LMFDB - Embedded newform 4368.2.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1373.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 8x - 5 $ x^3 - 8*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.43931$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.38955 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.38955 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.87862 q^{11} +1.00000 q^{13} -3.38955 q^{15} +4.00000 q^{17} -3.48907 q^{19} -1.00000 q^{21} -8.26818 q^{23} +6.48907 q^{25} -1.00000 q^{27} +5.48907 q^{29} +6.26818 q^{31} -4.87862 q^{33} +3.38955 q^{35} +8.77911 q^{37} -1.00000 q^{39} -0.0995171 q^{41} +2.26818 q^{43} +3.38955 q^{45} +10.1687 q^{47} +1.00000 q^{49} -4.00000 q^{51} -12.2682 q^{53} +16.5364 q^{55} +3.48907 q^{57} -6.87862 q^{59} +2.00000 q^{61} +1.00000 q^{63} +3.38955 q^{65} +6.77911 q^{67} +8.26818 q^{69} -3.65773 q^{71} -11.0473 q^{73} -6.48907 q^{75} +4.87862 q^{77} -13.0473 q^{79} +1.00000 q^{81} +4.61045 q^{83} +13.5582 q^{85} -5.48907 q^{87} -13.1468 q^{89} +1.00000 q^{91} -6.26818 q^{93} -11.8264 q^{95} +8.26818 q^{97} +4.87862 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{13} - q^{15} + 12 q^{17} - 5 q^{19} - 3 q^{21} - q^{23} + 14 q^{25} - 3 q^{27} + 11 q^{29} - 5 q^{31} + q^{35} + 8 q^{37} - 3 q^{39} - 4 q^{41} - 17 q^{43} + q^{45} + 3 q^{47} + 3 q^{49} - 12 q^{51} - 13 q^{53} + 2 q^{55} + 5 q^{57} - 6 q^{59} + 6 q^{61} + 3 q^{63} + q^{65} + 2 q^{67} + q^{69} + 22 q^{71} + 9 q^{73} - 14 q^{75} + 3 q^{79} + 3 q^{81} + 23 q^{83} + 4 q^{85} - 11 q^{87} - q^{89} + 3 q^{91} + 5 q^{93} + 25 q^{95} + q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 + 3 * q^13 - q^15 + 12 * q^17 - 5 * q^19 - 3 * q^21 - q^23 + 14 * q^25 - 3 * q^27 + 11 * q^29 - 5 * q^31 + q^35 + 8 * q^37 - 3 * q^39 - 4 * q^41 - 17 * q^43 + q^45 + 3 * q^47 + 3 * q^49 - 12 * q^51 - 13 * q^53 + 2 * q^55 + 5 * q^57 - 6 * q^59 + 6 * q^61 + 3 * q^63 + q^65 + 2 * q^67 + q^69 + 22 * q^71 + 9 * q^73 - 14 * q^75 + 3 * q^79 + 3 * q^81 + 23 * q^83 + 4 * q^85 - 11 * q^87 - q^89 + 3 * q^91 + 5 * q^93 + 25 * q^95 + q^97

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
$5$	3.38955	1.51585	0.757927	−	0.652339i	$-0.226212\pi$
$5$	3.38955	1.51585	0.757927	+	0.652339i	$0.226212\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.87862	1.47096	0.735480	−	0.677546i	$-0.236957\pi$
$11$	4.87862	1.47096	0.735480	+	0.677546i	$0.236957\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.38955	−0.875179
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−3.48907	−0.800448	−0.400224	−	0.916417i	$-0.631068\pi$
$19$	−3.48907	−0.800448	−0.400224	+	0.916417i	$0.631068\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−8.26818	−1.72403	−0.862017	−	0.506879i	$-0.830799\pi$
$23$	−8.26818	−1.72403	−0.862017	+	0.506879i	$0.830799\pi$
$24$	0	0
$25$	6.48907	1.29781
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.48907	1.01929	0.509647	−	0.860383i	$-0.329776\pi$
$29$	5.48907	1.01929	0.509647	+	0.860383i	$0.329776\pi$
$30$	0	0
$31$	6.26818	1.12580	0.562899	−	0.826526i	$-0.309686\pi$
$31$	6.26818	1.12580	0.562899	+	0.826526i	$0.309686\pi$
$32$	0	0
$33$	−4.87862	−0.849259
$34$	0	0
$35$	3.38955	0.572939
$36$	0	0
$37$	8.77911	1.44328	0.721638	−	0.692271i	$-0.243390\pi$
$37$	8.77911	1.44328	0.721638	+	0.692271i	$0.243390\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−0.0995171	−0.0155420	−0.00777098	−	0.999970i	$-0.502474\pi$
$41$	−0.0995171	−0.0155420	−0.00777098	+	0.999970i	$0.502474\pi$
$42$	0	0
$43$	2.26818	0.345894	0.172947	−	0.984931i	$-0.444671\pi$
$43$	2.26818	0.345894	0.172947	+	0.984931i	$0.444671\pi$
$44$	0	0
$45$	3.38955	0.505285
$46$	0	0
$47$	10.1687	1.48325	0.741626	−	0.670814i	$-0.234055\pi$
$47$	10.1687	1.48325	0.741626	+	0.670814i	$0.234055\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−12.2682	−1.68516	−0.842582	−	0.538568i	$-0.818965\pi$
$53$	−12.2682	−1.68516	−0.842582	+	0.538568i	$0.818965\pi$
$54$	0	0
$55$	16.5364	2.22976
$56$	0	0
$57$	3.48907	0.462139
$58$	0	0
$59$	−6.87862	−0.895520	−0.447760	−	0.894154i	$-0.647778\pi$
$59$	−6.87862	−0.895520	−0.447760	+	0.894154i	$0.647778\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.38955	0.420422
$66$	0	0
$67$	6.77911	0.828200	0.414100	−	0.910231i	$-0.364096\pi$
$67$	6.77911	0.828200	0.414100	+	0.910231i	$0.364096\pi$
$68$	0	0
$69$	8.26818	0.995371
$70$	0	0
$71$	−3.65773	−0.434093	−0.217046	−	0.976161i	$-0.569642\pi$
$71$	−3.65773	−0.434093	−0.217046	+	0.976161i	$0.569642\pi$
$72$	0	0
$73$	−11.0473	−1.29299	−0.646493	−	0.762920i	$-0.723765\pi$
