LMFDB - Embedded newform 960.2.h.g.191.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(191,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("960.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.h (of order $2$, degree $1$, not 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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 $ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.4
Root			$0.599676 - 1.28078i$ of defining polynomial
Character	$\chi$	$=$	960.191
Dual form			960.2.h.g.191.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.468213 + 1.66757i) q^{3} -1.00000i q^{5} -0.936426i q^{7} +(-2.56155 - 1.56155i) q^{9} +O(q^{10})$	\(q+(-0.468213 + 1.66757i) q^{3} -1.00000i q^{5} -0.936426i q^{7} +(-2.56155 - 1.56155i) q^{9} +4.27156 q^{11} -3.12311 q^{13} +(1.66757 + 0.468213i) q^{15} -2.00000i q^{17} -4.27156i q^{19} +(1.56155 + 0.438447i) q^{21} +7.60669 q^{23} -1.00000 q^{25} +(3.80335 - 3.54042i) q^{27} -5.12311i q^{29} +2.39871i q^{31} +(-2.00000 + 7.12311i) q^{33} -0.936426 q^{35} +3.12311 q^{37} +(1.46228 - 5.20798i) q^{39} -7.12311i q^{41} +1.46228i q^{43} +(-1.56155 + 2.56155i) q^{45} -0.936426 q^{47} +6.12311 q^{49} +(3.33513 + 0.936426i) q^{51} +4.24621i q^{53} -4.27156i q^{55} +(7.12311 + 2.00000i) q^{57} +7.19612 q^{59} +5.12311 q^{61} +(-1.46228 + 2.39871i) q^{63} +3.12311i q^{65} +5.20798i q^{67} +(-3.56155 + 12.6847i) q^{69} +6.67026 q^{71} -8.24621 q^{73} +(0.468213 - 1.66757i) q^{75} -4.00000i q^{77} -9.06897i q^{79} +(4.12311 + 8.00000i) q^{81} +4.68213 q^{83} -2.00000 q^{85} +(8.54312 + 2.39871i) q^{87} +6.24621i q^{89} +2.92456i q^{91} +(-4.00000 - 1.12311i) q^{93} -4.27156 q^{95} -6.00000 q^{97} +(-10.9418 - 6.67026i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 8 q^{13} - 4 q^{21} - 8 q^{25} - 16 q^{33} - 8 q^{37} + 4 q^{45} + 16 q^{49} + 24 q^{57} + 8 q^{61} - 12 q^{69} - 16 q^{85} - 32 q^{93} - 48 q^{97}+O(q^{100}) $ 8 * q - 4 * q^9 + 8 * q^13 - 4 * q^21 - 8 * q^25 - 16 * q^33 - 8 * q^37 + 4 * q^45 + 16 * q^49 + 24 * q^57 + 8 * q^61 - 12 * q^69 - 16 * q^85 - 32 * q^93 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−0.468213	+	1.66757i	−0.270323	+	0.962770i
$3$	−0.468213	+	1.66757i	−0.270323	+	0.962770i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	0.936426i		−	0.353936i	−0.984217	−	0.176968i	$-0.943371\pi$
$7$		−	0.936426i		−	0.353936i	0.984217	−	0.176968i	$-0.0566289\pi$
$8$		0			0
$9$	−2.56155	−	1.56155i	−0.853851	−	0.520518i
$10$		0			0
$11$	4.27156			1.28792			0.643962	−	0.765058i	$-0.277290\pi$
$11$	4.27156			1.28792			0.643962	+	0.765058i	$0.277290\pi$
$12$		0			0
$13$	−3.12311			−0.866194			−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311			−0.866194			−0.433097	+	0.901347i	$0.642579\pi$
$14$		0			0
$15$	1.66757	+	0.468213i	0.430564	+	0.120892i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		−	4.27156i		−	0.979963i	−0.871733	−	0.489981i	$-0.837004\pi$
$19$		−	4.27156i		−	0.979963i	0.871733	−	0.489981i	$-0.162996\pi$
$20$		0			0
$21$	1.56155	+	0.438447i	0.340759	+	0.0956770i
$22$		0			0
$23$	7.60669			1.58610			0.793052	−	0.609154i	$-0.208491\pi$
$23$	7.60669			1.58610			0.793052	+	0.609154i	$0.208491\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	3.80335	−	3.54042i	0.731954	−	0.681354i
$28$		0			0
$29$		−	5.12311i		−	0.951337i	−0.879625	−	0.475668i	$-0.842206\pi$
$29$		−	5.12311i		−	0.951337i	0.879625	−	0.475668i	$-0.157794\pi$
$30$		0			0
$31$			2.39871i			0.430820i	0.976524	+	0.215410i	$0.0691088\pi$
$31$			2.39871i			0.430820i	−0.976524	+	0.215410i	$0.930891\pi$
$32$		0			0
$33$	−2.00000	+	7.12311i	−0.348155	+	1.23997i
$34$		0			0
$35$	−0.936426			−0.158285
$36$		0			0
$37$	3.12311			0.513435			0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311			0.513435			0.256718	+	0.966486i	$0.417359\pi$
$38$		0			0
$39$	1.46228	−	5.20798i	0.234152	−	0.833945i
$40$		0			0
$41$		−	7.12311i		−	1.11244i	−0.831034	−	0.556221i	$-0.812251\pi$
$41$		−	7.12311i		−	1.11244i	0.831034	−	0.556221i	$-0.187749\pi$
$42$		0			0
$43$			1.46228i			0.222995i	0.993765	+	0.111498i	$0.0355648\pi$
$43$			1.46228i			0.222995i	−0.993765	+	0.111498i	$0.964435\pi$
$44$		0			0
$45$	−1.56155	+	2.56155i	−0.232783	+	0.381854i
$46$		0			0
$47$	−0.936426			−0.136592			−0.0682959	−	0.997665i	$-0.521756\pi$
$47$	−0.936426			−0.136592			−0.0682959	+	0.997665i	$0.521756\pi$
$48$		0			0
$49$	6.12311			0.874729
$50$		0			0
$51$	3.33513	+	0.936426i	0.467012	+	0.131126i
$52$		0			0
$53$			4.24621i			0.583262i	0.956531	+	0.291631i	$0.0941979\pi$
$53$			4.24621i			0.583262i	−0.956531	+	0.291631i	$0.905802\pi$
$54$		0			0
$55$		−	4.27156i		−	0.575977i
$56$		0			0
$57$	7.12311	+	2.00000i	0.943478	+	0.264906i
$58$		0			0
$59$	7.19612			0.936855			0.468427	−	0.883502i	$-0.344821\pi$
$59$	7.19612			0.936855			0.468427	+	0.883502i	$0.344821\pi$
$60$		0			0
$61$	5.12311			0.655946			0.327973	−	0.944687i	$-0.393634\pi$
$61$	5.12311			0.655946			0.327973	+	0.944687i	$0.393634\pi$
$62$		0			0
$63$	−1.46228	+	2.39871i	−0.184230	+	0.302209i
$64$		0			0
$65$			3.12311i			0.387374i
$66$		0			0
$67$			5.20798i			0.636257i	0.948048	+	0.318128i	$0.103054\pi$
$67$			5.20798i			0.636257i	−0.948048	+	0.318128i	$0.896946\pi$
$68$		0			0
$69$	−3.56155	+	12.6847i	−0.428761	+	1.52705i
$70$		0			0
$71$	6.67026			0.791615			0.395807	−	0.918334i	$-0.370465\pi$
$71$	6.67026			0.791615			0.395807	+	0.918334i	$0.370465\pi$
$72$		0			0
$73$	−8.24621			−0.965146			−0.482573	−	0.875856i	$-0.660298\pi$
$73$	−8.24621			−0.965146			−0.482573	+	0.875856i	$0.660298\pi$
$74$		0			0
$75$	0.468213	−	1.66757i	0.0540646	−	0.192554i
