LMFDB - Newform orbit 1170.2.o.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(53,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1170.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.o (of order $4$, degree $2$, 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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 12 x^{10} - 16 x^{9} + 8 x^{8} - 8 x^{7} + 18 x^{6} + 32 x^{5} + 16 x^{4} + 36 x^{3} + \cdots + 2 $ x^12 - 4x^11 + 12x^10 - 16x^9 + 8x^8 - 8x^7 + 18x^6 + 32x^5 + 16x^4 + 36x^3 + 80x^2 - 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{2} + \beta_{9} q^{4} + (\beta_{11} + \beta_{4} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{10} - \beta_{8} + \cdots + 2 \beta_1) q^{7}+ \cdots + \beta_{7} q^{8}+O(q^{10}) $ q + b6 * q^2 + b9 * q^4 + (b11 + b4 + b3 + b1) * q^5 + (-b10 - b8 - b7 + b3 - b2 + 2b1) q^7 + b7 * q^8	$ q + \beta_{6} q^{2} + \beta_{9} q^{4} + (\beta_{11} + \beta_{4} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{10} - \beta_{8} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{9} - 3 \beta_{8} + \cdots - 1) q^{98}+O(q^{100}) $ q + b6 * q^2 + b9 * q^4 + (b11 + b4 + b3 + b1) * q^5 + (-b10 - b8 - b7 + b3 - b2 + 2b1) q^7 + b7 * q^8 - b5 * q^10 + (b11 - b10 + b5 + b3 - b2 + b1) * q^11 + b6 * q^13 + (b11 + b10 - b3 + 1) * q^14 - q^16 + (2b8 - b6 + b4 + 2b1) * q^17 + (b9 - b8 - b7 - b6 - b4 + b3) * q^19 + (b4 - b2 + b1) * q^20 + (b11 + b10 + b8 - b5 + b4 + b2) * q^22 + (b11 + b9 + 2b8 - b7 - b5 + b4 - b3 - 1) q^23 + (-2b9 + b8 + b7 - 3b6 - b5 + b4 - b3 + b2 + b1 + 1) * q^25 + b9 * q^26 + (-b11 + b8 + b6 - b5 + b1) * q^28 + (b11 + b10 + b7 - b6 + b4 + 3) * q^29 + (-b11 - b10 + b3 - b1 - 2) * q^31 - b6 * q^32 + (-b9 - b8 - 2b4 + 2b3) * q^34 + (b11 + b10 - b9 + 4b8 - b5 + b4 - 2b3 + b2 - 2) * q^35 + (-b9 + b8 + 2b7 + 2b3 - b1 + 1) * q^37 + (-b9 + b8 + b7 - b3 - b1 + 1) * q^38 + (b10 - b8) * q^40 + (2b11 - 2b10 + 2b9 + 2b8 - b7 - b6 + b5 + 3b4 - b3 - b2 + 2b1) * q^41 + (2b11 + b10 + b9 - b8 + 4b6 + 2b5 - 2b4 - b2 + 1) * q^43 + (-b11 - b10 - b5 + b3 - b2 + b1) * q^44 + (b7 - b6 - b5 + 2b4 + 2b3 - b2 + 3b1 + 1) q^46 + (-5b9 + b8 + 2b6 + b1 - 5) * q^47 + (-b11 + b10 - b9 + 4b8 + b7 + b6 + 2b4 - 3b3 - b1) q^49 + (-b10 - 3b9 - b8 - 2b7 + b6 - b4 + b3 - b2 + 2b1 - 1) q^50 + b7 * q^52 + (b11 + 3b9 + 2b8 - b5 + b4 - 3) * q^53 + (b10 + b9 + 2b8 - 2b7 - 4b6 - b5 + 2b4 - 2b3 + 2b2 - b1 - 3) * q^55 + (b9 - b8 + b5 - b4 + b3 - b2) * q^56 + (-b11 - b9 + 3b6 - b5 - 1) q^58 + (4b11 + 4b10 - b7 + b6 + 4b4 + 2b1 + 6) * q^59 + (-b7 + b6 + b4 + b3 + 2b1 - 2) q^61 + (b11 - b8 - 2b6 + b5 + b4 - b1) q^62 - b9 * q^64 - b5 * q^65 + (2b10 + b8 + 2b7 - 6b3 + 2b2 - 3b1) q^67 + (2b8 - b7 - b3 - 2b1) * q^68 + (-b11 - b10 - b7 - 2b6 - b5 + 4b3 - b2 + 3b1) q^70 + (2b11 - 2b10 + 4b9 + 2b8 - 2b7 - 2b6 + b5 + 2b4 - b2 + 2b1) * q^71 + (b11 + 2b10 - 4b8 + 5b6 + b5 + 2b4 - 2b2 - 2b1) * q^73 + (-b7 + b6 + b4 + b3 - 2b1 - 2) q^74 + (-b7 + b6 + b4 + b3 + b1 - 1) * q^76 + (2b11 - 3b9 + b8 + 4b6 + 2b5 + b1 - 3) * q^77 + (-b11 + b10 - 6b8 + 2b7 + 2b6 - b4 - b1) q^79 + (-b11 - b4 - b3 - b1) * q^80 + (2b11 + b10 - b9 - b8 + 2b7 - 2b5 + 2b4 + 2b3 + b2 + 4b1 + 1) * q^82 + (2b11 - b10 - 3b9 - b8 - 2b7 - 2b5 + 2b4 + 2b3 - b2 + 6b1 + 3) q^83 + (b11 + 2b10 + b9 + 2b7 - 6b6 + 3b5 - b3 - 5b1 + 2) q^85 + (-b11 + b10 + 4b9 + 4b8 + b7 + b6 - 2b5 - b3 + 2b2 - b1) * q^86 + (b11 + b10 - b8 + b5 + b4 - b2) * q^88 + (-2b11 - 2b10 + 2b7 - 2b6 + 2b3 + 4b1 + 6) * q^89 + (b11 + b10 - b3 + 1) * q^91 + (b10 - b9 - 2b8 + b6 - b4 - b2 - b1 - 1) q^92 + (2b9 - 5b7 - 5b6 - b4 + b3) q^94 + (-b11 - b10 - 3b9 + 2b8 + b7 + 2b6 + 2b4 + b1 - 1) * q^95 + (-2b11 + b10 + 4b9 - b7 + 2b5 - 2b4 - b3 + b2 - 5b1 - 4) q^97 + (-b11 + b9 - 3b8 - b7 + b5 - b4 + 4b3 + b1 - 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} - 4 q^{7}+O(q^{10}) $ 12 * q - 4 * q^5 - 4 * q^7	$ 12 q - 4 q^{5} - 4 q^{7} + 8 q^{14} - 12 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} - 12 q^{23} + 8 q^{25} - 4 q^{28} + 32 q^{29} - 16 q^{31} - 28 q^{35} + 16 q^{37} + 16 q^{38} - 4 q^{40} + 8 q^{43} - 64 q^{47} - 16 q^{50} - 36 q^{53} - 36 q^{55} - 12 q^{58} + 48 q^{59} - 32 q^{61} + 4 q^{62} + 4 q^{67} + 8 q^{68} - 8 q^{70} - 16 q^{74} - 16 q^{76} - 40 q^{77} + 4 q^{80} - 8 q^{82} + 16 q^{83} + 36 q^{85} - 4 q^{88} + 64 q^{89} + 8 q^{91} - 12 q^{92} - 12 q^{95} - 32 q^{97} - 16 q^{98}+O(q^{100}) $ 12 * q - 4 * q^5 - 4 * q^7 + 8 * q^14 - 12 * q^16 - 8 * q^17 - 4 * q^20 - 4 * q^22 - 12 * q^23 + 8 * q^25 - 4 * q^28 + 32 * q^29 - 16 * q^31 - 28 * q^35 + 16 * q^37 + 16 * q^38 - 4 * q^40 + 8 * q^43 - 64 * q^47 - 16 * q^50 - 36 * q^53 - 36 * q^55 - 12 * q^58 + 48 * q^59 - 32 * q^61 + 4 * q^62 + 4 * q^67 + 8 * q^68 - 8 * q^70 - 16 * q^74 - 16 * q^76 - 40 * q^77 + 4 * q^80 - 8 * q^82 + 16 * q^83 + 36 * q^85 - 4 * q^88 + 64 * q^89 + 8 * q^91 - 12 * q^92 - 12 * q^95 - 32 * q^97 - 16 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4 x^{11} + 12 x^{10} - 16 x^{9} + 8 x^{8} - 8 x^{7} + 18 x^{6} + 32 x^{5} + 16 x^{4} + 36 x^{3} + \cdots + 2 $ x^12 - 4*x^11 + 12*x^10 - 16*x^9 + 8*x^8 - 8*x^7 + 18*x^6 + 32*x^5 + 16*x^4 + 36*x^3 + 80*x^2 - 24*x + 2 :

