LMFDB - Newform orbit 1100.6.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(1,1100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1100.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$1100 = 2^{2} \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 3x^{7} - 1111x^{6} + 2926x^{5} + 349410x^{4} - 822682x^{3} - 25635603x^{2} + 144346581x - 191266515$ x^8 - 3x^7 - 1111x^6 + 2926x^5 + 349410x^4 - 822682x^3 - 25635603x^2 + 144346581*x - 191266515
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{7}\cdot 5^{2}$
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 + 3) q^{3} + (\beta_{3} + 12) q^{7} + (\beta_{2} + 6 \beta_1 + 45) q^{9} + 121 q^{11} + (\beta_{7} + \beta_{3} + \beta_{2} + \cdots + 39) q^{13} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots + 64) q^{17}+ \cdots + (121 \beta_{2} + 726 \beta_1 + 5445) q^{99}+O(q^{100})$ q + (b1 + 3) * q^3 + (b3 + 12) * q^7 + (b2 + 6b1 + 45) q^9 + 121 * q^11 + (b7 + b3 + b2 - 5b1 + 39) q^13 + (b7 + b4 - b3 + b1 + 64) * q^17 + (3b7 - b6 + 3b5 + b4 + b2 - 43) * q^19 + (-b6 - 3b5 + 2b4 + 9b3 - 2b2 - 11b1 - 70) q^21 + (-2b7 - b6 + 2b5 + 2b4 + 2b3 + 3b2 + 29b1 + 374) * q^23 + (-b7 - 2b6 - b5 - 2b4 - 4b3 + 10b2 + 22b1 + 1054) q^27 + (4b6 - 3b5 - 2b4 + 14b3 + 12b2 + 12b1 + 342) * q^29 + (4b7 + 5b6 + 3b5 - 3b4 + 13b3 - 6b2 - 47b1 + 493) q^31 + (121b1 + 363) q^33 + (7b7 + 5b5 - 4b4 - 33b3 + b2 + 110b1 + 1114) q^37 + (8b7 - 6b6 - 12b5 + 2b4 + 15b3 - 5b2 + 243b1 - 1349) q^39 + (-2b7 - 4b6 + 14b5 + 37b3 - 10b2 - 31b1 + 1113) * q^41 + (-11b7 + 4b6 - 13b5 - 8b4 + 55b3 + 39b2 - 82b1 + 1949) q^43 + (-11b7 + 21b6 - 7b5 + 5b4 - 39b3 + 7b2 + 185b1 + 995) q^47 + (-23b7 + 3b6 - 2b5 - 10b4 + 27b3 + 24b2 + 468b1 + 3246) q^49 + (8b7 - 5b6 - 9b5 - 12b4 + 51b3 - 16b2 + 228b1 + 503) q^51 + (-8b7 + 8b6 + 19b5 - 9b4 + 33b3 + 40b2 - 321b1 + 3948) q^53 + (9b7 - 19b6 + 3b5 - b4 - 39b3 - 12b2 + 383b1 + 83) * q^57 + (-19b7 + 7b6 + 28b5 + 20b4 - 6b3 - 10b2 + 538b1 + 2357) q^59 + (-29b7 - 17b6 - 9b5 + 11b4 + 110b3 + 6b2 + 280b1 + 1210) q^61 + (32b7 - 10b6 - 55b5 + 5b4 + 101b3 - 28b2 - 380b1 - 7759) q^63 + (-32b7 - 21b6 + 2b5 + 23b4 - 43b3 - 23b2 + 935b1 - 447) q^67 + (-31b7 - 12b6 + 22b5 - 23b4 + 16b3 + 20b2 + 1165b1 + 8624) q^69 + (-29b7 + 51b6 - 30b5 + 21b4 + 73b3 + 58b2 + 193b1 + 3931) q^71 + (43b7 - 5b6 + 70b5 + 30b4 - 71b3 - 87b2 - 1227b1 - 4202) q^73 + (121b3 + 1452) q^77 + (88b7 + 9b6 - 27b5 + 10b4 - 55b3 - 135b2 + 1255b1 - 9450) q^79 + (7b7 - 10b6 + 19b5 + 8b4 - 101b3 - 81b2 + 1704b1 - 1561) q^81 + (35b7 - 52b6 + 69b5 + 18b4 - 6b3 - 74b2 - 1915b1 - 6974) q^83 + (-18b7 - 21b6 - 93b5 + 33b3 - 16b2 + 2062b1 + 2961) * q^87 + (30b7 - 31b6 + 12b5 + 39b4 - 111b3 - 178b2 + 3201b1 - 3110) q^89 + (97b7 - 53b6 - 5b5 - 41b4 - 159b3 + 173b2 + 966b1 + 16210) q^91 + (-19b7 + 