LMFDB - Embedded newform 6384.2.a.cc.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1240016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 2x^{2} + 16x + 6 $ x^5 - x^4 - 8x^3 + 2x^2 + 16*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.77304$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.54204 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.54204 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.54608 q^{11} -1.83340 q^{13} -2.54204 q^{15} -3.88952 q^{17} -1.00000 q^{19} +1.00000 q^{21} -5.54608 q^{23} +1.46199 q^{25} -1.00000 q^{27} +8.08812 q^{29} +3.54608 q^{31} -5.54608 q^{33} -2.54204 q^{35} +5.73661 q^{37} +1.83340 q^{39} -2.48592 q^{41} -3.80947 q^{43} +2.54204 q^{45} +2.99597 q^{47} +1.00000 q^{49} +3.88952 q^{51} -3.31529 q^{53} +14.0984 q^{55} +1.00000 q^{57} +11.0922 q^{59} +8.43157 q^{61} -1.00000 q^{63} -4.66058 q^{65} +6.14424 q^{67} +5.54608 q^{69} -8.18491 q^{71} +8.31932 q^{73} -1.46199 q^{75} -5.54608 q^{77} +4.79676 q^{79} +1.00000 q^{81} +15.3234 q^{83} -9.88734 q^{85} -8.08812 q^{87} -12.6541 q^{89} +1.83340 q^{91} -3.54608 q^{93} -2.54204 q^{95} +15.1241 q^{97} +5.54608 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 2 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} + 5 q^{21} - 2 q^{23} + 11 q^{25} - 5 q^{27} - 8 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} - 20 q^{43} - 2 q^{45} + 26 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} + 5 q^{57} + 4 q^{59} + 10 q^{61} - 5 q^{63} - 4 q^{65} - 10 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} - 11 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 34 q^{83} - 36 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{93} + 2 q^{95} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 2 * q^5 - 5 * q^7 + 5 * q^9 + 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 + 5 * q^21 - 2 * q^23 + 11 * q^25 - 5 * q^27 - 8 * q^31 - 2 * q^33 + 2 * q^35 + 2 * q^37 - 8 * q^39 + 2 * q^41 - 20 * q^43 - 2 * q^45 + 26 * q^47 + 5 * q^49 + 2 * q^51 + 4 * q^53 + 4 * q^55 + 5 * q^57 + 4 * q^59 + 10 * q^61 - 5 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 - 10 * q^71 + 10 * q^73 - 11 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 34 * q^83 - 36 * q^85 + 10 * q^89 - 8 * q^91 + 8 * q^93 + 2 * q^95 - 16 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.54204	1.13684	0.568418	−	0.822740i	$-0.307556\pi$
$5$	2.54204	1.13684	0.568418	+	0.822740i	$0.307556\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.54608	1.67220	0.836102	−	0.548574i	$-0.184829\pi$
$11$	5.54608	1.67220	0.836102	+	0.548574i	$0.184829\pi$
$12$	0	0
$13$	−1.83340	−0.508494	−0.254247	−	0.967139i	$-0.581828\pi$
$13$	−1.83340	−0.508494	−0.254247	+	0.967139i	$0.581828\pi$
$14$	0	0
$15$	−2.54204	−0.656353
$16$	0	0
$17$	−3.88952	−0.943348	−0.471674	−	0.881773i	$-0.656350\pi$
$17$	−3.88952	−0.943348	−0.471674	+	0.881773i	$0.656350\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−5.54608	−1.15644	−0.578218	−	0.815882i	$-0.696252\pi$
$23$	−5.54608	−1.15644	−0.578218	+	0.815882i	$0.696252\pi$
$24$	0	0
$25$	1.46199	0.292397
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.08812	1.50193	0.750963	−	0.660344i	$-0.229590\pi$
$29$	8.08812	1.50193	0.750963	+	0.660344i	$0.229590\pi$
$30$	0	0
$31$	3.54608	0.636894	0.318447	−	0.947941i	$-0.396839\pi$
$31$	3.54608	0.636894	0.318447	+	0.947941i	$0.396839\pi$
$32$	0	0
$33$	−5.54608	−0.965448
$34$	0	0
$35$	−2.54204	−0.429684
$36$	0	0
$37$	5.73661	0.943093	0.471546	−	0.881841i	$-0.343696\pi$
$37$	5.73661	0.943093	0.471546	+	0.881841i	$0.343696\pi$
$38$	0	0
$39$	1.83340	0.293579
$40$	0	0
$41$	−2.48592	−0.388236	−0.194118	−	0.980978i	$-0.562184\pi$
$41$	−2.48592	−0.388236	−0.194118	+	0.980978i	$0.562184\pi$
$42$	0	0
$43$	−3.80947	−0.580938	−0.290469	−	0.956884i	$-0.593811\pi$
$43$	−3.80947	−0.580938	−0.290469	+	0.956884i	$0.593811\pi$
$44$	0	0
$45$	2.54204	0.378946
$46$	0	0
$47$	2.99597	0.437007	0.218503	−	0.975836i	$-0.429883\pi$
$47$	2.99597	0.437007	0.218503	+	0.975836i	$0.429883\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.88952	0.544642
$52$	0	0
$53$	−3.31529	−0.455390	−0.227695	−	0.973732i	$-0.573119\pi$
$53$	−3.31529	−0.455390	−0.227695	+	0.973732i	$0.573119\pi$
$54$	0	0
$55$	14.0984	1.90102
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	11.0922	1.44408	0.722038	−	0.691854i	$-0.243206\pi$
$59$	11.0922	1.44408	0.722038	+	0.691854i	$0.243206\pi$
$60$	0	0
$61$	8.43157	1.07955	0.539776	−	0.841809i	$-0.318509\pi$
$61$	8.43157	1.07955	0.539776	+	0.841809i	$0.318509\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−4.66058	−0.578074
$66$	0	0
$67$	6.14424	0.750639	0.375319	−	0.926896i	$-0.377533\pi$
$67$	6.14424	0.750639	0.375319	+	0.926896i	$0.377533\pi$
