LMFDB - Embedded newform 507.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -8.00000 q^{4} -5.19615 q^{5} -10.3923 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -8.00000 q^{4} -5.19615 q^{5} -10.3923 q^{7} +9.00000 q^{9} +51.9615 q^{11} +24.0000 q^{12} +15.5885 q^{15} +64.0000 q^{16} +117.000 q^{17} -24.2487 q^{19} +41.5692 q^{20} +31.1769 q^{21} -18.0000 q^{23} -98.0000 q^{25} -27.0000 q^{27} +83.1384 q^{28} -99.0000 q^{29} -193.990 q^{31} -155.885 q^{33} +54.0000 q^{35} -72.0000 q^{36} +112.583 q^{37} +36.3731 q^{41} +82.0000 q^{43} -415.692 q^{44} -46.7654 q^{45} -72.7461 q^{47} -192.000 q^{48} -235.000 q^{49} -351.000 q^{51} -261.000 q^{53} -270.000 q^{55} +72.7461 q^{57} +789.815 q^{59} -124.708 q^{60} -719.000 q^{61} -93.5307 q^{63} -512.000 q^{64} +703.213 q^{67} -936.000 q^{68} +54.0000 q^{69} -467.654 q^{71} +684.160 q^{73} +294.000 q^{75} +193.990 q^{76} -540.000 q^{77} -440.000 q^{79} -332.554 q^{80} +81.0000 q^{81} -1195.12 q^{83} -249.415 q^{84} -607.950 q^{85} +297.000 q^{87} +1517.28 q^{89} +144.000 q^{92} +581.969 q^{93} +126.000 q^{95} -1157.01 q^{97} +467.654 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 16 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 16 * q^4 + 18 * q^9	$ 2 q - 6 q^{3} - 16 q^{4} + 18 q^{9} + 48 q^{12} + 128 q^{16} + 234 q^{17} - 36 q^{23} - 196 q^{25} - 54 q^{27} - 198 q^{29} + 108 q^{35} - 144 q^{36} + 164 q^{43} - 384 q^{48} - 470 q^{49} - 702 q^{51} - 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} - 1872 q^{68} + 108 q^{69} + 588 q^{75} - 1080 q^{77} - 880 q^{79} + 162 q^{81} + 594 q^{87} + 288 q^{92} + 252 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 - 16 * q^4 + 18 * q^9 + 48 * q^12 + 128 * q^16 + 234 * q^17 - 36 * q^23 - 196 * q^25 - 54 * q^27 - 198 * q^29 + 108 * q^35 - 144 * q^36 + 164 * q^43 - 384 * q^48 - 470 * q^49 - 702 * q^51 - 522 * q^53 - 540 * q^55 - 1438 * q^61 - 1024 * q^64 - 1872 * q^68 + 108 * q^69 + 588 * q^75 - 1080 * q^77 - 880 * q^79 + 162 * q^81 + 594 * q^87 + 288 * q^92 + 252 * q^95

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−8.00000	−1.00000
$5$	−5.19615	−0.464758	−0.232379	−	0.972625i	$-0.574651\pi$
$5$	−5.19615	−0.464758	−0.232379	+	0.972625i	$0.574651\pi$
$6$	0	0
$7$	−10.3923	−0.561132	−0.280566	−	0.959835i	$-0.590522\pi$
$7$	−10.3923	−0.561132	−0.280566	+	0.959835i	$0.590522\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	51.9615	1.42427	0.712136	−	0.702042i	$-0.247728\pi$
$11$	51.9615	1.42427	0.712136	+	0.702042i	$0.247728\pi$
$12$	24.0000	0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	15.5885	0.268328
$16$	64.0000	1.00000
$17$	117.000	1.66922	0.834608	−	0.550845i	$-0.185694\pi$
$17$	117.000	1.66922	0.834608	+	0.550845i	$0.185694\pi$
$18$	0	0
$19$	−24.2487	−0.292791	−0.146396	−	0.989226i	$-0.546767\pi$
$19$	−24.2487	−0.292791	−0.146396	+	0.989226i	$0.546767\pi$
$20$	41.5692	0.464758
$21$	31.1769	0.323970
$22$	0	0
$23$	−18.0000	−0.163185	−0.0815926	−	0.996666i	$-0.526001\pi$
$23$	−18.0000	−0.163185	−0.0815926	+	0.996666i	$0.526001\pi$
$24$	0	0
$25$	−98.0000	−0.784000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	83.1384	0.561132
$29$	−99.0000	−0.633925	−0.316963	−	0.948438i	$-0.602663\pi$
$29$	−99.0000	−0.633925	−0.316963	+	0.948438i	$0.602663\pi$
$30$	0	0
$31$	−193.990	−1.12392	−0.561961	−	0.827164i	$-0.689953\pi$
$31$	−193.990	−1.12392	−0.561961	+	0.827164i	$0.689953\pi$
$32$	0	0
$33$	−155.885	−0.822304
$34$	0	0
$35$	54.0000	0.260790
$36$	−72.0000	−0.333333
