LMFDB - Embedded newform 6845.2.a.t.1.35

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6845,2,Mod(1,6845)] 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("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6845 = 5 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6845.a (trivial)

Newform invariants

comment:select newform

sage:traces = [36,-6,-6,30,36,-12,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.6576001836$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.35
Character	$\chi$	$=$	6845.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.34842 q^{2} -0.875531 q^{3} +3.51507 q^{4} +1.00000 q^{5} -2.05611 q^{6} +4.44366 q^{7} +3.55801 q^{8} -2.23345 q^{9} +2.34842 q^{10} -5.16764 q^{11} -3.07755 q^{12} -4.74066 q^{13} +10.4356 q^{14} -0.875531 q^{15} +1.32555 q^{16} +1.91127 q^{17} -5.24506 q^{18} -3.04389 q^{19} +3.51507 q^{20} -3.89056 q^{21} -12.1358 q^{22} -9.26416 q^{23} -3.11514 q^{24} +1.00000 q^{25} -11.1331 q^{26} +4.58204 q^{27} +15.6197 q^{28} +2.92861 q^{29} -2.05611 q^{30} -5.61699 q^{31} -4.00306 q^{32} +4.52443 q^{33} +4.48847 q^{34} +4.44366 q^{35} -7.85071 q^{36} -7.14833 q^{38} +4.15060 q^{39} +3.55801 q^{40} -0.764680 q^{41} -9.13665 q^{42} -0.369905 q^{43} -18.1646 q^{44} -2.23345 q^{45} -21.7561 q^{46} -11.3275 q^{47} -1.16056 q^{48} +12.7461 q^{49} +2.34842 q^{50} -1.67338 q^{51} -16.6637 q^{52} +4.63589 q^{53} +10.7605 q^{54} -5.16764 q^{55} +15.8106 q^{56} +2.66502 q^{57} +6.87760 q^{58} -4.46901 q^{59} -3.07755 q^{60} +2.75744 q^{61} -13.1910 q^{62} -9.92467 q^{63} -12.0520 q^{64} -4.74066 q^{65} +10.6252 q^{66} -4.51338 q^{67} +6.71825 q^{68} +8.11105 q^{69} +10.4356 q^{70} -4.37850 q^{71} -7.94661 q^{72} +2.68113 q^{73} -0.875531 q^{75} -10.6995 q^{76} -22.9632 q^{77} +9.74733 q^{78} -2.01016 q^{79} +1.32555 q^{80} +2.68862 q^{81} -1.79579 q^{82} +1.84972 q^{83} -13.6756 q^{84} +1.91127 q^{85} -0.868692 q^{86} -2.56409 q^{87} -18.3865 q^{88} -6.19246 q^{89} -5.24506 q^{90} -21.0659 q^{91} -32.5641 q^{92} +4.91785 q^{93} -26.6016 q^{94} -3.04389 q^{95} +3.50480 q^{96} -2.84876 q^{97} +29.9331 q^{98} +11.5416 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 6 q^{2} - 6 q^{3} + 30 q^{4} + 36 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8} + 30 q^{9} - 6 q^{10} - 12 q^{12} - 24 q^{13} + 24 q^{14} - 6 q^{15} + 18 q^{16} - 12 q^{17} - 30 q^{18} - 36 q^{19} + 30 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) $ 36 * q - 6 * q^2 - 6 * q^3 + 30 * q^4 + 36 * q^5 - 12 * q^6 - 12 * q^7 - 18 * q^8 + 30 * q^9 - 6 * q^10 - 12 * q^12 - 24 * q^13 + 24 * q^14 - 6 * q^15 + 18 * q^16 - 12 * q^17 - 30 * q^18 - 36 * q^19 + 30 * q^20 - 36 * q^22 - 24 * q^23 - 36 * q^24 + 36 * q^25 - 24 * q^27 - 24 * q^28 - 12 * q^29 - 12 * q^30 - 48 * q^31 - 42 * q^32 - 6 * q^33 - 12 * q^35 + 6 * q^36 - 24 * q^38 + 12 * q^39 - 18 * q^40 - 66 * q^42 - 108 * q^43 - 18 * q^44 + 30 * q^45 + 18 * q^46 - 66 * q^47 - 24 * q^48 + 42 * q^49 - 6 * q^50 - 72 * q^51 + 18 * q^52 - 24 * q^53 + 12 * q^54 + 36 * q^56 - 72 * q^57 - 18 * q^58 - 36 * q^59 - 12 * q^60 - 48 * q^61 - 42 * q^62 - 48 * q^63 - 24 * q^65 + 84 * q^66 + 36 * q^67 - 36 * q^68 - 60 * q^69 + 24 * q^70 - 12 * q^71 - 90 * q^72 - 12 * q^73 - 6 * q^75 + 48 * q^76 - 66 * q^77 + 6 * q^78 - 72 * q^79 + 18 * q^80 + 24 * q^81 - 96 * q^82 - 72 * q^83 - 12 * q^84 - 12 * q^85 + 36 * q^86 - 48 * q^87 - 84 * q^88 - 30 * q^90 - 96 * q^91 - 60 * q^92 + 60 * q^93 - 12 * q^94 - 36 * q^95 + 24 * q^96 - 72 * q^98 + 24 * 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 $

