LMFDB - Embedded newform 1850.2.a.x.1.2

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			1.2
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)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} +1.26795 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.46410 q^{11} +2.73205 q^{12} -1.46410 q^{13} +1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} +4.46410 q^{18} -4.19615 q^{19} +3.46410 q^{21} +1.46410 q^{22} +8.00000 q^{23} +2.73205 q^{24} -1.46410 q^{26} +4.00000 q^{27} +1.26795 q^{28} -8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.46410 q^{34} +4.46410 q^{36} -1.00000 q^{37} -4.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} +3.46410 q^{42} +6.92820 q^{43} +1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} +2.73205 q^{48} -5.39230 q^{49} +4.00000 q^{51} -1.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +1.26795 q^{56} -11.4641 q^{57} -8.92820 q^{58} +0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +5.66025 q^{63} +1.00000 q^{64} +4.00000 q^{66} -13.6603 q^{67} +1.46410 q^{68} +21.8564 q^{69} -10.9282 q^{71} +4.46410 q^{72} -12.9282 q^{73} -1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} -4.00000 q^{78} +5.26795 q^{79} -2.46410 q^{81} -2.00000 q^{82} +5.26795 q^{83} +3.46410 q^{84} +6.92820 q^{86} -24.3923 q^{87} +1.46410 q^{88} -2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} -7.46410 q^{93} +1.26795 q^{94} +2.73205 q^{96} +2.00000 q^{97} -5.39230 q^{98} +6.53590 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.73205	1.11536
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	1.00000	0.353553
$9$	4.46410	1.48803
$10$	0	0
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	2.73205	0.788675
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	1.26795	0.338874
$15$	0	0
$16$	1.00000	0.250000
$17$	1.46410	0.355097	0.177548	−	0.984112i	$-0.443183\pi$
$17$	1.46410	0.355097	0.177548	+	0.984112i	$0.443183\pi$
$18$	4.46410	1.05220
$19$	−4.19615	−0.962663	−0.481332	−	0.876539i	$-0.659847\pi$
$19$	−4.19615	−0.962663	−0.481332	+	0.876539i	$0.659847\pi$
$20$	0	0
$21$	3.46410	0.755929
$22$	1.46410	0.312148
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	2.73205	0.557678
$25$	0	0
$26$	−1.46410	−0.287134
$27$	4.00000	0.769800
$28$	1.26795	0.239620
$29$	−8.92820	−1.65793	−0.828963	−	0.559304i	$-0.811069\pi$
$29$	−8.92820	−1.65793	−0.828963	+	0.559304i	$0.811069\pi$
$30$	0	0
$31$	−2.73205	−0.490691	−0.245345	−	0.969436i	$-0.578901\pi$
$31$	−2.73205	−0.490691	−0.245345	+	0.969436i	$0.578901\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	1.46410	0.251091
$35$	0	0
$36$	4.46410	0.744017
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−4.19615	−0.680706
$39$	−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	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	1.46410	0.220722
$45$	0	0
$46$	8.00000	1.17954
$47$	1.26795	0.184949	0.0924747	−	0.995715i	$-0.470522\pi$
$47$	1.26795	0.184949	0.0924747	+	0.995715i	$0.470522\pi$
$48$	2.73205	0.394338
$49$	−5.39230	−0.770329
$50$	0	0
$51$	4.00000	0.560112
$52$	−1.46410	−0.203034
$53$	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
$56$	1.26795	0.169437
$57$	−11.4641	−1.51846
$58$	−8.92820	−1.17233
$59$	0.196152	0.0255369	0.0127684	−	0.999918i	$-0.495936\pi$
$59$	0.196152	0.0255369	0.0127684	+	0.999918i	$0.495936\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	−2.73205	−0.346971
$63$	5.66025	0.713125
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−13.6603	−1.66887	−0.834433	−	0.551110i	$-0.814205\pi$
$67$	−13.6603	−1.66887	−0.834433	+	0.551110i	$0.814205\pi$
$68$	1.46410	0.177548
$69$	21.8564	2.63120
$70$	0	0
$71$	−10.9282	−1.29694	−0.648470	−	0.761241i	$-0.724591\pi$
$71$	−10.9282	−1.29694	−0.648470	+	0.761241i	$0.724591\pi$
$72$	4.46410	0.526099
$73$	−12.9282	−1.51313	−0.756566	−	0.653917i	$-0.773124\pi$
$73$	−12.9282	−1.51313	−0.756566	+	0.653917i	$0.773124\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−4.19615	−0.481332
$77$	1.85641	0.211557
$78$	−4.00000	−0.452911
$79$	5.26795	0.592691	0.296345	−	0.955081i	$-0.404232\pi$
$79$	5.26795	0.592691	0.296345	+	0.955081i	$0.404232\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	−2.00000	−0.220863
$83$	5.26795	0.578233	0.289116	−	0.957294i	$-0.406639\pi$
$83$	5.26795	0.578233	0.289116	+	0.957294i	$0.406639\pi$
$84$	3.46410	0.377964
$85$	0	0
$86$	6.92820	0.747087
$87$	−24.3923	−2.61513
$88$	1.46410	0.156074
$89$	−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
$91$	−1.85641	−0.194604
$92$	8.00000	0.834058
$93$	−7.46410	−0.773991
$94$	1.26795	0.130779
$95$	0	0
$96$	2.73205	0.278839
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−5.39230	−0.544705
$99$	6.53590	0.656883
$100$	0	0
$101$	−2.53590	−0.252331	−0.126166	−	0.992009i	$-0.540267\pi$
