LMFDB - Embedded newform 1456.4.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,4,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1456.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1456.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:	$85.9067809684$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5364412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 27x^{2} - 24x + 76 $ x^4 - 27x^2 - 24x + 76
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.63459$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.33748 q^{3} +11.0031 q^{5} -7.00000 q^{7} +1.48874 q^{9} +O(q^{10})$	$q+5.33748 q^{3} +11.0031 q^{5} -7.00000 q^{7} +1.48874 q^{9} +11.7038 q^{11} -13.0000 q^{13} +58.7289 q^{15} -102.636 q^{17} -43.2825 q^{19} -37.3624 q^{21} +25.9576 q^{23} -3.93182 q^{25} -136.166 q^{27} -272.081 q^{29} +121.190 q^{31} +62.4686 q^{33} -77.0217 q^{35} +168.192 q^{37} -69.3873 q^{39} -451.218 q^{41} -94.6309 q^{43} +16.3807 q^{45} -50.6431 q^{47} +49.0000 q^{49} -547.817 q^{51} -398.509 q^{53} +128.778 q^{55} -231.020 q^{57} +686.474 q^{59} -75.3794 q^{61} -10.4212 q^{63} -143.040 q^{65} +336.720 q^{67} +138.548 q^{69} -427.161 q^{71} -134.191 q^{73} -20.9860 q^{75} -81.9263 q^{77} -253.005 q^{79} -766.980 q^{81} +193.536 q^{83} -1129.31 q^{85} -1452.23 q^{87} -996.000 q^{89} +91.0000 q^{91} +646.851 q^{93} -476.242 q^{95} +761.982 q^{97} +17.4238 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} - 36 q^{5} - 28 q^{7} + 21 q^{9}+O(q^{10}) $ 4 * q + 5 * q^3 - 36 * q^5 - 28 * q^7 + 21 * q^9	$ 4 q + 5 q^{3} - 36 q^{5} - 28 q^{7} + 21 q^{9} + 95 q^{11} - 52 q^{13} + 16 q^{15} - 146 q^{17} + 48 q^{19} - 35 q^{21} + 121 q^{23} + 506 q^{25} + 83 q^{27} - 440 q^{29} + 283 q^{31} + 227 q^{33} + 252 q^{35} - 209 q^{37} - 65 q^{39} - 93 q^{41} - 526 q^{43} - 768 q^{45} + 783 q^{47} + 196 q^{49} + 672 q^{51} - 340 q^{53} - 756 q^{55} - 1014 q^{57} + 922 q^{59} - 141 q^{61} - 147 q^{63} + 468 q^{65} + 523 q^{67} + 1595 q^{69} - 1468 q^{71} - 47 q^{73} + 1547 q^{75} - 665 q^{77} - 1025 q^{79} - 1772 q^{81} + 1190 q^{83} - 568 q^{85} - 720 q^{87} - 2962 q^{89} + 364 q^{91} - 763 q^{93} - 2082 q^{95} + 2715 q^{97} - 586 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 - 36 * q^5 - 28 * q^7 + 21 * q^9 + 95 * q^11 - 52 * q^13 + 16 * q^15 - 146 * q^17 + 48 * q^19 - 35 * q^21 + 121 * q^23 + 506 * q^25 + 83 * q^27 - 440 * q^29 + 283 * q^31 + 227 * q^33 + 252 * q^35 - 209 * q^37 - 65 * q^39 - 93 * q^41 - 526 * q^43 - 768 * q^45 + 783 * q^47 + 196 * q^49 + 672 * q^51 - 340 * q^53 - 756 * q^55 - 1014 * q^57 + 922 * q^59 - 141 * q^61 - 147 * q^63 + 468 * q^65 + 523 * q^67 + 1595 * q^69 - 1468 * q^71 - 47 * q^73 + 1547 * q^75 - 665 * q^77 - 1025 * q^79 - 1772 * q^81 + 1190 * q^83 - 568 * q^85 - 720 * q^87 - 2962 * q^89 + 364 * q^91 - 763 * q^93 - 2082 * q^95 + 2715 * q^97 - 586 * 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
$3$	5.33748	1.02720	0.513600	−	0.858030i	$-0.328312\pi$
$3$	5.33748	1.02720	0.513600	+	0.858030i	$0.328312\pi$
$4$	0	0
$5$	11.0031	0.984147	0.492074	−	0.870554i	$-0.336239\pi$
$5$	11.0031	0.984147	0.492074	+	0.870554i	$0.336239\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	1.48874	0.0551384
$10$	0	0
$11$	11.7038	0.320801	0.160401	−	0.987052i	$-0.448721\pi$
$11$	11.7038	0.320801	0.160401	+	0.987052i	$0.448721\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	58.7289	1.01092
$16$	0	0
$17$	−102.636	−1.46429	−0.732143	−	0.681151i	$-0.761480\pi$
$17$	−102.636	−1.46429	−0.732143	+	0.681151i	$0.761480\pi$
$18$	0	0
$19$	−43.2825	−0.522616	−0.261308	−	0.965256i	$-0.584154\pi$
$19$	−43.2825	−0.522616	−0.261308	+	0.965256i	$0.584154\pi$
$20$	0	0
$21$	−37.3624	−0.388245
$22$	0	0
$23$	25.9576	0.235327	0.117664	−	0.993054i	$-0.462460\pi$
$23$	25.9576	0.235327	0.117664	+	0.993054i	$0.462460\pi$
$24$	0	0
$25$	−3.93182	−0.0314546
$26$	0	0
$27$	−136.166	−0.970561
$28$	0	0
$29$	−272.081	−1.74221	−0.871106	−	0.491095i	$-0.836597\pi$
$29$	−272.081	−1.74221	−0.871106	+	0.491095i	$0.836597\pi$
$30$	0	0
$31$	121.190	0.702142	0.351071	−	0.936349i	$-0.385818\pi$
$31$	121.190	0.702142	0.351071	+	0.936349i	$0.385818\pi$
$32$	0	0
$33$	62.4686	0.329527
$34$	0	0
$35$	−77.0217	−0.371973
$36$	0	0
$37$	168.192	0.747314	0.373657	−	0.927567i	$-0.378104\pi$
$37$	168.192	0.747314	0.373657	+	0.927567i	$0.378104\pi$
$38$	0	0
$39$	−69.3873	−0.284894
$40$	0	0
$41$	−451.218	−1.71874	−0.859372	−	0.511351i	$-0.829145\pi$
$41$	−451.218	−1.71874	−0.859372	+	0.511351i	$0.829145\pi$
$42$	0	0
$43$	−94.6309	−0.335606	−0.167803	−	0.985821i	$-0.553667\pi$
$43$	−94.6309	−0.335606	−0.167803	+	0.985821i	$0.553667\pi$
$44$	0	0
$45$	16.3807	0.0542643
$46$	0	0
$47$	−50.6431	−0.157171	−0.0785857	−	0.996907i	$-0.525040\pi$
$47$	−50.6431	−0.157171	−0.0785857	+	0.996907i	$0.525040\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−547.817	−1.50411
