LMFDB - Embedded newform 1850.2.a.ba.1.3

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.67513 q^{3} +1.00000 q^{4} -2.67513 q^{6} +3.28726 q^{7} -1.00000 q^{8} +4.15633 q^{9} +4.83146 q^{11} +2.67513 q^{12} +5.67513 q^{13} -3.28726 q^{14} +1.00000 q^{16} -5.63752 q^{17} -4.15633 q^{18} +4.76845 q^{19} +8.79384 q^{21} -4.83146 q^{22} +2.38787 q^{23} -2.67513 q^{24} -5.67513 q^{26} +3.09332 q^{27} +3.28726 q^{28} -7.92478 q^{29} -7.44358 q^{31} -1.00000 q^{32} +12.9248 q^{33} +5.63752 q^{34} +4.15633 q^{36} +1.00000 q^{37} -4.76845 q^{38} +15.1817 q^{39} -10.3503 q^{41} -8.79384 q^{42} -11.4314 q^{43} +4.83146 q^{44} -2.38787 q^{46} -5.92478 q^{47} +2.67513 q^{48} +3.80606 q^{49} -15.0811 q^{51} +5.67513 q^{52} +9.53690 q^{53} -3.09332 q^{54} -3.28726 q^{56} +12.7562 q^{57} +7.92478 q^{58} -11.4314 q^{59} -4.89938 q^{61} +7.44358 q^{62} +13.6629 q^{63} +1.00000 q^{64} -12.9248 q^{66} +2.10062 q^{67} -5.63752 q^{68} +6.38787 q^{69} -4.89938 q^{71} -4.15633 q^{72} +10.1939 q^{73} -1.00000 q^{74} +4.76845 q^{76} +15.8822 q^{77} -15.1817 q^{78} +7.61213 q^{79} -4.19394 q^{81} +10.3503 q^{82} +13.9199 q^{83} +8.79384 q^{84} +11.4314 q^{86} -21.1998 q^{87} -4.83146 q^{88} +4.44358 q^{89} +18.6556 q^{91} +2.38787 q^{92} -19.9126 q^{93} +5.92478 q^{94} -2.67513 q^{96} +5.35026 q^{97} -3.80606 q^{98} +20.0811 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9} - q^{11} + 3 q^{12} + 12 q^{13} - 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{18} + 3 q^{19} + q^{22} + 8 q^{23} - 3 q^{24} - 12 q^{26}+ \cdots + 28 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 2 * q^9 - q^11 + 3 * q^12 + 12 * q^13 - 4 * q^14 + 3 * q^16 - q^17 - 2 * q^18 + 3 * q^19 + q^22 + 8 * q^23 - 3 * q^24 - 12 * q^26 + 3 * q^27 + 4 * q^28 - 2 * q^29 - 6 * q^31 - 3 * q^32 + 17 * q^33 + q^34 + 2 * q^36 + 3 * q^37 - 3 * q^38 + 20 * q^39 - 21 * q^41 + 8 * q^43 - q^44 - 8 * q^46 + 4 * q^47 + 3 * q^48 + 11 * q^49 - 13 * q^51 + 12 * q^52 + 6 * q^53 - 3 * q^54 - 4 * q^56 + q^57 + 2 * q^58 + 8 * q^59 - 8 * q^61 + 6 * q^62 + 10 * q^63 + 3 * q^64 - 17 * q^66 + 13 * q^67 - q^68 + 20 * q^69 - 8 * q^71 - 2 * q^72 + 31 * q^73 - 3 * q^74 + 3 * q^76 + 2 * q^77 - 20 * q^78 + 22 * q^79 - 13 * q^81 + 21 * q^82 + 7 * q^83 - 8 * q^86 - 10 * q^87 + q^88 - 3 * q^89 + 12 * q^91 + 8 * q^92 - 12 * q^93 - 4 * q^94 - 3 * q^96 + 6 * q^97 - 11 * q^98 + 28 * 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
$3$	2.67513	1.54449	0.772244	−	0.635326i	$-0.219134\pi$
$3$	2.67513	1.54449	0.772244	+	0.635326i	$0.219134\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.67513	−1.09212
$7$	3.28726	1.24247	0.621233	−	0.783626i	$-0.286632\pi$
$7$	3.28726	1.24247	0.621233	+	0.783626i	$0.286632\pi$
$8$	−1.00000	−0.353553
$9$	4.15633	1.38544
$10$	0	0
$11$	4.83146	1.45674	0.728369	−	0.685185i	$-0.240279\pi$
$11$	4.83146	1.45674	0.728369	+	0.685185i	$0.240279\pi$
$12$	2.67513	0.772244
$13$	5.67513	1.57400	0.786999	−	0.616954i	$-0.211634\pi$
$13$	5.67513	1.57400	0.786999	+	0.616954i	$0.211634\pi$
$14$	−3.28726	−0.878557
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.63752	−1.36730	−0.683650	−	0.729810i	$-0.739608\pi$
$17$	−5.63752	−1.36730	−0.683650	+	0.729810i	$0.739608\pi$
$18$	−4.15633	−0.979655
$19$	4.76845	1.09396	0.546979	−	0.837146i	$-0.315778\pi$
$19$	4.76845	1.09396	0.546979	+	0.837146i	$0.315778\pi$
$20$	0	0
$21$	8.79384	1.91897
$22$	−4.83146	−1.03007
$23$	2.38787	0.497906	0.248953	−	0.968516i	$-0.419914\pi$
$23$	2.38787	0.497906	0.248953	+	0.968516i	$0.419914\pi$
$24$	−2.67513	−0.546059
$25$	0	0
$26$	−5.67513	−1.11298
$27$	3.09332	0.595310
$28$	3.28726	0.621233
$29$	−7.92478	−1.47159	−0.735797	−	0.677202i	$-0.763192\pi$
$29$	−7.92478	−1.47159	−0.735797	+	0.677202i	$0.763192\pi$
$30$	0	0
$31$	−7.44358	−1.33691	−0.668453	−	0.743754i	$-0.733043\pi$
$31$	−7.44358	−1.33691	−0.668453	+	0.743754i	$0.733043\pi$
$32$	−1.00000	−0.176777
$33$	12.9248	2.24991
$34$	5.63752	0.966827
$35$	0	0
$36$	4.15633	0.692721
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−4.76845	−0.773545
$39$	15.1817	2.43102
$40$	0	0
$41$	−10.3503	−1.61644	−0.808220	−	0.588881i	$-0.799569\pi$
$41$	−10.3503	−1.61644	−0.808220	+	0.588881i	$0.799569\pi$
$42$	−8.79384	−1.35692
$43$	−11.4314	−1.74327	−0.871633	−	0.490158i	$-0.836939\pi$
$43$	−11.4314	−1.74327	−0.871633	+	0.490158i	$0.836939\pi$
$44$	4.83146	0.728369
$45$	0	0
$46$	−2.38787	−0.352073
$47$	−5.92478	−0.864218	−0.432109	−	0.901821i	$-0.642230\pi$
$47$	−5.92478	−0.864218	−0.432109	+	0.901821i	$0.642230\pi$
$48$	2.67513	0.386122
$49$	3.80606	0.543723
$50$	0	0
$51$	−15.0811	−2.11178
$52$	5.67513	0.786999
$53$	9.53690	1.30999	0.654997	−	0.755631i	$-0.272670\pi$
$53$	9.53690	1.30999	0.654997	+	0.755631i	$0.272670\pi$
$54$	−3.09332	−0.420948
$55$	0	0
$56$	−3.28726	−0.439278
$57$	12.7562	1.68960
$58$	7.92478	1.04057
$59$	−11.4314	−1.48824	−0.744118	−	0.668048i	$-0.767130\pi$
$59$	−11.4314	−1.48824	−0.744118	+	0.668048i	$0.767130\pi$
$60$	0	0
$61$	−4.89938	−0.627302	−0.313651	−	0.949538i	$-0.601552\pi$
$61$	−4.89938	−0.627302	−0.313651	+	0.949538i	$0.601552\pi$
$62$	7.44358	0.945336
$63$	13.6629	1.72137
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.9248	−1.59093
$67$	2.10062	0.256631	0.128316	−	0.991733i	$-0.459043\pi$