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$		−	9.06897i		−	1.02034i	−0.860074	−	0.510169i	$-0.829583\pi$
$79$		−	9.06897i		−	1.02034i	0.860074	−	0.510169i	$-0.170417\pi$
$80$		0			0
$81$	4.12311	+	8.00000i	0.458123	+	0.888889i
$82$		0			0
$83$	4.68213			0.513931			0.256965	−	0.966421i	$-0.417277\pi$
$83$	4.68213			0.513931			0.256965	+	0.966421i	$0.417277\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	8.54312	+	2.39871i	0.915918	+	0.257168i
$88$		0			0
$89$			6.24621i			0.662097i	0.943614	+	0.331049i	$0.107402\pi$
$89$			6.24621i			0.662097i	−0.943614	+	0.331049i	$0.892598\pi$
$90$		0			0
$91$			2.92456i			0.306577i
$92$		0			0
$93$	−4.00000	−	1.12311i	−0.414781	−	0.116461i
$94$		0			0
$95$	−4.27156			−0.438253
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$	−10.9418	−	6.67026i	−1.09969	−	0.670387i
$100$		0			0
$101$		−	9.12311i		−	0.907783i	−0.891057	−	0.453891i	$-0.850035\pi$
$101$		−	9.12311i		−	0.907783i	0.891057	−	0.453891i	$-0.149965\pi$
$102$		0			0
$103$			12.4041i			1.22221i	0.791549	+	0.611106i	$0.209275\pi$
$103$			12.4041i			1.22221i	−0.791549	+	0.611106i	$0.790725\pi$
$104$		0			0
$105$	0.438447	−	1.56155i	0.0427881	−	0.152392i
$106$		0			0
$107$	−0.936426			−0.0905278			−0.0452639	−	0.998975i	$-0.514413\pi$
$107$	−0.936426			−0.0905278			−0.0452639	+	0.998975i	$0.514413\pi$
$108$		0			0
$109$	9.12311			0.873835			0.436918	−	0.899502i	$-0.356070\pi$
$109$	9.12311			0.873835			0.436918	+	0.899502i	$0.356070\pi$
$110$		0			0
$111$	−1.46228	+	5.20798i	−0.138793	+	0.494320i
$112$		0			0
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$		−	7.60669i		−	0.709328i
$116$		0			0
$117$	8.00000	+	4.87689i	0.739600	+	0.450869i
$118$		0			0
$119$	−1.87285			−0.171684
$120$		0			0
$121$	7.24621			0.658746
$122$		0			0
$123$	11.8782	+	3.33513i	1.07103	+	0.300719i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			4.68213i			0.415472i	0.978185	+	0.207736i	$0.0666095\pi$
$127$			4.68213i			0.415472i	−0.978185	+	0.207736i	$0.933391\pi$
$128$		0			0
$129$	−2.43845	−	0.684658i	−0.214693	−	0.0602808i
$130$		0			0
$131$	−17.6121			−1.53878			−0.769388	−	0.638782i	$-0.779438\pi$
$131$	−17.6121			−1.53878			−0.769388	+	0.638782i	$0.779438\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$		0			0
$135$	−3.54042	−	3.80335i	−0.304711	−	0.327340i
$136$		0			0
$137$		−	8.24621i		−	0.704521i	−0.935902	−	0.352261i	$-0.885413\pi$
$137$		−	8.24621i		−	0.704521i	0.935902	−	0.352261i	$-0.114587\pi$
$138$		0			0
$139$		−	13.8664i		−	1.17613i	−0.808813	−	0.588066i	$-0.799890\pi$
$139$		−	13.8664i		−	1.17613i	0.808813	−	0.588066i	$-0.200110\pi$
$140$		0			0
$141$	0.438447	−	1.56155i	0.0369239	−	0.131506i
$142$		0			0
$143$	−13.3405			−1.11559
$144$		0			0
$145$	−5.12311			−0.425451
$146$		0			0
$147$	−2.86692	+	10.2107i	−0.236459	+	0.842163i
$148$		0			0
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$		−	6.14441i		−	0.500025i	−0.968243	−	0.250013i	$-0.919565\pi$
$151$		−	6.14441i		−	0.500025i	0.968243	−	0.250013i	$-0.0804347\pi$
$152$		0			0
$153$	−3.12311	+	5.12311i	−0.252488	+	0.414179i
$154$		0			0
$155$	2.39871			0.192669
$156$		0			0
$157$	−21.3693			−1.70546			−0.852729	−	0.522354i	$-0.825054\pi$
$157$	−21.3693			−1.70546			−0.852729	+	0.522354i	$0.825054\pi$
$158$		0			0
$159$	−7.08084	−	1.98813i	−0.561547	−	0.157669i
$160$		0			0
$161$		−	7.12311i		−	0.561379i
$162$		0			0
$163$			24.1671i			1.89291i	0.322834	+	0.946456i	$0.395364\pi$
$163$			24.1671i			1.89291i	−0.322834	+	0.946456i	$0.604636\pi$
$164$		0			0
$165$	7.12311	+	2.00000i	0.554533	+	0.155700i
$166$		0			0
$167$	2.80928			0.217389			0.108694	−	0.994075i	$-0.465333\pi$
$167$	2.80928			0.217389			0.108694	+	0.994075i	$0.465333\pi$
$168$		0			0
$169$	−3.24621			−0.249709
$170$		0			0
$171$	−6.67026	+	10.9418i	−0.510088	+	0.836742i
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$			0.936426i			0.0707872i
$176$		0			0
$177$	−3.36932	+	12.0000i	−0.253253	+	0.901975i
$178$		0			0
$179$	−14.6875			−1.09780			−0.548899	−	0.835889i	$-0.684953\pi$
$179$	−14.6875			−1.09780			−0.548899	+	0.835889i	$0.684953\pi$
$180$		0			0
$181$	−4.24621			−0.315618			−0.157809	−	0.987470i	$-0.550443\pi$
$181$	−4.24621			−0.315618			−0.157809	+	0.987470i	$0.550443\pi$
$182$		0			0
$183$	−2.39871	+	8.54312i	−0.177317	+	0.631525i
$184$		0			0
$185$		−	3.12311i		−	0.229615i
$186$		0			0
$187$		−	8.54312i		−	0.624735i
$188$		0			0
$189$	−3.31534	−	3.56155i	−0.241156	−	0.259065i
$190$		0			0
$191$	−7.72197			−0.558742			−0.279371	−	0.960183i	$-0.590126\pi$
$191$	−7.72197			−0.558742			−0.279371	+	0.960183i	$0.590126\pi$
$192$		0			0
$193$	16.2462			1.16943			0.584714	−	0.811240i	$-0.301207\pi$
$193$	16.2462			1.16943			0.584714	+	0.811240i	$0.301207\pi$
$194$		0			0
$195$	−5.20798	−	1.46228i	−0.372952	−	0.104716i
$196$		0			0
$197$		−	12.2462i		−	0.872506i	−0.899824	−	0.436253i	$-0.856305\pi$
$197$		−	12.2462i		−	0.872506i	0.899824	−	0.436253i	$-0.143695\pi$
$198$		0			0
$199$		−	17.6121i		−	1.24849i	−0.781230	−	0.624244i	$-0.785407\pi$
$199$		−	17.6121i		−	1.24849i	0.781230	−	0.624244i	$-0.214593\pi$
$200$		0			0
$201$	−8.68466	−	2.43845i	−0.612569	−	0.171995i
$202$		0			0
$203$	−4.79741			−0.336712
$204$		0			0
$205$	−7.12311			−0.497499
$206$		0			0
$207$	−19.4849	−	11.8782i	−1.35430	−	0.825595i
$208$		0			0
$209$		−	18.2462i		−	1.26212i
$210$		0			0
$211$			1.34700i			0.0927313i	0.998925	+	0.0463656i	$0.0147639\pi$
$211$			1.34700i			0.0927313i	−0.998925	+	0.0463656i	$0.985236\pi$
$212$		0			0
$213$	−3.12311	+	11.1231i	−0.213992	+	0.762143i
$214$		0			0
$215$	1.46228			0.0997266
