Properties

 Label 1150.4.b.n Level $1150$ Weight $4$ Character orbit 1150.b Analytic conductor $67.852$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1150.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$67.8521965066$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 136 x^{6} + 5308 x^{4} + 58833 x^{2} + 116964$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} -4 q^{4} + ( 2 - 2 \beta_{3} ) q^{6} + ( -2 \beta_{1} + \beta_{4} ) q^{7} -8 \beta_{2} q^{8} + ( -8 + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} -4 q^{4} + ( 2 - 2 \beta_{3} ) q^{6} + ( -2 \beta_{1} + \beta_{4} ) q^{7} -8 \beta_{2} q^{8} + ( -8 + \beta_{6} ) q^{9} + ( -10 + \beta_{3} + \beta_{6} + \beta_{7} ) q^{11} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{12} + ( 8 \beta_{1} - 5 \beta_{2} + \beta_{5} ) q^{13} + ( 4 \beta_{3} - 2 \beta_{7} ) q^{14} + 16 q^{16} + ( -13 \beta_{1} + 6 \beta_{2} - \beta_{4} + \beta_{5} ) q^{17} + ( -16 \beta_{2} + 2 \beta_{5} ) q^{18} + ( -14 - 2 \beta_{3} - 2 \beta_{6} + 3 \beta_{7} ) q^{19} + ( 74 + \beta_{3} - 5 \beta_{6} + 4 \beta_{7} ) q^{21} + ( 2 \beta_{1} - 20 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{22} -23 \beta_{2} q^{23} + ( -8 + 8 \beta_{3} ) q^{24} + ( 10 - 16 \beta_{3} - 2 \beta_{6} ) q^{26} + ( \beta_{1} - 35 \beta_{2} + 3 \beta_{4} ) q^{27} + ( 8 \beta_{1} - 4 \beta_{4} ) q^{28} + ( -41 + 14 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{29} + ( 97 + 16 \beta_{3} + 5 \beta_{6} ) q^{31} + 32 \beta_{2} q^{32} + ( -26 \beta_{1} + 22 \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{33} + ( -12 + 26 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{34} + ( 32 - 4 \beta_{6} ) q^{36} + ( 18 \beta_{1} - 116 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{37} + ( -4 \beta_{1} - 28 \beta_{2} + 6 \beta_{4} - 4 \beta_{5} ) q^{38} + ( -261 + 15 \beta_{3} + 8 \beta_{6} - 3 \beta_{7} ) q^{39} + ( 121 - \beta_{3} + 10 \beta_{6} ) q^{41} + ( 2 \beta_{1} + 148 \beta_{2} + 8 \beta_{4} - 10 \beta_{5} ) q^{42} + ( 24 \beta_{1} + 222 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} ) q^{43} + ( 40 - 4 \beta_{3} - 4 \beta_{6} - 4 \beta_{7} ) q^{44} + 46 q^{46} + ( -44 \beta_{1} + 67 \beta_{2} - 3 \beta_{4} - 11 \beta_{5} ) q^{47} + ( 16 \beta_{1} - 16 \beta_{2} ) q^{48} + ( -411 + 49 \beta_{3} + 9 \beta_{6} + \beta_{7} ) q^{49} + ( 458 + 26 \beta_{3} - 10 \beta_{6} - 7 \beta_{7} ) q^{51} + ( -32 \beta_{1} + 20 \beta_{2} - 4 \beta_{5} ) q^{52} + ( -40 \beta_{1} + 146 \beta_{2} - 8 \beta_{4} - 8 \beta_{5} ) q^{53} + ( 70 - 2 \beta_{3} - 6 \beta_{7} ) q^{54} + ( -16 \beta_{3} + 8 \beta_{7} ) q^{56} + ( 23 \beta_{1} - 40 \beta_{2} - 18 \beta_{4} + 11 \beta_{5} ) q^{57} + ( 28 \beta_{1} - 82 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} ) q^{58} + ( 20 - 10 \beta_{3} + 2 \beta_{6} + 14 \beta_{7} ) q^{59} + ( 290 + 33 \beta_{3} + 19 \beta_{6} - 7 \beta_{7} ) q^{61} + ( 32 \beta_{1} + 194 \beta_{2} + 10 \beta_{5} ) q^{62} + ( 115 \beta_{1} + 16 \beta_{2} - 4 \beta_{4} + 11 \beta_{5} ) q^{63} -64 q^{64} + ( -44 + 52 \beta_{3} - 4 \beta_{6} + 2 \beta_{7} ) q^{66} + ( -28 \beta_{1} + 368 \beta_{2} - 12 \beta_{5} ) q^{67} + ( 52 \beta_{1} - 24 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} ) q^{68} + ( -23 + 23 \beta_{3} ) q^{69} + ( 57 + 39 \beta_{3} - 4 \beta_{6} - 28 \beta_{7} ) q^{71} + ( 