LMFDB - Embedded newform 1850.2.a.t.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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +3.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} -1.38197 q^{11} +1.61803 q^{12} +2.85410 q^{13} -3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +0.381966 q^{18} +4.47214 q^{19} +5.23607 q^{21} +1.38197 q^{22} -2.85410 q^{23} -1.61803 q^{24} -2.85410 q^{26} -5.47214 q^{27} +3.23607 q^{28} -9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -2.23607 q^{33} -4.47214 q^{34} -0.381966 q^{36} +1.00000 q^{37} -4.47214 q^{38} +4.61803 q^{39} +9.61803 q^{41} -5.23607 q^{42} +5.23607 q^{43} -1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +1.61803 q^{48} +3.47214 q^{49} +7.23607 q^{51} +2.85410 q^{52} -0.472136 q^{53} +5.47214 q^{54} -3.23607 q^{56} +7.23607 q^{57} +9.32624 q^{58} -4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} -1.23607 q^{63} +1.00000 q^{64} +2.23607 q^{66} -1.09017 q^{67} +4.47214 q^{68} -4.61803 q^{69} +2.94427 q^{71} +0.381966 q^{72} -7.09017 q^{73} -1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -4.61803 q^{78} -8.56231 q^{79} -7.70820 q^{81} -9.61803 q^{82} +14.4721 q^{83} +5.23607 q^{84} -5.23607 q^{86} -15.0902 q^{87} +1.38197 q^{88} -1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} +11.9443 q^{93} -1.23607 q^{94} -1.61803 q^{96} +0.472136 q^{97} -3.47214 q^{98} +0.527864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 - 5 * q^11 + q^12 - q^13 - 2 * q^14 + 2 * q^16 + 3 * q^18 + 6 * q^21 + 5 * q^22 + q^23 - q^24 + q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 + 17 * q^31 - 2 * q^32 - 3 * q^36 + 2 * q^37 + 7 * q^39 + 17 * q^41 - 6 * q^42 + 6 * q^43 - 5 * q^44 - q^46 - 2 * q^47 + q^48 - 2 * q^49 + 10 * q^51 - q^52 + 8 * q^53 + 2 * q^54 - 2 * q^56 + 10 * q^57 + 3 * q^58 - 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^63 + 2 * q^64 + 9 * q^67 - 7 * q^69 - 12 * q^71 + 3 * q^72 - 3 * q^73 - 2 * q^74 - 7 * q^78 + 3 * q^79 - 2 * q^81 - 17 * q^82 + 20 * q^83 + 6 * q^84 - 6 * q^86 - 19 * q^87 + 5 * q^88 - 12 * q^89 + 14 * q^91 + q^92 + 6 * q^93 + 2 * q^94 - q^96 - 8 * q^97 + 2 * q^98 + 10 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	−1.00000	−0.353553
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−1.38197	−0.416678	−0.208339	−	0.978057i	$-0.566806\pi$
$11$	−1.38197	−0.416678	−0.208339	+	0.978057i	$0.566806\pi$
$12$	1.61803	0.467086
$13$	2.85410	0.791585	0.395793	−	0.918340i	$-0.370470\pi$
$13$	2.85410	0.791585	0.395793	+	0.918340i	$0.370470\pi$
$14$	−3.23607	−0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0.381966	0.0900303
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	5.23607	1.14260
$22$	1.38197	0.294636
$23$	−2.85410	−0.595121	−0.297561	−	0.954703i	$-0.596173\pi$
$23$	−2.85410	−0.595121	−0.297561	+	0.954703i	$0.596173\pi$
$24$	−1.61803	−0.330280
$25$	0	0
$26$	−2.85410	−0.559735
$27$	−5.47214	−1.05311
$28$	3.23607	0.611559
$29$	−9.32624	−1.73184	−0.865919	−	0.500183i	$-0.833266\pi$
$29$	−9.32624	−1.73184	−0.865919	+	0.500183i	$0.833266\pi$
$30$	0	0
$31$	7.38197	1.32584	0.662920	−	0.748690i	$-0.269317\pi$
$31$	7.38197	1.32584	0.662920	+	0.748690i	$0.269317\pi$
$32$	−1.00000	−0.176777
$33$	−2.23607	−0.389249
$34$	−4.47214	−0.766965
$35$	0	0
$36$	−0.381966	−0.0636610
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−4.47214	−0.725476
$39$	4.61803	0.739477
$40$	0	0
$41$	9.61803	1.50208	0.751042	−	0.660254i	$-0.229551\pi$
$41$	9.61803	1.50208	0.751042	+	0.660254i	$0.229551\pi$
$42$	−5.23607	−0.807943
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	−1.38197	−0.208339
$45$	0	0
$46$	2.85410	0.420814
$47$	1.23607	0.180299	0.0901495	−	0.995928i	$-0.471266\pi$
$47$	1.23607	0.180299	0.0901495	+	0.995928i	$0.471266\pi$
$48$	1.61803	0.233543
$49$	3.47214	0.496019
$50$	0	0
$51$	7.23607	1.01325
$52$	2.85410	0.395793
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	−3.23607	−0.432438
$57$	7.23607	0.958441
$58$	9.32624	1.22460
$59$	−4.76393	−0.620211	−0.310106	−	0.950702i	$-0.600364\pi$
$59$	−4.76393	−0.620211	−0.310106	+	0.950702i	$0.600364\pi$
$60$	0	0
$61$	10.6180	1.35950	0.679750	−	0.733444i	$-0.262088\pi$
$61$	10.6180	1.35950	0.679750	+	0.733444i	$0.262088\pi$
$62$	−7.38197	−0.937511
$63$	−1.23607	−0.155730
$64$	1.00000	0.125000
$65$	0	0
$66$	2.23607	0.275241
$67$	−1.09017	−0.133185	−0.0665927	−	0.997780i	$-0.521213\pi$
$67$	−1.09017	−0.133185	−0.0665927	+	0.997780i	$0.521213\pi$
$68$	4.47214	0.542326
$69$	−4.61803	−0.555946
$70$	0	0
$71$	2.94427	0.349421	0.174710	−	0.984620i	$-0.444101\pi$
$71$	2.94427	0.349421	0.174710	+	0.984620i	$0.444101\pi$
$72$	0.381966	0.0450151
$73$	−7.09017	−0.829842	−0.414921	−	0.909858i	$-0.636191\pi$
$73$	−7.09017	−0.829842	−0.414921	+	0.909858i	$0.636191\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	4.47214	0.512989
$77$	−4.47214	−0.509647
$78$	−4.61803	−0.522889
$79$	−8.56231	−0.963335	−0.481667	−	0.876354i	$-0.659969\pi$
$79$	−8.56231	−0.963335	−0.481667	+	0.876354i	$0.659969\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	−9.61803	−1.06213
$83$	14.4721	1.58852	0.794262	−	0.607576i	$-0.207858\pi$
