Properties

 Label 35.5.c.e Level $35$ Weight $5$ Character orbit 35.c Analytic conductor $3.618$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 110 x^{6} + 7113 x^{4} + 190880 x^{2} + 4177936$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + 3 \beta_{3} q^{3} + ( -22 + \beta_{2} ) q^{4} -\beta_{7} q^{5} + 3 \beta_{1} q^{6} + ( 5 \beta_{3} - \beta_{6} + \beta_{7} ) q^{7} + ( \beta_{1} - \beta_{3} - 11 \beta_{4} + 2 \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + 3 \beta_{3} q^{3} + ( -22 + \beta_{2} ) q^{4} -\beta_{7} q^{5} + 3 \beta_{1} q^{6} + ( 5 \beta_{3} - \beta_{6} + \beta_{7} ) q^{7} + ( \beta_{1} - \beta_{3} - 11 \beta_{4} + 2 \beta_{6} ) q^{8} + 9 q^{9} + ( -3 \beta_{1} - 19 \beta_{3} + \beta_{5} - \beta_{7} ) q^{10} + ( -44 - 6 \beta_{2} ) q^{11} + ( -3 \beta_{1} - 66 \beta_{3} - 6 \beta_{7} ) q^{12} + ( 3 \beta_{1} + 28 \beta_{3} + 6 \beta_{7} ) q^{13} + ( -11 + 7 \beta_{1} + 16 \beta_{2} - \beta_{5} ) q^{14} + ( 15 \beta_{2} + 15 \beta_{4} ) q^{15} + ( 88 - 27 \beta_{2} ) q^{16} + ( -2 \beta_{1} + 100 \beta_{3} - 4 \beta_{7} ) q^{17} + 9 \beta_{4} q^{18} + ( -17 \beta_{1} - 2 \beta_{5} ) q^{19} + ( -3 \beta_{1} + 106 \beta_{3} + \beta_{5} + 21 \beta_{7} ) q^{20} + ( 135 - 15 \beta_{2} - 3 \beta_{5} ) q^{21} + ( -6 \beta_{1} + 6 \beta_{3} - 14 \beta_{4} - 12 \beta_{6} ) q^{22} + ( -5 \beta_{1} + 5 \beta_{3} + 94 \beta_{4} - 10 \beta_{6} ) q^{23} + ( -33 \beta_{1} + 6 \beta_{5} ) q^{24} + ( 435 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 25 \beta_{4} + 10 \beta_{6} ) q^{25} + ( 43 \beta_{1} - 6 \beta_{5} ) q^{26} -216 \beta_{3} q^{27} + ( -7 \beta_{1} - 213 \beta_{3} - 91 \beta_{4} + 16 \beta_{6} - 30 \beta_{7} ) q^{28} + ( -6 - 4 \beta_{2} ) q^{29} + ( -570 + 15 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} - 75 \beta_{4} + 30 \beta_{6} ) q^{30} + ( -20 \beta_{1} + 8 \beta_{5} ) q^{31} + ( -11 \beta_{1} + 11 \beta_{3} + 47 \beta_{4} - 22 \beta_{6} ) q^{32} + ( 18 \beta_{1} - 132 \beta_{3} + 36 \beta_{7} ) q^{33} + ( 90 \beta_{1} + 4 \beta_{5} ) q^{34} + ( -530 - 56 \beta_{1} + 20 \beta_{2} - 3 \beta_{3} + 35 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 16 \beta_{7} ) q^{35} + ( -198 + 9 \beta_{2} ) q^{36} + ( -10 \beta_{1} + 10 \beta_{3} - 134 \beta_{4} - 20 \beta_{6} ) q^{37} + ( -15 \beta_{1} + 624 \beta_{3} - 30 \beta_{7} ) q^{38} + ( 840 - 90 \beta_{2} ) q^{39} + ( 140 \beta_{1} + 220 \beta_{3} - 5 \beta_{5} + 43 \beta_{7} ) q^{40} + ( -36 \beta_{1} + 2 \beta_{5} ) q^{41} + ( -63 \beta_{1} - 18 \beta_{3} + 210 \beta_{4} - 30 \beta_{6} - 96 \beta_{7} ) q^{42} + ( 25 \beta_{1} - 25 \beta_{3} - 56 \beta_{4} + 50 \beta_{6} ) q^{43} + ( -304 + 82 \beta_{2} ) q^{44} -9 \beta_{7} q^{45} + ( -3682 + 254 \beta_{2} ) q^{46} + ( 3 \beta_{1} + 209 \beta_{3} + 6 \beta_{7} ) q^{47} + ( 81 \beta_{1} + 264 \beta_{3} + 162 \beta_{7} ) q^{48} + ( -936 + 91 \beta_{1} - 85 \beta_{2} - 