LMFDB - Embedded newform 1850.2.a.z.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:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.59774$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +2.59774 q^{7} -1.00000 q^{8} +8.19547 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +2.59774 q^{7} -1.00000 q^{8} +8.19547 q^{9} +4.74823 q^{11} +3.34596 q^{12} -6.69193 q^{13} -2.59774 q^{14} +1.00000 q^{16} +0.748228 q^{17} -8.19547 q^{18} -3.34596 q^{19} +8.69193 q^{21} -4.74823 q^{22} -1.49646 q^{23} -3.34596 q^{24} +6.69193 q^{26} +17.3839 q^{27} +2.59774 q^{28} +3.94370 q^{29} +7.79321 q^{31} -1.00000 q^{32} +15.8874 q^{33} -0.748228 q^{34} +8.19547 q^{36} +1.00000 q^{37} +3.34596 q^{38} -22.3909 q^{39} -6.44724 q^{41} -8.69193 q^{42} -1.94370 q^{43} +4.74823 q^{44} +1.49646 q^{46} -1.84951 q^{47} +3.34596 q^{48} -0.251772 q^{49} +2.50354 q^{51} -6.69193 q^{52} -10.4472 q^{53} -17.3839 q^{54} -2.59774 q^{56} -11.1955 q^{57} -3.94370 q^{58} +5.84951 q^{59} +7.94370 q^{61} -7.79321 q^{62} +21.2897 q^{63} +1.00000 q^{64} -15.8874 q^{66} -1.84951 q^{67} +0.748228 q^{68} -5.00709 q^{69} -3.88740 q^{71} -8.19547 q^{72} +7.49646 q^{73} -1.00000 q^{74} -3.34596 q^{76} +12.3346 q^{77} +22.3909 q^{78} -16.5414 q^{79} +33.5793 q^{81} +6.44724 q^{82} -15.2334 q^{83} +8.69193 q^{84} +1.94370 q^{86} +13.1955 q^{87} -4.74823 q^{88} +6.00000 q^{89} -17.3839 q^{91} -1.49646 q^{92} +26.0758 q^{93} +1.84951 q^{94} -3.34596 q^{96} +10.4472 q^{97} +0.251772 q^{98} +38.9140 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9	$ 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9} + 11 q^{11} - q^{14} + 3 q^{16} - q^{17} - 11 q^{18} + 6 q^{21} - 11 q^{22} + 2 q^{23} + 12 q^{27} + q^{28} - 5 q^{29} + 3 q^{31} - 3 q^{32} + 14 q^{33} + q^{34} + 11 q^{36} + 3 q^{37} - 40 q^{39} - 9 q^{41} - 6 q^{42} + 11 q^{43} + 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} + 14 q^{51} - 21 q^{53} - 12 q^{54} - q^{56} - 20 q^{57} + 5 q^{58} + 14 q^{59} + 7 q^{61} - 3 q^{62} + 37 q^{63} + 3 q^{64} - 14 q^{66} - 2 q^{67} - q^{68} - 28 q^{69} + 22 q^{71} - 11 q^{72} + 16 q^{73} - 3 q^{74} - 7 q^{77} + 40 q^{78} - 26 q^{79} + 47 q^{81} + 9 q^{82} - 2 q^{83} + 6 q^{84} - 11 q^{86} + 26 q^{87} - 11 q^{88} + 18 q^{89} - 12 q^{91} + 2 q^{92} + 18 q^{93} + 2 q^{94} + 21 q^{97} + 4 q^{98} + 19 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9 + 11 * q^11 - q^14 + 3 * q^16 - q^17 - 11 * q^18 + 6 * q^21 - 11 * q^22 + 2 * q^23 + 12 * q^27 + q^28 - 5 * q^29 + 3 * q^31 - 3 * q^32 + 14 * q^33 + q^34 + 11 * q^36 + 3 * q^37 - 40 * q^39 - 9 * q^41 - 6 * q^42 + 11 * q^43 + 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 + 14 * q^51 - 21 * q^53 - 12 * q^54 - q^56 - 20 * q^57 + 5 * q^58 + 14 * q^59 + 7 * q^61 - 3 * q^62 + 37 * q^63 + 3 * q^64 - 14 * q^66 - 2 * q^67 - q^68 - 28 * q^69 + 22 * q^71 - 11 * q^72 + 16 * q^73 - 3 * q^74 - 7 * q^77 + 40 * q^78 - 26 * q^79 + 47 * q^81 + 9 * q^82 - 2 * q^83 + 6 * q^84 - 11 * q^86 + 26 * q^87 - 11 * q^88 + 18 * q^89 - 12 * q^91 + 2 * q^92 + 18 * q^93 + 2 * q^94 + 21 * q^97 + 4 * q^98 + 19 * q^99

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
$3$	3.34596	1.93179	0.965896	−	0.258929i	$-0.0833695\pi$
$3$	3.34596	1.93179	0.965896	+	0.258929i	$0.0833695\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.34596	−1.36598
$7$	2.59774	0.981852	0.490926	−	0.871201i	$-0.336659\pi$
$7$	2.59774	0.981852	0.490926	+	0.871201i	$0.336659\pi$
$8$	−1.00000	−0.353553
$9$	8.19547	2.73182
$10$	0	0
$11$	4.74823	1.43164	0.715822	−	0.698282i	$-0.246052\pi$
$11$	4.74823	1.43164	0.715822	+	0.698282i	$0.246052\pi$
$12$	3.34596	0.965896
$13$	−6.69193	−1.85601	−0.928003	−	0.372572i	$-0.878476\pi$
$13$	−6.69193	−1.85601	−0.928003	+	0.372572i	$0.878476\pi$
$14$	−2.59774	−0.694274
$15$	0	0
$16$	1.00000	0.250000
$17$	0.748228	0.181472	0.0907360	−	0.995875i	$-0.471078\pi$
$17$	0.748228	0.181472	0.0907360	+	0.995875i	$0.471078\pi$
$18$	−8.19547	−1.93169
$19$	−3.34596	−0.767617	−0.383808	−	0.923413i	$-0.625388\pi$
$19$	−3.34596	−0.767617	−0.383808	+	0.923413i	$0.625388\pi$
$20$	0	0
$21$	8.69193	1.89673
$22$	−4.74823	−1.01233
$23$	−1.49646	−0.312033	−0.156016	−	0.987754i	$-0.549865\pi$
$23$	−1.49646	−0.312033	−0.156016	+	0.987754i	$0.549865\pi$
$24$	−3.34596	−0.682992
$25$	0	0
$26$	6.69193	1.31239
$27$	17.3839	3.34552
$28$	2.59774	0.490926
$29$	3.94370	0.732326	0.366163	−	0.930551i	$-0.380671\pi$
$29$	3.94370	0.732326	0.366163	+	0.930551i	$0.380671\pi$
$30$	0	0
$31$	7.79321	1.39970	0.699851	−	0.714289i	$-0.253250\pi$
$31$	7.79321	1.39970	0.699851	+	0.714289i	$0.253250\pi$
$32$	−1.00000	−0.176777
$33$	15.8874	2.76564
$34$	−0.748228	−0.128320
$35$	0	0
$36$	8.19547	1.36591
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	3.34596	0.542787
$39$	−22.3909	−3.58542
$40$	0	0
$41$	−6.44724	−1.00689	−0.503445	−	0.864027i	$-0.667934\pi$
$41$	−6.44724	−1.00689	−0.503445	+	0.864027i	$0.667934\pi$
$42$	−8.69193	−1.34119
$43$	−1.94370	−0.296411	−0.148206	−	0.988957i	$-0.547350\pi$
$43$	−1.94370	−0.296411	−0.148206	+	0.988957i	$0.547350\pi$
$44$	4.74823	0.715822
$45$	0	0
$46$	1.49646	0.220640
$47$	−1.84951	−0.269778	−0.134889	−	0.990861i	$-0.543068\pi$
$47$	−1.84951	−0.269778	−0.134889	+	0.990861i	$0.543068\pi$
$48$	3.34596	0.482948
$49$	−0.251772	−0.0359674
$50$	0	0
$51$	2.50354	0.350566
$52$	−6.69193	−0.928003
$53$	−10.4472	−1.43504	−0.717520	−	0.696538i	$-0.754723\pi$
$53$	−10.4472	−1.43504	−0.717520	+	0.696538i	$0.754723\pi$
