LMFDB - Embedded newform 3432.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3432.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.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:	$27.4046579737$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.70164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 10x - 2 $ x^4 - x^3 - 10x^2 + 10x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.20666$ of defining polynomial
Character	$\chi$	$=$	3432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.20666 q^{5} -1.53005 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.20666 q^{5} -1.53005 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.20666 q^{15} +2.73671 q^{17} -7.78961 q^{19} +1.53005 q^{21} -6.88326 q^{23} +5.28265 q^{25} -1.00000 q^{27} +1.22974 q^{29} -8.11301 q^{31} -1.00000 q^{33} -4.90635 q^{35} -6.41331 q^{37} +1.00000 q^{39} +3.53005 q^{41} +3.98234 q^{43} +3.20666 q^{45} -2.93990 q^{47} -4.65894 q^{49} -2.73671 q^{51} -10.2260 q^{53} +3.20666 q^{55} +7.78961 q^{57} +12.6729 q^{59} -6.30573 q^{61} -1.53005 q^{63} -3.20666 q^{65} -5.80727 q^{67} +6.88326 q^{69} +10.2260 q^{71} -15.4486 q^{73} -5.28265 q^{75} -1.53005 q^{77} -1.98411 q^{79} +1.00000 q^{81} +5.16591 q^{83} +8.77568 q^{85} -1.22974 q^{87} -4.11301 q^{89} +1.53005 q^{91} +8.11301 q^{93} -24.9786 q^{95} +15.9185 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} - 6 q^{17} + q^{21} - 9 q^{23} + q^{25} - 4 q^{27} - q^{29} - 8 q^{31} - 4 q^{33} - 7 q^{35} - 2 q^{37} + 4 q^{39} + 9 q^{41} - 5 q^{43} + q^{45} - 22 q^{47} + 9 q^{49} + 6 q^{51} + 8 q^{53} + q^{55} + q^{59} - 11 q^{61} - q^{63} - q^{65} - 13 q^{67} + 9 q^{69} - 8 q^{71} - 3 q^{73} - q^{75} - q^{77} - 6 q^{79} + 4 q^{81} - 18 q^{83} + 26 q^{85} + q^{87} + 8 q^{89} + q^{91} + 8 q^{93} - 36 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - q^15 - 6 * q^17 + q^21 - 9 * q^23 + q^25 - 4 * q^27 - q^29 - 8 * q^31 - 4 * q^33 - 7 * q^35 - 2 * q^37 + 4 * q^39 + 9 * q^41 - 5 * q^43 + q^45 - 22 * q^47 + 9 * q^49 + 6 * q^51 + 8 * q^53 + q^55 + q^59 - 11 * q^61 - q^63 - q^65 - 13 * q^67 + 9 * q^69 - 8 * q^71 - 3 * q^73 - q^75 - q^77 - 6 * q^79 + 4 * q^81 - 18 * q^83 + 26 * q^85 + q^87 + 8 * q^89 + q^91 + 8 * q^93 - 36 * q^95 + 10 * q^97 + 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.20666	1.43406	0.717030	−	0.697042i	$-0.245501\pi$
$5$	3.20666	1.43406	0.717030	+	0.697042i	$0.245501\pi$
$6$	0	0
$7$	−1.53005	−0.578305	−0.289152	−	0.957283i	$-0.593373\pi$
$7$	−1.53005	−0.578305	−0.289152	+	0.957283i	$0.593373\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.20666	−0.827955
$16$	0	0
$17$	2.73671	0.663749	0.331875	−	0.943324i	$-0.392319\pi$
$17$	2.73671	0.663749	0.331875	+	0.943324i	$0.392319\pi$
$18$	0	0
$19$	−7.78961	−1.78706	−0.893530	−	0.449004i	$-0.851779\pi$
$19$	−7.78961	−1.78706	−0.893530	+	0.449004i	$0.851779\pi$
$20$	0	0
$21$	1.53005	0.333885
$22$	0	0
$23$	−6.88326	−1.43526	−0.717630	−	0.696425i	$-0.754773\pi$
$23$	−6.88326	−1.43526	−0.717630	+	0.696425i	$0.754773\pi$
$24$	0	0
$25$	5.28265	1.05653
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.22974	0.228358	0.114179	−	0.993460i	$-0.463576\pi$
$29$	1.22974	0.228358	0.114179	+	0.993460i	$0.463576\pi$
$30$	0	0
$31$	−8.11301	−1.45714	−0.728569	−	0.684972i	$-0.759814\pi$
$31$	−8.11301	−1.45714	−0.728569	+	0.684972i	$0.759814\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−4.90635	−0.829324
$36$	0	0
$37$	−6.41331	−1.05434	−0.527171	−	0.849759i	$-0.676747\pi$
$37$	−6.41331	−1.05434	−0.527171	+	0.849759i	$0.676747\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.53005	0.551301	0.275651	−	0.961258i	$-0.411107\pi$
$41$	3.53005	0.551301	0.275651	+	0.961258i	$0.411107\pi$
$42$	0	0
$43$	3.98234	0.607301	0.303650	−	0.952784i	$-0.401795\pi$
$43$	3.98234	0.607301	0.303650	+	0.952784i	$0.401795\pi$
$44$	0	0
$45$	3.20666	0.478020
$46$	0	0
$47$	−2.93990	−0.428828	−0.214414	−	0.976743i	$-0.568784\pi$
$47$	−2.93990	−0.428828	−0.214414	+	0.976743i	$0.568784\pi$
$48$	0	0
$49$	−4.65894	−0.665563
$50$	0	0
$51$	−2.73671	−0.383216
$52$	0	0
$53$	−10.2260	−1.40465	−0.702325	−	0.711856i	$-0.747855\pi$
$53$	−10.2260	−1.40465	−0.702325	+	0.711856i	$0.747855\pi$
$54$	0	0
$55$	3.20666	0.432385
$56$	0	0
$57$	7.78961	1.03176
$58$	0	0
$59$	12.6729	1.64987	0.824934	−	0.565229i	$-0.191212\pi$
$59$	12.6729	1.64987	0.824934	+	0.565229i	$0.191212\pi$
$60$	0	0
$61$	−6.30573	−0.807366	−0.403683	−	0.914899i	$-0.632270\pi$
$61$	−6.30573	−0.807366	−0.403683	+	0.914899i	$0.632270\pi$
$62$	0	0
$63$	−1.53005	−0.192768
$64$	0	0
$65$	−3.20666	−0.397737
$66$	0	0
$67$	−5.80727	−0.709471	−0.354736	−	0.934967i	$-0.615429\pi$
