# Properties

 Label 380.2.r.a Level $380$ Weight $2$ Character orbit 380.r Analytic conductor $3.034$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(49,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("380.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.r (of order $$6$$, 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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + 124617 x^{4} - 24768 x^{2} + 4096$$ x^20 - 20*x^18 + 261*x^16 - 1994*x^14 + 11074*x^12 - 39211*x^10 + 99376*x^8 - 134299*x^6 + 124617*x^4 - 24768*x^2 + 4096 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{11} + \beta_1) q^{3} + (\beta_{13} - \beta_{7}) q^{5} + ( - \beta_{18} + \beta_{15}) q^{7} + (\beta_{10} - \beta_{3} + 1) q^{9}+O(q^{10})$$ q + (-b11 + b1) * q^3 + (b13 - b7) * q^5 + (-b18 + b15) * q^7 + (b10 - b3 + 1) * q^9 $$q + ( - \beta_{11} + \beta_1) q^{3} + (\beta_{13} - \beta_{7}) q^{5} + ( - \beta_{18} + \beta_{15}) q^{7} + (\beta_{10} - \beta_{3} + 1) q^{9} - \beta_{6} q^{11} + (\beta_{16} + \beta_{9}) q^{13} + (\beta_{12} + \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{15} + ( - \beta_{18} + \beta_{17} - \beta_{14} - \beta_{13} - \beta_{9} + \beta_{7}) q^{17} + (\beta_{12} - \beta_{10} - \beta_{8} - \beta_{7} + 1) q^{19} + (\beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{10} + \beta_{7} - \beta_{3} + 2 \beta_{2}) q^{21} + ( - 2 \beta_{16} + \beta_{9} + \beta_1) q^{23} + ( - \beta_{16} + \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_1 + 1) q^{25} + ( - \beta_{19} - \beta_{18} + \beta_{16} + \beta_{15}) q^{27} + (\beta_{12} - \beta_{10} + \beta_{5} + 2 \beta_{3} - 2) q^{29} + (\beta_{14} - \beta_{13} - \beta_{8} + \beta_{6} - \beta_{5} + 2 \beta_{2} + 1) q^{31} + (\beta_{19} + 2 \beta_{17} + \beta_{11} - 2 \beta_{9} - \beta_1) q^{33} + (\beta_{19} - \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - \beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{2}) q^{35} + ( - \beta_{19} - \beta_{17} + \beta_{16} - \beta_{11}) q^{37} + ( - \beta_{14} + \beta_{13} + \beta_{8} + \beta_{5} + \beta_{2} - 1) q^{39} + (2 \beta_{14} - 2 \beta_{13} + 2 \beta_{7} + \beta_{6} - \beta_{4} + 3 \beta_{3}) q^{41} + ( - 3 \beta_{17} + \beta_{14} + \beta_{13} + 3 \beta_{9} - \beta_{7}) q^{43} + (\beta_{19} + \beta_{18} - \beta_{17} - \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{8} + \beta_{5} - 2) q^{45} + (\beta_{16} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_1) q^{47} + ( - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{8} - 3 \beta_{6} + \beta_{5} - 2 \beta_{2} + \cdots - 3) q^{49}+ \cdots + (\beta_{12} - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + \beta_{5} + 2 \beta_{3} - 2) q^{99}+O(q^{100})$$ q + (-b11 + b1) * q^3 + (b13 - b7) * q^5 + (-b18 + b15) * q^7 + (b10 - b3 + 1) * q^9 - b6 * q^11 + (b16 + b9) * q^13 + (b12 + b9 + b8 + b5 - b4 + b3 - 1) * q^15 + (-b18 + b17 - b14 - b13 - b9 + b7) * q^17 + (b12 - b10 - b8 - b7 + 1) * q^19 + (b14 - b13 - b12 + 2*b10 + b7 - b3 + 2*b2) * q^21 + (-2*b16 + b9 + b1) * q^23 + (-b16 + b9 - b7 + b4 - b3 + b1 + 1) * q^25 + (-b19 - b18 + b16 + b15) * q^27 + (b12 - b10 + b5 + 2*b3 - 2) * q^29 + (b14 - b13 - b8 + b6 - b5 + 2*b2 + 1) * q^31 + (b19 + 2*b17 + b11 - 2*b9 - b1) * q^33 + (b19 - b14 + b13 + b12 - 2*b10 - b7 - b6 + b4 - 2*b2) * q^35 + (-b19 - b17 + b16 - b11) * q^37 + (-b14 + b13 + b8 + b5 + b2 - 1) * q^39 + (2*b14 - 2*b13 + 2*b7 + b6 - b4 + 3*b3) * q^41 + (-3*b17 + b14 + b13 + 3*b9 - b7) * q^43 + (b19 + b18 - b17 - b16 - b15 + b14 + b13 - b8 + b5 - 2) * q^45 + (b16 - b9 - b8 + b7 - b1) * q^47 + (-2*b14 + 2*b13 + 2*b8 - 3*b6 + b5 - 2*b2 - 3) * q^49 + (-3*b12 + 2*b10 - b8 - b7 - 3*b5 + b4 - 4*b3 + 4) * q^51 + (-b16 - 2*b9 + b8 - b7 + b1) * q^53 + (b18 + b17 - 2*b14 + b12 - b11 - b9 + b6 - b4 - b3 + b1) * q^55 + (b19 + b18 + 2*b17 - b16 - 2*b15 - b14 - b13 + b11 - 2*b9 - b8 + 2*b7 - b1) * q^57 + (-b12 - b10 - b2) * q^59 + (-2*b12 - 2*b5 - b3 + 1) * q^61 + (2*b16 + b8 - b7 - 3*b1) * q^63 + (-2*b19 + b18 + 2*b16 - b15 + b13 + b11 - 2*b2 - 1) * q^65 + (-b16 + 3*b15 + b9 + b8 - b7 + b1) * q^67 + (-b14 + b13 + b8 + 3*b6 - 2*b5 - 3*b2 + 3) * q^69 + (3*b14 - 3*b13 - b12 + 2*b10 + 3*b7 + 2*b2) * q^71 + (-2*b19 + b17 + b11 - b9 - b1) * q^73 + (-b19 - b17 + b16 - 2*b14 + b13 - 2*b11 + 2*b8 + b6 - 2*b2 + 2) * q^75 + (b19 + 2*b18 - 2*b17 - b16 - 2*b15 - b14 - b13 + b8) * q^77 + (b14 - b13 + b12 + b10 + b7 + 3*b6 - 3*b4 - b3 + b2) * q^79 + (b14 - b13 - 2*b12 + b7 - b6 + b4 + 2*b3) * q^81 + (-2*b19 + b18 - b17 + 2*b16 - b15 + 2*b14 + 2*b13 - 2*b8) * q^83 + (b16 + b12 - 3*b10 - 2*b8 - b7 + b5 + 4*b3 - 2*b1 - 4) * q^85 + (b19 + 2*b18 - 2*b17 - b16 - 2*b15 + 2*b14 + 2*b13 + 2*b11 - 2*b8) * q^87 + (-b12 - b10 - 3*b8 - 3*b7 - b5 + 3*b4 + 3*b3 - 3) * q^89 + (3*b8 + 3*b7 - 3*b4 + b3 - 1) * q^91 + (b19 + b18 + 2*b17 - 3*b14 - 3*b13 + 4*b11 - 2*b9 + 3*b7 - 4*b1) * q^93 + (-b18 + 2*b17 + 2*b15 - b14 + b12 + b11 + b10 - b9 + b6 - 2*b1 - 2) * q^95 + (b19 - b18 - b17 + 2*b14 + 2*b13 + b9 - 2*b7) * q^97 + (b12 - 2*b10 - 2*b8 - 2*b7 + b5 + 2*b3 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + q^{5} + 10 q^{9}+O(q^{10})$$ 20 * q + q^5 + 10 * q^9 $$20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65} + 48 q^{69} - 2 q^{71} + 46 q^{75} - 16 q^{79} + 26 q^{81} - 39 q^{85} - 40 q^{89} - 4 q^{91} - 43 q^{95} - 20 q^{99}+O(q^{100})$$ 20 * q + q^5 + 10 * q^9 - 5 * q^15 + 14 * q^19 - 8 * q^21 + 9 * q^25 - 16 * q^29 + 8 * q^31 - 2 * q^35 - 8 * q^39 + 26 * q^41 - 32 * q^45 - 44 * q^49 + 26 * q^51 - 12 * q^55 + 4 * q^59 + 2 * q^61 - 18 * q^65 + 48 * q^69 - 2 * q^71 + 46 * q^75 - 16 * q^79 + 26 * q^81 - 39 * q^85 - 40 * q^89 - 4 * q^91 - 43 * q^95 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + 124617 x^{4} - 24768 x^{2} + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 257269301703 \nu^{18} - 4826223210105 \nu^{16} + 61286511972345 \nu^{14} - 438571056316700 \nu^{12} + \cdots + 63\!