$83$	14.4721	1.58852	0.794262	+	0.607576i	$0.207858\pi$
$84$	5.23607	0.571302
$85$	0	0
$86$	−5.23607	−0.564620
$87$	−15.0902	−1.61784
$88$	1.38197	0.147318
$89$	−1.52786	−0.161953	−0.0809766	−	0.996716i	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	0	0
$91$	9.23607	0.968203
$92$	−2.85410	−0.297561
$93$	11.9443	1.23856
$94$	−1.23607	−0.127491
$95$	0	0
$96$	−1.61803	−0.165140
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	−3.47214	−0.350739
$99$	0.527864	0.0530523
$100$	0	0
$101$	3.52786	0.351036	0.175518	−	0.984476i	$-0.443840\pi$
$101$	3.52786	0.351036	0.175518	+	0.984476i	$0.443840\pi$
$102$	−7.23607	−0.716477
$103$	−17.2705	−1.70171	−0.850857	−	0.525397i	$-0.823917\pi$
$103$	−17.2705	−1.70171	−0.850857	+	0.525397i	$0.823917\pi$
$104$	−2.85410	−0.279868
$105$	0	0
$106$	0.472136	0.0458579
$107$	7.32624	0.708254	0.354127	−	0.935197i	$-0.384778\pi$
$107$	7.32624	0.708254	0.354127	+	0.935197i	$0.384778\pi$
$108$	−5.47214	−0.526557
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	1.61803	0.153577
$112$	3.23607	0.305780
$113$	6.94427	0.653262	0.326631	−	0.945152i	$-0.394087\pi$
$113$	6.94427	0.653262	0.326631	+	0.945152i	$0.394087\pi$
$114$	−7.23607	−0.677720
$115$	0	0
$116$	−9.32624	−0.865919
$117$	−1.09017	−0.100786
$118$	4.76393	0.438555
$119$	14.4721	1.32666
$120$	0	0
$121$	−9.09017	−0.826379
$122$	−10.6180	−0.961312
$123$	15.5623	1.40321
$124$	7.38197	0.662920
$125$	0	0
$126$	1.23607	0.110118
$127$	8.47214	0.751780	0.375890	−	0.926664i	$-0.377337\pi$
$127$	8.47214	0.751780	0.375890	+	0.926664i	$0.377337\pi$
$128$	−1.00000	−0.0883883
$129$	8.47214	0.745930
$130$	0	0
$131$	−22.6525	−1.97916	−0.989578	−	0.143998i	$-0.954004\pi$
$131$	−22.6525	−1.97916	−0.989578	+	0.143998i	$0.954004\pi$
$132$	−2.23607	−0.194625
$133$	14.4721	1.25489
$134$	1.09017	0.0941763
$135$	0	0
$136$	−4.47214	−0.383482
$137$	−3.67376	−0.313871	−0.156935	−	0.987609i	$-0.550161\pi$
$137$	−3.67376	−0.313871	−0.156935	+	0.987609i	$0.550161\pi$
$138$	4.61803	0.393113
$139$	4.85410	0.411720	0.205860	−	0.978581i	$-0.434001\pi$
$139$	4.85410	0.411720	0.205860	+	0.978581i	$0.434001\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	−2.94427	−0.247078
$143$	−3.94427	−0.329837
$144$	−0.381966	−0.0318305
$145$	0	0
$146$	7.09017	0.586787
$147$	5.61803	0.463368
$148$	1.00000	0.0821995
$149$	16.1803	1.32555	0.662773	−	0.748821i	$-0.269380\pi$
$149$	16.1803	1.32555	0.662773	+	0.748821i	$0.269380\pi$
$150$	0	0
$151$	4.29180	0.349261	0.174631	−	0.984634i	$-0.444127\pi$
$151$	4.29180	0.349261	0.174631	+	0.984634i	$0.444127\pi$
$152$	−4.47214	−0.362738
$153$	−1.70820	−0.138100
$154$	4.47214	0.360375
$155$	0	0
$156$	4.61803	0.369739
$157$	16.4721	1.31462	0.657310	−	0.753620i	$-0.271694\pi$
$157$	16.4721	1.31462	0.657310	+	0.753620i	$0.271694\pi$
$158$	8.56231	0.681180
$159$	−0.763932	−0.0605838
$160$	0	0
$161$	−9.23607	−0.727904
$162$	7.70820	0.605614
$163$	3.52786	0.276324	0.138162	−	0.990410i	$-0.455881\pi$
$163$	3.52786	0.276324	0.138162	+	0.990410i	$0.455881\pi$
$164$	9.61803	0.751042
$165$	0	0
$166$	−14.4721	−1.12326
$167$	−13.8541	−1.07206	−0.536031	−	0.844198i	$-0.680077\pi$
$167$	−13.8541	−1.07206	−0.536031	+	0.844198i	$0.680077\pi$
$168$	−5.23607	−0.403971
$169$	−4.85410	−0.373392
$170$	0	0
$171$	−1.70820	−0.130630
$172$	5.23607	0.399246
$173$	0.472136	0.0358958	0.0179479	−	0.999839i	$-0.494287\pi$
$173$	0.472136	0.0358958	0.0179479	+	0.999839i	$0.494287\pi$
$174$	15.0902	1.14398
$175$	0	0
$176$	−1.38197	−0.104170
$177$	−7.70820	−0.579384
$178$	1.52786	0.114518
$179$	−12.6525	−0.945690	−0.472845	−	0.881146i	$-0.656773\pi$
$179$	−12.6525	−0.945690	−0.472845	+	0.881146i	$0.656773\pi$
$180$	0	0
$181$	14.4721	1.07571	0.537853	−	0.843039i	$-0.319236\pi$
$181$	14.4721	1.07571	0.537853	+	0.843039i	$0.319236\pi$
$182$	−9.23607	−0.684623
$183$	17.1803	1.27001
$184$	2.85410	0.210407
$185$	0	0
$186$	−11.9443	−0.875797
$187$	−6.18034	−0.451951
$188$	1.23607	0.0901495
$189$	−17.7082	−1.28808
$190$	0	0
$191$	−7.09017	−0.513027	−0.256513	−	0.966541i	$-0.582574\pi$
$191$	−7.09017	−0.513027	−0.256513	+	0.966541i	$0.582574\pi$
$192$	1.61803	0.116772
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−0.472136	−0.0338974
$195$	0	0
$196$	3.47214	0.248010
$197$	7.52786	0.536338	0.268169	−	0.963372i	$-0.413581\pi$
$197$	7.52786	0.536338	0.268169	+	0.963372i	$0.413581\pi$
$198$	−0.527864	−0.0375137
$199$	3.05573	0.216615	0.108307	−	0.994117i	$-0.465457\pi$
$199$	3.05573	0.216615	0.108307	+	0.994117i	$0.465457\pi$
$200$	0	0
$201$	−1.76393	−0.124418
$202$	−3.52786	−0.248220
$203$	−30.1803	−2.11824
$204$	7.23607	0.506626
$205$	0	0
$206$	17.2705	1.20329
$207$	1.09017	0.0757720
$208$	2.85410	0.197896
$209$	−6.18034	−0.427503
$210$	0	0
$211$	11.2705	0.775894	0.387947	−	0.921682i	$-0.373184\pi$
$211$	11.2705	0.775894	0.387947	+	0.921682i	$0.373184\pi$
$212$	−0.472136	−0.0324264
$213$	4.76393	0.326419
$214$	−7.32624	−0.500811
$215$	0	0
$216$	5.47214	0.372332
$217$	23.8885	1.62166
$218$	−2.94427	−0.199411
$219$	−11.4721	−0.775215
$220$	0	0
$221$	12.7639	0.858595
$222$	−1.61803	−0.108595
$223$	−14.1803	−0.949586	−0.474793	−	0.880098i	$-0.657477\pi$
$223$	−14.1803	−0.949586	−0.474793	+	0.880098i	$0.657477\pi$
