LMFDB - Embedded newform 7744.2.a.dv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, 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("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.19898000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 $ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.935683$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0643165 q^{3} -2.31301 q^{5} +1.63066 q^{7} -2.99586 q^{9} +1.15644 q^{13} -0.148765 q^{15} +3.36056 q^{17} +5.83478 q^{19} +0.104878 q^{21} -7.86209 q^{23} +0.350020 q^{25} -0.385633 q^{27} +5.92691 q^{29} -3.61053 q^{31} -3.77174 q^{35} +0.362348 q^{37} +0.0743784 q^{39} -4.40970 q^{41} -5.15478 q^{43} +6.92946 q^{45} -1.15852 q^{47} -4.34095 q^{49} +0.216140 q^{51} -6.14411 q^{53} +0.375273 q^{57} +8.12241 q^{59} +8.82601 q^{61} -4.88524 q^{63} -2.67486 q^{65} +12.6763 q^{67} -0.505662 q^{69} -15.0072 q^{71} +13.0022 q^{73} +0.0225121 q^{75} +2.07633 q^{79} +8.96279 q^{81} -8.94922 q^{83} -7.77302 q^{85} +0.381198 q^{87} +13.9653 q^{89} +1.88577 q^{91} -0.232217 q^{93} -13.4959 q^{95} +0.00798799 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 2 q^{7} + 7 q^{9} - 4 q^{13} - 8 q^{15} - 9 q^{17} + 5 q^{19} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 26 q^{27} - 10 q^{29} - 12 q^{31} - 26 q^{35} + 12 q^{37} - 2 q^{39} - 17 q^{41} + 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) $ 6 * q + 5 * q^3 - 2 * q^7 + 7 * q^9 - 4 * q^13 - 8 * q^15 - 9 * q^17 + 5 * q^19 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 26 * q^27 - 10 * q^29 - 12 * q^31 - 26 * q^35 + 12 * q^37 - 2 * q^39 - 17 * q^41 + 11 * q^43 - 14 * q^45 - 22 * q^47 - 6 * q^51 - 18 * q^53 - 6 * q^57 + 7 * q^59 - 18 * q^61 - 26 * q^63 - 26 * q^65 + 31 * q^67 - 16 * q^69 - 22 * q^71 + q^73 + 11 * q^75 + 28 * q^79 + 34 * q^81 - 13 * q^83 - 2 * q^85 - 38 * q^87 - q^89 + 26 * q^91 - 34 * q^93 - 2 * q^95 - 21 * q^97

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
$3$	0.0643165	0.0371332	0.0185666	−	0.999828i	$-0.494090\pi$
$3$	0.0643165	0.0371332	0.0185666	+	0.999828i	$0.494090\pi$
$4$	0	0
$5$	−2.31301	−1.03441	−0.517205	−	0.855862i	$-0.673028\pi$
$5$	−2.31301	−1.03441	−0.517205	+	0.855862i	$0.673028\pi$
$6$	0	0
$7$	1.63066	0.616332	0.308166	−	0.951333i	$-0.400285\pi$
$7$	1.63066	0.616332	0.308166	+	0.951333i	$0.400285\pi$
$8$	0	0
$9$	−2.99586	−0.998621
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.15644	0.320739	0.160370	−	0.987057i	$-0.448731\pi$
$13$	1.15644	0.320739	0.160370	+	0.987057i	$0.448731\pi$
$14$	0	0
$15$	−0.148765	−0.0384109
$16$	0	0
$17$	3.36056	0.815057	0.407528	−	0.913193i	$-0.366391\pi$
$17$	3.36056	0.815057	0.407528	+	0.913193i	$0.366391\pi$
$18$	0	0
$19$	5.83478	1.33859	0.669295	−	0.742996i	$-0.266596\pi$
$19$	5.83478	1.33859	0.669295	+	0.742996i	$0.266596\pi$
$20$	0	0
$21$	0.104878	0.0228863
$22$	0	0
$23$	−7.86209	−1.63936	−0.819680	−	0.572822i	$-0.805848\pi$
$23$	−7.86209	−1.63936	−0.819680	+	0.572822i	$0.805848\pi$
$24$	0	0
$25$	0.350020	0.0700040
$26$	0	0
$27$	−0.385633	−0.0742151
$28$	0	0
$29$	5.92691	1.10060	0.550300	−	0.834967i	$-0.314513\pi$
$29$	5.92691	1.10060	0.550300	+	0.834967i	$0.314513\pi$
$30$	0	0
$31$	−3.61053	−0.648470	−0.324235	−	0.945977i	$-0.605107\pi$
$31$	−3.61053	−0.648470	−0.324235	+	0.945977i	$0.605107\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.77174	−0.637540
$36$	0	0
$37$	0.362348	0.0595696	0.0297848	−	0.999556i	$-0.490518\pi$
$37$	0.362348	0.0595696	0.0297848	+	0.999556i	$0.490518\pi$
$38$	0	0
$39$	0.0743784	0.0119101
$40$	0	0
$41$	−4.40970	−0.688679	−0.344340	−	0.938845i	$-0.611897\pi$
$41$	−4.40970	−0.688679	−0.344340	+	0.938845i	$0.611897\pi$
$42$	0	0
$43$	−5.15478	−0.786097	−0.393049	−	0.919518i	$-0.628580\pi$
$43$	−5.15478	−0.786097	−0.393049	+	0.919518i	$0.628580\pi$
$44$	0	0
$45$	6.92946	1.03298
$46$	0	0
$47$	−1.15852	−0.168988	−0.0844941	−	0.996424i	$-0.526927\pi$
$47$	−1.15852	−0.168988	−0.0844941	+	0.996424i	$0.526927\pi$
$48$	0	0
$49$	−4.34095	−0.620135
$50$	0	0
$51$	0.216140	0.0302656
$52$	0	0
$53$	−6.14411	−0.843959	−0.421980	−	0.906605i	$-0.638665\pi$
$53$	−6.14411	−0.843959	−0.421980	+	0.906605i	$0.638665\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.375273	0.0497061
$58$	0	0
$59$	8.12241	1.05745	0.528724	−	0.848794i	$-0.322671\pi$
$59$	8.12241	1.05745	0.528724	+	0.848794i	$0.322671\pi$
$60$	0	0
$61$	8.82601	1.13005	0.565027	−	0.825072i	$-0.308866\pi$
$61$	8.82601	1.13005	0.565027	+	0.825072i	$0.308866\pi$
$62$	0	0
$63$	−4.88524	−0.615482
$64$	0	0
$65$	−2.67486	−0.331776
$66$	0	0
$67$	12.6763	1.54865	0.774326	−	0.632787i	$-0.218089\pi$
$67$	12.6763	1.54865	0.774326	+	0.632787i	$0.218089\pi$
$68$	0	0
$69$	−0.505662	−0.0608746
$70$	0	0
$71$	−15.0072	−1.78103	−0.890515	−	0.454955i	$-0.849656\pi$
