LMFDB - Embedded newform 1850.2.a.bd.1.5

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:	$5$
Coefficient field:	5.5.1791440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} + 13x - 4 $ x^5 - 9x^3 + 13x - 4
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			1.5
Root			$2.72987$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} +4.14336 q^{7} -1.00000 q^{8} +4.45216 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} +4.14336 q^{7} -1.00000 q^{8} +4.45216 q^{9} -4.76853 q^{11} +2.72987 q^{12} +3.91744 q^{13} -4.14336 q^{14} +1.00000 q^{16} -3.31637 q^{17} -4.45216 q^{18} -1.85109 q^{19} +11.3108 q^{21} +4.76853 q^{22} +1.54229 q^{23} -2.72987 q^{24} -3.91744 q^{26} +3.96421 q^{27} +4.14336 q^{28} +8.87323 q^{29} +9.75286 q^{31} -1.00000 q^{32} -13.0174 q^{33} +3.31637 q^{34} +4.45216 q^{36} -1.00000 q^{37} +1.85109 q^{38} +10.6941 q^{39} -5.06977 q^{41} -11.3108 q^{42} +9.99446 q^{43} -4.76853 q^{44} -1.54229 q^{46} -4.82700 q^{47} +2.72987 q^{48} +10.1675 q^{49} -9.05324 q^{51} +3.91744 q^{52} +5.13611 q^{53} -3.96421 q^{54} -4.14336 q^{56} -5.05324 q^{57} -8.87323 q^{58} -1.05324 q^{59} -4.14422 q^{61} -9.75286 q^{62} +18.4469 q^{63} +1.00000 q^{64} +13.0174 q^{66} -1.00811 q^{67} -3.31637 q^{68} +4.21025 q^{69} +6.45248 q^{71} -4.45216 q^{72} -10.7714 q^{73} +1.00000 q^{74} -1.85109 q^{76} -19.7578 q^{77} -10.6941 q^{78} -1.19856 q^{79} -2.53473 q^{81} +5.06977 q^{82} -10.6932 q^{83} +11.3108 q^{84} -9.99446 q^{86} +24.2227 q^{87} +4.76853 q^{88} +7.29569 q^{89} +16.2314 q^{91} +1.54229 q^{92} +26.6240 q^{93} +4.82700 q^{94} -2.72987 q^{96} -14.8988 q^{97} -10.1675 q^{98} -21.2303 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * 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
$3$	2.72987	1.57609	0.788044	−	0.615619i	$-0.211094\pi$
$3$	2.72987	1.57609	0.788044	+	0.615619i	$0.211094\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.72987	−1.11446
$7$	4.14336	1.56604	0.783022	−	0.621994i	$-0.213677\pi$
$7$	4.14336	1.56604	0.783022	+	0.621994i	$0.213677\pi$
$8$	−1.00000	−0.353553
$9$	4.45216	1.48405
$10$	0	0
$11$	−4.76853	−1.43777	−0.718883	−	0.695131i	$-0.755346\pi$
$11$	−4.76853	−1.43777	−0.718883	+	0.695131i	$0.755346\pi$
$12$	2.72987	0.788044
$13$	3.91744	1.08650	0.543251	−	0.839570i	$-0.317193\pi$
$13$	3.91744	1.08650	0.543251	+	0.839570i	$0.317193\pi$
$14$	−4.14336	−1.10736
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.31637	−0.804337	−0.402169	−	0.915566i	$-0.631743\pi$
$17$	−3.31637	−0.804337	−0.402169	+	0.915566i	$0.631743\pi$
$18$	−4.45216	−1.04939
$19$	−1.85109	−0.424670	−0.212335	−	0.977197i	$-0.568107\pi$
$19$	−1.85109	−0.424670	−0.212335	+	0.977197i	$0.568107\pi$
$20$	0	0
$21$	11.3108	2.46822
$22$	4.76853	1.01665
$23$	1.54229	0.321590	0.160795	−	0.986988i	$-0.448594\pi$
$23$	1.54229	0.321590	0.160795	+	0.986988i	$0.448594\pi$
$24$	−2.72987	−0.557231
$25$	0	0
$26$	−3.91744	−0.768273
$27$	3.96421	0.762913
$28$	4.14336	0.783022
$29$	8.87323	1.64772	0.823859	−	0.566795i	$-0.191817\pi$
$29$	8.87323	1.64772	0.823859	+	0.566795i	$0.191817\pi$
$30$	0	0
$31$	9.75286	1.75166	0.875832	−	0.482615i	$-0.160313\pi$
$31$	9.75286	1.75166	0.875832	+	0.482615i	$0.160313\pi$
$32$	−1.00000	−0.176777
$33$	−13.0174	−2.26605
$34$	3.31637	0.568752
$35$	0	0
$36$	4.45216	0.742027
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	1.85109	0.300287
$39$	10.6941	1.71242
$40$	0	0
$41$	−5.06977	−0.791764	−0.395882	−	0.918301i	$-0.629561\pi$
$41$	−5.06977	−0.791764	−0.395882	+	0.918301i	$0.629561\pi$
$42$	−11.3108	−1.74530
$43$	9.99446	1.52414	0.762070	−	0.647494i	$-0.224183\pi$
$43$	9.99446	1.52414	0.762070	+	0.647494i	$0.224183\pi$
$44$	−4.76853	−0.718883
$45$	0	0
$46$	−1.54229	−0.227399
$47$	−4.82700	−0.704090	−0.352045	−	0.935983i	$-0.614514\pi$
$47$	−4.82700	−0.704090	−0.352045	+	0.935983i	$0.614514\pi$
$48$	2.72987	0.394022
$49$	10.1675	1.45249
$50$	0	0
$51$	−9.05324	−1.26771
$52$	3.91744	0.543251
$53$	5.13611	0.705499	0.352750	−	0.935718i	$-0.385247\pi$
$53$	5.13611	0.705499	0.352750	+	0.935718i	$0.385247\pi$
$54$	−3.96421	−0.539461
$55$	0	0
$56$	−4.14336	−0.553680
$57$	−5.05324	−0.669317
$58$	−8.87323	−1.16511
$59$	−1.05324	−0.137120	−0.0685598	−	0.997647i	$-0.521840\pi$
$59$	−1.05324	−0.137120	−0.0685598	+	0.997647i	$0.521840\pi$
$60$	0	0
$61$	−4.14422	−0.530613	−0.265306	−	0.964164i	$-0.585473\pi$
$61$	−4.14422	−0.530613	−0.265306	+	0.964164i	$0.585473\pi$
$62$	−9.75286	−1.23861
$63$	18.4469	2.32410
$64$	1.00000	0.125000
$65$	0	0
$66$	13.0174	1.60234
$67$	−1.00811	−0.123160	−0.0615800	−	0.998102i	$-0.519614\pi$
$67$	−1.00811	−0.123160	−0.0615800	+	0.998102i	$0.519614\pi$
$68$	−3.31637	−0.402169
$69$	4.21025	0.506855
$70$	0	0
$71$	6.45248	0.765768	0.382884	−	0.923796i	$-0.374931\pi$
$71$	6.45248	0.765768	0.382884	+	0.923796i	$0.374931\pi$
$72$	−4.45216	−0.524693
$73$	−10.7714	−1.26070	−0.630349	−	0.776312i	$-0.717088\pi$
$73$	−10.7714	−1.26070	−0.630349	+	0.776312i	$0.717088\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−1.85109	−0.212335
$77$	−19.7578	−2.25161
$78$	−10.6941	−1.21087