$37$	112.583	0.500232	0.250116	−	0.968216i	$-0.419531\pi$
$37$	112.583	0.500232	0.250116	+	0.968216i	$0.419531\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	36.3731	0.138549	0.0692746	−	0.997598i	$-0.477932\pi$
$41$	36.3731	0.138549	0.0692746	+	0.997598i	$0.477932\pi$
$42$	0	0
$43$	82.0000	0.290811	0.145406	−	0.989372i	$-0.453551\pi$
$43$	82.0000	0.290811	0.145406	+	0.989372i	$0.453551\pi$
$44$	−415.692	−1.42427
$45$	−46.7654	−0.154919
$46$	0	0
$47$	−72.7461	−0.225768	−0.112884	−	0.993608i	$-0.536009\pi$
$47$	−72.7461	−0.225768	−0.112884	+	0.993608i	$0.536009\pi$
$48$	−192.000	−0.577350
$49$	−235.000	−0.685131
$50$	0	0
$51$	−351.000	−0.963722
$52$	0	0
$53$	−261.000	−0.676436	−0.338218	−	0.941068i	$-0.609824\pi$
$53$	−261.000	−0.676436	−0.338218	+	0.941068i	$0.609824\pi$
$54$	0	0
$55$	−270.000	−0.661942
$56$	0	0
$57$	72.7461	0.169043
$58$	0	0
$59$	789.815	1.74280	0.871400	−	0.490574i	$-0.163213\pi$
$59$	789.815	1.74280	0.871400	+	0.490574i	$0.163213\pi$
$60$	−124.708	−0.268328
$61$	−719.000	−1.50916	−0.754578	−	0.656210i	$-0.772158\pi$
$61$	−719.000	−1.50916	−0.754578	+	0.656210i	$0.772158\pi$
$62$	0	0
$63$	−93.5307	−0.187044
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	703.213	1.28226	0.641128	−	0.767434i	$-0.278467\pi$
$67$	703.213	1.28226	0.641128	+	0.767434i	$0.278467\pi$
$68$	−936.000	−1.66922
$69$	54.0000	0.0942150
$70$	0	0
$71$	−467.654	−0.781694	−0.390847	−	0.920456i	$-0.627818\pi$
$71$	−467.654	−0.781694	−0.390847	+	0.920456i	$0.627818\pi$
$72$	0	0
$73$	684.160	1.09692	0.548458	−	0.836178i	$-0.315215\pi$
$73$	684.160	1.09692	0.548458	+	0.836178i	$0.315215\pi$
$74$	0	0
$75$	294.000	0.452643
$76$	193.990	0.292791
$77$	−540.000	−0.799204
$78$	0	0
$79$	−440.000	−0.626631	−0.313316	−	0.949649i	$-0.601440\pi$
$79$	−440.000	−0.626631	−0.313316	+	0.949649i	$0.601440\pi$
$80$	−332.554	−0.464758
$81$	81.0000	0.111111
$82$	0	0
$83$	−1195.12	−1.58049	−0.790247	−	0.612789i	$-0.790048\pi$
$83$	−1195.12	−1.58049	−0.790247	+	0.612789i	$0.790048\pi$
$84$	−249.415	−0.323970
$85$	−607.950	−0.775781
$86$	0	0
$87$	297.000	0.365997
$88$	0	0
$89$	1517.28	1.80709	0.903545	−	0.428493i	$-0.140955\pi$
$89$	1517.28	1.80709	0.903545	+	0.428493i	$0.140955\pi$
$90$	0	0
$91$	0	0
$92$	144.000	0.163185
$93$	581.969	0.648897
$94$	0	0
$95$	126.000	0.136077
$96$	0	0
$97$	−1157.01	−1.21110	−0.605549	−	0.795808i	$-0.707047\pi$
$97$	−1157.01	−1.21110	−0.605549	+	0.795808i	$0.707047\pi$
$98$	0	0
$99$	467.654	0.474757
$100$	784.000	0.784000
$101$	−1575.00	−1.55167	−0.775833	−	0.630938i	$-0.782670\pi$
$101$	−1575.00	−1.55167	−0.775833	+	0.630938i	$0.782670\pi$
$102$	0	0
$103$	−794.000	−0.759565	−0.379782	−	0.925076i	$-0.624001\pi$
$103$	−794.000	−0.759565	−0.379782	+	0.925076i	$0.624001\pi$
$104$	0	0
$105$	−162.000	−0.150567
$106$	0	0
$107$	450.000	0.406571	0.203286	−	0.979119i	$-0.434838\pi$
$107$	450.000	0.406571	0.203286	+	0.979119i	$0.434838\pi$
$108$	216.000	0.192450
$109$	−595.825	−0.523576	−0.261788	−	0.965125i	$-0.584312\pi$
$109$	−595.825	−0.523576	−0.261788	+	0.965125i	$0.584312\pi$
$110$	0	0
$111$	−337.750	−0.288809
$112$	−665.108	−0.561132
$113$	−1701.00	−1.41608	−0.708038	−	0.706174i	$-0.750420\pi$
$113$	−1701.00	−1.41608	−0.708038	+	0.706174i	$0.750420\pi$
$114$	0	0
$115$	93.5307	0.0758416
$116$	792.000	0.633925
$117$	0	0
$118$	0	0
$119$	−1215.90	−0.936650
$120$	0	0
$121$	1369.00	1.02855
$122$	0	0
$123$	−109.119	−0.0799914
$124$	1551.92	1.12392
$125$	1158.74	0.829128
$126$	0	0
$127$	1664.00	1.16265	0.581323	−	0.813673i	$-0.302535\pi$
