# Properties

 Label 3584.2.b.i Level $3584$ Weight $2$ Character orbit 3584.b Analytic conductor $28.618$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3584 = 2^{9} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3584.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6183840844$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + 108 x^{3} + 68 x^{2} + 32 x + 8$$ x^12 - 4*x^11 + 4*x^10 + 8*x^9 - 24*x^8 + 24*x^7 + 176*x^6 + 160*x^5 + 145*x^4 + 108*x^3 + 68*x^2 + 32*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{3} + \beta_{7} q^{5} - q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10})$$ q - b6 * q^3 + b7 * q^5 - q^7 + (b5 - 2) * q^9 $$q - \beta_{6} q^{3} + \beta_{7} q^{5} - q^{7} + (\beta_{5} - 2) q^{9} + (\beta_{7} - \beta_{3}) q^{11} + \beta_{2} q^{13} + ( - \beta_{10} + \beta_1 - 1) q^{15} - \beta_{10} q^{17} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{19} + \beta_{6} q^{21} + (\beta_{9} + \beta_{8}) q^{23} + (\beta_{8} + \beta_1) q^{25} + ( - \beta_{11} - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2}) q^{27} + (\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{29} + (\beta_{9} + \beta_1 - 1) q^{31} + ( - \beta_{10} + \beta_{9} - \beta_{8} + \beta_1 + 1) q^{33} - \beta_{7} q^{35} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2}) q^{37} + \beta_{9} q^{39} + (\beta_{10} + \beta_{5} - \beta_1) q^{41} + (\beta_{11} - \beta_{7} + \beta_{4}) q^{43} + (\beta_{11} - 2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}) q^{45} + (\beta_{10} + \beta_{8} - \beta_{5} + \beta_1 + 2) q^{47} + q^{49} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4}) q^{51} + (\beta_{11} - 3 \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{53} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - 3) q^{55} + ( - 2 \beta_{9} + \beta_{8} + \beta_1 + 1) q^{57} + ( - \beta_{11} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2}) q^{59} + (\beta_{11} + 3 \beta_{7} - \beta_{6} - \beta_{3} - 2 \beta_{2}) q^{61} + ( - \beta_{5} + 2) q^{63} + ( - 2 \beta_{8} - \beta_{5} - 1) q^{65} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{67} + (2 \beta_{11} + \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 3 \beta_{2}) q^{69} + (\beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{5} + \beta_1 + 1) q^{71} + ( - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - \beta_{5} + \beta_1 - 2) q^{73} + (2 \beta_{11} - 3 \beta_{7} + \beta_{4} + \beta_{2}) q^{75} + ( - \beta_{7} + \beta_{3}) q^{77} + (\beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_1 - 5) q^{79} + (2 \beta_{9} + 3 \beta_{8} + \beta_1 + 2) q^{81} + ( - \beta_{11} + 