LMFDB - Embedded newform 1668.2.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1668,2,Mod(1,1668)] 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("1668.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1668, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1668 = 2^{2} \cdot 3 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1668.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,7,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.3190470571$
Analytic rank:	$0$
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} - 28x^{5} - x^{4} + 217x^{3} + 51x^{2} - 456x - 300 $ x^7 - 28x^5 - x^4 + 217x^3 + 51x^2 - 456x - 300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11277$ of defining polynomial
Character	$\chi$	$=$	1668.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.11277 q^{5} +1.19918 q^{7} +1.00000 q^{9} +1.19006 q^{11} -1.11595 q^{13} -2.11277 q^{15} -0.502011 q^{17} +5.69468 q^{19} +1.19918 q^{21} +1.07025 q^{23} -0.536209 q^{25} +1.00000 q^{27} -0.465962 q^{29} +2.84599 q^{31} +1.19006 q^{33} -2.53360 q^{35} +8.89704 q^{37} -1.11595 q^{39} +6.00455 q^{41} +4.86788 q^{43} -2.11277 q^{45} +4.97115 q^{47} -5.56196 q^{49} -0.502011 q^{51} -13.2689 q^{53} -2.51432 q^{55} +5.69468 q^{57} +6.82158 q^{59} +1.95691 q^{61} +1.19918 q^{63} +2.35774 q^{65} +15.2513 q^{67} +1.07025 q^{69} +3.36027 q^{71} +9.76731 q^{73} -0.536209 q^{75} +1.42710 q^{77} +0.260986 q^{79} +1.00000 q^{81} -10.7862 q^{83} +1.06063 q^{85} -0.465962 q^{87} +0.709412 q^{89} -1.33823 q^{91} +2.84599 q^{93} -12.0315 q^{95} +3.11334 q^{97} +1.19006 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + q^{7} + 7 q^{9} + 8 q^{13} + 27 q^{17} + q^{21} - 8 q^{23} + 21 q^{25} + 7 q^{27} + 6 q^{29} + 7 q^{31} + 6 q^{35} + 14 q^{37} + 8 q^{39} + q^{41} + 27 q^{43} - 6 q^{47} + 32 q^{49} + 27 q^{51}+ \cdots + 7 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 + q^7 + 7 * q^9 + 8 * q^13 + 27 * q^17 + q^21 - 8 * q^23 + 21 * q^25 + 7 * q^27 + 6 * q^29 + 7 * q^31 + 6 * q^35 + 14 * q^37 + 8 * q^39 + q^41 + 27 * q^43 - 6 * q^47 + 32 * q^49 + 27 * q^51 + 18 * q^53 + 13 * q^55 + 17 * q^59 + 13 * q^61 + q^63 - 19 * q^65 + 31 * q^67 - 8 * q^69 + 6 * q^73 + 21 * q^75 - 34 * q^77 + 2 * q^79 + 7 * q^81 - 2 * q^83 + 37 * q^85 + 6 * q^87 + 13 * q^89 + 39 * q^91 + 7 * q^93 - 36 * q^95 + 7 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.11277	−0.944859	−0.472429	−	0.881369i	$-0.656623\pi$
$5$	−2.11277	−0.944859	−0.472429	+	0.881369i	$0.656623\pi$
$6$	0	0
$7$	1.19918	0.453249	0.226624	−	0.973982i	$-0.427231\pi$
$7$	1.19918	0.453249	0.226624	+	0.973982i	$0.427231\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.19006	0.358816	0.179408	−	0.983775i	$-0.442582\pi$
$11$	1.19006	0.358816	0.179408	+	0.983775i	$0.442582\pi$
$12$	0	0
$13$	−1.11595	−0.309509	−0.154754	−	0.987953i	$-0.549459\pi$
$13$	−1.11595	−0.309509	−0.154754	+	0.987953i	$0.549459\pi$
$14$	0	0
$15$	−2.11277	−0.545514
$16$	0	0
$17$	−0.502011	−0.121755	−0.0608777	−	0.998145i	$-0.519390\pi$
$17$	−0.502011	−0.121755	−0.0608777	+	0.998145i	$0.519390\pi$
$18$	0	0
$19$	5.69468	1.30645	0.653225	−	0.757164i	$-0.273416\pi$
$19$	5.69468	1.30645	0.653225	+	0.757164i	$0.273416\pi$
$20$	0	0
$21$	1.19918	0.261683
$22$	0	0
$23$	1.07025	0.223162	0.111581	−	0.993755i	$-0.464409\pi$
$23$	1.07025	0.223162	0.111581	+	0.993755i	$0.464409\pi$
$24$	0	0
$25$	−0.536209	−0.107242
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.465962	−0.0865270	−0.0432635	−	0.999064i	$-0.513775\pi$
$29$	−0.465962	−0.0865270	−0.0432635	+	0.999064i	$0.513775\pi$
$30$	0	0
$31$	2.84599	0.511155	0.255577	−	0.966789i	$-0.417734\pi$
$31$	2.84599	0.511155	0.255577	+	0.966789i	$0.417734\pi$
$32$	0	0
$33$	1.19006	0.207163
$34$	0	0
$35$	−2.53360	−0.428256
$36$	0	0
$37$	8.89704	1.46267	0.731333	−	0.682021i	$-0.238899\pi$
$37$	8.89704	1.46267	0.731333	+	0.682021i	$0.238899\pi$
$38$	0	0
$39$	−1.11595	−0.178695
$40$	0	0
$41$	6.00455	0.937753	0.468877	−	0.883264i	$-0.344659\pi$
$41$	6.00455	0.937753	0.468877	+	0.883264i	$0.344659\pi$
$42$	0	0
$43$	4.86788	0.742345	0.371173	−	0.928564i	$-0.378956\pi$
$43$	4.86788	0.742345	0.371173	+	0.928564i	$0.378956\pi$
$44$	0	0
$45$	−2.11277	−0.314953
$46$	0	0
$47$	4.97115	0.725117	0.362559	−	0.931961i	$-0.381903\pi$
$47$	4.97115	0.725117	0.362559	+	0.931961i	$0.381903\pi$
$48$	0	0
$49$	−5.56196	−0.794565
$50$	0	0
$51$	−0.502011	−0.0702956
$52$	0	0
$53$	−13.2689	−1.82263	−0.911315	−	0.411710i	$-0.864932\pi$
$53$	−13.2689	−1.82263	−0.911315	+	0.411710i	$0.864932\pi$
$54$	0	0
$55$	−2.51432	−0.339030
$56$	0	0
$57$	5.69468	0.754279
$58$	0	0
$59$	6.82158	0.888093	0.444047	−	0.896004i	$-0.353542\pi$
