LMFDB - Embedded newform 8450.2.a.by.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,-1,3,0,-1,-4,3,-4,0,2,-1,0,-4,0,3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.24698 q^{6} -2.80194 q^{7} +1.00000 q^{8} -1.44504 q^{9} +0.554958 q^{11} +1.24698 q^{12} -2.80194 q^{14} +1.00000 q^{16} +3.24698 q^{17} -1.44504 q^{18} -3.24698 q^{19} -3.49396 q^{21} +0.554958 q^{22} -4.04892 q^{23} +1.24698 q^{24} -5.54288 q^{27} -2.80194 q^{28} +0.384043 q^{29} +3.13706 q^{31} +1.00000 q^{32} +0.692021 q^{33} +3.24698 q^{34} -1.44504 q^{36} +11.3666 q^{37} -3.24698 q^{38} -1.96077 q^{41} -3.49396 q^{42} -8.32304 q^{43} +0.554958 q^{44} -4.04892 q^{46} +3.78986 q^{47} +1.24698 q^{48} +0.850855 q^{49} +4.04892 q^{51} -7.92692 q^{53} -5.54288 q^{54} -2.80194 q^{56} -4.04892 q^{57} +0.384043 q^{58} +9.04892 q^{59} -9.04892 q^{61} +3.13706 q^{62} +4.04892 q^{63} +1.00000 q^{64} +0.692021 q^{66} -0.396125 q^{67} +3.24698 q^{68} -5.04892 q^{69} +1.54288 q^{71} -1.44504 q^{72} -4.03684 q^{73} +11.3666 q^{74} -3.24698 q^{76} -1.55496 q^{77} -16.6136 q^{79} -2.57673 q^{81} -1.96077 q^{82} -15.3720 q^{83} -3.49396 q^{84} -8.32304 q^{86} +0.478894 q^{87} +0.554958 q^{88} -11.4668 q^{89} -4.04892 q^{92} +3.91185 q^{93} +3.78986 q^{94} +1.24698 q^{96} -10.1196 q^{97} +0.850855 q^{98} -0.801938 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 5 q^{17} - 4 q^{18} - 5 q^{19} - q^{21} + 2 q^{22} - 3 q^{23} - q^{24} + 2 q^{27} - 4 q^{28}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 5 * q^17 - 4 * q^18 - 5 * q^19 - q^21 + 2 * q^22 - 3 * q^23 - q^24 + 2 * q^27 - 4 * q^28 - 9 * q^29 + 4 * q^31 + 3 * q^32 - 3 * q^33 + 5 * q^34 - 4 * q^36 + 8 * q^37 - 5 * q^38 + 7 * q^41 - q^42 - 5 * q^43 + 2 * q^44 - 3 * q^46 - 12 * q^47 - q^48 - 11 * q^49 + 3 * q^51 + 5 * q^53 + 2 * q^54 - 4 * q^56 - 3 * q^57 - 9 * q^58 + 18 * q^59 - 18 * q^61 + 4 * q^62 + 3 * q^63 + 3 * q^64 - 3 * q^66 - 10 * q^67 + 5 * q^68 - 6 * q^69 - 14 * q^71 - 4 * q^72 + 16 * q^73 + 8 * q^74 - 5 * q^76 - 5 * q^77 - 19 * q^79 - 5 * q^81 + 7 * q^82 - 17 * q^83 - q^84 - 5 * q^86 + 17 * q^87 + 2 * q^88 - 31 * q^89 - 3 * q^92 + 8 * q^93 - 12 * q^94 - q^96 - 9 * q^97 - 11 * q^98 + 2 * q^99

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
$3$	1.24698	0.719944	0.359972	−	0.932963i	$-0.382786\pi$
$3$	1.24698	0.719944	0.359972	+	0.932963i	$0.382786\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.24698	0.509077
$7$	−2.80194	−1.05903	−0.529516	−	0.848300i	$-0.677627\pi$
$7$	−2.80194	−1.05903	−0.529516	+	0.848300i	$0.677627\pi$
$8$	1.00000	0.353553
$9$	−1.44504	−0.481681
$10$	0	0
$11$	0.554958	0.167326	0.0836631	−	0.996494i	$-0.473338\pi$
$11$	0.554958	0.167326	0.0836631	+	0.996494i	$0.473338\pi$
$12$	1.24698	0.359972
$13$	0	0
$13$	0	0
$14$	−2.80194	−0.748849
$15$	0	0
$16$	1.00000	0.250000
$17$	3.24698	0.787508	0.393754	−	0.919216i	$-0.371176\pi$
$17$	3.24698	0.787508	0.393754	+	0.919216i	$0.371176\pi$
$18$	−1.44504	−0.340600
$19$	−3.24698	−0.744908	−0.372454	−	0.928051i	$-0.621484\pi$
$19$	−3.24698	−0.744908	−0.372454	+	0.928051i	$0.621484\pi$
$20$	0	0
$21$	−3.49396	−0.762444
$22$	0.554958	0.118317
$23$	−4.04892	−0.844258	−0.422129	−	0.906536i	$-0.638717\pi$
$23$	−4.04892	−0.844258	−0.422129	+	0.906536i	$0.638717\pi$
$24$	1.24698	0.254539
$25$	0	0
$26$	0	0
$27$	−5.54288	−1.06673
$28$	−2.80194	−0.529516
$29$	0.384043	0.0713150	0.0356575	−	0.999364i	$-0.488647\pi$
$29$	0.384043	0.0713150	0.0356575	+	0.999364i	$0.488647\pi$
$30$	0	0
$31$	3.13706	0.563433	0.281717	−	0.959498i	$-0.409096\pi$
$31$	3.13706	0.563433	0.281717	+	0.959498i	$0.409096\pi$
$32$	1.00000	0.176777
$33$	0.692021	0.120465
$34$	3.24698	0.556852
$35$	0	0
$36$	−1.44504	−0.240840
$37$	11.3666	1.86865	0.934327	−	0.356416i	$-0.116001\pi$
$37$	11.3666	1.86865	0.934327	+	0.356416i	$0.116001\pi$
$38$	−3.24698	−0.526730
$39$	0	0
$40$	0	0
$41$	−1.96077	−0.306221	−0.153111	−	0.988209i	$-0.548929\pi$
$41$	−1.96077	−0.306221	−0.153111	+	0.988209i	$0.548929\pi$
$42$	−3.49396	−0.539130
$43$	−8.32304	−1.26925	−0.634626	−	0.772819i	$-0.718846\pi$
$43$	−8.32304	−1.26925	−0.634626	+	0.772819i	$0.718846\pi$
$44$	0.554958	0.0836631
$45$	0	0
$46$	−4.04892	−0.596980
$47$	3.78986	0.552807	0.276404	−	0.961042i	$-0.410857\pi$
$47$	3.78986	0.552807	0.276404	+	0.961042i	$0.410857\pi$
$48$	1.24698	0.179986
$49$	0.850855	0.121551
$50$	0	0
$51$	4.04892	0.566962
$52$	0	0
$53$	−7.92692	−1.08885	−0.544423	−	0.838811i	$-0.683251\pi$
$53$	−7.92692	−1.08885	−0.544423	+	0.838811i	$0.683251\pi$
$54$	−5.54288	−0.754290
$55$	0	0
$56$	−2.80194	−0.374425
$57$	−4.04892	−0.536292
$58$	0.384043	0.0504273
$59$	9.04892	1.17807	0.589034	−	0.808108i	$-0.299508\pi$
$59$	9.04892	1.17807	0.589034	+	0.808108i	$0.299508\pi$
$60$	0	0
$61$	−9.04892	−1.15860	−0.579298	−	0.815116i	$-0.696673\pi$
$61$	−9.04892	−1.15860	−0.579298	+	0.815116i	$0.696673\pi$
$62$	3.13706	0.398407
$63$	4.04892	0.510116
$64$	1.00000	0.125000
$65$	0	0
$66$	0.692021	0.0851820
$67$	−0.396125	−0.0483943	−0.0241972	−	0.999707i	$-0.507703\pi$
$67$	−0.396125	−0.0483943	−0.0241972	+	0.999707i	$0.507703\pi$
$68$	3.24698	0.393754
$69$	−5.04892	−0.607818
$70$	0	0
$71$	1.54288	0.183106	0.0915529	−	0.995800i	$-0.470817\pi$
