LMFDB - Embedded newform 1850.2.a.s.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)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} +0.449490 q^{7} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} -2.44949 q^{12} -4.00000 q^{13} -0.449490 q^{14} +1.00000 q^{16} -4.89898 q^{17} -3.00000 q^{18} +8.44949 q^{19} -1.10102 q^{21} +4.89898 q^{22} -0.898979 q^{23} +2.44949 q^{24} +4.00000 q^{26} +0.449490 q^{28} -6.44949 q^{31} -1.00000 q^{32} +12.0000 q^{33} +4.89898 q^{34} +3.00000 q^{36} -1.00000 q^{37} -8.44949 q^{38} +9.79796 q^{39} +2.00000 q^{41} +1.10102 q^{42} -4.00000 q^{43} -4.89898 q^{44} +0.898979 q^{46} +0.449490 q^{47} -2.44949 q^{48} -6.79796 q^{49} +12.0000 q^{51} -4.00000 q^{52} -7.79796 q^{53} -0.449490 q^{56} -20.6969 q^{57} -8.44949 q^{59} +12.0000 q^{61} +6.44949 q^{62} +1.34847 q^{63} +1.00000 q^{64} -12.0000 q^{66} +10.4495 q^{67} -4.89898 q^{68} +2.20204 q^{69} -4.89898 q^{71} -3.00000 q^{72} -4.00000 q^{73} +1.00000 q^{74} +8.44949 q^{76} -2.20204 q^{77} -9.79796 q^{78} +1.55051 q^{79} -9.00000 q^{81} -2.00000 q^{82} +14.4495 q^{83} -1.10102 q^{84} +4.00000 q^{86} +4.89898 q^{88} -3.79796 q^{89} -1.79796 q^{91} -0.898979 q^{92} +15.7980 q^{93} -0.449490 q^{94} +2.44949 q^{96} +2.00000 q^{97} +6.79796 q^{98} -14.6969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9} - 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}+ \cdots - 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

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
$7$	0.449490	0.169891	0.0849456	−	0.996386i	$-0.472928\pi$
$7$	0.449490	0.169891	0.0849456	+	0.996386i	$0.472928\pi$
$8$	−1.00000	−0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	−2.44949	−0.707107
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−0.449490	−0.120131
$15$	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
$19$	8.44949	1.93845	0.969223	−	0.246185i	$-0.0791770\pi$
$19$	8.44949	1.93845	0.969223	+	0.246185i	$0.0791770\pi$
$20$	0	0
$21$	−1.10102	−0.240262
$22$	4.89898	1.04447
$23$	−0.898979	−0.187450	−0.0937251	−	0.995598i	$-0.529877\pi$
$23$	−0.898979	−0.187450	−0.0937251	+	0.995598i	$0.529877\pi$
$24$	2.44949	0.500000
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	0.449490	0.0849456
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−6.44949	−1.15836	−0.579181	−	0.815199i	$-0.696628\pi$
$31$	−6.44949	−1.15836	−0.579181	+	0.815199i	$0.696628\pi$
$32$	−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
$38$	−8.44949	−1.37069
$39$	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	$0.450084\pi$
$42$	1.10102	0.169891
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	0.898979	0.132547
$47$	0.449490	0.0655648	0.0327824	−	0.999463i	$-0.489563\pi$
$47$	0.449490	0.0655648	0.0327824	+	0.999463i	$0.489563\pi$
$48$	−2.44949	−0.353553
$49$	−6.79796	−0.971137
$50$	0	0
$51$	12.0000	1.68034
$52$	−4.00000	−0.554700
$53$	−7.79796	−1.07113	−0.535566	−	0.844493i	$-0.679902\pi$
$53$	−7.79796	−1.07113	−0.535566	+	0.844493i	$0.679902\pi$
$54$	0	0
$55$	0	0
$56$	−0.449490	−0.0600656
$57$	−20.6969	−2.74138
$58$	0	0
$59$	−8.44949	−1.10003	−0.550015	−	0.835155i	$-0.685378\pi$
$59$	−8.44949	−1.10003	−0.550015	+	0.835155i	$0.685378\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	6.44949	0.819086
$63$	1.34847	0.169891
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	10.4495	1.27661	0.638304	−	0.769784i	$-0.279636\pi$
$67$	10.4495	1.27661	0.638304	+	0.769784i	$0.279636\pi$
$68$	−4.89898	−0.594089
$69$	2.20204	0.265095
$70$	0	0
$71$	−4.89898	−0.581402	−0.290701	−	0.956814i	$-0.593888\pi$
$71$	−4.89898	−0.581402	−0.290701	+	0.956814i	$0.593888\pi$
$72$	−3.00000	−0.353553
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	8.44949	0.969223
$77$	−2.20204	−0.250946
$78$	−9.79796	−1.10940
$79$	1.55051	0.174446	0.0872230	−	0.996189i	$-0.472201\pi$
$79$	1.55051	0.174446	0.0872230	+	0.996189i	$0.472201\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	−2.00000	−0.220863
$83$	14.4495	1.58604	0.793019	−	0.609197i	$-0.208508\pi$
$83$	14.4495	1.58604	0.793019	+	0.609197i	$0.208508\pi$
$84$	−1.10102	−0.120131
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	4.89898	0.522233
$89$	−3.79796	−0.402583	−0.201291	−	0.979531i	$-0.564514\pi$
$89$	−3.79796	−0.402583	−0.201291	+	0.979531i	$0.564514\pi$
$90$	0	0
$91$	−1.79796	−0.188477
$92$	−0.898979	−0.0937251
$93$	15.7980	1.63817
$94$	−0.449490	−0.0463613
$95$	0	0
$96$	2.44949	0.250000
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	6.79796	0.686698
$99$	−14.6969	−1.47710
$100$	0	0
