LMFDB - Newform orbit 378.4.f.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(127,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.f (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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
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} - x^{11} + 3 x^{10} + 27 x^{9} + 159 x^{8} + 2001 x^{7} + 12876 x^{6} + 4140 x^{5} + \cdots + 256562019 $ x^12 - x^11 + 3x^10 + 27x^9 + 159x^8 + 2001x^7 + 12876x^6 + 4140x^5 + 332841x^4 + 2657720x^3 + 20571241x^2 + 107277462x + 256562019
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{15} $
Twist minimal:	no (minimal twist has level 126)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{8} - \beta_{4} + \beta_1 + 2) q^{5} + 7 \beta_1 q^{7} - 8 q^{8}+O(q^{10}) $ q - 2b1 q^2 + (-4b1 - 4) q^4 + (b8 - b4 + b1 + 2) * q^5 + 7b1 q^7 - 8 * q^8	$ q - 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{8} - \beta_{4} + \beta_1 + 2) q^{5} + 7 \beta_1 q^{7} - 8 q^{8} + (2 \beta_{8} + 4) q^{10} + ( - \beta_{5} + \beta_{4} - 6 \beta_1) q^{11} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 4) q^{13}+ \cdots - 98 q^{98}+O(q^{100}) $ q - 2b1 q^2 + (-4b1 - 4) q^4 + (b8 - b4 + b1 + 2) * q^5 + 7b1 q^7 - 8 * q^8 + (2b8 + 4) q^10 + (-b5 + b4 - 6b1) q^11 + (-b10 - b8 - b6 + b4 - 3b1 - 4) q^13 + (14b1 + 14) q^14 + 16b1 q^16 + (-b8 + b6 + b3 + b2 - 15) * q^17 + (-b11 + b9 + 2b7 + b3 + b2 + 16) q^19 + (4b4 - 4b1) * q^20 + (-2b8 + 2b7 - 2b5 + 2b4 - 12b1 - 14) q^22 + (b11 + b10 + 2b8 + b6 - 2b4 + b3 + 27b1 + 29) q^23 + (3b9 - b4 + b2 + 51b1) * q^25 + (-2b8 - 2b6 - 8) * q^26 + 28 * q^28 + (2b10 + 2b9 - b5 + b4 + b2 - 19b1) q^29 + (3b11 + 2b10 + 5b8 + 2b7 + 2b6 - 2b5 - 5b4 + 2b3 - 24b1 - 19) q^31 + (32b1 + 32) q^32 + (-2b10 - 2b4 + 2b2 + 28b1) * q^34 + (-7b8 - 14) q^35 + (-b11 + b9 - 11b8 + 4b3 + 4b2 + 73) q^37 + (2b9 + 4b5 + 2b2 - 32b1) * q^38 + (-8b8 + 8b4 - 8b1 - 16) q^40 + (b8 + 3b7 - 3b5 - b4 - 2b3 + 42b1 + 43) * q^41 + (-b10 + 4b9 - b5 + 12b4 + 2b2 + 163b1) * q^43 + (-4b8 + 4b7 - 28) * q^44 + (2b11 - 2b9 + 4b8 + 2b6 + 2b3 + 2b2 + 58) * q^46 + (-5b10 - 5b9 + 3b5 + 15b4 + b2 - 6b1) q^47 + (-49b1 - 49) q^49 + (6b11 + 2b8 - 2b4 - 2b3 + 102b1 + 104) q^50 + (4b10 - 4b4 + 12b1) q^52 + (2b11 - 2b9 - 14b8 + b7 + b6 - b3 - b2 - 22) q^53 + (-b11 + b9 + 6b8 + 5b7 - 6b6 + b3 + b2 + 119) q^55 - 56b1 q^56 + (4b11 + 4b10 - 2b8 + 2b7 + 4b6 - 2b5 + 2b4 - 2b3 - 38b1 - 40) q^58 + (b11 - 5b10 - 4b8 + 6b7 - 5b6 - 6b5 + 4b4 - 3b3 + 76b1 + 72) * q^59 + (7b10 - b9 - 7b5 + 32b4 - 3b2 + 179b1) q^61 + (6b11 - 6b9 + 10b8 + 4b7 + 4b6 + 4b3 + 4b2 - 38) q^62 + 64 * q^64 + (7b9 + 12b5 - 15b4 + 4b2 - 132b1) q^65 + (2b11 - 5b10 + 27b8 + 6b7 - 5b6 - 6b5 - 27b4 + 3b3 - 124b1 - 97) q^67 + (-4b10 + 4b8 - 4b6 - 4b4 - 4b3 + 56b1 + 60) * q^68 + (-14b4 + 14b1) * q^70 + (-b11 + b9 + 27b8 + 8b7 - 2b6 + 5b3 + 5b2 - 71) q^71 + (-4b11 + 4b9 + 14b8 - 12b7 + 6b6 + 3b3 + 3b2 + 31) q^73 + (2b9 - 22b4 + 8b2 - 168b1) * q^74 + (4b11 - 8b7 + 8b5 - 4b3 - 64b1 - 64) q^76 + (7b8 - 7b7 + 7b5 - 7b4 + 42b1 + 49) q^77 + (-11b10 - 8b5 - 9b4 + 9b2 + 97b1) q^79 + (-16b8 - 32) q^80 + (2b8 + 6b7 - 4b3 - 4b2 + 86) * q^82 + (4b10 - 3b9 - 17b5 + 14b4 - b2 - 79b1) q^83 + (-10b11 + 3b10 - 27b8 + 10b7 + 3b6 - 10b5 + 27b4 + 12b3 - 247b1 - 274) q^85 + (8b11 - 2b10 - 24b8 + 2b7 - 2b6 - 2b5 + 24b4 - 4b3 + 326b1 + 302) q^86 + (8b5 - 8b4 + 48b1) q^88 + (7b11 - 7b9 + 28b8 + 2b7 - 5b6 - 14b3 - 14b2 - 49) q^89 + (7b8 + 7b6 + 28) * q^91 + (-4b10 - 4b9 + 8b4 + 4b2 - 108b1) q^92 + (-10b11 - 10b10 - 30b8 - 6b7 - 10b6 + 6b5 + 30b4 - 2b3 - 12b1 - 42) q^94 + (4b11 + 21b10 - 26b8 - 15b7 + 21b6 + 15b5 + 26b4 + 2b3 + 154b1 + 128) q^95 + (5b10 + 5b9 + 20b5 + 4b4 - 9b2 + 384b1) * q^97 - 98 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} - 24 q^{4} + 9 q^{5} - 42 q^{7} - 96 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 - 24 * q^4 + 9 * q^5 - 42 * q^7 - 96 * q^8	$ 12 q + 12 q^{2} - 24 q^{4} + 9 q^{5} - 42 q^{7} - 96 q^{8} + 36 q^{10} + 39 q^{11} - 21 q^{13} + 84 q^{14} - 96 q^{16} - 174 q^{17} + 192 q^{19} + 36 q^{20} - 78 q^{22} + 168 q^{23} - 309 q^{25} - 84 q^{26} + 336 q^{28} + 117 q^{29} - 129 q^{31} + 192 q^{32} - 174 q^{34} - 126 q^{35} + 942 q^{37} + 192 q^{38} - 72 q^{40} + 255 q^{41} - 942 q^{43} - 312 q^{44} + 672 q^{46} + 81 q^{47} - 294 q^{49} + 618 q^{50} - 84 q^{52} - 180 q^{53} + 1392 q^{55} + 336 q^{56} - 234 q^{58} + 444 q^{59} - 978 q^{61} - 516 q^{62} + 768 q^{64} + 747 q^{65} - 663 q^{67} + 348 q^{68} - 126 q^{70} - 1014 q^{71} + 288 q^{73} + 942 q^{74} - 384 q^{76} + 273 q^{77} - 609 q^{79} - 288 q^{80} + 1020 q^{82} + 516 q^{83} - 1563 q^{85} + 1884 q^{86} - 312 q^{88} - 756 q^{89} + 294 q^{91} + 672 q^{92} - 162 q^{94} + 846 q^{95} - 2292 q^{97} - 1176 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 - 24 * q^4 + 9 * q^5 - 42 * q^7 - 96 * q^8 + 36 * q^10 + 39 * q^11 - 21 * q^13 + 84 * q^14 - 96 * q^16 - 174 * q^17 + 192 * q^19 + 36 * q^20 - 78 * q^22 + 168 * q^23 - 309 * q^25 - 84 * q^26 + 336 * q^28 + 117 * q^29 - 129 * q^31 + 192 * q^32 - 174 * q^34 - 126 * q^35 + 942 * q^37 + 192 * q^38 - 72 * q^40 + 255 * q^41 - 942 * q^43 - 312 * q^44 + 672 * q^46 + 81 * q^47 - 294 * q^49 + 618 * q^50 - 84 * q^52 - 180 * q^53 + 1392 * q^55 + 336 * q^56 - 234 * q^58 + 444 * q^59 - 978 * q^61 - 516 * q^62 + 768 * q^64 + 747 * q^65 - 663 * q^67 + 348 * q^68 - 126 * q^70 - 1014 * q^71 + 288 * q^73 + 942 * q^74 - 384 * q^76 + 273 * q^77 - 609 * q^79 - 288 * q^80 + 1020 * q^82 + 516 * q^83 - 1563 * q^85 + 1884 * q^86 - 312 * q^88 - 756 * q^89 + 294 * q^91 + 672 * q^92 - 162 * q^94 + 846 * q^95 - 2292 * q^97 - 1176 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 3 x^{10} + 27 x^{9} + 159 x^{8} + 2001 x^{7} + 12876 x^{6} + 4140 x^{5} + \cdots + 256562019 $ x^12 - x^11 + 3*x^10 + 27*x^9 + 159*x^8 + 2001*x^7 + 12876*x^6 + 4140*x^5 + 332841*x^4 + 2657720*x^3 + 20571241*x^2 + 107277462*x + 256562019 :

