LMFDB - Newform orbit 810.3.g.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,3,Mod(163,810)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("810.163"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$810 = 2 \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	810.g (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,12,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$22.0709014132$
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:	$x^{12} - 54 x^{9} + 921 x^{8} - 1350 x^{7} + 1458 x^{6} - 18792 x^{5} + 231804 x^{4} - 552420 x^{3} + \cdots + 656100$ x^12 - 54x^9 + 921x^8 - 1350x^7 + 1458x^6 - 18792x^5 + 231804x^4 - 552420x^3 + 583200x^2 + 874800*x + 656100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{4}$
Twist minimal:	no (minimal twist has level 90)
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.

$f(q)$	$=$	$q + (\beta_1 + 1) q^{2} + 2 \beta_1 q^{4} + ( - \beta_{7} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{7} + (2 \beta_1 - 2) q^{8} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{10} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{11}+ \cdots + (4 \beta_{11} - 2 \beta_{10} + \cdots - 3) q^{98}+O(q^{100})$ q + (b1 + 1) * q^2 + 2b1 q^4 + (-b7 + b1) * q^5 + (-b5 - b4 + b2 + 1) * q^7 + (2b1 - 2) q^8 + (-b7 + b4 + b1 - 1) * q^10 + (b7 - b6 - b5 - b1 + 1) * q^11 + (-b11 + b10 - b9 + b7 - b5 + b4 - b1) * q^13 + (-b10 + b9 - b7 - b5 - b4 + b2 + b1) * q^14 - 4 * q^16 + (2b9 + b8 + 2b7 + b6 - b5 - b4 + b3 - b2 - 3b1 - 2) q^17 + (-b11 - b10 - b8 - 2b7 - 2b5 - b3 + b2 - 3b1) q^19 + (2b4 - 2) q^20 + (b9 - b8 + b7 - b6 - b5 - b4 + 1) * q^22 + (-b10 + 2b9 - b8 - 2b7 + b6 - b5 + b4 - 4b1 + 3) q^23 + (-3b11 - b10 + b9 + b7 + b6 - b5 + 2b4 - 2b2 - b1 - 6) q^25 + (-b11 + b10 + 2b7 - 2b5 + b3 + b2 - 2b1) q^26 + (-2b10 + 2b9 - 2b7 + 2b1 - 2) * q^28 + (b11 + 4b10 + 2b9 + 2b7 + 2b5 - 2b4 + b3 - 4b2 + 5b1) q^29 + (b11 - b10 + b9 - b6 + b4 - b3 - b2 + 5) * q^31 + (-4b1 - 4) q^32 + (b11 + b10 + 3b9 + 2b8 + b7 + b5 - 3b4 + b3 - b2 - 3b1) * q^34 + (b11 - 2b10 + 2b9 + b8 - 2b7 - b6 + b5 - 3b3 + b2 + 2b1 - 15) q^35 + (b9 + b8 + b7 + b6 - 4b5 - 4b4 + b3 + 3b2 + 3b1 + 7) * q^37 + (-2b11 - 2b10 + 2b9 - b8 - 2b7 + b6 - 2b5 + 2b4 - 3b1 + 1) q^38 + (2b7 + 2b4 - 2b1 - 2) q^40 + (3b11 + b10 + 4b9 - b6 + 4b4 - 3b3 + b2 - 6) * q^41 + (5b11 + 3b10 - 2b9 + 2b7 - 2b5 + 2b4 + 7b1 - 9) q^43 + (2b9 - 2b8 - 2b4 + 2b1) * q^44 + (-b10 + 3b9 - b7 + 2b6 + b5 + 3b4 - b2 + b1 + 6) q^46 + (2b9 - b8 + 2b7 - b6 - 2b5 - 2b4 - 5b3 + 8b2 - 6b1 - 4) q^47 + (2b11 - b10 + 2b9 + b8 - 4b7 - 4b5 - 2b4 + 2b3 + b2 - b1) * q^49 + (-3b11 + b10 + 2b9 + b8 + 3b7 + b6 + b4 + 3b3 - 3b2 - 6b1 - 6) * q^50 + (2b9 + 2b7 - 2b5 - 2b4 + 2b3 + 2b2 - 2b1) q^52 + (2b11 + 4b10 + 5b9 - 5b7 + b5 - b4 - 13b1 + 14) q^53 + (8b11 + 7b10 + b9 - 2b6 + 3b5 - b4 + b3 - 6b2 - 8b1 - 23) * q^55 + (-2b10 + 2b9 - 2b7 + 2b5 + 2b4 - 2b2 + 2b1 - 4) q^56 + (2b11 + 8b10 + 4b5 - 4b4 + 7b1 - 3) q^58 + (-4b11 - b10 + 3b9 - 3b4 - 4b3 + b2 + 26b1) q^59 + (b11 - b10 - b6 - b3 - b2 + 38) * q^61 + (b9 - b8 + b7 - b6 + b5 + b4 - 2b3 - 2b2 + 6b1 + 5) q^62 - 8b1 q^64 + (3b11 + 9b9 + b8 + b7 + 2b6 - 4b5 - 3b4 + 5b3 - 25) * q^65 + (-b9 + 3b8 - b7 + 3b6 - b5 - b4 - b3 - 4b2 - b1) q^67 + (2b11 + 2b10 + 2b9 + 2b8 - 2b7 - 2b6 + 4b5 - 4b4 + 4) * q^68 + (-2b11 - 3b10 + b9 - 2b7 - 2b6 + 3b5 + 2b4 - 4b3 - b2 - 11b1 - 16) * q^70 + (-5b11 - 2b10 + 10b9 + b6 + 10b4 + 5b3 - 2b2 - 17) * q^71 + (-6b11 - 11b9 + b8 + 11b7 - b6 - b5 + b4 + 12b1 - 13) * q^73 + (b11 - 3b10 + 5b9 + 2b8 - 3b7 - 3b5 - 5b4 + b3 + 3b2 + 11b1) * q^74 + (-2b11 - 2b10 + 4b9 + 2b6 + 4b4 + 2b3 - 2b2 + 2) q^76 + (10b9 + 10b7 + 15b3 - 7b2 - b1 - 1) * q^77 + (-2b11 - 4b10 - 3b9 - 4b8 - 6b7 - 6b5 + 3b4 - 2b3 + 4b2 + 2b1) * q^79 + (4b7 - 4b1) * q^80 + (4b9 - b8 + 4b7 - b6 + 4b5 + 4b4 - 6b3 + 2b2 - 2b1 - 6) q^82 + (-4b11 - b10 + 6b9 + 2b8 - 6b7 - 2b6 + 3b5 - 3b4 - 16b1 + 19) * q^83 + (-13b10 - 3b9 - b8 - 6b7 + b6 + 2b5 - b4 + b3 + 4b2 - 23b1 - 25) * q^85 + (5b11 + 3b10 + 4b7 - 4b5 - 5b3 + 3b2 - 4b1 - 18) q^86 + (2b9 - 2b8 - 2b7 + 2b6 + 2b5 - 2b4 + 4b1 - 2) q^88 + (2b11 - 6b10 + 3b9 - 4b7 - 4b5 - 3b4 + 2b3 + 6b2 + 31b1) q^89 + (-5b11 - 3b10 - 4b9 + 7b7 - 3b6 - 7b5 - 4b4 + 5b3 - 3b2 - 7b1 + 55) * q^91 + (2b9 + 2b8 + 2b7 + 2b6 + 4b5 + 4b4 - 2b2 + 10b1 + 6) * q^92 + (-5b11 - 8b10 + 4b9 - 2b8 - 4b4 - 5b3 + 8b2 - 8b1) * q^94 + (-5b11 + 2b10 + 5b9 - 2b8 + b7 - b6 - 5b5 - 5b4 - 5b3 + 4b2 - 8b1 - 36) q^95 + (-11b9 - 2b8 - 11b7 - 2b6 + 4b5 + 4b4 + 6b3 - 6b2 + 16b1 + 12) q^97 + (4b11 - 2b10 + 6b9 + b8 - 6b7 - b6 - 2b5 + 2b4 + b1 - 3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 12 q^{2} + 6 q^{7} - 24 q^{8} - 6 q^{10} + 12 q^{11} - 48 q^{16} - 18 q^{17} - 12 q^{20} + 12 q^{22} + 54 q^{23} - 54 q^{25} - 12 q^{28} + 72 q^{31} - 48 q^{32} - 168 q^{35} + 66 q^{37} + 36 q^{38}+ \cdots + 12 q^{98}+O(q^{100})$ 12 * q + 12 * q^2 + 6 * q^7 - 24 * q^8 - 6 * q^10 + 12 * q^11 - 48 * q^16 - 18 * q^17 - 12 * q^20 + 12 * q^22 + 54 * q^23 - 54 * q^25 - 12 * q^28 + 72 * q^31 - 48 * q^32 - 168 * q^35 + 66 * q^37 + 36 * q^38 - 12 * q^40 - 24 * q^41 - 108 * q^43 + 108 * q^46 - 48 * q^47 - 54 * q^50 + 192 * q^53 - 276 * q^55 - 24 * q^56 - 60 * q^58 + 456 * q^61 + 72 * q^62 - 264 * q^65 - 12 * q^67 + 36 * q^68 - 174 * q^70 - 84 * q^71 - 216 * q^73 + 72 * q^76 + 48 * q^77 - 24 * q^82 + 246 * q^83 - 324 * q^85 - 216 * q^86 - 24 * q^88 + 612 * q^91 + 108 * q^92 - 432 * q^95 + 102 * q^97 + 12 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 54 x^{9} + 921 x^{8} - 1350 x^{7} + 1458 x^{6} - 18792 x^{5} + 231804 x^{4} - 552420 x^{3} + \cdots + 656100$ x^12 - 54*x^9 + 921*x^8 - 1350*x^7 + 1458*x^6 - 18792*x^5 + 231804*x^4 - 552420*x^3 + 583200*x^2 + 874800*x + 656100 :

