LMFDB - Embedded newform 2601.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84776 q^{2} +1.41421 q^{4} -1.61313 q^{5} -0.152241 q^{7} -1.08239 q^{8} +O(q^{10})$	$q+1.84776 q^{2} +1.41421 q^{4} -1.61313 q^{5} -0.152241 q^{7} -1.08239 q^{8} -2.98067 q^{10} +5.02734 q^{11} -3.94495 q^{13} -0.281305 q^{14} -4.82843 q^{16} -6.57900 q^{19} -2.28130 q^{20} +9.28931 q^{22} +3.44834 q^{23} -2.39782 q^{25} -7.28931 q^{26} -0.215301 q^{28} -2.24264 q^{29} +2.57446 q^{31} -6.75699 q^{32} +0.245584 q^{35} -10.6762 q^{37} -12.1564 q^{38} +1.74603 q^{40} -0.276769 q^{41} +6.34277 q^{43} +7.10973 q^{44} +6.37170 q^{46} -9.82164 q^{47} -6.97682 q^{49} -4.43060 q^{50} -5.57900 q^{52} +2.12612 q^{53} -8.10973 q^{55} +0.164784 q^{56} -4.14386 q^{58} -1.32381 q^{59} -8.27836 q^{61} +4.75699 q^{62} -2.82843 q^{64} +6.36370 q^{65} +7.10973 q^{67} +0.453780 q^{70} -6.54712 q^{71} -8.32638 q^{73} -19.7270 q^{74} -9.30411 q^{76} -0.765367 q^{77} -0.532327 q^{79} +7.78886 q^{80} -0.511402 q^{82} -2.20345 q^{83} +11.7199 q^{86} -5.44155 q^{88} +7.64847 q^{89} +0.600582 q^{91} +4.87669 q^{92} -18.1480 q^{94} +10.6128 q^{95} -3.60634 q^{97} -12.8915 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 8 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 8 * q^7	$ 4 q + 4 q^{5} - 8 q^{7} - 8 q^{10} + 4 q^{11} - 4 q^{13} + 8 q^{14} - 8 q^{16} - 12 q^{19} + 8 q^{22} + 12 q^{23} + 8 q^{29} - 8 q^{31} - 16 q^{35} - 24 q^{37} - 16 q^{38} + 12 q^{41} + 4 q^{43} + 8 q^{44} - 8 q^{46} - 8 q^{47} - 4 q^{49} - 16 q^{50} - 8 q^{52} - 8 q^{53} - 12 q^{55} - 8 q^{56} - 24 q^{59} - 24 q^{61} - 8 q^{62} + 12 q^{65} + 8 q^{67} + 24 q^{70} - 24 q^{71} - 8 q^{73} - 8 q^{74} - 8 q^{76} - 8 q^{80} + 8 q^{82} - 8 q^{83} + 16 q^{86} - 16 q^{89} + 8 q^{91} - 16 q^{94} - 12 q^{95} + 16 q^{97} - 32 q^{98}+O(q^{100}) $ 4 * q + 4 * q^5 - 8 * q^7 - 8 * q^10 + 4 * q^11 - 4 * q^13 + 8 * q^14 - 8 * q^16 - 12 * q^19 + 8 * q^22 + 12 * q^23 + 8 * q^29 - 8 * q^31 - 16 * q^35 - 24 * q^37 - 16 * q^38 + 12 * q^41 + 4 * q^43 + 8 * q^44 - 8 * q^46 - 8 * q^47 - 4 * q^49 - 16 * q^50 - 8 * q^52 - 8 * q^53 - 12 * q^55 - 8 * q^56 - 24 * q^59 - 24 * q^61 - 8 * q^62 + 12 * q^65 + 8 * q^67 + 24 * q^70 - 24 * q^71 - 8 * q^73 - 8 * q^74 - 8 * q^76 - 8 * q^80 + 8 * q^82 - 8 * q^83 + 16 * q^86 - 16 * q^89 + 8 * q^91 - 16 * q^94 - 12 * q^95 + 16 * q^97 - 32 * q^98

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 $

$2$	1.84776	1.30656	0.653281	−	0.757115i	$-0.273392\pi$
$2$	1.84776	1.30656	0.653281	+	0.757115i	$0.273392\pi$
$3$	0	0
$3$	0	0
$4$	1.41421	0.707107
$5$	−1.61313	−0.721412	−0.360706	−	0.932680i	$-0.617464\pi$
$5$	−1.61313	−0.721412	−0.360706	+	0.932680i	$0.617464\pi$
$6$	0	0
$7$	−0.152241	−0.0575417	−0.0287708	−	0.999586i	$-0.509159\pi$
$7$	−0.152241	−0.0575417	−0.0287708	+	0.999586i	$0.509159\pi$
$8$	−1.08239	−0.382683
$9$	0	0
$10$	−2.98067	−0.942570
$11$	5.02734	1.51580	0.757900	−	0.652371i	$-0.226226\pi$
$11$	5.02734	1.51580	0.757900	+	0.652371i	$0.226226\pi$
$12$	0	0
$13$	−3.94495	−1.09413	−0.547066	−	0.837090i	$-0.684255\pi$
$13$	−3.94495	−1.09413	−0.547066	+	0.837090i	$0.684255\pi$
$14$	−0.281305	−0.0751818
$15$	0	0
$16$	−4.82843	−1.20711
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−6.57900	−1.50933	−0.754663	−	0.656113i	$-0.772200\pi$
$19$	−6.57900	−1.50933	−0.754663	+	0.656113i	$0.772200\pi$
$20$	−2.28130	−0.510115
$21$	0	0
$22$	9.28931	1.98049
$23$	3.44834	0.719029	0.359514	−	0.933140i	$-0.382942\pi$
$23$	3.44834	0.719029	0.359514	+	0.933140i	$0.382942\pi$
$24$	0	0
$25$	−2.39782	−0.479565
$26$	−7.28931	−1.42955
$27$	0	0
$28$	−0.215301	−0.0406881
$29$	−2.24264	−0.416448	−0.208224	−	0.978081i	$-0.566768\pi$
$29$	−2.24264	−0.416448	−0.208224	+	0.978081i	$0.566768\pi$
$30$	0	0
$31$	2.57446	0.462387	0.231194	−	0.972908i	$-0.425737\pi$
$31$	2.57446	0.462387	0.231194	+	0.972908i	$0.425737\pi$
$32$	−6.75699	−1.19448
$33$	0	0
$34$	0	0
$35$	0.245584	0.0415112
$36$	0	0
$37$	−10.6762	−1.75515	−0.877577	−	0.479436i	$-0.840841\pi$
$37$	−10.6762	−1.75515	−0.877577	+	0.479436i	$0.840841\pi$
$38$	−12.1564	−1.97203
$39$	0	0
$40$	1.74603	0.276072
$41$	−0.276769	−0.0432240	−0.0216120	−	0.999766i	$-0.506880\pi$
$41$	−0.276769	−0.0432240	−0.0216120	+	0.999766i	$0.506880\pi$
$42$	0	0
$43$	6.34277	0.967264	0.483632	−	0.875272i	$-0.339317\pi$
$43$	6.34277	0.967264	0.483632	+	0.875272i	$0.339317\pi$
$44$	7.10973	1.07183
$45$	0	0
$46$	6.37170	0.939457
$47$	−9.82164	−1.43263	−0.716317	−	0.697775i	$-0.754173\pi$
$47$	−9.82164	−1.43263	−0.716317	+	0.697775i	$0.754173\pi$
$48$	0	0
$49$	−6.97682	−0.996689
$50$	−4.43060	−0.626582
$51$	0	0
$52$	−5.57900	−0.773668
$53$	2.12612	0.292045	0.146023	−	0.989281i	$-0.453353\pi$
$53$	2.12612	0.292045	0.146023	+	0.989281i	$0.453353\pi$
$54$	0	0
$55$	−8.10973	−1.09352
$56$	0.164784	0.0220202
$57$	0	0
$58$	−4.14386	−0.544115
$59$	−1.32381	−0.172346	−0.0861729	−	0.996280i	$-0.527464\pi$
$59$	−1.32381	−0.172346	−0.0861729	+	0.996280i	$0.527464\pi$
$60$	0	0
$61$	−8.27836	−1.05994	−0.529968	−	0.848018i	$-0.677796\pi$
$61$	−8.27836	−1.05994	−0.529968	+	0.848018i	$0.677796\pi$
$62$	4.75699	0.604138
$63$	0	0
$64$	−2.82843	−0.353553
$65$	6.36370	0.789319
