LMFDB - Embedded newform 5571.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5571,2,Mod(1,5571)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5571, 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("5571.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5571 = 3^{2} \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5571.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:	$44.4846589661$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 619)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	5571.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46595 q^{2} +4.08090 q^{4} -2.37765 q^{5} +4.70879 q^{7} -5.13138 q^{8} +O(q^{10})$	\(q-2.46595 q^{2} +4.08090 q^{4} -2.37765 q^{5} +4.70879 q^{7} -5.13138 q^{8} +5.86316 q^{10} +3.89648 q^{11} -6.62279 q^{13} -11.6116 q^{14} +4.49192 q^{16} +5.95137 q^{17} -0.395246 q^{19} -9.70294 q^{20} -9.60851 q^{22} +1.84938 q^{23} +0.653219 q^{25} +16.3315 q^{26} +19.2161 q^{28} -10.6320 q^{29} +7.56721 q^{31} -0.814076 q^{32} -14.6758 q^{34} -11.1958 q^{35} -5.85939 q^{37} +0.974656 q^{38} +12.2006 q^{40} -1.13182 q^{41} +3.18315 q^{43} +15.9011 q^{44} -4.56048 q^{46} -0.625030 q^{47} +15.1727 q^{49} -1.61080 q^{50} -27.0269 q^{52} -12.8766 q^{53} -9.26447 q^{55} -24.1626 q^{56} +26.2180 q^{58} +0.141655 q^{59} -3.91754 q^{61} -18.6603 q^{62} -6.97637 q^{64} +15.7467 q^{65} +0.202947 q^{67} +24.2869 q^{68} +27.6084 q^{70} -6.99318 q^{71} +0.952512 q^{73} +14.4490 q^{74} -1.61296 q^{76} +18.3477 q^{77} -8.10807 q^{79} -10.6802 q^{80} +2.79100 q^{82} -9.46455 q^{83} -14.1503 q^{85} -7.84947 q^{86} -19.9943 q^{88} -10.3042 q^{89} -31.1853 q^{91} +7.54714 q^{92} +1.54129 q^{94} +0.939757 q^{95} -12.2470 q^{97} -37.4150 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) $ 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8	$ 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) $ 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8 + 5 * q^10 - 23 * q^11 + 9 * q^13 - 7 * q^14 + 35 * q^16 - 4 * q^17 - q^19 - 29 * q^20 - 4 * q^23 + 35 * q^25 - q^26 - 13 * q^28 - 90 * q^29 + 2 * q^31 - 43 * q^32 - 9 * q^34 - 9 * q^35 + 19 * q^37 - 5 * q^38 - 12 * q^40 - 59 * q^41 - 4 * q^43 - 52 * q^44 - q^46 - 4 * q^47 + 30 * q^49 - 31 * q^50 - 12 * q^52 - 34 * q^53 - 17 * q^55 - 2 * q^56 + 6 * q^58 - 13 * q^59 + 16 * q^61 - 28 * q^62 + 37 * q^64 - 31 * q^65 - 11 * q^67 + 52 * q^68 - 40 * q^70 - 42 * q^71 - 4 * q^73 - 16 * q^74 - 42 * q^76 - 29 * q^77 + 3 * q^79 - 21 * q^80 - 43 * q^82 + 11 * q^83 + 19 * q^85 + 11 * q^86 - 47 * q^88 - 58 * q^89 - 39 * q^91 + 7 * q^92 - 46 * q^94 - 23 * q^95 - 9 * q^97

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 $

$2$	−2.46595	−1.74369	−0.871844	−	0.489784i	$-0.837076\pi$
$2$	−2.46595	−1.74369	−0.871844	+	0.489784i	$0.837076\pi$
$3$	0	0
$3$	0	0
$4$	4.08090	2.04045
$5$	−2.37765	−1.06332	−0.531659	−	0.846959i	$-0.678431\pi$
$5$	−2.37765	−1.06332	−0.531659	+	0.846959i	$0.678431\pi$
$6$	0	0
$7$	4.70879	1.77975	0.889877	−	0.456200i	$-0.150790\pi$
$7$	4.70879	1.77975	0.889877	+	0.456200i	$0.150790\pi$
$8$	−5.13138	−1.81422
$9$	0	0
$10$	5.86316	1.85409
$11$	3.89648	1.17483	0.587416	−	0.809285i	$-0.300145\pi$
$11$	3.89648	1.17483	0.587416	+	0.809285i	$0.300145\pi$
$12$	0	0
$13$	−6.62279	−1.83683	−0.918416	−	0.395617i	$-0.870531\pi$
$13$	−6.62279	−1.83683	−0.918416	+	0.395617i	$0.870531\pi$
$14$	−11.6116	−3.10334
$15$	0	0
$16$	4.49192	1.12298
$17$	5.95137	1.44342	0.721710	−	0.692195i	$-0.243356\pi$
$17$	5.95137	1.44342	0.721710	+	0.692195i	$0.243356\pi$
$18$	0	0
$19$	−0.395246	−0.0906757	−0.0453378	−	0.998972i	$-0.514436\pi$
$19$	−0.395246	−0.0906757	−0.0453378	+	0.998972i	$0.514436\pi$
$20$	−9.70294	−2.16964
$21$	0	0
$22$	−9.60851	−2.04854
$23$	1.84938	0.385623	0.192812	−	0.981236i	$-0.438239\pi$
$23$	1.84938	0.385623	0.192812	+	0.981236i	$0.438239\pi$
$24$	0	0
$25$	0.653219	0.130644
$26$	16.3315	3.20286
$27$	0	0
$28$	19.2161	3.63150
$29$	−10.6320	−1.97432	−0.987159	−	0.159743i	$-0.948933\pi$
$29$	−10.6320	−1.97432	−0.987159	+	0.159743i	$0.948933\pi$
$30$	0	0
$31$	7.56721	1.35911	0.679556	−	0.733624i	$-0.262173\pi$
$31$	7.56721	1.35911	0.679556	+	0.733624i	$0.262173\pi$
$32$	−0.814076	−0.143910
$33$	0	0
$34$	−14.6758	−2.51687
$35$	−11.1958	−1.89244
$36$	0	0
$37$	−5.85939	−0.963278	−0.481639	−	0.876370i	$-0.659958\pi$
$37$	−5.85939	−0.963278	−0.481639	+	0.876370i	$0.659958\pi$
$38$	0.974656	0.158110
$39$	0	0
$40$	12.2006	1.92909
$41$	−1.13182	−0.176760	−0.0883801	−	0.996087i	$-0.528169\pi$
$41$	−1.13182	−0.176760	−0.0883801	+	0.996087i	$0.528169\pi$
$42$	0	0
$43$	3.18315	0.485425	0.242713	−	0.970098i	$-0.421963\pi$
$43$	3.18315	0.485425	0.242713	+	0.970098i	$0.421963\pi$
$44$	15.9011	2.39719
$45$	0	0
$46$	−4.56048	−0.672406
$47$	−0.625030	−0.0911700	−0.0455850	−	0.998960i	$-0.514515\pi$
$47$	−0.625030	−0.0911700	−0.0455850	+	0.998960i	$0.514515\pi$
$48$	0	0
$49$	15.1727	2.16753
$50$	−1.61080	−0.227802
$51$	0	0
$52$	−27.0269	−3.74796
$53$	−12.8766	−1.76874	−0.884368	−	0.466790i	$-0.845411\pi$
$53$	−12.8766	−1.76874	−0.884368	+	0.466790i	$0.845411\pi$
