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Properties

Label 45.5.g.a.28.1
Level $45$
Weight $5$
Character 45.28
Analytic conductor $4.652$
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] = [45,5,Mod(28,45)] 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("45.28"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.65164833877\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 28.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 45.28
Dual form 45.5.g.a.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-5.00000 + 5.00000i) q^{2} -34.0000i q^{4} +25.0000 q^{5} +(40.0000 - 40.0000i) q^{7} +(90.0000 + 90.0000i) q^{8} +(-125.000 + 125.000i) q^{10} -100.000 q^{11} +(205.000 + 205.000i) q^{13} +400.000i q^{14} -356.000 q^{16} +(235.000 - 235.000i) q^{17} -72.0000i q^{19} -850.000i q^{20} +(500.000 - 500.000i) q^{22} +(340.000 + 340.000i) q^{23} +625.000 q^{25} -2050.00 q^{26} +(-1360.00 - 1360.00i) q^{28} -450.000i q^{29} +428.000 q^{31} +(340.000 - 340.000i) q^{32} +2350.00i q^{34} +(1000.00 - 1000.00i) q^{35} +(-755.000 + 755.000i) q^{37} +(360.000 + 360.000i) q^{38} +(2250.00 + 2250.00i) q^{40} +950.000 q^{41} +(-1220.00 - 1220.00i) q^{43} +3400.00i q^{44} -3400.00 q^{46} +(-320.000 + 320.000i) q^{47} -799.000i q^{49} +(-3125.00 + 3125.00i) q^{50} +(6970.00 - 6970.00i) q^{52} +(505.000 + 505.000i) q^{53} -2500.00 q^{55} +7200.00 q^{56} +(2250.00 + 2250.00i) q^{58} -6300.00i q^{59} -3808.00 q^{61} +(-2140.00 + 2140.00i) q^{62} -2296.00i q^{64} +(5125.00 + 5125.00i) q^{65} +(340.000 - 340.000i) q^{67} +(-7990.00 - 7990.00i) q^{68} +10000.0i q^{70} -3400.00 q^{71} +(415.000 + 415.000i) q^{73} -7550.00i q^{74} -2448.00 q^{76} +(-4000.00 + 4000.00i) q^{77} +6732.00i q^{79} -8900.00 q^{80} +(-4750.00 + 4750.00i) q^{82} +(-680.000 - 680.000i) q^{83} +(5875.00 - 5875.00i) q^{85} +12200.0 q^{86} +(-9000.00 - 9000.00i) q^{88} -2250.00i q^{89} +16400.0 q^{91} +(11560.0 - 11560.0i) q^{92} -3200.00i q^{94} -1800.00i q^{95} +(1615.00 - 1615.00i) q^{97} +(3995.00 + 3995.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8} - 250 q^{10} - 200 q^{11} + 410 q^{13} - 712 q^{16} + 470 q^{17} + 1000 q^{22} + 680 q^{23} + 1250 q^{25} - 4100 q^{26} - 2720 q^{28} + 856 q^{31} + 680 q^{32}+ \cdots + 7990 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 + 5.00000i −1.25000 + 1.25000i −0.294281 + 0.955719i \(0.595080\pi\)
−0.955719 + 0.294281i \(0.904920\pi\)
\(3\) 0 0
\(4\) − 34.0000i − 2.12500i
\(5\) 25.0000 1.00000
\(6\) 0 0
\(7\) 40.0000 − 40.0000i 0.816327 − 0.816327i −0.169247 − 0.985574i \(-0.554134\pi\)
0.985574 + 0.169247i \(0.0541336\pi\)
\(8\) 90.0000 + 90.0000i 1.40625 + 1.40625i
\(9\) 0 0
\(10\) −125.000 + 125.000i −1.25000 + 1.25000i
\(11\) −100.000 −0.826446 −0.413223 − 0.910630i \(-0.635597\pi\)
−0.413223 + 0.910630i \(0.635597\pi\)
\(12\) 0 0
\(13\) 205.000 + 205.000i 1.21302 + 1.21302i 0.970029 + 0.242989i \(0.0781278\pi\)
0.242989 + 0.970029i \(0.421872\pi\)
\(14\) 400.000i 2.04082i
\(15\) 0 0
\(16\) −356.000 −1.39062
\(17\) 235.000 − 235.000i 0.813149 − 0.813149i −0.171956 − 0.985105i \(-0.555009\pi\)
0.985105 + 0.171956i \(0.0550086\pi\)
\(18\) 0 0
