LMFDB - Embedded newform 7616.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

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:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{3} +3.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})$	$q+2.30278 q^{3} +3.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} -2.60555 q^{11} +4.00000 q^{13} +7.60555 q^{15} +1.00000 q^{17} -2.60555 q^{19} +2.30278 q^{21} +6.00000 q^{23} +5.90833 q^{25} -1.60555 q^{27} +1.39445 q^{29} -5.90833 q^{31} -6.00000 q^{33} +3.30278 q^{35} -2.60555 q^{37} +9.21110 q^{39} +4.30278 q^{41} -6.30278 q^{43} +7.60555 q^{45} +5.21110 q^{47} +1.00000 q^{49} +2.30278 q^{51} +11.3028 q^{53} -8.60555 q^{55} -6.00000 q^{57} +9.21110 q^{59} +8.30278 q^{61} +2.30278 q^{63} +13.2111 q^{65} +4.51388 q^{67} +13.8167 q^{69} +6.60555 q^{71} +4.90833 q^{73} +13.6056 q^{75} -2.60555 q^{77} -13.2111 q^{79} -10.6056 q^{81} +3.81665 q^{83} +3.30278 q^{85} +3.21110 q^{87} +2.78890 q^{89} +4.00000 q^{91} -13.6056 q^{93} -8.60555 q^{95} +0.697224 q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 3 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q + q^3 + 3 * q^5 + 2 * q^7 + q^9	$ 2 q + q^{3} + 3 q^{5} + 2 q^{7} + q^{9} + 2 q^{11} + 8 q^{13} + 8 q^{15} + 2 q^{17} + 2 q^{19} + q^{21} + 12 q^{23} + q^{25} + 4 q^{27} + 10 q^{29} - q^{31} - 12 q^{33} + 3 q^{35} + 2 q^{37} + 4 q^{39} + 5 q^{41} - 9 q^{43} + 8 q^{45} - 4 q^{47} + 2 q^{49} + q^{51} + 19 q^{53} - 10 q^{55} - 12 q^{57} + 4 q^{59} + 13 q^{61} + q^{63} + 12 q^{65} - 9 q^{67} + 6 q^{69} + 6 q^{71} - q^{73} + 20 q^{75} + 2 q^{77} - 12 q^{79} - 14 q^{81} - 14 q^{83} + 3 q^{85} - 8 q^{87} + 20 q^{89} + 8 q^{91} - 20 q^{93} - 10 q^{95} + 5 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + q^3 + 3 * q^5 + 2 * q^7 + q^9 + 2 * q^11 + 8 * q^13 + 8 * q^15 + 2 * q^17 + 2 * q^19 + q^21 + 12 * q^23 + q^25 + 4 * q^27 + 10 * q^29 - q^31 - 12 * q^33 + 3 * q^35 + 2 * q^37 + 4 * q^39 + 5 * q^41 - 9 * q^43 + 8 * q^45 - 4 * q^47 + 2 * q^49 + q^51 + 19 * q^53 - 10 * q^55 - 12 * q^57 + 4 * q^59 + 13 * q^61 + q^63 + 12 * q^65 - 9 * q^67 + 6 * q^69 + 6 * q^71 - q^73 + 20 * q^75 + 2 * q^77 - 12 * q^79 - 14 * q^81 - 14 * q^83 + 3 * q^85 - 8 * q^87 + 20 * q^89 + 8 * q^91 - 20 * q^93 - 10 * q^95 + 5 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.30278	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$3$	2.30278	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$4$	0	0
$5$	3.30278	1.47705	0.738523	−	0.674228i	$-0.235524\pi$
$5$	3.30278	1.47705	0.738523	+	0.674228i	$0.235524\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	−2.60555	−0.785603	−0.392802	−	0.919623i	$-0.628494\pi$
$11$	−2.60555	−0.785603	−0.392802	+	0.919623i	$0.628494\pi$
$12$	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
$15$	7.60555	1.96374
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.60555	−0.597754	−0.298877	−	0.954292i	$-0.596612\pi$
$19$	−2.60555	−0.597754	−0.298877	+	0.954292i	$0.596612\pi$
$20$	0	0
$21$	2.30278	0.502507
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	5.90833	1.18167
$26$	0	0
$27$	−1.60555	−0.308988
$28$	0	0
$29$	1.39445	0.258943	0.129471	−	0.991583i	$-0.458672\pi$
$29$	1.39445	0.258943	0.129471	+	0.991583i	$0.458672\pi$
$30$	0	0
$31$	−5.90833	−1.06117	−0.530583	−	0.847633i	$-0.678027\pi$
$31$	−5.90833	−1.06117	−0.530583	+	0.847633i	$0.678027\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	3.30278	0.558271
$36$	0	0
$37$	−2.60555	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$37$	−2.60555	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$38$	0	0
$39$	9.21110	1.47496
$40$	0	0
$41$	4.30278	0.671981	0.335990	−	0.941865i	$-0.390929\pi$
$41$	4.30278	0.671981	0.335990	+	0.941865i	$0.390929\pi$
$42$	0	0
$43$	−6.30278	−0.961164	−0.480582	−	0.876950i	$-0.659575\pi$
$43$	−6.30278	−0.961164	−0.480582	+	0.876950i	$0.659575\pi$
$44$	0	0
$45$	7.60555	1.13377
$46$	0	0
$47$	5.21110	0.760117	0.380059	−	0.924962i	$-0.375904\pi$
$47$	5.21110	0.760117	0.380059	+	0.924962i	$0.375904\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.30278	0.322453
$52$	0	0
$53$	11.3028	1.55256	0.776278	−	0.630391i	$-0.217105\pi$
$53$	11.3028	1.55256	0.776278	+	0.630391i	$0.217105\pi$
$54$	0	0
$55$	−8.60555	−1.16037
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	9.21110	1.19918	0.599592	−	0.800306i	$-0.295330\pi$
$59$	9.21110	1.19918	0.599592	+	0.800306i	$0.295330\pi$
$60$	0	0
$61$	8.30278	1.06306	0.531531	−	0.847039i	$-0.321617\pi$
$61$	8.30278	1.06306	0.531531	+	0.847039i	$0.321617\pi$
$62$	0	0
$63$	2.30278	0.290122
$64$	0	0
$65$	13.2111	1.63864
$66$	0	0
