LMFDB - Embedded newform 1445.4.a.r.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,4,2,48,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$85.2577599583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 $ x^8 - 4x^7 - 48x^6 + 112x^5 + 767x^4 - 496x^3 - 3404x^2 + 576*x + 4128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.42383$ of defining polynomial
Character	$\chi$	$=$	1445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.42383 q^{2} +1.34922 q^{3} -2.12506 q^{4} -5.00000 q^{5} +3.27027 q^{6} -30.7032 q^{7} -24.5414 q^{8} -25.1796 q^{9} -12.1191 q^{10} -41.4735 q^{11} -2.86717 q^{12} -82.0224 q^{13} -74.4192 q^{14} -6.74609 q^{15} -42.4836 q^{16} -61.0310 q^{18} +82.3819 q^{19} +10.6253 q^{20} -41.4253 q^{21} -100.525 q^{22} -27.7228 q^{23} -33.1117 q^{24} +25.0000 q^{25} -198.808 q^{26} -70.4017 q^{27} +65.2461 q^{28} +139.844 q^{29} -16.3514 q^{30} -198.067 q^{31} +93.3582 q^{32} -55.9569 q^{33} +153.516 q^{35} +53.5082 q^{36} -313.534 q^{37} +199.679 q^{38} -110.666 q^{39} +122.707 q^{40} +142.004 q^{41} -100.408 q^{42} +285.706 q^{43} +88.1338 q^{44} +125.898 q^{45} -67.1952 q^{46} -481.359 q^{47} -57.3197 q^{48} +599.685 q^{49} +60.5957 q^{50} +174.303 q^{52} -130.417 q^{53} -170.642 q^{54} +207.368 q^{55} +753.499 q^{56} +111.151 q^{57} +338.957 q^{58} -168.650 q^{59} +14.3359 q^{60} -201.968 q^{61} -480.079 q^{62} +773.094 q^{63} +566.153 q^{64} +410.112 q^{65} -135.630 q^{66} +362.238 q^{67} -37.4041 q^{69} +372.096 q^{70} +771.485 q^{71} +617.943 q^{72} -522.658 q^{73} -759.952 q^{74} +33.7305 q^{75} -175.067 q^{76} +1273.37 q^{77} -268.236 q^{78} -1343.65 q^{79} +212.418 q^{80} +584.862 q^{81} +344.193 q^{82} -123.943 q^{83} +88.0313 q^{84} +692.502 q^{86} +188.680 q^{87} +1017.82 q^{88} -191.502 q^{89} +305.155 q^{90} +2518.35 q^{91} +58.9126 q^{92} -267.235 q^{93} -1166.73 q^{94} -411.909 q^{95} +125.961 q^{96} -1312.98 q^{97} +1453.53 q^{98} +1044.29 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) $ 8 * q + 4 * q^2 + 2 * q^3 + 48 * q^4 - 40 * q^5 - 44 * q^6 - 32 * q^7 - 36 * q^8 + 162 * q^9 - 20 * q^10 + 44 * q^11 + 36 * q^12 + 86 * q^13 - 146 * q^14 - 10 * q^15 + 348 * q^16 - 8 * q^18 + 288 * q^19 - 240 * q^20 + 180 * q^21 - 562 * q^22 - 160 * q^23 - 92 * q^24 + 200 * q^25 + 386 * q^26 + 110 * q^27 - 150 * q^28 + 642 * q^29 + 220 * q^30 + 182 * q^31 - 444 * q^32 + 22 * q^33 + 160 * q^35 - 268 * q^36 - 490 * q^37 - 746 * q^38 - 1034 * q^39 + 180 * q^40 + 72 * q^41 + 466 * q^42 + 1168 * q^43 + 1814 * q^44 - 810 * q^45 + 810 * q^46 + 216 * q^47 - 1948 * q^48 + 1202 * q^49 + 100 * q^50 + 2166 * q^52 + 508 * q^53 - 604 * q^54 - 220 * q^55 - 2386 * q^56 + 70 * q^57 - 1464 * q^58 - 918 * q^59 - 