LMFDB - Embedded newform 1456.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,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, 2, names="a")

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

chi := DirichletCharacter("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
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:	$11.6262185343$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.56155 q^{3} -2.56155 q^{5} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} -2.56155 q^{5} +1.00000 q^{7} -0.561553 q^{9} -0.438447 q^{11} +1.00000 q^{13} -4.00000 q^{15} -4.00000 q^{17} -5.68466 q^{19} +1.56155 q^{21} -5.00000 q^{23} +1.56155 q^{25} -5.56155 q^{27} +1.43845 q^{29} +2.12311 q^{31} -0.684658 q^{33} -2.56155 q^{35} +9.56155 q^{37} +1.56155 q^{39} -4.43845 q^{41} -8.80776 q^{43} +1.43845 q^{45} -3.00000 q^{47} +1.00000 q^{49} -6.24621 q^{51} +3.68466 q^{53} +1.12311 q^{55} -8.87689 q^{57} +3.12311 q^{59} -1.31534 q^{61} -0.561553 q^{63} -2.56155 q^{65} -0.438447 q^{67} -7.80776 q^{69} -14.2462 q^{71} -5.24621 q^{73} +2.43845 q^{75} -0.438447 q^{77} -1.87689 q^{79} -7.00000 q^{81} -1.43845 q^{83} +10.2462 q^{85} +2.24621 q^{87} -12.5616 q^{89} +1.00000 q^{91} +3.31534 q^{93} +14.5616 q^{95} +14.3693 q^{97} +0.246211 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9	$ 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 8 q^{15} - 8 q^{17} + q^{19} - q^{21} - 10 q^{23} - q^{25} - 7 q^{27} + 7 q^{29} - 4 q^{31} + 11 q^{33} - q^{35} + 15 q^{37} - q^{39} - 13 q^{41} + 3 q^{43} + 7 q^{45} - 6 q^{47} + 2 q^{49} + 4 q^{51} - 5 q^{53} - 6 q^{55} - 26 q^{57} - 2 q^{59} - 15 q^{61} + 3 q^{63} - q^{65} - 5 q^{67} + 5 q^{69} - 12 q^{71} + 6 q^{73} + 9 q^{75} - 5 q^{77} - 12 q^{79} - 14 q^{81} - 7 q^{83} + 4 q^{85} - 12 q^{87} - 21 q^{89} + 2 q^{91} + 19 q^{93} + 25 q^{95} + 4 q^{97} - 16 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 8 * q^15 - 8 * q^17 + q^19 - q^21 - 10 * q^23 - q^25 - 7 * q^27 + 7 * q^29 - 4 * q^31 + 11 * q^33 - q^35 + 15 * q^37 - q^39 - 13 * q^41 + 3 * q^43 + 7 * q^45 - 6 * q^47 + 2 * q^49 + 4 * q^51 - 5 * q^53 - 6 * q^55 - 26 * q^57 - 2 * q^59 - 15 * q^61 + 3 * q^63 - q^65 - 5 * q^67 + 5 * q^69 - 12 * q^71 + 6 * q^73 + 9 * q^75 - 5 * q^77 - 12 * q^79 - 14 * q^81 - 7 * q^83 + 4 * q^85 - 12 * q^87 - 21 * q^89 + 2 * q^91 + 19 * q^93 + 25 * q^95 + 4 * q^97 - 16 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−2.56155	−1.14556	−0.572781	−	0.819709i	$-0.694135\pi$
$5$	−2.56155	−1.14556	−0.572781	+	0.819709i	$0.694135\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−0.438447	−0.132197	−0.0660984	−	0.997813i	$-0.521055\pi$
$11$	−0.438447	−0.132197	−0.0660984	+	0.997813i	$0.521055\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−5.68466	−1.30415	−0.652075	−	0.758154i	$-0.726101\pi$
$19$	−5.68466	−1.30415	−0.652075	+	0.758154i	$0.726101\pi$
$20$	0	0
$21$	1.56155	0.340759
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	1.43845	0.267113	0.133556	−	0.991041i	$-0.457360\pi$
$29$	1.43845	0.267113	0.133556	+	0.991041i	$0.457360\pi$
$30$	0	0
$31$	2.12311	0.381321	0.190661	−	0.981656i	$-0.438937\pi$
$31$	2.12311	0.381321	0.190661	+	0.981656i	$0.438937\pi$
$32$	0	0
$33$	−0.684658	−0.119184
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	9.56155	1.57191	0.785955	−	0.618284i	$-0.212172\pi$
$37$	9.56155	1.57191	0.785955	+	0.618284i	$0.212172\pi$
$38$	0	0
$39$	1.56155	0.250049
$40$	0	0
$41$	−4.43845	−0.693169	−0.346584	−	0.938019i	$-0.612659\pi$
$41$	−4.43845	−0.693169	−0.346584	+	0.938019i	$0.612659\pi$
$42$	0	0
$43$	−8.80776	−1.34317	−0.671586	−	0.740927i	$-0.734386\pi$
$43$	−8.80776	−1.34317	−0.671586	+	0.740927i	$0.734386\pi$
$44$	0	0
$45$	1.43845	0.214431
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.24621	−0.874645
$52$	0	0
$53$	3.68466	0.506127	0.253063	−	0.967450i	$-0.418562\pi$
$53$	3.68466	0.506127	0.253063	+	0.967450i	$0.418562\pi$
$54$	0	0
$55$	1.12311	0.151440
$56$	0	0
$57$	−8.87689	−1.17577
$58$	0	0
$59$	3.12311	0.406594	0.203297	−	0.979117i	$-0.434834\pi$
$59$	3.12311	0.406594	0.203297	+	0.979117i	$0.434834\pi$
$60$	0	0
$61$	−1.31534	−0.168412	−0.0842061	−	0.996448i	$-0.526835\pi$
$61$	−1.31534	−0.168412	−0.0842061	+	0.996448i	$0.526835\pi$
$62$	0	0
$63$	−0.561553	−0.0707490
$64$	0	0
$65$	−2.56155	−0.317722
$66$	0	0
$67$	−0.438447	−0.0535648	−0.0267824	−	0.999641i	$-0.508526\pi$
$67$	−0.438447	−0.0535648	−0.0267824	+	0.999641i	$0.508526\pi$
$68$	0	0
$69$	−7.80776	−0.939944
$70$	0	0
$71$	−14.2462	−1.69071	−0.845357	−	0.534202i	$-0.820612\pi$
$71$	−14.2462	−1.69071	−0.845357	+	0.534202i	$0.820612\pi$
