LMFDB - Embedded newform 3328.2.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,8,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.10323968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 8 $ x^6 - 10x^4 + 25x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86678$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.86678 q^{3} +1.48486 q^{5} -2.55248 q^{7} +0.484862 q^{9} -1.59083 q^{11} +1.00000 q^{13} -2.77191 q^{15} +4.76491 q^{17} -3.51413 q^{19} +4.76491 q^{21} -3.73356 q^{23} -2.79518 q^{25} +4.69521 q^{27} +2.00000 q^{29} +0.685698 q^{31} +2.96972 q^{33} -3.79008 q^{35} -1.73463 q^{37} -1.86678 q^{39} -7.52982 q^{41} +4.69521 q^{43} +0.719953 q^{45} -5.38090 q^{47} -0.484862 q^{49} -8.89503 q^{51} -5.52982 q^{53} -2.36216 q^{55} +6.56009 q^{57} +10.9812 q^{59} +12.4995 q^{61} -1.23760 q^{63} +1.48486 q^{65} +2.96222 q^{67} +6.96972 q^{69} +11.3910 q^{71} +9.46927 q^{73} +5.21799 q^{75} +4.06055 q^{77} -6.91521 q^{79} -10.2195 q^{81} -4.06603 q^{83} +7.07523 q^{85} -3.73356 q^{87} -0.969724 q^{89} -2.55248 q^{91} -1.28005 q^{93} -5.21799 q^{95} +3.93945 q^{97} -0.771332 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q + 8 * q^5 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 4 * q^21 + 14 * q^25 + 12 * q^29 + 16 * q^33 + 24 * q^37 + 20 * q^41 + 36 * q^45 - 2 * q^49 + 32 * q^53 - 24 * q^57 + 8 * q^61 + 8 * q^65 + 40 * q^69 - 12 * q^73 + 28 * q^77 - 26 * q^81 - 20 * q^85 - 4 * q^89 + 24 * q^93 + 20 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.86678	−1.07779	−0.538893	−	0.842375i	$-0.681157\pi$
$3$	−1.86678	−1.07779	−0.538893	+	0.842375i	$0.681157\pi$
$4$	0	0
$5$	1.48486	0.664050	0.332025	−	0.943271i	$-0.392268\pi$
$5$	1.48486	0.664050	0.332025	+	0.943271i	$0.392268\pi$
$6$	0	0
$7$	−2.55248	−0.964746	−0.482373	−	0.875966i	$-0.660225\pi$
$7$	−2.55248	−0.964746	−0.482373	+	0.875966i	$0.660225\pi$
$8$	0	0
$9$	0.484862	0.161621
$10$	0	0
$11$	−1.59083	−0.479653	−0.239826	−	0.970816i	$-0.577091\pi$
$11$	−1.59083	−0.479653	−0.239826	+	0.970816i	$0.577091\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.77191	−0.715704
$16$	0	0
$17$	4.76491	1.15566	0.577830	−	0.816157i	$-0.303900\pi$
$17$	4.76491	1.15566	0.577830	+	0.816157i	$0.303900\pi$
$18$	0	0
$19$	−3.51413	−0.806196	−0.403098	−	0.915157i	$-0.632067\pi$
$19$	−3.51413	−0.806196	−0.403098	+	0.915157i	$0.632067\pi$
$20$	0	0
$21$	4.76491	1.03979
$22$	0	0
$23$	−3.73356	−0.778500	−0.389250	−	0.921132i	$-0.627266\pi$
$23$	−3.73356	−0.778500	−0.389250	+	0.921132i	$0.627266\pi$
$24$	0	0
$25$	−2.79518	−0.559037
$26$	0	0
$27$	4.69521	0.903593
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0.685698	0.123155	0.0615775	−	0.998102i	$-0.480387\pi$
$31$	0.685698	0.123155	0.0615775	+	0.998102i	$0.480387\pi$
$32$	0	0
$33$	2.96972	0.516963
$34$	0	0
$35$	−3.79008	−0.640640
$36$	0	0
$37$	−1.73463	−0.285172	−0.142586	−	0.989782i	$-0.545542\pi$
$37$	−1.73463	−0.285172	−0.142586	+	0.989782i	$0.545542\pi$
$38$	0	0
$39$	−1.86678	−0.298924
$40$	0	0
$41$	−7.52982	−1.17596	−0.587980	−	0.808875i	$-0.700077\pi$
$41$	−7.52982	−1.17596	−0.587980	+	0.808875i	$0.700077\pi$
$42$	0	0
$43$	4.69521	0.716012	0.358006	−	0.933719i	$-0.383457\pi$
$43$	4.69521	0.716012	0.358006	+	0.933719i	$0.383457\pi$
$44$	0	0
$45$	0.719953	0.107324
$46$	0	0
$47$	−5.38090	−0.784886	−0.392443	−	0.919776i	$-0.628370\pi$
$47$	−5.38090	−0.784886	−0.392443	+	0.919776i	$0.628370\pi$
$48$	0	0
$49$	−0.484862	−0.0692660
$50$	0	0
$51$	−8.89503	−1.24555
$52$	0	0
$53$	−5.52982	−0.759579	−0.379789	−	0.925073i	$-0.624004\pi$
$53$	−5.52982	−0.759579	−0.379789	+	0.925073i	$0.624004\pi$
$54$	0	0
$55$	−2.36216	−0.318514
$56$	0	0
$57$	6.56009	0.868906
$58$	0	0
$59$	10.9812	1.42964	0.714818	−	0.699311i	$-0.246510\pi$
$59$	10.9812	1.42964	0.714818	+	0.699311i	$0.246510\pi$
$60$	0	0
$61$	12.4995	1.60040	0.800201	−	0.599732i	$-0.204726\pi$
$61$	12.4995	1.60040	0.800201	+	0.599732i	$0.204726\pi$
$62$	0	0
$63$	−1.23760	−0.155923
$64$	0	0
$65$	1.48486	0.184174
$66$	0	0
$67$	2.96222	0.361893	0.180947	−	0.983493i	$-0.442084\pi$
$67$	2.96222	0.361893	0.180947	+	0.983493i	$0.442084\pi$
$68$	0	0
$69$	6.96972	0.839056
$70$	0	0
$71$	11.3910	1.35186	0.675931	−	0.736965i	$-0.263742\pi$
$71$	11.3910	1.35186	0.675931	+	0.736965i	$0.263742\pi$
$72$	0	0
$73$	9.46927	1.10829	0.554147	−	0.832419i	$-0.313045\pi$
$73$	9.46927	1.10829	0.554147	+	0.832419i	$0.313045\pi$
$74$	0	0
$75$	5.21799	0.602522
$76$	0	0
$77$	4.06055	0.462743
$78$	0	0
$79$	−6.91521	−0.778022	−0.389011	−	0.921233i	$-0.627183\pi$