$127$	1664.00	1.16265	0.581323	+	0.813673i	$0.302535\pi$
$128$	0	0
$129$	−246.000	−0.167900
$130$	0	0
$131$	−1476.00	−0.984418	−0.492209	−	0.870477i	$-0.663810\pi$
$131$	−1476.00	−0.984418	−0.492209	+	0.870477i	$0.663810\pi$
$132$	1247.08	0.822304
$133$	252.000	0.164295
$134$	0	0
$135$	140.296	0.0894427
$136$	0	0
$137$	−1013.25	−0.631882	−0.315941	−	0.948779i	$-0.602320\pi$
$137$	−1013.25	−0.631882	−0.315941	+	0.948779i	$0.602320\pi$
$138$	0	0
$139$	1124.00	0.685874	0.342937	−	0.939358i	$-0.388578\pi$
$139$	1124.00	0.685874	0.342937	+	0.939358i	$0.388578\pi$
$140$	−432.000	−0.260790
$141$	218.238	0.130347
$142$	0	0
$143$	0	0
$144$	576.000	0.333333
$145$	514.419	0.294622
$146$	0	0
$147$	705.000	0.395561
$148$	−900.666	−0.500232
$149$	3268.38	1.79702	0.898510	−	0.438952i	$-0.144650\pi$
$149$	3268.38	1.79702	0.898510	+	0.438952i	$0.144650\pi$
$150$	0	0
$151$	−1638.52	−0.883052	−0.441526	−	0.897248i	$-0.645563\pi$
$151$	−1638.52	−0.883052	−0.441526	+	0.897248i	$0.645563\pi$
$152$	0	0
$153$	1053.00	0.556405
$154$	0	0
$155$	1008.00	0.522352
$156$	0	0
$157$	1259.00	0.639995	0.319997	−	0.947418i	$-0.396318\pi$
$157$	1259.00	0.639995	0.319997	+	0.947418i	$0.396318\pi$
$158$	0	0
$159$	783.000	0.390540
$160$	0	0
$161$	187.061	0.0915684
$162$	0	0
$163$	−2951.41	−1.41824	−0.709118	−	0.705089i	$-0.750907\pi$
$163$	−2951.41	−1.41824	−0.709118	+	0.705089i	$0.750907\pi$
$164$	−290.985	−0.138549
$165$	810.000	0.382172
$166$	0	0
$167$	−3138.48	−1.45427	−0.727133	−	0.686496i	$-0.759148\pi$
$167$	−3138.48	−1.45427	−0.727133	+	0.686496i	$0.759148\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−218.238	−0.0975971
$172$	−656.000	−0.290811
$173$	−4266.00	−1.87479	−0.937393	−	0.348273i	$-0.886768\pi$
$173$	−4266.00	−1.87479	−0.937393	+	0.348273i	$0.886768\pi$
$174$	0	0
$175$	1018.45	0.439927
$176$	3325.54	1.42427
$177$	−2369.45	−1.00621
$178$	0	0
$179$	−3006.00	−1.25519	−0.627595	−	0.778540i	$-0.715961\pi$
$179$	−3006.00	−1.25519	−0.627595	+	0.778540i	$0.715961\pi$
$180$	374.123	0.154919
$181$	1873.00	0.769166	0.384583	−	0.923090i	$-0.374345\pi$
$181$	1873.00	0.769166	0.384583	+	0.923090i	$0.374345\pi$
$182$	0	0
$183$	2157.00	0.871312
$184$	0	0
$185$	−585.000	−0.232487
$186$	0	0
$187$	6079.50	2.37742
$188$	581.969	0.225768
$189$	280.592	0.107990
$190$	0	0
$191$	−2736.00	−1.03649	−0.518246	−	0.855232i	$-0.673415\pi$
$191$	−2736.00	−1.03649	−0.518246	+	0.855232i	$0.673415\pi$
$192$	1536.00	0.577350
$193$	−2603.27	−0.970920	−0.485460	−	0.874259i	$-0.661348\pi$
$193$	−2603.27	−0.970920	−0.485460	+	0.874259i	$0.661348\pi$
$194$	0	0
$195$	0	0
$196$	1880.00	0.685131
$197$	−3720.45	−1.34554	−0.672768	−	0.739853i	$-0.734895\pi$
$197$	−3720.45	−1.34554	−0.672768	+	0.739853i	$0.734895\pi$
$198$	0	0
$199$	−1198.00	−0.426754	−0.213377	−	0.976970i	$-0.568446\pi$
$199$	−1198.00	−0.426754	−0.213377	+	0.976970i	$0.568446\pi$
$200$	0	0
$201$	−2109.64	−0.740310
$202$	0	0
$203$	1028.84	0.355716
$204$	2808.00	0.963722
$205$	−189.000	−0.0643919
$206$	0	0
$207$	−162.000	−0.0543951
$208$	0	0
$209$	−1260.00	−0.417014
$210$	0	0
$211$	2392.00	0.780436	0.390218	−	0.920722i	$-0.372400\pi$
$211$	2392.00	0.780436	0.390218	+	0.920722i	$0.372400\pi$
$212$	2088.00	0.676436
$213$	1402.96	0.451311
$214$	0	0
$215$	−426.084	−0.135157
$216$	0	0
$217$	2016.00	0.630668
$218$	0	0
$219$	−2052.48	−0.633305
$220$	2160.00	0.661942
$221$	0	0
$222$	0	0
$223$	2036.89	0.611661	0.305830	−	0.952086i	$-0.401066\pi$
$223$	2036.89	0.611661	0.305830	+	0.952086i	$0.401066\pi$
$224$	0	0
$225$	−882.000	−0.261333
$226$	0	0
