LMFDB - Embedded newform 7448.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786 q^{3} +1.00000 q^{5} -2.65685 q^{9} +O(q^{10})$	$q-0.585786 q^{3} +1.00000 q^{5} -2.65685 q^{9} +5.24264 q^{11} +0.585786 q^{13} -0.585786 q^{15} +1.17157 q^{17} +1.00000 q^{19} +1.58579 q^{23} -4.00000 q^{25} +3.31371 q^{27} -6.58579 q^{29} -7.41421 q^{31} -3.07107 q^{33} -9.41421 q^{37} -0.343146 q^{39} -5.65685 q^{41} +5.24264 q^{43} -2.65685 q^{45} -7.24264 q^{47} -0.686292 q^{51} -2.58579 q^{53} +5.24264 q^{55} -0.585786 q^{57} -4.82843 q^{59} +5.00000 q^{61} +0.585786 q^{65} +8.82843 q^{67} -0.928932 q^{69} -16.2426 q^{71} +7.00000 q^{73} +2.34315 q^{75} -15.8995 q^{79} +6.02944 q^{81} +4.89949 q^{83} +1.17157 q^{85} +3.85786 q^{87} +17.5563 q^{89} +4.34315 q^{93} +1.00000 q^{95} -5.65685 q^{97} -13.9289 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9	$ 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 - 4 * q^15 + 8 * q^17 + 2 * q^19 + 6 * q^23 - 8 * q^25 - 16 * q^27 - 16 * q^29 - 12 * q^31 + 8 * q^33 - 16 * q^37 - 12 * q^39 + 2 * q^43 + 6 * q^45 - 6 * q^47 - 24 * q^51 - 8 * q^53 + 2 * q^55 - 4 * q^57 - 4 * q^59 + 10 * q^61 + 4 * q^65 + 12 * q^67 - 16 * q^69 - 24 * q^71 + 14 * q^73 + 16 * q^75 - 12 * q^79 + 46 * q^81 - 10 * q^83 + 8 * q^85 + 36 * q^87 + 4 * q^89 + 20 * q^93 + 2 * q^95 - 42 * 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
$3$	−0.585786	−0.338204	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	−0.585786	−0.338204	−0.169102	+	0.985599i	$0.554087\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	5.24264	1.58072	0.790358	−	0.612646i	$-0.209895\pi$
$11$	5.24264	1.58072	0.790358	+	0.612646i	$0.209895\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	−0.585786	−0.151249
$16$	0	0
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.58579	0.330659	0.165330	−	0.986238i	$-0.447131\pi$
$23$	1.58579	0.330659	0.165330	+	0.986238i	$0.447131\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	3.31371	0.637723
$28$	0	0
$29$	−6.58579	−1.22295	−0.611475	−	0.791264i	$-0.709423\pi$
$29$	−6.58579	−1.22295	−0.611475	+	0.791264i	$0.709423\pi$
$30$	0	0
$31$	−7.41421	−1.33163	−0.665816	−	0.746116i	$-0.731916\pi$
$31$	−7.41421	−1.33163	−0.665816	+	0.746116i	$0.731916\pi$
$32$	0	0
$33$	−3.07107	−0.534604
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.41421	−1.54769	−0.773844	−	0.633377i	$-0.781668\pi$
$37$	−9.41421	−1.54769	−0.773844	+	0.633377i	$0.781668\pi$
$38$	0	0
$39$	−0.343146	−0.0549473
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	5.24264	0.799495	0.399748	−	0.916625i	$-0.369098\pi$
$43$	5.24264	0.799495	0.399748	+	0.916625i	$0.369098\pi$
$44$	0	0
$45$	−2.65685	−0.396060
$46$	0	0
$47$	−7.24264	−1.05645	−0.528224	−	0.849105i	$-0.677142\pi$
$47$	−7.24264	−1.05645	−0.528224	+	0.849105i	$0.677142\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.686292	−0.0961000
$52$	0	0
$53$	−2.58579	−0.355185	−0.177593	−	0.984104i	$-0.556831\pi$
$53$	−2.58579	−0.355185	−0.177593	+	0.984104i	$0.556831\pi$
$54$	0	0
$55$	5.24264	0.706918
$56$	0	0
$57$	−0.585786	−0.0775893
$58$	0	0
$59$	−4.82843	−0.628608	−0.314304	−	0.949322i	$-0.601771\pi$
$59$	−4.82843	−0.628608	−0.314304	+	0.949322i	$0.601771\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.585786	0.0726579
$66$	0	0
$67$	8.82843	1.07856	0.539282	−	0.842125i	$-0.318696\pi$
$67$	8.82843	1.07856	0.539282	+	0.842125i	$0.318696\pi$
$68$	0	0
$69$	−0.928932	−0.111830
$70$	0	0
$71$	−16.2426	−1.92765	−0.963823	−	0.266542i	$-0.914119\pi$
$71$	−16.2426	−1.92765	−0.963823	+	0.266542i	$0.914119\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	2.34315	0.270563
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.8995	−1.78883	−0.894416	−	0.447235i	$-0.852409\pi$
$79$	−15.8995	−1.78883	−0.894416	+	0.447235i	$0.852409\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	4.89949	0.537789	0.268895	−	0.963170i	$-0.413342\pi$
$83$	4.89949	0.537789	0.268895	+	0.963170i	$0.413342\pi$
$84$	0	0
$85$	1.17157	0.127075
$86$	0	0
$87$	3.85786	0.413606
$88$	0	0
$89$	17.5563	1.86097	0.930485	−	0.366331i	$-0.119386\pi$
$89$	17.5563	1.86097	0.930485	+	0.366331i	$0.119386\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.34315	0.450363
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−5.65685	−0.574367	−0.287183	−	0.957876i	$-0.592719\pi$