$67$	2.10062	0.256631	0.128316	+	0.991733i	$0.459043\pi$
$68$	−5.63752	−0.683650
$69$	6.38787	0.769010
$70$	0	0
$71$	−4.89938	−0.581450	−0.290725	−	0.956807i	$-0.593896\pi$
$71$	−4.89938	−0.581450	−0.290725	+	0.956807i	$0.593896\pi$
$72$	−4.15633	−0.489828
$73$	10.1939	1.19311	0.596555	−	0.802572i	$-0.296536\pi$
$73$	10.1939	1.19311	0.596555	+	0.802572i	$0.296536\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	4.76845	0.546979
$77$	15.8822	1.80995
$78$	−15.1817	−1.71899
$79$	7.61213	0.856431	0.428216	−	0.903677i	$-0.359142\pi$
$79$	7.61213	0.856431	0.428216	+	0.903677i	$0.359142\pi$
$80$	0	0
$81$	−4.19394	−0.465993
$82$	10.3503	1.14300
$83$	13.9199	1.52790	0.763951	−	0.645274i	$-0.223257\pi$
$83$	13.9199	1.52790	0.763951	+	0.645274i	$0.223257\pi$
$84$	8.79384	0.959487
$85$	0	0
$86$	11.4314	1.23268
$87$	−21.1998	−2.27286
$88$	−4.83146	−0.515035
$89$	4.44358	0.471019	0.235509	−	0.971872i	$-0.424324\pi$
$89$	4.44358	0.471019	0.235509	+	0.971872i	$0.424324\pi$
$90$	0	0
$91$	18.6556	1.95564
$92$	2.38787	0.248953
$93$	−19.9126	−2.06484
$94$	5.92478	0.611094
$95$	0	0
$96$	−2.67513	−0.273029
$97$	5.35026	0.543237	0.271618	−	0.962405i	$-0.412441\pi$
$97$	5.35026	0.543237	0.271618	+	0.962405i	$0.412441\pi$
$98$	−3.80606	−0.384470
$99$	20.0811	2.01823
$100$	0	0
$101$	−8.12601	−0.808568	−0.404284	−	0.914634i	$-0.632479\pi$
$101$	−8.12601	−0.808568	−0.404284	+	0.914634i	$0.632479\pi$
$102$	15.0811	1.49325
$103$	−1.90668	−0.187871	−0.0939353	−	0.995578i	$-0.529945\pi$
$103$	−1.90668	−0.187871	−0.0939353	+	0.995578i	$0.529945\pi$
$104$	−5.67513	−0.556492
$105$	0	0
$106$	−9.53690	−0.926306
$107$	−2.41327	−0.233299	−0.116650	−	0.993173i	$-0.537215\pi$
$107$	−2.41327	−0.233299	−0.116650	+	0.993173i	$0.537215\pi$
$108$	3.09332	0.297655
$109$	−1.73813	−0.166483	−0.0832416	−	0.996529i	$-0.526527\pi$
$109$	−1.73813	−0.166483	−0.0832416	+	0.996529i	$0.526527\pi$
$110$	0	0
$111$	2.67513	0.253912
$112$	3.28726	0.310617
$113$	−5.09332	−0.479139	−0.239570	−	0.970879i	$-0.577006\pi$
$113$	−5.09332	−0.479139	−0.239570	+	0.970879i	$0.577006\pi$
$114$	−12.7562	−1.19473
$115$	0	0
$116$	−7.92478	−0.735797
$117$	23.5877	2.18068
$118$	11.4314	1.05234
$119$	−18.5320	−1.69882
$120$	0	0
$121$	12.3430	1.12209
$122$	4.89938	0.443569
$123$	−27.6883	−2.49657
$124$	−7.44358	−0.668453
$125$	0	0
$126$	−13.6629	−1.21719
$127$	−7.59991	−0.674383	−0.337191	−	0.941436i	$-0.609477\pi$
$127$	−7.59991	−0.674383	−0.337191	+	0.941436i	$0.609477\pi$
$128$	−1.00000	−0.0883883
$129$	−30.5804	−2.69245
$130$	0	0
$131$	13.3806	1.16907	0.584533	−	0.811370i	$-0.301278\pi$
$131$	13.3806	1.16907	0.584533	+	0.811370i	$0.301278\pi$
$132$	12.9248	1.12496
$133$	15.6751	1.35921
$134$	−2.10062	−0.181466
$135$	0	0
$136$	5.63752	0.483413
$137$	11.1866	0.955739	0.477870	−	0.878431i	$-0.341409\pi$
$137$	11.1866	0.955739	0.477870	+	0.878431i	$0.341409\pi$
$138$	−6.38787	−0.543772
$139$	−17.0689	−1.44776	−0.723882	−	0.689924i	$-0.757644\pi$
$139$	−17.0689	−1.44776	−0.723882	+	0.689924i	$0.757644\pi$
$140$	0	0
$141$	−15.8496	−1.33477
$142$	4.89938	0.411147
$143$	27.4191	2.29290
$144$	4.15633	0.346360
$145$	0	0
$146$	−10.1939	−0.843656
$147$	10.1817	0.839774
$148$	1.00000	0.0821995
$149$	−7.56959	−0.620125	−0.310063	−	0.950716i	$-0.600350\pi$
$149$	−7.56959	−0.620125	−0.310063	+	0.950716i	$0.600350\pi$
$150$	0	0
$151$	0.649738	0.0528749	0.0264375	−	0.999650i	$-0.491584\pi$
$151$	0.649738	0.0528749	0.0264375	+	0.999650i	$0.491584\pi$
$152$	−4.76845	−0.386773
$153$	−23.4314	−1.89431
$154$	−15.8822	−1.27983
$155$	0	0
$156$	15.1817	1.21551
$157$	−3.36836	−0.268824	−0.134412	−	0.990926i	$-0.542915\pi$
$157$	−3.36836	−0.268824	−0.134412	+	0.990926i	$0.542915\pi$
$158$	−7.61213	−0.605588
$159$	25.5125	2.02327
$160$	0	0
$161$	7.84955	0.618632
$162$	4.19394	0.329507
$163$	−4.31994	−0.338364	−0.169182	−	0.985585i	$-0.554113\pi$
$163$	−4.31994	−0.338364	−0.169182	+	0.985585i	$0.554113\pi$
$164$	−10.3503	−0.808220
$165$	0	0
$166$	−13.9199	−1.08039
$167$	14.6678	1.13503	0.567516	−	0.823363i	$-0.307905\pi$
$167$	14.6678	1.13503	0.567516	+	0.823363i	$0.307905\pi$
$168$	−8.79384	−0.678460
$169$	19.2071	1.47747
$170$	0	0
$171$	19.8192	1.51561
$172$	−11.4314	−0.871633
$173$	−8.05079	−0.612090	−0.306045	−	0.952017i	$-0.599006\pi$
$173$	−8.05079	−0.612090	−0.306045	+	0.952017i	$0.599006\pi$
$174$	21.1998	1.60715
$175$	0	0
$176$	4.83146	0.364185
$177$	−30.5804	−2.29856
$178$	−4.44358	−0.333061
$179$	−5.05079	−0.377513	−0.188757	−	0.982024i	$-0.560446\pi$
$179$	−5.05079	−0.377513	−0.188757	+	0.982024i	$0.560446\pi$
$180$	0	0
$181$	3.73813	0.277853	0.138927	−	0.990303i	$-0.455635\pi$
$181$	3.73813	0.277853	0.138927	+	0.990303i	$0.455635\pi$
$182$	−18.6556	−1.38285
$183$	−13.1065	−0.968860
$184$	−2.38787	−0.176036
$185$	0	0
$186$	19.9126	1.46006
$187$	−27.2374	−1.99180
$188$	−5.92478	−0.432109
$189$	10.1685	0.739653
$190$	0	0
$191$	−12.5501	−0.908092	−0.454046	−	0.890978i	$-0.650020\pi$
$191$	−12.5501	−0.908092	−0.454046	+	0.890978i	$0.650020\pi$
$192$	2.67513	0.193061
$193$	20.8822	1.50314	0.751568	−	0.659655i	$-0.229298\pi$
$193$	20.8822	1.50314	0.751568	+	0.659655i	$0.229298\pi$
$194$	−5.35026	−0.384126
$195$	0	0
$196$	3.80606	0.271862
$197$	−1.07030	−0.0762556	−0.0381278	−	0.999273i	$-0.512139\pi$
$197$	−1.07030	−0.0762556	−0.0381278	+	0.999273i	$0.512139\pi$
