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Properties

Label 45.5.h.a.29.19
Level $45$
Weight $5$
Character 45.29
Analytic conductor $4.652$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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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(14,45)] 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("45.14"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.65164833877\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 29.19
Character \(\chi\) \(=\) 45.29
Dual form 45.5.h.a.14.19

$q$-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+(2.87361 - 4.97724i) q^{2} +(8.30470 + 3.46872i) q^{3} +(-8.51525 - 14.7488i) q^{4} +(24.3310 + 5.74467i) q^{5} +(41.1291 - 31.3667i) q^{6} +(-41.4151 - 23.9110i) q^{7} -5.92246 q^{8} +(56.9359 + 57.6134i) q^{9} +(98.5104 - 104.593i) q^{10} +(-73.2455 - 42.2883i) q^{11} +(-19.5569 - 152.022i) q^{12} +(-193.485 + 111.709i) q^{13} +(-238.021 + 137.422i) q^{14} +(182.135 + 132.105i) q^{15} +(119.225 - 206.504i) q^{16} -434.998 q^{17} +(450.367 - 117.825i) q^{18} +378.461 q^{19} +(-122.457 - 407.772i) q^{20} +(-260.999 - 342.231i) q^{21} +(-420.958 + 243.040i) q^{22} +(326.825 + 566.078i) q^{23} +(-49.1842 - 20.5434i) q^{24} +(558.998 + 279.547i) q^{25} +1284.03i q^{26} +(272.991 + 675.956i) q^{27} +814.433i q^{28} +(-430.201 - 248.377i) q^{29} +(1180.90 - 526.910i) q^{30} +(151.498 + 262.402i) q^{31} +(-732.592 - 1268.89i) q^{32} +(-461.595 - 605.260i) q^{33} +(-1250.01 + 2165.09i) q^{34} +(-870.311 - 819.695i) q^{35} +(364.907 - 1330.33i) q^{36} -55.6914i q^{37} +(1087.55 - 1883.69i) q^{38} +(-1994.32 + 256.560i) q^{39} +(-144.099 - 34.0226i) q^{40} +(411.405 - 237.525i) q^{41} +(-2453.37 + 315.616i) q^{42} +(-707.073 - 408.229i) q^{43} +1440.38i q^{44} +(1054.34 + 1728.87i) q^{45} +3756.67 q^{46} +(435.232 - 753.844i) q^{47} +(1706.43 - 1301.39i) q^{48} +(-57.0270 - 98.7737i) q^{49} +(2997.71 - 1978.95i) q^{50} +(-3612.52 - 1508.89i) q^{51} +(3295.14 + 1902.45i) q^{52} -2339.28 q^{53} +(4148.86 + 583.694i) q^{54} +(-1539.21 - 1449.69i) q^{55} +(245.279 + 141.612i) q^{56} +(3143.00 + 1312.77i) q^{57} +(-2472.46 + 1427.47i) q^{58} +(-1019.68 + 588.710i) q^{59} +(397.475 - 3811.19i) q^{60} +(3456.52 - 5986.86i) q^{61} +1741.38 q^{62} +(-980.413 - 3747.46i) q^{63} -4605.53 q^{64} +(-5349.42 + 1606.48i) q^{65} +(-4338.96 + 558.188i) q^{66} +(3356.29 - 1937.75i) q^{67} +(3704.11 + 6415.71i) q^{68} +(750.618 + 5834.77i) q^{69} +(-6580.75 + 1976.26i) q^{70} +5822.30i q^{71} +(-337.201 - 341.213i) q^{72} -6443.82i q^{73} +(-277.189 - 160.035i) q^{74} +(3672.63 + 4260.56i) q^{75} +(-3222.69 - 5581.85i) q^{76} +(2022.31 + 3502.75i) q^{77} +(-4453.93 + 10663.5i) q^{78} +(-2446.72 + 4237.84i) q^{79} +(4087.17 - 4339.54i) q^{80} +(-77.5982 + 6560.54i) q^{81} -2730.21i q^{82} +(-3257.36 + 5641.91i) q^{83} +(-2825.04 + 6763.61i) q^{84} +(-10583.9 - 2498.92i) q^{85} +(-4063.70 + 2346.18i) q^{86} +(-2711.14 - 3554.94i) q^{87} +(433.793 + 250.451i) q^{88} -13898.7i q^{89} +(11634.8 - 279.602i) q^{90} +10684.3 q^{91} +(5566.00 - 9640.59i) q^{92} +(347.944 + 2704.67i) q^{93} +(-2501.37 - 4332.50i) q^{94} +(9208.33 + 2174.13i) q^{95} +(-1682.54 - 13078.9i) q^{96} +(11681.7 + 6744.42i) q^{97} -655.493 q^{98} +(-1733.93 - 6627.64i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 162 q^{4} + 6 q^{5} - 36 q^{6} - 126 q^{9} + 28 q^{10} + 228 q^{11} + 282 q^{14} + 450 q^{15} - 1058 q^{16} - 8 q^{19} - 2196 q^{20} + 1056 q^{21} + 2970 q^{24} - 148 q^{25} + 2370 q^{29} + 1068 q^{30}+ \cdots + 44568 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.87361 − 4.97724i 0.718402 − 1.24431i −0.243231 − 0.969968i \(-0.578207\pi\)
0.961633 − 0.274340i \(-0.0884595\pi\)
\(3\) 8.30470 + 3.46872i 0.922744 + 0.385414i
\(4\) −8.51525 − 14.7488i −0.532203 − 0.921802i
\(5\) 24.3310 + 5.74467i 0.973241 + 0.229787i
\(6\) 41.1291 − 31.3667i 1.14247 − 0.871297i
\(7\) −41.4151 − 23.9110i −0.845206 − 0.487980i 0.0138245 − 0.999904i \(-0.495599\pi\)
−0.859030 + 0.511925i \(0.828933\pi\)
\(8\) −5.92246 −0.0925384
\(9\) 56.9359 + 57.6134i 0.702913 + 0.711276i
\(10\) 98.5104 − 104.593i 0.985104 − 1.04593i
\(11\) −73.2455 − 42.2883i −0.605335 − 0.349490i 0.165803 − 0.986159i \(-0.446979\pi\)
−0.771137 + 0.636669i \(0.780312\pi\)
\(12\) −19.5569 − 152.022i −0.135812 − 1.05571i
\(13\) −193.485 + 111.709i −1.14488 + 0.660998i −0.947635 − 0.319356i \(-0.896533\pi\)
−0.197247 + 0.980354i \(0.563200\pi\)
\(14\) −238.021 + 137.422i −1.21440 + 0.701131i
\(15\) 182.135 + 132.105i 0.809489 + 0.587135i
\(16\) 119.225 − 206.504i 0.465723 − 0.806656i
\(17\) −434.998 −1.50518 −0.752591 − 0.658488i \(-0.771196\pi\)
−0.752591 + 0.658488i \(0.771196\pi\)
\(18\) 450.367 − 117.825i 1.39002 − 0.363658i
\(19\) 378.461 1.04837 0.524184 − 0.851605i \(-0.324371\pi\)
0.524184 + 0.851605i \(0.324371\pi\)
\(20\) −122.457 − 407.772i −0.306144 − 1.01943i
\(21\) −260.999 − 342.231i −0.591835 − 0.776034i
\(22\) −420.958 + 243.040i −0.869747 + 0.502149i
\(23\) 326.825 + 566.078i 0.617818 + 1.07009i 0.989883 + 0.141885i \(0.0453163\pi\)
−0.372066 + 0.928206i \(0.621350\pi\)
\(24\) −49.1842 − 20.5434i −0.0853893 − 0.0356656i
\(25\) 558.998 + 279.547i 0.894396 + 0.447276i
\(26\) 1284.03i 1.89945i
\(27\) 272.991 + 675.956i 0.374473 + 0.927238i