$216$		0			0
$217$	2.24621			0.152483
$218$		0			0
$219$	3.86098	−	13.7511i	0.260901	−	0.929213i
$220$		0			0
$221$			6.24621i			0.420166i
$222$		0			0
$223$			18.0227i			1.20689i	0.797406	+	0.603443i	$0.206205\pi$
$223$			18.0227i			1.20689i	−0.797406	+	0.603443i	$0.793795\pi$
$224$		0			0
$225$	2.56155	+	1.56155i	0.170770	+	0.104104i
$226$		0			0
$227$	−8.65840			−0.574678			−0.287339	−	0.957829i	$-0.592771\pi$
$227$	−8.65840			−0.574678			−0.287339	+	0.957829i	$0.592771\pi$
$228$		0			0
$229$	−0.246211			−0.0162701			−0.00813505	−	0.999967i	$-0.502589\pi$
$229$	−0.246211			−0.0162701			−0.00813505	+	0.999967i	$0.502589\pi$
$230$		0			0
$231$	6.67026	+	1.87285i	0.438871	+	0.123225i
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$			0.936426i			0.0610857i
$236$		0			0
$237$	15.1231	+	4.24621i	0.982351	+	0.275821i
$238$		0			0
$239$	−20.8319			−1.34751			−0.673753	−	0.738957i	$-0.735319\pi$
$239$	−20.8319			−1.34751			−0.673753	+	0.738957i	$0.735319\pi$
$240$		0			0
$241$	−11.3693			−0.732362			−0.366181	−	0.930544i	$-0.619335\pi$
$241$	−11.3693			−0.732362			−0.366181	+	0.930544i	$0.619335\pi$
$242$		0			0
$243$	−15.2710	+	3.12985i	−0.979636	+	0.200780i
$244$		0			0
$245$		−	6.12311i		−	0.391191i
$246$		0			0
$247$			13.3405i			0.848837i
$248$		0			0
$249$	−2.19224	+	7.80776i	−0.138927	+	0.494797i
$250$		0			0
$251$	25.1035			1.58452			0.792259	−	0.610184i	$-0.208905\pi$
$251$	25.1035			1.58452			0.792259	+	0.610184i	$0.208905\pi$
$252$		0			0
$253$	32.4924			2.04278
$254$		0			0
$255$	0.936426	−	3.33513i	0.0586413	−	0.208854i
$256$		0			0
$257$			2.49242i			0.155473i	0.996974	+	0.0777365i	$0.0247693\pi$
$257$			2.49242i			0.155473i	−0.996974	+	0.0777365i	$0.975231\pi$
$258$		0			0
$259$		−	2.92456i		−	0.181723i
$260$		0			0
$261$	−8.00000	+	13.1231i	−0.495188	+	0.812300i
$262$		0			0
$263$	15.0981			0.930989			0.465494	−	0.885051i	$-0.345877\pi$
$263$	15.0981			0.930989			0.465494	+	0.885051i	$0.345877\pi$
$264$		0			0
$265$	4.24621			0.260843
$266$		0			0
$267$	−10.4160	−	2.92456i	−0.637447	−	0.178980i
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$			31.7738i			1.93012i	0.262032	+	0.965059i	$0.415608\pi$
$271$			31.7738i			1.93012i	−0.262032	+	0.965059i	$0.584392\pi$
$272$		0			0
$273$	−4.87689	−	1.36932i	−0.295163	−	0.0828748i
$274$		0			0
$275$	−4.27156			−0.257585
$276$		0			0
$277$	1.36932			0.0822743			0.0411371	−	0.999154i	$-0.486902\pi$
$277$	1.36932			0.0822743			0.0411371	+	0.999154i	$0.486902\pi$
$278$		0			0
$279$	3.74571	−	6.14441i	0.224250	−	0.367856i
$280$		0			0
$281$			27.6155i			1.64740i	0.567023	+	0.823702i	$0.308095\pi$
$281$			27.6155i			1.64740i	−0.567023	+	0.823702i	$0.691905\pi$
$282$		0			0
$283$			4.38684i			0.260770i	0.991463	+	0.130385i	$0.0416214\pi$
$283$			4.38684i			0.260770i	−0.991463	+	0.130385i	$0.958379\pi$
$284$		0			0
$285$	2.00000	−	7.12311i	0.118470	−	0.421936i
$286$		0			0
$287$	−6.67026			−0.393733
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	2.80928	−	10.0054i	0.164683	−	0.586527i
$292$		0			0
$293$		−	30.4924i		−	1.78139i	−0.454605	−	0.890693i	$-0.650220\pi$
$293$		−	30.4924i		−	1.78139i	0.454605	−	0.890693i	$-0.349780\pi$
$294$		0			0
$295$		−	7.19612i		−	0.418974i
$296$		0			0
$297$	16.2462	−	15.1231i	0.942701	−	0.877532i
$298$		0			0
$299$	−23.7565			−1.37387
$300$		0			0
$301$	1.36932			0.0789261
$302$		0			0
$303$	15.2134	+	4.27156i	0.873986	+	0.245395i
$304$		0			0
$305$		−	5.12311i		−	0.293348i
$306$		0			0
$307$			8.13254i			0.464149i	0.972698	+	0.232074i	$0.0745513\pi$
$307$			8.13254i			0.464149i	−0.972698	+	0.232074i	$0.925449\pi$
$308$		0			0
$309$	−20.6847	−	5.80776i	−1.17671	−	0.330392i
$310$		0			0
$311$	−14.1617			−0.803035			−0.401517	−	0.915851i	$-0.631517\pi$
$311$	−14.1617			−0.803035			−0.401517	+	0.915851i	$0.631517\pi$
$312$		0			0
$313$	10.4924			0.593067			0.296533	−	0.955022i	$-0.404169\pi$
$313$	10.4924			0.593067			0.296533	+	0.955022i	$0.404169\pi$
$314$		0			0
$315$	2.39871	+	1.46228i	0.135152	+	0.0823901i
$316$		0			0
$317$			32.7386i			1.83878i	0.393342	+	0.919392i	$0.371319\pi$
$317$			32.7386i			1.83878i	−0.393342	+	0.919392i	$0.628681\pi$
$318$		0			0
$319$		−	21.8836i		−	1.22525i
$320$		0			0
$321$	0.438447	−	1.56155i	0.0244717	−	0.0871574i
$322$		0			0
$323$	−8.54312			−0.475352
$324$		0			0
$325$	3.12311			0.173239
$326$		0			0
$327$	−4.27156	+	15.2134i	−0.236218	+	0.841302i
$328$		0			0
$329$			0.876894i			0.0483448i
$330$		0			0
$331$			28.0281i			1.54056i	0.637705	+	0.770281i	$0.279884\pi$
$331$			28.0281i			1.54056i	−0.637705	+	0.770281i	$0.720116\pi$
$332$		0			0
$333$	−8.00000	−	4.87689i	−0.438397	−	0.267252i
$334$		0			0
$335$	5.20798			0.284543
$336$		0			0
$337$	−34.4924			−1.87892			−0.939461	−	0.342656i	$-0.888674\pi$
$337$	−34.4924			−1.87892			−0.939461	+	0.342656i	$0.888674\pi$
$338$		0			0
$339$	23.3459	+	6.55498i	1.26798	+	0.356018i
$340$		0			0
$341$			10.2462i			0.554863i
$342$		0			0
$343$		−	12.2888i		−	0.663534i
$344$		0			0
$345$	12.6847	+	3.56155i	0.682919	+	0.191748i
$346$		0			0
$347$	23.8718			1.28150			0.640752	−	0.767748i	$-0.278623\pi$
$347$	23.8718			1.28150			0.640752	+	0.767748i	$0.278623\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	−11.8782	+	11.0571i	−0.634014	+	0.590184i
$352$		0			0
$353$			3.75379i			0.199794i	0.994998	+	0.0998970i	$0.0318513\pi$
$353$			3.75379i			0.199794i	−0.994998	+	0.0998970i	$0.968149\pi$
$354$		0			0
$355$		−	6.67026i		−	0.354021i
$356$		0			0
$357$	0.876894	−	3.12311i	0.0464102	−	0.165292i
$358$		0			0
$359$	1.05171			0.0555069			0.0277535	−	0.999615i	$-0.491165\pi$
