LMFDB - Embedded newform 1850.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\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:	no (minimal twist has level 370)
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +4.89898 q^{11} -2.44949 q^{12} +4.00000 q^{13} +4.44949 q^{14} +1.00000 q^{16} -4.89898 q^{17} +3.00000 q^{18} +3.55051 q^{19} -10.8990 q^{21} +4.89898 q^{22} -8.89898 q^{23} -2.44949 q^{24} +4.00000 q^{26} +4.44949 q^{28} -1.55051 q^{31} +1.00000 q^{32} -12.0000 q^{33} -4.89898 q^{34} +3.00000 q^{36} +1.00000 q^{37} +3.55051 q^{38} -9.79796 q^{39} +2.00000 q^{41} -10.8990 q^{42} +4.00000 q^{43} +4.89898 q^{44} -8.89898 q^{46} +4.44949 q^{47} -2.44949 q^{48} +12.7980 q^{49} +12.0000 q^{51} +4.00000 q^{52} -11.7980 q^{53} +4.44949 q^{56} -8.69694 q^{57} -3.55051 q^{59} +12.0000 q^{61} -1.55051 q^{62} +13.3485 q^{63} +1.00000 q^{64} -12.0000 q^{66} -5.55051 q^{67} -4.89898 q^{68} +21.7980 q^{69} +4.89898 q^{71} +3.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} +3.55051 q^{76} +21.7980 q^{77} -9.79796 q^{78} +6.44949 q^{79} -9.00000 q^{81} +2.00000 q^{82} -9.55051 q^{83} -10.8990 q^{84} +4.00000 q^{86} +4.89898 q^{88} +15.7980 q^{89} +17.7980 q^{91} -8.89898 q^{92} +3.79796 q^{93} +4.44949 q^{94} -2.44949 q^{96} -2.00000 q^{97} +12.7980 q^{98} +14.6969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 8 q^{13} + 4 q^{14} + 2 q^{16} + 6 q^{18} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 8 q^{26} + 4 q^{28} - 8 q^{31} + 2 q^{32} - 24 q^{33} + 6 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{41} - 12 q^{42} + 8 q^{43} - 8 q^{46} + 4 q^{47} + 6 q^{49} + 24 q^{51} + 8 q^{52} - 4 q^{53} + 4 q^{56} + 12 q^{57} - 12 q^{59} + 24 q^{61} - 8 q^{62} + 12 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 24 q^{69} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 12 q^{76} + 24 q^{77} + 8 q^{79} - 18 q^{81} + 4 q^{82} - 24 q^{83} - 12 q^{84} + 8 q^{86} + 12 q^{89} + 16 q^{91} - 8 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9 + 8 * q^13 + 4 * q^14 + 2 * q^16 + 6 * q^18 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 8 * q^26 + 4 * q^28 - 8 * q^31 + 2 * q^32 - 24 * q^33 + 6 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^41 - 12 * q^42 + 8 * q^43 - 8 * q^46 + 4 * q^47 + 6 * q^49 + 24 * q^51 + 8 * q^52 - 4 * q^53 + 4 * q^56 + 12 * q^57 - 12 * q^59 + 24 * q^61 - 8 * q^62 + 12 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 24 * q^69 + 6 * q^72 + 8 * q^73 + 2 * q^74 + 12 * q^76 + 24 * q^77 + 8 * q^79 - 18 * q^81 + 4 * q^82 - 24 * q^83 - 12 * q^84 + 8 * q^86 + 12 * q^89 + 16 * q^91 - 8 * q^92 - 12 * q^93 + 4 * q^94 - 4 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.44949	−1.00000
$7$	4.44949	1.68175	0.840875	−	0.541230i	$-0.182041\pi$
$7$	4.44949	1.68175	0.840875	+	0.541230i	$0.182041\pi$
$8$	1.00000	0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	−2.44949	−0.707107
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	4.44949	1.18918
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.89898	−1.18818	−0.594089	−	0.804400i	$-0.702487\pi$
$17$	−4.89898	−1.18818	−0.594089	+	0.804400i	$0.702487\pi$
$18$	3.00000	0.707107
$19$	3.55051	0.814543	0.407271	−	0.913307i	$-0.366480\pi$
$19$	3.55051	0.814543	0.407271	+	0.913307i	$0.366480\pi$
$20$	0	0
$21$	−10.8990	−2.37835
$22$	4.89898	1.04447
$23$	−8.89898	−1.85557	−0.927783	−	0.373121i	$-0.878288\pi$
$23$	−8.89898	−1.85557	−0.927783	+	0.373121i	$0.878288\pi$
$24$	−2.44949	−0.500000
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	4.44949	0.840875
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.55051	−0.278480	−0.139240	−	0.990259i	$-0.544466\pi$
$31$	−1.55051	−0.278480	−0.139240	+	0.990259i	$0.544466\pi$
$32$	1.00000	0.176777
$33$	−12.0000	−2.08893
$34$	−4.89898	−0.840168
$35$	0	0
$36$	3.00000	0.500000
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	3.55051	0.575969
$39$	−9.79796	−1.56893
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−10.8990	−1.68175
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.89898	0.738549
$45$	0	0
$46$	−8.89898	−1.31208
$47$	4.44949	0.649025	0.324512	−	0.945881i	$-0.394800\pi$
$47$	4.44949	0.649025	0.324512	+	0.945881i	$0.394800\pi$
$48$	−2.44949	−0.353553
$49$	12.7980	1.82828
$50$	0	0
$51$	12.0000	1.68034
$52$	4.00000	0.554700
$53$	−11.7980	−1.62057	−0.810287	−	0.586033i	$-0.800689\pi$
$53$	−11.7980	−1.62057	−0.810287	+	0.586033i	$0.800689\pi$
$54$	0	0
$55$	0	0
$56$	4.44949	0.594588
$57$	−8.69694	−1.15194
$58$	0	0
$59$	−3.55051	−0.462237	−0.231119	−	0.972926i	$-0.574239\pi$
$59$	−3.55051	−0.462237	−0.231119	+	0.972926i	$0.574239\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	−1.55051	−0.196915
$63$	13.3485	1.68175
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	−5.55051	−0.678103	−0.339051	−	0.940768i	$-0.610106\pi$
$67$	−5.55051	−0.678103	−0.339051	+	0.940768i	$0.610106\pi$
$68$	−4.89898	−0.594089
$69$	21.7980	2.62417
$70$	0	0