$101$	−18.6969	−1.86041	−0.930207	−	0.367034i	$-0.880373\pi$
$101$	−18.6969	−1.86041	−0.930207	+	0.367034i	$0.880373\pi$
$102$	−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
$106$	7.79796	0.757405
$107$	10.4495	1.01019	0.505095	−	0.863064i	$-0.331457\pi$
$107$	10.4495	1.01019	0.505095	+	0.863064i	$0.331457\pi$
$108$	0	0
$109$	13.7980	1.32160	0.660802	−	0.750560i	$-0.270216\pi$
$109$	13.7980	1.32160	0.660802	+	0.750560i	$0.270216\pi$
$110$	0	0
$111$	2.44949	0.232495
$112$	0.449490	0.0424728
$113$	12.8990	1.21343	0.606717	−	0.794918i	$-0.292486\pi$
$113$	12.8990	1.21343	0.606717	+	0.794918i	$0.292486\pi$
$114$	20.6969	1.93845
$115$	0	0
$116$	0	0
$117$	−12.0000	−1.10940
$118$	8.44949	0.777839
$119$	−2.20204	−0.201861
$120$	0	0
$121$	13.0000	1.18182
$122$	−12.0000	−1.08643
$123$	−4.89898	−0.441726
$124$	−6.44949	−0.579181
$125$	0	0
$126$	−1.34847	−0.120131
$127$	17.3485	1.53943	0.769714	−	0.638389i	$-0.220399\pi$
$127$	17.3485	1.53943	0.769714	+	0.638389i	$0.220399\pi$
$128$	−1.00000	−0.0883883
$129$	9.79796	0.862662
$130$	0	0
$131$	14.2474	1.24481	0.622403	−	0.782697i	$-0.286157\pi$
$131$	14.2474	1.24481	0.622403	+	0.782697i	$0.286157\pi$
$132$	12.0000	1.04447
$133$	3.79796	0.329325
$134$	−10.4495	−0.902698
$135$	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	$0.184251\pi$
$138$	−2.20204	−0.187450
$139$	13.7980	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$139$	13.7980	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$140$	0	0
$141$	−1.10102	−0.0927227
$142$	4.89898	0.411113
$143$	19.5959	1.63869
$144$	3.00000	0.250000
$145$	0	0
$146$	4.00000	0.331042
$147$	16.6515	1.37340
$148$	−1.00000	−0.0821995
$149$	10.6969	0.876327	0.438164	−	0.898895i	$-0.355629\pi$
$149$	10.6969	0.876327	0.438164	+	0.898895i	$0.355629\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−8.44949	−0.685344
$153$	−14.6969	−1.18818
$154$	2.20204	0.177446
$155$	0	0
$156$	9.79796	0.784465
$157$	−11.7980	−0.941580	−0.470790	−	0.882245i	$-0.656031\pi$
$157$	−11.7980	−0.941580	−0.470790	+	0.882245i	$0.656031\pi$
$158$	−1.55051	−0.123352
$159$	19.1010	1.51481
$160$	0	0
$161$	−0.404082	−0.0318461
$162$	9.00000	0.707107
$163$	2.20204	0.172477	0.0862386	−	0.996275i	$-0.472515\pi$
$163$	2.20204	0.172477	0.0862386	+	0.996275i	$0.472515\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−14.4495	−1.12150
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	1.10102	0.0849456
$169$	3.00000	0.230769
$170$	0	0
$171$	25.3485	1.93845
$172$	−4.00000	−0.304997
$173$	−0.202041	−0.0153609	−0.00768045	−	0.999971i	$-0.502445\pi$
$173$	−0.202041	−0.0153609	−0.00768045	+	0.999971i	$0.502445\pi$
$174$	0	0
$175$	0	0
$176$	−4.89898	−0.369274
$177$	20.6969	1.55568
$178$	3.79796	0.284669
$179$	5.34847	0.399763	0.199882	−	0.979820i	$-0.435944\pi$
$179$	5.34847	0.399763	0.199882	+	0.979820i	$0.435944\pi$
$180$	0	0
$181$	−18.6969	−1.38973	−0.694866	−	0.719139i	$-0.744536\pi$
$181$	−18.6969	−1.38973	−0.694866	+	0.719139i	$0.744536\pi$
$182$	1.79796	0.133274
$183$	−29.3939	−2.17286
$184$	0.898979	0.0662736
$185$	0	0
$186$	−15.7980	−1.15836
$187$	24.0000	1.75505
$188$	0.449490	0.0327824
$189$	0	0
$190$	0	0
$191$	−3.34847	−0.242287	−0.121143	−	0.992635i	$-0.538656\pi$
$191$	−3.34847	−0.242287	−0.121143	+	0.992635i	$0.538656\pi$
$192$	−2.44949	−0.176777
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−6.79796	−0.485568
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	14.6969	1.04447
$199$	1.55051	0.109913	0.0549564	−	0.998489i	$-0.482498\pi$
$199$	1.55051	0.109913	0.0549564	+	0.998489i	$0.482498\pi$
$200$	0	0
$201$	−25.5959	−1.80540
$202$	18.6969	1.31551
$203$	0	0
$204$	12.0000	0.840168
$205$	0	0
$206$	−9.79796	−0.682656
$207$	−2.69694	−0.187450
$208$	−4.00000	−0.277350
$209$	−41.3939	−2.86327
$210$	0	0
$211$	22.6969	1.56252	0.781261	−	0.624205i	$-0.214577\pi$
$211$	22.6969	1.56252	0.781261	+	0.624205i	$0.214577\pi$
$212$	−7.79796	−0.535566
$213$	12.0000	0.822226
$214$	−10.4495	−0.714312
$215$	0	0
$216$	0	0
$217$	−2.89898	−0.196796
$218$	−13.7980	−0.934516
$219$	9.79796	0.662085
$220$	0	0
$221$	19.5959	1.31816
$222$	−2.44949	−0.164399
$223$	4.44949	0.297960	0.148980	−	0.988840i	$-0.452401\pi$
$223$	4.44949	0.297960	0.148980	+	0.988840i	$0.452401\pi$
$224$	−0.449490	−0.0300328
$225$	0	0
$226$	−12.8990	−0.858027
$227$	15.1010	1.00229	0.501145	−	0.865363i	$-0.332912\pi$
$227$	15.1010	1.00229	0.501145	+	0.865363i	$0.332912\pi$
$228$	−20.6969	−1.37069
$229$	−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
$231$	5.39388	0.354891
$232$	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