$52$	0	0
$53$	−398.509	−1.03282	−0.516410	−	0.856341i	$-0.672732\pi$
$53$	−398.509	−1.03282	−0.516410	+	0.856341i	$0.672732\pi$
$54$	0	0
$55$	128.778	0.315716
$56$	0	0
$57$	−231.020	−0.536830
$58$	0	0
$59$	686.474	1.51477	0.757384	−	0.652970i	$-0.226477\pi$
$59$	686.474	1.51477	0.757384	+	0.652970i	$0.226477\pi$
$60$	0	0
$61$	−75.3794	−0.158219	−0.0791094	−	0.996866i	$-0.525208\pi$
$61$	−75.3794	−0.158219	−0.0791094	+	0.996866i	$0.525208\pi$
$62$	0	0
$63$	−10.4212	−0.0208403
$64$	0	0
$65$	−143.040	−0.272953
$66$	0	0
$67$	336.720	0.613983	0.306991	−	0.951712i	$-0.400678\pi$
$67$	336.720	0.613983	0.306991	+	0.951712i	$0.400678\pi$
$68$	0	0
$69$	138.548	0.241728
$70$	0	0
$71$	−427.161	−0.714009	−0.357005	−	0.934103i	$-0.616202\pi$
$71$	−427.161	−0.714009	−0.357005	+	0.934103i	$0.616202\pi$
$72$	0	0
$73$	−134.191	−0.215148	−0.107574	−	0.994197i	$-0.534308\pi$
$73$	−134.191	−0.215148	−0.107574	+	0.994197i	$0.534308\pi$
$74$	0	0
$75$	−20.9860	−0.0323101
$76$	0	0
$77$	−81.9263	−0.121251
$78$	0	0
$79$	−253.005	−0.360319	−0.180160	−	0.983637i	$-0.557661\pi$
$79$	−253.005	−0.360319	−0.180160	+	0.983637i	$0.557661\pi$
$80$	0	0
$81$	−766.980	−1.05210
$82$	0	0
$83$	193.536	0.255944	0.127972	−	0.991778i	$-0.459153\pi$
$83$	193.536	0.255944	0.127972	+	0.991778i	$0.459153\pi$
$84$	0	0
$85$	−1129.31	−1.44107
$86$	0	0
$87$	−1452.23	−1.78960
$88$	0	0
$89$	−996.000	−1.18624	−0.593122	−	0.805112i	$-0.702105\pi$
$89$	−996.000	−1.18624	−0.593122	+	0.805112i	$0.702105\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	0	0
$93$	646.851	0.721240
$94$	0	0
$95$	−476.242	−0.514331
$96$	0	0
$97$	761.982	0.797604	0.398802	−	0.917037i	$-0.369426\pi$
$97$	761.982	0.797604	0.398802	+	0.917037i	$0.369426\pi$
$98$	0	0
$99$	17.4238	0.0176885
$100$	0	0
$101$	822.058	0.809879	0.404940	−	0.914343i	$-0.367293\pi$
$101$	822.058	0.809879	0.404940	+	0.914343i	$0.367293\pi$
$102$	0	0
$103$	−794.608	−0.760146	−0.380073	−	0.924956i	$-0.624101\pi$
$103$	−794.608	−0.760146	−0.380073	+	0.924956i	$0.624101\pi$
$104$	0	0
$105$	−411.102	−0.382090
$106$	0	0
$107$	−2115.28	−1.91114	−0.955571	−	0.294761i	$-0.904760\pi$
$107$	−2115.28	−1.91114	−0.955571	+	0.294761i	$0.904760\pi$
$108$	0	0
$109$	1364.17	1.19875	0.599377	−	0.800467i	$-0.295415\pi$
$109$	1364.17	1.19875	0.599377	+	0.800467i	$0.295415\pi$
$110$	0	0
$111$	897.722	0.767640
$112$	0	0
$113$	1277.27	1.06332	0.531660	−	0.846958i	$-0.321568\pi$
$113$	1277.27	1.06332	0.531660	+	0.846958i	$0.321568\pi$
$114$	0	0
$115$	285.614	0.231597
$116$	0	0
$117$	−19.3536	−0.0152926
$118$	0	0
$119$	718.451	0.553448
$120$	0	0
$121$	−1194.02	−0.897087
$122$	0	0
$123$	−2408.37	−1.76549
$124$	0	0
$125$	−1418.65	−1.01510
$126$	0	0
$127$	−1278.83	−0.893526	−0.446763	−	0.894652i	$-0.647423\pi$
$127$	−1278.83	−0.893526	−0.446763	+	0.894652i	$0.647423\pi$
$128$	0	0
$129$	−505.091	−0.344735
$130$	0	0
$131$	2865.00	1.91081	0.955405	−	0.295297i	$-0.0954187\pi$
$131$	2865.00	1.91081	0.955405	+	0.295297i	$0.0954187\pi$
$132$	0	0
$133$	302.978	0.197530
$134$	0	0
$135$	−1498.25	−0.955175
$136$	0	0
$137$	1494.96	0.932283	0.466141	−	0.884710i	$-0.345644\pi$
$137$	1494.96	0.932283	0.466141	+	0.884710i	$0.345644\pi$
$138$	0	0
$139$	1783.85	1.08852	0.544260	−	0.838917i	$-0.316811\pi$
$139$	1783.85	1.08852	0.544260	+	0.838917i	$0.316811\pi$
$140$	0	0
$141$	−270.307	−0.161446
$142$	0	0
$143$	−152.149	−0.0889743
$144$	0	0
$145$	−2993.73	−1.71459
$146$	0	0
$147$	261.537	0.146743
$148$	0	0
$149$	1727.01	0.949547	0.474774	−	0.880108i	$-0.342530\pi$
$149$	1727.01	0.949547	0.474774	+	0.880108i	$0.342530\pi$
$150$	0	0
$151$	−1499.77	−0.808277	−0.404139	−	0.914698i	$-0.632429\pi$
$151$	−1499.77	−0.808277	−0.404139	+	0.914698i	$0.632429\pi$
$152$	0	0
$153$	−152.798	−0.0807383
$154$	0	0
$155$	1333.47	0.691011
$156$	0	0
$157$	−1021.82	−0.519430	−0.259715	−	0.965685i	$-0.583629\pi$
$157$	−1021.82	−0.519430	−0.259715	+	0.965685i	$0.583629\pi$
$158$	0	0
$159$	−2127.04	−1.06091
$160$	0	0
$161$	−181.703	−0.0889454
$162$	0	0
$163$	3847.87	1.84901	0.924504	−	0.381172i	$-0.124480\pi$
$163$	3847.87	1.84901	0.924504	+	0.381172i	$0.124480\pi$
$164$	0	0
$165$	687.348	0.324303
$166$	0	0
$167$	424.130	0.196528	0.0982639	−	0.995160i	$-0.468671\pi$
$167$	424.130	0.196528	0.0982639	+	0.995160i	$0.468671\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−64.4363	−0.0288162
$172$	0	0
$173$	−3117.51	−1.37006	−0.685029	−	0.728516i	$-0.740210\pi$
$173$	−3117.51	−1.37006	−0.685029	+	0.728516i	$0.740210\pi$
$174$	0	0
$175$	27.5228	0.0118887
$176$	0	0
$177$	3664.05	1.55597
$178$	0	0
$179$	−2962.21	−1.23690	−0.618452	−	0.785823i	$-0.712240\pi$
$179$	−2962.21	−1.23690	−0.618452	+	0.785823i	$0.712240\pi$
$180$	0	0
$181$	−1198.27	−0.492081	−0.246041	−	0.969259i	$-0.579130\pi$
$181$	−1198.27	−0.492081	−0.246041	+	0.969259i	$0.579130\pi$
$182$	0	0
$183$	−402.336	−0.162522