$79$	−1.19856	−0.134849	−0.0674243	−	0.997724i	$-0.521478\pi$
$79$	−1.19856	−0.134849	−0.0674243	+	0.997724i	$0.521478\pi$
$80$	0	0
$81$	−2.53473	−0.281636
$82$	5.06977	0.559862
$83$	−10.6932	−1.17373	−0.586867	−	0.809683i	$-0.699639\pi$
$83$	−10.6932	−1.17373	−0.586867	+	0.809683i	$0.699639\pi$
$84$	11.3108	1.23411
$85$	0	0
$86$	−9.99446	−1.07773
$87$	24.2227	2.59695
$88$	4.76853	0.508327
$89$	7.29569	0.773342	0.386671	−	0.922218i	$-0.373625\pi$
$89$	7.29569	0.773342	0.386671	+	0.922218i	$0.373625\pi$
$90$	0	0
$91$	16.2314	1.70151
$92$	1.54229	0.160795
$93$	26.6240	2.76078
$94$	4.82700	0.497867
$95$	0	0
$96$	−2.72987	−0.278616
$97$	−14.8988	−1.51274	−0.756371	−	0.654143i	$-0.773030\pi$
$97$	−14.8988	−1.51274	−0.756371	+	0.654143i	$0.773030\pi$
$98$	−10.1675	−1.02707
$99$	−21.2303	−2.13372
$100$	0	0
$101$	18.5903	1.84980	0.924902	−	0.380206i	$-0.124147\pi$
$101$	18.5903	1.84980	0.924902	+	0.380206i	$0.124147\pi$
$102$	9.05324	0.896404
$103$	13.3033	1.31081	0.655404	−	0.755278i	$-0.272498\pi$
$103$	13.3033	1.31081	0.655404	+	0.755278i	$0.272498\pi$
$104$	−3.91744	−0.384136
$105$	0	0
$106$	−5.13611	−0.498863
$107$	−12.4763	−1.20613	−0.603066	−	0.797691i	$-0.706054\pi$
$107$	−12.4763	−1.20613	−0.603066	+	0.797691i	$0.706054\pi$
$108$	3.96421	0.381457
$109$	−4.35447	−0.417083	−0.208541	−	0.978014i	$-0.566872\pi$
$109$	−4.35447	−0.417083	−0.208541	+	0.978014i	$0.566872\pi$
$110$	0	0
$111$	−2.72987	−0.259107
$112$	4.14336	0.391511
$113$	−9.54261	−0.897693	−0.448846	−	0.893609i	$-0.648165\pi$
$113$	−9.54261	−0.897693	−0.448846	+	0.893609i	$0.648165\pi$
$114$	5.05324	0.473279
$115$	0	0
$116$	8.87323	0.823859
$117$	17.4411	1.61243
$118$	1.05324	0.0969582
$119$	−13.7409	−1.25963
$120$	0	0
$121$	11.7389	1.06717
$122$	4.14422	0.375200
$123$	−13.8398	−1.24789
$124$	9.75286	0.875832
$125$	0	0
$126$	−18.4469	−1.64338
$127$	−8.33492	−0.739605	−0.369802	−	0.929110i	$-0.620575\pi$
$127$	−8.33492	−0.739605	−0.369802	+	0.929110i	$0.620575\pi$
$128$	−1.00000	−0.0883883
$129$	27.2835	2.40218
$130$	0	0
$131$	3.68597	0.322045	0.161022	−	0.986951i	$-0.448521\pi$
$131$	3.68597	0.322045	0.161022	+	0.986951i	$0.448521\pi$
$132$	−13.0174	−1.13302
$133$	−7.66975	−0.665052
$134$	1.00811	0.0870873
$135$	0	0
$136$	3.31637	0.284376
$137$	12.3027	1.05109	0.525546	−	0.850765i	$-0.323861\pi$
$137$	12.3027	1.05109	0.525546	+	0.850765i	$0.323861\pi$
$138$	−4.21025	−0.358400
$139$	−0.842442	−0.0714550	−0.0357275	−	0.999362i	$-0.511375\pi$
$139$	−0.842442	−0.0714550	−0.0357275	+	0.999362i	$0.511375\pi$
$140$	0	0
$141$	−13.1770	−1.10971
$142$	−6.45248	−0.541480
$143$	−18.6804	−1.56214
$144$	4.45216	0.371014
$145$	0	0
$146$	10.7714	0.891448
$147$	27.7558	2.28926
$148$	−1.00000	−0.0821995
$149$	−16.7723	−1.37404	−0.687019	−	0.726640i	$-0.741081\pi$
$149$	−16.7723	−1.37404	−0.687019	+	0.726640i	$0.741081\pi$
$150$	0	0
$151$	−7.59755	−0.618280	−0.309140	−	0.951017i	$-0.600041\pi$
$151$	−7.59755	−0.618280	−0.309140	+	0.951017i	$0.600041\pi$
$152$	1.85109	0.150143
$153$	−14.7650	−1.19368
$154$	19.7578	1.59213
$155$	0	0
$156$	10.6941	0.856211
$157$	−4.45631	−0.355652	−0.177826	−	0.984062i	$-0.556906\pi$
$157$	−4.45631	−0.355652	−0.177826	+	0.984062i	$0.556906\pi$
$158$	1.19856	0.0953523
$159$	14.0209	1.11193
$160$	0	0
$161$	6.39028	0.503624
$162$	2.53473	0.199147
$163$	−19.4599	−1.52422	−0.762110	−	0.647447i	$-0.775837\pi$
$163$	−19.4599	−1.52422	−0.762110	+	0.647447i	$0.775837\pi$
$164$	−5.06977	−0.395882
$165$	0	0
$166$	10.6932	0.829955
$167$	−7.13813	−0.552365	−0.276183	−	0.961105i	$-0.589069\pi$
$167$	−7.13813	−0.552365	−0.276183	+	0.961105i	$0.589069\pi$
$168$	−11.3108	−0.872649
$169$	2.34632	0.180486
$170$	0	0
$171$	−8.24137	−0.630233
$172$	9.99446	0.762070
$173$	11.6199	0.883448	0.441724	−	0.897151i	$-0.354367\pi$
$173$	11.6199	0.883448	0.441724	+	0.897151i	$0.354367\pi$
$174$	−24.2227	−1.83632
$175$	0	0
$176$	−4.76853	−0.359442
$177$	−2.87519	−0.216113
$178$	−7.29569	−0.546835
$179$	−17.2224	−1.28726	−0.643631	−	0.765336i	$-0.722573\pi$
$179$	−17.2224	−1.28726	−0.643631	+	0.765336i	$0.722573\pi$
$180$	0	0
$181$	−23.5098	−1.74747	−0.873733	−	0.486405i	$-0.838308\pi$
$181$	−23.5098	−1.74747	−0.873733	+	0.486405i	$0.838308\pi$
$182$	−16.2314	−1.20315
$183$	−11.3132	−0.836293
$184$	−1.54229	−0.113699
$185$	0	0
$186$	−26.6240	−1.95217
$187$	15.8142	1.15645
$188$	−4.82700	−0.352045
$189$	16.4252	1.19476
$190$	0	0
$191$	1.16660	0.0844125	0.0422063	−	0.999109i	$-0.486561\pi$
$191$	1.16660	0.0844125	0.0422063	+	0.999109i	$0.486561\pi$
$192$	2.72987	0.197011
$193$	9.36727	0.674271	0.337135	−	0.941456i	$-0.390542\pi$
$193$	9.36727	0.674271	0.337135	+	0.941456i	$0.390542\pi$
$194$	14.8988	1.06967
$195$	0	0
$196$	10.1675	0.726247
$197$	−3.21836	−0.229299	−0.114649	−	0.993406i	$-0.536574\pi$
$197$	−3.21836	−0.229299	−0.114649	+	0.993406i	$0.536574\pi$
$198$	21.2303	1.50877
$199$	−2.99104	−0.212029	−0.106014	−	0.994365i	$-0.533809\pi$
$199$	−2.99104	−0.212029	−0.106014	+	0.994365i	$0.533809\pi$
$200$	0	0
$201$	−2.75200	−0.194111
$202$	−18.5903	−1.30801
$203$	36.7650	2.58040
$204$	−9.05324	−0.633853
$205$	0	0
$206$	−13.3033	−0.926882
