LMFDB - Embedded newform 1850.2.a.r.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} -3.44949 q^{3} +1.00000 q^{4} +3.44949 q^{6} +2.44949 q^{7} -1.00000 q^{8} +8.89898 q^{9} -1.44949 q^{11} -3.44949 q^{12} -4.44949 q^{13} -2.44949 q^{14} +1.00000 q^{16} +1.44949 q^{17} -8.89898 q^{18} -5.00000 q^{19} -8.44949 q^{21} +1.44949 q^{22} -2.00000 q^{23} +3.44949 q^{24} +4.44949 q^{26} -20.3485 q^{27} +2.44949 q^{28} +8.89898 q^{29} -0.449490 q^{31} -1.00000 q^{32} +5.00000 q^{33} -1.44949 q^{34} +8.89898 q^{36} +1.00000 q^{37} +5.00000 q^{38} +15.3485 q^{39} +1.00000 q^{41} +8.44949 q^{42} +10.8990 q^{43} -1.44949 q^{44} +2.00000 q^{46} +9.79796 q^{47} -3.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} -4.44949 q^{52} -6.00000 q^{53} +20.3485 q^{54} -2.44949 q^{56} +17.2474 q^{57} -8.89898 q^{58} -2.00000 q^{59} -1.55051 q^{61} +0.449490 q^{62} +21.7980 q^{63} +1.00000 q^{64} -5.00000 q^{66} -9.44949 q^{67} +1.44949 q^{68} +6.89898 q^{69} -12.4495 q^{71} -8.89898 q^{72} +6.79796 q^{73} -1.00000 q^{74} -5.00000 q^{76} -3.55051 q^{77} -15.3485 q^{78} -11.7980 q^{79} +43.4949 q^{81} -1.00000 q^{82} +1.44949 q^{83} -8.44949 q^{84} -10.8990 q^{86} -30.6969 q^{87} +1.44949 q^{88} -0.348469 q^{89} -10.8990 q^{91} -2.00000 q^{92} +1.55051 q^{93} -9.79796 q^{94} +3.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -12.8990 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * 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
$3$	−3.44949	−1.99156	−0.995782	−	0.0917517i	$-0.970753\pi$
$3$	−3.44949	−1.99156	−0.995782	+	0.0917517i	$0.970753\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.44949	1.40825
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	−1.00000	−0.353553
$9$	8.89898	2.96633
$10$	0	0
$11$	−1.44949	−0.437038	−0.218519	−	0.975833i	$-0.570122\pi$
$11$	−1.44949	−0.437038	−0.218519	+	0.975833i	$0.570122\pi$
$12$	−3.44949	−0.995782
$13$	−4.44949	−1.23407	−0.617033	−	0.786937i	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	−2.44949	−0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	1.44949	0.351553	0.175776	−	0.984430i	$-0.443756\pi$
$17$	1.44949	0.351553	0.175776	+	0.984430i	$0.443756\pi$
$18$	−8.89898	−2.09751
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−8.44949	−1.84383
$22$	1.44949	0.309032
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	3.44949	0.704124
$25$	0	0
$26$	4.44949	0.872617
$27$	−20.3485	−3.91606
$28$	2.44949	0.462910
$29$	8.89898	1.65250	0.826250	−	0.563304i	$-0.190470\pi$
$29$	8.89898	1.65250	0.826250	+	0.563304i	$0.190470\pi$
$30$	0	0
$31$	−0.449490	−0.0807307	−0.0403654	−	0.999185i	$-0.512852\pi$
$31$	−0.449490	−0.0807307	−0.0403654	+	0.999185i	$0.512852\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	−1.44949	−0.248585
$35$	0	0
$36$	8.89898	1.48316
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	5.00000	0.811107
$39$	15.3485	2.45772
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	8.44949	1.30378
$43$	10.8990	1.66208	0.831039	−	0.556214i	$-0.187746\pi$
$43$	10.8990	1.66208	0.831039	+	0.556214i	$0.187746\pi$
$44$	−1.44949	−0.218519
$45$	0	0
$46$	2.00000	0.294884
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	−3.44949	−0.497891
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−5.00000	−0.700140
$52$	−4.44949	−0.617033
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	20.3485	2.76908
$55$	0	0
$56$	−2.44949	−0.327327
$57$	17.2474	2.28448
$58$	−8.89898	−1.16849
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−1.55051	−0.198522	−0.0992612	−	0.995061i	$-0.531648\pi$
$61$	−1.55051	−0.198522	−0.0992612	+	0.995061i	$0.531648\pi$
$62$	0.449490	0.0570853
$63$	21.7980	2.74628
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	−9.44949	−1.15444	−0.577219	−	0.816589i	$-0.695862\pi$
$67$	−9.44949	−1.15444	−0.577219	+	0.816589i	$0.695862\pi$
$68$	1.44949	0.175776
$69$	6.89898	0.830540
$70$	0	0
$71$	−12.4495	−1.47748	−0.738741	−	0.673989i	$-0.764579\pi$
$71$	−12.4495	−1.47748	−0.738741	+	0.673989i	$0.764579\pi$
$72$	−8.89898	−1.04875
$73$	6.79796	0.795641	0.397820	−	0.917463i	$-0.369767\pi$
$73$	6.79796	0.795641	0.397820	+	0.917463i	$0.369767\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−3.55051	−0.404618
$78$	−15.3485	−1.73787
$79$	−11.7980	−1.32737	−0.663687	−	0.748010i	$-0.731009\pi$
$79$	−11.7980	−1.32737	−0.663687	+	0.748010i	$0.731009\pi$
$80$	0	0
$81$	43.4949	4.83277
$82$	−1.00000	−0.110432
$83$	1.44949	0.159102	0.0795511	−	0.996831i	$-0.474651\pi$
$83$	1.44949	0.159102	0.0795511	+	0.996831i	$0.474651\pi$
$84$	−8.44949	−0.921915
$85$	0	0
$86$	−10.8990	−1.17527
$87$	−30.6969	−3.29106
$88$	1.44949	0.154516
$89$	−0.348469	−0.0369377	−0.0184688	−	0.999829i	$-0.505879\pi$
$89$	−0.348469	−0.0369377	−0.0184688	+	0.999829i	$0.505879\pi$
$90$	0	0
$91$	−10.8990	−1.14252
$92$	−2.00000	−0.208514
$93$	1.55051	0.160780
$94$	−9.79796	−1.01058
