# Properties

 Label 3150.2.d.d Level 3150 Weight 2 Character orbit 3150.d Analytic conductor 25.153 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.7442857984.4 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta_{2} q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta_{2} q^{7} + q^{8} -\beta_{3} q^{11} + ( -1 + \beta_{7} ) q^{13} -\beta_{2} q^{14} + q^{16} + ( \beta_{3} - \beta_{6} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{19} -\beta_{3} q^{22} + ( -1 - \beta_{1} - \beta_{2} + \beta_{7} ) q^{23} + ( -1 + \beta_{7} ) q^{26} -\beta_{2} q^{28} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{31} + q^{32} + ( \beta_{3} - \beta_{6} ) q^{34} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} ) q^{37} + ( \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{38} + ( -1 + \beta_{7} ) q^{41} + ( -\beta_{3} + \beta_{5} + \beta_{6} ) q^{43} -\beta_{3} q^{44} + ( -1 - \beta_{1} - \beta_{2} + \beta_{7} ) q^{46} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{47} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{49} + ( -1 + \beta_{7} ) q^{52} + ( -1 + \beta_{1} + \beta_{2} + \beta_{7} ) q^{53} -\beta_{2} q^{56} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{58} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{59} + ( \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{61} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{62} + q^{64} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{67} + ( \beta_{3} - \beta_{6} ) q^{68} + ( \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{71} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} ) q^{74} + ( \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{76} + ( -\beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{77} + ( -1 - 3 \beta_{1} - 3 \beta_{2} - \beta_{7} ) q^{79} + ( -1 + \beta_{7} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} ) q^{83} + ( -\beta_{3} + \beta_{5} + \beta_{6} ) q^{86} -\beta_{3} q^{88} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{89} + ( \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} + ( -1 - \beta_{1} - \beta_{2} + \beta_{7} ) q^{92} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{94} + ( -3 - \beta_{1} - \beta_{2} + 3 \beta_{7} ) q^{97} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{2} + 8q^{4} + 8q^{8} + O(q^{10})$$ $$8q + 8q^{2} + 8q^{4} + 8q^{8} - 8q^{13} + 8q^{16} - 8q^{23} - 8q^{26} + 8q^{32} - 8q^{41} - 8q^{46} - 4q^{49} - 8q^{52} - 8q^{53} + 8q^{64} + 48q^{73} - 4q^{77} - 8q^{79} - 8q^{82} + 8q^{89} - 4q^{91} - 8q^{92} - 24q^{97} - 4q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 26 x^{6} + 205 x^{4} + 540 x^{2} + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 26 \nu^{5} + 18 \nu^{4} + 223 \nu^{3} + 234 \nu^{2} + 774 \nu + 324$$$$)/216$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 26 \nu^{5} + 18 \nu^{4} - 223 \nu^{3} + 234 \nu^{2} - 774 \nu + 324$$$$)/216$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 20 \nu^{5} + 73 \nu^{3} - 54 \nu$$$$)/72$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 26 \nu^{5} + 18 \nu^{4} - 223 \nu^{3} + 450 \nu^{2} - 558 \nu + 1836$$$$)/216$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} - 112 \nu^{5} - 665 \nu^{3} - 954 \nu$$$$)/216$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 26 \nu^{5} + 187 \nu^{3} + 306 \nu$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 22 \nu^{4} + 123 \nu^{2} + 138$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{3} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} - 14$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{6} - 13 \beta_{5} - 13 \beta_{3} + 7 \beta_{2} - 7 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$13 \beta_{5} - 26 \beta_{4} + 13 \beta_{3} + 25 \beta_{2} + 25 \beta_{1} + 146$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$50 \beta_{6} + 187 \beta_{5} + 163 \beta_{3} - 73 \beta_{2} + 73 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$24 \beta_{7} - 163 \beta_{5} + 326 \beta_{4} - 163 \beta_{3} - 427 \beta_{2} - 427 \beta_{1} - 1766$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-854 \beta_{6} - 2737 \beta_{5} - 2113 \beta_{3} + 895 \beta_{2} - 895 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3149.1