$71$	1.54288	0.183106	0.0915529	+	0.995800i	$0.470817\pi$
$72$	−1.44504	−0.170300
$73$	−4.03684	−0.472476	−0.236238	−	0.971695i	$-0.575914\pi$
$73$	−4.03684	−0.472476	−0.236238	+	0.971695i	$0.575914\pi$
$74$	11.3666	1.32134
$75$	0	0
$76$	−3.24698	−0.372454
$77$	−1.55496	−0.177204
$78$	0	0
$79$	−16.6136	−1.86917	−0.934586	−	0.355737i	$-0.884230\pi$
$79$	−16.6136	−1.86917	−0.934586	+	0.355737i	$0.884230\pi$
$80$	0	0
$81$	−2.57673	−0.286303
$82$	−1.96077	−0.216531
$83$	−15.3720	−1.68729	−0.843646	−	0.536900i	$-0.819595\pi$
$83$	−15.3720	−1.68729	−0.843646	+	0.536900i	$0.819595\pi$
$84$	−3.49396	−0.381222
$85$	0	0
$86$	−8.32304	−0.897497
$87$	0.478894	0.0513428
$88$	0.554958	0.0591587
$89$	−11.4668	−1.21548	−0.607740	−	0.794136i	$-0.707924\pi$
$89$	−11.4668	−1.21548	−0.607740	+	0.794136i	$0.707924\pi$
$90$	0	0
$91$	0	0
$92$	−4.04892	−0.422129
$93$	3.91185	0.405640
$94$	3.78986	0.390894
$95$	0	0
$96$	1.24698	0.127269
$97$	−10.1196	−1.02749	−0.513745	−	0.857943i	$-0.671742\pi$
$97$	−10.1196	−1.02749	−0.513745	+	0.857943i	$0.671742\pi$
$98$	0.850855	0.0859493
$99$	−0.801938	−0.0805978
$100$	0	0
$101$	−16.7778	−1.66945	−0.834725	−	0.550666i	$-0.814374\pi$
$101$	−16.7778	−1.66945	−0.834725	+	0.550666i	$0.814374\pi$
$102$	4.04892	0.400903
$103$	−9.89440	−0.974924	−0.487462	−	0.873144i	$-0.662077\pi$
$103$	−9.89440	−0.974924	−0.487462	+	0.873144i	$0.662077\pi$
$104$	0	0
$105$	0	0
$106$	−7.92692	−0.769930
$107$	8.56465	0.827976	0.413988	−	0.910282i	$-0.364136\pi$
$107$	8.56465	0.827976	0.413988	+	0.910282i	$0.364136\pi$
$108$	−5.54288	−0.533364
$109$	3.40044	0.325703	0.162851	−	0.986651i	$-0.447931\pi$
$109$	3.40044	0.325703	0.162851	+	0.986651i	$0.447931\pi$
$110$	0	0
$111$	14.1739	1.34533
$112$	−2.80194	−0.264758
$113$	3.87263	0.364306	0.182153	−	0.983270i	$-0.441693\pi$
$113$	3.87263	0.364306	0.182153	+	0.983270i	$0.441693\pi$
$114$	−4.04892	−0.379216
$115$	0	0
$116$	0.384043	0.0356575
$117$	0	0
$118$	9.04892	0.833020
$119$	−9.09783	−0.833997
$120$	0	0
$121$	−10.6920	−0.972002
$122$	−9.04892	−0.819250
$123$	−2.44504	−0.220462
$124$	3.13706	0.281717
$125$	0	0
$126$	4.04892	0.360706
$127$	9.85623	0.874599	0.437300	−	0.899316i	$-0.355935\pi$
$127$	9.85623	0.874599	0.437300	+	0.899316i	$0.355935\pi$
$128$	1.00000	0.0883883
$129$	−10.3787	−0.913791
$130$	0	0
$131$	−10.2403	−0.894697	−0.447348	−	0.894360i	$-0.647632\pi$
$131$	−10.2403	−0.894697	−0.447348	+	0.894360i	$0.647632\pi$
$132$	0.692021	0.0602327
$133$	9.09783	0.788882
$134$	−0.396125	−0.0342199
$135$	0	0
$136$	3.24698	0.278426
$137$	−21.5036	−1.83718	−0.918590	−	0.395211i	$-0.870671\pi$
$137$	−21.5036	−1.83718	−0.918590	+	0.395211i	$0.870671\pi$
$138$	−5.04892	−0.429792
$139$	10.0954	0.856284	0.428142	−	0.903711i	$-0.359168\pi$
$139$	10.0954	0.856284	0.428142	+	0.903711i	$0.359168\pi$
$140$	0	0
$141$	4.72587	0.397990
$142$	1.54288	0.129475
$143$	0	0
$144$	−1.44504	−0.120420
$145$	0	0
$146$	−4.03684	−0.334091
$147$	1.06100	0.0875097
$148$	11.3666	0.934327
$149$	−19.0489	−1.56055	−0.780274	−	0.625438i	$-0.784920\pi$
$149$	−19.0489	−1.56055	−0.780274	+	0.625438i	$0.784920\pi$
$150$	0	0
$151$	5.17092	0.420803	0.210402	−	0.977615i	$-0.432523\pi$
$151$	5.17092	0.420803	0.210402	+	0.977615i	$0.432523\pi$
$152$	−3.24698	−0.263365
$153$	−4.69202	−0.379327
$154$	−1.55496	−0.125302
$155$	0	0
$156$	0	0
$157$	10.9608	0.874765	0.437382	−	0.899276i	$-0.355906\pi$
$157$	10.9608	0.874765	0.437382	+	0.899276i	$0.355906\pi$
$158$	−16.6136	−1.32170
$159$	−9.88471	−0.783908
$160$	0	0
$161$	11.3448	0.894097
$162$	−2.57673	−0.202447
$163$	10.4601	0.819299	0.409649	−	0.912243i	$-0.365651\pi$
$163$	10.4601	0.819299	0.409649	+	0.912243i	$0.365651\pi$
$164$	−1.96077	−0.153111
$165$	0	0
$166$	−15.3720	−1.19310
$167$	9.83340	0.760931	0.380466	−	0.924795i	$-0.375764\pi$
$167$	9.83340	0.760931	0.380466	+	0.924795i	$0.375764\pi$
$168$	−3.49396	−0.269565
$169$	0	0
$170$	0	0
$171$	4.69202	0.358808
$172$	−8.32304	−0.634626
$173$	−8.63102	−0.656204	−0.328102	−	0.944642i	$-0.606409\pi$
$173$	−8.63102	−0.656204	−0.328102	+	0.944642i	$0.606409\pi$
$174$	0.478894	0.0363048
$175$	0	0
$176$	0.554958	0.0418315
$177$	11.2838	0.848143
$178$	−11.4668	−0.859474
$179$	16.5472	1.23679	0.618397	−	0.785865i	$-0.287782\pi$
$179$	16.5472	1.23679	0.618397	+	0.785865i	$0.287782\pi$
$180$	0	0
$181$	−21.0858	−1.56729	−0.783646	−	0.621208i	$-0.786642\pi$
$181$	−21.0858	−1.56729	−0.783646	+	0.621208i	$0.786642\pi$
$182$	0	0
$183$	−11.2838	−0.834124
$184$	−4.04892	−0.298490
$185$	0	0
$186$	3.91185	0.286831
$187$	1.80194	0.131771
$188$	3.78986	0.276404
$189$	15.5308	1.12970
$190$	0	0
$191$	−4.55496	−0.329585	−0.164793	−	0.986328i	$-0.552695\pi$
$191$	−4.55496	−0.329585	−0.164793	+	0.986328i	$0.552695\pi$
$192$	1.24698	0.0899930
$193$	14.6896	1.05738	0.528691	−	0.848814i	$-0.322683\pi$
$193$	14.6896	1.05738	0.528691	+	0.848814i	$0.322683\pi$
$194$	−10.1196	−0.726545
$195$	0	0
$196$	0.850855	0.0607754
$197$	−9.97823	−0.710919	−0.355460	−	0.934692i	$-0.615676\pi$
$197$	−9.97823	−0.710919	−0.355460	+	0.934692i	$0.615676\pi$
$198$	−0.801938	−0.0569912
$199$	−27.3448	−1.93842	−0.969211	−	0.246231i	$-0.920808\pi$