$359$	1.05171			0.0555069			0.0277535	+	0.999615i	$0.491165\pi$
$360$		0			0
$361$	0.753789			0.0396731
$362$		0			0
$363$	−3.39277	+	12.0835i	−0.178074	+	0.634221i
$364$		0			0
$365$			8.24621i			0.431626i
$366$		0			0
$367$		−	26.5658i		−	1.38672i	−0.720590	−	0.693361i	$-0.756129\pi$
$367$		−	26.5658i		−	1.38672i	0.720590	−	0.693361i	$-0.243871\pi$
$368$		0			0
$369$	−11.1231	+	18.2462i	−0.579046	+	0.949860i
$370$		0			0
$371$	3.97626			0.206437
$372$		0			0
$373$	0.876894			0.0454039			0.0227019	−	0.999742i	$-0.492773\pi$
$373$	0.876894			0.0454039			0.0227019	+	0.999742i	$0.492773\pi$
$374$		0			0
$375$	−1.66757	−	0.468213i	−0.0861127	−	0.0241784i
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$		−	25.1035i		−	1.28948i	−0.764402	−	0.644740i	$-0.776966\pi$
$379$		−	25.1035i		−	1.28948i	0.764402	−	0.644740i	$-0.223034\pi$
$380$		0			0
$381$	−7.80776	−	2.19224i	−0.400004	−	0.112312i
$382$		0			0
$383$	4.68213			0.239246			0.119623	−	0.992819i	$-0.461831\pi$
$383$	4.68213			0.239246			0.119623	+	0.992819i	$0.461831\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$	2.28343	−	3.74571i	0.116073	−	0.190405i
$388$		0			0
$389$			28.7386i			1.45711i	0.684989	+	0.728553i	$0.259807\pi$
$389$			28.7386i			1.45711i	−0.684989	+	0.728553i	$0.740193\pi$
$390$		0			0
$391$		−	15.2134i		−	0.769374i
$392$		0			0
$393$	8.24621	−	29.3693i	0.415966	−	1.48149i
$394$		0			0
$395$	−9.06897			−0.456309
$396$		0			0
$397$	−23.1231			−1.16052			−0.580258	−	0.814433i	$-0.697048\pi$
$397$	−23.1231			−1.16052			−0.580258	+	0.814433i	$0.697048\pi$
$398$		0			0
$399$	1.87285	−	6.67026i	0.0937599	−	0.333931i
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$		0			0
$403$		−	7.49141i		−	0.373174i
$404$		0			0
$405$	8.00000	−	4.12311i	0.397523	−	0.204879i
$406$		0			0
$407$	13.3405			0.661265
$408$		0			0
$409$	0.630683			0.0311853			0.0155926	−	0.999878i	$-0.495037\pi$
$409$	0.630683			0.0311853			0.0155926	+	0.999878i	$0.495037\pi$
$410$		0			0
$411$	13.7511	+	3.86098i	0.678292	+	0.190448i
$412$		0			0
$413$		−	6.73863i		−	0.331586i
$414$		0			0
$415$		−	4.68213i		−	0.229837i
$416$		0			0
$417$	23.1231	+	6.49242i	1.13234	+	0.317935i
$418$		0			0
$419$	6.14441			0.300174			0.150087	−	0.988673i	$-0.452045\pi$
$419$	6.14441			0.300174			0.150087	+	0.988673i	$0.452045\pi$
$420$		0			0
$421$	0.630683			0.0307376			0.0153688	−	0.999882i	$-0.495108\pi$
$421$	0.630683			0.0307376			0.0153688	+	0.999882i	$0.495108\pi$
$422$		0			0
$423$	2.39871	+	1.46228i	0.116629	+	0.0710985i
$424$		0			0
$425$			2.00000i			0.0970143i
$426$		0			0
$427$		−	4.79741i		−	0.232163i
$428$		0			0
$429$	6.24621	−	22.2462i	0.301570	−	1.07406i
$430$		0			0
$431$	36.0453			1.73624			0.868121	−	0.496353i	$-0.165328\pi$
$431$	36.0453			1.73624			0.868121	+	0.496353i	$0.165328\pi$
$432$		0			0
$433$	18.0000			0.865025			0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000			0.865025			0.432512	+	0.901628i	$0.357627\pi$
$434$		0			0
$435$	2.39871	−	8.54312i	0.115009	−	0.409611i
$436$		0			0
$437$		−	32.4924i		−	1.55432i
$438$		0			0
$439$			29.9009i			1.42709i	0.700608	+	0.713546i	$0.252912\pi$
$439$			29.9009i			1.42709i	−0.700608	+	0.713546i	$0.747088\pi$
$440$		0			0
$441$	−15.6847	−	9.56155i	−0.746888	−	0.455312i
$442$		0			0
$443$	25.7446			1.22316			0.611582	−	0.791181i	$-0.290533\pi$
$443$	25.7446			1.22316			0.611582	+	0.791181i	$0.290533\pi$
$444$		0			0
$445$	6.24621			0.296099
$446$		0			0
$447$	23.3459	+	6.55498i	1.10422	+	0.310040i
$448$		0			0
$449$			2.63068i			0.124150i	0.998071	+	0.0620748i	$0.0197717\pi$
$449$			2.63068i			0.124150i	−0.998071	+	0.0620748i	$0.980228\pi$
$450$		0			0
$451$		−	30.4268i		−	1.43274i
$452$		0			0
$453$	10.2462	+	2.87689i	0.481409	+	0.135168i
$454$		0			0
$455$	2.92456			0.137105
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$	−7.08084	−	7.60669i	−0.330505	−	0.355050i
$460$		0			0
$461$		−	15.8617i		−	0.738755i	−0.929279	−	0.369377i	$-0.879571\pi$
$461$		−	15.8617i		−	0.738755i	0.929279	−	0.369377i	$-0.120429\pi$
$462$		0			0
$463$			0.936426i			0.0435194i	0.999763	+	0.0217597i	$0.00692688\pi$
$463$			0.936426i			0.0435194i	−0.999763	+	0.0217597i	$0.993073\pi$
$464$		0			0
$465$	−1.12311	+	4.00000i	−0.0520828	+	0.185496i
$466$		0			0
$467$	16.1498			0.747324			0.373662	−	0.927565i	$-0.378102\pi$
$467$	16.1498			0.747324			0.373662	+	0.927565i	$0.378102\pi$
$468$		0			0
$469$	4.87689			0.225194
$470$		0			0
$471$	10.0054	−	35.6347i	0.461024	−	1.64196i
$472$		0			0
$473$			6.24621i			0.287201i
$474$		0			0
$475$			4.27156i			0.195993i
$476$		0			0
$477$	6.63068	−	10.8769i	0.303598	−	0.498019i
$478$		0			0
$479$	22.9354			1.04794			0.523971	−	0.851736i	$-0.324450\pi$
$479$	22.9354			1.04794			0.523971	+	0.851736i	$0.324450\pi$
$480$		0			0
$481$	−9.75379			−0.444734
$482$		0			0
$483$	11.8782	+	3.33513i	0.540479	+	0.151754i
$484$		0			0
$485$			6.00000i			0.272446i
$486$		0			0
$487$			15.3287i			0.694608i	0.937753	+	0.347304i	$0.112903\pi$
$487$			15.3287i			0.694608i	−0.937753	+	0.347304i	$0.887097\pi$
$488$		0			0
$489$	−40.3002	−	11.3153i	−1.82244	−	0.511697i
$490$		0			0
$491$	−18.6638			−0.842285			−0.421143	−	0.906994i	$-0.638371\pi$
$491$	−18.6638			−0.842285			−0.421143	+	0.906994i	$0.638371\pi$
$492$		0			0
$493$	−10.2462			−0.461466
$494$		0			0
$495$	−6.67026	+	10.9418i	−0.299806	+	0.491798i
$496$		0			0
$497$		−	6.24621i		−	0.280181i
$498$		0			0
$499$			1.57756i			0.0706212i	0.999376	+	0.0353106i	$0.0112421\pi$
$499$			1.57756i			0.0706212i	−0.999376	+	0.0353106i	$0.988758\pi$
$500$		0			0
$501$	−1.31534	+	4.68466i	−0.0587651	+	0.209295i
