LMFDB - Embedded newform 760.2.d.d.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,2,Mod(609,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("760.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.d (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:	$6.06863055362$
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^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	760.609
Dual form			760.2.d.d.609.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} +(1.00000 + 2.00000i) q^{5} +0.828427i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{3} +(1.00000 + 2.00000i) q^{5} +0.828427i q^{7} +1.00000 q^{9} -0.828427 q^{11} +3.41421i q^{13} +(2.82843 - 1.41421i) q^{15} +2.82843i q^{17} +1.00000 q^{19} +1.17157 q^{21} +3.65685i q^{23} +(-3.00000 + 4.00000i) q^{25} -5.65685i q^{27} +7.65685 q^{29} +1.17157 q^{31} +1.17157i q^{33} +(-1.65685 + 0.828427i) q^{35} +3.41421i q^{37} +4.82843 q^{39} -4.82843 q^{41} +3.17157i q^{43} +(1.00000 + 2.00000i) q^{45} -4.82843i q^{47} +6.31371 q^{49} +4.00000 q^{51} -7.89949i q^{53} +(-0.828427 - 1.65685i) q^{55} -1.41421i q^{57} +1.17157 q^{59} +5.65685 q^{61} +0.828427i q^{63} +(-6.82843 + 3.41421i) q^{65} -9.89949i q^{67} +5.17157 q^{69} +8.48528 q^{71} +6.82843i q^{73} +(5.65685 + 4.24264i) q^{75} -0.686292i q^{77} +8.48528 q^{79} -5.00000 q^{81} -4.34315i q^{83} +(-5.65685 + 2.82843i) q^{85} -10.8284i q^{87} -9.31371 q^{89} -2.82843 q^{91} -1.65685i q^{93} +(1.00000 + 2.00000i) q^{95} -13.0711i q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{5} + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 12 q^{25} + 8 q^{29} + 16 q^{31} + 16 q^{35} + 8 q^{39} - 8 q^{41} + 4 q^{45} - 20 q^{49} + 16 q^{51} + 8 q^{55} + 16 q^{59} - 16 q^{65} + 32 q^{69} - 20 q^{81} + 8 q^{89} + 4 q^{95} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^9 + 8 * q^11 + 4 * q^19 + 16 * q^21 - 12 * q^25 + 8 * q^29 + 16 * q^31 + 16 * q^35 + 8 * q^39 - 8 * q^41 + 4 * q^45 - 20 * q^49 + 16 * q^51 + 8 * q^55 + 16 * q^59 - 16 * q^65 + 32 * q^69 - 20 * q^81 + 8 * q^89 + 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.41421i		−	0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$		−	1.41421i		−	0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$		0			0
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$			0.828427i			0.313116i	0.987669	+	0.156558i	$0.0500398\pi$
$7$			0.828427i			0.313116i	−0.987669	+	0.156558i	$0.949960\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		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
$13$			3.41421i			0.946932i	0.880812	+	0.473466i	$0.156997\pi$
$13$			3.41421i			0.946932i	−0.880812	+	0.473466i	$0.843003\pi$
$14$		0			0
$15$	2.82843	−	1.41421i	0.730297	−	0.365148i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$	1.17157			0.255658
$22$		0			0
$23$			3.65685i			0.762507i	0.924471	+	0.381253i	$0.124507\pi$
$23$			3.65685i			0.762507i	−0.924471	+	0.381253i	$0.875493\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		−	5.65685i		−	1.08866i
$28$		0			0
$29$	7.65685			1.42184			0.710921	−	0.703272i	$-0.248278\pi$
$29$	7.65685			1.42184			0.710921	+	0.703272i	$0.248278\pi$
$30$		0			0
$31$	1.17157			0.210421			0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157			0.210421			0.105210	+	0.994450i	$0.466448\pi$
$32$		0			0
$33$			1.17157i			0.203945i
$34$		0			0
$35$	−1.65685	+	0.828427i	−0.280059	+	0.140030i
$36$		0			0
$37$			3.41421i			0.561293i	0.959811	+	0.280647i	$0.0905489\pi$
$37$			3.41421i			0.561293i	−0.959811	+	0.280647i	$0.909451\pi$
$38$		0			0
$39$	4.82843			0.773167
$40$		0			0
$41$	−4.82843			−0.754074			−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843			−0.754074			−0.377037	+	0.926198i	$0.623057\pi$
$42$		0			0
$43$			3.17157i			0.483660i	0.970319	+	0.241830i	$0.0777477\pi$
$43$			3.17157i			0.483660i	−0.970319	+	0.241830i	$0.922252\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$		−	4.82843i		−	0.704298i	−0.935944	−	0.352149i	$-0.885451\pi$
$47$		−	4.82843i		−	0.704298i	0.935944	−	0.352149i	$-0.114549\pi$
$48$		0			0
$49$	6.31371			0.901958
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$		−	7.89949i		−	1.08508i	−0.840030	−	0.542540i	$-0.817463\pi$
$53$		−	7.89949i		−	1.08508i	0.840030	−	0.542540i	$-0.182537\pi$
$54$		0			0
$55$	−0.828427	−	1.65685i	−0.111705	−	0.223410i
$56$		0			0
$57$		−	1.41421i		−	0.187317i
$58$		0			0
$59$	1.17157			0.152526			0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157			0.152526			0.0762629	+	0.997088i	$0.475701\pi$
$60$		0			0
$61$	5.65685			0.724286			0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685			0.724286			0.362143	+	0.932123i	$0.382045\pi$
$62$		0			0
$63$			0.828427i			0.104372i
$64$		0			0
$65$	−6.82843	+	3.41421i	−0.846962	+	0.423481i
$66$		0			0
$67$		−	9.89949i		−	1.20942i	−0.796447	−	0.604708i	$-0.793290\pi$
$67$		−	9.89949i		−	1.20942i	0.796447	−	0.604708i	$-0.206710\pi$
$68$		0			0
$69$	5.17157			0.622584
$70$		0			0
$71$	8.48528			1.00702			0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528			1.00702			0.503509	+	0.863990i	$0.332042\pi$
$72$		0			0
$73$			6.82843i			0.799207i	0.916688	+	0.399603i	$0.130852\pi$
$73$			6.82843i			0.799207i	−0.916688	+	0.399603i	$0.869148\pi$
$74$		0			0
$75$	5.65685	+	4.24264i	0.653197	+	0.489898i
$76$		0			0
$77$		−	0.686292i		−	0.0782102i
$78$		0			0
