LMFDB - Embedded newform 6354.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6354,2,Mod(1,6354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6354.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6354 = 2 \cdot 3^{2} \cdot 353 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6354.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:	$50.7369454443$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 164x^{6} - 186x^{5} - 510x^{4} + 200x^{3} + 579x^{2} + 108x - 27 $ x^10 - 2x^9 - 23x^8 + 40x^7 + 164x^6 - 186x^5 - 510x^4 + 200x^3 + 579x^2 + 108*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 706)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.10004$ of defining polynomial
Character	$\chi$	$=$	6354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.34822 q^{5} -4.10004 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.34822 q^{5} -4.10004 q^{7} -1.00000 q^{8} +3.34822 q^{10} -4.00504 q^{11} +2.12088 q^{13} +4.10004 q^{14} +1.00000 q^{16} -6.87458 q^{17} +7.58785 q^{19} -3.34822 q^{20} +4.00504 q^{22} +0.995753 q^{23} +6.21061 q^{25} -2.12088 q^{26} -4.10004 q^{28} +7.36349 q^{29} -7.45536 q^{31} -1.00000 q^{32} +6.87458 q^{34} +13.7279 q^{35} +6.72089 q^{37} -7.58785 q^{38} +3.34822 q^{40} +4.39959 q^{41} -1.03348 q^{43} -4.00504 q^{44} -0.995753 q^{46} +8.16828 q^{47} +9.81036 q^{49} -6.21061 q^{50} +2.12088 q^{52} +5.15036 q^{53} +13.4098 q^{55} +4.10004 q^{56} -7.36349 q^{58} -3.93074 q^{59} -6.26811 q^{61} +7.45536 q^{62} +1.00000 q^{64} -7.10118 q^{65} -16.2312 q^{67} -6.87458 q^{68} -13.7279 q^{70} +13.5373 q^{71} +12.8412 q^{73} -6.72089 q^{74} +7.58785 q^{76} +16.4209 q^{77} -7.00958 q^{79} -3.34822 q^{80} -4.39959 q^{82} +3.06803 q^{83} +23.0176 q^{85} +1.03348 q^{86} +4.00504 q^{88} -15.1986 q^{89} -8.69570 q^{91} +0.995753 q^{92} -8.16828 q^{94} -25.4058 q^{95} +10.7932 q^{97} -9.81036 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 12 q^{7} - 10 q^{8}+O(q^{10}) $ 10 * q - 10 * q^2 + 10 * q^4 - 3 * q^5 - 12 * q^7 - 10 * q^8	$ 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 12 q^{7} - 10 q^{8} + 3 q^{10} - 15 q^{11} + 11 q^{13} + 12 q^{14} + 10 q^{16} - 7 q^{17} + 4 q^{19} - 3 q^{20} + 15 q^{22} - 10 q^{23} + 27 q^{25} - 11 q^{26} - 12 q^{28} - 14 q^{29} - q^{31} - 10 q^{32} + 7 q^{34} + 20 q^{35} + 9 q^{37} - 4 q^{38} + 3 q^{40} - 7 q^{41} + 19 q^{43} - 15 q^{44} + 10 q^{46} + 22 q^{47} - 6 q^{49} - 27 q^{50} + 11 q^{52} + 2 q^{53} - 33 q^{55} + 12 q^{56} + 14 q^{58} + 6 q^{59} + 6 q^{61} + q^{62} + 10 q^{64} + 9 q^{65} - 20 q^{67} - 7 q^{68} - 20 q^{70} + 11 q^{71} - 9 q^{73} - 9 q^{74} + 4 q^{76} + 20 q^{77} - 10 q^{79} - 3 q^{80} + 7 q^{82} + 7 q^{83} - 21 q^{85} - 19 q^{86} + 15 q^{88} - 18 q^{89} - 8 q^{91} - 10 q^{92} - 22 q^{94} - 2 q^{95} + 19 q^{97} + 6 q^{98}+O(q^{100}) $ 10 * q - 10 * q^2 + 10 * q^4 - 3 * q^5 - 12 * q^7 - 10 * q^8 + 3 * q^10 - 15 * q^11 + 11 * q^13 + 12 * q^14 + 10 * q^16 - 7 * q^17 + 4 * q^19 - 3 * q^20 + 15 * q^22 - 10 * q^23 + 27 * q^25 - 11 * q^26 - 12 * q^28 - 14 * q^29 - q^31 - 10 * q^32 + 7 * q^34 + 20 * q^35 + 9 * q^37 - 4 * q^38 + 3 * q^40 - 7 * q^41 + 19 * q^43 - 15 * q^44 + 10 * q^46 + 22 * q^47 - 6 * q^49 - 27 * q^50 + 11 * q^52 + 2 * q^53 - 33 * q^55 + 12 * q^56 + 14 * q^58 + 6 * q^59 + 6 * q^61 + q^62 + 10 * q^64 + 9 * q^65 - 20 * q^67 - 7 * q^68 - 20 * q^70 + 11 * q^71 - 9 * q^73 - 9 * q^74 + 4 * q^76 + 20 * q^77 - 10 * q^79 - 3 * q^80 + 7 * q^82 + 7 * q^83 - 21 * q^85 - 19 * q^86 + 15 * q^88 - 18 * q^89 - 8 * q^91 - 10 * q^92 - 22 * q^94 - 2 * q^95 + 19 * q^97 + 6 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.34822	−1.49737	−0.748686	−	0.662925i	$-0.769315\pi$
$5$	−3.34822	−1.49737	−0.748686	+	0.662925i	$0.769315\pi$
$6$	0	0
$7$	−4.10004	−1.54967	−0.774836	−	0.632163i	$-0.782167\pi$
$7$	−4.10004	−1.54967	−0.774836	+	0.632163i	$0.782167\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.34822	1.05880
$11$	−4.00504	−1.20757	−0.603783	−	0.797149i	$-0.706341\pi$
$11$	−4.00504	−1.20757	−0.603783	+	0.797149i	$0.706341\pi$
$12$	0	0
$13$	2.12088	0.588226	0.294113	−	0.955771i	$-0.404976\pi$
$13$	2.12088	0.588226	0.294113	+	0.955771i	$0.404976\pi$
$14$	4.10004	1.09578
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.87458	−1.66733	−0.833665	−	0.552271i	$-0.813762\pi$
$17$	−6.87458	−1.66733	−0.833665	+	0.552271i	$0.813762\pi$
$18$	0	0
$19$	7.58785	1.74077	0.870386	−	0.492371i	$-0.163869\pi$
$19$	7.58785	1.74077	0.870386	+	0.492371i	$0.163869\pi$
$20$	−3.34822	−0.748686
$21$	0	0
$22$	4.00504	0.853878
$23$	0.995753	0.207629	0.103814	−	0.994597i	$-0.466895\pi$
$23$	0.995753	0.207629	0.103814	+	0.994597i	$0.466895\pi$
$24$	0	0
$25$	6.21061	1.24212
$26$	−2.12088	−0.415939
$27$	0	0
$28$	−4.10004	−0.774836
$29$	7.36349	1.36737	0.683683	−	0.729779i	$-0.260377\pi$
$29$	7.36349	1.36737	0.683683	+	0.729779i	$0.260377\pi$
$30$	0	0
$31$	−7.45536	−1.33902	−0.669511	−	0.742802i	$-0.733496\pi$
$31$	−7.45536	−1.33902	−0.669511	+	0.742802i	$0.733496\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.87458	1.17898
$35$	13.7279	2.32043
$36$	0	0
$37$	6.72089	1.10491	0.552454	−	0.833543i	$-0.313692\pi$
$37$	6.72089	1.10491	0.552454	+	0.833543i	$0.313692\pi$
$38$	−7.58785	−1.23091
$39$	0	0
$40$	3.34822	0.529401
$41$	4.39959	0.687101	0.343551	−	0.939134i	$-0.388370\pi$
