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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,2,Mod(2,145)] 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("145.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([7, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.o (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(13\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 113.10
Character \(\chi\) \(=\) 145.113
Dual form 145.2.o.a.77.10

$q$-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+(0.344569 + 1.50965i) q^{2} +(1.22921 - 2.55249i) q^{3} +(-0.358388 + 0.172591i) q^{4} +(-1.41763 - 1.72926i) q^{5} +(4.27692 + 0.976180i) q^{6} +(-1.66951 + 0.584188i) q^{7} +(1.54688 + 1.93972i) q^{8} +(-3.13377 - 3.92962i) q^{9} +(2.12211 - 2.73598i) q^{10} +(2.69639 - 0.303810i) q^{11} +1.12693i q^{12} +(0.424852 + 3.77067i) q^{13} +(-1.45718 - 2.31909i) q^{14} +(-6.15648 + 1.49286i) q^{15} +(-2.89133 + 3.62561i) q^{16} +2.23658 q^{17} +(4.85257 - 6.08493i) q^{18} +(-1.39983 + 4.00049i) q^{19} +(0.806515 + 0.375075i) q^{20} +(-0.561056 + 4.97951i) q^{21} +(1.38774 + 3.96593i) q^{22} +(-4.93689 + 3.10205i) q^{23} +(6.85257 - 1.56405i) q^{24} +(-0.980653 + 4.90289i) q^{25} +(-5.54601 + 1.94063i) q^{26} +(-5.59633 + 1.27733i) q^{27} +(0.497508 - 0.497508i) q^{28} +(-1.45500 - 5.18488i) q^{29} +(-4.37503 - 8.77976i) q^{30} +(3.69811 - 5.88551i) q^{31} +(-1.99907 - 0.962702i) q^{32} +(2.53897 - 7.25596i) q^{33} +(0.770656 + 3.37646i) q^{34} +(3.37696 + 2.05885i) q^{35} +(1.80132 + 0.867471i) q^{36} +(-2.74653 + 2.19028i) q^{37} +(-6.52169 - 0.734818i) q^{38} +(10.1468 + 3.55053i) q^{39} +(1.16138 - 5.42476i) q^{40} +(8.11700 - 8.11700i) q^{41} +(-7.71065 + 0.868782i) q^{42} +(-5.95142 - 1.35837i) q^{43} +(-0.913920 + 0.574254i) q^{44} +(-2.35280 + 10.9898i) q^{45} +(-6.38412 - 6.38412i) q^{46} +(-5.62099 - 4.48259i) q^{47} +(5.70028 + 11.8367i) q^{48} +(-3.02682 + 2.41381i) q^{49} +(-7.73956 + 0.208935i) q^{50} +(2.74924 - 5.70886i) q^{51} +(-0.803044 - 1.27804i) q^{52} +(-2.27196 + 3.61581i) q^{53} +(-3.85664 - 8.00840i) q^{54} +(-4.34785 - 4.23206i) q^{55} +(-3.71570 - 2.33473i) q^{56} +(8.49052 + 8.49052i) q^{57} +(7.32602 - 3.98310i) q^{58} -11.5100i q^{59} +(1.94876 - 1.59757i) q^{60} +(3.49977 + 10.0018i) q^{61} +(10.1593 + 3.55491i) q^{62} +(7.52750 + 4.72984i) q^{63} +(-1.29928 + 5.69251i) q^{64} +(5.91817 - 6.08009i) q^{65} +(11.8288 + 1.33279i) q^{66} +(0.188891 - 1.67645i) q^{67} +(-0.801565 + 0.386013i) q^{68} +(1.84946 + 16.4144i) q^{69} +(-1.94456 + 5.80746i) q^{70} +(-4.89761 - 3.90572i) q^{71} +(2.77482 - 12.1573i) q^{72} +(-1.63166 + 7.14875i) q^{73} +(-4.25294 - 3.39160i) q^{74} +(11.3091 + 