$227$	2151.21	0.628990	0.314495	−	0.949259i	$-0.398165\pi$
$227$	2151.21	0.628990	0.314495	+	0.949259i	$0.398165\pi$
$228$	−581.969	−0.169043
$229$	3471.03	1.00162	0.500812	−	0.865556i	$-0.333035\pi$
$229$	3471.03	1.00162	0.500812	+	0.865556i	$0.333035\pi$
$230$	0	0
$231$	1620.00	0.461421
$232$	0	0
$233$	1854.00	0.521286	0.260643	−	0.965435i	$-0.416065\pi$
$233$	1854.00	0.521286	0.260643	+	0.965435i	$0.416065\pi$
$234$	0	0
$235$	378.000	0.104928
$236$	−6318.52	−1.74280
$237$	1320.00	0.361786
$238$	0	0
$239$	4458.30	1.20662	0.603312	−	0.797505i	$-0.293847\pi$
$239$	4458.30	1.20662	0.603312	+	0.797505i	$0.293847\pi$
$240$	997.661	0.268328
$241$	417.424	0.111571	0.0557856	−	0.998443i	$-0.482234\pi$
$241$	417.424	0.111571	0.0557856	+	0.998443i	$0.482234\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	5752.00	1.50916
$245$	1221.10	0.318420
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3585.35	0.912498
$250$	0	0
$251$	−4104.00	−1.03204	−0.516020	−	0.856576i	$-0.672587\pi$
$251$	−4104.00	−1.03204	−0.516020	+	0.856576i	$0.672587\pi$
$252$	748.246	0.187044
$253$	−935.307	−0.232420
$254$	0	0
$255$	1823.85	0.447898
$256$	4096.00	1.00000
$257$	−1989.00	−0.482764	−0.241382	−	0.970430i	$-0.577601\pi$
$257$	−1989.00	−0.482764	−0.241382	+	0.970430i	$0.577601\pi$
$258$	0	0
$259$	−1170.00	−0.280696
$260$	0	0
$261$	−891.000	−0.211308
$262$	0	0
$263$	738.000	0.173031	0.0865153	−	0.996251i	$-0.472427\pi$
$263$	738.000	0.173031	0.0865153	+	0.996251i	$0.472427\pi$
$264$	0	0
$265$	1356.20	0.314379
$266$	0	0
$267$	−4551.83	−1.04332
$268$	−5625.70	−1.28226
$269$	−2106.00	−0.477342	−0.238671	−	0.971100i	$-0.576712\pi$
$269$	−2106.00	−0.477342	−0.238671	+	0.971100i	$0.576712\pi$
$270$	0	0
$271$	685.892	0.153745	0.0768727	−	0.997041i	$-0.475507\pi$
$271$	685.892	0.153745	0.0768727	+	0.997041i	$0.475507\pi$
$272$	7488.00	1.66922
$273$	0	0
$274$	0	0
$275$	−5092.23	−1.11663
$276$	−432.000	−0.0942150
$277$	−3665.00	−0.794977	−0.397488	−	0.917607i	$-0.630118\pi$
$277$	−3665.00	−0.794977	−0.397488	+	0.917607i	$0.630118\pi$
$278$	0	0
$279$	−1745.91	−0.374641
$280$	0	0
$281$	−1719.93	−0.365132	−0.182566	−	0.983194i	$-0.558440\pi$
$281$	−1719.93	−0.365132	−0.182566	+	0.983194i	$0.558440\pi$
$282$	0	0
$283$	1826.00	0.383549	0.191775	−	0.981439i	$-0.438576\pi$
$283$	1826.00	0.383549	0.191775	+	0.981439i	$0.438576\pi$
$284$	3741.23	0.781694
$285$	−378.000	−0.0785642
$286$	0	0
$287$	−378.000	−0.0777444
$288$	0	0
$289$	8776.00	1.78628
$290$	0	0
$291$	3471.03	0.699228
$292$	−5473.28	−1.09692
$293$	504.027	0.100497	0.0502484	−	0.998737i	$-0.483999\pi$
$293$	504.027	0.100497	0.0502484	+	0.998737i	$0.483999\pi$
$294$	0	0
$295$	−4104.00	−0.809980
$296$	0	0
$297$	−1402.96	−0.274101
$298$	0	0
$299$	0	0
$300$	−2352.00	−0.452643
$301$	−852.169	−0.163183
$302$	0	0
$303$	4725.00	0.895855
$304$	−1551.92	−0.292791
$305$	3736.03	0.701392
$306$	0	0
$307$	1950.29	0.362570	0.181285	−	0.983431i	$-0.441974\pi$
$307$	1950.29	0.362570	0.181285	+	0.983431i	$0.441974\pi$
$308$	4320.00	0.799204
$309$	2382.00	0.438535
$310$	0	0
$311$	3798.00	0.692491	0.346246	−	0.938144i	$-0.387456\pi$
$311$	3798.00	0.692491	0.346246	+	0.938144i	$0.387456\pi$
$312$	0	0
$313$	1378.00	0.248847	0.124424	−	0.992229i	$-0.460292\pi$
$313$	1378.00	0.248847	0.124424	+	0.992229i	$0.460292\pi$
$314$	0	0
$315$	486.000	0.0869302
$316$	3520.00	0.626631
$317$	−7103.14	−1.25852	−0.629262	−	0.777193i	$-0.716643\pi$
$317$	−7103.14	−1.25852	−0.629262	+	0.777193i	$0.716643\pi$
$318$	0	0
$319$	−5144.19	−0.902882
