LMFDB - Embedded newform 6032.2.a.bb.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 10x^{8} + 21x^{7} + 28x^{6} - 67x^{5} - 20x^{4} + 76x^{3} - 8x^{2} - 26x + 8 $ x^10 - 2x^9 - 10x^8 + 21x^7 + 28x^6 - 67x^5 - 20x^4 + 76x^3 - 8x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.12886$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39322 q^{3} -2.03033 q^{5} +4.56146 q^{7} +2.72750 q^{9} +O(q^{10})$	$q+2.39322 q^{3} -2.03033 q^{5} +4.56146 q^{7} +2.72750 q^{9} -1.62248 q^{11} -1.00000 q^{13} -4.85902 q^{15} +7.80302 q^{17} +6.68107 q^{19} +10.9166 q^{21} -5.09377 q^{23} -0.877765 q^{25} -0.652163 q^{27} +1.00000 q^{29} -3.04242 q^{31} -3.88296 q^{33} -9.26127 q^{35} -0.865819 q^{37} -2.39322 q^{39} +4.42022 q^{41} +5.81334 q^{43} -5.53771 q^{45} -3.77978 q^{47} +13.8070 q^{49} +18.6743 q^{51} +11.2927 q^{53} +3.29418 q^{55} +15.9893 q^{57} -1.79350 q^{59} +4.30884 q^{61} +12.4414 q^{63} +2.03033 q^{65} +8.09327 q^{67} -12.1905 q^{69} -1.76333 q^{71} +0.476101 q^{73} -2.10068 q^{75} -7.40091 q^{77} +14.6884 q^{79} -9.74326 q^{81} -0.448338 q^{83} -15.8427 q^{85} +2.39322 q^{87} -17.0217 q^{89} -4.56146 q^{91} -7.28117 q^{93} -13.5648 q^{95} +1.41271 q^{97} -4.42532 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * 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
$3$	2.39322	1.38173	0.690863	−	0.722986i	$-0.257231\pi$
$3$	2.39322	1.38173	0.690863	+	0.722986i	$0.257231\pi$
$4$	0	0
$5$	−2.03033	−0.907991	−0.453995	−	0.891004i	$-0.650002\pi$
$5$	−2.03033	−0.907991	−0.453995	+	0.891004i	$0.650002\pi$
$6$	0	0
$7$	4.56146	1.72407	0.862036	−	0.506848i	$-0.169189\pi$
$7$	4.56146	1.72407	0.862036	+	0.506848i	$0.169189\pi$
$8$	0	0
$9$	2.72750	0.909165
$10$	0	0
$11$	−1.62248	−0.489198	−0.244599	−	0.969624i	$-0.578656\pi$
$11$	−1.62248	−0.489198	−0.244599	+	0.969624i	$0.578656\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.85902	−1.25459
$16$	0	0
$17$	7.80302	1.89251	0.946255	−	0.323420i	$-0.104833\pi$
$17$	7.80302	1.89251	0.946255	+	0.323420i	$0.104833\pi$
$18$	0	0
$19$	6.68107	1.53274	0.766371	−	0.642398i	$-0.222060\pi$
$19$	6.68107	1.53274	0.766371	+	0.642398i	$0.222060\pi$
$20$	0	0
$21$	10.9166	2.38219
$22$	0	0
$23$	−5.09377	−1.06212	−0.531062	−	0.847333i	$-0.678207\pi$
$23$	−5.09377	−1.06212	−0.531062	+	0.847333i	$0.678207\pi$
$24$	0	0
$25$	−0.877765	−0.175553
$26$	0	0
$27$	−0.652163	−0.125509
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.04242	−0.546434	−0.273217	−	0.961952i	$-0.588088\pi$
$31$	−3.04242	−0.546434	−0.273217	+	0.961952i	$0.588088\pi$
$32$	0	0
$33$	−3.88296	−0.675937
$34$	0	0
$35$	−9.26127	−1.56544
$36$	0	0
$37$	−0.865819	−0.142340	−0.0711699	−	0.997464i	$-0.522673\pi$
$37$	−0.865819	−0.142340	−0.0711699	+	0.997464i	$0.522673\pi$
$38$	0	0
$39$	−2.39322	−0.383222
$40$	0	0
$41$	4.42022	0.690322	0.345161	−	0.938543i	$-0.387824\pi$
$41$	4.42022	0.690322	0.345161	+	0.938543i	$0.387824\pi$
$42$	0	0
$43$	5.81334	0.886526	0.443263	−	0.896392i	$-0.353821\pi$
$43$	5.81334	0.886526	0.443263	+	0.896392i	$0.353821\pi$
$44$	0	0
$45$	−5.53771	−0.825513
$46$	0	0
$47$	−3.77978	−0.551337	−0.275669	−	0.961253i	$-0.588899\pi$
$47$	−3.77978	−0.551337	−0.275669	+	0.961253i	$0.588899\pi$
$48$	0	0
$49$	13.8070	1.97242
$50$	0	0
$51$	18.6743	2.61493
$52$	0	0
$53$	11.2927	1.55118	0.775589	−	0.631239i	$-0.217453\pi$
$53$	11.2927	1.55118	0.775589	+	0.631239i	$0.217453\pi$
$54$	0	0
$55$	3.29418	0.444187
$56$	0	0
$57$	15.9893	2.11783
$58$	0	0
$59$	−1.79350	−0.233494	−0.116747	−	0.993162i	$-0.537247\pi$
$59$	−1.79350	−0.233494	−0.116747	+	0.993162i	$0.537247\pi$
$60$	0	0
$61$	4.30884	0.551690	0.275845	−	0.961202i	$-0.411042\pi$
$61$	4.30884	0.551690	0.275845	+	0.961202i	$0.411042\pi$
$62$	0	0
$63$	12.4414	1.56747
$64$	0	0
$65$	2.03033	0.251831
$66$	0	0
$67$	8.09327	0.988750	0.494375	−	0.869249i	$-0.335397\pi$
$67$	8.09327	0.988750	0.494375	+	0.869249i	$0.335397\pi$
$68$	0	0
$69$	−12.1905	−1.46756
$70$	0	0
$71$	−1.76333	−0.209269	−0.104635	−	0.994511i	$-0.533367\pi$
$71$	−1.76333	−0.209269	−0.104635	+	0.994511i	$0.533367\pi$
$72$	0	0
$73$	0.476101	0.0557234	0.0278617	−	0.999612i	$-0.491130\pi$
$73$	0.476101	0.0557234	0.0278617	+	0.999612i	$0.491130\pi$
$74$	0	0
$75$	−2.10068	−0.242566
$76$	0	0
$77$	−7.40091	−0.843411
$78$	0	0
$79$	14.6884	1.65257	0.826285	−	0.563252i	$-0.190450\pi$
$79$	14.6884	1.65257	0.826285	+	0.563252i	$0.190450\pi$
$80$	0	0
$81$	−9.74326	−1.08258
$82$	0	0
