# Properties

 Label 1950.2.bc.i Level $1950$ Weight $2$ Character orbit 1950.bc Analytic conductor $15.571$ 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] = [1950,2,Mod(751,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("1950.751");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.bc (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: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197$$ x^12 - 2*x^11 - 8*x^10 + 34*x^9 + 8*x^8 - 134*x^7 + 98*x^6 + 154*x^5 + 104*x^4 + 190*x^3 - 1196*x^2 - 338*x + 2197 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 203419 \nu^{11} - 163110633 \nu^{10} + 591783880 \nu^{9} + 97338749 \nu^{8} + \cdots + 81183629852 ) / 63907274600$$ (-203419*v^11 - 163110633*v^10 + 591783880*v^9 + 97338749*v^8 - 4513282461*v^7 + 6489146722*v^6 + 4655211661*v^5 - 5740233327*v^4 - 8165110694*v^3 - 33307875431*v^2 + 64277898945*v + 81183629852) / 63907274600 $$\beta_{3}$$ $$=$$ $$( 60968787 \nu^{11} - 2097063441 \nu^{10} + 2300362460 \nu^{9} + 17147379373 \nu^{8} + \cdots + 1269272187404 ) / 830794569800$$ (60968787*v^11 - 2097063441*v^10 + 2300362460*v^9 + 17147379373*v^8 - 55879543947*v^7 - 372787906*v^6 + 135156205247*v^5 - 272345430479*v^4 + 414518069862*v^3 - 554153957987*v^2 - 833086363785*v + 1269272187404) / 830794569800 $$\beta_{4}$$ $$=$$ $$( - 120243408 \nu^{11} + 454030419 \nu^{10} + 2051209685 \nu^{9} - 7036618932 \nu^{8} + \cdots + 618192439839 ) / 830794569800$$ (-120243408*v^11 + 454030419*v^10 + 2051209685*v^9 - 7036618932*v^8 + 2513656023*v^7 + 29967292429*v^6 - 18436259198*v^5 + 35262141211*v^4 + 8309942667*v^3 + 28687118858*v^2 + 98318215515*v + 618192439839) / 830794569800 $$\beta_{5}$$ $$=$$ $$( 16426431 \nu^{11} + 83789417 \nu^{10} - 226795620 \nu^{9} + 267354099 \nu^{8} + \cdots + 20321135952 ) / 63907274600$$ (16426431*v^11 + 83789417*v^10 - 226795620*v^9 + 267354099*v^8 + 1065744289*v^7 - 511723478*v^6 + 4136894311*v^5 + 1601173623*v^4 + 3964105106*v^3 - 3499453881*v^2 + 44426935995*v + 20321135952) / 63907274600 $$\beta_{6}$$ $$=$$ $$( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200$$ (4036*v^11 - 37023*v^10 + 57555*v^9 + 233294*v^8 - 817691*v^7 + 35557*v^6 + 1844066*v^5 - 940887*v^4 + 77961*v^3 - 7481036*v^2 - 4439955*v + 11867687) / 11796200 $$\beta_{7}$$ $$=$$ $$( - 20638 \nu^{11} + 28887 \nu^{10} - 18833 \nu^{9} - 91862 \nu^{8} - 291763 \nu^{7} + \cdots + 21764327 ) / 47610004$$ (-20638*v^11 + 28887*v^10 - 18833*v^9 - 91862*v^8 - 291763*v^7 - 1390803*v^6 + 3193024*v^5 - 2792113*v^4 - 7615595*v^3 - 12957312*v^2 + 2476201*v + 21764327) / 47610004 $$\beta_{8}$$ $$=$$ $$( 5665538 \nu^{11} - 7145579 \nu^{10} - 9226575 \nu^{9} + 110201552 \nu^{8} - 247897203 \nu^{7} + \cdots - 40106339 ) / 12781454920$$ (5665538*v^11 - 7145579*v^10 - 9226575*v^9 + 110201552*v^8 - 247897203*v^7 + 32851611*v^6 + 1496635908*v^5 - 1228577871*v^4 - 1963928377*v^3 + 3156976042*v^2 + 8732837145*v - 40106339) / 12781454920 $$\beta_{9}$$ $$=$$ $$( - 680246389 \nu^{11} - 845264698 \nu^{10} + 11881898255 \nu^{9} - 20291622981 \nu^{8} + \cdots + 1108736562037 ) / 830794569800$$ (-680246389*v^11 - 845264698*v^10 + 11881898255*v^9 - 20291622981*v^8 - 56431077566*v^7 + 138431742257*v^6 - 46377269759*v^5 - 85505629812*v^4 - 58320816489*v^3 - 633066289511*v^2 + 550784663520*v + 