LMFDB - Embedded newform 175.4.b.a.99.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,4,Mod(99,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.99"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,14,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.a.99.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -8.00000i q^{3} +7.00000 q^{4} -8.00000 q^{6} -7.00000i q^{7} -15.0000i q^{8} -37.0000 q^{9} +12.0000 q^{11} -56.0000i q^{12} -78.0000i q^{13} -7.00000 q^{14} +41.0000 q^{16} +94.0000i q^{17} +37.0000i q^{18} -40.0000 q^{19} -56.0000 q^{21} -12.0000i q^{22} +32.0000i q^{23} -120.000 q^{24} -78.0000 q^{26} +80.0000i q^{27} -49.0000i q^{28} +50.0000 q^{29} -248.000 q^{31} -161.000i q^{32} -96.0000i q^{33} +94.0000 q^{34} -259.000 q^{36} +434.000i q^{37} +40.0000i q^{38} -624.000 q^{39} +402.000 q^{41} +56.0000i q^{42} -68.0000i q^{43} +84.0000 q^{44} +32.0000 q^{46} -536.000i q^{47} -328.000i q^{48} -49.0000 q^{49} +752.000 q^{51} -546.000i q^{52} +22.0000i q^{53} +80.0000 q^{54} -105.000 q^{56} +320.000i q^{57} -50.0000i q^{58} +560.000 q^{59} -278.000 q^{61} +248.000i q^{62} +259.000i q^{63} +167.000 q^{64} -96.0000 q^{66} +164.000i q^{67} +658.000i q^{68} +256.000 q^{69} +672.000 q^{71} +555.000i q^{72} +82.0000i q^{73} +434.000 q^{74} -280.000 q^{76} -84.0000i q^{77} +624.000i q^{78} +1000.00 q^{79} -359.000 q^{81} -402.000i q^{82} -448.000i q^{83} -392.000 q^{84} -68.0000 q^{86} -400.000i q^{87} -180.000i q^{88} +870.000 q^{89} -546.000 q^{91} +224.000i q^{92} +1984.00i q^{93} -536.000 q^{94} -1288.00 q^{96} -1026.00i q^{97} +49.0000i q^{98} -444.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} - 16 q^{6} - 74 q^{9} + 24 q^{11} - 14 q^{14} + 82 q^{16} - 80 q^{19} - 112 q^{21} - 240 q^{24} - 156 q^{26} + 100 q^{29} - 496 q^{31} + 188 q^{34} - 518 q^{36} - 1248 q^{39} + 804 q^{41}+ \cdots - 888 q^{99}+O(q^{100}) $ 2 * q + 14 * q^4 - 16 * q^6 - 74 * q^9 + 24 * q^11 - 14 * q^14 + 82 * q^16 - 80 * q^19 - 112 * q^21 - 240 * q^24 - 156 * q^26 + 100 * q^29 - 496 * q^31 + 188 * q^34 - 518 * q^36 - 1248 * q^39 + 804 * q^41 + 168 * q^44 + 64 * q^46 - 98 * q^49 + 1504 * q^51 + 160 * q^54 - 210 * q^56 + 1120 * q^59 - 556 * q^61 + 334 * q^64 - 192 * q^66 + 512 * q^69 + 1344 * q^71 + 868 * q^74 - 560 * q^76 + 2000 * q^79 - 718 * q^81 - 784 * q^84 - 136 * q^86 + 1740 * q^89 - 1092 * q^91 - 1072 * q^94 - 2576 * q^96 - 888 * q^99

Character values

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

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.353553i	−0.984251	−	0.176777i	$-0.943433\pi$
$2$	− 1.00000i	− 0.353553i	0.984251	−	0.176777i	$-0.0565670\pi$
$3$	− 8.00000i	− 1.53960i	−0.638285	−	0.769800i	$-0.720356\pi$
$3$	− 8.00000i	− 1.53960i	0.638285	−	0.769800i	$-0.279644\pi$
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	−8.00000	−0.544331
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 15.0000i	− 0.662913i
$9$	−37.0000	−1.37037
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	− 56.0000i	− 1.34715i
$13$	− 78.0000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 78.0000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	−7.00000	−0.133631
$15$	0	0
$16$	41.0000	0.640625
$17$	94.0000i	1.34108i	0.741874	+	0.670540i	$0.233937\pi$
$17$	94.0000i	1.34108i	−0.741874	+	0.670540i	$0.766063\pi$
$18$	37.0000i	0.484499i
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	0	0
$21$	−56.0000	−0.581914
$22$	− 12.0000i	− 0.116291i
$23$	32.0000i	0.290107i	0.989424	+	0.145054i	$0.0463354\pi$
$23$	32.0000i	0.290107i	−0.989424	+	0.145054i	$0.953665\pi$
$24$	−120.000	−1.02062
$25$	0	0
$26$	−78.0000	−0.588348
$27$	80.0000i	0.570222i
$28$	− 49.0000i	− 0.330719i
$29$	50.0000	0.320164	0.160082	−	0.987104i	$-0.448824\pi$
$29$	50.0000	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$30$	0	0
$31$	−248.000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−248.000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 161.000i	− 0.889408i
$33$	− 96.0000i	− 0.506408i
$34$	94.0000	0.474143
$35$	0	0
$36$	−259.000	−1.19907
$37$	434.000i	1.92836i	0.265257	+	0.964178i	$0.414543\pi$
$37$	434.000i	1.92836i	−0.265257	+	0.964178i	$0.585457\pi$
$38$	40.0000i	0.170759i
$39$	−624.000	−2.56205
$40$	0	0
$41$	402.000	1.53126	0.765632	−	0.643278i	$-0.222426\pi$
$41$	402.000	1.53126	0.765632	+	0.643278i	$0.222426\pi$
$42$	56.0000i	0.205738i
$43$	− 68.0000i	− 0.241161i	−0.992704	−	0.120580i	$-0.961524\pi$
$43$	− 68.0000i	− 0.241161i	0.992704	−	0.120580i	$-0.0384755\pi$
$44$	84.0000	0.287806
$45$	0	0
$46$	32.0000	0.102568
$47$	− 536.000i	− 1.66348i	−0.555164	−	0.831741i	$-0.687345\pi$
$47$	− 536.000i	− 1.66348i	0.555164	−	0.831741i	$-0.312655\pi$
$48$	− 328.000i	− 0.986307i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	752.000	2.06473
$52$	− 546.000i	− 1.45609i
$53$	22.0000i	0.0570176i	0.999594	+	0.0285088i	$0.00907586\pi$
$53$	22.0000i	0.0570176i	−0.999594	+	0.0285088i	$0.990924\pi$
$54$	80.0000	0.201604
$55$	0	0
$56$	−105.000	−0.250557
$57$	320.000i	0.743597i
$58$	− 50.0000i	− 0.113195i
$59$	560.000	1.23569	0.617846	−	0.786299i	$-0.288006\pi$
$59$	560.000	1.23569	0.617846	+	0.786299i	$0.288006\pi$
$60$	0	0
$61$	−278.000	−0.583512	−0.291756	−	0.956493i	$-0.594240\pi$
$61$	−278.000	−0.583512	−0.291756	+	0.956493i	$0.594240\pi$
$62$	248.000i	0.508001i
$63$	259.000i	0.517951i
$64$	167.000	0.326172
$65$	0	0
$66$	−96.0000	−0.179042
$67$	164.000i	0.299042i	0.988759	+	0.149521i	$0.0477731\pi$
$67$	164.000i	0.299042i	−0.988759	+	0.149521i	$0.952227\pi$
$68$	658.000i	1.17344i
$69$	256.000	0.446649
$70$	0	0
