LMFDB - Embedded newform 4368.2.a.bb.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.12311 q^{11} +1.00000 q^{13} +3.56155 q^{15} +1.12311 q^{17} +1.56155 q^{19} -1.00000 q^{21} +1.56155 q^{23} +7.68466 q^{25} -1.00000 q^{27} -7.56155 q^{29} +5.56155 q^{31} +3.12311 q^{33} -3.56155 q^{35} +1.12311 q^{37} -1.00000 q^{39} -2.00000 q^{41} +1.56155 q^{43} -3.56155 q^{45} +8.68466 q^{47} +1.00000 q^{49} -1.12311 q^{51} -1.31534 q^{53} +11.1231 q^{55} -1.56155 q^{57} -2.24621 q^{59} +6.00000 q^{61} +1.00000 q^{63} -3.56155 q^{65} -0.876894 q^{67} -1.56155 q^{69} +10.2462 q^{71} -0.438447 q^{73} -7.68466 q^{75} -3.12311 q^{77} -0.684658 q^{79} +1.00000 q^{81} +12.6847 q^{83} -4.00000 q^{85} +7.56155 q^{87} -10.6847 q^{89} +1.00000 q^{91} -5.56155 q^{93} -5.56155 q^{95} -14.6847 q^{97} -3.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 3 q^{15} - 6 q^{17} - q^{19} - 2 q^{21} - q^{23} + 3 q^{25} - 2 q^{27} - 11 q^{29} + 7 q^{31} - 2 q^{33} - 3 q^{35} - 6 q^{37} - 2 q^{39} - 4 q^{41} - q^{43} - 3 q^{45} + 5 q^{47} + 2 q^{49} + 6 q^{51} - 15 q^{53} + 14 q^{55} + q^{57} + 12 q^{59} + 12 q^{61} + 2 q^{63} - 3 q^{65} - 10 q^{67} + q^{69} + 4 q^{71} - 5 q^{73} - 3 q^{75} + 2 q^{77} + 11 q^{79} + 2 q^{81} + 13 q^{83} - 8 q^{85} + 11 q^{87} - 9 q^{89} + 2 q^{91} - 7 q^{93} - 7 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 3 * q^15 - 6 * q^17 - q^19 - 2 * q^21 - q^23 + 3 * q^25 - 2 * q^27 - 11 * q^29 + 7 * q^31 - 2 * q^33 - 3 * q^35 - 6 * q^37 - 2 * q^39 - 4 * q^41 - q^43 - 3 * q^45 + 5 * q^47 + 2 * q^49 + 6 * q^51 - 15 * q^53 + 14 * q^55 + q^57 + 12 * q^59 + 12 * q^61 + 2 * q^63 - 3 * q^65 - 10 * q^67 + q^69 + 4 * q^71 - 5 * q^73 - 3 * q^75 + 2 * q^77 + 11 * q^79 + 2 * q^81 + 13 * q^83 - 8 * q^85 + 11 * q^87 - 9 * q^89 + 2 * q^91 - 7 * q^93 - 7 * q^95 - 17 * q^97 + 2 * 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
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.56155	0.919589
$16$	0	0
$17$	1.12311	0.272393	0.136197	−	0.990682i	$-0.456512\pi$
$17$	1.12311	0.272393	0.136197	+	0.990682i	$0.456512\pi$
$18$	0	0
$19$	1.56155	0.358245	0.179122	−	0.983827i	$-0.442674\pi$
$19$	1.56155	0.358245	0.179122	+	0.983827i	$0.442674\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.56155	0.325606	0.162803	−	0.986659i	$-0.447946\pi$
$23$	1.56155	0.325606	0.162803	+	0.986659i	$0.447946\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.56155	−1.40415	−0.702073	−	0.712105i	$-0.747742\pi$
$29$	−7.56155	−1.40415	−0.702073	+	0.712105i	$0.747742\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	3.12311	0.543663
$34$	0	0
$35$	−3.56155	−0.602012
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.56155	0.238135	0.119067	−	0.992886i	$-0.462010\pi$
$43$	1.56155	0.238135	0.119067	+	0.992886i	$0.462010\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	8.68466	1.26679	0.633394	−	0.773830i	$-0.281661\pi$
$47$	8.68466	1.26679	0.633394	+	0.773830i	$0.281661\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.12311	−0.157266
$52$	0	0
$53$	−1.31534	−0.180676	−0.0903380	−	0.995911i	$-0.528795\pi$
$53$	−1.31534	−0.180676	−0.0903380	+	0.995911i	$0.528795\pi$
$54$	0	0
$55$	11.1231	1.49984
$56$	0	0
$57$	−1.56155	−0.206833
$58$	0	0
$59$	−2.24621	−0.292432	−0.146216	−	0.989253i	$-0.546709\pi$
$59$	−2.24621	−0.292432	−0.146216	+	0.989253i	$0.546709\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.56155	−0.441756
$66$	0	0
$67$	−0.876894	−0.107130	−0.0535648	−	0.998564i	$-0.517058\pi$
$67$	−0.876894	−0.107130	−0.0535648	+	0.998564i	$0.517058\pi$
$68$	0	0
$69$	−1.56155	−0.187989
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	0	0
$73$	−0.438447	−0.0513164	−0.0256582	−	0.999671i	$-0.508168\pi$
$73$	−0.438447	−0.0513164	−0.0256582	+	0.999671i	$0.508168\pi$
$74$	0	0
$75$	−7.68466	−0.887348
$76$	0	0
$77$	−3.12311	−0.355911
$78$	0	0
$79$	−0.684658	−0.0770301	−0.0385150	−	0.999258i	$-0.512263\pi$
$79$	−0.684658	−0.0770301	−0.0385150	+	0.999258i	$0.512263\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.6847	1.39232	0.696161	−	0.717886i	$-0.254890\pi$
$83$	12.6847	1.39232	0.696161	+	0.717886i	$0.254890\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	7.56155	0.810684
$88$	0	0
$89$	−10.6847	−1.13257	−0.566286	−	0.824209i	$-0.691620\pi$
$89$	−10.6847	−1.13257	−0.566286	+	0.824209i	$0.691620\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−5.56155	−0.576706
