LMFDB - Embedded newform 960.2.h.e.191.2

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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	960.191
Dual form			960.2.h.e.191.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.41421i) q^{3} +1.00000i q^{5} -0.828427i q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	\(q+(1.00000 - 1.41421i) q^{3} +1.00000i q^{5} -0.828427i q^{7} +(-1.00000 - 2.82843i) q^{9} -0.828427 q^{11} -6.82843 q^{13} +(1.41421 + 1.00000i) q^{15} -4.82843i q^{17} -6.00000i q^{19} +(-1.17157 - 0.828427i) q^{21} +4.82843 q^{23} -1.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} -6.00000i q^{29} -2.00000i q^{31} +(-0.828427 + 1.17157i) q^{33} +0.828427 q^{35} -1.17157 q^{37} +(-6.82843 + 9.65685i) q^{39} -1.65685i q^{41} +6.82843i q^{43} +(2.82843 - 1.00000i) q^{45} +8.82843 q^{47} +6.31371 q^{49} +(-6.82843 - 4.82843i) q^{51} -3.65685i q^{53} -0.828427i q^{55} +(-8.48528 - 6.00000i) q^{57} +7.17157 q^{59} -9.31371 q^{61} +(-2.34315 + 0.828427i) q^{63} -6.82843i q^{65} +12.4853i q^{67} +(4.82843 - 6.82843i) q^{69} -11.3137 q^{71} -2.00000 q^{73} +(-1.00000 + 1.41421i) q^{75} +0.686292i q^{77} -6.00000i q^{79} +(-7.00000 + 5.65685i) q^{81} +17.3137 q^{83} +4.82843 q^{85} +(-8.48528 - 6.00000i) q^{87} +5.65685i q^{91} +(-2.82843 - 2.00000i) q^{93} +6.00000 q^{95} -6.00000 q^{97} +(0.828427 + 2.34315i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{9} + 8 q^{11} - 16 q^{13} - 16 q^{21} + 8 q^{23} - 4 q^{25} - 20 q^{27} + 8 q^{33} - 8 q^{35} - 16 q^{37} - 16 q^{39} + 24 q^{47} - 20 q^{49} - 16 q^{51} + 40 q^{59} + 8 q^{61} - 32 q^{63} + 8 q^{69} - 8 q^{73} - 4 q^{75} - 28 q^{81} + 24 q^{83} + 8 q^{85} + 24 q^{95} - 24 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^9 + 8 * q^11 - 16 * q^13 - 16 * q^21 + 8 * q^23 - 4 * q^25 - 20 * q^27 + 8 * q^33 - 8 * q^35 - 16 * q^37 - 16 * q^39 + 24 * q^47 - 20 * q^49 - 16 * q^51 + 40 * q^59 + 8 * q^61 - 32 * q^63 + 8 * q^69 - 8 * q^73 - 4 * q^75 - 28 * q^81 + 24 * q^83 + 8 * q^85 + 24 * q^95 - 24 * q^97 - 8 * q^99

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
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	0.828427i		−	0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$		−	0.828427i		−	0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$		0			0
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$	−0.828427			−0.249780			−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427			−0.249780			−0.124890	+	0.992171i	$0.539858\pi$
$12$		0			0
$13$	−6.82843			−1.89386			−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843			−1.89386			−0.946932	+	0.321433i	$0.895836\pi$
$14$		0			0
$15$	1.41421	+	1.00000i	0.365148	+	0.258199i
$16$		0			0
$17$		−	4.82843i		−	1.17107i	−0.810649	−	0.585533i	$-0.800885\pi$
$17$		−	4.82843i		−	1.17107i	0.810649	−	0.585533i	$-0.199115\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$	−1.17157	−	0.828427i	−0.255658	−	0.180778i
$22$		0			0
$23$	4.82843			1.00680			0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843			1.00680			0.503398	+	0.864054i	$0.332083\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		0			0
$33$	−0.828427	+	1.17157i	−0.144211	+	0.203945i
$34$		0			0
$35$	0.828427			0.140030
$36$		0			0
$37$	−1.17157			−0.192605			−0.0963027	−	0.995352i	$-0.530702\pi$
$37$	−1.17157			−0.192605			−0.0963027	+	0.995352i	$0.530702\pi$
$38$		0			0
$39$	−6.82843	+	9.65685i	−1.09342	+	1.54633i
$40$		0			0
$41$		−	1.65685i		−	0.258757i	−0.991595	−	0.129379i	$-0.958702\pi$
$41$		−	1.65685i		−	0.258757i	0.991595	−	0.129379i	$-0.0412982\pi$
$42$		0			0
$43$			6.82843i			1.04133i	0.853762	+	0.520663i	$0.174315\pi$
$43$			6.82843i			1.04133i	−0.853762	+	0.520663i	$0.825685\pi$
$44$		0			0
$45$	2.82843	−	1.00000i	0.421637	−	0.149071i
$46$		0			0
$47$	8.82843			1.28776			0.643879	−	0.765127i	$-0.277324\pi$
$47$	8.82843			1.28776			0.643879	+	0.765127i	$0.277324\pi$
$48$		0			0
$49$	6.31371			0.901958
$50$		0			0
$51$	−6.82843	−	4.82843i	−0.956171	−	0.676115i
$52$		0			0
$53$		−	3.65685i		−	0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$		−	3.65685i		−	0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$		0			0
$55$		−	0.828427i		−	0.111705i
$56$		0			0
$57$	−8.48528	−	6.00000i	−1.12390	−	0.794719i
$58$		0			0
$59$	7.17157			0.933659			0.466830	−	0.884347i	$-0.345396\pi$
$59$	7.17157			0.933659			0.466830	+	0.884347i	$0.345396\pi$
$60$		0			0
$61$	−9.31371			−1.19250			−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371			−1.19250			−0.596249	+	0.802799i	$0.703343\pi$
$62$		0			0
$63$	−2.34315	+	0.828427i	−0.295209	+	0.104372i
$64$		0			0
$65$		−	6.82843i		−	0.846962i
$66$		0			0
$67$			12.4853i			1.52532i	0.646800	+	0.762660i	$0.276107\pi$
$67$			12.4853i			1.52532i	−0.646800	+	0.762660i	$0.723893\pi$
$68$		0			0
$69$	4.82843	−	6.82843i	0.581274	−	0.822046i
$70$		0			0
$71$	−11.3137			−1.34269			−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137			−1.34269			−0.671345	+	0.741145i	$0.734283\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$	−1.00000	+	1.41421i	−0.115470	+	0.163299i
$76$		0			0
$77$			0.686292i			0.0782102i
$78$		0			0
$79$		−	6.00000i		−	0.675053i	−0.941316	−	0.337526i	$-0.890410\pi$
$79$		−	6.00000i		−	0.675053i	0.941316	−	0.337526i	$-0.109590\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	17.3137			1.90043			0.950213	−	0.311601i	$-0.100865\pi$
$83$	17.3137			1.90043			0.950213	+	0.311601i	$0.100865\pi$
$84$		0			0
$85$	4.82843			0.523716
$86$		0			0
$87$	−8.48528	−	6.00000i	−0.909718	−	0.643268i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$			5.65685i			0.592999i