$71$	672.000	1.12326	0.561632	−	0.827387i	$-0.310174\pi$
$71$	672.000	1.12326	0.561632	+	0.827387i	$0.310174\pi$
$72$	555.000i	0.908436i
$73$	82.0000i	0.131471i	0.997837	+	0.0657354i	$0.0209393\pi$
$73$	82.0000i	0.131471i	−0.997837	+	0.0657354i	$0.979061\pi$
$74$	434.000	0.681777
$75$	0	0
$76$	−280.000	−0.422608
$77$	− 84.0000i	− 0.124321i
$78$	624.000i	0.905822i
$79$	1000.00	1.42416	0.712081	−	0.702097i	$-0.247753\pi$
$79$	1000.00	1.42416	0.712081	+	0.702097i	$0.247753\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	− 402.000i	− 0.541384i
$83$	− 448.000i	− 0.592463i	−0.955116	−	0.296231i	$-0.904270\pi$
$83$	− 448.000i	− 0.592463i	0.955116	−	0.296231i	$-0.0957299\pi$
$84$	−392.000	−0.509175
$85$	0	0
$86$	−68.0000	−0.0852631
$87$	− 400.000i	− 0.492925i
$88$	− 180.000i	− 0.218046i
$89$	870.000	1.03618	0.518089	−	0.855327i	$-0.326644\pi$
$89$	870.000	1.03618	0.518089	+	0.855327i	$0.326644\pi$
$90$	0	0
$91$	−546.000	−0.628971
$92$	224.000i	0.253844i
$93$	1984.00i	2.21216i
$94$	−536.000	−0.588130
$95$	0	0
$96$	−1288.00	−1.36933
$97$	− 1026.00i	− 1.07396i	−0.843594	−	0.536982i	$-0.819564\pi$
$97$	− 1026.00i	− 1.07396i	0.843594	−	0.536982i	$-0.180436\pi$
$98$	49.0000i	0.0505076i
$99$	−444.000	−0.450744
$100$	0	0
$101$	482.000	0.474859	0.237430	−	0.971405i	$-0.423695\pi$
$101$	482.000	0.474859	0.237430	+	0.971405i	$0.423695\pi$
$102$	− 752.000i	− 0.729991i
$103$	272.000i	0.260203i	0.991501	+	0.130102i	$0.0415304\pi$
$103$	272.000i	0.260203i	−0.991501	+	0.130102i	$0.958470\pi$
$104$	−1170.00	−1.10315
$105$	0	0
$106$	22.0000	0.0201588
$107$	444.000i	0.401150i	0.979678	+	0.200575i	$0.0642811\pi$
$107$	444.000i	0.401150i	−0.979678	+	0.200575i	$0.935719\pi$
$108$	560.000i	0.498945i
$109$	1170.00	1.02813	0.514063	−	0.857753i	$-0.328140\pi$
$109$	1170.00	1.02813	0.514063	+	0.857753i	$0.328140\pi$
$110$	0	0
$111$	3472.00	2.96890
$112$	− 287.000i	− 0.242133i
$113$	− 798.000i	− 0.664332i	−0.943221	−	0.332166i	$-0.892221\pi$
$113$	− 798.000i	− 0.664332i	0.943221	−	0.332166i	$-0.107779\pi$
$114$	320.000	0.262901
$115$	0	0
$116$	350.000	0.280144
$117$	2886.00i	2.28043i
$118$	− 560.000i	− 0.436883i
$119$	658.000	0.506880
$120$	0	0
$121$	−1187.00	−0.891811
$122$	278.000i	0.206303i
$123$	− 3216.00i	− 2.35754i
$124$	−1736.00	−1.25724
$125$	0	0
$126$	259.000	0.183123
$127$	− 776.000i	− 0.542196i	−0.962552	−	0.271098i	$-0.912613\pi$
$127$	− 776.000i	− 0.542196i	0.962552	−	0.271098i	$-0.0873867\pi$
$128$	− 1455.00i	− 1.00473i
$129$	−544.000	−0.371291
$130$	0	0
$131$	1112.00	0.741648	0.370824	−	0.928703i	$-0.379075\pi$
$131$	1112.00	0.741648	0.370824	+	0.928703i	$0.379075\pi$
$132$	− 672.000i	− 0.443107i
$133$	280.000i	0.182549i
$134$	164.000	0.105727
$135$	0	0
$136$	1410.00	0.889018
$137$	694.000i	0.432791i	0.976306	+	0.216396i	$0.0694301\pi$
$137$	694.000i	0.432791i	−0.976306	+	0.216396i	$0.930570\pi$
$138$	− 256.000i	− 0.157914i
$139$	−360.000	−0.219675	−0.109837	−	0.993950i	$-0.535033\pi$
$139$	−360.000	−0.219675	−0.109837	+	0.993950i	$0.535033\pi$
$140$	0	0
$141$	−4288.00	−2.56110
$142$	− 672.000i	− 0.397134i
$143$	− 936.000i	− 0.547358i
$144$	−1517.00	−0.877894
$145$	0	0
$146$	82.0000	0.0464820
$147$	392.000i	0.219943i
$148$	3038.00i	1.68731i
$149$	−2270.00	−1.24809	−0.624046	−	0.781388i	$-0.714512\pi$
$149$	−2270.00	−1.24809	−0.624046	+	0.781388i	$0.714512\pi$
$150$	0	0
$151$	632.000	0.340606	0.170303	−	0.985392i	$-0.445525\pi$
$151$	632.000	0.340606	0.170303	+	0.985392i	$0.445525\pi$
$152$	600.000i	0.320174i
$153$	− 3478.00i	− 1.83778i
$154$	−84.0000	−0.0439540
$155$	0	0
$156$	−4368.00	−2.24179
$157$	734.000i	0.373118i	0.982444	+	0.186559i	$0.0597336\pi$
$157$	734.000i	0.373118i	−0.982444	+	0.186559i	$0.940266\pi$
$158$	− 1000.00i	− 0.503517i
$159$	176.000	0.0877843
$160$	0	0
$161$	224.000	0.109650
$162$	359.000i	0.174109i
$163$	2532.00i	1.21670i	0.793670	+	0.608348i	$0.208168\pi$
$163$	2532.00i	1.21670i	−0.793670	+	0.608348i	$0.791832\pi$
$164$	2814.00	1.33986
$165$	0	0
$166$	−448.000	−0.209467
$167$	− 416.000i	− 0.192761i	−0.995345	−	0.0963804i	$-0.969273\pi$
$167$	− 416.000i	− 0.192761i	0.995345	−	0.0963804i	$-0.0307265\pi$
$168$	840.000i	0.385758i
$169$	−3887.00	−1.76923
$170$	0	0
$171$	1480.00	0.661862
$172$	− 476.000i	− 0.211015i
$173$	3042.00i	1.33687i	0.743769	+	0.668436i	$0.233036\pi$
$173$	3042.00i	1.33687i	−0.743769	+	0.668436i	$0.766964\pi$
$174$	−400.000	−0.174275
$175$	0	0
$176$	492.000	0.210715
$177$	− 4480.00i	− 1.90247i
$178$	− 870.000i	− 0.366344i
$179$	180.000	0.0751611	0.0375805	−	0.999294i	$-0.488035\pi$
$179$	180.000	0.0751611	0.0375805	+	0.999294i	$0.488035\pi$
$180$	0	0
$181$	−1958.00	−0.804072	−0.402036	−	0.915624i	$-0.631697\pi$
$181$	−1958.00	−0.804072	−0.402036	+	0.915624i	$0.631697\pi$
$182$	546.000i	0.222375i
$183$	2224.00i	0.898376i
$184$	480.000	0.192316
$185$	0	0
$186$	1984.00	0.782118
$187$	1128.00i	0.441110i
$188$	− 3752.00i	− 1.45555i
$189$	560.000	0.215524
$190$	0	0
$191$	−2888.00	−1.09408	−0.547038	−	0.837108i	$-0.684245\pi$
$191$	−2888.00	−1.09408	−0.547038	+	0.837108i	$0.684245\pi$
$192$	− 1336.00i	− 0.502174i
$193$	1602.00i	0.597484i	0.954334	+	0.298742i	$0.0965671\pi$
$193$	1602.00i	0.597484i	−0.954334	+	0.298742i	$0.903433\pi$
$194$	−1026.00	−0.379704
$195$	0	0
$196$	−343.000	−0.125000
$197$	4794.00i	1.73380i	0.498483	+	0.866899i	$0.333891\pi$
$197$	4794.00i	1.73380i	−0.498483	+	0.866899i	$0.666109\pi$
$198$	444.000i	0.159362i
$199$	−1280.00	−0.455964	−0.227982	−	0.973665i	$-0.573213\pi$
$199$	−1280.00	−0.455964	−0.227982	+	0.973665i	$0.573213\pi$
$200$	0	0
$201$	1312.00	0.460405
