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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-3042] 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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{4729})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2365x^{2} + 1397124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(34.8839i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.e.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+43.8839i q^{2} -81.0000i q^{3} -1413.79 q^{4} +3554.59 q^{6} -7861.50i q^{7} -39574.2i q^{8} -6561.00 q^{9} -49373.3 q^{11} +114517. i q^{12} -24250.7i q^{13} +344993. q^{14} +1.01281e6 q^{16} +268222. i q^{17} -287922. i q^{18} +168364. q^{19} -636781. q^{21} -2.16669e6i q^{22} +2.12200e6i q^{23} -3.20551e6 q^{24} +1.06422e6 q^{26} +531441. i q^{27} +1.11145e7i q^{28} -389624. q^{29} +90532.2 q^{31} +2.41838e7i q^{32} +3.99924e6i q^{33} -1.17706e7 q^{34} +9.27590e6 q^{36} -3.31991e6i q^{37} +7.38848e6i q^{38} -1.96431e6 q^{39} +2.32694e7 q^{41} -2.79444e7i q^{42} -1.91140e7i q^{43} +6.98036e7 q^{44} -9.31215e7 q^{46} +6.28153e7i q^{47} -8.20372e7i q^{48} -2.14495e7 q^{49} +2.17260e7 q^{51} +3.42855e7i q^{52} +180207. i q^{53} -2.33217e7 q^{54} -3.11112e8 q^{56} -1.36375e7i q^{57} -1.70982e7i q^{58} -3.84564e7 q^{59} -553620. q^{61} +3.97290e6i q^{62} +5.15793e7i q^{63} -5.42724e8 q^{64} -1.75502e8 q^{66} -2.39163e8i q^{67} -3.79211e8i q^{68} +1.71882e8 q^{69} +1.28653e8 q^{71} +2.59646e8i q^{72} +2.39376e8i q^{73} +1.45690e8 q^{74} -2.38033e8 q^{76} +3.88148e8i q^{77} -8.62015e7i q^{78} +5.28027e8 q^{79} +4.30467e7 q^{81} +1.02115e9i q^{82} -2.12210e8i q^{83} +9.00277e8 q^{84} +8.38797e8 q^{86} +3.15595e7i q^{87} +1.95391e9i q^{88} +2.07724e8 q^{89} -1.90647e8 q^{91} -3.00007e9i q^{92} -7.33311e6i q^{93} -2.75658e9 q^{94} +1.95889e9 q^{96} +1.70780e9i q^{97} -9.41288e8i q^{98} +3.23938e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3042 q^{4} + 3078 q^{6} - 26244 q^{9} + 70976 q^{11} + 490392 q^{14} + 1414530 q^{16} + 806592 q^{19} - 1923264 q^{21} - 8042814 q^{24} - 3815092 q^{26} + 149144 q^{29} - 10054256 q^{31} - 17721764 q^{34}+ \cdots - 465673536 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 43.8839i 1.93941i 0.244277 + 0.969706i \(0.421449\pi\)
−0.244277 + 0.969706i \(0.578551\pi\)
\(3\) − 81.0000i − 0.577350i
\(4\) −1413.79 −2.76132
\(5\) 0 0
\(6\) 3554.59 1.11972
\(7\) − 7861.50i − 1.23755i −0.785567 − 0.618777i \(-0.787629\pi\)
0.785567 − 0.618777i \(-0.212371\pi\)
\(8\) − 39574.2i − 3.41591i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) −49373.3 −1.01678 −0.508388 − 0.861128i \(-0.669758\pi\)
−0.508388 + 0.861128i \(0.669758\pi\)
\(12\) 114517.i 1.59425i
\(13\) − 24250.7i − 0.235494i −0.993044 − 0.117747i \(-0.962433\pi\)
0.993044 − 0.117747i \(-0.0375672\pi\)
\(14\) 344993. 2.40013
\(15\) 0 0
\(16\) 1.01281e6 3.86355
\(17\) 268222.i 0.778888i 0.921050 + 0.389444i \(0.127333\pi\)
−0.921050 + 0.389444i \(0.872667\pi\)
\(18\) − 287922.i − 0.646470i
\(19\) 168364. 0.296387 0.148193 − 0.988958i \(-0.452654\pi\)
0.148193 + 0.988958i \(0.452654\pi\)
\(20\) 0 0
\(21\) −636781. −0.714502
\(22\) − 2.16669e6i − 1.97195i
\(23\) 2.12200e6i 1.58114i 0.612373 + 0.790569i \(0.290215\pi\)
−0.612373 + 0.790569i \(0.709785\pi\)
\(24\) −3.20551e6 −1.97218
\(25\) 0 0
\(26\) 1.06422e6 0.456720
\(27\) 531441.i 0.192450i
\(28\) 1.11145e7i 3.41728i
\(29\) −389624. −0.102295 −0.0511475 − 0.998691i \(-0.516288\pi\)
−0.0511475 + 0.998691i \(0.516288\pi\)
