LMFDB - Newform orbit 1224.2.bq.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1224,2,Mod(145,1224)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1224, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1224.145"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1224.bq (of order $8$ , degree $4$ , minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64$ x^16 + 36x^14 + 466x^12 + 2956x^10 + 10049x^8 + 18032x^6 + 14800x^4 + 3200*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{14} + \beta_{7} + \beta_{4}) q^{5} + ( - \beta_{10} + \beta_{6} + \cdots + \beta_1) q^{7} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_{2}) q^{11} + (\beta_{15} + \beta_{14} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + 2) q^{97}+O(q^{100})$ q + (-b14 + b7 + b4) * q^5 + (-b10 + b6 - b5 + b4 + b1) * q^7 + (-b14 + b13 + b11 + b8 - b6 + b4 + b3 - b2) * q^11 + (b15 + b14 + b12 - b9 - b4 + b3) * q^13 + (-b15 + b13 + b12 + b11 - b10 + b8 + b7 + b5 + b4 + b3 + b2 + 1) * q^17 + (b15 - b11 + b10 - b7 - b5 - b3 + b1 - 1) * q^19 + (-b14 + b13 + b11 - b9 + b8 - b6 + b4 + b3 - b2) * q^23 + (b15 + b12 + b11 - b5 - b4 + b3 - b2 - 1) * q^25 + (-b15 - b13 - b10 - b8 + b7 + 3b6 + b4 + b3 + b1 - 1) q^29 + (-b15 + b13 - 3b12 - b10 - 2b3 + 2b1 + 1) q^31 + (b15 - b14 - 2b12 - 2b9 + b7 - b6 - b2 + b1 - 1) * q^35 + (-b15 + b13 - b12 + 2b11 - b9 - b7 + b4 + 1) q^37 + (-b15 + b14 - b12 + b11 + b10 - 2b8 - 3b7 - 2) * q^41 + (-b15 - 2b12 - b11 + b10 - b9 - b8 + 3b6 + b5 - 2b4 - b3 - b2 - 2) q^43 + (b15 + b14 + b13 - b12 + b11 - b10 + b9 + b8 - 4b4) q^47 + (b15 - b14 - b13 + b12 - b11 + 2b10 - b9 + 2b8 + b7 + b5 - b4 + b3 + 2) * q^49 + (-2b12 - 2b10 - b9 + b8 + 2b7 - b5 - b3 + 4b1 + 3) * q^53 + (-b15 - b14 + 2b13 + b12 + 2b11 + b10 - b9 - b8 - 2b7 - 2b6 + b4 + b3 - b2 - b1 - 1) * q^55 + (2b15 - 3b14 + 3b13 - b12 + 2b11 + 3b10 - 2b9 + b8 - b5 + b4 + b3 - 2b2 - 1) q^59 + (-2b15 + b14 + b11 - 5b10 - b8 + b7 + 4b6 + b5 + 4b4 + 3b3 + 3b2 - b1 + 2) * q^61 + (-b15 + b13 - 2b12 - b11 + 2b10 - 3b9 - b7 + 4b6 + b5 + b4 - b2 + 3) * q^65 + (2b15 - 2b14 - b12 - b9 + 5b7 - 5b6 + 2b5 - b2 + b1 + 1) q^67 + (3b12 - b10 + b9 + 2b6 - 2b5 - 2b3 + 2b2 + 2b1 + 2) * q^71 + (-b15 - b13 + 2b10 + 3b9 + 2b8 - b7 - b6 + b5 - b4 + b2 + 3) q^73 + (b14 - b13 + b12 - b10 - 3b9 - 4b8 - 2b6 - b5 - b4 + b3 - 2b2 - 1) * q^77 + (-2b14 + 2b12 + 2b11 - b9 + 6b7 + b6 - b5 - b4 - b1 - 6) * q^79 + (b15 + b14 + b13 + b12 - b11 + 2b10 - b8 - 4b7 + 2b1) q^83 + (-b15 + b14 - 4b12 + 2b10 - 2b9 + b8 - b7 - b6 + 3b5 + 2b4 + b3 - b1 - 1) q^85 + (b15 + b14 + 2b13 + 3b12 + 2b11 - 4b10 - 3b9 + 4b8 + 3b7 + 3b6 + 4b4 + 2b3 + 2b2 + 2b1 + 2) * q^89 + (-b14 + 2b13 + b11 + 4b9 + b8 - b7 - 4b6 + 4b4 - b3 + b2) * q^91 + (b15 - b14 - b11 - b10 + b8 + b7 + 2b6 - b5 + 2b4 + b1) * q^95 + (-b15 - 2b14 - b13 - 4b10 - 3b9 + 2b8 + 2b7 - 2b6 - 3b5 + 2b4 + 2b3 - 3b2 + 2b1 + 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{11} - 8 q^{19} + 8 q^{23} - 8 q^{25} - 32 q^{29} + 8 q^{31} - 16 q^{35} + 8 q^{37} - 40 q^{41} - 24 q^{43} + 24 q^{49} + 24 q^{53} + 16 q^{59} + 64 q^{65} + 16 q^{71} + 24 q^{73} - 96 q^{79}+ \cdots + 56 q^{97}+O(q^{100})$ 16 * q + 8 * q^11 - 8 * q^19 + 8 * q^23 - 8 * q^25 - 32 * q^29 + 8 * q^31 - 16 * q^35 + 8 * q^37 - 40 * q^41 - 24 * q^43 + 24 * q^49 + 24 * q^53 + 16 * q^59 + 64 * q^65 + 16 * q^71 + 24 * q^73 - 96 * q^79 - 32 * q^85 + 8 * q^95 + 56 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64$ x^16 + 36*x^14 + 466*x^12 + 2956*x^10 + 10049*x^8 + 18032*x^6 + 14800*x^4 + 3200*x^2 + 64 :

