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Properties

Label 49.7.h.a.3.6
Level $49$
Weight $7$
Character 49.3
Analytic conductor $11.273$
Analytic rank $0$
Dimension $324$
Inner twists $2$

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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(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 3.6
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.6

$q$-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+(-3.59619 + 9.16295i) q^{2} +(6.31557 + 0.473286i) q^{3} +(-24.1117 - 22.3724i) q^{4} +(88.0567 - 129.155i) q^{5} +(-27.0487 + 56.1672i) q^{6} +(-147.850 + 309.499i) q^{7} +(-275.882 + 132.858i) q^{8} +(-681.195 - 102.674i) q^{9} +(866.776 + 1271.33i) q^{10} +(-1318.45 + 198.725i) q^{11} +(-141.690 - 152.706i) q^{12} +(-949.505 + 757.205i) q^{13} +(-2304.22 - 2467.76i) q^{14} +(617.256 - 774.014i) q^{15} +(-382.559 - 5104.89i) q^{16} +(-689.667 + 2235.85i) q^{17} +(3390.50 - 5872.52i) q^{18} +(2959.56 - 1708.70i) q^{19} +(-5012.71 + 1144.12i) q^{20} +(-1080.24 + 1884.68i) q^{21} +(2920.50 - 12795.6i) q^{22} +(-11223.1 + 3461.88i) q^{23} +(-1805.23 + 708.501i) q^{24} +(-3218.70 - 8201.12i) q^{25} +(-3523.63 - 11423.3i) q^{26} +(-8754.74 - 1998.21i) q^{27} +(10489.1 - 4154.78i) q^{28} +(-4908.74 - 21506.6i) q^{29} +(4872.48 + 8439.38i) q^{30} +(-7277.80 - 4201.84i) q^{31} +(29425.0 + 9076.42i) q^{32} +(-8420.82 + 631.054i) q^{33} +(-18006.8 - 14359.9i) q^{34} +(26954.3 + 46349.1i) q^{35} +(14127.7 + 17715.6i) q^{36} +(-45332.7 + 42062.6i) q^{37} +(5013.60 + 33263.1i) q^{38} +(-6355.04 + 4332.79i) q^{39} +(-7133.96 + 47330.7i) q^{40} +(17248.7 + 35817.3i) q^{41} +(-13384.5 - 16675.8i) q^{42} +(96082.4 + 46270.9i) q^{43} +(36236.0 + 24705.3i) q^{44} +(-73244.7 + 78939.0i) q^{45} +(8639.53 - 115287. i) q^{46} +(152057. + 59678.0i) q^{47} -32421.3i q^{48} +(-73929.9 - 91518.6i) q^{49} +86721.5 q^{50} +(-5413.84 + 13794.2i) q^{51} +(39834.6 + 2985.19i) q^{52} +(-154787. - 143622. i) q^{53} +(49793.2 - 73033.3i) q^{54} +(-90432.2 + 187784. i) q^{55} +(-330.208 - 105028. i) q^{56} +(19500.0 - 9390.70i) q^{57} +(214717. + 32363.3i) q^{58} +(-101329. - 148622. i) q^{59} +(-32199.6 + 4853.31i) q^{60} +(120141. + 129481. i) q^{61} +(64673.6 - 51575.5i) q^{62} +(132492. - 195649. i) q^{63} +(15288.5 - 19171.2i) q^{64} +(14186.9 + 189311. i) q^{65} +(24500.6 - 79429.0i) q^{66} +(-149585. + 259089. i) q^{67} +(66650.2 - 38480.5i) q^{68} +(-72518.9 + 16552.0i) q^{69} +(-521627. + 80300.6i) q^{70} +(17277.5 - 75697.9i) q^{71} +(201571. - 