$79$	−6.91521	−0.778022	−0.389011	+	0.921233i	$0.627183\pi$
$80$	0	0
$81$	−10.2195	−1.13550
$82$	0	0
$83$	−4.06603	−0.446304	−0.223152	−	0.974784i	$-0.571635\pi$
$83$	−4.06603	−0.446304	−0.223152	+	0.974784i	$0.571635\pi$
$84$	0	0
$85$	7.07523	0.767417
$86$	0	0
$87$	−3.73356	−0.400279
$88$	0	0
$89$	−0.969724	−0.102791	−0.0513953	−	0.998678i	$-0.516367\pi$
$89$	−0.969724	−0.102791	−0.0513953	+	0.998678i	$0.516367\pi$
$90$	0	0
$91$	−2.55248	−0.267572
$92$	0	0
$93$	−1.28005	−0.132735
$94$	0	0
$95$	−5.21799	−0.535355
$96$	0	0
$97$	3.93945	0.399990	0.199995	−	0.979797i	$-0.435907\pi$
$97$	3.93945	0.399990	0.199995	+	0.979797i	$0.435907\pi$
$98$	0	0
$99$	−0.771332	−0.0775218
$100$	0	0
$101$	18.5601	1.84680	0.923399	−	0.383841i	$-0.125399\pi$
$101$	18.5601	1.84680	0.923399	+	0.383841i	$0.125399\pi$
$102$	0	0
$103$	−6.47635	−0.638134	−0.319067	−	0.947732i	$-0.603369\pi$
$103$	−6.47635	−0.638134	−0.319067	+	0.947732i	$0.603369\pi$
$104$	0	0
$105$	7.07523	0.690472
$106$	0	0
$107$	0.0856337	0.00827852	0.00413926	−	0.999991i	$-0.498682\pi$
$107$	0.0856337	0.00827852	0.00413926	+	0.999991i	$0.498682\pi$
$108$	0	0
$109$	8.82546	0.845326	0.422663	−	0.906287i	$-0.361095\pi$
$109$	8.82546	0.845326	0.422663	+	0.906287i	$0.361095\pi$
$110$	0	0
$111$	3.23818	0.307354
$112$	0	0
$113$	−15.4693	−1.45523	−0.727613	−	0.685988i	$-0.759370\pi$
$113$	−15.4693	−1.45523	−0.727613	+	0.685988i	$0.759370\pi$
$114$	0	0
$115$	−5.54382	−0.516964
$116$	0	0
$117$	0.484862	0.0448255
$118$	0	0
$119$	−12.1623	−1.11492
$120$	0	0
$121$	−8.46927	−0.769933
$122$	0	0
$123$	14.0565	1.26743
$124$	0	0
$125$	−11.5748	−1.03528
$126$	0	0
$127$	7.58015	0.672630	0.336315	−	0.941750i	$-0.390819\pi$
$127$	7.58015	0.672630	0.336315	+	0.941750i	$0.390819\pi$
$128$	0	0
$129$	−8.76491	−0.771707
$130$	0	0
$131$	−2.33305	−0.203839	−0.101920	−	0.994793i	$-0.532498\pi$
$131$	−2.33305	−0.203839	−0.101920	+	0.994793i	$0.532498\pi$
$132$	0	0
$133$	8.96972	0.777774
$134$	0	0
$135$	6.97173	0.600031
$136$	0	0
$137$	0.0605522	0.00517332	0.00258666	−	0.999997i	$-0.499177\pi$
$137$	0.0605522	0.00517332	0.00258666	+	0.999997i	$0.499177\pi$
$138$	0	0
$139$	6.41983	0.544523	0.272262	−	0.962223i	$-0.412228\pi$
$139$	6.41983	0.544523	0.272262	+	0.962223i	$0.412228\pi$
$140$	0	0
$141$	10.0450	0.845938
$142$	0	0
$143$	−1.59083	−0.133032
$144$	0	0
$145$	2.96972	0.246622
$146$	0	0
$147$	0.905130	0.0746539
$148$	0	0
$149$	7.93945	0.650425	0.325212	−	0.945641i	$-0.394564\pi$
$149$	7.93945	0.650425	0.325212	+	0.945641i	$0.394564\pi$
$150$	0	0
$151$	2.63811	0.214686	0.107343	−	0.994222i	$-0.465766\pi$
$151$	2.63811	0.214686	0.107343	+	0.994222i	$0.465766\pi$
$152$	0	0
$153$	2.31032	0.186779
$154$	0	0
$155$	1.01817	0.0817812
$156$	0	0
$157$	11.5298	0.920180	0.460090	−	0.887872i	$-0.347817\pi$
$157$	11.5298	0.920180	0.460090	+	0.887872i	$0.347817\pi$
$158$	0	0
$159$	10.3229	0.818663
$160$	0	0
$161$	9.52982	0.751055
$162$	0	0
$163$	17.8965	1.40176	0.700879	−	0.713280i	$-0.252791\pi$
$163$	17.8965	1.40176	0.700879	+	0.713280i	$0.252791\pi$
$164$	0	0
$165$	4.40963	0.343289
$166$	0	0
$167$	19.9054	1.54032	0.770162	−	0.637848i	$-0.220175\pi$
$167$	19.9054	1.54032	0.770162	+	0.637848i	$0.220175\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.70387	−0.130298
$172$	0	0
$173$	23.4693	1.78434	0.892168	−	0.451704i	$-0.149184\pi$
$173$	23.4693	1.78434	0.892168	+	0.451704i	$0.149184\pi$
$174$	0	0
$175$	7.13464	0.539328
$176$	0	0
$177$	−20.4995	−1.54084
$178$	0	0
$179$	−24.6488	−1.84233	−0.921167	−	0.389167i	$-0.872763\pi$
$179$	−24.6488	−1.84233	−0.921167	+	0.389167i	$0.872763\pi$
$180$	0	0
$181$	−18.4995	−1.37506	−0.687530	−	0.726156i	$-0.741305\pi$
$181$	−18.4995	−1.37506	−0.687530	+	0.726156i	$0.741305\pi$
$182$	0	0
$183$	−23.3339	−1.72489
$184$	0	0
$185$	−2.57569	−0.189369
$186$	0	0
$187$	−7.58015	−0.554316
$188$	0	0
$189$	−11.9844	−0.871737
$190$	0	0
$191$	2.36216	0.170920	0.0854600	−	0.996342i	$-0.472764\pi$
$191$	2.36216	0.170920	0.0854600	+	0.996342i	$0.472764\pi$
$192$	0	0
$193$	0.0605522	0.00435864	0.00217932	−	0.999998i	$-0.499306\pi$
$193$	0.0605522	0.00435864	0.00217932	+	0.999998i	$0.499306\pi$
$194$	0	0
$195$	−2.77191	−0.198500
$196$	0	0
$197$	15.7952	1.12536	0.562680	−	0.826675i	$-0.309770\pi$
$197$	15.7952	1.12536	0.562680	+	0.826675i	$0.309770\pi$
$198$	0	0
$199$	22.5144	1.59600	0.798001	−	0.602656i	$-0.205891\pi$
$199$	22.5144	1.59600	0.798001	+	0.602656i	$0.205891\pi$
$200$	0	0
$201$	−5.52982	−0.390043
$202$	0	0
$203$	−5.10495	−0.358298
$204$	0	0
$205$	−11.1807	−0.780897