$184$	0	0
$185$	1850.63	0.735466
$186$	0	0
$187$	−1201.22	−0.469745
$188$	0	0
$189$	953.162	0.366838
$190$	0	0
$191$	3940.99	1.49298	0.746492	−	0.665394i	$-0.231737\pi$
$191$	3940.99	1.49298	0.746492	+	0.665394i	$0.231737\pi$
$192$	0	0
$193$	−1974.13	−0.736276	−0.368138	−	0.929771i	$-0.620005\pi$
$193$	−1974.13	−0.736276	−0.368138	+	0.929771i	$0.620005\pi$
$194$	0	0
$195$	−763.475	−0.280377
$196$	0	0
$197$	2772.61	1.00274	0.501372	−	0.865232i	$-0.332829\pi$
$197$	2772.61	1.00274	0.501372	+	0.865232i	$0.332829\pi$
$198$	0	0
$199$	919.870	0.327678	0.163839	−	0.986487i	$-0.447612\pi$
$199$	919.870	0.327678	0.163839	+	0.986487i	$0.447612\pi$
$200$	0	0
$201$	1797.24	0.630683
$202$	0	0
$203$	1904.57	0.658494
$204$	0	0
$205$	−4964.80	−1.69150
$206$	0	0
$207$	38.6440	0.0129756
$208$	0	0
$209$	−506.568	−0.167656
$210$	0	0
$211$	−936.550	−0.305568	−0.152784	−	0.988260i	$-0.548824\pi$
$211$	−936.550	−0.305568	−0.152784	+	0.988260i	$0.548824\pi$
$212$	0	0
$213$	−2279.96	−0.733430
$214$	0	0
$215$	−1041.23	−0.330286
$216$	0	0
$217$	−848.332	−0.265385
$218$	0	0
$219$	−716.241	−0.221000
$220$	0	0
$221$	1334.27	0.406120
$222$	0	0
$223$	−5243.34	−1.57453	−0.787264	−	0.616616i	$-0.788503\pi$
$223$	−5243.34	−1.57453	−0.787264	+	0.616616i	$0.788503\pi$
$224$	0	0
$225$	−5.85345	−0.00173435
$226$	0	0
$227$	3821.36	1.11732	0.558662	−	0.829396i	$-0.311315\pi$
$227$	3821.36	1.11732	0.558662	+	0.829396i	$0.311315\pi$
$228$	0	0
$229$	2144.86	0.618936	0.309468	−	0.950910i	$-0.399849\pi$
$229$	2144.86	0.618936	0.309468	+	0.950910i	$0.399849\pi$
$230$	0	0
$231$	−437.280	−0.124549
$232$	0	0
$233$	−3473.06	−0.976513	−0.488257	−	0.872700i	$-0.662367\pi$
$233$	−3473.06	−0.976513	−0.488257	+	0.872700i	$0.662367\pi$
$234$	0	0
$235$	−557.231	−0.154680
$236$	0	0
$237$	−1350.41	−0.370120
$238$	0	0
$239$	−3691.00	−0.998959	−0.499479	−	0.866326i	$-0.666475\pi$
$239$	−3691.00	−0.998959	−0.499479	+	0.866326i	$0.666475\pi$
$240$	0	0
$241$	−914.807	−0.244514	−0.122257	−	0.992498i	$-0.539013\pi$
$241$	−914.807	−0.244514	−0.122257	+	0.992498i	$0.539013\pi$
$242$	0	0
$243$	−417.260	−0.110153
$244$	0	0
$245$	539.152	0.140592
$246$	0	0
$247$	562.673	0.144948
$248$	0	0
$249$	1032.99	0.262905
$250$	0	0
$251$	−5817.09	−1.46283	−0.731417	−	0.681930i	$-0.761141\pi$
$251$	−5817.09	−1.46283	−0.731417	+	0.681930i	$0.761141\pi$
$252$	0	0
$253$	303.801	0.0754933
$254$	0	0
$255$	−6027.69	−1.48027
$256$	0	0
$257$	3066.16	0.744210	0.372105	−	0.928191i	$-0.378636\pi$
$257$	3066.16	0.744210	0.372105	+	0.928191i	$0.378636\pi$
$258$	0	0
$259$	−1177.34	−0.282458
$260$	0	0
$261$	−405.057	−0.0960627
$262$	0	0
$263$	−1215.01	−0.284869	−0.142435	−	0.989804i	$-0.545493\pi$
$263$	−1215.01	−0.284869	−0.142435	+	0.989804i	$0.545493\pi$
$264$	0	0
$265$	−4384.84	−1.01645
$266$	0	0
$267$	−5316.13	−1.21851
$268$	0	0
$269$	3689.75	0.836311	0.418156	−	0.908375i	$-0.362677\pi$
$269$	3689.75	0.836311	0.418156	+	0.908375i	$0.362677\pi$
$270$	0	0
$271$	−5672.09	−1.27142	−0.635710	−	0.771928i	$-0.719292\pi$
$271$	−5672.09	−1.27142	−0.635710	+	0.771928i	$0.719292\pi$
$272$	0	0
$273$	485.711	0.107680
$274$	0	0
$275$	−46.0171	−0.0100907
$276$	0	0
$277$	6266.61	1.35929	0.679647	−	0.733539i	$-0.262133\pi$
$277$	6266.61	1.35929	0.679647	+	0.733539i	$0.262133\pi$
$278$	0	0
$279$	180.420	0.0387150
$280$	0	0
$281$	−7122.92	−1.51216	−0.756082	−	0.654477i	$-0.772889\pi$
$281$	−7122.92	−1.51216	−0.756082	+	0.654477i	$0.772889\pi$
$282$	0	0
$283$	2576.18	0.541123	0.270561	−	0.962703i	$-0.412791\pi$
$283$	2576.18	0.541123	0.270561	+	0.962703i	$0.412791\pi$
$284$	0	0
$285$	−2541.93	−0.528320
$286$	0	0
$287$	3158.53	0.649624
$288$	0	0
$289$	5621.12	1.14413
$290$	0	0
$291$	4067.07	0.819298
$292$	0	0
$293$	−6955.17	−1.38678	−0.693388	−	0.720565i	$-0.743883\pi$
$293$	−6955.17	−1.38678	−0.693388	+	0.720565i	$0.743883\pi$
$294$	0	0
$295$	7553.34	1.49075
$296$	0	0
$297$	−1593.65	−0.311357
$298$	0	0
$299$	−337.448	−0.0652681
$300$	0	0
$301$	662.416	0.126847
$302$	0	0
$303$	4387.72	0.831908
$304$	0	0
$305$	−829.407	−0.155710
$306$	0	0
$307$	−1690.34	−0.314244	−0.157122	−	0.987579i	$-0.550222\pi$
$307$	−1690.34	−0.314244	−0.157122	+	0.987579i	$0.550222\pi$
$308$	0	0
$309$	−4241.21	−0.780822
$310$	0	0
$311$	−7291.22	−1.32941	−0.664706	−	0.747105i	$-0.731443\pi$
$311$	−7291.22	−1.32941	−0.664706	+	0.747105i	$0.731443\pi$
$312$	0	0
$313$	−4730.37	−0.854237	−0.427118	−	0.904196i	$-0.640471\pi$
$313$	−4730.37	−0.854237	−0.427118	+	0.904196i	$0.640471\pi$
$314$	0	0
$315$	−114.665	−0.0205100
$316$	0	0
$317$	8225.90	1.45745	0.728726	−	0.684805i	$-0.240113\pi$
$317$	8225.90	1.45745	0.728726	+	0.684805i	$0.240113\pi$
$318$	0	0
$319$	−3184.37	−0.558904
$320$	0	0
$321$	−11290.3	−1.96312
$322$	0	0
$323$	4442.34	0.765258
$324$	0	0
$325$	51.1137	0.00872393
$326$	0	0
$327$	7281.25	1.23136
$328$	0	0
$329$	354.502	0.0594052
$330$	0	0