$73$	−11.0473	−1.29299	−0.646493	+	0.762920i	$0.723765\pi$
$74$	0	0
$75$	−6.48907	−0.749293
$76$	0	0
$77$	4.87862	0.555971
$78$	0	0
$79$	−13.0473	−1.46793	−0.733967	−	0.679185i	$-0.762333\pi$
$79$	−13.0473	−1.46793	−0.733967	+	0.679185i	$0.762333\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.61045	0.506062	0.253031	−	0.967458i	$-0.418572\pi$
$83$	4.61045	0.506062	0.253031	+	0.967458i	$0.418572\pi$
$84$	0	0
$85$	13.5582	1.47059
$86$	0	0
$87$	−5.48907	−0.588490
$88$	0	0
$89$	−13.1468	−1.39356	−0.696779	−	0.717286i	$-0.745384\pi$
$89$	−13.1468	−1.39356	−0.696779	+	0.717286i	$0.745384\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−6.26818	−0.649980
$94$	0	0
$95$	−11.8264	−1.21336
$96$	0	0
$97$	8.26818	0.839506	0.419753	−	0.907638i	$-0.362117\pi$
$97$	8.26818	0.839506	0.419753	+	0.907638i	$0.362117\pi$
$98$	0	0
$99$	4.87862	0.490320
$100$	0	0
$101$	1.22089	0.121483	0.0607417	−	0.998154i	$-0.480653\pi$
$101$	1.22089	0.121483	0.0607417	+	0.998154i	$0.480653\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−3.38955	−0.330787
$106$	0	0
$107$	18.5364	1.79198	0.895988	−	0.444077i	$-0.146468\pi$
$107$	18.5364	1.79198	0.895988	+	0.444077i	$0.146468\pi$
$108$	0	0
$109$	3.75725	0.359879	0.179939	−	0.983678i	$-0.442410\pi$
$109$	3.75725	0.359879	0.179939	+	0.983678i	$0.442410\pi$
$110$	0	0
$111$	−8.77911	−0.833276
$112$	0	0
$113$	−19.0473	−1.79182	−0.895909	−	0.444238i	$-0.853474\pi$
$113$	−19.0473	−1.79182	−0.895909	+	0.444238i	$0.853474\pi$
$114$	0	0
$115$	−28.0254	−2.61338
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	12.8010	1.16372
$122$	0	0
$123$	0.0995171	0.00897316
$124$	0	0
$125$	5.04728	0.451443
$126$	0	0
$127$	0.199034	0.0176614	0.00883072	−	0.999961i	$-0.497189\pi$
$127$	0.199034	0.0176614	0.00883072	+	0.999961i	$0.497189\pi$
$128$	0	0
$129$	−2.26818	−0.199702
$130$	0	0
$131$	−1.75725	−0.153531	−0.0767657	−	0.997049i	$-0.524459\pi$
$131$	−1.75725	−0.153531	−0.0767657	+	0.997049i	$0.524459\pi$
$132$	0	0
$133$	−3.48907	−0.302541
$134$	0	0
$135$	−3.38955	−0.291726
$136$	0	0
$137$	6.09952	0.521117	0.260558	−	0.965458i	$-0.416093\pi$
$137$	6.09952	0.521117	0.260558	+	0.965458i	$0.416093\pi$
$138$	0	0
$139$	−6.97814	−0.591878	−0.295939	−	0.955207i	$-0.595633\pi$
$139$	−6.97814	−0.591878	−0.295939	+	0.955207i	$0.595633\pi$
$140$	0	0
$141$	−10.1687	−0.856356
$142$	0	0
$143$	4.87862	0.407971
$144$	0	0
$145$	18.6055	1.54510
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−12.8786	−1.05506	−0.527529	−	0.849537i	$-0.676881\pi$
$149$	−12.8786	−1.05506	−0.527529	+	0.849537i	$0.676881\pi$
$150$	0	0
$151$	13.5582	1.10335	0.551676	−	0.834059i	$-0.313989\pi$
$151$	13.5582	1.10335	0.551676	+	0.834059i	$0.313989\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	21.2463	1.70655
$156$	0	0
$157$	−14.5364	−1.16013	−0.580064	−	0.814571i	$-0.696972\pi$
$157$	−14.5364	−1.16013	−0.580064	+	0.814571i	$0.696972\pi$
$158$	0	0
$159$	12.2682	0.972930
$160$	0	0
$161$	−8.26818	−0.651624
$162$	0	0
$163$	−13.7572	−1.07755	−0.538775	−	0.842449i	$-0.681113\pi$
$163$	−13.7572	−1.07755	−0.538775	+	0.842449i	$0.681113\pi$
$164$	0	0
$165$	−16.5364	−1.28735
$166$	0	0
$167$	−2.36769	−0.183218	−0.0916088	−	0.995795i	$-0.529201\pi$
$167$	−2.36769	−0.183218	−0.0916088	+	0.995795i	$0.529201\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.48907	−0.266816
$172$	0	0
$173$	0.199034	0.0151323	0.00756615	−	0.999971i	$-0.497592\pi$
$173$	0.199034	0.0151323	0.00756615	+	0.999971i	$0.497592\pi$
$174$	0	0
$175$	6.48907	0.490528
$176$	0	0
$177$	6.87862	0.517029
$178$	0	0
$179$	23.0473	1.72263	0.861317	−	0.508067i	$-0.169640\pi$
$179$	23.0473	1.72263	0.861317	+	0.508067i	$0.169640\pi$
$180$	0	0
$181$	−11.5582	−0.859115	−0.429558	−	0.903039i	$-0.641330\pi$
$181$	−11.5582	−0.859115	−0.429558	+	0.903039i	$0.641330\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	29.7572	2.18780
$186$	0	0
$187$	19.5145	1.42704
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	16.7791	1.21409	0.607047	−	0.794666i	$-0.292354\pi$
$191$	16.7791	1.21409	0.607047	+	0.794666i	$0.292354\pi$
$192$	0	0
$193$	25.3155	1.82225	0.911123	−	0.412134i	$-0.135216\pi$
$193$	25.3155	1.82225	0.911123	+	0.412134i	$0.135216\pi$
$194$	0	0
$195$	−3.38955	−0.242731
$196$	0	0
$197$	0.878623	0.0625993	0.0312997	−	0.999510i	$-0.490035\pi$
$197$	0.878623	0.0625993	0.0312997	+	0.999510i	$0.490035\pi$
$198$	0	0
$199$	−13.5582	−0.961116	−0.480558	−	0.876963i	$-0.659566\pi$
$199$	−13.5582	−0.961116	−0.480558	+	0.876963i	$0.659566\pi$
$200$	0	0
$201$	−6.77911	−0.478161
$202$	0	0
$203$	5.48907	0.385257
$204$	0	0