$236$	−8.44949	−0.550015
$237$	−3.79796	−0.246704
$238$	2.20204	0.142737
$239$	26.0454	1.68474	0.842369	−	0.538902i	$-0.181161\pi$
$239$	26.0454	1.68474	0.842369	+	0.538902i	$0.181161\pi$
$240$	0	0
$241$	−11.7980	−0.759973	−0.379987	−	0.924992i	$-0.624071\pi$
$241$	−11.7980	−0.759973	−0.379987	+	0.924992i	$0.624071\pi$
$242$	−13.0000	−0.835672
$243$	22.0454	1.41421
$244$	12.0000	0.768221
$245$	0	0
$246$	4.89898	0.312348
$247$	−33.7980	−2.15051
$248$	6.44949	0.409543
$249$	−35.3939	−2.24300
$250$	0	0
$251$	6.65153	0.419841	0.209920	−	0.977718i	$-0.432679\pi$
$251$	6.65153	0.419841	0.209920	+	0.977718i	$0.432679\pi$
$252$	1.34847	0.0849456
$253$	4.40408	0.276882
$254$	−17.3485	−1.08854
$255$	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
$259$	−0.449490	−0.0279299
$260$	0	0
$261$	0	0
$262$	−14.2474	−0.880210
$263$	−26.2474	−1.61849	−0.809244	−	0.587473i	$-0.800123\pi$
$263$	−26.2474	−1.61849	−0.809244	+	0.587473i	$0.800123\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−3.79796	−0.232868
$267$	9.30306	0.569338
$268$	10.4495	0.638304
$269$	10.6969	0.652204	0.326102	−	0.945335i	$-0.394265\pi$
$269$	10.6969	0.652204	0.326102	+	0.945335i	$0.394265\pi$
$270$	0	0
$271$	16.4949	1.00199	0.500997	−	0.865449i	$-0.332967\pi$
$271$	16.4949	1.00199	0.500997	+	0.865449i	$0.332967\pi$
$272$	−4.89898	−0.297044
$273$	4.40408	0.266547
$274$	−19.5959	−1.18383
$275$	0	0
$276$	2.20204	0.132547
$277$	19.5959	1.17740	0.588702	−	0.808350i	$-0.299639\pi$
$277$	19.5959	1.17740	0.588702	+	0.808350i	$0.299639\pi$
$278$	−13.7980	−0.827547
$279$	−19.3485	−1.15836
$280$	0	0
$281$	8.20204	0.489293	0.244646	−	0.969612i	$-0.421328\pi$
$281$	8.20204	0.489293	0.244646	+	0.969612i	$0.421328\pi$
$282$	1.10102	0.0655648
$283$	23.5959	1.40263	0.701316	−	0.712851i	$-0.252596\pi$
$283$	23.5959	1.40263	0.701316	+	0.712851i	$0.252596\pi$
$284$	−4.89898	−0.290701
$285$	0	0
$286$	−19.5959	−1.15873
$287$	0.898979	0.0530651
$288$	−3.00000	−0.176777
$289$	7.00000	0.411765
$290$	0	0
$291$	−4.89898	−0.287183
$292$	−4.00000	−0.234082
$293$	13.5959	0.794282	0.397141	−	0.917758i	$-0.370002\pi$
$293$	13.5959	0.794282	0.397141	+	0.917758i	$0.370002\pi$
$294$	−16.6515	−0.971137
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	−10.6969	−0.619657
$299$	3.59592	0.207957
$300$	0	0
$301$	−1.79796	−0.103633
$302$	8.00000	0.460348
$303$	45.7980	2.63102
$304$	8.44949	0.484611
$305$	0	0
$306$	14.6969	0.840168
$307$	21.1464	1.20689	0.603445	−	0.797404i	$-0.293794\pi$
$307$	21.1464	1.20689	0.603445	+	0.797404i	$0.293794\pi$
$308$	−2.20204	−0.125473
$309$	−24.0000	−1.36531
$310$	0	0
$311$	7.34847	0.416693	0.208347	−	0.978055i	$-0.433192\pi$
$311$	7.34847	0.416693	0.208347	+	0.978055i	$0.433192\pi$
$312$	−9.79796	−0.554700
$313$	−21.5959	−1.22067	−0.610337	−	0.792142i	$-0.708966\pi$
$313$	−21.5959	−1.22067	−0.610337	+	0.792142i	$0.708966\pi$
$314$	11.7980	0.665797
$315$	0	0
$316$	1.55051	0.0872230
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−19.1010	−1.07113
$319$	0	0
$320$	0	0
$321$	−25.5959	−1.42862
$322$	0.404082	0.0225186
$323$	−41.3939	−2.30322
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−2.20204	−0.121960
$327$	−33.7980	−1.86903
$328$	−2.00000	−0.110432
$329$	0.202041	0.0111389
$330$	0	0
$331$	−10.2474	−0.563251	−0.281625	−	0.959524i	$-0.590874\pi$
$331$	−10.2474	−0.563251	−0.281625	+	0.959524i	$0.590874\pi$
$332$	14.4495	0.793019
$333$	−3.00000	−0.164399
$334$	8.00000	0.437741
$335$	0	0
$336$	−1.10102	−0.0600656
$337$	−35.5959	−1.93903	−0.969517	−	0.245026i	$-0.921204\pi$
$337$	−35.5959	−1.93903	−0.969517	+	0.245026i	$0.921204\pi$
$338$	−3.00000	−0.163178
$339$	−31.5959	−1.71605
$340$	0	0
$341$	31.5959	1.71101
$342$	−25.3485	−1.37069
$343$	−6.20204	−0.334879
$344$	4.00000	0.215666
$345$	0	0
$346$	0.202041	0.0108618
$347$	−21.7980	−1.17018	−0.585088	−	0.810970i	$-0.698940\pi$
$347$	−21.7980	−1.17018	−0.585088	+	0.810970i	$0.698940\pi$
$348$	0	0
$349$	16.8990	0.904582	0.452291	−	0.891871i	$-0.350607\pi$
$349$	16.8990	0.904582	0.452291	+	0.891871i	$0.350607\pi$
$350$	0	0
$351$	0	0
$352$	4.89898	0.261116
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−20.6969	−1.10003
$355$	0	0
$356$	−3.79796	−0.201291
$357$	5.39388	0.285474
$358$	−5.34847	−0.282675
$359$	−3.10102	−0.163666	−0.0818328	−	0.996646i	$-0.526077\pi$
$359$	−3.10102	−0.163666	−0.0818328	+	0.996646i	$0.526077\pi$
$360$	0	0
$361$	52.3939	2.75757
$362$	18.6969	0.982689
$363$	−31.8434	−1.67134
$364$	−1.79796	−0.0942387
$365$	0	0
$366$	29.3939	1.53644