$71$	4.89898	0.581402	0.290701	−	0.956814i	$-0.406112\pi$
$71$	4.89898	0.581402	0.290701	+	0.956814i	$0.406112\pi$
$72$	3.00000	0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	3.55051	0.407271
$77$	21.7980	2.48411
$78$	−9.79796	−1.10940
$79$	6.44949	0.725624	0.362812	−	0.931862i	$-0.381817\pi$
$79$	6.44949	0.725624	0.362812	+	0.931862i	$0.381817\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	2.00000	0.220863
$83$	−9.55051	−1.04830	−0.524152	−	0.851625i	$-0.675618\pi$
$83$	−9.55051	−1.04830	−0.524152	+	0.851625i	$0.675618\pi$
$84$	−10.8990	−1.18918
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	4.89898	0.522233
$89$	15.7980	1.67458	0.837290	−	0.546759i	$-0.184139\pi$
$89$	15.7980	1.67458	0.837290	+	0.546759i	$0.184139\pi$
$90$	0	0
$91$	17.7980	1.86573
$92$	−8.89898	−0.927783
$93$	3.79796	0.393830
$94$	4.44949	0.458930
$95$	0	0
$96$	−2.44949	−0.250000
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	12.7980	1.29279
$99$	14.6969	1.47710
$100$	0	0
$101$	10.6969	1.06439	0.532193	−	0.846623i	$-0.321368\pi$
$101$	10.6969	1.06439	0.532193	+	0.846623i	$0.321368\pi$
$102$	12.0000	1.18818
$103$	9.79796	0.965422	0.482711	−	0.875780i	$-0.339652\pi$
$103$	9.79796	0.965422	0.482711	+	0.875780i	$0.339652\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−11.7980	−1.14592
$107$	−5.55051	−0.536588	−0.268294	−	0.963337i	$-0.586460\pi$
$107$	−5.55051	−0.536588	−0.268294	+	0.963337i	$0.586460\pi$
$108$	0	0
$109$	−5.79796	−0.555344	−0.277672	−	0.960676i	$-0.589563\pi$
$109$	−5.79796	−0.555344	−0.277672	+	0.960676i	$0.589563\pi$
$110$	0	0
$111$	−2.44949	−0.232495
$112$	4.44949	0.420437
$113$	−3.10102	−0.291719	−0.145860	−	0.989305i	$-0.546595\pi$
$113$	−3.10102	−0.291719	−0.145860	+	0.989305i	$0.546595\pi$
$114$	−8.69694	−0.814543
$115$	0	0
$116$	0	0
$117$	12.0000	1.10940
$118$	−3.55051	−0.326851
$119$	−21.7980	−1.99822
$120$	0	0
$121$	13.0000	1.18182
$122$	12.0000	1.08643
$123$	−4.89898	−0.441726
$124$	−1.55051	−0.139240
$125$	0	0
$126$	13.3485	1.18918
$127$	−2.65153	−0.235285	−0.117643	−	0.993056i	$-0.537534\pi$
$127$	−2.65153	−0.235285	−0.117643	+	0.993056i	$0.537534\pi$
$128$	1.00000	0.0883883
$129$	−9.79796	−0.862662
$130$	0	0
$131$	−10.2474	−0.895324	−0.447662	−	0.894203i	$-0.647743\pi$
$131$	−10.2474	−0.895324	−0.447662	+	0.894203i	$0.647743\pi$
$132$	−12.0000	−1.04447
$133$	15.7980	1.36986
$134$	−5.55051	−0.479491
$135$	0	0
$136$	−4.89898	−0.420084
$137$	19.5959	1.67419	0.837096	−	0.547056i	$-0.184251\pi$
$137$	19.5959	1.67419	0.837096	+	0.547056i	$0.184251\pi$
$138$	21.7980	1.85557
$139$	−5.79796	−0.491776	−0.245888	−	0.969298i	$-0.579080\pi$
$139$	−5.79796	−0.491776	−0.245888	+	0.969298i	$0.579080\pi$
$140$	0	0
$141$	−10.8990	−0.917860
$142$	4.89898	0.411113
$143$	19.5959	1.63869
$144$	3.00000	0.250000
$145$	0	0
$146$	4.00000	0.331042
$147$	−31.3485	−2.58558
$148$	1.00000	0.0821995
$149$	−18.6969	−1.53171	−0.765856	−	0.643012i	$-0.777685\pi$
$149$	−18.6969	−1.53171	−0.765856	+	0.643012i	$0.777685\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	3.55051	0.287984
$153$	−14.6969	−1.18818
$154$	21.7980	1.75653
$155$	0	0
$156$	−9.79796	−0.784465
$157$	−7.79796	−0.622345	−0.311172	−	0.950353i	$-0.600722\pi$
$157$	−7.79796	−0.622345	−0.311172	+	0.950353i	$0.600722\pi$
$158$	6.44949	0.513094
$159$	28.8990	2.29184
$160$	0	0
$161$	−39.5959	−3.12060
$162$	−9.00000	−0.707107
$163$	−21.7980	−1.70735	−0.853674	−	0.520808i	$-0.825631\pi$
$163$	−21.7980	−1.70735	−0.853674	+	0.520808i	$0.825631\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−9.55051	−0.741263
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−10.8990	−0.840875
$169$	3.00000	0.230769
$170$	0	0
$171$	10.6515	0.814543
$172$	4.00000	0.304997
$173$	19.7980	1.50521	0.752605	−	0.658472i	$-0.228797\pi$
$173$	19.7980	1.50521	0.752605	+	0.658472i	$0.228797\pi$
$174$	0	0
$175$	0	0
$176$	4.89898	0.369274
$177$	8.69694	0.653702
$178$	15.7980	1.18411
$179$	−9.34847	−0.698737	−0.349369	−	0.936985i	$-0.613604\pi$
$179$	−9.34847	−0.698737	−0.349369	+	0.936985i	$0.613604\pi$
$180$	0	0
$181$	10.6969	0.795097	0.397549	−	0.917581i	$-0.369861\pi$
$181$	10.6969	0.795097	0.397549	+	0.917581i	$0.369861\pi$
$182$	17.7980	1.31927
$183$	−29.3939	−2.17286
$184$	−8.89898	−0.656041
$185$	0	0
$186$	3.79796	0.278480
$187$	−24.0000	−1.75505
$188$	4.44949	0.324512
$189$	0	0
$190$	0	0
$191$	11.3485	0.821146	0.410573	−	0.911828i	$-0.365329\pi$
$191$	11.3485	0.821146	0.410573	+	0.911828i	$0.365329\pi$
$192$	−2.44949	−0.176777
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	12.7980	0.914140
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	14.6969	1.04447
$199$	6.44949	0.457192	0.228596	−	0.973521i	$-0.426586\pi$
$199$	6.44949	0.457192	0.228596	+	0.973521i	$0.426586\pi$
$200$	0	0
$201$	13.5959	0.958982
$202$	10.6969	0.752634
$203$	0	0
$204$	12.0000	0.840168
$205$	0	0
$206$	9.79796	0.682656
$207$	−26.6969	−1.85557
$208$	4.00000	0.277350
$209$	17.3939	1.20316
$210$	0	0
