LMFDB - Embedded newform 3380.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80194 q^{3} +1.00000 q^{5} +1.80194 q^{7} +0.246980 q^{9} +O(q^{10})$	$q-1.80194 q^{3} +1.00000 q^{5} +1.80194 q^{7} +0.246980 q^{9} -1.55496 q^{11} -1.80194 q^{15} +0.911854 q^{17} -1.69202 q^{19} -3.24698 q^{21} +0.246980 q^{23} +1.00000 q^{25} +4.96077 q^{27} -5.29590 q^{29} +1.08815 q^{31} +2.80194 q^{33} +1.80194 q^{35} -3.00000 q^{37} -6.96077 q^{41} +2.26875 q^{43} +0.246980 q^{45} -8.52111 q^{47} -3.75302 q^{49} -1.64310 q^{51} +6.92692 q^{53} -1.55496 q^{55} +3.04892 q^{57} +14.7017 q^{59} -9.55496 q^{61} +0.445042 q^{63} -5.21983 q^{67} -0.445042 q^{69} +5.74094 q^{71} -14.5918 q^{73} -1.80194 q^{75} -2.80194 q^{77} +7.75063 q^{79} -9.67994 q^{81} -4.94869 q^{83} +0.911854 q^{85} +9.54288 q^{87} +11.9095 q^{89} -1.96077 q^{93} -1.69202 q^{95} +9.17390 q^{97} -0.384043 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10}) $ 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9	$ 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} - 5 q^{11} - q^{15} - q^{17} - 5 q^{21} - 4 q^{23} + 3 q^{25} + 2 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} + q^{35} - 9 q^{37} - 8 q^{41} - q^{43} - 4 q^{45} - 10 q^{47} - 16 q^{49} - 9 q^{51} - 8 q^{53} - 5 q^{55} + 17 q^{59} - 29 q^{61} + q^{63} - 17 q^{67} - q^{69} + 3 q^{71} - 16 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 5 q^{81} + 17 q^{83} - q^{85} + 10 q^{87} - 9 q^{89} + 7 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9 - 5 * q^11 - q^15 - q^17 - 5 * q^21 - 4 * q^23 + 3 * q^25 + 2 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 + q^35 - 9 * q^37 - 8 * q^41 - q^43 - 4 * q^45 - 10 * q^47 - 16 * q^49 - 9 * q^51 - 8 * q^53 - 5 * q^55 + 17 * q^59 - 29 * q^61 + q^63 - 17 * q^67 - q^69 + 3 * q^71 - 16 * q^73 - q^75 - 4 * q^77 - 13 * q^79 - 5 * q^81 + 17 * q^83 - q^85 + 10 * q^87 - 9 * q^89 + 7 * q^93 - 6 * q^97 + 9 * 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
$3$	−1.80194	−1.04035	−0.520175	−	0.854060i	$-0.674133\pi$
$3$	−1.80194	−1.04035	−0.520175	+	0.854060i	$0.674133\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.80194	0.681068	0.340534	−	0.940232i	$-0.389392\pi$
$7$	1.80194	0.681068	0.340534	+	0.940232i	$0.389392\pi$
$8$	0	0
$9$	0.246980	0.0823265
$10$	0	0
$11$	−1.55496	−0.468838	−0.234419	−	0.972136i	$-0.575319\pi$
$11$	−1.55496	−0.468838	−0.234419	+	0.972136i	$0.575319\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.80194	−0.465258
$16$	0	0
$17$	0.911854	0.221157	0.110579	−	0.993867i	$-0.464730\pi$
$17$	0.911854	0.221157	0.110579	+	0.993867i	$0.464730\pi$
$18$	0	0
$19$	−1.69202	−0.388176	−0.194088	−	0.980984i	$-0.562175\pi$
$19$	−1.69202	−0.388176	−0.194088	+	0.980984i	$0.562175\pi$
$20$	0	0
$21$	−3.24698	−0.708549
$22$	0	0
$23$	0.246980	0.0514988	0.0257494	−	0.999668i	$-0.491803\pi$
$23$	0.246980	0.0514988	0.0257494	+	0.999668i	$0.491803\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.96077	0.954701
$28$	0	0
$29$	−5.29590	−0.983423	−0.491712	−	0.870758i	$-0.663629\pi$
$29$	−5.29590	−0.983423	−0.491712	+	0.870758i	$0.663629\pi$
$30$	0	0
$31$	1.08815	0.195437	0.0977184	−	0.995214i	$-0.468846\pi$
$31$	1.08815	0.195437	0.0977184	+	0.995214i	$0.468846\pi$
$32$	0	0
$33$	2.80194	0.487755
$34$	0	0
$35$	1.80194	0.304583
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.96077	−1.08709	−0.543545	−	0.839380i	$-0.682918\pi$
$41$	−6.96077	−1.08709	−0.543545	+	0.839380i	$0.682918\pi$
$42$	0	0
$43$	2.26875	0.345981	0.172991	−	0.984923i	$-0.444657\pi$
$43$	2.26875	0.345981	0.172991	+	0.984923i	$0.444657\pi$
$44$	0	0
$45$	0.246980	0.0368175
$46$	0	0
$47$	−8.52111	−1.24293	−0.621466	−	0.783441i	$-0.713462\pi$
$47$	−8.52111	−1.24293	−0.621466	+	0.783441i	$0.713462\pi$
$48$	0	0
$49$	−3.75302	−0.536146
$50$	0	0
$51$	−1.64310	−0.230081
$52$	0	0
$53$	6.92692	0.951486	0.475743	−	0.879584i	$-0.342179\pi$
$53$	6.92692	0.951486	0.475743	+	0.879584i	$0.342179\pi$
$54$	0	0
$55$	−1.55496	−0.209671
$56$	0	0
$57$	3.04892	0.403839
$58$	0	0
$59$	14.7017	1.91400	0.957000	−	0.290089i	$-0.0936848\pi$
$59$	14.7017	1.91400	0.957000	+	0.290089i	$0.0936848\pi$
$60$	0	0
$61$	−9.55496	−1.22339	−0.611694	−	0.791095i	$-0.709511\pi$
$61$	−9.55496	−1.22339	−0.611694	+	0.791095i	$0.709511\pi$
$62$	0	0
$63$	0.445042	0.0560700
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.21983	−0.637704	−0.318852	−	0.947805i	$-0.603297\pi$
$67$	−5.21983	−0.637704	−0.318852	+	0.947805i	$0.603297\pi$
$68$	0	0
$69$	−0.445042	−0.0535767
$70$	0	0
$71$	5.74094	0.681324	0.340662	−	0.940186i	$-0.389349\pi$
$71$	5.74094	0.681324	0.340662	+	0.940186i	$0.389349\pi$
$72$	0	0