$67$	−5.80727	−0.709471	−0.354736	+	0.934967i	$0.615429\pi$
$68$	0	0
$69$	6.88326	0.828647
$70$	0	0
$71$	10.2260	1.21360	0.606802	−	0.794853i	$-0.292452\pi$
$71$	10.2260	1.21360	0.606802	+	0.794853i	$0.292452\pi$
$72$	0	0
$73$	−15.4486	−1.80812	−0.904058	−	0.427409i	$-0.859426\pi$
$73$	−15.4486	−1.80812	−0.904058	+	0.427409i	$0.859426\pi$
$74$	0	0
$75$	−5.28265	−0.609987
$76$	0	0
$77$	−1.53005	−0.174366
$78$	0	0
$79$	−1.98411	−0.223230	−0.111615	−	0.993752i	$-0.535602\pi$
$79$	−1.98411	−0.223230	−0.111615	+	0.993752i	$0.535602\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.16591	0.567032	0.283516	−	0.958967i	$-0.408499\pi$
$83$	5.16591	0.567032	0.283516	+	0.958967i	$0.408499\pi$
$84$	0	0
$85$	8.77568	0.951856
$86$	0	0
$87$	−1.22974	−0.131842
$88$	0	0
$89$	−4.11301	−0.435978	−0.217989	−	0.975951i	$-0.569950\pi$
$89$	−4.11301	−0.435978	−0.217989	+	0.975951i	$0.569950\pi$
$90$	0	0
$91$	1.53005	0.160393
$92$	0	0
$93$	8.11301	0.841279
$94$	0	0
$95$	−24.9786	−2.56275
$96$	0	0
$97$	15.9185	1.61628	0.808140	−	0.588991i	$-0.200475\pi$
$97$	15.9185	1.61628	0.808140	+	0.588991i	$0.200475\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	9.82858	0.977981	0.488990	−	0.872289i	$-0.337365\pi$
$101$	9.82858	0.977981	0.488990	+	0.872289i	$0.337365\pi$
$102$	0	0
$103$	−10.3057	−1.01545	−0.507727	−	0.861518i	$-0.669514\pi$
$103$	−10.3057	−1.01545	−0.507727	+	0.861518i	$0.669514\pi$
$104$	0	0
$105$	4.90635	0.478811
$106$	0	0
$107$	2.86018	0.276504	0.138252	−	0.990397i	$-0.455852\pi$
$107$	2.86018	0.276504	0.138252	+	0.990397i	$0.455852\pi$
$108$	0	0
$109$	−10.4364	−0.999626	−0.499813	−	0.866133i	$-0.666598\pi$
$109$	−10.4364	−0.999626	−0.499813	+	0.866133i	$0.666598\pi$
$110$	0	0
$111$	6.41331	0.608725
$112$	0	0
$113$	0.906349	0.0852621	0.0426311	−	0.999091i	$-0.486426\pi$
$113$	0.906349	0.0852621	0.0426311	+	0.999091i	$0.486426\pi$
$114$	0	0
$115$	−22.0723	−2.05825
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.18730	−0.383849
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.53005	−0.318294
$124$	0	0
$125$	0.906349	0.0810663
$126$	0	0
$127$	11.9026	1.05619	0.528093	−	0.849186i	$-0.322907\pi$
$127$	11.9026	1.05619	0.528093	+	0.849186i	$0.322907\pi$
$128$	0	0
$129$	−3.98234	−0.350625
$130$	0	0
$131$	−19.1324	−1.67160	−0.835801	−	0.549033i	$-0.814996\pi$
$131$	−19.1324	−1.67160	−0.835801	+	0.549033i	$0.814996\pi$
$132$	0	0
$133$	11.9185	1.03347
$134$	0	0
$135$	−3.20666	−0.275985
$136$	0	0
$137$	5.02113	0.428984	0.214492	−	0.976726i	$-0.431190\pi$
$137$	5.02113	0.428984	0.214492	+	0.976726i	$0.431190\pi$
$138$	0	0
$139$	14.1361	1.19901	0.599504	−	0.800372i	$-0.295365\pi$
$139$	14.1361	1.19901	0.599504	+	0.800372i	$0.295365\pi$
$140$	0	0
$141$	2.93990	0.247584
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	3.94336	0.327479
$146$	0	0
$147$	4.65894	0.384263
$148$	0	0
$149$	7.03702	0.576495	0.288247	−	0.957556i	$-0.406927\pi$
$149$	7.03702	0.576495	0.288247	+	0.957556i	$0.406927\pi$
$150$	0	0
$151$	10.8497	0.882937	0.441469	−	0.897277i	$-0.354458\pi$
$151$	10.8497	0.882937	0.441469	+	0.897277i	$0.354458\pi$
$152$	0	0
$153$	2.73671	0.221250
$154$	0	0
$155$	−26.0156	−2.08962
$156$	0	0
$157$	−16.6624	−1.32981	−0.664903	−	0.746930i	$-0.731527\pi$
$157$	−16.6624	−1.32981	−0.664903	+	0.746930i	$0.731527\pi$
$158$	0	0
$159$	10.2260	0.810975
$160$	0	0
$161$	10.5317	0.830018
$162$	0	0
$163$	−9.95925	−0.780069	−0.390034	−	0.920800i	$-0.627537\pi$
$163$	−9.95925	−0.780069	−0.390034	+	0.920800i	$0.627537\pi$
$164$	0	0
$165$	−3.20666	−0.249638
$166$	0	0
$167$	−6.68034	−0.516940	−0.258470	−	0.966019i	$-0.583218\pi$
$167$	−6.68034	−0.516940	−0.258470	+	0.966019i	$0.583218\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.78961	−0.595686
$172$	0	0
$173$	−0.169641	−0.0128976	−0.00644878	−	0.999979i	$-0.502053\pi$
$173$	−0.169641	−0.0128976	−0.00644878	+	0.999979i	$0.502053\pi$
$174$	0	0
$175$	−8.08272	−0.610996
$176$	0	0
$177$	−12.6729	−0.952551
$178$	0	0
$179$	−7.66941	−0.573238	−0.286619	−	0.958045i	$-0.592531\pi$
$179$	−7.66941	−0.573238	−0.286619	+	0.958045i	$0.592531\pi$
$180$	0	0
$181$	1.03702	0.0770808	0.0385404	−	0.999257i	$-0.487729\pi$
$181$	1.03702	0.0770808	0.0385404	+	0.999257i	$0.487729\pi$
$182$	0	0
$183$	6.30573	0.466133
$184$	0	0
$185$	−20.5653	−1.51199
$186$	0	0
$187$	2.73671	0.200128
$188$	0	0
$189$	1.53005	0.111295
$190$	0	0
$191$	−0.199927	−0.0144662	−0.00723311	−	0.999974i	$-0.502302\pi$
$191$	−0.199927	−0.0144662	−0.00723311	+	0.999974i	$0.502302\pi$
$192$	0	0
$193$	20.5069	1.47612	0.738059	−	0.674736i	$-0.235743\pi$
$193$	20.5069	1.47612	0.738059	+	0.674736i	$0.235743\pi$
$194$	0	0