$68$	0	0
$69$	5.54608	0.667669
$70$	0	0
$71$	−8.18491	−0.971370	−0.485685	−	0.874134i	$-0.661430\pi$
$71$	−8.18491	−0.971370	−0.485685	+	0.874134i	$0.661430\pi$
$72$	0	0
$73$	8.31932	0.973703	0.486851	−	0.873485i	$-0.338145\pi$
$73$	8.31932	0.973703	0.486851	+	0.873485i	$0.338145\pi$
$74$	0	0
$75$	−1.46199	−0.168816
$76$	0	0
$77$	−5.54608	−0.632034
$78$	0	0
$79$	4.79676	0.539678	0.269839	−	0.962905i	$-0.413030\pi$
$79$	4.79676	0.539678	0.269839	+	0.962905i	$0.413030\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.3234	1.68196	0.840978	−	0.541069i	$-0.181980\pi$
$83$	15.3234	1.68196	0.840978	+	0.541069i	$0.181980\pi$
$84$	0	0
$85$	−9.88734	−1.07243
$86$	0	0
$87$	−8.08812	−0.867137
$88$	0	0
$89$	−12.6541	−1.34133	−0.670666	−	0.741760i	$-0.733992\pi$
$89$	−12.6541	−1.34133	−0.670666	+	0.741760i	$0.733992\pi$
$90$	0	0
$91$	1.83340	0.192193
$92$	0	0
$93$	−3.54608	−0.367711
$94$	0	0
$95$	−2.54204	−0.260808
$96$	0	0
$97$	15.1241	1.53562	0.767812	−	0.640675i	$-0.221345\pi$
$97$	15.1241	1.53562	0.767812	+	0.640675i	$0.221345\pi$
$98$	0	0
$99$	5.54608	0.557402
$100$	0	0
$101$	14.9817	1.49073	0.745366	−	0.666655i	$-0.232275\pi$
$101$	14.9817	1.49073	0.745366	+	0.666655i	$0.232275\pi$
$102$	0	0
$103$	−6.36176	−0.626843	−0.313421	−	0.949614i	$-0.601475\pi$
$103$	−6.36176	−0.626843	−0.313421	+	0.949614i	$0.601475\pi$
$104$	0	0
$105$	2.54204	0.248078
$106$	0	0
$107$	−17.9296	−1.73332	−0.866659	−	0.498901i	$-0.833737\pi$
$107$	−17.9296	−1.73332	−0.866659	+	0.498901i	$0.833737\pi$
$108$	0	0
$109$	−4.00806	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$109$	−4.00806	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$110$	0	0
$111$	−5.73661	−0.544495
$112$	0	0
$113$	1.96781	0.185116	0.0925581	−	0.995707i	$-0.470496\pi$
$113$	1.96781	0.185116	0.0925581	+	0.995707i	$0.470496\pi$
$114$	0	0
$115$	−14.0984	−1.31468
$116$	0	0
$117$	−1.83340	−0.169498
$118$	0	0
$119$	3.88952	0.356552
$120$	0	0
$121$	19.7590	1.79627
$122$	0	0
$123$	2.48592	0.224148
$124$	0	0
$125$	−8.99378	−0.804428
$126$	0	0
$127$	−7.44928	−0.661017	−0.330509	−	0.943803i	$-0.607220\pi$
$127$	−7.44928	−0.661017	−0.330509	+	0.943803i	$0.607220\pi$
$128$	0	0
$129$	3.80947	0.335405
$130$	0	0
$131$	15.4356	1.34861	0.674307	−	0.738451i	$-0.264442\pi$
$131$	15.4356	1.34861	0.674307	+	0.738451i	$0.264442\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−2.54204	−0.218784
$136$	0	0
$137$	−15.4539	−1.32032	−0.660158	−	0.751127i	$-0.729511\pi$
$137$	−15.4539	−1.32032	−0.660158	+	0.751127i	$0.729511\pi$
$138$	0	0
$139$	10.3193	0.875273	0.437637	−	0.899152i	$-0.355816\pi$
$139$	10.3193	0.875273	0.437637	+	0.899152i	$0.355816\pi$
$140$	0	0
$141$	−2.99597	−0.252306
$142$	0	0
$143$	−10.1682	−0.850306
$144$	0	0
$145$	20.5604	1.70744
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	12.0081	0.983739	0.491869	−	0.870669i	$-0.336314\pi$
$149$	12.0081	0.983739	0.491869	+	0.870669i	$0.336314\pi$
$150$	0	0
$151$	−19.0463	−1.54996	−0.774982	−	0.631983i	$-0.782241\pi$
$151$	−19.0463	−1.54996	−0.774982	+	0.631983i	$0.782241\pi$
$152$	0	0
$153$	−3.88952	−0.314449
$154$	0	0
$155$	9.01428	0.724044
$156$	0	0
$157$	5.93019	0.473281	0.236640	−	0.971597i	$-0.423954\pi$
$157$	5.93019	0.473281	0.236640	+	0.971597i	$0.423954\pi$
$158$	0	0
$159$	3.31529	0.262920
$160$	0	0
$161$	5.54608	0.437092
$162$	0	0
$163$	2.31084	0.180999	0.0904995	−	0.995896i	$-0.471154\pi$
$163$	2.31084	0.180999	0.0904995	+	0.995896i	$0.471154\pi$
$164$	0	0
$165$	−14.0984	−1.09756
$166$	0	0
$167$	9.41729	0.728732	0.364366	−	0.931256i	$-0.381286\pi$
$167$	9.41729	0.728732	0.364366	+	0.931256i	$0.381286\pi$
$168$	0	0
$169$	−9.63864	−0.741434
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−13.5781	−1.03232	−0.516161	−	0.856492i	$-0.672639\pi$
$173$	−13.5781	−1.03232	−0.516161	+	0.856492i	$0.672639\pi$
$174$	0	0
$175$	−1.46199	−0.110516
$176$	0	0
$177$	−11.0922	−0.833737
$178$	0	0
$179$	−1.55976	−0.116582	−0.0582910	−	0.998300i	$-0.518565\pi$
$179$	−1.55976	−0.116582	−0.0582910	+	0.998300i	$0.518565\pi$
$180$	0	0
$181$	18.5334	1.37757	0.688787	−	0.724963i	$-0.258143\pi$
$181$	18.5334	1.37757	0.688787	+	0.724963i	$0.258143\pi$
$182$	0	0
$183$	−8.43157	−0.623279
$184$	0	0
$185$	14.5827	1.07214
$186$	0	0
$187$	−21.5716	−1.57747
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	23.3811	1.69179	0.845897	−	0.533347i	$-0.179066\pi$
$191$	23.3811	1.69179	0.845897	+	0.533347i	$0.179066\pi$
$192$	0	0
$193$	16.8288	1.21136	0.605680	−	0.795708i	$-0.292901\pi$
$193$	16.8288	1.21136	0.605680	+	0.795708i	$0.292901\pi$
$194$	0	0
$195$	4.66058	0.333751
$196$	0	0
$197$	−23.1207	−1.64728	−0.823641	−	0.567111i	$-0.808061\pi$
$197$	−23.1207	−1.64728	−0.823641	+	0.567111i	$0.808061\pi$
