LMFDB - Newform orbit 208.10.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$208 = 2^{4} \cdot 13$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	208.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:	$107.127453922$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580$ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{6}\cdot 3$
Twist minimal:	no (minimal twist has level 13)
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 $1,\beta_1,\beta_2,\beta_3,\beta_4$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{3} - 32) q^{3} + ( - 2 \beta_{4} + \beta_{3} + \cdots + 361) q^{5} + (5 \beta_{4} - 10 \beta_{3} + \cdots - 2020) q^{7} + (2 \beta_{4} - 57 \beta_{3} + \cdots + 12206) q^{9} + (39 \beta_{4} + 123 \beta_{3} + \cdots - 24324) q^{11}+ \cdots + (8179 \beta_{4} + 3898245 \beta_{3} + \cdots - 602456384) q^{99}+O(q^{100})$ q + (b3 - 32) * q^3 + (-2b4 + b3 - b2 + 3b1 + 361) * q^5 + (5b4 - 10b3 - 9b2 - b1 - 2020) q^7 + (2b4 - 57b3 - 27b2 - 52b1 + 12206) * q^9 + (39b4 + 123b3 - 3b2 + 142b1 - 24324) * q^11 + 28561 * q^13 + (-302b4 + 231b3 + 318b2 - 218b1 - 21212) * q^15 + (-178b4 - 547b3 + 639b2 - 465b1 - 99371) * q^17 + (605b4 + 1017b3 + 235b2 + 1425b1 + 168304) * q^19 + (978b4 - 179b3 + 1539b2 - 2406b1 - 320237) * q^21 + (-1484b4 - 3356b3 + 616b2 + 1407b1 + 117636) * q^23 + (-3218b4 + 5799b3 - 1459b2 - 663b1 + 335524) * q^25 + (2066b4 + 2733b3 + 4050b2 - 13b1 - 1369840) * q^27 + (4956b4 + 7884b3 + 616b2 + 8385b1 + 2137090) * q^29 + (2213b4 - 8921b3 - 4777b2 - 7166b1 - 2577888) * q^31 + (2884b4 - 38526b3 + 882b2 - 24155b1 + 3447778) * q^33 + (12484b4 - 54317b3 - 3688b2 - 1926b1 - 1686272) * q^35 + (15998b4 - 8667b3 - 8789b2 + 8802b1 + 1434389) * q^37 + (28561b3 - 913952) q^39 + (-5800b4 - 86290b3 + 27782b2 - 11721b1 + 1835988) * q^41 + (-12814b4 + 51621b3 + 7846b2 - 132291b1 - 2557640) * q^43 + (-7664b4 - 82608b3 - 36000b2 + 71383b1 + 5217718) * q^45 + (2641b4 - 225218b3 + 21387b2 - 53948b1 - 8720164) * q^47 + (-43010b4 - 5429b3 - 15471b2 + 50164b1 + 5051906) * q^49 + (-36806b4 - 152217b3 - 100998b2 + 249850b1 - 3391424) * q^51 + (29856b4 - 123750b3 + 49890b2 - 163839b1 + 18611628) * q^53 + (58378b4 - 207564b3 + 55370b2 - 142491b1 - 19942356) * q^55 + (49504b4 - 10286b3 - 36714b2 - 198944b1 + 18039962) * q^57 + (-5337b4 + 283791b3 + 123029b2 + 472432b1 - 49267084) * q^59 + (17072b4 + 45402b3 - 3830b2 + 391989b1 - 26436676) * q^61 + (35063b4 - 159861b3 - 193779b2 + 441818b1 + 83397824) * q^63 + (-57122b4 + 28561b3 - 28561b2 + 85683b1 + 10310521) * q^65 + (-57515b4 - 438627b3 + 95987b2 - 66981b1 + 73770784) * q^67 + (-259356b4 - 28932b3 + 93048b2 + 449829b1 - 116021748) * q^69 + (-247085b4 - 374018b3 - 52135b2 - 923707b1 - 42494468) * q^71 + (22424b4 - 1528432b3 - 81948b2 - 871951b1 - 50835258) * q^73 + (-372538b4 + 458314b3 + 176142b2 - 108862b1 + 150547992) * q^75 + (-287088b4 + 91530b3 + 218658b2 - 973973b1 + 89907798) * q^77 + (96094b4 + 411610b3 - 278862b2 + 339742b1 + 249592440) * q^79 + (121868b4 - 958764b3 - 285444b2 + 1560455b1 - 63677287) * q^81 + (342997b4 - 490445b3 + 303179b2 - 125193b1 - 339537584) * q^83 + (219262b4 + 2615559b3 + 251129b2 - 160095b1 - 154599867) * q^85 + (512284b4 + 1174890b3 - 191064b2 - 1676597b1 + 123179284) * q^87 + (629308b4 - 5708b3 - 447420b2 + 866958b1 - 150682534) * q^89 + (142805b4 - 285610b3 - 257049b2 - 28561b1 - 57693220) * q^91 + (583736b4 - 956564b3 + 691428b2 - 509473b1 - 178886532) * q^93 + (106690b4 - 1868360b3 + 274758b2 - 160202b1 - 288955216) * q^95 + (-395056b4 + 3922068b3 - 1171136b2 - 1208163b1 + 1783562) * q^97 + (8179b4 + 3898245b3 - 19683b2 + 836251b1 - 602456384) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27}+ \cdots - 3016199848 q^{99}+O(q^{100})$ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580$ x^5 - 1438*x^3 - 4164*x^2 + 396957*x - 59580 :