$205$	−0.337319	−0.0235594
$206$	0	0
$207$	−8.26818	−0.574678
$208$	0	0
$209$	−17.0219	−1.17743
$210$	0	0
$211$	−27.0036	−1.85900	−0.929501	−	0.368820i	$-0.879762\pi$
$211$	−27.0036	−1.85900	−0.929501	+	0.368820i	$0.879762\pi$
$212$	0	0
$213$	3.65773	0.250623
$214$	0	0
$215$	7.68810	0.524324
$216$	0	0
$217$	6.26818	0.425512
$218$	0	0
$219$	11.0473	0.746506
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	29.0473	1.94515	0.972575	−	0.232590i	$-0.0747199\pi$
$223$	29.0473	1.94515	0.972575	+	0.232590i	$0.0747199\pi$
$224$	0	0
$225$	6.48907	0.432605
$226$	0	0
$227$	7.07766	0.469761	0.234880	−	0.972024i	$-0.424530\pi$
$227$	7.07766	0.469761	0.234880	+	0.972024i	$0.424530\pi$
$228$	0	0
$229$	−9.80097	−0.647666	−0.323833	−	0.946114i	$-0.604972\pi$
$229$	−9.80097	−0.647666	−0.323833	+	0.946114i	$0.604972\pi$
$230$	0	0
$231$	−4.87862	−0.320990
$232$	0	0
$233$	−2.51093	−0.164496	−0.0822482	−	0.996612i	$-0.526210\pi$
$233$	−2.51093	−0.164496	−0.0822482	+	0.996612i	$0.526210\pi$
$234$	0	0
$235$	34.4672	2.24839
$236$	0	0
$237$	13.0473	0.847512
$238$	0	0
$239$	0.342270	0.0221396	0.0110698	−	0.999939i	$-0.496476\pi$
$239$	0.342270	0.0221396	0.0110698	+	0.999939i	$0.496476\pi$
$240$	0	0
$241$	18.5109	1.19239	0.596197	−	0.802838i	$-0.296678\pi$
$241$	18.5109	1.19239	0.596197	+	0.802838i	$0.296678\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.38955	0.216551
$246$	0	0
$247$	−3.48907	−0.222004
$248$	0	0
$249$	−4.61045	−0.292175
$250$	0	0
$251$	14.7791	0.932849	0.466424	−	0.884561i	$-0.345542\pi$
$251$	14.7791	0.932849	0.466424	+	0.884561i	$0.345542\pi$
$252$	0	0
$253$	−40.3373	−2.53599
$254$	0	0
$255$	−13.5582	−0.849048
$256$	0	0
$257$	22.7791	1.42092	0.710461	−	0.703737i	$-0.248487\pi$
$257$	22.7791	1.42092	0.710461	+	0.703737i	$0.248487\pi$
$258$	0	0
$259$	8.77911	0.545507
$260$	0	0
$261$	5.48907	0.339765
$262$	0	0
$263$	−7.24632	−0.446827	−0.223414	−	0.974724i	$-0.571720\pi$
$263$	−7.24632	−0.446827	−0.223414	+	0.974724i	$0.571720\pi$
$264$	0	0
$265$	−41.5836	−2.55446
$266$	0	0
$267$	13.1468	0.804571
$268$	0	0
$269$	−1.75725	−0.107141	−0.0535706	−	0.998564i	$-0.517060\pi$
$269$	−1.75725	−0.107141	−0.0535706	+	0.998564i	$0.517060\pi$
$270$	0	0
$271$	13.7572	0.835693	0.417847	−	0.908518i	$-0.362785\pi$
$271$	13.7572	0.835693	0.417847	+	0.908518i	$0.362785\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	31.6577	1.90903
$276$	0	0
$277$	−11.2463	−0.675726	−0.337863	−	0.941195i	$-0.609704\pi$
$277$	−11.2463	−0.675726	−0.337863	+	0.941195i	$0.609704\pi$
$278$	0	0
$279$	6.26818	0.375266
$280$	0	0
$281$	−30.4368	−1.81571	−0.907855	−	0.419285i	$-0.862281\pi$
$281$	−30.4368	−1.81571	−0.907855	+	0.419285i	$0.862281\pi$
$282$	0	0
$283$	−17.5582	−1.04373	−0.521864	−	0.853029i	$-0.674763\pi$
$283$	−17.5582	−1.04373	−0.521864	+	0.853029i	$0.674763\pi$
$284$	0	0
$285$	11.8264	0.700535
$286$	0	0
$287$	−0.0995171	−0.00587431
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.26818	−0.484689
$292$	0	0
$293$	15.1905	0.887440	0.443720	−	0.896166i	$-0.353659\pi$
$293$	15.1905	0.887440	0.443720	+	0.896166i	$0.353659\pi$
$294$	0	0
$295$	−23.3155	−1.35748
$296$	0	0
$297$	−4.87862	−0.283086
$298$	0	0
$299$	−8.26818	−0.478161
$300$	0	0
$301$	2.26818	0.130736
$302$	0	0
$303$	−1.22089	−0.0701385
$304$	0	0
$305$	6.77911	0.388170
$306$	0	0
$307$	3.29004	0.187772	0.0938861	−	0.995583i	$-0.470071\pi$
$307$	3.29004	0.187772	0.0938861	+	0.995583i	$0.470071\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−10.7791	−0.611227	−0.305614	−	0.952156i	$-0.598862\pi$
$311$	−10.7791	−0.611227	−0.305614	+	0.952156i	$0.598862\pi$
$312$	0	0
$313$	25.1164	1.41966	0.709832	−	0.704371i	$-0.248771\pi$
$313$	25.1164	1.41966	0.709832	+	0.704371i	$0.248771\pi$
$314$	0	0
$315$	3.38955	0.190980
$316$	0	0
$317$	23.8568	1.33993	0.669965	−	0.742393i	$-0.266309\pi$
$317$	23.8568	1.33993	0.669965	+	0.742393i	$0.266309\pi$
$318$	0	0
$319$	26.7791	1.49934
$320$	0	0
$321$	−18.5364	−1.03460
$322$	0	0
$323$	−13.9563	−0.776548
$324$	0	0
$325$	6.48907	0.359949
$326$	0	0
$327$	−3.75725	−0.207776
$328$	0	0
$329$	10.1687	0.560616
$330$	0	0
$331$	2.97814	0.163693	0.0818467	−	0.996645i	$-0.473918\pi$
$331$	2.97814	0.163693	0.0818467	+	0.996645i	$0.473918\pi$
$332$	0	0
$333$	8.77911	0.481092
$334$	0	0
$335$	22.9781	1.25543
$336$	0	0
$337$	−29.0036	−1.57992	−0.789962	−	0.613155i	$-0.789900\pi$
$337$	−29.0036	−1.57992	−0.789962	+	0.613155i	$0.789900\pi$
$338$	0	0
$339$	19.0473	1.03451
$340$	0	0
$341$	30.5801	1.65600
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	28.0254	1.50884
$346$	0	0
