LMFDB - Embedded newform 1216.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.961.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 10x + 8 $ x^3 - x^2 - 10*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.786802$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.786802 q^{3} -3.29707 q^{5} -2.08387 q^{7} -2.38094 q^{9} -1.29707 q^{11} -1.21320 q^{13} +2.59414 q^{15} +4.08387 q^{17} +1.00000 q^{19} +1.63959 q^{21} -8.95455 q^{23} +5.87067 q^{25} +4.23374 q^{27} +9.38094 q^{29} +1.02054 q^{33} +6.87067 q^{35} +2.00000 q^{37} +0.954547 q^{39} +3.57360 q^{41} -7.72347 q^{43} +7.85014 q^{45} +9.46482 q^{47} -2.65748 q^{49} -3.21320 q^{51} +11.9751 q^{53} +4.27653 q^{55} -0.786802 q^{57} +7.21320 q^{59} -4.87067 q^{61} +4.96158 q^{63} +4.00000 q^{65} -11.3809 q^{67} +7.04545 q^{69} -9.02054 q^{71} +5.65748 q^{73} -4.61906 q^{75} +2.70293 q^{77} +9.57360 q^{79} +3.81172 q^{81} -10.7619 q^{83} -13.4648 q^{85} -7.38094 q^{87} +11.0205 q^{89} +2.52815 q^{91} -3.29707 q^{95} -8.59414 q^{97} +3.08825 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + 5 q^{11} - 5 q^{13} - 10 q^{15} + 2 q^{17} + 3 q^{19} + 9 q^{21} - 5 q^{23} + 6 q^{25} - q^{27} + 9 q^{29} - 12 q^{33} + 9 q^{35} + 6 q^{37} - 19 q^{39} + 8 q^{41}+ \cdots + 51 q^{99}+O(q^{100}) $ 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9 + 5 * q^11 - 5 * q^13 - 10 * q^15 + 2 * q^17 + 3 * q^19 + 9 * q^21 - 5 * q^23 + 6 * q^25 - q^27 + 9 * q^29 - 12 * q^33 + 9 * q^35 + 6 * q^37 - 19 * q^39 + 8 * q^41 - 17 * q^43 + 27 * q^45 - q^47 + 5 * q^49 - 11 * q^51 - q^53 + 19 * q^55 - q^57 + 23 * q^59 - 3 * q^61 + 47 * q^63 + 12 * q^65 - 15 * q^67 + 43 * q^69 - 12 * q^71 + 4 * q^73 - 33 * q^75 + 17 * q^77 + 26 * q^79 + 47 * q^81 + 6 * q^83 - 11 * q^85 - 3 * q^87 + 18 * q^89 - 17 * q^91 - q^95 - 8 * q^97 + 51 * q^99

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
$3$	−0.786802	−0.454260	−0.227130	−	0.973864i	$-0.572934\pi$
$3$	−0.786802	−0.454260	−0.227130	+	0.973864i	$0.572934\pi$
$4$	0	0
$5$	−3.29707	−1.47449	−0.737247	−	0.675623i	$-0.763875\pi$
$5$	−3.29707	−1.47449	−0.737247	+	0.675623i	$0.763875\pi$
$6$	0	0
$7$	−2.08387	−0.787630	−0.393815	−	0.919190i	$-0.628845\pi$
$7$	−2.08387	−0.787630	−0.393815	+	0.919190i	$0.628845\pi$
$8$	0	0
$9$	−2.38094	−0.793648
$10$	0	0
$11$	−1.29707	−0.391081	−0.195541	−	0.980696i	$-0.562646\pi$
$11$	−1.29707	−0.391081	−0.195541	+	0.980696i	$0.562646\pi$
$12$	0	0
$13$	−1.21320	−0.336481	−0.168240	−	0.985746i	$-0.553808\pi$
$13$	−1.21320	−0.336481	−0.168240	+	0.985746i	$0.553808\pi$
$14$	0	0
$15$	2.59414	0.669804
$16$	0	0
$17$	4.08387	0.990485	0.495242	−	0.868755i	$-0.335079\pi$
$17$	4.08387	0.990485	0.495242	+	0.868755i	$0.335079\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.63959	0.357789
$22$	0	0
$23$	−8.95455	−1.86715	−0.933576	−	0.358379i	$-0.883329\pi$
$23$	−8.95455	−1.86715	−0.933576	+	0.358379i	$0.883329\pi$
$24$	0	0
$25$	5.87067	1.17413
$26$	0	0
$27$	4.23374	0.814783
$28$	0	0
$29$	9.38094	1.74200	0.870999	−	0.491285i	$-0.163473\pi$
$29$	9.38094	1.74200	0.870999	+	0.491285i	$0.163473\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.02054	0.177653
$34$	0	0
$35$	6.87067	1.16136
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0.954547	0.152850
$40$	0	0
$41$	3.57360	0.558103	0.279052	−	0.960276i	$-0.409980\pi$
$41$	3.57360	0.558103	0.279052	+	0.960276i	$0.409980\pi$
$42$	0	0
$43$	−7.72347	−1.17782	−0.588909	−	0.808199i	$-0.700442\pi$
$43$	−7.72347	−1.17782	−0.588909	+	0.808199i	$0.700442\pi$
$44$	0	0
$45$	7.85014	1.17023
$46$	0	0
$47$	9.46482	1.38059	0.690293	−	0.723530i	$-0.257482\pi$
$47$	9.46482	1.38059	0.690293	+	0.723530i	$0.257482\pi$
$48$	0	0
$49$	−2.65748	−0.379639
$50$	0	0
$51$	−3.21320	−0.449938
$52$	0	0
$53$	11.9751	1.64490	0.822452	−	0.568834i	$-0.192605\pi$
$53$	11.9751	1.64490	0.822452	+	0.568834i	$0.192605\pi$
$54$	0	0
$55$	4.27653	0.576648
$56$	0	0
$57$	−0.786802	−0.104214
$58$	0	0
$59$	7.21320	0.939078	0.469539	−	0.882912i	$-0.344420\pi$
$59$	7.21320	0.939078	0.469539	+	0.882912i	$0.344420\pi$
$60$	0	0
$61$	−4.87067	−0.623626	−0.311813	−	0.950144i	$-0.600936\pi$
$61$	−4.87067	−0.623626	−0.311813	+	0.950144i	$0.600936\pi$
$62$	0	0
$63$	4.96158	0.625100
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−11.3809	−1.39040	−0.695202	−	0.718815i	$-0.744685\pi$
$67$	−11.3809	−1.39040	−0.695202	+	0.718815i	$0.744685\pi$
$68$	0	0
$69$	7.04545	0.848173
$70$	0	0
$71$	−9.02054	−1.07054	−0.535270	−	0.844681i	$-0.679790\pi$
$71$	−9.02054	−1.07054	−0.535270	+	0.844681i	$0.679790\pi$
$72$	0	0
$73$	5.65748	0.662157	0.331079	−	0.943603i	$-0.392587\pi$
