LMFDB - Embedded newform 1920.2.b.g.191.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(191,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1920.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.6
Root			$1.18254 + 1.18254i$ of defining polynomial
Character	$\chi$	$=$	1920.191
Dual form			1920.2.b.g.191.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.759725 + 1.55654i) q^{3} -1.00000 q^{5} +0.480550i q^{7} +(-1.84564 + 2.36509i) q^{9} -4.73017i q^{11} +1.59363i q^{13} +(-0.759725 - 1.55654i) q^{15} -6.32380i q^{17} -2.09764 q^{19} +(-0.747995 + 0.365086i) q^{21} -5.84325 q^{23} +1.00000 q^{25} +(-5.08353 - 1.07599i) q^{27} -8.95633 q^{29} +8.32380i q^{31} +(7.36270 - 3.59363i) q^{33} -0.480550i q^{35} -3.67143i q^{37} +(-2.48055 + 1.21072i) q^{39} -4.53489i q^{41} -8.70671 q^{43} +(1.84564 - 2.36509i) q^{45} -0.578190 q^{47} +6.76907 q^{49} +(9.84325 - 4.80435i) q^{51} -7.26506 q^{53} +4.73017i q^{55} +(-1.59363 - 3.26506i) q^{57} -2.30873i q^{59} -13.9952i q^{61} +(-1.13654 - 0.886919i) q^{63} -1.59363i q^{65} +0.706711 q^{67} +(-4.43927 - 9.09526i) q^{69} +13.6087 q^{71} +1.80472 q^{73} +(0.759725 + 1.55654i) q^{75} +2.27308 q^{77} -6.54996i q^{79} +(-2.18726 - 8.73017i) q^{81} +3.92582i q^{83} +6.32380i q^{85} +(-6.80435 - 13.9409i) q^{87} -9.26506i q^{89} -0.765818 q^{91} +(-12.9563 + 6.32380i) q^{93} +2.09764 q^{95} -1.18726 q^{97} +(11.1873 + 8.73017i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 8 q^{5} - 4 q^{9} - 2 q^{15} - 8 q^{19} - 4 q^{21} + 12 q^{23} + 8 q^{25} + 14 q^{27} + 8 q^{29} - 8 q^{33} - 28 q^{39} - 36 q^{43} + 4 q^{45} - 4 q^{47} + 20 q^{51} + 16 q^{63} - 28 q^{67}+ \cdots + 64 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 - 8 * q^5 - 4 * q^9 - 2 * q^15 - 8 * q^19 - 4 * q^21 + 12 * q^23 + 8 * q^25 + 14 * q^27 + 8 * q^29 - 8 * q^33 - 28 * q^39 - 36 * q^43 + 4 * q^45 - 4 * q^47 + 20 * q^51 + 16 * q^63 - 28 * q^67 - 12 * q^69 + 24 * q^71 + 32 * q^73 + 2 * q^75 - 32 * q^77 + 8 * q^81 - 12 * q^87 - 40 * q^91 - 24 * q^93 + 8 * q^95 + 16 * q^97 + 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.759725	+	1.55654i	0.438628	+	0.898669i
$3$	0.759725	+	1.55654i	0.438628	+	0.898669i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			0.480550i			0.181631i	0.995868	+	0.0908153i	$0.0289473\pi$
$7$			0.480550i			0.181631i	−0.995868	+	0.0908153i	$0.971053\pi$
$8$		0			0
$9$	−1.84564	+	2.36509i	−0.615212	+	0.788362i
$10$		0			0
$11$		−	4.73017i		−	1.42620i	−0.701062	−	0.713100i	$-0.747290\pi$
$11$		−	4.73017i		−	1.42620i	0.701062	−	0.713100i	$-0.252710\pi$
$12$		0			0
$13$			1.59363i			0.441994i	0.975275	+	0.220997i	$0.0709310\pi$
$13$			1.59363i			0.441994i	−0.975275	+	0.220997i	$0.929069\pi$
$14$		0			0
$15$	−0.759725	−	1.55654i	−0.196160	−	0.401897i
$16$		0			0
$17$		−	6.32380i		−	1.53375i	−0.641798	−	0.766874i	$-0.721811\pi$
$17$		−	6.32380i		−	1.53375i	0.641798	−	0.766874i	$-0.278189\pi$
$18$		0			0
$19$	−2.09764			−0.481232			−0.240616	−	0.970620i	$-0.577349\pi$
$19$	−2.09764			−0.481232			−0.240616	+	0.970620i	$0.577349\pi$
$20$		0			0
$21$	−0.747995	+	0.365086i	−0.163226	+	0.0796682i
$22$		0			0
$23$	−5.84325			−1.21840			−0.609201	−	0.793016i	$-0.708510\pi$
$23$	−5.84325			−1.21840			−0.609201	+	0.793016i	$0.708510\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−5.08353	−	1.07599i	−0.978325	−	0.207074i
$28$		0			0
$29$	−8.95633			−1.66315			−0.831575	−	0.555413i	$-0.812560\pi$
$29$	−8.95633			−1.66315			−0.831575	+	0.555413i	$0.812560\pi$
$30$		0			0
$31$			8.32380i			1.49500i	0.664262	+	0.747499i	$0.268746\pi$
$31$			8.32380i			1.49500i	−0.664262	+	0.747499i	$0.731254\pi$
$32$		0			0
$33$	7.36270	−	3.59363i	1.28168	−	0.625571i
$34$		0			0
$35$		−	0.480550i		−	0.0812277i
$36$		0			0
$37$		−	3.67143i		−	0.603580i	−0.953374	−	0.301790i	$-0.902416\pi$
$37$		−	3.67143i		−	0.603580i	0.953374	−	0.301790i	$-0.0975841\pi$
$38$		0			0
$39$	−2.48055	+	1.21072i	−0.397206	+	0.193871i
$40$		0			0
$41$		−	4.53489i		−	0.708231i	−0.935202	−	0.354115i	$-0.884782\pi$
$41$		−	4.53489i		−	0.708231i	0.935202	−	0.354115i	$-0.115218\pi$
$42$		0			0
$43$	−8.70671			−1.32776			−0.663880	−	0.747839i	$-0.731092\pi$
$43$	−8.70671			−1.32776			−0.663880	+	0.747839i	$0.731092\pi$
$44$		0			0
$45$	1.84564	−	2.36509i	0.275131	−	0.352566i
$46$		0			0
$47$	−0.578190			−0.0843377			−0.0421688	−	0.999110i	$-0.513427\pi$
$47$	−0.578190			−0.0843377			−0.0421688	+	0.999110i	$0.513427\pi$
$48$		0			0
$49$	6.76907			0.967010
$50$		0			0
$51$	9.84325	−	4.80435i	1.37833	−	0.672744i
$52$		0			0
$53$	−7.26506			−0.997933			−0.498966	−	0.866621i	$-0.666287\pi$
$53$	−7.26506			−0.997933			−0.498966	+	0.866621i	$0.666287\pi$
$54$		0			0
$55$			4.73017i			0.637816i
$56$		0			0
$57$	−1.59363	−	3.26506i	−0.211081	−	0.432468i
$58$		0			0
$59$		−	2.30873i		−	0.300571i	−0.988643	−	0.150285i	$-0.951981\pi$
$59$		−	2.30873i		−	0.300571i	0.988643	−	0.150285i	$-0.0480193\pi$
$60$		0			0
$61$		−	13.9952i		−	1.79191i	−0.444149	−	0.895953i	$-0.646494\pi$
$61$		−	13.9952i		−	1.79191i	0.444149	−	0.895953i	$-0.353506\pi$
$62$		0			0
$63$	−1.13654	−	0.886919i	−0.143191	−	0.111741i
$64$		0			0
$65$		−	1.59363i		−	0.197666i
$66$		0			0
$67$	0.706711			0.0863385			0.0431692	−	0.999068i	$-0.486255\pi$
$67$	0.706711			0.0863385			0.0431692	+	0.999068i	$0.486255\pi$
$68$		0			0
$69$	−4.43927	−	9.09526i	−0.534425	−	1.09494i
$70$		0			0
$71$	13.6087			1.61506			0.807528	−	0.589829i	$-0.200805\pi$
$71$	13.6087			1.61506			0.807528	+	0.589829i	$0.200805\pi$
$72$		0			0
$73$	1.80472			0.211226			0.105613	−	0.994407i	$-0.466319\pi$
$73$	1.80472			0.211226			0.105613	+	0.994407i	$0.466319\pi$
$74$		0			0