$347$	−18.3373	−0.984399	−0.492199	−	0.870482i	$-0.663807\pi$
$347$	−18.3373	−0.984399	−0.492199	+	0.870482i	$0.663807\pi$
$348$	0	0
$349$	−33.0036	−1.76664	−0.883320	−	0.468770i	$-0.844697\pi$
$349$	−33.0036	−1.76664	−0.883320	+	0.468770i	$0.844697\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	23.4150	1.24625	0.623127	−	0.782121i	$-0.285862\pi$
$353$	23.4150	1.24625	0.623127	+	0.782121i	$0.285862\pi$
$354$	0	0
$355$	−12.3981	−0.658021
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	12.3423	0.651400	0.325700	−	0.945473i	$-0.394400\pi$
$359$	12.3423	0.651400	0.325700	+	0.945473i	$0.394400\pi$
$360$	0	0
$361$	−6.82639	−0.359284
$362$	0	0
$363$	−12.8010	−0.671876
$364$	0	0
$365$	−37.4454	−1.95998
$366$	0	0
$367$	−2.97814	−0.155458	−0.0777288	−	0.996975i	$-0.524767\pi$
$367$	−2.97814	−0.155458	−0.0777288	+	0.996975i	$0.524767\pi$
$368$	0	0
$369$	−0.0995171	−0.00518066
$370$	0	0
$371$	−12.2682	−0.636932
$372$	0	0
$373$	17.8010	0.921699	0.460850	−	0.887478i	$-0.347545\pi$
$373$	17.8010	0.921699	0.460850	+	0.887478i	$0.347545\pi$
$374$	0	0
$375$	−5.04728	−0.260641
$376$	0	0
$377$	5.48907	0.282702
$378$	0	0
$379$	7.46365	0.383382	0.191691	−	0.981455i	$-0.438603\pi$
$379$	7.46365	0.383382	0.191691	+	0.981455i	$0.438603\pi$
$380$	0	0
$381$	−0.199034	−0.0101968
$382$	0	0
$383$	14.6796	0.750092	0.375046	−	0.927006i	$-0.377627\pi$
$383$	14.6796	0.750092	0.375046	+	0.927006i	$0.377627\pi$
$384$	0	0
$385$	16.5364	0.842771
$386$	0	0
$387$	2.26818	0.115298
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−33.0727	−1.67256
$392$	0	0
$393$	1.75725	0.0886414
$394$	0	0
$395$	−44.2245	−2.22517
$396$	0	0
$397$	−10.8482	−0.544458	−0.272229	−	0.962232i	$-0.587761\pi$
$397$	−10.8482	−0.544458	−0.272229	+	0.962232i	$0.587761\pi$
$398$	0	0
$399$	3.48907	0.174672
$400$	0	0
$401$	5.90048	0.294656	0.147328	−	0.989088i	$-0.452933\pi$
$401$	5.90048	0.294656	0.147328	+	0.989088i	$0.452933\pi$
$402$	0	0
$403$	6.26818	0.312240
$404$	0	0
$405$	3.38955	0.168428
$406$	0	0
$407$	42.8300	2.12300
$408$	0	0
$409$	14.2245	0.703354	0.351677	−	0.936121i	$-0.385612\pi$
$409$	14.2245	0.703354	0.351677	+	0.936121i	$0.385612\pi$
$410$	0	0
$411$	−6.09952	−0.300867
$412$	0	0
$413$	−6.87862	−0.338475
$414$	0	0
$415$	15.6274	0.767117
$416$	0	0
$417$	6.97814	0.341721
$418$	0	0
$419$	18.2936	0.893701	0.446850	−	0.894609i	$-0.352546\pi$
$419$	18.2936	0.893701	0.446850	+	0.894609i	$0.352546\pi$
$420$	0	0
$421$	28.4926	1.38865	0.694323	−	0.719664i	$-0.255704\pi$
$421$	28.4926	1.38865	0.694323	+	0.719664i	$0.255704\pi$
$422$	0	0
$423$	10.1687	0.494417
$424$	0	0
$425$	25.9563	1.25906
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−4.87862	−0.235542
$430$	0	0
$431$	24.6796	1.18877	0.594387	−	0.804179i	$-0.297395\pi$
$431$	24.6796	1.18877	0.594387	+	0.804179i	$0.297395\pi$
$432$	0	0
$433$	−3.02186	−0.145221	−0.0726107	−	0.997360i	$-0.523133\pi$
$433$	−3.02186	−0.145221	−0.0726107	+	0.997360i	$0.523133\pi$
$434$	0	0
$435$	−18.6055	−0.892065
$436$	0	0
$437$	28.8482	1.38000
$438$	0	0
$439$	−32.1383	−1.53388	−0.766938	−	0.641721i	$-0.778221\pi$
$439$	−32.1383	−1.53388	−0.766938	+	0.641721i	$0.778221\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	30.8482	1.46564	0.732822	−	0.680420i	$-0.238203\pi$
$443$	30.8482	1.46564	0.732822	+	0.680420i	$0.238203\pi$
$444$	0	0
$445$	−44.5618	−2.11243
$446$	0	0
$447$	12.8786	0.609138
$448$	0	0
$449$	−5.70145	−0.269068	−0.134534	−	0.990909i	$-0.542954\pi$
$449$	−5.70145	−0.269068	−0.134534	+	0.990909i	$0.542954\pi$
$450$	0	0
$451$	−0.485507	−0.0228616
$452$	0	0
$453$	−13.5582	−0.637020
$454$	0	0
$455$	3.38955	0.158905
$456$	0	0
$457$	−2.33732	−0.109335	−0.0546676	−	0.998505i	$-0.517410\pi$
$457$	−2.33732	−0.109335	−0.0546676	+	0.998505i	$0.517410\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−24.9732	−1.16312	−0.581559	−	0.813504i	$-0.697557\pi$
$461$	−24.9732	−1.16312	−0.581559	+	0.813504i	$0.697557\pi$
$462$	0	0
$463$	38.2936	1.77965	0.889827	−	0.456298i	$-0.150825\pi$
$463$	38.2936	1.77965	0.889827	+	0.456298i	$0.150825\pi$
$464$	0	0
$465$	−21.2463	−0.985274
$466$	0	0
$467$	−36.3373	−1.68149	−0.840745	−	0.541431i	$-0.817883\pi$
$467$	−36.3373	−1.68149	−0.840745	+	0.541431i	$0.817883\pi$
$468$	0	0
$469$	6.77911	0.313030
$470$	0	0
$471$	14.5364	0.669800
$472$	0	0
$473$	11.0656	0.508796
$474$	0	0
$475$	−22.6408	−1.03883
$476$	0	0
$477$	−12.2682	−0.561721
$478$	0	0
$479$	14.9040	0.680983	0.340492	−	0.940248i	$-0.389407\pi$
$479$	14.9040	0.680983	0.340492	+	0.940248i	$0.389407\pi$
$480$	0	0
$481$	8.77911	0.400293
$482$	0	0
$483$	8.26818	0.376215