$101$	−2.53590	−0.252331	−0.126166	+	0.992009i	$0.540267\pi$
$102$	4.00000	0.396059
$103$	13.4641	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$103$	13.4641	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$104$	−1.46410	−0.143567
$105$	0	0
$106$	6.00000	0.582772
$107$	−6.73205	−0.650812	−0.325406	−	0.945574i	$-0.605501\pi$
$107$	−6.73205	−0.650812	−0.325406	+	0.945574i	$0.605501\pi$
$108$	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
$111$	−2.73205	−0.259315
$112$	1.26795	0.119810
$113$	17.4641	1.64288	0.821442	−	0.570292i	$-0.193170\pi$
$113$	17.4641	1.64288	0.821442	+	0.570292i	$0.193170\pi$
$114$	−11.4641	−1.07371
$115$	0	0
$116$	−8.92820	−0.828963
$117$	−6.53590	−0.604244
$118$	0.196152	0.0180573
$119$	1.85641	0.170177
$120$	0	0
$121$	−8.85641	−0.805128
$122$	8.92820	0.808322
$123$	−5.46410	−0.492681
$124$	−2.73205	−0.245345
$125$	0	0
$126$	5.66025	0.504256
$127$	13.6603	1.21215	0.606076	−	0.795407i	$-0.292743\pi$
$127$	13.6603	1.21215	0.606076	+	0.795407i	$0.292743\pi$
$128$	1.00000	0.0883883
$129$	18.9282	1.66654
$130$	0	0
$131$	12.5885	1.09986	0.549929	−	0.835211i	$-0.314655\pi$
$131$	12.5885	1.09986	0.549929	+	0.835211i	$0.314655\pi$
$132$	4.00000	0.348155
$133$	−5.32051	−0.461347
$134$	−13.6603	−1.18007
$135$	0	0
$136$	1.46410	0.125546
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	21.8564	1.86054
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	−10.9282	−0.917074
$143$	−2.14359	−0.179256
$144$	4.46410	0.372008
$145$	0	0
$146$	−12.9282	−1.06995
$147$	−14.7321	−1.21508
$148$	−1.00000	−0.0821995
$149$	−16.3923	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$149$	−16.3923	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$150$	0	0
$151$	8.39230	0.682956	0.341478	−	0.939890i	$-0.389073\pi$
$151$	8.39230	0.682956	0.341478	+	0.939890i	$0.389073\pi$
$152$	−4.19615	−0.340353
$153$	6.53590	0.528396
$154$	1.85641	0.149593
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−16.9282	−1.35102	−0.675509	−	0.737352i	$-0.736076\pi$
$157$	−16.9282	−1.35102	−0.675509	+	0.737352i	$0.736076\pi$
$158$	5.26795	0.419096
$159$	16.3923	1.29999
$160$	0	0
$161$	10.1436	0.799427
$162$	−2.46410	−0.193598
$163$	−23.3205	−1.82660	−0.913302	−	0.407284i	$-0.866476\pi$
$163$	−23.3205	−1.82660	−0.913302	+	0.407284i	$0.866476\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	5.26795	0.408872
$167$	−5.46410	−0.422825	−0.211412	−	0.977397i	$-0.567806\pi$
$167$	−5.46410	−0.422825	−0.211412	+	0.977397i	$0.567806\pi$
$168$	3.46410	0.267261
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−18.7321	−1.43248
$172$	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	$0.624125\pi$
$174$	−24.3923	−1.84918
$175$	0	0
$176$	1.46410	0.110361
$177$	0.535898	0.0402806
$178$	−2.00000	−0.149906
$179$	−17.6603	−1.31999	−0.659995	−	0.751270i	$-0.729441\pi$
$179$	−17.6603	−1.31999	−0.659995	+	0.751270i	$0.729441\pi$
$180$	0	0
$181$	−1.46410	−0.108826	−0.0544129	−	0.998519i	$-0.517329\pi$
$181$	−1.46410	−0.108826	−0.0544129	+	0.998519i	$0.517329\pi$
$182$	−1.85641	−0.137606
$183$	24.3923	1.80313
$184$	8.00000	0.589768
$185$	0	0
$186$	−7.46410	−0.547294
$187$	2.14359	0.156755
$188$	1.26795	0.0924747
$189$	5.07180	0.368919
$190$	0	0
$191$	−5.26795	−0.381175	−0.190588	−	0.981670i	$-0.561039\pi$
$191$	−5.26795	−0.381175	−0.190588	+	0.981670i	$0.561039\pi$
$192$	2.73205	0.197169
$193$	11.8564	0.853443	0.426721	−	0.904383i	$-0.359668\pi$
$193$	11.8564	0.853443	0.426721	+	0.904383i	$0.359668\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−5.39230	−0.385165
$197$	−18.7846	−1.33835	−0.669174	−	0.743106i	$-0.733352\pi$
$197$	−18.7846	−1.33835	−0.669174	+	0.743106i	$0.733352\pi$
$198$	6.53590	0.464486
$199$	26.0526	1.84682	0.923408	−	0.383819i	$-0.125391\pi$
$199$	26.0526	1.84682	0.923408	+	0.383819i	$0.125391\pi$
$200$	0	0
$201$	−37.3205	−2.63239
$202$	−2.53590	−0.178425
$203$	−11.3205	−0.794544
$204$	4.00000	0.280056
$205$	0	0
$206$	13.4641	0.938088
$207$	35.7128	2.48221
$208$	−1.46410	−0.101517
$209$	−6.14359	−0.424961
$210$	0	0
$211$	9.85641	0.678543	0.339272	−	0.940688i	$-0.389819\pi$
$211$	9.85641	0.678543	0.339272	+	0.940688i	$0.389819\pi$
$212$	6.00000	0.412082
$213$	−29.8564	−2.04573
$214$	−6.73205	−0.460194
$215$	0	0
$216$	4.00000	0.272166
$217$	−3.46410	−0.235159
$218$	2.00000	0.135457
$219$	−35.3205	−2.38674
$220$	0	0
$221$	−2.14359	−0.144194
$222$	−2.73205	−0.183363
$223$	22.0526	1.47675	0.738374	−	0.674391i	$-0.235594\pi$
$223$	22.0526	1.47675	0.738374	+	0.674391i	$0.235594\pi$
$224$	1.26795	0.0847184
$225$	0	0
$226$	17.4641	1.16169
$227$	−3.60770	−0.239451	−0.119726	−	0.992807i	$-0.538201\pi$
$227$	−3.60770	−0.239451	−0.119726	+	0.992807i	$0.538201\pi$
$228$	−11.4641	−0.759229
$229$	15.8564	1.04782	0.523910	−	0.851773i	$-0.324473\pi$
