LMFDB - Embedded newform 1850.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} -5.46410 q^{11} -0.732051 q^{12} +5.46410 q^{13} +4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} -2.46410 q^{18} +6.19615 q^{19} -3.46410 q^{21} -5.46410 q^{22} +8.00000 q^{23} -0.732051 q^{24} +5.46410 q^{26} +4.00000 q^{27} +4.73205 q^{28} +4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} +4.00000 q^{33} -5.46410 q^{34} -2.46410 q^{36} -1.00000 q^{37} +6.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} -3.46410 q^{42} -6.92820 q^{43} -5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} -0.732051 q^{48} +15.3923 q^{49} +4.00000 q^{51} +5.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +4.73205 q^{56} -4.53590 q^{57} +4.92820 q^{58} -10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} -11.6603 q^{63} +1.00000 q^{64} +4.00000 q^{66} +3.66025 q^{67} -5.46410 q^{68} -5.85641 q^{69} +2.92820 q^{71} -2.46410 q^{72} +0.928203 q^{73} -1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} -4.00000 q^{78} +8.73205 q^{79} +4.46410 q^{81} -2.00000 q^{82} +8.73205 q^{83} -3.46410 q^{84} -6.92820 q^{86} -3.60770 q^{87} -5.46410 q^{88} -2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} -0.535898 q^{93} +4.73205 q^{94} -0.732051 q^{96} +2.00000 q^{97} +15.3923 q^{98} +13.4641 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26}+ \cdots + 20 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 4 * q^13 + 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^22 + 16 * q^23 + 2 * q^24 + 4 * q^26 + 8 * q^27 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^33 - 4 * q^34 + 2 * q^36 - 2 * q^37 + 2 * q^38 - 8 * q^39 - 4 * q^41 - 4 * q^44 + 16 * q^46 + 6 * q^47 + 2 * q^48 + 10 * q^49 + 8 * q^51 + 4 * q^52 + 12 * q^53 + 8 * q^54 + 6 * q^56 - 16 * q^57 - 4 * q^58 - 10 * q^59 + 4 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 8 * q^66 - 10 * q^67 - 4 * q^68 + 16 * q^69 - 8 * q^71 + 2 * q^72 - 12 * q^73 - 2 * q^74 + 2 * q^76 - 24 * q^77 - 8 * q^78 + 14 * q^79 + 2 * q^81 - 4 * q^82 + 14 * q^83 - 28 * q^87 - 4 * q^88 - 4 * q^89 + 24 * q^91 + 16 * q^92 - 8 * q^93 + 6 * q^94 + 2 * q^96 + 4 * q^97 + 10 * q^98 + 20 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.732051	−0.298858
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	1.00000	0.353553
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	−0.732051	−0.211325
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	4.73205	1.26469
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.46410	−1.32524	−0.662620	−	0.748956i	$-0.730555\pi$
$17$	−5.46410	−1.32524	−0.662620	+	0.748956i	$0.730555\pi$
$18$	−2.46410	−0.580794
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	−5.46410	−1.16495
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−0.732051	−0.149429
$25$	0	0
$26$	5.46410	1.07160
$27$	4.00000	0.769800
$28$	4.73205	0.894274
$29$	4.92820	0.915144	0.457572	−	0.889172i	$-0.348719\pi$
$29$	4.92820	0.915144	0.457572	+	0.889172i	$0.348719\pi$
$30$	0	0
$31$	0.732051	0.131480	0.0657401	−	0.997837i	$-0.479059\pi$
$31$	0.732051	0.131480	0.0657401	+	0.997837i	$0.479059\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−5.46410	−0.937086
$35$	0	0
$36$	−2.46410	−0.410684
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	6.19615	1.00515
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−3.46410	−0.534522
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	−5.46410	−0.823744
$45$	0	0
$46$	8.00000	1.17954
$47$	4.73205	0.690241	0.345120	−	0.938558i	$-0.387838\pi$
$47$	4.73205	0.690241	0.345120	+	0.938558i	$0.387838\pi$
$48$	−0.732051	−0.105662
$49$	15.3923	2.19890
$50$	0	0
$51$	4.00000	0.560112
$52$	5.46410	0.757735
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	4.73205	0.632347
$57$	−4.53590	−0.600794
$58$	4.92820	0.647105
$59$	−10.1962	−1.32743	−0.663713	−	0.747987i	$-0.731020\pi$
$59$	−10.1962	−1.32743	−0.663713	+	0.747987i	$0.731020\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0.732051	0.0929705
$63$	−11.6603	−1.46905
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	3.66025	0.447171	0.223586	−	0.974684i	$-0.428224\pi$
$67$	3.66025	0.447171	0.223586	+	0.974684i	$0.428224\pi$
$68$	−5.46410	−0.662620
$69$	−5.85641	−0.705028
$70$	0	0
$71$	2.92820	0.347514	0.173757	−	0.984789i	$-0.444409\pi$
$71$	2.92820	0.347514	0.173757	+	0.984789i	$0.444409\pi$
$72$	−2.46410	−0.290397
$73$	0.928203	0.108638	0.0543190	−	0.998524i	$-0.482701\pi$
$73$	0.928203	0.108638	0.0543190	+	0.998524i	$0.482701\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	6.19615	0.710747
$77$	−25.8564	−2.94661
$78$	−4.00000	−0.452911
$79$	8.73205	0.982432	0.491216	−	0.871038i	$-0.336552\pi$
$79$	8.73205	0.982432	0.491216	+	0.871038i	$0.336552\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	−2.00000	−0.220863
$83$	8.73205	0.958467	0.479234	−	0.877687i	$-0.340915\pi$
$83$	8.73205	0.958467	0.479234	+	0.877687i	$0.340915\pi$
$84$	−3.46410	−0.377964
