LMFDB - Embedded newform 680.2.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [680,2,Mod(1,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 680 = 2^{3} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.42982733745$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68133 q^{3} -1.00000 q^{5} -0.508203 q^{7} +4.18953 q^{9} +5.87086 q^{11} -1.18953 q^{13} -2.68133 q^{15} +1.00000 q^{17} +0.810466 q^{19} -1.36266 q^{21} -0.508203 q^{23} +1.00000 q^{25} +3.18953 q^{27} -0.173127 q^{29} +9.06040 q^{31} +15.7417 q^{33} +0.508203 q^{35} -4.37907 q^{37} -3.18953 q^{39} +2.00000 q^{41} -1.36266 q^{43} -4.18953 q^{45} -10.1731 q^{47} -6.74173 q^{49} +2.68133 q^{51} +8.17313 q^{53} -5.87086 q^{55} +2.17313 q^{57} -9.91486 q^{59} -6.55220 q^{61} -2.12914 q^{63} +1.18953 q^{65} +2.37907 q^{67} -1.36266 q^{69} -11.6977 q^{71} +10.5522 q^{73} +2.68133 q^{75} -2.98359 q^{77} -4.85446 q^{79} -4.01641 q^{81} -15.7417 q^{83} -1.00000 q^{85} -0.464211 q^{87} -1.53579 q^{89} +0.604525 q^{91} +24.2939 q^{93} -0.810466 q^{95} +8.93126 q^{97} +24.5962 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 3 q^{5} + 4 q^{9} + 2 q^{11} + 5 q^{13} - q^{15} + 3 q^{17} + 11 q^{19} + 10 q^{21} + 3 q^{25} + q^{27} + 5 q^{29} + 3 q^{31} + 16 q^{33} + 4 q^{37} - q^{39} + 6 q^{41} + 10 q^{43} - 4 q^{45}+ \cdots + 30 q^{99}+O(q^{100}) $ 3 * q + q^3 - 3 * q^5 + 4 * q^9 + 2 * q^11 + 5 * q^13 - q^15 + 3 * q^17 + 11 * q^19 + 10 * q^21 + 3 * q^25 + q^27 + 5 * q^29 + 3 * q^31 + 16 * q^33 + 4 * q^37 - q^39 + 6 * q^41 + 10 * q^43 - 4 * q^45 - 25 * q^47 + 11 * q^49 + q^51 + 19 * q^53 - 2 * q^55 + q^57 + 7 * q^59 + 3 * q^61 - 22 * q^63 - 5 * q^65 - 10 * q^67 + 10 * q^69 - 25 * q^71 + 9 * q^73 + q^75 - 12 * q^77 - 2 * q^79 - 9 * q^81 - 16 * q^83 - 3 * q^85 - 21 * q^87 + 15 * q^89 + 22 * q^91 + 19 * q^93 - 11 * q^95 - 13 * q^97 + 30 * q^99

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
$3$	2.68133	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$3$	2.68133	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.508203	−0.192083	−0.0960414	−	0.995377i	$-0.530618\pi$
$7$	−0.508203	−0.192083	−0.0960414	+	0.995377i	$0.530618\pi$
$8$	0	0
$9$	4.18953	1.39651
$10$	0	0
$11$	5.87086	1.77013	0.885066	−	0.465465i	$-0.154113\pi$
$11$	5.87086	1.77013	0.885066	+	0.465465i	$0.154113\pi$
$12$	0	0
$13$	−1.18953	−0.329917	−0.164959	−	0.986300i	$-0.552749\pi$
$13$	−1.18953	−0.329917	−0.164959	+	0.986300i	$0.552749\pi$
$14$	0	0
$15$	−2.68133	−0.692317
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.810466	0.185934	0.0929668	−	0.995669i	$-0.470365\pi$
$19$	0.810466	0.185934	0.0929668	+	0.995669i	$0.470365\pi$
$20$	0	0
$21$	−1.36266	−0.297357
$22$	0	0
$23$	−0.508203	−0.105968	−0.0529839	−	0.998595i	$-0.516873\pi$
$23$	−0.508203	−0.105968	−0.0529839	+	0.998595i	$0.516873\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.18953	0.613826
$28$	0	0
$29$	−0.173127	−0.0321489	−0.0160745	−	0.999871i	$-0.505117\pi$
$29$	−0.173127	−0.0321489	−0.0160745	+	0.999871i	$0.505117\pi$
$30$	0	0
$31$	9.06040	1.62730	0.813648	−	0.581358i	$-0.197478\pi$
$31$	9.06040	1.62730	0.813648	+	0.581358i	$0.197478\pi$
$32$	0	0
$33$	15.7417	2.74028
$34$	0	0
$35$	0.508203	0.0859020
$36$	0	0
$37$	−4.37907	−0.719914	−0.359957	−	0.932969i	$-0.617209\pi$
$37$	−4.37907	−0.719914	−0.359957	+	0.932969i	$0.617209\pi$
$38$	0	0
$39$	−3.18953	−0.510734
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−1.36266	−0.207804	−0.103902	−	0.994588i	$-0.533133\pi$
$43$	−1.36266	−0.207804	−0.103902	+	0.994588i	$0.533133\pi$
$44$	0	0
$45$	−4.18953	−0.624539
$46$	0	0
$47$	−10.1731	−1.48390	−0.741952	−	0.670453i	$-0.766100\pi$
$47$	−10.1731	−1.48390	−0.741952	+	0.670453i	$0.766100\pi$
$48$	0	0
$49$	−6.74173	−0.963104
$50$	0	0
$51$	2.68133	0.375461
$52$	0	0
$53$	8.17313	1.12267	0.561333	−	0.827590i	$-0.310289\pi$
$53$	8.17313	1.12267	0.561333	+	0.827590i	$0.310289\pi$
$54$	0	0
$55$	−5.87086	−0.791627
$56$	0	0
$57$	2.17313	0.287838
$58$	0	0
$59$	−9.91486	−1.29080	−0.645402	−	0.763843i	$-0.723310\pi$
$59$	−9.91486	−1.29080	−0.645402	+	0.763843i	$0.723310\pi$
$60$	0	0
$61$	−6.55220	−0.838923	−0.419461	−	0.907773i	$-0.637781\pi$
$61$	−6.55220	−0.838923	−0.419461	+	0.907773i	$0.637781\pi$
$62$	0	0
$63$	−2.12914	−0.268246
$64$	0	0
$65$	1.18953	0.147544
$66$	0	0
$67$	2.37907	0.290649	0.145325	−	0.989384i	$-0.453577\pi$
$67$	2.37907	0.290649	0.145325	+	0.989384i	$0.453577\pi$
$68$	0	0
$69$	−1.36266	−0.164045
$70$	0	0
$71$	−11.6977	−1.38827	−0.694133	−	0.719847i	$-0.744212\pi$
$71$	−11.6977	−1.38827	−0.694133	+	0.719847i	$0.744212\pi$
$72$	0	0
$73$	10.5522	1.23504	0.617521	−	0.786555i	$-0.288137\pi$
