LMFDB - Embedded newform 2576.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, 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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.11948$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84074 q^{3} +2.40920 q^{5} -1.00000 q^{7} +0.388311 q^{9} +O(q^{10})$	$q+1.84074 q^{3} +2.40920 q^{5} -1.00000 q^{7} +0.388311 q^{9} -5.87722 q^{11} -6.24994 q^{13} +4.43471 q^{15} -5.42479 q^{17} +2.23897 q^{19} -1.84074 q^{21} +1.00000 q^{23} +0.804258 q^{25} -4.80743 q^{27} +0.642864 q^{29} -7.84074 q^{31} -10.8184 q^{33} -2.40920 q^{35} +0.557492 q^{37} -11.5045 q^{39} +2.56847 q^{41} +8.81841 q^{43} +0.935520 q^{45} -4.26766 q^{47} +1.00000 q^{49} -9.98561 q^{51} +3.01559 q^{53} -14.1594 q^{55} +4.12134 q^{57} -4.17024 q^{59} -0.148289 q^{61} -0.388311 q^{63} -15.0574 q^{65} -13.3396 q^{67} +1.84074 q^{69} +7.93141 q^{71} +4.28111 q^{73} +1.48043 q^{75} +5.87722 q^{77} +0.861628 q^{79} -10.0141 q^{81} +4.81841 q^{83} -13.0694 q^{85} +1.18334 q^{87} +6.32964 q^{89} +6.24994 q^{91} -14.4327 q^{93} +5.39412 q^{95} +10.4822 q^{97} -2.28219 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) $ 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9	$ 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9 + 4 * q^11 - 6 * q^13 - 10 * q^15 - 12 * q^17 - 6 * q^19 + 5 * q^23 + 19 * q^25 - 4 * q^29 - 30 * q^31 - 22 * q^33 + 4 * q^35 + 4 * q^37 - 16 * q^39 + 6 * q^41 + 12 * q^43 - 12 * q^45 - 10 * q^47 + 5 * q^49 + 4 * q^51 + 16 * q^53 - 18 * q^55 + 6 * q^57 - 22 * q^59 - 18 * q^61 - 11 * q^63 - 26 * q^65 + 2 * q^67 - 4 * q^71 - 2 * q^73 + 30 * q^75 - 4 * q^77 - 30 * q^79 - 3 * q^81 - 8 * q^83 - 12 * q^85 + 12 * q^87 - 20 * q^89 + 6 * q^91 - 26 * q^93 - 8 * q^95 - 12 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.84074	1.06275	0.531375	−	0.847137i	$-0.321676\pi$
$3$	1.84074	1.06275	0.531375	+	0.847137i	$0.321676\pi$
$4$	0	0
$5$	2.40920	1.07743	0.538714	−	0.842489i	$-0.318910\pi$
$5$	2.40920	1.07743	0.538714	+	0.842489i	$0.318910\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.388311	0.129437
$10$	0	0
$11$	−5.87722	−1.77205	−0.886024	−	0.463640i	$-0.846543\pi$
$11$	−5.87722	−1.77205	−0.886024	+	0.463640i	$0.846543\pi$
$12$	0	0
$13$	−6.24994	−1.73342	−0.866711	−	0.498811i	$-0.833770\pi$
$13$	−6.24994	−1.73342	−0.866711	+	0.498811i	$0.833770\pi$
$14$	0	0
$15$	4.43471	1.14504
$16$	0	0
$17$	−5.42479	−1.31570	−0.657852	−	0.753147i	$-0.728535\pi$
$17$	−5.42479	−1.31570	−0.657852	+	0.753147i	$0.728535\pi$
$18$	0	0
$19$	2.23897	0.513654	0.256827	−	0.966457i	$-0.417323\pi$
$19$	2.23897	0.513654	0.256827	+	0.966457i	$0.417323\pi$
$20$	0	0
$21$	−1.84074	−0.401682
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.804258	0.160852
$26$	0	0
$27$	−4.80743	−0.925191
$28$	0	0
$29$	0.642864	0.119377	0.0596884	−	0.998217i	$-0.480989\pi$
$29$	0.642864	0.119377	0.0596884	+	0.998217i	$0.480989\pi$
$30$	0	0
$31$	−7.84074	−1.40824	−0.704119	−	0.710082i	$-0.748658\pi$
$31$	−7.84074	−1.40824	−0.704119	+	0.710082i	$0.748658\pi$
$32$	0	0
$33$	−10.8184	−1.88324
$34$	0	0
$35$	−2.40920	−0.407230
$36$	0	0
$37$	0.557492	0.0916511	0.0458256	−	0.998949i	$-0.485408\pi$
$37$	0.557492	0.0916511	0.0458256	+	0.998949i	$0.485408\pi$
$38$	0	0
$39$	−11.5045	−1.84219
$40$	0	0
$41$	2.56847	0.401127	0.200564	−	0.979681i	$-0.435723\pi$
$41$	2.56847	0.401127	0.200564	+	0.979681i	$0.435723\pi$
$42$	0	0
$43$	8.81841	1.34479	0.672397	−	0.740191i	$-0.265265\pi$
$43$	8.81841	1.34479	0.672397	+	0.740191i	$0.265265\pi$
$44$	0	0
$45$	0.935520	0.139459
$46$	0	0
$47$	−4.26766	−0.622502	−0.311251	−	0.950328i	$-0.600748\pi$
$47$	−4.26766	−0.622502	−0.311251	+	0.950328i	$0.600748\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.98561	−1.39826
$52$	0	0
$53$	3.01559	0.414223	0.207111	−	0.978317i	$-0.433594\pi$
$53$	3.01559	0.414223	0.207111	+	0.978317i	$0.433594\pi$
$54$	0	0
$55$	−14.1594	−1.90925
$56$	0	0
$57$	4.12134	0.545885
$58$	0	0
$59$	−4.17024	−0.542919	−0.271459	−	0.962450i	$-0.587506\pi$
$59$	−4.17024	−0.542919	−0.271459	+	0.962450i	$0.587506\pi$
$60$	0	0
$61$	−0.148289	−0.0189865	−0.00949325	−	0.999955i	$-0.503022\pi$
$61$	−0.148289	−0.0189865	−0.00949325	+	0.999955i	$0.503022\pi$
$62$	0	0
$63$	−0.388311	−0.0489226
$64$	0	0
$65$	−15.0574	−1.86764
$66$	0	0
$67$	−13.3396	−1.62969	−0.814843	−	0.579681i	$-0.803177\pi$
$67$	−13.3396	−1.62969	−0.814843	+	0.579681i	$0.803177\pi$
$68$	0	0
$69$	1.84074	0.221599
$70$	0	0
$71$	7.93141	0.941285	0.470643	−	0.882324i	$-0.344022\pi$
$71$	7.93141	0.941285	0.470643	+	0.882324i	$0.344022\pi$
$72$	0	0