$54$	−17.3839	−2.36564
$55$	0	0
$56$	−2.59774	−0.347137
$57$	−11.1955	−1.48288
$58$	−3.94370	−0.517833
$59$	5.84951	0.761541	0.380770	−	0.924670i	$-0.375659\pi$
$59$	5.84951	0.761541	0.380770	+	0.924670i	$0.375659\pi$
$60$	0	0
$61$	7.94370	1.01709	0.508543	−	0.861036i	$-0.330184\pi$
$61$	7.94370	1.01709	0.508543	+	0.861036i	$0.330184\pi$
$62$	−7.79321	−0.989738
$63$	21.2897	2.68225
$64$	1.00000	0.125000
$65$	0	0
$66$	−15.8874	−1.95560
$67$	−1.84951	−0.225953	−0.112977	−	0.993598i	$-0.536039\pi$
$67$	−1.84951	−0.225953	−0.112977	+	0.993598i	$0.536039\pi$
$68$	0.748228	0.0907360
$69$	−5.00709	−0.602783
$70$	0	0
$71$	−3.88740	−0.461349	−0.230675	−	0.973031i	$-0.574093\pi$
$71$	−3.88740	−0.461349	−0.230675	+	0.973031i	$0.574093\pi$
$72$	−8.19547	−0.965845
$73$	7.49646	0.877394	0.438697	−	0.898635i	$-0.355440\pi$
$73$	7.49646	0.877394	0.438697	+	0.898635i	$0.355440\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−3.34596	−0.383808
$77$	12.3346	1.40566
$78$	22.3909	2.53527
$79$	−16.5414	−1.86106	−0.930528	−	0.366220i	$-0.880652\pi$
$79$	−16.5414	−1.86106	−0.930528	+	0.366220i	$0.880652\pi$
$80$	0	0
$81$	33.5793	3.73104
$82$	6.44724	0.711979
$83$	−15.2334	−1.67208	−0.836039	−	0.548670i	$-0.815134\pi$
$83$	−15.2334	−1.67208	−0.836039	+	0.548670i	$0.815134\pi$
$84$	8.69193	0.948367
$85$	0	0
$86$	1.94370	0.209594
$87$	13.1955	1.41470
$88$	−4.74823	−0.506163
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−17.3839	−1.82232
$92$	−1.49646	−0.156016
$93$	26.0758	2.70393
$94$	1.84951	0.190762
$95$	0	0
$96$	−3.34596	−0.341496
$97$	10.4472	1.06076	0.530378	−	0.847761i	$-0.322050\pi$
$97$	10.4472	1.06076	0.530378	+	0.847761i	$0.322050\pi$
$98$	0.251772	0.0254328
$99$	38.9140	3.91100
$100$	0	0
$101$	−12.1884	−1.21279	−0.606395	−	0.795164i	$-0.707385\pi$
$101$	−12.1884	−1.21279	−0.606395	+	0.795164i	$0.707385\pi$
$102$	−2.50354	−0.247888
$103$	1.30807	0.128888	0.0644442	−	0.997921i	$-0.479473\pi$
$103$	1.30807	0.128888	0.0644442	+	0.997921i	$0.479473\pi$
$104$	6.69193	0.656197
$105$	0	0
$106$	10.4472	1.01473
$107$	3.04498	0.294369	0.147185	−	0.989109i	$-0.452979\pi$
$107$	3.04498	0.294369	0.147185	+	0.989109i	$0.452979\pi$
$108$	17.3839	1.67276
$109$	−1.44015	−0.137942	−0.0689709	−	0.997619i	$-0.521972\pi$
$109$	−1.44015	−0.137942	−0.0689709	+	0.997619i	$0.521972\pi$
$110$	0	0
$111$	3.34596	0.317585
$112$	2.59774	0.245463
$113$	−11.1392	−1.04788	−0.523942	−	0.851754i	$-0.675539\pi$
$113$	−11.1392	−1.04788	−0.523942	+	0.851754i	$0.675539\pi$
$114$	11.1955	1.04855
$115$	0	0
$116$	3.94370	0.366163
$117$	−54.8435	−5.07028
$118$	−5.84951	−0.538491
$119$	1.94370	0.178179
$120$	0	0
$121$	11.5457	1.04961
$122$	−7.94370	−0.719189
$123$	−21.5722	−1.94510
$124$	7.79321	0.699851
$125$	0	0
$126$	−21.2897	−1.89663
$127$	8.84242	0.784638	0.392319	−	0.919829i	$-0.371673\pi$
$127$	8.84242	0.784638	0.392319	+	0.919829i	$0.371673\pi$
$128$	−1.00000	−0.0883883
$129$	−6.50354	−0.572605
$130$	0	0
$131$	6.15049	0.537371	0.268686	−	0.963228i	$-0.413411\pi$
$131$	6.15049	0.537371	0.268686	+	0.963228i	$0.413411\pi$
$132$	15.8874	1.38282
$133$	−8.69193	−0.753686
$134$	1.84951	0.159773
$135$	0	0
$136$	−0.748228	−0.0641600
$137$	−10.8945	−0.930779	−0.465389	−	0.885106i	$-0.654086\pi$
$137$	−10.8945	−0.930779	−0.465389	+	0.885106i	$0.654086\pi$
$138$	5.00709	0.426232
$139$	1.04921	0.0889932	0.0444966	−	0.999010i	$-0.485832\pi$
$139$	1.04921	0.0889932	0.0444966	+	0.999010i	$0.485832\pi$
$140$	0	0
$141$	−6.18838	−0.521156
$142$	3.88740	0.326223
$143$	−31.7748	−2.65714
$144$	8.19547	0.682956
$145$	0	0
$146$	−7.49646	−0.620411
$147$	−0.842420	−0.0694816
$148$	1.00000	0.0821995
$149$	−4.18838	−0.343126	−0.171563	−	0.985173i	$-0.554882\pi$
$149$	−4.18838	−0.343126	−0.171563	+	0.985173i	$0.554882\pi$
$150$	0	0
$151$	−6.69193	−0.544581	−0.272291	−	0.962215i	$-0.587781\pi$
$151$	−6.69193	−0.544581	−0.272291	+	0.962215i	$0.587781\pi$
$152$	3.34596	0.271393
$153$	6.13208	0.495749
$154$	−12.3346	−0.993954
$155$	0	0
$156$	−22.3909	−1.79271
$157$	3.94370	0.314741	0.157371	−	0.987540i	$-0.449698\pi$
$157$	3.94370	0.314741	0.157371	+	0.987540i	$0.449698\pi$
$158$	16.5414	1.31597
$159$	−34.9561	−2.77220
$160$	0	0
$161$	−3.88740	−0.306370
$162$	−33.5793	−2.63824
$163$	12.1463	0.951368	0.475684	−	0.879616i	$-0.342201\pi$
$163$	12.1463	0.951368	0.475684	+	0.879616i	$0.342201\pi$
$164$	−6.44724	−0.503445
$165$	0	0
$166$	15.2334	1.18234
$167$	−17.0829	−1.32191	−0.660956	−	0.750425i	$-0.729849\pi$
$167$	−17.0829	−1.32191	−0.660956	+	0.750425i	$0.729849\pi$
$168$	−8.69193	−0.670597
$169$	31.7819	2.44476
$170$	0	0
$171$	−27.4217	−2.09699
$172$	−1.94370	−0.148206
$173$	15.3417	1.16641	0.583205	−	0.812325i	$-0.301798\pi$
$173$	15.3417	1.16641	0.583205	+	0.812325i	$0.301798\pi$
$174$	−13.1955	−1.00035
$175$	0	0
$176$	4.74823	0.357911
$177$	19.5722	1.47114
$178$	−6.00000	−0.449719
$179$	2.45148	0.183232	0.0916161	−	0.995794i	$-0.470797\pi$
$179$	2.45148	0.183232	0.0916161	+	0.995794i	$0.470797\pi$
$180$	0	0
$181$	13.1955	0.980812	0.490406	−	0.871494i	$-0.336849\pi$
$181$	13.1955	0.980812	0.490406	+	0.871494i	$0.336849\pi$
$182$	17.3839	1.28858
$183$	26.5793	1.96480
$184$	1.49646	0.110320
$185$	0	0
$186$	−26.0758	−1.91197
$187$	3.55276	0.259803
$188$	−1.84951	−0.134889
$189$	45.1586	3.28481
$190$	0	0
$191$	−21.7790	−1.57588	−0.787938	−	0.615755i	$-0.788851\pi$
$191$	−21.7790	−1.57588	−0.787938	+	0.615755i	$0.788851\pi$
$192$	3.34596	0.241474
