LMFDB - Newform orbit 435.2.f.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [435,2,Mod(289,435)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(435, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("435.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$435 = 3 \cdot 5 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	435.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.47349248793$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{11} - 11 x^{9} + 55 x^{8} - 66 x^{7} + 328 x^{6} - 214 x^{5} + 207 x^{4} + 383 x^{3} + \cdots + 209$ x^12 - x^11 - 11x^9 + 55x^8 - 66x^7 + 328x^6 - 214x^5 + 207x^4 + 383x^3 + 16x^2 - 107*x + 209
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{6}$
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{5} q^{2} + q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{2} - 1) q^{5} - \beta_{5} q^{6} + \beta_{9} q^{7} + ( - \beta_{8} - 2 \beta_{5} + \beta_{3} + \cdots + 1) q^{8} + q^{9} + (\beta_{7} + \beta_{5}) q^{10}+ \cdots + ( - \beta_{11} - \beta_{2} + \beta_1) q^{99}+O(q^{100})$ q - b5 * q^2 + q^3 + (-b3 + 1) * q^4 + (b2 - 1) * q^5 - b5 * q^6 + b9 * q^7 + (-b8 - 2b5 + b3 - b2 + 1) q^8 + q^9 + (b7 + b5) * q^10 + (-b11 - b2 + b1) * q^11 + (-b3 + 1) * q^12 + (-b10 - b9 + b4 - b2 + b1) * q^13 + (b10 + 2b9 + b2 - b1) q^14 + (b2 - 1) * q^15 + (b8 - b7 - b6 + b5 - b3 + b2 + 2) * q^16 + (-b8 + b5 + b1 - 1) * q^17 - b5 * q^18 + (b11 - b10 - b9 - b4) * q^19 + (b9 + b8 + b6 + b5 - b4 + 2b2 - 2b1 - 2) * q^20 + b9 * q^21 + (b11 - b10 - b9 - b7 + b6) * q^22 + (-b10 + b4) * q^23 + (-b8 - 2b5 + b3 - b2 + 1) q^24 + (-b11 - b10 - b8 - b5 + b3 - b2 + 2) * q^25 + (-b11 - b9 - b7 + b6 - b4) * q^26 + q^27 + (b11 + 2b10 + 2b9 + b7 - b6 + b4 + b2 - b1) * q^28 + (b11 + b8 + b5 + b1) * q^29 + (b7 + b5) * q^30 + (-b11 - b10 + b9 - b7 + b6 + b4) * q^31 + (-b8 - 2b5 + 2b3 - 2b2 - b1 - 1) q^32 + (-b11 - b2 + b1) * q^33 + (b3 - 4) * q^34 + (-2b9 + b8 + b5 + 1) q^35 + (-b3 + 1) * q^36 + (-2b8 - b2 + b1 + 2) q^37 + (-2b11 - b10 - 2b9 - b4) * q^38 + (-b10 - b9 + b4 - b2 + b1) * q^39 + (b8 + b7 - b6 + 2b5 + b4 + 2b1 - 3) * q^40 + (2b11 + 2b10) * q^41 + (b10 + 2b9 + b2 - b1) q^42 + (-b7 - b6 + 4) * q^43 + (-3b9 + b7 - b6 + b4 - 3b2 + 3b1) q^44 + (b2 - 1) * q^45 + (-b11 + b10 + b9 - b4 + b2 - b1) * q^46 + (b8 + b5 - 2b3 - b1 - 3) q^47 + (b8 - b7 - b6 + b5 - b3 + b2 + 2) * q^48 + (-b8 + b7 + b6 + 3b5 + b2 + 2b1 - 2) * q^49 + (-b10 - b9 + b8 - b7 - b6 - b4 - 2b3 + 2b2 - b1 + 1) * q^50 + (-b8 + b5 + b1 - 1) * q^51 + (b11 - b10 - 4b9 + b7 - b6 - 4b2 + 4b1) q^52 + (2b11 + b7 - b6 + b2 - b1) q^53 - b5 * q^54 + (2b11 + b8 + b5 + b4 - b3 + b2 - b1 + 3) q^55 + (b11 + 2b10 + 4b9 + 3b2 - 3b1) * q^56 + (b11 - b10 - b9 - b4) * q^57 + (-b11 + b10 + b9 + 2b6 + b5 + b3 - 2) q^58 + (b7 + b6 + 2b3 - 2b2 - 2b1) q^59 + (b9 + b8 + b6 + b5 - b4 + 2b2 - 2b1 - 2) * q^60 + (2b10 + b7 - b6 + 2b4 - b2 + b1) * q^61 + (b10 + b7 - b6 + b4 - 3b2 + 3b1) * q^62 + b9 * q^63 + (4b5 - 2b3 - 1) * q^64 + (-2b11 + b10 + 2b9 + b8 + b5 + b4 - 2b3 + b2 - b1 + 1) q^65 + (b11 - b10 - b9 - b7 + b6) * q^66 + (2b11 + b9 - 2b4 + 2b2 - 2b1) * q^67 + (3b8 + 5b5 - b3 + b2 - 2b1 + 1) q^68 + (-b10 + b4) * q^69 + (-2b10 - 4b9 + b6 + b3 - 2b2 + 2b1 - 2) * q^70 + (2b8 - b7 - b6 + 2b5 - 2b3 + b2 - b1 - 2) q^71 + (-b8 - 2b5 + b3 - b2 + 1) q^72 + (-2b5 + b2 + b1 - 4) q^73 + (-b7 - b6 - 4b5 - 2) q^74 + (-b11 - b10 - b8 - b5 + b3 - b2 + 2) * q^75 + (-b11 - 3b10 - 5b9 + b4 - b2 + b1) * q^76 + (-3b8 - 3b5 - 2b2 + b1 + 1) q^77 + (-b11 - b9 - b7 + b6 - b4) * q^78 + (-b11 - b10 + 3b9 - b4 + 2b2 - 2b1) q^79 + (b10 + b9 - 2b8 - b7 + 2b6 + 2b5 + 2b3 + b2 - b1 - 1) * q^80 + q^81 + (2b10 + 2b9 + 2b4 - 2b2 + 2b1) q^82 + (2b11 + b10 - 2b9 + b7 - b6 - b4 + b2 - b1) * q^83 + (b11 + 2b10 + 2b9 + b7 - b6 + b4 + b2 - b1) * q^84 + (-b11 - b10 + b9 + b8 - 2b7 - b5 + b3 + 3) q^85 + (-2b8 - b7 - b6 - 6b5 + 2b3 - 3b2 - b1 + 2) * q^86 + (b11 + b8 + b5 + b1) * q^87 + (-2b11 - b9 - 2b7 + 2b6 - 2b4 + 2b2 - 2b1) * q^88 + (-2b11 + b10 + b9 + b7 - b6 - b4) q^89 + (b7 + b5) * q^90 + (-b8 - b7 - b6 - 5b5 - b2 + 3) q^91 + (2b11 + b10 + b7 - b6 - b4) q^92 + (-b11 - b10 + b9 - b7 + b6 + b4) * q^93 + (-2b8 - 2b5 + 3b3 - 2b2) * q^94 + (3b10 + 2b9 - 2b6 - 2b4 + b3 + 2b2 - 2b1 + 2) * q^95 + (-b8 - 2b5 + 2b3 - 2b2 - b1 - 1) q^96 + (2b8 + b7 + b6 - b2 - 3b1 - 2) * q^97 + (2b8 + 2b7 + 2b6 + 3b5 + b3 + 3b2 + b1 - 12) q^98 + (-b11 - b2 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 12 q^{3} + 16 q^{4} - 6 q^{5} + 12 q^{9} + 2 q^{10} + 16 q^{12} - 6 q^{15} + 32 q^{16} - 8 q^{17} - 20 q^{20} + 12 q^{25} + 12 q^{27} + 8 q^{29} + 2 q^{30} - 40 q^{32} - 52 q^{34} + 14 q^{35} + 16 q^{36}+ \cdots - 112 q^{98}+O(q^{100})$ 12 * q + 12 * q^3 + 16 * q^4 - 6 * q^5 + 12 * q^9 + 2 * q^10 + 16 * q^12 - 6 * q^15 + 32 * q^16 - 8 * q^17 - 20 * q^20 + 12 * q^25 + 12 * q^27 + 8 * q^29 + 2 * q^30 - 40 * q^32 - 52 * q^34 + 14 * q^35 + 16 * q^36 + 20 * q^37 - 22 * q^40 + 44 * q^43 - 6 * q^45 - 32 * q^47 + 32 * q^48 - 4 * q^49 + 24 * q^50 - 8 * q^51 + 42 * q^55 - 24 * q^58 - 28 * q^59 - 20 * q^60 - 4 * q^64 + 22 * q^65 + 16 * q^68 - 26 * q^70 - 16 * q^71 - 36 * q^73 - 28 * q^74 + 12 * q^75 - 22 * q^80 + 12 * q^81 + 30 * q^85 - 16 * q^86 + 8 * q^87 + 2 * q^90 + 24 * q^91 - 28 * q^94 + 16 * q^95 - 40 * q^96 - 40 * q^97 - 112 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - x^{11} - 11 x^{9} + 55 x^{8} - 66 x^{7} + 328 x^{6} - 214 x^{5} + 207 x^{4} + 383 x^{3} + \cdots + 209$ x^12 - x^11 - 11*x^9 + 55*x^8 - 66*x^7 + 328*x^6 - 214*x^5 + 207*x^4 + 383*x^3 + 16*x^2 - 107*x + 209 :