$97$	−5.65685	−0.574367	−0.287183	+	0.957876i	$0.592719\pi$
$98$	0	0
$99$	−13.9289	−1.39991
$100$	0	0
$101$	15.4853	1.54084	0.770422	−	0.637535i	$-0.220046\pi$
$101$	15.4853	1.54084	0.770422	+	0.637535i	$0.220046\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.92893	0.283151	0.141575	−	0.989927i	$-0.454783\pi$
$107$	2.92893	0.283151	0.141575	+	0.989927i	$0.454783\pi$
$108$	0	0
$109$	3.31371	0.317396	0.158698	−	0.987327i	$-0.449270\pi$
$109$	3.31371	0.317396	0.158698	+	0.987327i	$0.449270\pi$
$110$	0	0
$111$	5.51472	0.523434
$112$	0	0
$113$	−3.07107	−0.288902	−0.144451	−	0.989512i	$-0.546142\pi$
$113$	−3.07107	−0.288902	−0.144451	+	0.989512i	$0.546142\pi$
$114$	0	0
$115$	1.58579	0.147875
$116$	0	0
$117$	−1.55635	−0.143885
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.4853	1.49866
$122$	0	0
$123$	3.31371	0.298787
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	11.5563	1.02546	0.512730	−	0.858550i	$-0.328634\pi$
$127$	11.5563	1.02546	0.512730	+	0.858550i	$0.328634\pi$
$128$	0	0
$129$	−3.07107	−0.270392
$130$	0	0
$131$	9.65685	0.843723	0.421862	−	0.906660i	$-0.361377\pi$
$131$	9.65685	0.843723	0.421862	+	0.906660i	$0.361377\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.31371	0.285199
$136$	0	0
$137$	−10.6569	−0.910477	−0.455238	−	0.890370i	$-0.650446\pi$
$137$	−10.6569	−0.910477	−0.455238	+	0.890370i	$0.650446\pi$
$138$	0	0
$139$	−12.5563	−1.06502	−0.532508	−	0.846425i	$-0.678750\pi$
$139$	−12.5563	−1.06502	−0.532508	+	0.846425i	$0.678750\pi$
$140$	0	0
$141$	4.24264	0.357295
$142$	0	0
$143$	3.07107	0.256816
$144$	0	0
$145$	−6.58579	−0.546920
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.65685	0.381504	0.190752	−	0.981638i	$-0.438907\pi$
$149$	4.65685	0.381504	0.190752	+	0.981638i	$0.438907\pi$
$150$	0	0
$151$	−5.65685	−0.460348	−0.230174	−	0.973149i	$-0.573930\pi$
$151$	−5.65685	−0.460348	−0.230174	+	0.973149i	$0.573930\pi$
$152$	0	0
$153$	−3.11270	−0.251647
$154$	0	0
$155$	−7.41421	−0.595524
$156$	0	0
$157$	24.3137	1.94045	0.970223	−	0.242215i	$-0.0778739\pi$
$157$	24.3137	1.94045	0.970223	+	0.242215i	$0.0778739\pi$
$158$	0	0
$159$	1.51472	0.120125
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.0710678	−0.00556646	−0.00278323	−	0.999996i	$-0.500886\pi$
$163$	−0.0710678	−0.00556646	−0.00278323	+	0.999996i	$0.500886\pi$
$164$	0	0
$165$	−3.07107	−0.239082
$166$	0	0
$167$	−21.4142	−1.65708	−0.828541	−	0.559929i	$-0.810829\pi$
$167$	−21.4142	−1.65708	−0.828541	+	0.559929i	$0.810829\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	−2.65685	−0.203175
$172$	0	0
$173$	−1.65685	−0.125968	−0.0629841	−	0.998015i	$-0.520062\pi$
$173$	−1.65685	−0.125968	−0.0629841	+	0.998015i	$0.520062\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.82843	0.212598
$178$	0	0
$179$	−10.4853	−0.783707	−0.391853	−	0.920028i	$-0.628166\pi$
$179$	−10.4853	−0.783707	−0.391853	+	0.920028i	$0.628166\pi$
$180$	0	0
$181$	−10.7279	−0.797400	−0.398700	−	0.917081i	$-0.630539\pi$
$181$	−10.7279	−0.797400	−0.398700	+	0.917081i	$0.630539\pi$
$182$	0	0
$183$	−2.92893	−0.216513
$184$	0	0
$185$	−9.41421	−0.692147
$186$	0	0
$187$	6.14214	0.449157
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.757359	0.0548006	0.0274003	−	0.999625i	$-0.491277\pi$
$191$	0.757359	0.0548006	0.0274003	+	0.999625i	$0.491277\pi$
$192$	0	0
$193$	−6.24264	−0.449355	−0.224678	−	0.974433i	$-0.572133\pi$
$193$	−6.24264	−0.449355	−0.224678	+	0.974433i	$0.572133\pi$
$194$	0	0
$195$	−0.343146	−0.0245732
$196$	0	0
$197$	−17.4853	−1.24577	−0.622887	−	0.782312i	$-0.714041\pi$
$197$	−17.4853	−1.24577	−0.622887	+	0.782312i	$0.714041\pi$
$198$	0	0
$199$	12.0711	0.855695	0.427848	−	0.903851i	$-0.359272\pi$
$199$	12.0711	0.855695	0.427848	+	0.903851i	$0.359272\pi$
$200$	0	0
$201$	−5.17157	−0.364775
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5.65685	−0.395092
$206$	0	0
$207$	−4.21320	−0.292838
$208$	0	0
$209$	5.24264	0.362641
$210$	0	0
$211$	−16.1421	−1.11127	−0.555635	−	0.831426i	$-0.687525\pi$
$211$	−16.1421	−1.11127	−0.555635	+	0.831426i	$0.687525\pi$
$212$	0	0
$213$	9.51472	0.651938
$214$	0	0
$215$	5.24264	0.357545
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.10051	−0.277086
$220$	0	0
$221$	0.686292	0.0461650
$222$	0	0