$\beta_{1}$	$=$	$( 2 \nu^{15} + 12 \nu^{14} + 51 \nu^{13} + 393 \nu^{12} + 264 \nu^{11} + 4324 \nu^{10} - 1058 \nu^{9} + \cdots - 2168 ) / 1088$ (2v^15 + 12v^14 + 51v^13 + 393v^12 + 264v^11 + 4324v^10 - 1058v^9 + 21690v^8 - 12266v^7 + 52488v^6 - 34685v^5 + 54073v^4 - 35752v^3 + 11384v^2 - 9544*v - 2168) / 1088
$\beta_{2}$	$=$	$( 2 \nu^{15} - 12 \nu^{14} + 51 \nu^{13} - 393 \nu^{12} + 264 \nu^{11} - 4324 \nu^{10} - 1058 \nu^{9} + \cdots + 1080 ) / 1088$ (2v^15 - 12v^14 + 51v^13 - 393v^12 + 264v^11 - 4324v^10 - 1058v^9 - 21690v^8 - 12266v^7 - 52488v^6 - 34685v^5 - 54073v^4 - 35752v^3 - 11384v^2 - 9544*v + 1080) / 1088
$\beta_{3}$	$=$	$( - 23 \nu^{15} - 776 \nu^{13} - 9062 \nu^{11} - 50684 \nu^{9} - 150727 \nu^{7} - 238604 \nu^{5} + \cdots - 28576 \nu ) / 2176$ (-23v^15 - 776v^13 - 9062v^11 - 50684v^9 - 150727v^7 - 238604v^5 - 175352v^3 - 28576v) / 2176
$\beta_{4}$	$=$	$( 47 \nu^{15} + 1492 \nu^{13} + 15446 \nu^{11} + 69732 \nu^{9} + 137983 \nu^{7} + 75336 \nu^{5} + \cdots - 30720 \nu ) / 4352$ (47v^15 + 1492v^13 + 15446v^11 + 69732v^9 + 137983v^7 + 75336v^5 - 67080v^3 - 30720v) / 4352
$\beta_{5}$	$=$	$( - 39 \nu^{14} - 1288 \nu^{12} - 14390 \nu^{10} - 73964 \nu^{8} - 187047 \nu^{6} - 214076 \nu^{4} + \cdots - 7456 ) / 2176$ (-39v^14 - 1288v^12 - 14390v^10 - 73964v^8 - 187047v^6 - 214076v^4 - 75928*v^2 - 7456) / 2176
$\beta_{6}$	$=$	$( 15 \nu^{15} + 124 \nu^{14} + 454 \nu^{13} + 4032 \nu^{12} + 4174 \nu^{11} + 43768 \nu^{10} + \cdots + 6016 ) / 4352$ (15v^15 + 124v^14 + 454v^13 + 4032v^12 + 4174v^11 + 43768v^10 + 13464v^9 + 215056v^8 - 1993v^7 + 507420v^6 - 87270v^5 + 518960v^4 - 126360v^3 + 143520v^2 - 23408*v + 6016) / 4352
$\beta_{7}$	$=$	$( 15 \nu^{15} - 124 \nu^{14} + 454 \nu^{13} - 4032 \nu^{12} + 4174 \nu^{11} - 43768 \nu^{10} + \cdots - 6016 ) / 4352$ (15v^15 - 124v^14 + 454v^13 - 4032v^12 + 4174v^11 - 43768v^10 + 13464v^9 - 215056v^8 - 1993v^7 - 507420v^6 - 87270v^5 - 518960v^4 - 126360v^3 - 143520v^2 - 23408*v - 6016) / 4352
$\beta_{8}$	$=$	$( 29 \nu^{15} + 3 \nu^{14} + 957 \nu^{13} + 127 \nu^{12} + 10678 \nu^{11} + 1938 \nu^{10} + 54822 \nu^{9} + \cdots + 1048 ) / 1088$ (29v^15 + 3v^14 + 957v^13 + 127v^12 + 10678v^11 + 1938v^10 + 54822v^9 + 13274v^8 + 139121v^7 + 42407v^6 + 164425v^5 + 57531v^4 + 71632v^3 + 21920v^2 + 11048*v + 1048) / 1088