$\beta_{1}$	$=$	$( 1397998659 \nu^{11} + 2925295595 \nu^{10} - 7051159320 \nu^{9} - 16202543074 \nu^{8} + \cdots - 4008235081411 ) / 3213459622340$ (1397998659v^11 + 2925295595v^10 - 7051159320v^9 - 16202543074v^8 + 12622864171v^7 + 158262562347v^6 + 125074788459v^5 + 1327313721943v^4 - 1906016925784v^3 + 197941781608v^2 - 491520890193*v - 4008235081411) / 3213459622340
$\beta_{2}$	$=$	$( 438845871 \nu^{11} - 1612092568 \nu^{10} + 93438063 \nu^{9} - 3651448133 \nu^{8} + \cdots - 852415781250 ) / 642691924468$ (438845871v^11 - 1612092568v^10 + 93438063v^9 - 3651448133v^8 + 40975525123v^7 - 86600054135v^6 + 150413062959v^5 - 464110030659v^4 + 229217482380v^3 + 48797895497v^2 + 79140591340*v - 852415781250) / 642691924468
$\beta_{3}$	$=$	$( - 6774498506 \nu^{11} + 11302364650 \nu^{10} - 14173819980 \nu^{9} + 62634263811 \nu^{8} + \cdots - 3000675757541 ) / 3213459622340$ (-6774498506v^11 + 11302364650v^10 - 14173819980v^9 + 62634263811v^8 - 402715741289v^7 + 847834249232v^6 - 2762051697321v^5 + 2197300745838v^4 - 4695519142819v^3 - 4571103630662v^2 + 5453014482947*v - 3000675757541) / 3213459622340
$\beta_{4}$	$=$	$( 14834961346 \nu^{11} - 19830017805 \nu^{10} + 8761728055 \nu^{9} - 165086515566 \nu^{8} + \cdots - 802809213514 ) / 3213459622340$ (14834961346v^11 - 19830017805v^10 + 8761728055v^9 - 165086515566v^8 + 895774500149v^7 - 1313192606262v^6 + 5365102083441v^5 - 5209732834548v^4 + 6438006920199v^3 + 4454497821127v^2 + 5396904083698*v - 802809213514) / 3213459622340
$\beta_{5}$	$=$	$( - 15940407777 \nu^{11} + 23169062590 \nu^{10} + 3022410 \nu^{9} + 175231640332 \nu^{8} + \cdots + 2271334684498 ) / 3213459622340$ (-15940407777v^11 + 23169062590v^10 + 3022410v^9 + 175231640332v^8 - 951172524663v^7 + 1342953712874v^6 - 5371938150687v^5 + 5652569350406v^4 - 2115730805458v^3 - 5944128492144v^2 + 5714991881479*v + 2271334684498) / 3213459622340
$\beta_{6}$	$=$	$( - 9287830936 \nu^{11} + 18791211275 \nu^{10} - 7074036000 \nu^{9} + 97560593321 \nu^{8} + \cdots + 1390587069839 ) / 1606729811170$ (-9287830936v^11 + 18791211275v^10 - 7074036000v^9 + 97560593321v^8 - 611600487534v^7 + 1109826327087v^6 - 3560781919841v^5 + 4798956928898v^4 - 2668067717064v^3 - 1856218617732v^2 + 3566958207707*v + 1390587069839) / 1606729811170