$73$	10.5522	1.23504	0.617521	+	0.786555i	$0.288137\pi$
$74$	0	0
$75$	2.68133	0.309613
$76$	0	0
$77$	−2.98359	−0.340012
$78$	0	0
$79$	−4.85446	−0.546169	−0.273085	−	0.961990i	$-0.588044\pi$
$79$	−4.85446	−0.546169	−0.273085	+	0.961990i	$0.588044\pi$
$80$	0	0
$81$	−4.01641	−0.446267
$82$	0	0
$83$	−15.7417	−1.72788	−0.863940	−	0.503595i	$-0.832010\pi$
$83$	−15.7417	−1.72788	−0.863940	+	0.503595i	$0.832010\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−0.464211	−0.0497687
$88$	0	0
$89$	−1.53579	−0.162793	−0.0813966	−	0.996682i	$-0.525938\pi$
$89$	−1.53579	−0.162793	−0.0813966	+	0.996682i	$0.525938\pi$
$90$	0	0
$91$	0.604525	0.0633715
$92$	0	0
$93$	24.2939	2.51916
$94$	0	0
$95$	−0.810466	−0.0831521
$96$	0	0
$97$	8.93126	0.906832	0.453416	−	0.891299i	$-0.350205\pi$
$97$	8.93126	0.906832	0.453416	+	0.891299i	$0.350205\pi$
$98$	0	0
$99$	24.5962	2.47201
$100$	0	0
$101$	13.7417	1.36735	0.683677	−	0.729785i	$-0.260380\pi$
$101$	13.7417	1.36735	0.683677	+	0.729785i	$0.260380\pi$
$102$	0	0
$103$	2.03281	0.200299	0.100150	−	0.994972i	$-0.468068\pi$
$103$	2.03281	0.200299	0.100150	+	0.994972i	$0.468068\pi$
$104$	0	0
$105$	1.36266	0.132982
$106$	0	0
$107$	−13.8709	−1.34095	−0.670474	−	0.741933i	$-0.733909\pi$
$107$	−13.8709	−1.34095	−0.670474	+	0.741933i	$0.733909\pi$
$108$	0	0
$109$	8.93126	0.855460	0.427730	−	0.903907i	$-0.359313\pi$
$109$	8.93126	0.855460	0.427730	+	0.903907i	$0.359313\pi$
$110$	0	0
$111$	−11.7417	−1.11448
$112$	0	0
$113$	−6.55220	−0.616379	−0.308189	−	0.951325i	$-0.599723\pi$
$113$	−6.55220	−0.616379	−0.308189	+	0.951325i	$0.599723\pi$
$114$	0	0
$115$	0.508203	0.0473902
$116$	0	0
$117$	−4.98359	−0.460733
$118$	0	0
$119$	−0.508203	−0.0465869
$120$	0	0
$121$	23.4671	2.13337
$122$	0	0
$123$	5.36266	0.483535
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.53579	−0.668693	−0.334347	−	0.942450i	$-0.608516\pi$
$127$	−7.53579	−0.668693	−0.334347	+	0.942450i	$0.608516\pi$
$128$	0	0
$129$	−3.65375	−0.321694
$130$	0	0
$131$	−9.52461	−0.832169	−0.416085	−	0.909326i	$-0.636598\pi$
$131$	−9.52461	−0.832169	−0.416085	+	0.909326i	$0.636598\pi$
$132$	0	0
$133$	−0.411882	−0.0357147
$134$	0	0
$135$	−3.18953	−0.274511
$136$	0	0
$137$	−15.7745	−1.34771	−0.673855	−	0.738864i	$-0.735363\pi$
$137$	−15.7745	−1.34771	−0.673855	+	0.738864i	$0.735363\pi$
$138$	0	0
$139$	14.8873	1.26272	0.631361	−	0.775489i	$-0.282497\pi$
$139$	14.8873	1.26272	0.631361	+	0.775489i	$0.282497\pi$
$140$	0	0
$141$	−27.2775	−2.29718
$142$	0	0
$143$	−6.98359	−0.583997
$144$	0	0
$145$	0.173127	0.0143774
$146$	0	0
$147$	−18.0768	−1.49095
$148$	0	0
$149$	−5.65375	−0.463173	−0.231586	−	0.972814i	$-0.574392\pi$
$149$	−5.65375	−0.463173	−0.231586	+	0.972814i	$0.574392\pi$
$150$	0	0
$151$	20.8461	1.69643	0.848217	−	0.529650i	$-0.177677\pi$
$151$	20.8461	1.69643	0.848217	+	0.529650i	$0.177677\pi$
$152$	0	0
$153$	4.18953	0.338704
$154$	0	0
$155$	−9.06040	−0.727749
$156$	0	0
$157$	−10.6925	−0.853355	−0.426678	−	0.904404i	$-0.640316\pi$
$157$	−10.6925	−0.853355	−0.426678	+	0.904404i	$0.640316\pi$
$158$	0	0
$159$	21.9149	1.73796
$160$	0	0
$161$	0.258271	0.0203546
$162$	0	0
$163$	−20.3379	−1.59299	−0.796494	−	0.604646i	$-0.793315\pi$
$163$	−20.3379	−1.59299	−0.796494	+	0.604646i	$0.793315\pi$
$164$	0	0
$165$	−15.7417	−1.22549
$166$	0	0
$167$	11.2335	0.869276	0.434638	−	0.900605i	$-0.356876\pi$
$167$	11.2335	0.869276	0.434638	+	0.900605i	$0.356876\pi$
$168$	0	0
$169$	−11.5850	−0.891155
$170$	0	0
$171$	3.39547	0.259658
$172$	0	0
$173$	15.4506	1.17469	0.587345	−	0.809336i	$-0.300173\pi$
$173$	15.4506	1.17469	0.587345	+	0.809336i	$0.300173\pi$
$174$	0	0
$175$	−0.508203	−0.0384166
$176$	0	0
$177$	−26.5850	−1.99825
$178$	0	0
$179$	25.7745	1.92648	0.963240	−	0.268643i	$-0.0865751\pi$
$179$	25.7745	1.92648	0.963240	+	0.268643i	$0.0865751\pi$
$180$	0	0
$181$	−6.69251	−0.497450	−0.248725	−	0.968574i	$-0.580012\pi$
$181$	−6.69251	−0.497450	−0.248725	+	0.968574i	$0.580012\pi$
$182$	0	0
$183$	−17.5686	−1.29871
$184$	0	0
$185$	4.37907	0.321955
$186$	0	0
$187$	5.87086	0.429320
$188$	0	0
$189$	−1.62093	−0.117905
$190$	0	0
$191$	−18.7253	−1.35492	−0.677458	−	0.735561i	$-0.736919\pi$
$191$	−18.7253	−1.35492	−0.677458	+	0.735561i	$0.736919\pi$
$192$	0	0
$193$	−0.725323	−0.0522099	−0.0261049	−	0.999659i	$-0.508310\pi$
$193$	−0.725323	−0.0522099	−0.0261049	+	0.999659i	$0.508310\pi$
$194$	0	0
$195$	3.18953	0.228407
$196$	0	0
$197$	−21.0716	−1.50129	−0.750644	−	0.660707i	$-0.770257\pi$
$197$	−21.0716	−1.50129	−0.750644	+	0.660707i	$0.770257\pi$
$198$	0	0
$199$	2.68133	0.190074	0.0950372	−	0.995474i	$-0.469703\pi$
$199$	2.68133	0.190074	0.0950372	+	0.995474i	$0.469703\pi$
$200$	0	0
