# Properties

 Label 1170.2.bj.c Level $1170$ Weight $2$ Character orbit 1170.bj Analytic conductor $9.342$ 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] = [1170,2,Mod(199,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(6))

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("1170.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.bj (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: $$9.34249703649$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

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

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

 $$\beta_{1}$$ $$=$$ $$( 203419 \nu^{11} + 163110633 \nu^{10} - 591783880 \nu^{9} - 97338749 \nu^{8} + \cdots - 81183629852 ) / 63907274600$$ (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 - 128185173545*v - 81183629852) / 63907274600 $$\beta_{2}$$ $$=$$ $$( 99774878 \nu^{11} - 1446867854 \nu^{10} + 1099642740 \nu^{9} + 2834573637 \nu^{8} + \cdots - 399707750949 ) / 415397284900$$ (99774878*v^11 - 1446867854*v^10 + 1099642740*v^9 + 2834573637*v^8 - 11046550568*v^7 + 1368363861*v^6 - 68535056482*v^5 + 38321129974*v^4 + 51767136278*v^3 - 345701461353*v^2 - 110551992590*v - 399707750949) / 415397284900 $$\beta_{3}$$ $$=$$ $$( 120243408 \nu^{11} - 454030419 \nu^{10} - 2051209685 \nu^{9} + 7036618932 \nu^{8} + \cdots - 618192439839 ) / 415397284900$$ (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) / 415397284900 $$\beta_{4}$$ $$=$$ $$( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200$$ (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 $$\beta_{5}$$ $$=$$ $$( 164152294 \nu^{11} + 1358027358 \nu^{10} - 5882189980 \nu^{9} + 1843354751 \nu^{8} + \cdots - 1308690012227 ) / 415397284900$$ (164152294*v^11 + 1358027358*v^10 - 5882189980*v^9 + 1843354751*v^8 + 45609137136*v^7 - 49422223697*v^6 - 70650734586*v^5 + 94378041302*v^4 + 204958669994*v^3 + 189862006481*v^2 - 68215142970*v - 1308690012227) / 415397284900 $$\beta_{6}$$ $$=$$ $$( 480376508 \nu^{11} - 958108569 \nu^{10} - 1722573835 \nu^{9} + 8639610832 \nu^{8} + \cdots - 997979945989 ) / 415397284900$$ (480376508*v^11 - 958108569*v^10 - 1722573835*v^9 + 8639610832*v^8 + 2577608327*v^7 - 5697780079*v^6 - 37282009602*v^5 + 13460230639*v^4 + 124582190083*v^3 + 197417975542*v^2 - 141527922965*v - 997979945989) / 415397284900 $$\beta_{7}$$ $$=$$ $$( - 23159569 \nu^{11} + 44665642 \nu^{10} + 118225633 \nu^{9} - 603932751 \nu^{8} + \cdots + 21026586399 ) / 16615891396$$ (-23159569*v^11 + 44665642*v^10 + 118225633*v^9 - 603932751*v^8 - 6720278*v^7 + 1195412299*v^6 - 1392260235*v^5 - 486856544*v^4 - 1535131351*v^3 - 7949637689*v^2 - 2866001996*v + 21026586399) / 16615891396 $$\beta_{8}$$ $$=$$ $$( - 236542471 \nu^{11} + 346656978 \nu^{10} + 1242595595 \nu^{9} - 6207967149 \nu^{8} + \cdots + 460186493 ) / 166158913960$$ (-236542471*v^11 + 346656978*v^10 + 1242595595*v^9 - 6207967149*v^8 + 2310806556*v^7 + 6108999933*v^6 - 30747713501*v^5 + 45266458632*v^4 - 73287704061*v^3 - 120820074499*v^2 - 57773591330*v + 460186493) / 166158913960 $$\beta_{9}$$ $$=$$ $$( - 632187142 \nu^{11} + 553303131 \nu^{10} + 5382274615 \nu^{9} - 11841029268 \nu^{8} + \cdots + 880210105411 ) / 415397284900$$ (-632187142*v^11 + 553303131*v^10 + 5382274615*v^9 - 11841029268*v^8 - 18449268373*v^7 + 45918505121*v^6 + 41904457048*v^5 - 52841609661*v^4 - 129070461767*v^3 - 49363987858*v^2 + 441927290935*v + 880210105411) / 415397284900 $$\beta_{10}$$ $$=$$ $$( 320232543 \nu^{11} - 1004615314 \nu^{10} - 1143072025 \nu^{9} + 9975044447 \nu^{8} + \cdots - 144016360059 ) / 166158913960$$ (320232543*v^11 - 1004615314*v^10 - 1143072025*v^9 + 9975044447*v^8 - 4093093078*v^7 - 25465804119*v^6 + 21841076813*v^5 + 35457807644*v^4 + 27228013423*v^3 + 100602839757*v^2 - 286784071080*v - 144016360059) / 166158913960 $$\beta_{11}$$ $$=$$ $$( - 1027654074 \nu^{11} + 2460612632 \nu^{10} + 3624161555 \nu^{9} - 25210679196 \nu^{8} + \cdots + 27977691092 ) / 415397284900$$ (-1027654074*v^11 + 2460612632*v^10 + 3624161555*v^9 - 25210679196*v^8 + 10960029469*v^7 + 30148216612*v^6 - 9821523994*v^5 - 23732163042*v^4 - 402415418899*v^3 + 37700985074*v^2 + 390356046795*v + 27977691092) / 415397284900
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{5} - 2\beta _1 + 1 ) / 2$$ (b11 + b10 - b9 - b8 + 2*b7 + b5 - 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{10} + 2\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1$$ b10 + 2*b7 + b6 - b4 + b3 - b2 - b1 + 1 $$\nu^{3}$$ $$=$$ $$( 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 8 ) / 2$$ (4*b10 - 2*b9 - 4*b8 + 10*b7 - 2*b6 + 2*b5 - 2*b4 - b3 - 4*b1 - 8) / 2 $$\nu^{4}$$ $$=$$ $$- 7 \beta_{11} + 2 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + \cdots - 11$$ -7*b11 + 2*b9 + 7*b8 - 2*b7 - 3*b6 - 3*b5 - 7*b4 + 4*b3 - 5*b2 - b1 - 11 $$\nu^{5}$$ $$=$$ $$( - 6 \beta_{11} + 2 \beta_{10} - \beta_{9} - 10 \beta_{8} - 12 \beta_{7} - 34 \beta_{6} - 9 \beta_{5} + \cdots - 82 ) / 2$$ (-6*b11 + 2*b10 - b9 - 10*b8 - 12*b7 - 34*b6 - 9*b5 - 16*b4 - 19*b3 + 11*b2 + 8*b1 - 82) / 2 $$\nu^{6}$$ $$=$$ $$- 46 \beta_{11} - 40 \beta_{10} + 46 \beta_{9} + 42 \beta_{8} - 90 \beta_{7} - 13 \beta_{6} - 30 \beta_{5} + \cdots - 39$$ -46*b11 - 40*b10 + 46*b9 + 42*b8 - 90*b7 - 13*b6 - 30*b5 - 26*b4 + 13*b3 - 2*b2 + 46*b1 - 39 $$\nu^{7}$$ $$=$$ $$( 33 \beta_{11} - 67 \beta_{10} + 79 \beta_{9} - 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + 5 \beta_{5} + \cdots - 51 ) / 2$$ (33*b11 - 67*b10 + 79*b9 - 57*b8 - 234*b7 - 76*b6 + 5*b5 + 52*b4 - 156*b3 + 152*b2 + 146*b1 - 51) / 2 $$\nu^{8}$$ $$=$$ $$- 93 \beta_{11} - 174 \beta_{10} + 231 \beta_{9} + 181 \beta_{8} - 364 \beta_{7} + 130 \beta_{6} + \cdots + 362$$ -93*b11 - 174*b10 + 231*b9 + 181*b8 - 364*b7 + 130*b6 - 3*b5 + 107*b4 + 91*b3 - 23*b2 + 179*b1 + 362 $$\nu^{9}$$ $$=$$ $$( 1124 \beta_{11} + 316 \beta_{10} - 358 \beta_{9} - 776 \beta_{8} + 646 \beta_{7} + 682 \beta_{6} + \cdots + 1700 ) / 2$$ (1124*b11 + 316*b10 - 358*b9 - 776*b8 + 646*b7 + 682*b6 + 798*b5 + 1378*b4 - 861*b3 + 404*b2 - 248*b1 + 1700) / 2 $$\nu^{10}$$ $$=$$ $$246 \beta_{11} - 81 \beta_{10} - 196 \beta_{9} + 482 \beta_{8} + 630 \beta_{7} + 1314 \beta_{6} + \cdots + 2556$$ 246*b11 - 81*b10 - 196*b9 + 482*b8 + 630*b7 + 1314*b6 + 501*b5 + 1079*b4 + 580*b3 - 926*b2 - 469*b1 + 2556 $$\nu^{11}$$ $$=$$ $$( 7310 \beta_{11} + 5070 \beta_{10} - 9673 \beta_{9} - 6494 \beta_{8} + 12576 \beta_{7} + 2066 \beta_{6} + \cdots + 2574 ) / 2$$ (7310*b11 + 5070*b10 - 9673*b9 - 6494*b8 + 12576*b7 + 2066*b6 + 3475*b5 + 4992*b4 - 5163*b3 - 325*b2 - 6944*b1 + 2574) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$$.