$54$	0	0
$55$	−9.26447	−1.24922
$56$	−24.1626	−3.22886
$57$	0	0
$58$	26.2180	3.44259
$59$	0.141655	0.0184419	0.00922096	−	0.999957i	$-0.497065\pi$
$59$	0.141655	0.0184419	0.00922096	+	0.999957i	$0.497065\pi$
$60$	0	0
$61$	−3.91754	−0.501589	−0.250795	−	0.968040i	$-0.580692\pi$
$61$	−3.91754	−0.501589	−0.250795	+	0.968040i	$0.580692\pi$
$62$	−18.6603	−2.36987
$63$	0	0
$64$	−6.97637	−0.872046
$65$	15.7467	1.95313
$66$	0	0
$67$	0.202947	0.0247939	0.0123970	−	0.999923i	$-0.496054\pi$
$67$	0.202947	0.0247939	0.0123970	+	0.999923i	$0.496054\pi$
$68$	24.2869	2.94522
$69$	0	0
$70$	27.6084	3.29983
$71$	−6.99318	−0.829938	−0.414969	−	0.909836i	$-0.636208\pi$
$71$	−6.99318	−0.829938	−0.414969	+	0.909836i	$0.636208\pi$
$72$	0	0
$73$	0.952512	0.111483	0.0557415	−	0.998445i	$-0.482248\pi$
$73$	0.952512	0.111483	0.0557415	+	0.998445i	$0.482248\pi$
$74$	14.4490	1.67966
$75$	0	0
$76$	−1.61296	−0.185019
$77$	18.3477	2.09091
$78$	0	0
$79$	−8.10807	−0.912229	−0.456114	−	0.889921i	$-0.650759\pi$
$79$	−8.10807	−0.912229	−0.456114	+	0.889921i	$0.650759\pi$
$80$	−10.6802	−1.19408
$81$	0	0
$82$	2.79100	0.308215
$83$	−9.46455	−1.03887	−0.519435	−	0.854510i	$-0.673858\pi$
$83$	−9.46455	−1.03887	−0.519435	+	0.854510i	$0.673858\pi$
$84$	0	0
$85$	−14.1503	−1.53481
$86$	−7.84947	−0.846430
$87$	0	0
$88$	−19.9943	−2.13140
$89$	−10.3042	−1.09224	−0.546121	−	0.837706i	$-0.683896\pi$
$89$	−10.3042	−1.09224	−0.546121	+	0.837706i	$0.683896\pi$
$90$	0	0
$91$	−31.1853	−3.26911
$92$	7.54714	0.786844
$93$	0	0
$94$	1.54129	0.158972
$95$	0.939757	0.0964170
$96$	0	0
$97$	−12.2470	−1.24350	−0.621749	−	0.783216i	$-0.713578\pi$
$97$	−12.2470	−1.24350	−0.621749	+	0.783216i	$0.713578\pi$
$98$	−37.4150	−3.77949
$99$	0	0
$100$	2.66572	0.266572
$101$	−14.8308	−1.47572	−0.737858	−	0.674956i	$-0.764162\pi$
$101$	−14.8308	−1.47572	−0.737858	+	0.674956i	$0.764162\pi$
$102$	0	0
$103$	−3.30296	−0.325451	−0.162725	−	0.986671i	$-0.552028\pi$
$103$	−3.30296	−0.325451	−0.162725	+	0.986671i	$0.552028\pi$
$104$	33.9840	3.33241
$105$	0	0
$106$	31.7530	3.08413
$107$	2.29757	0.222115	0.111057	−	0.993814i	$-0.464576\pi$
$107$	2.29757	0.222115	0.111057	+	0.993814i	$0.464576\pi$
$108$	0	0
$109$	9.44203	0.904383	0.452191	−	0.891921i	$-0.350642\pi$
$109$	9.44203	0.904383	0.452191	+	0.891921i	$0.350642\pi$
$110$	22.8457	2.17825
$111$	0	0
$112$	21.1515	1.99863
$113$	10.3934	0.977725	0.488862	−	0.872361i	$-0.337412\pi$
$113$	10.3934	0.977725	0.488862	+	0.872361i	$0.337412\pi$
$114$	0	0
$115$	−4.39719	−0.410040
$116$	−43.3882	−4.02849
$117$	0	0
$118$	−0.349314	−0.0321570
$119$	28.0238	2.56893
$120$	0	0
$121$	4.18256	0.380232
$122$	9.66044	0.874615
$123$	0	0
$124$	30.8810	2.77320
$125$	10.3351	0.924402
$126$	0	0
$127$	−3.47652	−0.308491	−0.154246	−	0.988033i	$-0.549295\pi$
$127$	−3.47652	−0.308491	−0.154246	+	0.988033i	$0.549295\pi$
$128$	18.8315	1.66449
$129$	0	0
$130$	−38.8305	−3.40566
$131$	0.615745	0.0537979	0.0268989	−	0.999638i	$-0.491437\pi$
$131$	0.615745	0.0537979	0.0268989	+	0.999638i	$0.491437\pi$
$132$	0	0
$133$	−1.86113	−0.161380
$134$	−0.500457	−0.0432329
$135$	0	0
$136$	−30.5388	−2.61868
$137$	−3.33652	−0.285058	−0.142529	−	0.989791i	$-0.545523\pi$
$137$	−3.33652	−0.285058	−0.142529	+	0.989791i	$0.545523\pi$
$138$	0	0
$139$	−2.51131	−0.213007	−0.106503	−	0.994312i	$-0.533965\pi$
$139$	−2.51131	−0.213007	−0.106503	+	0.994312i	$0.533965\pi$
$140$	−45.6891	−3.86143
$141$	0	0
$142$	17.2448	1.44715
$143$	−25.8056	−2.15797
$144$	0	0
$145$	25.2792	2.09933
$146$	−2.34884	−0.194392
$147$	0	0
$148$	−23.9116	−1.96552
$149$	−2.65229	−0.217284	−0.108642	−	0.994081i	$-0.534650\pi$
$149$	−2.65229	−0.217284	−0.108642	+	0.994081i	$0.534650\pi$
$150$	0	0
$151$	−7.64268	−0.621953	−0.310976	−	0.950418i	$-0.600656\pi$
$151$	−7.64268	−0.621953	−0.310976	+	0.950418i	$0.600656\pi$
$152$	2.02816	0.164505
$153$	0	0
$154$	−45.2445	−3.64590
$155$	−17.9922	−1.44517
$156$	0	0
$157$	−4.49351	−0.358621	−0.179311	−	0.983793i	$-0.557387\pi$
$157$	−4.49351	−0.358621	−0.179311	+	0.983793i	$0.557387\pi$
$158$	19.9941	1.59064
$159$	0	0
$160$	1.93559	0.153022
$161$	8.70835	0.686314
$162$	0	0
$163$	16.3285	1.27894	0.639472	−	0.768815i	$-0.279153\pi$
$163$	16.3285	1.27894	0.639472	+	0.768815i	$0.279153\pi$
$164$	−4.61883	−0.360670
$165$	0	0
$166$	23.3391	1.81146
$167$	21.8457	1.69047	0.845234	−	0.534396i	$-0.179461\pi$
$167$	21.8457	1.69047	0.845234	+	0.534396i	$0.179461\pi$
$168$	0	0
$169$	30.8613	2.37395
$170$	34.8939	2.67624
$171$	0	0
$172$	12.9901	0.990485
$173$	2.93795	0.223368	0.111684	−	0.993744i	$-0.464375\pi$
$173$	2.93795	0.223368	0.111684	+	0.993744i	$0.464375\pi$
$174$	0	0
$175$	3.07587	0.232514
$176$	17.5027	1.31931
$177$	0	0
$178$	25.4096	1.90453
$179$	−1.37373	−0.102677	−0.0513387	−	0.998681i	$-0.516349\pi$
$179$	−1.37373	−0.102677	−0.0513387	+	0.998681i	$0.516349\pi$
$180$	0	0
$181$	−12.3352	−0.916865	−0.458433	−	0.888729i	$-0.651589\pi$
$181$	−12.3352	−0.916865	−0.458433	+	0.888729i	$0.651589\pi$
$182$	76.9013	5.70031
$183$	0	0
$184$	−9.48989	−0.699604
$185$	13.9316	1.02427
$186$	0	0
$187$	23.1894	1.69578
$188$	−2.55068	−0.186028