$331$	7168.03	1.19030	0.595152	−	0.803613i	$-0.297092\pi$
$331$	7168.03	1.19030	0.595152	+	0.803613i	$0.297092\pi$
$332$	0	0
$333$	250.394	0.0412057
$334$	0	0
$335$	3704.96	0.604249
$336$	0	0
$337$	4566.95	0.738213	0.369107	−	0.929387i	$-0.379664\pi$
$337$	4566.95	0.738213	0.369107	+	0.929387i	$0.379664\pi$
$338$	0	0
$339$	6817.39	1.09224
$340$	0	0
$341$	1418.38	0.225248
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	1524.46	0.237896
$346$	0	0
$347$	4616.95	0.714267	0.357134	−	0.934053i	$-0.383754\pi$
$347$	4616.95	0.714267	0.357134	+	0.934053i	$0.383754\pi$
$348$	0	0
$349$	−7688.68	−1.17927	−0.589636	−	0.807669i	$-0.700729\pi$
$349$	−7688.68	−1.17927	−0.589636	+	0.807669i	$0.700729\pi$
$350$	0	0
$351$	1770.16	0.269185
$352$	0	0
$353$	−8037.67	−1.21190	−0.605952	−	0.795501i	$-0.707208\pi$
$353$	−8037.67	−1.21190	−0.605952	+	0.795501i	$0.707208\pi$
$354$	0	0
$355$	−4700.09	−0.702690
$356$	0	0
$357$	3834.72	0.568501
$358$	0	0
$359$	−170.112	−0.0250089	−0.0125044	−	0.999922i	$-0.503980\pi$
$359$	−170.112	−0.0250089	−0.0125044	+	0.999922i	$0.503980\pi$
$360$	0	0
$361$	−4985.62	−0.726873
$362$	0	0
$363$	−6373.07	−0.921487
$364$	0	0
$365$	−1476.51	−0.211738
$366$	0	0
$367$	−6547.73	−0.931304	−0.465652	−	0.884968i	$-0.654180\pi$
$367$	−6547.73	−0.931304	−0.465652	+	0.884968i	$0.654180\pi$
$368$	0	0
$369$	−671.745	−0.0947687
$370$	0	0
$371$	2789.57	0.390369
$372$	0	0
$373$	−12845.2	−1.78311	−0.891554	−	0.452915i	$-0.850384\pi$
$373$	−12845.2	−1.78311	−0.891554	+	0.452915i	$0.850384\pi$
$374$	0	0
$375$	−7572.02	−1.04271
$376$	0	0
$377$	3537.05	0.483203
$378$	0	0
$379$	−9494.88	−1.28686	−0.643429	−	0.765505i	$-0.722489\pi$
$379$	−9494.88	−1.28686	−0.643429	+	0.765505i	$0.722489\pi$
$380$	0	0
$381$	−6825.74	−0.917829
$382$	0	0
$383$	−6344.95	−0.846506	−0.423253	−	0.906012i	$-0.639112\pi$
$383$	−6344.95	−0.846506	−0.423253	+	0.906012i	$0.639112\pi$
$384$	0	0
$385$	−901.443	−0.119329
$386$	0	0
$387$	−140.880	−0.0185048
$388$	0	0
$389$	−13196.7	−1.72005	−0.860026	−	0.510250i	$-0.829553\pi$
$389$	−13196.7	−1.72005	−0.860026	+	0.510250i	$0.829553\pi$
$390$	0	0
$391$	−2664.18	−0.344586
$392$	0	0
$393$	15291.9	1.96278
$394$	0	0
$395$	−2783.83	−0.354607
$396$	0	0
$397$	−32.4042	−0.00409653	−0.00204826	−	0.999998i	$-0.500652\pi$
$397$	−32.4042	−0.00409653	−0.00204826	+	0.999998i	$0.500652\pi$
$398$	0	0
$399$	1617.14	0.202903
$400$	0	0
$401$	3853.07	0.479834	0.239917	−	0.970793i	$-0.422880\pi$
$401$	3853.07	0.479834	0.239917	+	0.970793i	$0.422880\pi$
$402$	0	0
$403$	−1575.47	−0.194739
$404$	0	0
$405$	−8439.15	−1.03542
$406$	0	0
$407$	1968.48	0.239739
$408$	0	0
$409$	10221.7	1.23577	0.617886	−	0.786268i	$-0.287989\pi$
$409$	10221.7	1.23577	0.617886	+	0.786268i	$0.287989\pi$
$410$	0	0
$411$	7979.31	0.957640
$412$	0	0
$413$	−4805.32	−0.572529
$414$	0	0
$415$	2129.49	0.251886
$416$	0	0
$417$	9521.27	1.11813
$418$	0	0
$419$	12153.6	1.41704	0.708521	−	0.705689i	$-0.249363\pi$
$419$	12153.6	1.41704	0.708521	+	0.705689i	$0.249363\pi$
$420$	0	0
$421$	13630.9	1.57798	0.788990	−	0.614405i	$-0.210604\pi$
$421$	13630.9	1.57798	0.788990	+	0.614405i	$0.210604\pi$
$422$	0	0
$423$	−75.3942	−0.00866618
$424$	0	0
$425$	403.546	0.0460585
$426$	0	0
$427$	527.656	0.0598011
$428$	0	0
$429$	−812.092	−0.0913943
$430$	0	0
$431$	5959.31	0.666009	0.333004	−	0.942925i	$-0.391938\pi$
$431$	5959.31	0.666009	0.333004	+	0.942925i	$0.391938\pi$
$432$	0	0
$433$	13019.1	1.44494	0.722470	−	0.691402i	$-0.243007\pi$
$433$	13019.1	1.44494	0.722470	+	0.691402i	$0.243007\pi$
$434$	0	0
$435$	−15979.0	−1.76123
$436$	0	0
$437$	−1123.51	−0.122986
$438$	0	0
$439$	892.739	0.0970572	0.0485286	−	0.998822i	$-0.484547\pi$
$439$	892.739	0.0970572	0.0485286	+	0.998822i	$0.484547\pi$
$440$	0	0
$441$	72.9481	0.00787691
$442$	0	0
$443$	8779.74	0.941621	0.470810	−	0.882234i	$-0.343962\pi$
$443$	8779.74	0.941621	0.470810	+	0.882234i	$0.343962\pi$
$444$	0	0
$445$	−10959.1	−1.16744
$446$	0	0
$447$	9217.91	0.975374
$448$	0	0
$449$	13122.2	1.37923	0.689617	−	0.724174i	$-0.257779\pi$
$449$	13122.2	1.37923	0.689617	+	0.724174i	$0.257779\pi$
$450$	0	0
$451$	−5280.95	−0.551375
$452$	0	0
$453$	−8005.02	−0.830262
$454$	0	0
$455$	1001.28	0.103167
$456$	0	0
$457$	−9100.56	−0.931524	−0.465762	−	0.884910i	$-0.654220\pi$
$457$	−9100.56	−0.931524	−0.465762	+	0.884910i	$0.654220\pi$
$458$	0	0
$459$	13975.5	1.42118
$460$	0	0
$461$	−11153.6	−1.12685	−0.563423	−	0.826169i	$-0.690516\pi$
$461$	−11153.6	−1.12685	−0.563423	+	0.826169i	$0.690516\pi$
$462$	0	0
$463$	10006.3	1.00439	0.502197	−	0.864753i	$-0.332525\pi$
$463$	10006.3	1.00439	0.502197	+	0.864753i	$0.332525\pi$
$464$	0	0
$465$	7117.37	0.709806
$466$	0	0
$467$	−10455.5	−1.03603	−0.518014	−	0.855372i	$-0.673328\pi$
$467$	−10455.5	−1.03603	−0.518014	+	0.855372i	$0.673328\pi$
$468$	0	0
$469$	−2357.04	−0.232064
$470$	0	0
$471$	−5453.97	−0.533558
$472$	0	0
