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 1.6
Root \(-2.22172\) 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)
 
\(f(q)\) \(=\) \(q+3.22172 q^{2} -7.57054 q^{3} +2.37948 q^{4} -5.00000 q^{5} -24.3902 q^{6} +26.0836 q^{7} -18.1077 q^{8} +30.3130 q^{9} -16.1086 q^{10} -52.7821 q^{11} -18.0139 q^{12} +35.9814 q^{13} +84.0341 q^{14} +37.8527 q^{15} -77.3739 q^{16} +97.6601 q^{18} +37.6405 q^{19} -11.8974 q^{20} -197.467 q^{21} -170.049 q^{22} -169.894 q^{23} +137.085 q^{24} +25.0000 q^{25} +115.922 q^{26} -25.0816 q^{27} +62.0654 q^{28} -81.7265 q^{29} +121.951 q^{30} -179.682 q^{31} -104.415 q^{32} +399.589 q^{33} -130.418 q^{35} +72.1292 q^{36} +261.719 q^{37} +121.267 q^{38} -272.399 q^{39} +90.5387 q^{40} -366.907 q^{41} -636.184 q^{42} +284.695 q^{43} -125.594 q^{44} -151.565 q^{45} -547.351 q^{46} +417.036 q^{47} +585.762 q^{48} +337.356 q^{49} +80.5430 q^{50} +85.6170 q^{52} -580.839 q^{53} -80.8057 q^{54} +263.910 q^{55} -472.316 q^{56} -284.959 q^{57} -263.300 q^{58} -676.651 q^{59} +90.0696 q^{60} -321.754 q^{61} -578.887 q^{62} +790.674 q^{63} +282.595 q^{64} -179.907 q^{65} +1287.36 q^{66} -304.772 q^{67} +1286.19 q^{69} -420.171 q^{70} -73.8008 q^{71} -548.901 q^{72} +235.833 q^{73} +843.184 q^{74} -189.263 q^{75} +89.5647 q^{76} -1376.75 q^{77} -877.592 q^{78} +436.884 q^{79} +386.870 q^{80} -628.571 q^{81} -1182.07 q^{82} +1358.62 q^{83} -469.869 q^{84} +917.207 q^{86} +618.714 q^{87} +955.765 q^{88} +1418.60 q^{89} -488.301 q^{90} +938.526 q^{91} -404.259 q^{92} +1360.29 q^{93} +1343.57 q^{94} -188.202 q^{95} +790.478 q^{96} +152.268 q^{97} +1086.87 q^{98} -1599.99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.22172 1.13905 0.569525 − 0.821974i \(-0.307127\pi\)
0.569525 + 0.821974i \(0.307127\pi\)
\(3\) −7.57054 −1.45695 −0.728475 − 0.685072i \(-0.759771\pi\)
−0.728475 + 0.685072i \(0.759771\pi\)
\(4\) 2.37948 0.297435
\(5\) −5.00000 −0.447214
\(6\) −24.3902 −1.65954
\(7\) 26.0836 1.40838 0.704192 − 0.710010i \(-0.251310\pi\)
0.704192 + 0.710010i \(0.251310\pi\)
\(8\) −18.1077 −0.800257
\(9\) 30.3130 1.12271
\(10\) −16.1086 −0.509399
\(11\) −52.7821 −1.44676 −0.723382 − 0.690448i \(-0.757413\pi\)
−0.723382 + 0.690448i \(0.757413\pi\)
\(12\) −18.0139 −0.433348
\(13\) 35.9814 0.767650 0.383825 − 0.923406i \(-0.374607\pi\)
0.383825 + 0.923406i \(0.374607\pi\)
\(14\) 84.0341 1.60422
\(15\) 37.8527 0.651568
\(16\) −77.3739 −1.20897
\(17\) 0 0
\(18\) 97.6601 1.27882
\(19\) 37.6405 0.454490 0.227245 − 0.973838i \(-0.427028\pi\)
0.227245 + 0.973838i \(0.427028\pi\)
\(20\) −11.8974 −0.133017
\(21\) −197.467 −2.05195
\(22\) −170.049 −1.64794
\(23\) −169.894 −1.54023 −0.770116 − 0.637903i \(-0.779802\pi\)
−0.770116 + 0.637903i \(0.779802\pi\)
\(24\) 137.085 1.16593
\(25\) 25.0000 0.200000
\(26\) 115.922 0.874391
\(27\) −25.0816 −0.178776
\(28\) 62.0654 0.418902
\(29\) −81.7265 −0.523318 −0.261659 − 0.965160i \(-0.584270\pi\)
−0.261659 + 0.965160i \(0.584270\pi\)
\(30\) 121.951 0.742169
