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Properties

Label 1445.4.a.e.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,-1,8,-7,5] 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +8.00000 q^{3} -7.00000 q^{4} +5.00000 q^{5} -8.00000 q^{6} +14.0000 q^{7} +15.0000 q^{8} +37.0000 q^{9} -5.00000 q^{10} -20.0000 q^{11} -56.0000 q^{12} -58.0000 q^{13} -14.0000 q^{14} +40.0000 q^{15} +41.0000 q^{16} -37.0000 q^{18} -80.0000 q^{19} -35.0000 q^{20} +112.000 q^{21} +20.0000 q^{22} -118.000 q^{23} +120.000 q^{24} +25.0000 q^{25} +58.0000 q^{26} +80.0000 q^{27} -98.0000 q^{28} +126.000 q^{29} -40.0000 q^{30} +70.0000 q^{31} -161.000 q^{32} -160.000 q^{33} +70.0000 q^{35} -259.000 q^{36} -134.000 q^{37} +80.0000 q^{38} -464.000 q^{39} +75.0000 q^{40} +100.000 q^{41} -112.000 q^{42} -272.000 q^{43} +140.000 q^{44} +185.000 q^{45} +118.000 q^{46} -464.000 q^{47} +328.000 q^{48} -147.000 q^{49} -25.0000 q^{50} +406.000 q^{52} -642.000 q^{53} -80.0000 q^{54} -100.000 q^{55} +210.000 q^{56} -640.000 q^{57} -126.000 q^{58} +180.000 q^{59} -280.000 q^{60} -110.000 q^{61} -70.0000 q^{62} +518.000 q^{63} -167.000 q^{64} -290.000 q^{65} +160.000 q^{66} -924.000 q^{67} -944.000 q^{69} -70.0000 q^{70} +90.0000 q^{71} +555.000 q^{72} +828.000 q^{73} +134.000 q^{74} +200.000 q^{75} +560.000 q^{76} -280.000 q^{77} +464.000 q^{78} +1334.00 q^{79} +205.000 q^{80} -359.000 q^{81} -100.000 q^{82} -552.000 q^{83} -784.000 q^{84} +272.000 q^{86} +1008.00 q^{87} -300.000 q^{88} +1490.00 q^{89} -185.000 q^{90} -812.000 q^{91} +826.000 q^{92} +560.000 q^{93} +464.000 q^{94} -400.000 q^{95} -1288.00 q^{96} +1376.00 q^{97} +147.000 q^{98} -740.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 − 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 8.00000 1.53960 0.769800 − 0.638285i \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) −7.00000 −0.875000
\(5\) 5.00000 0.447214
\(6\) −8.00000 −0.544331
\(7\) 14.0000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 15.0000 0.662913
\(9\) 37.0000 1.37037
\(10\) −5.00000 −0.158114
\(11\) −20.0000 −0.548202 −0.274101 − 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) −56.0000 −1.34715
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) −14.0000 −0.267261
\(15\) 40.0000 0.688530
\(16\) 41.0000 0.640625
\(17\) 0 0
\(18\) −37.0000 −0.484499
\(19\) −80.0000 −0.965961 −0.482980 − 0.875631i \(-0.660446\pi\)
−0.482980 + 0.875631i \(0.660446\pi\)
\(20\) −35.0000 −0.391312
\(21\) 112.000 1.16383
\(22\) 20.0000 0.193819
\(23\) −118.000 −1.06977 −0.534885 − 0.844925i \(-0.679645\pi\)
−0.534885 + 0.844925i \(0.679645\pi\)
\(24\) 120.000 1.02062
\(25\) 25.0000 0.200000
\(26\) 58.0000 0.437490
\(27\) 80.0000 0.570222
\(28\) −98.0000 −0.661438
\(29\) 126.000 0.806814 0.403407 − 0.915021i \(-0.367826\pi\)
0.403407 + 0.915021i \(0.367826\pi\)
\(30\) −40.0000 −0.243432
\(31\) 70.0000 0.405560 0.202780 − 0.979224i \(-0.435002\pi\)
0.202780 + 0.979224i \(0.435002\pi\)
\(32\) −161.000 −0.889408
\(33\) −160.000 −0.844013
\(34\) 0 0
\(35\) 70.0000 0.338062
\(36\) −259.000 −1.19907
\(37\) −134.000 −0.595391 −0.297695 − 0.954661i \(-0.596218\pi\)
−0.297695 + 0.954661i \(0.596218\pi\)
\(38\) 80.0000 0.341519
\(39\) −464.000 −1.90511