$199$	−27.3448	−1.93842	−0.969211	+	0.246231i	$0.920808\pi$
$200$	0	0
$201$	−0.493959	−0.0348412
$202$	−16.7778	−1.18048
$203$	−1.07606	−0.0755249
$204$	4.04892	0.283481
$205$	0	0
$206$	−9.89440	−0.689375
$207$	5.85086	0.406663
$208$	0	0
$209$	−1.80194	−0.124643
$210$	0	0
$211$	18.7168	1.28852	0.644258	−	0.764808i	$-0.277166\pi$
$211$	18.7168	1.28852	0.644258	+	0.764808i	$0.277166\pi$
$212$	−7.92692	−0.544423
$213$	1.92394	0.131826
$214$	8.56465	0.585467
$215$	0	0
$216$	−5.54288	−0.377145
$217$	−8.78986	−0.596694
$218$	3.40044	0.230307
$219$	−5.03385	−0.340156
$220$	0	0
$221$	0	0
$222$	14.1739	0.951290
$223$	−11.8485	−0.793432	−0.396716	−	0.917941i	$-0.629850\pi$
$223$	−11.8485	−0.793432	−0.396716	+	0.917941i	$0.629850\pi$
$224$	−2.80194	−0.187212
$225$	0	0
$226$	3.87263	0.257603
$227$	−0.510353	−0.0338733	−0.0169366	−	0.999857i	$-0.505391\pi$
$227$	−0.510353	−0.0338733	−0.0169366	+	0.999857i	$0.505391\pi$
$228$	−4.04892	−0.268146
$229$	−17.2295	−1.13856	−0.569279	−	0.822144i	$-0.692778\pi$
$229$	−17.2295	−1.13856	−0.569279	+	0.822144i	$0.692778\pi$
$230$	0	0
$231$	−1.93900	−0.127577
$232$	0.384043	0.0252137
$233$	−10.4222	−0.682781	−0.341391	−	0.939921i	$-0.610898\pi$
$233$	−10.4222	−0.682781	−0.341391	+	0.939921i	$0.610898\pi$
$234$	0	0
$235$	0	0
$236$	9.04892	0.589034
$237$	−20.7168	−1.34570
$238$	−9.09783	−0.589725
$239$	28.5448	1.84641	0.923205	−	0.384309i	$-0.125560\pi$
$239$	28.5448	1.84641	0.923205	+	0.384309i	$0.125560\pi$
$240$	0	0
$241$	15.7071	1.01178	0.505891	−	0.862597i	$-0.331164\pi$
$241$	15.7071	1.01178	0.505891	+	0.862597i	$0.331164\pi$
$242$	−10.6920	−0.687309
$243$	13.4155	0.860605
$244$	−9.04892	−0.579298
$245$	0	0
$246$	−2.44504	−0.155890
$247$	0	0
$248$	3.13706	0.199204
$249$	−19.1685	−1.21476
$250$	0	0
$251$	7.49157	0.472863	0.236432	−	0.971648i	$-0.424022\pi$
$251$	7.49157	0.472863	0.236432	+	0.971648i	$0.424022\pi$
$252$	4.04892	0.255058
$253$	−2.24698	−0.141266
$254$	9.85623	0.618435
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.1793	1.25875	0.629374	−	0.777102i	$-0.283311\pi$
$257$	20.1793	1.25875	0.629374	+	0.777102i	$0.283311\pi$
$258$	−10.3787	−0.646148
$259$	−31.8485	−1.97897
$260$	0	0
$261$	−0.554958	−0.0343510
$262$	−10.2403	−0.632646
$263$	3.30559	0.203831	0.101916	−	0.994793i	$-0.467503\pi$
$263$	3.30559	0.203831	0.101916	+	0.994793i	$0.467503\pi$
$264$	0.692021	0.0425910
$265$	0	0
$266$	9.09783	0.557824
$267$	−14.2989	−0.875077
$268$	−0.396125	−0.0241972
$269$	10.8092	0.659051	0.329525	−	0.944147i	$-0.393111\pi$
$269$	10.8092	0.659051	0.329525	+	0.944147i	$0.393111\pi$
$270$	0	0
$271$	21.8659	1.32826	0.664130	−	0.747617i	$-0.268802\pi$
$271$	21.8659	1.32826	0.664130	+	0.747617i	$0.268802\pi$
$272$	3.24698	0.196877
$273$	0	0
$274$	−21.5036	−1.29908
$275$	0	0
$276$	−5.04892	−0.303909
$277$	8.14914	0.489635	0.244817	−	0.969569i	$-0.421272\pi$
$277$	8.14914	0.489635	0.244817	+	0.969569i	$0.421272\pi$
$278$	10.0954	0.605484
$279$	−4.53319	−0.271395
$280$	0	0
$281$	23.6189	1.40899	0.704494	−	0.709710i	$-0.251174\pi$
$281$	23.6189	1.40899	0.704494	+	0.709710i	$0.251174\pi$
$282$	4.72587	0.281422
$283$	1.24698	0.0741252	0.0370626	−	0.999313i	$-0.488200\pi$
$283$	1.24698	0.0741252	0.0370626	+	0.999313i	$0.488200\pi$
$284$	1.54288	0.0915529
$285$	0	0
$286$	0	0
$287$	5.49396	0.324298
$288$	−1.44504	−0.0851499
$289$	−6.45712	−0.379831
$290$	0	0
$291$	−12.6189	−0.739735
$292$	−4.03684	−0.236238
$293$	17.5144	1.02320	0.511601	−	0.859223i	$-0.329053\pi$
$293$	17.5144	1.02320	0.511601	+	0.859223i	$0.329053\pi$
$294$	1.06100	0.0618787
$295$	0	0
$296$	11.3666	0.660669
$297$	−3.07606	−0.178491
$298$	−19.0489	−1.10347
$299$	0	0
$300$	0	0
$301$	23.3207	1.34418
$302$	5.17092	0.297553
$303$	−20.9215	−1.20191
$304$	−3.24698	−0.186227
$305$	0	0
$306$	−4.69202	−0.268225
$307$	8.42626	0.480912	0.240456	−	0.970660i	$-0.422703\pi$
$307$	8.42626	0.480912	0.240456	+	0.970660i	$0.422703\pi$
$308$	−1.55496	−0.0886020
$309$	−12.3381	−0.701891
$310$	0	0
$311$	4.63342	0.262737	0.131368	−	0.991334i	$-0.458063\pi$
$311$	4.63342	0.262737	0.131368	+	0.991334i	$0.458063\pi$
$312$	0	0
$313$	−31.8485	−1.80018	−0.900091	−	0.435702i	$-0.856500\pi$
$313$	−31.8485	−1.80018	−0.900091	+	0.435702i	$0.856500\pi$
$314$	10.9608	0.618552
$315$	0	0
$316$	−16.6136	−0.934586
$317$	−16.0532	−0.901639	−0.450820	−	0.892615i	$-0.648868\pi$
$317$	−16.0532	−0.901639	−0.450820	+	0.892615i	$0.648868\pi$
$318$	−9.88471	−0.554307
$319$	0.213128	0.0119329
$320$	0	0
$321$	10.6799	0.596096
$322$	11.3448	0.632222
$323$	−10.5429	−0.586621
$324$	−2.57673	−0.143152
$325$	0	0
$326$	10.4601	0.579332
$327$	4.24027	0.234488
$328$	−1.96077	−0.108265
$329$	−10.6189	−0.585441
$330$	0	0
$331$	−1.87369	−0.102987	−0.0514937	−	0.998673i	$-0.516398\pi$
$331$	−1.87369	−0.102987	−0.0514937	+	0.998673i	$0.516398\pi$
$332$	−15.3720	−0.843646
$333$	−16.4252	−0.900095
$334$	9.83340	0.538060
$335$	0	0
$336$	−3.49396	−0.190611
$337$	7.10454	0.387009	0.193504	−	0.981099i	$-0.438015\pi$
$337$	7.10454	0.387009	0.193504	+	0.981099i	$0.438015\pi$
$338$	0	0
$339$	4.82908	0.262280
$340$	0	0
$341$	1.74094	0.0942771
$342$	4.69202	0.253715
$343$	17.2295	0.930307