3b6 - 51b5 + 47b4 - 204b3 - 197b2 - 1719b1 - 11411) q^93 + (97b7 + 70b6 + 215b5 - 66b4 - 45b3 - 152b2 + 890b1 + 23063) q^97 + (121b2 + 726b1 + 5445) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 27 q^{3} + 95 q^{7} + 377 q^{9} + 968 q^{11} + 294 q^{13} + 515 q^{17} - 354 q^{19} - 588 q^{21} + 3073 q^{23} + 8502 q^{27} + 2743 q^{29} + 3768 q^{31} + 3267 q^{33} + 9252 q^{37} - 10027 q^{39}+ \cdots + 45617 q^{99}+O(q^{100})$ 8 * q + 27 * q^3 + 95 * q^7 + 377 * q^9 + 968 * q^11 + 294 * q^13 + 515 * q^17 - 354 * q^19 - 588 * q^21 + 3073 * q^23 + 8502 * q^27 + 2743 * q^29 + 3768 * q^31 + 3267 * q^33 + 9252 * q^37 - 10027 * q^39 + 8756 * q^41 + 15290 * q^43 + 8516 * q^47 + 27341 * q^49 + 4707 * q^51 + 30475 * q^53 + 1903 * q^57 + 20400 * q^59 + 10511 * q^61 - 63122 * q^63 - 616 * q^67 + 72452 * q^69 + 31862 * q^71 - 37377 * q^73 + 11495 * q^77 - 71679 * q^79 - 7228 * q^81 - 61543 * q^83 + 30217 * q^87 - 14961 * q^89 + 132641 * q^91 - 95881 * q^93 + 186419 * q^97 + 45617 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 3x^{7} - 1111x^{6} + 2926x^{5} + 349410x^{4} - 822682x^{3} - 25635603x^{2} + 144346581x - 191266515$ x^8 - 3*x^7 - 1111*x^6 + 2926*x^5 + 349410*x^4 - 822682*x^3 - 25635603*x^2 + 144346581*x - 191266515 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 279$ v^2 - 279
$\beta_{3}$	$=$	$( 825345 \nu^{7} + 783568 \nu^{6} - 910438423 \nu^{5} - 1580309025 \nu^{4} + \cdots + 31044799336977 ) / 47468672334$ (825345v^7 + 783568v^6 - 910438423v^5 - 1580309025v^4 + 275678966755v^3 + 671886638675v^2 - 16319765874834*v + 31044799336977) / 47468672334
$\beta_{4}$	$=$	$( 4095023 \nu^{7} + 12041912 \nu^{6} - 4641936961 \nu^{5} - 13781533885 \nu^{4} + \cdots + 195753145100283 ) / 94937344668$ (4095023v^7 + 12041912v^6 - 4641936961v^5 - 13781533885v^4 + 1460445668915v^3 + 3986297635667v^2 - 92113826874360*v + 195753145100283) / 94937344668
$\beta_{5}$	$=$	$( 6134627 \nu^{7} + 9108780 \nu^{6} - 6618941513 \nu^{5} - 10741456993 \nu^{4} + \cdots + 287736307934139 ) / 94937344668$ (6134627v^7 + 9108780v^6 - 6618941513v^5 - 10741456993v^4 + 1980311929143v^3 + 3431549975707v^2 - 126133447636788*v + 287736307934139) / 94937344668
$\beta_{6}$	$=$	$( - 6828901 \nu^{7} - 6879444 \nu^{6} + 7638226531 \nu^{5} + 11107254299 \nu^{4} + \cdots - 371975085413541 ) / 94937344668$ (-6828901v^7 - 6879444v^6 + 7638226531v^5 + 11107254299v^4 - 2413663076469v^3 - 4126191447425v^2 + 172333798853208*v - 371975085413541) / 94937344668
$\beta_{7}$	$=$	$( - 7269631 \nu^{7} - 25702260 \nu^{6} + 7909869757 \nu^{5} + 28732488365 \nu^{4} + \cdots - 212606712127179 ) / 94937344668$ (-7269631v^7 - 25702260v^6 + 7909869757v^5 + 28732488365v^4 - 2374246192743v^3 - 8431918116923v^2 + 141916493463072*v - 212606712127179) / 94937344668

