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Properties

Label 49.7.h.a.3.5
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 3.5
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.5

$q$-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+(-4.21887 + 10.7495i) q^{2} +(22.3732 + 1.67664i) q^{3} +(-50.8375 - 47.1703i) q^{4} +(-46.9023 + 68.7930i) q^{5} +(-112.413 + 233.427i) q^{6} +(65.6154 - 336.665i) q^{7} +(55.6673 - 26.8080i) q^{8} +(-223.109 - 33.6282i) q^{9} +(-541.616 - 794.405i) q^{10} +(-1437.72 + 216.702i) q^{11} +(-1058.31 - 1140.59i) q^{12} +(-2691.13 + 2146.11i) q^{13} +(3342.16 + 2125.68i) q^{14} +(-1164.70 + 1460.48i) q^{15} +(-278.365 - 3714.53i) q^{16} +(1400.51 - 4540.35i) q^{17} +(1302.75 - 2256.43i) q^{18} +(-1179.71 + 681.106i) q^{19} +(5629.38 - 1284.87i) q^{20} +(2032.49 - 7422.27i) q^{21} +(3736.13 - 16369.0i) q^{22} +(13479.9 - 4157.99i) q^{23} +(1290.40 - 506.446i) q^{24} +(3175.80 + 8091.80i) q^{25} +(-11716.0 - 37982.5i) q^{26} +(-20881.0 - 4765.95i) q^{27} +(-19216.3 + 14020.1i) q^{28} +(4121.71 + 18058.4i) q^{29} +(-10785.8 - 18681.5i) q^{30} +(-21077.7 - 12169.2i) q^{31} +(44882.3 + 13844.4i) q^{32} +(-32529.8 + 2437.77i) q^{33} +(42897.9 + 34209.9i) q^{34} +(20082.7 + 20304.3i) q^{35} +(9756.04 + 12233.7i) q^{36} +(-26576.2 + 24659.1i) q^{37} +(-2344.51 - 15554.8i) q^{38} +(-63807.5 + 43503.2i) q^{39} +(-766.724 + 5086.88i) q^{40} +(44372.6 + 92140.7i) q^{41} +(71210.8 + 53161.8i) q^{42} +(-105445. - 50779.5i) q^{43} +(83312.2 + 56801.3i) q^{44} +(12777.7 - 13771.1i) q^{45} +(-12173.5 + 162444. i) q^{46} +(-113948. - 44721.4i) q^{47} -83572.5i q^{48} +(-109038. - 44180.9i) q^{49} -100381. q^{50} +(38946.5 - 99234.0i) q^{51} +(238043. + 17838.9i) q^{52} +(121034. + 112303. i) q^{53} +(139326. - 204353. i) q^{54} +(52525.0 - 109069. i) q^{55} +(-5372.68 - 20500.3i) q^{56} +(-27535.9 + 13260.6i) q^{57} +(-211507. - 31879.6i) q^{58} +(130438. + 191317. i) q^{59} +(128102. - 19308.2i) q^{60} +(50970.6 + 54933.2i) q^{61} +(219737. - 175235. i) q^{62} +(-25960.8 + 72906.5i) q^{63} +(-189535. + 237669. i) q^{64} +(-21416.9 - 285789. i) q^{65} +(111034. - 359964. i) q^{66} +(-247053. + 427909. i) q^{67} +(-285368. + 164757. i) q^{68} +(308560. - 70426.7i) q^{69} +(-302987. + 130218. i) q^{70} +(7737.36 - 33899.6i) q^{71} +(-13321.4 + 4109.10i) q^{72} +(221585. - 86965.5i) q^{73} +(-152952. - 389715. i) q^{74} +(57485.7 + 186364. i) q^{75} +(92101.5 + 21021.6i) q^{76} +(-21380.7 + 498251. i) q^{77} +(-198442. - 869433. i) q^{78} +(-317552. - 550016. i) q^{79} +(268589. + 155070. i) q^{80} +(-302008. - 93157.1i) q^{81} +(-1.17767e6 + 88254.1i) q^{82} +(-555604. - 443079. i) q^{83} +(-453438. + 281456. i) q^{84} +(246657. + 309298. i) q^{85} +(990711. - 919245. i) q^{86} +(61938.4 + 410934. i) q^{87} +(-74224.9 + 50605.7i) q^{88} +(-8277.34 + 54916.6i) q^{89} +(94124.8 + 195452. i) q^{90} +(545941. + 1.04683e6i) q^{91} +(-881418. - 424468. i) q^{92} +(-451173. - 307604. i) q^{93} +(961465. - 1.03621e6i) q^{94} +(8475.77 - 113101. i) q^{95} +(980948. + 384994. i) q^{96} +905298. i q^{97} +(934940. - 985713. i) q^{98} +328056. 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.21887 + 10.7495i −0.527358 + 1.34369i 0.380644 + 0.924722i \(0.375702\pi\)
−0.908002 + 0.418965i \(0.862393\pi\)