$207$	6.86654	0.477257
$208$	3.91744	0.271625
$209$	8.82700	0.610576
$210$	0	0
$211$	4.75340	0.327237	0.163619	−	0.986524i	$-0.447683\pi$
$211$	4.75340	0.327237	0.163619	+	0.986524i	$0.447683\pi$
$212$	5.13611	0.352750
$213$	17.6144	1.20692
$214$	12.4763	0.852864
$215$	0	0
$216$	−3.96421	−0.269731
$217$	40.4096	2.74318
$218$	4.35447	0.294922
$219$	−29.4045	−1.98697
$220$	0	0
$221$	−12.9917	−0.873914
$222$	2.72987	0.183217
$223$	6.95635	0.465832	0.232916	−	0.972497i	$-0.425173\pi$
$223$	6.95635	0.465832	0.232916	+	0.972497i	$0.425173\pi$
$224$	−4.14336	−0.276840
$225$	0	0
$226$	9.54261	0.634765
$227$	22.2980	1.47997	0.739986	−	0.672622i	$-0.234832\pi$
$227$	22.2980	1.47997	0.739986	+	0.672622i	$0.234832\pi$
$228$	−5.05324	−0.334659
$229$	12.9648	0.856739	0.428370	−	0.903604i	$-0.359088\pi$
$229$	12.9648	0.856739	0.428370	+	0.903604i	$0.359088\pi$
$230$	0	0
$231$	−53.9360	−3.54873
$232$	−8.87323	−0.582556
$233$	10.4327	0.683468	0.341734	−	0.939797i	$-0.388986\pi$
$233$	10.4327	0.683468	0.341734	+	0.939797i	$0.388986\pi$
$234$	−17.4411	−1.14016
$235$	0	0
$236$	−1.05324	−0.0685598
$237$	−3.27191	−0.212533
$238$	13.7409	0.890691
$239$	1.58711	0.102661	0.0513307	−	0.998682i	$-0.483654\pi$
$239$	1.58711	0.102661	0.0513307	+	0.998682i	$0.483654\pi$
$240$	0	0
$241$	16.2576	1.04724	0.523622	−	0.851951i	$-0.324581\pi$
$241$	16.2576	1.04724	0.523622	+	0.851951i	$0.324581\pi$
$242$	−11.7389	−0.754604
$243$	−18.8121	−1.20680
$244$	−4.14422	−0.265306
$245$	0	0
$246$	13.8398	0.882392
$247$	−7.25154	−0.461405
$248$	−9.75286	−0.619307
$249$	−29.1911	−1.84991
$250$	0	0
$251$	14.9486	0.943547	0.471774	−	0.881720i	$-0.343614\pi$
$251$	14.9486	0.943547	0.471774	+	0.881720i	$0.343614\pi$
$252$	18.4469	1.16205
$253$	−7.35447	−0.462372
$254$	8.33492	0.522979
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.9968	−0.935474	−0.467737	−	0.883868i	$-0.654931\pi$
$257$	−14.9968	−0.935474	−0.467737	+	0.883868i	$0.654931\pi$
$258$	−27.2835	−1.69860
$259$	−4.14336	−0.257456
$260$	0	0
$261$	39.5051	2.44530
$262$	−3.68597	−0.227720
$263$	−22.9610	−1.41583	−0.707917	−	0.706295i	$-0.750365\pi$
$263$	−22.9610	−1.41583	−0.707917	+	0.706295i	$0.750365\pi$
$264$	13.0174	0.801169
$265$	0	0
$266$	7.66975	0.470263
$267$	19.9163	1.21885
$268$	−1.00811	−0.0615800
$269$	10.0896	0.615175	0.307588	−	0.951520i	$-0.400478\pi$
$269$	10.0896	0.615175	0.307588	+	0.951520i	$0.400478\pi$
$270$	0	0
$271$	3.30678	0.200872	0.100436	−	0.994943i	$-0.467976\pi$
$271$	3.30678	0.200872	0.100436	+	0.994943i	$0.467976\pi$
$272$	−3.31637	−0.201084
$273$	44.3095	2.68173
$274$	−12.3027	−0.743234
$275$	0	0
$276$	4.21025	0.253427
$277$	15.9046	0.955617	0.477809	−	0.878464i	$-0.341431\pi$
$277$	15.9046	0.955617	0.477809	+	0.878464i	$0.341431\pi$
$278$	0.842442	0.0505263
$279$	43.4213	2.59957
$280$	0	0
$281$	11.3609	0.677732	0.338866	−	0.940835i	$-0.389957\pi$
$281$	11.3609	0.677732	0.338866	+	0.940835i	$0.389957\pi$
$282$	13.1770	0.784682
$283$	−20.1530	−1.19797	−0.598984	−	0.800761i	$-0.704429\pi$
$283$	−20.1530	−1.19797	−0.598984	+	0.800761i	$0.704429\pi$
$284$	6.45248	0.382884
$285$	0	0
$286$	18.6804	1.10460
$287$	−21.0059	−1.23994
$288$	−4.45216	−0.262346
$289$	−6.00171	−0.353042
$290$	0	0
$291$	−40.6717	−2.38422
$292$	−10.7714	−0.630349
$293$	21.0517	1.22986	0.614928	−	0.788583i	$-0.289185\pi$
$293$	21.0517	1.22986	0.614928	+	0.788583i	$0.289185\pi$
$294$	−27.7558	−1.61875
$295$	0	0
$296$	1.00000	0.0581238
$297$	−18.9035	−1.09689
$298$	16.7723	0.971591
$299$	6.04184	0.349408
$300$	0	0
$301$	41.4107	2.38687
$302$	7.59755	0.437190
$303$	50.7490	2.91545
$304$	−1.85109	−0.106167
$305$	0	0
$306$	14.7650	0.844059
$307$	−17.0946	−0.975638	−0.487819	−	0.872945i	$-0.662207\pi$
$307$	−17.0946	−0.975638	−0.487819	+	0.872945i	$0.662207\pi$
$308$	−19.7578	−1.12580
$309$	36.3161	2.06595
$310$	0	0
$311$	20.0503	1.13695	0.568473	−	0.822702i	$-0.307534\pi$
$311$	20.0503	1.13695	0.568473	+	0.822702i	$0.307534\pi$
$312$	−10.6941	−0.605433
$313$	−13.3400	−0.754019	−0.377010	−	0.926209i	$-0.623048\pi$
$313$	−13.3400	−0.754019	−0.377010	+	0.926209i	$0.623048\pi$
$314$	4.45631	0.251484
$315$	0	0
$316$	−1.19856	−0.0674243
$317$	−12.8528	−0.721885	−0.360943	−	0.932588i	$-0.617545\pi$
$317$	−12.8528	−0.721885	−0.360943	+	0.932588i	$0.617545\pi$
$318$	−14.0209	−0.786252
$319$	−42.3123	−2.36903
$320$	0	0
$321$	−34.0587	−1.90097
$322$	−6.39028	−0.356116
$323$	6.13890	0.341578
$324$	−2.53473	−0.140818
$325$	0	0
$326$	19.4599	1.07779
$327$	−11.8871	−0.657359
$328$	5.06977	0.279931
$329$	−20.0000	−1.10264
$330$	0	0
$331$	−23.5835	−1.29627	−0.648134	−	0.761526i	$-0.724451\pi$
$331$	−23.5835	−1.29627	−0.648134	+	0.761526i	$0.724451\pi$
$332$	−10.6932	−0.586867
$333$	−4.45216	−0.243977
$334$	7.13813	0.390581
$335$	0	0
$336$	11.3108	0.617056
$337$	−26.6074	−1.44940	−0.724698	−	0.689067i	$-0.758021\pi$
$337$	−26.6074	−1.44940	−0.724698	+	0.689067i	$0.758021\pi$
$338$	−2.34632	−0.127623
$339$	−26.0500	−1.41484
$340$	0	0
$341$	−46.5068	−2.51848
$342$	8.24137	0.445642
$343$	13.1239	0.708626
$344$	−9.99446	−0.538865
$345$	0	0
$346$	−11.6199	−0.624692