\cdots\!36 ) / 16\!\cdots\!37$$ (257269301703*v^18 - 4826223210105*v^16 + 61286511972345*v^14 - 438571056316700*v^12 + 2323089270554580*v^10 - 7266718287702885*v^8 + 16741977055075822*v^6 - 10869496174421115*v^4 + 2170497176992320*v^2 + 63024290250218436) / 16690142463557537 $$\beta_{3}$$ $$=$$ $$( 58379368812683 \nu^{18} + \cdots - 23\!\cdots\!96 ) / 10\!\cdots\!68$$ (58379368812683*v^18 - 1151122140944668*v^16 + 14928136974663543*v^14 - 112486124646259822*v^12 + 618424582627382742*v^10 - 2140435717198619993*v^8 + 5336438184716201168*v^6 - 6768804320649661609*v^4 + 6579414048167166051*v^2 - 238859269757341696) / 1068169117667682368 $$\beta_{4}$$ $$=$$ $$( 63\!\cdots\!85 \nu^{18} + \cdots - 25\!\cdots\!84 ) / 88\!\cdots\!44$$ (6366481800502185*v^18 - 137623959156754484*v^16 + 1868654652948801741*v^14 - 15323421807441242074*v^12 + 89902370458361971154*v^10 - 349276396009867994211*v^8 + 944354362189538749232*v^6 - 1468016942861277361043*v^4 + 1259578493369127028273*v^2 - 250780428775971323584) / 88658036766417636544 $$\beta_{5}$$ $$=$$ $$( - 108802723986440 \nu^{18} + \cdots - 49\!\cdots\!91 ) / 13\!\cdots\!71$$ (-108802723986440*v^18 + 2093074375676352*v^16 - 26579215713710528*v^14 + 192433368253136232*v^12 - 1007495594987342592*v^10 + 3151487442894034624*v^8 - 6782358791543175575*v^6 + 4713968444633542976*v^4 - 941318257748307968*v^2 - 4955504751551263591) / 1385281824475275571 $$\beta_{6}$$ $$=$$ $$( - 123113231093090 \nu^{18} + \cdots - 29\!\cdots\!75 ) / 13\!\cdots\!71$$ (-123113231093090*v^18 + 2391532877467530*v^16 - 30369235310188170*v^14 + 220945270298193633*v^12 - 1151157774093599880*v^10 + 3600868617089492610*v^8 - 7533337923107099056*v^6 + 5386149030199971390*v^4 - 1075543988176763520*v^2 - 2918597117977935675) / 1385281824475275571 $$\beta_{7}$$ $$=$$ $$( 112685518955184 \nu^{19} + \cdots + 22\!\cdots\!52 ) / 88\!\cdots\!44$$ (112685518955184*v^19 - 9174325379691717*v^18 - 1633888394360592*v^17 + 183358087220694436*v^16 + 20748174355631888*v^15 - 2387658497653929081*v^14 - 126231930067659184*v^13 + 18206740636302532114*v^12 + 786467685596365632*v^11 - 100804631860325583978*v^10 - 2460103098942103504*v^9 + 356441852003984207511*v^8 + 10217679424117915920*v^7 - 901408307418290504720*v^6 - 3679801550568386096*v^5 + 1230925670964922837447*v^4 + 734808564190512128*v^3 - 1127871072781709528461*v^2 + 182780307164276557072*v + 224153346284272802752) / 88658036766417636544 $$\beta_{8}$$ $$=$$ $$( - 112685518955184 \nu^{19} + \cdots + 22\!