\(30\) 0 0
\(31\) 90532.2 0.0176066 0.00880330 − 0.999961i \(-0.497198\pi\)
0.00880330 + 0.999961i \(0.497198\pi\)
\(32\) 2.41838e7i 4.07709i
\(33\) 3.99924e6i 0.587036i
\(34\) −1.17706e7 −1.51058
\(35\) 0 0
\(36\) 9.27590e6 0.920438
\(37\) − 3.31991e6i − 0.291218i −0.989342 − 0.145609i \(-0.953486\pi\)
0.989342 − 0.145609i \(-0.0465142\pi\)
\(38\) 7.38848e6i 0.574816i
\(39\) −1.96431e6 −0.135963
\(40\) 0 0
\(41\) 2.32694e7 1.28605 0.643024 − 0.765846i \(-0.277679\pi\)
0.643024 + 0.765846i \(0.277679\pi\)
\(42\) − 2.79444e7i − 1.38571i
\(43\) − 1.91140e7i − 0.852597i −0.904583 − 0.426298i \(-0.859817\pi\)
0.904583 − 0.426298i \(-0.140183\pi\)
\(44\) 6.98036e7 2.80764
\(45\) 0 0
\(46\) −9.31215e7 −3.06648
\(47\) 6.28153e7i 1.87770i 0.344332 + 0.938848i \(0.388105\pi\)
−0.344332 + 0.938848i \(0.611895\pi\)
\(48\) − 8.20372e7i − 2.23062i
\(49\) −2.14495e7 −0.531539
\(50\) 0 0
\(51\) 2.17260e7 0.449691
\(52\) 3.42855e7i 0.650273i
\(53\) 180207.i 0.00313711i 0.999999 + 0.00156856i \(0.000499287\pi\)
−0.999999 + 0.00156856i \(0.999501\pi\)
\(54\) −2.33217e7 −0.373240
\(55\) 0 0
\(56\) −3.11112e8 −4.22738
\(57\) − 1.36375e7i − 0.171119i
\(58\) − 1.70982e7i − 0.198392i
\(59\) −3.84564e7 −0.413175 −0.206588 − 0.978428i \(-0.566236\pi\)
−0.206588 + 0.978428i \(0.566236\pi\)
\(60\) 0 0
\(61\) −553620. −0.00511950 −0.00255975 − 0.999997i \(-0.500815\pi\)
−0.00255975 + 0.999997i \(0.500815\pi\)
\(62\) 3.97290e6i 0.0341464i
\(63\) 5.15793e7i 0.412518i
\(64\) −5.42724e8 −4.04361
\(65\) 0 0
\(66\) −1.75502e8 −1.13850
\(67\) − 2.39163e8i − 1.44996i −0.688768 − 0.724982i \(-0.741848\pi\)
0.688768 − 0.724982i \(-0.258152\pi\)
\(68\) − 3.79211e8i − 2.15076i
\(69\) 1.71882e8 0.912871
\(70\) 0 0
\(71\) 1.28653e8 0.600838 0.300419 − 0.953807i \(-0.402873\pi\)
0.300419 + 0.953807i \(0.402873\pi\)
\(72\) 2.59646e8i 1.13864i
\(73\) 2.39376e8i 0.986569i 0.869868 + 0.493284i \(0.164204\pi\)
−0.869868 + 0.493284i \(0.835796\pi\)
\(74\) 1.45690e8 0.564792
\(75\) 0 0
\(76\) −2.38033e8 −0.818418
\(77\) 3.88148e8i 1.25831i
\(78\) − 8.62015e7i − 0.263687i
\(79\) 5.28027e8 1.52523 0.762613 − 0.646855i \(-0.223916\pi\)
0.762613 + 0.646855i \(0.223916\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 1.02115e9i 2.49418i
\(83\) − 2.12210e8i − 0.490812i −0.969420 − 0.245406i \(-0.921079\pi\)
0.969420 − 0.245406i \(-0.0789213\pi\)
\(84\) 9.00277e8 1.97297
\(85\) 0 0
\(86\) 8.38797e8 1.65354
\(87\) 3.15595e7i 0.0590601i
\(88\) 1.95391e9i 3.47322i
\(89\) 2.07724e8 0.350939 0.175469 − 0.984485i \(-0.443856\pi\)
0.175469 + 0.984485i \(0.443856\pi\)
\(90\) 0 0
\(91\) −1.90647e8 −0.291436
\(92\) − 3.00007e9i − 4.36602i
\(93\) − 7.33311e6i − 0.0101652i
\(94\) −2.75658e9 −3.64162
\(95\) 0 0
\(96\) 1.95889e9 2.35391
\(97\) 1.70780e9i 1.95868i 0.202214 + 0.979341i \(0.435186\pi\)
−0.202214 + 0.979341i \(0.564814\pi\)
\(98\) − 9.41288e8i − 1.03087i
\(99\) 3.23938e8 0.338925
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 75.10.b.e.49.4 4
3.2 odd 2 225.10.b.g.199.1 4
5.2 odd 4 75.10.a.g.1.1 2
5.3 odd 4 15.10.a.c.1.2 ✓ 2
5.4 even 2 inner 75.10.b.e.49.1 4
15.2 even 4 225.10.a.j.1.2 2
15.8 even 4 45.10.a.e.1.1 2
15.14 odd 2 225.10.b.g.199.4 4
20.3 even 4 240.10.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.c.1.2 ✓ 2 5.3 odd 4
45.10.a.e.1.1 2 15.8 even 4
75.10.a.g.1.1 2 5.2 odd 4
75.10.b.e.49.1 4 5.4 even 2 inner
75.10.b.e.49.4 4 1.1 even 1 trivial
225.10.a.j.1.2 2 15.2 even 4
225.10.b.g.199.1 4 3.2 odd 2
225.10.b.g.199.4 4 15.14 odd 2
240.10.a.m.1.2 2 20.3 even 4