$79$	8.48528			0.954669			0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528			0.954669			0.477334	+	0.878722i	$0.341603\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$		−	4.34315i		−	0.476722i	−0.971177	−	0.238361i	$-0.923390\pi$
$83$		−	4.34315i		−	0.476722i	0.971177	−	0.238361i	$-0.0766102\pi$
$84$		0			0
$85$	−5.65685	+	2.82843i	−0.613572	+	0.306786i
$86$		0			0
$87$		−	10.8284i		−	1.16093i
$88$		0			0
$89$	−9.31371			−0.987251			−0.493626	−	0.869675i	$-0.664329\pi$
$89$	−9.31371			−0.987251			−0.493626	+	0.869675i	$0.664329\pi$
$90$		0			0
$91$	−2.82843			−0.296500
$92$		0			0
$93$		−	1.65685i		−	0.171808i
$94$		0			0
$95$	1.00000	+	2.00000i	0.102598	+	0.205196i
$96$		0			0
$97$		−	13.0711i		−	1.32717i	−0.748103	−	0.663583i	$-0.769035\pi$
$97$		−	13.0711i		−	1.32717i	0.748103	−	0.663583i	$-0.230965\pi$
$98$		0			0
$99$	−0.828427			−0.0832601
$100$		0			0
$101$	−8.00000			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$102$		0			0
$103$			9.89949i			0.975426i	0.873004	+	0.487713i	$0.162169\pi$
$103$			9.89949i			0.975426i	−0.873004	+	0.487713i	$0.837831\pi$
$104$		0			0
$105$	1.17157	+	2.34315i	0.114334	+	0.228668i
$106$		0			0
$107$		−	7.75736i		−	0.749932i	−0.927039	−	0.374966i	$-0.877654\pi$
$107$		−	7.75736i		−	0.749932i	0.927039	−	0.374966i	$-0.122346\pi$
$108$		0			0
$109$	−6.48528			−0.621177			−0.310589	−	0.950544i	$-0.600526\pi$
$109$	−6.48528			−0.621177			−0.310589	+	0.950544i	$0.600526\pi$
$110$		0			0
$111$	4.82843			0.458294
$112$		0			0
$113$			8.58579i			0.807683i	0.914829	+	0.403841i	$0.132325\pi$
$113$			8.58579i			0.807683i	−0.914829	+	0.403841i	$0.867675\pi$
$114$		0			0
$115$	−7.31371	+	3.65685i	−0.682007	+	0.341003i
$116$		0			0
$117$			3.41421i			0.315644i
$118$		0			0
$119$	−2.34315			−0.214796
$120$		0			0
$121$	−10.3137			−0.937610
$122$		0			0
$123$			6.82843i			0.615699i
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$			12.7279i			1.12942i	0.825289	+	0.564710i	$0.191012\pi$
$127$			12.7279i			1.12942i	−0.825289	+	0.564710i	$0.808988\pi$
$128$		0			0
$129$	4.48528			0.394907
$130$		0			0
$131$	−6.34315			−0.554203			−0.277102	−	0.960841i	$-0.589374\pi$
$131$	−6.34315			−0.554203			−0.277102	+	0.960841i	$0.589374\pi$
$132$		0			0
$133$			0.828427i			0.0718337i
$134$		0			0
$135$	11.3137	−	5.65685i	0.973729	−	0.486864i
$136$		0			0
$137$			2.82843i			0.241649i	0.992674	+	0.120824i	$0.0385538\pi$
$137$			2.82843i			0.241649i	−0.992674	+	0.120824i	$0.961446\pi$
$138$		0			0
$139$	−20.8284			−1.76664			−0.883322	−	0.468767i	$-0.844699\pi$
$139$	−20.8284			−1.76664			−0.883322	+	0.468767i	$0.844699\pi$
$140$		0			0
$141$	−6.82843			−0.575057
$142$		0			0
$143$		−	2.82843i		−	0.236525i
$144$		0			0
$145$	7.65685	+	15.3137i	0.635867	+	1.27173i
$146$		0			0
$147$		−	8.92893i		−	0.736446i
$148$		0			0
$149$	−4.00000			−0.327693			−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000			−0.327693			−0.163846	+	0.986486i	$0.552390\pi$
$150$		0			0
$151$	−13.6569			−1.11138			−0.555690	−	0.831390i	$-0.687546\pi$
$151$	−13.6569			−1.11138			−0.555690	+	0.831390i	$0.687546\pi$
$152$		0			0
$153$			2.82843i			0.228665i
$154$		0			0
$155$	1.17157	+	2.34315i	0.0941030	+	0.188206i
$156$		0			0
$157$		−	12.4853i		−	0.996434i	−0.867052	−	0.498217i	$-0.833988\pi$
$157$		−	12.4853i		−	0.996434i	0.867052	−	0.498217i	$-0.166012\pi$
$158$		0			0
$159$	−11.1716			−0.885963
$160$		0			0
$161$	−3.02944			−0.238753
$162$		0			0
$163$		−	2.48528i		−	0.194662i	−0.995252	−	0.0973311i	$-0.968969\pi$
$163$		−	2.48528i		−	0.194662i	0.995252	−	0.0973311i	$-0.0310306\pi$
$164$		0			0
$165$	−2.34315	+	1.17157i	−0.182414	+	0.0912068i
$166$		0			0
$167$		−	11.0711i		−	0.856705i	−0.903612	−	0.428352i	$-0.859094\pi$
$167$		−	11.0711i		−	0.856705i	0.903612	−	0.428352i	$-0.140906\pi$
$168$		0			0
$169$	1.34315			0.103319
$170$		0			0
$171$	1.00000			0.0764719
$172$		0			0
$173$		−	11.4142i		−	0.867807i	−0.900959	−	0.433903i	$-0.857136\pi$
$173$		−	11.4142i		−	0.867807i	0.900959	−	0.433903i	$-0.142864\pi$
$174$		0			0
$175$	−3.31371	−	2.48528i	−0.250493	−	0.187870i
$176$		0			0
$177$		−	1.65685i		−	0.124537i
$178$		0			0
$179$	−17.1716			−1.28346			−0.641732	−	0.766929i	$-0.721784\pi$
$179$	−17.1716			−1.28346			−0.641732	+	0.766929i	$0.721784\pi$
$180$		0			0
$181$	25.3137			1.88155			0.940777	−	0.339027i	$-0.110098\pi$
$181$	25.3137			1.88155			0.940777	+	0.339027i	$0.110098\pi$
$182$		0			0
$183$		−	8.00000i		−	0.591377i
$184$		0			0
$185$	−6.82843	+	3.41421i	−0.502036	+	0.251018i
$186$		0			0
$187$		−	2.34315i		−	0.171348i
$188$		0			0
$189$	4.68629			0.340878
$190$		0			0
$191$	−3.31371			−0.239772			−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371			−0.239772			−0.119886	+	0.992788i	$0.538253\pi$
$192$		0			0
$193$			13.0711i			0.940876i	0.882433	+	0.470438i	$0.155904\pi$
$193$			13.0711i			0.940876i	−0.882433	+	0.470438i	$0.844096\pi$
$194$		0			0
$195$	4.82843	+	9.65685i	0.345771	+	0.691542i
$196$		0			0
$197$		−	21.6569i		−	1.54299i	−0.636237	−	0.771493i	$-0.719510\pi$
$197$		−	21.6569i		−	1.54299i	0.636237	−	0.771493i	$-0.280490\pi$
$198$		0			0
$199$	−11.3137			−0.802008			−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137			−0.802008			−0.401004	+	0.916076i	$0.631339\pi$
$200$		0			0
$201$	−14.0000			−0.987484
$202$		0			0
$203$			6.34315i			0.445202i
$204$		0			0