$73$	−14.5918	−1.70784	−0.853920	−	0.520404i	$-0.825781\pi$
$73$	−14.5918	−1.70784	−0.853920	+	0.520404i	$0.825781\pi$
$74$	0	0
$75$	−1.80194	−0.208070
$76$	0	0
$77$	−2.80194	−0.319310
$78$	0	0
$79$	7.75063	0.872014	0.436007	−	0.899943i	$-0.356392\pi$
$79$	7.75063	0.872014	0.436007	+	0.899943i	$0.356392\pi$
$80$	0	0
$81$	−9.67994	−1.07555
$82$	0	0
$83$	−4.94869	−0.543189	−0.271595	−	0.962412i	$-0.587551\pi$
$83$	−4.94869	−0.543189	−0.271595	+	0.962412i	$0.587551\pi$
$84$	0	0
$85$	0.911854	0.0989045
$86$	0	0
$87$	9.54288	1.02310
$88$	0	0
$89$	11.9095	1.26240	0.631200	−	0.775620i	$-0.282563\pi$
$89$	11.9095	1.26240	0.631200	+	0.775620i	$0.282563\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.96077	−0.203323
$94$	0	0
$95$	−1.69202	−0.173598
$96$	0	0
$97$	9.17390	0.931468	0.465734	−	0.884925i	$-0.345790\pi$
$97$	9.17390	0.931468	0.465734	+	0.884925i	$0.345790\pi$
$98$	0	0
$99$	−0.384043	−0.0385978
$100$	0	0
$101$	−2.18060	−0.216978	−0.108489	−	0.994098i	$-0.534601\pi$
$101$	−2.18060	−0.216978	−0.108489	+	0.994098i	$0.534601\pi$
$102$	0	0
$103$	−9.58211	−0.944153	−0.472076	−	0.881558i	$-0.656495\pi$
$103$	−9.58211	−0.944153	−0.472076	+	0.881558i	$0.656495\pi$
$104$	0	0
$105$	−3.24698	−0.316873
$106$	0	0
$107$	6.03684	0.583603	0.291801	−	0.956479i	$-0.405745\pi$
$107$	6.03684	0.583603	0.291801	+	0.956479i	$0.405745\pi$
$108$	0	0
$109$	10.6703	1.02202	0.511012	−	0.859573i	$-0.329271\pi$
$109$	10.6703	1.02202	0.511012	+	0.859573i	$0.329271\pi$
$110$	0	0
$111$	5.40581	0.513097
$112$	0	0
$113$	−17.6136	−1.65694	−0.828472	−	0.560030i	$-0.810790\pi$
$113$	−17.6136	−1.65694	−0.828472	+	0.560030i	$0.810790\pi$
$114$	0	0
$115$	0.246980	0.0230310
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.64310	0.150623
$120$	0	0
$121$	−8.58211	−0.780191
$122$	0	0
$123$	12.5429	1.13095
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−21.4644	−1.90466	−0.952330	−	0.305071i	$-0.901320\pi$
$127$	−21.4644	−1.90466	−0.952330	+	0.305071i	$0.901320\pi$
$128$	0	0
$129$	−4.08815	−0.359941
$130$	0	0
$131$	7.94869	0.694480	0.347240	−	0.937776i	$-0.387119\pi$
$131$	7.94869	0.694480	0.347240	+	0.937776i	$0.387119\pi$
$132$	0	0
$133$	−3.04892	−0.264375
$134$	0	0
$135$	4.96077	0.426955
$136$	0	0
$137$	1.39075	0.118820	0.0594098	−	0.998234i	$-0.481078\pi$
$137$	1.39075	0.118820	0.0594098	+	0.998234i	$0.481078\pi$
$138$	0	0
$139$	20.1008	1.70493	0.852465	−	0.522785i	$-0.175107\pi$
$139$	20.1008	1.70493	0.852465	+	0.522785i	$0.175107\pi$
$140$	0	0
$141$	15.3545	1.29308
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.29590	−0.439800
$146$	0	0
$147$	6.76271	0.557779
$148$	0	0
$149$	−7.34050	−0.601357	−0.300679	−	0.953725i	$-0.597213\pi$
$149$	−7.34050	−0.601357	−0.300679	+	0.953725i	$0.597213\pi$
$150$	0	0
$151$	−8.46011	−0.688474	−0.344237	−	0.938883i	$-0.611862\pi$
$151$	−8.46011	−0.688474	−0.344237	+	0.938883i	$0.611862\pi$
$152$	0	0
$153$	0.225209	0.0182071
$154$	0	0
$155$	1.08815	0.0874020
$156$	0	0
$157$	−17.7681	−1.41805	−0.709024	−	0.705185i	$-0.750864\pi$
$157$	−17.7681	−1.41805	−0.709024	+	0.705185i	$0.750864\pi$
$158$	0	0
$159$	−12.4819	−0.989877
$160$	0	0
$161$	0.445042	0.0350742
$162$	0	0
$163$	−11.6286	−0.910825	−0.455412	−	0.890281i	$-0.650508\pi$
$163$	−11.6286	−0.910825	−0.455412	+	0.890281i	$0.650508\pi$
$164$	0	0
$165$	2.80194	0.218131
$166$	0	0
$167$	−9.20237	−0.712101	−0.356051	−	0.934467i	$-0.615877\pi$
$167$	−9.20237	−0.712101	−0.356051	+	0.934467i	$0.615877\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−0.417895	−0.0319572
$172$	0	0
$173$	−15.4601	−1.17541	−0.587705	−	0.809075i	$-0.699968\pi$
$173$	−15.4601	−1.17541	−0.587705	+	0.809075i	$0.699968\pi$
$174$	0	0
$175$	1.80194	0.136214
$176$	0	0
$177$	−26.4916	−1.99123
$178$	0	0
$179$	−21.0519	−1.57349	−0.786746	−	0.617276i	$-0.788236\pi$
$179$	−21.0519	−1.57349	−0.786746	+	0.617276i	$0.788236\pi$
$180$	0	0
$181$	−14.9608	−1.11203	−0.556013	−	0.831174i	$-0.687669\pi$
$181$	−14.9608	−1.11203	−0.556013	+	0.831174i	$0.687669\pi$
$182$	0	0
$183$	17.2174	1.27275
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−1.41789	−0.103687
$188$	0	0
$189$	8.93900	0.650217
$190$	0	0
$191$	3.15452	0.228253	0.114127	−	0.993466i	$-0.463593\pi$
$191$	3.15452	0.228253	0.114127	+	0.993466i	$0.463593\pi$
$192$	0	0
$193$	4.04892	0.291447	0.145724	−	0.989325i	$-0.453449\pi$
$193$	4.04892	0.291447	0.145724	+	0.989325i	$0.453449\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.20775	−0.371037	−0.185518	−	0.982641i	$-0.559396\pi$
$197$	−5.20775	−0.371037	−0.185518	+	0.982641i	$0.559396\pi$
$198$	0	0
$199$	−22.5013	−1.59507	−0.797536	−	0.603272i	$-0.793864\pi$