$66$	0	0
$67$	7.10973	0.868592	0.434296	−	0.900770i	$-0.356997\pi$
$67$	7.10973	0.868592	0.434296	+	0.900770i	$0.356997\pi$
$68$	0	0
$69$	0	0
$70$	0.453780	0.0542370
$71$	−6.54712	−0.777000	−0.388500	−	0.921449i	$-0.627007\pi$
$71$	−6.54712	−0.777000	−0.388500	+	0.921449i	$0.627007\pi$
$72$	0	0
$73$	−8.32638	−0.974529	−0.487265	−	0.873254i	$-0.662005\pi$
$73$	−8.32638	−0.974529	−0.487265	+	0.873254i	$0.662005\pi$
$74$	−19.7270	−2.29322
$75$	0	0
$76$	−9.30411	−1.06725
$77$	−0.765367	−0.0872216
$78$	0	0
$79$	−0.532327	−0.0598914	−0.0299457	−	0.999552i	$-0.509533\pi$
$79$	−0.532327	−0.0598914	−0.0299457	+	0.999552i	$0.509533\pi$
$80$	7.78886	0.870821
$81$	0	0
$82$	−0.511402	−0.0564749
$83$	−2.20345	−0.241860	−0.120930	−	0.992661i	$-0.538588\pi$
$83$	−2.20345	−0.241860	−0.120930	+	0.992661i	$0.538588\pi$
$84$	0	0
$85$	0	0
$86$	11.7199	1.26379
$87$	0	0
$88$	−5.44155	−0.580072
$89$	7.64847	0.810737	0.405368	−	0.914153i	$-0.367143\pi$
$89$	7.64847	0.810737	0.405368	+	0.914153i	$0.367143\pi$
$90$	0	0
$91$	0.600582	0.0629581
$92$	4.87669	0.508430
$93$	0	0
$94$	−18.1480	−1.87183
$95$	10.6128	1.08885
$96$	0	0
$97$	−3.60634	−0.366168	−0.183084	−	0.983097i	$-0.558608\pi$
$97$	−3.60634	−0.366168	−0.183084	+	0.983097i	$0.558608\pi$
$98$	−12.8915	−1.30224
$99$	0	0
$100$	−3.39104	−0.339104
$101$	−9.13707	−0.909173	−0.454586	−	0.890703i	$-0.650213\pi$
$101$	−9.13707	−0.909173	−0.454586	+	0.890703i	$0.650213\pi$
$102$	0	0
$103$	7.57862	0.746744	0.373372	−	0.927682i	$-0.378202\pi$
$103$	7.57862	0.746744	0.373372	+	0.927682i	$0.378202\pi$
$104$	4.26998	0.418706
$105$	0	0
$106$	3.92856	0.381575
$107$	18.4666	1.78524	0.892619	−	0.450812i	$-0.148866\pi$
$107$	18.4666	1.78524	0.892619	+	0.450812i	$0.148866\pi$
$108$	0	0
$109$	−10.7392	−1.02863	−0.514317	−	0.857600i	$-0.671954\pi$
$109$	−10.7392	−1.02863	−0.514317	+	0.857600i	$0.671954\pi$
$110$	−14.9848	−1.42875
$111$	0	0
$112$	0.735084	0.0694589
$113$	1.67497	0.157568	0.0787838	−	0.996892i	$-0.474896\pi$
$113$	1.67497	0.157568	0.0787838	+	0.996892i	$0.474896\pi$
$114$	0	0
$115$	−5.56261	−0.518716
$116$	−3.17157	−0.294473
$117$	0	0
$118$	−2.44609	−0.225181
$119$	0	0
$120$	0	0
$121$	14.2741	1.29765
$122$	−15.2964	−1.38487
$123$	0	0
$124$	3.64084	0.326957
$125$	11.9336	1.06738
$126$	0	0
$127$	−12.1812	−1.08090	−0.540452	−	0.841375i	$-0.681747\pi$
$127$	−12.1812	−1.08090	−0.540452	+	0.841375i	$0.681747\pi$
$128$	8.28772	0.732538
$129$	0	0
$130$	11.7586	1.03130
$131$	10.4512	0.913122	0.456561	−	0.889692i	$-0.349081\pi$
$131$	10.4512	0.913122	0.456561	+	0.889692i	$0.349081\pi$
$132$	0	0
$133$	1.00159	0.0868491
$134$	13.1371	1.13487
$135$	0	0
$136$	0	0
$137$	−13.4928	−1.15276	−0.576382	−	0.817180i	$-0.695536\pi$
$137$	−13.4928	−1.15276	−0.576382	+	0.817180i	$0.695536\pi$
$138$	0	0
$139$	−5.32798	−0.451913	−0.225957	−	0.974137i	$-0.572551\pi$
$139$	−5.32798	−0.451913	−0.225957	+	0.974137i	$0.572551\pi$
$140$	0.347308	0.0293529
$141$	0	0
$142$	−12.0975	−1.01520
$143$	−19.8326	−1.65848
$144$	0	0
$145$	3.61766	0.300430
$146$	−15.3852	−1.27328
$147$	0	0
$148$	−15.0984	−1.24108
$149$	−9.31890	−0.763434	−0.381717	−	0.924279i	$-0.624667\pi$
$149$	−9.31890	−0.763434	−0.381717	+	0.924279i	$0.624667\pi$
$150$	0	0
$151$	11.9632	0.973555	0.486778	−	0.873526i	$-0.338172\pi$
$151$	11.9632	0.973555	0.486778	+	0.873526i	$0.338172\pi$
$152$	7.12106	0.577594
$153$	0	0
$154$	−1.41421	−0.113961
$155$	−4.15293	−0.333571
$156$	0	0
$157$	−18.1548	−1.44891	−0.724456	−	0.689321i	$-0.757909\pi$
$157$	−18.1548	−1.44891	−0.724456	+	0.689321i	$0.757909\pi$
$158$	−0.983611	−0.0782519
$159$	0	0
$160$	10.8999	0.861710
$161$	−0.524979	−0.0413741
$162$	0	0
$163$	20.7225	1.62311	0.811555	−	0.584276i	$-0.198621\pi$
$163$	20.7225	1.62311	0.811555	+	0.584276i	$0.198621\pi$
$164$	−0.391410	−0.0305640
$165$	0	0
$166$	−4.07144	−0.316005
$167$	−8.08881	−0.625931	−0.312965	−	0.949765i	$-0.601322\pi$
$167$	−8.08881	−0.625931	−0.312965	+	0.949765i	$0.601322\pi$
$168$	0	0
$169$	2.56261	0.197124
$170$	0	0
$171$	0	0
$172$	8.97003	0.683959
$173$	1.16062	0.0882405	0.0441202	−	0.999026i	$-0.485952\pi$
$173$	1.16062	0.0882405	0.0441202	+	0.999026i	$0.485952\pi$
$174$	0	0
$175$	0.365047	0.0275950
$176$	−24.2741	−1.82973
$177$	0	0
$178$	14.1325	1.05928
$179$	−2.89668	−0.216508	−0.108254	−	0.994123i	$-0.534526\pi$
$179$	−2.89668	−0.216508	−0.108254	+	0.994123i	$0.534526\pi$
$180$	0	0
$181$	−0.842695	−0.0626370	−0.0313185	−	0.999509i	$-0.509971\pi$
$181$	−0.842695	−0.0626370	−0.0313185	+	0.999509i	$0.509971\pi$
$182$	1.10973	0.0822588
$183$	0	0
$184$	−3.73246	−0.275160
$185$	17.2220	1.26619
$186$	0	0
$187$	0	0
$188$	−13.8899	−1.01302
$189$	0	0
$190$	19.6098	1.42265
$191$	−9.97069	−0.721454	−0.360727	−	0.932671i	$-0.617471\pi$
$191$	−9.97069	−0.721454	−0.360727	+	0.932671i	$0.617471\pi$
$192$	0	0
$193$	3.73418	0.268792	0.134396	−	0.990928i	$-0.457091\pi$
$193$	3.73418	0.268792	0.134396	+	0.990928i	$0.457091\pi$
$194$	−6.66364	−0.478422
$195$	0	0
$196$	−9.86672	−0.704766
$197$	16.2053	1.15458	0.577291	−	0.816539i	$-0.304110\pi$
$197$	16.2053	1.15458	0.577291	+	0.816539i	$0.304110\pi$
$198$	0	0
$199$	16.4734	1.16777	0.583885	−	0.811836i	$-0.301532\pi$
