LMFDB - Embedded newform 1850.2.a.q.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$3.37228$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.37228 q^{11} -2.00000 q^{12} -6.74456 q^{13} -4.37228 q^{14} +1.00000 q^{16} -0.372281 q^{17} -1.00000 q^{18} -2.00000 q^{19} -8.74456 q^{21} -2.37228 q^{22} +4.74456 q^{23} +2.00000 q^{24} +6.74456 q^{26} +4.00000 q^{27} +4.37228 q^{28} -9.11684 q^{29} -8.37228 q^{31} -1.00000 q^{32} -4.74456 q^{33} +0.372281 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +13.4891 q^{39} -0.372281 q^{41} +8.74456 q^{42} -1.62772 q^{43} +2.37228 q^{44} -4.74456 q^{46} -2.74456 q^{47} -2.00000 q^{48} +12.1168 q^{49} +0.744563 q^{51} -6.74456 q^{52} +4.37228 q^{53} -4.00000 q^{54} -4.37228 q^{56} +4.00000 q^{57} +9.11684 q^{58} +1.25544 q^{59} +0.372281 q^{61} +8.37228 q^{62} +4.37228 q^{63} +1.00000 q^{64} +4.74456 q^{66} +6.74456 q^{67} -0.372281 q^{68} -9.48913 q^{69} +4.74456 q^{71} -1.00000 q^{72} +2.74456 q^{73} +1.00000 q^{74} -2.00000 q^{76} +10.3723 q^{77} -13.4891 q^{78} +6.74456 q^{79} -11.0000 q^{81} +0.372281 q^{82} -10.7446 q^{83} -8.74456 q^{84} +1.62772 q^{86} +18.2337 q^{87} -2.37228 q^{88} +10.0000 q^{89} -29.4891 q^{91} +4.74456 q^{92} +16.7446 q^{93} +2.74456 q^{94} +2.00000 q^{96} -17.1168 q^{97} -12.1168 q^{98} +2.37228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{21} + q^{22} - 2 q^{23} + 4 q^{24}+ \cdots - q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - q^11 - 4 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 - 2 * q^18 - 4 * q^19 - 6 * q^21 + q^22 - 2 * q^23 + 4 * q^24 + 2 * q^26 + 8 * q^27 + 3 * q^28 - q^29 - 11 * q^31 - 2 * q^32 + 2 * q^33 - 5 * q^34 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 + 5 * q^41 + 6 * q^42 - 9 * q^43 - q^44 + 2 * q^46 + 6 * q^47 - 4 * q^48 + 7 * q^49 - 10 * q^51 - 2 * q^52 + 3 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 + q^58 + 14 * q^59 - 5 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + 5 * q^68 + 4 * q^69 - 2 * q^71 - 2 * q^72 - 6 * q^73 + 2 * q^74 - 4 * q^76 + 15 * q^77 - 4 * q^78 + 2 * q^79 - 22 * q^81 - 5 * q^82 - 10 * q^83 - 6 * q^84 + 9 * q^86 + 2 * q^87 + q^88 + 20 * q^89 - 36 * q^91 - 2 * q^92 + 22 * q^93 - 6 * q^94 + 4 * q^96 - 17 * q^97 - 7 * q^98 - 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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	4.37228	1.65257	0.826284	−	0.563254i	$-0.190451\pi$
$7$	4.37228	1.65257	0.826284	+	0.563254i	$0.190451\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.37228	0.715270	0.357635	−	0.933862i	$-0.383583\pi$
$11$	2.37228	0.715270	0.357635	+	0.933862i	$0.383583\pi$
$12$	−2.00000	−0.577350
$13$	−6.74456	−1.87061	−0.935303	−	0.353849i	$-0.884873\pi$
$13$	−6.74456	−1.87061	−0.935303	+	0.353849i	$0.884873\pi$
$14$	−4.37228	−1.16854
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.372281	−0.0902915	−0.0451457	−	0.998980i	$-0.514375\pi$
$17$	−0.372281	−0.0902915	−0.0451457	+	0.998980i	$0.514375\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−8.74456	−1.90822
$22$	−2.37228	−0.505772
$23$	4.74456	0.989310	0.494655	−	0.869090i	$-0.335294\pi$
$23$	4.74456	0.989310	0.494655	+	0.869090i	$0.335294\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	6.74456	1.32272
$27$	4.00000	0.769800
$28$	4.37228	0.826284
$29$	−9.11684	−1.69296	−0.846478	−	0.532424i	$-0.821281\pi$
$29$	−9.11684	−1.69296	−0.846478	+	0.532424i	$0.821281\pi$
$30$	0	0
$31$	−8.37228	−1.50371	−0.751853	−	0.659331i	$-0.770840\pi$
$31$	−8.37228	−1.50371	−0.751853	+	0.659331i	$0.770840\pi$
$32$	−1.00000	−0.176777
$33$	−4.74456	−0.825922
$34$	0.372281	0.0638457
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	2.00000	0.324443
$39$	13.4891	2.15999
$40$	0	0
$41$	−0.372281	−0.0581406	−0.0290703	−	0.999577i	$-0.509255\pi$
$41$	−0.372281	−0.0581406	−0.0290703	+	0.999577i	$0.509255\pi$
$42$	8.74456	1.34932
$43$	−1.62772	−0.248225	−0.124112	−	0.992268i	$-0.539608\pi$
$43$	−1.62772	−0.248225	−0.124112	+	0.992268i	$0.539608\pi$
$44$	2.37228	0.357635
$45$	0	0
$46$	−4.74456	−0.699548
$47$	−2.74456	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$47$	−2.74456	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$48$	−2.00000	−0.288675
$49$	12.1168	1.73098
$50$	0	0
$51$	0.744563	0.104260
$52$	−6.74456	−0.935303
$53$	4.37228	0.600579	0.300290	−	0.953848i	$-0.402917\pi$
$53$	4.37228	0.600579	0.300290	+	0.953848i	$0.402917\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−4.37228	−0.584271
$57$	4.00000	0.529813
$58$	9.11684	1.19710
$59$	1.25544	0.163444	0.0817220	−	0.996655i	$-0.473958\pi$
$59$	1.25544	0.163444	0.0817220	+	0.996655i	$0.473958\pi$
$60$	0	0
$61$	0.372281	0.0476657	0.0238329	−	0.999716i	$-0.492413\pi$
$61$	0.372281	0.0476657	0.0238329	+	0.999716i	$0.492413\pi$
$62$	8.37228	1.06328
$63$	4.37228	0.550856
$64$	1.00000	0.125000
$65$	0	0
$66$	4.74456	0.584015
$67$	6.74456	0.823979	0.411990	−	0.911188i	$-0.364834\pi$
$67$	6.74456	0.823979	0.411990	+	0.911188i	$0.364834\pi$
$68$	−0.372281	−0.0451457
$69$	−9.48913	−1.14236
$70$	0	0
