# Properties

 Label 13.5.d.a Level $13$ Weight $5$ Character orbit 13.d Analytic conductor $1.344$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,5,Mod(5,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("13.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 13.d (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: $$1.34380952009$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.53039932416.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} - 12x^{3} + 529x^{2} - 1334x + 1682$$ x^6 - 2*x^5 + 2*x^4 - 12*x^3 + 529*x^2 - 1334*x + 1682 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + \cdots - 3) q^{5}+ \cdots + (\beta_{4} + 9 \beta_{3} + 9 \beta_1 - 16) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b4 - 1) * q^3 + (b5 + b3 - b2 - b1) * q^4 + (-b5 + b4 + 3*b2 + b1 - 3) * q^5 + (2*b5 - 2*b4 - 5*b2 + 3*b1 + 5) * q^6 + (-5*b5 - 5*b4 - b3 + 10*b2 + 10) * q^7 + (-b5 - b4 - 11*b3 - 12*b2 - 12) * q^8 + (b4 + 9*b3 + 9*b1 - 16) * q^9 $$q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + \cdots - 3) q^{5}+ \cdots + ( - 8 \beta_{5} - 8 \beta_{4} + \cdots + 3314) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b4 - 1) * q^3 + (b5 + b3 - b2 - b1) * q^4 + (-b5 + b4 + 3*b2 + b1 - 3) * q^5 + (2*b5 - 2*b4 - 5*b2 + 3*b1 + 5) * q^6 + (-5*b5 - 5*b4 - b3 + 10*b2 + 10) * q^7 + (-b5 - b4 - 11*b3 - 12*b2 - 12) * q^8 + (b4 + 9*b3 + 9*b1 - 16) * q^9 + (3*b5 - 6*b3 + 7*b2 + 6*b1) * q^10 + (7*b5 + 7*b4 + 22*b3 - 15*b2 - 15) * q^11 + (5*b5 + 6*b3 + 55*b2 - 6*b1) * q^12 + (-13*b5 - 13*b3 - 39*b2 - 26*b1 + 13) * q^13 + (19*b4 - 21*b3 - 21*b1 - 33) * q^14 + (-b5 + b4 - 62*b2 + 15*b1 + 62) * q^15 + (-23*b4 - 17*b3 - 17*b1 + 193) * q^16 + (9*b5 + 57*b3 + 51*b2 - 57*b1) * q^17 + (-7*b5 + 7*b4 + 148*b2 + 36*b1 - 148) * q^18 + (6*b5 - 6*b4 - 94*b2 + 90*b1 + 94) * q^19 + (4*b5 + 4*b4 + 3*b3 + 69*b2 + 69) * q^20 + (-2*b5 - 2*b4 - 87*b3 - 325*b2 - 325) * q^21 + (-6*b4 + 51*b3 + 51*b1 - 304) * q^22 + (-10*b5 - 13*b3 + 216*b2 + 13*b1) * q^23 + (-36*b5 - 36*b4 + 15*b3 + 3*b2 + 3) * q^24 + (-b5 - 7*b3 + 482*b2 + 7*b1) * q^25 + (52*b5 + 13*b4 + 26*b3 - 507*b2 - 52*b1 + 156) * q^26 + (-59*b4 - 18*b3 - 18*b1 + 71) * q^27 + (-21*b5 + 21*b4 - 292*b2 + 45*b1 + 292) * q^28 + (82*b4 - 57*b3 - 57*b1 + 708) * q^29 + (-11*b5 + 45*b3 + 245*b2 - 45*b1) * q^30 + (32*b5 - 32*b4 + 548*b2 - 58*b1 - 548) * q^31 + (-45*b5 + 45*b4 - 366*b2 - 97*b1 + 366) * q^32 + (43*b5 + 43*b4 + 60*b3 + 353*b2 + 353) * q^33 + (39*b5 + 39*b4 + 81*b3 - 924*b2 - 924) * q^34 + (-3*b4 - 64*b3 - 64*b1 - 657) * q^35 + (8*b5 - 54*b3 + 286*b2 + 54*b1) * q^36 + (57*b5 + 57*b4 + 33*b3 - 323*b2 - 323) * q^37 + (-114*b5 + 16*b3 + 1590*b2 - 16*b1) * q^38 + (-39*b5 - 65*b4 - 78*b3 - 858*b2 + 195*b1 + 182) * q^39 + (-61*b4 + 38*b3 + 38*b1 - 123) * q^40 + (113*b5 - 113*b4 - 876*b2 - 140*b1 + 876) * q^41 + (-79*b4 + 234*b3 + 234*b1 + 1459) * q^42 + (-169*b5 - 370*b3 + 537*b2 + 370*b1) * q^43 + (49*b5 - 49*b4 + 657*b2 + 42*b1 - 657) * q^44 + (-13*b5 + 13*b4 - 170*b2 - 108*b1 + 170) * q^45 + (7*b5 + 7*b4 - 262*b3 + 171*b2 + 171) * q^46 + (-107*b5 - 107*b4 + 361*b3 - 1230*b2 - 1230) * q^47 + (79*b4 - 156*b3 - 156*b1 - 1495) * q^48 + (-11*b5 + 409*b3 + 916*b2 - 409*b1) * q^49 + (-5*b5 - 5*b4 - 498*b3 + 114*b2 + 114) * q^50 + (297*b5 - 90*b3 - 45*b2 + 90*b1) * q^51 + (-26*b5 + 104*b4 + 273*b3 - 481*b2 + 52*b1 + 507) * q^52 + (310*b4 + 363*b3 + 363*b1 + 6) * q^53 + (-100*b5 + 100*b4 - 11*b2 - 225*b1 + 11) * q^54 + (64*b4 - 35*b3 - 35*b1 + 1070) * q^55 + (343*b5 - 131*b3 + 27*b2 + 131*b1) * q^56 + (-262*b5 + 262*b4 + 928*b2 - 378*b1 - 928) * q^57 + (221*b5 - 221*b4 - 1379*b2 - 658*b1 + 1379) * q^58 + (-240*b5 - 240*b4 + 284*b3 + 714*b2 + 714) * q^59 + (83*b5 + 83*b4 + 63*b3 + 172*b2 + 172) * q^60 + (-120*b4 - 318*b3 - 318*b1 + 1250) * q^61 + (-70*b5 - 426*b3 - 666*b2 + 426*b1) * q^62 + (-98*b5 - 98*b4 + 306*b3 - 178*b2 - 178) * q^63 + (-91*b5 + 101*b3 + 989*b2 - 101*b1) * q^64 + (-91*b4 - 325*b3 - 429*b2 + 247*b1 - 780) * q^65 + (-112*b4 - 207*b3 - 207*b1 - 590) * q^66 + (-163*b5 + 163*b4 + 209*b2 + 500*b1 - 209) * q^67 + (-219*b4 + 171*b3 + 171*b1 - 1803) * q^68 + (144*b5 - 51*b3 - 726*b2 + 51*b1) * q^69 + (58*b5 - 58*b4 - 1073*b2 + 523*b1 + 1073) * q^70 + (-41*b5 + 41*b4 + 2730*b2 + 813*b1 - 2730) * q^71 + (42*b5 + 42*b4 + 198*b3 - 1410*b2 - 1410) * q^72 + (45*b5 + 45*b4 + 162*b3 + 910*b2 + 910) * q^73 + (-195*b4 + 470*b3 + 470*b1 + 9) * q^74 + (452*b5 + 12*b3 - 476*b2 - 12*b1) * q^75 + (148*b5 + 148*b4 - 346*b3 + 662*b2 + 662) * q^76 + (-366*b5 - 139*b3 - 3984*b2 + 139*b1) * q^77 + (-247*b5 + 130*b4 + 507*b3 + 3445*b2 - 195*b1 + 806) * q^78 + (-72*b4 - 465*b3 - 465*b1 + 770) * q^79 + (-96*b5 + 96*b4 + 2055*b2 + 29*b1 - 2055) * q^80 + (-200*b4 - 1206*b3 - 1206*b1 - 2371) * q^81 + (-312*b5 + 1242*b3 - 1250*b2 - 1242*b1) * q^82 + (501*b5 - 