# Properties

 Label 76.2.i.a Level $76$ Weight $2$ Character orbit 76.i Analytic conductor $0.607$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,2,Mod(5,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 16]))

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

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

chi := DirichletCharacter("76.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 76.i (of order $$9$$, degree $$6$$, 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: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161$$ x^12 - 6*x^11 - 3*x^10 + 70*x^9 - 15*x^8 - 426*x^7 + 64*x^6 + 1659*x^5 + 267*x^4 - 3969*x^3 - 2088*x^2 + 4446*x + 4161 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} + \beta_{9} + \beta_{7} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{11} + \beta_{10} - \beta_{8} + \cdots + 1) q^{9}+O(q^{10})$$ q + (b11 + b9 + b7 - b5 - 1) * q^3 + (-b9 - b7 - b6 + b4 - b3 + b2 + b1) * q^5 + (b9 + b6 - b5 + b3) * q^7 + (-b11 + b10 - b8 - b6 + 3*b5 - b2 - b1 + 1) * q^9 $$q + (\beta_{11} + \beta_{9} + \beta_{7} + \cdots - 1) q^{3}+ \cdots + (\beta_{11} + \beta_{10} + 5 \beta_{9} + \cdots - 1) q^{99}+O(q^{100})$$ q + (b11 + b9 + b7 - b5 - 1) * q^3 + (-b9 - b7 - b6 + b4 - b3 + b2 + b1) * q^5 + (b9 + b6 - b5 + b3) * q^7 + (-b11 + b10 - b8 - b6 + 3*b5 - b2 - b1 + 1) * q^9 + (-b10 - b9 + b8 - b7) * q^11 + (-b10 + b8 + 2*b7 + b6 - 2*b5 - b4 - b1 - 1) * q^13 + (-b11 + b10 - 3*b9 + b7 + 3*b6 - 4*b4 + b3 - b2 + 1) * q^15 + (-b10 + 3*b9 - b8 - 2*b7 - b6 + 3*b5 + b4 - b3 - b2 + b1 - 3) * q^17 + (-b11 + b10 - b9 - b8 - 3*b7 - b6 + 2*b5 - b1 + 1) * q^19 + (-b9 + b8 + 2*b7 + b6 - b5 + 2*b4 + b3 + b2 - b1) * q^21 + (-b11 + 2*b9 - b7 - b6 + 3*b4 + b2 + b1 - 2) * q^23 + (b10 - 3*b9 + 2*b7 - 3*b6 - 2*b5 + b1) * q^25 + (b11 - b10 + b9 + b8 - 3*b7 - 2*b6 - b5 + b2 - 2) * q^27 + (b10 - b9 + 2*b6 + 4*b5 - 2*b4 - b3 + 3) * q^29 + (2*b11 - b10 + 3*b9 - 2*b8 + b7 + 2*b6 - 2*b5 - b3 - b2 - 2) * q^31 + (b11 - b10 + 3*b9 - b8 + 4*b7 + 2*b6 - 5*b5 - b4 + b3 - b2 + 2) * q^33 + (b11 - b9 + b8 + 2*b5 + b4 - b3 + b1 + 2) * q^35 + (b11 + b10 - b9 + 2*b8 - b7 + 4*b6 - b5 - 2*b4 + b2 + b1 - 1) * q^37 + (2*b7 - 2*b6 - b5 - b4 - 2*b1 + 6) * q^39 + (-b11 - 6*b6 - b4 + b3 + b2 + 1) * q^41 + (-b11 + b10 - b9 + 4*b7 - 3*b6 - b5 + 2*b4 + b3 - b2 - b1 + 4) * q^43 + (b11 - 2*b10 + 4*b9 - b8 - 8*b7 + 7*b4 - b3 + b2 - 1) * q^45 + (2*b11 - b10 - 3*b9 + b8 + b6 + 3*b5 + 4*b4 - b3 + b2 + 2*b1 - 2) * q^47 + (-b11 + 2*b10 - 2*b8 + b7 - b6 + b5 - b4 + b3 - b2 - b1 + 3) * q^49 + (-b11 + 3*b10 + 5*b9 - b8 + 4*b7 + 4*b6 - 4*b5 - 5*b4 - b3 + 3*b1 - 5) * q^51 + (b11 - b10 - 3*b9 - 4*b7 + 3*b6 - 2*b4 - b3 + b2 - 1) * q^53 + (3*b9 + b8 + b7 + b6 + 3*b5 - 4*b4 + b3 + b2 - b1 - 3) * q^55 + (-b10 - 4*b9 + 2*b8 - 3*b7 - 2*b6 - 3*b5 + b3 + b2 + 2*b1 - 3) * q^57 + (-b10 - 3*b9 - 2*b8 + 3*b7 - 2*b6 - b5 + 5*b4 - 2*b3 - 3) * q^59 + (2*b11 - b10 + 4*b9 + 4*b4 - b3 - 3*b2 - b1 - 4) * q^61 + (-b11 - b10 - 6*b9 + b8 - 5*b6 + 2*b4 - b3 - b1 + 2) * q^63 + (-2*b11 + b10 + 7*b9 - b8 - 2*b7 + 5*b5 - 5*b4 + 2*b3 - 2*b2 - 2*b1 - 4) * q^65 + (-b11 - 2*b10 - b9 + b8 + 4*b6 + 2*b5 - 3*b4 + 3*b3 + b2 - b1 + 3) * q^67 + (-3*b11 + 3*b10 - 8*b9 + 3*b8 + 4*b7 + 4*b6 - 4*b5 - 4*b4 + b3 + 3) * q^69 + (-3*b11 + 3*b10 - b9 + b8 - 3*b6 + b5 - b1 + 3) * q^71 + (b9 - b8 + 4*b6 + 3*b5 - b4 + b3 - b1 + 3) * q^73 + (-b11 - 3*b10 + 3*b9 - 4*b8 - 7*b7 + 2*b6 + 9*b5 + 3*b4 - 3*b2 - 2*b1 + 2) * q^75 + (-b10 + b9 - b8 - 4*b7 + 3*b6 - 2*b5 - 3*b4 - b2 + 2*b1) * q^77 + (-3*b11 - b8 - b7 - 4*b6 + 3*b5 + b4 - b3 - 2*b2 - b1 + 1) * q^79 + (b11 - b10 - 4*b9 + 4*b7 + 2*b5 + b4 + b3 - b2 - b1 + 2) * q^81 + (-b11 + 3*b10 + b8 + 4*b7 + b6 - b5 - 2*b4 - b3 - 2*b2 + 1) * q^83 + (-2*b10 - 2*b9 + 4*b8 - b6 - 5*b5 + b4 + 2*b3 + 4*b2 - 3) * q^85 + (2*b11 + b9 - 8*b7 - 5*b6 + b5 + b4 - 4*b3 + 2*b2 + 4*b1 - 5) * q^87 + (2*b11 - b10 + b9 + 9*b7 + b6 - 9*b5 - 2*b4 + 2*b3 - b1 - 2) * q^89 + (-b10 - 7*b9 - 4*b7 + 4*b6 - 5*b4 - b3 + 3*b2 + b1 + 1) * q^91 + (3*b10 + 3*b9 + b8 + 6*b7 + b6 + 4*b5 - 4*b4 + b3 + 2*b2 - 2*b1 - 5) * q^93 + (4*b11 - b10 + 5*b9 + b8 - b7 - 6*b5 + 7*b4 - b3 - b2 - 4) * q^95 + (b10 + 3*b9 - 2*b8 - b7 - 2*b6 + b5 + 7*b4 - 2*b3 - 4*b2 + 4*b1 - 3) * q^97 + (b11 + b10 + 5*b9 + 6*b7 - b6 + 2*b4 + b3 + 2*b2 - 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{3} + 3 q^{7} - 3 q^{9}+O(q^{10})$$ 12 * q - 3 * q^3 + 3 * q^7 - 3 * q^9 $$12 q - 3 q^{3} + 3 q^{7} - 3 q^{9} + 3 q^{11} - 9 q^{13} - 15 q^{15} - 3 q^{17} - 12 q^{19} - 15 q^{21} - 12 q^{23} - 18 q^{25} - 9 q^{27} + 27 q^{29} + 6 q^{31} + 48 q^{33} + 33 q^{35} - 12 q^{37} + 60 q^{39} + 3 q^{41} + 27 q^{43} + 24 q^{45} - 15 q^{47} + 9 q^{49} - 33 