$206$	0	0
$207$	−1.81026	−0.125822
$208$	0	0
$209$	5.59037	0.386694
$210$	0	0
$211$	24.8474	1.71057	0.855284	−	0.518160i	$-0.173383\pi$
$211$	24.8474	1.71057	0.855284	+	0.518160i	$0.173383\pi$
$212$	0	0
$213$	−21.2645	−1.45702
$214$	0	0
$215$	6.97173	0.475468
$216$	0	0
$217$	−1.75023	−0.118813
$218$	0	0
$219$	−17.6770	−1.19450
$220$	0	0
$221$	4.76491	0.320522
$222$	0	0
$223$	15.3232	1.02612	0.513059	−	0.858353i	$-0.328512\pi$
$223$	15.3232	1.02612	0.513059	+	0.858353i	$0.328512\pi$
$224$	0	0
$225$	−1.35528	−0.0903519
$226$	0	0
$227$	−6.25692	−0.415286	−0.207643	−	0.978205i	$-0.566579\pi$
$227$	−6.25692	−0.415286	−0.207643	+	0.978205i	$0.566579\pi$
$228$	0	0
$229$	21.0752	1.39269	0.696345	−	0.717707i	$-0.254808\pi$
$229$	21.0752	1.39269	0.696345	+	0.717707i	$0.254808\pi$
$230$	0	0
$231$	−7.58015	−0.498737
$232$	0	0
$233$	3.01468	0.197498	0.0987491	−	0.995112i	$-0.468516\pi$
$233$	3.01468	0.197498	0.0987491	+	0.995112i	$0.468516\pi$
$234$	0	0
$235$	−7.98990	−0.521204
$236$	0	0
$237$	12.9092	0.838541
$238$	0	0
$239$	18.3062	1.18413	0.592065	−	0.805890i	$-0.298313\pi$
$239$	18.3062	1.18413	0.592065	+	0.805890i	$0.298313\pi$
$240$	0	0
$241$	28.0294	1.80553	0.902765	−	0.430134i	$-0.141534\pi$
$241$	28.0294	1.80553	0.902765	+	0.430134i	$0.141534\pi$
$242$	0	0
$243$	4.99192	0.320232
$244$	0	0
$245$	−0.719953	−0.0459961
$246$	0	0
$247$	−3.51413	−0.223598
$248$	0	0
$249$	7.59037	0.481020
$250$	0	0
$251$	−9.85668	−0.622148	−0.311074	−	0.950386i	$-0.600689\pi$
$251$	−9.85668	−0.622148	−0.311074	+	0.950386i	$0.600689\pi$
$252$	0	0
$253$	5.93945	0.373410
$254$	0	0
$255$	−13.2079	−0.827110
$256$	0	0
$257$	−0.204815	−0.0127760	−0.00638800	−	0.999980i	$-0.502033\pi$
$257$	−0.204815	−0.0127760	−0.00638800	+	0.999980i	$0.502033\pi$
$258$	0	0
$259$	4.42761	0.275118
$260$	0	0
$261$	0.969724	0.0600244
$262$	0	0
$263$	15.2018	0.937385	0.468692	−	0.883361i	$-0.344725\pi$
$263$	15.2018	0.937385	0.468692	+	0.883361i	$0.344725\pi$
$264$	0	0
$265$	−8.21102	−0.504399
$266$	0	0
$267$	1.81026	0.110786
$268$	0	0
$269$	9.09083	0.554278	0.277139	−	0.960830i	$-0.410614\pi$
$269$	9.09083	0.554278	0.277139	+	0.960830i	$0.410614\pi$
$270$	0	0
$271$	−4.67445	−0.283952	−0.141976	−	0.989870i	$-0.545346\pi$
$271$	−4.67445	−0.283952	−0.141976	+	0.989870i	$0.545346\pi$
$272$	0	0
$273$	4.76491	0.288385
$274$	0	0
$275$	4.44666	0.268144
$276$	0	0
$277$	−2.06055	−0.123807	−0.0619033	−	0.998082i	$-0.519717\pi$
$277$	−2.06055	−0.123807	−0.0619033	+	0.998082i	$0.519717\pi$
$278$	0	0
$279$	0.332469	0.0199044
$280$	0	0
$281$	−17.5904	−1.04935	−0.524677	−	0.851302i	$-0.675814\pi$
$281$	−17.5904	−1.04935	−0.524677	+	0.851302i	$0.675814\pi$
$282$	0	0
$283$	−14.5810	−0.866750	−0.433375	−	0.901214i	$-0.642678\pi$
$283$	−14.5810	−0.866750	−0.433375	+	0.901214i	$0.642678\pi$
$284$	0	0
$285$	9.74083	0.576997
$286$	0	0
$287$	19.2197	1.13450
$288$	0	0
$289$	5.70436	0.335550
$290$	0	0
$291$	−7.35408	−0.431104
$292$	0	0
$293$	14.8860	0.869650	0.434825	−	0.900515i	$-0.356810\pi$
$293$	14.8860	0.869650	0.434825	+	0.900515i	$0.356810\pi$
$294$	0	0
$295$	16.3056	0.949350
$296$	0	0
$297$	−7.46927	−0.433411
$298$	0	0
$299$	−3.73356	−0.215917
$300$	0	0
$301$	−11.9844	−0.690770
$302$	0	0
$303$	−34.6476	−1.99045
$304$	0	0
$305$	18.5601	1.06275
$306$	0	0
$307$	17.3030	0.987536	0.493768	−	0.869594i	$-0.335619\pi$
$307$	17.3030	0.987536	0.493768	+	0.869594i	$0.335619\pi$
$308$	0	0
$309$	12.0899	0.687771
$310$	0	0
$311$	4.55305	0.258180	0.129090	−	0.991633i	$-0.458794\pi$
$311$	4.55305	0.258180	0.129090	+	0.991633i	$0.458794\pi$
$312$	0	0
$313$	−14.5445	−0.822104	−0.411052	−	0.911612i	$-0.634838\pi$
$313$	−14.5445	−0.822104	−0.411052	+	0.911612i	$0.634838\pi$
$314$	0	0
$315$	−1.83766	−0.103541
$316$	0	0
$317$	21.0596	1.18283	0.591413	−	0.806368i	$-0.298570\pi$
$317$	21.0596	1.18283	0.591413	+	0.806368i	$0.298570\pi$
$318$	0	0
$319$	−3.18166	−0.178139
$320$	0	0
$321$	−0.159859	−0.00892247
$322$	0	0
$323$	−16.7445	−0.931688
$324$	0	0
$325$	−2.79518	−0.155049
$326$	0	0
$327$	−16.4752	−0.911080
$328$	0	0
$329$	13.7346	0.757215
$330$	0	0
$331$	−17.1900	−0.944848	−0.472424	−	0.881372i	$-0.656621\pi$
$331$	−17.1900	−0.944848	−0.472424	+	0.881372i	$0.656621\pi$
$332$	0	0
$333$	−0.841057	−0.0460897
$334$	0	0
$335$	4.39850	0.240315
$336$	0	0
$337$	−4.10551	−0.223641	−0.111821	−	0.993728i	$-0.535668\pi$
$337$	−4.10551	−0.223641	−0.111821	+	0.993728i	$0.535668\pi$
$338$	0	0
$339$	28.8777	1.56842
$340$	0	0
$341$	−1.09083	−0.0590717
$342$	0	0
$343$	19.1049	1.03157
$344$	0	0