$73$	4.28111	0.501066	0.250533	−	0.968108i	$-0.419394\pi$
$73$	4.28111	0.501066	0.250533	+	0.968108i	$0.419394\pi$
$74$	0	0
$75$	1.48043	0.170945
$76$	0	0
$77$	5.87722	0.669771
$78$	0	0
$79$	0.861628	0.0969407	0.0484704	−	0.998825i	$-0.484565\pi$
$79$	0.861628	0.0969407	0.0484704	+	0.998825i	$0.484565\pi$
$80$	0	0
$81$	−10.0141	−1.11268
$82$	0	0
$83$	4.81841	0.528889	0.264444	−	0.964401i	$-0.414811\pi$
$83$	4.81841	0.528889	0.264444	+	0.964401i	$0.414811\pi$
$84$	0	0
$85$	−13.0694	−1.41758
$86$	0	0
$87$	1.18334	0.126868
$88$	0	0
$89$	6.32964	0.670941	0.335470	−	0.942051i	$-0.391105\pi$
$89$	6.32964	0.670941	0.335470	+	0.942051i	$0.391105\pi$
$90$	0	0
$91$	6.24994	0.655172
$92$	0	0
$93$	−14.4327	−1.49660
$94$	0	0
$95$	5.39412	0.553425
$96$	0	0
$97$	10.4822	1.06430	0.532151	−	0.846649i	$-0.321384\pi$
$97$	10.4822	1.06430	0.532151	+	0.846649i	$0.321384\pi$
$98$	0	0
$99$	−2.28219	−0.229369
$100$	0	0
$101$	−1.57308	−0.156527	−0.0782636	−	0.996933i	$-0.524938\pi$
$101$	−1.57308	−0.156527	−0.0782636	+	0.996933i	$0.524938\pi$
$102$	0	0
$103$	−5.01559	−0.494201	−0.247100	−	0.968990i	$-0.579478\pi$
$103$	−5.01559	−0.494201	−0.247100	+	0.968990i	$0.579478\pi$
$104$	0	0
$105$	−4.43471	−0.432783
$106$	0	0
$107$	5.81204	0.561872	0.280936	−	0.959727i	$-0.409355\pi$
$107$	5.81204	0.561872	0.280936	+	0.959727i	$0.409355\pi$
$108$	0	0
$109$	7.65030	0.732766	0.366383	−	0.930464i	$-0.380596\pi$
$109$	7.65030	0.732766	0.366383	+	0.930464i	$0.380596\pi$
$110$	0	0
$111$	1.02620	0.0974022
$112$	0	0
$113$	10.4999	0.987745	0.493873	−	0.869534i	$-0.335581\pi$
$113$	10.4999	0.987745	0.493873	+	0.869534i	$0.335581\pi$
$114$	0	0
$115$	2.40920	0.224659
$116$	0	0
$117$	−2.42692	−0.224369
$118$	0	0
$119$	5.42479	0.497290
$120$	0	0
$121$	23.5417	2.14015
$122$	0	0
$123$	4.72787	0.426298
$124$	0	0
$125$	−10.1084	−0.904122
$126$	0	0
$127$	17.9732	1.59486	0.797432	−	0.603409i	$-0.206191\pi$
$127$	17.9732	1.59486	0.797432	+	0.603409i	$0.206191\pi$
$128$	0	0
$129$	16.2324	1.42918
$130$	0	0
$131$	13.0641	1.14142	0.570708	−	0.821153i	$-0.306669\pi$
$131$	13.0641	1.14142	0.570708	+	0.821153i	$0.306669\pi$
$132$	0	0
$133$	−2.23897	−0.194143
$134$	0	0
$135$	−11.5821	−0.996826
$136$	0	0
$137$	−15.0262	−1.28377	−0.641887	−	0.766799i	$-0.721848\pi$
$137$	−15.0262	−1.28377	−0.641887	+	0.766799i	$0.721848\pi$
$138$	0	0
$139$	12.7987	1.08557	0.542786	−	0.839871i	$-0.317369\pi$
$139$	12.7987	1.08557	0.542786	+	0.839871i	$0.317369\pi$
$140$	0	0
$141$	−7.85563	−0.661564
$142$	0	0
$143$	36.7322	3.07170
$144$	0	0
$145$	1.54879	0.128620
$146$	0	0
$147$	1.84074	0.151821
$148$	0	0
$149$	−21.1303	−1.73106	−0.865532	−	0.500854i	$-0.833019\pi$
$149$	−21.1303	−1.73106	−0.865532	+	0.500854i	$0.833019\pi$
$150$	0	0
$151$	−19.5264	−1.58904	−0.794520	−	0.607239i	$-0.792277\pi$
$151$	−19.5264	−1.58904	−0.794520	+	0.607239i	$0.792277\pi$
$152$	0	0
$153$	−2.10651	−0.170301
$154$	0	0
$155$	−18.8899	−1.51728
$156$	0	0
$157$	−4.60638	−0.367630	−0.183815	−	0.982961i	$-0.558845\pi$
$157$	−4.60638	−0.367630	−0.183815	+	0.982961i	$0.558845\pi$
$158$	0	0
$159$	5.55090	0.440215
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−16.3229	−1.27851	−0.639254	−	0.768996i	$-0.720757\pi$
$163$	−16.3229	−1.27851	−0.639254	+	0.768996i	$0.720757\pi$
$164$	0	0
$165$	−26.0637	−2.02906
$166$	0	0
$167$	−16.7254	−1.29425	−0.647125	−	0.762384i	$-0.724029\pi$
$167$	−16.7254	−1.29425	−0.647125	+	0.762384i	$0.724029\pi$
$168$	0	0
$169$	26.0617	2.00475
$170$	0	0
$171$	0.869415	0.0664858
$172$	0	0
$173$	9.27650	0.705279	0.352640	−	0.935759i	$-0.385284\pi$
$173$	9.27650	0.705279	0.352640	+	0.935759i	$0.385284\pi$
$174$	0	0
$175$	−0.804258	−0.0607962
$176$	0	0
$177$	−7.67631	−0.576987
$178$	0	0
$179$	7.48254	0.559272	0.279636	−	0.960106i	$-0.409786\pi$
$179$	7.48254	0.559272	0.279636	+	0.960106i	$0.409786\pi$
$180$	0	0
$181$	−10.6482	−0.791472	−0.395736	−	0.918364i	$-0.629510\pi$
$181$	−10.6482	−0.791472	−0.395736	+	0.918364i	$0.629510\pi$
$182$	0	0
$183$	−0.272962	−0.0201779
$184$	0	0
$185$	1.34311	0.0987475
$186$	0	0
$187$	31.8827	2.33149
$188$	0	0
$189$	4.80743	0.349689
$190$	0	0
$191$	−3.30294	−0.238992	−0.119496	−	0.992835i	$-0.538128\pi$
$191$	−3.30294	−0.238992	−0.119496	+	0.992835i	$0.538128\pi$
$192$	0	0
$193$	9.38315	0.675414	0.337707	−	0.941251i	$-0.390349\pi$
$193$	9.38315	0.675414	0.337707	+	0.941251i	$0.390349\pi$
$194$	0	0
$195$	−27.7167	−1.98483
$196$	0	0
$197$	−23.0929	−1.64530	−0.822651	−	0.568547i	$-0.807506\pi$
$197$	−23.0929	−1.64530	−0.822651	+	0.568547i	$0.807506\pi$
$198$	0	0
$199$	−21.6149	−1.53224	−0.766118	−	0.642699i	$-0.777814\pi$
$199$	−21.6149	−1.53224	−0.766118	+	0.642699i	$0.777814\pi$