$224$	−3.23607	−0.216219
$225$	0	0
$226$	−6.94427	−0.461926
$227$	4.29180	0.284857	0.142428	−	0.989805i	$-0.454509\pi$
$227$	4.29180	0.284857	0.142428	+	0.989805i	$0.454509\pi$
$228$	7.23607	0.479220
$229$	−23.1246	−1.52812	−0.764059	−	0.645147i	$-0.776796\pi$
$229$	−23.1246	−1.52812	−0.764059	+	0.645147i	$0.776796\pi$
$230$	0	0
$231$	−7.23607	−0.476098
$232$	9.32624	0.612298
$233$	6.56231	0.429911	0.214955	−	0.976624i	$-0.431039\pi$
$233$	6.56231	0.429911	0.214955	+	0.976624i	$0.431039\pi$
$234$	1.09017	0.0712666
$235$	0	0
$236$	−4.76393	−0.310106
$237$	−13.8541	−0.899921
$238$	−14.4721	−0.938089
$239$	9.85410	0.637409	0.318704	−	0.947854i	$-0.396752\pi$
$239$	9.85410	0.637409	0.318704	+	0.947854i	$0.396752\pi$
$240$	0	0
$241$	−1.52786	−0.0984184	−0.0492092	−	0.998788i	$-0.515670\pi$
$241$	−1.52786	−0.0984184	−0.0492092	+	0.998788i	$0.515670\pi$
$242$	9.09017	0.584338
$243$	3.94427	0.253025
$244$	10.6180	0.679750
$245$	0	0
$246$	−15.5623	−0.992216
$247$	12.7639	0.812150
$248$	−7.38197	−0.468755
$249$	23.4164	1.48395
$250$	0	0
$251$	−20.9443	−1.32199	−0.660995	−	0.750390i	$-0.729866\pi$
$251$	−20.9443	−1.32199	−0.660995	+	0.750390i	$0.729866\pi$
$252$	−1.23607	−0.0778650
$253$	3.94427	0.247974
$254$	−8.47214	−0.531589
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.05573	0.0658545	0.0329273	−	0.999458i	$-0.489517\pi$
$257$	1.05573	0.0658545	0.0329273	+	0.999458i	$0.489517\pi$
$258$	−8.47214	−0.527452
$259$	3.23607	0.201079
$260$	0	0
$261$	3.56231	0.220501
$262$	22.6525	1.39947
$263$	13.2361	0.816171	0.408085	−	0.912944i	$-0.366197\pi$
$263$	13.2361	0.816171	0.408085	+	0.912944i	$0.366197\pi$
$264$	2.23607	0.137620
$265$	0	0
$266$	−14.4721	−0.887344
$267$	−2.47214	−0.151292
$268$	−1.09017	−0.0665927
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	12.9443	0.786309	0.393154	−	0.919473i	$-0.371384\pi$
$271$	12.9443	0.786309	0.393154	+	0.919473i	$0.371384\pi$
$272$	4.47214	0.271163
$273$	14.9443	0.904468
$274$	3.67376	0.221940
$275$	0	0
$276$	−4.61803	−0.277973
$277$	−16.7984	−1.00932	−0.504658	−	0.863319i	$-0.668381\pi$
$277$	−16.7984	−1.00932	−0.504658	+	0.863319i	$0.668381\pi$
$278$	−4.85410	−0.291130
$279$	−2.81966	−0.168809
$280$	0	0
$281$	29.8885	1.78300	0.891501	−	0.453020i	$-0.149653\pi$
$281$	29.8885	1.78300	0.891501	+	0.453020i	$0.149653\pi$
$282$	−2.00000	−0.119098
$283$	−6.76393	−0.402074	−0.201037	−	0.979584i	$-0.564431\pi$
$283$	−6.76393	−0.402074	−0.201037	+	0.979584i	$0.564431\pi$
$284$	2.94427	0.174710
$285$	0	0
$286$	3.94427	0.233230
$287$	31.1246	1.83723
$288$	0.381966	0.0225076
$289$	3.00000	0.176471
$290$	0	0
$291$	0.763932	0.0447825
$292$	−7.09017	−0.414921
$293$	12.6525	0.739166	0.369583	−	0.929198i	$-0.379501\pi$
$293$	12.6525	0.739166	0.369583	+	0.929198i	$0.379501\pi$
$294$	−5.61803	−0.327650
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	7.56231	0.438809
$298$	−16.1803	−0.937302
$299$	−8.14590	−0.471089
$300$	0	0
$301$	16.9443	0.976652
$302$	−4.29180	−0.246965
$303$	5.70820	0.327928
$304$	4.47214	0.256495
$305$	0	0
$306$	1.70820	0.0976515
$307$	12.8541	0.733622	0.366811	−	0.930295i	$-0.380450\pi$
$307$	12.8541	0.733622	0.366811	+	0.930295i	$0.380450\pi$
$308$	−4.47214	−0.254824
$309$	−27.9443	−1.58969
$310$	0	0
$311$	−27.0344	−1.53298	−0.766491	−	0.642255i	$-0.777999\pi$
$311$	−27.0344	−1.53298	−0.766491	+	0.642255i	$0.777999\pi$
$312$	−4.61803	−0.261445
$313$	−28.1803	−1.59285	−0.796423	−	0.604739i	$-0.793277\pi$
$313$	−28.1803	−1.59285	−0.796423	+	0.604739i	$0.793277\pi$
$314$	−16.4721	−0.929576
$315$	0	0
$316$	−8.56231	−0.481667
$317$	−20.9443	−1.17635	−0.588174	−	0.808735i	$-0.700153\pi$
$317$	−20.9443	−1.17635	−0.588174	+	0.808735i	$0.700153\pi$
$318$	0.763932	0.0428392
$319$	12.8885	0.721620
$320$	0	0
$321$	11.8541	0.661631
$322$	9.23607	0.514706
$323$	20.0000	1.11283
$324$	−7.70820	−0.428234
$325$	0	0
$326$	−3.52786	−0.195390
$327$	4.76393	0.263446
$328$	−9.61803	−0.531067
$329$	4.00000	0.220527
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	14.4721	0.794262
$333$	−0.381966	−0.0209316
$334$	13.8541	0.758063
$335$	0	0
$336$	5.23607	0.285651
$337$	−12.0344	−0.655558	−0.327779	−	0.944754i	$-0.606300\pi$
$337$	−12.0344	−0.655558	−0.327779	+	0.944754i	$0.606300\pi$
$338$	4.85410	0.264028
$339$	11.2361	0.610259
$340$	0	0
$341$	−10.2016	−0.552449
$342$	1.70820	0.0923691
$343$	−11.4164	−0.616428
$344$	−5.23607	−0.282310
$345$	0	0
$346$	−0.472136	−0.0253822
$347$	−17.2361	−0.925281	−0.462640	−	0.886546i	$-0.653098\pi$
$347$	−17.2361	−0.925281	−0.462640	+	0.886546i	$0.653098\pi$
$348$	−15.0902	−0.808918
$349$	−10.1803	−0.544941	−0.272471	−	0.962164i	$-0.587841\pi$
$349$	−10.1803	−0.544941	−0.272471	+	0.962164i	$0.587841\pi$
$350$	0	0
$351$	−15.6180	−0.833629
$352$	1.38197	0.0736590
$353$	−16.2918	−0.867125	−0.433562	−	0.901124i	$-0.642744\pi$
$353$	−16.2918	−0.867125	−0.433562	+	0.901124i	$0.642744\pi$
$354$	7.70820	0.409686
$355$	0	0
$356$	−1.52786	−0.0809766
$357$	23.4164	1.23933
$358$	12.6525	0.668704
$359$	−4.47214	−0.236030	−0.118015	−	0.993012i	$-0.537653\pi$
$359$	−4.47214	−0.236030	−0.118015	+	0.993012i	$0.537653\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.4721	−0.760639
