LMFDB - Embedded newform 5684.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5684,2,Mod(1,5684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5684, 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("5684.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5684 = 2^{2} \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5684.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:	$45.3869685089$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 10x^{4} + 18x^{3} + 38x^{2} - 12 $ x^6 - 3x^5 - 10x^4 + 18x^3 + 38x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 812)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30512$ of defining polynomial
Character	$\chi$	$=$	5684.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30512 q^{3} +4.07275 q^{5} +2.31358 q^{9} +O(q^{10})$	$q-2.30512 q^{3} +4.07275 q^{5} +2.31358 q^{9} +5.14700 q^{11} -2.16843 q^{13} -9.38819 q^{15} -3.88739 q^{17} -5.39931 q^{19} -7.99658 q^{23} +11.5873 q^{25} +1.58227 q^{27} +1.00000 q^{29} -0.632441 q^{31} -11.8645 q^{33} +3.63094 q^{37} +4.99850 q^{39} -0.841881 q^{41} +7.16393 q^{43} +9.42266 q^{45} -4.32205 q^{47} +8.96091 q^{51} +11.9614 q^{53} +20.9625 q^{55} +12.4461 q^{57} +6.40661 q^{59} +8.31093 q^{61} -8.83149 q^{65} -1.64744 q^{67} +18.4331 q^{69} -4.10919 q^{71} +13.4322 q^{73} -26.7102 q^{75} +0.649367 q^{79} -10.5881 q^{81} +15.7595 q^{83} -15.8324 q^{85} -2.30512 q^{87} +11.1910 q^{89} +1.45785 q^{93} -21.9900 q^{95} -5.82284 q^{97} +11.9080 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + q^{5} + 11 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 + q^5 + 11 * q^9	$ 6 q - 3 q^{3} + q^{5} + 11 q^{9} + q^{11} + 3 q^{13} + 13 q^{15} - 12 q^{19} + 17 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} - 17 q^{33} + 4 q^{37} + 13 q^{39} + 14 q^{41} + 29 q^{43} + 8 q^{45} - 31 q^{47} + 14 q^{51} - 7 q^{53} - 3 q^{55} + 18 q^{57} + 2 q^{59} + 6 q^{61} + 5 q^{65} + 6 q^{67} + 48 q^{69} - 10 q^{73} - 22 q^{75} + 19 q^{79} - 18 q^{81} + 14 q^{83} - 12 q^{85} - 3 q^{87} + 16 q^{89} - 9 q^{93} - 34 q^{95} + 6 q^{97} - 22 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + q^5 + 11 * q^9 + q^11 + 3 * q^13 + 13 * q^15 - 12 * q^19 + 17 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 - 17 * q^33 + 4 * q^37 + 13 * q^39 + 14 * q^41 + 29 * q^43 + 8 * q^45 - 31 * q^47 + 14 * q^51 - 7 * q^53 - 3 * q^55 + 18 * q^57 + 2 * q^59 + 6 * q^61 + 5 * q^65 + 6 * q^67 + 48 * q^69 - 10 * q^73 - 22 * q^75 + 19 * q^79 - 18 * q^81 + 14 * q^83 - 12 * q^85 - 3 * q^87 + 16 * q^89 - 9 * q^93 - 34 * q^95 + 6 * q^97 - 22 * 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
$3$	−2.30512	−1.33086	−0.665431	−	0.746459i	$-0.731752\pi$
$3$	−2.30512	−1.33086	−0.665431	+	0.746459i	$0.731752\pi$
$4$	0	0
$5$	4.07275	1.82139	0.910695	−	0.413079i	$-0.135547\pi$
$5$	4.07275	1.82139	0.910695	+	0.413079i	$0.135547\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.31358	0.771195
$10$	0	0
$11$	5.14700	1.55188	0.775940	−	0.630807i	$-0.217276\pi$
$11$	5.14700	1.55188	0.775940	+	0.630807i	$0.217276\pi$
$12$	0	0
$13$	−2.16843	−0.601415	−0.300708	−	0.953716i	$-0.597223\pi$
$13$	−2.16843	−0.601415	−0.300708	+	0.953716i	$0.597223\pi$
$14$	0	0
$15$	−9.38819	−2.42402
$16$	0	0
$17$	−3.88739	−0.942831	−0.471416	−	0.881911i	$-0.656257\pi$
$17$	−3.88739	−0.942831	−0.471416	+	0.881911i	$0.656257\pi$
$18$	0	0
$19$	−5.39931	−1.23869	−0.619343	−	0.785121i	$-0.712601\pi$
$19$	−5.39931	−1.23869	−0.619343	+	0.785121i	$0.712601\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.99658	−1.66740	−0.833701	−	0.552216i	$-0.813782\pi$
$23$	−7.99658	−1.66740	−0.833701	+	0.552216i	$0.813782\pi$
$24$	0	0
$25$	11.5873	2.31746
$26$	0	0
$27$	1.58227	0.304508
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.632441	−0.113590	−0.0567949	−	0.998386i	$-0.518088\pi$
$31$	−0.632441	−0.113590	−0.0567949	+	0.998386i	$0.518088\pi$
$32$	0	0
$33$	−11.8645	−2.06534
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.63094	0.596923	0.298462	−	0.954422i	$-0.403526\pi$
$37$	3.63094	0.596923	0.298462	+	0.954422i	$0.403526\pi$
$38$	0	0
$39$	4.99850	0.800401
$40$	0	0
$41$	−0.841881	−0.131480	−0.0657399	−	0.997837i	$-0.520941\pi$
$41$	−0.841881	−0.131480	−0.0657399	+	0.997837i	$0.520941\pi$
$42$	0	0
$43$	7.16393	1.09249	0.546244	−	0.837626i	$-0.316057\pi$
$43$	7.16393	1.09249	0.546244	+	0.837626i	$0.316057\pi$
$44$	0	0
$45$	9.42266	1.40465
$46$	0	0
$47$	−4.32205	−0.630435	−0.315218	−	0.949019i	$-0.602078\pi$
$47$	−4.32205	−0.630435	−0.315218	+	0.949019i	$0.602078\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.96091	1.25478
$52$	0	0
$53$	11.9614	1.64302	0.821511	−	0.570193i	$-0.193131\pi$
$53$	11.9614	1.64302	0.821511	+	0.570193i	$0.193131\pi$
$54$	0	0
