LMFDB - Embedded newform 592.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,4,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("592.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	592.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:	$34.9291307234$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 86x^{3} + 77x^{2} + 1237x + 572 $ x^5 - x^4 - 86x^3 + 77x^2 + 1237*x + 572
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 148)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.484906$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.484906 q^{3} -9.57900 q^{5} -23.3325 q^{7} -26.7649 q^{9} +O(q^{10})$	$q-0.484906 q^{3} -9.57900 q^{5} -23.3325 q^{7} -26.7649 q^{9} -64.1485 q^{11} +23.4196 q^{13} +4.64492 q^{15} +55.5488 q^{17} -61.8345 q^{19} +11.3141 q^{21} -64.5776 q^{23} -33.2427 q^{25} +26.0709 q^{27} +272.604 q^{29} +160.229 q^{31} +31.1060 q^{33} +223.502 q^{35} +37.0000 q^{37} -11.3563 q^{39} +204.259 q^{41} +193.307 q^{43} +256.381 q^{45} -456.853 q^{47} +201.406 q^{49} -26.9359 q^{51} -597.045 q^{53} +614.478 q^{55} +29.9839 q^{57} -90.3342 q^{59} -887.293 q^{61} +624.491 q^{63} -224.336 q^{65} +644.171 q^{67} +31.3141 q^{69} +618.262 q^{71} +146.763 q^{73} +16.1196 q^{75} +1496.74 q^{77} -1081.28 q^{79} +710.009 q^{81} +1322.79 q^{83} -532.102 q^{85} -132.187 q^{87} +502.235 q^{89} -546.437 q^{91} -77.6961 q^{93} +592.313 q^{95} -364.609 q^{97} +1716.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 21 q^{5} - 22 q^{7} + 38 q^{9}+O(q^{10}) $ 5 * q + q^3 + 21 * q^5 - 22 * q^7 + 38 * q^9	$ 5 q + q^{3} + 21 q^{5} - 22 q^{7} + 38 q^{9} - 27 q^{11} + 85 q^{13} - 108 q^{15} + 246 q^{17} - 148 q^{19} + 280 q^{21} - 153 q^{23} + 434 q^{25} - 26 q^{27} + 297 q^{29} - 55 q^{31} + 174 q^{33} - 144 q^{35} + 185 q^{37} + 467 q^{39} + 225 q^{41} + 662 q^{43} - 360 q^{45} + 402 q^{47} + 51 q^{49} + 870 q^{51} + 48 q^{53} + 1017 q^{55} - 266 q^{57} + 702 q^{59} - 41 q^{61} + 1466 q^{63} - 18 q^{65} + 953 q^{67} - 705 q^{69} + 576 q^{71} - 29 q^{73} + 1402 q^{75} + 498 q^{77} + 1967 q^{79} - 487 q^{81} + 1032 q^{83} + 450 q^{85} + 1539 q^{87} + 1950 q^{89} + 2818 q^{91} + 454 q^{93} - 1782 q^{95} + 790 q^{97} + 1074 q^{99}+O(q^{100}) $ 5 * q + q^3 + 21 * q^5 - 22 * q^7 + 38 * q^9 - 27 * q^11 + 85 * q^13 - 108 * q^15 + 246 * q^17 - 148 * q^19 + 280 * q^21 - 153 * q^23 + 434 * q^25 - 26 * q^27 + 297 * q^29 - 55 * q^31 + 174 * q^33 - 144 * q^35 + 185 * q^37 + 467 * q^39 + 225 * q^41 + 662 * q^43 - 360 * q^45 + 402 * q^47 + 51 * q^49 + 870 * q^51 + 48 * q^53 + 1017 * q^55 - 266 * q^57 + 702 * q^59 - 41 * q^61 + 1466 * q^63 - 18 * q^65 + 953 * q^67 - 705 * q^69 + 576 * q^71 - 29 * q^73 + 1402 * q^75 + 498 * q^77 + 1967 * q^79 - 487 * q^81 + 1032 * q^83 + 450 * q^85 + 1539 * q^87 + 1950 * q^89 + 2818 * q^91 + 454 * q^93 - 1782 * q^95 + 790 * q^97 + 1074 * 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
$3$	−0.484906	−0.0933202	−0.0466601	−	0.998911i	$-0.514858\pi$
$3$	−0.484906	−0.0933202	−0.0466601	+	0.998911i	$0.514858\pi$
$4$	0	0
$5$	−9.57900	−0.856772	−0.428386	−	0.903596i	$-0.640918\pi$
$5$	−9.57900	−0.856772	−0.428386	+	0.903596i	$0.640918\pi$
$6$	0	0
$7$	−23.3325	−1.25984	−0.629918	−	0.776661i	$-0.716912\pi$
$7$	−23.3325	−1.25984	−0.629918	+	0.776661i	$0.716912\pi$
$8$	0	0
$9$	−26.7649	−0.991291
$10$	0	0
$11$	−64.1485	−1.75832	−0.879159	−	0.476529i	$-0.841895\pi$
$11$	−64.1485	−1.75832	−0.879159	+	0.476529i	$0.841895\pi$
$12$	0	0
$13$	23.4196	0.499648	0.249824	−	0.968291i	$-0.419627\pi$
$13$	23.4196	0.499648	0.249824	+	0.968291i	$0.419627\pi$
$14$	0	0
$15$	4.64492	0.0799541
$16$	0	0
$17$	55.5488	0.792503	0.396252	−	0.918142i	$-0.370311\pi$
$17$	55.5488	0.792503	0.396252	+	0.918142i	$0.370311\pi$
$18$	0	0
$19$	−61.8345	−0.746622	−0.373311	−	0.927706i	$-0.621778\pi$
$19$	−61.8345	−0.746622	−0.373311	+	0.927706i	$0.621778\pi$
$20$	0	0
$21$	11.3141	0.117568
$22$	0	0
$23$	−64.5776	−0.585450	−0.292725	−	0.956197i	$-0.594562\pi$
$23$	−64.5776	−0.585450	−0.292725	+	0.956197i	$0.594562\pi$
$24$	0	0
$25$	−33.2427	−0.265942
$26$	0	0
$27$	26.0709	0.185828
$28$	0	0
$29$	272.604	1.74556	0.872781	−	0.488112i	$-0.162314\pi$
$29$	272.604	1.74556	0.872781	+	0.488112i	$0.162314\pi$
$30$	0	0
$31$	160.229	0.928322	0.464161	−	0.885751i	$-0.346356\pi$
$31$	160.229	0.928322	0.464161	+	0.885751i	$0.346356\pi$
$32$	0	0
$33$	31.1060	0.164087
$34$	0	0
$35$	223.502	1.07939
$36$	0	0
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	0	0
$39$	−11.3563	−0.0466272
$40$	0	0
$41$	204.259	0.778046	0.389023	−	0.921228i	$-0.372813\pi$
$41$	204.259	0.778046	0.389023	+	0.921228i	$0.372813\pi$
$42$	0	0
$43$	193.307	0.685560	0.342780	−	0.939416i	$-0.388631\pi$
$43$	193.307	0.685560	0.342780	+	0.939416i	$0.388631\pi$
$44$	0	0
$45$	256.381	0.849311
$46$	0	0
$47$	−456.853	−1.41785	−0.708924	−	0.705285i	$-0.750819\pi$