\(31\) −179.682 −1.04103 −0.520515 − 0.853853i \(-0.674260\pi\)
−0.520515 + 0.853853i \(0.674260\pi\)
\(32\) −104.415 −0.576817
\(33\) 399.589 2.10786
\(34\) 0 0
\(35\) −130.418 −0.629848
\(36\) 72.1292 0.333932
\(37\) 261.719 1.16287 0.581436 − 0.813592i \(-0.302491\pi\)
0.581436 + 0.813592i \(0.302491\pi\)
\(38\) 121.267 0.517687
\(39\) −272.399 −1.11843
\(40\) 90.5387 0.357886
\(41\) −366.907 −1.39759 −0.698795 − 0.715322i \(-0.746280\pi\)
−0.698795 + 0.715322i \(0.746280\pi\)
\(42\) −636.184 −2.33727
\(43\) 284.695 1.00966 0.504832 − 0.863217i \(-0.331554\pi\)
0.504832 + 0.863217i \(0.331554\pi\)
\(44\) −125.594 −0.430318
\(45\) −151.565 −0.502089
\(46\) −547.351 −1.75440
\(47\) 417.036 1.29428 0.647138 − 0.762373i \(-0.275966\pi\)
0.647138 + 0.762373i \(0.275966\pi\)
\(48\) 585.762 1.76141
\(49\) 337.356 0.983544
\(50\) 80.5430 0.227810
\(51\) 0 0
\(52\) 85.6170 0.228326
\(53\) −580.839 −1.50537 −0.752683 − 0.658383i \(-0.771241\pi\)
−0.752683 + 0.658383i \(0.771241\pi\)
\(54\) −80.8057 −0.203635
\(55\) 263.910 0.647012
\(56\) −472.316 −1.12707
\(57\) −284.959 −0.662170
\(58\) −263.300 −0.596086
\(59\) −676.651 −1.49309 −0.746546 − 0.665334i \(-0.768289\pi\)
−0.746546 + 0.665334i \(0.768289\pi\)
\(60\) 90.0696 0.193799
\(61\) −321.754 −0.675351 −0.337676 − 0.941263i \(-0.609641\pi\)
−0.337676 + 0.941263i \(0.609641\pi\)
\(62\) −578.887 −1.18578
\(63\) 790.674 1.58120
\(64\) 282.595 0.551944
\(65\) −179.907 −0.343303
\(66\) 1287.36 2.40096
\(67\) −304.772 −0.555729 −0.277865 − 0.960620i \(-0.589627\pi\)
−0.277865 + 0.960620i \(0.589627\pi\)
\(68\) 0 0
\(69\) 1286.19 2.24404
\(70\) −420.171 −0.717429
\(71\) −73.8008 −0.123360 −0.0616798 − 0.998096i \(-0.519646\pi\)
−0.0616798 + 0.998096i \(0.519646\pi\)
\(72\) −548.901 −0.898453
\(73\) 235.833 0.378111 0.189056 − 0.981966i \(-0.439457\pi\)
0.189056 + 0.981966i \(0.439457\pi\)
\(74\) 843.184 1.32457
\(75\) −189.263 −0.291390
\(76\) 89.5647 0.135181
\(77\) −1376.75 −2.03760
\(78\) −877.592 −1.27395
\(79\) 436.884 0.622194 0.311097 − 0.950378i \(-0.399304\pi\)
0.311097 + 0.950378i \(0.399304\pi\)
\(80\) 386.870 0.540667
\(81\) −628.571 −0.862238
\(82\) −1182.07 −1.59192
\(83\) 1358.62 1.79672 0.898360 − 0.439261i \(-0.144760\pi\)
0.898360 + 0.439261i \(0.144760\pi\)
\(84\) −469.869 −0.610320
\(85\) 0 0
\(86\) 917.207 1.15006
\(87\) 618.714 0.762449
\(88\) 955.765 1.15778
\(89\) 1418.60 1.68956 0.844782 − 0.535111i \(-0.179730\pi\)
0.844782 + 0.535111i \(0.179730\pi\)
\(90\) −488.301 −0.571905
\(91\) 938.526 1.08115
\(92\) −404.259 −0.458119
\(93\) 1360.29 1.51673
\(94\) 1343.57 1.47425
\(95\) −188.202 −0.203254
\(96\) 790.478 0.840394
\(97\) 152.268 0.159386 0.0796930 − 0.996819i \(-0.474606\pi\)
0.0796930 + 0.996819i \(0.474606\pi\)
\(98\) 1086.87 1.12031
\(99\) −1599.99 −1.62429
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.r.1.6 yes 8
17.16 even 2 1445.4.a.q.1.6 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.6 ✓ 8 17.16 even 2
1445.4.a.r.1.6 yes 8 1.1 even 1 trivial