$198$	−20.0811	−1.42710
$199$	−22.1016	−1.56674	−0.783369	−	0.621556i	$-0.786501\pi$
$199$	−22.1016	−1.56674	−0.783369	+	0.621556i	$0.786501\pi$
$200$	0	0
$201$	5.61942	0.396363
$202$	8.12601	0.571744
$203$	−26.0508	−1.82841
$204$	−15.0811	−1.05589
$205$	0	0
$206$	1.90668	0.132845
$207$	9.92478	0.689820
$208$	5.67513	0.393500
$209$	23.0386	1.59361
$210$	0	0
$211$	−16.3987	−1.12893	−0.564466	−	0.825456i	$-0.690918\pi$
$211$	−16.3987	−1.12893	−0.564466	+	0.825456i	$0.690918\pi$
$212$	9.53690	0.654997
$213$	−13.1065	−0.898042
$214$	2.41327	0.164967
$215$	0	0
$216$	−3.09332	−0.210474
$217$	−24.4690	−1.66106
$218$	1.73813	0.117721
$219$	27.2701	1.84274
$220$	0	0
$221$	−31.9937	−2.15213
$222$	−2.67513	−0.179543
$223$	18.4387	1.23474	0.617372	−	0.786671i	$-0.288197\pi$
$223$	18.4387	1.23474	0.617372	+	0.786671i	$0.288197\pi$
$224$	−3.28726	−0.219639
$225$	0	0
$226$	5.09332	0.338803
$227$	−11.4558	−0.760348	−0.380174	−	0.924915i	$-0.624136\pi$
$227$	−11.4558	−0.760348	−0.380174	+	0.924915i	$0.624136\pi$
$228$	12.7562	0.844802
$229$	25.6385	1.69424	0.847119	−	0.531403i	$-0.178335\pi$
$229$	25.6385	1.69424	0.847119	+	0.531403i	$0.178335\pi$
$230$	0	0
$231$	42.4871	2.79544
$232$	7.92478	0.520287
$233$	9.76845	0.639953	0.319976	−	0.947426i	$-0.396325\pi$
$233$	9.76845	0.639953	0.319976	+	0.947426i	$0.396325\pi$
$234$	−23.5877	−1.54198
$235$	0	0
$236$	−11.4314	−0.744118
$237$	20.3634	1.32275
$238$	18.5320	1.20125
$239$	16.2882	1.05360	0.526798	−	0.849990i	$-0.323392\pi$
$239$	16.2882	1.05360	0.526798	+	0.849990i	$0.323392\pi$
$240$	0	0
$241$	13.1744	0.848639	0.424320	−	0.905512i	$-0.360513\pi$
$241$	13.1744	0.848639	0.424320	+	0.905512i	$0.360513\pi$
$242$	−12.3430	−0.793436
$243$	−20.4993	−1.31503
$244$	−4.89938	−0.313651
$245$	0	0
$246$	27.6883	1.76534
$247$	27.0616	1.72189
$248$	7.44358	0.472668
$249$	37.2374	2.35983
$250$	0	0
$251$	13.0508	0.823758	0.411879	−	0.911238i	$-0.364873\pi$
$251$	13.0508	0.823758	0.411879	+	0.911238i	$0.364873\pi$
$252$	13.6629	0.860683
$253$	11.5369	0.725319
$254$	7.59991	0.476861
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.7005	1.29126	0.645632	−	0.763649i	$-0.276594\pi$
$257$	20.7005	1.29126	0.645632	+	0.763649i	$0.276594\pi$
$258$	30.5804	1.90385
$259$	3.28726	0.204260
$260$	0	0
$261$	−32.9380	−2.03881
$262$	−13.3806	−0.826655
$263$	−20.4387	−1.26030	−0.630151	−	0.776473i	$-0.717007\pi$
$263$	−20.4387	−1.26030	−0.630151	+	0.776473i	$0.717007\pi$
$264$	−12.9248	−0.795465
$265$	0	0
$266$	−15.6751	−0.961104
$267$	11.8872	0.727483
$268$	2.10062	0.128316
$269$	6.25202	0.381192	0.190596	−	0.981669i	$-0.438958\pi$
$269$	6.25202	0.381192	0.190596	+	0.981669i	$0.438958\pi$
$270$	0	0
$271$	−0.824162	−0.0500643	−0.0250321	−	0.999687i	$-0.507969\pi$
$271$	−0.824162	−0.0500643	−0.0250321	+	0.999687i	$0.507969\pi$
$272$	−5.63752	−0.341825
$273$	49.9062	3.02046
$274$	−11.1866	−0.675810
$275$	0	0
$276$	6.38787	0.384505
$277$	−18.1866	−1.09273	−0.546365	−	0.837547i	$-0.683989\pi$
$277$	−18.1866	−1.09273	−0.546365	+	0.837547i	$0.683989\pi$
$278$	17.0689	1.02372
$279$	−30.9380	−1.85221
$280$	0	0
$281$	25.0435	1.49397	0.746985	−	0.664841i	$-0.231501\pi$
$281$	25.0435	1.49397	0.746985	+	0.664841i	$0.231501\pi$
$282$	15.8496	0.943827
$283$	13.1055	0.779043	0.389522	−	0.921017i	$-0.372640\pi$
$283$	13.1055	0.779043	0.389522	+	0.921017i	$0.372640\pi$
$284$	−4.89938	−0.290725
$285$	0	0
$286$	−27.4191	−1.62133
$287$	−34.0240	−2.00837
$288$	−4.15633	−0.244914
$289$	14.7816	0.869507
$290$	0	0
$291$	14.3127	0.839022
$292$	10.1939	0.596555
$293$	9.68101	0.565571	0.282785	−	0.959183i	$-0.408742\pi$
$293$	9.68101	0.565571	0.282785	+	0.959183i	$0.408742\pi$
$294$	−10.1817	−0.593810
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	14.9452	0.867211
$298$	7.56959	0.438495
$299$	13.5515	0.783703
$300$	0	0
$301$	−37.5778	−2.16595
$302$	−0.649738	−0.0373882
$303$	−21.7381	−1.24882
$304$	4.76845	0.273489
$305$	0	0
$306$	23.4314	1.33948
$307$	−20.4337	−1.16621	−0.583107	−	0.812395i	$-0.698163\pi$
$307$	−20.4337	−1.16621	−0.583107	+	0.812395i	$0.698163\pi$
$308$	15.8822	0.904975
$309$	−5.10062	−0.290164
$310$	0	0
$311$	10.5320	0.597214	0.298607	−	0.954376i	$-0.403478\pi$
$311$	10.5320	0.597214	0.298607	+	0.954376i	$0.403478\pi$
$312$	−15.1817	−0.859496
$313$	9.96968	0.563520	0.281760	−	0.959485i	$-0.409082\pi$
$313$	9.96968	0.563520	0.281760	+	0.959485i	$0.409082\pi$
$314$	3.36836	0.190088
$315$	0	0
$316$	7.61213	0.428216
$317$	−24.3961	−1.37022	−0.685111	−	0.728438i	$-0.740246\pi$
$317$	−24.3961	−1.37022	−0.685111	+	0.728438i	$0.740246\pi$
$318$	−25.5125	−1.43067
$319$	−38.2882	−2.14373
$320$	0	0
$321$	−6.45580	−0.360328
$322$	−7.84955	−0.437439
$323$	−26.8822	−1.49577
$324$	−4.19394	−0.232996
$325$	0	0
$326$	4.31994	0.239260
$327$	−4.64974	−0.257131
$328$	10.3503	0.571498
$329$	−19.4763	−1.07376
$330$	0	0
$331$	34.3561	1.88838	0.944192	−	0.329395i	$-0.106845\pi$
$331$	34.3561	1.88838	0.944192	+	0.329395i	$0.106845\pi$
$332$	13.9199	0.763951
$333$	4.15633	0.227765
$334$	−14.6678	−0.802588
$335$	0	0
$336$	8.79384	0.479744
$337$	20.6385	1.12425	0.562125	−	0.827053i	$-0.309984\pi$
$337$	20.6385	1.12425	0.562125	+	0.827053i	$0.309984\pi$
$338$	−19.2071	−1.04473
$339$	−13.6253	−0.740025
$340$	0	0
$341$	−35.9633	−1.94752