$71$	4.74456	0.563076	0.281538	−	0.959550i	$-0.409155\pi$
$71$	4.74456	0.563076	0.281538	+	0.959550i	$0.409155\pi$
$72$	−1.00000	−0.117851
$73$	2.74456	0.321227	0.160613	−	0.987017i	$-0.448653\pi$
$73$	2.74456	0.321227	0.160613	+	0.987017i	$0.448653\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	10.3723	1.18203
$78$	−13.4891	−1.52734
$79$	6.74456	0.758823	0.379411	−	0.925228i	$-0.376127\pi$
$79$	6.74456	0.758823	0.379411	+	0.925228i	$0.376127\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0.372281	0.0411116
$83$	−10.7446	−1.17937	−0.589684	−	0.807634i	$-0.700748\pi$
$83$	−10.7446	−1.17937	−0.589684	+	0.807634i	$0.700748\pi$
$84$	−8.74456	−0.954110
$85$	0	0
$86$	1.62772	0.175521
$87$	18.2337	1.95486
$88$	−2.37228	−0.252886
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−29.4891	−3.09130
$92$	4.74456	0.494655
$93$	16.7446	1.73633
$94$	2.74456	0.283080
$95$	0	0
$96$	2.00000	0.204124
$97$	−17.1168	−1.73795	−0.868976	−	0.494854i	$-0.835222\pi$
$97$	−17.1168	−1.73795	−0.868976	+	0.494854i	$0.835222\pi$
$98$	−12.1168	−1.22399
$99$	2.37228	0.238423
$100$	0	0
$101$	11.4891	1.14321	0.571605	−	0.820529i	$-0.306321\pi$
$101$	11.4891	1.14321	0.571605	+	0.820529i	$0.306321\pi$
$102$	−0.744563	−0.0737227
$103$	−13.4891	−1.32912	−0.664562	−	0.747234i	$-0.731382\pi$
$103$	−13.4891	−1.32912	−0.664562	+	0.747234i	$0.731382\pi$
$104$	6.74456	0.661359
$105$	0	0
$106$	−4.37228	−0.424674
$107$	−19.4891	−1.88408	−0.942042	−	0.335494i	$-0.891097\pi$
$107$	−19.4891	−1.88408	−0.942042	+	0.335494i	$0.891097\pi$
$108$	4.00000	0.384900
$109$	−17.1168	−1.63950	−0.819748	−	0.572724i	$-0.805887\pi$
$109$	−17.1168	−1.63950	−0.819748	+	0.572724i	$0.805887\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	4.37228	0.413142
$113$	−11.6277	−1.09384	−0.546922	−	0.837184i	$-0.684201\pi$
$113$	−11.6277	−1.09384	−0.546922	+	0.837184i	$0.684201\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−9.11684	−0.846478
$117$	−6.74456	−0.623535
$118$	−1.25544	−0.115572
$119$	−1.62772	−0.149213
$120$	0	0
$121$	−5.37228	−0.488389
$122$	−0.372281	−0.0337048
$123$	0.744563	0.0671350
$124$	−8.37228	−0.751853
$125$	0	0
$126$	−4.37228	−0.389514
$127$	5.25544	0.466345	0.233172	−	0.972435i	$-0.425089\pi$
$127$	5.25544	0.466345	0.233172	+	0.972435i	$0.425089\pi$
$128$	−1.00000	−0.0883883
$129$	3.25544	0.286625
$130$	0	0
$131$	−14.7446	−1.28824	−0.644119	−	0.764925i	$-0.722776\pi$
$131$	−14.7446	−1.28824	−0.644119	+	0.764925i	$0.722776\pi$
$132$	−4.74456	−0.412961
$133$	−8.74456	−0.758250
$134$	−6.74456	−0.582641
$135$	0	0
$136$	0.372281	0.0319229
$137$	−5.25544	−0.449002	−0.224501	−	0.974474i	$-0.572075\pi$
$137$	−5.25544	−0.449002	−0.224501	+	0.974474i	$0.572075\pi$
$138$	9.48913	0.807768
$139$	−19.1168	−1.62147	−0.810735	−	0.585414i	$-0.800932\pi$
$139$	−19.1168	−1.62147	−0.810735	+	0.585414i	$0.800932\pi$
$140$	0	0
$141$	5.48913	0.462268
$142$	−4.74456	−0.398155
$143$	−16.0000	−1.33799
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.74456	−0.227142
$147$	−24.2337	−1.99876
$148$	−1.00000	−0.0821995
$149$	11.4891	0.941226	0.470613	−	0.882340i	$-0.344033\pi$
$149$	11.4891	0.941226	0.470613	+	0.882340i	$0.344033\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	2.00000	0.162221
$153$	−0.372281	−0.0300972
$154$	−10.3723	−0.835822
$155$	0	0
$156$	13.4891	1.07999
$157$	−11.6277	−0.927993	−0.463996	−	0.885837i	$-0.653585\pi$
$157$	−11.6277	−0.927993	−0.463996	+	0.885837i	$0.653585\pi$
$158$	−6.74456	−0.536569
$159$	−8.74456	−0.693489
$160$	0	0
$161$	20.7446	1.63490
$162$	11.0000	0.864242
$163$	−13.6277	−1.06741	−0.533703	−	0.845672i	$-0.679200\pi$
$163$	−13.6277	−1.06741	−0.533703	+	0.845672i	$0.679200\pi$
$164$	−0.372281	−0.0290703
$165$	0	0
$166$	10.7446	0.833940
$167$	21.4891	1.66288	0.831439	−	0.555616i	$-0.187517\pi$
$167$	21.4891	1.66288	0.831439	+	0.555616i	$0.187517\pi$
$168$	8.74456	0.674658
$169$	32.4891	2.49916
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−1.62772	−0.124112
$173$	5.86141	0.445634	0.222817	−	0.974860i	$-0.428475\pi$
$173$	5.86141	0.445634	0.222817	+	0.974860i	$0.428475\pi$
$174$	−18.2337	−1.38229
$175$	0	0
$176$	2.37228	0.178817
$177$	−2.51087	−0.188729
$178$	−10.0000	−0.749532
$179$	8.23369	0.615415	0.307707	−	0.951481i	$-0.400438\pi$
$179$	8.23369	0.615415	0.307707	+	0.951481i	$0.400438\pi$
$180$	0	0
$181$	−15.4891	−1.15130	−0.575649	−	0.817697i	$-0.695250\pi$
$181$	−15.4891	−1.15130	−0.575649	+	0.817697i	$0.695250\pi$
$182$	29.4891	2.18588
$183$	−0.744563	−0.0550397
$184$	−4.74456	−0.349774
$185$	0	0
$186$	−16.7446	−1.22777
$187$	−0.883156	−0.0645828
$188$	−2.74456	−0.200168
$189$	17.4891	1.27215
$190$	0	0
$191$	24.3723	1.76352	0.881758	−	0.471702i	$-0.156360\pi$
$191$	24.3723	1.76352	0.881758	+	0.471702i	$0.156360\pi$
$192$	−2.00000	−0.144338
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	17.1168	1.22892
$195$	0	0
$196$	12.1168	0.865489
$197$	−4.51087	−0.321387	−0.160693	−	0.987004i	$-0.551373\pi$
$197$	−4.51087	−0.321387	−0.160693	+	0.987004i	$0.551373\pi$
$198$	−2.37228	−0.168591
$199$	−22.7446	−1.61232	−0.806160	−	0.591698i	$-0.798458\pi$
