LMFDB - Embedded newform 7942.2.a.bi.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7942,2,Mod(1,7942)] 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("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-3,0,-6] 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:	$63.4171892853$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} -2.66908 q^{6} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} -4.12398 q^{10} -1.00000 q^{11} -2.66908 q^{12} +2.21417 q^{13} -4.21417 q^{14} +11.0072 q^{15} +1.00000 q^{16} -3.45490 q^{17} +4.12398 q^{18} -4.12398 q^{20} +11.2480 q^{21} -1.00000 q^{22} -5.45490 q^{23} -2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} -3.00000 q^{27} -4.21417 q^{28} -5.57889 q^{29} +11.0072 q^{30} -7.00724 q^{31} +1.00000 q^{32} +2.66908 q^{33} -3.45490 q^{34} +17.3792 q^{35} +4.12398 q^{36} +2.90981 q^{37} -5.90981 q^{39} -4.12398 q^{40} +11.9170 q^{41} +11.2480 q^{42} +1.46214 q^{43} -1.00000 q^{44} -17.0072 q^{45} -5.45490 q^{46} +7.58612 q^{47} -2.66908 q^{48} +10.7593 q^{49} +12.0072 q^{50} +9.22141 q^{51} +2.21417 q^{52} +13.2214 q^{53} -3.00000 q^{54} +4.12398 q^{55} -4.21417 q^{56} -5.57889 q^{58} -4.79306 q^{59} +11.0072 q^{60} +8.90981 q^{61} -7.00724 q^{62} -17.3792 q^{63} +1.00000 q^{64} -9.13122 q^{65} +2.66908 q^{66} -1.30437 q^{67} -3.45490 q^{68} +14.5596 q^{69} +17.3792 q^{70} +6.80030 q^{71} +4.12398 q^{72} -1.45490 q^{73} +2.90981 q^{74} -32.0483 q^{75} +4.21417 q^{77} -5.90981 q^{78} +9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} -13.2142 q^{83} +11.2480 q^{84} +14.2480 q^{85} +1.46214 q^{86} +14.8905 q^{87} -1.00000 q^{88} -8.24797 q^{89} -17.0072 q^{90} -9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} +7.58612 q^{94} -2.66908 q^{96} -2.18038 q^{97} +10.7593 q^{98} -4.12398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 6 q^{14} + 9 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} + 15 q^{21} - 3 q^{22} - 15 q^{23} + 12 q^{25} - 9 q^{27}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 - 6 * q^14 + 9 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^20 + 15 * q^21 - 3 * q^22 - 15 * q^23 + 12 * q^25 - 9 * q^27 - 6 * q^28 - 6 * q^29 + 9 * q^30 + 3 * q^31 + 3 * q^32 - 9 * q^34 + 3 * q^36 + 6 * q^37 - 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 + 9 * q^53 - 9 * q^54 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 - 9 * q^68 - 3 * q^69 - 21 * q^71 + 3 * q^72 - 3 * q^73 + 6 * q^74 - 36 * q^75 + 6 * q^77 - 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 + 15 * q^84 + 24 * q^85 - 21 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 27 * q^90 - 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 12 * q^97 + 27 * q^98 - 3 * 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
$3$	−2.66908	−1.54099	−0.770497	−	0.637444i	$-0.779992\pi$
$3$	−2.66908	−1.54099	−0.770497	+	0.637444i	$0.779992\pi$
$4$	1.00000	0.500000
$5$	−4.12398	−1.84430	−0.922151	−	0.386831i	$-0.873570\pi$
$5$	−4.12398	−1.84430	−0.922151	+	0.386831i	$0.873570\pi$
$6$	−2.66908	−1.08965
$7$	−4.21417	−1.59281	−0.796404	−	0.604765i	$-0.793267\pi$
$7$	−4.21417	−1.59281	−0.796404	+	0.604765i	$0.793267\pi$
$8$	1.00000	0.353553
$9$	4.12398	1.37466
$10$	−4.12398	−1.30412
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.66908	−0.770497
