Properties

Label 33.6.a.d
Level $33$
Weight $6$
Character orbit 33.a
Self dual yes
Analytic conductor $5.293$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,6,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{313}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 9 q^{3} + (\beta + 46) q^{4} + (10 \beta - 24) q^{5} - 9 \beta q^{6} + (2 \beta - 10) q^{7} + (15 \beta + 78) q^{8} + 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 9 q^{3} + (\beta + 46) q^{4} + (10 \beta - 24) q^{5} - 9 \beta q^{6} + (2 \beta - 10) q^{7} + (15 \beta + 78) q^{8} + 81 q^{9} + ( - 14 \beta + 780) q^{10} + 121 q^{11} + ( - 9 \beta - 414) q^{12} + ( - 106 \beta + 20) q^{13} + ( - 8 \beta + 156) q^{14} + ( - 90 \beta + 216) q^{15} + (61 \beta - 302) q^{16} + (4 \beta - 462) q^{17} + 81 \beta q^{18} + (4 \beta - 1468) q^{19} + (446 \beta - 324) q^{20} + ( - 18 \beta + 90) q^{21} + 121 \beta q^{22} + (26 \beta + 2610) q^{23} + ( - 135 \beta - 702) q^{24} + ( - 380 \beta + 5251) q^{25} + ( - 86 \beta - 8268) q^{26} - 729 q^{27} + (84 \beta - 304) q^{28} + ( - 132 \beta - 6234) q^{29} + (126 \beta - 7020) q^{30} + (608 \beta + 4664) q^{31} + ( - 721 \beta + 2262) q^{32} - 1089 q^{33} + ( - 458 \beta + 312) q^{34} + ( - 128 \beta + 1800) q^{35} + (81 \beta + 3726) q^{36} + ( - 320 \beta + 3158) q^{37} + ( - 1464 \beta + 312) q^{38} + (954 \beta - 180) q^{39} + (570 \beta + 9828) q^{40} + ( - 728 \beta + 12486) q^{41} + (72 \beta - 1404) q^{42} + (1240 \beta + 9560) q^{43} + (121 \beta + 5566) q^{44} + (810 \beta - 1944) q^{45} + (2636 \beta + 2028) q^{46} + ( - 778 \beta - 2514) q^{47} + ( - 549 \beta + 2718) q^{48} + ( - 36 \beta - 16395) q^{49} + (4871 \beta - 29640) q^{50} + ( - 36 \beta + 4158) q^{51} + ( - 4962 \beta - 7348) q^{52} + (594 \beta + 20088) q^{53} - 729 \beta q^{54} + (1210 \beta - 2904) q^{55} + (36 \beta + 1560) q^{56} + ( - 36 \beta + 13212) q^{57} + ( - 6366 \beta - 10296) q^{58} + ( - 3676 \beta + 10944) q^{59} + ( - 4014 \beta + 2916) q^{60} + (2746 \beta - 7072) q^{61} + (5272 \beta + 47424) q^{62} + (162 \beta - 810) q^{63} + ( - 411 \beta - 46574) q^{64} + (1684 \beta - 83160) q^{65} - 1089 \beta q^{66} + (768 \beta + 32300) q^{67} + ( - 274 \beta - 20940) q^{68} + ( - 234 \beta - 23490) q^{69} + (1672 \beta - 9984) q^{70} + ( - 3102 \beta + 32274) q^{71} + (1215 \beta + 6318) q^{72} + (320 \beta + 26546) q^{73} + (2838 \beta - 24960) q^{74} + (3420 \beta - 47259) q^{75} + ( - 1280 \beta - 67216) q^{76} + (242 \beta - 1210) q^{77} + (774 \beta + 74412) q^{78} + ( - 2130 \beta + 9626) q^{79} + ( - 3874 \beta + 54828) q^{80} + 6561 q^{81} + (11758 \beta - 56784) q^{82} + ( - 3528 \beta - 5388) q^{83} + ( - 756 \beta + 2736) q^{84} + ( - 4676 \beta + 14208) q^{85} + (10800 \beta + 96720) q^{86} + (1188 \beta + 56106) q^{87} + (1815 \beta + 9438) q^{88} + (3024 \beta - 30582) q^{89} + ( - 1134 \beta + 63180) q^{90} + (888 \beta - 16736) q^{91} + (3832 \beta + 122088) q^{92} + ( - 5472 \beta - 41976) q^{93} + ( - 3292 \beta - 60684) q^{94} + ( - 14736 \beta + 38352) q^{95} + (6489 \beta - 20358) q^{96} + (1092 \beta - 92074) q^{97} + ( - 16431 \beta - 2808) q^{98} + 9801 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 18 q^{3} + 93 q^{4} - 38 q^{5} - 9 q^{6} - 18 q^{7} + 171 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 18 q^{3} + 93 q^{4} - 38 q^{5} - 9 q^{6} - 18 q^{7} + 171 q^{8} + 162 q^{9} + 1546 q^{10} + 242 q^{11} - 837 q^{12} - 66 q^{13} + 304 q^{14} + 342 q^{15} - 543 q^{16} - 920 q^{17} + 81 q^{18} - 2932 q^{19} - 202 q^{20} + 162 q^{21} + 121 q^{22} + 5246 q^{23} - 1539 