$199$	−22.5013	−1.59507	−0.797536	+	0.603272i	$0.793864\pi$
$200$	0	0
$201$	9.40581	0.663435
$202$	0	0
$203$	−9.54288	−0.669779
$204$	0	0
$205$	−6.96077	−0.486161
$206$	0	0
$207$	0.0609989	0.00423972
$208$	0	0
$209$	2.63102	0.181992
$210$	0	0
$211$	−0.0435405	−0.00299745	−0.00149873	−	0.999999i	$-0.500477\pi$
$211$	−0.0435405	−0.00299745	−0.00149873	+	0.999999i	$0.500477\pi$
$212$	0	0
$213$	−10.3448	−0.708815
$214$	0	0
$215$	2.26875	0.154727
$216$	0	0
$217$	1.96077	0.133106
$218$	0	0
$219$	26.2935	1.77675
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0911	1.07754	0.538771	−	0.842452i	$-0.318889\pi$
$223$	16.0911	1.07754	0.538771	+	0.842452i	$0.318889\pi$
$224$	0	0
$225$	0.246980	0.0164653
$226$	0	0
$227$	1.08575	0.0720640	0.0360320	−	0.999351i	$-0.488528\pi$
$227$	1.08575	0.0720640	0.0360320	+	0.999351i	$0.488528\pi$
$228$	0	0
$229$	25.2620	1.66936	0.834681	−	0.550733i	$-0.185652\pi$
$229$	25.2620	1.66936	0.834681	+	0.550733i	$0.185652\pi$
$230$	0	0
$231$	5.04892	0.332194
$232$	0	0
$233$	−23.9879	−1.57150	−0.785750	−	0.618544i	$-0.787723\pi$
$233$	−23.9879	−1.57150	−0.785750	+	0.618544i	$0.787723\pi$
$234$	0	0
$235$	−8.52111	−0.555856
$236$	0	0
$237$	−13.9661	−0.907199
$238$	0	0
$239$	−3.88769	−0.251474	−0.125737	−	0.992064i	$-0.540130\pi$
$239$	−3.88769	−0.251474	−0.125737	+	0.992064i	$0.540130\pi$
$240$	0	0
$241$	−24.5894	−1.58394	−0.791971	−	0.610558i	$-0.790945\pi$
$241$	−24.5894	−1.58394	−0.791971	+	0.610558i	$0.790945\pi$
$242$	0	0
$243$	2.56033	0.164246
$244$	0	0
$245$	−3.75302	−0.239772
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.91723	0.565107
$250$	0	0
$251$	−14.5090	−0.915802	−0.457901	−	0.889003i	$-0.651399\pi$
$251$	−14.5090	−0.915802	−0.457901	+	0.889003i	$0.651399\pi$
$252$	0	0
$253$	−0.384043	−0.0241446
$254$	0	0
$255$	−1.64310	−0.102895
$256$	0	0
$257$	−5.19700	−0.324180	−0.162090	−	0.986776i	$-0.551823\pi$
$257$	−5.19700	−0.324180	−0.162090	+	0.986776i	$0.551823\pi$
$258$	0	0
$259$	−5.40581	−0.335901
$260$	0	0
$261$	−1.30798	−0.0809618
$262$	0	0
$263$	7.86294	0.484849	0.242425	−	0.970170i	$-0.422057\pi$
$263$	7.86294	0.484849	0.242425	+	0.970170i	$0.422057\pi$
$264$	0	0
$265$	6.92692	0.425517
$266$	0	0
$267$	−21.4601	−1.31334
$268$	0	0
$269$	6.81700	0.415640	0.207820	−	0.978167i	$-0.433363\pi$
$269$	6.81700	0.415640	0.207820	+	0.978167i	$0.433363\pi$
$270$	0	0
$271$	−16.2228	−0.985466	−0.492733	−	0.870181i	$-0.664002\pi$
$271$	−16.2228	−0.985466	−0.492733	+	0.870181i	$0.664002\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.55496	−0.0937675
$276$	0	0
$277$	2.15644	0.129568	0.0647840	−	0.997899i	$-0.479364\pi$
$277$	2.15644	0.129568	0.0647840	+	0.997899i	$0.479364\pi$
$278$	0	0
$279$	0.268750	0.0160896
$280$	0	0
$281$	−10.7235	−0.639709	−0.319855	−	0.947467i	$-0.603634\pi$
$281$	−10.7235	−0.639709	−0.319855	+	0.947467i	$0.603634\pi$
$282$	0	0
$283$	−5.50902	−0.327478	−0.163739	−	0.986504i	$-0.552355\pi$
$283$	−5.50902	−0.327478	−0.163739	+	0.986504i	$0.552355\pi$
$284$	0	0
$285$	3.04892	0.180602
$286$	0	0
$287$	−12.5429	−0.740383
$288$	0	0
$289$	−16.1685	−0.951090
$290$	0	0
$291$	−16.5308	−0.969052
$292$	0	0
$293$	−1.80864	−0.105662	−0.0528310	−	0.998603i	$-0.516824\pi$
$293$	−1.80864	−0.105662	−0.0528310	+	0.998603i	$0.516824\pi$
$294$	0	0
$295$	14.7017	0.855967
$296$	0	0
$297$	−7.71379	−0.447600
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.08815	0.235637
$302$	0	0
$303$	3.92931	0.225733
$304$	0	0
$305$	−9.55496	−0.547115
$306$	0	0
$307$	−10.9414	−0.624458	−0.312229	−	0.950007i	$-0.601076\pi$
$307$	−10.9414	−0.624458	−0.312229	+	0.950007i	$0.601076\pi$
$308$	0	0
$309$	17.2664	0.982249
$310$	0	0
$311$	−2.95646	−0.167645	−0.0838227	−	0.996481i	$-0.526713\pi$
$311$	−2.95646	−0.167645	−0.0838227	+	0.996481i	$0.526713\pi$
$312$	0	0
$313$	−1.17523	−0.0664278	−0.0332139	−	0.999448i	$-0.510574\pi$
$313$	−1.17523	−0.0664278	−0.0332139	+	0.999448i	$0.510574\pi$
$314$	0	0
$315$	0.445042	0.0250753
$316$	0	0
$317$	17.4620	0.980765	0.490383	−	0.871507i	$-0.336857\pi$
$317$	17.4620	0.980765	0.490383	+	0.871507i	$0.336857\pi$
$318$	0	0
$319$	8.23490	0.461066
$320$	0	0
$321$	−10.8780	−0.607151
$322$	0	0
$323$	−1.54288	−0.0858479
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−19.2271	−1.06326
$328$	0	0
$329$	−15.3545	−0.846521
$330$	0	0
$331$	11.7289	0.644676	0.322338	−	0.946625i	$-0.395531\pi$
$331$	11.7289	0.644676	0.322338	+	0.946625i	$0.395531\pi$
$332$	0	0
$333$	−0.740939	−0.0406032
$334$	0	0
$335$	−5.21983	−0.285190
$336$	0	0
$337$	−28.1879	−1.53549	−0.767746	−	0.640754i	$-0.778622\pi$
$337$	−28.1879	−1.53549	−0.767746	+	0.640754i	$0.778622\pi$
$338$	0	0
$339$	31.7385	1.72380
$340$	0	0
