LMFDB - Newform orbit 624.4.q.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(289,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("624.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$624 = 2^{4} \cdot 3 \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	624.q (of order $3$ , degree $2$ , not minimal)

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:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + 120 x^{8} - 979 x^{7} + 14252 x^{6} - 68003 x^{5} + 352315 x^{4} - 602502 x^{3} + \cdots + 12873744$ x^10 - x^9 + 120x^8 - 979x^7 + 14252x^6 - 68003x^5 + 352315x^4 - 602502x^3 + 2179576x^2 - 1643304x + 12873744
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{10}$
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 3 \beta_{2} + 3) q^{3} + ( - \beta_{8} - \beta_1 - 2) q^{5} + (\beta_{7} + \beta_{2}) q^{7} - 9 \beta_{2} q^{9} + (\beta_{5} - \beta_{3} + 9 \beta_{2} - 9) q^{11} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1) q^{13}+ \cdots + ( - 9 \beta_{9} + 9 \beta_{6} + \cdots + 81) q^{99}+O(q^{100})$ q + (-3b2 + 3) q^3 + (-b8 - b1 - 2) * q^5 + (b7 + b2) * q^7 - 9b2 q^9 + (b5 - b3 + 9b2 - 9) q^11 + (b9 - b8 - b7 - b4 - b3 - 4b2 - b1 - 1) q^13 + (6b2 - 3b1 - 6) * q^15 + (b9 + 4b8 + 4b7 - 2b2) q^17 + (b9 + 3b8 - 2b6 + 2b3 - 32b2) * q^19 + (3b7 + 3b4 + 3) * q^21 + (-2b5 - b4 + 2b3 - 4b2 + 3b1 + 4) * q^23 + (-5b9 + 8b8 + 3b7 + 3b6 - 5b5 + 3b4 + 8b1 + 32) q^25 - 27 * q^27 + (-2b5 - 4b4 - 3b3 + 25b2 - 6b1 - 25) q^29 + (b9 - 12b8 - b7 + 3b6 + b5 - b4 - 12b1 + 42) q^31 + (-3b9 + 3b6 - 3b3 + 27b2) * q^33 + (2b9 - 6b8 - 3b7 + b6 - b3 + 47b2) * q^35 + (-6b5 + 7b4 - 2b3 - 52b2 + 2b1 + 52) q^37 + (3b9 + 3b6 + 3b5 - 3b4 - 3b3 + 3b2 - 3b1 - 15) q^39 + (2b5 - 3b4 - 11b3 + 31b2 + 3b1 - 31) q^41 + (2b9 - 7b8 - 15b7 - 3b6 + 3b3 - 54b2) * q^43 + (9b8 + 18b2) * q^45 + (b9 + 17b8 + 6b6 + b5 + 17b1 + 30) q^47 + (-7b5 + 9b4 - 6b3 - 58b2 + 5b1 + 58) q^49 + (3b9 + 12b8 + 12b7 + 3b5 + 12b4 + 12b1 - 6) * q^51 + (-15b9 + 4b8 - 5b7 + 7b6 - 15b5 - 5b4 + 4b1 + 113) q^53 + (b5 + 10b4 + 5b3 - 15b2 + 42b1 + 15) * q^55 + (3b9 + 9b8 - 6b6 + 3b5 + 9b1 - 96) q^57 + (-5b9 + 20b8 - 7b7 + 8b6 - 8b3 + 8b2) * q^59 + (b9 - 37b8 + 14b7 + 16b6 - 16b3 - 21b2) q^61 + (9b4 - 9b2 + 9) * q^63 + (-3b9 + 45b8 + 19b7 + 3b6 - 10b5 + 3b4 + 3b3 - 51b2 + 16b1 + 134) q^65 + (4b5 + 11b4 + b3 + 222b2 + 5b1 - 222) * q^67 + (6b9 - 9b8 + 3b7 - 6b6 + 6b3 - 12b2) * q^69 + (14b9 - b8 + 23b7 - 2b6 + 2b3 - 172b2) q^71 + (-16b9 + 19b8 + 30b7 + 21b6 - 16b5 + 30b4 + 19b1 + 46) q^73 + (-15b5 + 9b4 + 9b3 - 96b2 + 24b1 + 96) q^75 + (6b9 - 5b8 - 2b7 - b6 + 6b5 - 2b4 - 5b1 + 21) * q^77 + (-5b9 - 54b8 - 13b7 + 13b6 - 5b5 - 13b4 - 54b1 + 22) q^79 + (81b2 - 81) q^81 + (-2b9 - 46b8 + 23b7 + b6 - 2b5 + 23b4 - 46b1 + 217) * q^83 + (32b9 - 13b8 - 11b7 - 7b6 + 7b3 - 443b2) * q^85 + (6b9 + 18b8 + 12b7 + 9b6 - 9b3 + 75b2) * q^87 + (21b5 + 53b4 - 16b3 + 26b2 + 34b1 - 26) q^89 + (-5b9 - b8 - 14b7 - 6b6 - 6b5 - 47b4 + 5b3 - 270b2 + 5b1 + 94) * q^91 + (3b5 - 3b4 + 9b3 - 126b2 - 36b1 + 126) q^93 + (13b9 + 51b8 - 2b7 - 370b2) * q^95 + (19b9 + 39b8 - 5b7 + 22b6 - 22b3 + 519b2) * q^97 + (-9b9 + 9b6 - 9b5 + 81) q^99
$\operatorname{Tr}(f)(q)$	$=$	$10 q + 15 q^{3} - 22 q^{5} + 4 q^{7} - 45 q^{9} - 46 q^{11} - 31 q^{13} - 33 q^{15} - 11 q^{17} - 158 q^{19} + 24 q^{21} + 26 q^{23} + 340 q^{25} - 270 q^{27} - 125 q^{29} + 396 q^{31} + 138 q^{33} + 230 q^{35}+ \cdots + 828 q^{99}+O(q^{100})$ 10 * q + 15 * q^3 - 22 * q^5 + 4 * q^7 - 45 * q^9 - 46 * q^11 - 31 * q^13 - 33 * q^15 - 11 * q^17 - 158 * q^19 + 24 * q^21 + 26 * q^23 + 340 * q^25 - 270 * q^27 - 125 * q^29 + 396 * q^31 + 138 * q^33 + 230 * q^35 + 261 * q^37 - 141 * q^39 - 151 * q^41 - 264 * q^43 + 99 * q^45 + 332 * q^47 + 293 * q^49 - 66 * q^51 + 1178 * q^53 + 106 * q^55 - 948 * q^57 + 72 * q^59 - 157 * q^61 + 36 * q^63 + 1137 * q^65 - 1120 * q^67 - 78 * q^69 - 898 * q^71 + 470 * q^73 + 510 * q^75 + 192 * q^77 + 148 * q^79 - 405 * q^81 + 2036 * q^83 - 2249 * q^85 + 375 * q^87 - 170 * q^89 - 334 * q^91 + 594 * q^93 - 1810 * q^95 + 2620 * q^97 + 828 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{10} - x^{9} + 120 x^{8} - 979 x^{7} + 14252 x^{6} - 68003 x^{5} + 352315 x^{4} - 602502 x^{3} + \cdots + 12873744$ x^10 - x^9 + 120*x^8 - 979*x^7 + 14252*x^6 - 68003*x^5 + 352315*x^4 - 602502*x^3 + 2179576*x^2 - 1643304*x + 12873744 :