$47$	−456.853	−1.41785	−0.708924	+	0.705285i	$0.750819\pi$
$48$	0	0
$49$	201.406	0.587189
$50$	0	0
$51$	−26.9359	−0.0739566
$52$	0	0
$53$	−597.045	−1.54737	−0.773683	−	0.633573i	$-0.781588\pi$
$53$	−597.045	−1.54737	−0.773683	+	0.633573i	$0.781588\pi$
$54$	0	0
$55$	614.478	1.50648
$56$	0	0
$57$	29.9839	0.0696749
$58$	0	0
$59$	−90.3342	−0.199331	−0.0996653	−	0.995021i	$-0.531777\pi$
$59$	−90.3342	−0.199331	−0.0996653	+	0.995021i	$0.531777\pi$
$60$	0	0
$61$	−887.293	−1.86240	−0.931199	−	0.364512i	$-0.881236\pi$
$61$	−887.293	−1.86240	−0.931199	+	0.364512i	$0.881236\pi$
$62$	0	0
$63$	624.491	1.24887
$64$	0	0
$65$	−224.336	−0.428084
$66$	0	0
$67$	644.171	1.17460	0.587299	−	0.809370i	$-0.300191\pi$
$67$	644.171	1.17460	0.587299	+	0.809370i	$0.300191\pi$
$68$	0	0
$69$	31.3141	0.0546343
$70$	0	0
$71$	618.262	1.03344	0.516719	−	0.856155i	$-0.327153\pi$
$71$	618.262	1.03344	0.516719	+	0.856155i	$0.327153\pi$
$72$	0	0
$73$	146.763	0.235305	0.117652	−	0.993055i	$-0.462463\pi$
$73$	146.763	0.235305	0.117652	+	0.993055i	$0.462463\pi$
$74$	0	0
$75$	16.1196	0.0248177
$76$	0	0
$77$	1496.74	2.21519
$78$	0	0
$79$	−1081.28	−1.53992	−0.769962	−	0.638090i	$-0.779725\pi$
$79$	−1081.28	−1.53992	−0.769962	+	0.638090i	$0.779725\pi$
$80$	0	0
$81$	710.009	0.973950
$82$	0	0
$83$	1322.79	1.74933	0.874667	−	0.484724i	$-0.161080\pi$
$83$	1322.79	1.74933	0.874667	+	0.484724i	$0.161080\pi$
$84$	0	0
$85$	−532.102	−0.678995
$86$	0	0
$87$	−132.187	−0.162896
$88$	0	0
$89$	502.235	0.598166	0.299083	−	0.954227i	$-0.403319\pi$
$89$	502.235	0.598166	0.299083	+	0.954227i	$0.403319\pi$
$90$	0	0
$91$	−546.437	−0.629474
$92$	0	0
$93$	−77.6961	−0.0866312
$94$	0	0
$95$	592.313	0.639685
$96$	0	0
$97$	−364.609	−0.381654	−0.190827	−	0.981624i	$-0.561117\pi$
$97$	−364.609	−0.381654	−0.190827	+	0.981624i	$0.561117\pi$
$98$	0	0
$99$	1716.93	1.74301
$100$	0	0
$101$	−462.820	−0.455963	−0.227982	−	0.973665i	$-0.573213\pi$
$101$	−462.820	−0.455963	−0.227982	+	0.973665i	$0.573213\pi$
$102$	0	0
$103$	−1752.71	−1.67670	−0.838350	−	0.545133i	$-0.816479\pi$
$103$	−1752.71	−1.67670	−0.838350	+	0.545133i	$0.816479\pi$
$104$	0	0
$105$	−108.378	−0.100729
$106$	0	0
$107$	72.3356	0.0653546	0.0326773	−	0.999466i	$-0.489597\pi$
$107$	72.3356	0.0653546	0.0326773	+	0.999466i	$0.489597\pi$
$108$	0	0
$109$	254.898	0.223989	0.111995	−	0.993709i	$-0.464276\pi$
$109$	254.898	0.223989	0.111995	+	0.993709i	$0.464276\pi$
$110$	0	0
$111$	−17.9415	−0.0153417
$112$	0	0
$113$	412.290	0.343230	0.171615	−	0.985164i	$-0.445102\pi$
$113$	412.290	0.343230	0.171615	+	0.985164i	$0.445102\pi$
$114$	0	0
$115$	618.589	0.501597
$116$	0	0
$117$	−626.822	−0.495296
$118$	0	0
$119$	−1296.09	−0.998425
$120$	0	0
$121$	2784.03	2.09168
$122$	0	0
$123$	−99.0464	−0.0726074
$124$	0	0
$125$	1515.81	1.08462
$126$	0	0
$127$	−268.364	−0.187507	−0.0937537	−	0.995595i	$-0.529887\pi$
$127$	−268.364	−0.187507	−0.0937537	+	0.995595i	$0.529887\pi$
$128$	0	0
$129$	−93.7359	−0.0639766
$130$	0	0
$131$	−1467.11	−0.978487	−0.489243	−	0.872147i	$-0.662727\pi$
$131$	−1467.11	−0.978487	−0.489243	+	0.872147i	$0.662727\pi$
$132$	0	0
$133$	1442.75	0.940622
$134$	0	0
$135$	−249.733	−0.159212
$136$	0	0
$137$	−602.005	−0.375422	−0.187711	−	0.982224i	$-0.560107\pi$
$137$	−602.005	−0.375422	−0.187711	+	0.982224i	$0.560107\pi$
$138$	0	0
$139$	2485.24	1.51651	0.758255	−	0.651958i	$-0.226052\pi$
$139$	2485.24	1.51651	0.758255	+	0.651958i	$0.226052\pi$
$140$	0	0
$141$	221.531	0.132314
$142$	0	0
$143$	−1502.33	−0.878539
$144$	0	0
$145$	−2611.27	−1.49555
$146$	0	0
$147$	−97.6628	−0.0547966
$148$	0	0
$149$	2407.82	1.32387	0.661934	−	0.749562i	$-0.269736\pi$
$149$	2407.82	1.32387	0.661934	+	0.749562i	$0.269736\pi$
$150$	0	0
$151$	1158.64	0.624427	0.312214	−	0.950012i	$-0.398930\pi$
$151$	1158.64	0.624427	0.312214	+	0.950012i	$0.398930\pi$
$152$	0	0
$153$	−1486.76	−0.785602
$154$	0	0
$155$	−1534.83	−0.795361
$156$	0	0
$157$	1908.44	0.970127	0.485064	−	0.874479i	$-0.338796\pi$
$157$	1908.44	0.970127	0.485064	+	0.874479i	$0.338796\pi$
$158$	0	0
$159$	289.511	0.144400
$160$	0	0
$161$	1506.76	0.737572
$162$	0	0
$163$	1317.57	0.633131	0.316566	−	0.948571i	$-0.397470\pi$
$163$	1317.57	0.633131	0.316566	+	0.948571i	$0.397470\pi$
$164$	0	0
$165$	−297.964	−0.140585
$166$	0	0
$167$	−1010.21	−0.468096	−0.234048	−	0.972225i	$-0.575197\pi$
$167$	−1010.21	−0.468096	−0.234048	+	0.972225i	$0.575197\pi$
$168$	0	0
$169$	−1648.52	−0.750352
$170$	0	0
$171$	1654.99	0.740120
$172$	0	0
$173$	1340.34	0.589042	0.294521	−	0.955645i	$-0.404840\pi$
$173$	1340.34	0.589042	0.294521	+	0.955645i	$0.404840\pi$
$174$	0	0
$175$	775.636	0.335043
$176$	0	0
$177$	43.8036	0.0186016
$178$	0	0
$179$	763.114	0.318647	0.159324	−	0.987226i	$-0.449069\pi$
$179$	763.114	0.318647	0.159324	+	0.987226i	$0.449069\pi$
$180$	0	0
$181$	−311.718	−0.128010	−0.0640049	−	0.997950i	$-0.520387\pi$