$95$	0	0
$96$	3.44949	0.352062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	−12.8990	−1.29640
$100$	0	0
$101$	10.8990	1.08449	0.542244	−	0.840221i	$-0.317575\pi$
$101$	10.8990	1.08449	0.542244	+	0.840221i	$0.317575\pi$
$102$	5.00000	0.495074
$103$	−1.55051	−0.152776	−0.0763882	−	0.997078i	$-0.524339\pi$
$103$	−1.55051	−0.152776	−0.0763882	+	0.997078i	$0.524339\pi$
$104$	4.44949	0.436308
$105$	0	0
$106$	6.00000	0.582772
$107$	13.4495	1.30021	0.650106	−	0.759844i	$-0.274725\pi$
$107$	13.4495	1.30021	0.650106	+	0.759844i	$0.274725\pi$
$108$	−20.3485	−1.95803
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−3.44949	−0.327411
$112$	2.44949	0.231455
$113$	9.44949	0.888933	0.444467	−	0.895795i	$-0.353393\pi$
$113$	9.44949	0.888933	0.444467	+	0.895795i	$0.353393\pi$
$114$	−17.2474	−1.61537
$115$	0	0
$116$	8.89898	0.826250
$117$	−39.5959	−3.66064
$118$	2.00000	0.184115
$119$	3.55051	0.325475
$120$	0	0
$121$	−8.89898	−0.808998
$122$	1.55051	0.140377
$123$	−3.44949	−0.311030
$124$	−0.449490	−0.0403654
$125$	0	0
$126$	−21.7980	−1.94192
$127$	−8.24745	−0.731843	−0.365921	−	0.930646i	$-0.619246\pi$
$127$	−8.24745	−0.731843	−0.365921	+	0.930646i	$0.619246\pi$
$128$	−1.00000	−0.0883883
$129$	−37.5959	−3.31014
$130$	0	0
$131$	−20.6969	−1.80830	−0.904150	−	0.427215i	$-0.859495\pi$
$131$	−20.6969	−1.80830	−0.904150	+	0.427215i	$0.859495\pi$
$132$	5.00000	0.435194
$133$	−12.2474	−1.06199
$134$	9.44949	0.816312
$135$	0	0
$136$	−1.44949	−0.124293
$137$	19.6969	1.68282	0.841412	−	0.540395i	$-0.181725\pi$
$137$	19.6969	1.68282	0.841412	+	0.540395i	$0.181725\pi$
$138$	−6.89898	−0.587280
$139$	−17.4495	−1.48005	−0.740023	−	0.672581i	$-0.765186\pi$
$139$	−17.4495	−1.48005	−0.740023	+	0.672581i	$0.765186\pi$
$140$	0	0
$141$	−33.7980	−2.84630
$142$	12.4495	1.04474
$143$	6.44949	0.539333
$144$	8.89898	0.741582
$145$	0	0
$146$	−6.79796	−0.562603
$147$	3.44949	0.284509
$148$	1.00000	0.0821995
$149$	−19.3485	−1.58509	−0.792544	−	0.609814i	$-0.791244\pi$
$149$	−19.3485	−1.58509	−0.792544	+	0.609814i	$0.791244\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	5.00000	0.405554
$153$	12.8990	1.04282
$154$	3.55051	0.286108
$155$	0	0
$156$	15.3485	1.22886
$157$	−11.5505	−0.921831	−0.460916	−	0.887444i	$-0.652479\pi$
$157$	−11.5505	−0.921831	−0.460916	+	0.887444i	$0.652479\pi$
$158$	11.7980	0.938595
$159$	20.6969	1.64137
$160$	0	0
$161$	−4.89898	−0.386094
$162$	−43.4949	−3.41728
$163$	−19.8990	−1.55861	−0.779304	−	0.626646i	$-0.784427\pi$
$163$	−19.8990	−1.55861	−0.779304	+	0.626646i	$0.784427\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	−1.44949	−0.112502
$167$	5.55051	0.429511	0.214756	−	0.976668i	$-0.431104\pi$
$167$	5.55051	0.429511	0.214756	+	0.976668i	$0.431104\pi$
$168$	8.44949	0.651892
$169$	6.79796	0.522920
$170$	0	0
$171$	−44.4949	−3.40261
$172$	10.8990	0.831039
$173$	−22.6969	−1.72562	−0.862808	−	0.505532i	$-0.831296\pi$
$173$	−22.6969	−1.72562	−0.862808	+	0.505532i	$0.831296\pi$
$174$	30.6969	2.32713
$175$	0	0
$176$	−1.44949	−0.109259
$177$	6.89898	0.518559
$178$	0.348469	0.0261189
$179$	18.7980	1.40503	0.702513	−	0.711671i	$-0.252061\pi$
$179$	18.7980	1.40503	0.702513	+	0.711671i	$0.252061\pi$
$180$	0	0
$181$	−2.89898	−0.215479	−0.107740	−	0.994179i	$-0.534361\pi$
$181$	−2.89898	−0.215479	−0.107740	+	0.994179i	$0.534361\pi$
$182$	10.8990	0.807886
$183$	5.34847	0.395370
$184$	2.00000	0.147442
$185$	0	0
$186$	−1.55051	−0.113689
$187$	−2.10102	−0.153642
$188$	9.79796	0.714590
$189$	−49.8434	−3.62557
$190$	0	0
$191$	11.7980	0.853670	0.426835	−	0.904329i	$-0.359628\pi$
$191$	11.7980	0.853670	0.426835	+	0.904329i	$0.359628\pi$
$192$	−3.44949	−0.248945
$193$	7.44949	0.536226	0.268113	−	0.963387i	$-0.413600\pi$
$193$	7.44949	0.536226	0.268113	+	0.963387i	$0.413600\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	2.65153	0.188914	0.0944569	−	0.995529i	$-0.469889\pi$
$197$	2.65153	0.188914	0.0944569	+	0.995529i	$0.469889\pi$
$198$	12.8990	0.916691
$199$	23.5959	1.67267	0.836335	−	0.548219i	$-0.184694\pi$
$199$	23.5959	1.67267	0.836335	+	0.548219i	$0.184694\pi$
$200$	0	0
$201$	32.5959	2.29914
$202$	−10.8990	−0.766850
$203$	21.7980	1.52992
$204$	−5.00000	−0.350070
$205$	0	0
$206$	1.55051	0.108029
$207$	−17.7980	−1.23704
$208$	−4.44949	−0.308517
$209$	7.24745	0.501317
$210$	0	0
$211$	−9.44949	−0.650530	−0.325265	−	0.945623i	$-0.605453\pi$
$211$	−9.44949	−0.650530	−0.325265	+	0.945623i	$0.605453\pi$
$212$	−6.00000	−0.412082
$213$	42.9444	2.94250
$214$	−13.4495	−0.919388
$215$	0	0
$216$	20.3485	1.38454
$217$	−1.10102	−0.0747421
$218$	−14.0000	−0.948200
$219$	−23.4495	−1.58457
$220$	0	0
$221$	−6.44949	−0.433840
$222$	3.44949	0.231515
$223$	−2.20204	−0.147460	−0.0737298	−	0.997278i	$-0.523490\pi$
$223$	−2.20204	−0.147460	−0.0737298	+	0.997278i	$0.523490\pi$
$224$	−2.44949	−0.163663
$225$	0	0
$226$	−9.44949	−0.628571