$83$	−0.448338	−0.0492114	−0.0246057	−	0.999697i	$-0.507833\pi$
$83$	−0.448338	−0.0492114	−0.0246057	+	0.999697i	$0.507833\pi$
$84$	0	0
$85$	−15.8427	−1.71838
$86$	0	0
$87$	2.39322	0.256580
$88$	0	0
$89$	−17.0217	−1.80429	−0.902146	−	0.431430i	$-0.858009\pi$
$89$	−17.0217	−1.80429	−0.902146	+	0.431430i	$0.858009\pi$
$90$	0	0
$91$	−4.56146	−0.478171
$92$	0	0
$93$	−7.28117	−0.755022
$94$	0	0
$95$	−13.5648	−1.39172
$96$	0	0
$97$	1.41271	0.143439	0.0717193	−	0.997425i	$-0.477151\pi$
$97$	1.41271	0.143439	0.0717193	+	0.997425i	$0.477151\pi$
$98$	0	0
$99$	−4.42532	−0.444761
$100$	0	0
$101$	7.28267	0.724652	0.362326	−	0.932051i	$-0.381983\pi$
$101$	7.28267	0.724652	0.362326	+	0.932051i	$0.381983\pi$
$102$	0	0
$103$	−7.47390	−0.736426	−0.368213	−	0.929742i	$-0.620030\pi$
$103$	−7.47390	−0.736426	−0.368213	+	0.929742i	$0.620030\pi$
$104$	0	0
$105$	−22.1642	−2.16301
$106$	0	0
$107$	1.17271	0.113370	0.0566851	−	0.998392i	$-0.481947\pi$
$107$	1.17271	0.113370	0.0566851	+	0.998392i	$0.481947\pi$
$108$	0	0
$109$	−4.50850	−0.431836	−0.215918	−	0.976411i	$-0.569274\pi$
$109$	−4.50850	−0.431836	−0.215918	+	0.976411i	$0.569274\pi$
$110$	0	0
$111$	−2.07209	−0.196675
$112$	0	0
$113$	−7.24168	−0.681240	−0.340620	−	0.940201i	$-0.610637\pi$
$113$	−7.24168	−0.681240	−0.340620	+	0.940201i	$0.610637\pi$
$114$	0	0
$115$	10.3420	0.964399
$116$	0	0
$117$	−2.72750	−0.252157
$118$	0	0
$119$	35.5932	3.26282
$120$	0	0
$121$	−8.36754	−0.760686
$122$	0	0
$123$	10.5785	0.953835
$124$	0	0
$125$	11.9338	1.06739
$126$	0	0
$127$	−13.2627	−1.17687	−0.588436	−	0.808544i	$-0.700256\pi$
$127$	−13.2627	−1.17687	−0.588436	+	0.808544i	$0.700256\pi$
$128$	0	0
$129$	13.9126	1.22494
$130$	0	0
$131$	−1.05643	−0.0923003	−0.0461502	−	0.998935i	$-0.514695\pi$
$131$	−1.05643	−0.0923003	−0.0461502	+	0.998935i	$0.514695\pi$
$132$	0	0
$133$	30.4755	2.64256
$134$	0	0
$135$	1.32411	0.113961
$136$	0	0
$137$	3.94882	0.337370	0.168685	−	0.985670i	$-0.446048\pi$
$137$	3.94882	0.337370	0.168685	+	0.985670i	$0.446048\pi$
$138$	0	0
$139$	13.5022	1.14524	0.572621	−	0.819820i	$-0.305927\pi$
$139$	13.5022	1.14524	0.572621	+	0.819820i	$0.305927\pi$
$140$	0	0
$141$	−9.04584	−0.761797
$142$	0	0
$143$	1.62248	0.135679
$144$	0	0
$145$	−2.03033	−0.168610
$146$	0	0
$147$	33.0431	2.72535
$148$	0	0
$149$	−2.50461	−0.205185	−0.102593	−	0.994723i	$-0.532714\pi$
$149$	−2.50461	−0.205185	−0.102593	+	0.994723i	$0.532714\pi$
$150$	0	0
$151$	7.82896	0.637111	0.318556	−	0.947904i	$-0.396802\pi$
$151$	7.82896	0.637111	0.318556	+	0.947904i	$0.396802\pi$
$152$	0	0
$153$	21.2827	1.72060
$154$	0	0
$155$	6.17711	0.496157
$156$	0	0
$157$	−3.93494	−0.314042	−0.157021	−	0.987595i	$-0.550189\pi$
$157$	−3.93494	−0.314042	−0.157021	+	0.987595i	$0.550189\pi$
$158$	0	0
$159$	27.0260	2.14330
$160$	0	0
$161$	−23.2350	−1.83118
$162$	0	0
$163$	23.8893	1.87116	0.935578	−	0.353120i	$-0.114879\pi$
$163$	23.8893	1.87116	0.935578	+	0.353120i	$0.114879\pi$
$164$	0	0
$165$	7.88369	0.613744
$166$	0	0
$167$	−15.6162	−1.20841	−0.604207	−	0.796827i	$-0.706510\pi$
$167$	−15.6162	−1.20841	−0.604207	+	0.796827i	$0.706510\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.2226	1.39352
$172$	0	0
$173$	4.63173	0.352144	0.176072	−	0.984377i	$-0.443661\pi$
$173$	4.63173	0.352144	0.176072	+	0.984377i	$0.443661\pi$
$174$	0	0
$175$	−4.00389	−0.302666
$176$	0	0
$177$	−4.29225	−0.322625
$178$	0	0
$179$	0.616224	0.0460587	0.0230294	−	0.999735i	$-0.492669\pi$
$179$	0.616224	0.0460587	0.0230294	+	0.999735i	$0.492669\pi$
$180$	0	0
$181$	12.6763	0.942224	0.471112	−	0.882074i	$-0.343853\pi$
$181$	12.6763	0.942224	0.471112	+	0.882074i	$0.343853\pi$
$182$	0	0
$183$	10.3120	0.762284
$184$	0	0
$185$	1.75790	0.129243
$186$	0	0
$187$	−12.6603	−0.925812
$188$	0	0
$189$	−2.97482	−0.216386
$190$	0	0
$191$	−16.5216	−1.19546	−0.597731	−	0.801697i	$-0.703931\pi$
$191$	−16.5216	−1.19546	−0.597731	+	0.801697i	$0.703931\pi$
$192$	0	0
$193$	16.6766	1.20041	0.600206	−	0.799846i	$-0.295085\pi$
$193$	16.6766	1.20041	0.600206	+	0.799846i	$0.295085\pi$
$194$	0	0
$195$	4.85902	0.347962
$196$	0	0
$197$	1.39529	0.0994101	0.0497050	−	0.998764i	$-0.484172\pi$
$197$	1.39529	0.0994101	0.0497050	+	0.998764i	$0.484172\pi$
$198$	0	0
$199$	−23.6336	−1.67534	−0.837670	−	0.546177i	$-0.816083\pi$
$199$	−23.6336	−1.67534	−0.837670	+	0.546177i	$0.816083\pi$
$200$	0	0
$201$	19.3690	1.36618
$202$	0	0
$203$	4.56146	0.320152
$204$	0	0
$205$	−8.97449	−0.626806
$206$	0	0
$207$	−13.8932	−0.965646