$320$	2660.43	0.464758
$321$	−1350.00	−0.234734
$322$	0	0
$323$	−2837.10	−0.488732
$324$	−648.000	−0.111111
$325$	0	0
$326$	0	0
$327$	1787.48	0.302286
$328$	0	0
$329$	756.000	0.126686
$330$	0	0
$331$	10073.6	1.67280	0.836398	−	0.548122i	$-0.184657\pi$
$331$	10073.6	1.67280	0.836398	+	0.548122i	$0.184657\pi$
$332$	9560.92	1.58049
$333$	1013.25	0.166744
$334$	0	0
$335$	−3654.00	−0.595938
$336$	1995.32	0.323970
$337$	−9001.00	−1.45494	−0.727471	−	0.686138i	$-0.759305\pi$
$337$	−9001.00	−1.45494	−0.727471	+	0.686138i	$0.759305\pi$
$338$	0	0
$339$	5103.00	0.817572
$340$	4863.60	0.775781
$341$	−10080.0	−1.60077
$342$	0	0
$343$	6006.75	0.945581
$344$	0	0
$345$	−280.592	−0.0437872
$346$	0	0
$347$	3294.00	0.509600	0.254800	−	0.966994i	$-0.417990\pi$
$347$	3294.00	0.509600	0.254800	+	0.966994i	$0.417990\pi$
$348$	−2376.00	−0.365997
$349$	−10544.7	−1.61732	−0.808662	−	0.588273i	$-0.799808\pi$
$349$	−10544.7	−1.61732	−0.808662	+	0.588273i	$0.799808\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2478.56	0.373713	0.186856	−	0.982387i	$-0.440170\pi$
$353$	2478.56	0.373713	0.186856	+	0.982387i	$0.440170\pi$
$354$	0	0
$355$	2430.00	0.363299
$356$	−12138.2	−1.80709
$357$	3647.70	0.540775
$358$	0	0
$359$	−5414.39	−0.795991	−0.397995	−	0.917387i	$-0.630294\pi$
$359$	−5414.39	−0.795991	−0.397995	+	0.917387i	$0.630294\pi$
$360$	0	0
$361$	−6271.00	−0.914273
$362$	0	0
$363$	−4107.00	−0.593834
$364$	0	0
$365$	−3555.00	−0.509801
$366$	0	0
$367$	−9946.00	−1.41465	−0.707326	−	0.706888i	$-0.750099\pi$
$367$	−9946.00	−1.41465	−0.707326	+	0.706888i	$0.750099\pi$
$368$	−1152.00	−0.163185
$369$	327.358	0.0461831
$370$	0	0
$371$	2712.39	0.379570
$372$	−4655.75	−0.648897
$373$	−7301.00	−1.01349	−0.506745	−	0.862096i	$-0.669151\pi$
$373$	−7301.00	−1.01349	−0.506745	+	0.862096i	$0.669151\pi$
$374$	0	0
$375$	−3476.23	−0.478697
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3422.53	0.463862	0.231931	−	0.972732i	$-0.425496\pi$
$379$	3422.53	0.463862	0.231931	+	0.972732i	$0.425496\pi$
$380$	−1008.00	−0.136077
$381$	−4992.00	−0.671254
$382$	0	0
$383$	−5778.12	−0.770883	−0.385442	−	0.922732i	$-0.625951\pi$
$383$	−5778.12	−0.770883	−0.385442	+	0.922732i	$0.625951\pi$
$384$	0	0
$385$	2805.92	0.371436
$386$	0	0
$387$	738.000	0.0969371
$388$	9256.08	1.21110
$389$	−9153.00	−1.19300	−0.596498	−	0.802614i	$-0.703442\pi$
$389$	−9153.00	−1.19300	−0.596498	+	0.802614i	$0.703442\pi$
$390$	0	0
$391$	−2106.00	−0.272391
$392$	0	0
$393$	4428.00	0.568354
$394$	0	0
$395$	2286.31	0.291232
$396$	−3741.23	−0.474757
$397$	−2023.04	−0.255751	−0.127876	−	0.991790i	$-0.540816\pi$
$397$	−2023.04	−0.255751	−0.127876	+	0.991790i	$0.540816\pi$
$398$	0	0
$399$	−756.000	−0.0948555
$400$	−6272.00	−0.784000
$401$	8308.65	1.03470	0.517349	−	0.855774i	$-0.326919\pi$
$401$	8308.65	1.03470	0.517349	+	0.855774i	$0.326919\pi$
$402$	0	0
$403$	0	0
$404$	12600.0	1.55167
$405$	−420.888	−0.0516398
$406$	0	0
$407$	5850.00	0.712466
$408$	0	0
$409$	−10418.3	−1.25954	−0.629769	−	0.776782i	$-0.716850\pi$
$409$	−10418.3	−1.25954	−0.629769	+	0.776782i	$0.716850\pi$
$410$	0	0
$411$	3039.75	0.364817
$412$	6352.00	0.759565
$413$	−8208.00	−0.977940
$414$	0	0
$415$	6210.00	0.734547
$416$	0	0
$417$	−3372.00	−0.395989
$418$	0	0
$419$	4176.00	0.486900	0.243450	−	0.969913i	$-0.421721\pi$
$419$	4176.00	0.486900	0.243450	+	0.969913i	$0.421721\pi$
$420$	1296.00	0.150567
$421$	−14471.3	−1.67527	−0.837633	−	0.546233i	$-0.816061\pi$
$421$	−14471.3	−1.67527	−0.837633	+	0.546233i	$0.816061\pi$
$422$	0	0
$423$	−654.715	−0.0752561