$502$		0			0
$503$	−19.8955			−0.887097			−0.443549	−	0.896250i	$-0.646281\pi$
$503$	−19.8955			−0.887097			−0.443549	+	0.896250i	$0.646281\pi$
$504$		0			0
$505$	−9.12311			−0.405973
$506$		0			0
$507$	1.51992	−	5.41327i	0.0675020	−	0.240412i
$508$		0			0
$509$			2.87689i			0.127516i	0.997965	+	0.0637581i	$0.0203086\pi$
$509$			2.87689i			0.127516i	−0.997965	+	0.0637581i	$0.979691\pi$
$510$		0			0
$511$			7.72197i			0.341600i
$512$		0			0
$513$	−15.1231	−	16.2462i	−0.667701	−	0.717288i
$514$		0			0
$515$	12.4041			0.546590
$516$		0			0
$517$	−4.00000			−0.175920
$518$		0			0
$519$	−3.33513	−	0.936426i	−0.146396	−	0.0411046i
$520$		0			0
$521$			21.7538i			0.953051i	0.879161	+	0.476525i	$0.158104\pi$
$521$			21.7538i			0.953051i	−0.879161	+	0.476525i	$0.841896\pi$
$522$		0			0
$523$		−	0.641132i		−	0.0280348i	−0.999902	−	0.0140174i	$-0.995538\pi$
$523$		−	0.641132i		−	0.0280348i	0.999902	−	0.0140174i	$-0.00446202\pi$
$524$		0			0
$525$	−1.56155	−	0.438447i	−0.0681518	−	0.0191354i
$526$		0			0
$527$	4.79741			0.208979
$528$		0			0
$529$	34.8617			1.51573
$530$		0			0
$531$	−18.4332	−	11.2371i	−0.799934	−	0.487649i
$532$		0			0
$533$			22.2462i			0.963590i
$534$		0			0
$535$			0.936426i			0.0404852i
$536$		0			0
$537$	6.87689	−	24.4924i	0.296760	−	1.05693i
$538$		0			0
$539$	26.1552			1.12658
$540$		0			0
$541$	−38.9848			−1.67609			−0.838045	−	0.545602i	$-0.816301\pi$
$541$	−38.9848			−1.67609			−0.838045	+	0.545602i	$0.816301\pi$
$542$		0			0
$543$	1.98813	−	7.08084i	0.0853189	−	0.303868i
$544$		0			0
$545$		−	9.12311i		−	0.390791i
$546$		0			0
$547$			25.2188i			1.07828i	0.842217	+	0.539139i	$0.181250\pi$
$547$			25.2188i			1.07828i	−0.842217	+	0.539139i	$0.818750\pi$
$548$		0			0
$549$	−13.1231	−	8.00000i	−0.560080	−	0.341432i
$550$		0			0
$551$	−21.8836			−0.932275
$552$		0			0
$553$	−8.49242			−0.361135
$554$		0			0
$555$	5.20798	+	1.46228i	0.221067	+	0.0620703i
$556$		0			0
$557$			19.7538i			0.836995i	0.908218	+	0.418497i	$0.137443\pi$
$557$			19.7538i			0.836995i	−0.908218	+	0.418497i	$0.862557\pi$
$558$		0			0
$559$		−	4.56685i		−	0.193157i
$560$		0			0
$561$	14.2462	+	4.00000i	0.601476	+	0.168880i
$562$		0			0
$563$	36.1606			1.52399			0.761994	−	0.647584i	$-0.224221\pi$
$563$	36.1606			1.52399			0.761994	+	0.647584i	$0.224221\pi$
$564$		0			0
$565$	−14.0000			−0.588984
$566$		0			0
$567$	7.49141	−	3.86098i	0.314610	−	0.162146i
$568$		0			0
$569$			4.87689i			0.204450i	0.994761	+	0.102225i	$0.0325962\pi$
$569$			4.87689i			0.204450i	−0.994761	+	0.102225i	$0.967404\pi$
$570$		0			0
$571$			16.7909i			0.702679i	0.936248	+	0.351339i	$0.114274\pi$
$571$			16.7909i			0.702679i	−0.936248	+	0.351339i	$0.885726\pi$
$572$		0			0
$573$	3.61553	−	12.8769i	0.151041	−	0.537940i
$574$		0			0
$575$	−7.60669			−0.317221
$576$		0			0
$577$	−15.7538			−0.655839			−0.327919	−	0.944706i	$-0.606347\pi$
$577$	−15.7538			−0.655839			−0.327919	+	0.944706i	$0.606347\pi$
$578$		0			0
$579$	−7.60669	+	27.0916i	−0.316123	+	1.12589i
$580$		0			0
$581$		−	4.38447i		−	0.181899i
$582$		0			0
$583$			18.1379i			0.751197i
$584$		0			0
$585$	4.87689	−	8.00000i	0.201635	−	0.330759i
$586$		0			0
$587$	−38.0335			−1.56981			−0.784904	−	0.619617i	$-0.787288\pi$
$587$	−38.0335			−1.56981			−0.784904	+	0.619617i	$0.787288\pi$
$588$		0			0
$589$	10.2462			0.422188
$590$		0			0
$591$	20.4214	+	5.73384i	0.840023	+	0.235859i
$592$		0			0
$593$			8.24621i			0.338631i	0.985562	+	0.169316i	$0.0541557\pi$
$593$			8.24621i			0.338631i	−0.985562	+	0.169316i	$0.945844\pi$
$594$		0			0
$595$			1.87285i			0.0767795i
$596$		0			0
$597$	29.3693	+	8.24621i	1.20201	+	0.337495i
$598$		0			0
$599$	36.8665			1.50632			0.753162	−	0.657836i	$-0.228528\pi$
$599$	36.8665			1.50632			0.753162	+	0.657836i	$0.228528\pi$
$600$		0			0
$601$	14.8769			0.606841			0.303421	−	0.952857i	$-0.401871\pi$
$601$	14.8769			0.606841			0.303421	+	0.952857i	$0.401871\pi$
$602$		0			0
$603$	8.13254	−	13.3405i	0.331183	−	0.543268i
$604$		0			0
$605$		−	7.24621i		−	0.294600i
$606$		0			0
$607$		−	29.4903i		−	1.19698i	−0.801132	−	0.598488i	$-0.795768\pi$
$607$		−	29.4903i		−	1.19698i	0.801132	−	0.598488i	$-0.204232\pi$
$608$		0			0
$609$	2.24621	−	8.00000i	0.0910211	−	0.324176i
$610$		0			0
$611$	2.92456			0.118315
$612$		0			0
$613$	−0.876894			−0.0354174			−0.0177087	−	0.999843i	$-0.505637\pi$
$613$	−0.876894			−0.0354174			−0.0177087	+	0.999843i	$0.505637\pi$
$614$		0			0
$615$	3.33513	−	11.8782i	0.134486	−	0.478977i
$616$		0			0
$617$			14.0000i			0.563619i	0.959470	+	0.281809i	$0.0909346\pi$
$617$			14.0000i			0.563619i	−0.959470	+	0.281809i	$0.909065\pi$
$618$		0			0
$619$			20.3061i			0.816171i	0.912944	+	0.408085i	$0.133803\pi$
$619$			20.3061i			0.816171i	−0.912944	+	0.408085i	$0.866197\pi$
$620$		0			0
$621$	28.9309	−	26.9309i	1.16096	−	1.08070i
$622$		0			0
$623$	5.84912			0.234340
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	30.4268	+	8.54312i	1.21513	+	0.341179i
$628$		0			0
$629$		−	6.24621i		−	0.249053i
$630$		0			0
$631$		−	30.1315i		−	1.19951i	−0.800182	−	0.599757i	$-0.795264\pi$
$631$		−	30.1315i		−	1.19951i	0.800182	−	0.599757i	$-0.204736\pi$
$632$		0			0
$633$	−2.24621	−	0.630683i	−0.0892789	−	0.0250674i
$634$		0			0
$635$	4.68213			0.185805
$636$		0			0
$637$	−19.1231			−0.757685
$638$		0			0
$639$	−17.0862	−	10.4160i	−0.675921	−	0.412049i
$640$		0			0
$641$			47.6155i			1.88070i	0.340208	+	0.940350i	$0.389502\pi$
$641$			47.6155i			1.88070i	−0.340208	+	0.940350i	$0.610498\pi$
$642$		0			0
$643$		−	20.4214i		−	0.805340i	−0.915345	−	0.402670i	$-0.868082\pi$