$73$	5.65748	0.662157	0.331079	+	0.943603i	$0.392587\pi$
$74$	0	0
$75$	−4.61906	−0.533363
$76$	0	0
$77$	2.70293	0.308027
$78$	0	0
$79$	9.57360	1.07711	0.538557	−	0.842589i	$-0.318970\pi$
$79$	9.57360	1.07711	0.538557	+	0.842589i	$0.318970\pi$
$80$	0	0
$81$	3.81172	0.423524
$82$	0	0
$83$	−10.7619	−1.18127	−0.590635	−	0.806939i	$-0.701123\pi$
$83$	−10.7619	−1.18127	−0.590635	+	0.806939i	$0.701123\pi$
$84$	0	0
$85$	−13.4648	−1.46046
$86$	0	0
$87$	−7.38094	−0.791320
$88$	0	0
$89$	11.0205	1.16817	0.584087	−	0.811691i	$-0.301453\pi$
$89$	11.0205	1.16817	0.584087	+	0.811691i	$0.301453\pi$
$90$	0	0
$91$	2.52815	0.265022
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.29707	−0.338272
$96$	0	0
$97$	−8.59414	−0.872603	−0.436301	−	0.899801i	$-0.643712\pi$
$97$	−8.59414	−0.872603	−0.436301	+	0.899801i	$0.643712\pi$
$98$	0	0
$99$	3.08825	0.310381
$100$	0	0
$101$	−9.14721	−0.910181	−0.455091	−	0.890445i	$-0.650393\pi$
$101$	−9.14721	−0.910181	−0.455091	+	0.890445i	$0.650393\pi$
$102$	0	0
$103$	17.3560	1.71014	0.855070	−	0.518512i	$-0.173514\pi$
$103$	17.3560	1.71014	0.855070	+	0.518512i	$0.173514\pi$
$104$	0	0
$105$	−5.40586	−0.527558
$106$	0	0
$107$	0.786802	0.0760630	0.0380315	−	0.999277i	$-0.487891\pi$
$107$	0.786802	0.0760630	0.0380315	+	0.999277i	$0.487891\pi$
$108$	0	0
$109$	13.5487	1.29773	0.648864	−	0.760904i	$-0.275244\pi$
$109$	13.5487	1.29773	0.648864	+	0.760904i	$0.275244\pi$
$110$	0	0
$111$	−1.57360	−0.149360
$112$	0	0
$113$	−1.14721	−0.107920	−0.0539601	−	0.998543i	$-0.517184\pi$
$113$	−1.14721	−0.107920	−0.0539601	+	0.998543i	$0.517184\pi$
$114$	0	0
$115$	29.5238	2.75311
$116$	0	0
$117$	2.88856	0.267047
$118$	0	0
$119$	−8.51027	−0.780135
$120$	0	0
$121$	−9.31761	−0.847055
$122$	0	0
$123$	−2.81172	−0.253524
$124$	0	0
$125$	−2.87067	−0.256761
$126$	0	0
$127$	−1.02054	−0.0905581	−0.0452790	−	0.998974i	$-0.514418\pi$
$127$	−1.02054	−0.0905581	−0.0452790	+	0.998974i	$0.514418\pi$
$128$	0	0
$129$	6.07684	0.535036
$130$	0	0
$131$	7.72347	0.674802	0.337401	−	0.941361i	$-0.390452\pi$
$131$	7.72347	0.674802	0.337401	+	0.941361i	$0.390452\pi$
$132$	0	0
$133$	−2.08387	−0.180695
$134$	0	0
$135$	−13.9589	−1.20139
$136$	0	0
$137$	15.2311	1.30128	0.650639	−	0.759387i	$-0.274501\pi$
$137$	15.2311	1.30128	0.650639	+	0.759387i	$0.274501\pi$
$138$	0	0
$139$	−16.0590	−1.36210	−0.681051	−	0.732236i	$-0.738477\pi$
$139$	−16.0590	−1.36210	−0.681051	+	0.732236i	$0.738477\pi$
$140$	0	0
$141$	−7.44693	−0.627145
$142$	0	0
$143$	1.57360	0.131591
$144$	0	0
$145$	−30.9296	−2.56857
$146$	0	0
$147$	2.09091	0.172455
$148$	0	0
$149$	2.10879	0.172759	0.0863793	−	0.996262i	$-0.472470\pi$
$149$	2.10879	0.172759	0.0863793	+	0.996262i	$0.472470\pi$
$150$	0	0
$151$	17.3560	1.41241	0.706207	−	0.708006i	$-0.250405\pi$
$151$	17.3560	1.41241	0.706207	+	0.708006i	$0.250405\pi$
$152$	0	0
$153$	−9.72347	−0.786096
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−9.42202	−0.747215
$160$	0	0
$161$	18.6601	1.47062
$162$	0	0
$163$	0.852793	0.0667959	0.0333979	−	0.999442i	$-0.489367\pi$
$163$	0.852793	0.0667959	0.0333979	+	0.999442i	$0.489367\pi$
$164$	0	0
$165$	−3.36478	−0.261948
$166$	0	0
$167$	−18.5941	−1.43886	−0.719429	−	0.694566i	$-0.755596\pi$
$167$	−18.5941	−1.43886	−0.719429	+	0.694566i	$0.755596\pi$
$168$	0	0
$169$	−11.5282	−0.886781
$170$	0	0
$171$	−2.38094	−0.182075
$172$	0	0
$173$	−2.85279	−0.216894	−0.108447	−	0.994102i	$-0.534588\pi$
$173$	−2.85279	−0.216894	−0.108447	+	0.994102i	$0.534588\pi$
$174$	0	0
$175$	−12.2337	−0.924783
$176$	0	0
$177$	−5.67536	−0.426586
$178$	0	0
$179$	26.0768	1.94907	0.974537	−	0.224226i	$-0.0719854\pi$
$179$	26.0768	1.94907	0.974537	+	0.224226i	$0.0719854\pi$
$180$	0	0
$181$	−3.18828	−0.236983	−0.118492	−	0.992955i	$-0.537806\pi$
$181$	−3.18828	−0.236983	−0.118492	+	0.992955i	$0.537806\pi$
$182$	0	0
$183$	3.83226	0.283288
$184$	0	0
$185$	−6.59414	−0.484811
$186$	0	0
$187$	−5.29707	−0.387360
$188$	0	0
$189$	−8.82256	−0.641747
$190$	0	0
$191$	7.95720	0.575763	0.287881	−	0.957666i	$-0.407049\pi$
$191$	7.95720	0.575763	0.287881	+	0.957666i	$0.407049\pi$
$192$	0	0
$193$	20.9296	1.50655	0.753274	−	0.657707i	$-0.228473\pi$
$193$	20.9296	1.50655	0.753274	+	0.657707i	$0.228473\pi$
$194$	0	0
$195$	−3.14721	−0.225376
$196$	0	0
$197$	3.57360	0.254609	0.127304	−	0.991864i	$-0.459368\pi$
$197$	3.57360	0.254609	0.127304	+	0.991864i	$0.459368\pi$
$198$	0	0
$199$	−26.4194	−1.87282	−0.936409	−	0.350909i	$-0.885873\pi$
$199$	−26.4194	−1.87282	−0.936409	+	0.350909i	$0.885873\pi$
$200$	0	0
$201$	8.95455	0.631605
$202$	0	0
$203$	−19.5487	−1.37205
$204$	0	0
$205$	−11.7824	−0.822920
$206$	0	0
$207$	21.3203	1.48186