$\beta_{1}$	$=$	$ ( - 625425286 \nu^{11} + 3033938907 \nu^{10} - 10725340440 \nu^{9} + 20241336346 \nu^{8} + \cdots - 57763499875 ) / 77980266441 $ (-625425286v^11 + 3033938907v^10 - 10725340440v^9 + 20241336346v^8 - 24945220030v^7 + 20784209632v^6 - 17034896078v^5 - 7411966855v^4 - 8080942664v^3 - 49325153210v^2 + 14020678592*v - 57763499875) / 77980266441
$\beta_{2}$	$=$	$ ( - 6737292673 \nu^{11} + 25026555636 \nu^{10} - 73928856315 \nu^{9} + 85766012794 \nu^{8} + \cdots - 64039357747 ) / 77980266441 $ (-6737292673v^11 + 25026555636v^10 - 73928856315v^9 + 85766012794v^8 - 25955572453v^7 + 34338328480v^6 - 104832215951v^5 - 214938551287v^4 - 219672014822v^3 - 265182847574v^2 - 727989651220*v - 64039357747) / 77980266441
$\beta_{3}$	$=$	$ ( - 7268050759 \nu^{11} + 27959757561 \nu^{10} - 81550465833 \nu^{9} + 96832758022 \nu^{8} + \cdots + 129738871973 ) / 77980266441 $ (-7268050759v^11 + 27959757561v^10 - 81550465833v^9 + 96832758022v^8 - 19143403339v^7 + 10548134530v^6 - 78732040796v^5 - 277807377877v^4 - 126350436215v^3 - 266341243094v^2 - 623587467682*v + 129738871973) / 77980266441
$\beta_{4}$	$=$	$ ( 2572378693 \nu^{11} - 9834119364 \nu^{10} + 29121239430 \nu^{9} - 35836638376 \nu^{8} + \cdots - 15789694427 ) / 25993422147 $ (2572378693v^11 - 9834119364v^10 + 29121239430v^9 - 35836638376v^8 + 13721421787v^7 - 16871357605v^6 + 41448332678v^5 + 93283493029v^4 + 39979575605v^3 + 124065869666v^2 + 198288447304*v - 15789694427) / 25993422147
$\beta_{5}$	$=$	$ ( 8703384511 \nu^{11} - 35680665159 \nu^{10} + 108693080247 \nu^{9} - 153300506821 \nu^{8} + \cdots - 270631661933 ) / 77980266441 $ (8703384511v^11 - 35680665159v^10 + 108693080247v^9 - 153300506821v^8 + 95491969681v^7 - 100904727634v^6 + 195310154267v^5 + 221766811657v^4 + 137993837510v^3 + 359637589070v^2 + 706294170550*v - 270631661933) / 77980266441
$\beta_{6}$	$=$	$ ( 9610634717 \nu^{11} - 36549040230 \nu^{10} + 108280934466 \nu^{9} - 132852939296 \nu^{8} + \cdots - 66590352715 ) / 77980266441 $ (9610634717v^11 - 36549040230v^10 + 108280934466v^9 - 132852939296v^8 + 51542053568v^7 - 66507289529v^6 + 157098208192v^5 + 340293521102v^4 + 214213197487v^3 + 440257320484v^2 + 836910488846*v - 66590352715) / 77980266441
$\beta_{7}$	$=$	$ ( 10108383283 \nu^{11} - 39808107846 \nu^{10} + 118266660489 \nu^{9} - 151008792088 \nu^{8} + \cdots - 178641610943 ) / 77980266441 $ (10108383283v^11 - 39808107846v^10 + 118266660489v^9 - 151008792088v^8 + 60625729918v^7 - 55921846234v^6 + 161166689462v^5 + 340503161134v^4 + 169146099383v^3 + 371982740852v^2 + 857995815850*v - 178641610943) / 77980266441
$\beta_{8}$	$=$	$ ( - 10234580326 \nu^{11} + 39295944054 \nu^{10} - 116393352654 \nu^{9} + 144219203128 \nu^{8} + \cdots + 125875666691 ) / 77980266441 $ (-10234580326v^11 + 39295944054v^10 - 116393352654v^9 + 144219203128v^8 - 53894297140v^7 + 57724064281v^6 - 135431286818v^5 - 408648452635v^4 - 155782404458v^3 - 426686167742v^2 - 869241599896*v + 125875666691) / 77980266441
$\beta_{9}$	$=$	$ ( 1826434 \nu^{11} - 7023690 \nu^{10} + 20794782 \nu^{9} - 25913272 \nu^{8} + 10290232 \nu^{7} + \cdots - 22129493 ) / 9929997 $ (1826434v^11 - 7023690v^10 + 20794782v^9 - 25913272v^8 + 10290232v^7 - 12505639v^6 + 29744588v^5 + 66773884v^4 + 36262778v^3 + 70172192v^2 + 153286036*v - 22129493) / 9929997
$\beta_{10}$	$=$	$ ( 19319829736 \nu^{11} - 73000978128 \nu^{10} + 216188029551 \nu^{9} - 265191617383 \nu^{8} + \cdots - 203053430738 ) / 77980266441 $ (19319829736v^11 - 73000978128v^10 + 216188029551v^9 - 265191617383v^8 + 109763426119v^7 - 163838396536v^6 + 360570855599v^5 + 636357078988v^4 + 553208595068v^3 + 721207315034v^2 + 1733006003572*v - 203053430738) / 77980266441
$\beta_{11}$	$=$	$ ( - 3145405997 \nu^{11} + 12244574228 \nu^{10} - 36343668871 \nu^{9} + 46269195588 \nu^{8} + \cdots + 44619961392 ) / 8664474049 $ (-3145405997v^11 + 12244574228v^10 - 36343668871v^9 + 46269195588v^8 - 20236067410v^7 + 25278810047v^6 - 60143619313v^5 - 101354600870v^4 - 60763743503v^3 - 114426275888v^2 - 261328230890*v + 44619961392) / 8664474049