$\beta_{9}$	$=$	$( 35 \nu^{15} + 67 \nu^{14} + 1138 \nu^{13} + 2194 \nu^{12} + 12294 \nu^{11} + 24086 \nu^{10} + \cdots - 1680 ) / 2176$ (35v^15 + 67v^14 + 1138v^13 + 2194v^12 + 12294v^11 + 24086v^10 + 58960v^9 + 119744v^8 + 127515v^7 + 283851v^6 + 90246v^5 + 282326v^4 - 32088v^3 + 58472v^2 - 8656*v - 1680) / 2176
$\beta_{10}$	$=$	$( - 29 \nu^{15} + 3 \nu^{14} - 957 \nu^{13} + 127 \nu^{12} - 10678 \nu^{11} + 1938 \nu^{10} + \cdots + 1048 ) / 1088$ (-29v^15 + 3v^14 - 957v^13 + 127v^12 - 10678v^11 + 1938v^10 - 54822v^9 + 13274v^8 - 139121v^7 + 42407v^6 - 164425v^5 + 57531v^4 - 71632v^3 + 21920v^2 - 11048*v + 1048) / 1088
$\beta_{11}$	$=$	$( - 81 \nu^{15} - 123 \nu^{14} - 2770 \nu^{13} - 3952 \nu^{12} - 32850 \nu^{11} - 41918 \nu^{10} + \cdots + 11680 ) / 4352$ (-81v^15 - 123v^14 - 2770v^13 - 3952v^12 - 32850v^11 - 41918v^10 - 183512v^9 - 198028v^8 - 521385v^7 - 435451v^6 - 723262v^5 - 378004v^4 - 404024v^3 - 30904v^2 - 54576*v + 11680) / 4352
$\beta_{12}$	$=$	$( - 35 \nu^{15} + 67 \nu^{14} - 1138 \nu^{13} + 2194 \nu^{12} - 12294 \nu^{11} + 24086 \nu^{10} + \cdots - 1680 ) / 2176$ (-35v^15 + 67v^14 - 1138v^13 + 2194v^12 - 12294v^11 + 24086v^10 - 58960v^9 + 119744v^8 - 127515v^7 + 283851v^6 - 90246v^5 + 282326v^4 + 32088v^3 + 58472v^2 + 8656*v - 1680) / 2176
$\beta_{13}$	$=$	$( - 81 \nu^{15} + 123 \nu^{14} - 2770 \nu^{13} + 3952 \nu^{12} - 32850 \nu^{11} + 41918 \nu^{10} + \cdots - 11680 ) / 4352$ (-81v^15 + 123v^14 - 2770v^13 + 3952v^12 - 32850v^11 + 41918v^10 - 183512v^9 + 198028v^8 - 521385v^7 + 435451v^6 - 723262v^5 + 378004v^4 - 404024v^3 + 30904v^2 - 54576*v - 11680) / 4352
$\beta_{14}$	$=$	$( - 22 \nu^{15} + 245 \nu^{14} - 770 \nu^{13} + 8104 \nu^{12} - 9652 \nu^{11} + 90498 \nu^{10} + \cdots + 352 ) / 4352$ (-22v^15 + 245v^14 - 770v^13 + 8104v^12 - 9652v^11 + 90498v^10 - 60684v^9 + 460036v^8 - 213774v^7 + 1119125v^6 - 417802v^5 + 1143652v^4 - 383424v^3 + 242952v^2 - 81104*v + 352) / 4352
$\beta_{15}$	$=$	$( - 22 \nu^{15} - 245 \nu^{14} - 770 \nu^{13} - 8104 \nu^{12} - 9652 \nu^{11} - 90498 \nu^{10} + \cdots - 352 ) / 4352$ (-22v^15 - 245v^14 - 770v^13 - 8104v^12 - 9652v^11 - 90498v^10 - 60684v^9 - 460036v^8 - 213774v^7 - 1119125v^6 - 417802v^5 - 1143652v^4 - 383424v^3 - 242952v^2 - 81104*v - 352) / 4352