$344$	−8.32304	−0.448748
$345$	0	0
$346$	−8.63102	−0.464007
$347$	−22.6112	−1.21383	−0.606916	−	0.794766i	$-0.707593\pi$
$347$	−22.6112	−1.21383	−0.606916	+	0.794766i	$0.707593\pi$
$348$	0.478894	0.0256714
$349$	−21.6082	−1.15666	−0.578330	−	0.815803i	$-0.696295\pi$
$349$	−21.6082	−1.15666	−0.578330	+	0.815803i	$0.696295\pi$
$350$	0	0
$351$	0	0
$352$	0.554958	0.0295794
$353$	−1.10992	−0.0590749	−0.0295374	−	0.999564i	$-0.509403\pi$
$353$	−1.10992	−0.0590749	−0.0295374	+	0.999564i	$0.509403\pi$
$354$	11.2838	0.599728
$355$	0	0
$356$	−11.4668	−0.607740
$357$	−11.3448	−0.600431
$358$	16.5472	0.874546
$359$	−23.9342	−1.26320	−0.631600	−	0.775295i	$-0.717601\pi$
$359$	−23.9342	−1.26320	−0.631600	+	0.775295i	$0.717601\pi$
$360$	0	0
$361$	−8.45712	−0.445112
$362$	−21.0858	−1.10824
$363$	−13.3327	−0.699787
$364$	0	0
$365$	0	0
$366$	−11.2838	−0.589814
$367$	14.7114	0.767929	0.383964	−	0.923348i	$-0.374559\pi$
$367$	14.7114	0.767929	0.383964	+	0.923348i	$0.374559\pi$
$368$	−4.04892	−0.211064
$369$	2.83340	0.147501
$370$	0	0
$371$	22.2107	1.15312
$372$	3.91185	0.202820
$373$	14.7278	0.762576	0.381288	−	0.924456i	$-0.375481\pi$
$373$	14.7278	0.762576	0.381288	+	0.924456i	$0.375481\pi$
$374$	1.80194	0.0931760
$375$	0	0
$376$	3.78986	0.195447
$377$	0	0
$378$	15.5308	0.798818
$379$	−34.5937	−1.77696	−0.888480	−	0.458916i	$-0.848238\pi$
$379$	−34.5937	−1.77696	−0.888480	+	0.458916i	$0.848238\pi$
$380$	0	0
$381$	12.2905	0.629662
$382$	−4.55496	−0.233052
$383$	−9.69069	−0.495171	−0.247586	−	0.968866i	$-0.579637\pi$
$383$	−9.69069	−0.495171	−0.247586	+	0.968866i	$0.579637\pi$
$384$	1.24698	0.0636347
$385$	0	0
$386$	14.6896	0.747682
$387$	12.0271	0.611374
$388$	−10.1196	−0.513745
$389$	−29.1511	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$389$	−29.1511	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$390$	0	0
$391$	−13.1468	−0.664860
$392$	0.850855	0.0429747
$393$	−12.7694	−0.644132
$394$	−9.97823	−0.502696
$395$	0	0
$396$	−0.801938	−0.0402989
$397$	1.31767	0.0661318	0.0330659	−	0.999453i	$-0.489473\pi$
$397$	1.31767	0.0661318	0.0330659	+	0.999453i	$0.489473\pi$
$398$	−27.3448	−1.37067
$399$	11.3448	0.567951
$400$	0	0
$401$	−27.3327	−1.36493	−0.682466	−	0.730918i	$-0.739092\pi$
$401$	−27.3327	−1.36493	−0.682466	+	0.730918i	$0.739092\pi$
$402$	−0.493959	−0.0246364
$403$	0	0
$404$	−16.7778	−0.834725
$405$	0	0
$406$	−1.07606	−0.0534042
$407$	6.30798	0.312675
$408$	4.04892	0.200451
$409$	28.8340	1.42575	0.712874	−	0.701292i	$-0.247393\pi$
$409$	28.8340	1.42575	0.712874	+	0.701292i	$0.247393\pi$
$410$	0	0
$411$	−26.8146	−1.32267
$412$	−9.89440	−0.487462
$413$	−25.3545	−1.24761
$414$	5.85086	0.287554
$415$	0	0
$416$	0	0
$417$	12.5888	0.616477
$418$	−1.80194	−0.0881357
$419$	25.4480	1.24322	0.621609	−	0.783328i	$-0.286479\pi$
$419$	25.4480	1.24322	0.621609	+	0.783328i	$0.286479\pi$
$420$	0	0
$421$	11.1220	0.542053	0.271027	−	0.962572i	$-0.412637\pi$
$421$	11.1220	0.542053	0.271027	+	0.962572i	$0.412637\pi$
$422$	18.7168	0.911118
$423$	−5.47650	−0.266277
$424$	−7.92692	−0.384965
$425$	0	0
$426$	1.92394	0.0932150
$427$	25.3545	1.22699
$428$	8.56465	0.413988
$429$	0	0
$430$	0	0
$431$	−22.4373	−1.08077	−0.540383	−	0.841419i	$-0.681721\pi$
$431$	−22.4373	−1.08077	−0.540383	+	0.841419i	$0.681721\pi$
$432$	−5.54288	−0.266682
$433$	40.3008	1.93673	0.968366	−	0.249533i	$-0.0802769\pi$
$433$	40.3008	1.93673	0.968366	+	0.249533i	$0.0802769\pi$
$434$	−8.78986	−0.421927
$435$	0	0
$436$	3.40044	0.162851
$437$	13.1468	0.628894
$438$	−5.03385	−0.240527
$439$	0.346142	0.0165205	0.00826023	−	0.999966i	$-0.497371\pi$
$439$	0.346142	0.0165205	0.00826023	+	0.999966i	$0.497371\pi$
$440$	0	0
$441$	−1.22952	−0.0585486
$442$	0	0
$443$	−30.0881	−1.42953	−0.714765	−	0.699364i	$-0.753467\pi$
$443$	−30.0881	−1.42953	−0.714765	+	0.699364i	$0.753467\pi$
$444$	14.1739	0.672663
$445$	0	0
$446$	−11.8485	−0.561041
$447$	−23.7536	−1.12351
$448$	−2.80194	−0.132379
$449$	36.4825	1.72171	0.860857	−	0.508847i	$-0.169928\pi$
$449$	36.4825	1.72171	0.860857	+	0.508847i	$0.169928\pi$
$450$	0	0
$451$	−1.08815	−0.0512388
$452$	3.87263	0.182153
$453$	6.44803	0.302955
$454$	−0.510353	−0.0239520
$455$	0	0
$456$	−4.04892	−0.189608
$457$	17.6571	0.825965	0.412982	−	0.910739i	$-0.364487\pi$
$457$	17.6571	0.825965	0.412982	+	0.910739i	$0.364487\pi$
$458$	−17.2295	−0.805083
$459$	−17.9976	−0.840056
$460$	0	0
$461$	−6.97823	−0.325009	−0.162504	−	0.986708i	$-0.551957\pi$
$461$	−6.97823	−0.325009	−0.162504	+	0.986708i	$0.551957\pi$
$462$	−1.93900	−0.0902105
$463$	40.5526	1.88464	0.942319	−	0.334717i	$-0.108641\pi$
$463$	40.5526	1.88464	0.942319	+	0.334717i	$0.108641\pi$
$464$	0.384043	0.0178287
$465$	0	0
$466$	−10.4222	−0.482799
$467$	38.5676	1.78470	0.892349	−	0.451347i	$-0.149056\pi$
$467$	38.5676	1.78470	0.892349	+	0.451347i	$0.149056\pi$
$468$	0	0
$469$	1.10992	0.0512512
$470$	0	0
$471$	13.6679	0.629782
$472$	9.04892	0.416510
$473$	−4.61894	−0.212379
$474$	−20.7168	−0.951553
$475$	0	0
$476$	−9.09783	−0.416999
$477$	11.4547	0.524476
$478$	28.5448	1.30561
$479$	5.73125	0.261868	0.130934	−	0.991391i	$-0.458202\pi$
$479$	5.73125	0.261868	0.130934	+	0.991391i	$0.458202\pi$