$424$	0	0
$425$	−11466.0	−1.30867
$426$	0	0
$427$	7472.07	0.846835
$428$	−3600.00	−0.406571
$429$	0	0
$430$	0	0
$431$	6578.33	0.735190	0.367595	−	0.929986i	$-0.380181\pi$
$431$	6578.33	0.735190	0.367595	+	0.929986i	$0.380181\pi$
$432$	−1728.00	−0.192450
$433$	6605.00	0.733062	0.366531	−	0.930406i	$-0.380545\pi$
$433$	6605.00	0.733062	0.366531	+	0.930406i	$0.380545\pi$
$434$	0	0
$435$	−1543.26	−0.170100
$436$	4766.60	0.523576
$437$	436.477	0.0477792
$438$	0	0
$439$	8542.00	0.928673	0.464336	−	0.885659i	$-0.346293\pi$
$439$	8542.00	0.928673	0.464336	+	0.885659i	$0.346293\pi$
$440$	0	0
$441$	−2115.00	−0.228377
$442$	0	0
$443$	−14328.0	−1.53667	−0.768334	−	0.640049i	$-0.778914\pi$
$443$	−14328.0	−1.53667	−0.768334	+	0.640049i	$0.778914\pi$
$444$	2702.00	0.288809
$445$	−7884.00	−0.839859
$446$	0	0
$447$	−9805.14	−1.03751
$448$	5320.86	0.561132
$449$	−3013.77	−0.316767	−0.158384	−	0.987378i	$-0.550628\pi$
$449$	−3013.77	−0.316767	−0.158384	+	0.987378i	$0.550628\pi$
$450$	0	0
$451$	1890.00	0.197332
$452$	13608.0	1.41608
$453$	4915.56	0.509830
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2887.33	0.295544	0.147772	−	0.989021i	$-0.452790\pi$
$457$	2887.33	0.295544	0.147772	+	0.989021i	$0.452790\pi$
$458$	0	0
$459$	−3159.00	−0.321241
$460$	−748.246	−0.0758416
$461$	3600.93	0.363801	0.181900	−	0.983317i	$-0.441775\pi$
$461$	3600.93	0.363801	0.181900	+	0.983317i	$0.441775\pi$
$462$	0	0
$463$	2677.75	0.268781	0.134391	−	0.990928i	$-0.457092\pi$
$463$	2677.75	0.268781	0.134391	+	0.990928i	$0.457092\pi$
$464$	−6336.00	−0.633925
$465$	−3024.00	−0.301580
$466$	0	0
$467$	−13878.0	−1.37515	−0.687577	−	0.726111i	$-0.741326\pi$
$467$	−13878.0	−1.37515	−0.687577	+	0.726111i	$0.741326\pi$
$468$	0	0
$469$	−7308.00	−0.719514
$470$	0	0
$471$	−3777.00	−0.369501
$472$	0	0
$473$	4260.84	0.414194
$474$	0	0
$475$	2376.37	0.229548
$476$	9727.20	0.936650
$477$	−2349.00	−0.225479
$478$	0	0
$479$	1101.58	0.105079	0.0525393	−	0.998619i	$-0.483269\pi$
$479$	1101.58	0.105079	0.0525393	+	0.998619i	$0.483269\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−561.184	−0.0528670
$484$	−10952.0	−1.02855
$485$	6012.00	0.562868
$486$	0	0
$487$	−17123.1	−1.59326	−0.796632	−	0.604464i	$-0.793387\pi$
$487$	−17123.1	−1.59326	−0.796632	+	0.604464i	$0.793387\pi$
$488$	0	0
$489$	8854.24	0.818820
$490$	0	0
$491$	−450.000	−0.0413609	−0.0206805	−	0.999786i	$-0.506583\pi$
$491$	−450.000	−0.0413609	−0.0206805	+	0.999786i	$0.506583\pi$
$492$	872.954	0.0799914
$493$	−11583.0	−1.05816
$494$	0	0
$495$	−2430.00	−0.220647
$496$	−12415.3	−1.12392
$497$	4860.00	0.438633
$498$	0	0
$499$	−13219.0	−1.18590	−0.592950	−	0.805239i	$-0.702037\pi$
$499$	−13219.0	−1.18590	−0.592950	+	0.805239i	$0.702037\pi$
$500$	−9269.94	−0.829128
$501$	9415.43	0.839621
$502$	0	0
$503$	5346.00	0.473889	0.236945	−	0.971523i	$-0.423854\pi$
$503$	5346.00	0.473889	0.236945	+	0.971523i	$0.423854\pi$
$504$	0	0
$505$	8183.94	0.721150
$506$	0	0
$507$	0	0
$508$	−13312.0	−1.16265
$509$	5866.46	0.510857	0.255428	−	0.966828i	$-0.417783\pi$
$509$	5866.46	0.510857	0.255428	+	0.966828i	$0.417783\pi$
$510$	0	0
$511$	−7110.00	−0.615514
$512$	0	0
$513$	654.715	0.0563477
$514$	0	0
$515$	4125.75	0.353014
$516$	1968.00	0.167900
$517$	−3780.00	−0.321556
$518$	0	0
$519$	12798.0	1.08241
$520$	0	0
$521$	−9657.00	−0.812055	−0.406028	−	0.913861i	$-0.633086\pi$
$521$	−9657.00	−0.812055	−0.406028	+	0.913861i	$0.633086\pi$
$522$	0	0
$523$	21626.0	1.80811	0.904053	−	0.427421i	$-0.140578\pi$
$523$	21626.0	1.80811	0.904053	+	0.427421i	$0.140578\pi$