$208$	0	0
$209$	−10.8399	−0.749814
$210$	0	0
$211$	24.3826	1.67857	0.839283	−	0.543695i	$-0.182975\pi$
$211$	24.3826	1.67857	0.839283	+	0.543695i	$0.182975\pi$
$212$	0	0
$213$	−4.22004	−0.289153
$214$	0	0
$215$	−11.8030	−0.804958
$216$	0	0
$217$	−13.8779	−0.942091
$218$	0	0
$219$	1.13941	0.0769945
$220$	0	0
$221$	−7.80302	−0.524888
$222$	0	0
$223$	15.9516	1.06820	0.534100	−	0.845421i	$-0.320651\pi$
$223$	15.9516	1.06820	0.534100	+	0.845421i	$0.320651\pi$
$224$	0	0
$225$	−2.39410	−0.159607
$226$	0	0
$227$	22.8569	1.51706	0.758532	−	0.651636i	$-0.225917\pi$
$227$	22.8569	1.51706	0.758532	+	0.651636i	$0.225917\pi$
$228$	0	0
$229$	−20.4037	−1.34831	−0.674156	−	0.738589i	$-0.735493\pi$
$229$	−20.4037	−1.34831	−0.674156	+	0.738589i	$0.735493\pi$
$230$	0	0
$231$	−17.7120	−1.16536
$232$	0	0
$233$	−27.2605	−1.78589	−0.892947	−	0.450162i	$-0.851366\pi$
$233$	−27.2605	−1.78589	−0.892947	+	0.450162i	$0.851366\pi$
$234$	0	0
$235$	7.67419	0.500609
$236$	0	0
$237$	35.1525	2.28340
$238$	0	0
$239$	4.23860	0.274172	0.137086	−	0.990559i	$-0.456226\pi$
$239$	4.23860	0.274172	0.137086	+	0.990559i	$0.456226\pi$
$240$	0	0
$241$	−14.1547	−0.911787	−0.455893	−	0.890034i	$-0.650680\pi$
$241$	−14.1547	−0.911787	−0.455893	+	0.890034i	$0.650680\pi$
$242$	0	0
$243$	−21.3613	−1.37032
$244$	0	0
$245$	−28.0327	−1.79094
$246$	0	0
$247$	−6.68107	−0.425106
$248$	0	0
$249$	−1.07297	−0.0679967
$250$	0	0
$251$	28.4389	1.79505	0.897524	−	0.440967i	$-0.145364\pi$
$251$	28.4389	1.79505	0.897524	+	0.440967i	$0.145364\pi$
$252$	0	0
$253$	8.26456	0.519588
$254$	0	0
$255$	−37.9150	−2.37433
$256$	0	0
$257$	17.3827	1.08430	0.542151	−	0.840281i	$-0.317610\pi$
$257$	17.3827	1.08430	0.542151	+	0.840281i	$0.317610\pi$
$258$	0	0
$259$	−3.94940	−0.245404
$260$	0	0
$261$	2.72750	0.168828
$262$	0	0
$263$	8.98588	0.554093	0.277047	−	0.960857i	$-0.410644\pi$
$263$	8.98588	0.554093	0.277047	+	0.960857i	$0.410644\pi$
$264$	0	0
$265$	−22.9280	−1.40845
$266$	0	0
$267$	−40.7366	−2.49304
$268$	0	0
$269$	−2.70259	−0.164780	−0.0823898	−	0.996600i	$-0.526255\pi$
$269$	−2.70259	−0.164780	−0.0823898	+	0.996600i	$0.526255\pi$
$270$	0	0
$271$	−11.8100	−0.717406	−0.358703	−	0.933452i	$-0.616781\pi$
$271$	−11.8100	−0.717406	−0.358703	+	0.933452i	$0.616781\pi$
$272$	0	0
$273$	−10.9166	−0.660702
$274$	0	0
$275$	1.42416	0.0858801
$276$	0	0
$277$	−4.39613	−0.264138	−0.132069	−	0.991241i	$-0.542162\pi$
$277$	−4.39613	−0.264138	−0.132069	+	0.991241i	$0.542162\pi$
$278$	0	0
$279$	−8.29818	−0.496799
$280$	0	0
$281$	9.25394	0.552044	0.276022	−	0.961151i	$-0.410984\pi$
$281$	9.25394	0.552044	0.276022	+	0.961151i	$0.410984\pi$
$282$	0	0
$283$	0.0720654	0.00428384	0.00214192	−	0.999998i	$-0.499318\pi$
$283$	0.0720654	0.00428384	0.00214192	+	0.999998i	$0.499318\pi$
$284$	0	0
$285$	−32.4634	−1.92297
$286$	0	0
$287$	20.1627	1.19016
$288$	0	0
$289$	43.8871	2.58160
$290$	0	0
$291$	3.38091	0.198193
$292$	0	0
$293$	20.3792	1.19057	0.595284	−	0.803515i	$-0.297040\pi$
$293$	20.3792	1.19057	0.595284	+	0.803515i	$0.297040\pi$
$294$	0	0
$295$	3.64140	0.212011
$296$	0	0
$297$	1.05812	0.0613986
$298$	0	0
$299$	5.09377	0.294580
$300$	0	0
$301$	26.5174	1.52843
$302$	0	0
$303$	17.4290	1.00127
$304$	0	0
$305$	−8.74836	−0.500929
$306$	0	0
$307$	−27.8478	−1.58936	−0.794680	−	0.607029i	$-0.792361\pi$
$307$	−27.8478	−1.58936	−0.794680	+	0.607029i	$0.792361\pi$
$308$	0	0
$309$	−17.8867	−1.01754
$310$	0	0
$311$	−5.42168	−0.307435	−0.153718	−	0.988115i	$-0.549125\pi$
$311$	−5.42168	−0.307435	−0.153718	+	0.988115i	$0.549125\pi$
$312$	0	0
$313$	16.1702	0.913992	0.456996	−	0.889469i	$-0.348925\pi$
$313$	16.1702	0.913992	0.456996	+	0.889469i	$0.348925\pi$
$314$	0	0
$315$	−25.2601	−1.42324
$316$	0	0
$317$	−32.3265	−1.81564	−0.907818	−	0.419363i	$-0.862253\pi$
$317$	−32.3265	−1.81564	−0.907818	+	0.419363i	$0.862253\pi$
$318$	0	0
$319$	−1.62248	−0.0908417
$320$	0	0
$321$	2.80655	0.156646
$322$	0	0
$323$	52.1325	2.90073
$324$	0	0
$325$	0.877765	0.0486896
$326$	0	0
$327$	−10.7898	−0.596679
$328$	0	0
$329$	−17.2413	−0.950545
$330$	0	0
$331$	15.9773	0.878193	0.439096	−	0.898440i	$-0.355299\pi$
$331$	15.9773	0.878193	0.439096	+	0.898440i	$0.355299\pi$
$332$	0	0
$333$	−2.36152	−0.129410
$334$	0	0
$335$	−16.4320	−0.897776
$336$	0	0
$337$	−16.0006	−0.871611	−0.435805	−	0.900041i	$-0.643536\pi$
$337$	−16.0006	−0.871611	−0.435805	+	0.900041i	$0.643536\pi$
$338$	0	0
$339$	−17.3309	−0.941287
$340$	0	0
$341$	4.93627	0.267314
