# Properties

 Label 435.2.f.e Level $435$ Weight $2$ Character orbit 435.f Analytic conductor $3.473$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(289,435)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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 - 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} - q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{2} - 1) q^{5} - \beta_{5} q^{6} + \beta_{9} q^{7} + (\beta_{8} + 2 \beta_{5} - \beta_{3} + \cdots - 1) q^{8}+ \cdots + q^{9}+O(q^{10})$$ q + b5 * q^2 - q^3 + (-b3 + 1) * q^4 + (b2 - 1) * q^5 - b5 * q^6 + b9 * q^7 + (b8 + 2*b5 - b3 + b2 - 1) * q^8 + q^9 $$q + \beta_{5} q^{2} - q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{2} - 1) q^{5} - \beta_{5} q^{6} + \beta_{9} q^{7} + (\beta_{8} + 2 \beta_{5} - \beta_{3} + \cdots - 1) q^{8}+ \cdots + (\beta_{11} + \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + b5 * q^2 - q^3 + (-b3 + 1) * q^4 + (b2 - 1) * q^5 - b5 * q^6 + b9 * q^7 + (b8 + 2*b5 - 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 - 2*b9 - 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 + 2*b2 - 2*b1 - 2) * q^20 - b9 * q^21 + (b11 - b10 - b9 - b7 + b6) * q^22 + (-b10 + b4) * q^23 + (-b8 - 2*b5 + b3 - b2 + 1) * q^24 + (-b11 - b10 - b8 - b5 + b3 - b2 + 2) * q^25 + (b11 + b9 + b7 - b6 + b4) * q^26 - q^27 + (b11 + 2*b10 + 2*b9 + b7 - b6 + b4 + b2 - b1) * q^28 + (-b11 + b8 + b5 + 2*b2 - b1) * q^29 + (b7 + b5) * q^30 + (b11 + b10 - b9 + b7 - b6 - b4) * q^31 + (b8 + 2*b5 - 2*b3 + 2*b2 + b1 + 1) * q^32 + (-b11 - b2 + b1) * q^33 + (b3 - 4) * q^34 + (-2*b9 + b8 + b5 + 1) * q^35 + (-b3 + 1) * q^36 + (2*b8 + b2 - b1 - 2) * q^37 + (-2*b11 - b10 - 2*b9 - b4) * q^38 + (b10 + b9 - b4 + b2 - b1) * q^39 + (-b8 - b7 + b6 - 2*b5 - b4 - 2*b1 + 3) * q^40 + (-2*b11 - 2*b10) * q^41 + (b10 + 2*b9 + b2 - b1) * q^42 + (b7 + b6 - 4) * q^43 + (3*b9 - b7 + b6 - b4 + 3*b2 - 3*b1) * q^44 + (b2 - 1) * q^45 + (b11 - b10 - b9 + b4 - b2 + b1) * q^46 + (-b8 - b5 + 2*b3 + b1 + 3) * q^47 + (-b8 + b7 + b6 - b5 + b3 - b2 - 2) * q^48 + (-b8 + b7 + b6 + 3*b5 + b2 + 2*b1 - 2) * q^49 + (b10 + b9 - b8 + b7 + b6 + b4 + 2*b3 - 2*b2 + b1 - 1) * q^50 + (-b8 + b5 + b1 - 1) * q^51 + (b11 - b10 - 4*b9 + b7 - b6 - 4*b2 + 4*b1) * q^52 + (2*b11 + b7 - b6 + b2 - b1) * q^53 - b5 * q^54 + (-2*b11 - b8 - b5 - b4 + b3 - b2 + b1 - 3) * q^55 + (-b11 - 2*b10 - 4*b9 - 3*b2 + 3*b1) * q^56 + (b11 - b10 - b9 - b4) * q^57 + (-b11 + b10 + b9 - 2*b7 - b5 - b3 + 2) * q^58 + (b7 + b6 + 2*b3 - 2*b2 - 2*b1) * q^59 + (-b9 - b8 - b6 - b5 + b4 - 2*b2 + 2*b1 + 2) * q^60 + (-2*b10 - b7 + b6 - 2*b4 + b2 - b1) * q^61 + (b10 + b7 - b6 + b4 - 3*b2 + 3*b1) * q^62 + b9 * q^63 + (4*b5 - 2*b3 - 1) * q^64 + (-2*b11 + b10 + 2*b9 + b8 + b5 + b4 - 2*b3 + b2 - b1 + 1) * q^65 + (-b11 + b10 + b9 + b7 - b6) * q^66 + (2*b11 + b9 - 2*b4 + 2*b2 - 2*b1) * q^67 + (-3*b8 - 5*b5 + b3 - b2 + 2*b1 - 1) * q^68 + (b10 - b4) * q^69 + (2*b10 + 4*b9 - b6 - b3 + 2*b2 - 2*b1 + 2) * q^70 + (2*b8 - b7 - b6 + 2*b5 - 2*b3 + b2 - b1 - 2) * q^71 + (b8 + 2*b5 - b3 + b2 - 1) * q^72 + (2*b5 - b2 - b1 + 4) * q^73 + (-b7 - b6 - 4*b5 - 2) * q^74 + (b11 + b10 + b8 + b5 - b3 + b2 - 2) * q^75 + (b11 + 3*b10 + 5*b9 - b4 + b2 - b1) * q^76 + (3*b8 + 3*b5 + 2*b2 - b1 - 1) * q^77 + (-b11 - b9 - b7 + b6 - b4) * q^78 + (b11 + b10 - 3*b9 + b4 - 2*b2 + 2*b1) * q^79 + (b10 + b9 - 2*b8 - b7 + 2*b6 + 2*b5 + 2*b3 + b2 - b1 - 1) * q^80 + q^81 + (2*b10 + 2*b9 + 2*b4 - 2*b2 + 2*b1) * q^82 + (2*b11 + b10 - 2*b9 + b7 - b6 - b4 + b2 - b1) * q^83 + (-b11 - 2*b10 - 2*b9 - b7 + b6 - b4 - b2 + b1) * q^84 + (b11 + b10 - b9 - b8 + 2*b7 + b5 - b3 - 3) * q^85 + (-2*b8 - b7 - b6 - 6*b5 + 2*b3 - 3*b2 - b1 + 2) * q^86 + (b11 - b8 - b5 - 2*b2 + b1) * q^87 + (-2*b11 - b9 - 2*b7 + 2*b6 - 2*b4 + 2*b2 - 2*b1) * q^88 + (2*b11 - b10 - b9 - b7 + b6 + b4) * q^89 + (-b7 - b5) * q^90 + (-b8 - b7 - b6 - 5*b5 - b2 + 3) * q^91 + (2*b11 + b10 + b7 - b6 - b4) * q^92 + (-b11 - b10 + b9 - b7 + b6 + b4) * q^93 + (-2*b8 - 2*b5 + 3*b3 - 2*b2) * q^94 + (-3*b10 - 2*b9 + 2*b6 + 2*b4 - b3 - 2*b2 + 2*b1 - 2) * q^95 + (-b8 - 2*b5 + 2*b3 - 2*b2 - b1 - 1) * q^96 + (-2*b8 - b7 - b6 + b2 + 3*b1 + 2) * q^97 + (-2*b8 - 2*b7 - 2*b6 - 3*b5 - b3 - 3*b2 - b1 + 12) * q^98 + (b11 + b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{3} + 16 q^{4} - 6 q^{5} + 12 q^{9}+O(q^{10})$$ 12 * q - 12 * q^3 + 16 * q^4 - 6 * q^5 + 12 * q^9 $$12 q - 12 q^{3} + 16 q^{4} - 6 q^{5} + 12 q^{9} - 2 q^{10} - 16 q^{12} + 6 q^{15} + 32 q^{16} + 8 q^{17} - 20 q^{20} + 12 q^{25} - 12 q^{27} + 8 q^{29} + 2 q^{30} + 40 q^{32} - 52 q^{34} + 14 q^{35} + 16 q^{36} - 20 q^{37} + 22 q^{40} - 44 q^{43} - 6 q^{45} + 32 q^{47} - 32 q^{48} - 4 q^{49} - 24 q^{50} - 8 q^{51} - 42 q^{55} + 24 q^{58} - 28 q^{59} + 20 q^{60} - 4 q^{64} + 22 q^{65} - 16 q^{68} + 26 q^{70} - 16 q^{71} + 36 q^{73} - 28 q^{74} - 12 q^{75} - 22 q^{80} + 12 q^{81} - 30 q^{85} - 16 q^{86} - 8 q^{87} - 2 q^{90} + 24 q^{91} - 28 q^{94} - 16 q^{95} - 40 q^{96} + 40 q^{97} + 112 q^{98}+O(q^{100})$$ 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 $$x^{12} - x^{11} - 11 x^{9} + 55 x^{8} - 66 x^{7} + 328 x^{6} - 214 x^{5} + 207 x^{4} + 383 x^{3} + \cdots + 209$$ :

