# Properties

 Label 10000.2.a.bo Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $1$ Dimension $12$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$10000 = 2^{4} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 10000.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.8504020213$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + 15 x^{3} - 285 x^{2} + 45$$ x^12 - x^11 - 20*x^10 + 11*x^9 + 144*x^8 - 29*x^7 - 440*x^6 + 4*x^5 + 556*x^4 + 15*x^3 - 285*x^2 + 45 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 5000) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{3} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9}+O(q^{10})$$ q - b6 * q^3 + (b5 - b2 + b1 + 1) * q^7 + (-b10 - b9 - b7 + b6 - b5 - b4 + 2) * q^9 $$q - \beta_{6} q^{3} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9} + (\beta_{10} + \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{10} - \beta_{8} + 2 \beta_{7} + \beta_{2}) q^{17} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{2} - 2) q^{19} + (2 \beta_{11} - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{2} - \beta_1 - 1) q^{21} + ( - \beta_{11} - \beta_{10} - 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{10} + \beta_{9} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + \beta_1 - 1) q^{27} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{29} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{5} + \beta_{2} - 4) q^{31} + ( - \beta_{11} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{33} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{37} + ( - 2 \beta_{11} - \beta_{10} - 3 \beta_{8} + \beta_{7} - 3 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{39}+ \cdots + (\beta_{8} + \beta_{7} + 4 \beta_{6} - 7 \beta_{5} - \beta_{4} - 3 \beta_{3} + 11 \beta_{2} + \cdots - 8) q^{99}+O(q^{100})$$ q - b6 * q^3 + (b5 - b2 + b1 + 1) * q^7 + (-b10 - b9 - b7 + b6 - b5 - b4 + 2) * q^9 + (b10 + b8 + b6 - b3 - b2 + b1) * q^11 + (b9 + b8 + b7 + b6 - b5 + 2*b2 - 2) * q^13 + (-b10 - b8 + 2*b7 + b2) * q^17 + (b10 + b9 + b8 + b7 + b6 + b2 - 2) * q^19 + (2*b11 - b9 + b8 - 3*b7 + b6 - 2*b5 + b2 - b1 - 1) * q^21 + (-b11 - b10 - 2*b8 - b7 - b6 - b2 - b1 + 1) * q^23 + (b10 + b9 - 2*b6 + 2*b5 - 2*b2 + b1 - 1) * q^27 + (-b11 - b8 + b7 - b6 + 2*b5 + b4 + b3 - b1 + 1) * q^29 + (b11 + b10 - b9 - b7 - 2*b5 + b2 - 4) * q^31 + (-b11 - 2*b8 - b7 - 2*b6 + 2*b3 - 2*b2 - b1 - 1) * q^33 + (b10 + b9 + b8 - b7 + b6 + b4 - 2*b3 - b1 - 2) * q^37 + (-2*b11 - b10 - 3*b8 + b7 - 3*b6 + 5*b5 + 2*b4 + 2*b3 - b2 - 2*b1 + 1) * q^39 + (-b11 - 2*b8 + b7 - b4 + b3 + 3) * q^41 + (b11 + b8 + 2*b7 + b4 + 2*b2 - 5) * q^43 + (-b9 + b7 - 3*b6 - 2*b5 + b4 + b3 + b2 - b1 + 1) * q^47 + (-b11 + 2*b7 + b5 + b4 - b3 - b1 + 1) * q^49 + (2*b11 + b8 - 2*b7 + b6 + 2*b4 - b2 - 3*b1 - 1) * q^51 + (2*b11 - b9 + 2*b6 + b5 - b4 - b2 - b1 - 1) * q^53 + (-2*b11 - 2*b8 + b7 - b6 + 2*b5 + 2*b4 + b3 + 2*b2 - b1 - 4) * q^57 + (-b10 - b9 - 2*b7 + b6 - b5 - 2*b4 - 2*b3 - b2 - 2) * q^59 + (-b11 + b10 + b9 + b8 + b7 - b6 + 2*b5 + b3 + b2 + b1 - 1) * q^61 + (-2*b11 - b10 + b9 - 2*b8 + 4*b7 - 2*b6 + 6*b5 - 2*b4 - b2 + 2*b1 + 3) * q^63 + (b11 - b8 - b7 - 3*b6 + 2*b5 + 3*b4 + 3*b3 + b2 - 2*b1 - 4) * q^67 + (b11 + b10 + b9 + 2*b8 + 3*b6 - 4*b5 - 2*b4 - 3*b3 - 4*b2 + b1 + 1) * q^69 + (-b9 - b8 - b6 - 3*b5 - b4 + b3 - 3*b2) * q^71 + (b11 - b8 - b7 - 2*b6 + b5 + 2*b4 + 2*b3 + 4*b2 - 3*b1 - 2) * q^73 + (-2*b11 + b10 + b9 - b8 - b6 + 4*b5 + b4 + b3 - 7*b2 + b1 + 5) * q^77 + (-b10 - b7 + 3*b6 - 4*b5 - b4 - b3 + 4*b2 + b1 - 8) * q^79 + (2*b11 + b10 - b9 + b8 - 4*b7 + 5*b6 - 3*b5 + b4 - 2*b3 + 2*b2 - b1 + 1) * q^81 + (b9 - b8 - b7 - b6 + 2*b4 + 3*b2 - 3*b1 - 4) * q^83 + (b11 + 2*b10 + 2*b9 + 4*b8 + 6*b7 + 2*b6 - 5*b5 + b4 - b3 + 8*b2 + b1 - 10) * q^87 + (-2*b11 - b10 + b9 - 3*b8 - 2*b7 - b6 + b3 - 6*b2 + 7) * q^89 + (b9 - 3*b7 - b6 + 2*b5 + 3*b3 - b2 - b1 - 4) * q^91 + (-2*b11 + b10 - b8 + 2*b6 + b5 - b4 + b3 + 3*b1 - 1) * q^93 + (2*b11 - b9 + 2*b8 - 3*b7 + 3*b6 - 3*b5 - 2*b4 - b3 - b2 - 3) * q^97 + (b8 + b7 + 4*b6 - 7*b5 - b4 - 3*b3 + 11*b2 + b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{3} + 26 q^{9}+O(q^{10})$$ 12 * q - 4 * q^3 + 26 * q^9 $$12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100})$$ 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * q^97 - 5 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + 15 x^{3} - 285 x^{2} + 45$$ :

 $$\beta_{1}$$ $$=$$ $$( 307 \nu^{11} - 8733 \nu^{10} - 678 \nu^{9} + 180827 \nu^{8} - 27635 \nu^{7} - 1274457 \nu^{6} + 159642 \nu^{5} + 3473001 \nu^{4} - 410492 \nu^{3} - 3087181 \nu^{2} + \cdots + 648531 ) / 63003$$ (307*v^11 - 8733*v^10 - 678*v^9 + 180827*v^8 - 27635*v^7 - 1274457*v^6 + 159642*v^5 + 3473001*v^4 - 410492*v^3 - 3087181*v^2 + 455907*v + 648531) / 63003 $$\beta_{2}$$ $$=$$ $$( - 802 \nu^{11} - 6738 \nu^{10} + 24756 \nu^{9} + 135204 \nu^{8} - 214775 \nu^{7} - 939928 \nu^{6} + 642787 \nu^{5} + 2541583 \nu^{4} - 590618 \nu^{3} + \cdots + 568359 ) / 63003$$ (-802*v^11 - 6738*v^10 + 24756*v^9 + 135204*v^8 - 214775*v^7 - 939928*v^6 + 