$\beta_{1}$	$=$	$( - 22705040150959 \nu^{9} + \cdots - 22\!\cdots\!00 ) / 52\!\cdots\!10$ (-22705040150959v^9 - 8869585653476875v^8 - 36227560450782926v^7 - 975658460292756135v^6 + 4070581174274678818v^5 - 77939722727738700965v^4 + 57889277434541639377v^3 - 346979687375955796360v^2 - 3997870794130462705476*v - 2295159629608930944600) / 529633358081702752410
$\beta_{2}$	$=$	$( - 40025142237939 \nu^{9} - 229951117700615 \nu^{8} + \cdots - 42\!\cdots\!84 ) / 56\!\cdots\!04$ (-40025142237939v^9 - 229951117700615v^8 - 4277560778258186v^7 + 6441434294656781v^6 - 284569200354047262v^5 - 1370243297138987323v^4 + 6952490734722105077v^3 - 84866115938377551592v^2 + 42242054389784226904*v - 42984454325870852184) / 564942248620482935904
$\beta_{3}$	$=$	$( - 963657775984429 \nu^{9} + \cdots - 45\!\cdots\!20 ) / 84\!\cdots\!60$ (-963657775984429v^9 - 145362637221139225v^8 - 643804378886399606v^7 - 15513913850264246445v^6 + 60860760783084152158v^5 - 1267589213100904025285v^4 + 1030515799973497806187v^3 - 6824666737090956015640v^2 - 29435766973011663729816*v - 45841454617938201934920) / 8474133729307244038560
$\beta_{4}$	$=$	$( 277783668470921 \nu^{9} + \cdots + 64\!\cdots\!36 ) / 42\!\cdots\!28$ (277783668470921v^9 + 11777718211073405v^8 + 76649267496543046v^7 + 1096273016646869313v^6 - 2955655859533788062v^5 + 88411848058020536209v^4 - 111102555443935722983v^3 + 937160377390304661944v^2 + 90828678625312702392*v + 6491409211931651093736) / 423706686465362201928
$\beta_{5}$	$=$	$( - 13\!\cdots\!19 \nu^{9} + \cdots - 23\!\cdots\!80 ) / 42\!\cdots\!80$ (-13456608753972919v^9 - 230740255320803275v^8 - 2022008496500234066v^7 - 15027782914947416775v^6 - 21373502961772140662v^5 - 1864861736321704130015v^4 + 3284552393793583373857v^3 - 33778899629413458510280v^2 - 13691474641143730310376*v - 238628028254677566200280) / 4237066864653622019280
$\beta_{6}$	$=$	$( 428509431163265 \nu^{9} - 701042128200067 \nu^{8} + \cdots - 80\!\cdots\!48 ) / 92\!\cdots\!80$ (428509431163265v^9 - 701042128200067v^8 + 43684601335207750v^7 - 474041278574202543v^6 + 5459475219966950770v^5 - 28089310741738821791v^4 + 85902258922480635985v^3 - 172746738389094099424v^2 + 145100349127095440520*v - 806615013952046724048) / 92110149231600478680
$\beta_{7}$	$=$	$( - 21\!\cdots\!85 \nu^{9} + \cdots + 13\!\cdots\!00 ) / 42\!\cdots\!28$ (-2160657980322085v^9 - 9111756698245417v^8 - 242775469015864046v^7 + 1086889017670940715v^6 - 17805936391761408218v^5 + 18407725755053378515v^4 - 99971499469251139733v^3 - 280229591797830532408v^2 - 642624256537207467672*v + 134757620759394336600) / 423706686465362201928
$\beta_{8}$	$=$	$( 13\!\cdots\!41 \nu^{9} + \cdots - 14\!\cdots\!04 ) / 21\!\cdots\!40$ (13996180976437241v^9 + 11472529310888549v^8 + 1640803771857192454v^7 - 11288352468351524439v^6 + 170666701033324449538v^5 - 650104486983261855263v^4 + 3378915008284192194697v^3 - 4527454640568076512832v^2 + 20960160166435720393464*v - 14982074891156746469304) / 2118533432326811009640
$\beta_{9}$	$=$	$( 54\!\cdots\!49 \nu^{9} + \cdots - 12\!\cdots\!56 ) / 42\!\cdots\!80$ (54795576648662849v^9 + 163649163798954461v^8 + 6202709581898459566v^7 - 30416800124638828671v^6 + 543855871581332333002v^5 - 823339788169032143207v^4 + 4369595356394762876713v^3 + 17246701678456595422472v^2 + 27577859512382122572216*v - 12892723108336589891256) / 4237066864653622019280