$524$	11808.0	0.984418
$525$	−3055.34	−0.253992
$526$	0	0
$527$	−22696.8	−1.87607
$528$	−9976.61	−0.822304
$529$	−11843.0	−0.973371
$530$	0	0
$531$	7108.34	0.580933
$532$	−2016.00	−0.164295
$533$	0	0
$534$	0	0
$535$	−2338.27	−0.188957
$536$	0	0
$537$	9018.00	0.724684
$538$	0	0
$539$	−12211.0	−0.975813
$540$	−1122.37	−0.0894427
$541$	−5371.09	−0.426841	−0.213421	−	0.976960i	$-0.568460\pi$
$541$	−5371.09	−0.426841	−0.213421	+	0.976960i	$0.568460\pi$
$542$	0	0
$543$	−5619.00	−0.444078
$544$	0	0
$545$	3096.00	0.243336
$546$	0	0
$547$	16946.0	1.32460	0.662302	−	0.749237i	$-0.269579\pi$
$547$	16946.0	1.32460	0.662302	+	0.749237i	$0.269579\pi$
$548$	8106.00	0.631882
$549$	−6471.00	−0.503052
$550$	0	0
$551$	2400.62	0.185608
$552$	0	0
$553$	4572.61	0.351623
$554$	0	0
$555$	1755.00	0.134226
$556$	−8992.00	−0.685874
$557$	3860.74	0.293689	0.146845	−	0.989160i	$-0.453088\pi$
$557$	3860.74	0.293689	0.146845	+	0.989160i	$0.453088\pi$
$558$	0	0
$559$	0	0
$560$	3456.00	0.260790
$561$	−18238.5	−1.37260
$562$	0	0
$563$	−21672.0	−1.62232	−0.811160	−	0.584825i	$-0.801163\pi$
$563$	−21672.0	−1.62232	−0.811160	+	0.584825i	$0.801163\pi$
$564$	−1745.91	−0.130347
$565$	8838.66	0.658133
$566$	0	0
$567$	−841.777	−0.0623480
$568$	0	0
$569$	1386.00	0.102116	0.0510581	−	0.998696i	$-0.483741\pi$
$569$	1386.00	0.102116	0.0510581	+	0.998696i	$0.483741\pi$
$570$	0	0
$571$	−1162.00	−0.0851632	−0.0425816	−	0.999093i	$-0.513558\pi$
$571$	−1162.00	−0.0851632	−0.0425816	+	0.999093i	$0.513558\pi$
$572$	0	0
$573$	8208.00	0.598419
$574$	0	0
$575$	1764.00	0.127937
$576$	−4608.00	−0.333333
$577$	8045.38	0.580474	0.290237	−	0.956955i	$-0.406266\pi$
$577$	8045.38	0.580474	0.290237	+	0.956955i	$0.406266\pi$
$578$	0	0
$579$	7809.82	0.560561
$580$	−4115.35	−0.294622
$581$	12420.0	0.886865
$582$	0	0
$583$	−13562.0	−0.963429
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27622.7	1.94227	0.971135	−	0.238530i	$-0.0766654\pi$
$587$	27622.7	1.94227	0.971135	+	0.238530i	$0.0766654\pi$
$588$	−5640.00	−0.395561
$589$	4704.00	0.329075
$590$	0	0
$591$	11161.3	0.776846
$592$	7205.33	0.500232
$593$	−275.396	−0.0190711	−0.00953555	−	0.999955i	$-0.503035\pi$
$593$	−275.396	−0.0190711	−0.00953555	+	0.999955i	$0.503035\pi$
$594$	0	0
$595$	6318.00	0.435316
$596$	−26147.0	−1.79702
$597$	3594.00	0.246386
$598$	0	0
$599$	22356.0	1.52494	0.762472	−	0.647021i	$-0.223986\pi$
$599$	22356.0	1.52494	0.762472	+	0.647021i	$0.223986\pi$
$600$	0	0
$601$	−18083.0	−1.22732	−0.613661	−	0.789569i	$-0.710304\pi$
$601$	−18083.0	−1.22732	−0.613661	+	0.789569i	$0.710304\pi$
$602$	0	0
$603$	6328.91	0.427418
$604$	13108.2	0.883052
$605$	−7113.53	−0.478027
$606$	0	0
$607$	5480.00	0.366435	0.183218	−	0.983072i	$-0.441349\pi$
$607$	5480.00	0.366435	0.183218	+	0.983072i	$0.441349\pi$
$608$	0	0
$609$	−3086.51	−0.205373
$610$	0	0
$611$	0	0
$612$	−8424.00	−0.556405
$613$	−17737.9	−1.16872	−0.584362	−	0.811493i	$-0.698655\pi$
$613$	−17737.9	−1.16872	−0.584362	+	0.811493i	$0.698655\pi$
$614$	0	0
$615$	567.000	0.0371767
$616$	0	0
$617$	9867.49	0.643842	0.321921	−	0.946767i	$-0.395672\pi$
$617$	9867.49	0.643842	0.321921	+	0.946767i	$0.395672\pi$
$618$	0	0
$619$	4115.35	0.267221	0.133611	−	0.991034i	$-0.457343\pi$
$619$	4115.35	0.267221	0.133611	+	0.991034i	$0.457343\pi$
$620$	−8064.00	−0.522352
$621$	486.000	0.0314050
$622$	0	0
$623$	−15768.0	−1.01402
$624$	0	0
$625$	6229.00	0.398656
$626$	0	0
$627$	3780.00	0.240763
$628$	−10072.0	−0.639995
$629$	13172.2	0.834995
$630$	0	0