$\beta_{7}$	$=$	$( - 1814963223 \nu^{11} + 1826616705 \nu^{10} + 1608454605 \nu^{9} + 19032062228 \nu^{8} + \cdots + 285575480212 ) / 292132692940$ (-1814963223v^11 + 1826616705v^10 + 1608454605v^9 + 19032062228v^8 - 101532722792v^7 + 100334854386v^6 - 510067673738v^5 + 320704835404v^4 + 45144095103v^3 - 924345305991v^2 + 557664272341*v + 285575480212) / 292132692940
$\beta_{8}$	$=$	$( 20053088776 \nu^{11} - 23152421330 \nu^{10} - 35040242445 \nu^{9} - 226992732701 \nu^{8} + \cdots - 11720209576834 ) / 3213459622340$ (20053088776v^11 - 23152421330v^10 - 35040242445v^9 - 226992732701v^8 + 1159961702574v^7 - 1016934126937v^6 + 5245556591026v^5 - 5041089270723v^4 - 6462366964626v^3 + 2965828611677v^2 - 1205718754777*v - 11720209576834) / 3213459622340
$\beta_{9}$	$=$	$( 26830096707 \nu^{11} - 18124014990 \nu^{10} + 14433023630 \nu^{9} - 344432223437 \nu^{8} + \cdots + 409628743007 ) / 3213459622340$ (26830096707v^11 - 18124014990v^10 + 14433023630v^9 - 344432223437v^8 + 1369333675898v^7 - 1543138855564v^6 + 9859505847302v^5 - 5376325857826v^4 + 11294588209803v^3 - 1361527673526v^2 + 5652711329436*v + 409628743007) / 3213459622340
$\beta_{10}$	$=$	$( - 42580783394 \nu^{11} - 6474185680 \nu^{10} + 34133697850 \nu^{9} + 503681618779 \nu^{8} + \cdots + 1277078933251 ) / 3213459622340$ (-42580783394v^11 - 6474185680v^10 + 34133697850v^9 + 503681618779v^8 - 1768024819631v^7 + 265055664228v^6 - 11852083344259v^5 - 5491812347758v^4 - 1630787402881v^3 - 17148624270588v^2 - 12536890370677*v + 1277078933251) / 3213459622340
$\beta_{11}$	$=$	$( 47064798624 \nu^{11} - 16749257055 \nu^{10} - 16699358825 \nu^{9} - 544078396489 \nu^{8} + \cdots - 1811594761361 ) / 3213459622340$ (47064798624v^11 - 16749257055v^10 - 16699358825v^9 - 544078396489v^8 + 2254521501956v^7 - 1580138940648v^6 + 14335461220584v^5 - 1579398287302v^4 + 8076137926236v^3 + 17521889223703v^2 + 15002068011897*v - 1811594761361) / 3213459622340