$92$		0			0
$93$	−2.82843	−	2.00000i	−0.293294	−	0.207390i
$94$		0			0
$95$	6.00000			0.615587
$96$		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
$99$	0.828427	+	2.34315i	0.0832601	+	0.235495i
$100$		0			0
$101$			3.65685i			0.363871i	0.983311	+	0.181935i	$0.0582361\pi$
$101$			3.65685i			0.363871i	−0.983311	+	0.181935i	$0.941764\pi$
$102$		0			0
$103$			4.82843i			0.475759i	0.971295	+	0.237880i	$0.0764523\pi$
$103$			4.82843i			0.475759i	−0.971295	+	0.237880i	$0.923548\pi$
$104$		0			0
$105$	0.828427	−	1.17157i	0.0808462	−	0.114334i
$106$		0			0
$107$	−17.3137			−1.67378			−0.836890	−	0.547372i	$-0.815628\pi$
$107$	−17.3137			−1.67378			−0.836890	+	0.547372i	$0.815628\pi$
$108$		0			0
$109$	3.65685			0.350263			0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685			0.350263			0.175132	+	0.984545i	$0.443965\pi$
$110$		0			0
$111$	−1.17157	+	1.65685i	−0.111201	+	0.157262i
$112$		0			0
$113$			12.8284i			1.20680i	0.797440	+	0.603398i	$0.206187\pi$
$113$			12.8284i			1.20680i	−0.797440	+	0.603398i	$0.793813\pi$
$114$		0			0
$115$			4.82843i			0.450253i
$116$		0			0
$117$	6.82843	+	19.3137i	0.631288	+	1.78555i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	−10.3137			−0.937610
$122$		0			0
$123$	−2.34315	−	1.65685i	−0.211274	−	0.149394i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	10.4853i		−	0.930418i	−0.885201	−	0.465209i	$-0.845979\pi$
$127$		−	10.4853i		−	0.930418i	0.885201	−	0.465209i	$-0.154021\pi$
$128$		0			0
$129$	9.65685	+	6.82843i	0.850239	+	0.601209i
$130$		0			0
$131$	14.4853			1.26558			0.632792	−	0.774321i	$-0.281909\pi$
$131$	14.4853			1.26558			0.632792	+	0.774321i	$0.281909\pi$
$132$		0			0
$133$	−4.97056			−0.431002
$134$		0			0
$135$	1.41421	−	5.00000i	0.121716	−	0.430331i
$136$		0			0
$137$			12.1421i			1.03737i	0.854965	+	0.518686i	$0.173579\pi$
$137$			12.1421i			1.03737i	−0.854965	+	0.518686i	$0.826421\pi$
$138$		0			0
$139$		−	1.31371i		−	0.111427i	−0.998447	−	0.0557137i	$-0.982257\pi$
$139$		−	1.31371i		−	0.111427i	0.998447	−	0.0557137i	$-0.0177434\pi$
$140$		0			0
$141$	8.82843	−	12.4853i	0.743488	−	1.05145i
$142$		0			0
$143$	5.65685			0.473050
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$	6.31371	−	8.92893i	0.520746	−	0.736446i
$148$		0			0
$149$		−	23.6569i		−	1.93805i	−0.246972	−	0.969023i	$-0.579436\pi$
$149$		−	23.6569i		−	1.93805i	0.246972	−	0.969023i	$-0.420564\pi$
$150$		0			0
$151$		−	3.65685i		−	0.297591i	−0.988868	−	0.148795i	$-0.952460\pi$
$151$		−	3.65685i		−	0.297591i	0.988868	−	0.148795i	$-0.0475395\pi$
$152$		0			0
$153$	−13.6569	+	4.82843i	−1.10409	+	0.390355i
$154$		0			0
$155$	2.00000			0.160644
$156$		0			0
$157$	16.4853			1.31567			0.657834	−	0.753163i	$-0.271473\pi$
$157$	16.4853			1.31567			0.657834	+	0.753163i	$0.271473\pi$
$158$		0			0
$159$	−5.17157	−	3.65685i	−0.410132	−	0.290007i
$160$		0			0
$161$		−	4.00000i		−	0.315244i
$162$		0			0
$163$		−	16.4853i		−	1.29123i	−0.763664	−	0.645613i	$-0.776602\pi$
$163$		−	16.4853i		−	1.29123i	0.763664	−	0.645613i	$-0.223398\pi$
$164$		0			0
$165$	−1.17157	−	0.828427i	−0.0912068	−	0.0644930i
$166$		0			0
$167$	19.1716			1.48354			0.741770	−	0.670654i	$-0.233986\pi$
$167$	19.1716			1.48354			0.741770	+	0.670654i	$0.233986\pi$
$168$		0			0
$169$	33.6274			2.58672
$170$		0			0
$171$	−16.9706	+	6.00000i	−1.29777	+	0.458831i
$172$		0			0
$173$			22.9706i			1.74642i	0.487345	+	0.873210i	$0.337966\pi$
$173$			22.9706i			1.74642i	−0.487345	+	0.873210i	$0.662034\pi$
$174$		0			0
$175$			0.828427i			0.0626232i
$176$		0			0
$177$	7.17157	−	10.1421i	0.539048	−	0.762330i
$178$		0			0
$179$	8.82843			0.659868			0.329934	−	0.944004i	$-0.392974\pi$
$179$	8.82843			0.659868			0.329934	+	0.944004i	$0.392974\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	−9.31371	+	13.1716i	−0.688489	+	0.973671i
$184$		0			0
$185$		−	1.17157i		−	0.0861358i
$186$		0			0
$187$			4.00000i			0.292509i
$188$		0			0
$189$	−1.17157	+	4.14214i	−0.0852194	+	0.301296i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$	−14.9706			−1.07760			−0.538802	−	0.842432i	$-0.681123\pi$
$193$	−14.9706			−1.07760			−0.538802	+	0.842432i	$0.681123\pi$
$194$		0			0
$195$	−9.65685	−	6.82843i	−0.691542	−	0.488994i
$196$		0			0
$197$			9.31371i			0.663574i	0.943354	+	0.331787i	$0.107652\pi$
$197$			9.31371i			0.663574i	−0.943354	+	0.331787i	$0.892348\pi$
$198$		0			0
$199$			3.65685i			0.259228i	0.991565	+	0.129614i	$0.0413737\pi$
$199$			3.65685i			0.259228i	−0.991565	+	0.129614i	$0.958626\pi$
$200$		0			0
$201$	17.6569	+	12.4853i	1.24542	+	0.880644i
$202$		0			0
$203$	−4.97056			−0.348865
$204$		0			0
$205$	1.65685			0.115720
$206$		0			0
$207$	−4.82843	−	13.6569i	−0.335599	−	0.949217i
$208$		0			0
$209$			4.97056i			0.343821i
$210$		0			0
$211$		−	26.9706i		−	1.85673i	−0.371670	−	0.928365i	$-0.621215\pi$
$211$		−	26.9706i		−	1.85673i	0.371670	−	0.928365i	$-0.378785\pi$
$212$		0			0
$213$	−11.3137	+	16.0000i	−0.775203	+	1.09630i
$214$		0			0
$215$	−6.82843			−0.465695
$216$		0			0
$217$	−1.65685			−0.112475
$218$		0			0
$219$	−2.00000	+	2.82843i	−0.135147	+	0.191127i
$220$		0			0
$221$			32.9706i			2.21784i
$222$		0			0
$223$		−	10.4853i		−	0.702146i	−0.936348	−	0.351073i	$-0.885817\pi$
$223$		−	10.4853i		−	0.702146i	0.936348	−	0.351073i	$-0.114183\pi$
$224$		0			0
$225$	1.00000	+	2.82843i	0.0666667	+	0.188562i
$226$		0			0
$227$	0.343146			0.0227754			0.0113877	−	0.999935i	$-0.496375\pi$
$227$	0.343146			0.0227754			0.0113877	+	0.999935i	$0.496375\pi$
$228$		0			0
$229$	14.9706			0.989283			0.494641	−	0.869097i	$-0.335299\pi$
