LMFDB - Embedded newform 2850.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(1,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.35026 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.35026 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.61213 q^{11} -1.00000 q^{12} +1.35026 q^{13} +3.35026 q^{14} +1.00000 q^{16} -6.96239 q^{17} -1.00000 q^{18} -1.00000 q^{19} +3.35026 q^{21} +1.61213 q^{22} +1.35026 q^{23} +1.00000 q^{24} -1.35026 q^{26} -1.00000 q^{27} -3.35026 q^{28} +3.61213 q^{29} -2.31265 q^{31} -1.00000 q^{32} +1.61213 q^{33} +6.96239 q^{34} +1.00000 q^{36} -11.2750 q^{37} +1.00000 q^{38} -1.35026 q^{39} -3.35026 q^{41} -3.35026 q^{42} +10.3127 q^{43} -1.61213 q^{44} -1.35026 q^{46} +4.57452 q^{47} -1.00000 q^{48} +4.22425 q^{49} +6.96239 q^{51} +1.35026 q^{52} -11.9248 q^{53} +1.00000 q^{54} +3.35026 q^{56} +1.00000 q^{57} -3.61213 q^{58} +1.03761 q^{59} +2.00000 q^{61} +2.31265 q^{62} -3.35026 q^{63} +1.00000 q^{64} -1.61213 q^{66} +9.92478 q^{67} -6.96239 q^{68} -1.35026 q^{69} -0.775746 q^{71} -1.00000 q^{72} +3.22425 q^{73} +11.2750 q^{74} -1.00000 q^{76} +5.40105 q^{77} +1.35026 q^{78} +14.3127 q^{79} +1.00000 q^{81} +3.35026 q^{82} -10.8872 q^{83} +3.35026 q^{84} -10.3127 q^{86} -3.61213 q^{87} +1.61213 q^{88} -2.57452 q^{89} -4.52373 q^{91} +1.35026 q^{92} +2.31265 q^{93} -4.57452 q^{94} +1.00000 q^{96} +1.16362 q^{97} -4.22425 q^{98} -1.61213 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 4 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{18} - 3 q^{19} + 4 q^{22} - 6 q^{23} + 3 q^{24} + 6 q^{26} - 3 q^{27} + 10 q^{29} + 14 q^{31} - 3 q^{32} + 4 q^{33} + 10 q^{34} + 3 q^{36} - 2 q^{37} + 3 q^{38} + 6 q^{39} + 10 q^{43} - 4 q^{44} + 6 q^{46} + 2 q^{47} - 3 q^{48} + 11 q^{49} + 10 q^{51} - 6 q^{52} - 14 q^{53} + 3 q^{54} + 3 q^{57} - 10 q^{58} + 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 4 q^{66} + 8 q^{67} - 10 q^{68} + 6 q^{69} - 4 q^{71} - 3 q^{72} + 8 q^{73} + 2 q^{74} - 3 q^{76} - 24 q^{77} - 6 q^{78} + 22 q^{79} + 3 q^{81} - 10 q^{86} - 10 q^{87} + 4 q^{88} + 4 q^{89} - 32 q^{91} - 6 q^{92} - 14 q^{93} - 2 q^{94} + 3 q^{96} + 6 q^{97} - 11 q^{98} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 4 * q^11 - 3 * q^12 - 6 * q^13 + 3 * q^16 - 10 * q^17 - 3 * q^18 - 3 * q^19 + 4 * q^22 - 6 * q^23 + 3 * q^24 + 6 * q^26 - 3 * q^27 + 10 * q^29 + 14 * q^31 - 3 * q^32 + 4 * q^33 + 10 * q^34 + 3 * q^36 - 2 * q^37 + 3 * q^38 + 6 * q^39 + 10 * q^43 - 4 * q^44 + 6 * q^46 + 2 * q^47 - 3 * q^48 + 11 * q^49 + 10 * q^51 - 6 * q^52 - 14 * q^53 + 3 * q^54 + 3 * q^57 - 10 * q^58 + 14 * q^59 + 6 * q^61 - 14 * q^62 + 3 * q^64 - 4 * q^66 + 8 * q^67 - 10 * q^68 + 6 * q^69 - 4 * q^71 - 3 * q^72 + 8 * q^73 + 2 * q^74 - 3 * q^76 - 24 * q^77 - 6 * q^78 + 22 * q^79 + 3 * q^81 - 10 * q^86 - 10 * q^87 + 4 * q^88 + 4 * q^89 - 32 * q^91 - 6 * q^92 - 14 * q^93 - 2 * q^94 + 3 * q^96 + 6 * q^97 - 11 * q^98 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−3.35026	−1.26628	−0.633140	−	0.774037i	$-0.718234\pi$
$7$	−3.35026	−1.26628	−0.633140	+	0.774037i	$0.718234\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.61213	−0.486075	−0.243037	−	0.970017i	$-0.578144\pi$
$11$	−1.61213	−0.486075	−0.243037	+	0.970017i	$0.578144\pi$
$12$	−1.00000	−0.288675
$13$	1.35026	0.374495	0.187248	−	0.982313i	$-0.440043\pi$
$13$	1.35026	0.374495	0.187248	+	0.982313i	$0.440043\pi$
$14$	3.35026	0.895395
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.96239	−1.68863	−0.844314	−	0.535849i	$-0.819992\pi$
$17$	−6.96239	−1.68863	−0.844314	+	0.535849i	$0.819992\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.35026	0.731087
$22$	1.61213	0.343707
$23$	1.35026	0.281549	0.140775	−	0.990042i	$-0.455041\pi$
$23$	1.35026	0.281549	0.140775	+	0.990042i	$0.455041\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.35026	−0.264808
$27$	−1.00000	−0.192450
$28$	−3.35026	−0.633140
$29$	3.61213	0.670755	0.335378	−	0.942084i	$-0.391136\pi$
$29$	3.61213	0.670755	0.335378	+	0.942084i	$0.391136\pi$
$30$	0	0
$31$	−2.31265	−0.415364	−0.207682	−	0.978196i	$-0.566592\pi$
$31$	−2.31265	−0.415364	−0.207682	+	0.978196i	$0.566592\pi$
$32$	−1.00000	−0.176777
$33$	1.61213	0.280635
$34$	6.96239	1.19404
$35$	0	0
$36$	1.00000	0.166667
$37$	−11.2750	−1.85360	−0.926802	−	0.375549i	$-0.877454\pi$
$37$	−11.2750	−1.85360	−0.926802	+	0.375549i	$0.877454\pi$
$38$	1.00000	0.162221
$39$	−1.35026	−0.216215
$40$	0	0
$41$	−3.35026	−0.523223	−0.261611	−	0.965173i	$-0.584254\pi$
$41$	−3.35026	−0.523223	−0.261611	+	0.965173i	$0.584254\pi$
$42$	−3.35026	−0.516957
$43$	10.3127	1.57266	0.786332	−	0.617804i	$-0.211977\pi$
$43$	10.3127	1.57266	0.786332	+	0.617804i	$0.211977\pi$
$44$	−1.61213	−0.243037
$45$	0	0
$46$	−1.35026	−0.199085
$47$	4.57452	0.667262	0.333631	−	0.942704i	$-0.391726\pi$
$47$	4.57452	0.667262	0.333631	+	0.942704i	$0.391726\pi$
$48$	−1.00000	−0.144338
$49$	4.22425	0.603465
$50$	0	0
$51$	6.96239	0.974929
$52$	1.35026	0.187248
$53$	−11.9248	−1.63799	−0.818997	−	0.573798i	$-0.805470\pi$
$53$	−11.9248	−1.63799	−0.818997	+	0.573798i	$0.805470\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.35026	0.447698
$57$	1.00000	0.132453
$58$	−3.61213	−0.474295
$59$	1.03761	0.135085	0.0675427	−	0.997716i	$-0.478484\pi$
$59$	1.03761	0.135085	0.0675427	+	0.997716i	$0.478484\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.31265	0.293707
$63$	−3.35026	−0.422093
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.61213	−0.198439
