Properties

Label 33.8.a.e
Level $33$
Weight $8$
Character orbit 33.a
Self dual yes
Analytic conductor $10.309$
Analytic rank $0$
Dimension $4$
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] = [33,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 510x^{2} - 1544x + 28880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 4) q^{2} + 27 q^{3} + (\beta_{2} - 2 \beta_1 + 142) q^{4} + (\beta_{3} + 10 \beta_1 + 74) q^{5} + ( - 27 \beta_1 + 108) q^{6} + ( - 2 \beta_{3} - 2 \beta_{2} - 32 \beta_1 + 230) q^{7} + ( - 6 \beta_{3} + \beta_{2} - 126 \beta_1 + 646) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 4) q^{2} + 27 q^{3} + (\beta_{2} - 2 \beta_1 + 142) q^{4} + (\beta_{3} + 10 \beta_1 + 74) q^{5} + ( - 27 \beta_1 + 108) q^{6} + ( - 2 \beta_{3} - 2 \beta_{2} - 32 \beta_1 + 230) q^{7} + ( - 6 \beta_{3} + \beta_{2} - 126 \beta_1 + 646) q^{8} + 729 q^{9} + (14 \beta_{3} - 20 \beta_{2} - 70 \beta_1 - 2224) q^{10} + 1331 q^{11} + (27 \beta_{2} - 54 \beta_1 + 3834) q^{12} + (15 \beta_{3} - 28 \beta_{2} + 206 \beta_1 - 514) q^{13} + ( - 16 \beta_{3} + 54 \beta_{2} + 18 \beta_1 + 8844) q^{14} + (27 \beta_{3} + 270 \beta_1 + 1998) q^{15} + ( - 90 \beta_{3} + 57 \beta_{2} - 398 \beta_1 + 16374) q^{16} + ( - 21 \beta_{3} - 14 \beta_{2} + 842 \beta_1 + 8000) q^{17} + ( - 729 \beta_1 + 2916) q^{18} + (43 \beta_{3} - 126 \beta_{2} + 810 \beta_1 - 3430) q^{19} + (188 \beta_{3} - 50 \beta_{2} + 3740 \beta_1 - 1948) q^{20} + ( - 54 \beta_{3} - 54 \beta_{2} - 864 \beta_1 + 6210) q^{21} + ( - 1331 \beta_1 + 5324) q^{22} + ( - 149 \beta_{3} + 12 \beta_{2} + 3414 \beta_1 + 27864) q^{23} + ( - 162 \beta_{3} + 27 \beta_{2} - 3402 \beta_1 + 17442) q^{24} + (106 \beta_{3} + 280 \beta_{2} - 1260 \beta_1 + 18599) q^{25} + (378 \beta_{3} - 328 \beta_{2} + 3710 \beta_1 - 56376) q^{26} + 19683 q^{27} + ( - 292 \beta_{3} + 344 \beta_{2} - 11432 \beta_1 + 5472) q^{28} + ( - 349 \beta_{3} - 110 \beta_{2} - 3254 \beta_1 - 25452) q^{29} + (378 \beta_{3} - 540 \beta_{2} - 1890 \beta_1 - 60048) q^{30} + (532 \beta_{3} + 744 \beta_{2} + 3832 \beta_1 - 7016) q^{31} + ( - 834 \beta_{3} + 1113 \beta_{2} - 8222 \beta_1 + 86774) q^{32} + 35937 q^{33} + ( - 210 \beta_{3} - 618 \beta_{2} - 8564 \beta_1 - 183436) q^{34} + (234 \beta_{3} - 780 \beta_{2} - 100 \beta_1 - 180884) q^{35} + (729 \beta_{2} - 1458 \beta_1 + 103518) q^{36} + ( - 610 \beta_{3} - 364 \beta_{2} + 8708 \beta_1 - 126982) q^{37} + (1358 \beta_{3} - 1114 \beta_{2} + 17458 \beta_1 - 228932) q^{38} + (405 \beta_{3} - 756 \beta_{2} + 5562 \beta_1 - 13878) q^{39} + (1140 \beta_{3} - 3010 \beta_{2} + 13740 \beta_1 - 673420) q^{40} + ( - 1477 \beta_{3} + 310 \beta_{2} + 13602 \beta_1 + 180316) q^{41} + ( - 432 \beta_{3} + 1458 \beta_{2} + 486 \beta_1 + 238788) q^{42} + (119 \beta_{3} + 2814 \beta_{2} + 25034 \beta_1 - 56034) q^{43} + (1331 \beta_{2} - 