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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(48,49)] 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("49.48"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 48.2
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.48
Dual form 49.7.b.b.48.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.242641 q^{2} +37.0395i q^{3} -63.9411 q^{4} +165.776i q^{5} -8.98729i q^{6} +31.0437 q^{8} -642.926 q^{9} -40.2239i q^{10} -173.082 q^{11} -2368.35i q^{12} -1963.14i q^{13} -6140.25 q^{15} +4084.70 q^{16} +3693.93i q^{17} +156.000 q^{18} -4564.66i q^{19} -10599.9i q^{20} +41.9967 q^{22} -15791.5 q^{23} +1149.84i q^{24} -11856.6 q^{25} +476.337i q^{26} +3188.14i q^{27} -23782.2 q^{29} +1489.88 q^{30} +2061.80i q^{31} -2977.91 q^{32} -6410.88i q^{33} -896.298i q^{34} +41109.4 q^{36} +48510.9 q^{37} +1107.57i q^{38} +72713.6 q^{39} +5146.30i q^{40} +26437.9i q^{41} +68471.6 q^{43} +11067.1 q^{44} -106582. i q^{45} +3831.66 q^{46} +140362. i q^{47} +151295. i q^{48} +2876.89 q^{50} -136821. q^{51} +125525. i q^{52} -254130. q^{53} -773.573i q^{54} -28692.8i q^{55} +169073. q^{57} +5770.54 q^{58} -97200.4i q^{59} +392615. q^{60} -17392.5i q^{61} -500.276i q^{62} -260698. q^{64} +325440. q^{65} +1555.54i q^{66} -120682. q^{67} -236194. i q^{68} -584910. i q^{69} -339555. q^{71} -19958.8 q^{72} +111351. i q^{73} -11770.7 q^{74} -439163. i q^{75} +291869. i q^{76} -17643.3 q^{78} -615838. q^{79} +677144. i q^{80} -586780. q^{81} -6414.91i q^{82} -383668. i q^{83} -612364. q^{85} -16614.0 q^{86} -880882. i q^{87} -5373.11 q^{88} +771685. i q^{89} +25861.0i q^{90} +1.00973e6 q^{92} -76367.9 q^{93} -34057.5i q^{94} +756709. q^{95} -110300. i q^{96} +292263. i q^{97} +111279. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 120 q^{4} - 928 q^{8} - 1248 q^{9} - 3764 q^{11} - 16500 q^{15} - 496 q^{16} + 624 q^{18} - 28088 q^{22} - 4940 q^{23} - 5000 q^{25} - 34544 q^{29} - 31800 q^{30} - 14016 q^{32} + 82368 q^{36}+ \cdots + 157872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.242641 −0.0303301 −0.0151650 − 0.999885i \(-0.504827\pi\)
−0.0151650 + 0.999885i \(0.504827\pi\)
\(3\) 37.0395i 1.37183i 0.727680 + 0.685917i \(0.240599\pi\)
−0.727680 + 0.685917i \(0.759401\pi\)
\(4\) −63.9411 −0.999080
\(5\) 165.776i 1.32621i 0.748528 + 0.663103i \(0.230761\pi\)
−0.748528 + 0.663103i \(0.769239\pi\)
\(6\) − 8.98729i − 0.0416078i
\(7\) 0 0
\(8\) 31.0437 0.0606323
\(9\) −642.926 −0.881929
\(10\) − 40.2239i − 0.0402239i
\(11\) −173.082 −0.130039 −0.0650195 − 0.997884i \(-0.520711\pi\)
−0.0650195 + 0.997884i \(0.520711\pi\)
\(12\) − 2368.35i − 1.37057i
\(13\) − 1963.14i − 0.893553i −0.894646 − 0.446777i \(-0.852572\pi\)
0.894646 − 0.446777i \(-0.147428\pi\)
\(14\) 0 0
\(15\) −6140.25 −1.81933
\(16\) 4084.70 0.997241
\(17\) 3693.93i 0.751868i 0.926646 + 0.375934i \(0.122678\pi\)
−0.926646 + 0.375934i \(0.877322\pi\)
\(18\) 156.000 0.0267490
\(19\) − 4564.66i − 0.665499i −0.943015 − 0.332749i \(-0.892024\pi\)
0.943015 − 0.332749i \(-0.107976\pi\)
\(20\) − 10599.9i − 1.32499i
\(21\) 0 0
\(22\) 41.9967 0.00394410
\(23\) −15791.5 −1.29790 −0.648948 − 0.760833i \(-0.724791\pi\)
−0.648948 + 0.760833i \(0.724791\pi\)
\(24\) 1149.84i 0.0831774i
\(25\) −11856.6 −0.758823
\(26\) 476.337i 0.0271015i
\(27\) 3188.14i 0.161974i
\(28\) 0 0
\(29\) −23782.2 −0.975121 −0.487561 − 0.873089i \(-0.662113\pi\)
−0.487561 + 0.873089i \(0.662113\pi\)
\(30\) 1489.88 0.0551806
\(31\) 2061.80i 0.0692087i 0.999401 + 0.0346044i \(0.0110171\pi\)