$341$	−1.69202	−0.0916281
$342$	0	0
$343$	−19.3763	−1.04622
$344$	0	0
$345$	−0.445042	−0.0239602
$346$	0	0
$347$	4.92394	0.264331	0.132165	−	0.991228i	$-0.457807\pi$
$347$	4.92394	0.264331	0.132165	+	0.991228i	$0.457807\pi$
$348$	0	0
$349$	14.3220	0.766638	0.383319	−	0.923616i	$-0.374781\pi$
$349$	14.3220	0.766638	0.383319	+	0.923616i	$0.374781\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.57242	−0.243365	−0.121683	−	0.992569i	$-0.538829\pi$
$353$	−4.57242	−0.243365	−0.121683	+	0.992569i	$0.538829\pi$
$354$	0	0
$355$	5.74094	0.304697
$356$	0	0
$357$	−2.96077	−0.156701
$358$	0	0
$359$	29.0315	1.53222	0.766111	−	0.642708i	$-0.222189\pi$
$359$	29.0315	1.53222	0.766111	+	0.642708i	$0.222189\pi$
$360$	0	0
$361$	−16.1371	−0.849319
$362$	0	0
$363$	15.4644	0.811672
$364$	0	0
$365$	−14.5918	−0.763769
$366$	0	0
$367$	27.8297	1.45270	0.726349	−	0.687326i	$-0.241216\pi$
$367$	27.8297	1.45270	0.726349	+	0.687326i	$0.241216\pi$
$368$	0	0
$369$	−1.71917	−0.0894963
$370$	0	0
$371$	12.4819	0.648027
$372$	0	0
$373$	−16.1588	−0.836673	−0.418336	−	0.908292i	$-0.637387\pi$
$373$	−16.1588	−0.836673	−0.418336	+	0.908292i	$0.637387\pi$
$374$	0	0
$375$	−1.80194	−0.0930517
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.05861	0.208477	0.104238	−	0.994552i	$-0.466760\pi$
$379$	4.05861	0.208477	0.104238	+	0.994552i	$0.466760\pi$
$380$	0	0
$381$	38.6775	1.98151
$382$	0	0
$383$	12.9269	0.660535	0.330267	−	0.943887i	$-0.392861\pi$
$383$	12.9269	0.660535	0.330267	+	0.943887i	$0.392861\pi$
$384$	0	0
$385$	−2.80194	−0.142800
$386$	0	0
$387$	0.560335	0.0284834
$388$	0	0
$389$	10.2784	0.521138	0.260569	−	0.965455i	$-0.416090\pi$
$389$	10.2784	0.521138	0.260569	+	0.965455i	$0.416090\pi$
$390$	0	0
$391$	0.225209	0.0113893
$392$	0	0
$393$	−14.3230	−0.722502
$394$	0	0
$395$	7.75063	0.389976
$396$	0	0
$397$	−15.0694	−0.756309	−0.378155	−	0.925742i	$-0.623441\pi$
$397$	−15.0694	−0.756309	−0.378155	+	0.925742i	$0.623441\pi$
$398$	0	0
$399$	5.49396	0.275042
$400$	0	0
$401$	−1.25129	−0.0624865	−0.0312433	−	0.999512i	$-0.509947\pi$
$401$	−1.25129	−0.0624865	−0.0312433	+	0.999512i	$0.509947\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−9.67994	−0.481000
$406$	0	0
$407$	4.66487	0.231229
$408$	0	0
$409$	−13.4849	−0.666783	−0.333392	−	0.942788i	$-0.608193\pi$
$409$	−13.4849	−0.666783	−0.333392	+	0.942788i	$0.608193\pi$
$410$	0	0
$411$	−2.50604	−0.123614
$412$	0	0
$413$	26.4916	1.30356
$414$	0	0
$415$	−4.94869	−0.242922
$416$	0	0
$417$	−36.2204	−1.77372
$418$	0	0
$419$	−8.84846	−0.432276	−0.216138	−	0.976363i	$-0.569346\pi$
$419$	−8.84846	−0.432276	−0.216138	+	0.976363i	$0.569346\pi$
$420$	0	0
$421$	−4.39373	−0.214137	−0.107069	−	0.994252i	$-0.534146\pi$
$421$	−4.39373	−0.214137	−0.107069	+	0.994252i	$0.534146\pi$
$422$	0	0
$423$	−2.10454	−0.102326
$424$	0	0
$425$	0.911854	0.0442314
$426$	0	0
$427$	−17.2174	−0.833210
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.7235	0.853710	0.426855	−	0.904320i	$-0.359621\pi$
$431$	17.7235	0.853710	0.426855	+	0.904320i	$0.359621\pi$
$432$	0	0
$433$	−7.81402	−0.375518	−0.187759	−	0.982215i	$-0.560122\pi$
$433$	−7.81402	−0.375518	−0.187759	+	0.982215i	$0.560122\pi$
$434$	0	0
$435$	9.54288	0.457546
$436$	0	0
$437$	−0.417895	−0.0199906
$438$	0	0
$439$	−5.64178	−0.269267	−0.134634	−	0.990895i	$-0.542986\pi$
$439$	−5.64178	−0.269267	−0.134634	+	0.990895i	$0.542986\pi$
$440$	0	0
$441$	−0.926919	−0.0441390
$442$	0	0
$443$	26.3491	1.25188	0.625942	−	0.779869i	$-0.284715\pi$
$443$	26.3491	1.25188	0.625942	+	0.779869i	$0.284715\pi$
$444$	0	0
$445$	11.9095	0.564563
$446$	0	0
$447$	13.2271	0.625622
$448$	0	0
$449$	39.0941	1.84497	0.922483	−	0.386039i	$-0.126157\pi$
$449$	39.0941	1.84497	0.922483	+	0.386039i	$0.126157\pi$
$450$	0	0
$451$	10.8237	0.509669
$452$	0	0
$453$	15.2446	0.716253
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.6025	−0.916968	−0.458484	−	0.888703i	$-0.651607\pi$
$457$	−19.6025	−0.916968	−0.458484	+	0.888703i	$0.651607\pi$
$458$	0	0
$459$	4.52350	0.211139
$460$	0	0
$461$	−40.5435	−1.88830	−0.944149	−	0.329520i	$-0.893113\pi$
$461$	−40.5435	−1.88830	−0.944149	+	0.329520i	$0.893113\pi$
$462$	0	0
$463$	31.4959	1.46374	0.731869	−	0.681446i	$-0.238648\pi$
$463$	31.4959	1.46374	0.731869	+	0.681446i	$0.238648\pi$
$464$	0	0
$465$	−1.96077	−0.0909286
$466$	0	0
$467$	35.6189	1.64825	0.824124	−	0.566409i	$-0.191668\pi$
$467$	35.6189	1.64825	0.824124	+	0.566409i	$0.191668\pi$
$468$	0	0
$469$	−9.40581	−0.434320
$470$	0	0
$471$	32.0170	1.47526
$472$	0	0
$473$	−3.52781	−0.162209
$474$	0	0
$475$	−1.69202	−0.0776353
$476$	0	0
$477$	1.71081	0.0783325
$478$	0	0
$479$	0.698726	0.0319256	0.0159628	−	0.999873i	$-0.494919\pi$