 $$\beta_{1}$$ $$=$$ $$( 1397998659 \nu^{11} + 2925295595 \nu^{10} - 7051159320 \nu^{9} - 16202543074 \nu^{8} + \cdots - 4008235081411 ) / 3213459622340$$ (1397998659*v^11 + 2925295595*v^10 - 7051159320*v^9 - 16202543074*v^8 + 12622864171*v^7 + 158262562347*v^6 + 125074788459*v^5 + 1327313721943*v^4 - 1906016925784*v^3 + 197941781608*v^2 - 491520890193*v - 4008235081411) / 3213459622340 $$\beta_{2}$$ $$=$$ $$( 438845871 \nu^{11} - 1612092568 \nu^{10} + 93438063 \nu^{9} - 3651448133 \nu^{8} + \cdots - 852415781250 ) / 642691924468$$ (438845871*v^11 - 1612092568*v^10 + 93438063*v^9 - 3651448133*v^8 + 40975525123*v^7 - 86600054135*v^6 + 150413062959*v^5 - 464110030659*v^4 + 229217482380*v^3 + 48797895497*v^2 + 79140591340*v - 852415781250) / 642691924468 $$\beta_{3}$$ $$=$$ $$( - 6774498506 \nu^{11} + 11302364650 \nu^{10} - 14173819980 \nu^{9} + 62634263811 \nu^{8} + \cdots - 3000675757541 ) / 3213459622340$$ (-6774498506*v^11 + 11302364650*v^10 - 14173819980*v^9 + 62634263811*v^8 - 402715741289*v^7 + 847834249232*v^6 - 2762051697321*v^5 + 2197300745838*v^4 - 4695519142819*v^3 - 4571103630662*v^2 + 5453014482947*v - 3000675757541) / 3213459622340 $$\beta_{4}$$ $$=$$ $$( 14834961346 \nu^{11} - 19830017805 \nu^{10} + 8761728055 \nu^{9} - 165086515566 \nu^{8} + \cdots - 802809213514 ) / 3213459622340$$ (14834961346*v^11 - 19830017805*v^10 + 8761728055*v^9 - 165086515566*v^8 + 895774500149*v^7 - 1313192606262*v^6 + 5365102083441*v^5 - 5209732834548*v^4 + 6438006920199*v^3 + 4454497821127*v^2 + 5396904083698*v - 802809213514) / 3213459622340 $$\beta_{5}$$ $$=$$ $$( - 15940407777 \nu^{11} + 23169062590 \nu^{10} + 3022410 \nu^{9} + 175231640332 \nu^{8} + \cdots + 2271334684498 ) / 3213459622340$$ (-15940407777*v^11 + 23169062590*v^10 + 3022410*v^9 + 175231640332*v^8 - 951172524663*v^7 + 1342953712874*v^6 - 5371938150687*v^5 + 5652569350406*v^4 - 2115730805458*v^3 - 5944128492144*v^2 + 5714991881479*v + 2271334684498) / 3213459622340 $$\beta_{6}$$ $$=$$ $$( - 9287830936 \nu^{11} + 18791211275 \nu^{10} - 7074036000 \nu^{9} + 97560593321 \nu^{8} + \cdots + 1390587069839 ) / 1606729811170$$ (-9287830936*v^11 + 18791211275*v^10 - 7074036000*v^9 + 97560593321*v^8 - 611600487534*v^7 + 1109826327087*v^6 - 3560781919841*v^5 + 4798956928898*v^4 - 2668067717064*v^3 - 1856218617732*v^2 + 3566958207707*v + 1390587069839) / 1606729811170 $$\beta_{7}$$ $$=$$ $$( - 1814963223 \nu^{11} + 1826616705 \nu^{10} + 1608454605 \nu^{9} + 19032062228 \nu^{8} + \cdots + 285575480212 ) / 292132692940$$ (-1814963223*v^11 + 1826616705*v^10 + 1608454605*v^9 + 19032062228*v^8 - 101532722792*v^7 + 100334854386*v^6 - 510067673738*v^5 + 320704835404*v^4 + 45144095103*v^3 - 