$\beta_{1}$	$=$	$ ( - 549265 \nu^{11} - 1650591 \nu^{10} + 28908834 \nu^{9} - 267393729 \nu^{8} + \cdots - 155590600968711 ) / 78506715704076 $ (-549265v^11 - 1650591v^10 + 28908834v^9 - 267393729v^8 + 1583466489v^7 - 8898227688v^6 + 1426521402v^5 - 3261562938v^4 - 197335147809v^3 - 2259702255926v^2 - 7469121208482*v - 155590600968711) / 78506715704076
$\beta_{2}$	$=$	$ ( 3608779 \nu^{11} + 62972154 \nu^{10} - 174314472 \nu^{9} + 3118665387 \nu^{8} + \cdots + 18\!\cdots\!60 ) / 43614842057820 $ (3608779v^11 + 62972154v^10 - 174314472v^9 + 3118665387v^8 - 17977232394v^7 + 45572076750v^6 + 122295863556v^5 + 1128967614702v^4 + 6305147536845v^3 + 61031266452569v^2 + 305312087564397*v + 1851795828156960) / 43614842057820
$\beta_{3}$	$=$	$ ( - 60717364 \nu^{11} - 332637651 \nu^{10} + 722539770 \nu^{9} - 21152574516 \nu^{8} + \cdots - 16\!\cdots\!41 ) / 392533578520380 $ (-60717364v^11 - 332637651v^10 + 722539770v^9 - 21152574516v^8 + 57571611273v^7 - 889552159338v^6 + 991545368634v^5 - 219909941100v^4 - 42331190199006v^3 - 633233192067629v^2 - 4630942726191549*v - 16849314100237941) / 392533578520380
$\beta_{4}$	$=$	$ ( 114956438 \nu^{11} - 433180197 \nu^{10} + 873780306 \nu^{9} + 3580475874 \nu^{8} + \cdots + 68\!\cdots\!05 ) / 392533578520380 $ (114956438v^11 - 433180197v^10 + 873780306v^9 + 3580475874v^8 - 4617503253v^7 + 333213715590v^6 - 297181511118v^5 + 6796789126704v^4 - 5099171340900v^3 + 291640104399283v^2 + 1711031508147339*v + 6877171947702105) / 392533578520380
$\beta_{5}$	$=$	$ ( 10247792 \nu^{11} + 69421482 \nu^{10} - 790961826 \nu^{9} + 4876259361 \nu^{8} + \cdots + 17\!\cdots\!65 ) / 32711131543365 $ (10247792v^11 + 69421482v^10 - 790961826v^9 + 4876259361v^8 - 13865454207v^7 + 53747231280v^6 + 211904392518v^5 - 503033992614v^4 - 258814947360v^3 + 33165615557752v^2 + 196069402258326*v + 1724549809225365) / 32711131543365
$\beta_{6}$	$=$	$ ( - 111337 \nu^{11} - 106245 \nu^{10} + 2640498 \nu^{9} - 10299117 \nu^{8} + \cdots - 10010494728729 ) / 197153982180 $ (-111337v^11 - 106245v^10 + 2640498v^9 - 10299117v^8 - 1564677v^7 - 87780612v^6 - 2343309318v^5 + 5273498322v^4 - 26486007249v^3 - 613188867752v^2 - 1150621677324*v - 10010494728729) / 197153982180
$\beta_{7}$	$=$	$ ( - 40048 \nu^{11} + 236385 \nu^{10} - 1251363 \nu^{9} + 4980507 \nu^{8} - 9686223 \nu^{7} + \cdots - 1565690641101 ) / 49288495545 $ (-40048v^11 + 236385v^10 - 1251363v^9 + 4980507v^8 - 9686223v^7 - 121404738v^6 + 360051588v^5 - 1034914032v^4 - 15845880756v^3 - 26647430273v^2 - 699322689036*v - 1565690641101) / 49288495545
$\beta_{8}$	$=$	$ ( 172621 \nu^{11} - 971055 \nu^{10} + 3718146 \nu^{9} - 3648939 \nu^{8} + 14363001 \nu^{7} + \cdots + 7168034708757 ) / 197153982180 $ (172621v^11 - 971055v^10 + 3718146v^9 - 3648939v^8 + 14363001v^7 + 327108216v^6 + 289429254v^5 + 86706594v^4 + 47110474497v^3 + 247552250936v^2 + 2177903279412*v + 7168034708757) / 197153982180
$\beta_{9}$	$=$	$ ( - 390129904 \nu^{11} + 624570291 \nu^{10} + 959177022 \nu^{9} - 57580338432 \nu^{8} + \cdots - 42\!\cdots\!15 ) / 392533578520380 $ (-390129904v^11 + 624570291v^10 + 959177022v^9 - 57580338432v^8 + 320175048699v^7 - 2483099520870v^6 + 4186354775274v^5 - 14572863171792v^4 - 125459326115130v^3 - 1286843775174059v^2 - 3953878649905707*v - 42158765221014015) / 392533578520380
$\beta_{10}$	$=$	$ ( 497582731 \nu^{11} + 421097346 \nu^{10} - 16130014728 \nu^{9} + 123131442063 \nu^{8} + \cdots + 53\!\cdots\!60 ) / 392533578520380 $ (497582731v^11 + 421097346v^10 - 16130014728v^9 + 123131442063v^8 - 334331628246v^7 + 2311817834010v^6 + 637391084184v^5 + 6763864436478v^4 + 182668415374785v^3 + 2000594025304961v^2 + 7286572802084913*v + 53982175251501360) / 392533578520380
$\beta_{11}$	$=$	$ ( 172902613 \nu^{11} - 1063790118 \nu^{10} + 6355493940 \nu^{9} - 32374640643 \nu^{8} + \cdots + 46\!\cdots\!92 ) / 130844526173460 $ (172902613v^11 - 1063790118v^10 + 6355493940v^9 - 32374640643v^8 + 157481999334v^7 - 108588755994v^6 + 1868171046372v^5 - 476507399310v^4 + 60403559933667v^3 + 53092925162903v^2 + 2741580373239783*v + 4684165425001692) / 130844526173460

