Properties

Label 1296.4.a
Level 12961296
Weight 44
Character orbit 1296.a
Rep. character χ1296(1,)\chi_{1296}(1,\cdot)
Character field Q\Q
Dimension 7070
Newform subspaces 3030
Sturm bound 864864
Trace bound 77

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Defining parameters

Level: N N == 1296=2434 1296 = 2^{4} \cdot 3^{4}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1296.a (trivial)
Character field: Q\Q
Newform subspaces: 30 30
Sturm bound: 864864
Trace bound: 77
Distinguishing TpT_p: 55

Dimensions

The following table gives the dimensions of various subspaces of M4(Γ0(1296))M_{4}(\Gamma_0(1296)).

Total New Old
Modular forms 684 74 610
Cusp forms 612 70 542
Eisenstein series 72 4 68

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

2233FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
++++++174174191915515515615619191371371818001818
++--168168171715115115015017171331331818001818
-++-168168181815015015015016161341341818221616
--++174174202015415415615618181381381818221616
Plus space++348348393930930931231237372752753636223434
Minus space-336336353530130130030033332672673636223434

Trace form

70q2q7+2q13+4q19+1552q25182q31400q37+250q43+2844q49246q55+470q611226q67+824q73362q79+1584q85+1298q9134q97+O(q100) 70 q - 2 q^{7} + 2 q^{13} + 4 q^{19} + 1552 q^{25} - 182 q^{31} - 400 q^{37} + 250 q^{43} + 2844 q^{49} - 246 q^{55} + 470 q^{61} - 1226 q^{67} + 824 q^{73} - 362 q^{79} + 1584 q^{85} + 1298 q^{91} - 34 q^{97}+O(q^{100}) Copy content Toggle raw display

Decomposition of S4new(Γ0(1296))S_{4}^{\mathrm{new}}(\Gamma_0(1296)) into newform subspaces