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

$\nu$	$=$	$( \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4$ (b3 - b2 - b1 + 1) / 4
$\nu^{2}$	$=$	$-\beta_{9} + 2\beta_{6} - 2\beta_{3} - 48\beta_{2}$ -b9 + 2b6 - 2b3 - 48*b2
$\nu^{3}$	$=$	$( 22\beta_{9} + 41\beta_{8} - 22\beta_{7} - 125\beta_{6} + 22\beta_{5} - 22\beta_{4} + 41\beta _1 + 1023 ) / 4$ (22b9 + 41b8 - 22b7 - 125b6 + 22b5 - 22b4 + 41*b1 + 1023) / 4
$\nu^{4}$	$=$	$-119\beta_{5} + 15\beta_{4} + 338\beta_{3} + 4964\beta_{2} - 55\beta _1 - 4964$ -119b5 + 15b4 + 338b3 + 4964b2 - 55*b1 - 4964
$\nu^{5}$	$=$	$( -4338\beta_{9} - 4201\beta_{8} + 2558\beta_{7} + 17457\beta_{6} - 17457\beta_{3} - 193395\beta_{2} ) / 4$ (-4338b9 - 4201b8 + 2558b7 + 17457b6 - 17457b3 - 193395b2) / 4
$\nu^{6}$	$=$	$15638 \beta_{9} + 10225 \beta_{8} - 4165 \beta_{7} - 51201 \beta_{6} + 15638 \beta_{5} - 4165 \beta_{4} + \cdots + 661687$ 15638b9 + 10225b8 - 4165b7 - 51201b6 + 15638b5 - 4165b4 + 10225*b1 + 661687
$\nu^{7}$	$=$	$( -690566\beta_{5} + 310606\beta_{4} + 2528873\beta_{3} + 30110771\beta_{2} - 561021\beta _1 - 30110771 ) / 4$ (-690566b5 + 310606b4 + 2528873b3 + 30110771b2 - 561021*b1 - 30110771) / 4
$\nu^{8}$	$=$	$-2209435\beta_{9} - 1591140\beta_{8} + 742465\beta_{7} + 7589745\beta_{6} - 7589745\beta_{3} - 94565381\beta_{2}$ -2209435b9 - 1591140b8 + 742465b7 + 7589745b6 - 7589745b3 - 94565381b2
$\nu^{9}$	$=$	$( 103985970 \beta_{9} + 80137621 \beta_{8} - 41751750 \beta_{7} - 370168021 \beta_{6} + 103985970 \beta_{5} + \cdots + 4498166639 ) / 4$ (103985970b9 + 80137621b8 - 41751750b7 - 370168021b6 + 103985970b5 - 41751750b4 + 80137621*b1 + 4498166639) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$ .

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-\beta_{2}$	$1$	$1$

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}

289.1

−1.19909	−	2.07689i
2.57048	+	4.45220i
−6.06267	−	10.5009i
3.45416	+	5.98279i
1.73712	+	3.00877i
−1.19909	+	2.07689i
2.57048	−	4.45220i
−6.06267	+	10.5009i
3.45416	−	5.98279i
1.73712	−	3.00877i