$59$	6.82158	0.888093	0.444047	+	0.896004i	$0.353542\pi$
$60$	0	0
$61$	1.95691	0.250557	0.125278	−	0.992122i	$-0.460018\pi$
$61$	1.95691	0.250557	0.125278	+	0.992122i	$0.460018\pi$
$62$	0	0
$63$	1.19918	0.151083
$64$	0	0
$65$	2.35774	0.292442
$66$	0	0
$67$	15.2513	1.86324	0.931621	−	0.363433i	$-0.118395\pi$
$67$	15.2513	1.86324	0.931621	+	0.363433i	$0.118395\pi$
$68$	0	0
$69$	1.07025	0.128843
$70$	0	0
$71$	3.36027	0.398790	0.199395	−	0.979919i	$-0.436102\pi$
$71$	3.36027	0.398790	0.199395	+	0.979919i	$0.436102\pi$
$72$	0	0
$73$	9.76731	1.14318	0.571588	−	0.820541i	$-0.306327\pi$
$73$	9.76731	1.14318	0.571588	+	0.820541i	$0.306327\pi$
$74$	0	0
$75$	−0.536209	−0.0619161
$76$	0	0
$77$	1.42710	0.162633
$78$	0	0
$79$	0.260986	0.0293632	0.0146816	−	0.999892i	$-0.495327\pi$
$79$	0.260986	0.0293632	0.0146816	+	0.999892i	$0.495327\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.7862	−1.18394	−0.591970	−	0.805960i	$-0.701650\pi$
$83$	−10.7862	−1.18394	−0.591970	+	0.805960i	$0.701650\pi$
$84$	0	0
$85$	1.06063	0.115042
$86$	0	0
$87$	−0.465962	−0.0499564
$88$	0	0
$89$	0.709412	0.0751975	0.0375987	−	0.999293i	$-0.488029\pi$
$89$	0.709412	0.0751975	0.0375987	+	0.999293i	$0.488029\pi$
$90$	0	0
$91$	−1.33823	−0.140284
$92$	0	0
$93$	2.84599	0.295115
$94$	0	0
$95$	−12.0315	−1.23441
$96$	0	0
$97$	3.11334	0.316112	0.158056	−	0.987430i	$-0.449477\pi$
$97$	3.11334	0.316112	0.158056	+	0.987430i	$0.449477\pi$
$98$	0	0
$99$	1.19006	0.119605
$100$	0	0
$101$	−7.15598	−0.712046	−0.356023	−	0.934477i	$-0.615868\pi$
$101$	−7.15598	−0.712046	−0.356023	+	0.934477i	$0.615868\pi$
$102$	0	0
$103$	4.89306	0.482128	0.241064	−	0.970509i	$-0.422504\pi$
$103$	4.89306	0.482128	0.241064	+	0.970509i	$0.422504\pi$
$104$	0	0
$105$	−2.53360	−0.247254
$106$	0	0
$107$	0.314564	0.0304100	0.0152050	−	0.999884i	$-0.495160\pi$
$107$	0.314564	0.0304100	0.0152050	+	0.999884i	$0.495160\pi$
$108$	0	0
$109$	15.2321	1.45897	0.729485	−	0.683997i	$-0.239760\pi$
$109$	15.2321	1.45897	0.729485	+	0.683997i	$0.239760\pi$
$110$	0	0
$111$	8.89704	0.844470
$112$	0	0
$113$	18.5556	1.74556	0.872782	−	0.488110i	$-0.162313\pi$
$113$	18.5556	1.74556	0.872782	+	0.488110i	$0.162313\pi$
$114$	0	0
$115$	−2.26118	−0.210857
$116$	0	0
$117$	−1.11595	−0.103170
$118$	0	0
$119$	−0.602003	−0.0551855
$120$	0	0
$121$	−9.58376	−0.871251
$122$	0	0
$123$	6.00455	0.541412
$124$	0	0
$125$	11.6967	1.04619
$126$	0	0
$127$	2.35319	0.208812	0.104406	−	0.994535i	$-0.466706\pi$
$127$	2.35319	0.208812	0.104406	+	0.994535i	$0.466706\pi$
$128$	0	0
$129$	4.86788	0.428593
$130$	0	0
$131$	−2.70410	−0.236258	−0.118129	−	0.992998i	$-0.537690\pi$
$131$	−2.70410	−0.236258	−0.118129	+	0.992998i	$0.537690\pi$
$132$	0	0
$133$	6.82897	0.592147
$134$	0	0
$135$	−2.11277	−0.181838
$136$	0	0
$137$	−12.8789	−1.10032	−0.550159	−	0.835060i	$-0.685433\pi$
$137$	−12.8789	−1.10032	−0.550159	+	0.835060i	$0.685433\pi$
$138$	0	0
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	0	0
$141$	4.97115	0.418647
$142$	0	0
$143$	−1.32804	−0.111057
$144$	0	0
$145$	0.984470	0.0817558
$146$	0	0
$147$	−5.56196	−0.458743
$148$	0	0
$149$	−8.88213	−0.727652	−0.363826	−	0.931467i	$-0.618530\pi$
$149$	−8.88213	−0.727652	−0.363826	+	0.931467i	$0.618530\pi$
$150$	0	0
$151$	−12.4113	−1.01002	−0.505008	−	0.863114i	$-0.668511\pi$
$151$	−12.4113	−1.01002	−0.505008	+	0.863114i	$0.668511\pi$
$152$	0	0
$153$	−0.502011	−0.0405852
$154$	0	0
$155$	−6.01292	−0.482969
$156$	0	0
$157$	−23.7549	−1.89585	−0.947923	−	0.318500i	$-0.896821\pi$
$157$	−23.7549	−1.89585	−0.947923	+	0.318500i	$0.896821\pi$
$158$	0	0
$159$	−13.2689	−1.05230
$160$	0	0
$161$	1.28342	0.101148
$162$	0	0
$163$	17.5968	1.37829	0.689146	−	0.724623i	$-0.257986\pi$
$163$	17.5968	1.37829	0.689146	+	0.724623i	$0.257986\pi$
$164$	0	0
$165$	−2.51432	−0.195739
$166$	0	0
$167$	−11.3179	−0.875805	−0.437903	−	0.899022i	$-0.644279\pi$
$167$	−11.3179	−0.875805	−0.437903	+	0.899022i	$0.644279\pi$
$168$	0	0
$169$	−11.7547	−0.904204
$170$	0	0
$171$	5.69468	0.435483
$172$	0	0
$173$	0.870493	0.0661824	0.0330912	−	0.999452i	$-0.489465\pi$
$173$	0.870493	0.0661824	0.0330912	+	0.999452i	$0.489465\pi$
$174$	0	0
$175$	−0.643014	−0.0486073
$176$	0	0
$177$	6.82158	0.512741
$178$	0	0
$179$	6.75535	0.504919	0.252459	−	0.967608i	$-0.418761\pi$
$179$	6.75535	0.504919	0.252459	+	0.967608i	$0.418761\pi$
$180$	0	0
$181$	−9.31707	−0.692532	−0.346266	−	0.938136i	$-0.612551\pi$
$181$	−9.31707	−0.692532	−0.346266	+	0.938136i	$0.612551\pi$
$182$	0	0
$183$	1.95691	0.144659
$184$	0	0
$185$	−18.7974	−1.38201
$186$	0	0
$187$	−0.597422	−0.0436878
$188$	0	0
$189$	1.19918	0.0872278
$190$	0	0
$191$	−22.8965	−1.65673	−0.828367	−	0.560186i	$-0.810730\pi$
$191$	−22.8965	−1.65673	−0.828367	+	0.560186i	$0.810730\pi$
