LMFDB - Embedded newform 5610.2.a.ci.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5610,2,Mod(1,5610)] 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("5610.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,5,-5,5,5,-5,2,5,5,5,5,-5,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.7960755339$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.18569692.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 23x^{3} - 32x^{2} + 26x - 4 $ x^5 - 23x^3 - 32x^2 + 26*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.228960$ of defining polynomial
Character	$\chi$	$=$	5610.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.06482 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +3.41051 q^{13} +3.06482 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +3.41051 q^{19} +1.00000 q^{20} -3.06482 q^{21} +1.00000 q^{22} -7.25619 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.41051 q^{26} -1.00000 q^{27} +3.06482 q^{28} +4.34569 q^{29} -1.00000 q^{30} +9.71411 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.06482 q^{35} +1.00000 q^{36} -3.86843 q^{37} +3.41051 q^{38} -3.41051 q^{39} +1.00000 q^{40} +12.3210 q^{41} -3.06482 q^{42} -11.8160 q^{43} +1.00000 q^{44} +1.00000 q^{45} -7.25619 q^{46} +12.3210 q^{47} -1.00000 q^{48} +2.39310 q^{49} +1.00000 q^{50} +1.00000 q^{51} +3.41051 q^{52} -1.14930 q^{53} -1.00000 q^{54} +1.00000 q^{55} +3.06482 q^{56} -3.41051 q^{57} +4.34569 q^{58} -6.36310 q^{59} -1.00000 q^{60} -12.1894 q^{61} +9.71411 q^{62} +3.06482 q^{63} +1.00000 q^{64} +3.41051 q^{65} -1.00000 q^{66} -5.08222 q^{67} -1.00000 q^{68} +7.25619 q^{69} +3.06482 q^{70} +4.45792 q^{71} +1.00000 q^{72} +3.14930 q^{73} -3.86843 q^{74} -1.00000 q^{75} +3.41051 q^{76} +3.06482 q^{77} -3.41051 q^{78} -10.7789 q^{79} +1.00000 q^{80} +1.00000 q^{81} +12.3210 q^{82} +13.2736 q^{83} -3.06482 q^{84} -1.00000 q^{85} -11.8160 q^{86} -4.34569 q^{87} +1.00000 q^{88} +2.69138 q^{89} +1.00000 q^{90} +10.4526 q^{91} -7.25619 q^{92} -9.71411 q^{93} +12.3210 q^{94} +3.41051 q^{95} -1.00000 q^{96} -16.5351 q^{97} +2.39310 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} + 2 q^{7} + 5 q^{8} + 5 q^{9} + 5 q^{10} + 5 q^{11} - 5 q^{12} - q^{13} + 2 q^{14} - 5 q^{15} + 5 q^{16} - 5 q^{17} + 5 q^{18} - q^{19} + 5 q^{20}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 + 2 * q^7 + 5 * q^8 + 5 * q^9 + 5 * q^10 + 5 * q^11 - 5 * q^12 - q^13 + 2 * q^14 - 5 * q^15 + 5 * q^16 - 5 * q^17 + 5 * q^18 - q^19 + 5 * q^20 - 2 * q^21 + 5 * q^22 - 5 * q^24 + 5 * q^25 - q^26 - 5 * q^27 + 2 * q^28 + 17 * q^29 - 5 * q^30 + 10 * q^31 + 5 * q^32 - 5 * q^33 - 5 * q^34 + 2 * q^35 + 5 * q^36 + q^37 - q^38 + q^39 + 5 * q^40 + 12 * q^41 - 2 * q^42 + 7 * q^43 + 5 * q^44 + 5 * q^45 + 12 * q^47 - 5 * q^48 + 23 * q^49 + 5 * q^50 + 5 * q^51 - q^52 + 6 * q^53 - 5 * q^54 + 5 * q^55 + 2 * q^56 + q^57 + 17 * q^58 + 2 * q^59 - 5 * q^60 + 9 * q^61 + 10 * q^62 + 2 * q^63 + 5 * q^64 - q^65 - 5 * q^66 + 17 * q^67 - 5 * q^68 + 2 * q^70 + 20 * q^71 + 5 * q^72 + 4 * q^73 + q^74 - 5 * q^75 - q^76 + 2 * q^77 + q^78 - 2 * q^79 + 5 * q^80 + 5 * q^81 + 12 * q^82 + q^83 - 2 * q^84 - 5 * q^85 + 7 * q^86 - 17 * q^87 + 5 * q^88 + 4 * q^89 + 5 * q^90 + 23 * q^91 - 10 * q^93 + 12 * q^94 - q^95 - 5 * q^96 - 8 * q^97 + 23 * q^98 + 5 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	3.06482	1.15839	0.579196	−	0.815188i	$-0.303367\pi$