$94$	0	0
$95$	−5.56155	−0.570603
$96$	0	0
$97$	−14.6847	−1.49100	−0.745501	−	0.666505i	$-0.767790\pi$
$97$	−14.6847	−1.49100	−0.745501	+	0.666505i	$0.767790\pi$
$98$	0	0
$99$	−3.12311	−0.313884
$100$	0	0
$101$	−18.4924	−1.84006	−0.920032	−	0.391842i	$-0.871838\pi$
$101$	−18.4924	−1.84006	−0.920032	+	0.391842i	$0.871838\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	3.56155	0.347572
$106$	0	0
$107$	6.24621	0.603844	0.301922	−	0.953333i	$-0.402372\pi$
$107$	6.24621	0.603844	0.301922	+	0.953333i	$0.402372\pi$
$108$	0	0
$109$	4.24621	0.406713	0.203357	−	0.979105i	$-0.434815\pi$
$109$	4.24621	0.406713	0.203357	+	0.979105i	$0.434815\pi$
$110$	0	0
$111$	−1.12311	−0.106600
$112$	0	0
$113$	−6.68466	−0.628840	−0.314420	−	0.949284i	$-0.601810\pi$
$113$	−6.68466	−0.628840	−0.314420	+	0.949284i	$0.601810\pi$
$114$	0	0
$115$	−5.56155	−0.518617
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	1.12311	0.102955
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−9.56155	−0.855211
$126$	0	0
$127$	−14.2462	−1.26415	−0.632073	−	0.774909i	$-0.717796\pi$
$127$	−14.2462	−1.26415	−0.632073	+	0.774909i	$0.717796\pi$
$128$	0	0
$129$	−1.56155	−0.137487
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	1.56155	0.135404
$134$	0	0
$135$	3.56155	0.306530
$136$	0	0
$137$	−10.8769	−0.929276	−0.464638	−	0.885501i	$-0.653816\pi$
$137$	−10.8769	−0.929276	−0.464638	+	0.885501i	$0.653816\pi$
$138$	0	0
$139$	15.1231	1.28273	0.641363	−	0.767238i	$-0.278369\pi$
$139$	15.1231	1.28273	0.641363	+	0.767238i	$0.278369\pi$
$140$	0	0
$141$	−8.68466	−0.731380
$142$	0	0
$143$	−3.12311	−0.261167
$144$	0	0
$145$	26.9309	2.23649
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	1.12311	0.0907977
$154$	0	0
$155$	−19.8078	−1.59100
$156$	0	0
$157$	−11.3693	−0.907370	−0.453685	−	0.891162i	$-0.649891\pi$
$157$	−11.3693	−0.907370	−0.453685	+	0.891162i	$0.649891\pi$
$158$	0	0
$159$	1.31534	0.104313
$160$	0	0
$161$	1.56155	0.123068
$162$	0	0
$163$	−18.2462	−1.42915	−0.714577	−	0.699557i	$-0.753381\pi$
$163$	−18.2462	−1.42915	−0.714577	+	0.699557i	$0.753381\pi$
$164$	0	0
$165$	−11.1231	−0.865933
$166$	0	0
$167$	21.5616	1.66848	0.834242	−	0.551399i	$-0.185906\pi$
$167$	21.5616	1.66848	0.834242	+	0.551399i	$0.185906\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.56155	0.119415
$172$	0	0
$173$	13.1231	0.997731	0.498866	−	0.866679i	$-0.333750\pi$
$173$	13.1231	0.997731	0.498866	+	0.866679i	$0.333750\pi$
$174$	0	0
$175$	7.68466	0.580906
$176$	0	0
$177$	2.24621	0.168836
$178$	0	0
$179$	−24.6847	−1.84502	−0.922509	−	0.385976i	$-0.873865\pi$
$179$	−24.6847	−1.84502	−0.922509	+	0.385976i	$0.873865\pi$
$180$	0	0
$181$	12.2462	0.910254	0.455127	−	0.890427i	$-0.349594\pi$
$181$	12.2462	0.910254	0.455127	+	0.890427i	$0.349594\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−3.50758	−0.256499
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−18.2462	−1.32025	−0.660125	−	0.751156i	$-0.729497\pi$
$191$	−18.2462	−1.32025	−0.660125	+	0.751156i	$0.729497\pi$
$192$	0	0
$193$	−4.24621	−0.305649	−0.152824	−	0.988253i	$-0.548837\pi$
$193$	−4.24621	−0.305649	−0.152824	+	0.988253i	$0.548837\pi$
$194$	0	0
$195$	3.56155	0.255048
$196$	0	0
$197$	4.24621	0.302530	0.151265	−	0.988493i	$-0.451665\pi$
$197$	4.24621	0.302530	0.151265	+	0.988493i	$0.451665\pi$
$198$	0	0
$199$	−14.2462	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$199$	−14.2462	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$200$	0	0
$201$	0.876894	0.0618514
$202$	0	0
$203$	−7.56155	−0.530717
$204$	0	0
$205$	7.12311	0.497499
$206$	0	0
$207$	1.56155	0.108535
$208$	0	0
$209$	−4.87689	−0.337342
$210$	0	0
$211$	−0.192236	−0.0132341	−0.00661703	−	0.999978i	$-0.502106\pi$
$211$	−0.192236	−0.0132341	−0.00661703	+	0.999978i	$0.502106\pi$
$212$	0	0
$213$	−10.2462	−0.702059
$214$	0	0
$215$	−5.56155	−0.379295
$216$	0	0
$217$	5.56155	0.377543
$218$	0	0
$219$	0.438447	0.0296275
$220$	0	0
$221$	1.12311	0.0755483
$222$	0	0
$223$	8.68466	0.581568	0.290784	−	0.956789i	$-0.406084\pi$
$223$	8.68466	0.581568	0.290784	+	0.956789i	$0.406084\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	0	0
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	24.7386	1.63477	0.817387	−	0.576088i	$-0.195422\pi$
$229$	24.7386	1.63477	0.817387	+	0.576088i	$0.195422\pi$
$230$	0	0
$231$	3.12311	0.205485
$232$	0	0
$233$	23.1771	1.51838	0.759191	−	0.650868i	$-0.225595\pi$