$67$	4.51388	0.551458	0.275729	−	0.961235i	$-0.411081\pi$
$67$	4.51388	0.551458	0.275729	+	0.961235i	$0.411081\pi$
$68$	0	0
$69$	13.8167	1.66333
$70$	0	0
$71$	6.60555	0.783935	0.391967	−	0.919979i	$-0.371795\pi$
$71$	6.60555	0.783935	0.391967	+	0.919979i	$0.371795\pi$
$72$	0	0
$73$	4.90833	0.574476	0.287238	−	0.957859i	$-0.407263\pi$
$73$	4.90833	0.574476	0.287238	+	0.957859i	$0.407263\pi$
$74$	0	0
$75$	13.6056	1.57103
$76$	0	0
$77$	−2.60555	−0.296930
$78$	0	0
$79$	−13.2111	−1.48637	−0.743183	−	0.669089i	$-0.766685\pi$
$79$	−13.2111	−1.48637	−0.743183	+	0.669089i	$0.766685\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	3.81665	0.418932	0.209466	−	0.977816i	$-0.432827\pi$
$83$	3.81665	0.418932	0.209466	+	0.977816i	$0.432827\pi$
$84$	0	0
$85$	3.30278	0.358236
$86$	0	0
$87$	3.21110	0.344266
$88$	0	0
$89$	2.78890	0.295623	0.147811	−	0.989016i	$-0.452777\pi$
$89$	2.78890	0.295623	0.147811	+	0.989016i	$0.452777\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−13.6056	−1.41083
$94$	0	0
$95$	−8.60555	−0.882911
$96$	0	0
$97$	0.697224	0.0707924	0.0353962	−	0.999373i	$-0.488731\pi$
$97$	0.697224	0.0707924	0.0353962	+	0.999373i	$0.488731\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	5.39445	0.536768	0.268384	−	0.963312i	$-0.413510\pi$
$101$	5.39445	0.536768	0.268384	+	0.963312i	$0.413510\pi$
$102$	0	0
$103$	−15.8167	−1.55846	−0.779231	−	0.626737i	$-0.784390\pi$
$103$	−15.8167	−1.55846	−0.779231	+	0.626737i	$0.784390\pi$
$104$	0	0
$105$	7.60555	0.742226
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−11.2111	−1.07383	−0.536914	−	0.843637i	$-0.680410\pi$
$109$	−11.2111	−1.07383	−0.536914	+	0.843637i	$0.680410\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−13.2111	−1.24280	−0.621398	−	0.783495i	$-0.713435\pi$
$113$	−13.2111	−1.24280	−0.621398	+	0.783495i	$0.713435\pi$
$114$	0	0
$115$	19.8167	1.84791
$116$	0	0
$117$	9.21110	0.851567
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−4.21110	−0.382828
$122$	0	0
$123$	9.90833	0.893404
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	13.7250	1.21790	0.608948	−	0.793210i	$-0.291592\pi$
$127$	13.7250	1.21790	0.608948	+	0.793210i	$0.291592\pi$
$128$	0	0
$129$	−14.5139	−1.27788
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−2.60555	−0.225930
$134$	0	0
$135$	−5.30278	−0.456390
$136$	0	0
$137$	−11.0917	−0.947626	−0.473813	−	0.880626i	$-0.657123\pi$
$137$	−11.0917	−0.947626	−0.473813	+	0.880626i	$0.657123\pi$
$138$	0	0
$139$	−4.51388	−0.382862	−0.191431	−	0.981506i	$-0.561313\pi$
$139$	−4.51388	−0.382862	−0.191431	+	0.981506i	$0.561313\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	−10.4222	−0.871549
$144$	0	0
$145$	4.60555	0.382470
$146$	0	0
$147$	2.30278	0.189930
$148$	0	0
$149$	23.9361	1.96092	0.980460	−	0.196718i	$-0.0630282\pi$
$149$	23.9361	1.96092	0.980460	+	0.196718i	$0.0630282\pi$
$150$	0	0
$151$	−18.9083	−1.53874	−0.769369	−	0.638805i	$-0.779429\pi$
$151$	−18.9083	−1.53874	−0.769369	+	0.638805i	$0.779429\pi$
$152$	0	0
$153$	2.30278	0.186168
$154$	0	0
$155$	−19.5139	−1.56739
$156$	0	0
$157$	16.6056	1.32527	0.662634	−	0.748944i	$-0.269439\pi$
$157$	16.6056	1.32527	0.662634	+	0.748944i	$0.269439\pi$
$158$	0	0
$159$	26.0278	2.06414
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	8.78890	0.688400	0.344200	−	0.938896i	$-0.388150\pi$
$163$	8.78890	0.688400	0.344200	+	0.938896i	$0.388150\pi$
$164$	0	0
$165$	−19.8167	−1.54272
$166$	0	0
$167$	24.1194	1.86642	0.933209	−	0.359335i	$-0.116996\pi$
$167$	24.1194	1.86642	0.933209	+	0.359335i	$0.116996\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	−12.9083	−0.981402	−0.490701	−	0.871328i	$-0.663259\pi$
$173$	−12.9083	−0.981402	−0.490701	+	0.871328i	$0.663259\pi$
$174$	0	0
$175$	5.90833	0.446628
$176$	0	0
$177$	21.2111	1.59432
$178$	0	0
$179$	−10.9083	−0.815327	−0.407663	−	0.913132i	$-0.633656\pi$
$179$	−10.9083	−0.815327	−0.407663	+	0.913132i	$0.633656\pi$
$180$	0	0
$181$	17.6333	1.31067	0.655337	−	0.755337i	$-0.272527\pi$
$181$	17.6333	1.31067	0.655337	+	0.755337i	$0.272527\pi$
$182$	0	0
$183$	19.1194	1.41335
$184$	0	0
$185$	−8.60555	−0.632693
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	−1.60555	−0.116787
$190$	0	0
$191$	16.6972	1.20817	0.604084	−	0.796920i	$-0.293539\pi$
$191$	16.6972	1.20817	0.604084	+	0.796920i	$0.293539\pi$
$192$	0	0
$193$	6.78890	0.488676	0.244338	−	0.969690i	$-0.421429\pi$
$193$	6.78890	0.488676	0.244338	+	0.969690i	$0.421429\pi$
$194$	0	0
$195$	30.4222	2.17858
$196$	0	0
$197$	−24.4222	−1.74001	−0.870005	−	0.493043i	$-0.835885\pi$
$197$	−24.4222	−1.74001	−0.870005	+	0.493043i	$0.835885\pi$
$198$	0	0