$71$	−15.0072	−1.78103	−0.890515	+	0.454955i	$0.849656\pi$
$72$	0	0
$73$	13.0022	1.52179	0.760896	−	0.648874i	$-0.224760\pi$
$73$	13.0022	1.52179	0.760896	+	0.648874i	$0.224760\pi$
$74$	0	0
$75$	0.0225121	0.00259947
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.07633	0.233606	0.116803	−	0.993155i	$-0.462735\pi$
$79$	2.07633	0.233606	0.116803	+	0.993155i	$0.462735\pi$
$80$	0	0
$81$	8.96279	0.995865
$82$	0	0
$83$	−8.94922	−0.982305	−0.491152	−	0.871074i	$-0.663424\pi$
$83$	−8.94922	−0.982305	−0.491152	+	0.871074i	$0.663424\pi$
$84$	0	0
$85$	−7.77302	−0.843103
$86$	0	0
$87$	0.381198	0.0408687
$88$	0	0
$89$	13.9653	1.48032	0.740160	−	0.672431i	$-0.234750\pi$
$89$	13.9653	1.48032	0.740160	+	0.672431i	$0.234750\pi$
$90$	0	0
$91$	1.88577	0.197682
$92$	0	0
$93$	−0.232217	−0.0240797
$94$	0	0
$95$	−13.4959	−1.38465
$96$	0	0
$97$	0.00798799	0.000811057 0	0.000405529	−	1.00000i	$-0.499871\pi$
$97$	0.00798799	0.000811057 0	0.000405529		1.00000i	$0.499871\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0557	−1.00058	−0.500292	−	0.865857i	$-0.666774\pi$
$101$	−10.0557	−1.00058	−0.500292	+	0.865857i	$0.666774\pi$
$102$	0	0
$103$	−4.87601	−0.480447	−0.240224	−	0.970718i	$-0.577221\pi$
$103$	−4.87601	−0.480447	−0.240224	+	0.970718i	$0.577221\pi$
$104$	0	0
$105$	−0.242585	−0.0236739
$106$	0	0
$107$	5.65316	0.546512	0.273256	−	0.961941i	$-0.411899\pi$
$107$	5.65316	0.546512	0.273256	+	0.961941i	$0.411899\pi$
$108$	0	0
$109$	−8.01561	−0.767756	−0.383878	−	0.923384i	$-0.625412\pi$
$109$	−8.01561	−0.767756	−0.383878	+	0.923384i	$0.625412\pi$
$110$	0	0
$111$	0.0233050	0.00221201
$112$	0	0
$113$	−9.62436	−0.905384	−0.452692	−	0.891667i	$-0.649536\pi$
$113$	−9.62436	−0.905384	−0.452692	+	0.891667i	$0.649536\pi$
$114$	0	0
$115$	18.1851	1.69577
$116$	0	0
$117$	−3.46454	−0.320297
$118$	0	0
$119$	5.47994	0.502345
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.283616	−0.0255728
$124$	0	0
$125$	10.7555	0.961997
$126$	0	0
$127$	20.6659	1.83380	0.916900	−	0.399116i	$-0.130683\pi$
$127$	20.6659	1.83380	0.916900	+	0.399116i	$0.130683\pi$
$128$	0	0
$129$	−0.331538	−0.0291903
$130$	0	0
$131$	17.9382	1.56727	0.783633	−	0.621224i	$-0.213364\pi$
$131$	17.9382	1.56727	0.783633	+	0.621224i	$0.213364\pi$
$132$	0	0
$133$	9.51455	0.825016
$134$	0	0
$135$	0.891973	0.0767688
$136$	0	0
$137$	−4.44854	−0.380064	−0.190032	−	0.981778i	$-0.560859\pi$
$137$	−4.44854	−0.380064	−0.190032	+	0.981778i	$0.560859\pi$
$138$	0	0
$139$	5.52337	0.468486	0.234243	−	0.972178i	$-0.424739\pi$
$139$	5.52337	0.468486	0.234243	+	0.972178i	$0.424739\pi$
$140$	0	0
$141$	−0.0745123	−0.00627506
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−13.7090	−1.13847
$146$	0	0
$147$	−0.279195	−0.0230276
$148$	0	0
$149$	−12.1307	−0.993788	−0.496894	−	0.867811i	$-0.665526\pi$
$149$	−12.1307	−0.993788	−0.496894	+	0.867811i	$0.665526\pi$
$150$	0	0
$151$	−15.1453	−1.23251	−0.616255	−	0.787547i	$-0.711351\pi$
$151$	−15.1453	−1.23251	−0.616255	+	0.787547i	$0.711351\pi$
$152$	0	0
$153$	−10.0678	−0.813933
$154$	0	0
$155$	8.35119	0.670784
$156$	0	0
$157$	−16.4555	−1.31329	−0.656646	−	0.754199i	$-0.728026\pi$
$157$	−16.4555	−1.31329	−0.656646	+	0.754199i	$0.728026\pi$
$158$	0	0
$159$	−0.395168	−0.0313389
$160$	0	0
$161$	−12.8204	−1.01039
$162$	0	0
$163$	−12.4470	−0.974924	−0.487462	−	0.873144i	$-0.662077\pi$
$163$	−12.4470	−0.974924	−0.487462	+	0.873144i	$0.662077\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.8104	1.22344	0.611722	−	0.791073i	$-0.290477\pi$
$167$	15.8104	1.22344	0.611722	+	0.791073i	$0.290477\pi$
$168$	0	0
$169$	−11.6626	−0.897126
$170$	0	0
$171$	−17.4802	−1.33675
$172$	0	0
$173$	−0.828750	−0.0630087	−0.0315044	−	0.999504i	$-0.510030\pi$
$173$	−0.828750	−0.0630087	−0.0315044	+	0.999504i	$0.510030\pi$
$174$	0	0
$175$	0.570764	0.0431457
$176$	0	0
$177$	0.522405	0.0392664
$178$	0	0
$179$	−16.8994	−1.26312	−0.631561	−	0.775326i	$-0.717585\pi$
$179$	−16.8994	−1.26312	−0.631561	+	0.775326i	$0.717585\pi$
$180$	0	0
$181$	−11.2573	−0.836747	−0.418374	−	0.908275i	$-0.637400\pi$
$181$	−11.2573	−0.836747	−0.418374	+	0.908275i	$0.637400\pi$
$182$	0	0
$183$	0.567658	0.0419625
$184$	0	0
$185$	−0.838115	−0.0616194
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−0.628837	−0.0457411
$190$	0	0
$191$	−25.9575	−1.87822	−0.939108	−	0.343622i	$-0.888346\pi$
$191$	−25.9575	−1.87822	−0.939108	+	0.343622i	$0.888346\pi$
$192$	0	0
$193$	0.630257	0.0453669	0.0226835	−	0.999743i	$-0.492779\pi$
$193$	0.630257	0.0453669	0.0226835	+	0.999743i	$0.492779\pi$
$194$	0	0
$195$	−0.172038	−0.0123199
$196$	0	0
$197$	−22.7866	−1.62348	−0.811741	−	0.584018i	$-0.801480\pi$
$197$	−22.7866	−1.62348	−0.811741	+	0.584018i	$0.801480\pi$
$198$	0	0
$199$	−17.6443	−1.25077	−0.625386	−	0.780316i	$-0.715058\pi$