$233$	23.1771	1.51838	0.759191	+	0.650868i	$0.225595\pi$
$234$	0	0
$235$	−30.9309	−2.01771
$236$	0	0
$237$	0.684658	0.0444733
$238$	0	0
$239$	−8.49242	−0.549329	−0.274665	−	0.961540i	$-0.588567\pi$
$239$	−8.49242	−0.549329	−0.274665	+	0.961540i	$0.588567\pi$
$240$	0	0
$241$	−27.1771	−1.75063	−0.875315	−	0.483553i	$-0.839346\pi$
$241$	−27.1771	−1.75063	−0.875315	+	0.483553i	$0.839346\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.56155	−0.227539
$246$	0	0
$247$	1.56155	0.0993592
$248$	0	0
$249$	−12.6847	−0.803858
$250$	0	0
$251$	0.876894	0.0553491	0.0276745	−	0.999617i	$-0.491190\pi$
$251$	0.876894	0.0553491	0.0276745	+	0.999617i	$0.491190\pi$
$252$	0	0
$253$	−4.87689	−0.306608
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−24.2462	−1.51244	−0.756219	−	0.654319i	$-0.772955\pi$
$257$	−24.2462	−1.51244	−0.756219	+	0.654319i	$0.772955\pi$
$258$	0	0
$259$	1.12311	0.0697864
$260$	0	0
$261$	−7.56155	−0.468048
$262$	0	0
$263$	−3.31534	−0.204433	−0.102216	−	0.994762i	$-0.532593\pi$
$263$	−3.31534	−0.204433	−0.102216	+	0.994762i	$0.532593\pi$
$264$	0	0
$265$	4.68466	0.287776
$266$	0	0
$267$	10.6847	0.653890
$268$	0	0
$269$	−10.8769	−0.663176	−0.331588	−	0.943424i	$-0.607584\pi$
$269$	−10.8769	−0.663176	−0.331588	+	0.943424i	$0.607584\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	−24.0000	−1.44725
$276$	0	0
$277$	−8.93087	−0.536604	−0.268302	−	0.963335i	$-0.586462\pi$
$277$	−8.93087	−0.536604	−0.268302	+	0.963335i	$0.586462\pi$
$278$	0	0
$279$	5.56155	0.332961
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	5.56155	0.329438
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	14.6847	0.860830
$292$	0	0
$293$	−11.5616	−0.675433	−0.337717	−	0.941248i	$-0.609655\pi$
$293$	−11.5616	−0.675433	−0.337717	+	0.941248i	$0.609655\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	3.12311	0.181221
$298$	0	0
$299$	1.56155	0.0903069
$300$	0	0
$301$	1.56155	0.0900064
$302$	0	0
$303$	18.4924	1.06236
$304$	0	0
$305$	−21.3693	−1.22360
$306$	0	0
$307$	0.192236	0.0109715	0.00548574	−	0.999985i	$-0.498254\pi$
$307$	0.192236	0.0109715	0.00548574	+	0.999985i	$0.498254\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.1231	−0.630733	−0.315367	−	0.948970i	$-0.602128\pi$
$311$	−11.1231	−0.630733	−0.315367	+	0.948970i	$0.602128\pi$
$312$	0	0
$313$	3.75379	0.212177	0.106088	−	0.994357i	$-0.466167\pi$
$313$	3.75379	0.212177	0.106088	+	0.994357i	$0.466167\pi$
$314$	0	0
$315$	−3.56155	−0.200671
$316$	0	0
$317$	7.36932	0.413902	0.206951	−	0.978351i	$-0.433646\pi$
$317$	7.36932	0.413902	0.206951	+	0.978351i	$0.433646\pi$
$318$	0	0
$319$	23.6155	1.32222
$320$	0	0
$321$	−6.24621	−0.348630
$322$	0	0
$323$	1.75379	0.0975834
$324$	0	0
$325$	7.68466	0.426268
$326$	0	0
$327$	−4.24621	−0.234816
$328$	0	0
$329$	8.68466	0.478801
$330$	0	0
$331$	−13.3693	−0.734844	−0.367422	−	0.930054i	$-0.619760\pi$
$331$	−13.3693	−0.734844	−0.367422	+	0.930054i	$0.619760\pi$
$332$	0	0
$333$	1.12311	0.0615458
$334$	0	0
$335$	3.12311	0.170633
$336$	0	0
$337$	3.06913	0.167186	0.0835931	−	0.996500i	$-0.473360\pi$
$337$	3.06913	0.167186	0.0835931	+	0.996500i	$0.473360\pi$
$338$	0	0
$339$	6.68466	0.363061
$340$	0	0
$341$	−17.3693	−0.940601
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	5.56155	0.299424
$346$	0	0
$347$	−32.9848	−1.77072	−0.885360	−	0.464907i	$-0.846088\pi$
$347$	−32.9848	−1.77072	−0.885360	+	0.464907i	$0.846088\pi$
$348$	0	0
$349$	3.56155	0.190646	0.0953228	−	0.995446i	$-0.469612\pi$
$349$	3.56155	0.190646	0.0953228	+	0.995446i	$0.469612\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	32.7386	1.74250	0.871251	−	0.490838i	$-0.163309\pi$
$353$	32.7386	1.74250	0.871251	+	0.490838i	$0.163309\pi$
$354$	0	0
$355$	−36.4924	−1.93682
$356$	0	0
$357$	−1.12311	−0.0594411
$358$	0	0
$359$	28.9848	1.52976	0.764881	−	0.644172i	$-0.222798\pi$
$359$	28.9848	1.52976	0.764881	+	0.644172i	$0.222798\pi$
$360$	0	0
$361$	−16.5616	−0.871661
$362$	0	0
$363$	1.24621	0.0654091
$364$	0	0
$365$	1.56155	0.0817354
$366$	0	0
$367$	4.87689	0.254572	0.127286	−	0.991866i	$-0.459373\pi$
$367$	4.87689	0.254572	0.127286	+	0.991866i	$0.459373\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−1.31534	−0.0682891
$372$	0	0
$373$	−22.4924	−1.16461	−0.582307	−	0.812969i	$-0.697850\pi$
$373$	−22.4924	−1.16461	−0.582307	+	0.812969i	$0.697850\pi$
$374$	0	0
$375$	9.56155	0.493756
