# Properties

 Label 1875.2.b.g Level $1875$ Weight $2$ Character orbit 1875.b Analytic conductor $14.972$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256$$ x^16 + 25*x^14 + 246*x^12 + 1220*x^10 + 3281*x^8 + 4880*x^6 + 3936*x^4 + 1600*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + \beta_{11} q^{3} + ( - \beta_{8} + \beta_{7} - \beta_{3} - 1) q^{4} + ( - \beta_{13} + \beta_{9} + \beta_{6} + 1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{8} - q^{9}+O(q^{10})$$ q + (b2 + b1) * q^2 + b11 * q^3 + (-b8 + b7 - b3 - 1) * q^4 + (-b13 + b9 + b6 + 1) * q^6 + (-b11 + b10 + b2) * q^7 + (-b5 + b4 - 2*b2) * q^8 - q^9 $$q + (\beta_{2} + \beta_1) q^{2} + \beta_{11} q^{3} + ( - \beta_{8} + \beta_{7} - \beta_{3} - 1) q^{4} + ( - \beta_{13} + \beta_{9} + \beta_{6} + 1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{8} - q^{9} + (\beta_{15} + \beta_{13} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{3} + 2) q^{11} + ( - \beta_{14} - \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{5}) q^{12} + ( - \beta_{12} + \beta_{11} + \beta_{5} + \beta_{2} + \beta_1) q^{13} + (\beta_{13} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3} - 2) q^{14} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} + 4 \beta_{3} + 1) q^{16} + (\beta_{14} - \beta_{12} + \beta_{10} - \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{2} - \beta_1) q^{18} + (\beta_{15} - 3 \beta_{13} - \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{19} + ( - 2 \beta_{13} - \beta_{8} + \beta_{6} + 3) q^{21} + ( - 2 \beta_{14} - \beta_{11} - \beta_{10} - \beta_{5} + \beta_{4} + 5 \beta_{2} + 3 \beta_1) q^{22} + ( - 2 \beta_{14} - 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{5} - \beta_{4} - \beta_{2} - 3 \beta_1) q^{23} + ( - \beta_{15} + 3 \beta_{13} - 3 \beta_{6} - 3) q^{24} + (\beta_{13} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3} - 4) q^{26} - \beta_{11} q^{27} + (\beta_{14} + \beta_{12} - \beta_{11} - 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{28} + (2 \beta_{15} + \beta_{13} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3}) q^{29} + (\beta_{15} + \beta_{13} + \beta_{9} + 3 \beta_{7} + \beta_{6} - \beta_{3} + 1) q^{31} + (\beta_{11} + \beta_{10} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{32} + ( - \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{4}) q^{33} + (\beta_{15} + \beta_{13} + 2 \beta_{8} - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{3} + 5) q^{34} + (\beta_{8} - \beta_{7} + \beta_{3} + 