# Properties

 Label 1850.2.b.n Level $1850$ Weight $2$ Character orbit 1850.b Analytic conductor $14.772$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( - \beta_{5} + \beta_{3}) q^{3} - q^{4} + ( - \beta_1 - 1) q^{6} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{7} - \beta_{3} q^{8} + ( - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10})$$ q + b3 * q^2 + (-b5 + b3) * q^3 - q^4 + (-b1 - 1) * q^6 + (-b5 + 2*b4 - 2*b3) * q^7 - b3 * q^8 + (-b2 - b1 - 1) * q^9 $$q + \beta_{3} q^{2} + ( - \beta_{5} + \beta_{3}) q^{3} - q^{4} + ( - \beta_1 - 1) q^{6} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{7} - \beta_{3} q^{8} + ( - \beta_{2} - \beta_1 - 1) q^{9} + (\beta_{2} + 2 \beta_1) q^{11} + (\beta_{5} - \beta_{3}) q^{12} + ( - \beta_{5} + 4 \beta_{3}) q^{13} + (2 \beta_{2} - \beta_1 + 2) q^{14} + q^{16} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{17} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{18} + ( - 3 \beta_{2} + \beta_1 - 2) q^{19} + (3 \beta_{2} + 2 \beta_1 + 1) q^{21} + ( - 2 \beta_{5} - \beta_{4}) q^{22} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{23} + (\beta_1 + 1) q^{24} + ( - \beta_1 - 4) q^{26} + ( - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3}) q^{27} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{28} + (4 \beta_{2} + 2) q^{29} + ( - 3 \beta_{2} - 3) q^{31} + \beta_{3} q^{32} + ( - 4 \beta_{4} + 7 \beta_{3}) q^{33} + ( - 2 \beta_{2} - \beta_1 - 1) q^{34} + (\beta_{2} + \beta_1 + 1) q^{36} - \beta_{3} q^{37} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3}) q^{38} + ( - \beta_{2} - 4 \beta_1 - 7) q^{39} + ( - 2 \beta_1 - 7) q^{41} + ( - 2 \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{42} + (3 \beta_{5} + 5 \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{2} - 2 \beta_1 - 2) q^{46} - 4 \beta_{4} q^{47} + ( - \beta_{5} + \beta_{3}) q^{48} + ( - \beta_{2} + \beta_1 - 4) q^{49} + ( - 5 \beta_{2} - \beta_1 - 6) q^{51} + (\beta_{5} - 4 \beta_{3}) q^{52} + (2 \beta_{5} - 6 \beta_{4} + 4 \beta_{3}) q^{53} + (3 \beta_{2} - 2 \beta_1 + 2) q^{54} + ( - 2 \beta_{2} + \beta_1 - 2) q^{56} + (2 \beta_{5} + 5 \beta_{4} - 2 \beta_{3}) q^{57} + ( - 4 \beta_{4} + 2 \beta_{3}) q^{58} + (5 \beta_{2} + 3 \beta_1 - 1) q^{59} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{61} + (3 \beta_{4} - 3 \beta_{3}) q^{62} + ( - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3}) q^{63} - q^{64} + ( - 4 \beta_{2} - 7) q^{66} + (3 \beta_{5} - 4 \beta_{4} - 3 \beta_{3}) q^{67} + (\beta_{5} + 2 \beta_{4} - \beta_{3}) q^{68} + (2 \beta_{2} - 2 \beta_1 - 6) q^{69} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{71} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{72} + ( - \beta_{5} + \beta_{4} + 10 \beta_{3}) q^{73} + q^{74} + (3 \beta_{2} - \beta_1 + 2) q^{76} + (8 \beta_{5} + \beta_{4} - \beta_{3}) q^{77} + (4 \beta_{5} + \beta_{4} - 7 \beta_{3}) q^{78} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{79} + (\beta_{2} - \beta_1 - 4) q^{81} + (2 \beta_{5} - 7 \beta_{3}) q^{82} + ( - 8 \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{83} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{84} + (5 \beta_{2} + 3 \beta_1 - 1) q^{86} + ( - 2 \beta_{5} - 8 \beta_{4} + 6 \beta_{3}) q^{87} + (2 \beta_{5} + \beta_{4}) q^{88} - 3 \beta_{2} q^{89} + (9 \beta_{2} - \beta_1 + 7) q^{91} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{92} + (3 \beta_{5} + 6 \beta_{4} - 6 \beta_{3}) q^{93} - 4 \beta_{2} q^{94} + ( - \beta_1 - 1) q^{96} + (2 \beta_{5} - 2 \beta_{3}) q^{97} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3}) q^{98} + ( - 5 \beta_{2} - \beta_1 - 11) q^{99}+O(q^{100})$$ q + b3 * q^2 + (-b5 + b3) * q^3 - q^4 + (-b1 - 1) * q^6 + (-b5 + 2*b4 - 2*b3) * q^7 - b3 * q^8 + (-b2 - b1 - 1) * q^9 + (b2 + 2*b1) * q^11 + (b5 - b3) * q^12 + (-b5 + 4*b3) * q^13 + (2*b2 - b1 + 2) * q^14 + q^16 + (-b5 - 2*b4 + b3) * q^17 + (b5 + b4 - b3) * q^18 + (-3*b2 + b1 - 2) * q^19 + (3*b2 + 2*b1 + 1) * q^21 + (-2*b5 - b4) * q^22 + (-2*b5 + 2*b4 + 2*b3) * q^23 + (b1 + 1) * q^24 + (-b1 - 4) * q^26 + (-2*b5 + 3*b4 - 2*b3) * q^27 + (b5 - 2*b4 + 2*b3) * q^28 + (4*b2 + 2) * q^29 + (-3*b2 - 3) * q^31 + b3 * q^32 + (-4*b4 + 7*b3) * q^33 + (-2*b2 - b1 - 1) * q^34 + (b2 + b1 + 1) * q^36 - b3 * q^37 + (-b5 + 3*b4 - 2*b3) * q^38 + (-b2 - 4*b1 - 7) * q^39 + (-2*b1 - 7) * q^41 + (-2*b5 - 3*b4 + b3) * q^42 + (3*b5 + 5*b4 + b3) * q^43 + (-b2 - 2*b1) * q^44 + (2*b2 - 2*b1 - 2) * q^46 - 4*b4 * q^47 + (-b5 + b3) * q^48 + (-b2 + b1 - 4) * q^49 + (-5*b2 - b1 - 6) * q^51 + (b5 - 4*b3) * q^52 + (2*b5 - 6*b4 + 4*b3) * q^53 + (3*b2 - 2*b1 + 2) * q^54 + (-2*b2 + b1 - 2) * q^56 + (2*b5 + 5*b4 - 2*b3) * q^57 + (-4*b4 + 2*b3) * q^58 + (5*b2 + 3*b1 - 1) * q^59 + (-4*b2 + 3*b1 - 4) * q^61 + (3*b4 - 3*b3) * q^62 + (-4*b5 - 2*b4 + 4*b3) * q^63 - q^64 + (-4*b2 - 7) * q^66 + (3*b5 - 4*b4 - 3*b3) * q^67 + (b5 + 2*b4 - b3) * q^68 + (2*b2 - 2*b1 - 6) * q^69 + (-4*b2 + 3*b1 - 4) * q^71 + (-b5 - b4 + b3) * q^72 + (-b5 + b4 + 10*b3) * q^73 + q^74 + (3*b2 - b1 + 2) * q^76 + (8*b5 + b4 - b3) * q^77 + (4*b5 + b4 - 7*b3) * q^78 + (-2*b2 + 2*b1 - 8) * q^79 + (b2 - b1 - 4) * q^81 + (2*b5 - 7*b3) * q^82 + (-8*b5 + b4 + 2*b3) * q^83 + (-3*b2 - 2*b1 - 1) * q^84 + (5*b2 + 3*b1 - 1) * q^86 + (-2*b5 - 8*b4 + 6*b3) * q^87 + (2*b5 + b4) * q^88 - 3*b2 * q^89 + (9*b2 - b1 + 7) * q^91 + (2*b5 - 2*b4 - 2*b3) * q^92 + (3*b5 + 6*b4 - 6*b3) * q^93 - 4*b2 * q^94 + (-b1 - 1) * q^96 + (2*b5 - 2*b3) * q^97 + (-b5 + b4 - 4*b3) * q^98 + (-5*b2 - b1 - 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 $$6 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 2 q^{11} + 8 q^{14} + 6 q^{16} - 6 q^{19} + 6 q^{24} - 24 q^{26} + 4 q^{29} - 12 q^{31} - 2 q^{34} + 4 q^{36} - 40 q^{39} - 42 q^{41} + 2 q^{44} - 16 q^{46} - 22 q^{49} - 26 q^{51} + 6 q^{54} - 8 q^{56} - 16 q^{59} - 16 q^{61} - 6 q^{64} - 34 q^{66} - 40 q^{69} - 16 q^{71} + 6 q^{74} + 6 q^{76} - 44 q^{79} - 26 q^{81} - 16 q^{86} + 6 q^{89} + 24 q^{91} + 8 