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{3} ) / 2$ (b12 - b10 - b9 + b8 + 2*b3) / 2
$\nu^{2}$	$=$	$- \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 4$ -b13 + 2b12 + b11 - b10 + 2b9 - b8 + b7 - b6 - b5 + b2 - b1 - 4
$\nu^{3}$	$=$	$( - 6 \beta_{15} - 6 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 2 \beta_{11} + 15 \beta_{10} + 11 \beta_{9} + \cdots - 6 ) / 2$ (-6b15 - 6b14 - 2b13 - 11b12 - 2b11 + 15b10 + 11b9 - 15b8 + 8b7 + 8b6 + 12b4 - 14b3 - 6b2 - 6b1 - 6) / 2
$\nu^{4}$	$=$	$- 5 \beta_{15} + 5 \beta_{14} + 16 \beta_{13} - 44 \beta_{12} - 16 \beta_{11} + 14 \beta_{10} + \cdots + 39$ -5b15 + 5b14 + 16b13 - 44b12 - 16b11 + 14b10 - 44b9 + 14b8 - 21b7 + 21b6 + 19b5 - 14b2 + 14*b1 + 39
$\nu^{5}$	$=$	$( 126 \beta_{15} + 126 \beta_{14} + 50 \beta_{13} + 167 \beta_{12} + 50 \beta_{11} - 257 \beta_{10} + \cdots + 108 ) / 2$ (126b15 + 126b14 + 50b13 + 167b12 + 50b11 - 257b10 - 167b9 + 257b8 - 168b7 - 168b6 - 244b4 + 170b3 + 108b2 + 108b1 + 108) / 2
$\nu^{6}$	$=$	$108 \beta_{15} - 108 \beta_{14} - 275 \beta_{13} + 796 \beta_{12} + 275 \beta_{11} - 225 \beta_{10} + \cdots - 532$ 108b15 - 108b14 - 275b13 + 796b12 + 275b11 - 225b10 + 796b9 - 225b8 + 365b7 - 365b6 - 315b5 + 223b2 - 223*b1 - 532
$\nu^{7}$	$=$	$( - 2258 \beta_{15} - 2258 \beta_{14} - 926 \beta_{13} - 2785 \beta_{12} - 926 \beta_{11} + 4393 \beta_{10} + \cdots - 1806 ) / 2$ (-2258b15 - 2258b14 - 926b13 - 2785b12 - 926b11 + 4393b10 + 2785b9 - 4393b8 + 2976b7 + 2976b6 + 4244b4 - 2610b3 - 1806b2 - 1806b1 - 1806) / 2
$\nu^{8}$	$=$	$- 1933 \beta_{15} + 1933 \beta_{14} + 4718 \beta_{13} - 13740 \beta_{12} - 4718 \beta_{11} + \cdots + 8355$ -1933b15 + 1933b14 + 4718b13 - 13740b12 - 4718b11 + 3772b10 - 13740b9 + 3772b8 - 6199b7 + 6199b6 + 5233b5 - 3696b2 + 3696*b1 + 8355
$\nu^{9}$	$=$	$( 38890 \beta_{15} + 38890 \beta_{14} + 16070 \beta_{13} + 47145 \beta_{12} + 16070 \beta_{11} + \cdots + 30256 ) / 2$ (38890b15 + 38890b14 + 16070b13 + 47145b12 + 16070b11 - 74667b10 - 47145b9 + 74667b8 - 50872b7 - 50872b6 - 72108b4 + 42974b3 + 30256b2 + 30256b1 + 30256) / 2