$199$	−17.6443	−1.25077	−0.625386	+	0.780316i	$0.715058\pi$
$200$	0	0
$201$	0.815293	0.0575063
$202$	0	0
$203$	9.66478	0.678334
$204$	0	0
$205$	10.1997	0.712377
$206$	0	0
$207$	23.5537	1.63710
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.73572	−0.601392	−0.300696	−	0.953720i	$-0.597219\pi$
$211$	−8.73572	−0.601392	−0.300696	+	0.953720i	$0.597219\pi$
$212$	0	0
$213$	−0.965212	−0.0661352
$214$	0	0
$215$	11.9231	0.813147
$216$	0	0
$217$	−5.88755	−0.399673
$218$	0	0
$219$	0.836256	0.0565089
$220$	0	0
$221$	3.88630	0.261421
$222$	0	0
$223$	8.23913	0.551733	0.275867	−	0.961196i	$-0.411035\pi$
$223$	8.23913	0.551733	0.275867	+	0.961196i	$0.411035\pi$
$224$	0	0
$225$	−1.04861	−0.0699074
$226$	0	0
$227$	3.16827	0.210285	0.105143	−	0.994457i	$-0.466470\pi$
$227$	3.16827	0.210285	0.105143	+	0.994457i	$0.466470\pi$
$228$	0	0
$229$	8.01670	0.529758	0.264879	−	0.964282i	$-0.414668\pi$
$229$	8.01670	0.529758	0.264879	+	0.964282i	$0.414668\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.3572	−1.20262	−0.601311	−	0.799015i	$-0.705355\pi$
$233$	−18.3572	−1.20262	−0.601311	+	0.799015i	$0.705355\pi$
$234$	0	0
$235$	2.67968	0.174803
$236$	0	0
$237$	0.133543	0.00867452
$238$	0	0
$239$	−14.9536	−0.967266	−0.483633	−	0.875271i	$-0.660683\pi$
$239$	−14.9536	−0.967266	−0.483633	+	0.875271i	$0.660683\pi$
$240$	0	0
$241$	−24.4586	−1.57552	−0.787758	−	0.615984i	$-0.788758\pi$
$241$	−24.4586	−1.57552	−0.787758	+	0.615984i	$0.788758\pi$
$242$	0	0
$243$	1.73335	0.111195
$244$	0	0
$245$	10.0407	0.641474
$246$	0	0
$247$	6.74759	0.429339
$248$	0	0
$249$	−0.575583	−0.0364761
$250$	0	0
$251$	7.09048	0.447547	0.223773	−	0.974641i	$-0.428162\pi$
$251$	7.09048	0.447547	0.223773	+	0.974641i	$0.428162\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.499934	−0.0313071
$256$	0	0
$257$	19.0829	1.19036	0.595181	−	0.803592i	$-0.297080\pi$
$257$	19.0829	1.19036	0.595181	+	0.803592i	$0.297080\pi$
$258$	0	0
$259$	0.590867	0.0367147
$260$	0	0
$261$	−17.7562	−1.09908
$262$	0	0
$263$	−16.3340	−1.00720	−0.503598	−	0.863938i	$-0.667991\pi$
$263$	−16.3340	−1.00720	−0.503598	+	0.863938i	$0.667991\pi$
$264$	0	0
$265$	14.2114	0.873000
$266$	0	0
$267$	0.898200	0.0549689
$268$	0	0
$269$	−17.7091	−1.07974	−0.539871	−	0.841748i	$-0.681527\pi$
$269$	−17.7091	−1.07974	−0.539871	+	0.841748i	$0.681527\pi$
$270$	0	0
$271$	9.21399	0.559710	0.279855	−	0.960042i	$-0.409714\pi$
$271$	9.21399	0.559710	0.279855	+	0.960042i	$0.409714\pi$
$272$	0	0
$273$	0.121286	0.00734055
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.7217	−1.78581	−0.892903	−	0.450249i	$-0.851335\pi$
$277$	−29.7217	−1.78581	−0.892903	+	0.450249i	$0.851335\pi$
$278$	0	0
$279$	10.8167	0.647576
$280$	0	0
$281$	−14.4799	−0.863797	−0.431898	−	0.901922i	$-0.642156\pi$
$281$	−14.4799	−0.863797	−0.431898	+	0.901922i	$0.642156\pi$
$282$	0	0
$283$	22.3729	1.32993	0.664965	−	0.746875i	$-0.268447\pi$
$283$	22.3729	1.32993	0.664965	+	0.746875i	$0.268447\pi$
$284$	0	0
$285$	−0.868010	−0.0514165
$286$	0	0
$287$	−7.19072	−0.424455
$288$	0	0
$289$	−5.70661	−0.335683
$290$	0	0
$291$	0.000513759 0	3.01171e−5 0
$292$	0	0
$293$	−23.2823	−1.36017	−0.680084	−	0.733134i	$-0.738057\pi$
$293$	−23.2823	−1.36017	−0.680084	+	0.733134i	$0.738057\pi$
$294$	0	0
$295$	−18.7872	−1.09384
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.09206	−0.525807
$300$	0	0
$301$	−8.40570	−0.484497
$302$	0	0
$303$	−0.646750	−0.0371548
$304$	0	0
$305$	−20.4147	−1.16894
$306$	0	0
$307$	−23.0645	−1.31636	−0.658179	−	0.752861i	$-0.728673\pi$
$307$	−23.0645	−1.31636	−0.658179	+	0.752861i	$0.728673\pi$
$308$	0	0
$309$	−0.313608	−0.0178405
$310$	0	0
$311$	2.61858	0.148486	0.0742431	−	0.997240i	$-0.476346\pi$
$311$	2.61858	0.148486	0.0742431	+	0.997240i	$0.476346\pi$
$312$	0	0
$313$	−31.5910	−1.78563	−0.892815	−	0.450424i	$-0.851273\pi$
$313$	−31.5910	−1.78563	−0.892815	+	0.450424i	$0.851273\pi$
$314$	0	0
$315$	11.2996	0.636661
$316$	0	0
$317$	11.4254	0.641715	0.320857	−	0.947128i	$-0.396029\pi$
$317$	11.4254	0.641715	0.320857	+	0.947128i	$0.396029\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0.363592	0.0202937
$322$	0	0
$323$	19.6082	1.09103
$324$	0	0
$325$	0.404778	0.0224530
$326$	0	0
$327$	−0.515536	−0.0285092
$328$	0	0
$329$	−1.88916	−0.104153
$330$	0	0
$331$	19.5632	1.07529	0.537645	−	0.843171i	$-0.319314\pi$
$331$	19.5632	1.07529	0.537645	+	0.843171i	$0.319314\pi$
$332$	0	0
$333$	−1.08555	−0.0594875
$334$	0	0
$335$	−29.3203	−1.60194
$336$	0	0
$337$	−3.95997	−0.215713	−0.107857	−	0.994166i	$-0.534399\pi$
$337$	−3.95997	−0.215713	−0.107857	+	0.994166i	$0.534399\pi$
$338$	0	0
$339$	−0.619006	−0.0336198
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.4932	−0.998541
$344$	0	0
$345$	1.16960	0.0629693
$346$	0	0
$347$	1.89515	0.101737	0.0508686	−	0.998705i	$-0.483801\pi$
