Properties

Label 1445.4.a.u.1.20
Level $1445$
Weight $4$
Character 1445.1
Self dual yes
Analytic conductor $85.258$
Analytic rank $1$
Dimension $21$
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 = [21,-6,-18,60,105] 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: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
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)
 
\(f(q)\) \(=\) \(q+4.66766 q^{2} +1.54092 q^{3} +13.7871 q^{4} +5.00000 q^{5} +7.19249 q^{6} -13.8059 q^{7} +27.0121 q^{8} -24.6256 q^{9} +23.3383 q^{10} -13.0024 q^{11} +21.2448 q^{12} -27.9728 q^{13} -64.4415 q^{14} +7.70459 q^{15} +15.7869 q^{16} -114.944 q^{18} -22.1833 q^{19} +68.9354 q^{20} -21.2738 q^{21} -60.6907 q^{22} +42.6199 q^{23} +41.6235 q^{24} +25.0000 q^{25} -130.568 q^{26} -79.5508 q^{27} -190.344 q^{28} +122.952 q^{29} +35.9624 q^{30} -262.933 q^{31} -142.409 q^{32} -20.0356 q^{33} -69.0297 q^{35} -339.515 q^{36} -51.7682 q^{37} -103.544 q^{38} -43.1038 q^{39} +135.061 q^{40} +398.693 q^{41} -99.2991 q^{42} -438.916 q^{43} -179.265 q^{44} -123.128 q^{45} +198.935 q^{46} -299.505 q^{47} +24.3264 q^{48} -152.396 q^{49} +116.692 q^{50} -385.663 q^{52} -99.4462 q^{53} -371.316 q^{54} -65.0118 q^{55} -372.928 q^{56} -34.1827 q^{57} +573.897 q^{58} -681.257 q^{59} +106.224 q^{60} +427.816 q^{61} -1227.28 q^{62} +339.979 q^{63} -791.013 q^{64} -139.864 q^{65} -93.5194 q^{66} +67.4700 q^{67} +65.6737 q^{69} -322.207 q^{70} -715.168 q^{71} -665.189 q^{72} -365.675 q^{73} -241.637 q^{74} +38.5230 q^{75} -305.843 q^{76} +179.510 q^{77} -201.194 q^{78} -55.7752 q^{79} +78.9346 q^{80} +542.309 q^{81} +1860.96 q^{82} +162.398 q^{83} -293.304 q^{84} -2048.71 q^{86} +189.459 q^{87} -351.222 q^{88} +1324.52 q^{89} -574.719 q^{90} +386.191 q^{91} +587.603 q^{92} -405.158 q^{93} -1397.99 q^{94} -110.917 q^{95} -219.441 q^{96} +1300.59 q^{97} -711.333 q^{98} +320.191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - 6 q^{2} - 18 q^{3} + 60 q^{4} + 105 q^{5} + 45 q^{6} - 42 q^{7} - 21 q^{8} + 135 q^{9} - 30 q^{10} - 30 q^{11} - 216 q^{12} - 132 q^{13} + 162 q^{14} - 90 q^{15} + 48 q^{16} - 126 q^{18} - 405 q^{19}+ \cdots + 9066 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.66766 1.65027 0.825134 − 0.564937i \(-0.191100\pi\)
0.825134 + 0.564937i \(0.191100\pi\)
\(3\) 1.54092 0.296550 0.148275 − 0.988946i \(-0.452628\pi\)
0.148275 + 0.988946i \(0.452628\pi\)
\(4\) 13.7871 1.72338
\(5\) 5.00000 0.447214
\(6\) 7.19249 0.489387
\(7\) −13.8059 −0.745451 −0.372725 − 0.927942i \(-0.621577\pi\)
−0.372725 + 0.927942i \(0.621577\pi\)
\(8\) 27.0121 1.19378
\(9\) −24.6256 −0.912058
\(10\) 23.3383 0.738022
\(11\) −13.0024 −0.356396 −0.178198 − 0.983995i \(-0.557027\pi\)
−0.178198 + 0.983995i \(0.557027\pi\)
\(12\) 21.2448 0.511070
\(13\) −27.9728 −0.596789 −0.298394 − 0.954443i \(-0.596451\pi\)
−0.298394 + 0.954443i \(0.596451\pi\)
\(14\) −64.4415 −1.23019
\(15\) 7.70459 0.132621
\(16\) 15.7869 0.246671
\(17\) 0 0
\(18\) −114.944 −1.50514
\(19\) −22.1833 −0.267853 −0.133927 − 0.990991i \(-0.542759\pi\)
−0.133927 + 0.990991i \(0.542759\pi\)
\(20\) 68.9354 0.770721
\(21\) −21.2738 −0.221063
\(22\) −60.6907 −0.588150
\(23\) 42.6199 0.386385 0.193192 − 0.981161i \(-0.438116\pi\)
0.193192 + 0.981161i \(0.438116\pi\)
\(24\) 41.6235 0.354015