$75$	0.759725	+	1.55654i	0.0877255	+	0.179734i
$76$		0			0
$77$	2.27308			0.259042
$78$		0			0
$79$		−	6.54996i		−	0.736928i	−0.929642	−	0.368464i	$-0.879884\pi$
$79$		−	6.54996i		−	0.736928i	0.929642	−	0.368464i	$-0.120116\pi$
$80$		0			0
$81$	−2.18726	−	8.73017i	−0.243029	−	0.970019i
$82$		0			0
$83$			3.92582i			0.430915i	0.976513	+	0.215457i	$0.0691242\pi$
$83$			3.92582i			0.430915i	−0.976513	+	0.215457i	$0.930876\pi$
$84$		0			0
$85$			6.32380i			0.685913i
$86$		0			0
$87$	−6.80435	−	13.9409i	−0.729503	−	1.49462i
$88$		0			0
$89$		−	9.26506i		−	0.982095i	−0.871133	−	0.491047i	$-0.836614\pi$
$89$		−	9.26506i		−	0.982095i	0.871133	−	0.491047i	$-0.163386\pi$
$90$		0			0
$91$	−0.765818			−0.0802796
$92$		0			0
$93$	−12.9563	+	6.32380i	−1.34351	+	0.655748i
$94$		0			0
$95$	2.09764			0.215213
$96$		0			0
$97$	−1.18726			−0.120548			−0.0602740	−	0.998182i	$-0.519197\pi$
$97$	−1.18726			−0.120548			−0.0602740	+	0.998182i	$0.519197\pi$
$98$		0			0
$99$	11.1873	+	8.73017i	1.12436	+	0.877415i
$100$		0			0
$101$	−6.87853			−0.684439			−0.342220	−	0.939620i	$-0.611179\pi$
$101$	−6.87853			−0.684439			−0.342220	+	0.939620i	$0.611179\pi$
$102$		0			0
$103$		−	1.94089i		−	0.191242i	−0.995418	−	0.0956209i	$-0.969516\pi$
$103$		−	1.94089i		−	0.191242i	0.995418	−	0.0956209i	$-0.0304837\pi$
$104$		0			0
$105$	0.747995	−	0.365086i	0.0729968	−	0.0356287i
$106$		0			0
$107$			7.45672i			0.720868i	0.932785	+	0.360434i	$0.117371\pi$
$107$			7.45672i			0.720868i	−0.932785	+	0.360434i	$0.882629\pi$
$108$		0			0
$109$			8.11271i			0.777057i	0.921437	+	0.388528i	$0.127016\pi$
$109$			8.11271i			0.777057i	−0.921437	+	0.388528i	$0.872984\pi$
$110$		0			0
$111$	5.71473	−	2.78928i	0.542418	−	0.264747i
$112$		0			0
$113$			3.94928i			0.371517i	0.982595	+	0.185759i	$0.0594742\pi$
$113$			3.94928i			0.371517i	−0.982595	+	0.185759i	$0.940526\pi$
$114$		0			0
$115$	5.84325			0.544886
$116$		0			0
$117$	−3.76907	−	2.94126i	−0.348451	−	0.271920i
$118$		0			0
$119$	3.03890			0.278576
$120$		0			0
$121$	−11.3745			−1.03405
$122$		0			0
$123$	7.05874	−	3.44527i	0.636465	−	0.310650i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			3.32417i			0.294972i	0.989064	+	0.147486i	$0.0471182\pi$
$127$			3.32417i			0.294972i	−0.989064	+	0.147486i	$0.952882\pi$
$128$		0			0
$129$	−6.61471	−	13.5523i	−0.582393	−	1.19322i
$130$		0			0
$131$			6.61269i			0.577754i	0.957366	+	0.288877i	$0.0932818\pi$
$131$			6.61269i			0.577754i	−0.957366	+	0.288877i	$0.906718\pi$
$132$		0			0
$133$		−	1.00802i		−	0.0874065i
$134$		0			0
$135$	5.08353	+	1.07599i	0.437520	+	0.0926065i
$136$		0			0
$137$		−	8.86346i		−	0.757256i	−0.925549	−	0.378628i	$-0.876396\pi$
$137$		−	8.86346i		−	0.757256i	0.925549	−	0.378628i	$-0.123604\pi$
$138$		0			0
$139$	−20.6278			−1.74962			−0.874812	−	0.484462i	$-0.839015\pi$
$139$	−20.6278			−1.74962			−0.874812	+	0.484462i	$0.839015\pi$
$140$		0			0
$141$	−0.439266	−	0.899976i	−0.0369928	−	0.0757917i
$142$		0			0
$143$	7.53814			0.630371
$144$		0			0
$145$	8.95633			0.743783
$146$		0			0
$147$	5.14263	+	10.5363i	0.424157	+	0.869022i
$148$		0			0
$149$	18.8429			1.54367			0.771835	−	0.635823i	$-0.219339\pi$
$149$	18.8429			1.54367			0.771835	+	0.635823i	$0.219339\pi$
$150$		0			0
$151$			15.2143i			1.23813i	0.785341	+	0.619063i	$0.212487\pi$
$151$			15.2143i			1.23813i	−0.785341	+	0.619063i	$0.787513\pi$
$152$		0			0
$153$	14.9563	+	11.6714i	1.20915	+	0.943579i
$154$		0			0
$155$		−	8.32380i		−	0.668584i
$156$		0			0
$157$		−	16.9762i		−	1.35485i	−0.735594	−	0.677423i	$-0.763097\pi$
$157$		−	16.9762i		−	1.35485i	0.735594	−	0.677423i	$-0.236903\pi$
$158$		0			0
$159$	−5.51945	−	11.3084i	−0.437721	−	0.896811i
$160$		0			0
$161$		−	2.80797i		−	0.221299i
$162$		0			0
$163$	7.89397			0.618304			0.309152	−	0.951013i	$-0.399955\pi$
$163$	7.89397			0.618304			0.309152	+	0.951013i	$0.399955\pi$
$164$		0			0
$165$	−7.36270	+	3.59363i	−0.573186	+	0.279764i
$166$		0			0
$167$	−13.0305			−1.00833			−0.504166	−	0.863607i	$-0.668200\pi$
$167$	−13.0305			−1.00833			−0.504166	+	0.863607i	$0.668200\pi$
$168$		0			0
$169$	10.4603			0.804642
$170$		0			0
$171$	3.87148	−	4.96110i	0.296059	−	0.379385i
$172$		0			0
$173$	−10.1556			−0.772116			−0.386058	−	0.922474i	$-0.626164\pi$
$173$	−10.1556			−0.772116			−0.386058	+	0.922474i	$0.626164\pi$
$174$		0			0
$175$			0.480550i			0.0363261i
$176$		0			0
$177$	3.59363	−	1.75400i	0.270114	−	0.131839i
$178$		0			0
$179$			3.76907i			0.281714i	0.990030	+	0.140857i	$0.0449857\pi$
$179$			3.76907i			0.281714i	−0.990030	+	0.140857i	$0.955014\pi$
$180$		0			0
$181$		−	8.45232i		−	0.628256i	−0.949381	−	0.314128i	$-0.898288\pi$
$181$		−	8.45232i		−	0.628256i	0.949381	−	0.314128i	$-0.101712\pi$
$182$		0			0
$183$	21.7841	−	10.6325i	1.61033	−	0.785979i
$184$		0			0
$185$			3.67143i			0.269929i
$186$		0			0
$187$	−29.9127			−2.18743
$188$		0			0
$189$	0.517067	−	2.44289i	0.0376111	−	0.177694i
$190$		0			0
$191$	−21.5690			−1.56068			−0.780340	−	0.625356i	$-0.784954\pi$
$191$	−21.5690			−1.56068			−0.780340	+	0.625356i	$0.784954\pi$
$192$		0			0
$193$	15.7174			1.13136			0.565681	−	0.824624i	$-0.308613\pi$
$193$	15.7174			1.13136			0.565681	+	0.824624i	$0.308613\pi$
$194$		0			0
$195$	2.48055	−	1.21072i	0.177636	−	0.0867015i
$196$		0			0
$197$	−22.0301			−1.56958			−0.784791	−	0.619760i	$-0.787230\pi$
$197$	−22.0301			−1.56958			−0.784791	+	0.619760i	$0.787230\pi$
$198$		0			0
$199$			5.08962i			0.360794i	0.983594	+	0.180397i	$0.0577382\pi$
$199$			5.08962i			0.360794i	−0.983594	+	0.180397i	$0.942262\pi$
$200$		0			0
$201$	0.536906	+	1.10002i	0.0378704	+	0.0775897i
$202$		0			0
$203$		−	4.30396i		−	0.302079i
$204$		0			0
$205$			4.53489i			0.316730i
$206$		0			0
