# Properties

 Label 33.8.a.b Level $33$ Weight $8$ Character orbit 33.a Self dual yes Analytic conductor $10.309$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 44$$ x^2 - x - 44 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 $$\beta = \frac{1}{2}(1 + \sqrt{177})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 9) q^{2} + 27 q^{3} + (19 \beta - 3) q^{4} + ( - 38 \beta + 2) q^{5} + ( - 27 \beta - 243) q^{6} + (154 \beta - 160) q^{7} + ( - 59 \beta + 343) q^{8} + 729 q^{9}+O(q^{10})$$ q + (-b - 9) * q^2 + 27 * q^3 + (19*b - 3) * q^4 + (-38*b + 2) * q^5 + (-27*b - 243) * q^6 + (154*b - 160) * q^7 + (-59*b + 343) * q^8 + 729 * q^9 $$q + ( - \beta - 9) q^{2} + 27 q^{3} + (19 \beta - 3) q^{4} + ( - 38 \beta + 2) q^{5} + ( - 27 \beta - 243) q^{6} + (154 \beta - 160) q^{7} + ( - 59 \beta + 343) q^{8} + 729 q^{9} + (378 \beta + 1654) q^{10} - 1331 q^{11} + (513 \beta - 81) q^{12} + (878 \beta - 6774) q^{13} + ( - 1380 \beta - 5336) q^{14} + ( - 1026 \beta + 54) q^{15} + ( - 2185 \beta - 107) q^{16} + ( - 996 \beta - 30674) q^{17} + ( - 729 \beta - 6561) q^{18} + (852 \beta - 19416) q^{19} + ( - 570 \beta - 31774) q^{20} + (4158 \beta - 4320) q^{21} + (1331 \beta + 11979) q^{22} + (6330 \beta - 48508) q^{23} + ( - 1593 \beta + 9261) q^{24} + (1292 \beta - 14585) q^{25} + ( - 2006 \beta + 22334) q^{26} + 19683 q^{27} + ( - 576 \beta + 129224) q^{28} + ( - 29644 \beta + 21818) q^{29} + (10206 \beta + 44658) q^{30} + ( - 17608 \beta + 131304) q^{31} + (29509 \beta + 53199) q^{32} - 35937 q^{33} + (40634 \beta + 319890) q^{34} + (536 \beta - 257808) q^{35} + (13851 \beta - 2187) q^{36} + ( - 27472 \beta + 177662) q^{37} + (10896 \beta + 137256) q^{38} + (23706 \beta - 182898) q^{39} + ( - 10910 \beta + 99334) q^{40} + ( - 87896 \beta - 94018) q^{41} + ( - 37260 \beta - 144072) q^{42} + (70936 \beta - 157520) q^{43} + ( - 25289 \beta + 3993) q^{44} + ( - 27702 \beta + 1458) q^{45} + ( - 14792 \beta + 158052) q^{46} + (74886 \beta + 231020) q^{47} + ( - 58995 \beta - 2889) q^{48} + ( - 25564 \beta + 245561) q^{49} + (1665 \beta + 74417) q^{50} + ( - 26892 \beta - 828198) q^{51} + ( - 114658 \beta + 754330) q^{52} + (98834 \beta - 960358) q^{53} + ( - 19683 \beta - 177147) q^{54} + (50578 \beta - 2662) q^{55} + (53176 \beta - 454664) q^{56} + (23004 \beta - 524232) q^{57} + (274622 \beta + 1107974) q^{58} + (189540 \beta + 1156244) q^{59} + ( - 15390 \beta - 857898) q^{60} + ( - 201934 \beta - 994582) q^{61} + (44776 \beta - 406984) q^{62} + (112266 \beta - 116640) q^{63} + ( - 68609 \beta - 1763491) q^{64} + (225804 \beta - 1481564) q^{65} + (35937 \beta + 323433) q^{66} + ( - 464104 \beta + 1069444) q^{67} + ( - 598742 \beta - 740634) q^{68} + (170910 \beta - 1309716) q^{69} + (252448 \beta + 2296688) q^{70} + ( - 36334 \beta - 165988) q^{71} + ( - 43011 \beta + 