LMFDB - Embedded newform 3920.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +4.82843 q^{11} +2.00000 q^{13} +2.41421 q^{15} +3.65685 q^{17} -5.65685 q^{19} +8.41421 q^{23} +1.00000 q^{25} -0.414214 q^{27} -2.17157 q^{29} +4.82843 q^{31} +11.6569 q^{33} -5.65685 q^{37} +4.82843 q^{39} +0.171573 q^{41} -12.8995 q^{43} +2.82843 q^{45} -0.343146 q^{47} +8.82843 q^{51} +5.65685 q^{53} +4.82843 q^{55} -13.6569 q^{57} +4.00000 q^{59} -4.65685 q^{61} +2.00000 q^{65} -6.89949 q^{67} +20.3137 q^{69} +12.0000 q^{71} -7.65685 q^{73} +2.41421 q^{75} +4.00000 q^{79} -9.48528 q^{81} +13.2426 q^{83} +3.65685 q^{85} -5.24264 q^{87} -16.6569 q^{89} +11.6569 q^{93} -5.65685 q^{95} +6.00000 q^{97} +13.6569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{11} + 4 q^{13} + 2 q^{15} - 4 q^{17} + 14 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 4 q^{31} + 12 q^{33} + 4 q^{39} + 6 q^{41} - 6 q^{43} - 12 q^{47} + 12 q^{51} + 4 q^{55} - 16 q^{57} + 8 q^{59} + 2 q^{61} + 4 q^{65} + 6 q^{67} + 18 q^{69} + 24 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{79} - 2 q^{81} + 18 q^{83} - 4 q^{85} - 2 q^{87} - 22 q^{89} + 12 q^{93} + 12 q^{97} + 16 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^11 + 4 * q^13 + 2 * q^15 - 4 * q^17 + 14 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 4 * q^31 + 12 * q^33 + 4 * q^39 + 6 * q^41 - 6 * q^43 - 12 * q^47 + 12 * q^51 + 4 * q^55 - 16 * q^57 + 8 * q^59 + 2 * q^61 + 4 * q^65 + 6 * q^67 + 18 * q^69 + 24 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^79 - 2 * q^81 + 18 * q^83 - 4 * q^85 - 2 * q^87 - 22 * q^89 + 12 * q^93 + 12 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.41421	0.623347
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.41421	1.75448	0.877242	−	0.480048i	$-0.159381\pi$
$23$	8.41421	1.75448	0.877242	+	0.480048i	$0.159381\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	−2.17157	−0.403251	−0.201625	−	0.979463i	$-0.564622\pi$
$29$	−2.17157	−0.403251	−0.201625	+	0.979463i	$0.564622\pi$
$30$	0	0
$31$	4.82843	0.867211	0.433606	−	0.901103i	$-0.357241\pi$
$31$	4.82843	0.867211	0.433606	+	0.901103i	$0.357241\pi$
$32$	0	0
$33$	11.6569	2.02920
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.65685	−0.929981	−0.464991	−	0.885316i	$-0.653942\pi$
$37$	−5.65685	−0.929981	−0.464991	+	0.885316i	$0.653942\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	0.171573	0.0267952	0.0133976	−	0.999910i	$-0.495735\pi$
$41$	0.171573	0.0267952	0.0133976	+	0.999910i	$0.495735\pi$
$42$	0	0
$43$	−12.8995	−1.96715	−0.983577	−	0.180488i	$-0.942232\pi$
$43$	−12.8995	−1.96715	−0.983577	+	0.180488i	$0.942232\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.82843	1.23623
$52$	0	0
$53$	5.65685	0.777029	0.388514	−	0.921443i	$-0.372988\pi$
$53$	5.65685	0.777029	0.388514	+	0.921443i	$0.372988\pi$
$54$	0	0
$55$	4.82843	0.651065
$56$	0	0
$57$	−13.6569	−1.80889
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−4.65685	−0.596249	−0.298125	−	0.954527i	$-0.596361\pi$
$61$	−4.65685	−0.596249	−0.298125	+	0.954527i	$0.596361\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−6.89949	−0.842907	−0.421454	−	0.906850i	$-0.638480\pi$
$67$	−6.89949	−0.842907	−0.421454	+	0.906850i	$0.638480\pi$
$68$	0	0
$69$	20.3137	2.44548
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	0	0
$75$	2.41421	0.278769
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	13.2426	1.45357	0.726784	−	0.686866i	$-0.241014\pi$
$83$	13.2426	1.45357	0.726784	+	0.686866i	$0.241014\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	−5.24264	−0.562070
$88$	0	0
$89$	−16.6569	−1.76562	−0.882812	−	0.469727i	$-0.844352\pi$
$89$	−16.6569	−1.76562	−0.882812	+	0.469727i	$0.844352\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.6569	1.20876
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	13.6569	1.37257
$100$	0	0
$101$	5.48528	0.545806	0.272903	−	0.962042i	$-0.412016\pi$
$101$	5.48528	0.545806	0.272903	+	0.962042i	$0.412016\pi$
$102$	0	0
$103$	−10.4142	−1.02614	−0.513071	−	0.858346i	$-0.671492\pi$
$103$	−10.4142	−1.02614	−0.513071	+	0.858346i	$0.671492\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.41421	0.813433	0.406716	−	0.913554i	$-0.366674\pi$
$107$	8.41421	0.813433	0.406716	+	0.913554i	$0.366674\pi$
$108$	0	0
