LMFDB - Embedded newform 5312.2.a.br.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5312,2,Mod(1,5312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5312 = 2^{6} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5312.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:	$42.4165335537$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 18x^{6} + 33x^{5} + 87x^{4} - 127x^{3} - 126x^{2} + 100x - 16 $ x^8 - 2x^7 - 18x^6 + 33x^5 + 87x^4 - 127x^3 - 126x^2 + 100*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.98233$ of defining polynomial
Character	$\chi$	$=$	5312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.98233 q^{3} +0.181960 q^{5} +2.34566 q^{7} +0.929627 q^{9} +O(q^{10})$	$q+1.98233 q^{3} +0.181960 q^{5} +2.34566 q^{7} +0.929627 q^{9} +4.25499 q^{11} -3.38918 q^{13} +0.360704 q^{15} +4.77248 q^{17} +6.08050 q^{19} +4.64988 q^{21} +7.42236 q^{23} -4.96689 q^{25} -4.10416 q^{27} -7.12665 q^{29} +4.10542 q^{31} +8.43478 q^{33} +0.426817 q^{35} -2.07037 q^{37} -6.71846 q^{39} +12.2090 q^{41} +3.35546 q^{43} +0.169155 q^{45} -0.219957 q^{47} -1.49786 q^{49} +9.46063 q^{51} -12.2651 q^{53} +0.774237 q^{55} +12.0536 q^{57} -10.3126 q^{59} -7.12665 q^{61} +2.18059 q^{63} -0.616694 q^{65} +10.7305 q^{67} +14.7136 q^{69} +7.60191 q^{71} +6.18461 q^{73} -9.84601 q^{75} +9.98077 q^{77} -1.88639 q^{79} -10.9247 q^{81} +1.00000 q^{83} +0.868400 q^{85} -14.1274 q^{87} -9.29359 q^{89} -7.94987 q^{91} +8.13830 q^{93} +1.10641 q^{95} -13.6575 q^{97} +3.95555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9	$ 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9} - 10 q^{11} - 7 q^{13} - 14 q^{15} + 15 q^{17} - 7 q^{19} + 5 q^{21} + 4 q^{23} + 27 q^{25} - 5 q^{27} - 16 q^{29} + 3 q^{31} + 21 q^{33} - 2 q^{35} - 8 q^{37} + 10 q^{39} + 24 q^{41} + q^{43} - 29 q^{45} - 2 q^{47} + 3 q^{49} + 9 q^{51} - 7 q^{53} - 30 q^{55} + 4 q^{57} + 2 q^{59} - 16 q^{61} - 10 q^{63} + 2 q^{65} + 25 q^{67} + 6 q^{69} + 8 q^{71} + 14 q^{73} + 30 q^{75} + 7 q^{77} - 4 q^{79} + 52 q^{81} + 8 q^{83} + 21 q^{85} - 4 q^{87} + 20 q^{89} + 45 q^{91} + 47 q^{93} + 8 q^{95} + 2 q^{97} - 10 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9 - 10 * q^11 - 7 * q^13 - 14 * q^15 + 15 * q^17 - 7 * q^19 + 5 * q^21 + 4 * q^23 + 27 * q^25 - 5 * q^27 - 16 * q^29 + 3 * q^31 + 21 * q^33 - 2 * q^35 - 8 * q^37 + 10 * q^39 + 24 * q^41 + q^43 - 29 * q^45 - 2 * q^47 + 3 * q^49 + 9 * q^51 - 7 * q^53 - 30 * q^55 + 4 * q^57 + 2 * q^59 - 16 * q^61 - 10 * q^63 + 2 * q^65 + 25 * q^67 + 6 * q^69 + 8 * q^71 + 14 * q^73 + 30 * q^75 + 7 * q^77 - 4 * q^79 + 52 * q^81 + 8 * q^83 + 21 * q^85 - 4 * q^87 + 20 * q^89 + 45 * q^91 + 47 * q^93 + 8 * q^95 + 2 * q^97 - 10 * 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
$3$	1.98233	1.14450	0.572249	−	0.820080i	$-0.306071\pi$
$3$	1.98233	1.14450	0.572249	+	0.820080i	$0.306071\pi$
$4$	0	0
$5$	0.181960	0.0813749	0.0406875	−	0.999172i	$-0.487045\pi$
$5$	0.181960	0.0813749	0.0406875	+	0.999172i	$0.487045\pi$
$6$	0	0
$7$	2.34566	0.886578	0.443289	−	0.896379i	$-0.353812\pi$
$7$	2.34566	0.886578	0.443289	+	0.896379i	$0.353812\pi$
$8$	0	0
$9$	0.929627	0.309876
$10$	0	0
$11$	4.25499	1.28293	0.641464	−	0.767153i	$-0.278328\pi$
$11$	4.25499	1.28293	0.641464	+	0.767153i	$0.278328\pi$
$12$	0	0
$13$	−3.38918	−0.939988	−0.469994	−	0.882670i	$-0.655744\pi$
$13$	−3.38918	−0.939988	−0.469994	+	0.882670i	$0.655744\pi$
$14$	0	0
$15$	0.360704	0.0931334
$16$	0	0
$17$	4.77248	1.15750	0.578748	−	0.815506i	$-0.303541\pi$
$17$	4.77248	1.15750	0.578748	+	0.815506i	$0.303541\pi$
$18$	0	0
$19$	6.08050	1.39496	0.697482	−	0.716603i	$-0.254304\pi$
$19$	6.08050	1.39496	0.697482	+	0.716603i	$0.254304\pi$
$20$	0	0
$21$	4.64988	1.01469
$22$	0	0
$23$	7.42236	1.54767	0.773834	−	0.633388i	$-0.218336\pi$
$23$	7.42236	1.54767	0.773834	+	0.633388i	$0.218336\pi$
$24$	0	0
$25$	−4.96689	−0.993378
$26$	0	0
$27$	−4.10416	−0.789846
$28$	0	0
$29$	−7.12665	−1.32339	−0.661693	−	0.749775i	$-0.730162\pi$
$29$	−7.12665	−1.32339	−0.661693	+	0.749775i	$0.730162\pi$
$30$	0	0
$31$	4.10542	0.737356	0.368678	−	0.929557i	$-0.379811\pi$
$31$	4.10542	0.737356	0.368678	+	0.929557i	$0.379811\pi$
$32$	0	0
$33$	8.43478	1.46831
$34$	0	0
$35$	0.426817	0.0721452
$36$	0	0
$37$	−2.07037	−0.340367	−0.170184	−	0.985412i	$-0.554436\pi$
$37$	−2.07037	−0.340367	−0.170184	+	0.985412i	$0.554436\pi$
$38$	0	0
$39$	−6.71846	−1.07581
$40$	0	0
$41$	12.2090	1.90672	0.953361	−	0.301832i	$-0.0975982\pi$
$41$	12.2090	1.90672	0.953361	+	0.301832i	$0.0975982\pi$
$42$	0	0
$43$	3.35546	0.511703	0.255851	−	0.966716i	$-0.417644\pi$
$43$	3.35546	0.511703	0.255851	+	0.966716i	$0.417644\pi$
$44$	0	0
$45$	0.169155	0.0252161
$46$	0	0
$47$	−0.219957	−0.0320841	−0.0160420	−	0.999871i	$-0.505107\pi$
$47$	−0.219957	−0.0320841	−0.0160420	+	0.999871i	$0.505107\pi$