$211$	−6.69694	−0.461036	−0.230518	−	0.973068i	$-0.574042\pi$
$211$	−6.69694	−0.461036	−0.230518	+	0.973068i	$0.574042\pi$
$212$	−11.7980	−0.810287
$213$	−12.0000	−0.822226
$214$	−5.55051	−0.379425
$215$	0	0
$216$	0	0
$217$	−6.89898	−0.468333
$218$	−5.79796	−0.392687
$219$	−9.79796	−0.662085
$220$	0	0
$221$	−19.5959	−1.31816
$222$	−2.44949	−0.164399
$223$	0.449490	0.0301001	0.0150500	−	0.999887i	$-0.495209\pi$
$223$	0.449490	0.0301001	0.0150500	+	0.999887i	$0.495209\pi$
$224$	4.44949	0.297294
$225$	0	0
$226$	−3.10102	−0.206277
$227$	−24.8990	−1.65260	−0.826302	−	0.563228i	$-0.809559\pi$
$227$	−24.8990	−1.65260	−0.826302	+	0.563228i	$0.809559\pi$
$228$	−8.69694	−0.575969
$229$	−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
$231$	−53.3939	−3.51306
$232$	0	0
$233$	9.79796	0.641886	0.320943	−	0.947099i	$-0.396000\pi$
$233$	9.79796	0.641886	0.320943	+	0.947099i	$0.396000\pi$
$234$	12.0000	0.784465
$235$	0	0
$236$	−3.55051	−0.231119
$237$	−15.7980	−1.02619
$238$	−21.7980	−1.41295
$239$	−18.0454	−1.16726	−0.583630	−	0.812020i	$-0.698368\pi$
$239$	−18.0454	−1.16726	−0.583630	+	0.812020i	$0.698368\pi$
$240$	0	0
$241$	7.79796	0.502311	0.251155	−	0.967947i	$-0.419189\pi$
$241$	7.79796	0.502311	0.251155	+	0.967947i	$0.419189\pi$
$242$	13.0000	0.835672
$243$	22.0454	1.41421
$244$	12.0000	0.768221
$245$	0	0
$246$	−4.89898	−0.312348
$247$	14.2020	0.903654
$248$	−1.55051	−0.0984575
$249$	23.3939	1.48253
$250$	0	0
$251$	21.3485	1.34750	0.673752	−	0.738958i	$-0.264682\pi$
$251$	21.3485	1.34750	0.673752	+	0.738958i	$0.264682\pi$
$252$	13.3485	0.840875
$253$	−43.5959	−2.74085
$254$	−2.65153	−0.166372
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.89898	−0.305590	−0.152795	−	0.988258i	$-0.548827\pi$
$257$	−4.89898	−0.305590	−0.152795	+	0.988258i	$0.548827\pi$
$258$	−9.79796	−0.609994
$259$	4.44949	0.276478
$260$	0	0
$261$	0	0
$262$	−10.2474	−0.633089
$263$	1.75255	0.108067	0.0540335	−	0.998539i	$-0.482792\pi$
$263$	1.75255	0.108067	0.0540335	+	0.998539i	$0.482792\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	15.7980	0.968635
$267$	−38.6969	−2.36821
$268$	−5.55051	−0.339051
$269$	−18.6969	−1.13997	−0.569986	−	0.821654i	$-0.693051\pi$
$269$	−18.6969	−1.13997	−0.569986	+	0.821654i	$0.693051\pi$
$270$	0	0
$271$	−32.4949	−1.97392	−0.986962	−	0.160952i	$-0.948544\pi$
$271$	−32.4949	−1.97392	−0.986962	+	0.160952i	$0.948544\pi$
$272$	−4.89898	−0.297044
$273$	−43.5959	−2.63854
$274$	19.5959	1.18383
$275$	0	0
$276$	21.7980	1.31208
$277$	19.5959	1.17740	0.588702	−	0.808350i	$-0.299639\pi$
$277$	19.5959	1.17740	0.588702	+	0.808350i	$0.299639\pi$
$278$	−5.79796	−0.347738
$279$	−4.65153	−0.278480
$280$	0	0
$281$	27.7980	1.65829	0.829144	−	0.559036i	$-0.188829\pi$
$281$	27.7980	1.65829	0.829144	+	0.559036i	$0.188829\pi$
$282$	−10.8990	−0.649025
$283$	15.5959	0.927081	0.463541	−	0.886076i	$-0.346579\pi$
$283$	15.5959	0.927081	0.463541	+	0.886076i	$0.346579\pi$
$284$	4.89898	0.290701
$285$	0	0
$286$	19.5959	1.15873
$287$	8.89898	0.525290
$288$	3.00000	0.176777
$289$	7.00000	0.411765
$290$	0	0
$291$	4.89898	0.287183
$292$	4.00000	0.234082
$293$	25.5959	1.49533	0.747665	−	0.664076i	$-0.231175\pi$
$293$	25.5959	1.49533	0.747665	+	0.664076i	$0.231175\pi$
$294$	−31.3485	−1.82828
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	−18.6969	−1.08308
$299$	−35.5959	−2.05857
$300$	0	0
$301$	17.7980	1.02586
$302$	−8.00000	−0.460348
$303$	−26.2020	−1.50527
$304$	3.55051	0.203636
$305$	0	0
$306$	−14.6969	−0.840168
$307$	13.1464	0.750306	0.375153	−	0.926963i	$-0.377590\pi$
$307$	13.1464	0.750306	0.375153	+	0.926963i	$0.377590\pi$
$308$	21.7980	1.24205
$309$	−24.0000	−1.36531
$310$	0	0
$311$	−7.34847	−0.416693	−0.208347	−	0.978055i	$-0.566808\pi$
$311$	−7.34847	−0.416693	−0.208347	+	0.978055i	$0.566808\pi$
$312$	−9.79796	−0.554700
$313$	−17.5959	−0.994580	−0.497290	−	0.867584i	$-0.665672\pi$
$313$	−17.5959	−0.994580	−0.497290	+	0.867584i	$0.665672\pi$
$314$	−7.79796	−0.440064
$315$	0	0
$316$	6.44949	0.362812
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	28.8990	1.62057
$319$	0	0
$320$	0	0
$321$	13.5959	0.758850
$322$	−39.5959	−2.20659
$323$	−17.3939	−0.967821
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−21.7980	−1.20728
$327$	14.2020	0.785375
$328$	2.00000	0.110432
$329$	19.7980	1.09150
$330$	0	0
$331$	14.2474	0.783111	0.391555	−	0.920155i	$-0.371937\pi$
$331$	14.2474	0.783111	0.391555	+	0.920155i	$0.371937\pi$
$332$	−9.55051	−0.524152
$333$	3.00000	0.164399
$334$	8.00000	0.437741
$335$	0	0
$336$	−10.8990	−0.594588
$337$	−3.59592	−0.195882	−0.0979411	−	0.995192i	$-0.531226\pi$
$337$	−3.59592	−0.195882	−0.0979411	+	0.995192i	$0.531226\pi$
$338$	3.00000	0.163178
$339$	7.59592	0.412554
$340$	0	0
$341$	−7.59592	−0.411342
$342$	10.6515	0.575969
$343$	25.7980	1.39296
$344$	4.00000	0.215666
$345$	0	0
$346$	19.7980	1.06434
$347$	2.20204	0.118212	0.0591059	−	0.998252i	$-0.481175\pi$
$347$	2.20204	0.118212	0.0591059	+	0.998252i	$0.481175\pi$
$348$	0	0
