LMFDB - Embedded newform 668.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(1,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("668.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.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:	$5.33400685502$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.38961$ of defining polynomial
Character	$\chi$	$=$	668.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.06430 q^{3} -0.836956 q^{5} +1.84506 q^{7} +6.38996 q^{9} +O(q^{10})$	$q-3.06430 q^{3} -0.836956 q^{5} +1.84506 q^{7} +6.38996 q^{9} -1.84716 q^{11} -2.10214 q^{13} +2.56469 q^{15} +6.13803 q^{17} +1.96578 q^{19} -5.65383 q^{21} -4.81683 q^{23} -4.29950 q^{25} -10.3879 q^{27} +1.45602 q^{29} -9.36690 q^{31} +5.66027 q^{33} -1.54424 q^{35} -7.19237 q^{37} +6.44159 q^{39} +8.94304 q^{41} -8.96556 q^{43} -5.34811 q^{45} -11.7486 q^{47} -3.59575 q^{49} -18.8088 q^{51} -2.57494 q^{53} +1.54599 q^{55} -6.02376 q^{57} +0.603581 q^{59} -6.89878 q^{61} +11.7899 q^{63} +1.75940 q^{65} -3.81742 q^{67} +14.7602 q^{69} +3.05806 q^{71} +2.63554 q^{73} +13.1750 q^{75} -3.40813 q^{77} +4.98598 q^{79} +12.6617 q^{81} +6.92379 q^{83} -5.13727 q^{85} -4.46168 q^{87} +12.0058 q^{89} -3.87858 q^{91} +28.7030 q^{93} -1.64528 q^{95} +7.38811 q^{97} -11.8033 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.06430	−1.76918	−0.884588	−	0.466373i	$-0.845561\pi$
$3$	−3.06430	−1.76918	−0.884588	+	0.466373i	$0.845561\pi$
$4$	0	0
$5$	−0.836956	−0.374298	−0.187149	−	0.982332i	$-0.559925\pi$
$5$	−0.836956	−0.374298	−0.187149	+	0.982332i	$0.559925\pi$
$6$	0	0
$7$	1.84506	0.697368	0.348684	−	0.937240i	$-0.386629\pi$
$7$	1.84506	0.697368	0.348684	+	0.937240i	$0.386629\pi$
$8$	0	0
$9$	6.38996	2.12999
$10$	0	0
$11$	−1.84716	−0.556941	−0.278470	−	0.960445i	$-0.589827\pi$
$11$	−1.84716	−0.556941	−0.278470	+	0.960445i	$0.589827\pi$
$12$	0	0
$13$	−2.10214	−0.583029	−0.291514	−	0.956566i	$-0.594159\pi$
$13$	−2.10214	−0.583029	−0.291514	+	0.956566i	$0.594159\pi$
$14$	0	0
$15$	2.56469	0.662200
$16$	0	0
$17$	6.13803	1.48869	0.744346	−	0.667794i	$-0.232761\pi$
$17$	6.13803	1.48869	0.744346	+	0.667794i	$0.232761\pi$
$18$	0	0
$19$	1.96578	0.450982	0.225491	−	0.974245i	$-0.427601\pi$
$19$	1.96578	0.450982	0.225491	+	0.974245i	$0.427601\pi$
$20$	0	0
$21$	−5.65383	−1.23377
$22$	0	0
$23$	−4.81683	−1.00438	−0.502189	−	0.864758i	$-0.667472\pi$
$23$	−4.81683	−1.00438	−0.502189	+	0.864758i	$0.667472\pi$
$24$	0	0
$25$	−4.29950	−0.859901
$26$	0	0
$27$	−10.3879	−1.99914
$28$	0	0
$29$	1.45602	0.270375	0.135188	−	0.990820i	$-0.456836\pi$
$29$	1.45602	0.270375	0.135188	+	0.990820i	$0.456836\pi$
$30$	0	0
$31$	−9.36690	−1.68234	−0.841172	−	0.540767i	$-0.818134\pi$
$31$	−9.36690	−1.68234	−0.841172	+	0.540767i	$0.818134\pi$
$32$	0	0
$33$	5.66027	0.985327
$34$	0	0
$35$	−1.54424	−0.261024
$36$	0	0
$37$	−7.19237	−1.18242	−0.591209	−	0.806518i	$-0.701349\pi$
$37$	−7.19237	−1.18242	−0.591209	+	0.806518i	$0.701349\pi$
$38$	0	0
$39$	6.44159	1.03148
$40$	0	0
$41$	8.94304	1.39667	0.698334	−	0.715772i	$-0.253925\pi$
$41$	8.94304	1.39667	0.698334	+	0.715772i	$0.253925\pi$
$42$	0	0
$43$	−8.96556	−1.36724	−0.683618	−	0.729840i	$-0.739594\pi$
$43$	−8.96556	−1.36724	−0.683618	+	0.729840i	$0.739594\pi$
$44$	0	0
$45$	−5.34811	−0.797250
$46$	0	0
$47$	−11.7486	−1.71371	−0.856856	−	0.515555i	$-0.827586\pi$
$47$	−11.7486	−1.71371	−0.856856	+	0.515555i	$0.827586\pi$
$48$	0	0
$49$	−3.59575	−0.513678
$50$	0	0
$51$	−18.8088	−2.63376
$52$	0	0
$53$	−2.57494	−0.353696	−0.176848	−	0.984238i	$-0.556590\pi$
$53$	−2.57494	−0.353696	−0.176848	+	0.984238i	$0.556590\pi$
$54$	0	0
$55$	1.54599	0.208462
$56$	0	0
$57$	−6.02376	−0.797866
$58$	0	0
$59$	0.603581	0.0785795	0.0392898	−	0.999228i	$-0.487490\pi$
$59$	0.603581	0.0785795	0.0392898	+	0.999228i	$0.487490\pi$
$60$	0	0
$61$	−6.89878	−0.883298	−0.441649	−	0.897188i	$-0.645606\pi$
$61$	−6.89878	−0.883298	−0.441649	+	0.897188i	$0.645606\pi$
$62$	0	0
$63$	11.7899	1.48538
$64$	0	0
$65$	1.75940	0.218227
$66$	0	0
$67$	−3.81742	−0.466372	−0.233186	−	0.972432i	$-0.574915\pi$
$67$	−3.81742	−0.466372	−0.233186	+	0.972432i	$0.574915\pi$
$68$	0	0
$69$	14.7602	1.77692
$70$	0	0
$71$	3.05806	0.362925	0.181462	−	0.983398i	$-0.441917\pi$
$71$	3.05806	0.362925	0.181462	+	0.983398i	$0.441917\pi$
$72$	0	0
$73$	2.63554	0.308466	0.154233	−	0.988034i	$-0.450709\pi$
$73$	2.63554	0.308466	0.154233	+	0.988034i	$0.450709\pi$
$74$	0	0
$75$	13.1750	1.52132
$76$	0	0
$77$	−3.40813	−0.388393
