LMFDB - Embedded newform 4675.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4675,2,Mod(1,4675)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4675 = 5^{2} \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4675.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$37.3300629449$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 935)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50848$ of defining polynomial
Character	$\chi$	$=$	4675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.78400 q^{2} -1.50848 q^{3} +1.18264 q^{4} -2.69113 q^{6} -1.32584 q^{7} -1.45816 q^{8} -0.724484 q^{9} +O(q^{10})$	\(q+1.78400 q^{2} -1.50848 q^{3} +1.18264 q^{4} -2.69113 q^{6} -1.32584 q^{7} -1.45816 q^{8} -0.724484 q^{9} -1.00000 q^{11} -1.78400 q^{12} +2.54184 q^{13} -2.36529 q^{14} -4.96664 q^{16} +1.00000 q^{17} -1.29248 q^{18} -5.38225 q^{19} +2.00000 q^{21} -1.78400 q^{22} -2.64080 q^{23} +2.19961 q^{24} +4.53463 q^{26} +5.61831 q^{27} -1.56799 q^{28} +0.584390 q^{29} +0.916320 q^{31} -5.94415 q^{32} +1.50848 q^{33} +1.78400 q^{34} -0.856807 q^{36} +8.04312 q^{37} -9.60192 q^{38} -3.83432 q^{39} +2.54793 q^{41} +3.56799 q^{42} -1.32274 q^{43} -1.18264 q^{44} -4.71119 q^{46} +5.03945 q^{47} +7.49208 q^{48} -5.24216 q^{49} -1.50848 q^{51} +3.00609 q^{52} +5.95577 q^{53} +10.0231 q^{54} +1.93328 q^{56} +8.11902 q^{57} +1.04255 q^{58} +6.90302 q^{59} +3.20570 q^{61} +1.63471 q^{62} +0.960548 q^{63} -0.671065 q^{64} +2.69113 q^{66} +9.91079 q^{67} +1.18264 q^{68} +3.98360 q^{69} -9.13599 q^{71} +1.05641 q^{72} -1.91079 q^{73} +14.3489 q^{74} -6.36529 q^{76} +1.32584 q^{77} -6.84041 q^{78} -11.9077 q^{79} -6.30167 q^{81} +4.54550 q^{82} -12.5090 q^{83} +2.36529 q^{84} -2.35976 q^{86} -0.881541 q^{87} +1.45816 q^{88} +13.1329 q^{89} -3.37007 q^{91} -3.12313 q^{92} -1.38225 q^{93} +8.99037 q^{94} +8.96664 q^{96} +19.6250 q^{97} -9.35199 q^{98} +0.724484 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) $ 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9	$ 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9} - 4 q^{11} + 13 q^{13} - 4 q^{14} - 10 q^{16} + 4 q^{17} + 9 q^{18} - 2 q^{19} + 8 q^{21} - 5 q^{23} - 8 q^{24} - 6 q^{26} + 4 q^{27} + 8 q^{28} + 12 q^{29} - 2 q^{31} - q^{32} - q^{33} - 5 q^{36} + q^{37} + 4 q^{38} - 4 q^{39} + 2 q^{41} + 7 q^{43} - 2 q^{44} - 3 q^{46} + 19 q^{47} - q^{48} - 11 q^{49} + q^{51} + q^{52} + 17 q^{53} + 15 q^{54} - 12 q^{56} + 18 q^{57} + 11 q^{58} + 6 q^{59} - 15 q^{61} + 12 q^{62} + 5 q^{63} + q^{64} + q^{66} + 7 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{71} - 11 q^{72} + 25 q^{73} + 28 q^{74} - 20 q^{76} + q^{77} - 5 q^{78} - 7 q^{79} - 8 q^{81} - 9 q^{82} - 5 q^{83} + 4 q^{84} + 23 q^{86} + 20 q^{87} + 3 q^{88} + 16 q^{89} - 16 q^{91} - 17 q^{92} + 14 q^{93} - 10 q^{94} + 26 q^{96} + 11 q^{97} - 16 q^{98} + 3 q^{99}+O(q^{100}) $ 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9 - 4 * q^11 + 13 * q^13 - 4 * q^14 - 10 * q^16 + 4 * q^17 + 9 * q^18 - 2 * q^19 + 8 * q^21 - 5 * q^23 - 8 * q^24 - 6 * q^26 + 4 * q^27 + 8 * q^28 + 12 * q^29 - 2 * q^31 - q^32 - q^33 - 5 * q^36 + q^37 + 4 * q^38 - 4 * q^39 + 2 * q^41 + 7 * q^43 - 2 * q^44 - 3 * q^46 + 19 * q^47 - q^48 - 11 * q^49 + q^51 + q^52 + 17 * q^53 + 15 * q^54 - 12 * q^56 + 18 * q^57 + 11 * q^58 + 6 * q^59 - 15 * q^61 + 12 * q^62 + 5 * q^63 + q^64 + q^66 + 7 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^71 - 11 * q^72 + 25 * q^73 + 28 * q^74 - 20 * q^76 + q^77 - 5 * q^78 - 7 * q^79 - 8 * q^81 - 9 * q^82 - 5 * q^83 + 4 * q^84 + 23 * q^86 + 20 * q^87 + 3 * q^88 + 16 * q^89 - 16 * q^91 - 17 * q^92 + 14 * q^93 - 10 * q^94 + 26 * q^96 + 11 * q^97 - 16 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.78400	1.26148	0.630738	−	0.775996i	$-0.282752\pi$
$2$	1.78400	1.26148	0.630738	+	0.775996i	$0.282752\pi$
$3$	−1.50848	−0.870922	−0.435461	−	0.900208i	$-0.643415\pi$
$3$	−1.50848	−0.870922	−0.435461	+	0.900208i	$0.643415\pi$
$4$	1.18264	0.591322
$5$	0	0
$5$	0	0
$6$	−2.69113	−1.09865
$7$	−1.32584	−0.501119	−0.250560	−	0.968101i	$-0.580615\pi$
$7$	−1.32584	−0.501119	−0.250560	+	0.968101i	$0.580615\pi$
$8$	−1.45816	−0.515537
$9$	−0.724484	−0.241495
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.78400	−0.514995
$13$	2.54184	0.704980	0.352490	−	0.935816i	$-0.385335\pi$
$13$	2.54184	0.704980	0.352490	+	0.935816i	$0.385335\pi$
$14$	−2.36529	−0.632150
$15$	0	0
$16$	−4.96664	−1.24166
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.29248	−0.304640
$19$	−5.38225	−1.23477	−0.617387	−	0.786660i	$-0.711809\pi$
$19$	−5.38225	−1.23477	−0.617387	+	0.786660i	$0.711809\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	−1.78400	−0.380349
$23$	−2.64080	−0.550646	−0.275323	−	0.961352i	$-0.588785\pi$
$23$	−2.64080	−0.550646	−0.275323	+	0.961352i	$0.588785\pi$
$24$	2.19961	0.448993
$25$	0	0
$26$	4.53463	0.889315
$27$	5.61831	1.08125
$28$	−1.56799	−0.296323
$29$	0.584390	0.108518	0.0542592	−	0.998527i	$-0.482720\pi$
$29$	0.584390	0.108518	0.0542592	+	0.998527i	$0.482720\pi$
$30$	0	0
$31$	0.916320	0.164576	0.0822879	−	0.996609i	$-0.473777\pi$
$31$	0.916320	0.164576	0.0822879	+	0.996609i	$0.473777\pi$
$32$	−5.94415	−1.05079
$33$	1.50848	0.262593
$34$	1.78400	0.305953
$35$	0	0
$36$	−0.856807	−0.142801
$37$	8.04312	1.32228	0.661140	−	0.750263i	$-0.270073\pi$
$37$	8.04312	1.32228	0.661140	+	0.750263i	$0.270073\pi$
$38$	−9.60192	−1.55764
$39$	−3.83432	−0.613982
$40$	0	0
$41$	2.54793	0.397920	0.198960	−	0.980008i	$-0.436244\pi$
$41$	2.54793	0.397920	0.198960	+	0.980008i	$0.436244\pi$
$42$	3.56799	0.550553