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

$\nu$	$=$	$ ( -\beta_{11} + 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 5 ) / 27 $ (-b11 + 2b9 + b8 - 2b7 + b5 + b4 - b3 + b2 + 5*b1 + 5) / 27
$\nu^{2}$	$=$	$ ( - 4 \beta_{11} + 2 \beta_{9} + 4 \beta_{8} + \beta_{7} - 9 \beta_{6} - 11 \beta_{5} + 7 \beta_{4} + \cdots - 25 ) / 27 $ (-4b11 + 2b9 + 4b8 + b7 - 9b6 - 11b5 + 7b4 - 4b3 + b2 - 28b1 - 25) / 27
$\nu^{3}$	$=$	$ ( 17 \beta_{11} + 45 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 2 \beta_{7} + 18 \beta_{6} - 56 \beta_{5} + \cdots - 346 ) / 27 $ (17b11 + 45b10 + 8b9 - 8b8 - 2b7 + 18b6 - 56b5 - 137b4 + 17b3 + 22b2 - 352*b1 - 346) / 27
$\nu^{4}$	$=$	$ ( 188 \beta_{11} + 189 \beta_{10} + 47 \beta_{9} - 377 \beta_{8} + 169 \beta_{7} + 216 \beta_{6} + \cdots - 4828 ) / 27 $ (188b11 + 189b10 + 47b9 - 377b8 + 169b7 + 216b6 - 323b5 + 775b4 + 224b3 + 217b2 - 5575*b1 - 4828) / 27
$\nu^{5}$	$=$	$ ( 1202 \beta_{11} + 162 \beta_{10} + 641 \beta_{9} - 392 \beta_{8} + 1738 \beta_{7} + 153 \beta_{6} + \cdots - 67291 ) / 27 $ (1202b11 + 162b10 + 641b9 - 392b8 + 1738b7 + 153b6 - 1082b5 + 1447b4 + 284b3 + 496b2 - 86221*b1 - 67291) / 27
$\nu^{6}$	$=$	$ ( 6866 \beta_{11} + 1584 \beta_{10} - 1243 \beta_{9} - 8153 \beta_{8} + 8080 \beta_{7} + 2691 \beta_{6} + \cdots - 431977 ) / 27 $ (6866b11 + 1584b10 - 1243b9 - 8153b8 + 8080b7 + 2691b6 - 8066b5 + 6595b4 - 5554b3 - 8042b2 - 472942*b1 - 431977) / 27
$\nu^{7}$	$=$	$ ( 44702 \beta_{11} + 27162 \beta_{10} - 21373 \beta_{9} - 35900 \beta_{8} + 72970 \beta_{7} + \cdots - 142024 ) / 27 $ (44702b11 + 27162b10 - 21373b9 - 35900b8 + 72970b7 + 36018b6 - 35873b5 + 18523b4 - 42445b3 - 54503b2 - 10957*b1 - 142024) / 27
$\nu^{8}$	$=$	$ ( 125420 \beta_{11} + 70308 \beta_{10} - 165931 \beta_{9} + 173632 \beta_{8} + 445924 \beta_{7} + \cdots - 1200625 ) / 27 $ (125420b11 + 70308b10 - 165931b9 + 173632b8 + 445924b7 + 268830b6 + 4516b5 - 137783b4 - 176197b3 - 86219b2 + 5552315*b1 - 1200625) / 27
$\nu^{9}$	$=$	$ ( 725057 \beta_{11} - 936558 \beta_{10} - 2013481 \beta_{9} + 381169 \beta_{8} + 1796299 \beta_{7} + \cdots - 20728048 ) / 27 $ (725057b11 - 936558b10 - 2013481b9 + 381169b8 + 1796299b7 + 617724b6 + 919393b5 - 1462736b4 - 872722b3 + 227056b2 + 44432612*b1 - 20728048) / 27
$\nu^{10}$	$=$	$ ( 4494401 \beta_{11} - 5966676 \beta_{10} - 13463845 \beta_{9} - 10573235 \beta_{8} + 1568455 \beta_{7} + \cdots - 216184153 ) / 27 $ (4494401b11 - 5966676b10 - 13463845b9 - 10573235b8 + 1568455b7 - 41175b6 + 12206557b5 - 9715157b4 - 1326034b3 + 1088929b2 + 360533048*b1 - 216184153) / 27
$\nu^{11}$	$=$	$ ( 7190042 \beta_{11} - 29814885 \beta_{10} - 63126781 \beta_{9} - 30067277 \beta_{8} - 20478419 \beta_{7} + \cdots - 903351877 ) / 27 $ (7190042b11 - 29814885b10 - 63126781b9 - 30067277b8 - 20478419b7 - 8321301b6 + 87645796b5 - 28177382b4 + 34794923b3 - 1808891b2 + 2306059118*b1 - 903351877) / 27

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1 - \beta_{1}$	$1$

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} $

127.1

−1.48269	+	4.56467i
−3.70580	−	2.20351i
5.63850	−	2.34186i
3.71435	+	4.43083i
0.291003	−	5.14756i
−3.95537	+	1.56344i
−1.48269	−	4.56467i
−3.70580	+	2.20351i
5.63850	+	2.34186i
3.71435	−	4.43083i
0.291003	+	5.14756i
−3.95537	−	1.56344i