$78$	0	0
$79$	4.98598	0.560966	0.280483	−	0.959859i	$-0.409505\pi$
$79$	4.98598	0.560966	0.280483	+	0.959859i	$0.409505\pi$
$80$	0	0
$81$	12.6617	1.40685
$82$	0	0
$83$	6.92379	0.759985	0.379992	−	0.924990i	$-0.375927\pi$
$83$	6.92379	0.759985	0.379992	+	0.924990i	$0.375927\pi$
$84$	0	0
$85$	−5.13727	−0.557215
$86$	0	0
$87$	−4.46168	−0.478342
$88$	0	0
$89$	12.0058	1.27261	0.636304	−	0.771438i	$-0.280462\pi$
$89$	12.0058	1.27261	0.636304	+	0.771438i	$0.280462\pi$
$90$	0	0
$91$	−3.87858	−0.406585
$92$	0	0
$93$	28.7030	2.97636
$94$	0	0
$95$	−1.64528	−0.168802
$96$	0	0
$97$	7.38811	0.750149	0.375075	−	0.926995i	$-0.377617\pi$
$97$	7.38811	0.750149	0.375075	+	0.926995i	$0.377617\pi$
$98$	0	0
$99$	−11.8033	−1.18628
$100$	0	0
$101$	−5.97159	−0.594196	−0.297098	−	0.954847i	$-0.596019\pi$
$101$	−5.97159	−0.594196	−0.297098	+	0.954847i	$0.596019\pi$
$102$	0	0
$103$	4.33003	0.426650	0.213325	−	0.976981i	$-0.431571\pi$
$103$	4.33003	0.426650	0.213325	+	0.976981i	$0.431571\pi$
$104$	0	0
$105$	4.73201	0.461797
$106$	0	0
$107$	−3.39346	−0.328058	−0.164029	−	0.986456i	$-0.552449\pi$
$107$	−3.39346	−0.328058	−0.164029	+	0.986456i	$0.552449\pi$
$108$	0	0
$109$	−6.45093	−0.617887	−0.308944	−	0.951080i	$-0.599975\pi$
$109$	−6.45093	−0.617887	−0.308944	+	0.951080i	$0.599975\pi$
$110$	0	0
$111$	22.0396	2.09191
$112$	0	0
$113$	−13.7067	−1.28942	−0.644711	−	0.764427i	$-0.723022\pi$
$113$	−13.7067	−1.28942	−0.644711	+	0.764427i	$0.723022\pi$
$114$	0	0
$115$	4.03148	0.375937
$116$	0	0
$117$	−13.4326	−1.24184
$118$	0	0
$119$	11.3251	1.03817
$120$	0	0
$121$	−7.58799	−0.689817
$122$	0	0
$123$	−27.4042	−2.47095
$124$	0	0
$125$	7.78328	0.696158
$126$	0	0
$127$	0.889000	0.0788860	0.0394430	−	0.999222i	$-0.487442\pi$
$127$	0.889000	0.0788860	0.0394430	+	0.999222i	$0.487442\pi$
$128$	0	0
$129$	27.4732	2.41888
$130$	0	0
$131$	−8.92540	−0.779816	−0.389908	−	0.920854i	$-0.627493\pi$
$131$	−8.92540	−0.779816	−0.389908	+	0.920854i	$0.627493\pi$
$132$	0	0
$133$	3.62699	0.314500
$134$	0	0
$135$	8.69418	0.748276
$136$	0	0
$137$	−2.98067	−0.254656	−0.127328	−	0.991861i	$-0.540640\pi$
$137$	−2.98067	−0.254656	−0.127328	+	0.991861i	$0.540640\pi$
$138$	0	0
$139$	−11.7522	−0.996808	−0.498404	−	0.866945i	$-0.666080\pi$
$139$	−11.7522	−0.996808	−0.498404	+	0.866945i	$0.666080\pi$
$140$	0	0
$141$	36.0013	3.03186
$142$	0	0
$143$	3.88300	0.324712
$144$	0	0
$145$	−1.21862	−0.101201
$146$	0	0
$147$	11.0185	0.908787
$148$	0	0
$149$	9.58053	0.784868	0.392434	−	0.919780i	$-0.371633\pi$
$149$	9.58053	0.784868	0.392434	+	0.919780i	$0.371633\pi$
$150$	0	0
$151$	−13.8371	−1.12604	−0.563022	−	0.826442i	$-0.690361\pi$
$151$	−13.8371	−1.12604	−0.563022	+	0.826442i	$0.690361\pi$
$152$	0	0
$153$	39.2218	3.17089
$154$	0	0
$155$	7.83968	0.629699
$156$	0	0
$157$	−10.2839	−0.820744	−0.410372	−	0.911918i	$-0.634601\pi$
$157$	−10.2839	−0.820744	−0.410372	+	0.911918i	$0.634601\pi$
$158$	0	0
$159$	7.89041	0.625750
$160$	0	0
$161$	−8.88735	−0.700421
$162$	0	0
$163$	3.25127	0.254659	0.127330	−	0.991860i	$-0.459359\pi$
$163$	3.25127	0.254659	0.127330	+	0.991860i	$0.459359\pi$
$164$	0	0
$165$	−4.73740	−0.368806
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−8.58101	−0.660078
$170$	0	0
$171$	12.5613	0.960585
$172$	0	0
$173$	−20.8691	−1.58665	−0.793326	−	0.608797i	$-0.791652\pi$
$173$	−20.8691	−1.58665	−0.793326	+	0.608797i	$0.791652\pi$
$174$	0	0
$175$	−7.93285	−0.599667
$176$	0	0
$177$	−1.84955	−0.139021
$178$	0	0
$179$	18.1238	1.35463	0.677316	−	0.735692i	$-0.263143\pi$
$179$	18.1238	1.35463	0.677316	+	0.735692i	$0.263143\pi$
$180$	0	0
$181$	20.6317	1.53355	0.766773	−	0.641919i	$-0.221861\pi$
$181$	20.6317	1.53355	0.766773	+	0.641919i	$0.221861\pi$
$182$	0	0
$183$	21.1400	1.56271
$184$	0	0
$185$	6.01970	0.442577
$186$	0	0
$187$	−11.3380	−0.829113
$188$	0	0
$189$	−19.1662	−1.39414
$190$	0	0
$191$	−17.5084	−1.26686	−0.633430	−	0.773800i	$-0.718354\pi$
$191$	−17.5084	−1.26686	−0.633430	+	0.773800i	$0.718354\pi$
$192$	0	0
$193$	−23.9477	−1.72379	−0.861897	−	0.507083i	$-0.830724\pi$
$193$	−23.9477	−1.72379	−0.861897	+	0.507083i	$0.830724\pi$
$194$	0	0
$195$	−5.39133	−0.386081
$196$	0	0
$197$	−3.71937	−0.264994	−0.132497	−	0.991183i	$-0.542300\pi$
$197$	−3.71937	−0.264994	−0.132497	+	0.991183i	$0.542300\pi$
$198$	0	0
$199$	4.24301	0.300779	0.150389	−	0.988627i	$-0.451947\pi$
$199$	4.24301	0.300779	0.150389	+	0.988627i	$0.451947\pi$
$200$	0	0
$201$	11.6977	0.825095
$202$	0	0
$203$	2.68644	0.188551
$204$	0	0
