LMFDB - Embedded newform 7406.2.a.bd.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,4,0,4] 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:	$59.1372077370$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.449797.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - x + 4 $ x^4 - 10*x^2 - x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.14867$ of defining polynomial
Character	$\chi$	$=$	7406.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +3.14867 q^{3} +1.00000 q^{4} +4.14867 q^{5} +3.14867 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.91413 q^{9} +4.14867 q^{10} -2.87829 q^{11} +3.14867 q^{12} -6.91413 q^{13} +1.00000 q^{14} +13.0628 q^{15} +1.00000 q^{16} +4.02697 q^{17} +6.91413 q^{18} -0.270377 q^{19} +4.14867 q^{20} +3.14867 q^{21} -2.87829 q^{22} +3.14867 q^{24} +12.2115 q^{25} -6.91413 q^{26} +12.3243 q^{27} +1.00000 q^{28} -3.02697 q^{29} +13.0628 q^{30} -2.76546 q^{31} +1.00000 q^{32} -9.06280 q^{33} +4.02697 q^{34} +4.14867 q^{35} +6.91413 q^{36} -0.765459 q^{37} -0.270377 q^{38} -21.7703 q^{39} +4.14867 q^{40} -1.38321 q^{41} +3.14867 q^{42} +10.5677 q^{43} -2.87829 q^{44} +28.6845 q^{45} -6.18451 q^{47} +3.14867 q^{48} +1.00000 q^{49} +12.2115 q^{50} +12.6796 q^{51} -6.91413 q^{52} -1.02697 q^{53} +12.3243 q^{54} -11.9411 q^{55} +1.00000 q^{56} -0.851329 q^{57} -3.02697 q^{58} -8.79242 q^{59} +13.0628 q^{60} +2.97303 q^{61} -2.76546 q^{62} +6.91413 q^{63} +1.00000 q^{64} -28.6845 q^{65} -9.06280 q^{66} +3.03584 q^{67} +4.02697 q^{68} +4.14867 q^{70} -4.58095 q^{71} +6.91413 q^{72} -0.459246 q^{73} -0.765459 q^{74} +38.4499 q^{75} -0.270377 q^{76} -2.87829 q^{77} -21.7703 q^{78} +7.53188 q^{79} +4.14867 q^{80} +18.0628 q^{81} -1.38321 q^{82} -1.91413 q^{83} +3.14867 q^{84} +16.7066 q^{85} +10.5677 q^{86} -9.53092 q^{87} -2.87829 q^{88} -8.33318 q^{89} +28.6845 q^{90} -6.91413 q^{91} -8.70752 q^{93} -6.18451 q^{94} -1.12171 q^{95} +3.14867 q^{96} -13.8641 q^{97} +1.00000 q^{98} -19.9009 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 4 q^{7} + 4 q^{8} + 8 q^{9} + 4 q^{10} - 3 q^{11} - 8 q^{13} + 4 q^{14} + 20 q^{15} + 4 q^{16} - 5 q^{17} + 8 q^{18} + 3 q^{19} + 4 q^{20} - 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots - 14 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 4 * q^7 + 4 * q^8 + 8 * q^9 + 4 * q^10 - 3 * q^11 - 8 * q^13 + 4 * q^14 + 20 * q^15 + 4 * q^16 - 5 * q^17 + 8 * q^18 + 3 * q^19 + 4 * q^20 - 3 * q^22 + 4 * q^25 - 8 * q^26 + 3 * q^27 + 4 * q^28 + 9 * q^29 + 20 * q^30 - 4 * q^31 + 4 * q^32 - 4 * q^33 - 5 * q^34 + 4 * q^35 + 8 * q^36 + 4 * q^37 + 3 * q^38 - 3 * q^39 + 4 * q^40 + 13 * q^43 - 3 * q^44 + 