$200$	0	0
$201$	−24.5546	−1.73195
$202$	0	0
$203$	−0.642864	−0.0451202
$204$	0	0
$205$	6.18796	0.432186
$206$	0	0
$207$	0.388311	0.0269895
$208$	0	0
$209$	−13.1589	−0.910219
$210$	0	0
$211$	−15.4579	−1.06416	−0.532081	−	0.846693i	$-0.678590\pi$
$211$	−15.4579	−1.06416	−0.532081	+	0.846693i	$0.678590\pi$
$212$	0	0
$213$	14.5996	1.00035
$214$	0	0
$215$	21.2453	1.44892
$216$	0	0
$217$	7.84074	0.532264
$218$	0	0
$219$	7.88040	0.532508
$220$	0	0
$221$	33.9046	2.28067
$222$	0	0
$223$	11.0708	0.741357	0.370679	−	0.928761i	$-0.379125\pi$
$223$	11.0708	0.741357	0.370679	+	0.928761i	$0.379125\pi$
$224$	0	0
$225$	0.312302	0.0208201
$226$	0	0
$227$	2.44676	0.162397	0.0811984	−	0.996698i	$-0.474125\pi$
$227$	2.44676	0.162397	0.0811984	+	0.996698i	$0.474125\pi$
$228$	0	0
$229$	−5.74097	−0.379374	−0.189687	−	0.981845i	$-0.560747\pi$
$229$	−5.74097	−0.379374	−0.189687	+	0.981845i	$0.560747\pi$
$230$	0	0
$231$	10.8184	0.711799
$232$	0	0
$233$	8.66481	0.567651	0.283825	−	0.958876i	$-0.408396\pi$
$233$	8.66481	0.567651	0.283825	+	0.958876i	$0.408396\pi$
$234$	0	0
$235$	−10.2817	−0.670701
$236$	0	0
$237$	1.58603	0.103024
$238$	0	0
$239$	0.995387	0.0643862	0.0321931	−	0.999482i	$-0.489751\pi$
$239$	0.995387	0.0643862	0.0321931	+	0.999482i	$0.489751\pi$
$240$	0	0
$241$	−13.9247	−0.896967	−0.448483	−	0.893791i	$-0.648036\pi$
$241$	−13.9247	−0.896967	−0.448483	+	0.893791i	$0.648036\pi$
$242$	0	0
$243$	−4.01111	−0.257313
$244$	0	0
$245$	2.40920	0.153918
$246$	0	0
$247$	−13.9934	−0.890378
$248$	0	0
$249$	8.86941	0.562076
$250$	0	0
$251$	0.229739	0.0145010	0.00725049	−	0.999974i	$-0.497692\pi$
$251$	0.229739	0.0145010	0.00725049	+	0.999974i	$0.497692\pi$
$252$	0	0
$253$	−5.87722	−0.369497
$254$	0	0
$255$	−24.0574	−1.50653
$256$	0	0
$257$	−4.24568	−0.264838	−0.132419	−	0.991194i	$-0.542274\pi$
$257$	−4.24568	−0.264838	−0.132419	+	0.991194i	$0.542274\pi$
$258$	0	0
$259$	−0.557492	−0.0346409
$260$	0	0
$261$	0.249631	0.0154518
$262$	0	0
$263$	−2.99220	−0.184507	−0.0922535	−	0.995736i	$-0.529407\pi$
$263$	−2.99220	−0.184507	−0.0922535	+	0.995736i	$0.529407\pi$
$264$	0	0
$265$	7.26516	0.446295
$266$	0	0
$267$	11.6512	0.713042
$268$	0	0
$269$	−8.92878	−0.544397	−0.272199	−	0.962241i	$-0.587751\pi$
$269$	−8.92878	−0.544397	−0.272199	+	0.962241i	$0.587751\pi$
$270$	0	0
$271$	−20.9773	−1.27428	−0.637140	−	0.770748i	$-0.719883\pi$
$271$	−20.9773	−1.27428	−0.637140	+	0.770748i	$0.719883\pi$
$272$	0	0
$273$	11.5045	0.696284
$274$	0	0
$275$	−4.72680	−0.285036
$276$	0	0
$277$	5.82584	0.350041	0.175020	−	0.984565i	$-0.444001\pi$
$277$	5.82584	0.350041	0.175020	+	0.984565i	$0.444001\pi$
$278$	0	0
$279$	−3.04464	−0.182278
$280$	0	0
$281$	−21.1877	−1.26395	−0.631976	−	0.774988i	$-0.717756\pi$
$281$	−21.1877	−1.26395	−0.631976	+	0.774988i	$0.717756\pi$
$282$	0	0
$283$	12.8049	0.761173	0.380587	−	0.924745i	$-0.375722\pi$
$283$	12.8049	0.761173	0.380587	+	0.924745i	$0.375722\pi$
$284$	0	0
$285$	9.92916	0.588152
$286$	0	0
$287$	−2.56847	−0.151612
$288$	0	0
$289$	12.4283	0.731079
$290$	0	0
$291$	19.2949	1.13109
$292$	0	0
$293$	−3.11921	−0.182226	−0.0911132	−	0.995841i	$-0.529043\pi$
$293$	−3.11921	−0.182226	−0.0911132	+	0.995841i	$0.529043\pi$
$294$	0	0
$295$	−10.0469	−0.584956
$296$	0	0
$297$	28.2543	1.63948
$298$	0	0
$299$	−6.24994	−0.361443
$300$	0	0
$301$	−8.81841	−0.508284
$302$	0	0
$303$	−2.89562	−0.166349
$304$	0	0
$305$	−0.357259	−0.0204566
$306$	0	0
$307$	16.1152	0.919746	0.459873	−	0.887985i	$-0.347895\pi$
$307$	16.1152	0.919746	0.459873	+	0.887985i	$0.347895\pi$
$308$	0	0
$309$	−9.23237	−0.525211
$310$	0	0
$311$	−30.2428	−1.71491	−0.857457	−	0.514556i	$-0.827957\pi$
$311$	−30.2428	−1.71491	−0.857457	+	0.514556i	$0.827957\pi$
$312$	0	0
$313$	−20.4956	−1.15848	−0.579241	−	0.815156i	$-0.696651\pi$
$313$	−20.4956	−1.15848	−0.579241	+	0.815156i	$0.696651\pi$
$314$	0	0
$315$	−0.935520	−0.0527106
$316$	0	0
$317$	2.95042	0.165712	0.0828559	−	0.996562i	$-0.473596\pi$
$317$	2.95042	0.165712	0.0828559	+	0.996562i	$0.473596\pi$
$318$	0	0
$319$	−3.77825	−0.211541
$320$	0	0
$321$	10.6984	0.597129
$322$	0	0
$323$	−12.1459	−0.675817
$324$	0	0
$325$	−5.02656	−0.278823
$326$	0	0
$327$	14.0822	0.778747
$328$	0	0
$329$	4.26766	0.235284
$330$	0	0
$331$	21.2497	1.16799	0.583994	−	0.811758i	$-0.301489\pi$
$331$	21.2497	1.16799	0.583994	+	0.811758i	$0.301489\pi$
$332$	0	0
$333$	0.216480	0.0118630
$334$	0	0
$335$	−32.1377	−1.75587
$336$	0	0
$337$	−24.8118	−1.35158	−0.675792	−	0.737092i	$-0.736198\pi$
$337$	−24.8118	−1.35158	−0.675792	+	0.737092i	$0.736198\pi$
$338$	0	0
$339$	19.3275	1.04973
$340$	0	0
$341$	46.0817	2.49546