$227$	−12.6969	−0.842725	−0.421363	−	0.906892i	$-0.638448\pi$
$227$	−12.6969	−0.842725	−0.421363	+	0.906892i	$0.638448\pi$
$228$	17.2474	1.14224
$229$	−13.7980	−0.911795	−0.455897	−	0.890032i	$-0.650682\pi$
$229$	−13.7980	−0.911795	−0.455897	+	0.890032i	$0.650682\pi$
$230$	0	0
$231$	12.2474	0.805823
$232$	−8.89898	−0.584247
$233$	−20.4949	−1.34267	−0.671333	−	0.741156i	$-0.734278\pi$
$233$	−20.4949	−1.34267	−0.671333	+	0.741156i	$0.734278\pi$
$234$	39.5959	2.58847
$235$	0	0
$236$	−2.00000	−0.130189
$237$	40.6969	2.64355
$238$	−3.55051	−0.230145
$239$	−8.89898	−0.575627	−0.287814	−	0.957686i	$-0.592928\pi$
$239$	−8.89898	−0.575627	−0.287814	+	0.957686i	$0.592928\pi$
$240$	0	0
$241$	−7.44949	−0.479864	−0.239932	−	0.970790i	$-0.577125\pi$
$241$	−7.44949	−0.479864	−0.239932	+	0.970790i	$0.577125\pi$
$242$	8.89898	0.572048
$243$	−88.9898	−5.70870
$244$	−1.55051	−0.0992612
$245$	0	0
$246$	3.44949	0.219931
$247$	22.2474	1.41557
$248$	0.449490	0.0285426
$249$	−5.00000	−0.316862
$250$	0	0
$251$	−11.2020	−0.707067	−0.353533	−	0.935422i	$-0.615020\pi$
$251$	−11.2020	−0.707067	−0.353533	+	0.935422i	$0.615020\pi$
$252$	21.7980	1.37314
$253$	2.89898	0.182257
$254$	8.24745	0.517491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.89898	−0.180833	−0.0904167	−	0.995904i	$-0.528820\pi$
$257$	−2.89898	−0.180833	−0.0904167	+	0.995904i	$0.528820\pi$
$258$	37.5959	2.34062
$259$	2.44949	0.152204
$260$	0	0
$261$	79.1918	4.90185
$262$	20.6969	1.27866
$263$	−19.7980	−1.22079	−0.610397	−	0.792095i	$-0.708990\pi$
$263$	−19.7980	−1.22079	−0.610397	+	0.792095i	$0.708990\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	12.2474	0.750939
$267$	1.20204	0.0735637
$268$	−9.44949	−0.577219
$269$	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
$271$	−14.0454	−0.853198	−0.426599	−	0.904441i	$-0.640288\pi$
$271$	−14.0454	−0.853198	−0.426599	+	0.904441i	$0.640288\pi$
$272$	1.44949	0.0878882
$273$	37.5959	2.27541
$274$	−19.6969	−1.18994
$275$	0	0
$276$	6.89898	0.415270
$277$	13.7980	0.829039	0.414520	−	0.910040i	$-0.363950\pi$
$277$	13.7980	0.829039	0.414520	+	0.910040i	$0.363950\pi$
$278$	17.4495	1.04655
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−0.898979	−0.0536286	−0.0268143	−	0.999640i	$-0.508536\pi$
$281$	−0.898979	−0.0536286	−0.0268143	+	0.999640i	$0.508536\pi$
$282$	33.7980	2.01264
$283$	−17.6969	−1.05197	−0.525987	−	0.850493i	$-0.676304\pi$
$283$	−17.6969	−1.05197	−0.525987	+	0.850493i	$0.676304\pi$
$284$	−12.4495	−0.738741
$285$	0	0
$286$	−6.44949	−0.381366
$287$	2.44949	0.144589
$288$	−8.89898	−0.524377
$289$	−14.8990	−0.876411
$290$	0	0
$291$	48.2929	2.83098
$292$	6.79796	0.397820
$293$	−16.0454	−0.937383	−0.468691	−	0.883362i	$-0.655274\pi$
$293$	−16.0454	−0.937383	−0.468691	+	0.883362i	$0.655274\pi$
$294$	−3.44949	−0.201178
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	29.4949	1.71147
$298$	19.3485	1.12083
$299$	8.89898	0.514641
$300$	0	0
$301$	26.6969	1.53879
$302$	14.0000	0.805609
$303$	−37.5959	−2.15983
$304$	−5.00000	−0.286770
$305$	0	0
$306$	−12.8990	−0.737386
$307$	−22.3485	−1.27549	−0.637747	−	0.770246i	$-0.720134\pi$
$307$	−22.3485	−1.27549	−0.637747	+	0.770246i	$0.720134\pi$
$308$	−3.55051	−0.202309
$309$	5.34847	0.304264
$310$	0	0
$311$	−3.55051	−0.201331	−0.100665	−	0.994920i	$-0.532097\pi$
$311$	−3.55051	−0.201331	−0.100665	+	0.994920i	$0.532097\pi$
$312$	−15.3485	−0.868936
$313$	0.898979	0.0508133	0.0254067	−	0.999677i	$-0.491912\pi$
$313$	0.898979	0.0508133	0.0254067	+	0.999677i	$0.491912\pi$
$314$	11.5505	0.651833
$315$	0	0
$316$	−11.7980	−0.663687
$317$	14.4495	0.811564	0.405782	−	0.913970i	$-0.366999\pi$
$317$	14.4495	0.811564	0.405782	+	0.913970i	$0.366999\pi$
$318$	−20.6969	−1.16063
$319$	−12.8990	−0.722204
$320$	0	0
$321$	−46.3939	−2.58945
$322$	4.89898	0.273009
$323$	−7.24745	−0.403259
$324$	43.4949	2.41638
$325$	0	0
$326$	19.8990	1.10210
$327$	−48.2929	−2.67060
$328$	−1.00000	−0.0552158
$329$	24.0000	1.32316
$330$	0	0
$331$	10.1010	0.555202	0.277601	−	0.960696i	$-0.410461\pi$
$331$	10.1010	0.555202	0.277601	+	0.960696i	$0.410461\pi$
$332$	1.44949	0.0795511
$333$	8.89898	0.487661
$334$	−5.55051	−0.303710
$335$	0	0
$336$	−8.44949	−0.460957
$337$	13.6969	0.746120	0.373060	−	0.927807i	$-0.378309\pi$
$337$	13.6969	0.746120	0.373060	+	0.927807i	$0.378309\pi$
$338$	−6.79796	−0.369760
$339$	−32.5959	−1.77037
$340$	0	0
$341$	0.651531	0.0352824
$342$	44.4949	2.40601
$343$	−19.5959	−1.05808
$344$	−10.8990	−0.587634
$345$	0	0
$346$	22.6969	1.22019
$347$	−0.797959	−0.0428367	−0.0214183	−	0.999771i	$-0.506818\pi$
$347$	−0.797959	−0.0428367	−0.0214183	+	0.999771i	$0.506818\pi$
$348$	−30.6969	−1.64553
$349$	−7.55051	−0.404170	−0.202085	−	0.979368i	$-0.564772\pi$
$349$	−7.55051	−0.404170	−0.202085	+	0.979368i	$0.564772\pi$
$350$	0	0
$351$	90.5403	4.83268
$352$	1.44949	0.0772581
$353$	−5.10102	−0.271500	−0.135750	−	0.990743i	$-0.543344\pi$