11 * q^45 - q^47 + 4 * q^49 + 4 * q^50 + 24 * q^51 - 8 * q^52 + 17 * q^53 + 3 * q^54 - 7 * q^55 + 4 * q^56 - 16 * q^57 + 9 * q^58 - 7 * q^59 + 20 * q^60 + 33 * q^61 - 4 * q^62 + 8 * q^63 + 4 * q^64 - 11 * q^65 - 4 * q^66 + q^67 - 5 * q^68 + 4 * q^70 - 35 * q^71 + 8 * q^72 - 10 * q^73 + 4 * q^74 + 43 * q^75 + 3 * q^76 - 3 * q^77 - 3 * q^78 + 12 * q^79 + 4 * q^80 + 40 * q^81 + 12 * q^83 + 19 * q^85 + 13 * q^86 - 24 * q^87 - 3 * q^88 + 3 * q^89 + 11 * q^90 - 8 * q^91 + 17 * q^93 - q^94 - 13 * q^95 - 5 * q^97 + 4 * q^98 - 14 * 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
$3$	3.14867	1.81789	0.908943	−	0.416920i	$-0.136891\pi$
$3$	3.14867	1.81789	0.908943	+	0.416920i	$0.136891\pi$
$4$	1.00000	0.500000
$5$	4.14867	1.85534	0.927671	−	0.373398i	$-0.121808\pi$
$5$	4.14867	1.85534	0.927671	+	0.373398i	$0.121808\pi$
$6$	3.14867	1.28544
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	6.91413	2.30471
$10$	4.14867	1.31192
$11$	−2.87829	−0.867838	−0.433919	−	0.900952i	$-0.642870\pi$
$11$	−2.87829	−0.867838	−0.433919	+	0.900952i	$0.642870\pi$
$12$	3.14867	0.908943
$13$	−6.91413	−1.91763	−0.958817	−	0.284024i	$-0.908331\pi$
$13$	−6.91413	−1.91763	−0.958817	+	0.284024i	$0.908331\pi$
$14$	1.00000	0.267261
$15$	13.0628	3.37280
$16$	1.00000	0.250000
$17$	4.02697	0.976682	0.488341	−	0.872653i	$-0.337602\pi$
$17$	4.02697	0.976682	0.488341	+	0.872653i	$0.337602\pi$
$18$	6.91413	1.62968
$19$	−0.270377	−0.0620288	−0.0310144	−	0.999519i	$-0.509874\pi$
$19$	−0.270377	−0.0620288	−0.0310144	+	0.999519i	$0.509874\pi$
$20$	4.14867	0.927671
$21$	3.14867	0.687096
$22$	−2.87829	−0.613654
$23$	0	0
$23$	0	0
$24$	3.14867	0.642720
$25$	12.2115	2.44229
$26$	−6.91413	−1.35597
$27$	12.3243	2.37181
$28$	1.00000	0.188982
$29$	−3.02697	−0.562093	−0.281047	−	0.959694i	$-0.590682\pi$
$29$	−3.02697	−0.562093	−0.281047	+	0.959694i	$0.590682\pi$
$30$	13.0628	2.38493
$31$	−2.76546	−0.496691	−0.248346	−	0.968672i	$-0.579887\pi$
$31$	−2.76546	−0.496691	−0.248346	+	0.968672i	$0.579887\pi$
$32$	1.00000	0.176777
$33$	−9.06280	−1.57763
$34$	4.02697	0.690619
$35$	4.14867	0.701253
$36$	6.91413	1.15235
$37$	−0.765459	−0.125841	−0.0629203	−	0.998019i	$-0.520041\pi$
$37$	−0.765459	−0.125841	−0.0629203	+	0.998019i	$0.520041\pi$
$38$	−0.270377	−0.0438610
$39$	−21.7703	−3.48604
$40$	4.14867	0.655962
$41$	−1.38321	−0.216021	−0.108011	−	0.994150i	$-0.534448\pi$
$41$	−1.38321	−0.216021	−0.108011	+	0.994150i	$0.534448\pi$
$42$	3.14867	0.485850
$43$	10.5677	1.61156	0.805781	−	0.592214i	$-0.201746\pi$
$43$	10.5677	1.61156	0.805781	+	0.592214i	$0.201746\pi$
$44$	−2.87829	−0.433919
$45$	28.6845	4.27603
$46$	0	0
$47$	−6.18451	−0.902103	−0.451052	−	0.892498i	$-0.648951\pi$