$201$	6.37907	0.449945
$202$	0	0
$203$	0.0879839	0.00617526
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−2.12914	−0.147985
$208$	0	0
$209$	4.75814	0.329127
$210$	0	0
$211$	4.76647	0.328138	0.164069	−	0.986449i	$-0.447538\pi$
$211$	4.76647	0.328138	0.164069	+	0.986449i	$0.447538\pi$
$212$	0	0
$213$	−31.3655	−2.14913
$214$	0	0
$215$	1.36266	0.0929327
$216$	0	0
$217$	−4.60453	−0.312576
$218$	0	0
$219$	28.2939	1.91193
$220$	0	0
$221$	−1.18953	−0.0800167
$222$	0	0
$223$	−10.2611	−0.687135	−0.343567	−	0.939128i	$-0.611635\pi$
$223$	−10.2611	−0.687135	−0.343567	+	0.939128i	$0.611635\pi$
$224$	0	0
$225$	4.18953	0.279302
$226$	0	0
$227$	10.0768	0.668821	0.334411	−	0.942428i	$-0.391463\pi$
$227$	10.0768	0.668821	0.334411	+	0.942428i	$0.391463\pi$
$228$	0	0
$229$	27.7089	1.83106	0.915528	−	0.402253i	$-0.131773\pi$
$229$	27.7089	1.83106	0.915528	+	0.402253i	$0.131773\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−12.5194	−0.820172	−0.410086	−	0.912047i	$-0.634501\pi$
$233$	−12.5194	−0.820172	−0.410086	+	0.912047i	$0.634501\pi$
$234$	0	0
$235$	10.1731	0.663622
$236$	0	0
$237$	−13.0164	−0.845506
$238$	0	0
$239$	−21.8625	−1.41417	−0.707085	−	0.707129i	$-0.749990\pi$
$239$	−21.8625	−1.41417	−0.707085	+	0.707129i	$0.749990\pi$
$240$	0	0
$241$	−3.96719	−0.255549	−0.127774	−	0.991803i	$-0.540783\pi$
$241$	−3.96719	−0.255549	−0.127774	+	0.991803i	$0.540783\pi$
$242$	0	0
$243$	−20.3379	−1.30468
$244$	0	0
$245$	6.74173	0.430713
$246$	0	0
$247$	−0.964077	−0.0613427
$248$	0	0
$249$	−42.2088	−2.67487
$250$	0	0
$251$	14.0328	0.885743	0.442872	−	0.896585i	$-0.353960\pi$
$251$	14.0328	0.885743	0.442872	+	0.896585i	$0.353960\pi$
$252$	0	0
$253$	−2.98359	−0.187577
$254$	0	0
$255$	−2.68133	−0.167911
$256$	0	0
$257$	23.7089	1.47892	0.739461	−	0.673200i	$-0.235081\pi$
$257$	23.7089	1.47892	0.739461	+	0.673200i	$0.235081\pi$
$258$	0	0
$259$	2.22546	0.138283
$260$	0	0
$261$	−0.725323	−0.0448963
$262$	0	0
$263$	−11.1895	−0.689976	−0.344988	−	0.938607i	$-0.612117\pi$
$263$	−11.1895	−0.689976	−0.344988	+	0.938607i	$0.612117\pi$
$264$	0	0
$265$	−8.17313	−0.502071
$266$	0	0
$267$	−4.11796	−0.252015
$268$	0	0
$269$	10.1403	0.618266	0.309133	−	0.951019i	$-0.399961\pi$
$269$	10.1403	0.618266	0.309133	+	0.951019i	$0.399961\pi$
$270$	0	0
$271$	−20.7581	−1.26097	−0.630483	−	0.776203i	$-0.717143\pi$
$271$	−20.7581	−1.26097	−0.630483	+	0.776203i	$0.717143\pi$
$272$	0	0
$273$	1.62093	0.0981033
$274$	0	0
$275$	5.87086	0.354026
$276$	0	0
$277$	25.4835	1.53115	0.765576	−	0.643345i	$-0.222454\pi$
$277$	25.4835	1.53115	0.765576	+	0.643345i	$0.222454\pi$
$278$	0	0
$279$	37.9588	2.27254
$280$	0	0
$281$	4.17313	0.248948	0.124474	−	0.992223i	$-0.460276\pi$
$281$	4.17313	0.248948	0.124474	+	0.992223i	$0.460276\pi$
$282$	0	0
$283$	31.5275	1.87411	0.937056	−	0.349179i	$-0.113539\pi$
$283$	31.5275	1.87411	0.937056	+	0.349179i	$0.113539\pi$
$284$	0	0
$285$	−2.17313	−0.128725
$286$	0	0
$287$	−1.01641	−0.0599966
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	23.9477	1.40384
$292$	0	0
$293$	16.1731	0.944844	0.472422	−	0.881372i	$-0.343380\pi$
$293$	16.1731	0.944844	0.472422	+	0.881372i	$0.343380\pi$
$294$	0	0
$295$	9.91486	0.577265
$296$	0	0
$297$	18.7253	1.08655
$298$	0	0
$299$	0.604525	0.0349606
$300$	0	0
$301$	0.692509	0.0399156
$302$	0	0
$303$	36.8461	2.11675
$304$	0	0
$305$	6.55220	0.375178
$306$	0	0
$307$	−19.4835	−1.11198	−0.555990	−	0.831189i	$-0.687661\pi$
$307$	−19.4835	−1.11198	−0.555990	+	0.831189i	$0.687661\pi$
$308$	0	0
$309$	5.45065	0.310076
$310$	0	0
$311$	−1.11273	−0.0630970	−0.0315485	−	0.999502i	$-0.510044\pi$
$311$	−1.11273	−0.0630970	−0.0315485	+	0.999502i	$0.510044\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	2.12914	0.119963
$316$	0	0
$317$	−27.8625	−1.56492	−0.782458	−	0.622704i	$-0.786034\pi$
$317$	−27.8625	−1.56492	−0.782458	+	0.622704i	$0.786034\pi$
$318$	0	0
$319$	−1.01641	−0.0569079
$320$	0	0
$321$	−37.1924	−2.07588
$322$	0	0
$323$	0.810466	0.0450955
$324$	0	0
$325$	−1.18953	−0.0659835
$326$	0	0
$327$	23.9477	1.32431
$328$	0	0
$329$	5.17002	0.285032
$330$	0	0
$331$	23.1895	1.27461	0.637306	−	0.770611i	$-0.280049\pi$
$331$	23.1895	1.27461	0.637306	+	0.770611i	$0.280049\pi$
$332$	0	0
$333$	−18.3463	−1.00537
$334$	0	0
$335$	−2.37907	−0.129982
$336$	0	0
$337$	20.5850	1.12134	0.560668	−	0.828040i	$-0.310544\pi$
$337$	20.5850	1.12134	0.560668	+	0.828040i	$0.310544\pi$
$338$	0	0
$339$	−17.5686	−0.954195
$340$	0	0
$341$	53.1924	2.88053
$342$	0	0
$343$	6.98359	0.377079
$344$	0	0
$345$	1.36266	0.0733632
$346$	0	0
$347$	22.4231	1.20373	0.601866	−	0.798597i	$-0.294424\pi$
$347$	22.4231	1.20373	0.601866	+	0.798597i	$0.294424\pi$