$643$		−	20.4214i		−	0.805340i	0.915345	−	0.402670i	$-0.131918\pi$
$644$		0			0
$645$	−0.684658	+	2.43845i	−0.0269584	+	0.0960138i
$646$		0			0
$647$	3.63043			0.142727			0.0713634	−	0.997450i	$-0.477265\pi$
$647$	3.63043			0.142727			0.0713634	+	0.997450i	$0.477265\pi$
$648$		0			0
$649$	30.7386			1.20660
$650$		0			0
$651$	−1.05171	+	3.74571i	−0.0412196	+	0.146806i
$652$		0			0
$653$			26.9848i			1.05600i	0.849245	+	0.527999i	$0.177058\pi$
$653$			26.9848i			1.05600i	−0.849245	+	0.527999i	$0.822942\pi$
$654$		0			0
$655$			17.6121i			0.688161i
$656$		0			0
$657$	21.1231	+	12.8769i	0.824091	+	0.502375i
$658$		0			0
$659$	26.9764			1.05085			0.525425	−	0.850840i	$-0.323906\pi$
$659$	26.9764			1.05085			0.525425	+	0.850840i	$0.323906\pi$
$660$		0			0
$661$	46.1080			1.79339			0.896696	−	0.442647i	$-0.145961\pi$
$661$	46.1080			1.79339			0.896696	+	0.442647i	$0.145961\pi$
$662$		0			0
$663$	−10.4160	−	2.92456i	−0.404523	−	0.113580i
$664$		0			0
$665$			4.00000i			0.155113i
$666$		0			0
$667$		−	38.9699i		−	1.50892i
$668$		0			0
$669$	−30.0540	−	8.43845i	−1.16195	−	0.326249i
$670$		0			0
$671$	21.8836			0.844809
$672$		0			0
$673$	−10.4924			−0.404453			−0.202227	−	0.979339i	$-0.564818\pi$
$673$	−10.4924			−0.404453			−0.202227	+	0.979339i	$0.564818\pi$
$674$		0			0
$675$	−3.80335	+	3.54042i	−0.146391	+	0.136271i
$676$		0			0
$677$			34.4924i			1.32565i	0.748774	+	0.662826i	$0.230643\pi$
$677$			34.4924i			1.32565i	−0.748774	+	0.662826i	$0.769357\pi$
$678$		0			0
$679$			5.61856i			0.215620i
$680$		0			0
$681$	4.05398	−	14.4384i	0.155349	−	0.553282i
$682$		0			0
$683$	−36.1606			−1.38365			−0.691823	−	0.722067i	$-0.743192\pi$
$683$	−36.1606			−1.38365			−0.691823	+	0.722067i	$0.743192\pi$
$684$		0			0
$685$	−8.24621			−0.315072
$686$		0			0
$687$	0.115279	−	0.410574i	0.00439818	−	0.0156644i
$688$		0			0
$689$		−	13.2614i		−	0.505218i
$690$		0			0
$691$		−	29.0798i		−	1.10625i	−0.833100	−	0.553123i	$-0.813436\pi$
$691$		−	29.0798i		−	1.10625i	0.833100	−	0.553123i	$-0.186564\pi$
$692$		0			0
$693$	−6.24621	+	10.2462i	−0.237274	+	0.389221i
$694$		0			0
$695$	−13.8664			−0.525982
$696$		0			0
$697$	−14.2462			−0.539614
$698$		0			0
$699$	−16.6757	−	4.68213i	−0.630731	−	0.177094i
$700$		0			0
$701$			50.4924i			1.90707i	0.301278	+	0.953536i	$0.402587\pi$
$701$			50.4924i			1.90707i	−0.301278	+	0.953536i	$0.597413\pi$
$702$		0			0
$703$		−	13.3405i		−	0.503148i
$704$		0			0
$705$	−1.56155	−	0.438447i	−0.0588115	−	0.0165129i
$706$		0			0
$707$	−8.54312			−0.321297
$708$		0			0
$709$	26.4924			0.994944			0.497472	−	0.867480i	$-0.334262\pi$
$709$	26.4924			0.994944			0.497472	+	0.867480i	$0.334262\pi$
$710$		0			0
$711$	−14.1617	+	23.2306i	−0.531104	+	0.871217i
$712$		0			0
$713$			18.2462i			0.683326i
$714$		0			0
$715$			13.3405i			0.498907i
$716$		0			0
$717$	9.75379	−	34.7386i	0.364262	−	1.29734i
$718$		0			0
$719$	−5.84912			−0.218135			−0.109068	−	0.994034i	$-0.534787\pi$
$719$	−5.84912			−0.218135			−0.109068	+	0.994034i	$0.534787\pi$
$720$		0			0
$721$	11.6155			0.432585
$722$		0			0
$723$	5.32326	−	18.9591i	0.197974	−	0.705096i
$724$		0			0
$725$			5.12311i			0.190267i
$726$		0			0
$727$			26.5658i			0.985270i	0.870236	+	0.492635i	$0.163966\pi$
$727$			26.5658i			0.985270i	−0.870236	+	0.492635i	$0.836034\pi$
$728$		0			0
$729$	1.93087	−	26.9309i	0.0715137	−	0.997440i
$730$		0			0
$731$	2.92456			0.108169
$732$		0			0
$733$	35.1231			1.29730			0.648651	−	0.761086i	$-0.275334\pi$
$733$	35.1231			1.29730			0.648651	+	0.761086i	$0.275334\pi$
$734$		0			0
$735$	10.2107	+	2.86692i	0.376627	+	0.105748i
$736$		0			0
$737$			22.2462i			0.819450i
$738$		0			0
$739$			18.6638i			0.686559i	0.939233	+	0.343279i	$0.111538\pi$
$739$			18.6638i			0.686559i	−0.939233	+	0.343279i	$0.888462\pi$
$740$		0			0
$741$	−22.2462	−	6.24621i	−0.817235	−	0.229460i
$742$		0			0
$743$	−12.4041			−0.455062			−0.227531	−	0.973771i	$-0.573065\pi$
$743$	−12.4041			−0.455062			−0.227531	+	0.973771i	$0.573065\pi$
$744$		0			0
$745$	−14.0000			−0.512920
$746$		0			0
$747$	−11.9935	−	7.31140i	−0.438820	−	0.267510i
$748$		0			0
$749$			0.876894i			0.0320410i
$750$		0			0
$751$			15.7392i			0.574333i	0.957881	+	0.287166i	$0.0927133\pi$
$751$			15.7392i			0.574333i	−0.957881	+	0.287166i	$0.907287\pi$
$752$		0			0
$753$	−11.7538	+	41.8617i	−0.428332	+	1.52553i
$754$		0			0
$755$	−6.14441			−0.223618
$756$		0			0
$757$	−19.1231			−0.695041			−0.347521	−	0.937672i	$-0.612976\pi$
$757$	−19.1231			−0.695041			−0.347521	+	0.937672i	$0.612976\pi$
$758$		0			0
$759$	−15.2134	+	54.1833i	−0.552211	+	1.96673i
$760$		0			0
$761$		−	51.2311i		−	1.85712i	−0.371177	−	0.928562i	$-0.621046\pi$
$761$		−	51.2311i		−	1.85712i	0.371177	−	0.928562i	$-0.378954\pi$
$762$		0			0
$763$		−	8.54312i		−	0.309282i
$764$		0			0
$765$	5.12311	+	3.12311i	0.185226	+	0.112916i
$766$		0			0
$767$	−22.4742			−0.811498
$768$		0			0
$769$	−26.9848			−0.973098			−0.486549	−	0.873653i	$-0.661745\pi$
$769$	−26.9848			−0.973098			−0.486549	+	0.873653i	$0.661745\pi$
$770$		0			0
$771$	−4.15628	−	1.16699i	−0.149685	−	0.0420279i
$772$		0			0
$773$			16.2462i			0.584336i	0.956367	+	0.292168i	$0.0943766\pi$
$773$			16.2462i			0.584336i	−0.956367	+	0.292168i	$0.905623\pi$
$774$		0			0
$775$		−	2.39871i		−	0.0861641i
$776$		0			0
$777$	4.87689	+	1.36932i	0.174958	+	0.0491240i
$778$		0			0
$779$	−30.4268			−1.09015
$780$		0			0
$781$	28.4924			1.01954
$782$		0			0
$783$	−18.1379	−	19.4849i	−0.648197	−	0.696335i
$784$		0			0
$785$			21.3693i			0.762704i
$786$		0			0
$787$			13.9817i			0.498392i	0.968453	+	0.249196i	$0.0801664\pi$
$787$			13.9817i			0.498392i	−0.968453	+	0.249196i	$0.919834\pi$