\(28\) 814.433i 1.03882i
\(29\) −430.201 − 248.377i −0.511535 − 0.295335i 0.221929 − 0.975063i \(-0.428765\pi\)
−0.733464 + 0.679728i \(0.762098\pi\)
\(30\) 1180.90 − 526.910i 1.31212 − 0.585456i
\(31\) 151.498 + 262.402i 0.157646 + 0.273051i 0.934019 − 0.357222i \(-0.116276\pi\)
−0.776373 + 0.630273i \(0.782943\pi\)
\(32\) −732.592 − 1268.89i −0.715422 − 1.23915i
\(33\) −461.595 − 605.260i −0.423871 − 0.555794i
\(34\) −1250.01 + 2165.09i −1.08133 + 1.87291i
\(35\) −870.311 − 819.695i −0.710458 − 0.669139i
\(36\) 364.907 − 1330.33i 0.281564 − 1.02649i
\(37\) − 55.6914i − 0.0406804i −0.999793 − 0.0203402i \(-0.993525\pi\)
0.999793 − 0.0203402i \(-0.00647493\pi\)
\(38\) 1087.55 − 1883.69i 0.753149 − 1.30449i
\(39\) −1994.32 + 256.560i −1.31119 + 0.168679i
\(40\) −144.099 − 34.0226i −0.0900622 − 0.0212641i
\(41\) 411.405 − 237.525i 0.244738 − 0.141300i −0.372614 − 0.927986i \(-0.621539\pi\)
0.617353 + 0.786687i \(0.288205\pi\)
\(42\) −2453.37 + 315.616i −1.39080 + 0.178920i
\(43\) −707.073 − 408.229i −0.382409 − 0.220784i 0.296457 − 0.955046i \(-0.404195\pi\)
−0.678866 + 0.734262i \(0.737528\pi\)
\(44\) 1440.38i 0.743999i
\(45\) 1054.34 + 1728.87i 0.520662 + 0.853763i
\(46\) 3756.67 1.77537
\(47\) 435.232 − 753.844i 0.197027 − 0.341260i −0.750536 − 0.660829i \(-0.770205\pi\)
0.947563 + 0.319569i \(0.103538\pi\)
\(48\) 1706.43 − 1301.39i 0.740639 − 0.564841i
\(49\) −57.0270 − 98.7737i −0.0237514 − 0.0411386i
\(50\) 2997.71 − 1978.95i 1.19909 − 0.791581i
\(51\) −3612.52 − 1508.89i −1.38890 − 0.580118i
\(52\) 3295.14 + 1902.45i 1.21862 + 0.703570i
\(53\) −2339.28 −0.832780 −0.416390 − 0.909186i \(-0.636705\pi\)
−0.416390 + 0.909186i \(0.636705\pi\)
\(54\) 4148.86 + 583.694i 1.42279 + 0.200169i
\(55\) −1539.21 − 1449.69i −0.508828 − 0.479236i
\(56\) 245.279 + 141.612i 0.0782140 + 0.0451569i
\(57\) 3143.00 + 1312.77i 0.967375 + 0.404055i
\(58\) −2472.46 + 1427.47i −0.734976 + 0.424338i
\(59\) −1019.68 + 588.710i −0.292926 + 0.169121i −0.639261 − 0.768990i \(-0.720760\pi\)
0.346335 + 0.938111i \(0.387426\pi\)
\(60\) 397.475 − 3811.19i 0.110410 − 1.05866i
\(61\) 3456.52 − 5986.86i 0.928921 − 1.60894i 0.143792 − 0.989608i \(-0.454070\pi\)
0.785130 − 0.619331i \(-0.212596\pi\)
\(62\) 1741.38 0.453013
\(63\) −980.413 − 3747.46i −0.247018 − 0.944182i
\(64\) −4605.53 −1.12440
\(65\) −5349.42 + 1606.48i −1.26613 + 0.380231i
\(66\) −4338.96 + 558.188i −0.996089 + 0.128142i
\(67\) 3356.29 − 1937.75i 0.747669 − 0.431667i −0.0771820 − 0.997017i \(-0.524592\pi\)
0.824851 + 0.565350i \(0.191259\pi\)
\(68\) 3704.11 + 6415.71i 0.801063 + 1.38748i
\(69\) 750.618 + 5834.77i 0.157660 + 1.22554i