$229$	15.8564	1.04782	0.523910	+	0.851773i	$0.324473\pi$
$230$	0	0
$231$	5.07180	0.333700
$232$	−8.92820	−0.586165
$233$	−15.0718	−0.987386	−0.493693	−	0.869636i	$-0.664353\pi$
$233$	−15.0718	−0.987386	−0.493693	+	0.869636i	$0.664353\pi$
$234$	−6.53590	−0.427265
$235$	0	0
$236$	0.196152	0.0127684
$237$	14.3923	0.934881
$238$	1.85641	0.120333
$239$	−17.2679	−1.11697	−0.558485	−	0.829514i	$-0.688617\pi$
$239$	−17.2679	−1.11697	−0.558485	+	0.829514i	$0.688617\pi$
$240$	0	0
$241$	−8.92820	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$241$	−8.92820	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$242$	−8.85641	−0.569311
$243$	−18.7321	−1.20166
$244$	8.92820	0.571570
$245$	0	0
$246$	−5.46410	−0.348378
$247$	6.14359	0.390907
$248$	−2.73205	−0.173485
$249$	14.3923	0.912075
$250$	0	0
$251$	22.7321	1.43483	0.717417	−	0.696644i	$-0.245324\pi$
$251$	22.7321	1.43483	0.717417	+	0.696644i	$0.245324\pi$
$252$	5.66025	0.356562
$253$	11.7128	0.736378
$254$	13.6603	0.857121
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.4641	1.58841	0.794204	−	0.607652i	$-0.207888\pi$
$257$	25.4641	1.58841	0.794204	+	0.607652i	$0.207888\pi$
$258$	18.9282	1.17842
$259$	−1.26795	−0.0787865
$260$	0	0
$261$	−39.8564	−2.46705
$262$	12.5885	0.777717
$263$	−30.0526	−1.85312	−0.926560	−	0.376147i	$-0.877249\pi$
$263$	−30.0526	−1.85312	−0.926560	+	0.376147i	$0.877249\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	−5.32051	−0.326221
$267$	−5.46410	−0.334398
$268$	−13.6603	−0.834433
$269$	−0.392305	−0.0239192	−0.0119596	−	0.999928i	$-0.503807\pi$
$269$	−0.392305	−0.0239192	−0.0119596	+	0.999928i	$0.503807\pi$
$270$	0	0
$271$	16.7846	1.01959	0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846	1.01959	0.509796	+	0.860295i	$0.329721\pi$
$272$	1.46410	0.0887742
$273$	−5.07180	−0.306959
$274$	−2.00000	−0.120824
$275$	0	0
$276$	21.8564	1.31560
$277$	26.2487	1.57713	0.788566	−	0.614950i	$-0.210824\pi$
$277$	26.2487	1.57713	0.788566	+	0.614950i	$0.210824\pi$
$278$	−6.92820	−0.415526
$279$	−12.1962	−0.730165
$280$	0	0
$281$	4.92820	0.293992	0.146996	−	0.989137i	$-0.453040\pi$
$281$	4.92820	0.293992	0.146996	+	0.989137i	$0.453040\pi$
$282$	3.46410	0.206284
$283$	4.39230	0.261095	0.130548	−	0.991442i	$-0.458326\pi$
$283$	4.39230	0.261095	0.130548	+	0.991442i	$0.458326\pi$
$284$	−10.9282	−0.648470
$285$	0	0
$286$	−2.14359	−0.126753
$287$	−2.53590	−0.149689
$288$	4.46410	0.263050
$289$	−14.8564	−0.873906
$290$	0	0
$291$	5.46410	0.320311
$292$	−12.9282	−0.756566
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−14.7321	−0.859191
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	5.85641	0.339823
$298$	−16.3923	−0.949581
$299$	−11.7128	−0.677369
$300$	0	0
$301$	8.78461	0.506336
$302$	8.39230	0.482923
$303$	−6.92820	−0.398015
$304$	−4.19615	−0.240666
$305$	0	0
$306$	6.53590	0.373632
$307$	12.5885	0.718461	0.359231	−	0.933249i	$-0.383039\pi$
$307$	12.5885	0.718461	0.359231	+	0.933249i	$0.383039\pi$
$308$	1.85641	0.105779
$309$	36.7846	2.09260
$310$	0	0
$311$	27.1244	1.53808	0.769041	−	0.639200i	$-0.220734\pi$
$311$	27.1244	1.53808	0.769041	+	0.639200i	$0.220734\pi$
$312$	−4.00000	−0.226455
$313$	−3.85641	−0.217977	−0.108988	−	0.994043i	$-0.534761\pi$
$313$	−3.85641	−0.217977	−0.108988	+	0.994043i	$0.534761\pi$
$314$	−16.9282	−0.955314
$315$	0	0
$316$	5.26795	0.296345
$317$	31.8564	1.78923	0.894617	−	0.446834i	$-0.147448\pi$
$317$	31.8564	1.78923	0.894617	+	0.446834i	$0.147448\pi$
$318$	16.3923	0.919235
$319$	−13.0718	−0.731880
$320$	0	0
$321$	−18.3923	−1.02656
$322$	10.1436	0.565280
$323$	−6.14359	−0.341839
$324$	−2.46410	−0.136895
$325$	0	0
$326$	−23.3205	−1.29160
$327$	5.46410	0.302166
$328$	−2.00000	−0.110432
$329$	1.60770	0.0886351
$330$	0	0
$331$	−8.87564	−0.487850	−0.243925	−	0.969794i	$-0.578435\pi$
$331$	−8.87564	−0.487850	−0.243925	+	0.969794i	$0.578435\pi$
$332$	5.26795	0.289116
$333$	−4.46410	−0.244631
$334$	−5.46410	−0.298982
$335$	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	$0.750472\pi$
$338$	−10.8564	−0.590511
$339$	47.7128	2.59140
$340$	0	0
$341$	−4.00000	−0.216612
$342$	−18.7321	−1.01291
$343$	−15.7128	−0.848412
$344$	6.92820	0.373544
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−17.0718	−0.916462	−0.458231	−	0.888833i	$-0.651517\pi$
$347$	−17.0718	−0.916462	−0.458231	+	0.888833i	$0.651517\pi$
$348$	−24.3923	−1.30756
$349$	19.3205	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$349$	19.3205	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$350$	0	0
$351$	−5.85641	−0.312592
$352$	1.46410	0.0780369
$353$	−15.8564	−0.843951	−0.421976	−	0.906607i	$-0.638663\pi$
$353$	−15.8564	−0.843951	−0.421976	+	0.906607i	$0.638663\pi$
$354$	0.535898	0.0284827
$355$	0	0
$356$	−2.00000	−0.106000
$357$	5.07180	0.268428
$358$	−17.6603	−0.933373
$359$	8.39230	0.442929	0.221464	−	0.975168i	$-0.428916\pi$
