Properties

 Label 2736.2.bm.t Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.bm (of order $$6$$, 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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 28x^{14} + 542x^{12} + 5488x^{10} + 40451x^{8} + 151312x^{6} + 395134x^{4} + 52164x^{2} + 6561$$ x^16 + 28*x^14 + 542*x^12 + 5488*x^10 + 40451*x^8 + 151312*x^6 + 395134*x^4 + 52164*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{8} q^{5} + (\beta_{5} + \beta_{3}) q^{7}+O(q^{10})$$ q - b8 * q^5 + (b5 + b3) * q^7 $$q - \beta_{8} q^{5} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{12} + \beta_{11}) q^{11} + (\beta_{6} + \beta_1) q^{13} + ( - \beta_{10} - \beta_{9} + \beta_{8}) q^{17} + \beta_{7} q^{19} - \beta_{15} q^{23} + (\beta_{6} - \beta_1) q^{25} + 2 \beta_{10} q^{29} + ( - \beta_{4} + \beta_{3}) q^{31} + (\beta_{15} - \beta_{14}) q^{35} + (2 \beta_{2} + 1) q^{37} + (2 \beta_{13} - \beta_{10} + \beta_{9} + \beta_{8}) q^{41} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{15} + \beta_{11}) q^{47} + ( - \beta_1 - 1) q^{49} + ( - 2 \beta_{13} + \beta_{10} + 2 \beta_{8}) q^{53} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{55} + (\beta_{12} - 2 \beta_{11}) q^{59} + ( - \beta_{6} - 2 \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{13} + 3 \beta_{9} - 4 \beta_{8}) q^{65} + (2 \beta_{7} + \beta_{3}) q^{67} + (\beta_{15} + \beta_{14} - \beta_{12} + 2 \beta_{11}) q^{71} + ( - 4 \beta_{6} - 3 \beta_{2}) q^{73} + ( - \beta_{13} - 4 \beta_{10} + 2 \beta_{9}) q^{77} + ( - \beta_{7} + 3 \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{79} + (\beta_{14} + \beta_{12} - \beta_{11}) q^{83} + ( - 4 \beta_{6} + 2 \beta_{2} + 4 \beta_1 + 2) q^{85} + (\beta_{13} + 2 \beta_{10} - \beta_{8}) q^{89} + (3 \beta_{7} + \beta_{5} + 2 \beta_{3}) q^{91} + (\beta_{15} - \beta_{12} - \beta_{11}) q^{95} + (3 \beta_{6} - 3 \beta_{2} - 6 \beta_1 - 6) q^{97}+O(q^{100})$$ q - b8 * q^5 + (b5 + b3) * q^7 + (-b12 + b11) * q^11 + (b6 + b1) * q^13 + (-b10 - b9 + b8) * q^17 + b7 * q^19 - b15 * q^23 + (b6 - b1) * q^25 + 2*b10 * q^29 + (-b4 + b3) * q^31 + (b15 - b14) * q^35 + (2*b2 + 1) * q^37 + (2*b13 - b10 + b9 + b8) * q^41 + (b7 - b5 - b4 + b3) * q^43 + (-b15 + b11) * q^47 + (-b1 - 1) * q^49 + (-2*b13 + b10 + 2*b8) * q^53 + (b7 - b4 + b3) * q^55 + (b12 - 2*b11) * q^59 + (-b6 - 2*b2 + b1 - 2) * q^61 + (-2*b13 + 3*b9 - 4*b8) * q^65 + (2*b7 + b3) * q^67 + (b15 + b14 - b12 + 2*b11) * q^71 + (-4*b6 - 3*b2) * q^73 + (-b13 - 4*b10 + 2*b9) * q^77 + (-b7 + 3*b5 - b4 + 5*b3) * q^79 + (b14 + b12 - b11) * q^83 + (-4*b6 + 2*b2 + 4*b1 + 2) * q^85 + (b13 + 2*b10 - b8) * q^89 + (3*b7 + b5 + 2*b3) * q^91 + (b15 - b12 - b11) * q^95 + (3*b6 - 3*b2 - 6*b1 - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q - 16 q^{49} - 16 q^{61} + 24 q^{73} + 16 q^{85} - 72 q^{97}+O(q^{100})$$ 16 * q - 16 * q^49 - 16 * q^61 + 24 * q^73 + 16 * q^85 - 72 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 28x^{14} + 542x^{12} + 5488x^{10} + 40451x^{8} + 151312x^{6} + 395134x^{4} + 52164x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( - 10164 \nu^{14} - 255509 \nu^{12} - 4754946 \nu^{10} - 41215272 \nu^{8} - 266357336 \nu^{6} - 462486696 \nu^{4} - 61069302 \nu^{2} + \cdots + 4905120953 ) / 1856785928$$ (-10164*v^14 - 255509*v^12 - 4754946*v^10 - 41215272*v^8 - 266357336*v^6 - 462486696*v^4 - 61069302*v^2 + 4905120953) / 1856785928 $$\beta_{2}$$ $$=$$ $$( - 3806323948 \nu^{14} - 105837336802 \nu^{12} - 2044580895560 \nu^{10} - 20543041875661 \nu^{8} - 150969971356232 \nu^{6} + \cdots - 194108471710191 ) / 184747182558867$$ (-3806323948*v^14 - 105837336802*v^12 - 2044580895560*v^10 - 20543041875661*v^8 - 150969971356232*v^6 - 555207640209094*v^4 - 1470348324238444*v^2 - 194108471710191) / 184747182558867 $$\beta_{3}$$ $$=$$ $$( - 96213205709 \nu^{14} - 2754784164545 \nu^{12} - 53992628470114 \nu^{10} - 561812335230251 \nu^{8} + \cdots - 11\!\cdots\!28 ) / 29\!\cdots\!72$$ (-96213205709*v^14 - 2754784164545*v^12 - 53992628470114*v^10 - 561812335230251*v^8 - 4226571858284230*v^6 - 16618957602281825*v^4 - 44301633706386407*v^2 - 11417555800756728) / 2955954920941872 $$\beta_{4}$$ $$=$$ $$( - 120384307763 \nu^{14} - 3391806056156 \nu^{12} - 65300409366145 \nu^{10} - 659826753342743 \nu^{8} + \cdots - 15\!\cdots\!11 ) / 29\!\cdots\!72$$ (-120384307763*v^14 - 3391806056156*v^12 - 65300409366145*v^10 - 659826753342743*v^8 - 4778910298762015*v^6 - 17718801475523981*v^4 - 44446863176182604*v^2 - 15031274091976611) / 2955954920941872 $$\beta_{5}$$ $$=$$ $$( - 132220047635 \nu^{14} - 3686953004252 \nu^{12} - 70837432581553 \nu^{10} - 707820971971799 \nu^{8} + \cdots + 52\!\cdots\!45 ) / 29\!\cdots\!72$$ (-132220047635*v^14 - 3686953004252*v^12 - 70837432581553*v^10 - 707820971971799*v^8 - 5110667882932927*v^6 - 18257356397613389*v^4 - 44517976947595100*v^2 + 5260837691851245) / 2955954920941872 $$\beta_{6}$$ $$=$$ $$( 77189178416 \nu^{14} + 2176005166862 \nu^{12} + 42180465284398 \nu^{10} + 430495021839779 \nu^{8} + \cdots + 41\!\cdots\!01 ) / 14\!