$229$	14.9706			0.989283			0.494641	+	0.869097i	$0.335299\pi$
$230$		0			0
$231$	0.970563	+	0.686292i	0.0638583	+	0.0451547i
$232$		0			0
$233$			15.1716i			0.993923i	0.867773	+	0.496961i	$0.165551\pi$
$233$			15.1716i			0.993923i	−0.867773	+	0.496961i	$0.834449\pi$
$234$		0			0
$235$			8.82843i			0.575903i
$236$		0			0
$237$	−8.48528	−	6.00000i	−0.551178	−	0.389742i
$238$		0			0
$239$	16.9706			1.09773			0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706			1.09773			0.548867	+	0.835910i	$0.315059\pi$
$240$		0			0
$241$	6.00000			0.386494			0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000			0.386494			0.193247	+	0.981150i	$0.438098\pi$
$242$		0			0
$243$	1.00000	+	15.5563i	0.0641500	+	0.997940i
$244$		0			0
$245$			6.31371i			0.403368i
$246$		0			0
$247$			40.9706i			2.60689i
$248$		0			0
$249$	17.3137	−	24.4853i	1.09721	−	1.55169i
$250$		0			0
$251$	21.7990			1.37594			0.687970	−	0.725739i	$-0.258502\pi$
$251$	21.7990			1.37594			0.687970	+	0.725739i	$0.258502\pi$
$252$		0			0
$253$	−4.00000			−0.251478
$254$		0			0
$255$	4.82843	−	6.82843i	0.302368	−	0.427613i
$256$		0			0
$257$		−	9.51472i		−	0.593512i	−0.954953	−	0.296756i	$-0.904095\pi$
$257$		−	9.51472i		−	0.593512i	0.954953	−	0.296756i	$-0.0959048\pi$
$258$		0			0
$259$			0.970563i			0.0603078i
$260$		0			0
$261$	−16.9706	+	6.00000i	−1.05045	+	0.371391i
$262$		0			0
$263$	−14.4853			−0.893201			−0.446600	−	0.894734i	$-0.647365\pi$
$263$	−14.4853			−0.893201			−0.446600	+	0.894734i	$0.647365\pi$
$264$		0			0
$265$	3.65685			0.224639
$266$		0			0
$267$		0			0
$268$		0			0
$269$			7.65685i			0.466847i	0.972375	+	0.233423i	$0.0749928\pi$
$269$			7.65685i			0.466847i	−0.972375	+	0.233423i	$0.925007\pi$
$270$		0			0
$271$			18.9706i			1.15238i	0.817316	+	0.576190i	$0.195461\pi$
$271$			18.9706i			1.15238i	−0.817316	+	0.576190i	$0.804539\pi$
$272$		0			0
$273$	8.00000	+	5.65685i	0.484182	+	0.342368i
$274$		0			0
$275$	0.828427			0.0499560
$276$		0			0
$277$	−20.4853			−1.23084			−0.615421	−	0.788199i	$-0.711014\pi$
$277$	−20.4853			−1.23084			−0.615421	+	0.788199i	$0.711014\pi$
$278$		0			0
$279$	−5.65685	+	2.00000i	−0.338667	+	0.119737i
$280$		0			0
$281$			0.686292i			0.0409407i	0.999790	+	0.0204704i	$0.00651637\pi$
$281$			0.686292i			0.0409407i	−0.999790	+	0.0204704i	$0.993484\pi$
$282$		0			0
$283$		−	22.8284i		−	1.35701i	−0.734596	−	0.678505i	$-0.762628\pi$
$283$		−	22.8284i		−	1.35701i	0.734596	−	0.678505i	$-0.237372\pi$
$284$		0			0
$285$	6.00000	−	8.48528i	0.355409	−	0.502625i
$286$		0			0
$287$	−1.37258			−0.0810210
$288$		0			0
$289$	−6.31371			−0.371395
$290$		0			0
$291$	−6.00000	+	8.48528i	−0.351726	+	0.497416i
$292$		0			0
$293$		−	11.6569i		−	0.681001i	−0.940244	−	0.340500i	$-0.889404\pi$
$293$		−	11.6569i		−	0.681001i	0.940244	−	0.340500i	$-0.110596\pi$
$294$		0			0
$295$			7.17157i			0.417545i
$296$		0			0
$297$	4.14214	+	1.17157i	0.240351	+	0.0679816i
$298$		0			0
$299$	−32.9706			−1.90674
$300$		0			0
$301$	5.65685			0.326056
$302$		0			0
$303$	5.17157	+	3.65685i	0.297099	+	0.210081i
$304$		0			0
$305$		−	9.31371i		−	0.533301i
$306$		0			0
$307$			3.51472i			0.200596i	0.994957	+	0.100298i	$0.0319796\pi$
$307$			3.51472i			0.200596i	−0.994957	+	0.100298i	$0.968020\pi$
$308$		0			0
$309$	6.82843	+	4.82843i	0.388456	+	0.274680i
$310$		0			0
$311$	−11.3137			−0.641542			−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137			−0.641542			−0.320771	+	0.947157i	$0.603942\pi$
$312$		0			0
$313$	2.68629			0.151838			0.0759191	−	0.997114i	$-0.475811\pi$
$313$	2.68629			0.151838			0.0759191	+	0.997114i	$0.475811\pi$
$314$		0			0
$315$	−0.828427	−	2.34315i	−0.0466766	−	0.132021i
$316$		0			0
$317$		−	17.3137i		−	0.972435i	−0.873838	−	0.486217i	$-0.838376\pi$
$317$		−	17.3137i		−	0.972435i	0.873838	−	0.486217i	$-0.161624\pi$
$318$		0			0
$319$			4.97056i			0.278298i
$320$		0			0
$321$	−17.3137	+	24.4853i	−0.966357	+	1.36664i
$322$		0			0
$323$	−28.9706			−1.61197
$324$		0			0
$325$	6.82843			0.378773
$326$		0			0
$327$	3.65685	−	5.17157i	0.202225	−	0.285989i
$328$		0			0
$329$		−	7.31371i		−	0.403218i
$330$		0			0
$331$			22.9706i			1.26258i	0.775548	+	0.631288i	$0.217473\pi$
$331$			22.9706i			1.26258i	−0.775548	+	0.631288i	$0.782527\pi$
$332$		0			0
$333$	1.17157	+	3.31371i	0.0642018	+	0.181590i
$334$		0			0
$335$	−12.4853			−0.682144
$336$		0			0
$337$	1.02944			0.0560770			0.0280385	−	0.999607i	$-0.491074\pi$
$337$	1.02944			0.0560770			0.0280385	+	0.999607i	$0.491074\pi$
$338$		0			0
$339$	18.1421	+	12.8284i	0.985346	+	0.696745i
$340$		0			0
$341$			1.65685i			0.0897237i
$342$		0			0
$343$		−	11.0294i		−	0.595534i
$344$		0			0
$345$	6.82843	+	4.82843i	0.367630	+	0.259954i
$346$		0			0
$347$	9.31371			0.499986			0.249993	−	0.968248i	$-0.419572\pi$
$347$	9.31371			0.499986			0.249993	+	0.968248i	$0.419572\pi$
$348$		0			0
$349$	22.9706			1.22959			0.614793	−	0.788688i	$-0.289240\pi$
$349$	22.9706			1.22959			0.614793	+	0.788688i	$0.289240\pi$
$350$		0			0
$351$	34.1421	+	9.65685i	1.82237	+	0.515445i
$352$		0			0
$353$		−	11.1716i		−	0.594603i	−0.954784	−	0.297301i	$-0.903913\pi$
$353$		−	11.1716i		−	0.594603i	0.954784	−	0.297301i	$-0.0960866\pi$
$354$		0			0
$355$		−	11.3137i		−	0.600469i
$356$		0			0
$357$	−4.00000	+	5.65685i	−0.211702	+	0.299392i
$358$		0			0
$359$	8.68629			0.458445			0.229222	−	0.973374i	$-0.426382\pi$
$359$	8.68629			0.458445			0.229222	+	0.973374i	$0.426382\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−10.3137	+	14.5858i	−0.541329	+	0.765555i
$364$		0			0
$365$		−	2.00000i		−	0.104685i
$366$		0			0