$\beta_{1}$	$=$	$( - 1875243545 \nu^{11} + 2732664981 \nu^{10} + 2774149290 \nu^{9} + 104197645485 \nu^{8} + \cdots - 866523189915450 ) / 20\!\cdots\!70$ (-1875243545v^11 + 2732664981v^10 + 2774149290v^9 + 104197645485v^8 - 1873864921524v^7 + 4981515908166v^6 - 3891751338015v^5 + 38281615930188v^4 - 488606626316952v^3 + 1738554979846284v^2 - 1976060739252510*v - 866523189915450) / 2035536381234270
$\beta_{2}$	$=$	$( 801101591 \nu^{11} - 10289055060 \nu^{10} - 84332616198 \nu^{9} - 293504336784 \nu^{8} + \cdots - 514673191072440 ) / 452341418052060$ (801101591v^11 - 10289055060v^10 - 84332616198v^9 - 293504336784v^8 + 1034227795947v^7 - 4340847960756v^6 - 36538823429478v^5 - 96577053626916v^4 + 302362724947734v^3 - 635889801951828v^2 - 3374651669136840*v - 514673191072440) / 452341418052060
$\beta_{3}$	$=$	$( - 801101591 \nu^{11} + 10289055060 \nu^{10} + 84332616198 \nu^{9} + \cdots + 514673191072440 ) / 452341418052060$ (-801101591v^11 + 10289055060v^10 + 84332616198v^9 + 293504336784v^8 - 1034227795947v^7 + 4340847960756v^6 + 36538823429478v^5 + 96577053626916v^4 - 302362724947734v^3 + 635889801951828v^2 + 4731675923293020*v + 514673191072440) / 452341418052060
$\beta_{4}$	$=$	$( 10813781488 \nu^{11} - 18832079700 \nu^{10} - 253052027046 \nu^{9} - 1619639566146 \nu^{8} + \cdots - 13\!\cdots\!70 ) / 20\!\cdots\!70$ (10813781488v^11 - 18832079700v^10 - 253052027046v^9 - 1619639566146v^8 + 8313264642702v^7 - 18728224777575v^6 - 112486169698992v^5 - 595414451140695v^4 + 1963277075631168v^3 - 4781616191780274v^2 - 10660201940923830*v - 13941613256310270) / 2035536381234270
$\beta_{5}$	$=$	$( 3622731026 \nu^{11} - 25387385657 \nu^{10} - 103969950084 \nu^{9} - 404288852031 \nu^{8} + \cdots + 809857776786660 ) / 678512127078090$ (3622731026v^11 - 25387385657v^10 - 103969950084v^9 - 404288852031v^8 + 5012953239183v^7 - 19655990449719v^6 - 33937589503557v^5 - 156345435879993v^4 + 1453610115391716v^3 - 4792172232064266v^2 + 834168523429260*v + 809857776786660) / 678512127078090
$\beta_{6}$	$=$	$( - 16157749007 \nu^{11} - 16169837742 \nu^{10} + 245744928891 \nu^{9} + \cdots + 42\!\cdots\!00 ) / 20\!\cdots\!70$ (-16157749007v^11 - 16169837742v^10 + 245744928891v^9 + 2957021682174v^8 - 7713673859184v^7 + 84861549207v^6 + 63915908241492v^5 + 1087697114716545v^4 - 1588967847139608v^3 + 2479719205493370v^2 + 3365547885870840*v + 42423545727242100) / 2035536381234270
