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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,5,Mod(757,900)] 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("900.757"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 900.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-110,0,0,0,300,0,360,0,0,0,960,0,0,0,0,0,-810,0, 0,0,0,0,0,0,-836] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.0329667755\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 757.2
Root \(8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 900.757
Dual form 900.5.l.a.793.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.21374 - 4.21374i) q^{7} +152.621 q^{11} +(136.573 - 136.573i) q^{13} +(348.669 + 348.669i) q^{17} -527.725i q^{19} +(-70.5445 + 70.5445i) q^{23} -68.9670i q^{29} -1373.31 q^{31} +(1003.31 + 1003.31i) q^{37} -663.379 q^{41} +(-1632.52 + 1632.52i) q^{43} +(438.809 + 438.809i) q^{47} -2365.49i q^{49} +(712.338 - 712.338i) q^{53} -2918.34i q^{59} +1395.51 q^{61} +(1691.51 + 1691.51i) q^{67} +3282.28 q^{71} +(1465.81 - 1465.81i) q^{73} +(-643.104 - 643.104i) q^{77} +3389.80i q^{79} +(8690.02 - 8690.02i) q^{83} +4274.77i q^{89} -1150.96 q^{91} +(-7215.47 - 7215.47i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 110 q^{7} + 300 q^{11} + 360 q^{13} + 960 q^{17} - 810 q^{23} - 836 q^{31} + 660 q^{37} - 2964 q^{41} - 3270 q^{43} - 2250 q^{47} + 1980 q^{53} + 15828 q^{61} + 4810 q^{67} + 5988 q^{71} + 14060 q^{73}+ \cdots + 3180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.21374 − 4.21374i −0.0859947 − 0.0859947i 0.662801 − 0.748796i \(-0.269368\pi\)
−0.748796 + 0.662801i \(0.769368\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 152.621 1.26133 0.630665 − 0.776055i \(-0.282782\pi\)
0.630665 + 0.776055i \(0.282782\pi\)
\(12\) 0 0
\(13\) 136.573 − 136.573i 0.808121 − 0.808121i −0.176228 − 0.984349i \(-0.556390\pi\)
0.984349 + 0.176228i \(0.0563896\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 348.669 + 348.669i 1.20647 + 1.20647i 0.972163 + 0.234305i \(0.0752813\pi\)
0.234305 + 0.972163i \(0.424719\pi\)
\(18\) 0 0
\(19\) − 527.725i − 1.46184i −0.682462 − 0.730921i \(-0.739091\pi\)
0.682462 − 0.730921i \(-0.260909\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −70.5445 + 70.5445i −0.133354 + 0.133354i −0.770633 − 0.637279i \(-0.780060\pi\)
0.637279 + 0.770633i \(0.280060\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 68.9670i − 0.0820059i −0.999159 − 0.0410030i \(-0.986945\pi\)
0.999159 − 0.0410030i \(-0.0130553\pi\)
\(30\) 0 0
\(31\) −1373.31 −1.42905 −0.714523 − 0.699612i \(-0.753356\pi\)
−0.714523 + 0.699612i \(0.753356\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1003.31 + 1003.31i 0.732875 + 0.732875i 0.971188 − 0.238314i \(-0.0765946\pi\)
−0.238314 + 0.971188i \(0.576595\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −663.379 −0.394634 −0.197317 − 0.980340i \(-0.563223\pi\)
−0.197317 + 0.980340i \(0.563223\pi\)
\(42\) 0 0
\(43\) −1632.52 + 1632.52i −0.882920 + 0.882920i −0.993830 − 0.110910i \(-0.964623\pi\)
0.110910 + 0.993830i \(0.464623\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 438.809 + 438.809i 0.198646 + 0.198646i 0.799419 − 0.600773i \(-0.205141\pi\)
−0.600773 + 0.799419i \(0.705141\pi\)
\(48\) 0 0
\(49\) − 2365.49i − 0.985210i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 712.338 − 712.338i 0.253591 − 0.253591i −0.568850 − 0.822441i \(-0.692611\pi\)
0.822441 + 0.568850i \(0.192611\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2918.34i − 0.838363i −0.907903 − 0.419181i \(-0.862317\pi\)
0.907903 − 0.419181i \(-0.137683\pi\)
\(60\) 0 0
\(61\) 1395.51 0.375037 0.187518 − 0.982261i \(-0.439956\pi\)
0.187518 + 0.982261i \(0.439956\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1691.51 + 1691.51i 0.376813 + 0.376813i 0.869951 − 0.493138i \(-0.164150\pi\)
−0.493138 + 0.869951i \(0.664150\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3282.28 0.651117 0.325558 − 0.945522i \(-0.394448\pi\)
0.325558 + 0.945522i \(0.394448\pi\)
\(72\) 0 0
\(73\) 1465.81 − 1465.81i 0.275063 − 0.275063i −0.556072 − 0.831134i \(-0.687692\pi\)
0.831134 + 0.556072i \(0.187692\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −643.104 − 643.104i −0.108468 − 0.108468i
\(78\) 0 0
\(79\) 3389.80i 0.543150i 0.962417 + 0.271575i \(0.0875446\pi\)
−0.962417 + 0.271575i \(0.912455\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8690.02 − 8690.02i 1.26143 − 1.26143i 0.311035 − 0.950398i \(-0.399324\pi\)
0.950398 − 0.311035i \(-0.100676\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4274.77i 0.539675i 0.962906 + 0.269838i \(0.0869701\pi\)
−0.962906 + 0.269838i \(0.913030\pi\)
\(90\) 0 0
\(91\) −1150.96 −0.138988
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7215.47 − 7215.47i −0.766869 − 0.766869i 0.210685 − 0.977554i \(-0.432431\pi\)
−0.977554 + 0.210685i \(0.932431\pi\)
\(98\) 0 0
\(99\) 0 0
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 900.5.l.a.757.2 4
3.2 odd 2 100.5.f.c.57.1 4
5.2 odd 4 180.5.l.a.73.1 4
5.3 odd 4 inner 900.5.l.a.793.2 4
5.4 even 2 180.5.l.a.37.1 4
12.11 even 2 400.5.p.h.257.2 4
15.2 even 4 20.5.f.a.13.2 ✓ 4
15.8 even 4 100.5.f.c.93.1 4
15.14 odd 2 20.5.f.a.17.2 yes 4
60.23 odd 4 400.5.p.h.193.2 4
60.47 odd 4 80.5.p.e.33.1 4
60.59 even 2 80.5.p.e.17.1 4
120.29 odd 2 320.5.p.m.257.1 4
120.59 even 2 320.5.p.l.257.2 4
120.77 even 4 320.5.p.m.193.1 4
120.107 odd 4 320.5.p.l.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.2 ✓ 4 15.2 even 4
20.5.f.a.17.2 yes 4 15.14 odd 2
80.5.p.e.17.1 4 60.59 even 2
80.5.p.e.33.1 4 60.47 odd 4
100.5.f.c.57.1 4 3.2 odd 2
100.5.f.c.93.1 4 15.8 even 4
180.5.l.a.37.1 4 5.4 even 2
180.5.l.a.73.1 4 5.2 odd 4
320.5.p.l.193.2 4 120.107 odd 4
320.5.p.l.257.2 4 120.59 even 2
320.5.p.m.193.1 4 120.77 even 4
320.5.p.m.257.1 4 120.29 odd 2
400.5.p.h.193.2 4 60.23 odd 4
400.5.p.h.257.2 4 12.11 even 2
900.5.l.a.757.2 4 1.1 even 1 trivial
900.5.l.a.793.2 4 5.3 odd 4 inner