Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,2,Mod(12,145)] 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("145.12"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.e (of order \(4\), degree \(2\), 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: \(26\)
Relative dimension: \(13\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 133.4
Character \(\chi\) \(=\) 145.133
Dual form 145.2.e.a.12.10

$q$-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-1.26373i q^{2} -0.913274 q^{3} +0.402981 q^{4} +(-1.78577 - 1.34575i) q^{5} +1.15413i q^{6} +(2.03055 - 2.03055i) q^{7} -3.03672i q^{8} -2.16593 q^{9} +(-1.70067 + 2.25673i) q^{10} +(-1.13644 + 1.13644i) q^{11} -0.368032 q^{12} +(3.45320 - 3.45320i) q^{13} +(-2.56607 - 2.56607i) q^{14} +(1.63089 + 1.22904i) q^{15} -3.03165 q^{16} +4.10729i q^{17} +2.73716i q^{18} +(1.88844 + 1.88844i) q^{19} +(-0.719629 - 0.542311i) q^{20} +(-1.85445 + 1.85445i) q^{21} +(1.43615 + 1.43615i) q^{22} +(0.0950030 + 0.0950030i) q^{23} +2.77336i q^{24} +(1.37792 + 4.80639i) q^{25} +(-4.36392 - 4.36392i) q^{26} +4.71791 q^{27} +(0.818271 - 0.818271i) q^{28} +(4.16339 + 3.41558i) q^{29} +(1.55318 - 2.06101i) q^{30} +(3.63014 - 3.63014i) q^{31} -2.24226i q^{32} +(1.03788 - 1.03788i) q^{33} +5.19051 q^{34} +(-6.35869 + 0.893474i) q^{35} -0.872828 q^{36} +2.07797 q^{37} +(2.38649 - 2.38649i) q^{38} +(-3.15372 + 3.15372i) q^{39} +(-4.08667 + 5.42288i) q^{40} +(6.10064 + 6.10064i) q^{41} +(2.34352 + 2.34352i) q^{42} -9.37469 q^{43} +(-0.457962 + 0.457962i) q^{44} +(3.86784 + 2.91480i) q^{45} +(0.120058 - 0.120058i) q^{46} +3.01418 q^{47} +2.76872 q^{48} -1.24624i q^{49} +(6.07398 - 1.74132i) q^{50} -3.75108i q^{51} +(1.39157 - 1.39157i) q^{52} +(-5.40841 - 5.40841i) q^{53} -5.96218i q^{54} +(3.55877 - 0.500051i) q^{55} +(-6.16621 - 6.16621i) q^{56} +(-1.72467 - 1.72467i) q^{57} +(4.31637 - 5.26141i) q^{58} +4.15848i q^{59} +(0.657219 + 0.495279i) q^{60} +(-2.53415 + 2.53415i) q^{61} +(-4.58752 - 4.58752i) q^{62} +(-4.39802 + 4.39802i) q^{63} -8.89691 q^{64} +(-10.8137 + 1.51947i) q^{65} +(-1.31160 - 1.31160i) q^{66} +(-3.18760 - 3.18760i) q^{67} +1.65516i q^{68} +(-0.0867638 - 0.0867638i) q^{69} +(1.12911 + 8.03568i) q^{70} -11.5554i q^{71} +6.57733i q^{72} +14.4237i q^{73} -2.62599i q^{74} +(-1.25842 - 4.38955i) q^{75} +(0.761006 + 0.761006i) q^{76} +4.61517i q^{77} +(3.98546 + 3.98546i) q^{78} +(-7.72601 - 7.72601i) q^{79} +(5.41381 + 4.07983i) q^{80} +2.18904 q^{81} +(7.70957 - 7.70957i) q^{82} +(-11.0085 - 11.0085i) q^{83} +(-0.747306 + 0.747306i) q^{84} +(5.52738 - 7.33465i) q^{85} +11.8471i q^{86} +(-3.80232 - 3.11936i) q^{87} +(3.45104 + 3.45104i) q^{88} +(9.63459 + 9.63459i) q^{89} +(3.68352 - 4.88792i) q^{90} -14.0238i q^{91} +(0.0382844 + 0.0382844i) q^{92} +(-3.31531 + 3.31531i) q^{93} -3.80912i q^{94} +(-0.830946 - 5.91369i) q^{95} +2.04780i q^{96} +1.01973 q^{97} -1.57492 q^{98} +(2.46144 - 2.46144i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 4 q^{3} - 22 q^{4} - 4 q^{7} + 10 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 14 q^{13} + 4 q^{14} - 10 q^{15} + 6 q^{16} - 20 q^{20} + 16 q^{21} + 8 q^{22} - 4 q^{23} + 10 q^{25} + 6 q^{26} - 4 q^{27}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.26373i − 0.893594i −0.894635 − 0.446797i \(-0.852565\pi\)