$\beta_{7}$	$=$	$( - 5903758925 \nu^{11} - 33865345891 \nu^{10} - 67696741206 \nu^{9} + \cdots + 10\!\cdots\!90 ) / 678512127078090$ (-5903758925v^11 - 33865345891v^10 - 67696741206v^9 + 347221587906v^8 - 2763963071277v^7 - 15956005799406v^6 - 14930684399106v^5 + 140717902199823v^4 - 348415585518852v^3 - 1979731934181828v^2 + 3303539986850910*v + 1007919072636390) / 678512127078090
$\beta_{8}$	$=$	$( - 24025140695 \nu^{11} + 169173040263 \nu^{10} + 438081208245 \nu^{9} + \cdots - 17\!\cdots\!00 ) / 20\!\cdots\!70$ (-24025140695v^11 + 169173040263v^10 + 438081208245v^9 + 1746363892779v^8 - 34924088358249v^7 + 134503746792381v^6 + 29616968614149v^5 + 630410445269325v^4 - 9875460177547812v^3 + 33458088034208070v^2 - 39403384557775110*v - 17294039977500900) / 2035536381234270
$\beta_{9}$	$=$	$( 28593720413 \nu^{11} + 60064310034 \nu^{10} - 33351976611 \nu^{9} - 2346815029725 \nu^{8} + \cdots - 98\!\cdots\!70 ) / 20\!\cdots\!70$ (28593720413v^11 + 60064310034v^10 - 33351976611v^9 - 2346815029725v^8 + 19824337365018v^7 + 12716029565406v^6 - 8816718099276v^5 - 870310936875519v^4 + 4144343169038616v^3 - 3848717489839236v^2 - 1323689487856290*v - 9850536006124770) / 2035536381234270
$\beta_{10}$	$=$	$( 25717796021 \nu^{11} + 60311648106 \nu^{10} + 62594041086 \nu^{9} - 1808354936952 \nu^{8} + \cdots + 34\!\cdots\!80 ) / 13\!\cdots\!80$ (25717796021v^11 + 60311648106v^10 + 62594041086v^9 - 1808354936952v^8 + 17343352267269v^7 + 8386289534322v^6 + 27037590897042v^5 - 641292252822288v^4 + 3582765378266994v^3 - 4392386617669956v^2 + 1361235364554240*v + 3485658627091080) / 1357024254156180
$\beta_{11}$	$=$	$( 31183125983 \nu^{11} + 65859946686 \nu^{10} + 68463029196 \nu^{9} - 2101886170110 \nu^{8} + \cdots + 59\!\cdots\!80 ) / 13\!\cdots\!80$ (31183125983v^11 + 65859946686v^10 + 68463029196v^9 - 2101886170110v^8 + 22243226512101v^7 + 6070997035512v^6 + 33121669362138v^5 - 749127596045832v^4 + 4988031259701762v^3 - 6157224025286976v^2 + 2909115091055340*v + 5946353206840080) / 1357024254156180

