# Properties

 Label 3150.2.m.i Level 3150 Weight 2 Character orbit 3150.m 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.1698758656.6 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{4} q^{4} + \beta_{2} q^{7} + \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{4} q^{4} + \beta_{2} q^{7} + \beta_{2} q^{8} + ( -\beta_{1} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( 1 + \beta_{4} + \beta_{6} ) q^{13} - q^{14} - q^{16} + ( \beta_{1} + \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{1} + 4 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{22} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{23} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{26} + \beta_{3} q^{28} + ( -4 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{29} + ( \beta_{4} + \beta_{5} - \beta_{7} ) q^{31} + \beta_{3} q^{32} + ( \beta_{4} + \beta_{5} - \beta_{6} ) q^{34} + ( 1 + \beta_{1} + 4 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 7 \beta_{4} - \beta_{5} + \beta_{6} ) q^{41} + ( \beta_{1} + 4 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{43} + ( -4 - \beta_{4} - \beta_{5} + \beta_{7} ) q^{44} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{46} + ( \beta_{1} - 2 \beta_{6} + \beta_{7} ) q^{47} -\beta_{4} q^{49} + ( -1 - \beta_{5} ) q^{52} + ( -4 + \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{53} -\beta_{4} q^{56} + ( -1 + \beta_{1} + 4 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{58} + ( -4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{59} + ( \beta_{6} + \beta_{7} ) q^{61} + ( -\beta_{1} - \beta_{7} ) q^{62} -\beta_{4} q^{64} + ( -4 - \beta_{1} + 4 \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{67} + ( \beta_{1} - \beta_{6} - \beta_{7} ) q^{68} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{71} + ( -5 - \beta_{1} - 5 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{73} + ( -4 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{74} + ( \beta_{1} + 4 \beta_{3} + \beta_{7} ) q^{77} + ( -4 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{79} + ( 1 - \beta_{1} + 8 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{82} + ( -8 - 2 \beta_{1} + 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 3 \beta_{1} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{86} + ( \beta_{1} + 4 \beta_{3} + \beta_{7} ) q^{88} + ( -8 + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{89} + ( \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{91} + 2 \beta_{6} q^{92} + ( 3 \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{94} + ( 3 + 3 \beta_{1} + 2 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{97} -\beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{13} - 8q^{14} - 8q^{16} + 4q^{22} - 8q^{23} - 24q^{29} - 8q^{31} + 4q^{37} + 12q^{43} - 24q^{44} + 12q^{47} - 4q^{52} - 32q^{53} - 12q^{58} - 16q^{59} - 4q^{62} - 20q^{67} - 36q^{73} - 40q^{74} + 4q^{77} + 12q^{82} - 56q^{83} + 4q^{88} - 72q^{89} - 8q^{92} + 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 10 \nu^{5} + 15 \nu^{3} + 120 \nu$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + 18 \nu^{5} + 28 \nu^{4} + 89 \nu^{3} + 74 \nu^{2} + 104 \nu - 16$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 18 \nu^{5} + 28 \nu^{4} - 89 \nu^{3} + 74 \nu^{2} - 104 \nu - 16$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + 46 \nu^{5} + 179 \nu^{3} + 168 \nu$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 14 \nu^{4} - 45 \nu^{2} + 8 \nu - 32$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} - 8 \nu^{6} - 46 \nu^{5} - 112 \nu^{4} - 179 \nu^{3} - 360 \nu^{2} - 232 \nu - 256$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} + 4 \nu^{6} + 46 \nu^{5} + 72 \nu^{4} + 179 \nu^{3} + 356 \nu^{2} + 232 \nu + 416$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{5} + 13 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + 4 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$8 \beta_{7} + 17 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + 44 \beta_{3} + 44 \beta_{2} + 74$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-37 \beta_{6} + 37 \beta_{5} - 141 \beta_{4} - 112 \beta_{3} + 112 \beta_{2} - 44 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-112 \beta_{7} - 201 \beta_{6} - 89 \beta_{5} - 89 \beta_{4} - 436 \beta_{3} - 436 \beta_{2} - 650$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$325 \beta_{6} - 325 \beta_{5} + 1485 \beta_{4} + 1240 \beta_{3} - 1240 \beta_{2} + 436 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1457.1
 3.16053i − 2.16053i 1.69230i − 0.692297i 2.16053i − 3.16053i 0.692297i − 1.69230i
−0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.2 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.3 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
1457.4 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
2843.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.2 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.3 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
2843.4 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2843.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.m.i 8
3.b odd 2 1 3150.2.m.j 8
5.b even 2 1 630.2.m.c 8
5.c odd 4 1 630.2.m.d yes 8
5.c odd 4 1 3150.2.m.j 8
15.d odd 2 1 630.2.m.d yes 8
15.e even 4 1 630.2.m.c 8
15.e even 4 1 inner 3150.2.m.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.m.c 8 5.b even 2 1
630.2.m.c 8 15.e even 4 1
630.2.m.d yes 8 5.c odd 4 1
630.2.m.d yes 8 15.d odd 2 1
3150.2.m.i 8 1.a even 1 1 trivial
3150.2.m.i 8 15.e even 4 1 inner
3150.2.m.j 8 3.b odd 2 1
3150.2.m.j 8 5.c odd 4 1