$85$	0	0
$86$	−6.92820	−0.747087
$87$	−3.60770	−0.386786
$88$	−5.46410	−0.582475
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	25.8564	2.71049
$92$	8.00000	0.834058
$93$	−0.535898	−0.0555701
$94$	4.73205	0.488074
$95$	0	0
$96$	−0.732051	−0.0747146
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	15.3923	1.55486
$99$	13.4641	1.35319
$100$	0	0
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	4.00000	0.396059
$103$	6.53590	0.644001	0.322001	−	0.946739i	$-0.395645\pi$
$103$	6.53590	0.644001	0.322001	+	0.946739i	$0.395645\pi$
$104$	5.46410	0.535799
$105$	0	0
$106$	6.00000	0.582772
$107$	−3.26795	−0.315925	−0.157962	−	0.987445i	$-0.550492\pi$
$107$	−3.26795	−0.315925	−0.157962	+	0.987445i	$0.550492\pi$
$108$	4.00000	0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0.732051	0.0694832
$112$	4.73205	0.447137
$113$	10.5359	0.991134	0.495567	−	0.868570i	$-0.334960\pi$
$113$	10.5359	0.991134	0.495567	+	0.868570i	$0.334960\pi$
$114$	−4.53590	−0.424826
$115$	0	0
$116$	4.92820	0.457572
$117$	−13.4641	−1.24476
$118$	−10.1962	−0.938632
$119$	−25.8564	−2.37025
$120$	0	0
$121$	18.8564	1.71422
$122$	−4.92820	−0.446179
$123$	1.46410	0.132014
$124$	0.732051	0.0657401
$125$	0	0
$126$	−11.6603	−1.03878
$127$	−3.66025	−0.324795	−0.162398	−	0.986725i	$-0.551923\pi$
$127$	−3.66025	−0.324795	−0.162398	+	0.986725i	$0.551923\pi$
$128$	1.00000	0.0883883
$129$	5.07180	0.446547
$130$	0	0
$131$	−18.5885	−1.62408	−0.812041	−	0.583601i	$-0.801643\pi$
$131$	−18.5885	−1.62408	−0.812041	+	0.583601i	$0.801643\pi$
$132$	4.00000	0.348155
$133$	29.3205	2.54241
$134$	3.66025	0.316198
$135$	0	0
$136$	−5.46410	−0.468543
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−5.85641	−0.498530
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	2.92820	0.245729
$143$	−29.8564	−2.49672
$144$	−2.46410	−0.205342
$145$	0	0
$146$	0.928203	0.0768186
$147$	−11.2679	−0.929365
$148$	−1.00000	−0.0821995
$149$	4.39230	0.359832	0.179916	−	0.983682i	$-0.442417\pi$
$149$	4.39230	0.359832	0.179916	+	0.983682i	$0.442417\pi$
$150$	0	0
$151$	−12.3923	−1.00847	−0.504236	−	0.863566i	$-0.668226\pi$
$151$	−12.3923	−1.00847	−0.504236	+	0.863566i	$0.668226\pi$
$152$	6.19615	0.502574
$153$	13.4641	1.08851
$154$	−25.8564	−2.08357
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−3.07180	−0.245156	−0.122578	−	0.992459i	$-0.539116\pi$
$157$	−3.07180	−0.245156	−0.122578	+	0.992459i	$0.539116\pi$
$158$	8.73205	0.694685
$159$	−4.39230	−0.348332
$160$	0	0
$161$	37.8564	2.98350
$162$	4.46410	0.350733
$163$	11.3205	0.886691	0.443345	−	0.896351i	$-0.353792\pi$
$163$	11.3205	0.886691	0.443345	+	0.896351i	$0.353792\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	8.73205	0.677739
$167$	1.46410	0.113296	0.0566478	−	0.998394i	$-0.481959\pi$
$167$	1.46410	0.113296	0.0566478	+	0.998394i	$0.481959\pi$
$168$	−3.46410	−0.267261
$169$	16.8564	1.29665
$170$	0	0
$171$	−15.2679	−1.16757
$172$	−6.92820	−0.528271
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	−3.60770	−0.273499
$175$	0	0
$176$	−5.46410	−0.411872
$177$	7.46410	0.561036
$178$	−2.00000	−0.149906
$179$	−0.339746	−0.0253938	−0.0126969	−	0.999919i	$-0.504042\pi$
$179$	−0.339746	−0.0253938	−0.0126969	+	0.999919i	$0.504042\pi$
$180$	0	0
$181$	5.46410	0.406143	0.203072	−	0.979164i	$-0.434908\pi$
$181$	5.46410	0.406143	0.203072	+	0.979164i	$0.434908\pi$
$182$	25.8564	1.91660
$183$	3.60770	0.266688
$184$	8.00000	0.589768
$185$	0	0
$186$	−0.535898	−0.0392940
$187$	29.8564	2.18332
$188$	4.73205	0.345120
$189$	18.9282	1.37682
$190$	0	0
$191$	−8.73205	−0.631829	−0.315915	−	0.948788i	$-0.602311\pi$
$191$	−8.73205	−0.631829	−0.315915	+	0.948788i	$0.602311\pi$
$192$	−0.732051	−0.0528312
$193$	−15.8564	−1.14137	−0.570685	−	0.821169i	$-0.693322\pi$
$193$	−15.8564	−1.14137	−0.570685	+	0.821169i	$0.693322\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	15.3923	1.09945
$197$	22.7846	1.62334	0.811668	−	0.584119i	$-0.198560\pi$
$197$	22.7846	1.62334	0.811668	+	0.584119i	$0.198560\pi$
$198$	13.4641	0.956852
$199$	−12.0526	−0.854383	−0.427192	−	0.904161i	$-0.640497\pi$
$199$	−12.0526	−0.854383	−0.427192	+	0.904161i	$0.640497\pi$
$200$	0	0
$201$	−2.67949	−0.188997
$202$	−9.46410	−0.665892
$203$	23.3205	1.63678
$204$	4.00000	0.280056
$205$	0	0
$206$	6.53590	0.455378
$207$	−19.7128	−1.37014
$208$	5.46410	0.378867
$209$	−33.8564	−2.34190
$210$	0	0
$211$	−17.8564	−1.22929	−0.614643	−	0.788806i	$-0.710700\pi$
$211$	−17.8564	−1.22929	−0.614643	+	0.788806i	$0.710700\pi$
$212$	6.00000	0.412082
$213$	−2.14359	−0.146877
$214$	−3.26795	−0.223392
$215$	0	0
$216$	4.00000	0.272166
$217$	3.46410	0.235159
$218$	2.00000	0.135457
$219$	−0.679492	−0.0459158
$220$	0	0
$221$	−29.8564	−2.00836
$222$	0.732051	0.0491320
$223$	−16.0526	−1.07496	−0.537479	−	0.843277i	$-0.680623\pi$
$223$	−16.0526	−1.07496	−0.537479	+	0.843277i	$0.680623\pi$
$224$	4.73205	0.316173
$225$	0	0
$226$	10.5359	0.700838
$227$	−24.3923	−1.61897	−0.809487	−	0.587138i	$-0.800255\pi$
