# Properties

 Label 390.2.t.a Level $390$ Weight $2$ Character orbit 390.t 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(307,390)]

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, 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("390.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.t (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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} - \beta_{7} q^{3} - q^{4} + ( - \beta_{8} + \beta_{7} + \beta_{3} + \cdots - 1) q^{5}+ \cdots - \beta_{4} q^{9}+O(q^{10})$$ q - b4 * q^2 - b7 * q^3 - q^4 + (-b8 + b7 + b3 + b1 - 1) * q^5 + b2 * q^6 + (b7 - b5 + b2) * q^7 + b4 * q^8 - b4 * q^9 $$q - \beta_{4} q^{2} - \beta_{7} q^{3} - q^{4} + ( - \beta_{8} + \beta_{7} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} - \beta_{9} + \cdots + \beta_{3}) q^{99}+O(q^{100})$$ q - b4 * q^2 - b7 * q^3 - q^4 + (-b8 + b7 + b3 + b1 - 1) * q^5 + b2 * q^6 + (b7 - b5 + b2) * q^7 + b4 * q^8 - b4 * q^9 - b11 * q^10 + (-b10 + b8 - 2*b7 - b4 + 1) * q^11 + b7 * q^12 + (-b11 + b6 - b4 + b3 + b2 + 2) * q^13 + (b7 + b3 - b2) * q^14 - b10 * q^15 + q^16 + (-b10 + b8 - b6 - b4 + 1) * q^17 - q^18 + (-b11 - b9 + 2*b7 + 3*b6 - 2*b5 - b4 + 2*b3 + 1) * q^19 + (b8 - b7 - b3 - b1 + 1) * q^20 + (-b6 + b4 - 1) * q^21 + (b11 - b9 + b5 + b3) * q^22 + (-b10 - b8 + b5 - b4 + b3 + 2*b2 + 3*b1 - 1) * q^23 - b2 * q^24 + (b9 + b8 + b7 + b6 + b5) * q^25 + (b8 + b5 - 2*b4 - b3) * q^26 + b2 * q^27 + (-b7 + b5 - b2) * q^28 + (-b11 + b10 + 2*b9 - 2*b8 + 3*b7 - b5 - 3*b4 - b3 + b1 - 2) * q^29 + (-b9 + b6 + b3 - b2 + 1) * q^30 + (b4 + 1) * q^31 - b4 * q^32 + (-b11 + b10 - b5 - b4 + b2 - b1) * q^33 + (b11 - b9 + b5 + b3 - 2*b2 - b1) * q^34 + (b10 - b8 + b6 - b4 + b3 - 2*b2 - b1 + 1) * q^35 + b4 * q^36 + (b11 + b10 + 2*b9 + 2*b8 + 2*b7 - 2*b6 - b5 + b4 - 2*b3 + 3*b2 - b1 - 2) * q^37 + (b10 + b8 + b5 + b3 - 2*b2 + b1) * q^38 + (-b9 - 2*b7 + b6 - b1) * q^39 + b11 * q^40 + (2*b11 - 2*b10 - 2*b9 - 2*b8 + 2*b5 + 2*b3 - 2*b2 + 4*b1) * q^41 + (b4 - b1 + 1) * q^42 + (b10 + b8 - 2*b5 + 2*b4 - 2*b3 + 4*b2 - 2*b1 + 2) * q^43 + (b10 - b8 + 2*b7 + b4 - 1) * q^44 - b11 * q^45 + (-b11 - b9 + 2*b7 - b6) * q^46 + (b11 + b10 + b9 + b8 - 3*b6 + b5 + b4 - b3 + b1 + 1) * q^47 - b7 * q^48 + (b11 + b10 - b7 + b6 + b5 + b4 - 2*b2 - 2*b1 - 1) * q^49 + (b11 - b10 - b7 - b6 + 2*b5 - b3 - 2*b2 + 2*b1) * q^50 + (-b11 + b10 - b5 + b4 + b3 + b2 - b1) * q^51 + (b11 - b6 + b4 - b3 - b2 - 2) * q^52 + (-2*b7 - 2*b6 + b5 + 2*b4 - b3 - 2) * q^53 + b7 * q^54 + (-b11 - b10 - 2*b8 - 3*b7 + b6 - b5 + 2*b4 - 1) * q^55 + (-b7 - b3 + b2) * q^56 + (-b9 + b8 - b7 - b6 + 2*b4 - 2*b3 - 2*b1 + 1) * q^57 + (-2*b11 - 2*b10 + b9 + b8 - 3*b7 - b5 - 2*b4 - b3 + b1 - 3) * q^58 + (b11 - 2*b10 - b9 - 2*b8 + 2*b4 - 4*b2 + 2) * q^59 + b10 * q^60 + (2*b11 + 2*b10 + b6 + 2*b4 - 2*b2 - 3*b1 + 2) * q^61 + (-b4 + 1) * q^62 + (b7 + b3 - b2) * q^63 - q^64 + (-b11 - b10 - b9 - b8 + 2*b6 - b4 + 2*b3 - 3*b2 + 4*b1 - 3) * q^65 + (b9 + b8 - b3 + b2 - b1 - 1) * q^66 + (-b9 + b8 - 4*b7 - 2*b6 - b3 + 3*b2 - 3*b1 + 1) * q^67 + (b10 - b8 + b6 + b4 - 1) * q^68 + (-b11 - b10 + b6 - 2*b5 - b4 + b2 - 2) * q^69 + (-b11 + b9 - 2*b7 + b6 - 2*b4 - b3 + 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 + q^72 + (2*b11 - 2*b10 - 2*b9 + 2*b8 - b7 - b6 + 2*b5 + 2*b4 - 3*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 + (b10 - b8 + b7 + b6 - b5 + 2*b4 - b3 - b1) * q^75 + (b11 + b9 - 2*b7 - 3*b6 + 2*b5 + b4 - 2*b3 - 1) * q^76 + (b11 - b10 + b9 + b8 - 2*b7 - 2*b6 + 2*b5 - b4 - 2*b3 + 1) * q^77 + (b10 + b7 + b6 - b5 + b4 + 2*b2) * q^78 + (b11 - b10 + 2*b9 - 2*b8 + 3*b7 - b6 + b5 + 3*b4 + 2*b3 - 2*b2 + 2*b1 - 2) * q^79 + (-b8 + b7 + b3 + b1 - 1) * q^80 - q^81 + (-2*b11 + 2*b10 - 2*b9 - 2*b8 + 2*b7 - 2*b5 + 2*b3) * q^82 + (b11 + b10 + 2*b9 + 2*b8 - b7 - 2*b6 - b5 + b4 - 2*b3 - b1 + 2) * q^83 + (b6 - b4 + 1) * q^84 + (-b11 - 3*b8 - 4*b7 + b6 + b5 + b4 + b2 + b1) * q^85 + (b11 + b9 + 4*b7 - b5 - b4 + b3 + 1) * q^86 + (b11 - 2*b10 - b9 - 2*b8 + b5 + b4 + b3 + 2*b2 + 3*b1 + 1) * q^87 + (-b11 + b9 - b5 - b3) * q^88 + (2*b11 + b10 - 2*b9 + b8 + 3*b5 + 2*b4 + 3*b3 - 6*b2 + 2) * q^89 + (b8 - b7 - b3 - b1 + 1) * q^90 + (-2*b11 + b9 - b8 + 4*b7 + b6 - 3*b5 - 4*b4 + 5*b2 + b1 - 3) * q^91 + (b10 + b8 - b5 + b4 - b3 - 2*b2 - 3*b1 + 1) * q^92 + (-b7 - b2) * q^93 + (b11 - b10 + b9 - b8 - 3*b6 + b5 - b4 - b3 - b1 - 1) * q^94 + (-2*b11 + 2*b10 + 2*b9 - b7 + 3*b6 - 4*b5 - 3*b4 - 5*b2 - 1) * q^95 + b2 * q^96 + (-2*b11 + 2*b10 - 2*b9 + 2*b8 - 3*b7 + 5*b6 - 2*b5 - 2*b4 + b3 + 3*b2 + b1 + 2) * q^97 + (b9 - b8 - b7 + b6 + b4 - b3 + 2*b2 + 2*b1 - 1) * q^98 + (b11 - b9 + b5 + b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{4} - 4 q^{5}+O(q^{10})$$ 12 * q - 12 * q^4 - 4 * q^5 $$12 q - 12 q^{4} - 4 q^{5} + 4 q^{11} + 20 q^{13} - 4 q^{15} + 12 q^{16} + 8 q^{17} - 12 