$\nu^{10}$	$=$	$33206 \beta_{15} - 33206 \beta_{14} - 80351 \beta_{13} + 233944 \beta_{12} + 80351 \beta_{11} + \cdots - 137862$ 33206b15 - 33206b14 - 80351b13 + 233944b12 + 80351b11 - 63753b10 + 233944b9 - 63753b8 + 104923b7 - 104923b6 - 87797b5 + 62051b2 - 62051*b1 - 137862
$\nu^{11}$	$=$	$( - 661806 \beta_{15} - 661806 \beta_{14} - 273970 \beta_{13} - 798691 \beta_{12} - 273970 \beta_{11} + \cdots - 509542 ) / 2$ (-661806b15 - 661806b14 - 273970b13 - 798691b12 - 273970b11 + 1265463b10 + 798691b9 - 1265463b8 + 862576b7 + 862576b6 + 1220572b4 - 721230b3 - 509542b2 - 509542b1 - 509542) / 2
$\nu^{12}$	$=$	$- 564289 \beta_{15} + 564289 \beta_{14} + 1362980 \beta_{13} - 3966012 \beta_{12} - 1362980 \beta_{11} + \cdots + 2312571$ -564289b15 + 564289b14 + 1362980b13 - 3966012b12 - 1362980b11 + 1078714b10 - 3966012b9 + 1078714b8 - 1775005b7 + 1775005b6 + 1480443b5 - 1046674b2 + 1046674*b1 + 2312571
$\nu^{13}$	$=$	$( 11218030 \beta_{15} + 11218030 \beta_{14} + 4646018 \beta_{13} + 13521263 \beta_{12} + 4646018 \beta_{11} + \cdots + 8604244 ) / 2$ (11218030b15 + 11218030b14 + 4646018b13 + 13521263b12 + 4646018b11 - 21421777b10 - 13521263b9 + 21421777b8 - 14598824b7 - 14598824b6 - 20648756b4 + 12170410b3 + 8604244b2 + 8604244b1 + 8604244) / 2
$\nu^{14}$	$=$	$9559272 \beta_{15} - 9559272 \beta_{14} - 23080535 \beta_{13} + 67138788 \beta_{12} + 23080535 \beta_{11} + \cdots - 39006928$ 9559272b15 - 9559272b14 - 23080535b13 + 67138788b12 + 23080535b11 - 18251429b10 + 67138788b9 - 18251429b8 + 30026021b7 - 30026021b6 - 25013827b5 + 17686739b2 - 17686739*b1 - 39006928
$\nu^{15}$	$=$	$( - 189898978 \beta_{15} - 189898978 \beta_{14} - 78656174 \beta_{13} - 228801361 \beta_{12} + \cdots - 145453174 ) / 2$ (-189898978b15 - 189898978b14 - 78656174b13 - 228801361b12 - 78656174b11 + 362463969b10 + 228801361b9 - 362463969b8 + 246979968b7 + 246979968b6 + 349294036b4 - 205709474b3 - 145453174b2 - 145453174b1 - 145453174) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$ .