$189$	0	0
$190$	−2.31739	−0.168121
$191$	13.4793	0.975329	0.487665	−	0.873031i	$-0.337849\pi$
$191$	13.4793	0.975329	0.487665	+	0.873031i	$0.337849\pi$
$192$	0	0
$193$	12.6210	0.908483	0.454241	−	0.890879i	$-0.349910\pi$
$193$	12.6210	0.908483	0.454241	+	0.890879i	$0.349910\pi$
$194$	30.2006	2.16827
$195$	0	0
$196$	61.9181	4.42272
$197$	−5.34673	−0.380938	−0.190469	−	0.981693i	$-0.561001\pi$
$197$	−5.34673	−0.380938	−0.190469	+	0.981693i	$0.561001\pi$
$198$	0	0
$199$	−27.4088	−1.94296	−0.971478	−	0.237129i	$-0.923794\pi$
$199$	−27.4088	−1.94296	−0.971478	+	0.237129i	$0.923794\pi$
$200$	−3.35191	−0.237016
$201$	0	0
$202$	36.5719	2.57319
$203$	−50.0639	−3.51380
$204$	0	0
$205$	2.69107	0.187952
$206$	8.14493	0.567484
$207$	0	0
$208$	−29.7490	−2.06272
$209$	−1.54007	−0.106529
$210$	0	0
$211$	−0.948427	−0.0652924	−0.0326462	−	0.999467i	$-0.510393\pi$
$211$	−0.948427	−0.0652924	−0.0326462	+	0.999467i	$0.510393\pi$
$212$	−52.5481	−3.60902
$213$	0	0
$214$	−5.66569	−0.387299
$215$	−7.56841	−0.516161
$216$	0	0
$217$	35.6324	2.41888
$218$	−23.2836	−1.57696
$219$	0	0
$220$	−37.8073	−2.54897
$221$	−39.4147	−2.65132
$222$	0	0
$223$	−23.6431	−1.58326	−0.791631	−	0.611000i	$-0.790768\pi$
$223$	−23.6431	−1.58326	−0.791631	+	0.611000i	$0.790768\pi$
$224$	−3.83331	−0.256124
$225$	0	0
$226$	−25.6295	−1.70485
$227$	−0.104193	−0.00691552	−0.00345776	−	0.999994i	$-0.501101\pi$
$227$	−0.104193	−0.00691552	−0.00345776	+	0.999994i	$0.501101\pi$
$228$	0	0
$229$	15.8886	1.04995	0.524973	−	0.851119i	$-0.324076\pi$
$229$	15.8886	1.04995	0.524973	+	0.851119i	$0.324076\pi$
$230$	10.8432	0.714981
$231$	0	0
$232$	54.5569	3.58184
$233$	−27.7194	−1.81596	−0.907978	−	0.419018i	$-0.862374\pi$
$233$	−27.7194	−1.81596	−0.907978	+	0.419018i	$0.862374\pi$
$234$	0	0
$235$	1.48610	0.0969426
$236$	0.578080	0.0376298
$237$	0	0
$238$	−69.1051	−4.47942
$239$	−2.94936	−0.190778	−0.0953892	−	0.995440i	$-0.530410\pi$
$239$	−2.94936	−0.190778	−0.0953892	+	0.995440i	$0.530410\pi$
$240$	0	0
$241$	−11.3904	−0.733722	−0.366861	−	0.930276i	$-0.619568\pi$
$241$	−11.3904	−0.733722	−0.366861	+	0.930276i	$0.619568\pi$
$242$	−10.3140	−0.663007
$243$	0	0
$244$	−15.9871	−1.02347
$245$	−36.0753	−2.30477
$246$	0	0
$247$	2.61763	0.166556
$248$	−38.8302	−2.46572
$249$	0	0
$250$	−25.4859	−1.61187
$251$	16.0716	1.01443	0.507214	−	0.861820i	$-0.330675\pi$
$251$	16.0716	1.01443	0.507214	+	0.861820i	$0.330675\pi$
$252$	0	0
$253$	7.20608	0.453043
$254$	8.57292	0.537913
$255$	0	0
$256$	−32.4848	−2.03030
$257$	18.2023	1.13543	0.567714	−	0.823226i	$-0.307828\pi$
$257$	18.2023	1.13543	0.567714	+	0.823226i	$0.307828\pi$
$258$	0	0
$259$	−27.5906	−1.71440
$260$	64.2605	3.98527
$261$	0	0
$262$	−1.51839	−0.0938067
$263$	−9.11949	−0.562332	−0.281166	−	0.959659i	$-0.590721\pi$
$263$	−9.11949	−0.562332	−0.281166	+	0.959659i	$0.590721\pi$
$264$	0	0
$265$	30.6160	1.88073
$266$	4.58945	0.281397
$267$	0	0
$268$	0.828206	0.0505907
$269$	14.0578	0.857119	0.428559	−	0.903514i	$-0.359021\pi$
$269$	14.0578	0.857119	0.428559	+	0.903514i	$0.359021\pi$
$270$	0	0
$271$	−7.13801	−0.433603	−0.216801	−	0.976216i	$-0.569562\pi$
$271$	−7.13801	−0.433603	−0.216801	+	0.976216i	$0.569562\pi$
$272$	26.7331	1.62093
$273$	0	0
$274$	8.22767	0.497052
$275$	2.54525	0.153485
$276$	0	0
$277$	−4.64697	−0.279210	−0.139605	−	0.990207i	$-0.544583\pi$
$277$	−4.64697	−0.279210	−0.139605	+	0.990207i	$0.544583\pi$
$278$	6.19276	0.371417
$279$	0	0
$280$	57.4501	3.43330
$281$	24.4344	1.45763	0.728817	−	0.684708i	$-0.240070\pi$
$281$	24.4344	1.45763	0.728817	+	0.684708i	$0.240070\pi$
$282$	0	0
$283$	−5.51364	−0.327752	−0.163876	−	0.986481i	$-0.552400\pi$
$283$	−5.51364	−0.327752	−0.163876	+	0.986481i	$0.552400\pi$
$284$	−28.5384	−1.69344
$285$	0	0
$286$	63.6352	3.76283
$287$	−5.32949	−0.314590
$288$	0	0
$289$	18.4189	1.08346
$290$	−62.3372	−3.66057
$291$	0	0
$292$	3.88710	0.227475
$293$	0.925316	0.0540576	0.0270288	−	0.999635i	$-0.491395\pi$
$293$	0.925316	0.0540576	0.0270288	+	0.999635i	$0.491395\pi$
$294$	0	0
$295$	−0.336806	−0.0196096
$296$	30.0668	1.74760
$297$	0	0
$298$	6.54041	0.378876
$299$	−12.2481	−0.708324
$300$	0	0
$301$	14.9888	0.863938
$302$	18.8465	1.08449
$303$	0	0
$304$	−1.77541	−0.101827
$305$	9.31453	0.533349
$306$	0	0
$307$	−27.4310	−1.56557	−0.782785	−	0.622292i	$-0.786202\pi$
$307$	−27.4310	−1.56557	−0.782785	+	0.622292i	$0.786202\pi$
$308$	74.8750	4.26640
$309$	0	0
$310$	44.3678	2.51992
$311$	22.0958	1.25294	0.626468	−	0.779447i	$-0.284500\pi$
$311$	22.0958	1.25294	0.626468	+	0.779447i	$0.284500\pi$
$312$	0	0
$313$	−12.0460	−0.680881	−0.340440	−	0.940266i	$-0.610576\pi$
$313$	−12.0460	−0.680881	−0.340440	+	0.940266i	$0.610576\pi$
$314$	11.0808	0.625323
$315$	0	0
$316$	−33.0882	−1.86136
$317$	−7.65816	−0.430125	−0.215063	−	0.976600i	$-0.568996\pi$
$317$	−7.65816	−0.430125	−0.215063	+	0.976600i	$0.568996\pi$
$318$	0	0
$319$	−41.4275	−2.31949
$320$	16.5874	0.927262
$321$	0	0
$322$	−21.4743	−1.19672
$323$	−2.35226	−0.130883
$324$	0	0
$325$	−4.32613	−0.239971
$326$	−40.2651	−2.23008
$327$	0	0
$328$	5.80779	0.320681
$329$	−2.94313	−0.162260