$43$	−1.32274	−0.201716	−0.100858	−	0.994901i	$-0.532159\pi$
$43$	−1.32274	−0.201716	−0.100858	+	0.994901i	$0.532159\pi$
$44$	−1.18264	−0.178290
$45$	0	0
$46$	−4.71119	−0.694626
$47$	5.03945	0.735080	0.367540	−	0.930008i	$-0.380200\pi$
$47$	5.03945	0.735080	0.367540	+	0.930008i	$0.380200\pi$
$48$	7.49208	1.08139
$49$	−5.24216	−0.748880
$50$	0	0
$51$	−1.50848	−0.211230
$52$	3.00609	0.416870
$53$	5.95577	0.818088	0.409044	−	0.912515i	$-0.365862\pi$
$53$	5.95577	0.818088	0.409044	+	0.912515i	$0.365862\pi$
$54$	10.0231	1.36397
$55$	0	0
$56$	1.93328	0.258346
$57$	8.11902	1.07539
$58$	1.04255	0.136894
$59$	6.90302	0.898697	0.449348	−	0.893357i	$-0.351656\pi$
$59$	6.90302	0.898697	0.449348	+	0.893357i	$0.351656\pi$
$60$	0	0
$61$	3.20570	0.410448	0.205224	−	0.978715i	$-0.434208\pi$
$61$	3.20570	0.410448	0.205224	+	0.978715i	$0.434208\pi$
$62$	1.63471	0.207609
$63$	0.960548	0.121018
$64$	−0.671065	−0.0838831
$65$	0	0
$66$	2.69113	0.331255
$67$	9.91079	1.21080	0.605398	−	0.795923i	$-0.293014\pi$
$67$	9.91079	1.21080	0.605398	+	0.795923i	$0.293014\pi$
$68$	1.18264	0.143417
$69$	3.98360	0.479569
$70$	0	0
$71$	−9.13599	−1.08424	−0.542121	−	0.840300i	$-0.682379\pi$
$71$	−9.13599	−1.08424	−0.542121	+	0.840300i	$0.682379\pi$
$72$	1.05641	0.124500
$73$	−1.91079	−0.223641	−0.111821	−	0.993728i	$-0.535668\pi$
$73$	−1.91079	−0.223641	−0.111821	+	0.993728i	$0.535668\pi$
$74$	14.3489	1.66802
$75$	0	0
$76$	−6.36529	−0.730149
$77$	1.32584	0.151093
$78$	−6.84041	−0.774524
$79$	−11.9077	−1.33972	−0.669860	−	0.742487i	$-0.733646\pi$
$79$	−11.9077	−1.33972	−0.669860	+	0.742487i	$0.733646\pi$
$80$	0	0
$81$	−6.30167	−0.700185
$82$	4.54550	0.501967
$83$	−12.5090	−1.37305	−0.686523	−	0.727108i	$-0.740864\pi$
$83$	−12.5090	−1.37305	−0.686523	+	0.727108i	$0.740864\pi$
$84$	2.36529	0.258074
$85$	0	0
$86$	−2.35976	−0.254460
$87$	−0.881541	−0.0945112
$88$	1.45816	0.155440
$89$	13.1329	1.39208	0.696042	−	0.718001i	$-0.254943\pi$
$89$	13.1329	1.39208	0.696042	+	0.718001i	$0.254943\pi$
$90$	0	0
$91$	−3.37007	−0.353279
$92$	−3.12313	−0.325609
$93$	−1.38225	−0.143333
$94$	8.99037	0.927285
$95$	0	0
$96$	8.96664	0.915154
$97$	19.6250	1.99261	0.996307	−	0.0858616i	$-0.0273643\pi$
$97$	19.6250	1.99261	0.996307	+	0.0858616i	$0.0273643\pi$
$98$	−9.35199	−0.944694
$99$	0.724484	0.0728134
$100$	0	0
$101$	2.89794	0.288356	0.144178	−	0.989552i	$-0.453946\pi$
$101$	2.89794	0.288356	0.144178	+	0.989552i	$0.453946\pi$
$102$	−2.69113	−0.266461
$103$	16.2094	1.59716	0.798578	−	0.601891i	$-0.205586\pi$
$103$	16.2094	1.59716	0.798578	+	0.601891i	$0.205586\pi$
$104$	−3.70641	−0.363443
$105$	0	0
$106$	10.6251	1.03200
$107$	5.24074	0.506641	0.253321	−	0.967382i	$-0.418477\pi$
$107$	5.24074	0.506641	0.253321	+	0.967382i	$0.418477\pi$
$108$	6.64447	0.639364
$109$	1.33304	0.127682	0.0638412	−	0.997960i	$-0.479665\pi$
$109$	1.33304	0.127682	0.0638412	+	0.997960i	$0.479665\pi$
$110$	0	0
$111$	−12.1329	−1.15160
$112$	6.58496	0.622220
$113$	6.36828	0.599078	0.299539	−	0.954084i	$-0.403167\pi$
$113$	6.36828	0.599078	0.299539	+	0.954084i	$0.403167\pi$
$114$	14.4843	1.35658
$115$	0	0
$116$	0.691126	0.0641694
$117$	−1.84152	−0.170249
$118$	12.3150	1.13368
$119$	−1.32584	−0.121539
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.71896	0.517770
$123$	−3.84351	−0.346558
$124$	1.08368	0.0973174
$125$	0	0
$126$	1.71361	0.152661
$127$	3.17965	0.282148	0.141074	−	0.989999i	$-0.454944\pi$
$127$	3.17965	0.282148	0.141074	+	0.989999i	$0.454944\pi$
$128$	10.6911	0.944971
$129$	1.99533	0.175679
$130$	0	0
$131$	−15.4151	−1.34682	−0.673410	−	0.739269i	$-0.735171\pi$
$131$	−15.4151	−1.34682	−0.673410	+	0.739269i	$0.735171\pi$
$132$	1.78400	0.155277
$133$	7.13599	0.618769
$134$	17.6808	1.52739
$135$	0	0
$136$	−1.45816	−0.125036
$137$	10.7932	0.922124	0.461062	−	0.887368i	$-0.347469\pi$
$137$	10.7932	0.922124	0.461062	+	0.887368i	$0.347469\pi$
$138$	7.10674	0.604966
$139$	1.98051	0.167984	0.0839921	−	0.996466i	$-0.473233\pi$
$139$	1.98051	0.167984	0.0839921	+	0.996466i	$0.473233\pi$
$140$	0	0
$141$	−7.60192	−0.640197
$142$	−16.2986	−1.36775
$143$	−2.54184	−0.212559
$144$	3.59825	0.299855
$145$	0	0
$146$	−3.40885	−0.282118
$147$	7.90769	0.652216
$148$	9.51215	0.781894
$149$	10.2197	0.837228	0.418614	−	0.908164i	$-0.362516\pi$
$149$	10.2197	0.837228	0.418614	+	0.908164i	$0.362516\pi$
$150$	0	0
$151$	5.41618	0.440762	0.220381	−	0.975414i	$-0.429270\pi$
$151$	5.41618	0.440762	0.220381	+	0.975414i	$0.429270\pi$
$152$	7.84818	0.636572
$153$	−0.724484	−0.0585711
$154$	2.36529	0.190600
$155$	0	0
$156$	−4.53463	−0.363061
$157$	−13.0468	−1.04125	−0.520623	−	0.853787i	$-0.674300\pi$
$157$	−13.0468	−1.04125	−0.520623	+	0.853787i	$0.674300\pi$
$158$	−21.2433	−1.69003
$159$	−8.98417	−0.712491
$160$	0	0
$161$	3.50128	0.275939
$162$	−11.2422	−0.883267
$163$	−7.14686	−0.559785	−0.279893	−	0.960031i	$-0.590299\pi$
$163$	−7.14686	−0.559785	−0.279893	+	0.960031i	$0.590299\pi$
$164$	3.01330	0.235299
$165$	0	0
$166$	−22.3161	−1.73207
$167$	18.7501	1.45093	0.725463	−	0.688261i	$-0.241626\pi$
$167$	18.7501	1.45093	0.725463	+	0.688261i	$0.241626\pi$
$168$	−2.91632	−0.224999
$169$	−6.53905	−0.503004
$170$	0	0
$171$	3.89936	0.298191
$172$	−1.56433	−0.119279
$173$	−8.32839	−0.633196	−0.316598	−	0.948560i	$-0.602541\pi$
$173$	−8.32839	−0.633196	−0.316598	+	0.948560i	$0.602541\pi$
$174$	−1.57267	−0.119224
$175$	0	0
$176$	4.96664	0.374375
$177$	−10.4131	−0.782695
$178$	23.4290	1.75608
$179$	5.02858	0.375854	0.187927	−	0.982183i	$-0.439823\pi$
$179$	5.02858	0.375854	0.187927	+	0.982183i	$0.439823\pi$
$180$	0	0