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −2.39378 − 0.0429626i 1.75374 − 1.62986i −0.330925 − 1.46916i −1.44229 − 0.433312i 2.00607 + 1.30680i 1.40719 + 0.536449i −2.39378 + 0.0429626i 1.75374 + 1.62986i −0.330925 + 1.46916i −1.44229 + 0.433312i 2.00607 − 1.30680i 1.40719 − 0.536449i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −2.10012 0.767774i 0 0.823063 + 1.42559i 1.00000 0 1.71497 1.43487i
199.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.40066 + 1.74303i 0 −0.763837 1.32301i 1.00000 0 −0.809179 2.08452i
199.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.571769 2.16173i 0 0.603137 + 1.04466i 1.00000 0 2.15800 + 0.585699i
199.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.230377 + 2.22417i 0 −0.432713 0.749482i 1.00000 0 −1.81100 1.31160i
199.5 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.26873 1.84128i 0 2.17283 + 3.76344i 1.00000 0 0.960230 + 2.01940i
199.6 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.03420 0.928463i 0 −1.40247 2.42916i 1.00000 0 −0.213026 + 2.22590i
829.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −2.10012 + 0.767774i 0 0.823063 1.42559i 1.00000 0 1.71497 + 1.43487i
829.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.40066 1.74303i 0 −0.763837 + 1.32301i 1.00000 0 −0.809179 + 2.08452i
829.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.571769 + 2.16173i 0 0.603137 1.04466i 1.00000 0 2.15800 0.585699i
829.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.230377 2.22417i 0 −0.432713 + 0.749482i 1.00000 0 −1.81100 + 1.31160i
829.5 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.26873 + 1.84128i 0 2.17283 3.76344i 1.00000 0 0.960230 2.01940i
829.6 −0.500000 0.866025i 0 −0.500000 + 0.866025i 2.03420 + 0.928463i 0 −1.40247 + 2.42916i 1.00000 0 −0.213026 2.22590i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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.l even 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 15.d odd 2 1
390.2.x.a 12 39.h odd 6 1
390.2.x.b yes 12 3.b odd 2 1
390.2.x.b yes 12 195.y odd 6 1
1170.2.bj.c 12 1.a even 1 1 trivial
1170.2.bj.c 12 65.l even 6 1 inner
1170.2.bj.d 12 5.b even 2 1
1170.2.bj.d 12 13.e even 6 1
1950.2.bc.i 12 15.e even 4 1
1950.2.bc.i 12 195.bf even 12 1
1950.2.bc.j 12 15.e even 4 1
1950.2.bc.j 12 195.bf even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1170, [\chi])$$:

 $$T_{7}^{12} - 2 T_{7}^{11} + 19 T_{7}^{10} + 6 T_{7}^{9} + 205 T_{7}^{8} - 20 T_{7}^{7} + 708 T_{7}^{6} + \cdots + 1024$$ T7^12 - 2*T7^11 + 19*T7^10 + 6*T7^9 + 205*T7^8 - 20*T7^7 + 708*T7^6 + 112*T7^5 + 1648*T7^4 + 64*T7^3 + 1664*T7^2 + 512*T7 + 1024 $$T_{17}^{12} - 18 T_{17}^{11} + 95 T_{17}^{10} + 234 T_{17}^{9} - 2695 T_{17}^{8} - 4632 T_{17}^{7} + \cdots + 65536$$ T17^12 - 18*T17^11 + 95*T17^10 + 234*T17^9 - 2695*T17^8 - 4632*T17^7 + 75744*T17^6 - 85248*T17^5 - 300288*T17^4 + 307200*T17^3 + 1269760*T17^2 + 491520*T17 + 65536

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{6}$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 2 T^{11} + \cdots + 15625$$
$7$ $$T^{12} - 2 T^{11} + \cdots + 1024$$
$11$ $$T^{12} + 6 T^{11} + \cdots + 16$$
$13$ $$T^{12} - 8 T^{11} + \cdots + 4826809$$
$17$ $$T^{12} - 18 T^{11} + \cdots + 65536$$
$19$ $$T^{12} + 6 T^{11} + \cdots + 1982464$$
$23$ $$T^{12} + \cdots + 190660864$$
$29$ $$T^{12} + 14 T^{11} + \cdots + 21904$$
$31$ $$T^{12} + \cdots + 177209344$$
$37$ $$T^{12} + \cdots + 227195329$$
$41$ $$T^{12} - 18 T^{11} + \cdots + 65536$$
$43$ $$T^{12} + \cdots + 349241344$$
$47$ $$(T^{6} - 8 T^{5} + \cdots + 5956)^{2}$$
$53$ $$T^{12} + \cdots + 2473271824$$
$59$ $$T^{12} + \cdots + 4983230464$$
$61$ $$T^{12} - 10 T^{11} + \cdots + 89718784$$
$67$ $$T^{12} + 4 T^{11} + \cdots + 83759104$$
$71$ $$T^{12} - 12 T^{11} + \cdots + 4194304$$
$73$ $$(T^{6} + 14 T^{5} + \cdots - 230528)^{2}$$
$79$ $$(T^{6} - 2 T^{5} + \cdots - 29312)^{2}$$
$83$ $$(T^{6} - 36 T^{5} + \cdots - 6912)^{2}$$
$89$ $$T^{12} + \cdots + 1341001056256$$
$97$ $$T^{12} + \cdots + 415519473664$$