$347$	17.2626	0.926707	0.463353	−	0.886174i	$-0.346646\pi$
$347$	17.2626	0.926707	0.463353	+	0.886174i	$0.346646\pi$
$348$	24.2227	1.29847
$349$	11.1619	0.597484	0.298742	−	0.954334i	$-0.403433\pi$
$349$	11.1619	0.597484	0.298742	+	0.954334i	$0.403433\pi$
$350$	0	0
$351$	15.5296	0.828906
$352$	4.76853	0.254164
$353$	12.6563	0.673628	0.336814	−	0.941571i	$-0.390651\pi$
$353$	12.6563	0.673628	0.336814	+	0.941571i	$0.390651\pi$
$354$	2.87519	0.152815
$355$	0	0
$356$	7.29569	0.386671
$357$	−37.5108	−1.98528
$358$	17.2224	0.910232
$359$	−23.2778	−1.22855	−0.614277	−	0.789091i	$-0.710552\pi$
$359$	−23.2778	−1.22855	−0.614277	+	0.789091i	$0.710552\pi$
$360$	0	0
$361$	−15.5735	−0.819655
$362$	23.5098	1.23565
$363$	32.0456	1.68196
$364$	16.2314	0.850755
$365$	0	0
$366$	11.3132	0.591348
$367$	9.09417	0.474712	0.237356	−	0.971423i	$-0.423719\pi$
$367$	9.09417	0.474712	0.237356	+	0.971423i	$0.423719\pi$
$368$	1.54229	0.0803976
$369$	−22.5714	−1.17502
$370$	0	0
$371$	21.2808	1.10484
$372$	26.6240	1.38039
$373$	−5.75976	−0.298229	−0.149114	−	0.988820i	$-0.547642\pi$
$373$	−5.75976	−0.298229	−0.149114	+	0.988820i	$0.547642\pi$
$374$	−15.8142	−0.817733
$375$	0	0
$376$	4.82700	0.248933
$377$	34.7603	1.79025
$378$	−16.4252	−0.844820
$379$	18.0902	0.929229	0.464614	−	0.885513i	$-0.346193\pi$
$379$	18.0902	0.929229	0.464614	+	0.885513i	$0.346193\pi$
$380$	0	0
$381$	−22.7532	−1.16568
$382$	−1.16660	−0.0596887
$383$	−3.54752	−0.181270	−0.0906350	−	0.995884i	$-0.528890\pi$
$383$	−3.54752	−0.181270	−0.0906350	+	0.995884i	$0.528890\pi$
$384$	−2.72987	−0.139308
$385$	0	0
$386$	−9.36727	−0.476781
$387$	44.4970	2.26191
$388$	−14.8988	−0.756371
$389$	25.6053	1.29824	0.649119	−	0.760687i	$-0.275138\pi$
$389$	25.6053	1.29824	0.649119	+	0.760687i	$0.275138\pi$
$390$	0	0
$391$	−5.11481	−0.258667
$392$	−10.1675	−0.513534
$393$	10.0622	0.507571
$394$	3.21836	0.162139
$395$	0	0
$396$	−21.2303	−1.06686
$397$	−31.4597	−1.57892	−0.789459	−	0.613803i	$-0.789639\pi$
$397$	−31.4597	−1.57892	−0.789459	+	0.613803i	$0.789639\pi$
$398$	2.99104	0.149927
$399$	−20.9374	−1.04818
$400$	0	0
$401$	−22.9940	−1.14826	−0.574132	−	0.818763i	$-0.694660\pi$
$401$	−22.9940	−1.14826	−0.574132	+	0.818763i	$0.694660\pi$
$402$	2.75200	0.137257
$403$	38.2062	1.90319
$404$	18.5903	0.924902
$405$	0	0
$406$	−36.7650	−1.82462
$407$	4.76853	0.236367
$408$	9.05324	0.448202
$409$	−33.3828	−1.65067	−0.825337	−	0.564640i	$-0.809015\pi$
$409$	−33.3828	−1.65067	−0.825337	+	0.564640i	$0.809015\pi$
$410$	0	0
$411$	33.5848	1.65661
$412$	13.3033	0.655404
$413$	−4.36394	−0.214735
$414$	−6.86654	−0.337472
$415$	0	0
$416$	−3.91744	−0.192068
$417$	−2.29975	−0.112619
$418$	−8.82700	−0.431743
$419$	8.32565	0.406734	0.203367	−	0.979103i	$-0.434811\pi$
$419$	8.32565	0.406734	0.203367	+	0.979103i	$0.434811\pi$
$420$	0	0
$421$	−8.41521	−0.410132	−0.205066	−	0.978748i	$-0.565741\pi$
$421$	−8.41521	−0.410132	−0.205066	+	0.978748i	$0.565741\pi$
$422$	−4.75340	−0.231392
$423$	−21.4906	−1.04491
$424$	−5.13611	−0.249432
$425$	0	0
$426$	−17.6144	−0.853420
$427$	−17.1710	−0.830963
$428$	−12.4763	−0.603066
$429$	−50.9950	−2.46206
$430$	0	0
$431$	3.04769	0.146802	0.0734011	−	0.997303i	$-0.476615\pi$
$431$	3.04769	0.146802	0.0734011	+	0.997303i	$0.476615\pi$
$432$	3.96421	0.190728
$433$	−22.9439	−1.10262	−0.551308	−	0.834302i	$-0.685871\pi$
$433$	−22.9439	−1.10262	−0.551308	+	0.834302i	$0.685871\pi$
$434$	−40.4096	−1.93972
$435$	0	0
$436$	−4.35447	−0.208541
$437$	−2.85493	−0.136570
$438$	29.4045	1.40500
$439$	10.3033	0.491751	0.245875	−	0.969301i	$-0.420925\pi$
$439$	10.3033	0.491751	0.245875	+	0.969301i	$0.420925\pi$
$440$	0	0
$441$	45.2672	2.15558
$442$	12.9917	0.617950
$443$	4.84236	0.230067	0.115034	−	0.993362i	$-0.463302\pi$
$443$	4.84236	0.230067	0.115034	+	0.993362i	$0.463302\pi$
$444$	−2.72987	−0.129554
$445$	0	0
$446$	−6.95635	−0.329393
$447$	−45.7860	−2.16560
$448$	4.14336	0.195756
$449$	25.3149	1.19468	0.597341	−	0.801987i	$-0.296224\pi$
$449$	25.3149	1.19468	0.597341	+	0.801987i	$0.296224\pi$
$450$	0	0
$451$	24.1753	1.13837
$452$	−9.54261	−0.448846
$453$	−20.7403	−0.974464
$454$	−22.2980	−1.04650
$455$	0	0
$456$	5.05324	0.236639
$457$	−11.9945	−0.561077	−0.280539	−	0.959843i	$-0.590513\pi$
$457$	−11.9945	−0.561077	−0.280539	+	0.959843i	$0.590513\pi$
$458$	−12.9648	−0.605806
$459$	−13.1468	−0.613639
$460$	0	0
$461$	−23.6570	−1.10182	−0.550909	−	0.834565i	$-0.685719\pi$
$461$	−23.6570	−1.10182	−0.550909	+	0.834565i	$0.685719\pi$
$462$	53.9360	2.50933
$463$	17.3667	0.807099	0.403550	−	0.914958i	$-0.367776\pi$
$463$	17.3667	0.807099	0.403550	+	0.914958i	$0.367776\pi$
$464$	8.87323	0.411929
$465$	0	0
$466$	−10.4327	−0.483285
$467$	20.4476	0.946200	0.473100	−	0.881009i	$-0.343135\pi$
$467$	20.4476	0.946200	0.473100	+	0.881009i	$0.343135\pi$
$468$	17.4411	0.806214
$469$	−4.17696	−0.192874
$470$	0	0
$471$	−12.1651	−0.560539
$472$	1.05324	0.0484791
$473$	−47.6589	−2.19136
$474$	3.27191	0.150284
$475$	0	0
$476$	−13.7409	−0.629814
$477$	22.8668	1.04700
$478$	−1.58711	−0.0725925
$479$	−34.1325	−1.55955	−0.779777	−	0.626057i	$-0.784668\pi$