$484$	0	0
$485$	28.0254	1.27257
$486$	0	0
$487$	25.0219	1.13385	0.566924	−	0.823770i	$-0.308133\pi$
$487$	25.0219	1.13385	0.566924	+	0.823770i	$0.308133\pi$
$488$	0	0
$489$	13.7572	0.622124
$490$	0	0
$491$	−38.0508	−1.71721	−0.858605	−	0.512637i	$-0.828669\pi$
$491$	−38.0508	−1.71721	−0.858605	+	0.512637i	$0.828669\pi$
$492$	0	0
$493$	21.9563	0.988861
$494$	0	0
$495$	16.5364	0.743254
$496$	0	0
$497$	−3.65773	−0.164072
$498$	0	0
$499$	1.95628	0.0875752	0.0437876	−	0.999041i	$-0.486058\pi$
$499$	1.95628	0.0875752	0.0437876	+	0.999041i	$0.486058\pi$
$500$	0	0
$501$	2.36769	0.105781
$502$	0	0
$503$	−9.22089	−0.411139	−0.205570	−	0.978642i	$-0.565905\pi$
$503$	−9.22089	−0.411139	−0.205570	+	0.978642i	$0.565905\pi$
$504$	0	0
$505$	4.13828	0.184151
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−16.1249	−0.714725	−0.357363	−	0.933966i	$-0.616324\pi$
$509$	−16.1249	−0.714725	−0.357363	+	0.933966i	$0.616324\pi$
$510$	0	0
$511$	−11.0473	−0.488703
$512$	0	0
$513$	3.48907	0.154046
$514$	0	0
$515$	−13.5582	−0.597446
$516$	0	0
$517$	49.6091	2.18180
$518$	0	0
$519$	−0.199034	−0.00873663
$520$	0	0
$521$	−23.3155	−1.02147	−0.510734	−	0.859739i	$-0.670626\pi$
$521$	−23.3155	−1.02147	−0.510734	+	0.859739i	$0.670626\pi$
$522$	0	0
$523$	−34.0946	−1.49085	−0.745426	−	0.666589i	$-0.767754\pi$
$523$	−34.0946	−1.49085	−0.745426	+	0.666589i	$0.767754\pi$
$524$	0	0
$525$	−6.48907	−0.283206
$526$	0	0
$527$	25.0727	1.09218
$528$	0	0
$529$	45.3627	1.97229
$530$	0	0
$531$	−6.87862	−0.298507
$532$	0	0
$533$	−0.0995171	−0.00431057
$534$	0	0
$535$	62.8300	2.71638
$536$	0	0
$537$	−23.0473	−0.994564
$538$	0	0
$539$	4.87862	0.210137
$540$	0	0
$541$	20.0946	0.863933	0.431966	−	0.901890i	$-0.357820\pi$
$541$	20.0946	0.863933	0.431966	+	0.901890i	$0.357820\pi$
$542$	0	0
$543$	11.5582	0.496010
$544$	0	0
$545$	12.7354	0.545524
$546$	0	0
$547$	10.8045	0.461968	0.230984	−	0.972958i	$-0.425805\pi$
$547$	10.8045	0.461968	0.230984	+	0.972958i	$0.425805\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−19.1518	−0.815892
$552$	0	0
$553$	−13.0473	−0.554827
$554$	0	0
$555$	−29.7572	−1.26312
$556$	0	0
$557$	45.6140	1.93273	0.966364	−	0.257179i	$-0.0827929\pi$
$557$	45.6140	1.93273	0.966364	+	0.257179i	$0.0827929\pi$
$558$	0	0
$559$	2.26818	0.0959336
$560$	0	0
$561$	−19.5145	−0.823903
$562$	0	0
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	0	0
$565$	−64.5618	−2.71613
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−6.02542	−0.252599	−0.126299	−	0.991992i	$-0.540310\pi$
$569$	−6.02542	−0.252599	−0.126299	+	0.991992i	$0.540310\pi$
$570$	0	0
$571$	−19.0910	−0.798934	−0.399467	−	0.916748i	$-0.630805\pi$
$571$	−19.0910	−0.798934	−0.399467	+	0.916748i	$0.630805\pi$
$572$	0	0
$573$	−16.7791	−0.700957
$574$	0	0
$575$	−53.6528	−2.23748
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−25.3155	−1.05207
$580$	0	0
$581$	4.61045	0.191274
$582$	0	0
$583$	−59.8518	−2.47881
$584$	0	0
$585$	3.38955	0.140141
$586$	0	0
$587$	−24.4623	−1.00967	−0.504833	−	0.863217i	$-0.668446\pi$
$587$	−24.4623	−1.00967	−0.504833	+	0.863217i	$0.668446\pi$
$588$	0	0
$589$	−21.8701	−0.901142
$590$	0	0
$591$	−0.878623	−0.0361417
$592$	0	0
$593$	16.9478	0.695961	0.347981	−	0.937502i	$-0.386867\pi$
$593$	16.9478	0.695961	0.347981	+	0.937502i	$0.386867\pi$
$594$	0	0
$595$	13.5582	0.555833
$596$	0	0
$597$	13.5582	0.554901
$598$	0	0
$599$	33.2900	1.36019	0.680097	−	0.733122i	$-0.261938\pi$
$599$	33.2900	1.36019	0.680097	+	0.733122i	$0.261938\pi$
$600$	0	0
$601$	33.1164	1.35085	0.675424	−	0.737430i	$-0.263961\pi$
$601$	33.1164	1.35085	0.675424	+	0.737430i	$0.263961\pi$
$602$	0	0
$603$	6.77911	0.276067
$604$	0	0
$605$	43.3896	1.76404
$606$	0	0
$607$	−26.5801	−1.07885	−0.539426	−	0.842033i	$-0.681359\pi$
$607$	−26.5801	−1.07885	−0.539426	+	0.842033i	$0.681359\pi$
$608$	0	0
$609$	−5.48907	−0.222428
$610$	0	0
$611$	10.1687	0.411380
$612$	0	0
$613$	7.41993	0.299688	0.149844	−	0.988710i	$-0.452123\pi$
$613$	7.41993	0.299688	0.149844	+	0.988710i	$0.452123\pi$
$614$	0	0
$615$	0.337319	0.0136020
$616$	0	0
$617$	24.5922	0.990043	0.495021	−	0.868881i	$-0.335160\pi$
$617$	24.5922	0.990043	0.495021	+	0.868881i	$0.335160\pi$
$618$	0	0
$619$	−25.3592	−1.01927	−0.509636	−	0.860390i	$-0.670220\pi$
$619$	−25.3592	−1.01927	−0.509636	+	0.860390i	$0.670220\pi$
$620$	0	0
$621$	8.26818	0.331790
$622$	0	0
$623$	−13.1468	−0.526715
$624$	0	0
$625$	−15.3373	−0.613493
$626$	0	0
$627$	17.0219	0.679788
$628$	0	0
$629$	35.1164	1.40018
$630$	0	0
$631$	−26.7791	−1.06606	−0.533030	−	0.846097i	$-0.678947\pi$
$631$	−26.7791	−1.06606	−0.533030	+	0.846097i	$0.678947\pi$