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.43471	0.238757
$346$	0	0
$347$	−11.8158	−0.634304	−0.317152	−	0.948375i	$-0.602726\pi$
$347$	−11.8158	−0.634304	−0.317152	+	0.948375i	$0.602726\pi$
$348$	0	0
$349$	−12.1614	−0.650984	−0.325492	−	0.945545i	$-0.605530\pi$
$349$	−12.1614	−0.650984	−0.325492	+	0.945545i	$0.605530\pi$
$350$	0	0
$351$	30.0462	1.60375
$352$	0	0
$353$	17.5264	0.932838	0.466419	−	0.884564i	$-0.345544\pi$
$353$	17.5264	0.932838	0.466419	+	0.884564i	$0.345544\pi$
$354$	0	0
$355$	19.1084	1.01417
$356$	0	0
$357$	9.98561	0.528494
$358$	0	0
$359$	−7.70275	−0.406535	−0.203268	−	0.979123i	$-0.565156\pi$
$359$	−7.70275	−0.406535	−0.203268	+	0.979123i	$0.565156\pi$
$360$	0	0
$361$	−13.9870	−0.736160
$362$	0	0
$363$	43.3340	2.27445
$364$	0	0
$365$	10.3141	0.539863
$366$	0	0
$367$	−15.0819	−0.787271	−0.393635	−	0.919267i	$-0.628783\pi$
$367$	−15.0819	−0.787271	−0.393635	+	0.919267i	$0.628783\pi$
$368$	0	0
$369$	0.997364	0.0519207
$370$	0	0
$371$	−3.01559	−0.156561
$372$	0	0
$373$	−21.5089	−1.11369	−0.556843	−	0.830618i	$-0.687988\pi$
$373$	−21.5089	−1.11369	−0.556843	+	0.830618i	$0.687988\pi$
$374$	0	0
$375$	−18.6069	−0.960856
$376$	0	0
$377$	−4.01786	−0.206930
$378$	0	0
$379$	−34.7117	−1.78302	−0.891511	−	0.452999i	$-0.850354\pi$
$379$	−34.7117	−1.78302	−0.891511	+	0.452999i	$0.850354\pi$
$380$	0	0
$381$	33.0839	1.69494
$382$	0	0
$383$	10.2963	0.526119	0.263059	−	0.964780i	$-0.415268\pi$
$383$	10.2963	0.526119	0.263059	+	0.964780i	$0.415268\pi$
$384$	0	0
$385$	14.1594	0.721630
$386$	0	0
$387$	3.42428	0.174066
$388$	0	0
$389$	−26.1594	−1.32633	−0.663167	−	0.748471i	$-0.730788\pi$
$389$	−26.1594	−1.32633	−0.663167	+	0.748471i	$0.730788\pi$
$390$	0	0
$391$	−5.42479	−0.274343
$392$	0	0
$393$	24.0476	1.21304
$394$	0	0
$395$	2.07584	0.104447
$396$	0	0
$397$	−4.52380	−0.227043	−0.113522	−	0.993536i	$-0.536213\pi$
$397$	−4.52380	−0.227043	−0.113522	+	0.993536i	$0.536213\pi$
$398$	0	0
$399$	−4.12134	−0.206325
$400$	0	0
$401$	−18.0222	−0.899987	−0.449994	−	0.893032i	$-0.648574\pi$
$401$	−18.0222	−0.899987	−0.449994	+	0.893032i	$0.648574\pi$
$402$	0	0
$403$	49.0041	2.44107
$404$	0	0
$405$	−24.1261	−1.19884
$406$	0	0
$407$	−3.27650	−0.162410
$408$	0	0
$409$	4.67260	0.231045	0.115523	−	0.993305i	$-0.463146\pi$
$409$	4.67260	0.231045	0.115523	+	0.993305i	$0.463146\pi$
$410$	0	0
$411$	−27.6593	−1.36433
$412$	0	0
$413$	4.17024	0.205204
$414$	0	0
$415$	11.6085	0.569840
$416$	0	0
$417$	23.5591	1.15369
$418$	0	0
$419$	12.7674	0.623728	0.311864	−	0.950127i	$-0.399047\pi$
$419$	12.7674	0.623728	0.311864	+	0.950127i	$0.399047\pi$
$420$	0	0
$421$	28.1367	1.37130	0.685649	−	0.727932i	$-0.259518\pi$
$421$	28.1367	1.37130	0.685649	+	0.727932i	$0.259518\pi$
$422$	0	0
$423$	−1.65718	−0.0805748
$424$	0	0
$425$	−4.36293	−0.211633
$426$	0	0
$427$	0.148289	0.00717622
$428$	0	0
$429$	67.6144	3.26445
$430$	0	0
$431$	5.49352	0.264613	0.132307	−	0.991209i	$-0.457762\pi$
$431$	5.49352	0.264613	0.132307	+	0.991209i	$0.457762\pi$
$432$	0	0
$433$	−21.0177	−1.01005	−0.505023	−	0.863106i	$-0.668516\pi$
$433$	−21.0177	−1.01005	−0.505023	+	0.863106i	$0.668516\pi$
$434$	0	0
$435$	2.85091	0.136691
$436$	0	0
$437$	2.23897	0.107104
$438$	0	0
$439$	−39.2674	−1.87413	−0.937066	−	0.349153i	$-0.886469\pi$
$439$	−39.2674	−1.87413	−0.937066	+	0.349153i	$0.886469\pi$
$440$	0	0
$441$	0.388311	0.0184910
$442$	0	0
$443$	−14.5489	−0.691240	−0.345620	−	0.938375i	$-0.612331\pi$
$443$	−14.5489	−0.691240	−0.345620	+	0.938375i	$0.612331\pi$
$444$	0	0
$445$	15.2494	0.722890
$446$	0	0
$447$	−38.8954	−1.83969
$448$	0	0
$449$	11.3660	0.536395	0.268197	−	0.963364i	$-0.413572\pi$
$449$	11.3660	0.536395	0.268197	+	0.963364i	$0.413572\pi$
$450$	0	0
$451$	−15.0954	−0.710816
$452$	0	0
$453$	−35.9430	−1.68875
$454$	0	0
$455$	15.0574	0.705900
$456$	0	0
$457$	−9.42744	−0.440997	−0.220499	−	0.975387i	$-0.570768\pi$
$457$	−9.42744	−0.440997	−0.220499	+	0.975387i	$0.570768\pi$
$458$	0	0
$459$	26.0793	1.21728
$460$	0	0
$461$	−28.7697	−1.33994	−0.669968	−	0.742390i	$-0.733692\pi$
$461$	−28.7697	−1.33994	−0.669968	+	0.742390i	$0.733692\pi$
$462$	0	0
$463$	4.65928	0.216535	0.108268	−	0.994122i	$-0.465470\pi$
$463$	4.65928	0.216535	0.108268	+	0.994122i	$0.465470\pi$
$464$	0	0
$465$	−34.7714	−1.61248
$466$	0	0
$467$	−7.82926	−0.362295	−0.181147	−	0.983456i	$-0.557981\pi$
$467$	−7.82926	−0.362295	−0.181147	+	0.983456i	$0.557981\pi$
$468$	0	0
$469$	13.3396	0.615964
$470$	0	0
$471$	−8.47914	−0.390698
$472$	0	0
$473$	−51.8277	−2.38304
$474$	0	0
$475$	1.80070	0.0826220
$476$	0	0
$477$	1.17099	0.0536158
$478$	0	0
