# Properties

 Label 13.7.d.a Level $13$ Weight $7$ Character orbit 13.d Analytic conductor $2.991$ 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] = [13,7,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, 7, names="a")

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

chi := DirichletCharacter("13.5");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$7$$ 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: $$2.99070308706$$ 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: $$x^{12} - 6 x^{11} + 18 x^{10} + 488 x^{9} + 36205 x^{8} - 155430 x^{7} + 399962 x^{6} + 9502784 x^{5} + \cdots + 56070144$$ x^12 - 6*x^11 + 18*x^10 + 488*x^9 + 36205*x^8 - 155430*x^7 + 399962*x^6 + 9502784*x^5 + 275595012*x^4 - 541321656*x^3 + 523196552*x^2 - 242221824*x + 56070144 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 5^{2}\cdot 13^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{3}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + 159) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b7 - b3 + b2 + b1) * q^3 + (-b9 - b4 - 26*b3 - b2 + b1 - 1) * q^4 + (-b9 + b7 - b6 - 7*b3 - 4*b2 + 7) * q^5 + (2*b9 - b8 - 2*b7 - b5 + b4 + 49*b3 + 7*b2 - 49) * q^6 + (b11 + b10 - b9 - b7 + 36*b3 - 4*b1 + 36) * q^7 + (-b10 - 3*b9 - 2*b8 - 3*b7 - 2*b4 - 83*b3 + 17*b1 - 83) * q^8 + (-b11 + b10 + 4*b8 + 3*b7 + b6 + b5 + 13*b3 - 13*b2 - 13*b1 + 159) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{3}+ \cdots + (2014 \beta_{11} + 1103 \beta_{10} + \cdots + 247870) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b7 - b3 + b2 + b1) * q^3 + (-b9 - b4 - 26*b3 - b2 + b1 - 1) * q^4 + (-b9 + b7 - b6 - 7*b3 - 4*b2 + 7) * q^5 + (2*b9 - b8 - 2*b7 - b5 + b4 + 49*b3 + 7*b2 - 49) * q^6 + (b11 + b10 - b9 - b7 + 36*b3 - 4*b1 + 36) * q^7 + (-b10 - 3*b9 - 2*b8 - 3*b7 - 2*b4 - 83*b3 + 17*b1 - 83) * q^8 + (-b11 + b10 + 4*b8 + 3*b7 + b6 + b5 + 13*b3 - 13*b2 - 13*b1 + 159) * q^9 + (-3*b10 - 3*b9 + 3*b5 - 5*b4 - 307*b3 - 22*b2 + 22*b1 - 22) * q^10 + (-8*b11 - b10 + 9*b9 + 8*b8 + 9*b7 + 8*b4 + 145*b3 - 26*b1 + 145) * q^11 + (8*b11 + 2*b10 + 43*b9 + 8*b6 - 2*b5 + 15*b4 + 561*b3 + 74*b2 - 74*b1 + 74) * q^12 + (-b11 + 2*b10 - 37*b9 - 12*b8 - 10*b7 + 8*b6 + 3*b5 - 8*b4 - 110*b3 + 33*b2 + 61*b1 - 336) * q^13 + (8*b11 + b10 + 11*b8 + 93*b7 - 8*b6 + b5 + 57*b3 - 57*b2 - 57*b1 + 428) * q^14 + (11*b9 - 40*b8 - 11*b7 - 7*b6 - 6*b5 + 40*b4 + 1256*b3 - 4*b2 - 1256) * q^15 + (-8*b11 - 3*b10 + 37*b8 - 57*b7 + 8*b6 - 3*b5 - 155*b3 + 155*b2 + 155*b1 + 178) * q^16 + (-7*b11 + 59*b9 - 7*b6 - 52*b4 - 1386*b3 + 297*b2 - 297*b1 + 297) * q^17 + (-117*b9 + 22*b8 + 117*b7 - 16*b6 - b5 - 22*b4 - 1022*b3 - 416*b2 + 1022) * q^18 + (122*b9 - 122*b7 - 8*b6 + 9*b5 - 50*b3 - 284*b2 + 50) * q^19 + (16*b11 - 3*b10 - 248*b9 - 45*b8 - 248*b7 - 45*b4 - 1716*b3 + 463*b1 - 1716) * q^20 + (b11 + 4*b10 - 76*b9 - 36*b8 - 76*b7 - 36*b4 - 135*b3 + 918*b1 - 135) * q^21 + (-8*b11 - 7*b10 + 28*b8 - 10*b7 + 8*b6 - 7*b5 + 521*b3 - 521*b2 - 521*b1 + 2389) * q^22 + (7*b11 + 28*b10 + 190*b9 + 7*b6 - 28*b5 + 56*b4 + 1652*b3 + 8*b2 - 8*b1 + 8) * q^23 + (32*b11 + 11*b10 + 168*b9 + 45*b8 + 168*b7 + 45*b4 + 