$2$	2.34842	1.66058	0.830291	−	0.557330i	$-0.188174\pi$
$2$	2.34842	1.66058	0.830291	+	0.557330i	$0.188174\pi$
$3$	−0.875531	−0.505488	−0.252744	−	0.967533i	$-0.581333\pi$
$3$	−0.875531	−0.505488	−0.252744	+	0.967533i	$0.581333\pi$
$4$	3.51507	1.75753
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.05611	−0.839404
$7$	4.44366	1.67954	0.839772	−	0.542939i	$-0.182689\pi$
$7$	4.44366	1.67954	0.839772	+	0.542939i	$0.182689\pi$
$8$	3.55801	1.25795
$9$	−2.23345	−0.744482
$10$	2.34842	0.742635
$11$	−5.16764	−1.55810	−0.779051	−	0.626961i	$-0.784299\pi$
$11$	−5.16764	−1.55810	−0.779051	+	0.626961i	$0.784299\pi$
$12$	−3.07755	−0.888411
$13$	−4.74066	−1.31482	−0.657412	−	0.753532i	$-0.728349\pi$
$13$	−4.74066	−1.31482	−0.657412	+	0.753532i	$0.728349\pi$
$14$	10.4356	2.78902
$15$	−0.875531	−0.226061
$16$	1.32555	0.331388
$17$	1.91127	0.463552	0.231776	−	0.972769i	$-0.425546\pi$
$17$	1.91127	0.463552	0.231776	+	0.972769i	$0.425546\pi$
$18$	−5.24506	−1.23627
$19$	−3.04389	−0.698317	−0.349159	−	0.937064i	$-0.613533\pi$
$19$	−3.04389	−0.698317	−0.349159	+	0.937064i	$0.613533\pi$
$20$	3.51507	0.785992
$21$	−3.89056	−0.848989
$22$	−12.1358	−2.58735
$23$	−9.26416	−1.93171	−0.965855	−	0.259083i	$-0.916580\pi$
$23$	−9.26416	−1.93171	−0.965855	+	0.259083i	$0.916580\pi$
$24$	−3.11514	−0.635876
$25$	1.00000	0.200000
$26$	−11.1331	−2.18337
$27$	4.58204	0.881814
$28$	15.6197	2.95185
$29$	2.92861	0.543830	0.271915	−	0.962321i	$-0.412343\pi$
$29$	2.92861	0.543830	0.271915	+	0.962321i	$0.412343\pi$
$30$	−2.05611	−0.375393
$31$	−5.61699	−1.00884	−0.504421	−	0.863458i	$-0.668294\pi$
$31$	−5.61699	−1.00884	−0.504421	+	0.863458i	$0.668294\pi$
$32$	−4.00306	−0.707648
$33$	4.52443	0.787601
$34$	4.48847	0.769766
$35$	4.44366	0.751115
$36$	−7.85071	−1.30845
$37$	0	0
$37$	0	0
$38$	−7.14833	−1.15961
$39$	4.15060	0.664627
$40$	3.55801	0.562570
$41$	−0.764680	−0.119423	−0.0597115	−	0.998216i	$-0.519018\pi$
$41$	−0.764680	−0.119423	−0.0597115	+	0.998216i	$0.519018\pi$
$42$	−9.13665	−1.40982
$43$	−0.369905	−0.0564100	−0.0282050	−	0.999602i	$-0.508979\pi$
$43$	−0.369905	−0.0564100	−0.0282050	+	0.999602i	$0.508979\pi$
$44$	−18.1646	−2.73841
$45$	−2.23345	−0.332942
$46$	−21.7561	−3.20776