8.52981i) q^{75} +(-0.188764 - 1.67533i) q^{76} +(-4.32418 + 2.08241i) q^{77} +(-1.86379 + 16.5416i) q^{78} +(4.08405 + 0.460162i) q^{79} +(10.3684 - 0.139926i) q^{80} +(-0.263444 + 1.15422i) q^{81} +(15.0507 + 9.45699i) q^{82} +(3.57990 + 1.25266i) q^{83} +(-0.658341 - 1.88143i) q^{84} +(-3.17065 - 3.86762i) q^{85} -9.45263i q^{86} +(-15.0229 - 2.65944i) q^{87} +(4.76030 + 4.76030i) q^{88} +(11.0910 + 6.96892i) q^{89} +(-17.4015 + 0.234841i) q^{90} +(-2.91207 - 6.04698i) q^{91} +(1.23394 - 1.96380i) q^{92} +(-10.4769 - 16.6740i) q^{93} +(4.83034 - 10.0303i) q^{94} +(8.90232 - 3.25055i) q^{95} +(-4.91458 + 3.91924i) q^{96} +(-5.52565 - 11.4741i) q^{97} +(-4.68696 - 3.73773i) q^{98} +(-9.64373 - 9.64373i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 156 q - 8 q^{2} - 14 q^{3} - 22 q^{4} - 14 q^{5} - 28 q^{6} - 10 q^{7} + 4 q^{8} + 10 q^{9} - 8 q^{10} - 20 q^{11} + 4 q^{14} - 24 q^{15} - 34 q^{16} - 48 q^{17} - 94 q^{18} + 6 q^{20} - 16 q^{21} - 6 q^{22}+ \cdots + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.344569 + 1.50965i 0.243647 + 1.06749i 0.937668 + 0.347532i \(0.112980\pi\)
−0.694021 + 0.719954i \(0.744163\pi\)
\(3\) 1.22921 − 2.55249i 0.709687 − 1.47368i −0.163626 − 0.986522i \(-0.552319\pi\)
0.873314 − 0.487158i \(-0.161967\pi\)
\(4\) −0.358388 + 0.172591i −0.179194 + 0.0862953i
\(5\) −1.41763 − 1.72926i −0.633983 − 0.773347i
\(6\) 4.27692 + 0.976180i 1.74605 + 0.398524i
\(7\) −1.66951 + 0.584188i −0.631017 + 0.220802i −0.626789 − 0.779189i \(-0.715631\pi\)
−0.00422731 + 0.999991i \(0.501346\pi\)
\(8\) 1.54688 + 1.93972i 0.546904 + 0.685796i
\(9\) −3.13377 − 3.92962i −1.04459 − 1.30987i
\(10\) 2.12211 − 2.73598i 0.671069 − 0.865192i
\(11\) 2.69639 − 0.303810i 0.812993 − 0.0916023i 0.304328 − 0.952567i \(-0.401568\pi\)
0.508664 + 0.860965i \(0.330139\pi\)
\(12\) 1.12693i 0.325318i
\(13\) 0.424852 + 3.77067i 0.117833 + 1.04579i 0.903927 + 0.427688i \(0.140672\pi\)
−0.786094 + 0.618107i \(0.787900\pi\)
\(14\) −1.45718 − 2.31909i −0.389448 − 0.619804i
\(15\) −6.15648 + 1.49286i −1.58960 + 0.385454i
\(16\) −2.89133 + 3.62561i −0.722832 + 0.906403i
\(17\) 2.23658 0.542451 0.271225 − 0.962516i \(-0.412571\pi\)
0.271225 + 0.962516i \(0.412571\pi\)
\(18\) 4.85257 − 6.08493i 1.14376 − 1.43423i
\(19\) −1.39983 + 4.00049i −0.321144 + 0.917776i 0.664254 + 0.747507i \(0.268749\pi\)
−0.985398 + 0.170269i \(0.945536\pi\)
\(20\) 0.806515 + 0.375075i 0.180342 + 0.0838694i
\(21\) −0.561056 + 4.97951i −0.122432 + 1.08662i
\(22\) 1.38774 + 3.96593i 0.295867 + 0.845540i
\(23\) −4.93689 + 3.10205i −1.02941 + 0.646822i −0.937155 − 0.348913i \(-0.886551\pi\)
−0.0922568 + 0.995735i \(0.529408\pi\)
\(24\) 6.85257 − 1.56405i 1.39877 − 0.319261i
