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Properties

Label 175.4.x.a.3.43
Level $175$
Weight $4$
Character 175.3
Analytic conductor $10.325$
Analytic rank $0$
Dimension $928$
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] = [175,4,Mod(3,175)] 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("175.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([21, 10])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.x (of order \(60\), degree \(16\), 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(928\)
Relative dimension: \(58\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 3.43
Character \(\chi\) \(=\) 175.3
Dual form 175.4.x.a.117.43

$q$-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+(1.66660 - 2.56634i) q^{2} +(3.28687 - 8.56260i) q^{3} +(-0.554650 - 1.24576i) q^{4} +(5.95503 + 9.46244i) q^{5} +(-16.4966 - 22.7057i) q^{6} +(16.8267 - 7.73716i) q^{7} +(20.0573 + 3.17676i) q^{8} +(-42.4496 - 38.2218i) q^{9} +(34.2085 + 0.487482i) q^{10} +(-18.2776 - 20.2994i) q^{11} +(-12.4900 + 0.654575i) q^{12} +(-30.7491 - 15.6674i) q^{13} +(8.18715 - 56.0777i) q^{14} +(100.596 - 19.8887i) q^{15} +(48.8798 - 54.2865i) q^{16} +(-33.3321 + 41.1617i) q^{17} +(-168.837 + 45.2397i) q^{18} +(17.3924 + 7.74361i) q^{19} +(8.48500 - 12.6669i) q^{20} +(-10.9431 - 169.511i) q^{21} +(-82.5567 + 13.0757i) q^{22} +(91.8109 + 59.6227i) q^{23} +(93.1269 - 161.301i) q^{24} +(-54.0754 + 112.698i) q^{25} +(-91.4544 + 52.8012i) q^{26} +(-246.157 + 125.423i) q^{27} +(-18.9716 - 16.6706i) q^{28} +(-142.281 + 195.833i) q^{29} +(116.613 - 291.311i) q^{30} +(112.275 - 11.8006i) q^{31} +(-15.8073 - 58.9935i) q^{32} +(-233.892 + 89.7826i) q^{33} +(50.0836 + 154.141i) q^{34} +(173.416 + 113.146i) q^{35} +(-24.0707 + 74.0819i) q^{36} +(9.15272 + 174.644i) q^{37} +(48.8590 - 31.7294i) q^{38} +(-235.222 + 211.795i) q^{39} +(89.3816 + 208.708i) q^{40} +(-190.983 + 62.0541i) q^{41} +(-453.260 - 254.423i) q^{42} +(-131.557 - 131.557i) q^{43} +(-15.1505 + 34.0287i) q^{44} +(108.883 - 629.289i) q^{45} +(306.024 - 136.251i) q^{46} +(440.124 - 356.405i) q^{47} +(-304.172 - 596.971i) q^{48} +(223.273 - 260.381i) q^{49} +(199.100 + 326.598i) q^{50} +(242.893 + 420.702i) q^{51} +(-2.46296 + 46.9960i) q^{52} +(-275.106 - 105.603i) q^{53} +(-88.3668 + 840.754i) q^{54} +(83.2378 - 293.834i) q^{55} +(362.076 - 101.732i) q^{56} +(123.472 - 123.472i) q^{57} +(265.448 + 691.516i) q^{58} +(425.060 + 90.3494i) q^{59} +(-80.5723 - 114.288i) q^{60} +(14.1121 + 66.3922i) q^{61} +(156.833 - 307.803i) q^{62} +(-1010.01 - 314.706i) q^{63} +(378.053 + 122.837i) q^{64} +(-34.8594 - 384.261i) q^{65} +(-159.391 + 749.877i) q^{66} +(594.519 + 