LMFDB - Newform orbit 576.7.b.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(415,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.415");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.b (of order $2$, degree $1$, 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 232 x^{14} + 20234 x^{12} - 800820 x^{10} + 13254755 x^{8} - 52108188 x^{6} + 68921090 x^{4} + \cdots + 6115729 $ x^16 - 232x^14 + 20234x^12 - 800820x^10 + 13254755x^8 - 52108188x^6 + 68921090x^4 + 44919956*x^2 + 6115729
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{114}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{9} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7}+O(q^{10}) $ q + b9 * q^5 + (-b10 - b7) * q^7	\( q + \beta_{9} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7} - \beta_{2} q^{11} - \beta_{5} q^{13} + \beta_{14} q^{17} + ( - \beta_{13} + 3 \beta_{12}) q^{19} + \beta_{6} q^{23} + ( - 5 \beta_1 + 1945) q^{25} + (\beta_{11} - 120 \beta_{9}) q^{29} + (73 \beta_{10} + 52 \beta_{7}) q^{31} + ( - 9 \beta_{3} + 5 \beta_{2}) q^{35} + ( - 53 \beta_{5} - 3 \beta_{4}) q^{37} + (\beta_{15} - 2 \beta_{14}) q^{41} + ( - 12 \beta_{13} + 25 \beta_{12}) q^{43} + ( - \beta_{8} - \beta_{6}) q^{47} + (11 \beta_1 + 4509) q^{49} + ( - 11 \beta_{11} - 782 \beta_{9}) q^{53} + ( - 80 \beta_{10} - 555 \beta_{7}) q^{55} + ( - 11 \beta_{3} + 10 \beta_{2}) q^{59} + (153 \beta_{5} - 55 \beta_{4}) q^{61} + (\beta_{15} + 10 \beta_{14}) q^{65} + ( - 5 \beta_{13} - 122 \beta_{12}) q^{67} + (\beta_{8} + 16 \beta_{6}) q^{71} + ( - 37 \beta_1 + 83998) q^{73} + (47 \beta_{11} + 713 \beta_{9}) q^{77} + ( - 855 \beta_{10} - 4331 \beta_{7}) q^{79} + ( - 26 \beta_{3} - 85 \beta_{2}) q^{83} + (320 \beta_{5} - 305 \beta_{4}) q^{85} + ( - 7 \beta_{15} + 35 \beta_{14}) q^{89} + ( - 199 \beta_{13} - 45 \beta_{12}) q^{91} + (4 \beta_{8} - 17 \beta_{6}) q^{95} + (406 \beta_1 + 157346) q^{97}+O(q^{100}) \) q + b9 * q^5 + (-b10 - b7) * q^7 - b2 * q^11 - b5 * q^13 + b14 * q^17 + (-b13 + 3b12) q^19 + b6 * q^23 + (-5b1 + 1945) q^25 + (b11 - 120b9) q^29 + (73b10 + 52b7) * q^31 + (-9b3 + 5b2) * q^35 + (-53b5 - 3b4) * q^37 + (b15 - 2b14) q^41 + (-12b13 + 25b12) * q^43 + (-b8 - b6) * q^47 + (11b1 + 4509) q^49 + (-11b11 - 782b9) * q^53 + (-80b10 - 555b7) * q^55 + (-11b3 + 10b2) * q^59 + (153b5 - 55b4) * q^61 + (b15 + 10b14) q^65 + (-5b13 - 122b12) * q^67 + (b8 + 16b6) q^71 + (-37b1 + 83998) q^73 + (47b11 + 713b9) * q^77 + (-855b10 - 4331b7) * q^79 + (-26b3 - 85b2) * q^83 + (320b5 - 305b4) * q^85 + (-7b15 + 35b14) * q^89 + (-199b13 - 45b12) * q^91 + (4b8 - 17b6) * q^95 + (406b1 + 157346) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 31120 q^{25} + 72144 q^{49} + 1343968 q^{73} + 2517536 q^{97}+O(q^{100}) $ 16 * q + 31120 * q^25 + 72144 * q^49 + 1343968 * q^73 + 2517536 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 232 x^{14} + 20234 x^{12} - 800820 x^{10} + 13254755 x^{8} - 52108188 x^{6} + 68921090 x^{4} + \cdots + 6115729 $ x^16 - 232*x^14 + 20234*x^12 - 800820*x^10 + 13254755*x^8 - 52108188*x^6 + 68921090*x^4 + 44919956*x^2 + 6115729 :