$342$	−19.8192	−1.07170
$343$	−10.4993	−0.566909
$344$	11.4314	0.616338
$345$	0	0
$346$	8.05079	0.432813
$347$	4.48612	0.240827	0.120414	−	0.992724i	$-0.461578\pi$
$347$	4.48612	0.240827	0.120414	+	0.992724i	$0.461578\pi$
$348$	−21.1998	−1.13643
$349$	−20.1949	−1.08101	−0.540504	−	0.841342i	$-0.681766\pi$
$349$	−20.1949	−1.08101	−0.540504	+	0.841342i	$0.681766\pi$
$350$	0	0
$351$	17.5550	0.937017
$352$	−4.83146	−0.257517
$353$	−25.9102	−1.37906	−0.689530	−	0.724257i	$-0.742183\pi$
$353$	−25.9102	−1.37906	−0.689530	+	0.724257i	$0.742183\pi$
$354$	30.5804	1.62533
$355$	0	0
$356$	4.44358	0.235509
$357$	−49.5755	−2.62381
$358$	5.05079	0.266942
$359$	4.43866	0.234263	0.117132	−	0.993116i	$-0.462630\pi$
$359$	4.43866	0.234263	0.117132	+	0.993116i	$0.462630\pi$
$360$	0	0
$361$	3.73813	0.196744
$362$	−3.73813	−0.196472
$363$	33.0191	1.73305
$364$	18.6556	0.977820
$365$	0	0
$366$	13.1065	0.685087
$367$	28.2858	1.47651	0.738254	−	0.674522i	$-0.235650\pi$
$367$	28.2858	1.47651	0.738254	+	0.674522i	$0.235650\pi$
$368$	2.38787	0.124476
$369$	−43.0191	−2.23948
$370$	0	0
$371$	31.3503	1.62762
$372$	−19.9126	−1.03242
$373$	−2.71862	−0.140765	−0.0703825	−	0.997520i	$-0.522422\pi$
$373$	−2.71862	−0.140765	−0.0703825	+	0.997520i	$0.522422\pi$
$374$	27.2374	1.40841
$375$	0	0
$376$	5.92478	0.305547
$377$	−44.9741	−2.31629
$378$	−10.1685	−0.523013
$379$	18.1514	0.932375	0.466187	−	0.884686i	$-0.345627\pi$
$379$	18.1514	0.932375	0.466187	+	0.884686i	$0.345627\pi$
$380$	0	0
$381$	−20.3307	−1.04158
$382$	12.5501	0.642118
$383$	−3.90668	−0.199622	−0.0998110	−	0.995006i	$-0.531824\pi$
$383$	−3.90668	−0.199622	−0.0998110	+	0.995006i	$0.531824\pi$
$384$	−2.67513	−0.136515
$385$	0	0
$386$	−20.8822	−1.06288
$387$	−47.5125	−2.41519
$388$	5.35026	0.271618
$389$	5.28726	0.268075	0.134037	−	0.990976i	$-0.457206\pi$
$389$	5.28726	0.268075	0.134037	+	0.990976i	$0.457206\pi$
$390$	0	0
$391$	−13.4617	−0.680786
$392$	−3.80606	−0.192235
$393$	35.7948	1.80561
$394$	1.07030	0.0539208
$395$	0	0
$396$	20.0811	1.00911
$397$	4.15396	0.208481	0.104241	−	0.994552i	$-0.466759\pi$
$397$	4.15396	0.208481	0.104241	+	0.994552i	$0.466759\pi$
$398$	22.1016	1.10785
$399$	41.9330	2.09928
$400$	0	0
$401$	−22.1817	−1.10770	−0.553851	−	0.832616i	$-0.686842\pi$
$401$	−22.1817	−1.10770	−0.553851	+	0.832616i	$0.686842\pi$
$402$	−5.61942	−0.280271
$403$	−42.2433	−2.10429
$404$	−8.12601	−0.404284
$405$	0	0
$406$	26.0508	1.29288
$407$	4.83146	0.239486
$408$	15.0811	0.746626
$409$	0.650693	0.0321747	0.0160874	−	0.999871i	$-0.494879\pi$
$409$	0.650693	0.0321747	0.0160874	+	0.999871i	$0.494879\pi$
$410$	0	0
$411$	29.9257	1.47613
$412$	−1.90668	−0.0939353
$413$	−37.5778	−1.84908
$414$	−9.92478	−0.487776
$415$	0	0
$416$	−5.67513	−0.278246
$417$	−45.6615	−2.23605
$418$	−23.0386	−1.12685
$419$	10.8169	0.528439	0.264219	−	0.964463i	$-0.414886\pi$
$419$	10.8169	0.528439	0.264219	+	0.964463i	$0.414886\pi$
$420$	0	0
$421$	2.05079	0.0999492	0.0499746	−	0.998750i	$-0.484086\pi$
$421$	2.05079	0.0999492	0.0499746	+	0.998750i	$0.484086\pi$
$422$	16.3987	0.798275
$423$	−24.6253	−1.19732
$424$	−9.53690	−0.463153
$425$	0	0
$426$	13.1065	0.635012
$427$	−16.1055	−0.779402
$428$	−2.41327	−0.116650
$429$	73.3498	3.54136
$430$	0	0
$431$	15.4861	0.745940	0.372970	−	0.927843i	$-0.378339\pi$
$431$	15.4861	0.745940	0.372970	+	0.927843i	$0.378339\pi$
$432$	3.09332	0.148827
$433$	35.8773	1.72415	0.862077	−	0.506777i	$-0.169163\pi$
$433$	35.8773	1.72415	0.862077	+	0.506777i	$0.169163\pi$
$434$	24.4690	1.17455
$435$	0	0
$436$	−1.73813	−0.0832416
$437$	11.3865	0.544688
$438$	−27.2701	−1.30302
$439$	−26.7005	−1.27435	−0.637173	−	0.770721i	$-0.719896\pi$
$439$	−26.7005	−1.27435	−0.637173	+	0.770721i	$0.719896\pi$
$440$	0	0
$441$	15.8192	0.753297
$442$	31.9937	1.52178
$443$	5.75765	0.273554	0.136777	−	0.990602i	$-0.456326\pi$
$443$	5.75765	0.273554	0.136777	+	0.990602i	$0.456326\pi$
$444$	2.67513	0.126956
$445$	0	0
$446$	−18.4387	−0.873096
$447$	−20.2496	−0.957775
$448$	3.28726	0.155308
$449$	−31.1803	−1.47149	−0.735745	−	0.677259i	$-0.763168\pi$
$449$	−31.1803	−1.47149	−0.735745	+	0.677259i	$0.763168\pi$
$450$	0	0
$451$	−50.0068	−2.35473
$452$	−5.09332	−0.239570
$453$	1.73813	0.0816647
$454$	11.4558	0.537647
$455$	0	0
$456$	−12.7562	−0.597365
$457$	7.27011	0.340082	0.170041	−	0.985437i	$-0.445610\pi$
$457$	7.27011	0.340082	0.170041	+	0.985437i	$0.445610\pi$
$458$	−25.6385	−1.19801
$459$	−17.4387	−0.813967
$460$	0	0
$461$	−18.6253	−0.867467	−0.433733	−	0.901041i	$-0.642804\pi$
$461$	−18.6253	−0.867467	−0.433733	+	0.901041i	$0.642804\pi$
$462$	−42.4871	−1.97668
$463$	1.89209	0.0879329	0.0439664	−	0.999033i	$-0.486001\pi$
$463$	1.89209	0.0879329	0.0439664	+	0.999033i	$0.486001\pi$
$464$	−7.92478	−0.367899
$465$	0	0
$466$	−9.76845	−0.452515
$467$	1.21440	0.0561959	0.0280980	−	0.999605i	$-0.491055\pi$
$467$	1.21440	0.0561959	0.0280980	+	0.999605i	$0.491055\pi$
$468$	23.5877	1.09034
$469$	6.90526	0.318855
$470$	0	0
$471$	−9.01080	−0.415196
$472$	11.4314	0.526171
$473$	−55.2301	−2.53948
$474$	−20.3634	−0.935324
$475$	0	0
$476$	−18.5320	−0.849412
$477$	39.6385	1.81492
$478$	−16.2882	−0.745006
$479$	−40.7694	−1.86280	−0.931401	−	0.363995i	$-0.881412\pi$
$479$	−40.7694	−1.86280	−0.931401	+	0.363995i	$0.881412\pi$
$480$	0	0