481.432i) q^{67} +(69.7653 + 18.6936i) q^{68} +(812.296 - 590.168i) q^{69} +(579.386 - 256.474i) q^{70} +(-785.349 - 570.590i) q^{71} +(-730.001 - 901.477i) q^{72} +(-540.459 - 28.3242i) q^{73} +(463.451 + 267.573i) q^{74} +(787.249 + 833.450i) q^{75} -25.9618i q^{76} +(-464.611 - 200.154i) q^{77} +(151.517 + 956.638i) q^{78} +(-708.371 - 74.4528i) q^{79} +(804.763 + 139.244i) q^{80} +(103.650 + 986.161i) q^{81} +(-159.040 + 593.546i) q^{82} +(-185.606 + 1171.87i) q^{83} +(-205.101 + 107.652i) q^{84} +(-587.983 - 70.2837i) q^{85} +(-556.874 + 118.367i) q^{86} +(1209.18 + 1861.97i) q^{87} +(-302.113 - 465.214i) q^{88} +(819.936 - 174.283i) q^{89} +(-1433.50 - 1328.20i) q^{90} +(-638.626 - 25.7202i) q^{91} +(23.3529 - 147.444i) q^{92} +(267.990 - 1000.15i) q^{93} +(-181.146 - 1723.49i) q^{94} +(30.2990 + 210.688i) q^{95} +(-557.094 - 58.5530i) q^{96} +(223.240 + 1409.48i) q^{97} +(-296.119 - 1006.95i) q^{98} +1560.31i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9} - 96 q^{10} - 6 q^{11} - 72 q^{12} - 20 q^{14} - 368 q^{15} - 1670 q^{16} + 120 q^{17} - 14 q^{18} - 30 q^{19} - 12 q^{21} - 880 q^{22}+ \cdots - 19478 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.66660 − 2.56634i 0.589232 − 0.907338i −0.410758 − 0.911744i \(-0.634736\pi\)
0.999990 + 0.00440625i \(0.00140256\pi\)
\(3\) 3.28687 − 8.56260i 0.632559 − 1.64787i −0.122698 − 0.992444i \(-0.539155\pi\)
0.755257 − 0.655428i \(-0.227512\pi\)
\(4\) −0.554650 − 1.24576i −0.0693312 − 0.155720i
\(5\) 5.95503 + 9.46244i 0.532634 + 0.846346i
\(6\) −16.4966 − 22.7057i −1.12245 − 1.54492i
\(7\) 16.8267 − 7.73716i 0.908554 − 0.417767i
\(8\) 20.0573 + 3.17676i 0.886414 + 0.140394i
\(9\) −42.4496 − 38.2218i −1.57221 − 1.41562i
\(10\) 34.2085 + 0.487482i 1.08177 + 0.0154155i
\(11\) −18.2776 − 20.2994i −0.500993 − 0.556409i 0.438609 − 0.898678i \(-0.355471\pi\)
−0.939602 + 0.342269i \(0.888805\pi\)
\(12\) −12.4900 + 0.654575i −0.300463 + 0.0157466i
\(13\) −30.7491 − 15.6674i −0.656020 − 0.334259i 0.0940979 − 0.995563i \(-0.470003\pi\)
−0.750118 + 0.661304i \(0.770003\pi\)
\(14\) 8.18715 − 56.0777i 0.156293 − 1.07053i
\(15\) 100.596 − 19.8887i 1.73159 − 0.342349i
\(16\) 48.8798 − 54.2865i 0.763747 − 0.848227i
\(17\) −33.3321 + 41.1617i −0.475542 + 0.587246i −0.957314 − 0.289051i \(-0.906660\pi\)
0.481772 + 0.876297i \(0.339994\pi\)
\(18\) −168.837 + 45.2397i −2.21084 + 0.592394i
\(19\) 17.3924 + 7.74361i 0.210005 + 0.0935003i 0.509044 − 0.860741i \(-0.329999\pi\)
−0.299038 + 0.954241i \(0.596666\pi\)
\(20\) 8.48500 − 12.6669i 0.0948652 − 0.141620i
\(21\) −10.9431 − 169.511i −0.113713 − 1.76144i
\(22\) −82.5567 + 13.0757i −0.800052 + 0.126716i
