Properties

Label 33.4.a.c
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: \(\Q(\sqrt{97}) \)
Defining polynomial: \(x^{2} - x - 24\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -3 q^{3} + ( 16 + \beta ) q^{4} + ( -6 - 2 \beta ) q^{5} -3 \beta q^{6} + ( 14 - 4 \beta ) q^{7} + ( 24 + 9 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta q^{2} -3 q^{3} + ( 16 + \beta ) q^{4} + ( -6 - 2 \beta ) q^{5} -3 \beta q^{6} + ( 14 - 4 \beta ) q^{7} + ( 24 + 9 \beta ) q^{8} + 9 q^{9} + ( -48 - 8 \beta ) q^{10} -11 q^{11} + ( -48 - 3 \beta ) q^{12} + ( 14 + 2 \beta ) q^{13} + ( -96 + 10 \beta ) q^{14} + ( 18 + 6 \beta ) q^{15} + ( 88 + 25 \beta ) q^{16} + ( 60 - 14 \beta ) q^{17} + 9 \beta q^{18} + ( 26 - 2 \beta ) q^{19} + ( -144 - 40 \beta ) q^{20} + ( -42 + 12 \beta ) q^{21} -11 \beta q^{22} + ( 72 - 10 \beta ) q^{23} + ( -72 - 27 \beta ) q^{24} + ( 7 + 28 \beta ) q^{25} + ( 48 + 16 \beta ) q^{26} -27 q^{27} + ( 128 - 54 \beta ) q^{28} + ( -96 - 6 \beta ) q^{29} + ( 144 + 24 \beta ) q^{30} + ( 176 + 8 \beta ) q^{31} + ( 408 + 41 \beta ) q^{32} + 33 q^{33} + ( -336 + 46 \beta ) q^{34} + ( 108 + 4 \beta ) q^{35} + ( 144 + 9 \beta ) q^{36} + ( -190 + 52 \beta ) q^{37} + ( -48 + 24 \beta ) q^{38} + ( -42 - 6 \beta ) q^{39} + ( -576 - 120 \beta ) q^{40} + ( -384 - 14 \beta ) q^{41} + ( 288 - 30 \beta ) q^{42} + ( 206 - 26 \beta ) q^{43} + ( -176 - 11 \beta ) q^{44} + ( -54 - 18 \beta ) q^{45} + ( -240 + 62 \beta ) q^{46} + ( 96 + 74 \beta ) q^{47} + ( -264 - 75 \beta ) q^{48} + ( 237 - 96 \beta ) q^{49} + ( 672 + 35 \beta ) q^{50} + ( -180 + 42 \beta ) q^{51} + ( 272 + 48 \beta ) q^{52} + ( -234 - 54 \beta ) q^{53} -27 \beta q^{54} + ( 66 + 22 \beta ) q^{55} + ( -528 - 6 \beta ) q^{56} + ( -78 + 6 \beta ) q^{57} + ( -144 - 102 \beta ) q^{58} + ( -36 - 100 \beta ) q^{59} + ( 432 + 120 \beta ) q^{60} + ( -406 + 34 \beta ) q^{61} + ( 192 + 184 \beta ) q^{62} + ( 126 - 36 \beta ) q^{63} + ( 280 + 249 \beta ) q^{64} + ( -180 - 44 \beta ) q^{65} + 33 \beta q^{66} + ( -340 - 96 \beta ) q^{67} + ( 624 - 178 \beta ) q^{68} + ( -216 + 30 \beta ) q^{69} + ( 96 + 112 \beta ) q^{70} + ( 288 + 54 \beta ) q^{71} + ( 216 + 81 \beta ) q^{72} + ( 662 - 28 \beta ) q^{73} + ( 1248 - 138 \beta ) q^{74} + ( -21 - 84 \beta ) q^{75} + ( 368 - 8 \beta ) q^{76} + ( -154 + 44 \beta ) q^{77} + ( -144 - 48 \beta ) q^{78} + ( 254 + 144 \beta ) q^{79} + ( -1728 - 376 \beta ) q^{80} + 81 q^{81} + ( -336 - 398 \beta ) q^{82} + ( -240 + 156 \beta ) q^{83} + ( -384 + 162 \beta ) q^{84} + ( 312 - 8 \beta ) q^{85} + ( -624 + 180 \beta ) q^{86} + ( 288 + 18 \beta ) q^{87} + ( -264 - 99 \beta ) q^{88} + ( -414 + 72 \beta ) q^{89} + ( -432 - 72 \beta ) q^{90} + ( 4 - 36 \beta ) q^{91} + ( 912 - 98 \beta ) q^{92} + ( -528 - 24 \beta ) q^{93} + ( 1776 + 170 \beta ) q^{94} + ( -60 - 36 \beta ) q^{95} + ( -1224 - 123 \beta ) q^{96} + ( -322 + 192 \beta ) q^{97} + ( -2304 + 141 \beta ) q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 6q^{3} + 33q^{4} - 14q^{5} - 3q^{6} + 24q^{7} + 57q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + q^{2} - 6q^{3} + 33q^{4} - 14q^{5} - 3q^{6} + 24q^{7} + 57q^{8} + 18q^{9} - 104q^{10} - 22q^{11} - 99q^{12} + 30q^{13} - 182q^{14} + 42q^{15} + 201q^{16} + 106q^{17} + 9q^{18} + 50q^{19} - 328q^{20} - 72q^{21} - 11q^{22} + 134q^{23} - 171q^{24} + 42q^{25} + 112q^{26} - 54q^{27} + 202q^{28} - 198q^{29} + 312q^{30} + 360q^{31} + 857q^{32} + 66q^{33} - 626q^{34} + 220q^{35} + 297q^{36} - 328q^{37} - 72q^{38} - 90q^{39} - 1272q^{40} - 782q^{41} + 546q^{42} + 386q^{43} - 363q^{44} - 126q^{45} - 418q^{46} + 266q^{47} - 