$223$	7.75736	0.519471	0.259736	−	0.965680i	$-0.416365\pi$
$223$	7.75736	0.519471	0.259736	+	0.965680i	$0.416365\pi$
$224$	0	0
$225$	10.6274	0.708494
$226$	0	0
$227$	1.41421	0.0938647	0.0469323	−	0.998898i	$-0.485055\pi$
$227$	1.41421	0.0938647	0.0469323	+	0.998898i	$0.485055\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.6569	−1.15674	−0.578369	−	0.815775i	$-0.696311\pi$
$233$	−17.6569	−1.15674	−0.578369	+	0.815775i	$0.696311\pi$
$234$	0	0
$235$	−7.24264	−0.472458
$236$	0	0
$237$	9.31371	0.604990
$238$	0	0
$239$	−15.6569	−1.01276	−0.506379	−	0.862311i	$-0.669016\pi$
$239$	−15.6569	−1.01276	−0.506379	+	0.862311i	$0.669016\pi$
$240$	0	0
$241$	16.7279	1.07754	0.538770	−	0.842453i	$-0.318889\pi$
$241$	16.7279	1.07754	0.538770	+	0.842453i	$0.318889\pi$
$242$	0	0
$243$	−13.4731	−0.864299
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.585786	0.0372727
$248$	0	0
$249$	−2.87006	−0.181883
$250$	0	0
$251$	−20.2132	−1.27585	−0.637923	−	0.770100i	$-0.720206\pi$
$251$	−20.2132	−1.27585	−0.637923	+	0.770100i	$0.720206\pi$
$252$	0	0
$253$	8.31371	0.522678
$254$	0	0
$255$	−0.686292	−0.0429772
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	17.4975	1.08307
$262$	0	0
$263$	−26.8284	−1.65431	−0.827156	−	0.561973i	$-0.810043\pi$
$263$	−26.8284	−1.65431	−0.827156	+	0.561973i	$0.810043\pi$
$264$	0	0
$265$	−2.58579	−0.158844
$266$	0	0
$267$	−10.2843	−0.629387
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−27.7279	−1.68435	−0.842176	−	0.539203i	$-0.818725\pi$
$271$	−27.7279	−1.68435	−0.842176	+	0.539203i	$0.818725\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.9706	−1.26457
$276$	0	0
$277$	5.97056	0.358736	0.179368	−	0.983782i	$-0.442595\pi$
$277$	5.97056	0.358736	0.179368	+	0.983782i	$0.442595\pi$
$278$	0	0
$279$	19.6985	1.17932
$280$	0	0
$281$	1.89949	0.113314	0.0566572	−	0.998394i	$-0.481956\pi$
$281$	1.89949	0.113314	0.0566572	+	0.998394i	$0.481956\pi$
$282$	0	0
$283$	16.5563	0.984173	0.492086	−	0.870546i	$-0.336234\pi$
$283$	16.5563	0.984173	0.492086	+	0.870546i	$0.336234\pi$
$284$	0	0
$285$	−0.585786	−0.0346990
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	3.31371	0.194253
$292$	0	0
$293$	1.65685	0.0967945	0.0483972	−	0.998828i	$-0.484589\pi$
$293$	1.65685	0.0967945	0.0483972	+	0.998828i	$0.484589\pi$
$294$	0	0
$295$	−4.82843	−0.281122
$296$	0	0
$297$	17.3726	1.00806
$298$	0	0
$299$	0.928932	0.0537215
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−9.07107	−0.521119
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	26.6274	1.51971	0.759853	−	0.650094i	$-0.225271\pi$
$307$	26.6274	1.51971	0.759853	+	0.650094i	$0.225271\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	27.1421	1.53416	0.767082	−	0.641549i	$-0.221708\pi$
$313$	27.1421	1.53416	0.767082	+	0.641549i	$0.221708\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.75736	−0.0987031	−0.0493516	−	0.998781i	$-0.515715\pi$
$317$	−1.75736	−0.0987031	−0.0493516	+	0.998781i	$0.515715\pi$
$318$	0	0
$319$	−34.5269	−1.93314
$320$	0	0
$321$	−1.71573	−0.0957626
$322$	0	0
$323$	1.17157	0.0651881
$324$	0	0
$325$	−2.34315	−0.129974
$326$	0	0
$327$	−1.94113	−0.107344
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.51472	−0.0832565	−0.0416282	−	0.999133i	$-0.513255\pi$
$331$	−1.51472	−0.0832565	−0.0416282	+	0.999133i	$0.513255\pi$
$332$	0	0
$333$	25.0122	1.37066
$334$	0	0
$335$	8.82843	0.482349
$336$	0	0
$337$	−9.07107	−0.494133	−0.247066	−	0.968999i	$-0.579467\pi$
$337$	−9.07107	−0.494133	−0.247066	+	0.968999i	$0.579467\pi$
$338$	0	0
$339$	1.79899	0.0977077
$340$	0	0
$341$	−38.8701	−2.10493
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.928932	−0.0500120
$346$	0	0
$347$	9.58579	0.514592	0.257296	−	0.966333i	$-0.417168\pi$
$347$	9.58579	0.514592	0.257296	+	0.966333i	$0.417168\pi$
$348$	0	0
$349$	5.65685	0.302804	0.151402	−	0.988472i	$-0.451621\pi$
$349$	5.65685	0.302804	0.151402	+	0.988472i	$0.451621\pi$
$350$	0	0
$351$	1.94113	0.103610
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−16.2426	−0.862070
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.6985	−1.19798	−0.598990	−	0.800756i	$-0.704431\pi$
$359$	−22.6985	−1.19798	−0.598990	+	0.800756i	$0.704431\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.65685	−0.506853
$364$	0	0
$365$	7.00000	0.366397
$366$	0	0