\(19\) − 72.0000i − 0.199446i −0.995015 − 0.0997230i \(-0.968204\pi\)
0.995015 − 0.0997230i \(-0.0317957\pi\)
\(20\) − 850.000i − 2.12500i
\(21\) 0 0
\(22\) 500.000 − 500.000i 1.03306 − 1.03306i
\(23\) 340.000 + 340.000i 0.642722 + 0.642722i 0.951224 − 0.308502i \(-0.0998275\pi\)
−0.308502 + 0.951224i \(0.599828\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) −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
\(35\) 1000.00 − 1000.00i 0.816327 − 0.816327i
\(36\) 0 0
\(37\) −755.000 + 755.000i −0.551497 + 0.551497i −0.926873 − 0.375375i \(-0.877514\pi\)
0.375375 + 0.926873i \(0.377514\pi\)
\(38\) 360.000 + 360.000i 0.249307 + 0.249307i
\(39\) 0 0
\(40\) 2250.00 + 2250.00i 1.40625 + 1.40625i
\(41\) 950.000 0.565140 0.282570 − 0.959247i \(-0.408813\pi\)
0.282570 + 0.959247i \(0.408813\pi\)
\(42\) 0 0
\(43\) −1220.00 − 1220.00i −0.659816 − 0.659816i 0.295520 − 0.955336i \(-0.404507\pi\)
−0.955336 + 0.295520i \(0.904507\pi\)
\(44\) 3400.00i 1.75620i
\(45\) 0 0
\(46\) −3400.00 −1.60681
\(47\) −320.000 + 320.000i −0.144862 + 0.144862i −0.775818 − 0.630956i \(-0.782663\pi\)
0.630956 + 0.775818i \(0.282663\pi\)
\(48\) 0 0
\(49\) − 799.000i − 0.332778i
\(50\) −3125.00 + 3125.00i −1.25000 + 1.25000i
\(51\) 0 0
\(52\) 6970.00 − 6970.00i 2.57766 − 2.57766i
\(53\) 505.000 + 505.000i 0.179779 + 0.179779i 0.791260 − 0.611480i \(-0.209426\pi\)
−0.611480 + 0.791260i \(0.709426\pi\)
\(54\) 0 0
\(55\) −2500.00 −0.826446
\(56\) 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
\(65\) 5125.00 + 5125.00i 1.21302 + 1.21302i
\(66\) 0 0
\(67\) 340.000 − 340.000i 0.0757407 − 0.0757407i −0.668222 − 0.743962i \(-0.732944\pi\)
0.743962 + 0.668222i \(0.232944\pi\)
\(68\) −7990.00 − 7990.00i −1.72794 − 1.72794i
\(69\) 0 0
\(70\) 10000.0i 2.04082i
\(71\) −3400.00 −0.674469 −0.337235 − 0.941421i \(-0.609492\pi\)
−0.337235 + 0.941421i \(0.609492\pi\)
\(72\) 0 0
\(73\) 415.000 + 415.000i 0.0778758 + 0.0778758i 0.744972 − 0.667096i \(-0.232463\pi\)
−0.667096 + 0.744972i \(0.732463\pi\)
\(74\) − 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 \(0.181325\pi\)
−0.842090 + 0.539337i \(0.818675\pi\)
\(80\) −8900.00 −1.39062
\(81\) 0 0
\(82\) −4750.00 + 4750.00i −0.706425 + 0.706425i
\(83\) −680.000 − 680.000i −0.0987081 − 0.0987081i 0.656028 − 0.754736i \(-0.272235\pi\)
−0.754736 + 0.656028i \(0.772235\pi\)
\(84\) 0 0
\(85\) 5875.00 − 5875.00i 0.813149 − 0.813149i
\(86\) 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
\(95\) − 1800.00i − 0.199446i
\(96\) 0 0
\(97\) 1615.00 − 1615.00i 0.171644 − 0.171644i −0.616057 − 0.787701i \(-0.711271\pi\)
0.787701 + 0.616057i \(0.211271\pi\)
\(98\) 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 45.5.g.a.28.1 ✓ 2
3.2 odd 2 45.5.g.c.28.1 yes 2
5.2 odd 4 inner 45.5.g.a.37.1 yes 2
5.3 odd 4 225.5.g.c.82.1 2
5.4 even 2 225.5.g.c.118.1 2
15.2 even 4 45.5.g.c.37.1 yes 2
15.8 even 4 225.5.g.a.82.1 2
15.14 odd 2 225.5.g.a.118.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.g.a.28.1 ✓ 2 1.1 even 1 trivial
45.5.g.a.37.1 yes 2 5.2 odd 4 inner
45.5.g.c.28.1 yes 2 3.2 odd 2
45.5.g.c.37.1 yes 2 15.2 even 4
225.5.g.a.82.1 2 15.8 even 4
225.5.g.a.118.1 2 15.14 odd 2
225.5.g.c.82.1 2 5.3 odd 4
225.5.g.c.118.1 2 5.4 even 2