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

$\nu$	$=$	$( \beta_{3} + \beta_{2} ) / 3$ (b3 + b2) / 3
$\nu^{2}$	$=$	$\beta_{10} + \beta_{7} + \beta_{5} - \beta_{2} + 15\beta_1$ b10 + b7 + b5 - b2 + 15*b1
$\nu^{3}$	$=$	$6\beta_{11} - 8\beta_{10} - 3\beta_{5} + 3\beta_{4} - 15\beta _1 + 12$ 6b11 - 8b10 - 3b5 + 3b4 - 15*b1 + 12
$\nu^{4}$	$=$	$- 5 \beta_{11} + 29 \beta_{10} - 29 \beta_{9} - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 29 \beta_{4} + \cdots - 278$ -5b11 + 29b10 - 29b9 - 4b7 - 4b6 + 4b5 - 29b4 + 5b3 + 29b2 + 4b1 - 278
$\nu^{5}$	$=$	$126 \beta_{9} + 9 \beta_{8} + 126 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} - 128 \beta_{3} + \cdots + 495$ 126b9 + 9b8 + 126b7 + 9b6 + 9b5 + 9b4 - 128b3 - 200b2 + 504*b1 + 495
$\nu^{6}$	$=$	$207 \beta_{11} - 789 \beta_{10} - 261 \beta_{9} - 180 \beta_{8} - 834 \beta_{7} - 834 \beta_{5} + \cdots - 7074 \beta_1$ 207b11 - 789b10 - 261b9 - 180b8 - 834b7 - 834b5 + 261b4 + 207b3 + 789b2 - 7074b1
$\nu^{7}$	$=$	$- 2988 \beta_{11} + 5340 \beta_{10} - 540 \beta_{9} + 513 \beta_{8} + 540 \beta_{7} - 513 \beta_{6} + \cdots - 15597$ -2988b11 + 5340b10 - 540b9 + 513b8 + 540b7 - 513b6 + 4365b5 - 4365b4 + 19962*b1 - 15597
$\nu^{8}$	$=$	$6771 \beta_{11} - 21387 \beta_{10} + 24357 \beta_{9} + 11076 \beta_{7} + 6378 \beta_{6} - 11076 \beta_{5} + \cdots + 152454$ 6771b11 - 21387b10 + 24357b9 + 11076b7 + 6378b6 - 11076b5 + 24357b4 - 6771b3 - 21387b2 - 11076b1 + 152454
$\nu^{9}$	$=$	$- 141048 \beta_{9} - 20817 \beta_{8} - 141048 \beta_{7} - 20817 \beta_{6} - 21303 \beta_{5} + \cdots - 456705$ -141048b9 - 20817b8 - 141048b7 - 20817b6 - 21303b5 - 21303b4 + 74442b3 + 147666b2 - 478008*b1 - 456705
$\nu^{10}$	$=$	$- 206037 \beta_{11} + 586557 \beta_{10} + 397683 \beta_{9} + 208872 \beta_{8} + 718722 \beta_{7} + \cdots + 4667490 \beta_1$ -206037b11 + 586557b10 + 397683b9 + 208872b8 + 718722b7 + 718722b5 - 397683b4 - 206037b3 - 586557b2 + 4667490b1
$\nu^{11}$	$=$	$1946268 \beta_{11} - 4162068 \beta_{10} + 717012 \beta_{9} - 735885 \beta_{8} - 717012 \beta_{7} + \cdots + 13089789$ 1946268b11 - 4162068b10 + 717012b9 - 735885b8 - 717012b7 + 735885b6 - 4405131b5 + 4405131b4 - 17494920*b1 + 13089789

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$ .