\cdots\!52 ) / 88\!\cdots\!44$$ (-112685518955184*v^19 - 9174325379691717*v^18 + 1633888394360592*v^17 + 183358087220694436*v^16 - 20748174355631888*v^15 - 2387658497653929081*v^14 + 126231930067659184*v^13 + 18206740636302532114*v^12 - 786467685596365632*v^11 - 100804631860325583978*v^10 + 2460103098942103504*v^9 + 356441852003984207511*v^8 - 10217679424117915920*v^7 - 901408307418290504720*v^6 + 3679801550568386096*v^5 + 1230925670964922837447*v^4 - 734808564190512128*v^3 - 1127871072781709528461*v^2 - 182780307164276557072*v + 224153346284272802752) / 88658036766417636544 $$\beta_{9}$$ $$=$$ $$( 149437719142871 \nu^{19} + \cdots + 22\!\cdots\!45 \nu ) / 55\!\cdots\!84$$ (149437719142871*v^19 - 2750374974599151*v^17 + 34926044957117039*v^15 - 247970499250483465*v^13 + 1323885430768152396*v^11 - 4141167794335228387*v^9 + 9657228077683790013*v^7 - 6194324001019140413*v^5 + 1236926030594408384*v^3 + 22508069497503397645*v) / 5541127297901102284 $$\beta_{10}$$ $$=$$ $$( 54263059985435 \nu^{18} + \cdots - 12\!\cdots\!72 ) / 26\!\cdots\!92$$ (54263059985435*v^18 - 1073902569582988*v^16 + 13947552783106023*v^14 - 105468987745192622*v^12 + 581255154298509462*v^10 - 2024168224595373833*v^8 + 5068566551834988016*v^6 - 6594892381858923769*v^4 + 6277643813918368339*v^2 - 1247247913760836672) / 267042279416920592 $$\beta_{11}$$ $$=$$ $$( 58379368812683 \nu^{19} + \cdots - 23\!\cdots\!96 \nu ) / 10\!\cdots\!68$$ (58379368812683*v^19 - 1151122140944668*v^17 + 14928136974663543*v^15 - 112486124646259822*v^13 + 618424582627382742*v^11 - 2140435717198619993*v^9 + 5336438184716201168*v^7 - 6768804320649661609*v^5 + 6579414048167166051*v^3 - 238859269757341696*v) / 1068169117667682368 $$\beta_{12}$$ $$=$$ $$( 21\!\cdots\!13 \nu^{18} + \cdots - 91\!\cdots\!84 ) / 88\!\cdots\!44$$ (21472706330772813*v^18 - 429531870127860356*v^16 + 5597440381244918081*v^14 - 42637019206442019714*v^12 + 235405967494108868890*v^10 - 823296813298088269295*v^8 + 2040723648133655490800*v^6 - 2590582498362261966815*v^4 + 2114820325419353216181*v^2 - 91430995209659510784) / 88658036766417636544 $$\beta_{13}$$ $$=$$ $$( 41\!\cdots\!91 \nu^{19} + \cdots + 70\!\cdots\!92 ) / 35\!\cdots\!76$$ (41145391020481691*v^19 + 9113271949323008*v^18 - 807651770702036060*v^17 - 169488444796903296*v^16 + 10458402250723591943*v^15 + 2152274179832962944*v^14 - 78527538519393341710*v^13 - 15365184231760305024*v^12 + 431134182000265878518*v^11 + 81582796826776290816*v^10 - 1487157977486890963977*v^9 - 255194326441686202752*v^8 + 3714706244185375280592*v^7 + 577191879275290897152*v^6 - 4710527327865639831641*v^5 - 381717529862951442048*v^4 + 4520388542025619161715*v^3 + 76224031701280088064*v^2 - 166218017713291595776*v + 709395219183111220992) / 354632147065670546176 $$\beta_{14}$$ $$=$$ $$( 40\!