LMFDB - Embedded newform 1920.2.m.v.959.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(959,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1920.959");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			959.1
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	1920.959
Dual form			1920.2.m.v.959.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.41421i) q^{3} +(-1.73205 - 1.41421i) q^{5} +3.46410 q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(1.00000 - 1.41421i) q^{3} +(-1.73205 - 1.41421i) q^{5} +3.46410 q^{7} +(-1.00000 - 2.82843i) q^{9} +4.00000 q^{13} +(-3.73205 + 1.03528i) q^{15} +6.92820 q^{17} +6.92820 q^{19} +(3.46410 - 4.89898i) q^{21} +4.89898i q^{23} +(1.00000 + 4.89898i) q^{25} +(-5.00000 - 1.41421i) q^{27} -3.46410 q^{29} +8.48528i q^{31} +(-6.00000 - 4.89898i) q^{35} -4.00000 q^{37} +(4.00000 - 5.65685i) q^{39} -5.65685i q^{41} -8.48528i q^{43} +(-2.26795 + 6.31319i) q^{45} -4.89898i q^{47} +5.00000 q^{49} +(6.92820 - 9.79796i) q^{51} +2.82843i q^{53} +(6.92820 - 9.79796i) q^{57} -9.79796i q^{59} +9.79796i q^{61} +(-3.46410 - 9.79796i) q^{63} +(-6.92820 - 5.65685i) q^{65} -8.48528i q^{67} +(6.92820 + 4.89898i) q^{69} +9.79796i q^{73} +(7.92820 + 3.48477i) q^{75} -8.48528i q^{79} +(-7.00000 + 5.65685i) q^{81} -6.00000 q^{83} +(-12.0000 - 9.79796i) q^{85} +(-3.46410 + 4.89898i) q^{87} +11.3137i q^{89} +13.8564 q^{91} +(12.0000 + 8.48528i) q^{93} +(-12.0000 - 9.79796i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 16 q^{13} - 8 q^{15} + 4 q^{25} - 20 q^{27} - 24 q^{35} - 16 q^{37} + 16 q^{39} - 16 q^{45} + 20 q^{49} + 4 q^{75} - 28 q^{81} - 24 q^{83} - 48 q^{85} + 48 q^{93} - 48 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^9 + 16 * q^13 - 8 * q^15 + 4 * q^25 - 20 * q^27 - 24 * q^35 - 16 * q^37 + 16 * q^39 - 16 * q^45 + 20 * q^49 + 4 * q^75 - 28 * q^81 - 24 * q^83 - 48 * q^85 + 48 * q^93 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$4$		0			0
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$6$		0			0
$7$	3.46410			1.30931			0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410			1.30931			0.654654	+	0.755929i	$0.272814\pi$
$8$		0			0
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$	−3.73205	+	1.03528i	−0.963611	+	0.267307i
$16$		0			0
$17$	6.92820			1.68034			0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820			1.68034			0.840168	+	0.542326i	$0.182456\pi$
$18$		0			0
$19$	6.92820			1.58944			0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820			1.58944			0.794719	+	0.606977i	$0.207618\pi$
$20$		0			0
$21$	3.46410	−	4.89898i	0.755929	−	1.06904i
$22$		0			0
$23$			4.89898i			1.02151i	0.859727	+	0.510754i	$0.170634\pi$
$23$			4.89898i			1.02151i	−0.859727	+	0.510754i	$0.829366\pi$
$24$		0			0
$25$	1.00000	+	4.89898i	0.200000	+	0.979796i
$26$		0			0
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$	−3.46410			−0.643268			−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410			−0.643268			−0.321634	+	0.946864i	$0.604232\pi$
$30$		0			0
$31$			8.48528i			1.52400i	0.647576	+	0.762001i	$0.275783\pi$
$31$			8.48528i			1.52400i	−0.647576	+	0.762001i	$0.724217\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000	−	4.89898i	−1.01419	−	0.828079i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$	4.00000	−	5.65685i	0.640513	−	0.905822i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$		−	8.48528i		−	1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$		−	8.48528i		−	1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$	−2.26795	+	6.31319i	−0.338086	+	0.941115i
$46$		0			0
$47$		−	4.89898i		−	0.714590i	−0.933992	−	0.357295i	$-0.883699\pi$
$47$		−	4.89898i		−	0.714590i	0.933992	−	0.357295i	$-0.116301\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$	6.92820	−	9.79796i	0.970143	−	1.37199i
$52$		0			0
$53$			2.82843i			0.388514i	0.980951	+	0.194257i	$0.0622296\pi$
$53$			2.82843i			0.388514i	−0.980951	+	0.194257i	$0.937770\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.92820	−	9.79796i	0.917663	−	1.29777i
$58$		0			0
$59$		−	9.79796i		−	1.27559i	−0.770208	−	0.637793i	$-0.779848\pi$
$59$		−	9.79796i		−	1.27559i	0.770208	−	0.637793i	$-0.220152\pi$
$60$		0			0
$61$			9.79796i			1.25450i	0.778818	+	0.627250i	$0.215820\pi$
$61$			9.79796i			1.25450i	−0.778818	+	0.627250i	$0.784180\pi$
$62$		0			0
$63$	−3.46410	−	9.79796i	−0.436436	−	1.23443i
$64$		0			0
$65$	−6.92820	−	5.65685i	−0.859338	−	0.701646i
$66$		0			0
$67$		−	8.48528i		−	1.03664i	−0.855186	−	0.518321i	$-0.826557\pi$
$67$		−	8.48528i		−	1.03664i	0.855186	−	0.518321i	$-0.173443\pi$
$68$		0			0
$69$	6.92820	+	4.89898i	0.834058	+	0.589768i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			9.79796i			1.14676i	0.819288	+	0.573382i	$0.194369\pi$
$73$			9.79796i			1.14676i	−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$	7.92820	+	3.48477i	0.915470	+	0.402386i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	8.48528i		−	0.954669i	−0.878722	−	0.477334i	$-0.841603\pi$
$79$		−	8.48528i		−	0.954669i	0.878722	−	0.477334i	$-0.158397\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	−12.0000	−	9.79796i	−1.30158	−	1.06274i
$86$		0			0
$87$	−3.46410	+	4.89898i	−0.371391	+	0.525226i
$88$		0			0
$89$			11.3137i			1.19925i	0.800281	+	0.599625i	$0.204684\pi$
