# Properties

 Label 390.2.j.a Level $390$ Weight $2$ Character orbit 390.j Analytic conductor $3.114$ 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] = [390,2,Mod(73,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("390.73");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2$$ x^12 + 6*x^10 - 24*x^9 + 18*x^8 + 40*x^7 - 82*x^6 + 12*x^5 + 228*x^4 - 284*x^3 + 124*x^2 - 16*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( 47412266 \nu^{11} + 206226196 \nu^{10} + 498918647 \nu^{9} + 275321548 \nu^{8} + \cdots - 957747072 ) / 13913005925$$ (47412266*v^11 + 206226196*v^10 + 498918647*v^9 + 275321548*v^8 - 2696897549*v^7 + 1486501171*v^6 + 4787559064*v^5 - 7076479199*v^4 + 8756643254*v^3 + 27337175830*v^2 + 5076012014*v - 957747072) / 13913005925 $$\beta_{2}$$ $$=$$ $$( - 478873536 \nu^{11} - 47412266 \nu^{10} - 3079467412 \nu^{9} + 10994046217 \nu^{8} + \cdots + 16498970487 ) / 13913005925$$ (-478873536*v^11 - 47412266*v^10 - 3079467412*v^9 + 10994046217*v^8 - 8895045196*v^7 - 16458043891*v^6 + 37781128781*v^5 - 10534041496*v^4 - 102106687009*v^3 + 127243440970*v^2 - 86717494294*v + 16498970487) / 13913005925 $$\beta_{3}$$ $$=$$ $$( - 562969191 \nu^{11} - 625242846 \nu^{10} - 3845754997 \nu^{9} + 9502475052 \nu^{8} + \cdots - 278134328 ) / 13913005925$$ (-562969191*v^11 - 625242846*v^10 - 3845754997*v^9 + 9502475052*v^8 + 1876460724*v^7 - 24232626521*v^6 + 16912165086*v^5 + 23524932649*v^4 - 115866301754*v^3 + 30598072970*v^2 + 7037733136*v - 278134328) / 13913005925 $$\beta_{4}$$ $$=$$ $$( - 65564 \nu^{11} + 54701 \nu^{10} - 348926 \nu^{9} + 1928638 \nu^{8} - 2217584 \nu^{7} + \cdots + 1239927 ) / 1476181$$ (-65564*v^11 + 54701*v^10 - 348926*v^9 + 1928638*v^8 - 2217584*v^7 - 2535511*v^6 + 7789630*v^5 - 3089126*v^4 - 16693786*v^3 + 29984596*v^2 - 14276912*v + 1239927) / 1476181 $$\beta_{5}$$ $$=$$ $$( 642662513 \nu^{11} + 232175953 \nu^{10} + 3952388821 \nu^{9} - 13970332336 \nu^{8} + \cdots - 4015649046 ) / 13913005925$$ (642662513*v^11 + 232175953*v^10 + 3952388821*v^9 - 13970332336*v^8 + 6804053268*v^7 + 28228005853*v^6 - 41483355748*v^5 - 9709359957*v^4 + 142813240272*v^3 - 124797081160*v^2 + 15569923552*v - 4015649046) / 13913005925 $$\beta_{6}$$ $$=$$ $$( 1158219438 \nu^{11} + 651192603 \nu^{10} + 7299225171 \nu^{9} - 23748128936 \nu^{8} + \cdots - 2779767646 ) / 13913005925$$ (1158219438*v^11 + 651192603*v^10 + 7299225171*v^9 - 23748128936*v^8 + 7624490093*v^7 + 50974131203*v^6 - 63183079898*v^5 - 26157813407*v^4 + 249922898772*v^3 - 182732329960*v^2 + 31282190252*v - 2779767646) / 13913005925 $$\beta_{7}$$ $$=$$ $$( 2007824523 \nu^{11} + 642662513 \nu^{10} + 12279123091 \nu^{9} - 44235399731 \nu^{8} + \cdots - 2642262891 ) / 13913005925$$ (2007824523*v^11 + 642662513*v^10 + 12279123091*v^9 - 44235399731*v^8 + 22170509078*v^7 + 87117034188*v^6 - 136413605033*v^5 - 17389461472*v^4 + 448074631287*v^3 - 427408924260*v^2 + 124173159692*v - 2642262891) / 13913005925 $$\beta_{8}$$ $$=$$ $$( 2024040388 \nu^{11} - 481621272 \nu^{10} + 11262741596 \nu^{9} - 52099333786 \nu^{8} + \cdots + 4694601129 ) / 13913005925$$ (2024040388*v^11 - 481621272*v^10 + 11262741596*v^9 - 52099333786*v^8 + 42254553243*v^7 + 89411069453*v^6 - 188844016148*v^5 + 27091101493*v^4 + 500988367497*v^3 - 653662361260*v^2 + 209022556802*v + 4694601129) / 13913005925 $$\beta_{9}$$ $$=$$ $$( 2281630388 \nu^{11} + 1661658328 \nu^{10} + 14577758021 \nu^{9} - 44298602736 \nu^{8} + \cdots + 27875512029 ) / 13913005925$$ (2281630388*v^11 + 1661658328*v^10 + 14577758021*v^9 - 44298602736*v^8 + 6809317318*v^7 + 102670257953*v^6 - 115495753223*v^5 - 73779180757*v^4 + 487889800972*v^3 - 279624133460*v^2 - 2375444498*v + 27875512029) / 13913005925 $$\beta_{10}$$ $$=$$ $$( - 511017007 \nu^{11} - 120344967 \nu^{10} - 3187847414 \nu^{9} + 11403406359 \nu^{8} + \cdots + 1016773599 ) / 2782601185$$ (-511017007*v^11 - 120344967*v^10 - 3187847414*v^9 + 11403406359*v^8 - 7135610307*v^7 - 20604017287*v^6 + 37582763392*v^5 - 1934833757*v^4 - 113438767513*v^3 + 124243790620*v^2 - 52095832808*v + 1016773599) / 2782601185 $$\beta_{11}$$ $$=$$ $$( 771058311 \nu^{11} + 121340051 \nu^{10} + 4638436737 \nu^{9} - 17846443417 \nu^{8} + \cdots - 7445776057 ) / 2782601185$$ (771058311*v^11 + 121340051*v^10 + 4638436737*v^9 - 17846443417*v^8 + 10922555446*v^7 + 32171693006*v^6 - 57280878911*v^5 + 349625841*v^4 + 174316524684*v^3 - 191644762680*v^2 + 67894160584*v - 7445776057) / 2782601185
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 ) / 2$$ (b6 - b5 + b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{5} - \beta_{4} - 3\beta_{3} - 4\beta_{2} - 4\beta _1 - 1 ) / 2$$ (-b11 + b10 + b9 + b8 - 3*b5 - b4 - 3*b3 - 4*b2 - 4*b1 - 1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{9} - \beta_{8} + 4\beta_{7} - 5\beta_{6} + 5\beta_{5} - \beta_{4} - 2\beta_{3} + \beta_{2} + 3\beta _1 + 3$$ b10 - b9 - b8 + 4*b7 - 5*b6 + 5*b5 - b4 - 2*b3 + b2 + 3*b1 + 3 $$\nu^{4}$$ $$=$$ $$5 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 5 \beta_{5} + \cdots + 3$$ 5*b11 - 5*b10 - 3*b9 + 3*b8 - 13*b7 + 13*b6 + 5*b5 + 17*b4 + 20*b3 + 5*b2 + 15*b1 + 3 $$\nu^{5}$$ $$=$$ $$- 13 \beta_{11} + 17 \beta_{9} + 13 \beta_{8} - 25 \beta_{7} + 27 \beta_{6} - 72 \beta_{5} - 25 \beta_{4} + \cdots - 45$$ -13*b11 + 17*b9 + 13*b8 - 25*b7 + 27*b6 - 72*b5 - 25*b4 - 44*b3 - 28*b2 - 72*b1 - 45 $$\nu^{6}$$ $$=$$ $$- 27 \beta_{11} + 59 \beta_{10} - 27 \beta_{9} - 59 \beta_{8} + 234 \beta_{7} - 218 \beta_{6} + \cdots + 100$$ -27*b11 + 59*b10 - 27*b9 - 59*b8 + 234*b7 - 218*b6 + 130*b5 - 100*b4 - 130*b3 + 100 $$\nu^{7}$$ $$=$$ $$218 \beta_{11} - 157 \beta_{10} - 157 \beta_{9} - 301 \beta_{7} + 301 \beta_{6} + 519 \beta_{5} + \cdots + 301$$ 218*b11 - 157*b10 - 157*b9 - 301*b7 + 301*b6 + 519*b5 + 564*b4 + 867*b3 + 346*b2 + 867*b1 + 301 $$\nu^{8}$$ $$=$$ $$- 301 \beta_{11} - 301 \beta_{10} + 710 \beta_{9} + 710 \beta_{8} - 1925 \beta_{7} + 1889 \beta_{6} + \cdots - 2036$$ -301*b11 - 301*b10 + 710*b9 + 710*b8 - 1925*b7 + 1889*b6 - 2961*b5 - 301*b4 - 710*b3 - 914*b2 - 2298*b1 - 2036 $$\nu^{9}$$ $$=$$ $$- 1889 \beta_{11} + 2660 \beta_{10} - 1889 \beta_{8} + 8736 \beta_{7} - 8557 \beta_{6} + 1674 \beta_{5} + \cdots + 1735$$ -1889*b11 + 2660*b10 - 1889*b8 + 8736*b7 - 8557*b6 + 1674*b5 - 6076*b4 - 8557*b3 - 1735*b2 - 4334*b1 + 1735 $$\nu^{10}$$ $$=$$ $$8557 \beta_{11} - 3563 \beta_{10} - 8557 \beta_{9} - 3563 \beta_{8} + 31302 \beta_{5} + 19663 \beta_{4} + \cdots + 19663$$ 8557*b11 - 3563*b10 - 8557*b9 - 3563*b8 + 31302*b5 + 19663*b4 + 31302*b3 + 15794*b2 + 39260*b1 + 19663 $$\nu^{11}$$ $$=$$ $$- 22745 \beta_{10} + 22745 \beta_{9} + 32134 \beta_{8} - 105363 \beta_{7} + 103116 \beta_{6} + \cdots - 73229$$ -22745*b10 + 22745*b9 + 32134*b8 - 105363*b7 + 103116*b6 - 103116*b5 + 20910*b4 + 20017*b3 - 20910*b2 - 52151*b1 - 73229

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 0.563963 − 1.36153i 0.0572576 − 0.138232i −1.32833 + 3.20687i 1.52752 + 0.632721i −1.46953 − 0.608701i 0.649118 + 0.268874i 0.563963 + 1.36153i 0.0572576 + 0.138232i −1.32833 − 3.20687i 1.52752 − 0.632721i −1.46953 + 0.608701i 0.649118 − 0.268874i
−1.00000 −0.707107 0.707107i 1.00000 −1.97581 + 1.04698i 0.707107 + 0.707107i 2.54214i −1.00000 1.00000i 1.97581 1.04698i
73.2 −1.00000 −0.707107 0.707107i 1.00000 0.383740 2.20289i 0.707107 + 0.707107i 1.52873i −1.00000 1.00000i −0.383740 + 2.20289i
73.3 −1.00000 −0.707107 0.707107i 1.00000 1.59207 + 1.57013i 0.707107 + 0.707107i 1.24244i −1.00000 1.00000i −1.59207 1.57013i
73.4 −1.00000 0.707107 + 0.707107i 1.00000 −2.14724 0.623976i −0.707107 0.707107i 1.64083i −1.00000 1.00000i 2.14724 + 0.623976i
73.5 −1.00000 0.707107 + 0.707107i 1.00000 −0.0440169 2.23563i −0.707107 0.707107i 4.35328i −1.00000 1.00000i 0.0440169 + 2.23563i
73.6 −1.00000 0.707107 + 0.707107i 1.00000 2.19126 + 0.445397i −0.707107 0.707107i 0.115977i −1.00000 1.00000i −2.19126 0.445397i
187.1 −1.00000 −0.707107 + 0.707107i 1.00000 −1.97581 1.04698i 0.707107 0.707107i 2.54214i −1.00000 1.00000i 1.97581 + 1.04698i