## Hecke kernels

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

 $$T_{11}^{8} + 76 T_{11}^{6} + 1668 T_{11}^{4} + 9728 T_{11}^{2} + 16384$$ $$T_{13}^{8} - \cdots$$ $$T_{17}^{8} - 64 T_{17}^{5} + 1040 T_{17}^{4} - 2304 T_{17}^{3} + 2048 T_{17}^{2} + 8192 T_{17} + 16384$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 1
$5$ 1
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 31460 T^{10} + 585640 T^{12} - 21258732 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 4 T + 8 T^{2} - 28 T^{3} + 80 T^{4} - 372 T^{5} + 1240 T^{6} - 6412 T^{7} + 34174 T^{8} - 83356 T^{9} + 209560 T^{10} - 817284 T^{11} + 2284880 T^{12} - 10396204 T^{13} + 38614472 T^{14} - 250994068 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 64 T^{3} - 252 T^{4} + 960 T^{5} + 2048 T^{6} + 6016 T^{7} - 17786 T^{8} + 102272 T^{9} + 591872 T^{10} + 4716480 T^{11} - 21047292 T^{12} - 90870848 T^{13} + 6975757441 T^{16}$$
$19$ $$( 1 - 19 T^{2} )^{8}$$
$23$ $$1 + 8 T + 32 T^{2} - 8 T^{3} - 924 T^{4} - 5032 T^{5} - 10656 T^{6} + 28456 T^{7} + 420614 T^{8} + 654488 T^{9} - 5637024 T^{10} - 61224344 T^{11} - 258573084 T^{12} - 51490744 T^{13} + 4737148448 T^{14} + 27238603576 T^{15} + 78310985281 T^{16}$$
$29$ $$( 1 + 12 T + 78 T^{2} + 252 T^{3} + 738 T^{4} + 7308 T^{5} + 65598 T^{6} + 292668 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 11036 T^{5} + 105710 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 - 4 T + 8 T^{2} - 220 T^{3} - 48 T^{4} + 8684 T^{5} - 10152 T^{6} + 208404 T^{7} - 3196994 T^{8} + 7710948 T^{9} - 13898088 T^{10} + 439870652 T^{11} - 89959728 T^{12} - 15255670540 T^{13} + 20525811272 T^{14} - 379727508532 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 531095140 T^{10} + 15485170280 T^{12} - 361007922316 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 - 12 T + 72 T^{2} - 300 T^{3} - 1008 T^{4} + 17988 T^{5} - 98280 T^{6} + 509316 T^{7} - 2377282 T^{8} + 21900588 T^{9} - 181719720 T^{10} + 1430171916 T^{11} - 3446151408 T^{12} - 44102532900 T^{13} + 455138139528 T^{14} - 3261823333284 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 12 T + 72 T^{2} - 220 T^{3} - 2352 T^{4} + 10116 T^{5} + 72152 T^{6} - 1599596 T^{7} + 18271774 T^{8} - 75181012 T^{9} + 159383768 T^{10} + 1050273468 T^{11} - 11477009712 T^{12} - 50455901540 T^{13} + 776103503688 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 32 T + 512 T^{2} + 5792 T^{3} + 60388 T^{4} + 619872 T^{5} + 5690880 T^{6} + 44817632 T^{7} + 328258854 T^{8} + 2375334496 T^{9} + 15985681920 T^{10} + 92284683744 T^{11} + 476490366628 T^{12} + 2422188295456 T^{13} + 11348152898048 T^{14} + 37590756474784 T^{15} + 62259690411361 T^{16}$$
$59$ $$( 1 + 8 T + 148 T^{2} + 520 T^{3} + 8710 T^{4} + 30680 T^{5} + 515188 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 976 T^{5} + 840946 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 20 T + 200 T^{2} + 1332 T^{3} + 9104 T^{4} + 102596 T^{5} + 1118232 T^{6} + 8938244 T^{7} + 69729150 T^{8} + 598862348 T^{9} + 5019743448 T^{10} + 30857080748 T^{11} + 183455805584 T^{12} + 1798366642524 T^{13} + 18091676433800 T^{14} + 121214232106460 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 16887954920 T^{10} + 797621843228 T^{12} - 27669661326936 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 + 36 T + 648 T^{2} + 8620 T^{3} + 98192 T^{4} + 980116 T^{5} + 8807960 T^{6} + 73999132 T^{7} + 618591198 T^{8} + 5401936636 T^{9} + 46937618840 T^{10} + 381281785972 T^{11} + 2788480080272 T^{12} + 17869877131660 T^{13} + 98064578635272 T^{14} + 397706346687492 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 11950216872 T^{10} + 1274914051292 T^{12} - 68064487545880 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 + 56 T + 1568 T^{2} + 30056 T^{3} + 449924 T^{4} + 5649064 T^{5} + 62548320 T^{6} + 632947128 T^{7} + 5962291750 T^{8} + 52534611624 T^{9} + 430895376480 T^{10} + 3230061357368 T^{11} + 21352637617604 T^{12} + 118391805566008 T^{13} + 512642505442592 T^{14} + 1519618855419112 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 36 T + 734 T^{2} + 10004 T^{3} + 106562 T^{4} + 890356 T^{5} + 5814014 T^{6} + 25378884 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 4 T + 8 T^{2} - 1852 T^{3} + 7856 T^{4} + 48156 T^{5} + 1459480 T^{6} - 10407580 T^{7} - 90680354 T^{8} - 1009535260 T^{9} + 13732247320 T^{10} + 43950680988 T^{11} + 695486031536 T^{12} - 15903754155964 T^{13} + 6663776039432 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16}$$