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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,22,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-11982368] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(209.608008215\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.22.b.a.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2844.00i q^{2} +59049.0i q^{3} -5.99118e6 q^{4} +1.67935e8 q^{6} +3.63304e8i q^{7} +1.10746e10i q^{8} -3.48678e9 q^{9} +1.45818e10 q^{11} -3.53773e11i q^{12} -1.13351e11i q^{13} +1.03324e12 q^{14} +1.89318e13 q^{16} -8.58939e12i q^{17} +9.91641e12i q^{18} +2.92029e13 q^{19} -2.14527e13 q^{21} -4.14707e13i q^{22} +1.55899e14i q^{23} -6.53946e14 q^{24} -3.22370e14 q^{26} -2.05891e14i q^{27} -2.17662e15i q^{28} -2.40079e15 q^{29} +2.23982e15 q^{31} -3.06169e16i q^{32} +8.61043e14i q^{33} -2.44282e16 q^{34} +2.08900e16 q^{36} -3.07851e16i q^{37} -8.30532e16i q^{38} +6.69325e15 q^{39} -1.03208e17 q^{41} +6.10116e16i q^{42} +1.65557e17i q^{43} -8.73624e16 q^{44} +4.43377e17 q^{46} -6.65872e16i q^{47} +1.11790e18i q^{48} +4.26556e17 q^{49} +5.07195e17 q^{51} +6.79105e17i q^{52} -4.35423e17i q^{53} -5.85554e17 q^{54} -4.02346e18 q^{56} +1.72440e18i q^{57} +6.82784e18i q^{58} -5.53437e18 q^{59} -7.17621e18 q^{61} -6.37005e18i q^{62} -1.26676e18i q^{63} -4.73716e19 q^{64} +2.44881e18 q^{66} -1.57554e19i q^{67} +5.14606e19i q^{68} -9.20569e18 q^{69} +2.64579e19 q^{71} -3.86148e19i q^{72} -1.34712e19i q^{73} -8.75527e19 q^{74} -1.74960e20 q^{76} +5.29764e18i q^{77} -1.90356e19i q^{78} +1.68861e19 q^{79} +1.21577e19 q^{81} +2.93522e20i q^{82} +1.70688e20i q^{83} +1.28527e20 q^{84} +4.70845e20 q^{86} -1.41764e20i q^{87} +1.61488e20i q^{88} +3.12592e20 q^{89} +4.11808e19 q^{91} -9.34021e20i q^{92} +1.32259e20i q^{93} -1.89374e20 q^{94} +1.80790e21 q^{96} +9.49015e20i q^{97} -1.21313e21i q^{98} -5.08437e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9} + 29163666312 q^{11} + 2066472696960 q^{14} + 37863631405568 q^{16} + 58405878547592 q^{19} - 42905466344160 q^{21} - 13\!\cdots\!84 q^{24} - 644739297512976 q^{26}+ \cdots - 10\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2844.00i − 1.96388i −0.189196 − 0.981939i \(-0.560588\pi\)
0.189196 − 0.981939i \(-0.439412\pi\)
\(3\) 59049.0i 0.577350i
\(4\) −5.99118e6 −2.85682
\(5\) 0 0
\(6\) 1.67935e8 1.13385
\(7\) 3.63304e8i 0.486117i 0.970012 + 0.243058i \(0.0781507\pi\)
−0.970012 + 0.243058i \(0.921849\pi\)
\(8\) 1.10746e10i 3.64657i
\(9\) −3.48678e9 −0.333333
\(10\) 0 0
\(11\) 1.45818e10 0.169508 0.0847538 − 0.996402i \(-0.472990\pi\)
0.0847538 + 0.996402i \(0.472990\pi\)
\(12\) − 3.53773e11i − 1.64939i
\(13\) − 1.13351e11i − 0.228044i −0.993478 − 0.114022i \(-0.963627\pi\)
0.993478 − 0.114022i \(-0.0363735\pi\)
\(14\) 1.03324e12 0.954674
\(15\) 0 0
\(16\) 1.89318e13 4.30460
\(17\) − 8.58939e12i − 1.03335i −0.856181 − 0.516676i \(-0.827169\pi\)
0.856181 − 0.516676i \(-0.172831\pi\)
\(18\) 9.91641e12i 0.654626i
\(19\) 2.92029e13 1.09273 0.546366 − 0.837546i \(-0.316011\pi\)
0.546366 + 0.837546i \(0.316011\pi\)
\(20\) 0 0
\(21\) −2.14527e13 −0.280660
\(22\) − 4.14707e13i − 0.332892i
\(23\) 1.55899e14i 0.784696i 0.919817 + 0.392348i \(0.128337\pi\)
−0.919817 + 0.392348i \(0.871663\pi\)
\(24\) −6.53946e14 −2.10535
\(25\) 0 0
\(26\) −3.22370e14 −0.447851
\(27\) − 2.05891e14i − 0.192450i
\(28\) − 2.17662e15i − 1.38875i