$181$	15.4563	1.14886	0.574429	−	0.818555i	$-0.305224\pi$
$181$	15.4563	1.14886	0.574429	+	0.818555i	$0.305224\pi$
$182$	−6.01219	−0.445653
$183$	−4.83574	−0.357468
$184$	3.85071	0.283878
$185$	0	0
$186$	−2.46593	−0.180811
$187$	−1.00000	−0.0731272
$188$	5.95988	0.434669
$189$	−7.44897	−0.541833
$190$	0	0
$191$	−5.25968	−0.380577	−0.190289	−	0.981728i	$-0.560942\pi$
$191$	−5.25968	−0.380577	−0.190289	+	0.981728i	$0.560942\pi$
$192$	1.01229	0.0730557
$193$	−14.1651	−1.01963	−0.509814	−	0.860284i	$-0.670286\pi$
$193$	−14.1651	−1.01963	−0.509814	+	0.860284i	$0.670286\pi$
$194$	35.0109	2.51364
$195$	0	0
$196$	−6.19961	−0.442829
$197$	−23.3277	−1.66203	−0.831016	−	0.556249i	$-0.812240\pi$
$197$	−23.3277	−1.66203	−0.831016	+	0.556249i	$0.812240\pi$
$198$	1.29248	0.0918524
$199$	7.63584	0.541291	0.270645	−	0.962679i	$-0.412763\pi$
$199$	7.63584	0.541291	0.270645	+	0.962679i	$0.412763\pi$
$200$	0	0
$201$	−14.9502	−1.05451
$202$	5.16991	0.363754
$203$	−0.774806	−0.0543807
$204$	−1.78400	−0.124905
$205$	0	0
$206$	28.9175	2.01477
$207$	1.91322	0.132978
$208$	−12.6244	−0.875345
$209$	5.38225	0.372298
$210$	0	0
$211$	9.68026	0.666416	0.333208	−	0.942853i	$-0.391869\pi$
$211$	9.68026	0.666416	0.333208	+	0.942853i	$0.391869\pi$
$212$	7.04356	0.483754
$213$	13.7815	0.944290
$214$	9.34946	0.639116
$215$	0	0
$216$	−8.19240	−0.557422
$217$	−1.21489	−0.0824721
$218$	2.37814	0.161068
$219$	2.88239	0.194774
$220$	0	0
$221$	2.54184	0.170983
$222$	−21.6450	−1.45272
$223$	24.1080	1.61439	0.807195	−	0.590285i	$-0.200985\pi$
$223$	24.1080	1.61439	0.807195	+	0.590285i	$0.200985\pi$
$224$	7.88098	0.526570
$225$	0	0
$226$	11.3610	0.755722
$227$	−6.29049	−0.417515	−0.208757	−	0.977967i	$-0.566942\pi$
$227$	−6.29049	−0.417515	−0.208757	+	0.977967i	$0.566942\pi$
$228$	9.60192	0.635903
$229$	6.22875	0.411608	0.205804	−	0.978593i	$-0.434019\pi$
$229$	6.22875	0.411608	0.205804	+	0.978593i	$0.434019\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	−0.852134	−0.0559453
$233$	5.34758	0.350331	0.175166	−	0.984539i	$-0.443954\pi$
$233$	5.34758	0.350331	0.175166	+	0.984539i	$0.443954\pi$
$234$	−3.28527	−0.214765
$235$	0	0
$236$	8.16382	0.531419
$237$	17.9625	1.16679
$238$	−2.36529	−0.153319
$239$	28.0379	1.81362	0.906810	−	0.421539i	$-0.138510\pi$
$239$	28.0379	1.81362	0.906810	+	0.421539i	$0.138510\pi$
$240$	0	0
$241$	5.38282	0.346738	0.173369	−	0.984857i	$-0.444535\pi$
$241$	5.38282	0.346738	0.173369	+	0.984857i	$0.444535\pi$
$242$	1.78400	0.114680
$243$	−7.34900	−0.471438
$244$	3.79120	0.242707
$245$	0	0
$246$	−6.85681	−0.437174
$247$	−13.6808	−0.870490
$248$	−1.33614	−0.0848450
$249$	18.8697	1.19582
$250$	0	0
$251$	21.0559	1.32903	0.664517	−	0.747273i	$-0.268637\pi$
$251$	21.0559	1.32903	0.664517	+	0.747273i	$0.268637\pi$
$252$	1.13599	0.0715604
$253$	2.64080	0.166026
$254$	5.67248	0.355923
$255$	0	0
$256$	20.4151	1.27594
$257$	6.57352	0.410045	0.205022	−	0.978757i	$-0.434273\pi$
$257$	6.57352	0.410045	0.205022	+	0.978757i	$0.434273\pi$
$258$	3.55966	0.221615
$259$	−10.6639	−0.662620
$260$	0	0
$261$	−0.423381	−0.0262067
$262$	−27.5004	−1.69898
$263$	−8.71119	−0.537155	−0.268577	−	0.963258i	$-0.586553\pi$
$263$	−8.71119	−0.537155	−0.268577	+	0.963258i	$0.586553\pi$
$264$	−2.19961	−0.135376
$265$	0	0
$266$	12.7306	0.780562
$267$	−19.8107	−1.21240
$268$	11.7209	0.715971
$269$	−4.40541	−0.268603	−0.134301	−	0.990941i	$-0.542879\pi$
$269$	−4.40541	−0.268603	−0.134301	+	0.990941i	$0.542879\pi$
$270$	0	0
$271$	−4.68224	−0.284426	−0.142213	−	0.989836i	$-0.545422\pi$
$271$	−4.68224	−0.284426	−0.142213	+	0.989836i	$0.545422\pi$
$272$	−4.96664	−0.301147
$273$	5.08368	0.307678
$274$	19.2550	1.16324
$275$	0	0
$276$	4.71119	0.283580
$277$	14.3332	0.861201	0.430601	−	0.902543i	$-0.358302\pi$
$277$	14.3332	0.861201	0.430601	+	0.902543i	$0.358302\pi$
$278$	3.53322	0.211908
$279$	−0.663859	−0.0397442
$280$	0	0
$281$	7.98895	0.476581	0.238290	−	0.971194i	$-0.423413\pi$
$281$	7.98895	0.476581	0.238290	+	0.971194i	$0.423413\pi$
$282$	−13.5618	−0.807593
$283$	−28.8569	−1.71537	−0.857683	−	0.514178i	$-0.828097\pi$
$283$	−28.8569	−1.71537	−0.857683	+	0.514178i	$0.828097\pi$
$284$	−10.8046	−0.641136
$285$	0	0
$286$	−4.53463	−0.268139
$287$	−3.37814	−0.199406
$288$	4.30645	0.253760
$289$	1.00000	0.0588235
$290$	0	0
$291$	−29.6039	−1.73541
$292$	−2.25979	−0.132244
$293$	−4.11182	−0.240215	−0.120108	−	0.992761i	$-0.538324\pi$
$293$	−4.11182	−0.240215	−0.120108	+	0.992761i	$0.538324\pi$
$294$	14.1073	0.822755
$295$	0	0
$296$	−11.7281	−0.681685
$297$	−5.61831	−0.326008
$298$	18.2319	1.05614
$299$	−6.71250	−0.388194
$300$	0	0
$301$	1.75374	0.101084
$302$	9.66244	0.556011
$303$	−4.37148	−0.251135
$304$	26.7317	1.53317
$305$	0	0
$306$	−1.29248	−0.0738860
$307$	−0.766025	−0.0437193	−0.0218597	−	0.999761i	$-0.506959\pi$
$307$	−0.766025	−0.0437193	−0.0218597	+	0.999761i	$0.506959\pi$
$308$	1.56799	0.0893447
$309$	−24.4515	−1.39100
$310$	0	0
$311$	6.81681	0.386546	0.193273	−	0.981145i	$-0.438090\pi$
$311$	6.81681	0.386546	0.193273	+	0.981145i	$0.438090\pi$
$312$	5.59105	0.316531
$313$	22.4649	1.26979	0.634896	−	0.772597i	$-0.281043\pi$
$313$	22.4649	1.26979	0.634896	+	0.772597i	$0.281043\pi$
$314$	−23.2754	−1.31351
$315$	0	0
$316$	−14.0826	−0.792206
$317$	−26.3253	−1.47858	−0.739288	−	0.673390i	$-0.764838\pi$
$317$	−26.3253	−1.47858	−0.739288	+	0.673390i	$0.764838\pi$
$318$	−16.0277	−0.898790
$319$	−0.584390	−0.0327196
$320$	0	0
$321$	−7.90555	−0.441245
$322$	6.24626	0.348091
$323$	−5.38225	−0.299476
$324$	−7.45263	−0.414035
$325$	0	0
$326$	−12.7500	−0.706155
$327$	−2.01087	−0.111201