$202$	− 482.000i	− 0.167888i
$203$	− 350.000i	− 0.121011i
$204$	5264.00	1.80664
$205$	0	0
$206$	272.000	0.0919958
$207$	− 1184.00i	− 0.397554i
$208$	− 3198.00i	− 1.06606i
$209$	−480.000	−0.158863
$210$	0	0
$211$	−68.0000	−0.0221863	−0.0110932	−	0.999938i	$-0.503531\pi$
$211$	−68.0000	−0.0221863	−0.0110932	+	0.999938i	$0.503531\pi$
$212$	154.000i	0.0498904i
$213$	− 5376.00i	− 1.72938i
$214$	444.000	0.141828
$215$	0	0
$216$	1200.00	0.378008
$217$	1736.00i	0.543075i
$218$	− 1170.00i	− 0.363497i
$219$	656.000	0.202413
$220$	0	0
$221$	7332.00	2.23169
$222$	− 3472.00i	− 1.04966i
$223$	− 1728.00i	− 0.518903i	−0.965756	−	0.259452i	$-0.916458\pi$
$223$	− 1728.00i	− 0.518903i	0.965756	−	0.259452i	$-0.0835418\pi$
$224$	−1127.00	−0.336165
$225$	0	0
$226$	−798.000	−0.234877
$227$	4864.00i	1.42218i	0.703101	+	0.711090i	$0.251798\pi$
$227$	4864.00i	1.42218i	−0.703101	+	0.711090i	$0.748202\pi$
$228$	2240.00i	0.650647i
$229$	5510.00	1.59000	0.795002	−	0.606606i	$-0.207470\pi$
$229$	5510.00	1.59000	0.795002	+	0.606606i	$0.207470\pi$
$230$	0	0
$231$	−672.000	−0.191404
$232$	− 750.000i	− 0.212241i
$233$	5322.00i	1.49638i	0.663486	+	0.748188i	$0.269076\pi$
$233$	5322.00i	1.49638i	−0.663486	+	0.748188i	$0.730924\pi$
$234$	2886.00	0.806255
$235$	0	0
$236$	3920.00	1.08123
$237$	− 8000.00i	− 2.19264i
$238$	− 658.000i	− 0.179209i
$239$	1840.00	0.497990	0.248995	−	0.968505i	$-0.419900\pi$
$239$	1840.00	0.497990	0.248995	+	0.968505i	$0.419900\pi$
$240$	0	0
$241$	−438.000	−0.117071	−0.0585354	−	0.998285i	$-0.518643\pi$
$241$	−438.000	−0.117071	−0.0585354	+	0.998285i	$0.518643\pi$
$242$	1187.00i	0.315303i
$243$	5032.00i	1.32841i
$244$	−1946.00	−0.510573
$245$	0	0
$246$	−3216.00	−0.833515
$247$	3120.00i	0.803728i
$248$	3720.00i	0.952501i
$249$	−3584.00	−0.912156
$250$	0	0
$251$	5592.00	1.40623	0.703115	−	0.711076i	$-0.251792\pi$
$251$	5592.00	1.40623	0.703115	+	0.711076i	$0.251792\pi$
$252$	1813.00i	0.453207i
$253$	384.000i	0.0954224i
$254$	−776.000	−0.191695
$255$	0	0
$256$	−119.000	−0.0290527
$257$	1974.00i	0.479123i	0.970881	+	0.239562i	$0.0770038\pi$
$257$	1974.00i	0.479123i	−0.970881	+	0.239562i	$0.922996\pi$
$258$	544.000i	0.131271i
$259$	3038.00	0.728850
$260$	0	0
$261$	−1850.00	−0.438744
$262$	− 1112.00i	− 0.262212i
$263$	− 728.000i	− 0.170686i	−0.996352	−	0.0853430i	$-0.972801\pi$
$263$	− 728.000i	− 0.170686i	0.996352	−	0.0853430i	$-0.0271986\pi$
$264$	−1440.00	−0.335704
$265$	0	0
$266$	280.000	0.0645410
$267$	− 6960.00i	− 1.59530i
$268$	1148.00i	0.261661i
$269$	−5810.00	−1.31688	−0.658442	−	0.752631i	$-0.728784\pi$
$269$	−5810.00	−1.31688	−0.658442	+	0.752631i	$0.728784\pi$
$270$	0	0
$271$	−6528.00	−1.46328	−0.731638	−	0.681693i	$-0.761244\pi$
$271$	−6528.00	−1.46328	−0.731638	+	0.681693i	$0.761244\pi$
$272$	3854.00i	0.859129i
$273$	4368.00i	0.968364i
$274$	694.000	0.153015
$275$	0	0
$276$	1792.00	0.390818
$277$	− 5126.00i	− 1.11188i	−0.831222	−	0.555941i	$-0.812358\pi$
$277$	− 5126.00i	− 1.11188i	0.831222	−	0.555941i	$-0.187642\pi$
$278$	360.000i	0.0776668i
$279$	9176.00	1.96901
$280$	0	0
$281$	−2358.00	−0.500592	−0.250296	−	0.968169i	$-0.580528\pi$
$281$	−2358.00	−0.500592	−0.250296	+	0.968169i	$0.580528\pi$
$282$	4288.00i	0.905485i
$283$	392.000i	0.0823392i	0.999152	+	0.0411696i	$0.0131084\pi$
$283$	392.000i	0.0823392i	−0.999152	+	0.0411696i	$0.986892\pi$
$284$	4704.00	0.982856
$285$	0	0
$286$	−936.000	−0.193520
$287$	− 2814.00i	− 0.578764i
$288$	5957.00i	1.21882i
$289$	−3923.00	−0.798494
$290$	0	0
$291$	−8208.00	−1.65348
$292$	574.000i	0.115037i
$293$	1202.00i	0.239664i	0.992794	+	0.119832i	$0.0382356\pi$
$293$	1202.00i	0.239664i	−0.992794	+	0.119832i	$0.961764\pi$
$294$	392.000	0.0777616
$295$	0	0
$296$	6510.00	1.27833
$297$	960.000i	0.187558i
$298$	2270.00i	0.441267i
$299$	2496.00	0.482767
$300$	0	0
$301$	−476.000	−0.0911501
$302$	− 632.000i	− 0.120422i
$303$	− 3856.00i	− 0.731094i
$304$	−1640.00	−0.309409
$305$	0	0
$306$	−3478.00	−0.649752
$307$	6384.00i	1.18682i	0.804900	+	0.593411i	$0.202219\pi$
$307$	6384.00i	1.18682i	−0.804900	+	0.593411i	$0.797781\pi$
$308$	− 588.000i	− 0.108781i
$309$	2176.00	0.400609
$310$	0	0
$311$	−4968.00	−0.905818	−0.452909	−	0.891557i	$-0.649614\pi$
$311$	−4968.00	−0.905818	−0.452909	+	0.891557i	$0.649614\pi$
$312$	9360.00i	1.69842i
$313$	− 2758.00i	− 0.498056i	−0.968496	−	0.249028i	$-0.919889\pi$
$313$	− 2758.00i	− 0.498056i	0.968496	−	0.249028i	$-0.0801110\pi$
$314$	734.000	0.131917
$315$	0	0
$316$	7000.00	1.24614
$317$	6274.00i	1.11162i	0.831310	+	0.555809i	$0.187591\pi$
$317$	6274.00i	1.11162i	−0.831310	+	0.555809i	$0.812409\pi$
$318$	− 176.000i	− 0.0310364i
$319$	600.000	0.105309
$320$	0	0
$321$	3552.00	0.617612
$322$	− 224.000i	− 0.0387672i
$323$	− 3760.00i	− 0.647715i
$324$	−2513.00	−0.430898
$325$	0	0
$326$	2532.00	0.430167
$327$	− 9360.00i	− 1.58290i
$328$	− 6030.00i	− 1.01509i
$329$	−3752.00	−0.628737
$330$	0	0
$331$	1932.00	0.320823	0.160411	−	0.987050i	$-0.448718\pi$
$331$	1932.00	0.320823	0.160411	+	0.987050i	$0.448718\pi$
$332$	− 3136.00i	− 0.518405i
$333$	− 16058.0i	− 2.64256i
$334$	−416.000	−0.0681512
$335$	0	0
$336$	−2296.00	−0.372789
$337$	− 2386.00i	− 0.385679i	−0.981230	−	0.192839i	$-0.938230\pi$
$337$	− 2386.00i	− 0.385679i	0.981230	−	0.192839i	$-0.0617696\pi$
$338$	3887.00i	0.625518i
$339$	−6384.00	−1.02281
$340$	0	0
$341$	−2976.00	−0.472608
$342$	− 1480.00i	− 0.234004i
$343$	343.000i	0.0539949i
$344$	−1020.00	−0.159868
$345$	0	0
$346$	3042.00	0.472656
$347$	− 6076.00i	− 0.939991i	−0.882669	−	0.469995i	$-0.844256\pi$
$347$	− 6076.00i	− 0.939991i	0.882669	−	0.469995i	$-0.155744\pi$