$198$	0	0
$199$	−12.1762	−0.863151	−0.431575	−	0.902077i	$-0.642042\pi$
$199$	−12.1762	−0.863151	−0.431575	+	0.902077i	$0.642042\pi$
$200$	0	0
$201$	−6.14424	−0.433381
$202$	0	0
$203$	−8.08812	−0.567675
$204$	0	0
$205$	−6.31932	−0.441361
$206$	0	0
$207$	−5.54608	−0.385479
$208$	0	0
$209$	−5.54608	−0.383630
$210$	0	0
$211$	6.95597	0.478869	0.239434	−	0.970913i	$-0.423038\pi$
$211$	6.95597	0.478869	0.239434	+	0.970913i	$0.423038\pi$
$212$	0	0
$213$	8.18491	0.560821
$214$	0	0
$215$	−9.68383	−0.660432
$216$	0	0
$217$	−3.54608	−0.240723
$218$	0	0
$219$	−8.31932	−0.562168
$220$	0	0
$221$	7.13105	0.479686
$222$	0	0
$223$	28.4092	1.90242	0.951211	−	0.308542i	$-0.0998411\pi$
$223$	28.4092	1.90242	0.951211	+	0.308542i	$0.0998411\pi$
$224$	0	0
$225$	1.46199	0.0974658
$226$	0	0
$227$	−3.31311	−0.219899	−0.109949	−	0.993937i	$-0.535069\pi$
$227$	−3.31311	−0.219899	−0.109949	+	0.993937i	$0.535069\pi$
$228$	0	0
$229$	1.88776	0.124746	0.0623732	−	0.998053i	$-0.480133\pi$
$229$	1.88776	0.124746	0.0623732	+	0.998053i	$0.480133\pi$
$230$	0	0
$231$	5.54608	0.364905
$232$	0	0
$233$	23.7227	1.55413	0.777064	−	0.629422i	$-0.216708\pi$
$233$	23.7227	1.55413	0.777064	+	0.629422i	$0.216708\pi$
$234$	0	0
$235$	7.61588	0.496805
$236$	0	0
$237$	−4.79676	−0.311583
$238$	0	0
$239$	−21.9159	−1.41762	−0.708811	−	0.705399i	$-0.750768\pi$
$239$	−21.9159	−1.41762	−0.708811	+	0.705399i	$0.750768\pi$
$240$	0	0
$241$	−17.2445	−1.11081	−0.555406	−	0.831579i	$-0.687437\pi$
$241$	−17.2445	−1.11081	−0.555406	+	0.831579i	$0.687437\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.54204	0.162405
$246$	0	0
$247$	1.83340	0.116656
$248$	0	0
$249$	−15.3234	−0.971078
$250$	0	0
$251$	3.72636	0.235206	0.117603	−	0.993061i	$-0.462479\pi$
$251$	3.72636	0.235206	0.117603	+	0.993061i	$0.462479\pi$
$252$	0	0
$253$	−30.7590	−1.93380
$254$	0	0
$255$	9.88734	0.619169
$256$	0	0
$257$	4.42999	0.276335	0.138168	−	0.990409i	$-0.455879\pi$
$257$	4.42999	0.276335	0.138168	+	0.990409i	$0.455879\pi$
$258$	0	0
$259$	−5.73661	−0.356456
$260$	0	0
$261$	8.08812	0.500642
$262$	0	0
$263$	−2.23297	−0.137691	−0.0688454	−	0.997627i	$-0.521932\pi$
$263$	−2.23297	−0.137691	−0.0688454	+	0.997627i	$0.521932\pi$
$264$	0	0
$265$	−8.42761	−0.517704
$266$	0	0
$267$	12.6541	0.774418
$268$	0	0
$269$	4.87505	0.297237	0.148619	−	0.988895i	$-0.452517\pi$
$269$	4.87505	0.297237	0.148619	+	0.988895i	$0.452517\pi$
$270$	0	0
$271$	16.5519	1.00545	0.502727	−	0.864445i	$-0.332330\pi$
$271$	16.5519	1.00545	0.502727	+	0.864445i	$0.332330\pi$
$272$	0	0
$273$	−1.83340	−0.110962
$274$	0	0
$275$	8.10829	0.488948
$276$	0	0
$277$	28.0841	1.68741	0.843704	−	0.536808i	$-0.180370\pi$
$277$	28.0841	1.68741	0.843704	+	0.536808i	$0.180370\pi$
$278$	0	0
$279$	3.54608	0.212298
$280$	0	0
$281$	11.2674	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$281$	11.2674	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$282$	0	0
$283$	−15.7952	−0.938925	−0.469463	−	0.882952i	$-0.655552\pi$
$283$	−15.7952	−0.938925	−0.469463	+	0.882952i	$0.655552\pi$
$284$	0	0
$285$	2.54204	0.150578
$286$	0	0
$287$	2.48592	0.146739
$288$	0	0
$289$	−1.87161	−0.110095
$290$	0	0
$291$	−15.1241	−0.886593
$292$	0	0
$293$	26.4573	1.54565	0.772827	−	0.634617i	$-0.218842\pi$
$293$	26.4573	1.54565	0.772827	+	0.634617i	$0.218842\pi$
$294$	0	0
$295$	28.1967	1.64168
$296$	0	0
$297$	−5.54608	−0.321816
$298$	0	0
$299$	10.1682	0.588041
$300$	0	0
$301$	3.80947	0.219574
$302$	0	0
$303$	−14.9817	−0.860675
$304$	0	0
$305$	21.4334	1.22727
$306$	0	0
$307$	−11.7875	−0.672750	−0.336375	−	0.941728i	$-0.609201\pi$
$307$	−11.7875	−0.672750	−0.336375	+	0.941728i	$0.609201\pi$
$308$	0	0
$309$	6.36176	0.361908
$310$	0	0
$311$	−2.39938	−0.136056	−0.0680281	−	0.997683i	$-0.521671\pi$
$311$	−2.39938	−0.136056	−0.0680281	+	0.997683i	$0.521671\pi$
$312$	0	0
$313$	6.66058	0.376478	0.188239	−	0.982123i	$-0.439722\pi$
$313$	6.66058	0.376478	0.188239	+	0.982123i	$0.439722\pi$
$314$	0	0
$315$	−2.54204	−0.143228
$316$	0	0
$317$	−18.3376	−1.02994	−0.514972	−	0.857207i	$-0.672198\pi$
$317$	−18.3376	−1.02994	−0.514972	+	0.857207i	$0.672198\pi$
$318$	0	0
$319$	44.8573	2.51153
$320$	0	0
$321$	17.9296	1.00073
$322$	0	0
$323$	3.88952	0.216419
$324$	0	0
$325$	−2.68041	−0.148682
$326$	0	0
$327$	4.00806	0.221646
$328$	0	0
$329$	−2.99597	−0.165173
$330$	0	0
$331$	3.09099	0.169896	0.0849482	−	0.996385i	$-0.472928\pi$
$331$	3.09099	0.169896	0.0849482	+	0.996385i	$0.472928\pi$
$332$	0	0
$333$	5.73661	0.314364
$334$	0	0
$335$	15.6189	0.853353
$336$	0	0
$337$	16.8233	0.916425	0.458213	−	0.888843i	$-0.348490\pi$
$337$	16.8233	0.916425	0.458213	+	0.888843i	$0.348490\pi$
$338$	0	0
$339$	−1.96781	−0.106877
$340$	0	0
$341$	19.6668	1.06502