1) q^{36} + ( - \beta_{14} - \beta_{12} + \beta_{11} - \beta_{5} + \beta_{4}) q^{37} + ( - 6 \beta_{14} - 4 \beta_{12} - 7 \beta_{11} - \beta_{10} + 8 \beta_{5} - 2 \beta_{4} + \cdots - 4 \beta_1) q^{38}+ \cdots + ( - \beta_{15} - \beta_{13} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - 2) q^{99}+O(q^{100})$$ q + (b2 + b1) * q^2 + b11 * q^3 + (-b8 + b7 - b3 - 1) * q^4 + (-b13 + b9 + b6 + 1) * q^6 + (-b11 + b10 + b2) * q^7 + (-b5 + b4 - 2*b2) * q^8 - q^9 + (b15 + b13 + b8 - b7 + b6 - b3 + 2) * q^11 + (-b14 - b12 - 2*b11 - b10 + b5) * q^12 + (-b12 + b11 + b5 + b2 + b1) * q^13 + (b13 - b8 + 2*b7 + 2*b6 - 2*b3 - 2) * q^14 + (-b9 + b8 - 2*b7 - b6 + 4*b3 + 1) * q^16 + (b14 - b12 + b10 - b5 - 2*b2 - 2*b1) * q^17 + (-b2 - b1) * q^18 + (b15 - 3*b13 - b7 + b6 + 2*b3) * q^19 + (-2*b13 - b8 + b6 + 3) * q^21 + (-2*b14 - b11 - b10 - b5 + b4 + 5*b2 + 3*b1) * q^22 + (-2*b14 - 2*b12 - b11 - b10 + b5 - b4 - b2 - 3*b1) * q^23 + (-b15 + 3*b13 - 3*b6 - 3) * q^24 + (b13 + 2*b9 - b8 + 2*b7 + 2*b6 - 2*b3 - 4) * q^26 - b11 * q^27 + (b14 + b12 - b11 - 5*b5 + 2*b4 - 2*b2 - 2*b1) * q^28 + (2*b15 + b13 - b9 - b8 - b7 - b6 + b3) * q^29 + (b15 + b13 + b9 + 3*b7 + b6 - b3 + 1) * q^31 + (b11 + b10 + 4*b5 - 2*b4 + 2*b1) * q^32 + (-b14 + b12 + 2*b11 + b10 + b4) * q^33 + (b15 + b13 + 2*b8 - 2*b7 + 3*b6 + 2*b3 + 5) * q^34 + (b8 - b7 + b3 + 1) * q^36 + (-b14 - b12 + b11 - b5 + b4) * q^37 + (-6*b14 - 4*b12 - 7*b11 - b10 + 8*b5 - 2*b4 - 4*b2 - 4*b1) * q^38 + (-2*b13 + b9 - b7 + b6) * q^39 + (b15 - b13 - b8 - b7 + b6 + b3) * q^41 + (-2*b14 - 2*b12 - 5*b11 - b10 + 4*b5 - 2*b2) * q^42 + (-b14 - 2*b12 + b11 - b10 + 3*b5 - 2*b4 - 2*b2 - 4*b1) * q^43 + (5*b13 - 3*b9 - 2*b8 - 5*b6 - 2*b3 - 9) * q^44 + (-2*b15 + 7*b13 - b9 - 5*b6 - 2*b3 - 1) * q^46 + (-4*b14 + 3*b11 + b10 + b5 - b4) * q^47 + (4*b14 + 2*b12 + 3*b11 + b10 - 2*b5 + b2 + b1) * q^48 + (b15 + 2*b13 + 3*b9 + 2*b8 + 2*b7 - b3 - 5) * q^49 + (2*b13 - 2*b9 - b8 - b7 - 2*b6 - b3 - 1) * q^51 + (b14 - b12 - b11 - 2*b10 - 3*b5 + 2*b4 - 4*b2 - 2*b1) * q^52 + (2*b11 + 2*b10 - 3*b5 - b4 + b2 + b1) * q^53 + (b13 - b9 - b6 - 1) * q^54 + (b15 - 6*b9 + 2*b8 - 5*b7 - b6 + 9*b3 + 6) * q^56 + (2*b14 + b12 - b11 - 3*b5 + b4) * q^57 + (-5*b14 - b12 - 2*b11 + b10 + 6*b5 - b4 - 4*b2 - 2*b1) * q^58 + (-2*b15 + b13 - b9 - b8 - b7 + b6 - b3 - 2) * q^59 + (2*b15 + 3*b13 - 3*b9 - b8 - b7 - 4*b6 - b3) * q^61 + (-2*b14 + 3*b11 - b10 - 7*b5 + b4 + b2 + b1) * q^62 + (b11 - b10 - b2) * q^63 + (-b13 + 4*b9 + 3*b7 + 6*b6 - 3*b3 - 3) * q^64 + (-b15 - 3*b13 + 3*b9 + b8 + 4*b6 + 2*b3 + 4) * q^66 + (-2*b12 - 7*b11 + 7*b5 - 2*b4 - 4*b2 + b1) * q^67 + (-2*b12 - 3*b11 - b10 - 2*b4 + 7*b2 + 3*b1) * q^68 + (b15 + b13 - 3*b9 + b8 - 2*b7 + 2*b3) * q^69 + (-2*b15 + 3*b13 + 3*b9 - b6 - 2*b3 + 1) * q^71 + (b5 - b4 + 2*b2) * q^72 + (3*b14 - b12 - 2*b11 - 2*b5 + 2*b2) * q^73 + (-b15 + 2*b13 + b8 - 2*b7 - b6 + 4*b3 - 1) * q^74 + (-4*b15 + 15*b13 + 2*b8 + 3*b7 - 6*b6 - 5*b3 - 11) * q^76 + (-2*b14 - 6*b12 + b5 - 3*b4 - 2*b2 - 2*b1) * q^77 + (-2*b14 - 2*b12 - 7*b11 - b10 + 4*b5 - 2*b2 - 2*b1) * q^78 + (-3*b15 + 2*b13 + 5*b9 - b8 + 4*b7 + 2*b6 - 2*b3 - 3) * q^79 + q^81 + (-4*b14 - 2*b12 - 7*b11 - b10 + 7*b5 - b4 - 5*b2 - 3*b1) * q^82 + (-2*b12 - 6*b11 + 2*b5 - 3*b2 + b1) * q^83 + (-2*b15 + 7*b13 - 2*b9 + b7 - 4*b6 - b3 - 3) * q^84 + (-b15 + 4*b13 + 3*b9 + 2*b7 - 6*b3 + 4) * q^86 + (b14 + b12 - b10 - b5 + 2*b4 + 3*b2 + b1) * q^87 + (b14 + 5*b12 + 10*b11 + 3*b10 - 5*b5 + 4*b4) * q^88 + (-b9 + b8 - 4*b7 + 3*b6 + 3) * q^89 + (-b13 - b9 - 3*b8 + 5*b7 + 4*b6 - b3 - 1) * q^91 + (9*b14 + 5*b12 + 14*b11 + 3*b10 - 9*b5 + 6*b2 + 2*b1) * q^92 + (-b14 - 3*b12 + b5 + b4 - b1) * q^93 + (-4*b15 + b13 + 5*b9 - b8 + 3*b7 + 3*b6 - 5*b3 - 2) * q^94 + (2*b15 - 5*b13 + 2*b9 - b8 + 2*b6 + 2) * q^96 + (-5*b11 - 2*b10 + 3*b5 - b2 - 2*b1) * q^97 + (-b14 + b12 + 3*b11 - 10*b5 + b4 + 2*b2 - 2*b1) * q^98 + (-b15 - b13 - b8 + b7 - b6 + b3 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10})$$ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 $$16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100})$$ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{15} + 25\nu^{13} + 246\nu^{11} + 1220\nu^{9} + 3281\nu^{7} + 4880\nu^{5} + 3936\nu^{3} + 1472\nu ) / 128$$ (v^15 + 25*v^13 + 246*v^11 + 1220*v^9 + 3281*v^7 + 4880*v^5 + 3936*v^3 + 1472*v) / 128 $$\beta_{3}$$ $$=$$ $$( \nu^{14} + 25\nu^{12} + 246\nu^{10} + 1220\nu^{8} + 3281\nu^{6} + 4880\nu^{4} + 3872\nu^{2} + 1216 ) / 64$$ (v^14 + 25*v^12 + 246*v^10 + 1220*v^8 + 3281*v^6 + 4880*v^4 + 3872*v^2 + 1216) / 64 $$\beta_{4}$$ $$=$$ $$( -3\nu^{15} - 75\nu^{13} - 738\nu^{11} - 3660\nu^{9} - 9843\nu^{7} - 14640\nu^{5} - 11680\nu^{3} - 3904\nu ) / 128$$ (-3*v^15 - 75*v^13 - 738*v^11 - 3660*v^9 - 9843*v^7 - 14640*v^5 - 11680*v^3 - 3904*v) / 128 $$\beta_{5}$$ $$=$$ $$( 4\nu^{15} + 99\nu^{13} + 959\nu^{11} + 4634\nu^{9} + 11904\nu^{7} + 16239\nu^{5} + 10896\nu^{3} + 2752\nu ) / 32$$ (4*v^15 + 99*v^13 + 959*v^11 + 4634*v^9 + 11904*v^7 + 16239*v^5 + 10896*v^3 + 2752*v) / 32 $$\beta_{6}$$ $$=$$ $$( -13\nu^{14} - 321\nu^{12} - 3098\nu^{10} - 14876\nu^{8} - 37773\nu^{6} - 50380\nu^{4} - 32416\nu^{2} - 7744 ) / 64$$ (-13*v^14 - 321*v^12 - 3098*v^10 - 14876*v^8 - 37773*v^6 - 50380*v^4 - 32416*v^2 - 7744) / 64 $$\beta_{7}$$ $$=$$ $$( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18592\nu^{2} + 4032 ) / 32$$ (9*v^14 + 221*v^12 + 2114*v^10 + 9996*v^8 + 24681*v^6 + 31308*v^4 + 18592*v^2 + 4032) / 32 $$\beta_{8}$$ $$=$$ $$( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18624\nu^{2} + 4128 ) / 32$$ (9*v^14 + 221*v^12 + 2114*v^10 + 9996*v^8 + 24681*v^6 + 31308*v^4 + 18624*v^2 + 4128) / 32 $$\beta_{9}$$ $$=$$ $$( -5\nu^{14} - 123\nu^{12} - 1180\nu^{10} - 5608\nu^{8} - 13981\nu^{6} - 18078\nu^{4} - 11120\nu^{2} - 2528 ) / 16$$ (-5*v^14 - 123*v^12 - 1180*v^10 - 5608*v^8 - 13981*v^6 - 18078*v^4 - 11120*v^2 - 2528) / 16 $$\beta_{10}$$ $$=$$ $$( 21\nu^{15} + 519\nu^{13} + 5016\nu^{11} + 24144\nu^{9} + 61581\nu^{7} + 82858\nu^{5} + 54208\nu^{3} + 13248\nu ) / 64$$ (21*v^15 + 519*v^13 + 5016*v^11 + 24144*v^9 + 61581*v^7 + 82858*v^5 + 54208*v^3 + 13248*v) / 64 $$\beta_{11}$$ $$=$$ $$( 79 \nu^{15} + 1935 \nu^{13} + 18450 \nu^{11} + 86940 \nu^{9} + 214335 \nu^{7} + 273672 \nu^{5} + 166320 \nu^{3} + 37440 \nu ) / 128$$ (79*v^15 + 1935*v^13 + 18450*v^11 + 86940*v^9 + 214335*v^7 + 273672*v^5 + 166320*v^3 + 37440*v) / 128 $$\beta_{12}$$ $$=$$ $$( 23\nu^{15} + 561\nu^{13} + 5316\nu^{11} + 24816\nu^{9} + 60319\nu^{7} + 75490\nu^{5} + 44712\nu^{3} + 9792\nu ) / 32$$ (23*v^15 + 561*v^13 + 5316*v^11 + 24816*v^9 + 60319*v^7 + 75490*v^5 + 44712*v^3 + 9792*v) / 32 $$\beta_{13}$$ $$=$$ $$( 23\nu^{14} + 561\nu^{12} + 5316\nu^{10} + 24816\nu^{8} + 60319\nu^{6} + 75490\nu^{4} + 44712\nu^{2} + 9824 ) / 32$$ (23*v^14 + 561*v^12 + 5316*v^10 + 24816*v^8 + 60319*v^6 + 75490*v^4 + 44712*v^2 + 9824) / 32 $$\beta_{14}$$ $$=$$ $$( 151 \nu^{15} + 3683 \nu^{13} + 34910 \nu^{11} + 163124 \nu^{9} + 397463 \nu^{7} + 500052 \nu^{5} + 298912 \nu^{3} + 66304 \nu ) / 128$$ (151*v^15 + 3683*v^13 + 34910*v^11 + 163124*v^9 + 397463*v^7 + 500052*v^5 + 298912*v^3 + 66304*v) / 128 $$\beta_{15}$$ $$=$$ $$( -23\nu^{14} - 560\nu^{12} - 5295\nu^{10} - 24654\nu^{8} - 59763\nu^{6} - 74673\nu^{4} - 44268\nu^{2} - 9744 ) / 16$$ (-23*v^14 - 560*v^12 - 5295*v^10 - 24654*v^8 - 59763*v^6 - 74673*v^4 - 44268*v^2 - 9744) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} - 3$$ b8 - b7 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{4} + 3\beta_{2} - 4\beta_1$$ b4 + 3*b2 - 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{9} - 7\beta_{8} + 8\beta_{7} + 2\beta_{3} + 15$$ b9 - 7*b8 + 8*b7 + 2*b3 + 15 $$\nu^{5}$$ $$=$$ $$\beta_{10} - 3\beta_{5} - 10\beta_{4} - 24\beta_{2} + 22\beta_1$$ b10 - 3*b5 - 10*b4 - 24*b2 + 22*b1 $$\nu^{6}$$ $$=$$ $$-14\beta_{9} + 46\beta_{8} - 59\beta_{7} + 2\beta_{6} - 20\beta_{3} - 90$$ -14*b9 + 46*b8 - 59*b7 + 2*b6 - 20*b3 - 90 $$\nu^{7}$$ $$=$$ $$-\beta_{12} + 2\beta_{11} - 16\beta_{10} + 42\beta_{5} + 79\beta_{4} + 171\beta_{2} - 136\beta_1$$ -b12 + 2*b11 - 16*b10 + 42*b5 + 79*b4 + 171*b2 - 136*b1 $$\nu^{8}$$ $$=$$ $$\beta_{15} + 3\beta_{13} + 139\beta_{9} - 307\beta_{8} + 427\beta_{7} - 32\beta_{6} + 158\beta_{3} + 577$$ b15 + 3*b13 + 139*b9 - 307*b8 + 427*b7 - 32*b6 + 158*b3 + 577 $$\nu^{9}$$ $$=$$ $$- 2 \beta_{14} + 23 \beta_{12} - 38 \beta_{11} + 172 \beta_{10} - 412 \beta_{5} - 585 \beta_{4} - 1199 \beta_{2} + 887 \beta_1$$ -2*b14 + 23*b12 - 38*b11 + 172*b10 - 412*b5 - 585*b4 - 1199*b2 + 887*b1 $$\nu^{10}$$ $$=$$ $$-25\beta_{15} - 71\beta_{13} - 1207\beta_{9} + 2086\beta_{8} - 3060\beta_{7} + 344\beta_{6} - 1170\beta_{3} - 3814$$ -25*b15 - 71*b13 - 1207*b9 + 2086*b8 - 3060*b7 + 344*b6 - 1170*b3 - 3814 $$\nu^{11}$$ $$=$$ $$50 \beta_{14} - 329 \beta_{12} + 466 \beta_{11} - 1576 \beta_{10} + 3524 \beta_{5} + 4230 \beta_{4} + 8402 \beta_{2} - 5971 \beta_1$$ 50*b14 - 329*b12 + 466*b11 - 1576*b10 + 3524*b5 + 4230*b4 + 8402*b2 - 5971*b1 $$\nu^{12}$$ $$=$$ $$379 \beta_{15} + 1037 \beta_{13} + 9796 \beta_{9} - 14373 \beta_{8} + 21798 \beta_{7} - 3152 \beta_{6} + 8460 \beta_{3} + 25657$$ 379*b15 + 1037*b13 + 9796*b9 - 14373*b8 + 21798*b7 - 3152*b6 + 8460*b3 + 25657 $$\nu^{13}$$ $$=$$ $$- 758 \beta_{14} + 3787 \beta_{12} - 4742 \beta_{11} + 13327 \beta_{10} - 28177 \beta_{5} - 30258 \beta_{4} - 59004 \beta_{2} + 41067 \beta_1$$ -758*b14 + 3787*b12 - 4742*b11 + 13327*b10 - 28177*b5 - 30258*b4 - 59004*b2 + 41067*b1 $$\nu^{14}$$ $$=$$ $$- 4545 \beta_{15} - 12119 \beta_{13} - 76504 \beta_{9} + 100071 \beta_{8} - 154719 \beta_{7} + 26654 \beta_{6} - 60516 \beta_{3} - 174631$$ -4545*b15 - 12119*b13 - 76504*b9 + 100071*b8 - 154719*b7 + 26654*b6 - 60516*b3 - 174631 $$\nu^{15}$$ $$=$$ $$9090 \beta_{14} - 38520 \beta_{12} + 43712 \beta_{11} - 107703 \beta_{10} + 216999 \beta_{5} + 215235 \beta_{4} + 415377 \beta_{2} - 286821 \beta_1$$ 9090*b14 - 38520*b12 + 43712*b11 - 107703*b10 + 216999*b5 + 215235*b4 + 415377*b2 - 286821*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$$.