\cdots\!55 \nu^{19} + \cdots + 18\!\cdots\!16 ) / 35\!\cdots\!76$$ (40694648944660955*v^19 - 45810573468089876*v^18 - 801116217124593692*v^17 + 902920793679681040*v^16 + 10375409553301064391*v^15 - 11702908170448679268*v^14 - 78022610799122704974*v^13 + 88192146776970433480*v^12 + 427988311257880415990*v^11 - 484801324268078626728*v^10 - 1477317565091122549961*v^9 + 1680961734457623032796*v^8 + 3673835526488903616912*v^7 - 4182825108948452916032*v^6 - 4695808121663366287257*v^5 + 5305420213722642791836*v^4 + 4517449307768857113203*v^3 - 4587708322828118201908*v^2 - 897339246370397824064*v + 187218165953979990016) / 354632147065670546176 $$\beta_{15}$$ $$=$$ $$( - 352957477246353 \nu^{19} + \cdots - 15\!\cdots\!77 \nu ) / 27\!\cdots\!42$$ (-352957477246353*v^19 + 6732287676656781*v^17 - 85490954590084109*v^15 + 617058244498298531*v^13 - 3240568160043291876*v^11 + 10136629792755534697*v^9 - 21944735732755401673*v^7 + 15162285696465178103*v^5 - 3027711475567210304*v^3 - 15112667902972457477*v) / 2770563648950551142 $$\beta_{16}$$ $$=$$ $$( 395664181329051 \nu^{19} + \cdots + 31\!\cdots\!37 \nu ) / 27\!\cdots\!42$$ (395664181329051*v^19 - 7533440729534211*v^17 + 95664515577493379*v^15 - 689861039846870731*v^13 + 3626200978955352156*v^11 - 11342905028514213607*v^9 + 24723903923897988125*v^7 - 16966622061419083193*v^5 + 3388014006947935424*v^3 + 31115827382409820137*v) / 2770563648950551142 $$\beta_{17}$$ $$=$$ $$( 86\!\cdots\!67 \nu^{19} + \cdots - 35\!\cdots\!92 \nu ) / 35\!\cdots\!76$$ (86955964488571567*v^19 - 1710572564381717100*v^17 + 22161310421172271211*v^15 - 166719685296363775190*v^13 + 915935506268344505246*v^11 - 3168119711944513996773*v^9 + 7897531353133828196624*v^7 - 10015947541588282623477*v^5 + 9108096864853737363623*v^3 - 353436183667271585792*v) / 354632147065670546176 $$\beta_{18}$$ $$=$$ $$( 44\!\cdots\!75 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 17\!\cdots\!88$$ (44403267011197975*v^19 - 905638902720961484*v^17 + 11924107064037836787*v^15 - 92792312405128286118*v^13 + 522399676445550876270*v^11 - 1902277734940496224477*v^9 + 4916800551459821697744*v^7 - 7052914632463443149421*v^5 + 6287552361196403142031*v^3 - 1250374162009682137408*v) / 177316073532835273088 $$\beta_{19}$$ $$=$$ $$( - 99\!\cdots\!71 \nu^{19} + \cdots + 25\!\cdots\!64 \nu ) / 17\!\cdots\!88$$ (-99815889287337571*v^19 + 2000882759717695292*v^17 - 26141423587608532335*v^15 + 200169113650494448030*v^13 - 1113670060332052229382*v^11 + 3956950094181766287265*v^9 - 10054026219204032528144*v^7 + 13679166208473456186225*v^5 - 12640273317629698848059*v^3 + 2512480201371985676864*v) / 177316073532835273088