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	14.0984	0.759031
$346$	0	0
$347$	26.8877	1.44341	0.721705	−	0.692201i	$-0.243359\pi$
$347$	26.8877	1.44341	0.721705	+	0.692201i	$0.243359\pi$
$348$	0	0
$349$	−4.80367	−0.257134	−0.128567	−	0.991701i	$-0.541038\pi$
$349$	−4.80367	−0.257134	−0.128567	+	0.991701i	$0.541038\pi$
$350$	0	0
$351$	1.83340	0.0978597
$352$	0	0
$353$	−12.5775	−0.669432	−0.334716	−	0.942319i	$-0.608640\pi$
$353$	−12.5775	−0.669432	−0.334716	+	0.942319i	$0.608640\pi$
$354$	0	0
$355$	−20.8064	−1.10429
$356$	0	0
$357$	−3.88952	−0.205855
$358$	0	0
$359$	31.3332	1.65370	0.826851	−	0.562421i	$-0.190130\pi$
$359$	31.3332	1.65370	0.826851	+	0.562421i	$0.190130\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−19.7590	−1.03708
$364$	0	0
$365$	21.1481	1.10694
$366$	0	0
$367$	21.0841	1.10058	0.550290	−	0.834973i	$-0.314517\pi$
$367$	21.0841	1.10058	0.550290	+	0.834973i	$0.314517\pi$
$368$	0	0
$369$	−2.48592	−0.129412
$370$	0	0
$371$	3.31529	0.172121
$372$	0	0
$373$	−23.6097	−1.22246	−0.611231	−	0.791452i	$-0.709325\pi$
$373$	−23.6097	−1.22246	−0.611231	+	0.791452i	$0.709325\pi$
$374$	0	0
$375$	8.99378	0.464437
$376$	0	0
$377$	−14.8288	−0.763720
$378$	0	0
$379$	−7.13803	−0.366656	−0.183328	−	0.983052i	$-0.558687\pi$
$379$	−7.13803	−0.366656	−0.183328	+	0.983052i	$0.558687\pi$
$380$	0	0
$381$	7.44928	0.381638
$382$	0	0
$383$	9.41729	0.481201	0.240600	−	0.970624i	$-0.422656\pi$
$383$	9.41729	0.481201	0.240600	+	0.970624i	$0.422656\pi$
$384$	0	0
$385$	−14.0984	−0.718519
$386$	0	0
$387$	−3.80947	−0.193646
$388$	0	0
$389$	−28.0721	−1.42331	−0.711655	−	0.702529i	$-0.752054\pi$
$389$	−28.0721	−1.42331	−0.711655	+	0.702529i	$0.752054\pi$
$390$	0	0
$391$	21.5716	1.09092
$392$	0	0
$393$	−15.4356	−0.778623
$394$	0	0
$395$	12.1936	0.613526
$396$	0	0
$397$	−6.59078	−0.330782	−0.165391	−	0.986228i	$-0.552889\pi$
$397$	−6.59078	−0.330782	−0.165391	+	0.986228i	$0.552889\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	34.1865	1.70719	0.853596	−	0.520936i	$-0.174417\pi$
$401$	34.1865	1.70719	0.853596	+	0.520936i	$0.174417\pi$
$402$	0	0
$403$	−6.50138	−0.323857
$404$	0	0
$405$	2.54204	0.126315
$406$	0	0
$407$	31.8157	1.57704
$408$	0	0
$409$	−33.4555	−1.65427	−0.827134	−	0.562005i	$-0.810030\pi$
$409$	−33.4555	−1.65427	−0.827134	+	0.562005i	$0.810030\pi$
$410$	0	0
$411$	15.4539	0.762285
$412$	0	0
$413$	−11.0922	−0.545809
$414$	0	0
$415$	38.9526	1.91211
$416$	0	0
$417$	−10.3193	−0.505339
$418$	0	0
$419$	30.4418	1.48718	0.743590	−	0.668636i	$-0.233121\pi$
$419$	30.4418	1.48718	0.743590	+	0.668636i	$0.233121\pi$
$420$	0	0
$421$	−17.4678	−0.851328	−0.425664	−	0.904881i	$-0.639959\pi$
$421$	−17.4678	−0.851328	−0.425664	+	0.904881i	$0.639959\pi$
$422$	0	0
$423$	2.99597	0.145669
$424$	0	0
$425$	−5.68643	−0.275833
$426$	0	0
$427$	−8.43157	−0.408032
$428$	0	0
$429$	10.1682	0.490924
$430$	0	0
$431$	−24.8969	−1.19924	−0.599621	−	0.800284i	$-0.704682\pi$
$431$	−24.8969	−1.19924	−0.599621	+	0.800284i	$0.704682\pi$
$432$	0	0
$433$	−11.3177	−0.543895	−0.271948	−	0.962312i	$-0.587668\pi$
$433$	−11.3177	−0.543895	−0.271948	+	0.962312i	$0.587668\pi$
$434$	0	0
$435$	−20.5604	−0.985794
$436$	0	0
$437$	5.54608	0.265305
$438$	0	0
$439$	−12.7424	−0.608162	−0.304081	−	0.952646i	$-0.598349\pi$
$439$	−12.7424	−0.608162	−0.304081	+	0.952646i	$0.598349\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	22.6140	1.07443	0.537213	−	0.843447i	$-0.319477\pi$
$443$	22.6140	1.07443	0.537213	+	0.843447i	$0.319477\pi$
$444$	0	0
$445$	−32.1673	−1.52487
$446$	0	0
$447$	−12.0081	−0.567962
$448$	0	0
$449$	5.06578	0.239069	0.119534	−	0.992830i	$-0.461860\pi$
$449$	5.06578	0.239069	0.119534	+	0.992830i	$0.461860\pi$
$450$	0	0
$451$	−13.7871	−0.649210
$452$	0	0
$453$	19.0463	0.894872
$454$	0	0
$455$	4.66058	0.218492
$456$	0	0
$457$	−7.79559	−0.364662	−0.182331	−	0.983237i	$-0.558364\pi$
$457$	−7.79559	−0.364662	−0.182331	+	0.983237i	$0.558364\pi$
$458$	0	0
$459$	3.88952	0.181547
$460$	0	0
$461$	6.89680	0.321216	0.160608	−	0.987018i	$-0.448655\pi$
$461$	6.89680	0.321216	0.160608	+	0.987018i	$0.448655\pi$
$462$	0	0
$463$	−10.6950	−0.497037	−0.248518	−	0.968627i	$-0.579944\pi$
$463$	−10.6950	−0.497037	−0.248518	+	0.968627i	$0.579944\pi$
$464$	0	0
$465$	−9.01428	−0.418027
$466$	0	0
$467$	−11.2146	−0.518952	−0.259476	−	0.965750i	$-0.583550\pi$
$467$	−11.2146	−0.518952	−0.259476	+	0.965750i	$0.583550\pi$
$468$	0	0
$469$	−6.14424	−0.283715
$470$	0	0
$471$	−5.93019	−0.273249
$472$	0	0
$473$	−21.1276	−0.971447
$474$	0	0
$475$	−1.46199	−0.0670806
$476$	0	0
$477$	−3.31529	−0.151797
$478$	0	0
$479$	35.4607	1.62024	0.810120	−	0.586264i	$-0.199402\pi$
$479$	35.4607	1.62024	0.810120	+	0.586264i	$0.199402\pi$
$480$	0	0
$481$	−10.5175	−0.479557