$481$	5.67513	0.258764
$482$	−13.1744	−0.600079
$483$	20.9986	0.955469
$484$	12.3430	0.561044
$485$	0	0
$486$	20.4993	0.929867
$487$	−23.8578	−1.08110	−0.540550	−	0.841312i	$-0.681784\pi$
$487$	−23.8578	−1.08110	−0.540550	+	0.841312i	$0.681784\pi$
$488$	4.89938	0.221785
$489$	−11.5564	−0.522599
$490$	0	0
$491$	36.0362	1.62629	0.813145	−	0.582061i	$-0.197753\pi$
$491$	36.0362	1.62629	0.813145	+	0.582061i	$0.197753\pi$
$492$	−27.6883	−1.24829
$493$	44.6761	2.01211
$494$	−27.0616	−1.21756
$495$	0	0
$496$	−7.44358	−0.334227
$497$	−16.1055	−0.722432
$498$	−37.2374	−1.66865
$499$	−31.4558	−1.40816	−0.704078	−	0.710123i	$-0.748639\pi$
$499$	−31.4558	−1.40816	−0.704078	+	0.710123i	$0.748639\pi$
$500$	0	0
$501$	39.2384	1.75304
$502$	−13.0508	−0.582485
$503$	3.07381	0.137054	0.0685272	−	0.997649i	$-0.478170\pi$
$503$	3.07381	0.137054	0.0685272	+	0.997649i	$0.478170\pi$
$504$	−13.6629	−0.608594
$505$	0	0
$506$	−11.5369	−0.512878
$507$	51.3815	2.28193
$508$	−7.59991	−0.337191
$509$	−3.01951	−0.133838	−0.0669188	−	0.997758i	$-0.521317\pi$
$509$	−3.01951	−0.133838	−0.0669188	+	0.997758i	$0.521317\pi$
$510$	0	0
$511$	33.5101	1.48240
$512$	−1.00000	−0.0441942
$513$	14.7504	0.651244
$514$	−20.7005	−0.913061
$515$	0	0
$516$	−30.5804	−1.34623
$517$	−28.6253	−1.25894
$518$	−3.28726	−0.144434
$519$	−21.5369	−0.945365
$520$	0	0
$521$	28.8700	1.26482	0.632409	−	0.774634i	$-0.282066\pi$
$521$	28.8700	1.26482	0.632409	+	0.774634i	$0.282066\pi$
$522$	32.9380	1.44165
$523$	3.37328	0.147503	0.0737517	−	0.997277i	$-0.476503\pi$
$523$	3.37328	0.147503	0.0737517	+	0.997277i	$0.476503\pi$
$524$	13.3806	0.584533
$525$	0	0
$526$	20.4387	0.891168
$527$	41.9633	1.82795
$528$	12.9248	0.562479
$529$	−17.2981	−0.752090
$530$	0	0
$531$	−47.5125	−2.06187
$532$	15.6751	0.679603
$533$	−58.7391	−2.54427
$534$	−11.8872	−0.514408
$535$	0	0
$536$	−2.10062	−0.0907328
$537$	−13.5115	−0.583065
$538$	−6.25202	−0.269544
$539$	18.3888	0.792063
$540$	0	0
$541$	10.3004	0.442850	0.221425	−	0.975177i	$-0.428929\pi$
$541$	10.3004	0.442850	0.221425	+	0.975177i	$0.428929\pi$
$542$	0.824162	0.0354008
$543$	10.0000	0.429141
$544$	5.63752	0.241707
$545$	0	0
$546$	−49.9062	−2.13579
$547$	15.3576	0.656642	0.328321	−	0.944566i	$-0.393517\pi$
$547$	15.3576	0.656642	0.328321	+	0.944566i	$0.393517\pi$
$548$	11.1866	0.477870
$549$	−20.3634	−0.869090
$550$	0	0
$551$	−37.7889	−1.60986
$552$	−6.38787	−0.271886
$553$	25.0230	1.06409
$554$	18.1866	0.772676
$555$	0	0
$556$	−17.0689	−0.723882
$557$	14.8119	0.627602	0.313801	−	0.949489i	$-0.398397\pi$
$557$	14.8119	0.627602	0.313801	+	0.949489i	$0.398397\pi$
$558$	30.9380	1.30971
$559$	−64.8745	−2.74390
$560$	0	0
$561$	−72.8637	−3.07631
$562$	−25.0435	−1.05640
$563$	3.24472	0.136749	0.0683744	−	0.997660i	$-0.478219\pi$
$563$	3.24472	0.136749	0.0683744	+	0.997660i	$0.478219\pi$
$564$	−15.8496	−0.667387
$565$	0	0
$566$	−13.1055	−0.550867
$567$	−13.7866	−0.578981
$568$	4.89938	0.205574
$569$	−17.1587	−0.719330	−0.359665	−	0.933082i	$-0.617109\pi$
$569$	−17.1587	−0.719330	−0.359665	+	0.933082i	$0.617109\pi$
$570$	0	0
$571$	0.0244376	0.00102268	0.000511340	−	1.00000i	$-0.499837\pi$
$571$	0.0244376	0.00102268	0.000511340		1.00000i	$0.499837\pi$
$572$	27.4191	1.14645
$573$	−33.5731	−1.40254
$574$	34.0240	1.42013
$575$	0	0
$576$	4.15633	0.173180
$577$	−34.5002	−1.43626	−0.718132	−	0.695907i	$-0.755003\pi$
$577$	−34.5002	−1.43626	−0.718132	+	0.695907i	$0.755003\pi$
$578$	−14.7816	−0.614835
$579$	55.8627	2.32158
$580$	0	0
$581$	45.7581	1.89837
$582$	−14.3127	−0.593278
$583$	46.0771	1.90832
$584$	−10.1939	−0.421828
$585$	0	0
$586$	−9.68101	−0.399919
$587$	−4.84367	−0.199920	−0.0999599	−	0.994991i	$-0.531871\pi$
$587$	−4.84367	−0.199920	−0.0999599	+	0.994991i	$0.531871\pi$
$588$	10.1817	0.419887
$589$	−35.4944	−1.46252
$590$	0	0
$591$	−2.86319	−0.117776
$592$	1.00000	0.0410997
$593$	31.0263	1.27410	0.637050	−	0.770823i	$-0.280155\pi$
$593$	31.0263	1.27410	0.637050	+	0.770823i	$0.280155\pi$
$594$	−14.9452	−0.613211
$595$	0	0
$596$	−7.56959	−0.310063
$597$	−59.1246	−2.41981
$598$	−13.5515	−0.554162
$599$	−32.0240	−1.30846	−0.654232	−	0.756294i	$-0.727008\pi$
$599$	−32.0240	−1.30846	−0.654232	+	0.756294i	$0.727008\pi$
$600$	0	0
$601$	−9.14174	−0.372899	−0.186450	−	0.982465i	$-0.559698\pi$
$601$	−9.14174	−0.372899	−0.186450	+	0.982465i	$0.559698\pi$
$602$	37.5778	1.53156
$603$	8.73084	0.355547
$604$	0.649738	0.0264375
$605$	0	0
$606$	21.7381	0.883051
$607$	36.8691	1.49647	0.748235	−	0.663434i	$-0.230902\pi$
$607$	36.8691	1.49647	0.748235	+	0.663434i	$0.230902\pi$
$608$	−4.76845	−0.193386
$609$	−69.6893	−2.82395
$610$	0	0
$611$	−33.6239	−1.36028
$612$	−23.4314	−0.947157
$613$	−27.9902	−1.13051	−0.565256	−	0.824916i	$-0.691223\pi$
$613$	−27.9902	−1.13051	−0.565256	+	0.824916i	$0.691223\pi$
$614$	20.4337	0.824638
$615$	0	0
$616$	−15.8822	−0.639914
$617$	−22.5804	−0.909052	−0.454526	−	0.890733i	$-0.650191\pi$
$617$	−22.5804	−0.909052	−0.454526	+	0.890733i	$0.650191\pi$
$618$	5.10062	0.205177
$619$	40.0362	1.60919	0.804595	−	0.593824i	$-0.202382\pi$
$619$	40.0362	1.60919	0.804595	+	0.593824i	$0.202382\pi$
$620$	0	0
$621$	7.38646	0.296408
$622$	−10.5320	−0.422294
$623$	14.6072	0.585225
$624$	15.1817	0.607755
$625$	0	0
$626$	−9.96968	−0.398469
$627$	61.6312	2.46131
$628$	−3.36836	−0.134412
