LMFDB - Embedded newform 3325.2.a.y.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3325,2,Mod(1,3325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3325 = 5^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-2,3,12,0,9,-8,0,7,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5502586721$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 24x^{5} + 36x^{4} - 70x^{3} - 20x^{2} + 32x + 3 $ x^8 - 2x^7 - 12x^6 + 24x^5 + 36x^4 - 70x^3 - 20x^2 + 32*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 665)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.84638$ of defining polynomial
Character	$\chi$	$=$	3325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.84638 q^{2} -1.55242 q^{3} +1.40913 q^{4} +2.86637 q^{6} -1.00000 q^{7} +1.09097 q^{8} -0.589984 q^{9} -0.635141 q^{11} -2.18756 q^{12} -5.57250 q^{13} +1.84638 q^{14} -4.83261 q^{16} -2.22744 q^{17} +1.08934 q^{18} +1.00000 q^{19} +1.55242 q^{21} +1.17271 q^{22} -6.31775 q^{23} -1.69365 q^{24} +10.2890 q^{26} +5.57317 q^{27} -1.40913 q^{28} -7.00533 q^{29} -7.19570 q^{31} +6.74091 q^{32} +0.986008 q^{33} +4.11270 q^{34} -0.831364 q^{36} +5.78472 q^{37} -1.84638 q^{38} +8.65087 q^{39} -7.94318 q^{41} -2.86637 q^{42} -3.12028 q^{43} -0.894996 q^{44} +11.6650 q^{46} +6.01342 q^{47} +7.50226 q^{48} +1.00000 q^{49} +3.45792 q^{51} -7.85237 q^{52} +3.72017 q^{53} -10.2902 q^{54} -1.09097 q^{56} -1.55242 q^{57} +12.9345 q^{58} -12.0350 q^{59} +2.35446 q^{61} +13.2860 q^{62} +0.589984 q^{63} -2.78107 q^{64} -1.82055 q^{66} -13.3961 q^{67} -3.13875 q^{68} +9.80782 q^{69} +5.20182 q^{71} -0.643657 q^{72} -1.04516 q^{73} -10.6808 q^{74} +1.40913 q^{76} +0.635141 q^{77} -15.9728 q^{78} -1.52502 q^{79} -6.88197 q^{81} +14.6661 q^{82} -14.9418 q^{83} +2.18756 q^{84} +5.76123 q^{86} +10.8752 q^{87} -0.692923 q^{88} +6.46580 q^{89} +5.57250 q^{91} -8.90253 q^{92} +11.1708 q^{93} -11.1031 q^{94} -10.4647 q^{96} +0.286485 q^{97} -1.84638 q^{98} +0.374723 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 3 q^{3} + 12 q^{4} + 9 q^{6} - 8 q^{7} + 7 q^{9} + 16 q^{11} + 19 q^{12} - 2 q^{13} + 2 q^{14} + 24 q^{16} + 7 q^{17} + 5 q^{18} + 8 q^{19} - 3 q^{21} + 10 q^{22} - 7 q^{23} + q^{24} - 6 q^{26}+ \cdots + 60 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 + 3 * q^3 + 12 * q^4 + 9 * q^6 - 8 * q^7 + 7 * q^9 + 16 * q^11 + 19 * q^12 - 2 * q^13 + 2 * q^14 + 24 * q^16 + 7 * q^17 + 5 * q^18 + 8 * q^19 - 3 * q^21 + 10 * q^22 - 7 * q^23 + q^24 - 6 * q^26 + 15 * q^27 - 12 * q^28 + 6 * q^29 + 6 * q^31 + 18 * q^32 + 6 * q^33 - 5 * q^34 + q^36 - 3 * q^37 - 2 * q^38 + 8 * q^39 - 7 * q^41 - 9 * q^42 - 23 * q^43 + 40 * q^44 + 13 * q^46 + 9 * q^48 + 8 * q^49 + 7 * q^51 + 2 * q^52 + 11 * q^53 - 5 * q^54 + 3 * q^57 - 52 * q^58 + 4 * q^59 + 18 * q^61 + 24 * q^62 - 7 * q^63 + 32 * q^64 + 12 * q^66 - 4 * q^67 + 13 * q^68 - 41 * q^69 + 13 * q^71 + 13 * q^72 + q^73 - 37 * q^74 + 12 * q^76 - 16 * q^77 - 28 * q^78 + 10 * q^79 + 12 * q^81 + 7 * q^82 + 20 * q^83 - 19 * q^84 + 7 * q^86 - 6 * q^87 + 70 * q^88 - 2 * q^89 + 2 * q^91 + 43 * q^92 + 16 * q^93 - 36 * q^94 + 33 * q^96 + 42 * q^97 - 2 * q^98 + 60 * 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 $

$2$	−1.84638	−1.30559	−0.652795	−	0.757535i	$-0.726404\pi$
$2$	−1.84638	−1.30559	−0.652795	+	0.757535i	$0.726404\pi$
$3$	−1.55242	−0.896292	−0.448146	−	0.893960i	$-0.647915\pi$
$3$	−1.55242	−0.896292	−0.448146	+	0.893960i	$0.647915\pi$
$4$	1.40913	0.704565
$5$	0	0
$5$	0	0
$6$	2.86637	1.17019
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.09097	0.385717
$9$	−0.589984	−0.196661
$10$	0	0
$11$	−0.635141	−0.191502	−0.0957512	−	0.995405i	$-0.530525\pi$
$11$	−0.635141	−0.191502	−0.0957512	+	0.995405i	$0.530525\pi$
$12$	−2.18756	−0.631495
$13$	−5.57250	−1.54553	−0.772766	−	0.634691i	$-0.781128\pi$
$13$	−5.57250	−1.54553	−0.772766	+	0.634691i	$0.781128\pi$
$14$	1.84638	0.493467
$15$	0	0
$16$	−4.83261	−1.20815
$17$	−2.22744	−0.540233	−0.270116	−	0.962828i	$-0.587062\pi$
$17$	−2.22744	−0.540233	−0.270116	+	0.962828i	$0.587062\pi$
$18$	1.08934	0.256759
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.55242	0.338766
$22$	1.17271	0.250023
$23$	−6.31775	−1.31734	−0.658671	−	0.752431i	$-0.728881\pi$
$23$	−6.31775	−1.31734	−0.658671	+	0.752431i	$0.728881\pi$
$24$	−1.69365	−0.345715
$25$	0	0
$26$	10.2890	2.01783
$27$	5.57317	1.07256
$28$	−1.40913	−0.266300
$29$	−7.00533	−1.30086	−0.650428	−	0.759568i	$-0.725411\pi$
$29$	−7.00533	−1.30086	−0.650428	+	0.759568i	$0.725411\pi$
$30$	0	0
$31$	−7.19570	−1.29239	−0.646193	−	0.763174i	$-0.723640\pi$
$31$	−7.19570	−1.29239	−0.646193	+	0.763174i	$0.723640\pi$
$32$	6.74091	1.19164
$33$	0.986008	0.171642
$34$	4.11270	0.705323
$35$	0	0
$36$	−0.831364	−0.138561
$37$	5.78472	0.951002	0.475501	−	0.879715i	$-0.342267\pi$
$37$	5.78472	0.951002	0.475501	+	0.879715i	$0.342267\pi$
$38$	−1.84638	−0.299523
$39$	8.65087	1.38525
$40$	0	0
$41$	−7.94318	−1.24052	−0.620258	−	0.784398i	$-0.712972\pi$
$41$	−7.94318	−1.24052	−0.620258	+	0.784398i	$0.712972\pi$
$42$	−2.86637	−0.442290
$43$	−3.12028	−0.475838	−0.237919	−	0.971285i	$-0.576465\pi$
$43$	−3.12028	−0.475838	−0.237919	+	0.971285i	$0.576465\pi$
$44$	−0.894996	−0.134926
$45$	0	0
$46$	11.6650	1.71991
$47$	6.01342	0.877147	0.438574	−	0.898695i	$-0.355484\pi$
$47$	6.01342	0.877147	0.438574	+	0.898695i	$0.355484\pi$
$48$	7.50226	1.08286
$49$	1.00000	0.142857
$50$	0	0
$51$	3.45792	0.484206
$52$	−7.85237	−1.08893