$72$	0	0
$73$	−5.24621	−0.614023	−0.307011	−	0.951706i	$-0.599329\pi$
$73$	−5.24621	−0.614023	−0.307011	+	0.951706i	$0.599329\pi$
$74$	0	0
$75$	2.43845	0.281568
$76$	0	0
$77$	−0.438447	−0.0499657
$78$	0	0
$79$	−1.87689	−0.211167	−0.105584	−	0.994410i	$-0.533671\pi$
$79$	−1.87689	−0.211167	−0.105584	+	0.994410i	$0.533671\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−1.43845	−0.157890	−0.0789450	−	0.996879i	$-0.525155\pi$
$83$	−1.43845	−0.157890	−0.0789450	+	0.996879i	$0.525155\pi$
$84$	0	0
$85$	10.2462	1.11136
$86$	0	0
$87$	2.24621	0.240819
$88$	0	0
$89$	−12.5616	−1.33152	−0.665761	−	0.746165i	$-0.731893\pi$
$89$	−12.5616	−1.33152	−0.665761	+	0.746165i	$0.731893\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	3.31534	0.343785
$94$	0	0
$95$	14.5616	1.49398
$96$	0	0
$97$	14.3693	1.45898	0.729492	−	0.683990i	$-0.239757\pi$
$97$	14.3693	1.45898	0.729492	+	0.683990i	$0.239757\pi$
$98$	0	0
$99$	0.246211	0.0247452
$100$	0	0
$101$	−8.43845	−0.839657	−0.419828	−	0.907603i	$-0.637910\pi$
$101$	−8.43845	−0.839657	−0.419828	+	0.907603i	$0.637910\pi$
$102$	0	0
$103$	−4.87689	−0.480535	−0.240267	−	0.970707i	$-0.577235\pi$
$103$	−4.87689	−0.480535	−0.240267	+	0.970707i	$0.577235\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	8.24621	0.789844	0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621	0.789844	0.394922	+	0.918715i	$0.370772\pi$
$110$	0	0
$111$	14.9309	1.41718
$112$	0	0
$113$	−9.24621	−0.869810	−0.434905	−	0.900476i	$-0.643218\pi$
$113$	−9.24621	−0.869810	−0.434905	+	0.900476i	$0.643218\pi$
$114$	0	0
$115$	12.8078	1.19433
$116$	0	0
$117$	−0.561553	−0.0519156
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.8078	−0.982524
$122$	0	0
$123$	−6.93087	−0.624935
$124$	0	0
$125$	8.80776	0.787790
$126$	0	0
$127$	3.31534	0.294189	0.147094	−	0.989122i	$-0.453008\pi$
$127$	3.31534	0.294189	0.147094	+	0.989122i	$0.453008\pi$
$128$	0	0
$129$	−13.7538	−1.21095
$130$	0	0
$131$	18.2462	1.59418	0.797089	−	0.603861i	$-0.206372\pi$
$131$	18.2462	1.59418	0.797089	+	0.603861i	$0.206372\pi$
$132$	0	0
$133$	−5.68466	−0.492922
$134$	0	0
$135$	14.2462	1.22612
$136$	0	0
$137$	17.3693	1.48396	0.741980	−	0.670422i	$-0.233887\pi$
$137$	17.3693	1.48396	0.741980	+	0.670422i	$0.233887\pi$
$138$	0	0
$139$	−17.1231	−1.45236	−0.726181	−	0.687503i	$-0.758707\pi$
$139$	−17.1231	−1.45236	−0.726181	+	0.687503i	$0.758707\pi$
$140$	0	0
$141$	−4.68466	−0.394519
$142$	0	0
$143$	−0.438447	−0.0366648
$144$	0	0
$145$	−3.68466	−0.305994
$146$	0	0
$147$	1.56155	0.128795
$148$	0	0
$149$	14.6847	1.20301	0.601507	−	0.798867i	$-0.294567\pi$
$149$	14.6847	1.20301	0.601507	+	0.798867i	$0.294567\pi$
$150$	0	0
$151$	22.2462	1.81037	0.905185	−	0.425017i	$-0.139732\pi$
$151$	22.2462	1.81037	0.905185	+	0.425017i	$0.139732\pi$
$152$	0	0
$153$	2.24621	0.181595
$154$	0	0
$155$	−5.43845	−0.436827
$156$	0	0
$157$	7.80776	0.623127	0.311564	−	0.950225i	$-0.399147\pi$
$157$	7.80776	0.623127	0.311564	+	0.950225i	$0.399147\pi$
$158$	0	0
$159$	5.75379	0.456305
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	−1.75379	−0.137367	−0.0686837	−	0.997638i	$-0.521880\pi$
$163$	−1.75379	−0.137367	−0.0686837	+	0.997638i	$0.521880\pi$
$164$	0	0
$165$	1.75379	0.136532
$166$	0	0
$167$	−6.56155	−0.507748	−0.253874	−	0.967237i	$-0.581705\pi$
$167$	−6.56155	−0.507748	−0.253874	+	0.967237i	$0.581705\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.19224	0.244116
$172$	0	0
$173$	−25.3693	−1.92879	−0.964397	−	0.264460i	$-0.914806\pi$
$173$	−25.3693	−1.92879	−0.964397	+	0.264460i	$0.914806\pi$
$174$	0	0
$175$	1.56155	0.118042
$176$	0	0
$177$	4.87689	0.366570
$178$	0	0
$179$	24.5616	1.83582	0.917908	−	0.396793i	$-0.129877\pi$
$179$	24.5616	1.83582	0.917908	+	0.396793i	$0.129877\pi$
$180$	0	0
$181$	13.8078	1.02632	0.513162	−	0.858292i	$-0.328474\pi$
$181$	13.8078	1.02632	0.513162	+	0.858292i	$0.328474\pi$
$182$	0	0
$183$	−2.05398	−0.151834
$184$	0	0
$185$	−24.4924	−1.80072
$186$	0	0
$187$	1.75379	0.128250
$188$	0	0
$189$	−5.56155	−0.404543
$190$	0	0
$191$	−20.4924	−1.48278	−0.741390	−	0.671075i	$-0.765833\pi$
$191$	−20.4924	−1.48278	−0.741390	+	0.671075i	$0.765833\pi$
$192$	0	0
$193$	17.1231	1.23255	0.616274	−	0.787532i	$-0.288641\pi$
$193$	17.1231	1.23255	0.616274	+	0.787532i	$0.288641\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−9.80776	−0.698774	−0.349387	−	0.936978i	$-0.613610\pi$
$197$	−9.80776	−0.698774	−0.349387	+	0.936978i	$0.613610\pi$
$198$	0	0
$199$	−2.24621	−0.159230	−0.0796148	−	0.996826i	$-0.525369\pi$
$199$	−2.24621	−0.159230	−0.0796148	+	0.996826i	$0.525369\pi$
$200$	0	0