$55$	20.9625	2.82658
$56$	0	0
$57$	12.4461	1.64852
$58$	0	0
$59$	6.40661	0.834070	0.417035	−	0.908890i	$-0.363069\pi$
$59$	6.40661	0.834070	0.417035	+	0.908890i	$0.363069\pi$
$60$	0	0
$61$	8.31093	1.06411	0.532053	−	0.846711i	$-0.321421\pi$
$61$	8.31093	1.06411	0.532053	+	0.846711i	$0.321421\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.83149	−1.09541
$66$	0	0
$67$	−1.64744	−0.201267	−0.100634	−	0.994924i	$-0.532087\pi$
$67$	−1.64744	−0.201267	−0.100634	+	0.994924i	$0.532087\pi$
$68$	0	0
$69$	18.4331	2.21908
$70$	0	0
$71$	−4.10919	−0.487671	−0.243835	−	0.969817i	$-0.578406\pi$
$71$	−4.10919	−0.487671	−0.243835	+	0.969817i	$0.578406\pi$
$72$	0	0
$73$	13.4322	1.57212	0.786060	−	0.618150i	$-0.212118\pi$
$73$	13.4322	1.57212	0.786060	+	0.618150i	$0.212118\pi$
$74$	0	0
$75$	−26.7102	−3.08422
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.649367	0.0730595	0.0365297	−	0.999333i	$-0.488370\pi$
$79$	0.649367	0.0730595	0.0365297	+	0.999333i	$0.488370\pi$
$80$	0	0
$81$	−10.5881	−1.17645
$82$	0	0
$83$	15.7595	1.72983	0.864916	−	0.501916i	$-0.167371\pi$
$83$	15.7595	1.72983	0.864916	+	0.501916i	$0.167371\pi$
$84$	0	0
$85$	−15.8324	−1.71726
$86$	0	0
$87$	−2.30512	−0.247135
$88$	0	0
$89$	11.1910	1.18625	0.593123	−	0.805112i	$-0.297895\pi$
$89$	11.1910	1.18625	0.593123	+	0.805112i	$0.297895\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.45785	0.151172
$94$	0	0
$95$	−21.9900	−2.25613
$96$	0	0
$97$	−5.82284	−0.591220	−0.295610	−	0.955309i	$-0.595523\pi$
$97$	−5.82284	−0.591220	−0.295610	+	0.955309i	$0.595523\pi$
$98$	0	0
$99$	11.9080	1.19680
$100$	0	0
$101$	1.60447	0.159651	0.0798254	−	0.996809i	$-0.474564\pi$
$101$	1.60447	0.159651	0.0798254	+	0.996809i	$0.474564\pi$
$102$	0	0
$103$	−16.1983	−1.59607	−0.798034	−	0.602612i	$-0.794127\pi$
$103$	−16.1983	−1.59607	−0.798034	+	0.602612i	$0.794127\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.8090	−1.43164	−0.715820	−	0.698285i	$-0.753947\pi$
$107$	−14.8090	−1.43164	−0.715820	+	0.698285i	$0.753947\pi$
$108$	0	0
$109$	3.36640	0.322443	0.161221	−	0.986918i	$-0.448457\pi$
$109$	3.36640	0.322443	0.161221	+	0.986918i	$0.448457\pi$
$110$	0	0
$111$	−8.36977	−0.794423
$112$	0	0
$113$	7.96781	0.749549	0.374774	−	0.927116i	$-0.377720\pi$
$113$	7.96781	0.749549	0.374774	+	0.927116i	$0.377720\pi$
$114$	0	0
$115$	−32.5681	−3.03699
$116$	0	0
$117$	−5.01685	−0.463808
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.4916	1.40833
$122$	0	0
$123$	1.94064	0.174981
$124$	0	0
$125$	26.8285	2.39961
$126$	0	0
$127$	−1.88739	−0.167479	−0.0837395	−	0.996488i	$-0.526686\pi$
$127$	−1.88739	−0.167479	−0.0837395	+	0.996488i	$0.526686\pi$
$128$	0	0
$129$	−16.5137	−1.45395
$130$	0	0
$131$	5.94187	0.519144	0.259572	−	0.965724i	$-0.416419\pi$
$131$	5.94187	0.519144	0.259572	+	0.965724i	$0.416419\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.44420	0.554629
$136$	0	0
$137$	1.96941	0.168258	0.0841288	−	0.996455i	$-0.473189\pi$
$137$	1.96941	0.168258	0.0841288	+	0.996455i	$0.473189\pi$
$138$	0	0
$139$	19.8123	1.68046	0.840230	−	0.542229i	$-0.182420\pi$
$139$	19.8123	1.68046	0.840230	+	0.542229i	$0.182420\pi$
$140$	0	0
$141$	9.96284	0.839023
$142$	0	0
$143$	−11.1609	−0.933324
$144$	0	0
$145$	4.07275	0.338224
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.38768	−0.605223	−0.302611	−	0.953114i	$-0.597858\pi$
$149$	−7.38768	−0.605223	−0.302611	+	0.953114i	$0.597858\pi$
$150$	0	0
$151$	16.3863	1.33350	0.666748	−	0.745284i	$-0.267686\pi$
$151$	16.3863	1.33350	0.666748	+	0.745284i	$0.267686\pi$
$152$	0	0
$153$	−8.99381	−0.727107
$154$	0	0
$155$	−2.57578	−0.206891
$156$	0	0
$157$	18.5256	1.47851	0.739254	−	0.673427i	$-0.235178\pi$
$157$	18.5256	1.47851	0.739254	+	0.673427i	$0.235178\pi$
$158$	0	0
$159$	−27.5724	−2.18664
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.99843	−0.313181	−0.156591	−	0.987664i	$-0.550050\pi$
$163$	−3.99843	−0.313181	−0.156591	+	0.987664i	$0.550050\pi$
$164$	0	0
$165$	−48.3210	−3.76179
$166$	0	0
$167$	−2.07576	−0.160627	−0.0803136	−	0.996770i	$-0.525592\pi$
$167$	−2.07576	−0.160627	−0.0803136	+	0.996770i	$0.525592\pi$
$168$	0	0
$169$	−8.29789	−0.638300
$170$	0	0
$171$	−12.4917	−0.955268
$172$	0	0
$173$	24.8492	1.88925	0.944627	−	0.328147i	$-0.106424\pi$
$173$	24.8492	1.88925	0.944627	+	0.328147i	$0.106424\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.7680	−1.11003
$178$	0	0
$179$	3.46167	0.258737	0.129369	−	0.991597i	$-0.458705\pi$
$179$	3.46167	0.258737	0.129369	+	0.991597i	$0.458705\pi$
$180$	0	0
$181$	−16.2558	−1.20829	−0.604143	−	0.796876i	$-0.706484\pi$
$181$	−16.2558	−1.20829	−0.604143	+	0.796876i	$0.706484\pi$
$182$	0	0
$183$	−19.1577	−1.41618
$184$	0	0
$185$	14.7879	1.08723
$186$	0	0
$187$	−20.0084	−1.46316
