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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(3,49)] 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("49.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.h (of order \(42\), degree \(12\), 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(27\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 33.8
Character \(\chi\) \(=\) 49.33
Dual form 49.7.h.a.3.8

$q$-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+(-3.12007 - 7.94981i) q^{2} +(45.1855 - 3.38619i) q^{3} +(-6.54926 + 6.07683i) q^{4} +(24.2678 + 35.5943i) q^{5} +(-167.901 - 348.651i) q^{6} +(341.606 - 30.8894i) q^{7} +(-423.699 - 204.043i) q^{8} +(1309.41 - 197.361i) q^{9} +(207.251 - 303.981i) q^{10} +(-168.252 - 25.3599i) q^{11} +(-275.354 + 296.762i) q^{12} +(3059.27 + 2439.69i) q^{13} +(-1311.40 - 2619.33i) q^{14} +(1217.08 + 1526.17i) q^{15} +(-342.859 + 4575.14i) q^{16} +(-1062.08 - 3443.19i) q^{17} +(-5654.42 - 9793.74i) q^{18} +(-8548.47 - 4935.46i) q^{19} +(-375.236 - 85.6453i) q^{20} +(15331.1 - 2552.50i) q^{21} +(323.352 + 1416.70i) q^{22} +(-9463.30 - 2919.04i) q^{23} +(-19836.0 - 7785.04i) q^{24} +(5030.42 - 12817.3i) q^{25} +(9849.90 - 31932.6i) q^{26} +(26293.5 - 6001.31i) q^{27} +(-2049.56 + 2278.19i) q^{28} +(-4344.19 + 19033.1i) q^{29} +(8335.39 - 14437.3i) q^{30} +(-34575.5 + 19962.2i) q^{31} +(8681.09 - 2677.76i) q^{32} +(-7688.43 - 576.168i) q^{33} +(-24058.9 + 19186.4i) q^{34} +(9389.52 + 11409.6i) q^{35} +(-7376.31 + 9249.60i) q^{36} +(20168.9 + 18714.0i) q^{37} +(-12564.1 + 83357.7i) q^{38} +(146496. + 99879.2i) q^{39} +(-3019.48 - 20032.9i) q^{40} +(-22456.2 + 46630.8i) q^{41} +(-68125.8 - 113915. i) q^{42} +(63711.1 - 30681.7i) q^{43} +(1256.03 - 856.350i) q^{44} +(38801.3 + 41817.9i) q^{45} +(6320.33 + 84339.0i) q^{46} +(-102780. + 40338.2i) q^{47} +207891. i q^{48} +(115741. - 21104.0i) q^{49} -117590. q^{50} +(-59650.1 - 151986. i) q^{51} +(-34861.5 + 2612.51i) q^{52} +(187716. - 174175. i) q^{53} +(-129747. - 190304. i) q^{54} +(-3180.44 - 6604.24i) q^{55} +(-151041. - 56614.4i) q^{56} +(-402979. - 194065. i) q^{57} +(164864. - 24849.2i) q^{58} +(-31492.8 + 46191.5i) q^{59} +(-17245.3 - 2599.31i) q^{60} +(-198817. + 214274. i) q^{61} +(266573. + 212585. i) q^{62} +(441205. - 107867. i) q^{63} +(134702. + 168911. i) q^{64} +(-12597.2 + 168098. i) q^{65} +(19408.0 + 62919.2i) q^{66} +(109141. + 189038. i) q^{67} +(27879.6 + 16096.3i) q^{68} +(-437488. - 99853.8i) q^{69} +(61408.3 - 110244. i) q^{70} +(104969. + 459898. i) q^{71} +(-595064. - 183553. i) q^{72} +(169408. + 66487.6i) q^{73} +(85844.5 - 218728. i) q^{74} +(183900. - 596191. i) q^{75} +(85978.1 - 19623.9i) q^{76} +(-58259.3 - 3465.90i) q^{77} +(336943. - 1.47624e6i) q^{78} +(-72238.1 + 125120. i) q^{79} +(-171169. + 98824.7i) q^{80} +(245311. - 75668.3i) q^{81} +(440771. + 33031.2i) q^{82} +(220938. - 176192. i) q^{83} +(-84896.0 + 109881. i) q^{84} +(96783.6 - 121363. i) q^{85} +(-442696. - 410762. i) q^{86} +(-131845. + 874733. i) q^{87} +(66113.7 + 45075.5i) q^{88} +(-108096. - 717170. i) q^{89} +(211381. - 438938. i) q^{90} +(1.12043e6 + 738913. i) q^{91} +(79716.1 - 38389.3i) q^{92} +(-1.49472e6 + 1.01908e6i) q^{93} +(641362. + 691223. i) q^{94} +(-31778.1 - 424050. i) q^{95} +(383192. - 150392. i) q^{96} -380942. i q^{97} +(-528892. - 854270. i) q^{98} -225315. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 324 q - 13 q^{2} - 11 q^{3} + 819 q^{4} - 179 q^{5} + 770 q^{6} + 392 q^{7} + 828 q^{8} - 1160 q^{9} - 2594 q^{10} - 5305 q^{11} + 7497 q^{12} - 14 q^{13} - 11403 q^{14} - 6196 q^{15} + 27903 q^{16} - 5107 q^{17}+ \cdots - 4449616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{41}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.12007 − 7.94981i −0.390009 − 0.993726i −0.982185 − 0.187917i \(-0.939826\pi\)
0.592176 − 0.805808i \(-0.298269\pi\)
\(3\) 45.1855 − 3.38619i 1.67354 − 0.125414i 0.796119 − 0.605140i \(-0.206883\pi\)
0.877418 + 0.479726i \(0.159264\pi\)
\(4\) −6.54926 + 6.07683i −0.102332 + 0.0949504i
\(5\) 24.2678 + 35.5943i 0.194142 + 0.284754i 0.910950 − 0.412516i \(-0.135350\pi\)
−0.716808 + 0.697271i \(0.754398\pi\)
\(6\) −167.901 − 348.651i −0.777321 − 1.61412i
\(7\) 341.606 − 30.8894i 0.995937 − 0.0900566i
\(8\) −423.699 − 204.043i −0.827537 − 0.398521i
\(9\) 1309.41 − 197.361i 1.79617 − 0.270729i
\(10\) 207.251 − 303.981i 0.207251 − 0.303981i
\(11\) −168.252 − 25.3599i −0.126410 − 0.0190533i 0.0855324 − 0.996335i \(-0.472741\pi\)
−0.211943 + 0.977282i \(0.567979\pi\)
\(12\) −275.354 + 296.762i −0.159349 + 0.171737i
\(13\) 3059.27 + 2439.69i 1.39248 + 1.11046i 0.979889 + 0.199545i \(0.0639464\pi\)
0.412588 + 0.910918i \(0.364625\pi\)
\(14\) −1311.40 − 2619.33i −0.477915 − 0.954565i
\(15\) 1217.08 + 1526.17i 0.360617 + 0.452199i
\(16\) −342.859 + 4575.14i −0.0837059 + 1.11698i
\(17\) −1062.08 − 3443.19i −0.216178 − 0.700833i −0.997093 − 0.0761953i \(-0.975723\pi\)
0.780915 − 0.624638i \(-0.214753\pi\)
\(18\) −5654.42 − 9793.74i −0.969551 − 1.67931i
\(19\) −8548.47 − 4935.46i −1.24631 − 0.719560i −0.275942 − 0.961174i \(-0.588990\pi\)
−0.970372 + 0.241614i \(0.922323\pi\)
\(20\) −375.236 − 85.6453i −0.0469046 − 0.0107057i
\(21\) 15331.1 − 2552.50i 1.65544 − 0.275618i