$7$	3.06482	1.15839	0.579196	+	0.815188i	$0.303367\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	3.41051	0.945905	0.472952	−	0.881088i	$-0.343188\pi$
$13$	3.41051	0.945905	0.472952	+	0.881088i	$0.343188\pi$
$14$	3.06482	0.819107
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	3.41051	0.782424	0.391212	−	0.920301i	$-0.372056\pi$
$19$	3.41051	0.782424	0.391212	+	0.920301i	$0.372056\pi$
$20$	1.00000	0.223607
$21$	−3.06482	−0.668798
$22$	1.00000	0.213201
$23$	−7.25619	−1.51302	−0.756511	−	0.653981i	$-0.773097\pi$
$23$	−7.25619	−1.51302	−0.756511	+	0.653981i	$0.773097\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	3.41051	0.668856
$27$	−1.00000	−0.192450
$28$	3.06482	0.579196
$29$	4.34569	0.806975	0.403487	−	0.914985i	$-0.367798\pi$
$29$	4.34569	0.806975	0.403487	+	0.914985i	$0.367798\pi$
$30$	−1.00000	−0.182574
$31$	9.71411	1.74471	0.872353	−	0.488876i	$-0.162593\pi$
$31$	9.71411	1.74471	0.872353	+	0.488876i	$0.162593\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	3.06482	0.518049
$36$	1.00000	0.166667
$37$	−3.86843	−0.635966	−0.317983	−	0.948096i	$-0.603005\pi$
$37$	−3.86843	−0.635966	−0.317983	+	0.948096i	$0.603005\pi$
$38$	3.41051	0.553258
$39$	−3.41051	−0.546118
$40$	1.00000	0.158114
$41$	12.3210	1.92422	0.962109	−	0.272664i	$-0.0879047\pi$
$41$	12.3210	1.92422	0.962109	+	0.272664i	$0.0879047\pi$
$42$	−3.06482	−0.472912
$43$	−11.8160	−1.80192	−0.900962	−	0.433898i	$-0.857138\pi$
$43$	−11.8160	−1.80192	−0.900962	+	0.433898i	$0.857138\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−7.25619	−1.06987
$47$	12.3210	1.79720	0.898602	−	0.438764i	$-0.144584\pi$
$47$	12.3210	1.79720	0.898602	+	0.438764i	$0.144584\pi$
$48$	−1.00000	−0.144338
$49$	2.39310	0.341872
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	3.41051	0.472952
$53$	−1.14930	−0.157869	−0.0789344	−	0.996880i	$-0.525152\pi$
$53$	−1.14930	−0.157869	−0.0789344	+	0.996880i	$0.525152\pi$
$54$	−1.00000	−0.136083
$55$	1.00000	0.134840
$56$	3.06482	0.409553
$57$	−3.41051	−0.451733
$58$	4.34569	0.570617
$59$	−6.36310	−0.828405	−0.414202	−	0.910185i	$-0.635939\pi$
$59$	−6.36310	−0.828405	−0.414202	+	0.910185i	$0.635939\pi$
$60$	−1.00000	−0.129099
$61$	−12.1894	−1.56070	−0.780349	−	0.625344i	$-0.784959\pi$
$61$	−12.1894	−1.56070	−0.780349	+	0.625344i	$0.784959\pi$
$62$	9.71411	1.23369
$63$	3.06482	0.386131
$64$	1.00000	0.125000
$65$	3.41051	0.423021
$66$	−1.00000	−0.123091
$67$	−5.08222	−0.620892	−0.310446	−	0.950591i	$-0.600478\pi$
$67$	−5.08222	−0.620892	−0.310446	+	0.950591i	$0.600478\pi$
$68$	−1.00000	−0.121268
$69$	7.25619	0.873543
$70$	3.06482	0.366316
$71$	4.45792	0.529058	0.264529	−	0.964378i	$-0.414784\pi$
$71$	4.45792	0.529058	0.264529	+	0.964378i	$0.414784\pi$
$72$	1.00000	0.117851
$73$	3.14930	0.368598	0.184299	−	0.982870i	$-0.440999\pi$
$73$	3.14930	0.368598	0.184299	+	0.982870i	$0.440999\pi$
$74$	−3.86843	−0.449696
$75$	−1.00000	−0.115470
$76$	3.41051	0.391212
$77$	3.06482	0.349268
$78$	−3.41051	−0.386164
$79$	−10.7789	−1.21272	−0.606362	−	0.795189i	$-0.707372\pi$
$79$	−10.7789	−1.21272	−0.606362	+	0.795189i	$0.707372\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	12.3210	1.36063
$83$	13.2736	1.45697	0.728483	−	0.685063i	$-0.240226\pi$
$83$	13.2736	1.45697	0.728483	+	0.685063i	$0.240226\pi$
$84$	−3.06482	−0.334399
$85$	−1.00000	−0.108465