$227$	−24.3923	−1.61897	−0.809487	+	0.587138i	$0.800255\pi$
$228$	−4.53590	−0.300397
$229$	−11.8564	−0.783493	−0.391747	−	0.920073i	$-0.628129\pi$
$229$	−11.8564	−0.783493	−0.391747	+	0.920073i	$0.628129\pi$
$230$	0	0
$231$	18.9282	1.24538
$232$	4.92820	0.323552
$233$	−28.9282	−1.89515	−0.947575	−	0.319534i	$-0.896474\pi$
$233$	−28.9282	−1.89515	−0.947575	+	0.319534i	$0.896474\pi$
$234$	−13.4641	−0.880176
$235$	0	0
$236$	−10.1962	−0.663713
$237$	−6.39230	−0.415225
$238$	−25.8564	−1.67602
$239$	−20.7321	−1.34104	−0.670522	−	0.741889i	$-0.733930\pi$
$239$	−20.7321	−1.34104	−0.670522	+	0.741889i	$0.733930\pi$
$240$	0	0
$241$	4.92820	0.317453	0.158727	−	0.987323i	$-0.449261\pi$
$241$	4.92820	0.317453	0.158727	+	0.987323i	$0.449261\pi$
$242$	18.8564	1.21214
$243$	−15.2679	−0.979439
$244$	−4.92820	−0.315496
$245$	0	0
$246$	1.46410	0.0933477
$247$	33.8564	2.15423
$248$	0.732051	0.0464853
$249$	−6.39230	−0.405096
$250$	0	0
$251$	19.2679	1.21618	0.608091	−	0.793867i	$-0.291936\pi$
$251$	19.2679	1.21618	0.608091	+	0.793867i	$0.291936\pi$
$252$	−11.6603	−0.734527
$253$	−43.7128	−2.74820
$254$	−3.66025	−0.229665
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.5359	1.15624	0.578119	−	0.815953i	$-0.303787\pi$
$257$	18.5359	1.15624	0.578119	+	0.815953i	$0.303787\pi$
$258$	5.07180	0.315756
$259$	−4.73205	−0.294035
$260$	0	0
$261$	−12.1436	−0.751670
$262$	−18.5885	−1.14840
$263$	8.05256	0.496542	0.248271	−	0.968691i	$-0.420138\pi$
$263$	8.05256	0.496542	0.248271	+	0.968691i	$0.420138\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	29.3205	1.79776
$267$	1.46410	0.0896016
$268$	3.66025	0.223586
$269$	20.3923	1.24334	0.621670	−	0.783279i	$-0.286454\pi$
$269$	20.3923	1.24334	0.621670	+	0.783279i	$0.286454\pi$
$270$	0	0
$271$	−24.7846	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$271$	−24.7846	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$272$	−5.46410	−0.331310
$273$	−18.9282	−1.14559
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−5.85641	−0.352514
$277$	−22.2487	−1.33680	−0.668398	−	0.743804i	$-0.733020\pi$
$277$	−22.2487	−1.33680	−0.668398	+	0.743804i	$0.733020\pi$
$278$	6.92820	0.415526
$279$	−1.80385	−0.107994
$280$	0	0
$281$	−8.92820	−0.532612	−0.266306	−	0.963889i	$-0.585803\pi$
$281$	−8.92820	−0.532612	−0.266306	+	0.963889i	$0.585803\pi$
$282$	−3.46410	−0.206284
$283$	−16.3923	−0.974421	−0.487211	−	0.873284i	$-0.661986\pi$
$283$	−16.3923	−0.974421	−0.487211	+	0.873284i	$0.661986\pi$
$284$	2.92820	0.173757
$285$	0	0
$286$	−29.8564	−1.76545
$287$	−9.46410	−0.558648
$288$	−2.46410	−0.145199
$289$	12.8564	0.756259
$290$	0	0
$291$	−1.46410	−0.0858272
$292$	0.928203	0.0543190
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−11.2679	−0.657160
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	−21.8564	−1.26824
$298$	4.39230	0.254439
$299$	43.7128	2.52798
$300$	0	0
$301$	−32.7846	−1.88967
$302$	−12.3923	−0.713097
$303$	6.92820	0.398015
$304$	6.19615	0.355374
$305$	0	0
$306$	13.4641	0.769691
$307$	−18.5885	−1.06090	−0.530450	−	0.847716i	$-0.677977\pi$
$307$	−18.5885	−1.06090	−0.530450	+	0.847716i	$0.677977\pi$
$308$	−25.8564	−1.47331
$309$	−4.78461	−0.272187
$310$	0	0
$311$	2.87564	0.163063	0.0815314	−	0.996671i	$-0.474019\pi$
$311$	2.87564	0.163063	0.0815314	+	0.996671i	$0.474019\pi$
$312$	−4.00000	−0.226455
$313$	23.8564	1.34844	0.674222	−	0.738529i	$-0.264479\pi$
$313$	23.8564	1.34844	0.674222	+	0.738529i	$0.264479\pi$
$314$	−3.07180	−0.173352
$315$	0	0
$316$	8.73205	0.491216
$317$	4.14359	0.232727	0.116364	−	0.993207i	$-0.462876\pi$
$317$	4.14359	0.232727	0.116364	+	0.993207i	$0.462876\pi$
$318$	−4.39230	−0.246308
$319$	−26.9282	−1.50769
$320$	0	0
$321$	2.39230	0.133525
$322$	37.8564	2.10966
$323$	−33.8564	−1.88382
$324$	4.46410	0.248006
$325$	0	0
$326$	11.3205	0.626985
$327$	−1.46410	−0.0809650
$328$	−2.00000	−0.110432
$329$	22.3923	1.23453
$330$	0	0
$331$	−33.1244	−1.82068	−0.910340	−	0.413862i	$-0.864180\pi$
$331$	−33.1244	−1.82068	−0.910340	+	0.413862i	$0.864180\pi$
$332$	8.73205	0.479234
$333$	2.46410	0.135032
$334$	1.46410	0.0801121
$335$	0	0
$336$	−3.46410	−0.188982
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	16.8564	0.916868
$339$	−7.71281	−0.418902
$340$	0	0
$341$	−4.00000	−0.216612
$342$	−15.2679	−0.825596
$343$	39.7128	2.14429
$344$	−6.92820	−0.373544
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−30.9282	−1.66031	−0.830156	−	0.557530i	$-0.811749\pi$
$347$	−30.9282	−1.66031	−0.830156	+	0.557530i	$0.811749\pi$
$348$	−3.60770	−0.193393
$349$	−15.3205	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$349$	−15.3205	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$350$	0	0
$351$	21.8564	1.16661
$352$	−5.46410	−0.291238
$353$	11.8564	0.631053	0.315526	−	0.948917i	$-0.397819\pi$
$353$	11.8564	0.631053	0.315526	+	0.948917i	$0.397819\pi$
$354$	7.46410	0.396713
$355$	0	0
$356$	−2.00000	−0.106000
$357$	18.9282	1.00179
$358$	−0.339746	−0.0179561
