}^{7} - 280 T_{3}^{6} - 48 T_{3}^{5} + \cdots + 4 acting on S2new(650,[χ])S_{2}^{\mathrm{new}}(650, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−T2+1)3 (T^{4} - T^{2} + 1)^{3} Copy content Toggle raw display
33 T12+12T10+⋯+4 T^{12} + 12 T^{10} + \cdots + 4 Copy content Toggle raw display
55 T12 T^{12} Copy content Toggle raw display
77 T12+21T10+⋯+169 T^{12} + 21 T^{10} + \cdots + 169 Copy content Toggle raw display
1111 T12−6T11+⋯+6889 T^{12} - 6 T^{11} + \cdots + 6889 Copy content Toggle raw display
1313 T12−12T11+⋯+4826809 T^{12} - 12 T^{11} + \cdots + 4826809 Copy content Toggle raw display
1717 T12+12T11+⋯+4 T^{12} + 12 T^{11} + \cdots + 4 Copy content Toggle raw display
1919 T12−36T11+⋯+221841 T^{12} - 36 T^{11} + \cdots + 221841 Copy content Toggle raw display
2323 T12+6T11+⋯+23078416 T^{12} + 6 T^{11} + \cdots + 23078416 Copy content Toggle raw display
2929 T12−6T11+⋯+8248384 T^{12} - 6 T^{11} + \cdots + 8248384 Copy content Toggle raw display
3131 T12+24T11+⋯+26152996 T^{12} + 24 T^{11} + \cdots + 26152996 Copy content Toggle raw display
3737 T12+93T10+⋯+769129 T^{12} + 93 T^{10} + \cdots + 769129 Copy content Toggle raw display
4141 T12−18T11+⋯+1690000 T^{12} - 18 T^{11} + \cdots + 1690000 Copy content Toggle raw display
4343 T12−36T10+⋯+22886656 T^{12} - 36 T^{10} + \cdots + 22886656 Copy content Toggle raw display
4747 (T6+6T5+⋯−6047)2 (T^{6} + 6 T^{5} + \cdots - 6047)^{2} Copy content Toggle raw display
5353 T12+18T11+⋯+36881329 T^{12} + 18 T^{11} + \cdots + 36881329 Copy content Toggle raw display
5959 T12−18T11+⋯+576 T^{12} - 18 T^{11} + \cdots + 576 Copy content Toggle raw display
6161 T12+⋯+307721764 T^{12} + \cdots + 307721764 Copy content Toggle raw display
6767 T12+12T11+⋯+2304 T^{12} + 12 T^{11} + \cdots + 2304 Copy content Toggle raw display
7171 T12+⋯+880427584 T^{12} + \cdots + 880427584 Copy content Toggle raw display
7373 T12+⋯+704902500 T^{12} + \cdots + 704902500 Copy content Toggle raw display
7979 T12+⋯+1427177284 T^{12} + \cdots + 1427177284 Copy content Toggle raw display
8383 (T6−24T5+⋯−31050)2 (T^{6} - 24 T^{5} + \cdots - 31050)^{2} Copy content Toggle raw display
8989 T12−6T11+⋯+39828721 T^{12} - 6 T^{11} + \cdots + 39828721 Copy content Toggle raw display
9797 T12+⋯+553217613796 T^{12} + \cdots + 553217613796 Copy content Toggle raw display
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