\(3\) 22.3732 + 1.67664i 0.828637 + 0.0620978i 0.482293 − 0.876010i \(-0.339804\pi\)
0.346344 + 0.938108i \(0.387423\pi\)
\(4\) −50.8375 − 47.1703i −0.794336 − 0.737036i
\(5\) −46.9023 + 68.7930i −0.375218 + 0.550344i −0.966680 − 0.255989i \(-0.917599\pi\)
0.591461 + 0.806333i \(0.298551\pi\)
\(6\) −112.413 + 233.427i −0.520429 + 1.08068i
\(7\) 65.6154 − 336.665i 0.191299 − 0.981532i
\(8\) 55.6673 − 26.8080i 0.108725 − 0.0523593i
\(9\) −223.109 − 33.6282i −0.306048 − 0.0461293i
\(10\) −541.616 − 794.405i −0.541616 − 0.794405i
\(11\) −1437.72 + 216.702i −1.08018 + 0.162811i −0.664938 − 0.746898i \(-0.731542\pi\)
−0.415245 + 0.909710i \(0.636304\pi\)
\(12\) −1058.31 − 1140.59i −0.612448 − 0.660062i
\(13\) −2691.13 + 2146.11i −1.22491 + 0.976836i −0.224916 + 0.974378i \(0.572211\pi\)
−0.999997 + 0.00245756i \(0.999218\pi\)
\(14\) 3342.16 + 2125.68i 1.21799 + 0.774664i
\(15\) −1164.70 + 1460.48i −0.345095 + 0.432735i
\(16\) −278.365 − 3714.53i −0.0679603 − 0.906867i
\(17\) 1400.51 − 4540.35i 0.285063 − 0.924150i −0.693893 − 0.720078i \(-0.744106\pi\)
0.978956 − 0.204072i \(-0.0654178\pi\)
\(18\) 1302.75 − 2256.43i 0.223380 − 0.386906i
\(19\) −1179.71 + 681.106i −0.171995 + 0.0993011i −0.583526 − 0.812094i \(-0.698327\pi\)
0.411531 + 0.911396i \(0.364994\pi\)
\(20\) 5629.38 − 1284.87i 0.703673 − 0.160609i
\(21\) 2032.49 − 7422.27i 0.219468 − 0.801454i
\(22\) 3736.13 − 16369.0i 0.350876 − 1.53729i
\(23\) 13479.9 − 4157.99i 1.10791 − 0.341744i 0.313779 − 0.949496i \(-0.398405\pi\)
0.794126 + 0.607753i \(0.207929\pi\)
\(24\) 1290.40 − 506.446i 0.0933452 − 0.0366353i
\(25\) 3175.80 + 8091.80i 0.203251 + 0.517875i
\(26\) −11716.0 − 37982.5i −0.666593 − 2.16104i
\(27\) −20881.0 − 4765.95i −1.06086 − 0.242135i
\(28\) −19216.3 + 14020.1i −0.875380 + 0.638672i
\(29\) 4121.71 + 18058.4i 0.168999 + 0.740431i 0.986400 + 0.164363i \(0.0525567\pi\)
−0.817401 + 0.576069i \(0.804586\pi\)
\(30\) −10785.8 − 18681.5i −0.399472 − 0.691906i
\(31\) −21077.7 − 12169.2i −0.707520 − 0.408487i 0.102622 − 0.994720i \(-0.467277\pi\)
−0.810142 + 0.586234i \(0.800610\pi\)
\(32\) 44882.3 + 13844.4i 1.36970 + 0.422496i
\(33\) −32529.8 + 2437.77i −0.905190 + 0.0678346i
\(34\) 42897.9 + 34209.9i 1.09144 + 0.870393i
\(35\) 20082.7 + 20304.3i 0.468402 + 0.473569i
\(36\) 9756.04 + 12233.7i 0.209106 + 0.262210i
\(37\) −26576.2 + 24659.1i −0.524673 + 0.486825i −0.897463 − 0.441091i \(-0.854592\pi\)
0.372790 + 0.927916i \(0.378401\pi\)
\(38\) −2344.51 − 15554.8i −0.0427268 − 0.283474i
\(39\) −63807.5 + 43503.2i −1.07567 + 0.733378i
\(40\) −766.724 + 5086.88i −0.0119801 + 0.0794825i
\(41\) 44372.6 + 92140.7i 0.643819 + 1.33690i 0.925993 + 0.377540i \(0.123230\pi\)
−0.282174 + 0.959363i \(0.591056\pi\)
\(42\) 71210.8 + 53161.8i 0.961166 + 0.717550i
\(43\) −105445. − 50779.5i −1.32623 − 0.638679i −0.369386 − 0.929276i \(-0.620432\pi\)
−0.956846 + 0.290597i \(0.906146\pi\)
\(44\) 83312.2 + 56801.3i 0.978026 + 0.666807i
\(45\) 12777.7 − 13771.1i 0.140222 − 0.151123i
\(46\) −12173.5 + 162444.i −0.125067 + 1.66890i
\(47\) −113948. − 44721.4i −1.09752 − 0.430747i −0.253655 − 0.967295i \(-0.581633\pi\)
−0.843870 + 0.536548i \(0.819728\pi\)
\(48\) − 83572.5i − 0.755683i
\(49\) −109038. − 44180.9i −0.926810 − 0.375531i
\(50\) −100381. −0.803048
\(51\) 38946.5 − 99234.0i 0.293601 − 0.748083i