\(22\) 323.352 + 1416.70i 0.0303674 + 0.133048i
\(23\) −9463.30 − 2919.04i −0.777784 − 0.239915i −0.119651 − 0.992816i \(-0.538177\pi\)
−0.658133 + 0.752901i \(0.728654\pi\)
\(24\) −19836.0 − 7785.04i −1.43489 − 0.563154i
\(25\) 5030.42 − 12817.3i 0.321947 − 0.820308i
\(26\) 9849.90 − 31932.6i 0.560418 − 1.81683i
\(27\) 26293.5 − 6001.31i 1.33585 − 0.304898i
\(28\) −2049.56 + 2278.19i −0.0933655 + 0.103780i
\(29\) −4344.19 + 19033.1i −0.178121 + 0.780399i 0.804376 + 0.594120i \(0.202500\pi\)
−0.982497 + 0.186278i \(0.940357\pi\)
\(30\) 8335.39 − 14437.3i 0.308718 − 0.534715i
\(31\) −34575.5 + 19962.2i −1.16060 + 0.670074i −0.951448 − 0.307809i \(-0.900404\pi\)
−0.209154 + 0.977883i \(0.567071\pi\)
\(32\) 8681.09 − 2677.76i 0.264926 − 0.0817188i
\(33\) −7688.43 − 576.168i −0.213942 − 0.0160327i
\(34\) −24058.9 + 19186.4i −0.612124 + 0.488153i
\(35\) 9389.52 + 11409.6i 0.218997 + 0.266114i
\(36\) −7376.31 + 9249.60i −0.158100 + 0.198251i
\(37\) 20168.9 + 18714.0i 0.398179 + 0.369456i 0.853696 − 0.520771i \(-0.174355\pi\)
−0.455518 + 0.890227i \(0.650546\pi\)
\(38\) −12564.1 + 83357.7i −0.228972 + 1.51913i
\(39\) 146496. + 99879.2i 2.46963 + 1.68376i
\(40\) −3019.48 − 20032.9i −0.0471793 − 0.313014i
\(41\) −22456.2 + 46630.8i −0.325825 + 0.676584i −0.997960 − 0.0638457i \(-0.979663\pi\)
0.672134 + 0.740429i \(0.265378\pi\)
\(42\) −68125.8 − 113915.i −0.919525 − 1.53756i
\(43\) 63711.1 − 30681.7i 0.801327 − 0.385899i 0.0120436 − 0.999927i \(-0.496166\pi\)
0.789284 + 0.614029i \(0.210452\pi\)
\(44\) 1256.03 − 856.350i 0.0147450 − 0.0100529i
\(45\) 38801.3 + 41817.9i 0.425803 + 0.458907i
\(46\) 6320.33 + 84339.0i 0.0649331 + 0.866473i
\(47\) −102780. + 40338.2i −0.989954 + 0.388528i −0.804389 − 0.594103i \(-0.797507\pi\)
−0.185566 + 0.982632i \(0.559412\pi\)
\(48\) 207891.i 1.87980i
\(49\) 115741. − 21104.0i 0.983780 − 0.179381i
\(50\) −117590. −0.940723
\(51\) −59650.1 − 151986.i −0.449677 − 1.14576i
\(52\) −34861.5 + 2612.51i −0.247934 + 0.0185801i
\(53\) 187716. − 174175.i 1.26088 − 1.16992i 0.283372 − 0.959010i \(-0.408547\pi\)
0.977505 − 0.210913i \(-0.0676437\pi\)
\(54\) −129747. − 190304.i −0.823977 − 1.20855i
\(55\) −3180.44 − 6604.24i −0.0191161 − 0.0396949i
\(56\) −151041. − 56614.4i −0.860063 − 0.322376i
\(57\) −402979. − 194065.i −2.17600 − 1.04790i
\(58\) 164864. − 24849.2i 0.844971 − 0.127359i
\(59\) −31492.8 + 46191.5i −0.153340 + 0.224909i −0.895220 − 0.445624i \(-0.852982\pi\)
0.741880 + 0.670532i \(0.233934\pi\)
\(60\) −17245.3 − 2599.31i −0.0798392 − 0.0120338i
\(61\) −198817. + 214274.i −0.875919 + 0.944017i −0.998828 − 0.0484013i \(-0.984587\pi\)