$479$	−34.1325	−1.55955	−0.779777	+	0.626057i	$0.784668\pi$
$480$	0	0
$481$	−3.91744	−0.178620
$482$	−16.2576	−0.740513
$483$	17.4446	0.793757
$484$	11.7389	0.533586
$485$	0	0
$486$	18.8121	0.853334
$487$	−6.16917	−0.279552	−0.139776	−	0.990183i	$-0.544638\pi$
$487$	−6.16917	−0.279552	−0.139776	+	0.990183i	$0.544638\pi$
$488$	4.14422	0.187600
$489$	−53.1230	−2.40231
$490$	0	0
$491$	−11.8290	−0.533836	−0.266918	−	0.963719i	$-0.586005\pi$
$491$	−11.8290	−0.533836	−0.266918	+	0.963719i	$0.586005\pi$
$492$	−13.8398	−0.623945
$493$	−29.4269	−1.32532
$494$	7.25154	0.326262
$495$	0	0
$496$	9.75286	0.437916
$497$	26.7350	1.19923
$498$	29.1911	1.30808
$499$	31.4004	1.40568	0.702839	−	0.711349i	$-0.251916\pi$
$499$	31.4004	1.40568	0.702839	+	0.711349i	$0.251916\pi$
$500$	0	0
$501$	−19.4861	−0.870577
$502$	−14.9486	−0.667189
$503$	−39.7743	−1.77345	−0.886724	−	0.462299i	$-0.847025\pi$
$503$	−39.7743	−1.77345	−0.886724	+	0.462299i	$0.847025\pi$
$504$	−18.4469	−0.821692
$505$	0	0
$506$	7.35447	0.326946
$507$	6.40514	0.284462
$508$	−8.33492	−0.369802
$509$	−6.82812	−0.302651	−0.151326	−	0.988484i	$-0.548354\pi$
$509$	−6.82812	−0.302651	−0.151326	+	0.988484i	$0.548354\pi$
$510$	0	0
$511$	−44.6299	−1.97431
$512$	−1.00000	−0.0441942
$513$	−7.33813	−0.323986
$514$	14.9968	0.661480
$515$	0	0
$516$	27.2835	1.20109
$517$	23.0177	1.01232
$518$	4.14336	0.182049
$519$	31.7209	1.39239
$520$	0	0
$521$	−30.6638	−1.34341	−0.671703	−	0.740821i	$-0.734437\pi$
$521$	−30.6638	−1.34341	−0.671703	+	0.740821i	$0.734437\pi$
$522$	−39.5051	−1.72909
$523$	−12.1618	−0.531800	−0.265900	−	0.964001i	$-0.585669\pi$
$523$	−12.1618	−0.531800	−0.265900	+	0.964001i	$0.585669\pi$
$524$	3.68597	0.161022
$525$	0	0
$526$	22.9610	1.00115
$527$	−32.3441	−1.40893
$528$	−13.0174	−0.566512
$529$	−20.6213	−0.896580
$530$	0	0
$531$	−4.68918	−0.203493
$532$	−7.66975	−0.332526
$533$	−19.8605	−0.860253
$534$	−19.9163	−0.861861
$535$	0	0
$536$	1.00811	0.0435437
$537$	−47.0148	−2.02884
$538$	−10.0896	−0.434995
$539$	−48.4839	−2.08835
$540$	0	0
$541$	10.5469	0.453446	0.226723	−	0.973959i	$-0.427199\pi$
$541$	10.5469	0.453446	0.226723	+	0.973959i	$0.427199\pi$
$542$	−3.30678	−0.142038
$543$	−64.1785	−2.75416
$544$	3.31637	0.142188
$545$	0	0
$546$	−44.3095	−1.89627
$547$	−4.09517	−0.175097	−0.0875484	−	0.996160i	$-0.527903\pi$
$547$	−4.09517	−0.175097	−0.0875484	+	0.996160i	$0.527903\pi$
$548$	12.3027	0.525546
$549$	−18.4507	−0.787459
$550$	0	0
$551$	−16.4252	−0.699736
$552$	−4.21025	−0.179200
$553$	−4.96607	−0.211179
$554$	−15.9046	−0.675723
$555$	0	0
$556$	−0.842442	−0.0357275
$557$	2.50711	0.106230	0.0531149	−	0.998588i	$-0.483085\pi$
$557$	2.50711	0.106230	0.0531149	+	0.998588i	$0.483085\pi$
$558$	−43.4213	−1.83817
$559$	39.1527	1.65598
$560$	0	0
$561$	43.1706	1.82267
$562$	−11.3609	−0.479229
$563$	−9.87532	−0.416195	−0.208097	−	0.978108i	$-0.566727\pi$
$563$	−9.87532	−0.416195	−0.208097	+	0.978108i	$0.566727\pi$
$564$	−13.1770	−0.554854
$565$	0	0
$566$	20.1530	0.847092
$567$	−10.5023	−0.441055
$568$	−6.45248	−0.270740
$569$	12.5198	0.524858	0.262429	−	0.964951i	$-0.415477\pi$
$569$	12.5198	0.524858	0.262429	+	0.964951i	$0.415477\pi$
$570$	0	0
$571$	20.3156	0.850179	0.425090	−	0.905151i	$-0.360243\pi$
$571$	20.3156	0.850179	0.425090	+	0.905151i	$0.360243\pi$
$572$	−18.6804	−0.781068
$573$	3.18467	0.133042
$574$	21.0059	0.876769
$575$	0	0
$576$	4.45216	0.185507
$577$	−20.0756	−0.835759	−0.417880	−	0.908502i	$-0.637227\pi$
$577$	−20.0756	−0.835759	−0.417880	+	0.908502i	$0.637227\pi$
$578$	6.00171	0.249638
$579$	25.5714	1.06271
$580$	0	0
$581$	−44.3059	−1.83812
$582$	40.6717	1.68590
$583$	−24.4917	−1.01434
$584$	10.7714	0.445724
$585$	0	0
$586$	−21.0517	−0.869639
$587$	−3.71178	−0.153201	−0.0766007	−	0.997062i	$-0.524407\pi$
$587$	−3.71178	−0.153201	−0.0766007	+	0.997062i	$0.524407\pi$
$588$	27.7558	1.14463
$589$	−18.0534	−0.743879
$590$	0	0
$591$	−8.78569	−0.361395
$592$	−1.00000	−0.0410997
$593$	−36.2899	−1.49025	−0.745124	−	0.666926i	$-0.767610\pi$
$593$	−36.2899	−1.49025	−0.745124	+	0.666926i	$0.767610\pi$
$594$	18.9035	0.775619
$595$	0	0
$596$	−16.7723	−0.687019
$597$	−8.16512	−0.334176
$598$	−6.04184	−0.247069
$599$	23.7528	0.970515	0.485257	−	0.874371i	$-0.338726\pi$
$599$	23.7528	0.970515	0.485257	+	0.874371i	$0.338726\pi$
$600$	0	0
$601$	15.9718	0.651502	0.325751	−	0.945456i	$-0.394383\pi$
$601$	15.9718	0.651502	0.325751	+	0.945456i	$0.394383\pi$
$602$	−41.4107	−1.68777
$603$	−4.48827	−0.182776
$604$	−7.59755	−0.309140
$605$	0	0
$606$	−50.7490	−2.06154
$607$	−38.0139	−1.54294	−0.771469	−	0.636267i	$-0.780478\pi$
$607$	−38.0139	−1.54294	−0.771469	+	0.636267i	$0.780478\pi$
$608$	1.85109	0.0750717
$609$	100.364	4.06694
$610$	0	0
$611$	−18.9095	−0.764995
$612$	−14.7650	−0.596840
$613$	8.40204	0.339355	0.169678	−	0.985500i	$-0.445727\pi$
$613$	8.40204	0.339355	0.169678	+	0.985500i	$0.445727\pi$
$614$	17.0946	0.689880
$615$	0	0
$616$	19.7578	0.796063
$617$	6.72707	0.270822	0.135411	−	0.990790i	$-0.456765\pi$
$617$	6.72707	0.270822	0.135411	+	0.990790i	$0.456765\pi$
$618$	−36.3161	−1.46085