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

$\nu$	$=$	$( 3\beta_{11} + 2\beta_{10} - \beta_{9} + 2\beta_{8} + 3\beta_{7} + \beta_{6} - 2\beta_{3} - 2\beta_{2} ) / 8$ (3b11 + 2b10 - b9 + 2b8 + 3b7 + b6 - 2b3 - 2b2) / 8
$\nu^{2}$	$=$	$( - 5 \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + 5 \beta_{6} - 8 \beta_{5} + \cdots + 8 \beta_1 ) / 8$ (-5b11 - 2b10 - b9 - 2b8 - b7 + 5b6 - 8b5 + 12b4 + 2b3 - 6b2 + 8*b1) / 8
$\nu^{3}$	$=$	$( 5 \beta_{11} + 5 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \cdots + 4 ) / 2$ (5b11 + 5b10 + b9 - 2b8 - 3b7 + b6 - 3b4 + 2b3 - b2 + 7*b1 + 4) / 2
$\nu^{4}$	$=$	$( - 7 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 38 \beta_{8} - 7 \beta_{7} + 27 \beta_{6} + 40 \beta_{5} + \cdots - 96 ) / 8$ (-7b11 + 10b10 + 5b9 + 38b8 - 7b7 + 27b6 + 40b5 + 44b4 - 62b3 - 90b2 - 12*b1 - 96) / 8
$\nu^{5}$	$=$	$( - 85 \beta_{11} - 30 \beta_{10} + 7 \beta_{9} - 130 \beta_{8} - 145 \beta_{7} - 83 \beta_{6} + \cdots + 128 ) / 8$ (-85b11 - 30b10 + 7b9 - 130b8 - 145b7 - 83b6 + 32b5 + 176b4 + 226b3 - 114b2 + 276*b1 + 128) / 8
$\nu^{6}$	$=$	$68 \beta_{11} + 48 \beta_{10} - 2 \beta_{8} - 12 \beta_{7} - 20 \beta_{6} + 22 \beta_{5} - 77 \beta_{4} + \cdots - 75$ 68b11 + 48b10 - 2b8 - 12b7 - 20b6 + 22b5 - 77b4 + 15b3 - 37b2 - 37b1 - 75
$\nu^{7}$	$=$	$( - 1577 \beta_{11} - 1166 \beta_{10} + 43 \beta_{9} + 226 \beta_{8} - 401 \beta_{7} - 155 \beta_{6} + \cdots - 1768 ) / 8$ (-1577b11 - 1166b10 + 43b9 + 226b8 - 401b7 - 155b6 + 824b5 + 1544b4 - 170b3 - 434b2 - 1408*b1 - 1768) / 8
$\nu^{8}$	$=$	$( 1631 \beta_{11} + 310 \beta_{10} - 381 \beta_{9} - 3146 \beta_{8} - 1533 \beta_{7} - 3951 \beta_{6} + \cdots + 5088 ) / 8$ (1631b11 + 310b10 - 381b9 - 3146b8 - 1533b7 - 3951b6 + 1064b5 - 3612b4 + 5170b3 + 3914b2 + 1952*b1 + 5088) / 8
$\nu^{9}$	$=$	$( 929 \beta_{11} - 85 \beta_{10} - 305 \beta_{9} + 1956 \beta_{8} + 2047 \beta_{7} + 59 \beta_{6} + \cdots - 5514 ) / 2$ (929b11 - 85b10 - 305b9 + 1956b8 + 2047b7 + 59b6 + 538b5 - 2657b4 - 2906b3 + 309b2 - 6289*b1 - 5514) / 2
$\nu^{10}$	$=$	$( - 42507 \beta_{11} - 35934 \beta_{10} - 911 \beta_{9} - 11842 \beta_{8} - 1675 \beta_{7} - 7345 \beta_{6} + \cdots + 34784 ) / 8$ (-42507b11 - 35934b10 - 911b9 - 11842b8 - 1675b7 - 7345b6 - 7336b5 + 32452b4 + 17314b3 + 35766b2 + 2124*b1 + 34784) / 8
$\nu^{11}$	$=$	$( 122527 \beta_{11} + 65570 \beta_{10} - 9181 \beta_{9} - 6818 \beta_{8} + 45979 \beta_{7} - 22831 \beta_{6} + \cdots + 83272 ) / 8$ (122527b11 + 65570b10 - 9181b9 - 6818b8 + 45979b7 - 22831b6 - 28536b5 - 183160b4 + 1578b3 + 104526b2 - 20348*b1 + 83272) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$ .

$n$	$31$	$146$	$262$
$\chi(n)$	$-1$	$1$	$-1$

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}

289.1

−0.177521	+	2.06715i
−0.177521	−	2.06715i
0.469890	−	0.575682i
0.469890	+	0.575682i
−2.21342	+	2.00212i
−2.21342	−	2.00212i
0.730544	−	1.10073i
0.730544	+	1.10073i
2.44773	+	1.33046i
2.44773	−	1.33046i
−0.757215	−	0.394074i
−0.757215	+	0.394074i