$89$			11.3137i			1.19925i	−0.800281	+	0.599625i	$0.795316\pi$
$90$		0			0
$91$	13.8564			1.45255
$92$		0			0
$93$	12.0000	+	8.48528i	1.24434	+	0.879883i
$94$		0			0
$95$	−12.0000	−	9.79796i	−1.23117	−	1.00525i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3923			−1.03407			−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923			−1.03407			−0.517036	+	0.855963i	$0.672965\pi$
$102$		0			0
$103$	3.46410			0.341328			0.170664	−	0.985329i	$-0.445409\pi$
$103$	3.46410			0.341328			0.170664	+	0.985329i	$0.445409\pi$
$104$		0			0
$105$	−12.9282	+	3.58630i	−1.26166	+	0.349987i
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$		−	9.79796i		−	0.938474i	−0.883072	−	0.469237i	$-0.844529\pi$
$109$		−	9.79796i		−	0.938474i	0.883072	−	0.469237i	$-0.155471\pi$
$110$		0			0
$111$	−4.00000	+	5.65685i	−0.379663	+	0.536925i
$112$		0			0
$113$	−6.92820			−0.651751			−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820			−0.651751			−0.325875	+	0.945413i	$0.605659\pi$
$114$		0			0
$115$	6.92820	−	8.48528i	0.646058	−	0.791257i
$116$		0			0
$117$	−4.00000	−	11.3137i	−0.369800	−	1.04595i
$118$		0			0
$119$	24.0000			2.20008
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−8.00000	−	5.65685i	−0.721336	−	0.510061i
$124$		0			0
$125$	5.19615	−	9.89949i	0.464758	−	0.885438i
$126$		0			0
$127$	−17.3205			−1.53695			−0.768473	−	0.639882i	$-0.778983\pi$
$127$	−17.3205			−1.53695			−0.768473	+	0.639882i	$0.778983\pi$
$128$		0			0
$129$	−12.0000	−	8.48528i	−1.05654	−	0.747087i
$130$		0			0
$131$			9.79796i			0.856052i	0.903767	+	0.428026i	$0.140791\pi$
$131$			9.79796i			0.856052i	−0.903767	+	0.428026i	$0.859209\pi$
$132$		0			0
$133$	24.0000			2.08106
$134$		0			0
$135$	6.66025	+	9.52056i	0.573223	+	0.819399i
$136$		0			0
$137$	−6.92820			−0.591916			−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820			−0.591916			−0.295958	+	0.955201i	$0.595639\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$	−6.92820	−	4.89898i	−0.583460	−	0.412568i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	+	4.89898i	0.498273	+	0.406838i
$146$		0			0
$147$	5.00000	−	7.07107i	0.412393	−	0.583212i
$148$		0			0
$149$	3.46410			0.283790			0.141895	−	0.989882i	$-0.454680\pi$
$149$	3.46410			0.283790			0.141895	+	0.989882i	$0.454680\pi$
$150$		0			0
$151$		−	8.48528i		−	0.690522i	−0.938507	−	0.345261i	$-0.887790\pi$
$151$		−	8.48528i		−	0.690522i	0.938507	−	0.345261i	$-0.112210\pi$
$152$		0			0
$153$	−6.92820	−	19.5959i	−0.560112	−	1.58424i
$154$		0			0
$155$	12.0000	−	14.6969i	0.963863	−	1.18049i
$156$		0			0
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$		0			0
$159$	4.00000	+	2.82843i	0.317221	+	0.224309i
$160$		0			0
$161$			16.9706i			1.33747i
$162$		0			0
$163$			8.48528i			0.664619i	0.943170	+	0.332309i	$0.107828\pi$
$163$			8.48528i			0.664619i	−0.943170	+	0.332309i	$0.892172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.6969i			1.13728i	0.822585	+	0.568642i	$0.192531\pi$
$167$			14.6969i			1.13728i	−0.822585	+	0.568642i	$0.807469\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−6.92820	−	19.5959i	−0.529813	−	1.49854i
$172$		0			0
$173$		−	2.82843i		−	0.215041i	−0.994203	−	0.107521i	$-0.965709\pi$
$173$		−	2.82843i		−	0.215041i	0.994203	−	0.107521i	$-0.0342912\pi$
$174$		0			0
$175$	3.46410	+	16.9706i	0.261861	+	1.28285i
$176$		0			0
$177$	−13.8564	−	9.79796i	−1.04151	−	0.736460i
$178$		0			0
$179$		−	9.79796i		−	0.732334i	−0.930549	−	0.366167i	$-0.880670\pi$
$179$		−	9.79796i		−	0.732334i	0.930549	−	0.366167i	$-0.119330\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	13.8564	+	9.79796i	1.02430	+	0.724286i
$184$		0			0
$185$	6.92820	+	5.65685i	0.509372	+	0.415900i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−17.3205	−	4.89898i	−1.25988	−	0.356348i
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			9.79796i			0.705273i	0.935760	+	0.352636i	$0.114715\pi$
$193$			9.79796i			0.705273i	−0.935760	+	0.352636i	$0.885285\pi$
$194$		0			0
$195$	−14.9282	+	4.14110i	−1.06903	+	0.296551i
$196$		0			0
$197$			19.7990i			1.41062i	0.708899	+	0.705310i	$0.249192\pi$
$197$			19.7990i			1.41062i	−0.708899	+	0.705310i	$0.750808\pi$
$198$		0			0
$199$		−	25.4558i		−	1.80452i	−0.431196	−	0.902258i	$-0.641908\pi$
$199$		−	25.4558i		−	1.80452i	0.431196	−	0.902258i	$-0.358092\pi$
$200$		0			0
$201$	−12.0000	−	8.48528i	−0.846415	−	0.598506i
$202$		0			0
$203$	−12.0000			−0.842235
$204$		0			0
$205$	−8.00000	+	9.79796i	−0.558744	+	0.684319i
$206$		0			0
$207$	13.8564	−	4.89898i	0.963087	−	0.340503i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.92820			−0.476957			−0.238479	−	0.971148i	$-0.576649\pi$
$211$	−6.92820			−0.476957			−0.238479	+	0.971148i	$0.576649\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.0000	+	14.6969i	−0.818393	+	1.00232i
$216$		0			0
$217$			29.3939i			1.99539i
$218$		0			0
$219$	13.8564	+	9.79796i	0.936329	+	0.662085i
$220$		0			0
$221$	27.7128			1.86417
$222$		0			0
$223$	−10.3923			−0.695920			−0.347960	−	0.937509i	$-0.613126\pi$
$223$	−10.3923			−0.695920			−0.347960	+	0.937509i	$0.613126\pi$
$224$		0			0
$225$	12.8564	−	7.72741i	0.857094	−	0.515160i
$226$		0			0