1108736562037) / 830794569800 $$\beta_{10}$$ $$=$$ $$( - 1202455216 \nu^{11} + 4870077513 \nu^{10} - 1280213355 \nu^{9} - 35058222714 \nu^{8} + \cdots - 1182431604497 ) / 830794569800$$ (-1202455216*v^11 + 4870077513*v^10 - 1280213355*v^9 - 35058222714*v^8 + 75140023171*v^7 - 22116401667*v^6 - 93428783546*v^5 + 206023103197*v^4 - 574822497041*v^3 + 859741244016*v^2 + 783010043855*v - 1182431604497) / 830794569800 $$\beta_{11}$$ $$=$$ $$( - 78524401 \nu^{11} + 161812093 \nu^{10} + 556127095 \nu^{9} - 2253760279 \nu^{8} + \cdots + 16572567908 ) / 31953637300$$ (-78524401*v^11 + 161812093*v^10 + 556127095*v^9 - 2253760279*v^8 - 695335944*v^7 + 7168384238*v^6 - 1904719281*v^5 - 3795941083*v^4 - 26424465301*v^3 - 15605355549*v^2 + 99935137380*v + 16572567908) / 31953637300
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + 2\beta_{10} + 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{5} - 3\beta_{4} + \beta_{3} + 2$$ -b11 + 2*b10 + 2*b9 + b8 - 2*b7 + b5 - 3*b4 + b3 + 2 $$\nu^{3}$$ $$=$$ $$- 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \cdots - 5$$ -2*b11 + b10 + 3*b9 + b8 + 2*b7 - b6 + b5 + 4*b4 + b3 - b2 + 3*b1 - 5 $$\nu^{4}$$ $$=$$ $$-7\beta_{11} + 7\beta_{10} + 12\beta_{9} + \beta_{8} + 2\beta_{7} + 9\beta_{5} - 6\beta_{4} + \beta_{2} - 3$$ -7*b11 + 7*b10 + 12*b9 + b8 + 2*b7 + 9*b5 - 6*b4 + b2 - 3 $$\nu^{5}$$ $$=$$ $$- 4 \beta_{11} - 5 \beta_{10} - 8 \beta_{9} + 4 \beta_{8} + 29 \beta_{7} - 12 \beta_{6} - 7 \beta_{5} + \cdots - 42$$ -4*b11 - 5*b10 - 8*b9 + 4*b8 + 29*b7 - 12*b6 - 7*b5 + 47*b4 - b3 + 12*b2 - 42 $$\nu^{6}$$ $$=$$ $$- 6 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 24 \beta_{6} + 18 \beta_{5} - 20 \beta_{4} + \cdots - 7$$ -6*b11 + 8*b10 - 2*b9 - 10*b8 - 24*b6 + 18*b5 - 20*b4 + 6*b3 + 38*b2 - 32*b1 - 7 $$\nu^{7}$$ $$=$$ $$50 \beta_{11} - 70 \beta_{10} - 162 \beta_{9} - 50 \beta_{8} + 54 \beta_{7} - 36 \beta_{6} - 90 \beta_{5} + \cdots - 104$$ 50*b11 - 70*b10 - 162*b9 - 50*b8 + 54*b7 - 36*b6 - 90*b5 + 202*b4 - 8*b3 + 76*b2 - 11*b1 - 104 $$\nu^{8}$$ $$=$$ $$81 \beta_{11} + 80 \beta_{10} - 90 \beta_{9} - 155 \beta_{8} - 382 \beta_{7} + 114 \beta_{6} + \cdots + 388$$ 81*b11 + 80*b10 - 90*b9 - 155*b8 - 382*b7 + 114*b6 + 113*b5 - 491*b4 + 73*b3 + 64*b2 - 112*b1 + 388 $$\nu^{9}$$ $$=$$ $$404 \beta_{11} - 223 \beta_{10} - 627 \beta_{9} - 213 \beta_{8} - 540 \beta_{7} + 459 \beta_{6} + \cdots + 249$$ 404*b11 - 223*b10 - 627*b9 - 213*b8 - 540*b7 + 459*b6 - 365*b5 + 328*b4 + 7*b3 - 231*b2 + 163*b1 + 249 $$\nu^{10}$$ $$=$$ $$327 \beta_{11} + 649 \beta_{10} + 866 \beta_{9} - 57 \beta_{8} - 2616 \beta_{7} + 1480 \beta_{6} + \cdots + 2981$$ 327*b11 + 649*b10 + 866*b9 - 57*b8 - 2616*b7 + 1480*b6 + 1181*b5 - 3528*b4 + 248*b3 - 1421*b2 - 202*b1 + 2981 $$\nu^{11}$$ $$=$$ $$1120 \beta_{11} - 2139 \beta_{10} - 614 \beta_{9} + 1672 \beta_{8} + 195 \beta_{7} + 1784 \beta_{6} + \cdots - 288$$ 1120*b11 - 2139*b10 - 614*b9 + 1672*b8 + 195*b7 + 1784*b6 - 1755*b5 + 3931*b4 - 1427*b3 - 3608*b2 + 1392*b1 - 288

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 1.75374 − 1.62986i −2.39378 − 0.0429626i 2.00607 + 1.30680i −0.330925 − 1.46916i −1.44229 − 0.433312i 1.40719 + 0.536449i 1.75374 + 1.62986i −2.39378 + 0.0429626i 2.00607 − 1.30680i −0.330925 + 1.46916i −1.44229 + 0.433312i 1.40719 − 0.536449i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i −1.32301 0.763837i 1.00000i −0.500000 + 0.866025i 0