$480$	0	0
$481$	0	0
$482$	15.7071	0.715438
$483$	14.1468	0.643700
$484$	−10.6920	−0.486001
$485$	0	0
$486$	13.4155	0.608540
$487$	27.6692	1.25381	0.626905	−	0.779096i	$-0.284321\pi$
$487$	27.6692	1.25381	0.626905	+	0.779096i	$0.284321\pi$
$488$	−9.04892	−0.409625
$489$	13.0435	0.589849
$490$	0	0
$491$	7.25906	0.327597	0.163798	−	0.986494i	$-0.447625\pi$
$491$	7.25906	0.327597	0.163798	+	0.986494i	$0.447625\pi$
$492$	−2.44504	−0.110231
$493$	1.24698	0.0561611
$494$	0	0
$495$	0	0
$496$	3.13706	0.140858
$497$	−4.32304	−0.193915
$498$	−19.1685	−0.858962
$499$	−8.19136	−0.366696	−0.183348	−	0.983048i	$-0.558693\pi$
$499$	−8.19136	−0.366696	−0.183348	+	0.983048i	$0.558693\pi$
$500$	0	0
$501$	12.2620	0.547828
$502$	7.49157	0.334365
$503$	36.6655	1.63483	0.817416	−	0.576048i	$-0.195406\pi$
$503$	36.6655	1.63483	0.817416	+	0.576048i	$0.195406\pi$
$504$	4.04892	0.180353
$505$	0	0
$506$	−2.24698	−0.0998904
$507$	0	0
$508$	9.85623	0.437300
$509$	−1.74392	−0.0772980	−0.0386490	−	0.999253i	$-0.512305\pi$
$509$	−1.74392	−0.0772980	−0.0386490	+	0.999253i	$0.512305\pi$
$510$	0	0
$511$	11.3110	0.500368
$512$	1.00000	0.0441942
$513$	17.9976	0.794614
$514$	20.1793	0.890070
$515$	0	0
$516$	−10.3787	−0.456895
$517$	2.10321	0.0924991
$518$	−31.8485	−1.39934
$519$	−10.7627	−0.472430
$520$	0	0
$521$	−15.2664	−0.668831	−0.334416	−	0.942426i	$-0.608539\pi$
$521$	−15.2664	−0.668831	−0.334416	+	0.942426i	$0.608539\pi$
$522$	−0.554958	−0.0242899
$523$	−2.29291	−0.100262	−0.0501310	−	0.998743i	$-0.515964\pi$
$523$	−2.29291	−0.100262	−0.0501310	+	0.998743i	$0.515964\pi$
$524$	−10.2403	−0.447348
$525$	0	0
$526$	3.30559	0.144130
$527$	10.1860	0.443708
$528$	0.692021	0.0301164
$529$	−6.60627	−0.287229
$530$	0	0
$531$	−13.0761	−0.567453
$532$	9.09783	0.394441
$533$	0	0
$534$	−14.2989	−0.618773
$535$	0	0
$536$	−0.396125	−0.0171100
$537$	20.6340	0.890423
$538$	10.8092	0.466019
$539$	0.472189	0.0203386
$540$	0	0
$541$	28.6612	1.23224	0.616120	−	0.787653i	$-0.288704\pi$
$541$	28.6612	1.23224	0.616120	+	0.787653i	$0.288704\pi$
$542$	21.8659	0.939222
$543$	−26.2935	−1.12836
$544$	3.24698	0.139213
$545$	0	0
$546$	0	0
$547$	−6.16554	−0.263619	−0.131810	−	0.991275i	$-0.542079\pi$
$547$	−6.16554	−0.263619	−0.131810	+	0.991275i	$0.542079\pi$
$548$	−21.5036	−0.918590
$549$	13.0761	0.558073
$550$	0	0
$551$	−1.24698	−0.0531231
$552$	−5.04892	−0.214896
$553$	46.5502	1.97951
$554$	8.14914	0.346224
$555$	0	0
$556$	10.0954	0.428142
$557$	7.97823	0.338048	0.169024	−	0.985612i	$-0.445938\pi$
$557$	7.97823	0.338048	0.169024	+	0.985612i	$0.445938\pi$
$558$	−4.53319	−0.191905
$559$	0	0
$560$	0	0
$561$	2.24698	0.0948676
$562$	23.6189	0.996305
$563$	7.92453	0.333979	0.166989	−	0.985959i	$-0.446595\pi$
$563$	7.92453	0.333979	0.166989	+	0.985959i	$0.446595\pi$
$564$	4.72587	0.198995
$565$	0	0
$566$	1.24698	0.0524145
$567$	7.21983	0.303204
$568$	1.54288	0.0647377
$569$	−36.1250	−1.51444	−0.757219	−	0.653161i	$-0.773442\pi$
$569$	−36.1250	−1.51444	−0.757219	+	0.653161i	$0.773442\pi$
$570$	0	0
$571$	−31.4336	−1.31545	−0.657727	−	0.753257i	$-0.728482\pi$
$571$	−31.4336	−1.31545	−0.657727	+	0.753257i	$0.728482\pi$
$572$	0	0
$573$	−5.67994	−0.237283
$574$	5.49396	0.229313
$575$	0	0
$576$	−1.44504	−0.0602101
$577$	−17.6886	−0.736385	−0.368192	−	0.929750i	$-0.620023\pi$
$577$	−17.6886	−0.736385	−0.368192	+	0.929750i	$0.620023\pi$
$578$	−6.45712	−0.268581
$579$	18.3177	0.761256
$580$	0	0
$581$	43.0713	1.78690
$582$	−12.6189	−0.523072
$583$	−4.39911	−0.182192
$584$	−4.03684	−0.167045
$585$	0	0
$586$	17.5144	0.723513
$587$	35.1497	1.45078	0.725392	−	0.688336i	$-0.241658\pi$
$587$	35.1497	1.45078	0.725392	+	0.688336i	$0.241658\pi$
$588$	1.06100	0.0437549
$589$	−10.1860	−0.419706
$590$	0	0
$591$	−12.4426	−0.511822
$592$	11.3666	0.467164
$593$	10.2553	0.421136	0.210568	−	0.977579i	$-0.432469\pi$
$593$	10.2553	0.421136	0.210568	+	0.977579i	$0.432469\pi$
$594$	−3.07606	−0.126212
$595$	0	0
$596$	−19.0489	−0.780274
$597$	−34.0984	−1.39556
$598$	0	0
$599$	−16.6843	−0.681700	−0.340850	−	0.940118i	$-0.610715\pi$
$599$	−16.6843	−0.681700	−0.340850	+	0.940118i	$0.610715\pi$
$600$	0	0
$601$	12.7144	0.518630	0.259315	−	0.965793i	$-0.416503\pi$
$601$	12.7144	0.518630	0.259315	+	0.965793i	$0.416503\pi$
$602$	23.3207	0.950479
$603$	0.572417	0.0233106
$604$	5.17092	0.210402
$605$	0	0
$606$	−20.9215	−0.849880
$607$	14.2155	0.576990	0.288495	−	0.957481i	$-0.406845\pi$
$607$	14.2155	0.576990	0.288495	+	0.957481i	$0.406845\pi$
$608$	−3.24698	−0.131682
$609$	−1.34183	−0.0543737
$610$	0	0
$611$	0	0
$612$	−4.69202	−0.189664
$613$	7.10992	0.287167	0.143583	−	0.989638i	$-0.454137\pi$
$613$	7.10992	0.287167	0.143583	+	0.989638i	$0.454137\pi$
$614$	8.42626	0.340056
$615$	0	0
$616$	−1.55496	−0.0626510
$617$	−21.7590	−0.875984	−0.437992	−	0.898979i	$-0.644310\pi$
$617$	−21.7590	−0.875984	−0.437992	+	0.898979i	$0.644310\pi$
$618$	−12.3381	−0.496312
$619$	−7.92021	−0.318340	−0.159170	−	0.987251i	$-0.550882\pi$
$619$	−7.92021	−0.318340	−0.159170	+	0.987251i	$0.550882\pi$
$620$	0	0
$621$	22.4426	0.900592
$622$	4.63342	0.185783
$623$	32.1293	1.28723
$624$	0	0
$625$	0	0
$626$	−31.8485	−1.27292
$627$	−2.24698	−0.0897357