$199$	−26.3028	−1.86455	−0.932277	−	0.361745i	$-0.882181\pi$
$199$	−26.3028	−1.86455	−0.932277	+	0.361745i	$0.882181\pi$
$200$	0	0
$201$	10.3944	0.733168
$202$	0	0
$203$	1.39445	0.0978711
$204$	0	0
$205$	14.2111	0.992546
$206$	0	0
$207$	13.8167	0.960324
$208$	0	0
$209$	6.78890	0.469598
$210$	0	0
$211$	13.8167	0.951178	0.475589	−	0.879668i	$-0.342235\pi$
$211$	13.8167	0.951178	0.475589	+	0.879668i	$0.342235\pi$
$212$	0	0
$213$	15.2111	1.04225
$214$	0	0
$215$	−20.8167	−1.41968
$216$	0	0
$217$	−5.90833	−0.401083
$218$	0	0
$219$	11.3028	0.763771
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−4.18335	−0.280138	−0.140069	−	0.990142i	$-0.544732\pi$
$223$	−4.18335	−0.280138	−0.140069	+	0.990142i	$0.544732\pi$
$224$	0	0
$225$	13.6056	0.907037
$226$	0	0
$227$	−28.5139	−1.89253	−0.946266	−	0.323388i	$-0.895178\pi$
$227$	−28.5139	−1.89253	−0.946266	+	0.323388i	$0.895178\pi$
$228$	0	0
$229$	−11.8167	−0.780866	−0.390433	−	0.920631i	$-0.627675\pi$
$229$	−11.8167	−0.780866	−0.390433	+	0.920631i	$0.627675\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	28.2389	1.84999	0.924995	−	0.379980i	$-0.124069\pi$
$233$	28.2389	1.84999	0.924995	+	0.379980i	$0.124069\pi$
$234$	0	0
$235$	17.2111	1.12273
$236$	0	0
$237$	−30.4222	−1.97613
$238$	0	0
$239$	−9.33053	−0.603542	−0.301771	−	0.953380i	$-0.597578\pi$
$239$	−9.33053	−0.603542	−0.301771	+	0.953380i	$0.597578\pi$
$240$	0	0
$241$	−16.5139	−1.06375	−0.531876	−	0.846822i	$-0.678513\pi$
$241$	−16.5139	−1.06375	−0.531876	+	0.846822i	$0.678513\pi$
$242$	0	0
$243$	−19.6056	−1.25770
$244$	0	0
$245$	3.30278	0.211007
$246$	0	0
$247$	−10.4222	−0.663149
$248$	0	0
$249$	8.78890	0.556974
$250$	0	0
$251$	−1.39445	−0.0880168	−0.0440084	−	0.999031i	$-0.514013\pi$
$251$	−1.39445	−0.0880168	−0.0440084	+	0.999031i	$0.514013\pi$
$252$	0	0
$253$	−15.6333	−0.982858
$254$	0	0
$255$	7.60555	0.476278
$256$	0	0
$257$	−10.4222	−0.650119	−0.325060	−	0.945694i	$-0.605384\pi$
$257$	−10.4222	−0.650119	−0.325060	+	0.945694i	$0.605384\pi$
$258$	0	0
$259$	−2.60555	−0.161901
$260$	0	0
$261$	3.21110	0.198762
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	37.3305	2.29320
$266$	0	0
$267$	6.42221	0.393033
$268$	0	0
$269$	−19.2111	−1.17132	−0.585661	−	0.810556i	$-0.699165\pi$
$269$	−19.2111	−1.17132	−0.585661	+	0.810556i	$0.699165\pi$
$270$	0	0
$271$	24.4222	1.48354	0.741772	−	0.670653i	$-0.233986\pi$
$271$	24.4222	1.48354	0.741772	+	0.670653i	$0.233986\pi$
$272$	0	0
$273$	9.21110	0.557481
$274$	0	0
$275$	−15.3944	−0.928320
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	−13.6056	−0.814543
$280$	0	0
$281$	−12.6972	−0.757453	−0.378726	−	0.925509i	$-0.623638\pi$
$281$	−12.6972	−0.757453	−0.378726	+	0.925509i	$0.623638\pi$
$282$	0	0
$283$	3.30278	0.196330	0.0981648	−	0.995170i	$-0.468703\pi$
$283$	3.30278	0.196330	0.0981648	+	0.995170i	$0.468703\pi$
$284$	0	0
$285$	−19.8167	−1.17384
$286$	0	0
$287$	4.30278	0.253985
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.60555	0.0941191
$292$	0	0
$293$	−15.6333	−0.913308	−0.456654	−	0.889644i	$-0.650952\pi$
$293$	−15.6333	−0.913308	−0.456654	+	0.889644i	$0.650952\pi$
$294$	0	0
$295$	30.4222	1.77125
$296$	0	0
$297$	4.18335	0.242742
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−6.30278	−0.363286
$302$	0	0
$303$	12.4222	0.713637
$304$	0	0
$305$	27.4222	1.57019
$306$	0	0
$307$	20.6056	1.17602	0.588010	−	0.808853i	$-0.299911\pi$
$307$	20.6056	1.17602	0.588010	+	0.808853i	$0.299911\pi$
$308$	0	0
$309$	−36.4222	−2.07199
$310$	0	0
$311$	−19.3028	−1.09456	−0.547280	−	0.836950i	$-0.684337\pi$
$311$	−19.3028	−1.09456	−0.547280	+	0.836950i	$0.684337\pi$
$312$	0	0
$313$	−22.9361	−1.29642	−0.648212	−	0.761460i	$-0.724483\pi$
$313$	−22.9361	−1.29642	−0.648212	+	0.761460i	$0.724483\pi$
$314$	0	0
$315$	7.60555	0.428524
$316$	0	0
$317$	6.42221	0.360707	0.180353	−	0.983602i	$-0.442276\pi$
$317$	6.42221	0.360707	0.180353	+	0.983602i	$0.442276\pi$
$318$	0	0
$319$	−3.63331	−0.203426
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.60555	−0.144977
$324$	0	0
$325$	23.6333	1.31094
$326$	0	0
$327$	−25.8167	−1.42766
$328$	0	0
$329$	5.21110	0.287297
$330$	0	0
$331$	−6.09167	−0.334829	−0.167414	−	0.985887i	$-0.553542\pi$
$331$	−6.09167	−0.334829	−0.167414	+	0.985887i	$0.553542\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	14.9083	0.814529
$336$	0	0
$337$	−31.6333	−1.72318	−0.861588	−	0.507608i	$-0.830530\pi$
$337$	−31.6333	−1.72318	−0.861588	+	0.507608i	$0.830530\pi$
$338$	0	0
$339$	−30.4222	−1.65231
$340$	0	0
$341$	15.3944	0.833656
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	45.6333	2.45681
$346$	0	0
$347$	−21.6333	−1.16134	−0.580668	−	0.814140i	$-0.697209\pi$