$349$	7.10102	0.380109	0.190054	−	0.981774i	$-0.439134\pi$
$349$	7.10102	0.380109	0.190054	+	0.981774i	$0.439134\pi$
$350$	0	0
$351$	0	0
$352$	4.89898	0.261116
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	8.69694	0.462237
$355$	0	0
$356$	15.7980	0.837290
$357$	53.3939	2.82590
$358$	−9.34847	−0.494082
$359$	−12.8990	−0.680782	−0.340391	−	0.940284i	$-0.610559\pi$
$359$	−12.8990	−0.680782	−0.340391	+	0.940284i	$0.610559\pi$
$360$	0	0
$361$	−6.39388	−0.336520
$362$	10.6969	0.562219
$363$	−31.8434	−1.67134
$364$	17.7980	0.932867
$365$	0	0
$366$	−29.3939	−1.53644
$367$	−2.65153	−0.138409	−0.0692044	−	0.997603i	$-0.522046\pi$
$367$	−2.65153	−0.138409	−0.0692044	+	0.997603i	$0.522046\pi$
$368$	−8.89898	−0.463891
$369$	6.00000	0.312348
$370$	0	0
$371$	−52.4949	−2.72540
$372$	3.79796	0.196915
$373$	−11.7980	−0.610875	−0.305438	−	0.952212i	$-0.598803\pi$
$373$	−11.7980	−0.610875	−0.305438	+	0.952212i	$0.598803\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	4.44949	0.229465
$377$	0	0
$378$	0	0
$379$	−31.5959	−1.62297	−0.811487	−	0.584371i	$-0.801341\pi$
$379$	−31.5959	−1.62297	−0.811487	+	0.584371i	$0.801341\pi$
$380$	0	0
$381$	6.49490	0.332744
$382$	11.3485	0.580638
$383$	−34.6969	−1.77293	−0.886465	−	0.462795i	$-0.846847\pi$
$383$	−34.6969	−1.77293	−0.886465	+	0.462795i	$0.846847\pi$
$384$	−2.44949	−0.125000
$385$	0	0
$386$	14.0000	0.712581
$387$	12.0000	0.609994
$388$	−2.00000	−0.101535
$389$	−25.7980	−1.30801	−0.654004	−	0.756491i	$-0.726912\pi$
$389$	−25.7980	−1.30801	−0.654004	+	0.756491i	$0.726912\pi$
$390$	0	0
$391$	43.5959	2.20474
$392$	12.7980	0.646395
$393$	25.1010	1.26618
$394$	−2.00000	−0.100759
$395$	0	0
$396$	14.6969	0.738549
$397$	−16.2020	−0.813157	−0.406579	−	0.913616i	$-0.633278\pi$
$397$	−16.2020	−0.813157	−0.406579	+	0.913616i	$0.633278\pi$
$398$	6.44949	0.323284
$399$	−38.6969	−1.93727
$400$	0	0
$401$	−23.7980	−1.18841	−0.594207	−	0.804312i	$-0.702534\pi$
$401$	−23.7980	−1.18841	−0.594207	+	0.804312i	$0.702534\pi$
$402$	13.5959	0.678103
$403$	−6.20204	−0.308946
$404$	10.6969	0.532193
$405$	0	0
$406$	0	0
$407$	4.89898	0.242833
$408$	12.0000	0.594089
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−48.0000	−2.36767
$412$	9.79796	0.482711
$413$	−15.7980	−0.777367
$414$	−26.6969	−1.31208
$415$	0	0
$416$	4.00000	0.196116
$417$	14.2020	0.695477
$418$	17.3939	0.850762
$419$	5.79796	0.283249	0.141624	−	0.989920i	$-0.454767\pi$
$419$	5.79796	0.283249	0.141624	+	0.989920i	$0.454767\pi$
$420$	0	0
$421$	−19.5959	−0.955047	−0.477523	−	0.878619i	$-0.658465\pi$
$421$	−19.5959	−0.955047	−0.477523	+	0.878619i	$0.658465\pi$
$422$	−6.69694	−0.326002
$423$	13.3485	0.649025
$424$	−11.7980	−0.572960
$425$	0	0
$426$	−12.0000	−0.581402
$427$	53.3939	2.58391
$428$	−5.55051	−0.268294
$429$	−48.0000	−2.31746
$430$	0	0
$431$	−15.7526	−0.758774	−0.379387	−	0.925238i	$-0.623865\pi$
$431$	−15.7526	−0.758774	−0.379387	+	0.925238i	$0.623865\pi$
$432$	0	0
$433$	21.3939	1.02812	0.514062	−	0.857753i	$-0.328140\pi$
$433$	21.3939	1.02812	0.514062	+	0.857753i	$0.328140\pi$
$434$	−6.89898	−0.331162
$435$	0	0
$436$	−5.79796	−0.277672
$437$	−31.5959	−1.51144
$438$	−9.79796	−0.468165
$439$	20.6515	0.985644	0.492822	−	0.870130i	$-0.335965\pi$
$439$	20.6515	0.985644	0.492822	+	0.870130i	$0.335965\pi$
$440$	0	0
$441$	38.3939	1.82828
$442$	−19.5959	−0.932083
$443$	24.6515	1.17123	0.585615	−	0.810589i	$-0.300853\pi$
$443$	24.6515	1.17123	0.585615	+	0.810589i	$0.300853\pi$
$444$	−2.44949	−0.116248
$445$	0	0
$446$	0.449490	0.0212840
$447$	45.7980	2.16617
$448$	4.44949	0.210219
$449$	−15.7980	−0.745552	−0.372776	−	0.927921i	$-0.621594\pi$
$449$	−15.7980	−0.745552	−0.372776	+	0.927921i	$0.621594\pi$
$450$	0	0
$451$	9.79796	0.461368
$452$	−3.10102	−0.145860
$453$	19.5959	0.920697
$454$	−24.8990	−1.16857
$455$	0	0
$456$	−8.69694	−0.407271
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	−33.7980	−1.57413	−0.787064	−	0.616871i	$-0.788400\pi$
$461$	−33.7980	−1.57413	−0.787064	+	0.616871i	$0.788400\pi$
$462$	−53.3939	−2.48411
$463$	−20.4949	−0.952479	−0.476239	−	0.879316i	$-0.658000\pi$
$463$	−20.4949	−0.952479	−0.476239	+	0.879316i	$0.658000\pi$
$464$	0	0
$465$	0	0
$466$	9.79796	0.453882
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	12.0000	0.554700
$469$	−24.6969	−1.14040
$470$	0	0
$471$	19.1010	0.880129
$472$	−3.55051	−0.163425
$473$	19.5959	0.901021
$474$	−15.7980	−0.725624
$475$	0	0
$476$	−21.7980	−0.999108
$477$	−35.3939	−1.62057
$478$	−18.0454	−0.825378
$479$	5.14643	0.235146	0.117573	−	0.993064i	$-0.462489\pi$
$479$	5.14643	0.235146	0.117573	+	0.993064i	$0.462489\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	7.79796	0.355187
$483$	96.9898	4.41319
$484$	13.0000	0.590909
$485$	0	0
$486$	22.0454	1.00000
$487$	−29.3939	−1.33196	−0.665982	−	0.745968i	$-0.731987\pi$
$487$	−29.3939	−1.33196	−0.665982	+	0.745968i	$0.731987\pi$
$488$	12.0000	0.543214
$489$	53.3939	2.41455
$490$	0	0
$491$	−9.30306	−0.419841	−0.209921	−	0.977718i	$-0.567321\pi$