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.8345	1.36282	0.681408	−	0.731903i	$-0.261368\pi$
$191$	18.8345	1.36282	0.681408	+	0.731903i	$0.261368\pi$
$192$	0	0
$193$	0.653106	0.0470116	0.0235058	−	0.999724i	$-0.492517\pi$
$193$	0.653106	0.0470116	0.0235058	+	0.999724i	$0.492517\pi$
$194$	0	0
$195$	20.3577	1.45784
$196$	0	0
$197$	20.0992	1.43201	0.716005	−	0.698095i	$-0.245969\pi$
$197$	20.0992	1.43201	0.716005	+	0.698095i	$0.245969\pi$
$198$	0	0
$199$	11.0460	0.783034	0.391517	−	0.920171i	$-0.371950\pi$
$199$	11.0460	0.783034	0.391517	+	0.920171i	$0.371950\pi$
$200$	0	0
$201$	3.79756	0.267859
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.42877	−0.239476
$206$	0	0
$207$	−18.5008	−1.28589
$208$	0	0
$209$	−27.7902	−1.92229
$210$	0	0
$211$	−4.76102	−0.327762	−0.163881	−	0.986480i	$-0.552401\pi$
$211$	−4.76102	−0.327762	−0.163881	+	0.986480i	$0.552401\pi$
$212$	0	0
$213$	9.47217	0.649023
$214$	0	0
$215$	29.1769	1.98985
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−30.9629	−2.09228
$220$	0	0
$221$	8.42955	0.567033
$222$	0	0
$223$	−29.4364	−1.97121	−0.985605	−	0.169064i	$-0.945925\pi$
$223$	−29.4364	−1.97121	−0.985605	+	0.169064i	$0.945925\pi$
$224$	0	0
$225$	26.8082	1.78722
$226$	0	0
$227$	−12.1196	−0.804404	−0.402202	−	0.915551i	$-0.631755\pi$
$227$	−12.1196	−0.804404	−0.402202	+	0.915551i	$0.631755\pi$
$228$	0	0
$229$	13.6129	0.899565	0.449783	−	0.893138i	$-0.351501\pi$
$229$	13.6129	0.899565	0.449783	+	0.893138i	$0.351501\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.0486	−1.24792	−0.623959	−	0.781457i	$-0.714477\pi$
$233$	−19.0486	−1.24792	−0.623959	+	0.781457i	$0.714477\pi$
$234$	0	0
$235$	−17.6026	−1.14827
$236$	0	0
$237$	−1.49687	−0.0972321
$238$	0	0
$239$	8.23217	0.532495	0.266248	−	0.963905i	$-0.414216\pi$
$239$	8.23217	0.532495	0.266248	+	0.963905i	$0.414216\pi$
$240$	0	0
$241$	−10.0299	−0.646082	−0.323041	−	0.946385i	$-0.604705\pi$
$241$	−10.0299	−0.646082	−0.323041	+	0.946385i	$0.604705\pi$
$242$	0	0
$243$	19.6600	1.26119
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.7080	0.744965
$248$	0	0
$249$	−36.3276	−2.30217
$250$	0	0
$251$	12.1122	0.764513	0.382257	−	0.924056i	$-0.375147\pi$
$251$	12.1122	0.764513	0.382257	+	0.924056i	$0.375147\pi$
$252$	0	0
$253$	−41.1584	−2.58761
$254$	0	0
$255$	36.4956	2.28544
$256$	0	0
$257$	−20.9893	−1.30927	−0.654637	−	0.755943i	$-0.727179\pi$
$257$	−20.9893	−1.30927	−0.654637	+	0.755943i	$0.727179\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.31358	0.143207
$262$	0	0
$263$	−26.5190	−1.63523	−0.817616	−	0.575764i	$-0.804705\pi$
$263$	−26.5190	−1.63523	−0.817616	+	0.575764i	$0.804705\pi$
$264$	0	0
$265$	48.7158	2.99258
$266$	0	0
$267$	−25.7967	−1.57873
$268$	0	0
$269$	1.71854	0.104781	0.0523905	−	0.998627i	$-0.483316\pi$
$269$	1.71854	0.104781	0.0523905	+	0.998627i	$0.483316\pi$
$270$	0	0
$271$	−6.44221	−0.391336	−0.195668	−	0.980670i	$-0.562688\pi$
$271$	−6.44221	−0.391336	−0.195668	+	0.980670i	$0.562688\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	59.6399	3.59642
$276$	0	0
$277$	−10.8802	−0.653728	−0.326864	−	0.945071i	$-0.605992\pi$
$277$	−10.8802	−0.653728	−0.326864	+	0.945071i	$0.605992\pi$
$278$	0	0
$279$	−1.46321	−0.0875998
$280$	0	0
$281$	20.4050	1.21726	0.608629	−	0.793455i	$-0.291720\pi$
$281$	20.4050	1.21726	0.608629	+	0.793455i	$0.291720\pi$
$282$	0	0
$283$	2.61554	0.155478	0.0777389	−	0.996974i	$-0.475230\pi$
$283$	2.61554	0.155478	0.0777389	+	0.996974i	$0.475230\pi$
$284$	0	0
$285$	50.6897	3.00260
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.88817	−0.111069
$290$	0	0
$291$	13.4224	0.786833
$292$	0	0
$293$	23.9176	1.39728	0.698640	−	0.715474i	$-0.253789\pi$
$293$	23.9176	1.39728	0.698640	+	0.715474i	$0.253789\pi$
$294$	0	0
$295$	26.0925	1.51917
$296$	0	0
$297$	8.14396	0.472560
$298$	0	0
$299$	17.3401	1.00280
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.69850	−0.212473
$304$	0	0
$305$	33.8484	1.93815
$306$	0	0
$307$	−26.2143	−1.49613	−0.748065	−	0.663625i	$-0.769017\pi$
$307$	−26.2143	−1.49613	−0.748065	+	0.663625i	$0.769017\pi$
$308$	0	0
$309$	37.3391	2.12415
$310$	0	0
$311$	27.7309	1.57247	0.786237	−	0.617926i	$-0.212027\pi$
$311$	27.7309	1.57247	0.786237	+	0.617926i	$0.212027\pi$
$312$	0	0
$313$	−8.85909	−0.500746	−0.250373	−	0.968149i	$-0.580553\pi$
$313$	−8.85909	−0.500746	−0.250373	+	0.968149i	$0.580553\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.7748	1.27916	0.639581	−	0.768723i	$-0.279108\pi$
$317$	22.7748	1.27916	0.639581	+	0.768723i	$0.279108\pi$
$318$	0	0
$319$	5.14700	0.288177
$320$	0	0
$321$	34.1365	1.90531
$322$	0	0
$323$	20.9892	1.16787
$324$	0	0
$325$	−25.1263	−1.39376
$326$	0	0
$327$	−7.75997	−0.429127
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.7357	1.08477	0.542385	−	0.840130i	$-0.317521\pi$