$195$	3.20666	0.229633
$196$	0	0
$197$	−10.4826	−0.746852	−0.373426	−	0.927660i	$-0.621817\pi$
$197$	−10.4826	−0.746852	−0.373426	+	0.927660i	$0.621817\pi$
$198$	0	0
$199$	−20.1184	−1.42616	−0.713079	−	0.701084i	$-0.752700\pi$
$199$	−20.1184	−1.42616	−0.713079	+	0.701084i	$0.752700\pi$
$200$	0	0
$201$	5.80727	0.409613
$202$	0	0
$203$	−1.88157	−0.132060
$204$	0	0
$205$	11.3197	0.790599
$206$	0	0
$207$	−6.88326	−0.478420
$208$	0	0
$209$	−7.78961	−0.538819
$210$	0	0
$211$	−16.0546	−1.10524	−0.552622	−	0.833432i	$-0.686373\pi$
$211$	−16.0546	−1.10524	−0.552622	+	0.833432i	$0.686373\pi$
$212$	0	0
$213$	−10.2260	−0.700675
$214$	0	0
$215$	12.7700	0.870906
$216$	0	0
$217$	12.4133	0.842671
$218$	0	0
$219$	15.4486	1.04392
$220$	0	0
$221$	−2.73671	−0.184091
$222$	0	0
$223$	−15.5124	−1.03879	−0.519393	−	0.854535i	$-0.673842\pi$
$223$	−15.5124	−1.03879	−0.519393	+	0.854535i	$0.673842\pi$
$224$	0	0
$225$	5.28265	0.352176
$226$	0	0
$227$	2.02309	0.134277	0.0671385	−	0.997744i	$-0.478613\pi$
$227$	2.02309	0.134277	0.0671385	+	0.997744i	$0.478613\pi$
$228$	0	0
$229$	7.19946	0.475754	0.237877	−	0.971295i	$-0.423549\pi$
$229$	7.19946	0.475754	0.237877	+	0.971295i	$0.423549\pi$
$230$	0	0
$231$	1.53005	0.100670
$232$	0	0
$233$	−16.7367	−1.09646	−0.548229	−	0.836328i	$-0.684698\pi$
$233$	−16.7367	−1.09646	−0.548229	+	0.836328i	$0.684698\pi$
$234$	0	0
$235$	−9.42724	−0.614965
$236$	0	0
$237$	1.98411	0.128882
$238$	0	0
$239$	8.23647	0.532773	0.266387	−	0.963866i	$-0.414170\pi$
$239$	8.23647	0.532773	0.266387	+	0.963866i	$0.414170\pi$
$240$	0	0
$241$	4.72082	0.304095	0.152047	−	0.988373i	$-0.451413\pi$
$241$	4.72082	0.304095	0.152047	+	0.988373i	$0.451413\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−14.9396	−0.954458
$246$	0	0
$247$	7.78961	0.495641
$248$	0	0
$249$	−5.16591	−0.327376
$250$	0	0
$251$	−11.6607	−0.736018	−0.368009	−	0.929822i	$-0.619960\pi$
$251$	−11.6607	−0.736018	−0.368009	+	0.929822i	$0.619960\pi$
$252$	0	0
$253$	−6.88326	−0.432747
$254$	0	0
$255$	−8.77568	−0.549554
$256$	0	0
$257$	−9.55314	−0.595908	−0.297954	−	0.954580i	$-0.596304\pi$
$257$	−9.55314	−0.595908	−0.297954	+	0.954580i	$0.596304\pi$
$258$	0	0
$259$	9.81270	0.609731
$260$	0	0
$261$	1.22974	0.0761192
$262$	0	0
$263$	−8.60062	−0.530337	−0.265168	−	0.964202i	$-0.585428\pi$
$263$	−8.60062	−0.530337	−0.265168	+	0.964202i	$0.585428\pi$
$264$	0	0
$265$	−32.7913	−2.01435
$266$	0	0
$267$	4.11301	0.251712
$268$	0	0
$269$	22.7913	1.38961	0.694805	−	0.719198i	$-0.255491\pi$
$269$	22.7913	1.38961	0.694805	+	0.719198i	$0.255491\pi$
$270$	0	0
$271$	10.2491	0.622588	0.311294	−	0.950314i	$-0.399238\pi$
$271$	10.2491	0.622588	0.311294	+	0.950314i	$0.399238\pi$
$272$	0	0
$273$	−1.53005	−0.0926029
$274$	0	0
$275$	5.28265	0.318556
$276$	0	0
$277$	−14.1184	−0.848294	−0.424147	−	0.905593i	$-0.639426\pi$
$277$	−14.1184	−0.848294	−0.424147	+	0.905593i	$0.639426\pi$
$278$	0	0
$279$	−8.11301	−0.485713
$280$	0	0
$281$	−22.4625	−1.34000	−0.670000	−	0.742361i	$-0.733706\pi$
$281$	−22.4625	−1.34000	−0.670000	+	0.742361i	$0.733706\pi$
$282$	0	0
$283$	27.1165	1.61191	0.805953	−	0.591979i	$-0.201653\pi$
$283$	27.1165	1.61191	0.805953	+	0.591979i	$0.201653\pi$
$284$	0	0
$285$	24.9786	1.47960
$286$	0	0
$287$	−5.40116	−0.318820
$288$	0	0
$289$	−9.51043	−0.559437
$290$	0	0
$291$	−15.9185	−0.933159
$292$	0	0
$293$	2.74391	0.160301	0.0801504	−	0.996783i	$-0.474460\pi$
$293$	2.74391	0.160301	0.0801504	+	0.996783i	$0.474460\pi$
$294$	0	0
$295$	40.6375	2.36601
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	6.88326	0.398069
$300$	0	0
$301$	−6.09318	−0.351205
$302$	0	0
$303$	−9.82858	−0.564637
$304$	0	0
$305$	−20.2203	−1.15781
$306$	0	0
$307$	15.1110	0.862433	0.431216	−	0.902248i	$-0.358085\pi$
$307$	15.1110	0.862433	0.431216	+	0.902248i	$0.358085\pi$
$308$	0	0
$309$	10.3057	0.586273
$310$	0	0
$311$	7.94159	0.450326	0.225163	−	0.974321i	$-0.427709\pi$
$311$	7.94159	0.450326	0.225163	+	0.974321i	$0.427709\pi$
$312$	0	0
$313$	15.2613	0.862617	0.431308	−	0.902205i	$-0.358052\pi$
$313$	15.2613	0.862617	0.431308	+	0.902205i	$0.358052\pi$
$314$	0	0
$315$	−4.90635	−0.276441
$316$	0	0
$317$	1.83905	0.103291	0.0516456	−	0.998665i	$-0.483553\pi$
$317$	1.83905	0.103291	0.0516456	+	0.998665i	$0.483553\pi$
$318$	0	0
$319$	1.22974	0.0688524
$320$	0	0
$321$	−2.86018	−0.159639
$322$	0	0
$323$	−21.3179	−1.18616
$324$	0	0
$325$	−5.28265	−0.293028
$326$	0	0
$327$	10.4364	0.577134
$328$	0	0
$329$	4.49819	0.247993
$330$	0	0
$331$	−30.6726	−1.68592	−0.842959	−	0.537977i	$-0.819189\pi$
$331$	−30.6726	−1.68592	−0.842959	+	0.537977i	$0.819189\pi$
$332$	0	0
$333$	−6.41331	−0.351447
$334$	0	0
$335$	−18.6219	−1.01742