$473$	−1107.54	−0.107663
$474$	0	0
$475$	170.179	0.0164387
$476$	0	0
$477$	−593.275	−0.0569480
$478$	0	0
$479$	11161.0	1.06463	0.532314	−	0.846547i	$-0.321322\pi$
$479$	11161.0	1.06463	0.532314	+	0.846547i	$0.321322\pi$
$480$	0	0
$481$	−2186.50	−0.207267
$482$	0	0
$483$	−969.837	−0.0913646
$484$	0	0
$485$	8384.17	0.784959
$486$	0	0
$487$	−3941.17	−0.366717	−0.183359	−	0.983046i	$-0.558697\pi$
$487$	−3941.17	−0.366717	−0.183359	+	0.983046i	$0.558697\pi$
$488$	0	0
$489$	20537.9	1.89930
$490$	0	0
$491$	−11636.5	−1.06955	−0.534775	−	0.844995i	$-0.679603\pi$
$491$	−11636.5	−1.06955	−0.534775	+	0.844995i	$0.679603\pi$
$492$	0	0
$493$	27925.3	2.55110
$494$	0	0
$495$	191.716	0.0174080
$496$	0	0
$497$	2990.13	0.269870
$498$	0	0
$499$	−2370.55	−0.212666	−0.106333	−	0.994331i	$-0.533911\pi$
$499$	−2370.55	−0.212666	−0.106333	+	0.994331i	$0.533911\pi$
$500$	0	0
$501$	2263.79	0.201873
$502$	0	0
$503$	−16697.1	−1.48010	−0.740048	−	0.672554i	$-0.765197\pi$
$503$	−16697.1	−1.48010	−0.740048	+	0.672554i	$0.765197\pi$
$504$	0	0
$505$	9045.18	0.797040
$506$	0	0
$507$	902.035	0.0790153
$508$	0	0
$509$	−12297.2	−1.07085	−0.535426	−	0.844582i	$-0.679849\pi$
$509$	−12297.2	−1.07085	−0.535426	+	0.844582i	$0.679849\pi$
$510$	0	0
$511$	939.335	0.0813185
$512$	0	0
$513$	5893.61	0.507230
$514$	0	0
$515$	−8743.15	−0.748096
$516$	0	0
$517$	−592.714	−0.0504208
$518$	0	0
$519$	−16639.7	−1.40732
$520$	0	0
$521$	18304.4	1.53922	0.769609	−	0.638516i	$-0.220451\pi$
$521$	18304.4	1.53922	0.769609	+	0.638516i	$0.220451\pi$
$522$	0	0
$523$	12673.2	1.05958	0.529788	−	0.848130i	$-0.322271\pi$
$523$	12673.2	1.05958	0.529788	+	0.848130i	$0.322271\pi$
$524$	0	0
$525$	146.902	0.0122121
$526$	0	0
$527$	−12438.5	−1.02814
$528$	0	0
$529$	−11493.2	−0.944621
$530$	0	0
$531$	1021.98	0.0835219
$532$	0	0
$533$	5865.84	0.476694
$534$	0	0
$535$	−23274.7	−1.88084
$536$	0	0
$537$	−15810.7	−1.27055
$538$	0	0
$539$	573.484	0.0458287
$540$	0	0
$541$	−21283.1	−1.69137	−0.845684	−	0.533684i	$-0.820807\pi$
$541$	−21283.1	−1.69137	−0.845684	+	0.533684i	$0.820807\pi$
$542$	0	0
$543$	−6395.75	−0.505466
$544$	0	0
$545$	15010.1	1.17975
$546$	0	0
$547$	19081.2	1.49150	0.745751	−	0.666225i	$-0.232091\pi$
$547$	19081.2	1.49150	0.745751	+	0.666225i	$0.232091\pi$
$548$	0	0
$549$	−112.220	−0.00872392
$550$	0	0
$551$	11776.4	0.910507
$552$	0	0
$553$	1771.03	0.136188
$554$	0	0
$555$	9877.73	0.755471
$556$	0	0
$557$	−13693.3	−1.04166	−0.520828	−	0.853661i	$-0.674377\pi$
$557$	−13693.3	−1.04166	−0.520828	+	0.853661i	$0.674377\pi$
$558$	0	0
$559$	1230.20	0.0930805
$560$	0	0
$561$	−6411.52	−0.482521
$562$	0	0
$563$	15762.2	1.17993	0.589964	−	0.807430i	$-0.299142\pi$
$563$	15762.2	1.17993	0.589964	+	0.807430i	$0.299142\pi$
$564$	0	0
$565$	14053.9	1.04646
$566$	0	0
$567$	5368.86	0.397656
$568$	0	0
$569$	22202.1	1.63578	0.817892	−	0.575372i	$-0.195143\pi$
$569$	22202.1	1.63578	0.817892	+	0.575372i	$0.195143\pi$
$570$	0	0
$571$	−21989.8	−1.61164	−0.805819	−	0.592162i	$-0.798275\pi$
$571$	−21989.8	−1.61164	−0.805819	+	0.592162i	$0.798275\pi$
$572$	0	0
$573$	21035.0	1.53359
$574$	0	0
$575$	−102.061	−0.00740212
$576$	0	0
$577$	−18405.1	−1.32793	−0.663965	−	0.747764i	$-0.731128\pi$
$577$	−18405.1	−1.32793	−0.663965	+	0.747764i	$0.731128\pi$
$578$	0	0
$579$	−10536.9	−0.756303
$580$	0	0
$581$	−1354.75	−0.0967376
$582$	0	0
$583$	−4664.05	−0.331330
$584$	0	0
$585$	−212.949	−0.0150502
$586$	0	0
$587$	13059.4	0.918260	0.459130	−	0.888369i	$-0.348161\pi$
$587$	13059.4	0.918260	0.459130	+	0.888369i	$0.348161\pi$
$588$	0	0
$589$	−5245.42	−0.366951
$590$	0	0
$591$	14798.8	1.03002
$592$	0	0
$593$	11511.8	0.797189	0.398595	−	0.917127i	$-0.369498\pi$
$593$	11511.8	0.797189	0.398595	+	0.917127i	$0.369498\pi$
$594$	0	0
$595$	7905.19	0.544674
$596$	0	0
$597$	4909.79	0.336590
$598$	0	0
$599$	−7113.35	−0.485215	−0.242607	−	0.970125i	$-0.578003\pi$
$599$	−7113.35	−0.485215	−0.242607	+	0.970125i	$0.578003\pi$
$600$	0	0
$601$	−7802.91	−0.529596	−0.264798	−	0.964304i	$-0.585305\pi$
$601$	−7802.91	−0.529596	−0.264798	+	0.964304i	$0.585305\pi$
$602$	0	0
$603$	501.287	0.0338540
$604$	0	0
$605$	−13137.9	−0.882865
$606$	0	0
$607$	13290.9	0.888732	0.444366	−	0.895845i	$-0.353429\pi$
$607$	13290.9	0.888732	0.444366	+	0.895845i	$0.353429\pi$
$608$	0	0
$609$	10165.6	0.676405
$610$	0	0
$611$	658.360	0.0435915
$612$	0	0
$613$	−3133.55	−0.206465	−0.103232	−	0.994657i	$-0.532919\pi$
$613$	−3133.55	−0.206465	−0.103232	+	0.994657i	$0.532919\pi$
$614$	0	0
$615$	−26499.5	−1.73750
$616$	0	0
$617$	−126.313	−0.00824175	−0.00412088	−	0.999992i	$-0.501312\pi$
$617$	−126.313	−0.00824175	−0.00412088	+	0.999992i	$0.501312\pi$
$618$	0	0
$619$	−5888.40	−0.382350	−0.191175	−	0.981556i	$-0.561230\pi$
$619$	−5888.40	−0.382350	−0.191175	+	0.981556i	$0.561230\pi$
$620$	0	0
$621$	−3534.54	−0.228400
$622$	0	0
$623$	6972.00	0.448358
$624$	0	0
$625$	−15118.1	−0.967556