1.50000

2.59808i

−21.8246

2.84961

−

4.93567i

−4.50000

7.79423i

289.2

1.50000

2.59808i

−6.40898

−14.7066

25.4726i

−4.50000

7.79423i

289.3

1.50000

2.59808i

−3.53523

0.928779

−

1.60869i

−4.50000

7.79423i

289.4

1.50000

2.59808i

5.41254

11.2944

−

19.5625i

−4.50000

7.79423i

289.5

1.50000

2.59808i

15.3563

1.63381

−

2.82985i

−4.50000

7.79423i

529.1

1.50000

−

2.59808i

−21.8246

2.84961

4.93567i

−4.50000

−

7.79423i

529.2

1.50000

−

2.59808i

−6.40898

−14.7066

−

25.4726i

−4.50000

−

7.79423i

529.3

1.50000

−

2.59808i

−3.53523

0.928779

1.60869i

−4.50000

−

7.79423i

529.4

1.50000

−

2.59808i

5.41254

11.2944

19.5625i

−4.50000

−

7.79423i

529.5

1.50000

−

2.59808i

15.3563

1.63381

2.82985i

−4.50000

−

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	624.4.q.m		10
4.b	odd	2	1		312.4.q.d	✓	10
13.c	even	3	1	inner	624.4.q.m		10
52.j	odd	6	1		312.4.q.d	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.4.q.d	✓	10	4.b	odd	2	1
312.4.q.d	✓	10	52.j	odd	6	1
624.4.q.m		10	1.a	even	1	1	trivial
624.4.q.m		10	13.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(624, [\chi])$ :

T_{5}^{5} + 11T_{5}^{4} - 337T_{5}^{3} - 1843T_{5}^{2} + 9652T_{5} + 41100

T_{7}^{10} - 4 T_{7}^{9} + 719 T_{7}^{8} - 11984 T_{7}^{7} + 547477 T_{7}^{6} - 5367218 T_{7}^{5} + \cdots + 528264256