$347$	−21.6333	−1.16134	−0.580668	+	0.814140i	$0.697209\pi$
$348$	0	0
$349$	−20.2389	−1.08336	−0.541681	−	0.840584i	$-0.682212\pi$
$349$	−20.2389	−1.08336	−0.541681	+	0.840584i	$0.682212\pi$
$350$	0	0
$351$	−6.42221	−0.342792
$352$	0	0
$353$	−9.39445	−0.500016	−0.250008	−	0.968244i	$-0.580433\pi$
$353$	−9.39445	−0.500016	−0.250008	+	0.968244i	$0.580433\pi$
$354$	0	0
$355$	21.8167	1.15791
$356$	0	0
$357$	2.30278	0.121876
$358$	0	0
$359$	−9.90833	−0.522941	−0.261471	−	0.965211i	$-0.584207\pi$
$359$	−9.90833	−0.522941	−0.261471	+	0.965211i	$0.584207\pi$
$360$	0	0
$361$	−12.2111	−0.642690
$362$	0	0
$363$	−9.69722	−0.508972
$364$	0	0
$365$	16.2111	0.848528
$366$	0	0
$367$	−15.5139	−0.809818	−0.404909	−	0.914357i	$-0.632697\pi$
$367$	−15.5139	−0.809818	−0.404909	+	0.914357i	$0.632697\pi$
$368$	0	0
$369$	9.90833	0.515807
$370$	0	0
$371$	11.3028	0.586811
$372$	0	0
$373$	27.3305	1.41512	0.707561	−	0.706653i	$-0.249796\pi$
$373$	27.3305	1.41512	0.707561	+	0.706653i	$0.249796\pi$
$374$	0	0
$375$	6.90833	0.356744
$376$	0	0
$377$	5.57779	0.287271
$378$	0	0
$379$	−14.6056	−0.750237	−0.375118	−	0.926977i	$-0.622398\pi$
$379$	−14.6056	−0.750237	−0.375118	+	0.926977i	$0.622398\pi$
$380$	0	0
$381$	31.6056	1.61920
$382$	0	0
$383$	−38.2389	−1.95391	−0.976957	−	0.213435i	$-0.931535\pi$
$383$	−38.2389	−1.95391	−0.976957	+	0.213435i	$0.931535\pi$
$384$	0	0
$385$	−8.60555	−0.438580
$386$	0	0
$387$	−14.5139	−0.737782
$388$	0	0
$389$	25.7250	1.30431	0.652154	−	0.758086i	$-0.273865\pi$
$389$	25.7250	1.30431	0.652154	+	0.758086i	$0.273865\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	18.4222	0.929277
$394$	0	0
$395$	−43.6333	−2.19543
$396$	0	0
$397$	−8.33053	−0.418097	−0.209049	−	0.977905i	$-0.567037\pi$
$397$	−8.33053	−0.418097	−0.209049	+	0.977905i	$0.567037\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−32.2389	−1.60993	−0.804966	−	0.593321i	$-0.797816\pi$
$401$	−32.2389	−1.60993	−0.804966	+	0.593321i	$0.797816\pi$
$402$	0	0
$403$	−23.6333	−1.17726
$404$	0	0
$405$	−35.0278	−1.74054
$406$	0	0
$407$	6.78890	0.336513
$408$	0	0
$409$	−15.3944	−0.761206	−0.380603	−	0.924738i	$-0.624284\pi$
$409$	−15.3944	−0.761206	−0.380603	+	0.924738i	$0.624284\pi$
$410$	0	0
$411$	−25.5416	−1.25988
$412$	0	0
$413$	9.21110	0.453249
$414$	0	0
$415$	12.6056	0.618782
$416$	0	0
$417$	−10.3944	−0.509018
$418$	0	0
$419$	−32.3305	−1.57945	−0.789725	−	0.613461i	$-0.789777\pi$
$419$	−32.3305	−1.57945	−0.789725	+	0.613461i	$0.789777\pi$
$420$	0	0
$421$	30.1194	1.46793	0.733966	−	0.679187i	$-0.237667\pi$
$421$	30.1194	1.46793	0.733966	+	0.679187i	$0.237667\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	5.90833	0.286596
$426$	0	0
$427$	8.30278	0.401799
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	0.605551	0.0291684	0.0145842	−	0.999894i	$-0.495358\pi$
$431$	0.605551	0.0291684	0.0145842	+	0.999894i	$0.495358\pi$
$432$	0	0
$433$	31.2111	1.49991	0.749955	−	0.661489i	$-0.230075\pi$
$433$	31.2111	1.49991	0.749955	+	0.661489i	$0.230075\pi$
$434$	0	0
$435$	10.6056	0.508497
$436$	0	0
$437$	−15.6333	−0.747843
$438$	0	0
$439$	−37.3305	−1.78169	−0.890845	−	0.454308i	$-0.849886\pi$
$439$	−37.3305	−1.78169	−0.890845	+	0.454308i	$0.849886\pi$
$440$	0	0
$441$	2.30278	0.109656
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	9.21110	0.436648
$446$	0	0
$447$	55.1194	2.60706
$448$	0	0
$449$	−11.2111	−0.529085	−0.264542	−	0.964374i	$-0.585221\pi$
$449$	−11.2111	−0.529085	−0.264542	+	0.964374i	$0.585221\pi$
$450$	0	0
$451$	−11.2111	−0.527910
$452$	0	0
$453$	−43.5416	−2.04576
$454$	0	0
$455$	13.2111	0.619346
$456$	0	0
$457$	36.5139	1.70805	0.854024	−	0.520234i	$-0.174155\pi$
$457$	36.5139	1.70805	0.854024	+	0.520234i	$0.174155\pi$
$458$	0	0
$459$	−1.60555	−0.0749407
$460$	0	0
$461$	3.21110	0.149556	0.0747780	−	0.997200i	$-0.476175\pi$
$461$	3.21110	0.149556	0.0747780	+	0.997200i	$0.476175\pi$
$462$	0	0
$463$	18.3028	0.850602	0.425301	−	0.905052i	$-0.360168\pi$
$463$	18.3028	0.850602	0.425301	+	0.905052i	$0.360168\pi$
$464$	0	0
$465$	−44.9361	−2.08386
$466$	0	0
$467$	7.21110	0.333690	0.166845	−	0.985983i	$-0.446642\pi$
$467$	7.21110	0.333690	0.166845	+	0.985983i	$0.446642\pi$
$468$	0	0
$469$	4.51388	0.208432
$470$	0	0
$471$	38.2389	1.76195
$472$	0	0
$473$	16.4222	0.755094
$474$	0	0
$475$	−15.3944	−0.706346
$476$	0	0
$477$	26.0278	1.19173
$478$	0	0
$479$	−4.11943	−0.188222	−0.0941108	−	0.995562i	$-0.530001\pi$
$479$	−4.11943	−0.188222	−0.0941108	+	0.995562i	$0.530001\pi$
$480$	0	0
$481$	−10.4222	−0.475212
$482$	0	0
$483$	13.8167	0.628680
$484$	0	0
$485$	2.30278	0.104564
$486$	0	0
$487$	−12.6056	−0.571212	−0.285606	−	0.958347i	$-0.592195\pi$