180 * q^60 + 1970 * q^61 - 2726 * q^62 - 568 * q^63 + 2252 * q^64 - 430 * q^65 + 1282 * q^66 + 2750 * q^67 + 2444 * q^69 + 730 * q^70 + 1368 * q^71 + 1792 * q^72 + 582 * q^73 + 4190 * q^74 + 50 * q^75 - 346 * q^76 - 2906 * q^77 + 1862 * q^78 + 1256 * q^79 - 1740 * q^80 + 1732 * q^81 - 3600 * q^82 + 2534 * q^83 - 2698 * q^84 + 552 * q^86 + 1856 * q^87 - 2202 * q^88 + 1200 * q^89 + 40 * q^90 + 3802 * q^91 + 6142 * q^92 - 214 * q^93 + 548 * q^94 - 1440 * q^95 - 372 * q^96 - 1698 * q^97 + 4168 * q^98 + 5258 * 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 $

$2$	2.42383	0.856952	0.428476	−	0.903553i	$-0.359051\pi$
$2$	2.42383	0.856952	0.428476	+	0.903553i	$0.359051\pi$
$3$	1.34922	0.259657	0.129829	−	0.991536i	$-0.458557\pi$
$3$	1.34922	0.259657	0.129829	+	0.991536i	$0.458557\pi$
$4$	−2.12506	−0.265633
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	3.27027	0.222514
$7$	−30.7032	−1.65782	−0.828908	−	0.559385i	$-0.811037\pi$
$7$	−30.7032	−1.65782	−0.828908	+	0.559385i	$0.811037\pi$
$8$	−24.5414	−1.08459
$9$	−25.1796	−0.932578
$10$	−12.1191	−0.383241
$11$	−41.4735	−1.13679	−0.568397	−	0.822754i	$-0.692436\pi$
$11$	−41.4735	−1.13679	−0.568397	+	0.822754i	$0.692436\pi$
$12$	−2.86717	−0.0689735
$13$	−82.0224	−1.74992	−0.874959	−	0.484197i	$-0.839112\pi$
$13$	−82.0224	−1.74992	−0.874959	+	0.484197i	$0.839112\pi$
$14$	−74.4192	−1.42067
$15$	−6.74609	−0.116122
$16$	−42.4836	−0.663807
$17$	0	0
$17$	0	0
$18$	−61.0310	−0.799175
$19$	82.3819	0.994721	0.497360	−	0.867544i	$-0.334303\pi$
$19$	82.3819	0.994721	0.497360	+	0.867544i	$0.334303\pi$
$20$	10.6253	0.118795
$21$	−41.4253	−0.430464
$22$	−100.525	−0.974179
$23$	−27.7228	−0.251330	−0.125665	−	0.992073i	$-0.540106\pi$
$23$	−27.7228	−0.251330	−0.125665	+	0.992073i	$0.540106\pi$
$24$	−33.1117	−0.281621
$25$	25.0000	0.200000
$26$	−198.808	−1.49960
$27$	−70.4017	−0.501808
$28$	65.2461	0.440370
$29$	139.844	0.895460	0.447730	−	0.894169i	$-0.352233\pi$
$29$	139.844	0.895460	0.447730	+	0.894169i	$0.352233\pi$
$30$	−16.3514	−0.0995112
$31$	−198.067	−1.14754	−0.573771	−	0.819016i	$-0.694520\pi$
$31$	−198.067	−1.14754	−0.573771	+	0.819016i	$0.694520\pi$
$32$	93.3582	0.515736
$33$	−55.9569	−0.295177
$34$	0	0
$35$	153.516	0.741398
$36$	53.5082	0.247723
$37$	−313.534	−1.39310	−0.696549	−	0.717509i	$-0.745282\pi$
$37$	−313.534	−1.39310	−0.696549	+	0.717509i	$0.745282\pi$
$38$	199.679	0.852428
$39$	−110.666	−0.454379
$40$	122.707	0.485042
$41$	142.004	0.540910	0.270455	−	0.962733i	$-0.412826\pi$
$41$	142.004	0.540910	0.270455	+	0.962733i	$0.412826\pi$
$42$	−100.408	−0.368887
$43$	285.706	1.01325	0.506625	−	0.862166i	$-0.330893\pi$
$43$	285.706	1.01325	0.506625	+	0.862166i	$0.330893\pi$
$44$	88.1338	0.301970