64 \beta_{2} - 8 \beta_{5} ) q^{72} + ( 44 \beta_{1} + 287 \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{73} + ( 232 - 36 \beta_{3} - 8 \beta_{6} + 4 \beta_{7} ) q^{74} + ( 56 + 8 \beta_{3} + 8 \beta_{6} - 12 \beta_{7} ) q^{76} + ( 61 \beta_{1} + 548 \beta_{2} - 16 \beta_{4} + 17 \beta_{5} ) q^{77} + ( 30 \beta_{1} - 522 \beta_{2} - 6 \beta_{4} + 16 \beta_{5} ) q^{78} + ( 232 + 72 \beta_{3} - 16 \beta_{6} - 20 \beta_{7} ) q^{79} + ( -267 + 31 \beta_{3} + 19 \beta_{6} + 12 \beta_{7} ) q^{81} + ( -2 \beta_{1} + 242 \beta_{2} + 20 \beta_{5} ) q^{82} + ( -26 \beta_{1} - 256 \beta_{2} - 24 \beta_{4} + 6 \beta_{5} ) q^{83} + ( -296 - 4 \beta_{3} + 20 \beta_{6} - 16 \beta_{7} ) q^{84} + ( -444 - 48 \beta_{3} + 12 \beta_{6} - 12 \beta_{7} ) q^{86} + ( 30 \beta_{1} + 547 \beta_{2} - 21 \beta_{4} - 5 \beta_{5} ) q^{87} + ( -8 \beta_{1} + 80 \beta_{2} - 8 \beta_{4} - 8 \beta_{5} ) q^{88} + ( 450 - 30 \beta_{3} - 34 \beta_{6} - 16 \beta_{7} ) q^{89} + ( 576 - 85 \beta_{3} - 51 \beta_{6} + 25 \beta_{7} ) q^{91} + 92 \beta_{2} q^{92} + ( 23 \beta_{1} + 367 \beta_{2} + 15 \beta_{4} - 16 \beta_{5} ) q^{93} + ( -134 + 88 \beta_{3} + 22 \beta_{6} + 6 \beta_{7} ) q^{94} + ( 32 - 32 \beta_{3} ) q^{96} + ( 137 \beta_{1} + 504 \beta_{2} + 31 \beta_{4} + \beta_{5} ) q^{97} + ( 98 \beta_{1} - 822 \beta_{2} + 2 \beta_{4} + 18 \beta_{5} ) q^{98} + ( 662 + 68 \beta_{3} + 4 \beta_{6} + 17 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} + O(q^{10})$$ $$8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} - 78 q^{11} - 4 q^{14} + 128 q^{16} - 106 q^{19} + 600 q^{21} - 64 q^{24} + 80 q^{26} - 322 q^{29} + 776 q^{31} - 92 q^{34} + 256 q^{36} - 2094 q^{39} + 968 q^{41} + 312 q^{44} + 368 q^{46} - 3286 q^{49} + 3650 q^{51} + 548 q^{54} + 16 q^{56} + 188 q^{59} + 2306 q^{61} - 512 q^{64} - 348 q^{66} - 184 q^{69} + 400 q^{71} + 1864 q^{74} + 424 q^{76} + 1816 q^{79} - 2112 q^{81} - 2400 q^{84} - 3576 q^{86} + 3568 q^{89} + 4658 q^{91} - 1060 q^{94} + 256 q^{96} + 5330 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 136 x^{6} + 5308 x^{4} + 58833 x^{2} + 116964$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 136 \nu^{5} + 4966 \nu^{3} + 35577 \nu$$$$)/37962$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} + 68 \nu^{2} + 342$$$$)/111$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 79 \nu^{5} + 1019 \nu^{3} + 89367 \nu$$$$)/6327$$ $$\beta_{5}$$ $$=$$ $$($$$$17 \nu^{7} + 2141 \nu^{5} + 72794 \nu^{3} + 584289 \nu$$$$)/18981$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{4} + 247 \nu^{2} + 4458$$$$)/111$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 118 \nu^{4} + 3409 \nu^{2} + 15102$$$$)/333$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{3} - 34$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{5} + 3 \beta_{4} - 84 \beta_{2} - 56 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-68 \beta_{6} + 247 \beta_{3} + 1970$$ $$\nu^{5}$$ $$=$$ $$-315 \beta_{5} - 204 \beta_{4} + 9486 \beta_{2} + 3688 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$333 \beta_{7} + 4615 \beta_{6} - 22328 \beta_{3} - 131656$$ $$\nu^{7}$$ $$=$$ $$27942 \beta_{5} + 12846 \beta_{4} - 834990 \beta_{2} - 259049 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
599.1
 − 8.73081i − 1.58997i 3.74869i 6.57209i − 6.57209i − 3.74869i 1.58997i 8.73081i