$201$	−0.684658	−0.0482921
$202$	0	0
$203$	1.43845	0.100959
$204$	0	0
$205$	11.3693	0.794068
$206$	0	0
$207$	2.80776	0.195153
$208$	0	0
$209$	2.49242	0.172404
$210$	0	0
$211$	2.80776	0.193294	0.0966472	−	0.995319i	$-0.469188\pi$
$211$	2.80776	0.193294	0.0966472	+	0.995319i	$0.469188\pi$
$212$	0	0
$213$	−22.2462	−1.52429
$214$	0	0
$215$	22.5616	1.53869
$216$	0	0
$217$	2.12311	0.144126
$218$	0	0
$219$	−8.19224	−0.553580
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−1.63068	−0.109199	−0.0545993	−	0.998508i	$-0.517388\pi$
$223$	−1.63068	−0.109199	−0.0545993	+	0.998508i	$0.517388\pi$
$224$	0	0
$225$	−0.876894	−0.0584596
$226$	0	0
$227$	−16.4924	−1.09464	−0.547320	−	0.836923i	$-0.684352\pi$
$227$	−16.4924	−1.09464	−0.547320	+	0.836923i	$0.684352\pi$
$228$	0	0
$229$	14.8769	0.983093	0.491546	−	0.870851i	$-0.336432\pi$
$229$	14.8769	0.983093	0.491546	+	0.870851i	$0.336432\pi$
$230$	0	0
$231$	−0.684658	−0.0450472
$232$	0	0
$233$	−7.24621	−0.474715	−0.237358	−	0.971422i	$-0.576281\pi$
$233$	−7.24621	−0.474715	−0.237358	+	0.971422i	$0.576281\pi$
$234$	0	0
$235$	7.68466	0.501292
$236$	0	0
$237$	−2.93087	−0.190380
$238$	0	0
$239$	13.3693	0.864789	0.432395	−	0.901684i	$-0.357669\pi$
$239$	13.3693	0.864789	0.432395	+	0.901684i	$0.357669\pi$
$240$	0	0
$241$	19.9309	1.28386	0.641930	−	0.766763i	$-0.278134\pi$
$241$	19.9309	1.28386	0.641930	+	0.766763i	$0.278134\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−2.56155	−0.163652
$246$	0	0
$247$	−5.68466	−0.361706
$248$	0	0
$249$	−2.24621	−0.142348
$250$	0	0
$251$	−13.8078	−0.871538	−0.435769	−	0.900058i	$-0.643524\pi$
$251$	−13.8078	−0.871538	−0.435769	+	0.900058i	$0.643524\pi$
$252$	0	0
$253$	2.19224	0.137825
$254$	0	0
$255$	16.0000	1.00196
$256$	0	0
$257$	11.3693	0.709199	0.354599	−	0.935018i	$-0.384617\pi$
$257$	11.3693	0.709199	0.354599	+	0.935018i	$0.384617\pi$
$258$	0	0
$259$	9.56155	0.594126
$260$	0	0
$261$	−0.807764	−0.0499993
$262$	0	0
$263$	−5.93087	−0.365713	−0.182857	−	0.983140i	$-0.558534\pi$
$263$	−5.93087	−0.365713	−0.182857	+	0.983140i	$0.558534\pi$
$264$	0	0
$265$	−9.43845	−0.579799
$266$	0	0
$267$	−19.6155	−1.20045
$268$	0	0
$269$	27.8078	1.69547	0.847735	−	0.530421i	$-0.177966\pi$
$269$	27.8078	1.69547	0.847735	+	0.530421i	$0.177966\pi$
$270$	0	0
$271$	−32.6847	−1.98545	−0.992726	−	0.120397i	$-0.961583\pi$
$271$	−32.6847	−1.98545	−0.992726	+	0.120397i	$0.961583\pi$
$272$	0	0
$273$	1.56155	0.0945095
$274$	0	0
$275$	−0.684658	−0.0412865
$276$	0	0
$277$	1.19224	0.0716345	0.0358173	−	0.999358i	$-0.488597\pi$
$277$	1.19224	0.0716345	0.0358173	+	0.999358i	$0.488597\pi$
$278$	0	0
$279$	−1.19224	−0.0713773
$280$	0	0
$281$	−9.12311	−0.544239	−0.272119	−	0.962263i	$-0.587725\pi$
$281$	−9.12311	−0.544239	−0.272119	+	0.962263i	$0.587725\pi$
$282$	0	0
$283$	9.56155	0.568375	0.284188	−	0.958769i	$-0.408276\pi$
$283$	9.56155	0.568375	0.284188	+	0.958769i	$0.408276\pi$
$284$	0	0
$285$	22.7386	1.34692
$286$	0	0
$287$	−4.43845	−0.261993
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	22.4384	1.31537
$292$	0	0
$293$	−5.68466	−0.332101	−0.166051	−	0.986117i	$-0.553102\pi$
$293$	−5.68466	−0.332101	−0.166051	+	0.986117i	$0.553102\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	2.43845	0.141493
$298$	0	0
$299$	−5.00000	−0.289157
$300$	0	0
$301$	−8.80776	−0.507671
$302$	0	0
$303$	−13.1771	−0.757004
$304$	0	0
$305$	3.36932	0.192927
$306$	0	0
$307$	−12.1771	−0.694983	−0.347491	−	0.937683i	$-0.612966\pi$
$307$	−12.1771	−0.694983	−0.347491	+	0.937683i	$0.612966\pi$
$308$	0	0
$309$	−7.61553	−0.433232
$310$	0	0
$311$	−12.2462	−0.694419	−0.347209	−	0.937788i	$-0.612871\pi$
$311$	−12.2462	−0.694419	−0.347209	+	0.937788i	$0.612871\pi$
$312$	0	0
$313$	−22.4924	−1.27135	−0.635673	−	0.771958i	$-0.719278\pi$
$313$	−22.4924	−1.27135	−0.635673	+	0.771958i	$0.719278\pi$
$314$	0	0
$315$	1.43845	0.0810473
$316$	0	0
$317$	26.0540	1.46334	0.731669	−	0.681661i	$-0.238742\pi$
$317$	26.0540	1.46334	0.731669	+	0.681661i	$0.238742\pi$
$318$	0	0
$319$	−0.630683	−0.0353115
$320$	0	0
$321$	−6.24621	−0.348630
$322$	0	0
$323$	22.7386	1.26521
$324$	0	0
$325$	1.56155	0.0866194
$326$	0	0
$327$	12.8769	0.712094
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	4.05398	0.222827	0.111413	−	0.993774i	$-0.464462\pi$
$331$	4.05398	0.222827	0.111413	+	0.993774i	$0.464462\pi$
$332$	0	0
$333$	−5.36932	−0.294237
$334$	0	0
$335$	1.12311	0.0613618
$336$	0	0
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	0	0
$339$	−14.4384	−0.784189
$340$	0	0
$341$	−0.930870	−0.0504094
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	20.0000	1.07676