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

$\nu$	$=$	$ ( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 $ (-b9 - b8 + b7 + b6 - b4 + b3 + b1 + 1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{11} - \beta_{10} - 2\beta_{9} + 6\beta_{6} - \beta_{5} - 3\beta_{4} + \beta_{2} - \beta _1 - 2 ) / 2 $ (-b11 - b10 - 2b9 + 6b6 - b5 - 3*b4 + b2 - b1 - 2) / 2
$\nu^{3}$	$=$	$ - \beta_{11} - 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} + \cdots - 4 $ -b11 - 2b10 + 3b9 + 2b8 - 4b7 + 3b6 - 3b4 - 2b3 - b2 - 2b1 - 4
$\nu^{4}$	$=$	$ 3 \beta_{11} - 3 \beta_{10} + 17 \beta_{9} + 7 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} + 5 \beta_{5} + \cdots + 3 \beta_1 $ 3b11 - 3b10 + 17b9 + 7b8 - 8b7 - 8b6 + 5b5 + 5b4 - 2b3 - 5b2 + 3*b1
$\nu^{5}$	$=$	$ 17 \beta_{11} + 10 \beta_{10} + 19 \beta_{9} + 10 \beta_{8} + 19 \beta_{7} - 32 \beta_{6} + 10 \beta_{5} + \cdots + 32 $ 17b11 + 10b10 + 19b9 + 10b8 + 19b7 - 32b6 + 10b5 + 30b4 + 10b3 + 20b1 + 32
$\nu^{6}$	$=$	$ 20 \beta_{11} + 40 \beta_{10} - 65 \beta_{9} - 22 \beta_{8} + 114 \beta_{7} - 20 \beta_{5} + 20 \beta_{4} + \cdots + 65 $ 20b11 + 40b10 - 65b9 - 22b8 + 114b7 - 20b5 + 20b4 + 48b3 + 40b2 + 22b1 + 65
$\nu^{7}$	$=$	$ - 82 \beta_{11} - 250 \beta_{9} - 98 \beta_{8} + 127 \beta_{7} + 250 \beta_{6} - 128 \beta_{5} + \cdots - 127 $ -82b11 - 250b9 - 98b8 + 127b7 + 250b6 - 128b5 - 180b4 + 59b3 + 82b2 - 95b1 - 127
$\nu^{8}$	$=$	$ - 308 \beta_{11} - 308 \beta_{10} - 515 \beta_{7} + 515 \beta_{6} - 141 \beta_{5} - 499 \beta_{4} + \cdots - 816 $ -308b11 - 308b10 - 515b7 + 515b6 - 141b5 - 499b4 - 191b3 - 141b2 - 364*b1 - 816
$\nu^{9}$	$=$	$ - 640 \beta_{10} + 1923 \beta_{9} + 754 \beta_{8} - 1923 \beta_{7} - 891 \beta_{6} + 640 \beta_{5} + \cdots - 891 $ -640b10 + 1923b9 + 754b8 - 1923b7 - 891b6 + 640b5 + 386b4 - 754b3 - 954b2 - 72b1 - 891
$\nu^{10}$	$=$	$ 2348 \beta_{11} + 1026 \beta_{10} + 3999 \beta_{9} + 1537 \beta_{8} - 6006 \beta_{6} + 2348 \beta_{5} + \cdots + 3999 $ 2348b11 + 1026b10 + 3999b9 + 1537b8 - 6006b6 + 2348b5 + 4794b4 - 1026b2 + 2563*b1 + 3999
$\nu^{11}$	$=$	$ 4911 \beta_{11} + 7142 \beta_{10} - 6457 \beta_{9} - 2689 \beta_{8} + 14673 \beta_{7} - 6457 \beta_{6} + \cdots + 14673 $ 4911b11 + 7142b10 - 6457b9 - 2689b8 + 14673b7 - 6457b6 + 7600b4 + 5779b3 + 4911b2 + 5779b1 + 14673

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$-1$	$-\beta_{9}$	$1$

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} $

53.1

0.461741	−	1.11474i
−0.514363	+	1.24178i
1.05262	−	2.54125i
1.84980	+	0.766212i
−0.986762	−	0.408730i
0.136963	+	0.0567319i
0.461741	+	1.11474i
−0.514363	−	1.24178i
1.05262	+	2.54125i
1.84980	−	0.766212i
−0.986762	+	0.408730i
0.136963	−	0.0567319i