$227$	−6.00000			−0.398234			−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000			−0.398234			−0.199117	+	0.979976i	$0.563807\pi$
$228$		0			0
$229$			19.5959i			1.29493i	0.762093	+	0.647467i	$0.224172\pi$
$229$			19.5959i			1.29493i	−0.762093	+	0.647467i	$0.775828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.92820			0.453882			0.226941	−	0.973909i	$-0.427128\pi$
$233$	6.92820			0.453882			0.226941	+	0.973909i	$0.427128\pi$
$234$		0			0
$235$	−6.92820	+	8.48528i	−0.451946	+	0.553519i
$236$		0			0
$237$	−12.0000	−	8.48528i	−0.779484	−	0.551178i
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	1.00000	+	15.5563i	0.0641500	+	0.997940i
$244$		0			0
$245$	−8.66025	−	7.07107i	−0.553283	−	0.451754i
$246$		0			0
$247$	27.7128			1.76332
$248$		0			0
$249$	−6.00000	+	8.48528i	−0.380235	+	0.537733i
$250$		0			0
$251$			19.5959i			1.23688i	0.785831	+	0.618442i	$0.212236\pi$
$251$			19.5959i			1.23688i	−0.785831	+	0.618442i	$0.787764\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−25.8564	+	7.17260i	−1.61919	+	0.449166i
$256$		0			0
$257$	−6.92820			−0.432169			−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820			−0.432169			−0.216085	+	0.976375i	$0.569329\pi$
$258$		0			0
$259$	−13.8564			−0.860995
$260$		0			0
$261$	3.46410	+	9.79796i	0.214423	+	0.606478i
$262$		0			0
$263$		−	4.89898i		−	0.302084i	−0.988527	−	0.151042i	$-0.951737\pi$
$263$		−	4.89898i		−	0.302084i	0.988527	−	0.151042i	$-0.0482629\pi$
$264$		0			0
$265$	4.00000	−	4.89898i	0.245718	−	0.300942i
$266$		0			0
$267$	16.0000	+	11.3137i	0.979184	+	0.692388i
$268$		0			0
$269$	17.3205			1.05605			0.528025	−	0.849229i	$-0.322933\pi$
$269$	17.3205			1.05605			0.528025	+	0.849229i	$0.322933\pi$
$270$		0			0
$271$		−	8.48528i		−	0.515444i	−0.966219	−	0.257722i	$-0.917028\pi$
$271$		−	8.48528i		−	0.515444i	0.966219	−	0.257722i	$-0.0829719\pi$
$272$		0			0
$273$	13.8564	−	19.5959i	0.838628	−	1.18600i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.00000			−0.240337			−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000			−0.240337			−0.120168	+	0.992754i	$0.538343\pi$
$278$		0			0
$279$	24.0000	−	8.48528i	1.43684	−	0.508001i
$280$		0			0
$281$		−	28.2843i		−	1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$		−	28.2843i		−	1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$		−	8.48528i		−	0.504398i	−0.967675	−	0.252199i	$-0.918846\pi$
$283$		−	8.48528i		−	0.504398i	0.967675	−	0.252199i	$-0.0811537\pi$
$284$		0			0
$285$	−25.8564	+	7.17260i	−1.53160	+	0.424868i
$286$		0			0
$287$		−	19.5959i		−	1.15671i
$288$		0			0
$289$	31.0000			1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.1421i			0.826192i	0.910687	+	0.413096i	$0.135553\pi$
$293$			14.1421i			0.826192i	−0.910687	+	0.413096i	$0.864447\pi$
$294$		0			0
$295$	−13.8564	+	16.9706i	−0.806751	+	0.988064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			19.5959i			1.13326i
$300$		0			0
$301$		−	29.3939i		−	1.69423i
$302$		0			0
$303$	−10.3923	+	14.6969i	−0.597022	+	0.844317i
$304$		0			0
$305$	13.8564	−	16.9706i	0.793416	−	0.971732i
$306$		0			0
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$	3.46410	−	4.89898i	0.197066	−	0.278693i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			9.79796i			0.553813i	0.960897	+	0.276907i	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$	−7.85641	+	21.8695i	−0.442658	+	1.23221i
$316$		0			0
$317$			2.82843i			0.158860i	0.996840	+	0.0794301i	$0.0253101\pi$
$317$			2.82843i			0.158860i	−0.996840	+	0.0794301i	$0.974690\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−6.00000	+	8.48528i	−0.334887	+	0.473602i
$322$		0			0
$323$	48.0000			2.67079
$324$		0			0
$325$	4.00000	+	19.5959i	0.221880	+	1.08699i
$326$		0			0
$327$	−13.8564	−	9.79796i	−0.766261	−	0.541828i
$328$		0			0
$329$		−	16.9706i		−	0.935617i
$330$		0			0
$331$	−20.7846			−1.14243			−0.571213	−	0.820802i	$-0.693527\pi$
$331$	−20.7846			−1.14243			−0.571213	+	0.820802i	$0.693527\pi$
$332$		0			0
$333$	4.00000	+	11.3137i	0.219199	+	0.619987i
$334$		0			0
$335$	−12.0000	+	14.6969i	−0.655630	+	0.802980i
$336$		0			0
$337$			29.3939i			1.60119i	0.599208	+	0.800593i	$0.295482\pi$
$337$			29.3939i			1.60119i	−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	−6.92820	+	9.79796i	−0.376288	+	0.532152i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.92820			−0.374088
$344$		0			0
$345$	−5.07180	−	18.2832i	−0.273056	−	0.984337i
$346$		0			0
$347$	18.0000			0.966291			0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000			0.966291			0.483145	+	0.875540i	$0.339494\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	−20.0000	−	5.65685i	−1.06752	−	0.301941i
$352$		0			0
$353$	20.7846			1.10625			0.553127	−	0.833097i	$-0.313435\pi$
$353$	20.7846			1.10625			0.553127	+	0.833097i	$0.313435\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	24.0000	−	33.9411i	1.27021	−	1.79635i
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	29.0000			1.52632
$362$		0			0
$363$	11.0000	−	15.5563i	0.577350	−	0.816497i
$364$		0			0
$365$	13.8564	−	16.9706i	0.725277	−	0.888280i
$366$		0			0
$367$	−10.3923			−0.542474			−0.271237	−	0.962513i	$-0.587433\pi$
$367$	−10.3923			−0.542474			−0.271237	+	0.962513i	$0.587433\pi$
$368$		0			0
$369$	−16.0000	+	5.65685i	−0.832927	+	0.294484i
$370$		0			0