501*b4 - 3549*b2 - 514*b1 + 3549) * q^83 + (-424*b5 + 424*b4 - 827*b2 + 243*b1 + 827) * q^84 + (240*b5 + 240*b4 - 489*b3 + 777*b2 + 777) * q^85 + (-32*b5 - 32*b4 - 1615*b3 + 5445*b2 + 5445) * q^86 + (644*b4 + 909*b3 + 909*b1 + 5110) * q^87 + (-334*b5 + 215*b3 - 3660*b2 - 215*b1) * q^88 + (149*b5 + 149*b4 + 1464*b3 - 702*b2 - 702) * q^89 + (160*b5 + 252*b3 - 1966*b2 - 252*b1) * q^90 + (468*b5 + 143*b4 - 832*b3 + 3406*b2 - 1456*b1 - 4433) * q^91 + (-130*b4 - 211*b3 - 211*b1 + 1068) * q^92 + (728*b5 - 728*b4 + 1210*b2 - 402*b1 - 1210) * q^93 + (789*b4 + 1377*b3 + 1377*b1 - 7207) * q^94 + (-434*b5 + 542*b3 + 642*b2 - 542*b1) * q^95 + (-262*b5 + 262*b4 - 2999*b2 + 1101*b1 + 2999) * q^96 + (202*b5 - 202*b4 + 4415*b2 + 1870*b1 - 4415) * q^97 + (431*b5 + 431*b4 - 120*b3 - 7008*b2 - 7008) * q^98 + (-8*b5 - 8*b4 - 1188*b3 + 3314*b2 + 3314) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - 4 q^{3} - 14 q^{5} + 32 q^{6} + 48 q^{7} - 96 q^{8} - 58 q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 - 4 * q^3 - 14 * q^5 + 32 * q^6 + 48 * q^7 - 96 * q^8 - 58 * q^9 $$6 q - 2 q^{2} - 4 q^{3} - 14 q^{5} + 32 q^{6} + 48 q^{7} - 96 q^{8} - 58 q^{9} - 32 q^{11} - 244 q^{14} + 404 q^{15} + 1044 q^{16} - 802 q^{18} + 732 q^{19} + 428 q^{20} - 2128 q^{21} - 1632 q^{22} - 24 q^{24} + 910 q^{26} + 236 q^{27} + 1884 q^{28} + 4184 q^{29} - 3468 q^{31} + 2092 q^{32} + 2324 q^{33} - 5304 q^{34} - 4204 q^{35} - 1758 q^{37} + 1196 q^{39} - 708 q^{40} + 4750 q^{41} + 9532 q^{42} - 3956 q^{44} + 830 q^{45} + 516 q^{46} - 6872 q^{47} - 9436 q^{48} - 322 q^{50} + 3900 q^{52} + 2108 q^{53} - 184 q^{54} + 6408 q^{55} - 5800 q^{57} + 6516 q^{58} + 4372 q^{59} + 1324 q^{60} + 5988 q^{61} - 652 q^{63} - 5018 q^{65} - 4592 q^{66} + 72 q^{67} - 10572 q^{68} + 7368 q^{70} - 14672 q^{71} - 7980 q^{72} + 5874 q^{73} + 1544 q^{74} + 3576 q^{76} + 5720 q^{78} + 2616 q^{79} - 12080 q^{80} - 19450 q^{81} + 19264 q^{83} + 6296 q^{84} + 4164 q^{85} + 29376 q^{86} + 35584 q^{87} - 986 q^{89} - 30888 q^{91} + 5304 q^{92} - 9520 q^{93} - 36156 q^{94} + 20720 q^{96} - 23154 q^{97} - 41426 q^{98} + 17492 q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 - 4 * q^3 - 14 * q^5 + 32 * q^6 + 48 * q^7 - 96 * q^8 - 58 * q^9 - 32 * q^11 - 244 * q^14 + 404 * q^15 + 1044 * q^16 - 802 * q^18 + 732 * q^19 + 428 * q^20 - 2128 * q^21 - 1632 * q^22 - 24 * q^24 + 910 * q^26 + 236 * q^27 + 1884 * q^28 + 