$363$	−14.7082	−0.771980
$364$	9.23607	0.484102
$365$	0	0
$366$	−17.1803	−0.898031
$367$	−13.1246	−0.685099	−0.342550	−	0.939500i	$-0.611290\pi$
$367$	−13.1246	−0.685099	−0.342550	+	0.939500i	$0.611290\pi$
$368$	−2.85410	−0.148780
$369$	−3.67376	−0.191248
$370$	0	0
$371$	−1.52786	−0.0793227
$372$	11.9443	0.619282
$373$	−27.7082	−1.43468	−0.717338	−	0.696725i	$-0.754640\pi$
$373$	−27.7082	−1.43468	−0.717338	+	0.696725i	$0.754640\pi$
$374$	6.18034	0.319578
$375$	0	0
$376$	−1.23607	−0.0637453
$377$	−26.6180	−1.37090
$378$	17.7082	0.910812
$379$	28.0902	1.44290	0.721448	−	0.692469i	$-0.243477\pi$
$379$	28.0902	1.44290	0.721448	+	0.692469i	$0.243477\pi$
$380$	0	0
$381$	13.7082	0.702293
$382$	7.09017	0.362765
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	−1.61803	−0.0825700
$385$	0	0
$386$	4.00000	0.203595
$387$	−2.00000	−0.101666
$388$	0.472136	0.0239691
$389$	−6.85410	−0.347517	−0.173758	−	0.984788i	$-0.555591\pi$
$389$	−6.85410	−0.347517	−0.173758	+	0.984788i	$0.555591\pi$
$390$	0	0
$391$	−12.7639	−0.645500
$392$	−3.47214	−0.175369
$393$	−36.6525	−1.84887
$394$	−7.52786	−0.379248
$395$	0	0
$396$	0.527864	0.0265262
$397$	−20.6525	−1.03652	−0.518259	−	0.855224i	$-0.673420\pi$
$397$	−20.6525	−1.03652	−0.518259	+	0.855224i	$0.673420\pi$
$398$	−3.05573	−0.153170
$399$	23.4164	1.17229
$400$	0	0
$401$	−4.76393	−0.237899	−0.118950	−	0.992900i	$-0.537953\pi$
$401$	−4.76393	−0.237899	−0.118950	+	0.992900i	$0.537953\pi$
$402$	1.76393	0.0879769
$403$	21.0689	1.04952
$404$	3.52786	0.175518
$405$	0	0
$406$	30.1803	1.49783
$407$	−1.38197	−0.0685015
$408$	−7.23607	−0.358239
$409$	−26.1803	−1.29453	−0.647267	−	0.762263i	$-0.724088\pi$
$409$	−26.1803	−1.29453	−0.647267	+	0.762263i	$0.724088\pi$
$410$	0	0
$411$	−5.94427	−0.293209
$412$	−17.2705	−0.850857
$413$	−15.4164	−0.758592
$414$	−1.09017	−0.0535789
$415$	0	0
$416$	−2.85410	−0.139934
$417$	7.85410	0.384617
$418$	6.18034	0.302290
$419$	−10.5623	−0.516002	−0.258001	−	0.966145i	$-0.583064\pi$
$419$	−10.5623	−0.516002	−0.258001	+	0.966145i	$0.583064\pi$
$420$	0	0
$421$	−31.0344	−1.51253	−0.756263	−	0.654268i	$-0.772977\pi$
$421$	−31.0344	−1.51253	−0.756263	+	0.654268i	$0.772977\pi$
$422$	−11.2705	−0.548640
$423$	−0.472136	−0.0229560
$424$	0.472136	0.0229289
$425$	0	0
$426$	−4.76393	−0.230813
$427$	34.3607	1.66283
$428$	7.32624	0.354127
$429$	−6.38197	−0.308124
$430$	0	0
$431$	12.3607	0.595393	0.297696	−	0.954661i	$-0.403782\pi$
$431$	12.3607	0.595393	0.297696	+	0.954661i	$0.403782\pi$
$432$	−5.47214	−0.263278
$433$	20.6738	0.993518	0.496759	−	0.867889i	$-0.334523\pi$
$433$	20.6738	0.993518	0.496759	+	0.867889i	$0.334523\pi$
$434$	−23.8885	−1.14669
$435$	0	0
$436$	2.94427	0.141005
$437$	−12.7639	−0.610582
$438$	11.4721	0.548160
$439$	7.79837	0.372196	0.186098	−	0.982531i	$-0.440416\pi$
$439$	7.79837	0.372196	0.186098	+	0.982531i	$0.440416\pi$
$440$	0	0
$441$	−1.32624	−0.0631542
$442$	−12.7639	−0.607118
$443$	−15.2705	−0.725524	−0.362762	−	0.931882i	$-0.618166\pi$
$443$	−15.2705	−0.725524	−0.362762	+	0.931882i	$0.618166\pi$
$444$	1.61803	0.0767885
$445$	0	0
$446$	14.1803	0.671459
$447$	26.1803	1.23829
$448$	3.23607	0.152890
$449$	−28.4721	−1.34368	−0.671842	−	0.740695i	$-0.734496\pi$
$449$	−28.4721	−1.34368	−0.671842	+	0.740695i	$0.734496\pi$
$450$	0	0
$451$	−13.2918	−0.625886
$452$	6.94427	0.326631
$453$	6.94427	0.326270
$454$	−4.29180	−0.201424
$455$	0	0
$456$	−7.23607	−0.338860
$457$	24.7639	1.15841	0.579204	−	0.815183i	$-0.303363\pi$
$457$	24.7639	1.15841	0.579204	+	0.815183i	$0.303363\pi$
$458$	23.1246	1.08054
$459$	−24.4721	−1.14226
$460$	0	0
$461$	−38.9443	−1.81382	−0.906908	−	0.421329i	$-0.861564\pi$
$461$	−38.9443	−1.81382	−0.906908	+	0.421329i	$0.861564\pi$
$462$	7.23607	0.336652
$463$	4.56231	0.212028	0.106014	−	0.994365i	$-0.466191\pi$
$463$	4.56231	0.212028	0.106014	+	0.994365i	$0.466191\pi$
$464$	−9.32624	−0.432960
$465$	0	0
$466$	−6.56231	−0.303993
$467$	24.3607	1.12728	0.563639	−	0.826021i	$-0.309401\pi$
$467$	24.3607	1.12728	0.563639	+	0.826021i	$0.309401\pi$
$468$	−1.09017	−0.0503931
$469$	−3.52786	−0.162902
$470$	0	0
$471$	26.6525	1.22808
$472$	4.76393	0.219278
$473$	−7.23607	−0.332715
$474$	13.8541	0.636340
$475$	0	0
$476$	14.4721	0.663329
$477$	0.180340	0.00825720
$478$	−9.85410	−0.450716
$479$	36.5623	1.67057	0.835287	−	0.549814i	$-0.185301\pi$
$479$	36.5623	1.67057	0.835287	+	0.549814i	$0.185301\pi$
$480$	0	0
$481$	2.85410	0.130136
$482$	1.52786	0.0695923
$483$	−14.9443	−0.679988
$484$	−9.09017	−0.413190
$485$	0	0
$486$	−3.94427	−0.178916
$487$	−37.3050	−1.69045	−0.845224	−	0.534412i	$-0.820533\pi$
$487$	−37.3050	−1.69045	−0.845224	+	0.534412i	$0.820533\pi$
$488$	−10.6180	−0.480656
$489$	5.70820	0.258134
$490$	0	0
$491$	−28.4508	−1.28397	−0.641984	−	0.766718i	$-0.721889\pi$
$491$	−28.4508	−1.28397	−0.641984	+	0.766718i	$0.721889\pi$
$492$	15.5623	0.701603
$493$	−41.7082	−1.87844
$494$	−12.7639	−0.574276
$495$	0	0
$496$	7.38197	0.331460
$497$	9.52786	0.427383
$498$	−23.4164	−1.04931
$499$	10.2918	0.460724	0.230362	−	0.973105i	$-0.426009\pi$
$499$	10.2918	0.460724	0.230362	+	0.973105i	$0.426009\pi$
$500$	0	0
$501$	−22.4164	−1.00149