$205$	−7.48493	−0.522770
$206$	0	0
$207$	−30.7793	−2.13931
$208$	0	0
$209$	−3.63112	−0.251170
$210$	0	0
$211$	28.7595	1.97988	0.989941	−	0.141479i	$-0.0451858\pi$
$211$	28.7595	1.97988	0.989941	+	0.141479i	$0.0451858\pi$
$212$	0	0
$213$	−9.37082	−0.642078
$214$	0	0
$215$	7.50378	0.511754
$216$	0	0
$217$	−17.2825	−1.17321
$218$	0	0
$219$	−8.07608	−0.545731
$220$	0	0
$221$	−12.9030	−0.867950
$222$	0	0
$223$	−8.73822	−0.585154	−0.292577	−	0.956242i	$-0.594513\pi$
$223$	−8.73822	−0.585154	−0.292577	+	0.956242i	$0.594513\pi$
$224$	0	0
$225$	−27.4736	−1.83158
$226$	0	0
$227$	−15.5400	−1.03142	−0.515712	−	0.856762i	$-0.672473\pi$
$227$	−15.5400	−1.03142	−0.515712	+	0.856762i	$0.672473\pi$
$228$	0	0
$229$	28.1261	1.85862	0.929312	−	0.369296i	$-0.120401\pi$
$229$	28.1261	1.85862	0.929312	+	0.369296i	$0.120401\pi$
$230$	0	0
$231$	10.4435	0.687135
$232$	0	0
$233$	−28.0403	−1.83698	−0.918492	−	0.395441i	$-0.870592\pi$
$233$	−28.0403	−1.83698	−0.918492	+	0.395441i	$0.870592\pi$
$234$	0	0
$235$	9.83308	0.641439
$236$	0	0
$237$	−15.2786	−0.992448
$238$	0	0
$239$	18.1404	1.17340	0.586702	−	0.809803i	$-0.300426\pi$
$239$	18.1404	1.17340	0.586702	+	0.809803i	$0.300426\pi$
$240$	0	0
$241$	−4.13886	−0.266608	−0.133304	−	0.991075i	$-0.542559\pi$
$241$	−4.13886	−0.266608	−0.133304	+	0.991075i	$0.542559\pi$
$242$	0	0
$243$	−7.63564	−0.489827
$244$	0	0
$245$	3.00948	0.192269
$246$	0	0
$247$	−4.13235	−0.262935
$248$	0	0
$249$	−21.2166	−1.34455
$250$	0	0
$251$	26.9927	1.70376	0.851881	−	0.523736i	$-0.175462\pi$
$251$	26.9927	1.70376	0.851881	+	0.523736i	$0.175462\pi$
$252$	0	0
$253$	8.89747	0.559379
$254$	0	0
$255$	15.7421	0.985811
$256$	0	0
$257$	22.6278	1.41148	0.705740	−	0.708471i	$-0.250615\pi$
$257$	22.6278	1.41148	0.705740	+	0.708471i	$0.250615\pi$
$258$	0	0
$259$	−13.2704	−0.824581
$260$	0	0
$261$	9.30388	0.575896
$262$	0	0
$263$	7.39244	0.455838	0.227919	−	0.973680i	$-0.426808\pi$
$263$	7.39244	0.455838	0.227919	+	0.973680i	$0.426808\pi$
$264$	0	0
$265$	2.15511	0.132388
$266$	0	0
$267$	−36.7893	−2.25147
$268$	0	0
$269$	−23.2255	−1.41608	−0.708042	−	0.706171i	$-0.750421\pi$
$269$	−23.2255	−1.41608	−0.708042	+	0.706171i	$0.750421\pi$
$270$	0	0
$271$	23.7200	1.44089	0.720444	−	0.693513i	$-0.243938\pi$
$271$	23.7200	1.44089	0.720444	+	0.693513i	$0.243938\pi$
$272$	0	0
$273$	11.8851	0.719321
$274$	0	0
$275$	7.94189	0.478914
$276$	0	0
$277$	0.265583	0.0159573	0.00797867	−	0.999968i	$-0.497460\pi$
$277$	0.265583	0.0159573	0.00797867	+	0.999968i	$0.497460\pi$
$278$	0	0
$279$	−59.8541	−3.58337
$280$	0	0
$281$	−8.45950	−0.504651	−0.252326	−	0.967642i	$-0.581195\pi$
$281$	−8.45950	−0.504651	−0.252326	+	0.967642i	$0.581195\pi$
$282$	0	0
$283$	0.282575	0.0167973	0.00839866	−	0.999965i	$-0.497327\pi$
$283$	0.282575	0.0167973	0.00839866	+	0.999965i	$0.497327\pi$
$284$	0	0
$285$	5.04162	0.298640
$286$	0	0
$287$	16.5005	0.973991
$288$	0	0
$289$	20.6755	1.21620
$290$	0	0
$291$	−22.6394	−1.32715
$292$	0	0
$293$	8.86742	0.518040	0.259020	−	0.965872i	$-0.416600\pi$
$293$	8.86742	0.518040	0.259020	+	0.965872i	$0.416600\pi$
$294$	0	0
$295$	−0.505171	−0.0294122
$296$	0	0
$297$	19.1881	1.11340
$298$	0	0
$299$	10.1256	0.585581
$300$	0	0
$301$	−16.5420	−0.953466
$302$	0	0
$303$	18.2988	1.05124
$304$	0	0
$305$	5.77398	0.330617
$306$	0	0
$307$	−9.06770	−0.517521	−0.258761	−	0.965941i	$-0.583314\pi$
$307$	−9.06770	−0.517521	−0.258761	+	0.965941i	$0.583314\pi$
$308$	0	0
$309$	−13.2685	−0.754820
$310$	0	0
$311$	8.84270	0.501423	0.250712	−	0.968062i	$-0.419335\pi$
$311$	8.84270	0.501423	0.250712	+	0.968062i	$0.419335\pi$
$312$	0	0
$313$	−22.5863	−1.27665	−0.638327	−	0.769766i	$-0.720373\pi$
$313$	−22.5863	−1.27665	−0.638327	+	0.769766i	$0.720373\pi$
$314$	0	0
$315$	−9.86760	−0.555976
$316$	0	0
$317$	12.3565	0.694012	0.347006	−	0.937863i	$-0.387198\pi$
$317$	12.3565	0.694012	0.347006	+	0.937863i	$0.387198\pi$
$318$	0	0
$319$	−2.68950	−0.150583
$320$	0	0
$321$	10.3986	0.580393
$322$	0	0
$323$	12.0660	0.671373
$324$	0	0
$325$	9.03816	0.501347
$326$	0	0
$327$	19.7676	1.09315
$328$	0	0
$329$	−21.6769	−1.19509
$330$	0	0
$331$	28.5673	1.57020	0.785102	−	0.619367i	$-0.212611\pi$
$331$	28.5673	1.57020	0.785102	+	0.619367i	$0.212611\pi$
$332$	0	0
$333$	−45.9589	−2.51853
$334$	0	0
$335$	3.19501	0.174562
$336$	0	0
$337$	1.38181	0.0752722	0.0376361	−	0.999292i	$-0.488017\pi$
$337$	1.38181	0.0752722	0.0376361	+	0.999292i	$0.488017\pi$
$338$	0	0
$339$	42.0016	2.28121
$340$	0	0
$341$	17.3022	0.936966
$342$	0	0