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	13.6847	0.732523	0.366261	−	0.930512i	$-0.380638\pi$
$349$	13.6847	0.732523	0.366261	+	0.930512i	$0.380638\pi$
$350$	0	0
$351$	−5.56155	−0.296854
$352$	0	0
$353$	−0.438447	−0.0233362	−0.0116681	−	0.999932i	$-0.503714\pi$
$353$	−0.438447	−0.0233362	−0.0116681	+	0.999932i	$0.503714\pi$
$354$	0	0
$355$	36.4924	1.93682
$356$	0	0
$357$	−6.24621	−0.330585
$358$	0	0
$359$	2.24621	0.118550	0.0592752	−	0.998242i	$-0.481121\pi$
$359$	2.24621	0.118550	0.0592752	+	0.998242i	$0.481121\pi$
$360$	0	0
$361$	13.3153	0.700807
$362$	0	0
$363$	−16.8769	−0.885807
$364$	0	0
$365$	13.4384	0.703400
$366$	0	0
$367$	−29.6155	−1.54592	−0.772959	−	0.634456i	$-0.781224\pi$
$367$	−29.6155	−1.54592	−0.772959	+	0.634456i	$0.781224\pi$
$368$	0	0
$369$	2.49242	0.129750
$370$	0	0
$371$	3.68466	0.191298
$372$	0	0
$373$	18.8769	0.977409	0.488704	−	0.872450i	$-0.337470\pi$
$373$	18.8769	0.977409	0.488704	+	0.872450i	$0.337470\pi$
$374$	0	0
$375$	13.7538	0.710243
$376$	0	0
$377$	1.43845	0.0740838
$378$	0	0
$379$	19.8617	1.02023	0.510115	−	0.860106i	$-0.329603\pi$
$379$	19.8617	1.02023	0.510115	+	0.860106i	$0.329603\pi$
$380$	0	0
$381$	5.17708	0.265230
$382$	0	0
$383$	−6.43845	−0.328989	−0.164495	−	0.986378i	$-0.552599\pi$
$383$	−6.43845	−0.328989	−0.164495	+	0.986378i	$0.552599\pi$
$384$	0	0
$385$	1.12311	0.0572388
$386$	0	0
$387$	4.94602	0.251421
$388$	0	0
$389$	−9.61553	−0.487527	−0.243763	−	0.969835i	$-0.578382\pi$
$389$	−9.61553	−0.487527	−0.243763	+	0.969835i	$0.578382\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	0	0
$393$	28.4924	1.43725
$394$	0	0
$395$	4.80776	0.241905
$396$	0	0
$397$	32.8078	1.64657	0.823287	−	0.567625i	$-0.192138\pi$
$397$	32.8078	1.64657	0.823287	+	0.567625i	$0.192138\pi$
$398$	0	0
$399$	−8.87689	−0.444401
$400$	0	0
$401$	4.49242	0.224341	0.112170	−	0.993689i	$-0.464220\pi$
$401$	4.49242	0.224341	0.112170	+	0.993689i	$0.464220\pi$
$402$	0	0
$403$	2.12311	0.105759
$404$	0	0
$405$	17.9309	0.890992
$406$	0	0
$407$	−4.19224	−0.207801
$408$	0	0
$409$	−31.3002	−1.54769	−0.773847	−	0.633372i	$-0.781670\pi$
$409$	−31.3002	−1.54769	−0.773847	+	0.633372i	$0.781670\pi$
$410$	0	0
$411$	27.1231	1.33788
$412$	0	0
$413$	3.12311	0.153678
$414$	0	0
$415$	3.68466	0.180873
$416$	0	0
$417$	−26.7386	−1.30940
$418$	0	0
$419$	15.1771	0.741449	0.370724	−	0.928743i	$-0.379109\pi$
$419$	15.1771	0.741449	0.370724	+	0.928743i	$0.379109\pi$
$420$	0	0
$421$	−36.3002	−1.76916	−0.884581	−	0.466386i	$-0.845556\pi$
$421$	−36.3002	−1.76916	−0.884581	+	0.466386i	$0.845556\pi$
$422$	0	0
$423$	1.68466	0.0819109
$424$	0	0
$425$	−6.24621	−0.302986
$426$	0	0
$427$	−1.31534	−0.0636538
$428$	0	0
$429$	−0.684658	−0.0330556
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	−7.12311	−0.342315	−0.171157	−	0.985244i	$-0.554751\pi$
$433$	−7.12311	−0.342315	−0.171157	+	0.985244i	$0.554751\pi$
$434$	0	0
$435$	−5.75379	−0.275873
$436$	0	0
$437$	28.4233	1.35967
$438$	0	0
$439$	7.75379	0.370068	0.185034	−	0.982732i	$-0.440760\pi$
$439$	7.75379	0.370068	0.185034	+	0.982732i	$0.440760\pi$
$440$	0	0
$441$	−0.561553	−0.0267406
$442$	0	0
$443$	5.93087	0.281784	0.140892	−	0.990025i	$-0.455003\pi$
$443$	5.93087	0.281784	0.140892	+	0.990025i	$0.455003\pi$
$444$	0	0
$445$	32.1771	1.52534
$446$	0	0
$447$	22.9309	1.08459
$448$	0	0
$449$	36.1080	1.70404	0.852020	−	0.523510i	$-0.175378\pi$
$449$	36.1080	1.70404	0.852020	+	0.523510i	$0.175378\pi$
$450$	0	0
$451$	1.94602	0.0916347
$452$	0	0
$453$	34.7386	1.63216
$454$	0	0
$455$	−2.56155	−0.120087
$456$	0	0
$457$	31.1231	1.45588	0.727939	−	0.685642i	$-0.240479\pi$
$457$	31.1231	1.45588	0.727939	+	0.685642i	$0.240479\pi$
$458$	0	0
$459$	22.2462	1.03836
$460$	0	0
$461$	−1.61553	−0.0752426	−0.0376213	−	0.999292i	$-0.511978\pi$
$461$	−1.61553	−0.0752426	−0.0376213	+	0.999292i	$0.511978\pi$
$462$	0	0
$463$	−1.12311	−0.0521951	−0.0260976	−	0.999659i	$-0.508308\pi$
$463$	−1.12311	−0.0521951	−0.0260976	+	0.999659i	$0.508308\pi$
$464$	0	0
$465$	−8.49242	−0.393827
$466$	0	0
$467$	23.8617	1.10419	0.552095	−	0.833781i	$-0.313829\pi$
$467$	23.8617	1.10419	0.552095	+	0.833781i	$0.313829\pi$
$468$	0	0
$469$	−0.438447	−0.0202456
$470$	0	0
$471$	12.1922	0.561789
$472$	0	0
$473$	3.86174	0.177563
$474$	0	0
$475$	−8.87689	−0.407300
$476$	0	0
$477$	−2.06913	−0.0947390
$478$	0	0
$479$	−6.56155	−0.299805	−0.149903	−	0.988701i	$-0.547896\pi$
$479$	−6.56155	−0.299805	−0.149903	+	0.988701i	$0.547896\pi$
$480$	0	0
$481$	9.56155	0.435969
$482$	0	0
$483$	−7.80776	−0.355266
$484$	0	0
$485$	−36.8078	−1.67135
$486$	0	0
$487$	−16.7386	−0.758500	−0.379250	−	0.925294i	$-0.623818\pi$