$205$	−4.82843	−	9.65685i	−0.337232	−	0.674464i
$206$		0			0
$207$			3.65685i			0.254169i
$208$		0			0
$209$	−0.828427			−0.0573035
$210$		0			0
$211$	22.6274			1.55774			0.778868	−	0.627188i	$-0.215794\pi$
$211$	22.6274			1.55774			0.778868	+	0.627188i	$0.215794\pi$
$212$		0			0
$213$		−	12.0000i		−	0.822226i
$214$		0			0
$215$	−6.34315	+	3.17157i	−0.432599	+	0.216299i
$216$		0			0
$217$			0.970563i			0.0658861i
$218$		0			0
$219$	9.65685			0.652550
$220$		0			0
$221$	−9.65685			−0.649590
$222$		0			0
$223$		−	9.89949i		−	0.662919i	−0.943469	−	0.331460i	$-0.892459\pi$
$223$		−	9.89949i		−	0.662919i	0.943469	−	0.331460i	$-0.107541\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$		−	19.0711i		−	1.26579i	−0.774237	−	0.632896i	$-0.781866\pi$
$227$		−	19.0711i		−	1.26579i	0.774237	−	0.632896i	$-0.218134\pi$
$228$		0			0
$229$	19.3137			1.27629			0.638143	−	0.769918i	$-0.279703\pi$
$229$	19.3137			1.27629			0.638143	+	0.769918i	$0.279703\pi$
$230$		0			0
$231$	−0.970563			−0.0638583
$232$		0			0
$233$		−	4.97056i		−	0.325632i	−0.986656	−	0.162816i	$-0.947942\pi$
$233$		−	4.97056i		−	0.325632i	0.986656	−	0.162816i	$-0.0520578\pi$
$234$		0			0
$235$	9.65685	−	4.82843i	0.629944	−	0.314972i
$236$		0			0
$237$		−	12.0000i		−	0.779484i
$238$		0			0
$239$	−24.9706			−1.61521			−0.807606	−	0.589723i	$-0.799237\pi$
$239$	−24.9706			−1.61521			−0.807606	+	0.589723i	$0.799237\pi$
$240$		0			0
$241$	8.82843			0.568689			0.284344	−	0.958722i	$-0.408224\pi$
$241$	8.82843			0.568689			0.284344	+	0.958722i	$0.408224\pi$
$242$		0			0
$243$		−	9.89949i		−	0.635053i
$244$		0			0
$245$	6.31371	+	12.6274i	0.403368	+	0.806736i
$246$		0			0
$247$			3.41421i			0.217241i
$248$		0			0
$249$	−6.14214			−0.389242
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		−	3.02944i		−	0.190459i
$254$		0			0
$255$	4.00000	+	8.00000i	0.250490	+	0.500979i
$256$		0			0
$257$			1.75736i			0.109621i	0.998497	+	0.0548105i	$0.0174555\pi$
$257$			1.75736i			0.109621i	−0.998497	+	0.0548105i	$0.982545\pi$
$258$		0			0
$259$	−2.82843			−0.175750
$260$		0			0
$261$	7.65685			0.473947
$262$		0			0
$263$			17.3137i			1.06761i	0.845608	+	0.533805i	$0.179238\pi$
$263$			17.3137i			1.06761i	−0.845608	+	0.533805i	$0.820762\pi$
$264$		0			0
$265$	15.7990	−	7.89949i	0.970524	−	0.485262i
$266$		0			0
$267$			13.1716i			0.806087i
$268$		0			0
$269$	8.14214			0.496435			0.248217	−	0.968704i	$-0.420155\pi$
$269$	8.14214			0.496435			0.248217	+	0.968704i	$0.420155\pi$
$270$		0			0
$271$	24.8284			1.50822			0.754110	−	0.656748i	$-0.228069\pi$
$271$	24.8284			1.50822			0.754110	+	0.656748i	$0.228069\pi$
$272$		0			0
$273$			4.00000i			0.242091i
$274$		0			0
$275$	2.48528	−	3.31371i	0.149868	−	0.199824i
$276$		0			0
$277$		−	14.8284i		−	0.890954i	−0.895293	−	0.445477i	$-0.853034\pi$
$277$		−	14.8284i		−	0.890954i	0.895293	−	0.445477i	$-0.146966\pi$
$278$		0			0
$279$	1.17157			0.0701402
$280$		0			0
$281$	−5.51472			−0.328981			−0.164490	−	0.986379i	$-0.552598\pi$
$281$	−5.51472			−0.328981			−0.164490	+	0.986379i	$0.552598\pi$
$282$		0			0
$283$			30.9706i			1.84101i	0.390732	+	0.920504i	$0.372222\pi$
$283$			30.9706i			1.84101i	−0.390732	+	0.920504i	$0.627778\pi$
$284$		0			0
$285$	2.82843	−	1.41421i	0.167542	−	0.0837708i
$286$		0			0
$287$		−	4.00000i		−	0.236113i
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	−18.4853			−1.08363
$292$		0			0
$293$		−	18.2426i		−	1.06575i	−0.846195	−	0.532873i	$-0.821112\pi$
$293$		−	18.2426i		−	1.06575i	0.846195	−	0.532873i	$-0.178888\pi$
$294$		0			0
$295$	1.17157	+	2.34315i	0.0682116	+	0.136423i
$296$		0			0
$297$			4.68629i			0.271926i
$298$		0			0
$299$	−12.4853			−0.722042
$300$		0			0
$301$	−2.62742			−0.151442
$302$		0			0
$303$			11.3137i			0.649956i
$304$		0			0
$305$	5.65685	+	11.3137i	0.323911	+	0.647821i
$306$		0			0
$307$		−	5.41421i		−	0.309005i	−0.987992	−	0.154503i	$-0.950622\pi$
$307$		−	5.41421i		−	0.309005i	0.987992	−	0.154503i	$-0.0493775\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	−28.8284			−1.63471			−0.817355	−	0.576134i	$-0.804561\pi$
$311$	−28.8284			−1.63471			−0.817355	+	0.576134i	$0.804561\pi$
$312$		0			0
$313$		−	12.9706i		−	0.733140i	−0.930391	−	0.366570i	$-0.880532\pi$
$313$		−	12.9706i		−	0.733140i	0.930391	−	0.366570i	$-0.119468\pi$
$314$		0			0
$315$	−1.65685	+	0.828427i	−0.0933532	+	0.0466766i
$316$		0			0
$317$		−	14.7279i		−	0.827203i	−0.910458	−	0.413601i	$-0.864271\pi$
$317$		−	14.7279i		−	0.827203i	0.910458	−	0.413601i	$-0.135729\pi$
$318$		0			0
$319$	−6.34315			−0.355148
$320$		0			0
$321$	−10.9706			−0.612317
$322$		0			0
$323$			2.82843i			0.157378i
$324$		0			0
$325$	−13.6569	−	10.2426i	−0.757546	−	0.568159i
$326$		0			0
$327$			9.17157i			0.507189i
$328$		0			0
$329$	4.00000			0.220527
$330$		0			0
$331$	−2.34315			−0.128791			−0.0643955	−	0.997924i	$-0.520512\pi$
$331$	−2.34315			−0.128791			−0.0643955	+	0.997924i	$0.520512\pi$
$332$		0			0
$333$			3.41421i			0.187098i
$334$		0			0
$335$	19.7990	−	9.89949i	1.08173	−	0.540867i
$336$		0			0
$337$		−	3.89949i		−	0.212419i	−0.994344	−	0.106210i	$-0.966129\pi$
$337$		−	3.89949i		−	0.212419i	0.994344	−	0.106210i	$-0.0338714\pi$
$338$		0			0
$339$	12.1421			0.659470
$340$		0			0
$341$	−0.970563			−0.0525589
$342$		0			0
$343$			11.0294i			0.595534i
$344$		0			0
$345$	5.17157	+	10.3431i	0.278428	+	0.556856i