$53$	3.72017	0.511005	0.255502	−	0.966808i	$-0.417759\pi$
$53$	3.72017	0.511005	0.255502	+	0.966808i	$0.417759\pi$
$54$	−10.2902	−1.40032
$55$	0	0
$56$	−1.09097	−0.145787
$57$	−1.55242	−0.205623
$58$	12.9345	1.69839
$59$	−12.0350	−1.56683	−0.783415	−	0.621499i	$-0.786524\pi$
$59$	−12.0350	−1.56683	−0.783415	+	0.621499i	$0.786524\pi$
$60$	0	0
$61$	2.35446	0.301457	0.150729	−	0.988575i	$-0.451838\pi$
$61$	2.35446	0.301457	0.150729	+	0.988575i	$0.451838\pi$
$62$	13.2860	1.68733
$63$	0.589984	0.0743310
$64$	−2.78107	−0.347633
$65$	0	0
$66$	−1.82055	−0.224094
$67$	−13.3961	−1.63659	−0.818296	−	0.574797i	$-0.805081\pi$
$67$	−13.3961	−1.63659	−0.818296	+	0.574797i	$0.805081\pi$
$68$	−3.13875	−0.380629
$69$	9.80782	1.18072
$70$	0	0
$71$	5.20182	0.617342	0.308671	−	0.951169i	$-0.400116\pi$
$71$	5.20182	0.617342	0.308671	+	0.951169i	$0.400116\pi$
$72$	−0.643657	−0.0758557
$73$	−1.04516	−0.122326	−0.0611632	−	0.998128i	$-0.519481\pi$
$73$	−1.04516	−0.122326	−0.0611632	+	0.998128i	$0.519481\pi$
$74$	−10.6808	−1.24162
$75$	0	0
$76$	1.40913	0.161638
$77$	0.635141	0.0723811
$78$	−15.9728	−1.80857
$79$	−1.52502	−0.171578	−0.0857888	−	0.996313i	$-0.527341\pi$
$79$	−1.52502	−0.171578	−0.0857888	+	0.996313i	$0.527341\pi$
$80$	0	0
$81$	−6.88197	−0.764663
$82$	14.6661	1.61961
$83$	−14.9418	−1.64008	−0.820041	−	0.572305i	$-0.806049\pi$
$83$	−14.9418	−1.64008	−0.820041	+	0.572305i	$0.806049\pi$
$84$	2.18756	0.238683
$85$	0	0
$86$	5.76123	0.621250
$87$	10.8752	1.16595
$88$	−0.692923	−0.0738658
$89$	6.46580	0.685373	0.342687	−	0.939450i	$-0.388663\pi$
$89$	6.46580	0.685373	0.342687	+	0.939450i	$0.388663\pi$
$90$	0	0
$91$	5.57250	0.584156
$92$	−8.90253	−0.928153
$93$	11.1708	1.15835
$94$	−11.1031	−1.14519
$95$	0	0
$96$	−10.4647	−1.06805
$97$	0.286485	0.0290881	0.0145441	−	0.999894i	$-0.495370\pi$
$97$	0.286485	0.0290881	0.0145441	+	0.999894i	$0.495370\pi$
$98$	−1.84638	−0.186513
$99$	0.374723	0.0376611
$100$	0	0
$101$	−5.92982	−0.590039	−0.295020	−	0.955491i	$-0.595326\pi$
$101$	−5.92982	−0.590039	−0.295020	+	0.955491i	$0.595326\pi$
$102$	−6.38465	−0.632175
$103$	−5.03528	−0.496141	−0.248070	−	0.968742i	$-0.579796\pi$
$103$	−5.03528	−0.496141	−0.248070	+	0.968742i	$0.579796\pi$
$104$	−6.07945	−0.596139
$105$	0	0
$106$	−6.86886	−0.667163
$107$	15.2709	1.47630	0.738148	−	0.674639i	$-0.235701\pi$
$107$	15.2709	1.47630	0.738148	+	0.674639i	$0.235701\pi$
$108$	7.85332	0.755686
$109$	−3.90048	−0.373598	−0.186799	−	0.982398i	$-0.559811\pi$
$109$	−3.90048	−0.373598	−0.186799	+	0.982398i	$0.559811\pi$
$110$	0	0
$111$	−8.98033	−0.852375
$112$	4.83261	0.456639
$113$	−11.5157	−1.08330	−0.541652	−	0.840603i	$-0.682201\pi$
$113$	−11.5157	−1.08330	−0.541652	+	0.840603i	$0.682201\pi$
$114$	2.86637	0.268460
$115$	0	0
$116$	−9.87141	−0.916538
$117$	3.28768	0.303947
$118$	22.2213	2.04564
$119$	2.22744	0.204189
$120$	0	0
$121$	−10.5966	−0.963327
$122$	−4.34723	−0.393579
$123$	12.3312	1.11186
$124$	−10.1397	−0.910569
$125$	0	0
$126$	−1.08934	−0.0970458
$127$	−9.21079	−0.817325	−0.408663	−	0.912686i	$-0.634005\pi$
$127$	−9.21079	−0.817325	−0.408663	+	0.912686i	$0.634005\pi$
$128$	−8.34690	−0.737769
$129$	4.84399	0.426490
$130$	0	0
$131$	4.05325	0.354134	0.177067	−	0.984199i	$-0.443339\pi$
$131$	4.05325	0.354134	0.177067	+	0.984199i	$0.443339\pi$
$132$	1.38941	0.120933
$133$	−1.00000	−0.0867110
$134$	24.7343	2.13672
$135$	0	0
$136$	−2.43008	−0.208377
$137$	−20.4577	−1.74782	−0.873911	−	0.486085i	$-0.838424\pi$
$137$	−20.4577	−1.74782	−0.873911	+	0.486085i	$0.838424\pi$
$138$	−18.1090	−1.54154
$139$	−14.0099	−1.18830	−0.594152	−	0.804353i	$-0.702512\pi$
$139$	−14.0099	−1.18830	−0.594152	+	0.804353i	$0.702512\pi$
$140$	0	0
$141$	−9.33537	−0.786180
$142$	−9.60454	−0.805995
$143$	3.53932	0.295973
$144$	2.85116	0.237597
$145$	0	0
$146$	1.92976	0.159708
$147$	−1.55242	−0.128042
$148$	8.15141	0.670042
$149$	3.46416	0.283795	0.141897	−	0.989881i	$-0.454680\pi$
$149$	3.46416	0.283795	0.141897	+	0.989881i	$0.454680\pi$
$150$	0	0
$151$	0.607812	0.0494631	0.0247315	−	0.999694i	$-0.492127\pi$
$151$	0.607812	0.0494631	0.0247315	+	0.999694i	$0.492127\pi$
$152$	1.09097	0.0884897
$153$	1.31415	0.106243
$154$	−1.17271	−0.0945000
$155$	0	0
$156$	12.1902	0.975997
$157$	5.18853	0.414090	0.207045	−	0.978331i	$-0.433615\pi$
$157$	5.18853	0.414090	0.207045	+	0.978331i	$0.433615\pi$
$158$	2.81576	0.224010
$159$	−5.77528	−0.458009
$160$	0	0
$161$	6.31775	0.497909
$162$	12.7067	0.998336
$163$	−17.5170	−1.37204	−0.686020	−	0.727582i	$-0.740644\pi$
$163$	−17.5170	−1.37204	−0.686020	+	0.727582i	$0.740644\pi$
$164$	−11.1930	−0.874024
$165$	0	0
$166$	27.5884	2.14127
$167$	21.9293	1.69694	0.848470	−	0.529244i	$-0.177524\pi$
$167$	21.9293	1.69694	0.848470	+	0.529244i	$0.177524\pi$
$168$	1.69365	0.130668
$169$	18.0527	1.38867
$170$	0	0
$171$	−0.589984	−0.0451172
$172$	−4.39688	−0.335259
$173$	−24.5830	−1.86901	−0.934505	−	0.355951i	$-0.884157\pi$
$173$	−24.5830	−1.86901	−0.934505	+	0.355951i	$0.884157\pi$
$174$	−20.0798	−1.52225
$175$	0	0
$176$	3.06939	0.231364
$177$	18.6835	1.40434
$178$	−11.9383	−0.894816
$179$	−0.0177631	−0.00132768	−0.000663840	−	1.00000i	$-0.500211\pi$
$179$	−0.0177631	−0.00132768	−0.000663840		1.00000i	$0.500211\pi$
$180$	0	0
$181$	17.7988	1.32297	0.661486	−	0.749957i	$-0.269926\pi$
$181$	17.7988	1.32297	0.661486	+	0.749957i	$0.269926\pi$
$182$	−10.2890	−0.762669
$183$	−3.65511	−0.270194
$184$	−6.89250	−0.508122
$185$	0	0
$186$	−20.6255	−1.51234