$45$	125.898	0.417062
$46$	−67.1952	−0.215378
$47$	−481.359	−1.49390	−0.746952	−	0.664878i	$-0.768484\pi$
$47$	−481.359	−1.49390	−0.746952	+	0.664878i	$0.768484\pi$
$48$	−57.3197	−0.172362
$49$	599.685	1.74835
$50$	60.5957	0.171390
$51$	0	0
$52$	174.303	0.464835
$53$	−130.417	−0.338003	−0.169002	−	0.985616i	$-0.554054\pi$
$53$	−130.417	−0.338003	−0.169002	+	0.985616i	$0.554054\pi$
$54$	−170.642	−0.430025
$55$	207.368	0.508390
$56$	753.499	1.79804
$57$	111.151	0.258286
$58$	338.957	0.767367
$59$	−168.650	−0.372141	−0.186071	−	0.982536i	$-0.559575\pi$
$59$	−168.650	−0.372141	−0.186071	+	0.982536i	$0.559575\pi$
$60$	14.3359	0.0308459
$61$	−201.968	−0.423924	−0.211962	−	0.977278i	$-0.567985\pi$
$61$	−201.968	−0.423924	−0.211962	+	0.977278i	$0.567985\pi$
$62$	−480.079	−0.983389
$63$	773.094	1.54604
$64$	566.153	1.10577
$65$	410.112	0.782587
$66$	−135.630	−0.252953
$67$	362.238	0.660514	0.330257	−	0.943891i	$-0.392865\pi$
$67$	362.238	0.660514	0.330257	+	0.943891i	$0.392865\pi$
$68$	0	0
$69$	−37.4041	−0.0652597
$70$	372.096	0.635342
$71$	771.485	1.28956	0.644778	−	0.764370i	$-0.276950\pi$
$71$	771.485	1.28956	0.644778	+	0.764370i	$0.276950\pi$
$72$	617.943	1.01146
$73$	−522.658	−0.837979	−0.418990	−	0.907991i	$-0.637616\pi$
$73$	−522.658	−0.837979	−0.418990	+	0.907991i	$0.637616\pi$
$74$	−759.952	−1.19382
$75$	33.7305	0.0519314
$76$	−175.067	−0.264230
$77$	1273.37	1.88460
$78$	−268.236	−0.389381
$79$	−1343.65	−1.91357	−0.956784	−	0.290799i	$-0.906079\pi$
$79$	−1343.65	−1.91357	−0.956784	+	0.290799i	$0.906079\pi$
$80$	212.418	0.296863
$81$	584.862	0.802280
$82$	344.193	0.463534
$83$	−123.943	−0.163910	−0.0819551	−	0.996636i	$-0.526116\pi$
$83$	−123.943	−0.163910	−0.0819551	+	0.996636i	$0.526116\pi$
$84$	88.0313	0.114345
$85$	0	0
$86$	692.502	0.868307
$87$	188.680	0.232513
$88$	1017.82	1.23295
$89$	−191.502	−0.228080	−0.114040	−	0.993476i	$-0.536379\pi$
$89$	−191.502	−0.228080	−0.114040	+	0.993476i	$0.536379\pi$
$90$	305.155	0.357402
$91$	2518.35	2.90104
$92$	58.9126	0.0667615
$93$	−267.235	−0.297968
$94$	−1166.73	−1.28020
$95$	−411.909	−0.444853
$96$	125.961	0.133915
$97$	−1312.98	−1.37436	−0.687182	−	0.726485i	$-0.741152\pi$
$97$	−1312.98	−1.37436	−0.687182	+	0.726485i	$0.741152\pi$
$98$	1453.53	1.49825
$99$	1044.29	1.06015

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	1445.4.a.r.1.5	yes	8
17.16	even	2		1445.4.a.q.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.4.a.q.1.5	✓	8	17.16	even	2