$359$	8.39230	0.442929	0.221464	+	0.975168i	$0.428916\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	−1.46410	−0.0769515
$363$	−24.1962	−1.26997
$364$	−1.85641	−0.0973021
$365$	0	0
$366$	24.3923	1.27501
$367$	21.6603	1.13066	0.565328	−	0.824866i	$-0.308750\pi$
$367$	21.6603	1.13066	0.565328	+	0.824866i	$0.308750\pi$
$368$	8.00000	0.417029
$369$	−8.92820	−0.464784
$370$	0	0
$371$	7.60770	0.394972
$372$	−7.46410	−0.386996
$373$	−30.7846	−1.59397	−0.796983	−	0.604001i	$-0.793572\pi$
$373$	−30.7846	−1.59397	−0.796983	+	0.604001i	$0.793572\pi$
$374$	2.14359	0.110843
$375$	0	0
$376$	1.26795	0.0653895
$377$	13.0718	0.673232
$378$	5.07180	0.260865
$379$	11.6077	0.596247	0.298124	−	0.954527i	$-0.403639\pi$
$379$	11.6077	0.596247	0.298124	+	0.954527i	$0.403639\pi$
$380$	0	0
$381$	37.3205	1.91199
$382$	−5.26795	−0.269532
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	2.73205	0.139419
$385$	0	0
$386$	11.8564	0.603475
$387$	30.9282	1.57217
$388$	2.00000	0.101535
$389$	15.8564	0.803952	0.401976	−	0.915650i	$-0.368324\pi$
$389$	15.8564	0.803952	0.401976	+	0.915650i	$0.368324\pi$
$390$	0	0
$391$	11.7128	0.592342
$392$	−5.39230	−0.272353
$393$	34.3923	1.73486
$394$	−18.7846	−0.946355
$395$	0	0
$396$	6.53590	0.328441
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	26.0526	1.30590
$399$	−14.5359	−0.727705
$400$	0	0
$401$	−19.0718	−0.952400	−0.476200	−	0.879337i	$-0.657986\pi$
$401$	−19.0718	−0.952400	−0.476200	+	0.879337i	$0.657986\pi$
$402$	−37.3205	−1.86138
$403$	4.00000	0.199254
$404$	−2.53590	−0.126166
$405$	0	0
$406$	−11.3205	−0.561827
$407$	−1.46410	−0.0725728
$408$	4.00000	0.198030
$409$	16.9282	0.837046	0.418523	−	0.908206i	$-0.362548\pi$
$409$	16.9282	0.837046	0.418523	+	0.908206i	$0.362548\pi$
$410$	0	0
$411$	−5.46410	−0.269524
$412$	13.4641	0.663329
$413$	0.248711	0.0122383
$414$	35.7128	1.75519
$415$	0	0
$416$	−1.46410	−0.0717835
$417$	−18.9282	−0.926918
$418$	−6.14359	−0.300493
$419$	−34.2487	−1.67316	−0.836580	−	0.547846i	$-0.815448\pi$
$419$	−34.2487	−1.67316	−0.836580	+	0.547846i	$0.815448\pi$
$420$	0	0
$421$	−27.8564	−1.35764	−0.678819	−	0.734306i	$-0.737508\pi$
$421$	−27.8564	−1.35764	−0.678819	+	0.734306i	$0.737508\pi$
$422$	9.85641	0.479802
$423$	5.66025	0.275211
$424$	6.00000	0.291386
$425$	0	0
$426$	−29.8564	−1.44655
$427$	11.3205	0.547838
$428$	−6.73205	−0.325406
$429$	−5.85641	−0.282750
$430$	0	0
$431$	8.19615	0.394795	0.197397	−	0.980324i	$-0.436751\pi$
$431$	8.19615	0.394795	0.197397	+	0.980324i	$0.436751\pi$
$432$	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
$437$	−33.5692	−1.60583
$438$	−35.3205	−1.68768
$439$	−31.5167	−1.50421	−0.752104	−	0.659044i	$-0.770961\pi$
$439$	−31.5167	−1.50421	−0.752104	+	0.659044i	$0.770961\pi$
$440$	0	0
$441$	−24.0718	−1.14628
$442$	−2.14359	−0.101960
$443$	39.1244	1.85885	0.929427	−	0.369006i	$-0.120302\pi$
$443$	39.1244	1.85885	0.929427	+	0.369006i	$0.120302\pi$
$444$	−2.73205	−0.129657
$445$	0	0
$446$	22.0526	1.04422
$447$	−44.7846	−2.11824
$448$	1.26795	0.0599050
$449$	33.7128	1.59101	0.795503	−	0.605950i	$-0.207207\pi$
$449$	33.7128	1.59101	0.795503	+	0.605950i	$0.207207\pi$
$450$	0	0
$451$	−2.92820	−0.137884
$452$	17.4641	0.821442
$453$	22.9282	1.07726
$454$	−3.60770	−0.169318
$455$	0	0
$456$	−11.4641	−0.536856
$457$	4.14359	0.193829	0.0969146	−	0.995293i	$-0.469103\pi$
$457$	4.14359	0.193829	0.0969146	+	0.995293i	$0.469103\pi$
$458$	15.8564	0.740921
$459$	5.85641	0.273354
$460$	0	0
$461$	−26.7846	−1.24748	−0.623742	−	0.781630i	$-0.714388\pi$
$461$	−26.7846	−1.24748	−0.623742	+	0.781630i	$0.714388\pi$
$462$	5.07180	0.235961
$463$	−5.07180	−0.235706	−0.117853	−	0.993031i	$-0.537601\pi$
$463$	−5.07180	−0.235706	−0.117853	+	0.993031i	$0.537601\pi$
$464$	−8.92820	−0.414481
$465$	0	0
$466$	−15.0718	−0.698188
$467$	−37.1769	−1.72034	−0.860171	−	0.510005i	$-0.829643\pi$
$467$	−37.1769	−1.72034	−0.860171	+	0.510005i	$0.829643\pi$
$468$	−6.53590	−0.302122
$469$	−17.3205	−0.799787
$470$	0	0
$471$	−46.2487	−2.13103
$472$	0.196152	0.00902865
$473$	10.1436	0.466403
$474$	14.3923	0.661060
$475$	0	0
$476$	1.85641	0.0850883
$477$	26.7846	1.22638
$478$	−17.2679	−0.789818
$479$	−34.0526	−1.55590	−0.777951	−	0.628325i	$-0.783741\pi$
$479$	−34.0526	−1.55590	−0.777951	+	0.628325i	$0.783741\pi$
$480$	0	0
$481$	1.46410	0.0667573
$482$	−8.92820	−0.406669
$483$	27.7128	1.26098
$484$	−8.85641	−0.402564
$485$	0	0
$486$	−18.7321	−0.849703
$487$	16.3923	0.742806	0.371403	−	0.928472i	$-0.378877\pi$
$487$	16.3923	0.742806	0.371403	+	0.928472i	$0.378877\pi$
$488$	8.92820	0.404161
$489$	−63.7128	−2.88119
$490$	0	0
$491$	−1.07180	−0.0483695	−0.0241848	−	0.999708i	$-0.507699\pi$
$491$	−1.07180	−0.0483695	−0.0241848	+	0.999708i	$0.507699\pi$
$492$	−5.46410	−0.246341
$493$	−13.0718	−0.588724
$494$	6.14359	0.276413