$336$	0	0
$337$	7.27918	0.396522	0.198261	−	0.980149i	$-0.436471\pi$
$337$	7.27918	0.396522	0.198261	+	0.980149i	$0.436471\pi$
$338$	0	0
$339$	−0.906349	−0.0492261
$340$	0	0
$341$	−8.11301	−0.439344
$342$	0	0
$343$	17.8388	0.963204
$344$	0	0
$345$	22.0723	1.18833
$346$	0	0
$347$	19.1197	1.02640	0.513201	−	0.858269i	$-0.328460\pi$
$347$	19.1197	1.02640	0.513201	+	0.858269i	$0.328460\pi$
$348$	0	0
$349$	−12.0387	−0.644417	−0.322209	−	0.946669i	$-0.604425\pi$
$349$	−12.0387	−0.644417	−0.322209	+	0.946669i	$0.604425\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−25.7066	−1.36823	−0.684113	−	0.729376i	$-0.739810\pi$
$353$	−25.7066	−1.36823	−0.684113	+	0.729376i	$0.739810\pi$
$354$	0	0
$355$	32.7913	1.74038
$356$	0	0
$357$	4.18730	0.221616
$358$	0	0
$359$	−21.7417	−1.14748	−0.573741	−	0.819037i	$-0.694508\pi$
$359$	−21.7417	−1.14748	−0.573741	+	0.819037i	$0.694508\pi$
$360$	0	0
$361$	41.6780	2.19358
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−49.5382	−2.59295
$366$	0	0
$367$	−27.5474	−1.43797	−0.718983	−	0.695028i	$-0.755392\pi$
$367$	−27.5474	−1.43797	−0.718983	+	0.695028i	$0.755392\pi$
$368$	0	0
$369$	3.53005	0.183767
$370$	0	0
$371$	15.6463	0.812316
$372$	0	0
$373$	6.98038	0.361430	0.180715	−	0.983535i	$-0.442159\pi$
$373$	6.98038	0.361430	0.180715	+	0.983535i	$0.442159\pi$
$374$	0	0
$375$	−0.906349	−0.0468036
$376$	0	0
$377$	−1.22974	−0.0633350
$378$	0	0
$379$	−26.2575	−1.34876	−0.674379	−	0.738385i	$-0.735589\pi$
$379$	−26.2575	−1.34876	−0.674379	+	0.738385i	$0.735589\pi$
$380$	0	0
$381$	−11.9026	−0.609789
$382$	0	0
$383$	−11.9647	−0.611366	−0.305683	−	0.952133i	$-0.598885\pi$
$383$	−11.9647	−0.611366	−0.305683	+	0.952133i	$0.598885\pi$
$384$	0	0
$385$	−4.90635	−0.250051
$386$	0	0
$387$	3.98234	0.202434
$388$	0	0
$389$	28.9324	1.46693	0.733466	−	0.679726i	$-0.237901\pi$
$389$	28.9324	1.46693	0.733466	+	0.679726i	$0.237901\pi$
$390$	0	0
$391$	−18.8375	−0.952652
$392$	0	0
$393$	19.1324	0.965100
$394$	0	0
$395$	−6.36237	−0.320126
$396$	0	0
$397$	1.13982	0.0572062	0.0286031	−	0.999591i	$-0.490894\pi$
$397$	1.13982	0.0572062	0.0286031	+	0.999591i	$0.490894\pi$
$398$	0	0
$399$	−11.9185	−0.596671
$400$	0	0
$401$	−26.8581	−1.34123	−0.670616	−	0.741805i	$-0.733970\pi$
$401$	−26.8581	−1.34123	−0.670616	+	0.741805i	$0.733970\pi$
$402$	0	0
$403$	8.11301	0.404138
$404$	0	0
$405$	3.20666	0.159340
$406$	0	0
$407$	−6.41331	−0.317896
$408$	0	0
$409$	−19.2891	−0.953785	−0.476893	−	0.878962i	$-0.658237\pi$
$409$	−19.2891	−0.953785	−0.476893	+	0.878962i	$0.658237\pi$
$410$	0	0
$411$	−5.02113	−0.247674
$412$	0	0
$413$	−19.3901	−0.954127
$414$	0	0
$415$	16.5653	0.813158
$416$	0	0
$417$	−14.1361	−0.692247
$418$	0	0
$419$	7.34097	0.358630	0.179315	−	0.983792i	$-0.442612\pi$
$419$	7.34097	0.358630	0.179315	+	0.983792i	$0.442612\pi$
$420$	0	0
$421$	−22.2665	−1.08520	−0.542601	−	0.839990i	$-0.682561\pi$
$421$	−22.2665	−1.08520	−0.542601	+	0.839990i	$0.682561\pi$
$422$	0	0
$423$	−2.93990	−0.142943
$424$	0	0
$425$	14.4571	0.701270
$426$	0	0
$427$	9.64809	0.466904
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−21.2474	−1.02345	−0.511726	−	0.859149i	$-0.670994\pi$
$431$	−21.2474	−1.02345	−0.511726	+	0.859149i	$0.670994\pi$
$432$	0	0
$433$	−39.5774	−1.90197	−0.950985	−	0.309236i	$-0.899927\pi$
$433$	−39.5774	−1.90197	−0.950985	+	0.309236i	$0.899927\pi$
$434$	0	0
$435$	−3.94336	−0.189070
$436$	0	0
$437$	53.6179	2.56489
$438$	0	0
$439$	3.16442	0.151030	0.0755148	−	0.997145i	$-0.475940\pi$
$439$	3.16442	0.151030	0.0755148	+	0.997145i	$0.475940\pi$
$440$	0	0
$441$	−4.65894	−0.221854
$442$	0	0
$443$	−16.8266	−0.799457	−0.399729	−	0.916634i	$-0.630896\pi$
$443$	−16.8266	−0.799457	−0.399729	+	0.916634i	$0.630896\pi$
$444$	0	0
$445$	−13.1890	−0.625218
$446$	0	0
$447$	−7.03702	−0.332839
$448$	0	0
$449$	31.3251	1.47832	0.739161	−	0.673529i	$-0.235222\pi$
$449$	31.3251	1.47832	0.739161	+	0.673529i	$0.235222\pi$
$450$	0	0
$451$	3.53005	0.166224
$452$	0	0
$453$	−10.8497	−0.509764
$454$	0	0
$455$	4.90635	0.230013
$456$	0	0
$457$	12.5209	0.585703	0.292851	−	0.956158i	$-0.405396\pi$
$457$	12.5209	0.585703	0.292851	+	0.956158i	$0.405396\pi$
$458$	0	0
$459$	−2.73671	−0.127739
$460$	0	0
$461$	−6.35845	−0.296143	−0.148071	−	0.988977i	$-0.547307\pi$
$461$	−6.35845	−0.296143	−0.148071	+	0.988977i	$0.547307\pi$
$462$	0	0
$463$	19.1974	0.892180	0.446090	−	0.894988i	$-0.352816\pi$
$463$	19.1974	0.892180	0.446090	+	0.894988i	$0.352816\pi$
$464$	0	0
$465$	26.0156	1.20645
$466$	0	0
$467$	−11.6376	−0.538525	−0.269263	−	0.963067i	$-0.586780\pi$
$467$	−11.6376	−0.538525	−0.269263	+	0.963067i	$0.586780\pi$
$468$	0	0
$469$	8.88542	0.410291