\(70\) −6580.75 + 1976.26i −1.34301 + 0.403318i
\(71\) 5822.30i 1.15499i 0.816394 + 0.577495i \(0.195970\pi\)
−0.816394 + 0.577495i \(0.804030\pi\)
\(72\) −337.201 − 341.213i −0.0650464 − 0.0658204i
\(73\) − 6443.82i − 1.20920i −0.796530 − 0.604599i \(-0.793333\pi\)
0.796530 − 0.604599i \(-0.206667\pi\)
\(74\) −277.189 − 160.035i −0.0506189 − 0.0292249i
\(75\) 3672.63 + 4260.56i 0.652912 + 0.757434i
\(76\) −3222.69 − 5581.85i −0.557944 − 0.966388i
\(77\) 2022.31 + 3502.75i 0.341088 + 0.590782i
\(78\) −4453.93 + 10663.5i −0.732073 + 1.75270i
\(79\) −2446.72 + 4237.84i −0.392039 + 0.679032i −0.992718 − 0.120459i \(-0.961564\pi\)
0.600679 + 0.799490i \(0.294897\pi\)
\(80\) 4087.17 − 4339.54i 0.638620 − 0.678054i
\(81\) −77.5982 + 6560.54i −0.0118272 + 0.999930i
\(82\) − 2730.21i − 0.406040i
\(83\) −3257.36 + 5641.91i −0.472835 + 0.818974i −0.999517 − 0.0310882i \(-0.990103\pi\)
0.526682 + 0.850063i \(0.323436\pi\)
\(84\) −2825.04 + 6763.61i −0.400374 + 0.958562i
\(85\) −10583.9 − 2498.92i −1.46491 − 0.345871i
\(86\) −4063.70 + 2346.18i −0.549446 + 0.317223i
\(87\) −2711.14 − 3554.94i −0.358190 − 0.469671i
\(88\) 433.793 + 250.451i 0.0560167 + 0.0323413i
\(89\) − 13898.7i − 1.75467i −0.479881 − 0.877333i \(-0.659320\pi\)
0.479881 − 0.877333i \(-0.340680\pi\)
\(90\) 11634.8 − 279.602i 1.43639 − 0.0345187i
\(91\) 10684.3 1.29021
\(92\) 5566.00 − 9640.59i 0.657609 − 1.13901i
\(93\) 347.944 + 2704.67i 0.0402294 + 0.312715i
\(94\) −2501.37 − 4332.50i −0.283089 − 0.490324i
\(95\) 9208.33 + 2174.13i 1.02031 + 0.240901i
\(96\) −1682.54 − 13078.9i −0.182567 − 1.41915i
\(97\) 11681.7 + 6744.42i 1.24154 + 0.716805i 0.969408 − 0.245454i \(-0.0789372\pi\)
0.272134 + 0.962259i \(0.412271\pi\)
\(98\) −655.493 −0.0682521
\(99\) −1733.93 − 6627.64i −0.176914 − 0.676221i
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.h.a.29.19 yes 44
3.2 odd 2 135.5.h.a.89.4 44
5.4 even 2 inner 45.5.h.a.29.4 yes 44
9.2 odd 6 405.5.d.a.404.38 44
9.4 even 3 135.5.h.a.44.19 44
9.5 odd 6 inner 45.5.h.a.14.4 ✓ 44
9.7 even 3 405.5.d.a.404.8 44
15.14 odd 2 135.5.h.a.89.19 44
45.4 even 6 135.5.h.a.44.4 44
45.14 odd 6 inner 45.5.h.a.14.19 yes 44
45.29 odd 6 405.5.d.a.404.7 44
45.34 even 6 405.5.d.a.404.37 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.h.a.14.4 ✓ 44 9.5 odd 6 inner
45.5.h.a.14.19 yes 44 45.14 odd 6 inner
45.5.h.a.29.4 yes 44 5.4 even 2 inner
45.5.h.a.29.19 yes 44 1.1 even 1 trivial
135.5.h.a.44.4 44 45.4 even 6
135.5.h.a.44.19 44 9.4 even 3
135.5.h.a.89.4 44 3.2 odd 2
135.5.h.a.89.19 44 15.14 odd 2
405.5.d.a.404.7 44 45.29 odd 6
405.5.d.a.404.8 44 9.7 even 3
405.5.d.a.404.37 44 45.34 even 6
405.5.d.a.404.38 44 9.2 odd 6