$628$	10.9608	0.437382
$629$	36.9071	1.47158
$630$	0	0
$631$	8.75973	0.348719	0.174360	−	0.984682i	$-0.444214\pi$
$631$	8.75973	0.348719	0.174360	+	0.984682i	$0.444214\pi$
$632$	−16.6136	−0.660852
$633$	23.3394	0.927659
$634$	−16.0532	−0.637555
$635$	0	0
$636$	−9.88471	−0.391954
$637$	0	0
$638$	0.213128	0.00843781
$639$	−2.22952	−0.0881985
$640$	0	0
$641$	−32.4916	−1.28334	−0.641670	−	0.766981i	$-0.721758\pi$
$641$	−32.4916	−1.28334	−0.641670	+	0.766981i	$0.721758\pi$
$642$	10.6799	0.421504
$643$	−36.6045	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$643$	−36.6045	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$644$	11.3448	0.447048
$645$	0	0
$646$	−10.5429	−0.414804
$647$	−9.09246	−0.357461	−0.178731	−	0.983898i	$-0.557199\pi$
$647$	−9.09246	−0.357461	−0.178731	+	0.983898i	$0.557199\pi$
$648$	−2.57673	−0.101223
$649$	5.02177	0.197122
$650$	0	0
$651$	−10.9608	−0.429586
$652$	10.4601	0.409649
$653$	35.4161	1.38594	0.692969	−	0.720967i	$-0.256302\pi$
$653$	35.4161	1.38594	0.692969	+	0.720967i	$0.256302\pi$
$654$	4.24027	0.165808
$655$	0	0
$656$	−1.96077	−0.0765553
$657$	5.83340	0.227582
$658$	−10.6189	−0.413969
$659$	0.783151	0.0305072	0.0152536	−	0.999884i	$-0.495144\pi$
$659$	0.783151	0.0305072	0.0152536	+	0.999884i	$0.495144\pi$
$660$	0	0
$661$	19.1159	0.743522	0.371761	−	0.928329i	$-0.378754\pi$
$661$	19.1159	0.743522	0.371761	+	0.928329i	$0.378754\pi$
$662$	−1.87369	−0.0728230
$663$	0	0
$664$	−15.3720	−0.596548
$665$	0	0
$666$	−16.4252	−0.636463
$667$	−1.55496	−0.0602082
$668$	9.83340	0.380466
$669$	−14.7748	−0.571226
$670$	0	0
$671$	−5.02177	−0.193863
$672$	−3.49396	−0.134782
$673$	6.98121	0.269106	0.134553	−	0.990906i	$-0.457040\pi$
$673$	6.98121	0.269106	0.134553	+	0.990906i	$0.457040\pi$
$674$	7.10454	0.273657
$675$	0	0
$676$	0	0
$677$	−43.0640	−1.65508	−0.827542	−	0.561404i	$-0.810261\pi$
$677$	−43.0640	−1.65508	−0.827542	+	0.561404i	$0.810261\pi$
$678$	4.82908	0.185460
$679$	28.3545	1.08815
$680$	0	0
$681$	−0.636399	−0.0243869
$682$	1.74094	0.0666640
$683$	10.1328	0.387719	0.193859	−	0.981029i	$-0.437899\pi$
$683$	10.1328	0.387719	0.193859	+	0.981029i	$0.437899\pi$
$684$	4.69202	0.179404
$685$	0	0
$686$	17.2295	0.657826
$687$	−21.4849	−0.819699
$688$	−8.32304	−0.317313
$689$	0	0
$690$	0	0
$691$	−17.4166	−0.662557	−0.331279	−	0.943533i	$-0.607480\pi$
$691$	−17.4166	−0.662557	−0.331279	+	0.943533i	$0.607480\pi$
$692$	−8.63102	−0.328102
$693$	2.24698	0.0853557
$694$	−22.6112	−0.858308
$695$	0	0
$696$	0.478894	0.0181524
$697$	−6.36658	−0.241152
$698$	−21.6082	−0.817882
$699$	−12.9963	−0.491564
$700$	0	0
$701$	27.3749	1.03394	0.516969	−	0.856004i	$-0.327060\pi$
$701$	27.3749	1.03394	0.516969	+	0.856004i	$0.327060\pi$
$702$	0	0
$703$	−36.9071	−1.39198
$704$	0.554958	0.0209158
$705$	0	0
$706$	−1.10992	−0.0417722
$707$	47.0103	1.76800
$708$	11.2838	0.424072
$709$	18.1390	0.681224	0.340612	−	0.940204i	$-0.389366\pi$
$709$	18.1390	0.681224	0.340612	+	0.940204i	$0.389366\pi$
$710$	0	0
$711$	24.0073	0.900344
$712$	−11.4668	−0.429737
$713$	−12.7017	−0.475683
$714$	−11.3448	−0.424569
$715$	0	0
$716$	16.5472	0.618397
$717$	35.5948	1.32931
$718$	−23.9342	−0.893217
$719$	−11.9444	−0.445450	−0.222725	−	0.974881i	$-0.571495\pi$
$719$	−11.9444	−0.445450	−0.222725	+	0.974881i	$0.571495\pi$
$720$	0	0
$721$	27.7235	1.03248
$722$	−8.45712	−0.314742
$723$	19.5864	0.728427
$724$	−21.0858	−0.783646
$725$	0	0
$726$	−13.3327	−0.494824
$727$	36.2954	1.34612	0.673061	−	0.739587i	$-0.264979\pi$
$727$	36.2954	1.34612	0.673061	+	0.739587i	$0.264979\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	0	0
$731$	−27.0248	−0.999547
$732$	−11.2838	−0.417062
$733$	7.17331	0.264952	0.132476	−	0.991186i	$-0.457707\pi$
$733$	7.17331	0.264952	0.132476	+	0.991186i	$0.457707\pi$
$734$	14.7114	0.543008
$735$	0	0
$736$	−4.04892	−0.149245
$737$	−0.219833	−0.00809764
$738$	2.83340	0.104299
$739$	−4.39134	−0.161538	−0.0807690	−	0.996733i	$-0.525738\pi$
$739$	−4.39134	−0.161538	−0.0807690	+	0.996733i	$0.525738\pi$
$740$	0	0
$741$	0	0
$742$	22.2107	0.815382
$743$	6.68233	0.245151	0.122576	−	0.992459i	$-0.460885\pi$
$743$	6.68233	0.245151	0.122576	+	0.992459i	$0.460885\pi$
$744$	3.91185	0.143416
$745$	0	0
$746$	14.7278	0.539223
$747$	22.2131	0.812736
$748$	1.80194	0.0658854
$749$	−23.9976	−0.876853
$750$	0	0
$751$	8.16182	0.297829	0.148914	−	0.988850i	$-0.452422\pi$
$751$	8.16182	0.297829	0.148914	+	0.988850i	$0.452422\pi$
$752$	3.78986	0.138202
$753$	9.34183	0.340435
$754$	0	0
$755$	0	0
$756$	15.5308	0.564850
$757$	11.5603	0.420168	0.210084	−	0.977683i	$-0.432626\pi$
$757$	11.5603	0.420168	0.210084	+	0.977683i	$0.432626\pi$
$758$	−34.5937	−1.25650
$759$	−2.80194	−0.101704
$760$	0	0
$761$	−27.1390	−0.983787	−0.491894	−	0.870655i	$-0.663695\pi$
$761$	−27.1390	−0.983787	−0.491894	+	0.870655i	$0.663695\pi$
$762$	12.2905	0.445239
$763$	−9.52781	−0.344930
$764$	−4.55496	−0.164793
$765$	0	0
$766$	−9.69069	−0.350139
$767$	0	0
$768$	1.24698	0.0449965
$769$	−25.7681	−0.929221	−0.464610	−	0.885515i	$-0.653806\pi$
$769$	−25.7681	−0.929221	−0.464610	+	0.885515i	$0.653806\pi$
$770$	0	0
$771$	25.1631	0.906229
$772$	14.6896	0.528691
