# Properties

 Label 3800.2.d.q Level $3800$ Weight $2$ Character orbit 3800.d Analytic conductor $30.343$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9$$ x^12 + 24*x^10 + 194*x^8 + 618*x^6 + 733*x^4 + 286*x^2 + 9 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{8} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b8 - b1) * q^7 + (b4 - b3 - 1) * q^9 $$q + \beta_1 q^{3} + (\beta_{8} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 1) q^{9} + ( - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{3}) q^{11} + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{13} + ( - \beta_{8} + 3 \beta_{7} + \beta_{5} - \beta_1) q^{17} + q^{19} + (\beta_{9} - \beta_{4} + \beta_{3} + 2) q^{21} + (\beta_{11} + \beta_{8} + 2 \beta_{7} - 2 \beta_{5} - \beta_1) q^{23} + (2 \beta_{8} - \beta_{7} - 2 \beta_{5} - 3 \beta_1) q^{27} + (\beta_{10} + \beta_{9} + \beta_{3} - 2) q^{29} + (2 \beta_{10} - \beta_{9} + \beta_{4} + 2 \beta_{3} + 1) q^{31} + ( - \beta_{11} - \beta_{8}) q^{33} + (2 \beta_{7} - \beta_{5} - 3 \beta_1) q^{37} + (\beta_{10} - 2 \beta_{9} + \beta_{4} + 3 \beta_{3} - 2) q^{39} + ( - \beta_{10} - \beta_{9} + \beta_{4} + \beta_{3}) q^{41} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{5} + \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{11} - \beta_{8} - \beta_{7}) q^{47} + ( - 3 \beta_{10} + 2 \beta_{6} - \beta_{4} - \beta_{3} - 4) q^{49} + ( - \beta_{10} - \beta_{9} + \beta_{6} - \beta_{4} - 3 \beta_{3} + 7) q^{51} + (\beta_{7} + \beta_{5} + 3 \beta_{2}) q^{53} + \beta_1 q^{57} + (\beta_{10} + 4 \beta_{3} - 1) q^{59} + ( - \beta_{9} - \beta_{4} + 4 \beta_{3} + 1) q^{61} + ( - \beta_{8} + 3 \beta_{7} + 3 \beta_{5} - \beta_{2} + 6 \beta_1) q^{63} + (2 \beta_{7} + 2 \beta_{5} - \beta_{2} - \beta_1) q^{67} + (\beta_{9} - 2 \beta_{4} - 3 \beta_{3} + 3) q^{69} + (2 \beta_{10} - \beta_{9} - \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 3) q^{71} + (\beta_{11} + 2 \beta_{7} - 2 \beta_{5} - 3 \beta_{2} - \beta_1) q^{73} + ( - \beta_{11} - 3 \beta_{8} + 6 \beta_{7} + 4 \beta_{5} + 2 \beta_{2}) q^{77} + ( - 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{4} + \beta_{3} + 1) q^{79} + (2 \beta_{10} + 2 \beta_{9} - 2 \beta_{6} + 3 \beta_{3} + 3) q^{81} + (\beta_{11} + 5 \beta_{7} + \beta_{5} + \beta_{2} + \beta_1) q^{83} + (\beta_{11} - 3 \beta_{8} + 5 \beta_{7} + 2 \beta_{5} - \beta_{2} + 2 \beta_1) q^{87} + (\beta_{10} - 2 \beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{89} + ( - 2 \beta_{10} - \beta_{4} - \beta_{3} + 7) q^{91} + (3 \beta_{11} + 2 \beta_{8} + 7 \beta_{7} - \beta_{5} + \beta_{2} - \beta_1) q^{93} + (\beta_{11} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{5} - \beta_{2} - 