$\beta_{1}$	$=$	$ ( 9320 \nu^{14} - 1903776 \nu^{12} + 134502760 \nu^{10} - 3508252784 \nu^{8} + \cdots + 312764385045184 ) / 117238332159 $ (9320v^14 - 1903776v^12 + 134502760v^10 - 3508252784v^8 + 14906293976v^6 - 20925587872v^4 - 13434297552*v^2 + 312764385045184) / 117238332159
$\beta_{2}$	$=$	$ ( 1305607 \nu^{14} - 303333565 \nu^{12} + 26263143082 \nu^{10} - 1013710767013 \nu^{8} + \cdots + 121769762940772 ) / 2579243307498 $ (1305607v^14 - 303333565v^12 + 26263143082v^10 - 1013710767013v^8 + 15628255040246v^6 - 41378079392357v^4 - 23617562196541*v^2 + 121769762940772) / 2579243307498
$\beta_{3}$	$=$	$ ( - 69861944 \nu^{14} + 16107894632 \nu^{12} - 1390279825616 \nu^{10} + 53899981709864 \nu^{8} + \cdots - 66\!\cdots\!48 ) / 1289621653749 $ (-69861944v^14 + 16107894632v^12 - 1390279825616v^10 + 53899981709864v^8 - 843901972955440v^6 + 2240556955258408v^4 + 1278355460098664*v^2 - 6615686843193248) / 1289621653749
$\beta_{4}$	$=$	$ ( - 5772544 \nu^{14} + 1340406784 \nu^{12} - 117074628352 \nu^{10} + 4646618341888 \nu^{8} + \cdots - 138755476501504 ) / 16785703935 $ (-5772544v^14 + 1340406784v^12 - 117074628352v^10 + 4646618341888v^8 - 77464684804352v^6 + 316990790497280v^4 - 473032728258560*v^2 - 138755476501504) / 16785703935
$\beta_{5}$	$=$	$ ( 2490873206 \nu^{14} - 578357285816 \nu^{12} + 50511749960738 \nu^{10} + \cdots + 59\!\cdots\!56 ) / 6448108268745 $ (2490873206v^14 - 578357285816v^12 + 50511749960738v^10 - 2004620147717732v^8 + 33416771871971278v^6 - 136741333859409640v^4 + 204051050283014800*v^2 + 59854757112290156) / 6448108268745
$\beta_{6}$	$=$	$ ( 2462304008 \nu^{14} - 572029040328 \nu^{12} + 50000286216064 \nu^{10} + \cdots + 60\!\cdots\!88 ) / 586191660795 $ (2462304008v^14 - 572029040328v^12 + 50000286216064v^10 - 1987218531013496v^8 + 33236550944847584v^6 - 138028059836044600v^4 + 205951409834700600*v^2 + 60405442429416688) / 586191660795
$\beta_{7}$	$=$	$ ( 270868892 \nu^{15} - 62750161080 \nu^{13} + 5459834921596 \nu^{11} + \cdots + 16\!\cdots\!96 \nu ) / 180522699040827 $ (270868892v^15 - 62750161080v^13 + 5459834921596v^11 - 215127334522700v^9 + 3521731625481596v^7 - 13058616834784684v^5 + 15873479597296032v^3 + 16348733697280396v) / 180522699040827
$\beta_{8}$	$=$	$ ( - 105617975744 \nu^{14} + 24534888736704 \nu^{12} + \cdots - 25\!\cdots\!64 ) / 1367780541855 $ (-105617975744v^14 + 24534888736704v^12 - 2144395781496832v^10 + 85219955449752128v^8 - 1425200809905827072v^6 + 5918608970822286400v^4 - 8831031726826889280*v^2 - 2590142459382388864) / 1367780541855
$\beta_{9}$	$=$	$ ( - 8454596961 \nu^{15} + 1962008863420 \nu^{13} - 171188405355101 \nu^{11} + \cdots - 53\!\cdots\!79 \nu ) / 11\!\cdots\!28 $ (-8454596961v^15 + 1962008863420v^13 - 171188405355101v^11 + 6779796927484845v^9 - 112343442048404869v^7 + 441766141264894449v^5 - 519147228799424864v^3 - 538942759115695379v) / 1159721581716828
$\beta_{10}$	$=$	$ ( - 192222556315 \nu^{15} + 44534136934512 \nu^{13} + \cdots - 11\!\cdots\!49 \nu ) / 12\!\cdots\!08 $ (-192222556315v^15 + 44534136934512v^13 - 3875204525010533v^11 + 152704323982907635v^9 - 2500066064437497493v^7 + 9270450356898545843v^5 - 11268949314190957266v^3 - 11606281769237451149v) / 12756937398885108
$\beta_{11}$	$=$	$ ( 37580846697 \nu^{15} - 8723769432220 \nu^{13} + 761432916740933 \nu^{11} + \cdots + 23\!\cdots\!35 \nu ) / 386573860572276 $ (37580846697v^15 - 8723769432220v^13 + 761432916740933v^11 - 30168185340468501v^9 + 500094876945033133v^7 - 1966683670387100025v^5 + 2311340605319343200v^3 + 2399420719928747435v) / 386573860572276
$\beta_{12}$	$=$	$ ( - 3740201646618 \nu^{15} + 868627661575298 \nu^{13} + \cdots - 89\!\cdots\!28 \nu ) / 33\!\cdots\!15 $ (-3740201646618v^15 + 868627661575298v^13 - 75888444887109304v^11 + 3013507340851845846v^9 - 50301655863730088024v^7 + 207035837951049116910v^5 - 308193215883471150550v^3 - 89984309401951354228v) / 3382521280007415
$\beta_{13}$	$=$	$ ( 51853955456 \nu^{15} - 12042604060416 \nu^{13} + \cdots + 12\!\cdots\!96 \nu ) / 38005857078735 $ (51853955456v^15 - 12042604060416v^13 + 1052113459339648v^11 - 41779169188917632v^9 + 697382434501505408v^7 - 2870347769183850880v^5 + 4272797054150568960v^3 + 1247544214700864896v) / 38005857078735
$\beta_{14}$	$=$	$ ( 23466126321164 \nu^{15} + \cdots + 56\!\cdots\!84 \nu ) / 85\!\cdots\!15 $ (23466126321164v^15 - 5450259220435224v^13 + 476234486146083412v^11 - 18916284050471423948v^9 + 315967153508127364052v^7 - 1305020418420140686300v^5 + 1953914707995829788120v^3 + 569984225664415469884v) / 8586400172326515
$\beta_{15}$	$=$	$ ( 725429925552716 \nu^{15} + \cdots + 17\!\cdots\!56 \nu ) / 74\!\cdots\!13 $ (725429925552716v^15 - 168489079684030872v^13 + 14722317331536631828v^11 - 584779567239545633804v^9 + 9767866048531850963348v^7 - 40343667670617728563996v^5 + 60403750205192485300824v^3 + 17620616559756449874556v) / 7441546816016313

