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Properties

Label 49.7.d.e.31.2
Level $49$
Weight $7$
Character 49.31
Analytic conductor $11.273$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.2
Character \(\chi\) \(=\) 49.31
Dual form 49.7.d.e.19.2

$q$-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+(-7.22006 - 12.5055i) q^{2} +(5.48749 + 3.16820i) q^{3} +(-72.2584 + 125.155i) q^{4} +(169.953 - 98.1223i) q^{5} -91.4984i q^{6} +1162.67 q^{8} +(-344.425 - 596.562i) q^{9} +(-2454.14 - 1416.90i) q^{10} +(388.746 - 673.327i) q^{11} +(-793.034 + 457.858i) q^{12} -1858.37i q^{13} +1243.49 q^{15} +(-3770.02 - 6529.86i) q^{16} +(1942.83 + 1121.69i) q^{17} +(-4973.54 + 8614.42i) q^{18} +(-5366.70 + 3098.47i) q^{19} +28360.6i q^{20} -11227.1 q^{22} +(662.628 + 1147.71i) q^{23} +(6380.15 + 3683.58i) q^{24} +(11443.5 - 19820.7i) q^{25} +(-23239.9 + 13417.6i) q^{26} -8984.07i q^{27} -19588.9 q^{29} +(-8978.03 - 15550.4i) q^{30} +(-43676.8 - 25216.8i) q^{31} +(-17234.0 + 29850.1i) q^{32} +(4266.47 - 2463.25i) q^{33} -32394.8i q^{34} +99550.4 q^{36} +(7033.78 + 12182.9i) q^{37} +(77495.8 + 44742.2i) q^{38} +(5887.70 - 10197.8i) q^{39} +(197599. - 114084. i) q^{40} -60374.1i q^{41} +59618.8 q^{43} +(56180.3 + 97307.1i) q^{44} +(-117072. - 67591.5i) q^{45} +(9568.43 - 16573.0i) q^{46} +(-155091. + 89541.8i) q^{47} -47776.7i q^{48} -330490. q^{50} +(7107.50 + 12310.6i) q^{51} +(232585. + 134283. i) q^{52} +(94548.6 - 163763. i) q^{53} +(-112350. + 64865.5i) q^{54} -152579. i q^{55} -39266.3 q^{57} +(141433. + 244969. i) q^{58} +(94989.9 + 54842.5i) q^{59} +(-89852.3 + 155629. i) q^{60} +(261423. - 150933. i) q^{61} +728267. i q^{62} +15158.7 q^{64} +(-182348. - 315836. i) q^{65} +(-61608.4 - 35569.6i) q^{66} +(-111524. + 193166. i) q^{67} +(-280772. + 162104. i) q^{68} +8397.36i q^{69} +667143. q^{71} +(-400453. - 693605. i) q^{72} +(-226606. - 130831. i) q^{73} +(101569. - 175922. i) q^{74} +(125592. - 72510.5i) q^{75} -895561. i q^{76} -170038. q^{78} +(252972. + 438160. i) q^{79} +(-1.28145e6 - 739846. i) q^{80} +(-222622. + 385593. i) q^{81} +(-755009. + 435905. i) q^{82} -230815. i q^{83} +440253. q^{85} +(-430451. - 745564. i) q^{86} +(-107494. - 62061.6i) q^{87} +(451984. - 782859. i) q^{88} +(697220. - 402540. i) q^{89} +1.95206e6i q^{90} -191522. q^{92} +(-159784. - 276754. i) q^{93} +(2.23953e6 + 1.29299e6i) q^{94} +(-608057. + 1.05319e6i) q^{95} +(-189142. + 109201. i) q^{96} +474359. i q^{97} -535575. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 20 q^{2} - 564 q^{4} + 1880 q^{8} + 1252 q^{9} + 3872 q^{11} + 44864 q^{15} - 37908 q^{16} + 19436 q^{18} + 159744 q^{22} - 40032 q^{23} + 28860 q^{25} + 68032 q^{29} - 21192 q^{30} + 82060 q^{32}+ \cdots - 8949984 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.22006 − 12.5055i −0.902507 − 1.56319i −0.824229 − 0.566256i \(-0.808391\pi\)
−0.0782778 − 0.996932i \(-0.524942\pi\)