$367$		−	0.142136i		−	0.00741942i	−0.999993	−	0.00370971i	$-0.998819\pi$
$367$		−	0.142136i		−	0.00741942i	0.999993	−	0.00370971i	$-0.00118084\pi$
$368$		0			0
$369$	−4.68629	+	1.65685i	−0.243959	+	0.0862524i
$370$		0			0
$371$	−3.02944			−0.157281
$372$		0			0
$373$	0.485281			0.0251269			0.0125635	−	0.999921i	$-0.496001\pi$
$373$	0.485281			0.0251269			0.0125635	+	0.999921i	$0.496001\pi$
$374$		0			0
$375$	−1.41421	−	1.00000i	−0.0730297	−	0.0516398i
$376$		0			0
$377$			40.9706i			2.11009i
$378$		0			0
$379$		−	33.3137i		−	1.71121i	−0.517629	−	0.855605i	$-0.673185\pi$
$379$		−	33.3137i		−	1.71121i	0.517629	−	0.855605i	$-0.326815\pi$
$380$		0			0
$381$	−14.8284	−	10.4853i	−0.759683	−	0.537177i
$382$		0			0
$383$	−4.14214			−0.211653			−0.105827	−	0.994385i	$-0.533749\pi$
$383$	−4.14214			−0.211653			−0.105827	+	0.994385i	$0.533749\pi$
$384$		0			0
$385$	−0.686292			−0.0349767
$386$		0			0
$387$	19.3137	−	6.82843i	0.981771	−	0.347108i
$388$		0			0
$389$		−	16.6274i		−	0.843044i	−0.906818	−	0.421522i	$-0.861496\pi$
$389$		−	16.6274i		−	0.843044i	0.906818	−	0.421522i	$-0.138504\pi$
$390$		0			0
$391$		−	23.3137i		−	1.17902i
$392$		0			0
$393$	14.4853	−	20.4853i	0.730686	−	1.03335i
$394$		0			0
$395$	6.00000			0.301893
$396$		0			0
$397$	7.51472			0.377153			0.188576	−	0.982059i	$-0.439613\pi$
$397$	7.51472			0.377153			0.188576	+	0.982059i	$0.439613\pi$
$398$		0			0
$399$	−4.97056	+	7.02944i	−0.248839	+	0.351912i
$400$		0			0
$401$		−	16.9706i		−	0.847469i	−0.905786	−	0.423735i	$-0.860719\pi$
$401$		−	16.9706i		−	0.847469i	0.905786	−	0.423735i	$-0.139281\pi$
$402$		0			0
$403$			13.6569i			0.680296i
$404$		0			0
$405$	−5.65685	−	7.00000i	−0.281091	−	0.347833i
$406$		0			0
$407$	0.970563			0.0481090
$408$		0			0
$409$	−33.3137			−1.64726			−0.823628	−	0.567130i	$-0.808054\pi$
$409$	−33.3137			−1.64726			−0.823628	+	0.567130i	$0.808054\pi$
$410$		0			0
$411$	17.1716	+	12.1421i	0.847011	+	0.598927i
$412$		0			0
$413$		−	5.94113i		−	0.292344i
$414$		0			0
$415$			17.3137i			0.849897i
$416$		0			0
$417$	−1.85786	−	1.31371i	−0.0909800	−	0.0643326i
$418$		0			0
$419$	11.1716			0.545767			0.272884	−	0.962047i	$-0.412023\pi$
$419$	11.1716			0.545767			0.272884	+	0.962047i	$0.412023\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		0			0
$423$	−8.82843	−	24.9706i	−0.429253	−	1.21411i
$424$		0			0
$425$			4.82843i			0.234213i
$426$		0			0
$427$			7.71573i			0.373390i
$428$		0			0
$429$	5.65685	−	8.00000i	0.273115	−	0.386244i
$430$		0			0
$431$	−6.34315			−0.305539			−0.152769	−	0.988262i	$-0.548819\pi$
$431$	−6.34315			−0.305539			−0.152769	+	0.988262i	$0.548819\pi$
$432$		0			0
$433$	−30.0000			−1.44171			−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000			−1.44171			−0.720854	+	0.693087i	$0.756250\pi$
$434$		0			0
$435$	6.00000	−	8.48528i	0.287678	−	0.406838i
$436$		0			0
$437$		−	28.9706i		−	1.38585i
$438$		0			0
$439$		−	30.2843i		−	1.44539i	−0.691167	−	0.722695i	$-0.742903\pi$
$439$		−	30.2843i		−	1.44539i	0.691167	−	0.722695i	$-0.257097\pi$
$440$		0			0
$441$	−6.31371	−	17.8579i	−0.300653	−	0.850374i
$442$		0			0
$443$	−8.34315			−0.396395			−0.198197	−	0.980162i	$-0.563509\pi$
$443$	−8.34315			−0.396395			−0.198197	+	0.980162i	$0.563509\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−33.4558	−	23.6569i	−1.58241	−	1.11893i
$448$		0			0
$449$		−	33.6569i		−	1.58837i	−0.607679	−	0.794183i	$-0.707899\pi$
$449$		−	33.6569i		−	1.58837i	0.607679	−	0.794183i	$-0.292101\pi$
$450$		0			0
$451$			1.37258i			0.0646324i
$452$		0			0
$453$	−5.17157	−	3.65685i	−0.242982	−	0.171814i
$454$		0			0
$455$	−5.65685			−0.265197
$456$		0			0
$457$	−1.02944			−0.0481550			−0.0240775	−	0.999710i	$-0.507665\pi$
$457$	−1.02944			−0.0481550			−0.0240775	+	0.999710i	$0.507665\pi$
$458$		0			0
$459$	−6.82843	+	24.1421i	−0.318724	+	1.12686i
$460$		0			0
$461$		−	9.31371i		−	0.433783i	−0.976196	−	0.216891i	$-0.930408\pi$
$461$		−	9.31371i		−	0.433783i	0.976196	−	0.216891i	$-0.0695917\pi$
$462$		0			0
$463$		−	12.8284i		−	0.596188i	−0.954537	−	0.298094i	$-0.903649\pi$
$463$		−	12.8284i		−	0.596188i	0.954537	−	0.298094i	$-0.0963508\pi$
$464$		0			0
$465$	2.00000	−	2.82843i	0.0927478	−	0.131165i
$466$		0			0
$467$	12.3431			0.571173			0.285586	−	0.958353i	$-0.407812\pi$
$467$	12.3431			0.571173			0.285586	+	0.958353i	$0.407812\pi$
$468$		0			0
$469$	10.3431			0.477602
$470$		0			0
$471$	16.4853	−	23.3137i	0.759602	−	1.07424i
$472$		0			0
$473$		−	5.65685i		−	0.260102i
$474$		0			0
$475$			6.00000i			0.275299i
$476$		0			0
$477$	−10.3431	+	3.65685i	−0.473580	+	0.167436i
$478$		0			0
$479$	−13.6569			−0.623998			−0.311999	−	0.950082i	$-0.600998\pi$
$479$	−13.6569			−0.623998			−0.311999	+	0.950082i	$0.600998\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$	−5.65685	−	4.00000i	−0.257396	−	0.182006i
$484$		0			0
$485$		−	6.00000i		−	0.272446i
$486$		0			0
$487$			27.4558i			1.24414i	0.782960	+	0.622072i	$0.213709\pi$
$487$			27.4558i			1.24414i	−0.782960	+	0.622072i	$0.786291\pi$
$488$		0			0
$489$	−23.3137	−	16.4853i	−1.05428	−	0.745490i
$490$		0			0
$491$	20.8284			0.939974			0.469987	−	0.882673i	$-0.344259\pi$
$491$	20.8284			0.939974			0.469987	+	0.882673i	$0.344259\pi$
$492$		0			0
$493$	−28.9706			−1.30477
$494$		0			0
$495$	−2.34315	+	0.828427i	−0.105317	+	0.0372350i
$496$		0			0
$497$			9.37258i			0.420418i
$498$		0			0
$499$		−	2.68629i		−	0.120255i	−0.998191	−	0.0601275i	$-0.980849\pi$
$499$		−	2.68629i		−	0.120255i	0.998191	−	0.0601275i	$-0.0191507\pi$
$500$		0			0
$501$	19.1716	−	27.1127i	0.856523	−	1.21131i
$502$		0			0
