Properties

Label 1445.4.a.u.1.19
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 1.19
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)
 
\(f(q)\) \(=\) \(q+4.56903 q^{2} -8.01532 q^{3} +12.8760 q^{4} +5.00000 q^{5} -36.6222 q^{6} +30.9023 q^{7} +22.2787 q^{8} +37.2454 q^{9} +22.8451 q^{10} +7.24233 q^{11} -103.205 q^{12} -88.1077 q^{13} +141.194 q^{14} -40.0766 q^{15} -1.21616 q^{16} +170.175 q^{18} -105.124 q^{19} +64.3801 q^{20} -247.692 q^{21} +33.0904 q^{22} -178.726 q^{23} -178.571 q^{24} +25.0000 q^{25} -402.567 q^{26} -82.1199 q^{27} +397.899 q^{28} +117.017 q^{29} -183.111 q^{30} +163.623 q^{31} -183.786 q^{32} -58.0496 q^{33} +154.512 q^{35} +479.572 q^{36} -217.628 q^{37} -480.316 q^{38} +706.212 q^{39} +111.394 q^{40} -194.788 q^{41} -1131.71 q^{42} -171.189 q^{43} +93.2525 q^{44} +186.227 q^{45} -816.602 q^{46} +322.604 q^{47} +9.74791 q^{48} +611.954 q^{49} +114.226 q^{50} -1134.48 q^{52} -31.6777 q^{53} -375.208 q^{54} +36.2117 q^{55} +688.464 q^{56} +842.606 q^{57} +534.653 q^{58} -123.752 q^{59} -516.027 q^{60} +66.9539 q^{61} +747.600 q^{62} +1150.97 q^{63} -829.996 q^{64} -440.539 q^{65} -265.230 q^{66} -261.016 q^{67} +1432.54 q^{69} +705.968 q^{70} -264.991 q^{71} +829.779 q^{72} -649.025 q^{73} -994.348 q^{74} -200.383 q^{75} -1353.58 q^{76} +223.805 q^{77} +3226.70 q^{78} -250.922 q^{79} -6.08080 q^{80} -347.407 q^{81} -889.991 q^{82} -1079.12 q^{83} -3189.29 q^{84} -782.169 q^{86} -937.927 q^{87} +161.350 q^{88} +516.445 q^{89} +850.876 q^{90} -2722.73 q^{91} -2301.27 q^{92} -1311.49 q^{93} +1473.99 q^{94} -525.622 q^{95} +1473.11 q^{96} +1011.13 q^{97} +2796.04 q^{98} +269.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - 6 q^{2} - 18 q^{3} + 60 q^{4} + 105 q^{5} + 45 q^{6} - 42 q^{7} - 21 q^{8} + 135 q^{9} - 30 q^{10} - 30 q^{11} - 216 q^{12} - 132 q^{13} + 162 q^{14} - 90 q^{15} + 48 q^{16} - 126 q^{18} - 405 q^{19}+ \cdots + 9066 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.56903 1.61540 0.807698 − 0.589597i \(-0.200713\pi\)
0.807698 + 0.589597i \(0.200713\pi\)
\(3\) −8.01532 −1.54255 −0.771275 − 0.636503i \(-0.780380\pi\)
−0.771275 + 0.636503i \(0.780380\pi\)
\(4\) 12.8760 1.60950
\(5\) 5.00000 0.447214
\(6\) −36.6222 −2.49183
\(7\) 30.9023 1.66857 0.834285 − 0.551334i \(-0.185881\pi\)
0.834285 + 0.551334i \(0.185881\pi\)
\(8\) 22.2787 0.984589
\(9\) 37.2454 1.37946
\(10\) 22.8451 0.722427
\(11\) 7.24233 0.198513 0.0992566 − 0.995062i \(-0.468354\pi\)
0.0992566 + 0.995062i \(0.468354\pi\)
\(12\) −103.205 −2.48274
\(13\) −88.1077 −1.87974 −0.939872 − 0.341526i \(-0.889056\pi\)
−0.939872 + 0.341526i \(0.889056\pi\)
\(14\) 141.194 2.69540
\(15\) −40.0766 −0.689849
\(16\) −1.21616 −0.0190025
\(17\) 0 0
\(18\) 170.175 2.22837
\(19\) −105.124 −1.26933 −0.634663 − 0.772789i \(-0.718861\pi\)
−0.634663 + 0.772789i \(0.718861\pi\)
\(20\) 64.3801 0.719792
\(21\) −247.692 −2.57385
\(22\) 33.0904 0.320677
\(23\) −178.726 −1.62030 −0.810149 − 0.586224i \(-0.800614\pi\)
−0.810149 + 0.586224i \(0.800614\pi\)
\(24\) −178.571 −1.51878
\(25\) 25.0000 0.200000
\(26\) −402.567 −3.03653
\(27\) −82.1199 −0.585333
\(28\) 397.899 2.68557