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{10} + 4\beta_{3} - \beta_{2}$$ -b10 + 4*b3 - b2 $$\nu^{3}$$ $$=$$ $$\beta_{19} + \beta_{18} - \beta_{16} - \beta_{15} + 6\beta_{11}$$ b19 + b18 - b16 - b15 + 6*b11 $$\nu^{4}$$ $$=$$ $$2\beta_{12} - 9\beta_{10} - \beta_{8} - \beta_{7} + 2\beta_{5} - \beta_{4} + 25\beta_{3} - 25$$ 2*b12 - 9*b10 - b8 - b7 + 2*b5 - b4 + 25*b3 - 25 $$\nu^{5}$$ $$=$$ $$10\beta_{19} + 11\beta_{18} + 3\beta_{14} + 3\beta_{13} + 40\beta_{11} - 3\beta_{7} - 40\beta_1$$ 10*b19 + 11*b18 + 3*b14 + 3*b13 + 40*b11 - 3*b7 - 40*b1 $$\nu^{6}$$ $$=$$ $$8\beta_{14} - 8\beta_{13} - 8\beta_{8} - 16\beta_{6} + 27\beta_{5} + 72\beta_{2} - 177$$ 8*b14 - 8*b13 - 8*b8 - 16*b6 + 27*b5 + 72*b2 - 177 $$\nu^{7}$$ $$=$$ $$88\beta_{16} + 99\beta_{15} + 6\beta_{9} + 46\beta_{8} - 46\beta_{7} - 283\beta_1$$ 88*b16 + 99*b15 + 6*b9 + 46*b8 - 46*b7 - 283*b1 $$\nu^{8}$$ $$=$$ $$47 \beta_{14} - 47 \beta_{13} - 279 \beta_{12} + 569 \beta_{10} + 47 \beta_{7} - 174 \beta_{6} + 174 \beta_{4} - 1329 \beta_{3} + 569 \beta_{2}$$ 47*b14 - 47*b13 - 279*b12 + 569*b10 + 47*b7 - 174*b6 + 174*b4 - 1329*b3 + 569*b2 $$\nu^{9}$$ $$=$$ $$- 743 \beta_{19} - 848 \beta_{18} + 116 \beta_{17} + 743 \beta_{16} + 848 \beta_{15} - 511 \beta_{14} - 511 \beta_{13} - 2083 \beta_{11} + 511 \beta_{8}$$ -743*b19 - 848*b18 + 116*b17 + 743*b16 + 848*b15 - 511*b14 - 511*b13 - 2083*b11 + 511*b8 $$\nu^{10}$$ $$=$$ $$- 2613 \beta_{12} + 4522 \beta_{10} + 221 \beta_{8} + 221 \beta_{7} - 2613 \beta_{5} + 1649 \beta_{4} - 10318 \beta_{3} + 10318$$ -2613*b12 + 4522*b10 + 221*b8 + 221*b7 - 2613*b5 + 1649*b4 - 10318*b3 + 10318 $$\nu^{11}$$ $$=$$ $$- 6171 \beta_{19} - 7135 \beta_{18} + 1486 \beta_{17} - 5005 \beta_{14} - 5005 \beta_{13} - 15785 \beta_{11} - 1486 \beta_{9} + 5005 \beta_{7} + 15785 \beta_1$$ -6171*b19 - 7135*b18 + 1486*b17 - 5005*b14 - 5005*b13 - 15785*b11 - 1486*b9 + 5005*b7 + 15785*b1 $$\nu^{12}$$ $$=$$ $$-644\beta_{14} + 644\beta_{13} + 644\beta_{8} + 14695\beta_{6} - 23316\beta_{5} - 36226\beta_{2} + 81771$$ -644*b14 + 644*b13 + 644*b8 + 14695*b6 - 23316*b5 - 36226*b2 + 81771 $$\nu^{13}$$ $$=$$ $$-50921\beta_{16} - 59542\beta_{15} - 15954\beta_{9} - 45988\beta_{8} + 45988\beta_{7} + 122286\beta_1$$ -50921*b16 - 59542*b15 - 15954*b9 - 45988*b8 + 45988*b7 + 122286*b1 $$\nu^{14}$$ $$=$$ $$2400 \beta_{14} - 2400 \beta_{13} + 202439 \beta_{12} - 292291 \beta_{10} + 2400 \beta_{7} + 126943 \beta_{6} - 126943 \beta_{4} + 656616 \beta_{3} - 292291 \beta_{2}$$ 2400*b14 - 2400*b13 + 202439*b12 - 292291*b10 + 2400*b7 + 126943*b6 - 126943*b4 + 656616*b3 - 292291*b2 $$\nu^{15}$$ $$=$$ $$419234 \beta_{19} + 494730 \beta_{18} - 155792 \beta_{17} - 419234 \beta_{16} - 494730 \beta_{15} + 407278 \beta_{14} + 407278 \beta_{13} + 963263 \beta_{11} - 407278 \beta_{8}$$ 419234*b19 + 494730*b18 - 155792*b17 - 419234*b16 - 494730*b15 + 407278*b14 + 407278*b13 + 963263*b11 - 407278*b8 $$\nu^{16}$$ $$=$$ $$1728520 \beta_{12} - 2371957 \beta_{10} + 68340 \beta_{8} + 68340 \beta_{7} + 1728520 \beta_{5} - 1077998 \beta_{4} + 5318130 \beta_{3} - 5318130$$ 1728520*b12 - 2371957*b10 + 68340*b8 + 68340*b7 + 1728520*b5 - 1077998*b4 + 5318130*b3 - 5318130 $$\nu^{17}$$ $$=$$ $$3449955 \beta_{19} + 4100477 \beta_{18} - 1437724 \beta_{17} + 3525380 \beta_{14} + 3525380 \beta_{13} + 7683002 \beta_{11} + 1437724 \beta_{9} - 3525380 \beta_{7} - 7683002 \beta_1$$ 3449955*b19 + 4100477*b18 - 1437724*b17 + 3525380*b14 + 3525380*b13 + 7683002*b11 + 1437724*b9 - 3525380*b7 - 7683002*b1 $$\nu^{18}$$ $$=$$ $$- 862627 \beta_{14} + 862627 \beta_{13} + 862627 \beta_{8} - 9062991 \beta_{6} + 14601192 \beta_{5} + 19333911 \beta_{2} - 43320969$$ -862627*b14 + 862627*b13 + 862627*b8 - 9062991*b6 + 14601192*b5 + 19333911*b2 - 43320969 $$\nu^{19}$$ $$=$$ $$28396902 \beta_{16} + 33935103 \beta_{15} + 12801656 \beta_{9} + 30065011 \beta_{8} - 30065011 \beta_{7} - 61849398 \beta_1$$ 28396902*b16 + 33935103*b15 + 12801656*b9 + 30065011*b8 - 30065011*b7 - 61849398*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −2.48777 + 1.43632i −2.10552 + 1.21562i −1.74361 + 1.00667i −1.08802 + 0.628167i −0.392182 + 0.226426i 0.392182 − 0.226426i 1.08802 − 0.628167i 1.74361 − 1.00667i 2.10552 − 1.21562i 2.48777 − 1.43632i −2.48777 − 1.43632i −2.10552 − 1.21562i −1.74361 − 1.00667i −1.08802 − 0.628167i −0.392182 − 0.226426i 0.392182 + 0.226426i 1.08802 + 0.628167i 1.74361 + 1.00667i 2.10552 + 1.21562i 2.48777 + 1.43632i
0 −2.48777 1.43632i 0 1.38776 + 1.75332i 0 3.54568i 0 2.62601 + 4.54838i 0
49.2 0 −2.10552 1.21562i 0 −0.896156 2.04863i 0 0.663818i 0 1.45548 + 2.52097i 0
49.3 0 −1.74361 1.00667i 0 −1.95659 + 1.08248i 0 1.34403i 0 0.526784 + 0.912416i 0
49.4 0 −1.08802 0.628167i 0 1.87507 + 1.21824i 0 4.97100i 0 −0.710812 1.23116i 0
49.5 0 −0.392182 0.226426i 0 1.82467 1.29251i 0 2.54366i 0 −1.39746 2.42048i 0
49.6 0 0.392182 + 0.226426i 0 0.207009 2.22647i 0 2.54366i 0 −1.39746 2.42048i 0
49.7 0 1.08802 + 0.628167i 0 −1.99256 1.01474i 0 4.97100i 0 −0.710812 1.23116i 0
49.8 0 1.74361 + 1.00667i 0 0.0408382 + 2.23570i 0 1.34403i 0 0.526784 + 0.912416i 0
49.9 0 2.10552 + 1.21562i 0 2.22225 0.248224i 0 0.663818i 0 1.45548 + 2.52097i 0
49.10 0 2.48777 + 1.43632i 0 −2.21230 0.325180i 0 3.54568i 0 2.62601 + 4.54838i 0
349.1 0 −2.48777 + 1.43632i 0 1.38776 1.75332i 0 3.54568i 0 2.62601 4.54838i 0
349.2 0 −2.10552 + 1.21562i 0 −0.896156 + 2.04863i 0 0.663818i 0 1.45548 2.52097i 0