$187$	1.41474	0.103456
$188$	8.47368	0.618007
$189$	−5.57317	−0.405389
$190$	0	0
$191$	19.6215	1.41976	0.709882	−	0.704321i	$-0.248748\pi$
$191$	19.6215	1.41976	0.709882	+	0.704321i	$0.248748\pi$
$192$	4.31739	0.311581
$193$	5.01539	0.361016	0.180508	−	0.983574i	$-0.442226\pi$
$193$	5.01539	0.361016	0.180508	+	0.983574i	$0.442226\pi$
$194$	−0.528961	−0.0379772
$195$	0	0
$196$	1.40913	0.100652
$197$	12.3267	0.878244	0.439122	−	0.898428i	$-0.355290\pi$
$197$	12.3267	0.878244	0.439122	+	0.898428i	$0.355290\pi$
$198$	−0.691883	−0.0491699
$199$	−10.7033	−0.758737	−0.379369	−	0.925246i	$-0.623859\pi$
$199$	−10.7033	−0.758737	−0.379369	+	0.925246i	$0.623859\pi$
$200$	0	0
$201$	20.7964	1.46686
$202$	10.9487	0.770349
$203$	7.00533	0.491678
$204$	4.87266	0.341155
$205$	0	0
$206$	9.29705	0.647756
$207$	3.72737	0.259070
$208$	26.9297	1.86724
$209$	−0.635141	−0.0439336
$210$	0	0
$211$	13.8781	0.955407	0.477704	−	0.878521i	$-0.341469\pi$
$211$	13.8781	0.955407	0.477704	+	0.878521i	$0.341469\pi$
$212$	5.24220	0.360036
$213$	−8.07542	−0.553319
$214$	−28.1960	−1.92744
$215$	0	0
$216$	6.08018	0.413704
$217$	7.19570	0.488476
$218$	7.20178	0.487766
$219$	1.62253	0.109640
$220$	0	0
$221$	12.4124	0.834948
$222$	16.5811	1.11285
$223$	−17.8488	−1.19525	−0.597623	−	0.801777i	$-0.703888\pi$
$223$	−17.8488	−1.19525	−0.597623	+	0.801777i	$0.703888\pi$
$224$	−6.74091	−0.450396
$225$	0	0
$226$	21.2623	1.41435
$227$	7.33763	0.487016	0.243508	−	0.969899i	$-0.421702\pi$
$227$	7.33763	0.487016	0.243508	+	0.969899i	$0.421702\pi$
$228$	−2.18756	−0.144875
$229$	11.4405	0.756010	0.378005	−	0.925804i	$-0.376610\pi$
$229$	11.4405	0.756010	0.378005	+	0.925804i	$0.376610\pi$
$230$	0	0
$231$	−0.986008	−0.0648746
$232$	−7.64263	−0.501763
$233$	24.5735	1.60987	0.804933	−	0.593366i	$-0.202201\pi$
$233$	24.5735	1.60987	0.804933	+	0.593366i	$0.202201\pi$
$234$	−6.07032	−0.396829
$235$	0	0
$236$	−16.9589	−1.10393
$237$	2.36747	0.153784
$238$	−4.11270	−0.266587
$239$	23.5206	1.52142	0.760712	−	0.649090i	$-0.224850\pi$
$239$	23.5206	1.52142	0.760712	+	0.649090i	$0.224850\pi$
$240$	0	0
$241$	−29.1440	−1.87733	−0.938666	−	0.344828i	$-0.887937\pi$
$241$	−29.1440	−1.87733	−0.938666	+	0.344828i	$0.887937\pi$
$242$	19.5654	1.25771
$243$	−6.03580	−0.387196
$244$	3.31773	0.212396
$245$	0	0
$246$	−22.7681	−1.45164
$247$	−5.57250	−0.354570
$248$	−7.85032	−0.498496
$249$	23.1961	1.46999
$250$	0	0
$251$	−2.36845	−0.149496	−0.0747478	−	0.997202i	$-0.523815\pi$
$251$	−2.36845	−0.149496	−0.0747478	+	0.997202i	$0.523815\pi$
$252$	0.831364	0.0523710
$253$	4.01267	0.252274
$254$	17.0066	1.06709
$255$	0	0
$256$	20.9737	1.31086
$257$	−1.77806	−0.110912	−0.0554562	−	0.998461i	$-0.517661\pi$
$257$	−1.77806	−0.110912	−0.0554562	+	0.998461i	$0.517661\pi$
$258$	−8.94387	−0.556821
$259$	−5.78472	−0.359445
$260$	0	0
$261$	4.13303	0.255828
$262$	−7.48385	−0.462354
$263$	24.8221	1.53060	0.765298	−	0.643677i	$-0.222592\pi$
$263$	24.8221	1.53060	0.765298	+	0.643677i	$0.222592\pi$
$264$	1.07571	0.0662053
$265$	0	0
$266$	1.84638	0.113209
$267$	−10.0377	−0.614294
$268$	−18.8768	−1.15308
$269$	−26.5502	−1.61879	−0.809397	−	0.587262i	$-0.800206\pi$
$269$	−26.5502	−1.61879	−0.809397	+	0.587262i	$0.800206\pi$
$270$	0	0
$271$	24.2958	1.47587	0.737933	−	0.674873i	$-0.235802\pi$
$271$	24.2958	1.47587	0.737933	+	0.674873i	$0.235802\pi$
$272$	10.7643	0.652684
$273$	−8.65087	−0.523575
$274$	37.7728	2.28194
$275$	0	0
$276$	13.8205	0.831896
$277$	5.95966	0.358082	0.179041	−	0.983842i	$-0.442701\pi$
$277$	5.95966	0.358082	0.179041	+	0.983842i	$0.442701\pi$
$278$	25.8676	1.55144
$279$	4.24535	0.254162
$280$	0	0
$281$	4.42446	0.263941	0.131971	−	0.991254i	$-0.457870\pi$
$281$	4.42446	0.263941	0.131971	+	0.991254i	$0.457870\pi$
$282$	17.2367	1.02643
$283$	−6.14101	−0.365045	−0.182522	−	0.983202i	$-0.558426\pi$
$283$	−6.14101	−0.365045	−0.182522	+	0.983202i	$0.558426\pi$
$284$	7.33003	0.434957
$285$	0	0
$286$	−6.53495	−0.386419
$287$	7.94318	0.468871
$288$	−3.97703	−0.234349
$289$	−12.0385	−0.708148
$290$	0	0
$291$	−0.444745	−0.0260714
$292$	−1.47276	−0.0861869
$293$	21.5171	1.25704	0.628520	−	0.777793i	$-0.283661\pi$
$293$	21.5171	1.25704	0.628520	+	0.777793i	$0.283661\pi$
$294$	2.86637	0.167170
$295$	0	0
$296$	6.31097	0.366818
$297$	−3.53975	−0.205397
$298$	−6.39616	−0.370520
$299$	35.2057	2.03600
$300$	0	0
$301$	3.12028	0.179850
$302$	−1.12225	−0.0645785
$303$	9.20559	0.528847
$304$	−4.83261	−0.277169
$305$	0	0
$306$	−2.42643	−0.138710
$307$	−17.3176	−0.988366	−0.494183	−	0.869358i	$-0.664533\pi$
$307$	−17.3176	−0.988366	−0.494183	+	0.869358i	$0.664533\pi$
$308$	0.894996	0.0509971
$309$	7.81688	0.444687
$310$	0	0
$311$	−15.4755	−0.877533	−0.438766	−	0.898601i	$-0.644584\pi$
$311$	−15.4755	−0.877533	−0.438766	+	0.898601i	$0.644584\pi$
$312$	9.43787	0.534314
$313$	13.0889	0.739830	0.369915	−	0.929066i	$-0.379387\pi$
$313$	13.0889	0.739830	0.369915	+	0.929066i	$0.379387\pi$
$314$	−9.58001	−0.540631
$315$	0	0
$316$	−2.14894	−0.120888
$317$	−8.99443	−0.505178	−0.252589	−	0.967574i	$-0.581282\pi$
$317$	−8.99443	−0.505178	−0.252589	+	0.967574i	$0.581282\pi$
$318$	10.6634	0.597972
$319$	4.44937	0.249117
$320$	0	0
$321$	−23.7069	−1.32319
$322$	−11.6650	−0.650064
$323$	−2.22744	−0.123938
$324$	−9.69758	−0.538754
$325$	0	0
$326$	32.3432	1.79132
$327$	6.05520	0.334853
$328$	−8.66580	−0.478489
$329$	−6.01342	−0.331530
$330$	0	0
$331$	21.5227	1.18299	0.591497	−	0.806307i	$-0.298537\pi$