$367$	17.3485	0.905583	0.452791	−	0.891617i	$-0.350428\pi$
$367$	17.3485	0.905583	0.452791	+	0.891617i	$0.350428\pi$
$368$	−0.898979	−0.0468625
$369$	6.00000	0.312348
$370$	0	0
$371$	−3.50510	−0.181976
$372$	15.7980	0.819086
$373$	−7.79796	−0.403763	−0.201882	−	0.979410i	$-0.564706\pi$
$373$	−7.79796	−0.403763	−0.201882	+	0.979410i	$0.564706\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	−0.449490	−0.0231807
$377$	0	0
$378$	0	0
$379$	7.59592	0.390176	0.195088	−	0.980786i	$-0.437501\pi$
$379$	7.59592	0.390176	0.195088	+	0.980786i	$0.437501\pi$
$380$	0	0
$381$	−42.4949	−2.17708
$382$	3.34847	0.171323
$383$	5.30306	0.270974	0.135487	−	0.990779i	$-0.456740\pi$
$383$	5.30306	0.270974	0.135487	+	0.990779i	$0.456740\pi$
$384$	2.44949	0.125000
$385$	0	0
$386$	14.0000	0.712581
$387$	−12.0000	−0.609994
$388$	2.00000	0.101535
$389$	−6.20204	−0.314456	−0.157228	−	0.987562i	$-0.550256\pi$
$389$	−6.20204	−0.314456	−0.157228	+	0.987562i	$0.550256\pi$
$390$	0	0
$391$	4.40408	0.222724
$392$	6.79796	0.343349
$393$	−34.8990	−1.76042
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−14.6969	−0.738549
$397$	35.7980	1.79665	0.898324	−	0.439334i	$-0.144785\pi$
$397$	35.7980	1.79665	0.898324	+	0.439334i	$0.144785\pi$
$398$	−1.55051	−0.0777201
$399$	−9.30306	−0.465736
$400$	0	0
$401$	−4.20204	−0.209840	−0.104920	−	0.994481i	$-0.533459\pi$
$401$	−4.20204	−0.209840	−0.104920	+	0.994481i	$0.533459\pi$
$402$	25.5959	1.27661
$403$	25.7980	1.28509
$404$	−18.6969	−0.930207
$405$	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
$413$	−3.79796	−0.186885
$414$	2.69694	0.132547
$415$	0	0
$416$	4.00000	0.196116
$417$	−33.7980	−1.65509
$418$	41.3939	2.02464
$419$	−13.7980	−0.674074	−0.337037	−	0.941491i	$-0.609425\pi$
$419$	−13.7980	−0.674074	−0.337037	+	0.941491i	$0.609425\pi$
$420$	0	0
$421$	19.5959	0.955047	0.477523	−	0.878619i	$-0.341535\pi$
$421$	19.5959	0.955047	0.477523	+	0.878619i	$0.341535\pi$
$422$	−22.6969	−1.10487
$423$	1.34847	0.0655648
$424$	7.79796	0.378702
$425$	0	0
$426$	−12.0000	−0.581402
$427$	5.39388	0.261028
$428$	10.4495	0.505095
$429$	−48.0000	−2.31746
$430$	0	0
$431$	−40.2474	−1.93865	−0.969326	−	0.245780i	$-0.920956\pi$
$431$	−40.2474	−1.93865	−0.969326	+	0.245780i	$0.920956\pi$
$432$	0	0
$433$	37.3939	1.79704	0.898518	−	0.438938i	$-0.144645\pi$
$433$	37.3939	1.79704	0.898518	+	0.438938i	$0.144645\pi$
$434$	2.89898	0.139155
$435$	0	0
$436$	13.7980	0.660802
$437$	−7.59592	−0.363362
$438$	−9.79796	−0.468165
$439$	35.3485	1.68709	0.843545	−	0.537058i	$-0.180464\pi$
$439$	35.3485	1.68709	0.843545	+	0.537058i	$0.180464\pi$
$440$	0	0
$441$	−20.3939	−0.971137
$442$	−19.5959	−0.932083
$443$	−39.3485	−1.86950	−0.934751	−	0.355303i	$-0.884378\pi$
$443$	−39.3485	−1.86950	−0.934751	+	0.355303i	$0.884378\pi$
$444$	2.44949	0.116248
$445$	0	0
$446$	−4.44949	−0.210689
$447$	−26.2020	−1.23931
$448$	0.449490	0.0212364
$449$	3.79796	0.179237	0.0896184	−	0.995976i	$-0.471435\pi$
$449$	3.79796	0.179237	0.0896184	+	0.995976i	$0.471435\pi$
$450$	0	0
$451$	−9.79796	−0.461368
$452$	12.8990	0.606717
$453$	19.5959	0.920697
$454$	−15.1010	−0.708726
$455$	0	0
$456$	20.6969	0.969223
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	−14.2020	−0.661455	−0.330727	−	0.943726i	$-0.607294\pi$
$461$	−14.2020	−0.661455	−0.330727	+	0.943726i	$0.607294\pi$
$462$	−5.39388	−0.250946
$463$	−28.4949	−1.32427	−0.662135	−	0.749384i	$-0.730350\pi$
$463$	−28.4949	−1.32427	−0.662135	+	0.749384i	$0.730350\pi$
$464$	0	0
$465$	0	0
$466$	−9.79796	−0.453882
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−12.0000	−0.554700
$469$	4.69694	0.216884
$470$	0	0
$471$	28.8990	1.33159
$472$	8.44949	0.388919
$473$	19.5959	0.901021
$474$	3.79796	0.174446
$475$	0	0
$476$	−2.20204	−0.100930
$477$	−23.3939	−1.07113
$478$	−26.0454	−1.19129
$479$	−29.1464	−1.33173	−0.665867	−	0.746070i	$-0.731938\pi$
$479$	−29.1464	−1.33173	−0.665867	+	0.746070i	$0.731938\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	11.7980	0.537382
$483$	0.989795	0.0450372
$484$	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
$489$	−5.39388	−0.243920
$490$	0	0
$491$	−38.6969	−1.74637	−0.873184	−	0.487390i	$-0.837949\pi$
$491$	−38.6969	−1.74637	−0.873184	+	0.487390i	$0.837949\pi$
$492$	−4.89898	−0.220863
$493$	0	0
$494$	33.7980	1.52064
$495$	0	0
$496$	−6.44949	−0.289591
$497$	−2.20204	−0.0987750
$498$	35.3939	1.58604
$499$	−25.3485	−1.13475	−0.567377	−	0.823458i	$-0.692042\pi$
$499$	−25.3485	−1.13475	−0.567377	+	0.823458i	$0.692042\pi$
$500$	0	0
$501$	19.5959	0.875481
$502$	−6.65153	−0.296872
$503$	−17.7980	−0.793572	−0.396786	−	0.917911i	$-0.629874\pi$
