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Properties

Label 300.3.t.a.19.2
Level $300$
Weight $3$
Character 300.19
Analytic conductor $8.174$
Analytic rank $0$
Dimension $240$
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] = [300,3,Mod(19,300)] 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("300.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 9])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.t (of order \(10\), degree \(4\), 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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 19.2
Character \(\chi\) \(=\) 300.19
Dual form 300.3.t.a.79.2

$q$-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+(-1.96691 - 0.362292i) q^{2} +(1.40126 - 1.01807i) q^{3} +(3.73749 + 1.42519i) q^{4} +(-0.557886 + 4.96878i) q^{5} +(-3.12499 + 1.49480i) q^{6} +12.1496 q^{7} +(-6.83498 - 4.15729i) q^{8} +(0.927051 - 2.85317i) q^{9} +(2.89746 - 9.57104i) q^{10} +(-0.624518 + 0.202918i) q^{11} +(6.68814 - 1.80798i) q^{12} +(2.73670 + 0.889208i) q^{13} +(-23.8972 - 4.40170i) q^{14} +(4.27684 + 7.53051i) q^{15} +(11.9377 + 10.6533i) q^{16} +(-6.80062 + 9.36025i) q^{17} +(-2.85711 + 5.27607i) q^{18} +(-8.53843 + 11.7521i) q^{19} +(-9.16656 + 17.7757i) q^{20} +(17.0247 - 12.3692i) q^{21} +(1.30189 - 0.172865i) q^{22} +(8.34118 + 25.6715i) q^{23} +(-13.8100 + 1.13308i) q^{24} +(-24.3775 - 5.54403i) q^{25} +(-5.06070 - 2.74048i) q^{26} +(-1.60570 - 4.94183i) q^{27} +(45.4090 + 17.3155i) q^{28} +(36.3973 - 26.4442i) q^{29} +(-5.68393 - 16.3613i) q^{30} +(3.78325 - 5.20720i) q^{31} +(-19.6207 - 25.2790i) q^{32} +(-0.668526 + 0.920147i) q^{33} +(16.7674 - 15.9470i) q^{34} +(-6.77810 + 60.3687i) q^{35} +(7.53116 - 9.34247i) q^{36} +(32.3862 + 10.5229i) q^{37} +(21.0521 - 20.0220i) q^{38} +(4.74010 - 1.54015i) q^{39} +(24.4698 - 31.6422i) q^{40} +(7.70786 - 23.7224i) q^{41} +(-37.9674 + 18.1612i) q^{42} +10.4531 q^{43} +(-2.62333 - 0.131653i) q^{44} +(13.6596 + 6.19806i) q^{45} +(-7.10580 - 53.5156i) q^{46} +(-40.9978 + 29.7867i) q^{47} +(27.5736 + 2.77458i) q^{48} +98.6129 q^{49} +(45.9399 + 19.7364i) q^{50} +20.0397i q^{51} +(8.96110 + 7.22373i) q^{52} +(26.6235 + 36.6441i) q^{53} +(1.36788 + 10.3019i) q^{54} +(-0.659846 - 3.21630i) q^{55} +(-83.0423 - 50.5094i) q^{56} +25.1605i q^{57} +(-81.1708 + 38.8270i) q^{58} +(84.6228 + 27.4956i) q^{59} +(5.25222 + 34.2405i) q^{60} +(-20.3992 - 62.7822i) q^{61} +(-9.32784 + 8.87146i) q^{62} +(11.2633 - 34.6649i) q^{63} +(29.4339 + 56.8300i) q^{64} +(-5.94504 + 13.1020i) q^{65} +(1.64829 - 1.56765i) q^{66} +(79.9693 + 58.1011i) q^{67} +(-38.7574 + 25.2916i) q^{68} +(37.8237 + 