$367$	28.6274	1.49434	0.747170	−	0.664634i	$-0.231412\pi$
$367$	28.6274	1.49434	0.747170	+	0.664634i	$0.231412\pi$
$368$	0	0
$369$	15.0294	0.782401
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.6569	−0.914237	−0.457119	−	0.889406i	$-0.651119\pi$
$373$	−17.6569	−0.914237	−0.457119	+	0.889406i	$0.651119\pi$
$374$	0	0
$375$	5.27208	0.272249
$376$	0	0
$377$	−3.85786	−0.198690
$378$	0	0
$379$	15.4142	0.791775	0.395887	−	0.918299i	$-0.370437\pi$
$379$	15.4142	0.791775	0.395887	+	0.918299i	$0.370437\pi$
$380$	0	0
$381$	−6.76955	−0.346815
$382$	0	0
$383$	12.2426	0.625570	0.312785	−	0.949824i	$-0.398738\pi$
$383$	12.2426	0.625570	0.312785	+	0.949824i	$0.398738\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.9289	−0.708047
$388$	0	0
$389$	7.31371	0.370820	0.185410	−	0.982661i	$-0.440639\pi$
$389$	7.31371	0.370820	0.185410	+	0.982661i	$0.440639\pi$
$390$	0	0
$391$	1.85786	0.0939562
$392$	0	0
$393$	−5.65685	−0.285351
$394$	0	0
$395$	−15.8995	−0.799990
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.6274	1.82909	0.914543	−	0.404489i	$-0.132551\pi$
$401$	36.6274	1.82909	0.914543	+	0.404489i	$0.132551\pi$
$402$	0	0
$403$	−4.34315	−0.216347
$404$	0	0
$405$	6.02944	0.299605
$406$	0	0
$407$	−49.3553	−2.44645
$408$	0	0
$409$	6.68629	0.330616	0.165308	−	0.986242i	$-0.447138\pi$
$409$	6.68629	0.330616	0.165308	+	0.986242i	$0.447138\pi$
$410$	0	0
$411$	6.24264	0.307927
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.89949	0.240507
$416$	0	0
$417$	7.35534	0.360193
$418$	0	0
$419$	−28.6985	−1.40201	−0.701006	−	0.713155i	$-0.747266\pi$
$419$	−28.6985	−1.40201	−0.701006	+	0.713155i	$0.747266\pi$
$420$	0	0
$421$	−27.4142	−1.33609	−0.668044	−	0.744122i	$-0.732868\pi$
$421$	−27.4142	−1.33609	−0.668044	+	0.744122i	$0.732868\pi$
$422$	0	0
$423$	19.2426	0.935609
$424$	0	0
$425$	−4.68629	−0.227319
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.79899	−0.0868560
$430$	0	0
$431$	5.89949	0.284169	0.142084	−	0.989855i	$-0.454620\pi$
$431$	5.89949	0.284169	0.142084	+	0.989855i	$0.454620\pi$
$432$	0	0
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	0	0
$435$	3.85786	0.184970
$436$	0	0
$437$	1.58579	0.0758585
$438$	0	0
$439$	−9.51472	−0.454113	−0.227056	−	0.973882i	$-0.572910\pi$
$439$	−9.51472	−0.454113	−0.227056	+	0.973882i	$0.572910\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−39.3137	−1.86785	−0.933925	−	0.357468i	$-0.883640\pi$
$443$	−39.3137	−1.86785	−0.933925	+	0.357468i	$0.883640\pi$
$444$	0	0
$445$	17.5563	0.832251
$446$	0	0
$447$	−2.72792	−0.129026
$448$	0	0
$449$	−14.6274	−0.690310	−0.345155	−	0.938546i	$-0.612174\pi$
$449$	−14.6274	−0.690310	−0.345155	+	0.938546i	$0.612174\pi$
$450$	0	0
$451$	−29.6569	−1.39649
$452$	0	0
$453$	3.31371	0.155692
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.171573	0.00802584	0.00401292	−	0.999992i	$-0.498723\pi$
$457$	0.171573	0.00802584	0.00401292	+	0.999992i	$0.498723\pi$
$458$	0	0
$459$	3.88225	0.181208
$460$	0	0
$461$	28.7990	1.34130	0.670651	−	0.741773i	$-0.266015\pi$
$461$	28.7990	1.34130	0.670651	+	0.741773i	$0.266015\pi$
$462$	0	0
$463$	−2.27208	−0.105592	−0.0527962	−	0.998605i	$-0.516813\pi$
$463$	−2.27208	−0.105592	−0.0527962	+	0.998605i	$0.516813\pi$
$464$	0	0
$465$	4.34315	0.201409
$466$	0	0
$467$	−17.7279	−0.820350	−0.410175	−	0.912007i	$-0.634532\pi$
$467$	−17.7279	−0.820350	−0.410175	+	0.912007i	$0.634532\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14.2426	−0.656266
$472$	0	0
$473$	27.4853	1.26377
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	6.87006	0.314558
$478$	0	0
$479$	24.0711	1.09984	0.549918	−	0.835219i	$-0.314659\pi$
$479$	24.0711	1.09984	0.549918	+	0.835219i	$0.314659\pi$
$480$	0	0
$481$	−5.51472	−0.251450
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.65685	−0.256865
$486$	0	0
$487$	−40.1421	−1.81901	−0.909507	−	0.415689i	$-0.863541\pi$
$487$	−40.1421	−1.81901	−0.909507	+	0.415689i	$0.863541\pi$
$488$	0	0
$489$	0.0416306	0.00188260
$490$	0	0
$491$	40.6985	1.83670	0.918348	−	0.395773i	$-0.129523\pi$
$491$	40.6985	1.83670	0.918348	+	0.395773i	$0.129523\pi$
$492$	0	0
$493$	−7.71573	−0.347499
$494$	0	0
$495$	−13.9289	−0.626059
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.24264	−0.413757	−0.206879	−	0.978367i	$-0.566331\pi$