2 \beta_1) q^{97} + ( - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3} - 1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b8 - b1) * q^7 + (b4 - b3 - 1) * q^9 + (-b10 + b9 + b6 - b3) * q^11 + (-b11 - b8 - b2) * q^13 + (-b8 + 3*b7 + b5 - b1) * q^17 + q^19 + (b9 - b4 + b3 + 2) * q^21 + (b11 + b8 + 2*b7 - 2*b5 - b1) * q^23 + (2*b8 - b7 - 2*b5 - 3*b1) * q^27 + (b10 + b9 + b3 - 2) * q^29 + (2*b10 - b9 + b4 + 2*b3 + 1) * q^31 + (-b11 - b8) * q^33 + (2*b7 - b5 - 3*b1) * q^37 + (b10 - 2*b9 + b4 + 3*b3 - 2) * q^39 + (-b10 - b9 + b4 + b3) * q^41 + (-b8 + 2*b7 + 2*b5 + b2 + 2*b1) * q^43 + (-b11 - b8 - b7) * q^47 + (-3*b10 + 2*b6 - b4 - b3 - 4) * q^49 + (-b10 - b9 + b6 - b4 - 3*b3 + 7) * q^51 + (b7 + b5 + 3*b2) * q^53 + b1 * q^57 + (b10 + 4*b3 - 1) * q^59 + (-b9 - b4 + 4*b3 + 1) * q^61 + (-b8 + 3*b7 + 3*b5 - b2 + 6*b1) * q^63 + (2*b7 + 2*b5 - b2 - b1) * q^67 + (b9 - 2*b4 - 3*b3 + 3) * q^69 + (2*b10 - b9 - b6 + 2*b4 + 2*b3 + 3) * q^71 + (b11 + 2*b7 - 2*b5 - 3*b2 - b1) * q^73 + (-b11 - 3*b8 + 6*b7 + 4*b5 + 2*b2) * q^77 + (-3*b10 + 2*b9 - 2*b4 + b3 + 1) * q^79 + (2*b10 + 2*b9 - 2*b6 + 3*b3 + 3) * q^81 + (b11 + 5*b7 + b5 + b2 + b1) * q^83 + (b11 - 3*b8 + 5*b7 + 2*b5 - b2 + 2*b1) * q^87 + (b10 - 2*b9 + b6 - b4 + b3 - 1) * q^89 + (-2*b10 - b4 - b3 + 7) * q^91 + (3*b11 + 2*b8 + 7*b7 - b5 + b2 - b1) * q^93 + (b11 + 2*b8 + 2*b7 - 4*b5 - b2 - 2*b1) * q^97 + (-b10 + 2*b9 + b6 + b4 + b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{9}+O(q^{10})$$ 12 * q - 12 * q^9 $$12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91} + 12 q^{99}+O(q^{100})$$ 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{11} + 27\nu^{9} + 260\nu^{7} + 1068\nu^{5} + 1702\nu^{3} + 637\nu ) / 60$$ (v^11 + 27*v^9 + 260*v^7 + 1068*v^5 + 1702*v^3 + 637*v) / 60 $$\beta_{3}$$ $$=$$ $$( -8\nu^{10} - 181\nu^{8} - 1320\nu^{6} - 3399\nu^{4} - 2321\nu^{2} - 126 ) / 270$$ (-8*v^10 - 181*v^8 - 1320*v^6 - 3399*v^4 - 2321*v^2 - 126) / 270 $$\beta_{4}$$ $$=$$ $$( -8\nu^{10} - 181\nu^{8} - 1320\nu^{6} - 3399\nu^{4} - 2051\nu^{2} + 954 ) / 270$$ (-8*v^10 - 181*v^8 - 1320*v^6 - 3399*v^4 - 2051*v^2 + 954) / 270 $$\beta_{5}$$ $$=$$ $$( -\nu^{9} - 23\nu^{7} - 171\nu^{5} - 450\nu^{3} - 313\nu ) / 12$$ (-v^9 - 23*v^7 - 171*v^5 - 450*v^3 - 313*v) / 12 $$\beta_{6}$$ $$=$$ $$( -19\nu^{10} - 413\nu^{8} - 2730\nu^{6} - 5052\nu^{4} + 1592\nu^{2} + 2367 ) / 540$$ (-19*v^10 - 413*v^8 - 2730*v^6 - 5052*v^4 + 1592*v^2 + 2367) / 540 $$\beta_{7}$$ $$=$$ $$( -14\nu^{11} - 328\nu^{9} - 2535\nu^{7} - 7332\nu^{5} - 6863\nu^{3} - 1683\nu ) / 270$$ (-14*v^11 - 328*v^9 - 2535*v^7 - 7332*v^5 - 6863*v^3 - 1683*v) / 270 $$\beta_{8}$$ $$=$$ $$( -14\nu^{11} - 373\nu^{9} - 3570\nu^{7} - 15027\nu^{5} - 26843\nu^{3} - 13338\nu ) / 540$$ (-14*v^11 - 373*v^9 - 3570*v^7 - 15027*v^5 - 26843*v^3 - 13338*v) / 540 $$\beta_{9}$$ $$=$$ $$( -37\nu^{10} - 854\nu^{8} - 6375\nu^{6} - 16581\nu^{4} - 9334\nu^{2} + 1206 ) / 540$$ (-37*v^10 - 854*v^8 - 6375*v^6 - 16581*v^4 - 9334*v^2 + 1206) / 540 $$\beta_{10}$$ $$=$$ $$( 14\nu^{10} + 328\nu^{8} + 2535\nu^{6} + 7332\nu^{4} + 6773\nu^{2} + 1053 ) / 180$$ (14*v^10 + 328*v^8 + 2535*v^6 + 7332*v^4 + 6773*v^2 + 1053) / 180 $$\beta_{11}$$ $$=$$ $$( 16\nu^{11} + 377\nu^{9} + 2940\nu^{7} + 8643\nu^{5} + 8377\nu^{3} + 2112\nu ) / 90$$ (16*v^11 + 377*v^9 + 2940*v^7 + 8643*v^5 + 8377*v^3 + 2112*v) / 90
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - 4$$ b4 - b3 - 4 $$\nu^{3}$$ $$=$$ $$2\beta_{8} - \beta_{7} - 2\beta_{5} - 9\beta_1$$ 2*b8 - b7 - 2*b5 - 9*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{10} + 2\beta_{9} - 2\beta_{6} - 9\beta_{4} + 12\beta_{3} + 30$$ 2*b10 + 2*b9 - 2*b6 - 9*b4 + 12*b3 + 30 $$\nu^{5}$$ $$=$$ $$3\beta_{11} - 24\beta_{8} + 21\beta_{7} + 22\beta_{5} - 4\beta_{2} + 84\beta_1$$ 3*b11 - 24*b8 + 21*b7 + 22*b5 - 4*b2 + 84*b1 $$\nu^{6}$$ $$=$$ $$-32\beta_{10} - 28\beta_{9} + 36\beta_{6} + 81\beta_{4} - 143\beta_{3} - 261$$ -32*b10 - 28*b9 + 36*b6 + 81*b4 - 143*b3 - 261 $$\nu^{7}$$ $$=$$ $$-62\beta_{11} + 250\beta_{8} - 317\beta_{7} - 222\beta_{5} + 64\beta_{2} - 816\beta_1$$ -62*b11 + 250*b8 - 317*b7 - 222*b5 + 64*b2 - 816*b1 $$\nu^{8}$$ $$=$$ $$410\beta_{10} + 314\beta_{9} - 474\beta_{6} - 754\beta_{4} + 1667\beta_{3} + 2420$$ 410*b10 + 314*b9 - 474*b6 - 754*b4 + 1667*b3 + 2420 $$\nu^{9}$$ $$=$$ $$913\beta_{11} - 2546\beta_{8} + 4150\beta_{7} + 2232\beta_{5} - 788\beta_{2} + 8141\beta_1$$ 913*b11 - 2546*b8 + 4150*b7 + 2232*b5 - 788*b2 + 8141*b1 $$\nu^{10}$$ $$=$$ $$-4846\beta_{10} - 3334\beta_{9} + 5634\beta_{6} + 7228\beta_{4} - 18963\beta_{3} - 23289$$ -4846*b10 - 3334*b9 + 5634*b6 + 7228*b4 - 18963*b3 - 23289 $$\nu^{11}$$ $$=$$ $$-11735\beta_{11} + 25970\beta_{8} - 50356\beta_{7} - 22636\beta_{5} + 8968\beta_{2} - 82678\beta_1$$ -11735*b11 + 25970*b8 - 50356*b7 - 22636*b5 + 8968*b2 - 82678*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3649.1
 − 3.26143i − 2.77008i − 1.93590i − 1.08999i − 0.848258i − 0.185519i 0.185519i 0.848258i 1.08999i 1.93590i 2.77008i 3.26143i
0 3.26143i 0 0 0 4.07225i 0 −7.63693 0
3649.2 0 2.77008i 0 0 0 2.31077i 0 −4.67334 0
3649.3 0 1.93590i 0 0 0 1.24708i 0 −0.747704 0
3649.4 0 1.08999i 0 0 0 4.19727i 0 1.81192 0
3649.5 0 0.848258i 0 0 0 1.74484i 0 2.28046 0
3649.6 0 0.185519i 0 0 0 4.45651i 0 2.96558 0
3649.7 0 0.185519i 0 0 0 4.45651i 0 2.96558 0