$192$	0	0
$193$	3.64334	0.262254	0.131127	−	0.991366i	$-0.458141\pi$
$193$	3.64334	0.262254	0.131127	+	0.991366i	$0.458141\pi$
$194$	0	0
$195$	2.35774	0.168841
$196$	0	0
$197$	8.31070	0.592113	0.296057	−	0.955170i	$-0.404328\pi$
$197$	8.31070	0.592113	0.296057	+	0.955170i	$0.404328\pi$
$198$	0	0
$199$	−3.62628	−0.257060	−0.128530	−	0.991706i	$-0.541026\pi$
$199$	−3.62628	−0.257060	−0.128530	+	0.991706i	$0.541026\pi$
$200$	0	0
$201$	15.2513	1.07574
$202$	0	0
$203$	−0.558774	−0.0392183
$204$	0	0
$205$	−12.6862	−0.886044
$206$	0	0
$207$	1.07025	0.0743873
$208$	0	0
$209$	6.77700	0.468775
$210$	0	0
$211$	7.94483	0.546945	0.273472	−	0.961880i	$-0.411828\pi$
$211$	7.94483	0.546945	0.273472	+	0.961880i	$0.411828\pi$
$212$	0	0
$213$	3.36027	0.230242
$214$	0	0
$215$	−10.2847	−0.701411
$216$	0	0
$217$	3.41287	0.231680
$218$	0	0
$219$	9.76731	0.660013
$220$	0	0
$221$	0.560219	0.0376844
$222$	0	0
$223$	−14.1544	−0.947848	−0.473924	−	0.880566i	$-0.657163\pi$
$223$	−14.1544	−0.947848	−0.473924	+	0.880566i	$0.657163\pi$
$224$	0	0
$225$	−0.536209	−0.0357473
$226$	0	0
$227$	−4.77631	−0.317015	−0.158508	−	0.987358i	$-0.550668\pi$
$227$	−4.77631	−0.317015	−0.158508	+	0.987358i	$0.550668\pi$
$228$	0	0
$229$	−4.38229	−0.289590	−0.144795	−	0.989462i	$-0.546252\pi$
$229$	−4.38229	−0.289590	−0.144795	+	0.989462i	$0.546252\pi$
$230$	0	0
$231$	1.42710	0.0938962
$232$	0	0
$233$	12.7514	0.835370	0.417685	−	0.908592i	$-0.362842\pi$
$233$	12.7514	0.835370	0.417685	+	0.908592i	$0.362842\pi$
$234$	0	0
$235$	−10.5029	−0.685133
$236$	0	0
$237$	0.260986	0.0169529
$238$	0	0
$239$	−21.8219	−1.41154	−0.705770	−	0.708441i	$-0.749399\pi$
$239$	−21.8219	−1.41154	−0.705770	+	0.708441i	$0.749399\pi$
$240$	0	0
$241$	14.7959	0.953090	0.476545	−	0.879150i	$-0.341889\pi$
$241$	14.7959	0.953090	0.476545	+	0.879150i	$0.341889\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	11.7511	0.750752
$246$	0	0
$247$	−6.35498	−0.404357
$248$	0	0
$249$	−10.7862	−0.683549
$250$	0	0
$251$	−10.8012	−0.681768	−0.340884	−	0.940105i	$-0.610726\pi$
$251$	−10.8012	−0.681768	−0.340884	+	0.940105i	$0.610726\pi$
$252$	0	0
$253$	1.27366	0.0800741
$254$	0	0
$255$	1.06063	0.0664194
$256$	0	0
$257$	17.2211	1.07422	0.537112	−	0.843511i	$-0.319515\pi$
$257$	17.2211	1.07422	0.537112	+	0.843511i	$0.319515\pi$
$258$	0	0
$259$	10.6692	0.662951
$260$	0	0
$261$	−0.465962	−0.0288423
$262$	0	0
$263$	−12.0566	−0.743439	−0.371720	−	0.928345i	$-0.621232\pi$
$263$	−12.0566	−0.743439	−0.371720	+	0.928345i	$0.621232\pi$
$264$	0	0
$265$	28.0342	1.72213
$266$	0	0
$267$	0.709412	0.0434153
$268$	0	0
$269$	17.9928	1.09704	0.548521	−	0.836136i	$-0.315191\pi$
$269$	17.9928	1.09704	0.548521	+	0.836136i	$0.315191\pi$
$270$	0	0
$271$	4.75787	0.289020	0.144510	−	0.989503i	$-0.453839\pi$
$271$	4.75787	0.289020	0.144510	+	0.989503i	$0.453839\pi$
$272$	0	0
$273$	−1.33823	−0.0809933
$274$	0	0
$275$	−0.638120	−0.0384801
$276$	0	0
$277$	19.1890	1.15296	0.576478	−	0.817113i	$-0.304427\pi$
$277$	19.1890	1.15296	0.576478	+	0.817113i	$0.304427\pi$
$278$	0	0
$279$	2.84599	0.170385
$280$	0	0
$281$	−2.87001	−0.171211	−0.0856053	−	0.996329i	$-0.527282\pi$
$281$	−2.87001	−0.171211	−0.0856053	+	0.996329i	$0.527282\pi$
$282$	0	0
$283$	−15.9261	−0.946705	−0.473353	−	0.880873i	$-0.656956\pi$
$283$	−15.9261	−0.946705	−0.473353	+	0.880873i	$0.656956\pi$
$284$	0	0
$285$	−12.0315	−0.712687
$286$	0	0
$287$	7.20056	0.425036
$288$	0	0
$289$	−16.7480	−0.985176
$290$	0	0
$291$	3.11334	0.182507
$292$	0	0
$293$	18.6193	1.08775	0.543875	−	0.839166i	$-0.316957\pi$
$293$	18.6193	1.08775	0.543875	+	0.839166i	$0.316957\pi$
$294$	0	0
$295$	−14.4124	−0.839123
$296$	0	0
$297$	1.19006	0.0690542
$298$	0	0
$299$	−1.19434	−0.0690706
$300$	0	0
$301$	5.83749	0.336467
$302$	0	0
$303$	−7.15598	−0.411100
$304$	0	0
$305$	−4.13450	−0.236741
$306$	0	0
$307$	−5.00418	−0.285604	−0.142802	−	0.989751i	$-0.545611\pi$
$307$	−5.00418	−0.285604	−0.142802	+	0.989751i	$0.545611\pi$
$308$	0	0
$309$	4.89306	0.278357
$310$	0	0
$311$	−8.21478	−0.465818	−0.232909	−	0.972499i	$-0.574824\pi$
$311$	−8.21478	−0.465818	−0.232909	+	0.972499i	$0.574824\pi$
$312$	0	0
$313$	−3.10467	−0.175486	−0.0877431	−	0.996143i	$-0.527965\pi$
$313$	−3.10467	−0.175486	−0.0877431	+	0.996143i	$0.527965\pi$
$314$	0	0
$315$	−2.53360	−0.142752
$316$	0	0
$317$	−4.28609	−0.240731	−0.120365	−	0.992730i	$-0.538407\pi$
$317$	−4.28609	−0.240731	−0.120365	+	0.992730i	$0.538407\pi$
$318$	0	0
$319$	−0.554522	−0.0310473
$320$	0	0
$321$	0.314564	0.0175572
$322$	0	0
$323$	−2.85879	−0.159067
$324$	0	0
$325$	0.598382	0.0331923
$326$	0	0
$327$	15.2321	0.842337
$328$	0	0
$329$	5.96133	0.328659
$330$	0	0
