Properties

Label 33.4.a.d
Level $33$
Weight $4$
Character orbit 33.a
Self dual yes
Analytic conductor $1.947$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) 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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} + \beta q^{4} + ( - 4 \beta + 10) q^{5} + 3 \beta q^{6} + ( - 2 \beta + 2) q^{7} + ( - 7 \beta + 8) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 3 q^{3} + \beta q^{4} + ( - 4 \beta + 10) q^{5} + 3 \beta q^{6} + ( - 2 \beta + 2) q^{7} + ( - 7 \beta + 8) q^{8} + 9 q^{9} + (6 \beta - 32) q^{10} + 11 q^{11} + 3 \beta q^{12} + (8 \beta - 42) q^{13} - 16 q^{14} + ( - 12 \beta + 30) q^{15} + ( - 7 \beta - 56) q^{16} + (30 \beta - 28) q^{17} + 9 \beta q^{18} + ( - 18 \beta - 18) q^{19} + (6 \beta - 32) q^{20} + ( - 6 \beta + 6) q^{21} + 11 \beta q^{22} + 112 q^{23} + ( - 21 \beta + 24) q^{24} + ( - 64 \beta + 103) q^{25} + ( - 34 \beta + 64) q^{26} + 27 q^{27} - 16 q^{28} + (46 \beta + 88) q^{29} + (18 \beta - 96) q^{30} + (104 \beta - 72) q^{31} + ( - 7 \beta - 120) q^{32} + 33 q^{33} + (2 \beta + 240) q^{34} + ( - 20 \beta + 84) q^{35} + 9 \beta q^{36} + (44 \beta - 46) q^{37} + ( - 36 \beta - 144) q^{38} + (24 \beta - 126) q^{39} + ( - 74 \beta + 304) q^{40} + (2 \beta - 248) q^{41} - 48 q^{42} + ( - 86 \beta + 10) q^{43} + 11 \beta q^{44} + ( - 36 \beta + 90) q^{45} + 112 \beta q^{46} + ( - 48 \beta - 8) q^{47} + ( - 21 \beta - 168) q^{48} + ( - 4 \beta - 307) q^{49} + (39 \beta - 512) q^{50} + (90 \beta - 84) q^{51} + ( - 34 \beta + 64) q^{52} + ( - 128 \beta + 22) q^{53} + 27 \beta q^{54} + ( - 44 \beta + 110) q^{55} + ( - 16 \beta + 128) q^{56} + ( - 54 \beta - 54) q^{57} + (134 \beta + 368) q^{58} + 196 q^{59} + (18 \beta - 96) q^{60} + ( - 52 \beta - 526) q^{61} + (32 \beta + 832) q^{62} + ( - 18 \beta + 18) q^{63} + ( - 71 \beta + 392) q^{64} + (216 \beta - 676) q^{65} + 33 \beta q^{66} + (152 \beta + 388) q^{67} + (2 \beta + 240) q^{68} + 336 q^{69} + (64 \beta - 160) q^{70} + (184 \beta + 136) q^{71} + ( - 63 \beta + 72) q^{72} + ( - 252 \beta - 170) q^{73} + ( - 2 \beta + 352) q^{74} + ( - 192 \beta + 309) q^{75} + ( - 36 \beta - 144) q^{76} + ( - 22 \beta + 22) q^{77} + ( - 102 \beta + 192) q^{78} + ( - 74 \beta - 78) q^{79} + (182 \beta - 336) q^{80} + 81 q^{81} + ( - 246 \beta + 16) q^{82} + ( - 324 \beta + 336) q^{83} - 48 q^{84} + (292 \beta - 1240) q^{85} + ( - 76 \beta - 688) q^{86} + (138 \beta + 264) q^{87} + ( - 77 \beta + 88) q^{88} + (8 \beta + 482) q^{89} + (54 \beta - 288) q^{90} + (84 \beta - 212) q^{91} + 112 \beta q^{92} + (312 \beta - 216) q^{93} + ( - 56 \beta - 384) q^{94} + ( - 36 \beta + 396) q^{95} + ( - 21 \beta - 360) q^{96} + (420 \beta - 802) q^{97} + ( - 311 \beta - 32) q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9} - 58 q^{10} + 22 q^{11} + 3 q^{12} - 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} - 26 q^{17} + 9 q^{18} - 54 q^{19} - 58 q^{20} + 6 q^{21} + 11 q^{22} + 224 q^{23} + 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} - 32 q^{28} + 222 q^{29} - 174 q^{30} - 40 q^{31} - 247 q^{32} + 66 q^{33} + 482 q^{34} + 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} - 