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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,5,Mod(82,225)] 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("225.82"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,12,0,0,0,0,72,-264,0,0,24,0,-144,0,0,-2320] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.2582416939\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 82.2
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 225.82
Dual form 225.5.g.l.118.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(5.44949 + 5.44949i) q^{2} +43.3939i q^{4} +(19.2247 + 19.2247i) q^{7} +(-149.283 + 149.283i) q^{8} -185.060 q^{11} +(48.5074 - 48.5074i) q^{13} +209.530i q^{14} -932.727 q^{16} +(147.551 + 147.551i) q^{17} +140.242i q^{19} +(-1008.48 - 1008.48i) q^{22} +(-177.267 + 177.267i) q^{23} +528.681 q^{26} +(-834.236 + 834.236i) q^{28} +588.756i q^{29} +1413.57 q^{31} +(-2694.36 - 2694.36i) q^{32} +1608.15i q^{34} +(-763.868 - 763.868i) q^{37} +(-764.246 + 764.246i) q^{38} -995.850 q^{41} +(657.277 - 657.277i) q^{43} -8030.48i q^{44} -1932.03 q^{46} +(-102.436 - 102.436i) q^{47} -1661.82i q^{49} +(2104.92 + 2104.92i) q^{52} +(-365.928 + 365.928i) q^{53} -5739.84 q^{56} +(-3208.42 + 3208.42i) q^{58} +5805.99i q^{59} +6282.87 q^{61} +(7703.26 + 7703.26i) q^{62} -14442.2i q^{64} +(5494.12 + 5494.12i) q^{67} +(-6402.79 + 6402.79i) q^{68} +7666.03 q^{71} +(-6406.59 + 6406.59i) q^{73} -8325.39i q^{74} -6085.64 q^{76} +(-3557.74 - 3557.74i) q^{77} +5699.38i q^{79} +(-5426.87 - 5426.87i) q^{82} +(999.189 - 999.189i) q^{83} +7163.65 q^{86} +(27626.3 - 27626.3i) q^{88} -3090.84i q^{89} +1865.08 q^{91} +(-7692.29 - 7692.29i) q^{92} -1116.45i q^{94} +(5031.88 + 5031.88i) q^{97} +(9056.06 - 9056.06i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{2} + 72 q^{7} - 264 q^{8} + 24 q^{11} - 144 q^{13} - 2320 q^{16} + 600 q^{17} - 1800 q^{22} + 888 q^{23} + 792 q^{26} - 1152 q^{28} + 3244 q^{31} - 6192 q^{32} - 2448 q^{37} - 1068 q^{38} - 6864 q^{41}+ \cdots + 20136 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.44949 + 5.44949i 1.36237 + 1.36237i 0.870884 + 0.491488i \(0.163547\pi\)
0.491488 + 0.870884i \(0.336453\pi\)
\(3\) 0 0
\(4\) 43.3939i 2.71212i
\(5\) 0 0
\(6\) 0 0
\(7\) 19.2247 + 19.2247i 0.392342 + 0.392342i 0.875521 − 0.483180i \(-0.160518\pi\)
−0.483180 + 0.875521i \(0.660518\pi\)
\(8\) −149.283 + 149.283i −2.33254 + 2.33254i
\(9\) 0 0
\(10\) 0 0
\(11\) −185.060 −1.52942 −0.764712 − 0.644373i \(-0.777119\pi\)
−0.764712 + 0.644373i \(0.777119\pi\)
\(12\) 0 0
\(13\) 48.5074 − 48.5074i 0.287026 − 0.287026i −0.548877 − 0.835903i \(-0.684944\pi\)
0.835903 + 0.548877i \(0.184944\pi\)
\(14\) 209.530i 1.06903i
\(15\) 0 0
\(16\) −932.727 −3.64346
\(17\) 147.551 + 147.551i 0.510555 + 0.510555i 0.914697 − 0.404141i \(-0.132430\pi\)
−0.404141 + 0.914697i \(0.632430\pi\)
\(18\) 0 0
\(19\) 140.242i 0.388482i 0.980954 + 0.194241i \(0.0622243\pi\)
−0.980954 + 0.194241i \(0.937776\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1008.48 − 1008.48i −2.08364 − 2.08364i
\(23\) −177.267 + 177.267i −0.335098 + 0.335098i −0.854519 − 0.519421i \(-0.826148\pi\)
0.519421 + 0.854519i \(0.326148\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 528.681 0.782073
\(27\) 0 0
\(28\) −834.236 + 834.236i −1.06408 + 1.06408i
\(29\) 588.756i 0.700067i 0.936737 + 0.350033i \(0.113830\pi\)
−0.936737 + 0.350033i \(0.886170\pi\)
\(30\) 0 0
\(31\) 1413.57 1.47094 0.735471 − 0.677557i \(-0.236961\pi\)