$331$	21.5227	1.18299	0.591497	+	0.806307i	$0.298537\pi$
$332$	−21.0550	−1.15554
$333$	−3.41289	−0.187025
$334$	−40.4899	−2.21551
$335$	0	0
$336$	−7.50226	−0.409282
$337$	−3.82310	−0.208258	−0.104129	−	0.994564i	$-0.533205\pi$
$337$	−3.82310	−0.208258	−0.104129	+	0.994564i	$0.533205\pi$
$338$	−33.3323	−1.81304
$339$	17.8772	0.970956
$340$	0	0
$341$	4.57029	0.247495
$342$	1.08934	0.0589046
$343$	−1.00000	−0.0539949
$344$	−3.40414	−0.183539
$345$	0	0
$346$	45.3896	2.44016
$347$	8.26780	0.443839	0.221919	−	0.975065i	$-0.428768\pi$
$347$	8.26780	0.443839	0.221919	+	0.975065i	$0.428768\pi$
$348$	15.3246	0.821485
$349$	22.3122	1.19434	0.597172	−	0.802113i	$-0.296291\pi$
$349$	22.3122	1.19434	0.597172	+	0.802113i	$0.296291\pi$
$350$	0	0
$351$	−31.0565	−1.65767
$352$	−4.28143	−0.228201
$353$	5.55432	0.295627	0.147813	−	0.989015i	$-0.452776\pi$
$353$	5.55432	0.295627	0.147813	+	0.989015i	$0.452776\pi$
$354$	−34.4969	−1.83349
$355$	0	0
$356$	9.11114	0.482890
$357$	−3.45792	−0.183013
$358$	0.0327975	0.00173340
$359$	27.8244	1.46852	0.734258	−	0.678871i	$-0.237530\pi$
$359$	27.8244	1.46852	0.734258	+	0.678871i	$0.237530\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−32.8634	−1.72726
$363$	16.4504	0.863422
$364$	7.85237	0.411576
$365$	0	0
$366$	6.74873	0.352762
$367$	11.4375	0.597033	0.298516	−	0.954405i	$-0.403508\pi$
$367$	11.4375	0.597033	0.298516	+	0.954405i	$0.403508\pi$
$368$	30.5313	1.59155
$369$	4.68635	0.243962
$370$	0	0
$371$	−3.72017	−0.193142
$372$	15.7411	0.816135
$373$	−33.1968	−1.71887	−0.859433	−	0.511249i	$-0.829183\pi$
$373$	−33.1968	−1.71887	−0.859433	+	0.511249i	$0.829183\pi$
$374$	−2.61215	−0.135071
$375$	0	0
$376$	6.56048	0.338331
$377$	39.0372	2.01052
$378$	10.2902	0.529271
$379$	8.87233	0.455741	0.227871	−	0.973691i	$-0.426824\pi$
$379$	8.87233	0.455741	0.227871	+	0.973691i	$0.426824\pi$
$380$	0	0
$381$	14.2990	0.732562
$382$	−36.2289	−1.85363
$383$	7.04228	0.359844	0.179922	−	0.983681i	$-0.442415\pi$
$383$	7.04228	0.359844	0.179922	+	0.983681i	$0.442415\pi$
$384$	12.9579	0.661256
$385$	0	0
$386$	−9.26033	−0.471338
$387$	1.84092	0.0935790
$388$	0.403694	0.0204945
$389$	9.52351	0.482861	0.241430	−	0.970418i	$-0.422383\pi$
$389$	9.52351	0.482861	0.241430	+	0.970418i	$0.422383\pi$
$390$	0	0
$391$	14.0724	0.711672
$392$	1.09097	0.0551025
$393$	−6.29235	−0.317407
$394$	−22.7599	−1.14663
$395$	0	0
$396$	0.528033	0.0265347
$397$	−18.0259	−0.904694	−0.452347	−	0.891842i	$-0.649413\pi$
$397$	−18.0259	−0.904694	−0.452347	+	0.891842i	$0.649413\pi$
$398$	19.7624	0.990600
$399$	1.55242	0.0777183
$400$	0	0
$401$	17.1013	0.854000	0.427000	−	0.904252i	$-0.359570\pi$
$401$	17.1013	0.854000	0.427000	+	0.904252i	$0.359570\pi$
$402$	−38.3981	−1.91512
$403$	40.0980	1.99742
$404$	−8.35588	−0.415721
$405$	0	0
$406$	−12.9345	−0.641929
$407$	−3.67411	−0.182119
$408$	3.77250	0.186767
$409$	−30.6842	−1.51724	−0.758618	−	0.651535i	$-0.774125\pi$
$409$	−30.6842	−1.51724	−0.758618	+	0.651535i	$0.774125\pi$
$410$	0	0
$411$	31.7591	1.56656
$412$	−7.09536	−0.349563
$413$	12.0350	0.592206
$414$	−6.88216	−0.338240
$415$	0	0
$416$	−37.5637	−1.84171
$417$	21.7493	1.06507
$418$	1.17271	0.0573593
$419$	18.3213	0.895056	0.447528	−	0.894270i	$-0.352304\pi$
$419$	18.3213	0.895056	0.447528	+	0.894270i	$0.352304\pi$
$420$	0	0
$421$	−39.7775	−1.93864	−0.969318	−	0.245810i	$-0.920946\pi$
$421$	−39.7775	−1.93864	−0.969318	+	0.245810i	$0.920946\pi$
$422$	−25.6243	−1.24737
$423$	−3.54782	−0.172501
$424$	4.05861	0.197104
$425$	0	0
$426$	14.9103	0.722407
$427$	−2.35446	−0.113940
$428$	21.5187	1.04015
$429$	−5.49453	−0.265278
$430$	0	0
$431$	−7.40688	−0.356777	−0.178388	−	0.983960i	$-0.557088\pi$
$431$	−7.40688	−0.356777	−0.178388	+	0.983960i	$0.557088\pi$
$432$	−26.9330	−1.29581
$433$	−2.89348	−0.139052	−0.0695260	−	0.997580i	$-0.522149\pi$
$433$	−2.89348	−0.139052	−0.0695260	+	0.997580i	$0.522149\pi$
$434$	−13.2860	−0.637749
$435$	0	0
$436$	−5.49628	−0.263224
$437$	−6.31775	−0.302219
$438$	−2.99580	−0.143145
$439$	13.4021	0.639648	0.319824	−	0.947477i	$-0.396376\pi$
$439$	13.4021	0.639648	0.319824	+	0.947477i	$0.396376\pi$
$440$	0	0
$441$	−0.589984	−0.0280945
$442$	−22.9180	−1.09010
$443$	30.1848	1.43413	0.717063	−	0.697009i	$-0.245486\pi$
$443$	30.1848	1.43413	0.717063	+	0.697009i	$0.245486\pi$
$444$	−12.6544	−0.600553
$445$	0	0
$446$	32.9558	1.56050
$447$	−5.37784	−0.254363
$448$	2.78107	0.131393
$449$	−4.25344	−0.200732	−0.100366	−	0.994951i	$-0.532001\pi$
$449$	−4.25344	−0.200732	−0.100366	+	0.994951i	$0.532001\pi$
$450$	0	0
$451$	5.04504	0.237562
$452$	−16.2271	−0.763257
$453$	−0.943581	−0.0443333
$454$	−13.5481	−0.635843
$455$	0	0
$456$	−1.69365	−0.0793125
$457$	−13.1584	−0.615523	−0.307762	−	0.951463i	$-0.599580\pi$
$457$	−13.1584	−0.615523	−0.307762	+	0.951463i	$0.599580\pi$
$458$	−21.1236	−0.987039
$459$	−12.4139	−0.579431
$460$	0	0
$461$	−10.4037	−0.484546	−0.242273	−	0.970208i	$-0.577893\pi$
$461$	−10.4037	−0.484546	−0.242273	+	0.970208i	$0.577893\pi$
$462$	1.82055	0.0846995
$463$	21.8835	1.01701	0.508506	−	0.861059i	$-0.330198\pi$
$463$	21.8835	1.01701	0.508506	+	0.861059i	$0.330198\pi$
$464$	33.8540	1.57163
$465$	0	0
$466$	−45.3721	−2.10182
$467$	30.8015	1.42532	0.712661	−	0.701509i	$-0.247490\pi$
$467$	30.8015	1.42532	0.712661	+	0.701509i	$0.247490\pi$
$468$	4.63277	0.214150
$469$	13.3961	0.618574
$470$	0	0
$471$	−8.05479	−0.371145
$472$	−13.1299	−0.604354
$473$	1.98182	0.0911241
$474$	−4.37125	−0.200778