62176.3i) q^{72} +(-609730. + 239302. i) q^{73} +(-222392. - 566647. i) q^{74} +(-16446.4 - 53318.1i) q^{75} +(-109588. - 25012.6i) q^{76} +(133428. - 437441. i) q^{77} +(-16847.2 - 73812.4i) q^{78} +(413726. + 716594. i) q^{79} +(-693011. - 400110. i) q^{80} +(425544. + 131263. i) q^{81} +(-390222. + 29243.1i) q^{82} +(170769. + 136184. i) q^{83} +(68211.2 - 21275.4i) q^{84} +(228042. + 285956. i) q^{85} +(-769508. + 713999. i) q^{86} +(-20822.7 - 138150. i) q^{87} +(337335. - 229991. i) q^{88} +(61163.0 - 405790. i) q^{89} +(-459912. - 955017. i) q^{90} +(-93969.8 - 405823. i) q^{91} +(348059. + 167616. i) q^{92} +(-43974.8 - 29981.5i) q^{93} +(-1.09365e6 + 1.17868e6i) q^{94} +(39920.8 - 532706. i) q^{95} +(181540. + 71249.2i) q^{96} +716182. i q^{97} +(1.10445e6 - 348297. i) q^{98} +918527. 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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.59619 + 9.16295i −0.449524 + 1.14537i 0.509104 + 0.860705i \(0.329977\pi\)
−0.958628 + 0.284663i \(0.908118\pi\)
\(3\) 6.31557 + 0.473286i 0.233910 + 0.0175291i 0.191169 − 0.981557i \(-0.438772\pi\)
0.0427406 + 0.999086i \(0.486391\pi\)
\(4\) −24.1117 − 22.3724i −0.376745 − 0.349568i
\(5\) 88.0567 − 129.155i 0.704454 − 1.03324i −0.292577 − 0.956242i \(-0.594513\pi\)
0.997031 − 0.0770020i \(-0.0245348\pi\)
\(6\) −27.0487 + 56.1672i −0.125225 + 0.260033i
\(7\) −147.850 + 309.499i −0.431049 + 0.902329i
\(8\) −275.882 + 132.858i −0.538832 + 0.259488i
\(9\) −681.195 − 102.674i −0.934424 − 0.140842i
\(10\) 866.776 + 1271.33i 0.866776 + 1.27133i
\(11\) −1318.45 + 198.725i −0.990572 + 0.149305i −0.624285 − 0.781197i \(-0.714609\pi\)
−0.366288 + 0.930502i \(0.619371\pi\)
\(12\) −141.690 − 152.706i −0.0819967 − 0.0883715i
\(13\) −949.505 + 757.205i −0.432183 + 0.344654i −0.815293 − 0.579048i \(-0.803424\pi\)
0.383111 + 0.923702i \(0.374853\pi\)
\(14\) −2304.22 − 2467.76i −0.839732 − 0.899328i
\(15\) 617.256 − 774.014i 0.182891 − 0.229338i
\(16\) −382.559 − 5104.89i −0.0933981 − 1.24631i
\(17\) −689.667 + 2235.85i −0.140376 + 0.455088i −0.998274 − 0.0587326i \(-0.981294\pi\)
0.857898 + 0.513821i \(0.171770\pi\)
\(18\) 3390.50 − 5872.52i 0.581362 − 1.00695i
\(19\) 2959.56 − 1708.70i 0.431485 − 0.249118i −0.268494 − 0.963281i \(-0.586526\pi\)
0.699979 + 0.714163i \(0.253193\pi\)
\(20\) −5012.71 + 1144.12i −0.626589 + 0.143015i
\(21\) −1080.24 + 1884.68i −0.116644 + 0.203508i
\(22\) 2920.50 − 12795.6i 0.274277 − 1.20169i
\(23\) −11223.1 + 3461.88i −0.922424 + 0.284530i −0.719362 − 0.694636i \(-0.755566\pi\)