$48$	0	0
$49$	−1.49786	−0.213980
$50$	0	0
$51$	9.46063	1.32475
$52$	0	0
$53$	−12.2651	−1.68474	−0.842372	−	0.538897i	$-0.818841\pi$
$53$	−12.2651	−1.68474	−0.842372	+	0.538897i	$0.818841\pi$
$54$	0	0
$55$	0.774237	0.104398
$56$	0	0
$57$	12.0536	1.59653
$58$	0	0
$59$	−10.3126	−1.34258	−0.671290	−	0.741195i	$-0.734260\pi$
$59$	−10.3126	−1.34258	−0.671290	+	0.741195i	$0.734260\pi$
$60$	0	0
$61$	−7.12665	−0.912474	−0.456237	−	0.889858i	$-0.650803\pi$
$61$	−7.12665	−0.912474	−0.456237	+	0.889858i	$0.650803\pi$
$62$	0	0
$63$	2.18059	0.274729
$64$	0	0
$65$	−0.616694	−0.0764915
$66$	0	0
$67$	10.7305	1.31094	0.655469	−	0.755222i	$-0.272471\pi$
$67$	10.7305	1.31094	0.655469	+	0.755222i	$0.272471\pi$
$68$	0	0
$69$	14.7136	1.77130
$70$	0	0
$71$	7.60191	0.902180	0.451090	−	0.892478i	$-0.351035\pi$
$71$	7.60191	0.902180	0.451090	+	0.892478i	$0.351035\pi$
$72$	0	0
$73$	6.18461	0.723854	0.361927	−	0.932206i	$-0.382119\pi$
$73$	6.18461	0.723854	0.361927	+	0.932206i	$0.382119\pi$
$74$	0	0
$75$	−9.84601	−1.13692
$76$	0	0
$77$	9.98077	1.13741
$78$	0	0
$79$	−1.88639	−0.212235	−0.106118	−	0.994354i	$-0.533842\pi$
$79$	−1.88639	−0.212235	−0.106118	+	0.994354i	$0.533842\pi$
$80$	0	0
$81$	−10.9247	−1.21385
$82$	0	0
$83$	1.00000	0.109764
$83$	1.00000	0.109764
$84$	0	0
$85$	0.868400	0.0941912
$86$	0	0
$87$	−14.1274	−1.51461
$88$	0	0
$89$	−9.29359	−0.985119	−0.492559	−	0.870279i	$-0.663939\pi$
$89$	−9.29359	−0.985119	−0.492559	+	0.870279i	$0.663939\pi$
$90$	0	0
$91$	−7.94987	−0.833373
$92$	0	0
$93$	8.13830	0.843902
$94$	0	0
$95$	1.10641	0.113515
$96$	0	0
$97$	−13.6575	−1.38671	−0.693355	−	0.720596i	$-0.743868\pi$
$97$	−13.6575	−1.38671	−0.693355	+	0.720596i	$0.743868\pi$
$98$	0	0
$99$	3.95555	0.397548
$100$	0	0
$101$	13.4343	1.33677	0.668383	−	0.743817i	$-0.266987\pi$
$101$	13.4343	1.33677	0.668383	+	0.743817i	$0.266987\pi$
$102$	0	0
$103$	−11.5050	−1.13362	−0.566812	−	0.823847i	$-0.691823\pi$
$103$	−11.5050	−1.13362	−0.566812	+	0.823847i	$0.691823\pi$
$104$	0	0
$105$	0.846091	0.0825700
$106$	0	0
$107$	5.57379	0.538839	0.269419	−	0.963023i	$-0.413168\pi$
$107$	5.57379	0.538839	0.269419	+	0.963023i	$0.413168\pi$
$108$	0	0
$109$	3.70449	0.354826	0.177413	−	0.984136i	$-0.443227\pi$
$109$	3.70449	0.354826	0.177413	+	0.984136i	$0.443227\pi$
$110$	0	0
$111$	−4.10416	−0.389550
$112$	0	0
$113$	13.8006	1.29825	0.649127	−	0.760680i	$-0.275135\pi$
$113$	13.8006	1.29825	0.649127	+	0.760680i	$0.275135\pi$
$114$	0	0
$115$	1.35057	0.125941
$116$	0	0
$117$	−3.15067	−0.291279
$118$	0	0
$119$	11.1946	1.02621
$120$	0	0
$121$	7.10492	0.645902
$122$	0	0
$123$	24.2022	2.18224
$124$	0	0
$125$	−1.81357	−0.162211
$126$	0	0
$127$	21.6879	1.92449	0.962247	−	0.272177i	$-0.0877437\pi$
$127$	21.6879	1.92449	0.962247	+	0.272177i	$0.0877437\pi$
$128$	0	0
$129$	6.65162	0.585643
$130$	0	0
$131$	2.90887	0.254149	0.127075	−	0.991893i	$-0.459441\pi$
$131$	2.90887	0.254149	0.127075	+	0.991893i	$0.459441\pi$
$132$	0	0
$133$	14.2628	1.23674
$134$	0	0
$135$	−0.746792	−0.0642737
$136$	0	0
$137$	−18.3436	−1.56720	−0.783599	−	0.621267i	$-0.786618\pi$
$137$	−18.3436	−1.56720	−0.783599	+	0.621267i	$0.786618\pi$
$138$	0	0
$139$	−2.03595	−0.172687	−0.0863435	−	0.996265i	$-0.527518\pi$
$139$	−2.03595	−0.172687	−0.0863435	+	0.996265i	$0.527518\pi$
$140$	0	0
$141$	−0.436028	−0.0367202
$142$	0	0
$143$	−14.4209	−1.20594
$144$	0	0
$145$	−1.29676	−0.107690
$146$	0	0
$147$	−2.96925	−0.244900
$148$	0	0
$149$	9.19865	0.753583	0.376791	−	0.926298i	$-0.377027\pi$
$149$	9.19865	0.753583	0.376791	+	0.926298i	$0.377027\pi$
$150$	0	0
$151$	23.0908	1.87911	0.939553	−	0.342403i	$-0.111241\pi$
$151$	23.0908	1.87911	0.939553	+	0.342403i	$0.111241\pi$
$152$	0	0
$153$	4.43663	0.358680
$154$	0	0
$155$	0.747022	0.0600023
$156$	0	0
$157$	−5.09120	−0.406322	−0.203161	−	0.979145i	$-0.565121\pi$
$157$	−5.09120	−0.406322	−0.203161	+	0.979145i	$0.565121\pi$
$158$	0	0
$159$	−24.3135	−1.92819
$160$	0	0
$161$	17.4104	1.37213
$162$	0	0
$163$	−19.9498	−1.56259	−0.781293	−	0.624164i	$-0.785440\pi$
$163$	−19.9498	−1.56259	−0.781293	+	0.624164i	$0.785440\pi$
$164$	0	0
$165$	1.53479	0.119483
$166$	0	0
$167$	−7.04284	−0.544991	−0.272496	−	0.962157i	$-0.587849\pi$
$167$	−7.04284	−0.544991	−0.272496	+	0.962157i	$0.587849\pi$
$168$	0	0
$169$	−1.51348	−0.116422
$170$	0	0
$171$	5.65260	0.432265
$172$	0	0
$173$	16.8873	1.28392	0.641960	−	0.766738i	$-0.278122\pi$
$173$	16.8873	1.28392	0.641960	+	0.766738i	$0.278122\pi$
$174$	0	0
$175$	−11.6507	−0.880707
$176$	0	0
$177$	−20.4429	−1.53658
$178$	0	0
$179$	3.23721	0.241960	0.120980	−	0.992655i	$-0.461396\pi$
$179$	3.23721	0.241960	0.120980	+	0.992655i	$0.461396\pi$
$180$	0	0
$181$	3.17876	0.236276	0.118138	−	0.992997i	$-0.462308\pi$
$181$	3.17876	0.236276	0.118138	+	0.992997i	$0.462308\pi$
$182$	0	0
$183$	−14.1274	−1.04432
$184$	0	0
$185$	−0.376725	−0.0276974