\(3\) 5.48749 + 3.16820i 0.203240 + 0.117341i 0.598166 − 0.801372i \(-0.295896\pi\)
−0.394926 + 0.918713i \(0.629230\pi\)
\(4\) −72.2584 + 125.155i −1.12904 + 1.95555i
\(5\) 169.953 − 98.1223i 1.35962 − 0.784978i 0.370050 − 0.929012i \(-0.379341\pi\)
0.989573 + 0.144034i \(0.0460073\pi\)
\(6\) − 91.4984i − 0.423604i
\(7\) 0 0
\(8\) 1162.67 2.27084
\(9\) −344.425 − 596.562i −0.472462 − 0.818329i
\(10\) −2454.14 − 1416.90i −2.45414 − 1.41690i
\(11\) 388.746 − 673.327i 0.292070 − 0.505881i −0.682229 − 0.731139i \(-0.738989\pi\)
0.974299 + 0.225258i \(0.0723225\pi\)
\(12\) −793.034 + 457.858i −0.458932 + 0.264964i
\(13\) − 1858.37i − 0.845868i −0.906160 − 0.422934i \(-0.861000\pi\)
0.906160 − 0.422934i \(-0.139000\pi\)
\(14\) 0 0
\(15\) 1243.49 0.368440
\(16\) −3770.02 − 6529.86i −0.920414 − 1.59420i
\(17\) 1942.83 + 1121.69i 0.395447 + 0.228311i 0.684517 − 0.728996i \(-0.260013\pi\)
−0.289071 + 0.957308i \(0.593346\pi\)
\(18\) −4973.54 + 8614.42i −0.852801 + 1.47709i
\(19\) −5366.70 + 3098.47i −0.782432 + 0.451737i −0.837292 − 0.546757i \(-0.815862\pi\)
0.0548594 + 0.998494i \(0.482529\pi\)
\(20\) 28360.6i 3.54508i
\(21\) 0 0
\(22\) −11227.1 −1.05438
\(23\) 662.628 + 1147.71i 0.0544611 + 0.0943294i 0.891971 − 0.452093i \(-0.149323\pi\)
−0.837510 + 0.546423i \(0.815989\pi\)
\(24\) 6380.15 + 3683.58i 0.461527 + 0.266463i
\(25\) 11443.5 − 19820.7i 0.732382 − 1.26852i
\(26\) −23239.9 + 13417.6i −1.32225 + 0.763402i
\(27\) − 8984.07i − 0.456438i
\(28\) 0 0
\(29\) −19588.9 −0.803186 −0.401593 − 0.915818i \(-0.631543\pi\)
−0.401593 + 0.915818i \(0.631543\pi\)
\(30\) −8978.03 − 15550.4i −0.332520 − 0.575941i
\(31\) −43676.8 − 25216.8i −1.46611 − 0.846457i −0.466824 − 0.884350i \(-0.654602\pi\)
−0.999282 + 0.0378935i \(0.987935\pi\)
\(32\) −17234.0 + 29850.1i −0.525939 + 0.910953i
\(33\) 4266.47 − 2463.25i 0.118721 − 0.0685436i
\(34\) − 32394.8i − 0.824210i
\(35\) 0 0
\(36\) 99550.4 2.13371
\(37\) 7033.78 + 12182.9i 0.138862 + 0.240516i 0.927066 − 0.374898i \(-0.122322\pi\)
−0.788204 + 0.615414i \(0.788989\pi\)
\(38\) 77495.8 + 44742.2i 1.41230 + 0.815392i
\(39\) 5887.70 − 10197.8i 0.0992548 − 0.171914i
\(40\) 197599. − 114084.i 3.08749 − 1.78256i
\(41\) − 60374.1i − 0.875990i −0.898977 − 0.437995i \(-0.855689\pi\)
0.898977 − 0.437995i \(-0.144311\pi\)
\(42\) 0 0
\(43\) 59618.8 0.749856 0.374928 − 0.927054i \(-0.377667\pi\)
0.374928 + 0.927054i \(0.377667\pi\)
\(44\) 56180.3 + 97307.1i 0.659517 + 1.14232i
\(45\) −117072. − 67591.5i −1.28474 − 0.741745i
\(46\) 9568.43 − 16573.0i 0.0983031 − 0.170266i
\(47\) −155091. + 89541.8i −1.49380 + 0.862447i −0.999975 − 0.00711370i \(-0.997736\pi\)
−0.493827 + 0.869560i \(0.664402\pi\)
\(48\) − 47776.7i − 0.432009i
\(49\) 0 0
\(50\) −330490. −2.64392
\(51\) 7107.50 + 12310.6i 0.0535805 + 0.0928041i
\(52\) 232585. + 134283.i 1.65414 + 0.955017i
\(53\) 94548.6 − 163763.i 0.635079 − 1.09999i −0.351420 − 0.936218i \(-0.614301\pi\)
0.986498 − 0.163771i \(-0.0523657\pi\)
