Properties

Label 1445.4.a.g.1.1
Level $1445$
Weight $4$
Character 1445.1
Self dual yes
Analytic conductor $85.258$
Analytic rank $1$
Dimension $1$
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 = [1,3,5,1,5] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +15.0000 q^{6} +22.0000 q^{7} -21.0000 q^{8} -2.00000 q^{9} +15.0000 q^{10} -60.0000 q^{11} +5.00000 q^{12} -31.0000 q^{13} +66.0000 q^{14} +25.0000 q^{15} -71.0000 q^{16} -6.00000 q^{18} -61.0000 q^{19} +5.00000 q^{20} +110.000 q^{21} -180.000 q^{22} +78.0000 q^{23} -105.000 q^{24} +25.0000 q^{25} -93.0000 q^{26} -145.000 q^{27} +22.0000 q^{28} -69.0000 q^{29} +75.0000 q^{30} +31.0000 q^{31} -45.0000 q^{32} -300.000 q^{33} +110.000 q^{35} -2.00000 q^{36} -56.0000 q^{37} -183.000 q^{38} -155.000 q^{39} -105.000 q^{40} +6.00000 q^{41} +330.000 q^{42} -538.000 q^{43} -60.0000 q^{44} -10.0000 q^{45} +234.000 q^{46} -465.000 q^{47} -355.000 q^{48} +141.000 q^{49} +75.0000 q^{50} -31.0000 q^{52} +723.000 q^{53} -435.000 q^{54} -300.000 q^{55} -462.000 q^{56} -305.000 q^{57} -207.000 q^{58} -753.000 q^{59} +25.0000 q^{60} -35.0000 q^{61} +93.0000 q^{62} -44.0000 q^{63} +433.000 q^{64} -155.000 q^{65} -900.000 q^{66} -322.000 q^{67} +390.000 q^{69} +330.000 q^{70} +99.0000 q^{71} +42.0000 q^{72} +1123.00 q^{73} -168.000 q^{74} +125.000 q^{75} -61.0000 q^{76} -1320.00 q^{77} -465.000 q^{78} -488.000 q^{79} -355.000 q^{80} -671.000 q^{81} +18.0000 q^{82} -852.000 q^{83} +110.000 q^{84} -1614.00 q^{86} -345.000 q^{87} +1260.00 q^{88} +1215.00 q^{89} -30.0000 q^{90} -682.000 q^{91} +78.0000 q^{92} +155.000 q^{93} -1395.00 q^{94} -305.000 q^{95} -225.000 q^{96} +601.000 q^{97} +423.000 q^{98} +120.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 − 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 5.00000 0.962250 0.481125 − 0.876652i \(-0.340228\pi\)
0.481125 + 0.876652i \(0.340228\pi\)
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 15.0000 1.02062
\(7\) 22.0000 1.18789 0.593944 − 0.804506i \(-0.297570\pi\)
0.593944 + 0.804506i \(0.297570\pi\)
\(8\) −21.0000 −0.928078
\(9\) −2.00000 −0.0740741
\(10\) 15.0000 0.474342
\(11\) −60.0000 −1.64461 −0.822304 − 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) 5.00000 0.120281
\(13\) −31.0000 −0.661373 −0.330687 − 0.943741i \(-0.607280\pi\)
−0.330687 + 0.943741i \(0.607280\pi\)
\(14\) 66.0000 1.25995
\(15\) 25.0000 0.430331
\(16\) −71.0000 −1.10938
\(17\) 0 0
\(18\) −6.00000 −0.0785674
\(19\) −61.0000 −0.736545 −0.368273 − 0.929718i \(-0.620051\pi\)
−0.368273 + 0.929718i \(0.620051\pi\)
\(20\) 5.00000 0.0559017
\(21\) 110.000 1.14305
\(22\) −180.000 −1.74437
\(23\) 78.0000 0.707136 0.353568 − 0.935409i \(-0.384968\pi\)
0.353568 + 0.935409i \(0.384968\pi\)
\(24\) −105.000 −0.893043
\(25\) 25.0000 0.200000
\(26\) −93.0000 −0.701492
\(27\) −145.000 −1.03353
\(28\) 22.0000 0.148486
\(29\) −69.0000 −0.441827 −0.220913 − 0.975293i \(-0.570904\pi\)
−0.220913 + 0.975293i \(0.570904\pi\)
\(30\) 75.0000 0.456435
\(31\) 31.0000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) −45.0000 −0.248592
\(33\) −300.000 −1.58252
\(34\) 0 0
\(35\) 110.000 0.531240