$208$	0	0
$209$	−1.29707	−0.0897202
$210$	0	0
$211$	23.5487	1.62116	0.810579	−	0.585629i	$-0.199152\pi$
$211$	23.5487	1.62116	0.810579	+	0.585629i	$0.199152\pi$
$212$	0	0
$213$	7.09738	0.486304
$214$	0	0
$215$	25.4648	1.73669
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.45131	−0.300792
$220$	0	0
$221$	−4.95455	−0.333279
$222$	0	0
$223$	1.02054	0.0683402	0.0341701	−	0.999416i	$-0.489121\pi$
$223$	1.02054	0.0683402	0.0341701	+	0.999416i	$0.489121\pi$
$224$	0	0
$225$	−13.9777	−0.931849
$226$	0	0
$227$	−3.04545	−0.202134	−0.101067	−	0.994880i	$-0.532226\pi$
$227$	−3.04545	−0.202134	−0.101067	+	0.994880i	$0.532226\pi$
$228$	0	0
$229$	4.01788	0.265509	0.132755	−	0.991149i	$-0.457618\pi$
$229$	4.01788	0.265509	0.132755	+	0.991149i	$0.457618\pi$
$230$	0	0
$231$	−2.12667	−0.139925
$232$	0	0
$233$	25.2062	1.65131	0.825655	−	0.564175i	$-0.190806\pi$
$233$	25.2062	1.65131	0.825655	+	0.564175i	$0.190806\pi$
$234$	0	0
$235$	−31.2062	−2.03567
$236$	0	0
$237$	−7.53253	−0.489290
$238$	0	0
$239$	−6.93667	−0.448696	−0.224348	−	0.974509i	$-0.572025\pi$
$239$	−6.93667	−0.448696	−0.224348	+	0.974509i	$0.572025\pi$
$240$	0	0
$241$	0.761886	0.0490774	0.0245387	−	0.999699i	$-0.492188\pi$
$241$	0.761886	0.0490774	0.0245387	+	0.999699i	$0.492188\pi$
$242$	0	0
$243$	−15.7003	−1.00717
$244$	0	0
$245$	8.76189	0.559776
$246$	0	0
$247$	−1.21320	−0.0771940
$248$	0	0
$249$	8.46747	0.536604
$250$	0	0
$251$	6.14986	0.388176	0.194088	−	0.980984i	$-0.437825\pi$
$251$	6.14986	0.388176	0.194088	+	0.980984i	$0.437825\pi$
$252$	0	0
$253$	11.6147	0.730209
$254$	0	0
$255$	10.5941	0.663431
$256$	0	0
$257$	−23.3560	−1.45691	−0.728454	−	0.685094i	$-0.759761\pi$
$257$	−23.3560	−1.45691	−0.728454	+	0.685094i	$0.759761\pi$
$258$	0	0
$259$	−4.16774	−0.258971
$260$	0	0
$261$	−22.3355	−1.38253
$262$	0	0
$263$	12.2765	0.757003	0.378502	−	0.925601i	$-0.376439\pi$
$263$	12.2765	0.757003	0.378502	+	0.925601i	$0.376439\pi$
$264$	0	0
$265$	−39.4827	−2.42540
$266$	0	0
$267$	−8.67098	−0.530655
$268$	0	0
$269$	7.74135	0.471998	0.235999	−	0.971753i	$-0.424164\pi$
$269$	7.74135	0.471998	0.235999	+	0.971753i	$0.424164\pi$
$270$	0	0
$271$	−0.954547	−0.0579846	−0.0289923	−	0.999580i	$-0.509230\pi$
$271$	−0.954547	−0.0579846	−0.0289923	+	0.999580i	$0.509230\pi$
$272$	0	0
$273$	−1.98915	−0.120389
$274$	0	0
$275$	−7.61468	−0.459182
$276$	0	0
$277$	−17.0384	−1.02374	−0.511870	−	0.859063i	$-0.671047\pi$
$277$	−17.0384	−1.02374	−0.511870	+	0.859063i	$0.671047\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.0974	−0.781324	−0.390662	−	0.920534i	$-0.627754\pi$
$281$	−13.0974	−0.781324	−0.390662	+	0.920534i	$0.627754\pi$
$282$	0	0
$283$	−1.29707	−0.0771028	−0.0385514	−	0.999257i	$-0.512274\pi$
$283$	−1.29707	−0.0771028	−0.0385514	+	0.999257i	$0.512274\pi$
$284$	0	0
$285$	2.59414	0.153664
$286$	0	0
$287$	−7.44693	−0.439579
$288$	0	0
$289$	−0.321987	−0.0189404
$290$	0	0
$291$	6.76189	0.396389
$292$	0	0
$293$	21.5487	1.25889	0.629444	−	0.777046i	$-0.283283\pi$
$293$	21.5487	1.25889	0.629444	+	0.777046i	$0.283283\pi$
$294$	0	0
$295$	−23.7824	−1.38467
$296$	0	0
$297$	−5.49145	−0.318646
$298$	0	0
$299$	10.8636	0.628260
$300$	0	0
$301$	16.0947	0.927684
$302$	0	0
$303$	7.19704	0.413459
$304$	0	0
$305$	16.0590	0.919533
$306$	0	0
$307$	9.18828	0.524403	0.262201	−	0.965013i	$-0.415551\pi$
$307$	9.18828	0.524403	0.262201	+	0.965013i	$0.415551\pi$
$308$	0	0
$309$	−13.6558	−0.776849
$310$	0	0
$311$	−7.48973	−0.424704	−0.212352	−	0.977193i	$-0.568112\pi$
$311$	−7.48973	−0.424704	−0.212352	+	0.977193i	$0.568112\pi$
$312$	0	0
$313$	−9.76626	−0.552022	−0.276011	−	0.961154i	$-0.589013\pi$
$313$	−9.76626	−0.552022	−0.276011	+	0.961154i	$0.589013\pi$
$314$	0	0
$315$	−16.3587	−0.921707
$316$	0	0
$317$	−13.7164	−0.770392	−0.385196	−	0.922835i	$-0.625866\pi$
$317$	−13.7164	−0.770392	−0.385196	+	0.922835i	$0.625866\pi$
$318$	0	0
$319$	−12.1677	−0.681263
$320$	0	0
$321$	−0.619057	−0.0345524
$322$	0	0
$323$	4.08387	0.227233
$324$	0	0
$325$	−7.12229	−0.395074
$326$	0	0
$327$	−10.6601	−0.589507
$328$	0	0
$329$	−19.7235	−1.08739
$330$	0	0
$331$	14.5282	0.798539	0.399270	−	0.916834i	$-0.369264\pi$
$331$	14.5282	0.798539	0.399270	+	0.916834i	$0.369264\pi$
$332$	0	0
$333$	−4.76189	−0.260950
$334$	0	0
$335$	37.5238	2.05014
$336$	0	0
$337$	9.31495	0.507418	0.253709	−	0.967281i	$-0.418349\pi$
$337$	9.31495	0.507418	0.253709	+	0.967281i	$0.418349\pi$
$338$	0	0
$339$	0.902625	0.0490238
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.1249	1.08665
$344$	0	0
$345$	−23.2294	−1.25063
$346$	0	0
$347$	9.29707	0.499093	0.249546	−	0.968363i	$-0.419718\pi$
