Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 299.5
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.c.99.5

$q$-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+(-0.657135 - 0.551402i) q^{2} +(1.35977 + 1.62051i) q^{3} +(-0.219514 - 1.24492i) q^{4} -1.81467i q^{6} +(-0.0421968 + 0.115935i) q^{7} +(-1.40003 + 2.42493i) q^{8} +(-0.256133 + 1.45260i) q^{9} +(2.92918 - 5.07348i) q^{11} +(1.71892 - 2.04853i) q^{12} +(-1.10194 - 6.24944i) q^{13} +(0.0916557 - 0.0529175i) q^{14} +(-0.118663 + 0.0431898i) q^{16} +(-1.20417 + 6.82921i) q^{17} +(0.969284 - 0.813325i) q^{18} +(-0.314377 - 0.374660i) q^{19} +(-0.245251 + 0.0892641i) q^{21} +(-4.72240 + 1.71881i) q^{22} +(-1.15250 - 1.99620i) q^{23} +(-5.83333 + 1.02857i) q^{24} +(-2.72183 + 4.71434i) q^{26} +(2.79379 - 1.61300i) q^{27} +(0.153593 + 0.0270826i) q^{28} +(-6.09229 - 3.51738i) q^{29} -5.43133i q^{31} +(5.36419 + 1.95241i) q^{32} +(12.2046 - 2.15200i) q^{33} +(4.55695 - 3.82373i) q^{34} +1.86461 q^{36} +(1.47377 - 5.90153i) q^{37} +0.419551i q^{38} +(8.62888 - 10.2835i) q^{39} +(0.155689 + 0.882953i) q^{41} +(0.210384 + 0.0765734i) q^{42} +3.16299 q^{43} +(-6.95910 - 2.53290i) q^{44} +(-0.343355 + 1.94727i) q^{46} +(7.72842 - 4.46201i) q^{47} +(-0.231343 - 0.133566i) q^{48} +(5.35065 + 4.48973i) q^{49} +(-12.7042 + 7.33476i) q^{51} +(-7.53818 + 2.74367i) q^{52} +(0.657192 + 1.80562i) q^{53} +(-2.72531 - 0.480545i) q^{54} +(-0.222057 - 0.264637i) q^{56} +(0.179660 - 1.01890i) q^{57} +(2.06397 + 5.67070i) q^{58} +(0.732827 + 2.01343i) q^{59} +(-7.50073 + 1.32258i) q^{61} +(-2.99485 + 3.56912i) q^{62} +(-0.157599 - 0.0909901i) q^{63} +(-2.32216 - 4.02210i) q^{64} +(-9.20671 - 5.31550i) q^{66} +(-3.55247 + 9.76033i) q^{67} +8.76618 q^{68} +(1.66771 - 4.58201i) q^{69} +(11.4875 - 9.63918i) q^{71} +(-3.16387 - 2.65480i) q^{72} -2.19090i q^{73} +(-4.22258 + 3.06546i) q^{74} +(-0.397414 + 0.473619i) q^{76} +(0.464592 + 0.553679i) q^{77} +(-11.3407 + 1.99967i) q^{78} +(1.44700 - 3.97561i) q^{79} +(10.5710 + 3.84751i) q^{81} +(0.384554 - 0.666067i) q^{82} +(-4.19839 - 0.740289i) q^{83} +(0.164963 + 0.285724i) q^{84} +(-2.07851 - 1.74408i) q^{86} +(-2.58415 - 14.6554i) q^{87} +(8.20189 + 14.2061i) q^{88} +(-2.85696 - 7.84942i) q^{89} +(0.771026 + 0.135953i) q^{91} +(-2.23212 + 1.87297i) q^{92} +(8.80152 - 7.38535i) q^{93} +(-7.53898 - 1.32933i) q^{94} +(4.13017 + 11.3475i) q^{96} +(-4.08476 - 7.07501i) q^{97} +(-1.04046 - 5.90072i) q^{98} +(6.61951 + 5.55442i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} + 6 q^{7} + 18 q^{8} + 42 q^{12} + 18 q^{13} + 18 q^{14} - 18 q^{16} - 12 q^{17} - 24 q^{18} + 18 q^{19} - 24 q^{22} + 24 q^{23} + 30 q^{24} + 48 q^{28} - 18 q^{29} - 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.657135 − 0.551402i −0.464665 − 0.389900i 0.380179 − 0.924913i \(-0.375862\pi\)
−0.844844 + 0.535013i \(0.820307\pi\)
\(3\) 1.35977 + 1.62051i 0.785062 + 0.935601i 0.999150 − 0.0412137i \(-0.0131224\pi\)
−0.214088 + 0.976814i \(0.568678\pi\)
\(4\) −0.219514 − 1.24492i −0.109757 − 0.622462i
\(5\) 0 0
\(6\) − 1.81467i − 0.740837i