$331$	−6.39906	−0.351724	−0.175862	−	0.984415i	$-0.556271\pi$
$331$	−6.39906	−0.351724	−0.175862	+	0.984415i	$0.556271\pi$
$332$	0	0
$333$	8.89704	0.487555
$334$	0	0
$335$	−32.2224	−1.76050
$336$	0	0
$337$	−18.7456	−1.02114	−0.510570	−	0.859836i	$-0.670566\pi$
$337$	−18.7456	−1.02114	−0.510570	+	0.859836i	$0.670566\pi$
$338$	0	0
$339$	18.5556	1.00780
$340$	0	0
$341$	3.38689	0.183411
$342$	0	0
$343$	−15.0641	−0.813385
$344$	0	0
$345$	−2.26118	−0.121738
$346$	0	0
$347$	4.35308	0.233685	0.116843	−	0.993150i	$-0.462723\pi$
$347$	4.35308	0.233685	0.116843	+	0.993150i	$0.462723\pi$
$348$	0	0
$349$	8.46849	0.453308	0.226654	−	0.973975i	$-0.427221\pi$
$349$	8.46849	0.453308	0.226654	+	0.973975i	$0.427221\pi$
$350$	0	0
$351$	−1.11595	−0.0595650
$352$	0	0
$353$	−28.6764	−1.52629	−0.763146	−	0.646226i	$-0.776346\pi$
$353$	−28.6764	−1.52629	−0.763146	+	0.646226i	$0.776346\pi$
$354$	0	0
$355$	−7.09946	−0.376800
$356$	0	0
$357$	−0.602003	−0.0318614
$358$	0	0
$359$	−27.7305	−1.46356	−0.731779	−	0.681542i	$-0.761310\pi$
$359$	−27.7305	−1.46356	−0.731779	+	0.681542i	$0.761310\pi$
$360$	0	0
$361$	13.4294	0.706810
$362$	0	0
$363$	−9.58376	−0.503017
$364$	0	0
$365$	−20.6361	−1.08014
$366$	0	0
$367$	15.6734	0.818143	0.409071	−	0.912502i	$-0.365853\pi$
$367$	15.6734	0.818143	0.409071	+	0.912502i	$0.365853\pi$
$368$	0	0
$369$	6.00455	0.312584
$370$	0	0
$371$	−15.9119	−0.826105
$372$	0	0
$373$	23.1918	1.20083	0.600414	−	0.799690i	$-0.295003\pi$
$373$	23.1918	1.20083	0.600414	+	0.799690i	$0.295003\pi$
$374$	0	0
$375$	11.6967	0.604016
$376$	0	0
$377$	0.519990	0.0267809
$378$	0	0
$379$	10.2330	0.525633	0.262817	−	0.964846i	$-0.415349\pi$
$379$	10.2330	0.525633	0.262817	+	0.964846i	$0.415349\pi$
$380$	0	0
$381$	2.35319	0.120558
$382$	0	0
$383$	−16.8075	−0.858823	−0.429411	−	0.903109i	$-0.641279\pi$
$383$	−16.8075	−0.858823	−0.429411	+	0.903109i	$0.641279\pi$
$384$	0	0
$385$	−3.01513	−0.153665
$386$	0	0
$387$	4.86788	0.247448
$388$	0	0
$389$	−1.31404	−0.0666243	−0.0333121	−	0.999445i	$-0.510606\pi$
$389$	−1.31404	−0.0666243	−0.0333121	+	0.999445i	$0.510606\pi$
$390$	0	0
$391$	−0.537276	−0.0271712
$392$	0	0
$393$	−2.70410	−0.136404
$394$	0	0
$395$	−0.551403	−0.0277441
$396$	0	0
$397$	−20.4801	−1.02786	−0.513932	−	0.857831i	$-0.671812\pi$
$397$	−20.4801	−1.02786	−0.513932	+	0.857831i	$0.671812\pi$
$398$	0	0
$399$	6.82897	0.341876
$400$	0	0
$401$	−14.5012	−0.724156	−0.362078	−	0.932148i	$-0.617933\pi$
$401$	−14.5012	−0.724156	−0.362078	+	0.932148i	$0.617933\pi$
$402$	0	0
$403$	−3.17598	−0.158207
$404$	0	0
$405$	−2.11277	−0.104984
$406$	0	0
$407$	10.5880	0.524828
$408$	0	0
$409$	21.0588	1.04129	0.520646	−	0.853773i	$-0.325691\pi$
$409$	21.0588	1.04129	0.520646	+	0.853773i	$0.325691\pi$
$410$	0	0
$411$	−12.8789	−0.635269
$412$	0	0
$413$	8.18032	0.402527
$414$	0	0
$415$	22.7888	1.11866
$416$	0	0
$417$	1.00000	0.0489702
$418$	0	0
$419$	3.09760	0.151328	0.0756638	−	0.997133i	$-0.475892\pi$
$419$	3.09760	0.151328	0.0756638	+	0.997133i	$0.475892\pi$
$420$	0	0
$421$	−2.54903	−0.124232	−0.0621161	−	0.998069i	$-0.519785\pi$
$421$	−2.54903	−0.124232	−0.0621161	+	0.998069i	$0.519785\pi$
$422$	0	0
$423$	4.97115	0.241706
$424$	0	0
$425$	0.269183	0.0130573
$426$	0	0
$427$	2.34669	0.113564
$428$	0	0
$429$	−1.32804	−0.0641186
$430$	0	0
$431$	−19.9387	−0.960411	−0.480206	−	0.877156i	$-0.659438\pi$
$431$	−19.9387	−0.960411	−0.480206	+	0.877156i	$0.659438\pi$
$432$	0	0
$433$	2.07107	0.0995292	0.0497646	−	0.998761i	$-0.484153\pi$
$433$	2.07107	0.0995292	0.0497646	+	0.998761i	$0.484153\pi$
$434$	0	0
$435$	0.984470	0.0472017
$436$	0	0
$437$	6.09472	0.291550
$438$	0	0
$439$	−6.68006	−0.318822	−0.159411	−	0.987212i	$-0.550959\pi$
$439$	−6.68006	−0.318822	−0.159411	+	0.987212i	$0.550959\pi$
$440$	0	0
$441$	−5.56196	−0.264855
$442$	0	0
$443$	−25.9330	−1.23211	−0.616056	−	0.787702i	$-0.711270\pi$
$443$	−25.9330	−1.23211	−0.616056	+	0.787702i	$0.711270\pi$
$444$	0	0
$445$	−1.49882	−0.0710510
$446$	0	0
$447$	−8.88213	−0.420110
$448$	0	0
$449$	−17.0078	−0.802646	−0.401323	−	0.915937i	$-0.631449\pi$
$449$	−17.0078	−0.802646	−0.401323	+	0.915937i	$0.631449\pi$
$450$	0	0
$451$	7.14576	0.336481
$452$	0	0
$453$	−12.4113	−0.583133
$454$	0	0
$455$	2.82737	0.132549
$456$	0	0
$457$	−21.9283	−1.02576	−0.512881	−	0.858460i	$-0.671422\pi$
$457$	−21.9283	−1.02576	−0.512881	+	0.858460i	$0.671422\pi$
$458$	0	0
$459$	−0.502011	−0.0234319
$460$	0	0
$461$	3.32185	0.154714	0.0773571	−	0.997003i	$-0.475352\pi$
$461$	3.32185	0.154714	0.0773571	+	0.997003i	$0.475352\pi$
$462$	0	0
$463$	−10.6288	−0.493962	−0.246981	−	0.969020i	$-0.579438\pi$
$463$	−10.6288	−0.493962	−0.246981	+	0.969020i	$0.579438\pi$
$464$	0	0
$465$	−6.01292	−0.278842
$466$	0	0