$n$	$487$	$731$
$\chi(n)$	$\beta_{1}$	$1$

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}

163.1

−3.83690	+	3.83690i
2.21548	−	2.21548i
2.49352	−	2.49352i
−0.509217	+	0.509217i
−3.24958	+	3.24958i
2.88670	−	2.88670i
−3.83690	−	3.83690i
2.21548	+	2.21548i
2.49352	+	2.49352i
−0.509217	−	0.509217i
−3.24958	−	3.24958i
2.88670	+	2.88670i

1.00000

−

1.00000i

−

2.00000i

−4.31004

2.53448i

−2.41180

2.41180i

−2.00000

−

2.00000i

−1.77556

6.84452i

163.2

1.00000

−

1.00000i

−

2.00000i

−4.07172

−

2.90192i

7.16337

−

7.16337i

−2.00000

−

2.00000i

−6.97363

1.16980i

163.3

1.00000

−

1.00000i

−

2.00000i

0.830091

4.93061i

0.952249

−

0.952249i

−2.00000

−

2.00000i

5.76070

4.10052i

163.4

1.00000

−

1.00000i

−

2.00000i

1.18786

−

4.85685i

6.55284

−

6.55284i

−2.00000

−

2.00000i

−3.66899

−

6.04471i

163.5

1.00000

−

1.00000i

−

2.00000i

1.76846

−

4.67681i

−4.32826

4.32826i

−2.00000

−

2.00000i

−2.90835

−

6.44527i

163.6

1.00000

−

1.00000i

−

2.00000i

4.59534

1.97049i

−4.92840

4.92840i

−2.00000

−

2.00000i

6.56583

−

2.62486i

487.1

1.00000

1.00000i

2.00000i

−4.31004

−

2.53448i

−2.41180

−

2.41180i

−2.00000

2.00000i

−1.77556

−

6.84452i

487.2

1.00000

1.00000i

2.00000i

−4.07172

2.90192i

7.16337

7.16337i

−2.00000

2.00000i

−6.97363

−

1.16980i

487.3

1.00000

1.00000i

2.00000i

0.830091

−

4.93061i

0.952249

0.952249i

−2.00000

2.00000i

5.76070

−

4.10052i

487.4

1.00000

1.00000i

2.00000i

1.18786

4.85685i

6.55284

6.55284i

−2.00000

2.00000i

−3.66899

6.04471i

487.5

1.00000

1.00000i

2.00000i

1.76846

4.67681i

−4.32826

−

4.32826i

−2.00000

2.00000i

−2.90835

6.44527i

487.6

1.00000

1.00000i

2.00000i

4.59534

−

1.97049i

−4.92840

−

4.92840i

−2.00000

2.00000i

6.56583

2.62486i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	810.3.g.k		12
3.b	odd	2	1		810.3.g.i		12
5.c	odd	4	1	inner	810.3.g.k		12
9.c	even	3	2		90.3.k.a	✓	24
9.d	odd	6	2		270.3.l.b		24
15.e	even	4	1		810.3.g.i		12
45.k	odd	12	2		90.3.k.a	✓	24
45.l	even	12	2		270.3.l.b		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.3.k.a	✓	24	9.c	even	3	2
90.3.k.a	✓	24	45.k	odd	12	2
270.3.l.b		24	9.d	odd	6	2
270.3.l.b		24	45.l	even	12	2
810.3.g.i		12	3.b	odd	2	1
810.3.g.i		12	15.e	even	4	1
810.3.g.k		12	1.a	even	1	1	trivial
810.3.g.k		12	5.c	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(810, [\chi])$ :