250047) q^{72} + ( - 436992 \beta - 1449806) q^{73} + (97058 \beta - 390190) q^{74} + (34884 \beta - 393795) q^{75} + ( - 355272 \beta + 770520) q^{76} + ( - 204974 \beta + 212960) q^{77} + ( - 54162 \beta + 603018) q^{78} + (708646 \beta - 1195632) q^{79} + (82726 \beta + 3653106) q^{80} + 531441 q^{81} + (972978 \beta + 4713586) q^{82} + ( - 461376 \beta - 3957564) q^{83} + ( - 15552 \beta + 3489048) q^{84} + (1201468 \beta + 1603964) q^{85} + ( - 551840 \beta - 1703504) q^{86} + ( - 800388 \beta + 589086) q^{87} + (78529 \beta - 456533) q^{88} + (903568 \beta + 4061818) q^{89} + (275562 \beta + 1205766) q^{90} + ( - 1048464 \beta + 7033168) q^{91} + ( - 820372 \beta + 5437404) q^{92} + ( - 475416 \beta + 3545208) q^{93} + ( - 979880 \beta - 5374164) q^{94} + (707136 \beta - 1463376) q^{95} + (796743 \beta + 1436373) q^{96} + (2748 \beta - 8353150) q^{97} + (10079 \beta - 1085233) q^{98} - 970299 q^{99}+O(q^{100})$$ q + (-b - 9) * q^2 + 27 * q^3 + (19*b - 3) * q^4 + (-38*b + 2) * q^5 + (-27*b - 243) * q^6 + (154*b - 160) * q^7 + (-59*b + 343) * q^8 + 729 * q^9 + (378*b + 1654) * q^10 - 1331 * q^11 + (513*b - 81) * q^12 + (878*b - 6774) * q^13 + (-1380*b - 5336) * q^14 + (-1026*b + 54) * q^15 + (-2185*b - 107) * q^16 + (-996*b - 30674) * q^17 + (-729*b - 6561) * q^18 + (852*b - 19416) * q^19 + (-570*b - 31774) * q^20 + (4158*b - 4320) * q^21 + (1331*b + 11979) * q^22 + (6330*b - 48508) * q^23 + (-1593*b + 9261) * q^24 + (1292*b - 14585) * q^25 + (-2006*b + 22334) * q^26 + 19683 * q^27 + (-576*b + 129224) * q^28 + (-29644*b + 21818) * q^29 + (10206*b + 44658) * q^30 + (-17608*b + 131304) * q^31 + (29509*b + 53199) * q^32 - 35937 * q^33 + (40634*b + 319890) * q^34 + (536*b - 257808) * q^35 + (13851*b - 2187) * q^36 + (-27472*b + 177662) * q^37 + (10896*b + 137256) * q^38 + (23706*b - 182898) * q^39 + (-10910*b + 99334) * q^40 + (-87896*b - 94018) * q^41 + (-37260*b - 144072) * q^42 + (70936*b - 157520) * q^43 + (-25289*b + 3993) * q^44 + (-27702*b + 1458) * q^45 + (-14792*b + 158052) * q^46 + (74886*b + 231020) * q^47 + (-58995*b - 2889) * q^48 + (-25564*b + 245561) * q^49 + (1665*b + 74417) * q^50 + (-26892*b - 828198) * q^51 + (-114658*b + 754330) * q^52 + (98834*b - 960358) * q^53 + (-19683*b - 177147) * q^54 + (50578*b - 2662) * q^55 + (53176*b - 454664) * q^56 + (23004*b - 524232) * q^57 + (274622*b + 1107974) * q^58 + (189540*b + 1156244) * q^59 + (-15390*b - 857898) * q^60 + (-201934*b - 994582) * q^61 + (44776*b - 406984) * q^62 + (112266*b - 116640) * q^63 + (-68609*b - 1763491) * q^64 + (225804*b - 1481564) * q^65 + (35937*b + 323433) * q^66 + (-464104*b + 1069444) * q^67 + (-598742*b - 740634) * q^68 + (170910*b - 1309716) * q^69 + (252448*b + 2296688) * q^70 + (-36334*b - 165988) * q^71 + (-43011*b + 250047) * q^72 + (-436992*b - 1449806) * q^73 + (97058*b - 390190) * q^74 + (34884*b - 393795) * q^75 + (-355272*b + 770520) * q^76 + (-204974*b + 212960) * q^77 + (-54162*b + 603018) * q^78 + (708646*b - 1195632) * q^79 + (82726*b + 3653106) * q^80 + 531441 * q^81 + (972978*b + 4713586) * q^82 + (-461376*b - 3957564) * q^83 + (-15552*b + 3489048) * q^84 + (1201468*b + 1603964) * q^85 + (-551840*b - 1703504) * q^86 + (-800388*b + 589086) * q^87 + (78529*b - 456533) * q^88 + (903568*b + 4061818) * q^89 + (275562*b + 1205766) * q^90 + (-1048464*b + 7033168) * q^91 + (-820372*b + 5437404) * q^92 + (-475416*b + 3545208) * q^93 + (-979880*b - 5374164) * q^94 + (707136*b - 1463376) * q^95 + (796743*b + 1436373) * q^96 + (2748*b - 8353150) * q^97 + (10079*b - 1085233) * q^98 - 970299 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 19 q^{2} + 54 q^{3} + 13 q^{4} - 34 q^{5} - 513 q^{6} - 166 q^{7} + 627 q^{8} + 1458 q^{9}+O(q^{10})$$ 2 * q - 19 * q^2 + 54 * q^3 + 13 * q^4 - 34 * q^5 - 513 * q^6 - 166 * q^7 + 627 * q^8 + 1458 * q^9 $$2 q - 19 q^{2} + 54 q^{3} + 13 q^{4} - 34 q^{5} - 513 q^{6} - 166 q^{7} + 627 q^{8} + 1458 q^{9} + 3686 q^{10} - 2662 q^{11} + 351 q^{12} - 12670 q^{13} - 12052 q^{14} - 918 q^{15} - 2399 q^{16} - 62344 q^{17} - 13851 q^{18} - 37980 q^{19} - 64118 q^{20} - 4482 q^{21} + 25289 q^{22} - 90686 q^{23} + 16929 q^{24} - 27878 q^{25} + 42662 q^{26} + 39366 q^{27} + 257872 q^{28} + 13992 q^{29} + 99522 q^{30} + 245000 q^{31} + 135907 q^{32} - 71874 q^{33} + 680414 q^{34} - 515080 q^{35} + 9477 q^{36} + 327852 q^{37} + 285408 q^{38} - 342090 q^{39} + 187758 q^{40} - 275932 q^{41} - 325404 q^{42} - 244104 q^{43} - 17303 q^{44} - 24786 q^{45} + 301312 q^{46} + 536926 q^{47} - 64773 q^{48} + 465558 q^{49} + 150499 q^{50} - 1683288 q^{51} + 1394002 q^{52} - 1821882 q^{53} - 373977 q^{54} + 45254 q^{55} - 856152 q^{56} - 1025460 q^{57} + 2490570 q^{58} + 2502028 q^{59} - 1731186 q^{60} - 2191098 q^{61} - 769192 q^{62} - 121014 q^{63} - 3595591 q^{64} - 2737324 q^{65} + 682803 q^{66} + 1674784 q^{67} - 2080010 q^{68} - 2448522 q^{69} + 4845824 q^{70} - 368310 q^{71} + 457083 q^{72} - 3336604 q^{73} - 683322 q^{74} - 752706 q^{75} + 1185768 q^{76} + 220946 q^{77} + 1151874 q^{78} - 1682618 q^{79} + 7388938 q^{80} + 1062882 q^{81} + 10400150 q^{82} - 8376504 q^{83} + 6962544 q^{84} + 4409396 q^{85} - 3958848 q^{86} + 377784 q^{87} - 834537 q^{88} + 9027204 q^{89} + 2687094 q^{90} + 13017872 q^{91} + 10054436 q^{92} + 6615000 q^{93} - 11728208 q^{94} - 2219616 q^{95} + 3669489 q^{96} - 16703552 q^{97} - 2160387 q^{98} - 1940598 q^{99}+O(q^{100})$$ 2 * q - 19 * q^2 + 54 * q^3 + 13 * q^4 - 34 * q^5 - 513 * q^6 - 166 * q^7 + 627 * q^8 + 1458 * q^9 + 3686 * q^10 - 2662 * q^11 + 351 * q^12 - 12670 * q^13 - 12052 * q^14 - 918 * q^15 - 2399 * q^16 - 62344 * q^17 - 13851 * q^18 - 37980 * q^19 - 64118 * q^20 - 4482 * q^21 + 25289 * q^22 - 90686 * q^23 + 16929 * q^24 - 27878 * q^25 + 42662 * q^26 + 39366 * q^27 + 257872 * q^28 + 13992 * q^29 + 99522 * q^30 + 245000 * q^31 + 135907 * q^32 - 71874 * q^33 + 680414 * q^34 - 515080 * q^35 + 9477 * q^36 + 327852 * q^37 + 285408 * q^38 - 342090 * q^39 + 187758 * q^40 - 275932 * q^41 - 325404 * q^42 - 244104 * q^43 - 17303 * q^44 - 24786 * q^45 + 301312 * q^46 + 536926 * q^47 - 64773 * q^48 + 465558 * q^49 + 150499 * q^50 - 1683288 * q^51 + 1394002 * q^52 - 1821882 * q^53 - 373977 * q^54 + 45254 * q^55 - 856152 * q^56 - 1025460 * q^57 + 2490570 * q^58 + 2502028 * q^59 - 1731186 * q^60 - 2191098 * q^61 - 769192 * q^62 - 121014 * q^63 - 3595591 * q^64 - 2737324 * q^65 + 682803 * q^66 + 1674784 * q^67 - 2080010 * q^68 - 2448522 * q^69 + 4845824 * q^70 - 368310 * q^71 + 457083 * q^72 - 3336604 * q^73 - 683322 * q^74 - 752706 * q^75 + 1185768 * q^76 + 220946 * q^77 + 1151874 * q^78 - 1682618 * q^79 + 7388938 * q^80 + 1062882 * q^81 + 10400150 * q^82 - 8376504 * q^83 + 6962544 * q^84 + 4409396 * q^85 - 3958848 * q^86 + 377784 * q^87 - 834537 * q^88 + 9027204 * q^89 + 2687094 * q^90 + 13017872 * q^91 + 10054436 * q^92 + 6615000 * q^93 - 11728208 * q^94 - 2219616 * q^95 + 3669489 * q^96 - 16703552 * q^97 - 2160387 * q^98 - 1940598 * q^99

## 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
 7.15207 −6.15207
−16.1521 27.0000 132.889 −269.779 −436.106 941.418 −78.9720 729.000 4357.48
1.2 −2.84793 27.0000 −119.889 235.779 −76.8942 −1107.42 705.972 729.000 −671.481
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 2
3.b odd 2 1 99.8.a.d 2
4.b odd 2 1 528.8.a.f 2
11.b odd 2 1 363.8.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.b 2 1.a even 1 1 trivial
99.8.a.d 2 3.b odd 2 1
363.8.a.d 2 11.b odd 2 1
528.8.a.f 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 19T_{2} + 46$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 19T + 46$$
$3$ $$(T - 27)^{2}$$
$5$ $$T^{2} + 34T - 63608$$
$7$ $$T^{2} + 166 T - 1042544$$
$11$ $$(T + 1331)^{2}$$
$13$ $$T^{2} + 12670 T + 6020608$$
$17$ $$T^{2} + 62344 T + 927796876$$
$19$ $$T^{2} + 37980 T + 328498848$$
$23$ $$T^{2} + 90686 T + 282938824$$
$29$ $$T^{2} - 13992 T - 38836484052$$
$31$ $$T^{2} - 245000 T + 1286906368$$
$37$ $$T^{2} - 327852 T - 6524218716$$
$41$ $$T^{2} + 275932 T - 322827909452$$
$43$ $$T^{2} + 244104 T - 207765596544$$
$47$ $$T^{2} - 536926 T - 176077767704$$
$53$ $$T^{2} + 1821882 T + 397572445128$$
$59$ $$T^{2} - 2502028 T - 24663435104$$
$61$ $$T^{2} + 2191098 T - 604169699352$$
$67$ $$T^{2} - 1674784 T - 8829893772944$$
$71$ $$T^{2} + 368310 T - 24503996328$$
$73$ $$T^{2} + 3336604 T - 5666837293628$$
$79$ $$T^{2} + 1682618 T - 21513626700752$$
$83$ $$T^{2} + 8376504 T + 8122054073616$$
$89$ $$T^{2} - 9027204 T - 15754651515708$$
$97$ $$T^{2} + 16703552 T + 69751828200124$$