$343$	−19.5498	−1.05559
$344$	0	0
$345$	−12.3537	−0.665099
$346$	0	0
$347$	−0.356068	−0.0191147	−0.00955737	−	0.999954i	$-0.503042\pi$
$347$	−0.356068	−0.0191147	−0.00955737	+	0.999954i	$0.503042\pi$
$348$	0	0
$349$	−9.24477	−0.494861	−0.247431	−	0.968906i	$-0.579586\pi$
$349$	−9.24477	−0.494861	−0.247431	+	0.968906i	$0.579586\pi$
$350$	0	0
$351$	21.8367	1.16556
$352$	0	0
$353$	4.09347	0.217874	0.108937	−	0.994049i	$-0.465255\pi$
$353$	4.09347	0.217874	0.108937	+	0.994049i	$0.465255\pi$
$354$	0	0
$355$	−2.55946	−0.135842
$356$	0	0
$357$	−34.7034	−1.83670
$358$	0	0
$359$	−14.6676	−0.774126	−0.387063	−	0.922053i	$-0.626510\pi$
$359$	−14.6676	−0.774126	−0.387063	+	0.922053i	$0.626510\pi$
$360$	0	0
$361$	−15.1357	−0.796615
$362$	0	0
$363$	23.2519	1.22041
$364$	0	0
$365$	−2.20583	−0.115458
$366$	0	0
$367$	−34.9339	−1.82354	−0.911768	−	0.410705i	$-0.865283\pi$
$367$	−34.9339	−1.82354	−0.911768	+	0.410705i	$0.865283\pi$
$368$	0	0
$369$	57.1456	2.97488
$370$	0	0
$371$	−4.75093	−0.246656
$372$	0	0
$373$	−12.5425	−0.649427	−0.324714	−	0.945812i	$-0.605268\pi$
$373$	−12.5425	−0.649427	−0.324714	+	0.945812i	$0.605268\pi$
$374$	0	0
$375$	−23.8503	−1.23163
$376$	0	0
$377$	−3.06075	−0.157637
$378$	0	0
$379$	−33.5625	−1.72399	−0.861996	−	0.506915i	$-0.830786\pi$
$379$	−33.5625	−1.72399	−0.861996	+	0.506915i	$0.830786\pi$
$380$	0	0
$381$	−2.72417	−0.139563
$382$	0	0
$383$	34.2413	1.74965	0.874824	−	0.484440i	$-0.160977\pi$
$383$	34.2413	1.74965	0.874824	+	0.484440i	$0.160977\pi$
$384$	0	0
$385$	2.85246	0.145375
$386$	0	0
$387$	−57.2896	−2.91219
$388$	0	0
$389$	22.5209	1.14186	0.570928	−	0.821000i	$-0.306583\pi$
$389$	22.5209	1.14186	0.570928	+	0.821000i	$0.306583\pi$
$390$	0	0
$391$	−29.5659	−1.49521
$392$	0	0
$393$	27.3501	1.37963
$394$	0	0
$395$	−4.17305	−0.209969
$396$	0	0
$397$	−1.99396	−0.100074	−0.0500369	−	0.998747i	$-0.515934\pi$
$397$	−1.99396	−0.100074	−0.0500369	+	0.998747i	$0.515934\pi$
$398$	0	0
$399$	−11.1142	−0.556406
$400$	0	0
$401$	12.6796	0.633187	0.316594	−	0.948561i	$-0.397461\pi$
$401$	12.6796	0.633187	0.316594	+	0.948561i	$0.397461\pi$
$402$	0	0
$403$	19.6905	0.980855
$404$	0	0
$405$	−10.5973	−0.526582
$406$	0	0
$407$	13.2855	0.658537
$408$	0	0
$409$	29.8391	1.47545	0.737724	−	0.675102i	$-0.235901\pi$
$409$	29.8391	1.47545	0.737724	+	0.675102i	$0.235901\pi$
$410$	0	0
$411$	9.13368	0.450531
$412$	0	0
$413$	1.11364	0.0547988
$414$	0	0
$415$	−5.79491	−0.284461
$416$	0	0
$417$	36.0123	1.76353
$418$	0	0
$419$	−15.9334	−0.778396	−0.389198	−	0.921154i	$-0.627248\pi$
$419$	−15.9334	−0.778396	−0.389198	+	0.921154i	$0.627248\pi$
$420$	0	0
$421$	−1.58627	−0.0773103	−0.0386551	−	0.999253i	$-0.512307\pi$
$421$	−1.58627	−0.0773103	−0.0386551	+	0.999253i	$0.512307\pi$
$422$	0	0
$423$	−75.0732	−3.65018
$424$	0	0
$425$	−26.3905	−1.28013
$426$	0	0
$427$	−12.7287	−0.615984
$428$	0	0
$429$	−11.8987	−0.574474
$430$	0	0
$431$	28.2245	1.35952	0.679762	−	0.733433i	$-0.262083\pi$
$431$	28.2245	1.35952	0.679762	+	0.733433i	$0.262083\pi$
$432$	0	0
$433$	36.7233	1.76481	0.882405	−	0.470491i	$-0.155923\pi$
$433$	36.7233	1.76481	0.882405	+	0.470491i	$0.155923\pi$
$434$	0	0
$435$	3.73423	0.179042
$436$	0	0
$437$	−9.46885	−0.452956
$438$	0	0
$439$	29.5279	1.40929	0.704645	−	0.709560i	$-0.251106\pi$
$439$	29.5279	1.40929	0.704645	+	0.709560i	$0.251106\pi$
$440$	0	0
$441$	−22.9767	−1.09413
$442$	0	0
$443$	21.4929	1.02116	0.510580	−	0.859830i	$-0.329431\pi$
$443$	21.4929	1.02116	0.510580	+	0.859830i	$0.329431\pi$
$444$	0	0
$445$	−10.0483	−0.476335
$446$	0	0
$447$	−29.3577	−1.38857
$448$	0	0
$449$	3.56001	0.168007	0.0840037	−	0.996465i	$-0.473229\pi$
$449$	3.56001	0.168007	0.0840037	+	0.996465i	$0.473229\pi$
$450$	0	0
$451$	−16.5192	−0.777861
$452$	0	0
$453$	42.4009	1.99217
$454$	0	0
$455$	3.24620	0.152184
$456$	0	0
$457$	11.5855	0.541949	0.270974	−	0.962587i	$-0.412654\pi$
$457$	11.5855	0.541949	0.270974	+	0.962587i	$0.412654\pi$
$458$	0	0
$459$	−63.7610	−2.97611
$460$	0	0
$461$	4.35982	0.203057	0.101529	−	0.994833i	$-0.467627\pi$
$461$	4.35982	0.203057	0.101529	+	0.994833i	$0.467627\pi$
$462$	0	0
$463$	−7.62484	−0.354356	−0.177178	−	0.984179i	$-0.556697\pi$
$463$	−7.62484	−0.354356	−0.177178	+	0.984179i	$0.556697\pi$
$464$	0	0
$465$	−24.0232	−1.11405
$466$	0	0
$467$	15.0977	0.698637	0.349318	−	0.937004i	$-0.386413\pi$
$467$	15.0977	0.698637	0.349318	+	0.937004i	$0.386413\pi$
$468$	0	0
$469$	−7.04338	−0.325233
$470$	0	0
$471$	31.5130	1.45204
$472$	0	0