\(40\) 75.0000 0.296464
\(41\) 100.000 0.380912 0.190456 − 0.981696i \(-0.439003\pi\)
0.190456 + 0.981696i \(0.439003\pi\)
\(42\) −112.000 −0.411476
\(43\) −272.000 −0.964642 −0.482321 − 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(44\) 140.000 0.479677
\(45\) 185.000 0.612848
\(46\) 118.000 0.378221
\(47\) −464.000 −1.44003 −0.720014 − 0.693959i \(-0.755865\pi\)
−0.720014 + 0.693959i \(0.755865\pi\)
\(48\) 328.000 0.986307
\(49\) −147.000 −0.428571
\(50\) −25.0000 −0.0707107
\(51\) 0 0
\(52\) 406.000 1.08273
\(53\) −642.000 −1.66388 −0.831939 − 0.554868i \(-0.812769\pi\)
−0.831939 + 0.554868i \(0.812769\pi\)
\(54\) −80.0000 −0.201604
\(55\) −100.000 −0.245164
\(56\) 210.000 0.501115
\(57\) −640.000 −1.48719
\(58\) −126.000 −0.285252
\(59\) 180.000 0.397187 0.198593 − 0.980082i \(-0.436363\pi\)
0.198593 + 0.980082i \(0.436363\pi\)
\(60\) −280.000 −0.602464
\(61\) −110.000 −0.230886 −0.115443 − 0.993314i \(-0.536829\pi\)
−0.115443 + 0.993314i \(0.536829\pi\)
\(62\) −70.0000 −0.143387
\(63\) 518.000 1.03590
\(64\) −167.000 −0.326172
\(65\) −290.000 −0.553386
\(66\) 160.000 0.298404
\(67\) −924.000 −1.68484 −0.842422 − 0.538818i \(-0.818871\pi\)
−0.842422 + 0.538818i \(0.818871\pi\)
\(68\) 0 0
\(69\) −944.000 −1.64702
\(70\) −70.0000 −0.119523
\(71\) 90.0000 0.150437 0.0752186 − 0.997167i \(-0.476035\pi\)
0.0752186 + 0.997167i \(0.476035\pi\)
\(72\) 555.000 0.908436
\(73\) 828.000 1.32754 0.663768 − 0.747939i \(-0.268956\pi\)
0.663768 + 0.747939i \(0.268956\pi\)
\(74\) 134.000 0.210502
\(75\) 200.000 0.307920
\(76\) 560.000 0.845216
\(77\) −280.000 −0.414402
\(78\) 464.000 0.673560
\(79\) 1334.00 1.89983 0.949916 − 0.312505i \(-0.101168\pi\)
0.949916 + 0.312505i \(0.101168\pi\)
\(80\) 205.000 0.286496
\(81\) −359.000 −0.492455
\(82\) −100.000 −0.134673
\(83\) −552.000 −0.729998 −0.364999 − 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) −784.000 −1.01835
\(85\) 0 0
\(86\) 272.000 0.341052
\(87\) 1008.00 1.24217
\(88\) −300.000 −0.363410
\(89\) 1490.00 1.77460 0.887302 − 0.461190i \(-0.152577\pi\)
0.887302 + 0.461190i \(0.152577\pi\)
\(90\) −185.000 −0.216675
\(91\) −812.000 −0.935393
\(92\) 826.000 0.936048
\(93\) 560.000 0.624401
\(94\) 464.000 0.509127
\(95\) −400.000 −0.431991
\(96\) −1288.00 −1.36933
\(97\) 1376.00 1.44033 0.720163 − 0.693805i \(-0.244067\pi\)
0.720163 + 0.693805i \(0.244067\pi\)
\(98\) 147.000 0.151523
\(99\) −740.000 −0.751240
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.e.1.1 1
17.4 even 4 85.4.d.a.16.1 ✓ 2
17.13 even 4 85.4.d.a.16.2 yes 2
17.16 even 2 1445.4.a.d.1.1 1
51.38 odd 4 765.4.g.a.271.2 2
51.47 odd 4 765.4.g.a.271.1 2
85.4 even 4 425.4.d.a.101.2 2
85.13 odd 4 425.4.c.a.424.1 2
85.38 odd 4 425.4.c.b.424.1 2
85.47 odd 4 425.4.c.b.424.2 2
85.64 even 4 425.4.d.a.101.1 2
85.72 odd 4 425.4.c.a.424.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.d.a.16.1 ✓ 2 17.4 even 4
85.4.d.a.16.2 yes 2 17.13 even 4
425.4.c.a.424.1 2 85.13 odd 4
425.4.c.a.424.2 2 85.72 odd 4
425.4.c.b.424.1 2 85.38 odd 4
425.4.c.b.424.2 2 85.47 odd 4
425.4.d.a.101.1 2 85.64 even 4
425.4.d.a.101.2 2 85.4 even 4
765.4.g.a.271.1 2 51.47 odd 4
765.4.g.a.271.2 2 51.38 odd 4
1445.4.a.d.1.1 1 17.16 even 2
1445.4.a.e.1.1 1 1.1 even 1 trivial