$629$	−5.63752	−0.224783
$630$	0	0
$631$	26.1768	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$631$	26.1768	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$632$	−7.61213	−0.302794
$633$	−43.8686	−1.74362
$634$	24.3961	0.968894
$635$	0	0
$636$	25.5125	1.01164
$637$	21.5999	0.855820
$638$	38.2882	1.51584
$639$	−20.3634	−0.805565
$640$	0	0
$641$	20.9525	0.827576	0.413788	−	0.910373i	$-0.364206\pi$
$641$	20.9525	0.827576	0.413788	+	0.910373i	$0.364206\pi$
$642$	6.45580	0.254790
$643$	17.7586	0.700331	0.350165	−	0.936688i	$-0.386125\pi$
$643$	17.7586	0.700331	0.350165	+	0.936688i	$0.386125\pi$
$644$	7.84955	0.309316
$645$	0	0
$646$	26.8822	1.05767
$647$	−11.5550	−0.454274	−0.227137	−	0.973863i	$-0.572937\pi$
$647$	−11.5550	−0.454274	−0.227137	+	0.973863i	$0.572937\pi$
$648$	4.19394	0.164753
$649$	−55.2301	−2.16797
$650$	0	0
$651$	−65.4577	−2.56549
$652$	−4.31994	−0.169182
$653$	7.78655	0.304711	0.152356	−	0.988326i	$-0.451314\pi$
$653$	7.78655	0.304711	0.152356	+	0.988326i	$0.451314\pi$
$654$	4.64974	0.181819
$655$	0	0
$656$	−10.3503	−0.404110
$657$	42.3693	1.65298
$658$	19.4763	0.759264
$659$	−13.7029	−0.533789	−0.266894	−	0.963726i	$-0.585998\pi$
$659$	−13.7029	−0.533789	−0.266894	+	0.963726i	$0.585998\pi$
$660$	0	0
$661$	42.0263	1.63464	0.817318	−	0.576187i	$-0.195460\pi$
$661$	42.0263	1.63464	0.817318	+	0.576187i	$0.195460\pi$
$662$	−34.3561	−1.33529
$663$	−85.5872	−3.32393
$664$	−13.9199	−0.540195
$665$	0	0
$666$	−4.15633	−0.161054
$667$	−18.9234	−0.732716
$668$	14.6678	0.567516
$669$	49.3258	1.90705
$670$	0	0
$671$	−23.6712	−0.913815
$672$	−8.79384	−0.339230
$673$	−29.0943	−1.12150	−0.560751	−	0.827985i	$-0.689487\pi$
$673$	−29.0943	−1.12150	−0.560751	+	0.827985i	$0.689487\pi$
$674$	−20.6385	−0.794964
$675$	0	0
$676$	19.2071	0.738735
$677$	25.3112	0.972790	0.486395	−	0.873739i	$-0.338312\pi$
$677$	25.3112	0.972790	0.486395	+	0.873739i	$0.338312\pi$
$678$	13.6253	0.523277
$679$	17.5877	0.674954
$680$	0	0
$681$	−30.6458	−1.17435
$682$	35.9633	1.37711
$683$	4.04349	0.154720	0.0773599	−	0.997003i	$-0.475351\pi$
$683$	4.04349	0.154720	0.0773599	+	0.997003i	$0.475351\pi$
$684$	19.8192	0.757807
$685$	0	0
$686$	10.4993	0.400865
$687$	68.5863	2.61673
$688$	−11.4314	−0.435817
$689$	54.1232	2.06193
$690$	0	0
$691$	7.92970	0.301660	0.150830	−	0.988560i	$-0.451805\pi$
$691$	7.92970	0.301660	0.150830	+	0.988560i	$0.451805\pi$
$692$	−8.05079	−0.306045
$693$	66.0118	2.50758
$694$	−4.48612	−0.170291
$695$	0	0
$696$	21.1998	0.803577
$697$	58.3498	2.21016
$698$	20.1949	0.764388
$699$	26.1319	0.988399
$700$	0	0
$701$	−48.3611	−1.82657	−0.913286	−	0.407319i	$-0.866464\pi$
$701$	−48.3611	−1.82657	−0.913286	+	0.407319i	$0.866464\pi$
$702$	−17.5550	−0.662571
$703$	4.76845	0.179846
$704$	4.83146	0.182092
$705$	0	0
$706$	25.9102	0.975143
$707$	−26.7123	−1.00462
$708$	−30.5804	−1.14928
$709$	7.45183	0.279859	0.139930	−	0.990161i	$-0.455312\pi$
$709$	7.45183	0.279859	0.139930	+	0.990161i	$0.455312\pi$
$710$	0	0
$711$	31.6385	1.18654
$712$	−4.44358	−0.166530
$713$	−17.7743	−0.665654
$714$	49.5755	1.85532
$715$	0	0
$716$	−5.05079	−0.188757
$717$	43.5731	1.62727
$718$	−4.43866	−0.165649
$719$	−6.45088	−0.240577	−0.120289	−	0.992739i	$-0.538382\pi$
$719$	−6.45088	−0.240577	−0.120289	+	0.992739i	$0.538382\pi$
$720$	0	0
$721$	−6.26774	−0.233423
$722$	−3.73813	−0.139119
$723$	35.2433	1.31071
$724$	3.73813	0.138927
$725$	0	0
$726$	−33.0191	−1.22545
$727$	37.1754	1.37876	0.689379	−	0.724401i	$-0.257883\pi$
$727$	37.1754	1.37876	0.689379	+	0.724401i	$0.257883\pi$
$728$	−18.6556	−0.691423
$729$	−42.2565	−1.56505
$730$	0	0
$731$	64.4445	2.38357
$732$	−13.1065	−0.484430
$733$	−37.1754	−1.37310	−0.686552	−	0.727081i	$-0.740877\pi$
$733$	−37.1754	−1.37310	−0.686552	+	0.727081i	$0.740877\pi$
$734$	−28.2858	−1.04405
$735$	0	0
$736$	−2.38787	−0.0880182
$737$	10.1490	0.373844
$738$	43.0191	1.58355
$739$	6.74940	0.248281	0.124140	−	0.992265i	$-0.460383\pi$
$739$	6.74940	0.248281	0.124140	+	0.992265i	$0.460383\pi$
$740$	0	0
$741$	72.3933	2.65943
$742$	−31.3503	−1.15090
$743$	−23.9248	−0.877715	−0.438857	−	0.898557i	$-0.644617\pi$
$743$	−23.9248	−0.877715	−0.438857	+	0.898557i	$0.644617\pi$
$744$	19.9126	0.730030
$745$	0	0
$746$	2.71862	0.0995358
$747$	57.8554	2.11682
$748$	−27.2374	−0.995899
$749$	−7.93303	−0.289866
$750$	0	0
$751$	−5.27504	−0.192489	−0.0962445	−	0.995358i	$-0.530683\pi$
$751$	−5.27504	−0.192489	−0.0962445	+	0.995358i	$0.530683\pi$
$752$	−5.92478	−0.216054
$753$	34.9126	1.27228
$754$	44.9741	1.63786
$755$	0	0
$756$	10.1685	0.369826
$757$	3.62672	0.131815	0.0659076	−	0.997826i	$-0.479006\pi$
$757$	3.62672	0.131815	0.0659076	+	0.997826i	$0.479006\pi$
$758$	−18.1514	−0.659289
$759$	30.8627	1.12025
$760$	0	0
$761$	−4.03173	−0.146150	−0.0730751	−	0.997326i	$-0.523281\pi$
$761$	−4.03173	−0.146150	−0.0730751	+	0.997326i	$0.523281\pi$
$762$	20.3307	0.736505
$763$	−5.71370	−0.206850
$764$	−12.5501	−0.454046
$765$	0	0
$766$	3.90668	0.141154
$767$	−64.8745	−2.34248
$768$	2.67513	0.0965305
$769$	−27.6072	−0.995541	−0.497771	−	0.867309i	$-0.665848\pi$
$769$	−27.6072	−0.995541	−0.497771	+	0.867309i	$0.665848\pi$
$770$	0	0
$771$	55.3766	1.99434
$772$	20.8822	0.751568
$773$	19.4798	0.700639	0.350319	−	0.936630i	$-0.386073\pi$