\(23\) 91.8109 + 59.6227i 0.832343 + 0.540530i 0.889023 − 0.457862i \(-0.151385\pi\)
−0.0566797 + 0.998392i \(0.518051\pi\)
\(24\) 93.1269 − 161.301i 0.792061 − 1.37189i
\(25\) −54.0754 + 112.698i −0.432603 + 0.901585i
\(26\) −91.4544 + 52.8012i −0.689834 + 0.398276i
\(27\) −246.157 + 125.423i −1.75455 + 0.893990i
\(28\) −18.9716 − 16.6706i −0.128046 − 0.112516i
\(29\) −142.281 + 195.833i −0.911065 + 1.25397i 0.0557366 + 0.998446i \(0.482249\pi\)
−0.966802 + 0.255528i \(0.917751\pi\)
\(30\) 116.613 − 291.311i 0.709684 − 1.77286i
\(31\) 112.275 − 11.8006i 0.650491 − 0.0683693i 0.226465 − 0.974019i \(-0.427283\pi\)
0.424026 + 0.905650i \(0.360617\pi\)
\(32\) −15.8073 − 58.9935i −0.0873237 − 0.325896i
\(33\) −233.892 + 89.7826i −1.23380 + 0.473611i
\(34\) 50.0836 + 154.141i 0.252626 + 0.777501i
\(35\) 173.416 + 113.146i 0.837502 + 0.546434i
\(36\) −24.0707 + 74.0819i −0.111438 + 0.342972i
\(37\) 9.15272 + 174.644i 0.0406675 + 0.775983i 0.940510 + 0.339767i \(0.110348\pi\)
−0.899842 + 0.436216i \(0.856318\pi\)
\(38\) 48.8590 − 31.7294i 0.208578 − 0.135452i
\(39\) −235.222 + 211.795i −0.965788 + 0.869599i
\(40\) 89.3816 + 208.708i 0.353312 + 0.824991i
\(41\) −190.983 + 62.0541i −0.727476 + 0.236371i −0.649261 − 0.760565i \(-0.724922\pi\)
−0.0782144 + 0.996937i \(0.524922\pi\)
\(42\) −453.260 − 254.423i −1.66523 − 0.934723i
\(43\) −131.557 − 131.557i −0.466565 − 0.466565i 0.434235 − 0.900800i \(-0.357019\pi\)
−0.900800 + 0.434235i \(0.857019\pi\)
\(44\) −15.1505 + 34.0287i −0.0519098 + 0.116591i
\(45\) 108.883 − 629.289i 0.360695 − 2.08464i
\(46\) 306.024 − 136.251i 0.980887 − 0.436719i
\(47\) 440.124 − 356.405i 1.36593 − 1.10611i 0.382671 − 0.923885i \(-0.375004\pi\)
0.983257 − 0.182223i \(-0.0583291\pi\)
\(48\) −304.172 − 596.971i −0.914654 − 1.79511i
\(49\) 223.273 − 260.381i 0.650941 − 0.759128i
\(50\) 199.100 + 326.598i 0.563139 + 0.923760i
\(51\) 242.893 + 420.702i 0.666897 + 1.15510i
\(52\) −2.46296 + 46.9960i −0.00656828 + 0.125330i
\(53\) −275.106 − 105.603i −0.712994 − 0.273693i −0.0252861 − 0.999680i \(-0.508050\pi\)
−0.687708 + 0.725987i \(0.741383\pi\)
\(54\) −88.3668 + 840.754i −0.222689 + 2.11874i
\(55\) 83.2378 − 293.834i 0.204069 − 0.720375i
\(56\) 362.076 − 101.732i 0.864007 − 0.242759i
\(57\) 123.472 − 123.472i 0.286917 − 0.286917i
\(58\) 265.448 + 691.516i 0.600949 + 1.56553i
\(59\) 425.060 + 90.3494i 0.937935 + 0.199364i 0.651425 − 0.758713i \(-0.274172\pi\)
0.286510 + 0.958077i \(0.407505\pi\)
\(60\) −80.5723 − 114.288i −0.173364 − 0.245909i
\(61\) 14.1121 + 66.3922i 0.0296208 + 0.139355i 0.990473 − 0.137706i \(-0.0439729\pi\)
−0.960852 + 0.277061i \(0.910640\pi\)
\(62\) 156.833 − 307.803i 0.321256 − 0.630500i