$207$	10.7845	−	13.8198i	0.749575	−	0.960542i
$208$		0			0
$209$			9.92220i			0.686333i
$210$		0			0
$211$	18.2365			1.25545			0.627725	−	0.778435i	$-0.283986\pi$
$211$	18.2365			1.25545			0.627725	+	0.778435i	$0.283986\pi$
$212$		0			0
$213$	10.3389	+	21.1825i	0.708408	+	1.45140i
$214$		0			0
$215$	8.70671			0.593793
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$	1.37109	+	2.80912i	0.0926497	+	0.189823i
$220$		0			0
$221$	10.0778			0.677906
$222$		0			0
$223$			24.1671i			1.61835i	0.587571	+	0.809173i	$0.300084\pi$
$223$			24.1671i			1.61835i	−0.587571	+	0.809173i	$0.699916\pi$
$224$		0			0
$225$	−1.84564	+	2.36509i	−0.123042	+	0.157672i
$226$		0			0
$227$			14.3473i			0.952261i	0.879375	+	0.476131i	$0.157961\pi$
$227$			14.3473i			0.952261i	−0.879375	+	0.476131i	$0.842039\pi$
$228$		0			0
$229$		−	10.5698i		−	0.698472i	−0.937035	−	0.349236i	$-0.886441\pi$
$229$		−	10.5698i		−	0.698472i	0.937035	−	0.349236i	$-0.113559\pi$
$230$		0			0
$231$	1.72692	+	3.53814i	0.113623	+	0.232793i
$232$		0			0
$233$		−	8.44128i		−	0.553007i	−0.961013	−	0.276503i	$-0.910824\pi$
$233$		−	8.44128i		−	0.553007i	0.961013	−	0.276503i	$-0.0891757\pi$
$234$		0			0
$235$	0.578190			0.0377170
$236$		0			0
$237$	10.1953	−	4.97617i	0.662255	−	0.323237i
$238$		0			0
$239$	11.8047			0.763584			0.381792	−	0.924248i	$-0.375307\pi$
$239$	11.8047			0.763584			0.381792	+	0.924248i	$0.375307\pi$
$240$		0			0
$241$	−27.6817			−1.78314			−0.891569	−	0.452885i	$-0.850395\pi$
$241$	−27.6817			−1.78314			−0.891569	+	0.452885i	$0.850395\pi$
$242$		0			0
$243$	11.9271	−	10.0371i	0.765127	−	0.643880i
$244$		0			0
$245$	−6.76907			−0.432460
$246$		0			0
$247$		−	3.34286i		−	0.212701i
$248$		0			0
$249$	−6.11070	+	2.98254i	−0.387250	+	0.189011i
$250$		0			0
$251$		−	4.73017i		−	0.298566i	−0.988795	−	0.149283i	$-0.952303\pi$
$251$		−	4.73017i		−	0.298566i	0.988795	−	0.149283i	$-0.0476965\pi$
$252$		0			0
$253$			27.6396i			1.73769i
$254$		0			0
$255$	−9.84325	+	4.80435i	−0.616408	+	0.300860i
$256$		0			0
$257$			22.9714i			1.43292i	0.697630	+	0.716458i	$0.254238\pi$
$257$			22.9714i			1.43292i	−0.697630	+	0.716458i	$0.745762\pi$
$258$		0			0
$259$	1.76431			0.109629
$260$		0			0
$261$	16.5301	−	21.1825i	1.02319	−	1.31116i
$262$		0			0
$263$	−19.6171			−1.20964			−0.604821	−	0.796362i	$-0.706755\pi$
$263$	−19.6171			−1.20964			−0.604821	+	0.796362i	$0.706755\pi$
$264$		0			0
$265$	7.26506			0.446289
$266$		0			0
$267$	14.4214	−	7.03890i	0.882578	−	0.430774i
$268$		0			0
$269$	−8.46836			−0.516325			−0.258163	−	0.966101i	$-0.583117\pi$
$269$	−8.46836			−0.516325			−0.258163	+	0.966101i	$0.583117\pi$
$270$		0			0
$271$		−	15.4714i		−	0.939820i	−0.882714	−	0.469910i	$-0.844286\pi$
$271$		−	15.4714i		−	0.939820i	0.882714	−	0.469910i	$-0.155714\pi$
$272$		0			0
$273$	−0.581812	−	1.19203i	−0.0352128	−	0.0721448i
$274$		0			0
$275$		−	4.73017i		−	0.285240i
$276$		0			0
$277$			24.7096i			1.48466i	0.670037	+	0.742328i	$0.266278\pi$
$277$			24.7096i			1.48466i	−0.670037	+	0.742328i	$0.733722\pi$
$278$		0			0
$279$	−19.6865	−	15.3627i	−1.17860	−	0.919741i
$280$		0			0
$281$		−	27.0033i		−	1.61088i	−0.592678	−	0.805440i	$-0.701929\pi$
$281$		−	27.0033i		−	1.61088i	0.592678	−	0.805440i	$-0.298071\pi$
$282$		0			0
$283$	−20.9020			−1.24249			−0.621247	−	0.783615i	$-0.713374\pi$
$283$	−20.9020			−1.24249			−0.621247	+	0.783615i	$0.713374\pi$
$284$		0			0
$285$	1.59363	+	3.26506i	0.0943985	+	0.193406i
$286$		0			0
$287$	2.17924			0.128636
$288$		0			0
$289$	−22.9905			−1.35238
$290$		0			0
$291$	−0.901992	−	1.84802i	−0.0528757	−	0.108333i
$292$		0			0
$293$	0.374521			0.0218797			0.0109399	−	0.999940i	$-0.496518\pi$
$293$	0.374521			0.0218797			0.0109399	+	0.999940i	$0.496518\pi$
$294$		0			0
$295$			2.30873i			0.134419i
$296$		0			0
$297$	−5.08962	+	24.0460i	−0.295330	+	1.39529i
$298$		0			0
$299$		−	9.31198i		−	0.538526i
$300$		0			0
$301$		−	4.18401i		−	0.241162i
$302$		0			0
$303$	−5.22579	−	10.7067i	−0.300214	−	0.615084i
$304$		0			0
$305$			13.9952i			0.801365i
$306$		0			0
$307$	−14.3623			−0.819702			−0.409851	−	0.912153i	$-0.634419\pi$
$307$	−14.3623			−0.819702			−0.409851	+	0.912153i	$0.634419\pi$
$308$		0			0
$309$	3.02108	−	1.47454i	0.171863	−	0.0838839i
$310$		0			0
$311$	22.2166			1.25979			0.629895	−	0.776681i	$-0.283098\pi$
$311$	22.2166			1.25979			0.629895	+	0.776681i	$0.283098\pi$
$312$		0			0
$313$	33.6697			1.90313			0.951563	−	0.307455i	$-0.0994772\pi$
$313$	33.6697			1.90313			0.951563	+	0.307455i	$0.0994772\pi$
$314$		0			0
$315$	1.13654	+	0.886919i	0.0640368	+	0.0499722i
$316$		0			0
$317$	29.7952			1.67346			0.836732	−	0.547612i	$-0.184463\pi$
$317$	29.7952			1.67346			0.836732	+	0.547612i	$0.184463\pi$
$318$		0			0
$319$			42.3650i			2.37198i
$320$		0			0
$321$	−11.6067	+	5.66506i	−0.647822	+	0.316193i
$322$		0			0
$323$			13.2651i			0.738088i
$324$		0			0
$325$			1.59363i			0.0883987i
$326$		0			0
$327$	−12.6278	+	6.16343i	−0.698317	+	0.340839i
$328$		0			0
$329$		−	0.277849i		−	0.0153183i
$330$		0			0
$331$	−23.7063			−1.30302			−0.651509	−	0.758641i	$-0.725864\pi$
$331$	−23.7063			−1.30302			−0.651509	+	0.758641i	$0.725864\pi$
$332$		0			0
$333$	8.68325	+	6.77612i	0.475839	+	0.371329i
$334$		0			0
$335$	−0.706711			−0.0386117
$336$		0			0
$337$	−27.9127			−1.52050			−0.760250	−	0.649631i	$-0.774924\pi$
$337$	−27.9127			−1.52050			−0.760250	+	0.649631i	$0.774924\pi$
$338$		0			0
$339$	−6.14721	+	3.00037i	−0.333871	+	0.162958i
$340$		0			0
$341$	39.3730			2.13217
$342$		0			0
$343$			6.61672i			0.357269i
$344$		0			0
$345$	4.43927	+	9.09526i	0.239002	+	0.489672i
$346$		0			0
$347$		−	14.4956i		−	0.778166i	−0.921203	−	0.389083i	$-0.872792\pi$