$199$	−22.7446	−1.61232	−0.806160	+	0.591698i	$0.798458\pi$
$200$	0	0
$201$	−13.4891	−0.951450
$202$	−11.4891	−0.808372
$203$	−39.8614	−2.79772
$204$	0.744563	0.0521298
$205$	0	0
$206$	13.4891	0.939832
$207$	4.74456	0.329770
$208$	−6.74456	−0.467651
$209$	−4.74456	−0.328188
$210$	0	0
$211$	3.11684	0.214572	0.107286	−	0.994228i	$-0.465784\pi$
$211$	3.11684	0.214572	0.107286	+	0.994228i	$0.465784\pi$
$212$	4.37228	0.300290
$213$	−9.48913	−0.650184
$214$	19.4891	1.33225
$215$	0	0
$216$	−4.00000	−0.272166
$217$	−36.6060	−2.48498
$218$	17.1168	1.15930
$219$	−5.48913	−0.370921
$220$	0	0
$221$	2.51087	0.168900
$222$	−2.00000	−0.134231
$223$	−4.37228	−0.292790	−0.146395	−	0.989226i	$-0.546767\pi$
$223$	−4.37228	−0.292790	−0.146395	+	0.989226i	$0.546767\pi$
$224$	−4.37228	−0.292135
$225$	0	0
$226$	11.6277	0.773464
$227$	5.62772	0.373525	0.186762	−	0.982405i	$-0.440201\pi$
$227$	5.62772	0.373525	0.186762	+	0.982405i	$0.440201\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−20.7446	−1.36489
$232$	9.11684	0.598550
$233$	28.2337	1.84965	0.924825	−	0.380392i	$-0.124211\pi$
$233$	28.2337	1.84965	0.924825	+	0.380392i	$0.124211\pi$
$234$	6.74456	0.440906
$235$	0	0
$236$	1.25544	0.0817220
$237$	−13.4891	−0.876213
$238$	1.62772	0.105509
$239$	−22.6060	−1.46226	−0.731129	−	0.682239i	$-0.761006\pi$
$239$	−22.6060	−1.46226	−0.731129	+	0.682239i	$0.761006\pi$
$240$	0	0
$241$	24.2337	1.56103	0.780515	−	0.625138i	$-0.214957\pi$
$241$	24.2337	1.56103	0.780515	+	0.625138i	$0.214957\pi$
$242$	5.37228	0.345343
$243$	10.0000	0.641500
$244$	0.372281	0.0238329
$245$	0	0
$246$	−0.744563	−0.0474716
$247$	13.4891	0.858292
$248$	8.37228	0.531640
$249$	21.4891	1.36182
$250$	0	0
$251$	−11.4891	−0.725187	−0.362594	−	0.931947i	$-0.618109\pi$
$251$	−11.4891	−0.725187	−0.362594	+	0.931947i	$0.618109\pi$
$252$	4.37228	0.275428
$253$	11.2554	0.707623
$254$	−5.25544	−0.329755
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.9783	−1.55810	−0.779050	−	0.626962i	$-0.784298\pi$
$257$	−24.9783	−1.55810	−0.779050	+	0.626962i	$0.784298\pi$
$258$	−3.25544	−0.202675
$259$	−4.37228	−0.271680
$260$	0	0
$261$	−9.11684	−0.564318
$262$	14.7446	0.910922
$263$	17.1168	1.05547	0.527735	−	0.849409i	$-0.323041\pi$
$263$	17.1168	1.05547	0.527735	+	0.849409i	$0.323041\pi$
$264$	4.74456	0.292008
$265$	0	0
$266$	8.74456	0.536164
$267$	−20.0000	−1.22398
$268$	6.74456	0.411990
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	4.74456	0.288212	0.144106	−	0.989562i	$-0.453969\pi$
$271$	4.74456	0.288212	0.144106	+	0.989562i	$0.453969\pi$
$272$	−0.372281	−0.0225729
$273$	58.9783	3.56953
$274$	5.25544	0.317493
$275$	0	0
$276$	−9.48913	−0.571178
$277$	10.7446	0.645578	0.322789	−	0.946471i	$-0.395380\pi$
$277$	10.7446	0.645578	0.322789	+	0.946471i	$0.395380\pi$
$278$	19.1168	1.14655
$279$	−8.37228	−0.501235
$280$	0	0
$281$	13.2554	0.790753	0.395377	−	0.918519i	$-0.370614\pi$
$281$	13.2554	0.790753	0.395377	+	0.918519i	$0.370614\pi$
$282$	−5.48913	−0.326873
$283$	17.4891	1.03962	0.519810	−	0.854282i	$-0.326003\pi$
$283$	17.4891	1.03962	0.519810	+	0.854282i	$0.326003\pi$
$284$	4.74456	0.281538
$285$	0	0
$286$	16.0000	0.946100
$287$	−1.62772	−0.0960812
$288$	−1.00000	−0.0589256
$289$	−16.8614	−0.991847
$290$	0	0
$291$	34.2337	2.00681
$292$	2.74456	0.160613
$293$	−9.86141	−0.576110	−0.288055	−	0.957614i	$-0.593009\pi$
$293$	−9.86141	−0.576110	−0.288055	+	0.957614i	$0.593009\pi$
$294$	24.2337	1.41334
$295$	0	0
$296$	1.00000	0.0581238
$297$	9.48913	0.550615
$298$	−11.4891	−0.665547
$299$	−32.0000	−1.85061
$300$	0	0
$301$	−7.11684	−0.410208
$302$	20.0000	1.15087
$303$	−22.9783	−1.32007
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0.372281	0.0212819
$307$	0.510875	0.0291572	0.0145786	−	0.999894i	$-0.495359\pi$
$307$	0.510875	0.0291572	0.0145786	+	0.999894i	$0.495359\pi$
$308$	10.3723	0.591016
$309$	26.9783	1.53474
$310$	0	0
$311$	4.37228	0.247929	0.123965	−	0.992287i	$-0.460439\pi$
$311$	4.37228	0.247929	0.123965	+	0.992287i	$0.460439\pi$
$312$	−13.4891	−0.763671
$313$	−19.4891	−1.10159	−0.550795	−	0.834640i	$-0.685675\pi$
$313$	−19.4891	−1.10159	−0.550795	+	0.834640i	$0.685675\pi$
$314$	11.6277	0.656190
$315$	0	0
$316$	6.74456	0.379411
$317$	−19.6277	−1.10240	−0.551201	−	0.834372i	$-0.685830\pi$
$317$	−19.6277	−1.10240	−0.551201	+	0.834372i	$0.685830\pi$
$318$	8.74456	0.490371
$319$	−21.6277	−1.21092
$320$	0	0
$321$	38.9783	2.17555
$322$	−20.7446	−1.15605
$323$	0.744563	0.0414286
$324$	−11.0000	−0.611111
$325$	0	0
$326$	13.6277	0.754770
$327$	34.2337	1.89313
$328$	0.372281	0.0205558
$329$	−12.0000	−0.661581
$330$	0	0
$331$	0.510875	0.0280802	0.0140401	−	0.999901i	$-0.495531\pi$
$331$	0.510875	0.0280802	0.0140401	+	0.999901i	$0.495531\pi$
$332$	−10.7446	−0.589684
$333$	−1.00000	−0.0547997
$334$	−21.4891	−1.17583
$335$	0	0
$336$	−8.74456	−0.477055
$337$	18.7446	1.02108	0.510541	−	0.859854i	$-0.329445\pi$
$337$	18.7446	1.02108	0.510541	+	0.859854i	$0.329445\pi$
$338$	−32.4891	−1.76718
$339$	23.2554	1.26306
$340$	0	0
$341$	−19.8614	−1.07556