$502$	20.9443	0.934789
$503$	−19.0902	−0.851189	−0.425594	−	0.904914i	$-0.639935\pi$
$503$	−19.0902	−0.851189	−0.425594	+	0.904914i	$0.639935\pi$
$504$	1.23607	0.0550588
$505$	0	0
$506$	−3.94427	−0.175344
$507$	−7.85410	−0.348813
$508$	8.47214	0.375890
$509$	−17.7082	−0.784902	−0.392451	−	0.919773i	$-0.628373\pi$
$509$	−17.7082	−0.784902	−0.392451	+	0.919773i	$0.628373\pi$
$510$	0	0
$511$	−22.9443	−1.01499
$512$	−1.00000	−0.0441942
$513$	−24.4721	−1.08047
$514$	−1.05573	−0.0465662
$515$	0	0
$516$	8.47214	0.372965
$517$	−1.70820	−0.0751267
$518$	−3.23607	−0.142185
$519$	0.763932	0.0335329
$520$	0	0
$521$	−1.41641	−0.0620540	−0.0310270	−	0.999519i	$-0.509878\pi$
$521$	−1.41641	−0.0620540	−0.0310270	+	0.999519i	$0.509878\pi$
$522$	−3.56231	−0.155918
$523$	−11.8197	−0.516838	−0.258419	−	0.966033i	$-0.583201\pi$
$523$	−11.8197	−0.516838	−0.258419	+	0.966033i	$0.583201\pi$
$524$	−22.6525	−0.989578
$525$	0	0
$526$	−13.2361	−0.577120
$527$	33.0132	1.43808
$528$	−2.23607	−0.0973124
$529$	−14.8541	−0.645831
$530$	0	0
$531$	1.81966	0.0789665
$532$	14.4721	0.627447
$533$	27.4508	1.18903
$534$	2.47214	0.106980
$535$	0	0
$536$	1.09017	0.0470882
$537$	−20.4721	−0.883438
$538$	−4.00000	−0.172452
$539$	−4.79837	−0.206681
$540$	0	0
$541$	−10.3262	−0.443960	−0.221980	−	0.975051i	$-0.571252\pi$
$541$	−10.3262	−0.443960	−0.221980	+	0.975051i	$0.571252\pi$
$542$	−12.9443	−0.556004
$543$	23.4164	1.00489
$544$	−4.47214	−0.191741
$545$	0	0
$546$	−14.9443	−0.639556
$547$	16.0689	0.687056	0.343528	−	0.939142i	$-0.388378\pi$
$547$	16.0689	0.687056	0.343528	+	0.939142i	$0.388378\pi$
$548$	−3.67376	−0.156935
$549$	−4.05573	−0.173094
$550$	0	0
$551$	−41.7082	−1.77683
$552$	4.61803	0.196557
$553$	−27.7082	−1.17827
$554$	16.7984	0.713695
$555$	0	0
$556$	4.85410	0.205860
$557$	−19.5623	−0.828882	−0.414441	−	0.910076i	$-0.636023\pi$
$557$	−19.5623	−0.828882	−0.414441	+	0.910076i	$0.636023\pi$
$558$	2.81966	0.119366
$559$	14.9443	0.632075
$560$	0	0
$561$	−10.0000	−0.422200
$562$	−29.8885	−1.26077
$563$	−7.88854	−0.332462	−0.166231	−	0.986087i	$-0.553160\pi$
$563$	−7.88854	−0.332462	−0.166231	+	0.986087i	$0.553160\pi$
$564$	2.00000	0.0842152
$565$	0	0
$566$	6.76393	0.284309
$567$	−24.9443	−1.04756
$568$	−2.94427	−0.123539
$569$	13.8885	0.582238	0.291119	−	0.956687i	$-0.405972\pi$
$569$	13.8885	0.582238	0.291119	+	0.956687i	$0.405972\pi$
$570$	0	0
$571$	−10.5623	−0.442019	−0.221009	−	0.975272i	$-0.570935\pi$
$571$	−10.5623	−0.442019	−0.221009	+	0.975272i	$0.570935\pi$
$572$	−3.94427	−0.164918
$573$	−11.4721	−0.479255
$574$	−31.1246	−1.29912
$575$	0	0
$576$	−0.381966	−0.0159153
$577$	−10.6525	−0.443468	−0.221734	−	0.975107i	$-0.571172\pi$
$577$	−10.6525	−0.443468	−0.221734	+	0.975107i	$0.571172\pi$
$578$	−3.00000	−0.124784
$579$	−6.47214	−0.268973
$580$	0	0
$581$	46.8328	1.94295
$582$	−0.763932	−0.0316660
$583$	0.652476	0.0270228
$584$	7.09017	0.293393
$585$	0	0
$586$	−12.6525	−0.522669
$587$	13.0557	0.538868	0.269434	−	0.963019i	$-0.413163\pi$
$587$	13.0557	0.538868	0.269434	+	0.963019i	$0.413163\pi$
$588$	5.61803	0.231684
$589$	33.0132	1.36028
$590$	0	0
$591$	12.1803	0.501032
$592$	1.00000	0.0410997
$593$	−17.5623	−0.721197	−0.360599	−	0.932721i	$-0.617428\pi$
$593$	−17.5623	−0.721197	−0.360599	+	0.932721i	$0.617428\pi$
$594$	−7.56231	−0.310285
$595$	0	0
$596$	16.1803	0.662773
$597$	4.94427	0.202356
$598$	8.14590	0.333111
$599$	−6.36068	−0.259890	−0.129945	−	0.991521i	$-0.541480\pi$
$599$	−6.36068	−0.259890	−0.129945	+	0.991521i	$0.541480\pi$
$600$	0	0
$601$	−24.6869	−1.00700	−0.503500	−	0.863995i	$-0.667955\pi$
$601$	−24.6869	−1.00700	−0.503500	+	0.863995i	$0.667955\pi$
$602$	−16.9443	−0.690597
$603$	0.416408	0.0169574
$604$	4.29180	0.174631
$605$	0	0
$606$	−5.70820	−0.231880
$607$	−35.0344	−1.42200	−0.711002	−	0.703190i	$-0.751758\pi$
$607$	−35.0344	−1.42200	−0.711002	+	0.703190i	$0.751758\pi$
$608$	−4.47214	−0.181369
$609$	−48.8328	−1.97881
$610$	0	0
$611$	3.52786	0.142722
$612$	−1.70820	−0.0690501
$613$	13.8197	0.558171	0.279085	−	0.960266i	$-0.409969\pi$
$613$	13.8197	0.558171	0.279085	+	0.960266i	$0.409969\pi$
$614$	−12.8541	−0.518749
$615$	0	0
$616$	4.47214	0.180187
$617$	−0.0901699	−0.00363011	−0.00181505	−	0.999998i	$-0.500578\pi$
$617$	−0.0901699	−0.00363011	−0.00181505	+	0.999998i	$0.500578\pi$
$618$	27.9443	1.12408
$619$	−15.2705	−0.613774	−0.306887	−	0.951746i	$-0.599287\pi$
$619$	−15.2705	−0.613774	−0.306887	+	0.951746i	$0.599287\pi$
$620$	0	0
$621$	15.6180	0.626730
$622$	27.0344	1.08398
$623$	−4.94427	−0.198088
$624$	4.61803	0.184869
$625$	0	0
$626$	28.1803	1.12631
$627$	−10.0000	−0.399362
$628$	16.4721	0.657310
$629$	4.47214	0.178316
$630$	0	0
$631$	−47.3951	−1.88677	−0.943385	−	0.331700i	$-0.892378\pi$
$631$	−47.3951	−1.88677	−0.943385	+	0.331700i	$0.892378\pi$
$632$	8.56231	0.340590
$633$	18.2361	0.724819
$634$	20.9443	0.831803
$635$	0	0
$636$	−0.763932	−0.0302919
$637$	9.90983	0.392642
$638$	−12.8885	−0.510262
$639$	−1.12461	−0.0444890
$640$	0	0
$641$	15.5066	0.612473	0.306236	−	0.951955i	$-0.400930\pi$
$641$	15.5066	0.612473	0.306236	+	0.951955i	$0.400930\pi$
$642$	−11.8541	−0.467844
$643$	28.7639	1.13434	0.567169	−	0.823601i	$-0.308038\pi$