$193$	12.9929	0.935250	0.467625	−	0.883927i	$-0.345110\pi$
$193$	12.9929	0.935250	0.467625	+	0.883927i	$0.345110\pi$
$194$	−10.4472	−0.750068
$195$	0	0
$196$	−0.251772	−0.0179837
$197$	−0.616147	−0.0438986	−0.0219493	−	0.999759i	$-0.506987\pi$
$197$	−0.616147	−0.0438986	−0.0219493	+	0.999759i	$0.506987\pi$
$198$	−38.9140	−2.76549
$199$	−1.54852	−0.109772	−0.0548859	−	0.998493i	$-0.517480\pi$
$199$	−1.54852	−0.109772	−0.0548859	+	0.998493i	$0.517480\pi$
$200$	0	0
$201$	−6.18838	−0.436495
$202$	12.1884	0.857572
$203$	10.2447	0.719036
$204$	2.50354	0.175283
$205$	0	0
$206$	−1.30807	−0.0911378
$207$	−12.2642	−0.852418
$208$	−6.69193	−0.464002
$209$	−15.8874	−1.09895
$210$	0	0
$211$	20.9366	1.44134	0.720668	−	0.693280i	$-0.243835\pi$
$211$	20.9366	1.44134	0.720668	+	0.693280i	$0.243835\pi$
$212$	−10.4472	−0.717520
$213$	−13.0071	−0.891231
$214$	−3.04498	−0.208150
$215$	0	0
$216$	−17.3839	−1.18282
$217$	20.2447	1.37430
$218$	1.44015	0.0975396
$219$	25.0829	1.69494
$220$	0	0
$221$	−5.00709	−0.336813
$222$	−3.34596	−0.224566
$223$	2.71034	0.181498	0.0907488	−	0.995874i	$-0.471074\pi$
$223$	2.71034	0.181498	0.0907488	+	0.995874i	$0.471074\pi$
$224$	−2.59774	−0.173568
$225$	0	0
$226$	11.1392	0.740966
$227$	−26.1321	−1.73445	−0.867224	−	0.497919i	$-0.834098\pi$
$227$	−26.1321	−1.73445	−0.867224	+	0.497919i	$0.834098\pi$
$228$	−11.1955	−0.741438
$229$	−24.7677	−1.63670	−0.818348	−	0.574723i	$-0.805110\pi$
$229$	−24.7677	−1.63670	−0.818348	+	0.574723i	$0.805110\pi$
$230$	0	0
$231$	41.2713	2.71545
$232$	−3.94370	−0.258916
$233$	12.2783	0.804381	0.402190	−	0.915556i	$-0.368249\pi$
$233$	12.2783	0.804381	0.402190	+	0.915556i	$0.368249\pi$
$234$	54.8435	3.58523
$235$	0	0
$236$	5.84951	0.380770
$237$	−55.3470	−3.59518
$238$	−1.94370	−0.125991
$239$	−17.1771	−1.11109	−0.555546	−	0.831486i	$-0.687491\pi$
$239$	−17.1771	−1.11109	−0.555546	+	0.831486i	$0.687491\pi$
$240$	0	0
$241$	−15.2713	−0.983708	−0.491854	−	0.870678i	$-0.663681\pi$
$241$	−15.2713	−0.983708	−0.491854	+	0.870678i	$0.663681\pi$
$242$	−11.5457	−0.742184
$243$	60.2036	3.86206
$244$	7.94370	0.508543
$245$	0	0
$246$	21.5722	1.37540
$247$	22.3909	1.42470
$248$	−7.79321	−0.494869
$249$	−50.9703	−3.23011
$250$	0	0
$251$	10.0379	0.633586	0.316793	−	0.948495i	$-0.397394\pi$
$251$	10.0379	0.633586	0.316793	+	0.948495i	$0.397394\pi$
$252$	21.2897	1.34112
$253$	−7.10552	−0.446720
$254$	−8.84242	−0.554823
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.9703	1.30809	0.654045	−	0.756456i	$-0.273071\pi$
$257$	20.9703	1.30809	0.654045	+	0.756456i	$0.273071\pi$
$258$	6.50354	0.404893
$259$	2.59774	0.161415
$260$	0	0
$261$	32.3205	2.00059
$262$	−6.15049	−0.379979
$263$	−22.3725	−1.37955	−0.689775	−	0.724024i	$-0.742290\pi$
$263$	−22.3725	−1.37955	−0.689775	+	0.724024i	$0.742290\pi$
$264$	−15.8874	−0.977802
$265$	0	0
$266$	8.69193	0.532936
$267$	20.0758	1.22862
$268$	−1.84951	−0.112977
$269$	−4.18838	−0.255370	−0.127685	−	0.991815i	$-0.540755\pi$
$269$	−4.18838	−0.255370	−0.127685	+	0.991815i	$0.540755\pi$
$270$	0	0
$271$	−12.1126	−0.735788	−0.367894	−	0.929868i	$-0.619921\pi$
$271$	−12.1126	−0.735788	−0.367894	+	0.929868i	$0.619921\pi$
$272$	0.748228	0.0453680
$273$	−58.1657	−3.52035
$274$	10.8945	0.658160
$275$	0	0
$276$	−5.00709	−0.301391
$277$	9.30807	0.559268	0.279634	−	0.960107i	$-0.409787\pi$
$277$	9.30807	0.559268	0.279634	+	0.960107i	$0.409787\pi$
$278$	−1.04921	−0.0629277
$279$	63.8690	3.82374
$280$	0	0
$281$	19.2713	1.14963	0.574813	−	0.818285i	$-0.305075\pi$
$281$	19.2713	1.14963	0.574813	+	0.818285i	$0.305075\pi$
$282$	6.18838	0.368513
$283$	13.6848	0.813479	0.406740	−	0.913544i	$-0.366666\pi$
$283$	13.6848	0.813479	0.406740	+	0.913544i	$0.366666\pi$
$284$	−3.88740	−0.230675
$285$	0	0
$286$	31.7748	1.87888
$287$	−16.7482	−0.988617
$288$	−8.19547	−0.482923
$289$	−16.4402	−0.967068
$290$	0	0
$291$	34.9561	2.04916
$292$	7.49646	0.438697
$293$	10.4472	0.610334	0.305167	−	0.952299i	$-0.401288\pi$
$293$	10.4472	0.610334	0.305167	+	0.952299i	$0.401288\pi$
$294$	0.842420	0.0491309
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	82.5425	4.78960
$298$	4.18838	0.242627
$299$	10.0142	0.579135
$300$	0	0
$301$	−5.04921	−0.291032
$302$	6.69193	0.385077
$303$	−40.7819	−2.34286
$304$	−3.34596	−0.191904
$305$	0	0
$306$	−6.13208	−0.350548
$307$	4.95502	0.282798	0.141399	−	0.989953i	$-0.454840\pi$
$307$	4.95502	0.282798	0.141399	+	0.989953i	$0.454840\pi$
$308$	12.3346	0.702831
$309$	4.37677	0.248985
$310$	0	0
$311$	3.68060	0.208708	0.104354	−	0.994540i	$-0.466723\pi$
$311$	3.68060	0.208708	0.104354	+	0.994540i	$0.466723\pi$
$312$	22.3909	1.26764
$313$	0.992912	0.0561227	0.0280614	−	0.999606i	$-0.491067\pi$
$313$	0.992912	0.0561227	0.0280614	+	0.999606i	$0.491067\pi$
$314$	−3.94370	−0.222556
$315$	0	0
$316$	−16.5414	−0.930528
$317$	6.55985	0.368438	0.184219	−	0.982885i	$-0.441024\pi$
$317$	6.55985	0.368438	0.184219	+	0.982885i	$0.441024\pi$
$318$	34.9561	1.96024
$319$	18.7256	1.04843
$320$	0	0
$321$	10.1884	0.568660
$322$	3.88740	0.216636
$323$	−2.50354	−0.139301
$324$	33.5793	1.86552
$325$	0	0
$326$	−12.1463	−0.672719
$327$	−4.81870	−0.266475
$328$	6.44724	0.355989
$329$	−4.80453	−0.264882
$330$	0	0
$331$	−18.0379	−0.991452	−0.495726	−	0.868479i	$-0.665098\pi$
$331$	−18.0379	−0.991452	−0.495726	+	0.868479i	$0.665098\pi$
$332$	−15.2334	−0.836039
$333$	8.19547	0.449109
$334$	17.0829	0.934733
$335$	0	0
$336$	8.69193	0.474183
$337$	14.8945	0.811354	0.405677	−	0.914016i	$-0.367036\pi$