228 q^{39} + 534 q^{40} - 494 q^{41} - 96 q^{42} - 66 q^{43} + 11 q^{44} + 144 q^{45} + 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} - 985 q^{50} - 78 q^{51} + 94 q^{52} - 84 q^{53} + 27 q^{54} + 176 q^{55} + 240 q^{56} - 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} - 1104 q^{61} + 1696 q^{62} + 18 q^{63} + 713 q^{64} - 1136 q^{65} + 33 q^{66} + 928 q^{67} + 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} + 81 q^{72} - 592 q^{73} + 702 q^{74} + 426 q^{75} - 324 q^{76} + 22 q^{77} + 282 q^{78} - 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} + 348 q^{83} - 96 q^{84} - 2188 q^{85} - 1452 q^{86} + 666 q^{87} + 99 q^{88} + 972 q^{89} - 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} - 824 q^{94} + 756 q^{95} - 741 q^{96} - 1184 q^{97} - 375 q^{98} + 198 q^{99}+O(q^{100}) \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.37228
3.37228
−2.37228 3.00000 −2.37228 19.4891 −7.11684 6.74456 24.6060 9.00000 −46.2337
1.2 3.37228 3.00000 3.37228 −3.48913 10.1168 −4.74456 −15.6060 9.00000 −11.7663
\(n\): 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 33.4.a.d 2
3.b odd 2 1 99.4.a.e 2
4.b odd 2 1 528.4.a.o 2
5.b even 2 1 825.4.a.k 2
5.c odd 4 2 825.4.c.i 4
7.b odd 2 1 1617.4.a.j 2
8.b even 2 1 2112.4.a.ba 2
8.d odd 2 1 2112.4.a.bh 2
11.b odd 2 1 363.4.a.j 2
12.b even 2 1 1584.4.a.x 2
15.d odd 2 1 2475.4.a.o 2
33.d even 2 1 1089.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 1.a even 1 1 trivial
99.4.a.e 2 3.b odd 2 1
363.4.a.j 2 11.b odd 2 1
528.4.a.o 2 4.b odd 2 1
825.4.a.k 2 5.b even 2 1
825.4.c.i 4 5.c odd 4 2
1089.4.a.t 2 33.d even 2 1
1584.4.a.x 2 12.b even 2 1
1617.4.a.j 2 7.b odd 2 1
2112.4.a.ba 2 8.b even 2 1
2112.4.a.bh 2 8.d odd 2 1
2475.4.a.o 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 16T - 68 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 76T + 916 \) Copy content Toggle raw display
$17$ \( T^{2} + 26T - 7256 \) Copy content Toggle raw display
$19$ \( T^{2} + 54T - 1944 \) Copy content Toggle raw display
$23$ \( (T - 112)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 222T - 5136 \) Copy content Toggle raw display
$31$ \( T^{2} + 40T - 88832 \) Copy content Toggle raw display
$37$ \( T^{2} + 48T - 15396 \) Copy content Toggle raw display
$41$ \( T^{2} + 494T + 60976 \) Copy content Toggle raw display
$43$ \( T^{2} + 66T - 59928 \) Copy content Toggle raw display
$47$ \( T^{2} + 64T - 17984 \) Copy content Toggle raw display
$53$ \( T^{2} + 84T - 133404 \) Copy content Toggle raw display
$59$ \( (T - 196)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 1104 T + 282396 \) Copy content Toggle raw display
$67$ \( T^{2} - 928T + 24688 \) Copy content Toggle raw display
$71$ \( T^{2} - 456T - 227328 \) Copy content Toggle raw display
$73$ \( T^{2} + 592T - 436292 \) Copy content Toggle raw display
$79$ \( T^{2} + 230T - 31952 \) Copy content Toggle raw display
$83$ \( T^{2} - 348T - 835776 \) Copy content Toggle raw display
$89$ \( T^{2} - 972T + 235668 \) Copy content Toggle raw display
$97$ \( T^{2} + 1184 T - 1104836 \) Copy content Toggle raw display
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