$475$	0	0
$476$	3.13875	0.143864
$477$	−2.19484	−0.100495
$478$	−43.4281	−1.98635
$479$	37.1170	1.69592	0.847960	−	0.530060i	$-0.177831\pi$
$479$	37.1170	1.69592	0.847960	+	0.530060i	$0.177831\pi$
$480$	0	0
$481$	−32.2353	−1.46980
$482$	53.8110	2.45103
$483$	−9.80782	−0.446271
$484$	−14.9320	−0.678726
$485$	0	0
$486$	11.1444	0.505520
$487$	−22.3662	−1.01351	−0.506754	−	0.862091i	$-0.669155\pi$
$487$	−22.3662	−1.01351	−0.506754	+	0.862091i	$0.669155\pi$
$488$	2.56865	0.116277
$489$	27.1938	1.22975
$490$	0	0
$491$	11.7030	0.528148	0.264074	−	0.964502i	$-0.414934\pi$
$491$	11.7030	0.528148	0.264074	+	0.964502i	$0.414934\pi$
$492$	17.3762	0.783380
$493$	15.6039	0.702766
$494$	10.2890	0.462922
$495$	0	0
$496$	34.7740	1.56140
$497$	−5.20182	−0.233333
$498$	−42.8288	−1.91920
$499$	13.1438	0.588397	0.294199	−	0.955744i	$-0.404947\pi$
$499$	13.1438	0.588397	0.294199	+	0.955744i	$0.404947\pi$
$500$	0	0
$501$	−34.0435	−1.52095
$502$	4.37307	0.195180
$503$	−43.2979	−1.93056	−0.965278	−	0.261225i	$-0.915874\pi$
$503$	−43.2979	−1.93056	−0.965278	+	0.261225i	$0.915874\pi$
$504$	0.643657	0.0286708
$505$	0	0
$506$	−7.40892	−0.329367
$507$	−28.0255	−1.24465
$508$	−12.9792	−0.575858
$509$	−24.1123	−1.06876	−0.534379	−	0.845245i	$-0.679454\pi$
$509$	−24.1123	−1.06876	−0.534379	+	0.845245i	$0.679454\pi$
$510$	0	0
$511$	1.04516	0.0462350
$512$	−22.0317	−0.973672
$513$	5.57317	0.246062
$514$	3.28298	0.144806
$515$	0	0
$516$	6.82581	0.300490
$517$	−3.81937	−0.167976
$518$	10.6808	0.469288
$519$	38.1632	1.67518
$520$	0	0
$521$	10.9385	0.479224	0.239612	−	0.970869i	$-0.422980\pi$
$521$	10.9385	0.479224	0.239612	+	0.970869i	$0.422980\pi$
$522$	−7.63116	−0.334007
$523$	40.8100	1.78450	0.892248	−	0.451546i	$-0.149127\pi$
$523$	40.8100	1.78450	0.892248	+	0.451546i	$0.149127\pi$
$524$	5.71155	0.249510
$525$	0	0
$526$	−45.8311	−1.99833
$527$	16.0280	0.698189
$528$	−4.76499	−0.207370
$529$	16.9140	0.735391
$530$	0	0
$531$	7.10049	0.308135
$532$	−1.40913	−0.0610935
$533$	44.2633	1.91726
$534$	18.5333	0.802016
$535$	0	0
$536$	−14.6148	−0.631262
$537$	0.0275759	0.00118999
$538$	49.0218	2.11348
$539$	−0.635141	−0.0273575
$540$	0	0
$541$	16.0217	0.688826	0.344413	−	0.938818i	$-0.388078\pi$
$541$	16.0217	0.688826	0.344413	+	0.938818i	$0.388078\pi$
$542$	−44.8594	−1.92688
$543$	−27.6312	−1.18577
$544$	−15.0149	−0.643761
$545$	0	0
$546$	15.9728	0.683574
$547$	−37.9792	−1.62387	−0.811936	−	0.583746i	$-0.801586\pi$
$547$	−37.9792	−1.62387	−0.811936	+	0.583746i	$0.801586\pi$
$548$	−28.8276	−1.23145
$549$	−1.38909	−0.0592850
$550$	0	0
$551$	−7.00533	−0.298437
$552$	10.7001	0.455425
$553$	1.52502	0.0648503
$554$	−11.0038	−0.467508
$555$	0	0
$556$	−19.7417	−0.837236
$557$	16.7176	0.708348	0.354174	−	0.935180i	$-0.384762\pi$
$557$	16.7176	0.708348	0.354174	+	0.935180i	$0.384762\pi$
$558$	−7.83854	−0.331832
$559$	17.3878	0.735424
$560$	0	0
$561$	−2.19627	−0.0927266
$562$	−8.16925	−0.344599
$563$	5.47540	0.230761	0.115380	−	0.993321i	$-0.463191\pi$
$563$	5.47540	0.230761	0.115380	+	0.993321i	$0.463191\pi$
$564$	−13.1547	−0.553914
$565$	0	0
$566$	11.3386	0.476599
$567$	6.88197	0.289015
$568$	5.67505	0.238120
$569$	−16.3600	−0.685847	−0.342924	−	0.939363i	$-0.611417\pi$
$569$	−16.3600	−0.685847	−0.342924	+	0.939363i	$0.611417\pi$
$570$	0	0
$571$	7.04971	0.295021	0.147511	−	0.989060i	$-0.452874\pi$
$571$	7.04971	0.295021	0.147511	+	0.989060i	$0.452874\pi$
$572$	4.98736	0.208532
$573$	−30.4609	−1.27252
$574$	−14.6661	−0.612153
$575$	0	0
$576$	1.64078	0.0683660
$577$	−18.0308	−0.750632	−0.375316	−	0.926897i	$-0.622466\pi$
$577$	−18.0308	−0.750632	−0.375316	+	0.926897i	$0.622466\pi$
$578$	22.2277	0.924551
$579$	−7.78600	−0.323575
$580$	0	0
$581$	14.9418	0.619892
$582$	0.821170	0.0340386
$583$	−2.36284	−0.0978586
$584$	−1.14024	−0.0471834
$585$	0	0
$586$	−39.7287	−1.64118
$587$	14.0660	0.580568	0.290284	−	0.956941i	$-0.406250\pi$
$587$	14.0660	0.580568	0.290284	+	0.956941i	$0.406250\pi$
$588$	−2.18756	−0.0902136
$589$	−7.19570	−0.296494
$590$	0	0
$591$	−19.1363	−0.787162
$592$	−27.9553	−1.14896
$593$	17.6566	0.725069	0.362534	−	0.931970i	$-0.381912\pi$
$593$	17.6566	0.725069	0.362534	+	0.931970i	$0.381912\pi$
$594$	6.53574	0.268165
$595$	0	0
$596$	4.88145	0.199952
$597$	16.6161	0.680050
$598$	−65.0031	−2.65818
$599$	−33.7765	−1.38007	−0.690034	−	0.723777i	$-0.742405\pi$
$599$	−33.7765	−1.38007	−0.690034	+	0.723777i	$0.742405\pi$
$600$	0	0
$601$	−22.3881	−0.913232	−0.456616	−	0.889664i	$-0.650939\pi$
$601$	−22.3881	−0.913232	−0.456616	+	0.889664i	$0.650939\pi$
$602$	−5.76123	−0.234810
$603$	7.90347	0.321854
$604$	0.856486	0.0348499
$605$	0	0
$606$	−16.9970	−0.690457
$607$	−33.7677	−1.37059	−0.685294	−	0.728267i	$-0.740326\pi$
$607$	−33.7677	−1.37059	−0.685294	+	0.728267i	$0.740326\pi$
$608$	6.74091	0.273380
$609$	−10.8752	−0.440687
$610$	0	0
$611$	−33.5098	−1.35566
$612$	1.85181	0.0748550
$613$	−2.12330	−0.0857592	−0.0428796	−	0.999080i	$-0.513653\pi$
$613$	−2.12330	−0.0857592	−0.0428796	+	0.999080i	$0.513653\pi$
$614$	31.9749	1.29040
$615$	0	0
$616$	0.692923	0.0279186
$617$	41.0771	1.65370	0.826851	−	0.562421i	$-0.190130\pi$
$617$	41.0771	1.65370	0.826851	+	0.562421i	$0.190130\pi$
$618$	−14.4330	−0.580579
$619$	4.48300	0.180187	0.0900934	−	0.995933i	$-0.471283\pi$
$619$	4.48300	0.180187	0.0900934	+	0.995933i	$0.471283\pi$
$620$	0	0
$621$	−35.2099	−1.41293
$622$	28.5736	1.14570
$623$	−6.46580	−0.259047
$624$	−41.8063	−1.67359
$625$	0	0