3649.8 0 0.848258i 0 0 0 1.74484i 0 2.28046 0
3649.9 0 1.08999i 0 0 0 4.19727i 0 1.81192 0
3649.10 0 1.93590i 0 0 0 1.24708i 0 −0.747704 0
3649.11 0 2.77008i 0 0 0 2.31077i 0 −4.67334 0
3649.12 0 3.26143i 0 0 0 4.07225i 0 −7.63693 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3649.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 3800.2.d.q 12
5.b even 2 1 inner 3800.2.d.q 12
5.c odd 4 1 3800.2.a.ba 6
5.c odd 4 1 3800.2.a.bc yes 6
20.e even 4 1 7600.2.a.ch 6
20.e even 4 1 7600.2.a.cl 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 5.c odd 4 1
3800.2.a.bc yes 6 5.c odd 4 1
3800.2.d.q 12 1.a even 1 1 trivial
3800.2.d.q 12 5.b even 2 1 inner
7600.2.a.ch 6 20.e even 4 1
7600.2.a.cl 6 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{12} + 24T_{3}^{10} + 194T_{3}^{8} + 618T_{3}^{6} + 733T_{3}^{4} + 286T_{3}^{2} + 9$$ T3^12 + 24*T3^10 + 194*T3^8 + 618*T3^6 + 733*T3^4 + 286*T3^2 + 9 $$T_{7}^{12} + 64T_{7}^{10} + 1538T_{7}^{8} + 17066T_{7}^{6} + 87493T_{7}^{4} + 194534T_{7}^{2} + 146689$$ T7^12 + 64*T7^10 + 1538*T7^8 + 17066*T7^6 + 87493*T7^4 + 194534*T7^2 + 146689 $$T_{11}^{6} - 3T_{11}^{5} - 40T_{11}^{4} + 139T_{11}^{3} + 10T_{11}^{2} - 208T_{11} - 88$$ T11^6 - 3*T11^5 - 40*T11^4 + 139*T11^3 + 10*T11^2 - 208*T11 - 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 24 T^{10} + 194 T^{8} + 618 T^{6} + \cdots + 9$$
$5$ $$T^{12}$$
$7$ $$T^{12} + 64 T^{10} + 1538 T^{8} + \cdots + 146689$$
$11$ $$(T^{6} - 3 T^{5} - 40 T^{4} + 139 T^{3} + \cdots - 88)^{2}$$
$13$ $$T^{12} + 131 T^{10} + 6021 T^{8} + \cdots + 680625$$
$17$ $$T^{12} + 152 T^{10} + 8082 T^{8} + \cdots + 9903609$$
$19$ $$(T - 1)^{12}$$
$23$ $$T^{12} + 196 T^{10} + 13806 T^{8} + \cdots + 6185169$$
$29$ $$(T^{6} + 9 T^{5} - 67 T^{4} - 705 T^{3} + \cdots + 21951)^{2}$$
$31$ $$(T^{6} - 5 T^{5} - 136 T^{4} + 599 T^{3} + \cdots + 11000)^{2}$$
$37$ $$T^{12} + 244 T^{10} + \cdots + 54243225$$
$41$ $$(T^{6} - 3 T^{5} - 180 T^{4} + \cdots - 113472)^{2}$$
$43$ $$T^{12} + 199 T^{10} + \cdots + 37454400$$
$47$ $$T^{12} + 120 T^{10} + 5136 T^{8} + \cdots + 3207681$$
$53$ $$T^{12} + 553 T^{10} + \cdots + 86211225$$
$59$ $$(T^{6} - 193 T^{4} - 255 T^{3} + \cdots - 32328)^{2}$$
$61$ $$(T^{6} - 9 T^{5} - 132 T^{4} + 443 T^{3} + \cdots + 12424)^{2}$$
$67$ $$T^{12} + 407 T^{10} + \cdots + 18504977089$$
$71$ $$(T^{6} - 19 T^{5} - 75 T^{4} + 3491 T^{3} + \cdots - 27576)^{2}$$
$73$ $$T^{12} + 619 T^{10} + \cdots + 549129907089$$
$79$ $$(T^{6} - 16 T^{5} - 285 T^{4} + \cdots - 553536)^{2}$$
$83$ $$T^{12} + 309 T^{10} + \cdots + 5063745600$$
$89$ $$(T^{6} + 14 T^{5} - 67 T^{4} - 727 T^{3} + \cdots + 3240)^{2}$$
$97$ $$T^{12} + 537 T^{10} + \cdots + 83400819264$$