$353$	−5.10102	−0.271500	−0.135750	+	0.990743i	$0.543344\pi$
$354$	−6.89898	−0.366677
$355$	0	0
$356$	−0.348469	−0.0184688
$357$	−12.2474	−0.648204
$358$	−18.7980	−0.993503
$359$	27.3939	1.44579	0.722897	−	0.690956i	$-0.242810\pi$
$359$	27.3939	1.44579	0.722897	+	0.690956i	$0.242810\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	2.89898	0.152367
$363$	30.6969	1.61117
$364$	−10.8990	−0.571262
$365$	0	0
$366$	−5.34847	−0.279569
$367$	−10.2474	−0.534912	−0.267456	−	0.963570i	$-0.586183\pi$
$367$	−10.2474	−0.534912	−0.267456	+	0.963570i	$0.586183\pi$
$368$	−2.00000	−0.104257
$369$	8.89898	0.463262
$370$	0	0
$371$	−14.6969	−0.763027
$372$	1.55051	0.0803902
$373$	−24.0454	−1.24502	−0.622512	−	0.782610i	$-0.713888\pi$
$373$	−24.0454	−1.24502	−0.622512	+	0.782610i	$0.713888\pi$
$374$	2.10102	0.108641
$375$	0	0
$376$	−9.79796	−0.505291
$377$	−39.5959	−2.03929
$378$	49.8434	2.56367
$379$	23.0454	1.18376	0.591882	−	0.806025i	$-0.298385\pi$
$379$	23.0454	1.18376	0.591882	+	0.806025i	$0.298385\pi$
$380$	0	0
$381$	28.4495	1.45751
$382$	−11.7980	−0.603636
$383$	−9.34847	−0.477684	−0.238842	−	0.971058i	$-0.576768\pi$
$383$	−9.34847	−0.477684	−0.238842	+	0.971058i	$0.576768\pi$
$384$	3.44949	0.176031
$385$	0	0
$386$	−7.44949	−0.379169
$387$	96.9898	4.93027
$388$	−14.0000	−0.710742
$389$	−21.1464	−1.07217	−0.536083	−	0.844165i	$-0.680097\pi$
$389$	−21.1464	−1.07217	−0.536083	+	0.844165i	$0.680097\pi$
$390$	0	0
$391$	−2.89898	−0.146608
$392$	1.00000	0.0505076
$393$	71.3939	3.60134
$394$	−2.65153	−0.133582
$395$	0	0
$396$	−12.8990	−0.648198
$397$	0.651531	0.0326994	0.0163497	−	0.999866i	$-0.494795\pi$
$397$	0.651531	0.0326994	0.0163497	+	0.999866i	$0.494795\pi$
$398$	−23.5959	−1.18276
$399$	42.2474	2.11502
$400$	0	0
$401$	−37.9444	−1.89485	−0.947426	−	0.319975i	$-0.896326\pi$
$401$	−37.9444	−1.89485	−0.947426	+	0.319975i	$0.896326\pi$
$402$	−32.5959	−1.62574
$403$	2.00000	0.0996271
$404$	10.8990	0.542244
$405$	0	0
$406$	−21.7980	−1.08181
$407$	−1.44949	−0.0718485
$408$	5.00000	0.247537
$409$	0.146428	0.00724041	0.00362020	−	0.999993i	$-0.498848\pi$
$409$	0.146428	0.00724041	0.00362020	+	0.999993i	$0.498848\pi$
$410$	0	0
$411$	−67.9444	−3.35145
$412$	−1.55051	−0.0763882
$413$	−4.89898	−0.241063
$414$	17.7980	0.874722
$415$	0	0
$416$	4.44949	0.218154
$417$	60.1918	2.94761
$418$	−7.24745	−0.354484
$419$	−16.5505	−0.808545	−0.404273	−	0.914639i	$-0.632475\pi$
$419$	−16.5505	−0.808545	−0.404273	+	0.914639i	$0.632475\pi$
$420$	0	0
$421$	23.1010	1.12587	0.562937	−	0.826500i	$-0.309671\pi$
$421$	23.1010	1.12587	0.562937	+	0.826500i	$0.309671\pi$
$422$	9.44949	0.459994
$423$	87.1918	4.23941
$424$	6.00000	0.291386
$425$	0	0
$426$	−42.9444	−2.08066
$427$	−3.79796	−0.183796
$428$	13.4495	0.650106
$429$	−22.2474	−1.07412
$430$	0	0
$431$	8.69694	0.418917	0.209458	−	0.977818i	$-0.432830\pi$
$431$	8.69694	0.418917	0.209458	+	0.977818i	$0.432830\pi$
$432$	−20.3485	−0.979016
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	1.10102	0.0528507
$435$	0	0
$436$	14.0000	0.670478
$437$	10.0000	0.478365
$438$	23.4495	1.12046
$439$	19.5959	0.935262	0.467631	−	0.883924i	$-0.345108\pi$
$439$	19.5959	0.935262	0.467631	+	0.883924i	$0.345108\pi$
$440$	0	0
$441$	−8.89898	−0.423761
$442$	6.44949	0.306771
$443$	12.5505	0.596293	0.298146	−	0.954520i	$-0.403632\pi$
$443$	12.5505	0.596293	0.298146	+	0.954520i	$0.403632\pi$
$444$	−3.44949	−0.163706
$445$	0	0
$446$	2.20204	0.104270
$447$	66.7423	3.15680
$448$	2.44949	0.115728
$449$	8.75255	0.413058	0.206529	−	0.978440i	$-0.433783\pi$
$449$	8.75255	0.413058	0.206529	+	0.978440i	$0.433783\pi$
$450$	0	0
$451$	−1.44949	−0.0682538
$452$	9.44949	0.444467
$453$	48.2929	2.26900
$454$	12.6969	0.595897
$455$	0	0
$456$	−17.2474	−0.807686
$457$	9.24745	0.432577	0.216289	−	0.976329i	$-0.430605\pi$
$457$	9.24745	0.432577	0.216289	+	0.976329i	$0.430605\pi$
$458$	13.7980	0.644736
$459$	−29.4949	−1.37670
$460$	0	0
$461$	38.6969	1.80230	0.901148	−	0.433511i	$-0.142726\pi$
$461$	38.6969	1.80230	0.901148	+	0.433511i	$0.142726\pi$
$462$	−12.2474	−0.569803
$463$	−14.4495	−0.671525	−0.335762	−	0.941947i	$-0.608994\pi$
$463$	−14.4495	−0.671525	−0.335762	+	0.941947i	$0.608994\pi$
$464$	8.89898	0.413125
$465$	0	0
$466$	20.4949	0.949408
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−39.5959	−1.83032
$469$	−23.1464	−1.06880
$470$	0	0
$471$	39.8434	1.83589
$472$	2.00000	0.0920575
$473$	−15.7980	−0.726391
$474$	−40.6969	−1.86927
$475$	0	0
$476$	3.55051	0.162737
$477$	−53.3939	−2.44474
$478$	8.89898	0.407030
$479$	−18.2474	−0.833747	−0.416874	−	0.908964i	$-0.636874\pi$
$479$	−18.2474	−0.833747	−0.416874	+	0.908964i	$0.636874\pi$
$480$	0	0
$481$	−4.44949	−0.202879
$482$	7.44949	0.339315
$483$	16.8990	0.768930
$484$	−8.89898	−0.404499
$485$	0	0
$486$	88.9898	4.03666
$487$	−42.7423	−1.93684	−0.968420	−	0.249323i	$-0.919792\pi$