$773$	19.4798	0.700639	0.350319	+	0.936630i	$0.386073\pi$
$774$	47.5125	1.70780
$775$	0	0
$776$	−5.35026	−0.192063
$777$	8.79384	0.315477
$778$	−5.28726	−0.189557
$779$	−49.3547	−1.76832
$780$	0	0
$781$	−23.6712	−0.847021
$782$	13.4617	0.481389
$783$	−24.5139	−0.876055
$784$	3.80606	0.135931
$785$	0	0
$786$	−35.7948	−1.27676
$787$	−14.1768	−0.505348	−0.252674	−	0.967551i	$-0.581310\pi$
$787$	−14.1768	−0.505348	−0.252674	+	0.967551i	$0.581310\pi$
$788$	−1.07030	−0.0381278
$789$	−54.6761	−1.94652
$790$	0	0
$791$	−16.7431	−0.595315
$792$	−20.0811	−0.713551
$793$	−27.8046	−0.987372
$794$	−4.15396	−0.147418
$795$	0	0
$796$	−22.1016	−0.783369
$797$	21.8764	0.774900	0.387450	−	0.921891i	$-0.373356\pi$
$797$	21.8764	0.774900	0.387450	+	0.921891i	$0.373356\pi$
$798$	−41.9330	−1.48441
$799$	33.4010	1.18164
$800$	0	0
$801$	18.4690	0.652569
$802$	22.1817	0.783264
$803$	49.2516	1.73805
$804$	5.61942	0.198182
$805$	0	0
$806$	42.2433	1.48796
$807$	16.7250	0.588747
$808$	8.12601	0.285872
$809$	−12.2130	−0.429386	−0.214693	−	0.976682i	$-0.568875\pi$
$809$	−12.2130	−0.429386	−0.214693	+	0.976682i	$0.568875\pi$
$810$	0	0
$811$	8.64974	0.303733	0.151867	−	0.988401i	$-0.451472\pi$
$811$	8.64974	0.303733	0.151867	+	0.988401i	$0.451472\pi$
$812$	−26.0508	−0.914203
$813$	−2.20474	−0.0773236
$814$	−4.83146	−0.169342
$815$	0	0
$816$	−15.0811	−0.527944
$817$	−54.5099	−1.90706
$818$	−0.650693	−0.0227510
$819$	77.5388	2.70943
$820$	0	0
$821$	−41.7972	−1.45873	−0.729366	−	0.684124i	$-0.760185\pi$
$821$	−41.7972	−1.45873	−0.729366	+	0.684124i	$0.760185\pi$
$822$	−29.9257	−1.04378
$823$	41.9633	1.46275	0.731375	−	0.681975i	$-0.238879\pi$
$823$	41.9633	1.46275	0.731375	+	0.681975i	$0.238879\pi$
$824$	1.90668	0.0664223
$825$	0	0
$826$	37.5778	1.30750
$827$	23.5950	0.820478	0.410239	−	0.911978i	$-0.365445\pi$
$827$	23.5950	0.820478	0.410239	+	0.911978i	$0.365445\pi$
$828$	9.92478	0.344910
$829$	39.4396	1.36979	0.684897	−	0.728640i	$-0.259847\pi$
$829$	39.4396	1.36979	0.684897	+	0.728640i	$0.259847\pi$
$830$	0	0
$831$	−48.6516	−1.68771
$832$	5.67513	0.196750
$833$	−21.4568	−0.743433
$834$	45.6615	1.58113
$835$	0	0
$836$	23.0386	0.796805
$837$	−23.0254	−0.795874
$838$	−10.8169	−0.373662
$839$	33.4495	1.15480	0.577402	−	0.816460i	$-0.304067\pi$
$839$	33.4495	1.15480	0.577402	+	0.816460i	$0.304067\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	−2.05079	−0.0706747
$843$	66.9946	2.30742
$844$	−16.3987	−0.564466
$845$	0	0
$846$	24.6253	0.846635
$847$	40.5745	1.39416
$848$	9.53690	0.327499
$849$	35.0590	1.20322
$850$	0	0
$851$	2.38787	0.0818552
$852$	−13.1065	−0.449021
$853$	−29.5101	−1.01041	−0.505203	−	0.863000i	$-0.668582\pi$
$853$	−29.5101	−1.01041	−0.505203	+	0.863000i	$0.668582\pi$
$854$	16.1055	0.551120
$855$	0	0
$856$	2.41327	0.0824837
$857$	36.4133	1.24385	0.621927	−	0.783075i	$-0.286350\pi$
$857$	36.4133	1.24385	0.621927	+	0.783075i	$0.286350\pi$
$858$	−73.3498	−2.50412
$859$	−50.7685	−1.73220	−0.866099	−	0.499873i	$-0.833380\pi$
$859$	−50.7685	−1.73220	−0.866099	+	0.499873i	$0.833380\pi$
$860$	0	0
$861$	−91.0186	−3.10191
$862$	−15.4861	−0.527459
$863$	44.6032	1.51831	0.759156	−	0.650909i	$-0.225612\pi$
$863$	44.6032	1.51831	0.759156	+	0.650909i	$0.225612\pi$
$864$	−3.09332	−0.105237
$865$	0	0
$866$	−35.8773	−1.21916
$867$	39.5428	1.34294
$868$	−24.4690	−0.830531
$869$	36.7777	1.24760
$870$	0	0
$871$	11.9213	0.403937
$872$	1.73813	0.0588607
$873$	22.2374	0.752623
$874$	−11.3865	−0.385153
$875$	0	0
$876$	27.2701	0.921372
$877$	−42.4079	−1.43201	−0.716006	−	0.698094i	$-0.754032\pi$
$877$	−42.4079	−1.43201	−0.716006	+	0.698094i	$0.754032\pi$
$878$	26.7005	0.901099
$879$	25.8980	0.873517
$880$	0	0
$881$	−11.0435	−0.372065	−0.186032	−	0.982544i	$-0.559563\pi$
$881$	−11.0435	−0.372065	−0.186032	+	0.982544i	$0.559563\pi$
$882$	−15.8192	−0.532661
$883$	3.52847	0.118742	0.0593712	−	0.998236i	$-0.481090\pi$
$883$	3.52847	0.118742	0.0593712	+	0.998236i	$0.481090\pi$
$884$	−31.9937	−1.07606
$885$	0	0
$886$	−5.75765	−0.193432
$887$	−11.8035	−0.396323	−0.198162	−	0.980169i	$-0.563497\pi$
$887$	−11.8035	−0.396323	−0.198162	+	0.980169i	$0.563497\pi$
$888$	−2.67513	−0.0897715
$889$	−24.9829	−0.837898
$890$	0	0
$891$	−20.2628	−0.678830
$892$	18.4387	0.617372
$893$	−28.2520	−0.945418
$894$	20.2496	0.677249
$895$	0	0
$896$	−3.28726	−0.109820
$897$	36.2520	1.21042
$898$	31.1803	1.04050
$899$	58.9887	1.96738
$900$	0	0
$901$	−53.7645	−1.79115
$902$	50.0068	1.66505
$903$	−100.526	−3.34528
$904$	5.09332	0.169401
$905$	0	0
$906$	−1.73813	−0.0577457
$907$	34.2579	1.13751	0.568757	−	0.822505i	$-0.307424\pi$
$907$	34.2579	1.13751	0.568757	+	0.822505i	$0.307424\pi$
$908$	−11.4558	−0.380174
$909$	−33.7743	−1.12022
$910$	0	0
$911$	31.0214	1.02779	0.513893	−	0.857854i	$-0.328203\pi$
$911$	31.0214	1.02779	0.513893	+	0.857854i	$0.328203\pi$
$912$	12.7562	0.422401
$913$	67.2532	2.22575
$914$	−7.27011	−0.240474
$915$	0	0
$916$	25.6385	0.847119
$917$	43.9854	1.45253
$918$	17.4387	0.575561
$919$	18.9986	0.626706	0.313353	−	0.949637i	$-0.398548\pi$
$919$	18.9986	0.626706	0.313353	+	0.949637i	$0.398548\pi$
$920$	0	0
$921$	−54.6629	−1.80120
$922$	18.6253	0.613392
$923$	−27.8046	−0.915201
$924$	42.4871	1.39772
$925$	0	0
$926$	−1.89209	−0.0621779
$927$	−7.92478	−0.260284
$928$	7.92478	0.260144
$929$	55.4530	1.81935	0.909677	−	0.415318i	$-0.136330\pi$