$109$	−4.31371	−0.413178	−0.206589	−	0.978428i	$-0.566236\pi$
$109$	−4.31371	−0.413178	−0.206589	+	0.978428i	$0.566236\pi$
$110$	0	0
$111$	−13.6569	−1.29625
$112$	0	0
$113$	−11.3137	−1.06430	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	−11.3137	−1.06430	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	8.41421	0.784629
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	0.414214	0.0373484
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	0	0
$129$	−31.1421	−2.74191
$130$	0	0
$131$	−2.34315	−0.204722	−0.102361	−	0.994747i	$-0.532640\pi$
$131$	−2.34315	−0.204722	−0.102361	+	0.994747i	$0.532640\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.414214	−0.0356498
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	14.4853	1.22863	0.614313	−	0.789063i	$-0.289433\pi$
$139$	14.4853	1.22863	0.614313	+	0.789063i	$0.289433\pi$
$140$	0	0
$141$	−0.828427	−0.0697661
$142$	0	0
$143$	9.65685	0.807547
$144$	0	0
$145$	−2.17157	−0.180339
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.65685	−0.545351	−0.272675	−	0.962106i	$-0.587908\pi$
$149$	−6.65685	−0.545351	−0.272675	+	0.962106i	$0.587908\pi$
$150$	0	0
$151$	16.8284	1.36948	0.684739	−	0.728788i	$-0.259916\pi$
$151$	16.8284	1.36948	0.684739	+	0.728788i	$0.259916\pi$
$152$	0	0
$153$	10.3431	0.836194
$154$	0	0
$155$	4.82843	0.387829
$156$	0	0
$157$	−21.3137	−1.70102	−0.850510	−	0.525960i	$-0.823706\pi$
$157$	−21.3137	−1.70102	−0.850510	+	0.525960i	$0.823706\pi$
$158$	0	0
$159$	13.6569	1.08306
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.34315	0.340181	0.170091	−	0.985428i	$-0.445594\pi$
$163$	4.34315	0.340181	0.170091	+	0.985428i	$0.445594\pi$
$164$	0	0
$165$	11.6569	0.907485
$166$	0	0
$167$	12.0711	0.934087	0.467044	−	0.884234i	$-0.345319\pi$
$167$	12.0711	0.934087	0.467044	+	0.884234i	$0.345319\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−16.0000	−1.22355
$172$	0	0
$173$	21.6569	1.64654	0.823270	−	0.567650i	$-0.192147\pi$
$173$	21.6569	1.64654	0.823270	+	0.567650i	$0.192147\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.65685	0.725854
$178$	0	0
$179$	−10.4853	−0.783707	−0.391853	−	0.920028i	$-0.628166\pi$
$179$	−10.4853	−0.783707	−0.391853	+	0.920028i	$0.628166\pi$
$180$	0	0
$181$	−9.82843	−0.730541	−0.365271	−	0.930901i	$-0.619024\pi$
$181$	−9.82843	−0.730541	−0.365271	+	0.930901i	$0.619024\pi$
$182$	0	0
$183$	−11.2426	−0.831080
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	17.6569	1.29120
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.4853	1.62698	0.813489	−	0.581580i	$-0.197565\pi$
$191$	22.4853	1.62698	0.813489	+	0.581580i	$0.197565\pi$
$192$	0	0
$193$	17.3137	1.24627	0.623134	−	0.782115i	$-0.285859\pi$
$193$	17.3137	1.24627	0.623134	+	0.782115i	$0.285859\pi$
$194$	0	0
$195$	4.82843	0.345771
$196$	0	0
$197$	11.6569	0.830516	0.415258	−	0.909704i	$-0.363691\pi$
$197$	11.6569	0.830516	0.415258	+	0.909704i	$0.363691\pi$
$198$	0	0
$199$	0.686292	0.0486499	0.0243250	−	0.999704i	$-0.492256\pi$
$199$	0.686292	0.0486499	0.0243250	+	0.999704i	$0.492256\pi$
$200$	0	0
$201$	−16.6569	−1.17488
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.171573	0.0119832
$206$	0	0
$207$	23.7990	1.65414
$208$	0	0
$209$	−27.3137	−1.88933
$210$	0	0
$211$	18.6274	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$211$	18.6274	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$212$	0	0
$213$	28.9706	1.98503
$214$	0	0
$215$	−12.8995	−0.879738
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−18.4853	−1.24912
$220$	0	0
$221$	7.31371	0.491973
$222$	0	0
$223$	−18.9706	−1.27036	−0.635181	−	0.772363i	$-0.719075\pi$
$223$	−18.9706	−1.27036	−0.635181	+	0.772363i	$0.719075\pi$
$224$	0	0
$225$	2.82843	0.188562
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.343146	0.0224802	0.0112401	−	0.999937i	$-0.496422\pi$
$233$	0.343146	0.0224802	0.0112401	+	0.999937i	$0.496422\pi$
$234$	0	0
$235$	−0.343146	−0.0223844
$236$	0	0
$237$	9.65685	0.627280
$238$	0	0
$239$	13.5147	0.874194	0.437097	−	0.899414i	$-0.356007\pi$
$239$	13.5147	0.874194	0.437097	+	0.899414i	$0.356007\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.3137	−0.719874
$248$	0	0
$249$	31.9706	2.02605
$250$	0	0
$251$	−23.4558	−1.48052	−0.740260	−	0.672321i	$-0.765298\pi$
$251$	−23.4558	−1.48052	−0.740260	+	0.672321i	$0.765298\pi$