$342$	2.00000	0.108148
$343$	22.3723	1.20799
$344$	1.62772	0.0877607
$345$	0	0
$346$	−5.86141	−0.315111
$347$	−22.9783	−1.23354	−0.616769	−	0.787145i	$-0.711559\pi$
$347$	−22.9783	−1.23354	−0.616769	+	0.787145i	$0.711559\pi$
$348$	18.2337	0.977428
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	−26.9783	−1.43999
$352$	−2.37228	−0.126443
$353$	−5.86141	−0.311971	−0.155986	−	0.987759i	$-0.549855\pi$
$353$	−5.86141	−0.311971	−0.155986	+	0.987759i	$0.549855\pi$
$354$	2.51087	0.133451
$355$	0	0
$356$	10.0000	0.529999
$357$	3.25544	0.172296
$358$	−8.23369	−0.435164
$359$	14.9783	0.790522	0.395261	−	0.918569i	$-0.370654\pi$
$359$	14.9783	0.790522	0.395261	+	0.918569i	$0.370654\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	15.4891	0.814090
$363$	10.7446	0.563943
$364$	−29.4891	−1.54565
$365$	0	0
$366$	0.744563	0.0389189
$367$	−21.1168	−1.10229	−0.551145	−	0.834409i	$-0.685809\pi$
$367$	−21.1168	−1.10229	−0.551145	+	0.834409i	$0.685809\pi$
$368$	4.74456	0.247327
$369$	−0.372281	−0.0193802
$370$	0	0
$371$	19.1168	0.992497
$372$	16.7446	0.868165
$373$	8.51087	0.440676	0.220338	−	0.975424i	$-0.429284\pi$
$373$	8.51087	0.440676	0.220338	+	0.975424i	$0.429284\pi$
$374$	0.883156	0.0456669
$375$	0	0
$376$	2.74456	0.141540
$377$	61.4891	3.16685
$378$	−17.4891	−0.899544
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−10.5109	−0.538488
$382$	−24.3723	−1.24699
$383$	−9.48913	−0.484872	−0.242436	−	0.970167i	$-0.577946\pi$
$383$	−9.48913	−0.484872	−0.242436	+	0.970167i	$0.577946\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	2.00000	0.101797
$387$	−1.62772	−0.0827416
$388$	−17.1168	−0.868976
$389$	−13.8614	−0.702801	−0.351401	−	0.936225i	$-0.614294\pi$
$389$	−13.8614	−0.702801	−0.351401	+	0.936225i	$0.614294\pi$
$390$	0	0
$391$	−1.76631	−0.0893262
$392$	−12.1168	−0.611993
$393$	29.4891	1.48753
$394$	4.51087	0.227255
$395$	0	0
$396$	2.37228	0.119212
$397$	20.9783	1.05287	0.526434	−	0.850216i	$-0.323529\pi$
$397$	20.9783	1.05287	0.526434	+	0.850216i	$0.323529\pi$
$398$	22.7446	1.14008
$399$	17.4891	0.875551
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	13.4891	0.672776
$403$	56.4674	2.81284
$404$	11.4891	0.571605
$405$	0	0
$406$	39.8614	1.97829
$407$	−2.37228	−0.117590
$408$	−0.744563	−0.0368613
$409$	−28.2337	−1.39607	−0.698033	−	0.716066i	$-0.745941\pi$
$409$	−28.2337	−1.39607	−0.698033	+	0.716066i	$0.745941\pi$
$410$	0	0
$411$	10.5109	0.518463
$412$	−13.4891	−0.664562
$413$	5.48913	0.270102
$414$	−4.74456	−0.233183
$415$	0	0
$416$	6.74456	0.330679
$417$	38.2337	1.87231
$418$	4.74456	0.232064
$419$	−9.48913	−0.463574	−0.231787	−	0.972767i	$-0.574457\pi$
$419$	−9.48913	−0.463574	−0.231787	+	0.972767i	$0.574457\pi$
$420$	0	0
$421$	15.4891	0.754894	0.377447	−	0.926031i	$-0.376802\pi$
$421$	15.4891	0.754894	0.377447	+	0.926031i	$0.376802\pi$
$422$	−3.11684	−0.151726
$423$	−2.74456	−0.133445
$424$	−4.37228	−0.212337
$425$	0	0
$426$	9.48913	0.459750
$427$	1.62772	0.0787708
$428$	−19.4891	−0.942042
$429$	32.0000	1.54497
$430$	0	0
$431$	4.37228	0.210605	0.105303	−	0.994440i	$-0.466419\pi$
$431$	4.37228	0.210605	0.105303	+	0.994440i	$0.466419\pi$
$432$	4.00000	0.192450
$433$	32.9783	1.58483	0.792417	−	0.609980i	$-0.208823\pi$
$433$	32.9783	1.58483	0.792417	+	0.609980i	$0.208823\pi$
$434$	36.6060	1.75714
$435$	0	0
$436$	−17.1168	−0.819748
$437$	−9.48913	−0.453926
$438$	5.48913	0.262281
$439$	−7.62772	−0.364051	−0.182026	−	0.983294i	$-0.558265\pi$
$439$	−7.62772	−0.364051	−0.182026	+	0.983294i	$0.558265\pi$
$440$	0	0
$441$	12.1168	0.576993
$442$	−2.51087	−0.119430
$443$	−28.9783	−1.37680	−0.688399	−	0.725332i	$-0.741686\pi$
$443$	−28.9783	−1.37680	−0.688399	+	0.725332i	$0.741686\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	4.37228	0.207034
$447$	−22.9783	−1.08683
$448$	4.37228	0.206571
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−0.883156	−0.0415862
$452$	−11.6277	−0.546922
$453$	40.0000	1.87936
$454$	−5.62772	−0.264122
$455$	0	0
$456$	−4.00000	−0.187317
$457$	9.86141	0.461297	0.230649	−	0.973037i	$-0.425915\pi$
$457$	9.86141	0.461297	0.230649	+	0.973037i	$0.425915\pi$
$458$	10.0000	0.467269
$459$	−1.48913	−0.0695064
$460$	0	0
$461$	24.0951	1.12222	0.561110	−	0.827741i	$-0.310374\pi$
$461$	24.0951	1.12222	0.561110	+	0.827741i	$0.310374\pi$
$462$	20.7446	0.965124
$463$	−17.4891	−0.812789	−0.406394	−	0.913698i	$-0.633214\pi$
$463$	−17.4891	−0.812789	−0.406394	+	0.913698i	$0.633214\pi$
$464$	−9.11684	−0.423239
$465$	0	0
$466$	−28.2337	−1.30790
$467$	21.3505	0.987985	0.493992	−	0.869466i	$-0.335537\pi$
$467$	21.3505	0.987985	0.493992	+	0.869466i	$0.335537\pi$
$468$	−6.74456	−0.311768
$469$	29.4891	1.36168
$470$	0	0
$471$	23.2554	1.07155
$472$	−1.25544	−0.0577862
$473$	−3.86141	−0.177548
$474$	13.4891	0.619576
$475$	0	0
$476$	−1.62772	−0.0746064
$477$	4.37228	0.200193
$478$	22.6060	1.03397
$479$	14.7446	0.673696	0.336848	−	0.941559i	$-0.390639\pi$
$479$	14.7446	0.673696	0.336848	+	0.941559i	$0.390639\pi$
$480$	0	0
$481$	6.74456	0.307526
$482$	−24.2337	−1.10381
$483$	−41.4891	−1.88782
$484$	−5.37228	−0.244195