−0.203062 + 0.979166i \(0.565089\pi\)
\(24\) −1805.23 + 708.501i −0.130587 + 0.0512516i
\(25\) −3218.70 − 8201.12i −0.205997 − 0.524871i
\(26\) −3523.63 − 11423.3i −0.200479 − 0.649938i
\(27\) −8754.74 − 1998.21i −0.444787 − 0.101520i
\(28\) 10489.1 − 4154.78i 0.477821 − 0.189267i
\(29\) −4908.74 − 21506.6i −0.201269 − 0.881816i −0.970166 − 0.242443i \(-0.922051\pi\)
0.768897 − 0.639373i \(-0.220806\pi\)
\(30\) 4872.48 + 8439.38i 0.180462 + 0.312570i
\(31\) −7277.80 − 4201.84i −0.244295 − 0.141044i 0.372854 − 0.927890i \(-0.378379\pi\)
−0.617149 + 0.786846i \(0.711713\pi\)
\(32\) 29425.0 + 9076.42i 0.897981 + 0.276990i
\(33\) −8420.82 + 631.054i −0.234322 + 0.0175600i
\(34\) −18006.8 − 14359.9i −0.458141 − 0.365355i
\(35\) 26954.3 + 46349.1i 0.628671 + 1.08103i
\(36\) 14127.7 + 17715.6i 0.302806 + 0.379706i
\(37\) −45332.7 + 42062.6i −0.894967 + 0.830408i −0.986223 − 0.165420i \(-0.947102\pi\)
0.0912567 + 0.995827i \(0.470912\pi\)
\(38\) 5013.60 + 33263.1i 0.0913690 + 0.606194i
\(39\) −6355.04 + 4332.79i −0.107133 + 0.0730422i
\(40\) −7133.96 + 47330.7i −0.111468 + 0.739543i
\(41\) 17248.7 + 35817.3i 0.250268 + 0.519686i 0.987819 − 0.155604i \(-0.0497325\pi\)
−0.737552 + 0.675291i \(0.764018\pi\)
\(42\) −13384.5 − 16675.8i −0.180657 − 0.225081i
\(43\) 96082.4 + 46270.9i 1.20848 + 0.581972i 0.926081 − 0.377325i \(-0.123156\pi\)
0.282397 + 0.959298i \(0.408870\pi\)
\(44\) 36236.0 + 24705.3i 0.425385 + 0.290023i
\(45\) −73244.7 + 78939.0i −0.803783 + 0.866272i
\(46\) 8639.53 − 115287.i 0.0887599 − 1.18442i
\(47\) 152057. + 59678.0i 1.46458 + 0.574805i 0.957842 − 0.287296i \(-0.0927563\pi\)
0.506737 + 0.862101i \(0.330852\pi\)
\(48\) − 32421.3i − 0.293162i
\(49\) −73929.9 − 91518.6i −0.628394 − 0.777896i
\(50\) 86721.5 0.693772
\(51\) −5413.84 + 13794.2i −0.0408126 + 0.103989i
\(52\) 39834.6 + 2985.19i 0.283303 + 0.0212306i
\(53\) −154787. − 143622.i −1.03970 − 0.964701i −0.0402951 − 0.999188i \(-0.512830\pi\)
−0.999405 + 0.0344869i \(0.989020\pi\)
\(54\) 49793.2 − 73033.3i 0.316220 − 0.463809i
\(55\) −90432.2 + 187784.i −0.543544 + 1.12868i
\(56\) −330.208 − 105028.i −0.00188028 − 0.598056i
\(57\) 19500.0 − 9390.70i 0.105295 − 0.0507076i
\(58\) 214717. + 32363.3i 1.10048 + 0.165871i
\(59\) −101329. − 148622.i −0.493376 − 0.723650i 0.496113 − 0.868258i \(-0.334760\pi\)
−0.989489 + 0.144608i \(0.953808\pi\)
\(60\) −32199.6 + 4853.31i −0.149072 + 0.0224690i
\(61\) 120141. + 129481.i 0.529300 + 0.570450i 0.939937 − 0.341349i \(-0.110884\pi\)
−0.410636 + 0.911799i \(0.634693\pi\)
\(62\) 64673.6 − 51575.5i 0.271364 − 0.216406i