\(25\) −0.980653 + 4.90289i −0.196131 + 0.980578i
\(26\) −5.54601 + 1.94063i −1.08766 + 0.380589i
\(27\) −5.59633 + 1.27733i −1.07701 + 0.245822i
\(28\) 0.497508 − 0.497508i 0.0940202 − 0.0940202i
\(29\) −1.45500 − 5.18488i −0.270188 − 0.962808i
\(30\) −4.37503 − 8.77976i −0.798767 − 1.60296i
\(31\) 3.69811 − 5.88551i 0.664201 − 1.05707i −0.329408 − 0.944188i \(-0.606849\pi\)
0.993608 − 0.112882i \(-0.0360081\pi\)
\(32\) −1.99907 − 0.962702i −0.353389 − 0.170183i
\(33\) 2.53897 − 7.25596i 0.441978 − 1.26310i
\(34\) 0.770656 + 3.37646i 0.132166 + 0.579059i
\(35\) 3.37696 + 2.05885i 0.570811 + 0.348010i
\(36\) 1.80132 + 0.867471i 0.300220 + 0.144578i
\(37\) −2.74653 + 2.19028i −0.451527 + 0.360080i −0.822693 − 0.568486i \(-0.807529\pi\)
0.371166 + 0.928566i \(0.378958\pi\)
\(38\) −6.52169 − 0.734818i −1.05796 − 0.119203i
\(39\) 10.1468 + 3.55053i 1.62479 + 0.568539i
\(40\) 1.16138 − 5.42476i 0.183630 − 0.857729i
\(41\) 8.11700 − 8.11700i 1.26766 − 1.26766i 0.320370 − 0.947293i \(-0.396193\pi\)
0.947293 − 0.320370i \(-0.103807\pi\)
\(42\) −7.71065 + 0.868782i −1.18978 + 0.134056i
\(43\) −5.95142 − 1.35837i −0.907583 − 0.207150i −0.256845 − 0.966453i \(-0.582683\pi\)
−0.650738 + 0.759303i \(0.725540\pi\)
\(44\) −0.913920 + 0.574254i −0.137779 + 0.0865720i
\(45\) −2.35280 + 10.9898i −0.350735 + 1.63827i
\(46\) −6.38412 − 6.38412i −0.941287 − 0.941287i
\(47\) −5.62099 − 4.48259i −0.819906 − 0.653853i 0.120950 − 0.992659i \(-0.461406\pi\)
−0.940857 + 0.338805i \(0.889977\pi\)
\(48\) 5.70028 + 11.8367i 0.822764 + 1.70849i
\(49\) −3.02682 + 2.41381i −0.432403 + 0.344830i
\(50\) −7.73956 + 0.208935i −1.09454 + 0.0295479i
\(51\) 2.74924 − 5.70886i 0.384971 − 0.799400i
\(52\) −0.803044 − 1.27804i −0.111362 − 0.177232i
\(53\) −2.27196 + 3.61581i −0.312078 + 0.496669i −0.965528 − 0.260301i \(-0.916178\pi\)
0.653450 + 0.756970i \(0.273321\pi\)
\(54\) −3.85664 − 8.00840i −0.524822 − 1.08980i
\(55\) −4.34785 − 4.23206i −0.586264 − 0.570651i
\(56\) −3.71570 − 2.33473i −0.496531 − 0.311991i
\(57\) 8.49052 + 8.49052i 1.12460 + 1.12460i
\(58\) 7.32602 − 3.98310i 0.961954 − 0.523007i
\(59\) − 11.5100i − 1.49848i −0.662301 − 0.749238i \(-0.730420\pi\)
0.662301 − 0.749238i \(-0.269580\pi\)
\(60\) 1.94876 − 1.59757i 0.251583 − 0.206246i
\(61\) 3.49977 + 10.0018i 0.448099 + 1.28059i 0.917545 + 0.397632i \(0.130168\pi\)
−0.469446 + 0.882961i \(0.655546\pi\)
\(62\) 10.1593 + 3.55491i 1.29024 + 0.451474i
\(63\) 7.52750 + 4.72984i 0.948376 + 0.595904i
\(64\) −1.29928 + 5.69251i −0.162410 + 0.711563i
\(65\) 5.91817 − 6.08009i 0.734058 − 0.754142i
\(66\) 11.8288 + 1.33279i 1.45603 + 0.164055i
\(67\) 0.188891 − 1.67645i 0.0230767 − 0.204811i −0.976883 − 0.213774i \(-0.931424\pi\)