$41$	4.39959	0.687101	0.343551	+	0.939134i	$0.388370\pi$
$42$	0	0
$43$	−1.03348	−0.157604	−0.0788019	−	0.996890i	$-0.525109\pi$
$43$	−1.03348	−0.157604	−0.0788019	+	0.996890i	$0.525109\pi$
$44$	−4.00504	−0.603783
$45$	0	0
$46$	−0.995753	−0.146816
$47$	8.16828	1.19147	0.595733	−	0.803183i	$-0.296862\pi$
$47$	8.16828	1.19147	0.595733	+	0.803183i	$0.296862\pi$
$48$	0	0
$49$	9.81036	1.40148
$50$	−6.21061	−0.878313
$51$	0	0
$52$	2.12088	0.294113
$53$	5.15036	0.707457	0.353728	−	0.935348i	$-0.384914\pi$
$53$	5.15036	0.707457	0.353728	+	0.935348i	$0.384914\pi$
$54$	0	0
$55$	13.4098	1.80817
$56$	4.10004	0.547891
$57$	0	0
$58$	−7.36349	−0.966874
$59$	−3.93074	−0.511739	−0.255870	−	0.966711i	$-0.582362\pi$
$59$	−3.93074	−0.511739	−0.255870	+	0.966711i	$0.582362\pi$
$60$	0	0
$61$	−6.26811	−0.802550	−0.401275	−	0.915958i	$-0.631433\pi$
$61$	−6.26811	−0.802550	−0.401275	+	0.915958i	$0.631433\pi$
$62$	7.45536	0.946831
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.10118	−0.880793
$66$	0	0
$67$	−16.2312	−1.98296	−0.991478	−	0.130275i	$-0.958414\pi$
$67$	−16.2312	−1.98296	−0.991478	+	0.130275i	$0.958414\pi$
$68$	−6.87458	−0.833665
$69$	0	0
$70$	−13.7279	−1.64079
$71$	13.5373	1.60658	0.803290	−	0.595588i	$-0.203081\pi$
$71$	13.5373	1.60658	0.803290	+	0.595588i	$0.203081\pi$
$72$	0	0
$73$	12.8412	1.50295	0.751475	−	0.659761i	$-0.229343\pi$
$73$	12.8412	1.50295	0.751475	+	0.659761i	$0.229343\pi$
$74$	−6.72089	−0.781288
$75$	0	0
$76$	7.58785	0.870386
$77$	16.4209	1.87133
$78$	0	0
$79$	−7.00958	−0.788640	−0.394320	−	0.918973i	$-0.629020\pi$
$79$	−7.00958	−0.788640	−0.394320	+	0.918973i	$0.629020\pi$
$80$	−3.34822	−0.374343
$81$	0	0
$82$	−4.39959	−0.485854
$83$	3.06803	0.336760	0.168380	−	0.985722i	$-0.446146\pi$
$83$	3.06803	0.336760	0.168380	+	0.985722i	$0.446146\pi$
$84$	0	0
$85$	23.0176	2.49661
$86$	1.03348	0.111443
$87$	0	0
$88$	4.00504	0.426939
$89$	−15.1986	−1.61105	−0.805524	−	0.592563i	$-0.798116\pi$
$89$	−15.1986	−1.61105	−0.805524	+	0.592563i	$0.798116\pi$
$90$	0	0
$91$	−8.69570	−0.911557
$92$	0.995753	0.103814
$93$	0	0
$94$	−8.16828	−0.842494
$95$	−25.4058	−2.60658
$96$	0	0
$97$	10.7932	1.09588	0.547942	−	0.836517i	$-0.315412\pi$
$97$	10.7932	1.09588	0.547942	+	0.836517i	$0.315412\pi$
$98$	−9.81036	−0.990996
$99$	0	0
$100$	6.21061	0.621061
$101$	−3.92331	−0.390384	−0.195192	−	0.980765i	$-0.562533\pi$
$101$	−3.92331	−0.390384	−0.195192	+	0.980765i	$0.562533\pi$
$102$	0	0
$103$	0.580451	0.0571936	0.0285968	−	0.999591i	$-0.490896\pi$
$103$	0.580451	0.0571936	0.0285968	+	0.999591i	$0.490896\pi$
$104$	−2.12088	−0.207969
$105$	0	0
$106$	−5.15036	−0.500247
$107$	14.4575	1.39766	0.698830	−	0.715288i	$-0.253704\pi$
$107$	14.4575	1.39766	0.698830	+	0.715288i	$0.253704\pi$
$108$	0	0
$109$	−6.80789	−0.652077	−0.326039	−	0.945356i	$-0.605714\pi$
$109$	−6.80789	−0.652077	−0.326039	+	0.945356i	$0.605714\pi$
$110$	−13.4098	−1.27857
$111$	0	0
$112$	−4.10004	−0.387418
$113$	10.2017	0.959698	0.479849	−	0.877351i	$-0.340691\pi$
$113$	10.2017	0.959698	0.479849	+	0.877351i	$0.340691\pi$
$114$	0	0
$115$	−3.33400	−0.310898
$116$	7.36349	0.683683
$117$	0	0
$118$	3.93074	0.361854
$119$	28.1861	2.58381
$120$	0	0
$121$	5.04037	0.458215
$122$	6.26811	0.567488
$123$	0	0
$124$	−7.45536	−0.669511
$125$	−4.05339	−0.362546
$126$	0	0
$127$	−4.93718	−0.438104	−0.219052	−	0.975713i	$-0.570296\pi$
$127$	−4.93718	−0.438104	−0.219052	+	0.975713i	$0.570296\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.10118	0.622815
$131$	−5.54886	−0.484806	−0.242403	−	0.970176i	$-0.577936\pi$
$131$	−5.54886	−0.484806	−0.242403	+	0.970176i	$0.577936\pi$
$132$	0	0
$133$	−31.1105	−2.69762
$134$	16.2312	1.40216
$135$	0	0
$136$	6.87458	0.589490
$137$	10.8916	0.930534	0.465267	−	0.885170i	$-0.345958\pi$
$137$	10.8916	0.930534	0.465267	+	0.885170i	$0.345958\pi$
$138$	0	0
$139$	3.74758	0.317865	0.158933	−	0.987289i	$-0.449195\pi$
$139$	3.74758	0.317865	0.158933	+	0.987289i	$0.449195\pi$
$140$	13.7279	1.16022
$141$	0	0
$142$	−13.5373	−1.13602
$143$	−8.49422	−0.710322
$144$	0	0
$145$	−24.6546	−2.04745
$146$	−12.8412	−1.06275
$147$	0	0
$148$	6.72089	0.552454
$149$	−22.3452	−1.83059	−0.915294	−	0.402787i	$-0.868042\pi$
$149$	−22.3452	−1.83059	−0.915294	+	0.402787i	$0.868042\pi$
$150$	0	0
$151$	12.8268	1.04383	0.521915	−	0.852997i	$-0.325218\pi$
$151$	12.8268	1.04383	0.521915	+	0.852997i	$0.325218\pi$
$152$	−7.58785	−0.615456
$153$	0	0
$154$	−16.4209	−1.32323
$155$	24.9622	2.00501
$156$	0	0
$157$	−16.9860	−1.35563	−0.677815	−	0.735233i	$-0.737073\pi$
$157$	−16.9860	−1.35563	−0.677815	+	0.735233i	$0.737073\pi$
$158$	7.00958	0.557652
$159$	0	0
$160$	3.34822	0.264700
$161$	−4.08263	−0.321756
$162$	0	0
$163$	2.64164	0.206909	0.103455	−	0.994634i	$-0.467010\pi$
$163$	2.64164	0.206909	0.103455	+	0.994634i	$0.467010\pi$
$164$	4.39959	0.343551
$165$	0	0
$166$	−3.06803	−0.238125
$167$	7.17467	0.555192	0.277596	−	0.960698i	$-0.410462\pi$
$167$	7.17467	0.555192	0.277596	+	0.960698i	$0.410462\pi$
$168$	0	0
$169$	−8.50187	−0.653990
$170$	−23.0176	−1.76537
$171$	0	0
$172$	−1.03348	−0.0788019
$173$	3.02558	0.230030	0.115015	−	0.993364i	$-0.463308\pi$
$173$	3.02558	0.230030	0.115015	+	0.993364i	$0.463308\pi$
$174$	0	0
$175$	−25.4638	−1.92488
$176$	−4.00504	−0.301891
$177$	0	0