3242*b3 - 1333*b1 + 3242) * q^24 + (-33*b11 - 23*b10 + 141*b9 - 33*b6 + 23*b5 + 20*b4 - 5041*b3 + 151*b2 - 151*b1 + 151) * q^25 + (-8*b11 - 23*b10 - 296*b9 - 70*b8 + 63*b7 - 40*b6 - 28*b5 + 27*b4 + 3709*b3 + 940*b2 + 670*b1 - 4716) * q^26 + (-40*b11 - 5*b10 - 8*b8 + 387*b7 + 40*b6 - 5*b5 - 629*b3 + 629*b2 + 629*b1 - 1860) * q^27 + (-295*b9 + 82*b8 + 295*b7 + 48*b6 + 61*b5 - 82*b4 + 510*b3 - 1439*b2 - 510) * q^28 + (41*b11 + 27*b10 + 24*b8 - 110*b7 - 41*b6 + 27*b5 - 956*b3 + 956*b2 + 956*b1 - 7444) * q^29 + (48*b11 - 3*b10 + 683*b9 + 48*b6 + 3*b5 + 105*b4 + 2952*b3 + 3335*b2 - 3335*b1 + 3335) * q^30 + (8*b9 - 72*b8 - 8*b7 + 156*b6 + 5*b5 + 72*b4 - 4668*b3 - 784*b2 + 4668) * q^31 + (241*b9 - 16*b8 - 241*b7 + 48*b6 - 25*b5 + 16*b4 + 6512*b3 - 2333*b2 - 6512) * q^32 + (-164*b11 - 36*b10 - 275*b9 + 140*b8 - 275*b7 + 140*b4 - 4063*b3 + 686*b1 - 4063) * q^33 + (31*b10 + 325*b9 + 238*b8 + 325*b7 + 238*b4 + 19825*b3 + 3301*b1 + 19825) * q^34 + (116*b11 + 2*b10 - 240*b8 - 21*b7 - 116*b6 + 2*b5 + 2747*b3 - 2747*b2 - 2747*b1 + 13396) * q^35 + (-56*b11 - 85*b10 - 360*b9 - 56*b6 + 85*b5 - 214*b4 - 21708*b3 - 2422*b2 + 2422*b1 - 2422) * q^36 + (65*b11 - 40*b10 - 159*b9 - 312*b8 - 159*b7 - 312*b4 - 1533*b3 - 3292*b1 - 1533) * q^37 + (-72*b11 + 106*b10 - 862*b9 - 72*b6 - 106*b5 - 346*b4 - 34282*b3 + 160*b2 - 160*b1 + 160) * q^38 + (124*b11 + 103*b10 + 285*b9 + 552*b8 + 395*b7 - 17*b6 + 83*b5 - 48*b4 + 24560*b3 + 2772*b2 + 80*b1 + 12596) * q^39 + (-24*b11 - 24*b10 - 235*b8 - 1187*b7 + 24*b6 - 24*b5 - 3942*b3 + 3942*b2 + 3942*b1 - 8233) * q^40 + (351*b9 + 356*b8 - 351*b7 - 100*b6 - 220*b5 - 356*b4 + 15910*b3 - 74*b2 - 15910) * q^41 + (32*b11 - 74*b10 - 965*b8 - 769*b7 - 32*b6 - 74*b5 - 2226*b3 + 2226*b2 + 2226*b1 - 79359) * q^42 + (-92*b11 + 33*b10 - 1447*b9 - 92*b6 - 33*b5 + 600*b4 + 4400*b3 + 2553*b2 - 2553*b1 + 2553) * q^43 + (671*b9 - 13*b8 - 671*b7 - 400*b6 + 22*b5 + 13*b4 - 40075*b3 - 2142*b2 + 40075) * q^44 + (-1347*b9 + 220*b8 + 1347*b7 - 16*b6 + 2*b5 - 220*b4 + 17308*b3 - 8342*b2 - 17308) * q^45 + (448*b11 + 218*b10 + 2865*b9 + 239*b8 + 2865*b7 + 239*b4 - 1155*b3 - 6790*b1 - 1155) * q^46 + (-69*b11 - 239*b10 - 831*b9 - 496*b8 - 831*b7 - 496*b4 + 43968*b3 + 6564*b1 + 43968) * q^47 + (-424*b11 + 104*b10 + 561*b8 - 55*b7 + 424*b6 + 104*b5 + 2076*b3 - 2076*b2 - 2076*b1 + 77109) * q^48 + (141*b11 + 26*b10 - 201*b9 + 141*b6 - 26*b5 + 356*b4 - 23077*b3 + 829*b2 - 829*b1 + 829) * q^49 + (-368*b11 + 9*b10 - 1751*b9 - 1751*b7 + 5118*b3 + 4426*b1 + 5118) * q^50 + (492*b11 - 231*b10 + 1167*b9 + 492*b6 + 231*b5 + 336*b4 - 90732*b3 + 591*b2 - 591*b1 + 591) * q^51 + (-472*b11 - 200*b10 + 2790*b9 - 347*b8 - 716*b7 + 344*b6 - b5 + 397*b4 + 81564*b3 + 5696*b2 - 1927*b1 - 44348) * q^52 + (399*b11 + 238*b10 + 744*b8 - 842*b7 - 399*b6 + 238*b5 + 1980*b3 - 1980*b2 - 1980*b1 + 42806) * q^53 + (600*b9 - 707*b8 - 600*b7 + 80*b6 + 227*b5 + 707*b4 + 71587*b3 + 1171*b2 - 71587) * q^54 + (-517*b11 - 13*b10 + 1800*b8 + 5228*b7 + 517*b6 - 13*b5 + 2630*b3 - 2630*b2 - 2630*b1 - 108028) * q^55 + (24*b11 - 135*b10 - 2119*b9 + 24*b6 + 135*b5 - 1217*b4 - 82658*b3 - 4089*b2 + 4089*b1 - 4089) * q^56 + (946*b9 - 312*b8 - 946*b7 + 222*b6 - 210*b5 + 312*b4 - 82636*b3 + 8000*b2 + 82636) * q^57 + (-1373*b9 - 729*b8 + 1373*b7 - 432*b6 + 54*b5 + 729*b4 + 81905*b3 + 6310*b2 - 81905) * q^58 + (-246*b11 - 287*b10 + 76*b9 - 704*b8 + 76*b7 - 704*b4 - 82546*b3 - 492*b1 - 82546) * q^59 + (400*b11 + 491*b10 + 2993*b9 + 910*b8 + 2993*b7 + 910*b4 + 207579*b3 - 16029*b1 + 207579) * q^60 + (622*b11 - 290*b10 - 1096*b8 + 456*b7 - 622*b6 - 290*b5 - 7284*b3 + 7284*b2 + 7284*b1 + 85006) * q^61 + (-40*b11 + 320*b10 - 1562*b9 - 40*b6 - 320*b5 - 514*b4 - 67472*b3 - 1278*b2 + 1278*b1 - 1278) * q^62 + (-236*b11 + 299*b10 + 2130*b9 + 1432*b8 + 2130*b7 + 1432*b4 + 71926*b3 - 1988*b1 + 71926) * q^63 + (-312*b11 + 145*b10 - 3261*b9 - 312*b6 - 145*b5 + 265*b4 - 209216*b3 - 2737*b2 + 2737*b1 - 2737) * q^64 + (631*b11 + 38*b10 - 872*b9 - 2100*b8 - 4753*b7 - 147*b6 - 450*b5 - 1400*b4 + 138284*b3 - 8473*b2 + 6359*b1 - 99113) * q^65 + (-288*b11 - 603*b10 - 786*b8 - 3848*b7 + 288*b6 - 603*b5 - 605*b3 + 605*b2 + 605*b1 - 33711) * q^66 + (3015*b9 - 512*b8 - 3015*b7 - 406*b6 + 479*b5 + 512*b4 + 115019*b3 + 7578*b2 - 115019) * q^67 + (696*b11 + 325*b10 + 689*b8 - 3037*b7 - 696*b6 + 325*b5 + 14873*b3 - 14873*b2 - 14873*b1 - 191406) * q^68 + (-335*b11 + 202*b10 - 4644*b9 - 335*b6 - 202*b5 - 1488*b4 - 134280*b3 - 29770*b2 + 29770*b1 - 29770) * q^69 + (-2326*b9 + 2635*b8 + 2326*b7 - 32*b6 + 443*b5 - 2635*b4 - 242139*b3 - 5745*b2 + 242139) * q^70 + (2019*b9 + 1144*b8 - 2019*b7 + 721*b6 + 229*b5 - 1144*b4 + 45872*b3 + 17388*b2 - 45872) * q^71 + (-336*b11 - 648*b10 - 3330*b9 - 1794*b8 - 3330*b7 - 1794*b4 - 147840*b3 + 14238*b1 - 147840) * q^72 + (-922*b11 + 182*b10 - 6153*b9 - 2524*b8 - 6153*b7 - 2524*b4 + 229338*b3 - 22074*b1 + 229338) * q^73 + (-320*b11 - 29*b10 + 2493*b8 - 1605*b7 + 320*b6 - 29*b5 - 18018*b3 + 18018*b2 + 18018*b1 + 280627) * q^74 + (-158*b11 - 335*b10 + 5884*b9 - 158*b6 + 335*b5 + 1440*b4 - 103044*b3 + 21168*b2 - 21168*b1 + 21168) * q^75 + (2208*b11 - 574*b10 + 1324*b9 + 378*b8 + 1324*b7 + 378*b4 + 3764*b3 + 33782*b1 + 3764) * q^76 + (-1113*b11 + 455*b10 + 8722*b9 - 1113*b6 - 455*b5 + 3448*b4 - 177068*b3 - 496*b2 + 496*b1 - 496) * q^77 + (160*b11 + 499*b10 - 5689*b9 + 1166*b8 + 10414*b7 - 1488*b6 + 677*b5 + 1657*b4 + 274610*b3 - 43487*b2 - 27713*b1 - 4545) * q^78 + (-1371*b11 + 115*b10 - 512*b8 + 9428*b7 + 1371*b6 + 115*b5 + 26970*b3 - 26970*b2 - 26970*b1 - 10124) * q^79 + (-8324*b9 - 1465*b8 + 8324*b7 + 1408*b6 - 1091*b5 + 1465*b4 + 225601*b3 + 2241*b2 - 225601) * q^80 + (998*b11 - 62*b10 + 376*b8 + 1752*b7 - 998*b6 - 62*b5 - 4172*b3 + 4172*b2 + 4172*b1 - 334743) * q^81 + (1760*b11 + 151*b10 + 20896*b9 + 1760*b6 - 151*b5 + 434*b4 - 58756*b3 + 5194*b2 - 5194*b1 + 5194) * q^82 + (-517*b9 - 4208*b8 + 517*b7 + 1280*b6 + 335*b5 + 4208*b4 - 91459*b3 + 32038*b2 + 91459) * q^83 + (6148*b9 - 1299*b8 - 6148*b7 + 1248*b6 - 897*b5 + 1299*b4 + 228107*b3 + 76703*b2 - 228107) * q^84 + (-665*b11 + 1784*b10 - 9938*b9 + 4060*b8 - 9938*b7 + 4060*b4 + 4811*b3 + 12014*b1 + 4811) * q^85 + (528*b11 - 1815*b10 + 5084*b9 + 3259*b8 + 5084*b7 + 3259*b4 + 319630*b3 - 27603*b1 + 319630) * q^86 + (-175*b11 + 1120*b10 - 920*b8 + 8144*b7 + 175*b6 + 1120*b5 - 8730*b3 + 8730*b2 + 8730*b1 + 173604) * q^87 + (-688*b11 - 577*b10 - 2282*b9 - 688*b6 + 577*b5 - 856*b4 - 133003*b3 - 6707*b2 + 6707*b1 - 6707) * q^88 + (-928*b11 - 688*b10 + 6811*b9 - 4660*b8 + 6811*b7 - 4660*b4 - 74302*b3 - 16574*b1 - 74302) * q^89 + (-16*b11 - 1379*b10 - 11616*b9 - 16*b6 + 1379*b5 - 8810*b4 - 651954*b3 - 9800*b2 + 9800*b1 - 9800) * q^90 + (-988*b11 - 806*b10 - 9464*b9 + 4888*b8 - 2171*b7 + 1586*b6 + 221*b5 - 1040*b4 + 137267*b3 - 22841*b2 + 36699*b1 + 59098) * q^91 + (1296*b11 + 1969*b10 + 5440*b8 + 19986*b7 - 1296*b6 + 1969*b5 + 34031*b3 - 34031*b2 - 34031*b1 + 299951) * q^92 + (6052*b9 + 5328*b8 - 6052*b7 - 490*b6 - 278*b5 - 5328*b4 + 5898*b3 + 23532*b2 - 5898) * q^93 + (-1912*b11 - 969*b10 - 9059*b8 - 31477*b7 + 1912*b6 - 969*b5 + 9571*b3 - 9571*b2 - 9571*b1 - 507944) * q^94 + (-1452*b11 - 751*b10 + 1770*b9 - 1452*b6 + 751*b5 + 8720*b4 + 41040*b3 - 34422*b2 + 34422*b1 - 34422) * q^95 + (-2878*b9 - 43*b8 + 2878*b7 + 384*b6 - 2455*b5 + 43*b4 - 62813*b3 - 20807*b2 + 62813) * q^96 + (6474*b9 + 376*b8 - 6474*b7 - 3766*b6 + 48*b5 - 376*b4 - 66213*b3 - 38064*b2 + 66213) * q^97 + (416*b11 + 363*b10 + 3005*b9 + 1482*b8 + 3005*b7 + 1482*b4 + 83406*b3 + 3812*b1 + 83406) * q^98 + (2014*b11 + 1103*b10 - 2988*b9 - 560*b8 - 2988*b7 - 560*b4 + 247870*b3 - 9212*b1 + 247870) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{2} - 4 q^{3} + 108 q^{5} - 640 q^{6} + 398 q^{7} - 912 q^{8} + 1940 q^{9}+O(q^{10})$$ 12 * q + 6 * q^2 - 4 * q^3 + 108 * q^5 - 640 * q^6 + 398 * q^7 - 912 * q^8 + 1940 * q^9 $$12 q + 6 q^{2} - 4 q^{3} + 108 q^{5} - 640 q^{6} + 398 q^{7} - 912 q^{8} + 1940 q^{9} + 1686 q^{11} - 3926 q^{13} + 5484 q^{14} - 15268 q^{15} + 2132 q^{16} + 15254 q^{18} + 1766 q^{19} - 19044 q^{20} + 3428 q^{21} + 28832 q^{22} + 31608 q^{24} - 58266 q^{26} - 20464 q^{27} + 4092 q^{28} - 90108 q^{29} + 61014 q^{31} - 64932 q^{32} - 44452 q^{33} + 259896 q^{34} + 158772 q^{35} - 40212 q^{37} + 137852 q^{39} - 104196 q^{40} - 190416 q^{41} - 959204 q^{42} + 489372 q^{44} - 151444 q^{45} - 44412 q^{46} + 562446 q^{47} + 930308 q^{48} + 82422 q^{50} - 578500 q^{52} + 509136 q^{53} - 871432 q^{54} - 1264036 q^{55} + 939908 q^{57} - 1019980 q^{58} - 994458 q^{59} + 2407804 q^{60} + 1013696 q^{61} + 865778 q^{63} - 1130064 q^{65} - 418352 q^{66} - 1442386 q^{67} - 2313132 q^{68} + 2958968 q^{70} - 655866 q^{71} - 1706508 q^{72} + 2588228 q^{73} + 3373752 q^{74} + 246984 q^{76} + 77480 q^{78} - 75316 q^{79} - 2685408 q^{80} - 4016140 q^{81} + 894966 q^{83} - 3220504 q^{84} + 105396 q^{85} + 3704832 q^{86} + 2109064 q^{87} - 977376 q^{89} + 1088750 q^{91} + 3682872 q^{92} - 216268 q^{93} - 6238300 q^{94} + 896384 q^{96} + 983388 q^{97} + 1039302 q^{98} + 2894714 q^{99}+O(q^{100})$$ 12 * q + 6 * q^2 - 4 * q^3 + 108 * q^5 - 640 * q^6 + 398 * q^7 - 912 * q^8 + 1940 * q^9 + 1686 * q^11 - 3926 * q^13 + 5484 * q^14 - 15268 * q^15 + 2132 * q^16 + 15254 * q^18 + 1766 * q^19 - 19044 * q^20 + 3428 * q^21 + 28832 * q^22 + 31608 * q^24 - 58266 * q^26 - 20464 * q^27 + 4092 * q^28 - 90108 * q^29 + 61014 * q^31 - 64932 * q^32 - 44452 * q^33 + 259896 * q^34 + 158772 * q^35 - 40212 * q^37 + 137852 * q^39 - 104196 * q^40 - 190416 * q^41 - 959204 * q^42 + 489372 * q^44 - 151444 * q^45 - 44412 * q^46 + 562446 * q^47 + 930308 * q^48 + 82422 * q^50 - 578500 * q^52 + 509136 * q^53 - 871432 * q^54 - 1264036 * q^55 + 939908 * q^57 - 1019980 * q^58 - 994458 * q^59 + 2407804 * q^60 + 1013696 * q^61 + 865778 * q^63 - 1130064 * q^65 - 418352 * q^66 - 1442386 * q^67 - 2313132 * q^68 + 2958968 * q^70 - 655866 * q^71 - 1706508 * q^72 + 2588228 * q^73 + 3373752 * q^74 + 246984 * q^76 + 77480 * q^78 - 75316 * q^79 - 2685408 * q^80 - 4016140 * q^81 + 894966 * q^83 - 3220504 * q^84 + 105396 * q^85 + 3704832 * q^86 + 2109064 * q^87 - 977376 * q^89 + 1088750 * q^91 + 3682872 * q^92 - 216268 * q^93 - 6238300 * q^94 + 896384 * q^96 + 983388 * q^97 + 1039302 * q^98 + 2894714 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 18 x^{10} + 488 x^{9} + 36205 x^{8} - 155430 x^{7} + 399962 x^{6} + 9502784 x^{5} + \cdots + 56070144$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 35\!\cdots\!91 \nu^{11} + \cdots - 84\!\cdots\!84 ) / 41\!\cdots\!00$$ (357114067457877400091*v^11 - 2128261731755613609450*v^10 + 6345490399526750340438*v^9 + 174860636719079419952536*v^8 + 12925952551033055488927871*v^7 - 54942822772347566670635754*v^6 + 140832422401403167276894318*v^5 + 3408416628369323032624656352*v^4 + 98035211725237854012633477804*v^3 - 188190316198958158128876499272*v^2 + 181983402678115217418930098200*v - 84271945543930352398314518784) / 41513260965910624743365635200 $$\beta_{3}$$ $$=$$ $$( 32\!\cdots\!99 \nu^{11} + \cdots - 39\!\cdots\!76 ) / 20\!\cdots\!00$$ (3241403613457336365199*v^11 - 17992588404576068826450*v^10 + 50993985296434053794382*v^9 + 1601071496665980448309304*v^8 + 118081786138293446450576419*v^7 - 450359340256545622236943506*v^6 + 1120793045650815378413957702*v^5 + 31218989329148360235860700128*v^4 + 907491573934919722127792205756*v^3 - 1337326456492378754493918632808*v^2 + 1299516318703631221526793457400*v - 395539390258771826127888936576) / 207566304829553123716828176000 $$\beta_{4}$$ $$=$$ $$( - 22\!\cdots\!13 \nu^{11} + \cdots + 29\!\cdots\!12 ) / 55\!\cdots\!00$$ (-22912445860669066253413*v^11 + 101527098802980281143875*v^10 - 256306667489467113090084*v^9 - 11528098898159150707367048*v^8 - 847855458750438196463051953*v^7 + 2230473399748094133745196847*v^6 - 5762775984044183221026812924*v^5 - 224937926601773064178639285136*v^4 - 6672498646502570675933302792572*v^3 + 1943965170319355491379614921596*v^2 - 9544084396769930498809171568000*v + 2904568872219137156875798874112) / 55351014621214166324487513600 $$\beta_{5}$$ $$=$$ $$( - 92\!\cdots\!59 \nu^{11} + \cdots + 26\!\cdots\!16 ) / 16\!\cdots\!00$$ (-92073754120669108051559*v^11 + 585294363815167122063150*v^10 - 1823874315754389341950512*v^9 - 44447331103509094213158064*v^8 - 3322262130728207350541254379*v^7 + 15527333843268327969982514046*v^6 - 40805763389300746844112120832*v^5 - 865648917115961028856086597448*v^4 - 25169942396198623513457361024396*v^3 + 59300502400341256538607672406728*v^2 - 57157125302550536238255245617600*v + 26428359035169628420344790358016) / 166053043863642498973462540800 $$\beta_{6}$$ $$=$$ $$( 12\!