LMFDB - Embedded newform 85.6.a.a.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [85,6,Mod(1,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	85.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.6326246841$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.29890$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.29890 q^{2} -29.9706 q^{3} -13.5195 q^{4} +25.0000 q^{5} +128.840 q^{6} +45.8012 q^{7} +195.684 q^{8} +655.235 q^{9} -107.472 q^{10} -504.686 q^{11} +405.186 q^{12} +513.142 q^{13} -196.895 q^{14} -749.264 q^{15} -408.601 q^{16} +289.000 q^{17} -2816.79 q^{18} -2084.02 q^{19} -337.987 q^{20} -1372.69 q^{21} +2169.60 q^{22} +4718.18 q^{23} -5864.75 q^{24} +625.000 q^{25} -2205.94 q^{26} -12354.9 q^{27} -619.207 q^{28} -3092.07 q^{29} +3221.01 q^{30} +4157.17 q^{31} -4505.34 q^{32} +15125.7 q^{33} -1242.38 q^{34} +1145.03 q^{35} -8858.42 q^{36} +4514.40 q^{37} +8958.99 q^{38} -15379.1 q^{39} +4892.09 q^{40} +7375.94 q^{41} +5901.04 q^{42} -17592.2 q^{43} +6823.09 q^{44} +16380.9 q^{45} -20283.0 q^{46} -14737.7 q^{47} +12246.0 q^{48} -14709.3 q^{49} -2686.81 q^{50} -8661.49 q^{51} -6937.40 q^{52} -1008.07 q^{53} +53112.5 q^{54} -12617.2 q^{55} +8962.53 q^{56} +62459.2 q^{57} +13292.5 q^{58} +496.132 q^{59} +10129.6 q^{60} -13046.2 q^{61} -17871.2 q^{62} +30010.5 q^{63} +32443.2 q^{64} +12828.5 q^{65} -65024.0 q^{66} +21128.0 q^{67} -3907.12 q^{68} -141406. q^{69} -4922.36 q^{70} -41206.2 q^{71} +128219. q^{72} +53173.8 q^{73} -19406.9 q^{74} -18731.6 q^{75} +28174.8 q^{76} -23115.2 q^{77} +66113.4 q^{78} +22083.6 q^{79} -10215.0 q^{80} +211061. q^{81} -31708.4 q^{82} -63967.7 q^{83} +18558.0 q^{84} +7225.00 q^{85} +75627.1 q^{86} +92671.1 q^{87} -98758.8 q^{88} -145034. q^{89} -70419.7 q^{90} +23502.5 q^{91} -63787.2 q^{92} -124593. q^{93} +63356.1 q^{94} -52100.5 q^{95} +135028. q^{96} -131574. q^{97} +63233.6 q^{98} -330688. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 7 q^{2} - 36 q^{3} + 43 q^{4} + 125 q^{5} - 105 q^{6} - 204 q^{7} - 63 q^{8} + 531 q^{9} - 175 q^{10} - 792 q^{11} + 785 q^{12} + 88 q^{13} + 860 q^{14} - 900 q^{15} - 2365 q^{16} + 1445 q^{17} - 2052 q^{18}+ \cdots - 535112 q^{99}+O(q^{100}) $ 5 * q - 7 * q^2 - 36 * q^3 + 43 * q^4 + 125 * q^5 - 105 * q^6 - 204 * q^7 - 63 * q^8 + 531 * q^9 - 175 * q^10 - 792 * q^11 + 785 * q^12 + 88 * q^13 + 860 * q^14 - 900 * q^15 - 2365 * q^16 + 1445 * q^17 - 2052 * q^18 - 5160 * q^19 + 1075 * q^20 - 6428 * q^21 - 3058 * q^22 - 6140 * q^23 - 11751 * q^24 + 3125 * q^25 - 9643 * q^26 - 13380 * q^27 - 4568 * q^28 - 7328 * q^29 - 2625 * q^30 - 8968 * q^31 - 4703 * q^32 + 13072 * q^33 - 2023 * q^34 - 5100 * q^35 + 2048 * q^36 + 9834 * q^37 + 10625 * q^38 + 10108 * q^39 - 1575 * q^40 - 5302 * q^41 + 16222 * q^42 - 4916 * q^43 - 31514 * q^44 + 13275 * q^45 - 16768 * q^46 - 20364 * q^47 + 45557 * q^48 - 26483 * q^49 - 4375 * q^50 - 10404 * q^51 - 11525 * q^52 + 56492 * q^53 + 64635 * q^54 - 19800 * q^55 + 73228 * q^56 + 67554 * q^57 + 54213 * q^58 - 96200 * q^59 + 19625 * q^60 - 7284 * q^61 + 122335 * q^62 + 64108 * q^63 + 28483 * q^64 + 2200 * q^65 + 115232 * q^66 - 60308 * q^67 + 12427 * q^68 - 89076 * q^69 + 21500 * q^70 - 25304 * q^71 + 118722 * q^72 + 163328 * q^73 - 82768 * q^74 - 22500 * q^75 + 92027 * q^76 + 19912 * q^77 + 134895 * q^78 - 50940 * q^79 - 59125 * q^80 + 92445 * q^81 + 191390 * q^82 - 179060 * q^83 - 128054 * q^84 + 36125 * q^85 + 87086 * q^86 + 55764 * q^87 + 167634 * q^88 - 387356 * q^89 - 51300 * q^90 - 115200 * q^91 - 293228 * q^92 - 202558 * q^93 + 100515 * q^94 - 129000 * q^95 + 428113 * q^96 - 106076 * q^97 + 49329 * q^98 - 535112 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.29890	−0.759945	−0.379973	−	0.924998i	$-0.624067\pi$
$2$	−4.29890	−0.759945	−0.379973	+	0.924998i	$0.624067\pi$
$3$	−29.9706	−1.92261	−0.961306	−	0.275482i	$-0.911163\pi$
$3$	−29.9706	−1.92261	−0.961306	+	0.275482i	$0.911163\pi$
$4$	−13.5195	−0.422483
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	128.840	1.46108
$7$	45.8012	0.353290	0.176645	−	0.984275i	$-0.443476\pi$
$7$	45.8012	0.353290	0.176645	+	0.984275i	$0.443476\pi$
$8$	195.684	1.08101
$9$	655.235	2.69644
$10$	−107.472	−0.339858
$11$	−504.686	−1.25759	−0.628796	−	0.777570i	$-0.716452\pi$
$11$	−504.686	−1.25759	−0.628796	+	0.777570i	$0.716452\pi$
$12$	405.186	0.812271
$13$	513.142	0.842130	0.421065	−	0.907031i	$-0.361656\pi$
$13$	513.142	0.842130	0.421065	+	0.907031i	$0.361656\pi$
$14$	−196.895	−0.268481
$15$	−749.264	−0.859818
$16$	−408.601	−0.399025
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	−2816.79	−2.04915
$19$	−2084.02	−1.32440	−0.662198	−	0.749329i	$-0.730376\pi$
$19$	−2084.02	−1.32440	−0.662198	+	0.749329i	$0.730376\pi$
$20$	−337.987	−0.188940
$21$	−1372.69	−0.679240
$22$	2169.60	0.955701
$23$	4718.18	1.85975	0.929875	−	0.367876i	$-0.119915\pi$
$23$	4718.18	1.85975	0.929875	+	0.367876i	$0.119915\pi$
$24$	−5864.75	−2.07836
$25$	625.000	0.200000
$26$	−2205.94	−0.639972
$27$	−12354.9	−3.26159
$28$	−619.207	−0.149259
$29$	−3092.07	−0.682739	−0.341369	−	0.939929i	$-0.610891\pi$
$29$	−3092.07	−0.682739	−0.341369	+	0.939929i	$0.610891\pi$
$30$	3221.01	0.653415
$31$	4157.17	0.776950	0.388475	−	0.921459i	$-0.373002\pi$
$31$	4157.17	0.776950	0.388475	+	0.921459i	$0.373002\pi$
$32$	−4505.34	−0.777772
$33$	15125.7	2.41786
$34$	−1242.38	−0.184314
$35$	1145.03	0.157996
$36$	−8858.42	−1.13920
$37$	4514.40	0.542120	0.271060	−	0.962562i	$-0.412626\pi$
$37$	4514.40	0.542120	0.271060	+	0.962562i	$0.412626\pi$
$38$	8958.99	1.00647
$39$	−15379.1	−1.61909
$40$	4892.09	0.483442
$41$	7375.94	0.685264	0.342632	−	0.939470i	$-0.388682\pi$
$41$	7375.94	0.685264	0.342632	+	0.939470i	$0.388682\pi$
$42$	5901.04	0.516185
$43$	−17592.2	−1.45094	−0.725469	−	0.688255i	$-0.758377\pi$
$43$	−17592.2	−1.45094	−0.725469	+	0.688255i	$0.758377\pi$
$44$	6823.09	0.531311
$45$	16380.9	1.20588
$46$	−20283.0	−1.41331
$47$	−14737.7	−0.973164	−0.486582	−	0.873635i	$-0.661757\pi$