$348$	− 2800.00i	− 0.431310i
$349$	−2210.00	−0.338964	−0.169482	−	0.985533i	$-0.554210\pi$
$349$	−2210.00	−0.338964	−0.169482	+	0.985533i	$0.554210\pi$
$350$	0	0
$351$	6240.00	0.948908
$352$	− 1932.00i	− 0.292545i
$353$	− 2598.00i	− 0.391721i	−0.980632	−	0.195861i	$-0.937250\pi$
$353$	− 2598.00i	− 0.391721i	0.980632	−	0.195861i	$-0.0627500\pi$
$354$	−4480.00	−0.672625
$355$	0	0
$356$	6090.00	0.906655
$357$	− 5264.00i	− 0.780393i
$358$	− 180.000i	− 0.0265735i
$359$	13320.0	1.95822	0.979112	−	0.203320i	$-0.0651731\pi$
$359$	13320.0	1.95822	0.979112	+	0.203320i	$0.0651731\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	1958.00i	0.284282i
$363$	9496.00i	1.37303i
$364$	−3822.00	−0.550350
$365$	0	0
$366$	2224.00	0.317624
$367$	− 10816.0i	− 1.53839i	−0.639012	−	0.769197i	$-0.720656\pi$
$367$	− 10816.0i	− 1.53839i	0.639012	−	0.769197i	$-0.279344\pi$
$368$	1312.00i	0.185850i
$369$	−14874.0	−2.09840
$370$	0	0
$371$	154.000	0.0215506
$372$	13888.0i	1.93564i
$373$	− 11098.0i	− 1.54057i	−0.637700	−	0.770285i	$-0.720114\pi$
$373$	− 11098.0i	− 1.54057i	0.637700	−	0.770285i	$-0.279886\pi$
$374$	1128.00	0.155956
$375$	0	0
$376$	−8040.00	−1.10274
$377$	− 3900.00i	− 0.532786i
$378$	− 560.000i	− 0.0761992i
$379$	−7100.00	−0.962276	−0.481138	−	0.876645i	$-0.659776\pi$
$379$	−7100.00	−0.962276	−0.481138	+	0.876645i	$0.659776\pi$
$380$	0	0
$381$	−6208.00	−0.834765
$382$	2888.00i	0.386814i
$383$	− 728.000i	− 0.0971255i	−0.998820	−	0.0485627i	$-0.984536\pi$
$383$	− 728.000i	− 0.0971255i	0.998820	−	0.0485627i	$-0.0154641\pi$
$384$	−11640.0	−1.54688
$385$	0	0
$386$	1602.00	0.211243
$387$	2516.00i	0.330479i
$388$	− 7182.00i	− 0.939719i
$389$	6810.00	0.887611	0.443806	−	0.896123i	$-0.353628\pi$
$389$	6810.00	0.887611	0.443806	+	0.896123i	$0.353628\pi$
$390$	0	0
$391$	−3008.00	−0.389057
$392$	735.000i	0.0947018i
$393$	− 8896.00i	− 1.14184i
$394$	4794.00	0.612990
$395$	0	0
$396$	−3108.00	−0.394401
$397$	574.000i	0.0725648i	0.999342	+	0.0362824i	$0.0115516\pi$
$397$	574.000i	0.0725648i	−0.999342	+	0.0362824i	$0.988448\pi$
$398$	1280.00i	0.161208i
$399$	2240.00	0.281053
$400$	0	0
$401$	6162.00	0.767371	0.383685	−	0.923464i	$-0.374655\pi$
$401$	6162.00	0.767371	0.383685	+	0.923464i	$0.374655\pi$
$402$	− 1312.00i	− 0.162778i
$403$	19344.0i	2.39105i
$404$	3374.00	0.415502
$405$	0	0
$406$	−350.000	−0.0427838
$407$	5208.00i	0.634278i
$408$	− 11280.0i	− 1.36873i
$409$	−8210.00	−0.992563	−0.496282	−	0.868162i	$-0.665302\pi$
$409$	−8210.00	−0.992563	−0.496282	+	0.868162i	$0.665302\pi$
$410$	0	0
$411$	5552.00	0.666326
$412$	1904.00i	0.227678i
$413$	− 3920.00i	− 0.467047i
$414$	−1184.00	−0.140557
$415$	0	0
$416$	−12558.0	−1.48006
$417$	2880.00i	0.338212i
$418$	480.000i	0.0561664i
$419$	−4800.00	−0.559655	−0.279827	−	0.960050i	$-0.590277\pi$
$419$	−4800.00	−0.559655	−0.279827	+	0.960050i	$0.590277\pi$
$420$	0	0
$421$	−9938.00	−1.15047	−0.575236	−	0.817988i	$-0.695090\pi$
$421$	−9938.00	−1.15047	−0.575236	+	0.817988i	$0.695090\pi$
$422$	68.0000i	0.00784405i
$423$	19832.0i	2.27959i
$424$	330.000	0.0377977
$425$	0	0
$426$	−5376.00	−0.611427
$427$	1946.00i	0.220547i
$428$	3108.00i	0.351007i
$429$	−7488.00	−0.842713
$430$	0	0
$431$	−9248.00	−1.03355	−0.516776	−	0.856121i	$-0.672868\pi$
$431$	−9248.00	−1.03355	−0.516776	+	0.856121i	$0.672868\pi$
$432$	3280.00i	0.365299i
$433$	− 1118.00i	− 0.124082i	−0.998074	−	0.0620412i	$-0.980239\pi$
$433$	− 1118.00i	− 0.124082i	0.998074	−	0.0620412i	$-0.0197610\pi$
$434$	1736.00	0.192006
$435$	0	0
$436$	8190.00	0.899610
$437$	− 1280.00i	− 0.140116i
$438$	− 656.000i	− 0.0715637i
$439$	11960.0	1.30027	0.650136	−	0.759818i	$-0.274712\pi$
$439$	11960.0	1.30027	0.650136	+	0.759818i	$0.274712\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	− 7332.00i	− 0.789022i
$443$	7332.00i	0.786352i	0.919463	+	0.393176i	$0.128624\pi$
$443$	7332.00i	0.786352i	−0.919463	+	0.393176i	$0.871376\pi$
$444$	24304.0	2.59779
$445$	0	0
$446$	−1728.00	−0.183460
$447$	18160.0i	1.92156i
$448$	− 1169.00i	− 0.123281i
$449$	−1890.00	−0.198652	−0.0993259	−	0.995055i	$-0.531669\pi$
$449$	−1890.00	−0.198652	−0.0993259	+	0.995055i	$0.531669\pi$
$450$	0	0
$451$	4824.00	0.503666
$452$	− 5586.00i	− 0.581291i
$453$	− 5056.00i	− 0.524396i
$454$	4864.00	0.502817
$455$	0	0
$456$	4800.00	0.492940
$457$	7014.00i	0.717945i	0.933348	+	0.358973i	$0.116873\pi$
$457$	7014.00i	0.717945i	−0.933348	+	0.358973i	$0.883127\pi$
$458$	− 5510.00i	− 0.562152i
$459$	−7520.00	−0.764714
$460$	0	0
$461$	−8318.00	−0.840364	−0.420182	−	0.907440i	$-0.638034\pi$
$461$	−8318.00	−0.840364	−0.420182	+	0.907440i	$0.638034\pi$
$462$	672.000i	0.0676716i
$463$	6432.00i	0.645616i	0.946464	+	0.322808i	$0.104627\pi$
$463$	6432.00i	0.645616i	−0.946464	+	0.322808i	$0.895373\pi$
$464$	2050.00	0.205105
$465$	0	0
$466$	5322.00	0.529049
$467$	10064.0i	0.997230i	0.866824	+	0.498615i	$0.166158\pi$
$467$	10064.0i	0.997230i	−0.866824	+	0.498615i	$0.833842\pi$
$468$	20202.0i	1.99538i
$469$	1148.00	0.113027
$470$	0	0
$471$	5872.00	0.574453
$472$	− 8400.00i	− 0.819155i
$473$	− 816.000i	− 0.0793229i
$474$	−8000.00	−0.775216
$475$	0	0
$476$	4606.00	0.443520
$477$	− 814.000i	− 0.0781352i
$478$	− 1840.00i	− 0.176066i
$479$	−1400.00	−0.133544	−0.0667721	−	0.997768i	$-0.521270\pi$
$479$	−1400.00	−0.133544	−0.0667721	+	0.997768i	$0.521270\pi$
$480$	0	0
$481$	33852.0	3.20898
$482$	438.000i	0.0413908i
$483$	− 1792.00i	− 0.168817i
$484$	−8309.00	−0.780334
$485$	0	0
$486$	5032.00	0.469663
$487$	− 13376.0i	− 1.24461i	−0.782775	−	0.622304i	$-0.786197\pi$
$487$	− 13376.0i	− 1.24461i	0.782775	−	0.622304i	$-0.213803\pi$