$359$	−12.3923	−0.654041	−0.327020	−	0.945017i	$-0.606045\pi$
$359$	−12.3923	−0.654041	−0.327020	+	0.945017i	$0.606045\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	5.46410	0.287187
$363$	−13.8038	−0.724514
$364$	25.8564	1.35524
$365$	0	0
$366$	3.60770	0.188577
$367$	4.33975	0.226533	0.113266	−	0.993565i	$-0.463869\pi$
$367$	4.33975	0.226533	0.113266	+	0.993565i	$0.463869\pi$
$368$	8.00000	0.417029
$369$	4.92820	0.256552
$370$	0	0
$371$	28.3923	1.47406
$372$	−0.535898	−0.0277850
$373$	10.7846	0.558406	0.279203	−	0.960232i	$-0.409930\pi$
$373$	10.7846	0.558406	0.279203	+	0.960232i	$0.409930\pi$
$374$	29.8564	1.54384
$375$	0	0
$376$	4.73205	0.244037
$377$	26.9282	1.38687
$378$	18.9282	0.973562
$379$	32.3923	1.66388	0.831940	−	0.554865i	$-0.187230\pi$
$379$	32.3923	1.66388	0.831940	+	0.554865i	$0.187230\pi$
$380$	0	0
$381$	2.67949	0.137275
$382$	−8.73205	−0.446771
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	−0.732051	−0.0373573
$385$	0	0
$386$	−15.8564	−0.807070
$387$	17.0718	0.867808
$388$	2.00000	0.101535
$389$	−11.8564	−0.601144	−0.300572	−	0.953759i	$-0.597178\pi$
$389$	−11.8564	−0.601144	−0.300572	+	0.953759i	$0.597178\pi$
$390$	0	0
$391$	−43.7128	−2.21065
$392$	15.3923	0.777429
$393$	13.6077	0.686417
$394$	22.7846	1.14787
$395$	0	0
$396$	13.4641	0.676597
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−12.0526	−0.604140
$399$	−21.4641	−1.07455
$400$	0	0
$401$	−32.9282	−1.64436	−0.822178	−	0.569230i	$-0.807241\pi$
$401$	−32.9282	−1.64436	−0.822178	+	0.569230i	$0.807241\pi$
$402$	−2.67949	−0.133641
$403$	4.00000	0.199254
$404$	−9.46410	−0.470857
$405$	0	0
$406$	23.3205	1.15738
$407$	5.46410	0.270845
$408$	4.00000	0.198030
$409$	3.07180	0.151891	0.0759453	−	0.997112i	$-0.475803\pi$
$409$	3.07180	0.151891	0.0759453	+	0.997112i	$0.475803\pi$
$410$	0	0
$411$	1.46410	0.0722188
$412$	6.53590	0.322001
$413$	−48.2487	−2.37416
$414$	−19.7128	−0.968832
$415$	0	0
$416$	5.46410	0.267900
$417$	−5.07180	−0.248367
$418$	−33.8564	−1.65597
$419$	14.2487	0.696095	0.348048	−	0.937477i	$-0.386845\pi$
$419$	14.2487	0.696095	0.348048	+	0.937477i	$0.386845\pi$
$420$	0	0
$421$	−0.143594	−0.00699832	−0.00349916	−	0.999994i	$-0.501114\pi$
$421$	−0.143594	−0.00699832	−0.00349916	+	0.999994i	$0.501114\pi$
$422$	−17.8564	−0.869236
$423$	−11.6603	−0.566941
$424$	6.00000	0.291386
$425$	0	0
$426$	−2.14359	−0.103857
$427$	−23.3205	−1.12856
$428$	−3.26795	−0.157962
$429$	21.8564	1.05524
$430$	0	0
$431$	−2.19615	−0.105785	−0.0528925	−	0.998600i	$-0.516844\pi$
$431$	−2.19615	−0.105785	−0.0528925	+	0.998600i	$0.516844\pi$
$432$	4.00000	0.192450
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	3.46410	0.166282
$435$	0	0
$436$	2.00000	0.0957826
$437$	49.5692	2.37122
$438$	−0.679492	−0.0324674
$439$	13.5167	0.645115	0.322558	−	0.946550i	$-0.395457\pi$
$439$	13.5167	0.645115	0.322558	+	0.946550i	$0.395457\pi$
$440$	0	0
$441$	−37.9282	−1.80610
$442$	−29.8564	−1.42012
$443$	14.8756	0.706763	0.353382	−	0.935479i	$-0.385032\pi$
$443$	14.8756	0.706763	0.353382	+	0.935479i	$0.385032\pi$
$444$	0.732051	0.0347416
$445$	0	0
$446$	−16.0526	−0.760111
$447$	−3.21539	−0.152083
$448$	4.73205	0.223568
$449$	−21.7128	−1.02469	−0.512345	−	0.858779i	$-0.671223\pi$
$449$	−21.7128	−1.02469	−0.512345	+	0.858779i	$0.671223\pi$
$450$	0	0
$451$	10.9282	0.514589
$452$	10.5359	0.495567
$453$	9.07180	0.426230
$454$	−24.3923	−1.14479
$455$	0	0
$456$	−4.53590	−0.212413
$457$	31.8564	1.49018	0.745090	−	0.666964i	$-0.232407\pi$
$457$	31.8564	1.49018	0.745090	+	0.666964i	$0.232407\pi$
$458$	−11.8564	−0.554013
$459$	−21.8564	−1.02017
$460$	0	0
$461$	14.7846	0.688588	0.344294	−	0.938862i	$-0.388118\pi$
$461$	14.7846	0.688588	0.344294	+	0.938862i	$0.388118\pi$
$462$	18.9282	0.880620
$463$	−18.9282	−0.879668	−0.439834	−	0.898079i	$-0.644963\pi$
$463$	−18.9282	−0.879668	−0.439834	+	0.898079i	$0.644963\pi$
$464$	4.92820	0.228786
$465$	0	0
$466$	−28.9282	−1.34007
$467$	25.1769	1.16505	0.582524	−	0.812813i	$-0.302065\pi$
$467$	25.1769	1.16505	0.582524	+	0.812813i	$0.302065\pi$
$468$	−13.4641	−0.622378
$469$	17.3205	0.799787
$470$	0	0
$471$	2.24871	0.103615
$472$	−10.1962	−0.469316
$473$	37.8564	1.74064
$474$	−6.39230	−0.293608
$475$	0	0
$476$	−25.8564	−1.18513
$477$	−14.7846	−0.676941
$478$	−20.7321	−0.948262
$479$	4.05256	0.185166	0.0925831	−	0.995705i	$-0.470488\pi$
$479$	4.05256	0.185166	0.0925831	+	0.995705i	$0.470488\pi$
$480$	0	0
$481$	−5.46410	−0.249142
$482$	4.92820	0.224474
$483$	−27.7128	−1.26098
$484$	18.8564	0.857109
$485$	0	0
$486$	−15.2679	−0.692568
$487$	−4.39230	−0.199034	−0.0995172	−	0.995036i	$-0.531730\pi$
$487$	−4.39230	−0.199034	−0.0995172	+	0.995036i	$0.531730\pi$
$488$	−4.92820	−0.223089
$489$	−8.28719	−0.374760
$490$	0	0
$491$	−14.9282	−0.673700	−0.336850	−	0.941558i	$-0.609362\pi$
$491$	−14.9282	−0.673700	−0.336850	+	0.941558i	$0.609362\pi$
$492$	1.46410	0.0660068
$493$	−26.9282	−1.21279
$494$	33.8564	1.52327
$495$	0	0