$186$	0	0
$187$	20.3068	1.48498
$188$	0	0
$189$	−9.62698	−0.700260
$190$	0	0
$191$	25.0909	1.81551	0.907757	−	0.419496i	$-0.137793\pi$
$191$	25.0909	1.81551	0.907757	+	0.419496i	$0.137793\pi$
$192$	0	0
$193$	9.55458	0.687754	0.343877	−	0.939015i	$-0.388260\pi$
$193$	9.55458	0.687754	0.343877	+	0.939015i	$0.388260\pi$
$194$	0	0
$195$	−1.22249	−0.0875443
$196$	0	0
$197$	8.55223	0.609321	0.304661	−	0.952461i	$-0.401457\pi$
$197$	8.55223	0.609321	0.304661	+	0.952461i	$0.401457\pi$
$198$	0	0
$199$	−15.0378	−1.06600	−0.533002	−	0.846114i	$-0.678936\pi$
$199$	−15.0378	−1.06600	−0.533002	+	0.846114i	$0.678936\pi$
$200$	0	0
$201$	21.2714	1.50037
$202$	0	0
$203$	−16.7167	−1.17328
$204$	0	0
$205$	2.22154	0.155159
$206$	0	0
$207$	6.90002	0.479585
$208$	0	0
$209$	25.8725	1.78964
$210$	0	0
$211$	−6.44067	−0.443394	−0.221697	−	0.975116i	$-0.571160\pi$
$211$	−6.44067	−0.443394	−0.221697	+	0.975116i	$0.571160\pi$
$212$	0	0
$213$	15.0695	1.03254
$214$	0	0
$215$	0.610559	0.0416398
$216$	0	0
$217$	9.62994	0.653723
$218$	0	0
$219$	12.2599	0.828450
$220$	0	0
$221$	−16.1748	−1.08803
$222$	0	0
$223$	−14.3433	−0.960496	−0.480248	−	0.877133i	$-0.659453\pi$
$223$	−14.3433	−0.960496	−0.480248	+	0.877133i	$0.659453\pi$
$224$	0	0
$225$	−4.61735	−0.307824
$226$	0	0
$227$	−1.94306	−0.128965	−0.0644826	−	0.997919i	$-0.520540\pi$
$227$	−1.94306	−0.128965	−0.0644826	+	0.997919i	$0.520540\pi$
$228$	0	0
$229$	−4.64229	−0.306771	−0.153386	−	0.988166i	$-0.549018\pi$
$229$	−4.64229	−0.306771	−0.153386	+	0.988166i	$0.549018\pi$
$230$	0	0
$231$	19.7852	1.30177
$232$	0	0
$233$	1.90633	0.124888	0.0624440	−	0.998048i	$-0.480111\pi$
$233$	1.90633	0.124888	0.0624440	+	0.998048i	$0.480111\pi$
$234$	0	0
$235$	−0.0400234	−0.00261084
$236$	0	0
$237$	−3.73944	−0.242903
$238$	0	0
$239$	14.5392	0.940460	0.470230	−	0.882544i	$-0.344171\pi$
$239$	14.5392	0.940460	0.470230	+	0.882544i	$0.344171\pi$
$240$	0	0
$241$	2.18211	0.140562	0.0702809	−	0.997527i	$-0.477610\pi$
$241$	2.18211	0.140562	0.0702809	+	0.997527i	$0.477610\pi$
$242$	0	0
$243$	−9.34381	−0.599406
$244$	0	0
$245$	−0.272550	−0.0174126
$246$	0	0
$247$	−20.6079	−1.31125
$248$	0	0
$249$	1.98233	0.125625
$250$	0	0
$251$	−18.1680	−1.14675	−0.573375	−	0.819293i	$-0.694366\pi$
$251$	−18.1680	−1.14675	−0.573375	+	0.819293i	$0.694366\pi$
$252$	0	0
$253$	31.5820	1.98555
$254$	0	0
$255$	1.72145	0.107802
$256$	0	0
$257$	−5.35026	−0.333740	−0.166870	−	0.985979i	$-0.553366\pi$
$257$	−5.35026	−0.333740	−0.166870	+	0.985979i	$0.553366\pi$
$258$	0	0
$259$	−4.85640	−0.301762
$260$	0	0
$261$	−6.62512	−0.410085
$262$	0	0
$263$	−3.27654	−0.202040	−0.101020	−	0.994884i	$-0.532211\pi$
$263$	−3.27654	−0.202040	−0.101020	+	0.994884i	$0.532211\pi$
$264$	0	0
$265$	−2.23176	−0.137096
$266$	0	0
$267$	−18.4230	−1.12747
$268$	0	0
$269$	−3.82404	−0.233156	−0.116578	−	0.993182i	$-0.537192\pi$
$269$	−3.82404	−0.233156	−0.116578	+	0.993182i	$0.537192\pi$
$270$	0	0
$271$	−5.88960	−0.357768	−0.178884	−	0.983870i	$-0.557249\pi$
$271$	−5.88960	−0.357768	−0.178884	+	0.983870i	$0.557249\pi$
$272$	0	0
$273$	−15.7593	−0.953793
$274$	0	0
$275$	−21.1341	−1.27443
$276$	0	0
$277$	−20.0501	−1.20469	−0.602346	−	0.798235i	$-0.705767\pi$
$277$	−20.0501	−1.20469	−0.602346	+	0.798235i	$0.705767\pi$
$278$	0	0
$279$	3.81651	0.228489
$280$	0	0
$281$	−30.4628	−1.81726	−0.908628	−	0.417606i	$-0.862869\pi$
$281$	−30.4628	−1.81726	−0.908628	+	0.417606i	$0.862869\pi$
$282$	0	0
$283$	24.3911	1.44990	0.724951	−	0.688800i	$-0.241862\pi$
$283$	24.3911	1.44990	0.724951	+	0.688800i	$0.241862\pi$
$284$	0	0
$285$	2.19326	0.129918
$286$	0	0
$287$	28.6382	1.69046
$288$	0	0
$289$	5.77657	0.339798
$290$	0	0
$291$	−27.0737	−1.58709
$292$	0	0
$293$	−0.300448	−0.0175524	−0.00877618	−	0.999961i	$-0.502794\pi$
$293$	−0.300448	−0.0175524	−0.00877618	+	0.999961i	$0.502794\pi$
$294$	0	0
$295$	−1.87647	−0.109252
$296$	0	0
$297$	−17.4632	−1.01331
$298$	0	0
$299$	−25.1557	−1.45479
$300$	0	0
$301$	7.87078	0.453664
$302$	0	0
$303$	26.6313	1.52993
$304$	0	0
$305$	−1.29676	−0.0742525
$306$	0	0
$307$	15.9338	0.909390	0.454695	−	0.890647i	$-0.349748\pi$
$307$	15.9338	0.909390	0.454695	+	0.890647i	$0.349748\pi$
$308$	0	0
$309$	−22.8067	−1.29743
$310$	0	0
$311$	15.7415	0.892618	0.446309	−	0.894879i	$-0.352738\pi$
$311$	15.7415	0.892618	0.446309	+	0.894879i	$0.352738\pi$
$312$	0	0
$313$	−23.2391	−1.31355	−0.656776	−	0.754086i	$-0.728080\pi$
$313$	−23.2391	−1.31355	−0.656776	+	0.754086i	$0.728080\pi$
$314$	0	0
$315$	0.396780	0.0223560
$316$	0	0
$317$	4.73037	0.265684	0.132842	−	0.991137i	$-0.457590\pi$
$317$	4.73037	0.265684	0.132842	+	0.991137i	$0.457590\pi$
$318$	0	0
$319$	−30.3238	−1.69781
$320$	0	0
$321$	11.0491	0.616700
$322$	0	0
$323$	29.0191	1.61467
$324$	0	0
$325$	16.8337	0.933764
$326$	0	0
$327$	7.34352	0.406098
$328$	0	0
$329$	−0.515946	−0.0284450