$337$	14.8945	0.811354	0.405677	+	0.914016i	$0.367036\pi$
$338$	−31.7819	−1.72871
$339$	−37.2713	−2.02430
$340$	0	0
$341$	37.0039	2.00387
$342$	27.4217	1.48280
$343$	−18.8382	−1.01717
$344$	1.94370	0.104797
$345$	0	0
$346$	−15.3417	−0.824776
$347$	9.38385	0.503752	0.251876	−	0.967760i	$-0.418953\pi$
$347$	9.38385	0.503752	0.251876	+	0.967760i	$0.418953\pi$
$348$	13.1955	0.707351
$349$	−32.9703	−1.76486	−0.882429	−	0.470446i	$-0.844093\pi$
$349$	−32.9703	−1.76486	−0.882429	+	0.470446i	$0.844093\pi$
$350$	0	0
$351$	−116.331	−6.20931
$352$	−4.74823	−0.253081
$353$	−27.4260	−1.45974	−0.729869	−	0.683587i	$-0.760419\pi$
$353$	−27.4260	−1.45974	−0.729869	+	0.683587i	$0.760419\pi$
$354$	−19.5722	−1.04025
$355$	0	0
$356$	6.00000	0.317999
$357$	6.50354	0.344204
$358$	−2.45148	−0.129565
$359$	−25.0829	−1.32382	−0.661912	−	0.749582i	$-0.730255\pi$
$359$	−25.0829	−1.32382	−0.661912	+	0.749582i	$0.730255\pi$
$360$	0	0
$361$	−7.80453	−0.410765
$362$	−13.1955	−0.693539
$363$	38.6314	2.02762
$364$	−17.3839	−0.911161
$365$	0	0
$366$	−26.5793	−1.38932
$367$	−2.48513	−0.129723	−0.0648614	−	0.997894i	$-0.520661\pi$
$367$	−2.48513	−0.129723	−0.0648614	+	0.997894i	$0.520661\pi$
$368$	−1.49646	−0.0780082
$369$	−52.8382	−2.75065
$370$	0	0
$371$	−27.1392	−1.40900
$372$	26.0758	1.35197
$373$	35.1586	1.82045	0.910223	−	0.414119i	$-0.135910\pi$
$373$	35.1586	1.82045	0.910223	+	0.414119i	$0.135910\pi$
$374$	−3.55276	−0.183709
$375$	0	0
$376$	1.84951	0.0953810
$377$	−26.3909	−1.35920
$378$	−45.1586	−2.32271
$379$	24.6778	1.26761	0.633805	−	0.773492i	$-0.281492\pi$
$379$	24.6778	1.26761	0.633805	+	0.773492i	$0.281492\pi$
$380$	0	0
$381$	29.5864	1.51576
$382$	21.7790	1.11431
$383$	13.3839	0.683883	0.341941	−	0.939721i	$-0.388916\pi$
$383$	13.3839	0.683883	0.341941	+	0.939721i	$0.388916\pi$
$384$	−3.34596	−0.170748
$385$	0	0
$386$	−12.9929	−0.661322
$387$	−15.9295	−0.809743
$388$	10.4472	0.530378
$389$	18.2220	0.923894	0.461947	−	0.886908i	$-0.347151\pi$
$389$	18.2220	0.923894	0.461947	+	0.886908i	$0.347151\pi$
$390$	0	0
$391$	−1.11969	−0.0566252
$392$	0.251772	0.0127164
$393$	20.5793	1.03809
$394$	0.616147	0.0310410
$395$	0	0
$396$	38.9140	1.95550
$397$	2.22521	0.111680	0.0558399	−	0.998440i	$-0.482216\pi$
$397$	2.22521	0.111680	0.0558399	+	0.998440i	$0.482216\pi$
$398$	1.54852	0.0776204
$399$	−29.0829	−1.45596
$400$	0	0
$401$	12.3909	0.618774	0.309387	−	0.950936i	$-0.399876\pi$
$401$	12.3909	0.618774	0.309387	+	0.950936i	$0.399876\pi$
$402$	6.18838	0.308648
$403$	−52.1516	−2.59785
$404$	−12.1884	−0.606395
$405$	0	0
$406$	−10.2447	−0.508435
$407$	4.74823	0.235361
$408$	−2.50354	−0.123944
$409$	7.27125	0.359540	0.179770	−	0.983709i	$-0.442465\pi$
$409$	7.27125	0.359540	0.179770	+	0.983709i	$0.442465\pi$
$410$	0	0
$411$	−36.4525	−1.79807
$412$	1.30807	0.0644442
$413$	15.1955	0.747720
$414$	12.2642	0.602751
$415$	0	0
$416$	6.69193	0.328099
$417$	3.51063	0.171916
$418$	15.8874	0.777078
$419$	28.4809	1.39138	0.695691	−	0.718341i	$-0.255098\pi$
$419$	28.4809	1.39138	0.695691	+	0.718341i	$0.255098\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−20.9366	−1.01918
$423$	−15.1576	−0.736987
$424$	10.4472	0.507363
$425$	0	0
$426$	13.0071	0.630195
$427$	20.6356	0.998628
$428$	3.04498	0.147185
$429$	−106.317	−5.13305
$430$	0	0
$431$	1.51487	0.0729686	0.0364843	−	0.999334i	$-0.488384\pi$
$431$	1.51487	0.0729686	0.0364843	+	0.999334i	$0.488384\pi$
$432$	17.3839	0.836381
$433$	−24.9929	−1.20108	−0.600541	−	0.799594i	$-0.705048\pi$
$433$	−24.9929	−1.20108	−0.600541	+	0.799594i	$0.705048\pi$
$434$	−20.2447	−0.971776
$435$	0	0
$436$	−1.44015	−0.0689709
$437$	5.00709	0.239521
$438$	−25.0829	−1.19851
$439$	−5.10128	−0.243471	−0.121735	−	0.992563i	$-0.538846\pi$
$439$	−5.10128	−0.243471	−0.121735	+	0.992563i	$0.538846\pi$
$440$	0	0
$441$	−2.06339	−0.0982566
$442$	5.00709	0.238163
$443$	−25.7369	−1.22280	−0.611399	−	0.791323i	$-0.709393\pi$
$443$	−25.7369	−1.22280	−0.611399	+	0.791323i	$0.709393\pi$
$444$	3.34596	0.158792
$445$	0	0
$446$	−2.71034	−0.128338
$447$	−14.0142	−0.662848
$448$	2.59774	0.122731
$449$	−17.7748	−0.838844	−0.419422	−	0.907791i	$-0.637767\pi$
$449$	−17.7748	−0.838844	−0.419422	+	0.907791i	$0.637767\pi$
$450$	0	0
$451$	−30.6130	−1.44151
$452$	−11.1392	−0.523942
$453$	−22.3909	−1.05202
$454$	26.1321	1.22644
$455$	0	0
$456$	11.1955	0.524276
$457$	−34.3346	−1.60611	−0.803053	−	0.595907i	$-0.796793\pi$
$457$	−34.3346	−1.60611	−0.803053	+	0.595907i	$0.796793\pi$
$458$	24.7677	1.15732
$459$	13.0071	0.607119
$460$	0	0
$461$	−16.9508	−0.789477	−0.394738	−	0.918794i	$-0.629165\pi$
$461$	−16.9508	−0.789477	−0.394738	+	0.918794i	$0.629165\pi$
$462$	−41.2713	−1.92011
$463$	−10.9929	−0.510884	−0.255442	−	0.966824i	$-0.582221\pi$
$463$	−10.9929	−0.510884	−0.255442	+	0.966824i	$0.582221\pi$
$464$	3.94370	0.183082
$465$	0	0
$466$	−12.2783	−0.568783
$467$	−37.4175	−1.73148	−0.865738	−	0.500498i	$-0.833150\pi$
$467$	−37.4175	−1.73148	−0.865738	+	0.500498i	$0.833150\pi$
$468$	−54.8435	−2.53514
$469$	−4.80453	−0.221853
$470$	0	0
$471$	13.1955	0.608015
$472$	−5.84951	−0.269245
$473$	−9.22912	−0.424356
$474$	55.3470	2.54217
$475$	0	0
$476$	1.94370	0.0890893
$477$	−85.6201	−3.92027
$478$	17.1771	0.785660
$479$	−0.465654	−0.0212763	−0.0106381	−	0.999943i	$-0.503386\pi$
$479$	−0.465654	−0.0212763	−0.0106381	+	0.999943i	$0.503386\pi$
$480$	0	0
$481$	−6.69193	−0.305126
$482$	15.2713	0.695586
$483$	−13.0071	−0.591843