$347$	9.29707	0.499093	0.249546	+	0.968363i	$0.419718\pi$
$348$	0	0
$349$	14.0590	0.752559	0.376279	−	0.926506i	$-0.377203\pi$
$349$	14.0590	0.752559	0.376279	+	0.926506i	$0.377203\pi$
$350$	0	0
$351$	−5.13636	−0.274159
$352$	0	0
$353$	−20.4783	−1.08995	−0.544975	−	0.838452i	$-0.683461\pi$
$353$	−20.4783	−1.08995	−0.544975	+	0.838452i	$0.683461\pi$
$354$	0	0
$355$	29.7413	1.57851
$356$	0	0
$357$	6.69589	0.354384
$358$	0	0
$359$	18.9724	1.00133	0.500663	−	0.865642i	$-0.333090\pi$
$359$	18.9724	1.00133	0.500663	+	0.865642i	$0.333090\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.33111	0.384784
$364$	0	0
$365$	−18.6531	−0.976348
$366$	0	0
$367$	−4.85279	−0.253314	−0.126657	−	0.991947i	$-0.540425\pi$
$367$	−4.85279	−0.253314	−0.126657	+	0.991947i	$0.540425\pi$
$368$	0	0
$369$	−8.50855	−0.442937
$370$	0	0
$371$	−24.9545	−1.29558
$372$	0	0
$373$	12.1071	0.626880	0.313440	−	0.949608i	$-0.398519\pi$
$373$	12.1071	0.626880	0.313440	+	0.949608i	$0.398519\pi$
$374$	0	0
$375$	2.25865	0.116636
$376$	0	0
$377$	−11.3809	−0.586148
$378$	0	0
$379$	2.02492	0.104013	0.0520065	−	0.998647i	$-0.483438\pi$
$379$	2.02492	0.104013	0.0520065	+	0.998647i	$0.483438\pi$
$380$	0	0
$381$	0.802961	0.0411369
$382$	0	0
$383$	−1.23811	−0.0632647	−0.0316323	−	0.999500i	$-0.510071\pi$
$383$	−1.23811	−0.0632647	−0.0316323	+	0.999500i	$0.510071\pi$
$384$	0	0
$385$	−8.91175	−0.454185
$386$	0	0
$387$	18.3891	0.934772
$388$	0	0
$389$	−12.8707	−0.652569	−0.326285	−	0.945272i	$-0.605797\pi$
$389$	−12.8707	−0.652569	−0.326285	+	0.945272i	$0.605797\pi$
$390$	0	0
$391$	−36.5692	−1.84939
$392$	0	0
$393$	−6.07684	−0.306536
$394$	0	0
$395$	−31.5648	−1.58820
$396$	0	0
$397$	−17.3739	−0.871971	−0.435986	−	0.899954i	$-0.643600\pi$
$397$	−17.3739	−0.871971	−0.435986	+	0.899954i	$0.643600\pi$
$398$	0	0
$399$	1.63959	0.0820824
$400$	0	0
$401$	19.0205	0.949840	0.474920	−	0.880029i	$-0.342477\pi$
$401$	19.0205	0.949840	0.474920	+	0.880029i	$0.342477\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−12.5675	−0.624484
$406$	0	0
$407$	−2.59414	−0.128587
$408$	0	0
$409$	20.4622	1.01179	0.505894	−	0.862595i	$-0.331163\pi$
$409$	20.4622	1.01179	0.505894	+	0.862595i	$0.331163\pi$
$410$	0	0
$411$	−11.9838	−0.591119
$412$	0	0
$413$	−15.0314	−0.739646
$414$	0	0
$415$	35.4827	1.74178
$416$	0	0
$417$	12.6352	0.618749
$418$	0	0
$419$	7.14721	0.349164	0.174582	−	0.984643i	$-0.444143\pi$
$419$	7.14721	0.349164	0.174582	+	0.984643i	$0.444143\pi$
$420$	0	0
$421$	−12.3604	−0.602409	−0.301205	−	0.953560i	$-0.597389\pi$
$421$	−12.3604	−0.602409	−0.301205	+	0.953560i	$0.597389\pi$
$422$	0	0
$423$	−22.5352	−1.09570
$424$	0	0
$425$	23.9751	1.16296
$426$	0	0
$427$	10.1499	0.491186
$428$	0	0
$429$	−1.23811	−0.0597767
$430$	0	0
$431$	−1.90909	−0.0919578	−0.0459789	−	0.998942i	$-0.514641\pi$
$431$	−1.90909	−0.0919578	−0.0459789	+	0.998942i	$0.514641\pi$
$432$	0	0
$433$	4.04107	0.194202	0.0971008	−	0.995275i	$-0.469043\pi$
$433$	4.04107	0.194202	0.0971008	+	0.995275i	$0.469043\pi$
$434$	0	0
$435$	24.3355	1.16680
$436$	0	0
$437$	−8.95455	−0.428354
$438$	0	0
$439$	14.0768	0.671851	0.335925	−	0.941889i	$-0.390951\pi$
$439$	14.0768	0.671851	0.335925	+	0.941889i	$0.390951\pi$
$440$	0	0
$441$	6.32730	0.301300
$442$	0	0
$443$	19.6736	0.934723	0.467361	−	0.884066i	$-0.345205\pi$
$443$	19.6736	0.934723	0.467361	+	0.884066i	$0.345205\pi$
$444$	0	0
$445$	−36.3355	−1.72247
$446$	0	0
$447$	−1.65920	−0.0784774
$448$	0	0
$449$	10.5531	0.498030	0.249015	−	0.968500i	$-0.419893\pi$
$449$	10.5531	0.498030	0.249015	+	0.968500i	$0.419893\pi$
$450$	0	0
$451$	−4.63522	−0.218264
$452$	0	0
$453$	−13.6558	−0.641603
$454$	0	0
$455$	−8.33549	−0.390774
$456$	0	0
$457$	2.72785	0.127603	0.0638016	−	0.997963i	$-0.479678\pi$
$457$	2.72785	0.127603	0.0638016	+	0.997963i	$0.479678\pi$
$458$	0	0
$459$	17.2900	0.807030
$460$	0	0
$461$	35.1153	1.63548	0.817740	−	0.575587i	$-0.195227\pi$
$461$	35.1153	1.63548	0.817740	+	0.575587i	$0.195227\pi$
$462$	0	0
$463$	17.1293	0.796067	0.398034	−	0.917371i	$-0.369693\pi$
$463$	17.1293	0.796067	0.398034	+	0.917371i	$0.369693\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.7029	−0.680370	−0.340185	−	0.940358i	$-0.610490\pi$
$467$	−14.7029	−0.680370	−0.340185	+	0.940358i	$0.610490\pi$
$468$	0	0
$469$	23.7164	1.09512
$470$	0	0
$471$	11.0152	0.507555
$472$	0	0
$473$	10.0179	0.460623
$474$	0	0
$475$	5.87067	0.269365
$476$	0	0
$477$	−28.5120	−1.30547
$478$	0	0
$479$	10.0411	0.458788	0.229394	−	0.973334i	$-0.426326\pi$
$479$	10.0411	0.458788	0.229394	+	0.973334i	$0.426326\pi$
$480$	0	0
$481$	−2.42640	−0.110634
$482$	0	0
$483$	−14.6818	−0.668046
$484$	0	0
$485$	28.3355	1.28665
$486$	0	0