\(7\) −0.0421968 + 0.115935i −0.0159489 + 0.0438192i −0.947412 − 0.320018i \(-0.896311\pi\)
0.931463 + 0.363837i \(0.118533\pi\)
\(8\) −1.40003 + 2.42493i −0.494986 + 0.857342i
\(9\) −0.256133 + 1.45260i −0.0853778 + 0.484201i
\(10\) 0 0
\(11\) 2.92918 − 5.07348i 0.883180 − 1.52971i 0.0353950 − 0.999373i \(-0.488731\pi\)
0.847785 − 0.530340i \(-0.177936\pi\)
\(12\) 1.71892 − 2.04853i 0.496210 − 0.591360i
\(13\) −1.10194 − 6.24944i −0.305625 − 1.73328i −0.620550 − 0.784167i \(-0.713091\pi\)
0.314926 − 0.949116i \(-0.398020\pi\)
\(14\) 0.0916557 − 0.0529175i 0.0244960 − 0.0141428i
\(15\) 0 0
\(16\) −0.118663 + 0.0431898i −0.0296657 + 0.0107974i
\(17\) −1.20417 + 6.82921i −0.292055 + 1.65633i 0.386881 + 0.922129i \(0.373552\pi\)
−0.678936 + 0.734197i \(0.737559\pi\)
\(18\) 0.969284 − 0.813325i 0.228462 − 0.191703i
\(19\) −0.314377 − 0.374660i −0.0721231 − 0.0859530i 0.728776 − 0.684752i \(-0.240089\pi\)
−0.800900 + 0.598799i \(0.795645\pi\)
\(20\) 0 0
\(21\) −0.245251 + 0.0892641i −0.0535182 + 0.0194790i
\(22\) −4.72240 + 1.71881i −1.00682 + 0.366452i
\(23\) −1.15250 − 1.99620i −0.240314 − 0.416236i 0.720490 − 0.693466i \(-0.243917\pi\)
−0.960804 + 0.277230i \(0.910584\pi\)
\(24\) −5.83333 + 1.02857i −1.19072 + 0.209957i
\(25\) 0 0
\(26\) −2.72183 + 4.71434i −0.533794 + 0.924559i
\(27\) 2.79379 − 1.61300i 0.537665 − 0.310421i
\(28\) 0.153593 + 0.0270826i 0.0290263 + 0.00511812i
\(29\) −6.09229 − 3.51738i −1.13131 − 0.653162i −0.187046 − 0.982351i \(-0.559891\pi\)
−0.944264 + 0.329189i \(0.893225\pi\)
\(30\) 0 0
\(31\) − 5.43133i − 0.975496i −0.872984 − 0.487748i \(-0.837818\pi\)
0.872984 − 0.487748i \(-0.162182\pi\)
\(32\) 5.36419 + 1.95241i 0.948265 + 0.345140i
\(33\) 12.2046 − 2.15200i 2.12455 − 0.374616i
\(34\) 4.55695 − 3.82373i 0.781510 − 0.655765i
\(35\) 0 0
\(36\) 1.86461 0.310768
\(37\) 1.47377 − 5.90153i 0.242286 − 0.970205i
\(38\) 0.419551i 0.0680602i
\(39\) 8.62888 − 10.2835i 1.38173 − 1.64668i
\(40\) 0 0
\(41\) 0.155689 + 0.882953i 0.0243145 + 0.137894i 0.994549 − 0.104275i \(-0.0332522\pi\)
−0.970234 + 0.242169i \(0.922141\pi\)
\(42\) 0.210384 + 0.0765734i 0.0324629 + 0.0118155i
\(43\) 3.16299 0.482352 0.241176 − 0.970481i \(-0.422467\pi\)
0.241176 + 0.970481i \(0.422467\pi\)
\(44\) −6.95910 − 2.53290i −1.04912 − 0.381850i
\(45\) 0 0
\(46\) −0.343355 + 1.94727i −0.0506250 + 0.287109i
\(47\) 7.72842 − 4.46201i 1.12731 − 0.650851i 0.184050 − 0.982917i \(-0.441079\pi\)
0.943256 + 0.332066i \(0.107746\pi\)
\(48\) −0.231343 − 0.133566i −0.0333916 − 0.0192786i
\(49\) 5.35065 + 4.48973i 0.764379 + 0.641390i
\(50\) 0 0
\(51\) −12.7042 + 7.33476i −1.77894 + 1.02707i
\(52\) −7.53818 + 2.74367i −1.04536 + 0.380479i
\(53\) 0.657192 + 1.80562i 0.0902723 + 0.248021i 0.976610 − 0.215018i \(-0.0689811\pi\)
−0.886338 + 0.463039i \(0.846759\pi\)
\(54\) −2.72531 − 0.480545i −0.370867 − 0.0653939i
\(55\) 0 0
\(56\) −0.222057 − 0.264637i −0.0296736 − 0.0353636i
\(57\) 0.179660 − 1.01890i 0.0237965 − 0.134957i
\(58\) 2.06397 + 5.67070i 0.271012 + 0.744599i
\(59\) 0.732827 + 2.01343i 0.0954060 + 0.262126i 0.978211 − 0.207614i \(-0.0665699\pi\)