$479$	0.698726	0.0319256	0.0159628	+	0.999873i	$0.494919\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.801938	−0.0364894
$484$	0	0
$485$	9.17390	0.416565
$486$	0	0
$487$	9.49263	0.430152	0.215076	−	0.976597i	$-0.431000\pi$
$487$	9.49263	0.430152	0.215076	+	0.976597i	$0.431000\pi$
$488$	0	0
$489$	20.9541	0.947576
$490$	0	0
$491$	23.0261	1.03915	0.519576	−	0.854424i	$-0.326090\pi$
$491$	23.0261	1.03915	0.519576	+	0.854424i	$0.326090\pi$
$492$	0	0
$493$	−4.82908	−0.217491
$494$	0	0
$495$	−0.384043	−0.0172614
$496$	0	0
$497$	10.3448	0.464028
$498$	0	0
$499$	−15.5109	−0.694365	−0.347183	−	0.937798i	$-0.612862\pi$
$499$	−15.5109	−0.694365	−0.347183	+	0.937798i	$0.612862\pi$
$500$	0	0
$501$	16.5821	0.740834
$502$	0	0
$503$	−27.8538	−1.24194	−0.620971	−	0.783834i	$-0.713261\pi$
$503$	−27.8538	−1.24194	−0.620971	+	0.783834i	$0.713261\pi$
$504$	0	0
$505$	−2.18060	−0.0970356
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.6474	0.915181	0.457590	−	0.889163i	$-0.348713\pi$
$509$	20.6474	0.915181	0.457590	+	0.889163i	$0.348713\pi$
$510$	0	0
$511$	−26.2935	−1.16316
$512$	0	0
$513$	−8.39373	−0.370592
$514$	0	0
$515$	−9.58211	−0.422238
$516$	0	0
$517$	13.2500	0.582733
$518$	0	0
$519$	27.8582	1.22284
$520$	0	0
$521$	27.5700	1.20786	0.603932	−	0.797036i	$-0.293600\pi$
$521$	27.5700	1.20786	0.603932	+	0.797036i	$0.293600\pi$
$522$	0	0
$523$	14.2427	0.622788	0.311394	−	0.950281i	$-0.399204\pi$
$523$	14.2427	0.622788	0.311394	+	0.950281i	$0.399204\pi$
$524$	0	0
$525$	−3.24698	−0.141710
$526$	0	0
$527$	0.992230	0.0432222
$528$	0	0
$529$	−22.9390	−0.997348
$530$	0	0
$531$	3.63102	0.157573
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.03684	0.260995
$536$	0	0
$537$	37.9342	1.63698
$538$	0	0
$539$	5.83579	0.251365
$540$	0	0
$541$	26.9667	1.15939	0.579695	−	0.814833i	$-0.303172\pi$
$541$	26.9667	1.15939	0.579695	+	0.814833i	$0.303172\pi$
$542$	0	0
$543$	26.9584	1.15689
$544$	0	0
$545$	10.6703	0.457063
$546$	0	0
$547$	−32.7294	−1.39941	−0.699705	−	0.714432i	$-0.746685\pi$
$547$	−32.7294	−1.39941	−0.699705	+	0.714432i	$0.746685\pi$
$548$	0	0
$549$	−2.35988	−0.100717
$550$	0	0
$551$	8.96077	0.381742
$552$	0	0
$553$	13.9661	0.593901
$554$	0	0
$555$	5.40581	0.229464
$556$	0	0
$557$	−41.7603	−1.76944	−0.884721	−	0.466122i	$-0.845651\pi$
$557$	−41.7603	−1.76944	−0.884721	+	0.466122i	$0.845651\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.55496	0.107870
$562$	0	0
$563$	24.8834	1.04871	0.524355	−	0.851500i	$-0.324307\pi$
$563$	24.8834	1.04871	0.524355	+	0.851500i	$0.324307\pi$
$564$	0	0
$565$	−17.6136	−0.741008
$566$	0	0
$567$	−17.4426	−0.732522
$568$	0	0
$569$	28.2271	1.18334	0.591671	−	0.806179i	$-0.298468\pi$
$569$	28.2271	1.18334	0.591671	+	0.806179i	$0.298468\pi$
$570$	0	0
$571$	22.4316	0.938735	0.469367	−	0.883003i	$-0.344482\pi$
$571$	22.4316	0.938735	0.469367	+	0.883003i	$0.344482\pi$
$572$	0	0
$573$	−5.68425	−0.237463
$574$	0	0
$575$	0.246980	0.0102998
$576$	0	0
$577$	36.8437	1.53382	0.766911	−	0.641753i	$-0.221793\pi$
$577$	36.8437	1.53382	0.766911	+	0.641753i	$0.221793\pi$
$578$	0	0
$579$	−7.29590	−0.303207
$580$	0	0
$581$	−8.91723	−0.369949
$582$	0	0
$583$	−10.7711	−0.446092
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−44.4480	−1.83457	−0.917283	−	0.398236i	$-0.869623\pi$
$587$	−44.4480	−1.83457	−0.917283	+	0.398236i	$0.869623\pi$
$588$	0	0
$589$	−1.84117	−0.0758639
$590$	0	0
$591$	9.38404	0.386008
$592$	0	0
$593$	−20.6276	−0.847073	−0.423536	−	0.905879i	$-0.639211\pi$
$593$	−20.6276	−0.847073	−0.423536	+	0.905879i	$0.639211\pi$
$594$	0	0
$595$	1.64310	0.0673607
$596$	0	0
$597$	40.5459	1.65943
$598$	0	0
$599$	31.2674	1.27755	0.638776	−	0.769393i	$-0.279441\pi$
$599$	31.2674	1.27755	0.638776	+	0.769393i	$0.279441\pi$
$600$	0	0
$601$	−7.94438	−0.324058	−0.162029	−	0.986786i	$-0.551804\pi$
$601$	−7.94438	−0.324058	−0.162029	+	0.986786i	$0.551804\pi$
$602$	0	0
$603$	−1.28919	−0.0525000
$604$	0	0
$605$	−8.58211	−0.348912
$606$	0	0
$607$	−3.97690	−0.161417	−0.0807087	−	0.996738i	$-0.525718\pi$
$607$	−3.97690	−0.161417	−0.0807087	+	0.996738i	$0.525718\pi$
$608$	0	0
$609$	17.1957	0.696804
$610$	0	0
$611$	0	0
$612$	0	0
$613$	36.0062	1.45428	0.727139	−	0.686490i	$-0.240850\pi$
$613$	36.0062	1.45428	0.727139	+	0.686490i	$0.240850\pi$
$614$	0	0
$615$	12.5429	0.505778
$616$	0	0
$617$	−3.14808	−0.126737	−0.0633685	−	0.997990i	$-0.520184\pi$
$617$	−3.14808	−0.126737	−0.0633685	+	0.997990i	$0.520184\pi$
$618$	0	0
$619$	24.9594	1.00320	0.501602	−	0.865098i	$-0.332744\pi$
$619$	24.9594	1.00320	0.501602	+	0.865098i	$0.332744\pi$
$620$	0	0
$621$	1.22521	0.0491660
$622$	0	0
$623$	21.4601	0.859781