$491$	−9.30306	−0.419841	−0.209921	+	0.977718i	$0.567321\pi$
$492$	−4.89898	−0.220863
$493$	0	0
$494$	14.2020	0.638980
$495$	0	0
$496$	−1.55051	−0.0696200
$497$	21.7980	0.977772
$498$	23.3939	1.04830
$499$	−10.6515	−0.476828	−0.238414	−	0.971164i	$-0.576628\pi$
$499$	−10.6515	−0.476828	−0.238414	+	0.971164i	$0.576628\pi$
$500$	0	0
$501$	−19.5959	−0.875481
$502$	21.3485	0.952829
$503$	−1.79796	−0.0801670	−0.0400835	−	0.999196i	$-0.512762\pi$
$503$	−1.79796	−0.0801670	−0.0400835	+	0.999196i	$0.512762\pi$
$504$	13.3485	0.594588
$505$	0	0
$506$	−43.5959	−1.93807
$507$	−7.34847	−0.326357
$508$	−2.65153	−0.117643
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	17.7980	0.787335
$512$	1.00000	0.0441942
$513$	0	0
$514$	−4.89898	−0.216085
$515$	0	0
$516$	−9.79796	−0.431331
$517$	21.7980	0.958673
$518$	4.44949	0.195499
$519$	−48.4949	−2.12869
$520$	0	0
$521$	17.7980	0.779743	0.389871	−	0.920869i	$-0.372519\pi$
$521$	17.7980	0.779743	0.389871	+	0.920869i	$0.372519\pi$
$522$	0	0
$523$	−8.89898	−0.389125	−0.194563	−	0.980890i	$-0.562329\pi$
$523$	−8.89898	−0.389125	−0.194563	+	0.980890i	$0.562329\pi$
$524$	−10.2474	−0.447662
$525$	0	0
$526$	1.75255	0.0764149
$527$	7.59592	0.330883
$528$	−12.0000	−0.522233
$529$	56.1918	2.44312
$530$	0	0
$531$	−10.6515	−0.462237
$532$	15.7980	0.684928
$533$	8.00000	0.346518
$534$	−38.6969	−1.67458
$535$	0	0
$536$	−5.55051	−0.239746
$537$	22.8990	0.988164
$538$	−18.6969	−0.806082
$539$	62.6969	2.70055
$540$	0	0
$541$	0.404082	0.0173728	0.00868642	−	0.999962i	$-0.497235\pi$
$541$	0.404082	0.0173728	0.00868642	+	0.999962i	$0.497235\pi$
$542$	−32.4949	−1.39578
$543$	−26.2020	−1.12444
$544$	−4.89898	−0.210042
$545$	0	0
$546$	−43.5959	−1.86573
$547$	−10.6969	−0.457368	−0.228684	−	0.973501i	$-0.573442\pi$
$547$	−10.6969	−0.457368	−0.228684	+	0.973501i	$0.573442\pi$
$548$	19.5959	0.837096
$549$	36.0000	1.53644
$550$	0	0
$551$	0	0
$552$	21.7980	0.927783
$553$	28.6969	1.22032
$554$	19.5959	0.832551
$555$	0	0
$556$	−5.79796	−0.245888
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	−4.65153	−0.196915
$559$	16.0000	0.676728
$560$	0	0
$561$	58.7878	2.48202
$562$	27.7980	1.17259
$563$	−34.6969	−1.46230	−0.731151	−	0.682216i	$-0.761016\pi$
$563$	−34.6969	−1.46230	−0.731151	+	0.682216i	$0.761016\pi$
$564$	−10.8990	−0.458930
$565$	0	0
$566$	15.5959	0.655545
$567$	−40.0454	−1.68175
$568$	4.89898	0.205557
$569$	21.5959	0.905348	0.452674	−	0.891676i	$-0.350470\pi$
$569$	21.5959	0.905348	0.452674	+	0.891676i	$0.350470\pi$
$570$	0	0
$571$	−25.3939	−1.06270	−0.531350	−	0.847152i	$-0.678315\pi$
$571$	−25.3939	−1.06270	−0.531350	+	0.847152i	$0.678315\pi$
$572$	19.5959	0.819346
$573$	−27.7980	−1.16128
$574$	8.89898	0.371436
$575$	0	0
$576$	3.00000	0.125000
$577$	−30.6969	−1.27793	−0.638965	−	0.769236i	$-0.720637\pi$
$577$	−30.6969	−1.27793	−0.638965	+	0.769236i	$0.720637\pi$
$578$	7.00000	0.291162
$579$	−34.2929	−1.42516
$580$	0	0
$581$	−42.4949	−1.76299
$582$	4.89898	0.203069
$583$	−57.7980	−2.39375
$584$	4.00000	0.165521
$585$	0	0
$586$	25.5959	1.05736
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−31.3485	−1.29279
$589$	−5.50510	−0.226834
$590$	0	0
$591$	4.89898	0.201517
$592$	1.00000	0.0410997
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	0	0
$596$	−18.6969	−0.765856
$597$	−15.7980	−0.646567
$598$	−35.5959	−1.45563
$599$	11.5959	0.473796	0.236898	−	0.971534i	$-0.423869\pi$
$599$	11.5959	0.473796	0.236898	+	0.971534i	$0.423869\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	17.7980	0.725391
$603$	−16.6515	−0.678103
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−26.2020	−1.06439
$607$	26.6969	1.08360	0.541798	−	0.840509i	$-0.317744\pi$
$607$	26.6969	1.08360	0.541798	+	0.840509i	$0.317744\pi$
$608$	3.55051	0.143992
$609$	0	0
$610$	0	0
$611$	17.7980	0.720028
$612$	−14.6969	−0.594089
$613$	39.7980	1.60742	0.803712	−	0.595018i	$-0.202855\pi$
$613$	39.7980	1.60742	0.803712	+	0.595018i	$0.202855\pi$
$614$	13.1464	0.530547
$615$	0	0
$616$	21.7980	0.878265
$617$	19.5959	0.788902	0.394451	−	0.918917i	$-0.370935\pi$
$617$	19.5959	0.788902	0.394451	+	0.918917i	$0.370935\pi$
$618$	−24.0000	−0.965422
$619$	−12.8990	−0.518454	−0.259227	−	0.965816i	$-0.583468\pi$
$619$	−12.8990	−0.518454	−0.259227	+	0.965816i	$0.583468\pi$
$620$	0	0
$621$	0	0
$622$	−7.34847	−0.294647
$623$	70.2929	2.81622
$624$	−9.79796	−0.392232
$625$	0	0
$626$	−17.5959	−0.703274
$627$	−42.6061	−1.70152
$628$	−7.79796	−0.311172
$629$	−4.89898	−0.195335
$630$	0	0
$631$	−20.2474	−0.806038	−0.403019	−	0.915192i	$-0.632039\pi$
$631$	−20.2474	−0.806038	−0.403019	+	0.915192i	$0.632039\pi$
$632$	6.44949	0.256547
$633$	16.4041	0.652004
$634$	18.0000	0.714871
$635$	0	0
$636$	28.8990	1.14592
$637$	51.1918	2.02829
$638$	0	0
$639$	14.6969	0.581402
$640$	0	0
$641$	17.7980	0.702977	0.351489	−	0.936192i	$-0.385676\pi$
$641$	17.7980	0.702977	0.351489	+	0.936192i	$0.385676\pi$
$642$	13.5959	0.536588
$643$	−23.1010	−0.911015	−0.455508	−	0.890232i	$-0.650542\pi$