\cdots\!41 \nu^{11} + \cdots - 13\!\cdots\!84 ) / 16\!\cdots\!00$$ (1280371007734425507571841*v^11 - 15841998837522733266745050*v^10 + 64140261566921047002107088*v^9 + 520679752698828203132183536*v^8 + 42255335220777824890377513821*v^7 - 498435274698423750267666347754*v^6 + 1493613446546286664243328987968*v^5 + 9974969093535582133925928569752*v^4 + 272725350192007604776556851298004*v^3 - 3020283937772783448085994684638872*v^2 + 2880417746017754970252043077313600*v - 1325333929538630898153683679504384) / 1660530438636424989734625408000 $$\beta_{7}$$ $$=$$ $$( - 88\!\cdots\!42 \nu^{11} + \cdots - 51\!\cdots\!92 ) / 92\!\cdots\!00$$ (-88005002947366457599642*v^11 + 442999245001071970552725*v^10 - 1114382541663357479277956*v^9 - 44209257227011221123982232*v^8 - 3228526341860911283090556402*v^7 + 10580426150111151354121985873*v^6 - 23433632862012524401290141516*v^5 - 863418945334969351238539131624*v^4 - 25079013543394492511478818033248*v^3 + 23822003024067271348959931592564*v^2 - 10940434783497474177671455075200*v - 519024354207783574933614920192) / 92251691035356943874145856000 $$\beta_{8}$$ $$=$$ $$( 88\!\cdots\!22 \nu^{11} + \cdots - 77\!\cdots\!28 ) / 92\!\cdots\!00$$ (88037053331792348246522*v^11 - 443182717922660954688725*v^10 + 1115691367884769143073796*v^9 + 44201785535279230498722712*v^8 + 3229778379044534211147241682*v^7 - 10584869781771561117273768593*v^6 + 23466607933025157737019703756*v^5 + 862566403154558956660818090984*v^4 + 25090398570457238614441836191968*v^3 - 23832797348704030556843055121524*v^2 + 10945387839612531410160577891200*v - 7735872675020737540437472892928) / 92251691035356943874145856000 $$\beta_{9}$$ $$=$$ $$( - 65\!\cdots\!49 \nu^{11} + \cdots + 78\!\cdots\!72 ) / 66\!\cdots\!32$$ (-6528610652683580397349*v^11 + 39294880871740546305399*v^10 - 115090599091069741302540*v^9 - 3199736340743872899054344*v^8 - 236264736620067253323865393*v^7 + 1020587240325504485325671955*v^6 - 2513817665804821796544677540*v^5 - 62372791415195418092622157136*v^4 - 1797190261476222261974023861020*v^3 + 3594756045422440548635250133644*v^2 - 2574841647580113523311974255072*v + 783763789746483243998432451072) / 6642121754545699958938501632 $$\beta_{10}$$ $$=$$ $$( 12\!\cdots\!07 \nu^{11} + \cdots - 69\!\cdots\!68 ) / 83\!\cdots\!00$$ (1206107845578338721073207*v^11 - 6674359625525494178037600*v^10 + 18300524098582539654248376*v^9 + 598492614910151512229725472*v^8 + 43942846036790063072591724667*v^7 - 167134087416288733042825165608*v^6 + 396565556869553713880142802136*v^5 + 11673245849259161340248963845304*v^4 + 337771952224323265068968287674108*v^3 - 497807711938793955881411855894544*v^2 + 378237620862801487477085843595200*v - 69974418310187098813340655034368) / 830265219318212494867312704000 $$\beta_{11}$$ $$=$$ $$( 18\!\cdots\!17 \nu^{11} + \cdots - 10\!\cdots\!08 ) / 16\!