1.00000

−

1.73205i

−2.00000

−

3.46410i

−9.01223

−

15.6096i

−3.50000

6.06218i

−8.00000

−36.0489

127.2

1.00000

−

1.73205i

−2.00000

−

3.46410i

−6.20795

−

10.7525i

−3.50000

6.06218i

−8.00000

−24.8318

127.3

1.00000

−

1.73205i

−2.00000

−

3.46410i

1.45017

2.51177i

−3.50000

6.06218i

−8.00000

5.80068

127.4

1.00000

−

1.73205i

−2.00000

−

3.46410i

3.48648

6.03876i

−3.50000

6.06218i

−8.00000

13.9459

127.5

1.00000

−

1.73205i

−2.00000

−

3.46410i

4.11639

7.12980i

−3.50000

6.06218i

−8.00000

16.4656

127.6

1.00000

−

1.73205i

−2.00000

−

3.46410i

10.6671

18.4760i

−3.50000

6.06218i

−8.00000

42.6686

253.1

1.00000

1.73205i

−2.00000

3.46410i

−9.01223

15.6096i

−3.50000

−

6.06218i

−8.00000

−36.0489

253.2

1.00000

1.73205i

−2.00000

3.46410i

−6.20795

10.7525i

−3.50000

−

6.06218i

−8.00000

−24.8318

253.3

1.00000

1.73205i

−2.00000

3.46410i

1.45017

−

2.51177i

−3.50000

−

6.06218i

−8.00000

5.80068

253.4

1.00000

1.73205i

−2.00000

3.46410i

3.48648

−

6.03876i

−3.50000

−

6.06218i

−8.00000

13.9459

253.5

1.00000

1.73205i

−2.00000

3.46410i

4.11639

−

7.12980i

−3.50000

−

6.06218i

−8.00000

16.4656

253.6

1.00000

1.73205i

−2.00000

3.46410i

10.6671

−

18.4760i

−3.50000

−

6.06218i

−8.00000

42.6686

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.4.f.d		12
3.b	odd	2	1		126.4.f.d	✓	12
9.c	even	3	1	inner	378.4.f.d		12
9.c	even	3	1		1134.4.a.s		6
9.d	odd	6	1		126.4.f.d	✓	12
9.d	odd	6	1		1134.4.a.t		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.4.f.d	✓	12	3.b	odd	2	1
126.4.f.d	✓	12	9.d	odd	6	1
378.4.f.d		12	1.a	even	1	1	trivial
378.4.f.d		12	9.c	even	3	1	inner
1134.4.a.s		6	9.c	even	3	1
1134.4.a.t		6	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} - 9 T_{5}^{11} + 570 T_{5}^{10} - 2979 T_{5}^{9} + 230607 T_{5}^{8} - 1467153 T_{5}^{7} + \cdots + 631920064356 $ T5^12 - 9*T5^11 + 570*T5^10 - 2979*T5^9 + 230607*T5^8 - 1467153*T5^7 + 31885029*T5^6 - 257864796*T5^5 + 3656444652*T5^4 - 23130961020*T5^3 + 138041924409*T5^2 - 328923815850*T5 + 631920064356 acting on $S_{4}^{\mathrm{new}}(378, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - 2 T + 4)^{6} $ (T^2 - 2*T + 4)^6
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + \cdots + 631920064356 $ T^12 - 9T^11 + 570T^10 - 2979T^9 + 230607T^8 - 1467153T^7 + 31885029T^6 - 257864796T^5 + 3656444652T^4 - 23130961020T^3 + 138041924409T^2 - 328923815850*T + 631920064356
$7$	$ (T^{2} + 7 T + 49)^{6} $ (T^2 + 7*T + 49)^6
$11$	$ T^{12} + \cdots + 12\!\cdots\!76 $ T^12 - 39T^11 + 5466T^10 - 109125T^9 + 19126692T^8 - 435885300T^7 + 21229580331T^6 - 89731422054T^5 + 6211911166029T^4 + 31215410940348T^3 + 2085740798635548T^2 + 13784843595709296*T + 123762905086626576
$13$	$ T^{12} + \cdots + 26\!\cdots\!44 $ T^12 + 21T^11 + 10524T^10 + 79843T^9 + 78472404T^8 + 441052083T^7 + 252065571819T^6 - 2680136318826T^5 + 536890897841328T^4 - 2783814597445352T^3 + 430547372113285056T^2 - 1194531167389434720*T + 262137248193115284544
$17$	$ (T^{6} + 87 T^{5} + \cdots + 19974003516)^{2} $ (T^6 + 87T^5 - 11091T^4 - 1163376T^3 - 1513773T^2 + 1609255620*T + 19974003516)^2
$19$	$ (T^{6} - 96 T^{5} + \cdots - 49162528397)^{2} $ (T^6 - 96T^5 - 24912T^4 + 1905245T^3 + 125787324T^2 - 3030221421*T - 49162528397)^2
$23$	$ T^{12} + \cdots + 29\!\cdots\!16 $ T^12 - 168T^11 + 42312T^10 - 3145194T^9 + 650738691T^8 - 41885290431T^7 + 6518356556481T^6 - 149162571008487T^5 + 17727387888655404T^4 + 226894162522546917T^3 + 45034330383050846661T^2 + 361060320451326250716*T + 2932653810212741619216
$29$	$ T^{12} + \cdots + 47\!\cdots\!04 $ T^12 - 117T^11 + 72417T^10 + 6979950T^9 + 2972510865T^8 + 133893941868T^7 + 30466552541517T^6 + 2379869161861968T^5 + 234865719422110620T^4 + 9752683984985644128T^3 + 326307070849552335504T^2 + 4490435438425214074944*T + 47078294328520061647104
$31$	$ T^{12} + \cdots + 54\!\cdots\!76 $ T^12 + 129T^11 + 166179T^10 + 7478896T^9 + 18530729487T^8 + 905631053850T^7 + 951880397930169T^6 + 25738390342042794T^5 + 34307854540667492760T^4 + 1681015377236533952584T^3 + 155144690083181085530016T^2 - 2489099567243143260523488*T + 54246409708449008143475776
$37$	$ (T^{6} + \cdots + 329496675669784)^{2} $ (T^6 - 471T^5 - 127704T^4 + 92087309T^3 - 6752138430T^2 - 2527911546420*T + 329496675669784)^2
$41$	$ T^{12} + \cdots + 12\!\cdots\!76 $ T^12 - 255T^11 + 143436T^10 - 272511T^9 + 8002559754T^8 - 465561560952T^7 + 148684258920123T^6 + 653463913379778T^5 + 961399333100493657T^4 - 29253219360104302704T^3 + 1945141252228819036224T^2 + 4803685226091338812416*T + 12379906149639428837376
$43$	$ T^{12} + \cdots + 25\!\cdots\!76 $ T^12 + 942T^11 + 746577T^10 + 257292988T^9 + 86263249554T^8 + 7598277967689T^7 + 3152539298755050T^6 - 79990704937936197T^5 + 180510857538735007197T^4 - 20338766343255098391800T^3 + 2948837760019982974112412T^2 - 91357621135471902168372096*T + 2502516693668500527066303376
$47$	$ T^{12} + \cdots + 71\!\cdots\!36 $ T^12 - 81T^11 + 447687T^10 + 381996T^9 + 148378351257T^8 + 1196753916834T^7 + 20899599513589593T^6 + 1988530456796812008T^5 + 2129839086030661491840T^4 + 70177135947401739446784T^3 + 14339677144148462626283520T^2 - 340214390357778122095558656*T + 71209698671127825299372507136
$53$	$ (T^{6} + \cdots + 16537522636368)^{2} $ (T^6 + 90T^5 - 136533T^4 - 22542345T^3 + 1808827884T^2 + 446488323228*T + 16537522636368)^2
$59$	$ T^{12} + \cdots + 13\!\cdots\!36 $ T^12 - 444T^11 + 567063T^10 + 12366810T^9 + 131099064570T^8 + 4408298044245T^7 + 16936067095862670T^6 + 1625646449688830703T^5 + 1325121030576494121465T^4 + 74307891928570381202832T^3 + 51863073205933104123292704T^2 + 2925835478655020619977369088*T + 1322987095644544709830754804736
$61$	$ T^{12} + \cdots + 99\!\cdots\!96 $ T^12 + 978T^11 + 1718604T^10 + 1253528206T^9 + 1646257720257T^8 + 1078799606806167T^7 + 853239482255498955T^6 + 339899279938031695941T^5 + 176777819842527292532166T^4 + 50444053685540526814651327T^3 + 23812959490476103768321102893T^2 + 4585434536647591988291375603076*T + 999763280712194849538884674385296
$67$	$ T^{12} + \cdots + 96\!\cdots\!24 $ T^12 + 663T^11 + 1459170T^10 + 484428265T^9 + 1338848732556T^8 + 499311344409786T^7 + 410016993474976197T^6 - 6690798207880901496T^5 + 19575308580263102415033T^4 + 1906700407726120193156152T^3 + 432249687978246492387213840T^2 + 2056701437696093083644143232*T + 9600178382203339248099361024