$643$	−23.1010	−0.911015	−0.455508	+	0.890232i	$0.650542\pi$
$644$	−39.5959	−1.56030
$645$	0	0
$646$	−17.3939	−0.684353
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	−9.00000	−0.353553
$649$	−17.3939	−0.682769
$650$	0	0
$651$	16.8990	0.662323
$652$	−21.7980	−0.853674
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	14.2020	0.555344
$655$	0	0
$656$	2.00000	0.0780869
$657$	12.0000	0.468165
$658$	19.7980	0.771805
$659$	31.5959	1.23080	0.615401	−	0.788214i	$-0.288994\pi$
$659$	31.5959	1.23080	0.615401	+	0.788214i	$0.288994\pi$
$660$	0	0
$661$	−31.1918	−1.21322	−0.606611	−	0.794999i	$-0.707471\pi$
$661$	−31.1918	−1.21322	−0.606611	+	0.794999i	$0.707471\pi$
$662$	14.2474	0.553743
$663$	48.0000	1.86417
$664$	−9.55051	−0.370632
$665$	0	0
$666$	3.00000	0.116248
$667$	0	0
$668$	8.00000	0.309529
$669$	−1.10102	−0.0425679
$670$	0	0
$671$	58.7878	2.26948
$672$	−10.8990	−0.420437
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	−3.59592	−0.138510
$675$	0	0
$676$	3.00000	0.115385
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	7.59592	0.291719
$679$	−8.89898	−0.341511
$680$	0	0
$681$	60.9898	2.33713
$682$	−7.59592	−0.290863
$683$	15.5959	0.596761	0.298381	−	0.954447i	$-0.403554\pi$
$683$	15.5959	0.596761	0.298381	+	0.954447i	$0.403554\pi$
$684$	10.6515	0.407271
$685$	0	0
$686$	25.7980	0.984971
$687$	24.4949	0.934539
$688$	4.00000	0.152499
$689$	−47.1918	−1.79787
$690$	0	0
$691$	37.7980	1.43790	0.718951	−	0.695061i	$-0.244623\pi$
$691$	37.7980	1.43790	0.718951	+	0.695061i	$0.244623\pi$
$692$	19.7980	0.752605
$693$	65.3939	2.48411
$694$	2.20204	0.0835883
$695$	0	0
$696$	0	0
$697$	−9.79796	−0.371124
$698$	7.10102	0.268778
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−13.7980	−0.521142	−0.260571	−	0.965455i	$-0.583911\pi$
$701$	−13.7980	−0.521142	−0.260571	+	0.965455i	$0.583911\pi$
$702$	0	0
$703$	3.55051	0.133910
$704$	4.89898	0.184637
$705$	0	0
$706$	−6.00000	−0.225813
$707$	47.5959	1.79003
$708$	8.69694	0.326851
$709$	5.79796	0.217747	0.108873	−	0.994056i	$-0.465276\pi$
$709$	5.79796	0.217747	0.108873	+	0.994056i	$0.465276\pi$
$710$	0	0
$711$	19.3485	0.725624
$712$	15.7980	0.592054
$713$	13.7980	0.516738
$714$	53.3939	1.99822
$715$	0	0
$716$	−9.34847	−0.349369
$717$	44.2020	1.65076
$718$	−12.8990	−0.481386
$719$	51.5959	1.92420	0.962102	−	0.272692i	$-0.0879138\pi$
$719$	51.5959	1.92420	0.962102	+	0.272692i	$0.0879138\pi$
$720$	0	0
$721$	43.5959	1.62360
$722$	−6.39388	−0.237955
$723$	−19.1010	−0.710375
$724$	10.6969	0.397549
$725$	0	0
$726$	−31.8434	−1.18182
$727$	0.898979	0.0333413	0.0166707	−	0.999861i	$-0.494693\pi$
$727$	0.898979	0.0333413	0.0166707	+	0.999861i	$0.494693\pi$
$728$	17.7980	0.659636
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−19.5959	−0.724781
$732$	−29.3939	−1.08643
$733$	−17.5959	−0.649920	−0.324960	−	0.945728i	$-0.605351\pi$
$733$	−17.5959	−0.649920	−0.324960	+	0.945728i	$0.605351\pi$
$734$	−2.65153	−0.0978698
$735$	0	0
$736$	−8.89898	−0.328021
$737$	−27.1918	−1.00162
$738$	6.00000	0.220863
$739$	−45.7980	−1.68471	−0.842353	−	0.538927i	$-0.818830\pi$
$739$	−45.7980	−1.68471	−0.842353	+	0.538927i	$0.818830\pi$
$740$	0	0
$741$	−34.7878	−1.27796
$742$	−52.4949	−1.92715
$743$	−13.7526	−0.504532	−0.252266	−	0.967658i	$-0.581176\pi$
$743$	−13.7526	−0.504532	−0.252266	+	0.967658i	$0.581176\pi$
$744$	3.79796	0.139240
$745$	0	0
$746$	−11.7980	−0.431954
$747$	−28.6515	−1.04830
$748$	−24.0000	−0.877527
$749$	−24.6969	−0.902406
$750$	0	0
$751$	30.6969	1.12015	0.560074	−	0.828443i	$-0.310773\pi$
$751$	30.6969	1.12015	0.560074	+	0.828443i	$0.310773\pi$
$752$	4.44949	0.162256
$753$	−52.2929	−1.90566
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.5959	−1.22106	−0.610532	−	0.791991i	$-0.709044\pi$
$757$	−33.5959	−1.22106	−0.610532	+	0.791991i	$0.709044\pi$
$758$	−31.5959	−1.14762
$759$	106.788	3.87615
$760$	0	0
$761$	25.1918	0.913203	0.456602	−	0.889671i	$-0.349066\pi$
$761$	25.1918	0.913203	0.456602	+	0.889671i	$0.349066\pi$
$762$	6.49490	0.235285
$763$	−25.7980	−0.933949
$764$	11.3485	0.410573
$765$	0	0
$766$	−34.6969	−1.25365
$767$	−14.2020	−0.512806
$768$	−2.44949	−0.0883883
$769$	−35.7980	−1.29091	−0.645454	−	0.763799i	$-0.723332\pi$
$769$	−35.7980	−1.29091	−0.645454	+	0.763799i	$0.723332\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	14.0000	0.503871
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	−10.8990	−0.390999
$778$	−25.7980	−0.924902
$779$	7.10102	0.254420
$780$	0	0
$781$	24.0000	0.858788
$782$	43.5959	1.55899
$783$	0	0
$784$	12.7980	0.457070
$785$	0	0
$786$	25.1010	0.895324
$787$	−28.7423	−1.02455	−0.512277	−	0.858820i	$-0.671198\pi$
$787$	−28.7423	−1.02455	−0.512277	+	0.858820i	$0.671198\pi$
$788$	−2.00000	−0.0712470
$789$	−4.29286	−0.152830
$790$	0	0
$791$	−13.7980	−0.490599
$792$	14.6969	0.522233
$793$	48.0000	1.70453
$794$	−16.2020	−0.574989
$795$	0	0
$796$	6.44949	0.228596
$797$	−43.5959	−1.54425	−0.772123	−	0.635473i	$-0.780805\pi$