$47$	−14737.7	−0.973164	−0.486582	+	0.873635i	$0.661757\pi$
$48$	12246.0	0.767170
$49$	−14709.3	−0.875186
$50$	−2686.81	−0.151989
$51$	−8661.49	−0.466302
$52$	−6937.40	−0.355786
$53$	−1008.07	−0.0492946	−0.0246473	−	0.999696i	$-0.507846\pi$
$53$	−1008.07	−0.0492946	−0.0246473	+	0.999696i	$0.507846\pi$
$54$	53112.5	2.47863
$55$	−12617.2	−0.562412
$56$	8962.53	0.381910
$57$	62459.2	2.54630
$58$	13292.5	0.518844
$59$	496.132	0.0185553	0.00927764	−	0.999957i	$-0.497047\pi$
$59$	496.132	0.0185553	0.00927764	+	0.999957i	$0.497047\pi$
$60$	10129.6	0.363259
$61$	−13046.2	−0.448910	−0.224455	−	0.974484i	$-0.572060\pi$
$61$	−13046.2	−0.448910	−0.224455	+	0.974484i	$0.572060\pi$
$62$	−17871.2	−0.590439
$63$	30010.5	0.952625
$64$	32443.2	0.990089
$65$	12828.5	0.376612
$66$	−65024.0	−1.83744
$67$	21128.0	0.575004	0.287502	−	0.957780i	$-0.407175\pi$
$67$	21128.0	0.575004	0.287502	+	0.957780i	$0.407175\pi$
$68$	−3907.12	−0.102467
$69$	−141406.	−3.57558
$70$	−4922.36	−0.120068
$71$	−41206.2	−0.970100	−0.485050	−	0.874487i	$-0.661199\pi$
$71$	−41206.2	−0.970100	−0.485050	+	0.874487i	$0.661199\pi$
$72$	128219.	2.91488
$73$	53173.8	1.16786	0.583929	−	0.811805i	$-0.301515\pi$
$73$	53173.8	1.16786	0.583929	+	0.811805i	$0.301515\pi$
$74$	−19406.9	−0.411981
$75$	−18731.6	−0.384523
$76$	28174.8	0.559535
$77$	−23115.2	−0.444295
$78$	66113.4	1.23042
$79$	22083.6	0.398110	0.199055	−	0.979988i	$-0.436213\pi$
$79$	22083.6	0.398110	0.199055	+	0.979988i	$0.436213\pi$
$80$	−10215.0	−0.178449
$81$	211061.	3.57434
$82$	−31708.4	−0.520763
$83$	−63967.7	−1.01922	−0.509608	−	0.860407i	$-0.670209\pi$
$83$	−63967.7	−1.01922	−0.509608	+	0.860407i	$0.670209\pi$
$84$	18558.0	0.286967
$85$	7225.00	0.108465
$86$	75627.1	1.10263
$87$	92671.1	1.31264
$88$	−98758.8	−1.35947
$89$	−145034.	−1.94087	−0.970433	−	0.241369i	$-0.922404\pi$
$89$	−145034.	−1.94087	−0.970433	+	0.241369i	$0.922404\pi$
$90$	−70419.7	−0.916406
$91$	23502.5	0.297516
$92$	−63787.2	−0.785713
$93$	−124593.	−1.49377
$94$	63356.1	0.739552
$95$	−52100.5	−0.592288
$96$	135028.	1.49535
$97$	−131574.	−1.41984	−0.709921	−	0.704282i	$-0.751269\pi$
$97$	−131574.	−1.41984	−0.709921	+	0.704282i	$0.751269\pi$
$98$	63233.6	0.665094
$99$	−330688.	−3.39102

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.6.a.a.1.3	✓	5
3.2	odd	2		765.6.a.g.1.3		5
5.4	even	2		425.6.a.d.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.3	✓	5	1.1	even	1	trivial
425.6.a.d.1.3		5	5.4	even	2
765.6.a.g.1.3		5	3.2	odd	2

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