$67$	9.92478	1.21250	0.606252	−	0.795272i	$-0.292672\pi$
$67$	9.92478	1.21250	0.606252	+	0.795272i	$0.292672\pi$
$68$	−6.96239	−0.844314
$69$	−1.35026	−0.162552
$70$	0	0
$71$	−0.775746	−0.0920641	−0.0460321	−	0.998940i	$-0.514658\pi$
$71$	−0.775746	−0.0920641	−0.0460321	+	0.998940i	$0.514658\pi$
$72$	−1.00000	−0.117851
$73$	3.22425	0.377370	0.188685	−	0.982038i	$-0.439577\pi$
$73$	3.22425	0.377370	0.188685	+	0.982038i	$0.439577\pi$
$74$	11.2750	1.31070
$75$	0	0
$76$	−1.00000	−0.114708
$77$	5.40105	0.615506
$78$	1.35026	0.152887
$79$	14.3127	1.61030	0.805149	−	0.593072i	$-0.202085\pi$
$79$	14.3127	1.61030	0.805149	+	0.593072i	$0.202085\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.35026	0.369975
$83$	−10.8872	−1.19502	−0.597511	−	0.801861i	$-0.703844\pi$
$83$	−10.8872	−1.19502	−0.597511	+	0.801861i	$0.703844\pi$
$84$	3.35026	0.365544
$85$	0	0
$86$	−10.3127	−1.11204
$87$	−3.61213	−0.387261
$88$	1.61213	0.171853
$89$	−2.57452	−0.272898	−0.136449	−	0.990647i	$-0.543569\pi$
$89$	−2.57452	−0.272898	−0.136449	+	0.990647i	$0.543569\pi$
$90$	0	0
$91$	−4.52373	−0.474216
$92$	1.35026	0.140775
$93$	2.31265	0.239811
$94$	−4.57452	−0.471825
$95$	0	0
$96$	1.00000	0.102062
$97$	1.16362	0.118148	0.0590738	−	0.998254i	$-0.481185\pi$
$97$	1.16362	0.118148	0.0590738	+	0.998254i	$0.481185\pi$
$98$	−4.22425	−0.426714
$99$	−1.61213	−0.162025
$100$	0	0
$101$	8.88717	0.884306	0.442153	−	0.896940i	$-0.354215\pi$
$101$	8.88717	0.884306	0.442153	+	0.896940i	$0.354215\pi$
$102$	−6.96239	−0.689379
$103$	7.03761	0.693436	0.346718	−	0.937969i	$-0.387296\pi$
$103$	7.03761	0.693436	0.346718	+	0.937969i	$0.387296\pi$
$104$	−1.35026	−0.132404
$105$	0	0
$106$	11.9248	1.15824
$107$	−0.775746	−0.0749942	−0.0374971	−	0.999297i	$-0.511938\pi$
$107$	−0.775746	−0.0749942	−0.0374971	+	0.999297i	$0.511938\pi$
$108$	−1.00000	−0.0962250
$109$	−20.1622	−1.93119	−0.965594	−	0.260052i	$-0.916260\pi$
$109$	−20.1622	−1.93119	−0.965594	+	0.260052i	$0.916260\pi$
$110$	0	0
$111$	11.2750	1.07018
$112$	−3.35026	−0.316570
$113$	11.1490	1.04881	0.524406	−	0.851468i	$-0.324287\pi$
$113$	11.1490	1.04881	0.524406	+	0.851468i	$0.324287\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	3.61213	0.335378
$117$	1.35026	0.124832
$118$	−1.03761	−0.0955199
$119$	23.3258	2.13827
$120$	0	0
$121$	−8.40105	−0.763732
$122$	−2.00000	−0.181071
$123$	3.35026	0.302083
$124$	−2.31265	−0.207682
$125$	0	0
$126$	3.35026	0.298465
$127$	13.7381	1.21906	0.609531	−	0.792762i	$-0.291358\pi$
$127$	13.7381	1.21906	0.609531	+	0.792762i	$0.291358\pi$
$128$	−1.00000	−0.0883883
$129$	−10.3127	−0.907978
$130$	0	0
$131$	5.61213	0.490334	0.245167	−	0.969481i	$-0.421157\pi$
$131$	5.61213	0.490334	0.245167	+	0.969481i	$0.421157\pi$
$132$	1.61213	0.140318
$133$	3.35026	0.290505
$134$	−9.92478	−0.857370
$135$	0	0
$136$	6.96239	0.597020
$137$	−17.6629	−1.50904	−0.754522	−	0.656275i	$-0.772131\pi$
$137$	−17.6629	−1.50904	−0.754522	+	0.656275i	$0.772131\pi$
$138$	1.35026	0.114942
$139$	10.7005	0.907607	0.453803	−	0.891102i	$-0.350067\pi$
$139$	10.7005	0.907607	0.453803	+	0.891102i	$0.350067\pi$
$140$	0	0
$141$	−4.57452	−0.385244
$142$	0.775746	0.0650992
$143$	−2.17679	−0.182033
$144$	1.00000	0.0833333
$145$	0	0
$146$	−3.22425	−0.266841
$147$	−4.22425	−0.348411
$148$	−11.2750	−0.926802
$149$	1.03761	0.0850044	0.0425022	−	0.999096i	$-0.486467\pi$
$149$	1.03761	0.0850044	0.0425022	+	0.999096i	$0.486467\pi$
$150$	0	0
$151$	1.16362	0.0946940	0.0473470	−	0.998879i	$-0.484923\pi$
$151$	1.16362	0.0946940	0.0473470	+	0.998879i	$0.484923\pi$
$152$	1.00000	0.0811107
$153$	−6.96239	−0.562876
$154$	−5.40105	−0.435229
$155$	0	0
$156$	−1.35026	−0.108107
$157$	10.9624	0.874894	0.437447	−	0.899244i	$-0.355883\pi$
$157$	10.9624	0.874894	0.437447	+	0.899244i	$0.355883\pi$
$158$	−14.3127	−1.13865
$159$	11.9248	0.945696
$160$	0	0
$161$	−4.52373	−0.356520
$162$	−1.00000	−0.0785674
$163$	21.0132	1.64588	0.822939	−	0.568129i	$-0.192333\pi$
$163$	21.0132	1.64588	0.822939	+	0.568129i	$0.192333\pi$
$164$	−3.35026	−0.261611
$165$	0	0
$166$	10.8872	0.845008
$167$	9.92478	0.768002	0.384001	−	0.923333i	$-0.374546\pi$
$167$	9.92478	0.768002	0.384001	+	0.923333i	$0.374546\pi$
$168$	−3.35026	−0.258478
$169$	−11.1768	−0.859753
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	10.3127	0.786332
$173$	14.6253	1.11194	0.555971	−	0.831202i	$-0.312347\pi$
$173$	14.6253	1.11194	0.555971	+	0.831202i	$0.312347\pi$
$174$	3.61213	0.273835
$175$	0	0
$176$	−1.61213	−0.121519
$177$	−1.03761	−0.0779916
$178$	2.57452	0.192968
$179$	11.7381	0.877349	0.438675	−	0.898646i	$-0.355448\pi$
$179$	11.7381	0.877349	0.438675	+	0.898646i	$0.355448\pi$
$180$	0	0
$181$	21.4617	1.59523	0.797617	−	0.603164i	$-0.206094\pi$
$181$	21.4617	1.59523	0.797617	+	0.603164i	$0.206094\pi$
$182$	4.52373	0.335321
$183$	−2.00000	−0.147844
$184$	−1.35026	−0.0995426
$185$	0	0
$186$	−2.31265	−0.169572
$187$	11.2243	0.820799
$188$	4.57452	0.333631
$189$	3.35026	0.243696
$190$	0	0
$191$	−21.2750	−1.53941	−0.769704	−	0.638401i	$-0.779596\pi$
$191$	−21.2750	−1.53941	−0.769704	+	0.638401i	$0.779596\pi$
$192$	−1.00000	−0.0721688
$193$	−14.1622	−1.01942	−0.509709	−	0.860347i	$-0.670247\pi$
$193$	−14.1622	−1.01942	−0.509709	+	0.860347i	$0.670247\pi$
$194$	−1.16362	−0.0835430
$195$	0	0
$196$	4.22425	0.301732
$197$	3.87399	0.276011	0.138005	−	0.990431i	$-0.455931\pi$
$197$	3.87399	0.276011	0.138005	+	0.990431i	$0.455931\pi$