924345305991*v^2 + 557664272341*v + 285575480212) / 292132692940 $$\beta_{8}$$ $$=$$ $$( 20053088776 \nu^{11} - 23152421330 \nu^{10} - 35040242445 \nu^{9} - 226992732701 \nu^{8} + \cdots - 11720209576834 ) / 3213459622340$$ (20053088776*v^11 - 23152421330*v^10 - 35040242445*v^9 - 226992732701*v^8 + 1159961702574*v^7 - 1016934126937*v^6 + 5245556591026*v^5 - 5041089270723*v^4 - 6462366964626*v^3 + 2965828611677*v^2 - 1205718754777*v - 11720209576834) / 3213459622340 $$\beta_{9}$$ $$=$$ $$( 26830096707 \nu^{11} - 18124014990 \nu^{10} + 14433023630 \nu^{9} - 344432223437 \nu^{8} + \cdots + 409628743007 ) / 3213459622340$$ (26830096707*v^11 - 18124014990*v^10 + 14433023630*v^9 - 344432223437*v^8 + 1369333675898*v^7 - 1543138855564*v^6 + 9859505847302*v^5 - 5376325857826*v^4 + 11294588209803*v^3 - 1361527673526*v^2 + 5652711329436*v + 409628743007) / 3213459622340 $$\beta_{10}$$ $$=$$ $$( - 42580783394 \nu^{11} - 6474185680 \nu^{10} + 34133697850 \nu^{9} + 503681618779 \nu^{8} + \cdots + 1277078933251 ) / 3213459622340$$ (-42580783394*v^11 - 6474185680*v^10 + 34133697850*v^9 + 503681618779*v^8 - 1768024819631*v^7 + 265055664228*v^6 - 11852083344259*v^5 - 5491812347758*v^4 - 1630787402881*v^3 - 17148624270588*v^2 - 12536890370677*v + 1277078933251) / 3213459622340 $$\beta_{11}$$ $$=$$ $$( 47064798624 \nu^{11} - 16749257055 \nu^{10} - 16699358825 \nu^{9} - 544078396489 \nu^{8} + \cdots - 1811594761361 ) / 3213459622340$$ (47064798624*v^11 - 16749257055*v^10 - 16699358825*v^9 - 544078396489*v^8 + 2254521501956*v^7 - 1580138940648*v^6 + 14335461220584*v^5 - 1579398287302*v^4 + 8076137926236*v^3 + 17521889223703*v^2 + 15002068011897*v - 1811594761361) / 3213459622340
 $$\nu$$ $$=$$ $$( 3\beta_{11} + 2\beta_{10} - \beta_{9} + 2\beta_{8} + 3\beta_{7} + \beta_{6} - 2\beta_{3} - 2\beta_{2} ) / 8$$ (3*b11 + 2*b10 - b9 + 2*b8 + 3*b7 + b6 - 2*b3 - 2*b2) / 8 $$\nu^{2}$$ $$=$$ $$( - 5 \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + 5 \beta_{6} - 8 \beta_{5} + \cdots + 8 \beta_1 ) / 8$$ (-5*b11 - 2*b10 - b9 - 2*b8 - b7 + 5*b6 - 8*b5 + 12*b4 + 2*b3 - 6*b2 + 8*b1) / 8 $$\nu^{3}$$ $$=$$ $$( 5 \beta_{11} + 5 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \cdots + 4 ) / 2$$ (5*b11 + 5*b10 + b9 - 2*b8 - 3*b7 + b6 - 3*b4 + 2*b3 - b2 + 7*b1 + 4) / 2 $$\nu^{4}$$ $$=$$ $$( - 7 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 38 \beta_{8} - 7 \beta_{7} + 27 \beta_{6} + 40 \beta_{5} + \cdots - 96 ) / 8$$ (-7*b11 + 10*b10 + 5*b9 + 38*b8 - 7*b7 + 27*b6 + 40*b5 + 44*b4 - 62*b3 - 90*b2 - 12*b1 - 96) / 8 $$\nu^{5}$$ $$=$$ $$( - 85 \beta_{11} - 30 \beta_{10} + 7 \beta_{9} - 