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$\beta_{6}$	$1$

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}

145.1

	−	4.11290i
		1.88319i
	−	0.149074i
	−	2.44964i
		1.58886i
	−	0.535137i
	−	1.94001i
		1.71472i
	−	1.58886i
		0.535137i
		1.94001i
	−	1.71472i
		4.11290i
	−	1.88319i
		0.149074i
		2.44964i

−1.19125

2.87594i

−0.497533

−

1.20115i

145.2

−1.10335

2.66372i

1.25868

3.03872i

145.3

0.325635

−

0.786153i

0.663444

1.60170i

145.4

0.554753

−

1.33929i

−0.0103761

−

0.0250502i

433.1

−2.39179

−

0.990712i

2.00524

−

0.830597i

433.2

0.429478

0.177896i

−1.62064

0.671292i

433.3

0.868455

0.359726i

−4.01891

1.66469i

433.4

2.50807

1.03888i

2.22010

−

0.919595i

865.1

−2.39179

0.990712i

2.00524

0.830597i

865.2

0.429478

−

0.177896i

−1.62064

−

0.671292i

865.3

0.868455

−

0.359726i

−4.01891

−

1.66469i

865.4

2.50807

−

1.03888i

2.22010

0.919595i

937.1

−1.19125

−

2.87594i

−0.497533

1.20115i

937.2

−1.10335

−

2.66372i

1.25868

−

3.03872i

937.3

0.325635

0.786153i

0.663444

−

1.60170i

937.4

0.554753

1.33929i

−0.0103761

0.0250502i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.d	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1224.2.bq.e		16
3.b	odd	2	1		408.2.ba.a	✓	16
12.b	even	2	1		816.2.bq.f		16
17.d	even	8	1	inner	1224.2.bq.e		16
51.g	odd	8	1		408.2.ba.a	✓	16
51.i	even	16	1		6936.2.a.bl		8
51.i	even	16	1		6936.2.a.bo		8
204.p	even	8	1		816.2.bq.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
408.2.ba.a	✓	16	3.b	odd	2	1
408.2.ba.a	✓	16	51.g	odd	8	1
816.2.bq.f		16	12.b	even	2	1
816.2.bq.f		16	204.p	even	8	1
1224.2.bq.e		16	1.a	even	1	1	trivial
1224.2.bq.e		16	17.d	even	8	1	inner
6936.2.a.bl		8	51.i	even	16	1
6936.2.a.bo		8	51.i	even	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{16} + 4 T_{5}^{14} - 32 T_{5}^{13} + 8 T_{5}^{12} + 56 T_{5}^{11} + 336 T_{5}^{10} + 72 T_{5}^{9} + \cdots + 1156$ T5^16 + 4*T5^14 - 32*T5^13 + 8*T5^12 + 56*T5^11 + 336*T5^10 + 72*T5^9 + 69*T5^8 - 2920*T5^7 + 14908*T5^6 - 34536*T5^5 + 46856*T5^4 - 41056*T5^3 + 23672*T5^2 - 7888*T5 + 1156 acting on $S_{2}^{\mathrm{new}}(1224, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16}$ T^16
$5$	$T^{16} + 4 T^{14} + \cdots + 1156$ T^16 + 4T^14 - 32T^13 + 8T^12 + 56T^11 + 336T^10 + 72T^9 + 69T^8 - 2920T^7 + 14908T^6 - 34536T^5 + 46856T^4 - 41056T^3 + 23672T^2 - 7888T + 1156
$7$	$T^{16} - 12 T^{14} + \cdots + 64$ T^16 - 12T^14 + 40T^13 + 72T^12 - 848T^11 + 1880T^10 + 144T^9 - 3948T^8 + 3264T^7 + 4608T^6 - 15616T^5 + 17536T^4 - 19456T^3 + 86656T^2 + 1792T + 64