\cdots\!36$$ (77189178416*v^14 + 2176005166862*v^12 + 42180465284398*v^10 + 430495021839779*v^8 + 3182021455043788*v^6 + 12113792985422063*v^4 + 31169326351674140*v^2 + 4114868769414801) / 1477977460470936 $$\beta_{7}$$ $$=$$ $$( - 65958342944 \nu^{14} - 1804014851009 \nu^{12} - 34720999750351 \nu^{10} - 343729127623994 \nu^{8} + \cdots + 17\!\cdots\!65 ) / 985318306980624$$ (-65958342944*v^14 - 1804014851009*v^12 - 34720999750351*v^10 - 343729127623994*v^8 - 2523799016893609*v^6 - 9102313901174438*v^4 - 24301148913260783*v^2 + 1702751241319065) / 985318306980624 $$\beta_{8}$$ $$=$$ $$( - 990175670 \nu^{15} - 25195594535 \nu^{13} - 463226273255 \nu^{11} - 4015186891660 \nu^{9} - 25480512543977 \nu^{7} + \cdots + 831702838632193 \nu ) / 109479811886736$$ (-990175670*v^15 - 25195594535*v^13 - 463226273255*v^11 - 4015186891660*v^9 - 25480512543977*v^7 - 45055398866380*v^5 - 5949364130685*v^3 + 831702838632193*v) / 109479811886736 $$\beta_{9}$$ $$=$$ $$( - 2834183765 \nu^{15} - 78805865495 \nu^{13} - 1518941639794 \nu^{11} - 15250342530713 \nu^{9} - 111711718597870 \nu^{7} + \cdots - 281697264791856 \nu ) / 83923010373744$$ (-2834183765*v^15 - 78805865495*v^13 - 1518941639794*v^11 - 15250342530713*v^9 - 111711718597870*v^7 - 414448034891111*v^5 - 1093376918915915*v^3 - 281697264791856*v) / 83923010373744 $$\beta_{10}$$ $$=$$ $$( 909972638167 \nu^{15} + 25176125585668 \nu^{13} + 486901477983041 \nu^{11} + \cdots - 41\!\cdots\!95 \nu ) / 26\!\cdots\!48$$ (909972638167*v^15 + 25176125585668*v^13 + 486901477983041*v^11 + 4881138908066497*v^9 + 36022974087467963*v^7 + 131904960593157655*v^5 + 346669798424140396*v^3 - 41317391409520995*v) / 26603594288476848 $$\beta_{11}$$ $$=$$ $$( - 1890306552809 \nu^{15} - 52424270976620 \nu^{13} + \cdots + 77\!\cdots\!89 \nu ) / 26\!\cdots\!48$$ (-1890306552809*v^15 - 52424270976620*v^13 - 1009971126798727*v^11 - 10090741797306707*v^9 - 73463991194917621*v^7 - 263761122113038781*v^5 - 655064719310154740*v^3 + 77578071609638889*v) / 26603594288476848 $$\beta_{12}$$ $$=$$ $$( - 3394373004184 \nu^{15} - 95065139318341 \nu^{13} + \cdots - 90\!\cdots\!71 \nu ) / 26\!\cdots\!48$$ (-3394373004184*v^15 - 95065139318341*v^13 - 1839250515565853*v^11 - 18615270407097682*v^9 - 136919912259057455*v^7 - 509947381143146086*v^5 - 1307808756467713693*v^3 - 90623477496149271*v) / 26603594288476848 $$\beta_{13}$$ $$=$$ $$( 46212447101 \nu^{15} + 1289258286323 \nu^{13} + 24906077016940 \nu^{11} + 250804054500443 \nu^{9} + \cdots + 114034611732426 \nu ) / 286060153639536$$ (46212447101*v^15 + 1289258286323*v^13 + 24906077016940*v^11 + 250804054500443*v^9 + 1839041801676268*v^7 + 6763265898390281*v^5 + 17340653478163769*v^3 + 114034611732426*v) / 286060153639536 $$\beta_{14}$$ $$=$$ $$( - 78658668889 \nu^{15} - 2212467122437 \nu^{13} - 42911401938458 \nu^{11} - 437082575176471 \nu^{9} + \cdots - 84\!