$198$	1.61213	0.114569
$199$	3.47627	0.246426	0.123213	−	0.992380i	$-0.460680\pi$
$199$	3.47627	0.246426	0.123213	+	0.992380i	$0.460680\pi$
$200$	0	0
$201$	−9.92478	−0.700040
$202$	−8.88717	−0.625299
$203$	−12.1016	−0.849364
$204$	6.96239	0.487465
$205$	0	0
$206$	−7.03761	−0.490334
$207$	1.35026	0.0938497
$208$	1.35026	0.0936238
$209$	1.61213	0.111513
$210$	0	0
$211$	9.92478	0.683250	0.341625	−	0.939836i	$-0.389023\pi$
$211$	9.92478	0.683250	0.341625	+	0.939836i	$0.389023\pi$
$212$	−11.9248	−0.818997
$213$	0.775746	0.0531533
$214$	0.775746	0.0530289
$215$	0	0
$216$	1.00000	0.0680414
$217$	7.74798	0.525967
$218$	20.1622	1.36556
$219$	−3.22425	−0.217875
$220$	0	0
$221$	−9.40105	−0.632383
$222$	−11.2750	−0.756731
$223$	7.03761	0.471273	0.235637	−	0.971841i	$-0.424282\pi$
$223$	7.03761	0.471273	0.235637	+	0.971841i	$0.424282\pi$
$224$	3.35026	0.223849
$225$	0	0
$226$	−11.1490	−0.741623
$227$	−14.5501	−0.965723	−0.482861	−	0.875697i	$-0.660402\pi$
$227$	−14.5501	−0.965723	−0.482861	+	0.875697i	$0.660402\pi$
$228$	1.00000	0.0662266
$229$	11.4010	0.753402	0.376701	−	0.926335i	$-0.377058\pi$
$229$	11.4010	0.753402	0.376701	+	0.926335i	$0.377058\pi$
$230$	0	0
$231$	−5.40105	−0.355363
$232$	−3.61213	−0.237148
$233$	21.9149	1.43569	0.717847	−	0.696201i	$-0.245128\pi$
$233$	21.9149	1.43569	0.717847	+	0.696201i	$0.245128\pi$
$234$	−1.35026	−0.0882694
$235$	0	0
$236$	1.03761	0.0675427
$237$	−14.3127	−0.929707
$238$	−23.3258	−1.51199
$239$	−13.2750	−0.858691	−0.429345	−	0.903140i	$-0.641256\pi$
$239$	−13.2750	−0.858691	−0.429345	+	0.903140i	$0.641256\pi$
$240$	0	0
$241$	21.3258	1.37372	0.686859	−	0.726791i	$-0.258989\pi$
$241$	21.3258	1.37372	0.686859	+	0.726791i	$0.258989\pi$
$242$	8.40105	0.540040
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−3.35026	−0.213605
$247$	−1.35026	−0.0859151
$248$	2.31265	0.146853
$249$	10.8872	0.689946
$250$	0	0
$251$	16.3127	1.02965	0.514823	−	0.857297i	$-0.327858\pi$
$251$	16.3127	1.02965	0.514823	+	0.857297i	$0.327858\pi$
$252$	−3.35026	−0.211047
$253$	−2.17679	−0.136854
$254$	−13.7381	−0.862007
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.5501	−1.53139	−0.765696	−	0.643203i	$-0.777605\pi$
$257$	−24.5501	−1.53139	−0.765696	+	0.643203i	$0.777605\pi$
$258$	10.3127	0.642038
$259$	37.7743	2.34718
$260$	0	0
$261$	3.61213	0.223585
$262$	−5.61213	−0.346718
$263$	−30.3488	−1.87139	−0.935695	−	0.352810i	$-0.885226\pi$
$263$	−30.3488	−1.87139	−0.935695	+	0.352810i	$0.885226\pi$
$264$	−1.61213	−0.0992195
$265$	0	0
$266$	−3.35026	−0.205418
$267$	2.57452	0.157558
$268$	9.92478	0.606252
$269$	14.3127	0.872658	0.436329	−	0.899787i	$-0.356278\pi$
$269$	14.3127	0.872658	0.436329	+	0.899787i	$0.356278\pi$
$270$	0	0
$271$	−7.32582	−0.445012	−0.222506	−	0.974931i	$-0.571424\pi$
$271$	−7.32582	−0.445012	−0.222506	+	0.974931i	$0.571424\pi$
$272$	−6.96239	−0.422157
$273$	4.52373	0.273789
$274$	17.6629	1.06706
$275$	0	0
$276$	−1.35026	−0.0812762
$277$	14.4387	0.867535	0.433767	−	0.901025i	$-0.357184\pi$
$277$	14.4387	0.867535	0.433767	+	0.901025i	$0.357184\pi$
$278$	−10.7005	−0.641775
$279$	−2.31265	−0.138455
$280$	0	0
$281$	−11.9756	−0.714402	−0.357201	−	0.934028i	$-0.616269\pi$
$281$	−11.9756	−0.714402	−0.357201	+	0.934028i	$0.616269\pi$
$282$	4.57452	0.272408
$283$	24.4894	1.45575	0.727873	−	0.685712i	$-0.240509\pi$
$283$	24.4894	1.45575	0.727873	+	0.685712i	$0.240509\pi$
$284$	−0.775746	−0.0460321
$285$	0	0
$286$	2.17679	0.128716
$287$	11.2243	0.662547
$288$	−1.00000	−0.0589256
$289$	31.4749	1.85146
$290$	0	0
$291$	−1.16362	−0.0682126
$292$	3.22425	0.188685
$293$	18.1016	1.05751	0.528753	−	0.848776i	$-0.322660\pi$
$293$	18.1016	1.05751	0.528753	+	0.848776i	$0.322660\pi$
$294$	4.22425	0.246363
$295$	0	0
$296$	11.2750	0.655348
$297$	1.61213	0.0935451
$298$	−1.03761	−0.0601072
$299$	1.82321	0.105439
$300$	0	0
$301$	−34.5501	−1.99143
$302$	−1.16362	−0.0669588
$303$	−8.88717	−0.510554
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	6.96239	0.398013
$307$	29.9248	1.70790	0.853949	−	0.520357i	$-0.174201\pi$
$307$	29.9248	1.70790	0.853949	+	0.520357i	$0.174201\pi$
$308$	5.40105	0.307753
$309$	−7.03761	−0.400356
$310$	0	0
$311$	−21.2750	−1.20640	−0.603198	−	0.797591i	$-0.706107\pi$
$311$	−21.2750	−1.20640	−0.603198	+	0.797591i	$0.706107\pi$
$312$	1.35026	0.0764435
$313$	17.4010	0.983565	0.491783	−	0.870718i	$-0.336345\pi$
$313$	17.4010	0.983565	0.491783	+	0.870718i	$0.336345\pi$
$314$	−10.9624	−0.618643
$315$	0	0
$316$	14.3127	0.805149
$317$	14.1016	0.792023	0.396012	−	0.918246i	$-0.370394\pi$
$317$	14.1016	0.792023	0.396012	+	0.918246i	$0.370394\pi$
$318$	−11.9248	−0.668708
$319$	−5.82321	−0.326037
$320$	0	0
$321$	0.775746	0.0432979
$322$	4.52373	0.252098
$323$	6.96239	0.387398
$324$	1.00000	0.0555556
$325$	0	0
$326$	−21.0132	−1.16381
$327$	20.1622	1.11497
$328$	3.35026	0.184987
$329$	−15.3258	−0.844940
$330$	0	0
$331$	6.85097	0.376563	0.188282	−	0.982115i	$-0.439708\pi$
$331$	6.85097	0.376563	0.188282	+	0.982115i	$0.439708\pi$
$332$	−10.8872	−0.597511
$333$	−11.2750	−0.617868
$334$	−9.92478	−0.543060
$335$	0	0
$336$	3.35026	0.182772
$337$	24.2374	1.32030	0.660148	−	0.751135i	$-0.270493\pi$
$337$	24.2374	1.32030	0.660148	+	0.751135i	$0.270493\pi$
$338$	11.1768	0.607937
$339$	−11.1490	−0.605532
$340$	0	0
$341$	3.72829	0.201898
$342$	1.00000	0.0540738
$343$	9.29948	0.502125
$344$	−10.3127	−0.556021
$345$	0	0
$346$	−14.6253	−0.786261
$347$	0.962389	0.0516637	0.0258319	−	0.999666i	$-0.491777\pi$