$330$	0	0
$331$	−11.7501	−0.645847	−0.322923	−	0.946425i	$-0.604666\pi$
$331$	−11.7501	−0.645847	−0.322923	+	0.946425i	$0.604666\pi$
$332$	0	0
$333$	−1.92467	−0.105471
$334$	0	0
$335$	1.95252	0.106678
$336$	0	0
$337$	24.1213	1.31397	0.656987	−	0.753902i	$-0.271831\pi$
$337$	24.1213	1.31397	0.656987	+	0.753902i	$0.271831\pi$
$338$	0	0
$339$	27.3574	1.48585
$340$	0	0
$341$	17.4685	0.945974
$342$	0	0
$343$	−19.9331	−1.07629
$344$	0	0
$345$	2.67728	0.144140
$346$	0	0
$347$	−12.1889	−0.654334	−0.327167	−	0.944967i	$-0.606094\pi$
$347$	−12.1889	−0.654334	−0.327167	+	0.944967i	$0.606094\pi$
$348$	0	0
$349$	−20.5051	−1.09761	−0.548806	−	0.835950i	$-0.684917\pi$
$349$	−20.5051	−1.09761	−0.548806	+	0.835950i	$0.684917\pi$
$350$	0	0
$351$	13.9097	0.742446
$352$	0	0
$353$	24.7330	1.31640	0.658201	−	0.752842i	$-0.271318\pi$
$353$	24.7330	1.31640	0.658201	+	0.752842i	$0.271318\pi$
$354$	0	0
$355$	1.38324	0.0734148
$356$	0	0
$357$	22.1914	1.17450
$358$	0	0
$359$	−35.6776	−1.88299	−0.941497	−	0.337021i	$-0.890581\pi$
$359$	−35.6776	−1.88299	−0.941497	+	0.337021i	$0.890581\pi$
$360$	0	0
$361$	17.9725	0.945923
$362$	0	0
$363$	14.0843	0.739234
$364$	0	0
$365$	1.12535	0.0589036
$366$	0	0
$367$	6.84904	0.357517	0.178758	−	0.983893i	$-0.442792\pi$
$367$	6.84904	0.357517	0.178758	+	0.983893i	$0.442792\pi$
$368$	0	0
$369$	11.3498	0.590847
$370$	0	0
$371$	−28.7699	−1.49366
$372$	0	0
$373$	19.8379	1.02717	0.513584	−	0.858039i	$-0.328317\pi$
$373$	19.8379	1.02717	0.513584	+	0.858039i	$0.328317\pi$
$374$	0	0
$375$	−3.59510	−0.185650
$376$	0	0
$377$	24.1535	1.24397
$378$	0	0
$379$	−5.27579	−0.270999	−0.135500	−	0.990777i	$-0.543264\pi$
$379$	−5.27579	−0.270999	−0.135500	+	0.990777i	$0.543264\pi$
$380$	0	0
$381$	42.9926	2.20258
$382$	0	0
$383$	18.3698	0.938651	0.469326	−	0.883025i	$-0.344497\pi$
$383$	18.3698	0.938651	0.469326	+	0.883025i	$0.344497\pi$
$384$	0	0
$385$	1.81610	0.0925570
$386$	0	0
$387$	3.11932	0.158564
$388$	0	0
$389$	6.35789	0.322358	0.161179	−	0.986925i	$-0.448470\pi$
$389$	6.35789	0.322358	0.161179	+	0.986925i	$0.448470\pi$
$390$	0	0
$391$	35.4231	1.79142
$392$	0	0
$393$	5.76634	0.290874
$394$	0	0
$395$	−0.343247	−0.0172706
$396$	0	0
$397$	5.14700	0.258321	0.129160	−	0.991624i	$-0.458772\pi$
$397$	5.14700	0.258321	0.129160	+	0.991624i	$0.458772\pi$
$398$	0	0
$399$	28.2736	1.41545
$400$	0	0
$401$	−4.12881	−0.206183	−0.103091	−	0.994672i	$-0.532873\pi$
$401$	−4.12881	−0.206183	−0.103091	+	0.994672i	$0.532873\pi$
$402$	0	0
$403$	−13.9140	−0.693106
$404$	0	0
$405$	−1.98785	−0.0987772
$406$	0	0
$407$	−8.80941	−0.436666
$408$	0	0
$409$	5.84153	0.288845	0.144423	−	0.989516i	$-0.453868\pi$
$409$	5.84153	0.288845	0.144423	+	0.989516i	$0.453868\pi$
$410$	0	0
$411$	−36.3630	−1.79365
$412$	0	0
$413$	−24.1898	−1.19030
$414$	0	0
$415$	0.181960	0.00893206
$416$	0	0
$417$	−4.03592	−0.197640
$418$	0	0
$419$	7.93548	0.387674	0.193837	−	0.981034i	$-0.437907\pi$
$419$	7.93548	0.387674	0.193837	+	0.981034i	$0.437907\pi$
$420$	0	0
$421$	2.74498	0.133782	0.0668910	−	0.997760i	$-0.478692\pi$
$421$	2.74498	0.133782	0.0668910	+	0.997760i	$0.478692\pi$
$422$	0	0
$423$	−0.204478	−0.00994208
$424$	0	0
$425$	−23.7044	−1.14983
$426$	0	0
$427$	−16.7167	−0.808979
$428$	0	0
$429$	−28.5870	−1.38019
$430$	0	0
$431$	−22.1573	−1.06728	−0.533640	−	0.845712i	$-0.679176\pi$
$431$	−22.1573	−1.06728	−0.533640	+	0.845712i	$0.679176\pi$
$432$	0	0
$433$	41.3392	1.98664	0.993318	−	0.115412i	$-0.0368187\pi$
$433$	41.3392	1.98664	0.993318	+	0.115412i	$0.0368187\pi$
$434$	0	0
$435$	−2.57061	−0.123251
$436$	0	0
$437$	45.1317	2.15894
$438$	0	0
$439$	−8.88344	−0.423984	−0.211992	−	0.977271i	$-0.567995\pi$
$439$	−8.88344	−0.423984	−0.211992	+	0.977271i	$0.567995\pi$
$440$	0	0
$441$	−1.39245	−0.0663072
$442$	0	0
$443$	−16.3325	−0.775981	−0.387990	−	0.921663i	$-0.626831\pi$
$443$	−16.3325	−0.775981	−0.387990	+	0.921663i	$0.626831\pi$
$444$	0	0
$445$	−1.69106	−0.0801640
$446$	0	0
$447$	18.2347	0.862474
$448$	0	0
$449$	−14.4604	−0.682427	−0.341214	−	0.939986i	$-0.610838\pi$
$449$	−14.4604	−0.682427	−0.341214	+	0.939986i	$0.610838\pi$
$450$	0	0
$451$	51.9491	2.44619
$452$	0	0
$453$	45.7736	2.15063
$454$	0	0
$455$	−1.44656	−0.0678156
$456$	0	0
$457$	4.97483	0.232713	0.116356	−	0.993208i	$-0.462879\pi$
$457$	4.97483	0.232713	0.116356	+	0.993208i	$0.462879\pi$
$458$	0	0
$459$	−19.5870	−0.914244
$460$	0	0
$461$	−12.5150	−0.582882	−0.291441	−	0.956589i	$-0.594135\pi$
$461$	−12.5150	−0.582882	−0.291441	+	0.956589i	$0.594135\pi$
$462$	0	0
$463$	−22.2650	−1.03474	−0.517372	−	0.855761i	$-0.673089\pi$
$463$	−22.2650	−1.03474	−0.517372	+	0.855761i	$0.673089\pi$
$464$	0	0
$465$	1.48084	0.0686725
$466$	0	0
$467$	−19.9821	−0.924659	−0.462330	−	0.886708i	$-0.652986\pi$
$467$	−19.9821	−0.924659	−0.462330	+	0.886708i	$0.652986\pi$
$468$	0	0
$469$	25.1701	1.16225