\(36\) −2.00000 −0.00925926
\(37\) −56.0000 −0.248820 −0.124410 − 0.992231i \(-0.539704\pi\)
−0.124410 + 0.992231i \(0.539704\pi\)
\(38\) −183.000 −0.781224
\(39\) −155.000 −0.636407
\(40\) −105.000 −0.415049
\(41\) 6.00000 0.0228547 0.0114273 − 0.999935i \(-0.496362\pi\)
0.0114273 + 0.999935i \(0.496362\pi\)
\(42\) 330.000 1.21238
\(43\) −538.000 −1.90801 −0.954003 − 0.299798i \(-0.903081\pi\)
−0.954003 + 0.299798i \(0.903081\pi\)
\(44\) −60.0000 −0.205576
\(45\) −10.0000 −0.0331269
\(46\) 234.000 0.750031
\(47\) −465.000 −1.44313 −0.721566 − 0.692345i \(-0.756577\pi\)
−0.721566 + 0.692345i \(0.756577\pi\)
\(48\) −355.000 −1.06750
\(49\) 141.000 0.411079
\(50\) 75.0000 0.212132
\(51\) 0 0
\(52\) −31.0000 −0.0826717
\(53\) 723.000 1.87381 0.936903 − 0.349590i \(-0.113679\pi\)
0.936903 + 0.349590i \(0.113679\pi\)
\(54\) −435.000 −1.09622
\(55\) −300.000 −0.735491
\(56\) −462.000 −1.10245
\(57\) −305.000 −0.708741
\(58\) −207.000 −0.468628
\(59\) −753.000 −1.66156 −0.830782 − 0.556598i \(-0.812106\pi\)
−0.830782 + 0.556598i \(0.812106\pi\)
\(60\) 25.0000 0.0537914
\(61\) −35.0000 −0.0734638 −0.0367319 − 0.999325i \(-0.511695\pi\)
−0.0367319 + 0.999325i \(0.511695\pi\)
\(62\) 93.0000 0.190500
\(63\) −44.0000 −0.0879917
\(64\) 433.000 0.845703
\(65\) −155.000 −0.295775
\(66\) −900.000 −1.67852
\(67\) −322.000 −0.587143 −0.293571 − 0.955937i \(-0.594844\pi\)
−0.293571 + 0.955937i \(0.594844\pi\)
\(68\) 0 0
\(69\) 390.000 0.680442
\(70\) 330.000 0.563465
\(71\) 99.0000 0.165481 0.0827404 − 0.996571i \(-0.473633\pi\)
0.0827404 + 0.996571i \(0.473633\pi\)
\(72\) 42.0000 0.0687465
\(73\) 1123.00 1.80051 0.900255 − 0.435363i \(-0.143380\pi\)
0.900255 + 0.435363i \(0.143380\pi\)
\(74\) −168.000 −0.263914
\(75\) 125.000 0.192450
\(76\) −61.0000 −0.0920682
\(77\) −1320.00 −1.95361
\(78\) −465.000 −0.675011
\(79\) −488.000 −0.694991 −0.347496 − 0.937682i \(-0.612968\pi\)
−0.347496 + 0.937682i \(0.612968\pi\)
\(80\) −355.000 −0.496128
\(81\) −671.000 −0.920439
\(82\) 18.0000 0.0242411
\(83\) −852.000 −1.12674 −0.563368 − 0.826206i \(-0.690495\pi\)
−0.563368 + 0.826206i \(0.690495\pi\)
\(84\) 110.000 0.142881
\(85\) 0 0
\(86\) −1614.00 −2.02375
\(87\) −345.000 −0.425148
\(88\) 1260.00 1.52632
\(89\) 1215.00 1.44708 0.723538 − 0.690285i \(-0.242515\pi\)
0.723538 + 0.690285i \(0.242515\pi\)
\(90\) −30.0000 −0.0351364
\(91\) −682.000 −0.785638
\(92\) 78.0000 0.0883920
\(93\) 155.000 0.172825
\(94\) −1395.00 −1.53067
\(95\) −305.000 −0.329393
\(96\) −225.000 −0.239208
\(97\) 601.000 0.629096 0.314548 − 0.949242i \(-0.398147\pi\)
0.314548 + 0.949242i \(0.398147\pi\)
\(98\) 423.000 0.436015
\(99\) 120.000 0.121823
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.g.1.1 1
17.16 even 2 85.4.a.b.1.1 ✓ 1
51.50 odd 2 765.4.a.c.1.1 1
68.67 odd 2 1360.4.a.g.1.1 1
85.33 odd 4 425.4.b.b.324.1 2
85.67 odd 4 425.4.b.b.324.2 2
85.84 even 2 425.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.b.1.1 ✓ 1 17.16 even 2
425.4.a.b.1.1 1 85.84 even 2
425.4.b.b.324.1 2 85.33 odd 4
425.4.b.b.324.2 2 85.67 odd 4
765.4.a.c.1.1 1 51.50 odd 2
1360.4.a.g.1.1 1 68.67 odd 2
1445.4.a.g.1.1 1 1.1 even 1 trivial