−2.67451

1.00000

5.15300

−0.347696

−

2.20887i

−2.67451

1.33684i

−8.43272

1.00000

0.929917

5.90764i

289.2

−2.67451

1.00000

5.15300

−0.347696

2.20887i

−2.67451

−

1.33684i

−8.43272

1.00000

0.929917

−

5.90764i

289.3

−1.60465

1.00000

0.574897

−2.22390

−

0.232983i

−1.60465

−

1.04058i

2.28679

1.00000

3.56857

0.373856i

289.4

−1.60465

1.00000

0.574897

−2.22390

0.232983i

−1.60465

1.04058i

2.28679

1.00000

3.56857

−

0.373856i

289.5

−0.334522

1.00000

−1.88809

1.95387

−

1.08739i

−0.334522

0.275019i

1.30066

1.00000

−0.653612

0.363756i

289.6

−0.334522

1.00000

−1.88809

1.95387

1.08739i

−0.334522

−

0.275019i

1.30066

1.00000

−0.653612

−

0.363756i

289.7

0.141254

1.00000

−1.98005

−1.62735

−

1.53353i

0.141254

4.11523i

−0.562197

1.00000

−0.229870

−

0.216617i

289.8

0.141254

1.00000

−1.98005

−1.62735

1.53353i

0.141254

−

4.11523i

−0.562197

1.00000

−0.229870

0.216617i

289.9

1.97264

1.00000

1.89129

1.38075

−

1.75885i

1.97264

0.520254i

−0.214447

1.00000

2.72371

−

3.46956i

289.10

1.97264

1.00000

1.89129

1.38075

1.75885i

1.97264

−

0.520254i

−0.214447

1.00000

2.72371

3.46956i

289.11

2.49979

1.00000

4.24896

−2.13567

−

0.662521i

2.49979

−

4.88350i

5.62192

1.00000

−5.33872

−

1.65616i

289.12

2.49979

1.00000

4.24896

−2.13567

0.662521i

2.49979

4.88350i

5.62192

1.00000

−5.33872

1.65616i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
145.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	435.2.f.f	yes	12
3.b	odd	2	1		1305.2.f.l		12
5.b	even	2	1		435.2.f.e	✓	12
5.c	odd	4	2		2175.2.d.j		24
15.d	odd	2	1		1305.2.f.k		12
29.b	even	2	1		435.2.f.e	✓	12
87.d	odd	2	1		1305.2.f.k		12
145.d	even	2	1	inner	435.2.f.f	yes	12
145.h	odd	4	2		2175.2.d.j		24
435.b	odd	2	1		1305.2.f.l		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
435.2.f.e	✓	12	5.b	even	2	1
435.2.f.e	✓	12	29.b	even	2	1
435.2.f.f	yes	12	1.a	even	1	1	trivial
435.2.f.f	yes	12	145.d	even	2	1	inner
1305.2.f.k		12	15.d	odd	2	1
1305.2.f.k		12	87.d	odd	2	1
1305.2.f.l		12	3.b	odd	2	1
1305.2.f.l		12	435.b	odd	2	1
2175.2.d.j		24	5.c	odd	4	2
2175.2.d.j		24	145.h	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{6} - 10T_{2}^{4} + 22T_{2}^{2} + 4T_{2} - 1$ T2^6 - 10*T2^4 + 22*T2^2 + 4*T2 - 1 acting on $S_{2}^{\mathrm{new}}(435, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{6} - 10 T^{4} + 22 T^{2} + \cdots - 1)^{2}$ (T^6 - 10T^4 + 22T^2 + 4*T - 1)^2
$3$	$(T - 1)^{12}$ (T - 1)^12
$5$	$T^{12} + 6 T^{11} + \cdots + 15625$ T^12 + 6T^11 + 12T^10 + 10T^9 + 27T^8 + 136T^7 + 384T^6 + 680T^5 + 675T^4 + 1250T^3 + 7500T^2 + 18750*T + 15625
$7$	$T^{12} + 44 T^{10} + \cdots + 16$ T^12 + 44T^10 + 538T^8 + 1420T^6 + 1221T^4 + 296*T^2 + 16