$487$	−23.6504	−1.07170	−0.535852	−	0.844312i	$-0.680009\pi$
$487$	−23.6504	−1.07170	−0.535852	+	0.844312i	$0.680009\pi$
$488$	0	0
$489$	−0.670979	−0.0303427
$490$	0	0
$491$	38.3766	1.73191	0.865955	−	0.500122i	$-0.166711\pi$
$491$	38.3766	1.73191	0.865955	+	0.500122i	$0.166711\pi$
$492$	0	0
$493$	38.3106	1.72542
$494$	0	0
$495$	−10.1822	−0.457655
$496$	0	0
$497$	18.7976	0.843190
$498$	0	0
$499$	−10.3176	−0.461880	−0.230940	−	0.972968i	$-0.574180\pi$
$499$	−10.3176	−0.461880	−0.230940	+	0.972968i	$0.574180\pi$
$500$	0	0
$501$	14.6299	0.653616
$502$	0	0
$503$	−20.4372	−0.911252	−0.455626	−	0.890171i	$-0.650584\pi$
$503$	−20.4372	−0.911252	−0.455626	+	0.890171i	$0.650584\pi$
$504$	0	0
$505$	30.1590	1.34206
$506$	0	0
$507$	9.07037	0.402829
$508$	0	0
$509$	15.4059	0.682853	0.341426	−	0.939909i	$-0.389090\pi$
$509$	15.4059	0.682853	0.341426	+	0.939909i	$0.389090\pi$
$510$	0	0
$511$	−11.7895	−0.521535
$512$	0	0
$513$	4.23374	0.186924
$514$	0	0
$515$	−57.2240	−2.52159
$516$	0	0
$517$	−12.2765	−0.539921
$518$	0	0
$519$	2.24458	0.0985262
$520$	0	0
$521$	36.0270	1.57837	0.789186	−	0.614154i	$-0.210503\pi$
$521$	36.0270	1.57837	0.789186	+	0.614154i	$0.210503\pi$
$522$	0	0
$523$	−44.7370	−1.95621	−0.978106	−	0.208109i	$-0.933269\pi$
$523$	−44.7370	−1.95621	−0.978106	+	0.208109i	$0.933269\pi$
$524$	0	0
$525$	9.62553	0.420092
$526$	0	0
$527$	0	0
$528$	0	0
$529$	57.1839	2.48626
$530$	0	0
$531$	−17.1742	−0.745297
$532$	0	0
$533$	−4.33549	−0.187791
$534$	0	0
$535$	−2.59414	−0.112154
$536$	0	0
$537$	−20.5173	−0.885387
$538$	0	0
$539$	3.44693	0.148470
$540$	0	0
$541$	37.5059	1.61250	0.806252	−	0.591572i	$-0.201493\pi$
$541$	37.5059	1.61250	0.806252	+	0.591572i	$0.201493\pi$
$542$	0	0
$543$	2.50855	0.107652
$544$	0	0
$545$	−44.6710	−1.91349
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	11.5968	0.494939
$550$	0	0
$551$	9.38094	0.399642
$552$	0	0
$553$	−19.9502	−0.848367
$554$	0	0
$555$	5.18828	0.220230
$556$	0	0
$557$	33.3381	1.41258	0.706291	−	0.707921i	$-0.250367\pi$
$557$	33.3381	1.41258	0.706291	+	0.707921i	$0.250367\pi$
$558$	0	0
$559$	9.37010	0.396313
$560$	0	0
$561$	4.16774	0.175962
$562$	0	0
$563$	−3.44693	−0.145271	−0.0726355	−	0.997359i	$-0.523141\pi$
$563$	−3.44693	−0.145271	−0.0726355	+	0.997359i	$0.523141\pi$
$564$	0	0
$565$	3.78242	0.159128
$566$	0	0
$567$	−7.94313	−0.333580
$568$	0	0
$569$	27.0205	1.13276	0.566380	−	0.824144i	$-0.308343\pi$
$569$	27.0205	1.13276	0.566380	+	0.824144i	$0.308343\pi$
$570$	0	0
$571$	5.70559	0.238771	0.119386	−	0.992848i	$-0.461908\pi$
$571$	5.70559	0.238771	0.119386	+	0.992848i	$0.461908\pi$
$572$	0	0
$573$	−6.26074	−0.261546
$574$	0	0
$575$	−52.5692	−2.19229
$576$	0	0
$577$	6.34252	0.264043	0.132021	−	0.991247i	$-0.457853\pi$
$577$	6.34252	0.264043	0.132021	+	0.991247i	$0.457853\pi$
$578$	0	0
$579$	−16.4675	−0.684365
$580$	0	0
$581$	22.4264	0.930404
$582$	0	0
$583$	−15.5325	−0.643292
$584$	0	0
$585$	−9.52377	−0.393759
$586$	0	0
$587$	37.5827	1.55121	0.775603	−	0.631222i	$-0.217446\pi$
$587$	37.5827	1.55121	0.775603	+	0.631222i	$0.217446\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.81172	−0.115659
$592$	0	0
$593$	−9.48270	−0.389408	−0.194704	−	0.980862i	$-0.562375\pi$
$593$	−9.48270	−0.389408	−0.194704	+	0.980862i	$0.562375\pi$
$594$	0	0
$595$	28.0590	1.15031
$596$	0	0
$597$	20.7868	0.850747
$598$	0	0
$599$	24.6710	1.00803	0.504014	−	0.863695i	$-0.331856\pi$
$599$	24.6710	1.00803	0.504014	+	0.863695i	$0.331856\pi$
$600$	0	0
$601$	41.4329	1.69008	0.845041	−	0.534702i	$-0.179576\pi$
$601$	41.4329	1.69008	0.845041	+	0.534702i	$0.179576\pi$
$602$	0	0
$603$	27.0974	1.10349
$604$	0	0
$605$	30.7208	1.24898
$606$	0	0
$607$	5.32026	0.215943	0.107971	−	0.994154i	$-0.465564\pi$
$607$	5.32026	0.215943	0.107971	+	0.994154i	$0.465564\pi$
$608$	0	0
$609$	15.3809	0.623267
$610$	0	0
$611$	−11.4827	−0.464540
$612$	0	0
$613$	27.2472	1.10051	0.550253	−	0.834998i	$-0.314531\pi$
$613$	27.2472	1.10051	0.550253	+	0.834998i	$0.314531\pi$
$614$	0	0
$615$	9.27043	0.373820
$616$	0	0
$617$	−30.9117	−1.24446	−0.622230	−	0.782834i	$-0.713773\pi$
$617$	−30.9117	−1.24446	−0.622230	+	0.782834i	$0.713773\pi$
$618$	0	0
$619$	24.2857	0.976123	0.488061	−	0.872809i	$-0.337704\pi$
$619$	24.2857	0.976123	0.488061	+	0.872809i	$0.337704\pi$
$620$	0	0
$621$	−37.9112	−1.52132
$622$	0	0
$623$	−22.9654	−0.920089
$624$	0	0
$625$	−19.8886	−0.795542
$626$	0	0
$627$	1.02054	0.0407563
$628$	0	0
$629$	8.16774	0.325669
$630$	0	0
$631$	24.9117	0.991721	0.495861	−	0.868402i	$-0.334853\pi$
$631$	24.9117	0.991721	0.495861	+	0.868402i	$0.334853\pi$
$632$	0	0
$633$	−18.5282	−0.736428
$634$	0	0