$345$	10.3491	0.557176
$346$	0	0
$347$	−3.79008	−0.203462	−0.101731	−	0.994812i	$-0.532438\pi$
$347$	−3.79008	−0.203462	−0.101731	+	0.994812i	$0.532438\pi$
$348$	0	0
$349$	−35.3931	−1.89455	−0.947275	−	0.320422i	$-0.896175\pi$
$349$	−35.3931	−1.89455	−0.947275	+	0.320422i	$0.896175\pi$
$350$	0	0
$351$	4.69521	0.250612
$352$	0	0
$353$	7.43991	0.395986	0.197993	−	0.980203i	$-0.436558\pi$
$353$	7.43991	0.395986	0.197993	+	0.980203i	$0.436558\pi$
$354$	0	0
$355$	16.9140	0.897704
$356$	0	0
$357$	22.7044	1.20164
$358$	0	0
$359$	5.63609	0.297462	0.148731	−	0.988878i	$-0.452481\pi$
$359$	5.63609	0.297462	0.148731	+	0.988878i	$0.452481\pi$
$360$	0	0
$361$	−6.65092	−0.350049
$362$	0	0
$363$	15.8102	0.829823
$364$	0	0
$365$	14.0606	0.735963
$366$	0	0
$367$	34.9734	1.82560	0.912799	−	0.408410i	$-0.133917\pi$
$367$	34.9734	1.82560	0.912799	+	0.408410i	$0.133917\pi$
$368$	0	0
$369$	−3.65092	−0.190059
$370$	0	0
$371$	14.1147	0.732800
$372$	0	0
$373$	−21.5904	−1.11791	−0.558953	−	0.829199i	$-0.688797\pi$
$373$	−21.5904	−1.11791	−0.558953	+	0.829199i	$0.688797\pi$
$374$	0	0
$375$	21.6075	1.11581
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−32.1242	−1.65011	−0.825055	−	0.565052i	$-0.808856\pi$
$379$	−32.1242	−1.65011	−0.825055	+	0.565052i	$0.808856\pi$
$380$	0	0
$381$	−14.1505	−0.724950
$382$	0	0
$383$	−3.62887	−0.185427	−0.0927134	−	0.995693i	$-0.529554\pi$
$383$	−3.62887	−0.185427	−0.0927134	+	0.995693i	$0.529554\pi$
$384$	0	0
$385$	6.02936	0.307285
$386$	0	0
$387$	2.27653	0.115722
$388$	0	0
$389$	−17.9688	−0.911055	−0.455527	−	0.890222i	$-0.650549\pi$
$389$	−17.9688	−0.911055	−0.455527	+	0.890222i	$0.650549\pi$
$390$	0	0
$391$	−17.7901	−0.899682
$392$	0	0
$393$	4.35528	0.219695
$394$	0	0
$395$	−10.2681	−0.516646
$396$	0	0
$397$	−30.9991	−1.55580	−0.777900	−	0.628388i	$-0.783715\pi$
$397$	−30.9991	−1.55580	−0.777900	+	0.628388i	$0.783715\pi$
$398$	0	0
$399$	−16.7445	−0.838273
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	0.685698	0.0341571
$404$	0	0
$405$	−15.1745	−0.754029
$406$	0	0
$407$	2.75950	0.136783
$408$	0	0
$409$	30.9991	1.53281	0.766403	−	0.642360i	$-0.222045\pi$
$409$	30.9991	1.53281	0.766403	+	0.642360i	$0.222045\pi$
$410$	0	0
$411$	−0.113038	−0.00557573
$412$	0	0
$413$	−28.0294	−1.37923
$414$	0	0
$415$	−6.03749	−0.296369
$416$	0	0
$417$	−11.9844	−0.586879
$418$	0	0
$419$	18.7243	0.914742	0.457371	−	0.889276i	$-0.348791\pi$
$419$	18.7243	0.914742	0.457371	+	0.889276i	$0.348791\pi$
$420$	0	0
$421$	−13.5748	−0.661594	−0.330797	−	0.943702i	$-0.607318\pi$
$421$	−13.5748	−0.661594	−0.330797	+	0.943702i	$0.607318\pi$
$422$	0	0
$423$	−2.60900	−0.126854
$424$	0	0
$425$	−13.3188	−0.646057
$426$	0	0
$427$	−31.9048	−1.54398
$428$	0	0
$429$	2.96972	0.143380
$430$	0	0
$431$	32.0951	1.54597	0.772983	−	0.634426i	$-0.218764\pi$
$431$	32.0951	1.54597	0.772983	+	0.634426i	$0.218764\pi$
$432$	0	0
$433$	20.0450	0.963299	0.481650	−	0.876364i	$-0.340038\pi$
$433$	20.0450	0.963299	0.481650	+	0.876364i	$0.340038\pi$
$434$	0	0
$435$	−5.54382	−0.265806
$436$	0	0
$437$	13.1202	0.627624
$438$	0	0
$439$	−35.7514	−1.70632	−0.853160	−	0.521649i	$-0.825317\pi$
$439$	−35.7514	−1.70632	−0.853160	+	0.521649i	$0.825317\pi$
$440$	0	0
$441$	−0.235091	−0.0111948
$442$	0	0
$443$	40.6011	1.92902	0.964509	−	0.264049i	$-0.0850579\pi$
$443$	40.6011	1.92902	0.964509	+	0.264049i	$0.0850579\pi$
$444$	0	0
$445$	−1.43991	−0.0682581
$446$	0	0
$447$	−14.8212	−0.701018
$448$	0	0
$449$	−23.4986	−1.10897	−0.554484	−	0.832194i	$-0.687084\pi$
$449$	−23.4986	−1.10897	−0.554484	+	0.832194i	$0.687084\pi$
$450$	0	0
$451$	11.9786	0.564052
$452$	0	0
$453$	−4.92477	−0.231386
$454$	0	0
$455$	−3.79008	−0.177681
$456$	0	0
$457$	32.0294	1.49827	0.749135	−	0.662417i	$-0.230469\pi$
$457$	32.0294	1.49827	0.749135	+	0.662417i	$0.230469\pi$
$458$	0	0
$459$	22.3722	1.04425
$460$	0	0
$461$	25.7034	1.19713	0.598564	−	0.801075i	$-0.295738\pi$
$461$	25.7034	1.19713	0.598564	+	0.801075i	$0.295738\pi$
$462$	0	0
$463$	−30.5542	−1.41997	−0.709986	−	0.704216i	$-0.751299\pi$
$463$	−30.5542	−1.41997	−0.709986	+	0.704216i	$0.751299\pi$
$464$	0	0
$465$	−1.90069	−0.0881425
$466$	0	0
$467$	18.9628	0.877493	0.438747	−	0.898611i	$-0.355423\pi$
$467$	18.9628	0.877493	0.438747	+	0.898611i	$0.355423\pi$
$468$	0	0
$469$	−7.56101	−0.349135
$470$	0	0
$471$	−21.5236	−0.991756
$472$	0	0
$473$	−7.46927	−0.343437
$474$	0	0
$475$	9.82263	0.450693
$476$	0	0
$477$	−2.68120	−0.122764
$478$	0	0
$479$	−33.0002	−1.50782	−0.753910	−	0.656978i	$-0.771834\pi$
$479$	−33.0002	−1.50782	−0.753910	+	0.656978i	$0.771834\pi$