$470$	0	0
$471$	16.6624	0.767763
$472$	0	0
$473$	3.98234	0.183108
$474$	0	0
$475$	−41.1498	−1.88808
$476$	0	0
$477$	−10.2260	−0.468217
$478$	0	0
$479$	30.3358	1.38608	0.693038	−	0.720901i	$-0.256272\pi$
$479$	30.3358	1.38608	0.693038	+	0.720901i	$0.256272\pi$
$480$	0	0
$481$	6.41331	0.292422
$482$	0	0
$483$	−10.5317	−0.479211
$484$	0	0
$485$	51.0452	2.31784
$486$	0	0
$487$	34.7136	1.57302	0.786512	−	0.617575i	$-0.211885\pi$
$487$	34.7136	1.57302	0.786512	+	0.617575i	$0.211885\pi$
$488$	0	0
$489$	9.95925	0.450373
$490$	0	0
$491$	17.4926	0.789428	0.394714	−	0.918804i	$-0.370844\pi$
$491$	17.4926	0.789428	0.394714	+	0.918804i	$0.370844\pi$
$492$	0	0
$493$	3.36545	0.151572
$494$	0	0
$495$	3.20666	0.144128
$496$	0	0
$497$	−15.6463	−0.701833
$498$	0	0
$499$	6.07946	0.272154	0.136077	−	0.990698i	$-0.456551\pi$
$499$	6.07946	0.272154	0.136077	+	0.990698i	$0.456551\pi$
$500$	0	0
$501$	6.68034	0.298456
$502$	0	0
$503$	−19.8514	−0.885130	−0.442565	−	0.896736i	$-0.645931\pi$
$503$	−19.8514	−0.885130	−0.442565	+	0.896736i	$0.645931\pi$
$504$	0	0
$505$	31.5169	1.40248
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	8.48406	0.376049	0.188025	−	0.982164i	$-0.439791\pi$
$509$	8.48406	0.376049	0.188025	+	0.982164i	$0.439791\pi$
$510$	0	0
$511$	23.6371	1.04564
$512$	0	0
$513$	7.78961	0.343920
$514$	0	0
$515$	−33.0469	−1.45622
$516$	0	0
$517$	−2.93990	−0.129297
$518$	0	0
$519$	0.169641	0.00744641
$520$	0	0
$521$	24.4041	1.06916	0.534581	−	0.845117i	$-0.320469\pi$
$521$	24.4041	1.06916	0.534581	+	0.845117i	$0.320469\pi$
$522$	0	0
$523$	27.4078	1.19846	0.599230	−	0.800577i	$-0.295474\pi$
$523$	27.4078	1.19846	0.599230	+	0.800577i	$0.295474\pi$
$524$	0	0
$525$	8.08272	0.352759
$526$	0	0
$527$	−22.2029	−0.967175
$528$	0	0
$529$	24.3793	1.05997
$530$	0	0
$531$	12.6729	0.549956
$532$	0	0
$533$	−3.53005	−0.152903
$534$	0	0
$535$	9.17160	0.396523
$536$	0	0
$537$	7.66941	0.330959
$538$	0	0
$539$	−4.65894	−0.200675
$540$	0	0
$541$	43.5826	1.87376	0.936881	−	0.349648i	$-0.113699\pi$
$541$	43.5826	1.87376	0.936881	+	0.349648i	$0.113699\pi$
$542$	0	0
$543$	−1.03702	−0.0445026
$544$	0	0
$545$	−33.4659	−1.43352
$546$	0	0
$547$	33.7767	1.44419	0.722094	−	0.691795i	$-0.243180\pi$
$547$	33.7767	1.44419	0.722094	+	0.691795i	$0.243180\pi$
$548$	0	0
$549$	−6.30573	−0.269122
$550$	0	0
$551$	−9.57922	−0.408089
$552$	0	0
$553$	3.03579	0.129095
$554$	0	0
$555$	20.5653	0.872948
$556$	0	0
$557$	16.5458	0.701066	0.350533	−	0.936550i	$-0.386000\pi$
$557$	16.5458	0.701066	0.350533	+	0.936550i	$0.386000\pi$
$558$	0	0
$559$	−3.98234	−0.168435
$560$	0	0
$561$	−2.73671	−0.115544
$562$	0	0
$563$	−24.6324	−1.03813	−0.519066	−	0.854734i	$-0.673720\pi$
$563$	−24.6324	−1.03813	−0.519066	+	0.854734i	$0.673720\pi$
$564$	0	0
$565$	2.90635	0.122271
$566$	0	0
$567$	−1.53005	−0.0642561
$568$	0	0
$569$	−7.34817	−0.308051	−0.154026	−	0.988067i	$-0.549224\pi$
$569$	−7.34817	−0.308051	−0.154026	+	0.988067i	$0.549224\pi$
$570$	0	0
$571$	39.9044	1.66995	0.834973	−	0.550290i	$-0.185483\pi$
$571$	39.9044	1.66995	0.834973	+	0.550290i	$0.185483\pi$
$572$	0	0
$573$	0.199927	0.00835208
$574$	0	0
$575$	−36.3618	−1.51639
$576$	0	0
$577$	−11.4626	−0.477193	−0.238596	−	0.971119i	$-0.576687\pi$
$577$	−11.4626	−0.477193	−0.238596	+	0.971119i	$0.576687\pi$
$578$	0	0
$579$	−20.5069	−0.852237
$580$	0	0
$581$	−7.90410	−0.327917
$582$	0	0
$583$	−10.2260	−0.423518
$584$	0	0
$585$	−3.20666	−0.132579
$586$	0	0
$587$	−29.4995	−1.21757	−0.608787	−	0.793333i	$-0.708344\pi$
$587$	−29.4995	−1.21757	−0.608787	+	0.793333i	$0.708344\pi$
$588$	0	0
$589$	63.1971	2.60399
$590$	0	0
$591$	10.4826	0.431195
$592$	0	0
$593$	−11.3763	−0.467169	−0.233584	−	0.972337i	$-0.575045\pi$
$593$	−11.3763	−0.467169	−0.233584	+	0.972337i	$0.575045\pi$
$594$	0	0
$595$	−13.4272	−0.550463
$596$	0	0
$597$	20.1184	0.823393
$598$	0	0
$599$	−26.2242	−1.07149	−0.535747	−	0.844379i	$-0.679970\pi$
$599$	−26.2242	−1.07149	−0.535747	+	0.844379i	$0.679970\pi$
$600$	0	0
$601$	−6.51912	−0.265920	−0.132960	−	0.991121i	$-0.542448\pi$
$601$	−6.51912	−0.265920	−0.132960	+	0.991121i	$0.542448\pi$
$602$	0	0
$603$	−5.80727	−0.236490
$604$	0	0
$605$	3.20666	0.130369
$606$	0	0
$607$	−3.34817	−0.135898	−0.0679491	−	0.997689i	$-0.521646\pi$
$607$	−3.34817	−0.135898	−0.0679491	+	0.997689i	$0.521646\pi$
$608$	0	0
$609$	1.88157	0.0762451
$610$	0	0
$611$	2.93990	0.118936
$612$	0	0
$613$	−0.948585	−0.0383130	−0.0191565	−	0.999816i	$-0.506098\pi$
$613$	−0.948585	−0.0383130	−0.0191565	+	0.999816i	$0.506098\pi$
$614$	0	0
$615$	−11.3197	−0.456453
$616$	0	0
$617$	−2.02066	−0.0813486	−0.0406743	−	0.999172i	$-0.512951\pi$
$617$	−2.02066	−0.0813486	−0.0406743	+	0.999172i	$0.512951\pi$