T_{7}^{12} - 6 T_{7}^{11} + 18 T_{7}^{10} + 544 T_{7}^{9} + 7479 T_{7}^{8} - 12756 T_{7}^{7} + \cdots + 338449609

T7^12 - 6*T7^11 + 18*T7^10 + 544*T7^9 + 7479*T7^8 - 12756*T7^7 + 89882*T7^6 + 2545176*T7^5 + 23901939*T7^4 + 53701934*T7^3 + 41259528*T7^2 - 167118348*T7 + 338449609

T_{11}^{6} - 6T_{11}^{5} - 621T_{11}^{4} + 5228T_{11}^{3} + 87432T_{11}^{2} - 913818T_{11} + 1018990

T11^6 - 6*T11^5 - 621*T11^4 + 5228*T11^3 + 87432*T11^2 - 913818*T11 + 1018990

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} - 2 T + 2)^{6}$ (T^2 - 2*T + 2)^6
$3$	$T^{12}$ T^12
$5$	$T^{12} + \cdots + 244140625$ T^12 + 27T^10 + 114T^9 + 327T^8 - 2034T^7 + 22850T^6 - 50850T^5 + 204375T^4 + 1781250T^3 + 10546875*T^2 + 244140625
$7$	$T^{12} + \cdots + 338449609$ T^12 - 6T^11 + 18T^10 + 544T^9 + 7479T^8 - 12756T^7 + 89882T^6 + 2545176T^5 + 23901939T^4 + 53701934T^3 + 41259528T^2 - 167118348*T + 338449609
$11$	$(T^{6} - 6 T^{5} + \cdots + 1018990)^{2}$ (T^6 - 6T^5 - 621T^4 + 5228T^3 + 87432T^2 - 913818*T + 1018990)^2
$13$	$T^{12} + \cdots + 3000191076$ T^12 + 1338T^9 + 160905T^8 + 478890T^7 + 895122T^6 + 27993096T^5 + 540336348T^4 + 2891553660T^3 + 8092973088T^2 + 6968567376*T + 3000191076
$17$	$T^{12} + \cdots + 9106503184$ T^12 + 18T^11 + 162T^10 - 8240T^9 + 731928T^8 + 10094172T^7 + 97071560T^6 - 4818641088T^5 + 76520325492T^4 - 355012584880T^3 + 815844722688T^2 + 121897436928*T + 9106503184
$19$	$T^{12} + \cdots + 87379360000$ T^12 + 1632T^10 + 783600T^8 + 142221884T^6 + 8347121124T^4 + 76425792384*T^2 + 87379360000
$23$	$T^{12} + \cdots + 34464707542225$ T^12 - 54T^11 + 1458T^10 - 2312T^9 + 130275T^8 - 11094768T^7 + 411849194T^6 + 1700766072T^5 + 14992518159T^4 - 621416734318T^3 + 15658525757448T^2 + 32853204129060*T + 34464707542225
$29$	$T^{12} + \cdots + 22\!\cdots\!61$ T^12 + 7770T^10 + 22113471T^8 + 28455692156T^6 + 16656579297951T^4 + 3600640152298890*T^2 + 22777957486589761
$31$	$(T^{6} - 36 T^{5} + \cdots - 18709310)^{2}$ (T^6 - 36T^5 - 597T^4 + 23530T^3 + 138432T^2 - 3677034*T - 18709310)^2
$37$	$T^{12} + \cdots + 88\!\cdots\!16$ T^12 - 66T^11 + 2178T^10 + 36432T^9 + 2307708T^8 - 152340804T^7 + 5691950352T^6 + 128608059096T^5 + 1509932591748T^4 - 2735594004528T^3 + 45865747227168T^2 + 901533304403424*T + 8860231742470416
$41$	$(T^{6} + 12 T^{5} + \cdots + 2668453777)^{2}$ (T^6 + 12T^5 - 5673T^4 - 141274T^3 + 6707571T^2 + 279113538*T + 2668453777)^2
$43$	$T^{12} + \cdots + 38\!\cdots\!84$ T^12 + 108T^11 + 5832T^10 - 33954T^9 + 5178117T^8 + 507211794T^7 + 25156532466T^6 - 16768659672T^5 - 270512548632T^4 + 100058277081348T^3 + 11321719742013192T^2 - 93100368414639384*T + 382789840962814884