$626$	0	0
$627$	−2703.80	−0.172216
$628$	0	0
$629$	−17262.5	−1.09428
$630$	0	0
$631$	21316.2	1.34483	0.672413	−	0.740176i	$-0.265258\pi$
$631$	21316.2	1.34483	0.672413	+	0.740176i	$0.265258\pi$
$632$	0	0
$633$	−4998.82	−0.313879
$634$	0	0
$635$	−14071.1	−0.879361
$636$	0	0
$637$	−637.000	−0.0396214
$638$	0	0
$639$	−635.930	−0.0393693
$640$	0	0
$641$	−11687.9	−0.720193	−0.360097	−	0.932915i	$-0.617256\pi$
$641$	−11687.9	−0.720193	−0.360097	+	0.932915i	$0.617256\pi$
$642$	0	0
$643$	20771.5	1.27395	0.636974	−	0.770885i	$-0.280186\pi$
$643$	20771.5	1.27395	0.636974	+	0.770885i	$0.280186\pi$
$644$	0	0
$645$	−5557.56	−0.339270
$646$	0	0
$647$	15685.1	0.953086	0.476543	−	0.879151i	$-0.341890\pi$
$647$	15685.1	0.953086	0.476543	+	0.879151i	$0.341890\pi$
$648$	0	0
$649$	8034.32	0.485940
$650$	0	0
$651$	−4527.96	−0.272603
$652$	0	0
$653$	5161.91	0.309343	0.154672	−	0.987966i	$-0.450568\pi$
$653$	5161.91	0.309343	0.154672	+	0.987966i	$0.450568\pi$
$654$	0	0
$655$	31523.9	1.88052
$656$	0	0
$657$	−199.775	−0.0118629
$658$	0	0
$659$	19647.1	1.16137	0.580683	−	0.814130i	$-0.302786\pi$
$659$	19647.1	1.16137	0.580683	+	0.814130i	$0.302786\pi$
$660$	0	0
$661$	25197.0	1.48268	0.741340	−	0.671130i	$-0.234191\pi$
$661$	25197.0	1.48268	0.741340	+	0.671130i	$0.234191\pi$
$662$	0	0
$663$	7121.62	0.417166
$664$	0	0
$665$	3333.69	0.194399
$666$	0	0
$667$	−7062.56	−0.409990
$668$	0	0
$669$	−27986.2	−1.61735
$670$	0	0
$671$	−882.222	−0.0507568
$672$	0	0
$673$	31227.3	1.78860	0.894298	−	0.447472i	$-0.147676\pi$
$673$	31227.3	1.78860	0.894298	+	0.447472i	$0.147676\pi$
$674$	0	0
$675$	535.381	0.0305286
$676$	0	0
$677$	−1846.13	−0.104804	−0.0524022	−	0.998626i	$-0.516688\pi$
$677$	−1846.13	−0.104804	−0.0524022	+	0.998626i	$0.516688\pi$
$678$	0	0
$679$	−5333.88	−0.301466
$680$	0	0
$681$	20396.4	1.14771
$682$	0	0
$683$	5829.61	0.326594	0.163297	−	0.986577i	$-0.447787\pi$
$683$	5829.61	0.326594	0.163297	+	0.986577i	$0.447787\pi$
$684$	0	0
$685$	16449.2	0.917503
$686$	0	0
$687$	11448.1	0.635770
$688$	0	0
$689$	5180.62	0.286453
$690$	0	0
$691$	12829.3	0.706294	0.353147	−	0.935568i	$-0.385112\pi$
$691$	12829.3	0.706294	0.353147	+	0.935568i	$0.385112\pi$
$692$	0	0
$693$	−121.967	−0.00668561
$694$	0	0
$695$	19627.9	1.07126
$696$	0	0
$697$	46311.2	2.51673
$698$	0	0
$699$	−18537.4	−1.00307
$700$	0	0
$701$	−30321.8	−1.63372	−0.816862	−	0.576834i	$-0.804288\pi$
$701$	−30321.8	−1.63372	−0.816862	+	0.576834i	$0.804288\pi$
$702$	0	0
$703$	−7279.78	−0.390558
$704$	0	0
$705$	−2974.21	−0.158887
$706$	0	0
$707$	−5754.41	−0.306106
$708$	0	0
$709$	13767.1	0.729246	0.364623	−	0.931155i	$-0.381198\pi$
$709$	13767.1	0.729246	0.364623	+	0.931155i	$0.381198\pi$
$710$	0	0
$711$	−376.657	−0.0198674
$712$	0	0
$713$	3145.81	0.165233
$714$	0	0
$715$	−1674.11	−0.0875638
$716$	0	0
$717$	−19700.7	−1.02613
$718$	0	0
$719$	25769.7	1.33664	0.668322	−	0.743872i	$-0.267013\pi$
$719$	25769.7	1.33664	0.668322	+	0.743872i	$0.267013\pi$
$720$	0	0
$721$	5562.26	0.287308
$722$	0	0
$723$	−4882.77	−0.251165
$724$	0	0
$725$	1069.77	0.0548006
$726$	0	0
$727$	−20584.7	−1.05013	−0.525064	−	0.851063i	$-0.675959\pi$
$727$	−20584.7	−1.05013	−0.525064	+	0.851063i	$0.675959\pi$
$728$	0	0
$729$	18481.3	0.938949
$730$	0	0
$731$	9712.52	0.491424
$732$	0	0
$733$	12560.0	0.632900	0.316450	−	0.948609i	$-0.397509\pi$
$733$	12560.0	0.632900	0.316450	+	0.948609i	$0.397509\pi$
$734$	0	0
$735$	2877.71	0.144416
$736$	0	0
$737$	3940.88	0.196966
$738$	0	0
$739$	−2247.01	−0.111850	−0.0559252	−	0.998435i	$-0.517811\pi$
$739$	−2247.01	−0.111850	−0.0559252	+	0.998435i	$0.517811\pi$
$740$	0	0
$741$	3003.26	0.148890
$742$	0	0
$743$	17437.7	0.861006	0.430503	−	0.902589i	$-0.358336\pi$
$743$	17437.7	0.861006	0.430503	+	0.902589i	$0.358336\pi$
$744$	0	0
$745$	19002.5	0.934494
$746$	0	0
$747$	288.124	0.0141123
$748$	0	0
$749$	14807.0	0.722344
$750$	0	0
$751$	13096.0	0.636323	0.318162	−	0.948036i	$-0.396935\pi$
$751$	13096.0	0.636323	0.318162	+	0.948036i	$0.396935\pi$
$752$	0	0
$753$	−31048.6	−1.50262
$754$	0	0
$755$	−16502.2	−0.795464
$756$	0	0
$757$	29343.6	1.40887	0.704433	−	0.709771i	$-0.251202\pi$
$757$	29343.6	1.40887	0.704433	+	0.709771i	$0.251202\pi$
$758$	0	0
$759$	1621.53	0.0775467
$760$	0	0
$761$	−23788.7	−1.13316	−0.566582	−	0.824005i	$-0.691735\pi$
$761$	−23788.7	−1.13316	−0.566582	+	0.824005i	$0.691735\pi$
$762$	0	0
$763$	−9549.21	−0.453086
$764$	0	0
$765$	−1681.25	−0.0794584
$766$	0	0
$767$	−8924.16	−0.420121
$768$	0	0
$769$	11235.6	0.526875	0.263437	−	0.964676i	$-0.415144\pi$
$769$	11235.6	0.526875	0.263437	+	0.964676i	$0.415144\pi$
$770$	0	0
$771$	16365.6	0.764452
$772$	0	0
$773$	−7742.41	−0.360252	−0.180126	−	0.983644i	$-0.557651\pi$
$773$	−7742.41	−0.360252	−0.180126	+	0.983644i	$0.557651\pi$
$774$	0	0
$775$	−476.499	−0.0220856
$776$	0	0
$777$	−6284.06	−0.290141
$778$	0	0
$779$	19529.9	0.898242