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

$\nu$	$=$	$ ( \beta_{15} - 27\beta_{14} - 18\beta_{13} + 576\beta_{7} ) / 36864 $ (b15 - 27b14 - 18b13 + 576*b7) / 36864
$\nu^{2}$	$=$	$ ( -\beta_{8} - 24\beta_{6} - 72\beta_{4} + 24\beta_{3} + 384\beta_{2} - 768\beta _1 + 2138112 ) / 73728 $ (-b8 - 24b6 - 72b4 + 24b3 + 384b2 - 768*b1 + 2138112) / 73728
$\nu^{3}$	$=$	$ ( 31 \beta_{15} - 1093 \beta_{14} - 648 \beta_{13} - 768 \beta_{12} + 24 \beta_{11} + \cdots + 32448 \beta_{7} ) / 24576 $ (31b15 - 1093b14 - 648b13 - 768b12 + 24b11 + 3072b10 + 648b9 + 32448b7) / 24576
$\nu^{4}$	$=$	$ ( - 71 \beta_{8} - 1320 \beta_{6} - 4608 \beta_{5} - 5364 \beta_{4} + 216 \beta_{3} + 21888 \beta_{2} + \cdots + 61544448 ) / 36864 $ (-71b8 - 1320b6 - 4608b5 - 5364b4 + 216b3 + 21888b2 - 23040*b1 + 61544448) / 36864
$\nu^{5}$	$=$	$ ( 5495 \beta_{15} - 195981 \beta_{14} - 177444 \beta_{13} - 218880 \beta_{12} + \cdots + 9433152 \beta_{7} ) / 73728 $ (5495b15 - 195981b14 - 177444b13 - 218880b12 + 13800b11 + 936960b10 + 188280b9 + 9433152b7) / 73728
$\nu^{6}$	$=$	$ ( - 2177 \beta_{8} - 40024 \beta_{6} - 222720 \beta_{5} - 250320 \beta_{4} + 5664 \beta_{3} + \cdots + 1239189504 ) / 12288 $ (-2177b8 - 40024b6 - 222720b5 - 250320b4 + 5664b3 + 615936b2 - 464512*b1 + 1239189504) / 12288
$\nu^{7}$	$=$	$ ( 167843 \beta_{15} - 5987121 \beta_{14} - 6983910 \beta_{13} - 8617728 \beta_{12} + \cdots + 408541824 \beta_{7} ) / 36864 $ (167843b15 - 5987121b14 - 6983910b13 - 8617728b12 + 832608b11 + 40664064b10 + 11126304b9 + 408541824b7) / 36864
$\nu^{8}$	$=$	$ ( - 91375 \beta_{8} - 1679208 \beta_{6} - 12011776 \beta_{5} - 13488770 \beta_{4} + 212296 \beta_{3} + \cdots + 38165885952 ) / 6144 $ (-91375b8 - 1679208b6 - 12011776b5 - 13488770b4 + 212296b3 + 23069824b2 - 14306304*b1 + 38165885952) / 6144
$\nu^{9}$	$=$	$ ( 20791031 \beta_{15} - 741622797 \beta_{14} - 1007367192 \beta_{13} - 1242991872 \beta_{12} + \cdots + 67029627456 \beta_{7} ) / 73728 $ (20791031b15 - 741622797b14 - 1007367192b13 - 1242991872b12 + 165770712b11 + 6672245760b10 + 2214092232b9 + 67029627456b7) / 73728
$\nu^{10}$	$=$	$ ( - 88459873 \beta_{8} - 1625616024 \beta_{6} - 13520378880 \beta_{5} - 15182578008 \beta_{4} + \cdots + 28444308221952 ) / 73728 $ (-88459873b8 - 1625616024b6 - 13520378880b5 - 15182578008b4 + 178237128b3 + 19365446784b2 - 10662163968*b1 + 28444308221952) / 73728
$\nu^{11}$	$=$	$ ( 430469787 \beta_{15} - 15354966361 \beta_{14} - 22893266100 \beta_{13} + \cdots + 1778823414336 \beta_{7} ) / 24576 $ (430469787b15 - 15354966361b14 - 22893266100b13 - 28247952128b12 + 4950593032b11 + 177067478016b10 + 66121210584b9 + 1778823414336b7) / 24576
$\nu^{12}$	$=$	$ ( - 870634667 \beta_{8} - 15999541224 \beta_{6} - 146007290880 \beta_{5} - 163957424400 \beta_{4} + \cdots + 220299053073408 ) / 9216 $ (-870634667b8 - 15999541224b6 - 146007290880b5 - 163957424400b4 + 1484951148b3 + 161337947328b2 - 82577636352*b1 + 220299053073408) / 9216
$\nu^{13}$	$=$	$ ( 39805369423 \beta_{15} - 1419867438933 \beta_{14} - 2241442303794 \beta_{13} + \cdots + 208054646542656 \beta_{7} ) / 36864 $ (39805369423b15 - 1419867438933b14 - 2241442303794b13 - 2765710411776b12 + 622894642032b11 + 20710156726272b10 + 8319517826256b9 + 208054646542656b7) / 36864
$\nu^{14}$	$=$	$ ( - 179432333197 \beta_{8} - 3297404915000 \beta_{6} - 31872337788928 \beta_{5} + \cdots + 35\!\cdots\!12 ) / 24576 $ (-179432333197b8 - 3297404915000b6 - 31872337788928b5 - 35790722659352b4 + 253325192504b3 + 27523426505600b2 - 13471086297344*b1 + 35937909400461312) / 24576
$\nu^{15}$	$=$	$ ( 4819586318665 \beta_{15} - 171915843143091 \beta_{14} - 281052584425656 \beta_{13} + \cdots + 31\!\cdots\!24 \beta_{7} ) / 73728 $ (4819586318665b15 - 171915843143091b14 - 281052584425656b13 - 346790123617536b12 + 99906820146792b11 + 3174633498080256b10 + 1334377477516536b9 + 31892431366165824b7) / 73728