$467$	−7.75100	−0.358673	−0.179337	−	0.983788i	$-0.557395\pi$
$467$	−7.75100	−0.358673	−0.179337	+	0.983788i	$0.557395\pi$
$468$	0	0
$469$	18.2891	0.844512
$470$	0	0
$471$	−23.7549	−1.09457
$472$	0	0
$473$	5.79306	0.266365
$474$	0	0
$475$	−3.05354	−0.140106
$476$	0	0
$477$	−13.2689	−0.607543
$478$	0	0
$479$	−0.505199	−0.0230831	−0.0115416	−	0.999933i	$-0.503674\pi$
$479$	−0.505199	−0.0230831	−0.0115416	+	0.999933i	$0.503674\pi$
$480$	0	0
$481$	−9.92865	−0.452708
$482$	0	0
$483$	1.28342	0.0583978
$484$	0	0
$485$	−6.57776	−0.298681
$486$	0	0
$487$	−5.98262	−0.271098	−0.135549	−	0.990771i	$-0.543280\pi$
$487$	−5.98262	−0.271098	−0.135549	+	0.990771i	$0.543280\pi$
$488$	0	0
$489$	17.5968	0.795757
$490$	0	0
$491$	12.0067	0.541855	0.270927	−	0.962600i	$-0.412670\pi$
$491$	12.0067	0.541855	0.270927	+	0.962600i	$0.412670\pi$
$492$	0	0
$493$	0.233918	0.0105351
$494$	0	0
$495$	−2.51432	−0.113010
$496$	0	0
$497$	4.02958	0.180751
$498$	0	0
$499$	9.83300	0.440186	0.220093	−	0.975479i	$-0.429364\pi$
$499$	9.83300	0.440186	0.220093	+	0.975479i	$0.429364\pi$
$500$	0	0
$501$	−11.3179	−0.505646
$502$	0	0
$503$	−39.1530	−1.74575	−0.872873	−	0.487948i	$-0.837746\pi$
$503$	−39.1530	−1.74575	−0.872873	+	0.487948i	$0.837746\pi$
$504$	0	0
$505$	15.1189	0.672783
$506$	0	0
$507$	−11.7547	−0.522043
$508$	0	0
$509$	8.66781	0.384194	0.192097	−	0.981376i	$-0.438471\pi$
$509$	8.66781	0.384194	0.192097	+	0.981376i	$0.438471\pi$
$510$	0	0
$511$	11.7128	0.518144
$512$	0	0
$513$	5.69468	0.251426
$514$	0	0
$515$	−10.3379	−0.455543
$516$	0	0
$517$	5.91596	0.260184
$518$	0	0
$519$	0.870493	0.0382104
$520$	0	0
$521$	−4.90845	−0.215043	−0.107522	−	0.994203i	$-0.534291\pi$
$521$	−4.90845	−0.215043	−0.107522	+	0.994203i	$0.534291\pi$
$522$	0	0
$523$	2.10262	0.0919412	0.0459706	−	0.998943i	$-0.485362\pi$
$523$	2.10262	0.0919412	0.0459706	+	0.998943i	$0.485362\pi$
$524$	0	0
$525$	−0.643014	−0.0280634
$526$	0	0
$527$	−1.42872	−0.0622359
$528$	0	0
$529$	−21.8546	−0.950199
$530$	0	0
$531$	6.82158	0.296031
$532$	0	0
$533$	−6.70077	−0.290243
$534$	0	0
$535$	−0.664600	−0.0287332
$536$	0	0
$537$	6.75535	0.291515
$538$	0	0
$539$	−6.61905	−0.285103
$540$	0	0
$541$	−27.9133	−1.20009	−0.600043	−	0.799968i	$-0.704850\pi$
$541$	−27.9133	−1.20009	−0.600043	+	0.799968i	$0.704850\pi$
$542$	0	0
$543$	−9.31707	−0.399834
$544$	0	0
$545$	−32.1819	−1.37852
$546$	0	0
$547$	−27.3619	−1.16991	−0.584954	−	0.811066i	$-0.698887\pi$
$547$	−27.3619	−1.16991	−0.584954	+	0.811066i	$0.698887\pi$
$548$	0	0
$549$	1.95691	0.0835188
$550$	0	0
$551$	−2.65351	−0.113043
$552$	0	0
$553$	0.312970	0.0133088
$554$	0	0
$555$	−18.7974	−0.797905
$556$	0	0
$557$	42.7990	1.81345	0.906726	−	0.421721i	$-0.138574\pi$
$557$	42.7990	1.81345	0.906726	+	0.421721i	$0.138574\pi$
$558$	0	0
$559$	−5.43231	−0.229762
$560$	0	0
$561$	−0.597422	−0.0252232
$562$	0	0
$563$	37.7543	1.59116	0.795578	−	0.605851i	$-0.207167\pi$
$563$	37.7543	1.59116	0.795578	+	0.605851i	$0.207167\pi$
$564$	0	0
$565$	−39.2037	−1.64931
$566$	0	0
$567$	1.19918	0.0503610
$568$	0	0
$569$	18.9386	0.793947	0.396974	−	0.917830i	$-0.370060\pi$
$569$	18.9386	0.793947	0.396974	+	0.917830i	$0.370060\pi$
$570$	0	0
$571$	−19.0719	−0.798134	−0.399067	−	0.916922i	$-0.630666\pi$
$571$	−19.0719	−0.798134	−0.399067	+	0.916922i	$0.630666\pi$
$572$	0	0
$573$	−22.8965	−0.956516
$574$	0	0
$575$	−0.573877	−0.0239323
$576$	0	0
$577$	9.93788	0.413719	0.206860	−	0.978371i	$-0.433676\pi$
$577$	9.93788	0.413719	0.206860	+	0.978371i	$0.433676\pi$
$578$	0	0
$579$	3.64334	0.151412
$580$	0	0
$581$	−12.9347	−0.536620
$582$	0	0
$583$	−15.7908	−0.653989
$584$	0	0
$585$	2.35774	0.0974807
$586$	0	0
$587$	−12.1933	−0.503272	−0.251636	−	0.967822i	$-0.580969\pi$
$587$	−12.1933	−0.503272	−0.251636	+	0.967822i	$0.580969\pi$
$588$	0	0
$589$	16.2070	0.667798
$590$	0	0
$591$	8.31070	0.341857
$592$	0	0
$593$	38.9972	1.60142	0.800712	−	0.599050i	$-0.204455\pi$
$593$	38.9972	1.60142	0.800712	+	0.599050i	$0.204455\pi$
$594$	0	0
$595$	1.27189	0.0521425
$596$	0	0
$597$	−3.62628	−0.148414
$598$	0	0
$599$	18.4967	0.755754	0.377877	−	0.925856i	$-0.376654\pi$
$599$	18.4967	0.755754	0.377877	+	0.925856i	$0.376654\pi$
$600$	0	0
$601$	42.2379	1.72292	0.861460	−	0.507825i	$-0.169550\pi$
$601$	42.2379	1.72292	0.861460	+	0.507825i	$0.169550\pi$
$602$	0	0
$603$	15.2513	0.621080
$604$	0	0
$605$	20.2483	0.823209
$606$	0	0
$607$	−25.7658	−1.04580	−0.522900	−	0.852394i	$-0.675150\pi$
$607$	−25.7658	−1.04580	−0.522900	+	0.852394i	$0.675150\pi$
$608$	0	0
$609$	−0.558774	−0.0226427
$610$	0	0
$611$	−5.54756	−0.224430
$612$	0	0
$613$	−23.1801	−0.936234	−0.468117	−	0.883666i	$-0.655067\pi$
$613$	−23.1801	−0.936234	−0.468117	+	0.883666i	$0.655067\pi$
$614$	0	0