$347$	0.962389	0.0516637	0.0258319	+	0.999666i	$0.491777\pi$
$348$	−3.61213	−0.193630
$349$	−1.37470	−0.0735860	−0.0367930	−	0.999323i	$-0.511714\pi$
$349$	−1.37470	−0.0735860	−0.0367930	+	0.999323i	$0.511714\pi$
$350$	0	0
$351$	−1.35026	−0.0720716
$352$	1.61213	0.0859267
$353$	28.7367	1.52950	0.764751	−	0.644326i	$-0.222862\pi$
$353$	28.7367	1.52950	0.764751	+	0.644326i	$0.222862\pi$
$354$	1.03761	0.0551484
$355$	0	0
$356$	−2.57452	−0.136449
$357$	−23.3258	−1.23453
$358$	−11.7381	−0.620380
$359$	31.1998	1.64666	0.823332	−	0.567561i	$-0.192113\pi$
$359$	31.1998	1.64666	0.823332	+	0.567561i	$0.192113\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−21.4617	−1.12800
$363$	8.40105	0.440941
$364$	−4.52373	−0.237108
$365$	0	0
$366$	2.00000	0.104542
$367$	20.1260	1.05057	0.525285	−	0.850927i	$-0.323959\pi$
$367$	20.1260	1.05057	0.525285	+	0.850927i	$0.323959\pi$
$368$	1.35026	0.0703873
$369$	−3.35026	−0.174408
$370$	0	0
$371$	39.9511	2.07416
$372$	2.31265	0.119905
$373$	−17.9756	−0.930739	−0.465370	−	0.885116i	$-0.654079\pi$
$373$	−17.9756	−0.930739	−0.465370	+	0.885116i	$0.654079\pi$
$374$	−11.2243	−0.580392
$375$	0	0
$376$	−4.57452	−0.235913
$377$	4.87732	0.251195
$378$	−3.35026	−0.172319
$379$	1.67276	0.0859240	0.0429620	−	0.999077i	$-0.486321\pi$
$379$	1.67276	0.0859240	0.0429620	+	0.999077i	$0.486321\pi$
$380$	0	0
$381$	−13.7381	−0.703826
$382$	21.2750	1.08853
$383$	31.8496	1.62744	0.813718	−	0.581260i	$-0.197440\pi$
$383$	31.8496	1.62744	0.813718	+	0.581260i	$0.197440\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.1622	0.720837
$387$	10.3127	0.524221
$388$	1.16362	0.0590738
$389$	−10.3371	−0.524111	−0.262056	−	0.965053i	$-0.584400\pi$
$389$	−10.3371	−0.524111	−0.262056	+	0.965053i	$0.584400\pi$
$390$	0	0
$391$	−9.40105	−0.475431
$392$	−4.22425	−0.213357
$393$	−5.61213	−0.283094
$394$	−3.87399	−0.195169
$395$	0	0
$396$	−1.61213	−0.0810124
$397$	−31.4372	−1.57779	−0.788895	−	0.614528i	$-0.789346\pi$
$397$	−31.4372	−1.57779	−0.788895	+	0.614528i	$0.789346\pi$
$398$	−3.47627	−0.174250
$399$	−3.35026	−0.167723
$400$	0	0
$401$	5.94921	0.297090	0.148545	−	0.988906i	$-0.452541\pi$
$401$	5.94921	0.297090	0.148545	+	0.988906i	$0.452541\pi$
$402$	9.92478	0.495003
$403$	−3.12268	−0.155552
$404$	8.88717	0.442153
$405$	0	0
$406$	12.1016	0.600591
$407$	18.1768	0.900990
$408$	−6.96239	−0.344690
$409$	−30.9986	−1.53278	−0.766391	−	0.642375i	$-0.777949\pi$
$409$	−30.9986	−1.53278	−0.766391	+	0.642375i	$0.777949\pi$
$410$	0	0
$411$	17.6629	0.871247
$412$	7.03761	0.346718
$413$	−3.47627	−0.171056
$414$	−1.35026	−0.0663617
$415$	0	0
$416$	−1.35026	−0.0662020
$417$	−10.7005	−0.524007
$418$	−1.61213	−0.0788517
$419$	−34.3390	−1.67757	−0.838785	−	0.544463i	$-0.816734\pi$
$419$	−34.3390	−1.67757	−0.838785	+	0.544463i	$0.816734\pi$
$420$	0	0
$421$	22.8627	1.11426	0.557131	−	0.830425i	$-0.311902\pi$
$421$	22.8627	1.11426	0.557131	+	0.830425i	$0.311902\pi$
$422$	−9.92478	−0.483131
$423$	4.57452	0.222421
$424$	11.9248	0.579118
$425$	0	0
$426$	−0.775746	−0.0375850
$427$	−6.70052	−0.324261
$428$	−0.775746	−0.0374971
$429$	2.17679	0.105097
$430$	0	0
$431$	11.5975	0.558634	0.279317	−	0.960199i	$-0.409892\pi$
$431$	11.5975	0.558634	0.279317	+	0.960199i	$0.409892\pi$
$432$	−1.00000	−0.0481125
$433$	27.8350	1.33766	0.668832	−	0.743414i	$-0.266795\pi$
$433$	27.8350	1.33766	0.668832	+	0.743414i	$0.266795\pi$
$434$	−7.74798	−0.371915
$435$	0	0
$436$	−20.1622	−0.965594
$437$	−1.35026	−0.0645918
$438$	3.22425	0.154061
$439$	−38.7875	−1.85123	−0.925613	−	0.378471i	$-0.876450\pi$
$439$	−38.7875	−1.85123	−0.925613	+	0.378471i	$0.876450\pi$
$440$	0	0
$441$	4.22425	0.201155
$442$	9.40105	0.447162
$443$	−29.2144	−1.38802	−0.694009	−	0.719966i	$-0.744157\pi$
$443$	−29.2144	−1.38802	−0.694009	+	0.719966i	$0.744157\pi$
$444$	11.2750	0.535090
$445$	0	0
$446$	−7.03761	−0.333241
$447$	−1.03761	−0.0490773
$448$	−3.35026	−0.158285
$449$	−20.4993	−0.967421	−0.483711	−	0.875228i	$-0.660711\pi$
$449$	−20.4993	−0.967421	−0.483711	+	0.875228i	$0.660711\pi$
$450$	0	0
$451$	5.40105	0.254325
$452$	11.1490	0.524406
$453$	−1.16362	−0.0546716
$454$	14.5501	0.682869
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	6.44851	0.301648	0.150824	−	0.988561i	$-0.451807\pi$
$457$	6.44851	0.301648	0.150824	+	0.988561i	$0.451807\pi$
$458$	−11.4010	−0.532736
$459$	6.96239	0.324976
$460$	0	0
$461$	−13.5125	−0.629338	−0.314669	−	0.949201i	$-0.601894\pi$
$461$	−13.5125	−0.629338	−0.314669	+	0.949201i	$0.601894\pi$
$462$	5.40105	0.251279
$463$	6.20123	0.288196	0.144098	−	0.989563i	$-0.453972\pi$
$463$	6.20123	0.288196	0.144098	+	0.989563i	$0.453972\pi$
$464$	3.61213	0.167689
$465$	0	0
$466$	−21.9149	−1.01519
$467$	−16.5599	−0.766302	−0.383151	−	0.923686i	$-0.625161\pi$
$467$	−16.5599	−0.766302	−0.383151	+	0.923686i	$0.625161\pi$
$468$	1.35026	0.0624159
$469$	−33.2506	−1.53537
$470$	0	0
$471$	−10.9624	−0.505120
$472$	−1.03761	−0.0477599
$473$	−16.6253	−0.764432
$474$	14.3127	0.657402
$475$	0	0
$476$	23.3258	1.06914
$477$	−11.9248	−0.545998
$478$	13.2750	0.607186
$479$	−30.5256	−1.39475	−0.697376	−	0.716705i	$-0.745649\pi$
$479$	−30.5256	−1.39475	−0.697376	+	0.716705i	$0.745649\pi$
$480$	0	0
$481$	−15.2243	−0.694166
$482$	−21.3258	−0.971365
$483$	4.52373	0.205837
$484$	−8.40105	−0.381866
$485$	0	0
$486$	1.00000	0.0453609
$487$	−2.51388	−0.113915	−0.0569574	−	0.998377i	$-0.518140\pi$
$487$	−2.51388	−0.113915	−0.0569574	+	0.998377i	$0.518140\pi$