$488$	4170.00i	0.386818i
$489$	20256.0	1.87323
$490$	0	0
$491$	7092.00	0.651848	0.325924	−	0.945396i	$-0.394325\pi$
$491$	7092.00	0.651848	0.325924	+	0.945396i	$0.394325\pi$
$492$	− 22512.0i	− 2.06284i
$493$	4700.00i	0.429366i
$494$	3120.00	0.284161
$495$	0	0
$496$	−10168.0	−0.920477
$497$	− 4704.00i	− 0.424554i
$498$	3584.00i	0.322496i
$499$	820.000	0.0735636	0.0367818	−	0.999323i	$-0.488289\pi$
$499$	820.000	0.0735636	0.0367818	+	0.999323i	$0.488289\pi$
$500$	0	0
$501$	−3328.00	−0.296775
$502$	− 5592.00i	− 0.497178i
$503$	− 4568.00i	− 0.404925i	−0.979290	−	0.202462i	$-0.935106\pi$
$503$	− 4568.00i	− 0.404925i	0.979290	−	0.202462i	$-0.0648944\pi$
$504$	3885.00	0.343356
$505$	0	0
$506$	384.000	0.0337369
$507$	31096.0i	2.72391i
$508$	− 5432.00i	− 0.474421i
$509$	−19810.0	−1.72507	−0.862537	−	0.505994i	$-0.831126\pi$
$509$	−19810.0	−1.72507	−0.862537	+	0.505994i	$0.831126\pi$
$510$	0	0
$511$	574.000	0.0496913
$512$	− 11521.0i	− 0.994455i
$513$	− 3200.00i	− 0.275406i
$514$	1974.00	0.169396
$515$	0	0
$516$	−3808.00	−0.324880
$517$	− 6432.00i	− 0.547155i
$518$	− 3038.00i	− 0.257687i
$519$	24336.0	2.05825
$520$	0	0
$521$	−1838.00	−0.154557	−0.0772785	−	0.997010i	$-0.524623\pi$
$521$	−1838.00	−0.154557	−0.0772785	+	0.997010i	$0.524623\pi$
$522$	1850.00i	0.155119i
$523$	2072.00i	0.173236i	0.996242	+	0.0866178i	$0.0276059\pi$
$523$	2072.00i	0.173236i	−0.996242	+	0.0866178i	$0.972394\pi$
$524$	7784.00	0.648942
$525$	0	0
$526$	−728.000	−0.0603466
$527$	− 23312.0i	− 1.92692i
$528$	− 3936.00i	− 0.324417i
$529$	11143.0	0.915838
$530$	0	0
$531$	−20720.0	−1.69335
$532$	1960.00i	0.159731i
$533$	− 31356.0i	− 2.54818i
$534$	−6960.00	−0.564024
$535$	0	0
$536$	2460.00	0.198238
$537$	− 1440.00i	− 0.115718i
$538$	5810.00i	0.465589i
$539$	−588.000	−0.0469888
$540$	0	0
$541$	−3498.00	−0.277987	−0.138993	−	0.990293i	$-0.544387\pi$
$541$	−3498.00	−0.277987	−0.138993	+	0.990293i	$0.544387\pi$
$542$	6528.00i	0.517346i
$543$	15664.0i	1.23795i
$544$	15134.0	1.19277
$545$	0	0
$546$	4368.00	0.342368
$547$	− 5076.00i	− 0.396772i	−0.980124	−	0.198386i	$-0.936430\pi$
$547$	− 5076.00i	− 0.396772i	0.980124	−	0.198386i	$-0.0635699\pi$
$548$	4858.00i	0.378692i
$549$	10286.0	0.799628
$550$	0	0
$551$	−2000.00	−0.154633
$552$	− 3840.00i	− 0.296089i
$553$	− 7000.00i	− 0.538283i
$554$	−5126.00	−0.393110
$555$	0	0
$556$	−2520.00	−0.192215
$557$	8674.00i	0.659837i	0.944009	+	0.329918i	$0.107021\pi$
$557$	8674.00i	0.659837i	−0.944009	+	0.329918i	$0.892979\pi$
$558$	− 9176.00i	− 0.696149i
$559$	−5304.00	−0.401315
$560$	0	0
$561$	9024.00	0.679133
$562$	2358.00i	0.176986i
$563$	16072.0i	1.20312i	0.798829	+	0.601558i	$0.205453\pi$
$563$	16072.0i	1.20312i	−0.798829	+	0.601558i	$0.794547\pi$
$564$	−30016.0	−2.24096
$565$	0	0
$566$	392.000	0.0291113
$567$	2513.00i	0.186131i
$568$	− 10080.0i	− 0.744626i
$569$	−2730.00	−0.201138	−0.100569	−	0.994930i	$-0.532066\pi$
$569$	−2730.00	−0.201138	−0.100569	+	0.994930i	$0.532066\pi$
$570$	0	0
$571$	19932.0	1.46082	0.730410	−	0.683009i	$-0.239329\pi$
$571$	19932.0	1.46082	0.730410	+	0.683009i	$0.239329\pi$
$572$	− 6552.00i	− 0.478939i
$573$	23104.0i	1.68444i
$574$	−2814.00	−0.204624
$575$	0	0
$576$	−6179.00	−0.446976
$577$	20054.0i	1.44690i	0.690379	+	0.723448i	$0.257444\pi$
$577$	20054.0i	1.44690i	−0.690379	+	0.723448i	$0.742556\pi$
$578$	3923.00i	0.282310i
$579$	12816.0	0.919887
$580$	0	0
$581$	−3136.00	−0.223930
$582$	8208.00i	0.584592i
$583$	264.000i	0.0187543i
$584$	1230.00	0.0871537
$585$	0	0
$586$	1202.00	0.0847341
$587$	2544.00i	0.178879i	0.995992	+	0.0894396i	$0.0285076\pi$
$587$	2544.00i	0.178879i	−0.995992	+	0.0894396i	$0.971492\pi$
$588$	2744.00i	0.192450i
$589$	9920.00	0.693967
$590$	0	0
$591$	38352.0	2.66936
$592$	17794.0i	1.23535i
$593$	14202.0i	0.983484i	0.870741	+	0.491742i	$0.163640\pi$
$593$	14202.0i	0.983484i	−0.870741	+	0.491742i	$0.836360\pi$
$594$	960.000	0.0663119
$595$	0	0
$596$	−15890.0	−1.09208
$597$	10240.0i	0.702002i
$598$	− 2496.00i	− 0.170684i
$599$	19600.0	1.33695	0.668476	−	0.743734i	$-0.266947\pi$
$599$	19600.0	1.33695	0.668476	+	0.743734i	$0.266947\pi$
$600$	0	0
$601$	−27078.0	−1.83783	−0.918914	−	0.394458i	$-0.870932\pi$
$601$	−27078.0	−1.83783	−0.918914	+	0.394458i	$0.870932\pi$
$602$	476.000i	0.0322264i
$603$	− 6068.00i	− 0.409798i
$604$	4424.00	0.298030
$605$	0	0
$606$	−3856.00	−0.258481
$607$	2704.00i	0.180811i	0.995905	+	0.0904053i	$0.0288162\pi$
$607$	2704.00i	0.180811i	−0.995905	+	0.0904053i	$0.971184\pi$
$608$	6440.00i	0.429567i
$609$	−2800.00	−0.186308
$610$	0	0
$611$	−41808.0	−2.76820
$612$	− 24346.0i	− 1.60805i
$613$	12702.0i	0.836915i	0.908236	+	0.418458i	$0.137429\pi$
$613$	12702.0i	0.836915i	−0.908236	+	0.418458i	$0.862571\pi$
$614$	6384.00	0.419605
$615$	0	0
$616$	−1260.00	−0.0824137
$617$	− 12666.0i	− 0.826441i	−0.910631	−	0.413220i	$-0.864404\pi$
$617$	− 12666.0i	− 0.826441i	0.910631	−	0.413220i	$-0.135596\pi$
$618$	− 2176.00i	− 0.141637i
$619$	−960.000	−0.0623355	−0.0311677	−	0.999514i	$-0.509923\pi$
$619$	−960.000	−0.0623355	−0.0311677	+	0.999514i	$0.509923\pi$
$620$	0	0
$621$	−2560.00	−0.165426
$622$	4968.00i	0.320255i
$623$	− 6090.00i	− 0.391638i
$624$	−25584.0	−1.64131
$625$	0	0
$626$	−2758.00	−0.176089
$627$	3840.00i	0.244585i
$628$	5138.00i	0.326479i
$629$	−40796.0	−2.58608
$630$	0	0
$631$	23232.0	1.46569	0.732846	−	0.680395i	$-0.238192\pi$
$631$	23232.0	1.46569	0.732846	+	0.680395i	$0.238192\pi$
$632$	− 15000.0i	− 0.944095i
$633$	544.000i	0.0341581i
$634$	6274.00	0.393016
$635$	0	0
$636$	1232.00	0.0768113
$637$	3822.00i	0.237729i
$638$	− 600.000i	− 0.0372323i