$503$	−17.7980	−0.793572	−0.396786	+	0.917911i	$0.629874\pi$
$504$	−1.34847	−0.0600656
$505$	0	0
$506$	−4.40408	−0.195785
$507$	−7.34847	−0.326357
$508$	17.3485	0.769714
$509$	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
$511$	−1.79796	−0.0795370
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.89898	0.216085
$515$	0	0
$516$	9.79796	0.431331
$517$	−2.20204	−0.0968457
$518$	0.449490	0.0197494
$519$	0.494897	0.0217236
$520$	0	0
$521$	−1.79796	−0.0787700	−0.0393850	−	0.999224i	$-0.512540\pi$
$521$	−1.79796	−0.0787700	−0.0393850	+	0.999224i	$0.512540\pi$
$522$	0	0
$523$	−0.898979	−0.0393096	−0.0196548	−	0.999807i	$-0.506257\pi$
$523$	−0.898979	−0.0393096	−0.0196548	+	0.999807i	$0.506257\pi$
$524$	14.2474	0.622403
$525$	0	0
$526$	26.2474	1.14444
$527$	31.5959	1.37634
$528$	12.0000	0.522233
$529$	−22.1918	−0.964862
$530$	0	0
$531$	−25.3485	−1.10003
$532$	3.79796	0.164662
$533$	−8.00000	−0.346518
$534$	−9.30306	−0.402583
$535$	0	0
$536$	−10.4495	−0.451349
$537$	−13.1010	−0.565351
$538$	−10.6969	−0.461178
$539$	33.3031	1.43446
$540$	0	0
$541$	39.5959	1.70236	0.851181	−	0.524873i	$-0.175887\pi$
$541$	39.5959	1.70236	0.851181	+	0.524873i	$0.175887\pi$
$542$	−16.4949	−0.708517
$543$	45.7980	1.96538
$544$	4.89898	0.210042
$545$	0	0
$546$	−4.40408	−0.188477
$547$	−18.6969	−0.799423	−0.399712	−	0.916641i	$-0.630890\pi$
$547$	−18.6969	−0.799423	−0.399712	+	0.916641i	$0.630890\pi$
$548$	19.5959	0.837096
$549$	36.0000	1.53644
$550$	0	0
$551$	0	0
$552$	−2.20204	−0.0937251
$553$	0.696938	0.0296368
$554$	−19.5959	−0.832551
$555$	0	0
$556$	13.7980	0.585164
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	19.3485	0.819086
$559$	16.0000	0.676728
$560$	0	0
$561$	−58.7878	−2.48202
$562$	−8.20204	−0.345982
$563$	5.30306	0.223497	0.111749	−	0.993736i	$-0.464355\pi$
$563$	5.30306	0.223497	0.111749	+	0.993736i	$0.464355\pi$
$564$	−1.10102	−0.0463613
$565$	0	0
$566$	−23.5959	−0.991810
$567$	−4.04541	−0.169891
$568$	4.89898	0.205557
$569$	−17.5959	−0.737659	−0.368830	−	0.929497i	$-0.620241\pi$
$569$	−17.5959	−0.737659	−0.368830	+	0.929497i	$0.620241\pi$
$570$	0	0
$571$	33.3939	1.39749	0.698745	−	0.715371i	$-0.253742\pi$
$571$	33.3939	1.39749	0.698745	+	0.715371i	$0.253742\pi$
$572$	19.5959	0.819346
$573$	8.20204	0.342645
$574$	−0.898979	−0.0375227
$575$	0	0
$576$	3.00000	0.125000
$577$	1.30306	0.0542472	0.0271236	−	0.999632i	$-0.491365\pi$
$577$	1.30306	0.0542472	0.0271236	+	0.999632i	$0.491365\pi$
$578$	−7.00000	−0.291162
$579$	34.2929	1.42516
$580$	0	0
$581$	6.49490	0.269454
$582$	4.89898	0.203069
$583$	38.2020	1.58217
$584$	4.00000	0.165521
$585$	0	0
$586$	−13.5959	−0.561642
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	16.6515	0.686698
$589$	−54.4949	−2.24542
$590$	0	0
$591$	−4.89898	−0.201517
$592$	−1.00000	−0.0410997
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	10.6969	0.438164
$597$	−3.79796	−0.155440
$598$	−3.59592	−0.147048
$599$	−27.5959	−1.12754	−0.563769	−	0.825932i	$-0.690649\pi$
$599$	−27.5959	−1.12754	−0.563769	+	0.825932i	$0.690649\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	1.79796	0.0732793
$603$	31.3485	1.27661
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−45.7980	−1.86041
$607$	2.69694	0.109465	0.0547327	−	0.998501i	$-0.482569\pi$
$607$	2.69694	0.109465	0.0547327	+	0.998501i	$0.482569\pi$
$608$	−8.44949	−0.342672
$609$	0	0
$610$	0	0
$611$	−1.79796	−0.0727376
$612$	−14.6969	−0.594089
$613$	−20.2020	−0.815953	−0.407976	−	0.912992i	$-0.633765\pi$
$613$	−20.2020	−0.815953	−0.407976	+	0.912992i	$0.633765\pi$
$614$	−21.1464	−0.853400
$615$	0	0
$616$	2.20204	0.0887228
$617$	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
$619$	−3.10102	−0.124641	−0.0623203	−	0.998056i	$-0.519850\pi$
$619$	−3.10102	−0.124641	−0.0623203	+	0.998056i	$0.519850\pi$
$620$	0	0
$621$	0	0
$622$	−7.34847	−0.294647
$623$	−1.70714	−0.0683953
$624$	9.79796	0.392232
$625$	0	0
$626$	21.5959	0.863146
$627$	101.394	4.04928
$628$	−11.7980	−0.470790
$629$	4.89898	0.195335
$630$	0	0
$631$	4.24745	0.169088	0.0845441	−	0.996420i	$-0.473057\pi$
$631$	4.24745	0.169088	0.0845441	+	0.996420i	$0.473057\pi$
$632$	−1.55051	−0.0616760
$633$	−55.5959	−2.20974
$634$	18.0000	0.714871
$635$	0	0
$636$	19.1010	0.757405
$637$	27.1918	1.07738
$638$	0	0
$639$	−14.6969	−0.581402
$640$	0	0
$641$	−1.79796	−0.0710151	−0.0355076	−	0.999369i	$-0.511305\pi$
$641$	−1.79796	−0.0710151	−0.0355076	+	0.999369i	$0.511305\pi$
$642$	25.5959	1.01019
$643$	32.8990	1.29741	0.648705	−	0.761040i	$-0.275311\pi$
$643$	32.8990	1.29741	0.648705	+	0.761040i	$0.275311\pi$