$348$	0	0
$349$	10.3463	0.553822	0.276911	−	0.960896i	$-0.410689\pi$
$349$	10.3463	0.553822	0.276911	+	0.960896i	$0.410689\pi$
$350$	0	0
$351$	−3.79406	−0.202512
$352$	0	0
$353$	−19.4506	−1.03525	−0.517627	−	0.855607i	$-0.673184\pi$
$353$	−19.4506	−1.03525	−0.517627	+	0.855607i	$0.673184\pi$
$354$	0	0
$355$	11.6977	0.620851
$356$	0	0
$357$	−1.36266	−0.0721197
$358$	0	0
$359$	20.1536	1.06367	0.531833	−	0.846849i	$-0.321503\pi$
$359$	20.1536	1.06367	0.531833	+	0.846849i	$0.321503\pi$
$360$	0	0
$361$	−18.3431	−0.965429
$362$	0	0
$363$	62.9229	3.30260
$364$	0	0
$365$	−10.5522	−0.552327
$366$	0	0
$367$	−9.20071	−0.480273	−0.240137	−	0.970739i	$-0.577192\pi$
$367$	−9.20071	−0.480273	−0.240137	+	0.970739i	$0.577192\pi$
$368$	0	0
$369$	8.37907	0.436197
$370$	0	0
$371$	−4.15361	−0.215645
$372$	0	0
$373$	22.4342	1.16160	0.580800	−	0.814046i	$-0.302740\pi$
$373$	22.4342	1.16160	0.580800	+	0.814046i	$0.302740\pi$
$374$	0	0
$375$	−2.68133	−0.138463
$376$	0	0
$377$	0.205941	0.0106065
$378$	0	0
$379$	14.8873	0.764708	0.382354	−	0.924016i	$-0.375114\pi$
$379$	14.8873	0.764708	0.382354	+	0.924016i	$0.375114\pi$
$380$	0	0
$381$	−20.2059	−1.03518
$382$	0	0
$383$	1.15672	0.0591057	0.0295528	−	0.999563i	$-0.490592\pi$
$383$	1.15672	0.0591057	0.0295528	+	0.999563i	$0.490592\pi$
$384$	0	0
$385$	2.98359	0.152058
$386$	0	0
$387$	−5.70892	−0.290201
$388$	0	0
$389$	12.8133	0.649660	0.324830	−	0.945772i	$-0.394693\pi$
$389$	12.8133	0.649660	0.324830	+	0.945772i	$0.394693\pi$
$390$	0	0
$391$	−0.508203	−0.0257009
$392$	0	0
$393$	−25.5386	−1.28825
$394$	0	0
$395$	4.85446	0.244254
$396$	0	0
$397$	3.62093	0.181729	0.0908647	−	0.995863i	$-0.471037\pi$
$397$	3.62093	0.181729	0.0908647	+	0.995863i	$0.471037\pi$
$398$	0	0
$399$	−1.10439	−0.0552887
$400$	0	0
$401$	9.07158	0.453013	0.226506	−	0.974010i	$-0.427270\pi$
$401$	9.07158	0.453013	0.226506	+	0.974010i	$0.427270\pi$
$402$	0	0
$403$	−10.7777	−0.536873
$404$	0	0
$405$	4.01641	0.199577
$406$	0	0
$407$	−25.7089	−1.27434
$408$	0	0
$409$	12.5850	0.622289	0.311144	−	0.950363i	$-0.399288\pi$
$409$	12.5850	0.622289	0.311144	+	0.950363i	$0.399288\pi$
$410$	0	0
$411$	−42.2968	−2.08635
$412$	0	0
$413$	5.03876	0.247941
$414$	0	0
$415$	15.7417	0.772731
$416$	0	0
$417$	39.9177	1.95478
$418$	0	0
$419$	−6.47539	−0.316343	−0.158172	−	0.987412i	$-0.550560\pi$
$419$	−6.47539	−0.316343	−0.158172	+	0.987412i	$0.550560\pi$
$420$	0	0
$421$	27.6043	1.34535	0.672675	−	0.739938i	$-0.265145\pi$
$421$	27.6043	1.34535	0.672675	+	0.739938i	$0.265145\pi$
$422$	0	0
$423$	−42.6207	−2.07229
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	3.32985	0.161143
$428$	0	0
$429$	−18.7253	−0.904067
$430$	0	0
$431$	−4.85446	−0.233831	−0.116916	−	0.993142i	$-0.537301\pi$
$431$	−4.85446	−0.233831	−0.116916	+	0.993142i	$0.537301\pi$
$432$	0	0
$433$	18.2583	0.877436	0.438718	−	0.898625i	$-0.355433\pi$
$433$	18.2583	0.877436	0.438718	+	0.898625i	$0.355433\pi$
$434$	0	0
$435$	0.464211	0.0222572
$436$	0	0
$437$	−0.411882	−0.0197030
$438$	0	0
$439$	−2.62900	−0.125475	−0.0627377	−	0.998030i	$-0.519983\pi$
$439$	−2.62900	−0.125475	−0.0627377	+	0.998030i	$0.519983\pi$
$440$	0	0
$441$	−28.2447	−1.34499
$442$	0	0
$443$	−38.7253	−1.83990	−0.919948	−	0.392041i	$-0.871769\pi$
$443$	−38.7253	−1.83990	−0.919948	+	0.392041i	$0.871769\pi$
$444$	0	0
$445$	1.53579	0.0728034
$446$	0	0
$447$	−15.1596	−0.717023
$448$	0	0
$449$	−11.0388	−0.520951	−0.260476	−	0.965480i	$-0.583879\pi$
$449$	−11.0388	−0.520951	−0.260476	+	0.965480i	$0.583879\pi$
$450$	0	0
$451$	11.7417	0.552896
$452$	0	0
$453$	55.8953	2.62619
$454$	0	0
$455$	−0.604525	−0.0283406
$456$	0	0
$457$	−34.4999	−1.61384	−0.806918	−	0.590664i	$-0.798866\pi$
$457$	−34.4999	−1.61384	−0.806918	+	0.590664i	$0.798866\pi$
$458$	0	0
$459$	3.18953	0.148875
$460$	0	0
$461$	36.1432	1.68335	0.841677	−	0.539981i	$-0.181569\pi$
$461$	36.1432	1.68335	0.841677	+	0.539981i	$0.181569\pi$
$462$	0	0
$463$	19.2775	0.895902	0.447951	−	0.894058i	$-0.352154\pi$
$463$	19.2775	0.895902	0.447951	+	0.894058i	$0.352154\pi$
$464$	0	0
$465$	−24.2939	−1.12660
$466$	0	0
$467$	10.2911	0.476215	0.238107	−	0.971239i	$-0.423473\pi$
$467$	10.2911	0.476215	0.238107	+	0.971239i	$0.423473\pi$
$468$	0	0
$469$	−1.20905	−0.0558288
$470$	0	0
$471$	−28.6702	−1.32105
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	0.810466	0.0371867
$476$	0	0
$477$	34.2416	1.56781
$478$	0	0
$479$	−42.7693	−1.95418	−0.977090	−	0.212827i	$-0.931733\pi$
$479$	−42.7693	−1.95418	−0.977090	+	0.212827i	$0.931733\pi$
$480$	0	0
$481$	5.20905	0.237512
$482$	0	0
$483$	0.692509	0.0315103
$484$	0	0
$485$	−8.93126	−0.405548
$486$	0	0