130 \beta_{8} - 145 \beta_{7} - 83 \beta_{6} + \cdots + 128 ) / 8$$ (-85*b11 - 30*b10 + 7*b9 - 130*b8 - 145*b7 - 83*b6 + 32*b5 + 176*b4 + 226*b3 - 114*b2 + 276*b1 + 128) / 8 $$\nu^{6}$$ $$=$$ $$68 \beta_{11} + 48 \beta_{10} - 2 \beta_{8} - 12 \beta_{7} - 20 \beta_{6} + 22 \beta_{5} - 77 \beta_{4} + \cdots - 75$$ 68*b11 + 48*b10 - 2*b8 - 12*b7 - 20*b6 + 22*b5 - 77*b4 + 15*b3 - 37*b2 - 37*b1 - 75 $$\nu^{7}$$ $$=$$ $$( - 1577 \beta_{11} - 1166 \beta_{10} + 43 \beta_{9} + 226 \beta_{8} - 401 \beta_{7} - 155 \beta_{6} + \cdots - 1768 ) / 8$$ (-1577*b11 - 1166*b10 + 43*b9 + 226*b8 - 401*b7 - 155*b6 + 824*b5 + 1544*b4 - 170*b3 - 434*b2 - 1408*b1 - 1768) / 8 $$\nu^{8}$$ $$=$$ $$( 1631 \beta_{11} + 310 \beta_{10} - 381 \beta_{9} - 3146 \beta_{8} - 1533 \beta_{7} - 3951 \beta_{6} + \cdots + 5088 ) / 8$$ (1631*b11 + 310*b10 - 381*b9 - 3146*b8 - 1533*b7 - 3951*b6 + 1064*b5 - 3612*b4 + 5170*b3 + 3914*b2 + 1952*b1 + 5088) / 8 $$\nu^{9}$$ $$=$$ $$( 929 \beta_{11} - 85 \beta_{10} - 305 \beta_{9} + 1956 \beta_{8} + 2047 \beta_{7} + 59 \beta_{6} + \cdots - 5514 ) / 2$$ (929*b11 - 85*b10 - 305*b9 + 1956*b8 + 2047*b7 + 59*b6 + 538*b5 - 2657*b4 - 2906*b3 + 309*b2 - 6289*b1 - 5514) / 2 $$\nu^{10}$$ $$=$$ $$( - 42507 \beta_{11} - 35934 \beta_{10} - 911 \beta_{9} - 11842 \beta_{8} - 1675 \beta_{7} - 7345 \beta_{6} + \cdots + 34784 ) / 8$$ (-42507*b11 - 35934*b10 - 911*b9 - 11842*b8 - 1675*b7 - 7345*b6 - 7336*b5 + 32452*b4 + 17314*b3 + 35766*b2 + 2124*b1 + 34784) / 8 $$\nu^{11}$$ $$=$$ $$( 122527 \beta_{11} + 65570 \beta_{10} - 9181 \beta_{9} - 6818 \beta_{8} + 45979 \beta_{7} - 22831 \beta_{6} + \cdots + 83272 ) / 8$$ (122527*b11 + 65570*b10 - 9181*b9 - 6818*b8 + 45979*b7 - 22831*b6 - 28536*b5 - 183160*b4 + 1578*b3 + 104526*b2 - 20348*b1 + 83272) / 8

## 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
 −0.757215 − 0.394074i −0.757215 + 0.394074i 2.44773 + 1.33046i 2.44773 − 1.33046i 0.730544 − 1.10073i 0.730544 + 1.10073i −2.21342 + 2.00212i −2.21342 − 2.00212i 0.469890 − 0.575682i 0.469890 + 0.575682i −0.177521 + 2.06715i −0.177521 − 2.06715i
−2.49979 −1.00000 4.24896 −2.13567 0.662521i 2.49979 4.88350i −5.62192 1.00000 5.33872 + 1.65616i
289.2 −2.49979 −1.00000 4.24896 −2.13567 + 0.662521i 2.49979 4.88350i −5.62192 1.00000 5.33872 1.65616i
289.3 −1.97264 −1.00000 1.89129 1.38075 1.75885i 1.97264 0.520254i 0.214447 1.00000 −2.72371 + 3.46956i
289.4 −1.97264 −1.00000 1.89129 1.38075 + 1.75885i 1.97264 0.520254i 0.214447 1.00000 −2.72371 3.46956i
289.5 −0.141254 −1.00000 −1.98005 −1.62735 1.53353i 0.141254 4.11523i 0.562197 1.00000 0.229870 + 0.216617i