603q^{48} + 378q^{49} + 1379q^{50} - 318q^{51} + 592q^{52} - 522q^{53} - 27q^{54} + 154q^{55} - 1062q^{56} - 150q^{57} - 390q^{58} - 172q^{59} + 984q^{60} - 778q^{61} + 568q^{62} + 216q^{63} + 809q^{64} - 404q^{65} + 33q^{66} - 776q^{67} + 1070q^{68} - 402q^{69} + 304q^{70} + 630q^{71} + 513q^{72} + 1296q^{73} + 2358q^{74} - 126q^{75} + 728q^{76} - 264q^{77} - 336q^{78} + 652q^{79} - 3832q^{80} + 162q^{81} - 1070q^{82} - 324q^{83} - 606q^{84} + 616q^{85} - 1068q^{86} + 594q^{87} - 627q^{88} - 756q^{89} - 936q^{90} - 28q^{91} + 1726q^{92} - 1080q^{93} + 3722q^{94} - 156q^{95} - 2571q^{96} - 452q^{97} - 4467q^{98} - 198q^{99} + O(q^{100}) \)

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
−4.42443
5.42443
−4.42443 −3.00000 11.5756 2.84886 13.2733 31.6977 −15.8199 9.00000 −12.6046
1.2 5.42443 −3.00000 21.4244 −16.8489 −16.2733 −7.69772 72.8199 9.00000 −91.3954
\(n\): 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.c 2
3.b odd 2 1 99.4.a.f 2
4.b odd 2 1 528.4.a.p 2
5.b even 2 1 825.4.a.l 2
5.c odd 4 2 825.4.c.h 4
7.b odd 2 1 1617.4.a.k 2
8.b even 2 1 2112.4.a.bn 2
8.d odd 2 1 2112.4.a.bg 2
11.b odd 2 1 363.4.a.i 2
12.b even 2 1 1584.4.a.bj 2
15.d odd 2 1 2475.4.a.p 2
33.d even 2 1 1089.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 1.a even 1 1 trivial
99.4.a.f 2 3.b odd 2 1
363.4.a.i 2 11.b odd 2 1
528.4.a.p 2 4.b odd 2 1
825.4.a.l 2 5.b even 2 1
825.4.c.h 4 5.c odd 4 2
1089.4.a.u 2 33.d even 2 1
1584.4.a.bj 2 12.b even 2 1
1617.4.a.k 2 7.b odd 2 1
2112.4.a.bg 2 8.d odd 2 1
2112.4.a.bn 2 8.b even 2 1
2475.4.a.p 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 8 T^{2} - 8 T^{3} + 64 T^{4} \)
$3$ \( ( 1 + 3 T )^{2} \)
$5$ \( 1 + 14 T + 202 T^{2} + 1750 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 24 T + 442 T^{2} - 8232 T^{3} + 117649 T^{4} \)
$11$ \( ( 1 + 11 T )^{2} \)
$13$ \( 1 - 30 T + 4522 T^{2} - 65910 T^{3} + 4826809 T^{4} \)
$17$ \( 1 - 106 T + 7882 T^{2} - 520778 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 50 T + 14246 T^{2} - 342950 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 134 T + 26398 T^{2} - 1630378 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 198 T + 57706 T^{2} + 4829022 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 360 T + 90430 T^{2} - 10724760 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 328 T + 62630 T^{2} + 16614184 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 782 T + 285970 T^{2} + 53896222 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 386 T + 179870 T^{2} - 30689702 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 266 T + 92542 T^{2} - 27616918 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 522 T + 295162 T^{2} + 77713794 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 172 T + 175654 T^{2} + 35325188 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 778 T + 577250 T^{2} + 176591218 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 776 T + 528582 T^{2} + 233392088 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 630 T + 744334 T^{2} - 225483930 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 1296 T + 1178926 T^{2} - 504166032 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 652 T + 589506 T^{2} - 321461428 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 324 T + 579670 T^{2} + 185258988 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 756 T + 1427110 T^{2} + 532956564 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 452 T + 982470 T^{2} + 412528196 T^{3} + 832972004929 T^{4} \)
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