$487$	0.184306	0.00835169	0.00417584	−	0.999991i	$-0.498671\pi$
$487$	0.184306	0.00835169	0.00417584	+	0.999991i	$0.498671\pi$
$488$	0	0
$489$	−54.5327	−2.46605
$490$	0	0
$491$	34.6074	1.56181	0.780904	−	0.624651i	$-0.214759\pi$
$491$	34.6074	1.56181	0.780904	+	0.624651i	$0.214759\pi$
$492$	0	0
$493$	−0.173127	−0.00779726
$494$	0	0
$495$	−24.5962	−1.10552
$496$	0	0
$497$	5.94483	0.266662
$498$	0	0
$499$	−34.7170	−1.55415	−0.777073	−	0.629411i	$-0.783296\pi$
$499$	−34.7170	−1.55415	−0.777073	+	0.629411i	$0.783296\pi$
$500$	0	0
$501$	30.1208	1.34570
$502$	0	0
$503$	−5.37100	−0.239481	−0.119741	−	0.992805i	$-0.538206\pi$
$503$	−5.37100	−0.239481	−0.119741	+	0.992805i	$0.538206\pi$
$504$	0	0
$505$	−13.7417	−0.611499
$506$	0	0
$507$	−31.0632	−1.37957
$508$	0	0
$509$	−15.8625	−0.703094	−0.351547	−	0.936170i	$-0.614344\pi$
$509$	−15.8625	−0.703094	−0.351547	+	0.936170i	$0.614344\pi$
$510$	0	0
$511$	−5.36266	−0.237230
$512$	0	0
$513$	2.58501	0.114131
$514$	0	0
$515$	−2.03281	−0.0895765
$516$	0	0
$517$	−59.7251	−2.62670
$518$	0	0
$519$	41.4283	1.81850
$520$	0	0
$521$	0.895609	0.0392374	0.0196187	−	0.999808i	$-0.493755\pi$
$521$	0.895609	0.0392374	0.0196187	+	0.999808i	$0.493755\pi$
$522$	0	0
$523$	29.7969	1.30293	0.651464	−	0.758680i	$-0.274155\pi$
$523$	29.7969	1.30293	0.651464	+	0.758680i	$0.274155\pi$
$524$	0	0
$525$	−1.36266	−0.0594714
$526$	0	0
$527$	9.06040	0.394677
$528$	0	0
$529$	−22.7417	−0.988771
$530$	0	0
$531$	−41.5386	−1.80262
$532$	0	0
$533$	−2.37907	−0.103049
$534$	0	0
$535$	13.8709	0.599690
$536$	0	0
$537$	69.1101	2.98232
$538$	0	0
$539$	−39.5798	−1.70482
$540$	0	0
$541$	19.5163	0.839070	0.419535	−	0.907739i	$-0.362193\pi$
$541$	19.5163	0.839070	0.419535	+	0.907739i	$0.362193\pi$
$542$	0	0
$543$	−17.9448	−0.770086
$544$	0	0
$545$	−8.93126	−0.382573
$546$	0	0
$547$	−3.28586	−0.140493	−0.0702465	−	0.997530i	$-0.522379\pi$
$547$	−3.28586	−0.140493	−0.0702465	+	0.997530i	$0.522379\pi$
$548$	0	0
$549$	−27.4506	−1.17156
$550$	0	0
$551$	−0.140314	−0.00597757
$552$	0	0
$553$	2.46705	0.104910
$554$	0	0
$555$	11.7417	0.498409
$556$	0	0
$557$	2.87609	0.121864	0.0609320	−	0.998142i	$-0.480593\pi$
$557$	2.87609	0.121864	0.0609320	+	0.998142i	$0.480593\pi$
$558$	0	0
$559$	1.62093	0.0685581
$560$	0	0
$561$	15.7417	0.664616
$562$	0	0
$563$	−41.5386	−1.75064	−0.875322	−	0.483540i	$-0.839351\pi$
$563$	−41.5386	−1.75064	−0.875322	+	0.483540i	$0.839351\pi$
$564$	0	0
$565$	6.55220	0.275653
$566$	0	0
$567$	2.04115	0.0857203
$568$	0	0
$569$	4.17313	0.174947	0.0874733	−	0.996167i	$-0.472121\pi$
$569$	4.17313	0.174947	0.0874733	+	0.996167i	$0.472121\pi$
$570$	0	0
$571$	31.6678	1.32525	0.662627	−	0.748949i	$-0.269441\pi$
$571$	31.6678	1.32525	0.662627	+	0.748949i	$0.269441\pi$
$572$	0	0
$573$	−50.2088	−2.09750
$574$	0	0
$575$	−0.508203	−0.0211935
$576$	0	0
$577$	29.3955	1.22375	0.611875	−	0.790955i	$-0.290416\pi$
$577$	29.3955	1.22375	0.611875	+	0.790955i	$0.290416\pi$
$578$	0	0
$579$	−1.94483	−0.0808244
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	47.9833	1.98727
$584$	0	0
$585$	4.98359	0.206046
$586$	0	0
$587$	12.0880	0.498924	0.249462	−	0.968385i	$-0.419746\pi$
$587$	12.0880	0.498924	0.249462	+	0.968385i	$0.419746\pi$
$588$	0	0
$589$	7.34314	0.302569
$590$	0	0
$591$	−56.4999	−2.32409
$592$	0	0
$593$	−35.4506	−1.45578	−0.727892	−	0.685692i	$-0.759500\pi$
$593$	−35.4506	−1.45578	−0.727892	+	0.685692i	$0.759500\pi$
$594$	0	0
$595$	0.508203	0.0208343
$596$	0	0
$597$	7.18953	0.294248
$598$	0	0
$599$	41.2804	1.68667	0.843335	−	0.537388i	$-0.180589\pi$
$599$	41.2804	1.68667	0.843335	+	0.537388i	$0.180589\pi$
$600$	0	0
$601$	−21.0716	−0.859528	−0.429764	−	0.902941i	$-0.641403\pi$
$601$	−21.0716	−0.859528	−0.429764	+	0.902941i	$0.641403\pi$
$602$	0	0
$603$	9.96719	0.405895
$604$	0	0
$605$	−23.4671	−0.954071
$606$	0	0
$607$	15.2992	0.620973	0.310487	−	0.950578i	$-0.399508\pi$
$607$	15.2992	0.620973	0.310487	+	0.950578i	$0.399508\pi$
$608$	0	0
$609$	0.235914	0.00955971
$610$	0	0
$611$	12.1013	0.489565
$612$	0	0
$613$	11.8269	0.477683	0.238841	−	0.971059i	$-0.423232\pi$
$613$	11.8269	0.477683	0.238841	+	0.971059i	$0.423232\pi$
$614$	0	0
$615$	−5.36266	−0.216243
$616$	0	0
$617$	−16.5850	−0.667687	−0.333844	−	0.942628i	$-0.608346\pi$
$617$	−16.5850	−0.667687	−0.333844	+	0.942628i	$0.608346\pi$
$618$	0	0
$619$	−47.9917	−1.92895	−0.964474	−	0.264178i	$-0.914899\pi$
$619$	−47.9917	−1.92895	−0.964474	+	0.264178i	$0.914899\pi$
$620$	0	0
$621$	−1.62093	−0.0650458
$622$	0	0
$623$	0.780493	0.0312698
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.7581	0.509511
$628$	0	0
$629$	−4.37907	−0.174605
$630$	0	0
$631$	20.4342	0.813474	0.406737	−	0.913545i	$-0.366667\pi$