$252$	0	0
$253$	40.6274	2.55422
$254$	0	0
$255$	8.82843	0.552858
$256$	0	0
$257$	−7.31371	−0.456217	−0.228108	−	0.973636i	$-0.573254\pi$
$257$	−7.31371	−0.456217	−0.228108	+	0.973636i	$0.573254\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.14214	−0.380189
$262$	0	0
$263$	−9.72792	−0.599849	−0.299925	−	0.953963i	$-0.596962\pi$
$263$	−9.72792	−0.599849	−0.299925	+	0.953963i	$0.596962\pi$
$264$	0	0
$265$	5.65685	0.347498
$266$	0	0
$267$	−40.2132	−2.46101
$268$	0	0
$269$	4.65685	0.283933	0.141967	−	0.989871i	$-0.454657\pi$
$269$	4.65685	0.283933	0.141967	+	0.989871i	$0.454657\pi$
$270$	0	0
$271$	−19.3137	−1.17322	−0.586612	−	0.809868i	$-0.699539\pi$
$271$	−19.3137	−1.17322	−0.586612	+	0.809868i	$0.699539\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.82843	0.291165
$276$	0	0
$277$	−16.6274	−0.999045	−0.499522	−	0.866301i	$-0.666491\pi$
$277$	−16.6274	−0.999045	−0.499522	+	0.866301i	$0.666491\pi$
$278$	0	0
$279$	13.6569	0.817614
$280$	0	0
$281$	25.3137	1.51009	0.755045	−	0.655673i	$-0.227615\pi$
$281$	25.3137	1.51009	0.755045	+	0.655673i	$0.227615\pi$
$282$	0	0
$283$	−18.0000	−1.06999	−0.534994	−	0.844856i	$-0.679686\pi$
$283$	−18.0000	−1.06999	−0.534994	+	0.844856i	$0.679686\pi$
$284$	0	0
$285$	−13.6569	−0.808962
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	14.4853	0.849142
$292$	0	0
$293$	−16.9706	−0.991431	−0.495715	−	0.868485i	$-0.665094\pi$
$293$	−16.9706	−0.991431	−0.495715	+	0.868485i	$0.665094\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	16.8284	0.973213
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.2426	0.760770
$304$	0	0
$305$	−4.65685	−0.266651
$306$	0	0
$307$	−13.2426	−0.755797	−0.377899	−	0.925847i	$-0.623353\pi$
$307$	−13.2426	−0.755797	−0.377899	+	0.925847i	$0.623353\pi$
$308$	0	0
$309$	−25.1421	−1.43029
$310$	0	0
$311$	−10.3431	−0.586506	−0.293253	−	0.956035i	$-0.594738\pi$
$311$	−10.3431	−0.586506	−0.293253	+	0.956035i	$0.594738\pi$
$312$	0	0
$313$	−12.9706	−0.733140	−0.366570	−	0.930391i	$-0.619468\pi$
$313$	−12.9706	−0.733140	−0.366570	+	0.930391i	$0.619468\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−10.4853	−0.587063
$320$	0	0
$321$	20.3137	1.13380
$322$	0	0
$323$	−20.6863	−1.15102
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−10.4142	−0.575907
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.51472	−0.522976	−0.261488	−	0.965207i	$-0.584213\pi$
$331$	−9.51472	−0.522976	−0.261488	+	0.965207i	$0.584213\pi$
$332$	0	0
$333$	−16.0000	−0.876795
$334$	0	0
$335$	−6.89949	−0.376960
$336$	0	0
$337$	−8.97056	−0.488658	−0.244329	−	0.969692i	$-0.578568\pi$
$337$	−8.97056	−0.488658	−0.244329	+	0.969692i	$0.578568\pi$
$338$	0	0
$339$	−27.3137	−1.48348
$340$	0	0
$341$	23.3137	1.26251
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.3137	1.09365
$346$	0	0
$347$	25.3848	1.36273	0.681363	−	0.731946i	$-0.261387\pi$
$347$	25.3848	1.36273	0.681363	+	0.731946i	$0.261387\pi$
$348$	0	0
$349$	4.17157	0.223299	0.111650	−	0.993748i	$-0.464387\pi$
$349$	4.17157	0.223299	0.111650	+	0.993748i	$0.464387\pi$
$350$	0	0
$351$	−0.828427	−0.0442182
$352$	0	0
$353$	22.3431	1.18921	0.594603	−	0.804020i	$-0.297309\pi$
$353$	22.3431	1.18921	0.594603	+	0.804020i	$0.297309\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.4853	−0.553392	−0.276696	−	0.960958i	$-0.589239\pi$
$359$	−10.4853	−0.553392	−0.276696	+	0.960958i	$0.589239\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	29.7279	1.56031
$364$	0	0
$365$	−7.65685	−0.400778
$366$	0	0
$367$	−24.4142	−1.27441	−0.637206	−	0.770694i	$-0.719910\pi$
$367$	−24.4142	−1.27441	−0.637206	+	0.770694i	$0.719910\pi$
$368$	0	0
$369$	0.485281	0.0252627
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	2.41421	0.124669
$376$	0	0
$377$	−4.34315	−0.223683
$378$	0	0
$379$	−27.3137	−1.40301	−0.701505	−	0.712664i	$-0.747488\pi$
$379$	−27.3137	−1.40301	−0.701505	+	0.712664i	$0.747488\pi$
$380$	0	0
$381$	37.7990	1.93650
$382$	0	0
$383$	−18.8995	−0.965719	−0.482860	−	0.875698i	$-0.660402\pi$
$383$	−18.8995	−0.965719	−0.482860	+	0.875698i	$0.660402\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−36.4853	−1.85465
$388$	0	0
$389$	−17.3137	−0.877840	−0.438920	−	0.898526i	$-0.644639\pi$
$389$	−17.3137	−0.877840	−0.438920	+	0.898526i	$0.644639\pi$