$347$	1.89515	0.101737	0.0508686	+	0.998705i	$0.483801\pi$
$348$	0	0
$349$	27.5241	1.47333	0.736664	−	0.676258i	$-0.236400\pi$
$349$	27.5241	1.47333	0.736664	+	0.676258i	$0.236400\pi$
$350$	0	0
$351$	−0.445962	−0.0238037
$352$	0	0
$353$	−17.3853	−0.925325	−0.462663	−	0.886534i	$-0.653106\pi$
$353$	−17.3853	−0.925325	−0.462663	+	0.886534i	$0.653106\pi$
$354$	0	0
$355$	34.7119	1.84231
$356$	0	0
$357$	0.352451	0.0186537
$358$	0	0
$359$	0.964744	0.0509172	0.0254586	−	0.999676i	$-0.491895\pi$
$359$	0.964744	0.0509172	0.0254586	+	0.999676i	$0.491895\pi$
$360$	0	0
$361$	15.0447	0.791825
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.0742	−1.57416
$366$	0	0
$367$	20.7760	1.08450	0.542249	−	0.840218i	$-0.317573\pi$
$367$	20.7760	1.08450	0.542249	+	0.840218i	$0.317573\pi$
$368$	0	0
$369$	13.2109	0.687729
$370$	0	0
$371$	−10.0190	−0.520159
$372$	0	0
$373$	2.62662	0.136001	0.0680006	−	0.997685i	$-0.478338\pi$
$373$	2.62662	0.136001	0.0680006	+	0.997685i	$0.478338\pi$
$374$	0	0
$375$	0.691753	0.0357220
$376$	0	0
$377$	6.85413	0.353006
$378$	0	0
$379$	4.79962	0.246540	0.123270	−	0.992373i	$-0.460662\pi$
$379$	4.79962	0.246540	0.123270	+	0.992373i	$0.460662\pi$
$380$	0	0
$381$	1.32916	0.0680948
$382$	0	0
$383$	−17.1461	−0.876125	−0.438062	−	0.898945i	$-0.644335\pi$
$383$	−17.1461	−0.876125	−0.438062	+	0.898945i	$0.644335\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.4430	0.785013
$388$	0	0
$389$	15.4901	0.785379	0.392690	−	0.919671i	$-0.371545\pi$
$389$	15.4901	0.785379	0.392690	+	0.919671i	$0.371545\pi$
$390$	0	0
$391$	−26.4211	−1.33617
$392$	0	0
$393$	1.15372	0.0581975
$394$	0	0
$395$	−4.80258	−0.241644
$396$	0	0
$397$	15.5364	0.779750	0.389875	−	0.920868i	$-0.372518\pi$
$397$	15.5364	0.779750	0.389875	+	0.920868i	$0.372518\pi$
$398$	0	0
$399$	0.611943	0.0306354
$400$	0	0
$401$	−18.7515	−0.936403	−0.468202	−	0.883622i	$-0.655098\pi$
$401$	−18.7515	−0.936403	−0.468202	+	0.883622i	$0.655098\pi$
$402$	0	0
$403$	−4.17537	−0.207990
$404$	0	0
$405$	−20.7310	−1.03013
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−36.7959	−1.81944	−0.909719	−	0.415224i	$-0.863703\pi$
$409$	−36.7959	−1.81944	−0.909719	+	0.415224i	$0.863703\pi$
$410$	0	0
$411$	−0.286115	−0.0141130
$412$	0	0
$413$	13.2449	0.651739
$414$	0	0
$415$	20.6996	1.01611
$416$	0	0
$417$	0.355244	0.0173964
$418$	0	0
$419$	−18.6797	−0.912565	−0.456283	−	0.889835i	$-0.650819\pi$
$419$	−18.6797	−0.912565	−0.456283	+	0.889835i	$0.650819\pi$
$420$	0	0
$421$	1.81263	0.0883420	0.0441710	−	0.999024i	$-0.485935\pi$
$421$	1.81263	0.0883420	0.0441710	+	0.999024i	$0.485935\pi$
$422$	0	0
$423$	3.47078	0.168755
$424$	0	0
$425$	1.17626	0.0570572
$426$	0	0
$427$	14.3922	0.696488
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.8525	0.956260	0.478130	−	0.878289i	$-0.341315\pi$
$431$	19.8525	0.956260	0.478130	+	0.878289i	$0.341315\pi$
$432$	0	0
$433$	16.6917	0.802153	0.401076	−	0.916045i	$-0.368636\pi$
$433$	16.6917	0.802153	0.401076	+	0.916045i	$0.368636\pi$
$434$	0	0
$435$	−0.881715	−0.0422750
$436$	0	0
$437$	−45.8736	−2.19443
$438$	0	0
$439$	33.3999	1.59409	0.797044	−	0.603921i	$-0.206396\pi$
$439$	33.3999	1.59409	0.797044	+	0.603921i	$0.206396\pi$
$440$	0	0
$441$	13.0049	0.619280
$442$	0	0
$443$	9.09056	0.431906	0.215953	−	0.976404i	$-0.430714\pi$
$443$	9.09056	0.431906	0.215953	+	0.976404i	$0.430714\pi$
$444$	0	0
$445$	−32.3019	−1.53126
$446$	0	0
$447$	−0.780206	−0.0369025
$448$	0	0
$449$	−5.13109	−0.242151	−0.121076	−	0.992643i	$-0.538634\pi$
$449$	−5.13109	−0.242151	−0.121076	+	0.992643i	$0.538634\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−0.974095	−0.0457670
$454$	0	0
$455$	−4.36180	−0.204484
$456$	0	0
$457$	−3.72570	−0.174281	−0.0871404	−	0.996196i	$-0.527773\pi$
$457$	−3.72570	−0.174281	−0.0871404	+	0.996196i	$0.527773\pi$
$458$	0	0
$459$	−1.29594	−0.0604895
$460$	0	0
$461$	0.439230	0.0204570	0.0102285	−	0.999948i	$-0.496744\pi$
$461$	0.439230	0.0204570	0.0102285	+	0.999948i	$0.496744\pi$
$462$	0	0
$463$	−26.8109	−1.24601	−0.623003	−	0.782219i	$-0.714088\pi$
$463$	−26.8109	−1.24601	−0.623003	+	0.782219i	$0.714088\pi$
$464$	0	0
$465$	0.537120	0.0249083
$466$	0	0
$467$	−16.9363	−0.783720	−0.391860	−	0.920025i	$-0.628168\pi$
$467$	−16.9363	−0.783720	−0.391860	+	0.920025i	$0.628168\pi$
$468$	0	0
$469$	20.6707	0.954483
$470$	0	0
$471$	−1.05836	−0.0487667
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.04229	0.0937067
$476$	0	0
$477$	18.4069	0.842795
$478$	0	0
$479$	−3.56180	−0.162743	−0.0813715	−	0.996684i	$-0.525930\pi$
$479$	−3.56180	−0.162743	−0.0813715	+	0.996684i	$0.525930\pi$
$480$	0	0
$481$	0.419035	0.0191063
$482$	0	0
$483$	−0.824563	−0.0375189
$484$	0	0
$485$	−0.0184763	−0.000838966 0
$486$	0	0
$487$	−17.9696	−0.814280	−0.407140	−	0.913366i	$-0.633474\pi$