$199$	16.4734	1.16777	0.583885	+	0.811836i	$0.301532\pi$
$200$	2.59539	0.183522
$201$	0	0
$202$	−16.8831	−1.18789
$203$	0.341422	0.0239631
$204$	0	0
$205$	0.446463	0.0311823
$206$	14.0035	0.975668
$207$	0	0
$208$	19.0479	1.32073
$209$	−33.0749	−2.28784
$210$	0	0
$211$	10.6887	0.735842	0.367921	−	0.929857i	$-0.380070\pi$
$211$	10.6887	0.735842	0.367921	+	0.929857i	$0.380070\pi$
$212$	3.00679	0.206507
$213$	0	0
$214$	34.1219	2.33253
$215$	−10.2317	−0.697795
$216$	0	0
$217$	−0.391939	−0.0266065
$218$	−19.8435	−1.34397
$219$	0	0
$220$	−11.4689	−0.773233
$221$	0	0
$222$	0	0
$223$	5.41650	0.362715	0.181358	−	0.983417i	$-0.441951\pi$
$223$	5.41650	0.362715	0.181358	+	0.983417i	$0.441951\pi$
$224$	1.02869	0.0687322
$225$	0	0
$226$	3.09494	0.205872
$227$	12.2987	0.816291	0.408146	−	0.912917i	$-0.366176\pi$
$227$	12.2987	0.816291	0.408146	+	0.912917i	$0.366176\pi$
$228$	0	0
$229$	30.1158	1.99011	0.995053	−	0.0993441i	$-0.0316745\pi$
$229$	30.1158	1.99011	0.995053	+	0.0993441i	$0.0316745\pi$
$230$	−10.2784	−0.677735
$231$	0	0
$232$	2.42742	0.159368
$233$	8.78523	0.575539	0.287770	−	0.957700i	$-0.407086\pi$
$233$	8.78523	0.575539	0.287770	+	0.957700i	$0.407086\pi$
$234$	0	0
$235$	15.8435	1.03352
$236$	−1.87216	−0.121867
$237$	0	0
$238$	0	0
$239$	−14.6501	−0.947634	−0.473817	−	0.880623i	$-0.657124\pi$
$239$	−14.6501	−0.947634	−0.473817	+	0.880623i	$0.657124\pi$
$240$	0	0
$241$	16.9552	1.09218	0.546091	−	0.837726i	$-0.316115\pi$
$241$	16.9552	1.09218	0.546091	+	0.837726i	$0.316115\pi$
$242$	26.3752	1.69546
$243$	0	0
$244$	−11.7074	−0.749488
$245$	11.2545	0.719023
$246$	0	0
$247$	25.9538	1.65140
$248$	−2.78658	−0.176948
$249$	0	0
$250$	22.0505	1.39459
$251$	13.9453	0.880217	0.440109	−	0.897945i	$-0.354940\pi$
$251$	13.9453	0.880217	0.440109	+	0.897945i	$0.354940\pi$
$252$	0	0
$253$	17.3360	1.08990
$254$	−22.5079	−1.41227
$255$	0	0
$256$	20.9706	1.31066
$257$	−19.6603	−1.22638	−0.613189	−	0.789936i	$-0.710113\pi$
$257$	−19.6603	−1.22638	−0.613189	+	0.789936i	$0.710113\pi$
$258$	0	0
$259$	1.62535	0.100994
$260$	8.99963	0.558133
$261$	0	0
$262$	19.3112	1.19305
$263$	27.8721	1.71867	0.859334	−	0.511415i	$-0.170879\pi$
$263$	27.8721	1.71867	0.859334	+	0.511415i	$0.170879\pi$
$264$	0	0
$265$	−3.42970	−0.210685
$266$	1.85070	0.113474
$267$	0	0
$268$	10.0547	0.614187
$269$	0.205327	0.0125190	0.00625951	−	0.999980i	$-0.498008\pi$
$269$	0.205327	0.0125190	0.00625951	+	0.999980i	$0.498008\pi$
$270$	0	0
$271$	−8.21077	−0.498768	−0.249384	−	0.968405i	$-0.580228\pi$
$271$	−8.21077	−0.498768	−0.249384	+	0.968405i	$0.580228\pi$
$272$	0	0
$273$	0	0
$274$	−24.9314	−1.50616
$275$	−12.0547	−0.726924
$276$	0	0
$277$	−23.5677	−1.41604	−0.708022	−	0.706190i	$-0.750412\pi$
$277$	−23.5677	−1.41604	−0.708022	+	0.706190i	$0.750412\pi$
$278$	−9.84482	−0.590453
$279$	0	0
$280$	−0.265818	−0.0158857
$281$	10.1532	0.605690	0.302845	−	0.953040i	$-0.402064\pi$
$281$	10.1532	0.605690	0.302845	+	0.953040i	$0.402064\pi$
$282$	0	0
$283$	9.47918	0.563479	0.281739	−	0.959491i	$-0.409089\pi$
$283$	9.47918	0.563479	0.281739	+	0.959491i	$0.409089\pi$
$284$	−9.25903	−0.549422
$285$	0	0
$286$	−36.6458	−2.16691
$287$	0.0421355	0.00248718
$288$	0	0
$289$	0	0
$290$	6.68457	0.392531
$291$	0	0
$292$	−11.7753	−0.689096
$293$	−9.28515	−0.542444	−0.271222	−	0.962517i	$-0.587428\pi$
$293$	−9.28515	−0.542444	−0.271222	+	0.962517i	$0.587428\pi$
$294$	0	0
$295$	2.13548	0.124332
$296$	11.5558	0.671668
$297$	0	0
$298$	−17.2191	−0.997475
$299$	−13.6035	−0.786712
$300$	0	0
$301$	−0.965630	−0.0556580
$302$	22.1052	1.27201
$303$	0	0
$304$	31.7662	1.82192
$305$	13.3540	0.764650
$306$	0	0
$307$	−27.1418	−1.54906	−0.774531	−	0.632536i	$-0.782014\pi$
$307$	−27.1418	−1.54906	−0.774531	+	0.632536i	$0.782014\pi$
$308$	−1.08239	−0.0616750
$309$	0	0
$310$	−7.67362	−0.435832
$311$	−2.58579	−0.146626	−0.0733132	−	0.997309i	$-0.523357\pi$
$311$	−2.58579	−0.146626	−0.0733132	+	0.997309i	$0.523357\pi$
$312$	0	0
$313$	−25.6105	−1.44759	−0.723796	−	0.690015i	$-0.757604\pi$
$313$	−25.6105	−1.44759	−0.723796	+	0.690015i	$0.757604\pi$
$314$	−33.5457	−1.89309
$315$	0	0
$316$	−0.752823	−0.0423496
$317$	25.5982	1.43774	0.718869	−	0.695146i	$-0.244660\pi$
$317$	25.5982	1.43774	0.718869	+	0.695146i	$0.244660\pi$
$318$	0	0
$319$	−11.2745	−0.631252
$320$	4.56261	0.255058
$321$	0	0
$322$	−0.970034	−0.0540579
$323$	0	0
$324$	0	0
$325$	9.45929	0.524707
$326$	38.2902	2.12070
$327$	0	0
$328$	0.299572	0.0165411
$329$	1.49526	0.0824361
$330$	0	0
$331$	−0.333172	−0.0183128	−0.00915639	−	0.999958i	$-0.502915\pi$
$331$	−0.333172	−0.0183128	−0.00915639	+	0.999958i	$0.502915\pi$
$332$	−3.11615	−0.171021
$333$	0	0
$334$	−14.9462	−0.817818
$335$	−11.4689	−0.626613
$336$	0	0
$337$	3.75539	0.204569	0.102285	−	0.994755i	$-0.467385\pi$
$337$	3.75539	0.204569	0.102285	+	0.994755i	$0.467385\pi$
$338$	4.73508	0.257555
$339$	0	0
$340$	0	0
$341$	12.9427	0.700886
$342$	0	0
$343$	2.12784	0.114893
$344$	−6.86537	−0.370156
$345$	0	0
$346$	2.14455	0.115292
$347$	3.30999	0.177690	0.0888449	−	0.996045i	$-0.471682\pi$
$347$	3.30999	0.177690	0.0888449	+	0.996045i	$0.471682\pi$
$348$	0	0
$349$	2.58488	0.138366	0.0691828	−	0.997604i	$-0.477961\pi$