\(63\) 132492. − 195649.i 0.529868 − 0.782448i
\(64\) 15288.5 − 19171.2i 0.0583211 − 0.0731323i
\(65\) 14186.9 + 189311.i 0.0516591 + 0.689343i
\(66\) 24500.6 − 79429.0i 0.0852206 − 0.276278i
\(67\) −149585. + 259089.i −0.497352 + 0.861438i −0.999995 − 0.00305547i \(-0.999027\pi\)
0.502644 + 0.864494i \(0.332361\pi\)
\(68\) 66650.2 − 38480.5i 0.211970 − 0.122381i
\(69\) −72518.9 + 16552.0i −0.220752 + 0.0503851i
\(70\) −521627. + 80300.6i −1.52078 + 0.234113i
\(71\) 17277.5 − 75697.9i 0.0482733 − 0.211499i −0.945039 − 0.326958i \(-0.893977\pi\)
0.993312 + 0.115458i \(0.0368337\pi\)
\(72\) 201571. − 62176.3i 0.540045 − 0.166582i
\(73\) −609730. + 239302.i −1.56736 + 0.615144i −0.980845 − 0.194792i \(-0.937597\pi\)
−0.586517 + 0.809937i \(0.699501\pi\)
\(74\) −222392. − 566647.i −0.548814 − 1.39835i
\(75\) −16446.4 − 53318.1i −0.0389842 − 0.126384i
\(76\) −109588. − 25012.6i −0.249644 − 0.0569795i
\(77\) 133428. − 437441.i 0.292263 − 0.958179i
\(78\) −16847.2 − 73812.4i −0.0355013 − 0.155541i
\(79\) 413726. + 716594.i 0.839134 + 1.45342i 0.890620 + 0.454748i \(0.150271\pi\)
−0.0514865 + 0.998674i \(0.516396\pi\)
\(80\) −693011. − 400110.i −1.35354 − 0.781465i
\(81\) 425544. + 131263.i 0.800736 + 0.246994i
\(82\) −390222. + 29243.1i −0.707733 + 0.0530373i
\(83\) 170769. + 136184.i 0.298658 + 0.238172i 0.761341 − 0.648352i \(-0.224542\pi\)
−0.462682 + 0.886524i \(0.653113\pi\)
\(84\) 68211.2 − 21275.4i 0.115085 − 0.0358956i
\(85\) 228042. + 285956.i 0.371328 + 0.465631i
\(86\) −769508. + 713999.i −1.20981 + 1.12254i
\(87\) −20822.7 − 138150.i −0.0316213 − 0.209794i
\(88\) 337335. − 229991.i 0.495010 − 0.337492i
\(89\) 61163.0 − 405790.i 0.0867598 − 0.575614i −0.902640 − 0.430396i \(-0.858374\pi\)
0.989400 − 0.145217i \(-0.0463881\pi\)
\(90\) −459912. − 955017.i −0.630881 − 1.31004i
\(91\) −93969.8 − 405823.i −0.124699 − 0.538533i
\(92\) 348059. + 167616.i 0.446981 + 0.215255i
\(93\) −43974.8 − 29981.5i −0.0546707 − 0.0372739i
\(94\) −1.09365e6 + 1.17868e6i −1.31673 + 1.41909i
\(95\) 39920.8 − 532706.i 0.0465616 − 0.621322i
\(96\) 181540. + 71249.2i 0.205191 + 0.0805316i
\(97\) 716182.i 0.784708i 0.919814 + 0.392354i \(0.128339\pi\)
−0.919814 + 0.392354i \(0.871661\pi\)
\(98\) 1.10445e6 − 348297.i 1.17345 − 0.370059i
\(99\) 918527. 0.946643
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.3.6 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.6 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.6 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.6 yes 324 49.33 odd 42 inner