−0.707107

−

0.707107i

1.00000i

−2.09188

−

0.789966i

2.04411

−

2.04411i

0.707107

−

0.707107i

0.920591

2.03777i

53.2

−0.707107

−

0.707107i

1.00000i

−1.84387

1.26497i

−3.59146

3.59146i

0.707107

−

0.707107i

2.19828

0.409343i

53.3

−0.707107

−

0.707107i

1.00000i

2.22864

−

0.182111i

−0.866866

0.866866i

0.707107

−

0.707107i

−1.70466

−

1.44711i

53.4

0.707107

0.707107i

1.00000i

−1.71335

1.43681i

1.77435

−

1.77435i

−0.707107

0.707107i

−2.22750

−

0.195549i

53.5

0.707107

0.707107i

1.00000i

0.314682

2.21381i

−2.69231

2.69231i

−0.707107

0.707107i

−1.34289

1.78792i

53.6

0.707107

0.707107i

1.00000i

1.10578

−

1.94352i

1.33218

−

1.33218i

−0.707107

0.707107i

2.15618

−

0.592368i

287.1

−0.707107

0.707107i

−

1.00000i

−2.09188

0.789966i

2.04411

2.04411i

0.707107

0.707107i

0.920591

−

2.03777i

287.2

−0.707107

0.707107i

−

1.00000i

−1.84387

−

1.26497i

−3.59146

−

3.59146i

0.707107

0.707107i

2.19828

−

0.409343i

287.3

−0.707107

0.707107i

−

1.00000i

2.22864

0.182111i

−0.866866

−

0.866866i

0.707107

0.707107i

−1.70466

1.44711i

287.4

0.707107

−

0.707107i

−

1.00000i

−1.71335

−

1.43681i

1.77435

1.77435i

−0.707107

−

0.707107i

−2.22750

0.195549i

287.5

0.707107

−

0.707107i

−

1.00000i

0.314682

−

2.21381i

−2.69231

−

2.69231i

−0.707107

−

0.707107i

−1.34289

−

1.78792i

287.6

0.707107

−

0.707107i

−

1.00000i

1.10578

1.94352i

1.33218

1.33218i

−0.707107

−

0.707107i

2.15618

0.592368i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1170.2.o.a	✓	12
3.b	odd	2	1		1170.2.o.c	yes	12
5.c	odd	4	1		1170.2.o.c	yes	12
15.e	even	4	1	inner	1170.2.o.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1170.2.o.a	✓	12	1.a	even	1	1	trivial
1170.2.o.a	✓	12	15.e	even	4	1	inner
1170.2.o.c	yes	12	3.b	odd	2	1
1170.2.o.c	yes	12	5.c	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1170, [\chi])$:

$ T_{7}^{12} + 4 T_{7}^{11} + 8 T_{7}^{10} - 56 T_{7}^{9} + 244 T_{7}^{8} + 384 T_{7}^{7} + 1152 T_{7}^{6} + \cdots + 104976 $