$788$		0			0
$789$	−7.06913	+	25.1771i	−0.251668	+	0.896328i
$790$		0			0
$791$	−13.1100			−0.466137
$792$		0			0
$793$	−16.0000			−0.568177
$794$		0			0
$795$	−1.98813	+	7.08084i	−0.0705118	+	0.251131i
$796$		0			0
$797$		−	36.7386i		−	1.30135i	−0.759357	−	0.650675i	$-0.774486\pi$
$797$		−	36.7386i		−	1.30135i	0.759357	−	0.650675i	$-0.225514\pi$
$798$		0			0
$799$			1.87285i			0.0662568i
$800$		0			0
$801$	9.75379	−	16.0000i	0.344633	−	0.565332i
$802$		0			0
$803$	−35.2242			−1.24303
$804$		0			0
$805$	−7.12311			−0.251056
$806$		0			0
$807$	−23.3459	−	6.55498i	−0.821815	−	0.230746i
$808$		0			0
$809$			46.2462i			1.62593i	0.582312	+	0.812965i	$0.302148\pi$
$809$			46.2462i			1.62593i	−0.582312	+	0.812965i	$0.697852\pi$
$810$		0			0
$811$			25.9246i			0.910337i	0.890405	+	0.455169i	$0.150421\pi$
$811$			25.9246i			0.910337i	−0.890405	+	0.455169i	$0.849579\pi$
$812$		0			0
$813$	−52.9848	−	14.8769i	−1.85826	−	0.521755i
$814$		0			0
$815$	24.1671			0.846536
$816$		0			0
$817$	6.24621			0.218527
$818$		0			0
$819$	4.56685	−	7.49141i	0.159579	−	0.261771i
$820$		0			0
$821$		−	29.2311i		−	1.02017i	−0.860124	−	0.510085i	$-0.829614\pi$
$821$		−	29.2311i		−	1.02017i	0.860124	−	0.510085i	$-0.170386\pi$
$822$		0			0
$823$		−	46.8071i		−	1.63159i	−0.578338	−	0.815797i	$-0.696299\pi$
$823$		−	46.8071i		−	1.63159i	0.578338	−	0.815797i	$-0.303701\pi$
$824$		0			0
$825$	2.00000	−	7.12311i	0.0696311	−	0.247995i
$826$		0			0
$827$	−13.2252			−0.459887			−0.229943	−	0.973204i	$-0.573854\pi$
$827$	−13.2252			−0.459887			−0.229943	+	0.973204i	$0.573854\pi$
$828$		0			0
$829$	17.1231			0.594710			0.297355	−	0.954767i	$-0.403896\pi$
$829$	17.1231			0.594710			0.297355	+	0.954767i	$0.403896\pi$
$830$		0			0
$831$	−0.641132	+	2.28343i	−0.0222406	+	0.0792112i
$832$		0			0
$833$		−	12.2462i		−	0.424306i
$834$		0			0
$835$		−	2.80928i		−	0.0972191i
$836$		0			0
$837$	8.49242	+	9.12311i	0.293541	+	0.315341i
$838$		0			0
$839$	−48.5647			−1.67664			−0.838320	−	0.545179i	$-0.816462\pi$
$839$	−48.5647			−1.67664			−0.838320	+	0.545179i	$0.816462\pi$
$840$		0			0
$841$	2.75379			0.0949582
$842$		0			0
$843$	−46.0507	−	12.9300i	−1.58607	−	0.445331i
$844$		0			0
$845$			3.24621i			0.111673i
$846$		0			0
$847$		−	6.78554i		−	0.233154i
$848$		0			0
$849$	−7.31534	−	2.05398i	−0.251062	−	0.0704923i
$850$		0			0
$851$	23.7565			0.814362
$852$		0			0
$853$	−49.8617			−1.70723			−0.853617	−	0.520902i	$-0.825596\pi$
$853$	−49.8617			−1.70723			−0.853617	+	0.520902i	$0.825596\pi$
$854$		0			0
$855$	10.9418	+	6.67026i	0.374202	+	0.228118i
$856$		0			0
$857$		−	28.7386i		−	0.981693i	−0.871246	−	0.490847i	$-0.836688\pi$
$857$		−	28.7386i		−	0.981693i	0.871246	−	0.490847i	$-0.163312\pi$
$858$		0			0
$859$			37.3923i			1.27581i	0.770115	+	0.637905i	$0.220199\pi$
$859$			37.3923i			1.27581i	−0.770115	+	0.637905i	$0.779801\pi$
$860$		0			0
$861$	3.12311	−	11.1231i	0.106435	−	0.379074i
$862$		0			0
$863$	27.6175			0.940110			0.470055	−	0.882637i	$-0.344234\pi$
$863$	27.6175			0.940110			0.470055	+	0.882637i	$0.344234\pi$
$864$		0			0
$865$	2.00000			0.0680020
$866$		0			0
$867$	−6.08677	+	21.6784i	−0.206718	+	0.736236i
$868$		0			0
$869$		−	38.7386i		−	1.31412i
$870$		0			0
$871$		−	16.2651i		−	0.551121i
$872$		0			0
$873$	15.3693	+	9.36932i	0.520173	+	0.317103i
$874$		0			0
$875$	0.936426			0.0316570
$876$		0			0
$877$	3.61553			0.122088			0.0610439	−	0.998135i	$-0.480557\pi$
$877$	3.61553			0.122088			0.0610439	+	0.998135i	$0.480557\pi$
$878$		0			0
$879$	50.8481	+	14.2770i	1.71506	+	0.481550i
$880$		0			0
$881$		−	25.3693i		−	0.854714i	−0.904083	−	0.427357i	$-0.859445\pi$
$881$		−	25.3693i		−	0.854714i	0.904083	−	0.427357i	$-0.140555\pi$
$882$		0			0
$883$			10.8265i			0.364342i	0.983267	+	0.182171i	$0.0583125\pi$
$883$			10.8265i			0.364342i	−0.983267	+	0.182171i	$0.941688\pi$
$884$		0			0
$885$	12.0000	+	3.36932i	0.403376	+	0.113258i
$886$		0			0
$887$	−53.4774			−1.79560			−0.897798	−	0.440408i	$-0.854834\pi$
$887$	−53.4774			−1.79560			−0.897798	+	0.440408i	$0.854834\pi$
$888$		0			0
$889$	4.38447			0.147050
$890$		0			0
$891$	17.6121	+	34.1725i	0.590027	+	1.14482i
$892$		0			0
$893$			4.00000i			0.133855i
$894$		0			0
$895$			14.6875i			0.490950i
$896$		0			0
$897$	11.1231	−	39.6155i	0.371390	−	1.32272i
$898$		0			0
$899$	12.2888			0.409855
$900$		0			0
$901$	8.49242			0.282924
$902$		0			0
$903$	−0.641132	+	2.28343i	−0.0213355	+	0.0759877i
$904$		0			0
$905$			4.24621i			0.141149i
$906$		0			0
$907$		−	35.8653i		−	1.19089i	−0.803397	−	0.595444i	$-0.796976\pi$
$907$		−	35.8653i		−	1.19089i	0.803397	−	0.595444i	$-0.203024\pi$
$908$		0			0
$909$	−14.2462	+	23.3693i	−0.472517	+	0.775111i
$910$		0			0
$911$	−41.8944			−1.38802			−0.694012	−	0.719963i	$-0.744159\pi$
$911$	−41.8944			−1.38802			−0.694012	+	0.719963i	$0.744159\pi$
$912$		0			0
$913$	20.0000			0.661903
$914$		0			0
$915$	8.54312	+	2.39871i	0.282427	+	0.0792988i
$916$		0			0
$917$			16.4924i			0.544628i
$918$		0			0
$919$			50.1423i			1.65404i	0.562172	+	0.827020i	$0.309966\pi$
$919$			50.1423i			1.65404i	−0.562172	+	0.827020i	$0.690034\pi$
$920$		0			0
$921$	−13.5616	−	3.80776i	−0.446868	−	0.125470i
$922$		0			0
$923$	−20.8319			−0.685692
$924$		0			0
$925$	−3.12311			−0.102687
$926$		0			0
$927$	19.3697	−	31.7738i	0.636183	−	1.04359i
$928$		0			0
$929$			24.8769i			0.816184i	0.912941	+	0.408092i	$0.133806\pi$
$929$			24.8769i			0.816184i	−0.912941	+	0.408092i	$0.866194\pi$
$930$		0			0
$931$		−	26.1552i		−	0.857202i
$932$		0			0
$933$	6.63068	−	23.6155i	0.217079	−	0.773138i
$934$		0			0
$935$	−8.54312			−0.279390