4184 * q^29 - 3468 * q^31 + 2092 * q^32 + 2324 * q^33 - 5304 * q^34 - 4204 * q^35 - 1758 * q^37 + 1196 * q^39 - 708 * q^40 + 4750 * q^41 + 9532 * q^42 - 3956 * q^44 + 830 * q^45 + 516 * q^46 - 6872 * q^47 - 9436 * q^48 - 322 * q^50 + 3900 * q^52 + 2108 * q^53 - 184 * q^54 + 6408 * q^55 - 5800 * q^57 + 6516 * q^58 + 4372 * q^59 + 1324 * q^60 + 5988 * q^61 - 652 * q^63 - 5018 * q^65 - 4592 * q^66 + 72 * q^67 - 10572 * q^68 + 7368 * q^70 - 14672 * q^71 - 7980 * q^72 + 5874 * q^73 + 1544 * q^74 + 3576 * q^76 + 5720 * q^78 + 2616 * q^79 - 12080 * q^80 - 19450 * q^81 + 19264 * q^83 + 6296 * q^84 + 4164 * q^85 + 29376 * q^86 + 35584 * q^87 - 986 * q^89 - 30888 * q^91 + 5304 * q^92 - 9520 * q^93 - 36156 * q^94 + 20720 * q^96 - 23154 * q^97 - 41426 * q^98 + 17492 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} - 12x^{3} + 529x^{2} - 1334x + 1682$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -581\nu^{5} + 437\nu^{4} + 114\nu^{3} - 11153\nu^{2} - 302999\nu + 392573 ) / 424531$$ (-581*v^5 + 437*v^4 + 114*v^3 - 11153*v^2 - 302999*v + 392573) / 424531 $$\beta_{3}$$ $$=$$ $$( -25\nu^{5} + 44\nu^{4} - 625\nu^{3} + 150\nu^{2} - 13189\nu + 33698 ) / 14639$$ (-25*v^5 + 44*v^4 - 625*v^3 + 150*v^2 - 13189*v + 33698) / 14639 $$\beta_{4}$$ $$=$$ $$( 31\nu^{5} + 531\nu^{4} + 775\nu^{3} - 186\nu^{2} - 1798\nu + 173115 ) / 14639$$ (31*v^5 + 531*v^4 + 775*v^3 - 186*v^2 - 1798*v + 173115) / 14639 $$\beta_{5}$$ $$=$$ $$( -9152\nu^{5} + 6153\nu^{4} + 20063\nu^{3} + 230580\nu^{2} - 4343971\nu + 5696499 ) / 424531$$ (-9152*v^5 + 6153*v^4 + 20063*v^3 + 230580*v^2 - 4343971*v + 5696499) / 424531
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{3} - 17\beta_{2} - \beta_1$$ b5 + b3 - 17*b2 - b1 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 21\beta_{3} + 12\beta_{2} + 12$$ b5 + b4 - 21*b3 + 12*b2 + 12 $$\nu^{4}$$ $$=$$ $$25\beta_{4} + 31\beta_{3} + 31\beta _1 - 367$$ 25*b4 + 31*b3 + 31*b1 - 367 $$\nu^{5}$$ $$=$$ $$-19\beta_{5} + 19\beta_{4} - 402\beta_{2} - 479\beta _1 + 402$$ -19*b5 + 19*b4 - 402*b2 - 479*b1 + 402

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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
 3.18200 − 3.18200i 1.30633 − 1.30633i −3.48832 + 3.48832i 3.18200 + 3.18200i 1.30633 + 1.30633i −3.48832 − 3.48832i
−3.18200 + 3.18200i −10.6142 4.25023i −9.43223 + 9.43223i 33.7744 33.7744i 54.8891 + 54.8891i −37.3878 37.3878i 31.6617 60.0267i