$328$	−3.71529	−0.205143
$329$	−6.68149	−0.368363
$330$	0	0
$331$	−2.79629	−0.153698	−0.0768489	−	0.997043i	$-0.524486\pi$
$331$	−2.79629	−0.153698	−0.0768489	+	0.997043i	$0.524486\pi$
$332$	−14.7938	−0.811913
$333$	−5.82711	−0.319324
$334$	33.4501	1.83031
$335$	0	0
$336$	−9.93328	−0.541905
$337$	3.41094	0.185806	0.0929028	−	0.995675i	$-0.470385\pi$
$337$	3.41094	0.185806	0.0929028	+	0.995675i	$0.470385\pi$
$338$	−11.6656	−0.634527
$339$	−9.60644	−0.521750
$340$	0	0
$341$	−0.916320	−0.0496215
$342$	6.95644	0.376161
$343$	16.2311	0.876397
$344$	1.92876	0.103992
$345$	0	0
$346$	−14.8578	−0.798761
$347$	13.1658	0.706777	0.353389	−	0.935477i	$-0.385029\pi$
$347$	13.1658	0.706777	0.353389	+	0.935477i	$0.385029\pi$
$348$	−1.04255	−0.0558865
$349$	0.578966	0.0309914	0.0154957	−	0.999880i	$-0.495067\pi$
$349$	0.578966	0.0309914	0.0154957	+	0.999880i	$0.495067\pi$
$350$	0	0
$351$	14.2809	0.762256
$352$	5.94415	0.316824
$353$	25.8924	1.37811	0.689057	−	0.724707i	$-0.258025\pi$
$353$	25.8924	1.37811	0.689057	+	0.724707i	$0.258025\pi$
$354$	−18.5769	−0.987351
$355$	0	0
$356$	15.5315	0.823170
$357$	2.00000	0.105851
$358$	8.97097	0.474131
$359$	14.1713	0.747934	0.373967	−	0.927442i	$-0.377997\pi$
$359$	14.1713	0.747934	0.373967	+	0.927442i	$0.377997\pi$
$360$	0	0
$361$	9.96863	0.524665
$362$	27.5740	1.44926
$363$	−1.50848	−0.0791747
$364$	−3.98559	−0.208902
$365$	0	0
$366$	−8.62694	−0.450937
$367$	−31.3167	−1.63472	−0.817358	−	0.576130i	$-0.804562\pi$
$367$	−31.3167	−1.63472	−0.817358	+	0.576130i	$0.804562\pi$
$368$	13.1159	0.683715
$369$	−1.84594	−0.0960957
$370$	0	0
$371$	−7.89638	−0.409960
$372$	−1.63471	−0.0847558
$373$	1.95502	0.101227	0.0506136	−	0.998718i	$-0.483882\pi$
$373$	1.95502	0.101227	0.0506136	+	0.998718i	$0.483882\pi$
$374$	−1.78400	−0.0922483
$375$	0	0
$376$	−7.34833	−0.378961
$377$	1.48543	0.0765033
$378$	−13.2889	−0.683509
$379$	−20.7195	−1.06429	−0.532145	−	0.846653i	$-0.678614\pi$
$379$	−20.7195	−1.06429	−0.532145	+	0.846653i	$0.678614\pi$
$380$	0	0
$381$	−4.79644	−0.245729
$382$	−9.38326	−0.480089
$383$	−32.8847	−1.68033	−0.840164	−	0.542333i	$-0.817541\pi$
$383$	−32.8847	−1.68033	−0.840164	+	0.542333i	$0.817541\pi$
$384$	−16.1274	−0.822996
$385$	0	0
$386$	−25.2706	−1.28624
$387$	0.958304	0.0487133
$388$	23.2094	1.17828
$389$	3.35142	0.169924	0.0849620	−	0.996384i	$-0.472923\pi$
$389$	3.35142	0.169924	0.0849620	+	0.996384i	$0.472923\pi$
$390$	0	0
$391$	−2.64080	−0.133551
$392$	7.64390	0.386075
$393$	23.2533	1.17298
$394$	−41.6166	−2.09661
$395$	0	0
$396$	0.856807	0.0430562
$397$	29.0157	1.45626	0.728129	−	0.685440i	$-0.240390\pi$
$397$	29.0157	1.45626	0.728129	+	0.685440i	$0.240390\pi$
$398$	13.6223	0.682825
$399$	−10.7645	−0.538899
$400$	0	0
$401$	−29.7037	−1.48333	−0.741666	−	0.670770i	$-0.765964\pi$
$401$	−29.7037	−1.48333	−0.741666	+	0.670770i	$0.765964\pi$
$402$	−26.6712	−1.33024
$403$	2.32914	0.116023
$404$	3.42723	0.170511
$405$	0	0
$406$	−1.38225	−0.0686000
$407$	−8.04312	−0.398682
$408$	2.19961	0.108897
$409$	−17.4040	−0.860572	−0.430286	−	0.902693i	$-0.641587\pi$
$409$	−17.4040	−0.860572	−0.430286	+	0.902693i	$0.641587\pi$
$410$	0	0
$411$	−16.2813	−0.803098
$412$	19.1699	0.944434
$413$	−9.15228	−0.450354
$414$	3.41318	0.167749
$415$	0	0
$416$	−15.1091	−0.740784
$417$	−2.98756	−0.146301
$418$	9.60192	0.469645
$419$	−33.4346	−1.63339	−0.816693	−	0.577073i	$-0.804195\pi$
$419$	−33.4346	−1.63339	−0.816693	+	0.577073i	$0.804195\pi$
$420$	0	0
$421$	11.1448	0.543163	0.271581	−	0.962415i	$-0.412453\pi$
$421$	11.1448	0.543163	0.271581	+	0.962415i	$0.412453\pi$
$422$	17.2695	0.840668
$423$	−3.65100	−0.177518
$424$	−8.68447	−0.421755
$425$	0	0
$426$	24.5861	1.19120
$427$	−4.25023	−0.205683
$428$	6.19793	0.299588
$429$	3.83432	0.185123
$430$	0	0
$431$	14.7995	0.712869	0.356434	−	0.934320i	$-0.383992\pi$
$431$	14.7995	0.712869	0.356434	+	0.934320i	$0.383992\pi$
$432$	−27.9042	−1.34254
$433$	−0.593034	−0.0284994	−0.0142497	−	0.999898i	$-0.504536\pi$
$433$	−0.593034	−0.0284994	−0.0142497	+	0.999898i	$0.504536\pi$
$434$	−2.16736	−0.104037
$435$	0	0
$436$	1.57652	0.0755014
$437$	14.2135	0.679923
$438$	5.14218	0.245703
$439$	−34.9071	−1.66603	−0.833014	−	0.553252i	$-0.813386\pi$
$439$	−34.9071	−1.66603	−0.833014	+	0.553252i	$0.813386\pi$
$440$	0	0
$441$	3.79786	0.180851
$442$	4.53463	0.215691
$443$	29.0970	1.38244	0.691220	−	0.722644i	$-0.257074\pi$
$443$	29.0970	1.38244	0.691220	+	0.722644i	$0.257074\pi$
$444$	−14.3489	−0.680968
$445$	0	0
$446$	43.0085	2.03651
$447$	−15.4162	−0.729160
$448$	0.889723	0.0420354
$449$	−4.89458	−0.230990	−0.115495	−	0.993308i	$-0.536845\pi$
$449$	−4.89458	−0.230990	−0.115495	+	0.993308i	$0.536845\pi$
$450$	0	0
$451$	−2.54793	−0.119977
$452$	7.53141	0.354248
$453$	−8.17020	−0.383869
$454$	−11.2222	−0.526685
$455$	0	0
$456$	−11.8388	−0.554404
$457$	−8.79742	−0.411526	−0.205763	−	0.978602i	$-0.565968\pi$
$457$	−8.79742	−0.411526	−0.205763	+	0.978602i	$0.565968\pi$
$458$	11.1121	0.519233
$459$	5.61831	0.262240
$460$	0	0
$461$	−9.49306	−0.442136	−0.221068	−	0.975258i	$-0.570954\pi$
$461$	−9.49306	−0.442136	−0.221068	+	0.975258i	$0.570954\pi$
$462$	−3.56799	−0.165998
$463$	−11.0794	−0.514902	−0.257451	−	0.966291i	$-0.582883\pi$
$463$	−11.0794	−0.514902	−0.257451	+	0.966291i	$0.582883\pi$
$464$	−2.90246	−0.134743
$465$	0	0
$466$	9.54006	0.441935
$467$	−33.4070	−1.54589	−0.772946	−	0.634472i	$-0.781218\pi$
$467$	−33.4070	−1.54589	−0.772946	+	0.634472i	$0.781218\pi$
$468$	−2.17787	−0.100672
$469$	−13.1401	−0.606753
$470$	0	0
$471$	19.6808	0.906844
$472$	−10.0657	−0.463312
$473$	1.32274	0.0608196