$349$	2.58488	0.138366	0.0691828	+	0.997604i	$0.477961\pi$
$350$	0.674519	0.0360546
$351$	0	0
$352$	−33.9697	−1.81059
$353$	−13.2848	−0.707079	−0.353539	−	0.935420i	$-0.615022\pi$
$353$	−13.2848	−0.707079	−0.353539	+	0.935420i	$0.615022\pi$
$354$	0	0
$355$	10.5613	0.560537
$356$	10.8166	0.573277
$357$	0	0
$358$	−5.35237	−0.282882
$359$	14.7281	0.777319	0.388659	−	0.921382i	$-0.372938\pi$
$359$	14.7281	0.777319	0.388659	+	0.921382i	$0.372938\pi$
$360$	0	0
$361$	24.2832	1.27806
$362$	−1.55710	−0.0818392
$363$	0	0
$364$	0.849352	0.0445181
$365$	13.4315	0.703037
$366$	0	0
$367$	−0.439960	−0.0229657	−0.0114829	−	0.999934i	$-0.503655\pi$
$367$	−0.439960	−0.0229657	−0.0114829	+	0.999934i	$0.503655\pi$
$368$	−16.6501	−0.867945
$369$	0	0
$370$	31.8222	1.65436
$371$	−0.323683	−0.0168048
$372$	0	0
$373$	−0.827899	−0.0428670	−0.0214335	−	0.999770i	$-0.506823\pi$
$373$	−0.827899	−0.0428670	−0.0214335	+	0.999770i	$0.506823\pi$
$374$	0	0
$375$	0	0
$376$	10.6309	0.548245
$377$	8.84710	0.455649
$378$	0	0
$379$	−19.8083	−1.01749	−0.508743	−	0.860918i	$-0.669890\pi$
$379$	−19.8083	−1.01749	−0.508743	+	0.860918i	$0.669890\pi$
$380$	15.0087	0.769930
$381$	0	0
$382$	−18.4234	−0.942625
$383$	16.6828	0.852453	0.426227	−	0.904616i	$-0.359843\pi$
$383$	16.6828	0.852453	0.426227	+	0.904616i	$0.359843\pi$
$384$	0	0
$385$	1.23463	0.0629227
$386$	6.89987	0.351194
$387$	0	0
$388$	−5.10013	−0.258920
$389$	−17.2980	−0.877042	−0.438521	−	0.898721i	$-0.644498\pi$
$389$	−17.2980	−0.877042	−0.438521	+	0.898721i	$0.644498\pi$
$390$	0	0
$391$	0	0
$392$	7.55166	0.381416
$393$	0	0
$394$	29.9435	1.50853
$395$	0.858710	0.0432064
$396$	0	0
$397$	−6.44065	−0.323247	−0.161623	−	0.986852i	$-0.551673\pi$
$397$	−6.44065	−0.323247	−0.161623	+	0.986852i	$0.551673\pi$
$398$	30.4389	1.52577
$399$	0	0
$400$	11.5777	0.578886
$401$	21.0724	1.05231	0.526153	−	0.850390i	$-0.323634\pi$
$401$	21.0724	1.05231	0.526153	+	0.850390i	$0.323634\pi$
$402$	0	0
$403$	−10.1561	−0.505912
$404$	−12.9218	−0.642882
$405$	0	0
$406$	0.630865	0.0313093
$407$	−53.6728	−2.66046
$408$	0	0
$409$	−27.6232	−1.36588	−0.682939	−	0.730475i	$-0.739299\pi$
$409$	−27.6232	−1.36588	−0.682939	+	0.730475i	$0.739299\pi$
$410$	0.824955	0.0407416
$411$	0	0
$412$	10.7178	0.528028
$413$	0.201539	0.00991707
$414$	0	0
$415$	3.55444	0.174481
$416$	26.6560	1.30692
$417$	0	0
$418$	−61.1144	−2.98920
$419$	−12.0510	−0.588728	−0.294364	−	0.955693i	$-0.595108\pi$
$419$	−12.0510	−0.588728	−0.294364	+	0.955693i	$0.595108\pi$
$420$	0	0
$421$	14.0183	0.683210	0.341605	−	0.939844i	$-0.389029\pi$
$421$	14.0183	0.683210	0.341605	+	0.939844i	$0.389029\pi$
$422$	19.7502	0.961425
$423$	0	0
$424$	−2.30130	−0.111761
$425$	0	0
$426$	0	0
$427$	1.26031	0.0609905
$428$	26.1158	1.26235
$429$	0	0
$430$	−18.9057	−0.911714
$431$	−3.44381	−0.165882	−0.0829411	−	0.996554i	$-0.526431\pi$
$431$	−3.44381	−0.165882	−0.0829411	+	0.996554i	$0.526431\pi$
$432$	0	0
$433$	5.56579	0.267475	0.133738	−	0.991017i	$-0.457302\pi$
$433$	5.56579	0.267475	0.133738	+	0.991017i	$0.457302\pi$
$434$	−0.724208	−0.0347631
$435$	0	0
$436$	−15.1876	−0.727354
$437$	−22.6866	−1.08525
$438$	0	0
$439$	−17.5864	−0.839353	−0.419676	−	0.907674i	$-0.637856\pi$
$439$	−17.5864	−0.839353	−0.419676	+	0.907674i	$0.637856\pi$
$440$	8.77791	0.418470
$441$	0	0
$442$	0	0
$443$	−5.87632	−0.279192	−0.139596	−	0.990209i	$-0.544580\pi$
$443$	−5.87632	−0.279192	−0.139596	+	0.990209i	$0.544580\pi$
$444$	0	0
$445$	−12.3380	−0.584875
$446$	10.0084	0.473911
$447$	0	0
$448$	0.430602	0.0203441
$449$	−39.9812	−1.88683	−0.943414	−	0.331617i	$-0.892406\pi$
$449$	−39.9812	−1.88683	−0.943414	+	0.331617i	$0.892406\pi$
$450$	0	0
$451$	−1.39141	−0.0655189
$452$	2.36876	0.111417
$453$	0	0
$454$	22.7250	1.06654
$455$	−0.968815	−0.0454188
$456$	0	0
$457$	−17.6906	−0.827533	−0.413767	−	0.910383i	$-0.635787\pi$
$457$	−17.6906	−0.827533	−0.413767	+	0.910383i	$0.635787\pi$
$458$	55.6467	2.60020
$459$	0	0
$460$	−7.86672	−0.366788
$461$	−29.8662	−1.39101	−0.695504	−	0.718522i	$-0.744819\pi$
$461$	−29.8662	−1.39101	−0.695504	+	0.718522i	$0.744819\pi$
$462$	0	0
$463$	9.45213	0.439278	0.219639	−	0.975581i	$-0.429512\pi$
$463$	9.45213	0.439278	0.219639	+	0.975581i	$0.429512\pi$
$464$	10.8284	0.502697
$465$	0	0
$466$	16.2330	0.751978
$467$	20.0094	0.925923	0.462961	−	0.886378i	$-0.346787\pi$
$467$	20.0094	0.925923	0.462961	+	0.886378i	$0.346787\pi$
$468$	0	0
$469$	−1.08239	−0.0499802
$470$	29.2750	1.35036
$471$	0	0
$472$	1.43289	0.0659539
$473$	31.8873	1.46618
$474$	0	0
$475$	15.7753	0.723820
$476$	0	0
$477$	0	0
$478$	−27.0698	−1.23814
$479$	−1.20570	−0.0550899	−0.0275449	−	0.999621i	$-0.508769\pi$
$479$	−1.20570	−0.0550899	−0.0275449	+	0.999621i	$0.508769\pi$
$480$	0	0
$481$	42.1170	1.92037
$482$	31.3292	1.42701
$483$	0	0
$484$	20.1867	0.917577
$485$	5.81748	0.264158
$486$	0	0
$487$	−11.9613	−0.542017	−0.271009	−	0.962577i	$-0.587357\pi$
$487$	−11.9613	−0.542017	−0.271009	+	0.962577i	$0.587357\pi$
$488$	8.96043	0.405620
$489$	0	0
$490$	20.7956	0.939449
$491$	−31.2632	−1.41089	−0.705444	−	0.708766i	$-0.749252\pi$
$491$	−31.2632	−1.41089	−0.705444	+	0.708766i	$0.749252\pi$
$492$	0	0
$493$	0	0