$342$	0	0
$343$	31.0497	1.67652
$344$	0	0
$345$	24.7507	1.33253
$346$	0	0
$347$	−35.1210	−1.88539	−0.942697	−	0.333650i	$-0.891719\pi$
$347$	−35.1210	−1.88539	−0.942697	+	0.333650i	$0.891719\pi$
$348$	0	0
$349$	10.5007	0.562091	0.281046	−	0.959694i	$-0.409319\pi$
$349$	10.5007	0.562091	0.281046	+	0.959694i	$0.409319\pi$
$350$	0	0
$351$	0.652163	0.0348099
$352$	0	0
$353$	15.4542	0.822544	0.411272	−	0.911513i	$-0.365085\pi$
$353$	15.4542	0.822544	0.411272	+	0.911513i	$0.365085\pi$
$354$	0	0
$355$	3.58015	0.190015
$356$	0	0
$357$	85.1823	4.50833
$358$	0	0
$359$	−6.83177	−0.360567	−0.180284	−	0.983615i	$-0.557702\pi$
$359$	−6.83177	−0.360567	−0.180284	+	0.983615i	$0.557702\pi$
$360$	0	0
$361$	25.6367	1.34930
$362$	0	0
$363$	−20.0254	−1.05106
$364$	0	0
$365$	−0.966642	−0.0505964
$366$	0	0
$367$	23.4891	1.22612	0.613061	−	0.790035i	$-0.289938\pi$
$367$	23.4891	1.22612	0.613061	+	0.790035i	$0.289938\pi$
$368$	0	0
$369$	12.0561	0.627617
$370$	0	0
$371$	51.5114	2.67434
$372$	0	0
$373$	15.6401	0.809811	0.404906	−	0.914359i	$-0.367304\pi$
$373$	15.6401	0.809811	0.404906	+	0.914359i	$0.367304\pi$
$374$	0	0
$375$	28.5602	1.47484
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−7.03272	−0.361247	−0.180623	−	0.983552i	$-0.557811\pi$
$379$	−7.03272	−0.361247	−0.180623	+	0.983552i	$0.557811\pi$
$380$	0	0
$381$	−31.7405	−1.62611
$382$	0	0
$383$	−14.7472	−0.753549	−0.376774	−	0.926305i	$-0.622967\pi$
$383$	−14.7472	−0.753549	−0.376774	+	0.926305i	$0.622967\pi$
$384$	0	0
$385$	15.0263	0.765810
$386$	0	0
$387$	15.8559	0.805999
$388$	0	0
$389$	−20.9395	−1.06167	−0.530836	−	0.847474i	$-0.678122\pi$
$389$	−20.9395	−1.06167	−0.530836	+	0.847474i	$0.678122\pi$
$390$	0	0
$391$	−39.7468	−2.01008
$392$	0	0
$393$	−2.52826	−0.127534
$394$	0	0
$395$	−29.8222	−1.50052
$396$	0	0
$397$	5.83663	0.292932	0.146466	−	0.989216i	$-0.453210\pi$
$397$	5.83663	0.292932	0.146466	+	0.989216i	$0.453210\pi$
$398$	0	0
$399$	72.9344	3.65129
$400$	0	0
$401$	−32.8744	−1.64167	−0.820834	−	0.571167i	$-0.806491\pi$
$401$	−32.8744	−1.64167	−0.820834	+	0.571167i	$0.806491\pi$
$402$	0	0
$403$	3.04242	0.151554
$404$	0	0
$405$	19.7820	0.982976
$406$	0	0
$407$	1.40478	0.0696323
$408$	0	0
$409$	−31.4110	−1.55317	−0.776587	−	0.630010i	$-0.783051\pi$
$409$	−31.4110	−1.55317	−0.776587	+	0.630010i	$0.783051\pi$
$410$	0	0
$411$	9.45038	0.466153
$412$	0	0
$413$	−8.18100	−0.402561
$414$	0	0
$415$	0.910273	0.0446835
$416$	0	0
$417$	32.3137	1.58241
$418$	0	0
$419$	31.6468	1.54605	0.773023	−	0.634378i	$-0.218744\pi$
$419$	31.6468	1.54605	0.773023	+	0.634378i	$0.218744\pi$
$420$	0	0
$421$	−22.8353	−1.11293	−0.556463	−	0.830872i	$-0.687842\pi$
$421$	−22.8353	−1.11293	−0.556463	+	0.830872i	$0.687842\pi$
$422$	0	0
$423$	−10.3093	−0.501257
$424$	0	0
$425$	−6.84922	−0.332236
$426$	0	0
$427$	19.6546	0.951153
$428$	0	0
$429$	3.88296	0.187471
$430$	0	0
$431$	−24.5711	−1.18355	−0.591773	−	0.806104i	$-0.701572\pi$
$431$	−24.5711	−1.18355	−0.591773	+	0.806104i	$0.701572\pi$
$432$	0	0
$433$	−9.64565	−0.463540	−0.231770	−	0.972771i	$-0.574452\pi$
$433$	−9.64565	−0.463540	−0.231770	+	0.972771i	$0.574452\pi$
$434$	0	0
$435$	−4.85902	−0.232972
$436$	0	0
$437$	−34.0318	−1.62796
$438$	0	0
$439$	9.57986	0.457222	0.228611	−	0.973518i	$-0.426582\pi$
$439$	9.57986	0.457222	0.228611	+	0.973518i	$0.426582\pi$
$440$	0	0
$441$	37.6584	1.79326
$442$	0	0
$443$	−9.45343	−0.449146	−0.224573	−	0.974457i	$-0.572099\pi$
$443$	−9.45343	−0.449146	−0.224573	+	0.974457i	$0.572099\pi$
$444$	0	0
$445$	34.5596	1.63828
$446$	0	0
$447$	−5.99407	−0.283510
$448$	0	0
$449$	−7.05660	−0.333022	−0.166511	−	0.986040i	$-0.553250\pi$
$449$	−7.05660	−0.333022	−0.166511	+	0.986040i	$0.553250\pi$
$450$	0	0
$451$	−7.17173	−0.337704
$452$	0	0
$453$	18.7364	0.880313
$454$	0	0
$455$	9.26127	0.434175
$456$	0	0
$457$	15.3254	0.716893	0.358447	−	0.933550i	$-0.383307\pi$
$457$	15.3254	0.716893	0.358447	+	0.933550i	$0.383307\pi$
$458$	0	0
$459$	−5.08884	−0.237527
$460$	0	0
$461$	−16.7745	−0.781264	−0.390632	−	0.920547i	$-0.627744\pi$
$461$	−16.7745	−0.781264	−0.390632	+	0.920547i	$0.627744\pi$
$462$	0	0
$463$	25.0195	1.16275	0.581377	−	0.813634i	$-0.302514\pi$
$463$	25.0195	1.16275	0.581377	+	0.813634i	$0.302514\pi$
$464$	0	0
$465$	14.7832	0.685553
$466$	0	0
$467$	−4.78458	−0.221404	−0.110702	−	0.993854i	$-0.535310\pi$
$467$	−4.78458	−0.221404	−0.110702	+	0.993854i	$0.535310\pi$
$468$	0	0
$469$	36.9172	1.70468
$470$	0	0
$471$	−9.41717	−0.433920