\(54\) −112350. + 64865.5i −0.713498 + 0.411939i
\(55\) − 152579.i − 0.917076i
\(56\) 0 0
\(57\) −39266.3 −0.212029
\(58\) 141433. + 244969.i 0.724881 + 1.25553i
\(59\) 94989.9 + 54842.5i 0.462510 + 0.267031i 0.713099 − 0.701063i \(-0.247291\pi\)
−0.250589 + 0.968094i \(0.580624\pi\)
\(60\) −89852.3 + 155629.i −0.415983 + 0.720503i
\(61\) 261423. − 150933.i 1.15174 − 0.664958i 0.202430 − 0.979297i \(-0.435116\pi\)
0.949311 + 0.314339i \(0.101783\pi\)
\(62\) 728267.i 3.05573i
\(63\) 0 0
\(64\) 15158.7 0.0578258
\(65\) −182348. − 315836.i −0.663988 − 1.15006i
\(66\) −61608.4 − 35569.6i −0.214293 − 0.123722i
\(67\) −111524. + 193166.i −0.370805 + 0.642253i −0.989690 − 0.143229i \(-0.954251\pi\)
0.618885 + 0.785482i \(0.287585\pi\)
\(68\) −280772. + 162104.i −0.892949 + 0.515544i
\(69\) 8397.36i 0.0255620i
\(70\) 0 0
\(71\) 667143. 1.86399 0.931996 − 0.362469i \(-0.118066\pi\)
0.931996 + 0.362469i \(0.118066\pi\)
\(72\) −400453. − 693605.i −1.07289 − 1.85830i
\(73\) −226606. − 130831.i −0.582510 − 0.336312i 0.179620 − 0.983736i \(-0.442513\pi\)
−0.762130 + 0.647424i \(0.775846\pi\)
\(74\) 101569. − 175922.i 0.250648 − 0.434135i
\(75\) 125592. − 72510.5i 0.297699 − 0.171877i
\(76\) − 895561.i − 2.04011i
\(77\) 0 0
\(78\) −170038. −0.358313
\(79\) 252972. + 438160.i 0.513087 + 0.888693i 0.999885 + 0.0151783i \(0.00483160\pi\)
−0.486798 + 0.873515i \(0.661835\pi\)
\(80\) −1.28145e6 − 739846.i −2.50283 − 1.44501i
\(81\) −222622. + 385593.i −0.418903 + 0.725562i
\(82\) −755009. + 435905.i −1.36934 + 0.790587i
\(83\) − 230815.i − 0.403673i −0.979419 − 0.201836i \(-0.935309\pi\)
0.979419 − 0.201836i \(-0.0646909\pi\)
\(84\) 0 0
\(85\) 440253. 0.716878
\(86\) −430451. − 745564.i −0.676751 − 1.17217i
\(87\) −107494. − 62061.6i −0.163240 − 0.0942465i
\(88\) 451984. − 782859.i 0.663246 − 1.14878i
\(89\) 697220. − 402540.i 0.989008 − 0.571004i 0.0840304 − 0.996463i \(-0.473221\pi\)
0.904978 + 0.425459i \(0.139887\pi\)
\(90\) 1.95206e6i 2.67772i
\(91\) 0 0
\(92\) −191522. −0.245955
\(93\) −159784. − 276754.i −0.198648 − 0.344068i
\(94\) 2.23953e6 + 1.29299e6i 2.69633 + 1.55673i
\(95\) −608057. + 1.05319e6i −0.709208 + 1.22838i
\(96\) −189142. + 109201.i −0.213784 + 0.123428i
\(97\) 474359.i 0.519747i 0.965643 + 0.259873i \(0.0836808\pi\)
−0.965643 + 0.259873i \(0.916319\pi\)
\(98\) 0 0
\(99\) −535575. −0.551969
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.d.e.31.2 24
7.2 even 3 inner 49.7.d.e.19.1 24
7.3 odd 6 49.7.b.c.48.12 yes 12
7.4 even 3 49.7.b.c.48.11 ✓ 12
7.5 odd 6 inner 49.7.d.e.19.2 24
7.6 odd 2 inner 49.7.d.e.31.1 24
21.11 odd 6 441.7.d.e.244.1 12
21.17 even 6 441.7.d.e.244.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.b.c.48.11 ✓ 12 7.4 even 3
49.7.b.c.48.12 yes 12 7.3 odd 6
49.7.d.e.19.1 24 7.2 even 3 inner
49.7.d.e.19.2 24 7.5 odd 6 inner
49.7.d.e.31.1 24 7.6 odd 2 inner
49.7.d.e.31.2 24 1.1 even 1 trivial
441.7.d.e.244.1 12 21.11 odd 6
441.7.d.e.244.2 12 21.17 even 6