$494$	47.9564	2.15766
$495$	0	0
$496$	−12.4306	−0.558151
$497$	0.996740	0.0447099
$498$	0	0
$499$	41.3590	1.85148	0.925741	−	0.378158i	$-0.123442\pi$
$499$	41.3590	1.85148	0.925741	+	0.378158i	$0.123442\pi$
$500$	16.8767	0.754749
$501$	0	0
$502$	25.7675	1.15006
$503$	3.08277	0.137454	0.0687269	−	0.997636i	$-0.478106\pi$
$503$	3.08277	0.137454	0.0687269	+	0.997636i	$0.478106\pi$
$504$	0	0
$505$	14.7392	0.655888
$506$	32.0327	1.42403
$507$	0	0
$508$	−17.2268	−0.764315
$509$	33.8077	1.49850	0.749249	−	0.662288i	$-0.230415\pi$
$509$	33.8077	1.49850	0.749249	+	0.662288i	$0.230415\pi$
$510$	0	0
$511$	1.26762	0.0560760
$512$	22.1731	0.979922
$513$	0	0
$514$	−36.3275	−1.60234
$515$	−12.2253	−0.538710
$516$	0	0
$517$	−49.3767	−2.17159
$518$	3.00326	0.131956
$519$	0	0
$520$	−6.88802	−0.302059
$521$	−40.9557	−1.79430	−0.897150	−	0.441725i	$-0.854367\pi$
$521$	−40.9557	−1.79430	−0.897150	+	0.441725i	$0.854367\pi$
$522$	0	0
$523$	−9.06788	−0.396511	−0.198255	−	0.980150i	$-0.563528\pi$
$523$	−9.06788	−0.396511	−0.198255	+	0.980150i	$0.563528\pi$
$524$	14.7802	0.645674
$525$	0	0
$526$	51.5009	2.24555
$527$	0	0
$528$	0	0
$529$	−11.1089	−0.482997
$530$	−6.33726	−0.275273
$531$	0	0
$532$	1.41647	0.0614116
$533$	1.09184	0.0472927
$534$	0	0
$535$	−29.7890	−1.28789
$536$	−7.69552	−0.332396
$537$	0	0
$538$	0.379395	0.0163569
$539$	−35.0749	−1.51078
$540$	0	0
$541$	−12.9420	−0.556420	−0.278210	−	0.960520i	$-0.589741\pi$
$541$	−12.9420	−0.556420	−0.278210	+	0.960520i	$0.589741\pi$
$542$	−15.1715	−0.651672
$543$	0	0
$544$	0	0
$545$	17.3238	0.742068
$546$	0	0
$547$	10.6598	0.455781	0.227891	−	0.973687i	$-0.426817\pi$
$547$	10.6598	0.455781	0.227891	+	0.973687i	$0.426817\pi$
$548$	−19.0816	−0.815127
$549$	0	0
$550$	−22.2741	−0.949773
$551$	14.7543	0.628556
$552$	0	0
$553$	0.0810419	0.00344625
$554$	−43.5474	−1.85015
$555$	0	0
$556$	−7.53490	−0.319551
$557$	30.1933	1.27933	0.639667	−	0.768653i	$-0.279072\pi$
$557$	30.1933	1.27933	0.639667	+	0.768653i	$0.279072\pi$
$558$	0	0
$559$	−25.0219	−1.05831
$560$	−1.18578	−0.0501085
$561$	0	0
$562$	18.7607	0.791372
$563$	29.9236	1.26113	0.630566	−	0.776136i	$-0.282823\pi$
$563$	29.9236	1.26113	0.630566	+	0.776136i	$0.282823\pi$
$564$	0	0
$565$	−2.70193	−0.113671
$566$	17.5152	0.736221
$567$	0	0
$568$	7.08655	0.297345
$569$	−25.6952	−1.07720	−0.538599	−	0.842562i	$-0.681046\pi$
$569$	−25.6952	−1.07720	−0.538599	+	0.842562i	$0.681046\pi$
$570$	0	0
$571$	−28.0143	−1.17236	−0.586181	−	0.810180i	$-0.699369\pi$
$571$	−28.0143	−1.17236	−0.586181	+	0.810180i	$0.699369\pi$
$572$	−28.0475	−1.17273
$573$	0	0
$574$	0.0778563	0.00324966
$575$	−8.26852	−0.344821
$576$	0	0
$577$	−11.8072	−0.491538	−0.245769	−	0.969328i	$-0.579041\pi$
$577$	−11.8072	−0.491538	−0.245769	+	0.969328i	$0.579041\pi$
$578$	0	0
$579$	0	0
$580$	5.11615	0.212436
$581$	0.335455	0.0139170
$582$	0	0
$583$	10.6887	0.442682
$584$	9.01241	0.372936
$585$	0	0
$586$	−17.1567	−0.708738
$587$	−7.44230	−0.307177	−0.153588	−	0.988135i	$-0.549083\pi$
$587$	−7.44230	−0.307177	−0.153588	+	0.988135i	$0.549083\pi$
$588$	0	0
$589$	−16.9374	−0.697893
$590$	3.94585	0.162448
$591$	0	0
$592$	51.5492	2.11866
$593$	−7.65194	−0.314228	−0.157114	−	0.987580i	$-0.550219\pi$
$593$	−7.65194	−0.314228	−0.157114	+	0.987580i	$0.550219\pi$
$594$	0	0
$595$	0	0
$596$	−13.1789	−0.539830
$597$	0	0
$598$	−25.1360	−1.02789
$599$	−16.1547	−0.660062	−0.330031	−	0.943970i	$-0.607059\pi$
$599$	−16.1547	−0.660062	−0.330031	+	0.943970i	$0.607059\pi$
$600$	0	0
$601$	−28.7176	−1.17141	−0.585707	−	0.810523i	$-0.699183\pi$
$601$	−28.7176	−1.17141	−0.585707	+	0.810523i	$0.699183\pi$
$602$	−1.78425	−0.0727206
$603$	0	0
$604$	16.9186	0.688407
$605$	−23.0260	−0.936140
$606$	0	0
$607$	−25.8129	−1.04771	−0.523856	−	0.851807i	$-0.675507\pi$
$607$	−25.8129	−1.04771	−0.523856	+	0.851807i	$0.675507\pi$
$608$	44.4542	1.80286
$609$	0	0
$610$	24.6750	0.999063
$611$	38.7458	1.56749
$612$	0	0
$613$	49.1769	1.98623	0.993117	−	0.117123i	$-0.0373670\pi$
$613$	49.1769	1.98623	0.993117	+	0.117123i	$0.0373670\pi$
$614$	−50.1514	−2.02395
$615$	0	0
$616$	0.828427	0.0333783
$617$	3.55073	0.142947	0.0714734	−	0.997443i	$-0.477230\pi$
$617$	3.55073	0.142947	0.0714734	+	0.997443i	$0.477230\pi$
$618$	0	0
$619$	35.2011	1.41485	0.707426	−	0.706787i	$-0.249856\pi$
$619$	35.2011	1.41485	0.707426	+	0.706787i	$0.249856\pi$
$620$	−5.87313	−0.235871
$621$	0	0
$622$	−4.77791	−0.191577
$623$	−1.16441	−0.0466511
$624$	0	0
$625$	−7.26131	−0.290453
$626$	−47.3220	−1.89137
$627$	0	0
$628$	−25.6748	−1.02454
$629$	0	0
$630$	0	0
$631$	−31.4131	−1.25054	−0.625268	−	0.780410i	$-0.715010\pi$
$631$	−31.4131	−1.25054	−0.625268	+	0.780410i	$0.715010\pi$
$632$	0.576186	0.0229195
$633$	0	0
$634$	47.2993	1.87850
$635$	19.6498	0.779777
$636$	0	0
$637$	27.5232	1.09051
$638$	−20.8326	−0.824770
$639$	0	0
$640$	−13.3691	−0.528461
$641$	−26.5755	−1.04967	−0.524834	−	0.851205i	$-0.675873\pi$
$641$	−26.5755	−1.04967	−0.524834	+	0.851205i	$0.675873\pi$
$642$	0	0
$643$	16.0247	0.631952	0.315976	−	0.948767i	$-0.397668\pi$
$643$	16.0247	0.631952	0.315976	+	0.948767i	$0.397668\pi$
$644$	−0.742432	−0.0292559
$645$	0	0
$646$	0	0
$647$	−41.6554	−1.63764	−0.818822	−	0.574048i	$-0.805372\pi$