3 \beta_{5} ) q^{49} + ( 1060 + 5 \beta_{1} - 185 \beta_{2} - 5 \beta_{3} + 410 \beta_{4} + 10 \beta_{6} ) q^{50} + ( 3000 + 60 \beta_{2} ) q^{51} + ( -91 \beta_{1} - 1252 \beta_{3} - 182 \beta_{7} ) q^{52} + ( 6 \beta_{1} - 6 \beta_{3} - 90 \beta_{4} + 12 \beta_{6} ) q^{53} -216 \beta_{1} q^{54} + ( 18 \beta_{1} - 636 \beta_{3} - 6 \beta_{5} + 50 \beta_{7} ) q^{55} + ( 3458 - 168 \beta_{1} - 91 \beta_{2} + 14 \beta_{5} ) q^{56} + ( -30 \beta_{1} + 30 \beta_{3} - 510 \beta_{4} - 60 \beta_{6} ) q^{57} + ( -4 \beta_{1} + 4 \beta_{3} + 14 \beta_{4} - 8 \beta_{6} ) q^{58} + ( -61 \beta_{1} - 26 \beta_{5} ) q^{59} + ( 3180 + 15 \beta_{1} - 315 \beta_{2} - 15 \beta_{3} - 405 \beta_{4} + 30 \beta_{6} ) q^{60} + ( -35 \beta_{1} + 36 \beta_{5} ) q^{61} + ( 148 \beta_{1} + 848 \beta_{3} + 296 \beta_{7} ) q^{62} + ( 45 \beta_{3} - 9 \beta_{6} + 9 \beta_{7} ) q^{63} + ( -620 - 33 \beta_{2} ) q^{64} + ( -3180 - 15 \beta_{1} + 125 \beta_{2} + 15 \beta_{3} + 215 \beta_{4} - 30 \beta_{6} ) q^{65} + ( -42 \beta_{1} - 36 \beta_{5} ) q^{66} + ( 5 \beta_{1} - 5 \beta_{3} + 716 \beta_{4} + 10 \beta_{6} ) q^{67} + ( -58 \beta_{1} - 1776 \beta_{3} - 116 \beta_{7} ) q^{68} + ( 282 \beta_{1} - 30 \beta_{5} ) q^{69} + ( -1385 - 28 \beta_{1} + 115 \beta_{2} + 1676 \beta_{3} - 630 \beta_{4} + 16 \beta_{5} + 40 \beta_{6} - 5 \beta_{7} ) q^{70} + ( -2344 + 184 \beta_{2} ) q^{71} + ( 9 \beta_{1} - 9 \beta_{3} - 99 \beta_{4} + 18 \beta_{6} ) q^{72} + ( -112 \beta_{1} + 956 \beta_{3} - 224 \beta_{7} ) q^{73} + ( 4872 + 186 \beta_{2} ) q^{74} + ( -90 \beta_{1} + 1305 \beta_{3} + 30 \beta_{5} - 30 \beta_{7} ) q^{75} + ( 277 \beta_{1} - 2 \beta_{5} ) q^{76} + ( 42 \beta_{1} + 398 \beta_{3} + 546 \beta_{4} + 80 \beta_{6} + 4 \beta_{7} ) q^{77} + ( -90 \beta_{1} + 90 \beta_{3} + 1290 \beta_{4} - 180 \beta_{6} ) q^{78} + ( -976 - 204 \beta_{2} ) q^{79} + ( 81 \beta_{1} - 2862 \beta_{3} - 27 \beta_{5} - 61 \beta_{7} ) q^{80} -7209 q^{81} + ( 68 \beta_{1} + 1390 \beta_{3} + 136 \beta_{7} ) q^{82} + ( 6 \beta_{1} - 109 \beta_{3} + 12 \beta_{7} ) q^{83} + ( -6150 - 273 \beta_{1} + 450 \beta_{2} + 48 \beta_{5} ) q^{84} + ( 2120 + 10 \beta_{1} + 510 \beta_{2} - 10 \beta_{3} + 450 \beta_{4} + 20 \beta_{6} ) q^{85} + ( 2678 - 856 \beta_{2} ) q^{86} + ( 12 \beta_{1} - 18 \beta_{3} + 24 \beta_{7} ) q^{87} + ( -14 \beta_{1} + 14 \beta_{3} - 938 \beta_{4} - 28 \beta_{6} ) q^{88} + ( 258 \beta_{1} + 60 \beta_{5} ) q^{89} + ( -27 \beta_{1} - 171 \beta_{3} + 9 \beta_{5} - 9 \beta_{7} ) q^{90} + ( 4440 + 273 \beta_{1} - 260 \beta_{2} - 10 \beta_{5} ) q^{91} + ( 174 \beta_{1} - 174 \beta_{3} - 3448 \beta_{4} + 348 \beta_{6} ) q^{92} + ( 120 \beta_{1} - 120 \beta_{3} - 600 \beta_{4} + 240 \beta_{6} ) q^{93} + ( 224 \beta_{1} - 6 \beta_{5} ) q^{94} + ( 3120 - 115 \beta_{1} + 75 \beta_{2} + 115 \beta_{3} - 485 \beta_{4} - 230 \beta_{6} ) q^{95} + ( 141 \beta_{1} - 66 \beta_{5} ) q^{96} + ( -138 \beta_{1} - 2060 \beta_{3} - 276 \beta_{7} ) q^{97} + ( -224 \beta_{1} - 3406 \beta_{3} - 511 \beta_{4} - 170 \beta_{6} - 278 \beta_{7} ) q^{98} + ( -396 - 54 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 172q^{4} + 72q^{9} + O(q^{10})$$ $$8q - 172q^{4} + 72q^{9} - 376q^{11} - 24q^{14} + 60q^{15} + 596q^{16} + 1020q^{21} + 3500q^{25} - 64q^{29} - 4500q^{30} - 4160q^{35} - 1548q^{36} + 6360q^{39} - 2104q^{44} - 28440q^{46} - 7828q^{49} + 7740q^{50} + 24240q^{51} + 27300q^{56} + 24180q^{60} - 5092q^{64} - 24940q^{65} - 10620q^{70} - 18016q^{71} + 39720q^{74} - 8624q^{79} - 57672q^{81} - 47400q^{84} + 19000q^{85} + 18000q^{86} + 34480q^{91} + 25260q^{95} - 3384q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 110 x^{6} + 7113 x^{4} + 190880 x^{2} + 4177936$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} + 125 \nu^{4} + 12513 \nu^{2} + 188720$$$$)/16898$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{6} - 125 \nu^{4} - 4064 \nu^{2} + 47852$$$$)/8449$$ $$\beta_{3}$$ $$=$$ $$($$$$-25 \nu^{7} - 2166 \nu^{5} - 141325 \nu^{3} - 1118204 \nu$$$$)/4934216$$ $$\beta_{4}$$ $$=$$ $$($$$$25 \nu^{7} + 2166 \nu^{5} + 141325 \nu^{3} + 6052420 \nu$$$$)/4934216$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{6} + 436 \nu^{4} + 22497 \nu^{2} + 517664$$$$)/1988$$ $$\beta_{6}$$ $$=$$ $$($$$$-113 \nu^{7} - 146 \nu^{6} + 783 \nu^{5} - 9125 \nu^{4} + 330432 \nu^{3} - 913449 \nu^{2} + 26249794 \nu - 13776560$$$$)/2467108$$ $$\beta_{7}$$ $$=$$ $$($$$$154 \nu^{7} - 146 \nu^{6} + 16867 \nu^{5} - 9125 \nu^{4} + 782451 \nu^{3} - 913449 \nu^{2} + 11977428 \nu - 13776560$$$$)/2467108$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{4} + \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} - 28$$ $$\nu^{3}$$ $$=$$ $$-6 \beta_{7} + 2 \beta_{6} - 13 \beta_{4} - 105 \beta_{3} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{5} - 15 \beta_{2} - 132 \beta_{1} - 524$$ $$\nu^{5}$$ $$=$$ $$550 \beta_{7} + 50 \beta_{6} - 1769 \beta_{4} + 4555 \beta_{3} + 300 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-500 \beta_{5} - 5319 \beta_{2} + 4186 \beta_{1} + 113572$$ $$\nu^{7}$$ $$=$$ $$-13734 \beta_{7} - 15638 \beta_{6} + 182027 \beta_{4} - 43177 \beta_{3} - 14686 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 −3.16228 − 7.21587i 3.16228 − 7.21587i −3.16228 − 4.78865i 3.16228 − 4.78865i −3.16228 + 4.78865i 3.16228 + 4.78865i −3.16228 + 7.21587i 3.16228 + 7.21587i
7.21587i −9.48683 −36.0688 22.2447 + 11.4093i 68.4558i −36.4750 32.7197i 144.814i 9.00000 82.3280 160.515i
34.2 7.21587i 9.48683 −36.0688 −22.2447 11.4093i 68.4558i 36.4750 32.7197i 144.814i 9.00000 −82.3280 + 160.515i
34.3 4.78865i −9.48683 −6.93120 −23.8259 + 7.57153i 45.4292i 9.59562 + 48.0513i 43.4273i 9.00000 36.2574 + 114.094i
34.4 4.78865i 9.48683 −6.93120 23.8259 7.57153i 45.4292i −9.59562 + 48.0513i 43.4273i 9.00000 −36.2574 114.094i