$472$	0	0
$473$	−9.43206	−0.433687
$474$	0	0
$475$	−5.86441	−0.269077
$476$	0	0
$477$	30.8009	1.41028
$478$	0	0
$479$	−12.3092	−0.562424	−0.281212	−	0.959646i	$-0.590736\pi$
$479$	−12.3092	−0.562424	−0.281212	+	0.959646i	$0.590736\pi$
$480$	0	0
$481$	0.865819	0.0394780
$482$	0	0
$483$	−55.6065	−2.53018
$484$	0	0
$485$	−2.86826	−0.130241
$486$	0	0
$487$	−17.1067	−0.775177	−0.387588	−	0.921833i	$-0.626692\pi$
$487$	−17.1067	−0.775177	−0.387588	+	0.921833i	$0.626692\pi$
$488$	0	0
$489$	57.1724	2.58542
$490$	0	0
$491$	−30.1414	−1.36026	−0.680132	−	0.733090i	$-0.738077\pi$
$491$	−30.1414	−1.36026	−0.680132	+	0.733090i	$0.738077\pi$
$492$	0	0
$493$	7.80302	0.351430
$494$	0	0
$495$	8.98485	0.403839
$496$	0	0
$497$	−8.04339	−0.360795
$498$	0	0
$499$	41.2540	1.84678	0.923391	−	0.383861i	$-0.125406\pi$
$499$	41.2540	1.84678	0.923391	+	0.383861i	$0.125406\pi$
$500$	0	0
$501$	−37.3729	−1.66970
$502$	0	0
$503$	13.3723	0.596243	0.298121	−	0.954528i	$-0.403640\pi$
$503$	13.3723	0.596243	0.298121	+	0.954528i	$0.403640\pi$
$504$	0	0
$505$	−14.7862	−0.657978
$506$	0	0
$507$	2.39322	0.106287
$508$	0	0
$509$	−22.2437	−0.985935	−0.492967	−	0.870048i	$-0.664088\pi$
$509$	−22.2437	−0.985935	−0.492967	+	0.870048i	$0.664088\pi$
$510$	0	0
$511$	2.17172	0.0960712
$512$	0	0
$513$	−4.35715	−0.192373
$514$	0	0
$515$	15.1745	0.668668
$516$	0	0
$517$	6.13263	0.269713
$518$	0	0
$519$	11.0847	0.486566
$520$	0	0
$521$	18.9549	0.830431	0.415216	−	0.909723i	$-0.363706\pi$
$521$	18.9549	0.830431	0.415216	+	0.909723i	$0.363706\pi$
$522$	0	0
$523$	−20.2659	−0.886167	−0.443083	−	0.896480i	$-0.646115\pi$
$523$	−20.2659	−0.886167	−0.443083	+	0.896480i	$0.646115\pi$
$524$	0	0
$525$	−9.58219	−0.418201
$526$	0	0
$527$	−23.7400	−1.03413
$528$	0	0
$529$	2.94647	0.128107
$530$	0	0
$531$	−4.89177	−0.212285
$532$	0	0
$533$	−4.42022	−0.191461
$534$	0	0
$535$	−2.38099	−0.102939
$536$	0	0
$537$	1.47476	0.0636405
$538$	0	0
$539$	−22.4016	−0.964904
$540$	0	0
$541$	−9.54727	−0.410469	−0.205235	−	0.978713i	$-0.565796\pi$
$541$	−9.54727	−0.410469	−0.205235	+	0.978713i	$0.565796\pi$
$542$	0	0
$543$	30.3372	1.30189
$544$	0	0
$545$	9.15374	0.392103
$546$	0	0
$547$	42.1433	1.80192	0.900959	−	0.433904i	$-0.142864\pi$
$547$	42.1433	1.80192	0.900959	+	0.433904i	$0.142864\pi$
$548$	0	0
$549$	11.7523	0.501577
$550$	0	0
$551$	6.68107	0.284623
$552$	0	0
$553$	67.0004	2.84915
$554$	0	0
$555$	4.20703	0.178579
$556$	0	0
$557$	9.49632	0.402372	0.201186	−	0.979553i	$-0.435520\pi$
$557$	9.49632	0.402372	0.201186	+	0.979553i	$0.435520\pi$
$558$	0	0
$559$	−5.81334	−0.245878
$560$	0	0
$561$	−30.2988	−1.27922
$562$	0	0
$563$	−21.2405	−0.895179	−0.447589	−	0.894239i	$-0.647717\pi$
$563$	−21.2405	−0.895179	−0.447589	+	0.894239i	$0.647717\pi$
$564$	0	0
$565$	14.7030	0.618560
$566$	0	0
$567$	−44.4435	−1.86645
$568$	0	0
$569$	33.8522	1.41916	0.709578	−	0.704627i	$-0.248886\pi$
$569$	33.8522	1.41916	0.709578	+	0.704627i	$0.248886\pi$
$570$	0	0
$571$	−1.29956	−0.0543847	−0.0271923	−	0.999630i	$-0.508657\pi$
$571$	−1.29956	−0.0543847	−0.0271923	+	0.999630i	$0.508657\pi$
$572$	0	0
$573$	−39.5398	−1.65180
$574$	0	0
$575$	4.47113	0.186459
$576$	0	0
$577$	−1.09671	−0.0456568	−0.0228284	−	0.999739i	$-0.507267\pi$
$577$	−1.09671	−0.0456568	−0.0228284	+	0.999739i	$0.507267\pi$
$578$	0	0
$579$	39.9109	1.65864
$580$	0	0
$581$	−2.04508	−0.0848440
$582$	0	0
$583$	−18.3223	−0.758832
$584$	0	0
$585$	5.53771	0.228956
$586$	0	0
$587$	−13.9230	−0.574662	−0.287331	−	0.957831i	$-0.592768\pi$
$587$	−13.9230	−0.574662	−0.287331	+	0.957831i	$0.592768\pi$
$588$	0	0
$589$	−20.3266	−0.837543
$590$	0	0
$591$	3.33923	0.137357
$592$	0	0
$593$	−27.6686	−1.13622	−0.568108	−	0.822954i	$-0.692324\pi$
$593$	−27.6686	−1.13622	−0.568108	+	0.822954i	$0.692324\pi$
$594$	0	0
$595$	−72.2659	−2.96261
$596$	0	0
$597$	−56.5603	−2.31486
$598$	0	0
$599$	−4.02262	−0.164360	−0.0821798	−	0.996618i	$-0.526188\pi$
$599$	−4.02262	−0.164360	−0.0821798	+	0.996618i	$0.526188\pi$
$600$	0	0
$601$	18.5134	0.755178	0.377589	−	0.925973i	$-0.376753\pi$
$601$	18.5134	0.755178	0.377589	+	0.925973i	$0.376753\pi$
$602$	0	0
$603$	22.0744	0.898937
$604$	0	0
$605$	16.9889	0.690696
$606$	0	0
$607$	−15.4965	−0.628984	−0.314492	−	0.949260i	$-0.601834\pi$
$607$	−15.4965	−0.628984	−0.314492	+	0.949260i	$0.601834\pi$
$608$	0	0
$609$	10.9166	0.442362
$610$	0	0
$611$	3.77978	0.152913
$612$	0	0
$613$	−22.3374	−0.902197	−0.451099	−	0.892474i	$-0.648968\pi$