$499$	−9.24264	−0.413757	−0.206879	+	0.978367i	$0.566331\pi$
$500$	0	0
$501$	12.5442	0.560432
$502$	0	0
$503$	3.72792	0.166220	0.0831099	−	0.996540i	$-0.473515\pi$
$503$	3.72792	0.166220	0.0831099	+	0.996540i	$0.473515\pi$
$504$	0	0
$505$	15.4853	0.689086
$506$	0	0
$507$	7.41421	0.329277
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.31371	0.146304
$514$	0	0
$515$	−13.6569	−0.601793
$516$	0	0
$517$	−37.9706	−1.66994
$518$	0	0
$519$	0.970563	0.0426030
$520$	0	0
$521$	2.10051	0.0920248	0.0460124	−	0.998941i	$-0.485349\pi$
$521$	2.10051	0.0920248	0.0460124	+	0.998941i	$0.485349\pi$
$522$	0	0
$523$	15.3137	0.669622	0.334811	−	0.942285i	$-0.391328\pi$
$523$	15.3137	0.669622	0.334811	+	0.942285i	$0.391328\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.68629	−0.378381
$528$	0	0
$529$	−20.4853	−0.890664
$530$	0	0
$531$	12.8284	0.556706
$532$	0	0
$533$	−3.31371	−0.143533
$534$	0	0
$535$	2.92893	0.126629
$536$	0	0
$537$	6.14214	0.265053
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.857864	−0.0368825	−0.0184412	−	0.999830i	$-0.505870\pi$
$541$	−0.857864	−0.0368825	−0.0184412	+	0.999830i	$0.505870\pi$
$542$	0	0
$543$	6.28427	0.269684
$544$	0	0
$545$	3.31371	0.141944
$546$	0	0
$547$	9.55635	0.408600	0.204300	−	0.978908i	$-0.434508\pi$
$547$	9.55635	0.408600	0.204300	+	0.978908i	$0.434508\pi$
$548$	0	0
$549$	−13.2843	−0.566959
$550$	0	0
$551$	−6.58579	−0.280564
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.51472	0.234087
$556$	0	0
$557$	18.5147	0.784494	0.392247	−	0.919860i	$-0.371698\pi$
$557$	18.5147	0.784494	0.392247	+	0.919860i	$0.371698\pi$
$558$	0	0
$559$	3.07107	0.129892
$560$	0	0
$561$	−3.59798	−0.151907
$562$	0	0
$563$	10.8701	0.458118	0.229059	−	0.973413i	$-0.426435\pi$
$563$	10.8701	0.458118	0.229059	+	0.973413i	$0.426435\pi$
$564$	0	0
$565$	−3.07107	−0.129201
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.3137	0.725828	0.362914	−	0.931823i	$-0.381782\pi$
$569$	17.3137	0.725828	0.362914	+	0.931823i	$0.381782\pi$
$570$	0	0
$571$	16.6985	0.698810	0.349405	−	0.936972i	$-0.386384\pi$
$571$	16.6985	0.698810	0.349405	+	0.936972i	$0.386384\pi$
$572$	0	0
$573$	−0.443651	−0.0185338
$574$	0	0
$575$	−6.34315	−0.264527
$576$	0	0
$577$	−33.8284	−1.40830	−0.704148	−	0.710053i	$-0.748671\pi$
$577$	−33.8284	−1.40830	−0.704148	+	0.710053i	$0.748671\pi$
$578$	0	0
$579$	3.65685	0.151974
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.5563	−0.561447
$584$	0	0
$585$	−1.55635	−0.0643471
$586$	0	0
$587$	34.0000	1.40333	0.701665	−	0.712507i	$-0.252440\pi$
$587$	34.0000	1.40333	0.701665	+	0.712507i	$0.252440\pi$
$588$	0	0
$589$	−7.41421	−0.305497
$590$	0	0
$591$	10.2426	0.421326
$592$	0	0
$593$	18.3137	0.752054	0.376027	−	0.926609i	$-0.377290\pi$
$593$	18.3137	0.752054	0.376027	+	0.926609i	$0.377290\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.07107	−0.289400
$598$	0	0
$599$	−21.2132	−0.866748	−0.433374	−	0.901214i	$-0.642677\pi$
$599$	−21.2132	−0.866748	−0.433374	+	0.901214i	$0.642677\pi$
$600$	0	0
$601$	−48.5269	−1.97945	−0.989727	−	0.142970i	$-0.954335\pi$
$601$	−48.5269	−1.97945	−0.989727	+	0.142970i	$0.954335\pi$
$602$	0	0
$603$	−23.4558	−0.955196
$604$	0	0
$605$	16.4853	0.670222
$606$	0	0
$607$	3.89949	0.158276	0.0791378	−	0.996864i	$-0.474783\pi$
$607$	3.89949	0.158276	0.0791378	+	0.996864i	$0.474783\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.24264	−0.171639
$612$	0	0
$613$	−15.5147	−0.626634	−0.313317	−	0.949649i	$-0.601440\pi$
$613$	−15.5147	−0.626634	−0.313317	+	0.949649i	$0.601440\pi$
$614$	0	0
$615$	3.31371	0.133622
$616$	0	0
$617$	−29.2843	−1.17894	−0.589470	−	0.807790i	$-0.700663\pi$
$617$	−29.2843	−1.17894	−0.589470	+	0.807790i	$0.700663\pi$
$618$	0	0
$619$	−28.8995	−1.16157	−0.580784	−	0.814057i	$-0.697254\pi$
$619$	−28.8995	−1.16157	−0.580784	+	0.814057i	$0.697254\pi$
$620$	0	0
$621$	5.25483	0.210869
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−3.07107	−0.122647
$628$	0	0
$629$	−11.0294	−0.439772
$630$	0	0
$631$	19.3848	0.771696	0.385848	−	0.922562i	$-0.373909\pi$
$631$	19.3848	0.771696	0.385848	+	0.922562i	$0.373909\pi$
$632$	0	0
$633$	9.45584	0.375836
$634$	0	0
$635$	11.5563	0.458600
$636$	0	0
$637$	0	0
$638$	0	0
$639$	43.1543	1.70716
$640$	0	0
$641$	33.3137	1.31581	0.657906	−	0.753100i	$-0.271442\pi$
