LMFDB - Newform orbit 304.8.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$304 = 2^{4} \cdot 19$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	304.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:	$94.9650477472$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312$ x^6 - 2x^5 - 8376x^4 + 135458x^3 + 16275767x^2 - 280013424*x - 6276171312
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 76)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 - 7) q^{3} + ( - \beta_{2} - \beta_1 + 47) q^{5} + (\beta_{4} + \beta_{2} - 5 \beta_1 + 262) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 662) q^{9} + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots - 1346) q^{11}+ \cdots + (7190 \beta_{5} - 6832 \beta_{4} + \cdots - 4064986) q^{99}+O(q^{100})$ q + (b1 - 7) * q^3 + (-b2 - b1 + 47) * q^5 + (b4 + b2 - 5b1 + 262) q^7 + (-b5 + b4 - b3 - 3b2 - 34b1 + 662) * q^9 + (-2b5 - 2b4 - 3b3 - b2 + 52b1 - 1346) * q^11 + (6b5 - b4 + 2b3 + 12b2 - 52b1 + 221) * q^13 + (9b5 - 7b4 + 13b3 + 3b2 + 217b1 - 4540) q^15 + (-19b5 - 7b4 + 12b3 + 16b2 - 46b1 + 3647) q^17 - 6859 * q^19 + (67b5 - 44b4 - 31b3 + 81b2 + 533b1 - 14601) q^21 + (-26b5 + b4 - 12b3 + 38b2 - 494b1 - 16649) * q^23 + (-108b5 + 42b4 + 123b3 - 93b2 + 12b1 + 62269) q^25 + (99b5 - 111b4 + 33b3 + 261b2 + 514b1 - 89359) q^27 + (171b5 - 90b4 - 165b3 + 143b2 + 1379b1 - 10315) q^29 + (-99b5 - 15b4 + 65b3 + 133b2 - 1281b1 - 66842) q^31 + (-241b5 + 111b4 - 27b3 - 27b2 - 2033b1 + 153504) q^33 + (-42b5 + 174b4 - 315b3 - 163b2 - 3832b1 - 75982) q^35 + (105b5 - 15b4 + 95b3 + 385b2 - 1059b1 + 135038) q^37 + (64b5 + 161b4 - 26b3 - 168b2 - 1624b1 - 125967) q^39 + (53b5 - 193b4 - 327b3 + 609b2 - 2693b1 + 93658) q^41 + (-528b5 - 36b4 + 59b3 - 95b2 - 96b1 - 203132) q^43 + (-474b5 + 664b4 + 68b3 - 63b2 - 15535b1 + 544345) q^45 + (982b5 + 124b4 + 114b3 + 985b2 + 333b1 - 320025) q^47 + (303b5 + 247b4 + 68b3 - 676b2 - 7868b1 + 343482) q^49 + (-1154b5 + 6b4 + 54b3 - 882b2 - 991b1 - 150417) q^51 + (337b5 - 464b4 + 1089b3 + 127b2 + 2955b1 + 84255) q^53 + (1158b5 - 264b4 + 1596b3 + 2925b2 + 123b1 - 303243) q^55 + (-6859b1 + 48013) q^57 + (697b5 - 209b4 - 651b3 + 141b2 - 11722b1 - 213157) q^59 + (492b5 + 532b4 + 1058b3 - 3229b2 + 4843b1 + 92723) q^61 + (-2220b5 + 2044b4 - 460b3 - 5571b2 - 39599b1 + 1209611) q^63 + (54b5 - 636b4 - 2994b3 - 38b2 + 8272b1 - 1035068) q^65 + (2616b5 - 340b4 - 1758b3 - 142b2 - 20179b1 + 13057) q^67 + (-508b5 - 717b4 - 198b3 + 2916b2 - 2012b1 - 1229577) q^69 + (-838b5 - 1684b4 - 246b3 - 3444b2 - 9992b1 + 242236) q^71 + (987b5 - 1089b4 - 16b3 + 382b2 + 15252b1 - 1864249) q^73 + (309b5 - 4641b4 + 771b3 - 4545b2 + 60880b1 - 675139) q^75 + (716b5 - 2866b4 - 1398b3 - 2113b2 + 42087b1 - 2932317) q^77 + (1395b5 + 695b4 - 25b3 + 5115b2 - 15445b1 + 1068724) q^79 + (-5248b5 + 6472b4 + 1898b3 - 6564b2 - 117862b1 + 1072103) q^81 + (-5164b5 - 2686b4 + 42b3 - 9308b2 + 45434b1 + 517244) q^83 + (1560b5 - 2352b4 - 1716b3 - 14307b2 + 13071b1 - 3248511) q^85 + (-3528b5 + 9689b4 - 1736b3 - 780b2 - 60914b1 + 4352219) q^87 + (-1783b5 - 1759b4 + 9039b3 + 2831b2 + 17253b1 - 2421858) q^89 + (-795b5 + 2055b4 + 5575b3 + 3455b2 + 50760b1 + 1604921) q^91 + (-3868b5 - 1686b4 + 234b3 - 240b2 - 64756b1 - 3033262) q^93 + (6859b2 + 6859b1 - 322373) * q^95 + (4998b5 + 2514b4 - 9084b3 + 168b2 - 25110b1 - 3557830) q^97 + (7190b5 - 6832b4 + 5047b3 + 21975b2 + 163550b1 - 4064986) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 40 q^{3} + 279 q^{5} + 1565 q^{7} + 3900 q^{9} - 7983 q^{11} + 1250 q^{13} - 26790 q^{15} + 21735 q^{17} - 41154 q^{19} - 86346 q^{21} - 100920 q^{23} + 373305 q^{25} - 534790 q^{27} - 58656 q^{29}+ \cdots - 24032935 q^{99}+O(q^{100})$ 6 * q - 40 * q^3 + 279 * q^5 + 1565 * q^7 + 3900 * q^9 - 7983 * q^11 + 1250 * q^13 - 26790 * q^15 + 21735 * q^17 - 41154 * q^19 - 86346 * q^21 - 100920 * q^23 + 373305 * q^25 - 534790 * q^27 - 58656 * q^29 - 403808 * q^31 + 916430 * q^33 - 463497 * q^35 + 808780 * q^37 - 758704 * q^39 + 556944 * q^41 - 1220735 * q^43 + 3234843 * q^45 - 1915305 * q^47 + 2045883 * q^49 - 908816 * q^51 + 511650 * q^53 - 1813341 * q^55 + 274360 * q^57 - 1300572 * q^59 + 565335 * q^61 + 7170325 * q^63 - 6195012 * q^65 + 45010 * q^67 - 7381528 * q^69 + 1424106 * q^71 - 11153825 * q^73 - 3941974 * q^75 - 17515425 * q^77 + 6392144 * q^79 + 6187530 * q^81 + 3164160 * q^83 - 19479255 * q^85 + 25999500 * q^87 - 14502678 * q^89 + 9736226 * q^91 - 18344300 * q^93 - 1913661 * q^95 - 21377010 * q^97 - 24032935 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312$ x^6 - 2*x^5 - 8376*x^4 + 135458*x^3 + 16275767*x^2 - 280013424*x - 6276171312 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -23\nu^{5} + 1337\nu^{4} + 331031\nu^{3} - 3798869\nu^{2} - 655626372\nu - 394340616 ) / 12622824$ (-23v^5 + 1337v^4 + 331031v^3 - 3798869v^2 - 655626372*v - 394340616) / 12622824
$\beta_{3}$	$=$	$( 51\nu^{5} + 2117\nu^{4} - 357983\nu^{3} - 9662021\nu^{2} + 548036696\nu + 11111121348 ) / 2103804$ (51v^5 + 2117v^4 - 357983v^3 - 9662021v^2 + 548036696*v + 11111121348) / 2103804
$\beta_{4}$	$=$	$( -269\nu^{5} - 12312\nu^{4} + 1628074\nu^{3} + 41365824\nu^{2} - 2202338405\nu - 33289998684 ) / 2103804$ (-269v^5 - 12312v^4 + 1628074v^3 + 41365824v^2 - 2202338405*v - 33289998684) / 2103804
$\beta_{5}$	$=$	$( -617\nu^{5} - 30195\nu^{4} + 3641083\nu^{3} + 101646951\nu^{2} - 4929275990\nu - 76626597048 ) / 4207608$ (-617v^5 - 30195v^4 + 3641083v^3 + 101646951v^2 - 4929275990*v - 76626597048) / 4207608