2 \beta_{7} + \beta_{3} + 2 \beta_{2}) q^{83} + (\beta_{11} + 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3}) q^{85} + ( - \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{5} + \beta_1 + 2) q^{87} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_1 - 3) q^{89} - \beta_{2} q^{91} + (2 \beta_{11} - 2 \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{93} + ( - \beta_{9} + 2 \beta_{8} + 2 \beta_1 - 2) q^{95} + ( - \beta_{9} - 3 \beta_{8} - \beta_{5} + 3) q^{97} + (\beta_{11} - \beta_{7} - \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{99}+O(q^{100})$$ q - b6 * q^3 + b7 * q^5 - q^7 + (b5 - 2) * q^9 + (b7 - b3) * q^11 + b2 * q^13 + (-b10 + b1 - 1) * q^15 - b10 * q^17 + (b7 - b4 - b2) * q^19 + b6 * q^21 + (b9 + b8) * q^23 + (b8 + b1) * q^25 + (-b11 - b7 + 2*b6 + b4 + b3 + b2) * q^27 + (b7 + b6 + b3 - b2) * q^29 + (b9 + b1 - 1) * q^31 + (-b10 + b9 - b8 + b1 + 1) * q^33 - b7 * q^35 + (-b7 + 2*b6 + b4 + b3 + b2) * q^37 + b9 * q^39 + (b10 + b5 - b1) * q^41 + (b11 - b7 + b4) * q^43 + (b11 - 2*b7 + b6 + b3 + b2) * q^45 + (b10 + b8 - b5 + b1 + 2) * q^47 + q^49 + (-2*b7 - b6 - b4) * q^51 + (b11 - 3*b7 - b6 + b4 + 2*b3 + b2) * q^53 + (b10 + b9 + b8 + b5 - 3) * q^55 + (-2*b9 + b8 + b1 + 1) * q^57 + (-b11 + b7 + b6 + b4 + b3 - b2) * q^59 + (b11 + 3*b7 - b6 - b3 - 2*b2) * q^61 + (-b5 + 2) * q^63 + (-2*b8 - b5 - 1) * q^65 + (b7 - b6 - b4 - b3 + 2*b2) * q^67 + (2*b11 + b7 - 2*b6 - 2*b4 - 3*b2) * q^69 + (b10 + b9 + 2*b8 + 2*b5 + b1 + 1) * q^71 + (-2*b10 - 2*b9 - 2*b8 - b5 + b1 - 2) * q^73 + (2*b11 - 3*b7 + b4 + b2) * q^75 + (-b7 + b3) * q^77 + (b10 + b9 + b8 + 2*b5 - b1 - 5) * q^79 + (2*b9 + 3*b8 + b1 + 2) * q^81 + (-b11 + 2*b7 + b3 + 2*b2) * q^83 + (b11 + 2*b7 + 2*b6 - b4 - b3) * q^85 + (-b10 - 2*b9 + b8 - b5 + b1 + 2) * q^87 + (b10 + b9 + b8 - b1 - 3) * q^89 - b2 * q^91 + (2*b11 - 2*b7 + b6 - b4 + 2*b3 - 2*b2) * q^93 + (-b9 + 2*b8 + 2*b1 - 2) * q^95 + (-b9 - 3*b8 - b5 + 3) * q^97 + (b11 - b7 - b6 - 2*b4 - 2*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{7} - 20 q^{9}+O(q^{10})$$ 12 * q - 12 * q^7 - 20 * q^9 $$12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100})$$ 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + 108 x^{3} + 68 x^{2} + 32 x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( 419637583 \nu^{11} - 1440544428 \nu^{10} + 635340644 \nu^{9} + 4866667572 \nu^{8} - 9783161750 \nu^{7} + 6711732869 \nu^{6} + \cdots + 11421355507 ) / 9640197931$$ (419637583*v^11 - 1440544428*v^10 + 635340644*v^9 + 4866667572*v^8 - 9783161750*v^7 + 6711732869*v^6 + 79227935542*v^5 + 102463056660*v^4 + 99225737137*v^3 + 72863841031*v^2 + 29581654498*v + 