$484$	11.5457	0.524803
$485$	0	0
$486$	−60.2036	−2.73089
$487$	−34.9561	−1.58401	−0.792006	−	0.610514i	$-0.790963\pi$
$487$	−34.9561	−1.58401	−0.792006	+	0.610514i	$0.790963\pi$
$488$	−7.94370	−0.359594
$489$	40.6409	1.83785
$490$	0	0
$491$	−24.7819	−1.11839	−0.559195	−	0.829036i	$-0.688890\pi$
$491$	−24.7819	−1.11839	−0.559195	+	0.829036i	$0.688890\pi$
$492$	−21.5722	−0.972552
$493$	2.95079	0.132897
$494$	−22.3909	−1.00742
$495$	0	0
$496$	7.79321	0.349925
$497$	−10.0984	−0.452976
$498$	50.9703	2.28403
$499$	−2.33888	−0.104702	−0.0523512	−	0.998629i	$-0.516672\pi$
$499$	−2.33888	−0.104702	−0.0523512	+	0.998629i	$0.516672\pi$
$500$	0	0
$501$	−57.1586	−2.55366
$502$	−10.0379	−0.448013
$503$	18.6161	0.830053	0.415026	−	0.909809i	$-0.363772\pi$
$503$	18.6161	0.830053	0.415026	+	0.909809i	$0.363772\pi$
$504$	−21.2897	−0.948317
$505$	0	0
$506$	7.10552	0.315879
$507$	106.341	4.72277
$508$	8.84242	0.392319
$509$	6.89448	0.305593	0.152796	−	0.988258i	$-0.451172\pi$
$509$	6.89448	0.305593	0.152796	+	0.988258i	$0.451172\pi$
$510$	0	0
$511$	19.4738	0.861471
$512$	−1.00000	−0.0441942
$513$	−58.1657	−2.56808
$514$	−20.9703	−0.924959
$515$	0	0
$516$	−6.50354	−0.286303
$517$	−8.78188	−0.386227
$518$	−2.59774	−0.114138
$519$	51.3329	2.25326
$520$	0	0
$521$	4.86083	0.212957	0.106478	−	0.994315i	$-0.466042\pi$
$521$	4.86083	0.212957	0.106478	+	0.994315i	$0.466042\pi$
$522$	−32.3205	−1.41463
$523$	−22.7677	−0.995562	−0.497781	−	0.867303i	$-0.665852\pi$
$523$	−22.7677	−0.995562	−0.497781	+	0.867303i	$0.665852\pi$
$524$	6.15049	0.268686
$525$	0	0
$526$	22.3725	0.975489
$527$	5.83110	0.254007
$528$	15.8874	0.691410
$529$	−20.7606	−0.902636
$530$	0	0
$531$	47.9395	2.08040
$532$	−8.69193	−0.376843
$533$	43.1445	1.86879
$534$	−20.0758	−0.868764
$535$	0	0
$536$	1.84951	0.0798865
$537$	8.20256	0.353967
$538$	4.18838	0.180574
$539$	−1.19547	−0.0514926
$540$	0	0
$541$	8.99291	0.386636	0.193318	−	0.981136i	$-0.438075\pi$
$541$	8.99291	0.386636	0.193318	+	0.981136i	$0.438075\pi$
$542$	12.1126	0.520281
$543$	44.1516	1.89472
$544$	−0.748228	−0.0320800
$545$	0	0
$546$	58.1657	2.48926
$547$	18.8382	0.805463	0.402731	−	0.915318i	$-0.368061\pi$
$547$	18.8382	0.805463	0.402731	+	0.915318i	$0.368061\pi$
$548$	−10.8945	−0.465389
$549$	65.1023	2.77850
$550$	0	0
$551$	−13.1955	−0.562146
$552$	5.00709	0.213116
$553$	−42.9703	−1.82728
$554$	−9.30807	−0.395462
$555$	0	0
$556$	1.04921	0.0444966
$557$	−3.69901	−0.156732	−0.0783661	−	0.996925i	$-0.524970\pi$
$557$	−3.69901	−0.156732	−0.0783661	+	0.996925i	$0.524970\pi$
$558$	−63.8690	−2.70379
$559$	13.0071	0.550141
$560$	0	0
$561$	11.8874	0.501886
$562$	−19.2713	−0.812909
$563$	14.3488	0.604730	0.302365	−	0.953192i	$-0.402224\pi$
$563$	14.3488	0.604730	0.302365	+	0.953192i	$0.402224\pi$
$564$	−6.18838	−0.260578
$565$	0	0
$566$	−13.6848	−0.575217
$567$	87.2302	3.66332
$568$	3.88740	0.163112
$569$	15.9858	0.670161	0.335080	−	0.942190i	$-0.391237\pi$
$569$	15.9858	0.670161	0.335080	+	0.942190i	$0.391237\pi$
$570$	0	0
$571$	−30.2079	−1.26416	−0.632080	−	0.774903i	$-0.717799\pi$
$571$	−30.2079	−1.26416	−0.632080	+	0.774903i	$0.717799\pi$
$572$	−31.7748	−1.32857
$573$	−72.8718	−3.04426
$574$	16.7482	0.699058
$575$	0	0
$576$	8.19547	0.341478
$577$	19.1813	0.798528	0.399264	−	0.916836i	$-0.369266\pi$
$577$	19.1813	0.798528	0.399264	+	0.916836i	$0.369266\pi$
$578$	16.4402	0.683820
$579$	43.4738	1.80671
$580$	0	0
$581$	−39.5722	−1.64173
$582$	−34.9561	−1.44898
$583$	−49.6059	−2.05447
$584$	−7.49646	−0.310206
$585$	0	0
$586$	−10.4472	−0.431572
$587$	26.6130	1.09844	0.549218	−	0.835679i	$-0.314926\pi$
$587$	26.6130	1.09844	0.549218	+	0.835679i	$0.314926\pi$
$588$	−0.842420	−0.0347408
$589$	−26.0758	−1.07443
$590$	0	0
$591$	−2.06160	−0.0848031
$592$	1.00000	0.0410997
$593$	−26.2642	−1.07854	−0.539270	−	0.842133i	$-0.681300\pi$
$593$	−26.2642	−1.07854	−0.539270	+	0.842133i	$0.681300\pi$
$594$	−82.5425	−3.38676
$595$	0	0
$596$	−4.18838	−0.171563
$597$	−5.18130	−0.212056
$598$	−10.0142	−0.409510
$599$	−38.2415	−1.56251	−0.781253	−	0.624215i	$-0.785419\pi$
$599$	−38.2415	−1.56251	−0.781253	+	0.624215i	$0.785419\pi$
$600$	0	0
$601$	30.2447	1.23371	0.616853	−	0.787078i	$-0.288407\pi$
$601$	30.2447	1.23371	0.616853	+	0.787078i	$0.288407\pi$
$602$	5.04921	0.205791
$603$	−15.1576	−0.617264
$604$	−6.69193	−0.272291
$605$	0	0
$606$	40.7819	1.65665
$607$	5.90157	0.239537	0.119769	−	0.992802i	$-0.461785\pi$
$607$	5.90157	0.239537	0.119769	+	0.992802i	$0.461785\pi$
$608$	3.34596	0.135697
$609$	34.2783	1.38903
$610$	0	0
$611$	12.3768	0.500710
$612$	6.13208	0.247875
$613$	−6.15473	−0.248587	−0.124294	−	0.992245i	$-0.539666\pi$
$613$	−6.15473	−0.248587	−0.124294	+	0.992245i	$0.539666\pi$
$614$	−4.95502	−0.199968
$615$	0	0
$616$	−12.3346	−0.496977
$617$	−16.5035	−0.664408	−0.332204	−	0.943208i	$-0.607792\pi$
$617$	−16.5035	−0.664408	−0.332204	+	0.943208i	$0.607792\pi$
$618$	−4.37677	−0.176059
$619$	9.04921	0.363719	0.181859	−	0.983325i	$-0.441788\pi$
$619$	9.04921	0.363719	0.181859	+	0.983325i	$0.441788\pi$
$620$	0	0
$621$	−26.0142	−1.04391
$622$	−3.68060	−0.147579
$623$	15.5864	0.624456
$624$	−22.3909	−0.896355
$625$	0	0
$626$	−0.992912	−0.0396848
$627$	−53.1586	−2.12295
$628$	3.94370	0.157371
$629$	0.748228	0.0298338
$630$	0	0
$631$	−0.507780	−0.0202144	−0.0101072	−	0.999949i	$-0.503217\pi$
$631$	−0.507780	−0.0202144	−0.0101072	+	0.999949i	$0.503217\pi$
$632$	16.5414	0.657983
$633$	70.0531	2.78436
$634$	−6.55985	−0.260525