$635$	3.36478	0.133527
$636$	0	0
$637$	3.22405	0.127741
$638$	0	0
$639$	21.4774	0.849632
$640$	0	0
$641$	4.25865	0.168207	0.0841033	−	0.996457i	$-0.473197\pi$
$641$	4.25865	0.168207	0.0841033	+	0.996457i	$0.473197\pi$
$642$	0	0
$643$	13.9323	0.549436	0.274718	−	0.961525i	$-0.411416\pi$
$643$	13.9323	0.549436	0.274718	+	0.961525i	$0.411416\pi$
$644$	0	0
$645$	−20.0358	−0.788907
$646$	0	0
$647$	5.58064	0.219398	0.109699	−	0.993965i	$-0.465011\pi$
$647$	5.58064	0.219398	0.109699	+	0.993965i	$0.465011\pi$
$648$	0	0
$649$	−9.35603	−0.367256
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.77330	0.0693945	0.0346973	−	0.999398i	$-0.488953\pi$
$653$	1.77330	0.0693945	0.0346973	+	0.999398i	$0.488953\pi$
$654$	0	0
$655$	−25.4648	−0.994993
$656$	0	0
$657$	−13.4701	−0.525520
$658$	0	0
$659$	1.12229	0.0437183	0.0218591	−	0.999761i	$-0.493041\pi$
$659$	1.12229	0.0437183	0.0218591	+	0.999761i	$0.493041\pi$
$660$	0	0
$661$	−21.9340	−0.853134	−0.426567	−	0.904456i	$-0.640277\pi$
$661$	−21.9340	−0.853134	−0.426567	+	0.904456i	$0.640277\pi$
$662$	0	0
$663$	3.89825	0.151395
$664$	0	0
$665$	6.87067	0.266433
$666$	0	0
$667$	−84.0021	−3.25257
$668$	0	0
$669$	−0.802961	−0.0310443
$670$	0	0
$671$	6.31761	0.243889
$672$	0	0
$673$	−8.24458	−0.317805	−0.158903	−	0.987294i	$-0.550796\pi$
$673$	−8.24458	−0.317805	−0.158903	+	0.987294i	$0.550796\pi$
$674$	0	0
$675$	24.8549	0.956665
$676$	0	0
$677$	5.76626	0.221616	0.110808	−	0.993842i	$-0.464656\pi$
$677$	5.76626	0.221616	0.110808	+	0.993842i	$0.464656\pi$
$678$	0	0
$679$	17.9091	0.687288
$680$	0	0
$681$	2.39617	0.0918214
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−50.2179	−1.91873
$686$	0	0
$687$	−3.16128	−0.120610
$688$	0	0
$689$	−14.5282	−0.553478
$690$	0	0
$691$	−23.0384	−0.876423	−0.438211	−	0.898872i	$-0.644388\pi$
$691$	−23.0384	−0.876423	−0.438211	+	0.898872i	$0.644388\pi$
$692$	0	0
$693$	−6.43552	−0.244465
$694$	0	0
$695$	52.9475	2.00841
$696$	0	0
$697$	14.5941	0.552793
$698$	0	0
$699$	−19.8323	−0.750125
$700$	0	0
$701$	3.90909	0.147644	0.0738222	−	0.997271i	$-0.476480\pi$
$701$	3.90909	0.147644	0.0738222	+	0.997271i	$0.476480\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	24.5531	0.924722
$706$	0	0
$707$	19.0616	0.716886
$708$	0	0
$709$	12.0411	0.452212	0.226106	−	0.974103i	$-0.427400\pi$
$709$	12.0411	0.452212	0.226106	+	0.974103i	$0.427400\pi$
$710$	0	0
$711$	−22.7942	−0.854849
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−5.18828	−0.194031
$716$	0	0
$717$	5.45778	0.203825
$718$	0	0
$719$	32.2927	1.20431	0.602157	−	0.798378i	$-0.294308\pi$
$719$	32.2927	1.20431	0.602157	+	0.798378i	$0.294308\pi$
$720$	0	0
$721$	−36.1677	−1.34696
$722$	0	0
$723$	−0.599453	−0.0222939
$724$	0	0
$725$	55.0725	2.04534
$726$	0	0
$727$	41.1812	1.52733	0.763664	−	0.645614i	$-0.223398\pi$
$727$	41.1812	1.52733	0.763664	+	0.645614i	$0.223398\pi$
$728$	0	0
$729$	0.917850	0.0339945
$730$	0	0
$731$	−31.5417	−1.16661
$732$	0	0
$733$	−1.27919	−0.0472479	−0.0236240	−	0.999721i	$-0.507520\pi$
$733$	−1.27919	−0.0472479	−0.0236240	+	0.999721i	$0.507520\pi$
$734$	0	0
$735$	−6.89387	−0.254284
$736$	0	0
$737$	14.7619	0.543761
$738$	0	0
$739$	−30.3534	−1.11657	−0.558283	−	0.829650i	$-0.688540\pi$
$739$	−30.3534	−1.11657	−0.558283	+	0.829650i	$0.688540\pi$
$740$	0	0
$741$	0.954547	0.0350661
$742$	0	0
$743$	−19.4827	−0.714751	−0.357375	−	0.933961i	$-0.616328\pi$
$743$	−19.4827	−0.714751	−0.357375	+	0.933961i	$0.616328\pi$
$744$	0	0
$745$	−6.95282	−0.254732
$746$	0	0
$747$	25.6234	0.937512
$748$	0	0
$749$	−1.63959	−0.0599095
$750$	0	0
$751$	−48.6710	−1.77603	−0.888015	−	0.459815i	$-0.847916\pi$
$751$	−48.6710	−1.77603	−0.888015	+	0.459815i	$0.847916\pi$
$752$	0	0
$753$	−4.83872	−0.176333
$754$	0	0
$755$	−57.2240	−2.08260
$756$	0	0
$757$	−41.7235	−1.51647	−0.758233	−	0.651984i	$-0.773937\pi$
$757$	−41.7235	−1.51647	−0.758233	+	0.651984i	$0.773937\pi$
$758$	0	0
$759$	−9.13845	−0.331705
$760$	0	0
$761$	−9.10441	−0.330035	−0.165017	−	0.986291i	$-0.552768\pi$
$761$	−9.10441	−0.330035	−0.165017	+	0.986291i	$0.552768\pi$
$762$	0	0
$763$	−28.2337	−1.02213
$764$	0	0
$765$	32.0590	1.15909
$766$	0	0
$767$	−8.75104	−0.315982
$768$	0	0
$769$	7.14548	0.257673	0.128836	−	0.991666i	$-0.458876\pi$
$769$	7.14548	0.257673	0.128836	+	0.991666i	$0.458876\pi$
$770$	0	0
$771$	18.3766	0.661816
$772$	0	0
$773$	28.9956	1.04290	0.521450	−	0.853282i	$-0.325391\pi$
$773$	28.9956	1.04290	0.521450	+	0.853282i	$0.325391\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.27919	0.117640
$778$	0	0
$779$	3.57360	0.128038
$780$	0	0
$781$	11.7003	0.418669
$782$	0	0
$783$	39.7164	1.41935
$784$	0	0