$ T_{17}^{12} + 8 T_{17}^{11} + 32 T_{17}^{10} - 96 T_{17}^{9} + 1136 T_{17}^{8} + 8016 T_{17}^{7} + \cdots + 2178576 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 4 T^{11} + \cdots + 15625 $ T^12 + 4T^11 + 4T^10 - 4T^9 - 5T^8 - 80T^7 - 328T^6 - 400T^5 - 125T^4 - 500T^3 + 2500T^2 + 12500*T + 15625
$7$	$ T^{12} + 4 T^{11} + \cdots + 104976 $ T^12 + 4T^11 + 8T^10 - 56T^9 + 244T^8 + 384T^7 + 1152T^6 - 9072T^5 + 29808T^4 - 15552*T^3 + 104976
$11$	$ T^{12} + 60 T^{10} + \cdots + 33856 $ T^12 + 60T^10 + 1180T^8 + 10336T^6 + 42352T^4 + 72384*T^2 + 33856
$13$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$17$	$ T^{12} + 8 T^{11} + \cdots + 2178576 $ T^12 + 8T^11 + 32T^10 - 96T^9 + 1136T^8 + 8016T^7 + 32384T^6 - 116768T^5 + 461152T^4 + 1371648T^3 + 2765952T^2 - 3471552*T + 2178576
$19$	$ T^{12} + 64 T^{10} + \cdots + 16 $ T^12 + 64T^10 + 592T^8 + 1560T^6 + 1216T^4 + 320*T^2 + 16
$23$	$ T^{12} + 12 T^{11} + \cdots + 38987536 $ T^12 + 12T^11 + 72T^10 + 208T^9 + 1140T^8 + 10032T^7 + 59936T^6 + 174224T^5 + 382320T^4 + 1540704T^3 + 9193472T^2 + 26774272*T + 38987536
$29$	$ (T^{6} - 16 T^{5} + \cdots - 36)^{2} $ (T^6 - 16T^5 + 86T^4 - 164T^3 + 32T^2 + 96*T - 36)^2
$31$	$ (T^{6} + 8 T^{5} + \cdots + 2056)^{2} $ (T^6 + 8T^5 - 22T^4 - 296T^3 - 452T^2 + 1008*T + 2056)^2
$37$	$ T^{12} - 16 T^{11} + \cdots + 29246464 $ T^12 - 16T^11 + 128T^10 - 656T^9 + 11328T^8 - 152192T^7 + 1200256T^6 - 5952512T^5 + 19838976T^4 - 44854784T^3 + 67837952T^2 - 62992384*T + 29246464
$41$	$ T^{12} + 376 T^{10} + \cdots + 97456384 $ T^12 + 376T^10 + 45552T^8 + 1969632T^6 + 35403648T^4 + 232669696*T^2 + 97456384
$43$	$ T^{12} + \cdots + 537497856 $ T^12 - 8T^11 + 32T^10 + 160T^9 + 22128T^8 - 163008T^7 + 608768T^6 + 1540736T^5 + 5922688T^4 - 32649216T^3 + 133171200T^2 + 378362880*T + 537497856
$47$	$ T^{12} + \cdots + 3403088896 $ T^12 + 64T^11 + 2048T^10 + 41840T^9 + 600304T^8 + 6302592T^7 + 49236096T^6 + 285405696T^5 + 1203787264T^4 + 3529727488T^3 + 6628149248T^2 + 6716573696*T + 3403088896
$53$	$ T^{12} + 36 T^{11} + \cdots + 1784896 $ T^12 + 36T^11 + 648T^10 + 6920T^9 + 47772T^8 + 209984T^7 + 546368T^6 + 537920T^5 - 567120T^4 - 841024T^3 + 5171328T^2 - 4296576*T + 1784896
$59$	$ (T^{6} - 24 T^{5} + \cdots + 39736)^{2} $ (T^6 - 24T^5 + 10T^4 + 3072T^3 - 19252T^2 + 21344*T + 39736)^2
$61$	$ (T^{6} + 16 T^{5} + \cdots + 32)^{2} $ (T^6 + 16T^5 + 64T^4 + 48T^3 - 96T^2 - 64*T + 32)^2
$67$	$ T^{12} + \cdots + 479434816 $ T^12 - 4T^11 + 8T^10 + 1544T^9 + 31020T^8 + 110240T^7 + 502848T^6 + 12540992T^5 + 200265648T^4 + 1172765376T^3 + 3548188800T^2 + 1844519040*T + 479434816
$71$	$ T^{12} + 336 T^{10} + \cdots + 9535744 $ T^12 + 336T^10 + 31600T^8 + 1037728T^6 + 11930944T^4 + 22639872*T^2 + 9535744
$73$	$ T^{12} + \cdots + 100623452944 $ T^12 + 456T^9 + 48600T^8 + 90848T^7 + 103968T^6 + 5356640T^5 + 520594320T^4 + 1336788128T^3 + 1516462592T^2 - 17469499264*T + 100623452944
$79$	$ T^{12} + \cdots + 11154739456 $ T^12 + 608T^10 + 146480T^8 + 17487456T^6 + 1036009024T^4 + 24517703424*T^2 + 11154739456
$83$	$ T^{12} + \cdots + 6629867776 $ T^12 - 16T^11 + 128T^10 - 304T^9 + 29168T^8 - 443776T^7 + 3413120T^6 - 10976512T^5 + 27997568T^4 - 176392960T^3 + 1490143232T^2 - 4445099008*T + 6629867776
$89$	$ (T^{6} - 32 T^{5} + \cdots + 22784)^{2} $ (T^6 - 32T^5 + 208T^4 + 1824T^3 - 16320T^2 - 12544*T + 22784)^2
$97$	$ T^{12} + \cdots + 30091147024 $ T^12 + 32T^11 + 512T^10 + 5224T^9 + 65944T^8 + 1135232T^7 + 16209184T^6 + 159341984T^5 + 1080942704T^4 + 5045055264T^3 + 15879045632T^2 + 30913385344*T + 30091147024
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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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + \beta_{9} q^{4} + (\beta_{11} + \beta_{4} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{10} - \beta_{8} + \cdots + 2 \beta_1) q^{7}+ \cdots + \beta_{7} q^{8}+O(q^{10}) \) q + b6 * q^2 + b9 * q^4 + (b11 + b4 + b3 + b1) * q^5 + (-b10 - b8 - b7 + b3 - b2 + 2b1) q^7 + b7 * q^8	\( q + \beta_{6} q^{2} + \beta_{9} q^{4} + (\beta_{11} + \beta_{4} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{10} - \beta_{8} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{9} - 3 \beta_{8} + \cdots - 1) q^{98}+O(q^{100}) \) q + b6 * q^2 + b9 * q^4 + (b11 + b4 + b3 + b1) * q^5 + (-b10 - b8 - b7 + b3 - b2 + 2b1) q^7 + b7 * q^8 - b5 * q^10 + (b11 - b10 + b5 + b3 - b2 + b1) * q^11 + b6 * q^13 + (b11 + b10 - b3 + 1) * q^14 - q^16 + (2b8 - b6 + b4 + 2b1) * q^17 + (b9 - b8 - b7 - b6 - b4 + b3) * q^19 + (b4 - b2 + b1) * q^20 + (b11 + b10 + b8 - b5 + b4 + b2) * q^22 + (b11 + b9 + 2b8 - b7 - b5 + b4 - b3 - 1) q^23 + (-2b9 + b8 + b7 - 3b6 - b5 + b4 - b3 + b2 + b1 + 1) * q^25 + b9 * q^26 + (-b11 + b8 + b6 - b5 + b1) * q^28 + (b11 + b10 + b7 - b6 + b4 + 3) * q^29 + (-b11 - b10 + b3 - b1 - 2) * q^31 - b6 * q^32 + (-b9 - b8 - 2b4 + 2b3) * q^34 + (b11 + b10 - b9 + 4b8 - b5 + b4 - 2b3 + b2 - 2) * q^35 + (-b9 + b8 + 2b7 + 2b3 - b1 + 1) * q^37 + (-b9 + b8 + b7 - b3 - b1 + 1) * q^38 + (b10 - b8) * q^40 + (2b11 - 2b10 + 2b9 + 2b8 - b7 - b6 + b5 + 3b4 - b3 - b2 + 2b1) * q^41 + (2b11 + b10 + b9 - b8 + 4b6 + 2b5 - 2b4 - b2 + 1) * q^43 + (-b11 - b10 - b5 + b3 - b2 + b1) * q^44 + (b7 - b6 - b5 + 2b4 + 2b3 - b2 + 3b1 + 1) q^46 + (-5b9 + b8 + 2b6 + b1 - 5) * q^47 + (-b11 + b10 - b9 + 4b8 + b7 + b6 + 2b4 - 3b3 - b1) q^49 + (-b10 - 3b9 - b8 - 2b7 + b6 - b4 + b3 - b2 + 2b1 - 1) q^50 + b7 * q^52 + (b11 + 3b9 + 2b8 - b5 + b4 - 3) * q^53 + (b10 + b9 + 2b8 - 2b7 - 4b6 - b5 + 2b4 - 2b3 + 2b2 - b1 - 3) * q^55 + (b9 - b8 + b5 - b4 + b3 - b2) * q^56 + (-b11 - b9 + 3b6 - b5 - 1) q^58 + (4b11 + 4b10 - b7 + b6 + 4b4 + 2b1 + 6) * q^59 + (-b7 + b6 + b4 + b3 + 2b1 - 2) q^61 + (b11 - b8 - 2b6 + b5 + b4 - b1) q^62 - b9 * q^64 - b5 * q^65 + (2b10 + b8 + 2b7 - 6b3 + 2b2 - 3b1) q^67 + (2b8 - b7 - b3 - 2b1) * q^68 + (-b11 - b10 - b7 - 2b6 - b5 + 4b3 - b2 + 3b1) q^70 + (2b11 - 2b10 + 4b9 + 2b8 - 2b7 - 2b6 + b5 + 2b4 - b2 + 2b1) * q^71 + (b11 + 2b10 - 4b8 + 5b6 + b5 + 2b4 - 2b2 - 2b1) * q^73 + (-b7 + b6 + b4 + b3 - 2b1 - 2) q^74 + (-b7 + b6 + b4 + b3 + b1 - 1) * q^76 + (2b11 - 3b9 + b8 + 4b6 + 2b5 + b1 - 3) * q^77 + (-b11 + b10 - 6b8 + 2b7 + 2b6 - b4 - b1) q^79 + (-b11 - b4 - b3 - b1) * q^80 + (2b11 + b10 - b9 - b8 + 2b7 - 2b5 + 2b4 + 2b3 + b2 + 4b1 + 1) * q^82 + (2b11 - b10 - 3b9 - b8 - 2b7 - 2b5 + 2b4 + 2b3 - b2 + 6b1 + 3) q^83 + (b11 + 2b10 + b9 + 2b7 - 6b6 + 3b5 - b3 - 5b1 + 2) q^85 + (-b11 + b10 + 4b9 + 4b8 + b7 + b6 - 2b5 - b3 + 2b2 - b1) * q^86 + (b11 + b10 - b8 + b5 + b4 - b2) * q^88 + (-2b11 - 2b10 + 2b7 - 2b6 + 2b3 + 4b1 + 6) * q^89 + (b11 + b10 - b3 + 1) * q^91 + (b10 - b9 - 2b8 + b6 - b4 - b2 - b1 - 1) q^92 + (2b9 - 5b7 - 5b6 - b4 + b3) q^94 + (-b11 - b10 - 3b9 + 2b8 + b7 + 2b6 + 2b4 + b1 - 1) * q^95 + (-2b11 + b10 + 4b9 - b7 + 2b5 - 2b4 - b3 + b2 - 5b1 - 4) q^97 + (-b11 + b9 - 3b8 - b7 + b5 - b4 + 4b3 + b1 - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 12 * q - 4 * q^5 - 4 * q^7	\( 12 q - 4 q^{5} - 4 q^{7} + 8 q^{14} - 12 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} - 12 q^{23} + 8 q^{25} - 4 q^{28} + 32 q^{29} - 16 q^{31} - 28 q^{35} + 16 q^{37} + 16 q^{38} - 4 q^{40} + 8 q^{43} - 64 q^{47} - 16 q^{50} - 36 q^{53} - 36 q^{55} - 12 q^{58} + 48 q^{59} - 32 q^{61} + 4 q^{62} + 4 q^{67} + 8 q^{68} - 8 q^{70} - 16 q^{74} - 16 q^{76} - 40 q^{77} + 4 q^{80} - 8 q^{82} + 16 q^{83} + 36 q^{85} - 4 q^{88} + 64 q^{89} + 8 q^{91} - 12 q^{92} - 12 q^{95} - 32 q^{97} - 16 q^{98}+O(q^{100}) \) 12 * q - 4 * q^5 - 4 * q^7 + 8 * q^14 - 12 * q^16 - 8 * q^17 - 4 * q^20 - 4 * q^22 - 12 * q^23 + 8 * q^25 - 4 * q^28 + 32 * q^29 - 16 * q^31 - 28 * q^35 + 16 * q^37 + 16 * q^38 - 4 * q^40 + 8 * q^43 - 64 * q^47 - 16 * q^50 - 36 * q^53 - 36 * q^55 - 12 * q^58 + 48 * q^59 - 32 * q^61 + 4 * q^62 + 4 * q^67 + 8 * q^68 - 8 * q^70 - 16 * q^74 - 16 * q^76 - 40 * q^77 + 4 * q^80 - 8 * q^82 + 16 * q^83 + 36 * q^85 - 4 * q^88 + 64 * q^89 + 8 * q^91 - 12 * q^92 - 12 * q^95 - 32 * q^97 - 16 * q^98