$474$	32.0451	1.47188
$475$	0	0
$476$	−1.56799	−0.0718689
$477$	−4.31486	−0.197564
$478$	50.0195	2.28784
$479$	−33.0978	−1.51228	−0.756138	−	0.654412i	$-0.772916\pi$
$479$	−33.0978	−1.51228	−0.756138	+	0.654412i	$0.772916\pi$
$480$	0	0
$481$	20.4443	0.932180
$482$	9.60293	0.437401
$483$	−5.28161	−0.240322
$484$	1.18264	0.0537566
$485$	0	0
$486$	−13.1106	−0.594708
$487$	−39.9742	−1.81140	−0.905701	−	0.423916i	$-0.860655\pi$
$487$	−39.9742	−1.81140	−0.905701	+	0.423916i	$0.860655\pi$
$488$	−4.67442	−0.211601
$489$	10.7809	0.487529
$490$	0	0
$491$	16.8592	0.760843	0.380422	−	0.924813i	$-0.375779\pi$
$491$	16.8592	0.760843	0.380422	+	0.924813i	$0.375779\pi$
$492$	−4.54550	−0.204927
$493$	0.584390	0.0263196
$494$	−24.4065	−1.09810
$495$	0	0
$496$	−4.55103	−0.204347
$497$	12.1128	0.543335
$498$	33.6634	1.50849
$499$	17.9209	0.802249	0.401125	−	0.916024i	$-0.368619\pi$
$499$	17.9209	0.802249	0.401125	+	0.916024i	$0.368619\pi$
$500$	0	0
$501$	−28.2842	−1.26364
$502$	37.5636	1.67655
$503$	−4.37672	−0.195148	−0.0975742	−	0.995228i	$-0.531108\pi$
$503$	−4.37672	−0.195148	−0.0975742	+	0.995228i	$0.531108\pi$
$504$	−1.40063	−0.0623891
$505$	0	0
$506$	4.71119	0.209438
$507$	9.86403	0.438077
$508$	3.76039	0.166841
$509$	12.0088	0.532280	0.266140	−	0.963934i	$-0.414252\pi$
$509$	12.0088	0.532280	0.266140	+	0.963934i	$0.414252\pi$
$510$	0	0
$511$	2.53340	0.112071
$512$	15.0382	0.664599
$513$	−30.2392	−1.33509
$514$	11.7271	0.517262
$515$	0	0
$516$	2.35976	0.103883
$517$	−5.03945	−0.221635
$518$	−19.0243	−0.835879
$519$	12.5632	0.551464
$520$	0	0
$521$	41.9562	1.83813	0.919066	−	0.394103i	$-0.128945\pi$
$521$	41.9562	1.83813	0.919066	+	0.394103i	$0.128945\pi$
$522$	−0.755311	−0.0330591
$523$	11.5849	0.506570	0.253285	−	0.967392i	$-0.418489\pi$
$523$	11.5849	0.506570	0.253285	+	0.967392i	$0.418489\pi$
$524$	−18.2305	−0.796405
$525$	0	0
$526$	−15.5407	−0.677608
$527$	0.916320	0.0399155
$528$	−7.49208	−0.326051
$529$	−16.0262	−0.696789
$530$	0	0
$531$	−5.00113	−0.217031
$532$	8.43933	0.365892
$533$	6.47644	0.280526
$534$	−35.3423	−1.52941
$535$	0	0
$536$	−14.4515	−0.624211
$537$	−7.58552	−0.327339
$538$	−7.85924	−0.338836
$539$	5.24216	0.225796
$540$	0	0
$541$	14.8657	0.639127	0.319563	−	0.947565i	$-0.396464\pi$
$541$	14.8657	0.639127	0.319563	+	0.947565i	$0.396464\pi$
$542$	−8.35310	−0.358796
$543$	−23.3155	−1.00057
$544$	−5.94415	−0.254853
$545$	0	0
$546$	9.06927	0.388129
$547$	16.6005	0.709786	0.354893	−	0.934907i	$-0.384517\pi$
$547$	16.6005	0.709786	0.354893	+	0.934907i	$0.384517\pi$
$548$	12.7645	0.545273
$549$	−2.32248	−0.0991210
$550$	0	0
$551$	−3.14533	−0.133996
$552$	−5.80873	−0.247236
$553$	15.7877	0.671360
$554$	25.5705	1.08638
$555$	0	0
$556$	2.34223	0.0993328
$557$	8.63341	0.365809	0.182905	−	0.983131i	$-0.441450\pi$
$557$	8.63341	0.365809	0.182905	+	0.983131i	$0.441450\pi$
$558$	−1.18432	−0.0501364
$559$	−3.36219	−0.142206
$560$	0	0
$561$	1.50848	0.0636881
$562$	14.2523	0.601195
$563$	20.7984	0.876549	0.438275	−	0.898841i	$-0.355590\pi$
$563$	20.7984	0.876549	0.438275	+	0.898841i	$0.355590\pi$
$564$	−8.99037	−0.378563
$565$	0	0
$566$	−51.4807	−2.16389
$567$	8.35498	0.350876
$568$	13.3217	0.558967
$569$	−15.5986	−0.653926	−0.326963	−	0.945037i	$-0.606025\pi$
$569$	−15.5986	−0.653926	−0.326963	+	0.945037i	$0.606025\pi$
$570$	0	0
$571$	−2.23831	−0.0936703	−0.0468351	−	0.998903i	$-0.514914\pi$
$571$	−2.23831	−0.0936703	−0.0468351	+	0.998903i	$0.514914\pi$
$572$	−3.00609	−0.125691
$573$	7.93414	0.331453
$574$	−6.02660	−0.251545
$575$	0	0
$576$	0.486176	0.0202573
$577$	2.05864	0.0857023	0.0428511	−	0.999081i	$-0.486356\pi$
$577$	2.05864	0.0857023	0.0428511	+	0.999081i	$0.486356\pi$
$578$	1.78400	0.0742045
$579$	21.3678	0.888017
$580$	0	0
$581$	16.5850	0.688060
$582$	−52.8133	−2.18918
$583$	−5.95577	−0.246663
$584$	2.78624	0.115295
$585$	0	0
$586$	−7.33547	−0.303026
$587$	40.5827	1.67503	0.837514	−	0.546415i	$-0.184008\pi$
$587$	40.5827	1.67503	0.837514	+	0.546415i	$0.184008\pi$
$588$	9.35199	0.385670
$589$	−4.93186	−0.203214
$590$	0	0
$591$	35.1894	1.44750
$592$	−39.9473	−1.64182
$593$	44.2617	1.81761	0.908805	−	0.417221i	$-0.136996\pi$
$593$	44.2617	1.81761	0.908805	+	0.417221i	$0.136996\pi$
$594$	−10.0231	−0.411251
$595$	0	0
$596$	12.0862	0.495071
$597$	−11.5185	−0.471422
$598$	−11.9751	−0.489698
$599$	24.2064	0.989045	0.494523	−	0.869165i	$-0.335343\pi$
$599$	24.2064	0.989045	0.494523	+	0.869165i	$0.335343\pi$
$600$	0	0
$601$	−42.7160	−1.74242	−0.871211	−	0.490909i	$-0.836665\pi$
$601$	−42.7160	−1.74242	−0.871211	+	0.490909i	$0.836665\pi$
$602$	3.12866	0.127515
$603$	−7.18022	−0.292401
$604$	6.40541	0.260632
$605$	0	0
$606$	−7.79871	−0.316801
$607$	25.4795	1.03418	0.517091	−	0.855930i	$-0.327015\pi$
$607$	25.4795	1.03418	0.517091	+	0.855930i	$0.327015\pi$
$608$	31.9929	1.29748
$609$	1.16878	0.0473614
$610$	0	0
$611$	12.8095	0.518216
$612$	−0.856807	−0.0346344
$613$	−29.0657	−1.17395	−0.586977	−	0.809604i	$-0.699682\pi$
$613$	−29.0657	−1.17395	−0.586977	+	0.809604i	$0.699682\pi$
$614$	−1.36659	−0.0551509
$615$	0	0
$616$	−1.93328	−0.0778942
$617$	20.4429	0.822999	0.411500	−	0.911410i	$-0.365005\pi$
$617$	20.4429	0.822999	0.411500	+	0.911410i	$0.365005\pi$
$618$	−43.6214	−1.75471
$619$	−6.04012	−0.242773	−0.121386	−	0.992605i	$-0.538734\pi$
$619$	−6.04012	−0.242773	−0.121386	+	0.992605i	$0.538734\pi$
$620$	0	0
$621$	−14.8369	−0.595383
$622$	12.1612	0.487618
$623$	−17.4121	−0.697600
$624$	19.0437	0.762357
$625$	0	0
$626$	40.0773	1.60181
$627$	−8.11902	−0.324243
$628$	−15.4297	−0.615712
$629$	8.04312	0.320700
$630$	0	0