$644$	−0.404082	−0.0159231
$645$	0	0
$646$	41.3939	1.62862
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	9.00000	0.353553
$649$	41.3939	1.62485
$650$	0	0
$651$	7.10102	0.278311
$652$	2.20204	0.0862386
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	33.7980	1.32160
$655$	0	0
$656$	2.00000	0.0780869
$657$	−12.0000	−0.468165
$658$	−0.202041	−0.00787638
$659$	−7.59592	−0.295895	−0.147947	−	0.988995i	$-0.547267\pi$
$659$	−7.59592	−0.295895	−0.147947	+	0.988995i	$0.547267\pi$
$660$	0	0
$661$	47.1918	1.83555	0.917775	−	0.397101i	$-0.129984\pi$
$661$	47.1918	1.83555	0.917775	+	0.397101i	$0.129984\pi$
$662$	10.2474	0.398278
$663$	−48.0000	−1.86417
$664$	−14.4495	−0.560749
$665$	0	0
$666$	3.00000	0.116248
$667$	0	0
$668$	−8.00000	−0.309529
$669$	−10.8990	−0.421379
$670$	0	0
$671$	−58.7878	−2.26948
$672$	1.10102	0.0424728
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	35.5959	1.37110
$675$	0	0
$676$	3.00000	0.115385
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	31.5959	1.21343
$679$	0.898979	0.0344997
$680$	0	0
$681$	−36.9898	−1.41745
$682$	−31.5959	−1.20987
$683$	23.5959	0.902873	0.451436	−	0.892303i	$-0.350912\pi$
$683$	23.5959	0.902873	0.451436	+	0.892303i	$0.350912\pi$
$684$	25.3485	0.969223
$685$	0	0
$686$	6.20204	0.236795
$687$	24.4949	0.934539
$688$	−4.00000	−0.152499
$689$	31.1918	1.18831
$690$	0	0
$691$	18.2020	0.692438	0.346219	−	0.938154i	$-0.387465\pi$
$691$	18.2020	0.692438	0.346219	+	0.938154i	$0.387465\pi$
$692$	−0.202041	−0.00768045
$693$	−6.60612	−0.250946
$694$	21.7980	0.827439
$695$	0	0
$696$	0	0
$697$	−9.79796	−0.371124
$698$	−16.8990	−0.639636
$699$	−24.0000	−0.907763
$700$	0	0
$701$	5.79796	0.218986	0.109493	−	0.993988i	$-0.465077\pi$
$701$	5.79796	0.218986	0.109493	+	0.993988i	$0.465077\pi$
$702$	0	0
$703$	−8.44949	−0.318679
$704$	−4.89898	−0.184637
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−8.40408	−0.316068
$708$	20.6969	0.777839
$709$	−13.7980	−0.518193	−0.259097	−	0.965851i	$-0.583425\pi$
$709$	−13.7980	−0.518193	−0.259097	+	0.965851i	$0.583425\pi$
$710$	0	0
$711$	4.65153	0.174446
$712$	3.79796	0.142335
$713$	5.79796	0.217135
$714$	−5.39388	−0.201861
$715$	0	0
$716$	5.34847	0.199882
$717$	−63.7980	−2.38258
$718$	3.10102	0.115729
$719$	12.4041	0.462594	0.231297	−	0.972883i	$-0.425703\pi$
$719$	12.4041	0.462594	0.231297	+	0.972883i	$0.425703\pi$
$720$	0	0
$721$	4.40408	0.164017
$722$	−52.3939	−1.94990
$723$	28.8990	1.07476
$724$	−18.6969	−0.694866
$725$	0	0
$726$	31.8434	1.18182
$727$	8.89898	0.330045	0.165022	−	0.986290i	$-0.447230\pi$
$727$	8.89898	0.330045	0.165022	+	0.986290i	$0.447230\pi$
$728$	1.79796	0.0666368
$729$	−27.0000	−1.00000
$730$	0	0
$731$	19.5959	0.724781
$732$	−29.3939	−1.08643
$733$	−21.5959	−0.797663	−0.398832	−	0.917024i	$-0.630584\pi$
$733$	−21.5959	−0.797663	−0.398832	+	0.917024i	$0.630584\pi$
$734$	−17.3485	−0.640344
$735$	0	0
$736$	0.898979	0.0331368
$737$	−51.1918	−1.88568
$738$	−6.00000	−0.220863
$739$	−26.2020	−0.963858	−0.481929	−	0.876210i	$-0.660064\pi$
$739$	−26.2020	−0.963858	−0.481929	+	0.876210i	$0.660064\pi$
$740$	0	0
$741$	82.7878	3.04128
$742$	3.50510	0.128676
$743$	38.2474	1.40316	0.701581	−	0.712589i	$-0.252478\pi$
$743$	38.2474	1.40316	0.701581	+	0.712589i	$0.252478\pi$
$744$	−15.7980	−0.579181
$745$	0	0
$746$	7.79796	0.285504
$747$	43.3485	1.58604
$748$	24.0000	0.877527
$749$	4.69694	0.171622
$750$	0	0
$751$	1.30306	0.0475494	0.0237747	−	0.999717i	$-0.492432\pi$
$751$	1.30306	0.0475494	0.0237747	+	0.999717i	$0.492432\pi$
$752$	0.449490	0.0163912
$753$	−16.2929	−0.593745
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−5.59592	−0.203387	−0.101694	−	0.994816i	$-0.532426\pi$
$757$	−5.59592	−0.203387	−0.101694	+	0.994816i	$0.532426\pi$
$758$	−7.59592	−0.275896
$759$	−10.7878	−0.391571
$760$	0	0
$761$	−53.1918	−1.92820	−0.964101	−	0.265535i	$-0.914451\pi$
$761$	−53.1918	−1.92820	−0.964101	+	0.265535i	$0.914451\pi$
$762$	42.4949	1.53943
$763$	6.20204	0.224529
$764$	−3.34847	−0.121143
$765$	0	0
$766$	−5.30306	−0.191607
$767$	33.7980	1.22037
$768$	−2.44949	−0.0883883
$769$	−16.2020	−0.584261	−0.292130	−	0.956379i	$-0.594364\pi$
$769$	−16.2020	−0.584261	−0.292130	+	0.956379i	$0.594364\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	−14.0000	−0.503871
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	1.10102	0.0394989
$778$	6.20204	0.222354
$779$	16.8990	0.605469
$780$	0	0
$781$	24.0000	0.858788
$782$	−4.40408	−0.157490
$783$	0	0
$784$	−6.79796	−0.242784
$785$	0	0
$786$	34.8990	1.24481