$635$	0	0
$636$	−34.9561	−1.38610
$637$	1.68484	0.0667558
$638$	−18.7256	−0.741353
$639$	−31.8590	−1.26032
$640$	0	0
$641$	−40.6356	−1.60501	−0.802505	−	0.596645i	$-0.796500\pi$
$641$	−40.6356	−1.60501	−0.802505	+	0.596645i	$0.796500\pi$
$642$	−10.1884	−0.402103
$643$	9.53011	0.375831	0.187915	−	0.982185i	$-0.439827\pi$
$643$	9.53011	0.375831	0.187915	+	0.982185i	$0.439827\pi$
$644$	−3.88740	−0.153185
$645$	0	0
$646$	2.50354	0.0985006
$647$	21.0071	0.825874	0.412937	−	0.910760i	$-0.364503\pi$
$647$	21.0071	0.825874	0.412937	+	0.910760i	$0.364503\pi$
$648$	−33.5793	−1.31912
$649$	27.7748	1.09026
$650$	0	0
$651$	67.7380	2.65486
$652$	12.1463	0.475684
$653$	45.5496	1.78249	0.891247	−	0.453519i	$-0.149832\pi$
$653$	45.5496	1.78249	0.891247	+	0.453519i	$0.149832\pi$
$654$	4.81870	0.188426
$655$	0	0
$656$	−6.44724	−0.251723
$657$	61.4370	2.39689
$658$	4.80453	0.187300
$659$	9.38385	0.365543	0.182772	−	0.983155i	$-0.441493\pi$
$659$	9.38385	0.365543	0.182772	+	0.983155i	$0.441493\pi$
$660$	0	0
$661$	4.05630	0.157772	0.0788859	−	0.996884i	$-0.474864\pi$
$661$	4.05630	0.157772	0.0788859	+	0.996884i	$0.474864\pi$
$662$	18.0379	0.701062
$663$	−16.7535	−0.650653
$664$	15.2334	0.591169
$665$	0	0
$666$	−8.19547	−0.317568
$667$	−5.90157	−0.228510
$668$	−17.0829	−0.660956
$669$	9.06869	0.350616
$670$	0	0
$671$	37.7185	1.45611
$672$	−8.69193	−0.335298
$673$	−7.60906	−0.293308	−0.146654	−	0.989188i	$-0.546850\pi$
$673$	−7.60906	−0.293308	−0.146654	+	0.989188i	$0.546850\pi$
$674$	−14.8945	−0.573714
$675$	0	0
$676$	31.7819	1.22238
$677$	−28.7677	−1.10563	−0.552816	−	0.833303i	$-0.686447\pi$
$677$	−28.7677	−1.10563	−0.552816	+	0.833303i	$0.686447\pi$
$678$	37.2713	1.43139
$679$	27.1392	1.04151
$680$	0	0
$681$	−87.4370	−3.35059
$682$	−37.0039	−1.41695
$683$	−0.0704767	−0.00269671	−0.00134836	−	0.999999i	$-0.500429\pi$
$683$	−0.0704767	−0.00269671	−0.00134836	+	0.999999i	$0.500429\pi$
$684$	−27.4217	−1.04850
$685$	0	0
$686$	18.8382	0.719245
$687$	−82.8718	−3.16176
$688$	−1.94370	−0.0741028
$689$	69.9122	2.66344
$690$	0	0
$691$	10.6498	0.405137	0.202569	−	0.979268i	$-0.435071\pi$
$691$	10.6498	0.405137	0.202569	+	0.979268i	$0.435071\pi$
$692$	15.3417	0.583205
$693$	101.088	3.84002
$694$	−9.38385	−0.356206
$695$	0	0
$696$	−13.1955	−0.500173
$697$	−4.82401	−0.182722
$698$	32.9703	1.24794
$699$	41.0829	1.55390
$700$	0	0
$701$	−7.98582	−0.301620	−0.150810	−	0.988563i	$-0.548188\pi$
$701$	−7.98582	−0.301620	−0.150810	+	0.988563i	$0.548188\pi$
$702$	116.331	4.39065
$703$	−3.34596	−0.126195
$704$	4.74823	0.178956
$705$	0	0
$706$	27.4260	1.03219
$707$	−31.6622	−1.19078
$708$	19.5722	0.735570
$709$	−37.2149	−1.39764	−0.698818	−	0.715299i	$-0.746290\pi$
$709$	−37.2149	−1.39764	−0.698818	+	0.715299i	$0.746290\pi$
$710$	0	0
$711$	−135.565	−5.08408
$712$	−6.00000	−0.224860
$713$	−11.6622	−0.436752
$714$	−6.50354	−0.243389
$715$	0	0
$716$	2.45148	0.0916161
$717$	−57.4738	−2.14640
$718$	25.0829	0.936084
$719$	30.1742	1.12531	0.562654	−	0.826692i	$-0.309780\pi$
$719$	30.1742	1.12531	0.562654	+	0.826692i	$0.309780\pi$
$720$	0	0
$721$	3.39803	0.126549
$722$	7.80453	0.290455
$723$	−51.0970	−1.90032
$724$	13.1955	0.490406
$725$	0	0
$726$	−38.6314	−1.43375
$727$	18.7677	0.696056	0.348028	−	0.937484i	$-0.386851\pi$
$727$	18.7677	0.696056	0.348028	+	0.937484i	$0.386851\pi$
$728$	17.3839	0.644288
$729$	100.701	3.72967
$730$	0	0
$731$	−1.45433	−0.0537903
$732$	26.5793	0.982400
$733$	−26.1094	−0.964374	−0.482187	−	0.876068i	$-0.660157\pi$
$733$	−26.1094	−0.964374	−0.482187	+	0.876068i	$0.660157\pi$
$734$	2.48513	0.0917279
$735$	0	0
$736$	1.49646	0.0551601
$737$	−8.78188	−0.323485
$738$	52.8382	1.94500
$739$	42.6130	1.56754	0.783772	−	0.621049i	$-0.213293\pi$
$739$	42.6130	1.56754	0.783772	+	0.621049i	$0.213293\pi$
$740$	0	0
$741$	74.9193	2.75223
$742$	27.1392	0.996310
$743$	35.4922	1.30208	0.651042	−	0.759042i	$-0.274332\pi$
$743$	35.4922	1.30208	0.651042	+	0.759042i	$0.274332\pi$
$744$	−26.0758	−0.955984
$745$	0	0
$746$	−35.1586	−1.28725
$747$	−124.845	−4.56782
$748$	3.55276	0.129902
$749$	7.91005	0.289027
$750$	0	0
$751$	21.5722	0.787182	0.393591	−	0.919286i	$-0.371233\pi$
$751$	21.5722	0.787182	0.393591	+	0.919286i	$0.371233\pi$
$752$	−1.84951	−0.0674446
$753$	33.5864	1.22396
$754$	26.3909	0.961101
$755$	0	0
$756$	45.1586	1.64240
$757$	23.7890	0.864625	0.432312	−	0.901724i	$-0.357698\pi$
$757$	23.7890	0.864625	0.432312	+	0.901724i	$0.357698\pi$
$758$	−24.6778	−0.896336
$759$	−23.7748	−0.862970
$760$	0	0
$761$	−18.6724	−0.676876	−0.338438	−	0.940989i	$-0.609898\pi$
$761$	−18.6724	−0.676876	−0.338438	+	0.940989i	$0.609898\pi$
$762$	−29.5864	−1.07180
$763$	−3.74114	−0.135438
$764$	−21.7790	−0.787938
$765$	0	0
$766$	−13.3839	−0.483578
$767$	−39.1445	−1.41342
$768$	3.34596	0.120737
$769$	−2.48937	−0.0897689	−0.0448845	−	0.998992i	$-0.514292\pi$
$769$	−2.48937	−0.0897689	−0.0448845	+	0.998992i	$0.514292\pi$
$770$	0	0
$771$	70.1657	2.52696
$772$	12.9929	0.467625
$773$	−0.950786	−0.0341974	−0.0170987	−	0.999854i	$-0.505443\pi$
$773$	−0.950786	−0.0341974	−0.0170987	+	0.999854i	$0.505443\pi$
$774$	15.9295	0.572575
$775$	0	0
$776$	−10.4472	−0.375034
$777$	8.69193	0.311821
$778$	−18.2220	−0.653292
$779$	21.5722	0.772906
$780$	0	0
$781$	−18.4582	−0.660488
$782$	1.11969	0.0400401
$783$	68.5567	2.45002
$784$	−0.251772	−0.00899185
$785$	0	0
$786$	−20.5793	−0.734040
$787$	29.4359	1.04928	0.524639	−	0.851325i	$-0.324200\pi$
$787$	29.4359	1.04928	0.524639	+	0.851325i	$0.324200\pi$