$331$	19.7357	1.08477	0.542385	+	0.840130i	$0.317521\pi$
$332$	0	0
$333$	8.40049	0.460344
$334$	0	0
$335$	−6.70963	−0.366586
$336$	0	0
$337$	0.155341	0.00846195	0.00423097	−	0.999991i	$-0.498653\pi$
$337$	0.155341	0.00846195	0.00423097	+	0.999991i	$0.498653\pi$
$338$	0	0
$339$	−18.3668	−0.997546
$340$	0	0
$341$	−3.25517	−0.176278
$342$	0	0
$343$	0	0
$344$	0	0
$345$	75.0734	4.04182
$346$	0	0
$347$	25.4716	1.36739	0.683694	−	0.729769i	$-0.260372\pi$
$347$	25.4716	1.36739	0.683694	+	0.729769i	$0.260372\pi$
$348$	0	0
$349$	6.19437	0.331577	0.165789	−	0.986161i	$-0.446983\pi$
$349$	6.19437	0.331577	0.165789	+	0.986161i	$0.446983\pi$
$350$	0	0
$351$	−3.43105	−0.183136
$352$	0	0
$353$	−23.4633	−1.24883	−0.624414	−	0.781094i	$-0.714662\pi$
$353$	−23.4633	−1.24883	−0.624414	+	0.781094i	$0.714662\pi$
$354$	0	0
$355$	−16.7357	−0.888239
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.2046	−0.696912	−0.348456	−	0.937325i	$-0.613294\pi$
$359$	−13.2046	−0.696912	−0.348456	+	0.937325i	$0.613294\pi$
$360$	0	0
$361$	10.1525	0.534342
$362$	0	0
$363$	−35.7101	−1.87429
$364$	0	0
$365$	54.7060	2.86345
$366$	0	0
$367$	25.8748	1.35066	0.675328	−	0.737518i	$-0.264002\pi$
$367$	25.8748	1.35066	0.675328	+	0.737518i	$0.264002\pi$
$368$	0	0
$369$	−1.94776	−0.101396
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.8903	0.978104	0.489052	−	0.872255i	$-0.337343\pi$
$373$	18.8903	0.978104	0.489052	+	0.872255i	$0.337343\pi$
$374$	0	0
$375$	−61.8429	−3.19356
$376$	0	0
$377$	−2.16843	−0.111680
$378$	0	0
$379$	−18.8906	−0.970344	−0.485172	−	0.874419i	$-0.661243\pi$
$379$	−18.8906	−0.970344	−0.485172	+	0.874419i	$0.661243\pi$
$380$	0	0
$381$	4.35067	0.222892
$382$	0	0
$383$	10.3078	0.526704	0.263352	−	0.964700i	$-0.415172\pi$
$383$	10.3078	0.526704	0.263352	+	0.964700i	$0.415172\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.5744	0.842522
$388$	0	0
$389$	−34.4861	−1.74851	−0.874257	−	0.485464i	$-0.838651\pi$
$389$	−34.4861	−1.74851	−0.874257	+	0.485464i	$0.838651\pi$
$390$	0	0
$391$	31.0858	1.57208
$392$	0	0
$393$	−13.6967	−0.690909
$394$	0	0
$395$	2.64471	0.133070
$396$	0	0
$397$	23.6377	1.18634	0.593170	−	0.805077i	$-0.297876\pi$
$397$	23.6377	1.18634	0.593170	+	0.805077i	$0.297876\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.00802	−0.0503382	−0.0251691	−	0.999683i	$-0.508012\pi$
$401$	−1.00802	−0.0503382	−0.0251691	+	0.999683i	$0.508012\pi$
$402$	0	0
$403$	1.37141	0.0683146
$404$	0	0
$405$	−43.1226	−2.14278
$406$	0	0
$407$	18.6885	0.926353
$408$	0	0
$409$	−13.0847	−0.646995	−0.323497	−	0.946229i	$-0.604859\pi$
$409$	−13.0847	−0.646995	−0.323497	+	0.946229i	$0.604859\pi$
$410$	0	0
$411$	−4.53972	−0.223928
$412$	0	0
$413$	0	0
$414$	0	0
$415$	64.1846	3.15070
$416$	0	0
$417$	−45.6699	−2.23646
$418$	0	0
$419$	20.6232	1.00751	0.503755	−	0.863846i	$-0.331951\pi$
$419$	20.6232	1.00751	0.503755	+	0.863846i	$0.331951\pi$
$420$	0	0
$421$	24.3330	1.18592	0.592959	−	0.805232i	$-0.297959\pi$
$421$	24.3330	1.18592	0.592959	+	0.805232i	$0.297959\pi$
$422$	0	0
$423$	−9.99942	−0.486188
$424$	0	0
$425$	−45.0444	−2.18498
$426$	0	0
$427$	0	0
$428$	0	0
$429$	25.7273	1.24213
$430$	0	0
$431$	34.2110	1.64789	0.823943	−	0.566673i	$-0.191770\pi$
$431$	34.2110	1.64789	0.823943	+	0.566673i	$0.191770\pi$
$432$	0	0
$433$	17.3184	0.832268	0.416134	−	0.909303i	$-0.363385\pi$
$433$	17.3184	0.832268	0.416134	+	0.909303i	$0.363385\pi$
$434$	0	0
$435$	−9.38819	−0.450129
$436$	0	0
$437$	43.1760	2.06539
$438$	0	0
$439$	−26.2612	−1.25338	−0.626690	−	0.779269i	$-0.715591\pi$
$439$	−26.2612	−1.25338	−0.626690	+	0.779269i	$0.715591\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−39.3086	−1.86761	−0.933803	−	0.357787i	$-0.883532\pi$
$443$	−39.3086	−1.86761	−0.933803	+	0.357787i	$0.883532\pi$
$444$	0	0
$445$	45.5782	2.16062
$446$	0	0
$447$	17.0295	0.805468
$448$	0	0
$449$	2.63152	0.124189	0.0620946	−	0.998070i	$-0.480222\pi$
$449$	2.63152	0.124189	0.0620946	+	0.998070i	$0.480222\pi$
$450$	0	0
$451$	−4.33316	−0.204041
$452$	0	0
$453$	−37.7723	−1.77470
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.69249	0.0791714	0.0395857	−	0.999216i	$-0.487396\pi$
$457$	1.69249	0.0791714	0.0395857	+	0.999216i	$0.487396\pi$
$458$	0	0
$459$	−6.15091	−0.287100
$460$	0	0
$461$	18.6532	0.868767	0.434384	−	0.900728i	$-0.356966\pi$
$461$	18.6532	0.868767	0.434384	+	0.900728i	$0.356966\pi$
$462$	0	0
$463$	−19.1435	−0.889676	−0.444838	−	0.895611i	$-0.646739\pi$
$463$	−19.1435	−0.889676	−0.444838	+	0.895611i	$0.646739\pi$
$464$	0	0
$465$	5.93747	0.275344
$466$	0	0
$467$	−37.3411	−1.72794	−0.863969	−	0.503544i	$-0.832029\pi$
$467$	−37.3411	−1.72794	−0.863969	+	0.503544i	$0.832029\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−42.7039	−1.96769