5.2 −1.30633 + 1.30633i 9.97438 12.5870i 9.28070 9.28070i −13.0298 + 13.0298i −46.1782 46.1782i −37.3440 37.3440i 18.4882 24.2472i
5.3 3.48832 3.48832i −1.36015 8.33680i −6.84848 + 6.84848i −4.74466 + 4.74466i 15.2891 + 15.2891i 26.7317 + 26.7317i −79.1500 47.7794i
8.1 −3.18200 3.18200i −10.6142 4.25023i −9.43223 9.43223i 33.7744 + 33.7744i 54.8891 54.8891i −37.3878 + 37.3878i 31.6617 60.0267i
8.2 −1.30633 1.30633i 9.97438 12.5870i 9.28070 + 9.28070i −13.0298 13.0298i −46.1782 + 46.1782i −37.3440 + 37.3440i 18.4882 24.2472i
8.3 3.48832 + 3.48832i −1.36015 8.33680i −6.84848 6.84848i −4.74466 4.74466i 15.2891 15.2891i 26.7317 26.7317i −79.1500 47.7794i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.3 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.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.5.d.a 6
3.b odd 2 1 117.5.j.a 6
4.b odd 2 1 208.5.t.c 6
13.b even 2 1 169.5.d.a 6
13.d odd 4 1 inner 13.5.d.a 6
13.d odd 4 1 169.5.d.a 6
39.f even 4 1 117.5.j.a 6
52.f even 4 1 208.5.t.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.d.a 6 1.a even 1 1 trivial
13.5.d.a 6 13.d odd 4 1 inner
117.5.j.a 6 3.b odd 2 1
117.5.j.a 6 39.f even 4 1
169.5.d.a 6 13.b even 2 1
169.5.d.a 6 13.d odd 4 1
208.5.t.c 6 4.b odd 2 1
208.5.t.c 6 52.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} + \cdots + 1682$$
$3$ $$(T^{3} + 2 T^{2} + \cdots - 144)^{2}$$
$5$ $$T^{6} + 14 T^{5} + \cdots + 2875202$$
$7$ $$T^{6} + \cdots + 12014360072$$
$11$ $$T^{6} + \cdots + 88136171552$$
$13$ $$T^{6} + \cdots + 23298085122481$$
$17$ $$T^{6} + \cdots + 11\!\cdots\!56$$
$19$ $$T^{6} + \cdots + 15\!\cdots\!68$$
$23$ $$T^{6} + \cdots + 26482633392384$$
$29$ $$(T^{3} - 2092 T^{2} + \cdots + 695715376)^{2}$$
$31$ $$T^{6} + \cdots + 15\!\cdots\!48$$
$37$ $$T^{6} + \cdots + 11\!\cdots\!18$$
$41$ $$T^{6} + \cdots + 10\!\cdots\!92$$
$43$ $$T^{6} + \cdots + 19\!\cdots\!24$$
$47$ $$T^{6} + \cdots + 16\!\cdots\!92$$
$53$ $$(T^{3} - 1054 T^{2} + \cdots - 7634592356)^{2}$$
$59$ $$T^{6} + \cdots + 10\!\cdots\!48$$
$61$ $$(T^{3} - 2994 T^{2} + \cdots + 1197889048)^{2}$$
$67$ $$T^{6} + \cdots + 55\!\cdots\!68$$
$71$ $$T^{6} + \cdots + 10\!\cdots\!68$$
$73$ $$T^{6} + \cdots + 22\!\cdots\!08$$
$79$ $$(T^{3} - 1308 T^{2} + \cdots - 7469664296)^{2}$$
$83$ $$T^{6} + \cdots + 75\!\cdots\!52$$
$89$ $$T^{6} + \cdots + 14\!\cdots\!12$$
$97$ $$T^{6} + \cdots + 20\!\cdots\!88$$