\cdots\!24 \nu ) / 286060153639536$$ (-78658668889*v^15 - 2212467122437*v^13 - 42911401938458*v^11 - 437082575176471*v^9 - 3236508022255694*v^7 - 12307020335183269*v^5 - 32608540003616059*v^3 - 8402395795060824*v) / 286060153639536 $$\beta_{15}$$ $$=$$ $$( 2811623585401 \nu^{15} + 78035033220436 \nu^{13} + \cdots - 12\!\cdots\!83 \nu ) / 88\!\cdots\!16$$ (2811623585401*v^15 + 78035033220436*v^13 + 1504847016559811*v^11 + 15062914749592717*v^9 + 110062989712353245*v^7 + 398499357853747327*v^5 + 1013107101355984660*v^3 - 120301723739013183*v) / 8867864762825616
 $$\nu$$ $$=$$ $$( -\beta_{15} - \beta_{14} + \beta_{12} - 2\beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{8} ) / 6$$ (-b15 - b14 + b12 - 2*b11 + b10 + b9 - 3*b8) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{5} + \beta_{3} - 7\beta_{2} - 7$$ b7 + b5 + b3 - 7*b2 - 7 $$\nu^{3}$$ $$=$$ $$( 22\beta_{15} - 11\beta_{14} - 45\beta_{13} + 5\beta_{12} + 5\beta_{11} - 34\beta_{10} + 17\beta_{9} ) / 6$$ (22*b15 - 11*b14 - 45*b13 + 5*b12 + 5*b11 - 34*b10 + 17*b9) / 6 $$\nu^{4}$$ $$=$$ $$-14\beta_{7} - 8\beta_{6} - 14\beta_{4} - 14\beta_{3} + 75\beta_{2}$$ -14*b7 - 8*b6 - 14*b4 - 14*b3 + 75*b2 $$\nu^{5}$$ $$=$$ $$( - 139 \beta_{15} + 278 \beta_{14} + 633 \beta_{13} - 14 \beta_{12} + 7 \beta_{11} + 223 \beta_{10} - 446 \beta_{9} + 633 \beta_{8} ) / 6$$ (-139*b15 + 278*b14 + 633*b13 - 14*b12 + 7*b11 + 223*b10 - 446*b9 + 633*b8) / 6 $$\nu^{6}$$ $$=$$ $$-197\beta_{5} + 181\beta_{4} + 16\beta_{3} + 168\beta _1 + 889$$ -197*b5 + 181*b4 + 16*b3 + 168*b1 + 889 $$\nu^{7}$$ $$=$$ $$( - 1829 \beta_{15} - 1829 \beta_{14} - 361 \beta_{12} + 722 \beta_{11} + 2819 \beta_{10} + 2819 \beta_{9} - 8955 \beta_{8} ) / 6$$ (-1829*b15 - 1829*b14 - 361*b12 + 722*b11 + 2819*b10 + 2819*b9 - 8955*b8) / 6 $$\nu^{8}$$ $$=$$ $$2324\beta_{7} + 2768\beta_{6} + 3220\beta_{5} + 2772\beta_{3} - 11169\beta_{2} - 2768\beta _1 - 11169$$ 2324*b7 + 2768*b6 + 3220*b5 + 2772*b3 - 11169*b2 - 2768*b1 - 11169 $$\nu^{9}$$ $$=$$ $$( 49154 \beta_{15} - 24577 \beta_{14} - 126291 \beta_{13} - 8663 \beta_{12} - 8663 \beta_{11} - 71666 \beta_{10} + 35833 \beta_{9} ) / 6$$ (49154*b15 - 24577*b14 - 126291*b13 - 8663*b12 - 8663*b11 - 71666*b10 + 35833*b9) / 6 $$\nu^{10}$$ $$=$$ $$-30205\beta_{7} - 42000\beta_{6} - 8672\beta_{5} - 