$71$	$ (T^{6} + \cdots - 28\!\cdots\!24)^{2} $ (T^6 + 507T^5 - 894789T^4 - 48186954T^3 + 133872603714T^2 - 1998177550083*T - 2860222732066224)^2
$73$	$ (T^{6} + \cdots + 50\!\cdots\!28)^{2} $ (T^6 - 144T^5 - 1206627T^4 + 486607379T^3 + 137643578343T^2 - 70614731374728*T + 5086061693234428)^2
$79$	$ T^{12} + \cdots + 21\!\cdots\!56 $ T^12 + 609T^11 + 2180538T^10 + 805495831T^9 + 2876167266273T^8 + 897312889612665T^7 + 2168725174977025347T^6 + 295665679829633318568T^5 + 1043083270724024006205534T^4 + 78671335479921658467674536T^3 + 275649763660068420545418871665T^2 - 53149385888700706606169488660362*T + 21276713869610323903239102953321956
$83$	$ T^{12} + \cdots + 16\!\cdots\!44 $ T^12 - 516T^11 + 1831629T^10 - 109106478T^9 + 1882182316725T^8 + 24267642507747T^7 + 1169005111856386713T^6 + 157227818031060080760T^5 + 488566412438145949559952T^4 + 45628753198564241202807936T^3 + 110522148945179582616645282048T^2 + 11308278947911119936196561575936*T + 16274807035583753997133827434348544
$89$	$ (T^{6} + \cdots + 12\!\cdots\!12)^{2} $ (T^6 + 378T^5 - 2341893T^4 - 1187169237T^3 + 1209054877866T^2 + 842962534116192*T + 127820616579964512)^2
$97$	$ T^{12} + \cdots + 97\!\cdots\!76 $ T^12 + 2292T^11 + 6935733T^10 + 8158264378T^9 + 17950399247538T^8 + 18749169815336235T^7 + 30178588437514195800T^6 + 18622213611565743726003T^5 + 18198979765266743722444509T^4 + 8569717181758490190027776320T^3 + 7332062704488821669116481612700T^2 + 2420955170651590766380784444978016*T + 977441807203397358833955124983788176
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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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{8} - \beta_{4} + \beta_1 + 2) q^{5} + 7 \beta_1 q^{7} - 8 q^{8}+O(q^{10}) \) q - 2b1 q^2 + (-4b1 - 4) q^4 + (b8 - b4 + b1 + 2) * q^5 + 7b1 q^7 - 8 * q^8	\( q - 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{8} - \beta_{4} + \beta_1 + 2) q^{5} + 7 \beta_1 q^{7} - 8 q^{8} + (2 \beta_{8} + 4) q^{10} + ( - \beta_{5} + \beta_{4} - 6 \beta_1) q^{11} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 4) q^{13}+ \cdots - 98 q^{98}+O(q^{100}) \) q - 2b1 q^2 + (-4b1 - 4) q^4 + (b8 - b4 + b1 + 2) * q^5 + 7b1 q^7 - 8 * q^8 + (2b8 + 4) q^10 + (-b5 + b4 - 6b1) q^11 + (-b10 - b8 - b6 + b4 - 3b1 - 4) q^13 + (14b1 + 14) q^14 + 16b1 q^16 + (-b8 + b6 + b3 + b2 - 15) * q^17 + (-b11 + b9 + 2b7 + b3 + b2 + 16) q^19 + (4b4 - 4b1) * q^20 + (-2b8 + 2b7 - 2b5 + 2b4 - 12b1 - 14) q^22 + (b11 + b10 + 2b8 + b6 - 2b4 + b3 + 27b1 + 29) q^23 + (3b9 - b4 + b2 + 51b1) * q^25 + (-2b8 - 2b6 - 8) * q^26 + 28 * q^28 + (2b10 + 2b9 - b5 + b4 + b2 - 19b1) q^29 + (3b11 + 2b10 + 5b8 + 2b7 + 2b6 - 2b5 - 5b4 + 2b3 - 24b1 - 19) q^31 + (32b1 + 32) q^32 + (-2b10 - 2b4 + 2b2 + 28b1) * q^34 + (-7b8 - 14) q^35 + (-b11 + b9 - 11b8 + 4b3 + 4b2 + 73) q^37 + (2b9 + 4b5 + 2b2 - 32b1) * q^38 + (-8b8 + 8b4 - 8b1 - 16) q^40 + (b8 + 3b7 - 3b5 - b4 - 2b3 + 42b1 + 43) * q^41 + (-b10 + 4b9 - b5 + 12b4 + 2b2 + 163b1) * q^43 + (-4b8 + 4b7 - 28) * q^44 + (2b11 - 2b9 + 4b8 + 2b6 + 2b3 + 2b2 + 58) * q^46 + (-5b10 - 5b9 + 3b5 + 15b4 + b2 - 6b1) q^47 + (-49b1 - 49) q^49 + (6b11 + 2b8 - 2b4 - 2b3 + 102b1 + 104) q^50 + (4b10 - 4b4 + 12b1) q^52 + (2b11 - 2b9 - 14b8 + b7 + b6 - b3 - b2 - 22) q^53 + (-b11 + b9 + 6b8 + 5b7 - 6b6 + b3 + b2 + 119) q^55 - 56b1 q^56 + (4b11 + 4b10 - 2b8 + 2b7 + 4b6 - 2b5 + 2b4 - 2b3 - 38b1 - 40) q^58 + (b11 - 5b10 - 4b8 + 6b7 - 5b6 - 6b5 + 4b4 - 3b3 + 76b1 + 72) * q^59 + (7b10 - b9 - 7b5 + 32b4 - 3b2 + 179b1) q^61 + (6b11 - 6b9 + 10b8 + 4b7 + 4b6 + 4b3 + 4b2 - 38) q^62 + 64 * q^64 + (7b9 + 12b5 - 15b4 + 4b2 - 132b1) q^65 + (2b11 - 5b10 + 27b8 + 6b7 - 5b6 - 6b5 - 27b4 + 3b3 - 124b1 - 97) q^67 + (-4b10 + 4b8 - 4b6 - 4b4 - 4b3 + 56b1 + 60) * q^68 + (-14b4 + 14b1) * q^70 + (-b11 + b9 + 27b8 + 8b7 - 2b6 + 5b3 + 5b2 - 71) q^71 + (-4b11 + 4b9 + 14b8 - 12b7 + 6b6 + 3b3 + 3b2 + 31) q^73 + (2b9 - 22b4 + 8b2 - 168b1) * q^74 + (4b11 - 8b7 + 8b5 - 4b3 - 64b1 - 64) q^76 + (7b8 - 7b7 + 7b5 - 7b4 + 42b1 + 49) q^77 + (-11b10 - 8b5 - 9b4 + 9b2 + 97b1) q^79 + (-16b8 - 32) q^80 + (2b8 + 6b7 - 4b3 - 4b2 + 86) * q^82 + (4b10 - 3b9 - 17b5 + 14b4 - b2 - 79b1) q^83 + (-10b11 + 3b10 - 27b8 + 10b7 + 3b6 - 10b5 + 27b4 + 12b3 - 247b1 - 274) q^85 + (8b11 - 2b10 - 24b8 + 2b7 - 2b6 - 2b5 + 24b4 - 4b3 + 326b1 + 302) q^86 + (8b5 - 8b4 + 48b1) q^88 + (7b11 - 7b9 + 28b8 + 2b7 - 5b6 - 14b3 - 14b2 - 49) q^89 + (7b8 + 7b6 + 28) * q^91 + (-4b10 - 4b9 + 8b4 + 4b2 - 108b1) q^92 + (-10b11 - 10b10 - 30b8 - 6b7 - 10b6 + 6b5 + 30b4 - 2b3 - 12b1 - 42) q^94 + (4b11 + 21b10 - 26b8 - 15b7 + 21b6 + 15b5 + 26b4 + 2b3 + 154b1 + 128) q^95 + (5b10 + 5b9 + 20b5 + 4b4 - 9b2 + 384b1) * q^97 - 98 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} - 24 q^{4} + 9 q^{5} - 42 q^{7} - 96 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 - 24 * q^4 + 9 * q^5 - 42 * q^7 - 96 * q^8	\( 12 q + 12 q^{2} - 24 q^{4} + 9 q^{5} - 42 q^{7} - 96 q^{8} + 36 q^{10} + 39 q^{11} - 21 q^{13} + 84 q^{14} - 96 q^{16} - 174 q^{17} + 192 q^{19} + 36 q^{20} - 78 q^{22} + 168 q^{23} - 309 q^{25} - 84 q^{26} + 336 q^{28} + 117 q^{29} - 129 q^{31} + 192 q^{32} - 174 q^{34} - 126 q^{35} + 942 q^{37} + 192 q^{38} - 72 q^{40} + 255 q^{41} - 942 q^{43} - 312 q^{44} + 672 q^{46} + 81 q^{47} - 294 q^{49} + 618 q^{50} - 84 q^{52} - 180 q^{53} + 1392 q^{55} + 336 q^{56} - 234 q^{58} + 444 q^{59} - 978 q^{61} - 516 q^{62} + 768 q^{64} + 747 q^{65} - 663 q^{67} + 348 q^{68} - 126 q^{70} - 1014 q^{71} + 288 q^{73} + 942 q^{74} - 384 q^{76} + 273 q^{77} - 609 q^{79} - 288 q^{80} + 1020 q^{82} + 516 q^{83} - 1563 q^{85} + 1884 q^{86} - 312 q^{88} - 756 q^{89} + 294 q^{91} + 672 q^{92} - 162 q^{94} + 846 q^{95} - 2292 q^{97} - 1176 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 - 24 * q^4 + 9 * q^5 - 42 * q^7 - 96 * q^8 + 36 * q^10 + 39 * q^11 - 21 * q^13 + 84 * q^14 - 96 * q^16 - 174 * q^17 + 192 * q^19 + 36 * q^20 - 78 * q^22 + 168 * q^23 - 309 * q^25 - 84 * q^26 + 336 * q^28 + 117 * q^29 - 129 * q^31 + 192 * q^32 - 174 * q^34 - 126 * q^35 + 942 * q^37 + 192 * q^38 - 72 * q^40 + 255 * q^41 - 942 * q^43 - 312 * q^44 + 672 * q^46 + 81 * q^47 - 294 * q^49 + 618 * q^50 - 84 * q^52 - 180 * q^53 + 1392 * q^55 + 336 * q^56 - 234 * q^58 + 444 * q^59 - 978 * q^61 - 516 * q^62 + 768 * q^64 + 747 * q^65 - 663 * q^67 + 348 * q^68 - 126 * q^70 - 1014 * q^71 + 288 * q^73 + 942 * q^74 - 384 * q^76 + 273 * q^77 - 609 * q^79 - 288 * q^80 + 1020 * q^82 + 516 * q^83 - 1563 * q^85 + 1884 * q^86 - 312 * q^88 - 756 * q^89 + 294 * q^91 + 672 * q^92 - 162 * q^94 + 846 * q^95 - 2292 * q^97 - 1176 * q^98