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	1170.2.o.a

Level	$1170$
Weight	$2$
Character orbit	1170.o
Analytic conductor	$9.342$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(9.34249703649\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4 x^{11} + 12 x^{10} - 16 x^{9} + 8 x^{8} - 8 x^{7} + 18 x^{6} + 32 x^{5} + 16 x^{4} + 36 x^{3} + \cdots + 2 \) x^12 - 4x^11 + 12x^10 - 16x^9 + 8x^8 - 8x^7 + 18x^6 + 32x^5 + 16x^4 + 36x^3 + 80x^2 - 24*x + 2
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 625425286 \nu^{11} + 3033938907 \nu^{10} - 10725340440 \nu^{9} + 20241336346 \nu^{8} + \cdots - 57763499875 ) / 77980266441 \) (-625425286v^11 + 3033938907v^10 - 10725340440v^9 + 20241336346v^8 - 24945220030v^7 + 20784209632v^6 - 17034896078v^5 - 7411966855v^4 - 8080942664v^3 - 49325153210v^2 + 14020678592*v - 57763499875) / 77980266441
\(\beta_{2}\)	\(=\)	\( ( - 6737292673 \nu^{11} + 25026555636 \nu^{10} - 73928856315 \nu^{9} + 85766012794 \nu^{8} + \cdots - 64039357747 ) / 77980266441 \) (-6737292673v^11 + 25026555636v^10 - 73928856315v^9 + 85766012794v^8 - 25955572453v^7 + 34338328480v^6 - 104832215951v^5 - 214938551287v^4 - 219672014822v^3 - 265182847574v^2 - 727989651220*v - 64039357747) / 77980266441
\(\beta_{3}\)	\(=\)	\( ( - 7268050759 \nu^{11} + 27959757561 \nu^{10} - 81550465833 \nu^{9} + 96832758022 \nu^{8} + \cdots + 129738871973 ) / 77980266441 \) (-7268050759v^11 + 27959757561v^10 - 81550465833v^9 + 96832758022v^8 - 19143403339v^7 + 10548134530v^6 - 78732040796v^5 - 277807377877v^4 - 126350436215v^3 - 266341243094v^2 - 623587467682*v + 129738871973) / 77980266441
\(\beta_{4}\)	\(=\)	\( ( 2572378693 \nu^{11} - 9834119364 \nu^{10} + 29121239430 \nu^{9} - 35836638376 \nu^{8} + \cdots - 15789694427 ) / 25993422147 \) (2572378693v^11 - 9834119364v^10 + 29121239430v^9 - 35836638376v^8 + 13721421787v^7 - 16871357605v^6 + 41448332678v^5 + 93283493029v^4 + 39979575605v^3 + 124065869666v^2 + 198288447304*v - 15789694427) / 25993422147
\(\beta_{5}\)	\(=\)	\( ( 8703384511 \nu^{11} - 35680665159 \nu^{10} + 108693080247 \nu^{9} - 153300506821 \nu^{8} + \cdots - 270631661933 ) / 77980266441 \) (8703384511v^11 - 35680665159v^10 + 108693080247v^9 - 153300506821v^8 + 95491969681v^7 - 100904727634v^6 + 195310154267v^5 + 221766811657v^4 + 137993837510v^3 + 359637589070v^2 + 706294170550*v - 270631661933) / 77980266441
\(\beta_{6}\)	\(=\)	\( ( 9610634717 \nu^{11} - 36549040230 \nu^{10} + 108280934466 \nu^{9} - 132852939296 \nu^{8} + \cdots - 66590352715 ) / 77980266441 \) (9610634717v^11 - 36549040230v^10 + 108280934466v^9 - 132852939296v^8 + 51542053568v^7 - 66507289529v^6 + 157098208192v^5 + 340293521102v^4 + 214213197487v^3 + 440257320484v^2 + 836910488846*v - 66590352715) / 77980266441
\(\beta_{7}\)	\(=\)	\( ( 10108383283 \nu^{11} - 39808107846 \nu^{10} + 118266660489 \nu^{9} - 151008792088 \nu^{8} + \cdots - 178641610943 ) / 77980266441 \) (10108383283v^11 - 39808107846v^10 + 118266660489v^9 - 151008792088v^8 + 60625729918v^7 - 55921846234v^6 + 161166689462v^5 + 340503161134v^4 + 169146099383v^3 + 371982740852v^2 + 857995815850*v - 178641610943) / 77980266441
\(\beta_{8}\)	\(=\)	\( ( - 10234580326 \nu^{11} + 39295944054 \nu^{10} - 116393352654 \nu^{9} + 144219203128 \nu^{8} + \cdots + 125875666691 ) / 77980266441 \) (-10234580326v^11 + 39295944054v^10 - 116393352654v^9 + 144219203128v^8 - 53894297140v^7 + 57724064281v^6 - 135431286818v^5 - 408648452635v^4 - 155782404458v^3 - 426686167742v^2 - 869241599896*v + 125875666691) / 77980266441
\(\beta_{9}\)	\(=\)	\( ( 1826434 \nu^{11} - 7023690 \nu^{10} + 20794782 \nu^{9} - 25913272 \nu^{8} + 10290232 \nu^{7} + \cdots - 22129493 ) / 9929997 \) (1826434v^11 - 7023690v^10 + 20794782v^9 - 25913272v^8 + 10290232v^7 - 12505639v^6 + 29744588v^5 + 66773884v^4 + 36262778v^3 + 70172192v^2 + 153286036*v - 22129493) / 9929997
\(\beta_{10}\)	\(=\)	\( ( 19319829736 \nu^{11} - 73000978128 \nu^{10} + 216188029551 \nu^{9} - 265191617383 \nu^{8} + \cdots - 203053430738 ) / 77980266441 \) (19319829736v^11 - 73000978128v^10 + 216188029551v^9 - 265191617383v^8 + 109763426119v^7 - 163838396536v^6 + 360570855599v^5 + 636357078988v^4 + 553208595068v^3 + 721207315034v^2 + 1733006003572*v - 203053430738) / 77980266441
\(\beta_{11}\)	\(=\)	\( ( - 3145405997 \nu^{11} + 12244574228 \nu^{10} - 36343668871 \nu^{9} + 46269195588 \nu^{8} + \cdots + 44619961392 ) / 8664474049 \) (-3145405997v^11 + 12244574228v^10 - 36343668871v^9 + 46269195588v^8 - 20236067410v^7 + 25278810047v^6 - 60143619313v^5 - 101354600870v^4 - 60763743503v^3 - 114426275888v^2 - 261328230890*v + 44619961392) / 8664474049

