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Properties

Label 225.5.g.a.82.1
Level $225$
Weight $5$
Character 225.82
Analytic conductor $23.258$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 = [2,-10,0,0,0,0,-80,180,0,0,200,0,-410,0,0,-712] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 82.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.82
Dual form 225.5.g.a.118.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-5.00000 - 5.00000i) q^{2} +34.0000i q^{4} +(-40.0000 - 40.0000i) q^{7} +(90.0000 - 90.0000i) q^{8} +100.000 q^{11} +(-205.000 + 205.000i) q^{13} +400.000i q^{14} -356.000 q^{16} +(235.000 + 235.000i) q^{17} +72.0000i q^{19} +(-500.000 - 500.000i) q^{22} +(340.000 - 340.000i) q^{23} +2050.00 q^{26} +(1360.00 - 1360.00i) q^{28} -450.000i q^{29} +428.000 q^{31} +(340.000 + 340.000i) q^{32} -2350.00i q^{34} +(755.000 + 755.000i) q^{37} +(360.000 - 360.000i) q^{38} -950.000 q^{41} +(1220.00 - 1220.00i) q^{43} +3400.00i q^{44} -3400.00 q^{46} +(-320.000 - 320.000i) q^{47} +799.000i q^{49} +(-6970.00 - 6970.00i) q^{52} +(505.000 - 505.000i) q^{53} -7200.00 q^{56} +(-2250.00 + 2250.00i) q^{58} -6300.00i q^{59} -3808.00 q^{61} +(-2140.00 - 2140.00i) q^{62} +2296.00i q^{64} +(-340.000 - 340.000i) q^{67} +(-7990.00 + 7990.00i) q^{68} +3400.00 q^{71} +(-415.000 + 415.000i) q^{73} -7550.00i q^{74} -2448.00 q^{76} +(-4000.00 - 4000.00i) q^{77} -6732.00i q^{79} +(4750.00 + 4750.00i) q^{82} +(-680.000 + 680.000i) q^{83} -12200.0 q^{86} +(9000.00 - 9000.00i) q^{88} -2250.00i q^{89} +16400.0 q^{91} +(11560.0 + 11560.0i) q^{92} +3200.00i q^{94} +(-1615.00 - 1615.00i) q^{97} +(3995.00 - 3995.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{2} - 80 q^{7} + 180 q^{8} + 200 q^{11} - 410 q^{13} - 712 q^{16} + 470 q^{17} - 1000 q^{22} + 680 q^{23} + 4100 q^{26} + 2720 q^{28} + 856 q^{31} + 680 q^{32} + 1510 q^{37} + 720 q^{38} - 1900 q^{41}+ \cdots + 7990 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 − 5.00000i −1.25000 − 1.25000i −0.955719 − 0.294281i \(-0.904920\pi\)
−0.294281 − 0.955719i \(-0.595080\pi\)
\(3\) 0 0
\(4\) 34.0000i 2.12500i
\(5\) 0 0
\(6\) 0 0
\(7\) −40.0000 − 40.0000i −0.816327 − 0.816327i 0.169247 − 0.985574i \(-0.445866\pi\)
−0.985574 + 0.169247i \(0.945866\pi\)
\(8\) 90.0000 − 90.0000i 1.40625 − 1.40625i
\(9\) 0 0
\(10\) 0 0
\(11\) 100.000 0.826446 0.413223 − 0.910630i \(-0.364403\pi\)
0.413223 + 0.910630i \(0.364403\pi\)
\(12\) 0 0
\(13\) −205.000 + 205.000i −1.21302 + 1.21302i −0.242989 + 0.970029i \(0.578128\pi\)
−0.970029 + 0.242989i \(0.921872\pi\)
\(14\) 400.000i 2.04082i
\(15\) 0 0
\(16\) −356.000 −1.39062
\(17\) 235.000 + 235.000i 0.813149 + 0.813149i 0.985105 − 0.171956i \(-0.0550086\pi\)
−0.171956 + 0.985105i \(0.555009\pi\)
\(18\) 0 0
\(19\) 72.0000i 0.199446i 0.995015 + 0.0997230i \(0.0317957\pi\)
−0.995015 + 0.0997230i \(0.968204\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −500.000 − 500.000i −1.03306 − 1.03306i
\(23\) 340.000 − 340.000i 0.642722 − 0.642722i −0.308502 − 0.951224i \(-0.599828\pi\)
0.951224 + 0.308502i \(0.0998275\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2050.00 3.03254
\(27\) 0 0
\(28\) 1360.00 − 1360.00i 1.73469 − 1.73469i
\(29\) − 450.000i − 0.535077i −0.963547 − 0.267539i \(-0.913790\pi\)
0.963547 − 0.267539i \(-0.0862103\pi\)
\(30\) 0 0
\(31\) 428.000 0.445369 0.222685 − 0.974891i \(-0.428518\pi\)
0.222685 + 0.974891i \(0.428518\pi\)