187.2 −1.00000 −0.707107 + 0.707107i 1.00000 0.383740 + 2.20289i 0.707107 0.707107i 1.52873i −1.00000 1.00000i −0.383740 2.20289i
187.3 −1.00000 −0.707107 + 0.707107i 1.00000 1.59207 1.57013i 0.707107 0.707107i 1.24244i −1.00000 1.00000i −1.59207 + 1.57013i
187.4 −1.00000 0.707107 0.707107i 1.00000 −2.14724 + 0.623976i −0.707107 + 0.707107i 1.64083i −1.00000 1.00000i 2.14724 0.623976i
187.5 −1.00000 0.707107 0.707107i 1.00000 −0.0440169 + 2.23563i −0.707107 + 0.707107i 4.35328i −1.00000 1.00000i 0.0440169 2.23563i
187.6 −1.00000 0.707107 0.707107i 1.00000 2.19126 0.445397i −0.707107 + 0.707107i 0.115977i −1.00000 1.00000i −2.19126 + 0.445397i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.j.a 12
3.b odd 2 1 1170.2.m.f 12
5.b even 2 1 1950.2.j.c 12
5.c odd 4 1 390.2.t.a yes 12
5.c odd 4 1 1950.2.t.b 12
13.d odd 4 1 390.2.t.a yes 12
15.e even 4 1 1170.2.w.f 12
39.f even 4 1 1170.2.w.f 12
65.f even 4 1 1950.2.j.c 12
65.g odd 4 1 1950.2.t.b 12
65.k even 4 1 inner 390.2.j.a 12
195.j odd 4 1 1170.2.m.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.a 12 1.a even 1 1 trivial
390.2.j.a 12 65.k even 4 1 inner
390.2.t.a yes 12 5.c odd 4 1
390.2.t.a yes 12 13.d odd 4 1
1170.2.m.f 12 3.b odd 2 1
1170.2.m.f 12 195.j odd 4 1
1170.2.w.f 12 15.e even 4 1
1170.2.w.f 12 39.f even 4 1
1950.2.j.c 12 5.b even 2 1
1950.2.j.c 12 65.f even 4 1
1950.2.t.b 12 5.c odd 4 1
1950.2.t.b 12 65.g odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} + 32T_{7}^{10} + 304T_{7}^{8} + 1176T_{7}^{6} + 1984T_{7}^{4} + 1216T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{12}$$
$3$ $$(T^{4} + 1)^{3}$$
$5$ $$T^{12} - 2 T^{10} + \cdots + 15625$$
$7$ $$T^{12} + 32 T^{10} + \cdots + 16$$
$11$ $$T^{12} - 4 T^{11} + \cdots + 141376$$
$13$ $$T^{12} - 4 T^{11} + \cdots + 4826809$$
$17$ $$T^{12} + 8 T^{11} + \cdots + 446224$$
$19$ $$T^{12} - 4 T^{11} + \cdots + 23309584$$
$23$ $$T^{12} - 104 T^{9} + \cdots + 38416$$
$29$ $$T^{12} + \cdots + 932203024$$
$31$ $$(T^{2} - 2 T + 2)^{6}$$
$37$ $$T^{12} + \cdots + 632220736$$
$41$ $$T^{12} + \cdots + 3237154816$$
$43$ $$T^{12} + \cdots + 289272064$$
$47$ $$T^{12} + 304 T^{10} + \cdots + 85525504$$
$53$ $$T^{12} + 16 T^{11} + \cdots + 1024$$
$59$ $$T^{12} + 20 T^{11} + \cdots + 94789696$$
$61$ $$(T^{6} - 8 T^{5} + \cdots - 33056)^{2}$$
$67$ $$(T^{6} + 8 T^{5} + \cdots - 122632)^{2}$$
$71$ $$T^{12} + \cdots + 209295270144$$
$73$ $$(T^{6} + 12 T^{5} + \cdots + 6628)^{2}$$
$79$ $$T^{12} + \cdots + 485409024$$
$83$ $$T^{12} + 320 T^{10} + \cdots + 2027776$$
$89$ $$T^{12} + \cdots + 24551129344$$
$97$ $$(T^{6} - 4 T^{5} + \cdots - 1546972)^{2}$$