LMFDB - Embedded newform 790.6.a.e.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [790,6,Mod(1,790)] 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("790.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 790 = 2 \cdot 5 \cdot 79 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	790.a (trivial)

Newform invariants

comment:select newform

sage:traces = [17,-68,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$126.703217652$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 2808 x^{15} + 13814 x^{14} + 3138420 x^{13} - 13599532 x^{12} + \cdots + 28\!\cdots\!40 $ x^17 - 5x^16 - 2808x^15 + 13814x^14 + 3138420x^13 - 13599532x^12 - 1794235241x^11 + 5842269565x^10 + 561598097426x^9 - 1037253761968x^8 - 93966714227253x^7 + 31556306807371x^6 + 7318090515571747x^5 + 6375769962622993x^4 - 170883861722576126x^3 - 55033009686120922x^2 + 422953390225755138x + 281395624894688340
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.27638$ of defining polynomial
Character	$\chi$	$=$	790.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +0.276377 q^{3} +16.0000 q^{4} +25.0000 q^{5} -1.10551 q^{6} -19.4017 q^{7} -64.0000 q^{8} -242.924 q^{9} -100.000 q^{10} +671.397 q^{11} +4.42203 q^{12} +849.686 q^{13} +77.6069 q^{14} +6.90943 q^{15} +256.000 q^{16} +2029.45 q^{17} +971.694 q^{18} +1852.80 q^{19} +400.000 q^{20} -5.36219 q^{21} -2685.59 q^{22} -2526.91 q^{23} -17.6881 q^{24} +625.000 q^{25} -3398.74 q^{26} -134.298 q^{27} -310.428 q^{28} +7976.79 q^{29} -27.6377 q^{30} -1797.10 q^{31} -1024.00 q^{32} +185.559 q^{33} -8117.81 q^{34} -485.043 q^{35} -3886.78 q^{36} +4200.41 q^{37} -7411.21 q^{38} +234.834 q^{39} -1600.00 q^{40} -6740.28 q^{41} +21.4488 q^{42} -5585.05 q^{43} +10742.3 q^{44} -6073.09 q^{45} +10107.6 q^{46} -20873.8 q^{47} +70.7525 q^{48} -16430.6 q^{49} -2500.00 q^{50} +560.894 q^{51} +13595.0 q^{52} +17654.7 q^{53} +537.193 q^{54} +16784.9 q^{55} +1241.71 q^{56} +512.072 q^{57} -31907.2 q^{58} +31989.6 q^{59} +110.551 q^{60} +17252.8 q^{61} +7188.39 q^{62} +4713.14 q^{63} +4096.00 q^{64} +21242.2 q^{65} -742.235 q^{66} +31852.2 q^{67} +32471.2 q^{68} -698.379 q^{69} +1940.17 q^{70} -37643.0 q^{71} +15547.1 q^{72} +89294.4 q^{73} -16801.6 q^{74} +172.736 q^{75} +29644.8 q^{76} -13026.3 q^{77} -939.335 q^{78} -6241.00 q^{79} +6400.00 q^{80} +58993.3 q^{81} +26961.1 q^{82} -111800. q^{83} -85.7951 q^{84} +50736.3 q^{85} +22340.2 q^{86} +2204.60 q^{87} -42969.4 q^{88} -140510. q^{89} +24292.4 q^{90} -16485.4 q^{91} -40430.5 q^{92} -496.677 q^{93} +83495.1 q^{94} +46320.1 q^{95} -283.010 q^{96} -134943. q^{97} +65722.3 q^{98} -163098. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 68 q^{2} - 22 q^{3} + 272 q^{4} + 425 q^{5} + 88 q^{6} - 4 q^{7} - 1088 q^{8} + 1537 q^{9} - 1700 q^{10} + 426 q^{11} - 352 q^{12} + 750 q^{13} + 16 q^{14} - 550 q^{15} + 4352 q^{16} + 2074 q^{17}+ \cdots + 126104 q^{99}+O(q^{100}) $ 17 * q - 68 * q^2 - 22 * q^3 + 272 * q^4 + 425 * q^5 + 88 * q^6 - 4 * q^7 - 1088 * q^8 + 1537 * q^9 - 1700 * q^10 + 426 * q^11 - 352 * q^12 + 750 * q^13 + 16 * q^14 - 550 * q^15 + 4352 * q^16 + 2074 * q^17 - 6148 * q^18 + 1546 * q^19 + 6800 * q^20 + 4478 * q^21 - 1704 * q^22 - 6366 * q^23 + 1408 * q^24 + 10625 * q^25 - 3000 * q^26 - 7066 * q^27 - 64 * q^28 + 13612 * q^29 + 2200 * q^30 + 4578 * q^31 - 17408 * q^32 + 1274 * q^33 - 8296 * q^34 - 100 * q^35 + 24592 * q^36 + 6908 * q^37 - 6184 * q^38 + 49306 * q^39 - 27200 * q^40 + 39394 * q^41 - 17912 * q^42 + 12302 * q^43 + 6816 * q^44 + 38425 * q^45 + 25464 * q^46 - 3814 * q^47 - 5632 * q^48 + 76123 * q^49 - 42500 * q^50 + 26042 * q^51 + 12000 * q^52 + 103642 * q^53 + 28264 * q^54 + 10650 * q^55 + 256 * q^56 + 35532 * q^57 - 54448 * q^58 + 88660 * q^59 - 8800 * q^60 + 75624 * q^61 - 18312 * q^62 + 80298 * q^63 + 69632 * q^64 + 18750 * q^65 - 5096 * q^66 - 82964 * q^67 + 33184 * q^68 + 77392 * q^69 + 400 * q^70 + 96816 * q^71 - 98368 * q^72 + 36940 * q^73 - 27632 * q^74 - 13750 * q^75 + 24736 * q^76 + 53744 * q^77 - 197224 * q^78 - 106097 * q^79 + 108800 * q^80 + 130149 * q^81 - 157576 * q^82 + 11764 * q^83 + 71648 * q^84 + 51850 * q^85 - 49208 * q^86 - 232276 * q^87 - 27264 * q^88 + 195846 * q^89 - 153700 * q^90 - 217334 * q^91 - 101856 * q^92 + 53668 * q^93 + 15256 * q^94 + 38650 * q^95 + 22528 * q^96 - 261292 * q^97 - 304492 * q^98 + 126104 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0.276377	0.0177296	0.00886480	−	0.999961i	$-0.497178\pi$
$3$	0.276377	0.0177296	0.00886480	+	0.999961i	$0.497178\pi$
$4$	16.0000	0.500000
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−1.10551	−0.0125367
$7$	−19.4017	−0.149656	−0.0748282	−	0.997196i	$-0.523841\pi$
$7$	−19.4017	−0.149656	−0.0748282	+	0.997196i	$0.523841\pi$
$8$	−64.0000	−0.353553
$9$	−242.924	−0.999686
$10$	−100.000	−0.316228
$11$	671.397	1.67301	0.836503	−	0.547963i	$-0.184596\pi$
$11$	671.397	1.67301	0.836503	+	0.547963i	$0.184596\pi$
$12$	4.42203	0.00886480
$13$	849.686	1.39444	0.697220	−	0.716857i	$-0.254420\pi$
$13$	849.686	1.39444	0.697220	+	0.716857i	$0.254420\pi$
$14$	77.6069	0.105823
$15$	6.90943	0.00792892
$16$	256.000	0.250000
$17$	2029.45	1.70316	0.851582	−	0.524221i	$-0.175643\pi$
$17$	2029.45	1.70316	0.851582	+	0.524221i	$0.175643\pi$
$18$	971.694	0.706885
$19$	1852.80	1.17746	0.588729	−	0.808331i	$-0.299629\pi$
$19$	1852.80	1.17746	0.588729	+	0.808331i	$0.299629\pi$
$20$	400.000	0.223607
$21$	−5.36219	−0.00265335
$22$	−2685.59	−1.18299
$23$	−2526.91	−0.996024	−0.498012	−	0.867170i	$-0.665936\pi$
$23$	−2526.91	−0.996024	−0.498012	+	0.867170i	$0.665936\pi$
$24$	−17.6881	−0.00626836
$25$	625.000	0.200000
$26$	−3398.74	−0.986019
$27$	−134.298	−0.0354536
$28$	−310.428	−0.0748282
$29$	7976.79	1.76130	0.880650	−	0.473767i	$-0.157106\pi$
$29$	7976.79	1.76130	0.880650	+	0.473767i	$0.157106\pi$
$30$	−27.6377	−0.00560659
$31$	−1797.10	−0.335867	−0.167934	−	0.985798i	$-0.553709\pi$
$31$	−1797.10	−0.335867	−0.167934	+	0.985798i	$0.553709\pi$
$32$	−1024.00	−0.176777
$33$	185.559	0.0296617
$34$	−8117.81	−1.20432
$35$	−485.043	−0.0669284
$36$	−3886.78	−0.499843
$37$	4200.41	0.504414	0.252207	−	0.967673i	$-0.418844\pi$
$37$	4200.41	0.504414	0.252207	+	0.967673i	$0.418844\pi$
$38$	−7411.21	−0.832588
$39$	234.834	0.0247229