Label Char Prim Dim AA Field CM Minimal twist Traces A-L signs Sato-Tate qq-expansion
a2a_{2} a3a_{3} a5a_{5} a7a_{7} 2 3
1296.4.a.a 1296.a 1.a 11 76.46676.466 Q\Q None 162.4.a.b 00 00 21-21 8-8 - ++ SU(2)\mathrm{SU}(2) q21q58q7+62q1172q13+q-21q^{5}-8q^{7}+6^{2}q^{11}-7^{2}q^{13}+\cdots
1296.4.a.b 1296.a 1.a 11 76.46676.466 Q\Q None 18.4.c.a 00 00 9-9 3131 - ++ SU(2)\mathrm{SU}(2) q9q5+31q7+15q1137q13+q-9q^{5}+31q^{7}+15q^{11}-37q^{13}+\cdots
1296.4.a.c 1296.a 1.a 11 76.46676.466 Q\Q None 648.4.a.a 00 00 5-5 36-36 ++ ++ SU(2)\mathrm{SU}(2) q5q562q726q1165q13+q-5q^{5}-6^{2}q^{7}-2^{6}q^{11}-65q^{13}+\cdots
1296.4.a.d 1296.a 1.a 11 76.46676.466 Q\Q None 324.4.a.a 00 00 3-3 44 - ++ SU(2)\mathrm{SU}(2) q3q5+4q724q1152q13+q-3q^{5}+4q^{7}-24q^{11}-5^{2}q^{13}+\cdots
1296.4.a.e 1296.a 1.a 11 76.46676.466 Q\Q None 324.4.a.a 00 00 33 44 - ++ SU(2)\mathrm{SU}(2) q+3q5+4q7+24q1152q13+q+3q^{5}+4q^{7}+24q^{11}-5^{2}q^{13}+\cdots
1296.4.a.f 1296.a 1.a 11 76.46676.466 Q\Q None 648.4.a.a 00 00 55 36-36 ++ ++ SU(2)\mathrm{SU}(2) q+5q562q7+26q1165q13+q+5q^{5}-6^{2}q^{7}+2^{6}q^{11}-65q^{13}+\cdots
1296.4.a.g 1296.a 1.a 11 76.46676.466 Q\Q None 18.4.c.a 00 00 99 3131 - - SU(2)\mathrm{SU}(2) q+9q5+31q715q1137q13+q+9q^{5}+31q^{7}-15q^{11}-37q^{13}+\cdots
1296.4.a.h 1296.a 1.a 11 76.46676.466 Q\Q None 162.4.a.b 00 00 2121 8-8 - ++ SU(2)\mathrm{SU}(2) q+21q58q762q1172q13+q+21q^{5}-8q^{7}-6^{2}q^{11}-7^{2}q^{13}+\cdots
1296.4.a.i 1296.a 1.a 22 76.46676.466 Q(33)\Q(\sqrt{33}) None 9.4.c.a 00 00 15-15 7-7 - - SU(2)\mathrm{SU}(2) q+(7β)q5+(5+3β)q7+(29+)q11+q+(-7-\beta )q^{5}+(-5+3\beta )q^{7}+(29+\cdots)q^{11}+\cdots
1296.4.a.j 1296.a 1.a 22 76.46676.466 Q(3)\Q(\sqrt{3}) None 162.4.a.e 00 00 12-12 16-16 - - SU(2)\mathrm{SU}(2) q+(6+β)q5+(8+2β)q7+(18+)q11+q+(-6+\beta )q^{5}+(-8+2\beta )q^{7}+(18+\cdots)q^{11}+\cdots
1296.4.a.k 1296.a 1.a 22 76.46676.466 Q(57)\Q(\sqrt{57}) None 81.4.a.b 00 00 12-12 10-10 - ++ SU(2)\mathrm{SU}(2) q+(6β)q5+(53β)q7+(21+)q11+q+(-6-\beta )q^{5}+(-5-3\beta )q^{7}+(21+\cdots)q^{11}+\cdots
1296.4.a.l 1296.a 1.a 22 76.46676.466 Q(105)\Q(\sqrt{105}) None 18.4.c.b 00 00 9-9 19-19 - ++ SU(2)\mathrm{SU}(2) q+(5β)q5+(9+β)q7+(112β)q11+q+(-5-\beta )q^{5}+(-9+\beta )q^{7}+(11-2\beta )q^{11}+\cdots
1296.4.a.m 1296.a 1.a 22 76.46676.466 Q(201)\Q(\sqrt{201}) None 648.4.a.c 00 00 8-8 3030 ++ ++ SU(2)\mathrm{SU}(2) q+(4β)q5+(15β)q7+(23+)q11+q+(-4-\beta )q^{5}+(15-\beta )q^{7}+(-23+\cdots)q^{11}+\cdots
1296.4.a.n 1296.a 1.a 22 76.46676.466 Q(129)\Q(\sqrt{129}) None 648.4.a.d 00 00 4-4 66 ++ ++ SU(2)\mathrm{SU}(2) q+(2β)q5+(3+β)q7+(5+3β)q11+q+(-2-\beta )q^{5}+(3+\beta )q^{7}+(5+3\beta )q^{11}+\cdots
1296.4.a.o 1296.a 1.a 22 76.46676.466 Q(3)\Q(\sqrt{3}) None 81.4.a.c 00 00 00 4444 - - SU(2)\mathrm{SU}(2) q+7βq5+22q7+34βq1172q13+q+7\beta q^{5}+22q^{7}+34\beta q^{11}-7^{2}q^{13}+\cdots
1296.4.a.p 1296.a 1.a 22 76.46676.466 Q(129)\Q(\sqrt{129}) None 648.4.a.d 00 00 44 66 ++ ++ SU(2)\mathrm{SU}(2) q+(2+β)q5+(3+β)q7+(53β)q11+q+(2+\beta )q^{5}+(3+\beta )q^{7}+(-5-3\beta )q^{11}+\cdots