q^{51} - 21 q^{53} - 27 q^{55} - 42 q^{57} - 48 q^{59} - 6 q^{61} - 9 q^{63} - 33 q^{65} + 24 q^{67} - 33 q^{69} + 30 q^{73} + 42 q^{75} + 24 q^{77} + 3 q^{79} + 3 q^{81} + 3 q^{83} - 42 q^{85} - 18 q^{87} - 18 q^{89} - 24 q^{91} - 78 q^{93} + 9 q^{95} + 12 q^{97} - 6 q^{99}+O(q^{100})$$ 12 * q - 3 * q^3 + 3 * q^7 - 3 * q^9 + 3 * q^11 - 9 * q^13 - 15 * q^15 - 3 * q^17 - 12 * q^19 - 15 * q^21 - 12 * q^23 - 18 * q^25 - 9 * q^27 + 27 * q^29 + 6 * q^31 + 48 * q^33 + 33 * q^35 - 12 * q^37 + 60 * q^39 + 3 * q^41 + 27 * q^43 + 24 * q^45 - 15 * q^47 + 9 * q^49 - 33 * q^51 - 21 * q^53 - 27 * q^55 - 42 * q^57 - 48 * q^59 - 6 * q^61 - 9 * q^63 - 33 * q^65 + 24 * q^67 - 33 * q^69 + 30 * q^73 + 42 * q^75 + 24 * q^77 + 3 * q^79 + 3 * q^81 + 3 * q^83 - 42 * q^85 - 18 * q^87 - 18 * q^89 - 24 * q^91 - 78 * q^93 + 9 * q^95 + 12 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 1861 \nu^{11} + 59764 \nu^{10} - 261706 \nu^{9} - 459237 \nu^{8} + 3512934 \nu^{7} + \cdots - 104999089 ) / 8740601$$ (-1861*v^11 + 59764*v^10 - 261706*v^9 - 459237*v^8 + 3512934*v^7 + 1223776*v^6 - 19762014*v^5 - 6600456*v^4 + 62490886*v^3 + 42461956*v^2 - 98039407*v - 104999089) / 8740601 $$\beta_{3}$$ $$=$$ $$( - 3014 \nu^{11} - 25876 \nu^{10} + 250822 \nu^{9} + 84243 \nu^{8} - 2839567 \nu^{7} + \cdots + 91337490 ) / 8740601$$ (-3014*v^11 - 25876*v^10 + 250822*v^9 + 84243*v^8 - 2839567*v^7 - 652331*v^6 + 18054126*v^5 + 5399484*v^4 - 59109957*v^3 - 38613723*v^2 + 94201422*v + 91337490) / 8740601 $$\beta_{4}$$ $$=$$ $$( - 7410 \nu^{11} + 92642 \nu^{10} - 242771 \nu^{9} - 590557 \nu^{8} + 2422014 \nu^{7} + \cdots - 39192874 ) / 8740601$$ (-7410*v^11 + 92642*v^10 - 242771*v^9 - 590557*v^8 + 2422014*v^7 + 2513886*v^6 - 10863570*v^5 - 10833438*v^4 + 27062496*v^3 + 32215080*v^2 - 29279114*v - 39192874) / 8740601 $$\beta_{5}$$ $$=$$ $$( 7410 \nu^{11} + 11132 \nu^{10} - 276099 \nu^{9} + 170744 \nu^{8} + 2370458 \nu^{7} + \cdots - 26703616 ) / 8740601$$ (7410*v^11 + 11132*v^10 - 276099*v^9 + 170744*v^8 + 2370458*v^7 - 1428976*v^6 - 11344066*v^5 + 2408876*v^4 + 33428312*v^3 + 6822812*v^2 - 44659861*v - 26703616) / 8740601 $$\beta_{6}$$ $$=$$ $$( 8405 \nu^{11} + 8018 \nu^{10} - 346679 \nu^{9} + 455105 \nu^{8} + 2627431 \nu^{7} + \cdots - 11328251 ) / 8740601$$ (8405*v^11 + 8018*v^10 - 346679*v^9 + 455105*v^8 + 2627431*v^7 - 