$487$	−12.6056	−0.571212	−0.285606	+	0.958347i	$0.592195\pi$
$488$	0	0
$489$	20.2389	0.915233
$490$	0	0
$491$	−3.48612	−0.157326	−0.0786632	−	0.996901i	$-0.525065\pi$
$491$	−3.48612	−0.157326	−0.0786632	+	0.996901i	$0.525065\pi$
$492$	0	0
$493$	1.39445	0.0628028
$494$	0	0
$495$	−19.8167	−0.890692
$496$	0	0
$497$	6.60555	0.296299
$498$	0	0
$499$	−27.0278	−1.20993	−0.604964	−	0.796253i	$-0.706813\pi$
$499$	−27.0278	−1.20993	−0.604964	+	0.796253i	$0.706813\pi$
$500$	0	0
$501$	55.5416	2.48142
$502$	0	0
$503$	10.0917	0.449965	0.224983	−	0.974363i	$-0.427767\pi$
$503$	10.0917	0.449965	0.224983	+	0.974363i	$0.427767\pi$
$504$	0	0
$505$	17.8167	0.792831
$506$	0	0
$507$	6.90833	0.306810
$508$	0	0
$509$	20.4222	0.905198	0.452599	−	0.891714i	$-0.350497\pi$
$509$	20.4222	0.905198	0.452599	+	0.891714i	$0.350497\pi$
$510$	0	0
$511$	4.90833	0.217132
$512$	0	0
$513$	4.18335	0.184699
$514$	0	0
$515$	−52.2389	−2.30192
$516$	0	0
$517$	−13.5778	−0.597151
$518$	0	0
$519$	−29.7250	−1.30478
$520$	0	0
$521$	22.1194	0.969070	0.484535	−	0.874772i	$-0.338989\pi$
$521$	22.1194	0.969070	0.484535	+	0.874772i	$0.338989\pi$
$522$	0	0
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	0	0
$525$	13.6056	0.593795
$526$	0	0
$527$	−5.90833	−0.257371
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	21.2111	0.920483
$532$	0	0
$533$	17.2111	0.745496
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−25.1194	−1.08398
$538$	0	0
$539$	−2.60555	−0.112229
$540$	0	0
$541$	−4.78890	−0.205891	−0.102945	−	0.994687i	$-0.532827\pi$
$541$	−4.78890	−0.205891	−0.102945	+	0.994687i	$0.532827\pi$
$542$	0	0
$543$	40.6056	1.74255
$544$	0	0
$545$	−37.0278	−1.58609
$546$	0	0
$547$	−10.1833	−0.435408	−0.217704	−	0.976015i	$-0.569857\pi$
$547$	−10.1833	−0.435408	−0.217704	+	0.976015i	$0.569857\pi$
$548$	0	0
$549$	19.1194	0.815997
$550$	0	0
$551$	−3.63331	−0.154784
$552$	0	0
$553$	−13.2111	−0.561793
$554$	0	0
$555$	−19.8167	−0.841170
$556$	0	0
$557$	37.6333	1.59457	0.797287	−	0.603600i	$-0.206268\pi$
$557$	37.6333	1.59457	0.797287	+	0.603600i	$0.206268\pi$
$558$	0	0
$559$	−25.2111	−1.06632
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	21.3944	0.901669	0.450834	−	0.892608i	$-0.351127\pi$
$563$	21.3944	0.901669	0.450834	+	0.892608i	$0.351127\pi$
$564$	0	0
$565$	−43.6333	−1.83567
$566$	0	0
$567$	−10.6056	−0.445391
$568$	0	0
$569$	32.9083	1.37959	0.689794	−	0.724005i	$-0.257701\pi$
$569$	32.9083	1.37959	0.689794	+	0.724005i	$0.257701\pi$
$570$	0	0
$571$	14.6056	0.611223	0.305612	−	0.952156i	$-0.401139\pi$
$571$	14.6056	0.611223	0.305612	+	0.952156i	$0.401139\pi$
$572$	0	0
$573$	38.4500	1.60627
$574$	0	0
$575$	35.4500	1.47837
$576$	0	0
$577$	16.4222	0.683665	0.341833	−	0.939761i	$-0.388952\pi$
$577$	16.4222	0.683665	0.341833	+	0.939761i	$0.388952\pi$
$578$	0	0
$579$	15.6333	0.649698
$580$	0	0
$581$	3.81665	0.158341
$582$	0	0
$583$	−29.4500	−1.21969
$584$	0	0
$585$	30.4222	1.25780
$586$	0	0
$587$	−21.6333	−0.892902	−0.446451	−	0.894808i	$-0.647312\pi$
$587$	−21.6333	−0.892902	−0.446451	+	0.894808i	$0.647312\pi$
$588$	0	0
$589$	15.3944	0.634317
$590$	0	0
$591$	−56.2389	−2.31336
$592$	0	0
$593$	−30.6056	−1.25682	−0.628410	−	0.777883i	$-0.716294\pi$
$593$	−30.6056	−1.25682	−0.628410	+	0.777883i	$0.716294\pi$
$594$	0	0
$595$	3.30278	0.135401
$596$	0	0
$597$	−60.5694	−2.47894
$598$	0	0
$599$	−7.88057	−0.321991	−0.160996	−	0.986955i	$-0.551471\pi$
$599$	−7.88057	−0.321991	−0.160996	+	0.986955i	$0.551471\pi$
$600$	0	0
$601$	−19.2111	−0.783637	−0.391819	−	0.920042i	$-0.628154\pi$
$601$	−19.2111	−0.783637	−0.391819	+	0.920042i	$0.628154\pi$
$602$	0	0
$603$	10.3944	0.423295
$604$	0	0
$605$	−13.9083	−0.565454
$606$	0	0
$607$	29.5416	1.19906	0.599529	−	0.800353i	$-0.295355\pi$
$607$	29.5416	1.19906	0.599529	+	0.800353i	$0.295355\pi$
$608$	0	0
$609$	3.21110	0.130120
$610$	0	0
$611$	20.8444	0.843275
$612$	0	0
$613$	2.09167	0.0844819	0.0422409	−	0.999107i	$-0.486550\pi$
$613$	2.09167	0.0844819	0.0422409	+	0.999107i	$0.486550\pi$
$614$	0	0
$615$	32.7250	1.31960
$616$	0	0
$617$	6.23886	0.251167	0.125584	−	0.992083i	$-0.459920\pi$
$617$	6.23886	0.251167	0.125584	+	0.992083i	$0.459920\pi$
$618$	0	0
$619$	−14.4222	−0.579677	−0.289839	−	0.957076i	$-0.593602\pi$
$619$	−14.4222	−0.579677	−0.289839	+	0.957076i	$0.593602\pi$
$620$	0	0
$621$	−9.63331	−0.386571
$622$	0	0
$623$	2.78890	0.111735
$624$	0	0
$625$	−19.6333	−0.785332
$626$	0	0
$627$	15.6333	0.624334
$628$	0	0
$629$	−2.60555	−0.103890
$630$	0	0
$631$	−44.5139	−1.77207	−0.886035	−	0.463619i	$-0.846551\pi$
$631$	−44.5139	−1.77207	−0.886035	+	0.463619i	$0.846551\pi$
$632$	0	0
