# Properties

 Label 1350.4.c.u Level $1350$ Weight $4$ Character orbit 1350.c Analytic conductor $79.653$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.6525785077$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{401})$$ Defining polynomial: $$x^{4} + 201x^{2} + 10000$$ x^4 + 201*x^2 + 10000 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 270) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta_1 q^{2} - 4 q^{4} - \beta_{2} q^{7} + 8 \beta_1 q^{8}+O(q^{10})$$ q - 2*b1 * q^2 - 4 * q^4 - b2 * q^7 + 8*b1 * q^8 $$q - 2 \beta_1 q^{2} - 4 q^{4} - \beta_{2} q^{7} + 8 \beta_1 q^{8} + ( - \beta_{3} - 16) q^{11} + ( - 2 \beta_{2} - 33 \beta_1) q^{13} + ( - 2 \beta_{3} + 2) q^{14} + 16 q^{16} + (\beta_{2} - 25 \beta_1) q^{17} + ( - \beta_{3} - 57) q^{19} + (2 \beta_{2} + 34 \beta_1) q^{22} + (5 \beta_{2} + 13 \beta_1) q^{23} + ( - 4 \beta_{3} - 62) q^{26} + 4 \beta_{2} q^{28} + ( - 7 \beta_{3} - 28) q^{29} + ( - 5 \beta_{3} + 150) q^{31} - 32 \beta_1 q^{32} + (2 \beta_{3} - 52) q^{34} + ( - 3 \beta_{2} + 68 \beta_1) q^{37} + (2 \beta_{2} + 116 \beta_1) q^{38} + (6 \beta_{3} + 318) q^{41} + ( - 5 \beta_{2} - 147 \beta_1) q^{43} + (4 \beta_{3} + 64) q^{44} + (10 \beta_{3} + 16) q^{46} + ( - 13 \beta_{2} - 95 \beta_1) q^{47} + (\beta_{3} - 560) q^{49} + (8 \beta_{2} + 132 \beta_1) q^{52} + ( - 2 \beta_{2} - 184 \beta_1) q^{53} + (8 \beta_{3} - 8) q^{56} + (14 \beta_{2} + 70 \beta_1) q^{58} + (8 \beta_{3} - 628) q^{59} + (3 \beta_{3} + 41) q^{61} + (10 \beta_{2} - 290 \beta_1) q^{62} - 64 q^{64} + (3 \beta_{2} + 494 \beta_1) q^{67} + ( - 4 \beta_{2} + 100 \beta_1) q^{68} + ( - 6 \beta_{3} + 750) q^{71} + ( - 3 \beta_{2} + 928 \beta_1) q^{73} + ( - 6 \beta_{3} + 142) q^{74} + (4 \beta_{3} + 228) q^{76} + (16 \beta_{2} + 902 \beta_1) q^{77} + (6 \beta_{3} - 635) q^{79} + ( - 12 \beta_{2} - 648 \beta_1) q^{82} + ( - 28 \beta_{2} + 526 \beta_1) q^{83} + ( - 10 \beta_{3} - 284) q^{86} + ( - 8 \beta_{2} - 136 \beta_1) q^{88} + (36 \beta_{3} - 360) q^{89} + ( - 31 \beta_{3} - 1773) q^{91} + ( - 20 \beta_{2} - 52 \beta_1) q^{92} + ( - 26 \beta_{3} - 164) q^{94} + ( - 17 \beta_{2} - 644 \beta_1) q^{97} + ( - 2 \beta_{2} + 1118 \beta_1) q^{98}+O(q^{100})$$ q - 2*b1 * q^2 - 4 * q^4 - b2 * q^7 + 8*b1 * q^8 + (-b3 - 16) * q^11 + (-2*b2 - 33*b1) * q^13 + (-2*b3 + 2) * q^14 + 16 * q^16 + (b2 - 25*b1) * q^17 + (-b3 - 57) * q^19 + (2*b2 + 34*b1) * q^22 + (5*b2 + 13*b1) * q^23 + (-4*b3 - 62) * q^26 + 4*b2 * q^28 + (-7*b3 - 28) * q^29 + (-5*b3 + 150) * q^31 - 32*b1 * q^32 + (2*b3 - 52) * q^34 + (-3*b2 + 68*b1) * q^37 + (2*b2 + 116*b1) * q^38 + (6*b3 + 318) * q^41 + (-5*b2 - 147*b1) * q^43 + (4*b3 + 64) * q^44 + (10*b3 + 16) * q^46 + (-13*b2 - 95*b1) * q^47 + (b3 - 560) * q^49 + (8*b2 + 132*b1) * q^52 + (-2*b2 - 184*b1) * q^53 + (8*b3 - 8) * q^56 + (14*b2 + 70*b1) * q^58 + (8*b3 - 628) * q^59 + (3*b3 + 41) * q^61 + (10*b2 - 290*b1) * q^62 - 64 * q^64 + (3*b2 + 494*b1) * q^67 + (-4*b2 + 100*b1) * q^68 + (-6*b3 + 750) * q^71 + (-3*b2 + 928*b1) * q^73 + (-6*b3 + 142) * q^74 + (4*b3 + 228) * q^76 + (16*b2 + 902*b1) * q^77 + (6*b3 - 635) * q^79 + (-12*b2 - 648*b1) * q^82 + (-28*b2 + 526*b1) * q^83 + (-10*b3 - 284) * q^86 + (-8*b2 - 136*b1) * q^88 + (36*b3 - 360) * q^89 + (-31*b3 - 1773) * q^91 + (-20*b2 - 52*b1) * q^92 + (-26*b3 - 164) * q^94 + (-17*b2 - 644*b1) * q^97 + (-2*b2 + 1118*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{4}+O(q^{10})$$ 4 * q - 16 * q^4 $$4 q - 16 q^{4} - 66 q^{11} + 4 q^{14} + 64 q^{16} - 230 q^{19} - 256 q^{26} - 126 q^{29} + 590 q^{31} - 204 q^{34} + 1284 q^{41} + 264 q^{44} + 84 q^{46} - 2238 q^{49} - 16 q^{56} - 2496 q^{59} + 170 q^{61} - 256 q^{64} + 2988 q^{71} + 556 q^{74} + 920 q^{76} - 2528 q^{79} - 1156 q^{86} - 1368 q^{89} - 7154 q^{91} - 708 q^{94}+O(q^{100})$$ 4 * q - 16 * q^4 - 66 * q^11 + 4 * q^14 + 64 * q^16 - 230 * q^19 - 256 * q^26 - 126 * q^29 + 590 * q^31 - 204 * q^34 + 1284 * q^41 + 264 * q^44 + 84 * q^46 - 2238 * q^49 - 16 * q^56 - 2496 * q^59 + 170 * q^61 - 256 * q^64 + 2988 * q^71 + 556 * q^74 + 920 * q^76 - 2528 * q^79 - 1156 * q^86 - 1368 * q^89 - 7154 * q^91 - 708 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 201x^{2} + 10000$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 101\nu ) / 100$$ (v^3 + 101*v) / 100 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 401\nu ) / 100$$ (v^3 + 401*v) / 100 $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 302$$ 3*v^2 + 302
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 3$$ (b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 302 ) / 3$$ (b3 - 302) / 3 $$\nu^{3}$$ $$=$$ $$( -101\beta_{2} + 401\beta_1 ) / 3$$ (-101*b2 + 401*b1) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 9.51249i − 10.5125i 10.5125i − 9.51249i
2.00000i 0 −4.00000 0 0 29.5375i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 30.5375i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 30.5375i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 29.5375i 8.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 1350.4.c.u 4
3.b odd 2 1 1350.4.c.bb 4
5.b even 2 1 inner 1350.4.c.u 4
5.c odd 4 1 270.4.a.m 2
5.c odd 4 1 1350.4.a.bm 2
15.d odd 2 1 1350.4.c.bb 4
15.e even 4 1 270.4.a.n yes 2
15.e even 4 1 1350.4.a.bf 2
20.e even 4 1 2160.4.a.w 2
45.k odd 12 2 810.4.e.bd 4
45.l even 12 2 810.4.e.z 4
60.l odd 4 1 2160.4.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.m 2 5.c odd 4 1
270.4.a.n yes 2 15.e even 4 1
810.4.e.z 4 45.l even 12 2
810.4.e.bd 4 45.k odd 12 2
1350.4.a.bf 2 15.e even 4 1
1350.4.a.bm 2 5.c odd 4 1
1350.4.c.u 4 1.a even 1 1 trivial
1350.4.c.u 4 5.b even 2 1 inner
1350.4.c.bb 4 3.b odd 2 1
1350.4.c.bb 4 15.d odd 2 1
2160.4.a.w 2 20.e even 4 1
2160.4.a.bb 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} + 1805T_{7}^{2} + 813604$$ T7^4 + 1805*T7^2 + 813604 $$T_{11}^{2} + 33T_{11} - 630$$ T11^2 + 33*T11 - 630

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1805 T^{2} + 813604$$
$11$ $$(T^{2} + 33 T - 630)^{2}$$
$13$ $$T^{4} + 9266 T^{2} + \cdots + 6682225$$
$17$ $$T^{4} + 3105 T^{2} + 63504$$
$19$ $$(T^{2} + 115 T + 2404)^{2}$$
$23$ $$T^{4} + 45333 T^{2} + \cdots + 503822916$$
$29$ $$(T^{2} + 63 T - 43218)^{2}$$
$31$ $$(T^{2} - 295 T - 800)^{2}$$
$37$ $$T^{4} + 25901 T^{2} + \cdots + 10824100$$
$41$ $$(T^{2} - 642 T + 70560)^{2}$$
$43$ $$T^{4} + 86873 T^{2} + \cdots + 2808976$$
$47$ $$T^{4} + 320625 T^{2} + \cdots + 20923043904$$
$53$ $$T^{4} + 74196 T^{2} + \cdots + 892814400$$
$59$ $$(T^{2} + 1248 T + 331632)^{2}$$
$61$ $$(T^{2} - 85 T - 6314)^{2}$$
$67$ $$T^{4} + 501353 T^{2} + \cdots + 54960238096$$
$71$ $$(T^{2} - 1494 T + 525528)^{2}$$
$73$ $$T^{4} + 1744181 T^{2} + \cdots + 732479222500$$
$79$ $$(T^{2} + 1264 T + 366943)^{2}$$
$83$ $$T^{4} + 1997928 T^{2} + \cdots + 172859703696$$
$89$ $$(T^{2} + 684 T - 1052352)^{2}$$
$97$ $$T^{4} + 1329221 T^{2} + \cdots + 20480472100$$