\(f(q)\)	\(=\)	\(q-4.29890 q^{2} -29.9706 q^{3} -13.5195 q^{4} +25.0000 q^{5} +128.840 q^{6} +45.8012 q^{7} +195.684 q^{8} +655.235 q^{9} -107.472 q^{10} -504.686 q^{11} +405.186 q^{12} +513.142 q^{13} -196.895 q^{14} -749.264 q^{15} -408.601 q^{16} +289.000 q^{17} -2816.79 q^{18} -2084.02 q^{19} -337.987 q^{20} -1372.69 q^{21} +2169.60 q^{22} +4718.18 q^{23} -5864.75 q^{24} +625.000 q^{25} -2205.94 q^{26} -12354.9 q^{27} -619.207 q^{28} -3092.07 q^{29} +3221.01 q^{30} +4157.17 q^{31} -4505.34 q^{32} +15125.7 q^{33} -1242.38 q^{34} +1145.03 q^{35} -8858.42 q^{36} +4514.40 q^{37} +8958.99 q^{38} -15379.1 q^{39} +4892.09 q^{40} +7375.94 q^{41} +5901.04 q^{42} -17592.2 q^{43} +6823.09 q^{44} +16380.9 q^{45} -20283.0 q^{46} -14737.7 q^{47} +12246.0 q^{48} -14709.3 q^{49} -2686.81 q^{50} -8661.49 q^{51} -6937.40 q^{52} -1008.07 q^{53} +53112.5 q^{54} -12617.2 q^{55} +8962.53 q^{56} +62459.2 q^{57} +13292.5 q^{58} +496.132 q^{59} +10129.6 q^{60} -13046.2 q^{61} -17871.2 q^{62} +30010.5 q^{63} +32443.2 q^{64} +12828.5 q^{65} -65024.0 q^{66} +21128.0 q^{67} -3907.12 q^{68} -141406. q^{69} -4922.36 q^{70} -41206.2 q^{71} +128219. q^{72} +53173.8 q^{73} -19406.9 q^{74} -18731.6 q^{75} +28174.8 q^{76} -23115.2 q^{77} +66113.4 q^{78} +22083.6 q^{79} -10215.0 q^{80} +211061. q^{81} -31708.4 q^{82} -63967.7 q^{83} +18558.0 q^{84} +7225.00 q^{85} +75627.1 q^{86} +92671.1 q^{87} -98758.8 q^{88} -145034. q^{89} -70419.7 q^{90} +23502.5 q^{91} -63787.2 q^{92} -124593. q^{93} +63356.1 q^{94} -52100.5 q^{95} +135028. q^{96} -131574. q^{97} +63233.6 q^{98} -330688. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 7 q^{2} - 36 q^{3} + 43 q^{4} + 125 q^{5} - 105 q^{6} - 204 q^{7} - 63 q^{8} + 531 q^{9} - 175 q^{10} - 792 q^{11} + 785 q^{12} + 88 q^{13} + 860 q^{14} - 900 q^{15} - 2365 q^{16} + 1445 q^{17} - 2052 q^{18}+ \cdots - 535112 q^{99}+O(q^{100}) \) 5 * q - 7 * q^2 - 36 * q^3 + 43 * q^4 + 125 * q^5 - 105 * q^6 - 204 * q^7 - 63 * q^8 + 531 * q^9 - 175 * q^10 - 792 * q^11 + 785 * q^12 + 88 * q^13 + 860 * q^14 - 900 * q^15 - 2365 * q^16 + 1445 * q^17 - 2052 * q^18 - 5160 * q^19 + 1075 * q^20 - 6428 * q^21 - 3058 * q^22 - 6140 * q^23 - 11751 * q^24 + 3125 * q^25 - 9643 * q^26 - 13380 * q^27 - 4568 * q^28 - 7328 * q^29 - 2625 * q^30 - 8968 * q^31 - 4703 * q^32 + 13072 * q^33 - 2023 * q^34 - 5100 * q^35 + 2048 * q^36 + 9834 * q^37 + 10625 * q^38 + 10108 * q^39 - 1575 * q^40 - 5302 * q^41 + 16222 * q^42 - 4916 * q^43 - 31514 * q^44 + 13275 * q^45 - 16768 * q^46 - 20364 * q^47 + 45557 * q^48 - 26483 * q^49 - 4375 * q^50 - 10404 * q^51 - 11525 * q^52 + 56492 * q^53 + 64635 * q^54 - 19800 * q^55 + 73228 * q^56 + 67554 * q^57 + 54213 * q^58 - 96200 * q^59 + 19625 * q^60 - 7284 * q^61 + 122335 * q^62 + 64108 * q^63 + 28483 * q^64 + 2200 * q^65 + 115232 * q^66 - 60308 * q^67 + 12427 * q^68 - 89076 * q^69 + 21500 * q^70 - 25304 * q^71 + 118722 * q^72 + 163328 * q^73 - 82768 * q^74 - 22500 * q^75 + 92027 * q^76 + 19912 * q^77 + 134895 * q^78 - 50940 * q^79 - 59125 * q^80 + 92445 * q^81 + 191390 * q^82 - 179060 * q^83 - 128054 * q^84 + 36125 * q^85 + 87086 * q^86 + 55764 * q^87 + 167634 * q^88 - 387356 * q^89 - 51300 * q^90 - 115200 * q^91 - 293228 * q^92 - 202558 * q^93 + 100515 * q^94 - 129000 * q^95 + 428113 * q^96 - 106076 * q^97 + 49329 * q^98 - 535112 * q^99

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