1296.4.a.q 1296.a 1.a 22 76.46676.466 Q(201)\Q(\sqrt{201}) None 648.4.a.c 00 00 88 3030 ++ ++ SU(2)\mathrm{SU}(2) q+(4+β)q5+(15β)q7+(23+3β)q11+q+(4+\beta )q^{5}+(15-\beta )q^{7}+(23+3\beta )q^{11}+\cdots
1296.4.a.r 1296.a 1.a 22 76.46676.466 Q(105)\Q(\sqrt{105}) None 18.4.c.b 00 00 99 19-19 - - SU(2)\mathrm{SU}(2) q+(5+β)q5+(9+β)q7+(11+2β)q11+q+(5+\beta )q^{5}+(-9+\beta )q^{7}+(-11+2\beta )q^{11}+\cdots
1296.4.a.s 1296.a 1.a 22 76.46676.466 Q(3)\Q(\sqrt{3}) None 162.4.a.e 00 00 1212 16-16 - - SU(2)\mathrm{SU}(2) q+(6+β)q5+(82β)q7+(18+)q11+q+(6+\beta )q^{5}+(-8-2\beta )q^{7}+(-18+\cdots)q^{11}+\cdots
1296.4.a.t 1296.a 1.a 22 76.46676.466 Q(57)\Q(\sqrt{57}) None 81.4.a.b 00 00 1212 10-10 - ++ SU(2)\mathrm{SU}(2) q+(6β)q5+(5+3β)q7+(21+)q11+q+(6-\beta )q^{5}+(-5+3\beta )q^{7}+(-21+\cdots)q^{11}+\cdots
1296.4.a.u 1296.a 1.a 22 76.46676.466 Q(33)\Q(\sqrt{33}) None 9.4.c.a 00 00 1515 7-7 - ++ SU(2)\mathrm{SU}(2) q+(8β)q5+(23β)q7+(37+)q11+q+(8-\beta )q^{5}+(-2-3\beta )q^{7}+(-37+\cdots)q^{11}+\cdots
1296.4.a.v 1296.a 1.a 33 76.46676.466 3.3.1509.1 None 36.4.e.a 00 00 6-6 6-6 - ++ SU(2)\mathrm{SU}(2) q+(2β2)q5+(2+β1β2)q7+q+(-2-\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots
1296.4.a.w 1296.a 1.a 33 76.46676.466 3.3.1509.1 None 36.4.e.a 00 00 66 6-6 - - SU(2)\mathrm{SU}(2) q+(2+β2)q5+(2+β1β2)q7+q+(2+\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots
1296.4.a.x 1296.a 1.a 44 76.46676.466 4.4.29952.1 None 648.4.a.g 00 00 8-8 00 ++ - SU(2)\mathrm{SU}(2) q+(2β1)q5+(β1+β2)q7+(2+)q11+q+(-2-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2+\cdots)q^{11}+\cdots
1296.4.a.y 1296.a 1.a 44 76.46676.466 4.4.72153.1 None 72.4.i.a 00 00 5-5 33 ++ - SU(2)\mathrm{SU}(2) q+(1β2)q5+(1+β3)q7+(4+)q11+q+(-1-\beta _{2})q^{5}+(1+\beta _{3})q^{7}+(-4+\cdots)q^{11}+\cdots
1296.4.a.z 1296.a 1.a 44 76.46676.466 Q(3,7)\Q(\sqrt{3}, \sqrt{7}) None 324.4.a.e 00 00 00 1616 - - SU(2)\mathrm{SU}(2) q+β2q5+(4β3)q7+(β1β2)q11+q+\beta _{2}q^{5}+(4-\beta _{3})q^{7}+(\beta _{1}-\beta _{2})q^{11}+\cdots
1296.4.a.ba 1296.a 1.a 44 76.46676.466 4.4.72153.1 None 72.4.i.a 00 00 55 33 ++ ++ SU(2)\mathrm{SU}(2) q+(1+β2)q5+(1+β3)q7+(4β1+)q11+q+(1+\beta _{2})q^{5}+(1+\beta _{3})q^{7}+(4-\beta _{1}+\cdots)q^{11}+\cdots
1296.4.a.bb 1296.a 1.a 44 76.46676.466 4.4.29952.1 None 648.4.a.g 00 00 88 00 ++ - SU(2)\mathrm{SU}(2) q+(2β1)q5+(β1β2)q7+(2+β1+)q11+q+(2-\beta _{1})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-2+\beta _{1}+\cdots)q^{11}+\cdots
1296.4.a.bc 1296.a 1.a 55 76.46676.466 Q[x]/(x5)\mathbb{Q}[x]/(x^{5} - \cdots) None 72.4.i.b 00 00 5-5 3-3 ++ - SU(2)\mathrm{SU}(2) q+(1β2)q5+(1β1)q7+(5+)q11+q+(-1-\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+(-5+\cdots)q^{11}+\cdots
1296.4.a.bd 1296.a 1.a 55 76.46676.466 Q[x]/(x5)\mathbb{Q}[x]/(x^{5} - \cdots) None 72.4.i.b 00 00 55 3-3 ++ ++ SU(2)\mathrm{SU}(2) q+(1+β2)q5+(1β1)q7+(52β2+)q11+q+(1+\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+(5-2\beta _{2}+\cdots)q^{11}+\cdots