$376$	0	0
$377$	−7.56155	−0.389440
$378$	0	0
$379$	−10.6307	−0.546062	−0.273031	−	0.962005i	$-0.588026\pi$
$379$	−10.6307	−0.546062	−0.273031	+	0.962005i	$0.588026\pi$
$380$	0	0
$381$	14.2462	0.729856
$382$	0	0
$383$	1.75379	0.0896144	0.0448072	−	0.998996i	$-0.485733\pi$
$383$	1.75379	0.0896144	0.0448072	+	0.998996i	$0.485733\pi$
$384$	0	0
$385$	11.1231	0.566886
$386$	0	0
$387$	1.56155	0.0793782
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	1.75379	0.0886929
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	2.43845	0.122692
$396$	0	0
$397$	32.0540	1.60874	0.804371	−	0.594127i	$-0.202502\pi$
$397$	32.0540	1.60874	0.804371	+	0.594127i	$0.202502\pi$
$398$	0	0
$399$	−1.56155	−0.0781754
$400$	0	0
$401$	−15.3693	−0.767507	−0.383754	−	0.923436i	$-0.625369\pi$
$401$	−15.3693	−0.767507	−0.383754	+	0.923436i	$0.625369\pi$
$402$	0	0
$403$	5.56155	0.277041
$404$	0	0
$405$	−3.56155	−0.176975
$406$	0	0
$407$	−3.50758	−0.173864
$408$	0	0
$409$	−16.4384	−0.812829	−0.406414	−	0.913689i	$-0.633221\pi$
$409$	−16.4384	−0.812829	−0.406414	+	0.913689i	$0.633221\pi$
$410$	0	0
$411$	10.8769	0.536518
$412$	0	0
$413$	−2.24621	−0.110529
$414$	0	0
$415$	−45.1771	−2.21766
$416$	0	0
$417$	−15.1231	−0.740582
$418$	0	0
$419$	40.1080	1.95940	0.979701	−	0.200465i	$-0.0642454\pi$
$419$	40.1080	1.95940	0.979701	+	0.200465i	$0.0642454\pi$
$420$	0	0
$421$	−23.8617	−1.16295	−0.581475	−	0.813564i	$-0.697524\pi$
$421$	−23.8617	−1.16295	−0.581475	+	0.813564i	$0.697524\pi$
$422$	0	0
$423$	8.68466	0.422263
$424$	0	0
$425$	8.63068	0.418650
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	3.12311	0.150785
$430$	0	0
$431$	0.876894	0.0422385	0.0211193	−	0.999777i	$-0.493277\pi$
$431$	0.876894	0.0422385	0.0211193	+	0.999777i	$0.493277\pi$
$432$	0	0
$433$	6.87689	0.330482	0.165241	−	0.986253i	$-0.447160\pi$
$433$	6.87689	0.330482	0.165241	+	0.986253i	$0.447160\pi$
$434$	0	0
$435$	−26.9309	−1.29124
$436$	0	0
$437$	2.43845	0.116647
$438$	0	0
$439$	−9.36932	−0.447173	−0.223587	−	0.974684i	$-0.571777\pi$
$439$	−9.36932	−0.447173	−0.223587	+	0.974684i	$0.571777\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−22.9309	−1.08948	−0.544739	−	0.838605i	$-0.683371\pi$
$443$	−22.9309	−1.08948	−0.544739	+	0.838605i	$0.683371\pi$
$444$	0	0
$445$	38.0540	1.80393
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	−23.3693	−1.10287	−0.551433	−	0.834219i	$-0.685919\pi$
$449$	−23.3693	−1.10287	−0.551433	+	0.834219i	$0.685919\pi$
$450$	0	0
$451$	6.24621	0.294123
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	−3.56155	−0.166968
$456$	0	0
$457$	−1.12311	−0.0525367	−0.0262683	−	0.999655i	$-0.508362\pi$
$457$	−1.12311	−0.0525367	−0.0262683	+	0.999655i	$0.508362\pi$
$458$	0	0
$459$	−1.12311	−0.0524221
$460$	0	0
$461$	−15.7538	−0.733727	−0.366864	−	0.930275i	$-0.619568\pi$
$461$	−15.7538	−0.733727	−0.366864	+	0.930275i	$0.619568\pi$
$462$	0	0
$463$	−15.6155	−0.725715	−0.362858	−	0.931845i	$-0.618199\pi$
$463$	−15.6155	−0.725715	−0.362858	+	0.931845i	$0.618199\pi$
$464$	0	0
$465$	19.8078	0.918563
$466$	0	0
$467$	40.1080	1.85597	0.927987	−	0.372612i	$-0.121538\pi$
$467$	40.1080	1.85597	0.927987	+	0.372612i	$0.121538\pi$
$468$	0	0
$469$	−0.876894	−0.0404912
$470$	0	0
$471$	11.3693	0.523870
$472$	0	0
$473$	−4.87689	−0.224240
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	−1.31534	−0.0602254
$478$	0	0
$479$	26.0540	1.19044	0.595218	−	0.803564i	$-0.297066\pi$
$479$	26.0540	1.19044	0.595218	+	0.803564i	$0.297066\pi$
$480$	0	0
$481$	1.12311	0.0512092
$482$	0	0
$483$	−1.56155	−0.0710531
$484$	0	0
$485$	52.3002	2.37483
$486$	0	0
$487$	9.36932	0.424564	0.212282	−	0.977208i	$-0.431910\pi$
$487$	9.36932	0.424564	0.212282	+	0.977208i	$0.431910\pi$
$488$	0	0
$489$	18.2462	0.825122
$490$	0	0
$491$	−14.2462	−0.642923	−0.321461	−	0.946923i	$-0.604174\pi$
$491$	−14.2462	−0.642923	−0.321461	+	0.946923i	$0.604174\pi$
$492$	0	0
$493$	−8.49242	−0.382479
$494$	0	0
$495$	11.1231	0.499946
$496$	0	0
$497$	10.2462	0.459605
$498$	0	0
$499$	24.4924	1.09643	0.548216	−	0.836337i	$-0.315307\pi$
$499$	24.4924	1.09643	0.548216	+	0.836337i	$0.315307\pi$
$500$	0	0
$501$	−21.5616	−0.963299
$502$	0	0
$503$	−31.6155	−1.40967	−0.704833	−	0.709373i	$-0.748978\pi$
$503$	−31.6155	−1.40967	−0.704833	+	0.709373i	$0.748978\pi$
$504$	0	0
$505$	65.8617	2.93081
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	20.4384	0.905918	0.452959	−	0.891531i	$-0.350368\pi$