$40$	−1600.00	−0.158114
$41$	−6740.28	−0.626208	−0.313104	−	0.949719i	$-0.601369\pi$
$41$	−6740.28	−0.626208	−0.313104	+	0.949719i	$0.601369\pi$
$42$	21.4488	0.00187620
$43$	−5585.05	−0.460634	−0.230317	−	0.973116i	$-0.573976\pi$
$43$	−5585.05	−0.460634	−0.230317	+	0.973116i	$0.573976\pi$
$44$	10742.3	0.836503
$45$	−6073.09	−0.447073
$46$	10107.6	0.704295
$47$	−20873.8	−1.37834	−0.689170	−	0.724600i	$-0.742025\pi$
$47$	−20873.8	−1.37834	−0.689170	+	0.724600i	$0.742025\pi$
$48$	70.7525	0.00443240
$49$	−16430.6	−0.977603
$50$	−2500.00	−0.141421
$51$	560.894	0.0301964
$52$	13595.0	0.697220
$53$	17654.7	0.863317	0.431658	−	0.902037i	$-0.357929\pi$
$53$	17654.7	0.863317	0.431658	+	0.902037i	$0.357929\pi$
$54$	537.193	0.0250695
$55$	16784.9	0.748191
$56$	1241.71	0.0529115
$57$	512.072	0.0208758
$58$	−31907.2	−1.24543
$59$	31989.6	1.19641	0.598203	−	0.801344i	$-0.295882\pi$
$59$	31989.6	1.19641	0.598203	+	0.801344i	$0.295882\pi$
$60$	110.551	0.00396446
$61$	17252.8	0.593657	0.296829	−	0.954931i	$-0.404071\pi$
$61$	17252.8	0.593657	0.296829	+	0.954931i	$0.404071\pi$
$62$	7188.39	0.237494
$63$	4713.14	0.149609
$64$	4096.00	0.125000
$65$	21242.2	0.623613
$66$	−742.235	−0.0209740
$67$	31852.2	0.866867	0.433434	−	0.901186i	$-0.357302\pi$
$67$	31852.2	0.866867	0.433434	+	0.901186i	$0.357302\pi$
$68$	32471.2	0.851582
$69$	−698.379	−0.0176591
$70$	1940.17	0.0473255
$71$	−37643.0	−0.886214	−0.443107	−	0.896469i	$-0.646124\pi$
$71$	−37643.0	−0.886214	−0.443107	+	0.896469i	$0.646124\pi$
$72$	15547.1	0.353442
$73$	89294.4	1.96118	0.980590	−	0.196072i	$-0.0628186\pi$
$73$	89294.4	1.96118	0.980590	+	0.196072i	$0.0628186\pi$
$74$	−16801.6	−0.356675
$75$	172.736	0.00354592
$76$	29644.8	0.588729
$77$	−13026.3	−0.250376
$78$	−939.335	−0.0174817
$79$	−6241.00	−0.112509
$79$	−6241.00	−0.112509
$80$	6400.00	0.111803
$81$	58993.3	0.999057
$82$	26961.1	0.442796
$83$	−111800.	−1.78135	−0.890673	−	0.454645i	$-0.849766\pi$
$83$	−111800.	−1.78135	−0.890673	+	0.454645i	$0.849766\pi$
$84$	−85.7951	−0.00132667
$85$	50736.3	0.761678
$86$	22340.2	0.325717
$87$	2204.60	0.0312271
$88$	−42969.4	−0.591497
$89$	−140510.	−1.88033	−0.940164	−	0.340724i	$-0.889328\pi$
$89$	−140510.	−1.88033	−0.940164	+	0.340724i	$0.889328\pi$
$90$	24292.4	0.316128
$91$	−16485.4	−0.208687
$92$	−40430.5	−0.498012
$93$	−496.677	−0.00595479
$94$	83495.1	0.974633
$95$	46320.1	0.526575
$96$	−283.010	−0.00313418
$97$	−134943.	−1.45620	−0.728102	−	0.685469i	$-0.759597\pi$
$97$	−134943.	−1.45620	−0.728102	+	0.685469i	$0.759597\pi$
$98$	65722.3	0.691270
$99$	−163098.	−1.67248

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	790.6.a.e.1.10	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