$619$	24.3689	0.979470	0.489735	−	0.871871i	$-0.337094\pi$
$619$	24.3689	0.979470	0.489735	+	0.871871i	$0.337094\pi$
$620$	0	0
$621$	6.11398	0.245345
$622$	−20.0503	−0.803943
$623$	30.2287	1.21109
$624$	10.6941	0.428106
$625$	0	0
$626$	13.3400	0.533172
$627$	24.0965	0.962322
$628$	−4.45631	−0.177826
$629$	3.31637	0.132232
$630$	0	0
$631$	34.7737	1.38432	0.692160	−	0.721744i	$-0.256659\pi$
$631$	34.7737	1.38432	0.692160	+	0.721744i	$0.256659\pi$
$632$	1.19856	0.0476762
$633$	12.9761	0.515755
$634$	12.8528	0.510450
$635$	0	0
$636$	14.0209	0.555964
$637$	39.8304	1.57814
$638$	42.3123	1.67516
$639$	28.7275	1.13644
$640$	0	0
$641$	−27.8544	−1.10018	−0.550091	−	0.835105i	$-0.685407\pi$
$641$	−27.8544	−1.10018	−0.550091	+	0.835105i	$0.685407\pi$
$642$	34.0587	1.34419
$643$	35.4013	1.39609	0.698045	−	0.716054i	$-0.254053\pi$
$643$	35.4013	1.39609	0.698045	+	0.716054i	$0.254053\pi$
$644$	6.39028	0.251812
$645$	0	0
$646$	−6.13890	−0.241532
$647$	−16.1343	−0.634305	−0.317152	−	0.948375i	$-0.602727\pi$
$647$	−16.1343	−0.634305	−0.317152	+	0.948375i	$0.602727\pi$
$648$	2.53473	0.0995734
$649$	5.02239	0.197146
$650$	0	0
$651$	110.313	4.32350
$652$	−19.4599	−0.762110
$653$	−11.9146	−0.466256	−0.233128	−	0.972446i	$-0.574896\pi$
$653$	−11.9146	−0.466256	−0.233128	+	0.972446i	$0.574896\pi$
$654$	11.8871	0.464823
$655$	0	0
$656$	−5.06977	−0.197941
$657$	−47.9561	−1.87095
$658$	20.0000	0.779681
$659$	−30.5124	−1.18859	−0.594297	−	0.804246i	$-0.702569\pi$
$659$	−30.5124	−1.18859	−0.594297	+	0.804246i	$0.702569\pi$
$660$	0	0
$661$	42.0973	1.63740	0.818698	−	0.574224i	$-0.194696\pi$
$661$	42.0973	1.63740	0.818698	+	0.574224i	$0.194696\pi$
$662$	23.5835	0.916600
$663$	−35.4655	−1.37737
$664$	10.6932	0.414977
$665$	0	0
$666$	4.45216	0.172518
$667$	13.6851	0.529890
$668$	−7.13813	−0.276183
$669$	18.9899	0.734192
$670$	0	0
$671$	19.7618	0.762897
$672$	−11.3108	−0.436324
$673$	41.1540	1.58637	0.793184	−	0.608982i	$-0.208422\pi$
$673$	41.1540	1.58637	0.793184	+	0.608982i	$0.208422\pi$
$674$	26.6074	1.02488
$675$	0	0
$676$	2.34632	0.0902431
$677$	10.3914	0.399373	0.199686	−	0.979860i	$-0.436008\pi$
$677$	10.3914	0.399373	0.199686	+	0.979860i	$0.436008\pi$
$678$	26.0500	1.00045
$679$	−61.7311	−2.36902
$680$	0	0
$681$	60.8706	2.33257
$682$	46.5068	1.78084
$683$	−5.05345	−0.193365	−0.0966824	−	0.995315i	$-0.530823\pi$
$683$	−5.05345	−0.193365	−0.0966824	+	0.995315i	$0.530823\pi$
$684$	−8.24137	−0.315117
$685$	0	0
$686$	−13.1239	−0.501074
$687$	35.3922	1.35030
$688$	9.99446	0.381035
$689$	20.1204	0.766526
$690$	0	0
$691$	13.7857	0.524432	0.262216	−	0.965009i	$-0.415547\pi$
$691$	13.7857	0.524432	0.262216	+	0.965009i	$0.415547\pi$
$692$	11.6199	0.441724
$693$	−87.9648	−3.34151
$694$	−17.2626	−0.655280
$695$	0	0
$696$	−24.2227	−0.918160
$697$	16.8132	0.636846
$698$	−11.1619	−0.422485
$699$	28.4798	1.07721
$700$	0	0
$701$	−2.20512	−0.0832862	−0.0416431	−	0.999133i	$-0.513259\pi$
$701$	−2.20512	−0.0832862	−0.0416431	+	0.999133i	$0.513259\pi$
$702$	−15.5296	−0.586125
$703$	1.85109	0.0698153
$704$	−4.76853	−0.179721
$705$	0	0
$706$	−12.6563	−0.476327
$707$	77.0264	2.89687
$708$	−2.87519	−0.108056
$709$	−12.7194	−0.477687	−0.238843	−	0.971058i	$-0.576768\pi$
$709$	−12.7194	−0.477687	−0.238843	+	0.971058i	$0.576768\pi$
$710$	0	0
$711$	−5.33619	−0.200123
$712$	−7.29569	−0.273418
$713$	15.0418	0.563318
$714$	37.5108	1.40381
$715$	0	0
$716$	−17.2224	−0.643631
$717$	4.33258	0.161803
$718$	23.2778	0.868718
$719$	34.9654	1.30399	0.651996	−	0.758223i	$-0.273932\pi$
$719$	34.9654	1.30399	0.651996	+	0.758223i	$0.273932\pi$
$720$	0	0
$721$	55.1202	2.05278
$722$	15.5735	0.579584
$723$	44.3810	1.65055
$724$	−23.5098	−0.873733
$725$	0	0
$726$	−32.0456	−1.18932
$727$	9.90635	0.367406	0.183703	−	0.982982i	$-0.441191\pi$
$727$	9.90635	0.367406	0.183703	+	0.982982i	$0.441191\pi$
$728$	−16.2314	−0.601575
$729$	−43.7503	−1.62038
$730$	0	0
$731$	−33.1453	−1.22592
$732$	−11.3132	−0.418146
$733$	−37.3826	−1.38076	−0.690379	−	0.723448i	$-0.742556\pi$
$733$	−37.3826	−1.38076	−0.690379	+	0.723448i	$0.742556\pi$
$734$	−9.09417	−0.335672
$735$	0	0
$736$	−1.54229	−0.0568497
$737$	4.80720	0.177075
$738$	22.5714	0.830866
$739$	14.4487	0.531506	0.265753	−	0.964041i	$-0.414380\pi$
$739$	14.4487	0.531506	0.265753	+	0.964041i	$0.414380\pi$
$740$	0	0
$741$	−19.7957	−0.727215
$742$	−21.2808	−0.781242
$743$	−45.1826	−1.65759	−0.828794	−	0.559553i	$-0.810973\pi$
$743$	−45.1826	−1.65759	−0.828794	+	0.559553i	$0.810973\pi$
$744$	−26.6240	−0.976083
$745$	0	0
$746$	5.75976	0.210880
$747$	−47.6080	−1.74188
$748$	15.8142	0.578224
$749$	−51.6939	−1.88886
$750$	0	0
$751$	49.3127	1.79945	0.899723	−	0.436461i	$-0.143768\pi$
$751$	49.3127	1.79945	0.899723	+	0.436461i	$0.143768\pi$
$752$	−4.82700	−0.176022
$753$	40.8077	1.48711
$754$	−34.7603	−1.26590
$755$	0	0
$756$	16.4252	0.597378
$757$	−21.5911	−0.784741	−0.392370	−	0.919807i	$-0.628345\pi$
$757$	−21.5911	−0.784741	−0.392370	+	0.919807i	$0.628345\pi$
$758$	−18.0902	−0.657064
$759$	−20.0767	−0.728738
$760$	0	0
$761$	18.4253	0.667917	0.333959	−	0.942588i	$-0.391615\pi$
$761$	18.4253	0.667917	0.333959	+	0.942588i	$0.391615\pi$