$390$	0	0
$391$	30.7696	1.55608
$392$	0	0
$393$	−5.65685	−0.285351
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	20.6274	1.03526	0.517630	−	0.855604i	$-0.326814\pi$
$397$	20.6274	1.03526	0.517630	+	0.855604i	$0.326814\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.68629	−0.483710	−0.241855	−	0.970312i	$-0.577756\pi$
$401$	−9.68629	−0.483710	−0.241855	+	0.970312i	$0.577756\pi$
$402$	0	0
$403$	9.65685	0.481042
$404$	0	0
$405$	−9.48528	−0.471327
$406$	0	0
$407$	−27.3137	−1.35389
$408$	0	0
$409$	3.14214	0.155369	0.0776843	−	0.996978i	$-0.475247\pi$
$409$	3.14214	0.155369	0.0776843	+	0.996978i	$0.475247\pi$
$410$	0	0
$411$	−9.65685	−0.476337
$412$	0	0
$413$	0	0
$414$	0	0
$415$	13.2426	0.650056
$416$	0	0
$417$	34.9706	1.71252
$418$	0	0
$419$	−7.31371	−0.357298	−0.178649	−	0.983913i	$-0.557173\pi$
$419$	−7.31371	−0.357298	−0.178649	+	0.983913i	$0.557173\pi$
$420$	0	0
$421$	38.6569	1.88402	0.942010	−	0.335585i	$-0.108934\pi$
$421$	38.6569	1.88402	0.942010	+	0.335585i	$0.108934\pi$
$422$	0	0
$423$	−0.970563	−0.0471904
$424$	0	0
$425$	3.65685	0.177383
$426$	0	0
$427$	0	0
$428$	0	0
$429$	23.3137	1.12560
$430$	0	0
$431$	−4.82843	−0.232577	−0.116289	−	0.993215i	$-0.537100\pi$
$431$	−4.82843	−0.232577	−0.116289	+	0.993215i	$0.537100\pi$
$432$	0	0
$433$	3.31371	0.159247	0.0796233	−	0.996825i	$-0.474628\pi$
$433$	3.31371	0.159247	0.0796233	+	0.996825i	$0.474628\pi$
$434$	0	0
$435$	−5.24264	−0.251365
$436$	0	0
$437$	−47.5980	−2.27692
$438$	0	0
$439$	29.6569	1.41544	0.707722	−	0.706491i	$-0.249723\pi$
$439$	29.6569	1.41544	0.707722	+	0.706491i	$0.249723\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.4142	0.874886	0.437443	−	0.899246i	$-0.355884\pi$
$443$	18.4142	0.874886	0.437443	+	0.899246i	$0.355884\pi$
$444$	0	0
$445$	−16.6569	−0.789611
$446$	0	0
$447$	−16.0711	−0.760135
$448$	0	0
$449$	9.48528	0.447638	0.223819	−	0.974631i	$-0.428148\pi$
$449$	9.48528	0.447638	0.223819	+	0.974631i	$0.428148\pi$
$450$	0	0
$451$	0.828427	0.0390091
$452$	0	0
$453$	40.6274	1.90884
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.97056	−0.232513	−0.116257	−	0.993219i	$-0.537089\pi$
$457$	−4.97056	−0.232513	−0.116257	+	0.993219i	$0.537089\pi$
$458$	0	0
$459$	−1.51472	−0.0707010
$460$	0	0
$461$	−21.3137	−0.992678	−0.496339	−	0.868129i	$-0.665323\pi$
$461$	−21.3137	−0.992678	−0.496339	+	0.868129i	$0.665323\pi$
$462$	0	0
$463$	4.89949	0.227699	0.113849	−	0.993498i	$-0.463682\pi$
$463$	4.89949	0.227699	0.113849	+	0.993498i	$0.463682\pi$
$464$	0	0
$465$	11.6569	0.540574
$466$	0	0
$467$	−15.8701	−0.734379	−0.367189	−	0.930146i	$-0.619680\pi$
$467$	−15.8701	−0.734379	−0.367189	+	0.930146i	$0.619680\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−51.4558	−2.37096
$472$	0	0
$473$	−62.2843	−2.86383
$474$	0	0
$475$	−5.65685	−0.259554
$476$	0	0
$477$	16.0000	0.732590
$478$	0	0
$479$	18.4853	0.844614	0.422307	−	0.906453i	$-0.361220\pi$
$479$	18.4853	0.844614	0.422307	+	0.906453i	$0.361220\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	−18.2843	−0.828539	−0.414270	−	0.910154i	$-0.635963\pi$
$487$	−18.2843	−0.828539	−0.414270	+	0.910154i	$0.635963\pi$
$488$	0	0
$489$	10.4853	0.474161
$490$	0	0
$491$	−18.4853	−0.834229	−0.417115	−	0.908854i	$-0.636959\pi$
$491$	−18.4853	−0.834229	−0.417115	+	0.908854i	$0.636959\pi$
$492$	0	0
$493$	−7.94113	−0.357650
$494$	0	0
$495$	13.6569	0.613830
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15.8579	−0.709896	−0.354948	−	0.934886i	$-0.615501\pi$
$499$	−15.8579	−0.709896	−0.354948	+	0.934886i	$0.615501\pi$
$500$	0	0
$501$	29.1421	1.30197
$502$	0	0
$503$	18.0711	0.805749	0.402875	−	0.915255i	$-0.368011\pi$
$503$	18.0711	0.805749	0.402875	+	0.915255i	$0.368011\pi$
$504$	0	0
$505$	5.48528	0.244092
$506$	0	0
$507$	−21.7279	−0.964971
$508$	0	0
$509$	−16.5147	−0.732002	−0.366001	−	0.930614i	$-0.619273\pi$
$509$	−16.5147	−0.732002	−0.366001	+	0.930614i	$0.619273\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.34315	0.103452
$514$	0	0
$515$	−10.4142	−0.458905
$516$	0	0
$517$	−1.65685	−0.0728684
$518$	0	0
$519$	52.2843	2.29502
$520$	0	0
$521$	−8.62742	−0.377974	−0.188987	−	0.981980i	$-0.560520\pi$
$521$	−8.62742	−0.377974	−0.188987	+	0.981980i	$0.560520\pi$
$522$	0	0
$523$	39.9411	1.74650	0.873252	−	0.487269i	$-0.162007\pi$