$788$	−0.616147	−0.0219493
$789$	−74.8577	−2.66500
$790$	0	0
$791$	−28.9366	−1.02887
$792$	−38.9140	−1.38275
$793$	−53.1586	−1.88772
$794$	−2.22521	−0.0789696
$795$	0	0
$796$	−1.54852	−0.0548859
$797$	14.1742	0.502076	0.251038	−	0.967977i	$-0.419228\pi$
$797$	14.1742	0.502076	0.251038	+	0.967977i	$0.419228\pi$
$798$	29.0829	1.02952
$799$	−1.38385	−0.0489572
$800$	0	0
$801$	49.1728	1.73744
$802$	−12.3909	−0.437539
$803$	35.5949	1.25612
$804$	−6.18838	−0.218247
$805$	0	0
$806$	52.1516	1.83696
$807$	−14.0142	−0.493322
$808$	12.1884	0.428786
$809$	−32.9929	−1.15997	−0.579985	−	0.814627i	$-0.696941\pi$
$809$	−32.9929	−1.15997	−0.579985	+	0.814627i	$0.696941\pi$
$810$	0	0
$811$	40.5567	1.42414	0.712069	−	0.702110i	$-0.247758\pi$
$811$	40.5567	1.42414	0.712069	+	0.702110i	$0.247758\pi$
$812$	10.2447	0.359518
$813$	−40.5283	−1.42139
$814$	−4.74823	−0.166425
$815$	0	0
$816$	2.50354	0.0876416
$817$	6.50354	0.227530
$818$	−7.27125	−0.254233
$819$	−142.469	−4.97826
$820$	0	0
$821$	−38.8661	−1.35644	−0.678219	−	0.734860i	$-0.737248\pi$
$821$	−38.8661	−1.35644	−0.678219	+	0.734860i	$0.737248\pi$
$822$	36.4525	1.27143
$823$	49.6243	1.72979	0.864897	−	0.501949i	$-0.167384\pi$
$823$	49.6243	1.72979	0.864897	+	0.501949i	$0.167384\pi$
$824$	−1.30807	−0.0455689
$825$	0	0
$826$	−15.1955	−0.528718
$827$	30.0195	1.04388	0.521940	−	0.852982i	$-0.325209\pi$
$827$	30.0195	1.04388	0.521940	+	0.852982i	$0.325209\pi$
$828$	−12.2642	−0.426209
$829$	10.5598	0.366759	0.183379	−	0.983042i	$-0.441296\pi$
$829$	10.5598	0.366759	0.183379	+	0.983042i	$0.441296\pi$
$830$	0	0
$831$	31.1445	1.08039
$832$	−6.69193	−0.232001
$833$	−0.188383	−0.00652708
$834$	−3.51063	−0.121563
$835$	0	0
$836$	−15.8874	−0.549477
$837$	135.476	4.68273
$838$	−28.4809	−0.983856
$839$	29.3839	1.01444	0.507222	−	0.861816i	$-0.330673\pi$
$839$	29.3839	1.01444	0.507222	+	0.861816i	$0.330673\pi$
$840$	0	0
$841$	−13.4472	−0.463698
$842$	−22.0000	−0.758170
$843$	64.4809	2.22084
$844$	20.9366	0.720668
$845$	0	0
$846$	15.1576	0.521128
$847$	29.9926	1.03056
$848$	−10.4472	−0.358760
$849$	45.7890	1.57147
$850$	0	0
$851$	−1.49646	−0.0512979
$852$	−13.0071	−0.445615
$853$	−6.22521	−0.213147	−0.106573	−	0.994305i	$-0.533988\pi$
$853$	−6.22521	−0.213147	−0.106573	+	0.994305i	$0.533988\pi$
$854$	−20.6356	−0.706137
$855$	0	0
$856$	−3.04498	−0.104075
$857$	3.65119	0.124722	0.0623611	−	0.998054i	$-0.480137\pi$
$857$	3.65119	0.124722	0.0623611	+	0.998054i	$0.480137\pi$
$858$	106.317	3.62961
$859$	−36.3162	−1.23909	−0.619547	−	0.784960i	$-0.712684\pi$
$859$	−36.3162	−1.23909	−0.619547	+	0.784960i	$0.712684\pi$
$860$	0	0
$861$	−56.0390	−1.90980
$862$	−1.51487	−0.0515966
$863$	−24.5751	−0.836546	−0.418273	−	0.908321i	$-0.637364\pi$
$863$	−24.5751	−0.836546	−0.418273	+	0.908321i	$0.637364\pi$
$864$	−17.3839	−0.591411
$865$	0	0
$866$	24.9929	0.849294
$867$	−55.0082	−1.86817
$868$	20.2447	0.687149
$869$	−78.5425	−2.66437
$870$	0	0
$871$	12.3768	0.419371
$872$	1.44015	0.0487698
$873$	85.6201	2.89780
$874$	−5.00709	−0.169367
$875$	0	0
$876$	25.0829	0.847472
$877$	−20.0563	−0.677253	−0.338627	−	0.940921i	$-0.609962\pi$
$877$	−20.0563	−0.677253	−0.338627	+	0.940921i	$0.609962\pi$
$878$	5.10128	0.172160
$879$	34.9561	1.17904
$880$	0	0
$881$	−15.0266	−0.506258	−0.253129	−	0.967433i	$-0.581460\pi$
$881$	−15.0266	−0.506258	−0.253129	+	0.967433i	$0.581460\pi$
$882$	2.06339	0.0694779
$883$	6.62145	0.222830	0.111415	−	0.993774i	$-0.464462\pi$
$883$	6.62145	0.222830	0.111415	+	0.993774i	$0.464462\pi$
$884$	−5.00709	−0.168407
$885$	0	0
$886$	25.7369	0.864648
$887$	37.5538	1.26093	0.630467	−	0.776216i	$-0.282863\pi$
$887$	37.5538	1.26093	0.630467	+	0.776216i	$0.282863\pi$
$888$	−3.34596	−0.112283
$889$	22.9703	0.770398
$890$	0	0
$891$	159.442	5.34152
$892$	2.71034	0.0907488
$893$	6.18838	0.207086
$894$	14.0142	0.468704
$895$	0	0
$896$	−2.59774	−0.0867842
$897$	33.5071	1.11877
$898$	17.7748	0.593153
$899$	30.7341	1.02504
$900$	0	0
$901$	−7.81692	−0.260419
$902$	30.6130	1.01930
$903$	−16.8945	−0.562213
$904$	11.1392	0.370483
$905$	0	0
$906$	22.3909	0.743889
$907$	−50.4667	−1.67572	−0.837860	−	0.545885i	$-0.816193\pi$
$907$	−50.4667	−1.67572	−0.837860	+	0.545885i	$0.816193\pi$
$908$	−26.1321	−0.867224
$909$	−99.8895	−3.31313
$910$	0	0
$911$	15.9395	0.528098	0.264049	−	0.964509i	$-0.414942\pi$
$911$	15.9395	0.528098	0.264049	+	0.964509i	$0.414942\pi$
$912$	−11.1955	−0.370719
$913$	−72.3315	−2.39382
$914$	34.3346	1.13569
$915$	0	0
$916$	−24.7677	−0.818348
$917$	15.9774	0.527619
$918$	−13.0071	−0.429298
$919$	32.5414	1.07344	0.536721	−	0.843760i	$-0.319663\pi$
$919$	32.5414	1.07344	0.536721	+	0.843760i	$0.319663\pi$
$920$	0	0
$921$	16.5793	0.546307
$922$	16.9508	0.558244
$923$	26.0142	0.856267
$924$	41.2713	1.35772
$925$	0	0
$926$	10.9929	0.361250
$927$	10.7203	0.352100
$928$	−3.94370	−0.129458
$929$	−34.1094	−1.11909	−0.559547	−	0.828799i	$-0.689025\pi$
$929$	−34.1094	−1.11909	−0.559547	+	0.828799i	$0.689025\pi$
$930$	0	0
$931$	0.842420	0.0276092
$932$	12.2783	0.402190
$933$	12.3152	0.403180
$934$	37.4175	1.22434
$935$	0	0
$936$	54.8435	1.79262
$937$	−0.0141751	−0.000463082 0	−0.000231541	−	1.00000i	$-0.500074\pi$
$937$	−0.0141751	−0.000463082 0	−0.000231541		1.00000i	$0.500074\pi$
$938$	4.80453	0.156873
$939$	3.32225	0.108417
$940$	0	0
$941$	11.1813	0.364500	0.182250	−	0.983252i	$-0.441662\pi$
$941$	11.1813	0.364500	0.182250	+	0.983252i	$0.441662\pi$
$942$	−13.1955	−0.429932