$632$	0	0
$633$	27.0036	1.07330
$634$	0	0
$635$	0.674637	0.0267722
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−3.65773	−0.144698
$640$	0	0
$641$	−9.68810	−0.382657	−0.191329	−	0.981526i	$-0.561280\pi$
$641$	−9.68810	−0.382657	−0.191329	+	0.981526i	$0.561280\pi$
$642$	0	0
$643$	−17.5582	−0.692428	−0.346214	−	0.938156i	$-0.612533\pi$
$643$	−17.5582	−0.692428	−0.346214	+	0.938156i	$0.612533\pi$
$644$	0	0
$645$	−7.68810	−0.302719
$646$	0	0
$647$	−5.41993	−0.213079	−0.106540	−	0.994308i	$-0.533977\pi$
$647$	−5.41993	−0.213079	−0.106540	+	0.994308i	$0.533977\pi$
$648$	0	0
$649$	−33.5582	−1.31728
$650$	0	0
$651$	−6.26818	−0.245669
$652$	0	0
$653$	13.8010	0.540074	0.270037	−	0.962850i	$-0.412964\pi$
$653$	13.8010	0.540074	0.270037	+	0.962850i	$0.412964\pi$
$654$	0	0
$655$	−5.95628	−0.232731
$656$	0	0
$657$	−11.0473	−0.430996
$658$	0	0
$659$	−19.5328	−0.760889	−0.380445	−	0.924804i	$-0.624229\pi$
$659$	−19.5328	−0.760889	−0.380445	+	0.924804i	$0.624229\pi$
$660$	0	0
$661$	3.58364	0.139387	0.0696936	−	0.997568i	$-0.477798\pi$
$661$	3.58364	0.139387	0.0696936	+	0.997568i	$0.477798\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−11.8264	−0.458608
$666$	0	0
$667$	−45.3846	−1.75730
$668$	0	0
$669$	−29.0473	−1.12303
$670$	0	0
$671$	9.75725	0.376674
$672$	0	0
$673$	8.75368	0.337430	0.168715	−	0.985665i	$-0.446038\pi$
$673$	8.75368	0.337430	0.168715	+	0.985665i	$0.446038\pi$
$674$	0	0
$675$	−6.48907	−0.249764
$676$	0	0
$677$	26.0946	1.00290	0.501448	−	0.865188i	$-0.332801\pi$
$677$	26.0946	1.00290	0.501448	+	0.865188i	$0.332801\pi$
$678$	0	0
$679$	8.26818	0.317303
$680$	0	0
$681$	−7.07766	−0.271216
$682$	0	0
$683$	18.9732	0.725989	0.362994	−	0.931791i	$-0.381754\pi$
$683$	18.9732	0.725989	0.362994	+	0.931791i	$0.381754\pi$
$684$	0	0
$685$	20.6746	0.789937
$686$	0	0
$687$	9.80097	0.373930
$688$	0	0
$689$	−12.2682	−0.467380
$690$	0	0
$691$	7.00356	0.266428	0.133214	−	0.991087i	$-0.457470\pi$
$691$	7.00356	0.266428	0.133214	+	0.991087i	$0.457470\pi$
$692$	0	0
$693$	4.87862	0.185324
$694$	0	0
$695$	−23.6528	−0.897201
$696$	0	0
$697$	−0.398069	−0.0150779
$698$	0	0
$699$	2.51093	0.0949721
$700$	0	0
$701$	9.82639	0.371138	0.185569	−	0.982631i	$-0.440587\pi$
$701$	9.82639	0.371138	0.185569	+	0.982631i	$0.440587\pi$
$702$	0	0
$703$	−30.6309	−1.15527
$704$	0	0
$705$	−34.4672	−1.29811
$706$	0	0
$707$	1.22089	0.0459164
$708$	0	0
$709$	−32.8300	−1.23295	−0.616477	−	0.787373i	$-0.711441\pi$
$709$	−32.8300	−1.23295	−0.616477	+	0.787373i	$0.711441\pi$
$710$	0	0
$711$	−13.0473	−0.489311
$712$	0	0
$713$	−51.8264	−1.94091
$714$	0	0
$715$	16.5364	0.618425
$716$	0	0
$717$	−0.342270	−0.0127823
$718$	0	0
$719$	33.8955	1.26409	0.632045	−	0.774932i	$-0.282216\pi$
$719$	33.8955	1.26409	0.632045	+	0.774932i	$0.282216\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−18.5109	−0.688429
$724$	0	0
$725$	35.6190	1.32286
$726$	0	0
$727$	−34.0946	−1.26450	−0.632249	−	0.774765i	$-0.717868\pi$
$727$	−34.0946	−1.26450	−0.632249	+	0.774765i	$0.717868\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.07271	0.335566
$732$	0	0
$733$	−47.0473	−1.73773	−0.868866	−	0.495048i	$-0.835150\pi$
$733$	−47.0473	−1.73773	−0.868866	+	0.495048i	$0.835150\pi$
$734$	0	0
$735$	−3.38955	−0.125026
$736$	0	0
$737$	33.0727	1.21825
$738$	0	0
$739$	40.2499	1.48062	0.740308	−	0.672268i	$-0.234680\pi$
$739$	40.2499	1.48062	0.740308	+	0.672268i	$0.234680\pi$
$740$	0	0
$741$	3.48907	0.128174
$742$	0	0
$743$	48.9295	1.79505	0.897524	−	0.440965i	$-0.145364\pi$
$743$	48.9295	1.79505	0.897524	+	0.440965i	$0.145364\pi$
$744$	0	0
$745$	−43.6528	−1.59931
$746$	0	0
$747$	4.61045	0.168687
$748$	0	0
$749$	18.5364	0.677304
$750$	0	0
$751$	−13.2463	−0.483365	−0.241682	−	0.970355i	$-0.577699\pi$
$751$	−13.2463	−0.483365	−0.241682	+	0.970355i	$0.577699\pi$
$752$	0	0
$753$	−14.7791	−0.538581
$754$	0	0
$755$	45.9563	1.67252
$756$	0	0
$757$	−49.0036	−1.78106	−0.890532	−	0.454920i	$-0.849668\pi$
$757$	−49.0036	−1.78106	−0.890532	+	0.454920i	$0.849668\pi$
$758$	0	0
$759$	40.3373	1.46415
$760$	0	0
$761$	7.58859	0.275086	0.137543	−	0.990496i	$-0.456079\pi$
$761$	7.58859	0.275086	0.137543	+	0.990496i	$0.456079\pi$
$762$	0	0
$763$	3.75725	0.136021
$764$	0	0
$765$	13.5582	0.490198
$766$	0	0
$767$	−6.87862	−0.248373
$768$	0	0
$769$	3.04728	0.109888	0.0549439	−	0.998489i	$-0.482502\pi$
$769$	3.04728	0.109888	0.0549439	+	0.998489i	$0.482502\pi$
$770$	0	0
$771$	−22.7791	−0.820369
$772$	0	0
$773$	29.8568	1.07387	0.536937	−	0.843623i	$-0.319581\pi$
$773$	29.8568	1.07387	0.536937	+	0.843623i	$0.319581\pi$
