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Properties

Label 45.5.g.c.37.1
Level $45$
Weight $5$
Character 45.37
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 37.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 45.37
Dual form 45.5.g.c.28.1

$q$-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 \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

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

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 \(0.0950802\pi\)
0.294281 + 0.955719i \(0.404920\pi\)
\(3\) 0 0
\(4\) 34.0000i 2.12500i
\(5\) −25.0000 −1.00000
\(6\) 0 0
\(7\) 40.0000 + 40.0000i 0.816327 + 0.816327i 0.985574 − 0.169247i \(-0.0541336\pi\)
−0.169247 + 0.985574i \(0.554134\pi\)
\(8\) −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 \(-0.364403\pi\)
0.413223 + 0.910630i \(0.364403\pi\)
\(12\) 0 0
\(13\) 205.000 − 205.000i 1.21302 − 1.21302i 0.242989 − 0.970029i \(-0.421872\pi\)
0.970029 − 0.242989i \(-0.0781278\pi\)
\(14\) 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 \(-0.444991\pi\)
−0.985105 + 0.171956i \(0.944991\pi\)
\(18\) 0 0
\(19\) 72.0000i 0.199446i 0.995015 + 0.0997230i \(0.0317957\pi\)
−0.995015 + 0.0997230i \(0.968204\pi\)
\(20\) − 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 \(-0.900172\pi\)
0.308502 + 0.951224i \(0.400172\pi\)
\(24\) 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 0.375375 − 0.926873i \(-0.377514\pi\)
−0.926873 + 0.375375i \(0.877514\pi\)
\(38\) −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 \(-0.591187\pi\)
−0.282570 + 0.959247i \(0.591187\pi\)
\(42\) 0 0
\(43\) −1220.00 + 1220.00i −0.659816 + 0.659816i −0.955336 − 0.295520i \(-0.904507\pi\)
0.295520 + 0.955336i \(0.404507\pi\)
\(44\) 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 \(-0.217337\pi\)
−0.630956 + 0.775818i \(0.717337\pi\)
\(48\) 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 \(-0.790574\pi\)
0.611480 + 0.791260i \(0.290574\pi\)
\(54\) 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 0.743962 − 0.668222i \(-0.232944\pi\)
−0.668222 + 0.743962i \(0.732944\pi\)
\(68\) 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 \(-0.390508\pi\)
0.337235 + 0.941421i \(0.390508\pi\)
\(72\) 0 0
\(73\) 415.000 − 415.000i 0.0778758 − 0.0778758i −0.667096 − 0.744972i \(-0.732463\pi\)
0.744972 + 0.667096i \(0.232463\pi\)
\(74\) − 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 \(-0.181325\pi\)
\(80\) 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 \(-0.727765\pi\)
0.754736 + 0.656028i \(0.227765\pi\)
\(84\) 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 0.787701 − 0.616057i \(-0.211271\pi\)
−0.616057 + 0.787701i \(0.711271\pi\)
\(98\) −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.c.37.1 yes 2
3.2 odd 2 45.5.g.a.37.1 yes 2
5.2 odd 4 225.5.g.a.118.1 2
5.3 odd 4 inner 45.5.g.c.28.1 yes 2
5.4 even 2 225.5.g.a.82.1 2
15.2 even 4 225.5.g.c.118.1 2
15.8 even 4 45.5.g.a.28.1 ✓ 2
15.14 odd 2 225.5.g.c.82.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.g.a.28.1 ✓ 2 15.8 even 4
45.5.g.a.37.1 yes 2 3.2 odd 2
45.5.g.c.28.1 yes 2 5.3 odd 4 inner
45.5.g.c.37.1 yes 2 1.1 even 1 trivial
225.5.g.a.82.1 2 5.4 even 2
225.5.g.a.118.1 2 5.2 odd 4
225.5.g.c.82.1 2 15.14 odd 2
225.5.g.c.118.1 2 15.2 even 4