$482$	0	0
$483$	−5.54608	−0.252355
$484$	0	0
$485$	38.4462	1.74575
$486$	0	0
$487$	−8.58734	−0.389129	−0.194565	−	0.980890i	$-0.562329\pi$
$487$	−8.58734	−0.389129	−0.194565	+	0.980890i	$0.562329\pi$
$488$	0	0
$489$	−2.31084	−0.104500
$490$	0	0
$491$	33.7384	1.52259	0.761297	−	0.648403i	$-0.224563\pi$
$491$	33.7384	1.52259	0.761297	+	0.648403i	$0.224563\pi$
$492$	0	0
$493$	−31.4589	−1.41684
$494$	0	0
$495$	14.0984	0.633674
$496$	0	0
$497$	8.18491	0.367143
$498$	0	0
$499$	10.9364	0.489581	0.244790	−	0.969576i	$-0.421281\pi$
$499$	10.9364	0.489581	0.244790	+	0.969576i	$0.421281\pi$
$500$	0	0
$501$	−9.41729	−0.420733
$502$	0	0
$503$	−22.9454	−1.02309	−0.511543	−	0.859258i	$-0.670926\pi$
$503$	−22.9454	−1.02309	−0.511543	+	0.859258i	$0.670926\pi$
$504$	0	0
$505$	38.0841	1.69472
$506$	0	0
$507$	9.63864	0.428067
$508$	0	0
$509$	7.57001	0.335535	0.167767	−	0.985827i	$-0.446344\pi$
$509$	7.57001	0.335535	0.167767	+	0.985827i	$0.446344\pi$
$510$	0	0
$511$	−8.31932	−0.368025
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−16.1719	−0.712618
$516$	0	0
$517$	16.6159	0.730765
$518$	0	0
$519$	13.5781	0.596011
$520$	0	0
$521$	−0.827184	−0.0362396	−0.0181198	−	0.999836i	$-0.505768\pi$
$521$	−0.827184	−0.0362396	−0.0181198	+	0.999836i	$0.505768\pi$
$522$	0	0
$523$	17.2414	0.753915	0.376958	−	0.926230i	$-0.376970\pi$
$523$	17.2414	0.753915	0.376958	+	0.926230i	$0.376970\pi$
$524$	0	0
$525$	1.46199	0.0638064
$526$	0	0
$527$	−13.7925	−0.600812
$528$	0	0
$529$	7.75895	0.337346
$530$	0	0
$531$	11.0922	0.481358
$532$	0	0
$533$	4.55769	0.197415
$534$	0	0
$535$	−45.5778	−1.97050
$536$	0	0
$537$	1.55976	0.0673087
$538$	0	0
$539$	5.54608	0.238886
$540$	0	0
$541$	34.1280	1.46728	0.733638	−	0.679540i	$-0.237821\pi$
$541$	34.1280	1.46728	0.733638	+	0.679540i	$0.237821\pi$
$542$	0	0
$543$	−18.5334	−0.795343
$544$	0	0
$545$	−10.1887	−0.436435
$546$	0	0
$547$	−30.0101	−1.28314	−0.641569	−	0.767066i	$-0.721716\pi$
$547$	−30.0101	−1.28314	−0.641569	+	0.767066i	$0.721716\pi$
$548$	0	0
$549$	8.43157	0.359850
$550$	0	0
$551$	−8.08812	−0.344565
$552$	0	0
$553$	−4.79676	−0.203979
$554$	0	0
$555$	−14.5827	−0.619002
$556$	0	0
$557$	−34.9066	−1.47904	−0.739521	−	0.673134i	$-0.764948\pi$
$557$	−34.9066	−1.47904	−0.739521	+	0.673134i	$0.764948\pi$
$558$	0	0
$559$	6.98428	0.295403
$560$	0	0
$561$	21.5716	0.910753
$562$	0	0
$563$	−20.8358	−0.878123	−0.439061	−	0.898457i	$-0.644689\pi$
$563$	−20.8358	−0.878123	−0.439061	+	0.898457i	$0.644689\pi$
$564$	0	0
$565$	5.00226	0.210447
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	32.3180	1.35484	0.677421	−	0.735595i	$-0.263097\pi$
$569$	32.3180	1.35484	0.677421	+	0.735595i	$0.263097\pi$
$570$	0	0
$571$	−20.4451	−0.855599	−0.427800	−	0.903874i	$-0.640711\pi$
$571$	−20.4451	−0.855599	−0.427800	+	0.903874i	$0.640711\pi$
$572$	0	0
$573$	−23.3811	−0.976757
$574$	0	0
$575$	−8.10829	−0.338139
$576$	0	0
$577$	20.1462	0.838699	0.419349	−	0.907825i	$-0.362258\pi$
$577$	20.1462	0.838699	0.419349	+	0.907825i	$0.362258\pi$
$578$	0	0
$579$	−16.8288	−0.699379
$580$	0	0
$581$	−15.3234	−0.635720
$582$	0	0
$583$	−18.3869	−0.761506
$584$	0	0
$585$	−4.66058	−0.192691
$586$	0	0
$587$	9.60605	0.396484	0.198242	−	0.980153i	$-0.436477\pi$
$587$	9.60605	0.396484	0.198242	+	0.980153i	$0.436477\pi$
$588$	0	0
$589$	−3.54608	−0.146113
$590$	0	0
$591$	23.1207	0.951059
$592$	0	0
$593$	36.4249	1.49579	0.747895	−	0.663817i	$-0.231064\pi$
$593$	36.4249	1.49579	0.747895	+	0.663817i	$0.231064\pi$
$594$	0	0
$595$	9.88734	0.405341
$596$	0	0
$597$	12.1762	0.498340
$598$	0	0
$599$	−12.6358	−0.516284	−0.258142	−	0.966107i	$-0.583110\pi$
$599$	−12.6358	−0.516284	−0.258142	+	0.966107i	$0.583110\pi$
$600$	0	0
$601$	−2.95144	−0.120392	−0.0601960	−	0.998187i	$-0.519173\pi$
$601$	−2.95144	−0.120392	−0.0601960	+	0.998187i	$0.519173\pi$
$602$	0	0
$603$	6.14424	0.250213
$604$	0	0
$605$	50.2281	2.04206
$606$	0	0
$607$	3.09215	0.125507	0.0627533	−	0.998029i	$-0.480012\pi$
$607$	3.09215	0.125507	0.0627533	+	0.998029i	$0.480012\pi$
$608$	0	0
$609$	8.08812	0.327747
$610$	0	0
$611$	−5.49281	−0.222215
$612$	0	0
$613$	−22.9195	−0.925709	−0.462855	−	0.886434i	$-0.653175\pi$
$613$	−22.9195	−0.925709	−0.462855	+	0.886434i	$0.653175\pi$
$614$	0	0
$615$	6.31932	0.254820
$616$	0	0
$617$	−13.8913	−0.559242	−0.279621	−	0.960110i	$-0.590209\pi$
$617$	−13.8913	−0.559242	−0.279621	+	0.960110i	$0.590209\pi$
$618$	0	0
$619$	42.8963	1.72415	0.862074	−	0.506783i	$-0.169165\pi$
$619$	42.8963	1.72415	0.862074	+	0.506783i	$0.169165\pi$
$620$	0	0
$621$	5.54608	0.222556
$622$	0	0
$623$	12.6541	0.506976
$624$	0	0
$625$	−30.1725	−1.20690
$626$	0	0
$627$	5.54608	0.221489
$628$	0	0
$629$	−22.3127	−0.889664
$630$	0	0
$631$	−39.0613	−1.55501	−0.777504	−	0.628878i	$-0.783514\pi$