27.4805i) q^{69} +(35.2030 - 116.284i) q^{70} +(-64.0626 - 88.1746i) q^{71} +(-18.1978 + 15.6473i) q^{72} +(-88.8778 + 28.8782i) q^{73} +(-59.8884 - 32.4309i) q^{74} +(-39.8034 + 17.0495i) q^{75} +(-48.6614 + 31.7546i) q^{76} +(-7.58765 + 2.46538i) q^{77} +(-9.88135 + 1.31205i) q^{78} +(26.4459 + 36.3997i) q^{79} +(-59.5936 + 53.3722i) q^{80} +(-7.28115 - 5.29007i) q^{81} +(-23.7551 + 43.8673i) q^{82} +(50.1589 + 36.4425i) q^{83} +(81.2582 - 21.9662i) q^{84} +(-42.7150 - 39.0127i) q^{85} +(-20.5603 - 3.78706i) q^{86} +(24.0799 - 74.1103i) q^{87} +(5.11216 + 1.20936i) q^{88} +(-28.9337 - 89.0487i) q^{89} +(-24.6217 - 17.1398i) q^{90} +(33.2498 + 10.8035i) q^{91} +(-5.41176 + 107.835i) q^{92} -11.1483i q^{93} +(91.4306 - 43.7346i) q^{94} +(-53.6303 - 48.9819i) q^{95} +(-53.2296 - 15.4470i) q^{96} +(-110.860 - 152.586i) q^{97} +(-193.963 - 35.7266i) q^{98} +1.96997i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 4 q^{5} - 30 q^{8} - 180 q^{9} + 22 q^{10} + 60 q^{14} - 60 q^{16} - 52 q^{20} - 150 q^{22} + 68 q^{25} - 100 q^{26} + 40 q^{29} - 60 q^{30} + 80 q^{34} + 20 q^{37} + 330 q^{38} + 296 q^{40} + 200 q^{41}+ \cdots - 140 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{9}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.96691 − 0.362292i −0.983456 − 0.181146i
\(3\) 1.40126 − 1.01807i 0.467086 − 0.339358i
\(4\) 3.73749 + 1.42519i 0.934372 + 0.356298i
\(5\) −0.557886 + 4.96878i −0.111577 + 0.993756i
\(6\) −3.12499 + 1.49480i −0.520832 + 0.249133i
\(7\) 12.1496 1.73566 0.867829 − 0.496863i \(-0.165515\pi\)
0.867829 + 0.496863i \(0.165515\pi\)
\(8\) −6.83498 − 4.15729i −0.854372 − 0.519661i
\(9\) 0.927051 − 2.85317i 0.103006 − 0.317019i
\(10\) 2.89746 − 9.57104i 0.289746 − 0.957104i
\(11\) −0.624518 + 0.202918i −0.0567744 + 0.0184471i −0.337267 − 0.941409i \(-0.609502\pi\)
0.280492 + 0.959856i \(0.409502\pi\)
\(12\) 6.68814 − 1.80798i 0.557345 − 0.150665i
\(13\) 2.73670 + 0.889208i 0.210515 + 0.0684006i 0.412376 − 0.911014i \(-0.364699\pi\)
−0.201861 + 0.979414i \(0.564699\pi\)
\(14\) −23.8972 − 4.40170i −1.70694 − 0.314407i
\(15\) 4.27684 + 7.53051i 0.285123 + 0.502034i
\(16\) 11.9377 + 10.6533i 0.746103 + 0.665830i
\(17\) −6.80062 + 9.36025i −0.400036 + 0.550603i −0.960753 − 0.277405i \(-0.910526\pi\)
0.560717 + 0.828008i \(0.310526\pi\)
\(18\) −2.85711 + 5.27607i −0.158728 + 0.293115i
\(19\) −8.53843 + 11.7521i −0.449391 + 0.618534i −0.972267 − 0.233875i \(-0.924859\pi\)
0.522875 + 0.852409i \(0.324859\pi\)
\(20\) −9.16656 + 17.7757i −0.458328 + 0.888783i
\(21\) 17.0247 − 12.3692i 0.810702 − 0.589009i
\(22\) 1.30189 − 0.172865i 0.0591767 − 0.00785749i