$479$	33.9427	1.55088	0.775440	−	0.631421i	$-0.217528\pi$
$479$	33.9427	1.55088	0.775440	+	0.631421i	$0.217528\pi$
$480$	0	0
$481$	−3.48429	−0.158870
$482$	0	0
$483$	−1.84074	−0.0837564
$484$	0	0
$485$	25.2536	1.14671
$486$	0	0
$487$	11.6880	0.529633	0.264817	−	0.964299i	$-0.414689\pi$
$487$	11.6880	0.529633	0.264817	+	0.964299i	$0.414689\pi$
$488$	0	0
$489$	−30.0462	−1.35873
$490$	0	0
$491$	19.6742	0.887885	0.443943	−	0.896055i	$-0.353579\pi$
$491$	19.6742	0.887885	0.443943	+	0.896055i	$0.353579\pi$
$492$	0	0
$493$	−3.48740	−0.157065
$494$	0	0
$495$	−5.49825	−0.247128
$496$	0	0
$497$	−7.93141	−0.355772
$498$	0	0
$499$	30.1698	1.35059	0.675294	−	0.737549i	$-0.264017\pi$
$499$	30.1698	1.35059	0.675294	+	0.737549i	$0.264017\pi$
$500$	0	0
$501$	−30.7870	−1.37546
$502$	0	0
$503$	−24.1110	−1.07506	−0.537529	−	0.843246i	$-0.680642\pi$
$503$	−24.1110	−1.07506	−0.537529	+	0.843246i	$0.680642\pi$
$504$	0	0
$505$	−3.78987	−0.168647
$506$	0	0
$507$	47.9728	2.13055
$508$	0	0
$509$	21.5774	0.956404	0.478202	−	0.878250i	$-0.341289\pi$
$509$	21.5774	0.956404	0.478202	+	0.878250i	$0.341289\pi$
$510$	0	0
$511$	−4.28111	−0.189385
$512$	0	0
$513$	−10.7637	−0.475228
$514$	0	0
$515$	−12.0836	−0.532466
$516$	0	0
$517$	25.0819	1.10310
$518$	0	0
$519$	17.0756	0.749535
$520$	0	0
$521$	−22.0862	−0.967613	−0.483807	−	0.875175i	$-0.660746\pi$
$521$	−22.0862	−0.967613	−0.483807	+	0.875175i	$0.660746\pi$
$522$	0	0
$523$	−5.79856	−0.253553	−0.126777	−	0.991931i	$-0.540463\pi$
$523$	−5.79856	−0.253553	−0.126777	+	0.991931i	$0.540463\pi$
$524$	0	0
$525$	−1.48043	−0.0646111
$526$	0	0
$527$	42.5343	1.85283
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.61935	−0.0702738
$532$	0	0
$533$	−16.0528	−0.695322
$534$	0	0
$535$	14.0024	0.605376
$536$	0	0
$537$	13.7734	0.594366
$538$	0	0
$539$	−5.87722	−0.253150
$540$	0	0
$541$	29.6582	1.27511	0.637553	−	0.770407i	$-0.279947\pi$
$541$	29.6582	1.27511	0.637553	+	0.770407i	$0.279947\pi$
$542$	0	0
$543$	−19.6005	−0.841137
$544$	0	0
$545$	18.4311	0.789502
$546$	0	0
$547$	7.24730	0.309872	0.154936	−	0.987924i	$-0.450483\pi$
$547$	7.24730	0.309872	0.154936	+	0.987924i	$0.450483\pi$
$548$	0	0
$549$	−0.0575824	−0.00245756
$550$	0	0
$551$	1.43935	0.0613183
$552$	0	0
$553$	−0.861628	−0.0366402
$554$	0	0
$555$	2.47231	0.104944
$556$	0	0
$557$	21.7634	0.922146	0.461073	−	0.887362i	$-0.347465\pi$
$557$	21.7634	0.922146	0.461073	+	0.887362i	$0.347465\pi$
$558$	0	0
$559$	−55.1145	−2.33109
$560$	0	0
$561$	58.6876	2.47779
$562$	0	0
$563$	26.9844	1.13726	0.568629	−	0.822594i	$-0.307474\pi$
$563$	26.9844	1.13726	0.568629	+	0.822594i	$0.307474\pi$
$564$	0	0
$565$	25.2963	1.06422
$566$	0	0
$567$	10.0141	0.420555
$568$	0	0
$569$	37.4420	1.56965	0.784826	−	0.619717i	$-0.212752\pi$
$569$	37.4420	1.56965	0.784826	+	0.619717i	$0.212752\pi$
$570$	0	0
$571$	3.32489	0.139142	0.0695711	−	0.997577i	$-0.477837\pi$
$571$	3.32489	0.139142	0.0695711	+	0.997577i	$0.477837\pi$
$572$	0	0
$573$	−6.07984	−0.253989
$574$	0	0
$575$	0.804258	0.0335399
$576$	0	0
$577$	−41.0573	−1.70924	−0.854618	−	0.519257i	$-0.826209\pi$
$577$	−41.0573	−1.70924	−0.854618	+	0.519257i	$0.826209\pi$
$578$	0	0
$579$	17.2719	0.717796
$580$	0	0
$581$	−4.81841	−0.199901
$582$	0	0
$583$	−17.7233	−0.734022
$584$	0	0
$585$	−5.84694	−0.241741
$586$	0	0
$587$	31.9182	1.31741	0.658703	−	0.752403i	$-0.271105\pi$
$587$	31.9182	1.31741	0.658703	+	0.752403i	$0.271105\pi$
$588$	0	0
$589$	−17.5551	−0.723347
$590$	0	0
$591$	−42.5080	−1.74854
$592$	0	0
$593$	36.5062	1.49913	0.749566	−	0.661930i	$-0.230262\pi$
$593$	36.5062	1.49913	0.749566	+	0.661930i	$0.230262\pi$
$594$	0	0
$595$	13.0694	0.535794
$596$	0	0
$597$	−39.7873	−1.62838
$598$	0	0
$599$	7.51073	0.306880	0.153440	−	0.988158i	$-0.450965\pi$
$599$	7.51073	0.306880	0.153440	+	0.988158i	$0.450965\pi$
$600$	0	0
$601$	−6.05329	−0.246919	−0.123460	−	0.992350i	$-0.539399\pi$
$601$	−6.05329	−0.246919	−0.123460	+	0.992350i	$0.539399\pi$
$602$	0	0
$603$	−5.17990	−0.210942
$604$	0	0
$605$	56.7166	2.30586
$606$	0	0
$607$	−24.4047	−0.990557	−0.495278	−	0.868734i	$-0.664934\pi$
$607$	−24.4047	−0.990557	−0.495278	+	0.868734i	$0.664934\pi$
$608$	0	0
$609$	−1.18334	−0.0479515
$610$	0	0
$611$	26.6726	1.07906
$612$	0	0
$613$	4.81180	0.194347	0.0971734	−	0.995267i	$-0.469020\pi$
$613$	4.81180	0.194347	0.0971734	+	0.995267i	$0.469020\pi$
$614$	0	0
$615$	11.3904	0.459305
$616$	0	0
$617$	−38.1414	−1.53552	−0.767758	−	0.640740i	$-0.778628\pi$
$617$	−38.1414	−1.53552	−0.767758	+	0.640740i	$0.778628\pi$
$618$	0	0
$619$	17.8340	0.716809	0.358404	−	0.933566i	$-0.383321\pi$
$619$	17.8340	0.716809	0.358404	+	0.933566i	$0.383321\pi$