$487$	−16.7386	−0.758500	−0.379250	+	0.925294i	$0.623818\pi$
$488$	0	0
$489$	−2.73863	−0.123845
$490$	0	0
$491$	34.2462	1.54551	0.772755	−	0.634705i	$-0.218878\pi$
$491$	34.2462	1.54551	0.772755	+	0.634705i	$0.218878\pi$
$492$	0	0
$493$	−5.75379	−0.259138
$494$	0	0
$495$	−0.630683	−0.0283471
$496$	0	0
$497$	−14.2462	−0.639030
$498$	0	0
$499$	−34.9309	−1.56372	−0.781860	−	0.623454i	$-0.785729\pi$
$499$	−34.9309	−1.56372	−0.781860	+	0.623454i	$0.785729\pi$
$500$	0	0
$501$	−10.2462	−0.457767
$502$	0	0
$503$	−18.8769	−0.841679	−0.420840	−	0.907135i	$-0.638264\pi$
$503$	−18.8769	−0.841679	−0.420840	+	0.907135i	$0.638264\pi$
$504$	0	0
$505$	21.6155	0.961878
$506$	0	0
$507$	1.56155	0.0693510
$508$	0	0
$509$	−9.19224	−0.407439	−0.203719	−	0.979029i	$-0.565303\pi$
$509$	−9.19224	−0.407439	−0.203719	+	0.979029i	$0.565303\pi$
$510$	0	0
$511$	−5.24621	−0.232079
$512$	0	0
$513$	31.6155	1.39586
$514$	0	0
$515$	12.4924	0.550482
$516$	0	0
$517$	1.31534	0.0578487
$518$	0	0
$519$	−39.6155	−1.73893
$520$	0	0
$521$	−32.8769	−1.44036	−0.720181	−	0.693786i	$-0.755941\pi$
$521$	−32.8769	−1.44036	−0.720181	+	0.693786i	$0.755941\pi$
$522$	0	0
$523$	−29.8078	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$523$	−29.8078	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$524$	0	0
$525$	2.43845	0.106423
$526$	0	0
$527$	−8.49242	−0.369936
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	−1.75379	−0.0761079
$532$	0	0
$533$	−4.43845	−0.192250
$534$	0	0
$535$	10.2462	0.442982
$536$	0	0
$537$	38.3542	1.65510
$538$	0	0
$539$	−0.438447	−0.0188853
$540$	0	0
$541$	23.1231	0.994140	0.497070	−	0.867710i	$-0.334409\pi$
$541$	23.1231	0.994140	0.497070	+	0.867710i	$0.334409\pi$
$542$	0	0
$543$	21.5616	0.925295
$544$	0	0
$545$	−21.1231	−0.904814
$546$	0	0
$547$	−5.43845	−0.232531	−0.116266	−	0.993218i	$-0.537092\pi$
$547$	−5.43845	−0.232531	−0.116266	+	0.993218i	$0.537092\pi$
$548$	0	0
$549$	0.738634	0.0315241
$550$	0	0
$551$	−8.17708	−0.348355
$552$	0	0
$553$	−1.87689	−0.0798137
$554$	0	0
$555$	−38.2462	−1.62346
$556$	0	0
$557$	−0.684658	−0.0290099	−0.0145050	−	0.999895i	$-0.504617\pi$
$557$	−0.684658	−0.0290099	−0.0145050	+	0.999895i	$0.504617\pi$
$558$	0	0
$559$	−8.80776	−0.372529
$560$	0	0
$561$	2.73863	0.115625
$562$	0	0
$563$	−10.9309	−0.460681	−0.230341	−	0.973110i	$-0.573984\pi$
$563$	−10.9309	−0.460681	−0.230341	+	0.973110i	$0.573984\pi$
$564$	0	0
$565$	23.6847	0.996421
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	−22.8617	−0.958414	−0.479207	−	0.877702i	$-0.659076\pi$
$569$	−22.8617	−0.958414	−0.479207	+	0.877702i	$0.659076\pi$
$570$	0	0
$571$	30.8078	1.28926	0.644632	−	0.764493i	$-0.277010\pi$
$571$	30.8078	1.28926	0.644632	+	0.764493i	$0.277010\pi$
$572$	0	0
$573$	−32.0000	−1.33682
$574$	0	0
$575$	−7.80776	−0.325606
$576$	0	0
$577$	−42.9848	−1.78948	−0.894741	−	0.446585i	$-0.852640\pi$
$577$	−42.9848	−1.78948	−0.894741	+	0.446585i	$0.852640\pi$
$578$	0	0
$579$	26.7386	1.11122
$580$	0	0
$581$	−1.43845	−0.0596768
$582$	0	0
$583$	−1.61553	−0.0669083
$584$	0	0
$585$	1.43845	0.0594725
$586$	0	0
$587$	−37.5464	−1.54971	−0.774853	−	0.632142i	$-0.782176\pi$
$587$	−37.5464	−1.54971	−0.774853	+	0.632142i	$0.782176\pi$
$588$	0	0
$589$	−12.0691	−0.497300
$590$	0	0
$591$	−15.3153	−0.629989
$592$	0	0
$593$	−23.9309	−0.982723	−0.491362	−	0.870956i	$-0.663501\pi$
$593$	−23.9309	−0.982723	−0.491362	+	0.870956i	$0.663501\pi$
$594$	0	0
$595$	10.2462	0.420054
$596$	0	0
$597$	−3.50758	−0.143556
$598$	0	0
$599$	26.6155	1.08748	0.543740	−	0.839253i	$-0.317008\pi$
$599$	26.6155	1.08748	0.543740	+	0.839253i	$0.317008\pi$
$600$	0	0
$601$	−34.4924	−1.40698	−0.703488	−	0.710708i	$-0.748375\pi$
$601$	−34.4924	−1.40698	−0.703488	+	0.710708i	$0.748375\pi$
$602$	0	0
$603$	0.246211	0.0100265
$604$	0	0
$605$	27.6847	1.12554
$606$	0	0
$607$	26.9848	1.09528	0.547641	−	0.836714i	$-0.315526\pi$
$607$	26.9848	1.09528	0.547641	+	0.836714i	$0.315526\pi$
$608$	0	0
$609$	2.24621	0.0910211
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	22.4384	0.906280	0.453140	−	0.891439i	$-0.350304\pi$
$613$	22.4384	0.906280	0.453140	+	0.891439i	$0.350304\pi$
$614$	0	0
$615$	17.7538	0.715902
$616$	0	0
$617$	−41.2311	−1.65990	−0.829950	−	0.557838i	$-0.811631\pi$
$617$	−41.2311	−1.65990	−0.829950	+	0.557838i	$0.811631\pi$
$618$	0	0
$619$	−24.8769	−0.999887	−0.499943	−	0.866058i	$-0.666646\pi$
$619$	−24.8769	−0.999887	−0.499943	+	0.866058i	$0.666646\pi$
$620$	0	0
$621$	27.8078	1.11589
$622$	0	0
$623$	−12.5616	−0.503268
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	3.89205	0.155433
$628$	0	0