$643$	28.7639	1.13434	0.567169	+	0.823601i	$0.308038\pi$
$644$	−9.23607	−0.363952
$645$	0	0
$646$	−20.0000	−0.786889
$647$	−30.0902	−1.18297	−0.591483	−	0.806317i	$-0.701457\pi$
$647$	−30.0902	−1.18297	−0.591483	+	0.806317i	$0.701457\pi$
$648$	7.70820	0.302807
$649$	6.58359	0.258429
$650$	0	0
$651$	38.6525	1.51491
$652$	3.52786	0.138162
$653$	38.2705	1.49764	0.748820	−	0.662773i	$-0.230621\pi$
$653$	38.2705	1.49764	0.748820	+	0.662773i	$0.230621\pi$
$654$	−4.76393	−0.186284
$655$	0	0
$656$	9.61803	0.375521
$657$	2.70820	0.105657
$658$	−4.00000	−0.155936
$659$	40.4508	1.57574	0.787871	−	0.615841i	$-0.211184\pi$
$659$	40.4508	1.57574	0.787871	+	0.615841i	$0.211184\pi$
$660$	0	0
$661$	17.3262	0.673913	0.336956	−	0.941520i	$-0.390603\pi$
$661$	17.3262	0.673913	0.336956	+	0.941520i	$0.390603\pi$
$662$	−28.0000	−1.08825
$663$	20.6525	0.802076
$664$	−14.4721	−0.561628
$665$	0	0
$666$	0.381966	0.0148009
$667$	26.6180	1.03065
$668$	−13.8541	−0.536031
$669$	−22.9443	−0.887077
$670$	0	0
$671$	−14.6738	−0.566474
$672$	−5.23607	−0.201986
$673$	−11.1459	−0.429643	−0.214821	−	0.976653i	$-0.568917\pi$
$673$	−11.1459	−0.429643	−0.214821	+	0.976653i	$0.568917\pi$
$674$	12.0344	0.463549
$675$	0	0
$676$	−4.85410	−0.186696
$677$	−34.6525	−1.33180	−0.665901	−	0.746040i	$-0.731953\pi$
$677$	−34.6525	−1.33180	−0.665901	+	0.746040i	$0.731953\pi$
$678$	−11.2361	−0.431519
$679$	1.52786	0.0586340
$680$	0	0
$681$	6.94427	0.266105
$682$	10.2016	0.390640
$683$	8.58359	0.328442	0.164221	−	0.986424i	$-0.447489\pi$
$683$	8.58359	0.328442	0.164221	+	0.986424i	$0.447489\pi$
$684$	−1.70820	−0.0653148
$685$	0	0
$686$	11.4164	0.435880
$687$	−37.4164	−1.42752
$688$	5.23607	0.199623
$689$	−1.34752	−0.0513366
$690$	0	0
$691$	35.7771	1.36102	0.680512	−	0.732737i	$-0.261757\pi$
$691$	35.7771	1.36102	0.680512	+	0.732737i	$0.261757\pi$
$692$	0.472136	0.0179479
$693$	1.70820	0.0648893
$694$	17.2361	0.654272
$695$	0	0
$696$	15.0902	0.571991
$697$	43.0132	1.62924
$698$	10.1803	0.385332
$699$	10.6180	0.401611
$700$	0	0
$701$	−42.9787	−1.62328	−0.811642	−	0.584155i	$-0.801426\pi$
$701$	−42.9787	−1.62328	−0.811642	+	0.584155i	$0.801426\pi$
$702$	15.6180	0.589465
$703$	4.47214	0.168670
$704$	−1.38197	−0.0520848
$705$	0	0
$706$	16.2918	0.613150
$707$	11.4164	0.429358
$708$	−7.70820	−0.289692
$709$	−8.21478	−0.308513	−0.154256	−	0.988031i	$-0.549298\pi$
$709$	−8.21478	−0.308513	−0.154256	+	0.988031i	$0.549298\pi$
$710$	0	0
$711$	3.27051	0.122654
$712$	1.52786	0.0572591
$713$	−21.0689	−0.789036
$714$	−23.4164	−0.876337
$715$	0	0
$716$	−12.6525	−0.472845
$717$	15.9443	0.595450
$718$	4.47214	0.166899
$719$	27.4164	1.02246	0.511230	−	0.859444i	$-0.329190\pi$
$719$	27.4164	1.02246	0.511230	+	0.859444i	$0.329190\pi$
$720$	0	0
$721$	−55.8885	−2.08140
$722$	−1.00000	−0.0372161
$723$	−2.47214	−0.0919397
$724$	14.4721	0.537853
$725$	0	0
$726$	14.7082	0.545873
$727$	29.8541	1.10723	0.553614	−	0.832774i	$-0.313248\pi$
$727$	29.8541	1.10723	0.553614	+	0.832774i	$0.313248\pi$
$728$	−9.23607	−0.342311
$729$	29.5066	1.09284
$730$	0	0
$731$	23.4164	0.866087
$732$	17.1803	0.635004
$733$	−27.5279	−1.01676	−0.508382	−	0.861131i	$-0.669756\pi$
$733$	−27.5279	−1.01676	−0.508382	+	0.861131i	$0.669756\pi$
$734$	13.1246	0.484438
$735$	0	0
$736$	2.85410	0.105204
$737$	1.50658	0.0554955
$738$	3.67376	0.135233
$739$	−10.9098	−0.401325	−0.200662	−	0.979660i	$-0.564309\pi$
$739$	−10.9098	−0.401325	−0.200662	+	0.979660i	$0.564309\pi$
$740$	0	0
$741$	20.6525	0.758688
$742$	1.52786	0.0560897
$743$	48.0689	1.76348	0.881738	−	0.471739i	$-0.156374\pi$
$743$	48.0689	1.76348	0.881738	+	0.471739i	$0.156374\pi$
$744$	−11.9443	−0.437898
$745$	0	0
$746$	27.7082	1.01447
$747$	−5.52786	−0.202254
$748$	−6.18034	−0.225976
$749$	23.7082	0.866279
$750$	0	0
$751$	−1.05573	−0.0385241	−0.0192620	−	0.999814i	$-0.506132\pi$
$751$	−1.05573	−0.0385241	−0.0192620	+	0.999814i	$0.506132\pi$
$752$	1.23607	0.0450748
$753$	−33.8885	−1.23497
$754$	26.6180	0.969372
$755$	0	0
$756$	−17.7082	−0.644041
$757$	4.14590	0.150685	0.0753426	−	0.997158i	$-0.475995\pi$
$757$	4.14590	0.150685	0.0753426	+	0.997158i	$0.475995\pi$
$758$	−28.0902	−1.02028
$759$	6.38197	0.231651
$760$	0	0
$761$	−19.1459	−0.694038	−0.347019	−	0.937858i	$-0.612806\pi$
$761$	−19.1459	−0.694038	−0.347019	+	0.937858i	$0.612806\pi$
$762$	−13.7082	−0.496596
$763$	9.52786	0.344932
$764$	−7.09017	−0.256513
$765$	0	0
$766$	17.8885	0.646339
$767$	−13.5967	−0.490950
$768$	1.61803	0.0583858
$769$	23.8885	0.861443	0.430721	−	0.902485i	$-0.358259\pi$
$769$	23.8885	0.861443	0.430721	+	0.902485i	$0.358259\pi$
$770$	0	0
$771$	1.70820	0.0615195
$772$	−4.00000	−0.143963
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	−0.472136	−0.0169487
$777$	5.23607	0.187843
$778$	6.85410	0.245731
$779$	43.0132	1.54111
$780$	0	0
$781$	−4.06888	−0.145596
$782$	12.7639	0.456437
$783$	51.0344	1.82382
$784$	3.47214	0.124005
$785$	0	0
$786$	36.6525	1.30735
$787$	25.5279	0.909970	0.454985	−	0.890499i	$-0.349645\pi$
$787$	25.5279	0.909970	0.454985	+	0.890499i	$0.349645\pi$
$788$	7.52786	0.268169
$789$	21.4164	0.762444
$790$	0	0
$791$	22.4721	0.799017
$792$	−0.527864	−0.0187568