$11$	$T^{16} - 8 T^{15} + \cdots + 73984$ T^16 - 8T^15 + 64T^14 - 312T^13 + 1472T^12 - 5560T^11 + 14896T^10 - 20680T^9 - 13567T^8 + 132160T^7 - 29360T^6 - 271744T^5 + 734336T^4 - 589056T^3 - 91904T^2 + 226304*T + 73984
$13$	$T^{16} + 100 T^{14} + \cdots + 150544$ T^16 + 100T^14 + 3470T^12 + 51356T^10 + 322209T^8 + 956728T^6 + 1360232T^4 + 836704*T^2 + 150544
$17$	$T^{16} + \cdots + 6975757441$ T^16 + 52T^14 + 80T^13 + 1398T^12 + 3328T^11 + 34788T^10 + 63664T^9 + 714594T^8 + 1082288T^7 + 10053732T^6 + 16350464T^5 + 116762358T^4 + 113588560T^3 + 1255153588*T^2 + 6975757441
$19$	$T^{16} + 8 T^{15} + \cdots + 64$ T^16 + 8T^15 + 32T^14 - 88T^13 + 274T^12 + 808T^11 + 1568T^10 - 2360T^9 + 6785T^8 + 16800T^7 + 21760T^6 - 4832T^5 + 10896T^4 + 23808T^3 + 25088T^2 + 1792*T + 64
$23$	$T^{16} - 8 T^{15} + \cdots + 4624$ T^16 - 8T^15 + 56T^14 - 136T^13 + 96T^12 - 312T^11 + 744T^10 + 2760T^9 + 5841T^8 - 132800T^7 + 599552T^6 - 1456800T^5 + 2075648T^4 - 1629888T^3 + 594304T^2 - 50048*T + 4624
$29$	$T^{16} + \cdots + 277155904$ T^16 + 32T^15 + 516T^14 + 5944T^13 + 57096T^12 + 466224T^11 + 3209416T^10 + 17954032T^9 + 79163156T^8 + 271528256T^7 + 720217216T^6 + 1461850112T^5 + 2233053312T^4 + 2509555712T^3 + 2001953664T^2 + 1044695296*T + 277155904
$31$	$T^{16} - 8 T^{15} + \cdots + 75203584$ T^16 - 8T^15 + 136T^14 - 1344T^13 + 12320T^12 - 65280T^11 + 102080T^10 + 1439104T^9 - 4187568T^8 + 14674560T^7 + 299725440T^6 + 220733952T^5 + 683303424T^4 + 1685881856T^3 + 1556046848T^2 + 563888128*T + 75203584
$37$	$T^{16} + \cdots + 8599223824$ T^16 - 8T^15 + 48T^14 - 400T^13 + 2304T^12 - 6032T^11 + 13744T^10 + 100160T^9 + 243144T^8 - 759456T^7 - 112576T^6 - 35504192T^5 + 242301056T^4 - 291706176T^3 + 1013300416T^2 + 7412625152*T + 8599223824
$41$	$T^{16} + 40 T^{15} + \cdots + 6728836$ T^16 + 40T^15 + 796T^14 + 10352T^13 + 97288T^12 + 693272T^11 + 3926112T^10 + 18541816T^9 + 75941797T^8 + 272342416T^7 + 813821092T^6 + 1782130936T^5 + 3774742216T^4 + 7266994416T^3 + 7298827320T^2 + 404684752*T + 6728836
$43$	$T^{16} + \cdots + 4773151744$ T^16 + 24T^15 + 288T^14 + 2600T^13 + 33154T^12 + 495944T^11 + 5734304T^10 + 46840552T^9 + 273397857T^8 + 1145506928T^7 + 3404129408T^6 + 6845948032T^5 + 8434616384T^4 + 4615941632T^3 + 90316800T^2 + 928542720*T + 4773151744
$47$	$T^{16} + \cdots + 33926692864$ T^16 + 320T^14 + 39600T^12 + 2430848T^10 + 78405440T^8 + 1295664128T^6 + 10049284096T^4 + 33415151616*T^2 + 33926692864