2662 \beta_1 + 189002) q^{44} + (729 \beta_{3} + 7290 \beta_1 + 53946) q^{45} + ( - 2158 \beta_{3} - 1936 \beta_{2} - 39660 \beta_1 - 757696) q^{46} + (1677 \beta_{3} + 3428 \beta_{2} + 11674 \beta_1 + 496912) q^{47} + ( - 2430 \beta_{3} + 1539 \beta_{2} - 10746 \beta_1 + 442098) q^{48} + ( - 1336 \beta_{3} + 540 \beta_{2} + 4216 \beta_1 - 193675) q^{49} + ( - 196 \beta_{3} - 80 \beta_{2} - 46015 \beta_1 + 419516) q^{50} + ( - 567 \beta_{3} - 378 \beta_{2} + 22734 \beta_1 + 216000) q^{51} + (5340 \beta_{3} - 3578 \beta_{2} + 69708 \beta_1 - 1121388) q^{52} + (1883 \beta_{3} - 3908 \beta_{2} - 74378 \beta_1 + 250110) q^{53} + ( - 19683 \beta_1 + 78732) q^{54} + (1331 \beta_{3} + 13310 \beta_1 + 98494) q^{55} + ( - 4104 \beta_{3} + 7096 \beta_{2} - 31824 \beta_1 + 1815952) q^{56} + (1161 \beta_{3} - 3402 \beta_{2} + 21870 \beta_1 - 92610) q^{57} + ( - 4226 \beta_{3} + 6854 \beta_{2} + 36344 \beta_1 + 708708) q^{58} + (582 \beta_{3} - 11944 \beta_{2} - 54068 \beta_1 + 348820) q^{59} + (5076 \beta_{3} - 1350 \beta_{2} + 100980 \beta_1 - 52596) q^{60} + ( - 1057 \beta_{3} + 56 \beta_{2} - 66010 \beta_1 + 1016210) q^{61} + (2984 \beta_{3} - 9896 \beta_{2} - 74184 \beta_1 - 929744) q^{62} + ( - 1458 \beta_{3} - 1458 \beta_{2} - 23328 \beta_1 + 167670) q^{63} + ( - 6834 \beta_{3} + 8153 \beta_{2} - 168510 \beta_1 + 414198) q^{64} + ( - 1322 \beta_{3} + 5600 \beta_{2} - 100100 \beta_1 + 1679452) q^{65} + ( - 35937 \beta_1 + 143748) q^{66} + (3760 \beta_{3} - 1048 \beta_{2} - 36688 \beta_1 + 439412) q^{67} + (3456 \beta_{3} + 13074 \beta_{2} + 159436 \beta_1 + 362636) q^{68} + ( - 4023 \beta_{3} + 324 \beta_{2} + 92178 \beta_1 + 752328) q^{69} + (7956 \beta_{3} - 1460 \beta_{2} + 277180 \beta_1 - 757416) q^{70} + ( - 7581 \beta_{3} + 4004 \beta_{2} + 62262 \beta_1 + 1378400) q^{71} + ( - 4374 \beta_{3} + 729 \beta_{2} - 91854 \beta_1 + 470934) q^{72} + ( - 6122 \beta_{3} + 8684 \beta_{2} - 94460 \beta_1 + 1425918) q^{73} + ( - 6356 \beta_{3} - 2244 \beta_{2} + 137150 \beta_1 - 2761808) q^{74} + (2862 \beta_{3} + 7560 \beta_{2} - 34020 \beta_1 + 502173) q^{75} + (20192 \beta_{3} - 13796 \beta_{2} + 252152 \beta_1 - 4975208) q^{76} + ( - 2662 \beta_{3} - 2662 \beta_{2} - 42592 \beta_1 + 306130) q^{77} + (10206 \beta_{3} - 8856 \beta_{2} + 100170 \beta_1 - 1522152) q^{78} + (16 \beta_{3} - 10722 \beta_{2} - 18508 \beta_1 - 1925610) q^{79} + (9956 \beta_{3} - 15730 \beta_{2} + 543740 \beta_1 - 6158316) q^{80} + 531441 q^{81} + ( - 22538 \beta_{3} + 858 \beta_{2} - 278928 \beta_1 - 2737764) q^{82} + (16670 \beta_{3} + 19220 \beta_{2} - 186716 \beta_1 + 909448) q^{83} + ( - 7884 \beta_{3} + 9288 \beta_{2} - 308664 \beta_1 + 147744) q^{84} + ( - 3556 \beta_{3} + 15300 \beta_{2} + 202160 \beta_1 + 1430656) q^{85} + ( - 15218 \beta_{3} - 29038 \beta_{2} - 317602 \beta_1 - 6349644) q^{86} + ( - 9423 \beta_{3} - 2970 \beta_{2} - 87858 \beta_1 - 687204) q^{87} + ( - 7986 \beta_{3} + 1331 \beta_{2} - 167706 \beta_1 + 859826) q^{88} + (6580 \beta_{3} - 18280 \beta_{2} + 107288 \beta_1 + 1121490) q^{89} + (10206 \beta_{3} - 14580 \beta_{2} - 51030 \beta_1 - 1621296) q^{90} + (6982 \beta_{3} - 26036 \beta_{2} + 243332 \beta_1 - 2325516) q^{91} + (476 \beta_{3} + 61640 \beta_{2} + 572808 \beta_1 + 3274352) q^{92} + (14364 \beta_{3} + 20088 \beta_{2} + 103464 \beta_1 - 189432) q^{93} + (2910 \beta_{3} - 31872 \beta_{2} - 877660 \beta_1 - 662912) q^{94} + ( - 9130 \beta_{3} + 20900 \beta_{2} - 398540 \beta_1 + 5561860) q^{95} + ( - 22518 \beta_{3} + 30051 \beta_{2} - 221994 \beta_1 + 2342898) q^{96} + (27384 \beta_{3} - 13116 \beta_{2} - 269112 \beta_1 + 414422) q^{97} + ( - 21944 \beta_{3} + 8604 \beta_{2} + 90539 \beta_1 - 1828004) q^{98} + 970299 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 15 q^{2} + 108 q^{3} + 565 q^{4} + 306 q^{5} + 405 q^{6} + 890 q^{7} + 2457 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 15 q^{2} + 108 q^{3} + 565 q^{4} + 306 q^{5} + 405 q^{6} + 890 q^{7} + 2457 q^{8} + 2916 q^{9} - 8946 q^{10} + 5324 q^{11} + 15255 q^{12} - 1822 q^{13} + 35340 q^{14} + 8262 q^{15} + 65041 q^{16} + 32856 q^{17} + 10935 q^{18} - 12784 q^{19} - 4002 q^{20} + 24030 q^{21} + 19965 q^{22} + 114858 q^{23} + 66339 q^{24} + 72856 q^{25} - 221466 q^{26} + 78732 q^{27} + 10112 q^{28} - 104952 q^{29} - 241542 q^{30} - 24976 q^{31} + 337761 q^{32} + 143748 q^{33} - 741690 q^{34} - 722856 q^{35} + 411885 q^{36} - 498856 q^{37} - 897156 q^{38} - 49194 q^{39} - 2676930 q^{40} + 734556 q^{41} + 954180 q^{42} - 201916 q^{43} + 752015 q^{44} + 223074 q^{45} - 3068508 q^{46} + 1995894 q^{47} + 1756107 q^{48} - 771024 q^{49} + 1632129 q^{50} + 887112 q^{51} - 4412266 q^{52} + 929970 q^{53} + 295245 q^{54} + 407286 q^{55} + 7224888 q^{56} - 345168 q^{57} + 2864322 q^{58} + 1353156 q^{59} - 108054 q^{60} + 3998774 q^{61} - 3783264 q^{62} + 648810 q^{63} + 1480129 q^{64} + 6612108 q^{65} + 539055 q^{66} + 1722008 q^{67} + 1596906 q^{68} + 3101166 q^{69} - 2751024 q^{70} + 5571858 q^{71} + 1791153 q^{72} + 5600528 q^{73} - 10907838 q^{74} + 1967112 q^{75} - 19634884 q^{76} + 1184590 q^{77} - 5979582 q^{78} - 7710226 q^{79} - 24073794 q^{80} + 2125764 q^{81} - 11230842 q^{82} + 3431856 q^{83} + 273024 q^{84} + 5909484 q^{85} - 25687140 q^{86} - 2833704 q^{87} + 3270267 q^{88} + 4611528 q^{89} - 6521634 q^{90} - 9032696 q^{91} + 13608576 q^{92} - 674352 q^{93} - 3497436 q^{94} + 21828000 q^{95} + 9119547 q^{96} + 1401692 q^{97} - 7230081 q^{98} + 3881196 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 510x^{2} - 1544x + 28880 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6\nu - 254 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 11\nu^{2} - 340\nu + 1352 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6\beta _1 + 254 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} + 11\beta_{2} + 406\beta _1 + 1442 \) 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