$503$	4.82843			0.215289			0.107644	−	0.994189i	$-0.465669\pi$
$503$	4.82843			0.215289			0.107644	+	0.994189i	$0.465669\pi$
$504$		0			0
$505$	−3.65685			−0.162728
$506$		0			0
$507$	33.6274	−	47.5563i	1.49345	−	2.11205i
$508$		0			0
$509$			23.6569i			1.04857i	0.851542	+	0.524286i	$0.175668\pi$
$509$			23.6569i			1.04857i	−0.851542	+	0.524286i	$0.824332\pi$
$510$		0			0
$511$			1.65685i			0.0732949i
$512$		0			0
$513$	−8.48528	+	30.0000i	−0.374634	+	1.32453i
$514$		0			0
$515$	−4.82843			−0.212766
$516$		0			0
$517$	−7.31371			−0.321657
$518$		0			0
$519$	32.4853	+	22.9706i	1.42595	+	1.00830i
$520$		0			0
$521$		−	12.0000i		−	0.525730i	−0.964833	−	0.262865i	$-0.915333\pi$
$521$		−	12.0000i		−	0.525730i	0.964833	−	0.262865i	$-0.0846673\pi$
$522$		0			0
$523$			12.4853i			0.545943i	0.962022	+	0.272972i	$0.0880065\pi$
$523$			12.4853i			0.545943i	−0.962022	+	0.272972i	$0.911993\pi$
$524$		0			0
$525$	1.17157	+	0.828427i	0.0511316	+	0.0361555i
$526$		0			0
$527$	−9.65685			−0.420659
$528$		0			0
$529$	0.313708			0.0136395
$530$		0			0
$531$	−7.17157	−	20.2843i	−0.311220	−	0.880262i
$532$		0			0
$533$			11.3137i			0.490051i
$534$		0			0
$535$		−	17.3137i		−	0.748537i
$536$		0			0
$537$	8.82843	−	12.4853i	0.380975	−	0.538780i
$538$		0			0
$539$	−5.23045			−0.225291
$540$		0			0
$541$	8.62742			0.370922			0.185461	−	0.982652i	$-0.440622\pi$
$541$	8.62742			0.370922			0.185461	+	0.982652i	$0.440622\pi$
$542$		0			0
$543$	2.00000	−	2.82843i	0.0858282	−	0.121379i
$544$		0			0
$545$			3.65685i			0.156642i
$546$		0			0
$547$			29.4558i			1.25944i	0.776822	+	0.629720i	$0.216831\pi$
$547$			29.4558i			1.25944i	−0.776822	+	0.629720i	$0.783169\pi$
$548$		0			0
$549$	9.31371	+	26.3431i	0.397499	+	1.12430i
$550$		0			0
$551$	−36.0000			−1.53365
$552$		0			0
$553$	−4.97056			−0.211370
$554$		0			0
$555$	−1.65685	−	1.17157i	−0.0703295	−	0.0497305i
$556$		0			0
$557$		−	34.9706i		−	1.48175i	−0.671643	−	0.740875i	$-0.734411\pi$
$557$		−	34.9706i		−	1.48175i	0.671643	−	0.740875i	$-0.265589\pi$
$558$		0			0
$559$		−	46.6274i		−	1.97213i
$560$		0			0
$561$	5.65685	+	4.00000i	0.238833	+	0.168880i
$562$		0			0
$563$	5.31371			0.223946			0.111973	−	0.993711i	$-0.464283\pi$
$563$	5.31371			0.223946			0.111973	+	0.993711i	$0.464283\pi$
$564$		0			0
$565$	−12.8284			−0.539696
$566$		0			0
$567$	4.68629	+	5.79899i	0.196806	+	0.243535i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	1.02944i		−	0.0430806i	−0.999768	−	0.0215403i	$-0.993143\pi$
$571$		−	1.02944i		−	0.0430806i	0.999768	−	0.0215403i	$-0.00685702\pi$
$572$		0			0
$573$	12.0000	−	16.9706i	0.501307	−	0.708955i
$574$		0			0
$575$	−4.82843			−0.201359
$576$		0			0
$577$	18.9706			0.789755			0.394877	−	0.918734i	$-0.370787\pi$
$577$	18.9706			0.789755			0.394877	+	0.918734i	$0.370787\pi$
$578$		0			0
$579$	−14.9706	+	21.1716i	−0.622155	+	0.879861i
$580$		0			0
$581$		−	14.3431i		−	0.595054i
$582$		0			0
$583$			3.02944i			0.125466i
$584$		0			0
$585$	−19.3137	+	6.82843i	−0.798524	+	0.282321i
$586$		0			0
$587$	−28.6274			−1.18158			−0.590790	−	0.806825i	$-0.701184\pi$
$587$	−28.6274			−1.18158			−0.590790	+	0.806825i	$0.701184\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	13.1716	+	9.31371i	0.541806	+	0.383115i
$592$		0			0
$593$		−	7.85786i		−	0.322684i	−0.986899	−	0.161342i	$-0.948418\pi$
$593$		−	7.85786i		−	0.322684i	0.986899	−	0.161342i	$-0.0515822\pi$
$594$		0			0
$595$		−	4.00000i		−	0.163984i
$596$		0			0
$597$	5.17157	+	3.65685i	0.211658	+	0.149665i
$598$		0			0
$599$	26.6274			1.08797			0.543983	−	0.839096i	$-0.316915\pi$
$599$	26.6274			1.08797			0.543983	+	0.839096i	$0.316915\pi$
$600$		0			0
$601$	8.62742			0.351920			0.175960	−	0.984397i	$-0.443697\pi$
$601$	8.62742			0.351920			0.175960	+	0.984397i	$0.443697\pi$
$602$		0			0
$603$	35.3137	−	12.4853i	1.43809	−	0.508440i
$604$		0			0
$605$		−	10.3137i		−	0.419312i
$606$		0			0
$607$			38.4853i			1.56207i	0.624488	+	0.781035i	$0.285308\pi$
$607$			38.4853i			1.56207i	−0.624488	+	0.781035i	$0.714692\pi$
$608$		0			0
$609$	−4.97056	+	7.02944i	−0.201417	+	0.284847i
$610$		0			0
$611$	−60.2843			−2.43884
$612$		0			0
$613$	25.4558			1.02815			0.514076	−	0.857745i	$-0.328135\pi$
$613$	25.4558			1.02815			0.514076	+	0.857745i	$0.328135\pi$
$614$		0			0
$615$	1.65685	−	2.34315i	0.0668108	−	0.0944848i
$616$		0			0
$617$			8.82843i			0.355419i	0.984083	+	0.177710i	$0.0568688\pi$
$617$			8.82843i			0.355419i	−0.984083	+	0.177710i	$0.943131\pi$
$618$		0			0
$619$		−	6.00000i		−	0.241160i	−0.992704	−	0.120580i	$-0.961525\pi$
$619$		−	6.00000i		−	0.241160i	0.992704	−	0.120580i	$-0.0384755\pi$
$620$		0			0
$621$	−24.1421	−	6.82843i	−0.968791	−	0.274015i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	7.02944	+	4.97056i	0.280729	+	0.198505i
$628$		0			0
$629$			5.65685i			0.225554i
$630$		0			0
$631$			10.6863i			0.425415i	0.977116	+	0.212707i	$0.0682281\pi$
$631$			10.6863i			0.425415i	−0.977116	+	0.212707i	$0.931772\pi$
$632$		0			0
$633$	−38.1421	−	26.9706i	−1.51601	−	1.07198i
$634$		0			0
$635$	10.4853			0.416096
$636$		0			0
$637$	−43.1127			−1.70819
$638$		0			0
$639$	11.3137	+	32.0000i	0.447563	+	1.26590i
$640$		0			0
$641$		−	30.6274i		−	1.20971i	−0.796336	−	0.604855i	$-0.793231\pi$
$641$		−	30.6274i		−	1.20971i	0.796336	−	0.604855i	$-0.206769\pi$
$642$		0			0
$643$			10.8284i			0.427031i	0.976940	+	0.213516i	$0.0684915\pi$
$643$			10.8284i			0.427031i	−0.976940	+	0.213516i	$0.931509\pi$
$644$		0			0
$645$	−6.82843	+	9.65685i	−0.268869	+	0.380238i
$646$		0			0
$647$	−20.8284			−0.818850			−0.409425	−	0.912344i	$-0.634271\pi$