$631$	18.2913	0.728164	0.364082	−	0.931367i	$-0.381383\pi$
$631$	18.2913	0.728164	0.364082	+	0.931367i	$0.381383\pi$
$632$	17.3633	0.690676
$633$	−14.6025	−0.580397
$634$	−46.9642	−1.86519
$635$	0	0
$636$	−10.6251	−0.421312
$637$	−13.3247	−0.527945
$638$	−1.04255	−0.0412749
$639$	6.61888	0.261839
$640$	0	0
$641$	−17.9005	−0.707027	−0.353513	−	0.935429i	$-0.615013\pi$
$641$	−17.9005	−0.707027	−0.353513	+	0.935429i	$0.615013\pi$
$642$	−14.1035	−0.556620
$643$	−45.7435	−1.80395	−0.901973	−	0.431793i	$-0.857881\pi$
$643$	−45.7435	−1.80395	−0.901973	+	0.431793i	$0.857881\pi$
$644$	4.14076	0.163169
$645$	0	0
$646$	−9.60192	−0.377782
$647$	12.1320	0.476957	0.238478	−	0.971148i	$-0.423351\pi$
$647$	12.1320	0.476957	0.238478	+	0.971148i	$0.423351\pi$
$648$	9.18884	0.360972
$649$	−6.90302	−0.270967
$650$	0	0
$651$	1.83264	0.0718268
$652$	−8.45219	−0.331013
$653$	11.1525	0.436433	0.218216	−	0.975900i	$-0.429976\pi$
$653$	11.1525	0.436433	0.218216	+	0.975900i	$0.429976\pi$
$654$	−3.58738	−0.140278
$655$	0	0
$656$	−12.6547	−0.494082
$657$	1.38434	0.0540082
$658$	−11.9198	−0.464681
$659$	−2.50889	−0.0977325	−0.0488663	−	0.998805i	$-0.515561\pi$
$659$	−2.50889	−0.0977325	−0.0488663	+	0.998805i	$0.515561\pi$
$660$	0	0
$661$	5.01417	0.195029	0.0975143	−	0.995234i	$-0.468911\pi$
$661$	5.01417	0.195029	0.0975143	+	0.995234i	$0.468911\pi$
$662$	−4.98856	−0.193886
$663$	−3.83432	−0.148913
$664$	18.2402	0.707857
$665$	0	0
$666$	−10.3955	−0.402819
$667$	−1.54326	−0.0597552
$668$	22.1747	0.857964
$669$	−36.3664	−1.40601
$670$	0	0
$671$	−3.20570	−0.123755
$672$	−11.8883	−0.458601
$673$	42.1950	1.62650	0.813248	−	0.581917i	$-0.197697\pi$
$673$	42.1950	1.62650	0.813248	+	0.581917i	$0.197697\pi$
$674$	6.08510	0.234389
$675$	0	0
$676$	−7.73337	−0.297437
$677$	−29.7833	−1.14467	−0.572333	−	0.820021i	$-0.693962\pi$
$677$	−29.7833	−1.14467	−0.572333	+	0.820021i	$0.693962\pi$
$678$	−17.1378	−0.658175
$679$	−26.0195	−0.998537
$680$	0	0
$681$	9.48909	0.363623
$682$	−1.63471	−0.0625963
$683$	23.2908	0.891198	0.445599	−	0.895233i	$-0.352991\pi$
$683$	23.2908	0.891198	0.445599	+	0.895233i	$0.352991\pi$
$684$	4.61155	0.176327
$685$	0	0
$686$	28.9562	1.10555
$687$	−9.39596	−0.358478
$688$	6.56957	0.250462
$689$	15.1386	0.576735
$690$	0	0
$691$	37.8216	1.43880	0.719400	−	0.694596i	$-0.244417\pi$
$691$	37.8216	1.43880	0.719400	+	0.694596i	$0.244417\pi$
$692$	−9.84952	−0.374423
$693$	−0.960548	−0.0364882
$694$	23.4878	0.891583
$695$	0	0
$696$	1.28543	0.0487240
$697$	2.54793	0.0965098
$698$	1.03287	0.0390949
$699$	−8.06672	−0.305111
$700$	0	0
$701$	−37.3845	−1.41199	−0.705997	−	0.708215i	$-0.749501\pi$
$701$	−37.3845	−1.41199	−0.705997	+	0.708215i	$0.749501\pi$
$702$	25.4770	0.961568
$703$	−43.2901	−1.63272
$704$	0.671065	0.0252917
$705$	0	0
$706$	46.1920	1.73846
$707$	−3.84219	−0.144501
$708$	−12.3150	−0.462825
$709$	12.6435	0.474835	0.237418	−	0.971408i	$-0.423699\pi$
$709$	12.6435	0.474835	0.237418	+	0.971408i	$0.423699\pi$
$710$	0	0
$711$	8.62694	0.323536
$712$	−19.1499	−0.717671
$713$	−2.41982	−0.0906230
$714$	3.56799	0.133529
$715$	0	0
$716$	5.94702	0.222251
$717$	−42.2946	−1.57952
$718$	25.2816	0.943501
$719$	−8.41274	−0.313742	−0.156871	−	0.987619i	$-0.550141\pi$
$719$	−8.41274	−0.313742	−0.156871	+	0.987619i	$0.550141\pi$
$720$	0	0
$721$	−21.4910	−0.800366
$722$	17.7840	0.661852
$723$	−8.11988	−0.301982
$724$	18.2793	0.679345
$725$	0	0
$726$	−2.69113	−0.0998770
$727$	6.89719	0.255803	0.127901	−	0.991787i	$-0.459176\pi$
$727$	6.89719	0.255803	0.127901	+	0.991787i	$0.459176\pi$
$728$	4.91409	0.182128
$729$	29.9908	1.11077
$730$	0	0
$731$	−1.32274	−0.0489233
$732$	−5.71896	−0.211379
$733$	9.02927	0.333504	0.166752	−	0.985999i	$-0.446672\pi$
$733$	9.02927	0.333504	0.166752	+	0.985999i	$0.446672\pi$
$734$	−55.8688	−2.06216
$735$	0	0
$736$	15.6973	0.578612
$737$	−9.91079	−0.365069
$738$	−3.29315	−0.121222
$739$	12.6745	0.466241	0.233120	−	0.972448i	$-0.425106\pi$
$739$	12.6745	0.466241	0.233120	+	0.972448i	$0.425106\pi$
$740$	0	0
$741$	20.6373	0.758129
$742$	−14.0871	−0.517154
$743$	−38.1848	−1.40086	−0.700432	−	0.713719i	$-0.747009\pi$
$743$	−38.1848	−1.40086	−0.700432	+	0.713719i	$0.747009\pi$
$744$	2.01554	0.0738934
$745$	0	0
$746$	3.48775	0.127696
$747$	9.06261	0.331584
$748$	−1.18264	−0.0432418
$749$	−6.94836	−0.253888
$750$	0	0
$751$	10.6129	0.387270	0.193635	−	0.981074i	$-0.437972\pi$
$751$	10.6129	0.387270	0.193635	+	0.981074i	$0.437972\pi$
$752$	−25.0291	−0.912719
$753$	−31.7624	−1.15749
$754$	2.65000	0.0965071
$755$	0	0
$756$	−8.80948	−0.320398
$757$	4.40407	0.160069	0.0800343	−	0.996792i	$-0.474497\pi$
$757$	4.40407	0.160069	0.0800343	+	0.996792i	$0.474497\pi$
$758$	−36.9636	−1.34258
$759$	−3.98360	−0.144596
$760$	0	0
$761$	4.77406	0.173059	0.0865297	−	0.996249i	$-0.472422\pi$
$761$	4.77406	0.173059	0.0865297	+	0.996249i	$0.472422\pi$
$762$	−8.55684	−0.309981
$763$	−1.76740	−0.0639841
$764$	−6.22034	−0.225044
$765$	0	0
$766$	−58.6661	−2.11969
$767$	17.5464	0.633563
$768$	−30.7957	−1.11125
$769$	−22.3099	−0.804514	−0.402257	−	0.915527i	$-0.631774\pi$
$769$	−22.3099	−0.804514	−0.402257	+	0.915527i	$0.631774\pi$
$770$	0	0
$771$	−9.91603	−0.357117
$772$	−16.7523	−0.602929
$773$	−8.72835	−0.313937	−0.156968	−	0.987604i	$-0.550172\pi$
$773$	−8.72835	−0.313937	−0.156968	+	0.987604i	$0.550172\pi$
$774$	1.70961	0.0614507
$775$	0	0
$776$	−28.6163	−1.02727
$777$	16.0862	0.577090
$778$	5.97893	0.214355
$779$	−13.7136	−0.491341
$780$	0	0
$781$	9.13599	0.326911
$782$	−4.71119	−0.168472
$783$	3.28329	0.117335
$784$	26.0359	0.929854
$785$	0	0
$786$	41.4839	1.47968