642787*v^5 + 2541583*v^4 - 590618*v^3 - 2266309*v^2 + 190755*v + 568359) / 63003 $$\beta_{3}$$ $$=$$ $$( - 1351 \nu^{11} - 13969 \nu^{10} + 48406 \nu^{9} + 261955 \nu^{8} - 433991 \nu^{7} - 1723439 \nu^{6} + 1296397 \nu^{5} + 4411093 \nu^{4} - 1154914 \nu^{3} + \cdots + 577755 ) / 63003$$ (-1351*v^11 - 13969*v^10 + 48406*v^9 + 261955*v^8 - 433991*v^7 - 1723439*v^6 + 1296397*v^5 + 4411093*v^4 - 1154914*v^3 - 3551140*v^2 + 347808*v + 577755) / 63003 $$\beta_{4}$$ $$=$$ $$( - 2231 \nu^{11} + 529 \nu^{10} + 42209 \nu^{9} + 21083 \nu^{8} - 281581 \nu^{7} - 323536 \nu^{6} + 746510 \nu^{5} + 1275695 \nu^{4} - 692225 \nu^{3} - 1604243 \nu^{2} + \cdots + 536010 ) / 63003$$ (-2231*v^11 + 529*v^10 + 42209*v^9 + 21083*v^8 - 281581*v^7 - 323536*v^6 + 746510*v^5 + 1275695*v^4 - 692225*v^3 - 1604243*v^2 + 62361*v + 536010) / 63003 $$\beta_{5}$$ $$=$$ $$( 3607 \nu^{11} - 10598 \nu^{10} - 56193 \nu^{9} + 160053 \nu^{8} + 293370 \nu^{7} - 848573 \nu^{6} - 513662 \nu^{5} + 1905609 \nu^{4} - 8724 \nu^{3} - 1635878 \nu^{2} + \cdots + 348303 ) / 63003$$ (3607*v^11 - 10598*v^10 - 56193*v^9 + 160053*v^8 + 293370*v^7 - 848573*v^6 - 513662*v^5 + 1905609*v^4 - 8724*v^3 - 1635878*v^2 + 282024*v + 348303) / 63003 $$\beta_{6}$$ $$=$$ $$( - 5601 \nu^{11} - 1703 \nu^{10} + 118948 \nu^{9} + 78063 \nu^{8} - 861089 \nu^{7} - 798674 \nu^{6} + 2409179 \nu^{5} + 2723982 \nu^{4} - 2248279 \nu^{3} + \cdots + 888045 ) / 63003$$ (-5601*v^11 - 1703*v^10 + 118948*v^9 + 78063*v^8 - 861089*v^7 - 798674*v^6 + 2409179*v^5 + 2723982*v^4 - 2248279*v^3 - 3041720*v^2 + 553611*v + 888045) / 63003 $$\beta_{7}$$ $$=$$ $$( - 7021 \nu^{11} + 11191 \nu^{10} + 129472 \nu^{9} - 144518 \nu^{8} - 854757 \nu^{7} + 587287 \nu^{6} + 2324195 \nu^{5} - 910206 \nu^{4} - 2310683 \nu^{3} + \cdots - 140112 ) / 63003$$ (-7021*v^11 + 11191*v^10 + 129472*v^9 - 144518*v^8 - 854757*v^7 + 587287*v^6 + 2324195*v^5 - 910206*v^4 - 2310683*v^3 + 592910*v^2 + 601323*v - 140112) / 63003 $$\beta_{8}$$ $$=$$ $$( 9252 \nu^{11} - 11720 \nu^{10} - 171681 \nu^{9} + 123435 \nu^{8} + 1136338 \nu^{7} - 263751 \nu^{6} - 3070705 \nu^{5} - 365489 \nu^{4} + 3065911 \nu^{3} + \cdots - 269892 ) / 63003$$ (9252*v^11 - 11720*v^10 - 171681*v^9 + 123435*v^8 + 1136338*v^7 - 263751*v^6 - 3070705*v^5 - 365489*v^4 + 3065911*v^3 + 948330*v^2 - 1041702*v - 269892) / 63003 $$\beta_{9}$$ $$=$$ $$( 11122 \nu^{11} + 4724 \nu^{10} - 234641 \nu^{9} - 183751 \nu^{8} + 1678758 \nu^{7} + 1757068 \nu^{6} - 4587698 \nu^{5} - 5542082 \nu^{4} + 4101418 \nu^{3} + \cdots - 