$626$	−24.1672	−0.965915
$627$	0.986008	0.0393774
$628$	7.31131	0.291753
$629$	−12.8851	−0.513762
$630$	0	0
$631$	−1.94680	−0.0775008	−0.0387504	−	0.999249i	$-0.512338\pi$
$631$	−1.94680	−0.0775008	−0.0387504	+	0.999249i	$0.512338\pi$
$632$	−1.66375	−0.0661805
$633$	−21.5447	−0.856324
$634$	16.6072	0.659555
$635$	0	0
$636$	−8.13811	−0.322697
$637$	−5.57250	−0.220790
$638$	−8.21525	−0.325245
$639$	−3.06899	−0.121407
$640$	0	0
$641$	−28.4963	−1.12553	−0.562767	−	0.826615i	$-0.690263\pi$
$641$	−28.4963	−1.12553	−0.562767	+	0.826615i	$0.690263\pi$
$642$	43.7721	1.72755
$643$	−8.76049	−0.345480	−0.172740	−	0.984967i	$-0.555262\pi$
$643$	−8.76049	−0.345480	−0.172740	+	0.984967i	$0.555262\pi$
$644$	8.90253	0.350809
$645$	0	0
$646$	4.11270	0.161812
$647$	13.6320	0.535928	0.267964	−	0.963429i	$-0.413649\pi$
$647$	13.6320	0.535928	0.267964	+	0.963429i	$0.413649\pi$
$648$	−7.50805	−0.294944
$649$	7.64396	0.300052
$650$	0	0
$651$	−11.1708	−0.437817
$652$	−24.6838	−0.966691
$653$	16.3713	0.640659	0.320330	−	0.947306i	$-0.396206\pi$
$653$	16.3713	0.640659	0.320330	+	0.947306i	$0.396206\pi$
$654$	−11.1802	−0.437181
$655$	0	0
$656$	38.3863	1.49873
$657$	0.616626	0.0240569
$658$	11.1031	0.432843
$659$	−6.46448	−0.251820	−0.125910	−	0.992042i	$-0.540185\pi$
$659$	−6.46448	−0.251820	−0.125910	+	0.992042i	$0.540185\pi$
$660$	0	0
$661$	−28.7000	−1.11630	−0.558149	−	0.829740i	$-0.688488\pi$
$661$	−28.7000	−1.11630	−0.558149	+	0.829740i	$0.688488\pi$
$662$	−39.7391	−1.54450
$663$	−19.2693	−0.748357
$664$	−16.3012	−0.632608
$665$	0	0
$666$	6.30150	0.244178
$667$	44.2579	1.71367
$668$	30.9012	1.19560
$669$	27.7089	1.07129
$670$	0	0
$671$	−1.49541	−0.0577297
$672$	10.4647	0.403686
$673$	37.2829	1.43715	0.718575	−	0.695450i	$-0.244795\pi$
$673$	37.2829	1.43715	0.718575	+	0.695450i	$0.244795\pi$
$674$	7.05891	0.271899
$675$	0	0
$676$	25.4386	0.978409
$677$	−39.2133	−1.50709	−0.753544	−	0.657397i	$-0.771657\pi$
$677$	−39.2133	−1.50709	−0.753544	+	0.657397i	$0.771657\pi$
$678$	−33.0081	−1.26767
$679$	−0.286485	−0.0109943
$680$	0	0
$681$	−11.3911	−0.436508
$682$	−8.43850	−0.323127
$683$	33.8564	1.29548	0.647739	−	0.761862i	$-0.275715\pi$
$683$	33.8564	1.29548	0.647739	+	0.761862i	$0.275715\pi$
$684$	−0.831364	−0.0317880
$685$	0	0
$686$	1.84638	0.0704952
$687$	−17.7605	−0.677606
$688$	15.0791	0.574886
$689$	−20.7307	−0.789775
$690$	0	0
$691$	−7.65402	−0.291173	−0.145586	−	0.989346i	$-0.546507\pi$
$691$	−7.65402	−0.291173	−0.145586	+	0.989346i	$0.546507\pi$
$692$	−34.6406	−1.31684
$693$	−0.374723	−0.0142346
$694$	−15.2655	−0.579471
$695$	0	0
$696$	11.8646	0.449726
$697$	17.6929	0.670168
$698$	−41.1968	−1.55932
$699$	−38.1485	−1.44291
$700$	0	0
$701$	−32.8682	−1.24141	−0.620707	−	0.784043i	$-0.713154\pi$
$701$	−32.8682	−1.24141	−0.620707	+	0.784043i	$0.713154\pi$
$702$	57.3422	2.16424
$703$	5.78472	0.218175
$704$	1.76637	0.0665726
$705$	0	0
$706$	−10.2554	−0.385967
$707$	5.92982	0.223014
$708$	26.3274	0.989446
$709$	17.6646	0.663408	0.331704	−	0.943384i	$-0.392377\pi$
$709$	17.6646	0.663408	0.331704	+	0.943384i	$0.392377\pi$
$710$	0	0
$711$	0.899735	0.0337427
$712$	7.05402	0.264360
$713$	45.4606	1.70251
$714$	6.38465	0.238940
$715$	0	0
$716$	−0.0250306	−0.000935436 0
$717$	−36.5140	−1.36364
$718$	−51.3745	−1.91728
$719$	−33.4548	−1.24765	−0.623827	−	0.781562i	$-0.714423\pi$
$719$	−33.4548	−1.24765	−0.623827	+	0.781562i	$0.714423\pi$
$720$	0	0
$721$	5.03528	0.187524
$722$	−1.84638	−0.0687152
$723$	45.2439	1.68264
$724$	25.0808	0.932120
$725$	0	0
$726$	−30.3737	−1.12727
$727$	−22.0141	−0.816458	−0.408229	−	0.912880i	$-0.633853\pi$
$727$	−22.0141	−0.816458	−0.408229	+	0.912880i	$0.633853\pi$
$728$	6.07945	0.225319
$729$	30.0160	1.11170
$730$	0	0
$731$	6.95023	0.257063
$732$	−5.15052	−0.190369
$733$	16.9719	0.626870	0.313435	−	0.949610i	$-0.398520\pi$
$733$	16.9719	0.626870	0.313435	+	0.949610i	$0.398520\pi$
$734$	−21.1180	−0.779480
$735$	0	0
$736$	−42.5874	−1.56979
$737$	8.50841	0.313411
$738$	−8.65279	−0.318514
$739$	13.1814	0.484887	0.242444	−	0.970165i	$-0.422051\pi$
$739$	13.1814	0.484887	0.242444	+	0.970165i	$0.422051\pi$
$740$	0	0
$741$	8.65087	0.317798
$742$	6.86886	0.252164
$743$	−41.5200	−1.52322	−0.761611	−	0.648035i	$-0.775591\pi$
$743$	−41.5200	−1.52322	−0.761611	+	0.648035i	$0.775591\pi$
$744$	12.1870	0.446797
$745$	0	0
$746$	61.2940	2.24413
$747$	8.81545	0.322541
$748$	1.99355	0.0728913
$749$	−15.2709	−0.557987
$750$	0	0
$751$	−9.25144	−0.337590	−0.168795	−	0.985651i	$-0.553988\pi$
$751$	−9.25144	−0.337590	−0.168795	+	0.985651i	$0.553988\pi$
$752$	−29.0605	−1.05973
$753$	3.67684	0.133992
$754$	−72.0776	−2.62491
$755$	0	0
$756$	−7.85332	−0.285622
$757$	−46.3097	−1.68315	−0.841577	−	0.540138i	$-0.818372\pi$
$757$	−46.3097	−1.68315	−0.841577	+	0.540138i	$0.818372\pi$
$758$	−16.3817	−0.595011
$759$	−6.22935	−0.226111
$760$	0	0
$761$	−13.8330	−0.501447	−0.250724	−	0.968059i	$-0.580669\pi$
$761$	−13.8330	−0.501447	−0.250724	+	0.968059i	$0.580669\pi$
$762$	−26.4015	−0.956425
$763$	3.90048	0.141207
$764$	27.6493	1.00032
$765$	0	0
$766$	−13.0027	−0.469808
$767$	67.0653	2.42159
$768$	−32.5601	−1.17491
$769$	−47.3912	−1.70897	−0.854486	−	0.519475i	$-0.826127\pi$
$769$	−47.3912	−1.70897	−0.854486	+	0.519475i	$0.826127\pi$
$770$	0	0
$771$	2.76030	0.0994098
$772$	7.06733	0.254359
$773$	2.71864	0.0977827	0.0488914	−	0.998804i	$-0.484431\pi$
$773$	2.71864	0.0977827	0.0488914	+	0.998804i	$0.484431\pi$