 − 3.73923i 3.73923i − 0.916813i 0.916813i − 1.91681i 1.91681i − 2.73923i 2.73923i
1.00000 0 1.00000 0 0 −2.64404 0.0951965i 1.00000 0 0
3149.2 1.00000 0 1.00000 0 0 −2.64404 + 0.0951965i 1.00000 0 0
3149.3 1.00000 0 1.00000 0 0 −0.648285 2.56510i 1.00000 0 0
3149.4 1.00000 0 1.00000 0 0 −0.648285 + 2.56510i 1.00000 0 0
3149.5 1.00000 0 1.00000 0 0 1.35539 2.27220i 1.00000 0 0
3149.6 1.00000 0 1.00000 0 0 1.35539 + 2.27220i 1.00000 0 0
3149.7 1.00000 0 1.00000 0 0 1.93693 1.80230i 1.00000 0 0
3149.8 1.00000 0 1.00000 0 0 1.93693 + 1.80230i 1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3149.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.d.d 8
3.b odd 2 1 3150.2.d.a 8
5.b even 2 1 3150.2.d.c 8
5.c odd 4 1 630.2.b.a 8
5.c odd 4 1 3150.2.b.e 8
7.b odd 2 1 3150.2.d.f 8
15.d odd 2 1 3150.2.d.f 8
15.e even 4 1 630.2.b.b yes 8
15.e even 4 1 3150.2.b.f 8
20.e even 4 1 5040.2.f.f 8
21.c even 2 1 3150.2.d.c 8
35.c odd 2 1 3150.2.d.a 8
35.f even 4 1 630.2.b.b yes 8
35.f even 4 1 3150.2.b.f 8
60.l odd 4 1 5040.2.f.i 8
105.g even 2 1 inner 3150.2.d.d 8
105.k odd 4 1 630.2.b.a 8
105.k odd 4 1 3150.2.b.e 8
140.j odd 4 1 5040.2.f.i 8
420.w even 4 1 5040.2.f.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.b.a 8 5.c odd 4 1
630.2.b.a 8 105.k odd 4 1
630.2.b.b yes 8 15.e even 4 1
630.2.b.b yes 8 35.f even 4 1
3150.2.b.e 8 5.c odd 4 1
3150.2.b.e 8 105.k odd 4 1
3150.2.b.f 8 15.e even 4 1
3150.2.b.f 8 35.f even 4 1
3150.2.d.a 8 3.b odd 2 1
3150.2.d.a 8 35.c odd 2 1
3150.2.d.c 8 5.b even 2 1
3150.2.d.c 8 21.c even 2 1
3150.2.d.d 8 1.a even 1 1 trivial
3150.2.d.d 8 105.g even 2 1 inner
3150.2.d.f 8 7.b odd 2 1
3150.2.d.f 8 15.d odd 2 1
5040.2.f.f 8 20.e even 4 1
5040.2.f.f 8 420.w even 4 1
5040.2.f.i 8 60.l odd 4 1
5040.2.f.i 8 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

 $$T_{11}^{8} + 52 T_{11}^{6} + 820 T_{11}^{4} + 4320 T_{11}^{2} + 5184$$ $$T_{13}^{4} + 4 T_{13}^{3} - 22 T_{13}^{2} - 24 T_{13} + 72$$ $$T_{23}^{4} + 4 T_{23}^{3} - 44 T_{23}^{2} - 96 T_{23} + 288$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{8}$$
$3$ 1
$5$ 1
$7$ $$1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 168 T^{5} + 98 T^{6} + 2401 T^{8}$$
$11$ $$1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + 4 T + 30 T^{2} + 132 T^{3} + 514 T^{4} + 1716 T^{5} + 5070 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 1550196 T^{10} + 20713208 T^{12} - 868952484 T^{14} + 6975757441 T^{16}$$
$19$ $$1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16}$$
$23$ $$( 1 + 4 T + 48 T^{2} + 180 T^{3} + 1438 T^{4} + 4140 T^{5} + 25392 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16}$$
$31$ $$1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16}$$
$37$ $$1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 540251208 T^{10} + 18749106644 T^{12} - 389990414168 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 4 T + 142 T^{2} + 468 T^{3} + 8354 T^{4} + 19188 T^{5} + 238702 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 2730092876 T^{10} + 79507636056 T^{12} - 1441270775172 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 + 68 T^{2} + 10040 T^{4} + 446124 T^{6} + 34404910 T^{8} + 985487916 T^{10} + 48991997240 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 + 4 T + 160 T^{2} + 636 T^{3} + 11630 T^{4} + 33708 T^{5} + 449440 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 8770410684 T^{10} + 461541275384 T^{12} - 16282508790420 T^{14} + 406067677556641 T^{16}$$
$71$ $$1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16}$$
$73$ $$( 1 - 24 T + 404 T^{2} - 4680 T^{3} + 45878 T^{4} - 341640 T^{5} + 2152916 T^{6} - 9336408 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 42028 T^{5} + 299568 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 78092987544 T^{10} + 4104765099932 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 116412 T^{5} + 934678 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 12 T + 176 T^{2} + 2148 T^{3} + 29438 T^{4} + 208356 T^{5} + 1655984 T^{6} + 10952076 T^{7} + 88529281 T^{8} )^{2}$$