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10}$ T^10
$3$	$(T^{2} - 3 T + 9)^{5}$ (T^2 - 3*T + 9)^5
$5$	$(T^{5} + 11 T^{4} + \cdots + 41100)^{2}$ (T^5 + 11T^4 - 337T^3 - 1843T^2 + 9652T + 41100)^2
$7$	$T^{10} + \cdots + 528264256$ T^10 - 4T^9 + 719T^8 - 11984T^7 + 547477T^6 - 5367218T^5 + 38178112T^4 - 142839544T^3 + 390517344T^2 - 544169184*T + 528264256
$11$	$T^{10} + \cdots + 32799353693184$ T^10 + 46T^9 + 4116T^8 + 77184T^7 + 7482928T^6 + 137347104T^5 + 7708969152T^4 - 11630963968T^3 + 651176631040T^2 + 2338386405888*T + 32799353693184
$13$	$T^{10} + \cdots + 51\!\cdots\!57$ T^10 + 31T^9 + 5355T^8 + 258674T^7 + 15125669T^6 + 880234641T^5 + 33231094793T^4 + 1248569991266T^3 + 56787094142415T^2 + 722240638796911*T + 51185893014090757
$17$	$T^{10} + \cdots + 37\!\cdots\!76$ T^10 + 11T^9 + 15962T^8 - 196093T^7 + 191821994T^6 - 2082263569T^5 + 941386610773T^4 - 18725156490156T^3 + 3487105209524832T^2 - 36067902504929856*T + 373773981198571776
$19$	$T^{10} + \cdots + 78\!\cdots\!00$ T^10 + 158T^9 + 24656T^8 + 1520008T^7 + 145709152T^6 + 8777602336T^5 + 594390214720T^4 + 21784699564544T^3 + 657604630262784T^2 + 8205132763176960*T + 78005183123046400
$23$	$T^{10} + \cdots + 14\!\cdots\!36$ T^10 - 26T^9 + 16980T^8 + 58496T^7 + 221939504T^6 - 1643583840T^5 + 795230469824T^4 - 30035825545984T^3 + 2583070856974080T^2 - 58046270772164096*T + 1424683480884315136
$29$	$T^{10} + \cdots + 73\!\cdots\!00$ T^10 + 125T^9 + 59674T^8 + 3481085T^7 + 2107419982T^6 + 126376297505T^5 + 34188221506681T^4 + 613090353668660T^3 + 277420165538717236T^2 + 10688100744491781840*T + 733665039823353459600
$31$	$(T^{5} - 198 T^{4} + \cdots - 106542921664)^{2}$ (T^5 - 198T^4 - 52711T^3 + 10406284T^2 + 541147152T - 106542921664)^2
$37$	$T^{10} + \cdots + 92\!\cdots\!76$ T^10 - 261T^9 + 190318T^8 - 42612961T^7 + 24950195338T^6 - 5010150910357T^5 + 1272425556502357T^4 - 85268526046091056T^3 + 11409380620191041836T^2 + 89661156992953888800*T + 92373285472979428700176
$41$	$T^{10} + \cdots + 95\!\cdots\!96$ T^10 + 151T^9 + 197910T^8 + 98861043T^7 + 47639138854T^6 + 12931580087967T^5 + 2655773403406689T^4 + 362667595842820424T^3 + 37129305495533536048T^2 + 2321885142633242289792*T + 95444245599783262730496
$43$	$T^{10} + \cdots + 12\!\cdots\!16$ T^10 + 264T^9 + 204911T^8 + 32077492T^7 + 25760636701T^6 + 3920324529122T^5 + 1316897640651952T^4 - 19012774974564008T^3 + 6010936611223003040T^2 + 167030325994761639648*T + 12934440817116588145216
$47$	$(T^{5} - 166 T^{4} + \cdots - 485225223264)^{2}$ (T^5 - 166T^4 - 187984T^3 + 21309520T^2 + 7858733040T - 485225223264)^2
$53$	$(T^{5} + \cdots - 2698227619776)^{2}$ (T^5 - 589T^4 - 371877T^3 + 155969729T^2 + 41809791192T - 2698227619776)^2
$59$	$T^{10} + \cdots + 17\!\cdots\!84$ T^10 - 72T^9 + 364868T^8 + 67680032T^7 + 97415521168T^6 + 16071298582016T^5 + 11690231891444224T^4 + 2364845353597636608T^3 + 1014532195061152235520T^2 + 127042850713453235601408*T + 17403787108171374437597184
$61$	$T^{10} + \cdots + 20\!\cdots\!25$ T^10 + 157T^9 + 1086531T^8 + 253472722T^7 + 933672954389T^6 + 197201164340927T^5 + 277965460314970381T^4 + 48650293592592662082T^3 + 61014631805937384491371T^2 + 10291116674153576659527885*T + 2057008578829461524195559225
$67$	$T^{10} + \cdots + 11\!\cdots\!96$ T^10 + 1120T^9 + 871143T^8 + 385281484T^7 + 131383537181T^6 + 27775624868690T^5 + 5182790183550448T^4 + 624020887097474328T^3 + 106797559564779578272T^2 + 9808360806893139282912*T + 1156965707525116781906496
$71$	$T^{10} + \cdots + 44\!\cdots\!00$ T^10 + 898T^9 + 1254708T^8 + 844560448T^7 + 914097477232T^6 + 552512984521056T^5 + 322066820693693376T^4 + 93616951054109195520T^3 + 22144820606530662433536T^2 + 322250671865884626777600*T + 4427086282282241516160000
$73$	$(T^{5} + \cdots - 107667757737387)^{2}$ (T^5 - 235T^4 - 1465762T^3 + 227012374T^2 + 403548940065T - 107667757737387)^2
$79$	$(T^{5} + \cdots - 48889825273536)^{2}$ (T^5 - 74T^4 - 1314351T^3 - 19749316T^2 + 299506426544T - 48889825273536)^2
$83$	$(T^{5} + \cdots - 48722741338752)^{2}$ (T^5 - 1018T^4 - 1045412T^3 + 745044776T^2 + 366167229792T - 48722741338752)^2
$89$	$T^{10} + \cdots + 30\!\cdots\!00$ T^10 + 170T^9 + 3180976T^8 - 506395048T^7 + 7530213063616T^6 - 1325509189010464T^5 + 7684072668842301376T^4 - 3523292780793446874880T^3 + 5789708119710213515008000T^2 - 1332177379885529541192192000*T + 306095262694052883801384960000
$97$	$T^{10} + \cdots + 13\!\cdots\!44$ T^10 - 2620T^9 + 7225639T^8 - 7478595792T^7 + 12267711589453T^6 - 8432839557479422T^5 + 14348055594809221768T^4 - 5462713004178531430296T^3 + 6034028824063599479823712T^2 + 1259916331214034087103386656*T + 1310907333543397169203215252544
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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 289.5
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	624.4.q.m

Level	$624$
Weight	$4$
Character orbit	624.q
Analytic conductor	$36.817$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$