$53$	$T^{16} + \cdots + 143598555136$ T^16 - 24T^15 + 288T^14 - 1168T^13 + 7672T^12 - 159136T^11 + 2291840T^10 - 8051648T^9 + 13889552T^8 - 154473472T^7 + 4190633984T^6 - 9943587840T^5 + 6255189504T^4 + 104875491328T^3 + 309237645312T^2 + 298013687808*T + 143598555136
$59$	$T^{16} + \cdots + 29302041354496$ T^16 - 16T^15 + 128T^14 + 80T^13 + 23196T^12 - 353504T^11 + 2690176T^10 + 715552T^9 + 140288516T^8 - 1923928448T^7 + 12964385792T^6 - 8258330752T^5 + 233189495616T^4 - 2728695135232T^3 + 15399918028800T^2 - 30041605647360*T + 29302041354496
$61$	$T^{16} + \cdots + 307387995210256$ T^16 + 72T^14 + 1024T^13 + 2592T^12 - 42400T^11 + 546496T^10 + 36814336T^9 + 758021576T^8 + 9728990848T^7 + 110819820832T^6 + 1067522693632T^5 + 8063181639680T^4 + 44834910822528T^3 + 166621597900288T^2 + 346522673526272T + 307387995210256
$67$	$(T^{8} - 316 T^{6} + \cdots + 2236384)^{2}$ (T^8 - 316T^6 + 304T^5 + 30268T^4 - 62560T^3 - 943776T^2 + 2796288T + 2236384)^2
$71$	$T^{16} + \cdots + 4226040064$ T^16 - 16T^15 + 256T^14 - 544T^13 - 7680T^12 - 44096T^11 + 381952T^10 + 6582144T^9 + 45016096T^8 - 146997504T^7 + 1715838976T^6 - 3044008448T^5 + 7750688768T^4 - 21619622912T^3 + 29688266752T^2 - 18098227200*T + 4226040064
$73$	$T^{16} + \cdots + 130269021184$ T^16 - 24T^15 + 204T^14 + 648T^13 - 29304T^12 + 153472T^11 + 2057512T^10 - 29806992T^9 + 139255940T^8 - 219413696T^7 - 352653376T^6 + 4301922048T^5 + 1301926400T^4 - 12482152448T^3 + 93848671232T^2 - 96601657344*T + 130269021184
$79$	$T^{16} + \cdots + 25305004646464$ T^16 + 96T^15 + 4436T^14 + 130344T^13 + 2703560T^12 + 41381680T^11 + 475248088T^10 + 4096327248T^9 + 26021726228T^8 + 114777377344T^7 + 292889510784T^6 + 67709121280T^5 - 1978040057728T^4 - 3032010739712T^3 + 18849119341696T^2 - 24085593504000*T + 25305004646464
$83$	$T^{16} + \cdots + 797549019136$ T^16 - 160T^13 + 28272T^12 - 23552T^11 + 12800T^10 - 1928960T^9 + 201764672T^8 - 399310848T^7 + 224100352T^6 + 14367764480T^5 + 151126026240T^4 + 150818947072T^3 + 47817162752T^2 - 276175781888*T + 797549019136
$89$	$T^{16} + \cdots + 47876546426944$ T^16 + 1040T^14 + 415148T^12 + 80091216T^10 + 7698035700T^8 + 337001054496T^6 + 4962618377632T^4 + 27801322739584*T^2 + 47876546426944
$97$	$T^{16} + \cdots + 343239946813696$ T^16 - 56T^15 + 1548T^14 - 30584T^13 + 514952T^12 - 7249936T^11 + 79544536T^10 - 482284720T^9 - 1194378748T^8 + 36979900256T^7 - 71415011680T^6 - 1812248538624T^5 + 30098729985152T^4 - 145437369324288T^3 + 532528882314496T^2 - 717986016227328*T + 343239946813696
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 145.4
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