$371$			9.79796i			0.508685i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$	−8.80385	−	17.2480i	−0.454629	−	0.890681i
$376$		0			0
$377$	−13.8564			−0.713641
$378$		0			0
$379$	−34.6410			−1.77939			−0.889695	−	0.456556i	$-0.849083\pi$
$379$	−34.6410			−1.77939			−0.889695	+	0.456556i	$0.849083\pi$
$380$		0			0
$381$	−17.3205	+	24.4949i	−0.887357	+	1.25491i
$382$		0			0
$383$		−	34.2929i		−	1.75228i	−0.482054	−	0.876142i	$-0.660109\pi$
$383$		−	34.2929i		−	1.75228i	0.482054	−	0.876142i	$-0.339891\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−24.0000	+	8.48528i	−1.21999	+	0.431331i
$388$		0			0
$389$	−31.1769			−1.58073			−0.790366	−	0.612635i	$-0.790110\pi$
$389$	−31.1769			−1.58073			−0.790366	+	0.612635i	$0.790110\pi$
$390$		0			0
$391$			33.9411i			1.71648i
$392$		0			0
$393$	13.8564	+	9.79796i	0.698963	+	0.494242i
$394$		0			0
$395$	−12.0000	+	14.6969i	−0.603786	+	0.739483i
$396$		0			0
$397$	4.00000			0.200754			0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000			0.200754			0.100377	+	0.994949i	$0.467995\pi$
$398$		0			0
$399$	24.0000	−	33.9411i	1.20150	−	1.69918i
$400$		0			0
$401$			11.3137i			0.564980i	0.959270	+	0.282490i	$0.0911603\pi$
$401$			11.3137i			0.564980i	−0.959270	+	0.282490i	$0.908840\pi$
$402$		0			0
$403$			33.9411i			1.69073i
$404$		0			0
$405$	20.1244	+	0.101536i	0.999987	+	0.00504536i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$		0			0
$411$	−6.92820	+	9.79796i	−0.341743	+	0.483298i
$412$		0			0
$413$		−	33.9411i		−	1.67013i
$414$		0			0
$415$	10.3923	+	8.48528i	0.510138	+	0.416526i
$416$		0			0
$417$	6.92820	−	9.79796i	0.339276	−	0.479808i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	9.79796i		−	0.477523i	−0.971078	−	0.238762i	$-0.923259\pi$
$421$		−	9.79796i		−	0.477523i	0.971078	−	0.238762i	$-0.0767415\pi$
$422$		0			0
$423$	−13.8564	+	4.89898i	−0.673722	+	0.238197i
$424$		0			0
$425$	6.92820	+	33.9411i	0.336067	+	1.64639i
$426$		0			0
$427$			33.9411i			1.64253i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$			19.5959i			0.941720i	0.882208	+	0.470860i	$0.156056\pi$
$433$			19.5959i			0.941720i	−0.882208	+	0.470860i	$0.843944\pi$
$434$		0			0
$435$	12.9282	−	3.58630i	0.619860	−	0.171950i
$436$		0			0
$437$			33.9411i			1.62362i
$438$		0			0
$439$		−	25.4558i		−	1.21494i	−0.794342	−	0.607471i	$-0.792184\pi$
$439$		−	25.4558i		−	1.21494i	0.794342	−	0.607471i	$-0.207816\pi$
$440$		0			0
$441$	−5.00000	−	14.1421i	−0.238095	−	0.673435i
$442$		0			0
$443$	−18.0000			−0.855206			−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000			−0.855206			−0.427603	+	0.903967i	$0.640642\pi$
$444$		0			0
$445$	16.0000	−	19.5959i	0.758473	−	0.928936i
$446$		0			0
$447$	3.46410	−	4.89898i	0.163846	−	0.231714i
$448$		0			0
$449$		−	28.2843i		−	1.33482i	−0.744692	−	0.667409i	$-0.767403\pi$
$449$		−	28.2843i		−	1.33482i	0.744692	−	0.667409i	$-0.232597\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−12.0000	−	8.48528i	−0.563809	−	0.398673i
$454$		0			0
$455$	−24.0000	−	19.5959i	−1.12514	−	0.918671i
$456$		0			0
$457$		−	19.5959i		−	0.916658i	−0.888783	−	0.458329i	$-0.848448\pi$
$457$		−	19.5959i		−	0.916658i	0.888783	−	0.458329i	$-0.151552\pi$
$458$		0			0
$459$	−34.6410	−	9.79796i	−1.61690	−	0.457330i
$460$		0			0
$461$	−31.1769			−1.45205			−0.726027	−	0.687666i	$-0.758635\pi$
$461$	−31.1769			−1.45205			−0.726027	+	0.687666i	$0.758635\pi$
$462$		0			0
$463$	38.1051			1.77090			0.885448	−	0.464739i	$-0.153852\pi$
$463$	38.1051			1.77090			0.885448	+	0.464739i	$0.153852\pi$
$464$		0			0
$465$	−8.78461	−	31.6675i	−0.407377	−	1.46855i
$466$		0			0
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		0			0
$469$		−	29.3939i		−	1.35728i
$470$		0			0
$471$	−4.00000	+	5.65685i	−0.184310	+	0.260654i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.92820	+	33.9411i	0.317888	+	1.55733i
$476$		0			0
$477$	8.00000	−	2.82843i	0.366295	−	0.129505i
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$	24.0000	+	16.9706i	1.09204	+	0.772187i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.46410			−0.156973			−0.0784867	−	0.996915i	$-0.525009\pi$
$487$	−3.46410			−0.156973			−0.0784867	+	0.996915i	$0.525009\pi$
$488$		0			0
$489$	12.0000	+	8.48528i	0.542659	+	0.383718i
$490$		0			0
$491$			9.79796i			0.442176i	0.975254	+	0.221088i	$0.0709608\pi$
$491$			9.79796i			0.442176i	−0.975254	+	0.221088i	$0.929039\pi$
$492$		0			0
$493$	−24.0000			−1.08091
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.92820			−0.310149			−0.155074	−	0.987903i	$-0.549562\pi$
$499$	−6.92820			−0.310149			−0.155074	+	0.987903i	$0.549562\pi$
$500$		0			0
$501$	20.7846	+	14.6969i	0.928588	+	0.656611i
$502$		0			0
$503$			14.6969i			0.655304i	0.944798	+	0.327652i	$0.106257\pi$
$503$			14.6969i			0.655304i	−0.944798	+	0.327652i	$0.893743\pi$
$504$		0			0
$505$	18.0000	+	14.6969i	0.800989	+	0.654005i
$506$		0			0
$507$	3.00000	−	4.24264i	0.133235	−	0.188422i
$508$		0			0
$509$	31.1769			1.38189			0.690946	−	0.722906i	$-0.257194\pi$
$509$	31.1769			1.38189			0.690946	+	0.722906i	$0.257194\pi$
$510$		0			0
$511$			33.9411i			1.50147i
$512$		0			0
$513$	−34.6410	−	9.79796i	−1.52944	−	0.432590i
$514$		0			0
$515$	−6.00000	−	4.89898i	−0.264392	−	0.215875i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−4.00000	−	2.82843i	−0.175581	−	0.124154i