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	16.6274			0.890045			0.445023	−	0.895519i	$-0.353196\pi$
$349$	16.6274			0.890045			0.445023	+	0.895519i	$0.353196\pi$
$350$		0			0
$351$	19.3137			1.03089
$352$		0			0
$353$			14.1421i			0.752710i	0.926476	+	0.376355i	$0.122823\pi$
$353$			14.1421i			0.752710i	−0.926476	+	0.376355i	$0.877177\pi$
$354$		0			0
$355$	8.48528	+	16.9706i	0.450352	+	0.900704i
$356$		0			0
$357$			3.31371i			0.175380i
$358$		0			0
$359$	−6.48528			−0.342280			−0.171140	−	0.985247i	$-0.554745\pi$
$359$	−6.48528			−0.342280			−0.171140	+	0.985247i	$0.554745\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			14.5858i			0.765555i
$364$		0			0
$365$	−13.6569	+	6.82843i	−0.714832	+	0.357416i
$366$		0			0
$367$			8.14214i			0.425016i	0.977159	+	0.212508i	$0.0681632\pi$
$367$			8.14214i			0.425016i	−0.977159	+	0.212508i	$0.931837\pi$
$368$		0			0
$369$	−4.82843			−0.251358
$370$		0			0
$371$	6.54416			0.339756
$372$		0			0
$373$			8.10051i			0.419428i	0.977763	+	0.209714i	$0.0672533\pi$
$373$			8.10051i			0.419428i	−0.977763	+	0.209714i	$0.932747\pi$
$374$		0			0
$375$	−2.82843	+	15.5563i	−0.146059	+	0.803326i
$376$		0			0
$377$			26.1421i			1.34639i
$378$		0			0
$379$	0.686292			0.0352524			0.0176262	−	0.999845i	$-0.494389\pi$
$379$	0.686292			0.0352524			0.0176262	+	0.999845i	$0.494389\pi$
$380$		0			0
$381$	18.0000			0.922168
$382$		0			0
$383$		−	10.5858i		−	0.540908i	−0.962733	−	0.270454i	$-0.912826\pi$
$383$		−	10.5858i		−	0.540908i	0.962733	−	0.270454i	$-0.0871739\pi$
$384$		0			0
$385$	1.37258	−	0.686292i	0.0699533	−	0.0349767i
$386$		0			0
$387$			3.17157i			0.161220i
$388$		0			0
$389$	−13.3137			−0.675032			−0.337516	−	0.941320i	$-0.609587\pi$
$389$	−13.3137			−0.675032			−0.337516	+	0.941320i	$0.609587\pi$
$390$		0			0
$391$	−10.3431			−0.523075
$392$		0			0
$393$			8.97056i			0.452505i
$394$		0			0
$395$	8.48528	+	16.9706i	0.426941	+	0.853882i
$396$		0			0
$397$		−	18.8284i		−	0.944972i	−0.881338	−	0.472486i	$-0.843357\pi$
$397$		−	18.8284i		−	0.944972i	0.881338	−	0.472486i	$-0.156643\pi$
$398$		0			0
$399$	1.17157			0.0586520
$400$		0			0
$401$	6.97056			0.348093			0.174047	−	0.984737i	$-0.444316\pi$
$401$	6.97056			0.348093			0.174047	+	0.984737i	$0.444316\pi$
$402$		0			0
$403$			4.00000i			0.199254i
$404$		0			0
$405$	−5.00000	−	10.0000i	−0.248452	−	0.496904i
$406$		0			0
$407$		−	2.82843i		−	0.140200i
$408$		0			0
$409$	31.1716			1.54134			0.770668	−	0.637237i	$-0.219923\pi$
$409$	31.1716			1.54134			0.770668	+	0.637237i	$0.219923\pi$
$410$		0			0
$411$	4.00000			0.197305
$412$		0			0
$413$			0.970563i			0.0477583i
$414$		0			0
$415$	8.68629	−	4.34315i	0.426393	−	0.213197i
$416$		0			0
$417$			29.4558i			1.44246i
$418$		0			0
$419$	−34.6274			−1.69166			−0.845830	−	0.533453i	$-0.820894\pi$
$419$	−34.6274			−1.69166			−0.845830	+	0.533453i	$0.820894\pi$
$420$		0			0
$421$	37.7990			1.84221			0.921105	−	0.389314i	$-0.127288\pi$
$421$	37.7990			1.84221			0.921105	+	0.389314i	$0.127288\pi$
$422$		0			0
$423$		−	4.82843i		−	0.234766i
$424$		0			0
$425$	−11.3137	−	8.48528i	−0.548795	−	0.411597i
$426$		0			0
$427$			4.68629i			0.226786i
$428$		0			0
$429$	−4.00000			−0.193122
$430$		0			0
$431$	37.9411			1.82756			0.913780	−	0.406210i	$-0.133150\pi$
$431$	37.9411			1.82756			0.913780	+	0.406210i	$0.133150\pi$
$432$		0			0
$433$		−	6.44365i		−	0.309662i	−0.987941	−	0.154831i	$-0.950517\pi$
$433$		−	6.44365i		−	0.309662i	0.987941	−	0.154831i	$-0.0494833\pi$
$434$		0			0
$435$	21.6569	−	10.8284i	1.03837	−	0.519183i
$436$		0			0
$437$			3.65685i			0.174931i
$438$		0			0
$439$	−11.5147			−0.549568			−0.274784	−	0.961506i	$-0.588606\pi$
$439$	−11.5147			−0.549568			−0.274784	+	0.961506i	$0.588606\pi$
$440$		0			0
$441$	6.31371			0.300653
$442$		0			0
$443$		−	28.6274i		−	1.36013i	−0.733152	−	0.680065i	$-0.761952\pi$
$443$		−	28.6274i		−	1.36013i	0.733152	−	0.680065i	$-0.238048\pi$
$444$		0			0
$445$	−9.31371	−	18.6274i	−0.441512	−	0.883024i
$446$		0			0
$447$			5.65685i			0.267560i
$448$		0			0
$449$	9.51472			0.449027			0.224514	−	0.974471i	$-0.427921\pi$
$449$	9.51472			0.449027			0.224514	+	0.974471i	$0.427921\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$		0			0
$453$			19.3137i			0.907437i
$454$		0			0
$455$	−2.82843	−	5.65685i	−0.132599	−	0.265197i
$456$		0			0
$457$		−	24.9706i		−	1.16807i	−0.811727	−	0.584037i	$-0.801472\pi$
$457$		−	24.9706i		−	1.16807i	0.811727	−	0.584037i	$-0.198528\pi$
$458$		0			0
$459$	16.0000			0.746816
$460$		0			0
$461$	17.3137			0.806380			0.403190	−	0.915116i	$-0.367901\pi$
$461$	17.3137			0.806380			0.403190	+	0.915116i	$0.367901\pi$
$462$		0			0
$463$		−	2.97056i		−	0.138054i	−0.997615	−	0.0690269i	$-0.978011\pi$
$463$		−	2.97056i		−	0.138054i	0.997615	−	0.0690269i	$-0.0219894\pi$
$464$		0			0
$465$	3.31371	−	1.65685i	0.153670	−	0.0768348i
$466$		0			0
$467$		−	8.34315i		−	0.386075i	−0.981191	−	0.193037i	$-0.938166\pi$
$467$		−	8.34315i		−	0.386075i	0.981191	−	0.193037i	$-0.0618339\pi$
$468$		0			0
$469$	8.20101			0.378687
$470$		0			0
$471$	−17.6569			−0.813585
$472$		0			0
$473$		−	2.62742i		−	0.120809i
$474$		0			0
$475$	−3.00000	+	4.00000i	−0.137649	+	0.183533i
$476$		0			0
$477$		−	7.89949i		−	0.361693i
$478$		0			0
$479$	7.17157			0.327678			0.163839	−	0.986487i	$-0.447612\pi$
$479$	7.17157			0.327678			0.163839	+	0.986487i	$0.447612\pi$
$480$		0			0
$481$	−11.6569			−0.531507