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	378.4.f.d

Level	$378$
Weight	$4$
Character orbit	378.f
Analytic conductor	$22.303$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(22.3027219822\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + 3 x^{10} + 27 x^{9} + 159 x^{8} + 2001 x^{7} + 12876 x^{6} + 4140 x^{5} + \cdots + 256562019 \) x^12 - x^11 + 3x^10 + 27x^9 + 159x^8 + 2001x^7 + 12876x^6 + 4140x^5 + 332841x^4 + 2657720x^3 + 20571241x^2 + 107277462x + 256562019
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{15} \)
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 549265 \nu^{11} - 1650591 \nu^{10} + 28908834 \nu^{9} - 267393729 \nu^{8} + \cdots - 155590600968711 ) / 78506715704076 \) (-549265v^11 - 1650591v^10 + 28908834v^9 - 267393729v^8 + 1583466489v^7 - 8898227688v^6 + 1426521402v^5 - 3261562938v^4 - 197335147809v^3 - 2259702255926v^2 - 7469121208482*v - 155590600968711) / 78506715704076
\(\beta_{2}\)	\(=\)	\( ( 3608779 \nu^{11} + 62972154 \nu^{10} - 174314472 \nu^{9} + 3118665387 \nu^{8} + \cdots + 18\!\cdots\!60 ) / 43614842057820 \) (3608779v^11 + 62972154v^10 - 174314472v^9 + 3118665387v^8 - 17977232394v^7 + 45572076750v^6 + 122295863556v^5 + 1128967614702v^4 + 6305147536845v^3 + 61031266452569v^2 + 305312087564397*v + 1851795828156960) / 43614842057820
\(\beta_{3}\)	\(=\)	\( ( - 60717364 \nu^{11} - 332637651 \nu^{10} + 722539770 \nu^{9} - 21152574516 \nu^{8} + \cdots - 16\!\cdots\!41 ) / 392533578520380 \) (-60717364v^11 - 332637651v^10 + 722539770v^9 - 21152574516v^8 + 57571611273v^7 - 889552159338v^6 + 991545368634v^5 - 219909941100v^4 - 42331190199006v^3 - 633233192067629v^2 - 4630942726191549*v - 16849314100237941) / 392533578520380
\(\beta_{4}\)	\(=\)	\( ( 114956438 \nu^{11} - 433180197 \nu^{10} + 873780306 \nu^{9} + 3580475874 \nu^{8} + \cdots + 68\!\cdots\!05 ) / 392533578520380 \) (114956438v^11 - 433180197v^10 + 873780306v^9 + 3580475874v^8 - 4617503253v^7 + 333213715590v^6 - 297181511118v^5 + 6796789126704v^4 - 5099171340900v^3 + 291640104399283v^2 + 1711031508147339*v + 6877171947702105) / 392533578520380
\(\beta_{5}\)	\(=\)	\( ( 10247792 \nu^{11} + 69421482 \nu^{10} - 790961826 \nu^{9} + 4876259361 \nu^{8} + \cdots + 17\!\cdots\!65 ) / 32711131543365 \) (10247792v^11 + 69421482v^10 - 790961826v^9 + 4876259361v^8 - 13865454207v^7 + 53747231280v^6 + 211904392518v^5 - 503033992614v^4 - 258814947360v^3 + 33165615557752v^2 + 196069402258326*v + 1724549809225365) / 32711131543365
\(\beta_{6}\)	\(=\)	\( ( - 111337 \nu^{11} - 106245 \nu^{10} + 2640498 \nu^{9} - 10299117 \nu^{8} + \cdots - 10010494728729 ) / 197153982180 \) (-111337v^11 - 106245v^10 + 2640498v^9 - 10299117v^8 - 1564677v^7 - 87780612v^6 - 2343309318v^5 + 5273498322v^4 - 26486007249v^3 - 613188867752v^2 - 1150621677324*v - 10010494728729) / 197153982180
\(\beta_{7}\)	\(=\)	\( ( - 40048 \nu^{11} + 236385 \nu^{10} - 1251363 \nu^{9} + 4980507 \nu^{8} - 9686223 \nu^{7} + \cdots - 1565690641101 ) / 49288495545 \) (-40048v^11 + 236385v^10 - 1251363v^9 + 4980507v^8 - 9686223v^7 - 121404738v^6 + 360051588v^5 - 1034914032v^4 - 15845880756v^3 - 26647430273v^2 - 699322689036*v - 1565690641101) / 49288495545
\(\beta_{8}\)	\(=\)	\( ( 172621 \nu^{11} - 971055 \nu^{10} + 3718146 \nu^{9} - 3648939 \nu^{8} + 14363001 \nu^{7} + \cdots + 7168034708757 ) / 197153982180 \) (172621v^11 - 971055v^10 + 3718146v^9 - 3648939v^8 + 14363001v^7 + 327108216v^6 + 289429254v^5 + 86706594v^4 + 47110474497v^3 + 247552250936v^2 + 2177903279412*v + 7168034708757) / 197153982180
\(\beta_{9}\)	\(=\)	\( ( - 390129904 \nu^{11} + 624570291 \nu^{10} + 959177022 \nu^{9} - 57580338432 \nu^{8} + \cdots - 42\!\cdots\!15 ) / 392533578520380 \) (-390129904v^11 + 624570291v^10 + 959177022v^9 - 57580338432v^8 + 320175048699v^7 - 2483099520870v^6 + 4186354775274v^5 - 14572863171792v^4 - 125459326115130v^3 - 1286843775174059v^2 - 3953878649905707*v - 42158765221014015) / 392533578520380
\(\beta_{10}\)	\(=\)	\( ( 497582731 \nu^{11} + 421097346 \nu^{10} - 16130014728 \nu^{9} + 123131442063 \nu^{8} + \cdots + 53\!\cdots\!60 ) / 392533578520380 \) (497582731v^11 + 421097346v^10 - 16130014728v^9 + 123131442063v^8 - 334331628246v^7 + 2311817834010v^6 + 637391084184v^5 + 6763864436478v^4 + 182668415374785v^3 + 2000594025304961v^2 + 7286572802084913*v + 53982175251501360) / 392533578520380
\(\beta_{11}\)	\(=\)	\( ( 172902613 \nu^{11} - 1063790118 \nu^{10} + 6355493940 \nu^{9} - 32374640643 \nu^{8} + \cdots + 46\!\cdots\!92 ) / 130844526173460 \) (172902613v^11 - 1063790118v^10 + 6355493940v^9 - 32374640643v^8 + 157481999334v^7 - 108588755994v^6 + 1868171046372v^5 - 476507399310v^4 + 60403559933667v^3 + 53092925162903v^2 + 2741580373239783*v + 4684165425001692) / 130844526173460