$633$	31.8167	1.26460
$634$	0	0
$635$	45.3305	1.79889
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	15.2111	0.601742
$640$	0	0
$641$	20.2389	0.799387	0.399693	−	0.916649i	$-0.369117\pi$
$641$	20.2389	0.799387	0.399693	+	0.916649i	$0.369117\pi$
$642$	0	0
$643$	23.7527	0.936717	0.468358	−	0.883539i	$-0.344846\pi$
$643$	23.7527	0.936717	0.468358	+	0.883539i	$0.344846\pi$
$644$	0	0
$645$	−47.9361	−1.88748
$646$	0	0
$647$	−11.8167	−0.464561	−0.232280	−	0.972649i	$-0.574619\pi$
$647$	−11.8167	−0.464561	−0.232280	+	0.972649i	$0.574619\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−13.6056	−0.533244
$652$	0	0
$653$	−4.42221	−0.173054	−0.0865271	−	0.996249i	$-0.527577\pi$
$653$	−4.42221	−0.173054	−0.0865271	+	0.996249i	$0.527577\pi$
$654$	0	0
$655$	26.4222	1.03240
$656$	0	0
$657$	11.3028	0.440963
$658$	0	0
$659$	−12.6972	−0.494614	−0.247307	−	0.968937i	$-0.579546\pi$
$659$	−12.6972	−0.494614	−0.247307	+	0.968937i	$0.579546\pi$
$660$	0	0
$661$	35.3944	1.37668	0.688342	−	0.725386i	$-0.258339\pi$
$661$	35.3944	1.37668	0.688342	+	0.725386i	$0.258339\pi$
$662$	0	0
$663$	9.21110	0.357730
$664$	0	0
$665$	−8.60555	−0.333709
$666$	0	0
$667$	8.36669	0.323960
$668$	0	0
$669$	−9.63331	−0.372445
$670$	0	0
$671$	−21.6333	−0.835145
$672$	0	0
$673$	−14.8444	−0.572210	−0.286105	−	0.958198i	$-0.592361\pi$
$673$	−14.8444	−0.572210	−0.286105	+	0.958198i	$0.592361\pi$
$674$	0	0
$675$	−9.48612	−0.365121
$676$	0	0
$677$	15.2111	0.584610	0.292305	−	0.956325i	$-0.405578\pi$
$677$	15.2111	0.584610	0.292305	+	0.956325i	$0.405578\pi$
$678$	0	0
$679$	0.697224	0.0267570
$680$	0	0
$681$	−65.6611	−2.51614
$682$	0	0
$683$	37.2666	1.42597	0.712984	−	0.701181i	$-0.247343\pi$
$683$	37.2666	1.42597	0.712984	+	0.701181i	$0.247343\pi$
$684$	0	0
$685$	−36.6333	−1.39969
$686$	0	0
$687$	−27.2111	−1.03817
$688$	0	0
$689$	45.2111	1.72241
$690$	0	0
$691$	−18.5416	−0.705357	−0.352679	−	0.935745i	$-0.614729\pi$
$691$	−18.5416	−0.705357	−0.352679	+	0.935745i	$0.614729\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−14.9083	−0.565505
$696$	0	0
$697$	4.30278	0.162979
$698$	0	0
$699$	65.0278	2.45958
$700$	0	0
$701$	−2.36669	−0.0893887	−0.0446944	−	0.999001i	$-0.514231\pi$
$701$	−2.36669	−0.0893887	−0.0446944	+	0.999001i	$0.514231\pi$
$702$	0	0
$703$	6.78890	0.256048
$704$	0	0
$705$	39.6333	1.49268
$706$	0	0
$707$	5.39445	0.202879
$708$	0	0
$709$	27.6333	1.03779	0.518895	−	0.854838i	$-0.326343\pi$
$709$	27.6333	1.03779	0.518895	+	0.854838i	$0.326343\pi$
$710$	0	0
$711$	−30.4222	−1.14092
$712$	0	0
$713$	−35.4500	−1.32761
$714$	0	0
$715$	−34.4222	−1.28732
$716$	0	0
$717$	−21.4861	−0.802414
$718$	0	0
$719$	−32.3583	−1.20676	−0.603380	−	0.797454i	$-0.706180\pi$
$719$	−32.3583	−1.20676	−0.603380	+	0.797454i	$0.706180\pi$
$720$	0	0
$721$	−15.8167	−0.589043
$722$	0	0
$723$	−38.0278	−1.41427
$724$	0	0
$725$	8.23886	0.305984
$726$	0	0
$727$	40.2389	1.49238	0.746188	−	0.665735i	$-0.231882\pi$
$727$	40.2389	1.49238	0.746188	+	0.665735i	$0.231882\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	−6.30278	−0.233117
$732$	0	0
$733$	−5.63331	−0.208071	−0.104035	−	0.994574i	$-0.533176\pi$
$733$	−5.63331	−0.208071	−0.104035	+	0.994574i	$0.533176\pi$
$734$	0	0
$735$	7.60555	0.280535
$736$	0	0
$737$	−11.7611	−0.433227
$738$	0	0
$739$	−18.5416	−0.682065	−0.341033	−	0.940051i	$-0.610777\pi$
$739$	−18.5416	−0.682065	−0.341033	+	0.940051i	$0.610777\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	−18.6056	−0.682571	−0.341286	−	0.939960i	$-0.610862\pi$
$743$	−18.6056	−0.682571	−0.341286	+	0.939960i	$0.610862\pi$
$744$	0	0
$745$	79.0555	2.89637
$746$	0	0
$747$	8.78890	0.321569
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.0278	0.913276	0.456638	−	0.889653i	$-0.349054\pi$
$751$	25.0278	0.913276	0.456638	+	0.889653i	$0.349054\pi$
$752$	0	0
$753$	−3.21110	−0.117019
$754$	0	0
$755$	−62.4500	−2.27279
$756$	0	0
$757$	23.6972	0.861290	0.430645	−	0.902521i	$-0.358286\pi$
$757$	23.6972	0.861290	0.430645	+	0.902521i	$0.358286\pi$
$758$	0	0
$759$	−36.0000	−1.30672
$760$	0	0
$761$	−35.4500	−1.28506	−0.642530	−	0.766260i	$-0.722115\pi$
$761$	−35.4500	−1.28506	−0.642530	+	0.766260i	$0.722115\pi$
$762$	0	0
$763$	−11.2111	−0.405869
$764$	0	0
$765$	7.60555	0.274979
$766$	0	0
$767$	36.8444	1.33037
$768$	0	0
$769$	0.605551	0.0218367	0.0109184	−	0.999940i	$-0.496525\pi$
$769$	0.605551	0.0218367	0.0109184	+	0.999940i	$0.496525\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	0	0
$773$	52.8444	1.90068	0.950341	−	0.311212i	$-0.100735\pi$
$773$	52.8444	1.90068	0.950341	+	0.311212i	$0.100735\pi$
$774$	0	0
$775$	−34.9083	−1.25394