$762$	22.7532	0.824262
$763$	−18.0422	−0.653170
$764$	1.16660	0.0422063
$765$	0	0
$766$	3.54752	0.128177
$767$	−4.12598	−0.148981
$768$	2.72987	0.0985055
$769$	6.50829	0.234695	0.117348	−	0.993091i	$-0.462561\pi$
$769$	6.50829	0.234695	0.117348	+	0.993091i	$0.462561\pi$
$770$	0	0
$771$	−40.9392	−1.47439
$772$	9.36727	0.337135
$773$	40.5853	1.45975	0.729875	−	0.683581i	$-0.239578\pi$
$773$	40.5853	1.45975	0.729875	+	0.683581i	$0.239578\pi$
$774$	−44.4970	−1.59941
$775$	0	0
$776$	14.8988	0.534835
$777$	−11.3108	−0.405774
$778$	−25.6053	−0.917993
$779$	9.38461	0.336239
$780$	0	0
$781$	−30.7688	−1.10100
$782$	5.11481	0.182905
$783$	35.1754	1.25707
$784$	10.1675	0.363124
$785$	0	0
$786$	−10.0622	−0.358907
$787$	37.5389	1.33812	0.669059	−	0.743210i	$-0.266698\pi$
$787$	37.5389	1.33812	0.669059	+	0.743210i	$0.266698\pi$
$788$	−3.21836	−0.114649
$789$	−62.6804	−2.23148
$790$	0	0
$791$	−39.5385	−1.40583
$792$	21.2303	0.754385
$793$	−16.2347	−0.576512
$794$	31.4597	1.11646
$795$	0	0
$796$	−2.99104	−0.106014
$797$	−28.8746	−1.02279	−0.511396	−	0.859345i	$-0.670871\pi$
$797$	−28.8746	−1.02279	−0.511396	+	0.859345i	$0.670871\pi$
$798$	20.9374	0.741176
$799$	16.0081	0.566326
$800$	0	0
$801$	32.4816	1.14768
$802$	22.9940	0.811945
$803$	51.3638	1.81259
$804$	−2.75200	−0.0970556
$805$	0	0
$806$	−38.2062	−1.34576
$807$	27.5433	0.969571
$808$	−18.5903	−0.654004
$809$	−8.98633	−0.315943	−0.157971	−	0.987444i	$-0.550495\pi$
$809$	−8.98633	−0.315943	−0.157971	+	0.987444i	$0.550495\pi$
$810$	0	0
$811$	−24.4949	−0.860134	−0.430067	−	0.902797i	$-0.641510\pi$
$811$	−24.4949	−0.860134	−0.430067	+	0.902797i	$0.641510\pi$
$812$	36.7650	1.29020
$813$	9.02706	0.316593
$814$	−4.76853	−0.167137
$815$	0	0
$816$	−9.05324	−0.316927
$817$	−18.5007	−0.647257
$818$	33.3828	1.16720
$819$	72.2647	2.52513
$820$	0	0
$821$	27.1346	0.947005	0.473502	−	0.880793i	$-0.342990\pi$
$821$	27.1346	0.947005	0.473502	+	0.880793i	$0.342990\pi$
$822$	−33.5848	−1.17140
$823$	10.7460	0.374584	0.187292	−	0.982304i	$-0.440029\pi$
$823$	10.7460	0.374584	0.187292	+	0.982304i	$0.440029\pi$
$824$	−13.3033	−0.463441
$825$	0	0
$826$	4.36394	0.151841
$827$	−19.1035	−0.664293	−0.332147	−	0.943228i	$-0.607773\pi$
$827$	−19.1035	−0.664293	−0.332147	+	0.943228i	$0.607773\pi$
$828$	6.86654	0.238629
$829$	51.1250	1.77565	0.887823	−	0.460185i	$-0.152217\pi$
$829$	51.1250	1.77565	0.887823	+	0.460185i	$0.152217\pi$
$830$	0	0
$831$	43.4175	1.50614
$832$	3.91744	0.135813
$833$	−33.7190	−1.16830
$834$	2.29975	0.0796340
$835$	0	0
$836$	8.82700	0.305288
$837$	38.6624	1.33637
$838$	−8.32565	−0.287605
$839$	−42.1947	−1.45672	−0.728361	−	0.685193i	$-0.759718\pi$
$839$	−42.1947	−1.45672	−0.728361	+	0.685193i	$0.759718\pi$
$840$	0	0
$841$	49.7342	1.71497
$842$	8.41521	0.290007
$843$	31.0136	1.06816
$844$	4.75340	0.163619
$845$	0	0
$846$	21.4906	0.738861
$847$	48.6385	1.67124
$848$	5.13611	0.176375
$849$	−55.0148	−1.88810
$850$	0	0
$851$	−1.54229	−0.0528691
$852$	17.6144	0.603459
$853$	−42.5489	−1.45685	−0.728423	−	0.685128i	$-0.759746\pi$
$853$	−42.5489	−1.45685	−0.728423	+	0.685128i	$0.759746\pi$
$854$	17.1710	0.587580
$855$	0	0
$856$	12.4763	0.426432
$857$	−29.7588	−1.01654	−0.508271	−	0.861197i	$-0.669715\pi$
$857$	−29.7588	−1.01654	−0.508271	+	0.861197i	$0.669715\pi$
$858$	50.9950	1.74094
$859$	8.40182	0.286666	0.143333	−	0.989674i	$-0.454218\pi$
$859$	8.40182	0.286666	0.143333	+	0.989674i	$0.454218\pi$
$860$	0	0
$861$	−57.3432	−1.95425
$862$	−3.04769	−0.103805
$863$	36.3794	1.23837	0.619185	−	0.785245i	$-0.287463\pi$
$863$	36.3794	1.23837	0.619185	+	0.785245i	$0.287463\pi$
$864$	−3.96421	−0.134865
$865$	0	0
$866$	22.9439	0.779667
$867$	−16.3839	−0.556425
$868$	40.4096	1.37159
$869$	5.71537	0.193881
$870$	0	0
$871$	−3.94920	−0.133814
$872$	4.35447	0.147461
$873$	−66.3318	−2.24499
$874$	2.85493	0.0965694
$875$	0	0
$876$	−29.4045	−0.993486
$877$	−41.3380	−1.39588	−0.697942	−	0.716154i	$-0.745901\pi$
$877$	−41.3380	−1.39588	−0.697942	+	0.716154i	$0.745901\pi$
$878$	−10.3033	−0.347720
$879$	57.4684	1.93836
$880$	0	0
$881$	21.1354	0.712071	0.356035	−	0.934472i	$-0.384128\pi$
$881$	21.1354	0.712071	0.356035	+	0.934472i	$0.384128\pi$
$882$	−45.2672	−1.52423
$883$	10.7365	0.361312	0.180656	−	0.983546i	$-0.442178\pi$
$883$	10.7365	0.361312	0.180656	+	0.983546i	$0.442178\pi$
$884$	−12.9917	−0.436957
$885$	0	0
$886$	−4.84236	−0.162682
$887$	11.1798	0.375379	0.187690	−	0.982228i	$-0.439900\pi$
$887$	11.1798	0.375379	0.187690	+	0.982228i	$0.439900\pi$
$888$	2.72987	0.0916083
$889$	−34.5346	−1.15825
$890$	0	0
$891$	12.0869	0.404927
$892$	6.95635	0.232916
$893$	8.93522	0.299006
$894$	45.7860	1.53131
$895$	0	0
$896$	−4.14336	−0.138420
$897$	16.4934	0.550698
$898$	−25.3149	−0.844768
$899$	86.5393	2.88625
$900$	0	0
$901$	−17.0332	−0.567459
$902$	−24.1753	−0.804951
$903$	113.046	3.76192
$904$	9.54261	0.317382
$905$	0	0
$906$	20.7403	0.689050
$907$	20.5343	0.681831	0.340915	−	0.940094i	$-0.389263\pi$
$907$	20.5343	0.681831	0.340915	+	0.940094i	$0.389263\pi$
$908$	22.2980	0.739986
$909$	82.7671	2.74521
$910$	0	0