30205\beta_{4} - 47549\beta_{3} + 145327\beta_{2}$$ -30205*b7 - 42000*b6 - 8672*b5 - 30205*b4 - 47549*b3 + 145327*b2 $$\nu^{11}$$ $$=$$ $$( - 334163 \beta_{15} + 668326 \beta_{14} + 1773045 \beta_{13} + 302198 \beta_{12} - 151099 \beta_{11} + 463361 \beta_{10} - 926722 \beta_{9} + 1773045 \beta_{8} ) / 6$$ (-334163*b15 + 668326*b14 + 1773045*b13 + 302198*b12 - 151099*b11 + 463361*b10 - 926722*b9 + 1773045*b8) / 6 $$\nu^{12}$$ $$=$$ $$-543466\beta_{5} + 398762\beta_{4} + 144704\beta_{3} + 613688\beta _1 + 1932699$$ -543466*b5 + 398762*b4 + 144704*b3 + 613688*b1 + 1932699 $$\nu^{13}$$ $$=$$ $$( - 4575547 \beta_{15} - 4575547 \beta_{14} - 2367377 \beta_{12} + 4734754 \beta_{11} + 6099895 \beta_{10} + 6099895 \beta_{9} - 24798585 \beta_{8} ) / 6$$ (-4575547*b15 - 4575547*b14 - 2367377*b12 + 4734754*b11 + 6099895*b10 + 6099895*b9 - 24798585*b8) / 6 $$\nu^{14}$$ $$=$$ $$5337721 \beta_{7} + 8788248 \beta_{6} + 9818329 \beta_{5} + 7578025 \beta_{3} - 26067265 \beta_{2} - 8788248 \beta _1 - 26067265$$ 5337721*b7 + 8788248*b6 + 9818329*b5 + 7578025*b3 - 26067265*b2 - 8788248*b1 - 26067265 $$\nu^{15}$$ $$=$$ $$( 125854618 \beta_{15} - 62927309 \beta_{14} - 345889539 \beta_{13} - 35270329 \beta_{12} - 35270329 \beta_{11} - 163023622 \beta_{10} + 81511811 \beta_{9} ) / 6$$ (125854618*b15 - 62927309*b14 - 345889539*b13 - 35270329*b12 - 35270329*b11 - 163023622*b10 + 81511811*b9) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$1 + \beta_{2}$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 −1.86197 + 3.22503i 0.181830 − 0.314939i −1.09560 + 1.89764i 1.51646 − 2.62659i 1.09560 − 1.89764i −1.51646 + 2.62659i 1.86197 − 3.22503i −0.181830 + 0.314939i 0.181830 + 0.314939i −1.86197 − 3.22503i 1.51646 + 2.62659i −1.09560 − 1.89764i −1.51646 − 2.62659i 1.09560 + 1.89764i −0.181830 − 0.314939i 1.86197 + 3.22503i
0 0 0 −1.38255 + 2.39464i 0 3.26278i 0 0 0
559.2 0 0 0 −1.38255 + 2.39464i 0 3.26278i 0 0 0
559.3 0 0 0 −0.767178 + 1.32879i 0 2.31392i 0 0 0
559.4 0 0 0 −0.767178 + 1.32879i 0 2.31392i 0 0 0
559.5 0 0 0 0.767178 1.32879i 0 2.31392i 0 0 0
559.6 0 0 0 0.767178 1.32879i 0 2.31392i 0 0 0
559.7 0 0 0 1.38255 2.39464i 0 3.26278i 0 0 0
559.8 0 0 0 1.38255 2.39464i 0 3.26278i 0 0 0
1855.1 0 0 0 −1.38255 2.39464i 0 3.26278i 0 0 0
1855.2 0 0 0 −1.38255 2.39464i 0 3.26278i 0 0 0
1855.3 0 0 0 −0.767178 1.32879i 0 2.31392i 0 0 0
1855.4 0 0 0 −0.767178 1.32879i 0 2.31392i 0 0 0
1855.5 0 0 0 0.767178 + 1.32879i 0 2.31392i 0 0 0
1855.6 0 0 0 0.767178 + 1.32879i 0 2.31392i 0 0 0
1855.7 0 0 0 1.38255 + 2.39464i 0 3.26278i 0 0 0