$178$	15.1986	1.13918
$179$	−21.7995	−1.62937	−0.814684	−	0.579904i	$-0.803090\pi$
$179$	−21.7995	−1.62937	−0.814684	+	0.579904i	$0.803090\pi$
$180$	0	0
$181$	13.3118	0.989455	0.494727	−	0.869048i	$-0.335268\pi$
$181$	13.3118	0.989455	0.494727	+	0.869048i	$0.335268\pi$
$182$	8.69570	0.644568
$183$	0	0
$184$	−0.995753	−0.0734079
$185$	−22.5031	−1.65446
$186$	0	0
$187$	27.5330	2.01341
$188$	8.16828	0.595733
$189$	0	0
$190$	25.4058	1.84313
$191$	−19.9533	−1.44377	−0.721885	−	0.692013i	$-0.756724\pi$
$191$	−19.9533	−1.44377	−0.721885	+	0.692013i	$0.756724\pi$
$192$	0	0
$193$	1.65872	0.119397	0.0596987	−	0.998216i	$-0.480986\pi$
$193$	1.65872	0.119397	0.0596987	+	0.998216i	$0.480986\pi$
$194$	−10.7932	−0.774906
$195$	0	0
$196$	9.81036	0.700740
$197$	12.2438	0.872331	0.436166	−	0.899866i	$-0.356336\pi$
$197$	12.2438	0.872331	0.436166	+	0.899866i	$0.356336\pi$
$198$	0	0
$199$	7.54659	0.534964	0.267482	−	0.963563i	$-0.413808\pi$
$199$	7.54659	0.534964	0.267482	+	0.963563i	$0.413808\pi$
$200$	−6.21061	−0.439156
$201$	0	0
$202$	3.92331	0.276043
$203$	−30.1906	−2.11897
$204$	0	0
$205$	−14.7308	−1.02885
$206$	−0.580451	−0.0404420
$207$	0	0
$208$	2.12088	0.147057
$209$	−30.3896	−2.10210
$210$	0	0
$211$	−27.8101	−1.91453	−0.957264	−	0.289217i	$-0.906605\pi$
$211$	−27.8101	−1.91453	−0.957264	+	0.289217i	$0.906605\pi$
$212$	5.15036	0.353728
$213$	0	0
$214$	−14.4575	−0.988295
$215$	3.46031	0.235991
$216$	0	0
$217$	30.5673	2.07504
$218$	6.80789	0.461088
$219$	0	0
$220$	13.4098	0.904087
$221$	−14.5802	−0.980767
$222$	0	0
$223$	16.4175	1.09940	0.549699	−	0.835363i	$-0.314742\pi$
$223$	16.4175	1.09940	0.549699	+	0.835363i	$0.314742\pi$
$224$	4.10004	0.273946
$225$	0	0
$226$	−10.2017	−0.678609
$227$	10.3544	0.687247	0.343624	−	0.939107i	$-0.388346\pi$
$227$	10.3544	0.687247	0.343624	+	0.939107i	$0.388346\pi$
$228$	0	0
$229$	2.16787	0.143257	0.0716283	−	0.997431i	$-0.477180\pi$
$229$	2.16787	0.143257	0.0716283	+	0.997431i	$0.477180\pi$
$230$	3.33400	0.219838
$231$	0	0
$232$	−7.36349	−0.483437
$233$	−15.6664	−1.02634	−0.513169	−	0.858288i	$-0.671528\pi$
$233$	−15.6664	−1.02634	−0.513169	+	0.858288i	$0.671528\pi$
$234$	0	0
$235$	−27.3492	−1.78407
$236$	−3.93074	−0.255870
$237$	0	0
$238$	−28.1861	−1.82703
$239$	−14.5865	−0.943519	−0.471760	−	0.881727i	$-0.656381\pi$
$239$	−14.5865	−0.943519	−0.471760	+	0.881727i	$0.656381\pi$
$240$	0	0
$241$	−8.35078	−0.537921	−0.268961	−	0.963151i	$-0.586680\pi$
$241$	−8.35078	−0.537921	−0.268961	+	0.963151i	$0.586680\pi$
$242$	−5.04037	−0.324007
$243$	0	0
$244$	−6.26811	−0.401275
$245$	−32.8473	−2.09854
$246$	0	0
$247$	16.0929	1.02397
$248$	7.45536	0.473416
$249$	0	0
$250$	4.05339	0.256359
$251$	−2.97212	−0.187599	−0.0937993	−	0.995591i	$-0.529901\pi$
$251$	−2.97212	−0.187599	−0.0937993	+	0.995591i	$0.529901\pi$
$252$	0	0
$253$	−3.98803	−0.250725
$254$	4.93718	0.309786
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.5696	1.71974	0.859872	−	0.510510i	$-0.170543\pi$
$257$	27.5696	1.71974	0.859872	+	0.510510i	$0.170543\pi$
$258$	0	0
$259$	−27.5560	−1.71224
$260$	−7.10118	−0.440397
$261$	0	0
$262$	5.54886	0.342810
$263$	−7.02368	−0.433098	−0.216549	−	0.976272i	$-0.569480\pi$
$263$	−7.02368	−0.433098	−0.216549	+	0.976272i	$0.569480\pi$
$264$	0	0
$265$	−17.2446	−1.05933
$266$	31.1105	1.90751
$267$	0	0
$268$	−16.2312	−0.991478
$269$	−5.10878	−0.311488	−0.155744	−	0.987797i	$-0.549777\pi$
$269$	−5.10878	−0.311488	−0.155744	+	0.987797i	$0.549777\pi$
$270$	0	0
$271$	−10.4798	−0.636601	−0.318300	−	0.947990i	$-0.603112\pi$
$271$	−10.4798	−0.636601	−0.318300	+	0.947990i	$0.603112\pi$
$272$	−6.87458	−0.416832
$273$	0	0
$274$	−10.8916	−0.657987
$275$	−24.8738	−1.49994
$276$	0	0
$277$	−1.61009	−0.0967407	−0.0483704	−	0.998829i	$-0.515403\pi$
$277$	−1.61009	−0.0967407	−0.0483704	+	0.998829i	$0.515403\pi$
$278$	−3.74758	−0.224765
$279$	0	0
$280$	−13.7279	−0.820397
$281$	26.1402	1.55939	0.779696	−	0.626158i	$-0.215373\pi$
$281$	26.1402	1.55939	0.779696	+	0.626158i	$0.215373\pi$
$282$	0	0
$283$	−7.59872	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$283$	−7.59872	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$284$	13.5373	0.803290
$285$	0	0
$286$	8.49422	0.502273
$287$	−18.0385	−1.06478
$288$	0	0
$289$	30.2598	1.77999
$290$	24.6546	1.44777
$291$	0	0
$292$	12.8412	0.751475
$293$	10.1035	0.590256	0.295128	−	0.955458i	$-0.404638\pi$
$293$	10.1035	0.590256	0.295128	+	0.955458i	$0.404638\pi$
$294$	0	0
$295$	13.1610	0.766264
$296$	−6.72089	−0.390644
$297$	0	0
$298$	22.3452	1.29442
$299$	2.11187	0.122133
$300$	0	0
$301$	4.23730	0.244234
$302$	−12.8268	−0.738099
$303$	0	0
$304$	7.58785	0.435193
$305$	20.9871	1.20172
$306$	0	0
$307$	10.5562	0.602473	0.301236	−	0.953549i	$-0.402601\pi$
$307$	10.5562	0.602473	0.301236	+	0.953549i	$0.402601\pi$
$308$	16.4209	0.935665
$309$	0	0
$310$	−24.9622	−1.41776
$311$	−20.9846	−1.18993	−0.594964	−	0.803752i	$-0.702834\pi$
$311$	−20.9846	−1.18993	−0.594964	+	0.803752i	$0.702834\pi$
$312$	0	0
$313$	−13.1620	−0.743961	−0.371981	−	0.928240i	$-0.621321\pi$
$313$	−13.1620	−0.743961	−0.371981	+	0.928240i	$0.621321\pi$
$314$	16.9860	0.958575
$315$	0	0
$316$	−7.00958	−0.394320
$317$	−11.1977	−0.628923	−0.314462	−	0.949270i	$-0.601824\pi$
$317$	−11.1977	−0.628923	−0.314462	+	0.949270i	$0.601824\pi$
$318$	0	0