$620$	0	0
$621$	−4.80743	−0.192916
$622$	0	0
$623$	−6.32964	−0.253592
$624$	0	0
$625$	−28.3745	−1.13498
$626$	0	0
$627$	−24.2220	−0.967335
$628$	0	0
$629$	−3.02428	−0.120586
$630$	0	0
$631$	−22.2906	−0.887377	−0.443688	−	0.896181i	$-0.646330\pi$
$631$	−22.2906	−0.887377	−0.443688	+	0.896181i	$0.646330\pi$
$632$	0	0
$633$	−28.4538	−1.13094
$634$	0	0
$635$	43.3011	1.71835
$636$	0	0
$637$	−6.24994	−0.247632
$638$	0	0
$639$	3.07986	0.121837
$640$	0	0
$641$	28.4927	1.12540	0.562698	−	0.826663i	$-0.309763\pi$
$641$	28.4927	1.12540	0.562698	+	0.826663i	$0.309763\pi$
$642$	0	0
$643$	−0.975186	−0.0384576	−0.0192288	−	0.999815i	$-0.506121\pi$
$643$	−0.975186	−0.0384576	−0.0192288	+	0.999815i	$0.506121\pi$
$644$	0	0
$645$	39.1070	1.53984
$646$	0	0
$647$	27.5193	1.08190	0.540948	−	0.841056i	$-0.318065\pi$
$647$	27.5193	1.08190	0.540948	+	0.841056i	$0.318065\pi$
$648$	0	0
$649$	24.5094	0.962078
$650$	0	0
$651$	14.4327	0.565663
$652$	0	0
$653$	−26.1590	−1.02368	−0.511840	−	0.859081i	$-0.671036\pi$
$653$	−26.1590	−1.02368	−0.511840	+	0.859081i	$0.671036\pi$
$654$	0	0
$655$	31.4741	1.22979
$656$	0	0
$657$	1.66240	0.0648566
$658$	0	0
$659$	2.67085	0.104042	0.0520208	−	0.998646i	$-0.483434\pi$
$659$	2.67085	0.104042	0.0520208	+	0.998646i	$0.483434\pi$
$660$	0	0
$661$	−18.4928	−0.719285	−0.359643	−	0.933090i	$-0.617101\pi$
$661$	−18.4928	−0.719285	−0.359643	+	0.933090i	$0.617101\pi$
$662$	0	0
$663$	62.4095	2.42378
$664$	0	0
$665$	−5.39412	−0.209175
$666$	0	0
$667$	0.642864	0.0248918
$668$	0	0
$669$	20.3785	0.787877
$670$	0	0
$671$	0.871528	0.0336450
$672$	0	0
$673$	47.5002	1.83100	0.915499	−	0.402321i	$-0.131797\pi$
$673$	47.5002	1.83100	0.915499	+	0.402321i	$0.131797\pi$
$674$	0	0
$675$	−3.86641	−0.148818
$676$	0	0
$677$	33.4288	1.28477	0.642386	−	0.766381i	$-0.277944\pi$
$677$	33.4288	1.28477	0.642386	+	0.766381i	$0.277944\pi$
$678$	0	0
$679$	−10.4822	−0.402268
$680$	0	0
$681$	4.50383	0.172587
$682$	0	0
$683$	−46.7168	−1.78757	−0.893784	−	0.448498i	$-0.851959\pi$
$683$	−46.7168	−1.78757	−0.893784	+	0.448498i	$0.851959\pi$
$684$	0	0
$685$	−36.2012	−1.38317
$686$	0	0
$687$	−10.5676	−0.403180
$688$	0	0
$689$	−18.8472	−0.718023
$690$	0	0
$691$	22.8670	0.869903	0.434952	−	0.900454i	$-0.356765\pi$
$691$	22.8670	0.869903	0.434952	+	0.900454i	$0.356765\pi$
$692$	0	0
$693$	2.28219	0.0866931
$694$	0	0
$695$	30.8347	1.16963
$696$	0	0
$697$	−13.9334	−0.527765
$698$	0	0
$699$	15.9496	0.603271
$700$	0	0
$701$	39.0885	1.47635	0.738177	−	0.674607i	$-0.235687\pi$
$701$	39.0885	1.47635	0.738177	+	0.674607i	$0.235687\pi$
$702$	0	0
$703$	1.24821	0.0470769
$704$	0	0
$705$	−18.9258	−0.712787
$706$	0	0
$707$	1.57308	0.0591617
$708$	0	0
$709$	18.3762	0.690132	0.345066	−	0.938578i	$-0.387856\pi$
$709$	18.3762	0.690132	0.345066	+	0.938578i	$0.387856\pi$
$710$	0	0
$711$	0.334580	0.0125477
$712$	0	0
$713$	−7.84074	−0.293638
$714$	0	0
$715$	88.4954	3.30954
$716$	0	0
$717$	1.83224	0.0684264
$718$	0	0
$719$	11.2232	0.418553	0.209276	−	0.977857i	$-0.432889\pi$
$719$	11.2232	0.418553	0.209276	+	0.977857i	$0.432889\pi$
$720$	0	0
$721$	5.01559	0.186790
$722$	0	0
$723$	−25.6316	−0.953251
$724$	0	0
$725$	0.517028	0.0192019
$726$	0	0
$727$	−7.71953	−0.286302	−0.143151	−	0.989701i	$-0.545723\pi$
$727$	−7.71953	−0.286302	−0.143151	+	0.989701i	$0.545723\pi$
$728$	0	0
$729$	22.6590	0.839224
$730$	0	0
$731$	−47.8380	−1.76935
$732$	0	0
$733$	−23.2083	−0.857217	−0.428609	−	0.903490i	$-0.640996\pi$
$733$	−23.2083	−0.857217	−0.428609	+	0.903490i	$0.640996\pi$
$734$	0	0
$735$	4.43471	0.163577
$736$	0	0
$737$	78.3995	2.88788
$738$	0	0
$739$	−8.36545	−0.307728	−0.153864	−	0.988092i	$-0.549172\pi$
$739$	−8.36545	−0.307728	−0.153864	+	0.988092i	$0.549172\pi$
$740$	0	0
$741$	−25.7582	−0.946249
$742$	0	0
$743$	−9.54735	−0.350258	−0.175129	−	0.984545i	$-0.556034\pi$
$743$	−9.54735	−0.350258	−0.175129	+	0.984545i	$0.556034\pi$
$744$	0	0
$745$	−50.9072	−1.86510
$746$	0	0
$747$	1.87104	0.0684578
$748$	0	0
$749$	−5.81204	−0.212367
$750$	0	0
$751$	−2.83381	−0.103407	−0.0517035	−	0.998662i	$-0.516465\pi$
$751$	−2.83381	−0.103407	−0.0517035	+	0.998662i	$0.516465\pi$
$752$	0	0
$753$	0.422889	0.0154109
$754$	0	0
$755$	−47.0432	−1.71208
$756$	0	0
$757$	49.7767	1.80916	0.904582	−	0.426300i	$-0.140183\pi$
$757$	49.7767	1.80916	0.904582	+	0.426300i	$0.140183\pi$
$758$	0	0
$759$	−10.8184	−0.392683
$760$	0	0
$761$	48.7458	1.76704	0.883518	−	0.468398i	$-0.155169\pi$
$761$	48.7458	1.76704	0.883518	+	0.468398i	$0.155169\pi$
$762$	0	0
$763$	−7.65030	−0.276959
$764$	0	0
$765$	−5.07500	−0.183487
$766$	0	0
$767$	26.0637	0.941107
$768$	0	0