$470$	0	0
$471$	−10.0924	−0.465035
$472$	0	0
$473$	14.2774	0.656477
$474$	0	0
$475$	−30.2012	−1.38573
$476$	0	0
$477$	−11.4020	−0.522061
$478$	0	0
$479$	−13.8323	−0.632015	−0.316008	−	0.948757i	$-0.602343\pi$
$479$	−13.8323	−0.632015	−0.316008	+	0.948757i	$0.602343\pi$
$480$	0	0
$481$	7.01686	0.319941
$482$	0	0
$483$	34.5131	1.57040
$484$	0	0
$485$	−2.48512	−0.112843
$486$	0	0
$487$	0.868364	0.0393493	0.0196747	−	0.999806i	$-0.493737\pi$
$487$	0.868364	0.0393493	0.0196747	+	0.999806i	$0.493737\pi$
$488$	0	0
$489$	−39.5470	−1.78838
$490$	0	0
$491$	−39.3004	−1.77360	−0.886800	−	0.462153i	$-0.847077\pi$
$491$	−39.3004	−1.77360	−0.886800	+	0.462153i	$0.847077\pi$
$492$	0	0
$493$	−34.0118	−1.53181
$494$	0	0
$495$	0.719751	0.0323504
$496$	0	0
$497$	17.8315	0.799853
$498$	0	0
$499$	17.1370	0.767159	0.383580	−	0.923508i	$-0.374691\pi$
$499$	17.1370	0.767159	0.383580	+	0.923508i	$0.374691\pi$
$500$	0	0
$501$	−13.9612	−0.623741
$502$	0	0
$503$	−25.0734	−1.11797	−0.558984	−	0.829178i	$-0.688809\pi$
$503$	−25.0734	−1.11797	−0.558984	+	0.829178i	$0.688809\pi$
$504$	0	0
$505$	2.44451	0.108779
$506$	0	0
$507$	−3.00022	−0.133245
$508$	0	0
$509$	−0.0916195	−0.00406096	−0.00203048	−	0.999998i	$-0.500646\pi$
$509$	−0.0916195	−0.00406096	−0.00203048	+	0.999998i	$0.500646\pi$
$510$	0	0
$511$	14.5070	0.641753
$512$	0	0
$513$	−24.9554	−1.10181
$514$	0	0
$515$	−2.09345	−0.0922485
$516$	0	0
$517$	−0.935916	−0.0411616
$518$	0	0
$519$	33.4762	1.46944
$520$	0	0
$521$	−17.7050	−0.775670	−0.387835	−	0.921729i	$-0.626777\pi$
$521$	−17.7050	−0.775670	−0.387835	+	0.921729i	$0.626777\pi$
$522$	0	0
$523$	8.91945	0.390020	0.195010	−	0.980801i	$-0.437526\pi$
$523$	8.91945	0.390020	0.195010	+	0.980801i	$0.437526\pi$
$524$	0	0
$525$	−23.0954	−1.00797
$526$	0	0
$527$	19.5931	0.853487
$528$	0	0
$529$	32.0914	1.39528
$530$	0	0
$531$	−9.58683	−0.416033
$532$	0	0
$533$	−41.3784	−1.79230
$534$	0	0
$535$	1.01421	0.0438480
$536$	0	0
$537$	6.41721	0.276923
$538$	0	0
$539$	−6.37338	−0.274521
$540$	0	0
$541$	6.24818	0.268630	0.134315	−	0.990939i	$-0.457117\pi$
$541$	6.24818	0.268630	0.134315	+	0.990939i	$0.457117\pi$
$542$	0	0
$543$	6.30135	0.270417
$544$	0	0
$545$	0.674069	0.0288739
$546$	0	0
$547$	−15.6769	−0.670295	−0.335147	−	0.942166i	$-0.608786\pi$
$547$	−15.6769	−0.670295	−0.335147	+	0.942166i	$0.608786\pi$
$548$	0	0
$549$	−6.62512	−0.282753
$550$	0	0
$551$	−43.3336	−1.84607
$552$	0	0
$553$	−4.42483	−0.188163
$554$	0	0
$555$	−0.746792	−0.0316996
$556$	0	0
$557$	−38.6265	−1.63666	−0.818328	−	0.574751i	$-0.805099\pi$
$557$	−38.6265	−1.63666	−0.818328	+	0.574751i	$0.805099\pi$
$558$	0	0
$559$	−11.3722	−0.480995
$560$	0	0
$561$	40.2548	1.69956
$562$	0	0
$563$	17.9612	0.756976	0.378488	−	0.925606i	$-0.376444\pi$
$563$	17.9612	0.756976	0.378488	+	0.925606i	$0.376444\pi$
$564$	0	0
$565$	2.51116	0.105645
$566$	0	0
$567$	−25.6256	−1.07617
$568$	0	0
$569$	−22.2552	−0.932987	−0.466494	−	0.884525i	$-0.654483\pi$
$569$	−22.2552	−0.932987	−0.466494	+	0.884525i	$0.654483\pi$
$570$	0	0
$571$	39.3458	1.64657	0.823285	−	0.567629i	$-0.192139\pi$
$571$	39.3458	1.64657	0.823285	+	0.567629i	$0.192139\pi$
$572$	0	0
$573$	49.7384	2.07785
$574$	0	0
$575$	−36.8660	−1.53742
$576$	0	0
$577$	−34.3430	−1.42972	−0.714859	−	0.699269i	$-0.753509\pi$
$577$	−34.3430	−1.42972	−0.714859	+	0.699269i	$0.753509\pi$
$578$	0	0
$579$	18.9403	0.787133
$580$	0	0
$581$	2.34566	0.0973145
$582$	0	0
$583$	−52.1879	−2.16140
$584$	0	0
$585$	−0.573295	−0.0237028
$586$	0	0
$587$	20.5324	0.847462	0.423731	−	0.905788i	$-0.360720\pi$
$587$	20.5324	0.847462	0.423731	+	0.905788i	$0.360720\pi$
$588$	0	0
$589$	24.9630	1.02858
$590$	0	0
$591$	16.9533	0.697367
$592$	0	0
$593$	−30.1714	−1.23899	−0.619496	−	0.785000i	$-0.712663\pi$
$593$	−30.1714	−1.23899	−0.619496	+	0.785000i	$0.712663\pi$
$594$	0	0
$595$	2.03697	0.0835078
$596$	0	0
$597$	−29.8099	−1.22004
$598$	0	0
$599$	17.8930	0.731088	0.365544	−	0.930794i	$-0.380883\pi$
$599$	17.8930	0.731088	0.365544	+	0.930794i	$0.380883\pi$
$600$	0	0
$601$	8.75261	0.357026	0.178513	−	0.983938i	$-0.442871\pi$
$601$	8.75261	0.357026	0.178513	+	0.983938i	$0.442871\pi$
$602$	0	0
$603$	9.97535	0.406228
$604$	0	0
$605$	1.29281	0.0525602
$606$	0	0
$607$	−0.353655	−0.0143544	−0.00717720	−	0.999974i	$-0.502285\pi$
$607$	−0.353655	−0.0143544	−0.00717720	+	0.999974i	$0.502285\pi$
$608$	0	0
$609$	−33.1380	−1.34282
$610$	0	0
$611$	0.745475	0.0301587
$612$	0	0
$613$	31.7084	1.28069	0.640345	−	0.768087i	$-0.278791\pi$
$613$	31.7084	1.28069	0.640345	+	0.768087i	$0.278791\pi$
$614$	0	0
$615$	4.40383	0.177580
$616$	0	0
$617$	−36.9917	−1.48923	−0.744615	−	0.667494i	$-0.767367\pi$
$617$	−36.9917	−1.48923	−0.744615	+	0.667494i	$0.767367\pi$
$618$	0	0
$619$	29.0717	1.16849	0.584244	−	0.811578i	$-0.301391\pi$
$619$	29.0717	1.16849	0.584244	+	0.811578i	$0.301391\pi$