$472$	0	0
$473$	36.8728	1.69541
$474$	0	0
$475$	−62.5634	−2.87061
$476$	0	0
$477$	27.6737	1.26709
$478$	0	0
$479$	17.8488	0.815533	0.407766	−	0.913086i	$-0.366308\pi$
$479$	17.8488	0.815533	0.407766	+	0.913086i	$0.366308\pi$
$480$	0	0
$481$	−7.87346	−0.358999
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.7150	−1.07684
$486$	0	0
$487$	−22.6075	−1.02445	−0.512223	−	0.858853i	$-0.671178\pi$
$487$	−22.6075	−1.02445	−0.512223	+	0.858853i	$0.671178\pi$
$488$	0	0
$489$	9.21687	0.416801
$490$	0	0
$491$	38.6701	1.74516	0.872579	−	0.488473i	$-0.162446\pi$
$491$	38.6701	1.74516	0.872579	+	0.488473i	$0.162446\pi$
$492$	0	0
$493$	−3.88739	−0.175079
$494$	0	0
$495$	48.4984	2.17984
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.3862	−1.44980	−0.724902	−	0.688852i	$-0.758115\pi$
$499$	−32.3862	−1.44980	−0.724902	+	0.688852i	$0.758115\pi$
$500$	0	0
$501$	4.78488	0.213773
$502$	0	0
$503$	3.00163	0.133836	0.0669179	−	0.997758i	$-0.478683\pi$
$503$	3.00163	0.133836	0.0669179	+	0.997758i	$0.478683\pi$
$504$	0	0
$505$	6.53461	0.290786
$506$	0	0
$507$	19.1277	0.849489
$508$	0	0
$509$	−1.49840	−0.0664155	−0.0332078	−	0.999448i	$-0.510572\pi$
$509$	−1.49840	−0.0664155	−0.0332078	+	0.999448i	$0.510572\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.54317	−0.377190
$514$	0	0
$515$	−65.9718	−2.90706
$516$	0	0
$517$	−22.2456	−0.978360
$518$	0	0
$519$	−57.2805	−2.51434
$520$	0	0
$521$	38.9150	1.70490	0.852449	−	0.522810i	$-0.175116\pi$
$521$	38.9150	1.70490	0.852449	+	0.522810i	$0.175116\pi$
$522$	0	0
$523$	24.8708	1.08753	0.543763	−	0.839239i	$-0.316999\pi$
$523$	24.8708	1.08753	0.543763	+	0.839239i	$0.316999\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.45855	0.107096
$528$	0	0
$529$	40.9453	1.78023
$530$	0	0
$531$	14.8222	0.643230
$532$	0	0
$533$	1.82556	0.0790739
$534$	0	0
$535$	−60.3134	−2.60757
$536$	0	0
$537$	−7.97957	−0.344344
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.7948	0.808050	0.404025	−	0.914748i	$-0.367611\pi$
$541$	18.7948	0.808050	0.404025	+	0.914748i	$0.367611\pi$
$542$	0	0
$543$	37.4716	1.60806
$544$	0	0
$545$	13.7105	0.587294
$546$	0	0
$547$	4.84707	0.207246	0.103623	−	0.994617i	$-0.466957\pi$
$547$	4.84707	0.207246	0.103623	+	0.994617i	$0.466957\pi$
$548$	0	0
$549$	19.2280	0.820633
$550$	0	0
$551$	−5.39931	−0.230018
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−34.0880	−1.44695
$556$	0	0
$557$	−1.12918	−0.0478448	−0.0239224	−	0.999714i	$-0.507615\pi$
$557$	−1.12918	−0.0478448	−0.0239224	+	0.999714i	$0.507615\pi$
$558$	0	0
$559$	−15.5345	−0.657040
$560$	0	0
$561$	46.1218	1.94727
$562$	0	0
$563$	−0.717650	−0.0302453	−0.0151227	−	0.999886i	$-0.504814\pi$
$563$	−0.717650	−0.0302453	−0.0151227	+	0.999886i	$0.504814\pi$
$564$	0	0
$565$	32.4509	1.36522
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.5330	−1.11232	−0.556161	−	0.831075i	$-0.687726\pi$
$569$	−26.5330	−1.11232	−0.556161	+	0.831075i	$0.687726\pi$
$570$	0	0
$571$	−11.9450	−0.499882	−0.249941	−	0.968261i	$-0.580411\pi$
$571$	−11.9450	−0.499882	−0.249941	+	0.968261i	$0.580411\pi$
$572$	0	0
$573$	−43.4158	−1.81372
$574$	0	0
$575$	−92.6589	−3.86414
$576$	0	0
$577$	14.5771	0.606853	0.303426	−	0.952855i	$-0.401869\pi$
$577$	14.5771	0.606853	0.303426	+	0.952855i	$0.401869\pi$
$578$	0	0
$579$	−1.50549	−0.0625659
$580$	0	0
$581$	0	0
$582$	0	0
$583$	61.5653	2.54977
$584$	0	0
$585$	−20.4324	−0.844776
$586$	0	0
$587$	−21.9916	−0.907689	−0.453845	−	0.891081i	$-0.649948\pi$
$587$	−21.9916	−0.907689	−0.453845	+	0.891081i	$0.649948\pi$
$588$	0	0
$589$	3.41474	0.140702
$590$	0	0
$591$	−46.3311	−1.90581
$592$	0	0
$593$	−23.4279	−0.962070	−0.481035	−	0.876701i	$-0.659739\pi$
$593$	−23.4279	−0.962070	−0.481035	+	0.876701i	$0.659739\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.4625	−1.04211
$598$	0	0
$599$	24.7602	1.01167	0.505837	−	0.862629i	$-0.331184\pi$
$599$	24.7602	1.01167	0.505837	+	0.862629i	$0.331184\pi$
$600$	0	0
$601$	32.7508	1.33593	0.667966	−	0.744192i	$-0.267165\pi$
$601$	32.7508	1.33593	0.667966	+	0.744192i	$0.267165\pi$
$602$	0	0
$603$	−3.81150	−0.155216
$604$	0	0
$605$	63.0936	2.56512
$606$	0	0
$607$	−14.8192	−0.601491	−0.300746	−	0.953704i	$-0.597236\pi$
$607$	−14.8192	−0.601491	−0.300746	+	0.953704i	$0.597236\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.37207	0.379153
$612$	0	0
$613$	−42.6331	−1.72193	−0.860967	−	0.508661i	$-0.830141\pi$
$613$	−42.6331	−1.72193	−0.860967	+	0.508661i	$0.830141\pi$
$614$	0	0
$615$	7.90374	0.318709
$616$	0	0
$617$	−1.41031	−0.0567771	−0.0283885	−	0.999597i	$-0.509038\pi$
$617$	−1.41031	−0.0567771	−0.0283885	+	0.999597i	$0.509038\pi$
$618$	0	0
$619$	33.5552	1.34870	0.674349	−	0.738413i	$-0.264425\pi$