\(29\) −2.40079e15 −1.05968 −0.529840 − 0.848097i \(-0.677748\pi\)
−0.529840 + 0.848097i \(0.677748\pi\)
\(30\) 0 0
\(31\) 2.23982e15 0.490812 0.245406 − 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) − 3.06169e16i − 4.80714i
\(33\) 8.61043e14i 0.0978652i
\(34\) −2.44282e16 −2.02938
\(35\) 0 0
\(36\) 2.08900e16 0.952273
\(37\) − 3.07851e16i − 1.05250i −0.850330 − 0.526250i \(-0.823598\pi\)
0.850330 − 0.526250i \(-0.176402\pi\)
\(38\) − 8.30532e16i − 2.14599i
\(39\) 6.69325e15 0.131661
\(40\) 0 0
\(41\) −1.03208e17 −1.20083 −0.600414 − 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) 6.10116e16i 0.551182i
\(43\) 1.65557e17i 1.16823i 0.811670 + 0.584117i \(0.198559\pi\)
−0.811670 + 0.584117i \(0.801441\pi\)
\(44\) −8.73624e16 −0.484252
\(45\) 0 0
\(46\) 4.43377e17 1.54105
\(47\) − 6.65872e16i − 0.184656i −0.995729 − 0.0923280i \(-0.970569\pi\)
0.995729 − 0.0923280i \(-0.0294308\pi\)
\(48\) 1.11790e18i 2.48526i
\(49\) 4.26556e17 0.763690
\(50\) 0 0
\(51\) 5.07195e17 0.596607
\(52\) 6.79105e17i 0.651482i
\(53\) − 4.35423e17i − 0.341991i −0.985272 − 0.170995i \(-0.945302\pi\)
0.985272 − 0.170995i \(-0.0546983\pi\)
\(54\) −5.85554e17 −0.377949
\(55\) 0 0
\(56\) −4.02346e18 −1.77266
\(57\) 1.72440e18i 0.630889i
\(58\) 6.82784e18i 2.08108i
\(59\) −5.53437e18 −1.40968 −0.704842 − 0.709364i \(-0.748982\pi\)
−0.704842 + 0.709364i \(0.748982\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 − 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) − 6.37005e18i − 0.963896i
\(63\) − 1.26676e18i − 0.162039i
\(64\) −4.73716e19 −5.13604
\(65\) 0 0
\(66\) 2.44881e18 0.192195
\(67\) − 1.57554e19i − 1.05595i −0.849258 − 0.527977i \(-0.822951\pi\)
0.849258 − 0.527977i \(-0.177049\pi\)
\(68\) 5.14606e19i 2.95210i
\(69\) −9.20569e18 −0.453045
\(70\) 0 0
\(71\) 2.64579e19 0.964588 0.482294 − 0.876009i \(-0.339804\pi\)
0.482294 + 0.876009i \(0.339804\pi\)
\(72\) − 3.86148e19i − 1.21552i
\(73\) − 1.34712e19i − 0.366875i −0.983031 − 0.183437i \(-0.941278\pi\)
0.983031 − 0.183437i \(-0.0587225\pi\)
\(74\) −8.75527e19 −2.06698
\(75\) 0 0
\(76\) −1.74960e20 −3.12174
\(77\) 5.29764e18i 0.0824005i
\(78\) − 1.90356e19i − 0.258567i
\(79\) 1.68861e19 0.200653 0.100326 − 0.994955i \(-0.468011\pi\)
0.100326 + 0.994955i \(0.468011\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 2.93522e20i 2.35828i
\(83\) 1.70688e20i 1.20749i 0.797178 + 0.603744i \(0.206325\pi\)
−0.797178 + 0.603744i \(0.793675\pi\)
\(84\) 1.28527e20 0.801794
\(85\) 0 0
\(86\) 4.70845e20 2.29427
\(87\) − 1.41764e20i − 0.611807i
\(88\) 1.61488e20i 0.618121i
\(89\) 3.12592e20 1.06263 0.531317 − 0.847173i \(-0.321697\pi\)
0.531317 + 0.847173i \(0.321697\pi\)
\(90\) 0 0
\(91\) 4.11808e19 0.110856
\(92\) − 9.34021e20i − 2.24174i
\(93\) 1.32259e20i 0.283371i
\(94\) −1.89374e20 −0.362642
\(95\) 0 0
\(96\) 1.80790e21 2.77540
\(97\) 9.49015e20i 1.30668i 0.757064 + 0.653341i \(0.226633\pi\)
−0.757064 + 0.653341i \(0.773367\pi\)
\(98\) − 1.21313e21i − 1.49980i
\(99\) −5.08437e19 −0.0565025
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 75.22.b.a.49.1 2
5.2 odd 4 75.22.a.c.1.1 1
5.3 odd 4 3.22.a.a.1.1 ✓ 1
5.4 even 2 inner 75.22.b.a.49.2 2
15.8 even 4 9.22.a.d.1.1 1
20.3 even 4 48.22.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 ✓ 1 5.3 odd 4
9.22.a.d.1.1 1 15.8 even 4
48.22.a.e.1.1 1 20.3 even 4
75.22.a.c.1.1 1 5.2 odd 4
75.22.b.a.49.1 2 1.1 even 1 trivial
75.22.b.a.49.2 2 5.4 even 2 inner