$620$	0	0
$621$	−30.4625	−1.22242
$622$	0	0
$623$	−21.7996	−0.873384
$624$	0	0
$625$	24.5045	0.980178
$626$	0	0
$627$	51.2877	2.04824
$628$	0	0
$629$	−9.88082	−0.393974
$630$	0	0
$631$	−11.4564	−0.456073	−0.228036	−	0.973653i	$-0.573230\pi$
$631$	−11.4564	−0.456073	−0.228036	+	0.973653i	$0.573230\pi$
$632$	0	0
$633$	−12.7675	−0.507463
$634$	0	0
$635$	3.94634	0.156606
$636$	0	0
$637$	5.07651	0.201139
$638$	0	0
$639$	7.06693	0.279564
$640$	0	0
$641$	−11.0192	−0.435234	−0.217617	−	0.976034i	$-0.569828\pi$
$641$	−11.0192	−0.435234	−0.217617	+	0.976034i	$0.569828\pi$
$642$	0	0
$643$	−42.9077	−1.69211	−0.846057	−	0.533093i	$-0.821030\pi$
$643$	−42.9077	−1.69211	−0.846057	+	0.533093i	$0.821030\pi$
$644$	0	0
$645$	1.21033	0.0476566
$646$	0	0
$647$	−1.60461	−0.0630839	−0.0315419	−	0.999502i	$-0.510042\pi$
$647$	−1.60461	−0.0630839	−0.0315419	+	0.999502i	$0.510042\pi$
$648$	0	0
$649$	−43.8798	−1.72243
$650$	0	0
$651$	19.0897	0.748185
$652$	0	0
$653$	−22.8290	−0.893369	−0.446685	−	0.894692i	$-0.647395\pi$
$653$	−22.8290	−0.893369	−0.446685	+	0.894692i	$0.647395\pi$
$654$	0	0
$655$	0.529298	0.0206814
$656$	0	0
$657$	5.74938	0.224305
$658$	0	0
$659$	−25.5572	−0.995567	−0.497784	−	0.867301i	$-0.665853\pi$
$659$	−25.5572	−0.995567	−0.497784	+	0.867301i	$0.665853\pi$
$660$	0	0
$661$	−24.2927	−0.944877	−0.472438	−	0.881364i	$-0.656626\pi$
$661$	−24.2927	−0.944877	−0.472438	+	0.881364i	$0.656626\pi$
$662$	0	0
$663$	−32.0637	−1.24525
$664$	0	0
$665$	2.59526	0.100640
$666$	0	0
$667$	−52.8965	−2.04816
$668$	0	0
$669$	−28.4331	−1.09929
$670$	0	0
$671$	−30.3238	−1.17064
$672$	0	0
$673$	−26.1202	−1.00686	−0.503430	−	0.864036i	$-0.667929\pi$
$673$	−26.1202	−1.00686	−0.503430	+	0.864036i	$0.667929\pi$
$674$	0	0
$675$	20.3849	0.784616
$676$	0	0
$677$	31.3551	1.20507	0.602537	−	0.798091i	$-0.294157\pi$
$677$	31.3551	1.20507	0.602537	+	0.798091i	$0.294157\pi$
$678$	0	0
$679$	−32.0359	−1.22943
$680$	0	0
$681$	−3.85178	−0.147600
$682$	0	0
$683$	14.8336	0.567593	0.283796	−	0.958885i	$-0.408406\pi$
$683$	14.8336	0.567593	0.283796	+	0.958885i	$0.408406\pi$
$684$	0	0
$685$	−3.33779	−0.127531
$686$	0	0
$687$	−9.20255	−0.351099
$688$	0	0
$689$	41.5687	1.58364
$690$	0	0
$691$	10.4716	0.398359	0.199179	−	0.979963i	$-0.436172\pi$
$691$	10.4716	0.398359	0.199179	+	0.979963i	$0.436172\pi$
$692$	0	0
$693$	9.27839	0.352457
$694$	0	0
$695$	−0.370461	−0.0140524
$696$	0	0
$697$	58.2671	2.20702
$698$	0	0
$699$	3.77898	0.142934
$700$	0	0
$701$	−29.5510	−1.11613	−0.558064	−	0.829798i	$-0.688455\pi$
$701$	−29.5510	−1.11613	−0.558064	+	0.829798i	$0.688455\pi$
$702$	0	0
$703$	−12.5889	−0.474800
$704$	0	0
$705$	−0.0793396	−0.00298810
$706$	0	0
$707$	31.5124	1.18515
$708$	0	0
$709$	−37.7757	−1.41870	−0.709348	−	0.704858i	$-0.751011\pi$
$709$	−37.7757	−1.41870	−0.709348	+	0.704858i	$0.751011\pi$
$710$	0	0
$711$	−1.75363	−0.0657664
$712$	0	0
$713$	30.4719	1.14118
$714$	0	0
$715$	−2.62403	−0.0981330
$716$	0	0
$717$	28.8214	1.07635
$718$	0	0
$719$	46.3761	1.72954	0.864768	−	0.502172i	$-0.167465\pi$
$719$	46.3761	1.72954	0.864768	+	0.502172i	$0.167465\pi$
$720$	0	0
$721$	−26.9869	−1.00505
$722$	0	0
$723$	4.32565	0.160873
$724$	0	0
$725$	35.3973	1.31462
$726$	0	0
$727$	−2.84018	−0.105336	−0.0526682	−	0.998612i	$-0.516773\pi$
$727$	−2.84018	−0.105336	−0.0526682	+	0.998612i	$0.516773\pi$
$728$	0	0
$729$	14.2515	0.527834
$730$	0	0
$731$	16.0139	0.592294
$732$	0	0
$733$	20.6265	0.761859	0.380929	−	0.924604i	$-0.375604\pi$
$733$	20.6265	0.761859	0.380929	+	0.924604i	$0.375604\pi$
$734$	0	0
$735$	−0.540284	−0.0199287
$736$	0	0
$737$	45.6581	1.68184
$738$	0	0
$739$	33.1934	1.22104	0.610520	−	0.792001i	$-0.290961\pi$
$739$	33.1934	1.22104	0.610520	+	0.792001i	$0.290961\pi$
$740$	0	0
$741$	−40.8516	−1.50072
$742$	0	0
$743$	−28.2253	−1.03549	−0.517743	−	0.855536i	$-0.673228\pi$
$743$	−28.2253	−1.03549	−0.517743	+	0.855536i	$0.673228\pi$
$744$	0	0
$745$	1.67378	0.0613227
$746$	0	0
$747$	0.929627	0.0340133
$748$	0	0
$749$	13.0742	0.477722
$750$	0	0
$751$	28.0631	1.02404	0.512019	−	0.858974i	$-0.328898\pi$
$751$	28.0631	1.02404	0.512019	+	0.858974i	$0.328898\pi$
$752$	0	0
$753$	−36.0148	−1.31245
$754$	0	0
$755$	4.20161	0.152912
$756$	0	0
$757$	−54.0719	−1.96528	−0.982638	−	0.185532i	$-0.940599\pi$
$757$	−54.0719	−1.96528	−0.982638	+	0.185532i	$0.940599\pi$
$758$	0	0
$759$	62.6060	2.27245
$760$	0	0
$761$	−33.3208	−1.20788	−0.603938	−	0.797031i	$-0.706403\pi$
$761$	−33.3208	−1.20788	−0.603938	+	0.797031i	$0.706403\pi$
$762$	0	0
$763$	8.68949	0.314581
$764$	0	0
$765$	0.807288	0.0291875
$766$	0	0
$767$	34.9511	1.26201
$768$	0	0
$769$	37.0283	1.33527	0.667637	−	0.744487i	$-0.267306\pi$
$769$	37.0283	1.33527	0.667637	+	0.744487i	$0.267306\pi$
$770$	0	0
$771$	−10.6060	−0.381965
$772$	0	0
$773$	−30.9032	−1.11151	−0.555756	−	0.831345i	$-0.687571\pi$