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	−11.2111	−0.401679
$780$	0	0
$781$	−17.2111	−0.615862
$782$	0	0
$783$	−2.23886	−0.0800103
$784$	0	0
$785$	54.8444	1.95748
$786$	0	0
$787$	10.7889	0.384583	0.192291	−	0.981338i	$-0.438408\pi$
$787$	10.7889	0.384583	0.192291	+	0.981338i	$0.438408\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.2111	−0.469733
$792$	0	0
$793$	33.2111	1.17936
$794$	0	0
$795$	85.9638	3.04882
$796$	0	0
$797$	31.8167	1.12700	0.563502	−	0.826115i	$-0.309454\pi$
$797$	31.8167	1.12700	0.563502	+	0.826115i	$0.309454\pi$
$798$	0	0
$799$	5.21110	0.184356
$800$	0	0
$801$	6.42221	0.226917
$802$	0	0
$803$	−12.7889	−0.451310
$804$	0	0
$805$	19.8167	0.698445
$806$	0	0
$807$	−44.2389	−1.55728
$808$	0	0
$809$	−0.788897	−0.0277362	−0.0138681	−	0.999904i	$-0.504414\pi$
$809$	−0.788897	−0.0277362	−0.0138681	+	0.999904i	$0.504414\pi$
$810$	0	0
$811$	16.4861	0.578906	0.289453	−	0.957192i	$-0.406527\pi$
$811$	16.4861	0.578906	0.289453	+	0.957192i	$0.406527\pi$
$812$	0	0
$813$	56.2389	1.97238
$814$	0	0
$815$	29.0278	1.01680
$816$	0	0
$817$	16.4222	0.574540
$818$	0	0
$819$	9.21110	0.321862
$820$	0	0
$821$	14.6056	0.509737	0.254869	−	0.966976i	$-0.417968\pi$
$821$	14.6056	0.509737	0.254869	+	0.966976i	$0.417968\pi$
$822$	0	0
$823$	5.02776	0.175257	0.0876283	−	0.996153i	$-0.472071\pi$
$823$	5.02776	0.175257	0.0876283	+	0.996153i	$0.472071\pi$
$824$	0	0
$825$	−35.4500	−1.23421
$826$	0	0
$827$	52.2389	1.81652	0.908262	−	0.418403i	$-0.137410\pi$
$827$	52.2389	1.81652	0.908262	+	0.418403i	$0.137410\pi$
$828$	0	0
$829$	−54.0555	−1.87743	−0.938713	−	0.344700i	$-0.887981\pi$
$829$	−54.0555	−1.87743	−0.938713	+	0.344700i	$0.887981\pi$
$830$	0	0
$831$	59.8722	2.07694
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	79.6611	2.75678
$836$	0	0
$837$	9.48612	0.327888
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	0	0
$843$	−29.2389	−1.00704
$844$	0	0
$845$	9.90833	0.340857
$846$	0	0
$847$	−4.21110	−0.144695
$848$	0	0
$849$	7.60555	0.261022
$850$	0	0
$851$	−15.6333	−0.535903
$852$	0	0
$853$	−36.0555	−1.23452	−0.617259	−	0.786760i	$-0.711757\pi$
$853$	−36.0555	−1.23452	−0.617259	+	0.786760i	$0.711757\pi$
$854$	0	0
$855$	−19.8167	−0.677715
$856$	0	0
$857$	−28.3305	−0.967753	−0.483876	−	0.875136i	$-0.660772\pi$
$857$	−28.3305	−0.967753	−0.483876	+	0.875136i	$0.660772\pi$
$858$	0	0
$859$	−16.8444	−0.574724	−0.287362	−	0.957822i	$-0.592778\pi$
$859$	−16.8444	−0.574724	−0.287362	+	0.957822i	$0.592778\pi$
$860$	0	0
$861$	9.90833	0.337675
$862$	0	0
$863$	19.3028	0.657074	0.328537	−	0.944491i	$-0.393444\pi$
$863$	19.3028	0.657074	0.328537	+	0.944491i	$0.393444\pi$
$864$	0	0
$865$	−42.6333	−1.44958
$866$	0	0
$867$	2.30278	0.0782064
$868$	0	0
$869$	34.4222	1.16769
$870$	0	0
$871$	18.0555	0.611788
$872$	0	0
$873$	1.60555	0.0543397
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	8.78890	0.296780	0.148390	−	0.988929i	$-0.452591\pi$
$877$	8.78890	0.296780	0.148390	+	0.988929i	$0.452591\pi$
$878$	0	0
$879$	−36.0000	−1.21425
$880$	0	0
$881$	29.7250	1.00146	0.500730	−	0.865604i	$-0.333065\pi$
$881$	29.7250	1.00146	0.500730	+	0.865604i	$0.333065\pi$
$882$	0	0
$883$	5.54163	0.186491	0.0932454	−	0.995643i	$-0.470276\pi$
$883$	5.54163	0.186491	0.0932454	+	0.995643i	$0.470276\pi$
$884$	0	0
$885$	70.0555	2.35489
$886$	0	0
$887$	28.1472	0.945090	0.472545	−	0.881307i	$-0.343335\pi$
$887$	28.1472	0.945090	0.472545	+	0.881307i	$0.343335\pi$
$888$	0	0
$889$	13.7250	0.460321
$890$	0	0
$891$	27.6333	0.925751
$892$	0	0
$893$	−13.5778	−0.454364
$894$	0	0
$895$	−36.0278	−1.20428
$896$	0	0
$897$	55.2666	1.84530
$898$	0	0
$899$	−8.23886	−0.274781
$900$	0	0
$901$	11.3028	0.376550
$902$	0	0
$903$	−14.5139	−0.482992
$904$	0	0
$905$	58.2389	1.93593
$906$	0	0
$907$	8.42221	0.279655	0.139827	−	0.990176i	$-0.455345\pi$
$907$	8.42221	0.279655	0.139827	+	0.990176i	$0.455345\pi$
$908$	0	0
$909$	12.4222	0.412019
$910$	0	0
$911$	−47.0278	−1.55810	−0.779050	−	0.626962i	$-0.784298\pi$
$911$	−47.0278	−1.55810	−0.779050	+	0.626962i	$0.784298\pi$
$912$	0	0
$913$	−9.94449	−0.329114
$914$	0	0
$915$	63.1472	2.08758
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	−10.3028	−0.339857	−0.169929	−	0.985456i	$-0.554354\pi$
$919$	−10.3028	−0.339857	−0.169929	+	0.985456i	$0.554354\pi$
$920$	0	0
$921$	47.4500	1.56353
$922$	0	0
$923$	26.4222	0.869697
$924$	0	0
$925$	−15.3944	−0.506166
$926$	0	0
$927$	−36.4222	−1.19626
$928$	0	0
$929$	36.5139	1.19798	0.598991	−	0.800756i	$-0.295569\pi$
$929$	36.5139	1.19798	0.598991	+	0.800756i	$0.295569\pi$
$930$	0	0
$931$	−2.60555	−0.0853935
$932$	0	0
$933$	−44.4500	−1.45523
$934$	0	0
$935$	−8.60555	−0.281432
$936$	0	0