\(\nu\)	\(=\)	\( ( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) (-b9 - b8 + b7 + b6 - b4 + b3 + b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{11} - \beta_{10} - 2\beta_{9} + 6\beta_{6} - \beta_{5} - 3\beta_{4} + \beta_{2} - \beta _1 - 2 ) / 2 \) (-b11 - b10 - 2b9 + 6b6 - b5 - 3*b4 + b2 - b1 - 2) / 2
\(\nu^{3}\)	\(=\)	\( - \beta_{11} - 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} + \cdots - 4 \) -b11 - 2b10 + 3b9 + 2b8 - 4b7 + 3b6 - 3b4 - 2b3 - b2 - 2b1 - 4
\(\nu^{4}\)	\(=\)	\( 3 \beta_{11} - 3 \beta_{10} + 17 \beta_{9} + 7 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} + 5 \beta_{5} + \cdots + 3 \beta_1 \) 3b11 - 3b10 + 17b9 + 7b8 - 8b7 - 8b6 + 5b5 + 5b4 - 2b3 - 5b2 + 3*b1
\(\nu^{5}\)	\(=\)	\( 17 \beta_{11} + 10 \beta_{10} + 19 \beta_{9} + 10 \beta_{8} + 19 \beta_{7} - 32 \beta_{6} + 10 \beta_{5} + \cdots + 32 \) 17b11 + 10b10 + 19b9 + 10b8 + 19b7 - 32b6 + 10b5 + 30b4 + 10b3 + 20b1 + 32
\(\nu^{6}\)	\(=\)	\( 20 \beta_{11} + 40 \beta_{10} - 65 \beta_{9} - 22 \beta_{8} + 114 \beta_{7} - 20 \beta_{5} + 20 \beta_{4} + \cdots + 65 \) 20b11 + 40b10 - 65b9 - 22b8 + 114b7 - 20b5 + 20b4 + 48b3 + 40b2 + 22b1 + 65
\(\nu^{7}\)	\(=\)	\( - 82 \beta_{11} - 250 \beta_{9} - 98 \beta_{8} + 127 \beta_{7} + 250 \beta_{6} - 128 \beta_{5} + \cdots - 127 \) -82b11 - 250b9 - 98b8 + 127b7 + 250b6 - 128b5 - 180b4 + 59b3 + 82b2 - 95b1 - 127
\(\nu^{8}\)	\(=\)	\( - 308 \beta_{11} - 308 \beta_{10} - 515 \beta_{7} + 515 \beta_{6} - 141 \beta_{5} - 499 \beta_{4} + \cdots - 816 \) -308b11 - 308b10 - 515b7 + 515b6 - 141b5 - 499b4 - 191b3 - 141b2 - 364*b1 - 816
\(\nu^{9}\)	\(=\)	\( - 640 \beta_{10} + 1923 \beta_{9} + 754 \beta_{8} - 1923 \beta_{7} - 891 \beta_{6} + 640 \beta_{5} + \cdots - 891 \) -640b10 + 1923b9 + 754b8 - 1923b7 - 891b6 + 640b5 + 386b4 - 754b3 - 954b2 - 72b1 - 891
\(\nu^{10}\)	\(=\)	\( 2348 \beta_{11} + 1026 \beta_{10} + 3999 \beta_{9} + 1537 \beta_{8} - 6006 \beta_{6} + 2348 \beta_{5} + \cdots + 3999 \) 2348b11 + 1026b10 + 3999b9 + 1537b8 - 6006b6 + 2348b5 + 4794b4 - 1026b2 + 2563*b1 + 3999
\(\nu^{11}\)	\(=\)	\( 4911 \beta_{11} + 7142 \beta_{10} - 6457 \beta_{9} - 2689 \beta_{8} + 14673 \beta_{7} - 6457 \beta_{6} + \cdots + 14673 \) 4911b11 + 7142b10 - 6457b9 - 2689b8 + 14673b7 - 6457b6 + 7600b4 + 5779b3 + 4911b2 + 5779b1 + 14673