$523$	39.9411	1.74650	0.873252	+	0.487269i	$0.162007\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.6569	0.769145
$528$	0	0
$529$	47.7990	2.07822
$530$	0	0
$531$	11.3137	0.490973
$532$	0	0
$533$	0.343146	0.0148633
$534$	0	0
$535$	8.41421	0.363778
$536$	0	0
$537$	−25.3137	−1.09237
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.51472	−0.280090	−0.140045	−	0.990145i	$-0.544725\pi$
$541$	−6.51472	−0.280090	−0.140045	+	0.990145i	$0.544725\pi$
$542$	0	0
$543$	−23.7279	−1.01826
$544$	0	0
$545$	−4.31371	−0.184779
$546$	0	0
$547$	27.7279	1.18556	0.592780	−	0.805364i	$-0.298030\pi$
$547$	27.7279	1.18556	0.592780	+	0.805364i	$0.298030\pi$
$548$	0	0
$549$	−13.1716	−0.562149
$550$	0	0
$551$	12.2843	0.523328
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13.6569	−0.579701
$556$	0	0
$557$	5.31371	0.225149	0.112575	−	0.993643i	$-0.464090\pi$
$557$	5.31371	0.225149	0.112575	+	0.993643i	$0.464090\pi$
$558$	0	0
$559$	−25.7990	−1.09118
$560$	0	0
$561$	42.6274	1.79973
$562$	0	0
$563$	−10.0711	−0.424445	−0.212222	−	0.977221i	$-0.568070\pi$
$563$	−10.0711	−0.424445	−0.212222	+	0.977221i	$0.568070\pi$
$564$	0	0
$565$	−11.3137	−0.475971
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.62742	0.361680	0.180840	−	0.983513i	$-0.442118\pi$
$569$	8.62742	0.361680	0.180840	+	0.983513i	$0.442118\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	0	0
$573$	54.2843	2.26776
$574$	0	0
$575$	8.41421	0.350897
$576$	0	0
$577$	34.2843	1.42727	0.713636	−	0.700516i	$-0.247047\pi$
$577$	34.2843	1.42727	0.713636	+	0.700516i	$0.247047\pi$
$578$	0	0
$579$	41.7990	1.73711
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27.3137	1.13122
$584$	0	0
$585$	5.65685	0.233882
$586$	0	0
$587$	45.3137	1.87030	0.935148	−	0.354256i	$-0.115266\pi$
$587$	45.3137	1.87030	0.935148	+	0.354256i	$0.115266\pi$
$588$	0	0
$589$	−27.3137	−1.12544
$590$	0	0
$591$	28.1421	1.15761
$592$	0	0
$593$	−37.9411	−1.55806	−0.779028	−	0.626990i	$-0.784287\pi$
$593$	−37.9411	−1.55806	−0.779028	+	0.626990i	$0.784287\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.65685	0.0678105
$598$	0	0
$599$	2.62742	0.107353	0.0536767	−	0.998558i	$-0.482906\pi$
$599$	2.62742	0.107353	0.0536767	+	0.998558i	$0.482906\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	−19.5147	−0.794701
$604$	0	0
$605$	12.3137	0.500623
$606$	0	0
$607$	−18.7574	−0.761338	−0.380669	−	0.924711i	$-0.624306\pi$
$607$	−18.7574	−0.761338	−0.380669	+	0.924711i	$0.624306\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.686292	−0.0277644
$612$	0	0
$613$	5.02944	0.203137	0.101569	−	0.994829i	$-0.467614\pi$
$613$	5.02944	0.203137	0.101569	+	0.994829i	$0.467614\pi$
$614$	0	0
$615$	0.414214	0.0167027
$616$	0	0
$617$	−35.3137	−1.42168	−0.710838	−	0.703356i	$-0.751684\pi$
$617$	−35.3137	−1.42168	−0.710838	+	0.703356i	$0.751684\pi$
$618$	0	0
$619$	11.4558	0.460449	0.230225	−	0.973138i	$-0.426054\pi$
$619$	11.4558	0.460449	0.230225	+	0.973138i	$0.426054\pi$
$620$	0	0
$621$	−3.48528	−0.139860
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−65.9411	−2.63343
$628$	0	0
$629$	−20.6863	−0.824816
$630$	0	0
$631$	18.4853	0.735887	0.367944	−	0.929848i	$-0.380062\pi$
$631$	18.4853	0.735887	0.367944	+	0.929848i	$0.380062\pi$
$632$	0	0
$633$	44.9706	1.78742
$634$	0	0
$635$	15.6569	0.621323
$636$	0	0
$637$	0	0
$638$	0	0
$639$	33.9411	1.34269
$640$	0	0
$641$	−48.1127	−1.90034	−0.950169	−	0.311736i	$-0.899089\pi$
$641$	−48.1127	−1.90034	−0.950169	+	0.311736i	$0.899089\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	−31.1421	−1.22622
$646$	0	0
$647$	29.2426	1.14965	0.574823	−	0.818277i	$-0.305071\pi$
$647$	29.2426	1.14965	0.574823	+	0.818277i	$0.305071\pi$
$648$	0	0
$649$	19.3137	0.758129
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.34315	−0.169960	−0.0849802	−	0.996383i	$-0.527083\pi$
$653$	−4.34315	−0.169960	−0.0849802	+	0.996383i	$0.527083\pi$
$654$	0	0
$655$	−2.34315	−0.0915543
$656$	0	0
$657$	−21.6569	−0.844914
$658$	0	0
$659$	−35.3137	−1.37563	−0.687813	−	0.725888i	$-0.741429\pi$
$659$	−35.3137	−1.37563	−0.687813	+	0.725888i	$0.741429\pi$
$660$	0	0
$661$	−27.6863	−1.07687	−0.538436	−	0.842666i	$-0.680985\pi$
$661$	−27.6863	−1.07687	−0.538436	+	0.842666i	$0.680985\pi$
$662$	0	0
$663$	17.6569	0.685735