$319$	−29.4911	−1.65118
$320$	−3.34822	−0.187171
$321$	0	0
$322$	4.08263	0.227516
$323$	−52.1632	−2.90244
$324$	0	0
$325$	13.1720	0.730649
$326$	−2.64164	−0.146307
$327$	0	0
$328$	−4.39959	−0.242927
$329$	−33.4903	−1.84638
$330$	0	0
$331$	6.01294	0.330501	0.165251	−	0.986252i	$-0.447157\pi$
$331$	6.01294	0.330501	0.165251	+	0.986252i	$0.447157\pi$
$332$	3.06803	0.168380
$333$	0	0
$334$	−7.17467	−0.392580
$335$	54.3457	2.96922
$336$	0	0
$337$	5.75107	0.313281	0.156640	−	0.987656i	$-0.449934\pi$
$337$	5.75107	0.313281	0.156640	+	0.987656i	$0.449934\pi$
$338$	8.50187	0.462441
$339$	0	0
$340$	23.0176	1.24831
$341$	29.8590	1.61696
$342$	0	0
$343$	−11.5226	−0.622163
$344$	1.03348	0.0557213
$345$	0	0
$346$	−3.02558	−0.162656
$347$	29.6443	1.59139	0.795695	−	0.605697i	$-0.207106\pi$
$347$	29.6443	1.59139	0.795695	+	0.605697i	$0.207106\pi$
$348$	0	0
$349$	16.5211	0.884355	0.442177	−	0.896928i	$-0.354206\pi$
$349$	16.5211	0.884355	0.442177	+	0.896928i	$0.354206\pi$
$350$	25.4638	1.36110
$351$	0	0
$352$	4.00504	0.213469
$353$	1.00000	0.0532246
$353$	1.00000	0.0532246
$354$	0	0
$355$	−45.3259	−2.40565
$356$	−15.1986	−0.805524
$357$	0	0
$358$	21.7995	1.15214
$359$	−9.34067	−0.492982	−0.246491	−	0.969145i	$-0.579278\pi$
$359$	−9.34067	−0.492982	−0.246491	+	0.969145i	$0.579278\pi$
$360$	0	0
$361$	38.5754	2.03028
$362$	−13.3118	−0.699650
$363$	0	0
$364$	−8.69570	−0.455779
$365$	−42.9953	−2.25048
$366$	0	0
$367$	−14.8704	−0.776229	−0.388115	−	0.921611i	$-0.626874\pi$
$367$	−14.8704	−0.776229	−0.388115	+	0.921611i	$0.626874\pi$
$368$	0.995753	0.0519072
$369$	0	0
$370$	22.5031	1.16988
$371$	−21.1167	−1.09633
$372$	0	0
$373$	−34.7972	−1.80173	−0.900867	−	0.434096i	$-0.857068\pi$
$373$	−34.7972	−1.80173	−0.900867	+	0.434096i	$0.857068\pi$
$374$	−27.5330	−1.42370
$375$	0	0
$376$	−8.16828	−0.421247
$377$	15.6171	0.804321
$378$	0	0
$379$	17.1231	0.879557	0.439778	−	0.898106i	$-0.355057\pi$
$379$	17.1231	0.879557	0.439778	+	0.898106i	$0.355057\pi$
$380$	−25.4058	−1.30329
$381$	0	0
$382$	19.9533	1.02090
$383$	−9.42211	−0.481447	−0.240724	−	0.970594i	$-0.577385\pi$
$383$	−9.42211	−0.481447	−0.240724	+	0.970594i	$0.577385\pi$
$384$	0	0
$385$	−54.9807	−2.80208
$386$	−1.65872	−0.0844267
$387$	0	0
$388$	10.7932	0.547942
$389$	−36.8405	−1.86789	−0.933943	−	0.357421i	$-0.883656\pi$
$389$	−36.8405	−1.86789	−0.933943	+	0.357421i	$0.883656\pi$
$390$	0	0
$391$	−6.84538	−0.346186
$392$	−9.81036	−0.495498
$393$	0	0
$394$	−12.2438	−0.616831
$395$	23.4697	1.18089
$396$	0	0
$397$	−2.11398	−0.106097	−0.0530487	−	0.998592i	$-0.516894\pi$
$397$	−2.11398	−0.106097	−0.0530487	+	0.998592i	$0.516894\pi$
$398$	−7.54659	−0.378276
$399$	0	0
$400$	6.21061	0.310530
$401$	20.3448	1.01597	0.507985	−	0.861366i	$-0.330390\pi$
$401$	20.3448	1.01597	0.507985	+	0.861366i	$0.330390\pi$
$402$	0	0
$403$	−15.8119	−0.787648
$404$	−3.92331	−0.195192
$405$	0	0
$406$	30.1906	1.49834
$407$	−26.9175	−1.33425
$408$	0	0
$409$	−4.91606	−0.243084	−0.121542	−	0.992586i	$-0.538784\pi$
$409$	−4.91606	−0.243084	−0.121542	+	0.992586i	$0.538784\pi$
$410$	14.7308	0.727504
$411$	0	0
$412$	0.580451	0.0285968
$413$	16.1162	0.793028
$414$	0	0
$415$	−10.2725	−0.504255
$416$	−2.12088	−0.103985
$417$	0	0
$418$	30.3896	1.48641
$419$	1.84085	0.0899312	0.0449656	−	0.998989i	$-0.485682\pi$
$419$	1.84085	0.0899312	0.0449656	+	0.998989i	$0.485682\pi$
$420$	0	0
$421$	−2.41138	−0.117524	−0.0587618	−	0.998272i	$-0.518715\pi$
$421$	−2.41138	−0.117524	−0.0587618	+	0.998272i	$0.518715\pi$
$422$	27.8101	1.35378
$423$	0	0
$424$	−5.15036	−0.250124
$425$	−42.6953	−2.07103
$426$	0	0
$427$	25.6995	1.24369
$428$	14.4575	0.698830
$429$	0	0
$430$	−3.46031	−0.166871
$431$	11.0418	0.531867	0.265933	−	0.963991i	$-0.414320\pi$
$431$	11.0418	0.531867	0.265933	+	0.963991i	$0.414320\pi$
$432$	0	0
$433$	9.77874	0.469936	0.234968	−	0.972003i	$-0.424501\pi$
$433$	9.77874	0.469936	0.234968	+	0.972003i	$0.424501\pi$
$434$	−30.5673	−1.46728
$435$	0	0
$436$	−6.80789	−0.326039
$437$	7.55562	0.361434
$438$	0	0
$439$	−23.8200	−1.13687	−0.568434	−	0.822729i	$-0.692451\pi$
$439$	−23.8200	−1.13687	−0.568434	+	0.822729i	$0.692451\pi$
$440$	−13.4098	−0.639286
$441$	0	0
$442$	14.5802	0.693507
$443$	−2.71213	−0.128857	−0.0644285	−	0.997922i	$-0.520522\pi$
$443$	−2.71213	−0.128857	−0.0644285	+	0.997922i	$0.520522\pi$
$444$	0	0
$445$	50.8883	2.41234
$446$	−16.4175	−0.777392
$447$	0	0
$448$	−4.10004	−0.193709
$449$	−11.8073	−0.557222	−0.278611	−	0.960404i	$-0.589874\pi$
$449$	−11.8073	−0.557222	−0.278611	+	0.960404i	$0.589874\pi$
$450$	0	0
$451$	−17.6206	−0.829720
$452$	10.2017	0.479849
$453$	0	0
$454$	−10.3544	−0.485957
$455$	29.1152	1.36494
$456$	0	0
$457$	9.86717	0.461567	0.230783	−	0.973005i	$-0.425871\pi$
$457$	9.86717	0.461567	0.230783	+	0.973005i	$0.425871\pi$
$458$	−2.16787	−0.101298
$459$	0	0
$460$	−3.33400	−0.155449
$461$	7.68914	0.358119	0.179060	−	0.983838i	$-0.442695\pi$
$461$	7.68914	0.358119	0.179060	+	0.983838i	$0.442695\pi$
$462$	0	0
$463$	−27.3927	−1.27305	−0.636524	−	0.771257i	$-0.719628\pi$
$463$	−27.3927	−1.27305	−0.636524	+	0.771257i	$0.719628\pi$
$464$	7.36349	0.341841
$465$	0	0
$466$	15.6664	0.725730
$467$	20.9222	0.968162	0.484081	−	0.875023i	$-0.339154\pi$
$467$	20.9222	0.968162	0.484081	+	0.875023i	$0.339154\pi$
$468$	0	0
$469$	66.5486	3.07293