$181$	−311.718	−0.128010	−0.0640049	+	0.997950i	$0.520387\pi$
$182$	0	0
$183$	430.254	0.173799
$184$	0	0
$185$	−354.423	−0.140852
$186$	0	0
$187$	−3563.37	−1.39347
$188$	0	0
$189$	−608.300	−0.234113
$190$	0	0
$191$	2216.63	0.839738	0.419869	−	0.907585i	$-0.362076\pi$
$191$	2216.63	0.839738	0.419869	+	0.907585i	$0.362076\pi$
$192$	0	0
$193$	−4842.72	−1.80615	−0.903074	−	0.429484i	$-0.858695\pi$
$193$	−4842.72	−1.80615	−0.903074	+	0.429484i	$0.858695\pi$
$194$	0	0
$195$	108.782	0.0399489
$196$	0	0
$197$	−4439.51	−1.60560	−0.802798	−	0.596252i	$-0.796656\pi$
$197$	−4439.51	−1.60560	−0.802798	+	0.596252i	$0.796656\pi$
$198$	0	0
$199$	2448.17	0.872092	0.436046	−	0.899924i	$-0.356378\pi$
$199$	2448.17	0.872092	0.436046	+	0.899924i	$0.356378\pi$
$200$	0	0
$201$	−312.362	−0.109614
$202$	0	0
$203$	−6360.53	−2.19912
$204$	0	0
$205$	−1956.60	−0.666608
$206$	0	0
$207$	1728.41	0.580352
$208$	0	0
$209$	3966.59	1.31280
$210$	0	0
$211$	3156.38	1.02983	0.514915	−	0.857241i	$-0.327824\pi$
$211$	3156.38	1.02983	0.514915	+	0.857241i	$0.327824\pi$
$212$	0	0
$213$	−299.799	−0.0964407
$214$	0	0
$215$	−1851.69	−0.587369
$216$	0	0
$217$	−3738.55	−1.16953
$218$	0	0
$219$	−71.1661	−0.0219587
$220$	0	0
$221$	1300.93	0.395972
$222$	0	0
$223$	−4037.16	−1.21232	−0.606162	−	0.795341i	$-0.707292\pi$
$223$	−4037.16	−1.21232	−0.606162	+	0.795341i	$0.707292\pi$
$224$	0	0
$225$	889.737	0.263626
$226$	0	0
$227$	2005.99	0.586530	0.293265	−	0.956031i	$-0.405258\pi$
$227$	2005.99	0.586530	0.293265	+	0.956031i	$0.405258\pi$
$228$	0	0
$229$	−2183.18	−0.629995	−0.314997	−	0.949093i	$-0.602004\pi$
$229$	−2183.18	−0.629995	−0.314997	+	0.949093i	$0.602004\pi$
$230$	0	0
$231$	−725.780	−0.206722
$232$	0	0
$233$	−2631.66	−0.739938	−0.369969	−	0.929044i	$-0.620632\pi$
$233$	−2631.66	−0.739938	−0.369969	+	0.929044i	$0.620632\pi$
$234$	0	0
$235$	4376.20	1.21477
$236$	0	0
$237$	524.321	0.143706
$238$	0	0
$239$	−2485.08	−0.672578	−0.336289	−	0.941759i	$-0.609172\pi$
$239$	−2485.08	−0.672578	−0.336289	+	0.941759i	$0.609172\pi$
$240$	0	0
$241$	−5725.37	−1.53030	−0.765152	−	0.643850i	$-0.777336\pi$
$241$	−5725.37	−1.53030	−0.765152	+	0.643850i	$0.777336\pi$
$242$	0	0
$243$	−1048.20	−0.276717
$244$	0	0
$245$	−1929.27	−0.503087
$246$	0	0
$247$	−1448.14	−0.373048
$248$	0	0
$249$	−641.428	−0.163248
$250$	0	0
$251$	2645.96	0.665383	0.332692	−	0.943036i	$-0.392043\pi$
$251$	2645.96	0.665383	0.332692	+	0.943036i	$0.392043\pi$
$252$	0	0
$253$	4142.55	1.02941
$254$	0	0
$255$	258.019	0.0633639
$256$	0	0
$257$	6386.09	1.55001	0.775007	−	0.631953i	$-0.217746\pi$
$257$	6386.09	1.55001	0.775007	+	0.631953i	$0.217746\pi$
$258$	0	0
$259$	−863.303	−0.207116
$260$	0	0
$261$	−7296.21	−1.73036
$262$	0	0
$263$	−325.063	−0.0762137	−0.0381069	−	0.999274i	$-0.512133\pi$
$263$	−325.063	−0.0762137	−0.0381069	+	0.999274i	$0.512133\pi$
$264$	0	0
$265$	5719.09	1.32574
$266$	0	0
$267$	−243.537	−0.0558210
$268$	0	0
$269$	−6208.73	−1.40726	−0.703630	−	0.710567i	$-0.748439\pi$
$269$	−6208.73	−1.40726	−0.703630	+	0.710567i	$0.748439\pi$
$270$	0	0
$271$	−2337.72	−0.524009	−0.262004	−	0.965067i	$-0.584384\pi$
$271$	−2337.72	−0.524009	−0.262004	+	0.965067i	$0.584384\pi$
$272$	0	0
$273$	264.971	0.0587427
$274$	0	0
$275$	2132.47	0.467610
$276$	0	0
$277$	6695.85	1.45240	0.726200	−	0.687483i	$-0.241285\pi$
$277$	6695.85	1.45240	0.726200	+	0.687483i	$0.241285\pi$
$278$	0	0
$279$	−4288.51	−0.920238
$280$	0	0
$281$	−1422.01	−0.301886	−0.150943	−	0.988542i	$-0.548231\pi$
$281$	−1422.01	−0.301886	−0.150943	+	0.988542i	$0.548231\pi$
$282$	0	0
$283$	−6070.16	−1.27503	−0.637515	−	0.770438i	$-0.720038\pi$
$283$	−6070.16	−1.27503	−0.637515	+	0.770438i	$0.720038\pi$
$284$	0	0
$285$	−287.216	−0.0596955
$286$	0	0
$287$	−4765.87	−0.980211
$288$	0	0
$289$	−1827.33	−0.371938
$290$	0	0
$291$	176.801	0.0356160
$292$	0	0
$293$	−2079.88	−0.414702	−0.207351	−	0.978267i	$-0.566484\pi$
$293$	−2079.88	−0.414702	−0.207351	+	0.978267i	$0.566484\pi$
$294$	0	0
$295$	865.311	0.170781
$296$	0	0
$297$	−1672.41	−0.326744
$298$	0	0
$299$	−1512.38	−0.292519
$300$	0	0
$301$	−4510.34	−0.863694
$302$	0	0
$303$	224.424	0.0425506
$304$	0	0
$305$	8499.38	1.59565
$306$	0	0
$307$	5335.03	0.991811	0.495906	−	0.868376i	$-0.334836\pi$
$307$	5335.03	0.991811	0.495906	+	0.868376i	$0.334836\pi$
$308$	0	0
$309$	849.901	0.156470
$310$	0	0
$311$	2418.90	0.441040	0.220520	−	0.975382i	$-0.429225\pi$
$311$	2418.90	0.441040	0.220520	+	0.975382i	$0.429225\pi$
$312$	0	0
$313$	−1859.62	−0.335821	−0.167910	−	0.985802i	$-0.553702\pi$
$313$	−1859.62	−0.335821	−0.167910	+	0.985802i	$0.553702\pi$
$314$	0	0
$315$	−5982.00	−1.06999
$316$	0	0
$317$	4536.78	0.803819	0.401910	−	0.915679i	$-0.368347\pi$
$317$	4536.78	0.803819	0.401910	+	0.915679i	$0.368347\pi$
$318$	0	0
$319$	−17487.1	−3.06925
$320$	0	0
$321$	−35.0760	−0.00609891
$322$	0	0
$323$	−3434.83	−0.591700
$324$	0	0