$629$	−38.2462	−1.52498
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	4.38447	0.174267
$634$	0	0
$635$	−8.49242	−0.337012
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−20.8617	−0.823989	−0.411995	−	0.911186i	$-0.635168\pi$
$641$	−20.8617	−0.823989	−0.411995	+	0.911186i	$0.635168\pi$
$642$	0	0
$643$	30.2462	1.19279	0.596397	−	0.802690i	$-0.296598\pi$
$643$	30.2462	1.19279	0.596397	+	0.802690i	$0.296598\pi$
$644$	0	0
$645$	35.2311	1.38722
$646$	0	0
$647$	15.3693	0.604230	0.302115	−	0.953271i	$-0.402307\pi$
$647$	15.3693	0.604230	0.302115	+	0.953271i	$0.402307\pi$
$648$	0	0
$649$	−1.36932	−0.0537504
$650$	0	0
$651$	3.31534	0.129938
$652$	0	0
$653$	−26.8769	−1.05177	−0.525887	−	0.850554i	$-0.676267\pi$
$653$	−26.8769	−1.05177	−0.525887	+	0.850554i	$0.676267\pi$
$654$	0	0
$655$	−46.7386	−1.82623
$656$	0	0
$657$	2.94602	0.114935
$658$	0	0
$659$	−39.7926	−1.55010	−0.775050	−	0.631900i	$-0.782275\pi$
$659$	−39.7926	−1.55010	−0.775050	+	0.631900i	$0.782275\pi$
$660$	0	0
$661$	−29.9309	−1.16418	−0.582088	−	0.813126i	$-0.697764\pi$
$661$	−29.9309	−1.16418	−0.582088	+	0.813126i	$0.697764\pi$
$662$	0	0
$663$	−6.24621	−0.242583
$664$	0	0
$665$	14.5616	0.564673
$666$	0	0
$667$	−7.19224	−0.278484
$668$	0	0
$669$	−2.54640	−0.0984494
$670$	0	0
$671$	0.576708	0.0222636
$672$	0	0
$673$	−28.8617	−1.11254	−0.556269	−	0.831002i	$-0.687768\pi$
$673$	−28.8617	−1.11254	−0.556269	+	0.831002i	$0.687768\pi$
$674$	0	0
$675$	−8.68466	−0.334273
$676$	0	0
$677$	−21.4233	−0.823364	−0.411682	−	0.911328i	$-0.635059\pi$
$677$	−21.4233	−0.823364	−0.411682	+	0.911328i	$0.635059\pi$
$678$	0	0
$679$	14.3693	0.551444
$680$	0	0
$681$	−25.7538	−0.986887
$682$	0	0
$683$	−10.0540	−0.384705	−0.192352	−	0.981326i	$-0.561612\pi$
$683$	−10.0540	−0.384705	−0.192352	+	0.981326i	$0.561612\pi$
$684$	0	0
$685$	−44.4924	−1.69997
$686$	0	0
$687$	23.2311	0.886320
$688$	0	0
$689$	3.68466	0.140374
$690$	0	0
$691$	−1.93087	−0.0734537	−0.0367269	−	0.999325i	$-0.511693\pi$
$691$	−1.93087	−0.0734537	−0.0367269	+	0.999325i	$0.511693\pi$
$692$	0	0
$693$	0.246211	0.00935279
$694$	0	0
$695$	43.8617	1.66377
$696$	0	0
$697$	17.7538	0.672473
$698$	0	0
$699$	−11.3153	−0.427986
$700$	0	0
$701$	−37.9309	−1.43263	−0.716315	−	0.697777i	$-0.754172\pi$
$701$	−37.9309	−1.43263	−0.716315	+	0.697777i	$0.754172\pi$
$702$	0	0
$703$	−54.3542	−2.05001
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	−8.43845	−0.317360
$708$	0	0
$709$	−39.1771	−1.47133	−0.735663	−	0.677348i	$-0.763129\pi$
$709$	−39.1771	−1.47133	−0.735663	+	0.677348i	$0.763129\pi$
$710$	0	0
$711$	1.05398	0.0395272
$712$	0	0
$713$	−10.6155	−0.397555
$714$	0	0
$715$	1.12311	0.0420018
$716$	0	0
$717$	20.8769	0.779662
$718$	0	0
$719$	−15.7538	−0.587517	−0.293759	−	0.955880i	$-0.594906\pi$
$719$	−15.7538	−0.587517	−0.293759	+	0.955880i	$0.594906\pi$
$720$	0	0
$721$	−4.87689	−0.181625
$722$	0	0
$723$	31.1231	1.15748
$724$	0	0
$725$	2.24621	0.0834222
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	35.2311	1.30307
$732$	0	0
$733$	−49.9309	−1.84424	−0.922119	−	0.386905i	$-0.873544\pi$
$733$	−49.9309	−1.84424	−0.922119	+	0.386905i	$0.873544\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	0.192236	0.00708110
$738$	0	0
$739$	0.630683	0.0232001	0.0116000	−	0.999933i	$-0.496308\pi$
$739$	0.630683	0.0232001	0.0116000	+	0.999933i	$0.496308\pi$
$740$	0	0
$741$	−8.87689	−0.326101
$742$	0	0
$743$	42.7386	1.56793	0.783964	−	0.620806i	$-0.213195\pi$
$743$	42.7386	1.56793	0.783964	+	0.620806i	$0.213195\pi$
$744$	0	0
$745$	−37.6155	−1.37813
$746$	0	0
$747$	0.807764	0.0295545
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	0	0
$753$	−21.5616	−0.785747
$754$	0	0
$755$	−56.9848	−2.07389
$756$	0	0
$757$	40.4233	1.46921	0.734605	−	0.678495i	$-0.237368\pi$
$757$	40.4233	1.46921	0.734605	+	0.678495i	$0.237368\pi$
$758$	0	0
$759$	3.42329	0.124258
$760$	0	0
$761$	−44.6155	−1.61731	−0.808656	−	0.588282i	$-0.799804\pi$
$761$	−44.6155	−1.61731	−0.808656	+	0.588282i	$0.799804\pi$
$762$	0	0
$763$	8.24621	0.298533
$764$	0	0
$765$	−5.75379	−0.208029
$766$	0	0
$767$	3.12311	0.112769
$768$	0	0
$769$	−9.38447	−0.338413	−0.169206	−	0.985581i	$-0.554120\pi$
$769$	−9.38447	−0.338413	−0.169206	+	0.985581i	$0.554120\pi$
$770$	0	0
$771$	17.7538	0.639387
$772$	0	0
$773$	−8.73863	−0.314307	−0.157153	−	0.987574i	$-0.550232\pi$
$773$	−8.73863	−0.314307	−0.157153	+	0.987574i	$0.550232\pi$
$774$	0	0
$775$	3.31534	0.119091
$776$	0	0
$777$	14.9309	0.535642
$778$	0	0
$779$	25.2311	0.903996
$780$	0	0
$781$	6.24621	0.223507
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	7.19224	0.256376	0.128188	−	0.991750i	$-0.459084\pi$