$470$	27.3492	1.26153
$471$	0	0
$472$	3.93074	0.180927
$473$	4.13912	0.190317
$474$	0	0
$475$	47.1251	2.16225
$476$	28.1861	1.29191
$477$	0	0
$478$	14.5865	0.667169
$479$	−6.23231	−0.284761	−0.142381	−	0.989812i	$-0.545476\pi$
$479$	−6.23231	−0.284761	−0.142381	+	0.989812i	$0.545476\pi$
$480$	0	0
$481$	14.2542	0.649936
$482$	8.35078	0.380368
$483$	0	0
$484$	5.04037	0.229108
$485$	−36.1381	−1.64094
$486$	0	0
$487$	33.0999	1.49990	0.749949	−	0.661495i	$-0.230078\pi$
$487$	33.0999	1.49990	0.749949	+	0.661495i	$0.230078\pi$
$488$	6.26811	0.283744
$489$	0	0
$490$	32.8473	1.48389
$491$	−4.65183	−0.209934	−0.104967	−	0.994476i	$-0.533474\pi$
$491$	−4.65183	−0.209934	−0.104967	+	0.994476i	$0.533474\pi$
$492$	0	0
$493$	−50.6209	−2.27985
$494$	−16.0929	−0.724054
$495$	0	0
$496$	−7.45536	−0.334755
$497$	−55.5035	−2.48967
$498$	0	0
$499$	32.3299	1.44729	0.723643	−	0.690174i	$-0.242466\pi$
$499$	32.3299	1.44729	0.723643	+	0.690174i	$0.242466\pi$
$500$	−4.05339	−0.181273
$501$	0	0
$502$	2.97212	0.132652
$503$	−11.1766	−0.498338	−0.249169	−	0.968460i	$-0.580158\pi$
$503$	−11.1766	−0.498338	−0.249169	+	0.968460i	$0.580158\pi$
$504$	0	0
$505$	13.1361	0.584550
$506$	3.98803	0.177290
$507$	0	0
$508$	−4.93718	−0.219052
$509$	12.9129	0.572356	0.286178	−	0.958176i	$-0.407615\pi$
$509$	12.9129	0.572356	0.286178	+	0.958176i	$0.407615\pi$
$510$	0	0
$511$	−52.6495	−2.32908
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−27.5696	−1.21604
$515$	−1.94348	−0.0856400
$516$	0	0
$517$	−32.7143	−1.43877
$518$	27.5560	1.21074
$519$	0	0
$520$	7.10118	0.311407
$521$	−3.53372	−0.154815	−0.0774075	−	0.997000i	$-0.524664\pi$
$521$	−3.53372	−0.154815	−0.0774075	+	0.997000i	$0.524664\pi$
$522$	0	0
$523$	−4.55706	−0.199266	−0.0996331	−	0.995024i	$-0.531767\pi$
$523$	−4.55706	−0.199266	−0.0996331	+	0.995024i	$0.531767\pi$
$524$	−5.54886	−0.242403
$525$	0	0
$526$	7.02368	0.306247
$527$	51.2524	2.23259
$528$	0	0
$529$	−22.0085	−0.956890
$530$	17.2446	0.749056
$531$	0	0
$532$	−31.1105	−1.34881
$533$	9.33101	0.404171
$534$	0	0
$535$	−48.4070	−2.09282
$536$	16.2312	0.701081
$537$	0	0
$538$	5.10878	0.220255
$539$	−39.2909	−1.69238
$540$	0	0
$541$	0.513007	0.0220559	0.0110279	−	0.999939i	$-0.496490\pi$
$541$	0.513007	0.0220559	0.0110279	+	0.999939i	$0.496490\pi$
$542$	10.4798	0.450145
$543$	0	0
$544$	6.87458	0.294745
$545$	22.7943	0.976402
$546$	0	0
$547$	34.2952	1.46636	0.733179	−	0.680035i	$-0.238036\pi$
$547$	34.2952	1.46636	0.733179	+	0.680035i	$0.238036\pi$
$548$	10.8916	0.465267
$549$	0	0
$550$	24.8738	1.06062
$551$	55.8730	2.38027
$552$	0	0
$553$	28.7396	1.22213
$554$	1.61009	0.0684060
$555$	0	0
$556$	3.74758	0.158933
$557$	−18.6595	−0.790627	−0.395314	−	0.918546i	$-0.629364\pi$
$557$	−18.6595	−0.790627	−0.395314	+	0.918546i	$0.629364\pi$
$558$	0	0
$559$	−2.19188	−0.0927067
$560$	13.7279	0.580108
$561$	0	0
$562$	−26.1402	−1.10266
$563$	13.7055	0.577616	0.288808	−	0.957387i	$-0.406741\pi$
$563$	13.7055	0.577616	0.288808	+	0.957387i	$0.406741\pi$
$564$	0	0
$565$	−34.1577	−1.43702
$566$	7.59872	0.319398
$567$	0	0
$568$	−13.5373	−0.568012
$569$	9.31889	0.390668	0.195334	−	0.980737i	$-0.437421\pi$
$569$	9.31889	0.390668	0.195334	+	0.980737i	$0.437421\pi$
$570$	0	0
$571$	27.5395	1.15249	0.576247	−	0.817276i	$-0.304517\pi$
$571$	27.5395	1.15249	0.576247	+	0.817276i	$0.304517\pi$
$572$	−8.49422	−0.355161
$573$	0	0
$574$	18.0385	0.752914
$575$	6.18423	0.257900
$576$	0	0
$577$	39.2944	1.63585	0.817924	−	0.575326i	$-0.195125\pi$
$577$	39.2944	1.63585	0.817924	+	0.575326i	$0.195125\pi$
$578$	−30.2598	−1.25864
$579$	0	0
$580$	−24.6546	−1.02373
$581$	−12.5791	−0.521867
$582$	0	0
$583$	−20.6274	−0.854300
$584$	−12.8412	−0.531373
$585$	0	0
$586$	−10.1035	−0.417374
$587$	−40.4303	−1.66874	−0.834368	−	0.551207i	$-0.814167\pi$
$587$	−40.4303	−1.66874	−0.834368	+	0.551207i	$0.814167\pi$
$588$	0	0
$589$	−56.5701	−2.33093
$590$	−13.1610	−0.541830
$591$	0	0
$592$	6.72089	0.276227
$593$	−29.5139	−1.21199	−0.605995	−	0.795468i	$-0.707225\pi$
$593$	−29.5139	−1.21199	−0.605995	+	0.795468i	$0.707225\pi$
$594$	0	0
$595$	−94.3733	−3.86893
$596$	−22.3452	−0.915294
$597$	0	0
$598$	−2.11187	−0.0863609
$599$	−31.4845	−1.28642	−0.643211	−	0.765689i	$-0.722398\pi$
$599$	−31.4845	−1.28642	−0.643211	+	0.765689i	$0.722398\pi$
$600$	0	0
$601$	−2.60831	−0.106395	−0.0531976	−	0.998584i	$-0.516941\pi$
$601$	−2.60831	−0.106395	−0.0531976	+	0.998584i	$0.516941\pi$
$602$	−4.23730	−0.172699
$603$	0	0
$604$	12.8268	0.521915
$605$	−16.8763	−0.686118
$606$	0	0
$607$	0.256177	0.0103979	0.00519894	−	0.999986i	$-0.498345\pi$
$607$	0.256177	0.0103979	0.00519894	+	0.999986i	$0.498345\pi$
$608$	−7.58785	−0.307728
$609$	0	0
$610$	−20.9871	−0.849741
$611$	17.3239	0.700852
$612$	0	0
$613$	21.5229	0.869300	0.434650	−	0.900599i	$-0.356872\pi$
$613$	21.5229	0.869300	0.434650	+	0.900599i	$0.356872\pi$
$614$	−10.5562	−0.426013
$615$	0	0
$616$	−16.4209	−0.661615
$617$	1.94502	0.0783036	0.0391518	−	0.999233i	$-0.487534\pi$
$617$	1.94502	0.0783036	0.0391518	+	0.999233i	$0.487534\pi$
$618$	0	0
$619$	−6.51838	−0.261996	−0.130998	−	0.991383i	$-0.541818\pi$
$619$	−6.51838	−0.261996	−0.130998	+	0.991383i	$0.541818\pi$
$620$	24.9622	1.00251
$621$	0	0
$622$	20.9846	0.841406
$623$	62.3149	2.49659
$624$	0	0
$625$	−17.4814	−0.699255