$496$	0.732051	0.0328701
$497$	13.8564	0.621545
$498$	−6.39230	−0.286446
$499$	9.41154	0.421319	0.210659	−	0.977560i	$-0.432439\pi$
$499$	9.41154	0.421319	0.210659	+	0.977560i	$0.432439\pi$
$500$	0	0
$501$	−1.07180	−0.0478843
$502$	19.2679	0.859971
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	−11.6603	−0.519389
$505$	0	0
$506$	−43.7128	−1.94327
$507$	−12.3397	−0.548027
$508$	−3.66025	−0.162398
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	4.39230	0.194304
$512$	1.00000	0.0441942
$513$	24.7846	1.09427
$514$	18.5359	0.817583
$515$	0	0
$516$	5.07180	0.223273
$517$	−25.8564	−1.13716
$518$	−4.73205	−0.207914
$519$	7.32051	0.321335
$520$	0	0
$521$	26.5359	1.16256	0.581279	−	0.813704i	$-0.302552\pi$
$521$	26.5359	1.16256	0.581279	+	0.813704i	$0.302552\pi$
$522$	−12.1436	−0.531511
$523$	−36.7846	−1.60848	−0.804239	−	0.594306i	$-0.797427\pi$
$523$	−36.7846	−1.60848	−0.804239	+	0.594306i	$0.797427\pi$
$524$	−18.5885	−0.812041
$525$	0	0
$526$	8.05256	0.351108
$527$	−4.00000	−0.174243
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	0	0
$531$	25.1244	1.09030
$532$	29.3205	1.27121
$533$	−10.9282	−0.473353
$534$	1.46410	0.0633579
$535$	0	0
$536$	3.66025	0.158099
$537$	0.248711	0.0107327
$538$	20.3923	0.879175
$539$	−84.1051	−3.62266
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−24.7846	−1.06459
$543$	−4.00000	−0.171656
$544$	−5.46410	−0.234271
$545$	0	0
$546$	−18.9282	−0.810052
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−2.00000	−0.0854358
$549$	12.1436	0.518276
$550$	0	0
$551$	30.5359	1.30087
$552$	−5.85641	−0.249265
$553$	41.3205	1.75713
$554$	−22.2487	−0.945257
$555$	0	0
$556$	6.92820	0.293821
$557$	−44.1051	−1.86879	−0.934397	−	0.356234i	$-0.884061\pi$
$557$	−44.1051	−1.86879	−0.934397	+	0.356234i	$0.884061\pi$
$558$	−1.80385	−0.0763630
$559$	−37.8564	−1.60116
$560$	0	0
$561$	−21.8564	−0.922778
$562$	−8.92820	−0.376614
$563$	−30.9282	−1.30347	−0.651734	−	0.758447i	$-0.725958\pi$
$563$	−30.9282	−1.30347	−0.651734	+	0.758447i	$0.725958\pi$
$564$	−3.46410	−0.145865
$565$	0	0
$566$	−16.3923	−0.689020
$567$	21.1244	0.887140
$568$	2.92820	0.122865
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−29.8564	−1.24836
$573$	6.39230	0.267042
$574$	−9.46410	−0.395024
$575$	0	0
$576$	−2.46410	−0.102671
$577$	24.3923	1.01546	0.507732	−	0.861515i	$-0.330484\pi$
$577$	24.3923	1.01546	0.507732	+	0.861515i	$0.330484\pi$
$578$	12.8564	0.534756
$579$	11.6077	0.482399
$580$	0	0
$581$	41.3205	1.71426
$582$	−1.46410	−0.0606890
$583$	−32.7846	−1.35780
$584$	0.928203	0.0384093
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−44.7846	−1.84846	−0.924229	−	0.381838i	$-0.875291\pi$
$587$	−44.7846	−1.84846	−0.924229	+	0.381838i	$0.875291\pi$
$588$	−11.2679	−0.464682
$589$	4.53590	0.186898
$590$	0	0
$591$	−16.6795	−0.686103
$592$	−1.00000	−0.0410997
$593$	34.7846	1.42843	0.714216	−	0.699925i	$-0.246783\pi$
$593$	34.7846	1.42843	0.714216	+	0.699925i	$0.246783\pi$
$594$	−21.8564	−0.896779
$595$	0	0
$596$	4.39230	0.179916
$597$	8.82309	0.361105
$598$	43.7128	1.78755
$599$	9.46410	0.386693	0.193346	−	0.981131i	$-0.438066\pi$
$599$	9.46410	0.386693	0.193346	+	0.981131i	$0.438066\pi$
$600$	0	0
$601$	27.6077	1.12614	0.563071	−	0.826409i	$-0.309620\pi$
$601$	27.6077	1.12614	0.563071	+	0.826409i	$0.309620\pi$
$602$	−32.7846	−1.33620
$603$	−9.01924	−0.367292
$604$	−12.3923	−0.504236
$605$	0	0
$606$	6.92820	0.281439
$607$	−0.784610	−0.0318463	−0.0159232	−	0.999873i	$-0.505069\pi$
$607$	−0.784610	−0.0318463	−0.0159232	+	0.999873i	$0.505069\pi$
$608$	6.19615	0.251287
$609$	−17.0718	−0.691784
$610$	0	0
$611$	25.8564	1.04604
$612$	13.4641	0.544254
$613$	−16.9282	−0.683724	−0.341862	−	0.939750i	$-0.611057\pi$
$613$	−16.9282	−0.683724	−0.341862	+	0.939750i	$0.611057\pi$
$614$	−18.5885	−0.750169
$615$	0	0
$616$	−25.8564	−1.04178
$617$	0.928203	0.0373681	0.0186840	−	0.999825i	$-0.494052\pi$
$617$	0.928203	0.0373681	0.0186840	+	0.999825i	$0.494052\pi$
$618$	−4.78461	−0.192465
$619$	−17.8564	−0.717710	−0.358855	−	0.933393i	$-0.616833\pi$
$619$	−17.8564	−0.717710	−0.358855	+	0.933393i	$0.616833\pi$
$620$	0	0
$621$	32.0000	1.28412
$622$	2.87564	0.115303
$623$	−9.46410	−0.379171
$624$	−4.00000	−0.160128
$625$	0	0
$626$	23.8564	0.953494
$627$	24.7846	0.989802
$628$	−3.07180	−0.122578
$629$	5.46410	0.217868
$630$	0	0
$631$	30.9808	1.23332	0.616662	−	0.787228i	$-0.288484\pi$
$631$	30.9808	1.23332	0.616662	+	0.787228i	$0.288484\pi$
$632$	8.73205	0.347342
$633$	13.0718	0.519557
$634$	4.14359	0.164563
$635$	0	0
$636$	−4.39230	−0.174166
$637$	84.1051	3.33237
$638$	−26.9282	−1.06610
$639$	−7.21539	−0.285436
$640$	0	0
$641$	−6.53590	−0.258152	−0.129076	−	0.991635i	$-0.541201\pi$
$641$	−6.53590	−0.258152	−0.129076	+	0.991635i	$0.541201\pi$
$642$	2.39230	0.0944167
$643$	−34.5359	−1.36196	−0.680981	−	0.732301i	$-0.738447\pi$
$643$	−34.5359	−1.36196	−0.680981	+	0.732301i	$0.738447\pi$
$644$	37.8564	1.49175
$645$	0	0
$646$	−33.8564	−1.33206