$647$	−20.8284			−0.818850			−0.409425	+	0.912344i	$0.634271\pi$
$648$		0			0
$649$	−5.94113			−0.233210
$650$		0			0
$651$	−1.65685	+	2.34315i	−0.0649372	+	0.0918351i
$652$		0			0
$653$			26.2843i			1.02858i	0.857615	+	0.514292i	$0.171945\pi$
$653$			26.2843i			1.02858i	−0.857615	+	0.514292i	$0.828055\pi$
$654$		0			0
$655$			14.4853i			0.565987i
$656$		0			0
$657$	2.00000	+	5.65685i	0.0780274	+	0.220695i
$658$		0			0
$659$	−45.7990			−1.78408			−0.892038	−	0.451961i	$-0.850725\pi$
$659$	−45.7990			−1.78408			−0.892038	+	0.451961i	$0.850725\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$		0			0
$663$	46.6274	+	32.9706i	1.81086	+	1.28047i
$664$		0			0
$665$		−	4.97056i		−	0.192750i
$666$		0			0
$667$		−	28.9706i		−	1.12174i
$668$		0			0
$669$	−14.8284	−	10.4853i	−0.573300	−	0.405384i
$670$		0			0
$671$	7.71573			0.297862
$672$		0			0
$673$	35.9411			1.38543			0.692714	−	0.721212i	$-0.256415\pi$
$673$	35.9411			1.38543			0.692714	+	0.721212i	$0.256415\pi$
$674$		0			0
$675$	5.00000	+	1.41421i	0.192450	+	0.0544331i
$676$		0			0
$677$		−	0.343146i		−	0.0131882i	−0.999978	−	0.00659408i	$-0.997901\pi$
$677$		−	0.343146i		−	0.0131882i	0.999978	−	0.00659408i	$-0.00209898\pi$
$678$		0			0
$679$			4.97056i			0.190753i
$680$		0			0
$681$	0.343146	−	0.485281i	0.0131494	−	0.0185960i
$682$		0			0
$683$	−20.3431			−0.778409			−0.389204	−	0.921151i	$-0.627250\pi$
$683$	−20.3431			−0.778409			−0.389204	+	0.921151i	$0.627250\pi$
$684$		0			0
$685$	−12.1421			−0.463927
$686$		0			0
$687$	14.9706	−	21.1716i	0.571163	−	0.807746i
$688$		0			0
$689$			24.9706i			0.951303i
$690$		0			0
$691$		−	9.02944i		−	0.343496i	−0.985141	−	0.171748i	$-0.945059\pi$
$691$		−	9.02944i		−	0.343496i	0.985141	−	0.171748i	$-0.0549415\pi$
$692$		0			0
$693$	1.94113	−	0.686292i	0.0737373	−	0.0260701i
$694$		0			0
$695$	1.31371			0.0498318
$696$		0			0
$697$	−8.00000			−0.303022
$698$		0			0
$699$	21.4558	+	15.1716i	0.811535	+	0.573842i
$700$		0			0
$701$			27.9411i			1.05532i	0.849455	+	0.527661i	$0.176931\pi$
$701$			27.9411i			1.05532i	−0.849455	+	0.527661i	$0.823069\pi$
$702$		0			0
$703$			7.02944i			0.265120i
$704$		0			0
$705$	12.4853	+	8.82843i	0.470223	+	0.332498i
$706$		0			0
$707$	3.02944			0.113934
$708$		0			0
$709$	−42.9706			−1.61379			−0.806897	−	0.590693i	$-0.798855\pi$
$709$	−42.9706			−1.61379			−0.806897	+	0.590693i	$0.798855\pi$
$710$		0			0
$711$	−16.9706	+	6.00000i	−0.636446	+	0.225018i
$712$		0			0
$713$		−	9.65685i		−	0.361652i
$714$		0			0
$715$			5.65685i			0.211554i
$716$		0			0
$717$	16.9706	−	24.0000i	0.633777	−	0.896296i
$718$		0			0
$719$	−43.3137			−1.61533			−0.807664	−	0.589642i	$-0.799269\pi$
$719$	−43.3137			−1.61533			−0.807664	+	0.589642i	$0.799269\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$		0			0
$723$	6.00000	−	8.48528i	0.223142	−	0.315571i
$724$		0			0
$725$			6.00000i			0.222834i
$726$		0			0
$727$		−	19.1716i		−	0.711034i	−0.934670	−	0.355517i	$-0.884305\pi$
$727$		−	19.1716i		−	0.711034i	0.934670	−	0.355517i	$-0.115695\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	32.9706			1.21946
$732$		0			0
$733$	30.8284			1.13867			0.569337	−	0.822104i	$-0.307200\pi$
$733$	30.8284			1.13867			0.569337	+	0.822104i	$0.307200\pi$
$734$		0			0
$735$	8.92893	+	6.31371i	0.329349	+	0.232885i
$736$		0			0
$737$		−	10.3431i		−	0.380995i
$738$		0			0
$739$			21.3137i			0.784037i	0.919957	+	0.392019i	$0.128223\pi$
$739$			21.3137i			0.784037i	−0.919957	+	0.392019i	$0.871777\pi$
$740$		0			0
$741$	57.9411	+	40.9706i	2.12852	+	1.50509i
$742$		0			0
$743$	19.4558			0.713766			0.356883	−	0.934149i	$-0.383840\pi$
$743$	19.4558			0.713766			0.356883	+	0.934149i	$0.383840\pi$
$744$		0			0
$745$	23.6569			0.866720
$746$		0			0
$747$	−17.3137	−	48.9706i	−0.633475	−	1.79174i
$748$		0			0
$749$			14.3431i			0.524087i
$750$		0			0
$751$			25.3137i			0.923710i	0.886955	+	0.461855i	$0.152816\pi$
$751$			25.3137i			0.923710i	−0.886955	+	0.461855i	$0.847184\pi$
$752$		0			0
$753$	21.7990	−	30.8284i	0.794399	−	1.12345i
$754$		0			0
$755$	3.65685			0.133087
$756$		0			0
$757$	10.1421			0.368622			0.184311	−	0.982868i	$-0.440995\pi$
$757$	10.1421			0.368622			0.184311	+	0.982868i	$0.440995\pi$
$758$		0			0
$759$	−4.00000	+	5.65685i	−0.145191	+	0.205331i
$760$		0			0
$761$			31.3137i			1.13512i	0.823332	+	0.567561i	$0.192113\pi$
$761$			31.3137i			1.13512i	−0.823332	+	0.567561i	$0.807887\pi$
$762$		0			0
$763$		−	3.02944i		−	0.109673i
$764$		0			0
$765$	−4.82843	−	13.6569i	−0.174572	−	0.493765i
$766$		0			0
$767$	−48.9706			−1.76822
$768$		0			0
$769$	10.6863			0.385358			0.192679	−	0.981262i	$-0.438282\pi$
$769$	10.6863			0.385358			0.192679	+	0.981262i	$0.438282\pi$
$770$		0			0
$771$	−13.4558	−	9.51472i	−0.484600	−	0.342664i
$772$		0			0
$773$		−	29.3137i		−	1.05434i	−0.849760	−	0.527170i	$-0.823253\pi$
$773$		−	29.3137i		−	1.05434i	0.849760	−	0.527170i	$-0.176747\pi$
$774$		0			0
$775$			2.00000i			0.0718421i
$776$		0			0
$777$	1.37258	+	0.970563i	0.0492411	+	0.0348187i
$778$		0			0
$779$	−9.94113			−0.356178
$780$		0			0
$781$	9.37258			0.335377
$782$		0			0
$783$	−8.48528	+	30.0000i	−0.303239	+	1.07211i
$784$		0			0
$785$			16.4853i			0.588385i
$786$		0			0
$787$			39.7990i			1.41868i	0.704866	+	0.709340i	$0.251007\pi$
$787$			39.7990i			1.41868i	−0.704866	+	0.709340i	$0.748993\pi$
$788$		0			0
$789$	−14.4853	+	20.4853i	−0.515690	+	0.729295i
$790$		0			0
$791$	10.6274			0.377867
$792$		0			0
$793$	63.5980			2.25843
$794$		0			0
$795$	3.65685	−	5.17157i	0.129695	−	0.183417i
$796$		0			0
$797$			48.6274i			1.72247i	0.508206	+	0.861236i	$0.330309\pi$