$619$	33.5552	1.34870	0.674349	+	0.738413i	$0.264425\pi$
$620$	0	0
$621$	−12.6528	−0.507738
$622$	0	0
$623$	0	0
$624$	0	0
$625$	51.3293	2.05317
$626$	0	0
$627$	64.0599	2.55830
$628$	0	0
$629$	−14.1149	−0.562798
$630$	0	0
$631$	8.64656	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$631$	8.64656	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$632$	0	0
$633$	10.9747	0.436206
$634$	0	0
$635$	−7.68689	−0.305045
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.50695	−0.376089
$640$	0	0
$641$	32.0067	1.26419	0.632094	−	0.774892i	$-0.282196\pi$
$641$	32.0067	1.26419	0.632094	+	0.774892i	$0.282196\pi$
$642$	0	0
$643$	3.13177	0.123505	0.0617526	−	0.998091i	$-0.480331\pi$
$643$	3.13177	0.123505	0.0617526	+	0.998091i	$0.480331\pi$
$644$	0	0
$645$	−67.2563	−2.64821
$646$	0	0
$647$	−46.8012	−1.83995	−0.919973	−	0.391983i	$-0.871789\pi$
$647$	−46.8012	−1.83995	−0.919973	+	0.391983i	$0.871789\pi$
$648$	0	0
$649$	32.9748	1.29438
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.2127	−0.556186	−0.278093	−	0.960554i	$-0.589702\pi$
$653$	−14.2127	−0.556186	−0.278093	+	0.960554i	$0.589702\pi$
$654$	0	0
$655$	24.1998	0.945564
$656$	0	0
$657$	31.0765	1.21241
$658$	0	0
$659$	−14.8238	−0.577453	−0.288727	−	0.957412i	$-0.593232\pi$
$659$	−14.8238	−0.577453	−0.288727	+	0.957412i	$0.593232\pi$
$660$	0	0
$661$	23.5420	0.915676	0.457838	−	0.889036i	$-0.348624\pi$
$661$	23.5420	0.915676	0.457838	+	0.889036i	$0.348624\pi$
$662$	0	0
$663$	−19.4311	−0.754643
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.99658	−0.309629
$668$	0	0
$669$	67.8546	2.62341
$670$	0	0
$671$	42.7764	1.65136
$672$	0	0
$673$	−21.9996	−0.848022	−0.424011	−	0.905657i	$-0.639378\pi$
$673$	−21.9996	−0.848022	−0.424011	+	0.905657i	$0.639378\pi$
$674$	0	0
$675$	18.3343	0.705687
$676$	0	0
$677$	−9.78886	−0.376217	−0.188108	−	0.982148i	$-0.560236\pi$
$677$	−9.78886	−0.376217	−0.188108	+	0.982148i	$0.560236\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	27.9371	1.07055
$682$	0	0
$683$	15.1716	0.580526	0.290263	−	0.956947i	$-0.406257\pi$
$683$	15.1716	0.580526	0.290263	+	0.956947i	$0.406257\pi$
$684$	0	0
$685$	8.02090	0.306463
$686$	0	0
$687$	−31.3794	−1.19720
$688$	0	0
$689$	−25.9375	−0.988139
$690$	0	0
$691$	−11.0637	−0.420882	−0.210441	−	0.977607i	$-0.567490\pi$
$691$	−11.0637	−0.420882	−0.210441	+	0.977607i	$0.567490\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	80.6908	3.06078
$696$	0	0
$697$	3.27272	0.123963
$698$	0	0
$699$	43.9094	1.66081
$700$	0	0
$701$	−10.4661	−0.395300	−0.197650	−	0.980273i	$-0.563331\pi$
$701$	−10.4661	−0.395300	−0.197650	+	0.980273i	$0.563331\pi$
$702$	0	0
$703$	−19.6046	−0.739401
$704$	0	0
$705$	40.5762	1.52819
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.87436	−0.333283	−0.166642	−	0.986018i	$-0.553292\pi$
$709$	−8.87436	−0.333283	−0.166642	+	0.986018i	$0.553292\pi$
$710$	0	0
$711$	1.50236	0.0563431
$712$	0	0
$713$	5.05736	0.189400
$714$	0	0
$715$	−45.4557	−1.69995
$716$	0	0
$717$	−18.9762	−0.708678
$718$	0	0
$719$	0.816360	0.0304451	0.0152226	−	0.999884i	$-0.495154\pi$
$719$	0.816360	0.0304451	0.0152226	+	0.999884i	$0.495154\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.1201	0.859846
$724$	0	0
$725$	11.5873	0.430342
$726$	0	0
$727$	9.63053	0.357176	0.178588	−	0.983924i	$-0.442847\pi$
$727$	9.63053	0.357176	0.178588	+	0.983924i	$0.442847\pi$
$728$	0	0
$729$	−13.5544	−0.502016
$730$	0	0
$731$	−27.8490	−1.03003
$732$	0	0
$733$	−16.7845	−0.619948	−0.309974	−	0.950745i	$-0.600320\pi$
$733$	−16.7845	−0.619948	−0.309974	+	0.950745i	$0.600320\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.47939	−0.312342
$738$	0	0
$739$	28.7959	1.05928	0.529638	−	0.848224i	$-0.322328\pi$
$739$	28.7959	1.05928	0.529638	+	0.848224i	$0.322328\pi$
$740$	0	0
$741$	−26.9884	−0.991445
$742$	0	0
$743$	−29.9895	−1.10021	−0.550104	−	0.835096i	$-0.685412\pi$
$743$	−29.9895	−1.10021	−0.550104	+	0.835096i	$0.685412\pi$
$744$	0	0
$745$	−30.0882	−1.10235
$746$	0	0
$747$	36.4610	1.33404
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.5075	−1.47814	−0.739070	−	0.673629i	$-0.764735\pi$
$751$	−40.5075	−1.47814	−0.739070	+	0.673629i	$0.764735\pi$
$752$	0	0
$753$	−27.9200	−1.01746
$754$	0	0
$755$	66.7372	2.42882
$756$	0	0
$757$	−3.36732	−0.122387	−0.0611937	−	0.998126i	$-0.519491\pi$
$757$	−3.36732	−0.122387	−0.0611937	+	0.998126i	$0.519491\pi$
$758$	0	0
$759$	94.8751	3.44375
$760$	0	0
$761$	5.53295	0.200569	0.100285	−	0.994959i	$-0.468025\pi$
$761$	5.53295	0.200569	0.100285	+	0.994959i	$0.468025\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−36.6296	−1.32434
$766$	0	0
$767$	−13.8923	−0.501622
$768$	0	0
$769$	−12.2593	−0.442083	−0.221042	−	0.975264i	$-0.570946\pi$
$769$	−12.2593	−0.442083	−0.221042	+	0.975264i	$0.570946\pi$
$770$	0	0