$509$	20.4384	0.905918	0.452959	+	0.891531i	$0.350368\pi$
$510$	0	0
$511$	−0.438447	−0.0193958
$512$	0	0
$513$	−1.56155	−0.0689442
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−27.1231	−1.19287
$518$	0	0
$519$	−13.1231	−0.576040
$520$	0	0
$521$	37.6155	1.64797	0.823983	−	0.566614i	$-0.191747\pi$
$521$	37.6155	1.64797	0.823983	+	0.566614i	$0.191747\pi$
$522$	0	0
$523$	−23.1231	−1.01110	−0.505551	−	0.862796i	$-0.668711\pi$
$523$	−23.1231	−1.01110	−0.505551	+	0.862796i	$0.668711\pi$
$524$	0	0
$525$	−7.68466	−0.335386
$526$	0	0
$527$	6.24621	0.272089
$528$	0	0
$529$	−20.5616	−0.893981
$530$	0	0
$531$	−2.24621	−0.0974773
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	−22.2462	−0.961788
$536$	0	0
$537$	24.6847	1.06522
$538$	0	0
$539$	−3.12311	−0.134522
$540$	0	0
$541$	12.6307	0.543035	0.271518	−	0.962433i	$-0.412474\pi$
$541$	12.6307	0.543035	0.271518	+	0.962433i	$0.412474\pi$
$542$	0	0
$543$	−12.2462	−0.525535
$544$	0	0
$545$	−15.1231	−0.647803
$546$	0	0
$547$	−9.56155	−0.408822	−0.204411	−	0.978885i	$-0.565528\pi$
$547$	−9.56155	−0.408822	−0.204411	+	0.978885i	$0.565528\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−11.8078	−0.503028
$552$	0	0
$553$	−0.684658	−0.0291146
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−5.12311	−0.217073	−0.108536	−	0.994092i	$-0.534616\pi$
$557$	−5.12311	−0.217073	−0.108536	+	0.994092i	$0.534616\pi$
$558$	0	0
$559$	1.56155	0.0660466
$560$	0	0
$561$	3.50758	0.148090
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	23.8078	1.00160
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	10.6847	0.447924	0.223962	−	0.974598i	$-0.428101\pi$
$569$	10.6847	0.447924	0.223962	+	0.974598i	$0.428101\pi$
$570$	0	0
$571$	6.05398	0.253351	0.126675	−	0.991944i	$-0.459569\pi$
$571$	6.05398	0.253351	0.126675	+	0.991944i	$0.459569\pi$
$572$	0	0
$573$	18.2462	0.762246
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−23.7538	−0.988883	−0.494442	−	0.869211i	$-0.664627\pi$
$577$	−23.7538	−0.988883	−0.494442	+	0.869211i	$0.664627\pi$
$578$	0	0
$579$	4.24621	0.176467
$580$	0	0
$581$	12.6847	0.526248
$582$	0	0
$583$	4.10795	0.170134
$584$	0	0
$585$	−3.56155	−0.147252
$586$	0	0
$587$	−19.3153	−0.797229	−0.398615	−	0.917118i	$-0.630509\pi$
$587$	−19.3153	−0.797229	−0.398615	+	0.917118i	$0.630509\pi$
$588$	0	0
$589$	8.68466	0.357845
$590$	0	0
$591$	−4.24621	−0.174666
$592$	0	0
$593$	−3.06913	−0.126034	−0.0630170	−	0.998012i	$-0.520072\pi$
$593$	−3.06913	−0.126034	−0.0630170	+	0.998012i	$0.520072\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	14.2462	0.583059
$598$	0	0
$599$	25.5616	1.04442	0.522208	−	0.852818i	$-0.325108\pi$
$599$	25.5616	1.04442	0.522208	+	0.852818i	$0.325108\pi$
$600$	0	0
$601$	34.9848	1.42706	0.713531	−	0.700624i	$-0.247095\pi$
$601$	34.9848	1.42706	0.713531	+	0.700624i	$0.247095\pi$
$602$	0	0
$603$	−0.876894	−0.0357099
$604$	0	0
$605$	4.43845	0.180449
$606$	0	0
$607$	17.3693	0.704999	0.352499	−	0.935812i	$-0.385332\pi$
$607$	17.3693	0.704999	0.352499	+	0.935812i	$0.385332\pi$
$608$	0	0
$609$	7.56155	0.306410
$610$	0	0
$611$	8.68466	0.351344
$612$	0	0
$613$	33.1231	1.33783	0.668915	−	0.743339i	$-0.266759\pi$
$613$	33.1231	1.33783	0.668915	+	0.743339i	$0.266759\pi$
$614$	0	0
$615$	−7.12311	−0.287231
$616$	0	0
$617$	−8.73863	−0.351804	−0.175902	−	0.984408i	$-0.556284\pi$
$617$	−8.73863	−0.351804	−0.175902	+	0.984408i	$0.556284\pi$
$618$	0	0
$619$	32.4924	1.30598	0.652990	−	0.757366i	$-0.273514\pi$
$619$	32.4924	1.30598	0.652990	+	0.757366i	$0.273514\pi$
$620$	0	0
$621$	−1.56155	−0.0626630
$622$	0	0
$623$	−10.6847	−0.428072
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	4.87689	0.194764
$628$	0	0
$629$	1.26137	0.0502940
$630$	0	0
$631$	20.8769	0.831096	0.415548	−	0.909571i	$-0.363590\pi$
$631$	20.8769	0.831096	0.415548	+	0.909571i	$0.363590\pi$
$632$	0	0
$633$	0.192236	0.00764069
$634$	0	0
$635$	50.7386	2.01350
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	10.2462	0.405334
$640$	0	0
$641$	−36.9309	−1.45868	−0.729341	−	0.684151i	$-0.760173\pi$
$641$	−36.9309	−1.45868	−0.729341	+	0.684151i	$0.760173\pi$
$642$	0	0
$643$	29.7538	1.17337	0.586687	−	0.809813i	$-0.300432\pi$
$643$	29.7538	1.17337	0.586687	+	0.809813i	$0.300432\pi$
$644$	0	0
$645$	5.56155	0.218986
$646$	0	0
$647$	7.61553	0.299397	0.149699	−	0.988732i	$-0.452170\pi$