$797$	−43.5959	−1.54425	−0.772123	+	0.635473i	$0.780805\pi$
$798$	−38.6969	−1.36986
$799$	−21.7980	−0.771156
$800$	0	0
$801$	47.3939	1.67458
$802$	−23.7980	−0.840335
$803$	19.5959	0.691525
$804$	13.5959	0.479491
$805$	0	0
$806$	−6.20204	−0.218458
$807$	45.7980	1.61216
$808$	10.6969	0.376317
$809$	−33.1918	−1.16696	−0.583481	−	0.812127i	$-0.698310\pi$
$809$	−33.1918	−1.16696	−0.583481	+	0.812127i	$0.698310\pi$
$810$	0	0
$811$	26.2020	0.920078	0.460039	−	0.887899i	$-0.347835\pi$
$811$	26.2020	0.920078	0.460039	+	0.887899i	$0.347835\pi$
$812$	0	0
$813$	79.5959	2.79155
$814$	4.89898	0.171709
$815$	0	0
$816$	12.0000	0.420084
$817$	14.2020	0.496867
$818$	−10.0000	−0.349642
$819$	53.3939	1.86573
$820$	0	0
$821$	−6.40408	−0.223504	−0.111752	−	0.993736i	$-0.535646\pi$
$821$	−6.40408	−0.223504	−0.111752	+	0.993736i	$0.535646\pi$
$822$	−48.0000	−1.67419
$823$	33.3485	1.16245	0.581227	−	0.813741i	$-0.302573\pi$
$823$	33.3485	1.16245	0.581227	+	0.813741i	$0.302573\pi$
$824$	9.79796	0.341328
$825$	0	0
$826$	−15.7980	−0.549681
$827$	−36.4949	−1.26905	−0.634526	−	0.772902i	$-0.718805\pi$
$827$	−36.4949	−1.26905	−0.634526	+	0.772902i	$0.718805\pi$
$828$	−26.6969	−0.927783
$829$	8.40408	0.291886	0.145943	−	0.989293i	$-0.453378\pi$
$829$	8.40408	0.291886	0.145943	+	0.989293i	$0.453378\pi$
$830$	0	0
$831$	−48.0000	−1.66510
$832$	4.00000	0.138675
$833$	−62.6969	−2.17232
$834$	14.2020	0.491776
$835$	0	0
$836$	17.3939	0.601580
$837$	0	0
$838$	5.79796	0.200287
$839$	14.2020	0.490309	0.245154	−	0.969484i	$-0.421161\pi$
$839$	14.2020	0.490309	0.245154	+	0.969484i	$0.421161\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−19.5959	−0.675320
$843$	−68.0908	−2.34517
$844$	−6.69694	−0.230518
$845$	0	0
$846$	13.3485	0.458930
$847$	57.8434	1.98752
$848$	−11.7980	−0.405144
$849$	−38.2020	−1.31109
$850$	0	0
$851$	−8.89898	−0.305053
$852$	−12.0000	−0.411113
$853$	−37.5959	−1.28726	−0.643630	−	0.765337i	$-0.722572\pi$
$853$	−37.5959	−1.28726	−0.643630	+	0.765337i	$0.722572\pi$
$854$	53.3939	1.82710
$855$	0	0
$856$	−5.55051	−0.189713
$857$	9.59592	0.327790	0.163895	−	0.986478i	$-0.447594\pi$
$857$	9.59592	0.327790	0.163895	+	0.986478i	$0.447594\pi$
$858$	−48.0000	−1.63869
$859$	−55.1464	−1.88157	−0.940786	−	0.339001i	$-0.889911\pi$
$859$	−55.1464	−1.88157	−0.940786	+	0.339001i	$0.889911\pi$
$860$	0	0
$861$	−21.7980	−0.742872
$862$	−15.7526	−0.536534
$863$	−42.7423	−1.45497	−0.727483	−	0.686126i	$-0.759310\pi$
$863$	−42.7423	−1.45497	−0.727483	+	0.686126i	$0.759310\pi$
$864$	0	0
$865$	0	0
$866$	21.3939	0.726994
$867$	−17.1464	−0.582323
$868$	−6.89898	−0.234167
$869$	31.5959	1.07182
$870$	0	0
$871$	−22.2020	−0.752287
$872$	−5.79796	−0.196344
$873$	−6.00000	−0.203069
$874$	−31.5959	−1.06875
$875$	0	0
$876$	−9.79796	−0.331042
$877$	15.3939	0.519814	0.259907	−	0.965634i	$-0.416308\pi$
$877$	15.3939	0.519814	0.259907	+	0.965634i	$0.416308\pi$
$878$	20.6515	0.696955
$879$	−62.6969	−2.11472
$880$	0	0
$881$	−5.39388	−0.181724	−0.0908622	−	0.995863i	$-0.528962\pi$
$881$	−5.39388	−0.181724	−0.0908622	+	0.995863i	$0.528962\pi$
$882$	38.3939	1.29279
$883$	42.6969	1.43687	0.718433	−	0.695596i	$-0.244860\pi$
$883$	42.6969	1.43687	0.718433	+	0.695596i	$0.244860\pi$
$884$	−19.5959	−0.659082
$885$	0	0
$886$	24.6515	0.828184
$887$	−20.0454	−0.673059	−0.336529	−	0.941673i	$-0.609253\pi$
$887$	−20.0454	−0.673059	−0.336529	+	0.941673i	$0.609253\pi$
$888$	−2.44949	−0.0821995
$889$	−11.7980	−0.395691
$890$	0	0
$891$	−44.0908	−1.47710
$892$	0.449490	0.0150500
$893$	15.7980	0.528659
$894$	45.7980	1.53171
$895$	0	0
$896$	4.44949	0.148647
$897$	87.1918	2.91125
$898$	−15.7980	−0.527185
$899$	0	0
$900$	0	0
$901$	57.7980	1.92553
$902$	9.79796	0.326236
$903$	−43.5959	−1.45078
$904$	−3.10102	−0.103138
$905$	0	0
$906$	19.5959	0.651031
$907$	0.898979	0.0298501	0.0149251	−	0.999889i	$-0.495249\pi$
$907$	0.898979	0.0298501	0.0149251	+	0.999889i	$0.495249\pi$
$908$	−24.8990	−0.826302
$909$	32.0908	1.06439
$910$	0	0
$911$	−31.8434	−1.05502	−0.527509	−	0.849550i	$-0.676874\pi$
$911$	−31.8434	−1.05502	−0.527509	+	0.849550i	$0.676874\pi$
$912$	−8.69694	−0.287984
$913$	−46.7878	−1.54845
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−45.5959	−1.50571
$918$	0	0
$919$	25.1464	0.829504	0.414752	−	0.909934i	$-0.363868\pi$
$919$	25.1464	0.829504	0.414752	+	0.909934i	$0.363868\pi$
$920$	0	0
$921$	−32.2020	−1.06109
$922$	−33.7980	−1.11308
$923$	19.5959	0.645007
$924$	−53.3939	−1.75653
$925$	0	0
$926$	−20.4949	−0.673504
$927$	29.3939	0.965422
$928$	0	0
$929$	−1.59592	−0.0523604	−0.0261802	−	0.999657i	$-0.508334\pi$
$929$	−1.59592	−0.0523604	−0.0261802	+	0.999657i	$0.508334\pi$
$930$	0	0
$931$	45.4393	1.48921
$932$	9.79796	0.320943
$933$	18.0000	0.589294
$934$	−12.0000	−0.392652
$935$	0	0
$936$	12.0000	0.392232
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	−24.6969	−0.806384
$939$	43.1010	1.40655
$940$	0	0
$941$	−38.2929	−1.24831	−0.624156	−	0.781300i	$-0.714557\pi$
$941$	−38.2929	−1.24831	−0.624156	+	0.781300i	$0.714557\pi$