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.74094	−0.189335
$628$	0	0
$629$	−2.73556	−0.109074
$630$	0	0
$631$	−1.34960	−0.0537267	−0.0268634	−	0.999639i	$-0.508552\pi$
$631$	−1.34960	−0.0537267	−0.0268634	+	0.999639i	$0.508552\pi$
$632$	0	0
$633$	0.0784573	0.00311840
$634$	0	0
$635$	−21.4644	−0.851789
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.41789	0.0560911
$640$	0	0
$641$	30.7808	1.21577	0.607883	−	0.794026i	$-0.292019\pi$
$641$	30.7808	1.21577	0.607883	+	0.794026i	$0.292019\pi$
$642$	0	0
$643$	−0.528402	−0.0208381	−0.0104191	−	0.999946i	$-0.503317\pi$
$643$	−0.528402	−0.0208381	−0.0104191	+	0.999946i	$0.503317\pi$
$644$	0	0
$645$	−4.08815	−0.160971
$646$	0	0
$647$	−12.7114	−0.499737	−0.249868	−	0.968280i	$-0.580387\pi$
$647$	−12.7114	−0.499737	−0.249868	+	0.968280i	$0.580387\pi$
$648$	0	0
$649$	−22.8605	−0.897355
$650$	0	0
$651$	−3.53319	−0.138477
$652$	0	0
$653$	16.2241	0.634900	0.317450	−	0.948275i	$-0.397173\pi$
$653$	16.2241	0.634900	0.317450	+	0.948275i	$0.397173\pi$
$654$	0	0
$655$	7.94869	0.310581
$656$	0	0
$657$	−3.60388	−0.140601
$658$	0	0
$659$	−36.5478	−1.42370	−0.711850	−	0.702332i	$-0.752142\pi$
$659$	−36.5478	−1.42370	−0.711850	+	0.702332i	$0.752142\pi$
$660$	0	0
$661$	7.94810	0.309145	0.154573	−	0.987981i	$-0.450600\pi$
$661$	7.94810	0.309145	0.154573	+	0.987981i	$0.450600\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.04892	−0.118232
$666$	0	0
$667$	−1.30798	−0.0506451
$668$	0	0
$669$	−28.9952	−1.12102
$670$	0	0
$671$	14.8576	0.573570
$672$	0	0
$673$	−10.0771	−0.388445	−0.194222	−	0.980958i	$-0.562218\pi$
$673$	−10.0771	−0.388445	−0.194222	+	0.980958i	$0.562218\pi$
$674$	0	0
$675$	4.96077	0.190940
$676$	0	0
$677$	−37.3236	−1.43446	−0.717232	−	0.696835i	$-0.754591\pi$
$677$	−37.3236	−1.43446	−0.717232	+	0.696835i	$0.754591\pi$
$678$	0	0
$679$	16.5308	0.634394
$680$	0	0
$681$	−1.95646	−0.0749717
$682$	0	0
$683$	−10.8237	−0.414158	−0.207079	−	0.978324i	$-0.566396\pi$
$683$	−10.8237	−0.414158	−0.207079	+	0.978324i	$0.566396\pi$
$684$	0	0
$685$	1.39075	0.0531377
$686$	0	0
$687$	−45.5206	−1.73672
$688$	0	0
$689$	0	0
$690$	0	0
$691$	24.4838	0.931408	0.465704	−	0.884941i	$-0.345801\pi$
$691$	24.4838	0.931408	0.465704	+	0.884941i	$0.345801\pi$
$692$	0	0
$693$	−0.692021	−0.0262877
$694$	0	0
$695$	20.1008	0.762468
$696$	0	0
$697$	−6.34721	−0.240418
$698$	0	0
$699$	43.2247	1.63491
$700$	0	0
$701$	30.9554	1.16917	0.584585	−	0.811333i	$-0.301258\pi$
$701$	30.9554	1.16917	0.584585	+	0.811333i	$0.301258\pi$
$702$	0	0
$703$	5.07606	0.191447
$704$	0	0
$705$	15.3545	0.578284
$706$	0	0
$707$	−3.92931	−0.147777
$708$	0	0
$709$	−21.6396	−0.812694	−0.406347	−	0.913719i	$-0.633198\pi$
$709$	−21.6396	−0.812694	−0.406347	+	0.913719i	$0.633198\pi$
$710$	0	0
$711$	1.91425	0.0717899
$712$	0	0
$713$	0.268750	0.0100648
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.00538	0.261621
$718$	0	0
$719$	−11.2556	−0.419763	−0.209882	−	0.977727i	$-0.567308\pi$
$719$	−11.2556	−0.419763	−0.209882	+	0.977727i	$0.567308\pi$
$720$	0	0
$721$	−17.2664	−0.643033
$722$	0	0
$723$	44.3086	1.64785
$724$	0	0
$725$	−5.29590	−0.196685
$726$	0	0
$727$	18.6485	0.691634	0.345817	−	0.938302i	$-0.387602\pi$
$727$	18.6485	0.691634	0.345817	+	0.938302i	$0.387602\pi$
$728$	0	0
$729$	24.4263	0.904676
$730$	0	0
$731$	2.06877	0.0765162
$732$	0	0
$733$	−5.01639	−0.185285	−0.0926424	−	0.995699i	$-0.529531\pi$
$733$	−5.01639	−0.185285	−0.0926424	+	0.995699i	$0.529531\pi$
$734$	0	0
$735$	6.76271	0.249446
$736$	0	0
$737$	8.11662	0.298980
$738$	0	0
$739$	26.4276	0.972154	0.486077	−	0.873916i	$-0.338427\pi$
$739$	26.4276	0.972154	0.486077	+	0.873916i	$0.338427\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.25129	−0.0825919	−0.0412959	−	0.999147i	$-0.513149\pi$
$743$	−2.25129	−0.0825919	−0.0412959	+	0.999147i	$0.513149\pi$
$744$	0	0
$745$	−7.34050	−0.268935
$746$	0	0
$747$	−1.22223	−0.0447189
$748$	0	0
$749$	10.8780	0.397474
$750$	0	0
$751$	29.7375	1.08514	0.542568	−	0.840012i	$-0.317452\pi$
$751$	29.7375	1.08514	0.542568	+	0.840012i	$0.317452\pi$
$752$	0	0
$753$	26.1444	0.952753
$754$	0	0
$755$	−8.46011	−0.307895
$756$	0	0
$757$	24.9764	0.907784	0.453892	−	0.891057i	$-0.350035\pi$
$757$	24.9764	0.907784	0.453892	+	0.891057i	$0.350035\pi$
$758$	0	0
$759$	0.692021	0.0251188
$760$	0	0
$761$	−7.37090	−0.267195	−0.133597	−	0.991036i	$-0.542653\pi$
$761$	−7.37090	−0.267195	−0.133597	+	0.991036i	$0.542653\pi$
$762$	0	0
$763$	19.2271	0.696069
$764$	0	0
$765$	0.225209	0.00814246
$766$	0	0
$767$	0	0
$768$	0	0
$769$	55.2780	1.99337	0.996687	−	0.0813304i	$-0.0259169\pi$
$769$	55.2780	1.99337	0.996687	+	0.0813304i	$0.0259169\pi$
$770$	0	0
$771$	9.36467	0.337260