0.894635 − 0.446797i \(-0.147435\pi\)
\(3\) −0.913274 −0.527279 −0.263640 − 0.964621i \(-0.584923\pi\)
−0.263640 + 0.964621i \(0.584923\pi\)
\(4\) 0.402981 0.201490
\(5\) −1.78577 − 1.34575i −0.798619 − 0.601837i
\(6\) 1.15413i 0.471173i
\(7\) 2.03055 − 2.03055i 0.767475 − 0.767475i −0.210187 − 0.977661i \(-0.567407\pi\)
0.977661 + 0.210187i \(0.0674071\pi\)
\(8\) − 3.03672i − 1.07364i
\(9\) −2.16593 −0.721977
\(10\) −1.70067 + 2.25673i −0.537798 + 0.713641i
\(11\) −1.13644 + 1.13644i −0.342648 + 0.342648i −0.857362 − 0.514714i \(-0.827898\pi\)
0.514714 + 0.857362i \(0.327898\pi\)
\(12\) −0.368032 −0.106242
\(13\) 3.45320 − 3.45320i 0.957746 − 0.957746i −0.0413972 − 0.999143i \(-0.513181\pi\)
0.999143 + 0.0413972i \(0.0131809\pi\)
\(14\) −2.56607 − 2.56607i −0.685811 − 0.685811i
\(15\) 1.63089 + 1.22904i 0.421095 + 0.317336i
\(16\) −3.03165 −0.757911
\(17\) 4.10729i 0.996163i 0.867130 + 0.498082i \(0.165962\pi\)
−0.867130 + 0.498082i \(0.834038\pi\)
\(18\) 2.73716i 0.645154i
\(19\) 1.88844 + 1.88844i 0.433239 + 0.433239i 0.889729 − 0.456490i \(-0.150894\pi\)
−0.456490 + 0.889729i \(0.650894\pi\)
\(20\) −0.719629 − 0.542311i −0.160914 − 0.121264i
\(21\) −1.85445 + 1.85445i −0.404673 + 0.404673i
\(22\) 1.43615 + 1.43615i 0.306188 + 0.306188i
\(23\) 0.0950030 + 0.0950030i 0.0198095 + 0.0198095i 0.716942 − 0.697133i \(-0.245541\pi\)
−0.697133 + 0.716942i \(0.745541\pi\)
\(24\) 2.77336i 0.566110i
\(25\) 1.37792 + 4.80639i 0.275584 + 0.961277i
\(26\) −4.36392 − 4.36392i −0.855835 − 0.855835i
\(27\) 4.71791 0.907962
\(28\) 0.818271 − 0.818271i 0.154639 − 0.154639i
\(29\) 4.16339 + 3.41558i 0.773123 + 0.634256i
\(30\) 1.55318 − 2.06101i 0.283570 − 0.376288i
\(31\) 3.63014 − 3.63014i 0.651992 − 0.651992i −0.301481 − 0.953472i \(-0.597481\pi\)
0.953472 + 0.301481i \(0.0974809\pi\)
\(32\) − 2.24226i − 0.396379i
\(33\) 1.03788 − 1.03788i 0.180671 − 0.180671i
\(34\) 5.19051 0.890165
\(35\) −6.35869 + 0.893474i −1.07481 + 0.151025i
\(36\) −0.872828 −0.145471
\(37\) 2.07797 0.341615 0.170808 − 0.985304i \(-0.445362\pi\)
0.170808 + 0.985304i \(0.445362\pi\)
\(38\) 2.38649 − 2.38649i 0.387139 − 0.387139i
\(39\) −3.15372 + 3.15372i −0.504999 + 0.504999i
\(40\) −4.08667 + 5.42288i −0.646159 + 0.857432i
\(41\) 6.10064 + 6.10064i 0.952760 + 0.952760i 0.998933 − 0.0461739i \(-0.0147028\pi\)
−0.0461739 + 0.998933i \(0.514703\pi\)
\(42\) 2.34352 + 2.34352i 0.361614 + 0.361614i
\(43\) −9.37469 −1.42963 −0.714814 − 0.699315i \(-0.753489\pi\)
−0.714814 + 0.699315i \(0.753489\pi\)
\(44\) −0.457962 + 0.457962i −0.0690403 + 0.0690403i
\(45\) 3.86784 + 2.91480i 0.576584 + 0.434512i
\(46\) 0.120058 − 0.120058i 0.0177016 − 0.0177016i
\(47\) 3.01418 0.439663 0.219832 − 0.975538i \(-0.429449\pi\)
0.219832 + 0.975538i \(0.429449\pi\)
\(48\) 2.76872 0.399631
\(49\) − 1.24624i − 0.178035i
\(50\) 6.07398 − 1.74132i 0.858991 − 0.246260i
\(51\) − 3.75108i − 0.525256i
\(52\) 1.39157 − 1.39157i 0.192977 − 0.192977i
\(53\) −5.40841 − 5.40841i −0.742902 − 0.742902i 0.230234 − 0.973135i \(-0.426051\pi\)
−0.973135 + 0.230234i \(0.926051\pi\)
\(54\) − 5.96218i − 0.811350i