$793$	30.3050	1.07616
$794$	20.6525	0.732929
$795$	0	0
$796$	3.05573	0.108307
$797$	−46.2705	−1.63899	−0.819493	−	0.573090i	$-0.805745\pi$
$797$	−46.2705	−1.63899	−0.819493	+	0.573090i	$0.805745\pi$
$798$	−23.4164	−0.828932
$799$	5.52786	0.195562
$800$	0	0
$801$	0.583592	0.0206202
$802$	4.76393	0.168220
$803$	9.79837	0.345777
$804$	−1.76393	−0.0622091
$805$	0	0
$806$	−21.0689	−0.742120
$807$	6.47214	0.227830
$808$	−3.52786	−0.124110
$809$	−13.1246	−0.461437	−0.230718	−	0.973021i	$-0.574108\pi$
$809$	−13.1246	−0.461437	−0.230718	+	0.973021i	$0.574108\pi$
$810$	0	0
$811$	−34.1033	−1.19753	−0.598765	−	0.800925i	$-0.704342\pi$
$811$	−34.1033	−1.19753	−0.598765	+	0.800925i	$0.704342\pi$
$812$	−30.1803	−1.05912
$813$	20.9443	0.734548
$814$	1.38197	0.0484379
$815$	0	0
$816$	7.23607	0.253313
$817$	23.4164	0.819236
$818$	26.1803	0.915374
$819$	−3.52786	−0.123274
$820$	0	0
$821$	−5.41641	−0.189034	−0.0945170	−	0.995523i	$-0.530131\pi$
$821$	−5.41641	−0.189034	−0.0945170	+	0.995523i	$0.530131\pi$
$822$	5.94427	0.207330
$823$	−1.88854	−0.0658305	−0.0329152	−	0.999458i	$-0.510479\pi$
$823$	−1.88854	−0.0658305	−0.0329152	+	0.999458i	$0.510479\pi$
$824$	17.2705	0.601647
$825$	0	0
$826$	15.4164	0.536405
$827$	2.06888	0.0719421	0.0359711	−	0.999353i	$-0.488548\pi$
$827$	2.06888	0.0719421	0.0359711	+	0.999353i	$0.488548\pi$
$828$	1.09017	0.0378860
$829$	55.7984	1.93796	0.968979	−	0.247144i	$-0.0794920\pi$
$829$	55.7984	1.93796	0.968979	+	0.247144i	$0.0794920\pi$
$830$	0	0
$831$	−27.1803	−0.942876
$832$	2.85410	0.0989482
$833$	15.5279	0.538009
$834$	−7.85410	−0.271965
$835$	0	0
$836$	−6.18034	−0.213752
$837$	−40.3951	−1.39626
$838$	10.5623	0.364869
$839$	−36.6525	−1.26538	−0.632692	−	0.774404i	$-0.718050\pi$
$839$	−36.6525	−1.26538	−0.632692	+	0.774404i	$0.718050\pi$
$840$	0	0
$841$	57.9787	1.99927
$842$	31.0344	1.06952
$843$	48.3607	1.66563
$844$	11.2705	0.387947
$845$	0	0
$846$	0.472136	0.0162324
$847$	−29.4164	−1.01076
$848$	−0.472136	−0.0162132
$849$	−10.9443	−0.375606
$850$	0	0
$851$	−2.85410	−0.0978374
$852$	4.76393	0.163210
$853$	−29.7426	−1.01837	−0.509184	−	0.860657i	$-0.670053\pi$
$853$	−29.7426	−1.01837	−0.509184	+	0.860657i	$0.670053\pi$
$854$	−34.3607	−1.17580
$855$	0	0
$856$	−7.32624	−0.250406
$857$	9.05573	0.309338	0.154669	−	0.987966i	$-0.450569\pi$
$857$	9.05573	0.309338	0.154669	+	0.987966i	$0.450569\pi$
$858$	6.38197	0.217877
$859$	53.4164	1.82254	0.911272	−	0.411805i	$-0.135101\pi$
$859$	53.4164	1.82254	0.911272	+	0.411805i	$0.135101\pi$
$860$	0	0
$861$	50.3607	1.71629
$862$	−12.3607	−0.421006
$863$	−19.4164	−0.660942	−0.330471	−	0.943816i	$-0.607208\pi$
$863$	−19.4164	−0.660942	−0.330471	+	0.943816i	$0.607208\pi$
$864$	5.47214	0.186166
$865$	0	0
$866$	−20.6738	−0.702523
$867$	4.85410	0.164854
$868$	23.8885	0.810830
$869$	11.8328	0.401401
$870$	0	0
$871$	−3.11146	−0.105428
$872$	−2.94427	−0.0997056
$873$	−0.180340	−0.00610358
$874$	12.7639	0.431746
$875$	0	0
$876$	−11.4721	−0.387608
$877$	16.8328	0.568404	0.284202	−	0.958764i	$-0.408271\pi$
$877$	16.8328	0.568404	0.284202	+	0.958764i	$0.408271\pi$
$878$	−7.79837	−0.263182
$879$	20.4721	0.690508
$880$	0	0
$881$	19.7426	0.665147	0.332573	−	0.943077i	$-0.392083\pi$
$881$	19.7426	0.665147	0.332573	+	0.943077i	$0.392083\pi$
$882$	1.32624	0.0446568
$883$	33.3050	1.12080	0.560400	−	0.828222i	$-0.310647\pi$
$883$	33.3050	1.12080	0.560400	+	0.828222i	$0.310647\pi$
$884$	12.7639	0.429297
$885$	0	0
$886$	15.2705	0.513023
$887$	−27.8885	−0.936406	−0.468203	−	0.883621i	$-0.655098\pi$
$887$	−27.8885	−0.936406	−0.468203	+	0.883621i	$0.655098\pi$
$888$	−1.61803	−0.0542977
$889$	27.4164	0.919517
$890$	0	0
$891$	10.6525	0.356871
$892$	−14.1803	−0.474793
$893$	5.52786	0.184983
$894$	−26.1803	−0.875602
$895$	0	0
$896$	−3.23607	−0.108109
$897$	−13.1803	−0.440079
$898$	28.4721	0.950127
$899$	−68.8460	−2.29614
$900$	0	0
$901$	−2.11146	−0.0703428
$902$	13.2918	0.442568
$903$	27.4164	0.912361
$904$	−6.94427	−0.230963
$905$	0	0
$906$	−6.94427	−0.230708
$907$	55.8885	1.85575	0.927874	−	0.372893i	$-0.121634\pi$
$907$	55.8885	1.85575	0.927874	+	0.372893i	$0.121634\pi$
$908$	4.29180	0.142428
$909$	−1.34752	−0.0446946
$910$	0	0
$911$	13.8885	0.460148	0.230074	−	0.973173i	$-0.426103\pi$
$911$	13.8885	0.460148	0.230074	+	0.973173i	$0.426103\pi$
$912$	7.23607	0.239610
$913$	−20.0000	−0.661903
$914$	−24.7639	−0.819118
$915$	0	0
$916$	−23.1246	−0.764059
$917$	−73.3050	−2.42074
$918$	24.4721	0.807701
$919$	−5.88854	−0.194245	−0.0971226	−	0.995272i	$-0.530964\pi$
$919$	−5.88854	−0.194245	−0.0971226	+	0.995272i	$0.530964\pi$
$920$	0	0
$921$	20.7984	0.685330
$922$	38.9443	1.28256
$923$	8.40325	0.276596
$924$	−7.23607	−0.238049
$925$	0	0
$926$	−4.56231	−0.149927
$927$	6.59675	0.216666
$928$	9.32624	0.306149
$929$	−27.4508	−0.900633	−0.450317	−	0.892869i	$-0.648689\pi$
$929$	−27.4508	−0.900633	−0.450317	+	0.892869i	$0.648689\pi$
$930$	0	0
$931$	15.5279	0.508905
$932$	6.56231	0.214955
$933$	−43.7426	−1.43207
$934$	−24.3607	−0.797106
$935$	0	0
$936$	1.09017	0.0356333
$937$	48.0476	1.56965	0.784823	−	0.619720i	$-0.212754\pi$
$937$	48.0476	1.56965	0.784823	+	0.619720i	$0.212754\pi$
$938$	3.52786	0.115189