$773$	−30.9032	−1.11151	−0.555756	+	0.831345i	$0.687571\pi$
$774$	0	0
$775$	−20.3912	−0.732473
$776$	0	0
$777$	−9.62698	−0.345366
$778$	0	0
$779$	74.2367	2.65981
$780$	0	0
$781$	32.3460	1.15743
$782$	0	0
$783$	29.2489	1.04527
$784$	0	0
$785$	−0.926394	−0.0330644
$786$	0	0
$787$	36.0669	1.28565	0.642823	−	0.766015i	$-0.277763\pi$
$787$	36.0669	1.28565	0.642823	+	0.766015i	$0.277763\pi$
$788$	0	0
$789$	−6.49519	−0.231235
$790$	0	0
$791$	32.3716	1.15100
$792$	0	0
$793$	24.1535	0.857715
$794$	0	0
$795$	−4.42408	−0.156906
$796$	0	0
$797$	32.0697	1.13597	0.567984	−	0.823040i	$-0.307724\pi$
$797$	32.0697	1.13597	0.567984	+	0.823040i	$0.307724\pi$
$798$	0	0
$799$	−1.04974	−0.0371372
$800$	0	0
$801$	−8.63957	−0.305264
$802$	0	0
$803$	26.3155	0.928652
$804$	0	0
$805$	3.16799	0.111657
$806$	0	0
$807$	−7.58050	−0.266846
$808$	0	0
$809$	20.9796	0.737602	0.368801	−	0.929508i	$-0.379768\pi$
$809$	20.9796	0.737602	0.368801	+	0.929508i	$0.379768\pi$
$810$	0	0
$811$	28.7107	1.00817	0.504085	−	0.863654i	$-0.331830\pi$
$811$	28.7107	1.00817	0.504085	+	0.863654i	$0.331830\pi$
$812$	0	0
$813$	−11.6751	−0.409465
$814$	0	0
$815$	−3.63006	−0.127155
$816$	0	0
$817$	20.4029	0.713807
$818$	0	0
$819$	−7.39041	−0.258242
$820$	0	0
$821$	−48.7879	−1.70271	−0.851355	−	0.524590i	$-0.824219\pi$
$821$	−48.7879	−1.70271	−0.851355	+	0.524590i	$0.824219\pi$
$822$	0	0
$823$	38.6346	1.34672	0.673358	−	0.739316i	$-0.264851\pi$
$823$	38.6346	1.34672	0.673358	+	0.739316i	$0.264851\pi$
$824$	0	0
$825$	−41.8947	−1.45858
$826$	0	0
$827$	−7.34970	−0.255574	−0.127787	−	0.991802i	$-0.540787\pi$
$827$	−7.34970	−0.255574	−0.127787	+	0.991802i	$0.540787\pi$
$828$	0	0
$829$	51.2594	1.78031	0.890155	−	0.455657i	$-0.150596\pi$
$829$	51.2594	1.78031	0.890155	+	0.455657i	$0.150596\pi$
$830$	0	0
$831$	−39.7458	−1.37877
$832$	0	0
$833$	−7.14851	−0.247681
$834$	0	0
$835$	−1.28151	−0.0443486
$836$	0	0
$837$	−16.8493	−0.582398
$838$	0	0
$839$	10.4577	0.361040	0.180520	−	0.983571i	$-0.442222\pi$
$839$	10.4577	0.361040	0.180520	+	0.983571i	$0.442222\pi$
$840$	0	0
$841$	21.7891	0.751349
$842$	0	0
$843$	−60.3872	−2.07985
$844$	0	0
$845$	−0.275393	−0.00947382
$846$	0	0
$847$	16.6658	0.572642
$848$	0	0
$849$	48.3513	1.65941
$850$	0	0
$851$	−15.3671	−0.526776
$852$	0	0
$853$	3.97184	0.135993	0.0679965	−	0.997686i	$-0.478339\pi$
$853$	3.97184	0.135993	0.0679965	+	0.997686i	$0.478339\pi$
$854$	0	0
$855$	1.02855	0.0351755
$856$	0	0
$857$	−42.7299	−1.45963	−0.729813	−	0.683646i	$-0.760393\pi$
$857$	−42.7299	−1.45963	−0.729813	+	0.683646i	$0.760393\pi$
$858$	0	0
$859$	−24.3538	−0.830941	−0.415470	−	0.909607i	$-0.636383\pi$
$859$	−24.3538	−0.830941	−0.415470	+	0.909607i	$0.636383\pi$
$860$	0	0
$861$	56.7702	1.93472
$862$	0	0
$863$	21.1082	0.718533	0.359267	−	0.933235i	$-0.383027\pi$
$863$	21.1082	0.718533	0.359267	+	0.933235i	$0.383027\pi$
$864$	0	0
$865$	3.07281	0.104479
$866$	0	0
$867$	11.4511	0.388899
$868$	0	0
$869$	−8.02655	−0.272282
$870$	0	0
$871$	−36.3675	−1.23227
$872$	0	0
$873$	−12.6964	−0.429708
$874$	0	0
$875$	−4.25404	−0.143813
$876$	0	0
$877$	−50.6080	−1.70891	−0.854455	−	0.519525i	$-0.826109\pi$
$877$	−50.6080	−1.70891	−0.854455	+	0.519525i	$0.826109\pi$
$878$	0	0
$879$	−0.595587	−0.0200886
$880$	0	0
$881$	−47.3988	−1.59691	−0.798454	−	0.602056i	$-0.794348\pi$
$881$	−47.3988	−1.59691	−0.798454	+	0.602056i	$0.794348\pi$
$882$	0	0
$883$	28.1227	0.946405	0.473202	−	0.880954i	$-0.343098\pi$
$883$	28.1227	0.946405	0.473202	+	0.880954i	$0.343098\pi$
$884$	0	0
$885$	−3.71978	−0.125039
$886$	0	0
$887$	2.23874	0.0751694	0.0375847	−	0.999293i	$-0.488034\pi$
$887$	2.23874	0.0751694	0.0375847	+	0.999293i	$0.488034\pi$
$888$	0	0
$889$	50.8726	1.70621
$890$	0	0
$891$	−46.4844	−1.55728
$892$	0	0
$893$	−1.33745	−0.0447561
$894$	0	0
$895$	0.589042	0.0196895
$896$	0	0
$897$	−49.8668	−1.66500
$898$	0	0
$899$	−29.2579	−0.975806
$900$	0	0
$901$	−58.5350	−1.95009
$902$	0	0
$903$	15.6025	0.519218
$904$	0	0
$905$	0.578407	0.0192269
$906$	0	0
$907$	25.4445	0.844872	0.422436	−	0.906393i	$-0.361175\pi$
$907$	25.4445	0.844872	0.422436	+	0.906393i	$0.361175\pi$
$908$	0	0
$909$	12.4889	0.414231
$910$	0	0
$911$	37.8779	1.25495	0.627475	−	0.778637i	$-0.284088\pi$
$911$	37.8779	1.25495	0.627475	+	0.778637i	$0.284088\pi$
$912$	0	0
$913$	4.25499	0.140820
$914$	0	0
$915$	−2.57061	−0.0849818
$916$	0	0
$917$	6.82324	0.225323
$918$	0	0
$919$	11.0390	0.364143	0.182071	−	0.983285i	$-0.441720\pi$
$919$	11.0390	0.364143	0.182071	+	0.983285i	$0.441720\pi$
$920$	0	0
$921$	31.5860	1.04079
$922$	0	0
$923$	−25.7642	−0.848039
$924$	0	0
$925$	10.2833	0.338113
$926$	0	0
$927$	−10.6954	−0.351282
$928$	0	0
$929$	22.8337	0.749151	0.374575	−	0.927196i	$-0.377788\pi$
$929$	22.8337	0.749151	0.374575	+	0.927196i	$0.377788\pi$
$930$	0	0
$931$	−9.10774	−0.298494
$932$	0	0
$933$	31.2048	1.02160