$47$	−6.18451	−0.902103	−0.451052	+	0.892498i	$0.648951\pi$
$48$	3.14867	0.454472
$49$	1.00000	0.142857
$50$	12.2115	1.72696
$51$	12.6796	1.77550
$52$	−6.91413	−0.958817
$53$	−1.02697	−0.141064	−0.0705322	−	0.997509i	$-0.522470\pi$
$53$	−1.02697	−0.141064	−0.0705322	+	0.997509i	$0.522470\pi$
$54$	12.3243	1.67713
$55$	−11.9411	−1.61014
$56$	1.00000	0.133631
$57$	−0.851329	−0.112761
$58$	−3.02697	−0.397460
$59$	−8.79242	−1.14468	−0.572338	−	0.820018i	$-0.693963\pi$
$59$	−8.79242	−1.14468	−0.572338	+	0.820018i	$0.693963\pi$
$60$	13.0628	1.68640
$61$	2.97303	0.380658	0.190329	−	0.981720i	$-0.439045\pi$
$61$	2.97303	0.380658	0.190329	+	0.981720i	$0.439045\pi$
$62$	−2.76546	−0.351214
$63$	6.91413	0.871098
$64$	1.00000	0.125000
$65$	−28.6845	−3.55787
$66$	−9.06280	−1.11555
$67$	3.03584	0.370886	0.185443	−	0.982655i	$-0.440628\pi$
$67$	3.03584	0.370886	0.185443	+	0.982655i	$0.440628\pi$
$68$	4.02697	0.488341
$69$	0	0
$70$	4.14867	0.495861
$71$	−4.58095	−0.543659	−0.271829	−	0.962345i	$-0.587629\pi$
$71$	−4.58095	−0.543659	−0.271829	+	0.962345i	$0.587629\pi$
$72$	6.91413	0.814838
$73$	−0.459246	−0.0537506	−0.0268753	−	0.999639i	$-0.508556\pi$
$73$	−0.459246	−0.0537506	−0.0268753	+	0.999639i	$0.508556\pi$
$74$	−0.765459	−0.0889828
$75$	38.4499	4.43981
$76$	−0.270377	−0.0310144
$77$	−2.87829	−0.328012
$78$	−21.7703	−2.46500
$79$	7.53188	0.847403	0.423702	−	0.905802i	$-0.360731\pi$
$79$	7.53188	0.847403	0.423702	+	0.905802i	$0.360731\pi$
$80$	4.14867	0.463836
$81$	18.0628	2.00698
$82$	−1.38321	−0.152750
$83$	−1.91413	−0.210103	−0.105052	−	0.994467i	$-0.533501\pi$
$83$	−1.91413	−0.210103	−0.105052	+	0.994467i	$0.533501\pi$
$84$	3.14867	0.343548
$85$	16.7066	1.81208
$86$	10.5677	1.13955
$87$	−9.53092	−1.02182
$88$	−2.87829	−0.306827
$89$	−8.33318	−0.883315	−0.441658	−	0.897184i	$-0.645609\pi$
$89$	−8.33318	−0.883315	−0.441658	+	0.897184i	$0.645609\pi$
$90$	28.6845	3.02361
$91$	−6.91413	−0.724798
$92$	0	0
$93$	−8.70752	−0.902928
$94$	−6.18451	−0.637883
$95$	−1.12171	−0.115085
$96$	3.14867	0.321360
$97$	−13.8641	−1.40769	−0.703843	−	0.710356i	$-0.748534\pi$
$97$	−13.8641	−1.40769	−0.703843	+	0.710356i	$0.748534\pi$
$98$	1.00000	0.101015
$99$	−19.9009	−2.00012

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	7406.2.a.bd.1.4	yes	4
23.22	odd	2		7406.2.a.ba.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.ba.1.4	✓	4	23.22	odd	2