$797$			48.6274i			1.72247i	−0.508206	+	0.861236i	$0.669691\pi$
$798$		0			0
$799$		−	42.6274i		−	1.50805i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.65685			0.0584691
$804$		0			0
$805$	4.00000			0.140981
$806$		0			0
$807$	10.8284	+	7.65685i	0.381179	+	0.269534i
$808$		0			0
$809$			5.65685i			0.198884i	0.995043	+	0.0994422i	$0.0317058\pi$
$809$			5.65685i			0.198884i	−0.995043	+	0.0994422i	$0.968294\pi$
$810$		0			0
$811$			18.2843i			0.642048i	0.947071	+	0.321024i	$0.104027\pi$
$811$			18.2843i			0.642048i	−0.947071	+	0.321024i	$0.895973\pi$
$812$		0			0
$813$	26.8284	+	18.9706i	0.940914	+	0.665327i
$814$		0			0
$815$	16.4853			0.577454
$816$		0			0
$817$	40.9706			1.43338
$818$		0			0
$819$	16.0000	−	5.65685i	0.559085	−	0.197666i
$820$		0			0
$821$			24.3431i			0.849582i	0.905292	+	0.424791i	$0.139652\pi$
$821$			24.3431i			0.849582i	−0.905292	+	0.424791i	$0.860348\pi$
$822$		0			0
$823$			11.4558i			0.399326i	0.979865	+	0.199663i	$0.0639847\pi$
$823$			11.4558i			0.399326i	−0.979865	+	0.199663i	$0.936015\pi$
$824$		0			0
$825$	0.828427	−	1.17157i	0.0288421	−	0.0407889i
$826$		0			0
$827$	−14.6863			−0.510692			−0.255346	−	0.966850i	$-0.582189\pi$
$827$	−14.6863			−0.510692			−0.255346	+	0.966850i	$0.582189\pi$
$828$		0			0
$829$	−44.9117			−1.55985			−0.779924	−	0.625875i	$-0.784742\pi$
$829$	−44.9117			−1.55985			−0.779924	+	0.625875i	$0.784742\pi$
$830$		0			0
$831$	−20.4853	+	28.9706i	−0.710627	+	1.00498i
$832$		0			0
$833$		−	30.4853i		−	1.05625i
$834$		0			0
$835$			19.1716i			0.663460i
$836$		0			0
$837$	−2.82843	+	10.0000i	−0.0977647	+	0.345651i
$838$		0			0
$839$	45.9411			1.58606			0.793032	−	0.609180i	$-0.208501\pi$
$839$	45.9411			1.58606			0.793032	+	0.609180i	$0.208501\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	0.970563	+	0.686292i	0.0334280	+	0.0236371i
$844$		0			0
$845$			33.6274i			1.15682i
$846$		0			0
$847$			8.54416i			0.293581i
$848$		0			0
$849$	−32.2843	−	22.8284i	−1.10799	−	0.783470i
$850$		0			0
$851$	−5.65685			−0.193914
$852$		0			0
$853$	31.5147			1.07904			0.539522	−	0.841972i	$-0.318605\pi$
$853$	31.5147			1.07904			0.539522	+	0.841972i	$0.318605\pi$
$854$		0			0
$855$	−6.00000	−	16.9706i	−0.205196	−	0.580381i
$856$		0			0
$857$		−	29.7990i		−	1.01791i	−0.860792	−	0.508957i	$-0.830031\pi$
$857$		−	29.7990i		−	1.01791i	0.860792	−	0.508957i	$-0.169969\pi$
$858$		0			0
$859$		−	22.0000i		−	0.750630i	−0.926897	−	0.375315i	$-0.877534\pi$
$859$		−	22.0000i		−	0.750630i	0.926897	−	0.375315i	$-0.122466\pi$
$860$		0			0
$861$	−1.37258	+	1.94113i	−0.0467775	+	0.0661534i
$862$		0			0
$863$	−18.2010			−0.619570			−0.309785	−	0.950807i	$-0.600257\pi$
$863$	−18.2010			−0.619570			−0.309785	+	0.950807i	$0.600257\pi$
$864$		0			0
$865$	−22.9706			−0.781023
$866$		0			0
$867$	−6.31371	+	8.92893i	−0.214425	+	0.303242i
$868$		0			0
$869$			4.97056i			0.168615i
$870$		0			0
$871$		−	85.2548i		−	2.88875i
$872$		0			0
$873$	6.00000	+	16.9706i	0.203069	+	0.574367i
$874$		0			0
$875$	−0.828427			−0.0280059
$876$		0			0
$877$	11.7990			0.398424			0.199212	−	0.979956i	$-0.436162\pi$
$877$	11.7990			0.398424			0.199212	+	0.979956i	$0.436162\pi$
$878$		0			0
$879$	−16.4853	−	11.6569i	−0.556035	−	0.393176i
$880$		0			0
$881$			39.5980i			1.33409i	0.745018	+	0.667045i	$0.232441\pi$
$881$			39.5980i			1.33409i	−0.745018	+	0.667045i	$0.767559\pi$
$882$		0			0
$883$		−	32.4853i		−	1.09322i	−0.837388	−	0.546608i	$-0.815919\pi$
$883$		−	32.4853i		−	1.09322i	0.837388	−	0.546608i	$-0.184081\pi$
$884$		0			0
$885$	10.1421	+	7.17157i	0.340924	+	0.241070i
$886$		0			0
$887$	35.1716			1.18095			0.590473	−	0.807057i	$-0.298941\pi$
$887$	35.1716			1.18095			0.590473	+	0.807057i	$0.298941\pi$
$888$		0			0
$889$	−8.68629			−0.291329
$890$		0			0
$891$	5.79899	−	4.68629i	0.194273	−	0.156997i
$892$		0			0
$893$		−	52.9706i		−	1.77259i
$894$		0			0
$895$			8.82843i			0.295102i
$896$		0			0
$897$	−32.9706	+	46.6274i	−1.10086	+	1.55684i
$898$		0			0
$899$	−12.0000			−0.400222
$900$		0			0
$901$	−17.6569			−0.588235
$902$		0			0
$903$	5.65685	−	8.00000i	0.188248	−	0.266223i
$904$		0			0
$905$			2.00000i			0.0664822i
$906$		0			0
$907$			5.17157i			0.171719i	0.996307	+	0.0858596i	$0.0273637\pi$
$907$			5.17157i			0.171719i	−0.996307	+	0.0858596i	$0.972636\pi$
$908$		0			0
$909$	10.3431	−	3.65685i	0.343060	−	0.121290i
$910$		0			0
$911$	−27.5980			−0.914362			−0.457181	−	0.889374i	$-0.651141\pi$
$911$	−27.5980			−0.914362			−0.457181	+	0.889374i	$0.651141\pi$
$912$		0			0
$913$	−14.3431			−0.474689
$914$		0			0
$915$	−13.1716	−	9.31371i	−0.435439	−	0.307902i
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$		−	2.68629i		−	0.0886126i	−0.999018	−	0.0443063i	$-0.985892\pi$
$919$		−	2.68629i		−	0.0886126i	0.999018	−	0.0443063i	$-0.0141077\pi$
$920$		0			0
$921$	4.97056	+	3.51472i	0.163786	+	0.115814i
$922$		0			0
$923$	77.2548			2.54287
$924$		0			0
$925$	1.17157			0.0385211
$926$		0			0
$927$	13.6569	−	4.82843i	0.448550	−	0.158586i
$928$		0			0
$929$			41.6569i			1.36672i	0.730083	+	0.683359i	$0.239481\pi$
$929$			41.6569i			1.36672i	−0.730083	+	0.683359i	$0.760519\pi$
$930$		0			0
$931$		−	37.8823i		−	1.24154i
$932$		0			0
$933$	−11.3137	+	16.0000i	−0.370394	+	0.523816i
$934$		0			0
$935$	−4.00000			−0.130814
$936$		0			0
$937$	−26.9706			−0.881090			−0.440545	−	0.897731i	$-0.645215\pi$
$937$	−26.9706			−0.881090			−0.440545	+	0.897731i	$0.645215\pi$
$938$		0			0
$939$	2.68629	−	3.79899i	0.0876638	−	0.123975i
$940$		0			0
$941$		−	37.5980i		−	1.22566i	−0.790215	−	0.612830i	$-0.790031\pi$