$485$	0	0
$486$	−10.0000	−0.453609
$487$	−19.7228	−0.893726	−0.446863	−	0.894602i	$-0.647459\pi$
$487$	−19.7228	−0.893726	−0.446863	+	0.894602i	$0.647459\pi$
$488$	−0.372281	−0.0168524
$489$	27.2554	1.23253
$490$	0	0
$491$	−30.9783	−1.39803	−0.699014	−	0.715108i	$-0.746378\pi$
$491$	−30.9783	−1.39803	−0.699014	+	0.715108i	$0.746378\pi$
$492$	0.744563	0.0335675
$493$	3.39403	0.152859
$494$	−13.4891	−0.606904
$495$	0	0
$496$	−8.37228	−0.375927
$497$	20.7446	0.930521
$498$	−21.4891	−0.962951
$499$	20.9783	0.939115	0.469558	−	0.882902i	$-0.344413\pi$
$499$	20.9783	0.939115	0.469558	+	0.882902i	$0.344413\pi$
$500$	0	0
$501$	−42.9783	−1.92013
$502$	11.4891	0.512785
$503$	−6.51087	−0.290306	−0.145153	−	0.989409i	$-0.546367\pi$
$503$	−6.51087	−0.290306	−0.145153	+	0.989409i	$0.546367\pi$
$504$	−4.37228	−0.194757
$505$	0	0
$506$	−11.2554	−0.500365
$507$	−64.9783	−2.88579
$508$	5.25544	0.233172
$509$	−22.7446	−1.00814	−0.504068	−	0.863664i	$-0.668164\pi$
$509$	−22.7446	−1.00814	−0.504068	+	0.863664i	$0.668164\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	24.9783	1.10174
$515$	0	0
$516$	3.25544	0.143313
$517$	−6.51087	−0.286348
$518$	4.37228	0.192107
$519$	−11.7228	−0.514574
$520$	0	0
$521$	−36.0951	−1.58135	−0.790677	−	0.612233i	$-0.790271\pi$
$521$	−36.0951	−1.58135	−0.790677	+	0.612233i	$0.790271\pi$
$522$	9.11684	0.399033
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−14.7446	−0.644119
$525$	0	0
$526$	−17.1168	−0.746330
$527$	3.11684	0.135772
$528$	−4.74456	−0.206481
$529$	−0.489125	−0.0212663
$530$	0	0
$531$	1.25544	0.0544813
$532$	−8.74456	−0.379125
$533$	2.51087	0.108758
$534$	20.0000	0.865485
$535$	0	0
$536$	−6.74456	−0.291321
$537$	−16.4674	−0.710620
$538$	−10.0000	−0.431131
$539$	28.7446	1.23812
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−4.74456	−0.203796
$543$	30.9783	1.32940
$544$	0.372281	0.0159614
$545$	0	0
$546$	−58.9783	−2.52404
$547$	−33.6277	−1.43782	−0.718909	−	0.695104i	$-0.755358\pi$
$547$	−33.6277	−1.43782	−0.718909	+	0.695104i	$0.755358\pi$
$548$	−5.25544	−0.224501
$549$	0.372281	0.0158886
$550$	0	0
$551$	18.2337	0.776781
$552$	9.48913	0.403884
$553$	29.4891	1.25401
$554$	−10.7446	−0.456493
$555$	0	0
$556$	−19.1168	−0.810735
$557$	2.74456	0.116291	0.0581454	−	0.998308i	$-0.481481\pi$
$557$	2.74456	0.116291	0.0581454	+	0.998308i	$0.481481\pi$
$558$	8.37228	0.354427
$559$	10.9783	0.464331
$560$	0	0
$561$	1.76631	0.0745738
$562$	−13.2554	−0.559147
$563$	30.0951	1.26836	0.634179	−	0.773187i	$-0.281338\pi$
$563$	30.0951	1.26836	0.634179	+	0.773187i	$0.281338\pi$
$564$	5.48913	0.231134
$565$	0	0
$566$	−17.4891	−0.735123
$567$	−48.0951	−2.01980
$568$	−4.74456	−0.199077
$569$	12.9783	0.544077	0.272038	−	0.962286i	$-0.412302\pi$
$569$	12.9783	0.544077	0.272038	+	0.962286i	$0.412302\pi$
$570$	0	0
$571$	12.6060	0.527543	0.263772	−	0.964585i	$-0.415033\pi$
$571$	12.6060	0.527543	0.263772	+	0.964585i	$0.415033\pi$
$572$	−16.0000	−0.668994
$573$	−48.7446	−2.03633
$574$	1.62772	0.0679397
$575$	0	0
$576$	1.00000	0.0416667
$577$	−15.4891	−0.644821	−0.322410	−	0.946600i	$-0.604493\pi$
$577$	−15.4891	−0.644821	−0.322410	+	0.946600i	$0.604493\pi$
$578$	16.8614	0.701342
$579$	4.00000	0.166234
$580$	0	0
$581$	−46.9783	−1.94899
$582$	−34.2337	−1.41903
$583$	10.3723	0.429576
$584$	−2.74456	−0.113571
$585$	0	0
$586$	9.86141	0.407371
$587$	26.8397	1.10779	0.553896	−	0.832586i	$-0.313141\pi$
$587$	26.8397	1.10779	0.553896	+	0.832586i	$0.313141\pi$
$588$	−24.2337	−0.999380
$589$	16.7446	0.689948
$590$	0	0
$591$	9.02175	0.371105
$592$	−1.00000	−0.0410997
$593$	−24.2337	−0.995158	−0.497579	−	0.867419i	$-0.665778\pi$
$593$	−24.2337	−0.995158	−0.497579	+	0.867419i	$0.665778\pi$
$594$	−9.48913	−0.389344
$595$	0	0
$596$	11.4891	0.470613
$597$	45.4891	1.86175
$598$	32.0000	1.30858
$599$	−5.48913	−0.224280	−0.112140	−	0.993692i	$-0.535770\pi$
$599$	−5.48913	−0.224280	−0.112140	+	0.993692i	$0.535770\pi$
$600$	0	0
$601$	0.372281	0.0151857	0.00759284	−	0.999971i	$-0.497583\pi$
$601$	0.372281	0.0151857	0.00759284	+	0.999971i	$0.497583\pi$
$602$	7.11684	0.290061
$603$	6.74456	0.274660
$604$	−20.0000	−0.813788
$605$	0	0
$606$	22.9783	0.933428
$607$	39.7228	1.61230	0.806150	−	0.591712i	$-0.201548\pi$
$607$	39.7228	1.61230	0.806150	+	0.591712i	$0.201548\pi$
$608$	2.00000	0.0811107
$609$	79.7228	3.23053
$610$	0	0
$611$	18.5109	0.748870
$612$	−0.372281	−0.0150486
$613$	20.0951	0.811633	0.405817	−	0.913955i	$-0.366987\pi$
$613$	20.0951	0.811633	0.405817	+	0.913955i	$0.366987\pi$
$614$	−0.510875	−0.0206172
$615$	0	0
$616$	−10.3723	−0.417911
$617$	−25.7228	−1.03556	−0.517781	−	0.855513i	$-0.673242\pi$
$617$	−25.7228	−1.03556	−0.517781	+	0.855513i	$0.673242\pi$
$618$	−26.9783	−1.08522
$619$	12.8832	0.517818	0.258909	−	0.965902i	$-0.416637\pi$
$619$	12.8832	0.517818	0.258909	+	0.965902i	$0.416637\pi$
$620$	0	0
$621$	18.9783	0.761571
$622$	−4.37228	−0.175313
$623$	43.7228	1.75172
$624$	13.4891	0.539997
$625$	0	0
$626$	19.4891	0.778942
$627$	9.48913	0.378959
$628$	−11.6277	−0.463996
$629$	0.372281	0.0148438
$630$	0	0