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.2721	−0.707498
$668$	0	0
$669$	−45.7990	−1.77069
$670$	0	0
$671$	−22.4853	−0.868035
$672$	0	0
$673$	5.65685	0.218056	0.109028	−	0.994039i	$-0.465226\pi$
$673$	5.65685	0.218056	0.109028	+	0.994039i	$0.465226\pi$
$674$	0	0
$675$	−0.414214	−0.0159431
$676$	0	0
$677$	−10.9706	−0.421633	−0.210816	−	0.977526i	$-0.567612\pi$
$677$	−10.9706	−0.421633	−0.210816	+	0.977526i	$0.567612\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−33.7990	−1.29518
$682$	0	0
$683$	16.0711	0.614942	0.307471	−	0.951557i	$-0.400517\pi$
$683$	16.0711	0.614942	0.307471	+	0.951557i	$0.400517\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	0	0
$687$	−33.7990	−1.28951
$688$	0	0
$689$	11.3137	0.431018
$690$	0	0
$691$	−30.7696	−1.17053	−0.585264	−	0.810842i	$-0.699009\pi$
$691$	−30.7696	−1.17053	−0.585264	+	0.810842i	$0.699009\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.4853	0.549458
$696$	0	0
$697$	0.627417	0.0237651
$698$	0	0
$699$	0.828427	0.0313340
$700$	0	0
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	−0.828427	−0.0312004
$706$	0	0
$707$	0	0
$708$	0	0
$709$	49.4264	1.85625	0.928124	−	0.372272i	$-0.121421\pi$
$709$	49.4264	1.85625	0.928124	+	0.372272i	$0.121421\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	0	0
$713$	40.6274	1.52151
$714$	0	0
$715$	9.65685	0.361146
$716$	0	0
$717$	32.6274	1.21849
$718$	0	0
$719$	−13.7990	−0.514615	−0.257308	−	0.966330i	$-0.582835\pi$
$719$	−13.7990	−0.514615	−0.257308	+	0.966330i	$0.582835\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−24.1421	−0.897856
$724$	0	0
$725$	−2.17157	−0.0806502
$726$	0	0
$727$	−23.9289	−0.887475	−0.443737	−	0.896157i	$-0.646348\pi$
$727$	−23.9289	−0.887475	−0.443737	+	0.896157i	$0.646348\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−47.1716	−1.74470
$732$	0	0
$733$	−3.65685	−0.135069	−0.0675345	−	0.997717i	$-0.521513\pi$
$733$	−3.65685	−0.135069	−0.0675345	+	0.997717i	$0.521513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.3137	−1.22713
$738$	0	0
$739$	−11.1716	−0.410953	−0.205476	−	0.978662i	$-0.565874\pi$
$739$	−11.1716	−0.410953	−0.205476	+	0.978662i	$0.565874\pi$
$740$	0	0
$741$	−27.3137	−1.00339
$742$	0	0
$743$	19.2426	0.705944	0.352972	−	0.935634i	$-0.385171\pi$
$743$	19.2426	0.705944	0.352972	+	0.935634i	$0.385171\pi$
$744$	0	0
$745$	−6.65685	−0.243888
$746$	0	0
$747$	37.4558	1.37044
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	−56.6274	−2.06362
$754$	0	0
$755$	16.8284	0.612449
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	98.0833	3.56020
$760$	0	0
$761$	23.9411	0.867865	0.433933	−	0.900945i	$-0.357126\pi$
$761$	23.9411	0.867865	0.433933	+	0.900945i	$0.357126\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10.3431	0.373957
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−47.2548	−1.70405	−0.852026	−	0.523499i	$-0.824626\pi$
$769$	−47.2548	−1.70405	−0.852026	+	0.523499i	$0.824626\pi$
$770$	0	0
$771$	−17.6569	−0.635896
$772$	0	0
$773$	35.6569	1.28249	0.641244	−	0.767337i	$-0.278419\pi$
$773$	35.6569	1.28249	0.641244	+	0.767337i	$0.278419\pi$
$774$	0	0
$775$	4.82843	0.173442
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.970563	−0.0347740
$780$	0	0
$781$	57.9411	2.07330
$782$	0	0
$783$	0.899495	0.0321453
$784$	0	0
$785$	−21.3137	−0.760719
$786$	0	0
$787$	4.41421	0.157350	0.0786749	−	0.996900i	$-0.474931\pi$
$787$	4.41421	0.157350	0.0786749	+	0.996900i	$0.474931\pi$
$788$	0	0
$789$	−23.4853	−0.836098
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.31371	−0.330739
$794$	0	0
$795$	13.6569	0.484359
$796$	0	0
$797$	−12.6863	−0.449372	−0.224686	−	0.974431i	$-0.572136\pi$
$797$	−12.6863	−0.449372	−0.224686	+	0.974431i	$0.572136\pi$
$798$	0	0
$799$	−1.25483	−0.0443928
$800$	0	0
$801$	−47.1127	−1.66465
$802$	0	0
$803$	−36.9706	−1.30466
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.2426	0.395760
$808$	0	0
$809$	−2.02944	−0.0713512	−0.0356756	−	0.999363i	$-0.511358\pi$
$809$	−2.02944	−0.0713512	−0.0356756	+	0.999363i	$0.511358\pi$
$810$	0	0
$811$	−23.8579	−0.837763	−0.418881	−	0.908041i	$-0.637578\pi$
$811$	−23.8579	−0.837763	−0.418881	+	0.908041i	$0.637578\pi$
$812$	0	0
$813$	−46.6274	−1.63529
$814$	0	0
$815$	4.34315	0.152134
$816$	0	0
$817$	72.9706	2.55292