1445.4.a.r.1.5	yes	8	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:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.42383 q^{2} +1.34922 q^{3} -2.12506 q^{4} -5.00000 q^{5} +3.27027 q^{6} -30.7032 q^{7} -24.5414 q^{8} -25.1796 q^{9} -12.1191 q^{10} -41.4735 q^{11} -2.86717 q^{12} -82.0224 q^{13} -74.4192 q^{14} -6.74609 q^{15} -42.4836 q^{16} -61.0310 q^{18} +82.3819 q^{19} +10.6253 q^{20} -41.4253 q^{21} -100.525 q^{22} -27.7228 q^{23} -33.1117 q^{24} +25.0000 q^{25} -198.808 q^{26} -70.4017 q^{27} +65.2461 q^{28} +139.844 q^{29} -16.3514 q^{30} -198.067 q^{31} +93.3582 q^{32} -55.9569 q^{33} +153.516 q^{35} +53.5082 q^{36} -313.534 q^{37} +199.679 q^{38} -110.666 q^{39} +122.707 q^{40} +142.004 q^{41} -100.408 q^{42} +285.706 q^{43} +88.1338 q^{44} +125.898 q^{45} -67.1952 q^{46} -481.359 q^{47} -57.3197 q^{48} +599.685 q^{49} +60.5957 q^{50} +174.303 q^{52} -130.417 q^{53} -170.642 q^{54} +207.368 q^{55} +753.499 q^{56} +111.151 q^{57} +338.957 q^{58} -168.650 q^{59} +14.3359 q^{60} -201.968 q^{61} -480.079 q^{62} +773.094 q^{63} +566.153 q^{64} +410.112 q^{65} -135.630 q^{66} +362.238 q^{67} -37.4041 q^{69} +372.096 q^{70} +771.485 q^{71} +617.943 q^{72} -522.658 q^{73} -759.952 q^{74} +33.7305 q^{75} -175.067 q^{76} +1273.37 q^{77} -268.236 q^{78} -1343.65 q^{79} +212.418 q^{80} +584.862 q^{81} +344.193 q^{82} -123.943 q^{83} +88.0313 q^{84} +692.502 q^{86} +188.680 q^{87} +1017.82 q^{88} -191.502 q^{89} +305.155 q^{90} +2518.35 q^{91} +58.9126 q^{92} -267.235 q^{93} -1166.73 q^{94} -411.909 q^{95} +125.961 q^{96} -1312.98 q^{97} +1453.53 q^{98} +1044.29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 8 * q + 4 * q^2 + 2 * q^3 + 48 * q^4 - 40 * q^5 - 44 * q^6 - 32 * q^7 - 36 * q^8 + 162 * q^9 - 20 * q^10 + 44 * q^11 + 36 * q^12 + 86 * q^13 - 146 * q^14 - 10 * q^15 + 348 * q^16 - 8 * q^18 + 288 * q^19 - 240 * q^20 + 180 * q^21 - 562 * q^22 - 160 * q^23 - 92 * q^24 + 200 * q^25 + 386 * q^26 + 110 * q^27 - 150 * q^28 + 642 * q^29 + 220 * q^30 + 182 * q^31 - 444 * q^32 + 22 * q^33 + 160 * q^35 - 268 * q^36 - 490 * q^37 - 746 * q^38 - 1034 * q^39 + 180 * q^40 + 72 * q^41 + 466 * q^42 + 1168 * q^43 + 1814 * q^44 - 810 * q^45 + 810 * q^46 + 216 * q^47 - 1948 * q^48 + 1202 * q^49 + 100 * q^50 + 2166 * q^52 + 508 * q^53 - 604 * q^54 - 220 * q^55 - 2386 * q^56 + 70 * q^57 - 1464 * q^58 - 918 * q^59 - 180 * q^60 + 1970 * q^61 - 2726 * q^62 - 568 * q^63 + 2252 * q^64 - 430 * q^65 + 1282 * q^66 + 2750 * q^67 + 2444 * q^69 + 730 * q^70 + 1368 * q^71 + 1792 * q^72 + 582 * q^73 + 4190 * q^74 + 50 * q^75 - 346 * q^76 - 2906 * q^77 + 1862 * q^78 + 1256 * q^79 - 1740 * q^80 + 1732 * q^81 - 3600 * q^82 + 2534 * q^83 - 2698 * q^84 + 552 * q^86 + 1856 * q^87 - 2202 * q^88 + 1200 * q^89 + 40 * q^90 + 3802 * q^91 + 6142 * q^92 - 214 * q^93 + 548 * q^94 - 1440 * q^95 - 372 * q^96 - 1698 * q^97 + 4168 * q^98 + 5258 * q^99

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