$\beta_{1}$	$=$	$8\nu$ 8*v
$\beta_{2}$	$=$	$( 3\nu^{4} - 29\nu^{3} - 2583\nu^{2} + 4861\nu + 29588 ) / 544$ (3v^4 - 29v^3 - 2583v^2 + 4861v + 29588) / 544
$\beta_{3}$	$=$	$( -\nu^{4} + 55\nu^{3} + 317\nu^{2} - 41559\nu + 189604 ) / 1088$ (-v^4 + 55v^3 + 317v^2 - 41559*v + 189604) / 1088
$\beta_{4}$	$=$	$( -7\nu^{4} + 113\nu^{3} + 7659\nu^{2} - 62161\nu - 1120772 ) / 1088$ (-7v^4 + 113v^3 + 7659v^2 - 62161v - 1120772) / 1088

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

$\nu$	$=$	$( \beta_1 ) / 8$ (b1) / 8
$\nu^{2}$	$=$	$( 4\beta_{4} - 4\beta_{3} + 4\beta_{2} + 5\beta _1 + 4600 ) / 8$ (4b4 - 4b3 + 4b2 + 5b1 + 4600) / 8
$\nu^{3}$	$=$	$( 48\beta_{4} + 144\beta_{3} + 80\beta_{2} + 941\beta _1 + 20000 ) / 8$ (48b4 + 144b3 + 80b2 + 941b1 + 20000) / 8
$\nu^{4}$	$=$	$( 3908\beta_{4} - 2052\beta_{3} + 5668\beta_{2} + 11781\beta _1 + 4075032 ) / 8$ (3908b4 - 2052b3 + 5668b2 + 11781b1 + 4075032) / 8