$943$	9.64802	0.314183
$944$	5.84951	0.190385
$945$	0	0
$946$	9.22912	0.300065
$947$	53.6427	1.74315	0.871577	−	0.490259i	$-0.163098\pi$
$947$	53.6427	1.74315	0.871577	+	0.490259i	$0.163098\pi$
$948$	−55.3470	−1.79759
$949$	−50.1657	−1.62845
$950$	0	0
$951$	21.9490	0.711745
$952$	−1.94370	−0.0629956
$953$	9.88740	0.320284	0.160142	−	0.987094i	$-0.448805\pi$
$953$	9.88740	0.320284	0.160142	+	0.987094i	$0.448805\pi$
$954$	85.6201	2.77205
$955$	0	0
$956$	−17.1771	−0.555546
$957$	62.6551	2.02535
$958$	0.465654	0.0150446
$959$	−28.3010	−0.913886
$960$	0	0
$961$	29.7341	0.959163
$962$	6.69193	0.215756
$963$	24.9550	0.804164
$964$	−15.2713	−0.491854
$965$	0	0
$966$	13.0071	0.418496
$967$	−51.6254	−1.66016	−0.830080	−	0.557644i	$-0.811705\pi$
$967$	−51.6254	−1.66016	−0.830080	+	0.557644i	$0.811705\pi$
$968$	−11.5457	−0.371092
$969$	−8.37677	−0.269100
$970$	0	0
$971$	20.5598	0.659797	0.329898	−	0.944016i	$-0.392986\pi$
$971$	20.5598	0.659797	0.329898	+	0.944016i	$0.392986\pi$
$972$	60.2036	1.93103
$973$	2.72558	0.0873781
$974$	34.9561	1.12007
$975$	0	0
$976$	7.94370	0.254272
$977$	−38.4472	−1.23004	−0.615018	−	0.788513i	$-0.710851\pi$
$977$	−38.4472	−1.23004	−0.615018	+	0.788513i	$0.710851\pi$
$978$	−40.6409	−1.29955
$979$	28.4894	0.910524
$980$	0	0
$981$	−11.8027	−0.376833
$982$	24.7819	0.790822
$983$	−11.8973	−0.379466	−0.189733	−	0.981836i	$-0.560762\pi$
$983$	−11.8973	−0.379466	−0.189733	+	0.981836i	$0.560762\pi$
$984$	21.5722	0.687698
$985$	0	0
$986$	−2.95079	−0.0939722
$987$	−16.0758	−0.511698
$988$	22.3909	0.712351
$989$	2.90866	0.0924900
$990$	0	0
$991$	−19.7932	−0.628752	−0.314376	−	0.949299i	$-0.601795\pi$
$991$	−19.7932	−0.628752	−0.314376	+	0.949299i	$0.601795\pi$
$992$	−7.79321	−0.247435
$993$	−60.3541	−1.91528
$994$	10.0984	0.320303
$995$	0	0
$996$	−50.9703	−1.61505
$997$	−5.88740	−0.186456	−0.0932279	−	0.995645i	$-0.529719\pi$
$997$	−5.88740	−0.186456	−0.0932279	+	0.995645i	$0.529719\pi$
$998$	2.33888	0.0740358
$999$	17.3839	0.550001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.z.1.3		3
5.2	odd	4		1850.2.b.o.149.1		6
5.3	odd	4		1850.2.b.o.149.6		6
5.4	even	2		370.2.a.g.1.1	✓	3
15.14	odd	2		3330.2.a.bg.1.1		3
20.19	odd	2		2960.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.1	✓	3	5.4	even	2
1850.2.a.z.1.3		3	1.1	even	1	trivial
1850.2.b.o.149.1		6	5.2	odd	4
1850.2.b.o.149.6		6	5.3	odd	4
2960.2.a.u.1.3		3	20.19	odd	2
3330.2.a.bg.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +2.59774 q^{7} -1.00000 q^{8} +8.19547 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -3.34596 q^{6} +2.59774 q^{7} -1.00000 q^{8} +8.19547 q^{9} +4.74823 q^{11} +3.34596 q^{12} -6.69193 q^{13} -2.59774 q^{14} +1.00000 q^{16} +0.748228 q^{17} -8.19547 q^{18} -3.34596 q^{19} +8.69193 q^{21} -4.74823 q^{22} -1.49646 q^{23} -3.34596 q^{24} +6.69193 q^{26} +17.3839 q^{27} +2.59774 q^{28} +3.94370 q^{29} +7.79321 q^{31} -1.00000 q^{32} +15.8874 q^{33} -0.748228 q^{34} +8.19547 q^{36} +1.00000 q^{37} +3.34596 q^{38} -22.3909 q^{39} -6.44724 q^{41} -8.69193 q^{42} -1.94370 q^{43} +4.74823 q^{44} +1.49646 q^{46} -1.84951 q^{47} +3.34596 q^{48} -0.251772 q^{49} +2.50354 q^{51} -6.69193 q^{52} -10.4472 q^{53} -17.3839 q^{54} -2.59774 q^{56} -11.1955 q^{57} -3.94370 q^{58} +5.84951 q^{59} +7.94370 q^{61} -7.79321 q^{62} +21.2897 q^{63} +1.00000 q^{64} -15.8874 q^{66} -1.84951 q^{67} +0.748228 q^{68} -5.00709 q^{69} -3.88740 q^{71} -8.19547 q^{72} +7.49646 q^{73} -1.00000 q^{74} -3.34596 q^{76} +12.3346 q^{77} +22.3909 q^{78} -16.5414 q^{79} +33.5793 q^{81} +6.44724 q^{82} -15.2334 q^{83} +8.69193 q^{84} +1.94370 q^{86} +13.1955 q^{87} -4.74823 q^{88} +6.00000 q^{89} -17.3839 q^{91} -1.49646 q^{92} +26.0758 q^{93} +1.84951 q^{94} -3.34596 q^{96} +10.4472 q^{97} +0.251772 q^{98} +38.9140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9	\( 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9} + 11 q^{11} - q^{14} + 3 q^{16} - q^{17} - 11 q^{18} + 6 q^{21} - 11 q^{22} + 2 q^{23} + 12 q^{27} + q^{28} - 5 q^{29} + 3 q^{31} - 3 q^{32} + 14 q^{33} + q^{34} + 11 q^{36} + 3 q^{37} - 40 q^{39} - 9 q^{41} - 6 q^{42} + 11 q^{43} + 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} + 14 q^{51} - 21 q^{53} - 12 q^{54} - q^{56} - 20 q^{57} + 5 q^{58} + 14 q^{59} + 7 q^{61} - 3 q^{62} + 37 q^{63} + 3 q^{64} - 14 q^{66} - 2 q^{67} - q^{68} - 28 q^{69} + 22 q^{71} - 11 q^{72} + 16 q^{73} - 3 q^{74} - 7 q^{77} + 40 q^{78} - 26 q^{79} + 47 q^{81} + 9 q^{82} - 2 q^{83} + 6 q^{84} - 11 q^{86} + 26 q^{87} - 11 q^{88} + 18 q^{89} - 12 q^{91} + 2 q^{92} + 18 q^{93} + 2 q^{94} + 21 q^{97} + 4 q^{98} + 19 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9 + 11 * q^11 - q^14 + 3 * q^16 - q^17 - 11 * q^18 + 6 * q^21 - 11 * q^22 + 2 * q^23 + 12 * q^27 + q^28 - 5 * q^29 + 3 * q^31 - 3 * q^32 + 14 * q^33 + q^34 + 11 * q^36 + 3 * q^37 - 40 * q^39 - 9 * q^41 - 6 * q^42 + 11 * q^43 + 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 + 14 * q^51 - 21 * q^53 - 12 * q^54 - q^56 - 20 * q^57 + 5 * q^58 + 14 * q^59 + 7 * q^61 - 3 * q^62 + 37 * q^63 + 3 * q^64 - 14 * q^66 - 2 * q^67 - q^68 - 28 * q^69 + 22 * q^71 - 11 * q^72 + 16 * q^73 - 3 * q^74 - 7 * q^77 + 40 * q^78 - 26 * q^79 + 47 * q^81 + 9 * q^82 - 2 * q^83 + 6 * q^84 - 11 * q^86 + 26 * q^87 - 11 * q^88 + 18 * q^89 - 12 * q^91 + 2 * q^92 + 18 * q^93 + 2 * q^94 + 21 * q^97 + 4 * q^98 + 19 * q^99

Introduction

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