$771$	48.3828	1.74246
$772$	0	0
$773$	−33.6246	−1.20939	−0.604697	−	0.796456i	$-0.706706\pi$
$773$	−33.6246	−1.20939	−0.604697	+	0.796456i	$0.706706\pi$
$774$	0	0
$775$	−7.32829	−0.263240
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.54557	0.162862
$780$	0	0
$781$	−21.1500	−0.756806
$782$	0	0
$783$	1.58227	0.0565458
$784$	0	0
$785$	75.4504	2.69294
$786$	0	0
$787$	−4.25957	−0.151837	−0.0759187	−	0.997114i	$-0.524189\pi$
$787$	−4.25957	−0.151837	−0.0759187	+	0.997114i	$0.524189\pi$
$788$	0	0
$789$	61.1295	2.17627
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.0217	−0.639969
$794$	0	0
$795$	−112.296	−3.98272
$796$	0	0
$797$	44.9518	1.59227	0.796137	−	0.605116i	$-0.206873\pi$
$797$	44.9518	1.59227	0.796137	+	0.605116i	$0.206873\pi$
$798$	0	0
$799$	16.8015	0.594394
$800$	0	0
$801$	25.8914	0.914826
$802$	0	0
$803$	69.1356	2.43974
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.96144	−0.139449
$808$	0	0
$809$	−13.3079	−0.467880	−0.233940	−	0.972251i	$-0.575162\pi$
$809$	−13.3079	−0.467880	−0.233940	+	0.972251i	$0.575162\pi$
$810$	0	0
$811$	−6.83379	−0.239967	−0.119983	−	0.992776i	$-0.538284\pi$
$811$	−6.83379	−0.239967	−0.119983	+	0.992776i	$0.538284\pi$
$812$	0	0
$813$	14.8501	0.520814
$814$	0	0
$815$	−16.2846	−0.570425
$816$	0	0
$817$	−38.6802	−1.35325
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.0323	−1.50184	−0.750918	−	0.660395i	$-0.770389\pi$
$821$	−43.0323	−1.50184	−0.750918	+	0.660395i	$0.770389\pi$
$822$	0	0
$823$	25.0665	0.873762	0.436881	−	0.899519i	$-0.356083\pi$
$823$	25.0665	0.873762	0.436881	+	0.899519i	$0.356083\pi$
$824$	0	0
$825$	−137.477	−4.78634
$826$	0	0
$827$	8.07674	0.280856	0.140428	−	0.990091i	$-0.455152\pi$
$827$	8.07674	0.280856	0.140428	+	0.990091i	$0.455152\pi$
$828$	0	0
$829$	−28.3861	−0.985891	−0.492946	−	0.870060i	$-0.664080\pi$
$829$	−28.3861	−0.985891	−0.492946	+	0.870060i	$0.664080\pi$
$830$	0	0
$831$	25.0802	0.870022
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.45406	−0.292565
$836$	0	0
$837$	−1.00069	−0.0345890
$838$	0	0
$839$	13.5027	0.466164	0.233082	−	0.972457i	$-0.425119\pi$
$839$	13.5027	0.466164	0.233082	+	0.972457i	$0.425119\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−47.0359	−1.62000
$844$	0	0
$845$	−33.7953	−1.16259
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.02914	−0.206920
$850$	0	0
$851$	−29.0351	−0.995311
$852$	0	0
$853$	−20.2972	−0.694965	−0.347482	−	0.937687i	$-0.612963\pi$
$853$	−20.2972	−0.694965	−0.347482	+	0.937687i	$0.612963\pi$
$854$	0	0
$855$	−50.8758	−1.73992
$856$	0	0
$857$	25.3276	0.865174	0.432587	−	0.901592i	$-0.357601\pi$
$857$	25.3276	0.865174	0.432587	+	0.901592i	$0.357601\pi$
$858$	0	0
$859$	44.4673	1.51720	0.758602	−	0.651555i	$-0.225883\pi$
$859$	44.4673	1.51720	0.758602	+	0.651555i	$0.225883\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.3403	0.828553	0.414276	−	0.910151i	$-0.364035\pi$
$863$	24.3403	0.828553	0.414276	+	0.910151i	$0.364035\pi$
$864$	0	0
$865$	101.205	3.44107
$866$	0	0
$867$	4.35247	0.147818
$868$	0	0
$869$	3.34229	0.113379
$870$	0	0
$871$	3.57237	0.121045
$872$	0	0
$873$	−13.4716	−0.455946
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.3013	−1.42841	−0.714206	−	0.699935i	$-0.753212\pi$
$877$	−42.3013	−1.42841	−0.714206	+	0.699935i	$0.753212\pi$
$878$	0	0
$879$	−55.1329	−1.85959
$880$	0	0
$881$	−11.7654	−0.396388	−0.198194	−	0.980163i	$-0.563508\pi$
$881$	−11.7654	−0.396388	−0.198194	+	0.980163i	$0.563508\pi$
$882$	0	0
$883$	13.3421	0.448999	0.224499	−	0.974474i	$-0.427925\pi$
$883$	13.3421	0.448999	0.224499	+	0.974474i	$0.427925\pi$
$884$	0	0
$885$	−60.1465	−2.02180
$886$	0	0
$887$	−45.6472	−1.53268	−0.766342	−	0.642433i	$-0.777925\pi$
$887$	−45.6472	−1.53268	−0.766342	+	0.642433i	$0.777925\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−54.4969	−1.82571
$892$	0	0
$893$	23.3361	0.780911
$894$	0	0
$895$	14.0985	0.471262
$896$	0	0
$897$	−39.9709	−1.33459
$898$	0	0
$899$	−0.632441	−0.0210931
$900$	0	0
$901$	−46.4986	−1.54909
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−66.2059	−2.20076
$906$	0	0
$907$	−4.04999	−0.134478	−0.0672389	−	0.997737i	$-0.521419\pi$
$907$	−4.04999	−0.134478	−0.0672389	+	0.997737i	$0.521419\pi$
$908$	0	0
$909$	3.71208	0.123122
$910$	0	0
$911$	17.3207	0.573862	0.286931	−	0.957951i	$-0.407365\pi$
$911$	17.3207	0.573862	0.286931	+	0.957951i	$0.407365\pi$
$912$	0	0
$913$	81.1143	2.68449
$914$	0	0
$915$	−78.0246	−2.57941
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.18033	−0.0719225	−0.0359612	−	0.999353i	$-0.511449\pi$
$919$	−2.18033	−0.0719225	−0.0359612	+	0.999353i	$0.511449\pi$
$920$	0	0
$921$	60.4272	1.99114
$922$	0	0
$923$	8.91050	0.293293
$924$	0	0
$925$	42.0729	1.38335
$926$	0	0
$927$	−37.4762	−1.23088