$473$	16.5609	0.761469
$474$	0	0
$475$	−8.45190	−0.387800
$476$	0	0
$477$	−16.4538	−0.753367
$478$	0	0
$479$	−17.5835	−0.803409	−0.401705	−	0.915769i	$-0.631582\pi$
$479$	−17.5835	−0.803409	−0.401705	+	0.915769i	$0.631582\pi$
$480$	0	0
$481$	15.1194	0.689384
$482$	0	0
$483$	27.2335	1.23917
$484$	0	0
$485$	−6.18353	−0.280780
$486$	0	0
$487$	−34.5966	−1.56772	−0.783861	−	0.620936i	$-0.786753\pi$
$487$	−34.5966	−1.56772	−0.783861	+	0.620936i	$0.786753\pi$
$488$	0	0
$489$	−9.96289	−0.450537
$490$	0	0
$491$	−16.0847	−0.725892	−0.362946	−	0.931810i	$-0.618229\pi$
$491$	−16.0847	−0.725892	−0.362946	+	0.931810i	$0.618229\pi$
$492$	0	0
$493$	8.93708	0.402506
$494$	0	0
$495$	9.87884	0.444021
$496$	0	0
$497$	5.64231	0.253092
$498$	0	0
$499$	−35.9801	−1.61069	−0.805344	−	0.592807i	$-0.798020\pi$
$499$	−35.9801	−1.61069	−0.805344	+	0.592807i	$0.798020\pi$
$500$	0	0
$501$	−3.06430	−0.136903
$502$	0	0
$503$	−5.67768	−0.253155	−0.126578	−	0.991957i	$-0.540399\pi$
$503$	−5.67768	−0.253155	−0.126578	+	0.991957i	$0.540399\pi$
$504$	0	0
$505$	4.99796	0.222406
$506$	0	0
$507$	26.2948	1.16779
$508$	0	0
$509$	−38.6178	−1.71170	−0.855851	−	0.517222i	$-0.826966\pi$
$509$	−38.6178	−1.71170	−0.855851	+	0.517222i	$0.826966\pi$
$510$	0	0
$511$	4.86273	0.215114
$512$	0	0
$513$	−20.4203	−0.901577
$514$	0	0
$515$	−3.62404	−0.159694
$516$	0	0
$517$	21.7016	0.954436
$518$	0	0
$519$	63.9494	2.80707
$520$	0	0
$521$	1.50918	0.0661185	0.0330593	−	0.999453i	$-0.489475\pi$
$521$	1.50918	0.0661185	0.0330593	+	0.999453i	$0.489475\pi$
$522$	0	0
$523$	33.8705	1.48106	0.740528	−	0.672026i	$-0.234576\pi$
$523$	33.8705	1.48106	0.740528	+	0.672026i	$0.234576\pi$
$524$	0	0
$525$	24.3087	1.06092
$526$	0	0
$527$	−57.4943	−2.50449
$528$	0	0
$529$	0.201851	0.00877615
$530$	0	0
$531$	3.85685	0.167373
$532$	0	0
$533$	−18.7995	−0.814297
$534$	0	0
$535$	2.84018	0.122792
$536$	0	0
$537$	−55.5367	−2.39658
$538$	0	0
$539$	6.64193	0.286088
$540$	0	0
$541$	−40.9806	−1.76189	−0.880946	−	0.473217i	$-0.843093\pi$
$541$	−40.9806	−1.76189	−0.880946	+	0.473217i	$0.843093\pi$
$542$	0	0
$543$	−63.2219	−2.71311
$544$	0	0
$545$	5.39915	0.231274
$546$	0	0
$547$	−0.964746	−0.0412495	−0.0206248	−	0.999787i	$-0.506566\pi$
$547$	−0.964746	−0.0412495	−0.0206248	+	0.999787i	$0.506566\pi$
$548$	0	0
$549$	−44.0829	−1.88141
$550$	0	0
$551$	2.86221	0.121934
$552$	0	0
$553$	9.19944	0.391200
$554$	0	0
$555$	−18.4462	−0.782997
$556$	0	0
$557$	22.2866	0.944315	0.472158	−	0.881514i	$-0.343475\pi$
$557$	22.2866	0.944315	0.472158	+	0.881514i	$0.343475\pi$
$558$	0	0
$559$	18.8469	0.797138
$560$	0	0
$561$	34.7429	1.46685
$562$	0	0
$563$	24.2578	1.02234	0.511172	−	0.859479i	$-0.329212\pi$
$563$	24.2578	1.02234	0.511172	+	0.859479i	$0.329212\pi$
$564$	0	0
$565$	11.4719	0.482628
$566$	0	0
$567$	23.3616	0.981094
$568$	0	0
$569$	−42.8159	−1.79494	−0.897469	−	0.441078i	$-0.854596\pi$
$569$	−42.8159	−1.79494	−0.897469	+	0.441078i	$0.854596\pi$
$570$	0	0
$571$	9.20950	0.385405	0.192703	−	0.981257i	$-0.438275\pi$
$571$	9.20950	0.385405	0.192703	+	0.981257i	$0.438275\pi$
$572$	0	0
$573$	53.6509	2.24130
$574$	0	0
$575$	20.7100	0.863666
$576$	0	0
$577$	−23.1471	−0.963628	−0.481814	−	0.876274i	$-0.660022\pi$
$577$	−23.1471	−0.963628	−0.481814	+	0.876274i	$0.660022\pi$
$578$	0	0
$579$	73.3831	3.04970
$580$	0	0
$581$	12.7748	0.529989
$582$	0	0
$583$	4.75634	0.196988
$584$	0	0
$585$	11.2425	0.464819
$586$	0	0
$587$	19.2365	0.793975	0.396988	−	0.917824i	$-0.370056\pi$
$587$	19.2365	0.793975	0.396988	+	0.917824i	$0.370056\pi$
$588$	0	0
$589$	−18.4133	−0.758707
$590$	0	0
$591$	11.3973	0.468822
$592$	0	0
$593$	41.8328	1.71787	0.858933	−	0.512088i	$-0.171128\pi$
$593$	41.8328	1.71787	0.858933	+	0.512088i	$0.171128\pi$
$594$	0	0
$595$	−9.47858	−0.388584
$596$	0	0
$597$	−13.0019	−0.532131
$598$	0	0
$599$	28.8763	1.17985	0.589927	−	0.807456i	$-0.299156\pi$
$599$	28.8763	1.17985	0.589927	+	0.807456i	$0.299156\pi$
$600$	0	0
$601$	−4.64122	−0.189319	−0.0946597	−	0.995510i	$-0.530176\pi$
$601$	−4.64122	−0.189319	−0.0946597	+	0.995510i	$0.530176\pi$
$602$	0	0
$603$	−24.3932	−0.993366
$604$	0	0
$605$	6.35081	0.258197
$606$	0	0
$607$	38.5844	1.56609	0.783047	−	0.621962i	$-0.213664\pi$
$607$	38.5844	1.56609	0.783047	+	0.621962i	$0.213664\pi$
$608$	0	0
$609$	−8.23207	−0.333580
$610$	0	0
$611$	24.6972	0.999143
$612$	0	0
$613$	−18.4803	−0.746413	−0.373206	−	0.927748i	$-0.621742\pi$
$613$	−18.4803	−0.746413	−0.373206	+	0.927748i	$0.621742\pi$
$614$	0	0
$615$	22.9361	0.924873