$641$	33.3137	1.31581	0.657906	+	0.753100i	$0.271442\pi$
$642$	0	0
$643$	−44.4853	−1.75433	−0.877164	−	0.480191i	$-0.840567\pi$
$643$	−44.4853	−1.75433	−0.877164	+	0.480191i	$0.840567\pi$
$644$	0	0
$645$	−3.07107	−0.120923
$646$	0	0
$647$	−42.0711	−1.65398	−0.826992	−	0.562213i	$-0.809950\pi$
$647$	−42.0711	−1.65398	−0.826992	+	0.562213i	$0.809950\pi$
$648$	0	0
$649$	−25.3137	−0.993650
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.85786	−0.0727039	−0.0363519	−	0.999339i	$-0.511574\pi$
$653$	−1.85786	−0.0727039	−0.0363519	+	0.999339i	$0.511574\pi$
$654$	0	0
$655$	9.65685	0.377325
$656$	0	0
$657$	−18.5980	−0.725576
$658$	0	0
$659$	−12.3848	−0.482442	−0.241221	−	0.970470i	$-0.577548\pi$
$659$	−12.3848	−0.482442	−0.241221	+	0.970470i	$0.577548\pi$
$660$	0	0
$661$	44.8701	1.74524	0.872621	−	0.488397i	$-0.162418\pi$
$661$	44.8701	1.74524	0.872621	+	0.488397i	$0.162418\pi$
$662$	0	0
$663$	−0.402020	−0.0156132
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.4437	−0.404380
$668$	0	0
$669$	−4.54416	−0.175687
$670$	0	0
$671$	26.2132	1.01195
$672$	0	0
$673$	4.62742	0.178374	0.0891869	−	0.996015i	$-0.471573\pi$
$673$	4.62742	0.178374	0.0891869	+	0.996015i	$0.471573\pi$
$674$	0	0
$675$	−13.2548	−0.510179
$676$	0	0
$677$	11.3137	0.434821	0.217411	−	0.976080i	$-0.430239\pi$
$677$	11.3137	0.434821	0.217411	+	0.976080i	$0.430239\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.828427	−0.0317454
$682$	0	0
$683$	4.14214	0.158494	0.0792472	−	0.996855i	$-0.474748\pi$
$683$	4.14214	0.158494	0.0792472	+	0.996855i	$0.474748\pi$
$684$	0	0
$685$	−10.6569	−0.407177
$686$	0	0
$687$	9.37258	0.357586
$688$	0	0
$689$	−1.51472	−0.0577062
$690$	0	0
$691$	−28.6274	−1.08904	−0.544519	−	0.838748i	$-0.683288\pi$
$691$	−28.6274	−1.08904	−0.544519	+	0.838748i	$0.683288\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.5563	−0.476289
$696$	0	0
$697$	−6.62742	−0.251031
$698$	0	0
$699$	10.3431	0.391214
$700$	0	0
$701$	18.7990	0.710028	0.355014	−	0.934861i	$-0.384476\pi$
$701$	18.7990	0.710028	0.355014	+	0.934861i	$0.384476\pi$
$702$	0	0
$703$	−9.41421	−0.355064
$704$	0	0
$705$	4.24264	0.159787
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26.5980	−0.998908	−0.499454	−	0.866340i	$-0.666466\pi$
$709$	−26.5980	−0.998908	−0.499454	+	0.866340i	$0.666466\pi$
$710$	0	0
$711$	42.2426	1.58422
$712$	0	0
$713$	−11.7574	−0.440317
$714$	0	0
$715$	3.07107	0.114851
$716$	0	0
$717$	9.17157	0.342519
$718$	0	0
$719$	19.3137	0.720280	0.360140	−	0.932898i	$-0.382729\pi$
$719$	19.3137	0.720280	0.360140	+	0.932898i	$0.382729\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.79899	−0.364428
$724$	0	0
$725$	26.3431	0.978360
$726$	0	0
$727$	5.24264	0.194439	0.0972194	−	0.995263i	$-0.469005\pi$
$727$	5.24264	0.194439	0.0972194	+	0.995263i	$0.469005\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	6.14214	0.227175
$732$	0	0
$733$	28.6863	1.05955	0.529776	−	0.848137i	$-0.322276\pi$
$733$	28.6863	1.05955	0.529776	+	0.848137i	$0.322276\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.2843	1.70490
$738$	0	0
$739$	−0.627417	−0.0230799	−0.0115400	−	0.999933i	$-0.503673\pi$
$739$	−0.627417	−0.0230799	−0.0115400	+	0.999933i	$0.503673\pi$
$740$	0	0
$741$	−0.343146	−0.0126058
$742$	0	0
$743$	46.7279	1.71428	0.857141	−	0.515083i	$-0.172239\pi$
$743$	46.7279	1.71428	0.857141	+	0.515083i	$0.172239\pi$
$744$	0	0
$745$	4.65685	0.170614
$746$	0	0
$747$	−13.0172	−0.476276
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22.4853	0.820500	0.410250	−	0.911973i	$-0.365442\pi$
$751$	22.4853	0.820500	0.410250	+	0.911973i	$0.365442\pi$
$752$	0	0
$753$	11.8406	0.431496
$754$	0	0
$755$	−5.65685	−0.205874
$756$	0	0
$757$	−54.3137	−1.97407	−0.987033	−	0.160520i	$-0.948683\pi$
$757$	−54.3137	−1.97407	−0.987033	+	0.160520i	$0.948683\pi$
$758$	0	0
$759$	−4.87006	−0.176772
$760$	0	0
$761$	−15.1421	−0.548902	−0.274451	−	0.961601i	$-0.588496\pi$
$761$	−15.1421	−0.548902	−0.274451	+	0.961601i	$0.588496\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.11270	−0.112540
$766$	0	0
$767$	−2.82843	−0.102129
$768$	0	0
$769$	54.4558	1.96373	0.981864	−	0.189587i	$-0.0607148\pi$
$769$	54.4558	1.96373	0.981864	+	0.189587i	$0.0607148\pi$
$770$	0	0
$771$	8.20101	0.295352
$772$	0	0
$773$	15.0711	0.542069	0.271034	−	0.962570i	$-0.412634\pi$