3363517*v^6 - 11636805*v^5 + 9113704*v^4 + 31536935*v^3 - 4569699*v^2 - 40761460*v - 11328251) / 8740601 $$\beta_{7}$$ $$=$$ $$( 15815 \nu^{11} - 89341 \nu^{10} - 80323 \nu^{9} + 1088115 \nu^{8} - 105905 \nu^{7} + \cdots + 1553197 ) / 8740601$$ (15815*v^11 - 89341*v^10 - 80323*v^9 + 1088115*v^8 - 105905*v^7 - 5646270*v^6 - 277950*v^5 + 17338641*v^4 + 8455587*v^3 - 27341345*v^2 - 22774844*v + 1553197) / 8740601 $$\beta_{8}$$ $$=$$ $$( 16036 \nu^{11} - 88198 \nu^{10} + 124775 \nu^{9} + 357074 \nu^{8} - 2014841 \nu^{7} + \cdots + 33629265 ) / 8740601$$ (16036*v^11 - 88198*v^10 + 124775*v^9 + 357074*v^8 - 2014841*v^7 + 1083449*v^6 + 10730801*v^5 - 7009805*v^4 - 37334292*v^3 + 5511793*v^2 + 68308502*v + 33629265) / 8740601 $$\beta_{9}$$ $$=$$ $$( - 1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + \cdots - 745423 ) / 514153$$ (-1506*v^11 + 8283*v^10 + 6301*v^9 - 90477*v^8 - 1422*v^7 + 485184*v^6 + 4698*v^5 - 1477233*v^4 - 388357*v^3 + 2375103*v^2 + 1084425*v - 745423) / 514153 $$\beta_{10}$$ $$=$$ $$( 32367 \nu^{11} - 175660 \nu^{10} + 8466 \nu^{9} + 1222749 \nu^{8} - 536869 \nu^{7} + \cdots + 20539212 ) / 8740601$$ (32367*v^11 - 175660*v^10 + 8466*v^9 + 1222749*v^8 - 536869*v^7 - 4743060*v^6 + 1737702*v^5 + 11702325*v^4 - 5650797*v^3 - 17409849*v^2 + 16526830*v + 20539212) / 8740601 $$\beta_{11}$$ $$=$$ $$( 382 \nu^{11} - 2101 \nu^{10} + 162 \nu^{9} + 12670 \nu^{8} + 605 \nu^{7} - 42146 \nu^{6} + \cdots - 476163 ) / 80189$$ (382*v^11 - 2101*v^10 + 162*v^9 + 12670*v^8 + 605*v^7 - 42146*v^6 - 39078*v^5 + 48159*v^4 + 203904*v^3 + 142404*v^2 - 273582*v - 476163) / 80189
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 3$$ -b10 - b9 + b8 - b7 + b6 - b5 - b4 + b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{4} + 3\beta _1 + 2$$ b11 - 3*b10 - b9 + b8 - 2*b7 + b6 + 2*b5 - 3*b4 + 3*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{11} - 9 \beta_{10} - 7 \beta_{9} + 6 \beta_{8} - 12 \beta_{7} + 8 \beta_{6} - 13 \beta_{4} + \cdots + 5$$ b11 - 9*b10 - 7*b9 + 6*b8 - 12*b7 + 8*b6 - 13*b4 - 2*b3 + 3*b2 + 6*b1 + 5 $$\nu^{5}$$ $$=$$ $$8 \beta_{11} - 28 \beta_{10} - 22 \beta_{9} + 10 \beta_{8} - 34 \beta_{7} + 11 \beta_{6} + 12 \beta_{5} + \cdots + 4$$ 8*b11 - 28*b10 - 22*b9 + 10*b8 - 34*b7 + 11*b6 + 12*b5 - 25*b4 - 5*b3 - 3*b2 + 9*b1 + 4 $$\nu^{6}$$ $$=$$ $$11 \beta_{11} - 62 \beta_{10} - 71 \beta_{9} + 26 \beta_{8} - 108 \beta_{7} + 26 \beta_{6} + 25 \beta_{5} + \cdots - 6$$ 11*b11 - 62*b10 - 71*b9 + 26*b8 - 108*b7 + 26*b6 + 25*b5 - 66*b4 - 33*b3 - 16*b2 + 24*b1 - 6 $$\nu^{7}$$ $$=$$ $$26 \beta_{11} - 152 \beta_{10} - 261 \beta_{9} + 42 \beta_{8} - 327 \beta_{7} - 12 \beta_{6} + 87 \beta_{5} + \cdots - 5$$ 26*b11 - 152*b10 - 261*b9 + 42*b8 - 327*b7 - 12*b6 + 87*b5 - 91*b4 - 98*b3 - 75*b2 + 45*b1 - 5 $$\nu^{8}$$ $$=$$ $$- 12 \beta_{11} - 288 \beta_{10} - 804 \beta_{9} + 76 \beta_{8} - 866 \beta_{7} - 236 \beta_{6} + \cdots + 18$$ -12*b11 - 288*b10 - 804*b9 + 76*b8 - 866*b7 - 236*b6 + 233*b5 - 51*b4 - 362*b3 - 270*b2 + 141*b1 + 18 $$\nu^{9}$$ $$=$$ $$- 236 \beta_{11} - 480 \beta_{10} - 2610 \beta_{9} + 88 \beta_{8} - 2462 \beta_{7} - 1334 \beta_{6} + \cdots + 384$$ -236*b11 - 480*b10 - 2610*b9 + 88*b8 - 2462*b7 - 1334*b6 + 761*b5 + 402*b4 - 1062*b3 - 762*b2 + 417*b1 + 384 $$\nu^{10}$$ $$=$$ $$- 1334 \beta_{11} - 495 \beta_{10} - 7824 \beta_{9} + 204 \beta_{8} - 6291 \beta_{7} - 5298 \beta_{6} + \cdots + 1889$$ -1334*b11 - 495*b10 - 7824*b9 + 204*b8 - 6291*b7 - 5298*b6 + 2331*b5 + 2500*b4 - 3136*b3 - 2046*b2 + 1327*b1 + 1889 $$\nu^{11}$$ $$=$$ $$- 5298 \beta_{11} + 678 \beta_{10} - 23140 \beta_{9} + 726 \beta_{8} - 16756 \beta_{7} - 18648 \beta_{6} + \cdots + 7815$$ -5298*b11 + 678*b10 - 23140*b9 + 726*b8 - 16756*b7 - 18648*b6 + 7923*b5 + 9189*b4 - 8536*b3 - 4679*b2 + 3928*b1 + 7815

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 2.75227 − 0.342020i −1.75227 − 0.342020i 2.26253 − 0.984808i −1.26253 − 0.984808i 2.26253 + 0.984808i −1.26253 + 0.984808i −1.25236 + 0.642788i 2.25236 + 0.642788i 2.75227 + 0.342020i −1.75227 + 0.342020i −1.25236 − 0.642788i 2.25236 − 0.642788i
0 −0.467454 + 2.65106i 0 −0.982226 0.824185i 0 1.67233 + 2.89656i 0 −3.99055 1.45244i 0
5.2 0 0.314751 1.78504i 0 0.216181 + 0.181398i 0 −0.579936 1.00448i 0 −0.268219 0.0976237i 0
9.1 0 −0.834130 + 0.699919i 0 3.00735 + 1.09458i 0 0.278396 0.482195i 0 −0.315057 + 1.78678i 0
9.2 0 1.86622 1.56594i 0 −2.06765 0.752564i 0 −1.48413 + 2.57059i 0 0.509650 2.89037i 0
17.1 0 −0.834130 0.699919i 0 3.00735 1.09458i 0 0.278396 + 0.482195i 0 −0.315057 1.78678i 0
17.2 0 1.86622 + 1.56594i 0 −2.06765 + 0.752564i 0 −1.48413 2.57059i 0 0.509650 + 2.89037i 0