1338864 ) / 63003$$ (11122*v^11 + 4724*v^10 - 234641*v^9 - 183751*v^8 + 1678758*v^7 + 1757068*v^6 - 4587698*v^5 - 5542082*v^4 + 4101418*v^3 + 5256201*v^2 - 1041531*v - 1338864) / 63003 $$\beta_{10}$$ $$=$$ $$( - 20291 \nu^{11} + 14554 \nu^{10} + 393757 \nu^{9} - 88219 \nu^{8} - 2649292 \nu^{7} - 451469 \nu^{6} + 6888818 \nu^{5} + 2888145 \nu^{4} - 5681070 \nu^{3} + \cdots + 608583 ) / 63003$$ (-20291*v^11 + 14554*v^10 + 393757*v^9 - 88219*v^8 - 2649292*v^7 - 451469*v^6 + 6888818*v^5 + 2888145*v^4 - 5681070*v^3 - 2909367*v^2 + 1236204*v + 608583) / 63003 $$\beta_{11}$$ $$=$$ $$( 28451 \nu^{11} - 22984 \nu^{10} - 557759 \nu^{9} + 196840 \nu^{8} + 3801976 \nu^{7} + 15784 \nu^{6} - 10129172 \nu^{5} - 2213534 \nu^{4} + 8855582 \nu^{3} + \cdots - 914526 ) / 63003$$ (28451*v^11 - 22984*v^10 - 557759*v^9 + 196840*v^8 + 3801976*v^7 + 15784*v^6 - 10129172*v^5 - 2213534*v^4 + 8855582*v^3 + 2716718*v^2 - 1951401*v - 914526) / 63003
 $$\nu$$ $$=$$ $$( 2\beta_{10} + \beta_{9} + 3\beta_{8} + 3\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 2 ) / 5$$ (2*b10 + b9 + 3*b8 + 3*b6 + 2*b5 - 2*b4 - 2*b3 - 4*b2 + 3*b1 + 2) / 5 $$\nu^{2}$$ $$=$$ $$( 3 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 2 \beta _1 + 18 ) / 5$$ (3*b10 + 4*b9 + 2*b8 + 2*b6 + 3*b5 + 2*b4 + 2*b3 - 6*b2 + 2*b1 + 18) / 5 $$\nu^{3}$$ $$=$$ $$3 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + 4 \beta _1 + 4$$ 3*b10 + 2*b9 + 5*b8 + b7 + 4*b6 + 3*b5 - b4 - 2*b3 - 6*b2 + 4*b1 + 4 $$\nu^{4}$$ $$=$$ $$( - 5 \beta_{11} + 32 \beta_{10} + 41 \beta_{9} + 33 \beta_{8} - 5 \beta_{7} + 28 \beta_{6} + 37 \beta_{5} + 18 \beta_{4} + 8 \beta_{3} - 74 \beta_{2} + 33 \beta _1 + 122 ) / 5$$ (-5*b11 + 32*b10 + 41*b9 + 33*b8 - 5*b7 + 28*b6 + 37*b5 + 18*b4 + 8*b3 - 74*b2 + 33*b1 + 122) / 5 $$\nu^{5}$$ $$=$$ $$( - 10 \beta_{11} + 133 \beta_{10} + 99 \beta_{9} + 207 \beta_{8} + 20 \beta_{7} + 147 \beta_{6} + 153 \beta_{5} + 2 \beta_{4} - 63 \beta_{3} - 306 \beta_{2} + 172 \beta _1 + 238 ) / 5$$ (-10*b11 + 133*b10 + 99*b9 + 207*b8 + 20*b7 + 147*b6 + 153*b5 + 2*b4 - 63*b3 - 306*b2 + 172*b1 + 238) / 5 $$\nu^{6}$$ $$=$$ $$- 15 \beta_{11} + 69 \beta_{10} + 80 \beta_{9} + 81 \beta_{8} - 16 \beta_{7} + 56 \beta_{6} + 88 \beta_{5} + 37 \beta_{4} + \beta_{3} - 164 \beta_{2} + 78 \beta _1 + 204$$ -15*b11 + 69*b10 + 80*b9 + 81*b8 - 16*b7 + 56*b6 + 88*b5 + 37*b4 + b3 - 164*b2 + 78*b1 + 204 $$\nu^{7}$$ $$=$$ $$( - 175 \beta_{11} + 1302 \beta_{10} + 1031 \beta_{9} + 1853 \beta_{8} - 65 \beta_{7} + 1168 \beta_{6} + 1647 \beta_{5} + 318 \beta_{4} - 477 \beta_{3} - 3129 \beta_{2} + 1588 \beta _1 + 2567 ) / 5$$ (-175*b11 + 1302*b10 + 1031*b9 + 1853*b8 - 65*b7 + 1168*b6 + 1647*b5 + 318*b4 - 477*b3 - 3129*b2 + 1588*b1 + 2567) / 5 $$\nu^{8}$$ $$=$$ $$( - 875 \beta_{11} + 3753 \beta_{10} + 3944 \beta_{9} + 4517 \beta_{8} - 1075 \beta_{7} + 2657 \beta_{6} + 5078 \beta_{5} + 2047 \beta_{4} - 423 \beta_{3} - 8866 \beta_{2} + 4212 \beta _1 + 9383 ) / 5$$ (-875*b11 + 3753*b10 + 3944*b9 + 4517*b8 - 1075*b7 + 2657*b6 + 5078*b5 + 2047*b4 - 423*b3 - 8866*b2 + 4212*b1 + 9383) / 5 $$\nu^{9}$$ $$=$$ $$- 463 \beta_{11} + 2662 \beta_{10} + 2182 \beta_{9} + 3516 \beta_{8} - 511 \beta_{7} + 1966 \beta_{6} + 3589 \beta_{5} + 1043 \beta_{4} - 833 \beta_{3} - 6398 \beta_{2} + 3055 \beta _1 + 5343$$ -463*b11 + 2662*b10 + 2182*b9 + 3516*b8 - 511*b7 + 1966*b6 + 3589*b5 + 1043*b4 - 833*b3 - 6398*b2 + 3055*b1 + 5343 $$\nu^{10}$$ $$=$$ $$( - 9580 \beta_{11} + 40577 \beta_{10} + 39526 \beta_{9} + 48468 \beta_{8} - 13430 \beta_{7} + 25073 \beta_{6} + 57032 \beta_{5} + 22963 \beta_{4} - 7372 \beta_{3} - 94789 \beta_{2} + 44143 \beta _1 + 90952 ) / 5$$ (-9580*b11 + 40577*b10 + 39526*b9 + 48468*b8 - 13430*b7 + 25073*b6 + 57032*b5 + 22963*b4 - 7372*b3 - 94789*b2 + 44143*b1 + 90952) / 5 $$\nu^{11}$$ $$=$$ $$( - 27580 \beta_{11} + 138588 \beta_{10} + 115539 \beta_{9} + 173197 \beta_{8} - 39595 \beta_{7} + 86647 \beta_{6} + 194933 \beta_{5} + 67367 \beta_{4} - 39963 \beta_{3} - 328846 \beta_{2} + \cdots + 275263 ) / 5$$ (-27580*b11 + 138588*b10 + 115539*b9 + 173197*b8 - 39595*b7 + 86647*b6 + 194933*b5 + 67367*b4 - 39963*b3 - 328846*b2 + 151142*b1 + 275263) / 5

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.504162 −2.17259 0.571452 2.75578 3.24312 1.91609 −2.46481 −1.99544 −0.969395 −1.14928 1.01267 0.756559
0 −3.42065 0 0 0 −3.76868 0 8.70082 0
1.2 0 −3.13445 0 0 0 −2.92885 0 6.82478 0
1.3 0 −2.94508 0 0 0 3.90152 0 5.67351 0
1.4 0 −2.81354 0 0 0 0.0468346 0 4.91602 0
1.5 0 −0.602175 0 0 0 3.10675 0 −2.63738 0
1.6 0 −0.444623 0 0 0 −2.92309 0 −2.80231 0
1.7 0 −0.208054 0 0 0 0.439180 0 −2.95671 0
1.8 0 0.0330070 0 0 0 2.74120 0 −2.99891 0
1.9 0 1.78053 0 0 0 −4.82418 0 0.170304 0
1.10 0 2.38971 0 0 0 2.31613 0 2.71069 0
1.11 0 2.62870 0 0 0 0.0349473 0 3.91007 0
1.12 0 2.73663 0 0 0 1.85824 0 4.48912 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10000.2.a.bo 12
4.b odd 2 1 5000.2.a.p yes 12
5.b even 2 1 10000.2.a.bp 12
20.d odd 2 1 5000.2.a.o 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5000.2.a.o 12 20.d odd 2 1