$47$	−11.3275	−1.65228	−0.826140	−	0.563465i	$-0.809468\pi$
$47$	−11.3275	−1.65228	−0.826140	+	0.563465i	$0.809468\pi$
$48$	−1.16056	−0.167513
$49$	12.7461	1.82087
$50$	2.34842	0.332116
$51$	−1.67338	−0.234320
$52$	−16.6637	−2.31084
$53$	4.63589	0.636788	0.318394	−	0.947958i	$-0.396856\pi$
$53$	4.63589	0.636788	0.318394	+	0.947958i	$0.396856\pi$
$54$	10.7605	1.46433
$55$	−5.16764	−0.696804
$56$	15.8106	2.11277
$57$	2.66502	0.352991
$58$	6.87760	0.903074
$59$	−4.46901	−0.581816	−0.290908	−	0.956751i	$-0.593957\pi$
$59$	−4.46901	−0.581816	−0.290908	+	0.956751i	$0.593957\pi$
$60$	−3.07755	−0.397310
$61$	2.75744	0.353054	0.176527	−	0.984296i	$-0.443514\pi$
$61$	2.75744	0.353054	0.176527	+	0.984296i	$0.443514\pi$
$62$	−13.1910	−1.67526
$63$	−9.92467	−1.25039
$64$	−12.0520	−1.50650
$65$	−4.74066	−0.588007
$66$	10.6252	1.30788
$67$	−4.51338	−0.551397	−0.275699	−	0.961244i	$-0.588909\pi$
$67$	−4.51338	−0.551397	−0.275699	+	0.961244i	$0.588909\pi$
$68$	6.71825	0.814707
$69$	8.11105	0.976456
$70$	10.4356	1.24729
$71$	−4.37850	−0.519632	−0.259816	−	0.965658i	$-0.583662\pi$
$71$	−4.37850	−0.519632	−0.259816	+	0.965658i	$0.583662\pi$
$72$	−7.94661	−0.936517
$73$	2.68113	0.313803	0.156902	−	0.987614i	$-0.449849\pi$
$73$	2.68113	0.313803	0.156902	+	0.987614i	$0.449849\pi$
$74$	0	0
$75$	−0.875531	−0.101098
$76$	−10.6995	−1.22731
$77$	−22.9632	−2.61690
$78$	9.74733	1.10367
$79$	−2.01016	−0.226161	−0.113080	−	0.993586i	$-0.536072\pi$
$79$	−2.01016	−0.226161	−0.113080	+	0.993586i	$0.536072\pi$
$80$	1.32555	0.148201
$81$	2.68862	0.298735
$82$	−1.79579	−0.198312
$83$	1.84972	0.203033	0.101517	−	0.994834i	$-0.467631\pi$
$83$	1.84972	0.203033	0.101517	+	0.994834i	$0.467631\pi$
$84$	−13.6756	−1.49213
$85$	1.91127	0.207307
$86$	−0.868692	−0.0936735
$87$	−2.56409	−0.274899
$88$	−18.3865	−1.96001
$89$	−6.19246	−0.656399	−0.328200	−	0.944608i	$-0.606442\pi$
$89$	−6.19246	−0.656399	−0.328200	+	0.944608i	$0.606442\pi$
$90$	−5.24506	−0.552878
$91$	−21.0659	−2.20830
$92$	−32.5641	−3.39504
$93$	4.91785	0.509957
$94$	−26.6016	−2.74375
$95$	−3.04389	−0.312297
$96$	3.50480	0.357707
$97$	−2.84876	−0.289248	−0.144624	−	0.989487i	$-0.546197\pi$
$97$	−2.84876	−0.289248	−0.144624	+	0.989487i	$0.546197\pi$
$98$	29.9331	3.02370
$99$	11.5416	1.15998