$785$	46.1590	1.64748
$786$	0	0
$787$	23.3311	0.831664	0.415832	−	0.909441i	$-0.363490\pi$
$787$	23.3311	0.831664	0.415832	+	0.909441i	$0.363490\pi$
$788$	0	0
$789$	−9.65920	−0.343877
$790$	0	0
$791$	2.39063	0.0850011
$792$	0	0
$793$	5.90909	0.209838
$794$	0	0
$795$	31.0651	1.10176
$796$	0	0
$797$	−19.2543	−0.682021	−0.341011	−	0.940059i	$-0.610769\pi$
$797$	−19.2543	−0.682021	−0.341011	+	0.940059i	$0.610769\pi$
$798$	0	0
$799$	38.6531	1.36745
$800$	0	0
$801$	−26.2393	−0.927119
$802$	0	0
$803$	−7.33815	−0.258958
$804$	0	0
$805$	−61.5238	−2.16843
$806$	0	0
$807$	−6.09091	−0.214410
$808$	0	0
$809$	27.9983	0.984367	0.492184	−	0.870491i	$-0.336199\pi$
$809$	27.9983	0.984367	0.492184	+	0.870491i	$0.336199\pi$
$810$	0	0
$811$	47.4631	1.66665	0.833327	−	0.552780i	$-0.186433\pi$
$811$	47.4631	1.66665	0.833327	+	0.552780i	$0.186433\pi$
$812$	0	0
$813$	0.751039	0.0263401
$814$	0	0
$815$	−2.81172	−0.0984902
$816$	0	0
$817$	−7.72347	−0.270210
$818$	0	0
$819$	−6.01938	−0.210334
$820$	0	0
$821$	−2.14455	−0.0748454	−0.0374227	−	0.999300i	$-0.511915\pi$
$821$	−2.14455	−0.0748454	−0.0374227	+	0.999300i	$0.511915\pi$
$822$	0	0
$823$	−32.8458	−1.14493	−0.572466	−	0.819929i	$-0.694013\pi$
$823$	−32.8458	−1.14493	−0.572466	+	0.819929i	$0.694013\pi$
$824$	0	0
$825$	5.99124	0.208588
$826$	0	0
$827$	−26.6103	−0.925331	−0.462665	−	0.886533i	$-0.653107\pi$
$827$	−26.6103	−0.925331	−0.462665	+	0.886533i	$0.653107\pi$
$828$	0	0
$829$	−50.6871	−1.76044	−0.880219	−	0.474569i	$-0.842604\pi$
$829$	−50.6871	−1.76044	−0.880219	+	0.474569i	$0.842604\pi$
$830$	0	0
$831$	13.4059	0.465044
$832$	0	0
$833$	−10.8528	−0.376027
$834$	0	0
$835$	61.3062	2.12159
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−51.2651	−1.76987	−0.884934	−	0.465716i	$-0.845797\pi$
$839$	−51.2651	−1.76987	−0.884934	+	0.465716i	$0.845797\pi$
$840$	0	0
$841$	59.0021	2.03455
$842$	0	0
$843$	10.3050	0.354924
$844$	0	0
$845$	38.0091	1.30755
$846$	0	0
$847$	19.4167	0.667166
$848$	0	0
$849$	1.02054	0.0350248
$850$	0	0
$851$	−17.9091	−0.613916
$852$	0	0
$853$	7.95893	0.272508	0.136254	−	0.990674i	$-0.456494\pi$
$853$	7.95893	0.272508	0.136254	+	0.990674i	$0.456494\pi$
$854$	0	0
$855$	7.85014	0.268469
$856$	0	0
$857$	−4.09091	−0.139743	−0.0698714	−	0.997556i	$-0.522259\pi$
$857$	−4.09091	−0.139743	−0.0698714	+	0.997556i	$0.522259\pi$
$858$	0	0
$859$	13.3328	0.454910	0.227455	−	0.973789i	$-0.426959\pi$
$859$	13.3328	0.454910	0.227455	+	0.973789i	$0.426959\pi$
$860$	0	0
$861$	5.85926	0.199683
$862$	0	0
$863$	−20.1677	−0.686518	−0.343259	−	0.939241i	$-0.611531\pi$
$863$	−20.1677	−0.686518	−0.343259	+	0.939241i	$0.611531\pi$
$864$	0	0
$865$	9.40586	0.319809
$866$	0	0
$867$	0.253340	0.00860386
$868$	0	0
$869$	−12.4176	−0.421240
$870$	0	0
$871$	13.8073	0.467844
$872$	0	0
$873$	20.4622	0.692539
$874$	0	0
$875$	5.98212	0.202233
$876$	0	0
$877$	−23.8572	−0.805599	−0.402800	−	0.915288i	$-0.631963\pi$
$877$	−23.8572	−0.805599	−0.402800	+	0.915288i	$0.631963\pi$
$878$	0	0
$879$	−16.9545	−0.571863
$880$	0	0
$881$	−47.9182	−1.61441	−0.807203	−	0.590274i	$-0.799020\pi$
$881$	−47.9182	−1.61441	−0.807203	+	0.590274i	$0.799020\pi$
$882$	0	0
$883$	28.5622	0.961194	0.480597	−	0.876942i	$-0.340420\pi$
$883$	28.5622	0.961194	0.480597	+	0.876942i	$0.340420\pi$
$884$	0	0
$885$	18.7121	0.628999
$886$	0	0
$887$	47.3150	1.58868	0.794340	−	0.607473i	$-0.207817\pi$
$887$	47.3150	1.58868	0.794340	+	0.607473i	$0.207817\pi$
$888$	0	0
$889$	2.12667	0.0713262
$890$	0	0
$891$	−4.94407	−0.165632
$892$	0	0
$893$	9.46482	0.316728
$894$	0	0
$895$	−85.9772	−2.87390
$896$	0	0
$897$	−8.54753	−0.285394
$898$	0	0
$899$	0	0
$900$	0	0
$901$	48.9047	1.62925
$902$	0	0
$903$	−12.6634	−0.421410
$904$	0	0
$905$	10.5120	0.349430
$906$	0	0
$907$	15.8982	0.527893	0.263946	−	0.964537i	$-0.414976\pi$
$907$	15.8982	0.527893	0.263946	+	0.964537i	$0.414976\pi$
$908$	0	0
$909$	21.7790	0.722363
$910$	0	0
$911$	−20.2997	−0.672560	−0.336280	−	0.941762i	$-0.609169\pi$
$911$	−20.2997	−0.672560	−0.336280	+	0.941762i	$0.609169\pi$
$912$	0	0
$913$	13.9589	0.461973
$914$	0	0
$915$	−12.6352	−0.417707
$916$	0	0
$917$	−16.0947	−0.531494
$918$	0	0
$919$	32.6191	1.07600	0.538002	−	0.842944i	$-0.319179\pi$
$919$	32.6191	1.07600	0.538002	+	0.842944i	$0.319179\pi$
$920$	0	0
$921$	−7.22936	−0.238215
$922$	0	0
$923$	10.9437	0.360216
$924$	0	0
$925$	11.7413	0.386053
$926$	0	0
$927$	−41.3237	−1.35725
$928$	0	0
$929$	53.4631	1.75407	0.877034	−	0.480429i	$-0.159519\pi$
$929$	53.4631	1.75407	0.877034	+	0.480429i	$0.159519\pi$
$930$	0	0
$931$	−2.65748	−0.0870953
$932$	0	0
$933$	5.89293	0.192926
$934$	0	0
$935$	17.4648	0.571161
$936$	0	0