7406.2.a.bd.1.4	yes	4	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:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.14867 q^{3} +1.00000 q^{4} +4.14867 q^{5} +3.14867 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.91413 q^{9} +4.14867 q^{10} -2.87829 q^{11} +3.14867 q^{12} -6.91413 q^{13} +1.00000 q^{14} +13.0628 q^{15} +1.00000 q^{16} +4.02697 q^{17} +6.91413 q^{18} -0.270377 q^{19} +4.14867 q^{20} +3.14867 q^{21} -2.87829 q^{22} +3.14867 q^{24} +12.2115 q^{25} -6.91413 q^{26} +12.3243 q^{27} +1.00000 q^{28} -3.02697 q^{29} +13.0628 q^{30} -2.76546 q^{31} +1.00000 q^{32} -9.06280 q^{33} +4.02697 q^{34} +4.14867 q^{35} +6.91413 q^{36} -0.765459 q^{37} -0.270377 q^{38} -21.7703 q^{39} +4.14867 q^{40} -1.38321 q^{41} +3.14867 q^{42} +10.5677 q^{43} -2.87829 q^{44} +28.6845 q^{45} -6.18451 q^{47} +3.14867 q^{48} +1.00000 q^{49} +12.2115 q^{50} +12.6796 q^{51} -6.91413 q^{52} -1.02697 q^{53} +12.3243 q^{54} -11.9411 q^{55} +1.00000 q^{56} -0.851329 q^{57} -3.02697 q^{58} -8.79242 q^{59} +13.0628 q^{60} +2.97303 q^{61} -2.76546 q^{62} +6.91413 q^{63} +1.00000 q^{64} -28.6845 q^{65} -9.06280 q^{66} +3.03584 q^{67} +4.02697 q^{68} +4.14867 q^{70} -4.58095 q^{71} +6.91413 q^{72} -0.459246 q^{73} -0.765459 q^{74} +38.4499 q^{75} -0.270377 q^{76} -2.87829 q^{77} -21.7703 q^{78} +7.53188 q^{79} +4.14867 q^{80} +18.0628 q^{81} -1.38321 q^{82} -1.91413 q^{83} +3.14867 q^{84} +16.7066 q^{85} +10.5677 q^{86} -9.53092 q^{87} -2.87829 q^{88} -8.33318 q^{89} +28.6845 q^{90} -6.91413 q^{91} -8.70752 q^{93} -6.18451 q^{94} -1.12171 q^{95} +3.14867 q^{96} -13.8641 q^{97} +1.00000 q^{98} -19.9009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 4 q^{7} + 4 q^{8} + 8 q^{9} + 4 q^{10} - 3 q^{11} - 8 q^{13} + 4 q^{14} + 20 q^{15} + 4 q^{16} - 5 q^{17} + 8 q^{18} + 3 q^{19} + 4 q^{20} - 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots - 14 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 4 * q^7 + 4 * q^8 + 8 * q^9 + 4 * q^10 - 3 * q^11 - 8 * q^13 + 4 * q^14 + 20 * q^15 + 4 * q^16 - 5 * q^17 + 8 * q^18 + 3 * q^19 + 4 * q^20 - 3 * q^22 + 4 * q^25 - 8 * q^26 + 3 * q^27 + 4 * q^28 + 9 * q^29 + 20 * q^30 - 4 * q^31 + 4 * q^32 - 4 * q^33 - 5 * q^34 + 4 * q^35 + 8 * q^36 + 4 * q^37 + 3 * q^38 - 3 * q^39 + 4 * q^40 + 13 * q^43 - 3 * q^44 + 11 * q^45 - q^47 + 4 * q^49 + 4 * q^50 + 24 * q^51 - 8 * q^52 + 17 * q^53 + 3 * q^54 - 7 * q^55 + 4 * q^56 - 16 * q^57 + 9 * q^58 - 7 * q^59 + 20 * q^60 + 33 * q^61 - 4 * q^62 + 8 * q^63 + 4 * q^64 - 11 * q^65 - 4 * q^66 + q^67 - 5 * q^68 + 4 * q^70 - 35 * q^71 + 8 * q^72 - 10 * q^73 + 4 * q^74 + 43 * q^75 + 3 * q^76 - 3 * q^77 - 3 * q^78 + 12 * q^79 + 4 * q^80 + 40 * q^81 + 12 * q^83 + 19 * q^85 + 13 * q^86 - 24 * q^87 - 3 * q^88 + 3 * q^89 + 11 * q^90 - 8 * q^91 + 17 * q^93 - q^94 - 13 * q^95 - 5 * q^97 + 4 * q^98 - 14 * q^99

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