$487$	−17.9696	−0.814280	−0.407140	+	0.913366i	$0.633474\pi$
$488$	0	0
$489$	−0.800548	−0.0362020
$490$	0	0
$491$	−12.2524	−0.552941	−0.276471	−	0.961022i	$-0.589165\pi$
$491$	−12.2524	−0.552941	−0.276471	+	0.961022i	$0.589165\pi$
$492$	0	0
$493$	19.9178	0.897051
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.4717	−1.09770
$498$	0	0
$499$	−20.2052	−0.904508	−0.452254	−	0.891889i	$-0.649380\pi$
$499$	−20.2052	−0.904508	−0.452254	+	0.891889i	$0.649380\pi$
$500$	0	0
$501$	1.01687	0.0454304
$502$	0	0
$503$	16.3770	0.730214	0.365107	−	0.930966i	$-0.381032\pi$
$503$	16.3770	0.730214	0.365107	+	0.930966i	$0.381032\pi$
$504$	0	0
$505$	23.2590	1.03501
$506$	0	0
$507$	−0.750100	−0.0333131
$508$	0	0
$509$	−4.61829	−0.204702	−0.102351	−	0.994748i	$-0.532637\pi$
$509$	−4.61829	−0.204702	−0.102351	+	0.994748i	$0.532637\pi$
$510$	0	0
$511$	21.2022	0.937928
$512$	0	0
$513$	−2.25008	−0.0993437
$514$	0	0
$515$	11.2783	0.496979
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−0.0533023	−0.00233971
$520$	0	0
$521$	−11.5905	−0.507789	−0.253894	−	0.967232i	$-0.581712\pi$
$521$	−11.5905	−0.507789	−0.253894	+	0.967232i	$0.581712\pi$
$522$	0	0
$523$	29.8079	1.30341	0.651705	−	0.758472i	$-0.274054\pi$
$523$	29.8079	1.30341	0.651705	+	0.758472i	$0.274054\pi$
$524$	0	0
$525$	0.0367095	0.00160213
$526$	0	0
$527$	−12.1334	−0.528540
$528$	0	0
$529$	38.8125	1.68750
$530$	0	0
$531$	−24.3336	−1.05599
$532$	0	0
$533$	−5.09956	−0.220887
$534$	0	0
$535$	−13.0758	−0.565317
$536$	0	0
$537$	−1.08691	−0.0469037
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.41990	0.104039	0.0520197	−	0.998646i	$-0.483434\pi$
$541$	2.41990	0.104039	0.0520197	+	0.998646i	$0.483434\pi$
$542$	0	0
$543$	−0.724029	−0.0310711
$544$	0	0
$545$	18.5402	0.794174
$546$	0	0
$547$	13.4096	0.573352	0.286676	−	0.958028i	$-0.407450\pi$
$547$	13.4096	0.573352	0.286676	+	0.958028i	$0.407450\pi$
$548$	0	0
$549$	−26.4415	−1.12850
$550$	0	0
$551$	34.5822	1.47325
$552$	0	0
$553$	3.38580	0.143979
$554$	0	0
$555$	−0.0539046	−0.00228812
$556$	0	0
$557$	−7.54076	−0.319512	−0.159756	−	0.987157i	$-0.551071\pi$
$557$	−7.54076	−0.319512	−0.159756	+	0.987157i	$0.551071\pi$
$558$	0	0
$559$	−5.96121	−0.252132
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.22795	0.0517519	0.0258759	−	0.999665i	$-0.491763\pi$
$563$	1.22795	0.0517519	0.0258759	+	0.999665i	$0.491763\pi$
$564$	0	0
$565$	22.2613	0.936538
$566$	0	0
$567$	14.6153	0.613783
$568$	0	0
$569$	13.4679	0.564605	0.282303	−	0.959325i	$-0.408902\pi$
$569$	13.4679	0.564605	0.282303	+	0.959325i	$0.408902\pi$
$570$	0	0
$571$	11.8746	0.496935	0.248467	−	0.968640i	$-0.420073\pi$
$571$	11.8746	0.496935	0.248467	+	0.968640i	$0.420073\pi$
$572$	0	0
$573$	−1.66949	−0.0697441
$574$	0	0
$575$	−2.75189	−0.114762
$576$	0	0
$577$	20.3792	0.848397	0.424199	−	0.905569i	$-0.360556\pi$
$577$	20.3792	0.848397	0.424199	+	0.905569i	$0.360556\pi$
$578$	0	0
$579$	0.0405359	0.00168462
$580$	0	0
$581$	−14.5931	−0.605425
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.01353	0.331319
$586$	0	0
$587$	−23.7047	−0.978399	−0.489199	−	0.872172i	$-0.662711\pi$
$587$	−23.7047	−0.978399	−0.489199	+	0.872172i	$0.662711\pi$
$588$	0	0
$589$	−21.0666	−0.868036
$590$	0	0
$591$	−1.46556	−0.0602850
$592$	0	0
$593$	−40.7647	−1.67400	−0.837002	−	0.547199i	$-0.815694\pi$
$593$	−40.7647	−1.67400	−0.837002	+	0.547199i	$0.815694\pi$
$594$	0	0
$595$	−12.6752	−0.519631
$596$	0	0
$597$	−1.13482	−0.0464451
$598$	0	0
$599$	23.4038	0.956252	0.478126	−	0.878291i	$-0.341316\pi$
$599$	23.4038	0.956252	0.478126	+	0.878291i	$0.341316\pi$
$600$	0	0
$601$	32.3471	1.31947	0.659733	−	0.751500i	$-0.270669\pi$
$601$	32.3471	1.31947	0.659733	+	0.751500i	$0.270669\pi$
$602$	0	0
$603$	−37.9763	−1.54652
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.0266	−1.13757	−0.568783	−	0.822488i	$-0.692586\pi$
$607$	−28.0266	−1.13757	−0.568783	+	0.822488i	$0.692586\pi$
$608$	0	0
$609$	0.621605	0.0251887
$610$	0	0
$611$	−1.33977	−0.0542012
$612$	0	0
$613$	−2.36787	−0.0956373	−0.0478186	−	0.998856i	$-0.515227\pi$
$613$	−2.36787	−0.0956373	−0.0478186	+	0.998856i	$0.515227\pi$
$614$	0	0
$615$	0.656008	0.0264528
$616$	0	0
$617$	−10.4196	−0.419477	−0.209738	−	0.977758i	$-0.567261\pi$
$617$	−10.4196	−0.419477	−0.209738	+	0.977758i	$0.567261\pi$
$618$	0	0
$619$	−32.8611	−1.32080	−0.660400	−	0.750914i	$-0.729613\pi$
$619$	−32.8611	−1.32080	−0.660400	+	0.750914i	$0.729613\pi$
$620$	0	0
$621$	3.03188	0.121665
$622$	0	0
$623$	22.7727	0.912368
$624$	0	0
$625$	−26.6276	−1.06510
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.21769	0.0485526
$630$	0	0
$631$	−20.7502	−0.826050	−0.413025	−	0.910720i	$-0.635528\pi$
$631$	−20.7502	−0.826050	−0.413025	+	0.910720i	$0.635528\pi$
$632$	0	0
$633$	−0.561851	−0.0223316