$647$	−41.6554	−1.63764	−0.818822	+	0.574048i	$0.805372\pi$
$648$	0	0
$649$	−6.65526	−0.261242
$650$	17.4785	0.685563
$651$	0	0
$652$	29.3060	1.14771
$653$	−12.3264	−0.482370	−0.241185	−	0.970479i	$-0.577536\pi$
$653$	−12.3264	−0.482370	−0.241185	+	0.970479i	$0.577536\pi$
$654$	0	0
$655$	−16.8590	−0.658737
$656$	1.33636	0.0521760
$657$	0	0
$658$	2.76287	0.107708
$659$	−3.46449	−0.134957	−0.0674786	−	0.997721i	$-0.521495\pi$
$659$	−3.46449	−0.134957	−0.0674786	+	0.997721i	$0.521495\pi$
$660$	0	0
$661$	−28.3233	−1.10165	−0.550825	−	0.834621i	$-0.685687\pi$
$661$	−28.3233	−1.10165	−0.550825	+	0.834621i	$0.685687\pi$
$662$	−0.615621	−0.0239268
$663$	0	0
$664$	2.38500	0.0925558
$665$	−1.61570	−0.0626540
$666$	0	0
$667$	−7.73339	−0.299438
$668$	−11.4393	−0.442600
$669$	0	0
$670$	−21.1918	−0.818709
$671$	−41.6181	−1.60665
$672$	0	0
$673$	28.4086	1.09507	0.547535	−	0.836783i	$-0.315566\pi$
$673$	28.4086	1.09507	0.547535	+	0.836783i	$0.315566\pi$
$674$	6.93906	0.267283
$675$	0	0
$676$	3.62408	0.139388
$677$	19.2983	0.741692	0.370846	−	0.928694i	$-0.379068\pi$
$677$	19.2983	0.741692	0.370846	+	0.928694i	$0.379068\pi$
$678$	0	0
$679$	0.549032	0.0210699
$680$	0	0
$681$	0	0
$682$	23.9150	0.915752
$683$	−19.4793	−0.745355	−0.372678	−	0.927961i	$-0.621560\pi$
$683$	−19.4793	−0.745355	−0.372678	+	0.927961i	$0.621560\pi$
$684$	0	0
$685$	21.7655	0.831618
$686$	3.93174	0.150115
$687$	0	0
$688$	−30.6256	−1.16759
$689$	−8.38743	−0.319536
$690$	0	0
$691$	7.53940	0.286812	0.143406	−	0.989664i	$-0.454194\pi$
$691$	7.53940	0.286812	0.143406	+	0.989664i	$0.454194\pi$
$692$	1.64137	0.0623954
$693$	0	0
$694$	6.11607	0.232163
$695$	8.59470	0.326015
$696$	0	0
$697$	0	0
$698$	4.77624	0.180783
$699$	0	0
$700$	0.516255	0.0195126
$701$	39.0875	1.47632	0.738158	−	0.674628i	$-0.235696\pi$
$701$	39.0875	1.47632	0.738158	+	0.674628i	$0.235696\pi$
$702$	0	0
$703$	70.2386	2.64910
$704$	−14.2195	−0.535916
$705$	0	0
$706$	−24.5471	−0.923843
$707$	1.39104	0.0523153
$708$	0	0
$709$	−6.71284	−0.252106	−0.126053	−	0.992024i	$-0.540231\pi$
$709$	−6.71284	−0.252106	−0.126053	+	0.992024i	$0.540231\pi$
$710$	19.5148	0.732377
$711$	0	0
$712$	−8.27865	−0.310255
$713$	8.87762	0.332470
$714$	0	0
$715$	31.9925	1.19645
$716$	−4.09653	−0.153094
$717$	0	0
$718$	27.2140	1.01562
$719$	7.40623	0.276206	0.138103	−	0.990418i	$-0.455900\pi$
$719$	7.40623	0.276206	0.138103	+	0.990418i	$0.455900\pi$
$720$	0	0
$721$	−1.15378	−0.0429689
$722$	44.8695	1.66987
$723$	0	0
$724$	−1.19175	−0.0442910
$725$	5.37746	0.199714
$726$	0	0
$727$	−6.36054	−0.235899	−0.117950	−	0.993020i	$-0.537632\pi$
$727$	−6.36054	−0.235899	−0.117950	+	0.993020i	$0.537632\pi$
$728$	−0.650066	−0.0240930
$729$	0	0
$730$	24.8182	0.918562
$731$	0	0
$732$	0	0
$733$	−41.3337	−1.52670	−0.763348	−	0.645988i	$-0.776446\pi$
$733$	−41.3337	−1.52670	−0.763348	+	0.645988i	$0.776446\pi$
$734$	−0.812941	−0.0300062
$735$	0	0
$736$	−23.3004	−0.858864
$737$	35.7430	1.31661
$738$	0	0
$739$	23.1653	0.852150	0.426075	−	0.904688i	$-0.359896\pi$
$739$	23.1653	0.852150	0.426075	+	0.904688i	$0.359896\pi$
$740$	24.3556	0.895331
$741$	0	0
$742$	−0.598087	−0.0219565
$743$	38.1505	1.39960	0.699802	−	0.714337i	$-0.253271\pi$
$743$	38.1505	1.39960	0.699802	+	0.714337i	$0.253271\pi$
$744$	0	0
$745$	15.0326	0.550751
$746$	−1.52976	−0.0560084
$747$	0	0
$748$	0	0
$749$	−2.81138	−0.102726
$750$	0	0
$751$	−34.2073	−1.24824	−0.624120	−	0.781328i	$-0.714543\pi$
$751$	−34.2073	−1.24824	−0.624120	+	0.781328i	$0.714543\pi$
$752$	47.4231	1.72934
$753$	0	0
$754$	16.3473	0.595334
$755$	−19.2982	−0.702334
$756$	0	0
$757$	−7.02190	−0.255215	−0.127608	−	0.991825i	$-0.540730\pi$
$757$	−7.02190	−0.255215	−0.127608	+	0.991825i	$0.540730\pi$
$758$	−36.6011	−1.32941
$759$	0	0
$760$	−11.4872	−0.416683
$761$	−47.2917	−1.71432	−0.857161	−	0.515048i	$-0.827774\pi$
$761$	−47.2917	−1.71432	−0.857161	+	0.515048i	$0.827774\pi$
$762$	0	0
$763$	1.63495	0.0591893
$764$	−14.1007	−0.510145
$765$	0	0
$766$	30.8259	1.11378
$767$	5.22238	0.188569
$768$	0	0
$769$	6.39156	0.230486	0.115243	−	0.993337i	$-0.463235\pi$
$769$	6.39156	0.230486	0.115243	+	0.993337i	$0.463235\pi$
$770$	2.28130	0.0822125
$771$	0	0
$772$	5.28093	0.190065
$773$	−3.67729	−0.132263	−0.0661315	−	0.997811i	$-0.521066\pi$
$773$	−3.67729	−0.132263	−0.0661315	+	0.997811i	$0.521066\pi$
$774$	0	0
$775$	−6.17311	−0.221745
$776$	3.90347	0.140126
$777$	0	0
$778$	−31.9625	−1.14591
$779$	1.82086	0.0652391
$780$	0	0
$781$	−32.9146	−1.17778
$782$	0	0
$783$	0	0
$784$	33.6871	1.20311
$785$	29.2860	1.04526
$786$	0	0
$787$	4.06409	0.144869	0.0724346	−	0.997373i	$-0.476923\pi$
$787$	4.06409	0.144869	0.0724346	+	0.997373i	$0.476923\pi$
$788$	22.9178	0.816413
$789$	0	0
$790$	1.58669	0.0564519
$791$	−0.254999	−0.00906670
$792$	0	0
$793$	32.6577	1.15971
$794$	−11.9008	−0.422343
$795$	0	0
$796$	23.2969	0.825738
$797$	13.8056	0.489020	0.244510	−	0.969647i	$-0.421373\pi$
$797$	13.8056	0.489020	0.244510	+	0.969647i	$0.421373\pi$
$798$	0	0
$799$	0	0
$800$	16.2021	0.572830
$801$	0	0
$802$	38.9368	1.37490
$803$	−41.8596	−1.47719
$804$	0	0
$805$	0.846857	0.0298478
$806$	−18.7661	−0.661006
$807$	0	0