$780$	0	0
$781$	−4999.38	−0.229055
$782$	0	0
$783$	37048.2	1.69092
$784$	0	0
$785$	−11243.2	−0.511195
$786$	0	0
$787$	−73.6706	−0.00333681	−0.00166841	−	0.999999i	$-0.500531\pi$
$787$	−73.6706	−0.00333681	−0.00166841	+	0.999999i	$0.500531\pi$
$788$	0	0
$789$	−6485.09	−0.292618
$790$	0	0
$791$	−8940.87	−0.401897
$792$	0	0
$793$	979.932	0.0438820
$794$	0	0
$795$	−23404.0	−1.04409
$796$	0	0
$797$	5805.43	0.258016	0.129008	−	0.991644i	$-0.458821\pi$
$797$	5805.43	0.258016	0.129008	+	0.991644i	$0.458821\pi$
$798$	0	0
$799$	5197.80	0.230144
$800$	0	0
$801$	−1482.78	−0.0654076
$802$	0	0
$803$	−1570.53	−0.0690199
$804$	0	0
$805$	−1999.30	−0.0875353
$806$	0	0
$807$	19694.0	0.859059
$808$	0	0
$809$	27095.9	1.17755	0.588776	−	0.808296i	$-0.299610\pi$
$809$	27095.9	1.17755	0.588776	+	0.808296i	$0.299610\pi$
$810$	0	0
$811$	28006.3	1.21262	0.606309	−	0.795229i	$-0.292650\pi$
$811$	28006.3	1.21262	0.606309	+	0.795229i	$0.292650\pi$
$812$	0	0
$813$	−30274.7	−1.30600
$814$	0	0
$815$	42338.5	1.81970
$816$	0	0
$817$	4095.87	0.175393
$818$	0	0
$819$	135.475	0.00578007
$820$	0	0
$821$	−16408.0	−0.697493	−0.348747	−	0.937217i	$-0.613393\pi$
$821$	−16408.0	−0.697493	−0.348747	+	0.937217i	$0.613393\pi$
$822$	0	0
$823$	−7614.47	−0.322507	−0.161254	−	0.986913i	$-0.551554\pi$
$823$	−7614.47	−0.322507	−0.161254	+	0.986913i	$0.551554\pi$
$824$	0	0
$825$	−245.615	−0.0103651
$826$	0	0
$827$	17850.7	0.750581	0.375290	−	0.926907i	$-0.377543\pi$
$827$	17850.7	0.750581	0.375290	+	0.926907i	$0.377543\pi$
$828$	0	0
$829$	−3802.11	−0.159292	−0.0796459	−	0.996823i	$-0.525379\pi$
$829$	−3802.11	−0.159292	−0.0796459	+	0.996823i	$0.525379\pi$
$830$	0	0
$831$	33448.0	1.39627
$832$	0	0
$833$	−5029.16	−0.209184
$834$	0	0
$835$	4666.74	0.193412
$836$	0	0
$837$	−16502.0	−0.681472
$838$	0	0
$839$	3538.68	0.145613	0.0728063	−	0.997346i	$-0.476805\pi$
$839$	3538.68	0.145613	0.0728063	+	0.997346i	$0.476805\pi$
$840$	0	0
$841$	49639.0	2.03530
$842$	0	0
$843$	−38018.5	−1.55329
$844$	0	0
$845$	1859.52	0.0757036
$846$	0	0
$847$	8358.16	0.339067
$848$	0	0
$849$	13750.3	0.555841
$850$	0	0
$851$	4365.86	0.175863
$852$	0	0
$853$	−21190.1	−0.850570	−0.425285	−	0.905059i	$-0.639826\pi$
$853$	−21190.1	−0.850570	−0.425285	+	0.905059i	$0.639826\pi$
$854$	0	0
$855$	−708.999	−0.0283594
$856$	0	0
$857$	−19184.4	−0.764673	−0.382337	−	0.924023i	$-0.624881\pi$
$857$	−19184.4	−0.764673	−0.382337	+	0.924023i	$0.624881\pi$
$858$	0	0
$859$	37876.3	1.50445	0.752226	−	0.658905i	$-0.228980\pi$
$859$	37876.3	1.50445	0.752226	+	0.658905i	$0.228980\pi$
$860$	0	0
$861$	16858.6	0.667293
$862$	0	0
$863$	−9543.13	−0.376422	−0.188211	−	0.982129i	$-0.560269\pi$
$863$	−9543.13	−0.376422	−0.188211	+	0.982129i	$0.560269\pi$
$864$	0	0
$865$	−34302.3	−1.34834
$866$	0	0
$867$	30002.6	1.17525
$868$	0	0
$869$	−2961.10	−0.115591
$870$	0	0
$871$	−4377.35	−0.170288
$872$	0	0
$873$	1134.39	0.0439786
$874$	0	0
$875$	9930.55	0.383673
$876$	0	0
$877$	27017.4	1.04027	0.520133	−	0.854085i	$-0.325882\pi$
$877$	27017.4	1.04027	0.520133	+	0.854085i	$0.325882\pi$
$878$	0	0
$879$	−37123.1	−1.42449
$880$	0	0
$881$	−31954.5	−1.22199	−0.610996	−	0.791634i	$-0.709231\pi$
$881$	−31954.5	−1.22199	−0.610996	+	0.791634i	$0.709231\pi$
$882$	0	0
$883$	−42055.4	−1.60280	−0.801402	−	0.598126i	$-0.795912\pi$
$883$	−42055.4	−1.60280	−0.801402	+	0.598126i	$0.795912\pi$
$884$	0	0
$885$	40315.8	1.53130
$886$	0	0
$887$	−34830.5	−1.31848	−0.659242	−	0.751931i	$-0.729123\pi$
$887$	−34830.5	−1.31848	−0.659242	+	0.751931i	$0.729123\pi$
$888$	0	0
$889$	8951.81	0.337721
$890$	0	0
$891$	−8976.54	−0.337514
$892$	0	0
$893$	2191.96	0.0821403
$894$	0	0
$895$	−32593.4	−1.21729
$896$	0	0
$897$	−1801.13	−0.0670433
$898$	0	0
$899$	−32973.6	−1.22328
$900$	0	0
$901$	40901.3	1.51234
$902$	0	0
$903$	3535.64	0.130297
$904$	0	0
$905$	−13184.7	−0.484281
$906$	0	0
$907$	−28035.7	−1.02636	−0.513180	−	0.858281i	$-0.671533\pi$
$907$	−28035.7	−1.02636	−0.513180	+	0.858281i	$0.671533\pi$
$908$	0	0
$909$	1223.83	0.0446554
$910$	0	0
$911$	−19163.8	−0.696954	−0.348477	−	0.937317i	$-0.613301\pi$
$911$	−19163.8	−0.696954	−0.348477	+	0.937317i	$0.613301\pi$
$912$	0	0
$913$	2265.10	0.0821070
$914$	0	0
$915$	−4426.95	−0.159946
$916$	0	0
$917$	−20055.0	−0.722219
$918$	0	0
$919$	34654.1	1.24389	0.621945	−	0.783061i	$-0.286343\pi$
$919$	34654.1	1.24389	0.621945	+	0.783061i	$0.286343\pi$
$920$	0	0
$921$	−9022.18	−0.322791
$922$	0	0
$923$	5553.09	0.198031
$924$	0	0
$925$	−661.301	−0.0235064
$926$	0	0
$927$	−1182.96	−0.0419132
$928$	0	0
$929$	42110.2	1.48718	0.743591	−	0.668635i	$-0.233121\pi$
$929$	42110.2	1.48718	0.743591	+	0.668635i	$0.233121\pi$
$930$	0	0
$931$	−2120.84	−0.0746594
$932$	0	0
$933$	−38916.8	−1.36557
$934$	0	0
$935$	−13217.2	−0.462298
$936$	0	0
$937$	−18514.6	−0.645514	−0.322757	−	0.946482i	$-0.604610\pi$
$937$	−18514.6	−0.645514	−0.322757	+	0.946482i	$0.604610\pi$