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 279$ b2 + 279
$\nu^{3}$	$=$	$-\beta_{7} - 2\beta_{6} - \beta_{5} - 2\beta_{4} - 4\beta_{3} + \beta_{2} + 481\beta _1 - 26$ -b7 - 2b6 - b5 - 2b4 - 4b3 + b2 + 481b1 - 26
$\nu^{4}$	$=$	$19\beta_{7} + 14\beta_{6} + 31\beta_{5} + 32\beta_{4} - 53\beta_{3} + 582\beta_{2} + 198\beta _1 + 134507$ 19b7 + 14b6 + 31b5 + 32b4 - 53b3 + 582b2 + 198*b1 + 134507
$\nu^{5}$	$=$	$- 860 \beta_{7} - 1309 \beta_{6} - 587 \beta_{5} - 1810 \beta_{4} - 2531 \beta_{3} - 232 \beta_{2} + \cdots + 46988$ -860b7 - 1309b6 - 587b5 - 1810b4 - 2531b3 - 232b2 + 255496*b1 + 46988
$\nu^{6}$	$=$	$12050 \beta_{7} + 12901 \beta_{6} + 24968 \beta_{5} + 29995 \beta_{4} - 60763 \beta_{3} + \cdots + 71560444$ 12050b7 + 12901b6 + 24968b5 + 29995b4 - 60763b3 + 329063b2 - 46281*b1 + 71560444
$\nu^{7}$	$=$	$- 589710 \beta_{7} - 761367 \beta_{6} - 277851 \beta_{5} - 1295784 \beta_{4} - 1442160 \beta_{3} + \cdots - 14616537$ -589710b7 - 761367b6 - 277851b5 - 1295784b4 - 1442160b3 - 602040b2 + 141372061*b1 - 14616537