$631$	44.0951	1.75540	0.877699	−	0.479212i	$-0.159078\pi$
$631$	44.0951	1.75540	0.877699	+	0.479212i	$0.159078\pi$
$632$	−6.74456	−0.268284
$633$	−6.23369	−0.247767
$634$	19.6277	0.779516
$635$	0	0
$636$	−8.74456	−0.346744
$637$	−81.7228	−3.23798
$638$	21.6277	0.856250
$639$	4.74456	0.187692
$640$	0	0
$641$	−9.39403	−0.371042	−0.185521	−	0.982640i	$-0.559397\pi$
$641$	−9.39403	−0.371042	−0.185521	+	0.982640i	$0.559397\pi$
$642$	−38.9783	−1.53835
$643$	2.37228	0.0935536	0.0467768	−	0.998905i	$-0.485105\pi$
$643$	2.37228	0.0935536	0.0467768	+	0.998905i	$0.485105\pi$
$644$	20.7446	0.817450
$645$	0	0
$646$	−0.744563	−0.0292944
$647$	−9.48913	−0.373056	−0.186528	−	0.982450i	$-0.559724\pi$
$647$	−9.48913	−0.373056	−0.186528	+	0.982450i	$0.559724\pi$
$648$	11.0000	0.432121
$649$	2.97825	0.116907
$650$	0	0
$651$	73.2119	2.86940
$652$	−13.6277	−0.533703
$653$	−23.4891	−0.919201	−0.459600	−	0.888126i	$-0.652007\pi$
$653$	−23.4891	−0.919201	−0.459600	+	0.888126i	$0.652007\pi$
$654$	−34.2337	−1.33864
$655$	0	0
$656$	−0.372281	−0.0145351
$657$	2.74456	0.107076
$658$	12.0000	0.467809
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−47.3505	−1.84172	−0.920861	−	0.389891i	$-0.872513\pi$
$661$	−47.3505	−1.84172	−0.920861	+	0.389891i	$0.872513\pi$
$662$	−0.510875	−0.0198557
$663$	−5.02175	−0.195029
$664$	10.7446	0.416970
$665$	0	0
$666$	1.00000	0.0387492
$667$	−43.2554	−1.67486
$668$	21.4891	0.831439
$669$	8.74456	0.338084
$670$	0	0
$671$	0.883156	0.0340939
$672$	8.74456	0.337329
$673$	31.4891	1.21382	0.606908	−	0.794772i	$-0.292410\pi$
$673$	31.4891	1.21382	0.606908	+	0.794772i	$0.292410\pi$
$674$	−18.7446	−0.722014
$675$	0	0
$676$	32.4891	1.24958
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	−23.2554	−0.893120
$679$	−74.8397	−2.87208
$680$	0	0
$681$	−11.2554	−0.431309
$682$	19.8614	0.760533
$683$	−14.3723	−0.549940	−0.274970	−	0.961453i	$-0.588668\pi$
$683$	−14.3723	−0.549940	−0.274970	+	0.961453i	$0.588668\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−22.3723	−0.854178
$687$	20.0000	0.763048
$688$	−1.62772	−0.0620562
$689$	−29.4891	−1.12345
$690$	0	0
$691$	−29.3505	−1.11655	−0.558273	−	0.829657i	$-0.688536\pi$
$691$	−29.3505	−1.11655	−0.558273	+	0.829657i	$0.688536\pi$
$692$	5.86141	0.222817
$693$	10.3723	0.394010
$694$	22.9783	0.872242
$695$	0	0
$696$	−18.2337	−0.691146
$697$	0.138593	0.00524960
$698$	22.0000	0.832712
$699$	−56.4674	−2.13579
$700$	0	0
$701$	42.4674	1.60397	0.801985	−	0.597344i	$-0.203777\pi$
$701$	42.4674	1.60397	0.801985	+	0.597344i	$0.203777\pi$
$702$	26.9783	1.01823
$703$	2.00000	0.0754314
$704$	2.37228	0.0894087
$705$	0	0
$706$	5.86141	0.220597
$707$	50.2337	1.88923
$708$	−2.51087	−0.0943645
$709$	22.8832	0.859395	0.429697	−	0.902973i	$-0.358620\pi$
$709$	22.8832	0.859395	0.429697	+	0.902973i	$0.358620\pi$
$710$	0	0
$711$	6.74456	0.252941
$712$	−10.0000	−0.374766
$713$	−39.7228	−1.48763
$714$	−3.25544	−0.121832
$715$	0	0
$716$	8.23369	0.307707
$717$	45.2119	1.68847
$718$	−14.9783	−0.558983
$719$	−23.2554	−0.867281	−0.433641	−	0.901086i	$-0.642771\pi$
$719$	−23.2554	−0.867281	−0.433641	+	0.901086i	$0.642771\pi$
$720$	0	0
$721$	−58.9783	−2.19646
$722$	15.0000	0.558242
$723$	−48.4674	−1.80252
$724$	−15.4891	−0.575649
$725$	0	0
$726$	−10.7446	−0.398768
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	29.4891	1.09294
$729$	13.0000	0.481481
$730$	0	0
$731$	0.605969	0.0224126
$732$	−0.744563	−0.0275198
$733$	23.6277	0.872710	0.436355	−	0.899775i	$-0.356269\pi$
$733$	23.6277	0.872710	0.436355	+	0.899775i	$0.356269\pi$
$734$	21.1168	0.779437
$735$	0	0
$736$	−4.74456	−0.174887
$737$	16.0000	0.589368
$738$	0.372281	0.0137039
$739$	−8.13859	−0.299383	−0.149691	−	0.988733i	$-0.547828\pi$
$739$	−8.13859	−0.299383	−0.149691	+	0.988733i	$0.547828\pi$
$740$	0	0
$741$	−26.9783	−0.991071
$742$	−19.1168	−0.701801
$743$	−0.372281	−0.0136577	−0.00682884	−	0.999977i	$-0.502174\pi$
$743$	−0.372281	−0.0136577	−0.00682884	+	0.999977i	$0.502174\pi$
$744$	−16.7446	−0.613885
$745$	0	0
$746$	−8.51087	−0.311605
$747$	−10.7446	−0.393123
$748$	−0.883156	−0.0322914
$749$	−85.2119	−3.11358
$750$	0	0
$751$	−0.744563	−0.0271695	−0.0135847	−	0.999908i	$-0.504324\pi$
$751$	−0.744563	−0.0271695	−0.0135847	+	0.999908i	$0.504324\pi$
$752$	−2.74456	−0.100084
$753$	22.9783	0.837374
$754$	−61.4891	−2.23930
$755$	0	0
$756$	17.4891	0.636073
$757$	47.9565	1.74301	0.871504	−	0.490388i	$-0.163145\pi$
$757$	47.9565	1.74301	0.871504	+	0.490388i	$0.163145\pi$
$758$	8.00000	0.290573
$759$	−22.5109	−0.817093
$760$	0	0
$761$	−0.372281	−0.0134952	−0.00674759	−	0.999977i	$-0.502148\pi$
$761$	−0.372281	−0.0134952	−0.00674759	+	0.999977i	$0.502148\pi$
$762$	10.5109	0.380769
$763$	−74.8397	−2.70938
$764$	24.3723	0.881758
$765$	0	0
$766$	9.48913	0.342856
$767$	−8.46738	−0.305739
$768$	−2.00000	−0.0721688
$769$	11.7663	0.424304	0.212152	−	0.977237i	$-0.431953\pi$
$769$	11.7663	0.424304	0.212152	+	0.977237i	$0.431953\pi$
$770$	0	0
$771$	49.9565	1.79914
$772$	−2.00000	−0.0719816
$773$	−27.3505	−0.983730	−0.491865	−	0.870671i	$-0.663685\pi$