\(23\) 8.34118 + 25.6715i 0.362660 + 1.11615i 0.951433 + 0.307855i \(0.0996111\pi\)
−0.588773 + 0.808298i \(0.700389\pi\)
\(24\) −13.8100 + 1.13308i −0.575417 + 0.0472116i
\(25\) −24.3775 − 5.54403i −0.975101 − 0.221761i
\(26\) −5.06070 − 2.74048i −0.194642 − 0.105403i
\(27\) −1.60570 − 4.94183i −0.0594703 − 0.183031i
\(28\) 45.4090 + 17.3155i 1.62175 + 0.618411i
\(29\) 36.3973 − 26.4442i 1.25508 − 0.911868i 0.256574 − 0.966525i \(-0.417406\pi\)
0.998505 + 0.0546564i \(0.0174063\pi\)
\(30\) −5.68393 − 16.3613i −0.189464 − 0.545377i
\(31\) 3.78325 − 5.20720i 0.122040 − 0.167974i −0.743626 − 0.668596i \(-0.766895\pi\)
0.865666 + 0.500622i \(0.166895\pi\)
\(32\) −19.6207 − 25.2790i −0.613148 − 0.789968i
\(33\) −0.668526 + 0.920147i −0.0202584 + 0.0278832i
\(34\) 16.7674 − 15.9470i 0.493158 − 0.469029i
\(35\) −6.77810 + 60.3687i −0.193660 + 1.72482i
\(36\) 7.53116 − 9.34247i 0.209199 − 0.259513i
\(37\) 32.3862 + 10.5229i 0.875302 + 0.284403i 0.712005 − 0.702174i \(-0.247787\pi\)
0.163297 + 0.986577i \(0.447787\pi\)
\(38\) 21.0521 − 20.0220i 0.554001 − 0.526896i
\(39\) 4.74010 − 1.54015i 0.121541 − 0.0394911i
\(40\) 24.4698 − 31.6422i 0.611745 − 0.791055i
\(41\) 7.70786 − 23.7224i 0.187997 − 0.578594i −0.811990 − 0.583671i \(-0.801616\pi\)
0.999987 + 0.00507648i \(0.00161590\pi\)
\(42\) −37.9674 + 18.1612i −0.903986 + 0.432410i
\(43\) 10.4531 0.243095 0.121547 − 0.992586i \(-0.461214\pi\)
0.121547 + 0.992586i \(0.461214\pi\)
\(44\) −2.62333 − 0.131653i −0.0596211 − 0.00299212i
\(45\) 13.6596 + 6.19806i 0.303546 + 0.137735i
\(46\) −7.10580 − 53.5156i −0.154474 − 1.16338i
\(47\) −40.9978 + 29.7867i −0.872295 + 0.633759i −0.931202 − 0.364504i \(-0.881238\pi\)
0.0589070 + 0.998263i \(0.481238\pi\)
\(48\) 27.5736 + 2.77458i 0.574449 + 0.0578038i
\(49\) 98.6129 2.01251
\(50\) 45.9399 + 19.7364i 0.918798 + 0.394728i
\(51\) 20.0397i 0.392935i
\(52\) 8.96110 + 7.22373i 0.172329 + 0.138918i
\(53\) 26.6235 + 36.6441i 0.502330 + 0.691398i 0.982602 − 0.185721i \(-0.0594621\pi\)
−0.480272 + 0.877119i \(0.659462\pi\)
\(54\) 1.36788 + 10.3019i 0.0253312 + 0.190776i
\(55\) −0.659846 − 3.21630i −0.0119972 − 0.0584782i
\(56\) −83.0423 − 50.5094i −1.48290 − 0.901954i
\(57\) 25.1605i 0.441413i
\(58\) −81.1708 + 38.8270i −1.39950 + 0.669430i
\(59\) 84.6228 + 27.4956i 1.43428 + 0.466027i 0.920111 − 0.391659i \(-0.128099\pi\)
0.514174 + 0.857686i \(0.328099\pi\)
\(60\) 5.25222 + 34.2405i 0.0875370 + 0.570676i
\(61\) −20.3992 − 62.7822i −0.334413 − 1.02922i −0.967011 − 0.254736i \(-0.918011\pi\)
0.632598 − 0.774480i \(-0.281989\pi\)
\(62\) −9.32784 + 8.87146i −0.150449 + 0.143088i
\(63\) 11.2633 − 34.6649i 0.178783 − 0.550236i