q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{21} - 4 q^{22} - 4 q^{25} - 4 q^{26} + 4 q^{30} + 12 q^{31} - 8 q^{34} + 12 q^{35} - 16 q^{37} + 4 q^{38} - 12 q^{39} + 8 q^{41} + 8 q^{42} + 16 q^{43} - 4 q^{44} + 32 q^{47} - 20 q^{49} + 8 q^{50} - 20 q^{52} - 16 q^{53} - 12 q^{55} - 40 q^{58} + 20 q^{59} + 4 q^{60} + 16 q^{61} + 12 q^{62} - 12 q^{64} - 32 q^{65} - 16 q^{66} - 8 q^{68} - 32 q^{69} - 20 q^{70} - 32 q^{71} + 12 q^{72} + 4 q^{76} + 16 q^{77} - 4 q^{80} - 12 q^{81} + 8 q^{82} + 32 q^{83} + 8 q^{84} + 12 q^{85} + 16 q^{86} + 20 q^{87} + 4 q^{88} + 16 q^{89} + 4 q^{90} - 28 q^{91} - 8 q^{95} - 4 q^{99}+O(q^{100})$$ 12 * q - 12 * q^4 - 4 * q^5 + 4 * q^11 + 20 * q^13 - 4 * q^15 + 12 * q^16 + 8 * q^17 - 12 * q^18 - 4 * q^19 + 4 * q^20 - 8 * q^21 - 4 * q^22 - 4 * q^25 - 4 * q^26 + 4 * q^30 + 12 * q^31 - 8 * q^34 + 12 * q^35 - 16 * q^37 + 4 * q^38 - 12 * q^39 + 8 * q^41 + 8 * q^42 + 16 * q^43 - 4 * q^44 + 32 * q^47 - 20 * q^49 + 8 * q^50 - 20 * q^52 - 16 * q^53 - 12 * q^55 - 40 * q^58 + 20 * q^59 + 4 * q^60 + 16 * q^61 + 12 * q^62 - 12 * q^64 - 32 * q^65 - 16 * q^66 - 8 * q^68 - 32 * q^69 - 20 * q^70 - 32 * q^71 + 12 * q^72 + 4 * q^76 + 16 * q^77 - 4 * q^80 - 12 * q^81 + 8 * q^82 + 32 * q^83 + 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 - 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 0.0572576 − 0.138232i 0.563963 − 1.36153i −1.32833 + 3.20687i −1.46953 − 0.608701i 1.52752 + 0.632721i 0.649118 + 0.268874i 0.0572576 + 0.138232i 0.563963 + 1.36153i −1.32833 − 3.20687i −1.46953 + 0.608701i 1.52752 − 0.632721i 0.649118 − 0.268874i
1.00000i −0.707107 + 0.707107i −1.00000 −2.20289 + 0.383740i 0.707107 + 0.707107i 1.52873 1.00000i 1.00000i 0.383740 + 2.20289i
307.2 1.00000i −0.707107 + 0.707107i −1.00000 1.04698 1.97581i 0.707107 + 0.707107i 2.54214 1.00000i 1.00000i −1.97581 1.04698i
307.3 1.00000i −0.707107 + 0.707107i −1.00000 1.57013 + 1.59207i 0.707107 + 0.707107i −1.24244 1.00000i 1.00000i 1.59207 1.57013i
307.4 1.00000i 0.707107 0.707107i −1.00000 −2.23563 0.0440169i −0.707107 0.707107i −4.35328 1.00000i 1.00000i −0.0440169 + 2.23563i
307.5 1.00000i 0.707107 0.707107i −1.00000 −0.623976 2.14724i −0.707107 0.707107i 1.64083 1.00000i 1.00000i −2.14724 + 0.623976i
307.6 1.00000i 0.707107 0.707107i −1.00000 0.445397 + 2.19126i −0.707107 0.707107i −0.115977 1.00000i 1.00000i 2.19126 0.445397i