$634$	0	0
$635$	−47.8004	−1.89690
$636$	0	0
$637$	−5.02006	−0.198902
$638$	0	0
$639$	44.9596	1.77857
$640$	0	0
$641$	−0.977465	−0.0386075	−0.0193038	−	0.999814i	$-0.506145\pi$
$641$	−0.977465	−0.0386075	−0.0193038	+	0.999814i	$0.506145\pi$
$642$	0	0
$643$	−38.2108	−1.50689	−0.753443	−	0.657514i	$-0.771608\pi$
$643$	−38.2108	−1.50689	−0.753443	+	0.657514i	$0.771608\pi$
$644$	0	0
$645$	0.766850	0.0301947
$646$	0	0
$647$	34.9743	1.37498	0.687491	−	0.726193i	$-0.258712\pi$
$647$	34.9743	1.37498	0.687491	+	0.726193i	$0.258712\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−0.378666	−0.0148411
$652$	0	0
$653$	−35.3919	−1.38499	−0.692496	−	0.721422i	$-0.743489\pi$
$653$	−35.3919	−1.38499	−0.692496	+	0.721422i	$0.743489\pi$
$654$	0	0
$655$	−41.4912	−1.62120
$656$	0	0
$657$	−38.9528	−1.51969
$658$	0	0
$659$	23.8852	0.930437	0.465219	−	0.885196i	$-0.345976\pi$
$659$	23.8852	0.930437	0.465219	+	0.885196i	$0.345976\pi$
$660$	0	0
$661$	28.4658	1.10719	0.553596	−	0.832785i	$-0.313255\pi$
$661$	28.4658	1.10719	0.553596	+	0.832785i	$0.313255\pi$
$662$	0	0
$663$	0.249953	0.00970738
$664$	0	0
$665$	−22.0073	−0.853405
$666$	0	0
$667$	−46.5979	−1.80428
$668$	0	0
$669$	0.529912	0.0204876
$670$	0	0
$671$	0	0
$672$	0	0
$673$	44.7473	1.72488	0.862440	−	0.506159i	$-0.168935\pi$
$673$	44.7473	1.72488	0.862440	+	0.506159i	$0.168935\pi$
$674$	0	0
$675$	−0.134979	−0.00519535
$676$	0	0
$677$	5.90113	0.226799	0.113399	−	0.993549i	$-0.463826\pi$
$677$	5.90113	0.226799	0.113399	+	0.993549i	$0.463826\pi$
$678$	0	0
$679$	0.0130257	0.000499880 0
$680$	0	0
$681$	0.203772	0.00780856
$682$	0	0
$683$	−35.5174	−1.35904	−0.679518	−	0.733659i	$-0.737811\pi$
$683$	−35.5174	−1.35904	−0.679518	+	0.733659i	$0.737811\pi$
$684$	0	0
$685$	10.2895	0.393142
$686$	0	0
$687$	0.515606	0.0196716
$688$	0	0
$689$	−7.10532	−0.270691
$690$	0	0
$691$	21.1563	0.804824	0.402412	−	0.915459i	$-0.368172\pi$
$691$	21.1563	0.804824	0.402412	+	0.915459i	$0.368172\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.7756	−0.484607
$696$	0	0
$697$	−14.8191	−0.561312
$698$	0	0
$699$	−1.18067	−0.0446571
$700$	0	0
$701$	19.3398	0.730455	0.365228	−	0.930918i	$-0.380991\pi$
$701$	19.3398	0.730455	0.365228	+	0.930918i	$0.380991\pi$
$702$	0	0
$703$	2.11422	0.0797394
$704$	0	0
$705$	0.172348	0.00649099
$706$	0	0
$707$	−16.3975	−0.616692
$708$	0	0
$709$	26.6296	1.00009	0.500047	−	0.865998i	$-0.333316\pi$
$709$	26.6296	1.00009	0.500047	+	0.865998i	$0.333316\pi$
$710$	0	0
$711$	−6.22041	−0.233284
$712$	0	0
$713$	28.3863	1.06308
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.961761	−0.0359176
$718$	0	0
$719$	34.2435	1.27707	0.638533	−	0.769594i	$-0.279541\pi$
$719$	34.2435	1.27707	0.638533	+	0.769594i	$0.279541\pi$
$720$	0	0
$721$	−7.95111	−0.296115
$722$	0	0
$723$	−1.57309	−0.0585039
$724$	0	0
$725$	2.07454	0.0770463
$726$	0	0
$727$	38.7547	1.43733	0.718666	−	0.695355i	$-0.244753\pi$
$727$	38.7547	1.43733	0.718666	+	0.695355i	$0.244753\pi$
$728$	0	0
$729$	−26.7769	−0.991736
$730$	0	0
$731$	−17.3230	−0.640714
$732$	0	0
$733$	27.9567	1.03260	0.516301	−	0.856407i	$-0.327308\pi$
$733$	27.9567	1.03260	0.516301	+	0.856407i	$0.327308\pi$
$734$	0	0
$735$	0.645780	0.0238200
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−0.0393633	−0.00144800	−0.000724001	−	1.00000i	$-0.500230\pi$
$739$	−0.0393633	−0.00144800	−0.000724001		1.00000i	$0.500230\pi$
$740$	0	0
$741$	0.433981	0.0159427
$742$	0	0
$743$	26.1740	0.960231	0.480116	−	0.877205i	$-0.340595\pi$
$743$	26.1740	0.960231	0.480116	+	0.877205i	$0.340595\pi$
$744$	0	0
$745$	28.0585	1.02798
$746$	0	0
$747$	26.8106	0.980950
$748$	0	0
$749$	9.21839	0.336833
$750$	0	0
$751$	−19.2932	−0.704017	−0.352009	−	0.935997i	$-0.614501\pi$
$751$	−19.2932	−0.704017	−0.352009	+	0.935997i	$0.614501\pi$
$752$	0	0
$753$	0.456035	0.0166188
$754$	0	0
$755$	35.0313	1.27492
$756$	0	0
$757$	−45.7863	−1.66413	−0.832065	−	0.554678i	$-0.812841\pi$
$757$	−45.7863	−1.66413	−0.832065	+	0.554678i	$0.812841\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.2108	−0.478891	−0.239445	−	0.970910i	$-0.576966\pi$
$761$	−13.2108	−0.478891	−0.239445	+	0.970910i	$0.576966\pi$
$762$	0	0
$763$	−13.0707	−0.473192
$764$	0	0
$765$	23.2869	0.841940
$766$	0	0
$767$	9.39311	0.339165
$768$	0	0
$769$	−3.36696	−0.121416	−0.0607078	−	0.998156i	$-0.519336\pi$
$769$	−3.36696	−0.121416	−0.0607078	+	0.998156i	$0.519336\pi$
$770$	0	0
$771$	1.22735	0.0442019
$772$	0	0
$773$	−48.2075	−1.73390	−0.866952	−	0.498392i	$-0.833924\pi$
$773$	−48.2075	−1.73390	−0.866952	+	0.498392i	$0.833924\pi$
$774$	0	0
$775$	−1.26376	−0.0453955
$776$	0	0
$777$	0.0380025	0.00136333
$778$	0	0
$779$	−25.7296	−0.921860
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.28561	−0.0816811
$784$	0	0
$785$	38.0618	1.35848
$786$	0	0