$773$	−27.3505	−0.983730	−0.491865	+	0.870671i	$0.663685\pi$
$774$	1.62772	0.0585071
$775$	0	0
$776$	17.1168	0.614459
$777$	8.74456	0.313709
$778$	13.8614	0.496956
$779$	0.744563	0.0266767
$780$	0	0
$781$	11.2554	0.402751
$782$	1.76631	0.0631632
$783$	−36.4674	−1.30324
$784$	12.1168	0.432744
$785$	0	0
$786$	−29.4891	−1.05184
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−4.51087	−0.160693
$789$	−34.2337	−1.21875
$790$	0	0
$791$	−50.8397	−1.80765
$792$	−2.37228	−0.0842953
$793$	−2.51087	−0.0891638
$794$	−20.9783	−0.744490
$795$	0	0
$796$	−22.7446	−0.806160
$797$	4.51087	0.159783	0.0798917	−	0.996804i	$-0.474543\pi$
$797$	4.51087	0.159783	0.0798917	+	0.996804i	$0.474543\pi$
$798$	−17.4891	−0.619108
$799$	1.02175	0.0361469
$800$	0	0
$801$	10.0000	0.353333
$802$	−10.0000	−0.353112
$803$	6.51087	0.229764
$804$	−13.4891	−0.475725
$805$	0	0
$806$	−56.4674	−1.98898
$807$	−20.0000	−0.704033
$808$	−11.4891	−0.404186
$809$	−28.5109	−1.00239	−0.501194	−	0.865335i	$-0.667106\pi$
$809$	−28.5109	−1.00239	−0.501194	+	0.865335i	$0.667106\pi$
$810$	0	0
$811$	26.5109	0.930923	0.465461	−	0.885068i	$-0.345888\pi$
$811$	26.5109	0.930923	0.465461	+	0.885068i	$0.345888\pi$
$812$	−39.8614	−1.39886
$813$	−9.48913	−0.332798
$814$	2.37228	0.0831484
$815$	0	0
$816$	0.744563	0.0260649
$817$	3.25544	0.113893
$818$	28.2337	0.987168
$819$	−29.4891	−1.03043
$820$	0	0
$821$	−21.2554	−0.741820	−0.370910	−	0.928669i	$-0.620954\pi$
$821$	−21.2554	−0.741820	−0.370910	+	0.928669i	$0.620954\pi$
$822$	−10.5109	−0.366609
$823$	−17.7228	−0.617778	−0.308889	−	0.951098i	$-0.599957\pi$
$823$	−17.7228	−0.617778	−0.308889	+	0.951098i	$0.599957\pi$
$824$	13.4891	0.469916
$825$	0	0
$826$	−5.48913	−0.190991
$827$	2.64947	0.0921310	0.0460655	−	0.998938i	$-0.485332\pi$
$827$	2.64947	0.0921310	0.0460655	+	0.998938i	$0.485332\pi$
$828$	4.74456	0.164885
$829$	11.3505	0.394220	0.197110	−	0.980381i	$-0.436844\pi$
$829$	11.3505	0.394220	0.197110	+	0.980381i	$0.436844\pi$
$830$	0	0
$831$	−21.4891	−0.745449
$832$	−6.74456	−0.233826
$833$	−4.51087	−0.156293
$834$	−38.2337	−1.32392
$835$	0	0
$836$	−4.74456	−0.164094
$837$	−33.4891	−1.15755
$838$	9.48913	0.327796
$839$	−6.51087	−0.224780	−0.112390	−	0.993664i	$-0.535851\pi$
$839$	−6.51087	−0.224780	−0.112390	+	0.993664i	$0.535851\pi$
$840$	0	0
$841$	54.1168	1.86610
$842$	−15.4891	−0.533791
$843$	−26.5109	−0.913083
$844$	3.11684	0.107286
$845$	0	0
$846$	2.74456	0.0943600
$847$	−23.4891	−0.807096
$848$	4.37228	0.150145
$849$	−34.9783	−1.20045
$850$	0	0
$851$	−4.74456	−0.162642
$852$	−9.48913	−0.325092
$853$	−12.5109	−0.428364	−0.214182	−	0.976794i	$-0.568709\pi$
$853$	−12.5109	−0.428364	−0.214182	+	0.976794i	$0.568709\pi$
$854$	−1.62772	−0.0556994
$855$	0	0
$856$	19.4891	0.666125
$857$	57.8614	1.97651	0.988254	−	0.152820i	$-0.0488356\pi$
$857$	57.8614	1.97651	0.988254	+	0.152820i	$0.0488356\pi$
$858$	−32.0000	−1.09246
$859$	37.7228	1.28709	0.643543	−	0.765410i	$-0.277464\pi$
$859$	37.7228	1.28709	0.643543	+	0.765410i	$0.277464\pi$
$860$	0	0
$861$	3.25544	0.110945
$862$	−4.37228	−0.148920
$863$	20.8397	0.709390	0.354695	−	0.934982i	$-0.384585\pi$
$863$	20.8397	0.709390	0.354695	+	0.934982i	$0.384585\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−32.9783	−1.12065
$867$	33.7228	1.14529
$868$	−36.6060	−1.24249
$869$	16.0000	0.542763
$870$	0	0
$871$	−45.4891	−1.54134
$872$	17.1168	0.579649
$873$	−17.1168	−0.579317
$874$	9.48913	0.320974
$875$	0	0
$876$	−5.48913	−0.185460
$877$	−43.6277	−1.47320	−0.736602	−	0.676327i	$-0.763571\pi$
$877$	−43.6277	−1.47320	−0.736602	+	0.676327i	$0.763571\pi$
$878$	7.62772	0.257423
$879$	19.7228	0.665234
$880$	0	0
$881$	−4.37228	−0.147306	−0.0736530	−	0.997284i	$-0.523466\pi$
$881$	−4.37228	−0.147306	−0.0736530	+	0.997284i	$0.523466\pi$
$882$	−12.1168	−0.407995
$883$	5.62772	0.189388	0.0946939	−	0.995506i	$-0.469813\pi$
$883$	5.62772	0.189388	0.0946939	+	0.995506i	$0.469813\pi$
$884$	2.51087	0.0844499
$885$	0	0
$886$	28.9783	0.973543
$887$	19.6277	0.659034	0.329517	−	0.944150i	$-0.393114\pi$
$887$	19.6277	0.659034	0.329517	+	0.944150i	$0.393114\pi$
$888$	−2.00000	−0.0671156
$889$	22.9783	0.770666
$890$	0	0
$891$	−26.0951	−0.874219
$892$	−4.37228	−0.146395
$893$	5.48913	0.183687
$894$	22.9783	0.768508
$895$	0	0
$896$	−4.37228	−0.146068
$897$	64.0000	2.13690
$898$	−18.0000	−0.600668
$899$	76.3288	2.54571
$900$	0	0
$901$	−1.62772	−0.0542272
$902$	0.883156	0.0294059
$903$	14.2337	0.473667
$904$	11.6277	0.386732
$905$	0	0
$906$	−40.0000	−1.32891
$907$	28.4674	0.945244	0.472622	−	0.881265i	$-0.343308\pi$
$907$	28.4674	0.945244	0.472622	+	0.881265i	$0.343308\pi$
$908$	5.62772	0.186762
$909$	11.4891	0.381070
$910$	0	0
$911$	52.2337	1.73058	0.865290	−	0.501272i	$-0.167134\pi$
$911$	52.2337	1.73058	0.865290	+	0.501272i	$0.167134\pi$
$912$	4.00000	0.132453
$913$	−25.4891	−0.843567
$914$	−9.86141	−0.326186
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−64.4674	−2.12890
$918$	1.48913	0.0491485
$919$	−49.7228	−1.64020	−0.820102	−	0.572217i	$-0.806083\pi$
$919$	−49.7228	−1.64020	−0.820102	+	0.572217i	$0.806083\pi$