$330$	0	0
$331$	16.9303	0.930575	0.465287	−	0.885160i	$-0.345951\pi$
$331$	16.9303	0.930575	0.465287	+	0.885160i	$0.345951\pi$
$332$	−38.6239	−2.11976
$333$	0	0
$334$	−53.8703	−2.94765
$335$	−0.482537	−0.0263638
$336$	0	0
$337$	9.35829	0.509779	0.254889	−	0.966970i	$-0.417961\pi$
$337$	9.35829	0.509779	0.254889	+	0.966970i	$0.417961\pi$
$338$	−76.1024	−4.13943
$339$	0	0
$340$	−57.7458	−3.13171
$341$	29.4855	1.59673
$342$	0	0
$343$	38.4834	2.07791
$344$	−16.3339	−0.880666
$345$	0	0
$346$	−7.24484	−0.389485
$347$	35.5930	1.91073	0.955367	−	0.295421i	$-0.0954600\pi$
$347$	35.5930	1.91073	0.955367	+	0.295421i	$0.0954600\pi$
$348$	0	0
$349$	−11.8428	−0.633930	−0.316965	−	0.948437i	$-0.602664\pi$
$349$	−11.8428	−0.633930	−0.316965	+	0.948437i	$0.602664\pi$
$350$	−7.58493	−0.405432
$351$	0	0
$352$	−3.17203	−0.169070
$353$	3.61444	0.192377	0.0961886	−	0.995363i	$-0.469335\pi$
$353$	3.61444	0.192377	0.0961886	+	0.995363i	$0.469335\pi$
$354$	0	0
$355$	16.6273	0.882487
$356$	−42.0504	−2.22866
$357$	0	0
$358$	3.38755	0.179037
$359$	−19.2576	−1.01638	−0.508188	−	0.861246i	$-0.669684\pi$
$359$	−19.2576	−1.01638	−0.508188	+	0.861246i	$0.669684\pi$
$360$	0	0
$361$	−18.8438	−0.991778
$362$	30.4179	1.59873
$363$	0	0
$364$	−127.264	−6.67045
$365$	−2.26474	−0.118542
$366$	0	0
$367$	29.0707	1.51748	0.758740	−	0.651393i	$-0.225815\pi$
$367$	29.0707	1.51748	0.758740	+	0.651393i	$0.225815\pi$
$368$	8.30728	0.433047
$369$	0	0
$370$	−34.3546	−1.78601
$371$	−60.6332	−3.14792
$372$	0	0
$373$	11.0922	0.574334	0.287167	−	0.957881i	$-0.407287\pi$
$373$	11.0922	0.574334	0.287167	+	0.957881i	$0.407287\pi$
$374$	−57.1839	−2.95691
$375$	0	0
$376$	3.20727	0.165402
$377$	70.4137	3.62649
$378$	0	0
$379$	28.5334	1.46566	0.732830	−	0.680412i	$-0.238199\pi$
$379$	28.5334	1.46566	0.732830	+	0.680412i	$0.238199\pi$
$380$	3.83505	0.196734
$381$	0	0
$382$	−33.2393	−1.70067
$383$	22.5616	1.15284	0.576422	−	0.817152i	$-0.304448\pi$
$383$	22.5616	1.15284	0.576422	+	0.817152i	$0.304448\pi$
$384$	0	0
$385$	−43.6244	−2.22331
$386$	−31.1228	−1.58411
$387$	0	0
$388$	−49.9789	−2.53729
$389$	7.21475	0.365802	0.182901	−	0.983131i	$-0.441451\pi$
$389$	7.21475	0.365802	0.182901	+	0.983131i	$0.441451\pi$
$390$	0	0
$391$	11.0064	0.556616
$392$	−77.8568	−3.93236
$393$	0	0
$394$	13.1847	0.664238
$395$	19.2781	0.969989
$396$	0	0
$397$	27.6931	1.38988	0.694939	−	0.719069i	$-0.255431\pi$
$397$	27.6931	1.38988	0.694939	+	0.719069i	$0.255431\pi$
$398$	67.5886	3.38791
$399$	0	0
$400$	2.93421	0.146710
$401$	−19.7647	−0.987002	−0.493501	−	0.869745i	$-0.664283\pi$
$401$	−19.7647	−0.987002	−0.493501	+	0.869745i	$0.664283\pi$
$402$	0	0
$403$	−50.1161	−2.49646
$404$	−60.5228	−3.01112
$405$	0	0
$406$	123.455	6.12697
$407$	−22.8310	−1.13169
$408$	0	0
$409$	−3.39180	−0.167714	−0.0838569	−	0.996478i	$-0.526724\pi$
$409$	−3.39180	−0.167714	−0.0838569	+	0.996478i	$0.526724\pi$
$410$	−6.63603	−0.327730
$411$	0	0
$412$	−13.4790	−0.664065
$413$	0.667024	0.0328221
$414$	0	0
$415$	22.5034	1.10465
$416$	5.39146	0.264338
$417$	0	0
$418$	3.79773	0.185753
$419$	−34.8756	−1.70379	−0.851893	−	0.523716i	$-0.824545\pi$
$419$	−34.8756	−1.70379	−0.851893	+	0.523716i	$0.824545\pi$
$420$	0	0
$421$	−19.5846	−0.954494	−0.477247	−	0.878769i	$-0.658365\pi$
$421$	−19.5846	−0.954494	−0.477247	+	0.878769i	$0.658365\pi$
$422$	2.33877	0.113850
$423$	0	0
$424$	66.0747	3.20887
$425$	3.88755	0.188574
$426$	0	0
$427$	−18.4469	−0.892706
$428$	9.37615	0.453214
$429$	0	0
$430$	18.6633	0.900024
$431$	−38.4490	−1.85202	−0.926012	−	0.377493i	$-0.876786\pi$
$431$	−38.4490	−1.85202	−0.926012	+	0.377493i	$0.876786\pi$
$432$	0	0
$433$	12.4697	0.599253	0.299627	−	0.954057i	$-0.403138\pi$
$433$	12.4697	0.599253	0.299627	+	0.954057i	$0.403138\pi$
$434$	−87.8676	−4.21778
$435$	0	0
$436$	38.5319	1.84535
$437$	−0.730962	−0.0349666
$438$	0	0
$439$	−20.4231	−0.974741	−0.487371	−	0.873195i	$-0.662044\pi$
$439$	−20.4231	−0.974741	−0.487371	+	0.873195i	$0.662044\pi$
$440$	47.5395	2.26636
$441$	0	0
$442$	97.1946	4.62307
$443$	−32.8537	−1.56093	−0.780463	−	0.625202i	$-0.785016\pi$
$443$	−32.8537	−1.56093	−0.780463	+	0.625202i	$0.785016\pi$
$444$	0	0
$445$	24.4998	1.16140
$446$	58.3027	2.76071
$447$	0	0
$448$	−32.8502	−1.55203
$449$	−17.9950	−0.849238	−0.424619	−	0.905372i	$-0.639592\pi$
$449$	−17.9950	−0.849238	−0.424619	+	0.905372i	$0.639592\pi$
$450$	0	0
$451$	−4.41011	−0.207664
$452$	42.4142	1.99500
$453$	0	0
$454$	0.256934	0.0120585
$455$	74.1477	3.47610
$456$	0	0
$457$	37.3590	1.74758	0.873791	−	0.486302i	$-0.161655\pi$
$457$	37.3590	1.74758	0.873791	+	0.486302i	$0.161655\pi$
$458$	−39.1803	−1.83078
$459$	0	0
$460$	−17.9445	−0.836665
$461$	−9.41308	−0.438411	−0.219205	−	0.975679i	$-0.570346\pi$
$461$	−9.41308	−0.438411	−0.219205	+	0.975679i	$0.570346\pi$
$462$	0	0
$463$	−29.5125	−1.37156	−0.685781	−	0.727808i	$-0.740539\pi$
$463$	−29.5125	−1.37156	−0.685781	+	0.727808i	$0.740539\pi$
$464$	−47.7582	−2.21712
$465$	0	0
$466$	68.3545	3.16646
$467$	−15.7255	−0.727690	−0.363845	−	0.931460i	$-0.618536\pi$
$467$	−15.7255	−0.727690	−0.363845	+	0.931460i	$0.618536\pi$