$647$	35.7128	1.40402	0.702008	−	0.712169i	$-0.252287\pi$
$647$	35.7128	1.40402	0.702008	+	0.712169i	$0.252287\pi$
$648$	4.46410	0.175366
$649$	55.7128	2.18692
$650$	0	0
$651$	−2.53590	−0.0993897
$652$	11.3205	0.443345
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−1.46410	−0.0572509
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−2.28719	−0.0892317
$658$	22.3923	0.872943
$659$	−6.92820	−0.269884	−0.134942	−	0.990853i	$-0.543085\pi$
$659$	−6.92820	−0.269884	−0.134942	+	0.990853i	$0.543085\pi$
$660$	0	0
$661$	−31.0718	−1.20855	−0.604276	−	0.796775i	$-0.706538\pi$
$661$	−31.0718	−1.20855	−0.604276	+	0.796775i	$0.706538\pi$
$662$	−33.1244	−1.28741
$663$	21.8564	0.848832
$664$	8.73205	0.338869
$665$	0	0
$666$	2.46410	0.0954820
$667$	39.4256	1.52657
$668$	1.46410	0.0566478
$669$	11.7513	0.454331
$670$	0	0
$671$	26.9282	1.03955
$672$	−3.46410	−0.133631
$673$	32.9282	1.26929	0.634644	−	0.772804i	$-0.281147\pi$
$673$	32.9282	1.26929	0.634644	+	0.772804i	$0.281147\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	16.8564	0.648323
$677$	4.14359	0.159251	0.0796256	−	0.996825i	$-0.474628\pi$
$677$	4.14359	0.159251	0.0796256	+	0.996825i	$0.474628\pi$
$678$	−7.71281	−0.296209
$679$	9.46410	0.363199
$680$	0	0
$681$	17.8564	0.684259
$682$	−4.00000	−0.153168
$683$	4.78461	0.183078	0.0915390	−	0.995801i	$-0.470821\pi$
$683$	4.78461	0.183078	0.0915390	+	0.995801i	$0.470821\pi$
$684$	−15.2679	−0.583785
$685$	0	0
$686$	39.7128	1.51624
$687$	8.67949	0.331143
$688$	−6.92820	−0.264135
$689$	32.7846	1.24899
$690$	0	0
$691$	−0.392305	−0.0149240	−0.00746199	−	0.999972i	$-0.502375\pi$
$691$	−0.392305	−0.0149240	−0.00746199	+	0.999972i	$0.502375\pi$
$692$	−10.0000	−0.380143
$693$	63.7128	2.42025
$694$	−30.9282	−1.17402
$695$	0	0
$696$	−3.60770	−0.136749
$697$	10.9282	0.413935
$698$	−15.3205	−0.579890
$699$	21.1769	0.800984
$700$	0	0
$701$	−11.8564	−0.447810	−0.223905	−	0.974611i	$-0.571881\pi$
$701$	−11.8564	−0.447810	−0.223905	+	0.974611i	$0.571881\pi$
$702$	21.8564	0.824917
$703$	−6.19615	−0.233692
$704$	−5.46410	−0.205936
$705$	0	0
$706$	11.8564	0.446222
$707$	−44.7846	−1.68430
$708$	7.46410	0.280518
$709$	43.8564	1.64706	0.823531	−	0.567271i	$-0.192001\pi$
$709$	43.8564	1.64706	0.823531	+	0.567271i	$0.192001\pi$
$710$	0	0
$711$	−21.5167	−0.806938
$712$	−2.00000	−0.0749532
$713$	5.85641	0.219324
$714$	18.9282	0.708370
$715$	0	0
$716$	−0.339746	−0.0126969
$717$	15.1769	0.566792
$718$	−12.3923	−0.462477
$719$	−12.3923	−0.462155	−0.231077	−	0.972935i	$-0.574225\pi$
$719$	−12.3923	−0.462155	−0.231077	+	0.972935i	$0.574225\pi$
$720$	0	0
$721$	30.9282	1.15183
$722$	19.3923	0.721707
$723$	−3.60770	−0.134172
$724$	5.46410	0.203072
$725$	0	0
$726$	−13.8038	−0.512309
$727$	−8.78461	−0.325803	−0.162902	−	0.986642i	$-0.552085\pi$
$727$	−8.78461	−0.325803	−0.162902	+	0.986642i	$0.552085\pi$
$728$	25.8564	0.958302
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	37.8564	1.40017
$732$	3.60770	0.133344
$733$	−27.8564	−1.02890	−0.514450	−	0.857520i	$-0.672004\pi$
$733$	−27.8564	−1.02890	−0.514450	+	0.857520i	$0.672004\pi$
$734$	4.33975	0.160183
$735$	0	0
$736$	8.00000	0.294884
$737$	−20.0000	−0.736709
$738$	4.92820	0.181410
$739$	6.92820	0.254858	0.127429	−	0.991848i	$-0.459327\pi$
$739$	6.92820	0.254858	0.127429	+	0.991848i	$0.459327\pi$
$740$	0	0
$741$	−24.7846	−0.910485
$742$	28.3923	1.04231
$743$	−17.1244	−0.628232	−0.314116	−	0.949385i	$-0.601708\pi$
$743$	−17.1244	−0.628232	−0.314116	+	0.949385i	$0.601708\pi$
$744$	−0.535898	−0.0196470
$745$	0	0
$746$	10.7846	0.394853
$747$	−21.5167	−0.787253
$748$	29.8564	1.09166
$749$	−15.4641	−0.565046
$750$	0	0
$751$	−20.3923	−0.744126	−0.372063	−	0.928208i	$-0.621349\pi$
$751$	−20.3923	−0.744126	−0.372063	+	0.928208i	$0.621349\pi$
$752$	4.73205	0.172560
$753$	−14.1051	−0.514019
$754$	26.9282	0.980667
$755$	0	0
$756$	18.9282	0.688412
$757$	−1.71281	−0.0622532	−0.0311266	−	0.999515i	$-0.509910\pi$
$757$	−1.71281	−0.0622532	−0.0311266	+	0.999515i	$0.509910\pi$
$758$	32.3923	1.17654
$759$	32.0000	1.16153
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	2.67949	0.0970678
$763$	9.46410	0.342623
$764$	−8.73205	−0.315915
$765$	0	0
$766$	8.00000	0.289052
$767$	−55.7128	−2.01167
$768$	−0.732051	−0.0264156
$769$	7.07180	0.255016	0.127508	−	0.991838i	$-0.459302\pi$
$769$	7.07180	0.255016	0.127508	+	0.991838i	$0.459302\pi$
$770$	0	0
$771$	−13.5692	−0.488683
$772$	−15.8564	−0.570685
$773$	−2.78461	−0.100155	−0.0500777	−	0.998745i	$-0.515947\pi$
$773$	−2.78461	−0.100155	−0.0500777	+	0.998745i	$0.515947\pi$
$774$	17.0718	0.613633
$775$	0	0
$776$	2.00000	0.0717958
$777$	3.46410	0.124274
$778$	−11.8564	−0.425073
$779$	−12.3923	−0.444000
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−43.7128	−1.56317
$783$	19.7128	0.704478
$784$	15.3923	0.549725
$785$	0	0
$786$	13.6077	0.485370
$787$	−22.1962	−0.791207	−0.395604	−	0.918421i	$-0.629465\pi$
$787$	−22.1962	−0.791207	−0.395604	+	0.918421i	$0.629465\pi$