$613$	−22.3374	−0.902197	−0.451099	+	0.892474i	$0.648968\pi$
$614$	0	0
$615$	−21.4779	−0.866074
$616$	0	0
$617$	0.942597	0.0379475	0.0189738	−	0.999820i	$-0.493960\pi$
$617$	0.942597	0.0379475	0.0189738	+	0.999820i	$0.493960\pi$
$618$	0	0
$619$	−30.5808	−1.22914	−0.614572	−	0.788861i	$-0.710671\pi$
$619$	−30.5808	−1.22914	−0.614572	+	0.788861i	$0.710671\pi$
$620$	0	0
$621$	3.32197	0.133306
$622$	0	0
$623$	−77.6437	−3.11073
$624$	0	0
$625$	−19.8407	−0.793628
$626$	0	0
$627$	−25.9423	−1.03604
$628$	0	0
$629$	−6.75601	−0.269380
$630$	0	0
$631$	−5.48620	−0.218402	−0.109201	−	0.994020i	$-0.534829\pi$
$631$	−5.48620	−0.218402	−0.109201	+	0.994020i	$0.534829\pi$
$632$	0	0
$633$	58.3529	2.31932
$634$	0	0
$635$	26.9276	1.06859
$636$	0	0
$637$	−13.8070	−0.547051
$638$	0	0
$639$	−4.80949	−0.190260
$640$	0	0
$641$	−18.6943	−0.738381	−0.369191	−	0.929354i	$-0.620365\pi$
$641$	−18.6943	−0.738381	−0.369191	+	0.929354i	$0.620365\pi$
$642$	0	0
$643$	−16.6695	−0.657381	−0.328691	−	0.944438i	$-0.606607\pi$
$643$	−16.6695	−0.657381	−0.328691	+	0.944438i	$0.606607\pi$
$644$	0	0
$645$	−28.2472	−1.11223
$646$	0	0
$647$	33.3917	1.31276	0.656381	−	0.754429i	$-0.272086\pi$
$647$	33.3917	1.31276	0.656381	+	0.754429i	$0.272086\pi$
$648$	0	0
$649$	2.90993	0.114225
$650$	0	0
$651$	−33.2128	−1.30171
$652$	0	0
$653$	30.8305	1.20649	0.603246	−	0.797555i	$-0.293874\pi$
$653$	30.8305	1.20649	0.603246	+	0.797555i	$0.293874\pi$
$654$	0	0
$655$	2.14489	0.0838078
$656$	0	0
$657$	1.29856	0.0506618
$658$	0	0
$659$	41.2997	1.60881	0.804403	−	0.594083i	$-0.202485\pi$
$659$	41.2997	1.60881	0.804403	+	0.594083i	$0.202485\pi$
$660$	0	0
$661$	−30.6778	−1.19323	−0.596614	−	0.802528i	$-0.703488\pi$
$661$	−30.6778	−1.19323	−0.596614	+	0.802528i	$0.703488\pi$
$662$	0	0
$663$	−18.6743	−0.725251
$664$	0	0
$665$	−61.8752	−2.39942
$666$	0	0
$667$	−5.09377	−0.197231
$668$	0	0
$669$	38.1757	1.47596
$670$	0	0
$671$	−6.99102	−0.269885
$672$	0	0
$673$	29.3015	1.12949	0.564744	−	0.825266i	$-0.308975\pi$
$673$	29.3015	1.12949	0.564744	+	0.825266i	$0.308975\pi$
$674$	0	0
$675$	0.572446	0.0220335
$676$	0	0
$677$	−17.9529	−0.689985	−0.344993	−	0.938605i	$-0.612119\pi$
$677$	−17.9529	−0.689985	−0.344993	+	0.938605i	$0.612119\pi$
$678$	0	0
$679$	6.44401	0.247298
$680$	0	0
$681$	54.7015	2.09617
$682$	0	0
$683$	−34.6798	−1.32699	−0.663493	−	0.748183i	$-0.730927\pi$
$683$	−34.6798	−1.32699	−0.663493	+	0.748183i	$0.730927\pi$
$684$	0	0
$685$	−8.01739	−0.306329
$686$	0	0
$687$	−48.8305	−1.86300
$688$	0	0
$689$	−11.2927	−0.430219
$690$	0	0
$691$	−41.4688	−1.57755	−0.788773	−	0.614684i	$-0.789283\pi$
$691$	−41.4688	−1.57755	−0.788773	+	0.614684i	$0.789283\pi$
$692$	0	0
$693$	−20.1859	−0.766800
$694$	0	0
$695$	−27.4139	−1.03987
$696$	0	0
$697$	34.4911	1.30644
$698$	0	0
$699$	−65.2403	−2.46761
$700$	0	0
$701$	−38.4632	−1.45273	−0.726367	−	0.687307i	$-0.758793\pi$
$701$	−38.4632	−1.45273	−0.726367	+	0.687307i	$0.758793\pi$
$702$	0	0
$703$	−5.78460	−0.218170
$704$	0	0
$705$	18.3660	0.691704
$706$	0	0
$707$	33.2196	1.24935
$708$	0	0
$709$	−46.4469	−1.74435	−0.872176	−	0.489193i	$-0.837291\pi$
$709$	−46.4469	−1.74435	−0.872176	+	0.489193i	$0.837291\pi$
$710$	0	0
$711$	40.0624	1.50246
$712$	0	0
$713$	15.4974	0.580381
$714$	0	0
$715$	−3.29418	−0.123195
$716$	0	0
$717$	10.1439	0.378831
$718$	0	0
$719$	0.0460189	0.00171622	0.000858108	−	1.00000i	$-0.499727\pi$
$719$	0.0460189	0.00171622	0.000858108		1.00000i	$0.499727\pi$
$720$	0	0
$721$	−34.0919	−1.26965
$722$	0	0
$723$	−33.8754	−1.25984
$724$	0	0
$725$	−0.877765	−0.0325994
$726$	0	0
$727$	38.1598	1.41527	0.707635	−	0.706578i	$-0.249762\pi$
$727$	38.1598	1.41527	0.707635	+	0.706578i	$0.249762\pi$
$728$	0	0
$729$	−21.8924	−0.810829
$730$	0	0
$731$	45.3616	1.67776
$732$	0	0
$733$	−2.89943	−0.107093	−0.0535464	−	0.998565i	$-0.517053\pi$
$733$	−2.89943	−0.107093	−0.0535464	+	0.998565i	$0.517053\pi$
$734$	0	0
$735$	−67.0883	−2.47459
$736$	0	0
$737$	−13.1312	−0.483694
$738$	0	0
$739$	−30.8861	−1.13616	−0.568082	−	0.822972i	$-0.692314\pi$
$739$	−30.8861	−1.13616	−0.568082	+	0.822972i	$0.692314\pi$
$740$	0	0
$741$	−15.9893	−0.587380
$742$	0	0
$743$	−46.2688	−1.69744	−0.848718	−	0.528845i	$-0.822625\pi$
$743$	−46.2688	−1.69744	−0.848718	+	0.528845i	$0.822625\pi$
$744$	0	0
$745$	5.08517	0.186306
$746$	0	0
$747$	−1.22284	−0.0447413
$748$	0	0
$749$	5.34927	0.195458
$750$	0	0
$751$	−12.6473	−0.461508	−0.230754	−	0.973012i	$-0.574119\pi$