\(63\) −1010.01 − 314.706i −2.01984 − 0.629353i
\(64\) 378.053 + 122.837i 0.738385 + 0.239916i
\(65\) −34.8594 − 384.261i −0.0665197 − 0.733258i
\(66\) −159.391 + 749.877i −0.297269 + 1.39854i
\(67\) 594.519 + 481.432i 1.08406 + 0.877855i 0.993015 − 0.117992i \(-0.0376456\pi\)
0.0910462 + 0.995847i \(0.470979\pi\)
\(68\) 69.7653 + 18.6936i 0.124416 + 0.0333372i
\(69\) 812.296 − 590.168i 1.41723 − 1.02968i
\(70\) 579.386 − 256.474i 0.989284 − 0.437921i
\(71\) −785.349 − 570.590i −1.31273 − 0.953754i −0.999992 − 0.00391469i \(-0.998754\pi\)
−0.312738 − 0.949840i \(-0.601246\pi\)
\(72\) −730.001 − 901.477i −1.19488 − 1.47556i
\(73\) −540.459 − 28.3242i −0.866519 − 0.0454123i −0.386106 − 0.922455i \(-0.626180\pi\)
−0.480413 + 0.877042i \(0.659513\pi\)
\(74\) 463.451 + 267.573i 0.728041 + 0.420335i
\(75\) 787.249 + 833.450i 1.21205 + 1.28318i
\(76\) − 25.9618i − 0.0391846i
\(77\) −464.611 − 200.154i −0.687628 − 0.296229i
\(78\) 151.517 + 956.638i 0.219947 + 1.38869i
\(79\) −708.371 − 74.4528i −1.00883 − 0.106033i −0.414332 − 0.910126i \(-0.635985\pi\)
−0.594503 + 0.804093i \(0.702651\pi\)
\(80\) 804.763 + 139.244i 1.12469 + 0.194600i
\(81\) 103.650 + 986.161i 0.142181 + 1.35276i
\(82\) −159.040 + 593.546i −0.214184 + 0.799344i
\(83\) −185.606 + 1171.87i −0.245457 + 1.54976i 0.489719 + 0.871880i \(0.337099\pi\)
−0.735176 + 0.677876i \(0.762901\pi\)
\(84\) −205.101 + 107.652i −0.266409 + 0.139830i
\(85\) −587.983 − 70.2837i −0.750303 − 0.0896864i
\(86\) −556.874 + 118.367i −0.698247 + 0.148417i
\(87\) 1209.18 + 1861.97i 1.49009 + 2.29453i
\(88\) −302.113 − 465.214i −0.365970 − 0.563545i
\(89\) 819.936 − 174.283i 0.976551 − 0.207572i 0.308121 − 0.951347i \(-0.400300\pi\)
0.668430 + 0.743775i \(0.266967\pi\)
\(90\) −1433.50 − 1328.20i −1.67894 − 1.55561i
\(91\) −638.626 − 25.7202i −0.735672 − 0.0296287i
\(92\) 23.3529 − 147.444i 0.0264642 − 0.167088i
\(93\) 267.990 − 1000.15i 0.298810 − 1.11517i
\(94\) −181.146 − 1723.49i −0.198764 − 1.89111i
\(95\) 30.2990 + 210.688i 0.0327222 + 0.227538i
\(96\) −557.094 − 58.5530i −0.592273 − 0.0622504i
\(97\) 223.240 + 1409.48i 0.233677 + 1.47538i 0.773607 + 0.633666i \(0.218451\pi\)
−0.539930 + 0.841710i \(0.681549\pi\)
\(98\) −296.119 − 1006.95i −0.305230 − 1.03793i
\(99\) 1560.31i 1.58401i
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 175.4.x.a.3.43 ✓ 928
7.5 odd 6 inner 175.4.x.a.103.16 yes 928
25.17 odd 20 inner 175.4.x.a.17.16 yes 928
175.117 even 60 inner 175.4.x.a.117.43 yes 928
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.x.a.3.43 ✓ 928 1.1 even 1 trivial
175.4.x.a.17.16 yes 928 25.17 odd 20 inner
175.4.x.a.103.16 yes 928 7.5 odd 6 inner
175.4.x.a.117.43 yes 928 175.117 even 60 inner