$480$	0	0
$481$	−1.73463	−0.0790924
$482$	0	0
$483$	−17.7901	−0.809476
$484$	0	0
$485$	5.84954	0.265614
$486$	0	0
$487$	13.3708	0.605889	0.302944	−	0.953008i	$-0.402030\pi$
$487$	13.3708	0.605889	0.302944	+	0.953008i	$0.402030\pi$
$488$	0	0
$489$	−33.4087	−1.51079
$490$	0	0
$491$	−33.5729	−1.51512	−0.757562	−	0.652763i	$-0.773610\pi$
$491$	−33.5729	−1.51512	−0.757562	+	0.652763i	$0.773610\pi$
$492$	0	0
$493$	9.52982	0.429201
$494$	0	0
$495$	−1.14532	−0.0514784
$496$	0	0
$497$	−29.0752	−1.30420
$498$	0	0
$499$	22.5625	1.01004	0.505019	−	0.863108i	$-0.331485\pi$
$499$	22.5625	1.01004	0.505019	+	0.863108i	$0.331485\pi$
$500$	0	0
$501$	−37.1589	−1.66014
$502$	0	0
$503$	−34.9152	−1.55679	−0.778395	−	0.627774i	$-0.783966\pi$
$503$	−34.9152	−1.55679	−0.778395	+	0.627774i	$0.783966\pi$
$504$	0	0
$505$	27.5592	1.22637
$506$	0	0
$507$	−1.86678	−0.0829065
$508$	0	0
$509$	16.9385	0.750787	0.375394	−	0.926865i	$-0.377508\pi$
$509$	16.9385	0.750787	0.375394	+	0.926865i	$0.377508\pi$
$510$	0	0
$511$	−24.1701	−1.06922
$512$	0	0
$513$	−16.4995	−0.728473
$514$	0	0
$515$	−9.61649	−0.423753
$516$	0	0
$517$	8.56009	0.376472
$518$	0	0
$519$	−43.8119	−1.92313
$520$	0	0
$521$	−8.10551	−0.355109	−0.177554	−	0.984111i	$-0.556819\pi$
$521$	−8.10551	−0.355109	−0.177554	+	0.984111i	$0.556819\pi$
$522$	0	0
$523$	41.8030	1.82792	0.913959	−	0.405808i	$-0.133010\pi$
$523$	41.8030	1.82792	0.913959	+	0.405808i	$0.133010\pi$
$524$	0	0
$525$	−13.3188	−0.581280
$526$	0	0
$527$	3.26729	0.142325
$528$	0	0
$529$	−9.06055	−0.393937
$530$	0	0
$531$	5.32439	0.231059
$532$	0	0
$533$	−7.52982	−0.326153
$534$	0	0
$535$	0.127154	0.00549736
$536$	0	0
$537$	46.0138	1.98564
$538$	0	0
$539$	0.771332	0.0332236
$540$	0	0
$541$	−22.2654	−0.957263	−0.478631	−	0.878016i	$-0.658867\pi$
$541$	−22.2654	−0.957263	−0.478631	+	0.878016i	$0.658867\pi$
$542$	0	0
$543$	34.5345	1.48202
$544$	0	0
$545$	13.1046	0.561339
$546$	0	0
$547$	8.74047	0.373716	0.186858	−	0.982387i	$-0.440170\pi$
$547$	8.74047	0.373716	0.186858	+	0.982387i	$0.440170\pi$
$548$	0	0
$549$	6.06055	0.258658
$550$	0	0
$551$	−7.02825	−0.299414
$552$	0	0
$553$	17.6509	0.750594
$554$	0	0
$555$	4.80824	0.204099
$556$	0	0
$557$	15.1745	0.642966	0.321483	−	0.946915i	$-0.395819\pi$
$557$	15.1745	0.642966	0.321483	+	0.946915i	$0.395819\pi$
$558$	0	0
$559$	4.69521	0.198586
$560$	0	0
$561$	14.1505	0.597433
$562$	0	0
$563$	−2.41868	−0.101935	−0.0509676	−	0.998700i	$-0.516231\pi$
$563$	−2.41868	−0.101935	−0.0509676	+	0.998700i	$0.516231\pi$
$564$	0	0
$565$	−22.9697	−0.966344
$566$	0	0
$567$	26.0850	1.09547
$568$	0	0
$569$	−5.61353	−0.235331	−0.117666	−	0.993053i	$-0.537541\pi$
$569$	−5.61353	−0.235331	−0.117666	+	0.993053i	$0.537541\pi$
$570$	0	0
$571$	−23.8293	−0.997223	−0.498612	−	0.866825i	$-0.666157\pi$
$571$	−23.8293	−0.997223	−0.498612	+	0.866825i	$0.666157\pi$
$572$	0	0
$573$	−4.40963	−0.184215
$574$	0	0
$575$	10.4360	0.435211
$576$	0	0
$577$	−34.4683	−1.43494	−0.717468	−	0.696591i	$-0.754699\pi$
$577$	−34.4683	−1.43494	−0.717468	+	0.696591i	$0.754699\pi$
$578$	0	0
$579$	−0.113038	−0.00469768
$580$	0	0
$581$	10.3784	0.430570
$582$	0	0
$583$	8.79699	0.364334
$584$	0	0
$585$	0.719953	0.0297664
$586$	0	0
$587$	−7.51528	−0.310189	−0.155094	−	0.987900i	$-0.549568\pi$
$587$	−7.51528	−0.310189	−0.155094	+	0.987900i	$0.549568\pi$
$588$	0	0
$589$	−2.40963	−0.0992871
$590$	0	0
$591$	−29.4861	−1.21290
$592$	0	0
$593$	−24.6812	−1.01354	−0.506768	−	0.862083i	$-0.669160\pi$
$593$	−24.6812	−1.01354	−0.506768	+	0.862083i	$0.669160\pi$
$594$	0	0
$595$	−18.0594	−0.740362
$596$	0	0
$597$	−42.0294	−1.72015
$598$	0	0
$599$	5.69837	0.232829	0.116415	−	0.993201i	$-0.462860\pi$
$599$	5.69837	0.232829	0.116415	+	0.993201i	$0.462860\pi$
$600$	0	0
$601$	−46.6732	−1.90384	−0.951919	−	0.306350i	$-0.900892\pi$
$601$	−46.6732	−1.90384	−0.951919	+	0.306350i	$0.900892\pi$
$602$	0	0
$603$	1.43627	0.0584894
$604$	0	0
$605$	−12.5757	−0.511275
$606$	0	0
$607$	13.3916	0.543547	0.271773	−	0.962361i	$-0.412390\pi$
$607$	13.3916	0.543547	0.271773	+	0.962361i	$0.412390\pi$
$608$	0	0
$609$	9.52982	0.386168
$610$	0	0
$611$	−5.38090	−0.217688
$612$	0	0
$613$	28.9385	1.16882	0.584408	−	0.811460i	$-0.301327\pi$
$613$	28.9385	1.16882	0.584408	+	0.811460i	$0.301327\pi$
$614$	0	0
$615$	20.8720	0.841639
$616$	0	0
$617$	7.65092	0.308015	0.154007	−	0.988070i	$-0.450782\pi$
$617$	7.65092	0.308015	0.154007	+	0.988070i	$0.450782\pi$
$618$	0	0
$619$	−30.4685	−1.22463	−0.612317	−	0.790613i	$-0.709762\pi$
$619$	−30.4685	−1.22463	−0.612317	+	0.790613i	$0.709762\pi$