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

16.7176
−27.7188
35.1685
0.150341
−24.3176

−250.479

1555.58

8329.39

43056.6

1.2

−194.269

−920.299

−5359.02

18057.4

1.3

−47.8784

−109.762

−5947.44

−17390.7

1.4

136.532

2554.62

−9399.91

−1041.89

1.5

195.094

−1277.14

2277.98

18378.5

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$13$	$-1$

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	208.10.a.h		5
4.b	odd	2	1		13.10.a.b	✓	5
12.b	even	2	1		117.10.a.e		5
20.d	odd	2	1		325.10.a.b		5
52.b	odd	2	1		169.10.a.b		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.10.a.b	✓	5	4.b	odd	2	1
117.10.a.e		5	12.b	even	2	1
169.10.a.b		5	52.b	odd	2	1
208.10.a.h		5	1.a	even	1	1	trivial
325.10.a.b		5	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{5} + 161T_{3}^{4} - 66777T_{3}^{3} - 7746921T_{3}^{2} + 1090724832T_{3} + 62057286864$ T3^5 + 161*T3^4 - 66777*T3^3 - 7746921*T3^2 + 1090724832*T3 + 62057286864 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(208))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{5}$ T^5
$3$	$T^{5} + \cdots + 62057286864$ T^5 + 161T^4 - 66777T^3 - 7746921T^2 + 1090724832T + 62057286864
$5$	$T^{5} + \cdots + 512670311383500$ T^5 - 1803T^4 - 4092589T^3 + 3475347315T^2 + 5098977459000T + 512670311383500
$7$	$T^{5} + \cdots + 56\!\cdots\!88$ T^5 + 10099T^4 - 62513861T^3 - 772947069643T^2 - 556625160811972T + 5684631812366849888
$11$	$T^{5} + \cdots + 19\!\cdots\!96$ T^5 + 121746T^4 + 2479670132T^3 - 100685267329800T^2 - 2162481301083825888T + 19450453762899214039296
$13$	$(T - 28561)^{5}$ (T - 28561)^5
$17$	$T^{5} + \cdots - 39\!\cdots\!92$ T^5 + 495669T^4 - 289191316885T^3 - 98882562477434877T^2 + 17126546902780079776656T - 399153090461429727080315892
$19$	$T^{5} + \cdots + 29\!\cdots\!00$ T^5 - 840738T^4 - 260984517612T^3 + 132037771414811912T^2 + 43590663172787565531168T + 2943821455732214929563897600
$23$	$T^{5} + \cdots + 42\!\cdots\!52$ T^5 - 592152T^4 - 3493027002432T^3 + 5203750196460916224T^2 - 2574986399360436795752448T + 428621968496899390019183050752
$29$	$T^{5} + \cdots - 44\!\cdots\!20$ T^5 - 10678182T^4 + 16260933487592T^3 + 72928748649832496976T^2 - 5368198298306611736512176T - 44459311911325939063349211345120
$31$	$T^{5} + \cdots - 48\!\cdots\!64$ T^5 + 12885296T^4 + 32338285004512T^3 - 105055140872487792640T^2 - 302260787091748018727890688T - 48102238044029854183074610622464
$37$	$T^{5} + \cdots - 51\!\cdots\!48$ T^5 - 7171823T^4 - 294915826692917T^3 + 2641277396215514731471T^2 + 6766082189398987208105351416T - 51510793797841272350692715200003748
$41$	$T^{5} + \cdots + 49\!\cdots\!12$ T^5 - 9294012T^4 - 1064915461247484T^3 + 3176296057078183644768T^2 + 138693246906168815606880689664T + 494782986122239059884653191807270912
$43$	$T^{5} + \cdots - 16\!\cdots\!08$ T^5 + 12831975T^4 - 2190268153760577T^3 + 1063158495960812067089T^2 + 1401924732869580555364611427320T - 16185929811522090497298671067442009008
$47$	$T^{5} + \cdots + 10\!\cdots\!64$ T^5 + 43354215T^4 - 3068542687682205T^3 - 118384348682900685487527T^2 + 1107959277012333917870538355356T + 10856493902269268859639670525864260864
$53$	$T^{5} + \cdots - 36\!\cdots\!12$ T^5 - 93231780T^4 - 2145547062601020T^3 + 129076946266905410876064T^2 + 1204843160261172809898664644096T - 36723519617807505912655816002716762112
$59$	$T^{5} + \cdots + 43\!\cdots\!00$ T^5 + 246496182T^4 - 10091472046384188T^3 - 5930341249581317084193432T^2 - 310469412650158482140152192248864T + 4372455985727469565995467194585904697600
$61$	$T^{5} + \cdots + 57\!\cdots\!08$ T^5 + 132232612T^4 - 6538173996828668T^3 - 1049889574460517545593856T^2 - 8951528105021696916307122275072T + 578110697319573188159158003609953783808
$67$	$T^{5} + \cdots + 85\!\cdots\!64$ T^5 - 369388534T^4 + 31699487729867620T^3 + 1402869766077392493965528T^2 - 273369432837307438768980109950752T + 8568994877041624094412008079312263608064
$71$	$T^{5} + \cdots + 41\!\cdots\!92$ T^5 + 212150457T^4 - 103163802673700829T^3 - 4880249511470949355321833T^2 + 1522066704005485905711115864136076T + 41821928126676484730584325399818860813792
$73$	$T^{5} + \cdots + 16\!\cdots\!16$ T^5 + 252729806T^4 - 180925206782132312T^3 - 44119689947951932040208272T^2 + 7409006510820583446681290412435536T + 1670294301045437304637381500483582889726816
$79$	$T^{5} + \cdots + 41\!\cdots\!40$ T^5 - 1247271728T^4 + 535977005216281984T^3 - 86820229442392262196871168T^2 + 2253193366828739192300052181815296T + 410024861234306597370828654020014035107840
$83$	$T^{5} + \cdots + 52\!\cdots\!24$ T^5 + 1696894296T^4 + 965304229131652832T^3 + 216125993836531517484173568T^2 + 18546842306111067162747956572418304T + 526809733080455133331026189945941188122624
$89$	$T^{5} + \cdots - 19\!\cdots\!60$ T^5 + 753854382T^4 - 360717372948090904T^3 - 170822820065517807458198544T^2 + 61162600944059411370271828202285904T - 1903086219907992929581753200277786862796960
$97$	$T^{5} + \cdots - 14\!\cdots\!32$ T^5 - 3824606T^4 - 2441997563116303880T^3 + 153757537448202660807707632T^2 + 1438608683131201717747160784795813904T - 142814432358561470129404623607552203752395232
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Elliptic curves over $\Q$
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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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