$787$	7.19224	0.256376	0.128188	+	0.991750i	$0.459084\pi$
$788$	0	0
$789$	−9.26137	−0.329713
$790$	0	0
$791$	−9.24621	−0.328757
$792$	0	0
$793$	−1.31534	−0.0467091
$794$	0	0
$795$	−14.7386	−0.522725
$796$	0	0
$797$	−12.4384	−0.440592	−0.220296	−	0.975433i	$-0.570702\pi$
$797$	−12.4384	−0.440592	−0.220296	+	0.975433i	$0.570702\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	7.05398	0.249240
$802$	0	0
$803$	2.30019	0.0811718
$804$	0	0
$805$	12.8078	0.451414
$806$	0	0
$807$	43.4233	1.52857
$808$	0	0
$809$	26.6695	0.937650	0.468825	−	0.883291i	$-0.344678\pi$
$809$	26.6695	0.937650	0.468825	+	0.883291i	$0.344678\pi$
$810$	0	0
$811$	−16.4924	−0.579127	−0.289564	−	0.957159i	$-0.593510\pi$
$811$	−16.4924	−0.579127	−0.289564	+	0.957159i	$0.593510\pi$
$812$	0	0
$813$	−51.0388	−1.79001
$814$	0	0
$815$	4.49242	0.157363
$816$	0	0
$817$	50.0691	1.75170
$818$	0	0
$819$	−0.561553	−0.0196222
$820$	0	0
$821$	34.4924	1.20379	0.601897	−	0.798574i	$-0.294412\pi$
$821$	34.4924	1.20379	0.601897	+	0.798574i	$0.294412\pi$
$822$	0	0
$823$	−5.94602	−0.207265	−0.103633	−	0.994616i	$-0.533047\pi$
$823$	−5.94602	−0.207265	−0.103633	+	0.994616i	$0.533047\pi$
$824$	0	0
$825$	−1.06913	−0.0372223
$826$	0	0
$827$	−2.73863	−0.0952316	−0.0476158	−	0.998866i	$-0.515162\pi$
$827$	−2.73863	−0.0952316	−0.0476158	+	0.998866i	$0.515162\pi$
$828$	0	0
$829$	−28.2462	−0.981031	−0.490516	−	0.871432i	$-0.663192\pi$
$829$	−28.2462	−0.981031	−0.490516	+	0.871432i	$0.663192\pi$
$830$	0	0
$831$	1.86174	0.0645830
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	16.8078	0.581657
$836$	0	0
$837$	−11.8078	−0.408136
$838$	0	0
$839$	−29.1771	−1.00730	−0.503652	−	0.863906i	$-0.668011\pi$
$839$	−29.1771	−1.00730	−0.503652	+	0.863906i	$0.668011\pi$
$840$	0	0
$841$	−26.9309	−0.928651
$842$	0	0
$843$	−14.2462	−0.490666
$844$	0	0
$845$	−2.56155	−0.0881201
$846$	0	0
$847$	−10.8078	−0.371359
$848$	0	0
$849$	14.9309	0.512426
$850$	0	0
$851$	−47.8078	−1.63883
$852$	0	0
$853$	33.9309	1.16177	0.580885	−	0.813985i	$-0.302706\pi$
$853$	33.9309	1.16177	0.580885	+	0.813985i	$0.302706\pi$
$854$	0	0
$855$	−8.17708	−0.279650
$856$	0	0
$857$	15.7538	0.538139	0.269070	−	0.963121i	$-0.413284\pi$
$857$	15.7538	0.538139	0.269070	+	0.963121i	$0.413284\pi$
$858$	0	0
$859$	−47.6695	−1.62646	−0.813231	−	0.581941i	$-0.802294\pi$
$859$	−47.6695	−1.62646	−0.813231	+	0.581941i	$0.802294\pi$
$860$	0	0
$861$	−6.93087	−0.236203
$862$	0	0
$863$	2.24621	0.0764619	0.0382310	−	0.999269i	$-0.487828\pi$
$863$	2.24621	0.0764619	0.0382310	+	0.999269i	$0.487828\pi$
$864$	0	0
$865$	64.9848	2.20955
$866$	0	0
$867$	−1.56155	−0.0530331
$868$	0	0
$869$	0.822919	0.0279156
$870$	0	0
$871$	−0.438447	−0.0148562
$872$	0	0
$873$	−8.06913	−0.273099
$874$	0	0
$875$	8.80776	0.297757
$876$	0	0
$877$	25.1771	0.850170	0.425085	−	0.905154i	$-0.360244\pi$
$877$	25.1771	0.850170	0.425085	+	0.905154i	$0.360244\pi$
$878$	0	0
$879$	−8.87689	−0.299410
$880$	0	0
$881$	42.9848	1.44820	0.724098	−	0.689697i	$-0.242256\pi$
$881$	42.9848	1.44820	0.724098	+	0.689697i	$0.242256\pi$
$882$	0	0
$883$	−16.8769	−0.567953	−0.283976	−	0.958831i	$-0.591654\pi$
$883$	−16.8769	−0.567953	−0.283976	+	0.958831i	$0.591654\pi$
$884$	0	0
$885$	−12.4924	−0.419928
$886$	0	0
$887$	−37.3693	−1.25474	−0.627369	−	0.778722i	$-0.715868\pi$
$887$	−37.3693	−1.25474	−0.627369	+	0.778722i	$0.715868\pi$
$888$	0	0
$889$	3.31534	0.111193
$890$	0	0
$891$	3.06913	0.102820
$892$	0	0
$893$	17.0540	0.570690
$894$	0	0
$895$	−62.9157	−2.10304
$896$	0	0
$897$	−7.80776	−0.260694
$898$	0	0
$899$	3.05398	0.101856
$900$	0	0
$901$	−14.7386	−0.491015
$902$	0	0
$903$	−13.7538	−0.457697
$904$	0	0
$905$	−35.3693	−1.17572
$906$	0	0
$907$	41.5464	1.37953	0.689763	−	0.724035i	$-0.257715\pi$
$907$	41.5464	1.37953	0.689763	+	0.724035i	$0.257715\pi$
$908$	0	0
$909$	4.73863	0.157171
$910$	0	0
$911$	−36.1771	−1.19860	−0.599300	−	0.800524i	$-0.704554\pi$
$911$	−36.1771	−1.19860	−0.599300	+	0.800524i	$0.704554\pi$
$912$	0	0
$913$	0.630683	0.0208726
$914$	0	0
$915$	5.26137	0.173935
$916$	0	0
$917$	18.2462	0.602543
$918$	0	0
$919$	55.9157	1.84449	0.922245	−	0.386607i	$-0.126353\pi$
$919$	55.9157	1.84449	0.922245	+	0.386607i	$0.126353\pi$
$920$	0	0
$921$	−19.0152	−0.626571
$922$	0	0
$923$	−14.2462	−0.468920
$924$	0	0
$925$	14.9309	0.490924
$926$	0	0
$927$	2.73863	0.0899485
$928$	0	0
$929$	33.3542	1.09431	0.547157	−	0.837030i	$-0.315710\pi$
$929$	33.3542	1.09431	0.547157	+	0.837030i	$0.315710\pi$
$930$	0	0
$931$	−5.68466	−0.186307
$932$	0	0
$933$	−19.1231	−0.626062
$934$	0	0
$935$	−4.49242	−0.146918