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} - 20\beta _1 + 2800$ -b5 + b4 - b3 - 3b2 - 20b1 + 2800
$\nu^{3}$	$=$	$78\beta_{5} - 90\beta_{4} + 12\beta_{3} + 198\beta_{2} + 4321\beta _1 - 60834$ 78b5 - 90b4 + 12b3 + 198b2 + 4321*b1 - 60834
$\nu^{4}$	$=$	$-9331\beta_{5} + 10219\beta_{4} - 4033\beta_{3} - 19821\beta_{2} - 212696\beta _1 + 12452470$ -9331b5 + 10219b4 - 4033b3 - 19821b2 - 212696*b1 + 12452470
$\nu^{5}$	$=$	$745380\beta_{5} - 866472\beta_{4} + 103440\beta_{3} + 1644228\beta_{2} + 24624409\beta _1 - 631311360$ 745380b5 - 866472b4 + 103440b3 + 1644228b2 + 24624409*b1 - 631311360

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

−84.8914
−46.2258
−13.7370
44.4724
49.5941
52.7878

−91.8914

432.300

1482.77

6257.03

1.2

−53.2258

88.7692

−1094.09

645.987

1.3

−20.7370

−501.411

1008.02

−1756.98

1.4

37.4724

534.866

−1051.36

−782.818

1.5

42.5941

51.7645

1212.43

−372.745

1.6

45.7878

−327.289

7.23734

−90.4805

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$19$	$+1$

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	304.8.a.i		6
4.b	odd	2	1		76.8.a.b	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.8.a.b	✓	6	4.b	odd	2	1
304.8.a.i		6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{6} + 40T_{3}^{5} - 7711T_{3}^{4} - 93190T_{3}^{3} + 16686996T_{3}^{2} - 43655400T_{3} - 7412317344$ T3^6 + 40*T3^5 - 7711*T3^4 - 93190*T3^3 + 16686996*T3^2 - 43655400*T3 - 7412317344 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(304))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} + \cdots - 7412317344$ T^6 + 40T^5 - 7711T^4 - 93190T^3 + 16686996T^2 - 43655400*T - 7412317344
$5$	$T^{6} + \cdots + 174360864384000$ T^6 - 279T^5 - 382107T^4 + 89339211T^3 + 31455689670T^2 - 5181395099400*T + 174360864384000
$7$	$T^{6} + \cdots + 15\!\cdots\!84$ T^6 - 1565T^5 - 2268958T^4 + 3630360070T^3 + 1278964317269T^2 - 2093958739148105*T + 15086336025908584
$11$	$T^{6} + \cdots - 47\!\cdots\!56$ T^6 + 7983T^5 - 37767781T^4 - 432916267239T^3 - 880155543756408T^2 - 385525041126730164*T - 47929083669332301456
$13$	$T^{6} + \cdots - 15\!\cdots\!56$ T^6 - 1250T^5 - 122286729T^4 - 6544577750T^3 + 3312548747774008T^2 + 589363572143005920*T - 15319634805440895689856
$17$	$T^{6} + \cdots + 81\!\cdots\!86$ T^6 - 21735T^5 - 1164308584T^4 + 36745972939470T^3 - 231907545730575987T^2 - 858871018484235580455*T + 8152361263409668732149786
$19$	$(T + 6859)^{6}$ (T + 6859)^6
$23$	$T^{6} + \cdots - 10\!\cdots\!64$ T^6 + 100920T^5 - 284311201T^4 - 151244309224680T^3 + 1502913205656366528T^2 + 7015145233308404129280*T - 10131803990133889037451264
$29$	$T^{6} + \cdots - 19\!\cdots\!56$ T^6 + 58656T^5 - 95711902329T^4 - 4494973587859440T^3 + 2578255904089960750008T^2 + 76523107161988774318618944*T - 19309171134099958491730368909456
$31$	$T^{6} + \cdots - 59\!\cdots\!24$ T^6 + 403808T^5 + 18356065296T^4 - 4120055684069632T^3 - 319372289042744119808T^2 - 7665008493910929005383680*T - 59469561500940745594498842624