5000.2.a.p yes 12 4.b odd 2 1
10000.2.a.bo 12 1.a even 1 1 trivial
10000.2.a.bp 12 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{12} + 4 T_{3}^{11} - 23 T_{3}^{10} - 93 T_{3}^{9} + 195 T_{3}^{8} + 773 T_{3}^{7} - 708 T_{3}^{6} - 2624 T_{3}^{5} + 726 T_{3}^{4} + 2760 T_{3}^{3} + 1165 T_{3}^{2} + 110 T_{3} - 5$$ T3^12 + 4*T3^11 - 23*T3^10 - 93*T3^9 + 195*T3^8 + 773*T3^7 - 708*T3^6 - 2624*T3^5 + 726*T3^4 + 2760*T3^3 + 1165*T3^2 + 110*T3 - 5 $$T_{7}^{12} - 48 T_{7}^{10} + 29 T_{7}^{9} + 824 T_{7}^{8} - 919 T_{7}^{7} - 5896 T_{7}^{6} + 9353 T_{7}^{5} + 13407 T_{7}^{4} - 30798 T_{7}^{3} + 12224 T_{7}^{2} - 848 T_{7} + 16$$ T7^12 - 48*T7^10 + 29*T7^9 + 824*T7^8 - 919*T7^7 - 5896*T7^6 + 9353*T7^5 + 13407*T7^4 - 30798*T7^3 + 12224*T7^2 - 848*T7 + 16 $$T_{11}^{12} - T_{11}^{11} - 68 T_{11}^{10} + 48 T_{11}^{9} + 1533 T_{11}^{8} - 1322 T_{11}^{7} - 14022 T_{11}^{6} + 17573 T_{11}^{5} + 44673 T_{11}^{4} - 78417 T_{11}^{3} + 4252 T_{11}^{2} + 29479 T_{11} - 8609$$ T11^12 - T11^11 - 68*T11^10 + 48*T11^9 + 1533*T11^8 - 1322*T11^7 - 14022*T11^6 + 17573*T11^5 + 44673*T11^4 - 78417*T11^3 + 4252*T11^2 + 29479*T11 - 8609

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 4 T^{11} - 23 T^{10} - 93 T^{9} + \cdots - 5$$
$5$ $$T^{12}$$
$7$ $$T^{12} - 48 T^{10} + 29 T^{9} + 824 T^{8} + \cdots + 16$$
$11$ $$T^{12} - T^{11} - 68 T^{10} + 48 T^{9} + \cdots - 8609$$
$13$ $$T^{12} + 4 T^{11} - 89 T^{10} + \cdots + 302256$$
$17$ $$T^{12} + 8 T^{11} - 106 T^{10} + \cdots - 4824725$$
$19$ $$T^{12} + 9 T^{11} - 73 T^{10} + \cdots + 4621$$
$23$ $$T^{12} - 143 T^{10} + 41 T^{9} + \cdots - 8514864$$
$29$ $$T^{12} - 8 T^{11} - 117 T^{10} + \cdots - 527024$$
$31$ $$T^{12} + 33 T^{11} + 319 T^{10} + \cdots + 9284400$$
$37$ $$T^{12} + 6 T^{11} - 283 T^{10} + \cdots + 650813616$$
$41$ $$T^{12} - 27 T^{11} + \cdots - 995145329$$
$43$ $$T^{12} + 50 T^{11} + \cdots + 347146281$$
$47$ $$T^{12} - 18 T^{11} - 154 T^{10} + \cdots - 26255664$$
$53$ $$T^{12} + 22 T^{11} + \cdots + 189235456$$
$59$ $$T^{12} + 33 T^{11} + 199 T^{10} + \cdots - 4622139$$
$61$ $$T^{12} + 8 T^{11} - 221 T^{10} + \cdots - 8891600$$
$67$ $$T^{12} + 41 T^{11} + \cdots - 23687194259$$
$71$ $$T^{12} + 19 T^{11} + \cdots - 6051744080$$
$73$ $$T^{12} - 5 T^{11} - 418 T^{10} + \cdots + 927999711$$
$79$ $$T^{12} + 58 T^{11} + \cdots + 39511336720$$
$83$ $$T^{12} + 18 T^{11} + \cdots - 4396075199$$
$89$ $$T^{12} - 44 T^{11} + \cdots - 82788670169$$
$97$ $$T^{12} + 22 T^{11} + \cdots + 171567421$$