$495$	0	0
$496$	−2.73205	−0.122673
$497$	−13.8564	−0.621545
$498$	14.3923	0.644935
$499$	40.5885	1.81699	0.908494	−	0.417897i	$-0.137233\pi$
$499$	40.5885	1.81699	0.908494	+	0.417897i	$0.137233\pi$
$500$	0	0
$501$	−14.9282	−0.666943
$502$	22.7321	1.01458
$503$	5.07180	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$503$	5.07180	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$504$	5.66025	0.252128
$505$	0	0
$506$	11.7128	0.520698
$507$	−29.6603	−1.31726
$508$	13.6603	0.606076
$509$	−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
$511$	−16.3923	−0.725153
$512$	1.00000	0.0441942
$513$	−16.7846	−0.741059
$514$	25.4641	1.12317
$515$	0	0
$516$	18.9282	0.833268
$517$	1.85641	0.0816447
$518$	−1.26795	−0.0557105
$519$	−27.3205	−1.19924
$520$	0	0
$521$	33.4641	1.46609	0.733044	−	0.680181i	$-0.238099\pi$
$521$	33.4641	1.46609	0.733044	+	0.680181i	$0.238099\pi$
$522$	−39.8564	−1.74447
$523$	4.78461	0.209216	0.104608	−	0.994514i	$-0.466641\pi$
$523$	4.78461	0.209216	0.104608	+	0.994514i	$0.466641\pi$
$524$	12.5885	0.549929
$525$	0	0
$526$	−30.0526	−1.31035
$527$	−4.00000	−0.174243
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	0	0
$531$	0.875644	0.0379997
$532$	−5.32051	−0.230673
$533$	2.92820	0.126835
$534$	−5.46410	−0.236455
$535$	0	0
$536$	−13.6603	−0.590033
$537$	−48.2487	−2.08209
$538$	−0.392305	−0.0169135
$539$	−7.89488	−0.340057
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	16.7846	0.720961
$543$	−4.00000	−0.171656
$544$	1.46410	0.0627728
$545$	0	0
$546$	−5.07180	−0.217053
$547$	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
$549$	39.8564	1.70103
$550$	0	0
$551$	37.4641	1.59602
$552$	21.8564	0.930270
$553$	6.67949	0.284041
$554$	26.2487	1.11520
$555$	0	0
$556$	−6.92820	−0.293821
$557$	32.1051	1.36034	0.680169	−	0.733056i	$-0.261906\pi$
$557$	32.1051	1.36034	0.680169	+	0.733056i	$0.261906\pi$
$558$	−12.1962	−0.516304
$559$	−10.1436	−0.429028
$560$	0	0
$561$	5.85641	0.247258
$562$	4.92820	0.207884
$563$	−17.0718	−0.719490	−0.359745	−	0.933051i	$-0.617136\pi$
$563$	−17.0718	−0.719490	−0.359745	+	0.933051i	$0.617136\pi$
$564$	3.46410	0.145865
$565$	0	0
$566$	4.39230	0.184622
$567$	−3.12436	−0.131211
$568$	−10.9282	−0.458537
$569$	−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	$0.580790\pi$
$572$	−2.14359	−0.0896281
$573$	−14.3923	−0.601247
$574$	−2.53590	−0.105846
$575$	0	0
$576$	4.46410	0.186004
$577$	3.60770	0.150190	0.0750952	−	0.997176i	$-0.476074\pi$
$577$	3.60770	0.150190	0.0750952	+	0.997176i	$0.476074\pi$
$578$	−14.8564	−0.617945
$579$	32.3923	1.34618
$580$	0	0
$581$	6.67949	0.277112
$582$	5.46410	0.226494
$583$	8.78461	0.363821
$584$	−12.9282	−0.534973
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−3.21539	−0.132713	−0.0663567	−	0.997796i	$-0.521138\pi$
$587$	−3.21539	−0.132713	−0.0663567	+	0.997796i	$0.521138\pi$
$588$	−14.7321	−0.607540
$589$	11.4641	0.472370
$590$	0	0
$591$	−51.3205	−2.11104
$592$	−1.00000	−0.0410997
$593$	−6.78461	−0.278611	−0.139305	−	0.990249i	$-0.544487\pi$
$593$	−6.78461	−0.278611	−0.139305	+	0.990249i	$0.544487\pi$
$594$	5.85641	0.240291
$595$	0	0
$596$	−16.3923	−0.671455
$597$	71.1769	2.91308
$598$	−11.7128	−0.478973
$599$	2.53590	0.103614	0.0518070	−	0.998657i	$-0.483502\pi$
$599$	2.53590	0.103614	0.0518070	+	0.998657i	$0.483502\pi$
$600$	0	0
$601$	48.3923	1.97396	0.986982	−	0.160833i	$-0.0514180\pi$
$601$	48.3923	1.97396	0.986982	+	0.160833i	$0.0514180\pi$
$602$	8.78461	0.358034
$603$	−60.9808	−2.48333
$604$	8.39230	0.341478
$605$	0	0
$606$	−6.92820	−0.281439
$607$	40.7846	1.65540	0.827698	−	0.561174i	$-0.189650\pi$
$607$	40.7846	1.65540	0.827698	+	0.561174i	$0.189650\pi$
$608$	−4.19615	−0.170176
$609$	−30.9282	−1.25327
$610$	0	0
$611$	−1.85641	−0.0751022
$612$	6.53590	0.264198
$613$	−3.07180	−0.124069	−0.0620344	−	0.998074i	$-0.519759\pi$
$613$	−3.07180	−0.124069	−0.0620344	+	0.998074i	$0.519759\pi$
$614$	12.5885	0.508029
$615$	0	0
$616$	1.85641	0.0747967
$617$	−12.9282	−0.520470	−0.260235	−	0.965545i	$-0.583800\pi$
$617$	−12.9282	−0.520470	−0.260235	+	0.965545i	$0.583800\pi$
$618$	36.7846	1.47969
$619$	9.85641	0.396162	0.198081	−	0.980186i	$-0.436529\pi$
$619$	9.85641	0.396162	0.198081	+	0.980186i	$0.436529\pi$
$620$	0	0
$621$	32.0000	1.28412
$622$	27.1244	1.08759
$623$	−2.53590	−0.101599
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−3.85641	−0.154133
$627$	−16.7846	−0.670313
$628$	−16.9282	−0.675509
$629$	−1.46410	−0.0583776
$630$	0	0
$631$	−20.9808	−0.835231	−0.417615	−	0.908624i	$-0.637134\pi$
$631$	−20.9808	−0.835231	−0.417615	+	0.908624i	$0.637134\pi$
$632$	5.26795	0.209548
$633$	26.9282	1.07030
$634$	31.8564	1.26518
$635$	0	0
$636$	16.3923	0.649997
$637$	7.89488	0.312807
$638$	−13.0718	−0.517517
$639$	−48.7846	−1.92989
$640$	0	0