$631$	20.4342	0.813474	0.406737	+	0.913545i	$0.366667\pi$
$632$	0	0
$633$	12.7805	0.507979
$634$	0	0
$635$	7.53579	0.299049
$636$	0	0
$637$	8.01952	0.317745
$638$	0	0
$639$	−49.0081	−1.93873
$640$	0	0
$641$	27.5163	1.08683	0.543414	−	0.839465i	$-0.317132\pi$
$641$	27.5163	1.08683	0.543414	+	0.839465i	$0.317132\pi$
$642$	0	0
$643$	9.61259	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$643$	9.61259	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$644$	0	0
$645$	3.65375	0.143866
$646$	0	0
$647$	34.3491	1.35040	0.675201	−	0.737634i	$-0.264057\pi$
$647$	34.3491	1.35040	0.675201	+	0.737634i	$0.264057\pi$
$648$	0	0
$649$	−58.2088	−2.28489
$650$	0	0
$651$	−12.3463	−0.483888
$652$	0	0
$653$	−19.9672	−0.781376	−0.390688	−	0.920523i	$-0.627763\pi$
$653$	−19.9672	−0.781376	−0.390688	+	0.920523i	$0.627763\pi$
$654$	0	0
$655$	9.52461	0.372157
$656$	0	0
$657$	44.2088	1.72475
$658$	0	0
$659$	0.898450	0.0349986	0.0174993	−	0.999847i	$-0.494430\pi$
$659$	0.898450	0.0349986	0.0174993	+	0.999847i	$0.494430\pi$
$660$	0	0
$661$	−26.4119	−1.02730	−0.513652	−	0.857999i	$-0.671708\pi$
$661$	−26.4119	−1.02730	−0.513652	+	0.857999i	$0.671708\pi$
$662$	0	0
$663$	−3.18953	−0.123871
$664$	0	0
$665$	0.411882	0.0159721
$666$	0	0
$667$	0.0879839	0.00340675
$668$	0	0
$669$	−27.5134	−1.06373
$670$	0	0
$671$	−38.4671	−1.48500
$672$	0	0
$673$	3.76125	0.144985	0.0724927	−	0.997369i	$-0.476905\pi$
$673$	3.76125	0.144985	0.0724927	+	0.997369i	$0.476905\pi$
$674$	0	0
$675$	3.18953	0.122765
$676$	0	0
$677$	10.0000	0.384331	0.192166	−	0.981363i	$-0.438449\pi$
$677$	10.0000	0.384331	0.192166	+	0.981363i	$0.438449\pi$
$678$	0	0
$679$	−4.53890	−0.174187
$680$	0	0
$681$	27.0192	1.03538
$682$	0	0
$683$	1.34103	0.0513129	0.0256565	−	0.999671i	$-0.491832\pi$
$683$	1.34103	0.0513129	0.0256565	+	0.999671i	$0.491832\pi$
$684$	0	0
$685$	15.7745	0.602714
$686$	0	0
$687$	74.2968	2.83460
$688$	0	0
$689$	−9.72221	−0.370387
$690$	0	0
$691$	−4.16195	−0.158328	−0.0791640	−	0.996862i	$-0.525225\pi$
$691$	−4.16195	−0.158328	−0.0791640	+	0.996862i	$0.525225\pi$
$692$	0	0
$693$	−12.4999	−0.474831
$694$	0	0
$695$	−14.8873	−0.564706
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	−33.5686	−1.26968
$700$	0	0
$701$	42.0880	1.58964	0.794821	−	0.606844i	$-0.207565\pi$
$701$	42.0880	1.58964	0.794821	+	0.606844i	$0.207565\pi$
$702$	0	0
$703$	−3.54909	−0.133856
$704$	0	0
$705$	27.2775	1.02733
$706$	0	0
$707$	−6.98359	−0.262645
$708$	0	0
$709$	18.5522	0.696742	0.348371	−	0.937357i	$-0.386735\pi$
$709$	18.5522	0.696742	0.348371	+	0.937357i	$0.386735\pi$
$710$	0	0
$711$	−20.3379	−0.762731
$712$	0	0
$713$	−4.60453	−0.172441
$714$	0	0
$715$	6.98359	0.261172
$716$	0	0
$717$	−58.6207	−2.18923
$718$	0	0
$719$	−3.69774	−0.137902	−0.0689512	−	0.997620i	$-0.521965\pi$
$719$	−3.69774	−0.137902	−0.0689512	+	0.997620i	$0.521965\pi$
$720$	0	0
$721$	−1.03308	−0.0384740
$722$	0	0
$723$	−10.6373	−0.395607
$724$	0	0
$725$	−0.173127	−0.00642979
$726$	0	0
$727$	15.2119	0.564178	0.282089	−	0.959388i	$-0.408973\pi$
$727$	15.2119	0.564178	0.282089	+	0.959388i	$0.408973\pi$
$728$	0	0
$729$	−42.4835	−1.57346
$730$	0	0
$731$	−1.36266	−0.0503998
$732$	0	0
$733$	13.3075	0.491523	0.245762	−	0.969330i	$-0.420962\pi$
$733$	13.3075	0.491523	0.245762	+	0.969330i	$0.420962\pi$
$734$	0	0
$735$	18.0768	0.666773
$736$	0	0
$737$	13.9672	0.514488
$738$	0	0
$739$	3.94767	0.145217	0.0726087	−	0.997361i	$-0.476868\pi$
$739$	3.94767	0.145217	0.0726087	+	0.997361i	$0.476868\pi$
$740$	0	0
$741$	−2.58501	−0.0949627
$742$	0	0
$743$	−50.3931	−1.84874	−0.924372	−	0.381493i	$-0.875410\pi$
$743$	−50.3931	−1.84874	−0.924372	+	0.381493i	$0.875410\pi$
$744$	0	0
$745$	5.65375	0.207137
$746$	0	0
$747$	−65.9505	−2.41300
$748$	0	0
$749$	7.04922	0.257573
$750$	0	0
$751$	−52.9781	−1.93320	−0.966599	−	0.256293i	$-0.917499\pi$
$751$	−52.9781	−1.93320	−0.966599	+	0.256293i	$0.917499\pi$
$752$	0	0
$753$	37.6266	1.37119
$754$	0	0
$755$	−20.8461	−0.758668
$756$	0	0
$757$	−25.3655	−0.921925	−0.460962	−	0.887420i	$-0.652496\pi$
$757$	−25.3655	−0.921925	−0.460962	+	0.887420i	$0.652496\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	47.1924	1.71072	0.855361	−	0.518032i	$-0.173335\pi$
$761$	47.1924	1.71072	0.855361	+	0.518032i	$0.173335\pi$
$762$	0	0
$763$	−4.53890	−0.164319
$764$	0	0
$765$	−4.18953	−0.151473
$766$	0	0
$767$	11.7941	0.425859
$768$	0	0
$769$	10.8105	0.389835	0.194918	−	0.980820i	$-0.437556\pi$
$769$	10.8105	0.389835	0.194918	+	0.980820i	$0.437556\pi$
$770$	0	0
$771$	63.5714	2.28947
$772$	0	0
$773$	49.9833	1.79778	0.898888	−	0.438179i	$-0.144376\pi$
$773$	49.9833	1.79778	0.898888	+	0.438179i	$0.144376\pi$