$631$	−39.0613	−1.55501	−0.777504	+	0.628878i	$0.783514\pi$
$632$	0	0
$633$	−6.95597	−0.276475
$634$	0	0
$635$	−18.9364	−0.751468
$636$	0	0
$637$	−1.83340	−0.0726420
$638$	0	0
$639$	−8.18491	−0.323790
$640$	0	0
$641$	49.8109	1.96741	0.983705	−	0.179789i	$-0.0575416\pi$
$641$	49.8109	1.96741	0.983705	+	0.179789i	$0.0575416\pi$
$642$	0	0
$643$	−19.9187	−0.785515	−0.392758	−	0.919642i	$-0.628479\pi$
$643$	−19.9187	−0.785515	−0.392758	+	0.919642i	$0.628479\pi$
$644$	0	0
$645$	9.68383	0.381300
$646$	0	0
$647$	38.5332	1.51490	0.757448	−	0.652896i	$-0.226446\pi$
$647$	38.5332	1.51490	0.757448	+	0.652896i	$0.226446\pi$
$648$	0	0
$649$	61.5179	2.41479
$650$	0	0
$651$	3.54608	0.138982
$652$	0	0
$653$	−1.47322	−0.0576515	−0.0288257	−	0.999584i	$-0.509177\pi$
$653$	−1.47322	−0.0576515	−0.0288257	+	0.999584i	$0.509177\pi$
$654$	0	0
$655$	39.2380	1.53315
$656$	0	0
$657$	8.31932	0.324568
$658$	0	0
$659$	−35.4063	−1.37923	−0.689616	−	0.724175i	$-0.742221\pi$
$659$	−35.4063	−1.37923	−0.689616	+	0.724175i	$0.742221\pi$
$660$	0	0
$661$	12.2390	0.476043	0.238022	−	0.971260i	$-0.423501\pi$
$661$	12.2390	0.476043	0.238022	+	0.971260i	$0.423501\pi$
$662$	0	0
$663$	−7.13105	−0.276947
$664$	0	0
$665$	2.54204	0.0985762
$666$	0	0
$667$	−44.8573	−1.73688
$668$	0	0
$669$	−28.4092	−1.09836
$670$	0	0
$671$	46.7621	1.80523
$672$	0	0
$673$	−39.6565	−1.52864	−0.764322	−	0.644835i	$-0.776926\pi$
$673$	−39.6565	−1.52864	−0.764322	+	0.644835i	$0.776926\pi$
$674$	0	0
$675$	−1.46199	−0.0562719
$676$	0	0
$677$	−12.7354	−0.489463	−0.244731	−	0.969591i	$-0.578700\pi$
$677$	−12.7354	−0.489463	−0.244731	+	0.969591i	$0.578700\pi$
$678$	0	0
$679$	−15.1241	−0.580412
$680$	0	0
$681$	3.31311	0.126958
$682$	0	0
$683$	10.2019	0.390366	0.195183	−	0.980767i	$-0.437470\pi$
$683$	10.2019	0.390366	0.195183	+	0.980767i	$0.437470\pi$
$684$	0	0
$685$	−39.2845	−1.50098
$686$	0	0
$687$	−1.88776	−0.0720224
$688$	0	0
$689$	6.07825	0.231563
$690$	0	0
$691$	16.3435	0.621736	0.310868	−	0.950453i	$-0.399380\pi$
$691$	16.3435	0.621736	0.310868	+	0.950453i	$0.399380\pi$
$692$	0	0
$693$	−5.54608	−0.210678
$694$	0	0
$695$	26.2322	0.995043
$696$	0	0
$697$	9.66905	0.366241
$698$	0	0
$699$	−23.7227	−0.897276
$700$	0	0
$701$	19.9807	0.754660	0.377330	−	0.926079i	$-0.376842\pi$
$701$	19.9807	0.754660	0.377330	+	0.926079i	$0.376842\pi$
$702$	0	0
$703$	−5.73661	−0.216360
$704$	0	0
$705$	−7.61588	−0.286831
$706$	0	0
$707$	−14.9817	−0.563444
$708$	0	0
$709$	−2.38955	−0.0897414	−0.0448707	−	0.998993i	$-0.514288\pi$
$709$	−2.38955	−0.0897414	−0.0448707	+	0.998993i	$0.514288\pi$
$710$	0	0
$711$	4.79676	0.179893
$712$	0	0
$713$	−19.6668	−0.736527
$714$	0	0
$715$	−25.8479	−0.966659
$716$	0	0
$717$	21.9159	0.818464
$718$	0	0
$719$	−48.8139	−1.82045	−0.910225	−	0.414113i	$-0.864092\pi$
$719$	−48.8139	−1.82045	−0.910225	+	0.414113i	$0.864092\pi$
$720$	0	0
$721$	6.36176	0.236924
$722$	0	0
$723$	17.2445	0.641328
$724$	0	0
$725$	11.8247	0.439159
$726$	0	0
$727$	27.4034	1.01634	0.508168	−	0.861258i	$-0.330323\pi$
$727$	27.4034	1.01634	0.508168	+	0.861258i	$0.330323\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.8170	0.548027
$732$	0	0
$733$	16.5229	0.610288	0.305144	−	0.952306i	$-0.401295\pi$
$733$	16.5229	0.610288	0.305144	+	0.952306i	$0.401295\pi$
$734$	0	0
$735$	−2.54204	−0.0937647
$736$	0	0
$737$	34.0764	1.25522
$738$	0	0
$739$	22.6771	0.834192	0.417096	−	0.908863i	$-0.363048\pi$
$739$	22.6771	0.834192	0.417096	+	0.908863i	$0.363048\pi$
$740$	0	0
$741$	−1.83340	−0.0673516
$742$	0	0
$743$	−15.2185	−0.558313	−0.279156	−	0.960246i	$-0.590055\pi$
$743$	−15.2185	−0.558313	−0.279156	+	0.960246i	$0.590055\pi$
$744$	0	0
$745$	30.5250	1.11835
$746$	0	0
$747$	15.3234	0.560652
$748$	0	0
$749$	17.9296	0.655133
$750$	0	0
$751$	12.3674	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$751$	12.3674	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$752$	0	0
$753$	−3.72636	−0.135796
$754$	0	0
$755$	−48.4165	−1.76206
$756$	0	0
$757$	−14.9271	−0.542536	−0.271268	−	0.962504i	$-0.587443\pi$
$757$	−14.9271	−0.542536	−0.271268	+	0.962504i	$0.587443\pi$
$758$	0	0
$759$	30.7590	1.11648
$760$	0	0
$761$	15.2563	0.553041	0.276520	−	0.961008i	$-0.410819\pi$
$761$	15.2563	0.553041	0.276520	+	0.961008i	$0.410819\pi$
$762$	0	0
$763$	4.00806	0.145102
$764$	0	0
$765$	−9.88734	−0.357477
$766$	0	0
$767$	−20.3364	−0.734303
$768$	0	0
$769$	−10.7563	−0.387883	−0.193941	−	0.981013i	$-0.562127\pi$
$769$	−10.7563	−0.387883	−0.193941	+	0.981013i	$0.562127\pi$
$770$	0	0
$771$	−4.42999	−0.159542
$772$	0	0
$773$	−13.8026	−0.496444	−0.248222	−	0.968703i	$-0.579846\pi$
$773$	−13.8026	−0.496444	−0.248222	+	0.968703i	$0.579846\pi$
$774$	0	0
$775$	5.18432	0.186226
$776$	0	0