$934$	0	0
$935$	3.69503	0.120840
$936$	0	0
$937$	−8.19679	−0.267777	−0.133889	−	0.990996i	$-0.542746\pi$
$937$	−8.19679	−0.267777	−0.133889	+	0.990996i	$0.542746\pi$
$938$	0	0
$939$	−46.0676	−1.50336
$940$	0	0
$941$	6.09849	0.198805	0.0994025	−	0.995047i	$-0.468307\pi$
$941$	6.09849	0.198805	0.0994025	+	0.995047i	$0.468307\pi$
$942$	0	0
$943$	90.6194	2.95097
$944$	0	0
$945$	−1.75172	−0.0569836
$946$	0	0
$947$	−34.8700	−1.13312	−0.566562	−	0.824019i	$-0.691727\pi$
$947$	−34.8700	−1.13312	−0.566562	+	0.824019i	$0.691727\pi$
$948$	0	0
$949$	−20.9607	−0.680415
$950$	0	0
$951$	9.37716	0.304075
$952$	0	0
$953$	−9.69210	−0.313958	−0.156979	−	0.987602i	$-0.550175\pi$
$953$	−9.69210	−0.313958	−0.156979	+	0.987602i	$0.550175\pi$
$954$	0	0
$955$	4.56554	0.147737
$956$	0	0
$957$	−60.1117	−1.94314
$958$	0	0
$959$	−43.0279	−1.38944
$960$	0	0
$961$	−14.1455	−0.456306
$962$	0	0
$963$	5.18154	0.166973
$964$	0	0
$965$	1.73855	0.0559659
$966$	0	0
$967$	5.91204	0.190118	0.0950592	−	0.995472i	$-0.469696\pi$
$967$	5.91204	0.190118	0.0950592	+	0.995472i	$0.469696\pi$
$968$	0	0
$969$	57.5254	1.84798
$970$	0	0
$971$	8.49257	0.272540	0.136270	−	0.990672i	$-0.456489\pi$
$971$	8.49257	0.272540	0.136270	+	0.990672i	$0.456489\pi$
$972$	0	0
$973$	−4.77566	−0.153100
$974$	0	0
$975$	33.3699	1.06869
$976$	0	0
$977$	−10.7246	−0.343110	−0.171555	−	0.985175i	$-0.554879\pi$
$977$	−10.7246	−0.343110	−0.171555	+	0.985175i	$0.554879\pi$
$978$	0	0
$979$	−39.5441	−1.26384
$980$	0	0
$981$	3.44379	0.109952
$982$	0	0
$983$	5.64078	0.179913	0.0899564	−	0.995946i	$-0.471327\pi$
$983$	5.64078	0.179913	0.0899564	+	0.995946i	$0.471327\pi$
$984$	0	0
$985$	1.55616	0.0495835
$986$	0	0
$987$	−1.02278	−0.0325553
$988$	0	0
$989$	24.9054	0.791946
$990$	0	0
$991$	−11.4731	−0.364454	−0.182227	−	0.983256i	$-0.558331\pi$
$991$	−11.4731	−0.364454	−0.182227	+	0.983256i	$0.558331\pi$
$992$	0	0
$993$	−23.2927	−0.739170
$994$	0	0
$995$	−2.73628	−0.0867460
$996$	0	0
$997$	46.9025	1.48542	0.742709	−	0.669615i	$-0.233541\pi$
$997$	46.9025	1.48542	0.742709	+	0.669615i	$0.233541\pi$
$998$	0	0
$999$	8.49714	0.268838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5312.2.a.br.1.7		8
4.3	odd	2		5312.2.a.bs.1.2		8
8.3	odd	2		1328.2.a.n.1.7		8
8.5	even	2		664.2.a.f.1.2	✓	8
24.5	odd	2		5976.2.a.s.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
664.2.a.f.1.2	✓	8	8.5	even	2
1328.2.a.n.1.7		8	8.3	odd	2
5312.2.a.br.1.7		8	1.1	even	1	trivial
5312.2.a.bs.1.2		8	4.3	odd	2
5976.2.a.s.1.6		8	24.5	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:	\( N \)	\(=\)	\( 5312 = 2^{6} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5312.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98233 q^{3} +0.181960 q^{5} +2.34566 q^{7} +0.929627 q^{9} +O(q^{10})\)	\(q+1.98233 q^{3} +0.181960 q^{5} +2.34566 q^{7} +0.929627 q^{9} +4.25499 q^{11} -3.38918 q^{13} +0.360704 q^{15} +4.77248 q^{17} +6.08050 q^{19} +4.64988 q^{21} +7.42236 q^{23} -4.96689 q^{25} -4.10416 q^{27} -7.12665 q^{29} +4.10542 q^{31} +8.43478 q^{33} +0.426817 q^{35} -2.07037 q^{37} -6.71846 q^{39} +12.2090 q^{41} +3.35546 q^{43} +0.169155 q^{45} -0.219957 q^{47} -1.49786 q^{49} +9.46063 q^{51} -12.2651 q^{53} +0.774237 q^{55} +12.0536 q^{57} -10.3126 q^{59} -7.12665 q^{61} +2.18059 q^{63} -0.616694 q^{65} +10.7305 q^{67} +14.7136 q^{69} +7.60191 q^{71} +6.18461 q^{73} -9.84601 q^{75} +9.98077 q^{77} -1.88639 q^{79} -10.9247 q^{81} +1.00000 q^{83} +0.868400 q^{85} -14.1274 q^{87} -9.29359 q^{89} -7.94987 q^{91} +8.13830 q^{93} +1.10641 q^{95} -13.6575 q^{97} +3.95555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9	\( 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9} - 10 q^{11} - 7 q^{13} - 14 q^{15} + 15 q^{17} - 7 q^{19} + 5 q^{21} + 4 q^{23} + 27 q^{25} - 5 q^{27} - 16 q^{29} + 3 q^{31} + 21 q^{33} - 2 q^{35} - 8 q^{37} + 10 q^{39} + 24 q^{41} + q^{43} - 29 q^{45} - 2 q^{47} + 3 q^{49} + 9 q^{51} - 7 q^{53} - 30 q^{55} + 4 q^{57} + 2 q^{59} - 16 q^{61} - 10 q^{63} + 2 q^{65} + 25 q^{67} + 6 q^{69} + 8 q^{71} + 14 q^{73} + 30 q^{75} + 7 q^{77} - 4 q^{79} + 52 q^{81} + 8 q^{83} + 21 q^{85} - 4 q^{87} + 20 q^{89} + 45 q^{91} + 47 q^{93} + 8 q^{95} + 2 q^{97} - 10 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9 - 10 * q^11 - 7 * q^13 - 14 * q^15 + 15 * q^17 - 7 * q^19 + 5 * q^21 + 4 * q^23 + 27 * q^25 - 5 * q^27 - 16 * q^29 + 3 * q^31 + 21 * q^33 - 2 * q^35 - 8 * q^37 + 10 * q^39 + 24 * q^41 + q^43 - 29 * q^45 - 2 * q^47 + 3 * q^49 + 9 * q^51 - 7 * q^53 - 30 * q^55 + 4 * q^57 + 2 * q^59 - 16 * q^61 - 10 * q^63 + 2 * q^65 + 25 * q^67 + 6 * q^69 + 8 * q^71 + 14 * q^73 + 30 * q^75 + 7 * q^77 - 4 * q^79 + 52 * q^81 + 8 * q^83 + 21 * q^85 - 4 * q^87 + 20 * q^89 + 45 * q^91 + 47 * q^93 + 8 * q^95 + 2 * q^97 - 10 * q^99

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