$47$	$T^{12} + \cdots + 26\!\cdots\!25$ T^12 + 48T^11 + 1152T^10 - 9018T^9 + 44347167T^8 + 2386826352T^7 + 63520390674T^6 - 2454135217488T^5 + 60777310787091T^4 + 483716039216004T^3 + 6854943422639538T^2 - 190879795866238650*T + 2657578204774400625
$53$	$T^{12} + \cdots + 23\!\cdots\!00$ T^12 - 192T^11 + 18432T^10 - 647392T^9 + 8825892T^8 - 171893472T^7 + 79882906112T^6 - 1951103452992T^5 + 17363153836416T^4 + 1120728970616320T^3 + 37646933635699200T^2 - 13243202996966400*T + 2329305585894400
$59$	$T^{12} + \cdots + 27\!\cdots\!44$ T^12 + 18222T^10 + 100471713T^8 + 215326823420T^6 + 166126216380792T^4 + 39408471950759808*T^2 + 27979925998528144
$61$	$(T^{6} - 228 T^{5} + \cdots + 1299208905)^{2}$ (T^6 - 228T^5 + 20583T^4 - 938946T^3 + 22738899T^2 - 275714334*T + 1299208905)^2
$67$	$T^{12} + \cdots + 37\!\cdots\!25$ T^12 + 12T^11 + 72T^10 - 524246T^9 + 141647631T^8 - 4951301304T^7 + 67802689178T^6 + 4884857775888T^5 + 419019460525731T^4 - 3807402078093616T^3 + 16876628329856418T^2 - 35614482413892990*T + 37578340081282225
$71$	$(T^{6} + 42 T^{5} + \cdots + 910683616)^{2}$ (T^6 + 42T^5 - 21156T^4 - 439454T^3 + 101355414T^2 - 709031088*T + 910683616)^2
$73$	$T^{12} + \cdots + 87\!\cdots\!56$ T^12 + 216T^11 + 23328T^10 + 2180648T^9 + 479677332T^8 + 89256135456T^7 + 10467025307552T^6 + 771097340824032T^5 + 36512298115178496T^4 + 1029400713360790528T^3 + 14617622495177884800T^2 + 5056729792660181760*T + 874646892968141056
$79$	$T^{12} + \cdots + 14\!\cdots\!04$ T^12 + 19494T^10 + 128591025T^8 + 366202975500T^6 + 444194211002400T^4 + 180292551752151936*T^2 + 14764327877994813504
$83$	$T^{12} + \cdots + 56\!\cdots\!29$ T^12 - 246T^11 + 30258T^10 - 1404732T^9 + 57516735T^8 - 8342302284T^7 + 1298500990146T^6 - 49631039459568T^5 + 1039236329382291T^4 - 55389762841188546T^3 + 12010444477951612032T^2 - 368416532005587322416*T + 5650529474749834615329
$89$	$T^{12} + \cdots + 10\!\cdots\!00$ T^12 + 25716T^10 + 244274910T^8 + 1101989485004T^6 + 2489043328047993T^4 + 2678684879715686376*T^2 + 1069896782237162890000
$97$	$T^{12} + \cdots + 52\!\cdots\!84$ T^12 - 102T^11 + 5202T^10 + 653346T^9 + 697866681T^8 - 64391234436T^7 + 3151033935768T^6 + 328467304327896T^5 + 15580579114678176T^4 + 72117760467168576T^3 + 310556914381246752T^2 + 57006572477475992256*T + 5232131624092994616384
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Overview	Random
Universe	Knowledge

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 163.6
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	810.3.g.k

Level	$810$
Weight	$3$
Character orbit	810.g
Analytic conductor	$22.071$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$