$626$	13.1620	0.526060
$627$	0	0
$628$	−16.9860	−0.677815
$629$	−46.2033	−1.84225
$630$	0	0
$631$	17.1064	0.680994	0.340497	−	0.940246i	$-0.389405\pi$
$631$	17.1064	0.680994	0.340497	+	0.940246i	$0.389405\pi$
$632$	7.00958	0.278826
$633$	0	0
$634$	11.1977	0.444716
$635$	16.5308	0.656005
$636$	0	0
$637$	20.8066	0.824388
$638$	29.4911	1.16756
$639$	0	0
$640$	3.34822	0.132350
$641$	−4.03337	−0.159309	−0.0796543	−	0.996823i	$-0.525382\pi$
$641$	−4.03337	−0.159309	−0.0796543	+	0.996823i	$0.525382\pi$
$642$	0	0
$643$	18.4857	0.729003	0.364502	−	0.931203i	$-0.381239\pi$
$643$	18.4857	0.729003	0.364502	+	0.931203i	$0.381239\pi$
$644$	−4.08263	−0.160878
$645$	0	0
$646$	52.1632	2.05234
$647$	24.4999	0.963190	0.481595	−	0.876394i	$-0.340058\pi$
$647$	24.4999	0.963190	0.481595	+	0.876394i	$0.340058\pi$
$648$	0	0
$649$	15.7428	0.617959
$650$	−13.1720	−0.516647
$651$	0	0
$652$	2.64164	0.103455
$653$	28.1439	1.10135	0.550677	−	0.834718i	$-0.314369\pi$
$653$	28.1439	1.10135	0.550677	+	0.834718i	$0.314369\pi$
$654$	0	0
$655$	18.5788	0.725935
$656$	4.39959	0.171775
$657$	0	0
$658$	33.4903	1.30559
$659$	−20.2203	−0.787671	−0.393835	−	0.919181i	$-0.628852\pi$
$659$	−20.2203	−0.787671	−0.393835	+	0.919181i	$0.628852\pi$
$660$	0	0
$661$	7.78497	0.302800	0.151400	−	0.988473i	$-0.451622\pi$
$661$	7.78497	0.302800	0.151400	+	0.988473i	$0.451622\pi$
$662$	−6.01294	−0.233700
$663$	0	0
$664$	−3.06803	−0.119063
$665$	104.165	4.03934
$666$	0	0
$667$	7.33222	0.283905
$668$	7.17467	0.277596
$669$	0	0
$670$	−54.3457	−2.09956
$671$	25.1041	0.969132
$672$	0	0
$673$	32.4940	1.25255	0.626275	−	0.779602i	$-0.284579\pi$
$673$	32.4940	1.25255	0.626275	+	0.779602i	$0.284579\pi$
$674$	−5.75107	−0.221523
$675$	0	0
$676$	−8.50187	−0.326995
$677$	15.7645	0.605879	0.302939	−	0.953010i	$-0.402032\pi$
$677$	15.7645	0.605879	0.302939	+	0.953010i	$0.402032\pi$
$678$	0	0
$679$	−44.2526	−1.69826
$680$	−23.0176	−0.882686
$681$	0	0
$682$	−29.8590	−1.14336
$683$	−21.9957	−0.841641	−0.420821	−	0.907144i	$-0.638258\pi$
$683$	−21.9957	−0.841641	−0.420821	+	0.907144i	$0.638258\pi$
$684$	0	0
$685$	−36.4676	−1.39335
$686$	11.5226	0.439936
$687$	0	0
$688$	−1.03348	−0.0394009
$689$	10.9233	0.416145
$690$	0	0
$691$	11.1807	0.425333	0.212667	−	0.977125i	$-0.431785\pi$
$691$	11.1807	0.425333	0.212667	+	0.977125i	$0.431785\pi$
$692$	3.02558	0.115015
$693$	0	0
$694$	−29.6443	−1.12528
$695$	−12.5477	−0.475963
$696$	0	0
$697$	−30.2453	−1.14562
$698$	−16.5211	−0.625333
$699$	0	0
$700$	−25.4638	−0.962440
$701$	−37.3330	−1.41005	−0.705024	−	0.709184i	$-0.749064\pi$
$701$	−37.3330	−1.41005	−0.705024	+	0.709184i	$0.749064\pi$
$702$	0	0
$703$	50.9971	1.92339
$704$	−4.00504	−0.150946
$705$	0	0
$706$	−1.00000	−0.0376355
$707$	16.0858	0.604967
$708$	0	0
$709$	13.0742	0.491013	0.245506	−	0.969395i	$-0.421046\pi$
$709$	13.0742	0.491013	0.245506	+	0.969395i	$0.421046\pi$
$710$	45.3259	1.70105
$711$	0	0
$712$	15.1986	0.569591
$713$	−7.42370	−0.278020
$714$	0	0
$715$	28.4405	1.06362
$716$	−21.7995	−0.814684
$717$	0	0
$718$	9.34067	0.348591
$719$	8.45108	0.315172	0.157586	−	0.987505i	$-0.449629\pi$
$719$	8.45108	0.315172	0.157586	+	0.987505i	$0.449629\pi$
$720$	0	0
$721$	−2.37988	−0.0886312
$722$	−38.5754	−1.43563
$723$	0	0
$724$	13.3118	0.494727
$725$	45.7318	1.69843
$726$	0	0
$727$	−23.9328	−0.887617	−0.443809	−	0.896122i	$-0.646373\pi$
$727$	−23.9328	−0.887617	−0.443809	+	0.896122i	$0.646373\pi$
$728$	8.69570	0.322284
$729$	0	0
$730$	42.9953	1.59133
$731$	7.10472	0.262777
$732$	0	0
$733$	−42.8071	−1.58111	−0.790557	−	0.612388i	$-0.790209\pi$
$733$	−42.8071	−1.58111	−0.790557	+	0.612388i	$0.790209\pi$
$734$	14.8704	0.548877
$735$	0	0
$736$	−0.995753	−0.0367039
$737$	65.0066	2.39455
$738$	0	0
$739$	−46.3345	−1.70444	−0.852222	−	0.523181i	$-0.824745\pi$
$739$	−46.3345	−1.70444	−0.852222	+	0.523181i	$0.824745\pi$
$740$	−22.5031	−0.827229
$741$	0	0
$742$	21.1167	0.775219
$743$	−32.0590	−1.17613	−0.588065	−	0.808814i	$-0.700110\pi$
$743$	−32.0590	−1.17613	−0.588065	+	0.808814i	$0.700110\pi$
$744$	0	0
$745$	74.8166	2.74107
$746$	34.7972	1.27402
$747$	0	0
$748$	27.5330	1.00671
$749$	−59.2764	−2.16591
$750$	0	0
$751$	−28.0855	−1.02486	−0.512428	−	0.858730i	$-0.671254\pi$
$751$	−28.0855	−1.02486	−0.512428	+	0.858730i	$0.671254\pi$
$752$	8.16828	0.297867
$753$	0	0
$754$	−15.6171	−0.568741
$755$	−42.9470	−1.56300
$756$	0	0
$757$	−42.9220	−1.56003	−0.780013	−	0.625764i	$-0.784787\pi$
$757$	−42.9220	−1.56003	−0.780013	+	0.625764i	$0.784787\pi$
$758$	−17.1231	−0.621940
$759$	0	0
$760$	25.4058	0.921566
$761$	12.6196	0.457459	0.228729	−	0.973490i	$-0.426543\pi$
$761$	12.6196	0.457459	0.228729	+	0.973490i	$0.426543\pi$
$762$	0	0
$763$	27.9126	1.01051
$764$	−19.9533	−0.721885
$765$	0	0
$766$	9.42211	0.340435
$767$	−8.33664	−0.301019
$768$	0	0
$769$	−27.3474	−0.986171	−0.493085	−	0.869981i	$-0.664131\pi$
$769$	−27.3474	−0.986171	−0.493085	+	0.869981i	$0.664131\pi$
$770$	54.9807	1.98137
$771$	0	0
$772$	1.65872	0.0596987
$773$	19.3733	0.696808	0.348404	−	0.937344i	$-0.386724\pi$
$773$	19.3733	0.696808	0.348404	+	0.937344i	$0.386724\pi$
$774$	0	0
$775$	−46.3023	−1.66323
$776$	−10.7932	−0.387453
$777$	0	0
$778$	36.8405	1.32080
$779$	33.3834	1.19609
$780$	0	0
$781$	−54.2174	−1.94005
$782$	6.84538	0.244790
$783$	0	0
$784$	9.81036	0.350370