$618$	0	0
$619$	29.2345	1.17503	0.587517	−	0.809212i	$-0.300105\pi$
$619$	29.2345	1.17503	0.587517	+	0.809212i	$0.300105\pi$
$620$	0	0
$621$	6.88326	0.276216
$622$	0	0
$623$	6.29311	0.252128
$624$	0	0
$625$	−23.5069	−0.940275
$626$	0	0
$627$	7.78961	0.311087
$628$	0	0
$629$	−17.5514	−0.699819
$630$	0	0
$631$	45.2253	1.80039	0.900195	−	0.435487i	$-0.143424\pi$
$631$	45.2253	1.80039	0.900195	+	0.435487i	$0.143424\pi$
$632$	0	0
$633$	16.0546	0.638113
$634$	0	0
$635$	38.1676	1.51463
$636$	0	0
$637$	4.65894	0.184594
$638$	0	0
$639$	10.2260	0.404535
$640$	0	0
$641$	−46.2629	−1.82728	−0.913638	−	0.406528i	$-0.866739\pi$
$641$	−46.2629	−1.82728	−0.913638	+	0.406528i	$0.866739\pi$
$642$	0	0
$643$	38.2506	1.50846	0.754228	−	0.656613i	$-0.228011\pi$
$643$	38.2506	1.50846	0.754228	+	0.656613i	$0.228011\pi$
$644$	0	0
$645$	−12.7700	−0.502818
$646$	0	0
$647$	35.4306	1.39292	0.696461	−	0.717595i	$-0.254757\pi$
$647$	35.4306	1.39292	0.696461	+	0.717595i	$0.254757\pi$
$648$	0	0
$649$	12.6729	0.497454
$650$	0	0
$651$	−12.4133	−0.486516
$652$	0	0
$653$	3.06704	0.120022	0.0600112	−	0.998198i	$-0.480886\pi$
$653$	3.06704	0.120022	0.0600112	+	0.998198i	$0.480886\pi$
$654$	0	0
$655$	−61.3509	−2.39718
$656$	0	0
$657$	−15.4486	−0.602705
$658$	0	0
$659$	44.3109	1.72611	0.863054	−	0.505112i	$-0.168549\pi$
$659$	44.3109	1.72611	0.863054	+	0.505112i	$0.168549\pi$
$660$	0	0
$661$	16.6746	0.648569	0.324284	−	0.945960i	$-0.394877\pi$
$661$	16.6746	0.648569	0.324284	+	0.945960i	$0.394877\pi$
$662$	0	0
$663$	2.73671	0.106285
$664$	0	0
$665$	38.2185	1.48205
$666$	0	0
$667$	−8.46464	−0.327752
$668$	0	0
$669$	15.5124	0.599744
$670$	0	0
$671$	−6.30573	−0.243430
$672$	0	0
$673$	−6.19815	−0.238921	−0.119461	−	0.992839i	$-0.538117\pi$
$673$	−6.19815	−0.238921	−0.119461	+	0.992839i	$0.538117\pi$
$674$	0	0
$675$	−5.28265	−0.203329
$676$	0	0
$677$	27.1465	1.04332	0.521662	−	0.853152i	$-0.325312\pi$
$677$	27.1465	1.04332	0.521662	+	0.853152i	$0.325312\pi$
$678$	0	0
$679$	−24.3561	−0.934702
$680$	0	0
$681$	−2.02309	−0.0775249
$682$	0	0
$683$	−8.55267	−0.327259	−0.163629	−	0.986522i	$-0.552320\pi$
$683$	−8.55267	−0.327259	−0.163629	+	0.986522i	$0.552320\pi$
$684$	0	0
$685$	16.1010	0.615189
$686$	0	0
$687$	−7.19946	−0.274676
$688$	0	0
$689$	10.2260	0.389580
$690$	0	0
$691$	−5.53331	−0.210497	−0.105249	−	0.994446i	$-0.533564\pi$
$691$	−5.53331	−0.210497	−0.105249	+	0.994446i	$0.533564\pi$
$692$	0	0
$693$	−1.53005	−0.0581218
$694$	0	0
$695$	45.3296	1.71945
$696$	0	0
$697$	9.66072	0.365926
$698$	0	0
$699$	16.7367	0.633040
$700$	0	0
$701$	46.6639	1.76247	0.881236	−	0.472677i	$-0.156712\pi$
$701$	46.6639	1.76247	0.881236	+	0.472677i	$0.156712\pi$
$702$	0	0
$703$	49.9572	1.88417
$704$	0	0
$705$	9.42724	0.355050
$706$	0	0
$707$	−15.0382	−0.565571
$708$	0	0
$709$	14.1001	0.529541	0.264770	−	0.964311i	$-0.414704\pi$
$709$	14.1001	0.529541	0.264770	+	0.964311i	$0.414704\pi$
$710$	0	0
$711$	−1.98411	−0.0744100
$712$	0	0
$713$	55.8439	2.09137
$714$	0	0
$715$	−3.20666	−0.119922
$716$	0	0
$717$	−8.23647	−0.307597
$718$	0	0
$719$	16.3127	0.608360	0.304180	−	0.952615i	$-0.401618\pi$
$719$	16.3127	0.608360	0.304180	+	0.952615i	$0.401618\pi$
$720$	0	0
$721$	15.7683	0.587242
$722$	0	0
$723$	−4.72082	−0.175569
$724$	0	0
$725$	6.49630	0.241266
$726$	0	0
$727$	−43.9533	−1.63014	−0.815069	−	0.579364i	$-0.803301\pi$
$727$	−43.9533	−1.63014	−0.815069	+	0.579364i	$0.803301\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.8985	0.403095
$732$	0	0
$733$	−40.2029	−1.48493	−0.742464	−	0.669885i	$-0.766343\pi$
$733$	−40.2029	−1.48493	−0.742464	+	0.669885i	$0.766343\pi$
$734$	0	0
$735$	14.9396	0.551057
$736$	0	0
$737$	−5.80727	−0.213914
$738$	0	0
$739$	46.1467	1.69753	0.848766	−	0.528768i	$-0.177346\pi$
$739$	46.1467	1.69753	0.848766	+	0.528768i	$0.177346\pi$
$740$	0	0
$741$	−7.78961	−0.286158
$742$	0	0
$743$	−40.8174	−1.49744	−0.748722	−	0.662884i	$-0.769332\pi$
$743$	−40.8174	−1.49744	−0.748722	+	0.662884i	$0.769332\pi$
$744$	0	0
$745$	22.5653	0.826728
$746$	0	0
$747$	5.16591	0.189011
$748$	0	0
$749$	−4.37621	−0.159903
$750$	0	0
$751$	19.9272	0.727154	0.363577	−	0.931564i	$-0.381555\pi$
$751$	19.9272	0.727154	0.363577	+	0.931564i	$0.381555\pi$
$752$	0	0
$753$	11.6607	0.424940
$754$	0	0
$755$	34.7913	1.26619
$756$	0	0
$757$	−1.62323	−0.0589974	−0.0294987	−	0.999565i	$-0.509391\pi$
$757$	−1.62323	−0.0589974	−0.0294987	+	0.999565i	$0.509391\pi$
$758$	0	0
$759$	6.88326	0.249847
$760$	0	0
$761$	16.5017	0.598187	0.299093	−	0.954224i	$-0.403316\pi$
$761$	16.5017	0.598187	0.299093	+	0.954224i	$0.403316\pi$
$762$	0	0
$763$	15.9682	0.578089
$764$	0	0