$774$	0	0
$775$	9.06040	0.325459
$776$	0	0
$777$	5.96719	0.214072
$778$	0	0
$779$	1.62093	0.0580759
$780$	0	0
$781$	−68.6758	−2.45741
$782$	0	0
$783$	−0.552195	−0.0197339
$784$	0	0
$785$	10.6925	0.381632
$786$	0	0
$787$	8.13198	0.289874	0.144937	−	0.989441i	$-0.453702\pi$
$787$	8.13198	0.289874	0.144937	+	0.989441i	$0.453702\pi$
$788$	0	0
$789$	−30.0028	−1.06813
$790$	0	0
$791$	3.32985	0.118396
$792$	0	0
$793$	7.79406	0.276775
$794$	0	0
$795$	−21.9149	−0.777240
$796$	0	0
$797$	49.7251	1.76135	0.880676	−	0.473719i	$-0.157089\pi$
$797$	49.7251	1.76135	0.880676	+	0.473719i	$0.157089\pi$
$798$	0	0
$799$	−10.1731	−0.359899
$800$	0	0
$801$	−6.43424	−0.227343
$802$	0	0
$803$	61.9505	2.18619
$804$	0	0
$805$	−0.258271	−0.00910285
$806$	0	0
$807$	27.1895	0.957117
$808$	0	0
$809$	13.4178	0.471746	0.235873	−	0.971784i	$-0.424205\pi$
$809$	13.4178	0.471746	0.235873	+	0.971784i	$0.424205\pi$
$810$	0	0
$811$	−13.4590	−0.472609	−0.236304	−	0.971679i	$-0.575936\pi$
$811$	−13.4590	−0.472609	−0.236304	+	0.971679i	$0.575936\pi$
$812$	0	0
$813$	−55.6594	−1.95206
$814$	0	0
$815$	20.3379	0.712406
$816$	0	0
$817$	−1.10439	−0.0386377
$818$	0	0
$819$	2.53268	0.0884990
$820$	0	0
$821$	−38.7282	−1.35162	−0.675811	−	0.737075i	$-0.736206\pi$
$821$	−38.7282	−1.35162	−0.675811	+	0.737075i	$0.736206\pi$
$822$	0	0
$823$	−29.4423	−1.02629	−0.513147	−	0.858301i	$-0.671520\pi$
$823$	−29.4423	−1.02629	−0.513147	+	0.858301i	$0.671520\pi$
$824$	0	0
$825$	15.7417	0.548057
$826$	0	0
$827$	−17.6126	−0.612450	−0.306225	−	0.951959i	$-0.599066\pi$
$827$	−17.6126	−0.612450	−0.306225	+	0.951959i	$0.599066\pi$
$828$	0	0
$829$	−0.895609	−0.0311058	−0.0155529	−	0.999879i	$-0.504951\pi$
$829$	−0.895609	−0.0311058	−0.0155529	+	0.999879i	$0.504951\pi$
$830$	0	0
$831$	68.3296	2.37033
$832$	0	0
$833$	−6.74173	−0.233587
$834$	0	0
$835$	−11.2335	−0.388752
$836$	0	0
$837$	28.8984	0.998877
$838$	0	0
$839$	−39.0276	−1.34738	−0.673691	−	0.739013i	$-0.735292\pi$
$839$	−39.0276	−1.34738	−0.673691	+	0.739013i	$0.735292\pi$
$840$	0	0
$841$	−28.9700	−0.998966
$842$	0	0
$843$	11.1895	0.385388
$844$	0	0
$845$	11.5850	0.398536
$846$	0	0
$847$	−11.9260	−0.409783
$848$	0	0
$849$	84.5355	2.90125
$850$	0	0
$851$	2.22546	0.0762877
$852$	0	0
$853$	28.2088	0.965850	0.482925	−	0.875662i	$-0.339574\pi$
$853$	28.2088	0.965850	0.482925	+	0.875662i	$0.339574\pi$
$854$	0	0
$855$	−3.39547	−0.116123
$856$	0	0
$857$	−33.9700	−1.16039	−0.580197	−	0.814476i	$-0.697024\pi$
$857$	−33.9700	−1.16039	−0.580197	+	0.814476i	$0.697024\pi$
$858$	0	0
$859$	−2.60737	−0.0889622	−0.0444811	−	0.999010i	$-0.514163\pi$
$859$	−2.60737	−0.0889622	−0.0444811	+	0.999010i	$0.514163\pi$
$860$	0	0
$861$	−2.72532	−0.0928787
$862$	0	0
$863$	−3.93437	−0.133928	−0.0669638	−	0.997755i	$-0.521331\pi$
$863$	−3.93437	−0.133928	−0.0669638	+	0.997755i	$0.521331\pi$
$864$	0	0
$865$	−15.4506	−0.525338
$866$	0	0
$867$	2.68133	0.0910628
$868$	0	0
$869$	−28.4999	−0.966792
$870$	0	0
$871$	−2.82998	−0.0958903
$872$	0	0
$873$	37.4178	1.26640
$874$	0	0
$875$	0.508203	0.0171804
$876$	0	0
$877$	−15.1044	−0.510039	−0.255020	−	0.966936i	$-0.582082\pi$
$877$	−15.1044	−0.510039	−0.255020	+	0.966936i	$0.582082\pi$
$878$	0	0
$879$	43.3655	1.46268
$880$	0	0
$881$	−28.9669	−0.975920	−0.487960	−	0.872866i	$-0.662259\pi$
$881$	−28.9669	−0.975920	−0.487960	+	0.872866i	$0.662259\pi$
$882$	0	0
$883$	−22.3134	−0.750907	−0.375454	−	0.926841i	$-0.622513\pi$
$883$	−22.3134	−0.750907	−0.375454	+	0.926841i	$0.622513\pi$
$884$	0	0
$885$	26.5850	0.893645
$886$	0	0
$887$	−14.4587	−0.485476	−0.242738	−	0.970092i	$-0.578046\pi$
$887$	−14.4587	−0.485476	−0.242738	+	0.970092i	$0.578046\pi$
$888$	0	0
$889$	3.82971	0.128444
$890$	0	0
$891$	−23.5798	−0.789952
$892$	0	0
$893$	−8.24497	−0.275908
$894$	0	0
$895$	−25.7745	−0.861548
$896$	0	0
$897$	1.62093	0.0541213
$898$	0	0
$899$	−1.56860	−0.0523158
$900$	0	0
$901$	8.17313	0.272286
$902$	0	0
$903$	1.85685	0.0617920
$904$	0	0
$905$	6.69251	0.222467
$906$	0	0
$907$	−28.1976	−0.936286	−0.468143	−	0.883653i	$-0.655077\pi$
$907$	−28.1976	−0.936286	−0.468143	+	0.883653i	$0.655077\pi$
$908$	0	0
$909$	57.5714	1.90952
$910$	0	0
$911$	22.0468	0.730444	0.365222	−	0.930920i	$-0.380993\pi$
$911$	22.0468	0.730444	0.365222	+	0.930920i	$0.380993\pi$
$912$	0	0
$913$	−92.4176	−3.05857
$914$	0	0
$915$	17.5686	0.580800
$916$	0	0
$917$	4.84044	0.159845
$918$	0	0
$919$	29.8625	0.985074	0.492537	−	0.870292i	$-0.336070\pi$
$919$	29.8625	0.985074	0.492537	+	0.870292i	$0.336070\pi$
$920$	0	0
$921$	−52.2416	−1.72142
$922$	0	0
$923$	13.9149	0.458013
$924$	0	0
$925$	−4.37907	−0.143983
$926$	0	0
$927$	8.51654	0.279720
$928$	0	0