$641$	−13.4641	−0.531800	−0.265900	−	0.964001i	$-0.585669\pi$
$641$	−13.4641	−0.531800	−0.265900	+	0.964001i	$0.585669\pi$
$642$	−18.3923	−0.725886
$643$	−41.4641	−1.63518	−0.817592	−	0.575798i	$-0.804692\pi$
$643$	−41.4641	−1.63518	−0.817592	+	0.575798i	$0.804692\pi$
$644$	10.1436	0.399714
$645$	0	0
$646$	−6.14359	−0.241716
$647$	−19.7128	−0.774991	−0.387495	−	0.921872i	$-0.626660\pi$
$647$	−19.7128	−0.774991	−0.387495	+	0.921872i	$0.626660\pi$
$648$	−2.46410	−0.0967991
$649$	0.287187	0.0112731
$650$	0	0
$651$	−9.46410	−0.370927
$652$	−23.3205	−0.913302
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	5.46410	0.213663
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−57.7128	−2.25159
$658$	1.60770	0.0626745
$659$	6.92820	0.269884	0.134942	−	0.990853i	$-0.456915\pi$
$659$	6.92820	0.269884	0.134942	+	0.990853i	$0.456915\pi$
$660$	0	0
$661$	−44.9282	−1.74750	−0.873752	−	0.486371i	$-0.838320\pi$
$661$	−44.9282	−1.74750	−0.873752	+	0.486371i	$0.838320\pi$
$662$	−8.87564	−0.344962
$663$	−5.85641	−0.227444
$664$	5.26795	0.204436
$665$	0	0
$666$	−4.46410	−0.172980
$667$	−71.4256	−2.76561
$668$	−5.46410	−0.211412
$669$	60.2487	2.32935
$670$	0	0
$671$	13.0718	0.504631
$672$	3.46410	0.133631
$673$	19.0718	0.735164	0.367582	−	0.929991i	$-0.380186\pi$
$673$	19.0718	0.735164	0.367582	+	0.929991i	$0.380186\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−10.8564	−0.417554
$677$	31.8564	1.22434	0.612171	−	0.790726i	$-0.290297\pi$
$677$	31.8564	1.22434	0.612171	+	0.790726i	$0.290297\pi$
$678$	47.7128	1.83240
$679$	2.53590	0.0973188
$680$	0	0
$681$	−9.85641	−0.377698
$682$	−4.00000	−0.153168
$683$	−36.7846	−1.40752	−0.703762	−	0.710436i	$-0.748498\pi$
$683$	−36.7846	−1.40752	−0.703762	+	0.710436i	$0.748498\pi$
$684$	−18.7321	−0.716238
$685$	0	0
$686$	−15.7128	−0.599918
$687$	43.3205	1.65278
$688$	6.92820	0.264135
$689$	−8.78461	−0.334667
$690$	0	0
$691$	20.3923	0.775760	0.387880	−	0.921710i	$-0.373208\pi$
$691$	20.3923	0.775760	0.387880	+	0.921710i	$0.373208\pi$
$692$	−10.0000	−0.380143
$693$	8.28719	0.314804
$694$	−17.0718	−0.648037
$695$	0	0
$696$	−24.3923	−0.924588
$697$	−2.92820	−0.110914
$698$	19.3205	0.731292
$699$	−41.1769	−1.55745
$700$	0	0
$701$	15.8564	0.598888	0.299444	−	0.954114i	$-0.403199\pi$
$701$	15.8564	0.598888	0.299444	+	0.954114i	$0.403199\pi$
$702$	−5.85641	−0.221036
$703$	4.19615	0.158261
$704$	1.46410	0.0551804
$705$	0	0
$706$	−15.8564	−0.596764
$707$	−3.21539	−0.120927
$708$	0.535898	0.0201403
$709$	16.1436	0.606285	0.303143	−	0.952945i	$-0.401964\pi$
$709$	16.1436	0.606285	0.303143	+	0.952945i	$0.401964\pi$
$710$	0	0
$711$	23.5167	0.881944
$712$	−2.00000	−0.0749532
$713$	−21.8564	−0.818529
$714$	5.07180	0.189807
$715$	0	0
$716$	−17.6603	−0.659995
$717$	−47.1769	−1.76185
$718$	8.39230	0.313198
$719$	8.39230	0.312980	0.156490	−	0.987680i	$-0.449982\pi$
$719$	8.39230	0.312980	0.156490	+	0.987680i	$0.449982\pi$
$720$	0	0
$721$	17.0718	0.635787
$722$	−1.39230	−0.0518162
$723$	−24.3923	−0.907160
$724$	−1.46410	−0.0544129
$725$	0	0
$726$	−24.1962	−0.898003
$727$	32.7846	1.21591	0.607957	−	0.793970i	$-0.291989\pi$
$727$	32.7846	1.21591	0.607957	+	0.793970i	$0.291989\pi$
$728$	−1.85641	−0.0688030
$729$	−43.7846	−1.62165
$730$	0	0
$731$	10.1436	0.375174
$732$	24.3923	0.901566
$733$	−0.143594	−0.00530375	−0.00265187	−	0.999996i	$-0.500844\pi$
$733$	−0.143594	−0.00530375	−0.00265187	+	0.999996i	$0.500844\pi$
$734$	21.6603	0.799495
$735$	0	0
$736$	8.00000	0.294884
$737$	−20.0000	−0.736709
$738$	−8.92820	−0.328652
$739$	−6.92820	−0.254858	−0.127429	−	0.991848i	$-0.540673\pi$
$739$	−6.92820	−0.254858	−0.127429	+	0.991848i	$0.540673\pi$
$740$	0	0
$741$	16.7846	0.616598
$742$	7.60770	0.279287
$743$	7.12436	0.261367	0.130684	−	0.991424i	$-0.458283\pi$
$743$	7.12436	0.261367	0.130684	+	0.991424i	$0.458283\pi$
$744$	−7.46410	−0.273647
$745$	0	0
$746$	−30.7846	−1.12710
$747$	23.5167	0.860430
$748$	2.14359	0.0783775
$749$	−8.53590	−0.311895
$750$	0	0
$751$	0.392305	0.0143154	0.00715770	−	0.999974i	$-0.497722\pi$
$751$	0.392305	0.0143154	0.00715770	+	0.999974i	$0.497722\pi$
$752$	1.26795	0.0462373
$753$	62.1051	2.26324
$754$	13.0718	0.476047
$755$	0	0
$756$	5.07180	0.184459
$757$	53.7128	1.95223	0.976113	−	0.217265i	$-0.0697135\pi$
$757$	53.7128	1.95223	0.976113	+	0.217265i	$0.0697135\pi$
$758$	11.6077	0.421610
$759$	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	$0.441986\pi$
$762$	37.3205	1.35198
$763$	2.53590	0.0918057
$764$	−5.26795	−0.190588
$765$	0	0
$766$	8.00000	0.289052
$767$	−0.287187	−0.0103697
$768$	2.73205	0.0985844
$769$	20.9282	0.754690	0.377345	−	0.926073i	$-0.376837\pi$
$769$	20.9282	0.754690	0.377345	+	0.926073i	$0.376837\pi$
$770$	0	0
$771$	69.5692	2.50547
$772$	11.8564	0.426721
$773$	38.7846	1.39499	0.697493	−	0.716592i	$-0.254299\pi$