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$-1$

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} $

415.1

−1.70900	−	0.500000i
1.70900	−	0.500000i
0.0230464	+	0.500000i
−0.0230464	+	0.500000i
6.63660	+	0.500000i
−6.63660	+	0.500000i
8.36865	−	0.500000i
−8.36865	−	0.500000i
−8.36865	+	0.500000i
8.36865	+	0.500000i
−6.63660	−	0.500000i
6.63660	−	0.500000i
−0.0230464	−	0.500000i
0.0230464	−	0.500000i
1.70900	+	0.500000i
−1.70900	+	0.500000i

−

164.374i

−

289.473i

415.2

−

164.374i

−

289.473i

415.3

−

164.374i

289.473i

415.4

−

164.374i

289.473i

415.5

−

18.4687i

−

377.473i

415.6

−

18.4687i

−

377.473i

415.7

−

18.4687i

377.473i

415.8

−

18.4687i

377.473i

415.9

18.4687i

−

377.473i

415.10

18.4687i

−

377.473i

415.11

18.4687i

377.473i

415.12

18.4687i

377.473i

415.13

164.374i

−

289.473i

415.14

164.374i

−

289.473i

415.15

164.374i

289.473i

415.16

164.374i

289.473i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
12.b	even	2	1	inner
24.f	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.7.b.h	✓	16
3.b	odd	2	1	inner	576.7.b.h	✓	16
4.b	odd	2	1	inner	576.7.b.h	✓	16
8.b	even	2	1	inner	576.7.b.h	✓	16
8.d	odd	2	1	inner	576.7.b.h	✓	16
12.b	even	2	1	inner	576.7.b.h	✓	16
24.f	even	2	1	inner	576.7.b.h	✓	16
24.h	odd	2	1	inner	576.7.b.h	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
576.7.b.h	✓	16	1.a	even	1	1	trivial
576.7.b.h	✓	16	3.b	odd	2	1	inner
576.7.b.h	✓	16	4.b	odd	2	1	inner
576.7.b.h	✓	16	8.b	even	2	1	inner
576.7.b.h	✓	16	8.d	odd	2	1	inner
576.7.b.h	✓	16	12.b	even	2	1	inner
576.7.b.h	✓	16	24.f	even	2	1	inner
576.7.b.h	✓	16	24.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} + 27360T_{5}^{2} + 9216000 $ T5^4 + 27360*T5^2 + 9216000 acting on $S_{7}^{\mathrm{new}}(576, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{4} + 27360 T^{2} + 9216000)^{4} $ (T^4 + 27360*T^2 + 9216000)^4
$7$	$ (T^{4} + 226280 T^{2} + 11939495824)^{4} $ (T^4 + 226280*T^2 + 11939495824)^4
$11$	$ (T^{4} - 6044544 T^{2} + 14161674240)^{4} $ (T^4 - 6044544*T^2 + 14161674240)^4
$13$	$ (T^{4} + \cdots + 1713627611136)^{4} $ (T^4 + 2719680*T^2 + 1713627611136)^4
$17$	$ (T^{4} + \cdots + 849346560000000)^{4} $ (T^4 - 78962688*T^2 + 849346560000000)^4
$19$	$ (T^{4} + \cdots + 22\!\cdots\!16)^{4} $ (T^4 - 97509120*T^2 + 2239721501884416)^4
$23$	$ (T^{4} + \cdots + 40\!\cdots\!60)^{4} $ (T^4 + 457408512*T^2 + 40954757287772160)^4
$29$	$ (T^{4} + \cdots + 14\!\cdots\!40)^{4} $ (T^4 + 777901536*T^2 + 142901252746813440)^4
$31$	$ (T^{4} + \cdots + 34\!\cdots\!56)^{4} $ (T^4 + 1198115432*T^2 + 343577116627194256)^4
$37$	$ (T^{4} + \cdots + 13\!\cdots\!00)^{4} $ (T^4 + 7743611328*T^2 + 13141518215178240000)^4
$41$	$ (T^{4} + \cdots + 78\!\cdots\!40)^{4} $ (T^4 - 19935977472*T^2 + 78241409434008944640)^4
$43$	$ (T^{4} + \cdots + 11\!\cdots\!24)^{4} $ (T^4 - 6672824064*T^2 + 11127748232091009024)^4
$47$	$ (T^{4} + \cdots + 64\!\cdots\!40)^{4} $ (T^4 + 54320136192*T^2 + 643035942575816048640)^4
$53$	$ (T^{4} + \cdots + 56\!\cdots\!60)^{4} $ (T^4 + 63118727136*T^2 + 569080725719367720960)^4