$785$	56.8729	2.02988
$786$	0	0
$787$	−12.3016	−0.438503	−0.219252	−	0.975668i	$-0.570362\pi$
$787$	−12.3016	−0.438503	−0.219252	+	0.975668i	$0.570362\pi$
$788$	12.2438	0.436166
$789$	0	0
$790$	−23.4697	−0.835013
$791$	−41.8275	−1.48722
$792$	0	0
$793$	−13.2939	−0.472081
$794$	2.11398	0.0750222
$795$	0	0
$796$	7.54659	0.267482
$797$	−30.6983	−1.08739	−0.543694	−	0.839283i	$-0.682975\pi$
$797$	−30.6983	−1.08739	−0.543694	+	0.839283i	$0.682975\pi$
$798$	0	0
$799$	−56.1535	−1.98657
$800$	−6.21061	−0.219578
$801$	0	0
$802$	−20.3448	−0.718400
$803$	−51.4296	−1.81491
$804$	0	0
$805$	13.6696	0.481789
$806$	15.8119	0.556951
$807$	0	0
$808$	3.92331	0.138022
$809$	53.8671	1.89387	0.946934	−	0.321429i	$-0.104163\pi$
$809$	53.8671	1.89387	0.946934	+	0.321429i	$0.104163\pi$
$810$	0	0
$811$	−46.0297	−1.61632	−0.808160	−	0.588963i	$-0.799536\pi$
$811$	−46.0297	−1.61632	−0.808160	+	0.588963i	$0.799536\pi$
$812$	−30.1906	−1.05948
$813$	0	0
$814$	26.9175	0.943456
$815$	−8.84482	−0.309820
$816$	0	0
$817$	−7.84186	−0.274352
$818$	4.91606	0.171886
$819$	0	0
$820$	−14.7308	−0.514423
$821$	−12.4212	−0.433504	−0.216752	−	0.976227i	$-0.569546\pi$
$821$	−12.4212	−0.433504	−0.216752	+	0.976227i	$0.569546\pi$
$822$	0	0
$823$	−35.8644	−1.25016	−0.625078	−	0.780562i	$-0.714933\pi$
$823$	−35.8644	−1.25016	−0.625078	+	0.780562i	$0.714933\pi$
$824$	−0.580451	−0.0202210
$825$	0	0
$826$	−16.1162	−0.560755
$827$	−21.6475	−0.752757	−0.376379	−	0.926466i	$-0.622831\pi$
$827$	−21.6475	−0.752757	−0.376379	+	0.926466i	$0.622831\pi$
$828$	0	0
$829$	−17.9120	−0.622110	−0.311055	−	0.950392i	$-0.600682\pi$
$829$	−17.9120	−0.622110	−0.311055	+	0.950392i	$0.600682\pi$
$830$	10.2725	0.356562
$831$	0	0
$832$	2.12088	0.0735283
$833$	−67.4421	−2.33673
$834$	0	0
$835$	−24.0224	−0.831329
$836$	−30.3896	−1.05105
$837$	0	0
$838$	−1.84085	−0.0635910
$839$	−30.6313	−1.05751	−0.528755	−	0.848775i	$-0.677341\pi$
$839$	−30.6313	−1.05751	−0.528755	+	0.848775i	$0.677341\pi$
$840$	0	0
$841$	25.2210	0.869689
$842$	2.41138	0.0831018
$843$	0	0
$844$	−27.8101	−0.957264
$845$	28.4662	0.979266
$846$	0	0
$847$	−20.6657	−0.710083
$848$	5.15036	0.176864
$849$	0	0
$850$	42.6953	1.46444
$851$	6.69235	0.229411
$852$	0	0
$853$	52.0422	1.78189	0.890945	−	0.454112i	$-0.150043\pi$
$853$	52.0422	1.78189	0.890945	+	0.454112i	$0.150043\pi$
$854$	−25.6995	−0.879420
$855$	0	0
$856$	−14.4575	−0.494147
$857$	−39.8012	−1.35958	−0.679791	−	0.733406i	$-0.737929\pi$
$857$	−39.8012	−1.35958	−0.679791	+	0.733406i	$0.737929\pi$
$858$	0	0
$859$	10.3436	0.352920	0.176460	−	0.984308i	$-0.443535\pi$
$859$	10.3436	0.352920	0.176460	+	0.984308i	$0.443535\pi$
$860$	3.46031	0.117996
$861$	0	0
$862$	−11.0418	−0.376087
$863$	−24.0236	−0.817772	−0.408886	−	0.912585i	$-0.634083\pi$
$863$	−24.0236	−0.817772	−0.408886	+	0.912585i	$0.634083\pi$
$864$	0	0
$865$	−10.1303	−0.344441
$866$	−9.77874	−0.332295
$867$	0	0
$868$	30.5673	1.03752
$869$	28.0737	0.952334
$870$	0	0
$871$	−34.4244	−1.16643
$872$	6.80789	0.230544
$873$	0	0
$874$	−7.55562	−0.255573
$875$	16.6191	0.561828
$876$	0	0
$877$	−38.3139	−1.29377	−0.646883	−	0.762589i	$-0.723928\pi$
$877$	−38.3139	−1.29377	−0.646883	+	0.762589i	$0.723928\pi$
$878$	23.8200	0.803888
$879$	0	0
$880$	13.4098	0.452044
$881$	−8.20685	−0.276496	−0.138248	−	0.990398i	$-0.544147\pi$
$881$	−8.20685	−0.276496	−0.138248	+	0.990398i	$0.544147\pi$
$882$	0	0
$883$	13.7485	0.462675	0.231337	−	0.972874i	$-0.425690\pi$
$883$	13.7485	0.462675	0.231337	+	0.972874i	$0.425690\pi$
$884$	−14.5802	−0.490384
$885$	0	0
$886$	2.71213	0.0911157
$887$	7.25776	0.243692	0.121846	−	0.992549i	$-0.461119\pi$
$887$	7.25776	0.243692	0.121846	+	0.992549i	$0.461119\pi$
$888$	0	0
$889$	20.2427	0.678917
$890$	−50.8883	−1.70578
$891$	0	0
$892$	16.4175	0.549699
$893$	61.9797	2.07407
$894$	0	0
$895$	72.9895	2.43977
$896$	4.10004	0.136973
$897$	0	0
$898$	11.8073	0.394016
$899$	−54.8975	−1.83093
$900$	0	0
$901$	−35.4066	−1.17956
$902$	17.6206	0.586700
$903$	0	0
$904$	−10.2017	−0.339304
$905$	−44.5707	−1.48158
$906$	0	0
$907$	−23.6659	−0.785814	−0.392907	−	0.919578i	$-0.628530\pi$
$907$	−23.6659	−0.785814	−0.392907	+	0.919578i	$0.628530\pi$
$908$	10.3544	0.343624
$909$	0	0
$910$	−29.1152	−0.965158
$911$	−55.5800	−1.84145	−0.920724	−	0.390216i	$-0.872400\pi$
$911$	−55.5800	−1.84145	−0.920724	+	0.390216i	$0.872400\pi$
$912$	0	0
$913$	−12.2876	−0.406660
$914$	−9.86717	−0.326377
$915$	0	0
$916$	2.16787	0.0716283
$917$	22.7506	0.751291
$918$	0	0
$919$	−29.1941	−0.963025	−0.481512	−	0.876439i	$-0.659912\pi$
$919$	−29.1941	−0.963025	−0.481512	+	0.876439i	$0.659912\pi$
$920$	3.33400	0.109919
$921$	0	0
$922$	−7.68914	−0.253228
$923$	28.7110	0.945033
$924$	0	0
$925$	41.7408	1.37243
$926$	27.3927	0.900181
$927$	0	0
$928$	−7.36349	−0.241718
$929$	−12.8461	−0.421468	−0.210734	−	0.977543i	$-0.567585\pi$
$929$	−12.8461	−0.421468	−0.210734	+	0.977543i	$0.567585\pi$
$930$	0	0
$931$	74.4395	2.43966
$932$	−15.6664	−0.513169
$933$	0	0
$934$	−20.9222	−0.684594
$935$	−92.1866	−3.01482
$936$	0	0
$937$	5.11607	0.167135	0.0835674	−	0.996502i	$-0.473369\pi$
$937$	5.11607	0.167135	0.0835674	+	0.996502i	$0.473369\pi$
$938$	−66.5486	−2.17289
$939$	0	0
$940$	−27.3492	−0.892034
$941$	60.0595	1.95788	0.978942	−	0.204139i	$-0.0654396\pi$
$941$	60.0595	1.95788	0.978942	+	0.204139i	$0.0654396\pi$