q^{94} - 6 q^{96} - 56 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 - 2 * q^11 + 8 * q^14 + 6 * q^16 - 6 * q^19 + 6 * q^24 - 24 * q^26 + 4 * q^29 - 12 * q^31 - 2 * q^34 + 4 * q^36 - 40 * q^39 - 42 * q^41 + 2 * q^44 - 16 * q^46 - 22 * q^49 - 26 * q^51 + 6 * q^54 - 8 * q^56 - 16 * q^59 - 16 * q^61 - 6 * q^64 - 34 * q^66 - 40 * q^69 - 16 * q^71 + 6 * q^74 + 6 * q^76 - 44 * q^79 - 26 * q^81 - 16 * q^86 + 6 * q^89 + 24 * q^91 + 8 * q^94 - 6 * q^96 - 56 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 0.403032 − 0.403032i −0.854638 + 0.854638i 1.45161 − 1.45161i 1.45161 + 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i
1.00000i 2.67513i −1.00000 0 −2.67513 3.28726i 1.00000i −4.15633 0
149.2 1.00000i 1.53919i −1.00000 0 −1.53919 2.87936i 1.00000i 0.630898 0
149.3 1.00000i 1.21432i −1.00000 0 1.21432 3.59210i 1.00000i 1.52543 0
149.4 1.00000i 1.21432i −1.00000 0 1.21432 3.59210i 1.00000i 1.52543 0
149.5 1.00000i 1.53919i −1.00000 0 −1.53919 2.87936i 1.00000i 0.630898 0
149.6 1.00000i 2.67513i −1.00000 0 −2.67513 3.28726i 1.00000i −4.15633 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.6 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 1850.2.b.n 6
5.b even 2 1 inner 1850.2.b.n 6
5.c odd 4 1 1850.2.a.ba 3
5.c odd 4 1 1850.2.a.bb yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.ba 3 5.c odd 4 1
1850.2.a.bb yes 3 5.c odd 4 1
1850.2.b.n 6 1.a even 1 1 trivial
1850.2.b.n 6 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1850, [\chi])$$:

 $$T_{3}^{6} + 11T_{3}^{4} + 31T_{3}^{2} + 25$$ T3^6 + 11*T3^4 + 31*T3^2 + 25 $$T_{7}^{6} + 32T_{7}^{4} + 336T_{7}^{2} + 1156$$ T7^6 + 32*T7^4 + 336*T7^2 + 1156 $$T_{13}^{6} + 56T_{13}^{4} + 832T_{13}^{2} + 2116$$ T13^6 + 56*T13^4 + 832*T13^2 + 2116

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6} + 11 T^{4} + 31 T^{2} + 25$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 32 T^{4} + 336 T^{2} + \cdots + 1156$$
$11$ $$(T^{3} + T^{2} - 23 T - 25)^{2}$$
$13$ $$T^{6} + 56 T^{4} + 832 T^{2} + \cdots + 2116$$
$17$ $$T^{6} + 43 T^{4} + 383 T^{2} + \cdots + 841$$
$19$ $$(T^{3} + 3 T^{2} - 25 T - 79)^{2}$$
$23$ $$T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024$$
$29$ $$(T^{3} - 2 T^{2} - 52 T + 40)^{2}$$
$31$ $$(T^{3} + 6 T^{2} - 18 T - 54)^{2}$$
$37$ $$(T^{2} + 1)^{3}$$
$41$ $$(T^{3} + 21 T^{2} + 131 T + 215)^{2}$$
$43$ $$T^{6} + 320 T^{4} + 33600 T^{2} + \cdots + 1157776$$
$47$ $$T^{6} + 112 T^{4} + 2816 T^{2} + \cdots + 4096$$
$53$ $$T^{6} + 236 T^{4} + 17584 T^{2} + \cdots + 399424$$
$59$ $$(T^{3} + 8 T^{2} - 128 T - 1076)^{2}$$
$61$ $$(T^{3} + 8 T^{2} - 44 T - 290)^{2}$$
$67$ $$T^{6} + 187 T^{4} + 1823 T^{2} + \cdots + 4489$$
$71$ $$(T^{3} + 8 T^{2} - 44 T - 290)^{2}$$
$73$ $$T^{6} + 331 T^{4} + 34187 T^{2} + \cdots + 1100401$$
$79$ $$(T^{3} + 22 T^{2} + 140 T + 232)^{2}$$
$83$ $$T^{6} + 503 T^{4} + 76995 T^{2} + \cdots + 3308761$$
$89$ $$(T^{3} - 3 T^{2} - 27 T + 27)^{2}$$
$97$ $$T^{6} + 44 T^{4} + 496 T^{2} + \cdots + 1600$$