$347$		−	14.4956i		−	0.778166i	0.921203	−	0.389083i	$-0.127208\pi$
$348$		0			0
$349$			2.73494i			0.146398i	0.997317	+	0.0731989i	$0.0233208\pi$
$349$			2.73494i			0.146398i	−0.997317	+	0.0731989i	$0.976679\pi$
$350$		0			0
$351$	1.71473	−	8.10126i	0.0915256	−	0.432413i
$352$		0			0
$353$			35.3936i			1.88381i	0.335879	+	0.941905i	$0.390967\pi$
$353$			35.3936i			1.88381i	−0.335879	+	0.941905i	$0.609033\pi$
$354$		0			0
$355$	−13.6087			−0.722275
$356$		0			0
$357$	2.30873	+	4.73017i	0.122191	+	0.250347i
$358$		0			0
$359$	−11.1777			−0.589938			−0.294969	−	0.955507i	$-0.595309\pi$
$359$	−11.1777			−0.589938			−0.294969	+	0.955507i	$0.595309\pi$
$360$		0			0
$361$	−14.5999			−0.768416
$362$		0			0
$363$	−8.64151	−	17.7049i	−0.453562	−	0.929266i
$364$		0			0
$365$	−1.80472			−0.0944633
$366$		0			0
$367$			8.18383i			0.427192i	0.976922	+	0.213596i	$0.0685177\pi$
$367$			8.18383i			0.427192i	−0.976922	+	0.213596i	$0.931482\pi$
$368$		0			0
$369$	10.7254	+	8.36975i	0.558342	+	0.435712i
$370$		0			0
$371$		−	3.49122i		−	0.181255i
$372$		0			0
$373$			14.2016i			0.735329i	0.929959	+	0.367664i	$0.119842\pi$
$373$			14.2016i			0.735329i	−0.929959	+	0.367664i	$0.880158\pi$
$374$		0			0
$375$	−0.759725	−	1.55654i	−0.0392320	−	0.0803794i
$376$		0			0
$377$		−	14.2731i		−	0.735101i
$378$		0			0
$379$	10.0976			0.518681			0.259340	−	0.965786i	$-0.416495\pi$
$379$	10.0976			0.518681			0.259340	+	0.965786i	$0.416495\pi$
$380$		0			0
$381$	−5.17420	+	2.52546i	−0.265082	+	0.129383i
$382$		0			0
$383$	−9.35125			−0.477827			−0.238913	−	0.971041i	$-0.576791\pi$
$383$	−9.35125			−0.477827			−0.238913	+	0.971041i	$0.576791\pi$
$384$		0			0
$385$	−2.27308			−0.115847
$386$		0			0
$387$	16.0694	−	20.5921i	0.816854	−	1.04676i
$388$		0			0
$389$	12.5301			0.635302			0.317651	−	0.948208i	$-0.397106\pi$
$389$	12.5301			0.635302			0.317651	+	0.948208i	$0.397106\pi$
$390$		0			0
$391$			36.9516i			1.86872i
$392$		0			0
$393$	−10.2929	+	5.02383i	−0.519209	+	0.253419i
$394$		0			0
$395$			6.54996i			0.329564i
$396$		0			0
$397$		−	28.3492i		−	1.42280i	−0.702785	−	0.711402i	$-0.748060\pi$
$397$		−	28.3492i		−	1.42280i	0.702785	−	0.711402i	$-0.251940\pi$
$398$		0			0
$399$	1.56902	−	0.765818i	0.0785495	−	0.0383389i
$400$		0			0
$401$			17.7571i			0.886745i	0.896337	+	0.443373i	$0.146218\pi$
$401$			17.7571i			0.886745i	−0.896337	+	0.443373i	$0.853782\pi$
$402$		0			0
$403$	−13.2651			−0.660780
$404$		0			0
$405$	2.18726	+	8.73017i	0.108686	+	0.433806i
$406$		0			0
$407$	−17.3665			−0.860825
$408$		0			0
$409$	−23.2294			−1.14862			−0.574310	−	0.818638i	$-0.694730\pi$
$409$	−23.2294			−1.14862			−0.574310	+	0.818638i	$0.694730\pi$
$410$		0			0
$411$	13.7963	−	6.73379i	0.680523	−	0.332154i
$412$		0			0
$413$	1.10946			0.0545929
$414$		0			0
$415$		−	3.92582i		−	0.192711i
$416$		0			0
$417$	−15.6714	−	32.1079i	−0.767433	−	1.57233i
$418$		0			0
$419$		−	36.4167i		−	1.77907i	−0.456866	−	0.889536i	$-0.651028\pi$
$419$		−	36.4167i		−	1.77907i	0.456866	−	0.889536i	$-0.348972\pi$
$420$		0			0
$421$		−	21.8301i		−	1.06393i	−0.846765	−	0.531967i	$-0.821453\pi$
$421$		−	21.8301i		−	1.06393i	0.846765	−	0.531967i	$-0.178547\pi$
$422$		0			0
$423$	1.06713	−	1.36747i	0.0518855	−	0.0664886i
$424$		0			0
$425$		−	6.32380i		−	0.306749i
$426$		0			0
$427$	6.72540			0.325465
$428$		0			0
$429$	5.72692	+	11.7334i	0.276498	+	0.566495i
$430$		0			0
$431$	−32.4355			−1.56237			−0.781183	−	0.624303i	$-0.785383\pi$
$431$	−32.4355			−1.56237			−0.781183	+	0.624303i	$0.785383\pi$
$432$		0			0
$433$	19.5618			0.940079			0.470040	−	0.882645i	$-0.344240\pi$
$433$	19.5618			0.940079			0.470040	+	0.882645i	$0.344240\pi$
$434$		0			0
$435$	6.80435	+	13.9409i	0.326244	+	0.668415i
$436$		0			0
$437$	12.2570			0.586334
$438$		0			0
$439$		−	9.89283i		−	0.472159i	−0.971734	−	0.236079i	$-0.924137\pi$
$439$		−	9.89283i		−	0.472159i	0.971734	−	0.236079i	$-0.0758625\pi$
$440$		0			0
$441$	−12.4932	+	16.0094i	−0.594916	+	0.762354i
$442$		0			0
$443$			39.2519i			1.86491i	0.361281	+	0.932457i	$0.382340\pi$
$443$			39.2519i			1.86491i	−0.361281	+	0.932457i	$0.617660\pi$
$444$		0			0
$445$			9.26506i			0.439206i
$446$		0			0
$447$	14.3154	+	29.3297i	0.677096	+	1.38725i
$448$		0			0
$449$			12.7302i			0.600774i	0.953817	+	0.300387i	$0.0971158\pi$
$449$			12.7302i			0.600774i	−0.953817	+	0.300387i	$0.902884\pi$
$450$		0			0
$451$	−21.4508			−1.01008
$452$		0			0
$453$	−23.6817	+	11.5587i	−1.11267	+	0.543076i
$454$		0			0
$455$	0.765818			0.0359021
$456$		0			0
$457$	−5.14758			−0.240794			−0.120397	−	0.992726i	$-0.538417\pi$
$457$	−5.14758			−0.240794			−0.120397	+	0.992726i	$0.538417\pi$
$458$		0			0
$459$	−6.80435	+	32.1472i	−0.317600	+	1.50050i
$460$		0			0
$461$	9.11194			0.424385			0.212193	−	0.977228i	$-0.431940\pi$
$461$	9.11194			0.424385			0.212193	+	0.977228i	$0.431940\pi$
$462$		0			0
$463$		−	36.4542i		−	1.69417i	−0.531456	−	0.847086i	$-0.678355\pi$
$463$		−	36.4542i		−	1.69417i	0.531456	−	0.847086i	$-0.321645\pi$
$464$		0			0
$465$	12.9563	−	6.32380i	0.600836	−	0.293259i
$466$		0			0
$467$		−	10.5425i		−	0.487851i	−0.969794	−	0.243925i	$-0.921565\pi$
$467$		−	10.5425i		−	0.487851i	0.969794	−	0.243925i	$-0.0784352\pi$
$468$		0			0
$469$			0.339610i			0.0156817i
$470$		0			0
$471$	26.4241	−	12.8972i	1.21756	−	0.594273i
$472$		0			0
$473$			41.1842i			1.89365i
$474$		0			0
$475$	−2.09764			−0.0962463
$476$		0			0
$477$	13.4087	−	17.1825i	0.613940	−	0.786732i
$478$		0			0
$479$	31.0221			1.41744			0.708719	−	0.705491i	$-0.249274\pi$
$479$	31.0221			1.41744			0.708719	+	0.705491i	$0.249274\pi$
$480$		0			0
$481$	5.85090			0.266778
$482$		0			0
$483$	4.37072	−	2.13329i	0.198875	−	0.0970679i