\(29\) 117.017 0.749292 0.374646 − 0.927168i \(-0.377764\pi\)
0.374646 + 0.927168i \(0.377764\pi\)
\(30\) −183.111 −1.11438
\(31\) 163.623 0.947988 0.473994 − 0.880528i \(-0.342812\pi\)
0.473994 + 0.880528i \(0.342812\pi\)
\(32\) −183.786 −1.01529
\(33\) −58.0496 −0.306216
\(34\) 0 0
\(35\) 154.512 0.746207
\(36\) 479.572 2.22024
\(37\) −217.628 −0.966968 −0.483484 − 0.875353i \(-0.660629\pi\)
−0.483484 + 0.875353i \(0.660629\pi\)
\(38\) −480.316 −2.05046
\(39\) 706.212 2.89960
\(40\) 111.394 0.440322
\(41\) −194.788 −0.741970 −0.370985 − 0.928639i \(-0.620980\pi\)
−0.370985 + 0.928639i \(0.620980\pi\)
\(42\) −1131.71 −4.15779
\(43\) −171.189 −0.607119 −0.303560 − 0.952812i \(-0.598175\pi\)
−0.303560 + 0.952812i \(0.598175\pi\)
\(44\) 93.2525 0.319508
\(45\) 186.227 0.616912
\(46\) −816.602 −2.61742
\(47\) 322.604 1.00120 0.500602 − 0.865678i \(-0.333112\pi\)
0.500602 + 0.865678i \(0.333112\pi\)
\(48\) 9.74791 0.0293123
\(49\) 611.954 1.78412
\(50\) 114.226 0.323079
\(51\) 0 0
\(52\) −1134.48 −3.02546
\(53\) −31.6777 −0.0820994 −0.0410497 − 0.999157i \(-0.513070\pi\)
−0.0410497 + 0.999157i \(0.513070\pi\)
\(54\) −375.208 −0.945544
\(55\) 36.2117 0.0887778
\(56\) 688.464 1.64286
\(57\) 842.606 1.95800
\(58\) 534.653 1.21040
\(59\) −123.752 −0.273070 −0.136535 − 0.990635i \(-0.543597\pi\)
−0.136535 + 0.990635i \(0.543597\pi\)
\(60\) −516.027 −1.11031
\(61\) 66.9539 0.140534 0.0702669 − 0.997528i \(-0.477615\pi\)
0.0702669 + 0.997528i \(0.477615\pi\)
\(62\) 747.600 1.53138
\(63\) 1150.97 2.30172
\(64\) −829.996 −1.62109
\(65\) −440.539 −0.840647
\(66\) −265.230 −0.494661
\(67\) −261.016 −0.475943 −0.237971 − 0.971272i \(-0.576482\pi\)
−0.237971 + 0.971272i \(0.576482\pi\)
\(68\) 0 0
\(69\) 1432.54 2.49939
\(70\) 705.968 1.20542
\(71\) −264.991 −0.442940 −0.221470 − 0.975167i \(-0.571085\pi\)
−0.221470 + 0.975167i \(0.571085\pi\)
\(72\) 829.779 1.35820
\(73\) −649.025 −1.04058 −0.520292 − 0.853989i \(-0.674177\pi\)
−0.520292 + 0.853989i \(0.674177\pi\)
\(74\) −994.348 −1.56204
\(75\) −200.383 −0.308510
\(76\) −1353.58 −2.04298
\(77\) 223.805 0.331233
\(78\) 3226.70 4.68400
\(79\) −250.922 −0.357354 −0.178677 − 0.983908i \(-0.557182\pi\)
−0.178677 + 0.983908i \(0.557182\pi\)
\(80\) −6.08080 −0.00849818
\(81\) −347.407 −0.476553
\(82\) −889.991 −1.19857
\(83\) −1079.12 −1.42709 −0.713547 − 0.700607i \(-0.752912\pi\)
−0.713547 + 0.700607i \(0.752912\pi\)
\(84\) −3189.29 −4.14262
\(85\) 0 0
\(86\) −782.169 −0.980738
\(87\) −937.927 −1.15582
\(88\) 161.350 0.195454
\(89\) 516.445 0.615090 0.307545 − 0.951534i \(-0.400493\pi\)
0.307545 + 0.951534i \(0.400493\pi\)
\(90\) 850.876 0.996558
\(91\) −2722.73 −3.13648
\(92\) −2301.27 −2.60787
\(93\) −1311.49 −1.46232
\(94\) 1473.99 1.61734
\(95\) −525.622 −0.567660
\(96\) 1473.11 1.56613
\(97\) 1011.13 1.05839 0.529197 − 0.848499i \(-0.322493\pi\)
0.529197 + 0.848499i \(0.322493\pi\)
\(98\) 2796.04 2.88207
\(99\) 269.743 0.273841
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.1.19 ✓ 21
17.16 even 2 1445.4.a.v.1.19 yes 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.u.1.19 ✓ 21 1.1 even 1 trivial
1445.4.a.v.1.19 yes 21 17.16 even 2