$468$	0	0
$469$	0.955634	0.0441271
$470$	−3.66465	−0.169038
$471$	0	0
$472$	−0.726886	−0.0334576
$473$	12.4031	0.570294
$474$	0	0
$475$	−0.258182	−0.0118462
$476$	114.362	5.24178
$477$	0	0
$478$	7.27297	0.332658
$479$	19.5493	0.893232	0.446616	−	0.894726i	$-0.352629\pi$
$479$	19.5493	0.893232	0.446616	+	0.894726i	$0.352629\pi$
$480$	0	0
$481$	38.8055	1.76938
$482$	28.0882	1.27938
$483$	0	0
$484$	17.0686	0.775844
$485$	29.1192	1.32223
$486$	0	0
$487$	−16.1545	−0.732030	−0.366015	−	0.930609i	$-0.619278\pi$
$487$	−16.1545	−0.732030	−0.366015	+	0.930609i	$0.619278\pi$
$488$	20.1024	0.909992
$489$	0	0
$490$	88.9598	4.01880
$491$	−28.8831	−1.30348	−0.651738	−	0.758444i	$-0.725960\pi$
$491$	−28.8831	−1.30348	−0.651738	+	0.758444i	$0.725960\pi$
$492$	0	0
$493$	−63.2752	−2.84977
$494$	−6.45494	−0.290422
$495$	0	0
$496$	33.9913	1.52625
$497$	−32.9294	−1.47709
$498$	0	0
$499$	−1.88279	−0.0842854	−0.0421427	−	0.999112i	$-0.513418\pi$
$499$	−1.88279	−0.0842854	−0.0421427	+	0.999112i	$0.513418\pi$
$500$	42.1766	1.88619
$501$	0	0
$502$	−39.6316	−1.76884
$503$	−19.1927	−0.855760	−0.427880	−	0.903836i	$-0.640739\pi$
$503$	−19.1927	−0.855760	−0.427880	+	0.903836i	$0.640739\pi$
$504$	0	0
$505$	35.2623	1.56915
$506$	−17.7698	−0.789965
$507$	0	0
$508$	−14.1873	−0.629461
$509$	−25.9305	−1.14935	−0.574674	−	0.818383i	$-0.694871\pi$
$509$	−25.9305	−1.14935	−0.574674	+	0.818383i	$0.694871\pi$
$510$	0	0
$511$	4.48517	0.198412
$512$	42.4427	1.87572
$513$	0	0
$514$	−44.8859	−1.97983
$515$	7.85329	0.346057
$516$	0	0
$517$	−2.43542	−0.107110
$518$	68.0371	2.98938
$519$	0	0
$520$	−80.8022	−3.54341
$521$	−21.8092	−0.955478	−0.477739	−	0.878502i	$-0.658544\pi$
$521$	−21.8092	−0.955478	−0.477739	+	0.878502i	$0.658544\pi$
$522$	0	0
$523$	27.0965	1.18485	0.592423	−	0.805627i	$-0.298172\pi$
$523$	27.0965	1.18485	0.592423	+	0.805627i	$0.298172\pi$
$524$	2.51279	0.109772
$525$	0	0
$526$	22.4882	0.980531
$527$	45.0353	1.96177
$528$	0	0
$529$	−19.5798	−0.851295
$530$	−75.4975	−3.27940
$531$	0	0
$532$	−7.59508	−0.329288
$533$	7.49579	0.324679
$534$	0	0
$535$	−5.46282	−0.236178
$536$	−1.04140	−0.0449816
$537$	0	0
$538$	−34.6658	−1.49455
$539$	59.1200	2.54648
$540$	0	0
$541$	17.7634	0.763707	0.381854	−	0.924223i	$-0.375286\pi$
$541$	17.7634	0.763707	0.381854	+	0.924223i	$0.375286\pi$
$542$	17.6019	0.756068
$543$	0	0
$544$	−4.84487	−0.207722
$545$	−22.4498	−0.961646
$546$	0	0
$547$	−30.2921	−1.29520	−0.647599	−	0.761982i	$-0.724227\pi$
$547$	−30.2921	−1.29520	−0.647599	+	0.761982i	$0.724227\pi$
$548$	−13.6160	−0.581646
$549$	0	0
$550$	−6.27646	−0.267629
$551$	4.20227	0.179023
$552$	0	0
$553$	−38.1792	−1.62354
$554$	11.4592	0.486854
$555$	0	0
$556$	−10.2484	−0.434629
$557$	−5.30744	−0.224883	−0.112442	−	0.993658i	$-0.535867\pi$
$557$	−5.30744	−0.224883	−0.112442	+	0.993658i	$0.535867\pi$
$558$	0	0
$559$	−21.0813	−0.891644
$560$	−50.2908	−2.12518
$561$	0	0
$562$	−60.2540	−2.54166
$563$	−0.877218	−0.0369703	−0.0184852	−	0.999829i	$-0.505884\pi$
$563$	−0.877218	−0.0369703	−0.0184852	+	0.999829i	$0.505884\pi$
$564$	0	0
$565$	−24.7118	−1.03963
$566$	13.5964	0.571498
$567$	0	0
$568$	35.8847	1.50569
$569$	−5.25733	−0.220399	−0.110199	−	0.993910i	$-0.535149\pi$
$569$	−5.25733	−0.220399	−0.110199	+	0.993910i	$0.535149\pi$
$570$	0	0
$571$	−35.4142	−1.48204	−0.741018	−	0.671485i	$-0.765657\pi$
$571$	−35.4142	−1.48204	−0.741018	+	0.671485i	$0.765657\pi$
$572$	−105.310	−4.40323
$573$	0	0
$574$	13.1422	0.548547
$575$	1.20805	0.0503793
$576$	0	0
$577$	−7.68971	−0.320127	−0.160063	−	0.987107i	$-0.551170\pi$
$577$	−7.68971	−0.320127	−0.160063	+	0.987107i	$0.551170\pi$
$578$	−45.4199	−1.88922
$579$	0	0
$580$	103.162	4.28356
$581$	−44.5666	−1.84893
$582$	0	0
$583$	−50.1734	−2.07797
$584$	−4.88770	−0.202254
$585$	0	0
$586$	−2.28178	−0.0942595
$587$	15.4388	0.637226	0.318613	−	0.947885i	$-0.396783\pi$
$587$	15.4388	0.637226	0.318613	+	0.947885i	$0.396783\pi$
$588$	0	0
$589$	−2.99091	−0.123238
$590$	0.830546	0.0341930
$591$	0	0
$592$	−26.3199	−1.08174
$593$	23.4716	0.963865	0.481932	−	0.876208i	$-0.339935\pi$
$593$	23.4716	0.963865	0.481932	+	0.876208i	$0.339935\pi$
$594$	0	0
$595$	−66.6307	−2.73159
$596$	−10.8237	−0.443357
$597$	0	0
$598$	30.2031	1.23510
$599$	4.63449	0.189360	0.0946801	−	0.995508i	$-0.469817\pi$
$599$	4.63449	0.189360	0.0946801	+	0.995508i	$0.469817\pi$
$600$	0	0
$601$	17.8220	0.726973	0.363486	−	0.931600i	$-0.381586\pi$
$601$	17.8220	0.726973	0.363486	+	0.931600i	$0.381586\pi$
$602$	−36.9615	−1.50644
$603$	0	0
$604$	−31.1890	−1.26906
$605$	−9.94465	−0.404308
$606$	0	0
$607$	46.4186	1.88407	0.942036	−	0.335511i	$-0.108909\pi$
$607$	46.4186	1.88407	0.942036	+	0.335511i	$0.108909\pi$
$608$	0.321761	0.0130491
$609$	0	0
$610$	−22.9691	−0.929993
$611$	4.13944	0.167464
$612$	0	0
$613$	7.84940	0.317034	0.158517	−	0.987356i	$-0.449329\pi$
$613$	7.84940	0.317034	0.158517	+	0.987356i	$0.449329\pi$
$614$	67.6434	2.72987
$615$	0	0
$616$	−94.1490	−3.79337
$617$	26.4714	1.06570	0.532849	−	0.846211i	$-0.321121\pi$
$617$	26.4714	1.06570	0.532849	+	0.846211i	$0.321121\pi$
$618$	0	0