$937$	−1.57779	−0.0515443	−0.0257722	−	0.999668i	$-0.508204\pi$
$937$	−1.57779	−0.0515443	−0.0257722	+	0.999668i	$0.508204\pi$
$938$	0	0
$939$	−52.8167	−1.72361
$940$	0	0
$941$	−49.7527	−1.62189	−0.810946	−	0.585120i	$-0.801047\pi$
$941$	−49.7527	−1.62189	−0.810946	+	0.585120i	$0.801047\pi$
$942$	0	0
$943$	25.8167	0.840706
$944$	0	0
$945$	−5.30278	−0.172499
$946$	0	0
$947$	5.39445	0.175296	0.0876480	−	0.996152i	$-0.472065\pi$
$947$	5.39445	0.175296	0.0876480	+	0.996152i	$0.472065\pi$
$948$	0	0
$949$	19.6333	0.637324
$950$	0	0
$951$	14.7889	0.479563
$952$	0	0
$953$	−45.3583	−1.46930	−0.734650	−	0.678447i	$-0.762653\pi$
$953$	−45.3583	−1.46930	−0.734650	+	0.678447i	$0.762653\pi$
$954$	0	0
$955$	55.1472	1.78452
$956$	0	0
$957$	−8.36669	−0.270457
$958$	0	0
$959$	−11.0917	−0.358169
$960$	0	0
$961$	3.90833	0.126075
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.4222	0.721796
$966$	0	0
$967$	−41.6972	−1.34089	−0.670446	−	0.741958i	$-0.733897\pi$
$967$	−41.6972	−1.34089	−0.670446	+	0.741958i	$0.733897\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	25.3944	0.814947	0.407473	−	0.913217i	$-0.366410\pi$
$971$	25.3944	0.814947	0.407473	+	0.913217i	$0.366410\pi$
$972$	0	0
$973$	−4.51388	−0.144708
$974$	0	0
$975$	54.4222	1.74291
$976$	0	0
$977$	19.3028	0.617551	0.308775	−	0.951135i	$-0.400081\pi$
$977$	19.3028	0.617551	0.308775	+	0.951135i	$0.400081\pi$
$978$	0	0
$979$	−7.26662	−0.232242
$980$	0	0
$981$	−25.8167	−0.824262
$982$	0	0
$983$	30.5694	0.975012	0.487506	−	0.873120i	$-0.337907\pi$
$983$	30.5694	0.975012	0.487506	+	0.873120i	$0.337907\pi$
$984$	0	0
$985$	−80.6611	−2.57008
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−37.8167	−1.20250
$990$	0	0
$991$	15.5778	0.494845	0.247422	−	0.968908i	$-0.420416\pi$
$991$	15.5778	0.494845	0.247422	+	0.968908i	$0.420416\pi$
$992$	0	0
$993$	−14.0278	−0.445157
$994$	0	0
$995$	−86.8722	−2.75403
$996$	0	0
$997$	−26.1194	−0.827211	−0.413605	−	0.910456i	$-0.635731\pi$
$997$	−26.1194	−0.827211	−0.413605	+	0.910456i	$0.635731\pi$
$998$	0	0
$999$	4.18335	0.132355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.x.1.2		2
4.3	odd	2		7616.2.a.s.1.1		2
8.3	odd	2		952.2.a.b.1.2	✓	2
8.5	even	2		1904.2.a.g.1.1		2
24.11	even	2		8568.2.a.t.1.2		2
56.27	even	2		6664.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.b.1.2	✓	2	8.3	odd	2
1904.2.a.g.1.1		2	8.5	even	2
6664.2.a.g.1.1		2	56.27	even	2
7616.2.a.s.1.1		2	4.3	odd	2
7616.2.a.x.1.2		2	1.1	even	1	trivial
8568.2.a.t.1.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{3} +3.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})\)	\(q+2.30278 q^{3} +3.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} -2.60555 q^{11} +4.00000 q^{13} +7.60555 q^{15} +1.00000 q^{17} -2.60555 q^{19} +2.30278 q^{21} +6.00000 q^{23} +5.90833 q^{25} -1.60555 q^{27} +1.39445 q^{29} -5.90833 q^{31} -6.00000 q^{33} +3.30278 q^{35} -2.60555 q^{37} +9.21110 q^{39} +4.30278 q^{41} -6.30278 q^{43} +7.60555 q^{45} +5.21110 q^{47} +1.00000 q^{49} +2.30278 q^{51} +11.3028 q^{53} -8.60555 q^{55} -6.00000 q^{57} +9.21110 q^{59} +8.30278 q^{61} +2.30278 q^{63} +13.2111 q^{65} +4.51388 q^{67} +13.8167 q^{69} +6.60555 q^{71} +4.90833 q^{73} +13.6056 q^{75} -2.60555 q^{77} -13.2111 q^{79} -10.6056 q^{81} +3.81665 q^{83} +3.30278 q^{85} +3.21110 q^{87} +2.78890 q^{89} +4.00000 q^{91} -13.6056 q^{93} -8.60555 q^{95} +0.697224 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q + q^3 + 3 * q^5 + 2 * q^7 + q^9	\( 2 q + q^{3} + 3 q^{5} + 2 q^{7} + q^{9} + 2 q^{11} + 8 q^{13} + 8 q^{15} + 2 q^{17} + 2 q^{19} + q^{21} + 12 q^{23} + q^{25} + 4 q^{27} + 10 q^{29} - q^{31} - 12 q^{33} + 3 q^{35} + 2 q^{37} + 4 q^{39} + 5 q^{41} - 9 q^{43} + 8 q^{45} - 4 q^{47} + 2 q^{49} + q^{51} + 19 q^{53} - 10 q^{55} - 12 q^{57} + 4 q^{59} + 13 q^{61} + q^{63} + 12 q^{65} - 9 q^{67} + 6 q^{69} + 6 q^{71} - q^{73} + 20 q^{75} + 2 q^{77} - 12 q^{79} - 14 q^{81} - 14 q^{83} + 3 q^{85} - 8 q^{87} + 20 q^{89} + 8 q^{91} - 20 q^{93} - 10 q^{95} + 5 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + q^3 + 3 * q^5 + 2 * q^7 + q^9 + 2 * q^11 + 8 * q^13 + 8 * q^15 + 2 * q^17 + 2 * q^19 + q^21 + 12 * q^23 + q^25 + 4 * q^27 + 10 * q^29 - q^31 - 12 * q^33 + 3 * q^35 + 2 * q^37 + 4 * q^39 + 5 * q^41 - 9 * q^43 + 8 * q^45 - 4 * q^47 + 2 * q^49 + q^51 + 19 * q^53 - 10 * q^55 - 12 * q^57 + 4 * q^59 + 13 * q^61 + q^63 + 12 * q^65 - 9 * q^67 + 6 * q^69 + 6 * q^71 - q^73 + 20 * q^75 + 2 * q^77 - 12 * q^79 - 14 * q^81 - 14 * q^83 + 3 * q^85 - 8 * q^87 + 20 * q^89 + 8 * q^91 - 20 * q^93 - 10 * q^95 + 5 * q^97 - 12 * q^99

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