$615$	−12.6862	−0.511558
$616$	0	0
$617$	4.15759	0.167378	0.0836892	−	0.996492i	$-0.473330\pi$
$617$	4.15759	0.167378	0.0836892	+	0.996492i	$0.473330\pi$
$618$	0	0
$619$	21.5497	0.866157	0.433079	−	0.901356i	$-0.357427\pi$
$619$	21.5497	0.866157	0.433079	+	0.901356i	$0.357427\pi$
$620$	0	0
$621$	1.07025	0.0429475
$622$	0	0
$623$	0.850715	0.0340832
$624$	0	0
$625$	−22.0314	−0.881257
$626$	0	0
$627$	6.77700	0.270647
$628$	0	0
$629$	−4.46641	−0.178087
$630$	0	0
$631$	25.3883	1.01069	0.505346	−	0.862917i	$-0.331365\pi$
$631$	25.3883	1.01069	0.505346	+	0.862917i	$0.331365\pi$
$632$	0	0
$633$	7.94483	0.315779
$634$	0	0
$635$	−4.97175	−0.197298
$636$	0	0
$637$	6.20686	0.245925
$638$	0	0
$639$	3.36027	0.132930
$640$	0	0
$641$	−9.89762	−0.390932	−0.195466	−	0.980710i	$-0.562622\pi$
$641$	−9.89762	−0.390932	−0.195466	+	0.980710i	$0.562622\pi$
$642$	0	0
$643$	18.1710	0.716594	0.358297	−	0.933608i	$-0.383358\pi$
$643$	18.1710	0.716594	0.358297	+	0.933608i	$0.383358\pi$
$644$	0	0
$645$	−10.2847	−0.404960
$646$	0	0
$647$	31.5024	1.23849	0.619243	−	0.785200i	$-0.287440\pi$
$647$	31.5024	1.23849	0.619243	+	0.785200i	$0.287440\pi$
$648$	0	0
$649$	8.11807	0.318662
$650$	0	0
$651$	3.41287	0.133761
$652$	0	0
$653$	6.77382	0.265080	0.132540	−	0.991178i	$-0.457687\pi$
$653$	6.77382	0.265080	0.132540	+	0.991178i	$0.457687\pi$
$654$	0	0
$655$	5.71314	0.223231
$656$	0	0
$657$	9.76731	0.381059
$658$	0	0
$659$	37.6072	1.46497	0.732484	−	0.680785i	$-0.238361\pi$
$659$	37.6072	1.46497	0.732484	+	0.680785i	$0.238361\pi$
$660$	0	0
$661$	−0.768246	−0.0298813	−0.0149407	−	0.999888i	$-0.504756\pi$
$661$	−0.768246	−0.0298813	−0.0149407	+	0.999888i	$0.504756\pi$
$662$	0	0
$663$	0.560219	0.0217571
$664$	0	0
$665$	−14.4280	−0.559495
$666$	0	0
$667$	−0.498695	−0.0193095
$668$	0	0
$669$	−14.1544	−0.547240
$670$	0	0
$671$	2.32884	0.0899037
$672$	0	0
$673$	−7.80025	−0.300678	−0.150339	−	0.988635i	$-0.548036\pi$
$673$	−7.80025	−0.300678	−0.150339	+	0.988635i	$0.548036\pi$
$674$	0	0
$675$	−0.536209	−0.0206387
$676$	0	0
$677$	27.4588	1.05533	0.527663	−	0.849454i	$-0.323068\pi$
$677$	27.4588	1.05533	0.527663	+	0.849454i	$0.323068\pi$
$678$	0	0
$679$	3.73347	0.143277
$680$	0	0
$681$	−4.77631	−0.183029
$682$	0	0
$683$	10.2505	0.392223	0.196112	−	0.980582i	$-0.437169\pi$
$683$	10.2505	0.392223	0.196112	+	0.980582i	$0.437169\pi$
$684$	0	0
$685$	27.2101	1.03965
$686$	0	0
$687$	−4.38229	−0.167195
$688$	0	0
$689$	14.8075	0.564120
$690$	0	0
$691$	18.7012	0.711429	0.355714	−	0.934595i	$-0.384238\pi$
$691$	18.7012	0.711429	0.355714	+	0.934595i	$0.384238\pi$
$692$	0	0
$693$	1.42710	0.0542110
$694$	0	0
$695$	−2.11277	−0.0801419
$696$	0	0
$697$	−3.01435	−0.114177
$698$	0	0
$699$	12.7514	0.482301
$700$	0	0
$701$	−22.6522	−0.855560	−0.427780	−	0.903883i	$-0.640704\pi$
$701$	−22.6522	−0.855560	−0.427780	+	0.903883i	$0.640704\pi$
$702$	0	0
$703$	50.6658	1.91090
$704$	0	0
$705$	−10.5029	−0.395562
$706$	0	0
$707$	−8.58133	−0.322734
$708$	0	0
$709$	−5.20135	−0.195341	−0.0976704	−	0.995219i	$-0.531139\pi$
$709$	−5.20135	−0.195341	−0.0976704	+	0.995219i	$0.531139\pi$
$710$	0	0
$711$	0.260986	0.00978774
$712$	0	0
$713$	3.04591	0.114070
$714$	0	0
$715$	2.80585	0.104933
$716$	0	0
$717$	−21.8219	−0.814953
$718$	0	0
$719$	32.5871	1.21529	0.607646	−	0.794208i	$-0.292114\pi$
$719$	32.5871	1.21529	0.607646	+	0.794208i	$0.292114\pi$
$720$	0	0
$721$	5.86768	0.218524
$722$	0	0
$723$	14.7959	0.550267
$724$	0	0
$725$	0.249853	0.00927932
$726$	0	0
$727$	−0.439204	−0.0162892	−0.00814458	−	0.999967i	$-0.502593\pi$
$727$	−0.439204	−0.0162892	−0.00814458	+	0.999967i	$0.502593\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.44373	−0.0903846
$732$	0	0
$733$	−21.0913	−0.779026	−0.389513	−	0.921021i	$-0.627357\pi$
$733$	−21.0913	−0.779026	−0.389513	+	0.921021i	$0.627357\pi$
$734$	0	0
$735$	11.7511	0.433447
$736$	0	0
$737$	18.1499	0.668561
$738$	0	0
$739$	3.38194	0.124407	0.0622034	−	0.998063i	$-0.480187\pi$
$739$	3.38194	0.124407	0.0622034	+	0.998063i	$0.480187\pi$
$740$	0	0
$741$	−6.35498	−0.233456
$742$	0	0
$743$	−23.7876	−0.872681	−0.436340	−	0.899782i	$-0.643726\pi$
$743$	−23.7876	−0.872681	−0.436340	+	0.899782i	$0.643726\pi$
$744$	0	0
$745$	18.7659	0.687529
$746$	0	0
$747$	−10.7862	−0.394647
$748$	0	0
$749$	0.377220	0.0137833
$750$	0	0
$751$	−28.1396	−1.02683	−0.513414	−	0.858141i	$-0.671619\pi$
$751$	−28.1396	−1.02683	−0.513414	+	0.858141i	$0.671619\pi$
$752$	0	0
$753$	−10.8012	−0.393619
$754$	0	0
$755$	26.2222	0.954323
$756$	0	0
$757$	2.18965	0.0795843	0.0397922	−	0.999208i	$-0.487330\pi$
$757$	2.18965	0.0795843	0.0397922	+	0.999208i	$0.487330\pi$
$758$	0	0
$759$	1.27366	0.0462308
$760$	0	0
$761$	25.1853	0.912966	0.456483	−	0.889732i	$-0.349109\pi$