$325$	−778.530	−0.132877
$326$	0	0
$327$	−123.602	−0.0209027
$328$	0	0
$329$	10659.5	1.78626
$330$	0	0
$331$	9746.40	1.61846	0.809231	−	0.587491i	$-0.199884\pi$
$331$	9746.40	1.61846	0.809231	+	0.587491i	$0.199884\pi$
$332$	0	0
$333$	−990.300	−0.162967
$334$	0	0
$335$	−6170.52	−1.00636
$336$	0	0
$337$	4289.74	0.693403	0.346702	−	0.937975i	$-0.387302\pi$
$337$	4289.74	0.693403	0.346702	+	0.937975i	$0.387302\pi$
$338$	0	0
$339$	−199.922	−0.0320302
$340$	0	0
$341$	−10278.5	−1.63229
$342$	0	0
$343$	3303.75	0.520075
$344$	0	0
$345$	−299.957	−0.0468092
$346$	0	0
$347$	4634.84	0.717036	0.358518	−	0.933523i	$-0.383282\pi$
$347$	4634.84	0.717036	0.358518	+	0.933523i	$0.383282\pi$
$348$	0	0
$349$	−2779.92	−0.426377	−0.213189	−	0.977011i	$-0.568385\pi$
$349$	−2779.92	−0.426377	−0.213189	+	0.977011i	$0.568385\pi$
$350$	0	0
$351$	610.569	0.0928484
$352$	0	0
$353$	226.965	0.0342214	0.0171107	−	0.999854i	$-0.494553\pi$
$353$	226.965	0.0342214	0.0171107	+	0.999854i	$0.494553\pi$
$354$	0	0
$355$	−5922.33	−0.885421
$356$	0	0
$357$	628.483	0.0931732
$358$	0	0
$359$	−3736.48	−0.549315	−0.274657	−	0.961542i	$-0.588564\pi$
$359$	−3736.48	−0.549315	−0.274657	+	0.961542i	$0.588564\pi$
$360$	0	0
$361$	−3035.49	−0.442556
$362$	0	0
$363$	−1349.99	−0.195196
$364$	0	0
$365$	−1405.84	−0.201603
$366$	0	0
$367$	−6498.56	−0.924311	−0.462155	−	0.886799i	$-0.652924\pi$
$367$	−6498.56	−0.924311	−0.462155	+	0.886799i	$0.652924\pi$
$368$	0	0
$369$	−5466.97	−0.771271
$370$	0	0
$371$	13930.5	1.94943
$372$	0	0
$373$	−3799.34	−0.527406	−0.263703	−	0.964604i	$-0.584944\pi$
$373$	−3799.34	−0.527406	−0.263703	+	0.964604i	$0.584944\pi$
$374$	0	0
$375$	−735.024	−0.101217
$376$	0	0
$377$	6384.27	0.872166
$378$	0	0
$379$	12317.1	1.66936	0.834682	−	0.550732i	$-0.185651\pi$
$379$	12317.1	1.66936	0.834682	+	0.550732i	$0.185651\pi$
$380$	0	0
$381$	130.131	0.0174982
$382$	0	0
$383$	8761.59	1.16892	0.584460	−	0.811423i	$-0.301307\pi$
$383$	8761.59	1.16892	0.584460	+	0.811423i	$0.301307\pi$
$384$	0	0
$385$	−14337.3	−1.89792
$386$	0	0
$387$	−5173.85	−0.679590
$388$	0	0
$389$	8508.13	1.10894	0.554472	−	0.832202i	$-0.312920\pi$
$389$	8508.13	1.10894	0.554472	+	0.832202i	$0.312920\pi$
$390$	0	0
$391$	−3587.21	−0.463971
$392$	0	0
$393$	711.409	0.0913126
$394$	0	0
$395$	10357.6	1.31936
$396$	0	0
$397$	10131.8	1.28085	0.640426	−	0.768020i	$-0.278758\pi$
$397$	10131.8	1.28085	0.640426	+	0.768020i	$0.278758\pi$
$398$	0	0
$399$	−699.600	−0.0877790
$400$	0	0
$401$	12587.0	1.56750	0.783750	−	0.621077i	$-0.213304\pi$
$401$	12587.0	1.56750	0.783750	+	0.621077i	$0.213304\pi$
$402$	0	0
$403$	3752.50	0.463834
$404$	0	0
$405$	−6801.18	−0.834453
$406$	0	0
$407$	−2373.49	−0.289066
$408$	0	0
$409$	−3127.51	−0.378106	−0.189053	−	0.981967i	$-0.560542\pi$
$409$	−3127.51	−0.378106	−0.189053	+	0.981967i	$0.560542\pi$
$410$	0	0
$411$	291.916	0.0350344
$412$	0	0
$413$	2107.72	0.251124
$414$	0	0
$415$	−12671.0	−1.49878
$416$	0	0
$417$	−1205.11	−0.141521
$418$	0	0
$419$	−2182.40	−0.254456	−0.127228	−	0.991873i	$-0.540608\pi$
$419$	−2182.40	−0.254456	−0.127228	+	0.991873i	$0.540608\pi$
$420$	0	0
$421$	8995.85	1.04140	0.520702	−	0.853738i	$-0.325670\pi$
$421$	8995.85	1.04140	0.520702	+	0.853738i	$0.325670\pi$
$422$	0	0
$423$	12227.6	1.40550
$424$	0	0
$425$	−1846.59	−0.210760
$426$	0	0
$427$	20702.8	2.34632
$428$	0	0
$429$	728.489	0.0819855
$430$	0	0
$431$	6327.43	0.707149	0.353575	−	0.935406i	$-0.384966\pi$
$431$	6327.43	0.707149	0.353575	+	0.935406i	$0.384966\pi$
$432$	0	0
$433$	2888.71	0.320607	0.160303	−	0.987068i	$-0.448753\pi$
$433$	2888.71	0.320607	0.160303	+	0.987068i	$0.448753\pi$
$434$	0	0
$435$	1266.22	0.139565
$436$	0	0
$437$	3993.12	0.437110
$438$	0	0
$439$	−11294.2	−1.22789	−0.613943	−	0.789351i	$-0.710417\pi$
$439$	−11294.2	−1.22789	−0.613943	+	0.789351i	$0.710417\pi$
$440$	0	0
$441$	−5390.60	−0.582075
$442$	0	0
$443$	16129.5	1.72987	0.864937	−	0.501881i	$-0.167358\pi$
$443$	16129.5	1.72987	0.864937	+	0.501881i	$0.167358\pi$
$444$	0	0
$445$	−4810.91	−0.512492
$446$	0	0
$447$	−1167.57	−0.123544
$448$	0	0
$449$	−7203.90	−0.757178	−0.378589	−	0.925565i	$-0.623591\pi$
$449$	−7203.90	−0.757178	−0.378589	+	0.925565i	$0.623591\pi$
$450$	0	0
$451$	−13102.9	−1.36805
$452$	0	0
$453$	−561.830	−0.0582717
$454$	0	0
$455$	5234.32	0.539316
$456$	0	0
$457$	−12751.9	−1.30527	−0.652633	−	0.757674i	$-0.726336\pi$
$457$	−12751.9	−1.30527	−0.652633	+	0.757674i	$0.726336\pi$
$458$	0	0
$459$	1448.21	0.147269
$460$	0	0
$461$	−2887.87	−0.291761	−0.145880	−	0.989302i	$-0.546601\pi$
$461$	−2887.87	−0.291761	−0.145880	+	0.989302i	$0.546601\pi$
$462$	0	0
$463$	−1214.03	−0.121859	−0.0609297	−	0.998142i	$-0.519407\pi$
$463$	−1214.03	−0.121859	−0.0609297	+	0.998142i	$0.519407\pi$
$464$	0	0
$465$	744.251	0.0742232
$466$	0	0
$467$	−10792.0	−1.06937	−0.534685	−	0.845052i	$-0.679570\pi$