$482$		0			0
$483$			4.28427i			0.194941i
$484$		0			0
$485$	26.1421	−	13.0711i	1.18705	−	0.593527i
$486$		0			0
$487$			32.7279i			1.48304i	0.670929	+	0.741522i	$0.265896\pi$
$487$			32.7279i			1.48304i	−0.670929	+	0.741522i	$0.734104\pi$
$488$		0			0
$489$	−3.51472			−0.158941
$490$		0			0
$491$	17.6569			0.796843			0.398421	−	0.917203i	$-0.369558\pi$
$491$	17.6569			0.796843			0.398421	+	0.917203i	$0.369558\pi$
$492$		0			0
$493$			21.6569i			0.975376i
$494$		0			0
$495$	−0.828427	−	1.65685i	−0.0372350	−	0.0744701i
$496$		0			0
$497$			7.02944i			0.315313i
$498$		0			0
$499$	35.4558			1.58722			0.793611	−	0.608426i	$-0.208199\pi$
$499$	35.4558			1.58722			0.793611	+	0.608426i	$0.208199\pi$
$500$		0			0
$501$	−15.6569			−0.699497
$502$		0			0
$503$		−	26.0000i		−	1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$		−	26.0000i		−	1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$		0			0
$505$	−8.00000	−	16.0000i	−0.355995	−	0.711991i
$506$		0			0
$507$		−	1.89949i		−	0.0843595i
$508$		0			0
$509$	−23.9411			−1.06117			−0.530586	−	0.847631i	$-0.678028\pi$
$509$	−23.9411			−1.06117			−0.530586	+	0.847631i	$0.678028\pi$
$510$		0			0
$511$	−5.65685			−0.250244
$512$		0			0
$513$		−	5.65685i		−	0.249756i
$514$		0			0
$515$	−19.7990	+	9.89949i	−0.872448	+	0.436224i
$516$		0			0
$517$			4.00000i			0.175920i
$518$		0			0
$519$	−16.1421			−0.708561
$520$		0			0
$521$	−35.6569			−1.56216			−0.781078	−	0.624434i	$-0.785330\pi$
$521$	−35.6569			−1.56216			−0.781078	+	0.624434i	$0.785330\pi$
$522$		0			0
$523$			37.4142i			1.63601i	0.575212	+	0.818005i	$0.304920\pi$
$523$			37.4142i			1.63601i	−0.575212	+	0.818005i	$0.695080\pi$
$524$		0			0
$525$	−3.51472	+	4.68629i	−0.153395	+	0.204527i
$526$		0			0
$527$			3.31371i			0.144347i
$528$		0			0
$529$	9.62742			0.418583
$530$		0			0
$531$	1.17157			0.0508419
$532$		0			0
$533$		−	16.4853i		−	0.714057i
$534$		0			0
$535$	15.5147	−	7.75736i	0.670760	−	0.335380i
$536$		0			0
$537$			24.2843i			1.04794i
$538$		0			0
$539$	−5.23045			−0.225291
$540$		0			0
$541$	4.00000			0.171973			0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000			0.171973			0.0859867	+	0.996296i	$0.472596\pi$
$542$		0			0
$543$		−	35.7990i		−	1.53628i
$544$		0			0
$545$	−6.48528	−	12.9706i	−0.277799	−	0.555598i
$546$		0			0
$547$		−	0.242641i		−	0.0103746i	−0.999987	−	0.00518728i	$-0.998349\pi$
$547$		−	0.242641i		−	0.0103746i	0.999987	−	0.00518728i	$-0.00165117\pi$
$548$		0			0
$549$	5.65685			0.241429
$550$		0			0
$551$	7.65685			0.326193
$552$		0			0
$553$			7.02944i			0.298922i
$554$		0			0
$555$	4.82843	+	9.65685i	0.204955	+	0.409911i
$556$		0			0
$557$		−	9.17157i		−	0.388612i	−0.980941	−	0.194306i	$-0.937755\pi$
$557$		−	9.17157i		−	0.388612i	0.980941	−	0.194306i	$-0.0622455\pi$
$558$		0			0
$559$	−10.8284			−0.457994
$560$		0			0
$561$	−3.31371			−0.139905
$562$		0			0
$563$		−	42.5858i		−	1.79478i	−0.441241	−	0.897388i	$-0.645462\pi$
$563$		−	42.5858i		−	1.79478i	0.441241	−	0.897388i	$-0.354538\pi$
$564$		0			0
$565$	−17.1716	+	8.58579i	−0.722414	+	0.361207i
$566$		0			0
$567$		−	4.14214i		−	0.173953i
$568$		0			0
$569$	39.4558			1.65408			0.827038	−	0.562147i	$-0.190024\pi$
$569$	39.4558			1.65408			0.827038	+	0.562147i	$0.190024\pi$
$570$		0			0
$571$	3.45584			0.144623			0.0723113	−	0.997382i	$-0.476962\pi$
$571$	3.45584			0.144623			0.0723113	+	0.997382i	$0.476962\pi$
$572$		0			0
$573$			4.68629i			0.195773i
$574$		0			0
$575$	−14.6274	−	10.9706i	−0.610005	−	0.457504i
$576$		0			0
$577$		−	17.6569i		−	0.735064i	−0.930011	−	0.367532i	$-0.880203\pi$
$577$		−	17.6569i		−	0.735064i	0.930011	−	0.367532i	$-0.119797\pi$
$578$		0			0
$579$	18.4853			0.768222
$580$		0			0
$581$	3.59798			0.149269
$582$		0			0
$583$			6.54416i			0.271031i
$584$		0			0
$585$	−6.82843	+	3.41421i	−0.282321	+	0.141160i
$586$		0			0
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$	1.17157			0.0482738
$590$		0			0
$591$	−30.6274			−1.25984
$592$		0			0
$593$			30.6274i			1.25772i	0.777520	+	0.628859i	$0.216478\pi$
$593$			30.6274i			1.25772i	−0.777520	+	0.628859i	$0.783522\pi$
$594$		0			0
$595$	−2.34315	−	4.68629i	−0.0960596	−	0.192119i
$596$		0			0
$597$			16.0000i			0.654836i
$598$		0			0
$599$	5.45584			0.222920			0.111460	−	0.993769i	$-0.464447\pi$
$599$	5.45584			0.222920			0.111460	+	0.993769i	$0.464447\pi$
$600$		0			0
$601$	−40.1421			−1.63743			−0.818716	−	0.574199i	$-0.805314\pi$
$601$	−40.1421			−1.63743			−0.818716	+	0.574199i	$0.805314\pi$
$602$		0			0
$603$		−	9.89949i		−	0.403139i
$604$		0			0
$605$	−10.3137	−	20.6274i	−0.419312	−	0.838624i
$606$		0			0
$607$		−	1.21320i		−	0.0492424i	−0.999697	−	0.0246212i	$-0.992162\pi$
$607$		−	1.21320i		−	0.0492424i	0.999697	−	0.0246212i	$-0.00783796\pi$
$608$		0			0
$609$	8.97056			0.363506
$610$		0			0
$611$	16.4853			0.666923
$612$		0			0
$613$		−	7.51472i		−	0.303517i	−0.988418	−	0.151758i	$-0.951506\pi$
$613$		−	7.51472i		−	0.303517i	0.988418	−	0.151758i	$-0.0484935\pi$
$614$		0			0
$615$	−13.6569	+	6.82843i	−0.550698	+	0.275349i
$616$		0			0
$617$			1.17157i			0.0471657i	0.999722	+	0.0235829i	$0.00750736\pi$
$617$			1.17157i			0.0471657i	−0.999722	+	0.0235829i	$0.992493\pi$
$618$		0			0
$619$	−36.8284			−1.48026			−0.740130	−	0.672464i	$-0.765236\pi$
$619$	−36.8284			−1.48026			−0.740130	+	0.672464i	$0.765236\pi$
$620$		0			0
$621$	20.6863			0.830112