$751$	−12.6473	−0.461508	−0.230754	+	0.973012i	$0.574119\pi$
$752$	0	0
$753$	68.0605	2.48026
$754$	0	0
$755$	−15.8954	−0.578491
$756$	0	0
$757$	−34.4885	−1.25351	−0.626754	−	0.779218i	$-0.715617\pi$
$757$	−34.4885	−1.25351	−0.626754	+	0.779218i	$0.715617\pi$
$758$	0	0
$759$	19.7789	0.717929
$760$	0	0
$761$	15.2737	0.553671	0.276836	−	0.960917i	$-0.410714\pi$
$761$	15.2737	0.553671	0.276836	+	0.960917i	$0.410714\pi$
$762$	0	0
$763$	−20.5654	−0.744517
$764$	0	0
$765$	−43.2109	−1.56229
$766$	0	0
$767$	1.79350	0.0647596
$768$	0	0
$769$	52.0828	1.87815	0.939077	−	0.343707i	$-0.111683\pi$
$769$	52.0828	1.87815	0.939077	+	0.343707i	$0.111683\pi$
$770$	0	0
$771$	41.6006	1.49821
$772$	0	0
$773$	−26.6507	−0.958559	−0.479280	−	0.877662i	$-0.659102\pi$
$773$	−26.6507	−0.958559	−0.479280	+	0.877662i	$0.659102\pi$
$774$	0	0
$775$	2.67053	0.0959281
$776$	0	0
$777$	−9.45179	−0.339081
$778$	0	0
$779$	29.5318	1.05809
$780$	0	0
$781$	2.86098	0.102374
$782$	0	0
$783$	−0.652163	−0.0233064
$784$	0	0
$785$	7.98922	0.285148
$786$	0	0
$787$	34.3770	1.22541	0.612703	−	0.790313i	$-0.290082\pi$
$787$	34.3770	1.22541	0.612703	+	0.790313i	$0.290082\pi$
$788$	0	0
$789$	21.5052	0.765604
$790$	0	0
$791$	−33.0327	−1.17451
$792$	0	0
$793$	−4.30884	−0.153011
$794$	0	0
$795$	−54.8717	−1.94610
$796$	0	0
$797$	22.8789	0.810411	0.405205	−	0.914226i	$-0.367200\pi$
$797$	22.8789	0.810411	0.405205	+	0.914226i	$0.367200\pi$
$798$	0	0
$799$	−29.4937	−1.04341
$800$	0	0
$801$	−46.4265	−1.64040
$802$	0	0
$803$	−0.772467	−0.0272598
$804$	0	0
$805$	47.1748	1.66269
$806$	0	0
$807$	−6.46788	−0.227680
$808$	0	0
$809$	1.34551	0.0473056	0.0236528	−	0.999720i	$-0.492470\pi$
$809$	1.34551	0.0473056	0.0236528	+	0.999720i	$0.492470\pi$
$810$	0	0
$811$	36.0980	1.26757	0.633787	−	0.773508i	$-0.281500\pi$
$811$	36.0980	1.26757	0.633787	+	0.773508i	$0.281500\pi$
$812$	0	0
$813$	−28.2639	−0.991258
$814$	0	0
$815$	−48.5032	−1.69899
$816$	0	0
$817$	38.8393	1.35882
$818$	0	0
$819$	−12.4414	−0.434737
$820$	0	0
$821$	−49.5116	−1.72797	−0.863984	−	0.503520i	$-0.832038\pi$
$821$	−49.5116	−1.72797	−0.863984	+	0.503520i	$0.832038\pi$
$822$	0	0
$823$	38.7279	1.34997	0.674985	−	0.737831i	$-0.264150\pi$
$823$	38.7279	1.34997	0.674985	+	0.737831i	$0.264150\pi$
$824$	0	0
$825$	3.40833	0.118663
$826$	0	0
$827$	−37.0580	−1.28863	−0.644316	−	0.764760i	$-0.722858\pi$
$827$	−37.0580	−1.28863	−0.644316	+	0.764760i	$0.722858\pi$
$828$	0	0
$829$	20.7871	0.721965	0.360983	−	0.932573i	$-0.382441\pi$
$829$	20.7871	0.721965	0.360983	+	0.932573i	$0.382441\pi$
$830$	0	0
$831$	−10.5209	−0.364966
$832$	0	0
$833$	107.736	3.73283
$834$	0	0
$835$	31.7059	1.09723
$836$	0	0
$837$	1.98415	0.0685823
$838$	0	0
$839$	8.58626	0.296431	0.148215	−	0.988955i	$-0.452647\pi$
$839$	8.58626	0.296431	0.148215	+	0.988955i	$0.452647\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	22.1467	0.762773
$844$	0	0
$845$	−2.03033	−0.0698454
$846$	0	0
$847$	−38.1682	−1.31148
$848$	0	0
$849$	0.172468	0.00591909
$850$	0	0
$851$	4.41028	0.151183
$852$	0	0
$853$	−36.9746	−1.26599	−0.632993	−	0.774158i	$-0.718174\pi$
$853$	−36.9746	−1.26599	−0.632993	+	0.774158i	$0.718174\pi$
$854$	0	0
$855$	−36.9978	−1.26530
$856$	0	0
$857$	−33.3236	−1.13831	−0.569156	−	0.822229i	$-0.692730\pi$
$857$	−33.3236	−1.13831	−0.569156	+	0.822229i	$0.692730\pi$
$858$	0	0
$859$	10.0642	0.343385	0.171693	−	0.985151i	$-0.445076\pi$
$859$	10.0642	0.343385	0.171693	+	0.985151i	$0.445076\pi$
$860$	0	0
$861$	48.2537	1.64448
$862$	0	0
$863$	−27.7952	−0.946158	−0.473079	−	0.881020i	$-0.656857\pi$
$863$	−27.7952	−0.946158	−0.473079	+	0.881020i	$0.656857\pi$
$864$	0	0
$865$	−9.40393	−0.319743
$866$	0	0
$867$	105.032	3.56706
$868$	0	0
$869$	−23.8316	−0.808433
$870$	0	0
$871$	−8.09327	−0.274230
$872$	0	0
$873$	3.85315	0.130409
$874$	0	0
$875$	54.4356	1.84026
$876$	0	0
$877$	−33.9615	−1.14680	−0.573399	−	0.819277i	$-0.694375\pi$
$877$	−33.9615	−1.14680	−0.573399	+	0.819277i	$0.694375\pi$
$878$	0	0
$879$	48.7720	1.64504
$880$	0	0
$881$	−13.7085	−0.461852	−0.230926	−	0.972971i	$-0.574175\pi$
$881$	−13.7085	−0.461852	−0.230926	+	0.972971i	$0.574175\pi$
$882$	0	0
$883$	25.6682	0.863805	0.431902	−	0.901920i	$-0.357843\pi$
$883$	25.6682	0.863805	0.431902	+	0.901920i	$0.357843\pi$
$884$	0	0
$885$	8.71467	0.292940
$886$	0	0
$887$	13.7011	0.460039	0.230020	−	0.973186i	$-0.426121\pi$
$887$	13.7011	0.460039	0.230020	+	0.973186i	$0.426121\pi$
$888$	0	0
$889$	−60.4972	−2.02901