$520$		0			0
$521$			22.6274i			0.991325i	0.868515	+	0.495663i	$0.165075\pi$
$521$			22.6274i			0.991325i	−0.868515	+	0.495663i	$0.834925\pi$
$522$		0			0
$523$		−	25.4558i		−	1.11311i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$		−	25.4558i		−	1.11311i	0.830812	−	0.556553i	$-0.187876\pi$
$524$		0			0
$525$	27.4641	+	12.0716i	1.19863	+	0.526847i
$526$		0			0
$527$			58.7878i			2.56083i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−27.7128	+	9.79796i	−1.20263	+	0.425195i
$532$		0			0
$533$		−	22.6274i		−	0.980102i
$534$		0			0
$535$	10.3923	+	8.48528i	0.449299	+	0.366851i
$536$		0			0
$537$	−13.8564	−	9.79796i	−0.597948	−	0.422813i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			19.5959i			0.842494i	0.906946	+	0.421247i	$0.138408\pi$
$541$			19.5959i			0.842494i	−0.906946	+	0.421247i	$0.861592\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.8564	+	16.9706i	−0.593543	+	0.726939i
$546$		0			0
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$		0			0
$549$	27.7128	−	9.79796i	1.18275	−	0.418167i
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$		−	29.3939i		−	1.24995i
$554$		0			0
$555$	14.9282	−	4.14110i	0.633667	−	0.175780i
$556$		0			0
$557$			31.1127i			1.31829i	0.752017	+	0.659144i	$0.229081\pi$
$557$			31.1127i			1.31829i	−0.752017	+	0.659144i	$0.770919\pi$
$558$		0			0
$559$		−	33.9411i		−	1.43556i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.00000			0.252870			0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000			0.252870			0.126435	+	0.991975i	$0.459647\pi$
$564$		0			0
$565$	12.0000	+	9.79796i	0.504844	+	0.412203i
$566$		0			0
$567$	−24.2487	+	19.5959i	−1.01835	+	0.822951i
$568$		0			0
$569$			28.2843i			1.18574i	0.805299	+	0.592869i	$0.202005\pi$
$569$			28.2843i			1.18574i	−0.805299	+	0.592869i	$0.797995\pi$
$570$		0			0
$571$	−20.7846			−0.869809			−0.434904	−	0.900477i	$-0.643218\pi$
$571$	−20.7846			−0.869809			−0.434904	+	0.900477i	$0.643218\pi$
$572$		0			0
$573$	−24.0000	+	33.9411i	−1.00261	+	1.41791i
$574$		0			0
$575$	−24.0000	+	4.89898i	−1.00087	+	0.204302i
$576$		0			0
$577$			9.79796i			0.407894i	0.978982	+	0.203947i	$0.0653771\pi$
$577$			9.79796i			0.407894i	−0.978982	+	0.203947i	$0.934623\pi$
$578$		0			0
$579$	13.8564	+	9.79796i	0.575853	+	0.407189i
$580$		0			0
$581$	−20.7846			−0.862291
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−9.07180	+	25.2528i	−0.375073	+	1.04407i
$586$		0			0
$587$	42.0000			1.73353			0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000			1.73353			0.866763	+	0.498721i	$0.166197\pi$
$588$		0			0
$589$			58.7878i			2.42231i
$590$		0			0
$591$	28.0000	+	19.7990i	1.15177	+	0.814422i
$592$		0			0
$593$	−34.6410			−1.42254			−0.711268	−	0.702921i	$-0.751879\pi$
$593$	−34.6410			−1.42254			−0.711268	+	0.702921i	$0.751879\pi$
$594$		0			0
$595$	−41.5692	−	33.9411i	−1.70417	−	1.39145i
$596$		0			0
$597$	−36.0000	−	25.4558i	−1.47338	−	1.04184i
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$	−24.0000	+	8.48528i	−0.977356	+	0.345547i
$604$		0			0
$605$	−19.0526	−	15.5563i	−0.774597	−	0.632456i
$606$		0			0
$607$	−10.3923			−0.421811			−0.210905	−	0.977506i	$-0.567641\pi$
$607$	−10.3923			−0.421811			−0.210905	+	0.977506i	$0.567641\pi$
$608$		0			0
$609$	−12.0000	+	16.9706i	−0.486265	+	0.687682i
$610$		0			0
$611$		−	19.5959i		−	0.792766i
$612$		0			0
$613$	−44.0000			−1.77714			−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000			−1.77714			−0.888572	+	0.458738i	$0.848302\pi$
$614$		0			0
$615$	5.85641	+	21.1117i	0.236153	+	0.851305i
$616$		0			0
$617$	34.6410			1.39459			0.697297	−	0.716782i	$-0.254386\pi$
$617$	34.6410			1.39459			0.697297	+	0.716782i	$0.254386\pi$
$618$		0			0
$619$	34.6410			1.39234			0.696170	−	0.717877i	$-0.254886\pi$
$619$	34.6410			1.39234			0.696170	+	0.717877i	$0.254886\pi$
$620$		0			0
$621$	6.92820	−	24.4949i	0.278019	−	0.982946i
$622$		0			0
$623$			39.1918i			1.57019i
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.7128			−1.10498
$630$		0			0
$631$			8.48528i			0.337794i	0.985634	+	0.168897i	$0.0540205\pi$
$631$			8.48528i			0.337794i	−0.985634	+	0.168897i	$0.945980\pi$
$632$		0			0
$633$	−6.92820	+	9.79796i	−0.275371	+	0.389434i
$634$		0			0
$635$	30.0000	+	24.4949i	1.19051	+	0.972050i
$636$		0			0
$637$	20.0000			0.792429
$638$		0			0
$639$		0			0
$640$		0			0
$641$			5.65685i			0.223432i	0.993740	+	0.111716i	$0.0356347\pi$
$641$			5.65685i			0.223432i	−0.993740	+	0.111716i	$0.964365\pi$
$642$		0			0
$643$			42.4264i			1.67313i	0.547865	+	0.836567i	$0.315441\pi$
$643$			42.4264i			1.67313i	−0.547865	+	0.836567i	$0.684559\pi$
$644$		0			0
$645$	8.78461	+	31.6675i	0.345894	+	1.24691i
$646$		0			0
$647$		−	44.0908i		−	1.73339i	−0.498839	−	0.866694i	$-0.666240\pi$
$647$		−	44.0908i		−	1.73339i	0.498839	−	0.866694i	$-0.333760\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	41.5692	+	29.3939i	1.62923	+	1.15204i
$652$		0			0
$653$		−	14.1421i		−	0.553425i	−0.960953	−	0.276712i	$-0.910755\pi$
$653$		−	14.1421i		−	0.553425i	0.960953	−	0.276712i	$-0.0892449\pi$
$654$		0			0
$655$	13.8564	−	16.9706i	0.541415	−	0.663095i
$656$		0			0
$657$	27.7128	−	9.79796i	1.08118	−	0.382255i