$639$	−24864.0	−1.53929
$640$	0	0
$641$	12162.0	0.749407	0.374704	−	0.927145i	$-0.377744\pi$
$641$	12162.0	0.749407	0.374704	+	0.927145i	$0.377744\pi$
$642$	− 3552.00i	− 0.218359i
$643$	− 488.000i	− 0.0299298i	−0.999888	−	0.0149649i	$-0.995236\pi$
$643$	− 488.000i	− 0.0299298i	0.999888	−	0.0149649i	$-0.00476365\pi$
$644$	1568.00	0.0959439
$645$	0	0
$646$	−3760.00	−0.229002
$647$	3984.00i	0.242082i	0.992647	+	0.121041i	$0.0386233\pi$
$647$	3984.00i	0.242082i	−0.992647	+	0.121041i	$0.961377\pi$
$648$	5385.00i	0.326455i
$649$	6720.00	0.406445
$650$	0	0
$651$	13888.0	0.836119
$652$	17724.0i	1.06461i
$653$	− 30538.0i	− 1.83008i	−0.403360	−	0.915042i	$-0.632158\pi$
$653$	− 30538.0i	− 1.83008i	0.403360	−	0.915042i	$-0.367842\pi$
$654$	−9360.00	−0.559641
$655$	0	0
$656$	16482.0	0.980966
$657$	− 3034.00i	− 0.180164i
$658$	3752.00i	0.222292i
$659$	−22740.0	−1.34420	−0.672098	−	0.740463i	$-0.734606\pi$
$659$	−22740.0	−1.34420	−0.672098	+	0.740463i	$0.734606\pi$
$660$	0	0
$661$	−18718.0	−1.10143	−0.550715	−	0.834693i	$-0.685645\pi$
$661$	−18718.0	−1.10143	−0.550715	+	0.834693i	$0.685645\pi$
$662$	− 1932.00i	− 0.113428i
$663$	− 58656.0i	− 3.43591i
$664$	−6720.00	−0.392751
$665$	0	0
$666$	−16058.0	−0.934287
$667$	1600.00i	0.0928819i
$668$	− 2912.00i	− 0.168666i
$669$	−13824.0	−0.798904
$670$	0	0
$671$	−3336.00	−0.191930
$672$	9016.00i	0.517559i
$673$	10802.0i	0.618702i	0.950948	+	0.309351i	$0.100112\pi$
$673$	10802.0i	0.618702i	−0.950948	+	0.309351i	$0.899888\pi$
$674$	−2386.00	−0.136358
$675$	0	0
$676$	−27209.0	−1.54808
$677$	− 346.000i	− 0.0196423i	−0.999952	−	0.00982117i	$-0.996874\pi$
$677$	− 346.000i	− 0.0196423i	0.999952	−	0.00982117i	$-0.00312622\pi$
$678$	6384.00i	0.361617i
$679$	−7182.00	−0.405920
$680$	0	0
$681$	38912.0	2.18959
$682$	2976.00i	0.167092i
$683$	− 11628.0i	− 0.651439i	−0.945466	−	0.325720i	$-0.894393\pi$
$683$	− 11628.0i	− 0.651439i	0.945466	−	0.325720i	$-0.105607\pi$
$684$	10360.0	0.579129
$685$	0	0
$686$	343.000	0.0190901
$687$	− 44080.0i	− 2.44797i
$688$	− 2788.00i	− 0.154493i
$689$	1716.00	0.0948830
$690$	0	0
$691$	2472.00	0.136092	0.0680458	−	0.997682i	$-0.478324\pi$
$691$	2472.00	0.136092	0.0680458	+	0.997682i	$0.478324\pi$
$692$	21294.0i	1.16976i
$693$	3108.00i	0.170365i
$694$	−6076.00	−0.332337
$695$	0	0
$696$	−6000.00	−0.326766
$697$	37788.0i	2.05355i
$698$	2210.00i	0.119842i
$699$	42576.0	2.30382
$700$	0	0
$701$	−2018.00	−0.108729	−0.0543643	−	0.998521i	$-0.517313\pi$
$701$	−2018.00	−0.108729	−0.0543643	+	0.998521i	$0.517313\pi$
$702$	− 6240.00i	− 0.335489i
$703$	− 17360.0i	− 0.931358i
$704$	2004.00	0.107285
$705$	0	0
$706$	−2598.00	−0.138494
$707$	− 3374.00i	− 0.179480i
$708$	− 31360.0i	− 1.66466i
$709$	−790.000	−0.0418464	−0.0209232	−	0.999781i	$-0.506661\pi$
$709$	−790.000	−0.0418464	−0.0209232	+	0.999781i	$0.506661\pi$
$710$	0	0
$711$	−37000.0	−1.95163
$712$	− 13050.0i	− 0.686895i
$713$	− 7936.00i	− 0.416838i
$714$	−5264.00	−0.275911
$715$	0	0
$716$	1260.00	0.0657659
$717$	− 14720.0i	− 0.766706i
$718$	− 13320.0i	− 0.692337i
$719$	−18200.0	−0.944013	−0.472007	−	0.881595i	$-0.656470\pi$
$719$	−18200.0	−0.944013	−0.472007	+	0.881595i	$0.656470\pi$
$720$	0	0
$721$	1904.00	0.0983477
$722$	5259.00i	0.271080i
$723$	3504.00i	0.180242i
$724$	−13706.0	−0.703563
$725$	0	0
$726$	9496.00	0.485440
$727$	− 29056.0i	− 1.48229i	−0.671343	−	0.741147i	$-0.734282\pi$
$727$	− 29056.0i	− 1.48229i	0.671343	−	0.741147i	$-0.265718\pi$
$728$	8190.00i	0.416953i
$729$	30563.0	1.55276
$730$	0	0
$731$	6392.00	0.323415
$732$	15568.0i	0.786079i
$733$	7082.00i	0.356862i	0.983952	+	0.178431i	$0.0571021\pi$
$733$	7082.00i	0.356862i	−0.983952	+	0.178431i	$0.942898\pi$
$734$	−10816.0	−0.543904
$735$	0	0
$736$	5152.00	0.258023
$737$	1968.00i	0.0983612i
$738$	14874.0i	0.741896i
$739$	−11060.0	−0.550539	−0.275270	−	0.961367i	$-0.588767\pi$
$739$	−11060.0	−0.550539	−0.275270	+	0.961367i	$0.588767\pi$
$740$	0	0
$741$	24960.0	1.23742
$742$	− 154.000i	− 0.00761930i
$743$	33072.0i	1.63297i	0.577369	+	0.816483i	$0.304079\pi$
$743$	33072.0i	1.63297i	−0.577369	+	0.816483i	$0.695921\pi$
$744$	29760.0	1.46647
$745$	0	0
$746$	−11098.0	−0.544674
$747$	16576.0i	0.811893i
$748$	7896.00i	0.385971i
$749$	3108.00	0.151621
$750$	0	0
$751$	29072.0	1.41259	0.706293	−	0.707919i	$-0.250366\pi$
$751$	29072.0	1.41259	0.706293	+	0.707919i	$0.250366\pi$
$752$	− 21976.0i	− 1.06567i
$753$	− 44736.0i	− 2.16503i
$754$	−3900.00	−0.188368
$755$	0	0
$756$	3920.00	0.188583
$757$	13234.0i	0.635400i	0.948191	+	0.317700i	$0.102911\pi$
$757$	13234.0i	0.635400i	−0.948191	+	0.317700i	$0.897089\pi$
$758$	7100.00i	0.340216i
$759$	3072.00	0.146912
$760$	0	0
$761$	−22398.0	−1.06692	−0.533460	−	0.845825i	$-0.679109\pi$
$761$	−22398.0	−1.06692	−0.533460	+	0.845825i	$0.679109\pi$
$762$	6208.00i	0.295134i
$763$	− 8190.00i	− 0.388595i
$764$	−20216.0	−0.957316
$765$	0	0
$766$	−728.000	−0.0343390
$767$	− 43680.0i	− 2.05631i
$768$	952.000i	0.0447296i
$769$	−6890.00	−0.323095	−0.161547	−	0.986865i	$-0.551648\pi$
$769$	−6890.00	−0.323095	−0.161547	+	0.986865i	$0.551648\pi$
$770$	0	0
$771$	15792.0	0.737659
$772$	11214.0i	0.522799i
$773$	16722.0i	0.778071i	0.921223	+	0.389035i	$0.127192\pi$
$773$	16722.0i	0.778071i	−0.921223	+	0.389035i	$0.872808\pi$
$774$	2516.00	0.116842
$775$	0	0
$776$	−15390.0	−0.711944
$777$	− 24304.0i	− 1.12214i
$778$	− 6810.00i	− 0.313818i
$779$	−16080.0	−0.739571
$780$	0	0
$781$	8064.00	0.369466
$782$	3008.00i	0.137552i
$783$	4000.00i	0.182565i
$784$	−2009.00	−0.0915179
$785$	0	0
$786$	−8896.00	−0.403702
$787$	32624.0i	1.47766i	0.673891	+	0.738831i	$0.264622\pi$