$808$	9.88989	0.347925
$809$	−18.8490	−0.662695	−0.331347	−	0.943509i	$-0.607503\pi$
$809$	−18.8490	−0.662695	−0.331347	+	0.943509i	$0.607503\pi$
$810$	0	0
$811$	−2.59955	−0.0912825	−0.0456413	−	0.998958i	$-0.514533\pi$
$811$	−2.59955	−0.0912825	−0.0456413	+	0.998958i	$0.514533\pi$
$812$	0.482843	0.0169445
$813$	0	0
$814$	−99.1744	−3.47606
$815$	−33.4280	−1.17093
$816$	0	0
$817$	−41.7291	−1.45992
$818$	−51.0410	−1.78461
$819$	0	0
$820$	0.631394	0.0220492
$821$	13.0073	0.453960	0.226980	−	0.973899i	$-0.427115\pi$
$821$	13.0073	0.453960	0.226980	+	0.973899i	$0.427115\pi$
$822$	0	0
$823$	21.6544	0.754824	0.377412	−	0.926046i	$-0.376814\pi$
$823$	21.6544	0.754824	0.377412	+	0.926046i	$0.376814\pi$
$824$	−8.20304	−0.285767
$825$	0	0
$826$	0.372395	0.0129573
$827$	−47.8742	−1.66475	−0.832375	−	0.554213i	$-0.813019\pi$
$827$	−47.8742	−1.66475	−0.832375	+	0.554213i	$0.813019\pi$
$828$	0	0
$829$	12.9906	0.451181	0.225590	−	0.974222i	$-0.427569\pi$
$829$	12.9906	0.451181	0.225590	+	0.974222i	$0.427569\pi$
$830$	6.56775	0.227970
$831$	0	0
$832$	11.1580	0.386834
$833$	0	0
$834$	0	0
$835$	13.0483	0.451554
$836$	−46.7749	−1.61774
$837$	0	0
$838$	−22.2673	−0.769210
$839$	−18.8727	−0.651557	−0.325779	−	0.945446i	$-0.605626\pi$
$839$	−18.8727	−0.651557	−0.325779	+	0.945446i	$0.605626\pi$
$840$	0	0
$841$	−23.9706	−0.826571
$842$	25.9024	0.892657
$843$	0	0
$844$	15.1161	0.520319
$845$	−4.13381	−0.142207
$846$	0	0
$847$	−2.17311	−0.0746689
$848$	−10.2658	−0.352530
$849$	0	0
$850$	0	0
$851$	−36.8151	−1.26201
$852$	0	0
$853$	6.03591	0.206665	0.103333	−	0.994647i	$-0.467049\pi$
$853$	6.03591	0.206665	0.103333	+	0.994647i	$0.467049\pi$
$854$	2.32874	0.0796879
$855$	0	0
$856$	−19.9881	−0.683181
$857$	16.0835	0.549401	0.274701	−	0.961530i	$-0.411421\pi$
$857$	16.0835	0.549401	0.274701	+	0.961530i	$0.411421\pi$
$858$	0	0
$859$	−57.8556	−1.97401	−0.987003	−	0.160699i	$-0.948625\pi$
$859$	−57.8556	−1.97401	−0.987003	+	0.160699i	$0.948625\pi$
$860$	−14.4698	−0.493416
$861$	0	0
$862$	−6.36332	−0.216736
$863$	7.56067	0.257368	0.128684	−	0.991686i	$-0.458925\pi$
$863$	7.56067	0.257368	0.128684	+	0.991686i	$0.458925\pi$
$864$	0	0
$865$	−1.87223	−0.0636577
$866$	10.2842	0.349473
$867$	0	0
$868$	−0.554285	−0.0188137
$869$	−2.67619	−0.0907834
$870$	0	0
$871$	−28.0475	−0.950354
$872$	11.6241	0.393641
$873$	0	0
$874$	−41.9194	−1.41795
$875$	−1.81679	−0.0614186
$876$	0	0
$877$	17.3660	0.586407	0.293204	−	0.956050i	$-0.405279\pi$
$877$	17.3660	0.586407	0.293204	+	0.956050i	$0.405279\pi$
$878$	−32.4954	−1.09667
$879$	0	0
$880$	39.1572	1.31999
$881$	41.1018	1.38476	0.692378	−	0.721535i	$-0.256563\pi$
$881$	41.1018	1.38476	0.692378	+	0.721535i	$0.256563\pi$
$882$	0	0
$883$	28.3729	0.954824	0.477412	−	0.878680i	$-0.341575\pi$
$883$	28.3729	0.954824	0.477412	+	0.878680i	$0.341575\pi$
$884$	0	0
$885$	0	0
$886$	−10.8580	−0.364782
$887$	21.5004	0.721914	0.360957	−	0.932583i	$-0.382450\pi$
$887$	21.5004	0.721914	0.360957	+	0.932583i	$0.382450\pi$
$888$	0	0
$889$	1.85447	0.0621970
$890$	−22.7976	−0.764176
$891$	0	0
$892$	7.66008	0.256479
$893$	64.6165	2.16231
$894$	0	0
$895$	4.67271	0.156192
$896$	−1.26173	−0.0421514
$897$	0	0
$898$	−73.8756	−2.46526
$899$	−5.77359	−0.192560
$900$	0	0
$901$	0	0
$902$	−2.57099	−0.0856046
$903$	0	0
$904$	−1.81297	−0.0602985
$905$	1.35937	0.0451871
$906$	0	0
$907$	−14.8069	−0.491654	−0.245827	−	0.969314i	$-0.579060\pi$
$907$	−14.8069	−0.491654	−0.245827	+	0.969314i	$0.579060\pi$
$908$	17.3929	0.577205
$909$	0	0
$910$	−1.79014	−0.0593425
$911$	−47.5209	−1.57444	−0.787220	−	0.616673i	$-0.788480\pi$
$911$	−47.5209	−1.57444	−0.787220	+	0.616673i	$0.788480\pi$
$912$	0	0
$913$	−11.0775	−0.366611
$914$	−32.6880	−1.08122
$915$	0	0
$916$	42.5901	1.40722
$917$	−1.59109	−0.0525425
$918$	0	0
$919$	−52.4090	−1.72881	−0.864407	−	0.502793i	$-0.832306\pi$
$919$	−52.4090	−1.72881	−0.864407	+	0.502793i	$0.832306\pi$
$920$	6.02092	0.198504
$921$	0	0
$922$	−55.1856	−1.81744
$923$	25.8281	0.850141
$924$	0	0
$925$	25.5996	0.841710
$926$	17.4653	0.573944
$927$	0	0
$928$	15.1535	0.497438
$929$	45.8975	1.50585	0.752924	−	0.658107i	$-0.228643\pi$
$929$	45.8975	1.50585	0.752924	+	0.658107i	$0.228643\pi$
$930$	0	0
$931$	45.9005	1.50433
$932$	12.4242	0.406968
$933$	0	0
$934$	36.9725	1.20978
$935$	0	0
$936$	0	0
$937$	21.7579	0.710799	0.355400	−	0.934714i	$-0.384345\pi$
$937$	21.7579	0.710799	0.355400	+	0.934714i	$0.384345\pi$
$938$	−2.00000	−0.0653023
$939$	0	0
$940$	22.4061	0.730808
$941$	−54.1022	−1.76368	−0.881841	−	0.471548i	$-0.843696\pi$
$941$	−54.1022	−1.76368	−0.881841	+	0.471548i	$0.843696\pi$
$942$	0	0
$943$	−0.954393	−0.0310793
$944$	6.39194	0.208040
$945$	0	0
$946$	58.9200	1.91565
$947$	−5.77608	−0.187697	−0.0938486	−	0.995586i	$-0.529917\pi$
$947$	−5.77608	−0.187697	−0.0938486	+	0.995586i	$0.529917\pi$
$948$	0	0
$949$	32.8471	1.06626
$950$	29.1489	0.945716
$951$	0	0
$952$	0	0
$953$	31.4698	1.01941	0.509704	−	0.860350i	$-0.329755\pi$
$953$	31.4698	1.01941	0.509704	+	0.860350i	$0.329755\pi$
$954$	0	0
$955$	16.0840	0.520466
$956$	−20.7183	−0.670078
$957$	0	0
$958$	−2.22784	−0.0719784
$959$	2.05415	0.0663320
$960$	0	0
$961$	−24.3721	−0.786198
$962$	77.8221	2.50908