$59$	$ (T^{4} + \cdots + 11\!\cdots\!00)^{4} $ (T^4 - 3730890240*T^2 + 1112710263865344000)^4
$61$	$ (T^{4} + \cdots + 93\!\cdots\!96)^{4} $ (T^4 + 63666340800*T^2 + 938994433498562052096)^4
$67$	$ (T^{4} + \cdots + 43\!\cdots\!96)^{4} $ (T^4 - 185515146240*T^2 + 4374211120066247786496)^4
$71$	$ (T^{4} + \cdots + 11\!\cdots\!40)^{4} $ (T^4 + 166075564032*T^2 + 114143424692449443840)^4
$73$	$ (T^{2} - 167996 T - 2687585660)^{8} $ (T^2 - 167996*T - 2687585660)^8
$79$	$ (T^{4} + \cdots + 34\!\cdots\!16)^{4} $ (T^4 + 206899644008*T^2 + 3497064863373502550416)^4
$83$	$ (T^{4} + \cdots + 88\!\cdots\!00)^{4} $ (T^4 - 67906592640*T^2 + 885216636374630400000)^4
$89$	$ (T^{4} + \cdots + 27\!\cdots\!00)^{4} $ (T^4 - 1059741204480*T^2 + 276888509482033741824000)^4
$97$	$ (T^{2} + \cdots - 1148389279100)^{8} $ (T^2 - 314692*T - 1148389279100)^8
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7}+O(q^{10}) \) q + b9 * q^5 + (-b10 - b7) * q^7	\( q + \beta_{9} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7} - \beta_{2} q^{11} - \beta_{5} q^{13} + \beta_{14} q^{17} + ( - \beta_{13} + 3 \beta_{12}) q^{19} + \beta_{6} q^{23} + ( - 5 \beta_1 + 1945) q^{25} + (\beta_{11} - 120 \beta_{9}) q^{29} + (73 \beta_{10} + 52 \beta_{7}) q^{31} + ( - 9 \beta_{3} + 5 \beta_{2}) q^{35} + ( - 53 \beta_{5} - 3 \beta_{4}) q^{37} + (\beta_{15} - 2 \beta_{14}) q^{41} + ( - 12 \beta_{13} + 25 \beta_{12}) q^{43} + ( - \beta_{8} - \beta_{6}) q^{47} + (11 \beta_1 + 4509) q^{49} + ( - 11 \beta_{11} - 782 \beta_{9}) q^{53} + ( - 80 \beta_{10} - 555 \beta_{7}) q^{55} + ( - 11 \beta_{3} + 10 \beta_{2}) q^{59} + (153 \beta_{5} - 55 \beta_{4}) q^{61} + (\beta_{15} + 10 \beta_{14}) q^{65} + ( - 5 \beta_{13} - 122 \beta_{12}) q^{67} + (\beta_{8} + 16 \beta_{6}) q^{71} + ( - 37 \beta_1 + 83998) q^{73} + (47 \beta_{11} + 713 \beta_{9}) q^{77} + ( - 855 \beta_{10} - 4331 \beta_{7}) q^{79} + ( - 26 \beta_{3} - 85 \beta_{2}) q^{83} + (320 \beta_{5} - 305 \beta_{4}) q^{85} + ( - 7 \beta_{15} + 35 \beta_{14}) q^{89} + ( - 199 \beta_{13} - 45 \beta_{12}) q^{91} + (4 \beta_{8} - 17 \beta_{6}) q^{95} + (406 \beta_1 + 157346) q^{97}+O(q^{100}) \) q + b9 * q^5 + (-b10 - b7) * q^7 - b2 * q^11 - b5 * q^13 + b14 * q^17 + (-b13 + 3b12) q^19 + b6 * q^23 + (-5b1 + 1945) q^25 + (b11 - 120b9) q^29 + (73b10 + 52b7) * q^31 + (-9b3 + 5b2) * q^35 + (-53b5 - 3b4) * q^37 + (b15 - 2b14) q^41 + (-12b13 + 25b12) * q^43 + (-b8 - b6) * q^47 + (11b1 + 4509) q^49 + (-11b11 - 782b9) * q^53 + (-80b10 - 555b7) * q^55 + (-11b3 + 10b2) * q^59 + (153b5 - 55b4) * q^61 + (b15 + 10b14) q^65 + (-5b13 - 122b12) * q^67 + (b8 + 16b6) q^71 + (-37b1 + 83998) q^73 + (47b11 + 713b9) * q^77 + (-855b10 - 4331b7) * q^79 + (-26b3 - 85b2) * q^83 + (320b5 - 305b4) * q^85 + (-7b15 + 35b14) * q^89 + (-199b13 - 45b12) * q^91 + (4b8 - 17b6) * q^95 + (406b1 + 157346) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 31120 q^{25} + 72144 q^{49} + 1343968 q^{73} + 2517536 q^{97}+O(q^{100}) \) 16 * q + 31120 * q^25 + 72144 * q^49 + 1343968 * q^73 + 2517536 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	576.7.b.h