289.6 −0.141254 −1.00000 −1.98005 −1.62735 + 1.53353i 0.141254 4.11523i 0.562197 1.00000 0.229870 0.216617i
289.7 0.334522 −1.00000 −1.88809 1.95387 1.08739i −0.334522 0.275019i −1.30066 1.00000 0.653612 0.363756i
289.8 0.334522 −1.00000 −1.88809 1.95387 + 1.08739i −0.334522 0.275019i −1.30066 1.00000 0.653612 + 0.363756i
289.9 1.60465 −1.00000 0.574897 −2.22390 0.232983i −1.60465 1.04058i −2.28679 1.00000 −3.56857 0.373856i
289.10 1.60465 −1.00000 0.574897 −2.22390 + 0.232983i −1.60465 1.04058i −2.28679 1.00000 −3.56857 + 0.373856i
289.11 2.67451 −1.00000 5.15300 −0.347696 2.20887i −2.67451 1.33684i 8.43272 1.00000 −0.929917 5.90764i
289.12 2.67451 −1.00000 5.15300 −0.347696 + 2.20887i −2.67451 1.33684i 8.43272 1.00000 −0.929917 + 5.90764i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.e 12
3.b odd 2 1 1305.2.f.k 12
5.b even 2 1 435.2.f.f yes 12
5.c odd 4 2 2175.2.d.j 24
15.d odd 2 1 1305.2.f.l 12
29.b even 2 1 435.2.f.f yes 12
87.d odd 2 1 1305.2.f.l 12
145.d even 2 1 inner 435.2.f.e 12
145.h odd 4 2 2175.2.d.j 24
435.b odd 2 1 1305.2.f.k 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.f.e 12 1.a even 1 1 trivial
435.2.f.e 12 145.d even 2 1 inner
435.2.f.f yes 12 5.b even 2 1
435.2.f.f yes 12 29.b even 2 1
1305.2.f.k 12 3.b odd 2 1
1305.2.f.k 12 435.b odd 2 1
1305.2.f.l 12 15.d odd 2 1
1305.2.f.l 12 87.d 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 $$T_{2}^{6} - 10T_{2}^{4} + 22T_{2}^{2} - 4T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(435, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - 10 T^{4} + 22 T^{2} + \cdots - 1)^{2}$$
$3$ $$(T + 1)^{12}$$
$5$ $$T^{12} + 6 T^{11} + \cdots + 15625$$
$7$ $$T^{12} + 44 T^{10} + \cdots + 16$$
$11$ $$T^{12} + 76 T^{10} + \cdots + 85264$$
$13$ $$T^{12} + 100 T^{10} + \cdots + 43264$$
$17$ $$(T^{6} - 4 T^{5} + \cdots - 1172)^{2}$$
$19$ $$T^{12} + 160 T^{10} + \cdots + 6801664$$
$23$ $$T^{12} + 96 T^{10} + \cdots + 43264$$
$29$ $$T^{12} + \cdots + 594823321$$
$31$ $$T^{12} + 248 T^{10} + \cdots + 63744256$$
$37$ $$(T^{6} + 10 T^{5} + \cdots + 9328)^{2}$$
$41$ $$T^{12} + \cdots + 881852416$$
$43$ $$(T^{6} + 22 T^{5} + \cdots - 704)^{2}$$
$47$ $$(T^{6} - 16 T^{5} + \cdots + 1408)^{2}$$
$53$ $$T^{12} + \cdots + 7721488384$$
$59$ $$(T^{6} + 14 T^{5} + \cdots + 135232)^{2}$$
$61$ $$T^{12} + \cdots + 104748027904$$
$67$ $$T^{12} + \cdots + 19142382736$$
$71$ $$(T^{6} + 8 T^{5} + \cdots - 43264)^{2}$$
$73$ $$(T^{6} - 18 T^{5} + \cdots + 2864)^{2}$$
$79$ $$T^{12} + \cdots + 342694144$$
$83$ $$T^{12} + \cdots + 405780736$$
$89$ $$T^{12} + \cdots + 1392185344$$
$97$ $$(T^{6} - 20 T^{5} + \cdots - 23504)^{2}$$