$941$		−	37.5980i		−	1.22566i	0.790215	−	0.612830i	$-0.209969\pi$
$942$		0			0
$943$		−	8.00000i		−	0.260516i
$944$		0			0
$945$	−4.14214	−	1.17157i	−0.134744	−	0.0381113i
$946$		0			0
$947$	10.2843			0.334194			0.167097	−	0.985940i	$-0.446561\pi$
$947$	10.2843			0.334194			0.167097	+	0.985940i	$0.446561\pi$
$948$		0			0
$949$	13.6569			0.443320
$950$		0			0
$951$	−24.4853	−	17.3137i	−0.793990	−	0.561435i
$952$		0			0
$953$			37.7990i			1.22443i	0.790692	+	0.612215i	$0.209721\pi$
$953$			37.7990i			1.22443i	−0.790692	+	0.612215i	$0.790279\pi$
$954$		0			0
$955$			12.0000i			0.388311i
$956$		0			0
$957$	7.02944	+	4.97056i	0.227229	+	0.160675i
$958$		0			0
$959$	10.0589			0.324818
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	17.3137	+	48.9706i	0.557926	+	1.57805i
$964$		0			0
$965$		−	14.9706i		−	0.481919i
$966$		0			0
$967$			1.51472i			0.0487101i	0.999703	+	0.0243550i	$0.00775321\pi$
$967$			1.51472i			0.0487101i	−0.999703	+	0.0243550i	$0.992247\pi$
$968$		0			0
$969$	−28.9706	+	40.9706i	−0.930669	+	1.31616i
$970$		0			0
$971$	−50.7696			−1.62927			−0.814636	−	0.579972i	$-0.803063\pi$
$971$	−50.7696			−1.62927			−0.814636	+	0.579972i	$0.803063\pi$
$972$		0			0
$973$	−1.08831			−0.0348897
$974$		0			0
$975$	6.82843	−	9.65685i	0.218685	−	0.309267i
$976$		0			0
$977$			9.79899i			0.313497i	0.987639	+	0.156749i	$0.0501013\pi$
$977$			9.79899i			0.313497i	−0.987639	+	0.156749i	$0.949899\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−3.65685	−	10.3431i	−0.116754	−	0.330231i
$982$		0			0
$983$	20.8284			0.664324			0.332162	−	0.943222i	$-0.392222\pi$
$983$	20.8284			0.664324			0.332162	+	0.943222i	$0.392222\pi$
$984$		0			0
$985$	−9.31371			−0.296759
$986$		0			0
$987$	−10.3431	−	7.31371i	−0.329226	−	0.232798i
$988$		0			0
$989$			32.9706i			1.04840i
$990$		0			0
$991$			14.2843i			0.453755i	0.973923	+	0.226877i	$0.0728517\pi$
$991$			14.2843i			0.453755i	−0.973923	+	0.226877i	$0.927148\pi$
$992$		0			0
$993$	32.4853	+	22.9706i	1.03089	+	0.728949i
$994$		0			0
$995$	−3.65685			−0.115930
$996$		0			0
$997$	44.4853			1.40886			0.704431	−	0.709772i	$-0.251202\pi$
$997$	44.4853			1.40886			0.704431	+	0.709772i	$0.251202\pi$
$998$		0			0
$999$	5.85786	+	1.65685i	0.185335	+	0.0524205i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.h.e.191.2		4
3.2	odd	2		960.2.h.a.191.1		4
4.3	odd	2		960.2.h.a.191.4		4
8.3	odd	2		480.2.h.c.191.1	yes	4
8.5	even	2		480.2.h.a.191.3	yes	4
12.11	even	2	inner	960.2.h.e.191.3		4
24.5	odd	2		480.2.h.c.191.4	yes	4
24.11	even	2		480.2.h.a.191.2	✓	4
40.3	even	4		2400.2.o.e.2399.4		4
40.13	odd	4		2400.2.o.f.2399.1		4
40.19	odd	2		2400.2.h.a.1151.4		4
40.27	even	4		2400.2.o.g.2399.1		4
40.29	even	2		2400.2.h.d.1151.1		4
40.37	odd	4		2400.2.o.d.2399.4		4
120.29	odd	2		2400.2.h.a.1151.1		4
120.53	even	4		2400.2.o.g.2399.2		4
120.59	even	2		2400.2.h.d.1151.4		4
120.77	even	4		2400.2.o.e.2399.3		4
120.83	odd	4		2400.2.o.d.2399.3		4
120.107	odd	4		2400.2.o.f.2399.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.h.a.191.2	✓	4	24.11	even	2
480.2.h.a.191.3	yes	4	8.5	even	2
480.2.h.c.191.1	yes	4	8.3	odd	2
480.2.h.c.191.4	yes	4	24.5	odd	2
960.2.h.a.191.1		4	3.2	odd	2
960.2.h.a.191.4		4	4.3	odd	2
960.2.h.e.191.2		4	1.1	even	1	trivial
960.2.h.e.191.3		4	12.11	even	2	inner
2400.2.h.a.1151.1		4	120.29	odd	2
2400.2.h.a.1151.4		4	40.19	odd	2
2400.2.h.d.1151.1		4	40.29	even	2
2400.2.h.d.1151.4		4	120.59	even	2
2400.2.o.d.2399.3		4	120.83	odd	4
2400.2.o.d.2399.4		4	40.37	odd	4
2400.2.o.e.2399.3		4	120.77	even	4
2400.2.o.e.2399.4		4	40.3	even	4
2400.2.o.f.2399.1		4	40.13	odd	4
2400.2.o.f.2399.2		4	120.107	odd	4
2400.2.o.g.2399.1		4	40.27	even	4
2400.2.o.g.2399.2		4	120.53	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.41421i) q^{3} +1.00000i q^{5} -0.828427i q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.41421i) q^{3} +1.00000i q^{5} -0.828427i q^{7} +(-1.00000 - 2.82843i) q^{9} -0.828427 q^{11} -6.82843 q^{13} +(1.41421 + 1.00000i) q^{15} -4.82843i q^{17} -6.00000i q^{19} +(-1.17157 - 0.828427i) q^{21} +4.82843 q^{23} -1.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} -6.00000i q^{29} -2.00000i q^{31} +(-0.828427 + 1.17157i) q^{33} +0.828427 q^{35} -1.17157 q^{37} +(-6.82843 + 9.65685i) q^{39} -1.65685i q^{41} +6.82843i q^{43} +(2.82843 - 1.00000i) q^{45} +8.82843 q^{47} +6.31371 q^{49} +(-6.82843 - 4.82843i) q^{51} -3.65685i q^{53} -0.828427i q^{55} +(-8.48528 - 6.00000i) q^{57} +7.17157 q^{59} -9.31371 q^{61} +(-2.34315 + 0.828427i) q^{63} -6.82843i q^{65} +12.4853i q^{67} +(4.82843 - 6.82843i) q^{69} -11.3137 q^{71} -2.00000 q^{73} +(-1.00000 + 1.41421i) q^{75} +0.686292i q^{77} -6.00000i q^{79} +(-7.00000 + 5.65685i) q^{81} +17.3137 q^{83} +4.82843 q^{85} +(-8.48528 - 6.00000i) q^{87} +5.65685i q^{91} +(-2.82843 - 2.00000i) q^{93} +6.00000 q^{95} -6.00000 q^{97} +(0.828427 + 2.34315i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{9} + 8 q^{11} - 16 q^{13} - 16 q^{21} + 8 q^{23} - 4 q^{25} - 20 q^{27} + 8 q^{33} - 8 q^{35} - 16 q^{37} - 16 q^{39} + 24 q^{47} - 20 q^{49} - 16 q^{51} + 40 q^{59} + 8 q^{61} - 32 q^{63} + 8 q^{69} - 8 q^{73} - 4 q^{75} - 28 q^{81} + 24 q^{83} + 8 q^{85} + 24 q^{95} - 24 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^9 + 8 * q^11 - 16 * q^13 - 16 * q^21 + 8 * q^23 - 4 * q^25 - 20 * q^27 + 8 * q^33 - 8 * q^35 - 16 * q^37 - 16 * q^39 + 24 * q^47 - 20 * q^49 - 16 * q^51 + 40 * q^59 + 8 * q^61 - 32 * q^63 + 8 * q^69 - 8 * q^73 - 4 * q^75 - 28 * q^81 + 24 * q^83 + 8 * q^85 + 24 * q^95 - 24 * q^97 - 8 * q^99

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