$942$	19.1010	0.622345
$943$	−17.7980	−0.579581
$944$	−3.55051	−0.115559
$945$	0	0
$946$	19.5959	0.637118
$947$	25.3939	0.825190	0.412595	−	0.910915i	$-0.364622\pi$
$947$	25.3939	0.825190	0.412595	+	0.910915i	$0.364622\pi$
$948$	−15.7980	−0.513094
$949$	16.0000	0.519382
$950$	0	0
$951$	−44.0908	−1.42974
$952$	−21.7980	−0.706476
$953$	−21.7980	−0.706105	−0.353053	−	0.935603i	$-0.614856\pi$
$953$	−21.7980	−0.706105	−0.353053	+	0.935603i	$0.614856\pi$
$954$	−35.3939	−1.14592
$955$	0	0
$956$	−18.0454	−0.583630
$957$	0	0
$958$	5.14643	0.166274
$959$	87.1918	2.81557
$960$	0	0
$961$	−28.5959	−0.922449
$962$	4.00000	0.128965
$963$	−16.6515	−0.536588
$964$	7.79796	0.251155
$965$	0	0
$966$	96.9898	3.12060
$967$	3.50510	0.112716	0.0563582	−	0.998411i	$-0.482051\pi$
$967$	3.50510	0.112716	0.0563582	+	0.998411i	$0.482051\pi$
$968$	13.0000	0.417836
$969$	42.6061	1.36871
$970$	0	0
$971$	−35.1010	−1.12645	−0.563223	−	0.826305i	$-0.690439\pi$
$971$	−35.1010	−1.12645	−0.563223	+	0.826305i	$0.690439\pi$
$972$	22.0454	0.707107
$973$	−25.7980	−0.827045
$974$	−29.3939	−0.941841
$975$	0	0
$976$	12.0000	0.384111
$977$	−10.4041	−0.332856	−0.166428	−	0.986054i	$-0.553223\pi$
$977$	−10.4041	−0.332856	−0.166428	+	0.986054i	$0.553223\pi$
$978$	53.3939	1.70735
$979$	77.3939	2.47352
$980$	0	0
$981$	−17.3939	−0.555344
$982$	−9.30306	−0.296873
$983$	4.94439	0.157701	0.0788507	−	0.996886i	$-0.474875\pi$
$983$	4.94439	0.157701	0.0788507	+	0.996886i	$0.474875\pi$
$984$	−4.89898	−0.156174
$985$	0	0
$986$	0	0
$987$	−48.4949	−1.54361
$988$	14.2020	0.451827
$989$	−35.5959	−1.13188
$990$	0	0
$991$	−31.8434	−1.01154	−0.505769	−	0.862669i	$-0.668791\pi$
$991$	−31.8434	−1.01154	−0.505769	+	0.862669i	$0.668791\pi$
$992$	−1.55051	−0.0492287
$993$	−34.8990	−1.10749
$994$	21.7980	0.691389
$995$	0	0
$996$	23.3939	0.741263
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−10.6515	−0.337168
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.v.1.1		2
5.2	odd	4		370.2.b.c.149.4	yes	4
5.3	odd	4		370.2.b.c.149.1	✓	4
5.4	even	2		1850.2.a.s.1.2		2
15.2	even	4		3330.2.d.m.1999.2		4
15.8	even	4		3330.2.d.m.1999.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.c.149.1	✓	4	5.3	odd	4
370.2.b.c.149.4	yes	4	5.2	odd	4
1850.2.a.s.1.2		2	5.4	even	2
1850.2.a.v.1.1		2	1.1	even	1	trivial
3330.2.d.m.1999.2		4	15.2	even	4
3330.2.d.m.1999.3		4	15.8	even	4

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +4.89898 q^{11} -2.44949 q^{12} +4.00000 q^{13} +4.44949 q^{14} +1.00000 q^{16} -4.89898 q^{17} +3.00000 q^{18} +3.55051 q^{19} -10.8990 q^{21} +4.89898 q^{22} -8.89898 q^{23} -2.44949 q^{24} +4.00000 q^{26} +4.44949 q^{28} -1.55051 q^{31} +1.00000 q^{32} -12.0000 q^{33} -4.89898 q^{34} +3.00000 q^{36} +1.00000 q^{37} +3.55051 q^{38} -9.79796 q^{39} +2.00000 q^{41} -10.8990 q^{42} +4.00000 q^{43} +4.89898 q^{44} -8.89898 q^{46} +4.44949 q^{47} -2.44949 q^{48} +12.7980 q^{49} +12.0000 q^{51} +4.00000 q^{52} -11.7980 q^{53} +4.44949 q^{56} -8.69694 q^{57} -3.55051 q^{59} +12.0000 q^{61} -1.55051 q^{62} +13.3485 q^{63} +1.00000 q^{64} -12.0000 q^{66} -5.55051 q^{67} -4.89898 q^{68} +21.7980 q^{69} +4.89898 q^{71} +3.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} +3.55051 q^{76} +21.7980 q^{77} -9.79796 q^{78} +6.44949 q^{79} -9.00000 q^{81} +2.00000 q^{82} -9.55051 q^{83} -10.8990 q^{84} +4.00000 q^{86} +4.89898 q^{88} +15.7980 q^{89} +17.7980 q^{91} -8.89898 q^{92} +3.79796 q^{93} +4.44949 q^{94} -2.44949 q^{96} -2.00000 q^{97} +12.7980 q^{98} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 8 q^{13} + 4 q^{14} + 2 q^{16} + 6 q^{18} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 8 q^{26} + 4 q^{28} - 8 q^{31} + 2 q^{32} - 24 q^{33} + 6 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{41} - 12 q^{42} + 8 q^{43} - 8 q^{46} + 4 q^{47} + 6 q^{49} + 24 q^{51} + 8 q^{52} - 4 q^{53} + 4 q^{56} + 12 q^{57} - 12 q^{59} + 24 q^{61} - 8 q^{62} + 12 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 24 q^{69} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 12 q^{76} + 24 q^{77} + 8 q^{79} - 18 q^{81} + 4 q^{82} - 24 q^{83} - 12 q^{84} + 8 q^{86} + 12 q^{89} + 16 q^{91} - 8 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9 + 8 * q^13 + 4 * q^14 + 2 * q^16 + 6 * q^18 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 8 * q^26 + 4 * q^28 - 8 * q^31 + 2 * q^32 - 24 * q^33 + 6 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^41 - 12 * q^42 + 8 * q^43 - 8 * q^46 + 4 * q^47 + 6 * q^49 + 24 * q^51 + 8 * q^52 - 4 * q^53 + 4 * q^56 + 12 * q^57 - 12 * q^59 + 24 * q^61 - 8 * q^62 + 12 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 24 * q^69 + 6 * q^72 + 8 * q^73 + 2 * q^74 + 12 * q^76 + 24 * q^77 + 8 * q^79 - 18 * q^81 + 4 * q^82 - 24 * q^83 - 12 * q^84 + 8 * q^86 + 12 * q^89 + 16 * q^91 - 8 * q^92 - 12 * q^93 + 4 * q^94 - 4 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

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