$658$		0			0
$659$			48.9898i			1.90837i	0.299215	+	0.954186i	$0.403275\pi$
$659$			48.9898i			1.90837i	−0.299215	+	0.954186i	$0.596725\pi$
$660$		0			0
$661$			9.79796i			0.381096i	0.981678	+	0.190548i	$0.0610266\pi$
$661$			9.79796i			0.381096i	−0.981678	+	0.190548i	$0.938973\pi$
$662$		0			0
$663$	27.7128	−	39.1918i	1.07628	−	1.52208i
$664$		0			0
$665$	−41.5692	−	33.9411i	−1.61199	−	1.31618i
$666$		0			0
$667$		−	16.9706i		−	0.657103i
$668$		0			0
$669$	−10.3923	+	14.6969i	−0.401790	+	0.568216i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	19.5959i		−	0.755367i	−0.925935	−	0.377684i	$-0.876721\pi$
$673$		−	19.5959i		−	0.755367i	0.925935	−	0.377684i	$-0.123279\pi$
$674$		0			0
$675$	1.92820	−	25.9091i	0.0742166	−	0.997242i
$676$		0			0
$677$			36.7696i			1.41317i	0.707629	+	0.706584i	$0.249765\pi$
$677$			36.7696i			1.41317i	−0.707629	+	0.706584i	$0.750235\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.00000	+	8.48528i	−0.229920	+	0.325157i
$682$		0			0
$683$	−30.0000			−1.14792			−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000			−1.14792			−0.573959	+	0.818884i	$0.694593\pi$
$684$		0			0
$685$	12.0000	+	9.79796i	0.458496	+	0.374361i
$686$		0			0
$687$	27.7128	+	19.5959i	1.05731	+	0.747631i
$688$		0			0
$689$			11.3137i			0.431018i
$690$		0			0
$691$	6.92820			0.263561			0.131781	−	0.991279i	$-0.457931\pi$
$691$	6.92820			0.263561			0.131781	+	0.991279i	$0.457931\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0000	−	9.79796i	−0.455186	−	0.371658i
$696$		0			0
$697$		−	39.1918i		−	1.48450i
$698$		0			0
$699$	6.92820	−	9.79796i	0.262049	−	0.370593i
$700$		0			0
$701$	38.1051			1.43921			0.719605	−	0.694383i	$-0.244323\pi$
$701$	38.1051			1.43921			0.719605	+	0.694383i	$0.244323\pi$
$702$		0			0
$703$	−27.7128			−1.04521
$704$		0			0
$705$	5.07180	+	18.2832i	0.191015	+	0.688587i
$706$		0			0
$707$	−36.0000			−1.35392
$708$		0			0
$709$		−	39.1918i		−	1.47188i	−0.677047	−	0.735940i	$-0.736740\pi$
$709$		−	39.1918i		−	1.47188i	0.677047	−	0.735940i	$-0.263260\pi$
$710$		0			0
$711$	−24.0000	+	8.48528i	−0.900070	+	0.318223i
$712$		0			0
$713$	−41.5692			−1.55678
$714$		0			0
$715$		0			0
$716$		0			0
$717$	24.0000	−	33.9411i	0.896296	−	1.26755i
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$	2.00000	−	2.82843i	0.0743808	−	0.105190i
$724$		0			0
$725$	−3.46410	−	16.9706i	−0.128654	−	0.630271i
$726$		0			0
$727$	−3.46410			−0.128476			−0.0642382	−	0.997935i	$-0.520462\pi$
$727$	−3.46410			−0.128476			−0.0642382	+	0.997935i	$0.520462\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$		−	58.7878i		−	2.17434i
$732$		0			0
$733$	−4.00000			−0.147743			−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000			−0.147743			−0.0738717	+	0.997268i	$0.523536\pi$
$734$		0			0
$735$	−18.6603	+	5.17638i	−0.688294	+	0.190934i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	20.7846			0.764574			0.382287	−	0.924044i	$-0.375137\pi$
$739$	20.7846			0.764574			0.382287	+	0.924044i	$0.375137\pi$
$740$		0			0
$741$	27.7128	−	39.1918i	1.01806	−	1.43975i
$742$		0			0
$743$		−	14.6969i		−	0.539178i	−0.962975	−	0.269589i	$-0.913112\pi$
$743$		−	14.6969i		−	0.539178i	0.962975	−	0.269589i	$-0.0868879\pi$
$744$		0			0
$745$	−6.00000	−	4.89898i	−0.219823	−	0.179485i
$746$		0			0
$747$	6.00000	+	16.9706i	0.219529	+	0.620920i
$748$		0			0
$749$	−20.7846			−0.759453
$750$		0			0
$751$		−	25.4558i		−	0.928897i	−0.885600	−	0.464448i	$-0.846253\pi$
$751$		−	25.4558i		−	0.928897i	0.885600	−	0.464448i	$-0.153747\pi$
$752$		0			0
$753$	27.7128	+	19.5959i	1.00991	+	0.714115i
$754$		0			0
$755$	−12.0000	+	14.6969i	−0.436725	+	0.534876i
$756$		0			0
$757$	44.0000			1.59921			0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000			1.59921			0.799604	+	0.600528i	$0.205043\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	11.3137i		−	0.410122i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$		−	11.3137i		−	0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$		0			0
$763$		−	33.9411i		−	1.22875i
$764$		0			0
$765$	−15.7128	+	43.7391i	−0.568098	+	1.58139i
$766$		0			0
$767$		−	39.1918i		−	1.41514i
$768$		0			0
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$	−6.92820	+	9.79796i	−0.249513	+	0.352865i
$772$		0			0
$773$		−	14.1421i		−	0.508657i	−0.967118	−	0.254329i	$-0.918146\pi$
$773$		−	14.1421i		−	0.508657i	0.967118	−	0.254329i	$-0.0818545\pi$
$774$		0			0
$775$	−41.5692	+	8.48528i	−1.49321	+	0.304800i
$776$		0			0
$777$	−13.8564	+	19.5959i	−0.497096	+	0.703000i
$778$		0			0
$779$		−	39.1918i		−	1.40419i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	17.3205	+	4.89898i	0.618984	+	0.175075i
$784$		0			0
$785$	6.92820	+	5.65685i	0.247278	+	0.201902i
$786$		0			0
$787$		−	25.4558i		−	0.907403i	−0.891154	−	0.453701i	$-0.850103\pi$
$787$		−	25.4558i		−	0.907403i	0.891154	−	0.453701i	$-0.149897\pi$
$788$		0			0
$789$	−6.92820	−	4.89898i	−0.246651	−	0.174408i
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$			39.1918i			1.39174i
$794$		0			0
$795$	−2.92820	−	10.5558i	−0.103853	−	0.374377i
$796$		0			0
$797$			2.82843i			0.100188i	0.998745	+	0.0500940i	$0.0159521\pi$
$797$			2.82843i			0.100188i	−0.998745	+	0.0500940i	$0.984048\pi$
$798$		0			0
$799$		−	33.9411i		−	1.20075i