23.3791
6.33327
−11.0316
−17.6807
−19.3791 27.0000 247.551 336.012 −523.236 −879.192 −2316.79 729.000 −6511.62
1.2 −2.33327 27.0000 −122.556 −27.4165 −62.9982 860.612 584.614 729.000 63.9699
1.3 15.0316 27.0000 97.9505 367.278 405.855 −91.9512 −451.694 729.000 5520.80
1.4 21.6807 27.0000 342.055 −369.874 585.380 1000.53 4640.87 729.000 −8019.14
\(n\): 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 33.8.a.e 4
3.b odd 2 1 99.8.a.f 4
4.b odd 2 1 528.8.a.r 4
11.b odd 2 1 363.8.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.e 4 1.a even 1 1 trivial
99.8.a.f 4 3.b odd 2 1
363.8.a.f 4 11.b odd 2 1
528.8.a.r 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 15T_{2}^{3} - 426T_{2}^{2} + 5416T_{2} + 14736 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 15 T^{3} - 426 T^{2} + \cdots + 14736 \) Copy content Toggle raw display
$3$ \( (T - 27)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 306 T^{3} + \cdots + 1251456000 \) Copy content Toggle raw display
$7$ \( T^{4} - 890 T^{3} + \cdots + 69611139776 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 161846909982208 \) Copy content Toggle raw display
$17$ \( T^{4} - 32856 T^{3} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{4} + 12784 T^{3} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} - 114858 T^{3} + \cdots - 49\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{4} + 104952 T^{3} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + 24976 T^{3} + \cdots + 59\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{4} + 498856 T^{3} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{4} - 734556 T^{3} + \cdots - 49\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{4} + 201916 T^{3} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} - 1995894 T^{3} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} - 929970 T^{3} + \cdots - 80\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{4} - 1353156 T^{3} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} - 3998774 T^{3} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{4} - 1722008 T^{3} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} - 5571858 T^{3} + \cdots - 50\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{4} - 5600528 T^{3} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{4} + 7710226 T^{3} + \cdots - 88\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{4} - 3431856 T^{3} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{4} - 4611528 T^{3} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} - 1401692 T^{3} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
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