$647$	7.61553	0.299397	0.149699	+	0.988732i	$0.452170\pi$
$648$	0	0
$649$	7.01515	0.275369
$650$	0	0
$651$	−5.56155	−0.217974
$652$	0	0
$653$	−16.2462	−0.635763	−0.317882	−	0.948130i	$-0.602971\pi$
$653$	−16.2462	−0.635763	−0.317882	+	0.948130i	$0.602971\pi$
$654$	0	0
$655$	14.2462	0.556646
$656$	0	0
$657$	−0.438447	−0.0171055
$658$	0	0
$659$	−27.8078	−1.08324	−0.541618	−	0.840625i	$-0.682188\pi$
$659$	−27.8078	−1.08324	−0.541618	+	0.840625i	$0.682188\pi$
$660$	0	0
$661$	30.3002	1.17854	0.589270	−	0.807936i	$-0.299415\pi$
$661$	30.3002	1.17854	0.589270	+	0.807936i	$0.299415\pi$
$662$	0	0
$663$	−1.12311	−0.0436178
$664$	0	0
$665$	−5.56155	−0.215668
$666$	0	0
$667$	−11.8078	−0.457198
$668$	0	0
$669$	−8.68466	−0.335768
$670$	0	0
$671$	−18.7386	−0.723397
$672$	0	0
$673$	12.0540	0.464647	0.232323	−	0.972639i	$-0.425367\pi$
$673$	12.0540	0.464647	0.232323	+	0.972639i	$0.425367\pi$
$674$	0	0
$675$	−7.68466	−0.295783
$676$	0	0
$677$	−28.2462	−1.08559	−0.542795	−	0.839865i	$-0.682634\pi$
$677$	−28.2462	−1.08559	−0.542795	+	0.839865i	$0.682634\pi$
$678$	0	0
$679$	−14.6847	−0.563545
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	24.9848	0.956019	0.478009	−	0.878355i	$-0.341359\pi$
$683$	24.9848	0.956019	0.478009	+	0.878355i	$0.341359\pi$
$684$	0	0
$685$	38.7386	1.48013
$686$	0	0
$687$	−24.7386	−0.943838
$688$	0	0
$689$	−1.31534	−0.0501105
$690$	0	0
$691$	−17.5616	−0.668073	−0.334036	−	0.942560i	$-0.608411\pi$
$691$	−17.5616	−0.668073	−0.334036	+	0.942560i	$0.608411\pi$
$692$	0	0
$693$	−3.12311	−0.118637
$694$	0	0
$695$	−53.8617	−2.04309
$696$	0	0
$697$	−2.24621	−0.0850813
$698$	0	0
$699$	−23.1771	−0.876638
$700$	0	0
$701$	0.822919	0.0310812	0.0155406	−	0.999879i	$-0.495053\pi$
$701$	0.822919	0.0310812	0.0155406	+	0.999879i	$0.495053\pi$
$702$	0	0
$703$	1.75379	0.0661454
$704$	0	0
$705$	30.9309	1.16492
$706$	0	0
$707$	−18.4924	−0.695479
$708$	0	0
$709$	30.9848	1.16366	0.581830	−	0.813310i	$-0.302337\pi$
$709$	30.9848	1.16366	0.581830	+	0.813310i	$0.302337\pi$
$710$	0	0
$711$	−0.684658	−0.0256767
$712$	0	0
$713$	8.68466	0.325243
$714$	0	0
$715$	11.1231	0.415981
$716$	0	0
$717$	8.49242	0.317155
$718$	0	0
$719$	4.87689	0.181877	0.0909387	−	0.995856i	$-0.471013\pi$
$719$	4.87689	0.181877	0.0909387	+	0.995856i	$0.471013\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	27.1771	1.01073
$724$	0	0
$725$	−58.1080	−2.15808
$726$	0	0
$727$	−29.8617	−1.10751	−0.553755	−	0.832679i	$-0.686806\pi$
$727$	−29.8617	−1.10751	−0.553755	+	0.832679i	$0.686806\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.75379	0.0648662
$732$	0	0
$733$	−13.4233	−0.495801	−0.247900	−	0.968786i	$-0.579741\pi$
$733$	−13.4233	−0.495801	−0.247900	+	0.968786i	$0.579741\pi$
$734$	0	0
$735$	3.56155	0.131370
$736$	0	0
$737$	2.73863	0.100879
$738$	0	0
$739$	−29.3693	−1.08037	−0.540184	−	0.841547i	$-0.681645\pi$
$739$	−29.3693	−1.08037	−0.540184	+	0.841547i	$0.681645\pi$
$740$	0	0
$741$	−1.56155	−0.0573651
$742$	0	0
$743$	−18.2462	−0.669389	−0.334694	−	0.942327i	$-0.608633\pi$
$743$	−18.2462	−0.669389	−0.334694	+	0.942327i	$0.608633\pi$
$744$	0	0
$745$	64.1080	2.34873
$746$	0	0
$747$	12.6847	0.464107
$748$	0	0
$749$	6.24621	0.228232
$750$	0	0
$751$	−30.9309	−1.12868	−0.564342	−	0.825541i	$-0.690870\pi$
$751$	−30.9309	−1.12868	−0.564342	+	0.825541i	$0.690870\pi$
$752$	0	0
$753$	−0.876894	−0.0319558
$754$	0	0
$755$	28.4924	1.03695
$756$	0	0
$757$	35.5616	1.29251	0.646253	−	0.763123i	$-0.276335\pi$
$757$	35.5616	1.29251	0.646253	+	0.763123i	$0.276335\pi$
$758$	0	0
$759$	4.87689	0.177020
$760$	0	0
$761$	−9.31534	−0.337681	−0.168840	−	0.985643i	$-0.554002\pi$
$761$	−9.31534	−0.337681	−0.168840	+	0.985643i	$0.554002\pi$
$762$	0	0
$763$	4.24621	0.153723
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	−2.24621	−0.0811060
$768$	0	0
$769$	−30.3002	−1.09265	−0.546326	−	0.837572i	$-0.683974\pi$
$769$	−30.3002	−1.09265	−0.546326	+	0.837572i	$0.683974\pi$
$770$	0	0
$771$	24.2462	0.873206
$772$	0	0
$773$	30.4924	1.09674	0.548368	−	0.836237i	$-0.315249\pi$
$773$	30.4924	1.09674	0.548368	+	0.836237i	$0.315249\pi$
$774$	0	0
$775$	42.7386	1.53522
$776$	0	0
$777$	−1.12311	−0.0402912
$778$	0	0
$779$	−3.12311	−0.111897
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	7.56155	0.270228
$784$	0	0
$785$	40.4924	1.44524
$786$	0	0
$787$	−31.4233	−1.12012	−0.560060	−	0.828452i	$-0.689222\pi$