$622$		0			0
$623$		−	7.71573i		−	0.309124i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			1.17157i			0.0467881i
$628$		0			0
$629$	−9.65685			−0.385044
$630$		0			0
$631$	17.7990			0.708567			0.354283	−	0.935138i	$-0.384725\pi$
$631$	17.7990			0.708567			0.354283	+	0.935138i	$0.384725\pi$
$632$		0			0
$633$		−	32.0000i		−	1.27189i
$634$		0			0
$635$	−25.4558	+	12.7279i	−1.01018	+	0.505092i
$636$		0			0
$637$			21.5563i			0.854094i
$638$		0			0
$639$	8.48528			0.335673
$640$		0			0
$641$	−40.1421			−1.58552			−0.792760	−	0.609535i	$-0.791356\pi$
$641$	−40.1421			−1.58552			−0.792760	+	0.609535i	$0.791356\pi$
$642$		0			0
$643$		−	22.9706i		−	0.905871i	−0.891543	−	0.452935i	$-0.850377\pi$
$643$		−	22.9706i		−	0.905871i	0.891543	−	0.452935i	$-0.149623\pi$
$644$		0			0
$645$	4.48528	+	8.97056i	0.176608	+	0.353216i
$646$		0			0
$647$			46.2843i			1.81962i	0.415022	+	0.909811i	$0.363774\pi$
$647$			46.2843i			1.81962i	−0.415022	+	0.909811i	$0.636226\pi$
$648$		0			0
$649$	−0.970563			−0.0380979
$650$		0			0
$651$	1.37258			0.0537958
$652$		0			0
$653$			2.82843i			0.110685i	0.998467	+	0.0553425i	$0.0176251\pi$
$653$			2.82843i			0.110685i	−0.998467	+	0.0553425i	$0.982375\pi$
$654$		0			0
$655$	−6.34315	−	12.6863i	−0.247847	−	0.495694i
$656$		0			0
$657$			6.82843i			0.266402i
$658$		0			0
$659$	15.5147			0.604368			0.302184	−	0.953250i	$-0.402284\pi$
$659$	15.5147			0.604368			0.302184	+	0.953250i	$0.402284\pi$
$660$		0			0
$661$	−23.4558			−0.912327			−0.456163	−	0.889896i	$-0.650777\pi$
$661$	−23.4558			−0.912327			−0.456163	+	0.889896i	$0.650777\pi$
$662$		0			0
$663$			13.6569i			0.530388i
$664$		0			0
$665$	−1.65685	+	0.828427i	−0.0642501	+	0.0321250i
$666$		0			0
$667$			28.0000i			1.08416i
$668$		0			0
$669$	−14.0000			−0.541271
$670$		0			0
$671$	−4.68629			−0.180912
$672$		0			0
$673$		−	30.2426i		−	1.16577i	−0.812555	−	0.582884i	$-0.801924\pi$
$673$		−	30.2426i		−	1.16577i	0.812555	−	0.582884i	$-0.198076\pi$
$674$		0			0
$675$	22.6274	+	16.9706i	0.870930	+	0.653197i
$676$		0			0
$677$			31.8995i			1.22600i	0.790084	+	0.612999i	$0.210037\pi$
$677$			31.8995i			1.22600i	−0.790084	+	0.612999i	$0.789963\pi$
$678$		0			0
$679$	10.8284			0.415557
$680$		0			0
$681$	−26.9706			−1.03351
$682$		0			0
$683$		−	20.9289i		−	0.800823i	−0.916335	−	0.400412i	$-0.868867\pi$
$683$		−	20.9289i		−	0.800823i	0.916335	−	0.400412i	$-0.131133\pi$
$684$		0			0
$685$	−5.65685	+	2.82843i	−0.216137	+	0.108069i
$686$		0			0
$687$		−	27.3137i		−	1.04208i
$688$		0			0
$689$	26.9706			1.02750
$690$		0			0
$691$	−34.7696			−1.32270			−0.661348	−	0.750079i	$-0.730015\pi$
$691$	−34.7696			−1.32270			−0.661348	+	0.750079i	$0.730015\pi$
$692$		0			0
$693$		−	0.686292i		−	0.0260701i
$694$		0			0
$695$	−20.8284	−	41.6569i	−0.790067	−	1.58013i
$696$		0			0
$697$		−	13.6569i		−	0.517290i
$698$		0			0
$699$	−7.02944			−0.265878
$700$		0			0
$701$	8.68629			0.328077			0.164038	−	0.986454i	$-0.447548\pi$
$701$	8.68629			0.328077			0.164038	+	0.986454i	$0.447548\pi$
$702$		0			0
$703$			3.41421i			0.128770i
$704$		0			0
$705$	−6.82843	−	13.6569i	−0.257173	−	0.514347i
$706$		0			0
$707$		−	6.62742i		−	0.249250i
$708$		0			0
$709$	−10.6863			−0.401332			−0.200666	−	0.979660i	$-0.564311\pi$
$709$	−10.6863			−0.401332			−0.200666	+	0.979660i	$0.564311\pi$
$710$		0			0
$711$	8.48528			0.318223
$712$		0			0
$713$			4.28427i			0.160447i
$714$		0			0
$715$	5.65685	−	2.82843i	0.211554	−	0.105777i
$716$		0			0
$717$			35.3137i			1.31881i
$718$		0			0
$719$	14.4853			0.540210			0.270105	−	0.962831i	$-0.412942\pi$
$719$	14.4853			0.540210			0.270105	+	0.962831i	$0.412942\pi$
$720$		0			0
$721$	−8.20101			−0.305422
$722$		0			0
$723$		−	12.4853i		−	0.464333i
$724$		0			0
$725$	−22.9706	+	30.6274i	−0.853105	+	1.13747i
$726$		0			0
$727$			19.4558i			0.721577i	0.932648	+	0.360789i	$0.117492\pi$
$727$			19.4558i			0.721577i	−0.932648	+	0.360789i	$0.882508\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	−8.97056			−0.331788
$732$		0			0
$733$			35.3137i			1.30434i	0.758072	+	0.652171i	$0.226142\pi$
$733$			35.3137i			1.30434i	−0.758072	+	0.652171i	$0.773858\pi$
$734$		0			0
$735$	17.8579	−	8.92893i	0.658697	−	0.329349i
$736$		0			0
$737$			8.20101i			0.302088i
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$	4.82843			0.177377
$742$		0			0
$743$			21.8995i			0.803415i	0.915768	+	0.401707i	$0.131583\pi$
$743$			21.8995i			0.803415i	−0.915768	+	0.401707i	$0.868417\pi$
$744$		0			0
$745$	−4.00000	−	8.00000i	−0.146549	−	0.293097i
$746$		0			0
$747$		−	4.34315i		−	0.158907i
$748$		0			0
$749$	6.42641			0.234816
$750$		0			0
$751$	−35.5980			−1.29899			−0.649494	−	0.760366i	$-0.725019\pi$
$751$	−35.5980			−1.29899			−0.649494	+	0.760366i	$0.725019\pi$
$752$		0			0
$753$		−	5.65685i		−	0.206147i
$754$		0			0
$755$	−13.6569	−	27.3137i	−0.497024	−	0.994048i
$756$		0			0
$757$		−	6.62742i		−	0.240878i	−0.992721	−	0.120439i	$-0.961570\pi$
$757$		−	6.62742i		−	0.240878i	0.992721	−	0.120439i	$-0.0384301\pi$
$758$		0			0
$759$	−4.28427			−0.155509
$760$		0			0
$761$	32.2843			1.17030			0.585152	−	0.810924i	$-0.301035\pi$
$761$	32.2843			1.17030			0.585152	+	0.810924i	$0.301035\pi$
$762$		0			0
$763$		−	5.37258i		−	0.194501i
$764$		0			0
$765$	−5.65685	+	2.82843i	−0.204524	+	0.102262i
$766$		0			0
$767$			4.00000i			0.144432i
$768$		0			0