\(52\) 238043. + 17838.9i 1.69296 + 0.126869i
\(53\) 121034. + 112303.i 0.812979 + 0.754334i 0.972511 − 0.232859i \(-0.0748080\pi\)
−0.159532 + 0.987193i \(0.550998\pi\)
\(54\) 139326. − 204353.i 0.884810 − 1.29778i
\(55\) 52525.0 − 109069.i 0.315702 − 0.655562i
\(56\) −5372.68 − 20500.3i −0.0305934 − 0.116734i
\(57\) −27535.9 + 13260.6i −0.148687 + 0.0716041i
\(58\) −211507. − 31879.6i −1.08403 − 0.163391i
\(59\) 130438. + 191317.i 0.635108 + 0.931533i 1.00000 0.000482931i \(0.000153722\pi\)
−0.364891 + 0.931050i \(0.618894\pi\)
\(60\) 128102. − 19308.2i 0.593063 − 0.0893899i
\(61\) 50970.6 + 54933.2i 0.224559 + 0.242017i 0.835275 − 0.549833i \(-0.185309\pi\)
−0.610716 + 0.791850i \(0.709118\pi\)
\(62\) 219737. − 175235.i 0.921995 − 0.735266i
\(63\) −25960.8 + 72906.5i −0.103824 + 0.291571i
\(64\) −189535. + 237669.i −0.723017 + 0.906635i
\(65\) −21416.9 − 285789.i −0.0779860 − 1.04065i
\(66\) 111034. − 359964.i 0.386211 − 1.25206i
\(67\) −247053. + 427909.i −0.821422 + 1.42274i 0.0832020 + 0.996533i \(0.473485\pi\)
−0.904624 + 0.426211i \(0.859848\pi\)
\(68\) −285368. + 164757.i −0.907567 + 0.523984i
\(69\) 308560. − 70426.7i 0.939273 − 0.214383i
\(70\) −302987. + 130218.i −0.883344 + 0.379645i
\(71\) 7737.36 − 33899.6i 0.0216181 − 0.0947152i −0.962968 − 0.269616i \(-0.913103\pi\)
0.984586 + 0.174901i \(0.0559604\pi\)
\(72\) −13321.4 + 4109.10i −0.0356904 + 0.0110090i
\(73\) 221585. − 86965.5i 0.569601 − 0.223552i −0.0630215 − 0.998012i \(-0.520074\pi\)
0.632623 + 0.774460i \(0.281978\pi\)
\(74\) −152952. − 389715.i −0.377450 − 0.961727i
\(75\) 57485.7 + 186364.i 0.136262 + 0.441752i
\(76\) 92101.5 + 21021.6i 0.209810 + 0.0478877i
\(77\) −21380.7 + 498251.i −0.0468328 + 1.09138i
\(78\) −198442. − 869433.i −0.418168 − 1.83211i
\(79\) −317552. − 550016.i −0.644070 − 1.11556i −0.984515 − 0.175298i \(-0.943911\pi\)
0.340445 − 0.940264i \(-0.389422\pi\)
\(80\) 268589. + 155070.i 0.524589 + 0.302871i
\(81\) −302008. − 93157.1i −0.568281 − 0.175292i
\(82\) −1.17767e6 + 88254.1i −2.13590 + 0.160064i
\(83\) −555604. − 443079.i −0.971697 − 0.774902i 0.00264012 − 0.999997i \(-0.499160\pi\)
−0.974337 + 0.225094i \(0.927731\pi\)
\(84\) −453438. + 281456.i −0.765032 + 0.474868i
\(85\) 246657. + 309298.i 0.401640 + 0.503641i
\(86\) 990711. − 919245.i 1.55758 − 1.44523i
\(87\) 61938.4 + 410934.i 0.0940594 + 0.624043i
\(88\) −74224.9 + 50605.7i −0.108918 + 0.0742594i
\(89\) −8277.34 + 54916.6i −0.0117414 + 0.0778993i −0.993934 − 0.109977i \(-0.964922\pi\)
0.982193 + 0.187876i \(0.0601604\pi\)
\(90\) 94124.8 + 195452.i 0.129115 + 0.268110i
\(91\) 545941. + 1.04683e6i 0.724471 + 1.38916i
\(92\) −881418. − 424468.i −1.13193 − 0.545107i
\(93\) −451173. − 307604.i −0.560911 − 0.382423i
\(94\) 961465. − 1.03621e6i 1.15758 − 1.24757i
\(95\) 8475.77 − 113101.i 0.00988572 − 0.131916i
\(96\) 980948. + 384994.i 1.10875 + 0.435151i
\(97\) 905298.i 0.991919i 0.868346 + 0.495960i \(0.165184\pi\)
−0.868346 + 0.495960i \(0.834816\pi\)
\(98\) 934940. − 985713.i 0.993357 − 1.04730i
\(99\) 328056. 0.338098
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.3.5 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.5 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.5 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.5 yes 324 49.33 odd 42 inner