$772$	0	0
$773$	−54.5120	−1.96066	−0.980330	−	0.197364i	$-0.936762\pi$
$773$	−54.5120	−1.96066	−0.980330	+	0.197364i	$0.936762\pi$
$774$	0	0
$775$	1.08815	0.0390874
$776$	0	0
$777$	9.74094	0.349454
$778$	0	0
$779$	11.7778	0.421983
$780$	0	0
$781$	−8.92692	−0.319430
$782$	0	0
$783$	−26.2717	−0.938875
$784$	0	0
$785$	−17.7681	−0.634170
$786$	0	0
$787$	−54.7493	−1.95160	−0.975801	−	0.218661i	$-0.929831\pi$
$787$	−54.7493	−1.95160	−0.975801	+	0.218661i	$0.929831\pi$
$788$	0	0
$789$	−14.1685	−0.504413
$790$	0	0
$791$	−31.7385	−1.12849
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−12.4819	−0.442687
$796$	0	0
$797$	33.9982	1.20428	0.602139	−	0.798391i	$-0.294315\pi$
$797$	33.9982	1.20428	0.602139	+	0.798391i	$0.294315\pi$
$798$	0	0
$799$	−7.77000	−0.274883
$800$	0	0
$801$	2.94139	0.103929
$802$	0	0
$803$	22.6896	0.800700
$804$	0	0
$805$	0.445042	0.0156857
$806$	0	0
$807$	−12.2838	−0.432411
$808$	0	0
$809$	−11.3532	−0.399156	−0.199578	−	0.979882i	$-0.563957\pi$
$809$	−11.3532	−0.399156	−0.199578	+	0.979882i	$0.563957\pi$
$810$	0	0
$811$	−32.6082	−1.14503	−0.572514	−	0.819895i	$-0.694032\pi$
$811$	−32.6082	−1.14503	−0.572514	+	0.819895i	$0.694032\pi$
$812$	0	0
$813$	29.2325	1.02523
$814$	0	0
$815$	−11.6286	−0.407333
$816$	0	0
$817$	−3.83877	−0.134302
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.6377	0.475960	0.237980	−	0.971270i	$-0.423515\pi$
$821$	13.6377	0.475960	0.237980	+	0.971270i	$0.423515\pi$
$822$	0	0
$823$	−33.4077	−1.16452	−0.582260	−	0.813003i	$-0.697831\pi$
$823$	−33.4077	−1.16452	−0.582260	+	0.813003i	$0.697831\pi$
$824$	0	0
$825$	2.80194	0.0975510
$826$	0	0
$827$	13.9801	0.486137	0.243069	−	0.970009i	$-0.421846\pi$
$827$	13.9801	0.486137	0.243069	+	0.970009i	$0.421846\pi$
$828$	0	0
$829$	36.8528	1.27995	0.639975	−	0.768396i	$-0.278945\pi$
$829$	36.8528	1.27995	0.639975	+	0.768396i	$0.278945\pi$
$830$	0	0
$831$	−3.88577	−0.134796
$832$	0	0
$833$	−3.42221	−0.118572
$834$	0	0
$835$	−9.20237	−0.318461
$836$	0	0
$837$	5.39804	0.186584
$838$	0	0
$839$	−3.58775	−0.123863	−0.0619314	−	0.998080i	$-0.519726\pi$
$839$	−3.58775	−0.123863	−0.0619314	+	0.998080i	$0.519726\pi$
$840$	0	0
$841$	−0.953476	−0.0328785
$842$	0	0
$843$	19.3230	0.665521
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.4644	−0.531364
$848$	0	0
$849$	9.92692	0.340691
$850$	0	0
$851$	−0.740939	−0.0253991
$852$	0	0
$853$	47.2723	1.61857	0.809287	−	0.587414i	$-0.199854\pi$
$853$	47.2723	1.61857	0.809287	+	0.587414i	$0.199854\pi$
$854$	0	0
$855$	−0.417895	−0.0142917
$856$	0	0
$857$	−20.5435	−0.701751	−0.350876	−	0.936422i	$-0.614116\pi$
$857$	−20.5435	−0.701751	−0.350876	+	0.936422i	$0.614116\pi$
$858$	0	0
$859$	20.0597	0.684427	0.342214	−	0.939622i	$-0.388823\pi$
$859$	20.0597	0.684427	0.342214	+	0.939622i	$0.388823\pi$
$860$	0	0
$861$	22.6015	0.770256
$862$	0	0
$863$	43.7103	1.48792	0.743959	−	0.668226i	$-0.232946\pi$
$863$	43.7103	1.48792	0.743959	+	0.668226i	$0.232946\pi$
$864$	0	0
$865$	−15.4601	−0.525659
$866$	0	0
$867$	29.1347	0.989465
$868$	0	0
$869$	−12.0519	−0.408833
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.26577	0.0766846
$874$	0	0
$875$	1.80194	0.0609166
$876$	0	0
$877$	23.6655	0.799126	0.399563	−	0.916706i	$-0.369162\pi$
$877$	23.6655	0.799126	0.399563	+	0.916706i	$0.369162\pi$
$878$	0	0
$879$	3.25906	0.109925
$880$	0	0
$881$	−33.5743	−1.13115	−0.565574	−	0.824698i	$-0.691345\pi$
$881$	−33.5743	−1.13115	−0.565574	+	0.824698i	$0.691345\pi$
$882$	0	0
$883$	−8.10513	−0.272759	−0.136380	−	0.990657i	$-0.543547\pi$
$883$	−8.10513	−0.272759	−0.136380	+	0.990657i	$0.543547\pi$
$884$	0	0
$885$	−26.4916	−0.890504
$886$	0	0
$887$	42.9687	1.44275	0.721373	−	0.692547i	$-0.243511\pi$
$887$	42.9687	1.44275	0.721373	+	0.692547i	$0.243511\pi$
$888$	0	0
$889$	−38.6775	−1.29720
$890$	0	0
$891$	15.0519	0.504258
$892$	0	0
$893$	14.4179	0.482476
$894$	0	0
$895$	−21.0519	−0.703687
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.76271	−0.192197
$900$	0	0
$901$	6.31634	0.210428
$902$	0	0
$903$	−7.36658	−0.245145
$904$	0	0
$905$	−14.9608	−0.497313
$906$	0	0
$907$	−26.9536	−0.894979	−0.447490	−	0.894289i	$-0.647682\pi$
$907$	−26.9536	−0.894979	−0.447490	+	0.894289i	$0.647682\pi$
$908$	0	0
$909$	−0.538565	−0.0178631
$910$	0	0
$911$	−30.1909	−1.00027	−0.500134	−	0.865948i	$-0.666716\pi$
$911$	−30.1909	−1.00027	−0.500134	+	0.865948i	$0.666716\pi$
$912$	0	0
$913$	7.69501	0.254668
$914$	0	0
$915$	17.2174	0.569191
$916$	0	0
$917$	14.3230	0.472989
$918$	0	0
$919$	−28.9336	−0.954432	−0.477216	−	0.878786i	$-0.658354\pi$
$919$	−28.9336	−0.954432	−0.477216	+	0.878786i	$0.658354\pi$
$920$	0	0
$921$	19.7157	0.649655