$787$	25.8460	0.921310	0.460655	−	0.887579i	$-0.347615\pi$
$787$	25.8460	0.921310	0.460655	+	0.887579i	$0.347615\pi$
$788$	0	0
$789$	−1.05054	−0.0374004
$790$	0	0
$791$	−15.6941	−0.558017
$792$	0	0
$793$	10.2068	0.362453
$794$	0	0
$795$	0.914028	0.0324172
$796$	0	0
$797$	−0.215683	−0.00763990	−0.00381995	−	0.999993i	$-0.501216\pi$
$797$	−0.215683	−0.00763990	−0.00381995	+	0.999993i	$0.501216\pi$
$798$	0	0
$799$	−3.89330	−0.137735
$800$	0	0
$801$	−41.8382	−1.47828
$802$	0	0
$803$	0	0
$804$	0	0
$805$	29.6537	1.04516
$806$	0	0
$807$	−1.13899	−0.0400942
$808$	0	0
$809$	−23.6384	−0.831082	−0.415541	−	0.909574i	$-0.636408\pi$
$809$	−23.6384	−0.831082	−0.415541	+	0.909574i	$0.636408\pi$
$810$	0	0
$811$	−37.8284	−1.32834	−0.664168	−	0.747584i	$-0.731214\pi$
$811$	−37.8284	−1.32834	−0.664168	+	0.747584i	$0.731214\pi$
$812$	0	0
$813$	0.592612	0.0207838
$814$	0	0
$815$	28.7901	1.00847
$816$	0	0
$817$	−30.0770	−1.05226
$818$	0	0
$819$	−5.64950	−0.197409
$820$	0	0
$821$	−26.7163	−0.932404	−0.466202	−	0.884678i	$-0.654378\pi$
$821$	−26.7163	−0.932404	−0.466202	+	0.884678i	$0.654378\pi$
$822$	0	0
$823$	−19.5534	−0.681589	−0.340794	−	0.940138i	$-0.610696\pi$
$823$	−19.5534	−0.681589	−0.340794	+	0.940138i	$0.610696\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.7772	0.931133	0.465567	−	0.885013i	$-0.345851\pi$
$827$	26.7772	0.931133	0.465567	+	0.885013i	$0.345851\pi$
$828$	0	0
$829$	−34.5078	−1.19851	−0.599253	−	0.800560i	$-0.704536\pi$
$829$	−34.5078	−1.19851	−0.599253	+	0.800560i	$0.704536\pi$
$830$	0	0
$831$	−1.91160	−0.0663126
$832$	0	0
$833$	−14.5880	−0.505445
$834$	0	0
$835$	−36.5696	−1.26554
$836$	0	0
$837$	1.39234	0.0481263
$838$	0	0
$839$	32.8263	1.13329	0.566645	−	0.823962i	$-0.308241\pi$
$839$	32.8263	1.13329	0.566645	+	0.823962i	$0.308241\pi$
$840$	0	0
$841$	6.12824	0.211319
$842$	0	0
$843$	−0.931295	−0.0320755
$844$	0	0
$845$	26.9758	0.927996
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.43895	0.0493845
$850$	0	0
$851$	−2.84881	−0.0976560
$852$	0	0
$853$	30.2939	1.03724	0.518621	−	0.855004i	$-0.326446\pi$
$853$	30.2939	1.03724	0.518621	+	0.855004i	$0.326446\pi$
$854$	0	0
$855$	40.4319	1.38274
$856$	0	0
$857$	−22.7328	−0.776538	−0.388269	−	0.921546i	$-0.626927\pi$
$857$	−22.7328	−0.776538	−0.388269	+	0.921546i	$0.626927\pi$
$858$	0	0
$859$	−18.4958	−0.631068	−0.315534	−	0.948914i	$-0.602184\pi$
$859$	−18.4958	−0.631068	−0.315534	+	0.948914i	$0.602184\pi$
$860$	0	0
$861$	−0.462482	−0.0157613
$862$	0	0
$863$	39.8463	1.35638	0.678192	−	0.734884i	$-0.262764\pi$
$863$	39.8463	1.35638	0.678192	+	0.734884i	$0.262764\pi$
$864$	0	0
$865$	1.91691	0.0651769
$866$	0	0
$867$	−0.367029	−0.0124650
$868$	0	0
$869$	0	0
$870$	0	0
$871$	14.6594	0.496714
$872$	0	0
$873$	−0.0239309	−0.000809939 0
$874$	0	0
$875$	17.5385	0.592909
$876$	0	0
$877$	−0.181186	−0.00611822	−0.00305911	−	0.999995i	$-0.500974\pi$
$877$	−0.181186	−0.00611822	−0.00305911	+	0.999995i	$0.500974\pi$
$878$	0	0
$879$	−1.49744	−0.0505073
$880$	0	0
$881$	−2.32439	−0.0783108	−0.0391554	−	0.999233i	$-0.512467\pi$
$881$	−2.32439	−0.0783108	−0.0391554	+	0.999233i	$0.512467\pi$
$882$	0	0
$883$	57.5471	1.93661	0.968307	−	0.249764i	$-0.0803530\pi$
$883$	57.5471	1.93661	0.968307	+	0.249764i	$0.0803530\pi$
$884$	0	0
$885$	−1.20833	−0.0406175
$886$	0	0
$887$	−9.19676	−0.308797	−0.154398	−	0.988009i	$-0.549344\pi$
$887$	−9.19676	−0.308797	−0.154398	+	0.988009i	$0.549344\pi$
$888$	0	0
$889$	33.6990	1.13023
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.75974	−0.226206
$894$	0	0
$895$	39.0885	1.30659
$896$	0	0
$897$	−0.584769	−0.0195249
$898$	0	0
$899$	−21.3993	−0.713706
$900$	0	0
$901$	−20.6477	−0.687874
$902$	0	0
$903$	−0.540626	−0.0179909
$904$	0	0
$905$	26.0382	0.865540
$906$	0	0
$907$	−21.3640	−0.709379	−0.354689	−	0.934984i	$-0.615413\pi$
$907$	−21.3640	−0.709379	−0.354689	+	0.934984i	$0.615413\pi$
$908$	0	0
$909$	30.1256	0.999204
$910$	0	0
$911$	7.88751	0.261325	0.130662	−	0.991427i	$-0.458290\pi$
$911$	7.88751	0.261325	0.130662	+	0.991427i	$0.458290\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.31300	−0.0434064
$916$	0	0
$917$	29.2511	0.965956
$918$	0	0
$919$	−44.1977	−1.45795	−0.728974	−	0.684541i	$-0.760003\pi$
$919$	−44.1977	−1.45795	−0.728974	+	0.684541i	$0.760003\pi$
$920$	0	0
$921$	−1.48343	−0.0488806
$922$	0	0
$923$	−17.3550	−0.571246
$924$	0	0
$925$	0.126829	0.00417011
$926$	0	0
$927$	14.6078	0.479785
$928$	0	0
$929$	37.4330	1.22814	0.614069	−	0.789253i	$-0.289532\pi$
$929$	37.4330	1.22814	0.614069	+	0.789253i	$0.289532\pi$
$930$	0	0
$931$	−25.3285	−0.830107
$932$	0	0
$933$	0.168418	0.00551376
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.0621	1.27610	0.638051	−	0.769994i	$-0.279741\pi$
$937$	39.0621	1.27610	0.638051	+	0.769994i	$0.279741\pi$
$938$	0	0