$788$	22.7846	0.811668
$789$	−5.89488	−0.209863
$790$	0	0
$791$	49.8564	1.77269
$792$	13.4641	0.478426
$793$	−26.9282	−0.956249
$794$	22.0000	0.780751
$795$	0	0
$796$	−12.0526	−0.427192
$797$	−10.5359	−0.373201	−0.186600	−	0.982436i	$-0.559747\pi$
$797$	−10.5359	−0.373201	−0.186600	+	0.982436i	$0.559747\pi$
$798$	−21.4641	−0.759821
$799$	−25.8564	−0.914734
$800$	0	0
$801$	4.92820	0.174129
$802$	−32.9282	−1.16274
$803$	−5.07180	−0.178980
$804$	−2.67949	−0.0944984
$805$	0	0
$806$	4.00000	0.140894
$807$	−14.9282	−0.525498
$808$	−9.46410	−0.332946
$809$	43.5692	1.53181	0.765906	−	0.642952i	$-0.222291\pi$
$809$	43.5692	1.53181	0.765906	+	0.642952i	$0.222291\pi$
$810$	0	0
$811$	−41.8564	−1.46978	−0.734889	−	0.678188i	$-0.762766\pi$
$811$	−41.8564	−1.46978	−0.734889	+	0.678188i	$0.762766\pi$
$812$	23.3205	0.818389
$813$	18.1436	0.636324
$814$	5.46410	0.191517
$815$	0	0
$816$	4.00000	0.140028
$817$	−42.9282	−1.50187
$818$	3.07180	0.107403
$819$	−63.7128	−2.22631
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	1.46410	0.0510664
$823$	−48.4449	−1.68868	−0.844341	−	0.535806i	$-0.820008\pi$
$823$	−48.4449	−1.68868	−0.844341	+	0.535806i	$0.820008\pi$
$824$	6.53590	0.227689
$825$	0	0
$826$	−48.2487	−1.67879
$827$	16.3923	0.570016	0.285008	−	0.958525i	$-0.408004\pi$
$827$	16.3923	0.570016	0.285008	+	0.958525i	$0.408004\pi$
$828$	−19.7128	−0.685068
$829$	−51.5692	−1.79107	−0.895537	−	0.444988i	$-0.853208\pi$
$829$	−51.5692	−1.79107	−0.895537	+	0.444988i	$0.853208\pi$
$830$	0	0
$831$	16.2872	0.564996
$832$	5.46410	0.189434
$833$	−84.1051	−2.91407
$834$	−5.07180	−0.175622
$835$	0	0
$836$	−33.8564	−1.17095
$837$	2.92820	0.101214
$838$	14.2487	0.492214
$839$	−8.78461	−0.303278	−0.151639	−	0.988436i	$-0.548455\pi$
$839$	−8.78461	−0.303278	−0.151639	+	0.988436i	$0.548455\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	−0.143594	−0.00494856
$843$	6.53590	0.225108
$844$	−17.8564	−0.614643
$845$	0	0
$846$	−11.6603	−0.400888
$847$	89.2295	3.06596
$848$	6.00000	0.206041
$849$	12.0000	0.411839
$850$	0	0
$851$	−8.00000	−0.274236
$852$	−2.14359	−0.0734383
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	−23.3205	−0.798011
$855$	0	0
$856$	−3.26795	−0.111696
$857$	31.8564	1.08819	0.544097	−	0.839022i	$-0.316872\pi$
$857$	31.8564	1.08819	0.544097	+	0.839022i	$0.316872\pi$
$858$	21.8564	0.746165
$859$	44.4449	1.51644	0.758220	−	0.651999i	$-0.226069\pi$
$859$	44.4449	1.51644	0.758220	+	0.651999i	$0.226069\pi$
$860$	0	0
$861$	6.92820	0.236113
$862$	−2.19615	−0.0748012
$863$	20.4449	0.695951	0.347976	−	0.937504i	$-0.386869\pi$
$863$	20.4449	0.695951	0.347976	+	0.937504i	$0.386869\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	34.0000	1.15537
$867$	−9.41154	−0.319633
$868$	3.46410	0.117579
$869$	−47.7128	−1.61855
$870$	0	0
$871$	20.0000	0.677674
$872$	2.00000	0.0677285
$873$	−4.92820	−0.166794
$874$	49.5692	1.67670
$875$	0	0
$876$	−0.679492	−0.0229579
$877$	9.21539	0.311182	0.155591	−	0.987822i	$-0.450272\pi$
$877$	9.21539	0.311182	0.155591	+	0.987822i	$0.450272\pi$
$878$	13.5167	0.456165
$879$	4.39230	0.148149
$880$	0	0
$881$	12.6795	0.427183	0.213591	−	0.976923i	$-0.431484\pi$
$881$	12.6795	0.427183	0.213591	+	0.976923i	$0.431484\pi$
$882$	−37.9282	−1.27711
$883$	−14.2487	−0.479507	−0.239754	−	0.970834i	$-0.577067\pi$
$883$	−14.2487	−0.479507	−0.239754	+	0.970834i	$0.577067\pi$
$884$	−29.8564	−1.00418
$885$	0	0
$886$	14.8756	0.499757
$887$	−9.80385	−0.329181	−0.164590	−	0.986362i	$-0.552630\pi$
$887$	−9.80385	−0.329181	−0.164590	+	0.986362i	$0.552630\pi$
$888$	0.732051	0.0245660
$889$	−17.3205	−0.580911
$890$	0	0
$891$	−24.3923	−0.817173
$892$	−16.0526	−0.537479
$893$	29.3205	0.981173
$894$	−3.21539	−0.107539
$895$	0	0
$896$	4.73205	0.158087
$897$	−32.0000	−1.06845
$898$	−21.7128	−0.724566
$899$	3.60770	0.120323
$900$	0	0
$901$	−32.7846	−1.09221
$902$	10.9282	0.363869
$903$	24.0000	0.798670
$904$	10.5359	0.350419
$905$	0	0
$906$	9.07180	0.301390
$907$	−54.2487	−1.80130	−0.900649	−	0.434546i	$-0.856909\pi$
$907$	−54.2487	−1.80130	−0.900649	+	0.434546i	$0.856909\pi$
$908$	−24.3923	−0.809487
$909$	23.3205	0.773492
$910$	0	0
$911$	37.9090	1.25598	0.627990	−	0.778221i	$-0.283878\pi$
$911$	37.9090	1.25598	0.627990	+	0.778221i	$0.283878\pi$
$912$	−4.53590	−0.150199
$913$	−47.7128	−1.57906
$914$	31.8564	1.05372
$915$	0	0
$916$	−11.8564	−0.391747
$917$	−87.9615	−2.90475
$918$	−21.8564	−0.721369
$919$	−16.7321	−0.551939	−0.275970	−	0.961166i	$-0.588999\pi$
$919$	−16.7321	−0.551939	−0.275970	+	0.961166i	$0.588999\pi$
$920$	0	0
$921$	13.6077	0.448389
$922$	14.7846	0.486905
$923$	16.0000	0.526646
$924$	18.9282	0.622692
$925$	0	0
$926$	−18.9282	−0.622019
$927$	−16.1051	−0.528961
$928$	4.92820	0.161776
$929$	11.8564	0.388996	0.194498	−	0.980903i	$-0.437692\pi$
$929$	11.8564	0.388996	0.194498	+	0.980903i	$0.437692\pi$
$930$	0	0
$931$	95.3731	3.12573
$932$	−28.9282	−0.947575
$933$	−2.10512	−0.0689185
$934$	25.1769	0.823814