$787$	32624.0i	1.47766i	−0.673891	+	0.738831i	$0.735378\pi$
$788$	33558.0i	1.51707i
$789$	−5824.00	−0.262788
$790$	0	0
$791$	−5586.00	−0.251094
$792$	6660.00i	0.298804i
$793$	21684.0i	0.971023i
$794$	574.000	0.0256555
$795$	0	0
$796$	−8960.00	−0.398968
$797$	− 11346.0i	− 0.504261i	−0.967693	−	0.252130i	$-0.918869\pi$
$797$	− 11346.0i	− 0.504261i	0.967693	−	0.252130i	$-0.0811312\pi$
$798$	− 2240.00i	− 0.0993673i
$799$	50384.0	2.23086
$800$	0	0
$801$	−32190.0	−1.41995
$802$	− 6162.00i	− 0.271306i
$803$	984.000i	0.0432436i
$804$	9184.00	0.402854
$805$	0	0
$806$	19344.0	0.845364
$807$	46480.0i	2.02748i
$808$	− 7230.00i	− 0.314790i
$809$	35190.0	1.52931	0.764657	−	0.644438i	$-0.222909\pi$
$809$	35190.0	1.52931	0.764657	+	0.644438i	$0.222909\pi$
$810$	0	0
$811$	30432.0	1.31765	0.658824	−	0.752297i	$-0.271054\pi$
$811$	30432.0	1.31765	0.658824	+	0.752297i	$0.271054\pi$
$812$	− 2450.00i	− 0.105884i
$813$	52224.0i	2.25286i
$814$	5208.00	0.224251
$815$	0	0
$816$	30832.0	1.32272
$817$	2720.00i	0.116476i
$818$	8210.00i	0.350924i
$819$	20202.0	0.861923
$820$	0	0
$821$	12702.0	0.539955	0.269977	−	0.962867i	$-0.412984\pi$
$821$	12702.0	0.539955	0.269977	+	0.962867i	$0.412984\pi$
$822$	− 5552.00i	− 0.235582i
$823$	16952.0i	0.717995i	0.933339	+	0.358997i	$0.116881\pi$
$823$	16952.0i	0.717995i	−0.933339	+	0.358997i	$0.883119\pi$
$824$	4080.00	0.172492
$825$	0	0
$826$	−3920.00	−0.165126
$827$	25404.0i	1.06818i	0.845428	+	0.534089i	$0.179345\pi$
$827$	25404.0i	1.06818i	−0.845428	+	0.534089i	$0.820655\pi$
$828$	− 8288.00i	− 0.347860i
$829$	−26250.0	−1.09976	−0.549879	−	0.835244i	$-0.685326\pi$
$829$	−26250.0	−1.09976	−0.549879	+	0.835244i	$0.685326\pi$
$830$	0	0
$831$	−41008.0	−1.71186
$832$	− 13026.0i	− 0.542783i
$833$	− 4606.00i	− 0.191583i
$834$	2880.00	0.119576
$835$	0	0
$836$	−3360.00	−0.139005
$837$	− 19840.0i	− 0.819320i
$838$	4800.00i	0.197868i
$839$	15360.0	0.632045	0.316023	−	0.948752i	$-0.397652\pi$
$839$	15360.0	0.632045	0.316023	+	0.948752i	$0.397652\pi$
$840$	0	0
$841$	−21889.0	−0.897495
$842$	9938.00i	0.406753i
$843$	18864.0i	0.770713i
$844$	−476.000	−0.0194130
$845$	0	0
$846$	19832.0	0.805955
$847$	8309.00i	0.337073i
$848$	902.000i	0.0365269i
$849$	3136.00	0.126769
$850$	0	0
$851$	−13888.0	−0.559430
$852$	− 37632.0i	− 1.51321i
$853$	10362.0i	0.415930i	0.978136	+	0.207965i	$0.0666840\pi$
$853$	10362.0i	0.415930i	−0.978136	+	0.207965i	$0.933316\pi$
$854$	1946.00	0.0779751
$855$	0	0
$856$	6660.00	0.265928
$857$	− 4506.00i	− 0.179606i	−0.995960	−	0.0898028i	$-0.971376\pi$
$857$	− 4506.00i	− 0.179606i	0.995960	−	0.0898028i	$-0.0286237\pi$
$858$	7488.00i	0.297944i
$859$	−24200.0	−0.961226	−0.480613	−	0.876933i	$-0.659586\pi$
$859$	−24200.0	−0.961226	−0.480613	+	0.876933i	$0.659586\pi$
$860$	0	0
$861$	−22512.0	−0.891065
$862$	9248.00i	0.365415i
$863$	− 37008.0i	− 1.45975i	−0.683579	−	0.729877i	$-0.739578\pi$
$863$	− 37008.0i	− 1.45975i	0.683579	−	0.729877i	$-0.260422\pi$
$864$	12880.0	0.507160
$865$	0	0
$866$	−1118.00	−0.0438697
$867$	31384.0i	1.22936i
$868$	12152.0i	0.475191i
$869$	12000.0	0.468437
$870$	0	0
$871$	12792.0	0.497635
$872$	− 17550.0i	− 0.681557i
$873$	37962.0i	1.47173i
$874$	−1280.00	−0.0495385
$875$	0	0
$876$	4592.00	0.177111
$877$	− 3446.00i	− 0.132683i	−0.997797	−	0.0663416i	$-0.978867\pi$
$877$	− 3446.00i	− 0.132683i	0.997797	−	0.0663416i	$-0.0211327\pi$
$878$	− 11960.0i	− 0.459716i
$879$	9616.00	0.368987
$880$	0	0
$881$	−16158.0	−0.617908	−0.308954	−	0.951077i	$-0.599979\pi$
$881$	−16158.0	−0.617908	−0.308954	+	0.951077i	$0.599979\pi$
$882$	− 1813.00i	− 0.0692142i
$883$	− 44708.0i	− 1.70390i	−0.523623	−	0.851950i	$-0.675420\pi$
$883$	− 44708.0i	− 1.70390i	0.523623	−	0.851950i	$-0.324580\pi$
$884$	51324.0	1.95273
$885$	0	0
$886$	7332.00	0.278017
$887$	23504.0i	0.889726i	0.895599	+	0.444863i	$0.146748\pi$
$887$	23504.0i	0.889726i	−0.895599	+	0.444863i	$0.853252\pi$
$888$	− 52080.0i	− 1.96812i
$889$	−5432.00	−0.204931
$890$	0	0
$891$	−4308.00	−0.161979
$892$	− 12096.0i	− 0.454040i
$893$	21440.0i	0.803429i
$894$	18160.0	0.679375
$895$	0	0
$896$	−10185.0	−0.379751
$897$	− 19968.0i	− 0.743269i
$898$	1890.00i	0.0702340i
$899$	−12400.0	−0.460026
$900$	0	0
$901$	−2068.00	−0.0764651
$902$	− 4824.00i	− 0.178073i
$903$	3808.00i	0.140335i
$904$	−11970.0	−0.440394
$905$	0	0
$906$	−5056.00	−0.185402
$907$	− 42436.0i	− 1.55354i	−0.629782	−	0.776772i	$-0.716856\pi$
$907$	− 42436.0i	− 1.55354i	0.629782	−	0.776772i	$-0.283144\pi$
$908$	34048.0i	1.24441i
$909$	−17834.0	−0.650733
$910$	0	0
$911$	−7968.00	−0.289782	−0.144891	−	0.989448i	$-0.546283\pi$
$911$	−7968.00	−0.289782	−0.144891	+	0.989448i	$0.546283\pi$
$912$	13120.0i	0.476367i
$913$	− 5376.00i	− 0.194874i
$914$	7014.00	0.253832
$915$	0	0
$916$	38570.0	1.39125
$917$	− 7784.00i	− 0.280317i
$918$	7520.00i	0.270367i
$919$	−14880.0	−0.534109	−0.267054	−	0.963681i	$-0.586050\pi$
$919$	−14880.0	−0.534109	−0.267054	+	0.963681i	$0.586050\pi$
$920$	0	0
$921$	51072.0	1.82723
$922$	8318.00i	0.297114i
$923$	− 52416.0i	− 1.86922i
$924$	−4704.00	−0.167479
$925$	0	0
$926$	6432.00	0.228260
$927$	− 10064.0i	− 0.356575i
$928$	− 8050.00i	− 0.284757i
$929$	−27610.0	−0.975086	−0.487543	−	0.873099i	$-0.662107\pi$
$929$	−27610.0	−0.975086	−0.487543	+	0.873099i	$0.662107\pi$
$930$	0	0
$931$	1960.00	0.0689972
$932$	37254.0i	1.30933i
$933$	39744.0i	1.39460i
$934$	10064.0	0.352574
$935$	0	0
$936$	43290.0	1.51173
$937$	28094.0i	0.979499i	0.871863	+	0.489750i	$0.162912\pi$
$937$	28094.0i	0.979499i	−0.871863	+	0.489750i	$0.837088\pi$
$938$	− 1148.00i	− 0.0399611i
$939$	−22064.0	−0.766807