$484$		0			0
$485$	1.18726			0.0539107
$486$		0			0
$487$		−	18.7536i		−	0.849808i	−0.905238	−	0.424904i	$-0.860308\pi$
$487$		−	18.7536i		−	0.849808i	0.905238	−	0.424904i	$-0.139692\pi$
$488$		0			0
$489$	5.99725	+	12.2873i	0.271205	+	0.555650i
$490$		0			0
$491$			16.9255i			0.763835i	0.924196	+	0.381917i	$0.124736\pi$
$491$			16.9255i			0.763835i	−0.924196	+	0.381917i	$0.875264\pi$
$492$		0			0
$493$			56.6381i			2.55085i
$494$		0			0
$495$	−11.1873	−	8.73017i	−0.502830	−	0.392392i
$496$		0			0
$497$			6.53966i			0.293344i
$498$		0			0
$499$	−5.90236			−0.264226			−0.132113	−	0.991235i	$-0.542176\pi$
$499$	−5.90236			−0.264226			−0.132113	+	0.991235i	$0.542176\pi$
$500$		0			0
$501$	−9.89961	−	20.2825i	−0.442282	−	0.906156i
$502$		0			0
$503$	−20.2647			−0.903558			−0.451779	−	0.892130i	$-0.649210\pi$
$503$	−20.2647			−0.903558			−0.451779	+	0.892130i	$0.649210\pi$
$504$		0			0
$505$	6.87853			0.306091
$506$		0			0
$507$	7.94699	+	16.2819i	0.352938	+	0.723107i
$508$		0			0
$509$	12.5040			0.554231			0.277115	−	0.960837i	$-0.410622\pi$
$509$	12.5040			0.554231			0.277115	+	0.960837i	$0.410622\pi$
$510$		0			0
$511$			0.867257i			0.0383652i
$512$		0			0
$513$	10.6634	+	2.25704i	0.470801	+	0.0996508i
$514$		0			0
$515$			1.94089i			0.0855259i
$516$		0			0
$517$			2.73494i			0.120282i
$518$		0			0
$519$	−7.71547	−	15.8076i	−0.338671	−	0.693877i
$520$		0			0
$521$			23.6697i			1.03699i	0.855081	+	0.518495i	$0.173507\pi$
$521$			23.6697i			1.03699i	−0.855081	+	0.518495i	$0.826493\pi$
$522$		0			0
$523$	43.1655			1.88750			0.943749	−	0.330664i	$-0.107273\pi$
$523$	43.1655			1.88750			0.943749	+	0.330664i	$0.107273\pi$
$524$		0			0
$525$	−0.747995	+	0.365086i	−0.0326452	+	0.0159336i
$526$		0			0
$527$	52.6381			2.29295
$528$		0			0
$529$	11.1436			0.484504
$530$		0			0
$531$	5.46034	+	4.26107i	0.236959	+	0.184915i
$532$		0			0
$533$	7.22694			0.313033
$534$		0			0
$535$		−	7.45672i		−	0.322382i
$536$		0			0
$537$	−5.86671	+	2.86346i	−0.253167	+	0.123567i
$538$		0			0
$539$		−	32.0189i		−	1.37915i
$540$		0			0
$541$		−	6.99198i		−	0.300609i	−0.988640	−	0.150304i	$-0.951975\pi$
$541$		−	6.99198i		−	0.300609i	0.988640	−	0.150304i	$-0.0480254\pi$
$542$		0			0
$543$	13.1564	−	6.42144i	0.564594	−	0.275570i
$544$		0			0
$545$		−	8.11271i		−	0.347510i
$546$		0			0
$547$	15.8227			0.676529			0.338264	−	0.941051i	$-0.390160\pi$
$547$	15.8227			0.676529			0.338264	+	0.941051i	$0.390160\pi$
$548$		0			0
$549$	33.0999	+	25.8301i	1.41267	+	1.10240i
$550$		0			0
$551$	18.7872			0.800360
$552$		0			0
$553$	3.14758			0.133849
$554$		0			0
$555$	−5.71473	+	2.78928i	−0.242577	+	0.118398i
$556$		0			0
$557$	16.3032			0.690790			0.345395	−	0.938457i	$-0.387745\pi$
$557$	16.3032			0.690790			0.345395	+	0.938457i	$0.387745\pi$
$558$		0			0
$559$		−	13.8753i		−	0.586862i
$560$		0			0
$561$	−22.7254	−	46.5603i	−0.959467	−	1.96578i
$562$		0			0
$563$			15.4567i			0.651423i	0.945469	+	0.325712i	$0.105604\pi$
$563$			15.4567i			0.651423i	−0.945469	+	0.325712i	$0.894396\pi$
$564$		0			0
$565$		−	3.94928i		−	0.166147i
$566$		0			0
$567$	4.19528	−	1.05109i	0.176185	−	0.0441415i
$568$		0			0
$569$			2.64284i			0.110793i	0.998464	+	0.0553967i	$0.0176424\pi$
$569$			2.64284i			0.110793i	−0.998464	+	0.0553967i	$0.982358\pi$
$570$		0			0
$571$	5.43326			0.227375			0.113687	−	0.993517i	$-0.463734\pi$
$571$	5.43326			0.227375			0.113687	+	0.993517i	$0.463734\pi$
$572$		0			0
$573$	−16.3865	−	33.5731i	−0.684557	−	1.40253i
$574$		0			0
$575$	−5.84325			−0.243680
$576$		0			0
$577$	16.7651			0.697940			0.348970	−	0.937134i	$-0.386532\pi$
$577$	16.7651			0.697940			0.348970	+	0.937134i	$0.386532\pi$
$578$		0			0
$579$	11.9409	+	24.4647i	0.496247	+	1.01672i
$580$		0			0
$581$	−1.88655			−0.0782673
$582$		0			0
$583$			34.3650i			1.42325i
$584$		0			0
$585$	3.76907	+	2.94126i	0.155832	+	0.121606i
$586$		0			0
$587$		−	4.87014i		−	0.201012i	−0.994936	−	0.100506i	$-0.967954\pi$
$587$		−	4.87014i		−	0.201012i	0.994936	−	0.100506i	$-0.0320462\pi$
$588$		0			0
$589$		−	17.4603i		−	0.719441i
$590$		0			0
$591$	−16.7369	−	34.2908i	−0.688462	−	1.41054i
$592$		0			0
$593$			23.5492i			0.967049i	0.875331	+	0.483525i	$0.160644\pi$
$593$			23.5492i			0.967049i	−0.875331	+	0.483525i	$0.839356\pi$
$594$		0			0
$595$	−3.03890			−0.124583
$596$		0			0
$597$	−7.92220	+	3.86671i	−0.324234	+	0.158254i
$598$		0			0
$599$	39.1160			1.59824			0.799118	−	0.601175i	$-0.205300\pi$
$599$	39.1160			1.59824			0.799118	+	0.601175i	$0.205300\pi$
$600$		0			0
$601$	−33.9945			−1.38666			−0.693332	−	0.720618i	$-0.743858\pi$
$601$	−33.9945			−1.38666			−0.693332	+	0.720618i	$0.743858\pi$
$602$		0			0
$603$	−1.30433	+	1.67143i	−0.0531164	+	0.0680660i
$604$		0			0
$605$	11.3745			0.462440
$606$		0			0
$607$			38.7441i			1.57257i	0.617861	+	0.786287i	$0.287999\pi$
$607$			38.7441i			1.57257i	−0.617861	+	0.786287i	$0.712001\pi$
$608$		0			0
$609$	6.69929	−	3.26983i	0.271469	−	0.132500i
$610$		0			0
$611$		−	0.921421i		−	0.0372767i
$612$		0			0
$613$			21.4190i			0.865104i	0.901609	+	0.432552i	$0.142387\pi$
$613$			21.4190i			0.865104i	−0.901609	+	0.432552i	$0.857613\pi$
$614$		0			0
$615$	−7.05874	+	3.44527i	−0.284636	+	0.138927i
$616$		0			0
$617$		−	10.4794i		−	0.421885i	−0.977498	−	0.210942i	$-0.932347\pi$
$617$		−	10.4794i		−	0.421885i	0.977498	−	0.210942i	$-0.0676533\pi$
$618$		0			0
$619$	23.4943			0.944315			0.472158	−	0.881514i	$-0.343475\pi$
$619$	23.4943			0.944315			0.472158	+	0.881514i	$0.343475\pi$
$620$		0			0
$621$	29.7043	+	6.28728i	1.19199	+	0.252300i
$622$		0			0
$623$	4.45232			0.178379
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−15.4443	+	7.53814i	−0.616786	+	0.301044i