$13$	2.21417	0.614102	0.307051	−	0.951693i	$-0.400658\pi$
$13$	2.21417	0.614102	0.307051	+	0.951693i	$0.400658\pi$
$14$	−4.21417	−1.12629
$15$	11.0072	2.84206
$16$	1.00000	0.250000
$17$	−3.45490	−0.837937	−0.418969	−	0.908001i	$-0.637608\pi$
$17$	−3.45490	−0.837937	−0.418969	+	0.908001i	$0.637608\pi$
$18$	4.12398	0.972032
$19$	0	0
$19$	0	0
$20$	−4.12398	−0.922151
$21$	11.2480	2.45451
$22$	−1.00000	−0.213201
$23$	−5.45490	−1.13743	−0.568713	−	0.822536i	$-0.692559\pi$
$23$	−5.45490	−1.13743	−0.568713	+	0.822536i	$0.692559\pi$
$24$	−2.66908	−0.544823
$25$	12.0072	2.40145
$26$	2.21417	0.434235
$27$	−3.00000	−0.577350
$28$	−4.21417	−0.796404
$29$	−5.57889	−1.03597	−0.517987	−	0.855389i	$-0.673318\pi$
$29$	−5.57889	−1.03597	−0.517987	+	0.855389i	$0.673318\pi$
$30$	11.0072	2.00964
$31$	−7.00724	−1.25854	−0.629268	−	0.777188i	$-0.716645\pi$
$31$	−7.00724	−1.25854	−0.629268	+	0.777188i	$0.716645\pi$
$32$	1.00000	0.176777
$33$	2.66908	0.464627
$34$	−3.45490	−0.592511
$35$	17.3792	2.93762
$36$	4.12398	0.687331
$37$	2.90981	0.478370	0.239185	−	0.970974i	$-0.423120\pi$
$37$	2.90981	0.478370	0.239185	+	0.970974i	$0.423120\pi$
$38$	0	0
$39$	−5.90981	−0.946327
$40$	−4.12398	−0.652059
$41$	11.9170	1.86113	0.930565	−	0.366127i	$-0.119316\pi$
$41$	11.9170	1.86113	0.930565	+	0.366127i	$0.119316\pi$
$42$	11.2480	1.73560
$43$	1.46214	0.222974	0.111487	−	0.993766i	$-0.464439\pi$
$43$	1.46214	0.222974	0.111487	+	0.993766i	$0.464439\pi$
$44$	−1.00000	−0.150756
$45$	−17.0072	−2.53529
$46$	−5.45490	−0.804282
$47$	7.58612	1.10655	0.553275	−	0.832999i	$-0.313378\pi$
$47$	7.58612	1.10655	0.553275	+	0.832999i	$0.313378\pi$
$48$	−2.66908	−0.385248
$49$	10.7593	1.53704
$50$	12.0072	1.69808
$51$	9.22141	1.29126
$52$	2.21417	0.307051
$53$	13.2214	1.81610	0.908050	−	0.418861i	$-0.137571\pi$
$53$	13.2214	1.81610	0.908050	+	0.418861i	$0.137571\pi$
$54$	−3.00000	−0.408248
$55$	4.12398	0.556078
$56$	−4.21417	−0.563143
$57$	0	0
$58$	−5.57889	−0.732544
$59$	−4.79306	−0.624004	−0.312002	−	0.950082i	$-0.600999\pi$
$59$	−4.79306	−0.624004	−0.312002	+	0.950082i	$0.600999\pi$
$60$	11.0072	1.42103
$61$	8.90981	1.14078	0.570392	−	0.821373i	$-0.306791\pi$
$61$	8.90981	1.14078	0.570392	+	0.821373i	$0.306791\pi$
$62$	−7.00724	−0.889920
$63$	−17.3792	−2.18957
$64$	1.00000	0.125000
$65$	−9.13122	−1.13259
$66$	2.66908	0.328541
$67$	−1.30437	−0.159354	−0.0796769	−	0.996821i	$-0.525389\pi$
$67$	−1.30437	−0.159354	−0.0796769	+	0.996821i	$0.525389\pi$
$68$	−3.45490	−0.418969
$69$	14.5596	1.75277
$70$	17.3792	2.07721
$71$	6.80030	0.807047	0.403524	−	0.914969i	$-0.367785\pi$
$71$	6.80030	0.807047	0.403524	+	0.914969i	$0.367785\pi$
$72$	4.12398	0.486016
$73$	−1.45490	−0.170284	−0.0851418	−	0.996369i	$-0.527134\pi$
$73$	−1.45490	−0.170284	−0.0851418	+	0.996369i	$0.527134\pi$
$74$	2.90981	0.338258
$75$	−32.0483	−3.70061
$76$	0	0
$77$	4.21417	0.480250
$78$	−5.90981	−0.669154
$79$	9.15777	1.03033	0.515165	−	0.857091i	$-0.327731\pi$
$79$	9.15777	1.03033	0.515165	+	0.857091i	$0.327731\pi$
$80$	−4.12398	−0.461075
$81$	−4.36471	−0.484968
$82$	11.9170	1.31602
$83$	−13.2142	−1.45044	−0.725222	−	0.688515i	$-0.758263\pi$
$83$	−13.2142	−1.45044	−0.725222	+	0.688515i	$0.758263\pi$
$84$	11.2480	1.22725
$85$	14.2480	1.54541
$86$	1.46214	0.157667
$87$	14.8905	1.59643
$88$	−1.00000	−0.106600
$89$	−8.24797	−0.874283	−0.437141	−	0.899393i	$-0.644009\pi$