$929$	55.4530	1.81935	0.909677	+	0.415318i	$0.136330\pi$
$930$	0	0
$931$	18.1490	0.594810
$932$	9.76845	0.319976
$933$	28.1744	0.922389
$934$	−1.21440	−0.0397365
$935$	0	0
$936$	−23.5877	−0.770988
$937$	27.9683	0.913683	0.456842	−	0.889548i	$-0.348981\pi$
$937$	27.9683	0.913683	0.456842	+	0.889548i	$0.348981\pi$
$938$	−6.90526	−0.225465
$939$	26.6702	0.870349
$940$	0	0
$941$	52.3390	1.70620	0.853101	−	0.521745i	$-0.174719\pi$
$941$	52.3390	1.70620	0.853101	+	0.521745i	$0.174719\pi$
$942$	9.01080	0.293588
$943$	−24.7151	−0.804835
$944$	−11.4314	−0.372059
$945$	0	0
$946$	55.2301	1.79569
$947$	−10.0752	−0.327401	−0.163700	−	0.986510i	$-0.552343\pi$
$947$	−10.0752	−0.327401	−0.163700	+	0.986510i	$0.552343\pi$
$948$	20.3634	0.661374
$949$	57.8519	1.87795
$950$	0	0
$951$	−65.2628	−2.11629
$952$	18.5320	0.600625
$953$	10.2809	0.333032	0.166516	−	0.986039i	$-0.446748\pi$
$953$	10.2809	0.333032	0.166516	+	0.986039i	$0.446748\pi$
$954$	−39.6385	−1.28334
$955$	0	0
$956$	16.2882	0.526798
$957$	−102.426	−3.31096
$958$	40.7694	1.31720
$959$	36.7734	1.18747
$960$	0	0
$961$	24.4069	0.787320
$962$	−5.67513	−0.182974
$963$	−10.0303	−0.323222
$964$	13.1744	0.424320
$965$	0	0
$966$	−20.9986	−0.675618
$967$	−30.7612	−0.989212	−0.494606	−	0.869117i	$-0.664688\pi$
$967$	−30.7612	−0.989212	−0.494606	+	0.869117i	$0.664688\pi$
$968$	−12.3430	−0.396718
$969$	−71.9135	−2.31019
$970$	0	0
$971$	52.9086	1.69792	0.848959	−	0.528459i	$-0.177230\pi$
$971$	52.9086	1.69792	0.848959	+	0.528459i	$0.177230\pi$
$972$	−20.4993	−0.657515
$973$	−56.1098	−1.79880
$974$	23.8578	0.764453
$975$	0	0
$976$	−4.89938	−0.156825
$977$	−16.5901	−0.530763	−0.265382	−	0.964143i	$-0.585498\pi$
$977$	−16.5901	−0.530763	−0.265382	+	0.964143i	$0.585498\pi$
$978$	11.5564	0.369533
$979$	21.4690	0.686151
$980$	0	0
$981$	−7.22425	−0.230653
$982$	−36.0362	−1.14996
$983$	36.3146	1.15825	0.579127	−	0.815237i	$-0.303393\pi$
$983$	36.3146	1.15825	0.579127	+	0.815237i	$0.303393\pi$
$984$	27.6883	0.882671
$985$	0	0
$986$	−44.6761	−1.42278
$987$	−52.1016	−1.65841
$988$	27.0616	0.860944
$989$	−27.2966	−0.867983
$990$	0	0
$991$	3.95272	0.125562	0.0627812	−	0.998027i	$-0.480003\pi$
$991$	3.95272	0.125562	0.0627812	+	0.998027i	$0.480003\pi$
$992$	7.44358	0.236334
$993$	91.9072	2.91659
$994$	16.1055	0.510837
$995$	0	0
$996$	37.2374	1.17991
$997$	−41.6625	−1.31946	−0.659732	−	0.751501i	$-0.729330\pi$
$997$	−41.6625	−1.31946	−0.659732	+	0.751501i	$0.729330\pi$
$998$	31.4558	0.995716
$999$	3.09332	0.0978684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.ba.1.3	✓	3
5.2	odd	4		1850.2.b.n.149.1		6
5.3	odd	4		1850.2.b.n.149.6		6
5.4	even	2		1850.2.a.bb.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.ba.1.3	✓	3	1.1	even	1	trivial
1850.2.a.bb.1.1	yes	3	5.4	even	2
1850.2.b.n.149.1		6	5.2	odd	4
1850.2.b.n.149.6		6	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.67513 q^{3} +1.00000 q^{4} -2.67513 q^{6} +3.28726 q^{7} -1.00000 q^{8} +4.15633 q^{9} +4.83146 q^{11} +2.67513 q^{12} +5.67513 q^{13} -3.28726 q^{14} +1.00000 q^{16} -5.63752 q^{17} -4.15633 q^{18} +4.76845 q^{19} +8.79384 q^{21} -4.83146 q^{22} +2.38787 q^{23} -2.67513 q^{24} -5.67513 q^{26} +3.09332 q^{27} +3.28726 q^{28} -7.92478 q^{29} -7.44358 q^{31} -1.00000 q^{32} +12.9248 q^{33} +5.63752 q^{34} +4.15633 q^{36} +1.00000 q^{37} -4.76845 q^{38} +15.1817 q^{39} -10.3503 q^{41} -8.79384 q^{42} -11.4314 q^{43} +4.83146 q^{44} -2.38787 q^{46} -5.92478 q^{47} +2.67513 q^{48} +3.80606 q^{49} -15.0811 q^{51} +5.67513 q^{52} +9.53690 q^{53} -3.09332 q^{54} -3.28726 q^{56} +12.7562 q^{57} +7.92478 q^{58} -11.4314 q^{59} -4.89938 q^{61} +7.44358 q^{62} +13.6629 q^{63} +1.00000 q^{64} -12.9248 q^{66} +2.10062 q^{67} -5.63752 q^{68} +6.38787 q^{69} -4.89938 q^{71} -4.15633 q^{72} +10.1939 q^{73} -1.00000 q^{74} +4.76845 q^{76} +15.8822 q^{77} -15.1817 q^{78} +7.61213 q^{79} -4.19394 q^{81} +10.3503 q^{82} +13.9199 q^{83} +8.79384 q^{84} +11.4314 q^{86} -21.1998 q^{87} -4.83146 q^{88} +4.44358 q^{89} +18.6556 q^{91} +2.38787 q^{92} -19.9126 q^{93} +5.92478 q^{94} -2.67513 q^{96} +5.35026 q^{97} -3.80606 q^{98} +20.0811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9} - q^{11} + 3 q^{12} + 12 q^{13} - 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{18} + 3 q^{19} + q^{22} + 8 q^{23} - 3 q^{24} - 12 q^{26}+ \cdots + 28 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 2 * q^9 - q^11 + 3 * q^12 + 12 * q^13 - 4 * q^14 + 3 * q^16 - q^17 - 2 * q^18 + 3 * q^19 + q^22 + 8 * q^23 - 3 * q^24 - 12 * q^26 + 3 * q^27 + 4 * q^28 - 2 * q^29 - 6 * q^31 - 3 * q^32 + 17 * q^33 + q^34 + 2 * q^36 + 3 * q^37 - 3 * q^38 + 20 * q^39 - 21 * q^41 + 8 * q^43 - q^44 - 8 * q^46 + 4 * q^47 + 3 * q^48 + 11 * q^49 - 13 * q^51 + 12 * q^52 + 6 * q^53 - 3 * q^54 - 4 * q^56 + q^57 + 2 * q^58 + 8 * q^59 - 8 * q^61 + 6 * q^62 + 10 * q^63 + 3 * q^64 - 17 * q^66 + 13 * q^67 - q^68 + 20 * q^69 - 8 * q^71 - 2 * q^72 + 31 * q^73 - 3 * q^74 + 3 * q^76 + 2 * q^77 - 20 * q^78 + 22 * q^79 - 13 * q^81 + 21 * q^82 + 7 * q^83 - 8 * q^86 - 10 * q^87 + q^88 - 3 * q^89 + 12 * q^91 + 8 * q^92 - 12 * q^93 - 4 * q^94 - 3 * q^96 + 6 * q^97 - 11 * q^98 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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