$777$	5.73661	0.205800
$778$	0	0
$779$	2.48592	0.0890674
$780$	0	0
$781$	−45.3941	−1.62433
$782$	0	0
$783$	−8.08812	−0.289046
$784$	0	0
$785$	15.0748	0.538043
$786$	0	0
$787$	19.6101	0.699023	0.349512	−	0.936932i	$-0.386347\pi$
$787$	19.6101	0.699023	0.349512	+	0.936932i	$0.386347\pi$
$788$	0	0
$789$	2.23297	0.0794958
$790$	0	0
$791$	−1.96781	−0.0699673
$792$	0	0
$793$	−15.4584	−0.548945
$794$	0	0
$795$	8.42761	0.298897
$796$	0	0
$797$	−25.2766	−0.895344	−0.447672	−	0.894198i	$-0.647747\pi$
$797$	−25.2766	−0.895344	−0.447672	+	0.894198i	$0.647747\pi$
$798$	0	0
$799$	−11.6529	−0.412250
$800$	0	0
$801$	−12.6541	−0.447111
$802$	0	0
$803$	46.1396	1.62823
$804$	0	0
$805$	14.0984	0.496902
$806$	0	0
$807$	−4.87505	−0.171610
$808$	0	0
$809$	36.3718	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$809$	36.3718	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$810$	0	0
$811$	−38.3409	−1.34633	−0.673165	−	0.739492i	$-0.735066\pi$
$811$	−38.3409	−1.34633	−0.673165	+	0.739492i	$0.735066\pi$
$812$	0	0
$813$	−16.5519	−0.580500
$814$	0	0
$815$	5.87426	0.205766
$816$	0	0
$817$	3.80947	0.133276
$818$	0	0
$819$	1.83340	0.0640642
$820$	0	0
$821$	7.02009	0.245003	0.122501	−	0.992468i	$-0.460908\pi$
$821$	7.02009	0.245003	0.122501	+	0.992468i	$0.460908\pi$
$822$	0	0
$823$	18.4767	0.644056	0.322028	−	0.946730i	$-0.395635\pi$
$823$	18.4767	0.644056	0.322028	+	0.946730i	$0.395635\pi$
$824$	0	0
$825$	−8.10829	−0.282294
$826$	0	0
$827$	30.2879	1.05321	0.526606	−	0.850109i	$-0.323464\pi$
$827$	30.2879	1.05321	0.526606	+	0.850109i	$0.323464\pi$
$828$	0	0
$829$	5.07259	0.176178	0.0880891	−	0.996113i	$-0.471924\pi$
$829$	5.07259	0.176178	0.0880891	+	0.996113i	$0.471924\pi$
$830$	0	0
$831$	−28.0841	−0.974226
$832$	0	0
$833$	−3.88952	−0.134764
$834$	0	0
$835$	23.9392	0.828449
$836$	0	0
$837$	−3.54608	−0.122570
$838$	0	0
$839$	−5.10458	−0.176230	−0.0881149	−	0.996110i	$-0.528084\pi$
$839$	−5.10458	−0.176230	−0.0881149	+	0.996110i	$0.528084\pi$
$840$	0	0
$841$	36.4177	1.25578
$842$	0	0
$843$	−11.2674	−0.388071
$844$	0	0
$845$	−24.5019	−0.842889
$846$	0	0
$847$	−19.7590	−0.678926
$848$	0	0
$849$	15.7952	0.542089
$850$	0	0
$851$	−31.8157	−1.09063
$852$	0	0
$853$	−38.4821	−1.31760	−0.658800	−	0.752318i	$-0.728936\pi$
$853$	−38.4821	−1.31760	−0.658800	+	0.752318i	$0.728936\pi$
$854$	0	0
$855$	−2.54204	−0.0869361
$856$	0	0
$857$	−47.8034	−1.63293	−0.816466	−	0.577394i	$-0.804070\pi$
$857$	−47.8034	−1.63293	−0.816466	+	0.577394i	$0.804070\pi$
$858$	0	0
$859$	10.5654	0.360486	0.180243	−	0.983622i	$-0.442312\pi$
$859$	10.5654	0.360486	0.180243	+	0.983622i	$0.442312\pi$
$860$	0	0
$861$	−2.48592	−0.0847200
$862$	0	0
$863$	−1.30828	−0.0445344	−0.0222672	−	0.999752i	$-0.507088\pi$
$863$	−1.30828	−0.0445344	−0.0222672	+	0.999752i	$0.507088\pi$
$864$	0	0
$865$	−34.5161	−1.17358
$866$	0	0
$867$	1.87161	0.0635633
$868$	0	0
$869$	26.6032	0.902452
$870$	0	0
$871$	−11.2649	−0.381695
$872$	0	0
$873$	15.1241	0.511875
$874$	0	0
$875$	8.99378	0.304045
$876$	0	0
$877$	47.8543	1.61592	0.807962	−	0.589235i	$-0.200571\pi$
$877$	47.8543	1.61592	0.807962	+	0.589235i	$0.200571\pi$
$878$	0	0
$879$	−26.4573	−0.892384
$880$	0	0
$881$	45.8583	1.54501	0.772503	−	0.635011i	$-0.219005\pi$
$881$	45.8583	1.54501	0.772503	+	0.635011i	$0.219005\pi$
$882$	0	0
$883$	−28.6217	−0.963196	−0.481598	−	0.876392i	$-0.659944\pi$
$883$	−28.6217	−0.963196	−0.481598	+	0.876392i	$0.659944\pi$
$884$	0	0
$885$	−28.1967	−0.947823
$886$	0	0
$887$	58.2209	1.95487	0.977434	−	0.211243i	$-0.0677511\pi$
$887$	58.2209	1.95487	0.977434	+	0.211243i	$0.0677511\pi$
$888$	0	0
$889$	7.44928	0.249841
$890$	0	0
$891$	5.54608	0.185801
$892$	0	0
$893$	−2.99597	−0.100256
$894$	0	0
$895$	−3.96498	−0.132535
$896$	0	0
$897$	−10.1682	−0.339506
$898$	0	0
$899$	28.6811	0.956568
$900$	0	0
$901$	12.8949	0.429591
$902$	0	0
$903$	−3.80947	−0.126771
$904$	0	0
$905$	47.1126	1.56608
$906$	0	0
$907$	34.9591	1.16080	0.580399	−	0.814332i	$-0.302896\pi$
$907$	34.9591	1.16080	0.580399	+	0.814332i	$0.302896\pi$
$908$	0	0
$909$	14.9817	0.496911
$910$	0	0
$911$	−14.7994	−0.490325	−0.245162	−	0.969482i	$-0.578841\pi$
$911$	−14.7994	−0.490325	−0.245162	+	0.969482i	$0.578841\pi$
$912$	0	0
$913$	84.9845	2.81258
$914$	0	0
$915$	−21.4334	−0.708567
$916$	0	0
$917$	−15.4356	−0.509728
$918$	0	0
$919$	−0.762136	−0.0251405	−0.0125703	−	0.999921i	$-0.504001\pi$
$919$	−0.762136	−0.0251405	−0.0125703	+	0.999921i	$0.504001\pi$
$920$	0	0
$921$	11.7875	0.388412
$922$	0	0
$923$	15.0062	0.493936
$924$	0	0
$925$	8.38685	0.275758
$926$	0	0
$927$	−6.36176	−0.208948
$928$	0	0
$929$	26.5680	0.871667	0.435833	−	0.900027i	$-0.356454\pi$
$929$	26.5680	0.871667	0.435833	+	0.900027i	$0.356454\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	2.39938	0.0785521