349.3 0 −1.74361 + 1.00667i 0 −1.95659 1.08248i 0 1.34403i 0 0.526784 0.912416i 0
349.4 0 −1.08802 + 0.628167i 0 1.87507 1.21824i 0 4.97100i 0 −0.710812 + 1.23116i 0
349.5 0 −0.392182 + 0.226426i 0 1.82467 + 1.29251i 0 2.54366i 0 −1.39746 + 2.42048i 0
349.6 0 0.392182 0.226426i 0 0.207009 + 2.22647i 0 2.54366i 0 −1.39746 + 2.42048i 0
349.7 0 1.08802 0.628167i 0 −1.99256 + 1.01474i 0 4.97100i 0 −0.710812 + 1.23116i 0
349.8 0 1.74361 1.00667i 0 0.0408382 2.23570i 0 1.34403i 0 0.526784 0.912416i 0
349.9 0 2.10552 1.21562i 0 2.22225 + 0.248224i 0 0.663818i 0 1.45548 2.52097i 0
349.10 0 2.48777 1.43632i 0 −2.21230 + 0.325180i 0 3.54568i 0 2.62601 4.54838i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.r.a 20
3.b odd 2 1 3420.2.bj.c 20
5.b even 2 1 inner 380.2.r.a 20
5.c odd 4 2 1900.2.i.g 20
15.d odd 2 1 3420.2.bj.c 20
19.c even 3 1 inner 380.2.r.a 20
57.h odd 6 1 3420.2.bj.c 20
95.i even 6 1 inner 380.2.r.a 20
95.m odd 12 2 1900.2.i.g 20
285.n odd 6 1 3420.2.bj.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.r.a 20 1.a even 1 1 trivial
380.2.r.a 20 5.b even 2 1 inner
380.2.r.a 20 19.c even 3 1 inner
380.2.r.a 20 95.i even 6 1 inner
1900.2.i.g 20 5.c odd 4 2
1900.2.i.g 20 95.m odd 12 2
3420.2.bj.c 20 3.b odd 2 1
3420.2.bj.c 20 15.d odd 2 1
3420.2.bj.c 20 57.h odd 6 1
3420.2.bj.c 20 285.n odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20} - 20 T^{18} + 261 T^{16} + \cdots + 4096$$
$5$ $$T^{20} - T^{19} - 4 T^{18} + \cdots + 9765625$$
$7$ $$(T^{10} + 46 T^{8} + 651 T^{6} + 3285 T^{4} + \cdots + 1600)^{2}$$
$11$ $$(T^{5} - 27 T^{3} - 29 T^{2} + 107 T + 148)^{4}$$
$13$ $$T^{20} - 78 T^{18} + \cdots + 13051691536$$
$17$ $$T^{20} - 98 T^{18} + \cdots + 9721171216$$
$19$ $$(T^{10} - 7 T^{9} + 48 T^{8} + \cdots + 2476099)^{2}$$
$23$ $$T^{20} - 211 T^{18} + \cdots + 39691260010000$$
$29$ $$(T^{10} + 8 T^{9} + 78 T^{8} + 38 T^{7} + \cdots + 28561)^{2}$$
$31$ $$(T^{5} - 2 T^{4} - 93 T^{3} + 251 T^{2} + \cdots + 388)^{4}$$
$37$ $$(T^{10} + 86 T^{8} + 2655 T^{6} + \cdots + 518400)^{2}$$
$41$ $$(T^{10} - 13 T^{9} + 186 T^{8} + \cdots + 1368900)^{2}$$
$43$ $$T^{20} + \cdots + 122963703210000$$
$47$ $$T^{20} - 81 T^{18} + \cdots + 2998219536$$
$53$ $$T^{20} + \cdots + 591687332741376$$
$59$ $$(T^{10} - 2 T^{9} + 62 T^{8} - 82 T^{7} + \cdots + 1)^{2}$$
$61$ $$(T^{10} - T^{9} + 115 T^{8} - 450 T^{7} + \cdots + 3073009)^{2}$$
$67$ $$T^{20} + \cdots + 805854925357056$$
$71$ $$(T^{10} + T^{9} + 164 T^{8} + \cdots + 640140601)^{2}$$
$73$ $$T^{20} - 227 T^{18} + \cdots + 25628906250000$$
$79$ $$(T^{10} + 8 T^{9} + 213 T^{8} + \cdots + 233967616)^{2}$$
$83$ $$(T^{10} + 323 T^{8} + 26698 T^{6} + \cdots + 54464400)^{2}$$
$89$ $$(T^{10} + 20 T^{9} + 479 T^{8} + \cdots + 5314993216)^{2}$$
$97$ $$T^{20} + \cdots + 385136700010000$$