$940$	0	0
$941$	−12198.0	−0.422575	−0.211288	−	0.977424i	$-0.567766\pi$
$941$	−12198.0	−0.422575	−0.211288	+	0.977424i	$0.567766\pi$
$942$	− 5872.00i	− 0.203100i
$943$	12864.0i	0.444231i
$944$	22960.0	0.791615
$945$	0	0
$946$	−816.000	−0.0280449
$947$	− 31316.0i	− 1.07459i	−0.843396	−	0.537293i	$-0.819447\pi$
$947$	− 31316.0i	− 1.07459i	0.843396	−	0.537293i	$-0.180553\pi$
$948$	− 56000.0i	− 1.91856i
$949$	6396.00	0.218781
$950$	0	0
$951$	50192.0	1.71145
$952$	− 9870.00i	− 0.336017i
$953$	27322.0i	0.928695i	0.885653	+	0.464348i	$0.153711\pi$
$953$	27322.0i	0.928695i	−0.885653	+	0.464348i	$0.846289\pi$
$954$	−814.000	−0.0276250
$955$	0	0
$956$	12880.0	0.435742
$957$	− 4800.00i	− 0.162134i
$958$	1400.00i	0.0472150i
$959$	4858.00	0.163580
$960$	0	0
$961$	31713.0	1.06452
$962$	− 33852.0i	− 1.13454i
$963$	− 16428.0i	− 0.549725i
$964$	−3066.00	−0.102437
$965$	0	0
$966$	−1792.00	−0.0596860
$967$	− 5296.00i	− 0.176120i	−0.996115	−	0.0880599i	$-0.971933\pi$
$967$	− 5296.00i	− 0.176120i	0.996115	−	0.0880599i	$-0.0280667\pi$
$968$	17805.0i	0.591193i
$969$	−30080.0	−0.997223
$970$	0	0
$971$	512.000	0.0169216	0.00846079	−	0.999964i	$-0.497307\pi$
$971$	512.000	0.0169216	0.00846079	+	0.999964i	$0.497307\pi$
$972$	35224.0i	1.16236i
$973$	2520.00i	0.0830293i
$974$	−13376.0	−0.440036
$975$	0	0
$976$	−11398.0	−0.373813
$977$	20734.0i	0.678955i	0.940614	+	0.339478i	$0.110250\pi$
$977$	20734.0i	0.678955i	−0.940614	+	0.339478i	$0.889750\pi$
$978$	− 20256.0i	− 0.662286i
$979$	10440.0	0.340821
$980$	0	0
$981$	−43290.0	−1.40891
$982$	− 7092.00i	− 0.230463i
$983$	− 61168.0i	− 1.98470i	−0.123472	−	0.992348i	$-0.539403\pi$
$983$	− 61168.0i	− 1.98470i	0.123472	−	0.992348i	$-0.460597\pi$
$984$	−48240.0	−1.56284
$985$	0	0
$986$	4700.00	0.151804
$987$	30016.0i	0.968004i
$988$	21840.0i	0.703262i
$989$	2176.00	0.0699624
$990$	0	0
$991$	−47928.0	−1.53631	−0.768155	−	0.640264i	$-0.778825\pi$
$991$	−47928.0	−1.53631	−0.768155	+	0.640264i	$0.778825\pi$
$992$	39928.0i	1.27794i
$993$	− 15456.0i	− 0.493939i
$994$	−4704.00	−0.150102
$995$	0	0
$996$	−25088.0	−0.798136
$997$	9454.00i	0.300312i	0.988662	+	0.150156i	$0.0479776\pi$
$997$	9454.00i	0.300312i	−0.988662	+	0.150156i	$0.952022\pi$
$998$	− 820.000i	− 0.0260087i
$999$	−34720.0	−1.09959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.a.99.1		2
5.2	odd	4		35.4.a.a.1.1	✓	1
5.3	odd	4		175.4.a.a.1.1		1
5.4	even	2	inner	175.4.b.a.99.2		2
15.2	even	4		315.4.a.c.1.1		1
15.8	even	4		1575.4.a.g.1.1		1
20.7	even	4		560.4.a.p.1.1		1
35.2	odd	12		245.4.e.e.116.1		2
35.12	even	12		245.4.e.b.116.1		2
35.13	even	4		1225.4.a.e.1.1		1
35.17	even	12		245.4.e.b.226.1		2
35.27	even	4		245.4.a.d.1.1		1
35.32	odd	12		245.4.e.e.226.1		2
40.27	even	4		2240.4.a.b.1.1		1
40.37	odd	4		2240.4.a.bk.1.1		1
105.62	odd	4		2205.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.a.1.1	✓	1	5.2	odd	4
175.4.a.a.1.1		1	5.3	odd	4
175.4.b.a.99.1		2	1.1	even	1	trivial
175.4.b.a.99.2		2	5.4	even	2	inner
245.4.a.d.1.1		1	35.27	even	4
245.4.e.b.116.1		2	35.12	even	12
245.4.e.b.226.1		2	35.17	even	12
245.4.e.e.116.1		2	35.2	odd	12
245.4.e.e.226.1		2	35.32	odd	12
315.4.a.c.1.1		1	15.2	even	4
560.4.a.p.1.1		1	20.7	even	4
1225.4.a.e.1.1		1	35.13	even	4
1575.4.a.g.1.1		1	15.8	even	4
2205.4.a.i.1.1		1	105.62	odd	4
2240.4.a.b.1.1		1	40.27	even	4
2240.4.a.bk.1.1		1	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -8.00000i q^{3} +7.00000 q^{4} -8.00000 q^{6} -7.00000i q^{7} -15.0000i q^{8} -37.0000 q^{9} +12.0000 q^{11} -56.0000i q^{12} -78.0000i q^{13} -7.00000 q^{14} +41.0000 q^{16} +94.0000i q^{17} +37.0000i q^{18} -40.0000 q^{19} -56.0000 q^{21} -12.0000i q^{22} +32.0000i q^{23} -120.000 q^{24} -78.0000 q^{26} +80.0000i q^{27} -49.0000i q^{28} +50.0000 q^{29} -248.000 q^{31} -161.000i q^{32} -96.0000i q^{33} +94.0000 q^{34} -259.000 q^{36} +434.000i q^{37} +40.0000i q^{38} -624.000 q^{39} +402.000 q^{41} +56.0000i q^{42} -68.0000i q^{43} +84.0000 q^{44} +32.0000 q^{46} -536.000i q^{47} -328.000i q^{48} -49.0000 q^{49} +752.000 q^{51} -546.000i q^{52} +22.0000i q^{53} +80.0000 q^{54} -105.000 q^{56} +320.000i q^{57} -50.0000i q^{58} +560.000 q^{59} -278.000 q^{61} +248.000i q^{62} +259.000i q^{63} +167.000 q^{64} -96.0000 q^{66} +164.000i q^{67} +658.000i q^{68} +256.000 q^{69} +672.000 q^{71} +555.000i q^{72} +82.0000i q^{73} +434.000 q^{74} -280.000 q^{76} -84.0000i q^{77} +624.000i q^{78} +1000.00 q^{79} -359.000 q^{81} -402.000i q^{82} -448.000i q^{83} -392.000 q^{84} -68.0000 q^{86} -400.000i q^{87} -180.000i q^{88} +870.000 q^{89} -546.000 q^{91} +224.000i q^{92} +1984.00i q^{93} -536.000 q^{94} -1288.00 q^{96} -1026.00i q^{97} +49.0000i q^{98} -444.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} - 16 q^{6} - 74 q^{9} + 24 q^{11} - 14 q^{14} + 82 q^{16} - 80 q^{19} - 112 q^{21} - 240 q^{24} - 156 q^{26} + 100 q^{29} - 496 q^{31} + 188 q^{34} - 518 q^{36} - 1248 q^{39} + 804 q^{41}+ \cdots - 888 q^{99}+O(q^{100}) \) 2 * q + 14 * q^4 - 16 * q^6 - 74 * q^9 + 24 * q^11 - 14 * q^14 + 82 * q^16 - 80 * q^19 - 112 * q^21 - 240 * q^24 - 156 * q^26 + 100 * q^29 - 496 * q^31 + 188 * q^34 - 518 * q^36 - 1248 * q^39 + 804 * q^41 + 168 * q^44 + 64 * q^46 - 98 * q^49 + 1504 * q^51 + 160 * q^54 - 210 * q^56 + 1120 * q^59 - 556 * q^61 + 334 * q^64 - 192 * q^66 + 512 * q^69 + 1344 * q^71 + 868 * q^74 - 560 * q^76 + 2000 * q^79 - 718 * q^81 - 784 * q^84 - 136 * q^86 + 1740 * q^89 - 1092 * q^91 - 1072 * q^94 - 2576 * q^96 - 888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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