$934$	0	0
$935$	−54.8359	−1.79333
$936$	0	0
$937$	−28.7590	−0.939514	−0.469757	−	0.882796i	$-0.655658\pi$
$937$	−28.7590	−0.939514	−0.469757	+	0.882796i	$0.655658\pi$
$938$	0	0
$939$	−6.66058	−0.217360
$940$	0	0
$941$	−34.1732	−1.11402	−0.557008	−	0.830507i	$-0.688051\pi$
$941$	−34.1732	−1.11402	−0.557008	+	0.830507i	$0.688051\pi$
$942$	0	0
$943$	13.7871	0.448970
$944$	0	0
$945$	2.54204	0.0826927
$946$	0	0
$947$	40.5626	1.31811	0.659054	−	0.752096i	$-0.270957\pi$
$947$	40.5626	1.31811	0.659054	+	0.752096i	$0.270957\pi$
$948$	0	0
$949$	−15.2526	−0.495122
$950$	0	0
$951$	18.3376	0.594638
$952$	0	0
$953$	−38.6589	−1.25229	−0.626143	−	0.779709i	$-0.715367\pi$
$953$	−38.6589	−1.25229	−0.626143	+	0.779709i	$0.715367\pi$
$954$	0	0
$955$	59.4357	1.92329
$956$	0	0
$957$	−44.8573	−1.45003
$958$	0	0
$959$	15.4539	0.499033
$960$	0	0
$961$	−18.4254	−0.594366
$962$	0	0
$963$	−17.9296	−0.577773
$964$	0	0
$965$	42.7794	1.37712
$966$	0	0
$967$	−6.83153	−0.219687	−0.109844	−	0.993949i	$-0.535035\pi$
$967$	−6.83153	−0.219687	−0.109844	+	0.993949i	$0.535035\pi$
$968$	0	0
$969$	−3.88952	−0.124949
$970$	0	0
$971$	−57.7215	−1.85237	−0.926186	−	0.377068i	$-0.876932\pi$
$971$	−57.7215	−1.85237	−0.926186	+	0.377068i	$0.876932\pi$
$972$	0	0
$973$	−10.3193	−0.330822
$974$	0	0
$975$	2.68041	0.0858418
$976$	0	0
$977$	38.7294	1.23906	0.619532	−	0.784972i	$-0.287323\pi$
$977$	38.7294	1.23906	0.619532	+	0.784972i	$0.287323\pi$
$978$	0	0
$979$	−70.1806	−2.24298
$980$	0	0
$981$	−4.00806	−0.127968
$982$	0	0
$983$	−44.7814	−1.42831	−0.714153	−	0.699990i	$-0.753188\pi$
$983$	−44.7814	−1.42831	−0.714153	+	0.699990i	$0.753188\pi$
$984$	0	0
$985$	−58.7739	−1.87269
$986$	0	0
$987$	2.99597	0.0953627
$988$	0	0
$989$	21.1276	0.671818
$990$	0	0
$991$	−22.1647	−0.704086	−0.352043	−	0.935984i	$-0.614513\pi$
$991$	−22.1647	−0.704086	−0.352043	+	0.935984i	$0.614513\pi$
$992$	0	0
$993$	−3.09099	−0.0980897
$994$	0	0
$995$	−30.9525	−0.981261
$996$	0	0
$997$	5.54303	0.175549	0.0877747	−	0.996140i	$-0.472024\pi$
$997$	5.54303	0.175549	0.0877747	+	0.996140i	$0.472024\pi$
$998$	0	0
$999$	−5.73661	−0.181498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cc.1.5		5
4.3	odd	2		399.2.a.f.1.4	✓	5
12.11	even	2		1197.2.a.p.1.2		5
20.19	odd	2		9975.2.a.bq.1.2		5
28.27	even	2		2793.2.a.be.1.4		5
76.75	even	2		7581.2.a.x.1.2		5
84.83	odd	2		8379.2.a.ce.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.f.1.4	✓	5	4.3	odd	2
1197.2.a.p.1.2		5	12.11	even	2
2793.2.a.be.1.4		5	28.27	even	2
6384.2.a.cc.1.5		5	1.1	even	1	trivial
7581.2.a.x.1.2		5	76.75	even	2
8379.2.a.ce.1.2		5	84.83	odd	2
9975.2.a.bq.1.2		5	20.19	odd	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.54204 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.54204 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.54608 q^{11} -1.83340 q^{13} -2.54204 q^{15} -3.88952 q^{17} -1.00000 q^{19} +1.00000 q^{21} -5.54608 q^{23} +1.46199 q^{25} -1.00000 q^{27} +8.08812 q^{29} +3.54608 q^{31} -5.54608 q^{33} -2.54204 q^{35} +5.73661 q^{37} +1.83340 q^{39} -2.48592 q^{41} -3.80947 q^{43} +2.54204 q^{45} +2.99597 q^{47} +1.00000 q^{49} +3.88952 q^{51} -3.31529 q^{53} +14.0984 q^{55} +1.00000 q^{57} +11.0922 q^{59} +8.43157 q^{61} -1.00000 q^{63} -4.66058 q^{65} +6.14424 q^{67} +5.54608 q^{69} -8.18491 q^{71} +8.31932 q^{73} -1.46199 q^{75} -5.54608 q^{77} +4.79676 q^{79} +1.00000 q^{81} +15.3234 q^{83} -9.88734 q^{85} -8.08812 q^{87} -12.6541 q^{89} +1.83340 q^{91} -3.54608 q^{93} -2.54204 q^{95} +15.1241 q^{97} +5.54608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} + 5 q^{21} - 2 q^{23} + 11 q^{25} - 5 q^{27} - 8 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} - 20 q^{43} - 2 q^{45} + 26 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} + 5 q^{57} + 4 q^{59} + 10 q^{61} - 5 q^{63} - 4 q^{65} - 10 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} - 11 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 34 q^{83} - 36 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{93} + 2 q^{95} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 2 * q^5 - 5 * q^7 + 5 * q^9 + 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 + 5 * q^21 - 2 * q^23 + 11 * q^25 - 5 * q^27 - 8 * q^31 - 2 * q^33 + 2 * q^35 + 2 * q^37 - 8 * q^39 + 2 * q^41 - 20 * q^43 - 2 * q^45 + 26 * q^47 + 5 * q^49 + 2 * q^51 + 4 * q^53 + 4 * q^55 + 5 * q^57 + 4 * q^59 + 10 * q^61 - 5 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 - 10 * q^71 + 10 * q^73 - 11 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 34 * q^83 - 36 * q^85 + 10 * q^89 - 8 * q^91 + 8 * q^93 + 2 * q^95 - 16 * q^97 + 2 * q^99

Introduction

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