$800$		0			0
$801$	32.0000	−	11.3137i	1.13066	−	0.399750i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	24.0000	−	29.3939i	0.845889	−	1.03600i
$806$		0			0
$807$	17.3205	−	24.4949i	0.609711	−	0.862261i
$808$		0			0
$809$		−	45.2548i		−	1.59108i	−0.605904	−	0.795538i	$-0.707189\pi$
$809$		−	45.2548i		−	1.59108i	0.605904	−	0.795538i	$-0.292811\pi$
$810$		0			0
$811$	20.7846			0.729846			0.364923	−	0.931038i	$-0.381095\pi$
$811$	20.7846			0.729846			0.364923	+	0.931038i	$0.381095\pi$
$812$		0			0
$813$	−12.0000	−	8.48528i	−0.420858	−	0.297592i
$814$		0			0
$815$	12.0000	−	14.6969i	0.420342	−	0.514811i
$816$		0			0
$817$		−	58.7878i		−	2.05672i
$818$		0			0
$819$	−13.8564	−	39.1918i	−0.484182	−	1.36947i
$820$		0			0
$821$	3.46410			0.120898			0.0604490	−	0.998171i	$-0.480747\pi$
$821$	3.46410			0.120898			0.0604490	+	0.998171i	$0.480747\pi$
$822$		0			0
$823$	−51.9615			−1.81126			−0.905632	−	0.424064i	$-0.860603\pi$
$823$	−51.9615			−1.81126			−0.905632	+	0.424064i	$0.860603\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−42.0000			−1.46048			−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000			−1.46048			−0.730242	+	0.683189i	$0.760592\pi$
$828$		0			0
$829$		−	29.3939i		−	1.02089i	−0.859910	−	0.510446i	$-0.829480\pi$
$829$		−	29.3939i		−	1.02089i	0.859910	−	0.510446i	$-0.170520\pi$
$830$		0			0
$831$	−4.00000	+	5.65685i	−0.138758	+	0.196234i
$832$		0			0
$833$	34.6410			1.20024
$834$		0			0
$835$	20.7846	−	25.4558i	0.719281	−	0.880936i
$836$		0			0
$837$	12.0000	−	42.4264i	0.414781	−	1.46647i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−17.0000			−0.586207
$842$		0			0
$843$	−40.0000	−	28.2843i	−1.37767	−	0.974162i
$844$		0			0
$845$	−5.19615	−	4.24264i	−0.178753	−	0.145951i
$846$		0			0
$847$	38.1051			1.30931
$848$		0			0
$849$	−12.0000	−	8.48528i	−0.411839	−	0.291214i
$850$		0			0
$851$		−	19.5959i		−	0.671739i
$852$		0			0
$853$	−44.0000			−1.50653			−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000			−1.50653			−0.753266	+	0.657716i	$0.771523\pi$
$854$		0			0
$855$	−15.7128	+	43.7391i	−0.537367	+	1.49585i
$856$		0			0
$857$	48.4974			1.65664			0.828320	−	0.560255i	$-0.189297\pi$
$857$	48.4974			1.65664			0.828320	+	0.560255i	$0.189297\pi$
$858$		0			0
$859$	20.7846			0.709162			0.354581	−	0.935025i	$-0.384624\pi$
$859$	20.7846			0.709162			0.354581	+	0.935025i	$0.384624\pi$
$860$		0			0
$861$	−27.7128	−	19.5959i	−0.944450	−	0.667827i
$862$		0			0
$863$			34.2929i			1.16734i	0.811990	+	0.583671i	$0.198384\pi$
$863$			34.2929i			1.16734i	−0.811990	+	0.583671i	$0.801616\pi$
$864$		0			0
$865$	−4.00000	+	4.89898i	−0.136004	+	0.166570i
$866$		0			0
$867$	31.0000	−	43.8406i	1.05282	−	1.48891i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	33.9411i		−	1.15005i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	18.0000	−	34.2929i	0.608511	−	1.15931i
$876$		0			0
$877$	−4.00000			−0.135070			−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000			−0.135070			−0.0675352	+	0.997717i	$0.521513\pi$
$878$		0			0
$879$	20.0000	+	14.1421i	0.674583	+	0.477002i
$880$		0			0
$881$		−	5.65685i		−	0.190584i	−0.995449	−	0.0952921i	$-0.969621\pi$
$881$		−	5.65685i		−	0.190584i	0.995449	−	0.0952921i	$-0.0303785\pi$
$882$		0			0
$883$			8.48528i			0.285552i	0.989755	+	0.142776i	$0.0456029\pi$
$883$			8.48528i			0.285552i	−0.989755	+	0.142776i	$0.954397\pi$
$884$		0			0
$885$	10.1436	+	36.5665i	0.340973	+	1.22917i
$886$		0			0
$887$		−	4.89898i		−	0.164492i	−0.996612	−	0.0822458i	$-0.973791\pi$
$887$		−	4.89898i		−	0.164492i	0.996612	−	0.0822458i	$-0.0262093\pi$
$888$		0			0
$889$	−60.0000			−2.01234
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	33.9411i		−	1.13580i
$894$		0			0
$895$	−13.8564	+	16.9706i	−0.463169	+	0.567263i
$896$		0			0
$897$	27.7128	+	19.5959i	0.925304	+	0.654289i
$898$		0			0
$899$		−	29.3939i		−	0.980341i
$900$		0			0
$901$			19.5959i			0.652835i
$902$		0			0
$903$	−41.5692	−	29.3939i	−1.38334	−	0.978167i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			25.4558i			0.845247i	0.906305	+	0.422624i	$0.138891\pi$
$907$			25.4558i			0.845247i	−0.906305	+	0.422624i	$0.861109\pi$
$908$		0			0
$909$	10.3923	+	29.3939i	0.344691	+	0.974933i
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−10.1436	−	36.5665i	−0.335337	−	1.20885i
$916$		0			0
$917$			33.9411i			1.12083i
$918$		0			0
$919$			25.4558i			0.839711i	0.907591	+	0.419855i	$0.137919\pi$
$919$			25.4558i			0.839711i	−0.907591	+	0.419855i	$0.862081\pi$
$920$		0			0
$921$	12.0000	+	8.48528i	0.395413	+	0.279600i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.00000	−	19.5959i	−0.131519	−	0.644310i
$926$		0			0
$927$	−3.46410	−	9.79796i	−0.113776	−	0.321807i
$928$		0			0
$929$			5.65685i			0.185595i	0.995685	+	0.0927977i	$0.0295810\pi$
$929$			5.65685i			0.185595i	−0.995685	+	0.0927977i	$0.970419\pi$
$930$		0			0
$931$	34.6410			1.13531
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			39.1918i			1.28034i	0.768233	+	0.640171i	$0.221136\pi$
$937$			39.1918i			1.28034i	−0.768233	+	0.640171i	$0.778864\pi$
$938$		0			0
$939$	13.8564	+	9.79796i	0.452187	+	0.319744i
$940$		0			0
$941$	3.46410			0.112926			0.0564632	−	0.998405i	$-0.482018\pi$
$941$	3.46410			0.112926			0.0564632	+	0.998405i	$0.482018\pi$