\cdots\!00$$ (18818270548927347405555317*v^11 - 104120432833520459028858600*v^10 + 286035933925434023882990856*v^9 + 9338203466932758402662828032*v^8 + 685623199938251875643472304577*v^7 - 2606835122282705077212062708448*v^6 + 6160360793754360550118500469416*v^5 + 182211381749628968703296147441224*v^4 + 5270192220960805727664242174976948*v^3 - 7767270297607681075250087377547664*v^2 + 5060507631507659690918549648507200*v - 1091811988682700188423951481643008) / 1660530438636424989734625408000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{4} + 90\beta_{3} - \beta_{2} + \beta _1 - 1$$ b9 + b4 + 90*b3 - b2 + b1 - 1 $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{5} + \beta_{4} + 203\beta_{3} - 145\beta_{2} - 203$$ -b8 + b5 + b4 + 203*b3 - 145*b2 - 203 $$\nu^{4}$$ $$=$$ $$- 8 \beta_{11} + \beta_{10} - 151 \beta_{8} - 237 \beta_{7} + 8 \beta_{6} + \beta_{5} + 229 \beta_{3} + \cdots - 12974$$ -8*b11 + b10 - 151*b8 - 237*b7 + 8*b6 + b5 + 229*b3 - 229*b2 - 229*b1 - 12974 $$\nu^{5}$$ $$=$$ $$- 32 \beta_{11} + 271 \beta_{10} - 196 \beta_{9} - 227 \beta_{8} - 196 \beta_{7} - 227 \beta_{4} + \cdots - 18442$$ -32*b11 + 271*b10 - 196*b9 - 227*b8 - 196*b7 - 227*b4 - 18442*b3 - 21909*b1 - 18442 $$\nu^{6}$$ $$=$$ $$- 2200 \beta_{11} + 531 \beta_{10} - 47025 \beta_{9} - 2200 \beta_{6} - 531 \beta_{5} - 23503 \beta_{4} + \cdots + 46621$$ -2200*b11 + 531*b10 - 47025*b9 - 2200*b6 - 531*b5 - 23503*b4 - 1971602*b3 + 46621*b2 - 46621*b1 + 46621 $$\nu^{7}$$ $$=$$ $$- 79632 \beta_{9} + 47607 \beta_{8} + 79632 \beta_{7} - 12896 \beta_{6} - 56887 \beta_{5} + \cdots + 7589847$$ -79632*b9 + 47607*b8 + 79632*b7 - 12896*b6 - 56887*b5 - 47607*b4 - 7589847*b3 + 3427153*b2 + 7589847 $$\nu^{8}$$ $$=$$ $$467992 \beta_{11} - 162311 \beta_{10} + 3755579 \beta_{8} + 8718005 \beta_{7} - 467992 \beta_{6} + \cdots + 311456790$$ 467992*b11 - 162311*b10 + 3755579*b8 + 8718005*b7 - 467992*b6 - 162311*b5 - 9165741*b3 + 9165741*b2 + 9165741*b1 + 311456790 $$\nu^{9}$$ $$=$$ $$3532960 \beta_{11} - 10914595 \beta_{10} + 22071372 \beta_{9} + 9671615 \beta_{8} + 22071372 \beta_{7} + \cdots + 854289306$$ 3532960*b11 - 10914595*b10 + 22071372*b9 + 9671615*b8 + 22071372*b7 + 9671615*b4 + 854289306*b3 + 550736589*b1 + 854289306 $$\nu^{10}$$ $$=$$ $$90849720 \beta_{11} - 40051887 \beta_{10} + 1567964169 \beta_{9} + 90849720 \beta_{6} + \cdots - 1779235357$$ 90849720*b11 - 40051887*b10 + 1567964169*b9 + 90849720*b6 + 40051887*b5 + 612691199*b4 + 50643771034*b3 - 1779235357*b2 + 1779235357*b1 - 1779235357 $$\nu^{11}$$ $$=$$ $$5203970696 \beta_{9} - 1928696959 \beta_{8} - 5203970696 \beta_{7} + 822529632 \beta_{6} + \cdots - 259296282423$$ 5203970696*b9 - 1928696959*b8 - 5203970696*b7 + 822529632*b6 + 2011466823*b5 + 1928696959*b4 + 259296282423*b3 - 90386183129*b2 - 259296282423

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{3}$$

## 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
 9.50218 + 9.50218i 7.85712 + 7.85712i 0.561862 + 0.561862i 0.388469 + 0.388469i −6.57888 − 6.57888i −8.73075 − 8.73075i 9.50218 − 9.50218i 7.85712 − 7.85712i 0.561862 − 0.561862i 0.388469 − 0.388469i −6.57888 + 6.57888i −8.73075 + 8.73075i
−8.50218 + 8.50218i −17.1101 80.5742i 73.6481 73.6481i 145.473 145.473i −214.232 214.232i 140.917 + 140.917i −436.244 1252.34i