$787$	−31.4233	−1.12012	−0.560060	+	0.828452i	$0.689222\pi$
$788$	0	0
$789$	3.31534	0.118029
$790$	0	0
$791$	−6.68466	−0.237679
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	−4.68466	−0.166148
$796$	0	0
$797$	−4.24621	−0.150409	−0.0752043	−	0.997168i	$-0.523961\pi$
$797$	−4.24621	−0.150409	−0.0752043	+	0.997168i	$0.523961\pi$
$798$	0	0
$799$	9.75379	0.345064
$800$	0	0
$801$	−10.6847	−0.377524
$802$	0	0
$803$	1.36932	0.0483221
$804$	0	0
$805$	−5.56155	−0.196019
$806$	0	0
$807$	10.8769	0.382885
$808$	0	0
$809$	4.05398	0.142530	0.0712651	−	0.997457i	$-0.477296\pi$
$809$	4.05398	0.142530	0.0712651	+	0.997457i	$0.477296\pi$
$810$	0	0
$811$	7.50758	0.263627	0.131813	−	0.991275i	$-0.457920\pi$
$811$	7.50758	0.263627	0.131813	+	0.991275i	$0.457920\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	64.9848	2.27632
$816$	0	0
$817$	2.43845	0.0853105
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−14.4924	−0.505789	−0.252895	−	0.967494i	$-0.581383\pi$
$821$	−14.4924	−0.505789	−0.252895	+	0.967494i	$0.581383\pi$
$822$	0	0
$823$	−24.9848	−0.870917	−0.435458	−	0.900209i	$-0.643414\pi$
$823$	−24.9848	−0.870917	−0.435458	+	0.900209i	$0.643414\pi$
$824$	0	0
$825$	24.0000	0.835573
$826$	0	0
$827$	20.4924	0.712591	0.356296	−	0.934373i	$-0.384040\pi$
$827$	20.4924	0.712591	0.356296	+	0.934373i	$0.384040\pi$
$828$	0	0
$829$	−56.2462	−1.95351	−0.976756	−	0.214355i	$-0.931235\pi$
$829$	−56.2462	−1.95351	−0.976756	+	0.214355i	$0.931235\pi$
$830$	0	0
$831$	8.93087	0.309808
$832$	0	0
$833$	1.12311	0.0389133
$834$	0	0
$835$	−76.7926	−2.65752
$836$	0	0
$837$	−5.56155	−0.192235
$838$	0	0
$839$	−32.9848	−1.13876	−0.569382	−	0.822073i	$-0.692817\pi$
$839$	−32.9848	−1.13876	−0.569382	+	0.822073i	$0.692817\pi$
$840$	0	0
$841$	28.1771	0.971623
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	−3.56155	−0.122521
$846$	0	0
$847$	−1.24621	−0.0428203
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	1.75379	0.0601191
$852$	0	0
$853$	−46.7926	−1.60215	−0.801074	−	0.598565i	$-0.795738\pi$
$853$	−46.7926	−1.60215	−0.801074	+	0.598565i	$0.795738\pi$
$854$	0	0
$855$	−5.56155	−0.190201
$856$	0	0
$857$	−34.9848	−1.19506	−0.597530	−	0.801847i	$-0.703851\pi$
$857$	−34.9848	−1.19506	−0.597530	+	0.801847i	$0.703851\pi$
$858$	0	0
$859$	26.6307	0.908627	0.454314	−	0.890842i	$-0.349885\pi$
$859$	26.6307	0.908627	0.454314	+	0.890842i	$0.349885\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	−8.49242	−0.289085	−0.144543	−	0.989499i	$-0.546171\pi$
$863$	−8.49242	−0.289085	−0.144543	+	0.989499i	$0.546171\pi$
$864$	0	0
$865$	−46.7386	−1.58916
$866$	0	0
$867$	15.7386	0.534512
$868$	0	0
$869$	2.13826	0.0725355
$870$	0	0
$871$	−0.876894	−0.0297124
$872$	0	0
$873$	−14.6847	−0.497000
$874$	0	0
$875$	−9.56155	−0.323239
$876$	0	0
$877$	−22.8769	−0.772498	−0.386249	−	0.922395i	$-0.626229\pi$
$877$	−22.8769	−0.772498	−0.386249	+	0.922395i	$0.626229\pi$
$878$	0	0
$879$	11.5616	0.389961
$880$	0	0
$881$	−3.75379	−0.126468	−0.0632342	−	0.997999i	$-0.520142\pi$
$881$	−3.75379	−0.126468	−0.0632342	+	0.997999i	$0.520142\pi$
$882$	0	0
$883$	−0.492423	−0.0165713	−0.00828567	−	0.999966i	$-0.502637\pi$
$883$	−0.492423	−0.0165713	−0.00828567	+	0.999966i	$0.502637\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−2.73863	−0.0919543	−0.0459772	−	0.998942i	$-0.514640\pi$
$887$	−2.73863	−0.0919543	−0.0459772	+	0.998942i	$0.514640\pi$
$888$	0	0
$889$	−14.2462	−0.477803
$890$	0	0
$891$	−3.12311	−0.104628
$892$	0	0
$893$	13.5616	0.453820
$894$	0	0
$895$	87.9157	2.93870
$896$	0	0
$897$	−1.56155	−0.0521387
$898$	0	0
$899$	−42.0540	−1.40258
$900$	0	0
$901$	−1.47727	−0.0492149
$902$	0	0
$903$	−1.56155	−0.0519652
$904$	0	0
$905$	−43.6155	−1.44983
$906$	0	0
$907$	−1.94602	−0.0646167	−0.0323083	−	0.999478i	$-0.510286\pi$
$907$	−1.94602	−0.0646167	−0.0323083	+	0.999478i	$0.510286\pi$
$908$	0	0
$909$	−18.4924	−0.613355
$910$	0	0
$911$	41.1771	1.36426	0.682129	−	0.731232i	$-0.261054\pi$
$911$	41.1771	1.36426	0.682129	+	0.731232i	$0.261054\pi$
$912$	0	0
$913$	−39.6155	−1.31108
$914$	0	0
$915$	21.3693	0.706448
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	52.4924	1.73157	0.865783	−	0.500420i	$-0.166821\pi$
$919$	52.4924	1.73157	0.865783	+	0.500420i	$0.166821\pi$
$920$	0	0
$921$	−0.192236	−0.00633439
$922$	0	0
$923$	10.2462	0.337258
$924$	0	0
$925$	8.63068	0.283775
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.5464	−1.06781	−0.533906	−	0.845544i	$-0.679277\pi$