1855.8 0 0 0 1.38255 + 2.39464i 0 3.26278i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner
76.f even 6 1 inner
228.n odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.t 16
3.b odd 2 1 inner 2736.2.bm.t 16
4.b odd 2 1 inner 2736.2.bm.t 16
12.b even 2 1 inner 2736.2.bm.t 16
19.d odd 6 1 inner 2736.2.bm.t 16
57.f even 6 1 inner 2736.2.bm.t 16
76.f even 6 1 inner 2736.2.bm.t 16
228.n odd 6 1 inner 2736.2.bm.t 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.bm.t 16 1.a even 1 1 trivial
2736.2.bm.t 16 3.b odd 2 1 inner
2736.2.bm.t 16 4.b odd 2 1 inner
2736.2.bm.t 16 12.b even 2 1 inner
2736.2.bm.t 16 19.d odd 6 1 inner
2736.2.bm.t 16 57.f even 6 1 inner
2736.2.bm.t 16 76.f even 6 1 inner
2736.2.bm.t 16 228.n odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{8} + 10T_{5}^{6} + 82T_{5}^{4} + 180T_{5}^{2} + 324$$ T5^8 + 10*T5^6 + 82*T5^4 + 180*T5^2 + 324 $$T_{7}^{4} + 16T_{7}^{2} + 57$$ T7^4 + 16*T7^2 + 57 $$T_{11}^{4} + 34T_{11}^{2} + 114$$ T11^4 + 34*T11^2 + 114 $$T_{23}^{8} - 94T_{23}^{6} + 7810T_{23}^{4} - 96444T_{23}^{2} + 1052676$$ T23^8 - 94*T23^6 + 7810*T23^4 - 96444*T23^2 + 1052676 $$T_{31}^{4} - 40T_{31}^{2} + 57$$ T31^4 - 40*T31^2 + 57

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} + 10 T^{6} + 82 T^{4} + 180 T^{2} + \cdots + 324)^{2}$$
$7$ $$(T^{4} + 16 T^{2} + 57)^{4}$$
$11$ $$(T^{4} + 34 T^{2} + 114)^{4}$$
$13$ $$(T^{4} - 21 T^{2} + 441)^{4}$$
$17$ $$(T^{8} + 52 T^{6} + 2056 T^{4} + \cdots + 419904)^{2}$$
$19$ $$(T^{8} + 24 T^{6} + 418 T^{4} + \cdots + 130321)^{2}$$
$23$ $$(T^{8} - 94 T^{6} + 7810 T^{4} + \cdots + 1052676)^{2}$$
$29$ $$(T^{8} - 72 T^{6} + 4896 T^{4} + \cdots + 82944)^{2}$$
$31$ $$(T^{4} - 40 T^{2} + 57)^{4}$$
$37$ $$(T^{2} + 3)^{8}$$
$41$ $$(T^{8} - 60 T^{6} + 2952 T^{4} + \cdots + 419904)^{2}$$
$43$ $$(T^{8} - 120 T^{6} + 13887 T^{4} + \cdots + 263169)^{2}$$
$47$ $$(T^{8} - 124 T^{6} + 14920 T^{4} + \cdots + 207936)^{2}$$
$53$ $$(T^{8} - 162 T^{6} + 19746 T^{4} + \cdots + 42224004)^{2}$$
$59$ $$(T^{8} + 102 T^{6} + 9378 T^{4} + \cdots + 1052676)^{2}$$
$61$ $$(T^{4} + 4 T^{3} + 19 T^{2} - 12 T + 9)^{4}$$
$67$ $$(T^{8} + 136 T^{6} + 13879 T^{4} + \cdots + 21316689)^{2}$$
$71$ $$(T^{8} + 396 T^{6} + 119880 T^{4} + \cdots + 1364268096)^{2}$$
$73$ $$(T^{4} - 6 T^{3} + 139 T^{2} + 618 T + 10609)^{4}$$
$79$ $$(T^{8} + 304 T^{6} + 71839 T^{4} + \cdots + 423412929)^{2}$$
$83$ $$(T^{4} + 124 T^{2} + 456)^{4}$$
$89$ $$(T^{8} - 78 T^{6} + 4626 T^{4} + \cdots + 2125764)^{2}$$
$97$ $$(T^{4} + 18 T^{3} - 54 T^{2} - 2916 T + 26244)^{4}$$