$769$	−20.2843			−0.731470			−0.365735	−	0.930719i	$-0.619182\pi$
$769$	−20.2843			−0.731470			−0.365735	+	0.930719i	$0.619182\pi$
$770$		0			0
$771$	2.48528			0.0895052
$772$		0			0
$773$			1.07107i			0.0385236i	0.999814	+	0.0192618i	$0.00613161\pi$
$773$			1.07107i			0.0385236i	−0.999814	+	0.0192618i	$0.993868\pi$
$774$		0			0
$775$	−3.51472	+	4.68629i	−0.126252	+	0.168337i
$776$		0			0
$777$			4.00000i			0.143499i
$778$		0			0
$779$	−4.82843			−0.172996
$780$		0			0
$781$	−7.02944			−0.251533
$782$		0			0
$783$		−	43.3137i		−	1.54791i
$784$		0			0
$785$	24.9706	−	12.4853i	0.891238	−	0.445619i
$786$		0			0
$787$		−	36.5269i		−	1.30204i	−0.759059	−	0.651022i	$-0.774341\pi$
$787$		−	36.5269i		−	1.30204i	0.759059	−	0.651022i	$-0.225659\pi$
$788$		0			0
$789$	24.4853			0.871699
$790$		0			0
$791$	−7.11270			−0.252898
$792$		0			0
$793$			19.3137i			0.685850i
$794$		0			0
$795$	−11.1716	−	22.3431i	−0.396215	−	0.792430i
$796$		0			0
$797$		−	10.0416i		−	0.355693i	−0.984058	−	0.177846i	$-0.943087\pi$
$797$		−	10.0416i		−	0.355693i	0.984058	−	0.177846i	$-0.0569130\pi$
$798$		0			0
$799$	13.6569			0.483145
$800$		0			0
$801$	−9.31371			−0.329084
$802$		0			0
$803$		−	5.65685i		−	0.199626i
$804$		0			0
$805$	−3.02944	−	6.05887i	−0.106774	−	0.213547i
$806$		0			0
$807$		−	11.5147i		−	0.405337i
$808$		0			0
$809$	−55.2548			−1.94266			−0.971328	−	0.237742i	$-0.923593\pi$
$809$	−55.2548			−1.94266			−0.971328	+	0.237742i	$0.923593\pi$
$810$		0			0
$811$	−21.1716			−0.743434			−0.371717	−	0.928346i	$-0.621231\pi$
$811$	−21.1716			−0.743434			−0.371717	+	0.928346i	$0.621231\pi$
$812$		0			0
$813$		−	35.1127i		−	1.23146i
$814$		0			0
$815$	4.97056	−	2.48528i	0.174111	−	0.0870556i
$816$		0			0
$817$			3.17157i			0.110959i
$818$		0			0
$819$	−2.82843			−0.0988332
$820$		0			0
$821$	38.9706			1.36008			0.680041	−	0.733174i	$-0.261962\pi$
$821$	38.9706			1.36008			0.680041	+	0.733174i	$0.261962\pi$
$822$		0			0
$823$			9.51472i			0.331662i	0.986154	+	0.165831i	$0.0530307\pi$
$823$			9.51472i			0.331662i	−0.986154	+	0.165831i	$0.946969\pi$
$824$		0			0
$825$	−4.68629	−	3.51472i	−0.163156	−	0.122367i
$826$		0			0
$827$			30.5858i			1.06357i	0.846879	+	0.531786i	$0.178479\pi$
$827$			30.5858i			1.06357i	−0.846879	+	0.531786i	$0.821521\pi$
$828$		0			0
$829$	−13.5147			−0.469386			−0.234693	−	0.972070i	$-0.575408\pi$
$829$	−13.5147			−0.469386			−0.234693	+	0.972070i	$0.575408\pi$
$830$		0			0
$831$	−20.9706			−0.727461
$832$		0			0
$833$			17.8579i			0.618738i
$834$		0			0
$835$	22.1421	−	11.0711i	0.766260	−	0.383130i
$836$		0			0
$837$		−	6.62742i		−	0.229077i
$838$		0			0
$839$	8.48528			0.292944			0.146472	−	0.989215i	$-0.453208\pi$
$839$	8.48528			0.292944			0.146472	+	0.989215i	$0.453208\pi$
$840$		0			0
$841$	29.6274			1.02164
$842$		0			0
$843$			7.79899i			0.268611i
$844$		0			0
$845$	1.34315	+	2.68629i	0.0462056	+	0.0924112i
$846$		0			0
$847$		−	8.54416i		−	0.293581i
$848$		0			0
$849$	43.7990			1.50318
$850$		0			0
$851$	−12.4853			−0.427990
$852$		0			0
$853$			24.6863i			0.845243i	0.906306	+	0.422621i	$0.138890\pi$
$853$			24.6863i			0.845243i	−0.906306	+	0.422621i	$0.861110\pi$
$854$		0			0
$855$	1.00000	+	2.00000i	0.0341993	+	0.0683986i
$856$		0			0
$857$			36.6690i			1.25259i	0.779586	+	0.626295i	$0.215430\pi$
$857$			36.6690i			1.25259i	−0.779586	+	0.626295i	$0.784570\pi$
$858$		0			0
$859$	−53.9411			−1.84045			−0.920224	−	0.391393i	$-0.871993\pi$
$859$	−53.9411			−1.84045			−0.920224	+	0.391393i	$0.871993\pi$
$860$		0			0
$861$	−5.65685			−0.192785
$862$		0			0
$863$		−	44.0416i		−	1.49919i	−0.661894	−	0.749597i	$-0.730247\pi$
$863$		−	44.0416i		−	1.49919i	0.661894	−	0.749597i	$-0.269753\pi$
$864$		0			0
$865$	22.8284	−	11.4142i	0.776190	−	0.388095i
$866$		0			0
$867$		−	12.7279i		−	0.432263i
$868$		0			0
$869$	−7.02944			−0.238457
$870$		0			0
$871$	33.7990			1.14524
$872$		0			0
$873$		−	13.0711i		−	0.442389i
$874$		0			0
$875$	1.65685	−	9.11270i	0.0560119	−	0.308065i
$876$		0			0
$877$			52.1838i			1.76212i	0.473004	+	0.881060i	$0.343170\pi$
$877$			52.1838i			1.76212i	−0.473004	+	0.881060i	$0.656830\pi$
$878$		0			0
$879$	−25.7990			−0.870178
$880$		0			0
$881$	−36.2843			−1.22245			−0.611224	−	0.791458i	$-0.709323\pi$
$881$	−36.2843			−1.22245			−0.611224	+	0.791458i	$0.709323\pi$
$882$		0			0
$883$			30.7696i			1.03548i	0.855539	+	0.517739i	$0.173226\pi$
$883$			30.7696i			1.03548i	−0.855539	+	0.517739i	$0.826774\pi$
$884$		0			0
$885$	3.31371	−	1.65685i	0.111389	−	0.0556945i
$886$		0			0
$887$			31.5563i			1.05956i	0.848136	+	0.529779i	$0.177725\pi$
$887$			31.5563i			1.05956i	−0.848136	+	0.529779i	$0.822275\pi$
$888$		0			0
$889$	−10.5442			−0.353640
$890$		0			0
$891$	4.14214			0.138767
$892$		0			0
$893$		−	4.82843i		−	0.161577i
$894$		0			0
$895$	−17.1716	−	34.3431i	−0.573982	−	1.14796i
$896$		0			0
$897$			17.6569i			0.589545i
$898$		0			0
$899$	8.97056			0.299185
$900$		0			0
$901$	22.3431			0.744358
$902$		0			0
$903$			3.71573i			0.123652i
$904$		0			0
$905$	25.3137	+	50.6274i	0.841456	+	1.68291i
$906$		0			0
$907$		−	11.0711i		−	0.367609i	−0.982963	−	0.183804i	$-0.941159\pi$
$907$		−	11.0711i		−	0.367609i	0.982963	−	0.183804i	$-0.0588413\pi$
$908$		0			0
$909$	−8.00000			−0.265343
$910$		0			0
$911$	33.6569			1.11510			0.557551	−	0.830143i	$-0.311741\pi$
$911$	33.6569			1.11510			0.557551	+	0.830143i	$0.311741\pi$