$487$	−42.7423	−1.93684	−0.968420	+	0.249323i	$0.919792\pi$
$488$	1.55051	0.0701883
$489$	68.6413	3.10407
$490$	0	0
$491$	−22.2020	−1.00196	−0.500982	−	0.865458i	$-0.667028\pi$
$491$	−22.2020	−1.00196	−0.500982	+	0.865458i	$0.667028\pi$
$492$	−3.44949	−0.155515
$493$	12.8990	0.580941
$494$	−22.2474	−1.00096
$495$	0	0
$496$	−0.449490	−0.0201827
$497$	−30.4949	−1.36788
$498$	5.00000	0.224055
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−19.1464	−0.855399
$502$	11.2020	0.499972
$503$	9.79796	0.436869	0.218435	−	0.975852i	$-0.429905\pi$
$503$	9.79796	0.436869	0.218435	+	0.975852i	$0.429905\pi$
$504$	−21.7980	−0.970958
$505$	0	0
$506$	−2.89898	−0.128875
$507$	−23.4495	−1.04143
$508$	−8.24745	−0.365921
$509$	14.6515	0.649418	0.324709	−	0.945814i	$-0.394734\pi$
$509$	14.6515	0.649418	0.324709	+	0.945814i	$0.394734\pi$
$510$	0	0
$511$	16.6515	0.736620
$512$	−1.00000	−0.0441942
$513$	101.742	4.49203
$514$	2.89898	0.127869
$515$	0	0
$516$	−37.5959	−1.65507
$517$	−14.2020	−0.624605
$518$	−2.44949	−0.107624
$519$	78.2929	3.43667
$520$	0	0
$521$	19.8990	0.871790	0.435895	−	0.899997i	$-0.356432\pi$
$521$	19.8990	0.871790	0.435895	+	0.899997i	$0.356432\pi$
$522$	−79.1918	−3.46613
$523$	−40.3939	−1.76630	−0.883150	−	0.469090i	$-0.844582\pi$
$523$	−40.3939	−1.76630	−0.883150	+	0.469090i	$0.844582\pi$
$524$	−20.6969	−0.904150
$525$	0	0
$526$	19.7980	0.863232
$527$	−0.651531	−0.0283811
$528$	5.00000	0.217597
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−17.7980	−0.772366
$532$	−12.2474	−0.530994
$533$	−4.44949	−0.192729
$534$	−1.20204	−0.0520174
$535$	0	0
$536$	9.44949	0.408156
$537$	−64.8434	−2.79820
$538$	−4.00000	−0.172452
$539$	1.44949	0.0624339
$540$	0	0
$541$	10.6515	0.457945	0.228973	−	0.973433i	$-0.426463\pi$
$541$	10.6515	0.457945	0.228973	+	0.973433i	$0.426463\pi$
$542$	14.0454	0.603302
$543$	10.0000	0.429141
$544$	−1.44949	−0.0621464
$545$	0	0
$546$	−37.5959	−1.60896
$547$	25.6969	1.09872	0.549361	−	0.835585i	$-0.314871\pi$
$547$	25.6969	1.09872	0.549361	+	0.835585i	$0.314871\pi$
$548$	19.6969	0.841412
$549$	−13.7980	−0.588883
$550$	0	0
$551$	−44.4949	−1.89555
$552$	−6.89898	−0.293640
$553$	−28.8990	−1.22891
$554$	−13.7980	−0.586219
$555$	0	0
$556$	−17.4495	−0.740023
$557$	2.20204	0.0933035	0.0466517	−	0.998911i	$-0.485145\pi$
$557$	2.20204	0.0933035	0.0466517	+	0.998911i	$0.485145\pi$
$558$	4.00000	0.169334
$559$	−48.4949	−2.05112
$560$	0	0
$561$	7.24745	0.305988
$562$	0.898979	0.0379212
$563$	1.59592	0.0672599	0.0336300	−	0.999434i	$-0.489293\pi$
$563$	1.59592	0.0672599	0.0336300	+	0.999434i	$0.489293\pi$
$564$	−33.7980	−1.42315
$565$	0	0
$566$	17.6969	0.743858
$567$	106.540	4.47427
$568$	12.4495	0.522369
$569$	−17.0454	−0.714581	−0.357290	−	0.933993i	$-0.616299\pi$
$569$	−17.0454	−0.714581	−0.357290	+	0.933993i	$0.616299\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	6.44949	0.269667
$573$	−40.6969	−1.70014
$574$	−2.44949	−0.102240
$575$	0	0
$576$	8.89898	0.370791
$577$	23.0454	0.959393	0.479696	−	0.877435i	$-0.340747\pi$
$577$	23.0454	0.959393	0.479696	+	0.877435i	$0.340747\pi$
$578$	14.8990	0.619716
$579$	−25.6969	−1.06793
$580$	0	0
$581$	3.55051	0.147300
$582$	−48.2929	−2.00180
$583$	8.69694	0.360190
$584$	−6.79796	−0.281302
$585$	0	0
$586$	16.0454	0.662830
$587$	35.6969	1.47337	0.736685	−	0.676236i	$-0.236390\pi$
$587$	35.6969	1.47337	0.736685	+	0.676236i	$0.236390\pi$
$588$	3.44949	0.142255
$589$	2.24745	0.0926045
$590$	0	0
$591$	−9.14643	−0.376234
$592$	1.00000	0.0410997
$593$	12.3939	0.508956	0.254478	−	0.967079i	$-0.418096\pi$
$593$	12.3939	0.508956	0.254478	+	0.967079i	$0.418096\pi$
$594$	−29.4949	−1.21019
$595$	0	0
$596$	−19.3485	−0.792544
$597$	−81.3939	−3.33123
$598$	−8.89898	−0.363906
$599$	−14.9444	−0.610611	−0.305306	−	0.952254i	$-0.598759\pi$
$599$	−14.9444	−0.610611	−0.305306	+	0.952254i	$0.598759\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	−26.6969	−1.08809
$603$	−84.0908	−3.42444
$604$	−14.0000	−0.569652
$605$	0	0
$606$	37.5959	1.52723
$607$	−44.7423	−1.81604	−0.908018	−	0.418931i	$-0.862405\pi$
$607$	−44.7423	−1.81604	−0.908018	+	0.418931i	$0.862405\pi$
$608$	5.00000	0.202777
$609$	−75.1918	−3.04693
$610$	0	0
$611$	−43.5959	−1.76370
$612$	12.8990	0.521410
$613$	−14.4949	−0.585443	−0.292722	−	0.956198i	$-0.594561\pi$
$613$	−14.4949	−0.585443	−0.292722	+	0.956198i	$0.594561\pi$
$614$	22.3485	0.901911
$615$	0	0
$616$	3.55051	0.143054
$617$	−4.89898	−0.197225	−0.0986127	−	0.995126i	$-0.531441\pi$
$617$	−4.89898	−0.197225	−0.0986127	+	0.995126i	$0.531441\pi$
$618$	−5.34847	−0.215147
$619$	−23.1010	−0.928508	−0.464254	−	0.885702i	$-0.653678\pi$
$619$	−23.1010	−0.928508	−0.464254	+	0.885702i	$0.653678\pi$
$620$	0	0
$621$	40.6969	1.63311
$622$	3.55051	0.142362
$623$	−0.853572	−0.0341976
$624$	15.3485	0.614431
$625$	0	0
$626$	−0.898979	−0.0359304
$627$	−25.0000	−0.998404
$628$	−11.5505	−0.460916