$773$	15.0711	0.542069	0.271034	+	0.962570i	$0.412634\pi$
$774$	0	0
$775$	29.6569	1.06531
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.65685	−0.202678
$780$	0	0
$781$	−85.1543	−3.04706
$782$	0	0
$783$	−21.8234	−0.779904
$784$	0	0
$785$	24.3137	0.867793
$786$	0	0
$787$	−20.2426	−0.721572	−0.360786	−	0.932649i	$-0.617492\pi$
$787$	−20.2426	−0.721572	−0.360786	+	0.932649i	$0.617492\pi$
$788$	0	0
$789$	15.7157	0.559495
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.92893	0.104009
$794$	0	0
$795$	1.51472	0.0537215
$796$	0	0
$797$	−53.4975	−1.89498	−0.947489	−	0.319789i	$-0.896388\pi$
$797$	−53.4975	−1.89498	−0.947489	+	0.319789i	$0.896388\pi$
$798$	0	0
$799$	−8.48528	−0.300188
$800$	0	0
$801$	−46.6447	−1.64811
$802$	0	0
$803$	36.6985	1.29506
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.37258	−0.329931
$808$	0	0
$809$	5.97056	0.209914	0.104957	−	0.994477i	$-0.466530\pi$
$809$	5.97056	0.209914	0.104957	+	0.994477i	$0.466530\pi$
$810$	0	0
$811$	52.4264	1.84094	0.920470	−	0.390813i	$-0.127806\pi$
$811$	52.4264	1.84094	0.920470	+	0.390813i	$0.127806\pi$
$812$	0	0
$813$	16.2426	0.569654
$814$	0	0
$815$	−0.0710678	−0.00248940
$816$	0	0
$817$	5.24264	0.183417
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.9706	1.46478	0.732391	−	0.680884i	$-0.238404\pi$
$821$	41.9706	1.46478	0.732391	+	0.680884i	$0.238404\pi$
$822$	0	0
$823$	−15.7279	−0.548241	−0.274120	−	0.961695i	$-0.588387\pi$
$823$	−15.7279	−0.548241	−0.274120	+	0.961695i	$0.588387\pi$
$824$	0	0
$825$	12.2843	0.427683
$826$	0	0
$827$	−23.3137	−0.810697	−0.405349	−	0.914162i	$-0.632850\pi$
$827$	−23.3137	−0.810697	−0.405349	+	0.914162i	$0.632850\pi$
$828$	0	0
$829$	−16.6863	−0.579539	−0.289769	−	0.957096i	$-0.593579\pi$
$829$	−16.6863	−0.579539	−0.289769	+	0.957096i	$0.593579\pi$
$830$	0	0
$831$	−3.49747	−0.121326
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−21.4142	−0.741069
$836$	0	0
$837$	−24.5685	−0.849213
$838$	0	0
$839$	−13.5147	−0.466580	−0.233290	−	0.972407i	$-0.574949\pi$
$839$	−13.5147	−0.466580	−0.233290	+	0.972407i	$0.574949\pi$
$840$	0	0
$841$	14.3726	0.495606
$842$	0	0
$843$	−1.11270	−0.0383234
$844$	0	0
$845$	−12.6569	−0.435409
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.69848	−0.332851
$850$	0	0
$851$	−14.9289	−0.511757
$852$	0	0
$853$	1.54416	0.0528709	0.0264354	−	0.999651i	$-0.491584\pi$
$853$	1.54416	0.0528709	0.0264354	+	0.999651i	$0.491584\pi$
$854$	0	0
$855$	−2.65685	−0.0908625
$856$	0	0
$857$	−40.9706	−1.39953	−0.699764	−	0.714374i	$-0.746711\pi$
$857$	−40.9706	−1.39953	−0.699764	+	0.714374i	$0.746711\pi$
$858$	0	0
$859$	−49.1838	−1.67813	−0.839064	−	0.544032i	$-0.816897\pi$
$859$	−49.1838	−1.67813	−0.839064	+	0.544032i	$0.816897\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.8701	0.982748	0.491374	−	0.870949i	$-0.336495\pi$
$863$	28.8701	0.982748	0.491374	+	0.870949i	$0.336495\pi$
$864$	0	0
$865$	−1.65685	−0.0563347
$866$	0	0
$867$	9.15433	0.310897
$868$	0	0
$869$	−83.3553	−2.82764
$870$	0	0
$871$	5.17157	0.175232
$872$	0	0
$873$	15.0294	0.508669
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16.8701	−0.569661	−0.284831	−	0.958578i	$-0.591937\pi$
$877$	−16.8701	−0.569661	−0.284831	+	0.958578i	$0.591937\pi$
$878$	0	0
$879$	−0.970563	−0.0327363
$880$	0	0
$881$	15.3137	0.515932	0.257966	−	0.966154i	$-0.416948\pi$
$881$	15.3137	0.515932	0.257966	+	0.966154i	$0.416948\pi$
$882$	0	0
$883$	−30.8284	−1.03746	−0.518730	−	0.854938i	$-0.673595\pi$
$883$	−30.8284	−1.03746	−0.518730	+	0.854938i	$0.673595\pi$
$884$	0	0
$885$	2.82843	0.0950765
$886$	0	0
$887$	−28.2843	−0.949693	−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843	−0.949693	−0.474846	+	0.880069i	$0.657496\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	31.6102	1.05898
$892$	0	0
$893$	−7.24264	−0.242366
$894$	0	0
$895$	−10.4853	−0.350484
$896$	0	0
$897$	−0.544156	−0.0181688
$898$	0	0
$899$	48.8284	1.62852
$900$	0	0
$901$	−3.02944	−0.100925
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.7279	−0.356608
$906$	0	0
$907$	−18.4437	−0.612411	−0.306206	−	0.951965i	$-0.599060\pi$
$907$	−18.4437	−0.612411	−0.306206	+	0.951965i	$0.599060\pi$
$908$	0	0
$909$	−41.1421	−1.36460
$910$	0	0
$911$	5.55635	0.184090	0.0920450	−	0.995755i	$-0.470660\pi$