$774$	−3.39903	−0.122176
$775$	0	0
$776$	0.312547	0.0112198
$777$	8.98033	0.322167
$778$	−17.5840	−0.630418
$779$	−7.94318	−0.284594
$780$	0	0
$781$	−3.30389	−0.118222
$782$	−25.9830	−0.929151
$783$	−39.0419	−1.39524
$784$	−4.83261	−0.172593
$785$	0	0
$786$	11.6181	0.414404
$787$	42.6584	1.52061	0.760304	−	0.649567i	$-0.225050\pi$
$787$	42.6584	1.52061	0.760304	+	0.649567i	$0.225050\pi$
$788$	17.3700	0.618779
$789$	−38.5344	−1.37186
$790$	0	0
$791$	11.5157	0.409450
$792$	0.408813	0.0145265
$793$	−13.1202	−0.465912
$794$	33.2827	1.18116
$795$	0	0
$796$	−15.0823	−0.534579
$797$	−1.20848	−0.0428064	−0.0214032	−	0.999771i	$-0.506813\pi$
$797$	−1.20848	−0.0428064	−0.0214032	+	0.999771i	$0.506813\pi$
$798$	−2.86637	−0.101468
$799$	−13.3945	−0.473864
$800$	0	0
$801$	−3.81472	−0.134786
$802$	−31.5756	−1.11497
$803$	0.663823	0.0234258
$804$	29.3048	1.03350
$805$	0	0
$806$	−74.0363	−2.60782
$807$	41.2171	1.45091
$808$	−6.46928	−0.227588
$809$	21.2138	0.745838	0.372919	−	0.927864i	$-0.378357\pi$
$809$	21.2138	0.745838	0.372919	+	0.927864i	$0.378357\pi$
$810$	0	0
$811$	15.5505	0.546051	0.273026	−	0.962007i	$-0.411976\pi$
$811$	15.5505	0.546051	0.273026	+	0.962007i	$0.411976\pi$
$812$	9.87141	0.346419
$813$	−37.7174	−1.32281
$814$	6.78382	0.237773
$815$	0	0
$816$	−16.7108	−0.584995
$817$	−3.12028	−0.109165
$818$	56.6548	1.98089
$819$	−3.28768	−0.114881
$820$	0	0
$821$	−8.59655	−0.300022	−0.150011	−	0.988684i	$-0.547931\pi$
$821$	−8.59655	−0.300022	−0.150011	+	0.988684i	$0.547931\pi$
$822$	−58.6394	−2.04528
$823$	−43.7337	−1.52446	−0.762230	−	0.647306i	$-0.775896\pi$
$823$	−43.7337	−1.52446	−0.762230	+	0.647306i	$0.775896\pi$
$824$	−5.49336	−0.191370
$825$	0	0
$826$	−22.2213	−0.773178
$827$	−1.50328	−0.0522740	−0.0261370	−	0.999658i	$-0.508321\pi$
$827$	−1.50328	−0.0522740	−0.0261370	+	0.999658i	$0.508321\pi$
$828$	5.25235	0.182532
$829$	−27.2420	−0.946153	−0.473076	−	0.881021i	$-0.656857\pi$
$829$	−27.2420	−0.946153	−0.473076	+	0.881021i	$0.656857\pi$
$830$	0	0
$831$	−9.25192	−0.320945
$832$	15.4975	0.537279
$833$	−2.22744	−0.0771761
$834$	−40.1575	−1.39054
$835$	0	0
$836$	−0.894996	−0.0309541
$837$	−40.1029	−1.38616
$838$	−33.8282	−1.16858
$839$	−14.1262	−0.487692	−0.243846	−	0.969814i	$-0.578409\pi$
$839$	−14.1262	−0.487692	−0.243846	+	0.969814i	$0.578409\pi$
$840$	0	0
$841$	20.0746	0.692228
$842$	73.4445	2.53106
$843$	−6.86864	−0.236568
$844$	19.5560	0.673146
$845$	0	0
$846$	6.55063	0.225215
$847$	10.5966	0.364103
$848$	−17.9782	−0.617372
$849$	9.53344	0.327187
$850$	0	0
$851$	−36.5464	−1.25279
$852$	−11.3793	−0.389849
$853$	−9.42263	−0.322625	−0.161312	−	0.986903i	$-0.551573\pi$
$853$	−9.42263	−0.322625	−0.161312	+	0.986903i	$0.551573\pi$
$854$	4.34723	0.148759
$855$	0	0
$856$	16.6602	0.569433
$857$	26.6568	0.910578	0.455289	−	0.890344i	$-0.349536\pi$
$857$	26.6568	0.910578	0.455289	+	0.890344i	$0.349536\pi$
$858$	10.1450	0.346345
$859$	18.0373	0.615425	0.307713	−	0.951479i	$-0.400436\pi$
$859$	18.0373	0.615425	0.307713	+	0.951479i	$0.400436\pi$
$860$	0	0
$861$	−12.3312	−0.420245
$862$	13.6759	0.465804
$863$	−49.8010	−1.69525	−0.847623	−	0.530600i	$-0.821967\pi$
$863$	−49.8010	−1.69525	−0.847623	+	0.530600i	$0.821967\pi$
$864$	37.5682	1.27810
$865$	0	0
$866$	5.34248	0.181545
$867$	18.6889	0.634707
$868$	10.1397	0.344163
$869$	0.968601	0.0328575
$870$	0	0
$871$	74.6496	2.52941
$872$	−4.25532	−0.144103
$873$	−0.169021	−0.00572051
$874$	11.6650	0.394574
$875$	0	0
$876$	2.28635	0.0772486
$877$	23.9045	0.807198	0.403599	−	0.914936i	$-0.367759\pi$
$877$	23.9045	0.807198	0.403599	+	0.914936i	$0.367759\pi$
$878$	−24.7454	−0.835118
$879$	−33.4036	−1.12667
$880$	0	0
$881$	−22.3593	−0.753305	−0.376652	−	0.926355i	$-0.622925\pi$
$881$	−22.3593	−0.753305	−0.376652	+	0.926355i	$0.622925\pi$
$882$	1.08934	0.0366799
$883$	−39.2226	−1.31994	−0.659972	−	0.751290i	$-0.729432\pi$
$883$	−39.2226	−1.31994	−0.659972	+	0.751290i	$0.729432\pi$
$884$	17.4907	0.588275
$885$	0	0
$886$	−55.7328	−1.87238
$887$	−26.7082	−0.896773	−0.448387	−	0.893840i	$-0.648001\pi$
$887$	−26.7082	−0.896773	−0.448387	+	0.893840i	$0.648001\pi$
$888$	−9.79730	−0.328776
$889$	9.21079	0.308920
$890$	0	0
$891$	4.37102	0.146435
$892$	−25.1513	−0.842128
$893$	6.01342	0.201231
$894$	9.92955	0.332094
$895$	0	0
$896$	8.34690	0.278850
$897$	−54.6541	−1.82485
$898$	7.85348	0.262074
$899$	50.4082	1.68121
$900$	0	0
$901$	−8.28645	−0.276062
$902$	−9.31508	−0.310158
$903$	−4.84399	−0.161198
$904$	−12.5633	−0.417849
$905$	0	0
$906$	1.74221	0.0578811
$907$	−12.1350	−0.402936	−0.201468	−	0.979495i	$-0.564571\pi$
$907$	−12.1350	−0.402936	−0.201468	+	0.979495i	$0.564571\pi$
$908$	10.3397	0.343134
$909$	3.49850	0.116038
$910$	0	0
$911$	20.5631	0.681286	0.340643	−	0.940193i	$-0.389355\pi$
$911$	20.5631	0.681286	0.340643	+	0.940193i	$0.389355\pi$
$912$	7.50226	0.248425
$913$	9.49019	0.314079
$914$	24.2954	0.803621
$915$	0	0
$916$	16.1212	0.532658
$917$	−4.05325	−0.133850
$918$	22.9208	0.756499
$919$	9.87993	0.325909	0.162954	−	0.986634i	$-0.447898\pi$
$919$	9.87993	0.325909	0.162954	+	0.986634i	$0.447898\pi$
$920$	0	0
$921$	26.8842	0.885865
$922$	19.2091	0.632619
$923$	−28.9871	−0.954122
$924$	−1.38941	−0.0457083
$925$	0	0
$926$	−40.4053	−1.32780
$927$	2.97073	0.0975717
$928$	−47.2223	−1.55015
$929$	−5.42109	−0.177860	−0.0889300	−	0.996038i	$-0.528345\pi$
$929$	−5.42109	−0.177860	−0.0889300	+	0.996038i	$0.528345\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	34.6273	1.13425