q^{24} + 10122 q^{25} - 16622 q^{26} - 1458 q^{27} - 524 q^{28} - 12600 q^{29} - 13914 q^{30} + 9936 q^{31} + 3803 q^{32} - 2178 q^{33} + 166 q^{34} + 3472 q^{35} + 7533 q^{36} + 5996 q^{37} - 840 q^{38} + 594 q^{39} + 20226 q^{40} + 24244 q^{41} - 2736 q^{42} + 20360 q^{43} + 11253 q^{44} - 3078 q^{45} + 6692 q^{46} - 5806 q^{47} + 4887 q^{48} - 32826 q^{49} - 54409 q^{50} + 8280 q^{51} - 19658 q^{52} + 40770 q^{53} - 729 q^{54} - 4598 q^{55} + 3156 q^{56} + 26388 q^{57} - 26958 q^{58} + 18212 q^{59} + 1818 q^{60} - 11398 q^{61} + 100120 q^{62} - 1458 q^{63} - 93559 q^{64} - 164636 q^{65} - 1089 q^{66} + 65368 q^{67} - 42154 q^{68} - 47214 q^{69} - 18296 q^{70} + 61446 q^{71} + 13851 q^{72} + 53412 q^{73} - 47082 q^{74} - 91098 q^{75} - 135712 q^{76} - 2178 q^{77} + 149598 q^{78} + 17122 q^{79} + 105782 q^{80} + 13122 q^{81} - 101810 q^{82} - 14304 q^{83} + 4716 q^{84} + 23740 q^{85} + 204240 q^{86} + 113400 q^{87} + 20691 q^{88} - 58140 q^{89} + 125226 q^{90} - 32584 q^{91} + 248008 q^{92} - 89424 q^{93} - 124660 q^{94} + 61968 q^{95} - 34227 q^{96} - 183056 q^{97} - 22047 q^{98} + 19602 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.34590
9.34590
−8.34590 −9.00000 37.6541 −107.459 75.1131 −26.6918 −47.1885 81.0000 896.843
1.2 9.34590 −9.00000 55.3459 69.4590 −84.1131 8.69181 218.189 81.0000 649.157
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

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 33.6.a.d 2
3.b odd 2 1 99.6.a.e 2
4.b odd 2 1 528.6.a.q 2
5.b even 2 1 825.6.a.d 2
11.b odd 2 1 363.6.a.g 2
33.d even 2 1 1089.6.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.d 2 1.a even 1 1 trivial
99.6.a.e 2 3.b odd 2 1
363.6.a.g 2 11.b odd 2 1
528.6.a.q 2 4.b odd 2 1
825.6.a.d 2 5.b even 2 1
1089.6.a.o 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 78 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 78 \) Copy content Toggle raw display
$3$ \( (T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 38T - 7464 \) Copy content Toggle raw display
$7$ \( T^{2} + 18T - 232 \) Copy content Toggle raw display
$11$ \( (T - 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 66T - 878128 \) Copy content Toggle raw display
$17$ \( T^{2} + 920T + 210348 \) Copy content Toggle raw display
$19$ \( T^{2} + 2932 T + 2147904 \) Copy content Toggle raw display
$23$ \( T^{2} - 5246 T + 6827232 \) Copy content Toggle raw display
$29$ \( T^{2} + 12600 T + 38326572 \) Copy content Toggle raw display
$31$ \( T^{2} - 9936 T - 4245184 \) Copy content Toggle raw display
$37$ \( T^{2} - 5996 T + 975204 \) Copy content Toggle raw display
$41$ \( T^{2} - 24244 T + 105471636 \) Copy content Toggle raw display
$43$ \( T^{2} - 20360 T - 16684800 \) Copy content Toggle raw display
$47$ \( T^{2} + 5806 T - 38936064 \) Copy content Toggle raw display
$53$ \( T^{2} - 40770 T + 387938808 \) Copy content Toggle raw display
$59$ \( T^{2} - 18212 T - 974471136 \) Copy content Toggle raw display
$61$ \( T^{2} + 11398 T - 557566776 \) Copy content Toggle raw display
$67$ \( T^{2} - 65368 T + 1022090128 \) Copy content Toggle raw display
$71$ \( T^{2} - 61446 T + 190949616 \) Copy content Toggle raw display
$73$ \( T^{2} - 53412 T + 705197636 \) Copy content Toggle raw display
$79$ \( T^{2} - 17122 T - 281721704 \) Copy content Toggle raw display
$83$ \( T^{2} + 14304 T - 922809744 \) Copy content Toggle raw display
$89$ \( T^{2} + 58140 T + 129501828 \) Copy content Toggle raw display
$97$ \( T^{2} + 183056 T + 8284064476 \) Copy content Toggle raw display
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