$467$	−10792.0	−1.06937	−0.534685	+	0.845052i	$0.679570\pi$
$468$	0	0
$469$	−15030.1	−1.47980
$470$	0	0
$471$	−925.414	−0.0905325
$472$	0	0
$473$	−12400.4	−1.20543
$474$	0	0
$475$	2055.55	0.198558
$476$	0	0
$477$	15979.8	1.53389
$478$	0	0
$479$	15080.2	1.43848	0.719240	−	0.694762i	$-0.244490\pi$
$479$	15080.2	1.43848	0.719240	+	0.694762i	$0.244490\pi$
$480$	0	0
$481$	866.524	0.0821416
$482$	0	0
$483$	−730.635	−0.0688303
$484$	0	0
$485$	3492.59	0.326990
$486$	0	0
$487$	−4018.77	−0.373938	−0.186969	−	0.982366i	$-0.559866\pi$
$487$	−4018.77	−0.373938	−0.186969	+	0.982366i	$0.559866\pi$
$488$	0	0
$489$	−638.900	−0.0590839
$490$	0	0
$491$	6038.83	0.555048	0.277524	−	0.960719i	$-0.410486\pi$
$491$	6038.83	0.555048	0.277524	+	0.960719i	$0.410486\pi$
$492$	0	0
$493$	15142.8	1.38336
$494$	0	0
$495$	−16446.4	−1.49336
$496$	0	0
$497$	−14425.6	−1.30196
$498$	0	0
$499$	2744.73	0.246235	0.123117	−	0.992392i	$-0.460711\pi$
$499$	2744.73	0.246235	0.123117	+	0.992392i	$0.460711\pi$
$500$	0	0
$501$	489.855	0.0436828
$502$	0	0
$503$	2399.54	0.212704	0.106352	−	0.994329i	$-0.466083\pi$
$503$	2399.54	0.212704	0.106352	+	0.994329i	$0.466083\pi$
$504$	0	0
$505$	4433.35	0.390657
$506$	0	0
$507$	799.379	0.0700230
$508$	0	0
$509$	3501.58	0.304921	0.152460	−	0.988310i	$-0.451280\pi$
$509$	3501.58	0.304921	0.152460	+	0.988310i	$0.451280\pi$
$510$	0	0
$511$	−3424.34	−0.296446
$512$	0	0
$513$	−1612.08	−0.138743
$514$	0	0
$515$	16789.2	1.43655
$516$	0	0
$517$	29306.4	2.49303
$518$	0	0
$519$	−649.940	−0.0549695
$520$	0	0
$521$	11158.8	0.938346	0.469173	−	0.883106i	$-0.344552\pi$
$521$	11158.8	0.938346	0.469173	+	0.883106i	$0.344552\pi$
$522$	0	0
$523$	−11672.6	−0.975920	−0.487960	−	0.872866i	$-0.662259\pi$
$523$	−11672.6	−0.975920	−0.487960	+	0.872866i	$0.662259\pi$
$524$	0	0
$525$	−376.110	−0.0312663
$526$	0	0
$527$	8900.53	0.735699
$528$	0	0
$529$	−7996.74	−0.657248
$530$	0	0
$531$	2417.78	0.197595
$532$	0	0
$533$	4783.66	0.388749
$534$	0	0
$535$	−692.903	−0.0559940
$536$	0	0
$537$	−370.039	−0.0297362
$538$	0	0
$539$	−12919.9	−1.03246
$540$	0	0
$541$	−1481.48	−0.117734	−0.0588668	−	0.998266i	$-0.518749\pi$
$541$	−1481.48	−0.117734	−0.0588668	+	0.998266i	$0.518749\pi$
$542$	0	0
$543$	151.154	0.0119459
$544$	0	0
$545$	−2441.67	−0.191908
$546$	0	0
$547$	−8257.01	−0.645419	−0.322710	−	0.946498i	$-0.604594\pi$
$547$	−8257.01	−0.645419	−0.322710	+	0.946498i	$0.604594\pi$
$548$	0	0
$549$	23748.3	1.84618
$550$	0	0
$551$	−16856.3	−1.30327
$552$	0	0
$553$	25229.1	1.94005
$554$	0	0
$555$	171.862	0.0131444
$556$	0	0
$557$	−1904.86	−0.144904	−0.0724519	−	0.997372i	$-0.523082\pi$
$557$	−1904.86	−0.144904	−0.0724519	+	0.997372i	$0.523082\pi$
$558$	0	0
$559$	4527.17	0.342539
$560$	0	0
$561$	1727.90	0.130039
$562$	0	0
$563$	17179.7	1.28603	0.643016	−	0.765853i	$-0.277683\pi$
$563$	17179.7	1.28603	0.643016	+	0.765853i	$0.277683\pi$
$564$	0	0
$565$	−3949.32	−0.294069
$566$	0	0
$567$	−16566.3	−1.22702
$568$	0	0
$569$	21666.1	1.59629	0.798147	−	0.602462i	$-0.205814\pi$
$569$	21666.1	1.59629	0.798147	+	0.602462i	$0.205814\pi$
$570$	0	0
$571$	17462.9	1.27986	0.639929	−	0.768434i	$-0.278964\pi$
$571$	17462.9	1.27986	0.639929	+	0.768434i	$0.278964\pi$
$572$	0	0
$573$	−1074.86	−0.0783646
$574$	0	0
$575$	2146.73	0.155696
$576$	0	0
$577$	25277.1	1.82374	0.911872	−	0.410474i	$-0.134637\pi$
$577$	25277.1	1.82374	0.911872	+	0.410474i	$0.134637\pi$
$578$	0	0
$579$	2348.26	0.168550
$580$	0	0
$581$	−30863.9	−2.20388
$582$	0	0
$583$	38299.5	2.72076
$584$	0	0
$585$	6004.33	0.424356
$586$	0	0
$587$	7904.39	0.555791	0.277895	−	0.960611i	$-0.410363\pi$
$587$	7904.39	0.555791	0.277895	+	0.960611i	$0.410363\pi$
$588$	0	0
$589$	−9907.69	−0.693106
$590$	0	0
$591$	2152.75	0.149834
$592$	0	0
$593$	22869.8	1.58373	0.791865	−	0.610696i	$-0.209110\pi$
$593$	22869.8	1.58373	0.791865	+	0.610696i	$0.209110\pi$
$594$	0	0
$595$	12415.3	0.855423
$596$	0	0
$597$	−1187.13	−0.0813838
$598$	0	0
$599$	−6645.52	−0.453303	−0.226652	−	0.973976i	$-0.572778\pi$
$599$	−6645.52	−0.453303	−0.226652	+	0.973976i	$0.572778\pi$
$600$	0	0
$601$	−14172.5	−0.961912	−0.480956	−	0.876745i	$-0.659710\pi$
$601$	−14172.5	−0.961912	−0.480956	+	0.876745i	$0.659710\pi$
$602$	0	0
$603$	−17241.2	−1.16437
$604$	0	0
$605$	−26668.2	−1.79209
$606$	0	0
$607$	20205.5	1.35110	0.675548	−	0.737316i	$-0.263907\pi$
$607$	20205.5	1.35110	0.675548	+	0.737316i	$0.263907\pi$
$608$	0	0
$609$	3084.26	0.205223
$610$	0	0
$611$	−10699.3	−0.708425
$612$	0	0
$613$	13227.4	0.871530	0.435765	−	0.900061i	$-0.356478\pi$
$613$	13227.4	0.871530	0.435765	+	0.900061i	$0.356478\pi$
$614$	0	0
$615$	948.766	0.0622080
$616$	0	0
$617$	24955.7	1.62833	0.814165	−	0.580634i	$-0.197195\pi$
$617$	24955.7	1.62833	0.814165	+	0.580634i	$0.197195\pi$
$618$	0	0
$619$	−25143.0	−1.63261	−0.816303	−	0.577624i	$-0.803980\pi$
$619$	−25143.0	−1.63261	−0.816303	+	0.577624i	$0.803980\pi$
$620$	0	0