 $$n$$ $$626$$ $$1252$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 0.741379i − 0.770071i − 0.895394i − 1.31354i − 1.52260i − 2.23365i − 2.59716i − 2.69767i 2.69767i 2.59716i 2.23365i 1.52260i 1.31354i 0.895394i 0.770071i 0.741379i
2.69767i 1.00000i −5.27745 0 −2.69767 3.56649i 8.84149i −1.00000 0
1249.2 2.59716i 1.00000i −4.74525 0 2.59716 3.28414i 7.12986i −1.00000 0
1249.3 2.23365i 1.00000i −2.98921 0 2.23365 1.03143i 2.20956i −1.00000 0
1249.4 1.52260i 1.00000i −0.318310 0 −1.52260 0.990985i 2.56054i −1.00000 0
1249.5 1.31354i 1.00000i 0.274605 0 1.31354 4.19091i 2.98779i −1.00000 0
1249.6 0.895394i 1.00000i 1.19827 0 −0.895394 5.08992i 2.86371i −1.00000 0
1249.7 0.770071i 1.00000i 1.40699 0 −0.770071 3.98808i 2.62363i −1.00000 0
1249.8 0.741379i 1.00000i 1.45036 0 0.741379 1.03586i 2.55802i −1.00000 0
1249.9 0.741379i 1.00000i 1.45036 0 0.741379 1.03586i 2.55802i −1.00000 0
1249.10 0.770071i 1.00000i 1.40699 0 −0.770071 3.98808i 2.62363i −1.00000 0
1249.11 0.895394i 1.00000i 1.19827 0 −0.895394 5.08992i 2.86371i −1.00000 0
1249.12 1.31354i 1.00000i 0.274605 0 1.31354 4.19091i 2.98779i −1.00000 0
1249.13 1.52260i 1.00000i −0.318310 0 −1.52260 0.990985i 2.56054i −1.00000 0
1249.14 2.23365i 1.00000i −2.98921 0 2.23365 1.03143i 2.20956i −1.00000 0
1249.15 2.59716i 1.00000i −4.74525 0 2.59716 3.28414i 7.12986i −1.00000 0
1249.16 2.69767i 1.00000i −5.27745 0 −2.69767 3.56649i 8.84149i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.2.b.g 16
5.b even 2 1 inner 1875.2.b.g 16
5.c odd 4 1 1875.2.a.n 8
5.c odd 4 1 1875.2.a.o yes 8
15.e even 4 1 5625.2.a.u 8
15.e even 4 1 5625.2.a.bc 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.n 8 5.c odd 4 1
1875.2.a.o yes 8 5.c odd 4 1
1875.2.b.g 16 1.a even 1 1 trivial
1875.2.b.g 16 5.b even 2 1 inner
5625.2.a.u 8 15.e even 4 1
5625.2.a.bc 8 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 25T_{2}^{14} + 246T_{2}^{12} + 1220T_{2}^{10} + 3281T_{2}^{8} + 4880T_{2}^{6} + 3936T_{2}^{4} + 1600T_{2}^{2} + 256$$ acting on $$S_{2}^{\mathrm{new}}(1875, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 25 T^{14} + 246 T^{12} + \cdots + 256$$
$3$ $$(T^{2} + 1)^{8}$$
$5$ $$T^{16}$$
$7$ $$T^{16} + 86 T^{14} + 2941 T^{12} + \cdots + 1113025$$
$11$ $$(T^{8} - 12 T^{7} + 2 T^{6} + 455 T^{5} + \cdots + 16)^{2}$$
$13$ $$T^{16} + 110 T^{14} + \cdots + 76230361$$
$17$ $$T^{16} + 177 T^{14} + \cdots + 74926336$$
$19$ $$(T^{8} + 16 T^{7} + 51 T^{6} - 554 T^{5} + \cdots - 14975)^{2}$$
$23$ $$T^{16} + 224 T^{14} + \cdots + 824838400$$
$29$ $$(T^{8} + 2 T^{7} - 126 T^{6} - 257 T^{5} + \cdots + 57520)^{2}$$
$31$ $$(T^{8} - 13 T^{7} - 96 T^{6} + \cdots + 801025)^{2}$$
$37$ $$T^{16} + 126 T^{14} + 5021 T^{12} + \cdots + 625$$
$41$ $$(T^{8} + 12 T^{7} - 6 T^{6} - 487 T^{5} + \cdots - 48080)^{2}$$
$43$ $$T^{16} + 388 T^{14} + \cdots + 5530748161$$
$47$ $$T^{16} + 367 T^{14} + \cdots + 65445918976$$
$53$ $$T^{16} + 334 T^{14} + \cdots + 824838400$$
$59$ $$(T^{8} + 14 T^{7} - 164 T^{6} + \cdots + 11920)^{2}$$
$61$ $$(T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 1093919)^{2}$$
$67$ $$T^{16} + \cdots + 408210546216841$$
$71$ $$(T^{8} - 21 T^{7} - 18 T^{6} + 2102 T^{5} + \cdots + 67696)^{2}$$
$73$ $$T^{16} + 319 T^{14} + \cdots + 144137919025$$
$79$ $$(T^{8} + 10 T^{7} - 320 T^{6} + \cdots + 6951025)^{2}$$
$83$ $$T^{16} + 465 T^{14} + \cdots + 12937697536$$
$89$ $$(T^{8} - 9 T^{7} - 294 T^{6} + \cdots - 12105680)^{2}$$
$97$ $$T^{16} + 402 T^{14} + \cdots + 161554959721$$