$939$	−45.5967	−1.48799
$940$	0	0
$941$	−26.1803	−0.853455	−0.426727	−	0.904380i	$-0.640334\pi$
$941$	−26.1803	−0.853455	−0.426727	+	0.904380i	$0.640334\pi$
$942$	−26.6525	−0.868385
$943$	−27.4508	−0.893923
$944$	−4.76393	−0.155053
$945$	0	0
$946$	7.23607	0.235265
$947$	−18.8328	−0.611984	−0.305992	−	0.952034i	$-0.598988\pi$
$947$	−18.8328	−0.611984	−0.305992	+	0.952034i	$0.598988\pi$
$948$	−13.8541	−0.449960
$949$	−20.2361	−0.656891
$950$	0	0
$951$	−33.8885	−1.09891
$952$	−14.4721	−0.469045
$953$	−11.4508	−0.370929	−0.185465	−	0.982651i	$-0.559379\pi$
$953$	−11.4508	−0.370929	−0.185465	+	0.982651i	$0.559379\pi$
$954$	−0.180340	−0.00583872
$955$	0	0
$956$	9.85410	0.318704
$957$	20.8541	0.674117
$958$	−36.5623	−1.18127
$959$	−11.8885	−0.383901
$960$	0	0
$961$	23.4934	0.757852
$962$	−2.85410	−0.0920199
$963$	−2.79837	−0.0901763
$964$	−1.52786	−0.0492092
$965$	0	0
$966$	14.9443	0.480824
$967$	−45.2705	−1.45580	−0.727901	−	0.685683i	$-0.759504\pi$
$967$	−45.2705	−1.45580	−0.727901	+	0.685683i	$0.759504\pi$
$968$	9.09017	0.292169
$969$	32.3607	1.03957
$970$	0	0
$971$	42.3262	1.35831	0.679157	−	0.733993i	$-0.262346\pi$
$971$	42.3262	1.35831	0.679157	+	0.733993i	$0.262346\pi$
$972$	3.94427	0.126513
$973$	15.7082	0.503582
$974$	37.3050	1.19533
$975$	0	0
$976$	10.6180	0.339875
$977$	−43.5279	−1.39258	−0.696290	−	0.717761i	$-0.745167\pi$
$977$	−43.5279	−1.39258	−0.696290	+	0.717761i	$0.745167\pi$
$978$	−5.70820	−0.182528
$979$	2.11146	0.0674824
$980$	0	0
$981$	−1.12461	−0.0359061
$982$	28.4508	0.907903
$983$	31.7771	1.01353	0.506766	−	0.862084i	$-0.330841\pi$
$983$	31.7771	1.01353	0.506766	+	0.862084i	$0.330841\pi$
$984$	−15.5623	−0.496108
$985$	0	0
$986$	41.7082	1.32826
$987$	6.47214	0.206010
$988$	12.7639	0.406075
$989$	−14.9443	−0.475200
$990$	0	0
$991$	33.1033	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$991$	33.1033	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$992$	−7.38197	−0.234378
$993$	45.3050	1.43771
$994$	−9.52786	−0.302205
$995$	0	0
$996$	23.4164	0.741977
$997$	17.7771	0.563006	0.281503	−	0.959560i	$-0.409167\pi$
$997$	17.7771	0.563006	0.281503	+	0.959560i	$0.409167\pi$
$998$	−10.2918	−0.325781
$999$	−5.47214	−0.173131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.t.1.2		2
5.2	odd	4		1850.2.b.j.149.1		4
5.3	odd	4		1850.2.b.j.149.4		4
5.4	even	2		74.2.a.b.1.1	✓	2
15.14	odd	2		666.2.a.i.1.1		2
20.19	odd	2		592.2.a.g.1.2		2
35.34	odd	2		3626.2.a.s.1.2		2
40.19	odd	2		2368.2.a.u.1.1		2
40.29	even	2		2368.2.a.y.1.2		2
55.54	odd	2		8954.2.a.j.1.1		2
60.59	even	2		5328.2.a.bc.1.1		2
185.184	even	2		2738.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.b.1.1	✓	2	5.4	even	2
592.2.a.g.1.2		2	20.19	odd	2
666.2.a.i.1.1		2	15.14	odd	2
1850.2.a.t.1.2		2	1.1	even	1	trivial
1850.2.b.j.149.1		4	5.2	odd	4
1850.2.b.j.149.4		4	5.3	odd	4
2368.2.a.u.1.1		2	40.19	odd	2
2368.2.a.y.1.2		2	40.29	even	2
2738.2.a.g.1.1		2	185.184	even	2
3626.2.a.s.1.2		2	35.34	odd	2
5328.2.a.bc.1.1		2	60.59	even	2
8954.2.a.j.1.1		2	55.54	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +3.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} -1.38197 q^{11} +1.61803 q^{12} +2.85410 q^{13} -3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +0.381966 q^{18} +4.47214 q^{19} +5.23607 q^{21} +1.38197 q^{22} -2.85410 q^{23} -1.61803 q^{24} -2.85410 q^{26} -5.47214 q^{27} +3.23607 q^{28} -9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -2.23607 q^{33} -4.47214 q^{34} -0.381966 q^{36} +1.00000 q^{37} -4.47214 q^{38} +4.61803 q^{39} +9.61803 q^{41} -5.23607 q^{42} +5.23607 q^{43} -1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +1.61803 q^{48} +3.47214 q^{49} +7.23607 q^{51} +2.85410 q^{52} -0.472136 q^{53} +5.47214 q^{54} -3.23607 q^{56} +7.23607 q^{57} +9.32624 q^{58} -4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} -1.23607 q^{63} +1.00000 q^{64} +2.23607 q^{66} -1.09017 q^{67} +4.47214 q^{68} -4.61803 q^{69} +2.94427 q^{71} +0.381966 q^{72} -7.09017 q^{73} -1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -4.61803 q^{78} -8.56231 q^{79} -7.70820 q^{81} -9.61803 q^{82} +14.4721 q^{83} +5.23607 q^{84} -5.23607 q^{86} -15.0902 q^{87} +1.38197 q^{88} -1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} +11.9443 q^{93} -1.23607 q^{94} -1.61803 q^{96} +0.472136 q^{97} -3.47214 q^{98} +0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 - 5 * q^11 + q^12 - q^13 - 2 * q^14 + 2 * q^16 + 3 * q^18 + 6 * q^21 + 5 * q^22 + q^23 - q^24 + q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 + 17 * q^31 - 2 * q^32 - 3 * q^36 + 2 * q^37 + 7 * q^39 + 17 * q^41 - 6 * q^42 + 6 * q^43 - 5 * q^44 - q^46 - 2 * q^47 + q^48 - 2 * q^49 + 10 * q^51 - q^52 + 8 * q^53 + 2 * q^54 - 2 * q^56 + 10 * q^57 + 3 * q^58 - 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^63 + 2 * q^64 + 9 * q^67 - 7 * q^69 - 12 * q^71 + 3 * q^72 - 3 * q^73 - 2 * q^74 - 7 * q^78 + 3 * q^79 - 2 * q^81 - 17 * q^82 + 20 * q^83 + 6 * q^84 - 6 * q^86 - 19 * q^87 + 5 * q^88 - 12 * q^89 + 14 * q^91 + q^92 + 6 * q^93 + 2 * q^94 - q^96 - 8 * q^97 + 2 * q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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