$939$	−2.03182	−0.0663060
$940$	0	0
$941$	−57.6873	−1.88055	−0.940277	−	0.340412i	$-0.889434\pi$
$941$	−57.6873	−1.88055	−0.940277	+	0.340412i	$0.889434\pi$
$942$	0	0
$943$	34.6694	1.12899
$944$	0	0
$945$	1.45451	0.0473151
$946$	0	0
$947$	−28.2809	−0.919007	−0.459503	−	0.888176i	$-0.651973\pi$
$947$	−28.2809	−0.919007	−0.459503	+	0.888176i	$0.651973\pi$
$948$	0	0
$949$	15.0363	0.488099
$950$	0	0
$951$	0.734842	0.0238289
$952$	0	0
$953$	−7.80122	−0.252706	−0.126353	−	0.991985i	$-0.540327\pi$
$953$	−7.80122	−0.252706	−0.126353	+	0.991985i	$0.540327\pi$
$954$	0	0
$955$	60.0399	1.94285
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.25406	−0.234246
$960$	0	0
$961$	−17.9641	−0.579487
$962$	0	0
$963$	−16.9361	−0.545758
$964$	0	0
$965$	−1.45779	−0.0469280
$966$	0	0
$967$	21.5958	0.694475	0.347237	−	0.937777i	$-0.387120\pi$
$967$	21.5958	0.694475	0.347237	+	0.937777i	$0.387120\pi$
$968$	0	0
$969$	1.26113	0.0405133
$970$	0	0
$971$	30.4357	0.976729	0.488364	−	0.872640i	$-0.337594\pi$
$971$	30.4357	0.976729	0.488364	+	0.872640i	$0.337594\pi$
$972$	0	0
$973$	9.00674	0.288743
$974$	0	0
$975$	0.0260339	0.000833752 0
$976$	0	0
$977$	−29.2066	−0.934403	−0.467201	−	0.884151i	$-0.654738\pi$
$977$	−29.2066	−0.934403	−0.467201	+	0.884151i	$0.654738\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	24.0137	0.766697
$982$	0	0
$983$	5.42324	0.172975	0.0864873	−	0.996253i	$-0.472436\pi$
$983$	5.42324	0.172975	0.0864873	+	0.996253i	$0.472436\pi$
$984$	0	0
$985$	52.7058	1.67935
$986$	0	0
$987$	−0.121504	−0.00386752
$988$	0	0
$989$	40.5274	1.28870
$990$	0	0
$991$	−27.9224	−0.886985	−0.443492	−	0.896278i	$-0.646261\pi$
$991$	−27.9224	−0.886985	−0.443492	+	0.896278i	$0.646261\pi$
$992$	0	0
$993$	1.25824	0.0399289
$994$	0	0
$995$	40.8115	1.29381
$996$	0	0
$997$	−12.8037	−0.405496	−0.202748	−	0.979231i	$-0.564987\pi$
$997$	−12.8037	−0.405496	−0.202748	+	0.979231i	$0.564987\pi$
$998$	0	0
$999$	−0.139733	−0.00442097

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.dv.1.3		6
4.3	odd	2		7744.2.a.du.1.4		6
8.3	odd	2		3872.2.a.bq.1.3		6
8.5	even	2		3872.2.a.bn.1.4		6
11.3	even	5		704.2.m.n.449.2		12
11.4	even	5		704.2.m.n.577.2		12
11.10	odd	2		7744.2.a.dw.1.3		6
44.3	odd	10		704.2.m.m.449.2		12
44.15	odd	10		704.2.m.m.577.2		12
44.43	even	2		7744.2.a.dt.1.4		6
88.3	odd	10		352.2.m.e.97.2	✓	12
88.21	odd	2		3872.2.a.bo.1.4		6
88.37	even	10		352.2.m.f.225.2	yes	12
88.43	even	2		3872.2.a.bp.1.3		6
88.59	odd	10		352.2.m.e.225.2	yes	12
88.69	even	10		352.2.m.f.97.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.e.97.2	✓	12	88.3	odd	10
352.2.m.e.225.2	yes	12	88.59	odd	10
352.2.m.f.97.2	yes	12	88.69	even	10
352.2.m.f.225.2	yes	12	88.37	even	10
704.2.m.m.449.2		12	44.3	odd	10
704.2.m.m.577.2		12	44.15	odd	10
704.2.m.n.449.2		12	11.3	even	5
704.2.m.n.577.2		12	11.4	even	5
3872.2.a.bn.1.4		6	8.5	even	2
3872.2.a.bo.1.4		6	88.21	odd	2
3872.2.a.bp.1.3		6	88.43	even	2
3872.2.a.bq.1.3		6	8.3	odd	2
7744.2.a.dt.1.4		6	44.43	even	2
7744.2.a.du.1.4		6	4.3	odd	2
7744.2.a.dv.1.3		6	1.1	even	1	trivial
7744.2.a.dw.1.3		6	11.10	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:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0643165 q^{3} -2.31301 q^{5} +1.63066 q^{7} -2.99586 q^{9} +1.15644 q^{13} -0.148765 q^{15} +3.36056 q^{17} +5.83478 q^{19} +0.104878 q^{21} -7.86209 q^{23} +0.350020 q^{25} -0.385633 q^{27} +5.92691 q^{29} -3.61053 q^{31} -3.77174 q^{35} +0.362348 q^{37} +0.0743784 q^{39} -4.40970 q^{41} -5.15478 q^{43} +6.92946 q^{45} -1.15852 q^{47} -4.34095 q^{49} +0.216140 q^{51} -6.14411 q^{53} +0.375273 q^{57} +8.12241 q^{59} +8.82601 q^{61} -4.88524 q^{63} -2.67486 q^{65} +12.6763 q^{67} -0.505662 q^{69} -15.0072 q^{71} +13.0022 q^{73} +0.0225121 q^{75} +2.07633 q^{79} +8.96279 q^{81} -8.94922 q^{83} -7.77302 q^{85} +0.381198 q^{87} +13.9653 q^{89} +1.88577 q^{91} -0.232217 q^{93} -13.4959 q^{95} +0.00798799 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 2 q^{7} + 7 q^{9} - 4 q^{13} - 8 q^{15} - 9 q^{17} + 5 q^{19} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 26 q^{27} - 10 q^{29} - 12 q^{31} - 26 q^{35} + 12 q^{37} - 2 q^{39} - 17 q^{41} + 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) \) 6 * q + 5 * q^3 - 2 * q^7 + 7 * q^9 - 4 * q^13 - 8 * q^15 - 9 * q^17 + 5 * q^19 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 26 * q^27 - 10 * q^29 - 12 * q^31 - 26 * q^35 + 12 * q^37 - 2 * q^39 - 17 * q^41 + 11 * q^43 - 14 * q^45 - 22 * q^47 - 6 * q^51 - 18 * q^53 - 6 * q^57 + 7 * q^59 - 18 * q^61 - 26 * q^63 - 26 * q^65 + 31 * q^67 - 16 * q^69 - 22 * q^71 + q^73 + 11 * q^75 + 28 * q^79 + 34 * q^81 - 13 * q^83 - 2 * q^85 - 38 * q^87 - q^89 + 26 * q^91 - 34 * q^93 - 2 * q^95 - 21 * q^97

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