$933$	24.0244	0.786525
$934$	−56.8713	−1.86089
$935$	0	0
$936$	3.58678	0.117237
$937$	−9.19384	−0.300350	−0.150175	−	0.988659i	$-0.547984\pi$
$937$	−9.19384	−0.300350	−0.150175	+	0.988659i	$0.547984\pi$
$938$	−24.7343	−0.807603
$939$	−20.3196	−0.663103
$940$	0	0
$941$	−38.0888	−1.24166	−0.620830	−	0.783945i	$-0.713204\pi$
$941$	−38.0888	−1.24166	−0.620830	+	0.783945i	$0.713204\pi$
$942$	14.8722	0.484563
$943$	50.1830	1.63418
$944$	58.1607	1.89297
$945$	0	0
$946$	−3.65920	−0.118971
$947$	40.9053	1.32924	0.664622	−	0.747179i	$-0.268592\pi$
$947$	40.9053	1.32924	0.664622	+	0.747179i	$0.268592\pi$
$948$	3.33607	0.108351
$949$	5.82414	0.189059
$950$	0	0
$951$	13.9632	0.452786
$952$	2.43008	0.0787592
$953$	27.3800	0.886925	0.443463	−	0.896293i	$-0.353750\pi$
$953$	27.3800	0.886925	0.443463	+	0.896293i	$0.353750\pi$
$954$	4.05252	0.131205
$955$	0	0
$956$	33.1436	1.07194
$957$	−6.90731	−0.223282
$958$	−68.5322	−2.21418
$959$	20.4577	0.660615
$960$	0	0
$961$	20.7781	0.670260
$962$	59.5187	1.91896
$963$	−9.00960	−0.290330
$964$	−41.0677	−1.32270
$965$	0	0
$966$	18.1090	0.582647
$967$	−4.50712	−0.144939	−0.0724696	−	0.997371i	$-0.523088\pi$
$967$	−4.50712	−0.144939	−0.0724696	+	0.997371i	$0.523088\pi$
$968$	−11.5606	−0.371572
$969$	3.45792	0.111085
$970$	0	0
$971$	24.6779	0.791952	0.395976	−	0.918261i	$-0.370406\pi$
$971$	24.6779	0.791952	0.395976	+	0.918261i	$0.370406\pi$
$972$	−8.50522	−0.272805
$973$	14.0099	0.449136
$974$	41.2965	1.32323
$975$	0	0
$976$	−11.3782	−0.364206
$977$	19.1163	0.611584	0.305792	−	0.952098i	$-0.401079\pi$
$977$	19.1163	0.611584	0.305792	+	0.952098i	$0.401079\pi$
$978$	−50.2103	−1.60555
$979$	−4.10670	−0.131251
$980$	0	0
$981$	2.30122	0.0734724
$982$	−21.6082	−0.689544
$983$	36.5423	1.16552	0.582760	−	0.812644i	$-0.301973\pi$
$983$	36.5423	1.16552	0.582760	+	0.812644i	$0.301973\pi$
$984$	13.4530	0.428865
$985$	0	0
$986$	−28.8108	−0.917524
$987$	9.33537	0.297148
$988$	−7.85237	−0.249817
$989$	19.7132	0.626842
$990$	0	0
$991$	47.7696	1.51745	0.758726	−	0.651409i	$-0.225822\pi$
$991$	47.7696	1.51745	0.758726	+	0.651409i	$0.225822\pi$
$992$	−48.5055	−1.54005
$993$	−33.4123	−1.06031
$994$	9.60454	0.304638
$995$	0	0
$996$	32.6863	1.03570
$997$	−48.9690	−1.55086	−0.775432	−	0.631432i	$-0.782468\pi$
$997$	−48.9690	−1.55086	−0.775432	+	0.631432i	$0.782468\pi$
$998$	−24.2685	−0.768205
$999$	32.2392	1.02000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3325.2.a.y.1.3		8
5.4	even	2		665.2.a.k.1.6	✓	8
15.14	odd	2		5985.2.a.bq.1.3		8
35.34	odd	2		4655.2.a.bi.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
665.2.a.k.1.6	✓	8	5.4	even	2
3325.2.a.y.1.3		8	1.1	even	1	trivial
4655.2.a.bi.1.6		8	35.34	odd	2
5985.2.a.bq.1.3		8	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3325 = 5^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3325.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84638 q^{2} -1.55242 q^{3} +1.40913 q^{4} +2.86637 q^{6} -1.00000 q^{7} +1.09097 q^{8} -0.589984 q^{9} -0.635141 q^{11} -2.18756 q^{12} -5.57250 q^{13} +1.84638 q^{14} -4.83261 q^{16} -2.22744 q^{17} +1.08934 q^{18} +1.00000 q^{19} +1.55242 q^{21} +1.17271 q^{22} -6.31775 q^{23} -1.69365 q^{24} +10.2890 q^{26} +5.57317 q^{27} -1.40913 q^{28} -7.00533 q^{29} -7.19570 q^{31} +6.74091 q^{32} +0.986008 q^{33} +4.11270 q^{34} -0.831364 q^{36} +5.78472 q^{37} -1.84638 q^{38} +8.65087 q^{39} -7.94318 q^{41} -2.86637 q^{42} -3.12028 q^{43} -0.894996 q^{44} +11.6650 q^{46} +6.01342 q^{47} +7.50226 q^{48} +1.00000 q^{49} +3.45792 q^{51} -7.85237 q^{52} +3.72017 q^{53} -10.2902 q^{54} -1.09097 q^{56} -1.55242 q^{57} +12.9345 q^{58} -12.0350 q^{59} +2.35446 q^{61} +13.2860 q^{62} +0.589984 q^{63} -2.78107 q^{64} -1.82055 q^{66} -13.3961 q^{67} -3.13875 q^{68} +9.80782 q^{69} +5.20182 q^{71} -0.643657 q^{72} -1.04516 q^{73} -10.6808 q^{74} +1.40913 q^{76} +0.635141 q^{77} -15.9728 q^{78} -1.52502 q^{79} -6.88197 q^{81} +14.6661 q^{82} -14.9418 q^{83} +2.18756 q^{84} +5.76123 q^{86} +10.8752 q^{87} -0.692923 q^{88} +6.46580 q^{89} +5.57250 q^{91} -8.90253 q^{92} +11.1708 q^{93} -11.1031 q^{94} -10.4647 q^{96} +0.286485 q^{97} -1.84638 q^{98} +0.374723 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 3 q^{3} + 12 q^{4} + 9 q^{6} - 8 q^{7} + 7 q^{9} + 16 q^{11} + 19 q^{12} - 2 q^{13} + 2 q^{14} + 24 q^{16} + 7 q^{17} + 5 q^{18} + 8 q^{19} - 3 q^{21} + 10 q^{22} - 7 q^{23} + q^{24} - 6 q^{26}+ \cdots + 60 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 + 3 * q^3 + 12 * q^4 + 9 * q^6 - 8 * q^7 + 7 * q^9 + 16 * q^11 + 19 * q^12 - 2 * q^13 + 2 * q^14 + 24 * q^16 + 7 * q^17 + 5 * q^18 + 8 * q^19 - 3 * q^21 + 10 * q^22 - 7 * q^23 + q^24 - 6 * q^26 + 15 * q^27 - 12 * q^28 + 6 * q^29 + 6 * q^31 + 18 * q^32 + 6 * q^33 - 5 * q^34 + q^36 - 3 * q^37 - 2 * q^38 + 8 * q^39 - 7 * q^41 - 9 * q^42 - 23 * q^43 + 40 * q^44 + 13 * q^46 + 9 * q^48 + 8 * q^49 + 7 * q^51 + 2 * q^52 + 11 * q^53 - 5 * q^54 + 3 * q^57 - 52 * q^58 + 4 * q^59 + 18 * q^61 + 24 * q^62 - 7 * q^63 + 32 * q^64 + 12 * q^66 - 4 * q^67 + 13 * q^68 - 41 * q^69 + 13 * q^71 + 13 * q^72 + q^73 - 37 * q^74 + 12 * q^76 - 16 * q^77 - 28 * q^78 + 10 * q^79 + 12 * q^81 + 7 * q^82 + 20 * q^83 - 19 * q^84 + 7 * q^86 - 6 * q^87 + 70 * q^88 - 2 * q^89 + 2 * q^91 + 43 * q^92 + 16 * q^93 - 36 * q^94 + 33 * q^96 + 42 * q^97 - 2 * q^98 + 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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