$616$	0	0
$617$	7.39109	0.297554	0.148777	−	0.988871i	$-0.452466\pi$
$617$	7.39109	0.297554	0.148777	+	0.988871i	$0.452466\pi$
$618$	0	0
$619$	27.8538	1.11954	0.559769	−	0.828649i	$-0.310890\pi$
$619$	27.8538	1.11954	0.559769	+	0.828649i	$0.310890\pi$
$620$	0	0
$621$	50.0365	2.00790
$622$	0	0
$623$	22.1514	0.887476
$624$	0	0
$625$	14.9833	0.599330
$626$	0	0
$627$	11.1269	0.444364
$628$	0	0
$629$	−44.1470	−1.76026
$630$	0	0
$631$	−0.228979	−0.00911552	−0.00455776	−	0.999990i	$-0.501451\pi$
$631$	−0.228979	−0.00911552	−0.00455776	+	0.999990i	$0.501451\pi$
$632$	0	0
$633$	−88.1277	−3.50276
$634$	0	0
$635$	−0.744054	−0.0295269
$636$	0	0
$637$	7.55876	0.299489
$638$	0	0
$639$	19.5409	0.773025
$640$	0	0
$641$	30.6516	1.21066	0.605332	−	0.795973i	$-0.293040\pi$
$641$	30.6516	1.21066	0.605332	+	0.795973i	$0.293040\pi$
$642$	0	0
$643$	13.7112	0.540718	0.270359	−	0.962760i	$-0.412858\pi$
$643$	13.7112	0.540718	0.270359	+	0.962760i	$0.412858\pi$
$644$	0	0
$645$	−22.9939	−0.905383
$646$	0	0
$647$	−41.1846	−1.61913	−0.809567	−	0.587027i	$-0.800298\pi$
$647$	−41.1846	−1.61913	−0.809567	+	0.587027i	$0.800298\pi$
$648$	0	0
$649$	−1.11491	−0.0437641
$650$	0	0
$651$	52.9589	2.07562
$652$	0	0
$653$	−3.83271	−0.149986	−0.0749928	−	0.997184i	$-0.523893\pi$
$653$	−3.83271	−0.149986	−0.0749928	+	0.997184i	$0.523893\pi$
$654$	0	0
$655$	7.47017	0.291884
$656$	0	0
$657$	16.8410	0.657029
$658$	0	0
$659$	9.60290	0.374076	0.187038	−	0.982353i	$-0.440111\pi$
$659$	9.60290	0.374076	0.187038	+	0.982353i	$0.440111\pi$
$660$	0	0
$661$	14.1757	0.551372	0.275686	−	0.961248i	$-0.411095\pi$
$661$	14.1757	0.551372	0.275686	+	0.961248i	$0.411095\pi$
$662$	0	0
$663$	39.5387	1.53556
$664$	0	0
$665$	−3.03563	−0.117717
$666$	0	0
$667$	−7.01338	−0.271559
$668$	0	0
$669$	26.7765	1.03524
$670$	0	0
$671$	12.7432	0.491945
$672$	0	0
$673$	48.3745	1.86470	0.932351	−	0.361556i	$-0.117754\pi$
$673$	48.3745	1.86470	0.932351	+	0.361556i	$0.117754\pi$
$674$	0	0
$675$	44.6626	1.71907
$676$	0	0
$677$	−1.91892	−0.0737500	−0.0368750	−	0.999320i	$-0.511740\pi$
$677$	−1.91892	−0.0737500	−0.0368750	+	0.999320i	$0.511740\pi$
$678$	0	0
$679$	13.6315	0.523130
$680$	0	0
$681$	47.6192	1.82477
$682$	0	0
$683$	−1.69206	−0.0647449	−0.0323724	−	0.999476i	$-0.510306\pi$
$683$	−1.69206	−0.0647449	−0.0323724	+	0.999476i	$0.510306\pi$
$684$	0	0
$685$	2.49469	0.0953172
$686$	0	0
$687$	−86.1868	−3.28823
$688$	0	0
$689$	5.41289	0.206215
$690$	0	0
$691$	10.4347	0.396954	0.198477	−	0.980106i	$-0.436401\pi$
$691$	10.4347	0.396954	0.198477	+	0.980106i	$0.436401\pi$
$692$	0	0
$693$	−21.7778	−0.827271
$694$	0	0
$695$	9.83607	0.373104
$696$	0	0
$697$	54.8927	2.07921
$698$	0	0
$699$	85.9241	3.24995
$700$	0	0
$701$	−14.4861	−0.547134	−0.273567	−	0.961853i	$-0.588204\pi$
$701$	−14.4861	−0.547134	−0.273567	+	0.961853i	$0.588204\pi$
$702$	0	0
$703$	−14.1386	−0.533249
$704$	0	0
$705$	−30.1315	−1.13482
$706$	0	0
$707$	−11.0180	−0.414373
$708$	0	0
$709$	32.3666	1.21555	0.607776	−	0.794108i	$-0.292062\pi$
$709$	32.3666	1.21555	0.607776	+	0.794108i	$0.292062\pi$
$710$	0	0
$711$	31.8602	1.19485
$712$	0	0
$713$	45.1188	1.68971
$714$	0	0
$715$	−3.24990	−0.121539
$716$	0	0
$717$	−55.5877	−2.07596
$718$	0	0
$719$	−11.7998	−0.440057	−0.220028	−	0.975493i	$-0.570615\pi$
$719$	−11.7998	−0.440057	−0.220028	+	0.975493i	$0.570615\pi$
$720$	0	0
$721$	7.98917	0.297532
$722$	0	0
$723$	12.6827	0.471676
$724$	0	0
$725$	−6.26015	−0.232496
$726$	0	0
$727$	−21.7915	−0.808200	−0.404100	−	0.914715i	$-0.632415\pi$
$727$	−21.7915	−0.808200	−0.404100	+	0.914715i	$0.632415\pi$
$728$	0	0
$729$	−14.5871	−0.540263
$730$	0	0
$731$	−55.0309	−2.03539
$732$	0	0
$733$	−47.1224	−1.74051	−0.870253	−	0.492604i	$-0.836045\pi$
$733$	−47.1224	−1.74051	−0.870253	+	0.492604i	$0.836045\pi$
$734$	0	0
$735$	−9.22197	−0.340157
$736$	0	0
$737$	7.05140	0.259742
$738$	0	0
$739$	5.22504	0.192206	0.0961032	−	0.995371i	$-0.469362\pi$
$739$	5.22504	0.192206	0.0961032	+	0.995371i	$0.469362\pi$
$740$	0	0
$741$	12.6628	0.465179
$742$	0	0
$743$	3.37497	0.123816	0.0619079	−	0.998082i	$-0.480281\pi$
$743$	3.37497	0.123816	0.0619079	+	0.998082i	$0.480281\pi$
$744$	0	0
$745$	−8.01848	−0.293775
$746$	0	0
$747$	44.2427	1.61876
$748$	0	0
$749$	−6.26114	−0.228777
$750$	0	0
$751$	15.9858	0.583332	0.291666	−	0.956520i	$-0.405791\pi$
$751$	15.9858	0.583332	0.291666	+	0.956520i	$0.405791\pi$
$752$	0	0
$753$	−82.7137	−3.01426
$754$	0	0
$755$	11.5810	0.421476
$756$	0	0