$787$	−17.8808	−0.637383	−0.318692	−	0.947858i	$-0.603243\pi$
$787$	−17.8808	−0.637383	−0.318692	+	0.947858i	$0.603243\pi$
$788$	−27.5884	−0.982796
$789$	13.1407	0.467820
$790$	0	0
$791$	−8.44330	−0.300209
$792$	−1.05641	−0.0375380
$793$	8.14838	0.289357
$794$	51.7640	1.83704
$795$	0	0
$796$	9.03049	0.320077
$797$	33.8956	1.20064	0.600322	−	0.799758i	$-0.295039\pi$
$797$	33.8956	1.20064	0.600322	+	0.799758i	$0.295039\pi$
$798$	−19.2038	−0.679808
$799$	5.03945	0.178283
$800$	0	0
$801$	−9.51457	−0.336181
$802$	−52.9913	−1.87119
$803$	1.91079	0.0674304
$804$	−17.6808	−0.623555
$805$	0	0
$806$	4.15517	0.146360
$807$	6.64548	0.233932
$808$	−4.22566	−0.148658
$809$	8.71408	0.306371	0.153185	−	0.988197i	$-0.451047\pi$
$809$	8.71408	0.306371	0.153185	+	0.988197i	$0.451047\pi$
$810$	0	0
$811$	−37.6295	−1.32135	−0.660675	−	0.750672i	$-0.729730\pi$
$811$	−37.6295	−1.32135	−0.660675	+	0.750672i	$0.729730\pi$
$812$	−0.916320	−0.0321565
$813$	7.06307	0.247713
$814$	−14.3489	−0.502928
$815$	0	0
$816$	7.49208	0.262275
$817$	7.11931	0.249073
$818$	−31.0487	−1.08559
$819$	2.44156	0.0853150
$820$	0	0
$821$	23.7988	0.830585	0.415293	−	0.909688i	$-0.363679\pi$
$821$	23.7988	0.830585	0.415293	+	0.909688i	$0.363679\pi$
$822$	−29.0458	−1.01309
$823$	16.5781	0.577875	0.288937	−	0.957348i	$-0.406698\pi$
$823$	16.5781	0.577875	0.288937	+	0.957348i	$0.406698\pi$
$824$	−23.6358	−0.823394
$825$	0	0
$826$	−16.3276	−0.568111
$827$	22.5975	0.785793	0.392896	−	0.919583i	$-0.371473\pi$
$827$	22.5975	0.785793	0.392896	+	0.919583i	$0.371473\pi$
$828$	2.26266	0.0786329
$829$	51.0697	1.77372	0.886862	−	0.462034i	$-0.152880\pi$
$829$	51.0697	1.77372	0.886862	+	0.462034i	$0.152880\pi$
$830$	0	0
$831$	−21.6214	−0.750039
$832$	−1.70574	−0.0591359
$833$	−5.24216	−0.181630
$834$	−5.32979	−0.184555
$835$	0	0
$836$	6.36529	0.220148
$837$	5.14817	0.177947
$838$	−59.6471	−2.06048
$839$	−28.2392	−0.974925	−0.487462	−	0.873144i	$-0.662077\pi$
$839$	−28.2392	−0.974925	−0.487462	+	0.873144i	$0.662077\pi$
$840$	0	0
$841$	−28.6585	−0.988224
$842$	19.8822	0.685187
$843$	−12.0512	−0.415065
$844$	11.4483	0.394067
$845$	0	0
$846$	−6.51338	−0.223935
$847$	−1.32584	−0.0455563
$848$	−29.5802	−1.01579
$849$	43.5301	1.49395
$850$	0	0
$851$	−21.2403	−0.728108
$852$	16.2986	0.558380
$853$	10.2993	0.352642	0.176321	−	0.984333i	$-0.443580\pi$
$853$	10.2993	0.352642	0.176321	+	0.984333i	$0.443580\pi$
$854$	−7.58240	−0.259465
$855$	0	0
$856$	−7.64183	−0.261192
$857$	32.4575	1.10873	0.554364	−	0.832275i	$-0.312962\pi$
$857$	32.4575	1.10873	0.554364	+	0.832275i	$0.312962\pi$
$858$	6.84041	0.233528
$859$	28.2292	0.963169	0.481585	−	0.876400i	$-0.340061\pi$
$859$	28.2292	0.963169	0.481585	+	0.876400i	$0.340061\pi$
$860$	0	0
$861$	5.09587	0.173667
$862$	26.4023	0.899267
$863$	−25.4869	−0.867583	−0.433791	−	0.901013i	$-0.642825\pi$
$863$	−25.4869	−0.867583	−0.433791	+	0.901013i	$0.642825\pi$
$864$	−33.3961	−1.13616
$865$	0	0
$866$	−1.05797	−0.0359513
$867$	−1.50848	−0.0512307
$868$	−1.43678	−0.0487676
$869$	11.9077	0.403941
$870$	0	0
$871$	25.1917	0.853587
$872$	−1.94379	−0.0658250
$873$	−14.2180	−0.481206
$874$	25.3568	0.857706
$875$	0	0
$876$	3.40885	0.115174
$877$	36.1131	1.21945	0.609726	−	0.792612i	$-0.291280\pi$
$877$	36.1131	1.21945	0.609726	+	0.792612i	$0.291280\pi$
$878$	−62.2742	−2.10165
$879$	6.20260	0.209209
$880$	0	0
$881$	14.3808	0.484503	0.242251	−	0.970214i	$-0.422114\pi$
$881$	14.3808	0.484503	0.242251	+	0.970214i	$0.422114\pi$
$882$	6.77537	0.228139
$883$	38.1234	1.28295	0.641477	−	0.767142i	$-0.278322\pi$
$883$	38.1234	1.28295	0.641477	+	0.767142i	$0.278322\pi$
$884$	3.00609	0.101106
$885$	0	0
$886$	51.9089	1.74392
$887$	9.30732	0.312509	0.156255	−	0.987717i	$-0.450058\pi$
$887$	9.30732	0.312509	0.156255	+	0.987717i	$0.450058\pi$
$888$	17.6917	0.593694
$889$	−4.21570	−0.141390
$890$	0	0
$891$	6.30167	0.211114
$892$	28.5112	0.954624
$893$	−27.1236	−0.907656
$894$	−27.5024	−0.919818
$895$	0	0
$896$	−14.1747	−0.473543
$897$	10.1257	0.338087
$898$	−8.73192	−0.291388
$899$	0.535488	0.0178595
$900$	0	0
$901$	5.95577	0.198416
$902$	−4.54550	−0.151349
$903$	−2.64548	−0.0880360
$904$	−9.28598	−0.308847
$905$	0	0
$906$	−14.5756	−0.484242
$907$	29.3750	0.975380	0.487690	−	0.873017i	$-0.337840\pi$
$907$	29.3750	0.975380	0.487690	+	0.873017i	$0.337840\pi$
$908$	−7.43942	−0.246886
$909$	−2.09951	−0.0696364
$910$	0	0
$911$	11.3627	0.376464	0.188232	−	0.982125i	$-0.439724\pi$
$911$	11.3627	0.376464	0.188232	+	0.982125i	$0.439724\pi$
$912$	−40.3243	−1.33527
$913$	12.5090	0.413989
$914$	−15.6946	−0.519130
$915$	0	0
$916$	7.36640	0.243393
$917$	20.4379	0.674918
$918$	10.0231	0.330810
$919$	28.4671	0.939042	0.469521	−	0.882921i	$-0.344427\pi$
$919$	28.4671	0.939042	0.469521	+	0.882921i	$0.344427\pi$
$920$	0	0
$921$	1.15553	0.0380761
$922$	−16.9356	−0.557744
$923$	−23.2222	−0.764369
$924$	−2.36529	−0.0778123
$925$	0	0
$926$	−19.7656	−0.649536
$927$	−11.7434	−0.385705
$928$	−3.47370	−0.114030
$929$	−19.8349	−0.650762	−0.325381	−	0.945583i	$-0.605493\pi$
$929$	−19.8349	−0.650762	−0.325381	+	0.945583i	$0.605493\pi$
$930$	0	0
$931$	28.2146	0.924696
$932$	6.32428	0.207159
$933$	−10.2830	−0.336651
$934$	−59.5980	−1.95011
$935$	0	0
$936$	2.68524	0.0877697
$937$	39.7024	1.29702	0.648510	−	0.761206i	$-0.275392\pi$
$937$	39.7024	1.29702	0.648510	+	0.761206i	$0.275392\pi$
$938$	−23.4419	−0.765405
$939$	−33.8879	−1.10589
$940$	0	0
$941$	−40.6077	−1.32377	−0.661886	−	0.749604i	$-0.730244\pi$
$941$	−40.6077	−1.32377	−0.661886	+	0.749604i	$0.730244\pi$
$942$	35.1105	1.14396
$943$	−6.72859	−0.219113