$942$		0			0
$943$	27.7128			0.902453
$944$		0			0
$945$	23.0718	+	32.9802i	0.750526	+	1.07285i
$946$		0			0
$947$	−18.0000			−0.584921			−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000			−0.584921			−0.292461	+	0.956278i	$0.594474\pi$
$948$		0			0
$949$			39.1918i			1.27222i
$950$		0			0
$951$	4.00000	+	2.82843i	0.129709	+	0.0917180i
$952$		0			0
$953$	48.4974			1.57099			0.785493	−	0.618871i	$-0.212410\pi$
$953$	48.4974			1.57099			0.785493	+	0.618871i	$0.212410\pi$
$954$		0			0
$955$	41.5692	+	33.9411i	1.34515	+	1.09831i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000			−0.775000
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$	6.00000	+	16.9706i	0.193347	+	0.546869i
$964$		0			0
$965$	13.8564	−	16.9706i	0.446054	−	0.546302i
$966$		0			0
$967$	−31.1769			−1.00258			−0.501291	−	0.865279i	$-0.667141\pi$
$967$	−31.1769			−1.00258			−0.501291	+	0.865279i	$0.667141\pi$
$968$		0			0
$969$	48.0000	−	67.8823i	1.54198	−	2.18069i
$970$		0			0
$971$		−	19.5959i		−	0.628863i	−0.949280	−	0.314431i	$-0.898186\pi$
$971$		−	19.5959i		−	0.628863i	0.949280	−	0.314431i	$-0.101814\pi$
$972$		0			0
$973$	24.0000			0.769405
$974$		0			0
$975$	31.7128	+	13.9391i	1.01562	+	0.446407i
$976$		0			0
$977$	−20.7846			−0.664959			−0.332479	−	0.943111i	$-0.607885\pi$
$977$	−20.7846			−0.664959			−0.332479	+	0.943111i	$0.607885\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−27.7128	+	9.79796i	−0.884802	+	0.312825i
$982$		0			0
$983$		−	14.6969i		−	0.468760i	−0.972145	−	0.234380i	$-0.924694\pi$
$983$		−	14.6969i		−	0.468760i	0.972145	−	0.234380i	$-0.0753059\pi$
$984$		0			0
$985$	28.0000	−	34.2929i	0.892154	−	1.09266i
$986$		0			0
$987$	−24.0000	−	16.9706i	−0.763928	−	0.540179i
$988$		0			0
$989$	41.5692			1.32182
$990$		0			0
$991$		−	8.48528i		−	0.269544i	−0.990877	−	0.134772i	$-0.956970\pi$
$991$		−	8.48528i		−	0.269544i	0.990877	−	0.134772i	$-0.0430302\pi$
$992$		0			0
$993$	−20.7846	+	29.3939i	−0.659580	+	0.932786i
$994$		0			0
$995$	−36.0000	+	44.0908i	−1.14128	+	1.39777i
$996$		0			0
$997$	−44.0000			−1.39349			−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000			−1.39349			−0.696747	+	0.717317i	$0.745370\pi$
$998$		0			0
$999$	20.0000	+	5.65685i	0.632772	+	0.178975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.m.v.959.1	yes	4
3.2	odd	2		1920.2.m.b.959.2	yes	4
4.3	odd	2		1920.2.m.b.959.3	yes	4
5.4	even	2		1920.2.m.a.959.3	yes	4
8.3	odd	2		1920.2.m.u.959.2	yes	4
8.5	even	2		1920.2.m.a.959.4	yes	4
12.11	even	2	inner	1920.2.m.v.959.4	yes	4
15.14	odd	2		1920.2.m.u.959.4	yes	4
20.19	odd	2		1920.2.m.u.959.1	yes	4
24.5	odd	2		1920.2.m.u.959.3	yes	4
24.11	even	2		1920.2.m.a.959.1	✓	4
40.19	odd	2		1920.2.m.b.959.4	yes	4
40.29	even	2	inner	1920.2.m.v.959.2	yes	4
60.59	even	2		1920.2.m.a.959.2	yes	4
120.29	odd	2		1920.2.m.b.959.1	yes	4
120.59	even	2	inner	1920.2.m.v.959.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.m.a.959.1	✓	4	24.11	even	2
1920.2.m.a.959.2	yes	4	60.59	even	2
1920.2.m.a.959.3	yes	4	5.4	even	2
1920.2.m.a.959.4	yes	4	8.5	even	2
1920.2.m.b.959.1	yes	4	120.29	odd	2
1920.2.m.b.959.2	yes	4	3.2	odd	2
1920.2.m.b.959.3	yes	4	4.3	odd	2
1920.2.m.b.959.4	yes	4	40.19	odd	2
1920.2.m.u.959.1	yes	4	20.19	odd	2
1920.2.m.u.959.2	yes	4	8.3	odd	2
1920.2.m.u.959.3	yes	4	24.5	odd	2
1920.2.m.u.959.4	yes	4	15.14	odd	2
1920.2.m.v.959.1	yes	4	1.1	even	1	trivial
1920.2.m.v.959.2	yes	4	40.29	even	2	inner
1920.2.m.v.959.3	yes	4	120.59	even	2	inner
1920.2.m.v.959.4	yes	4	12.11	even	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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.41421i) q^{3} +(-1.73205 - 1.41421i) q^{5} +3.46410 q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.41421i) q^{3} +(-1.73205 - 1.41421i) q^{5} +3.46410 q^{7} +(-1.00000 - 2.82843i) q^{9} +4.00000 q^{13} +(-3.73205 + 1.03528i) q^{15} +6.92820 q^{17} +6.92820 q^{19} +(3.46410 - 4.89898i) q^{21} +4.89898i q^{23} +(1.00000 + 4.89898i) q^{25} +(-5.00000 - 1.41421i) q^{27} -3.46410 q^{29} +8.48528i q^{31} +(-6.00000 - 4.89898i) q^{35} -4.00000 q^{37} +(4.00000 - 5.65685i) q^{39} -5.65685i q^{41} -8.48528i q^{43} +(-2.26795 + 6.31319i) q^{45} -4.89898i q^{47} +5.00000 q^{49} +(6.92820 - 9.79796i) q^{51} +2.82843i q^{53} +(6.92820 - 9.79796i) q^{57} -9.79796i q^{59} +9.79796i q^{61} +(-3.46410 - 9.79796i) q^{63} +(-6.92820 - 5.65685i) q^{65} -8.48528i q^{67} +(6.92820 + 4.89898i) q^{69} +9.79796i q^{73} +(7.92820 + 3.48477i) q^{75} -8.48528i q^{79} +(-7.00000 + 5.65685i) q^{81} -6.00000 q^{83} +(-12.0000 - 9.79796i) q^{85} +(-3.46410 + 4.89898i) q^{87} +11.3137i q^{89} +13.8564 q^{91} +(12.0000 + 8.48528i) q^{93} +(-12.0000 - 9.79796i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{9} + 16 q^{13} - 8 q^{15} + 4 q^{25} - 20 q^{27} - 24 q^{35} - 16 q^{37} + 16 q^{39} - 16 q^{45} + 20 q^{49} + 4 q^{75} - 28 q^{81} - 24 q^{83} - 48 q^{85} + 48 q^{93} - 48 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^9 + 16 * q^13 - 8 * q^15 + 4 * q^25 - 20 * q^27 - 24 * q^35 - 16 * q^37 + 16 * q^39 - 16 * q^45 + 20 * q^49 + 4 * q^75 - 28 * q^81 - 24 * q^83 - 48 * q^85 + 48 * q^93 - 48 * q^95

Introduction

L-functions

Modular forms

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