343.1 1.00000i −0.707107 0.707107i −1.00000 −2.20289 0.383740i 0.707107 0.707107i 1.52873 1.00000i 1.00000i 0.383740 2.20289i
343.2 1.00000i −0.707107 0.707107i −1.00000 1.04698 + 1.97581i 0.707107 0.707107i 2.54214 1.00000i 1.00000i −1.97581 + 1.04698i
343.3 1.00000i −0.707107 0.707107i −1.00000 1.57013 1.59207i 0.707107 0.707107i −1.24244 1.00000i 1.00000i 1.59207 + 1.57013i
343.4 1.00000i 0.707107 + 0.707107i −1.00000 −2.23563 + 0.0440169i −0.707107 + 0.707107i −4.35328 1.00000i 1.00000i −0.0440169 2.23563i
343.5 1.00000i 0.707107 + 0.707107i −1.00000 −0.623976 + 2.14724i −0.707107 + 0.707107i 1.64083 1.00000i 1.00000i −2.14724 0.623976i
343.6 1.00000i 0.707107 + 0.707107i −1.00000 0.445397 2.19126i −0.707107 + 0.707107i −0.115977 1.00000i 1.00000i 2.19126 + 0.445397i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.6 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.f 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.t.a yes 12
3.b odd 2 1 1170.2.w.f 12
5.b even 2 1 1950.2.t.b 12
5.c odd 4 1 390.2.j.a 12
5.c odd 4 1 1950.2.j.c 12
13.d odd 4 1 390.2.j.a 12
15.e even 4 1 1170.2.m.f 12
39.f even 4 1 1170.2.m.f 12
65.f even 4 1 inner 390.2.t.a yes 12
65.g odd 4 1 1950.2.j.c 12
65.k even 4 1 1950.2.t.b 12
195.u odd 4 1 1170.2.w.f 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} - 16T_{7}^{4} + 20T_{7}^{3} + 24T_{7}^{2} - 32T_{7} - 4$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{6}$$
$3$ $$(T^{4} + 1)^{3}$$
$5$ $$T^{12} + 4 T^{11} + \cdots + 15625$$
$7$ $$(T^{6} - 16 T^{4} + 20 T^{3} + \cdots - 4)^{2}$$
$11$ $$T^{12} - 4 T^{11} + \cdots + 141376$$
$13$ $$T^{12} - 20 T^{11} + \cdots + 4826809$$
$17$ $$T^{12} - 8 T^{11} + \cdots + 446224$$
$19$ $$T^{12} + 4 T^{11} + \cdots + 23309584$$
$23$ $$T^{12} + 104 T^{9} + \cdots + 38416$$
$29$ $$T^{12} + \cdots + 932203024$$
$31$ $$(T^{2} - 2 T + 2)^{6}$$
$37$ $$(T^{6} + 8 T^{5} + \cdots + 25144)^{2}$$
$41$ $$T^{12} + \cdots + 3237154816$$
$43$ $$T^{12} + \cdots + 289272064$$
$47$ $$(T^{6} - 16 T^{5} + \cdots - 9248)^{2}$$
$53$ $$T^{12} + 16 T^{11} + \cdots + 1024$$
$59$ $$T^{12} - 20 T^{11} + \cdots + 94789696$$
$61$ $$(T^{6} - 8 T^{5} + \cdots - 33056)^{2}$$
$67$ $$T^{12} + \cdots + 15038607424$$
$71$ $$T^{12} + \cdots + 209295270144$$
$73$ $$T^{12} + 536 T^{10} + \cdots + 43930384$$
$79$ $$T^{12} + \cdots + 485409024$$
$83$ $$(T^{6} - 16 T^{5} + \cdots - 1424)^{2}$$
$89$ $$T^{12} + \cdots + 24551129344$$
$97$ $$T^{12} + \cdots + 2393122368784$$