\(55\) 3.55877 − 0.500051i 0.479864 − 0.0674268i
\(56\) −6.16621 − 6.16621i −0.823995 − 0.823995i
\(57\) −1.72467 − 1.72467i −0.228438 − 0.228438i
\(58\) 4.31637 − 5.26141i 0.566768 − 0.690858i
\(59\) 4.15848i 0.541388i 0.962665 + 0.270694i \(0.0872531\pi\)
−0.962665 + 0.270694i \(0.912747\pi\)
\(60\) 0.657219 + 0.495279i 0.0848466 + 0.0639402i
\(61\) −2.53415 + 2.53415i −0.324464 + 0.324464i −0.850477 − 0.526013i \(-0.823686\pi\)
0.526013 + 0.850477i \(0.323686\pi\)
\(62\) −4.58752 − 4.58752i −0.582616 − 0.582616i
\(63\) −4.39802 + 4.39802i −0.554099 + 0.554099i
\(64\) −8.89691 −1.11211
\(65\) −10.8137 + 1.51947i −1.34128 + 0.188466i
\(66\) −1.31160 − 1.31160i −0.161447 − 0.161447i
\(67\) −3.18760 − 3.18760i −0.389428 − 0.389428i 0.485056 − 0.874483i \(-0.338799\pi\)
−0.874483 + 0.485056i \(0.838799\pi\)
\(68\) 1.65516i 0.200717i
\(69\) −0.0867638 − 0.0867638i −0.0104451 − 0.0104451i
\(70\) 1.12911 + 8.03568i 0.134955 + 0.960448i
\(71\) − 11.5554i − 1.37137i −0.727896 − 0.685687i \(-0.759502\pi\)
0.727896 − 0.685687i \(-0.240498\pi\)
\(72\) 6.57733i 0.775146i
\(73\) 14.4237i 1.68816i 0.536214 + 0.844082i \(0.319854\pi\)
−0.536214 + 0.844082i \(0.680146\pi\)
\(74\) − 2.62599i − 0.305265i
\(75\) −1.25842 − 4.38955i −0.145310 − 0.506861i
\(76\) 0.761006 + 0.761006i 0.0872934 + 0.0872934i
\(77\) 4.61517i 0.525948i
\(78\) 3.98546 + 3.98546i 0.451264 + 0.451264i
\(79\) −7.72601 − 7.72601i −0.869244 − 0.869244i 0.123145 − 0.992389i \(-0.460702\pi\)
−0.992389 + 0.123145i \(0.960702\pi\)
\(80\) 5.41381 + 4.07983i 0.605282 + 0.456139i
\(81\) 2.18904 0.243227
\(82\) 7.70957 − 7.70957i 0.851380 − 0.851380i
\(83\) −11.0085 − 11.0085i −1.20834 − 1.20834i −0.971564 − 0.236777i \(-0.923909\pi\)
−0.236777 − 0.971564i \(-0.576091\pi\)
\(84\) −0.747306 + 0.747306i −0.0815378 + 0.0815378i
\(85\) 5.52738 − 7.33465i 0.599528 − 0.795555i
\(86\) 11.8471i 1.27751i
\(87\) −3.80232 − 3.11936i −0.407652 − 0.334430i
\(88\) 3.45104 + 3.45104i 0.367882 + 0.367882i
\(89\) 9.63459 + 9.63459i 1.02126 + 1.02126i 0.999769 + 0.0214957i \(0.00684283\pi\)
0.0214957 + 0.999769i \(0.493157\pi\)
\(90\) 3.68352 − 4.88792i 0.388278 − 0.515232i
\(91\) − 14.0238i − 1.47009i
\(92\) 0.0382844 + 0.0382844i 0.00399142 + 0.00399142i
\(93\) −3.31531 + 3.31531i −0.343782 + 0.343782i
\(94\) − 3.80912i − 0.392880i
\(95\) −0.830946 − 5.91369i −0.0852533 − 0.606732i
\(96\) 2.04780i 0.209003i
\(97\) 1.01973 0.103538 0.0517689 − 0.998659i \(-0.483514\pi\)
0.0517689 + 0.998659i \(0.483514\pi\)
\(98\) −1.57492 −0.159091
\(99\) 2.46144 − 2.46144i 0.247384 − 0.247384i
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.e.a.133.4 yes 26
5.2 odd 4 145.2.j.a.17.10 yes 26
5.3 odd 4 725.2.j.c.307.4 26
5.4 even 2 725.2.e.c.568.10 26
29.12 odd 4 145.2.j.a.128.10 yes 26
145.12 even 4 inner 145.2.e.a.12.10 ✓ 26
145.99 odd 4 725.2.j.c.418.4 26
145.128 even 4 725.2.e.c.157.4 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.e.a.12.10 ✓ 26 145.12 even 4 inner
145.2.e.a.133.4 yes 26 1.1 even 1 trivial
145.2.j.a.17.10 yes 26 5.2 odd 4
145.2.j.a.128.10 yes 26 29.12 odd 4
725.2.e.c.157.4 26 145.128 even 4
725.2.e.c.568.10 26 5.4 even 2
725.2.j.c.307.4 26 5.3 odd 4
725.2.j.c.418.4 26 145.99 odd 4