5.2 −6.85712 + 6.85712i 38.7457 30.0401i −127.232 + 127.232i −265.684 + 265.684i 215.792 + 215.792i −232.867 232.867i 772.229 1744.89i
5.3 0.438138 0.438138i −27.8396 63.6161i −78.9164 + 78.9164i −12.1976 + 12.1976i −88.1833 88.1833i 55.9134 + 55.9134i 46.0426 69.1525i
5.4 0.611531 0.611531i 18.7379 63.2521i 134.607 134.607i 11.4588 11.4588i 241.226 + 241.226i 77.8186 + 77.8186i −377.891 164.633i
5.5 7.57888 7.57888i 26.7789 50.8787i −59.0322 + 59.0322i 202.954 202.954i −226.950 226.950i 99.4446 + 99.4446i −11.8881 894.795i
5.6 9.73075 9.73075i −41.3128 125.375i 110.925 110.925i −402.005 + 402.005i 271.347 + 271.347i −597.226 597.226i 977.751 2158.77i
8.1 −8.50218 8.50218i −17.1101 80.5742i 73.6481 + 73.6481i 145.473 + 145.473i −214.232 + 214.232i 140.917 140.917i −436.244 1252.34i
8.2 −6.85712 6.85712i 38.7457 30.0401i −127.232 127.232i −265.684 265.684i 215.792 215.792i −232.867 + 232.867i 772.229 1744.89i
8.3 0.438138 + 0.438138i −27.8396 63.6161i −78.9164 78.9164i −12.1976 12.1976i −88.1833 + 88.1833i 55.9134 55.9134i 46.0426 69.1525i
8.4 0.611531 + 0.611531i 18.7379 63.2521i 134.607 + 134.607i 11.4588 + 11.4588i 241.226 241.226i 77.8186 77.8186i −377.891 164.633i
8.5 7.57888 + 7.57888i 26.7789 50.8787i −59.0322 59.0322i 202.954 + 202.954i −226.950 + 226.950i 99.4446 99.4446i −11.8881 894.795i
8.6 9.73075 + 9.73075i −41.3128 125.375i 110.925 + 110.925i −402.005 402.005i 271.347 271.347i −597.226 + 597.226i 977.751 2158.77i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.6 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.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.7.d.a 12
3.b odd 2 1 117.7.j.b 12
4.b odd 2 1 208.7.t.c 12
13.d odd 4 1 inner 13.7.d.a 12
39.f even 4 1 117.7.j.b 12
52.f even 4 1 208.7.t.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.7.d.a 12 1.a even 1 1 trivial
13.7.d.a 12 13.d odd 4 1 inner
117.7.j.b 12 3.b odd 2 1
117.7.j.b 12 39.f even 4 1
208.7.t.c 12 4.b odd 2 1
208.7.t.c 12 52.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 6 T^{11} + \cdots + 84934656$$
$3$ $$(T^{6} + 2 T^{5} + \cdots - 382594752)^{2}$$
$5$ $$T^{12} + \cdots + 27\!\cdots\!00$$
$7$ $$T^{12} + \cdots + 23\!\cdots\!36$$
$11$ $$T^{12} + \cdots + 89\!\cdots\!16$$
$13$ $$T^{12} + \cdots + 12\!\cdots\!41$$
$17$ $$T^{12} + \cdots + 11\!\cdots\!00$$
$19$ $$T^{12} + \cdots + 66\!\cdots\!00$$
$23$ $$T^{12} + \cdots + 11\!\cdots\!00$$
$29$ $$(T^{6} + \cdots + 11\!\cdots\!00)^{2}$$
$31$ $$T^{12} + \cdots + 29\!\cdots\!00$$
$37$ $$T^{12} + \cdots + 25\!\cdots\!00$$
$41$ $$T^{12} + \cdots + 21\!\cdots\!76$$
$43$ $$T^{12} + \cdots + 13\!\cdots\!00$$
$47$ $$T^{12} + \cdots + 31\!\cdots\!56$$
$53$ $$(T^{6} + \cdots - 11\!\cdots\!00)^{2}$$
$59$ $$T^{12} + \cdots + 57\!\cdots\!84$$
$61$ $$(T^{6} + \cdots - 51\!\cdots\!36)^{2}$$
$67$ $$T^{12} + \cdots + 49\!\cdots\!24$$
$71$ $$T^{12} + \cdots + 48\!\cdots\!64$$
$73$ $$T^{12} + \cdots + 62\!\cdots\!00$$
$79$ $$(T^{6} + \cdots - 14\!\cdots\!68)^{2}$$
$83$ $$T^{12} + \cdots + 14\!\cdots\!36$$
$89$ $$T^{12} + \cdots + 32\!\cdots\!00$$
$97$ $$T^{12} + \cdots + 25\!\cdots\!64$$