$936$	0	0
$937$	38.7386	1.26554	0.632768	−	0.774341i	$-0.281919\pi$
$937$	38.7386	1.26554	0.632768	+	0.774341i	$0.281919\pi$
$938$	0	0
$939$	−35.1231	−1.14620
$940$	0	0
$941$	26.6695	0.869401	0.434700	−	0.900575i	$-0.356854\pi$
$941$	26.6695	0.869401	0.434700	+	0.900575i	$0.356854\pi$
$942$	0	0
$943$	22.1922	0.722679
$944$	0	0
$945$	14.2462	0.463429
$946$	0	0
$947$	5.12311	0.166479	0.0832393	−	0.996530i	$-0.473473\pi$
$947$	5.12311	0.166479	0.0832393	+	0.996530i	$0.473473\pi$
$948$	0	0
$949$	−5.24621	−0.170299
$950$	0	0
$951$	40.6847	1.31929
$952$	0	0
$953$	49.6847	1.60944	0.804722	−	0.593652i	$-0.202314\pi$
$953$	49.6847	1.60944	0.804722	+	0.593652i	$0.202314\pi$
$954$	0	0
$955$	52.4924	1.69861
$956$	0	0
$957$	−0.984845	−0.0318355
$958$	0	0
$959$	17.3693	0.560884
$960$	0	0
$961$	−26.4924	−0.854594
$962$	0	0
$963$	2.24621	0.0723831
$964$	0	0
$965$	−43.8617	−1.41196
$966$	0	0
$967$	56.2462	1.80876	0.904378	−	0.426732i	$-0.140335\pi$
$967$	56.2462	1.80876	0.904378	+	0.426732i	$0.140335\pi$
$968$	0	0
$969$	35.5076	1.14067
$970$	0	0
$971$	17.8078	0.571478	0.285739	−	0.958307i	$-0.407761\pi$
$971$	17.8078	0.571478	0.285739	+	0.958307i	$0.407761\pi$
$972$	0	0
$973$	−17.1231	−0.548942
$974$	0	0
$975$	2.43845	0.0780928
$976$	0	0
$977$	18.4924	0.591625	0.295813	−	0.955246i	$-0.404410\pi$
$977$	18.4924	0.591625	0.295813	+	0.955246i	$0.404410\pi$
$978$	0	0
$979$	5.50758	0.176023
$980$	0	0
$981$	−4.63068	−0.147846
$982$	0	0
$983$	−37.3002	−1.18969	−0.594846	−	0.803840i	$-0.702787\pi$
$983$	−37.3002	−1.18969	−0.594846	+	0.803840i	$0.702787\pi$
$984$	0	0
$985$	25.1231	0.800489
$986$	0	0
$987$	−4.68466	−0.149114
$988$	0	0
$989$	44.0388	1.40035
$990$	0	0
$991$	−50.5464	−1.60566	−0.802830	−	0.596209i	$-0.796673\pi$
$991$	−50.5464	−1.60566	−0.802830	+	0.596209i	$0.796673\pi$
$992$	0	0
$993$	6.33050	0.200892
$994$	0	0
$995$	5.75379	0.182407
$996$	0	0
$997$	5.56155	0.176136	0.0880681	−	0.996114i	$-0.471931\pi$
$997$	5.56155	0.176136	0.0880681	+	0.996114i	$0.471931\pi$
$998$	0	0
$999$	−53.1771	−1.68245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.o.1.2		2
4.3	odd	2		728.2.a.g.1.1	✓	2
8.3	odd	2		5824.2.a.bj.1.2		2
8.5	even	2		5824.2.a.bo.1.1		2
12.11	even	2		6552.2.a.bg.1.2		2
28.27	even	2		5096.2.a.n.1.2		2
52.51	odd	2		9464.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.g.1.1	✓	2	4.3	odd	2
1456.2.a.o.1.2		2	1.1	even	1	trivial
5096.2.a.n.1.2		2	28.27	even	2
5824.2.a.bj.1.2		2	8.3	odd	2
5824.2.a.bo.1.1		2	8.5	even	2
6552.2.a.bg.1.2		2	12.11	even	2
9464.2.a.r.1.1		2	52.51	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -2.56155 q^{5} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} -2.56155 q^{5} +1.00000 q^{7} -0.561553 q^{9} -0.438447 q^{11} +1.00000 q^{13} -4.00000 q^{15} -4.00000 q^{17} -5.68466 q^{19} +1.56155 q^{21} -5.00000 q^{23} +1.56155 q^{25} -5.56155 q^{27} +1.43845 q^{29} +2.12311 q^{31} -0.684658 q^{33} -2.56155 q^{35} +9.56155 q^{37} +1.56155 q^{39} -4.43845 q^{41} -8.80776 q^{43} +1.43845 q^{45} -3.00000 q^{47} +1.00000 q^{49} -6.24621 q^{51} +3.68466 q^{53} +1.12311 q^{55} -8.87689 q^{57} +3.12311 q^{59} -1.31534 q^{61} -0.561553 q^{63} -2.56155 q^{65} -0.438447 q^{67} -7.80776 q^{69} -14.2462 q^{71} -5.24621 q^{73} +2.43845 q^{75} -0.438447 q^{77} -1.87689 q^{79} -7.00000 q^{81} -1.43845 q^{83} +10.2462 q^{85} +2.24621 q^{87} -12.5616 q^{89} +1.00000 q^{91} +3.31534 q^{93} +14.5616 q^{95} +14.3693 q^{97} +0.246211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9	\( 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 8 q^{15} - 8 q^{17} + q^{19} - q^{21} - 10 q^{23} - q^{25} - 7 q^{27} + 7 q^{29} - 4 q^{31} + 11 q^{33} - q^{35} + 15 q^{37} - q^{39} - 13 q^{41} + 3 q^{43} + 7 q^{45} - 6 q^{47} + 2 q^{49} + 4 q^{51} - 5 q^{53} - 6 q^{55} - 26 q^{57} - 2 q^{59} - 15 q^{61} + 3 q^{63} - q^{65} - 5 q^{67} + 5 q^{69} - 12 q^{71} + 6 q^{73} + 9 q^{75} - 5 q^{77} - 12 q^{79} - 14 q^{81} - 7 q^{83} + 4 q^{85} - 12 q^{87} - 21 q^{89} + 2 q^{91} + 19 q^{93} + 25 q^{95} + 4 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 8 * q^15 - 8 * q^17 + q^19 - q^21 - 10 * q^23 - q^25 - 7 * q^27 + 7 * q^29 - 4 * q^31 + 11 * q^33 - q^35 + 15 * q^37 - q^39 - 13 * q^41 + 3 * q^43 + 7 * q^45 - 6 * q^47 + 2 * q^49 + 4 * q^51 - 5 * q^53 - 6 * q^55 - 26 * q^57 - 2 * q^59 - 15 * q^61 + 3 * q^63 - q^65 - 5 * q^67 + 5 * q^69 - 12 * q^71 + 6 * q^73 + 9 * q^75 - 5 * q^77 - 12 * q^79 - 14 * q^81 - 7 * q^83 + 4 * q^85 - 12 * q^87 - 21 * q^89 + 2 * q^91 + 19 * q^93 + 25 * q^95 + 4 * q^97 - 16 * q^99

Introduction

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