Level	$576$
Weight	$7$
Character orbit	576.b
Analytic conductor	$132.511$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(132.511152165\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 232 x^{14} + 20234 x^{12} - 800820 x^{10} + 13254755 x^{8} - 52108188 x^{6} + 68921090 x^{4} + \cdots + 6115729 \) x^16 - 232x^14 + 20234x^12 - 800820x^10 + 13254755x^8 - 52108188x^6 + 68921090x^4 + 44919956*x^2 + 6115729
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{114}\cdot 3^{12} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 9320 \nu^{14} - 1903776 \nu^{12} + 134502760 \nu^{10} - 3508252784 \nu^{8} + \cdots + 312764385045184 ) / 117238332159 \) (9320v^14 - 1903776v^12 + 134502760v^10 - 3508252784v^8 + 14906293976v^6 - 20925587872v^4 - 13434297552*v^2 + 312764385045184) / 117238332159
\(\beta_{2}\)	\(=\)	\( ( 1305607 \nu^{14} - 303333565 \nu^{12} + 26263143082 \nu^{10} - 1013710767013 \nu^{8} + \cdots + 121769762940772 ) / 2579243307498 \) (1305607v^14 - 303333565v^12 + 26263143082v^10 - 1013710767013v^8 + 15628255040246v^6 - 41378079392357v^4 - 23617562196541*v^2 + 121769762940772) / 2579243307498
\(\beta_{3}\)	\(=\)	\( ( - 69861944 \nu^{14} + 16107894632 \nu^{12} - 1390279825616 \nu^{10} + 53899981709864 \nu^{8} + \cdots - 66\!\cdots\!48 ) / 1289621653749 \) (-69861944v^14 + 16107894632v^12 - 1390279825616v^10 + 53899981709864v^8 - 843901972955440v^6 + 2240556955258408v^4 + 1278355460098664*v^2 - 6615686843193248) / 1289621653749
\(\beta_{4}\)	\(=\)	\( ( - 5772544 \nu^{14} + 1340406784 \nu^{12} - 117074628352 \nu^{10} + 4646618341888 \nu^{8} + \cdots - 138755476501504 ) / 16785703935 \) (-5772544v^14 + 1340406784v^12 - 117074628352v^10 + 4646618341888v^8 - 77464684804352v^6 + 316990790497280v^4 - 473032728258560*v^2 - 138755476501504) / 16785703935
\(\beta_{5}\)	\(=\)	\( ( 2490873206 \nu^{14} - 578357285816 \nu^{12} + 50511749960738 \nu^{10} + \cdots + 59\!\cdots\!56 ) / 6448108268745 \) (2490873206v^14 - 578357285816v^12 + 50511749960738v^10 - 2004620147717732v^8 + 33416771871971278v^6 - 136741333859409640v^4 + 204051050283014800*v^2 + 59854757112290156) / 6448108268745
\(\beta_{6}\)	\(=\)	\( ( 2462304008 \nu^{14} - 572029040328 \nu^{12} + 50000286216064 \nu^{10} + \cdots + 60\!\cdots\!88 ) / 586191660795 \) (2462304008v^14 - 572029040328v^12 + 50000286216064v^10 - 1987218531013496v^8 + 33236550944847584v^6 - 138028059836044600v^4 + 205951409834700600*v^2 + 60405442429416688) / 586191660795
\(\beta_{7}\)	\(=\)	\( ( 270868892 \nu^{15} - 62750161080 \nu^{13} + 5459834921596 \nu^{11} + \cdots + 16\!\cdots\!96 \nu ) / 180522699040827 \) (270868892v^15 - 62750161080v^13 + 5459834921596v^11 - 215127334522700v^9 + 3521731625481596v^7 - 13058616834784684v^5 + 15873479597296032v^3 + 16348733697280396v) / 180522699040827
\(\beta_{8}\)	\(=\)	\( ( - 105617975744 \nu^{14} + 24534888736704 \nu^{12} + \cdots - 25\!\cdots\!64 ) / 1367780541855 \) (-105617975744v^14 + 24534888736704v^12 - 2144395781496832v^10 + 85219955449752128v^8 - 1425200809905827072v^6 + 5918608970822286400v^4 - 8831031726826889280*v^2 - 2590142459382388864) / 1367780541855
\(\beta_{9}\)	\(=\)	\( ( - 8454596961 \nu^{15} + 1962008863420 \nu^{13} - 171188405355101 \nu^{11} + \cdots - 53\!\cdots\!79 \nu ) / 11\!\cdots\!28 \) (-8454596961v^15 + 1962008863420v^13 - 171188405355101v^11 + 6779796927484845v^9 - 112343442048404869v^7 + 441766141264894449v^5 - 519147228799424864v^3 - 538942759115695379v) / 1159721581716828
\(\beta_{10}\)	\(=\)	\( ( - 192222556315 \nu^{15} + 44534136934512 \nu^{13} + \cdots - 11\!\cdots\!49 \nu ) / 12\!\cdots\!08 \) (-192222556315v^15 + 44534136934512v^13 - 3875204525010533v^11 + 152704323982907635v^9 - 2500066064437497493v^7 + 9270450356898545843v^5 - 11268949314190957266v^3 - 11606281769237451149v) / 12756937398885108
\(\beta_{11}\)	\(=\)	\( ( 37580846697 \nu^{15} - 8723769432220 \nu^{13} + 761432916740933 \nu^{11} + \cdots + 23\!\cdots\!35 \nu ) / 386573860572276 \) (37580846697v^15 - 8723769432220v^13 + 761432916740933v^11 - 30168185340468501v^9 + 500094876945033133v^7 - 1966683670387100025v^5 + 2311340605319343200v^3 + 2399420719928747435v) / 386573860572276
\(\beta_{12}\)	\(=\)	\( ( - 3740201646618 \nu^{15} + 868627661575298 \nu^{13} + \cdots - 89\!\cdots\!28 \nu ) / 33\!\cdots\!15 \) (-3740201646618v^15 + 868627661575298v^13 - 75888444887109304v^11 + 3013507340851845846v^9 - 50301655863730088024v^7 + 207035837951049116910v^5 - 308193215883471150550v^3 - 89984309401951354228v) / 3382521280007415
\(\beta_{13}\)	\(=\)	\( ( 51853955456 \nu^{15} - 12042604060416 \nu^{13} + \cdots + 12\!\cdots\!96 \nu ) / 38005857078735 \) (51853955456v^15 - 12042604060416v^13 + 1052113459339648v^11 - 41779169188917632v^9 + 697382434501505408v^7 - 2870347769183850880v^5 + 4272797054150568960v^3 + 1247544214700864896v) / 38005857078735
\(\beta_{14}\)	\(=\)	\( ( 23466126321164 \nu^{15} + \cdots + 56\!\cdots\!84 \nu ) / 85\!\cdots\!15 \) (23466126321164v^15 - 5450259220435224v^13 + 476234486146083412v^11 - 18916284050471423948v^9 + 315967153508127364052v^7 - 1305020418420140686300v^5 + 1953914707995829788120v^3 + 569984225664415469884v) / 8586400172326515
\(\beta_{15}\)	\(=\)	\( ( 725429925552716 \nu^{15} + \cdots + 17\!\cdots\!56 \nu ) / 74\!\cdots\!13 \) (725429925552716v^15 - 168489079684030872v^13 + 14722317331536631828v^11 - 584779567239545633804v^9 + 9767866048531850963348v^7 - 40343667670617728563996v^5 + 60403750205192485300824v^3 + 17620616559756449874556v) / 7441546816016313