Decomposition of S4old(Γ0(1296))S_{4}^{\mathrm{old}}(\Gamma_0(1296)) into lower level spaces

S4old(Γ0(1296)) S_{4}^{\mathrm{old}}(\Gamma_0(1296)) \simeq S4new(Γ0(6))S_{4}^{\mathrm{new}}(\Gamma_0(6))16^{\oplus 16}\oplusS4new(Γ0(8))S_{4}^{\mathrm{new}}(\Gamma_0(8))10^{\oplus 10}\oplusS4new(Γ0(9))S_{4}^{\mathrm{new}}(\Gamma_0(9))15^{\oplus 15}\oplusS4new(Γ0(12))S_{4}^{\mathrm{new}}(\Gamma_0(12))12^{\oplus 12}\oplusS4new(Γ0(16))S_{4}^{\mathrm{new}}(\Gamma_0(16))5^{\oplus 5}\oplusS4new(Γ0(18))S_{4}^{\mathrm{new}}(\Gamma_0(18))12^{\oplus 12}\oplusS4new(Γ0(24))S_{4}^{\mathrm{new}}(\Gamma_0(24))8^{\oplus 8}\oplusS4new(Γ0(27))S_{4}^{\mathrm{new}}(\Gamma_0(27))10^{\oplus 10}\oplusS4new(Γ0(36))S_{4}^{\mathrm{new}}(\Gamma_0(36))9^{\oplus 9}\oplusS4new(Γ0(48))S_{4}^{\mathrm{new}}(\Gamma_0(48))4^{\oplus 4}\oplusS4new(Γ0(54))S_{4}^{\mathrm{new}}(\Gamma_0(54))8^{\oplus 8}\oplusS4new(Γ0(72))S_{4}^{\mathrm{new}}(\Gamma_0(72))6^{\oplus 6}\oplusS4new(Γ0(81))S_{4}^{\mathrm{new}}(\Gamma_0(81))5^{\oplus 5}\oplusS4new(Γ0(108))S_{4}^{\mathrm{new}}(\Gamma_0(108))6^{\oplus 6}\oplusS4new(Γ0(144))S_{4}^{\mathrm{new}}(\Gamma_0(144))3^{\oplus 3}\oplusS4new(Γ0(162))S_{4}^{\mathrm{new}}(\Gamma_0(162))4^{\oplus 4}\oplusS4new(Γ0(216))S_{4}^{\mathrm{new}}(\Gamma_0(216))4^{\oplus 4}\oplusS4new(Γ0(324))S_{4}^{\mathrm{new}}(\Gamma_0(324))3^{\oplus 3}\oplusS4new(Γ0(432))S_{4}^{\mathrm{new}}(\Gamma_0(432))2^{\oplus 2}\oplusS4new(Γ0(648))S_{4}^{\mathrm{new}}(\Gamma_0(648))2^{\oplus 2}