$911$	17.5197	0.580454	0.290227	−	0.956958i	$-0.406269\pi$
$911$	17.5197	0.580454	0.290227	+	0.956958i	$0.406269\pi$
$912$	−5.05324	−0.167329
$913$	50.9910	1.68755
$914$	11.9945	0.396741
$915$	0	0
$916$	12.9648	0.428370
$917$	15.2723	0.504336
$918$	13.1468	0.433909
$919$	43.8760	1.44734	0.723668	−	0.690149i	$-0.242455\pi$
$919$	43.8760	1.44734	0.723668	+	0.690149i	$0.242455\pi$
$920$	0	0
$921$	−46.6658	−1.53769
$922$	23.6570	0.779103
$923$	25.2772	0.832009
$924$	−53.9360	−1.77436
$925$	0	0
$926$	−17.3667	−0.570705
$927$	59.2283	1.94531
$928$	−8.87323	−0.291278
$929$	−21.9800	−0.721142	−0.360571	−	0.932732i	$-0.617418\pi$
$929$	−21.9800	−0.721142	−0.360571	+	0.932732i	$0.617418\pi$
$930$	0	0
$931$	−18.8209	−0.616831
$932$	10.4327	0.341734
$933$	54.7346	1.79193
$934$	−20.4476	−0.669065
$935$	0	0
$936$	−17.4411	−0.570079
$937$	9.62735	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$937$	9.62735	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$938$	4.17696	0.136383
$939$	−36.4163	−1.18840
$940$	0	0
$941$	−28.3548	−0.924340	−0.462170	−	0.886791i	$-0.652929\pi$
$941$	−28.3548	−0.924340	−0.462170	+	0.886791i	$0.652929\pi$
$942$	12.1651	0.396361
$943$	−7.81906	−0.254624
$944$	−1.05324	−0.0342799
$945$	0	0
$946$	47.6589	1.54952
$947$	12.5179	0.406778	0.203389	−	0.979098i	$-0.434804\pi$
$947$	12.5179	0.406778	0.203389	+	0.979098i	$0.434804\pi$
$948$	−3.27191	−0.106267
$949$	−42.1963	−1.36975
$950$	0	0
$951$	−35.0864	−1.13776
$952$	13.7409	0.445346
$953$	3.39730	0.110049	0.0550247	−	0.998485i	$-0.482476\pi$
$953$	3.39730	0.110049	0.0550247	+	0.998485i	$0.482476\pi$
$954$	−22.8668	−0.740340
$955$	0	0
$956$	1.58711	0.0513307
$957$	−115.507	−3.73380
$958$	34.1325	1.10277
$959$	50.9746	1.64606
$960$	0	0
$961$	64.1182	2.06833
$962$	3.91744	0.126303
$963$	−55.5466	−1.78997
$964$	16.2576	0.523622
$965$	0	0
$966$	−17.4446	−0.561271
$967$	−2.54955	−0.0819879	−0.0409940	−	0.999159i	$-0.513052\pi$
$967$	−2.54955	−0.0819879	−0.0409940	+	0.999159i	$0.513052\pi$
$968$	−11.7389	−0.377302
$969$	16.7584	0.538357
$970$	0	0
$971$	−35.2223	−1.13034	−0.565168	−	0.824976i	$-0.691189\pi$
$971$	−35.2223	−1.13034	−0.565168	+	0.824976i	$0.691189\pi$
$972$	−18.8121	−0.603398
$973$	−3.49054	−0.111902
$974$	6.16917	0.197673
$975$	0	0
$976$	−4.14422	−0.132653
$977$	5.79786	0.185490	0.0927450	−	0.995690i	$-0.470436\pi$
$977$	5.79786	0.185490	0.0927450	+	0.995690i	$0.470436\pi$
$978$	53.1230	1.69869
$979$	−34.7897	−1.11188
$980$	0	0
$981$	−19.3868	−0.618973
$982$	11.8290	0.377479
$983$	20.6825	0.659671	0.329835	−	0.944038i	$-0.393007\pi$
$983$	20.6825	0.659671	0.329835	+	0.944038i	$0.393007\pi$
$984$	13.8398	0.441196
$985$	0	0
$986$	29.4269	0.937143
$987$	−54.5973	−1.73785
$988$	−7.25154	−0.230702
$989$	15.4144	0.490149
$990$	0	0
$991$	49.7158	1.57927	0.789637	−	0.613574i	$-0.210269\pi$
$991$	49.7158	1.57927	0.789637	+	0.613574i	$0.210269\pi$
$992$	−9.75286	−0.309654
$993$	−64.3799	−2.04303
$994$	−26.7350	−0.847981
$995$	0	0
$996$	−29.1911	−0.924954
$997$	53.1550	1.68344	0.841718	−	0.539918i	$-0.181545\pi$
$997$	53.1550	1.68344	0.841718	+	0.539918i	$0.181545\pi$
$998$	−31.4004	−0.993964
$999$	−3.96421	−0.125422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bd.1.5		5
5.2	odd	4		370.2.b.d.149.1	✓	10
5.3	odd	4		370.2.b.d.149.10	yes	10
5.4	even	2		1850.2.a.be.1.1		5
15.2	even	4		3330.2.d.p.1999.10		10
15.8	even	4		3330.2.d.p.1999.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.d.149.1	✓	10	5.2	odd	4
370.2.b.d.149.10	yes	10	5.3	odd	4
1850.2.a.bd.1.5		5	1.1	even	1	trivial
1850.2.a.be.1.1		5	5.4	even	2
3330.2.d.p.1999.5		10	15.8	even	4
3330.2.d.p.1999.10		10	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} +4.14336 q^{7} -1.00000 q^{8} +4.45216 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} +4.14336 q^{7} -1.00000 q^{8} +4.45216 q^{9} -4.76853 q^{11} +2.72987 q^{12} +3.91744 q^{13} -4.14336 q^{14} +1.00000 q^{16} -3.31637 q^{17} -4.45216 q^{18} -1.85109 q^{19} +11.3108 q^{21} +4.76853 q^{22} +1.54229 q^{23} -2.72987 q^{24} -3.91744 q^{26} +3.96421 q^{27} +4.14336 q^{28} +8.87323 q^{29} +9.75286 q^{31} -1.00000 q^{32} -13.0174 q^{33} +3.31637 q^{34} +4.45216 q^{36} -1.00000 q^{37} +1.85109 q^{38} +10.6941 q^{39} -5.06977 q^{41} -11.3108 q^{42} +9.99446 q^{43} -4.76853 q^{44} -1.54229 q^{46} -4.82700 q^{47} +2.72987 q^{48} +10.1675 q^{49} -9.05324 q^{51} +3.91744 q^{52} +5.13611 q^{53} -3.96421 q^{54} -4.14336 q^{56} -5.05324 q^{57} -8.87323 q^{58} -1.05324 q^{59} -4.14422 q^{61} -9.75286 q^{62} +18.4469 q^{63} +1.00000 q^{64} +13.0174 q^{66} -1.00811 q^{67} -3.31637 q^{68} +4.21025 q^{69} +6.45248 q^{71} -4.45216 q^{72} -10.7714 q^{73} +1.00000 q^{74} -1.85109 q^{76} -19.7578 q^{77} -10.6941 q^{78} -1.19856 q^{79} -2.53473 q^{81} +5.06977 q^{82} -10.6932 q^{83} +11.3108 q^{84} -9.99446 q^{86} +24.2227 q^{87} +4.76853 q^{88} +7.29569 q^{89} +16.2314 q^{91} +1.54229 q^{92} +26.6240 q^{93} +4.82700 q^{94} -2.72987 q^{96} -14.8988 q^{97} -10.1675 q^{98} -21.2303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Introduction

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