\(64\) 29.4339 + 56.8300i 0.459905 + 0.887968i
\(65\) −5.94504 + 13.1020i −0.0914622 + 0.201569i
\(66\) 1.64829 − 1.56765i 0.0249741 − 0.0237522i
\(67\) 79.9693 + 58.1011i 1.19357 + 0.867181i 0.993637 − 0.112629i \(-0.0359270\pi\)
0.199935 + 0.979809i \(0.435927\pi\)
\(68\) −38.7574 + 25.2916i −0.569962 + 0.371936i
\(69\) 37.8237 + 27.4805i 0.548169 + 0.398268i
\(70\) 35.2030 − 116.284i 0.502900 − 1.66120i
\(71\) −64.0626 − 88.1746i −0.902291 − 1.24190i −0.969732 − 0.244173i \(-0.921483\pi\)
0.0674412 − 0.997723i \(-0.478517\pi\)
\(72\) −18.1978 + 15.6473i −0.252748 + 0.217324i
\(73\) −88.8778 + 28.8782i −1.21750 + 0.395591i −0.846173 − 0.532909i \(-0.821099\pi\)
−0.371332 + 0.928500i \(0.621099\pi\)
\(74\) −59.8884 − 32.4309i −0.809303 − 0.438255i
\(75\) −39.8034 + 17.0495i −0.530713 + 0.227327i
\(76\) −48.6614 + 31.7546i −0.640281 + 0.417824i
\(77\) −7.58765 + 2.46538i −0.0985409 + 0.0320179i
\(78\) −9.88135 + 1.31205i −0.126684 + 0.0168211i
\(79\) 26.4459 + 36.3997i 0.334758 + 0.460755i 0.942901 − 0.333072i \(-0.108085\pi\)
−0.608143 + 0.793827i \(0.708085\pi\)
\(80\) −59.5936 + 53.3722i −0.744921 + 0.667153i
\(81\) −7.28115 − 5.29007i −0.0898908 − 0.0653095i
\(82\) −23.7551 + 43.8673i −0.289696 + 0.534967i
\(83\) 50.1589 + 36.4425i 0.604324 + 0.439067i 0.847411 − 0.530938i \(-0.178160\pi\)
−0.243087 + 0.970004i \(0.578160\pi\)
\(84\) 81.2582 − 21.9662i 0.967360 − 0.261503i
\(85\) −42.7150 − 39.0127i −0.502530 − 0.458973i
\(86\) −20.5603 − 3.78706i −0.239073 − 0.0440356i
\(87\) 24.0799 − 74.1103i 0.276780 − 0.851842i
\(88\) 5.11216 + 1.20936i 0.0580927 + 0.0137427i
\(89\) −28.9337 − 89.0487i −0.325097 − 1.00055i −0.971397 − 0.237463i \(-0.923684\pi\)
0.646299 − 0.763084i \(-0.276316\pi\)
\(90\) −24.6217 − 17.1398i −0.273574 − 0.190442i
\(91\) 33.2498 + 10.8035i 0.365383 + 0.118720i
\(92\) −5.41176 + 107.835i −0.0588235 + 1.17212i
\(93\) − 11.1483i − 0.119874i
\(94\) 91.4306 − 43.7346i 0.972666 − 0.465262i
\(95\) −53.6303 − 48.9819i −0.564530 − 0.515599i
\(96\) −53.2296 − 15.4470i −0.554475 − 0.160907i
\(97\) −110.860 − 152.586i −1.14289 − 1.57305i −0.760877 − 0.648896i \(-0.775231\pi\)
−0.382013 − 0.924157i \(-0.624769\pi\)
\(98\) −193.963 − 35.7266i −1.97921 − 0.364557i
\(99\) 1.96997i 0.0198987i
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 300.3.t.a.19.2 ✓ 240
4.3 odd 2 inner 300.3.t.a.19.15 yes 240
25.4 even 10 inner 300.3.t.a.79.15 yes 240
100.79 odd 10 inner 300.3.t.a.79.2 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.t.a.19.2 ✓ 240 1.1 even 1 trivial
300.3.t.a.19.15 yes 240 4.3 odd 2 inner
300.3.t.a.79.2 yes 240 100.79 odd 10 inner
300.3.t.a.79.15 yes 240 25.4 even 10 inner