$890$	0	0
$891$	15.8083	0.529597
$892$	0	0
$893$	−25.2530	−0.845058
$894$	0	0
$895$	−1.25114	−0.0418209
$896$	0	0
$897$	12.1905	0.407029
$898$	0	0
$899$	−3.04242	−0.101470
$900$	0	0
$901$	88.1175	2.93562
$902$	0	0
$903$	63.4618	2.11188
$904$	0	0
$905$	−25.7371	−0.855530
$906$	0	0
$907$	−26.3398	−0.874599	−0.437299	−	0.899316i	$-0.644065\pi$
$907$	−26.3398	−0.874599	−0.437299	+	0.899316i	$0.644065\pi$
$908$	0	0
$909$	19.8634	0.658829
$910$	0	0
$911$	31.3410	1.03838	0.519188	−	0.854660i	$-0.326235\pi$
$911$	31.3410	1.03838	0.519188	+	0.854660i	$0.326235\pi$
$912$	0	0
$913$	0.727421	0.0240741
$914$	0	0
$915$	−20.9367	−0.692147
$916$	0	0
$917$	−4.81885	−0.159132
$918$	0	0
$919$	48.0051	1.58354	0.791770	−	0.610819i	$-0.209160\pi$
$919$	48.0051	1.58354	0.791770	+	0.610819i	$0.209160\pi$
$920$	0	0
$921$	−66.6459	−2.19606
$922$	0	0
$923$	1.76333	0.0580409
$924$	0	0
$925$	0.759986	0.0249882
$926$	0	0
$927$	−20.3850	−0.669533
$928$	0	0
$929$	−16.7100	−0.548239	−0.274119	−	0.961696i	$-0.588386\pi$
$929$	−16.7100	−0.548239	−0.274119	+	0.961696i	$0.588386\pi$
$930$	0	0
$931$	92.2452	3.02321
$932$	0	0
$933$	−12.9753	−0.424791
$934$	0	0
$935$	25.7045	0.840628
$936$	0	0
$937$	−49.7629	−1.62568	−0.812841	−	0.582485i	$-0.802080\pi$
$937$	−49.7629	−1.62568	−0.812841	+	0.582485i	$0.802080\pi$
$938$	0	0
$939$	38.6987	1.26289
$940$	0	0
$941$	48.4741	1.58021	0.790105	−	0.612971i	$-0.210026\pi$
$941$	48.4741	1.58021	0.790105	+	0.612971i	$0.210026\pi$
$942$	0	0
$943$	−22.5156	−0.733207
$944$	0	0
$945$	6.03986	0.196477
$946$	0	0
$947$	−37.4351	−1.21648	−0.608239	−	0.793754i	$-0.708124\pi$
$947$	−37.4351	−1.21648	−0.608239	+	0.793754i	$0.708124\pi$
$948$	0	0
$949$	−0.476101	−0.0154549
$950$	0	0
$951$	−77.3644	−2.50871
$952$	0	0
$953$	0.364386	0.0118036	0.00590182	−	0.999983i	$-0.498121\pi$
$953$	0.364386	0.0118036	0.00590182	+	0.999983i	$0.498121\pi$
$954$	0	0
$955$	33.5443	1.08547
$956$	0	0
$957$	−3.88296	−0.125518
$958$	0	0
$959$	18.0124	0.581650
$960$	0	0
$961$	−21.7437	−0.701410
$962$	0	0
$963$	3.19856	0.103072
$964$	0	0
$965$	−33.8591	−1.08996
$966$	0	0
$967$	−34.3640	−1.10507	−0.552536	−	0.833489i	$-0.686340\pi$
$967$	−34.3640	−1.10507	−0.552536	+	0.833489i	$0.686340\pi$
$968$	0	0
$969$	124.765	4.00801
$970$	0	0
$971$	21.4474	0.688278	0.344139	−	0.938919i	$-0.388171\pi$
$971$	21.4474	0.688278	0.344139	+	0.938919i	$0.388171\pi$
$972$	0	0
$973$	61.5898	1.97448
$974$	0	0
$975$	2.10068	0.0672757
$976$	0	0
$977$	−52.0906	−1.66653	−0.833263	−	0.552877i	$-0.813530\pi$
$977$	−52.0906	−1.66653	−0.833263	+	0.552877i	$0.813530\pi$
$978$	0	0
$979$	27.6174	0.882655
$980$	0	0
$981$	−12.2969	−0.392611
$982$	0	0
$983$	4.55793	0.145375	0.0726877	−	0.997355i	$-0.476842\pi$
$983$	4.55793	0.145375	0.0726877	+	0.997355i	$0.476842\pi$
$984$	0	0
$985$	−2.83289	−0.0902634
$986$	0	0
$987$	−41.2623	−1.31339
$988$	0	0
$989$	−29.6118	−0.941601
$990$	0	0
$991$	43.8418	1.39268	0.696340	−	0.717712i	$-0.254811\pi$
$991$	43.8418	1.39268	0.696340	+	0.717712i	$0.254811\pi$
$992$	0	0
$993$	38.2372	1.21342
$994$	0	0
$995$	47.9839	1.52119
$996$	0	0
$997$	−37.6133	−1.19123	−0.595613	−	0.803272i	$-0.703091\pi$
$997$	−37.6133	−1.19123	−0.595613	+	0.803272i	$0.703091\pi$
$998$	0	0
$999$	0.564655	0.0178649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bb.1.9		10
4.3	odd	2		3016.2.a.f.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.f.1.2	✓	10	4.3	odd	2
6032.2.a.bb.1.9		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39322 q^{3} -2.03033 q^{5} +4.56146 q^{7} +2.72750 q^{9} +O(q^{10})\)	\(q+2.39322 q^{3} -2.03033 q^{5} +4.56146 q^{7} +2.72750 q^{9} -1.62248 q^{11} -1.00000 q^{13} -4.85902 q^{15} +7.80302 q^{17} +6.68107 q^{19} +10.9166 q^{21} -5.09377 q^{23} -0.877765 q^{25} -0.652163 q^{27} +1.00000 q^{29} -3.04242 q^{31} -3.88296 q^{33} -9.26127 q^{35} -0.865819 q^{37} -2.39322 q^{39} +4.42022 q^{41} +5.81334 q^{43} -5.53771 q^{45} -3.77978 q^{47} +13.8070 q^{49} +18.6743 q^{51} +11.2927 q^{53} +3.29418 q^{55} +15.9893 q^{57} -1.79350 q^{59} +4.30884 q^{61} +12.4414 q^{63} +2.03033 q^{65} +8.09327 q^{67} -12.1905 q^{69} -1.76333 q^{71} +0.476101 q^{73} -2.10068 q^{75} -7.40091 q^{77} +14.6884 q^{79} -9.74326 q^{81} -0.448338 q^{83} -15.8427 q^{85} +2.39322 q^{87} -17.0217 q^{89} -4.56146 q^{91} -7.28117 q^{93} -13.5648 q^{95} +1.41271 q^{97} -4.42532 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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