$938$	0	0
$939$	−25248.3	−0.877471
$940$	0	0
$941$	−8119.45	−0.281282	−0.140641	−	0.990061i	$-0.544916\pi$
$941$	−8119.45	−0.281282	−0.140641	+	0.990061i	$0.544916\pi$
$942$	0	0
$943$	−11712.5	−0.404467
$944$	0	0
$945$	10487.7	0.361022
$946$	0	0
$947$	−49354.7	−1.69357	−0.846786	−	0.531934i	$-0.821466\pi$
$947$	−49354.7	−1.69357	−0.846786	+	0.531934i	$0.821466\pi$
$948$	0	0
$949$	1744.48	0.0596714
$950$	0	0
$951$	43905.6	1.49709
$952$	0	0
$953$	−300.006	−0.0101974	−0.00509872	−	0.999987i	$-0.501623\pi$
$953$	−300.006	−0.0101974	−0.00509872	+	0.999987i	$0.501623\pi$
$954$	0	0
$955$	43363.1	1.46932
$956$	0	0
$957$	−16996.5	−0.574106
$958$	0	0
$959$	−10464.7	−0.352370
$960$	0	0
$961$	−15103.9	−0.506996
$962$	0	0
$963$	−3149.10	−0.105377
$964$	0	0
$965$	−21721.6	−0.724604
$966$	0	0
$967$	20103.3	0.668539	0.334270	−	0.942477i	$-0.391510\pi$
$967$	20103.3	0.668539	0.334270	+	0.942477i	$0.391510\pi$
$968$	0	0
$969$	23710.9	0.786073
$970$	0	0
$971$	−39115.5	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$971$	−39115.5	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$972$	0	0
$973$	−12487.0	−0.411422
$974$	0	0
$975$	272.819	0.00896122
$976$	0	0
$977$	21590.6	0.707005	0.353502	−	0.935434i	$-0.384991\pi$
$977$	21590.6	0.707005	0.353502	+	0.935434i	$0.384991\pi$
$978$	0	0
$979$	−11656.9	−0.380549
$980$	0	0
$981$	2030.89	0.0660973
$982$	0	0
$983$	7689.51	0.249499	0.124749	−	0.992188i	$-0.460187\pi$
$983$	7689.51	0.249499	0.124749	+	0.992188i	$0.460187\pi$
$984$	0	0
$985$	30507.3	0.986847
$986$	0	0
$987$	1892.15	0.0610210
$988$	0	0
$989$	−2456.39	−0.0789774
$990$	0	0
$991$	−23913.5	−0.766537	−0.383269	−	0.923637i	$-0.625202\pi$
$991$	−23913.5	−0.766537	−0.383269	+	0.923637i	$0.625202\pi$
$992$	0	0
$993$	38259.3	1.22268
$994$	0	0
$995$	10121.4	0.322483
$996$	0	0
$997$	−33.8007	−0.00107370	−0.000536851	−	1.00000i	$-0.500171\pi$
$997$	−33.8007	−0.00107370	−0.000536851		1.00000i	$0.500171\pi$
$998$	0	0
$999$	−22902.0	−0.725314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.4.a.s.1.3		4
4.3	odd	2		91.4.a.b.1.2	✓	4
12.11	even	2		819.4.a.h.1.3		4
20.19	odd	2		2275.4.a.h.1.3		4
28.27	even	2		637.4.a.d.1.2		4
52.51	odd	2		1183.4.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.4.a.b.1.2	✓	4	4.3	odd	2
637.4.a.d.1.2		4	28.27	even	2
819.4.a.h.1.3		4	12.11	even	2
1183.4.a.e.1.3		4	52.51	odd	2
1456.4.a.s.1.3		4	1.1	even	1	trivial
2275.4.a.h.1.3		4	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+5.33748 q^{3} +11.0031 q^{5} -7.00000 q^{7} +1.48874 q^{9} +O(q^{10})\)	\(q+5.33748 q^{3} +11.0031 q^{5} -7.00000 q^{7} +1.48874 q^{9} +11.7038 q^{11} -13.0000 q^{13} +58.7289 q^{15} -102.636 q^{17} -43.2825 q^{19} -37.3624 q^{21} +25.9576 q^{23} -3.93182 q^{25} -136.166 q^{27} -272.081 q^{29} +121.190 q^{31} +62.4686 q^{33} -77.0217 q^{35} +168.192 q^{37} -69.3873 q^{39} -451.218 q^{41} -94.6309 q^{43} +16.3807 q^{45} -50.6431 q^{47} +49.0000 q^{49} -547.817 q^{51} -398.509 q^{53} +128.778 q^{55} -231.020 q^{57} +686.474 q^{59} -75.3794 q^{61} -10.4212 q^{63} -143.040 q^{65} +336.720 q^{67} +138.548 q^{69} -427.161 q^{71} -134.191 q^{73} -20.9860 q^{75} -81.9263 q^{77} -253.005 q^{79} -766.980 q^{81} +193.536 q^{83} -1129.31 q^{85} -1452.23 q^{87} -996.000 q^{89} +91.0000 q^{91} +646.851 q^{93} -476.242 q^{95} +761.982 q^{97} +17.4238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 36 q^{5} - 28 q^{7} + 21 q^{9}+O(q^{10}) \) 4 * q + 5 * q^3 - 36 * q^5 - 28 * q^7 + 21 * q^9	\( 4 q + 5 q^{3} - 36 q^{5} - 28 q^{7} + 21 q^{9} + 95 q^{11} - 52 q^{13} + 16 q^{15} - 146 q^{17} + 48 q^{19} - 35 q^{21} + 121 q^{23} + 506 q^{25} + 83 q^{27} - 440 q^{29} + 283 q^{31} + 227 q^{33} + 252 q^{35} - 209 q^{37} - 65 q^{39} - 93 q^{41} - 526 q^{43} - 768 q^{45} + 783 q^{47} + 196 q^{49} + 672 q^{51} - 340 q^{53} - 756 q^{55} - 1014 q^{57} + 922 q^{59} - 141 q^{61} - 147 q^{63} + 468 q^{65} + 523 q^{67} + 1595 q^{69} - 1468 q^{71} - 47 q^{73} + 1547 q^{75} - 665 q^{77} - 1025 q^{79} - 1772 q^{81} + 1190 q^{83} - 568 q^{85} - 720 q^{87} - 2962 q^{89} + 364 q^{91} - 763 q^{93} - 2082 q^{95} + 2715 q^{97} - 586 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 - 36 * q^5 - 28 * q^7 + 21 * q^9 + 95 * q^11 - 52 * q^13 + 16 * q^15 - 146 * q^17 + 48 * q^19 - 35 * q^21 + 121 * q^23 + 506 * q^25 + 83 * q^27 - 440 * q^29 + 283 * q^31 + 227 * q^33 + 252 * q^35 - 209 * q^37 - 65 * q^39 - 93 * q^41 - 526 * q^43 - 768 * q^45 + 783 * q^47 + 196 * q^49 + 672 * q^51 - 340 * q^53 - 756 * q^55 - 1014 * q^57 + 922 * q^59 - 141 * q^61 - 147 * q^63 + 468 * q^65 + 523 * q^67 + 1595 * q^69 - 1468 * q^71 - 47 * q^73 + 1547 * q^75 - 665 * q^77 - 1025 * q^79 - 1772 * q^81 + 1190 * q^83 - 568 * q^85 - 720 * q^87 - 2962 * q^89 + 364 * q^91 - 763 * q^93 - 2082 * q^95 + 2715 * q^97 - 586 * q^99

Introduction

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