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−73.4242	−2.94879
$621$	0	0
$622$	−54.4870	−2.18473
$623$	−48.5203	−1.94392
$624$	0	0
$625$	−27.8394	−1.11358
$626$	29.7048	1.18724
$627$	0	0
$628$	−18.3375	−0.731748
$629$	−34.8714	−1.39042
$630$	0	0
$631$	−25.2324	−1.00448	−0.502242	−	0.864727i	$-0.667491\pi$
$631$	−25.2324	−1.00448	−0.502242	+	0.864727i	$0.667491\pi$
$632$	41.6056	1.65498
$633$	0	0
$634$	18.8846	0.750004
$635$	8.26595	0.328024
$636$	0	0
$637$	−100.485	−3.98138
$638$	102.158	4.04447
$639$	0	0
$640$	−44.7747	−1.76988
$641$	−24.4105	−0.964159	−0.482079	−	0.876128i	$-0.660118\pi$
$641$	−24.4105	−0.964159	−0.482079	+	0.876128i	$0.660118\pi$
$642$	0	0
$643$	−5.52050	−0.217707	−0.108854	−	0.994058i	$-0.534718\pi$
$643$	−5.52050	−0.217707	−0.108854	+	0.994058i	$0.534718\pi$
$644$	35.5379	1.40039
$645$	0	0
$646$	5.80054	0.228219
$647$	−39.5404	−1.55449	−0.777247	−	0.629195i	$-0.783385\pi$
$647$	−39.5404	−1.55449	−0.777247	+	0.629195i	$0.783385\pi$
$648$	0	0
$649$	0.551956	0.0216662
$650$	10.6680	0.418434
$651$	0	0
$652$	66.6347	2.60962
$653$	−26.1438	−1.02308	−0.511542	−	0.859258i	$-0.670926\pi$
$653$	−26.1438	−1.02308	−0.511542	+	0.859258i	$0.670926\pi$
$654$	0	0
$655$	−1.46402	−0.0572042
$656$	−5.08403	−0.198498
$657$	0	0
$658$	7.25761	0.282931
$659$	−9.69906	−0.377822	−0.188911	−	0.981994i	$-0.560496\pi$
$659$	−9.69906	−0.377822	−0.188911	+	0.981994i	$0.560496\pi$
$660$	0	0
$661$	−3.26550	−0.127013	−0.0635067	−	0.997981i	$-0.520228\pi$
$661$	−3.26550	−0.127013	−0.0635067	+	0.997981i	$0.520228\pi$
$662$	−41.7493	−1.62263
$663$	0	0
$664$	48.5662	1.88473
$665$	4.42512	0.171599
$666$	0	0
$667$	−19.6627	−0.761342
$668$	89.1499	3.44931
$669$	0	0
$670$	1.18991	0.0459703
$671$	−15.2646	−0.589284
$672$	0	0
$673$	−22.2964	−0.859462	−0.429731	−	0.902957i	$-0.641392\pi$
$673$	−22.2964	−0.859462	−0.429731	+	0.902957i	$0.641392\pi$
$674$	−23.0771	−0.888895
$675$	0	0
$676$	125.942	4.84392
$677$	−5.74561	−0.220822	−0.110411	−	0.993886i	$-0.535217\pi$
$677$	−5.74561	−0.220822	−0.110411	+	0.993886i	$0.535217\pi$
$678$	0	0
$679$	−57.6687	−2.21312
$680$	72.6105	2.78448
$681$	0	0
$682$	−72.7097	−2.78420
$683$	−3.03267	−0.116042	−0.0580210	−	0.998315i	$-0.518479\pi$
$683$	−3.03267	−0.116042	−0.0580210	+	0.998315i	$0.518479\pi$
$684$	0	0
$685$	7.93307	0.303107
$686$	−94.8981	−3.62323
$687$	0	0
$688$	14.2984	0.545123
$689$	85.2790	3.24887
$690$	0	0
$691$	−12.7010	−0.483168	−0.241584	−	0.970380i	$-0.577667\pi$
$691$	−12.7010	−0.483168	−0.241584	+	0.970380i	$0.577667\pi$
$692$	11.9895	0.455772
$693$	0	0
$694$	−87.7705	−3.33172
$695$	5.97102	0.226494
$696$	0	0
$697$	−6.73587	−0.255139
$698$	29.2037	1.10538
$699$	0	0
$700$	12.5523	0.474432
$701$	−9.73745	−0.367778	−0.183889	−	0.982947i	$-0.558869\pi$
$701$	−9.73745	−0.367778	−0.183889	+	0.982947i	$0.558869\pi$
$702$	0	0
$703$	2.31590	0.0873459
$704$	−27.1833	−1.02451
$705$	0	0
$706$	−8.91302	−0.335446
$707$	−69.8349	−2.62641
$708$	0	0
$709$	−36.5973	−1.37444	−0.687220	−	0.726449i	$-0.741169\pi$
$709$	−36.5973	−1.37444	−0.687220	+	0.726449i	$0.741169\pi$
$710$	−41.0021	−1.53878
$711$	0	0
$712$	52.8747	1.98156
$713$	13.9947	0.524105
$714$	0	0
$715$	61.3566	2.29461
$716$	−5.60605	−0.209508
$717$	0	0
$718$	47.4881	1.77224
$719$	−8.17131	−0.304739	−0.152369	−	0.988324i	$-0.548690\pi$
$719$	−8.17131	−0.304739	−0.152369	+	0.988324i	$0.548690\pi$
$720$	0	0
$721$	−15.5530	−0.579222
$722$	46.4678	1.72935
$723$	0	0
$724$	−50.3385	−1.87082
$725$	−6.94504	−0.257932
$726$	0	0
$727$	1.52574	0.0565864	0.0282932	−	0.999600i	$-0.490993\pi$
$727$	1.52574	0.0565864	0.0282932	+	0.999600i	$0.490993\pi$
$728$	160.024	5.93087
$729$	0	0
$730$	5.58473	0.206700
$731$	18.9441	0.700673
$732$	0	0
$733$	−38.8518	−1.43503	−0.717513	−	0.696545i	$-0.754719\pi$
$733$	−38.8518	−1.43503	−0.717513	+	0.696545i	$0.754719\pi$
$734$	−71.6869	−2.64601
$735$	0	0
$736$	−1.50554	−0.0554949
$737$	0.790779	0.0291287
$738$	0	0
$739$	44.8177	1.64865	0.824323	−	0.566120i	$-0.191556\pi$
$739$	44.8177	1.64865	0.824323	+	0.566120i	$0.191556\pi$
$740$	56.8534	2.08997
$741$	0	0
$742$	149.518	5.48899
$743$	0.348330	0.0127790	0.00638950	−	0.999980i	$-0.497966\pi$
$743$	0.348330	0.0127790	0.00638950	+	0.999980i	$0.497966\pi$
$744$	0	0
$745$	6.30622	0.231042
$746$	−27.3528	−1.00146
$747$	0	0
$748$	94.6336	3.46015
$749$	10.8188	0.395310
$750$	0	0
$751$	6.42985	0.234628	0.117314	−	0.993095i	$-0.462572\pi$
$751$	6.42985	0.234628	0.117314	+	0.993095i	$0.462572\pi$
$752$	−2.80758	−0.102382
$753$	0	0
$754$	−173.636	−6.32346
$755$	18.1716	0.661333
$756$	0	0
$757$	−33.2343	−1.20792	−0.603960	−	0.797014i	$-0.706411\pi$
$757$	−33.2343	−1.20792	−0.603960	+	0.797014i	$0.706411\pi$
$758$	−70.3617	−2.55565
$759$	0	0
$760$	−4.82225	−0.174921
$761$	1.37647	0.0498969	0.0249484	−	0.999689i	$-0.492058\pi$
$761$	1.37647	0.0498969	0.0249484	+	0.999689i	$0.492058\pi$
$762$	0	0
$763$	44.4605	1.60958
$764$	55.0077	1.99011
$765$	0	0
$766$	−55.6357	−2.01020
$767$	−0.938152	−0.0338747
$768$	0	0
$769$	−4.55292	−0.164183	−0.0820913	−	0.996625i	$-0.526160\pi$
$769$	−4.55292	−0.164183	−0.0820913	+	0.996625i	$0.526160\pi$