$488$	−2.00000	−0.0905357
$489$	−21.0132	−0.950249
$490$	0	0
$491$	1.46168	0.0659648	0.0329824	−	0.999456i	$-0.489499\pi$
$491$	1.46168	0.0659648	0.0329824	+	0.999456i	$0.489499\pi$
$492$	3.35026	0.151041
$493$	−25.1490	−1.13266
$494$	1.35026	0.0607511
$495$	0	0
$496$	−2.31265	−0.103841
$497$	2.59895	0.116579
$498$	−10.8872	−0.487866
$499$	−5.55149	−0.248519	−0.124259	−	0.992250i	$-0.539656\pi$
$499$	−5.55149	−0.248519	−0.124259	+	0.992250i	$0.539656\pi$
$500$	0	0
$501$	−9.92478	−0.443406
$502$	−16.3127	−0.728069
$503$	6.90175	0.307734	0.153867	−	0.988092i	$-0.450827\pi$
$503$	6.90175	0.307734	0.153867	+	0.988092i	$0.450827\pi$
$504$	3.35026	0.149233
$505$	0	0
$506$	2.17679	0.0967703
$507$	11.1768	0.496379
$508$	13.7381	0.609531
$509$	8.76116	0.388331	0.194166	−	0.980969i	$-0.437800\pi$
$509$	8.76116	0.388331	0.194166	+	0.980969i	$0.437800\pi$
$510$	0	0
$511$	−10.8021	−0.477857
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	24.5501	1.08286
$515$	0	0
$516$	−10.3127	−0.453989
$517$	−7.37470	−0.324339
$518$	−37.7743	−1.65971
$519$	−14.6253	−0.641979
$520$	0	0
$521$	39.4518	1.72842	0.864208	−	0.503135i	$-0.167820\pi$
$521$	39.4518	1.72842	0.864208	+	0.503135i	$0.167820\pi$
$522$	−3.61213	−0.158098
$523$	−37.5026	−1.63987	−0.819937	−	0.572453i	$-0.805992\pi$
$523$	−37.5026	−1.63987	−0.819937	+	0.572453i	$0.805992\pi$
$524$	5.61213	0.245167
$525$	0	0
$526$	30.3488	1.32327
$527$	16.1016	0.701395
$528$	1.61213	0.0701588
$529$	−21.1768	−0.920730
$530$	0	0
$531$	1.03761	0.0450285
$532$	3.35026	0.145252
$533$	−4.52373	−0.195945
$534$	−2.57452	−0.111410
$535$	0	0
$536$	−9.92478	−0.428685
$537$	−11.7381	−0.506538
$538$	−14.3127	−0.617062
$539$	−6.81003	−0.293329
$540$	0	0
$541$	−30.7757	−1.32315	−0.661576	−	0.749878i	$-0.730112\pi$
$541$	−30.7757	−1.32315	−0.661576	+	0.749878i	$0.730112\pi$
$542$	7.32582	0.314671
$543$	−21.4617	−0.921009
$544$	6.96239	0.298510
$545$	0	0
$546$	−4.52373	−0.193598
$547$	−1.77433	−0.0758649	−0.0379325	−	0.999280i	$-0.512077\pi$
$547$	−1.77433	−0.0758649	−0.0379325	+	0.999280i	$0.512077\pi$
$548$	−17.6629	−0.754522
$549$	2.00000	0.0853579
$550$	0	0
$551$	−3.61213	−0.153882
$552$	1.35026	0.0574710
$553$	−47.9511	−2.03909
$554$	−14.4387	−0.613440
$555$	0	0
$556$	10.7005	0.453803
$557$	−8.90175	−0.377179	−0.188590	−	0.982056i	$-0.560392\pi$
$557$	−8.90175	−0.377179	−0.188590	+	0.982056i	$0.560392\pi$
$558$	2.31265	0.0979023
$559$	13.9248	0.588955
$560$	0	0
$561$	−11.2243	−0.473888
$562$	11.9756	0.505159
$563$	−10.9525	−0.461595	−0.230797	−	0.973002i	$-0.574133\pi$
$563$	−10.9525	−0.461595	−0.230797	+	0.973002i	$0.574133\pi$
$564$	−4.57452	−0.192622
$565$	0	0
$566$	−24.4894	−1.02937
$567$	−3.35026	−0.140698
$568$	0.775746	0.0325496
$569$	−10.2012	−0.427658	−0.213829	−	0.976871i	$-0.568594\pi$
$569$	−10.2012	−0.427658	−0.213829	+	0.976871i	$0.568594\pi$
$570$	0	0
$571$	−25.6531	−1.07355	−0.536774	−	0.843726i	$-0.680357\pi$
$571$	−25.6531	−1.07355	−0.536774	+	0.843726i	$0.680357\pi$
$572$	−2.17679	−0.0910163
$573$	21.2750	0.888778
$574$	−11.2243	−0.468491
$575$	0	0
$576$	1.00000	0.0416667
$577$	6.44851	0.268455	0.134227	−	0.990951i	$-0.457145\pi$
$577$	6.44851	0.268455	0.134227	+	0.990951i	$0.457145\pi$
$578$	−31.4749	−1.30918
$579$	14.1622	0.588561
$580$	0	0
$581$	36.4749	1.51323
$582$	1.16362	0.0482336
$583$	19.2243	0.796187
$584$	−3.22425	−0.133421
$585$	0	0
$586$	−18.1016	−0.747769
$587$	−41.8397	−1.72691	−0.863455	−	0.504426i	$-0.831704\pi$
$587$	−41.8397	−1.72691	−0.863455	+	0.504426i	$0.831704\pi$
$588$	−4.22425	−0.174205
$589$	2.31265	0.0952911
$590$	0	0
$591$	−3.87399	−0.159355
$592$	−11.2750	−0.463401
$593$	8.73672	0.358774	0.179387	−	0.983779i	$-0.442589\pi$
$593$	8.73672	0.358774	0.179387	+	0.983779i	$0.442589\pi$
$594$	−1.61213	−0.0661464
$595$	0	0
$596$	1.03761	0.0425022
$597$	−3.47627	−0.142274
$598$	−1.82321	−0.0745565
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	30.6253	1.24923	0.624616	−	0.780932i	$-0.285255\pi$
$601$	30.6253	1.24923	0.624616	+	0.780932i	$0.285255\pi$
$602$	34.5501	1.40816
$603$	9.92478	0.404168
$604$	1.16362	0.0473470
$605$	0	0
$606$	8.88717	0.361016
$607$	−2.51388	−0.102035	−0.0510176	−	0.998698i	$-0.516246\pi$
$607$	−2.51388	−0.102035	−0.0510176	+	0.998698i	$0.516246\pi$
$608$	1.00000	0.0405554
$609$	12.1016	0.490380
$610$	0	0
$611$	6.17679	0.249886
$612$	−6.96239	−0.281438
$613$	−2.96239	−0.119650	−0.0598249	−	0.998209i	$-0.519054\pi$
$613$	−2.96239	−0.119650	−0.0598249	+	0.998209i	$0.519054\pi$
$614$	−29.9248	−1.20767
$615$	0	0
$616$	−5.40105	−0.217614
$617$	−18.3371	−0.738223	−0.369112	−	0.929385i	$-0.620338\pi$
$617$	−18.3371	−0.738223	−0.369112	+	0.929385i	$0.620338\pi$
$618$	7.03761	0.283094
$619$	−40.7269	−1.63695	−0.818476	−	0.574541i	$-0.805180\pi$
$619$	−40.7269	−1.63695	−0.818476	+	0.574541i	$0.805180\pi$
$620$	0	0
$621$	−1.35026	−0.0541841
$622$	21.2750	0.853051
$623$	8.62530	0.345565
$624$	−1.35026	−0.0540537
$625$	0	0
$626$	−17.4010	−0.695486
$627$	−1.61213	−0.0643821
$628$	10.9624	0.437447
$629$	78.5012	3.13005
$630$	0	0
$631$	−5.92478	−0.235862	−0.117931	−	0.993022i	$-0.537626\pi$
$631$	−5.92478	−0.235862	−0.117931	+	0.993022i	$0.537626\pi$
$632$	−14.3127	−0.569327
$633$	−9.92478	−0.394474
$634$	−14.1016	−0.560045
$635$	0	0
$636$	11.9248	0.472848
$637$	5.70385	0.225995
$638$	5.82321	0.230543
$639$	−0.775746	−0.0306880
$640$	0	0
$641$	19.0494	0.752405	0.376202	−	0.926537i	$-0.377230\pi$
