Properties

Label 1445.4.a.q.1.5
Level $1445$
Weight $4$
Character 1445.1
Self dual yes
Analytic conductor $85.258$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content 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);
 
Copy content 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

Copy content comment:select newform
 
Copy content 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)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 \) Copy content Toggle raw display
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.42383 q^{2} -1.34922 q^{3} -2.12506 q^{4} +5.00000 q^{5} -3.27027 q^{6} +30.7032 q^{7} -24.5414 q^{8} -25.1796 q^{9} +12.1191 q^{10} +41.4735 q^{11} +2.86717 q^{12} -82.0224 q^{13} +74.4192 q^{14} -6.74609 q^{15} -42.4836 q^{16} -61.0310 q^{18} +82.3819 q^{19} -10.6253 q^{20} -41.4253 q^{21} +100.525 q^{22} +27.7228 q^{23} +33.1117 q^{24} +25.0000 q^{25} -198.808 q^{26} +70.4017 q^{27} -65.2461 q^{28} -139.844 q^{29} -16.3514 q^{30} +198.067 q^{31} +93.3582 q^{32} -55.9569 q^{33} +153.516 q^{35} +53.5082 q^{36} +313.534 q^{37} +199.679 q^{38} +110.666 q^{39} -122.707 q^{40} -142.004 q^{41} -100.408 q^{42} +285.706 q^{43} -88.1338 q^{44} -125.898 q^{45} +67.1952 q^{46} -481.359 q^{47} +57.3197 q^{48} +599.685 q^{49} +60.5957 q^{50} +174.303 q^{52} -130.417 q^{53} +170.642 q^{54} +207.368 q^{55} -753.499 q^{56} -111.151 q^{57} -338.957 q^{58} -168.650 q^{59} +14.3359 q^{60} +201.968 q^{61} +480.079 q^{62} -773.094 q^{63} +566.153 q^{64} -410.112 q^{65} -135.630 q^{66} +362.238 q^{67} -37.4041 q^{69} +372.096 q^{70} -771.485 q^{71} +617.943 q^{72} +522.658 q^{73} +759.952 q^{74} -33.7305 q^{75} -175.067 q^{76} +1273.37 q^{77} +268.236 q^{78} +1343.65 q^{79} -212.418 q^{80} +584.862 q^{81} -344.193 q^{82} -123.943 q^{83} +88.0313 q^{84} +692.502 q^{86} +188.680 q^{87} -1017.82 q^{88} -191.502 q^{89} -305.155 q^{90} -2518.35 q^{91} -58.9126 q^{92} -267.235 q^{93} -1166.73 q^{94} +411.909 q^{95} -125.961 q^{96} +1312.98 q^{97} +1453.53 q^{98} -1044.29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 2 q^{3} + 48 q^{4} + 40 q^{5} + 44 q^{6} + 32 q^{7} - 36 q^{8} + 162 q^{9} + 20 q^{10} - 44 q^{11} - 36 q^{12} + 86 q^{13} + 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} + 240 q^{20}+ \cdots - 5258 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.42383 0.856952 0.428476 − 0.903553i \(-0.359051\pi\)
0.428476 + 0.903553i \(0.359051\pi\)
\(3\) −1.34922 −0.259657 −0.129829 − 0.991536i \(-0.541443\pi\)
−0.129829 + 0.991536i \(0.541443\pi\)
\(4\) −2.12506 −0.265633
\(5\) 5.00000 0.447214
\(6\) −3.27027 −0.222514
\(7\) 30.7032 1.65782 0.828908 − 0.559385i \(-0.188963\pi\)
0.828908 + 0.559385i \(0.188963\pi\)
\(8\) −24.5414 −1.08459
\(9\) −25.1796 −0.932578
\(10\) 12.1191 0.383241
\(11\) 41.4735 1.13679 0.568397 − 0.822754i \(-0.307564\pi\)
0.568397 + 0.822754i \(0.307564\pi\)
\(12\) 2.86717 0.0689735
\(13\) −82.0224 −1.74992 −0.874959 − 0.484197i \(-0.839112\pi\)
−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
\(18\) −61.0310 −0.799175
\(19\) 82.3819 0.994721 0.497360 − 0.867544i \(-0.334303\pi\)
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 \(-0.459894\pi\)
0.125665 + 0.992073i \(0.459894\pi\)
\(24\) 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 \(-0.647767\pi\)
−0.447730 + 0.894169i \(0.647767\pi\)
\(30\) −16.3514 −0.0995112
\(31\) 198.067 1.14754 0.573771 − 0.819016i \(-0.305480\pi\)
0.573771 + 0.819016i \(0.305480\pi\)
\(32\) 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 \(-0.254718\pi\)
0.696549 + 0.717509i \(0.254718\pi\)
\(38\) 199.679 0.852428
\(39\) 110.666 0.454379
\(40\) −122.707 −0.485042
\(41\) −142.004 −0.540910 −0.270455 − 0.962733i \(-0.587174\pi\)
−0.270455 + 0.962733i \(0.587174\pi\)
\(42\) −100.408 −0.368887
\(43\) 285.706 1.01325 0.506625 − 0.862166i \(-0.330893\pi\)
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 \(-0.768484\pi\)
−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 \(-0.554054\pi\)
−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 \(-0.559575\pi\)
−0.186071 + 0.982536i \(0.559575\pi\)
\(60\) 14.3359 0.0308459
\(61\) 201.968 0.423924 0.211962 − 0.977278i \(-0.432015\pi\)
0.211962 + 0.977278i \(0.432015\pi\)
\(62\) 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 \(-0.392865\pi\)
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 \(-0.723050\pi\)
−0.644778 + 0.764370i \(0.723050\pi\)
\(72\) 617.943 1.01146
\(73\) 522.658 0.837979 0.418990 − 0.907991i \(-0.362384\pi\)
0.418990 + 0.907991i \(0.362384\pi\)
\(74\) 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 \(-0.0939210\pi\)
0.956784 + 0.290799i \(0.0939210\pi\)
\(80\) −212.418 −0.296863
\(81\) 584.862 0.802280
\(82\) −344.193 −0.463534
\(83\) −123.943 −0.163910 −0.0819551 − 0.996636i \(-0.526116\pi\)
−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 \(-0.536379\pi\)
−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 \(-0.258848\pi\)
0.687182 + 0.726485i \(0.258848\pi\)
\(98\) 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.q.1.5 ✓ 8
17.16 even 2 1445.4.a.r.1.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.5 ✓ 8 1.1 even 1 trivial
1445.4.a.r.1.5 yes 8 17.16 even 2