$761$	25.1853	0.912966	0.456483	+	0.889732i	$0.349109\pi$
$762$	0	0
$763$	18.2661	0.661277
$764$	0	0
$765$	1.06063	0.0383472
$766$	0	0
$767$	−7.61253	−0.274873
$768$	0	0
$769$	11.3221	0.408284	0.204142	−	0.978941i	$-0.434560\pi$
$769$	11.3221	0.408284	0.204142	+	0.978941i	$0.434560\pi$
$770$	0	0
$771$	17.2211	0.620203
$772$	0	0
$773$	30.5134	1.09749	0.548745	−	0.835990i	$-0.315106\pi$
$773$	30.5134	1.09749	0.548745	+	0.835990i	$0.315106\pi$
$774$	0	0
$775$	−1.52605	−0.0548172
$776$	0	0
$777$	10.6692	0.382755
$778$	0	0
$779$	34.1940	1.22513
$780$	0	0
$781$	3.99891	0.143092
$782$	0	0
$783$	−0.465962	−0.0166521
$784$	0	0
$785$	50.1886	1.79131
$786$	0	0
$787$	13.7369	0.489666	0.244833	−	0.969565i	$-0.421267\pi$
$787$	13.7369	0.489666	0.244833	+	0.969565i	$0.421267\pi$
$788$	0	0
$789$	−12.0566	−0.429225
$790$	0	0
$791$	22.2516	0.791175
$792$	0	0
$793$	−2.18381	−0.0775494
$794$	0	0
$795$	28.0342	0.994271
$796$	0	0
$797$	−27.6886	−0.980782	−0.490391	−	0.871503i	$-0.663146\pi$
$797$	−27.6886	−0.980782	−0.490391	+	0.871503i	$0.663146\pi$
$798$	0	0
$799$	−2.49557	−0.0882870
$800$	0	0
$801$	0.709412	0.0250658
$802$	0	0
$803$	11.6237	0.410190
$804$	0	0
$805$	−2.71158	−0.0955705
$806$	0	0
$807$	17.9928	0.633378
$808$	0	0
$809$	30.5363	1.07360	0.536799	−	0.843710i	$-0.319633\pi$
$809$	30.5363	1.07360	0.536799	+	0.843710i	$0.319633\pi$
$810$	0	0
$811$	−39.5194	−1.38771	−0.693857	−	0.720113i	$-0.744090\pi$
$811$	−39.5194	−1.38771	−0.693857	+	0.720113i	$0.744090\pi$
$812$	0	0
$813$	4.75787	0.166866
$814$	0	0
$815$	−37.1781	−1.30229
$816$	0	0
$817$	27.7210	0.969836
$818$	0	0
$819$	−1.33823	−0.0467615
$820$	0	0
$821$	−40.5082	−1.41375	−0.706874	−	0.707340i	$-0.749895\pi$
$821$	−40.5082	−1.41375	−0.706874	+	0.707340i	$0.749895\pi$
$822$	0	0
$823$	−19.9995	−0.697138	−0.348569	−	0.937283i	$-0.613332\pi$
$823$	−19.9995	−0.697138	−0.348569	+	0.937283i	$0.613332\pi$
$824$	0	0
$825$	−0.638120	−0.0222165
$826$	0	0
$827$	−15.7879	−0.548999	−0.274500	−	0.961587i	$-0.588512\pi$
$827$	−15.7879	−0.548999	−0.274500	+	0.961587i	$0.588512\pi$
$828$	0	0
$829$	52.8048	1.83399	0.916993	−	0.398903i	$-0.130609\pi$
$829$	52.8048	1.83399	0.916993	+	0.398903i	$0.130609\pi$
$830$	0	0
$831$	19.1890	0.665659
$832$	0	0
$833$	2.79216	0.0967427
$834$	0	0
$835$	23.9121	0.827512
$836$	0	0
$837$	2.84599	0.0983718
$838$	0	0
$839$	−27.6298	−0.953887	−0.476943	−	0.878934i	$-0.658255\pi$
$839$	−27.6298	−0.953887	−0.476943	+	0.878934i	$0.658255\pi$
$840$	0	0
$841$	−28.7829	−0.992513
$842$	0	0
$843$	−2.87001	−0.0988484
$844$	0	0
$845$	24.8349	0.854345
$846$	0	0
$847$	−11.4927	−0.394894
$848$	0	0
$849$	−15.9261	−0.546581
$850$	0	0
$851$	9.52204	0.326411
$852$	0	0
$853$	−34.4685	−1.18018	−0.590090	−	0.807337i	$-0.700908\pi$
$853$	−34.4685	−1.18018	−0.590090	+	0.807337i	$0.700908\pi$
$854$	0	0
$855$	−12.0315	−0.411470
$856$	0	0
$857$	6.77648	0.231480	0.115740	−	0.993280i	$-0.463076\pi$
$857$	6.77648	0.231480	0.115740	+	0.993280i	$0.463076\pi$
$858$	0	0
$859$	−25.4652	−0.868862	−0.434431	−	0.900705i	$-0.643051\pi$
$859$	−25.4652	−0.868862	−0.434431	+	0.900705i	$0.643051\pi$
$860$	0	0
$861$	7.20056	0.245394
$862$	0	0
$863$	49.5387	1.68632	0.843158	−	0.537665i	$-0.180694\pi$
$863$	49.5387	1.68632	0.843158	+	0.537665i	$0.180694\pi$
$864$	0	0
$865$	−1.83915	−0.0625330
$866$	0	0
$867$	−16.7480	−0.568791
$868$	0	0
$869$	0.310589	0.0105360
$870$	0	0
$871$	−17.0197	−0.576689
$872$	0	0
$873$	3.11334	0.105371
$874$	0	0
$875$	14.0265	0.474183
$876$	0	0
$877$	19.0971	0.644863	0.322432	−	0.946593i	$-0.395500\pi$
$877$	19.0971	0.644863	0.322432	+	0.946593i	$0.395500\pi$
$878$	0	0
$879$	18.6193	0.628013
$880$	0	0
$881$	−11.4428	−0.385519	−0.192760	−	0.981246i	$-0.561744\pi$
$881$	−11.4428	−0.385519	−0.192760	+	0.981246i	$0.561744\pi$
$882$	0	0
$883$	31.3477	1.05493	0.527467	−	0.849576i	$-0.323142\pi$
$883$	31.3477	1.05493	0.527467	+	0.849576i	$0.323142\pi$
$884$	0	0
$885$	−14.4124	−0.484468
$886$	0	0
$887$	39.7500	1.33467	0.667337	−	0.744756i	$-0.267434\pi$
$887$	39.7500	1.33467	0.667337	+	0.744756i	$0.267434\pi$
$888$	0	0
$889$	2.82191	0.0946439
$890$	0	0
$891$	1.19006	0.0398684
$892$	0	0
$893$	28.3091	0.947329
$894$	0	0
$895$	−14.2725	−0.477077
$896$	0	0
$897$	−1.19434	−0.0398779
$898$	0	0
$899$	−1.32612	−0.0442287
$900$	0	0
$901$	6.66115	0.221915
$902$	0	0
$903$	5.83749	0.194259
$904$	0	0
$905$	19.6848	0.654345
$906$	0	0
$907$	24.3288	0.807824	0.403912	−	0.914798i	$-0.367650\pi$
$907$	24.3288	0.807824	0.403912	+	0.914798i	$0.367650\pi$
$908$	0	0
$909$	−7.15598	−0.237349
$910$	0	0
$911$	−18.0914	−0.599394	−0.299697	−	0.954034i	$-0.596886\pi$
$911$	−18.0914	−0.599394	−0.299697	+	0.954034i	$0.596886\pi$
$912$	0	0