\(n\)	\(911\)	\(937\)	\(1081\)
\(\chi(n)\)	\(-1\)	\(-\beta_{9}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 53.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 4 T^{11} + \cdots + 15625 \) T^12 + 4T^11 + 4T^10 - 4T^9 - 5T^8 - 80T^7 - 328T^6 - 400T^5 - 125T^4 - 500T^3 + 2500T^2 + 12500*T + 15625
$7$	\( T^{12} + 4 T^{11} + \cdots + 104976 \) T^12 + 4T^11 + 8T^10 - 56T^9 + 244T^8 + 384T^7 + 1152T^6 - 9072T^5 + 29808T^4 - 15552*T^3 + 104976
$11$	\( T^{12} + 60 T^{10} + \cdots + 33856 \) T^12 + 60T^10 + 1180T^8 + 10336T^6 + 42352T^4 + 72384*T^2 + 33856
$13$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$17$	\( T^{12} + 8 T^{11} + \cdots + 2178576 \) T^12 + 8T^11 + 32T^10 - 96T^9 + 1136T^8 + 8016T^7 + 32384T^6 - 116768T^5 + 461152T^4 + 1371648T^3 + 2765952T^2 - 3471552*T + 2178576
$19$	\( T^{12} + 64 T^{10} + \cdots + 16 \) T^12 + 64T^10 + 592T^8 + 1560T^6 + 1216T^4 + 320*T^2 + 16
$23$	\( T^{12} + 12 T^{11} + \cdots + 38987536 \) T^12 + 12T^11 + 72T^10 + 208T^9 + 1140T^8 + 10032T^7 + 59936T^6 + 174224T^5 + 382320T^4 + 1540704T^3 + 9193472T^2 + 26774272*T + 38987536
$29$	\( (T^{6} - 16 T^{5} + \cdots - 36)^{2} \) (T^6 - 16T^5 + 86T^4 - 164T^3 + 32T^2 + 96*T - 36)^2
$31$	\( (T^{6} + 8 T^{5} + \cdots + 2056)^{2} \) (T^6 + 8T^5 - 22T^4 - 296T^3 - 452T^2 + 1008*T + 2056)^2
$37$	\( T^{12} - 16 T^{11} + \cdots + 29246464 \) T^12 - 16T^11 + 128T^10 - 656T^9 + 11328T^8 - 152192T^7 + 1200256T^6 - 5952512T^5 + 19838976T^4 - 44854784T^3 + 67837952T^2 - 62992384*T + 29246464
$41$	\( T^{12} + 376 T^{10} + \cdots + 97456384 \) T^12 + 376T^10 + 45552T^8 + 1969632T^6 + 35403648T^4 + 232669696*T^2 + 97456384
$43$	\( T^{12} + \cdots + 537497856 \) T^12 - 8T^11 + 32T^10 + 160T^9 + 22128T^8 - 163008T^7 + 608768T^6 + 1540736T^5 + 5922688T^4 - 32649216T^3 + 133171200T^2 + 378362880*T + 537497856
$47$	\( T^{12} + \cdots + 3403088896 \) T^12 + 64T^11 + 2048T^10 + 41840T^9 + 600304T^8 + 6302592T^7 + 49236096T^6 + 285405696T^5 + 1203787264T^4 + 3529727488T^3 + 6628149248T^2 + 6716573696*T + 3403088896
$53$	\( T^{12} + 36 T^{11} + \cdots + 1784896 \) T^12 + 36T^11 + 648T^10 + 6920T^9 + 47772T^8 + 209984T^7 + 546368T^6 + 537920T^5 - 567120T^4 - 841024T^3 + 5171328T^2 - 4296576*T + 1784896
$59$	\( (T^{6} - 24 T^{5} + \cdots + 39736)^{2} \) (T^6 - 24T^5 + 10T^4 + 3072T^3 - 19252T^2 + 21344*T + 39736)^2
$61$	\( (T^{6} + 16 T^{5} + \cdots + 32)^{2} \) (T^6 + 16T^5 + 64T^4 + 48T^3 - 96T^2 - 64*T + 32)^2
$67$	\( T^{12} + \cdots + 479434816 \) T^12 - 4T^11 + 8T^10 + 1544T^9 + 31020T^8 + 110240T^7 + 502848T^6 + 12540992T^5 + 200265648T^4 + 1172765376T^3 + 3548188800T^2 + 1844519040*T + 479434816
$71$	\( T^{12} + 336 T^{10} + \cdots + 9535744 \) T^12 + 336T^10 + 31600T^8 + 1037728T^6 + 11930944T^4 + 22639872*T^2 + 9535744
$73$	\( T^{12} + \cdots + 100623452944 \) T^12 + 456T^9 + 48600T^8 + 90848T^7 + 103968T^6 + 5356640T^5 + 520594320T^4 + 1336788128T^3 + 1516462592T^2 - 17469499264*T + 100623452944
$79$	\( T^{12} + \cdots + 11154739456 \) T^12 + 608T^10 + 146480T^8 + 17487456T^6 + 1036009024T^4 + 24517703424*T^2 + 11154739456
$83$	\( T^{12} + \cdots + 6629867776 \) T^12 - 16T^11 + 128T^10 - 304T^9 + 29168T^8 - 443776T^7 + 3413120T^6 - 10976512T^5 + 27997568T^4 - 176392960T^3 + 1490143232T^2 - 4445099008*T + 6629867776
$89$	\( (T^{6} - 32 T^{5} + \cdots + 22784)^{2} \) (T^6 - 32T^5 + 208T^4 + 1824T^3 - 16320T^2 - 12544*T + 22784)^2
$97$	\( T^{12} + \cdots + 30091147024 \) T^12 + 32T^11 + 512T^10 + 5224T^9 + 65944T^8 + 1135232T^7 + 16209184T^6 + 159341984T^5 + 1080942704T^4 + 5045055264T^3 + 15879045632T^2 + 30913385344*T + 30091147024
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