$89$	−8.24797	−0.874283	−0.437141	+	0.899393i	$0.644009\pi$
$90$	−17.0072	−1.79272
$91$	−9.33092	−0.978146
$92$	−5.45490	−0.568713
$93$	18.7029	1.93940
$94$	7.58612	0.782449
$95$	0	0
$96$	−2.66908	−0.272412
$97$	−2.18038	−0.221384	−0.110692	−	0.993855i	$-0.535307\pi$
$97$	−2.18038	−0.221384	−0.110692	+	0.993855i	$0.535307\pi$
$98$	10.7593	1.08685
$99$	−4.12398	−0.414476

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	7942.2.a.bi.1.1		3
19.18	odd	2		418.2.a.g.1.3	✓	3
57.56	even	2		3762.2.a.bg.1.3		3
76.75	even	2		3344.2.a.q.1.1		3
209.208	even	2		4598.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.3	✓	3	19.18	odd	2
3344.2.a.q.1.1		3	76.75	even	2
3762.2.a.bg.1.3		3	57.56	even	2
4598.2.a.bo.1.3		3	209.208	even	2
7942.2.a.bi.1.1		3	1.1	even	1	trivial

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Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} -2.66908 q^{6} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} -4.12398 q^{10} -1.00000 q^{11} -2.66908 q^{12} +2.21417 q^{13} -4.21417 q^{14} +11.0072 q^{15} +1.00000 q^{16} -3.45490 q^{17} +4.12398 q^{18} -4.12398 q^{20} +11.2480 q^{21} -1.00000 q^{22} -5.45490 q^{23} -2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} -3.00000 q^{27} -4.21417 q^{28} -5.57889 q^{29} +11.0072 q^{30} -7.00724 q^{31} +1.00000 q^{32} +2.66908 q^{33} -3.45490 q^{34} +17.3792 q^{35} +4.12398 q^{36} +2.90981 q^{37} -5.90981 q^{39} -4.12398 q^{40} +11.9170 q^{41} +11.2480 q^{42} +1.46214 q^{43} -1.00000 q^{44} -17.0072 q^{45} -5.45490 q^{46} +7.58612 q^{47} -2.66908 q^{48} +10.7593 q^{49} +12.0072 q^{50} +9.22141 q^{51} +2.21417 q^{52} +13.2214 q^{53} -3.00000 q^{54} +4.12398 q^{55} -4.21417 q^{56} -5.57889 q^{58} -4.79306 q^{59} +11.0072 q^{60} +8.90981 q^{61} -7.00724 q^{62} -17.3792 q^{63} +1.00000 q^{64} -9.13122 q^{65} +2.66908 q^{66} -1.30437 q^{67} -3.45490 q^{68} +14.5596 q^{69} +17.3792 q^{70} +6.80030 q^{71} +4.12398 q^{72} -1.45490 q^{73} +2.90981 q^{74} -32.0483 q^{75} +4.21417 q^{77} -5.90981 q^{78} +9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} -13.2142 q^{83} +11.2480 q^{84} +14.2480 q^{85} +1.46214 q^{86} +14.8905 q^{87} -1.00000 q^{88} -8.24797 q^{89} -17.0072 q^{90} -9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} +7.58612 q^{94} -2.66908 q^{96} -2.18038 q^{97} +10.7593 q^{98} -4.12398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 6 q^{14} + 9 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} + 15 q^{21} - 3 q^{22} - 15 q^{23} + 12 q^{25} - 9 q^{27}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 - 6 * q^14 + 9 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^20 + 15 * q^21 - 3 * q^22 - 15 * q^23 + 12 * q^25 - 9 * q^27 - 6 * q^28 - 6 * q^29 + 9 * q^30 + 3 * q^31 + 3 * q^32 - 9 * q^34 + 3 * q^36 + 6 * q^37 - 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 + 9 * q^53 - 9 * q^54 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 - 9 * q^68 - 3 * q^69 - 21 * q^71 + 3 * q^72 - 3 * q^73 + 6 * q^74 - 36 * q^75 + 6 * q^77 - 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 + 15 * q^84 + 24 * q^85 - 21 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 27 * q^90 - 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 12 * q^97 + 27 * q^98 - 3 * q^99

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