0.122908 + 0.992418i \(0.460778\pi\)
\(62\) 266573. + 212585.i 1.11851 + 0.891986i
\(63\) 441205. − 107867.i 1.76449 − 0.431385i
\(64\) 134702. + 168911.i 0.513848 + 0.644345i
\(65\) −12597.2 + 168098.i −0.0458707 + 0.612102i
\(66\) 19408.0 + 62919.2i 0.0675070 + 0.218852i
\(67\) 109141. + 189038.i 0.362882 + 0.628530i 0.988434 − 0.151653i \(-0.0484596\pi\)
−0.625552 + 0.780182i \(0.715126\pi\)
\(68\) 27879.6 + 16096.3i 0.0886664 + 0.0511916i
\(69\) −437488. − 99853.8i −1.33174 − 0.303961i
\(70\) 61408.3 − 110244.i 0.179033 − 0.321410i
\(71\) 104969. + 459898.i 0.293282 + 1.28495i 0.879928 + 0.475107i \(0.157591\pi\)
−0.586646 + 0.809843i \(0.699552\pi\)
\(72\) −595064. − 183553.i −1.59429 − 0.491772i
\(73\) 169408. + 66487.6i 0.435476 + 0.170912i 0.572944 − 0.819594i \(-0.305801\pi\)
−0.137468 + 0.990506i \(0.543896\pi\)
\(74\) 85844.5 − 218728.i 0.211845 − 0.539771i
\(75\) 183900. − 596191.i 0.435912 − 1.41319i
\(76\) 85978.1 − 19623.9i 0.195861 − 0.0447039i
\(77\) −58259.3 − 3465.90i −0.127612 − 0.00759178i
\(78\) 336943. − 1.47624e6i 0.710024 − 3.11082i
\(79\) −72238.1 + 125120.i −0.146516 + 0.253773i −0.929938 − 0.367718i \(-0.880139\pi\)
0.783421 + 0.621491i \(0.213473\pi\)
\(80\) −171169. + 98824.7i −0.334315 + 0.193017i
\(81\) 245311. − 75668.3i 0.461595 − 0.142383i
\(82\) 440771. + 33031.2i 0.799413 + 0.0599077i
\(83\) 220938. − 176192.i 0.386399 − 0.308143i −0.410954 − 0.911656i \(-0.634804\pi\)
0.797353 + 0.603513i \(0.206233\pi\)
\(84\) −84896.0 + 109881.i −0.143235 + 0.185390i
\(85\) 96783.6 − 121363.i 0.157596 − 0.197619i
\(86\) −442696. − 410762.i −0.696002 − 0.645796i
\(87\) −131845. + 874733.i −0.200219 + 1.32837i
\(88\) 66113.7 + 45075.5i 0.0970160 + 0.0661444i
\(89\) −108096. − 717170.i −0.153334 − 1.01731i −0.924640 − 0.380842i \(-0.875634\pi\)
0.771306 − 0.636465i \(-0.219604\pi\)
\(90\) 211381. − 438938.i 0.289961 − 0.602109i
\(91\) 1.12043e6 + 738913.i 1.48682 + 0.980549i
\(92\) 79716.1 − 38389.3i 0.102372 − 0.0492999i
\(93\) −1.49472e6 + 1.01908e6i −1.85827 + 1.26695i
\(94\) 641362. + 691223.i 0.772181 + 0.832214i
\(95\) −31778.1 − 424050.i −0.0370644 − 0.494591i
\(96\) 383192. − 150392.i 0.433114 − 0.169985i
\(97\) − 380942.i − 0.417391i −0.977981 − 0.208696i \(-0.933078\pi\)
0.977981 − 0.208696i \(-0.0669218\pi\)
\(98\) −528892. − 854270.i −0.561938 − 0.907647i
\(99\) −225315. −0.232212
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.h.a.33.8 yes 324
49.3 odd 42 inner 49.7.h.a.3.8 ✓ 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.8 ✓ 324 49.3 odd 42 inner
49.7.h.a.33.8 yes 324 1.1 even 1 trivial