$937$	−43.0135	−1.40519	−0.702595	−	0.711590i	$-0.747975\pi$
$937$	−43.0135	−1.40519	−0.702595	+	0.711590i	$0.747975\pi$
$938$	0	0
$939$	7.68411	0.250762
$940$	0	0
$941$	−25.3311	−0.825771	−0.412885	−	0.910783i	$-0.635479\pi$
$941$	−25.3311	−0.825771	−0.412885	+	0.910783i	$0.635479\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	29.0886	0.946253
$946$	0	0
$947$	−12.6710	−0.411751	−0.205876	−	0.978578i	$-0.566004\pi$
$947$	−12.6710	−0.411751	−0.205876	+	0.978578i	$0.566004\pi$
$948$	0	0
$949$	−6.86364	−0.222803
$950$	0	0
$951$	10.7921	0.349958
$952$	0	0
$953$	−31.2240	−1.01145	−0.505723	−	0.862696i	$-0.668774\pi$
$953$	−31.2240	−1.01145	−0.505723	+	0.862696i	$0.668774\pi$
$954$	0	0
$955$	−26.2355	−0.848959
$956$	0	0
$957$	9.57360	0.309471
$958$	0	0
$959$	−31.7396	−1.02493
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−1.87333	−0.0603672
$964$	0	0
$965$	−69.0065	−2.22140
$966$	0	0
$967$	6.29441	0.202415	0.101207	−	0.994865i	$-0.467729\pi$
$967$	6.29441	0.202415	0.101207	+	0.994865i	$0.467729\pi$
$968$	0	0
$969$	−3.21320	−0.103223
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	33.4648	1.07283
$974$	0	0
$975$	5.60383	0.179466
$976$	0	0
$977$	−34.1677	−1.09312	−0.546561	−	0.837419i	$-0.684064\pi$
$977$	−34.1677	−1.09312	−0.546561	+	0.837419i	$0.684064\pi$
$978$	0	0
$979$	−14.2944	−0.456851
$980$	0	0
$981$	−32.2587	−1.02994
$982$	0	0
$983$	30.4264	0.970451	0.485226	−	0.874389i	$-0.338737\pi$
$983$	30.4264	0.970451	0.485226	+	0.874389i	$0.338737\pi$
$984$	0	0
$985$	−11.7824	−0.375419
$986$	0	0
$987$	15.5185	0.493958
$988$	0	0
$989$	69.1601	2.19916
$990$	0	0
$991$	−24.1179	−0.766131	−0.383065	−	0.923721i	$-0.625132\pi$
$991$	−24.1179	−0.766131	−0.383065	+	0.923721i	$0.625132\pi$
$992$	0	0
$993$	−11.4308	−0.362745
$994$	0	0
$995$	87.1065	2.76146
$996$	0	0
$997$	−15.9323	−0.504581	−0.252290	−	0.967652i	$-0.581184\pi$
$997$	−15.9323	−0.504581	−0.252290	+	0.967652i	$0.581184\pi$
$998$	0	0
$999$	8.46747	0.267899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.u.1.2		3
4.3	odd	2		1216.2.a.v.1.2		3
8.3	odd	2		304.2.a.g.1.2		3
8.5	even	2		152.2.a.c.1.2	✓	3
24.5	odd	2		1368.2.a.n.1.1		3
24.11	even	2		2736.2.a.bd.1.1		3
40.13	odd	4		3800.2.d.j.3649.4		6
40.19	odd	2		7600.2.a.bv.1.2		3
40.29	even	2		3800.2.a.r.1.2		3
40.37	odd	4		3800.2.d.j.3649.3		6
56.13	odd	2		7448.2.a.bf.1.2		3
152.37	odd	2		2888.2.a.o.1.2		3
152.75	even	2		5776.2.a.bp.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.c.1.2	✓	3	8.5	even	2
304.2.a.g.1.2		3	8.3	odd	2
1216.2.a.u.1.2		3	1.1	even	1	trivial
1216.2.a.v.1.2		3	4.3	odd	2
1368.2.a.n.1.1		3	24.5	odd	2
2736.2.a.bd.1.1		3	24.11	even	2
2888.2.a.o.1.2		3	152.37	odd	2
3800.2.a.r.1.2		3	40.29	even	2
3800.2.d.j.3649.3		6	40.37	odd	4
3800.2.d.j.3649.4		6	40.13	odd	4
5776.2.a.bp.1.2		3	152.75	even	2
7448.2.a.bf.1.2		3	56.13	odd	2
7600.2.a.bv.1.2		3	40.19	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-0.786802 q^{3} -3.29707 q^{5} -2.08387 q^{7} -2.38094 q^{9} -1.29707 q^{11} -1.21320 q^{13} +2.59414 q^{15} +4.08387 q^{17} +1.00000 q^{19} +1.63959 q^{21} -8.95455 q^{23} +5.87067 q^{25} +4.23374 q^{27} +9.38094 q^{29} +1.02054 q^{33} +6.87067 q^{35} +2.00000 q^{37} +0.954547 q^{39} +3.57360 q^{41} -7.72347 q^{43} +7.85014 q^{45} +9.46482 q^{47} -2.65748 q^{49} -3.21320 q^{51} +11.9751 q^{53} +4.27653 q^{55} -0.786802 q^{57} +7.21320 q^{59} -4.87067 q^{61} +4.96158 q^{63} +4.00000 q^{65} -11.3809 q^{67} +7.04545 q^{69} -9.02054 q^{71} +5.65748 q^{73} -4.61906 q^{75} +2.70293 q^{77} +9.57360 q^{79} +3.81172 q^{81} -10.7619 q^{83} -13.4648 q^{85} -7.38094 q^{87} +11.0205 q^{89} +2.52815 q^{91} -3.29707 q^{95} -8.59414 q^{97} +3.08825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{5} + 4 q^{7} + 12 q^{9} + 5 q^{11} - 5 q^{13} - 10 q^{15} + 2 q^{17} + 3 q^{19} + 9 q^{21} - 5 q^{23} + 6 q^{25} - q^{27} + 9 q^{29} - 12 q^{33} + 9 q^{35} + 6 q^{37} - 19 q^{39} + 8 q^{41}+ \cdots + 51 q^{99}+O(q^{100}) \) 3 * q - q^3 - q^5 + 4 * q^7 + 12 * q^9 + 5 * q^11 - 5 * q^13 - 10 * q^15 + 2 * q^17 + 3 * q^19 + 9 * q^21 - 5 * q^23 + 6 * q^25 - q^27 + 9 * q^29 - 12 * q^33 + 9 * q^35 + 6 * q^37 - 19 * q^39 + 8 * q^41 - 17 * q^43 + 27 * q^45 - q^47 + 5 * q^49 - 11 * q^51 - q^53 + 19 * q^55 - q^57 + 23 * q^59 - 3 * q^61 + 47 * q^63 + 12 * q^65 - 15 * q^67 + 43 * q^69 - 12 * q^71 + 4 * q^73 - 33 * q^75 + 17 * q^77 + 26 * q^79 + 47 * q^81 + 6 * q^83 - 11 * q^85 - 3 * q^87 + 18 * q^89 - 17 * q^91 - q^95 - 8 * q^97 + 51 * q^99

Introduction

L-functions

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Database

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