$770$	107.575	3.87675
$771$	0	0
$772$	51.5052	1.85371
$773$	24.0249	0.864116	0.432058	−	0.901846i	$-0.357788\pi$
$773$	24.0249	0.864116	0.432058	+	0.901846i	$0.357788\pi$
$774$	0	0
$775$	4.94305	0.177559
$776$	62.8442	2.25598
$777$	0	0
$778$	−17.7912	−0.637845
$779$	0.447347	0.0160279
$780$	0	0
$781$	−27.2488	−0.975038
$782$	−27.1411	−0.970565
$783$	0	0
$784$	68.1544	2.43409
$785$	10.6840	0.381328
$786$	0	0
$787$	−8.01053	−0.285545	−0.142772	−	0.989756i	$-0.545602\pi$
$787$	−8.01053	−0.285545	−0.142772	+	0.989756i	$0.545602\pi$
$788$	−21.8194	−0.777285
$789$	0	0
$790$	−47.5389	−1.69136
$791$	48.9401	1.74011
$792$	0	0
$793$	25.9450	0.921335
$794$	−68.2898	−2.42351
$795$	0	0
$796$	−111.852	−3.96450
$797$	−17.7909	−0.630185	−0.315093	−	0.949061i	$-0.602036\pi$
$797$	−17.7909	−0.630185	−0.315093	+	0.949061i	$0.602036\pi$
$798$	0	0
$799$	−3.71979	−0.131597
$800$	−0.531770	−0.0188009
$801$	0	0
$802$	48.7387	1.72102
$803$	3.71144	0.130974
$804$	0	0
$805$	−20.7054	−0.729770
$806$	123.584	4.35305
$807$	0	0
$808$	76.1022	2.67727
$809$	−16.4208	−0.577325	−0.288663	−	0.957431i	$-0.593211\pi$
$809$	−16.4208	−0.577325	−0.288663	+	0.957431i	$0.593211\pi$
$810$	0	0
$811$	−41.4014	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$811$	−41.4014	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$812$	−204.306	−7.16972
$813$	0	0
$814$	56.3001	1.97332
$815$	−38.8234	−1.35992
$816$	0	0
$817$	−1.25813	−0.0440163
$818$	8.36401	0.292441
$819$	0	0
$820$	10.9820	0.383507
$821$	−3.47859	−0.121404	−0.0607018	−	0.998156i	$-0.519334\pi$
$821$	−3.47859	−0.121404	−0.0607018	+	0.998156i	$0.519334\pi$
$822$	0	0
$823$	−32.2803	−1.12522	−0.562610	−	0.826723i	$-0.690203\pi$
$823$	−32.2803	−1.12522	−0.562610	+	0.826723i	$0.690203\pi$
$824$	16.9488	0.590438
$825$	0	0
$826$	−1.64485	−0.0572315
$827$	32.1878	1.11928	0.559641	−	0.828735i	$-0.310939\pi$
$827$	32.1878	1.11928	0.559641	+	0.828735i	$0.310939\pi$
$828$	0	0
$829$	25.5266	0.886576	0.443288	−	0.896379i	$-0.353812\pi$
$829$	25.5266	0.886576	0.443288	+	0.896379i	$0.353812\pi$
$830$	−55.4922	−1.92616
$831$	0	0
$832$	46.2030	1.60180
$833$	90.2983	3.12865
$834$	0	0
$835$	−51.9413	−1.79750
$836$	−6.28486	−0.217366
$837$	0	0
$838$	86.0014	2.97087
$839$	−22.3785	−0.772592	−0.386296	−	0.922375i	$-0.626246\pi$
$839$	−22.3785	−0.772592	−0.386296	+	0.922375i	$0.626246\pi$
$840$	0	0
$841$	84.0399	2.89793
$842$	48.2945	1.66434
$843$	0	0
$844$	−3.87043	−0.133226
$845$	−73.3775	−2.52426
$846$	0	0
$847$	19.6948	0.676720
$848$	−57.8406	−1.98626
$849$	0	0
$850$	−9.58649	−0.328814
$851$	−10.8363	−0.371462
$852$	0	0
$853$	1.03212	0.0353391	0.0176696	−	0.999844i	$-0.494375\pi$
$853$	1.03212	0.0353391	0.0176696	+	0.999844i	$0.494375\pi$
$854$	45.4890	1.55660
$855$	0	0
$856$	−11.7897	−0.402964
$857$	−10.9674	−0.374639	−0.187320	−	0.982299i	$-0.559980\pi$
$857$	−10.9674	−0.374639	−0.187320	+	0.982299i	$0.559980\pi$
$858$	0	0
$859$	2.06472	0.0704474	0.0352237	−	0.999379i	$-0.488786\pi$
$859$	2.06472	0.0704474	0.0352237	+	0.999379i	$0.488786\pi$
$860$	−30.8859	−1.05320
$861$	0	0
$862$	94.8133	3.22935
$863$	−33.9733	−1.15647	−0.578233	−	0.815872i	$-0.696258\pi$
$863$	−33.9733	−1.15647	−0.578233	+	0.815872i	$0.696258\pi$
$864$	0	0
$865$	−6.98542	−0.237512
$866$	−30.7495	−1.04491
$867$	0	0
$868$	145.412	4.93561
$869$	−31.5929	−1.07172
$870$	0	0
$871$	−1.34408	−0.0455423
$872$	−48.4506	−1.64075
$873$	0	0
$874$	1.80251	0.0609709
$875$	48.6659	1.64521
$876$	0	0
$877$	13.8165	0.466549	0.233274	−	0.972411i	$-0.425056\pi$
$877$	13.8165	0.466549	0.233274	+	0.972411i	$0.425056\pi$
$878$	50.3623	1.69964
$879$	0	0
$880$	−41.6152	−1.40285
$881$	28.9080	0.973937	0.486968	−	0.873420i	$-0.338103\pi$
$881$	28.9080	0.973937	0.486968	+	0.873420i	$0.338103\pi$
$882$	0	0
$883$	−27.2983	−0.918662	−0.459331	−	0.888265i	$-0.651911\pi$
$883$	−27.2983	−0.918662	−0.459331	+	0.888265i	$0.651911\pi$
$884$	−160.847	−5.40988
$885$	0	0
$886$	81.0154	2.72177
$887$	−7.67238	−0.257613	−0.128807	−	0.991670i	$-0.541115\pi$
$887$	−7.67238	−0.257613	−0.128807	+	0.991670i	$0.541115\pi$
$888$	0	0
$889$	−16.3702	−0.549039
$890$	−60.4152	−2.02512
$891$	0	0
$892$	−96.4852	−3.23056
$893$	0.247041	0.00826690
$894$	0	0
$895$	3.26625	0.109179
$896$	88.6736	2.96238
$897$	0	0
$898$	44.3748	1.48081
$899$	−80.4548	−2.68332
$900$	0	0
$901$	−76.6335	−2.55303
$902$	10.8751	0.362101
$903$	0	0
$904$	−53.3323	−1.77380
$905$	29.3287	0.974919
$906$	0	0
$907$	12.5770	0.417612	0.208806	−	0.977957i	$-0.433042\pi$
$907$	12.5770	0.417612	0.208806	+	0.977957i	$0.433042\pi$
$908$	−0.425200	−0.0141108
$909$	0	0
$910$	−182.844	−6.06123
$911$	45.3692	1.50315	0.751574	−	0.659649i	$-0.229295\pi$
$911$	45.3692	1.50315	0.751574	+	0.659649i	$0.229295\pi$
$912$	0	0
$913$	−36.8784	−1.22050
$914$	−92.1254	−3.04724
$915$	0	0
$916$	64.8395	2.14236
$917$	2.89941	0.0957470
$918$	0	0
$919$	−7.95324	−0.262353	−0.131177	−	0.991359i	$-0.541875\pi$
$919$	−7.95324	−0.262353	−0.131177	+	0.991359i	$0.541875\pi$
$920$	22.5636	0.743901
$921$	0	0
$922$	23.2121	0.764451
$923$	46.3144	1.52446
$924$	0	0