$911$	5.55635	0.184090	0.0920450	+	0.995755i	$0.470660\pi$
$912$	0	0
$913$	25.6863	0.850092
$914$	0	0
$915$	−2.92893	−0.0968275
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8.41421	0.277559	0.138780	−	0.990323i	$-0.455682\pi$
$919$	8.41421	0.277559	0.138780	+	0.990323i	$0.455682\pi$
$920$	0	0
$921$	−15.5980	−0.513971
$922$	0	0
$923$	−9.51472	−0.313181
$924$	0	0
$925$	37.6569	1.23815
$926$	0	0
$927$	36.2843	1.19173
$928$	0	0
$929$	−9.62742	−0.315865	−0.157933	−	0.987450i	$-0.550483\pi$
$929$	−9.62742	−0.315865	−0.157933	+	0.987450i	$0.550483\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.68629	0.153422
$934$	0	0
$935$	6.14214	0.200869
$936$	0	0
$937$	−47.4853	−1.55128	−0.775638	−	0.631178i	$-0.782572\pi$
$937$	−47.4853	−1.55128	−0.775638	+	0.631178i	$0.782572\pi$
$938$	0	0
$939$	−15.8995	−0.518860
$940$	0	0
$941$	−24.5858	−0.801474	−0.400737	−	0.916193i	$-0.631246\pi$
$941$	−24.5858	−0.801474	−0.400737	+	0.916193i	$0.631246\pi$
$942$	0	0
$943$	−8.97056	−0.292122
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	0	0
$949$	4.10051	0.133108
$950$	0	0
$951$	1.02944	0.0333818
$952$	0	0
$953$	51.1543	1.65705	0.828526	−	0.559951i	$-0.189180\pi$
$953$	51.1543	1.65705	0.828526	+	0.559951i	$0.189180\pi$
$954$	0	0
$955$	0.757359	0.0245076
$956$	0	0
$957$	20.2254	0.653794
$958$	0	0
$959$	0	0
$960$	0	0
$961$	23.9706	0.773244
$962$	0	0
$963$	−7.78175	−0.250763
$964$	0	0
$965$	−6.24264	−0.200958
$966$	0	0
$967$	−16.9706	−0.545737	−0.272868	−	0.962051i	$-0.587972\pi$
$967$	−16.9706	−0.545737	−0.272868	+	0.962051i	$0.587972\pi$
$968$	0	0
$969$	−0.686292	−0.0220469
$970$	0	0
$971$	4.87006	0.156288	0.0781438	−	0.996942i	$-0.475101\pi$
$971$	4.87006	0.156288	0.0781438	+	0.996942i	$0.475101\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.37258	0.0439578
$976$	0	0
$977$	−5.31371	−0.170001	−0.0850003	−	0.996381i	$-0.527089\pi$
$977$	−5.31371	−0.170001	−0.0850003	+	0.996381i	$0.527089\pi$
$978$	0	0
$979$	92.0416	2.94166
$980$	0	0
$981$	−8.80404	−0.281091
$982$	0	0
$983$	−17.9411	−0.572233	−0.286117	−	0.958195i	$-0.592364\pi$
$983$	−17.9411	−0.572233	−0.286117	+	0.958195i	$0.592364\pi$
$984$	0	0
$985$	−17.4853	−0.557127
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.31371	0.264361
$990$	0	0
$991$	7.45584	0.236843	0.118421	−	0.992963i	$-0.462217\pi$
$991$	7.45584	0.236843	0.118421	+	0.992963i	$0.462217\pi$
$992$	0	0
$993$	0.887302	0.0281577
$994$	0	0
$995$	12.0711	0.382679
$996$	0	0
$997$	59.5980	1.88749	0.943743	−	0.330678i	$-0.107278\pi$
$997$	59.5980	1.88749	0.943743	+	0.330678i	$0.107278\pi$
$998$	0	0
$999$	−31.1960	−0.986996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.x.1.2		2
7.3	odd	6		1064.2.q.k.457.2	yes	4
7.5	odd	6		1064.2.q.k.305.2	✓	4
7.6	odd	2		7448.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.k.305.2	✓	4	7.5	odd	6
1064.2.q.k.457.2	yes	4	7.3	odd	6
7448.2.a.x.1.2		2	1.1	even	1	trivial
7448.2.a.bd.1.1		2	7.6	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.585786 q^{3} +1.00000 q^{5} -2.65685 q^{9} +O(q^{10})\)	\(q-0.585786 q^{3} +1.00000 q^{5} -2.65685 q^{9} +5.24264 q^{11} +0.585786 q^{13} -0.585786 q^{15} +1.17157 q^{17} +1.00000 q^{19} +1.58579 q^{23} -4.00000 q^{25} +3.31371 q^{27} -6.58579 q^{29} -7.41421 q^{31} -3.07107 q^{33} -9.41421 q^{37} -0.343146 q^{39} -5.65685 q^{41} +5.24264 q^{43} -2.65685 q^{45} -7.24264 q^{47} -0.686292 q^{51} -2.58579 q^{53} +5.24264 q^{55} -0.585786 q^{57} -4.82843 q^{59} +5.00000 q^{61} +0.585786 q^{65} +8.82843 q^{67} -0.928932 q^{69} -16.2426 q^{71} +7.00000 q^{73} +2.34315 q^{75} -15.8995 q^{79} +6.02944 q^{81} +4.89949 q^{83} +1.17157 q^{85} +3.85786 q^{87} +17.5563 q^{89} +4.34315 q^{93} +1.00000 q^{95} -5.65685 q^{97} -13.9289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9	\( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 - 4 * q^15 + 8 * q^17 + 2 * q^19 + 6 * q^23 - 8 * q^25 - 16 * q^27 - 16 * q^29 - 12 * q^31 + 8 * q^33 - 16 * q^37 - 12 * q^39 + 2 * q^43 + 6 * q^45 - 6 * q^47 - 24 * q^51 - 8 * q^53 + 2 * q^55 - 4 * q^57 - 4 * q^59 + 10 * q^61 + 4 * q^65 + 12 * q^67 - 16 * q^69 - 24 * q^71 + 14 * q^73 + 16 * q^75 - 12 * q^79 + 46 * q^81 - 10 * q^83 + 8 * q^85 + 36 * q^87 + 4 * q^89 + 20 * q^93 + 2 * q^95 - 42 * q^99

Introduction

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