$641$	19.0494	0.752405	0.376202	+	0.926537i	$0.377230\pi$
$642$	−0.775746	−0.0306163
$643$	−1.06205	−0.0418831	−0.0209416	−	0.999781i	$-0.506666\pi$
$643$	−1.06205	−0.0418831	−0.0209416	+	0.999781i	$0.506666\pi$
$644$	−4.52373	−0.178260
$645$	0	0
$646$	−6.96239	−0.273932
$647$	26.6497	1.04771	0.523855	−	0.851808i	$-0.324493\pi$
$647$	26.6497	1.04771	0.523855	+	0.851808i	$0.324493\pi$
$648$	−1.00000	−0.0392837
$649$	−1.67276	−0.0656616
$650$	0	0
$651$	−7.74798	−0.303667
$652$	21.0132	0.822939
$653$	−9.64832	−0.377568	−0.188784	−	0.982019i	$-0.560455\pi$
$653$	−9.64832	−0.377568	−0.188784	+	0.982019i	$0.560455\pi$
$654$	−20.1622	−0.788405
$655$	0	0
$656$	−3.35026	−0.130806
$657$	3.22425	0.125790
$658$	15.3258	0.597463
$659$	10.0654	0.392091	0.196046	−	0.980595i	$-0.437190\pi$
$659$	10.0654	0.392091	0.196046	+	0.980595i	$0.437190\pi$
$660$	0	0
$661$	13.5633	0.527549	0.263775	−	0.964584i	$-0.415032\pi$
$661$	13.5633	0.527549	0.263775	+	0.964584i	$0.415032\pi$
$662$	−6.85097	−0.266270
$663$	9.40105	0.365106
$664$	10.8872	0.422504
$665$	0	0
$666$	11.2750	0.436899
$667$	4.87732	0.188850
$668$	9.92478	0.384001
$669$	−7.03761	−0.272090
$670$	0	0
$671$	−3.22425	−0.124471
$672$	−3.35026	−0.129239
$673$	−5.93937	−0.228946	−0.114473	−	0.993426i	$-0.536518\pi$
$673$	−5.93937	−0.228946	−0.114473	+	0.993426i	$0.536518\pi$
$674$	−24.2374	−0.933591
$675$	0	0
$676$	−11.1768	−0.429877
$677$	−24.9525	−0.959004	−0.479502	−	0.877541i	$-0.659183\pi$
$677$	−24.9525	−0.959004	−0.479502	+	0.877541i	$0.659183\pi$
$678$	11.1490	0.428176
$679$	−3.89843	−0.149608
$680$	0	0
$681$	14.5501	0.557560
$682$	−3.72829	−0.142763
$683$	45.6239	1.74575	0.872875	−	0.487944i	$-0.162253\pi$
$683$	45.6239	1.74575	0.872875	+	0.487944i	$0.162253\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−9.29948	−0.355056
$687$	−11.4010	−0.434977
$688$	10.3127	0.393166
$689$	−16.1016	−0.613421
$690$	0	0
$691$	−11.7480	−0.446914	−0.223457	−	0.974714i	$-0.571734\pi$
$691$	−11.7480	−0.446914	−0.223457	+	0.974714i	$0.571734\pi$
$692$	14.6253	0.555971
$693$	5.40105	0.205169
$694$	−0.962389	−0.0365318
$695$	0	0
$696$	3.61213	0.136917
$697$	23.3258	0.883529
$698$	1.37470	0.0520331
$699$	−21.9149	−0.828899
$700$	0	0
$701$	−43.8105	−1.65470	−0.827350	−	0.561686i	$-0.810153\pi$
$701$	−43.8105	−1.65470	−0.827350	+	0.561686i	$0.810153\pi$
$702$	1.35026	0.0509623
$703$	11.2750	0.425246
$704$	−1.61213	−0.0607593
$705$	0	0
$706$	−28.7367	−1.08152
$707$	−29.7743	−1.11978
$708$	−1.03761	−0.0389958
$709$	−29.5975	−1.11156	−0.555779	−	0.831330i	$-0.687580\pi$
$709$	−29.5975	−1.11156	−0.555779	+	0.831330i	$0.687580\pi$
$710$	0	0
$711$	14.3127	0.536766
$712$	2.57452	0.0964840
$713$	−3.12268	−0.116945
$714$	23.3258	0.872947
$715$	0	0
$716$	11.7381	0.438675
$717$	13.2750	0.495765
$718$	−31.1998	−1.16437
$719$	7.45183	0.277906	0.138953	−	0.990299i	$-0.455626\pi$
$719$	7.45183	0.277906	0.138953	+	0.990299i	$0.455626\pi$
$720$	0	0
$721$	−23.5778	−0.878085
$722$	−1.00000	−0.0372161
$723$	−21.3258	−0.793116
$724$	21.4617	0.797617
$725$	0	0
$726$	−8.40105	−0.311792
$727$	−0.600863	−0.0222848	−0.0111424	−	0.999938i	$-0.503547\pi$
$727$	−0.600863	−0.0222848	−0.0111424	+	0.999938i	$0.503547\pi$
$728$	4.52373	0.167661
$729$	1.00000	0.0370370
$730$	0	0
$731$	−71.8007	−2.65564
$732$	−2.00000	−0.0739221
$733$	−36.0625	−1.33200	−0.666000	−	0.745952i	$-0.731995\pi$
$733$	−36.0625	−1.33200	−0.666000	+	0.745952i	$0.731995\pi$
$734$	−20.1260	−0.742865
$735$	0	0
$736$	−1.35026	−0.0497713
$737$	−16.0000	−0.589368
$738$	3.35026	0.123325
$739$	−14.1768	−0.521502	−0.260751	−	0.965406i	$-0.583970\pi$
$739$	−14.1768	−0.521502	−0.260751	+	0.965406i	$0.583970\pi$
$740$	0	0
$741$	1.35026	0.0496031
$742$	−39.9511	−1.46665
$743$	−24.9986	−0.917109	−0.458555	−	0.888666i	$-0.651633\pi$
$743$	−24.9986	−0.917109	−0.458555	+	0.888666i	$0.651633\pi$
$744$	−2.31265	−0.0847859
$745$	0	0
$746$	17.9756	0.658132
$747$	−10.8872	−0.398341
$748$	11.2243	0.410399
$749$	2.59895	0.0949637
$750$	0	0
$751$	−10.2111	−0.372608	−0.186304	−	0.982492i	$-0.559651\pi$
$751$	−10.2111	−0.372608	−0.186304	+	0.982492i	$0.559651\pi$
$752$	4.57452	0.166815
$753$	−16.3127	−0.594466
$754$	−4.87732	−0.177621
$755$	0	0
$756$	3.35026	0.121848
$757$	−44.4847	−1.61682	−0.808412	−	0.588617i	$-0.799673\pi$
$757$	−44.4847	−1.61682	−0.808412	+	0.588617i	$0.799673\pi$
$758$	−1.67276	−0.0607574
$759$	2.17679	0.0790126
$760$	0	0
$761$	27.1490	0.984152	0.492076	−	0.870552i	$-0.336238\pi$
$761$	27.1490	0.984152	0.492076	+	0.870552i	$0.336238\pi$
$762$	13.7381	0.497680
$763$	67.5487	2.44543
$764$	−21.2750	−0.769704
$765$	0	0
$766$	−31.8496	−1.15077
$767$	1.40105	0.0505889
$768$	−1.00000	−0.0360844
$769$	−31.4010	−1.13235	−0.566175	−	0.824285i	$-0.691578\pi$
$769$	−31.4010	−1.13235	−0.566175	+	0.824285i	$0.691578\pi$
$770$	0	0
$771$	24.5501	0.884149
$772$	−14.1622	−0.509709
$773$	−4.02635	−0.144818	−0.0724088	−	0.997375i	$-0.523069\pi$
$773$	−4.02635	−0.144818	−0.0724088	+	0.997375i	$0.523069\pi$
$774$	−10.3127	−0.370681
$775$	0	0
$776$	−1.16362	−0.0417715
$777$	−37.7743	−1.35515
$778$	10.3371	0.370603
$779$	3.35026	0.120036
$780$	0	0
$781$	1.25060	0.0447500
$782$	9.40105	0.336181
$783$	−3.61213	−0.129087
$784$	4.22425	0.150866
$785$	0	0
$786$	5.61213	0.200178
$787$	−18.2981	−0.652255	−0.326128	−	0.945326i	$-0.605744\pi$
$787$	−18.2981	−0.652255	−0.326128	+	0.945326i	$0.605744\pi$
$788$	3.87399	0.138005
$789$	30.3488	1.08045