$774$	0	0
$775$	40.6746	1.46108
$776$	0	0
$777$	−8.77911	−0.314949
$778$	0	0
$779$	0.347222	0.0124405
$780$	0	0
$781$	−17.8447	−0.638533
$782$	0	0
$783$	−5.48907	−0.196163
$784$	0	0
$785$	−49.2717	−1.75858
$786$	0	0
$787$	4.90900	0.174987	0.0874934	−	0.996165i	$-0.472114\pi$
$787$	4.90900	0.174987	0.0874934	+	0.996165i	$0.472114\pi$
$788$	0	0
$789$	7.24632	0.257976
$790$	0	0
$791$	−19.0473	−0.677243
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	41.5836	1.47482
$796$	0	0
$797$	−31.6528	−1.12120	−0.560599	−	0.828087i	$-0.689429\pi$
$797$	−31.6528	−1.12120	−0.560599	+	0.828087i	$0.689429\pi$
$798$	0	0
$799$	40.6746	1.43897
$800$	0	0
$801$	−13.1468	−0.464519
$802$	0	0
$803$	−53.8955	−1.90193
$804$	0	0
$805$	−28.0254	−0.987766
$806$	0	0
$807$	1.75725	0.0618580
$808$	0	0
$809$	14.0254	0.493108	0.246554	−	0.969129i	$-0.420702\pi$
$809$	14.0254	0.493108	0.246554	+	0.969129i	$0.420702\pi$
$810$	0	0
$811$	−47.5145	−1.66846	−0.834230	−	0.551417i	$-0.814087\pi$
$811$	−47.5145	−1.66846	−0.834230	+	0.551417i	$0.814087\pi$
$812$	0	0
$813$	−13.7572	−0.482488
$814$	0	0
$815$	−46.6309	−1.63341
$816$	0	0
$817$	−7.91383	−0.276870
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	9.07766	0.316812	0.158406	−	0.987374i	$-0.449364\pi$
$821$	9.07766	0.316812	0.158406	+	0.987374i	$0.449364\pi$
$822$	0	0
$823$	−53.0727	−1.85000	−0.924999	−	0.379969i	$-0.875935\pi$
$823$	−53.0727	−1.85000	−0.924999	+	0.379969i	$0.875935\pi$
$824$	0	0
$825$	−31.6577	−1.10218
$826$	0	0
$827$	−19.1722	−0.666684	−0.333342	−	0.942806i	$-0.608176\pi$
$827$	−19.1722	−0.666684	−0.333342	+	0.942806i	$0.608176\pi$
$828$	0	0
$829$	−12.0437	−0.418296	−0.209148	−	0.977884i	$-0.567069\pi$
$829$	−12.0437	−0.418296	−0.209148	+	0.977884i	$0.567069\pi$
$830$	0	0
$831$	11.2463	0.390130
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−8.02542	−0.277731
$836$	0	0
$837$	−6.26818	−0.216660
$838$	0	0
$839$	−43.9513	−1.51737	−0.758684	−	0.651459i	$-0.774157\pi$
$839$	−43.9513	−1.51737	−0.758684	+	0.651459i	$0.774157\pi$
$840$	0	0
$841$	1.12989	0.0389618
$842$	0	0
$843$	30.4368	1.04830
$844$	0	0
$845$	3.38955	0.116604
$846$	0	0
$847$	12.8010	0.439846
$848$	0	0
$849$	17.5582	0.602596
$850$	0	0
$851$	−72.5872	−2.48826
$852$	0	0
$853$	6.17361	0.211380	0.105690	−	0.994399i	$-0.466295\pi$
$853$	6.17361	0.211380	0.105690	+	0.994399i	$0.466295\pi$
$854$	0	0
$855$	−11.8264	−0.404454
$856$	0	0
$857$	−51.3663	−1.75464	−0.877320	−	0.479906i	$-0.840671\pi$
$857$	−51.3663	−1.75464	−0.877320	+	0.479906i	$0.840671\pi$
$858$	0	0
$859$	−20.0508	−0.684126	−0.342063	−	0.939677i	$-0.611126\pi$
$859$	−20.0508	−0.684126	−0.342063	+	0.939677i	$0.611126\pi$
$860$	0	0
$861$	0.0995171	0.00339153
$862$	0	0
$863$	17.6140	0.599588	0.299794	−	0.954004i	$-0.403082\pi$
$863$	17.6140	0.599588	0.299794	+	0.954004i	$0.403082\pi$
$864$	0	0
$865$	0.674637	0.0229384
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−63.6528	−2.15927
$870$	0	0
$871$	6.77911	0.229701
$872$	0	0
$873$	8.26818	0.279835
$874$	0	0
$875$	5.04728	0.170629
$876$	0	0
$877$	−46.5364	−1.57142	−0.785710	−	0.618594i	$-0.787702\pi$
$877$	−46.5364	−1.57142	−0.785710	+	0.618594i	$0.787702\pi$
$878$	0	0
$879$	−15.1905	−0.512363
$880$	0	0
$881$	−24.7354	−0.833356	−0.416678	−	0.909054i	$-0.636806\pi$
$881$	−24.7354	−0.833356	−0.416678	+	0.909054i	$0.636806\pi$
$882$	0	0
$883$	−46.8300	−1.57595	−0.787977	−	0.615705i	$-0.788871\pi$
$883$	−46.8300	−1.57595	−0.787977	+	0.615705i	$0.788871\pi$
$884$	0	0
$885$	23.3155	0.783741
$886$	0	0
$887$	29.0727	0.976166	0.488083	−	0.872797i	$-0.337696\pi$
$887$	29.0727	0.976166	0.488083	+	0.872797i	$0.337696\pi$
$888$	0	0
$889$	0.199034	0.00667540
$890$	0	0
$891$	4.87862	0.163440
$892$	0	0
$893$	−35.4792	−1.18727
$894$	0	0
$895$	78.1200	2.61126
$896$	0	0
$897$	8.26818	0.276066
$898$	0	0
$899$	34.4065	1.14752
$900$	0	0
$901$	−49.0727	−1.63485
$902$	0	0
$903$	−2.26818	−0.0754802
$904$	0	0
$905$	−39.1772	−1.30229
$906$	0	0
$907$	−32.5618	−1.08120	−0.540598	−	0.841281i	$-0.681802\pi$
$907$	−32.5618	−1.08120	−0.540598	+	0.841281i	$0.681802\pi$
$908$	0	0
$909$	1.22089	0.0404945
$910$	0	0
$911$	16.4672	0.545583	0.272792	−	0.962073i	$-0.412053\pi$
$911$	16.4672	0.545583	0.272792	+	0.962073i	$0.412053\pi$
$912$	0	0
$913$	22.4926	0.744398
$914$	0	0
$915$	−6.77911	−0.224110
$916$	0	0
$917$	−1.75725	−0.0580294
$918$	0	0
$919$	18.8300	0.621143	0.310571	−	0.950550i	$-0.399480\pi$
$919$	18.8300	0.621143	0.310571	+	0.950550i	$0.399480\pi$
$920$	0	0
$921$	−3.29004	−0.108410
$922$	0	0
$923$	−3.65773	−0.120396
$924$	0	0
$925$	56.9682	1.87310