$787$	−44.7423	−1.59489	−0.797446	−	0.603390i	$-0.793816\pi$
$787$	−44.7423	−1.59489	−0.797446	+	0.603390i	$0.793816\pi$
$788$	2.00000	0.0712470
$789$	64.2929	2.28889
$790$	0	0
$791$	5.79796	0.206152
$792$	14.6969	0.522233
$793$	−48.0000	−1.70453
$794$	−35.7980	−1.27042
$795$	0	0
$796$	1.55051	0.0549564
$797$	4.40408	0.156001	0.0780003	−	0.996953i	$-0.475146\pi$
$797$	4.40408	0.156001	0.0780003	+	0.996953i	$0.475146\pi$
$798$	9.30306	0.329325
$799$	−2.20204	−0.0779026
$800$	0	0
$801$	−11.3939	−0.402583
$802$	4.20204	0.148379
$803$	19.5959	0.691525
$804$	−25.5959	−0.902698
$805$	0	0
$806$	−25.7980	−0.908694
$807$	−26.2020	−0.922356
$808$	18.6969	0.657756
$809$	45.1918	1.58886	0.794430	−	0.607356i	$-0.207770\pi$
$809$	45.1918	1.58886	0.794430	+	0.607356i	$0.207770\pi$
$810$	0	0
$811$	45.7980	1.60818	0.804092	−	0.594505i	$-0.202652\pi$
$811$	45.7980	1.60818	0.804092	+	0.594505i	$0.202652\pi$
$812$	0	0
$813$	−40.4041	−1.41703
$814$	−4.89898	−0.171709
$815$	0	0
$816$	12.0000	0.420084
$817$	−33.7980	−1.18244
$818$	10.0000	0.349642
$819$	−5.39388	−0.188477
$820$	0	0
$821$	−45.5959	−1.59131	−0.795654	−	0.605751i	$-0.792873\pi$
$821$	−45.5959	−1.59131	−0.795654	+	0.605751i	$0.792873\pi$
$822$	48.0000	1.67419
$823$	−18.6515	−0.650151	−0.325076	−	0.945688i	$-0.605390\pi$
$823$	−18.6515	−0.650151	−0.325076	+	0.945688i	$0.605390\pi$
$824$	−9.79796	−0.341328
$825$	0	0
$826$	3.79796	0.132148
$827$	−12.4949	−0.434490	−0.217245	−	0.976117i	$-0.569707\pi$
$827$	−12.4949	−0.434490	−0.217245	+	0.976117i	$0.569707\pi$
$828$	−2.69694	−0.0937251
$829$	47.5959	1.65307	0.826537	−	0.562882i	$-0.190307\pi$
$829$	47.5959	1.65307	0.826537	+	0.562882i	$0.190307\pi$
$830$	0	0
$831$	−48.0000	−1.66510
$832$	−4.00000	−0.138675
$833$	33.3031	1.15388
$834$	33.7980	1.17033
$835$	0	0
$836$	−41.3939	−1.43164
$837$	0	0
$838$	13.7980	0.476643
$839$	33.7980	1.16684	0.583418	−	0.812172i	$-0.301715\pi$
$839$	33.7980	1.16684	0.583418	+	0.812172i	$0.301715\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−19.5959	−0.675320
$843$	−20.0908	−0.691964
$844$	22.6969	0.781261
$845$	0	0
$846$	−1.34847	−0.0463613
$847$	5.84337	0.200780
$848$	−7.79796	−0.267783
$849$	−57.7980	−1.98362
$850$	0	0
$851$	0.898979	0.0308166
$852$	12.0000	0.411113
$853$	−1.59592	−0.0546432	−0.0273216	−	0.999627i	$-0.508698\pi$
$853$	−1.59592	−0.0546432	−0.0273216	+	0.999627i	$0.508698\pi$
$854$	−5.39388	−0.184575
$855$	0	0
$856$	−10.4495	−0.357156
$857$	29.5959	1.01098	0.505489	−	0.862833i	$-0.331312\pi$
$857$	29.5959	1.01098	0.505489	+	0.862833i	$0.331312\pi$
$858$	48.0000	1.63869
$859$	−20.8536	−0.711515	−0.355757	−	0.934578i	$-0.615777\pi$
$859$	−20.8536	−0.711515	−0.355757	+	0.934578i	$0.615777\pi$
$860$	0	0
$861$	−2.20204	−0.0750454
$862$	40.2474	1.37083
$863$	−30.7423	−1.04648	−0.523241	−	0.852185i	$-0.675277\pi$
$863$	−30.7423	−1.04648	−0.523241	+	0.852185i	$0.675277\pi$
$864$	0	0
$865$	0	0
$866$	−37.3939	−1.27070
$867$	−17.1464	−0.582323
$868$	−2.89898	−0.0983978
$869$	−7.59592	−0.257674
$870$	0	0
$871$	−41.7980	−1.41627
$872$	−13.7980	−0.467258
$873$	6.00000	0.203069
$874$	7.59592	0.256936
$875$	0	0
$876$	9.79796	0.331042
$877$	43.3939	1.46531	0.732654	−	0.680602i	$-0.238282\pi$
$877$	43.3939	1.46531	0.732654	+	0.680602i	$0.238282\pi$
$878$	−35.3485	−1.19295
$879$	−33.3031	−1.12328
$880$	0	0
$881$	53.3939	1.79889	0.899443	−	0.437039i	$-0.143973\pi$
$881$	53.3939	1.79889	0.899443	+	0.437039i	$0.143973\pi$
$882$	20.3939	0.686698
$883$	−13.3031	−0.447684	−0.223842	−	0.974625i	$-0.571860\pi$
$883$	−13.3031	−0.447684	−0.223842	+	0.974625i	$0.571860\pi$
$884$	19.5959	0.659082
$885$	0	0
$886$	39.3485	1.32194
$887$	−24.0454	−0.807366	−0.403683	−	0.914899i	$-0.632270\pi$
$887$	−24.0454	−0.807366	−0.403683	+	0.914899i	$0.632270\pi$
$888$	−2.44949	−0.0821995
$889$	7.79796	0.261535
$890$	0	0
$891$	44.0908	1.47710
$892$	4.44949	0.148980
$893$	3.79796	0.127094
$894$	26.2020	0.876327
$895$	0	0
$896$	−0.449490	−0.0150164
$897$	−8.80816	−0.294096
$898$	−3.79796	−0.126740
$899$	0	0
$900$	0	0
$901$	38.2020	1.27269
$902$	9.79796	0.326236
$903$	4.40408	0.146559
$904$	−12.8990	−0.429014
$905$	0	0
$906$	−19.5959	−0.651031
$907$	8.89898	0.295486	0.147743	−	0.989026i	$-0.452799\pi$
$907$	8.89898	0.295486	0.147743	+	0.989026i	$0.452799\pi$
$908$	15.1010	0.501145
$909$	−56.0908	−1.86041
$910$	0	0
$911$	31.8434	1.05502	0.527509	−	0.849550i	$-0.323126\pi$
$911$	31.8434	1.05502	0.527509	+	0.849550i	$0.323126\pi$
$912$	−20.6969	−0.685344
$913$	−70.7878	−2.34273
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	6.40408	0.211481
$918$	0	0