$929$	−32.5464	−1.06781	−0.533906	+	0.845544i	$0.679277\pi$
$930$	0	0
$931$	1.56155	0.0511778
$932$	0	0
$933$	11.1231	0.364154
$934$	0	0
$935$	12.4924	0.408546
$936$	0	0
$937$	−49.1231	−1.60478	−0.802391	−	0.596799i	$-0.796439\pi$
$937$	−49.1231	−1.60478	−0.802391	+	0.596799i	$0.796439\pi$
$938$	0	0
$939$	−3.75379	−0.122500
$940$	0	0
$941$	28.0540	0.914533	0.457267	−	0.889330i	$-0.348828\pi$
$941$	28.0540	0.914533	0.457267	+	0.889330i	$0.348828\pi$
$942$	0	0
$943$	−3.12311	−0.101702
$944$	0	0
$945$	3.56155	0.115857
$946$	0	0
$947$	−39.6155	−1.28733	−0.643666	−	0.765307i	$-0.722587\pi$
$947$	−39.6155	−1.28733	−0.643666	+	0.765307i	$0.722587\pi$
$948$	0	0
$949$	−0.438447	−0.0142326
$950$	0	0
$951$	−7.36932	−0.238966
$952$	0	0
$953$	11.0691	0.358564	0.179282	−	0.983798i	$-0.442623\pi$
$953$	11.0691	0.358564	0.179282	+	0.983798i	$0.442623\pi$
$954$	0	0
$955$	64.9848	2.10286
$956$	0	0
$957$	−23.6155	−0.763382
$958$	0	0
$959$	−10.8769	−0.351233
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	6.24621	0.201281
$964$	0	0
$965$	15.1231	0.486830
$966$	0	0
$967$	17.3693	0.558560	0.279280	−	0.960210i	$-0.409904\pi$
$967$	17.3693	0.558560	0.279280	+	0.960210i	$0.409904\pi$
$968$	0	0
$969$	−1.75379	−0.0563398
$970$	0	0
$971$	−10.6307	−0.341155	−0.170577	−	0.985344i	$-0.554563\pi$
$971$	−10.6307	−0.341155	−0.170577	+	0.985344i	$0.554563\pi$
$972$	0	0
$973$	15.1231	0.484825
$974$	0	0
$975$	−7.68466	−0.246106
$976$	0	0
$977$	−39.7538	−1.27184	−0.635918	−	0.771756i	$-0.719378\pi$
$977$	−39.7538	−1.27184	−0.635918	+	0.771756i	$0.719378\pi$
$978$	0	0
$979$	33.3693	1.06649
$980$	0	0
$981$	4.24621	0.135571
$982$	0	0
$983$	−5.17708	−0.165123	−0.0825616	−	0.996586i	$-0.526310\pi$
$983$	−5.17708	−0.165123	−0.0825616	+	0.996586i	$0.526310\pi$
$984$	0	0
$985$	−15.1231	−0.481862
$986$	0	0
$987$	−8.68466	−0.276436
$988$	0	0
$989$	2.43845	0.0775381
$990$	0	0
$991$	−38.2462	−1.21493	−0.607465	−	0.794346i	$-0.707814\pi$
$991$	−38.2462	−1.21493	−0.607465	+	0.794346i	$0.707814\pi$
$992$	0	0
$993$	13.3693	0.424262
$994$	0	0
$995$	50.7386	1.60852
$996$	0	0
$997$	−57.6155	−1.82470	−0.912351	−	0.409409i	$-0.865735\pi$
$997$	−57.6155	−1.82470	−0.912351	+	0.409409i	$0.865735\pi$
$998$	0	0
$999$	−1.12311	−0.0355335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bb.1.1		2
4.3	odd	2		2184.2.a.q.1.1	✓	2
12.11	even	2		6552.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.q.1.1	✓	2	4.3	odd	2
4368.2.a.bb.1.1		2	1.1	even	1	trivial
6552.2.a.bl.1.2		2	12.11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.12311 q^{11} +1.00000 q^{13} +3.56155 q^{15} +1.12311 q^{17} +1.56155 q^{19} -1.00000 q^{21} +1.56155 q^{23} +7.68466 q^{25} -1.00000 q^{27} -7.56155 q^{29} +5.56155 q^{31} +3.12311 q^{33} -3.56155 q^{35} +1.12311 q^{37} -1.00000 q^{39} -2.00000 q^{41} +1.56155 q^{43} -3.56155 q^{45} +8.68466 q^{47} +1.00000 q^{49} -1.12311 q^{51} -1.31534 q^{53} +11.1231 q^{55} -1.56155 q^{57} -2.24621 q^{59} +6.00000 q^{61} +1.00000 q^{63} -3.56155 q^{65} -0.876894 q^{67} -1.56155 q^{69} +10.2462 q^{71} -0.438447 q^{73} -7.68466 q^{75} -3.12311 q^{77} -0.684658 q^{79} +1.00000 q^{81} +12.6847 q^{83} -4.00000 q^{85} +7.56155 q^{87} -10.6847 q^{89} +1.00000 q^{91} -5.56155 q^{93} -5.56155 q^{95} -14.6847 q^{97} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 3 q^{15} - 6 q^{17} - q^{19} - 2 q^{21} - q^{23} + 3 q^{25} - 2 q^{27} - 11 q^{29} + 7 q^{31} - 2 q^{33} - 3 q^{35} - 6 q^{37} - 2 q^{39} - 4 q^{41} - q^{43} - 3 q^{45} + 5 q^{47} + 2 q^{49} + 6 q^{51} - 15 q^{53} + 14 q^{55} + q^{57} + 12 q^{59} + 12 q^{61} + 2 q^{63} - 3 q^{65} - 10 q^{67} + q^{69} + 4 q^{71} - 5 q^{73} - 3 q^{75} + 2 q^{77} + 11 q^{79} + 2 q^{81} + 13 q^{83} - 8 q^{85} + 11 q^{87} - 9 q^{89} + 2 q^{91} - 7 q^{93} - 7 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 3 * q^15 - 6 * q^17 - q^19 - 2 * q^21 - q^23 + 3 * q^25 - 2 * q^27 - 11 * q^29 + 7 * q^31 - 2 * q^33 - 3 * q^35 - 6 * q^37 - 2 * q^39 - 4 * q^41 - q^43 - 3 * q^45 + 5 * q^47 + 2 * q^49 + 6 * q^51 - 15 * q^53 + 14 * q^55 + q^57 + 12 * q^59 + 12 * q^61 + 2 * q^63 - 3 * q^65 - 10 * q^67 + q^69 + 4 * q^71 - 5 * q^73 - 3 * q^75 + 2 * q^77 + 11 * q^79 + 2 * q^81 + 13 * q^83 - 8 * q^85 + 11 * q^87 - 9 * q^89 + 2 * q^91 - 7 * q^93 - 7 * q^95 - 17 * q^97 + 2 * q^99

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