$629$	1.44949	0.0577949
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	11.7980	0.469298
$633$	32.5959	1.29557
$634$	−14.4495	−0.573863
$635$	0	0
$636$	20.6969	0.820687
$637$	4.44949	0.176295
$638$	12.8990	0.510675
$639$	−110.788	−4.38270
$640$	0	0
$641$	−17.5959	−0.694997	−0.347498	−	0.937681i	$-0.612969\pi$
$641$	−17.5959	−0.694997	−0.347498	+	0.937681i	$0.612969\pi$
$642$	46.3939	1.83102
$643$	36.2929	1.43125	0.715625	−	0.698484i	$-0.246142\pi$
$643$	36.2929	1.43125	0.715625	+	0.698484i	$0.246142\pi$
$644$	−4.89898	−0.193047
$645$	0	0
$646$	7.24745	0.285147
$647$	15.7526	0.619297	0.309648	−	0.950851i	$-0.399789\pi$
$647$	15.7526	0.619297	0.309648	+	0.950851i	$0.399789\pi$
$648$	−43.4949	−1.70864
$649$	2.89898	0.113795
$650$	0	0
$651$	3.79796	0.148854
$652$	−19.8990	−0.779304
$653$	−34.0454	−1.33230	−0.666150	−	0.745818i	$-0.732059\pi$
$653$	−34.0454	−1.33230	−0.666150	+	0.745818i	$0.732059\pi$
$654$	48.2929	1.88840
$655$	0	0
$656$	1.00000	0.0390434
$657$	60.4949	2.36013
$658$	−24.0000	−0.935617
$659$	11.4495	0.446009	0.223004	−	0.974817i	$-0.428414\pi$
$659$	11.4495	0.446009	0.223004	+	0.974817i	$0.428414\pi$
$660$	0	0
$661$	6.20204	0.241231	0.120616	−	0.992699i	$-0.461513\pi$
$661$	6.20204	0.241231	0.120616	+	0.992699i	$0.461513\pi$
$662$	−10.1010	−0.392587
$663$	22.2474	0.864019
$664$	−1.44949	−0.0562511
$665$	0	0
$666$	−8.89898	−0.344828
$667$	−17.7980	−0.689140
$668$	5.55051	0.214756
$669$	7.59592	0.293675
$670$	0	0
$671$	2.24745	0.0867618
$672$	8.44949	0.325946
$673$	13.7980	0.531872	0.265936	−	0.963991i	$-0.414319\pi$
$673$	13.7980	0.531872	0.265936	+	0.963991i	$0.414319\pi$
$674$	−13.6969	−0.527586
$675$	0	0
$676$	6.79796	0.261460
$677$	4.40408	0.169263	0.0846313	−	0.996412i	$-0.473029\pi$
$677$	4.40408	0.169263	0.0846313	+	0.996412i	$0.473029\pi$
$678$	32.5959	1.25184
$679$	−34.2929	−1.31604
$680$	0	0
$681$	43.7980	1.67834
$682$	−0.651531	−0.0249484
$683$	7.00000	0.267848	0.133924	−	0.990992i	$-0.457242\pi$
$683$	7.00000	0.267848	0.133924	+	0.990992i	$0.457242\pi$
$684$	−44.4949	−1.70130
$685$	0	0
$686$	19.5959	0.748176
$687$	47.5959	1.81590
$688$	10.8990	0.415520
$689$	26.6969	1.01707
$690$	0	0
$691$	44.3485	1.68710	0.843548	−	0.537054i	$-0.180463\pi$
$691$	44.3485	1.68710	0.843548	+	0.537054i	$0.180463\pi$
$692$	−22.6969	−0.862808
$693$	−31.5959	−1.20023
$694$	0.797959	0.0302901
$695$	0	0
$696$	30.6969	1.16356
$697$	1.44949	0.0549033
$698$	7.55051	0.285791
$699$	70.6969	2.67400
$700$	0	0
$701$	−9.75255	−0.368349	−0.184174	−	0.982894i	$-0.558961\pi$
$701$	−9.75255	−0.368349	−0.184174	+	0.982894i	$0.558961\pi$
$702$	−90.5403	−3.41722
$703$	−5.00000	−0.188579
$704$	−1.44949	−0.0546297
$705$	0	0
$706$	5.10102	0.191979
$707$	26.6969	1.00404
$708$	6.89898	0.259280
$709$	7.10102	0.266684	0.133342	−	0.991070i	$-0.457429\pi$
$709$	7.10102	0.266684	0.133342	+	0.991070i	$0.457429\pi$
$710$	0	0
$711$	−104.990	−3.93742
$712$	0.348469	0.0130594
$713$	0.898979	0.0336670
$714$	12.2474	0.458349
$715$	0	0
$716$	18.7980	0.702513
$717$	30.6969	1.14640
$718$	−27.3939	−1.02233
$719$	−15.3485	−0.572401	−0.286201	−	0.958170i	$-0.592392\pi$
$719$	−15.3485	−0.572401	−0.286201	+	0.958170i	$0.592392\pi$
$720$	0	0
$721$	−3.79796	−0.141443
$722$	−6.00000	−0.223297
$723$	25.6969	0.955679
$724$	−2.89898	−0.107740
$725$	0	0
$726$	−30.6969	−1.13927
$727$	−36.0000	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$728$	10.8990	0.403943
$729$	176.485	6.53647
$730$	0	0
$731$	15.7980	0.584309
$732$	5.34847	0.197685
$733$	−23.5959	−0.871535	−0.435768	−	0.900059i	$-0.643523\pi$
$733$	−23.5959	−0.871535	−0.435768	+	0.900059i	$0.643523\pi$
$734$	10.2474	0.378240
$735$	0	0
$736$	2.00000	0.0737210
$737$	13.6969	0.504533
$738$	−8.89898	−0.327576
$739$	7.59592	0.279420	0.139710	−	0.990192i	$-0.455383\pi$
$739$	7.59592	0.279420	0.139710	+	0.990192i	$0.455383\pi$
$740$	0	0
$741$	−76.7423	−2.81920
$742$	14.6969	0.539542
$743$	45.5959	1.67275	0.836376	−	0.548156i	$-0.184670\pi$
$743$	45.5959	1.67275	0.836376	+	0.548156i	$0.184670\pi$
$744$	−1.55051	−0.0568445
$745$	0	0
$746$	24.0454	0.880365
$747$	12.8990	0.471949
$748$	−2.10102	−0.0768209
$749$	32.9444	1.20376
$750$	0	0
$751$	−1.30306	−0.0475494	−0.0237747	−	0.999717i	$-0.507568\pi$
$751$	−1.30306	−0.0475494	−0.0237747	+	0.999717i	$0.507568\pi$
$752$	9.79796	0.357295
$753$	38.6413	1.40817
$754$	39.5959	1.44200
$755$	0	0
$756$	−49.8434	−1.81279
$757$	−5.79796	−0.210730	−0.105365	−	0.994434i	$-0.533601\pi$
$757$	−5.79796	−0.210730	−0.105365	+	0.994434i	$0.533601\pi$
$758$	−23.0454	−0.837047
$759$	−10.0000	−0.362977
$760$	0	0
$761$	−43.6969	−1.58401	−0.792006	−	0.610513i	$-0.790963\pi$
$761$	−43.6969	−1.58401	−0.792006	+	0.610513i	$0.790963\pi$
$762$	−28.4495	−1.03062
$763$	34.2929	1.24148
$764$	11.7980	0.426835
$765$	0	0
$766$	9.34847	0.337774
$767$	8.89898	0.321324
$768$	−3.44949	−0.124473