11421355507) / 9640197931 $$\beta_{2}$$ $$=$$ $$( 638917040 \nu^{11} - 2376776737 \nu^{10} + 1975208857 \nu^{9} + 5237100284 \nu^{8} - 12962071145 \nu^{7} + 10962211405 \nu^{6} + \cdots + 36460193404 ) / 9640197931$$ (638917040*v^11 - 2376776737*v^10 + 1975208857*v^9 + 5237100284*v^8 - 12962071145*v^7 + 10962211405*v^6 + 114217992840*v^5 + 140673595658*v^4 + 135177226266*v^3 + 116563754298*v^2 + 109607514376*v + 36460193404) / 9640197931 $$\beta_{3}$$ $$=$$ $$( 8615178688 \nu^{11} - 52103353005 \nu^{10} + 115591508302 \nu^{9} - 43547709868 \nu^{8} - 308923880124 \nu^{7} + 722748596326 \nu^{6} + \cdots - 39713864336 ) / 96401979310$$ (8615178688*v^11 - 52103353005*v^10 + 115591508302*v^9 - 43547709868*v^8 - 308923880124*v^7 + 722748596326*v^6 + 842555137942*v^5 - 1519592667492*v^4 + 377243672422*v^3 + 74217323797*v^2 - 95818529098*v - 39713864336) / 96401979310 $$\beta_{4}$$ $$=$$ $$( - 894988184 \nu^{11} + 5710664665 \nu^{10} - 13387713576 \nu^{9} + 6452256724 \nu^{8} + 33852806542 \nu^{7} - 84141470418 \nu^{6} + \cdots + 8956064308 ) / 7415536870$$ (-894988184*v^11 + 5710664665*v^10 - 13387713576*v^9 + 6452256724*v^8 + 33852806542*v^7 - 84141470418*v^6 - 75031606216*v^5 + 205197370076*v^4 - 23800749796*v^3 + 5442408659*v^2 + 24125661214*v + 8956064308) / 7415536870 $$\beta_{5}$$ $$=$$ $$( 495332004 \nu^{11} - 2247729805 \nu^{10} + 2867331746 \nu^{9} + 3802764346 \nu^{8} - 15503170202 \nu^{7} + 17719026533 \nu^{6} + \cdots + 6862964757 ) / 3707768435$$ (495332004*v^11 - 2247729805*v^10 + 2867331746*v^9 + 3802764346*v^8 - 15503170202*v^7 + 17719026533*v^6 + 86145318266*v^5 + 23993059084*v^4 + 3174773026*v^3 + 17083504926*v^2 + 9421475556*v + 6862964757) / 3707768435 $$\beta_{6}$$ $$=$$ $$( - 1410256356 \nu^{11} + 5921534715 \nu^{10} - 6862437384 \nu^{9} - 9554539644 \nu^{8} + 34683435378 \nu^{7} - 39753339342 \nu^{6} + \cdots - 25484705788 ) / 7415536870$$ (-1410256356*v^11 + 5921534715*v^10 - 6862437384*v^9 - 9554539644*v^8 + 34683435378*v^7 - 39753339342*v^6 - 238677149144*v^5 - 183608962756*v^4 - 169228225624*v^3 - 96334816959*v^2 - 77503534254*v - 25484705788) / 7415536870 $$\beta_{7}$$ $$=$$ $$( - 13383971527 \nu^{11} + 62676653525 \nu^{10} - 94344403828 \nu^{9} - 51680040438 \nu^{8} + 369964045326 \nu^{7} + \cdots - 86421877996 ) / 48200989655$$ (-13383971527*v^11 + 62676653525*v^10 - 94344403828*v^9 - 51680040438*v^8 + 369964045326*v^7 - 568161416759*v^6 - 2013928908103*v^5 - 691682397522*v^4 - 1165040879498*v^3 - 478290812768*v^2 - 275960154468*v - 86421877996) / 48200989655 $$\beta_{8}$$ $$=$$ $$( - 6278889 \nu^{11} + 25797695 \nu^{10} - 26888646 \nu^{9} - 52239876 \nu^{8} + 