$928$	0	0
$929$	−9.72916	−0.319203	−0.159602	−	0.987181i	$-0.551021\pi$
$929$	−9.72916	−0.319203	−0.159602	+	0.987181i	$0.551021\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−63.9230	−2.09275
$934$	0	0
$935$	−81.4894	−2.66499
$936$	0	0
$937$	−26.2896	−0.858842	−0.429421	−	0.903104i	$-0.641282\pi$
$937$	−26.2896	−0.858842	−0.429421	+	0.903104i	$0.641282\pi$
$938$	0	0
$939$	20.4213	0.666424
$940$	0	0
$941$	−46.5174	−1.51642	−0.758212	−	0.652008i	$-0.773927\pi$
$941$	−46.5174	−1.51642	−0.758212	+	0.652008i	$0.773927\pi$
$942$	0	0
$943$	6.73217	0.219230
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.2106	0.364296	0.182148	−	0.983271i	$-0.441695\pi$
$947$	11.2106	0.364296	0.182148	+	0.983271i	$0.441695\pi$
$948$	0	0
$949$	−29.1268	−0.945497
$950$	0	0
$951$	−52.4988	−1.70239
$952$	0	0
$953$	−49.0298	−1.58823	−0.794115	−	0.607767i	$-0.792066\pi$
$953$	−49.0298	−1.58823	−0.794115	+	0.607767i	$0.792066\pi$
$954$	0	0
$955$	76.7083	2.48222
$956$	0	0
$957$	−11.8645	−0.383524
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.6000	−0.987097
$962$	0	0
$963$	−34.2619	−1.10407
$964$	0	0
$965$	2.65994	0.0856264
$966$	0	0
$967$	35.4337	1.13947	0.569735	−	0.821829i	$-0.307046\pi$
$967$	35.4337	1.13947	0.569735	+	0.821829i	$0.307046\pi$
$968$	0	0
$969$	−48.3827	−1.55428
$970$	0	0
$971$	43.4548	1.39453	0.697266	−	0.716813i	$-0.254400\pi$
$971$	43.4548	1.39453	0.697266	+	0.716813i	$0.254400\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	57.9192	1.85490
$976$	0	0
$977$	−16.6021	−0.531147	−0.265573	−	0.964091i	$-0.585561\pi$
$977$	−16.6021	−0.531147	−0.265573	+	0.964091i	$0.585561\pi$
$978$	0	0
$979$	57.6002	1.84091
$980$	0	0
$981$	7.78845	0.248666
$982$	0	0
$983$	−60.7571	−1.93785	−0.968925	−	0.247355i	$-0.920439\pi$
$983$	−60.7571	−1.93785	−0.968925	+	0.247355i	$0.920439\pi$
$984$	0	0
$985$	81.8592	2.60825
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−57.2869	−1.82162
$990$	0	0
$991$	6.19549	0.196806	0.0984032	−	0.995147i	$-0.468627\pi$
$991$	6.19549	0.196806	0.0984032	+	0.995147i	$0.468627\pi$
$992$	0	0
$993$	−45.4931	−1.44368
$994$	0	0
$995$	44.9878	1.42621
$996$	0	0
$997$	28.3949	0.899275	0.449637	−	0.893211i	$-0.351553\pi$
$997$	28.3949	0.899275	0.449637	+	0.893211i	$0.351553\pi$
$998$	0	0
$999$	5.74514	0.181768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5684.2.a.r.1.2		6
7.6	odd	2		812.2.a.e.1.5	✓	6
21.20	even	2		7308.2.a.o.1.6		6
28.27	even	2		3248.2.a.bc.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
812.2.a.e.1.5	✓	6	7.6	odd	2
3248.2.a.bc.1.2		6	28.27	even	2
5684.2.a.r.1.2		6	1.1	even	1	trivial
7308.2.a.o.1.6		6	21.20	even	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:	\( N \)	\(=\)	\( 5684 = 2^{2} \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5684.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30512 q^{3} +4.07275 q^{5} +2.31358 q^{9} +O(q^{10})\)	\(q-2.30512 q^{3} +4.07275 q^{5} +2.31358 q^{9} +5.14700 q^{11} -2.16843 q^{13} -9.38819 q^{15} -3.88739 q^{17} -5.39931 q^{19} -7.99658 q^{23} +11.5873 q^{25} +1.58227 q^{27} +1.00000 q^{29} -0.632441 q^{31} -11.8645 q^{33} +3.63094 q^{37} +4.99850 q^{39} -0.841881 q^{41} +7.16393 q^{43} +9.42266 q^{45} -4.32205 q^{47} +8.96091 q^{51} +11.9614 q^{53} +20.9625 q^{55} +12.4461 q^{57} +6.40661 q^{59} +8.31093 q^{61} -8.83149 q^{65} -1.64744 q^{67} +18.4331 q^{69} -4.10919 q^{71} +13.4322 q^{73} -26.7102 q^{75} +0.649367 q^{79} -10.5881 q^{81} +15.7595 q^{83} -15.8324 q^{85} -2.30512 q^{87} +11.1910 q^{89} +1.45785 q^{93} -21.9900 q^{95} -5.82284 q^{97} +11.9080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + q^{5} + 11 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 + q^5 + 11 * q^9	\( 6 q - 3 q^{3} + q^{5} + 11 q^{9} + q^{11} + 3 q^{13} + 13 q^{15} - 12 q^{19} + 17 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} - 17 q^{33} + 4 q^{37} + 13 q^{39} + 14 q^{41} + 29 q^{43} + 8 q^{45} - 31 q^{47} + 14 q^{51} - 7 q^{53} - 3 q^{55} + 18 q^{57} + 2 q^{59} + 6 q^{61} + 5 q^{65} + 6 q^{67} + 48 q^{69} - 10 q^{73} - 22 q^{75} + 19 q^{79} - 18 q^{81} + 14 q^{83} - 12 q^{85} - 3 q^{87} + 16 q^{89} - 9 q^{93} - 34 q^{95} + 6 q^{97} - 22 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + q^5 + 11 * q^9 + q^11 + 3 * q^13 + 13 * q^15 - 12 * q^19 + 17 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 - 17 * q^33 + 4 * q^37 + 13 * q^39 + 14 * q^41 + 29 * q^43 + 8 * q^45 - 31 * q^47 + 14 * q^51 - 7 * q^53 - 3 * q^55 + 18 * q^57 + 2 * q^59 + 6 * q^61 + 5 * q^65 + 6 * q^67 + 48 * q^69 - 10 * q^73 - 22 * q^75 + 19 * q^79 - 18 * q^81 + 14 * q^83 - 12 * q^85 - 3 * q^87 + 16 * q^89 - 9 * q^93 - 34 * q^95 + 6 * q^97 - 22 * q^99

Introduction

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