751.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 1.42559 + 0.823063i 1.00000i −0.500000 + 0.866025i 0
751.3 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 3.76344 + 2.17283i 1.00000i −0.500000 + 0.866025i 0
751.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i −1.04466 0.603137i 1.00000i −0.500000 + 0.866025i 0
751.5 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 0.749482 + 0.432713i 1.00000i −0.500000 + 0.866025i 0
751.6 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 2.42916 + 1.40247i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i −1.32301 + 0.763837i 1.00000i −0.500000 0.866025i 0
901.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 1.42559 0.823063i 1.00000i −0.500000 0.866025i 0
901.3 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 3.76344 2.17283i 1.00000i −0.500000 0.866025i 0
901.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −1.04466 + 0.603137i 1.00000i −0.500000 0.866025i 0
901.5 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0.749482 0.432713i 1.00000i −0.500000 0.866025i 0
901.6 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 2.42916 1.40247i 1.00000i −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 751.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.bc.i 12
5.b even 2 1 1950.2.bc.j 12
5.c odd 4 1 390.2.x.a 12
5.c odd 4 1 390.2.x.b yes 12
13.e even 6 1 inner 1950.2.bc.i 12
15.e even 4 1 1170.2.bj.c 12
15.e even 4 1 1170.2.bj.d 12
65.l even 6 1 1950.2.bc.j 12
65.r odd 12 1 390.2.x.a 12
65.r odd 12 1 390.2.x.b yes 12
195.bf even 12 1 1170.2.bj.c 12
195.bf even 12 1 1170.2.bj.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 5.c odd 4 1
390.2.x.a 12 65.r odd 12 1
390.2.x.b yes 12 5.c odd 4 1
390.2.x.b yes 12 65.r odd 12 1
1170.2.bj.c 12 15.e even 4 1
1170.2.bj.c 12 195.bf even 12 1
1170.2.bj.d 12 15.e even 4 1
1170.2.bj.d 12 195.bf even 12 1
1950.2.bc.i 12 1.a even 1 1 trivial
1950.2.bc.i 12 13.e even 6 1 inner
1950.2.bc.j 12 5.b even 2 1
1950.2.bc.j 12 65.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} - 12 T_{7}^{11} + 55 T_{7}^{10} - 84 T_{7}^{9} - 107 T_{7}^{8} + 384 T_{7}^{7} + 260 T_{7}^{6} + \cdots + 1024$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{3}$$
$3$ $$(T^{2} + T + 1)^{6}$$
$5$ $$T^{12}$$
$7$ $$T^{12} - 12 T^{11} + \cdots + 1024$$
$11$ $$T^{12} - 6 T^{11} + \cdots + 16$$
$13$ $$T^{12} + 4 T^{11} + \cdots + 4826809$$
$17$ $$T^{12} + 8 T^{11} + \cdots + 65536$$
$19$ $$T^{12} - 6 T^{11} + \cdots + 1982464$$
$23$ $$T^{12} + \cdots + 190660864$$
$29$ $$T^{12} + 14 T^{11} + \cdots + 21904$$
$31$ $$T^{12} + \cdots + 177209344$$
$37$ $$T^{12} + \cdots + 227195329$$
$41$ $$T^{12} + 18 T^{11} + \cdots + 65536$$
$43$ $$T^{12} + \cdots + 349241344$$
$47$ $$T^{12} + 260 T^{10} + \cdots + 35473936$$
$53$ $$(T^{6} - 254 T^{4} + \cdots + 49732)^{2}$$
$59$ $$T^{12} + \cdots + 4983230464$$
$61$ $$T^{12} - 10 T^{11} + \cdots + 89718784$$
$67$ $$T^{12} + 24 T^{11} + \cdots + 83759104$$
$71$ $$T^{12} + 12 T^{11} + \cdots + 4194304$$
$73$ $$T^{12} + \cdots + 53143158784$$
$79$ $$(T^{6} + 2 T^{5} + \cdots - 29312)^{2}$$
$83$ $$T^{12} + 576 T^{10} + \cdots + 47775744$$
$89$ $$T^{12} + \cdots + 1341001056256$$
$97$ $$T^{12} + \cdots + 415519473664$$