164047612 \nu^{7} - 164419198 \nu^{6} - 1118786356 \nu^{5} + \cdots - 96252252 ) / 22141015$$ (-6278889*v^11 + 25797695*v^10 - 26888646*v^9 - 52239876*v^8 + 164047612*v^7 - 164419198*v^6 - 1118786356*v^5 - 833215544*v^4 - 666378781*v^3 - 592838171*v^2 - 257611546*v - 96252252) / 22141015 $$\beta_{9}$$ $$=$$ $$( 2182781366 \nu^{11} - 9596353535 \nu^{10} + 11698649024 \nu^{9} + 16394832844 \nu^{8} - 62798712888 \nu^{7} + 69749861327 \nu^{6} + \cdots - 8063859872 ) / 6885855665$$ (2182781366*v^11 - 9596353535*v^10 + 11698649024*v^9 + 16394832844*v^8 - 62798712888*v^7 + 69749861327*v^6 + 381569431644*v^5 + 173135871256*v^4 + 94219524054*v^3 + 123211730674*v^2 + 58999925004*v - 8063859872) / 6885855665 $$\beta_{10}$$ $$=$$ $$( - 3571797907 \nu^{11} + 15225607627 \nu^{10} - 17186575048 \nu^{9} - 29155049066 \nu^{8} + 100637456110 \nu^{7} + \cdots - 63122961648 ) / 9640197931$$ (-3571797907*v^11 + 15225607627*v^10 - 17186575048*v^9 - 29155049066*v^8 + 100637456110*v^7 - 106624682470*v^6 - 631614135390*v^5 - 362909252140*v^4 - 247359961889*v^3 - 258213400303*v^2 - 117576838690*v - 63122961648) / 9640197931 $$\beta_{11}$$ $$=$$ $$( - 3742088406 \nu^{11} + 17147484255 \nu^{10} - 24666155409 \nu^{9} - 16843522124 \nu^{8} + 101347688343 \nu^{7} + \cdots - 34789559518 ) / 6885855665$$ (-3742088406*v^11 + 17147484255*v^10 - 24666155409*v^9 - 16843522124*v^8 + 101347688343*v^7 - 147316301692*v^6 - 580709702529*v^5 - 248659053356*v^4 - 356304558469*v^3 - 163881140269*v^2 - 108627487044*v - 34789559518) / 6885855665
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{6} + \beta_{5} + \beta_{3} + 2\beta _1 + 3 ) / 8$$ (b11 + b10 + b9 + b8 - 3*b6 + b5 + b3 + 2*b1 + 3) / 8 $$\nu^{2}$$ $$=$$ $$( 3\beta_{11} + 4\beta_{10} - 4\beta_{8} - 4\beta_{7} - \beta_{6} + 4\beta_{5} + 4\beta_{4} + 7\beta_{3} + 4 ) / 8$$ (3*b11 + 4*b10 - 4*b8 - 4*b7 - b6 + 4*b5 + 4*b4 + 7*b3 + 4) / 8 $$\nu^{3}$$ $$=$$ $$( - 3 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 15 \beta_{6} + 11 \beta_{5} + 22 \beta_{4} + 29 \beta_{3} + 4 \beta_{2} - 9 ) / 8$$ (-3*b11 + 7*b10 - 3*b9 - 7*b8 - 4*b7 + 15*b6 + 11*b5 + 22*b4 + 29*b3 + 4*b2 - 9) / 8 $$\nu^{4}$$ $$=$$ $$( -7\beta_{11} - 14\beta_{7} + 42\beta_{6} + 43\beta_{4} + 51\beta_{3} + 14\beta_{2} ) / 4$$ (-7*b11 - 14*b7 + 42*b6 + 43*b4 + 51*b3 + 14*b2) / 4 $$\nu^{5}$$ $$=$$ $$( - 11 \beta_{11} - 65 \beta_{10} + 9 \beta_{9} + 53 \beta_{8} - 68 \beta_{7} + 143 \beta_{6} - 85 \beta_{5} + 222 \beta_{4} + 293 \beta_{3} + 44 \beta_{2} - 12 \beta _1 - 13 ) / 8$$ (-11*b11 - 65*b10 + 9*b9 + 53*b8 - 68*b7 + 143*b6 - 85*b5 + 222*b4 + 293*b3 + 44*b2 - 12*b1 - 13) / 8 $$\nu^{6}$$ $$=$$ $$( 111 \beta_{11} - 340 \beta_{10} - 72 \beta_{9} + 188 \beta_{8} - 236 \beta_{7} + 67 \beta_{6} - 316 \beta_{5} + 468 \beta_{4} + 699 \beta_{3} + 24 \beta_{2} - 176 \beta _1 - 700 ) / 8$$ (111*b11 - 340*b10 - 72*b9 + 188*b8 - 236*b7 + 67*b6 - 316*b5 + 468*b4 + 699*b3 + 24*b2 - 176*b1 - 700) / 8 $$\nu^{7}$$ $$=$$ $$( 315 \beta_{11} - 1259 \beta_{10} - 615 \beta_{9} + 393 \beta_{8} - 420 \beta_{7} - 209 \beta_{6} - 839 \beta_{5} + 716 \beta_{4} + 1183 \beta_{3} - 68 \beta_{2} - 1102 \beta _1 - 4397 ) / 8$$ (315*b11 - 1259*b10 - 615*b9 + 393*b8 - 420*b7 - 209*b6 - 839*b5 + 716*b4 + 1183*b3 - 68*b2 - 1102*b1 - 4397) / 8 $$\nu^{8}$$ $$=$$ $$( -1064\beta_{10} - 594\beta_{9} + 279\beta_{8} - 625\beta_{5} - 995\beta _1 - 4114 ) / 2$$ (-1064*b10 - 594*b9 + 279*b8 - 625*b5 - 995*b1 - 4114) / 2 $$\nu^{9}$$ $$=$$ $$( - 2147 \beta_{11} - 12873 \beta_{10} - 5921 \beta_{9} + 4367 \beta_{8} + 3980 \beta_{7} - 383 \beta_{6} - 8893 \beta_{5} - 7892 \beta_{4} - 12071 \beta_{3} - 124 \beta_{2} - 10710 \beta _1 - 43091 ) / 8$$ (-2147*b11 - 12873*b10 - 5921*b9 + 4367*b8 + 3980*b7 - 383*b6 - 8893*b5 - 7892*b4 - 12071*b3 - 124*b2 - 10710*b1 - 43091) / 8 $$\nu^{10}$$ $$=$$ $$( - 5767 \beta_{11} - 32820 \beta_{10} - 10864 \beta_{9} + 14660 \beta_{8} + 21420 \beta_{7} - 18435 \beta_{6} - 26876 \beta_{5} - 50636 \beta_{4} - 71715 \beta_{3} - 5944 \beta_{2} - 22264 \beta _1 - 87764 ) / 8$$ (-5767*b11 - 32820*b10 - 10864*b9 + 14660*b8 + 21420*b7 - 18435*b6 - 26876*b5 - 50636*b4 - 71715*b3 - 5944*b2 - 22264*b1 - 87764) / 8 $$\nu^{11}$$ $$=$$ $$( - 5365 \beta_{11} - 57879 \beta_{10} - 15021 \beta_{9} + 29199 \beta_{8} + 82804 \beta_{7} - 119319 \beta_{6} - 51627 \beta_{5} - 221478 \beta_{4} - 299877 \beta_{3} - 38276 \beta_{2} + \cdots - 133047 ) / 8$$ (-5365*b11 - 57879*b10 - 15021*b9 + 29199*b8 + 82804*b7 - 119319*b6 - 51627*b5 - 221478*b4 - 299877*b3 - 38276*b2 - 34624*b1 - 133047) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1793.1
 0.960396 − 2.31860i −1.45042 − 0.600784i −0.232297 − 0.560814i −0.484138 − 0.200537i 2.93456 − 1.21553i 0.271901 + 0.656426i 0.271901 − 0.656426i 2.93456 + 1.21553i −0.484138 + 0.200537i −0.232297 + 0.560814i −1.45042 + 0.600784i 0.960396 + 2.31860i
0 3.22299i 0 0.725063i 0 −1.00000 0 −7.38766 0
1793.2 0 2.61578i 0 3.96111i 0 −1.00000 0 −3.84231 0
1793.3 0 2.53584i 0 1.47134i 0 −1.00000 0 −3.43049 0
1793.4 0 1.81529i 0 2.66965i 0 −1.00000 0 −0.295267 0
1793.5 0 1.01685i 0 1.29145i 0 −1.00000 0 1.96601 0
1793.6 0 0.101362i 0 2.19640i 0 −1.00000 0 2.98973 0
1793.7 0 0.101362i 0 2.19640i 0 −1.00000 0 2.98973 0
1793.8 0 1.01685i 0 1.29145i 0 −1.00000 0 1.96601 0