$769$	28.7526	1.03684	0.518422	−	0.855125i	$-0.326520\pi$
$769$	28.7526	1.03684	0.518422	+	0.855125i	$0.326520\pi$
$770$	0	0
$771$	10.0000	0.360141
$772$	7.44949	0.268113
$773$	22.0454	0.792918	0.396459	−	0.918052i	$-0.370239\pi$
$773$	22.0454	0.792918	0.396459	+	0.918052i	$0.370239\pi$
$774$	−96.9898	−3.48623
$775$	0	0
$776$	14.0000	0.502571
$777$	−8.44949	−0.303124
$778$	21.1464	0.758136
$779$	−5.00000	−0.179144
$780$	0	0
$781$	18.0454	0.645715
$782$	2.89898	0.103667
$783$	−181.081	−6.47129
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−71.3939	−2.54654
$787$	−1.30306	−0.0464491	−0.0232246	−	0.999730i	$-0.507393\pi$
$787$	−1.30306	−0.0464491	−0.0232246	+	0.999730i	$0.507393\pi$
$788$	2.65153	0.0944569
$789$	68.2929	2.43129
$790$	0	0
$791$	23.1464	0.822992
$792$	12.8990	0.458345
$793$	6.89898	0.244990
$794$	−0.651531	−0.0231220
$795$	0	0
$796$	23.5959	0.836335
$797$	20.2474	0.717201	0.358601	−	0.933491i	$-0.383254\pi$
$797$	20.2474	0.717201	0.358601	+	0.933491i	$0.383254\pi$
$798$	−42.2474	−1.49554
$799$	14.2020	0.502432
$800$	0	0
$801$	−3.10102	−0.109569
$802$	37.9444	1.33986
$803$	−9.85357	−0.347725
$804$	32.5959	1.14957
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	−13.7980	−0.485711
$808$	−10.8990	−0.383425
$809$	19.3939	0.681852	0.340926	−	0.940090i	$-0.389259\pi$
$809$	19.3939	0.681852	0.340926	+	0.940090i	$0.389259\pi$
$810$	0	0
$811$	−13.7980	−0.484512	−0.242256	−	0.970212i	$-0.577887\pi$
$811$	−13.7980	−0.484512	−0.242256	+	0.970212i	$0.577887\pi$
$812$	21.7980	0.764958
$813$	48.4495	1.69920
$814$	1.44949	0.0508046
$815$	0	0
$816$	−5.00000	−0.175035
$817$	−54.4949	−1.90654
$818$	−0.146428	−0.00511974
$819$	−96.9898	−3.38910
$820$	0	0
$821$	−40.0454	−1.39759	−0.698797	−	0.715320i	$-0.746281\pi$
$821$	−40.0454	−1.39759	−0.698797	+	0.715320i	$0.746281\pi$
$822$	67.9444	2.36983
$823$	29.8434	1.04027	0.520137	−	0.854083i	$-0.325881\pi$
$823$	29.8434	1.04027	0.520137	+	0.854083i	$0.325881\pi$
$824$	1.55051	0.0540146
$825$	0	0
$826$	4.89898	0.170457
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	−17.7980	−0.618522
$829$	6.65153	0.231017	0.115509	−	0.993306i	$-0.463150\pi$
$829$	6.65153	0.231017	0.115509	+	0.993306i	$0.463150\pi$
$830$	0	0
$831$	−47.5959	−1.65108
$832$	−4.44949	−0.154258
$833$	−1.44949	−0.0502218
$834$	−60.1918	−2.08427
$835$	0	0
$836$	7.24745	0.250658
$837$	9.14643	0.316147
$838$	16.5505	0.571728
$839$	13.3485	0.460840	0.230420	−	0.973091i	$-0.425990\pi$
$839$	13.3485	0.460840	0.230420	+	0.973091i	$0.425990\pi$
$840$	0	0
$841$	50.1918	1.73075
$842$	−23.1010	−0.796114
$843$	3.10102	0.106805
$844$	−9.44949	−0.325265
$845$	0	0
$846$	−87.1918	−2.99772
$847$	−21.7980	−0.748987
$848$	−6.00000	−0.206041
$849$	61.0454	2.09507
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	42.9444	1.47125
$853$	4.65153	0.159265	0.0796327	−	0.996824i	$-0.474625\pi$
$853$	4.65153	0.159265	0.0796327	+	0.996824i	$0.474625\pi$
$854$	3.79796	0.129963
$855$	0	0
$856$	−13.4495	−0.459694
$857$	−2.14643	−0.0733206	−0.0366603	−	0.999328i	$-0.511672\pi$
$857$	−2.14643	−0.0733206	−0.0366603	+	0.999328i	$0.511672\pi$
$858$	22.2474	0.759515
$859$	39.4949	1.34755	0.673774	−	0.738937i	$-0.264672\pi$
$859$	39.4949	1.34755	0.673774	+	0.738937i	$0.264672\pi$
$860$	0	0
$861$	−8.44949	−0.287958
$862$	−8.69694	−0.296219
$863$	6.24745	0.212666	0.106333	−	0.994331i	$-0.466089\pi$
$863$	6.24745	0.212666	0.106333	+	0.994331i	$0.466089\pi$
$864$	20.3485	0.692269
$865$	0	0
$866$	19.0000	0.645646
$867$	51.3939	1.74543
$868$	−1.10102	−0.0373711
$869$	17.1010	0.580112
$870$	0	0
$871$	42.0454	1.42465
$872$	−14.0000	−0.474100
$873$	−124.586	−4.21659
$874$	−10.0000	−0.338255
$875$	0	0
$876$	−23.4495	−0.792285
$877$	−19.3485	−0.653351	−0.326676	−	0.945136i	$-0.605928\pi$
$877$	−19.3485	−0.653351	−0.326676	+	0.945136i	$0.605928\pi$
$878$	−19.5959	−0.661330
$879$	55.3485	1.86686
$880$	0	0
$881$	−34.2929	−1.15536	−0.577678	−	0.816265i	$-0.696041\pi$
$881$	−34.2929	−1.15536	−0.577678	+	0.816265i	$0.696041\pi$
$882$	8.89898	0.299644
$883$	−35.8990	−1.20810	−0.604048	−	0.796948i	$-0.706447\pi$
$883$	−35.8990	−1.20810	−0.604048	+	0.796948i	$0.706447\pi$
$884$	−6.44949	−0.216920
$885$	0	0
$886$	−12.5505	−0.421643
$887$	−21.3031	−0.715287	−0.357643	−	0.933858i	$-0.616420\pi$
$887$	−21.3031	−0.715287	−0.357643	+	0.933858i	$0.616420\pi$
$888$	3.44949	0.115757
$889$	−20.2020	−0.677555
$890$	0	0
$891$	−63.0454	−2.11210
$892$	−2.20204	−0.0737298
$893$	−48.9898	−1.63938
$894$	−66.7423	−2.23220
$895$	0	0
$896$	−2.44949	−0.0818317
$897$	−30.6969	−1.02494
$898$	−8.75255	−0.292076
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−8.69694	−0.289737
$902$	1.44949	0.0482627
$903$	−92.0908	−3.06459
$904$	−9.44949	−0.314285
$905$	0	0
$906$	−48.2929	−1.60442
$907$	25.1010	0.833466	0.416733	−	0.909029i	$-0.363175\pi$
$907$	25.1010	0.833466	0.416733	+	0.909029i	$0.363175\pi$