$620$	0	0
$621$	−17.5298	−0.703447
$622$	0	0
$623$	2.47520	0.0991667
$624$	0	0
$625$	−3.21102	−0.128441
$626$	0	0
$627$	−10.4360	−0.416773
$628$	0	0
$629$	−8.26537	−0.329562
$630$	0	0
$631$	−2.25748	−0.0898688	−0.0449344	−	0.998990i	$-0.514308\pi$
$631$	−2.25748	−0.0898688	−0.0449344	+	0.998990i	$0.514308\pi$
$632$	0	0
$633$	−46.3846	−1.84362
$634$	0	0
$635$	11.2555	0.446660
$636$	0	0
$637$	−0.484862	−0.0192109
$638$	0	0
$639$	5.52306	0.218489
$640$	0	0
$641$	11.7115	0.462575	0.231288	−	0.972885i	$-0.425706\pi$
$641$	11.7115	0.462575	0.231288	+	0.972885i	$0.425706\pi$
$642$	0	0
$643$	22.0106	0.868015	0.434008	−	0.900909i	$-0.357099\pi$
$643$	22.0106	0.868015	0.434008	+	0.900909i	$0.357099\pi$
$644$	0	0
$645$	−13.0147	−0.512453
$646$	0	0
$647$	−15.1603	−0.596013	−0.298007	−	0.954564i	$-0.596322\pi$
$647$	−15.1603	−0.596013	−0.298007	+	0.954564i	$0.596322\pi$
$648$	0	0
$649$	−17.4693	−0.685729
$650$	0	0
$651$	3.26729	0.128055
$652$	0	0
$653$	−4.15046	−0.162420	−0.0812101	−	0.996697i	$-0.525878\pi$
$653$	−4.15046	−0.162420	−0.0812101	+	0.996697i	$0.525878\pi$
$654$	0	0
$655$	−3.46425	−0.135359
$656$	0	0
$657$	4.59129	0.179123
$658$	0	0
$659$	−17.7627	−0.691935	−0.345967	−	0.938247i	$-0.612449\pi$
$659$	−17.7627	−0.691935	−0.345967	+	0.938247i	$0.612449\pi$
$660$	0	0
$661$	−23.9394	−0.931137	−0.465568	−	0.885012i	$-0.654150\pi$
$661$	−23.9394	−0.931137	−0.465568	+	0.885012i	$0.654150\pi$
$662$	0	0
$663$	−8.89503	−0.345454
$664$	0	0
$665$	13.3188	0.516481
$666$	0	0
$667$	−7.46711	−0.289128
$668$	0	0
$669$	−28.6050	−1.10594
$670$	0	0
$671$	−19.8846	−0.767637
$672$	0	0
$673$	−25.6741	−0.989663	−0.494832	−	0.868989i	$-0.664770\pi$
$673$	−25.6741	−0.989663	−0.494832	+	0.868989i	$0.664770\pi$
$674$	0	0
$675$	−13.1240	−0.505142
$676$	0	0
$677$	13.0303	0.500794	0.250397	−	0.968143i	$-0.419439\pi$
$677$	13.0303	0.500794	0.250397	+	0.968143i	$0.419439\pi$
$678$	0	0
$679$	−10.0553	−0.385889
$680$	0	0
$681$	11.6803	0.447589
$682$	0	0
$683$	48.2171	1.84497	0.922487	−	0.386028i	$-0.126153\pi$
$683$	48.2171	1.84497	0.922487	+	0.386028i	$0.126153\pi$
$684$	0	0
$685$	0.0899116	0.00343535
$686$	0	0
$687$	−39.3428	−1.50102
$688$	0	0
$689$	−5.52982	−0.210669
$690$	0	0
$691$	−37.7644	−1.43662	−0.718312	−	0.695721i	$-0.755085\pi$
$691$	−37.7644	−1.43662	−0.718312	+	0.695721i	$0.755085\pi$
$692$	0	0
$693$	1.96881	0.0747888
$694$	0	0
$695$	9.53256	0.361591
$696$	0	0
$697$	−35.8789	−1.35901
$698$	0	0
$699$	−5.62774	−0.212861
$700$	0	0
$701$	−41.4381	−1.56509	−0.782547	−	0.622591i	$-0.786080\pi$
$701$	−41.4381	−1.56509	−0.782547	+	0.622591i	$0.786080\pi$
$702$	0	0
$703$	6.09572	0.229904
$704$	0	0
$705$	14.9154	0.561745
$706$	0	0
$707$	−47.3742	−1.78169
$708$	0	0
$709$	5.05964	0.190019	0.0950093	−	0.995476i	$-0.469712\pi$
$709$	5.05964	0.190019	0.0950093	+	0.995476i	$0.469712\pi$
$710$	0	0
$711$	−3.35292	−0.125744
$712$	0	0
$713$	−2.56009	−0.0958763
$714$	0	0
$715$	−2.36216	−0.0883398
$716$	0	0
$717$	−34.1736	−1.27624
$718$	0	0
$719$	−50.5726	−1.88604	−0.943019	−	0.332738i	$-0.892028\pi$
$719$	−50.5726	−1.88604	−0.943019	+	0.332738i	$0.892028\pi$
$720$	0	0
$721$	16.5307	0.615637
$722$	0	0
$723$	−52.3246	−1.94597
$724$	0	0
$725$	−5.59037	−0.207621
$726$	0	0
$727$	48.3650	1.79376	0.896879	−	0.442276i	$-0.145829\pi$
$727$	48.3650	1.79376	0.896879	+	0.442276i	$0.145829\pi$
$728$	0	0
$729$	21.3397	0.790359
$730$	0	0
$731$	22.3722	0.827467
$732$	0	0
$733$	−18.1055	−0.668742	−0.334371	−	0.942441i	$-0.608524\pi$
$733$	−18.1055	−0.668742	−0.334371	+	0.942441i	$0.608524\pi$
$734$	0	0
$735$	1.34399	0.0495739
$736$	0	0
$737$	−4.71239	−0.173583
$738$	0	0
$739$	−20.9236	−0.769685	−0.384843	−	0.922982i	$-0.625744\pi$
$739$	−20.9236	−0.769685	−0.384843	+	0.922982i	$0.625744\pi$
$740$	0	0
$741$	6.56009	0.240991
$742$	0	0
$743$	−23.8941	−0.876591	−0.438295	−	0.898831i	$-0.644418\pi$
$743$	−23.8941	−0.876591	−0.438295	+	0.898831i	$0.644418\pi$
$744$	0	0
$745$	11.7890	0.431915
$746$	0	0
$747$	−1.97146	−0.0721320
$748$	0	0
$749$	−0.218578	−0.00798667
$750$	0	0
$751$	48.8754	1.78349	0.891743	−	0.452541i	$-0.149482\pi$
$751$	48.8754	1.78349	0.891743	+	0.452541i	$0.149482\pi$
$752$	0	0
$753$	18.4002	0.670542
$754$	0	0
$755$	3.91723	0.142563
$756$	0	0
$757$	−26.1505	−0.950455	−0.475227	−	0.879863i	$-0.657634\pi$
$757$	−26.1505	−0.950455	−0.475227	+	0.879863i	$0.657634\pi$
$758$	0	0
$759$	−11.0876	−0.402456
$760$	0	0
$761$	35.9083	1.30167	0.650837	−	0.759218i	$-0.274418\pi$
$761$	35.9083	1.30167	0.650837	+	0.759218i	$0.274418\pi$