−0.999401 + 0.0346044i \(0.988983\pi\)
\(32\) −2977.91 −0.0908787
\(33\) − 6410.88i − 0.178392i
\(34\) − 896.298i − 0.0228042i
\(35\) 0 0
\(36\) 41109.4 0.881117
\(37\) 48510.9 0.957710 0.478855 − 0.877894i \(-0.341052\pi\)
0.478855 + 0.877894i \(0.341052\pi\)
\(38\) 1107.57i 0.0201846i
\(39\) 72713.6 1.22581
\(40\) 5146.30i 0.0804109i
\(41\) 26437.9i 0.383597i 0.981434 + 0.191799i \(0.0614320\pi\)
−0.981434 + 0.191799i \(0.938568\pi\)
\(42\) 0 0
\(43\) 68471.6 0.861202 0.430601 − 0.902542i \(-0.358302\pi\)
0.430601 + 0.902542i \(0.358302\pi\)
\(44\) 11067.1 0.129919
\(45\) − 106582.i − 1.16962i
\(46\) 3831.66 0.0393653
\(47\) 140362.i 1.35193i 0.736932 + 0.675967i \(0.236274\pi\)
−0.736932 + 0.675967i \(0.763726\pi\)
\(48\) 151295.i 1.36805i
\(49\) 0 0
\(50\) 2876.89 0.0230152
\(51\) −136821. −1.03144
\(52\) 125525.i 0.892731i
\(53\) −254130. −1.70698 −0.853489 − 0.521110i \(-0.825518\pi\)
−0.853489 + 0.521110i \(0.825518\pi\)
\(54\) − 773.573i − 0.00491270i
\(55\) − 28692.8i − 0.172459i
\(56\) 0 0
\(57\) 169073. 0.912954
\(58\) 5770.54 0.0295755
\(59\) − 97200.4i − 0.473273i −0.971598 − 0.236637i \(-0.923955\pi\)
0.971598 − 0.236637i \(-0.0760451\pi\)
\(60\) 392615. 1.81766
\(61\) − 17392.5i − 0.0766255i −0.999266 − 0.0383127i \(-0.987802\pi\)
0.999266 − 0.0383127i \(-0.0121983\pi\)
\(62\) − 500.276i − 0.00209911i
\(63\) 0 0
\(64\) −260698. −0.994485
\(65\) 325440. 1.18504
\(66\) 1555.54i 0.00541065i
\(67\) −120682. −0.401251 −0.200626 − 0.979668i \(-0.564297\pi\)
−0.200626 + 0.979668i \(0.564297\pi\)
\(68\) − 236194.i − 0.751177i
\(69\) − 584910.i − 1.78050i
\(70\) 0 0
\(71\) −339555. −0.948713 −0.474357 − 0.880333i \(-0.657319\pi\)
−0.474357 + 0.880333i \(0.657319\pi\)
\(72\) −19958.8 −0.0534733
\(73\) 111351.i 0.286237i 0.989706 + 0.143118i \(0.0457130\pi\)
−0.989706 + 0.143118i \(0.954287\pi\)
\(74\) −11770.7 −0.0290474
\(75\) − 439163.i − 1.04098i
\(76\) 291869.i 0.664886i
\(77\) 0 0
\(78\) −17643.3 −0.0371788
\(79\) −615838. −1.24907 −0.624533 − 0.780998i \(-0.714711\pi\)
−0.624533 + 0.780998i \(0.714711\pi\)
\(80\) 677144.i 1.32255i
\(81\) −586780. −1.10413
\(82\) − 6414.91i − 0.0116345i
\(83\) − 383668.i − 0.670999i −0.942040 − 0.335499i \(-0.891095\pi\)
0.942040 − 0.335499i \(-0.108905\pi\)
\(84\) 0 0
\(85\) −612364. −0.997133
\(86\) −16614.0 −0.0261203
\(87\) − 880882.i − 1.33770i
\(88\) −5373.11 −0.00788457
\(89\) 771685.i 1.09464i 0.836924 + 0.547318i \(0.184351\pi\)
−0.836924 + 0.547318i \(0.815649\pi\)
\(90\) 25861.0i 0.0354746i
\(91\) 0 0
\(92\) 1.00973e6 1.29670
\(93\) −76367.9 −0.0949429
\(94\) − 34057.5i − 0.0410042i
\(95\) 756709. 0.882588
\(96\) − 110300.i − 0.124670i
\(97\) 292263.i 0.320227i 0.987099 + 0.160113i \(0.0511860\pi\)
−0.987099 + 0.160113i \(0.948814\pi\)
\(98\) 0 0
\(99\) 111279. 0.114685
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 49.7.b.b.48.2 4
3.2 odd 2 441.7.d.b.244.3 4
7.2 even 3 49.7.d.c.31.2 4
7.3 odd 6 49.7.d.c.19.2 4
7.4 even 3 7.7.d.b.5.2 yes 4
7.5 odd 6 7.7.d.b.3.2 ✓ 4
7.6 odd 2 inner 49.7.b.b.48.1 4
21.5 even 6 63.7.m.b.10.1 4
21.11 odd 6 63.7.m.b.19.1 4
21.20 even 2 441.7.d.b.244.4 4
28.11 odd 6 112.7.s.b.33.1 4
28.19 even 6 112.7.s.b.17.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.2 ✓ 4 7.5 odd 6
7.7.d.b.5.2 yes 4 7.4 even 3
49.7.b.b.48.1 4 7.6 odd 2 inner
49.7.b.b.48.2 4 1.1 even 1 trivial
49.7.d.c.19.2 4 7.3 odd 6
49.7.d.c.31.2 4 7.2 even 3
63.7.m.b.10.1 4 21.5 even 6
63.7.m.b.19.1 4 21.11 odd 6
112.7.s.b.17.1 4 28.19 even 6
112.7.s.b.33.1 4 28.11 odd 6
441.7.d.b.244.3 4 3.2 odd 2
441.7.d.b.244.4 4 21.20 even 2