$765$	8.77568	0.317285
$766$	0	0
$767$	−12.6729	−0.457591
$768$	0	0
$769$	39.2699	1.41611	0.708055	−	0.706157i	$-0.249573\pi$
$769$	39.2699	1.41611	0.708055	+	0.706157i	$0.249573\pi$
$770$	0	0
$771$	9.55314	0.344048
$772$	0	0
$773$	30.2224	1.08702	0.543511	−	0.839402i	$-0.317094\pi$
$773$	30.2224	1.08702	0.543511	+	0.839402i	$0.317094\pi$
$774$	0	0
$775$	−42.8581	−1.53951
$776$	0	0
$777$	−9.81270	−0.352029
$778$	0	0
$779$	−27.4977	−0.985208
$780$	0	0
$781$	10.2260	0.365915
$782$	0	0
$783$	−1.22974	−0.0439474
$784$	0	0
$785$	−53.4306	−1.90702
$786$	0	0
$787$	38.6405	1.37739	0.688693	−	0.725053i	$-0.258185\pi$
$787$	38.6405	1.37739	0.688693	+	0.725053i	$0.258185\pi$
$788$	0	0
$789$	8.60062	0.306190
$790$	0	0
$791$	−1.38676	−0.0493075
$792$	0	0
$793$	6.30573	0.223923
$794$	0	0
$795$	32.7913	1.16299
$796$	0	0
$797$	4.00693	0.141933	0.0709664	−	0.997479i	$-0.477392\pi$
$797$	4.00693	0.141933	0.0709664	+	0.997479i	$0.477392\pi$
$798$	0	0
$799$	−8.04564	−0.284634
$800$	0	0
$801$	−4.11301	−0.145326
$802$	0	0
$803$	−15.4486	−0.545168
$804$	0	0
$805$	33.7717	1.19030
$806$	0	0
$807$	−22.7913	−0.802292
$808$	0	0
$809$	52.9409	1.86130	0.930651	−	0.365909i	$-0.119242\pi$
$809$	52.9409	1.86130	0.930651	+	0.365909i	$0.119242\pi$
$810$	0	0
$811$	21.6947	0.761802	0.380901	−	0.924616i	$-0.375614\pi$
$811$	21.6947	0.761802	0.380901	+	0.924616i	$0.375614\pi$
$812$	0	0
$813$	−10.2491	−0.359452
$814$	0	0
$815$	−31.9359	−1.11867
$816$	0	0
$817$	−31.0209	−1.08528
$818$	0	0
$819$	1.53005	0.0534643
$820$	0	0
$821$	−10.5497	−0.368186	−0.184093	−	0.982909i	$-0.558935\pi$
$821$	−10.5497	−0.368186	−0.184093	+	0.982909i	$0.558935\pi$
$822$	0	0
$823$	−10.3723	−0.361556	−0.180778	−	0.983524i	$-0.557861\pi$
$823$	−10.3723	−0.361556	−0.180778	+	0.983524i	$0.557861\pi$
$824$	0	0
$825$	−5.28265	−0.183918
$826$	0	0
$827$	41.3296	1.43717	0.718585	−	0.695439i	$-0.244790\pi$
$827$	41.3296	1.43717	0.718585	+	0.695439i	$0.244790\pi$
$828$	0	0
$829$	−40.4799	−1.40592	−0.702962	−	0.711227i	$-0.748140\pi$
$829$	−40.4799	−1.40592	−0.702962	+	0.711227i	$0.748140\pi$
$830$	0	0
$831$	14.1184	0.489763
$832$	0	0
$833$	−12.7502	−0.441767
$834$	0	0
$835$	−21.4216	−0.741323
$836$	0	0
$837$	8.11301	0.280426
$838$	0	0
$839$	40.6427	1.40314	0.701571	−	0.712600i	$-0.252482\pi$
$839$	40.6427	1.40314	0.701571	+	0.712600i	$0.252482\pi$
$840$	0	0
$841$	−27.4877	−0.947853
$842$	0	0
$843$	22.4625	0.773649
$844$	0	0
$845$	3.20666	0.110312
$846$	0	0
$847$	−1.53005	−0.0525732
$848$	0	0
$849$	−27.1165	−0.930635
$850$	0	0
$851$	44.1445	1.51325
$852$	0	0
$853$	−32.3815	−1.10872	−0.554361	−	0.832276i	$-0.687037\pi$
$853$	−32.3815	−1.10872	−0.554361	+	0.832276i	$0.687037\pi$
$854$	0	0
$855$	−24.9786	−0.854250
$856$	0	0
$857$	−36.8956	−1.26033	−0.630165	−	0.776461i	$-0.717013\pi$
$857$	−36.8956	−1.26033	−0.630165	+	0.776461i	$0.717013\pi$
$858$	0	0
$859$	23.5013	0.801853	0.400927	−	0.916110i	$-0.368688\pi$
$859$	23.5013	0.801853	0.400927	+	0.916110i	$0.368688\pi$
$860$	0	0
$861$	5.40116	0.184071
$862$	0	0
$863$	9.31042	0.316930	0.158465	−	0.987365i	$-0.449345\pi$
$863$	9.31042	0.316930	0.158465	+	0.987365i	$0.449345\pi$
$864$	0	0
$865$	−0.543980	−0.0184959
$866$	0	0
$867$	9.51043	0.322991
$868$	0	0
$869$	−1.98411	−0.0673064
$870$	0	0
$871$	5.80727	0.196772
$872$	0	0
$873$	15.9185	0.538760
$874$	0	0
$875$	−1.38676	−0.0468810
$876$	0	0
$877$	−15.3601	−0.518675	−0.259338	−	0.965787i	$-0.583504\pi$
$877$	−15.3601	−0.518675	−0.259338	+	0.965787i	$0.583504\pi$
$878$	0	0
$879$	−2.74391	−0.0925497
$880$	0	0
$881$	44.9232	1.51350	0.756750	−	0.653704i	$-0.226786\pi$
$881$	44.9232	1.51350	0.756750	+	0.653704i	$0.226786\pi$
$882$	0	0
$883$	26.2892	0.884702	0.442351	−	0.896842i	$-0.354145\pi$
$883$	26.2892	0.884702	0.442351	+	0.896842i	$0.354145\pi$
$884$	0	0
$885$	−40.6375	−1.36602
$886$	0	0
$887$	−31.5400	−1.05901	−0.529504	−	0.848307i	$-0.677622\pi$
$887$	−31.5400	−1.05901	−0.529504	+	0.848307i	$0.677622\pi$
$888$	0	0
$889$	−18.2116	−0.610798
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	22.9007	0.766341
$894$	0	0
$895$	−24.5931	−0.822058
$896$	0	0
$897$	−6.88326	−0.229825
$898$	0	0
$899$	−9.97691	−0.332749
$900$	0	0
$901$	−27.9856	−0.932336
$902$	0	0
$903$	6.09318	0.202768
$904$	0	0
$905$	3.32535	0.110538
$906$	0	0
$907$	−42.6502	−1.41618	−0.708088	−	0.706124i	$-0.750442\pi$
$907$	−42.6502	−1.41618	−0.708088	+	0.706124i	$0.750442\pi$
$908$	0	0
$909$	9.82858	0.325994
$910$	0	0
$911$	27.8052	0.921228	0.460614	−	0.887600i	$-0.347629\pi$
$911$	27.8052	0.921228	0.460614	+	0.887600i	$0.347629\pi$
$912$	0	0
$913$	5.16591	0.170967
$914$	0	0
$915$	20.2203	0.668463
$916$	0	0
$917$	29.2735	0.966696