$757$	10.8998	0.396161	0.198080	−	0.980186i	$-0.436529\pi$
$757$	10.8998	0.396161	0.198080	+	0.980186i	$0.436529\pi$
$758$	0	0
$759$	−27.2646	−0.989641
$760$	0	0
$761$	−46.3855	−1.68147	−0.840737	−	0.541444i	$-0.817878\pi$
$761$	−46.3855	−1.68147	−0.840737	+	0.541444i	$0.817878\pi$
$762$	0	0
$763$	−11.9024	−0.430895
$764$	0	0
$765$	−32.8269	−1.18686
$766$	0	0
$767$	−1.26881	−0.0458141
$768$	0	0
$769$	−41.7936	−1.50712	−0.753558	−	0.657382i	$-0.771664\pi$
$769$	−41.7936	−1.50712	−0.753558	+	0.657382i	$0.771664\pi$
$770$	0	0
$771$	−69.3383	−2.49716
$772$	0	0
$773$	−26.3536	−0.947873	−0.473937	−	0.880559i	$-0.657167\pi$
$773$	−26.3536	−0.947873	−0.473937	+	0.880559i	$0.657167\pi$
$774$	0	0
$775$	40.2730	1.44665
$776$	0	0
$777$	40.6644	1.45883
$778$	0	0
$779$	17.5801	0.629872
$780$	0	0
$781$	−5.64874	−0.202128
$782$	0	0
$783$	−15.1249	−0.540519
$784$	0	0
$785$	8.60717	0.307203
$786$	0	0
$787$	2.97833	0.106166	0.0530830	−	0.998590i	$-0.483095\pi$
$787$	2.97833	0.106166	0.0530830	+	0.998590i	$0.483095\pi$
$788$	0	0
$789$	−22.6527	−0.806457
$790$	0	0
$791$	−25.2898	−0.899201
$792$	0	0
$793$	14.5022	0.514988
$794$	0	0
$795$	−6.60393	−0.234217
$796$	0	0
$797$	46.3008	1.64006	0.820029	−	0.572322i	$-0.193957\pi$
$797$	46.3008	1.64006	0.820029	+	0.572322i	$0.193957\pi$
$798$	0	0
$799$	−72.1134	−2.55119
$800$	0	0
$801$	76.7163	2.71064
$802$	0	0
$803$	−4.86827	−0.171797
$804$	0	0
$805$	7.43832	0.262166
$806$	0	0
$807$	71.1700	2.50530
$808$	0	0
$809$	−5.58116	−0.196223	−0.0981116	−	0.995175i	$-0.531280\pi$
$809$	−5.58116	−0.196223	−0.0981116	+	0.995175i	$0.531280\pi$
$810$	0	0
$811$	−43.7584	−1.53657	−0.768283	−	0.640111i	$-0.778889\pi$
$811$	−43.7584	−1.53657	−0.768283	+	0.640111i	$0.778889\pi$
$812$	0	0
$813$	−72.6853	−2.54919
$814$	0	0
$815$	−2.72117	−0.0953185
$816$	0	0
$817$	−17.6244	−0.616598
$818$	0	0
$819$	−24.7839	−0.866021
$820$	0	0
$821$	23.8411	0.832059	0.416029	−	0.909351i	$-0.363421\pi$
$821$	23.8411	0.832059	0.416029	+	0.909351i	$0.363421\pi$
$822$	0	0
$823$	5.69795	0.198618	0.0993090	−	0.995057i	$-0.468337\pi$
$823$	5.69795	0.198618	0.0993090	+	0.995057i	$0.468337\pi$
$824$	0	0
$825$	−24.3364	−0.847283
$826$	0	0
$827$	41.3231	1.43695	0.718473	−	0.695555i	$-0.244842\pi$
$827$	41.3231	1.43695	0.718473	+	0.695555i	$0.244842\pi$
$828$	0	0
$829$	−34.6916	−1.20489	−0.602444	−	0.798161i	$-0.705806\pi$
$829$	−34.6916	−1.20489	−0.602444	+	0.798161i	$0.705806\pi$
$830$	0	0
$831$	−0.813827	−0.0282313
$832$	0	0
$833$	−22.0708	−0.764708
$834$	0	0
$835$	−0.836956	−0.0289641
$836$	0	0
$837$	97.3020	3.36325
$838$	0	0
$839$	−43.4533	−1.50017	−0.750087	−	0.661339i	$-0.769988\pi$
$839$	−43.4533	−1.50017	−0.750087	+	0.661339i	$0.769988\pi$
$840$	0	0
$841$	−26.8800	−0.926897
$842$	0	0
$843$	25.9225	0.892817
$844$	0	0
$845$	7.18193	0.247066
$846$	0	0
$847$	−14.0003	−0.481056
$848$	0	0
$849$	−0.865894	−0.0297174
$850$	0	0
$851$	34.6444	1.18760
$852$	0	0
$853$	−23.5987	−0.808006	−0.404003	−	0.914758i	$-0.632381\pi$
$853$	−23.5987	−0.808006	−0.404003	+	0.914758i	$0.632381\pi$
$854$	0	0
$855$	−10.5132	−0.359545
$856$	0	0
$857$	3.31375	0.113196	0.0565978	−	0.998397i	$-0.481975\pi$
$857$	3.31375	0.113196	0.0565978	+	0.998397i	$0.481975\pi$
$858$	0	0
$859$	35.7038	1.21820	0.609098	−	0.793095i	$-0.291532\pi$
$859$	35.7038	1.21820	0.609098	+	0.793095i	$0.291532\pi$
$860$	0	0
$861$	−50.5624	−1.72316
$862$	0	0
$863$	−28.8565	−0.982286	−0.491143	−	0.871079i	$-0.663421\pi$
$863$	−28.8565	−0.982286	−0.491143	+	0.871079i	$0.663421\pi$
$864$	0	0
$865$	17.4666	0.593881
$866$	0	0
$867$	−63.3559	−2.15168
$868$	0	0
$869$	−9.20992	−0.312425
$870$	0	0
$871$	8.02475	0.271908
$872$	0	0
$873$	47.2097	1.59781
$874$	0	0
$875$	14.3606	0.485478
$876$	0	0
$877$	13.7350	0.463800	0.231900	−	0.972740i	$-0.425506\pi$
$877$	13.7350	0.463800	0.231900	+	0.972740i	$0.425506\pi$
$878$	0	0
$879$	−27.1725	−0.916504
$880$	0	0
$881$	50.1991	1.69125	0.845626	−	0.533776i	$-0.179227\pi$
$881$	50.1991	1.69125	0.845626	+	0.533776i	$0.179227\pi$
$882$	0	0
$883$	−26.0039	−0.875099	−0.437550	−	0.899194i	$-0.644154\pi$
$883$	−26.0039	−0.875099	−0.437550	+	0.899194i	$0.644154\pi$
$884$	0	0
$885$	1.54800	0.0520353
$886$	0	0
$887$	−27.6749	−0.929232	−0.464616	−	0.885512i	$-0.653808\pi$
$887$	−27.6749	−0.929232	−0.464616	+	0.885512i	$0.653808\pi$
$888$	0	0
$889$	1.64026	0.0550126
$890$	0	0
$891$	−23.3882	−0.783533
$892$	0	0
$893$	−23.0952	−0.772853
$894$	0	0
$895$	−15.1688	−0.507037
$896$	0	0
$897$	−31.0281	−1.03600
$898$	0	0