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	6845.2.a.t.1.35		36
37.5	odd	36		185.2.w.a.136.1	✓	72
37.15	odd	36		185.2.w.a.151.1	yes	72
37.36	even	2		6845.2.a.u.1.2		36
185.42	even	36		925.2.ba.c.99.12		72
185.52	even	36		925.2.ba.b.299.1		72
185.79	odd	36		925.2.bb.b.876.12		72
185.89	odd	36		925.2.bb.b.151.12		72
185.153	even	36		925.2.ba.b.99.1		72
185.163	even	36		925.2.ba.c.299.12		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.w.a.136.1	✓	72	37.5	odd	36
185.2.w.a.151.1	yes	72	37.15	odd	36
925.2.ba.b.99.1		72	185.153	even	36
925.2.ba.b.299.1		72	185.52	even	36
925.2.ba.c.99.12		72	185.42	even	36
925.2.ba.c.299.12		72	185.163	even	36
925.2.bb.b.151.12		72	185.89	odd	36
925.2.bb.b.876.12		72	185.79	odd	36
6845.2.a.t.1.35		36	1.1	even	1	trivial
6845.2.a.u.1.2		36	37.36	even	2

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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6845 = 5 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6845.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34842 q^{2} -0.875531 q^{3} +3.51507 q^{4} +1.00000 q^{5} -2.05611 q^{6} +4.44366 q^{7} +3.55801 q^{8} -2.23345 q^{9} +2.34842 q^{10} -5.16764 q^{11} -3.07755 q^{12} -4.74066 q^{13} +10.4356 q^{14} -0.875531 q^{15} +1.32555 q^{16} +1.91127 q^{17} -5.24506 q^{18} -3.04389 q^{19} +3.51507 q^{20} -3.89056 q^{21} -12.1358 q^{22} -9.26416 q^{23} -3.11514 q^{24} +1.00000 q^{25} -11.1331 q^{26} +4.58204 q^{27} +15.6197 q^{28} +2.92861 q^{29} -2.05611 q^{30} -5.61699 q^{31} -4.00306 q^{32} +4.52443 q^{33} +4.48847 q^{34} +4.44366 q^{35} -7.85071 q^{36} -7.14833 q^{38} +4.15060 q^{39} +3.55801 q^{40} -0.764680 q^{41} -9.13665 q^{42} -0.369905 q^{43} -18.1646 q^{44} -2.23345 q^{45} -21.7561 q^{46} -11.3275 q^{47} -1.16056 q^{48} +12.7461 q^{49} +2.34842 q^{50} -1.67338 q^{51} -16.6637 q^{52} +4.63589 q^{53} +10.7605 q^{54} -5.16764 q^{55} +15.8106 q^{56} +2.66502 q^{57} +6.87760 q^{58} -4.46901 q^{59} -3.07755 q^{60} +2.75744 q^{61} -13.1910 q^{62} -9.92467 q^{63} -12.0520 q^{64} -4.74066 q^{65} +10.6252 q^{66} -4.51338 q^{67} +6.71825 q^{68} +8.11105 q^{69} +10.4356 q^{70} -4.37850 q^{71} -7.94661 q^{72} +2.68113 q^{73} -0.875531 q^{75} -10.6995 q^{76} -22.9632 q^{77} +9.74733 q^{78} -2.01016 q^{79} +1.32555 q^{80} +2.68862 q^{81} -1.79579 q^{82} +1.84972 q^{83} -13.6756 q^{84} +1.91127 q^{85} -0.868692 q^{86} -2.56409 q^{87} -18.3865 q^{88} -6.19246 q^{89} -5.24506 q^{90} -21.0659 q^{91} -32.5641 q^{92} +4.91785 q^{93} -26.6016 q^{94} -3.04389 q^{95} +3.50480 q^{96} -2.84876 q^{97} +29.9331 q^{98} +11.5416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 6 q^{2} - 6 q^{3} + 30 q^{4} + 36 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8} + 30 q^{9} - 6 q^{10} - 12 q^{12} - 24 q^{13} + 24 q^{14} - 6 q^{15} + 18 q^{16} - 12 q^{17} - 30 q^{18} - 36 q^{19} + 30 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) 36 * q - 6 * q^2 - 6 * q^3 + 30 * q^4 + 36 * q^5 - 12 * q^6 - 12 * q^7 - 18 * q^8 + 30 * q^9 - 6 * q^10 - 12 * q^12 - 24 * q^13 + 24 * q^14 - 6 * q^15 + 18 * q^16 - 12 * q^17 - 30 * q^18 - 36 * q^19 + 30 * q^20 - 36 * q^22 - 24 * q^23 - 36 * q^24 + 36 * q^25 - 24 * q^27 - 24 * q^28 - 12 * q^29 - 12 * q^30 - 48 * q^31 - 42 * q^32 - 6 * q^33 - 12 * q^35 + 6 * q^36 - 24 * q^38 + 12 * q^39 - 18 * q^40 - 66 * q^42 - 108 * q^43 - 18 * q^44 + 30 * q^45 + 18 * q^46 - 66 * q^47 - 24 * q^48 + 42 * q^49 - 6 * q^50 - 72 * q^51 + 18 * q^52 - 24 * q^53 + 12 * q^54 + 36 * q^56 - 72 * q^57 - 18 * q^58 - 36 * q^59 - 12 * q^60 - 48 * q^61 - 42 * q^62 - 48 * q^63 - 24 * q^65 + 84 * q^66 + 36 * q^67 - 36 * q^68 - 60 * q^69 + 24 * q^70 - 12 * q^71 - 90 * q^72 - 12 * q^73 - 6 * q^75 + 48 * q^76 - 66 * q^77 + 6 * q^78 - 72 * q^79 + 18 * q^80 + 24 * q^81 - 96 * q^82 - 72 * q^83 - 12 * q^84 - 12 * q^85 + 36 * q^86 - 48 * q^87 - 84 * q^88 - 30 * q^90 - 96 * q^91 - 60 * q^92 + 60 * q^93 - 12 * q^94 - 36 * q^95 + 24 * q^96 - 72 * q^98 + 24 * q^99

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