1793.9 0 1.81529i 0 2.66965i 0 −1.00000 0 −0.295267 0
1793.10 0 2.53584i 0 1.47134i 0 −1.00000 0 −3.43049 0
1793.11 0 2.61578i 0 3.96111i 0 −1.00000 0 −3.84231 0
1793.12 0 3.22299i 0 0.725063i 0 −1.00000 0 −7.38766 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1793.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.b.i 12
4.b odd 2 1 3584.2.b.k 12
8.b even 2 1 inner 3584.2.b.i 12
8.d odd 2 1 3584.2.b.k 12
16.e even 4 1 3584.2.a.f yes 6
16.e even 4 1 3584.2.a.l yes 6
16.f odd 4 1 3584.2.a.e 6
16.f odd 4 1 3584.2.a.k yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.e 6 16.f odd 4 1
3584.2.a.f yes 6 16.e even 4 1
3584.2.a.k yes 6 16.f odd 4 1
3584.2.a.l yes 6 16.e even 4 1
3584.2.b.i 12 1.a even 1 1 trivial
3584.2.b.i 12 8.b even 2 1 inner
3584.2.b.k 12 4.b odd 2 1
3584.2.b.k 12 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3584, [\chi])$$:

 $$T_{3}^{12} + 28T_{3}^{10} + 288T_{3}^{8} + 1328T_{3}^{6} + 2612T_{3}^{4} + 1584T_{3}^{2} + 16$$ T3^12 + 28*T3^10 + 288*T3^8 + 1328*T3^6 + 2612*T3^4 + 1584*T3^2 + 16 $$T_{23}^{6} - 76T_{23}^{4} - 96T_{23}^{3} + 1060T_{23}^{2} + 832T_{23} - 2848$$ T23^6 - 76*T23^4 - 96*T23^3 + 1060*T23^2 + 832*T23 - 2848

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 28 T^{10} + 288 T^{8} + \cdots + 16$$
$5$ $$T^{12} + 32 T^{10} + 348 T^{8} + \cdots + 1024$$
$7$ $$(T + 1)^{12}$$
$11$ $$T^{12} + 68 T^{10} + 1604 T^{8} + \cdots + 4096$$
$13$ $$T^{12} + 48 T^{10} + 668 T^{8} + \cdots + 1024$$
$17$ $$(T^{6} - 48 T^{4} + 32 T^{3} + 464 T^{2} + \cdots - 896)^{2}$$
$19$ $$T^{12} + 140 T^{10} + 5792 T^{8} + \cdots + 300304$$
$23$ $$(T^{6} - 76 T^{4} - 96 T^{3} + 1060 T^{2} + \cdots - 2848)^{2}$$
$29$ $$T^{12} + 168 T^{10} + 9808 T^{8} + \cdots + 16384$$
$31$ $$(T^{6} + 8 T^{5} - 72 T^{4} - 512 T^{3} + \cdots + 11776)^{2}$$
$37$ $$T^{12} + 168 T^{10} + 9296 T^{8} + \cdots + 16384$$
$41$ $$(T^{6} - 4 T^{5} - 84 T^{4} + 416 T^{3} + \cdots + 5696)^{2}$$
$43$ $$T^{12} + 292 T^{10} + \cdots + 2117472256$$
$47$ $$(T^{6} - 8 T^{5} - 152 T^{4} + \cdots - 179200)^{2}$$
$53$ $$T^{12} + 552 T^{10} + \cdots + 2390818816$$
$59$ $$T^{12} + 364 T^{10} + \cdots + 241740304$$
$61$ $$T^{12} + 448 T^{10} + \cdots + 2466512896$$
$67$ $$T^{12} + 356 T^{10} + \cdots + 18939904$$
$71$ $$(T^{6} - 8 T^{5} - 248 T^{4} + 800 T^{3} + \cdots - 51200)^{2}$$
$73$ $$(T^{6} + 16 T^{5} - 240 T^{4} + \cdots + 899200)^{2}$$
$79$ $$(T^{6} + 24 T^{5} + 64 T^{4} - 1696 T^{3} + \cdots - 21632)^{2}$$
$83$ $$T^{12} + 636 T^{10} + \cdots + 72755351824$$
$89$ $$(T^{6} + 16 T^{5} - 32 T^{4} - 1056 T^{3} + \cdots + 6272)^{2}$$
$97$ $$(T^{6} - 16 T^{5} - 96 T^{4} + \cdots + 232576)^{2}$$