$942$	0	0
$943$	4.38091	0.142662
$944$	−3.93074	−0.127935
$945$	0	0
$946$	−4.13912	−0.134574
$947$	−6.57438	−0.213639	−0.106819	−	0.994278i	$-0.534067\pi$
$947$	−6.57438	−0.213639	−0.106819	+	0.994278i	$0.534067\pi$
$948$	0	0
$949$	27.2347	0.884075
$950$	−47.1251	−1.52894
$951$	0	0
$952$	−28.1861	−0.913516
$953$	−44.0804	−1.42790	−0.713952	−	0.700195i	$-0.753097\pi$
$953$	−44.0804	−1.42790	−0.713952	+	0.700195i	$0.753097\pi$
$954$	0	0
$955$	66.8081	2.16186
$956$	−14.5865	−0.471760
$957$	0	0
$958$	6.23231	0.201357
$959$	−44.6561	−1.44202
$960$	0	0
$961$	24.5824	0.792980
$962$	−14.2542	−0.459574
$963$	0	0
$964$	−8.35078	−0.268961
$965$	−5.55378	−0.178782
$966$	0	0
$967$	31.7622	1.02140	0.510701	−	0.859758i	$-0.329386\pi$
$967$	31.7622	1.02140	0.510701	+	0.859758i	$0.329386\pi$
$968$	−5.04037	−0.162003
$969$	0	0
$970$	36.1381	1.16032
$971$	33.3761	1.07109	0.535544	−	0.844507i	$-0.320106\pi$
$971$	33.3761	1.07109	0.535544	+	0.844507i	$0.320106\pi$
$972$	0	0
$973$	−15.3652	−0.492587
$974$	−33.0999	−1.06059
$975$	0	0
$976$	−6.26811	−0.200637
$977$	60.7670	1.94411	0.972055	−	0.234753i	$-0.0754281\pi$
$977$	60.7670	1.94411	0.972055	+	0.234753i	$0.0754281\pi$
$978$	0	0
$979$	60.8710	1.94545
$980$	−32.8473	−1.04927
$981$	0	0
$982$	4.65183	0.148446
$983$	27.0839	0.863844	0.431922	−	0.901911i	$-0.357836\pi$
$983$	27.0839	0.863844	0.431922	+	0.901911i	$0.357836\pi$
$984$	0	0
$985$	−40.9948	−1.30620
$986$	50.6209	1.61210
$987$	0	0
$988$	16.0929	0.511984
$989$	−1.02909	−0.0327231
$990$	0	0
$991$	−1.92075	−0.0610146	−0.0305073	−	0.999535i	$-0.509712\pi$
$991$	−1.92075	−0.0610146	−0.0305073	+	0.999535i	$0.509712\pi$
$992$	7.45536	0.236708
$993$	0	0
$994$	55.5035	1.76046
$995$	−25.2677	−0.801039
$996$	0	0
$997$	−39.2469	−1.24296	−0.621481	−	0.783429i	$-0.713469\pi$
$997$	−39.2469	−1.24296	−0.621481	+	0.783429i	$0.713469\pi$
$998$	−32.3299	−1.02339
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6354.2.a.bk.1.3		10
3.2	odd	2		706.2.a.h.1.5	✓	10
12.11	even	2		5648.2.a.m.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
706.2.a.h.1.5	✓	10	3.2	odd	2
5648.2.a.m.1.6		10	12.11	even	2
6354.2.a.bk.1.3		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6354 = 2 \cdot 3^{2} \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6354.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.34822 q^{5} -4.10004 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.34822 q^{5} -4.10004 q^{7} -1.00000 q^{8} +3.34822 q^{10} -4.00504 q^{11} +2.12088 q^{13} +4.10004 q^{14} +1.00000 q^{16} -6.87458 q^{17} +7.58785 q^{19} -3.34822 q^{20} +4.00504 q^{22} +0.995753 q^{23} +6.21061 q^{25} -2.12088 q^{26} -4.10004 q^{28} +7.36349 q^{29} -7.45536 q^{31} -1.00000 q^{32} +6.87458 q^{34} +13.7279 q^{35} +6.72089 q^{37} -7.58785 q^{38} +3.34822 q^{40} +4.39959 q^{41} -1.03348 q^{43} -4.00504 q^{44} -0.995753 q^{46} +8.16828 q^{47} +9.81036 q^{49} -6.21061 q^{50} +2.12088 q^{52} +5.15036 q^{53} +13.4098 q^{55} +4.10004 q^{56} -7.36349 q^{58} -3.93074 q^{59} -6.26811 q^{61} +7.45536 q^{62} +1.00000 q^{64} -7.10118 q^{65} -16.2312 q^{67} -6.87458 q^{68} -13.7279 q^{70} +13.5373 q^{71} +12.8412 q^{73} -6.72089 q^{74} +7.58785 q^{76} +16.4209 q^{77} -7.00958 q^{79} -3.34822 q^{80} -4.39959 q^{82} +3.06803 q^{83} +23.0176 q^{85} +1.03348 q^{86} +4.00504 q^{88} -15.1986 q^{89} -8.69570 q^{91} +0.995753 q^{92} -8.16828 q^{94} -25.4058 q^{95} +10.7932 q^{97} -9.81036 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 12 q^{7} - 10 q^{8}+O(q^{10}) \) 10 * q - 10 * q^2 + 10 * q^4 - 3 * q^5 - 12 * q^7 - 10 * q^8	\( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 12 q^{7} - 10 q^{8} + 3 q^{10} - 15 q^{11} + 11 q^{13} + 12 q^{14} + 10 q^{16} - 7 q^{17} + 4 q^{19} - 3 q^{20} + 15 q^{22} - 10 q^{23} + 27 q^{25} - 11 q^{26} - 12 q^{28} - 14 q^{29} - q^{31} - 10 q^{32} + 7 q^{34} + 20 q^{35} + 9 q^{37} - 4 q^{38} + 3 q^{40} - 7 q^{41} + 19 q^{43} - 15 q^{44} + 10 q^{46} + 22 q^{47} - 6 q^{49} - 27 q^{50} + 11 q^{52} + 2 q^{53} - 33 q^{55} + 12 q^{56} + 14 q^{58} + 6 q^{59} + 6 q^{61} + q^{62} + 10 q^{64} + 9 q^{65} - 20 q^{67} - 7 q^{68} - 20 q^{70} + 11 q^{71} - 9 q^{73} - 9 q^{74} + 4 q^{76} + 20 q^{77} - 10 q^{79} - 3 q^{80} + 7 q^{82} + 7 q^{83} - 21 q^{85} - 19 q^{86} + 15 q^{88} - 18 q^{89} - 8 q^{91} - 10 q^{92} - 22 q^{94} - 2 q^{95} + 19 q^{97} + 6 q^{98}+O(q^{100}) \) 10 * q - 10 * q^2 + 10 * q^4 - 3 * q^5 - 12 * q^7 - 10 * q^8 + 3 * q^10 - 15 * q^11 + 11 * q^13 + 12 * q^14 + 10 * q^16 - 7 * q^17 + 4 * q^19 - 3 * q^20 + 15 * q^22 - 10 * q^23 + 27 * q^25 - 11 * q^26 - 12 * q^28 - 14 * q^29 - q^31 - 10 * q^32 + 7 * q^34 + 20 * q^35 + 9 * q^37 - 4 * q^38 + 3 * q^40 - 7 * q^41 + 19 * q^43 - 15 * q^44 + 10 * q^46 + 22 * q^47 - 6 * q^49 - 27 * q^50 + 11 * q^52 + 2 * q^53 - 33 * q^55 + 12 * q^56 + 14 * q^58 + 6 * q^59 + 6 * q^61 + q^62 + 10 * q^64 + 9 * q^65 - 20 * q^67 - 7 * q^68 - 20 * q^70 + 11 * q^71 - 9 * q^73 - 9 * q^74 + 4 * q^76 + 20 * q^77 - 10 * q^79 - 3 * q^80 + 7 * q^82 + 7 * q^83 - 21 * q^85 - 19 * q^86 + 15 * q^88 - 18 * q^89 - 8 * q^91 - 10 * q^92 - 22 * q^94 - 2 * q^95 + 19 * q^97 + 6 * q^98

Introduction

L-functions

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Database

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