$762$	0	0
$763$	−22.5268	−0.815524
$764$	0	0
$765$	3.43051	0.124030
$766$	0	0
$767$	10.9812	0.396510
$768$	0	0
$769$	17.7115	0.638692	0.319346	−	0.947638i	$-0.396537\pi$
$769$	17.7115	0.638692	0.319346	+	0.947638i	$0.396537\pi$
$770$	0	0
$771$	0.382344	0.0137698
$772$	0	0
$773$	−1.07523	−0.0386734	−0.0193367	−	0.999813i	$-0.506155\pi$
$773$	−1.07523	−0.0386734	−0.0193367	+	0.999813i	$0.506155\pi$
$774$	0	0
$775$	−1.91665	−0.0688482
$776$	0	0
$777$	−8.26537	−0.296518
$778$	0	0
$779$	26.4607	0.948054
$780$	0	0
$781$	−18.1211	−0.648424
$782$	0	0
$783$	9.39041	0.335586
$784$	0	0
$785$	17.1202	0.611046
$786$	0	0
$787$	25.6479	0.914248	0.457124	−	0.889403i	$-0.348880\pi$
$787$	25.6479	0.914248	0.457124	+	0.889403i	$0.348880\pi$
$788$	0	0
$789$	−28.3784	−1.01030
$790$	0	0
$791$	39.4849	1.40392
$792$	0	0
$793$	12.4995	0.443872
$794$	0	0
$795$	15.3281	0.543633
$796$	0	0
$797$	12.0294	0.426102	0.213051	−	0.977041i	$-0.431660\pi$
$797$	12.0294	0.426102	0.213051	+	0.977041i	$0.431660\pi$
$798$	0	0
$799$	−25.6395	−0.907061
$800$	0	0
$801$	−0.470182	−0.0166131
$802$	0	0
$803$	−15.0640	−0.531596
$804$	0	0
$805$	14.1505	0.498738
$806$	0	0
$807$	−16.9706	−0.597392
$808$	0	0
$809$	−24.1055	−0.847505	−0.423752	−	0.905778i	$-0.639287\pi$
$809$	−24.1055	−0.847505	−0.423752	+	0.905778i	$0.639287\pi$
$810$	0	0
$811$	−35.4189	−1.24373	−0.621863	−	0.783126i	$-0.713624\pi$
$811$	−35.4189	−1.24373	−0.621863	+	0.783126i	$0.713624\pi$
$812$	0	0
$813$	8.72615	0.306040
$814$	0	0
$815$	26.5738	0.930838
$816$	0	0
$817$	−16.4995	−0.577246
$818$	0	0
$819$	−1.23760	−0.0432452
$820$	0	0
$821$	−26.8936	−0.938592	−0.469296	−	0.883041i	$-0.655492\pi$
$821$	−26.8936	−0.938592	−0.469296	+	0.883041i	$0.655492\pi$
$822$	0	0
$823$	31.1983	1.08751	0.543753	−	0.839245i	$-0.317003\pi$
$823$	31.1983	1.08751	0.543753	+	0.839245i	$0.317003\pi$
$824$	0	0
$825$	−8.30093	−0.289001
$826$	0	0
$827$	39.8007	1.38401	0.692003	−	0.721895i	$-0.256728\pi$
$827$	39.8007	1.38401	0.692003	+	0.721895i	$0.256728\pi$
$828$	0	0
$829$	10.0294	0.348334	0.174167	−	0.984716i	$-0.444277\pi$
$829$	10.0294	0.348334	0.174167	+	0.984716i	$0.444277\pi$
$830$	0	0
$831$	3.84659	0.133437
$832$	0	0
$833$	−2.31032	−0.0800479
$834$	0	0
$835$	29.5567	1.02285
$836$	0	0
$837$	3.21949	0.111282
$838$	0	0
$839$	−43.7745	−1.51126	−0.755631	−	0.654998i	$-0.772670\pi$
$839$	−43.7745	−1.51126	−0.755631	+	0.654998i	$0.772670\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	32.8373	1.13098
$844$	0	0
$845$	1.48486	0.0510808
$846$	0	0
$847$	21.6176	0.742790
$848$	0	0
$849$	27.2195	0.934171
$850$	0	0
$851$	6.47635	0.222006
$852$	0	0
$853$	−38.6050	−1.32181	−0.660906	−	0.750469i	$-0.729828\pi$
$853$	−38.6050	−1.32181	−0.660906	+	0.750469i	$0.729828\pi$
$854$	0	0
$855$	−2.53001	−0.0865244
$856$	0	0
$857$	−3.93945	−0.134569	−0.0672845	−	0.997734i	$-0.521434\pi$
$857$	−3.93945	−0.134569	−0.0672845	+	0.997734i	$0.521434\pi$
$858$	0	0
$859$	8.65655	0.295358	0.147679	−	0.989035i	$-0.452820\pi$
$859$	8.65655	0.295358	0.147679	+	0.989035i	$0.452820\pi$
$860$	0	0
$861$	−35.8789	−1.22275
$862$	0	0
$863$	0.361584	0.0123085	0.00615423	−	0.999981i	$-0.498041\pi$
$863$	0.361584	0.0123085	0.00615423	+	0.999981i	$0.498041\pi$
$864$	0	0
$865$	34.8486	1.18489
$866$	0	0
$867$	−10.6488	−0.361651
$868$	0	0
$869$	11.0009	0.373181
$870$	0	0
$871$	2.96222	0.100371
$872$	0	0
$873$	1.91009	0.0646467
$874$	0	0
$875$	29.5443	0.998781
$876$	0	0
$877$	0.0761486	0.00257135	0.00128568	−	0.999999i	$-0.499591\pi$
$877$	0.0761486	0.00257135	0.00128568	+	0.999999i	$0.499591\pi$
$878$	0	0
$879$	−27.7889	−0.937296
$880$	0	0
$881$	47.3544	1.59541	0.797704	−	0.603049i	$-0.206047\pi$
$881$	47.3544	1.59541	0.797704	+	0.603049i	$0.206047\pi$
$882$	0	0
$883$	3.47837	0.117056	0.0585282	−	0.998286i	$-0.481359\pi$
$883$	3.47837	0.117056	0.0585282	+	0.998286i	$0.481359\pi$
$884$	0	0
$885$	−30.4390	−1.02320
$886$	0	0
$887$	−28.5519	−0.958678	−0.479339	−	0.877630i	$-0.659124\pi$
$887$	−28.5519	−0.958678	−0.479339	+	0.877630i	$0.659124\pi$
$888$	0	0
$889$	−19.3482	−0.648916
$890$	0	0
$891$	16.2575	0.544645
$892$	0	0
$893$	18.9092	0.632771
$894$	0	0
$895$	−36.6000	−1.22340
$896$	0	0
$897$	6.96972	0.232712
$898$	0	0
$899$	1.37140	0.0457386
$900$	0	0
$901$	−26.3491	−0.877815
$902$	0	0
$903$	22.3722	0.744501
$904$	0	0
$905$	−27.4693	−0.913109
$906$	0	0
$907$	−16.8284	−0.558778	−0.279389	−	0.960178i	$-0.590132\pi$
$907$	−16.8284	−0.558778	−0.279389	+	0.960178i	$0.590132\pi$
$908$	0	0
$909$	8.99908	0.298481
$910$	0	0
$911$	48.1522	1.59535	0.797677	−	0.603086i	$-0.206062\pi$