$908$	−12.6969	−0.421363
$909$	96.9898	3.21695
$910$	0	0
$911$	8.24745	0.273250	0.136625	−	0.990623i	$-0.456374\pi$
$911$	8.24745	0.273250	0.136625	+	0.990623i	$0.456374\pi$
$912$	17.2474	0.571120
$913$	−2.10102	−0.0695336
$914$	−9.24745	−0.305878
$915$	0	0
$916$	−13.7980	−0.455897
$917$	−50.6969	−1.67416
$918$	29.4949	0.973477
$919$	−32.2020	−1.06225	−0.531124	−	0.847294i	$-0.678230\pi$
$919$	−32.2020	−1.06225	−0.531124	+	0.847294i	$0.678230\pi$
$920$	0	0
$921$	77.0908	2.54023
$922$	−38.6969	−1.27442
$923$	55.3939	1.82331
$924$	12.2474	0.402911
$925$	0	0
$926$	14.4495	0.474840
$927$	−13.7980	−0.453184
$928$	−8.89898	−0.292123
$929$	13.7980	0.452696	0.226348	−	0.974046i	$-0.427321\pi$
$929$	13.7980	0.452696	0.226348	+	0.974046i	$0.427321\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−20.4949	−0.671333
$933$	12.2474	0.400963
$934$	−12.0000	−0.392652
$935$	0	0
$936$	39.5959	1.29423
$937$	24.5959	0.803514	0.401757	−	0.915746i	$-0.368400\pi$
$937$	24.5959	0.803514	0.401757	+	0.915746i	$0.368400\pi$
$938$	23.1464	0.755758
$939$	−3.10102	−0.101198
$940$	0	0
$941$	17.3939	0.567024	0.283512	−	0.958969i	$-0.408500\pi$
$941$	17.3939	0.567024	0.283512	+	0.958969i	$0.408500\pi$
$942$	−39.8434	−1.29817
$943$	−2.00000	−0.0651290
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	15.7980	0.513636
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	40.6969	1.32178
$949$	−30.2474	−0.981874
$950$	0	0
$951$	−49.8434	−1.61628
$952$	−3.55051	−0.115073
$953$	23.4949	0.761074	0.380537	−	0.924766i	$-0.375739\pi$
$953$	23.4949	0.761074	0.380537	+	0.924766i	$0.375739\pi$
$954$	53.3939	1.72869
$955$	0	0
$956$	−8.89898	−0.287814
$957$	44.4949	1.43832
$958$	18.2474	0.589548
$959$	48.2474	1.55799
$960$	0	0
$961$	−30.7980	−0.993483
$962$	4.44949	0.143457
$963$	119.687	3.85685
$964$	−7.44949	−0.239932
$965$	0	0
$966$	−16.8990	−0.543716
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	8.89898	0.286024
$969$	25.0000	0.803116
$970$	0	0
$971$	39.5403	1.26891	0.634454	−	0.772960i	$-0.281225\pi$
$971$	39.5403	1.26891	0.634454	+	0.772960i	$0.281225\pi$
$972$	−88.9898	−2.85435
$973$	−42.7423	−1.37026
$974$	42.7423	1.36955
$975$	0	0
$976$	−1.55051	−0.0496306
$977$	−49.7423	−1.59140	−0.795699	−	0.605692i	$-0.792896\pi$
$977$	−49.7423	−1.59140	−0.795699	+	0.605692i	$0.792896\pi$
$978$	−68.6413	−2.19491
$979$	0.505103	0.0161431
$980$	0	0
$981$	124.586	3.97772
$982$	22.2020	0.708496
$983$	30.4949	0.972636	0.486318	−	0.873782i	$-0.338340\pi$
$983$	30.4949	0.972636	0.486318	+	0.873782i	$0.338340\pi$
$984$	3.44949	0.109966
$985$	0	0
$986$	−12.8990	−0.410787
$987$	−82.7878	−2.63516
$988$	22.2474	0.707786
$989$	−21.7980	−0.693135
$990$	0	0
$991$	−11.5505	−0.366914	−0.183457	−	0.983028i	$-0.558729\pi$
$991$	−11.5505	−0.366914	−0.183457	+	0.983028i	$0.558729\pi$
$992$	0.449490	0.0142713
$993$	−34.8434	−1.10572
$994$	30.4949	0.967239
$995$	0	0
$996$	−5.00000	−0.158431
$997$	−44.4495	−1.40773	−0.703865	−	0.710334i	$-0.748544\pi$
$997$	−44.4495	−1.40773	−0.703865	+	0.710334i	$0.748544\pi$
$998$	−22.0000	−0.696398
$999$	−20.3485	−0.643797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.r.1.1	✓	2
5.2	odd	4		1850.2.b.k.149.2		4
5.3	odd	4		1850.2.b.k.149.3		4
5.4	even	2		1850.2.a.w.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.r.1.1	✓	2	1.1	even	1	trivial
1850.2.a.w.1.2	yes	2	5.4	even	2
1850.2.b.k.149.2		4	5.2	odd	4
1850.2.b.k.149.3		4	5.3	odd	4

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.44949 q^{3} +1.00000 q^{4} +3.44949 q^{6} +2.44949 q^{7} -1.00000 q^{8} +8.89898 q^{9} -1.44949 q^{11} -3.44949 q^{12} -4.44949 q^{13} -2.44949 q^{14} +1.00000 q^{16} +1.44949 q^{17} -8.89898 q^{18} -5.00000 q^{19} -8.44949 q^{21} +1.44949 q^{22} -2.00000 q^{23} +3.44949 q^{24} +4.44949 q^{26} -20.3485 q^{27} +2.44949 q^{28} +8.89898 q^{29} -0.449490 q^{31} -1.00000 q^{32} +5.00000 q^{33} -1.44949 q^{34} +8.89898 q^{36} +1.00000 q^{37} +5.00000 q^{38} +15.3485 q^{39} +1.00000 q^{41} +8.44949 q^{42} +10.8990 q^{43} -1.44949 q^{44} +2.00000 q^{46} +9.79796 q^{47} -3.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} -4.44949 q^{52} -6.00000 q^{53} +20.3485 q^{54} -2.44949 q^{56} +17.2474 q^{57} -8.89898 q^{58} -2.00000 q^{59} -1.55051 q^{61} +0.449490 q^{62} +21.7980 q^{63} +1.00000 q^{64} -5.00000 q^{66} -9.44949 q^{67} +1.44949 q^{68} +6.89898 q^{69} -12.4495 q^{71} -8.89898 q^{72} +6.79796 q^{73} -1.00000 q^{74} -5.00000 q^{76} -3.55051 q^{77} -15.3485 q^{78} -11.7980 q^{79} +43.4949 q^{81} -1.00000 q^{82} +1.44949 q^{83} -8.44949 q^{84} -10.8990 q^{86} -30.6969 q^{87} +1.44949 q^{88} -0.348469 q^{89} -10.8990 q^{91} -2.00000 q^{92} +1.55051 q^{93} -9.79796 q^{94} +3.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -12.8990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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