$790$	0	0
$791$	−37.3522	−1.32809
$792$	1.61213	0.0572844
$793$	2.70052	0.0958984
$794$	31.4372	1.11567
$795$	0	0
$796$	3.47627	0.123213
$797$	8.55008	0.302859	0.151430	−	0.988468i	$-0.451612\pi$
$797$	8.55008	0.302859	0.151430	+	0.988468i	$0.451612\pi$
$798$	3.35026	0.118598
$799$	−31.8496	−1.12676
$800$	0	0
$801$	−2.57452	−0.0909660
$802$	−5.94921	−0.210074
$803$	−5.19791	−0.183430
$804$	−9.92478	−0.350020
$805$	0	0
$806$	3.12268	0.109992
$807$	−14.3127	−0.503829
$808$	−8.88717	−0.312649
$809$	50.7269	1.78346	0.891731	−	0.452566i	$-0.149491\pi$
$809$	50.7269	1.78346	0.891731	+	0.452566i	$0.149491\pi$
$810$	0	0
$811$	10.3272	0.362638	0.181319	−	0.983424i	$-0.441963\pi$
$811$	10.3272	0.362638	0.181319	+	0.983424i	$0.441963\pi$
$812$	−12.1016	−0.424682
$813$	7.32582	0.256928
$814$	−18.1768	−0.637096
$815$	0	0
$816$	6.96239	0.243732
$817$	−10.3127	−0.360794
$818$	30.9986	1.08384
$819$	−4.52373	−0.158072
$820$	0	0
$821$	36.5139	1.27434	0.637172	−	0.770722i	$-0.280104\pi$
$821$	36.5139	1.27434	0.637172	+	0.770722i	$0.280104\pi$
$822$	−17.6629	−0.616065
$823$	−23.0982	−0.805154	−0.402577	−	0.915386i	$-0.631885\pi$
$823$	−23.0982	−0.805154	−0.402577	+	0.915386i	$0.631885\pi$
$824$	−7.03761	−0.245167
$825$	0	0
$826$	3.47627	0.120955
$827$	3.37470	0.117350	0.0586749	−	0.998277i	$-0.481312\pi$
$827$	3.37470	0.117350	0.0586749	+	0.998277i	$0.481312\pi$
$828$	1.35026	0.0469248
$829$	22.6399	0.786316	0.393158	−	0.919471i	$-0.371383\pi$
$829$	22.6399	0.786316	0.393158	+	0.919471i	$0.371383\pi$
$830$	0	0
$831$	−14.4387	−0.500872
$832$	1.35026	0.0468119
$833$	−29.4109	−1.01903
$834$	10.7005	0.370529
$835$	0	0
$836$	1.61213	0.0557566
$837$	2.31265	0.0799369
$838$	34.3390	1.18622
$839$	−9.02776	−0.311673	−0.155836	−	0.987783i	$-0.549807\pi$
$839$	−9.02776	−0.311673	−0.155836	+	0.987783i	$0.549807\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	−22.8627	−0.787902
$843$	11.9756	0.412460
$844$	9.92478	0.341625
$845$	0	0
$846$	−4.57452	−0.157275
$847$	28.1457	0.967098
$848$	−11.9248	−0.409499
$849$	−24.4894	−0.840476
$850$	0	0
$851$	−15.2243	−0.521881
$852$	0.775746	0.0265766
$853$	43.1852	1.47863	0.739317	−	0.673358i	$-0.235149\pi$
$853$	43.1852	1.47863	0.739317	+	0.673358i	$0.235149\pi$
$854$	6.70052	0.229287
$855$	0	0
$856$	0.775746	0.0265145
$857$	33.0249	1.12811	0.564055	−	0.825737i	$-0.309241\pi$
$857$	33.0249	1.12811	0.564055	+	0.825737i	$0.309241\pi$
$858$	−2.17679	−0.0743145
$859$	15.1754	0.517777	0.258889	−	0.965907i	$-0.416644\pi$
$859$	15.1754	0.517777	0.258889	+	0.965907i	$0.416644\pi$
$860$	0	0
$861$	−11.2243	−0.382522
$862$	−11.5975	−0.395014
$863$	39.4763	1.34379	0.671894	−	0.740647i	$-0.265481\pi$
$863$	39.4763	1.34379	0.671894	+	0.740647i	$0.265481\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−27.8350	−0.945871
$867$	−31.4749	−1.06894
$868$	7.74798	0.262984
$869$	−23.0738	−0.782725
$870$	0	0
$871$	13.4010	0.454077
$872$	20.1622	0.682778
$873$	1.16362	0.0393825
$874$	1.35026	0.0456733
$875$	0	0
$876$	−3.22425	−0.108937
$877$	52.5256	1.77366	0.886832	−	0.462091i	$-0.152901\pi$
$877$	52.5256	1.77366	0.886832	+	0.462091i	$0.152901\pi$
$878$	38.7875	1.30901
$879$	−18.1016	−0.610551
$880$	0	0
$881$	−41.8007	−1.40830	−0.704150	−	0.710051i	$-0.748672\pi$
$881$	−41.8007	−1.40830	−0.704150	+	0.710051i	$0.748672\pi$
$882$	−4.22425	−0.142238
$883$	−36.9643	−1.24395	−0.621974	−	0.783038i	$-0.713669\pi$
$883$	−36.9643	−1.24395	−0.621974	+	0.783038i	$0.713669\pi$
$884$	−9.40105	−0.316191
$885$	0	0
$886$	29.2144	0.981477
$887$	36.0724	1.21119	0.605596	−	0.795772i	$-0.292935\pi$
$887$	36.0724	1.21119	0.605596	+	0.795772i	$0.292935\pi$
$888$	−11.2750	−0.378366
$889$	−46.0263	−1.54367
$890$	0	0
$891$	−1.61213	−0.0540083
$892$	7.03761	0.235637
$893$	−4.57452	−0.153080
$894$	1.03761	0.0347029
$895$	0	0
$896$	3.35026	0.111924
$897$	−1.82321	−0.0608751
$898$	20.4993	0.684070
$899$	−8.35359	−0.278608
$900$	0	0
$901$	83.0249	2.76596
$902$	−5.40105	−0.179835
$903$	34.5501	1.14975
$904$	−11.1490	−0.370811
$905$	0	0
$906$	1.16362	0.0386587
$907$	−57.6239	−1.91337	−0.956685	−	0.291126i	$-0.905970\pi$
$907$	−57.6239	−1.91337	−0.956685	+	0.291126i	$0.905970\pi$
$908$	−14.5501	−0.482861
$909$	8.88717	0.294769
$910$	0	0
$911$	30.9234	1.02454	0.512268	−	0.858825i	$-0.328805\pi$
$911$	30.9234	1.02454	0.512268	+	0.858825i	$0.328805\pi$
$912$	1.00000	0.0331133
$913$	17.5515	0.580870
$914$	−6.44851	−0.213298
$915$	0	0
$916$	11.4010	0.376701
$917$	−18.8021	−0.620900
$918$	−6.96239	−0.229793
$919$	33.7743	1.11411	0.557056	−	0.830475i	$-0.311931\pi$
$919$	33.7743	1.11411	0.557056	+	0.830475i	$0.311931\pi$
$920$	0	0
$921$	−29.9248	−0.986055
$922$	13.5125	0.445009
$923$	−1.04746	−0.0344776
$924$	−5.40105	−0.177681
$925$	0	0
$926$	−6.20123	−0.203785
$927$	7.03761	0.231145
$928$	−3.61213	−0.118574
$929$	48.6516	1.59621	0.798104	−	0.602519i	$-0.205836\pi$
$929$	48.6516	1.59621	0.798104	+	0.602519i	$0.205836\pi$
$930$	0	0
$931$	−4.22425	−0.138444
$932$	21.9149	0.717847
$933$	21.2750	0.696514
$934$	16.5599	0.541857
$935$	0	0
$936$	−1.35026	−0.0441347
$937$	−18.7005	−0.610919	−0.305460	−	0.952205i	$-0.598810\pi$
$937$	−18.7005	−0.610919	−0.305460	+	0.952205i	$0.598810\pi$
$938$	33.2506	1.08567
$939$	−17.4010	−0.567862
$940$	0	0
$941$	38.4142	1.25227	0.626134	−	0.779716i	$-0.284636\pi$
$941$	38.4142	1.25227	0.626134	+	0.779716i	$0.284636\pi$
$942$	10.9624	0.357174
$943$	−4.52373	−0.147313
$944$	1.03761	0.0337714