$918$	0	0
$919$	−21.6547	−0.714324	−0.357162	−	0.934042i	$-0.616256\pi$
$919$	−21.6547	−0.714324	−0.357162	+	0.934042i	$0.616256\pi$
$920$	0	0
$921$	−15.1110	−0.497926
$922$	0	0
$923$	−10.2260	−0.336593
$924$	0	0
$925$	−33.8793	−1.11394
$926$	0	0
$927$	−10.3057	−0.338485
$928$	0	0
$929$	−48.8228	−1.60182	−0.800912	−	0.598782i	$-0.795652\pi$
$929$	−48.8228	−1.60182	−0.800912	+	0.598782i	$0.795652\pi$
$930$	0	0
$931$	36.2914	1.18940
$932$	0	0
$933$	−7.94159	−0.259996
$934$	0	0
$935$	8.77568	0.286995
$936$	0	0
$937$	−53.2181	−1.73856	−0.869279	−	0.494321i	$-0.835417\pi$
$937$	−53.2181	−1.73856	−0.869279	+	0.494321i	$0.835417\pi$
$938$	0	0
$939$	−15.2613	−0.498032
$940$	0	0
$941$	−27.8636	−0.908329	−0.454164	−	0.890918i	$-0.650062\pi$
$941$	−27.8636	−0.908329	−0.454164	+	0.890918i	$0.650062\pi$
$942$	0	0
$943$	−24.2983	−0.791260
$944$	0	0
$945$	4.90635	0.159604
$946$	0	0
$947$	−60.2359	−1.95740	−0.978702	−	0.205285i	$-0.934188\pi$
$947$	−60.2359	−1.95740	−0.978702	+	0.205285i	$0.934188\pi$
$948$	0	0
$949$	15.4486	0.501481
$950$	0	0
$951$	−1.83905	−0.0596352
$952$	0	0
$953$	45.0928	1.46070	0.730350	−	0.683073i	$-0.239357\pi$
$953$	45.0928	1.46070	0.730350	+	0.683073i	$0.239357\pi$
$954$	0	0
$955$	−0.641098	−0.0207454
$956$	0	0
$957$	−1.22974	−0.0397520
$958$	0	0
$959$	−7.68258	−0.248084
$960$	0	0
$961$	34.8209	1.12325
$962$	0	0
$963$	2.86018	0.0921679
$964$	0	0
$965$	65.7585	2.11684
$966$	0	0
$967$	23.8936	0.768368	0.384184	−	0.923257i	$-0.374483\pi$
$967$	23.8936	0.768368	0.384184	+	0.923257i	$0.374483\pi$
$968$	0	0
$969$	21.3179	0.684829
$970$	0	0
$971$	45.4503	1.45857	0.729285	−	0.684210i	$-0.239853\pi$
$971$	45.4503	1.45857	0.729285	+	0.684210i	$0.239853\pi$
$972$	0	0
$973$	−21.6289	−0.693392
$974$	0	0
$975$	5.28265	0.169180
$976$	0	0
$977$	19.6461	0.628533	0.314266	−	0.949335i	$-0.398241\pi$
$977$	19.6461	0.628533	0.314266	+	0.949335i	$0.398241\pi$
$978$	0	0
$979$	−4.11301	−0.131452
$980$	0	0
$981$	−10.4364	−0.333209
$982$	0	0
$983$	−29.6742	−0.946459	−0.473230	−	0.880939i	$-0.656912\pi$
$983$	−29.6742	−0.946459	−0.473230	+	0.880939i	$0.656912\pi$
$984$	0	0
$985$	−33.6140	−1.07103
$986$	0	0
$987$	−4.49819	−0.143179
$988$	0	0
$989$	−27.4115	−0.871634
$990$	0	0
$991$	2.55267	0.0810882	0.0405441	−	0.999178i	$-0.487091\pi$
$991$	2.55267	0.0810882	0.0405441	+	0.999178i	$0.487091\pi$
$992$	0	0
$993$	30.6726	0.973366
$994$	0	0
$995$	−64.5129	−2.04520
$996$	0	0
$997$	−32.7260	−1.03644	−0.518221	−	0.855247i	$-0.673406\pi$
$997$	−32.7260	−1.03644	−0.518221	+	0.855247i	$0.673406\pi$
$998$	0	0
$999$	6.41331	0.202908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.a.r.1.4	✓	4
4.3	odd	2		6864.2.a.ca.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.r.1.4	✓	4	1.1	even	1	trivial
6864.2.a.ca.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.20666 q^{5} -1.53005 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.20666 q^{5} -1.53005 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.20666 q^{15} +2.73671 q^{17} -7.78961 q^{19} +1.53005 q^{21} -6.88326 q^{23} +5.28265 q^{25} -1.00000 q^{27} +1.22974 q^{29} -8.11301 q^{31} -1.00000 q^{33} -4.90635 q^{35} -6.41331 q^{37} +1.00000 q^{39} +3.53005 q^{41} +3.98234 q^{43} +3.20666 q^{45} -2.93990 q^{47} -4.65894 q^{49} -2.73671 q^{51} -10.2260 q^{53} +3.20666 q^{55} +7.78961 q^{57} +12.6729 q^{59} -6.30573 q^{61} -1.53005 q^{63} -3.20666 q^{65} -5.80727 q^{67} +6.88326 q^{69} +10.2260 q^{71} -15.4486 q^{73} -5.28265 q^{75} -1.53005 q^{77} -1.98411 q^{79} +1.00000 q^{81} +5.16591 q^{83} +8.77568 q^{85} -1.22974 q^{87} -4.11301 q^{89} +1.53005 q^{91} +8.11301 q^{93} -24.9786 q^{95} +15.9185 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} - 6 q^{17} + q^{21} - 9 q^{23} + q^{25} - 4 q^{27} - q^{29} - 8 q^{31} - 4 q^{33} - 7 q^{35} - 2 q^{37} + 4 q^{39} + 9 q^{41} - 5 q^{43} + q^{45} - 22 q^{47} + 9 q^{49} + 6 q^{51} + 8 q^{53} + q^{55} + q^{59} - 11 q^{61} - q^{63} - q^{65} - 13 q^{67} + 9 q^{69} - 8 q^{71} - 3 q^{73} - q^{75} - q^{77} - 6 q^{79} + 4 q^{81} - 18 q^{83} + 26 q^{85} + q^{87} + 8 q^{89} + q^{91} + 8 q^{93} - 36 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - q^15 - 6 * q^17 + q^21 - 9 * q^23 + q^25 - 4 * q^27 - q^29 - 8 * q^31 - 4 * q^33 - 7 * q^35 - 2 * q^37 + 4 * q^39 + 9 * q^41 - 5 * q^43 + q^45 - 22 * q^47 + 9 * q^49 + 6 * q^51 + 8 * q^53 + q^55 + q^59 - 11 * q^61 - q^63 - q^65 - 13 * q^67 + 9 * q^69 - 8 * q^71 - 3 * q^73 - q^75 - q^77 - 6 * q^79 + 4 * q^81 - 18 * q^83 + 26 * q^85 + q^87 + 8 * q^89 + q^91 + 8 * q^93 - 36 * q^95 + 10 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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