$769$	−13.3190	−0.480296	−0.240148	−	0.970736i	$-0.577196\pi$
$769$	−13.3190	−0.480296	−0.240148	+	0.970736i	$0.577196\pi$
$770$	0	0
$771$	−7.81517	−0.281457
$772$	0	0
$773$	−13.9252	−0.500855	−0.250427	−	0.968135i	$-0.580571\pi$
$773$	−13.9252	−0.500855	−0.250427	+	0.968135i	$0.580571\pi$
$774$	0	0
$775$	−6.30597	−0.226517
$776$	0	0
$777$	−1.02620	−0.0368146
$778$	0	0
$779$	5.75071	0.206040
$780$	0	0
$781$	−46.6146	−1.66800
$782$	0	0
$783$	−3.09052	−0.110446
$784$	0	0
$785$	−11.0977	−0.396094
$786$	0	0
$787$	−45.7185	−1.62969	−0.814844	−	0.579680i	$-0.803178\pi$
$787$	−45.7185	−1.62969	−0.814844	+	0.579680i	$0.803178\pi$
$788$	0	0
$789$	−5.50785	−0.196085
$790$	0	0
$791$	−10.4999	−0.373333
$792$	0	0
$793$	0.926799	0.0329116
$794$	0	0
$795$	13.3732	0.474300
$796$	0	0
$797$	−11.4211	−0.404555	−0.202277	−	0.979328i	$-0.564834\pi$
$797$	−11.4211	−0.404555	−0.202277	+	0.979328i	$0.564834\pi$
$798$	0	0
$799$	23.1511	0.819029
$800$	0	0
$801$	2.45787	0.0868446
$802$	0	0
$803$	−25.1610	−0.887913
$804$	0	0
$805$	−2.40920	−0.0849132
$806$	0	0
$807$	−16.4355	−0.578558
$808$	0	0
$809$	17.6307	0.619864	0.309932	−	0.950759i	$-0.399694\pi$
$809$	17.6307	0.619864	0.309932	+	0.950759i	$0.399694\pi$
$810$	0	0
$811$	−30.9666	−1.08738	−0.543692	−	0.839285i	$-0.682974\pi$
$811$	−30.9666	−1.08738	−0.543692	+	0.839285i	$0.682974\pi$
$812$	0	0
$813$	−38.6137	−1.35424
$814$	0	0
$815$	−39.3252	−1.37750
$816$	0	0
$817$	19.7441	0.690759
$818$	0	0
$819$	2.42692	0.0848035
$820$	0	0
$821$	25.3060	0.883187	0.441593	−	0.897215i	$-0.354413\pi$
$821$	25.3060	0.883187	0.441593	+	0.897215i	$0.354413\pi$
$822$	0	0
$823$	−45.6057	−1.58972	−0.794858	−	0.606795i	$-0.792455\pi$
$823$	−45.6057	−1.58972	−0.794858	+	0.606795i	$0.792455\pi$
$824$	0	0
$825$	−8.70079	−0.302922
$826$	0	0
$827$	25.4338	0.884420	0.442210	−	0.896912i	$-0.354195\pi$
$827$	25.4338	0.884420	0.442210	+	0.896912i	$0.354195\pi$
$828$	0	0
$829$	−5.12771	−0.178093	−0.0890463	−	0.996027i	$-0.528382\pi$
$829$	−5.12771	−0.178093	−0.0890463	+	0.996027i	$0.528382\pi$
$830$	0	0
$831$	10.7238	0.372006
$832$	0	0
$833$	−5.42479	−0.187958
$834$	0	0
$835$	−40.2948	−1.39446
$836$	0	0
$837$	37.6938	1.30289
$838$	0	0
$839$	7.81839	0.269921	0.134960	−	0.990851i	$-0.456909\pi$
$839$	7.81839	0.269921	0.134960	+	0.990851i	$0.456909\pi$
$840$	0	0
$841$	−28.5867	−0.985749
$842$	0	0
$843$	−39.0010	−1.34327
$844$	0	0
$845$	62.7880	2.15997
$846$	0	0
$847$	−23.5417	−0.808901
$848$	0	0
$849$	23.5705	0.808937
$850$	0	0
$851$	0.557492	0.0191106
$852$	0	0
$853$	−36.6540	−1.25501	−0.627505	−	0.778613i	$-0.715924\pi$
$853$	−36.6540	−1.25501	−0.627505	+	0.778613i	$0.715924\pi$
$854$	0	0
$855$	2.09460	0.0716337
$856$	0	0
$857$	44.0599	1.50506	0.752529	−	0.658559i	$-0.228834\pi$
$857$	44.0599	1.50506	0.752529	+	0.658559i	$0.228834\pi$
$858$	0	0
$859$	6.04906	0.206391	0.103196	−	0.994661i	$-0.467093\pi$
$859$	6.04906	0.206391	0.103196	+	0.994661i	$0.467093\pi$
$860$	0	0
$861$	−4.72787	−0.161125
$862$	0	0
$863$	25.7890	0.877867	0.438933	−	0.898520i	$-0.355356\pi$
$863$	25.7890	0.877867	0.438933	+	0.898520i	$0.355356\pi$
$864$	0	0
$865$	22.3490	0.759888
$866$	0	0
$867$	22.8773	0.776954
$868$	0	0
$869$	−5.06397	−0.171784
$870$	0	0
$871$	83.3714	2.82493
$872$	0	0
$873$	4.07034	0.137760
$874$	0	0
$875$	10.1084	0.341726
$876$	0	0
$877$	9.59839	0.324114	0.162057	−	0.986781i	$-0.448187\pi$
$877$	9.59839	0.324114	0.162057	+	0.986781i	$0.448187\pi$
$878$	0	0
$879$	−5.74165	−0.193661
$880$	0	0
$881$	−9.38385	−0.316150	−0.158075	−	0.987427i	$-0.550529\pi$
$881$	−9.38385	−0.316150	−0.158075	+	0.987427i	$0.550529\pi$
$882$	0	0
$883$	4.98842	0.167874	0.0839368	−	0.996471i	$-0.473251\pi$
$883$	4.98842	0.167874	0.0839368	+	0.996471i	$0.473251\pi$
$884$	0	0
$885$	−18.4938	−0.621662
$886$	0	0
$887$	−36.6169	−1.22948	−0.614738	−	0.788732i	$-0.710738\pi$
$887$	−36.6169	−1.22948	−0.614738	+	0.788732i	$0.710738\pi$
$888$	0	0
$889$	−17.9732	−0.602802
$890$	0	0
$891$	58.8553	1.97173
$892$	0	0
$893$	−9.55514	−0.319750
$894$	0	0
$895$	18.0270	0.602575
$896$	0	0
$897$	−11.5045	−0.384124
$898$	0	0
$899$	−5.04052	−0.168111
$900$	0	0
$901$	−16.3589	−0.544995
$902$	0	0
$903$	−16.2324	−0.540179
$904$	0	0
$905$	−25.6536	−0.852754
$906$	0	0
$907$	−34.2361	−1.13679	−0.568395	−	0.822756i	$-0.692436\pi$
$907$	−34.2361	−1.13679	−0.568395	+	0.822756i	$0.692436\pi$
$908$	0	0
$909$	−0.610844	−0.0202604
$910$	0	0
$911$	−2.10958	−0.0698934	−0.0349467	−	0.999389i	$-0.511126\pi$
$911$	−2.10958	−0.0698934	−0.0349467	+	0.999389i	$0.511126\pi$
$912$	0	0
$913$	−28.3188	−0.937216
$914$	0	0
$915$	−0.657620	−0.0217402
$916$	0	0
$917$	−13.0641	−0.431415