$899$	−13.6384	−0.454865
$900$	0	0
$901$	−15.8051	−0.526544
$902$	0	0
$903$	50.6898	1.68685
$904$	0	0
$905$	−17.2679	−0.574003
$906$	0	0
$907$	19.1826	0.636949	0.318474	−	0.947931i	$-0.396830\pi$
$907$	19.1826	0.636949	0.318474	+	0.947931i	$0.396830\pi$
$908$	0	0
$909$	−38.1582	−1.26563
$910$	0	0
$911$	−45.4449	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$911$	−45.4449	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$912$	0	0
$913$	−12.7894	−0.423266
$914$	0	0
$915$	−17.6932	−0.584920
$916$	0	0
$917$	−16.4679	−0.543819
$918$	0	0
$919$	40.1393	1.32407	0.662036	−	0.749472i	$-0.269693\pi$
$919$	40.1393	1.32407	0.662036	+	0.749472i	$0.269693\pi$
$920$	0	0
$921$	27.7862	0.915586
$922$	0	0
$923$	−6.42847	−0.211596
$924$	0	0
$925$	30.9236	1.01676
$926$	0	0
$927$	27.6687	0.908759
$928$	0	0
$929$	−49.4565	−1.62262	−0.811308	−	0.584620i	$-0.801244\pi$
$929$	−49.4565	−1.62262	−0.811308	+	0.584620i	$0.801244\pi$
$930$	0	0
$931$	−7.06846	−0.231659
$932$	0	0
$933$	−27.0967	−0.887106
$934$	0	0
$935$	9.48937	0.310336
$936$	0	0
$937$	26.7647	0.874364	0.437182	−	0.899373i	$-0.355977\pi$
$937$	26.7647	0.874364	0.437182	+	0.899373i	$0.355977\pi$
$938$	0	0
$939$	69.2113	2.25862
$940$	0	0
$941$	−50.6621	−1.65154	−0.825769	−	0.564009i	$-0.809258\pi$
$941$	−50.6621	−1.65154	−0.825769	+	0.564009i	$0.809258\pi$
$942$	0	0
$943$	−43.0771	−1.40278
$944$	0	0
$945$	16.0413	0.521824
$946$	0	0
$947$	0.300235	0.00975631	0.00487816	−	0.999988i	$-0.498447\pi$
$947$	0.300235	0.00975631	0.00487816	+	0.999988i	$0.498447\pi$
$948$	0	0
$949$	−5.54027	−0.179845
$950$	0	0
$951$	−37.8642	−1.22783
$952$	0	0
$953$	24.2712	0.786220	0.393110	−	0.919491i	$-0.371399\pi$
$953$	24.2712	0.786220	0.393110	+	0.919491i	$0.371399\pi$
$954$	0	0
$955$	14.6537	0.474183
$956$	0	0
$957$	8.24145	0.266408
$958$	0	0
$959$	−5.49952	−0.177589
$960$	0	0
$961$	56.7388	1.83028
$962$	0	0
$963$	−21.6841	−0.698759
$964$	0	0
$965$	20.0432	0.645213
$966$	0	0
$967$	19.4799	0.626431	0.313215	−	0.949682i	$-0.398594\pi$
$967$	19.4799	0.626431	0.313215	+	0.949682i	$0.398594\pi$
$968$	0	0
$969$	−36.9740	−1.18778
$970$	0	0
$971$	38.9393	1.24962	0.624811	−	0.780776i	$-0.285176\pi$
$971$	38.9393	1.24962	0.624811	+	0.780776i	$0.285176\pi$
$972$	0	0
$973$	−21.6835	−0.695142
$974$	0	0
$975$	−27.6957	−0.886971
$976$	0	0
$977$	−39.0038	−1.24784	−0.623921	−	0.781487i	$-0.714461\pi$
$977$	−39.0038	−1.24784	−0.623921	+	0.781487i	$0.714461\pi$
$978$	0	0
$979$	−22.1766	−0.708768
$980$	0	0
$981$	−41.2212	−1.31609
$982$	0	0
$983$	−0.977616	−0.0311811	−0.0155906	−	0.999878i	$-0.504963\pi$
$983$	−0.977616	−0.0311811	−0.0155906	+	0.999878i	$0.504963\pi$
$984$	0	0
$985$	3.11295	0.0991869
$986$	0	0
$987$	66.4247	2.11432
$988$	0	0
$989$	43.1856	1.37322
$990$	0	0
$991$	−38.9190	−1.23630	−0.618151	−	0.786059i	$-0.712118\pi$
$991$	−38.9190	−1.23630	−0.618151	+	0.786059i	$0.712118\pi$
$992$	0	0
$993$	−87.5390	−2.77797
$994$	0	0
$995$	−3.55121	−0.112581
$996$	0	0
$997$	−30.1836	−0.955926	−0.477963	−	0.878380i	$-0.658625\pi$
$997$	−30.1836	−0.955926	−0.477963	+	0.878380i	$0.658625\pi$
$998$	0	0
$999$	74.7133	2.36382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.c.1.1	✓	7
3.2	odd	2		6012.2.a.g.1.4		7
4.3	odd	2		2672.2.a.k.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.1	✓	7	1.1	even	1	trivial
2672.2.a.k.1.7		7	4.3	odd	2
6012.2.a.g.1.4		7	3.2	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:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.a (trivial)

\(f(q)\)	\(=\)	\(q-3.06430 q^{3} -0.836956 q^{5} +1.84506 q^{7} +6.38996 q^{9} +O(q^{10})\)	\(q-3.06430 q^{3} -0.836956 q^{5} +1.84506 q^{7} +6.38996 q^{9} -1.84716 q^{11} -2.10214 q^{13} +2.56469 q^{15} +6.13803 q^{17} +1.96578 q^{19} -5.65383 q^{21} -4.81683 q^{23} -4.29950 q^{25} -10.3879 q^{27} +1.45602 q^{29} -9.36690 q^{31} +5.66027 q^{33} -1.54424 q^{35} -7.19237 q^{37} +6.44159 q^{39} +8.94304 q^{41} -8.96556 q^{43} -5.34811 q^{45} -11.7486 q^{47} -3.59575 q^{49} -18.8088 q^{51} -2.57494 q^{53} +1.54599 q^{55} -6.02376 q^{57} +0.603581 q^{59} -6.89878 q^{61} +11.7899 q^{63} +1.75940 q^{65} -3.81742 q^{67} +14.7602 q^{69} +3.05806 q^{71} +2.63554 q^{73} +13.1750 q^{75} -3.40813 q^{77} +4.98598 q^{79} +12.6617 q^{81} +6.92379 q^{83} -5.13727 q^{85} -4.46168 q^{87} +12.0058 q^{89} -3.87858 q^{91} +28.7030 q^{93} -1.64528 q^{95} +7.38811 q^{97} -11.8033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * q^99

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