$911$	48.1522	1.59535	0.797677	+	0.603086i	$0.206062\pi$
$912$	0	0
$913$	6.46835	0.214071
$914$	0	0
$915$	−34.6476	−1.14541
$916$	0	0
$917$	5.95504	0.196653
$918$	0	0
$919$	37.8292	1.24787	0.623936	−	0.781476i	$-0.285533\pi$
$919$	37.8292	1.24787	0.623936	+	0.781476i	$0.285533\pi$
$920$	0	0
$921$	−32.3009	−1.06435
$922$	0	0
$923$	11.3910	0.374939
$924$	0	0
$925$	4.84862	0.159422
$926$	0	0
$927$	−3.14014	−0.103136
$928$	0	0
$929$	−21.9688	−0.720773	−0.360387	−	0.932803i	$-0.617355\pi$
$929$	−21.9688	−0.720773	−0.360387	+	0.932803i	$0.617355\pi$
$930$	0	0
$931$	1.70387	0.0558419
$932$	0	0
$933$	−8.49954	−0.278263
$934$	0	0
$935$	−11.2555	−0.368094
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	27.1514	0.886051
$940$	0	0
$941$	−0.0449558	−0.00146552	−0.000732759	−	1.00000i	$-0.500233\pi$
$941$	−0.0449558	−0.00146552	−0.000732759		1.00000i	$0.500233\pi$
$942$	0	0
$943$	28.1130	0.915485
$944$	0	0
$945$	−17.7952	−0.578877
$946$	0	0
$947$	−37.7811	−1.22772	−0.613860	−	0.789415i	$-0.710384\pi$
$947$	−37.7811	−1.22772	−0.613860	+	0.789415i	$0.710384\pi$
$948$	0	0
$949$	9.46927	0.307385
$950$	0	0
$951$	−39.3137	−1.27483
$952$	0	0
$953$	22.8860	0.741351	0.370675	−	0.928763i	$-0.379126\pi$
$953$	22.8860	0.741351	0.370675	+	0.928763i	$0.379126\pi$
$954$	0	0
$955$	3.50748	0.113499
$956$	0	0
$957$	5.93945	0.191995
$958$	0	0
$959$	−0.154558	−0.00499094
$960$	0	0
$961$	−30.5298	−0.984833
$962$	0	0
$963$	0.0415205	0.00133798
$964$	0	0
$965$	0.0899116	0.00289436
$966$	0	0
$967$	−0.457912	−0.0147255	−0.00736273	−	0.999973i	$-0.502344\pi$
$967$	−0.457912	−0.0147255	−0.00736273	+	0.999973i	$0.502344\pi$
$968$	0	0
$969$	31.2582	1.00416
$970$	0	0
$971$	−3.15254	−0.101170	−0.0505849	−	0.998720i	$-0.516109\pi$
$971$	−3.15254	−0.101170	−0.0505849	+	0.998720i	$0.516109\pi$
$972$	0	0
$973$	−16.3865	−0.525326
$974$	0	0
$975$	5.21799	0.167109
$976$	0	0
$977$	9.87890	0.316054	0.158027	−	0.987435i	$-0.449487\pi$
$977$	9.87890	0.316054	0.158027	+	0.987435i	$0.449487\pi$
$978$	0	0
$979$	1.54266	0.0493038
$980$	0	0
$981$	4.27913	0.136622
$982$	0	0
$983$	44.7802	1.42827	0.714133	−	0.700010i	$-0.246821\pi$
$983$	44.7802	1.42827	0.714133	+	0.700010i	$0.246821\pi$
$984$	0	0
$985$	23.4537	0.747296
$986$	0	0
$987$	−25.6395	−0.816115
$988$	0	0
$989$	−17.5298	−0.557416
$990$	0	0
$991$	−12.2878	−0.390334	−0.195167	−	0.980770i	$-0.562525\pi$
$991$	−12.2878	−0.390334	−0.195167	+	0.980770i	$0.562525\pi$
$992$	0	0
$993$	32.0899	1.01834
$994$	0	0
$995$	33.4307	1.05983
$996$	0	0
$997$	23.0908	0.731294	0.365647	−	0.930754i	$-0.380848\pi$
$997$	23.0908	0.731294	0.365647	+	0.930754i	$0.380848\pi$
$998$	0	0
$999$	−8.14446	−0.257679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bp.1.2		6
4.3	odd	2	inner	3328.2.a.bp.1.5		6
8.3	odd	2		3328.2.a.bo.1.2		6
8.5	even	2		3328.2.a.bo.1.5		6
16.3	odd	4		1664.2.b.k.833.9	yes	12
16.5	even	4		1664.2.b.k.833.10	yes	12
16.11	odd	4		1664.2.b.k.833.4	yes	12
16.13	even	4		1664.2.b.k.833.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.k.833.3	✓	12	16.13	even	4
1664.2.b.k.833.4	yes	12	16.11	odd	4
1664.2.b.k.833.9	yes	12	16.3	odd	4
1664.2.b.k.833.10	yes	12	16.5	even	4
3328.2.a.bo.1.2		6	8.3	odd	2
3328.2.a.bo.1.5		6	8.5	even	2
3328.2.a.bp.1.2		6	1.1	even	1	trivial
3328.2.a.bp.1.5		6	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86678 q^{3} +1.48486 q^{5} -2.55248 q^{7} +0.484862 q^{9} -1.59083 q^{11} +1.00000 q^{13} -2.77191 q^{15} +4.76491 q^{17} -3.51413 q^{19} +4.76491 q^{21} -3.73356 q^{23} -2.79518 q^{25} +4.69521 q^{27} +2.00000 q^{29} +0.685698 q^{31} +2.96972 q^{33} -3.79008 q^{35} -1.73463 q^{37} -1.86678 q^{39} -7.52982 q^{41} +4.69521 q^{43} +0.719953 q^{45} -5.38090 q^{47} -0.484862 q^{49} -8.89503 q^{51} -5.52982 q^{53} -2.36216 q^{55} +6.56009 q^{57} +10.9812 q^{59} +12.4995 q^{61} -1.23760 q^{63} +1.48486 q^{65} +2.96222 q^{67} +6.96972 q^{69} +11.3910 q^{71} +9.46927 q^{73} +5.21799 q^{75} +4.06055 q^{77} -6.91521 q^{79} -10.2195 q^{81} -4.06603 q^{83} +7.07523 q^{85} -3.73356 q^{87} -0.969724 q^{89} -2.55248 q^{91} -1.28005 q^{93} -5.21799 q^{95} +3.93945 q^{97} -0.771332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q + 8 * q^5 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 4 * q^21 + 14 * q^25 + 12 * q^29 + 16 * q^33 + 24 * q^37 + 20 * q^41 + 36 * q^45 - 2 * q^49 + 32 * q^53 - 24 * q^57 + 8 * q^61 + 8 * q^65 + 40 * q^69 - 12 * q^73 + 28 * q^77 - 26 * q^81 - 20 * q^85 - 4 * q^89 + 24 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more