25.1 0 −2.83638 1.03236i 0 −0.658711 3.73574i 0 −0.0695116 + 0.120398i 0 4.68114 + 3.92794i 0
25.2 0 0.456991 + 0.166331i 0 0.485063 + 2.75093i 0 1.68285 2.91479i 0 −2.11696 1.77634i 0
61.1 0 −0.467454 2.65106i 0 −0.982226 + 0.824185i 0 1.67233 2.89656i 0 −3.99055 + 1.45244i 0
61.2 0 0.314751 + 1.78504i 0 0.216181 0.181398i 0 −0.579936 + 1.00448i 0 −0.268219 + 0.0976237i 0
73.1 0 −2.83638 + 1.03236i 0 −0.658711 + 3.73574i 0 −0.0695116 0.120398i 0 4.68114 3.92794i 0
73.2 0 0.456991 0.166331i 0 0.485063 2.75093i 0 1.68285 + 2.91479i 0 −2.11696 + 1.77634i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.i.a 12
3.b odd 2 1 684.2.bo.c 12
4.b odd 2 1 304.2.u.e 12
19.e even 9 1 inner 76.2.i.a 12
19.e even 9 1 1444.2.a.h 6
19.e even 9 2 1444.2.e.g 12
19.f odd 18 1 1444.2.a.g 6
19.f odd 18 2 1444.2.e.h 12
57.l odd 18 1 684.2.bo.c 12
76.k even 18 1 5776.2.a.by 6
76.l odd 18 1 304.2.u.e 12
76.l odd 18 1 5776.2.a.bw 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.i.a 12 1.a even 1 1 trivial
76.2.i.a 12 19.e even 9 1 inner
304.2.u.e 12 4.b odd 2 1
304.2.u.e 12 76.l odd 18 1
684.2.bo.c 12 3.b odd 2 1
684.2.bo.c 12 57.l odd 18 1
1444.2.a.g 6 19.f odd 18 1
1444.2.a.h 6 19.e even 9 1
1444.2.e.g 12 19.e even 9 2
1444.2.e.h 12 19.f odd 18 2
5776.2.a.bw 6 76.l odd 18 1
5776.2.a.by 6 76.k even 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 3 T^{11} + \cdots + 361$$
$5$ $$T^{12} + 9 T^{10} + \cdots + 729$$
$7$ $$T^{12} - 3 T^{11} + \cdots + 9$$
$11$ $$T^{12} - 3 T^{11} + \cdots + 729$$
$13$ $$T^{12} + 9 T^{11} + \cdots + 130321$$
$17$ $$T^{12} + 3 T^{11} + \cdots + 8982009$$
$19$ $$T^{12} + 12 T^{11} + \cdots + 47045881$$
$23$ $$T^{12} + \cdots + 151585344$$
$29$ $$T^{12} - 27 T^{11} + \cdots + 263169$$
$31$ $$T^{12} - 6 T^{11} + \cdots + 130321$$
$37$ $$(T^{6} + 6 T^{5} + \cdots + 11096)^{2}$$
$41$ $$T^{12} + \cdots + 360278361$$
$43$ $$T^{12} - 27 T^{11} + \cdots + 1203409$$
$47$ $$T^{12} + 15 T^{11} + \cdots + 729$$
$53$ $$T^{12} + 21 T^{11} + \cdots + 95004009$$
$59$ $$T^{12} + \cdots + 12621848409$$
$61$ $$T^{12} + \cdots + 1418049649$$
$67$ $$T^{12} + \cdots + 6080256576$$
$71$ $$T^{12} + 45 T^{10} + \cdots + 16842816$$
$73$ $$T^{12} - 30 T^{11} + \cdots + 36864$$
$79$ $$T^{12} + \cdots + 1548343801$$
$83$ $$T^{12} + \cdots + 11939714361$$
$89$ $$T^{12} + \cdots + 7695324729$$
$97$ $$T^{12} + \cdots + 16983563041$$