−0.882805 + 0.469740i \(0.844348\pi\)
\(60\) 0 0
\(61\) −7.50073 + 1.32258i −0.960371 + 0.169339i −0.631793 − 0.775138i \(-0.717681\pi\)
−0.328578 + 0.944477i \(0.606569\pi\)
\(62\) −2.99485 + 3.56912i −0.380346 + 0.453279i
\(63\) −0.157599 − 0.0909901i −0.0198557 − 0.0114637i
\(64\) −2.32216 − 4.02210i −0.290270 − 0.502763i
\(65\) 0 0
\(66\) −9.20671 − 5.31550i −1.13327 − 0.654292i
\(67\) −3.55247 + 9.76033i −0.434003 + 1.19241i 0.509331 + 0.860571i \(0.329893\pi\)
−0.943334 + 0.331844i \(0.892329\pi\)
\(68\) 8.76618 1.06306
\(69\) 1.66771 − 4.58201i 0.200769 − 0.551609i
\(70\) 0 0
\(71\) 11.4875 − 9.63918i 1.36332 − 1.14396i 0.388377 − 0.921500i \(-0.373036\pi\)
0.974942 − 0.222460i \(-0.0714087\pi\)
\(72\) −3.16387 − 2.65480i −0.372865 − 0.312871i
\(73\) − 2.19090i − 0.256425i −0.991747 − 0.128213i \(-0.959076\pi\)
0.991747 − 0.128213i \(-0.0409240\pi\)
\(74\) −4.22258 + 3.06546i −0.490865 + 0.356353i
\(75\) 0 0
\(76\) −0.397414 + 0.473619i −0.0455865 + 0.0543278i
\(77\) 0.464592 + 0.553679i 0.0529451 + 0.0630975i
\(78\) −11.3407 + 1.99967i −1.28408 + 0.226418i
\(79\) 1.44700 − 3.97561i 0.162801 − 0.447291i −0.831291 − 0.555838i \(-0.812398\pi\)
0.994091 + 0.108547i \(0.0346197\pi\)
\(80\) 0 0
\(81\) 10.5710 + 3.84751i 1.17455 + 0.427501i
\(82\) 0.384554 − 0.666067i 0.0424669 − 0.0735548i
\(83\) −4.19839 − 0.740289i −0.460833 − 0.0812573i −0.0615882 − 0.998102i \(-0.519617\pi\)
−0.399245 + 0.916844i \(0.630728\pi\)
\(84\) 0.164963 + 0.285724i 0.0179989 + 0.0311751i
\(85\) 0 0
\(86\) −2.07851 − 1.74408i −0.224132 − 0.188069i
\(87\) −2.58415 − 14.6554i −0.277050 − 1.57123i
\(88\) 8.20189 + 14.2061i 0.874324 + 1.51437i
\(89\) −2.85696 − 7.84942i −0.302837 − 0.832037i −0.994004 − 0.109342i \(-0.965126\pi\)
0.691167 − 0.722695i \(-0.257097\pi\)
\(90\) 0 0
\(91\) 0.771026 + 0.135953i 0.0808255 + 0.0142517i
\(92\) −2.23212 + 1.87297i −0.232715 + 0.195271i
\(93\) 8.80152 − 7.38535i 0.912675 − 0.765825i
\(94\) −7.53898 − 1.32933i −0.777586 − 0.137109i
\(95\) 0 0
\(96\) 4.13017 + 11.3475i 0.421533 + 1.15815i
\(97\) −4.08476 − 7.07501i −0.414744 − 0.718358i 0.580657 − 0.814148i \(-0.302796\pi\)
−0.995402 + 0.0957899i \(0.969462\pi\)
\(98\) −1.04046 − 5.90072i −0.105102 − 0.596063i
\(99\) 6.61951 + 5.55442i 0.665285 + 0.558241i
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 925.2.ba.c.299.5 72
5.2 odd 4 185.2.w.a.151.9 yes 72
5.3 odd 4 925.2.bb.b.151.4 72
5.4 even 2 925.2.ba.b.299.8 72
37.25 even 18 925.2.ba.b.99.8 72
185.32 even 36 6845.2.a.u.1.22 36
185.42 even 36 6845.2.a.t.1.15 36
185.62 odd 36 185.2.w.a.136.9 ✓ 72
185.99 even 18 inner 925.2.ba.c.99.5 72
185.173 odd 36 925.2.bb.b.876.4 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.9 ✓ 72 185.62 odd 36
185.2.w.a.151.9 yes 72 5.2 odd 4
925.2.ba.b.99.8 72 37.25 even 18
925.2.ba.b.299.8 72 5.4 even 2
925.2.ba.c.99.5 72 185.99 even 18 inner
925.2.ba.c.299.5 72 1.1 even 1 trivial
925.2.bb.b.151.4 72 5.3 odd 4
925.2.bb.b.876.4 72 185.173 odd 36
6845.2.a.t.1.15 36 185.42 even 36
6845.2.a.u.1.22 36 185.32 even 36