$919$	−9.14643	−0.301713	−0.150856	−	0.988556i	$-0.548203\pi$
$919$	−9.14643	−0.301713	−0.150856	+	0.988556i	$0.548203\pi$
$920$	0	0
$921$	−51.7980	−1.70680
$922$	14.2020	0.467719
$923$	19.5959	0.645007
$924$	5.39388	0.177446
$925$	0	0
$926$	28.4949	0.936400
$927$	29.3939	0.965422
$928$	0	0
$929$	37.5959	1.23348	0.616741	−	0.787166i	$-0.288453\pi$
$929$	37.5959	1.23348	0.616741	+	0.787166i	$0.288453\pi$
$930$	0	0
$931$	−57.4393	−1.88250
$932$	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	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	−4.69694	−0.153360
$939$	52.8990	1.72629
$940$	0	0
$941$	30.2929	0.987519	0.493759	−	0.869599i	$-0.335622\pi$
$941$	30.2929	0.987519	0.493759	+	0.869599i	$0.335622\pi$
$942$	−28.8990	−0.941580
$943$	−1.79796	−0.0585496
$944$	−8.44949	−0.275007
$945$	0	0
$946$	−19.5959	−0.637118
$947$	33.3939	1.08516	0.542578	−	0.840006i	$-0.317448\pi$
$947$	33.3939	1.08516	0.542578	+	0.840006i	$0.317448\pi$
$948$	−3.79796	−0.123352
$949$	16.0000	0.519382
$950$	0	0
$951$	44.0908	1.42974
$952$	2.20204	0.0713686
$953$	2.20204	0.0713311	0.0356656	−	0.999364i	$-0.488645\pi$
$953$	2.20204	0.0713311	0.0356656	+	0.999364i	$0.488645\pi$
$954$	23.3939	0.757405
$955$	0	0
$956$	26.0454	0.842369
$957$	0	0
$958$	29.1464	0.941678
$959$	8.80816	0.284430
$960$	0	0
$961$	10.5959	0.341804
$962$	−4.00000	−0.128965
$963$	31.3485	1.01019
$964$	−11.7980	−0.379987
$965$	0	0
$966$	−0.989795	−0.0318461
$967$	−52.4949	−1.68812	−0.844061	−	0.536247i	$-0.819842\pi$
$967$	−52.4949	−1.68812	−0.844061	+	0.536247i	$0.819842\pi$
$968$	−13.0000	−0.417836
$969$	101.394	3.25724
$970$	0	0
$971$	−44.8990	−1.44088	−0.720438	−	0.693519i	$-0.756059\pi$
$971$	−44.8990	−1.44088	−0.720438	+	0.693519i	$0.756059\pi$
$972$	22.0454	0.707107
$973$	6.20204	0.198828
$974$	29.3939	0.941841
$975$	0	0
$976$	12.0000	0.384111
$977$	49.5959	1.58671	0.793357	−	0.608757i	$-0.208331\pi$
$977$	49.5959	1.58671	0.793357	+	0.608757i	$0.208331\pi$
$978$	5.39388	0.172477
$979$	18.6061	0.594654
$980$	0	0
$981$	41.3939	1.32160
$982$	38.6969	1.23487
$983$	48.9444	1.56108	0.780542	−	0.625104i	$-0.214943\pi$
$983$	48.9444	1.56108	0.780542	+	0.625104i	$0.214943\pi$
$984$	4.89898	0.156174
$985$	0	0
$986$	0	0
$987$	−0.494897	−0.0157528
$988$	−33.7980	−1.07526
$989$	3.59592	0.114344
$990$	0	0
$991$	31.8434	1.01154	0.505769	−	0.862669i	$-0.331209\pi$
$991$	31.8434	1.01154	0.505769	+	0.862669i	$0.331209\pi$
$992$	6.44949	0.204772
$993$	25.1010	0.796557
$994$	2.20204	0.0698445
$995$	0	0
$996$	−35.3939	−1.12150
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	25.3485	0.802392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.s.1.1		2
5.2	odd	4		370.2.b.c.149.2	✓	4
5.3	odd	4		370.2.b.c.149.3	yes	4
5.4	even	2		1850.2.a.v.1.2		2
15.2	even	4		3330.2.d.m.1999.4		4
15.8	even	4		3330.2.d.m.1999.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.c.149.2	✓	4	5.2	odd	4
370.2.b.c.149.3	yes	4	5.3	odd	4
1850.2.a.s.1.1		2	1.1	even	1	trivial
1850.2.a.v.1.2		2	5.4	even	2
3330.2.d.m.1999.1		4	15.8	even	4
3330.2.d.m.1999.4		4	15.2	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} +0.449490 q^{7} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} -2.44949 q^{12} -4.00000 q^{13} -0.449490 q^{14} +1.00000 q^{16} -4.89898 q^{17} -3.00000 q^{18} +8.44949 q^{19} -1.10102 q^{21} +4.89898 q^{22} -0.898979 q^{23} +2.44949 q^{24} +4.00000 q^{26} +0.449490 q^{28} -6.44949 q^{31} -1.00000 q^{32} +12.0000 q^{33} +4.89898 q^{34} +3.00000 q^{36} -1.00000 q^{37} -8.44949 q^{38} +9.79796 q^{39} +2.00000 q^{41} +1.10102 q^{42} -4.00000 q^{43} -4.89898 q^{44} +0.898979 q^{46} +0.449490 q^{47} -2.44949 q^{48} -6.79796 q^{49} +12.0000 q^{51} -4.00000 q^{52} -7.79796 q^{53} -0.449490 q^{56} -20.6969 q^{57} -8.44949 q^{59} +12.0000 q^{61} +6.44949 q^{62} +1.34847 q^{63} +1.00000 q^{64} -12.0000 q^{66} +10.4495 q^{67} -4.89898 q^{68} +2.20204 q^{69} -4.89898 q^{71} -3.00000 q^{72} -4.00000 q^{73} +1.00000 q^{74} +8.44949 q^{76} -2.20204 q^{77} -9.79796 q^{78} +1.55051 q^{79} -9.00000 q^{81} -2.00000 q^{82} +14.4495 q^{83} -1.10102 q^{84} +4.00000 q^{86} +4.89898 q^{88} -3.79796 q^{89} -1.79796 q^{91} -0.898979 q^{92} +15.7980 q^{93} -0.449490 q^{94} +2.44949 q^{96} +2.00000 q^{97} +6.79796 q^{98} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9} - 8 q^{13} + 4 q^{14} + 2 q^{16} - 6 q^{18} + 12 q^{19} - 12 q^{21} + 8 q^{23} + 8 q^{26} - 4 q^{28} - 8 q^{31} - 2 q^{32} + 24 q^{33} + 6 q^{36} - 2 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

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