$37$	$T^{6} + \cdots + 12\!\cdots\!96$ T^6 - 808780T^5 + 176256645884T^4 - 9069720444468640T^3 - 347190860463975948304T^2 + 5191054539189353523141440*T + 124536595731119467187730296896
$41$	$T^{6} + \cdots + 23\!\cdots\!84$ T^6 - 556944T^5 - 326056713208T^4 + 174434267004322080T^3 + 2157162144421067749584T^2 - 1067514808032839309216904576*T + 23338992175248254326181410627584
$43$	$T^{6} + \cdots - 15\!\cdots\!44$ T^6 + 1220735T^5 + 158935767027T^4 - 142748983288435795T^3 - 46051523057753836668440T^2 - 4661658345451186652836838160*T - 154097329310339035579788080098944
$47$	$T^{6} + \cdots + 10\!\cdots\!56$ T^6 + 1915305T^5 - 384082342837T^4 - 2150474405962335525T^3 - 661005833538587861055696T^2 + 324720105707311102197786553920*T + 104713598761424722953743469567814656
$53$	$T^{6} + \cdots - 69\!\cdots\!76$ T^6 - 511650T^5 - 3740637014161T^4 + 1036250365853981250T^3 + 3965837149172783092356840T^2 - 189404486376422674701976467840*T - 695595985358363257740317072458258176
$59$	$T^{6} + \cdots - 86\!\cdots\!56$ T^6 + 1300572T^5 - 1547972074327T^4 - 1965619922082623202T^3 + 574513744255956073498212T^2 + 619638885049121808670044671976*T - 86009270406166492856689005529297056
$61$	$T^{6} + \cdots - 23\!\cdots\!64$ T^6 - 565335T^5 - 8735718697011T^4 + 3565037032223823767T^3 + 12147376867741010024327214T^2 - 3269585231287627013776097234316*T - 2301352032237925633469153645158227464
$67$	$T^{6} + \cdots - 13\!\cdots\!76$ T^6 - 45010T^5 - 16470947063367T^4 - 1501881992060699320T^3 + 85557155517434897219425360T^2 + 12206470804539541818915606522240*T - 132726823572930625244662477069019198976
$71$	$T^{6} + \cdots + 13\!\cdots\!04$ T^6 - 1424106T^5 - 14152150868392T^4 + 18728657102643413232T^3 + 41117713101441495009540432T^2 - 50907169907215768219104435032352*T + 13535981091827314004436212385699151104
$73$	$T^{6} + \cdots + 96\!\cdots\!26$ T^6 + 11153825T^5 + 46136474610476T^4 + 92710992204432255410T^3 + 96226770293526768419688329T^2 + 49054691402281892204791620009725*T + 9624457063719626672182251795785919226
$79$	$T^{6} + \cdots + 29\!\cdots\!76$ T^6 - 6392144T^5 + 2824710229340T^4 + 43513478257211180320T^3 - 79446304014369974498532160T^2 + 35993795123493685348231918829056*T + 2956440985861855561798670399576498176
$83$	$T^{6} + \cdots - 65\!\cdots\!64$ T^6 - 3164160T^5 - 103202687694196T^4 + 121042182434330223840T^3 + 2896490929079852479940429568T^2 + 1700840388442765245769356686922240*T - 6545218979767270128105855563739588824064
$89$	$T^{6} + \cdots + 18\!\cdots\!76$ T^6 + 14502678T^5 - 112082030411632T^4 - 2225414419024761179040T^3 - 1452412221116096437145650176T^2 + 69701367951459345963376571477455872*T + 188797913343229671059662130763365927755776
$97$	$T^{6} + \cdots + 48\!\cdots\!44$ T^6 + 21377010T^5 - 45190604396916T^4 - 2330897223876266942920T^3 - 1588463970838532305718360640T^2 + 28421264378688452504439526273536000*T + 48176140375807984342995060572820534943744
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Classical	Maass
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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
Embeddings:		e.g. 1-3 or 1.6
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