$773$	38.7846	1.39499	0.697493	+	0.716592i	$0.254299\pi$
$774$	30.9282	1.11169
$775$	0	0
$776$	2.00000	0.0717958
$777$	−3.46410	−0.124274
$778$	15.8564	0.568480
$779$	8.39230	0.300686
$780$	0	0
$781$	−16.0000	−0.572525
$782$	11.7128	0.418849
$783$	−35.7128	−1.27627
$784$	−5.39230	−0.192582
$785$	0	0
$786$	34.3923	1.22673
$787$	−11.8038	−0.420762	−0.210381	−	0.977620i	$-0.567470\pi$
$787$	−11.8038	−0.420762	−0.210381	+	0.977620i	$0.567470\pi$
$788$	−18.7846	−0.669174
$789$	−82.1051	−2.92302
$790$	0	0
$791$	22.1436	0.787336
$792$	6.53590	0.232243
$793$	−13.0718	−0.464193
$794$	22.0000	0.780751
$795$	0	0
$796$	26.0526	0.923408
$797$	−17.4641	−0.618610	−0.309305	−	0.950963i	$-0.600096\pi$
$797$	−17.4641	−0.618610	−0.309305	+	0.950963i	$0.600096\pi$
$798$	−14.5359	−0.514565
$799$	1.85641	0.0656749
$800$	0	0
$801$	−8.92820	−0.315463
$802$	−19.0718	−0.673449
$803$	−18.9282	−0.667962
$804$	−37.3205	−1.31619
$805$	0	0
$806$	4.00000	0.140894
$807$	−1.07180	−0.0377290
$808$	−2.53590	−0.0892126
$809$	−39.5692	−1.39118	−0.695590	−	0.718439i	$-0.744857\pi$
$809$	−39.5692	−1.39118	−0.695590	+	0.718439i	$0.744857\pi$
$810$	0	0
$811$	−14.1436	−0.496649	−0.248324	−	0.968677i	$-0.579880\pi$
$811$	−14.1436	−0.496649	−0.248324	+	0.968677i	$0.579880\pi$
$812$	−11.3205	−0.397272
$813$	45.8564	1.60825
$814$	−1.46410	−0.0513167
$815$	0	0
$816$	4.00000	0.140028
$817$	−29.0718	−1.01709
$818$	16.9282	0.591881
$819$	−8.28719	−0.289578
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−5.46410	−0.190582
$823$	10.4449	0.364085	0.182043	−	0.983291i	$-0.441729\pi$
$823$	10.4449	0.364085	0.182043	+	0.983291i	$0.441729\pi$
$824$	13.4641	0.469044
$825$	0	0
$826$	0.248711	0.00865377
$827$	−4.39230	−0.152735	−0.0763677	−	0.997080i	$-0.524332\pi$
$827$	−4.39230	−0.152735	−0.0763677	+	0.997080i	$0.524332\pi$
$828$	35.7128	1.24111
$829$	31.5692	1.09644	0.548222	−	0.836333i	$-0.315305\pi$
$829$	31.5692	1.09644	0.548222	+	0.836333i	$0.315305\pi$
$830$	0	0
$831$	71.7128	2.48769
$832$	−1.46410	−0.0507586
$833$	−7.89488	−0.273541
$834$	−18.9282	−0.655430
$835$	0	0
$836$	−6.14359	−0.212481
$837$	−10.9282	−0.377734
$838$	−34.2487	−1.18310
$839$	32.7846	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$839$	32.7846	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$840$	0	0
$841$	50.7128	1.74872
$842$	−27.8564	−0.959995
$843$	13.4641	0.463728
$844$	9.85641	0.339272
$845$	0	0
$846$	5.66025	0.194604
$847$	−11.2295	−0.385849
$848$	6.00000	0.206041
$849$	12.0000	0.411839
$850$	0	0
$851$	−8.00000	−0.274236
$852$	−29.8564	−1.02286
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	11.3205	0.387380
$855$	0	0
$856$	−6.73205	−0.230097
$857$	4.14359	0.141542	0.0707712	−	0.997493i	$-0.477454\pi$
$857$	4.14359	0.141542	0.0707712	+	0.997493i	$0.477454\pi$
$858$	−5.85641	−0.199934
$859$	−14.4449	−0.492852	−0.246426	−	0.969162i	$-0.579256\pi$
$859$	−14.4449	−0.492852	−0.246426	+	0.969162i	$0.579256\pi$
$860$	0	0
$861$	−6.92820	−0.236113
$862$	8.19615	0.279162
$863$	−38.4449	−1.30868	−0.654339	−	0.756201i	$-0.727053\pi$
$863$	−38.4449	−1.30868	−0.654339	+	0.756201i	$0.727053\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	34.0000	1.15537
$867$	−40.5885	−1.37846
$868$	−3.46410	−0.117579
$869$	7.71281	0.261639
$870$	0	0
$871$	20.0000	0.677674
$872$	2.00000	0.0677285
$873$	8.92820	0.302174
$874$	−33.5692	−1.13550
$875$	0	0
$876$	−35.3205	−1.19337
$877$	50.7846	1.71487	0.857437	−	0.514589i	$-0.172055\pi$
$877$	50.7846	1.71487	0.857437	+	0.514589i	$0.172055\pi$
$878$	−31.5167	−1.06364
$879$	−16.3923	−0.552899
$880$	0	0
$881$	47.3205	1.59427	0.797134	−	0.603802i	$-0.206348\pi$
$881$	47.3205	1.59427	0.797134	+	0.603802i	$0.206348\pi$
$882$	−24.0718	−0.810540
$883$	34.2487	1.15256	0.576280	−	0.817252i	$-0.304504\pi$
$883$	34.2487	1.15256	0.576280	+	0.817252i	$0.304504\pi$
$884$	−2.14359	−0.0720969
$885$	0	0
$886$	39.1244	1.31441
$887$	−20.1962	−0.678120	−0.339060	−	0.940765i	$-0.610109\pi$
$887$	−20.1962	−0.678120	−0.339060	+	0.940765i	$0.610109\pi$
$888$	−2.73205	−0.0916816
$889$	17.3205	0.580911
$890$	0	0
$891$	−3.60770	−0.120862
$892$	22.0526	0.738374
$893$	−5.32051	−0.178044
$894$	−44.7846	−1.49782
$895$	0	0
$896$	1.26795	0.0423592
$897$	−32.0000	−1.06845
$898$	33.7128	1.12501
$899$	24.3923	0.813529
$900$	0	0
$901$	8.78461	0.292658
$902$	−2.92820	−0.0974985
$903$	24.0000	0.798670
$904$	17.4641	0.580847
$905$	0	0
$906$	22.9282	0.761739
$907$	−5.75129	−0.190968	−0.0954842	−	0.995431i	$-0.530440\pi$
$907$	−5.75129	−0.190968	−0.0954842	+	0.995431i	$0.530440\pi$
$908$	−3.60770	−0.119726
$909$	−11.3205	−0.375478
$910$	0	0
$911$	−27.9090	−0.924665	−0.462333	−	0.886707i	$-0.652987\pi$
$911$	−27.9090	−0.924665	−0.462333	+	0.886707i	$0.652987\pi$
$912$	−11.4641	−0.379614
$913$	7.71281	0.255257
$914$	4.14359	0.137058
$915$	0	0
$916$	15.8564	0.523910