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	0	0
$927$	−2.36658	−0.0777288
$928$	0	0
$929$	9.34854	0.306715	0.153358	−	0.988171i	$-0.450991\pi$
$929$	9.34854	0.306715	0.153358	+	0.988171i	$0.450991\pi$
$930$	0	0
$931$	6.35019	0.208119
$932$	0	0
$933$	5.32736	0.174410
$934$	0	0
$935$	−1.41789	−0.0463701
$936$	0	0
$937$	−40.7808	−1.33225	−0.666125	−	0.745840i	$-0.732048\pi$
$937$	−40.7808	−1.33225	−0.666125	+	0.745840i	$0.732048\pi$
$938$	0	0
$939$	2.11769	0.0691081
$940$	0	0
$941$	37.4161	1.21973	0.609865	−	0.792505i	$-0.291224\pi$
$941$	37.4161	1.21973	0.609865	+	0.792505i	$0.291224\pi$
$942$	0	0
$943$	−1.71917	−0.0559838
$944$	0	0
$945$	8.93900	0.290786
$946$	0	0
$947$	12.4612	0.404934	0.202467	−	0.979289i	$-0.435104\pi$
$947$	12.4612	0.404934	0.202467	+	0.979289i	$0.435104\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−31.4655	−1.02034
$952$	0	0
$953$	−6.66978	−0.216055	−0.108028	−	0.994148i	$-0.534454\pi$
$953$	−6.66978	−0.216055	−0.108028	+	0.994148i	$0.534454\pi$
$954$	0	0
$955$	3.15452	0.102078
$956$	0	0
$957$	−14.8388	−0.479669
$958$	0	0
$959$	2.50604	0.0809243
$960$	0	0
$961$	−29.8159	−0.961804
$962$	0	0
$963$	1.49098	0.0480460
$964$	0	0
$965$	4.04892	0.130339
$966$	0	0
$967$	−45.9584	−1.47792	−0.738961	−	0.673748i	$-0.764683\pi$
$967$	−45.9584	−1.47792	−0.738961	+	0.673748i	$0.764683\pi$
$968$	0	0
$969$	2.78017	0.0893118
$970$	0	0
$971$	45.3991	1.45693	0.728463	−	0.685085i	$-0.240235\pi$
$971$	45.3991	1.45693	0.728463	+	0.685085i	$0.240235\pi$
$972$	0	0
$973$	36.2204	1.16117
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.2174	0.390871	0.195435	−	0.980717i	$-0.437388\pi$
$977$	12.2174	0.390871	0.195435	+	0.980717i	$0.437388\pi$
$978$	0	0
$979$	−18.5187	−0.591861
$980$	0	0
$981$	2.63533	0.0841398
$982$	0	0
$983$	52.9081	1.68751	0.843754	−	0.536731i	$-0.180341\pi$
$983$	52.9081	1.68751	0.843754	+	0.536731i	$0.180341\pi$
$984$	0	0
$985$	−5.20775	−0.165933
$986$	0	0
$987$	27.6679	0.880678
$988$	0	0
$989$	0.560335	0.0178176
$990$	0	0
$991$	−11.9011	−0.378051	−0.189025	−	0.981972i	$-0.560533\pi$
$991$	−11.9011	−0.378051	−0.189025	+	0.981972i	$0.560533\pi$
$992$	0	0
$993$	−21.1347	−0.670688
$994$	0	0
$995$	−22.5013	−0.713338
$996$	0	0
$997$	32.5937	1.03225	0.516127	−	0.856512i	$-0.327373\pi$
$997$	32.5937	1.03225	0.516127	+	0.856512i	$0.327373\pi$
$998$	0	0
$999$	−14.8823	−0.470856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.l.1.1	yes	3
13.5	odd	4		3380.2.f.g.3041.1		6
13.8	odd	4		3380.2.f.g.3041.2		6
13.12	even	2		3380.2.a.k.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.k.1.1	✓	3	13.12	even	2
3380.2.a.l.1.1	yes	3	1.1	even	1	trivial
3380.2.f.g.3041.1		6	13.5	odd	4
3380.2.f.g.3041.2		6	13.8	odd	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-1.80194 q^{3} +1.00000 q^{5} +1.80194 q^{7} +0.246980 q^{9} +O(q^{10})\)	\(q-1.80194 q^{3} +1.00000 q^{5} +1.80194 q^{7} +0.246980 q^{9} -1.55496 q^{11} -1.80194 q^{15} +0.911854 q^{17} -1.69202 q^{19} -3.24698 q^{21} +0.246980 q^{23} +1.00000 q^{25} +4.96077 q^{27} -5.29590 q^{29} +1.08815 q^{31} +2.80194 q^{33} +1.80194 q^{35} -3.00000 q^{37} -6.96077 q^{41} +2.26875 q^{43} +0.246980 q^{45} -8.52111 q^{47} -3.75302 q^{49} -1.64310 q^{51} +6.92692 q^{53} -1.55496 q^{55} +3.04892 q^{57} +14.7017 q^{59} -9.55496 q^{61} +0.445042 q^{63} -5.21983 q^{67} -0.445042 q^{69} +5.74094 q^{71} -14.5918 q^{73} -1.80194 q^{75} -2.80194 q^{77} +7.75063 q^{79} -9.67994 q^{81} -4.94869 q^{83} +0.911854 q^{85} +9.54288 q^{87} +11.9095 q^{89} -1.96077 q^{93} -1.69202 q^{95} +9.17390 q^{97} -0.384043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10}) \) 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9	\( 3 q - q^{3} + 3 q^{5} + q^{7} - 4 q^{9} - 5 q^{11} - q^{15} - q^{17} - 5 q^{21} - 4 q^{23} + 3 q^{25} + 2 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} + q^{35} - 9 q^{37} - 8 q^{41} - q^{43} - 4 q^{45} - 10 q^{47} - 16 q^{49} - 9 q^{51} - 8 q^{53} - 5 q^{55} + 17 q^{59} - 29 q^{61} + q^{63} - 17 q^{67} - q^{69} + 3 q^{71} - 16 q^{73} - q^{75} - 4 q^{77} - 13 q^{79} - 5 q^{81} + 17 q^{83} - q^{85} + 10 q^{87} - 9 q^{89} + 7 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 + q^7 - 4 * q^9 - 5 * q^11 - q^15 - q^17 - 5 * q^21 - 4 * q^23 + 3 * q^25 + 2 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 + q^35 - 9 * q^37 - 8 * q^41 - q^43 - 4 * q^45 - 10 * q^47 - 16 * q^49 - 9 * q^51 - 8 * q^53 - 5 * q^55 + 17 * q^59 - 29 * q^61 + q^63 - 17 * q^67 - q^69 + 3 * q^71 - 16 * q^73 - q^75 - 4 * q^77 - 13 * q^79 - 5 * q^81 + 17 * q^83 - q^85 + 10 * q^87 - 9 * q^89 + 7 * q^93 - 6 * q^97 + 9 * q^99

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