$945$	0	0
$946$	16.6253	0.540535
$947$	44.0362	1.43098	0.715492	−	0.698621i	$-0.246203\pi$
$947$	44.0362	1.43098	0.715492	+	0.698621i	$0.246203\pi$
$948$	−14.3127	−0.464853
$949$	4.35359	0.141323
$950$	0	0
$951$	−14.1016	−0.457275
$952$	−23.3258	−0.755994
$953$	−38.0724	−1.23329	−0.616643	−	0.787243i	$-0.711508\pi$
$953$	−38.0724	−1.23329	−0.616643	+	0.787243i	$0.711508\pi$
$954$	11.9248	0.386079
$955$	0	0
$956$	−13.2750	−0.429345
$957$	5.82321	0.188238
$958$	30.5256	0.986239
$959$	59.1754	1.91087
$960$	0	0
$961$	−25.6516	−0.827473
$962$	15.2243	0.490850
$963$	−0.775746	−0.0249981
$964$	21.3258	0.686859
$965$	0	0
$966$	−4.52373	−0.145549
$967$	42.5256	1.36753	0.683766	−	0.729701i	$-0.260341\pi$
$967$	42.5256	1.36753	0.683766	+	0.729701i	$0.260341\pi$
$968$	8.40105	0.270020
$969$	−6.96239	−0.223664
$970$	0	0
$971$	−22.1378	−0.710435	−0.355217	−	0.934784i	$-0.615593\pi$
$971$	−22.1378	−0.710435	−0.355217	+	0.934784i	$0.615593\pi$
$972$	−1.00000	−0.0320750
$973$	−35.8496	−1.14928
$974$	2.51388	0.0805499
$975$	0	0
$976$	2.00000	0.0640184
$977$	−21.7480	−0.695780	−0.347890	−	0.937535i	$-0.613102\pi$
$977$	−21.7480	−0.695780	−0.347890	+	0.937535i	$0.613102\pi$
$978$	21.0132	0.671927
$979$	4.15045	0.132649
$980$	0	0
$981$	−20.1622	−0.643730
$982$	−1.46168	−0.0466441
$983$	9.04746	0.288569	0.144285	−	0.989536i	$-0.453912\pi$
$983$	9.04746	0.288569	0.144285	+	0.989536i	$0.453912\pi$
$984$	−3.35026	−0.106802
$985$	0	0
$986$	25.1490	0.800908
$987$	15.3258	0.487826
$988$	−1.35026	−0.0429575
$989$	13.9248	0.442782
$990$	0	0
$991$	−26.0606	−0.827843	−0.413922	−	0.910313i	$-0.635841\pi$
$991$	−26.0606	−0.827843	−0.413922	+	0.910313i	$0.635841\pi$
$992$	2.31265	0.0734267
$993$	−6.85097	−0.217409
$994$	−2.59895	−0.0824338
$995$	0	0
$996$	10.8872	0.344973
$997$	28.2130	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$997$	28.2130	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$998$	5.55149	0.175729
$999$	11.2750	0.356726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.bk.1.1		3
3.2	odd	2		8550.2.a.cr.1.1		3
5.2	odd	4		570.2.d.d.229.1	✓	6
5.3	odd	4		570.2.d.d.229.4	yes	6
5.4	even	2		2850.2.a.bn.1.3		3
15.2	even	4		1710.2.d.e.1369.6		6
15.8	even	4		1710.2.d.e.1369.3		6
15.14	odd	2		8550.2.a.cf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.d.229.1	✓	6	5.2	odd	4
570.2.d.d.229.4	yes	6	5.3	odd	4
1710.2.d.e.1369.3		6	15.8	even	4
1710.2.d.e.1369.6		6	15.2	even	4
2850.2.a.bk.1.1		3	1.1	even	1	trivial
2850.2.a.bn.1.3		3	5.4	even	2
8550.2.a.cf.1.3		3	15.14	odd	2
8550.2.a.cr.1.1		3	3.2	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.35026 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.35026 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.61213 q^{11} -1.00000 q^{12} +1.35026 q^{13} +3.35026 q^{14} +1.00000 q^{16} -6.96239 q^{17} -1.00000 q^{18} -1.00000 q^{19} +3.35026 q^{21} +1.61213 q^{22} +1.35026 q^{23} +1.00000 q^{24} -1.35026 q^{26} -1.00000 q^{27} -3.35026 q^{28} +3.61213 q^{29} -2.31265 q^{31} -1.00000 q^{32} +1.61213 q^{33} +6.96239 q^{34} +1.00000 q^{36} -11.2750 q^{37} +1.00000 q^{38} -1.35026 q^{39} -3.35026 q^{41} -3.35026 q^{42} +10.3127 q^{43} -1.61213 q^{44} -1.35026 q^{46} +4.57452 q^{47} -1.00000 q^{48} +4.22425 q^{49} +6.96239 q^{51} +1.35026 q^{52} -11.9248 q^{53} +1.00000 q^{54} +3.35026 q^{56} +1.00000 q^{57} -3.61213 q^{58} +1.03761 q^{59} +2.00000 q^{61} +2.31265 q^{62} -3.35026 q^{63} +1.00000 q^{64} -1.61213 q^{66} +9.92478 q^{67} -6.96239 q^{68} -1.35026 q^{69} -0.775746 q^{71} -1.00000 q^{72} +3.22425 q^{73} +11.2750 q^{74} -1.00000 q^{76} +5.40105 q^{77} +1.35026 q^{78} +14.3127 q^{79} +1.00000 q^{81} +3.35026 q^{82} -10.8872 q^{83} +3.35026 q^{84} -10.3127 q^{86} -3.61213 q^{87} +1.61213 q^{88} -2.57452 q^{89} -4.52373 q^{91} +1.35026 q^{92} +2.31265 q^{93} -4.57452 q^{94} +1.00000 q^{96} +1.16362 q^{97} -4.22425 q^{98} -1.61213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 4 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{18} - 3 q^{19} + 4 q^{22} - 6 q^{23} + 3 q^{24} + 6 q^{26} - 3 q^{27} + 10 q^{29} + 14 q^{31} - 3 q^{32} + 4 q^{33} + 10 q^{34} + 3 q^{36} - 2 q^{37} + 3 q^{38} + 6 q^{39} + 10 q^{43} - 4 q^{44} + 6 q^{46} + 2 q^{47} - 3 q^{48} + 11 q^{49} + 10 q^{51} - 6 q^{52} - 14 q^{53} + 3 q^{54} + 3 q^{57} - 10 q^{58} + 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 4 q^{66} + 8 q^{67} - 10 q^{68} + 6 q^{69} - 4 q^{71} - 3 q^{72} + 8 q^{73} + 2 q^{74} - 3 q^{76} - 24 q^{77} - 6 q^{78} + 22 q^{79} + 3 q^{81} - 10 q^{86} - 10 q^{87} + 4 q^{88} + 4 q^{89} - 32 q^{91} - 6 q^{92} - 14 q^{93} - 2 q^{94} + 3 q^{96} + 6 q^{97} - 11 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 4 * q^11 - 3 * q^12 - 6 * q^13 + 3 * q^16 - 10 * q^17 - 3 * q^18 - 3 * q^19 + 4 * q^22 - 6 * q^23 + 3 * q^24 + 6 * q^26 - 3 * q^27 + 10 * q^29 + 14 * q^31 - 3 * q^32 + 4 * q^33 + 10 * q^34 + 3 * q^36 - 2 * q^37 + 3 * q^38 + 6 * q^39 + 10 * q^43 - 4 * q^44 + 6 * q^46 + 2 * q^47 - 3 * q^48 + 11 * q^49 + 10 * q^51 - 6 * q^52 - 14 * q^53 + 3 * q^54 + 3 * q^57 - 10 * q^58 + 14 * q^59 + 6 * q^61 - 14 * q^62 + 3 * q^64 - 4 * q^66 + 8 * q^67 - 10 * q^68 + 6 * q^69 - 4 * q^71 - 3 * q^72 + 8 * q^73 + 2 * q^74 - 3 * q^76 - 24 * q^77 - 6 * q^78 + 22 * q^79 + 3 * q^81 - 10 * q^86 - 10 * q^87 + 4 * q^88 + 4 * q^89 - 32 * q^91 - 6 * q^92 - 14 * q^93 - 2 * q^94 + 3 * q^96 + 6 * q^97 - 11 * q^98 - 4 * q^99

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