$628$		0			0
$629$	−23.2174			−0.925739
$630$		0			0
$631$		−	43.5809i		−	1.73493i	−0.497500	−	0.867464i	$-0.665749\pi$
$631$		−	43.5809i		−	1.73493i	0.497500	−	0.867464i	$-0.334251\pi$
$632$		0			0
$633$	13.8547	+	28.3858i	0.550675	+	1.12823i
$634$		0			0
$635$		−	3.32417i		−	0.131916i
$636$		0			0
$637$			10.7874i			0.427412i
$638$		0			0
$639$	−25.1167	+	32.1857i	−0.993601	+	1.27325i
$640$		0			0
$641$		−	6.29195i		−	0.248517i	−0.992250	−	0.124259i	$-0.960345\pi$
$641$		−	6.29195i		−	0.248517i	0.992250	−	0.124259i	$-0.0396552\pi$
$642$		0			0
$643$	−25.6583			−1.01186			−0.505932	−	0.862573i	$-0.668851\pi$
$643$	−25.6583			−1.01186			−0.505932	+	0.862573i	$0.668851\pi$
$644$		0			0
$645$	6.61471	+	13.5523i	0.260454	+	0.533623i
$646$		0			0
$647$	−6.22575			−0.244760			−0.122380	−	0.992483i	$-0.539053\pi$
$647$	−6.22575			−0.244760			−0.122380	+	0.992483i	$0.539053\pi$
$648$		0			0
$649$	−10.9207			−0.428674
$650$		0			0
$651$	−3.03890	−	6.22616i	−0.119104	−	0.244022i
$652$		0			0
$653$	−35.0382			−1.37115			−0.685575	−	0.728002i	$-0.740449\pi$
$653$	−35.0382			−1.37115			−0.685575	+	0.728002i	$0.740449\pi$
$654$		0			0
$655$		−	6.61269i		−	0.258379i
$656$		0			0
$657$	−3.33085	+	4.26832i	−0.129949	+	0.166523i
$658$		0			0
$659$			17.5166i			0.682350i	0.940000	+	0.341175i	$0.110825\pi$
$659$			17.5166i			0.682350i	−0.940000	+	0.341175i	$0.889175\pi$
$660$		0			0
$661$			14.5509i			0.565966i	0.959125	+	0.282983i	$0.0913239\pi$
$661$			14.5509i			0.565966i	−0.959125	+	0.282983i	$0.908676\pi$
$662$		0			0
$663$	7.65636	+	15.6865i	0.297348	+	0.609213i
$664$		0			0
$665$			1.00802i			0.0390894i
$666$		0			0
$667$	52.3341			2.02638
$668$		0			0
$669$	−37.6170	+	18.3603i	−1.45436	+	0.709851i
$670$		0			0
$671$	−66.1999			−2.55562
$672$		0			0
$673$	17.3429			0.668518			0.334259	−	0.942481i	$-0.391514\pi$
$673$	17.3429			0.668518			0.334259	+	0.942481i	$0.391514\pi$
$674$		0			0
$675$	−5.08353	−	1.07599i	−0.195665	−	0.0414149i
$676$		0			0
$677$	15.0382			0.577964			0.288982	−	0.957335i	$-0.406683\pi$
$677$	15.0382			0.577964			0.288982	+	0.957335i	$0.406683\pi$
$678$		0			0
$679$		−	0.570538i		−	0.0218952i
$680$		0			0
$681$	−22.3321	+	10.9000i	−0.855768	+	0.417688i
$682$		0			0
$683$		−	12.5566i		−	0.480467i	−0.970715	−	0.240233i	$-0.922776\pi$
$683$		−	12.5566i		−	0.480467i	0.970715	−	0.240233i	$-0.0772240\pi$
$684$		0			0
$685$			8.86346i			0.338655i
$686$		0			0
$687$	16.4523	−	8.03014i	0.627695	−	0.306369i
$688$		0			0
$689$		−	11.5778i		−	0.441080i
$690$		0			0
$691$	−34.2365			−1.30242			−0.651208	−	0.758899i	$-0.725738\pi$
$691$	−34.2365			−1.30242			−0.651208	+	0.758899i	$0.725738\pi$
$692$		0			0
$693$	−4.19528	+	5.37603i	−0.159366	+	0.204219i
$694$		0			0
$695$	20.6278			0.782456
$696$		0			0
$697$	−28.6777			−1.08625
$698$		0			0
$699$	13.1392	−	6.41305i	0.496970	−	0.242564i
$700$		0			0
$701$	−7.29521			−0.275536			−0.137768	−	0.990465i	$-0.543993\pi$
$701$	−7.29521			−0.275536			−0.137768	+	0.990465i	$0.543993\pi$
$702$		0			0
$703$			7.70134i			0.290462i
$704$		0			0
$705$	0.439266	+	0.899976i	0.0165437	+	0.0338951i
$706$		0			0
$707$		−	3.30548i		−	0.124315i
$708$		0			0
$709$		−	19.6396i		−	0.737580i	−0.929513	−	0.368790i	$-0.879772\pi$
$709$		−	19.6396i		−	0.737580i	0.929513	−	0.368790i	$-0.120228\pi$
$710$		0			0
$711$	15.4912	+	12.0888i	0.580966	+	0.453367i
$712$		0			0
$713$		−	48.6381i		−	1.82151i
$714$		0			0
$715$	−7.53814			−0.281911
$716$		0			0
$717$	8.96834	+	18.3745i	0.334929	+	0.686209i
$718$		0			0
$719$	45.1541			1.68396			0.841982	−	0.539506i	$-0.181389\pi$
$719$	45.1541			1.68396			0.841982	+	0.539506i	$0.181389\pi$
$720$		0			0
$721$	0.932695			0.0347354
$722$		0			0
$723$	−21.0305	−	43.0877i	−0.782133	−	1.60245i
$724$		0			0
$725$	−8.95633			−0.332630
$726$		0			0
$727$			5.93365i			0.220067i	0.993928	+	0.110033i	$0.0350958\pi$
$727$			5.93365i			0.220067i	−0.993928	+	0.110033i	$0.964904\pi$
$728$		0			0
$729$	24.6845	+	10.9397i	0.914240	+	0.405172i
$730$		0			0
$731$			55.0595i			2.03645i
$732$		0			0
$733$			26.0460i			0.962029i	0.876713	+	0.481015i	$0.159732\pi$
$733$			26.0460i			0.962029i	−0.876713	+	0.481015i	$0.840268\pi$
$734$		0			0
$735$	−5.14263	−	10.5363i	−0.189689	−	0.388639i
$736$		0			0
$737$		−	3.34286i		−	0.123136i
$738$		0			0
$739$	−1.11976			−0.0411912			−0.0205956	−	0.999788i	$-0.506556\pi$
$739$	−1.11976			−0.0411912			−0.0205956	+	0.999788i	$0.506556\pi$
$740$		0			0
$741$	5.20330	−	2.53966i	0.191148	−	0.0932966i
$742$		0			0
$743$	13.7250			0.503523			0.251761	−	0.967789i	$-0.418990\pi$
$743$	13.7250			0.503523			0.251761	+	0.967789i	$0.418990\pi$
$744$		0			0
$745$	−18.8429			−0.690350
$746$		0			0
$747$	−9.28490	−	7.24563i	−0.339717	−	0.265104i
$748$		0			0
$749$	−3.58332			−0.130932
$750$		0			0
$751$		−	41.2826i		−	1.50642i	−0.657778	−	0.753212i	$-0.728503\pi$
$751$		−	41.2826i		−	1.50642i	0.657778	−	0.753212i	$-0.271497\pi$
$752$		0			0
$753$	7.36270	−	3.59363i	0.268312	−	0.130959i
$754$		0			0
$755$		−	15.2143i		−	0.553707i
$756$		0			0
$757$		−	46.4270i		−	1.68742i	−0.536801	−	0.843709i	$-0.680368\pi$
$757$		−	46.4270i		−	1.68742i	0.536801	−	0.843709i	$-0.319632\pi$
$758$		0			0
$759$	−43.0221	+	20.9985i	−1.56160	+	0.762197i
$760$		0			0
$761$		−	49.4413i		−	1.79224i	−0.443807	−	0.896122i	$-0.646372\pi$
$761$		−	49.4413i		−	1.79224i	0.443807	−	0.896122i	$-0.353628\pi$
$762$		0			0
$763$	−3.89856			−0.141137
$764$		0			0
$765$	−14.9563	−	11.6714i	−0.540747	−	0.421982i
$766$		0			0
$767$	3.67926			0.132850
$768$		0			0
$769$	−11.3047			−0.407659			−0.203830	−	0.979006i	$-0.565339\pi$
$769$	−11.3047			−0.407659			−0.203830	+	0.979006i	$0.565339\pi$