0.999960 + 0.00896262i \(0.00285293\pi\)
\(68\) −0.801565 + 0.386013i −0.0972040 + 0.0468110i
\(69\) 1.84946 + 16.4144i 0.222649 + 1.97607i
\(70\) −1.94456 + 5.80746i −0.232419 + 0.694124i
\(71\) −4.89761 − 3.90572i −0.581240 − 0.463523i 0.288194 − 0.957572i \(-0.406945\pi\)
−0.869434 + 0.494049i \(0.835516\pi\)
\(72\) 2.77482 − 12.1573i 0.327016 − 1.43275i
\(73\) −1.63166 + 7.14875i −0.190971 + 0.836698i 0.785121 + 0.619342i \(0.212601\pi\)
−0.976092 + 0.217356i \(0.930257\pi\)
\(74\) −4.25294 − 3.39160i −0.494394 − 0.394266i
\(75\) 11.3091 + 8.52981i 1.30587 + 0.984938i
\(76\) −0.188764 − 1.67533i −0.0216527 − 0.192173i
\(77\) −4.32418 + 2.08241i −0.492786 + 0.237313i
\(78\) −1.86379 + 16.5416i −0.211033 + 1.87297i
\(79\) 4.08405 + 0.460162i 0.459492 + 0.0517723i 0.338677 − 0.940903i \(-0.390021\pi\)
0.120815 + 0.992675i \(0.461449\pi\)
\(80\) 10.3684 − 0.139926i 1.15923 − 0.0156442i
\(81\) −0.263444 + 1.15422i −0.0292715 + 0.128247i
\(82\) 15.0507 + 9.45699i 1.66207 + 1.04435i
\(83\) 3.57990 + 1.25266i 0.392945 + 0.137497i 0.519517 − 0.854460i \(-0.326112\pi\)
−0.126572 + 0.991957i \(0.540398\pi\)
\(84\) −0.658341 − 1.88143i −0.0718308 − 0.205281i
\(85\) −3.17065 − 3.86762i −0.343905 − 0.419503i
\(86\) − 9.45263i − 1.01930i
\(87\) −15.0229 − 2.65944i −1.61062 − 0.285122i
\(88\) 4.76030 + 4.76030i 0.507449 + 0.507449i
\(89\) 11.0910 + 6.96892i 1.17564 + 0.738704i 0.970753 − 0.240081i \(-0.0771741\pi\)
0.204888 + 0.978785i \(0.434317\pi\)
\(90\) −17.4015 + 0.234841i −1.83428 + 0.0247544i
\(91\) −2.91207 − 6.04698i −0.305268 − 0.633896i
\(92\) 1.23394 − 1.96380i 0.128647 − 0.204740i
\(93\) −10.4769 − 16.6740i −1.08641 − 1.72901i
\(94\) 4.83034 − 10.0303i 0.498212 − 1.03455i
\(95\) 8.90232 − 3.25055i 0.913359 − 0.333499i
\(96\) −4.91458 + 3.91924i −0.501592 + 0.400006i
\(97\) −5.52565 − 11.4741i −0.561045 − 1.16502i −0.967855 − 0.251509i \(-0.919073\pi\)
0.406810 − 0.913513i \(-0.366641\pi\)
\(98\) −4.68696 − 3.73773i −0.473455 − 0.377568i
\(99\) −9.64373 − 9.64373i −0.969231 − 0.969231i
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 145.2.o.a.113.10 yes 156
5.2 odd 4 145.2.t.a.142.4 yes 156
5.3 odd 4 725.2.bd.b.432.10 156
5.4 even 2 725.2.y.b.693.4 156
29.19 odd 28 145.2.t.a.48.4 yes 156
145.19 odd 28 725.2.bd.b.193.10 156
145.48 even 28 725.2.y.b.657.4 156
145.77 even 28 inner 145.2.o.a.77.10 ✓ 156
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.o.a.77.10 ✓ 156 145.77 even 28 inner
145.2.o.a.113.10 yes 156 1.1 even 1 trivial
145.2.t.a.48.4 yes 156 29.19 odd 28
145.2.t.a.142.4 yes 156 5.2 odd 4
725.2.y.b.657.4 156 145.48 even 28
725.2.y.b.693.4 156 5.4 even 2
725.2.bd.b.193.10 156 145.19 odd 28
725.2.bd.b.432.10 156 5.3 odd 4