0.735471 + 0.677557i \(0.236961\pi\)
\(32\) −2694.36 − 2694.36i −2.63121 − 2.63121i
\(33\) 0 0
\(34\) 1608.15i 1.39113i
\(35\) 0 0
\(36\) 0 0
\(37\) −763.868 − 763.868i −0.557975 − 0.557975i 0.370755 − 0.928731i \(-0.379099\pi\)
−0.928731 + 0.370755i \(0.879099\pi\)
\(38\) −764.246 + 764.246i −0.529257 + 0.529257i
\(39\) 0 0
\(40\) 0 0
\(41\) −995.850 −0.592415 −0.296208 − 0.955124i \(-0.595722\pi\)
−0.296208 + 0.955124i \(0.595722\pi\)
\(42\) 0 0
\(43\) 657.277 − 657.277i 0.355477 − 0.355477i −0.506666 − 0.862143i \(-0.669122\pi\)
0.862143 + 0.506666i \(0.169122\pi\)
\(44\) − 8030.48i − 4.14797i
\(45\) 0 0
\(46\) −1932.03 −0.913056
\(47\) −102.436 − 102.436i −0.0463722 − 0.0463722i 0.683540 − 0.729913i \(-0.260439\pi\)
−0.729913 + 0.683540i \(0.760439\pi\)
\(48\) 0 0
\(49\) − 1661.82i − 0.692136i
\(50\) 0 0
\(51\) 0 0
\(52\) 2104.92 + 2104.92i 0.778448 + 0.778448i
\(53\) −365.928 + 365.928i −0.130270 + 0.130270i −0.769235 − 0.638966i \(-0.779363\pi\)
0.638966 + 0.769235i \(0.279363\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5739.84 −1.83031
\(57\) 0 0
\(58\) −3208.42 + 3208.42i −0.953752 + 0.953752i
\(59\) 5805.99i 1.66791i 0.551833 + 0.833955i \(0.313929\pi\)
−0.551833 + 0.833955i \(0.686071\pi\)
\(60\) 0 0
\(61\) 6282.87 1.68849 0.844245 − 0.535957i \(-0.180049\pi\)
0.844245 + 0.535957i \(0.180049\pi\)
\(62\) 7703.26 + 7703.26i 2.00397 + 2.00397i
\(63\) 0 0
\(64\) − 14442.2i − 3.52592i
\(65\) 0 0
\(66\) 0 0
\(67\) 5494.12 + 5494.12i 1.22391 + 1.22391i 0.966231 + 0.257677i \(0.0829571\pi\)
0.257677 + 0.966231i \(0.417043\pi\)
\(68\) −6402.79 + 6402.79i −1.38469 + 1.38469i
\(69\) 0 0
\(70\) 0 0
\(71\) 7666.03 1.52074 0.760368 − 0.649493i \(-0.225019\pi\)
0.760368 + 0.649493i \(0.225019\pi\)
\(72\) 0 0
\(73\) −6406.59 + 6406.59i −1.20221 + 1.20221i −0.228721 + 0.973492i \(0.573454\pi\)
−0.973492 + 0.228721i \(0.926546\pi\)
\(74\) − 8325.39i − 1.52034i
\(75\) 0 0
\(76\) −6085.64 −1.05361
\(77\) −3557.74 − 3557.74i −0.600057 − 0.600057i
\(78\) 0 0
\(79\) 5699.38i 0.913215i 0.889668 + 0.456608i \(0.150936\pi\)
−0.889668 + 0.456608i \(0.849064\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5426.87 − 5426.87i −0.807090 − 0.807090i
\(83\) 999.189 − 999.189i 0.145041 − 0.145041i −0.630857 − 0.775899i \(-0.717297\pi\)
0.775899 + 0.630857i \(0.217297\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7163.65 0.968584
\(87\) 0 0
\(88\) 27626.3 − 27626.3i 3.56744 − 3.56744i
\(89\) − 3090.84i − 0.390208i −0.980783 − 0.195104i \(-0.937496\pi\)
0.980783 − 0.195104i \(-0.0625045\pi\)
\(90\) 0 0
\(91\) 1865.08 0.225225
\(92\) −7692.29 − 7692.29i −0.908825 − 0.908825i
\(93\) 0 0
\(94\) − 1116.45i − 0.126352i
\(95\) 0 0
\(96\) 0 0
\(97\) 5031.88 + 5031.88i 0.534794 + 0.534794i 0.921995 − 0.387201i \(-0.126558\pi\)
−0.387201 + 0.921995i \(0.626558\pi\)
\(98\) 9056.06 − 9056.06i 0.942947 − 0.942947i
\(99\) 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 225.5.g.l.82.2 4
3.2 odd 2 75.5.f.a.7.1 ✓ 4
5.2 odd 4 225.5.g.d.118.1 4
5.3 odd 4 inner 225.5.g.l.118.2 4
5.4 even 2 225.5.g.d.82.1 4
15.2 even 4 75.5.f.d.43.2 yes 4
15.8 even 4 75.5.f.a.43.1 yes 4
15.14 odd 2 75.5.f.d.7.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.f.a.7.1 ✓ 4 3.2 odd 2
75.5.f.a.43.1 yes 4 15.8 even 4
75.5.f.d.7.2 yes 4 15.14 odd 2
75.5.f.d.43.2 yes 4 15.2 even 4
225.5.g.d.82.1 4 5.4 even 2
225.5.g.d.118.1 4 5.2 odd 4
225.5.g.l.82.2 4 1.1 even 1 trivial
225.5.g.l.118.2 4 5.3 odd 4 inner