\(\nu\)	\(=\)	\( ( \beta_{15} - 27\beta_{14} - 18\beta_{13} + 576\beta_{7} ) / 36864 \) (b15 - 27b14 - 18b13 + 576*b7) / 36864
\(\nu^{2}\)	\(=\)	\( ( -\beta_{8} - 24\beta_{6} - 72\beta_{4} + 24\beta_{3} + 384\beta_{2} - 768\beta _1 + 2138112 ) / 73728 \) (-b8 - 24b6 - 72b4 + 24b3 + 384b2 - 768*b1 + 2138112) / 73728
\(\nu^{3}\)	\(=\)	\( ( 31 \beta_{15} - 1093 \beta_{14} - 648 \beta_{13} - 768 \beta_{12} + 24 \beta_{11} + \cdots + 32448 \beta_{7} ) / 24576 \) (31b15 - 1093b14 - 648b13 - 768b12 + 24b11 + 3072b10 + 648b9 + 32448b7) / 24576
\(\nu^{4}\)	\(=\)	\( ( - 71 \beta_{8} - 1320 \beta_{6} - 4608 \beta_{5} - 5364 \beta_{4} + 216 \beta_{3} + 21888 \beta_{2} + \cdots + 61544448 ) / 36864 \) (-71b8 - 1320b6 - 4608b5 - 5364b4 + 216b3 + 21888b2 - 23040*b1 + 61544448) / 36864
\(\nu^{5}\)	\(=\)	\( ( 5495 \beta_{15} - 195981 \beta_{14} - 177444 \beta_{13} - 218880 \beta_{12} + \cdots + 9433152 \beta_{7} ) / 73728 \) (5495b15 - 195981b14 - 177444b13 - 218880b12 + 13800b11 + 936960b10 + 188280b9 + 9433152b7) / 73728
\(\nu^{6}\)	\(=\)	\( ( - 2177 \beta_{8} - 40024 \beta_{6} - 222720 \beta_{5} - 250320 \beta_{4} + 5664 \beta_{3} + \cdots + 1239189504 ) / 12288 \) (-2177b8 - 40024b6 - 222720b5 - 250320b4 + 5664b3 + 615936b2 - 464512*b1 + 1239189504) / 12288
\(\nu^{7}\)	\(=\)	\( ( 167843 \beta_{15} - 5987121 \beta_{14} - 6983910 \beta_{13} - 8617728 \beta_{12} + \cdots + 408541824 \beta_{7} ) / 36864 \) (167843b15 - 5987121b14 - 6983910b13 - 8617728b12 + 832608b11 + 40664064b10 + 11126304b9 + 408541824b7) / 36864
\(\nu^{8}\)	\(=\)	\( ( - 91375 \beta_{8} - 1679208 \beta_{6} - 12011776 \beta_{5} - 13488770 \beta_{4} + 212296 \beta_{3} + \cdots + 38165885952 ) / 6144 \) (-91375b8 - 1679208b6 - 12011776b5 - 13488770b4 + 212296b3 + 23069824b2 - 14306304*b1 + 38165885952) / 6144
\(\nu^{9}\)	\(=\)	\( ( 20791031 \beta_{15} - 741622797 \beta_{14} - 1007367192 \beta_{13} - 1242991872 \beta_{12} + \cdots + 67029627456 \beta_{7} ) / 73728 \) (20791031b15 - 741622797b14 - 1007367192b13 - 1242991872b12 + 165770712b11 + 6672245760b10 + 2214092232b9 + 67029627456b7) / 73728
\(\nu^{10}\)	\(=\)	\( ( - 88459873 \beta_{8} - 1625616024 \beta_{6} - 13520378880 \beta_{5} - 15182578008 \beta_{4} + \cdots + 28444308221952 ) / 73728 \) (-88459873b8 - 1625616024b6 - 13520378880b5 - 15182578008b4 + 178237128b3 + 19365446784b2 - 10662163968*b1 + 28444308221952) / 73728
\(\nu^{11}\)	\(=\)	\( ( 430469787 \beta_{15} - 15354966361 \beta_{14} - 22893266100 \beta_{13} + \cdots + 1778823414336 \beta_{7} ) / 24576 \) (430469787b15 - 15354966361b14 - 22893266100b13 - 28247952128b12 + 4950593032b11 + 177067478016b10 + 66121210584b9 + 1778823414336b7) / 24576
\(\nu^{12}\)	\(=\)	\( ( - 870634667 \beta_{8} - 15999541224 \beta_{6} - 146007290880 \beta_{5} - 163957424400 \beta_{4} + \cdots + 220299053073408 ) / 9216 \) (-870634667b8 - 15999541224b6 - 146007290880b5 - 163957424400b4 + 1484951148b3 + 161337947328b2 - 82577636352*b1 + 220299053073408) / 9216
\(\nu^{13}\)	\(=\)	\( ( 39805369423 \beta_{15} - 1419867438933 \beta_{14} - 2241442303794 \beta_{13} + \cdots + 208054646542656 \beta_{7} ) / 36864 \) (39805369423b15 - 1419867438933b14 - 2241442303794b13 - 2765710411776b12 + 622894642032b11 + 20710156726272b10 + 8319517826256b9 + 208054646542656b7) / 36864
\(\nu^{14}\)	\(=\)	\( ( - 179432333197 \beta_{8} - 3297404915000 \beta_{6} - 31872337788928 \beta_{5} + \cdots + 35\!\cdots\!12 ) / 24576 \) (-179432333197b8 - 3297404915000b6 - 31872337788928b5 - 35790722659352b4 + 253325192504b3 + 27523426505600b2 - 13471086297344*b1 + 35937909400461312) / 24576
\(\nu^{15}\)	\(=\)	\( ( 4819586318665 \beta_{15} - 171915843143091 \beta_{14} - 281052584425656 \beta_{13} + \cdots + 31\!\cdots\!24 \beta_{7} ) / 73728 \) (4819586318665b15 - 171915843143091b14 - 281052584425656b13 - 346790123617536b12 + 99906820146792b11 + 3174633498080256b10 + 1334377477516536b9 + 31892431366165824b7) / 73728