2.00000i 7.73081i −4.00000 0 −15.4616 23.5622i 8.00000i −32.7654 0
599.2 2.00000i 0.589969i −4.00000 0 −1.17994 18.5077i 8.00000i 26.6519 0
599.3 2.00000i 4.74869i −4.00000 0 9.49738 29.3684i 8.00000i 4.44993 0
599.4 2.00000i 7.57209i −4.00000 0 15.1442 35.4229i 8.00000i −30.3365 0
599.5 2.00000i 7.57209i −4.00000 0 15.1442 35.4229i 8.00000i −30.3365 0
599.6 2.00000i 4.74869i −4.00000 0 9.49738 29.3684i 8.00000i 4.44993 0
599.7 2.00000i 0.589969i −4.00000 0 −1.17994 18.5077i 8.00000i 26.6519 0
599.8 2.00000i 7.73081i −4.00000 0 −15.4616 23.5622i 8.00000i −32.7654 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 599.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1150.4.b.n 8
5.b even 2 1 inner 1150.4.b.n 8
5.c odd 4 1 230.4.a.h 4
5.c odd 4 1 1150.4.a.p 4
15.e even 4 1 2070.4.a.bj 4
20.e even 4 1 1840.4.a.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.h 4 5.c odd 4 1
1150.4.a.p 4 5.c odd 4 1
1150.4.b.n 8 1.a even 1 1 trivial
1150.4.b.n 8 5.b even 2 1 inner
1840.4.a.m 4 20.e even 4 1
2070.4.a.bj 4 15.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 $$S_{4}^{\mathrm{new}}(1150, [\chi])$$:

 $$T_{3}^{8} + 140 T_{3}^{6} + 6116 T_{3}^{4} + 79385 T_{3}^{2} + 26896$$ $$T_{7}^{8} + 3015 T_{7}^{6} + 3173141 T_{7}^{4} + 1374197220 T_{7}^{2} + 205811024896$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + T^{2} )^{4}$$
$3$ $$26896 + 79385 T^{2} + 6116 T^{4} + 140 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$205811024896 + 1374197220 T^{2} + 3173141 T^{4} + 3015 T^{6} + T^{8}$$
$11$ $$( -977536 - 94420 T - 1771 T^{2} + 39 T^{3} + T^{4} )^{2}$$
$13$ $$9357493236004 + 30921165153 T^{2} + 29581468 T^{4} + 10296 T^{6} + T^{8}$$
$17$ $$1098689697856 + 378462071476 T^{2} + 231041117 T^{4} + 28939 T^{6} + T^{8}$$
$19$ $$( -78336 - 340080 T - 15029 T^{2} + 53 T^{3} + T^{4} )^{2}$$
$23$ $$( 529 + T^{2} )^{4}$$
$29$ $$( 1597064 - 1886712 T - 27260 T^{2} + 161 T^{3} + T^{4} )^{2}$$
$31$ $$( -397027152 + 5546709 T + 13076 T^{2} - 388 T^{3} + T^{4} )^{2}$$
$37$ $$62742621208576 + 1065658064896 T^{2} + 786594560 T^{4} + 141316 T^{6} + T^{8}$$
$41$ $$( 1506099394 + 17572043 T - 36232 T^{2} - 484 T^{3} + T^{4} )^{2}$$
$43$ $$29073917985322696704 + 2319170105769984 T^{2} + 57830464512 T^{4} + 465732 T^{6} + T^{8}$$
$47$ $$1705996182469100544 + 2496179747456784 T^{2} + 78759541516 T^{4} + 531005 T^{6} + T^{8}$$
$53$ $$10474877669735098624 + 5669291856436992 T^{2} + 96953199584 T^{4} + 546480 T^{6} + T^{8}$$
$59$ $$( 11673423616 + 20596976 T - 243460 T^{2} - 94 T^{3} + T^{4} )^{2}$$
$61$ $$( -42772329400 + 315388730 T - 54057 T^{2} - 1153 T^{3} + T^{4} )^{2}$$
$67$ $$10\!\cdots\!64$$$$+ 14235723146461184 T^{2} + 216109619456 T^{4} + 962720 T^{6} + T^{8}$$
$71$ $$( 274201266224 + 93278335 T - 1067322 T^{2} - 200 T^{3} + T^{4} )^{2}$$
$73$ $$22601538539917711936 + 3467966319547776 T^{2} + 96552759536 T^{4} + 603369 T^{6} + T^{8}$$
$79$ $$( 145785325568 + 272612224 T - 923664 T^{2} - 908 T^{3} + T^{4} )^{2}$$
$83$ $$31\!\cdots\!84$$$$+ 327375019180504064 T^{2} + 1186957522832 T^{4} + 1817192 T^{6} + T^{8}$$
$89$ $$( -969417760000 + 2249185600 T - 492084 T^{2} - 1784 T^{3} + T^{4} )^{2}$$
$97$ $$43\!\cdots\!76$$$$+ 6601831785785843608 T^{2} + 10443122214545 T^{4} + 5623807 T^{6} + T^{8}$$
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