\(\nu\)	\(=\)	\( ( -\beta_{11} + 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 5 ) / 27 \) (-b11 + 2b9 + b8 - 2b7 + b5 + b4 - b3 + b2 + 5*b1 + 5) / 27
\(\nu^{2}\)	\(=\)	\( ( - 4 \beta_{11} + 2 \beta_{9} + 4 \beta_{8} + \beta_{7} - 9 \beta_{6} - 11 \beta_{5} + 7 \beta_{4} + \cdots - 25 ) / 27 \) (-4b11 + 2b9 + 4b8 + b7 - 9b6 - 11b5 + 7b4 - 4b3 + b2 - 28b1 - 25) / 27
\(\nu^{3}\)	\(=\)	\( ( 17 \beta_{11} + 45 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 2 \beta_{7} + 18 \beta_{6} - 56 \beta_{5} + \cdots - 346 ) / 27 \) (17b11 + 45b10 + 8b9 - 8b8 - 2b7 + 18b6 - 56b5 - 137b4 + 17b3 + 22b2 - 352*b1 - 346) / 27
\(\nu^{4}\)	\(=\)	\( ( 188 \beta_{11} + 189 \beta_{10} + 47 \beta_{9} - 377 \beta_{8} + 169 \beta_{7} + 216 \beta_{6} + \cdots - 4828 ) / 27 \) (188b11 + 189b10 + 47b9 - 377b8 + 169b7 + 216b6 - 323b5 + 775b4 + 224b3 + 217b2 - 5575*b1 - 4828) / 27
\(\nu^{5}\)	\(=\)	\( ( 1202 \beta_{11} + 162 \beta_{10} + 641 \beta_{9} - 392 \beta_{8} + 1738 \beta_{7} + 153 \beta_{6} + \cdots - 67291 ) / 27 \) (1202b11 + 162b10 + 641b9 - 392b8 + 1738b7 + 153b6 - 1082b5 + 1447b4 + 284b3 + 496b2 - 86221*b1 - 67291) / 27
\(\nu^{6}\)	\(=\)	\( ( 6866 \beta_{11} + 1584 \beta_{10} - 1243 \beta_{9} - 8153 \beta_{8} + 8080 \beta_{7} + 2691 \beta_{6} + \cdots - 431977 ) / 27 \) (6866b11 + 1584b10 - 1243b9 - 8153b8 + 8080b7 + 2691b6 - 8066b5 + 6595b4 - 5554b3 - 8042b2 - 472942*b1 - 431977) / 27
\(\nu^{7}\)	\(=\)	\( ( 44702 \beta_{11} + 27162 \beta_{10} - 21373 \beta_{9} - 35900 \beta_{8} + 72970 \beta_{7} + \cdots - 142024 ) / 27 \) (44702b11 + 27162b10 - 21373b9 - 35900b8 + 72970b7 + 36018b6 - 35873b5 + 18523b4 - 42445b3 - 54503b2 - 10957*b1 - 142024) / 27
\(\nu^{8}\)	\(=\)	\( ( 125420 \beta_{11} + 70308 \beta_{10} - 165931 \beta_{9} + 173632 \beta_{8} + 445924 \beta_{7} + \cdots - 1200625 ) / 27 \) (125420b11 + 70308b10 - 165931b9 + 173632b8 + 445924b7 + 268830b6 + 4516b5 - 137783b4 - 176197b3 - 86219b2 + 5552315*b1 - 1200625) / 27
\(\nu^{9}\)	\(=\)	\( ( 725057 \beta_{11} - 936558 \beta_{10} - 2013481 \beta_{9} + 381169 \beta_{8} + 1796299 \beta_{7} + \cdots - 20728048 ) / 27 \) (725057b11 - 936558b10 - 2013481b9 + 381169b8 + 1796299b7 + 617724b6 + 919393b5 - 1462736b4 - 872722b3 + 227056b2 + 44432612*b1 - 20728048) / 27
\(\nu^{10}\)	\(=\)	\( ( 4494401 \beta_{11} - 5966676 \beta_{10} - 13463845 \beta_{9} - 10573235 \beta_{8} + 1568455 \beta_{7} + \cdots - 216184153 ) / 27 \) (4494401b11 - 5966676b10 - 13463845b9 - 10573235b8 + 1568455b7 - 41175b6 + 12206557b5 - 9715157b4 - 1326034b3 + 1088929b2 + 360533048*b1 - 216184153) / 27
\(\nu^{11}\)	\(=\)	\( ( 7190042 \beta_{11} - 29814885 \beta_{10} - 63126781 \beta_{9} - 30067277 \beta_{8} - 20478419 \beta_{7} + \cdots - 903351877 ) / 27 \) (7190042b11 - 29814885b10 - 63126781b9 - 30067277b8 - 20478419b7 - 8321301b6 + 87645796b5 - 28177382b4 + 34794923b3 - 1808891b2 + 2306059118*b1 - 903351877) / 27