$621$	−1683.60	−0.108793
$622$	0	0
$623$	−11718.4	−0.753592
$624$	0	0
$625$	−10364.6	−0.663333
$626$	0	0
$627$	−1923.42	−0.122511
$628$	0	0
$629$	2055.30	0.130287
$630$	0	0
$631$	−8850.38	−0.558365	−0.279182	−	0.960238i	$-0.590063\pi$
$631$	−8850.38	−0.558365	−0.279182	+	0.960238i	$0.590063\pi$
$632$	0	0
$633$	−1530.55	−0.0961039
$634$	0	0
$635$	2570.66	0.160651
$636$	0	0
$637$	4716.83	0.293387
$638$	0	0
$639$	−16547.7	−1.02444
$640$	0	0
$641$	−15760.1	−0.971115	−0.485557	−	0.874205i	$-0.661383\pi$
$641$	−15760.1	−0.971115	−0.485557	+	0.874205i	$0.661383\pi$
$642$	0	0
$643$	−17971.7	−1.10223	−0.551116	−	0.834429i	$-0.685798\pi$
$643$	−17971.7	−1.10223	−0.551116	+	0.834429i	$0.685798\pi$
$644$	0	0
$645$	897.896	0.0548134
$646$	0	0
$647$	−6492.83	−0.394528	−0.197264	−	0.980350i	$-0.563206\pi$
$647$	−6492.83	−0.394528	−0.197264	+	0.980350i	$0.563206\pi$
$648$	0	0
$649$	5794.80	0.350487
$650$	0	0
$651$	1812.84	0.109141
$652$	0	0
$653$	31042.0	1.86029	0.930143	−	0.367198i	$-0.119683\pi$
$653$	31042.0	1.86029	0.930143	+	0.367198i	$0.119683\pi$
$654$	0	0
$655$	14053.4	0.838340
$656$	0	0
$657$	−3928.08	−0.233256
$658$	0	0
$659$	714.997	0.0422645	0.0211323	−	0.999777i	$-0.493273\pi$
$659$	714.997	0.0422645	0.0211323	+	0.999777i	$0.493273\pi$
$660$	0	0
$661$	−10470.6	−0.616126	−0.308063	−	0.951366i	$-0.599681\pi$
$661$	−10470.6	−0.616126	−0.308063	+	0.951366i	$0.599681\pi$
$662$	0	0
$663$	−630.828	−0.0369522
$664$	0	0
$665$	−13820.1	−0.805898
$666$	0	0
$667$	−17604.1	−1.02194
$668$	0	0
$669$	1957.64	0.113134
$670$	0	0
$671$	56918.5	3.27469
$672$	0	0
$673$	7726.82	0.442566	0.221283	−	0.975210i	$-0.428975\pi$
$673$	7726.82	0.442566	0.221283	+	0.975210i	$0.428975\pi$
$674$	0	0
$675$	−866.668	−0.0494193
$676$	0	0
$677$	−18527.6	−1.05181	−0.525903	−	0.850544i	$-0.676273\pi$
$677$	−18527.6	−1.05181	−0.525903	+	0.850544i	$0.676273\pi$
$678$	0	0
$679$	8507.23	0.480821
$680$	0	0
$681$	−972.717	−0.0547351
$682$	0	0
$683$	17382.7	0.973839	0.486919	−	0.873447i	$-0.338121\pi$
$683$	17382.7	0.973839	0.486919	+	0.873447i	$0.338121\pi$
$684$	0	0
$685$	5766.61	0.321651
$686$	0	0
$687$	1058.64	0.0587912
$688$	0	0
$689$	−13982.5	−0.773138
$690$	0	0
$691$	6065.85	0.333945	0.166972	−	0.985962i	$-0.446601\pi$
$691$	6065.85	0.333945	0.166972	+	0.985962i	$0.446601\pi$
$692$	0	0
$693$	−40060.2	−2.19590
$694$	0	0
$695$	−23806.1	−1.29930
$696$	0	0
$697$	11346.3	0.616604
$698$	0	0
$699$	1276.11	0.0690512
$700$	0	0
$701$	−5438.17	−0.293005	−0.146503	−	0.989210i	$-0.546802\pi$
$701$	−5438.17	−0.293005	−0.146503	+	0.989210i	$0.546802\pi$
$702$	0	0
$703$	−2287.88	−0.122744
$704$	0	0
$705$	−2122.04	−0.113363
$706$	0	0
$707$	10798.7	0.574439
$708$	0	0
$709$	−36009.4	−1.90742	−0.953710	−	0.300727i	$-0.902771\pi$
$709$	−36009.4	−1.90742	−0.953710	+	0.300727i	$0.902771\pi$
$710$	0	0
$711$	28940.4	1.52651
$712$	0	0
$713$	−10347.2	−0.543487
$714$	0	0
$715$	14390.8	0.752708
$716$	0	0
$717$	1205.03	0.0627651
$718$	0	0
$719$	10405.9	0.539740	0.269870	−	0.962897i	$-0.413019\pi$
$719$	10405.9	0.539740	0.269870	+	0.962897i	$0.413019\pi$
$720$	0	0
$721$	40895.2	2.11237
$722$	0	0
$723$	2776.26	0.142808
$724$	0	0
$725$	−9062.10	−0.464218
$726$	0	0
$727$	−30988.8	−1.58090	−0.790448	−	0.612529i	$-0.790152\pi$
$727$	−30988.8	−1.58090	−0.790448	+	0.612529i	$0.790152\pi$
$728$	0	0
$729$	−18662.0	−0.948127
$730$	0	0
$731$	10738.0	0.543309
$732$	0	0
$733$	34079.9	1.71729	0.858644	−	0.512573i	$-0.171308\pi$
$733$	34079.9	1.71729	0.858644	+	0.512573i	$0.171308\pi$
$734$	0	0
$735$	935.512	0.0469482
$736$	0	0
$737$	−41322.6	−2.06532
$738$	0	0
$739$	−28405.2	−1.41394	−0.706970	−	0.707243i	$-0.749938\pi$
$739$	−28405.2	−1.41394	−0.706970	+	0.707243i	$0.749938\pi$
$740$	0	0
$741$	702.211	0.0348129
$742$	0	0
$743$	−13657.0	−0.674331	−0.337165	−	0.941445i	$-0.609468\pi$
$743$	−13657.0	−0.674331	−0.337165	+	0.941445i	$0.609468\pi$
$744$	0	0
$745$	−23064.5	−1.13425
$746$	0	0
$747$	−35404.2	−1.73410
$748$	0	0
$749$	−1687.77	−0.0823361
$750$	0	0
$751$	34917.9	1.69663	0.848317	−	0.529488i	$-0.177616\pi$
$751$	34917.9	1.69663	0.848317	+	0.529488i	$0.177616\pi$
$752$	0	0
$753$	−1283.04	−0.0620937
$754$	0	0
$755$	−11098.6	−0.534992
$756$	0	0
$757$	−236.604	−0.0113600	−0.00567999	−	0.999984i	$-0.501808\pi$
$757$	−236.604	−0.0113600	−0.00567999	+	0.999984i	$0.501808\pi$
$758$	0	0
$759$	−2008.75	−0.0960645
$760$	0	0
$761$	17460.3	0.831714	0.415857	−	0.909430i	$-0.363482\pi$
$761$	17460.3	0.831714	0.415857	+	0.909430i	$0.363482\pi$
$762$	0	0
$763$	−5947.41	−0.282190
$764$	0	0
$765$	14241.6	0.673082
$766$	0	0
$767$	−2115.59	−0.0995951
$768$	0	0
$769$	−14013.9	−0.657158	−0.328579	−	0.944477i	$-0.606570\pi$
$769$	−14013.9	−0.657158	−0.328579	+	0.944477i	$0.606570\pi$
$770$	0	0
$771$	−3096.65	−0.144648
$772$	0	0
$773$	5709.14	0.265645	0.132822	−	0.991140i	$-0.457596\pi$