$773$	−15.6176	−0.561726	−0.280863	−	0.959748i	$-0.590621\pi$
$773$	−15.6176	−0.561726	−0.280863	+	0.959748i	$0.590621\pi$
$774$	12.0271	0.432307
$775$	0	0
$776$	−10.1196	−0.363273
$777$	−39.7144	−1.42475
$778$	−29.1511	−1.04512
$779$	6.36658	0.228107
$780$	0	0
$781$	0.856232	0.0306384
$782$	−13.1468	−0.470127
$783$	−2.12870	−0.0760736
$784$	0.850855	0.0303877
$785$	0	0
$786$	−12.7694	−0.455470
$787$	3.21121	0.114467	0.0572336	−	0.998361i	$-0.481772\pi$
$787$	3.21121	0.114467	0.0572336	+	0.998361i	$0.481772\pi$
$788$	−9.97823	−0.355460
$789$	4.12200	0.146747
$790$	0	0
$791$	−10.8509	−0.385812
$792$	−0.801938	−0.0284956
$793$	0	0
$794$	1.31767	0.0467623
$795$	0	0
$796$	−27.3448	−0.969211
$797$	43.9197	1.55572	0.777859	−	0.628439i	$-0.216306\pi$
$797$	43.9197	1.55572	0.777859	+	0.628439i	$0.216306\pi$
$798$	11.3448	0.401602
$799$	12.3056	0.435340
$800$	0	0
$801$	16.5700	0.585473
$802$	−27.3327	−0.965152
$803$	−2.24027	−0.0790576
$804$	−0.493959	−0.0174206
$805$	0	0
$806$	0	0
$807$	13.4789	0.474480
$808$	−16.7778	−0.590240
$809$	−30.9385	−1.08774	−0.543870	−	0.839169i	$-0.683042\pi$
$809$	−30.9385	−1.08774	−0.543870	+	0.839169i	$0.683042\pi$
$810$	0	0
$811$	−23.4470	−0.823334	−0.411667	−	0.911334i	$-0.635053\pi$
$811$	−23.4470	−0.823334	−0.411667	+	0.911334i	$0.635053\pi$
$812$	−1.07606	−0.0377625
$813$	27.2664	0.956273
$814$	6.30798	0.221095
$815$	0	0
$816$	4.04892	0.141740
$817$	27.0248	0.945476
$818$	28.8340	1.00816
$819$	0	0
$820$	0	0
$821$	36.9976	1.29123	0.645613	−	0.763665i	$-0.276602\pi$
$821$	36.9976	1.29123	0.645613	+	0.763665i	$0.276602\pi$
$822$	−26.8146	−0.935267
$823$	26.4838	0.923167	0.461584	−	0.887097i	$-0.347281\pi$
$823$	26.4838	0.923167	0.461584	+	0.887097i	$0.347281\pi$
$824$	−9.89440	−0.344688
$825$	0	0
$826$	−25.3545	−0.882196
$827$	21.3806	0.743476	0.371738	−	0.928338i	$-0.378762\pi$
$827$	21.3806	0.743476	0.371738	+	0.928338i	$0.378762\pi$
$828$	5.85086	0.203331
$829$	−49.6233	−1.72349	−0.861743	−	0.507344i	$-0.830627\pi$
$829$	−49.6233	−1.72349	−0.861743	+	0.507344i	$0.830627\pi$
$830$	0	0
$831$	10.1618	0.352510
$832$	0	0
$833$	2.76271	0.0957222
$834$	12.5888	0.435915
$835$	0	0
$836$	−1.80194	−0.0623213
$837$	−17.3884	−0.601029
$838$	25.4480	0.879087
$839$	−47.8980	−1.65362	−0.826811	−	0.562480i	$-0.809847\pi$
$839$	−47.8980	−1.65362	−0.826811	+	0.562480i	$0.809847\pi$
$840$	0	0
$841$	−28.8525	−0.994914
$842$	11.1220	0.383289
$843$	29.4523	1.01439
$844$	18.7168	0.644258
$845$	0	0
$846$	−5.47650	−0.188286
$847$	29.9584	1.02938
$848$	−7.92692	−0.272212
$849$	1.55496	0.0533660
$850$	0	0
$851$	−46.0224	−1.57763
$852$	1.92394	0.0659129
$853$	−7.75063	−0.265376	−0.132688	−	0.991158i	$-0.542361\pi$
$853$	−7.75063	−0.265376	−0.132688	+	0.991158i	$0.542361\pi$
$854$	25.3545	0.867613
$855$	0	0
$856$	8.56465	0.292734
$857$	28.0054	0.956645	0.478323	−	0.878184i	$-0.341245\pi$
$857$	28.0054	0.956645	0.478323	+	0.878184i	$0.341245\pi$
$858$	0	0
$859$	−2.21014	−0.0754091	−0.0377046	−	0.999289i	$-0.512005\pi$
$859$	−2.21014	−0.0754091	−0.0377046	+	0.999289i	$0.512005\pi$
$860$	0	0
$861$	6.85086	0.233477
$862$	−22.4373	−0.764217
$863$	−26.9071	−0.915927	−0.457964	−	0.888971i	$-0.651421\pi$
$863$	−26.9071	−0.915927	−0.457964	+	0.888971i	$0.651421\pi$
$864$	−5.54288	−0.188572
$865$	0	0
$866$	40.3008	1.36948
$867$	−8.05190	−0.273457
$868$	−8.78986	−0.298347
$869$	−9.21983	−0.312761
$870$	0	0
$871$	0	0
$872$	3.40044	0.115153
$873$	14.6233	0.494922
$874$	13.1468	0.444696
$875$	0	0
$876$	−5.03385	−0.170078
$877$	12.5961	0.425340	0.212670	−	0.977124i	$-0.431784\pi$
$877$	12.5961	0.425340	0.212670	+	0.977124i	$0.431784\pi$
$878$	0.346142	0.0116817
$879$	21.8401	0.736648
$880$	0	0
$881$	−27.5883	−0.929475	−0.464737	−	0.885449i	$-0.653851\pi$
$881$	−27.5883	−0.929475	−0.464737	+	0.885449i	$0.653851\pi$
$882$	−1.22952	−0.0414001
$883$	−18.9506	−0.637739	−0.318869	−	0.947799i	$-0.603303\pi$
$883$	−18.9506	−0.637739	−0.318869	+	0.947799i	$0.603303\pi$
$884$	0	0
$885$	0	0
$886$	−30.0881	−1.01083
$887$	−8.36658	−0.280922	−0.140461	−	0.990086i	$-0.544859\pi$
$887$	−8.36658	−0.280922	−0.140461	+	0.990086i	$0.544859\pi$
$888$	14.1739	0.475645
$889$	−27.6165	−0.926229
$890$	0	0
$891$	−1.42998	−0.0479060
$892$	−11.8485	−0.396716
$893$	−12.3056	−0.411791
$894$	−23.7536	−0.794440
$895$	0	0
$896$	−2.80194	−0.0936062
$897$	0	0
$898$	36.4825	1.21744
$899$	1.20477	0.0401812
$900$	0	0
$901$	−25.7385	−0.857475
$902$	−1.08815	−0.0362313
$903$	29.0804	0.967734
$904$	3.87263	0.128802
$905$	0	0
$906$	6.44803	0.214221
$907$	−7.72992	−0.256668	−0.128334	−	0.991731i	$-0.540963\pi$
$907$	−7.72992	−0.256668	−0.128334	+	0.991731i	$0.540963\pi$
$908$	−0.510353	−0.0169366
$909$	24.2446	0.804142
$910$	0	0
$911$	−22.1140	−0.732668	−0.366334	−	0.930483i	$-0.619387\pi$
$911$	−22.1140	−0.732668	−0.366334	+	0.930483i	$0.619387\pi$
$912$	−4.04892	−0.134073
$913$	−8.53079	−0.282328
$914$	17.6571	0.584045
$915$	0	0
$916$	−17.2295	−0.569279
$917$	28.6926	0.947514
$918$	−17.9976	−0.594010
$919$	−16.2067	−0.534609	−0.267305	−	0.963612i	$-0.586133\pi$
$919$	−16.2067	−0.534609	−0.267305	+	0.963612i	$0.586133\pi$
$920$	0	0
$921$	10.5074	0.346230
$922$	−6.97823	−0.229816