$770$		0			0
$771$	−35.7559	+	17.4520i	−1.28772	+	0.628517i
$772$		0			0
$773$	−4.34438			−0.156256			−0.0781282	−	0.996943i	$-0.524894\pi$
$773$	−4.34438			−0.156256			−0.0781282	+	0.996943i	$0.524894\pi$
$774$		0			0
$775$			8.32380i			0.299000i
$776$		0			0
$777$	1.34039	+	2.74621i	0.0480861	+	0.0985198i
$778$		0			0
$779$			9.51257i			0.340823i
$780$		0			0
$781$		−	64.3715i		−	2.30339i
$782$		0			0
$783$	45.5298	+	9.63693i	1.62710	+	0.344396i
$784$		0			0
$785$			16.9762i			0.605906i
$786$		0			0
$787$	−5.58121			−0.198949			−0.0994743	−	0.995040i	$-0.531716\pi$
$787$	−5.58121			−0.198949			−0.0994743	+	0.995040i	$0.531716\pi$
$788$		0			0
$789$	−14.9036	−	30.5348i	−0.530582	−	1.08707i
$790$		0			0
$791$	−1.89783			−0.0674789
$792$		0			0
$793$	22.3032			0.792011
$794$		0			0
$795$	5.51945	+	11.3084i	0.195755	+	0.401066i
$796$		0			0
$797$	−20.7159			−0.733794			−0.366897	−	0.930262i	$-0.619580\pi$
$797$	−20.7159			−0.733794			−0.366897	+	0.930262i	$0.619580\pi$
$798$		0			0
$799$			3.65636i			0.129353i
$800$		0			0
$801$	21.9127	+	17.0999i	0.774246	+	0.604196i
$802$		0			0
$803$		−	8.53663i		−	0.301251i
$804$		0			0
$805$			2.80797i			0.0989680i
$806$		0			0
$807$	−6.43363	−	13.1813i	−0.226474	−	0.464005i
$808$		0			0
$809$		−	12.2570i		−	0.430935i	−0.976511	−	0.215467i	$-0.930873\pi$
$809$		−	12.2570i		−	0.430935i	0.976511	−	0.215467i	$-0.0691275\pi$
$810$		0			0
$811$	22.6285			0.794594			0.397297	−	0.917690i	$-0.369948\pi$
$811$	22.6285			0.794594			0.397297	+	0.917690i	$0.369948\pi$
$812$		0			0
$813$	24.0818	−	11.7540i	0.844587	−	0.412231i
$814$		0			0
$815$	−7.89397			−0.276514
$816$		0			0
$817$	18.2635			0.638961
$818$		0			0
$819$	1.41342	−	1.81123i	0.0493889	−	0.0632894i
$820$		0			0
$821$	−25.0618			−0.874663			−0.437331	−	0.899300i	$-0.644076\pi$
$821$	−25.0618			−0.874663			−0.437331	+	0.899300i	$0.644076\pi$
$822$		0			0
$823$		−	33.3139i		−	1.16125i	−0.814171	−	0.580625i	$-0.802808\pi$
$823$		−	33.3139i		−	1.16125i	0.814171	−	0.580625i	$-0.197192\pi$
$824$		0			0
$825$	7.36270	−	3.59363i	0.256336	−	0.125114i
$826$		0			0
$827$			17.9098i			0.622784i	0.950282	+	0.311392i	$0.100795\pi$
$827$			17.9098i			0.622784i	−0.950282	+	0.311392i	$0.899205\pi$
$828$		0			0
$829$		−	16.1127i		−	0.559618i	−0.960056	−	0.279809i	$-0.909729\pi$
$829$		−	16.1127i		−	0.559618i	0.960056	−	0.279809i	$-0.0902711\pi$
$830$		0			0
$831$	−38.4615	+	18.7725i	−1.33421	+	0.651211i
$832$		0			0
$833$		−	42.8063i		−	1.48315i
$834$		0			0
$835$	13.0305			0.450939
$836$		0			0
$837$	8.95633	−	42.3143i	0.309576	−	1.46260i
$838$		0			0
$839$	−12.8032			−0.442016			−0.221008	−	0.975272i	$-0.570935\pi$
$839$	−12.8032			−0.442016			−0.221008	+	0.975272i	$0.570935\pi$
$840$		0			0
$841$	51.2159			1.76607
$842$		0			0
$843$	42.0317	−	20.5151i	1.44765	−	0.706576i
$844$		0			0
$845$	−10.4603			−0.359847
$846$		0			0
$847$		−	5.46602i		−	0.187815i
$848$		0			0
$849$	−15.8798	−	32.5348i	−0.544992	−	1.11659i
$850$		0			0
$851$			21.4531i			0.735403i
$852$		0			0
$853$		−	32.9365i		−	1.12772i	−0.825869	−	0.563862i	$-0.809315\pi$
$853$		−	32.9365i		−	1.12772i	0.825869	−	0.563862i	$-0.190685\pi$
$854$		0			0
$855$	−3.87148	+	4.96110i	−0.132402	+	0.169666i
$856$		0			0
$857$		−	10.4889i		−	0.358295i	−0.983822	−	0.179148i	$-0.942666\pi$
$857$		−	10.4889i		−	0.358295i	0.983822	−	0.179148i	$-0.0573340\pi$
$858$		0			0
$859$	11.9405			0.407404			0.203702	−	0.979033i	$-0.434703\pi$
$859$	11.9405			0.407404			0.203702	+	0.979033i	$0.434703\pi$
$860$		0			0
$861$	1.65562	+	3.39207i	0.0564235	+	0.115602i
$862$		0			0
$863$	−38.1869			−1.29990			−0.649948	−	0.759978i	$-0.725210\pi$
$863$	−38.1869			−1.29990			−0.649948	+	0.759978i	$0.725210\pi$
$864$		0			0
$865$	10.1556			0.345301
$866$		0			0
$867$	−17.4664	−	35.7856i	−0.593191	−	1.21534i
$868$		0			0
$869$	−30.9824			−1.05101
$870$		0			0
$871$			1.12624i			0.0381610i
$872$		0			0
$873$	2.19125	−	2.80797i	0.0741626	−	0.0950355i
$874$		0			0
$875$		−	0.480550i		−	0.0162455i
$876$		0			0
$877$		−	3.83657i		−	0.129552i	−0.997900	−	0.0647758i	$-0.979367\pi$
$877$		−	3.83657i		−	0.129552i	0.997900	−	0.0647758i	$-0.0206332\pi$
$878$		0			0
$879$	0.284533	+	0.582957i	0.00959706	+	0.0196626i
$880$		0			0
$881$		−	15.0650i		−	0.507553i	−0.967263	−	0.253777i	$-0.918327\pi$
$881$		−	15.0650i		−	0.507553i	0.967263	−	0.253777i	$-0.0816728\pi$
$882$		0			0
$883$	39.9878			1.34570			0.672849	−	0.739780i	$-0.265071\pi$
$883$	39.9878			1.34570			0.672849	+	0.739780i	$0.265071\pi$
$884$		0			0
$885$	−3.59363	+	1.75400i	−0.120799	+	0.0589601i
$886$		0			0
$887$	−43.1289			−1.44813			−0.724064	−	0.689733i	$-0.757728\pi$
$887$	−43.1289			−1.44813			−0.724064	+	0.689733i	$0.757728\pi$
$888$		0			0
$889$	−1.59743			−0.0535760
$890$		0			0
$891$	−41.2952	+	10.3461i	−1.38344	+	0.346608i
$892$		0			0
$893$	1.21283			0.0405860
$894$		0			0
$895$		−	3.76907i		−	0.125986i
$896$		0			0
$897$	14.4945	−	7.07455i	0.483956	−	0.236212i
$898$		0			0
$899$		−	74.5507i		−	2.48641i
$900$		0			0
$901$			45.9428i			1.53058i
$902$		0			0
$903$	6.51258	−	3.17870i	0.216725	−	0.105780i
$904$		0			0
$905$			8.45232i			0.280965i
$906$		0			0
$907$	0.316149			0.0104976			0.00524878	−	0.999986i	$-0.498329\pi$
$907$	0.316149			0.0104976			0.00524878	+	0.999986i	$0.498329\pi$
$908$		0			0
$909$	12.6953	−	16.2683i	0.421075	−	0.539586i
$910$		0			0
$911$	−9.06254			−0.300255			−0.150128	−	0.988667i	$-0.547968\pi$
$911$	−9.06254			−0.300255			−0.150128	+	0.988667i	$0.547968\pi$
$912$		0			0
$913$	18.5698			0.614571
$914$		0			0
$915$	−21.7841	+	10.6325i	−0.720162	+	0.351501i
$916$		0			0
$917$	−3.17773			−0.104938
$918$		0			0