$773$	5709.14	0.265645	0.132822	+	0.991140i	$0.457596\pi$
$774$	0	0
$775$	−5326.45	−0.246880
$776$	0	0
$777$	418.621	0.0193281
$778$	0	0
$779$	−12630.3	−0.580906
$780$	0	0
$781$	−39660.5	−1.81711
$782$	0	0
$783$	7107.03	0.324374
$784$	0	0
$785$	−18280.9	−0.831178
$786$	0	0
$787$	−21313.1	−0.965350	−0.482675	−	0.875799i	$-0.660335\pi$
$787$	−21313.1	−0.965350	−0.482675	+	0.875799i	$0.660335\pi$
$788$	0	0
$789$	157.625	0.00711228
$790$	0	0
$791$	−9619.75	−0.432413
$792$	0	0
$793$	−20780.0	−0.930542
$794$	0	0
$795$	−2773.22	−0.123718
$796$	0	0
$797$	1300.68	0.0578075	0.0289038	−	0.999582i	$-0.490798\pi$
$797$	1300.68	0.0578075	0.0289038	+	0.999582i	$0.490798\pi$
$798$	0	0
$799$	−25377.6	−1.12365
$800$	0	0
$801$	−13442.2	−0.592957
$802$	0	0
$803$	−9414.60	−0.413741
$804$	0	0
$805$	−14433.2	−0.631931
$806$	0	0
$807$	3010.65	0.131326
$808$	0	0
$809$	3614.34	0.157075	0.0785373	−	0.996911i	$-0.474975\pi$
$809$	3614.34	0.157075	0.0785373	+	0.996911i	$0.474975\pi$
$810$	0	0
$811$	29368.7	1.27161	0.635804	−	0.771851i	$-0.280669\pi$
$811$	29368.7	1.27161	0.635804	+	0.771851i	$0.280669\pi$
$812$	0	0
$813$	1133.57	0.0489006
$814$	0	0
$815$	−12621.1	−0.542449
$816$	0	0
$817$	−11953.1	−0.511854
$818$	0	0
$819$	14625.3	0.623993
$820$	0	0
$821$	−25253.7	−1.07352	−0.536761	−	0.843734i	$-0.680352\pi$
$821$	−25253.7	−1.07352	−0.536761	+	0.843734i	$0.680352\pi$
$822$	0	0
$823$	−7503.21	−0.317795	−0.158898	−	0.987295i	$-0.550794\pi$
$823$	−7503.21	−0.317795	−0.158898	+	0.987295i	$0.550794\pi$
$824$	0	0
$825$	−1034.05	−0.0436375
$826$	0	0
$827$	2350.54	0.0988348	0.0494174	−	0.998778i	$-0.484264\pi$
$827$	2350.54	0.0988348	0.0494174	+	0.998778i	$0.484264\pi$
$828$	0	0
$829$	−4131.54	−0.173093	−0.0865465	−	0.996248i	$-0.527583\pi$
$829$	−4131.54	−0.173093	−0.0865465	+	0.996248i	$0.527583\pi$
$830$	0	0
$831$	−3246.86	−0.135538
$832$	0	0
$833$	11187.8	0.465349
$834$	0	0
$835$	9676.77	0.401052
$836$	0	0
$837$	4177.32	0.172508
$838$	0	0
$839$	11874.3	0.488613	0.244306	−	0.969698i	$-0.421440\pi$
$839$	11874.3	0.488613	0.244306	+	0.969698i	$0.421440\pi$
$840$	0	0
$841$	49923.9	2.04699
$842$	0	0
$843$	689.541	0.0281721
$844$	0	0
$845$	15791.2	0.642881
$846$	0	0
$847$	−64958.3	−2.63518
$848$	0	0
$849$	2943.46	0.118986
$850$	0	0
$851$	−2389.37	−0.0962474
$852$	0	0
$853$	43164.8	1.73263	0.866316	−	0.499496i	$-0.166481\pi$
$853$	43164.8	1.73263	0.866316	+	0.499496i	$0.166481\pi$
$854$	0	0
$855$	−15853.2	−0.634114
$856$	0	0
$857$	33308.9	1.32767	0.663834	−	0.747880i	$-0.268928\pi$
$857$	33308.9	1.32767	0.663834	+	0.747880i	$0.268928\pi$
$858$	0	0
$859$	806.867	0.0320488	0.0160244	−	0.999872i	$-0.494899\pi$
$859$	806.867	0.0320488	0.0160244	+	0.999872i	$0.494899\pi$
$860$	0	0
$861$	2311.00	0.0914735
$862$	0	0
$863$	43599.2	1.71974	0.859869	−	0.510515i	$-0.170545\pi$
$863$	43599.2	1.71974	0.859869	+	0.510515i	$0.170545\pi$
$864$	0	0
$865$	−12839.1	−0.504675
$866$	0	0
$867$	886.085	0.0347094
$868$	0	0
$869$	69362.8	2.70768
$870$	0	0
$871$	15086.2	0.586885
$872$	0	0
$873$	9758.70	0.378330
$874$	0	0
$875$	−35367.6	−1.36645
$876$	0	0
$877$	−49448.1	−1.90393	−0.951963	−	0.306214i	$-0.900938\pi$
$877$	−49448.1	−1.90393	−0.951963	+	0.306214i	$0.900938\pi$
$878$	0	0
$879$	1008.55	0.0387001
$880$	0	0
$881$	−46119.5	−1.76368	−0.881842	−	0.471545i	$-0.843697\pi$
$881$	−46119.5	−1.76368	−0.881842	+	0.471545i	$0.843697\pi$
$882$	0	0
$883$	−7874.72	−0.300119	−0.150060	−	0.988677i	$-0.547947\pi$
$883$	−7874.72	−0.300119	−0.150060	+	0.988677i	$0.547947\pi$
$884$	0	0
$885$	−419.595	−0.0159373
$886$	0	0
$887$	26640.4	1.00845	0.504227	−	0.863571i	$-0.331778\pi$
$887$	26640.4	1.00845	0.504227	+	0.863571i	$0.331778\pi$
$888$	0	0
$889$	6261.60	0.236229
$890$	0	0
$891$	−45546.0	−1.71251
$892$	0	0
$893$	28249.3	1.05860
$894$	0	0
$895$	−7309.87	−0.273008
$896$	0	0
$897$	733.362	0.0272979
$898$	0	0
$899$	43679.1	1.62044
$900$	0	0
$901$	−33165.1	−1.22629
$902$	0	0
$903$	2187.09	0.0806001
$904$	0	0
$905$	2985.94	0.109675
$906$	0	0
$907$	−7589.80	−0.277856	−0.138928	−	0.990303i	$-0.544366\pi$
$907$	−7589.80	−0.277856	−0.138928	+	0.990303i	$0.544366\pi$
$908$	0	0
$909$	12387.3	0.451992
$910$	0	0
$911$	−37969.2	−1.38088	−0.690438	−	0.723392i	$-0.742582\pi$
$911$	−37969.2	−1.38088	−0.690438	+	0.723392i	$0.742582\pi$
$912$	0	0
$913$	−84854.8	−3.07589
$914$	0	0
$915$	−4121.40	−0.148906
$916$	0	0
$917$	34231.3	1.23273
$918$	0	0
$919$	−34827.5	−1.25011	−0.625057	−	0.780579i	$-0.714924\pi$
$919$	−34827.5	−1.25011	−0.625057	+	0.780579i	$0.714924\pi$
$920$	0	0
$921$	−2586.99	−0.0925560
$922$	0	0
$923$	14479.4	0.516355
$924$	0	0
$925$	−1229.98	−0.0437205
$926$	0	0
$927$	46911.1	1.66210
$928$	0	0
$929$	46472.6	1.64124	0.820622	−	0.571471i	$-0.193627\pi$
$929$	46472.6	1.64124	0.820622	+	0.571471i	$0.193627\pi$
$930$	0	0
$931$	−12453.8	−0.438408
$932$	0	0
$933$	−1172.94	−0.0411580
$934$	0	0
$935$	34133.5	1.19389