$918$	0	0
$919$	−49.7639	−1.64156	−0.820779	−	0.571245i	$-0.806461\pi$
$919$	−49.7639	−1.64156	−0.820779	+	0.571245i	$0.806461\pi$
$920$	0	0
$921$	29.6639	0.977460
$922$	0	0
$923$	−49.5708	−1.63164
$924$	0	0
$925$	0.448367	0.0147422
$926$	0	0
$927$	−1.94761	−0.0639678
$928$	0	0
$929$	−45.0863	−1.47923	−0.739617	−	0.673028i	$-0.764993\pi$
$929$	−45.0863	−1.47923	−0.739617	+	0.673028i	$0.764993\pi$
$930$	0	0
$931$	2.23897	0.0733791
$932$	0	0
$933$	−55.6691	−1.82252
$934$	0	0
$935$	76.8118	2.51201
$936$	0	0
$937$	14.8779	0.486040	0.243020	−	0.970021i	$-0.421862\pi$
$937$	14.8779	0.486040	0.243020	+	0.970021i	$0.421862\pi$
$938$	0	0
$939$	−37.7271	−1.23118
$940$	0	0
$941$	30.3917	0.990742	0.495371	−	0.868681i	$-0.335032\pi$
$941$	30.3917	0.990742	0.495371	+	0.868681i	$0.335032\pi$
$942$	0	0
$943$	2.56847	0.0836408
$944$	0	0
$945$	11.5821	0.376765
$946$	0	0
$947$	26.5595	0.863068	0.431534	−	0.902097i	$-0.357972\pi$
$947$	26.5595	0.863068	0.431534	+	0.902097i	$0.357972\pi$
$948$	0	0
$949$	−26.7567	−0.868559
$950$	0	0
$951$	5.43094	0.176110
$952$	0	0
$953$	8.75467	0.283592	0.141796	−	0.989896i	$-0.454712\pi$
$953$	8.75467	0.283592	0.141796	+	0.989896i	$0.454712\pi$
$954$	0	0
$955$	−7.95745	−0.257497
$956$	0	0
$957$	−6.95476	−0.224815
$958$	0	0
$959$	15.0262	0.485221
$960$	0	0
$961$	30.4771	0.983134
$962$	0	0
$963$	2.25688	0.0727270
$964$	0	0
$965$	22.6059	0.727710
$966$	0	0
$967$	19.3848	0.623372	0.311686	−	0.950185i	$-0.399106\pi$
$967$	19.3848	0.623372	0.311686	+	0.950185i	$0.399106\pi$
$968$	0	0
$969$	−22.3574	−0.718224
$970$	0	0
$971$	17.9424	0.575799	0.287899	−	0.957661i	$-0.407043\pi$
$971$	17.9424	0.575799	0.287899	+	0.957661i	$0.407043\pi$
$972$	0	0
$973$	−12.7987	−0.410308
$974$	0	0
$975$	−9.25258	−0.296320
$976$	0	0
$977$	5.12609	0.163998	0.0819991	−	0.996632i	$-0.473870\pi$
$977$	5.12609	0.163998	0.0819991	+	0.996632i	$0.473870\pi$
$978$	0	0
$979$	−37.2007	−1.18894
$980$	0	0
$981$	2.97070	0.0948470
$982$	0	0
$983$	48.1332	1.53521	0.767606	−	0.640922i	$-0.221448\pi$
$983$	48.1332	1.53521	0.767606	+	0.640922i	$0.221448\pi$
$984$	0	0
$985$	−55.6355	−1.77269
$986$	0	0
$987$	7.85563	0.250048
$988$	0	0
$989$	8.81841	0.280409
$990$	0	0
$991$	6.50910	0.206769	0.103384	−	0.994641i	$-0.467033\pi$
$991$	6.50910	0.206769	0.103384	+	0.994641i	$0.467033\pi$
$992$	0	0
$993$	39.1151	1.24128
$994$	0	0
$995$	−52.0746	−1.65088
$996$	0	0
$997$	43.2321	1.36917	0.684587	−	0.728931i	$-0.259982\pi$
$997$	43.2321	1.36917	0.684587	+	0.728931i	$0.259982\pi$
$998$	0	0
$999$	−2.68010	−0.0847948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.bd.1.4		5
4.3	odd	2		161.2.a.d.1.4	✓	5
12.11	even	2		1449.2.a.r.1.2		5
20.19	odd	2		4025.2.a.p.1.2		5
28.27	even	2		1127.2.a.h.1.4		5
92.91	even	2		3703.2.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.4	✓	5	4.3	odd	2
1127.2.a.h.1.4		5	28.27	even	2
1449.2.a.r.1.2		5	12.11	even	2
2576.2.a.bd.1.4		5	1.1	even	1	trivial
3703.2.a.j.1.4		5	92.91	even	2
4025.2.a.p.1.2		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.84074 q^{3} +2.40920 q^{5} -1.00000 q^{7} +0.388311 q^{9} +O(q^{10})\)	\(q+1.84074 q^{3} +2.40920 q^{5} -1.00000 q^{7} +0.388311 q^{9} -5.87722 q^{11} -6.24994 q^{13} +4.43471 q^{15} -5.42479 q^{17} +2.23897 q^{19} -1.84074 q^{21} +1.00000 q^{23} +0.804258 q^{25} -4.80743 q^{27} +0.642864 q^{29} -7.84074 q^{31} -10.8184 q^{33} -2.40920 q^{35} +0.557492 q^{37} -11.5045 q^{39} +2.56847 q^{41} +8.81841 q^{43} +0.935520 q^{45} -4.26766 q^{47} +1.00000 q^{49} -9.98561 q^{51} +3.01559 q^{53} -14.1594 q^{55} +4.12134 q^{57} -4.17024 q^{59} -0.148289 q^{61} -0.388311 q^{63} -15.0574 q^{65} -13.3396 q^{67} +1.84074 q^{69} +7.93141 q^{71} +4.28111 q^{73} +1.48043 q^{75} +5.87722 q^{77} +0.861628 q^{79} -10.0141 q^{81} +4.81841 q^{83} -13.0694 q^{85} +1.18334 q^{87} +6.32964 q^{89} +6.24994 q^{91} -14.4327 q^{93} +5.39412 q^{95} +10.4822 q^{97} -2.28219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) \) 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9	\( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9 + 4 * q^11 - 6 * q^13 - 10 * q^15 - 12 * q^17 - 6 * q^19 + 5 * q^23 + 19 * q^25 - 4 * q^29 - 30 * q^31 - 22 * q^33 + 4 * q^35 + 4 * q^37 - 16 * q^39 + 6 * q^41 + 12 * q^43 - 12 * q^45 - 10 * q^47 + 5 * q^49 + 4 * q^51 + 16 * q^53 - 18 * q^55 + 6 * q^57 - 22 * q^59 - 18 * q^61 - 11 * q^63 - 26 * q^65 + 2 * q^67 - 4 * q^71 - 2 * q^73 + 30 * q^75 - 4 * q^77 - 30 * q^79 - 3 * q^81 - 8 * q^83 - 12 * q^85 + 12 * q^87 - 20 * q^89 + 6 * q^91 - 26 * q^93 - 8 * q^95 - 12 * q^97 + 8 * q^99

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