$929$	50.8685	1.66894	0.834470	−	0.551053i	$-0.185774\pi$
$929$	50.8685	1.66894	0.834470	+	0.551053i	$0.185774\pi$
$930$	0	0
$931$	−5.46394	−0.179073
$932$	0	0
$933$	−2.98359	−0.0976785
$934$	0	0
$935$	−5.87086	−0.191998
$936$	0	0
$937$	14.3463	0.468672	0.234336	−	0.972156i	$-0.424708\pi$
$937$	14.3463	0.468672	0.234336	+	0.972156i	$0.424708\pi$
$938$	0	0
$939$	5.36266	0.175004
$940$	0	0
$941$	−26.3819	−0.860026	−0.430013	−	0.902823i	$-0.641491\pi$
$941$	−26.3819	−0.860026	−0.430013	+	0.902823i	$0.641491\pi$
$942$	0	0
$943$	−1.01641	−0.0330988
$944$	0	0
$945$	1.62093	0.0527289
$946$	0	0
$947$	−33.6649	−1.09396	−0.546982	−	0.837145i	$-0.684223\pi$
$947$	−33.6649	−1.09396	−0.546982	+	0.837145i	$0.684223\pi$
$948$	0	0
$949$	−12.5522	−0.407462
$950$	0	0
$951$	−74.7086	−2.42259
$952$	0	0
$953$	−22.3296	−0.723326	−0.361663	−	0.932309i	$-0.617791\pi$
$953$	−22.3296	−0.723326	−0.361663	+	0.932309i	$0.617791\pi$
$954$	0	0
$955$	18.7253	0.605937
$956$	0	0
$957$	−2.72532	−0.0880972
$958$	0	0
$959$	8.01668	0.258872
$960$	0	0
$961$	51.0908	1.64809
$962$	0	0
$963$	−58.1125	−1.87265
$964$	0	0
$965$	0.725323	0.0233490
$966$	0	0
$967$	37.7745	1.21475	0.607374	−	0.794416i	$-0.292223\pi$
$967$	37.7745	1.21475	0.607374	+	0.794416i	$0.292223\pi$
$968$	0	0
$969$	2.17313	0.0698109
$970$	0	0
$971$	25.5030	0.818429	0.409215	−	0.912438i	$-0.365803\pi$
$971$	25.5030	0.818429	0.409215	+	0.912438i	$0.365803\pi$
$972$	0	0
$973$	−7.56576	−0.242547
$974$	0	0
$975$	−3.18953	−0.102147
$976$	0	0
$977$	7.17002	0.229389	0.114695	−	0.993401i	$-0.463411\pi$
$977$	7.17002	0.229389	0.114695	+	0.993401i	$0.463411\pi$
$978$	0	0
$979$	−9.01641	−0.288166
$980$	0	0
$981$	37.4178	1.19466
$982$	0	0
$983$	−32.1843	−1.02652	−0.513260	−	0.858233i	$-0.671562\pi$
$983$	−32.1843	−1.02652	−0.513260	+	0.858233i	$0.671562\pi$
$984$	0	0
$985$	21.0716	0.671396
$986$	0	0
$987$	13.8625	0.441249
$988$	0	0
$989$	0.692509	0.0220205
$990$	0	0
$991$	−12.3902	−0.393589	−0.196795	−	0.980445i	$-0.563053\pi$
$991$	−12.3902	−0.393589	−0.196795	+	0.980445i	$0.563053\pi$
$992$	0	0
$993$	62.1788	1.97318
$994$	0	0
$995$	−2.68133	−0.0850039
$996$	0	0
$997$	12.0328	0.381083	0.190542	−	0.981679i	$-0.438976\pi$
$997$	12.0328	0.381083	0.190542	+	0.981679i	$0.438976\pi$
$998$	0	0
$999$	−13.9672	−0.441902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	680.2.a.f.1.3	✓	3
3.2	odd	2		6120.2.a.br.1.2		3
4.3	odd	2		1360.2.a.r.1.1		3
5.2	odd	4		3400.2.e.k.2449.1		6
5.3	odd	4		3400.2.e.k.2449.6		6
5.4	even	2		3400.2.a.n.1.1		3
8.3	odd	2		5440.2.a.bt.1.3		3
8.5	even	2		5440.2.a.bo.1.1		3
20.19	odd	2		6800.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.a.f.1.3	✓	3	1.1	even	1	trivial
1360.2.a.r.1.1		3	4.3	odd	2
3400.2.a.n.1.1		3	5.4	even	2
3400.2.e.k.2449.1		6	5.2	odd	4
3400.2.e.k.2449.6		6	5.3	odd	4
5440.2.a.bo.1.1		3	8.5	even	2
5440.2.a.bt.1.3		3	8.3	odd	2
6120.2.a.br.1.2		3	3.2	odd	2
6800.2.a.bo.1.3		3	20.19	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:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68133 q^{3} -1.00000 q^{5} -0.508203 q^{7} +4.18953 q^{9} +5.87086 q^{11} -1.18953 q^{13} -2.68133 q^{15} +1.00000 q^{17} +0.810466 q^{19} -1.36266 q^{21} -0.508203 q^{23} +1.00000 q^{25} +3.18953 q^{27} -0.173127 q^{29} +9.06040 q^{31} +15.7417 q^{33} +0.508203 q^{35} -4.37907 q^{37} -3.18953 q^{39} +2.00000 q^{41} -1.36266 q^{43} -4.18953 q^{45} -10.1731 q^{47} -6.74173 q^{49} +2.68133 q^{51} +8.17313 q^{53} -5.87086 q^{55} +2.17313 q^{57} -9.91486 q^{59} -6.55220 q^{61} -2.12914 q^{63} +1.18953 q^{65} +2.37907 q^{67} -1.36266 q^{69} -11.6977 q^{71} +10.5522 q^{73} +2.68133 q^{75} -2.98359 q^{77} -4.85446 q^{79} -4.01641 q^{81} -15.7417 q^{83} -1.00000 q^{85} -0.464211 q^{87} -1.53579 q^{89} +0.604525 q^{91} +24.2939 q^{93} -0.810466 q^{95} +8.93126 q^{97} +24.5962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 3 q^{5} + 4 q^{9} + 2 q^{11} + 5 q^{13} - q^{15} + 3 q^{17} + 11 q^{19} + 10 q^{21} + 3 q^{25} + q^{27} + 5 q^{29} + 3 q^{31} + 16 q^{33} + 4 q^{37} - q^{39} + 6 q^{41} + 10 q^{43} - 4 q^{45}+ \cdots + 30 q^{99}+O(q^{100}) \) 3 * q + q^3 - 3 * q^5 + 4 * q^9 + 2 * q^11 + 5 * q^13 - q^15 + 3 * q^17 + 11 * q^19 + 10 * q^21 + 3 * q^25 + q^27 + 5 * q^29 + 3 * q^31 + 16 * q^33 + 4 * q^37 - q^39 + 6 * q^41 + 10 * q^43 - 4 * q^45 - 25 * q^47 + 11 * q^49 + q^51 + 19 * q^53 - 2 * q^55 + q^57 + 7 * q^59 + 3 * q^61 - 22 * q^63 - 5 * q^65 - 10 * q^67 + 10 * q^69 - 25 * q^71 + 9 * q^73 + q^75 - 12 * q^77 - 2 * q^79 - 9 * q^81 - 16 * q^83 - 3 * q^85 - 21 * q^87 + 15 * q^89 + 22 * q^91 + 19 * q^93 - 11 * q^95 - 13 * q^97 + 30 * q^99

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