790.6.a.e.1.10	✓	17	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 790 = 2 \cdot 5 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	790.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +0.276377 q^{3} +16.0000 q^{4} +25.0000 q^{5} -1.10551 q^{6} -19.4017 q^{7} -64.0000 q^{8} -242.924 q^{9} -100.000 q^{10} +671.397 q^{11} +4.42203 q^{12} +849.686 q^{13} +77.6069 q^{14} +6.90943 q^{15} +256.000 q^{16} +2029.45 q^{17} +971.694 q^{18} +1852.80 q^{19} +400.000 q^{20} -5.36219 q^{21} -2685.59 q^{22} -2526.91 q^{23} -17.6881 q^{24} +625.000 q^{25} -3398.74 q^{26} -134.298 q^{27} -310.428 q^{28} +7976.79 q^{29} -27.6377 q^{30} -1797.10 q^{31} -1024.00 q^{32} +185.559 q^{33} -8117.81 q^{34} -485.043 q^{35} -3886.78 q^{36} +4200.41 q^{37} -7411.21 q^{38} +234.834 q^{39} -1600.00 q^{40} -6740.28 q^{41} +21.4488 q^{42} -5585.05 q^{43} +10742.3 q^{44} -6073.09 q^{45} +10107.6 q^{46} -20873.8 q^{47} +70.7525 q^{48} -16430.6 q^{49} -2500.00 q^{50} +560.894 q^{51} +13595.0 q^{52} +17654.7 q^{53} +537.193 q^{54} +16784.9 q^{55} +1241.71 q^{56} +512.072 q^{57} -31907.2 q^{58} +31989.6 q^{59} +110.551 q^{60} +17252.8 q^{61} +7188.39 q^{62} +4713.14 q^{63} +4096.00 q^{64} +21242.2 q^{65} -742.235 q^{66} +31852.2 q^{67} +32471.2 q^{68} -698.379 q^{69} +1940.17 q^{70} -37643.0 q^{71} +15547.1 q^{72} +89294.4 q^{73} -16801.6 q^{74} +172.736 q^{75} +29644.8 q^{76} -13026.3 q^{77} -939.335 q^{78} -6241.00 q^{79} +6400.00 q^{80} +58993.3 q^{81} +26961.1 q^{82} -111800. q^{83} -85.7951 q^{84} +50736.3 q^{85} +22340.2 q^{86} +2204.60 q^{87} -42969.4 q^{88} -140510. q^{89} +24292.4 q^{90} -16485.4 q^{91} -40430.5 q^{92} -496.677 q^{93} +83495.1 q^{94} +46320.1 q^{95} -283.010 q^{96} -134943. q^{97} +65722.3 q^{98} -163098. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 68 q^{2} - 22 q^{3} + 272 q^{4} + 425 q^{5} + 88 q^{6} - 4 q^{7} - 1088 q^{8} + 1537 q^{9} - 1700 q^{10} + 426 q^{11} - 352 q^{12} + 750 q^{13} + 16 q^{14} - 550 q^{15} + 4352 q^{16} + 2074 q^{17}+ \cdots + 126104 q^{99}+O(q^{100}) \) 17 * q - 68 * q^2 - 22 * q^3 + 272 * q^4 + 425 * q^5 + 88 * q^6 - 4 * q^7 - 1088 * q^8 + 1537 * q^9 - 1700 * q^10 + 426 * q^11 - 352 * q^12 + 750 * q^13 + 16 * q^14 - 550 * q^15 + 4352 * q^16 + 2074 * q^17 - 6148 * q^18 + 1546 * q^19 + 6800 * q^20 + 4478 * q^21 - 1704 * q^22 - 6366 * q^23 + 1408 * q^24 + 10625 * q^25 - 3000 * q^26 - 7066 * q^27 - 64 * q^28 + 13612 * q^29 + 2200 * q^30 + 4578 * q^31 - 17408 * q^32 + 1274 * q^33 - 8296 * q^34 - 100 * q^35 + 24592 * q^36 + 6908 * q^37 - 6184 * q^38 + 49306 * q^39 - 27200 * q^40 + 39394 * q^41 - 17912 * q^42 + 12302 * q^43 + 6816 * q^44 + 38425 * q^45 + 25464 * q^46 - 3814 * q^47 - 5632 * q^48 + 76123 * q^49 - 42500 * q^50 + 26042 * q^51 + 12000 * q^52 + 103642 * q^53 + 28264 * q^54 + 10650 * q^55 + 256 * q^56 + 35532 * q^57 - 54448 * q^58 + 88660 * q^59 - 8800 * q^60 + 75624 * q^61 - 18312 * q^62 + 80298 * q^63 + 69632 * q^64 + 18750 * q^65 - 5096 * q^66 - 82964 * q^67 + 33184 * q^68 + 77392 * q^69 + 400 * q^70 + 96816 * q^71 - 98368 * q^72 + 36940 * q^73 - 27632 * q^74 - 13750 * q^75 + 24736 * q^76 + 53744 * q^77 - 197224 * q^78 - 106097 * q^79 + 108800 * q^80 + 130149 * q^81 - 157576 * q^82 + 11764 * q^83 + 71648 * q^84 + 51850 * q^85 - 49208 * q^86 - 232276 * q^87 - 27264 * q^88 + 195846 * q^89 - 153700 * q^90 - 217334 * q^91 - 101856 * q^92 + 53668 * q^93 + 15256 * q^94 + 38650 * q^95 + 22528 * q^96 - 261292 * q^97 - 304492 * q^98 + 126104 * q^99

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