\(32\) 340.000 + 340.000i 0.332031 + 0.332031i
\(33\) 0 0
\(34\) − 2350.00i − 2.03287i
\(35\) 0 0
\(36\) 0 0
\(37\) 755.000 + 755.000i 0.551497 + 0.551497i 0.926873 − 0.375375i \(-0.122486\pi\)
−0.375375 + 0.926873i \(0.622486\pi\)
\(38\) 360.000 − 360.000i 0.249307 − 0.249307i
\(39\) 0 0
\(40\) 0 0
\(41\) −950.000 −0.565140 −0.282570 − 0.959247i \(-0.591187\pi\)
−0.282570 + 0.959247i \(0.591187\pi\)
\(42\) 0 0
\(43\) 1220.00 − 1220.00i 0.659816 − 0.659816i −0.295520 − 0.955336i \(-0.595493\pi\)
0.955336 + 0.295520i \(0.0954930\pi\)
\(44\) 3400.00i 1.75620i
\(45\) 0 0
\(46\) −3400.00 −1.60681
\(47\) −320.000 − 320.000i −0.144862 − 0.144862i 0.630956 − 0.775818i \(-0.282663\pi\)
−0.775818 + 0.630956i \(0.782663\pi\)
\(48\) 0 0
\(49\) 799.000i 0.332778i
\(50\) 0 0
\(51\) 0 0
\(52\) −6970.00 − 6970.00i −2.57766 − 2.57766i
\(53\) 505.000 − 505.000i 0.179779 − 0.179779i −0.611480 − 0.791260i \(-0.709426\pi\)
0.791260 + 0.611480i \(0.209426\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7200.00 −2.29592
\(57\) 0 0
\(58\) −2250.00 + 2250.00i −0.668847 + 0.668847i
\(59\) − 6300.00i − 1.80982i −0.425598 − 0.904912i \(-0.639936\pi\)
0.425598 − 0.904912i \(-0.360064\pi\)
\(60\) 0 0
\(61\) −3808.00 −1.02338 −0.511690 − 0.859170i \(-0.670981\pi\)
−0.511690 + 0.859170i \(0.670981\pi\)
\(62\) −2140.00 − 2140.00i −0.556712 − 0.556712i
\(63\) 0 0
\(64\) 2296.00i 0.560547i
\(65\) 0 0
\(66\) 0 0
\(67\) −340.000 − 340.000i −0.0757407 − 0.0757407i 0.668222 − 0.743962i \(-0.267056\pi\)
−0.743962 + 0.668222i \(0.767056\pi\)
\(68\) −7990.00 + 7990.00i −1.72794 + 1.72794i
\(69\) 0 0
\(70\) 0 0
\(71\) 3400.00 0.674469 0.337235 − 0.941421i \(-0.390508\pi\)
0.337235 + 0.941421i \(0.390508\pi\)
\(72\) 0 0
\(73\) −415.000 + 415.000i −0.0778758 + 0.0778758i −0.744972 − 0.667096i \(-0.767537\pi\)
0.667096 + 0.744972i \(0.267537\pi\)
\(74\) − 7550.00i − 1.37874i
\(75\) 0 0
\(76\) −2448.00 −0.423823
\(77\) −4000.00 − 4000.00i −0.674650 − 0.674650i
\(78\) 0 0
\(79\) − 6732.00i − 1.07867i −0.842090 − 0.539337i \(-0.818675\pi\)
0.842090 − 0.539337i \(-0.181325\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4750.00 + 4750.00i 0.706425 + 0.706425i
\(83\) −680.000 + 680.000i −0.0987081 + 0.0987081i −0.754736 − 0.656028i \(-0.772235\pi\)
0.656028 + 0.754736i \(0.272235\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12200.0 −1.64954
\(87\) 0 0
\(88\) 9000.00 − 9000.00i 1.16219 − 1.16219i
\(89\) − 2250.00i − 0.284055i −0.989863 − 0.142028i \(-0.954638\pi\)
0.989863 − 0.142028i \(-0.0453622\pi\)
\(90\) 0 0
\(91\) 16400.0 1.98044
\(92\) 11560.0 + 11560.0i 1.36578 + 1.36578i
\(93\) 0 0
\(94\) 3200.00i 0.362155i
\(95\) 0 0
\(96\) 0 0
\(97\) −1615.00 − 1615.00i −0.171644 − 0.171644i 0.616057 − 0.787701i \(-0.288729\pi\)
−0.787701 + 0.616057i \(0.788729\pi\)
\(98\) 3995.00 − 3995.00i 0.415973 − 0.415973i
\(99\) 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.a.82.1 2
3.2 odd 2 225.5.g.c.82.1 2
5.2 odd 4 45.5.g.c.28.1 yes 2
5.3 odd 4 inner 225.5.g.a.118.1 2
5.4 even 2 45.5.g.c.37.1 yes 2
15.2 even 4 45.5.g.a.28.1 ✓ 2
15.8 even 4 225.5.g.c.118.1 2
15.14 odd 2 45.5.g.a.37.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.g.a.28.1 ✓ 2 15.2 even 4
45.5.g.a.37.1 yes 2 15.14 odd 2
45.5.g.c.28.1 yes 2 5.2 odd 4
45.5.g.c.37.1 yes 2 5.4 even 2
225.5.g.a.82.1 2 1.1 even 1 trivial
225.5.g.a.118.1 2 5.3 odd 4 inner
225.5.g.c.82.1 2 3.2 odd 2
225.5.g.c.118.1 2 15.8 even 4