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.6274	0.440700	0.220350	−	0.975421i	$-0.429280\pi$
$821$	12.6274	0.440700	0.220350	+	0.975421i	$0.429280\pi$
$822$	0	0
$823$	52.0122	1.81303	0.906516	−	0.422172i	$-0.138732\pi$
$823$	52.0122	1.81303	0.906516	+	0.422172i	$0.138732\pi$
$824$	0	0
$825$	11.6569	0.405840
$826$	0	0
$827$	51.5269	1.79177	0.895883	−	0.444290i	$-0.146544\pi$
$827$	51.5269	1.79177	0.895883	+	0.444290i	$0.146544\pi$
$828$	0	0
$829$	−17.3137	−0.601330	−0.300665	−	0.953730i	$-0.597209\pi$
$829$	−17.3137	−0.601330	−0.300665	+	0.953730i	$0.597209\pi$
$830$	0	0
$831$	−40.1421	−1.39252
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.0711	0.417737
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	8.14214	0.281098	0.140549	−	0.990074i	$-0.455113\pi$
$839$	8.14214	0.281098	0.140549	+	0.990074i	$0.455113\pi$
$840$	0	0
$841$	−24.2843	−0.837389
$842$	0	0
$843$	61.1127	2.10483
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−43.4558	−1.49140
$850$	0	0
$851$	−47.5980	−1.63164
$852$	0	0
$853$	35.3137	1.20912	0.604559	−	0.796560i	$-0.293349\pi$
$853$	35.3137	1.20912	0.604559	+	0.796560i	$0.293349\pi$
$854$	0	0
$855$	−16.0000	−0.547188
$856$	0	0
$857$	14.9706	0.511385	0.255692	−	0.966758i	$-0.417697\pi$
$857$	14.9706	0.511385	0.255692	+	0.966758i	$0.417697\pi$
$858$	0	0
$859$	−14.6274	−0.499081	−0.249541	−	0.968364i	$-0.580280\pi$
$859$	−14.6274	−0.499081	−0.249541	+	0.968364i	$0.580280\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.0122	−0.885465	−0.442733	−	0.896654i	$-0.645991\pi$
$863$	−26.0122	−0.885465	−0.442733	+	0.896654i	$0.645991\pi$
$864$	0	0
$865$	21.6569	0.736355
$866$	0	0
$867$	−8.75736	−0.297416
$868$	0	0
$869$	19.3137	0.655173
$870$	0	0
$871$	−13.7990	−0.467561
$872$	0	0
$873$	16.9706	0.574367
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.2843	0.955092	0.477546	−	0.878607i	$-0.341526\pi$
$877$	28.2843	0.955092	0.477546	+	0.878607i	$0.341526\pi$
$878$	0	0
$879$	−40.9706	−1.38190
$880$	0	0
$881$	24.4558	0.823938	0.411969	−	0.911198i	$-0.364841\pi$
$881$	24.4558	0.823938	0.411969	+	0.911198i	$0.364841\pi$
$882$	0	0
$883$	−29.3137	−0.986485	−0.493242	−	0.869892i	$-0.664188\pi$
$883$	−29.3137	−0.986485	−0.493242	+	0.869892i	$0.664188\pi$
$884$	0	0
$885$	9.65685	0.324612
$886$	0	0
$887$	−20.4142	−0.685442	−0.342721	−	0.939437i	$-0.611349\pi$
$887$	−20.4142	−0.685442	−0.342721	+	0.939437i	$0.611349\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−45.7990	−1.53432
$892$	0	0
$893$	1.94113	0.0649573
$894$	0	0
$895$	−10.4853	−0.350484
$896$	0	0
$897$	40.6274	1.35651
$898$	0	0
$899$	−10.4853	−0.349704
$900$	0	0
$901$	20.6863	0.689160
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.82843	−0.326708
$906$	0	0
$907$	−3.78680	−0.125739	−0.0628693	−	0.998022i	$-0.520025\pi$
$907$	−3.78680	−0.125739	−0.0628693	+	0.998022i	$0.520025\pi$
$908$	0	0
$909$	15.5147	0.514591
$910$	0	0
$911$	1.51472	0.0501849	0.0250924	−	0.999685i	$-0.492012\pi$
$911$	1.51472	0.0501849	0.0250924	+	0.999685i	$0.492012\pi$
$912$	0	0
$913$	63.9411	2.11614
$914$	0	0
$915$	−11.2426	−0.371670
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−57.2548	−1.88866	−0.944331	−	0.328996i	$-0.893290\pi$
$919$	−57.2548	−1.88866	−0.944331	+	0.328996i	$0.893290\pi$
$920$	0	0
$921$	−31.9706	−1.05347
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−5.65685	−0.185996
$926$	0	0
$927$	−29.4558	−0.967457
$928$	0	0
$929$	48.7990	1.60104	0.800521	−	0.599304i	$-0.204556\pi$
$929$	48.7990	1.60104	0.800521	+	0.599304i	$0.204556\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.9706	−0.817500
$934$	0	0
$935$	17.6569	0.577441
$936$	0	0
$937$	−26.6274	−0.869880	−0.434940	−	0.900459i	$-0.643230\pi$
$937$	−26.6274	−0.869880	−0.434940	+	0.900459i	$0.643230\pi$
$938$	0	0
$939$	−31.3137	−1.02188
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	1.44365	0.0470117
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.8995	−1.00410	−0.502049	−	0.864839i	$-0.667420\pi$
$947$	−30.8995	−1.00410	−0.502049	+	0.864839i	$0.667420\pi$
$948$	0	0
$949$	−15.3137	−0.497104
$950$	0	0
$951$	−53.1127	−1.72230
$952$	0	0
$953$	−9.37258	−0.303608	−0.151804	−	0.988411i	$-0.548508\pi$
$953$	−9.37258	−0.303608	−0.151804	+	0.988411i	$0.548508\pi$
$954$	0	0
$955$	22.4853	0.727607
$956$	0	0