\(25\) 25.0000 0.200000
\(26\) −130.568 −0.984862
\(27\) −79.5508 −0.567021
\(28\) −190.344 −1.28470
\(29\) 122.952 0.787295 0.393648 − 0.919261i \(-0.371213\pi\)
0.393648 + 0.919261i \(0.371213\pi\)
\(30\) 35.9624 0.218860
\(31\) −262.933 −1.52336 −0.761680 − 0.647954i \(-0.775625\pi\)
−0.761680 + 0.647954i \(0.775625\pi\)
\(32\) −142.409 −0.786706
\(33\) −20.0356 −0.105689
\(34\) 0 0
\(35\) −69.0297 −0.333376
\(36\) −339.515 −1.57183
\(37\) −51.7682 −0.230017 −0.115009 − 0.993364i \(-0.536690\pi\)
−0.115009 + 0.993364i \(0.536690\pi\)
\(38\) −103.544 −0.442029
\(39\) −43.1038 −0.176978
\(40\) 135.061 0.533874
\(41\) 398.693 1.51867 0.759334 − 0.650701i \(-0.225525\pi\)
0.759334 + 0.650701i \(0.225525\pi\)
\(42\) −99.2991 −0.364814
\(43\) −438.916 −1.55661 −0.778304 − 0.627888i \(-0.783920\pi\)
−0.778304 + 0.627888i \(0.783920\pi\)
\(44\) −179.265 −0.614208
\(45\) −123.128 −0.407885
\(46\) 198.935 0.637639
\(47\) −299.505 −0.929517 −0.464759 − 0.885437i \(-0.653859\pi\)
−0.464759 + 0.885437i \(0.653859\pi\)
\(48\) 24.3264 0.0731502
\(49\) −152.396 −0.444303
\(50\) 116.692 0.330054
\(51\) 0 0
\(52\) −385.663 −1.02850
\(53\) −99.4462 −0.257736 −0.128868 − 0.991662i \(-0.541134\pi\)
−0.128868 + 0.991662i \(0.541134\pi\)
\(54\) −371.316 −0.935736
\(55\) −65.0118 −0.159385
\(56\) −372.928 −0.889903
\(57\) −34.1827 −0.0794318
\(58\) 573.897 1.29925
\(59\) −681.257 −1.50326 −0.751628 − 0.659587i \(-0.770731\pi\)
−0.751628 + 0.659587i \(0.770731\pi\)
\(60\) 106.224 0.228557
\(61\) 427.816 0.897972 0.448986 − 0.893539i \(-0.351785\pi\)
0.448986 + 0.893539i \(0.351785\pi\)
\(62\) −1227.28 −2.51395
\(63\) 339.979 0.679894
\(64\) −791.013 −1.54495
\(65\) −139.864 −0.266892
\(66\) −93.5194 −0.174416
\(67\) 67.4700 0.123027 0.0615133 − 0.998106i \(-0.480407\pi\)
0.0615133 + 0.998106i \(0.480407\pi\)
\(68\) 0 0
\(69\) 65.6737 0.114582
\(70\) −322.207 −0.550159
\(71\) −715.168 −1.19542 −0.597710 − 0.801713i \(-0.703922\pi\)
−0.597710 + 0.801713i \(0.703922\pi\)
\(72\) −665.189 −1.08880
\(73\) −365.675 −0.586289 −0.293144 − 0.956068i \(-0.594702\pi\)
−0.293144 + 0.956068i \(0.594702\pi\)
\(74\) −241.637 −0.379590
\(75\) 38.5230 0.0593100
\(76\) −305.843 −0.461614
\(77\) 179.510 0.265676
\(78\) −201.194 −0.292061
\(79\) −55.7752 −0.0794329 −0.0397165 − 0.999211i \(-0.512645\pi\)
−0.0397165 + 0.999211i \(0.512645\pi\)
\(80\) 78.9346 0.110314
\(81\) 542.309 0.743908
\(82\) 1860.96 2.50621
\(83\) 162.398 0.214765 0.107382 − 0.994218i \(-0.465753\pi\)
0.107382 + 0.994218i \(0.465753\pi\)
\(84\) −293.304 −0.380977
\(85\) 0 0
\(86\) −2048.71 −2.56882
\(87\) 189.459 0.233472
\(88\) −351.222 −0.425459
\(89\) 1324.52 1.57752 0.788758 − 0.614704i \(-0.210725\pi\)
0.788758 + 0.614704i \(0.210725\pi\)
\(90\) −574.719 −0.673119
\(91\) 386.191 0.444877
\(92\) 587.603 0.665890
\(93\) −405.158 −0.451752
\(94\) −1397.99 −1.53395
\(95\) −110.917 −0.119788
\(96\) −219.441 −0.233298
\(97\) 1300.59 1.36140 0.680698 − 0.732564i \(-0.261677\pi\)
0.680698 + 0.732564i \(0.261677\pi\)
\(98\) −711.333 −0.733220
\(99\) 320.191 0.325054
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.u.1.20 ✓ 21
17.16 even 2 1445.4.a.v.1.20 yes 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.u.1.20 ✓ 21 1.1 even 1 trivial
1445.4.a.v.1.20 yes 21 17.16 even 2