$11$	$T^{12} + 76 T^{10} + \cdots + 85264$ T^12 + 76T^10 + 2082T^8 + 26092T^6 + 151005T^4 + 332680*T^2 + 85264
$13$	$T^{12} + 100 T^{10} + \cdots + 43264$ T^12 + 100T^10 + 3802T^8 + 66460T^6 + 491157T^4 + 852512*T^2 + 43264
$17$	$(T^{6} + 4 T^{5} + \cdots - 1172)^{2}$ (T^6 + 4T^5 - 38T^4 - 60T^3 + 389T^2 + 224*T - 1172)^2
$19$	$T^{12} + 160 T^{10} + \cdots + 6801664$ T^12 + 160T^10 + 9832T^8 + 286752T^6 + 3914192T^4 + 19981952*T^2 + 6801664
$23$	$T^{12} + 96 T^{10} + \cdots + 43264$ T^12 + 96T^10 + 2700T^8 + 20320T^6 + 62608T^4 + 85632*T^2 + 43264
$29$	$T^{12} + \cdots + 594823321$ T^12 - 8T^11 + 58T^10 + 8T^9 - 809T^8 + 10656T^7 - 42196T^6 + 309024T^5 - 680369T^4 + 195112T^3 + 41022298T^2 - 164089192*T + 594823321
$31$	$T^{12} + 248 T^{10} + \cdots + 63744256$ T^12 + 248T^10 + 21580T^8 + 795968T^6 + 12208912T^4 + 63542912*T^2 + 63744256
$37$	$(T^{6} - 10 T^{5} + \cdots + 9328)^{2}$ (T^6 - 10T^5 - 50T^4 + 828T^3 - 2308T^2 - 1616*T + 9328)^2
$41$	$T^{12} + \cdots + 881852416$ T^12 + 312T^10 + 36080T^8 + 1879936T^6 + 42675456T^4 + 356679680*T^2 + 881852416
$43$	$(T^{6} - 22 T^{5} + \cdots - 704)^{2}$ (T^6 - 22T^5 + 126T^4 + 292T^3 - 4492T^2 + 10080*T - 704)^2
$47$	$(T^{6} + 16 T^{5} + \cdots + 1408)^{2}$ (T^6 + 16T^5 + 10T^4 - 616T^3 - 2551T^2 - 2424*T + 1408)^2
$53$	$T^{12} + \cdots + 7721488384$ T^12 + 328T^10 + 42072T^8 + 2672032T^6 + 86976336T^4 + 1342226944*T^2 + 7721488384
$59$	$(T^{6} + 14 T^{5} + \cdots + 135232)^{2}$ (T^6 + 14T^5 - 150T^4 - 2188T^3 + 3604T^2 + 84384*T + 135232)^2
$61$	$T^{12} + \cdots + 104748027904$ T^12 + 536T^10 + 108888T^8 + 10577632T^6 + 516473680T^4 + 12011022848*T^2 + 104748027904
$67$	$T^{12} + \cdots + 19142382736$ T^12 + 388T^10 + 57522T^8 + 4205748T^6 + 159302893T^4 + 2910682216*T^2 + 19142382736
$71$	$(T^{6} + 8 T^{5} + \cdots - 43264)^{2}$ (T^6 + 8T^5 - 184T^4 - 1808T^3 + 1636T^2 + 28672*T - 43264)^2
$73$	$(T^{6} + 18 T^{5} + \cdots + 2864)^{2}$ (T^6 + 18T^5 + 58T^4 - 420T^3 - 1924T^2 + 560*T + 2864)^2
$79$	$T^{12} + \cdots + 342694144$ T^12 + 528T^10 + 94824T^8 + 6336224T^6 + 95491024T^4 + 413724288*T^2 + 342694144
$83$	$T^{12} + \cdots + 405780736$ T^12 + 376T^10 + 45084T^8 + 2028032T^6 + 32457616T^4 + 203317376*T^2 + 405780736
$89$	$T^{12} + \cdots + 1392185344$ T^12 + 468T^10 + 65202T^8 + 3382412T^6 + 64977533T^4 + 511769728*T^2 + 1392185344
$97$	$(T^{6} + 20 T^{5} + \cdots - 23504)^{2}$ (T^6 + 20T^5 - 92T^4 - 2696T^3 - 4876T^2 + 29856*T - 23504)^2
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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 $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 289.12
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	435.2.f.f

Level	$435$
Weight	$2$
Character orbit	435.f
Analytic conductor	$3.473$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$