$920$	0	0
$921$	−1.02175	−0.0336678
$922$	−24.0951	−0.793530
$923$	−32.0000	−1.05329
$924$	−20.7446	−0.682446
$925$	0	0
$926$	17.4891	0.574728
$927$	−13.4891	−0.443041
$928$	9.11684	0.299275
$929$	−32.0951	−1.05301	−0.526503	−	0.850173i	$-0.676497\pi$
$929$	−32.0951	−1.05301	−0.526503	+	0.850173i	$0.676497\pi$
$930$	0	0
$931$	−24.2337	−0.794227
$932$	28.2337	0.924825
$933$	−8.74456	−0.286284
$934$	−21.3505	−0.698611
$935$	0	0
$936$	6.74456	0.220453
$937$	8.97825	0.293307	0.146653	−	0.989188i	$-0.453150\pi$
$937$	8.97825	0.293307	0.146653	+	0.989188i	$0.453150\pi$
$938$	−29.4891	−0.962854
$939$	38.9783	1.27201
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−23.2554	−0.757703
$943$	−1.76631	−0.0575190
$944$	1.25544	0.0408610
$945$	0	0
$946$	3.86141	0.125545
$947$	26.0951	0.847977	0.423988	−	0.905668i	$-0.360630\pi$
$947$	26.0951	0.847977	0.423988	+	0.905668i	$0.360630\pi$
$948$	−13.4891	−0.438106
$949$	−18.5109	−0.600888
$950$	0	0
$951$	39.2554	1.27294
$952$	1.62772	0.0527547
$953$	−11.7663	−0.381148	−0.190574	−	0.981673i	$-0.561035\pi$
$953$	−11.7663	−0.381148	−0.190574	+	0.981673i	$0.561035\pi$
$954$	−4.37228	−0.141558
$955$	0	0
$956$	−22.6060	−0.731129
$957$	43.2554	1.39825
$958$	−14.7446	−0.476375
$959$	−22.9783	−0.742006
$960$	0	0
$961$	39.0951	1.26113
$962$	−6.74456	−0.217453
$963$	−19.4891	−0.628028
$964$	24.2337	0.780515
$965$	0	0
$966$	41.4891	1.33489
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	5.37228	0.172672
$969$	−1.48913	−0.0478376
$970$	0	0
$971$	20.6060	0.661277	0.330639	−	0.943757i	$-0.392736\pi$
$971$	20.6060	0.661277	0.330639	+	0.943757i	$0.392736\pi$
$972$	10.0000	0.320750
$973$	−83.5842	−2.67959
$974$	19.7228	0.631960
$975$	0	0
$976$	0.372281	0.0119164
$977$	29.1168	0.931530	0.465765	−	0.884908i	$-0.345779\pi$
$977$	29.1168	0.931530	0.465765	+	0.884908i	$0.345779\pi$
$978$	−27.2554	−0.871533
$979$	23.7228	0.758184
$980$	0	0
$981$	−17.1168	−0.546499
$982$	30.9783	0.988556
$983$	−2.13859	−0.0682105	−0.0341053	−	0.999418i	$-0.510858\pi$
$983$	−2.13859	−0.0682105	−0.0341053	+	0.999418i	$0.510858\pi$
$984$	−0.744563	−0.0237358
$985$	0	0
$986$	−3.39403	−0.108088
$987$	24.0000	0.763928
$988$	13.4891	0.429146
$989$	−7.72281	−0.245571
$990$	0	0
$991$	−14.6060	−0.463974	−0.231987	−	0.972719i	$-0.574523\pi$
$991$	−14.6060	−0.463974	−0.231987	+	0.972719i	$0.574523\pi$
$992$	8.37228	0.265820
$993$	−1.02175	−0.0324242
$994$	−20.7446	−0.657978
$995$	0	0
$996$	21.4891	0.680909
$997$	8.23369	0.260764	0.130382	−	0.991464i	$-0.458380\pi$
$997$	8.23369	0.260764	0.130382	+	0.991464i	$0.458380\pi$
$998$	−20.9783	−0.664055
$999$	−4.00000	−0.126554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.f.1.1	✓	2	5.4	even	2
1850.2.a.q.1.2		2	1.1	even	1	trivial
1850.2.b.m.149.2		4	5.2	odd	4
1850.2.b.m.149.3		4	5.3	odd	4
2960.2.a.o.1.2		2	20.19	odd	2
3330.2.a.bb.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.37228 q^{11} -2.00000 q^{12} -6.74456 q^{13} -4.37228 q^{14} +1.00000 q^{16} -0.372281 q^{17} -1.00000 q^{18} -2.00000 q^{19} -8.74456 q^{21} -2.37228 q^{22} +4.74456 q^{23} +2.00000 q^{24} +6.74456 q^{26} +4.00000 q^{27} +4.37228 q^{28} -9.11684 q^{29} -8.37228 q^{31} -1.00000 q^{32} -4.74456 q^{33} +0.372281 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +13.4891 q^{39} -0.372281 q^{41} +8.74456 q^{42} -1.62772 q^{43} +2.37228 q^{44} -4.74456 q^{46} -2.74456 q^{47} -2.00000 q^{48} +12.1168 q^{49} +0.744563 q^{51} -6.74456 q^{52} +4.37228 q^{53} -4.00000 q^{54} -4.37228 q^{56} +4.00000 q^{57} +9.11684 q^{58} +1.25544 q^{59} +0.372281 q^{61} +8.37228 q^{62} +4.37228 q^{63} +1.00000 q^{64} +4.74456 q^{66} +6.74456 q^{67} -0.372281 q^{68} -9.48913 q^{69} +4.74456 q^{71} -1.00000 q^{72} +2.74456 q^{73} +1.00000 q^{74} -2.00000 q^{76} +10.3723 q^{77} -13.4891 q^{78} +6.74456 q^{79} -11.0000 q^{81} +0.372281 q^{82} -10.7446 q^{83} -8.74456 q^{84} +1.62772 q^{86} +18.2337 q^{87} -2.37228 q^{88} +10.0000 q^{89} -29.4891 q^{91} +4.74456 q^{92} +16.7446 q^{93} +2.74456 q^{94} +2.00000 q^{96} -17.1168 q^{97} -12.1168 q^{98} +2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{21} + q^{22} - 2 q^{23} + 4 q^{24}+ \cdots - q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - q^11 - 4 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 - 2 * q^18 - 4 * q^19 - 6 * q^21 + q^22 - 2 * q^23 + 4 * q^24 + 2 * q^26 + 8 * q^27 + 3 * q^28 - q^29 - 11 * q^31 - 2 * q^32 + 2 * q^33 - 5 * q^34 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 + 5 * q^41 + 6 * q^42 - 9 * q^43 - q^44 + 2 * q^46 + 6 * q^47 - 4 * q^48 + 7 * q^49 - 10 * q^51 - 2 * q^52 + 3 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 + q^58 + 14 * q^59 - 5 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + 5 * q^68 + 4 * q^69 - 2 * q^71 - 2 * q^72 - 6 * q^73 + 2 * q^74 - 4 * q^76 + 15 * q^77 - 4 * q^78 + 2 * q^79 - 22 * q^81 - 5 * q^82 - 10 * q^83 - 6 * q^84 + 9 * q^86 + 2 * q^87 + q^88 + 20 * q^89 - 36 * q^91 - 2 * q^92 + 22 * q^93 - 6 * q^94 + 4 * q^96 - 17 * q^97 - 7 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more