$86$	−11.8160	−1.27415
$87$	−4.34569	−0.465907
$88$	1.00000	0.106600
$89$	2.69138	0.285286	0.142643	−	0.989774i	$-0.454440\pi$
$89$	2.69138	0.285286	0.142643	+	0.989774i	$0.454440\pi$
$90$	1.00000	0.105409
$91$	10.4526	1.09573
$92$	−7.25619	−0.756511
$93$	−9.71411	−1.00731
$94$	12.3210	1.27082
$95$	3.41051	0.349911
$96$	−1.00000	−0.102062
$97$	−16.5351	−1.67889	−0.839444	−	0.543446i	$-0.817119\pi$
$97$	−16.5351	−1.67889	−0.839444	+	0.543446i	$0.817119\pi$
$98$	2.39310	0.241740
$99$	1.00000	0.100504

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	5610.2.a.ci.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.ci.1.4	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.06482 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +3.41051 q^{13} +3.06482 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +3.41051 q^{19} +1.00000 q^{20} -3.06482 q^{21} +1.00000 q^{22} -7.25619 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.41051 q^{26} -1.00000 q^{27} +3.06482 q^{28} +4.34569 q^{29} -1.00000 q^{30} +9.71411 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.06482 q^{35} +1.00000 q^{36} -3.86843 q^{37} +3.41051 q^{38} -3.41051 q^{39} +1.00000 q^{40} +12.3210 q^{41} -3.06482 q^{42} -11.8160 q^{43} +1.00000 q^{44} +1.00000 q^{45} -7.25619 q^{46} +12.3210 q^{47} -1.00000 q^{48} +2.39310 q^{49} +1.00000 q^{50} +1.00000 q^{51} +3.41051 q^{52} -1.14930 q^{53} -1.00000 q^{54} +1.00000 q^{55} +3.06482 q^{56} -3.41051 q^{57} +4.34569 q^{58} -6.36310 q^{59} -1.00000 q^{60} -12.1894 q^{61} +9.71411 q^{62} +3.06482 q^{63} +1.00000 q^{64} +3.41051 q^{65} -1.00000 q^{66} -5.08222 q^{67} -1.00000 q^{68} +7.25619 q^{69} +3.06482 q^{70} +4.45792 q^{71} +1.00000 q^{72} +3.14930 q^{73} -3.86843 q^{74} -1.00000 q^{75} +3.41051 q^{76} +3.06482 q^{77} -3.41051 q^{78} -10.7789 q^{79} +1.00000 q^{80} +1.00000 q^{81} +12.3210 q^{82} +13.2736 q^{83} -3.06482 q^{84} -1.00000 q^{85} -11.8160 q^{86} -4.34569 q^{87} +1.00000 q^{88} +2.69138 q^{89} +1.00000 q^{90} +10.4526 q^{91} -7.25619 q^{92} -9.71411 q^{93} +12.3210 q^{94} +3.41051 q^{95} -1.00000 q^{96} -16.5351 q^{97} +2.39310 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} + 2 q^{7} + 5 q^{8} + 5 q^{9} + 5 q^{10} + 5 q^{11} - 5 q^{12} - q^{13} + 2 q^{14} - 5 q^{15} + 5 q^{16} - 5 q^{17} + 5 q^{18} - q^{19} + 5 q^{20}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 + 2 * q^7 + 5 * q^8 + 5 * q^9 + 5 * q^10 + 5 * q^11 - 5 * q^12 - q^13 + 2 * q^14 - 5 * q^15 + 5 * q^16 - 5 * q^17 + 5 * q^18 - q^19 + 5 * q^20 - 2 * q^21 + 5 * q^22 - 5 * q^24 + 5 * q^25 - q^26 - 5 * q^27 + 2 * q^28 + 17 * q^29 - 5 * q^30 + 10 * q^31 + 5 * q^32 - 5 * q^33 - 5 * q^34 + 2 * q^35 + 5 * q^36 + q^37 - q^38 + q^39 + 5 * q^40 + 12 * q^41 - 2 * q^42 + 7 * q^43 + 5 * q^44 + 5 * q^45 + 12 * q^47 - 5 * q^48 + 23 * q^49 + 5 * q^50 + 5 * q^51 - q^52 + 6 * q^53 - 5 * q^54 + 5 * q^55 + 2 * q^56 + q^57 + 17 * q^58 + 2 * q^59 - 5 * q^60 + 9 * q^61 + 10 * q^62 + 2 * q^63 + 5 * q^64 - q^65 - 5 * q^66 + 17 * q^67 - 5 * q^68 + 2 * q^70 + 20 * q^71 + 5 * q^72 + 4 * q^73 + q^74 - 5 * q^75 - q^76 + 2 * q^77 + q^78 - 2 * q^79 + 5 * q^80 + 5 * q^81 + 12 * q^82 + q^83 - 2 * q^84 - 5 * q^85 + 7 * q^86 - 17 * q^87 + 5 * q^88 + 4 * q^89 + 5 * q^90 + 23 * q^91 - 10 * q^93 + 12 * q^94 - q^95 - 5 * q^96 - 8 * q^97 + 23 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more