$935$	0	0
$936$	−13.4641	−0.440088
$937$	23.8564	0.779355	0.389677	−	0.920951i	$-0.372587\pi$
$937$	23.8564	0.779355	0.389677	+	0.920951i	$0.372587\pi$
$938$	17.3205	0.565535
$939$	−17.4641	−0.569919
$940$	0	0
$941$	7.60770	0.248004	0.124002	−	0.992282i	$-0.460427\pi$
$941$	7.60770	0.248004	0.124002	+	0.992282i	$0.460427\pi$
$942$	2.24871	0.0732670
$943$	−16.0000	−0.521032
$944$	−10.1962	−0.331856
$945$	0	0
$946$	37.8564	1.23082
$947$	−17.1769	−0.558175	−0.279087	−	0.960266i	$-0.590032\pi$
$947$	−17.1769	−0.558175	−0.279087	+	0.960266i	$0.590032\pi$
$948$	−6.39230	−0.207612
$949$	5.07180	0.164637
$950$	0	0
$951$	−3.03332	−0.0983622
$952$	−25.8564	−0.838011
$953$	−15.8564	−0.513639	−0.256820	−	0.966459i	$-0.582675\pi$
$953$	−15.8564	−0.513639	−0.256820	+	0.966459i	$0.582675\pi$
$954$	−14.7846	−0.478669
$955$	0	0
$956$	−20.7321	−0.670522
$957$	19.7128	0.637225
$958$	4.05256	0.130932
$959$	−9.46410	−0.305612
$960$	0	0
$961$	−30.4641	−0.982713
$962$	−5.46410	−0.176170
$963$	8.05256	0.259490
$964$	4.92820	0.158727
$965$	0	0
$966$	−27.7128	−0.891645
$967$	44.3923	1.42756	0.713780	−	0.700370i	$-0.246982\pi$
$967$	44.3923	1.42756	0.713780	+	0.700370i	$0.246982\pi$
$968$	18.8564	0.606068
$969$	24.7846	0.796196
$970$	0	0
$971$	1.07180	0.0343956	0.0171978	−	0.999852i	$-0.494526\pi$
$971$	1.07180	0.0343956	0.0171978	+	0.999852i	$0.494526\pi$
$972$	−15.2679	−0.489720
$973$	32.7846	1.05103
$974$	−4.39230	−0.140739
$975$	0	0
$976$	−4.92820	−0.157748
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	−8.28719	−0.264995
$979$	10.9282	0.349267
$980$	0	0
$981$	−4.92820	−0.157345
$982$	−14.9282	−0.476378
$983$	−20.4449	−0.652090	−0.326045	−	0.945354i	$-0.605716\pi$
$983$	−20.4449	−0.652090	−0.326045	+	0.945354i	$0.605716\pi$
$984$	1.46410	0.0466739
$985$	0	0
$986$	−26.9282	−0.857569
$987$	−16.3923	−0.521773
$988$	33.8564	1.07712
$989$	−55.4256	−1.76243
$990$	0	0
$991$	44.0526	1.39938	0.699688	−	0.714449i	$-0.253322\pi$
$991$	44.0526	1.39938	0.699688	+	0.714449i	$0.253322\pi$
$992$	0.732051	0.0232426
$993$	24.2487	0.769510
$994$	13.8564	0.439499
$995$	0	0
$996$	−6.39230	−0.202548
$997$	31.8564	1.00890	0.504451	−	0.863440i	$-0.331695\pi$
$997$	31.8564	1.00890	0.504451	+	0.863440i	$0.331695\pi$
$998$	9.41154	0.297917
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.x.1.1		2
5.2	odd	4		1850.2.b.l.149.4		4
5.3	odd	4		1850.2.b.l.149.1		4
5.4	even	2		370.2.a.e.1.2	✓	2
15.14	odd	2		3330.2.a.bd.1.1		2
20.19	odd	2		2960.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.e.1.2	✓	2	5.4	even	2
1850.2.a.x.1.1		2	1.1	even	1	trivial
1850.2.b.l.149.1		4	5.3	odd	4
1850.2.b.l.149.4		4	5.2	odd	4
2960.2.a.q.1.1		2	20.19	odd	2
3330.2.a.bd.1.1		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} -5.46410 q^{11} -0.732051 q^{12} +5.46410 q^{13} +4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} -2.46410 q^{18} +6.19615 q^{19} -3.46410 q^{21} -5.46410 q^{22} +8.00000 q^{23} -0.732051 q^{24} +5.46410 q^{26} +4.00000 q^{27} +4.73205 q^{28} +4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} +4.00000 q^{33} -5.46410 q^{34} -2.46410 q^{36} -1.00000 q^{37} +6.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} -3.46410 q^{42} -6.92820 q^{43} -5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} -0.732051 q^{48} +15.3923 q^{49} +4.00000 q^{51} +5.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +4.73205 q^{56} -4.53590 q^{57} +4.92820 q^{58} -10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} -11.6603 q^{63} +1.00000 q^{64} +4.00000 q^{66} +3.66025 q^{67} -5.46410 q^{68} -5.85641 q^{69} +2.92820 q^{71} -2.46410 q^{72} +0.928203 q^{73} -1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} -4.00000 q^{78} +8.73205 q^{79} +4.46410 q^{81} -2.00000 q^{82} +8.73205 q^{83} -3.46410 q^{84} -6.92820 q^{86} -3.60770 q^{87} -5.46410 q^{88} -2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} -0.535898 q^{93} +4.73205 q^{94} -0.732051 q^{96} +2.00000 q^{97} +15.3923 q^{98} +13.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26}+ \cdots + 20 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 4 * q^13 + 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^22 + 16 * q^23 + 2 * q^24 + 4 * q^26 + 8 * q^27 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^33 - 4 * q^34 + 2 * q^36 - 2 * q^37 + 2 * q^38 - 8 * q^39 - 4 * q^41 - 4 * q^44 + 16 * q^46 + 6 * q^47 + 2 * q^48 + 10 * q^49 + 8 * q^51 + 4 * q^52 + 12 * q^53 + 8 * q^54 + 6 * q^56 - 16 * q^57 - 4 * q^58 - 10 * q^59 + 4 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 8 * q^66 - 10 * q^67 - 4 * q^68 + 16 * q^69 - 8 * q^71 + 2 * q^72 - 12 * q^73 - 2 * q^74 + 2 * q^76 - 24 * q^77 - 8 * q^78 + 14 * q^79 + 2 * q^81 - 4 * q^82 + 14 * q^83 - 28 * q^87 - 4 * q^88 - 4 * q^89 + 24 * q^91 + 16 * q^92 - 8 * q^93 + 6 * q^94 + 2 * q^96 + 4 * q^97 + 10 * q^98 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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