34.5 4.78865i −9.48683 −6.93120 −23.8259 7.57153i 45.4292i 9.59562 48.0513i 43.4273i 9.00000 36.2574 114.094i
34.6 4.78865i 9.48683 −6.93120 23.8259 + 7.57153i 45.4292i −9.59562 48.0513i 43.4273i 9.00000 −36.2574 + 114.094i
34.7 7.21587i −9.48683 −36.0688 22.2447 11.4093i 68.4558i −36.4750 + 32.7197i 144.814i 9.00000 82.3280 + 160.515i
34.8 7.21587i 9.48683 −36.0688 −22.2447 + 11.4093i 68.4558i 36.4750 + 32.7197i 144.814i 9.00000 −82.3280 160.515i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.8 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
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.c.e 8
3.b odd 2 1 315.5.e.e 8
4.b odd 2 1 560.5.p.g 8
5.b even 2 1 inner 35.5.c.e 8
5.c odd 4 2 175.5.d.g 8
7.b odd 2 1 inner 35.5.c.e 8
15.d odd 2 1 315.5.e.e 8
20.d odd 2 1 560.5.p.g 8
21.c even 2 1 315.5.e.e 8
28.d even 2 1 560.5.p.g 8
35.c odd 2 1 inner 35.5.c.e 8
35.f even 4 2 175.5.d.g 8
105.g even 2 1 315.5.e.e 8
140.c even 2 1 560.5.p.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.e 8 1.a even 1 1 trivial
35.5.c.e 8 5.b even 2 1 inner
35.5.c.e 8 7.b odd 2 1 inner
35.5.c.e 8 35.c odd 2 1 inner
175.5.d.g 8 5.c odd 4 2
175.5.d.g 8 35.f even 4 2
315.5.e.e 8 3.b odd 2 1
315.5.e.e 8 15.d odd 2 1
315.5.e.e 8 21.c even 2 1
315.5.e.e 8 105.g even 2 1
560.5.p.g 8 4.b odd 2 1
560.5.p.g 8 20.d odd 2 1
560.5.p.g 8 28.d even 2 1
560.5.p.g 8 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}^{4} + 75 T_{2}^{2} + 1194$$ $$T_{3}^{2} - 90$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1194 + 75 T^{2} + T^{4} )^{2}$$
$3$ $$( -90 + T^{2} )^{4}$$
$5$ $$152587890625 - 683593750 T^{2} + 1525650 T^{4} - 1750 T^{6} + T^{8}$$
$7$ $$33232930569601 + 22563431114 T^{2} + 9225426 T^{4} + 3914 T^{6} + T^{8}$$
$11$ $$( -5432 + 94 T + T^{2} )^{4}$$
$13$ $$( 145926400 - 52250 T^{2} + T^{4} )^{2}$$
$17$ $$( 8745990400 - 221000 T^{2} + T^{4} )^{2}$$
$19$ $$( 10320458400 + 347850 T^{2} + T^{4} )^{2}$$
$23$ $$( 933058464 + 1011930 T^{2} + T^{4} )^{2}$$
$29$ $$( -3332 + 16 T + T^{2} )^{4}$$
$31$ $$( 1373776281600 + 2482080 T^{2} + T^{4} )^{2}$$
$37$ $$( 250900015104 + 2666340 T^{2} + T^{4} )^{2}$$
$41$ $$( 61553565600 + 1115820 T^{2} + T^{4} )^{2}$$
$43$ $$( 18960156666024 + 8717550 T^{2} + T^{4} )^{2}$$
$47$ $$( 169299331600 - 899330 T^{2} + T^{4} )^{2}$$
$53$ $$( 23031858816 + 1107108 T^{2} + T^{4} )^{2}$$
$59$ $$( 40047069962400 + 25445850 T^{2} + T^{4} )^{2}$$
$61$ $$( 499402073322600 + 44868270 T^{2} + T^{4} )^{2}$$
$67$ $$( 357890658714024 + 38701230 T^{2} + T^{4} )^{2}$$
$71$ $$( -2114432 + 4504 T + T^{2} )^{4}$$
$73$ $$( 268409242240000 - 73732160 T^{2} + T^{4} )^{2}$$
$79$ $$( -7670912 + 2156 T + T^{2} )^{4}$$
$83$ $$( 2403940900 - 403700 T^{2} + T^{4} )^{2}$$
$89$ $$( 17469797990400 + 169727400 T^{2} + T^{4} )^{2}$$
$97$ $$( 608524806400 - 160123400 T^{2} + T^{4} )^{2}$$
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