$944$	−34.2848	−1.11588
$945$	0	0
$946$	2.35976	0.0767225
$947$	−18.5478	−0.602723	−0.301362	−	0.953510i	$-0.597441\pi$
$947$	−18.5478	−0.602723	−0.301362	+	0.953510i	$0.597441\pi$
$948$	21.2433	0.689950
$949$	−4.85693	−0.157663
$950$	0	0
$951$	39.7112	1.28772
$952$	1.93328	0.0626580
$953$	57.7847	1.87183	0.935915	−	0.352226i	$-0.114575\pi$
$953$	57.7847	1.87183	0.935915	+	0.352226i	$0.114575\pi$
$954$	−7.69770	−0.249222
$955$	0	0
$956$	33.1589	1.07243
$957$	0.881541	0.0284962
$958$	−59.0463	−1.90770
$959$	−14.3100	−0.462094
$960$	0	0
$961$	−30.1604	−0.972915
$962$	36.4726	1.17592
$963$	−3.79683	−0.122351
$964$	6.36596	0.205034
$965$	0	0
$966$	−9.42237	−0.303160
$967$	−20.3827	−0.655462	−0.327731	−	0.944771i	$-0.606284\pi$
$967$	−20.3827	−0.655462	−0.327731	+	0.944771i	$0.606284\pi$
$968$	−1.45816	−0.0468670
$969$	8.11902	0.260821
$970$	0	0
$971$	−12.6339	−0.405441	−0.202721	−	0.979237i	$-0.564978\pi$
$971$	−12.6339	−0.405441	−0.202721	+	0.979237i	$0.564978\pi$
$972$	−8.69125	−0.278772
$973$	−2.62583	−0.0841802
$974$	−71.3138	−2.28504
$975$	0	0
$976$	−15.9216	−0.509637
$977$	−15.8492	−0.507060	−0.253530	−	0.967328i	$-0.581592\pi$
$977$	−15.8492	−0.507060	−0.253530	+	0.967328i	$0.581592\pi$
$978$	19.2331	0.615006
$979$	−13.1329	−0.419729
$980$	0	0
$981$	−0.965769	−0.0308346
$982$	30.0767	0.959785
$983$	−25.3225	−0.807662	−0.403831	−	0.914834i	$-0.632322\pi$
$983$	−25.3225	−0.807662	−0.403831	+	0.914834i	$0.632322\pi$
$984$	5.60445	0.178663
$985$	0	0
$986$	1.04255	0.0332016
$987$	10.0789	0.320815
$988$	−16.1795	−0.514740
$989$	3.49309	0.111074
$990$	0	0
$991$	−9.43172	−0.299608	−0.149804	−	0.988716i	$-0.547864\pi$
$991$	−9.43172	−0.299608	−0.149804	+	0.988716i	$0.547864\pi$
$992$	−5.44674	−0.172934
$993$	4.21814	0.133859
$994$	21.6092	0.685404
$995$	0	0
$996$	22.3161	0.707113
$997$	−19.6216	−0.621421	−0.310711	−	0.950505i	$-0.600567\pi$
$997$	−19.6216	−0.621421	−0.310711	+	0.950505i	$0.600567\pi$
$998$	31.9708	1.01202
$999$	45.1888	1.42971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4675.2.a.be.1.4		4
5.4	even	2		935.2.a.g.1.1	✓	4
15.14	odd	2		8415.2.a.ba.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
935.2.a.g.1.1	✓	4	5.4	even	2
4675.2.a.be.1.4		4	1.1	even	1	trivial
8415.2.a.ba.1.4		4	15.14	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4675 = 5^{2} \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.78400 q^{2} -1.50848 q^{3} +1.18264 q^{4} -2.69113 q^{6} -1.32584 q^{7} -1.45816 q^{8} -0.724484 q^{9} +O(q^{10})\)	\(q+1.78400 q^{2} -1.50848 q^{3} +1.18264 q^{4} -2.69113 q^{6} -1.32584 q^{7} -1.45816 q^{8} -0.724484 q^{9} -1.00000 q^{11} -1.78400 q^{12} +2.54184 q^{13} -2.36529 q^{14} -4.96664 q^{16} +1.00000 q^{17} -1.29248 q^{18} -5.38225 q^{19} +2.00000 q^{21} -1.78400 q^{22} -2.64080 q^{23} +2.19961 q^{24} +4.53463 q^{26} +5.61831 q^{27} -1.56799 q^{28} +0.584390 q^{29} +0.916320 q^{31} -5.94415 q^{32} +1.50848 q^{33} +1.78400 q^{34} -0.856807 q^{36} +8.04312 q^{37} -9.60192 q^{38} -3.83432 q^{39} +2.54793 q^{41} +3.56799 q^{42} -1.32274 q^{43} -1.18264 q^{44} -4.71119 q^{46} +5.03945 q^{47} +7.49208 q^{48} -5.24216 q^{49} -1.50848 q^{51} +3.00609 q^{52} +5.95577 q^{53} +10.0231 q^{54} +1.93328 q^{56} +8.11902 q^{57} +1.04255 q^{58} +6.90302 q^{59} +3.20570 q^{61} +1.63471 q^{62} +0.960548 q^{63} -0.671065 q^{64} +2.69113 q^{66} +9.91079 q^{67} +1.18264 q^{68} +3.98360 q^{69} -9.13599 q^{71} +1.05641 q^{72} -1.91079 q^{73} +14.3489 q^{74} -6.36529 q^{76} +1.32584 q^{77} -6.84041 q^{78} -11.9077 q^{79} -6.30167 q^{81} +4.54550 q^{82} -12.5090 q^{83} +2.36529 q^{84} -2.35976 q^{86} -0.881541 q^{87} +1.45816 q^{88} +13.1329 q^{89} -3.37007 q^{91} -3.12313 q^{92} -1.38225 q^{93} +8.99037 q^{94} +8.96664 q^{96} +19.6250 q^{97} -9.35199 q^{98} +0.724484 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9	\( 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9} - 4 q^{11} + 13 q^{13} - 4 q^{14} - 10 q^{16} + 4 q^{17} + 9 q^{18} - 2 q^{19} + 8 q^{21} - 5 q^{23} - 8 q^{24} - 6 q^{26} + 4 q^{27} + 8 q^{28} + 12 q^{29} - 2 q^{31} - q^{32} - q^{33} - 5 q^{36} + q^{37} + 4 q^{38} - 4 q^{39} + 2 q^{41} + 7 q^{43} - 2 q^{44} - 3 q^{46} + 19 q^{47} - q^{48} - 11 q^{49} + q^{51} + q^{52} + 17 q^{53} + 15 q^{54} - 12 q^{56} + 18 q^{57} + 11 q^{58} + 6 q^{59} - 15 q^{61} + 12 q^{62} + 5 q^{63} + q^{64} + q^{66} + 7 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{71} - 11 q^{72} + 25 q^{73} + 28 q^{74} - 20 q^{76} + q^{77} - 5 q^{78} - 7 q^{79} - 8 q^{81} - 9 q^{82} - 5 q^{83} + 4 q^{84} + 23 q^{86} + 20 q^{87} + 3 q^{88} + 16 q^{89} - 16 q^{91} - 17 q^{92} + 14 q^{93} - 10 q^{94} + 26 q^{96} + 11 q^{97} - 16 q^{98} + 3 q^{99}+O(q^{100}) \) 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9 - 4 * q^11 + 13 * q^13 - 4 * q^14 - 10 * q^16 + 4 * q^17 + 9 * q^18 - 2 * q^19 + 8 * q^21 - 5 * q^23 - 8 * q^24 - 6 * q^26 + 4 * q^27 + 8 * q^28 + 12 * q^29 - 2 * q^31 - q^32 - q^33 - 5 * q^36 + q^37 + 4 * q^38 - 4 * q^39 + 2 * q^41 + 7 * q^43 - 2 * q^44 - 3 * q^46 + 19 * q^47 - q^48 - 11 * q^49 + q^51 + q^52 + 17 * q^53 + 15 * q^54 - 12 * q^56 + 18 * q^57 + 11 * q^58 + 6 * q^59 - 15 * q^61 + 12 * q^62 + 5 * q^63 + q^64 + q^66 + 7 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^71 - 11 * q^72 + 25 * q^73 + 28 * q^74 - 20 * q^76 + q^77 - 5 * q^78 - 7 * q^79 - 8 * q^81 - 9 * q^82 - 5 * q^83 + 4 * q^84 + 23 * q^86 + 20 * q^87 + 3 * q^88 + 16 * q^89 - 16 * q^91 - 17 * q^92 + 14 * q^93 - 10 * q^94 + 26 * q^96 + 11 * q^97 - 16 * q^98 + 3 * q^99

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