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 415.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{4} + 27360 T^{2} + 9216000)^{4} \) (T^4 + 27360*T^2 + 9216000)^4
$7$	\( (T^{4} + 226280 T^{2} + 11939495824)^{4} \) (T^4 + 226280*T^2 + 11939495824)^4
$11$	\( (T^{4} - 6044544 T^{2} + 14161674240)^{4} \) (T^4 - 6044544*T^2 + 14161674240)^4
$13$	\( (T^{4} + \cdots + 1713627611136)^{4} \) (T^4 + 2719680*T^2 + 1713627611136)^4
$17$	\( (T^{4} + \cdots + 849346560000000)^{4} \) (T^4 - 78962688*T^2 + 849346560000000)^4
$19$	\( (T^{4} + \cdots + 22\!\cdots\!16)^{4} \) (T^4 - 97509120*T^2 + 2239721501884416)^4
$23$	\( (T^{4} + \cdots + 40\!\cdots\!60)^{4} \) (T^4 + 457408512*T^2 + 40954757287772160)^4
$29$	\( (T^{4} + \cdots + 14\!\cdots\!40)^{4} \) (T^4 + 777901536*T^2 + 142901252746813440)^4
$31$	\( (T^{4} + \cdots + 34\!\cdots\!56)^{4} \) (T^4 + 1198115432*T^2 + 343577116627194256)^4
$37$	\( (T^{4} + \cdots + 13\!\cdots\!00)^{4} \) (T^4 + 7743611328*T^2 + 13141518215178240000)^4
$41$	\( (T^{4} + \cdots + 78\!\cdots\!40)^{4} \) (T^4 - 19935977472*T^2 + 78241409434008944640)^4
$43$	\( (T^{4} + \cdots + 11\!\cdots\!24)^{4} \) (T^4 - 6672824064*T^2 + 11127748232091009024)^4
$47$	\( (T^{4} + \cdots + 64\!\cdots\!40)^{4} \) (T^4 + 54320136192*T^2 + 643035942575816048640)^4
$53$	\( (T^{4} + \cdots + 56\!\cdots\!60)^{4} \) (T^4 + 63118727136*T^2 + 569080725719367720960)^4
$59$	\( (T^{4} + \cdots + 11\!\cdots\!00)^{4} \) (T^4 - 3730890240*T^2 + 1112710263865344000)^4
$61$	\( (T^{4} + \cdots + 93\!\cdots\!96)^{4} \) (T^4 + 63666340800*T^2 + 938994433498562052096)^4
$67$	\( (T^{4} + \cdots + 43\!\cdots\!96)^{4} \) (T^4 - 185515146240*T^2 + 4374211120066247786496)^4
$71$	\( (T^{4} + \cdots + 11\!\cdots\!40)^{4} \) (T^4 + 166075564032*T^2 + 114143424692449443840)^4
$73$	\( (T^{2} - 167996 T - 2687585660)^{8} \) (T^2 - 167996*T - 2687585660)^8
$79$	\( (T^{4} + \cdots + 34\!\cdots\!16)^{4} \) (T^4 + 206899644008*T^2 + 3497064863373502550416)^4
$83$	\( (T^{4} + \cdots + 88\!\cdots\!00)^{4} \) (T^4 - 67906592640*T^2 + 885216636374630400000)^4
$89$	\( (T^{4} + \cdots + 27\!\cdots\!00)^{4} \) (T^4 - 1059741204480*T^2 + 276888509482033741824000)^4
$97$	\( (T^{2} + \cdots - 1148389279100)^{8} \) (T^2 - 314692*T - 1148389279100)^8
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\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)