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 127.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - 2 T + 4)^{6} \) (T^2 - 2*T + 4)^6
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + \cdots + 631920064356 \) T^12 - 9T^11 + 570T^10 - 2979T^9 + 230607T^8 - 1467153T^7 + 31885029T^6 - 257864796T^5 + 3656444652T^4 - 23130961020T^3 + 138041924409T^2 - 328923815850*T + 631920064356
$7$	\( (T^{2} + 7 T + 49)^{6} \) (T^2 + 7*T + 49)^6
$11$	\( T^{12} + \cdots + 12\!\cdots\!76 \) T^12 - 39T^11 + 5466T^10 - 109125T^9 + 19126692T^8 - 435885300T^7 + 21229580331T^6 - 89731422054T^5 + 6211911166029T^4 + 31215410940348T^3 + 2085740798635548T^2 + 13784843595709296*T + 123762905086626576
$13$	\( T^{12} + \cdots + 26\!\cdots\!44 \) T^12 + 21T^11 + 10524T^10 + 79843T^9 + 78472404T^8 + 441052083T^7 + 252065571819T^6 - 2680136318826T^5 + 536890897841328T^4 - 2783814597445352T^3 + 430547372113285056T^2 - 1194531167389434720*T + 262137248193115284544
$17$	\( (T^{6} + 87 T^{5} + \cdots + 19974003516)^{2} \) (T^6 + 87T^5 - 11091T^4 - 1163376T^3 - 1513773T^2 + 1609255620*T + 19974003516)^2
$19$	\( (T^{6} - 96 T^{5} + \cdots - 49162528397)^{2} \) (T^6 - 96T^5 - 24912T^4 + 1905245T^3 + 125787324T^2 - 3030221421*T - 49162528397)^2
$23$	\( T^{12} + \cdots + 29\!\cdots\!16 \) T^12 - 168T^11 + 42312T^10 - 3145194T^9 + 650738691T^8 - 41885290431T^7 + 6518356556481T^6 - 149162571008487T^5 + 17727387888655404T^4 + 226894162522546917T^3 + 45034330383050846661T^2 + 361060320451326250716*T + 2932653810212741619216
$29$	\( T^{12} + \cdots + 47\!\cdots\!04 \) T^12 - 117T^11 + 72417T^10 + 6979950T^9 + 2972510865T^8 + 133893941868T^7 + 30466552541517T^6 + 2379869161861968T^5 + 234865719422110620T^4 + 9752683984985644128T^3 + 326307070849552335504T^2 + 4490435438425214074944*T + 47078294328520061647104
$31$	\( T^{12} + \cdots + 54\!\cdots\!76 \) T^12 + 129T^11 + 166179T^10 + 7478896T^9 + 18530729487T^8 + 905631053850T^7 + 951880397930169T^6 + 25738390342042794T^5 + 34307854540667492760T^4 + 1681015377236533952584T^3 + 155144690083181085530016T^2 - 2489099567243143260523488*T + 54246409708449008143475776
$37$	\( (T^{6} + \cdots + 329496675669784)^{2} \) (T^6 - 471T^5 - 127704T^4 + 92087309T^3 - 6752138430T^2 - 2527911546420*T + 329496675669784)^2
$41$	\( T^{12} + \cdots + 12\!\cdots\!76 \) T^12 - 255T^11 + 143436T^10 - 272511T^9 + 8002559754T^8 - 465561560952T^7 + 148684258920123T^6 + 653463913379778T^5 + 961399333100493657T^4 - 29253219360104302704T^3 + 1945141252228819036224T^2 + 4803685226091338812416*T + 12379906149639428837376
$43$	\( T^{12} + \cdots + 25\!\cdots\!76 \) T^12 + 942T^11 + 746577T^10 + 257292988T^9 + 86263249554T^8 + 7598277967689T^7 + 3152539298755050T^6 - 79990704937936197T^5 + 180510857538735007197T^4 - 20338766343255098391800T^3 + 2948837760019982974112412T^2 - 91357621135471902168372096*T + 2502516693668500527066303376
$47$	\( T^{12} + \cdots + 71\!\cdots\!36 \) T^12 - 81T^11 + 447687T^10 + 381996T^9 + 148378351257T^8 + 1196753916834T^7 + 20899599513589593T^6 + 1988530456796812008T^5 + 2129839086030661491840T^4 + 70177135947401739446784T^3 + 14339677144148462626283520T^2 - 340214390357778122095558656*T + 71209698671127825299372507136
$53$	\( (T^{6} + \cdots + 16537522636368)^{2} \) (T^6 + 90T^5 - 136533T^4 - 22542345T^3 + 1808827884T^2 + 446488323228*T + 16537522636368)^2
$59$	\( T^{12} + \cdots + 13\!\cdots\!36 \) T^12 - 444T^11 + 567063T^10 + 12366810T^9 + 131099064570T^8 + 4408298044245T^7 + 16936067095862670T^6 + 1625646449688830703T^5 + 1325121030576494121465T^4 + 74307891928570381202832T^3 + 51863073205933104123292704T^2 + 2925835478655020619977369088*T + 1322987095644544709830754804736
$61$	\( T^{12} + \cdots + 99\!\cdots\!96 \) T^12 + 978T^11 + 1718604T^10 + 1253528206T^9 + 1646257720257T^8 + 1078799606806167T^7 + 853239482255498955T^6 + 339899279938031695941T^5 + 176777819842527292532166T^4 + 50444053685540526814651327T^3 + 23812959490476103768321102893T^2 + 4585434536647591988291375603076*T + 999763280712194849538884674385296
$67$	\( T^{12} + \cdots + 96\!\cdots\!24 \) T^12 + 663T^11 + 1459170T^10 + 484428265T^9 + 1338848732556T^8 + 499311344409786T^7 + 410016993474976197T^6 - 6690798207880901496T^5 + 19575308580263102415033T^4 + 1906700407726120193156152T^3 + 432249687978246492387213840T^2 + 2056701437696093083644143232*T + 9600178382203339248099361024
$71$	\( (T^{6} + \cdots - 28\!\cdots\!24)^{2} \) (T^6 + 507T^5 - 894789T^4 - 48186954T^3 + 133872603714T^2 - 1998177550083*T - 2860222732066224)^2
$73$	\( (T^{6} + \cdots + 50\!\cdots\!28)^{2} \) (T^6 - 144T^5 - 1206627T^4 + 486607379T^3 + 137643578343T^2 - 70614731374728*T + 5086061693234428)^2
$79$	\( T^{12} + \cdots + 21\!\cdots\!56 \) T^12 + 609T^11 + 2180538T^10 + 805495831T^9 + 2876167266273T^8 + 897312889612665T^7 + 2168725174977025347T^6 + 295665679829633318568T^5 + 1043083270724024006205534T^4 + 78671335479921658467674536T^3 + 275649763660068420545418871665T^2 - 53149385888700706606169488660362*T + 21276713869610323903239102953321956
$83$	\( T^{12} + \cdots + 16\!\cdots\!44 \) T^12 - 516T^11 + 1831629T^10 - 109106478T^9 + 1882182316725T^8 + 24267642507747T^7 + 1169005111856386713T^6 + 157227818031060080760T^5 + 488566412438145949559952T^4 + 45628753198564241202807936T^3 + 110522148945179582616645282048T^2 + 11308278947911119936196561575936*T + 16274807035583753997133827434348544
$89$	\( (T^{6} + \cdots + 12\!\cdots\!12)^{2} \) (T^6 + 378T^5 - 2341893T^4 - 1187169237T^3 + 1209054877866T^2 + 842962534116192*T + 127820616579964512)^2
$97$	\( T^{12} + \cdots + 97\!\cdots\!76 \) T^12 + 2292T^11 + 6935733T^10 + 8158264378T^9 + 17950399247538T^8 + 18749169815336235T^7 + 30178588437514195800T^6 + 18622213611565743726003T^5 + 18198979765266743722444509T^4 + 8569717181758490190027776320T^3 + 7332062704488821669116481612700T^2 + 2420955170651590766380784444978016*T + 977441807203397358833955124983788176
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\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(-1 - \beta_{1}\)	\(1\)