$917$	15.9615	0.527096
$918$	5.85641	0.193290
$919$	−13.2679	−0.437669	−0.218835	−	0.975762i	$-0.570226\pi$
$919$	−13.2679	−0.437669	−0.218835	+	0.975762i	$0.570226\pi$
$920$	0	0
$921$	34.3923	1.13326
$922$	−26.7846	−0.882104
$923$	16.0000	0.526646
$924$	5.07180	0.166850
$925$	0	0
$926$	−5.07180	−0.166670
$927$	60.1051	1.97411
$928$	−8.92820	−0.293083
$929$	−15.8564	−0.520232	−0.260116	−	0.965577i	$-0.583761\pi$
$929$	−15.8564	−0.520232	−0.260116	+	0.965577i	$0.583761\pi$
$930$	0	0
$931$	22.6269	0.741568
$932$	−15.0718	−0.493693
$933$	74.1051	2.42609
$934$	−37.1769	−1.21647
$935$	0	0
$936$	−6.53590	−0.213633
$937$	−3.85641	−0.125983	−0.0629917	−	0.998014i	$-0.520064\pi$
$937$	−3.85641	−0.125983	−0.0629917	+	0.998014i	$0.520064\pi$
$938$	−17.3205	−0.565535
$939$	−10.5359	−0.343826
$940$	0	0
$941$	28.3923	0.925563	0.462781	−	0.886472i	$-0.346852\pi$
$941$	28.3923	0.925563	0.462781	+	0.886472i	$0.346852\pi$
$942$	−46.2487	−1.50686
$943$	−16.0000	−0.521032
$944$	0.196152	0.00638422
$945$	0	0
$946$	10.1436	0.329797
$947$	45.1769	1.46805	0.734026	−	0.679121i	$-0.237639\pi$
$947$	45.1769	1.46805	0.734026	+	0.679121i	$0.237639\pi$
$948$	14.3923	0.467440
$949$	18.9282	0.614435
$950$	0	0
$951$	87.0333	2.82225
$952$	1.85641	0.0601665
$953$	11.8564	0.384067	0.192033	−	0.981388i	$-0.438492\pi$
$953$	11.8564	0.384067	0.192033	+	0.981388i	$0.438492\pi$
$954$	26.7846	0.867184
$955$	0	0
$956$	−17.2679	−0.558485
$957$	−35.7128	−1.15443
$958$	−34.0526	−1.10019
$959$	−2.53590	−0.0818884
$960$	0	0
$961$	−23.5359	−0.759223
$962$	1.46410	0.0472045
$963$	−30.0526	−0.968430
$964$	−8.92820	−0.287558
$965$	0	0
$966$	27.7128	0.891645
$967$	23.6077	0.759172	0.379586	−	0.925156i	$-0.376066\pi$
$967$	23.6077	0.759172	0.379586	+	0.925156i	$0.376066\pi$
$968$	−8.85641	−0.284656
$969$	−16.7846	−0.539199
$970$	0	0
$971$	14.9282	0.479069	0.239534	−	0.970888i	$-0.423005\pi$
$971$	14.9282	0.479069	0.239534	+	0.970888i	$0.423005\pi$
$972$	−18.7321	−0.600831
$973$	−8.78461	−0.281622
$974$	16.3923	0.525243
$975$	0	0
$976$	8.92820	0.285785
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	−63.7128	−2.03731
$979$	−2.92820	−0.0935858
$980$	0	0
$981$	8.92820	0.285056
$982$	−1.07180	−0.0342024
$983$	38.4449	1.22620	0.613100	−	0.790005i	$-0.289922\pi$
$983$	38.4449	1.22620	0.613100	+	0.790005i	$0.289922\pi$
$984$	−5.46410	−0.174189
$985$	0	0
$986$	−13.0718	−0.416291
$987$	4.39230	0.139809
$988$	6.14359	0.195454
$989$	55.4256	1.76243
$990$	0	0
$991$	5.94744	0.188927	0.0944633	−	0.995528i	$-0.469886\pi$
$991$	5.94744	0.188927	0.0944633	+	0.995528i	$0.469886\pi$
$992$	−2.73205	−0.0867427
$993$	−24.2487	−0.769510
$994$	−13.8564	−0.439499
$995$	0	0
$996$	14.3923	0.456038
$997$	4.14359	0.131229	0.0656145	−	0.997845i	$-0.479099\pi$
$997$	4.14359	0.131229	0.0656145	+	0.997845i	$0.479099\pi$
$998$	40.5885	1.28481
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.x.1.2		2
5.2	odd	4		1850.2.b.l.149.3		4
5.3	odd	4		1850.2.b.l.149.2		4
5.4	even	2		370.2.a.e.1.1	✓	2
15.14	odd	2		3330.2.a.bd.1.2		2
20.19	odd	2		2960.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.e.1.1	✓	2	5.4	even	2
1850.2.a.x.1.2		2	1.1	even	1	trivial
1850.2.b.l.149.2		4	5.3	odd	4
1850.2.b.l.149.3		4	5.2	odd	4
2960.2.a.q.1.2		2	20.19	odd	2
3330.2.a.bd.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +2.73205 q^{6} +1.26795 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.46410 q^{11} +2.73205 q^{12} -1.46410 q^{13} +1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} +4.46410 q^{18} -4.19615 q^{19} +3.46410 q^{21} +1.46410 q^{22} +8.00000 q^{23} +2.73205 q^{24} -1.46410 q^{26} +4.00000 q^{27} +1.26795 q^{28} -8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.46410 q^{34} +4.46410 q^{36} -1.00000 q^{37} -4.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} +3.46410 q^{42} +6.92820 q^{43} +1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} +2.73205 q^{48} -5.39230 q^{49} +4.00000 q^{51} -1.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +1.26795 q^{56} -11.4641 q^{57} -8.92820 q^{58} +0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +5.66025 q^{63} +1.00000 q^{64} +4.00000 q^{66} -13.6603 q^{67} +1.46410 q^{68} +21.8564 q^{69} -10.9282 q^{71} +4.46410 q^{72} -12.9282 q^{73} -1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} -4.00000 q^{78} +5.26795 q^{79} -2.46410 q^{81} -2.00000 q^{82} +5.26795 q^{83} +3.46410 q^{84} +6.92820 q^{86} -24.3923 q^{87} +1.46410 q^{88} -2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} -7.46410 q^{93} +1.26795 q^{94} +2.73205 q^{96} +2.00000 q^{97} -5.39230 q^{98} +6.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26}+ \cdots + 20 q^{99}+O(q^{100}) \) 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

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