$925$	−3.82747	−0.125846
$926$	72.7762	2.39157
$927$	0	0
$928$	8.65528	0.284123
$929$	−45.9188	−1.50655	−0.753273	−	0.657708i	$-0.771526\pi$
$929$	−45.9188	−1.50655	−0.753273	+	0.657708i	$0.771526\pi$
$930$	0	0
$931$	−5.99694	−0.196542
$932$	−113.120	−3.70536
$933$	0	0
$934$	38.7783	1.26886
$935$	−55.1363	−1.80315
$936$	0	0
$937$	−22.0280	−0.719623	−0.359812	−	0.933025i	$-0.617159\pi$
$937$	−22.0280	−0.719623	−0.359812	+	0.933025i	$0.617159\pi$
$938$	−2.35654	−0.0769439
$939$	0	0
$940$	6.06463	0.197806
$941$	−25.0867	−0.817804	−0.408902	−	0.912578i	$-0.634088\pi$
$941$	−25.0867	−0.817804	−0.408902	+	0.912578i	$0.634088\pi$
$942$	0	0
$943$	−2.09317	−0.0681628
$944$	0.636303	0.0207099
$945$	0	0
$946$	−30.5853	−0.994414
$947$	44.2883	1.43918	0.719588	−	0.694401i	$-0.244331\pi$
$947$	44.2883	1.43918	0.719588	+	0.694401i	$0.244331\pi$
$948$	0	0
$949$	−6.30828	−0.204776
$950$	0.636664	0.0206561
$951$	0	0
$952$	−143.801	−4.66060
$953$	−5.30431	−0.171824	−0.0859118	−	0.996303i	$-0.527380\pi$
$953$	−5.30431	−0.171824	−0.0859118	+	0.996303i	$0.527380\pi$
$954$	0	0
$955$	−32.0491	−1.03708
$956$	−12.0360	−0.389273
$957$	0	0
$958$	−48.2076	−1.55752
$959$	−15.7109	−0.507333
$960$	0	0
$961$	26.2627	0.847184
$962$	−95.6924	−3.08525
$963$	0	0
$964$	−46.4832	−1.49712
$965$	−30.0084	−0.966006
$966$	0	0
$967$	−4.04692	−0.130140	−0.0650701	−	0.997881i	$-0.520727\pi$
$967$	−4.04692	−0.130140	−0.0650701	+	0.997881i	$0.520727\pi$
$968$	−21.4623	−0.689824
$969$	0	0
$970$	−71.8064	−2.30556
$971$	−38.9428	−1.24973	−0.624867	−	0.780732i	$-0.714847\pi$
$971$	−38.9428	−1.24973	−0.624867	+	0.780732i	$0.714847\pi$
$972$	0	0
$973$	−11.8252	−0.379100
$974$	39.8361	1.27643
$975$	0	0
$976$	−17.5973	−0.563275
$977$	−46.3245	−1.48205	−0.741026	−	0.671476i	$-0.765661\pi$
$977$	−46.3245	−1.48205	−0.741026	+	0.671476i	$0.765661\pi$
$978$	0	0
$979$	−40.1501	−1.28320
$980$	−147.220	−4.70276
$981$	0	0
$982$	71.2242	2.27285
$983$	−1.07524	−0.0342949	−0.0171475	−	0.999853i	$-0.505458\pi$
$983$	−1.07524	−0.0342949	−0.0171475	+	0.999853i	$0.505458\pi$
$984$	0	0
$985$	12.7126	0.405058
$986$	156.033	4.96911
$987$	0	0
$988$	10.6823	0.339849
$989$	5.88686	0.187191
$990$	0	0
$991$	30.8827	0.981020	0.490510	−	0.871436i	$-0.336811\pi$
$991$	30.8827	0.981020	0.490510	+	0.871436i	$0.336811\pi$
$992$	−6.16029	−0.195589
$993$	0	0
$994$	81.2021	2.57558
$995$	65.1685	2.06598
$996$	0	0
$997$	40.1301	1.27093	0.635466	−	0.772129i	$-0.280808\pi$
$997$	40.1301	1.27093	0.635466	+	0.772129i	$0.280808\pi$
$998$	4.64287	0.146967
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5571.2.a.g.1.4		30
3.2	odd	2		619.2.a.b.1.27	✓	30
12.11	even	2		9904.2.a.n.1.16		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.27	✓	30	3.2	odd	2
5571.2.a.g.1.4		30	1.1	even	1	trivial
9904.2.a.n.1.16		30	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5571 = 3^{2} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5571.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46595 q^{2} +4.08090 q^{4} -2.37765 q^{5} +4.70879 q^{7} -5.13138 q^{8} +O(q^{10})\)	\(q-2.46595 q^{2} +4.08090 q^{4} -2.37765 q^{5} +4.70879 q^{7} -5.13138 q^{8} +5.86316 q^{10} +3.89648 q^{11} -6.62279 q^{13} -11.6116 q^{14} +4.49192 q^{16} +5.95137 q^{17} -0.395246 q^{19} -9.70294 q^{20} -9.60851 q^{22} +1.84938 q^{23} +0.653219 q^{25} +16.3315 q^{26} +19.2161 q^{28} -10.6320 q^{29} +7.56721 q^{31} -0.814076 q^{32} -14.6758 q^{34} -11.1958 q^{35} -5.85939 q^{37} +0.974656 q^{38} +12.2006 q^{40} -1.13182 q^{41} +3.18315 q^{43} +15.9011 q^{44} -4.56048 q^{46} -0.625030 q^{47} +15.1727 q^{49} -1.61080 q^{50} -27.0269 q^{52} -12.8766 q^{53} -9.26447 q^{55} -24.1626 q^{56} +26.2180 q^{58} +0.141655 q^{59} -3.91754 q^{61} -18.6603 q^{62} -6.97637 q^{64} +15.7467 q^{65} +0.202947 q^{67} +24.2869 q^{68} +27.6084 q^{70} -6.99318 q^{71} +0.952512 q^{73} +14.4490 q^{74} -1.61296 q^{76} +18.3477 q^{77} -8.10807 q^{79} -10.6802 q^{80} +2.79100 q^{82} -9.46455 q^{83} -14.1503 q^{85} -7.84947 q^{86} -19.9943 q^{88} -10.3042 q^{89} -31.1853 q^{91} +7.54714 q^{92} +1.54129 q^{94} +0.939757 q^{95} -12.2470 q^{97} -37.4150 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) \) 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8	\( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) \) 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8 + 5 * q^10 - 23 * q^11 + 9 * q^13 - 7 * q^14 + 35 * q^16 - 4 * q^17 - q^19 - 29 * q^20 - 4 * q^23 + 35 * q^25 - q^26 - 13 * q^28 - 90 * q^29 + 2 * q^31 - 43 * q^32 - 9 * q^34 - 9 * q^35 + 19 * q^37 - 5 * q^38 - 12 * q^40 - 59 * q^41 - 4 * q^43 - 52 * q^44 - q^46 - 4 * q^47 + 30 * q^49 - 31 * q^50 - 12 * q^52 - 34 * q^53 - 17 * q^55 - 2 * q^56 + 6 * q^58 - 13 * q^59 + 16 * q^61 - 28 * q^62 + 37 * q^64 - 31 * q^65 - 11 * q^67 + 52 * q^68 - 40 * q^70 - 42 * q^71 - 4 * q^73 - 16 * q^74 - 42 * q^76 - 29 * q^77 + 3 * q^79 - 21 * q^80 - 43 * q^82 + 11 * q^83 + 19 * q^85 + 11 * q^86 - 47 * q^88 - 58 * q^89 - 39 * q^91 + 7 * q^92 - 46 * q^94 - 23 * q^95 - 9 * q^97

Introduction

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