$957$	−25.3137	−0.818276
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7.68629	−0.247945
$962$	0	0
$963$	23.7990	0.766912
$964$	0	0
$965$	17.3137	0.557348
$966$	0	0
$967$	33.2426	1.06901	0.534506	−	0.845165i	$-0.320498\pi$
$967$	33.2426	1.06901	0.534506	+	0.845165i	$0.320498\pi$
$968$	0	0
$969$	−49.9411	−1.60434
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.82843	0.154633
$976$	0	0
$977$	−26.2843	−0.840908	−0.420454	−	0.907314i	$-0.638129\pi$
$977$	−26.2843	−0.840908	−0.420454	+	0.907314i	$0.638129\pi$
$978$	0	0
$979$	−80.4264	−2.57044
$980$	0	0
$981$	−12.2010	−0.389548
$982$	0	0
$983$	4.61522	0.147203	0.0736014	−	0.997288i	$-0.476551\pi$
$983$	4.61522	0.147203	0.0736014	+	0.997288i	$0.476551\pi$
$984$	0	0
$985$	11.6569	0.371418
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−108.539	−3.45134
$990$	0	0
$991$	0.828427	0.0263159	0.0131579	−	0.999913i	$-0.495812\pi$
$991$	0.828427	0.0263159	0.0131579	+	0.999913i	$0.495812\pi$
$992$	0	0
$993$	−22.9706	−0.728949
$994$	0	0
$995$	0.686292	0.0217569
$996$	0	0
$997$	−25.6569	−0.812561	−0.406280	−	0.913748i	$-0.633174\pi$
$997$	−25.6569	−0.812561	−0.406280	+	0.913748i	$0.633174\pi$
$998$	0	0
$999$	2.34315	0.0741339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.bz.1.2		2
4.3	odd	2		1960.2.a.p.1.1		2
7.2	even	3		560.2.q.j.81.1		4
7.4	even	3		560.2.q.j.401.1		4
7.6	odd	2		3920.2.a.bp.1.1		2
20.19	odd	2		9800.2.a.bz.1.2		2
28.3	even	6		1960.2.q.q.961.1		4
28.11	odd	6		280.2.q.d.121.2	yes	4
28.19	even	6		1960.2.q.q.361.1		4
28.23	odd	6		280.2.q.d.81.2	✓	4
28.27	even	2		1960.2.a.t.1.2		2
84.11	even	6		2520.2.bi.k.1801.1		4
84.23	even	6		2520.2.bi.k.361.1		4
140.23	even	12		1400.2.bh.g.249.4		8
140.39	odd	6		1400.2.q.h.401.1		4
140.67	even	12		1400.2.bh.g.849.4		8
140.79	odd	6		1400.2.q.h.1201.1		4
140.107	even	12		1400.2.bh.g.249.1		8
140.123	even	12		1400.2.bh.g.849.1		8
140.139	even	2		9800.2.a.br.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.d.81.2	✓	4	28.23	odd	6
280.2.q.d.121.2	yes	4	28.11	odd	6
560.2.q.j.81.1		4	7.2	even	3
560.2.q.j.401.1		4	7.4	even	3
1400.2.q.h.401.1		4	140.39	odd	6
1400.2.q.h.1201.1		4	140.79	odd	6
1400.2.bh.g.249.1		8	140.107	even	12
1400.2.bh.g.249.4		8	140.23	even	12
1400.2.bh.g.849.1		8	140.123	even	12
1400.2.bh.g.849.4		8	140.67	even	12
1960.2.a.p.1.1		2	4.3	odd	2
1960.2.a.t.1.2		2	28.27	even	2
1960.2.q.q.361.1		4	28.19	even	6
1960.2.q.q.961.1		4	28.3	even	6
2520.2.bi.k.361.1		4	84.23	even	6
2520.2.bi.k.1801.1		4	84.11	even	6
3920.2.a.bp.1.1		2	7.6	odd	2
3920.2.a.bz.1.2		2	1.1	even	1	trivial
9800.2.a.br.1.1		2	140.139	even	2
9800.2.a.bz.1.2		2	20.19	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +4.82843 q^{11} +2.00000 q^{13} +2.41421 q^{15} +3.65685 q^{17} -5.65685 q^{19} +8.41421 q^{23} +1.00000 q^{25} -0.414214 q^{27} -2.17157 q^{29} +4.82843 q^{31} +11.6569 q^{33} -5.65685 q^{37} +4.82843 q^{39} +0.171573 q^{41} -12.8995 q^{43} +2.82843 q^{45} -0.343146 q^{47} +8.82843 q^{51} +5.65685 q^{53} +4.82843 q^{55} -13.6569 q^{57} +4.00000 q^{59} -4.65685 q^{61} +2.00000 q^{65} -6.89949 q^{67} +20.3137 q^{69} +12.0000 q^{71} -7.65685 q^{73} +2.41421 q^{75} +4.00000 q^{79} -9.48528 q^{81} +13.2426 q^{83} +3.65685 q^{85} -5.24264 q^{87} -16.6569 q^{89} +11.6569 q^{93} -5.65685 q^{95} +6.00000 q^{97} +13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{11} + 4 q^{13} + 2 q^{15} - 4 q^{17} + 14 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 4 q^{31} + 12 q^{33} + 4 q^{39} + 6 q^{41} - 6 q^{43} - 12 q^{47} + 12 q^{51} + 4 q^{55} - 16 q^{57} + 8 q^{59} + 2 q^{61} + 4 q^{65} + 6 q^{67} + 18 q^{69} + 24 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{79} - 2 q^{81} + 18 q^{83} - 4 q^{85} - 2 q^{87} - 22 q^{89} + 12 q^{93} + 12 q^{97} + 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^11 + 4 * q^13 + 2 * q^15 - 4 * q^17 + 14 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 4 * q^31 + 12 * q^33 + 4 * q^39 + 6 * q^41 - 6 * q^43 - 12 * q^47 + 12 * q^51 + 4 * q^55 - 16 * q^57 + 8 * q^59 + 2 * q^61 + 4 * q^65 + 6 * q^67 + 18 * q^69 + 24 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^79 - 2 * q^81 + 18 * q^83 - 4 * q^85 - 2 * q^87 - 22 * q^89 + 12 * q^93 + 12 * q^97 + 16 * q^99

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