$912$		0			0
$913$			3.59798i			0.119076i
$914$		0			0
$915$	16.0000	−	8.00000i	0.528944	−	0.264472i
$916$		0			0
$917$		−	5.25483i		−	0.173530i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$	−7.65685			−0.252302
$922$		0			0
$923$			28.9706i			0.953578i
$924$		0			0
$925$	−13.6569	−	10.2426i	−0.449035	−	0.336776i
$926$		0			0
$927$			9.89949i			0.325142i
$928$		0			0
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$	6.31371			0.206923
$932$		0			0
$933$			40.7696i			1.33474i
$934$		0			0
$935$	4.68629	−	2.34315i	0.153258	−	0.0766291i
$936$		0			0
$937$		−	26.1421i		−	0.854026i	−0.904245	−	0.427013i	$-0.859566\pi$
$937$		−	26.1421i		−	0.854026i	0.904245	−	0.427013i	$-0.140434\pi$
$938$		0			0
$939$	−18.3431			−0.598606
$940$		0			0
$941$	−35.9411			−1.17165			−0.585824	−	0.810439i	$-0.699229\pi$
$941$	−35.9411			−1.17165			−0.585824	+	0.810439i	$0.699229\pi$
$942$		0			0
$943$		−	17.6569i		−	0.574986i
$944$		0			0
$945$	4.68629	+	9.37258i	0.152445	+	0.304890i
$946$		0			0
$947$		−	18.6863i		−	0.607223i	−0.952796	−	0.303611i	$-0.901808\pi$
$947$		−	18.6863i		−	0.607223i	0.952796	−	0.303611i	$-0.0981925\pi$
$948$		0			0
$949$	−23.3137			−0.756795
$950$		0			0
$951$	−20.8284			−0.675408
$952$		0			0
$953$			7.61522i			0.246681i	0.992364	+	0.123341i	$0.0393608\pi$
$953$			7.61522i			0.246681i	−0.992364	+	0.123341i	$0.960639\pi$
$954$		0			0
$955$	−3.31371	−	6.62742i	−0.107229	−	0.214458i
$956$		0			0
$957$			8.97056i			0.289977i
$958$		0			0
$959$	−2.34315			−0.0756641
$960$		0			0
$961$	−29.6274			−0.955723
$962$		0			0
$963$		−	7.75736i		−	0.249977i
$964$		0			0
$965$	−26.1421	+	13.0711i	−0.841545	+	0.420773i
$966$		0			0
$967$			39.6569i			1.27528i	0.770335	+	0.637639i	$0.220089\pi$
$967$			39.6569i			1.27528i	−0.770335	+	0.637639i	$0.779911\pi$
$968$		0			0
$969$	4.00000			0.128499
$970$		0			0
$971$	36.4853			1.17087			0.585434	−	0.810720i	$-0.300924\pi$
$971$	36.4853			1.17087			0.585434	+	0.810720i	$0.300924\pi$
$972$		0			0
$973$		−	17.2548i		−	0.553165i
$974$		0			0
$975$	−14.4853	+	19.3137i	−0.463900	+	0.618534i
$976$		0			0
$977$			2.92893i			0.0937048i	0.998902	+	0.0468524i	$0.0149191\pi$
$977$			2.92893i			0.0937048i	−0.998902	+	0.0468524i	$0.985081\pi$
$978$		0			0
$979$	7.71573			0.246596
$980$		0			0
$981$	−6.48528			−0.207059
$982$		0			0
$983$		−	60.5269i		−	1.93051i	−0.261312	−	0.965254i	$-0.584155\pi$
$983$		−	60.5269i		−	1.93051i	0.261312	−	0.965254i	$-0.415845\pi$
$984$		0			0
$985$	43.3137	−	21.6569i	1.38009	−	0.690045i
$986$		0			0
$987$		−	5.65685i		−	0.180060i
$988$		0			0
$989$	−11.5980			−0.368794
$990$		0			0
$991$	36.2843			1.15261			0.576304	−	0.817235i	$-0.304494\pi$
$991$	36.2843			1.15261			0.576304	+	0.817235i	$0.304494\pi$
$992$		0			0
$993$			3.31371i			0.105157i
$994$		0			0
$995$	−11.3137	−	22.6274i	−0.358669	−	0.717337i
$996$		0			0
$997$			45.4558i			1.43960i	0.694181	+	0.719801i	$0.255767\pi$
$997$			45.4558i			1.43960i	−0.694181	+	0.719801i	$0.744233\pi$
$998$		0			0
$999$	19.3137			0.611059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	760.2.d.d.609.2	✓	4
4.3	odd	2		1520.2.d.g.609.4		4
5.2	odd	4		3800.2.a.o.1.1		2
5.3	odd	4		3800.2.a.m.1.2		2
5.4	even	2	inner	760.2.d.d.609.3	yes	4
20.3	even	4		7600.2.a.bb.1.1		2
20.7	even	4		7600.2.a.z.1.2		2
20.19	odd	2		1520.2.d.g.609.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.d.609.2	✓	4	1.1	even	1	trivial
760.2.d.d.609.3	yes	4	5.4	even	2	inner
1520.2.d.g.609.1		4	20.19	odd	2
1520.2.d.g.609.4		4	4.3	odd	2
3800.2.a.m.1.2		2	5.3	odd	4
3800.2.a.o.1.1		2	5.2	odd	4
7600.2.a.z.1.2		2	20.7	even	4
7600.2.a.bb.1.1		2	20.3	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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} +(1.00000 + 2.00000i) q^{5} +0.828427i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{3} +(1.00000 + 2.00000i) q^{5} +0.828427i q^{7} +1.00000 q^{9} -0.828427 q^{11} +3.41421i q^{13} +(2.82843 - 1.41421i) q^{15} +2.82843i q^{17} +1.00000 q^{19} +1.17157 q^{21} +3.65685i q^{23} +(-3.00000 + 4.00000i) q^{25} -5.65685i q^{27} +7.65685 q^{29} +1.17157 q^{31} +1.17157i q^{33} +(-1.65685 + 0.828427i) q^{35} +3.41421i q^{37} +4.82843 q^{39} -4.82843 q^{41} +3.17157i q^{43} +(1.00000 + 2.00000i) q^{45} -4.82843i q^{47} +6.31371 q^{49} +4.00000 q^{51} -7.89949i q^{53} +(-0.828427 - 1.65685i) q^{55} -1.41421i q^{57} +1.17157 q^{59} +5.65685 q^{61} +0.828427i q^{63} +(-6.82843 + 3.41421i) q^{65} -9.89949i q^{67} +5.17157 q^{69} +8.48528 q^{71} +6.82843i q^{73} +(5.65685 + 4.24264i) q^{75} -0.686292i q^{77} +8.48528 q^{79} -5.00000 q^{81} -4.34315i q^{83} +(-5.65685 + 2.82843i) q^{85} -10.8284i q^{87} -9.31371 q^{89} -2.82843 q^{91} -1.65685i q^{93} +(1.00000 + 2.00000i) q^{95} -13.0711i q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{5} + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 12 q^{25} + 8 q^{29} + 16 q^{31} + 16 q^{35} + 8 q^{39} - 8 q^{41} + 4 q^{45} - 20 q^{49} + 16 q^{51} + 8 q^{55} + 16 q^{59} - 16 q^{65} + 32 q^{69} - 20 q^{81} + 8 q^{89} + 4 q^{95} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^9 + 8 * q^11 + 4 * q^19 + 16 * q^21 - 12 * q^25 + 8 * q^29 + 16 * q^31 + 16 * q^35 + 8 * q^39 - 8 * q^41 + 4 * q^45 - 20 * q^49 + 16 * q^51 + 8 * q^55 + 16 * q^59 - 16 * q^65 + 32 * q^69 - 20 * q^81 + 8 * q^89 + 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more