$936$		0			0
$937$	10.4924			0.342773			0.171386	−	0.985204i	$-0.445175\pi$
$937$	10.4924			0.342773			0.171386	+	0.985204i	$0.445175\pi$
$938$		0			0
$939$	−4.91269	+	17.4968i	−0.160320	+	0.570987i
$940$		0			0
$941$			9.12311i			0.297405i	0.988882	+	0.148702i	$0.0475096\pi$
$941$			9.12311i			0.297405i	−0.988882	+	0.148702i	$0.952490\pi$
$942$		0			0
$943$		−	54.1833i		−	1.76445i
$944$		0			0
$945$	−3.56155	+	3.31534i	−0.115857	+	0.107848i
$946$		0			0
$947$	25.5141			0.829096			0.414548	−	0.910027i	$-0.363940\pi$
$947$	25.5141			0.829096			0.414548	+	0.910027i	$0.363940\pi$
$948$		0			0
$949$	25.7538			0.836003
$950$		0			0
$951$	−54.5938	−	15.3287i	−1.77033	−	0.497066i
$952$		0			0
$953$		−	10.4924i		−	0.339883i	−0.985454	−	0.169941i	$-0.945642\pi$
$953$		−	10.4924i		−	0.339883i	0.985454	−	0.169941i	$-0.0543579\pi$
$954$		0			0
$955$			7.72197i			0.249877i
$956$		0			0
$957$	36.4924	+	10.2462i	1.17963	+	0.331213i
$958$		0			0
$959$	−7.72197			−0.249355
$960$		0			0
$961$	25.2462			0.814394
$962$		0			0
$963$	2.39871	+	1.46228i	0.0772972	+	0.0471213i
$964$		0			0
$965$		−	16.2462i		−	0.522984i
$966$		0			0
$967$			23.6412i			0.760250i	0.924935	+	0.380125i	$0.124119\pi$
$967$			23.6412i			0.760250i	−0.924935	+	0.380125i	$0.875881\pi$
$968$		0			0
$969$	4.00000	−	14.2462i	0.128499	−	0.457654i
$970$		0			0
$971$	−41.5991			−1.33498			−0.667490	−	0.744619i	$-0.732631\pi$
$971$	−41.5991			−1.33498			−0.667490	+	0.744619i	$0.732631\pi$
$972$		0			0
$973$	−12.9848			−0.416275
$974$		0			0
$975$	−1.46228	+	5.20798i	−0.0468304	+	0.166789i
$976$		0			0
$977$		−	48.2462i		−	1.54353i	−0.635906	−	0.771767i	$-0.719373\pi$
$977$		−	48.2462i		−	1.54353i	0.635906	−	0.771767i	$-0.280627\pi$
$978$		0			0
$979$			26.6811i			0.852730i
$980$		0			0
$981$	−23.3693	−	14.2462i	−0.746125	−	0.454847i
$982$		0			0
$983$	3.03984			0.0969558			0.0484779	−	0.998824i	$-0.484563\pi$
$983$	3.03984			0.0969558			0.0484779	+	0.998824i	$0.484563\pi$
$984$		0			0
$985$	−12.2462			−0.390197
$986$		0			0
$987$	−1.46228	−	0.410574i	−0.0465449	−	0.0130687i
$988$		0			0
$989$			11.1231i			0.353694i
$990$		0			0
$991$			16.7909i			0.533382i	0.963782	+	0.266691i	$0.0859303\pi$
$991$			16.7909i			0.533382i	−0.963782	+	0.266691i	$0.914070\pi$
$992$		0			0
$993$	−46.7386	−	13.1231i	−1.48321	−	0.416449i
$994$		0			0
$995$	−17.6121			−0.558341
$996$		0			0
$997$	−7.61553			−0.241186			−0.120593	−	0.992702i	$-0.538480\pi$
$997$	−7.61553			−0.241186			−0.120593	+	0.992702i	$0.538480\pi$
$998$		0			0
$999$	11.8782	−	11.0571i	0.375811	−	0.349831i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.h.g.191.4		8
3.2	odd	2	inner	960.2.h.g.191.6		8
4.3	odd	2	inner	960.2.h.g.191.5		8
8.3	odd	2		60.2.e.a.11.4	yes	8
8.5	even	2		60.2.e.a.11.6	yes	8
12.11	even	2	inner	960.2.h.g.191.3		8
24.5	odd	2		60.2.e.a.11.3	✓	8
24.11	even	2		60.2.e.a.11.5	yes	8
40.3	even	4		300.2.h.b.299.7		8
40.13	odd	4		300.2.h.b.299.6		8
40.19	odd	2		300.2.e.c.251.5		8
40.27	even	4		300.2.h.a.299.2		8
40.29	even	2		300.2.e.c.251.3		8
40.37	odd	4		300.2.h.a.299.3		8
120.29	odd	2		300.2.e.c.251.6		8
120.53	even	4		300.2.h.a.299.4		8
120.59	even	2		300.2.e.c.251.4		8
120.77	even	4		300.2.h.b.299.5		8
120.83	odd	4		300.2.h.a.299.1		8
120.107	odd	4		300.2.h.b.299.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.e.a.11.3	✓	8	24.5	odd	2
60.2.e.a.11.4	yes	8	8.3	odd	2
60.2.e.a.11.5	yes	8	24.11	even	2
60.2.e.a.11.6	yes	8	8.5	even	2
300.2.e.c.251.3		8	40.29	even	2
300.2.e.c.251.4		8	120.59	even	2
300.2.e.c.251.5		8	40.19	odd	2
300.2.e.c.251.6		8	120.29	odd	2
300.2.h.a.299.1		8	120.83	odd	4
300.2.h.a.299.2		8	40.27	even	4
300.2.h.a.299.3		8	40.37	odd	4
300.2.h.a.299.4		8	120.53	even	4
300.2.h.b.299.5		8	120.77	even	4
300.2.h.b.299.6		8	40.13	odd	4
300.2.h.b.299.7		8	40.3	even	4
300.2.h.b.299.8		8	120.107	odd	4
960.2.h.g.191.3		8	12.11	even	2	inner
960.2.h.g.191.4		8	1.1	even	1	trivial
960.2.h.g.191.5		8	4.3	odd	2	inner
960.2.h.g.191.6		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.468213 + 1.66757i) q^{3} -1.00000i q^{5} -0.936426i q^{7} +(-2.56155 - 1.56155i) q^{9} +O(q^{10})\)	\(q+(-0.468213 + 1.66757i) q^{3} -1.00000i q^{5} -0.936426i q^{7} +(-2.56155 - 1.56155i) q^{9} +4.27156 q^{11} -3.12311 q^{13} +(1.66757 + 0.468213i) q^{15} -2.00000i q^{17} -4.27156i q^{19} +(1.56155 + 0.438447i) q^{21} +7.60669 q^{23} -1.00000 q^{25} +(3.80335 - 3.54042i) q^{27} -5.12311i q^{29} +2.39871i q^{31} +(-2.00000 + 7.12311i) q^{33} -0.936426 q^{35} +3.12311 q^{37} +(1.46228 - 5.20798i) q^{39} -7.12311i q^{41} +1.46228i q^{43} +(-1.56155 + 2.56155i) q^{45} -0.936426 q^{47} +6.12311 q^{49} +(3.33513 + 0.936426i) q^{51} +4.24621i q^{53} -4.27156i q^{55} +(7.12311 + 2.00000i) q^{57} +7.19612 q^{59} +5.12311 q^{61} +(-1.46228 + 2.39871i) q^{63} +3.12311i q^{65} +5.20798i q^{67} +(-3.56155 + 12.6847i) q^{69} +6.67026 q^{71} -8.24621 q^{73} +(0.468213 - 1.66757i) q^{75} -4.00000i q^{77} -9.06897i q^{79} +(4.12311 + 8.00000i) q^{81} +4.68213 q^{83} -2.00000 q^{85} +(8.54312 + 2.39871i) q^{87} +6.24621i q^{89} +2.92456i q^{91} +(-4.00000 - 1.12311i) q^{93} -4.27156 q^{95} -6.00000 q^{97} +(-10.9418 - 6.67026i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 8 q^{13} - 4 q^{21} - 8 q^{25} - 16 q^{33} - 8 q^{37} + 4 q^{45} + 16 q^{49} + 24 q^{57} + 8 q^{61} - 12 q^{69} - 16 q^{85} - 32 q^{93} - 48 q^{97}+O(q^{100}) \) 8 * q - 4 * q^9 + 8 * q^13 - 4 * q^21 - 8 * q^25 - 16 * q^33 - 8 * q^37 + 4 * q^45 + 16 * q^49 + 24 * q^57 + 8 * q^61 - 12 * q^69 - 16 * q^85 - 32 * q^93 - 48 * q^97

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