$936$	0	0
$937$	33654.9	1.17338	0.586690	−	0.809812i	$-0.300431\pi$
$937$	33654.9	1.17338	0.586690	+	0.809812i	$0.300431\pi$
$938$	0	0
$939$	901.740	0.0313389
$940$	0	0
$941$	34709.3	1.20243	0.601217	−	0.799085i	$-0.294683\pi$
$941$	34709.3	1.20243	0.601217	+	0.799085i	$0.294683\pi$
$942$	0	0
$943$	−13190.6	−0.455507
$944$	0	0
$945$	5826.90	0.200581
$946$	0	0
$947$	39142.1	1.34313	0.671567	−	0.740944i	$-0.265621\pi$
$947$	39142.1	1.34313	0.671567	+	0.740944i	$0.265621\pi$
$948$	0	0
$949$	3437.12	0.117570
$950$	0	0
$951$	−2199.91	−0.0750126
$952$	0	0
$953$	−11179.4	−0.379997	−0.189999	−	0.981784i	$-0.560848\pi$
$953$	−11179.4	−0.379997	−0.189999	+	0.981784i	$0.560848\pi$
$954$	0	0
$955$	−21233.1	−0.719464
$956$	0	0
$957$	8479.62	0.286423
$958$	0	0
$959$	14046.3	0.472970
$960$	0	0
$961$	−4117.64	−0.138217
$962$	0	0
$963$	−1936.05	−0.0647855
$964$	0	0
$965$	46388.4	1.54746
$966$	0	0
$967$	−22436.9	−0.746146	−0.373073	−	0.927802i	$-0.621696\pi$
$967$	−22436.9	−0.746146	−0.373073	+	0.927802i	$0.621696\pi$
$968$	0	0
$969$	1665.57	0.0552176
$970$	0	0
$971$	−45998.3	−1.52024	−0.760122	−	0.649780i	$-0.774861\pi$
$971$	−45998.3	−1.52024	−0.760122	+	0.649780i	$0.774861\pi$
$972$	0	0
$973$	−57986.8	−1.91056
$974$	0	0
$975$	377.514	0.0124001
$976$	0	0
$977$	−42607.9	−1.39524	−0.697619	−	0.716469i	$-0.745757\pi$
$977$	−42607.9	−1.39524	−0.697619	+	0.716469i	$0.745757\pi$
$978$	0	0
$979$	−32217.6	−1.05177
$980$	0	0
$981$	−6822.32	−0.222039
$982$	0	0
$983$	8375.52	0.271757	0.135879	−	0.990725i	$-0.456614\pi$
$983$	8375.52	0.271757	0.135879	+	0.990725i	$0.456614\pi$
$984$	0	0
$985$	42526.1	1.37563
$986$	0	0
$987$	−5168.87	−0.166694
$988$	0	0
$989$	−12483.3	−0.401361
$990$	0	0
$991$	−30005.3	−0.961807	−0.480904	−	0.876773i	$-0.659691\pi$
$991$	−30005.3	−0.961807	−0.480904	+	0.876773i	$0.659691\pi$
$992$	0	0
$993$	−4726.09	−0.151035
$994$	0	0
$995$	−23451.0	−0.747184
$996$	0	0
$997$	−56412.8	−1.79199	−0.895993	−	0.444068i	$-0.853535\pi$
$997$	−56412.8	−1.79199	−0.895993	+	0.444068i	$0.853535\pi$
$998$	0	0
$999$	964.624	0.0305499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.4.a.h.1.3		5
4.3	odd	2		148.4.a.b.1.3	✓	5
8.3	odd	2		2368.4.a.p.1.3		5
8.5	even	2		2368.4.a.o.1.3		5
12.11	even	2		1332.4.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.4.a.b.1.3	✓	5	4.3	odd	2
592.4.a.h.1.3		5	1.1	even	1	trivial
1332.4.a.f.1.4		5	12.11	even	2
2368.4.a.o.1.3		5	8.5	even	2
2368.4.a.p.1.3		5	8.3	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-0.484906 q^{3} -9.57900 q^{5} -23.3325 q^{7} -26.7649 q^{9} +O(q^{10})\)	\(q-0.484906 q^{3} -9.57900 q^{5} -23.3325 q^{7} -26.7649 q^{9} -64.1485 q^{11} +23.4196 q^{13} +4.64492 q^{15} +55.5488 q^{17} -61.8345 q^{19} +11.3141 q^{21} -64.5776 q^{23} -33.2427 q^{25} +26.0709 q^{27} +272.604 q^{29} +160.229 q^{31} +31.1060 q^{33} +223.502 q^{35} +37.0000 q^{37} -11.3563 q^{39} +204.259 q^{41} +193.307 q^{43} +256.381 q^{45} -456.853 q^{47} +201.406 q^{49} -26.9359 q^{51} -597.045 q^{53} +614.478 q^{55} +29.9839 q^{57} -90.3342 q^{59} -887.293 q^{61} +624.491 q^{63} -224.336 q^{65} +644.171 q^{67} +31.3141 q^{69} +618.262 q^{71} +146.763 q^{73} +16.1196 q^{75} +1496.74 q^{77} -1081.28 q^{79} +710.009 q^{81} +1322.79 q^{83} -532.102 q^{85} -132.187 q^{87} +502.235 q^{89} -546.437 q^{91} -77.6961 q^{93} +592.313 q^{95} -364.609 q^{97} +1716.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + 21 q^{5} - 22 q^{7} + 38 q^{9}+O(q^{10}) \) 5 * q + q^3 + 21 * q^5 - 22 * q^7 + 38 * q^9	\( 5 q + q^{3} + 21 q^{5} - 22 q^{7} + 38 q^{9} - 27 q^{11} + 85 q^{13} - 108 q^{15} + 246 q^{17} - 148 q^{19} + 280 q^{21} - 153 q^{23} + 434 q^{25} - 26 q^{27} + 297 q^{29} - 55 q^{31} + 174 q^{33} - 144 q^{35} + 185 q^{37} + 467 q^{39} + 225 q^{41} + 662 q^{43} - 360 q^{45} + 402 q^{47} + 51 q^{49} + 870 q^{51} + 48 q^{53} + 1017 q^{55} - 266 q^{57} + 702 q^{59} - 41 q^{61} + 1466 q^{63} - 18 q^{65} + 953 q^{67} - 705 q^{69} + 576 q^{71} - 29 q^{73} + 1402 q^{75} + 498 q^{77} + 1967 q^{79} - 487 q^{81} + 1032 q^{83} + 450 q^{85} + 1539 q^{87} + 1950 q^{89} + 2818 q^{91} + 454 q^{93} - 1782 q^{95} + 790 q^{97} + 1074 q^{99}+O(q^{100}) \) 5 * q + q^3 + 21 * q^5 - 22 * q^7 + 38 * q^9 - 27 * q^11 + 85 * q^13 - 108 * q^15 + 246 * q^17 - 148 * q^19 + 280 * q^21 - 153 * q^23 + 434 * q^25 - 26 * q^27 + 297 * q^29 - 55 * q^31 + 174 * q^33 - 144 * q^35 + 185 * q^37 + 467 * q^39 + 225 * q^41 + 662 * q^43 - 360 * q^45 + 402 * q^47 + 51 * q^49 + 870 * q^51 + 48 * q^53 + 1017 * q^55 - 266 * q^57 + 702 * q^59 - 41 * q^61 + 1466 * q^63 - 18 * q^65 + 953 * q^67 - 705 * q^69 + 576 * q^71 - 29 * q^73 + 1402 * q^75 + 498 * q^77 + 1967 * q^79 - 487 * q^81 + 1032 * q^83 + 450 * q^85 + 1539 * q^87 + 1950 * q^89 + 2818 * q^91 + 454 * q^93 - 1782 * q^95 + 790 * q^97 + 1074 * q^99

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