Display $\beta_i$ in terms of $\nu^j$

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

−24.8760
−17.8741
−13.1518
2.15294
4.75672
6.06056
22.3000
23.6316

−21.8760

−48.9354

235.558

1.2

−14.8741

166.492

−21.7626

1.3

−10.1518

−122.703

−139.941

1.4

5.15294

47.7885

−216.447

1.5

7.75672

−86.6415

−182.833

1.6

9.06056

199.386

−160.906

1.7

25.3000

150.442

397.092

1.8

26.6316

−210.828

466.240

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$11$	$-1$

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	1100.6.a.k	yes	8
5.b	even	2	1		1100.6.a.h	✓	8
5.c	odd	4	2		1100.6.b.i		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1100.6.a.h	✓	8	5.b	even	2	1
1100.6.a.k	yes	8	1.a	even	1	1	trivial
1100.6.b.i		16	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{8} - 27 T_{3}^{7} - 796 T_{3}^{6} + 20845 T_{3}^{5} + 164040 T_{3}^{4} - 4174435 T_{3}^{3} + \cdots - 806019876$ T3^8 - 27*T3^7 - 796*T3^6 + 20845*T3^5 + 164040*T3^4 - 4174435*T3^3 - 1467489*T3^2 + 240983568*T3 - 806019876 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(1100))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} - 27 T^{7} + \cdots - 806019876$ T^8 - 27T^7 - 796T^6 + 20845T^5 + 164040T^4 - 4174435T^3 - 1467489T^2 + 240983568*T - 806019876
$5$	$T^{8}$ T^8
$7$	$T^{8} + \cdots + 26\!\cdots\!00$ T^8 - 95T^7 - 76386T^6 + 6185905T^5 + 1756812982T^4 - 88065177555T^3 - 14968227075303T^2 + 164273224522500*T + 26176650194520000
$11$	$(T - 121)^{8}$ (T - 121)^8
$13$	$T^{8} + \cdots - 89\!\cdots\!25$ T^8 - 294T^7 - 1323358T^6 + 464918866T^5 + 388997143812T^4 - 143029591687954T^3 - 28474750265694354T^2 + 12216568003289117670*T - 891554725245366703125
$17$	$T^{8} + \cdots + 75\!\cdots\!08$ T^8 - 515T^7 - 3288182T^6 + 1740210085T^5 + 2367813772018T^4 - 934748063229935T^3 - 693280164129571995T^2 + 138300137474625705900*T + 75451166504765876992608
$19$	$T^{8} + \cdots - 11\!\cdots\!89$ T^8 + 354T^7 - 11308006T^6 - 6119322094T^5 + 27781233806340T^4 + 2929891334233846T^3 - 22671404637401809946T^2 + 10065684003726792063894*T - 1100875006767821221712389
$23$	$T^{8} + \cdots - 35\!\cdots\!77$ T^8 - 3073T^7 - 18726231T^6 + 67845868092T^5 + 48623435090280T^4 - 304591910994998676T^3 + 139226681891868162867T^2 + 128384575613479113623217*T - 35166835227574862174662677
$29$	$T^{8} + \cdots - 14\!\cdots\!19$ T^8 - 2743T^7 - 57702581T^6 + 234480376864T^5 + 627468091772648T^4 - 4285929644426353184T^3 + 5650159866888782520999T^2 + 6020400749308731419943*T - 1498738735866810302369226819
$31$	$T^{8} + \cdots + 62\!\cdots\!25$ T^8 - 3768T^7 - 104259904T^6 + 245310001048T^5 + 3172583179546026T^4 - 1910619633030715528T^3 - 26578345674921150247704T^2 + 5356344429180936709930920*T + 62398456862040213736476862125
$37$	$T^{8} + \cdots - 30\!\cdots\!52$ T^8 - 9252T^7 - 165387211T^6 + 1565997842368T^5 + 1136005040472915T^4 - 29971549192906331764T^3 + 35918052416419725093447T^2 + 63728979648507703472676888*T - 30202095502531122305764166352
$41$	$T^{8} + \cdots - 31\!\cdots\!00$ T^8 - 8756T^7 - 355718271T^6 + 1881514716244T^5 + 45427691845325851T^4 - 35414197539837661836T^3 - 2197578108950692765072701T^2 - 7137093357563808706937791380*T - 3176394822850950034213626511200
$43$	$T^{8} + \cdots + 11\!\cdots\!68$ T^8 - 15290T^7 - 496314867T^6 + 8754041635140T^5 + 11932358096683548T^4 - 528077909036024210880T^3 + 800513345300721423669680T^2 + 4017575531063129340806323520*T + 1139991554103799167368045890368
$47$	$T^{8} + \cdots + 96\!\cdots\!00$ T^8 - 8516T^7 - 1480854843T^6 + 10969550334944T^5 + 719068338328011827T^4 - 3850620394267990129716T^3 - 141716504349169046944921209T^2 + 395333682452521696871444061720*T + 9655933159282724077380041334931200
$53$	$T^{8} + \cdots + 17\!\cdots\!00$ T^8 - 30475T^7 - 787500974T^6 + 20187800707225T^5 + 248702825642547950T^4 - 4241020576100112018875T^3 - 35605560445151727502854375T^2 + 274265346193025807043328275000*T + 1748574286895006237858952868312500
$59$	$T^{8} + \cdots + 75\!\cdots\!36$ T^8 - 20400T^7 - 3045878467T^6 + 42042272145388T^5 + 2794917475159660131T^4 - 24698950229484726777832T^3 - 877890243251729424281956737T^2 + 4714882853825962765748824452780*T + 75742589574397792813009044256775136
$61$	$T^{8} + \cdots + 30\!\cdots\!04$ T^8 - 10511T^7 - 2325908022T^6 + 20662392339609T^5 + 857491594857092358T^4 - 553477079673106513395T^3 - 73362305997996069462079927T^2 - 233550183469848442136119827892*T + 308388166900774117261899860564604
$67$	$T^{8} + \cdots - 39\!\cdots\!76$ T^8 + 616T^7 - 3808847268T^6 + 54997308380288T^5 + 3110641118561850560T^4 - 68814814833977141187072T^3 - 151220482566647757232873472T^2 + 9666507879284396970936956256256*T - 39111031037103055689397681003954176
$71$	$T^{8} + \cdots + 36\!\cdots\!12$ T^8 - 31862T^7 - 9670688481T^6 + 194430744120902T^5 + 31307686538856805472T^4 - 312266714764719636827016T^3 - 33282125696041971719016552192T^2 + 146420027406260792467432116955488*T + 3695111656175761302693393261408117312
$73$	$T^{8} + \cdots + 41\!\cdots\!88$ T^8 + 37377T^7 - 6892854346T^6 - 292301268765223T^5 + 4182185306431635990T^4 + 357329298231323318661229T^3 + 5743381766466651785998568617T^2 + 29478704272636680077884744404732*T + 4192075932330508556990811172861088
$79$	$T^{8} + \cdots - 89\!\cdots\!08$ T^8 + 71679T^7 - 13023850944T^6 - 1200494095111701T^5 + 25673362585007064552T^4 + 4410400647742499452536147T^3 + 41677761291387432295080271367T^2 - 4435923119017605774606329733741444*T - 89445275450693214466451475315715831008
$83$	$T^{8} + \cdots + 80\!\cdots\!47$ T^8 + 61543T^7 - 10157680649T^6 - 660787699980726T^5 + 22750415213698424970T^4 + 1499170697720190053645418T^3 - 19562728707774407047753824183T^2 - 959928203321723183366864439395613*T + 8021004783845317830976307954376717747
$89$	$T^{8} + \cdots - 34\!\cdots\!19$ T^8 + 14961T^7 - 20046089679T^6 + 455787474947838T^5 + 95473976846995236618T^4 - 5801515231120842185690202T^3 + 110070591243811703178840302661T^2 - 464126498744364926221067808401079*T - 3414853333833232004113270952982089619
$97$	$T^{8} + \cdots - 23\!\cdots\!41$ T^8 - 186419T^7 - 55909887869T^6 + 11483859970434310T^5 + 819489158162299283710T^4 - 214822449215943262000114610T^3 - 327575869326608602787706954531T^2 + 1124284916263302772062130425567128289*T - 23885276539636349215920923056418415828141
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
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