$913$	−12.8362	−0.424817
$914$	0	0
$915$	−4.13450	−0.136682
$916$	0	0
$917$	−3.24271	−0.107084
$918$	0	0
$919$	−25.0485	−0.826274	−0.413137	−	0.910669i	$-0.635567\pi$
$919$	−25.0485	−0.826274	−0.413137	+	0.910669i	$0.635567\pi$
$920$	0	0
$921$	−5.00418	−0.164893
$922$	0	0
$923$	−3.74989	−0.123429
$924$	0	0
$925$	−4.77068	−0.156859
$926$	0	0
$927$	4.89306	0.160709
$928$	0	0
$929$	41.8860	1.37424	0.687118	−	0.726546i	$-0.258875\pi$
$929$	41.8860	1.37424	0.687118	+	0.726546i	$0.258875\pi$
$930$	0	0
$931$	−31.6736	−1.03806
$932$	0	0
$933$	−8.21478	−0.268940
$934$	0	0
$935$	1.26221	0.0412788
$936$	0	0
$937$	−18.9061	−0.617634	−0.308817	−	0.951121i	$-0.599933\pi$
$937$	−18.9061	−0.617634	−0.308817	+	0.951121i	$0.599933\pi$
$938$	0	0
$939$	−3.10467	−0.101317
$940$	0	0
$941$	8.51032	0.277429	0.138714	−	0.990332i	$-0.455703\pi$
$941$	8.51032	0.277429	0.138714	+	0.990332i	$0.455703\pi$
$942$	0	0
$943$	6.42635	0.209271
$944$	0	0
$945$	−2.53360	−0.0824180
$946$	0	0
$947$	1.92725	0.0626272	0.0313136	−	0.999510i	$-0.490031\pi$
$947$	1.92725	0.0626272	0.0313136	+	0.999510i	$0.490031\pi$
$948$	0	0
$949$	−10.8998	−0.353823
$950$	0	0
$951$	−4.28609	−0.138986
$952$	0	0
$953$	22.9087	0.742087	0.371043	−	0.928616i	$-0.379000\pi$
$953$	22.9087	0.742087	0.371043	+	0.928616i	$0.379000\pi$
$954$	0	0
$955$	48.3750	1.56538
$956$	0	0
$957$	−0.554522	−0.0179252
$958$	0	0
$959$	−15.4442	−0.498718
$960$	0	0
$961$	−22.9003	−0.738721
$962$	0	0
$963$	0.314564	0.0101367
$964$	0	0
$965$	−7.69754	−0.247793
$966$	0	0
$967$	20.9662	0.674226	0.337113	−	0.941464i	$-0.390549\pi$
$967$	20.9662	0.674226	0.337113	+	0.941464i	$0.390549\pi$
$968$	0	0
$969$	−2.85879	−0.0918376
$970$	0	0
$971$	17.4702	0.560647	0.280323	−	0.959906i	$-0.409558\pi$
$971$	17.4702	0.560647	0.280323	+	0.959906i	$0.409558\pi$
$972$	0	0
$973$	1.19918	0.0384441
$974$	0	0
$975$	0.598382	0.0191636
$976$	0	0
$977$	−17.2262	−0.551115	−0.275557	−	0.961285i	$-0.588862\pi$
$977$	−17.2262	−0.551115	−0.275557	+	0.961285i	$0.588862\pi$
$978$	0	0
$979$	0.844241	0.0269821
$980$	0	0
$981$	15.2321	0.486323
$982$	0	0
$983$	53.8817	1.71856	0.859279	−	0.511507i	$-0.170912\pi$
$983$	53.8817	1.71856	0.859279	+	0.511507i	$0.170912\pi$
$984$	0	0
$985$	−17.5586	−0.559463
$986$	0	0
$987$	5.96133	0.189751
$988$	0	0
$989$	5.20984	0.165663
$990$	0	0
$991$	44.9597	1.42819	0.714096	−	0.700048i	$-0.246838\pi$
$991$	44.9597	1.42819	0.714096	+	0.700048i	$0.246838\pi$
$992$	0	0
$993$	−6.39906	−0.203068
$994$	0	0
$995$	7.66150	0.242886
$996$	0	0
$997$	34.2112	1.08348	0.541740	−	0.840546i	$-0.317766\pi$
$997$	34.2112	1.08348	0.541740	+	0.840546i	$0.317766\pi$
$998$	0	0
$999$	8.89704	0.281490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1668.2.a.h.1.3	✓	7
3.2	odd	2		5004.2.a.m.1.5		7
4.3	odd	2		6672.2.a.bn.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1668.2.a.h.1.3	✓	7	1.1	even	1	trivial
5004.2.a.m.1.5		7	3.2	odd	2
6672.2.a.bn.1.3		7	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1668 = 2^{2} \cdot 3 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1668.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.11277 q^{5} +1.19918 q^{7} +1.00000 q^{9} +1.19006 q^{11} -1.11595 q^{13} -2.11277 q^{15} -0.502011 q^{17} +5.69468 q^{19} +1.19918 q^{21} +1.07025 q^{23} -0.536209 q^{25} +1.00000 q^{27} -0.465962 q^{29} +2.84599 q^{31} +1.19006 q^{33} -2.53360 q^{35} +8.89704 q^{37} -1.11595 q^{39} +6.00455 q^{41} +4.86788 q^{43} -2.11277 q^{45} +4.97115 q^{47} -5.56196 q^{49} -0.502011 q^{51} -13.2689 q^{53} -2.51432 q^{55} +5.69468 q^{57} +6.82158 q^{59} +1.95691 q^{61} +1.19918 q^{63} +2.35774 q^{65} +15.2513 q^{67} +1.07025 q^{69} +3.36027 q^{71} +9.76731 q^{73} -0.536209 q^{75} +1.42710 q^{77} +0.260986 q^{79} +1.00000 q^{81} -10.7862 q^{83} +1.06063 q^{85} -0.465962 q^{87} +0.709412 q^{89} -1.33823 q^{91} +2.84599 q^{93} -12.0315 q^{95} +3.11334 q^{97} +1.19006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + q^{7} + 7 q^{9} + 8 q^{13} + 27 q^{17} + q^{21} - 8 q^{23} + 21 q^{25} + 7 q^{27} + 6 q^{29} + 7 q^{31} + 6 q^{35} + 14 q^{37} + 8 q^{39} + q^{41} + 27 q^{43} - 6 q^{47} + 32 q^{49} + 27 q^{51}+ \cdots + 7 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 + q^7 + 7 * q^9 + 8 * q^13 + 27 * q^17 + q^21 - 8 * q^23 + 21 * q^25 + 7 * q^27 + 6 * q^29 + 7 * q^31 + 6 * q^35 + 14 * q^37 + 8 * q^39 + q^41 + 27 * q^43 - 6 * q^47 + 32 * q^49 + 27 * q^51 + 18 * q^53 + 13 * q^55 + 17 * q^59 + 13 * q^61 + q^63 - 19 * q^65 + 31 * q^67 - 8 * q^69 + 6 * q^73 + 21 * q^75 - 34 * q^77 + 2 * q^79 + 7 * q^81 - 2 * q^83 + 37 * q^85 + 6 * q^87 + 13 * q^89 + 39 * q^91 + 7 * q^93 - 36 * q^95 + 7 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more