$919$		−	19.3326i		−	0.637722i	−0.947801	−	0.318861i	$-0.896700\pi$
$919$		−	19.3326i		−	0.637722i	0.947801	−	0.318861i	$-0.103300\pi$
$920$		0			0
$921$	−10.9114	−	22.3556i	−0.359544	−	0.736640i
$922$		0			0
$923$			21.6872i			0.713844i
$924$		0			0
$925$		−	3.67143i		−	0.120716i
$926$		0			0
$927$	4.59038	+	3.58218i	0.150768	+	0.117654i
$928$		0			0
$929$			15.8160i			0.518906i	0.965756	+	0.259453i	$0.0835422\pi$
$929$			15.8160i			0.518906i	−0.965756	+	0.259453i	$0.916458\pi$
$930$		0			0
$931$	−14.1991			−0.465356
$932$		0			0
$933$	16.8785	+	34.5811i	0.552578	+	1.13213i
$934$		0			0
$935$	29.9127			0.978249
$936$		0			0
$937$	16.8810			0.551478			0.275739	−	0.961232i	$-0.411077\pi$
$937$	16.8810			0.551478			0.275739	+	0.961232i	$0.411077\pi$
$938$		0			0
$939$	25.5797	+	52.4083i	0.834763	+	1.71028i
$940$		0			0
$941$	−20.6596			−0.673484			−0.336742	−	0.941597i	$-0.609325\pi$
$941$	−20.6596			−0.673484			−0.336742	+	0.941597i	$0.609325\pi$
$942$		0			0
$943$			26.4985i			0.862910i
$944$		0			0
$945$	−0.517067	+	2.44289i	−0.0168202	+	0.0794671i
$946$		0			0
$947$		−	41.1028i		−	1.33566i	−0.744313	−	0.667831i	$-0.767223\pi$
$947$		−	41.1028i		−	1.33566i	0.744313	−	0.667831i	$-0.232777\pi$
$948$		0			0
$949$			2.87605i			0.0933607i
$950$		0			0
$951$	22.6362	+	46.3774i	0.734028	+	1.50389i
$952$		0			0
$953$		−	10.4252i		−	0.337707i	−0.985641	−	0.168853i	$-0.945994\pi$
$953$		−	10.4252i		−	0.337707i	0.985641	−	0.168853i	$-0.0540064\pi$
$954$		0			0
$955$	21.5690			0.697957
$956$		0			0
$957$	−65.9428	+	32.1857i	−2.13163	+	1.04042i
$958$		0			0
$959$	4.25933			0.137541
$960$		0			0
$961$	−38.2857			−1.23502
$962$		0			0
$963$	−17.6358	−	13.7624i	−0.568305	−	0.443487i
$964$		0			0
$965$	−15.7174			−0.505960
$966$		0			0
$967$			30.7273i			0.988124i	0.869427	+	0.494062i	$0.164488\pi$
$967$			30.7273i			0.988124i	−0.869427	+	0.494062i	$0.835512\pi$
$968$		0			0
$969$	−20.6476	+	10.0778i	−0.663297	+	0.323746i
$970$		0			0
$971$		−	11.1047i		−	0.356367i	−0.983997	−	0.178183i	$-0.942978\pi$
$971$		−	11.1047i		−	0.356367i	0.983997	−	0.178183i	$-0.0570220\pi$
$972$		0			0
$973$		−	9.91266i		−	0.317785i
$974$		0			0
$975$	−2.48055	+	1.21072i	−0.0794412	+	0.0387741i
$976$		0			0
$977$		−	34.3143i		−	1.09781i	−0.835885	−	0.548905i	$-0.815045\pi$
$977$		−	34.3143i		−	1.09781i	0.835885	−	0.548905i	$-0.184955\pi$
$978$		0			0
$979$	−43.8253			−1.40066
$980$		0			0
$981$	−19.1873	−	14.9731i	−0.612602	−	0.478055i
$982$		0			0
$983$	−6.72729			−0.214567			−0.107284	−	0.994228i	$-0.534215\pi$
$983$	−6.72729			−0.214567			−0.107284	+	0.994228i	$0.534215\pi$
$984$		0			0
$985$	22.0301			0.701939
$986$		0			0
$987$	0.432483	−	0.211089i	0.0137661	−	0.00671903i
$988$		0			0
$989$	50.8755			1.61775
$990$		0			0
$991$		−	43.0191i		−	1.36655i	−0.730163	−	0.683273i	$-0.760556\pi$
$991$		−	43.0191i		−	1.36655i	0.730163	−	0.683273i	$-0.239444\pi$
$992$		0			0
$993$	−18.0103	−	36.8999i	−0.571540	−	1.17098i
$994$		0			0
$995$		−	5.08962i		−	0.161352i
$996$		0			0
$997$			24.9365i			0.789747i	0.918736	+	0.394873i	$0.129212\pi$
$997$			24.9365i			0.789747i	−0.918736	+	0.394873i	$0.870788\pi$
$998$		0			0
$999$	−3.95043	+	18.6638i	−0.124986	+	0.590497i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.b.g.191.6	yes	8
3.2	odd	2		1920.2.b.h.191.5	yes	8
4.3	odd	2		1920.2.b.a.191.3	✓	8
8.3	odd	2		1920.2.b.h.191.6	yes	8
8.5	even	2		1920.2.b.b.191.3	yes	8
12.11	even	2		1920.2.b.b.191.4	yes	8
24.5	odd	2		1920.2.b.a.191.4	yes	8
24.11	even	2	inner	1920.2.b.g.191.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.b.a.191.3	✓	8	4.3	odd	2
1920.2.b.a.191.4	yes	8	24.5	odd	2
1920.2.b.b.191.3	yes	8	8.5	even	2
1920.2.b.b.191.4	yes	8	12.11	even	2
1920.2.b.g.191.5	yes	8	24.11	even	2	inner
1920.2.b.g.191.6	yes	8	1.1	even	1	trivial
1920.2.b.h.191.5	yes	8	3.2	odd	2
1920.2.b.h.191.6	yes	8	8.3	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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.759725 + 1.55654i) q^{3} -1.00000 q^{5} +0.480550i q^{7} +(-1.84564 + 2.36509i) q^{9} -4.73017i q^{11} +1.59363i q^{13} +(-0.759725 - 1.55654i) q^{15} -6.32380i q^{17} -2.09764 q^{19} +(-0.747995 + 0.365086i) q^{21} -5.84325 q^{23} +1.00000 q^{25} +(-5.08353 - 1.07599i) q^{27} -8.95633 q^{29} +8.32380i q^{31} +(7.36270 - 3.59363i) q^{33} -0.480550i q^{35} -3.67143i q^{37} +(-2.48055 + 1.21072i) q^{39} -4.53489i q^{41} -8.70671 q^{43} +(1.84564 - 2.36509i) q^{45} -0.578190 q^{47} +6.76907 q^{49} +(9.84325 - 4.80435i) q^{51} -7.26506 q^{53} +4.73017i q^{55} +(-1.59363 - 3.26506i) q^{57} -2.30873i q^{59} -13.9952i q^{61} +(-1.13654 - 0.886919i) q^{63} -1.59363i q^{65} +0.706711 q^{67} +(-4.43927 - 9.09526i) q^{69} +13.6087 q^{71} +1.80472 q^{73} +(0.759725 + 1.55654i) q^{75} +2.27308 q^{77} -6.54996i q^{79} +(-2.18726 - 8.73017i) q^{81} +3.92582i q^{83} +6.32380i q^{85} +(-6.80435 - 13.9409i) q^{87} -9.26506i q^{89} -0.765818 q^{91} +(-12.9563 + 6.32380i) q^{93} +2.09764 q^{95} -1.18726 q^{97} +(11.1873 + 8.73017i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 8 q^{5} - 4 q^{9} - 2 q^{15} - 8 q^{19} - 4 q^{21} + 12 q^{23} + 8 q^{25} + 14 q^{27} + 8 q^{29} - 8 q^{33} - 28 q^{39} - 36 q^{43} + 4 q^{45} - 4 q^{47} + 20 q^{51} + 16 q^{63} - 28 q^{67}+ \cdots + 64 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 - 8 * q^5 - 4 * q^9 - 2 * q^15 - 8 * q^19 - 4 * q^21 + 12 * q^23 + 8 * q^25 + 14 * q^27 + 8 * q^29 - 8 * q^33 - 28 * q^39 - 36 * q^43 + 4 * q^45 - 4 * q^47 + 20 * q^51 + 16 * q^63 - 28 * q^67 - 12 * q^69 + 24 * q^71 + 32 * q^73 + 2 * q^75 - 32 * q^77 + 8 * q^81 - 12 * q^87 - 40 * q^91 - 24 * q^93 + 8 * q^95 + 16 * q^97 + 64 * q^99

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