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−41.8822	−1.37411	−0.687055	−	0.726605i	$-0.741097\pi$
$929$	−41.8822	−1.37411	−0.687055	+	0.726605i	$0.741097\pi$
$930$	0	0
$931$	−3.48907	−0.114350
$932$	0	0
$933$	10.7791	0.352892
$934$	0	0
$935$	66.1454	2.16319
$936$	0	0
$937$	−36.4926	−1.19216	−0.596081	−	0.802924i	$-0.703276\pi$
$937$	−36.4926	−1.19216	−0.596081	+	0.802924i	$0.703276\pi$
$938$	0	0
$939$	−25.1164	−0.819644
$940$	0	0
$941$	18.1687	0.592281	0.296141	−	0.955144i	$-0.404300\pi$
$941$	18.1687	0.592281	0.296141	+	0.955144i	$0.404300\pi$
$942$	0	0
$943$	0.822825	0.0267949
$944$	0	0
$945$	−3.38955	−0.110262
$946$	0	0
$947$	−30.0388	−0.976129	−0.488064	−	0.872808i	$-0.662297\pi$
$947$	−30.0388	−0.976129	−0.488064	+	0.872808i	$0.662297\pi$
$948$	0	0
$949$	−11.0473	−0.358610
$950$	0	0
$951$	−23.8568	−0.773609
$952$	0	0
$953$	17.1418	0.555279	0.277639	−	0.960685i	$-0.410448\pi$
$953$	17.1418	0.555279	0.277639	+	0.960685i	$0.410448\pi$
$954$	0	0
$955$	56.8737	1.84039
$956$	0	0
$957$	−26.7791	−0.865646
$958$	0	0
$959$	6.09952	0.196964
$960$	0	0
$961$	8.29004	0.267421
$962$	0	0
$963$	18.5364	0.597326
$964$	0	0
$965$	85.8081	2.76226
$966$	0	0
$967$	−39.3663	−1.26594	−0.632968	−	0.774178i	$-0.718163\pi$
$967$	−39.3663	−1.26594	−0.632968	+	0.774178i	$0.718163\pi$
$968$	0	0
$969$	13.9563	0.448340
$970$	0	0
$971$	−42.5801	−1.36646	−0.683230	−	0.730203i	$-0.739425\pi$
$971$	−42.5801	−1.36646	−0.683230	+	0.730203i	$0.739425\pi$
$972$	0	0
$973$	−6.97814	−0.223709
$974$	0	0
$975$	−6.48907	−0.207817
$976$	0	0
$977$	9.36413	0.299585	0.149793	−	0.988717i	$-0.452139\pi$
$977$	9.36413	0.299585	0.149793	+	0.988717i	$0.452139\pi$
$978$	0	0
$979$	−64.1383	−2.04987
$980$	0	0
$981$	3.75725	0.119960
$982$	0	0
$983$	30.9915	0.988475	0.494237	−	0.869327i	$-0.335447\pi$
$983$	30.9915	0.988475	0.494237	+	0.869327i	$0.335447\pi$
$984$	0	0
$985$	2.97814	0.0948914
$986$	0	0
$987$	−10.1687	−0.323672
$988$	0	0
$989$	−18.7537	−0.596332
$990$	0	0
$991$	45.5582	1.44720	0.723602	−	0.690217i	$-0.242485\pi$
$991$	45.5582	1.44720	0.723602	+	0.690217i	$0.242485\pi$
$992$	0	0
$993$	−2.97814	−0.0945084
$994$	0	0
$995$	−45.9563	−1.45691
$996$	0	0
$997$	−39.6091	−1.25443	−0.627216	−	0.778846i	$-0.715806\pi$
$997$	−39.6091	−1.25443	−0.627216	+	0.778846i	$0.715806\pi$
$998$	0	0
$999$	−8.77911	−0.277759

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bn.1.3		3
4.3	odd	2		1092.2.a.h.1.3	✓	3
12.11	even	2		3276.2.a.r.1.1		3
28.27	even	2		7644.2.a.t.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.h.1.3	✓	3	4.3	odd	2
3276.2.a.r.1.1		3	12.11	even	2
4368.2.a.bn.1.3		3	1.1	even	1	trivial
7644.2.a.t.1.1		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.38955 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.38955 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.87862 q^{11} +1.00000 q^{13} -3.38955 q^{15} +4.00000 q^{17} -3.48907 q^{19} -1.00000 q^{21} -8.26818 q^{23} +6.48907 q^{25} -1.00000 q^{27} +5.48907 q^{29} +6.26818 q^{31} -4.87862 q^{33} +3.38955 q^{35} +8.77911 q^{37} -1.00000 q^{39} -0.0995171 q^{41} +2.26818 q^{43} +3.38955 q^{45} +10.1687 q^{47} +1.00000 q^{49} -4.00000 q^{51} -12.2682 q^{53} +16.5364 q^{55} +3.48907 q^{57} -6.87862 q^{59} +2.00000 q^{61} +1.00000 q^{63} +3.38955 q^{65} +6.77911 q^{67} +8.26818 q^{69} -3.65773 q^{71} -11.0473 q^{73} -6.48907 q^{75} +4.87862 q^{77} -13.0473 q^{79} +1.00000 q^{81} +4.61045 q^{83} +13.5582 q^{85} -5.48907 q^{87} -13.1468 q^{89} +1.00000 q^{91} -6.26818 q^{93} -11.8264 q^{95} +8.26818 q^{97} +4.87862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{13} - q^{15} + 12 q^{17} - 5 q^{19} - 3 q^{21} - q^{23} + 14 q^{25} - 3 q^{27} + 11 q^{29} - 5 q^{31} + q^{35} + 8 q^{37} - 3 q^{39} - 4 q^{41} - 17 q^{43} + q^{45} + 3 q^{47} + 3 q^{49} - 12 q^{51} - 13 q^{53} + 2 q^{55} + 5 q^{57} - 6 q^{59} + 6 q^{61} + 3 q^{63} + q^{65} + 2 q^{67} + q^{69} + 22 q^{71} + 9 q^{73} - 14 q^{75} + 3 q^{79} + 3 q^{81} + 23 q^{83} + 4 q^{85} - 11 q^{87} - q^{89} + 3 q^{91} + 5 q^{93} + 25 q^{95} + q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 + 3 * q^13 - q^15 + 12 * q^17 - 5 * q^19 - 3 * q^21 - q^23 + 14 * q^25 - 3 * q^27 + 11 * q^29 - 5 * q^31 + q^35 + 8 * q^37 - 3 * q^39 - 4 * q^41 - 17 * q^43 + q^45 + 3 * q^47 + 3 * q^49 - 12 * q^51 - 13 * q^53 + 2 * q^55 + 5 * q^57 - 6 * q^59 + 6 * q^61 + 3 * q^63 + q^65 + 2 * q^67 + q^69 + 22 * q^71 + 9 * q^73 - 14 * q^75 + 3 * q^79 + 3 * q^81 + 23 * q^83 + 4 * q^85 - 11 * q^87 - q^89 + 3 * q^91 + 5 * q^93 + 25 * q^95 + q^97

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