$963$	0	0
$964$	23.9783	0.772290
$965$	−6.02371	−0.193910
$966$	0	0
$967$	−5.82126	−0.187199	−0.0935996	−	0.995610i	$-0.529837\pi$
$967$	−5.82126	−0.187199	−0.0935996	+	0.995610i	$0.529837\pi$
$968$	−15.4502	−0.496589
$969$	0	0
$970$	10.7493	0.345139
$971$	−21.9049	−0.702963	−0.351481	−	0.936195i	$-0.614322\pi$
$971$	−21.9049	−0.702963	−0.351481	+	0.936195i	$0.614322\pi$
$972$	0	0
$973$	0.811136	0.0260038
$974$	−22.1016	−0.708180
$975$	0	0
$976$	39.9715	1.27946
$977$	28.0473	0.897312	0.448656	−	0.893705i	$-0.351903\pi$
$977$	28.0473	0.897312	0.448656	+	0.893705i	$0.351903\pi$
$978$	0	0
$979$	38.4515	1.22891
$980$	15.9163	0.508426
$981$	0	0
$982$	−57.7668	−1.84341
$983$	−31.3271	−0.999179	−0.499590	−	0.866262i	$-0.666516\pi$
$983$	−31.3271	−0.999179	−0.499590	+	0.866262i	$0.666516\pi$
$984$	0	0
$985$	−26.1412	−0.832929
$986$	0	0
$987$	0	0
$988$	36.7042	1.16772
$989$	21.8720	0.695491
$990$	0	0
$991$	−17.9993	−0.571765	−0.285883	−	0.958265i	$-0.592287\pi$
$991$	−17.9993	−0.571765	−0.285883	+	0.958265i	$0.592287\pi$
$992$	−17.3956	−0.552311
$993$	0	0
$994$	1.84174	0.0584163
$995$	−26.5737	−0.842443
$996$	0	0
$997$	−55.5666	−1.75981	−0.879907	−	0.475147i	$-0.842395\pi$
$997$	−55.5666	−1.75981	−0.879907	+	0.475147i	$0.842395\pi$
$998$	76.4214	2.41908
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bd.1.4		4
3.2	odd	2		867.2.a.n.1.1		4
17.3	odd	16		153.2.l.e.145.2		8
17.6	odd	16		153.2.l.e.19.2		8
17.16	even	2		2601.2.a.bc.1.4		4
51.2	odd	8		867.2.e.i.616.1		8
51.5	even	16		867.2.h.b.688.2		8
51.8	odd	8		867.2.e.h.829.4		8
51.11	even	16		867.2.h.g.733.1		8
51.14	even	16		867.2.h.g.757.1		8
51.20	even	16		51.2.h.a.43.1	yes	8
51.23	even	16		51.2.h.a.19.1	✓	8
51.26	odd	8		867.2.e.i.829.4		8
51.29	even	16		867.2.h.f.688.2		8
51.32	odd	8		867.2.e.h.616.1		8
51.38	odd	4		867.2.d.e.577.7		8
51.41	even	16		867.2.h.b.712.2		8
51.44	even	16		867.2.h.f.712.2		8
51.47	odd	4		867.2.d.e.577.8		8
51.50	odd	2		867.2.a.m.1.1		4
204.23	odd	16		816.2.bq.a.529.1		8
204.71	odd	16		816.2.bq.a.145.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.h.a.19.1	✓	8	51.23	even	16
51.2.h.a.43.1	yes	8	51.20	even	16
153.2.l.e.19.2		8	17.6	odd	16
153.2.l.e.145.2		8	17.3	odd	16
816.2.bq.a.145.1		8	204.71	odd	16
816.2.bq.a.529.1		8	204.23	odd	16
867.2.a.m.1.1		4	51.50	odd	2
867.2.a.n.1.1		4	3.2	odd	2
867.2.d.e.577.7		8	51.38	odd	4
867.2.d.e.577.8		8	51.47	odd	4
867.2.e.h.616.1		8	51.32	odd	8
867.2.e.h.829.4		8	51.8	odd	8
867.2.e.i.616.1		8	51.2	odd	8
867.2.e.i.829.4		8	51.26	odd	8
867.2.h.b.688.2		8	51.5	even	16
867.2.h.b.712.2		8	51.41	even	16
867.2.h.f.688.2		8	51.29	even	16
867.2.h.f.712.2		8	51.44	even	16
867.2.h.g.733.1		8	51.11	even	16
867.2.h.g.757.1		8	51.14	even	16
2601.2.a.bc.1.4		4	17.16	even	2
2601.2.a.bd.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+1.84776 q^{2} +1.41421 q^{4} -1.61313 q^{5} -0.152241 q^{7} -1.08239 q^{8} +O(q^{10})\)	\(q+1.84776 q^{2} +1.41421 q^{4} -1.61313 q^{5} -0.152241 q^{7} -1.08239 q^{8} -2.98067 q^{10} +5.02734 q^{11} -3.94495 q^{13} -0.281305 q^{14} -4.82843 q^{16} -6.57900 q^{19} -2.28130 q^{20} +9.28931 q^{22} +3.44834 q^{23} -2.39782 q^{25} -7.28931 q^{26} -0.215301 q^{28} -2.24264 q^{29} +2.57446 q^{31} -6.75699 q^{32} +0.245584 q^{35} -10.6762 q^{37} -12.1564 q^{38} +1.74603 q^{40} -0.276769 q^{41} +6.34277 q^{43} +7.10973 q^{44} +6.37170 q^{46} -9.82164 q^{47} -6.97682 q^{49} -4.43060 q^{50} -5.57900 q^{52} +2.12612 q^{53} -8.10973 q^{55} +0.164784 q^{56} -4.14386 q^{58} -1.32381 q^{59} -8.27836 q^{61} +4.75699 q^{62} -2.82843 q^{64} +6.36370 q^{65} +7.10973 q^{67} +0.453780 q^{70} -6.54712 q^{71} -8.32638 q^{73} -19.7270 q^{74} -9.30411 q^{76} -0.765367 q^{77} -0.532327 q^{79} +7.78886 q^{80} -0.511402 q^{82} -2.20345 q^{83} +11.7199 q^{86} -5.44155 q^{88} +7.64847 q^{89} +0.600582 q^{91} +4.87669 q^{92} -18.1480 q^{94} +10.6128 q^{95} -3.60634 q^{97} -12.8915 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 8 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 8 * q^7	\( 4 q + 4 q^{5} - 8 q^{7} - 8 q^{10} + 4 q^{11} - 4 q^{13} + 8 q^{14} - 8 q^{16} - 12 q^{19} + 8 q^{22} + 12 q^{23} + 8 q^{29} - 8 q^{31} - 16 q^{35} - 24 q^{37} - 16 q^{38} + 12 q^{41} + 4 q^{43} + 8 q^{44} - 8 q^{46} - 8 q^{47} - 4 q^{49} - 16 q^{50} - 8 q^{52} - 8 q^{53} - 12 q^{55} - 8 q^{56} - 24 q^{59} - 24 q^{61} - 8 q^{62} + 12 q^{65} + 8 q^{67} + 24 q^{70} - 24 q^{71} - 8 q^{73} - 8 q^{74} - 8 q^{76} - 8 q^{80} + 8 q^{82} - 8 q^{83} + 16 q^{86} - 16 q^{89} + 8 q^{91} - 16 q^{94} - 12 q^{95} + 16 q^{97} - 32 q^{98}+O(q^{100}) \) 4 * q + 4 * q^5 - 8 * q^7 - 8 * q^10 + 4 * q^11 - 4 * q^13 + 8 * q^14 - 8 * q^16 - 12 * q^19 + 8 * q^22 + 12 * q^23 + 8 * q^29 - 8 * q^31 - 16 * q^35 - 24 * q^37 - 16 * q^38 + 12 * q^41 + 4 * q^43 + 8 * q^44 - 8 * q^46 - 8 * q^47 - 4 * q^49 - 16 * q^50 - 8 * q^52 - 8 * q^53 - 12 * q^55 - 8 * q^56 - 24 * q^59 - 24 * q^61 - 8 * q^62 + 12 * q^65 + 8 * q^67 + 24 * q^70 - 24 * q^71 - 8 * q^73 - 8 * q^74 - 8 * q^76 - 8 * q^80 + 8 * q^82 - 8 * q^83 + 16 * q^86 - 16 * q^89 + 8 * q^91 - 16 * q^94 - 12 * q^95 + 16 * q^97 - 32 * q^98

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