$923$	0	0
$924$	−1.93900	−0.0637885
$925$	0	0
$926$	40.5526	1.33264
$927$	14.2978	0.469602
$928$	0.384043	0.0126068
$929$	28.9541	0.949952	0.474976	−	0.879999i	$-0.342457\pi$
$929$	28.9541	0.949952	0.474976	+	0.879999i	$0.342457\pi$
$930$	0	0
$931$	−2.76271	−0.0905441
$932$	−10.4222	−0.341391
$933$	5.77777	0.189156
$934$	38.5676	1.26197
$935$	0	0
$936$	0	0
$937$	45.5400	1.48773	0.743864	−	0.668331i	$-0.232991\pi$
$937$	45.5400	1.48773	0.743864	+	0.668331i	$0.232991\pi$
$938$	1.10992	0.0362401
$939$	−39.7144	−1.29603
$940$	0	0
$941$	−25.6853	−0.837317	−0.418659	−	0.908144i	$-0.637500\pi$
$941$	−25.6853	−0.837317	−0.418659	+	0.908144i	$0.637500\pi$
$942$	13.6679	0.445323
$943$	7.93900	0.258529
$944$	9.04892	0.294517
$945$	0	0
$946$	−4.61894	−0.150175
$947$	−46.7434	−1.51896	−0.759479	−	0.650532i	$-0.774546\pi$
$947$	−46.7434	−1.51896	−0.759479	+	0.650532i	$0.774546\pi$
$948$	−20.7168	−0.672850
$949$	0	0
$950$	0	0
$951$	−20.0180	−0.649130
$952$	−9.09783	−0.294863
$953$	2.37973	0.0770871	0.0385435	−	0.999257i	$-0.487728\pi$
$953$	2.37973	0.0770871	0.0385435	+	0.999257i	$0.487728\pi$
$954$	11.4547	0.370861
$955$	0	0
$956$	28.5448	0.923205
$957$	0.265766	0.00859099
$958$	5.73125	0.185168
$959$	60.2519	1.94563
$960$	0	0
$961$	−21.1588	−0.682543
$962$	0	0
$963$	−12.3763	−0.398820
$964$	15.7071	0.505891
$965$	0	0
$966$	14.1468	0.455164
$967$	30.8388	0.991708	0.495854	−	0.868406i	$-0.334855\pi$
$967$	30.8388	0.991708	0.495854	+	0.868406i	$0.334855\pi$
$968$	−10.6920	−0.343655
$969$	−13.1468	−0.422335
$970$	0	0
$971$	13.2701	0.425857	0.212929	−	0.977068i	$-0.431700\pi$
$971$	13.2701	0.425857	0.212929	+	0.977068i	$0.431700\pi$
$972$	13.4155	0.430302
$973$	−28.2868	−0.906833
$974$	27.6692	0.886578
$975$	0	0
$976$	−9.04892	−0.289649
$977$	−32.0428	−1.02514	−0.512570	−	0.858645i	$-0.671307\pi$
$977$	−32.0428	−1.02514	−0.512570	+	0.858645i	$0.671307\pi$
$978$	13.0435	0.417086
$979$	−6.36360	−0.203382
$980$	0	0
$981$	−4.91377	−0.156885
$982$	7.25906	0.231646
$983$	9.85623	0.314365	0.157182	−	0.987570i	$-0.449759\pi$
$983$	9.85623	0.314365	0.157182	+	0.987570i	$0.449759\pi$
$984$	−2.44504	−0.0779451
$985$	0	0
$986$	1.24698	0.0397119
$987$	−13.2416	−0.421485
$988$	0	0
$989$	33.6993	1.07158
$990$	0	0
$991$	−16.5536	−0.525843	−0.262922	−	0.964817i	$-0.584686\pi$
$991$	−16.5536	−0.525843	−0.262922	+	0.964817i	$0.584686\pi$
$992$	3.13706	0.0996019
$993$	−2.33645	−0.0741451
$994$	−4.32304	−0.137119
$995$	0	0
$996$	−19.1685	−0.607378
$997$	−13.0406	−0.412999	−0.206499	−	0.978447i	$-0.566207\pi$
$997$	−13.0406	−0.412999	−0.206499	+	0.978447i	$0.566207\pi$
$998$	−8.19136	−0.259293
$999$	−63.0036	−1.99334

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.by.1.3	yes	3
5.4	even	2		8450.2.a.bw.1.1	yes	3
13.12	even	2		8450.2.a.bp.1.3	✓	3
65.64	even	2		8450.2.a.cf.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8450.2.a.bp.1.3	✓	3	13.12	even	2
8450.2.a.bw.1.1	yes	3	5.4	even	2
8450.2.a.by.1.3	yes	3	1.1	even	1	trivial
8450.2.a.cf.1.1	yes	3	65.64	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +1.24698 q^{6} -2.80194 q^{7} +1.00000 q^{8} -1.44504 q^{9} +0.554958 q^{11} +1.24698 q^{12} -2.80194 q^{14} +1.00000 q^{16} +3.24698 q^{17} -1.44504 q^{18} -3.24698 q^{19} -3.49396 q^{21} +0.554958 q^{22} -4.04892 q^{23} +1.24698 q^{24} -5.54288 q^{27} -2.80194 q^{28} +0.384043 q^{29} +3.13706 q^{31} +1.00000 q^{32} +0.692021 q^{33} +3.24698 q^{34} -1.44504 q^{36} +11.3666 q^{37} -3.24698 q^{38} -1.96077 q^{41} -3.49396 q^{42} -8.32304 q^{43} +0.554958 q^{44} -4.04892 q^{46} +3.78986 q^{47} +1.24698 q^{48} +0.850855 q^{49} +4.04892 q^{51} -7.92692 q^{53} -5.54288 q^{54} -2.80194 q^{56} -4.04892 q^{57} +0.384043 q^{58} +9.04892 q^{59} -9.04892 q^{61} +3.13706 q^{62} +4.04892 q^{63} +1.00000 q^{64} +0.692021 q^{66} -0.396125 q^{67} +3.24698 q^{68} -5.04892 q^{69} +1.54288 q^{71} -1.44504 q^{72} -4.03684 q^{73} +11.3666 q^{74} -3.24698 q^{76} -1.55496 q^{77} -16.6136 q^{79} -2.57673 q^{81} -1.96077 q^{82} -15.3720 q^{83} -3.49396 q^{84} -8.32304 q^{86} +0.478894 q^{87} +0.554958 q^{88} -11.4668 q^{89} -4.04892 q^{92} +3.91185 q^{93} +3.78986 q^{94} +1.24698 q^{96} -10.1196 q^{97} +0.850855 q^{98} -0.801938 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 5 q^{17} - 4 q^{18} - 5 q^{19} - q^{21} + 2 q^{22} - 3 q^{23} - q^{24} + 2 q^{27} - 4 q^{28}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 5 * q^17 - 4 * q^18 - 5 * q^19 - q^21 + 2 * q^22 - 3 * q^23 - q^24 + 2 * q^27 - 4 * q^28 - 9 * q^29 + 4 * q^31 + 3 * q^32 - 3 * q^33 + 5 * q^34 - 4 * q^36 + 8 * q^37 - 5 * q^38 + 7 * q^41 - q^42 - 5 * q^43 + 2 * q^44 - 3 * q^46 - 12 * q^47 - q^48 - 11 * q^49 + 3 * q^51 + 5 * q^53 + 2 * q^54 - 4 * q^56 - 3 * q^57 - 9 * q^58 + 18 * q^59 - 18 * q^61 + 4 * q^62 + 3 * q^63 + 3 * q^64 - 3 * q^66 - 10 * q^67 + 5 * q^68 - 6 * q^69 - 14 * q^71 - 4 * q^72 + 16 * q^73 + 8 * q^74 - 5 * q^76 - 5 * q^77 - 19 * q^79 - 5 * q^81 + 7 * q^82 - 17 * q^83 - q^84 - 5 * q^86 + 17 * q^87 + 2 * q^88 - 31 * q^89 - 3 * q^92 + 8 * q^93 - 12 * q^94 - q^96 - 9 * q^97 - 11 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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