$631$	12664.8	0.799011	0.399506	−	0.916731i	$-0.369182\pi$
$631$	12664.8	0.799011	0.399506	+	0.916731i	$0.369182\pi$
$632$	0	0
$633$	−7176.00	−0.450585
$634$	0	0
$635$	−8646.40	−0.540349
$636$	−6264.00	−0.390540
$637$	0	0
$638$	0	0
$639$	−4208.88	−0.260565
$640$	0	0
$641$	3789.00	0.233473	0.116737	−	0.993163i	$-0.462757\pi$
$641$	3789.00	0.233473	0.116737	+	0.993163i	$0.462757\pi$
$642$	0	0
$643$	16911.7	1.03722	0.518611	−	0.855010i	$-0.326449\pi$
$643$	16911.7	1.03722	0.518611	+	0.855010i	$0.326449\pi$
$644$	−1496.49	−0.0915684
$645$	1278.25	0.0780328
$646$	0	0
$647$	27792.0	1.68874	0.844371	−	0.535759i	$-0.179974\pi$
$647$	27792.0	1.68874	0.844371	+	0.535759i	$0.179974\pi$
$648$	0	0
$649$	41040.0	2.48222
$650$	0	0
$651$	−6048.00	−0.364116
$652$	23611.3	1.41824
$653$	−594.000	−0.0355973	−0.0177986	−	0.999842i	$-0.505666\pi$
$653$	−594.000	−0.0355973	−0.0177986	+	0.999842i	$0.505666\pi$
$654$	0	0
$655$	7669.52	0.457516
$656$	2327.88	0.138549
$657$	6157.44	0.365639
$658$	0	0
$659$	17748.0	1.04911	0.524555	−	0.851376i	$-0.324232\pi$
$659$	17748.0	1.04911	0.524555	+	0.851376i	$0.324232\pi$
$660$	−6480.00	−0.382172
$661$	15791.1	0.929203	0.464601	−	0.885520i	$-0.346198\pi$
$661$	15791.1	0.929203	0.464601	+	0.885520i	$0.346198\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1309.43	−0.0763572
$666$	0	0
$667$	1782.00	0.103447
$668$	25107.8	1.45427
$669$	−6110.68	−0.353143
$670$	0	0
$671$	−37360.3	−2.14945
$672$	0	0
$673$	−20933.0	−1.19897	−0.599486	−	0.800385i	$-0.704628\pi$
$673$	−20933.0	−1.19897	−0.599486	+	0.800385i	$0.704628\pi$
$674$	0	0
$675$	2646.00	0.150881
$676$	0	0
$677$	3402.00	0.193131	0.0965653	−	0.995327i	$-0.469214\pi$
$677$	3402.00	0.193131	0.0965653	+	0.995327i	$0.469214\pi$
$678$	0	0
$679$	12024.0	0.679586
$680$	0	0
$681$	−6453.62	−0.363147
$682$	0	0
$683$	−24983.1	−1.39964	−0.699818	−	0.714321i	$-0.746736\pi$
$683$	−24983.1	−1.39964	−0.699818	+	0.714321i	$0.746736\pi$
$684$	1745.91	0.0975971
$685$	5265.00	0.293672
$686$	0	0
$687$	−10413.1	−0.578288
$688$	5248.00	0.290811
$689$	0	0
$690$	0	0
$691$	−13866.8	−0.763412	−0.381706	−	0.924284i	$-0.624663\pi$
$691$	−13866.8	−0.763412	−0.381706	+	0.924284i	$0.624663\pi$
$692$	34128.0	1.87479
$693$	−4860.00	−0.266401
$694$	0	0
$695$	−5840.48	−0.318765
$696$	0	0
$697$	4255.65	0.231269
$698$	0	0
$699$	−5562.00	−0.300964
$700$	−8147.57	−0.439927
$701$	21906.0	1.18028	0.590141	−	0.807300i	$-0.299072\pi$
$701$	21906.0	1.18028	0.590141	+	0.807300i	$0.299072\pi$
$702$	0	0
$703$	−2730.00	−0.146464
$704$	−26604.3	−1.42427
$705$	−1134.00	−0.0605800
$706$	0	0
$707$	16367.9	0.870690
$708$	18955.6	1.00621
$709$	13057.9	0.691680	0.345840	−	0.938294i	$-0.387594\pi$
$709$	13057.9	0.691680	0.345840	+	0.938294i	$0.387594\pi$
$710$	0	0
$711$	−3960.00	−0.208877
$712$	0	0
$713$	3491.81	0.183407
$714$	0	0
$715$	0	0
$716$	24048.0	1.25519
$717$	−13374.9	−0.696645
$718$	0	0
$719$	14220.0	0.737575	0.368788	−	0.929514i	$-0.379773\pi$
$719$	14220.0	0.737575	0.368788	+	0.929514i	$0.379773\pi$
$720$	−2992.98	−0.154919
$721$	8251.49	0.426216
$722$	0	0
$723$	−1252.27	−0.0644157
$724$	−14984.0	−0.769166
$725$	9702.00	0.496998
$726$	0	0
$727$	5282.00	0.269462	0.134731	−	0.990882i	$-0.456983\pi$
$727$	5282.00	0.269462	0.134731	+	0.990882i	$0.456983\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	9594.00	0.485427
$732$	−17256.0	−0.871312
$733$	−11419.4	−0.575424	−0.287712	−	0.957717i	$-0.592895\pi$
$733$	−11419.4	−0.575424	−0.287712	+	0.957717i	$0.592895\pi$
$734$	0	0
$735$	−3663.29	−0.183840
$736$	0	0
$737$	36540.0	1.82628
$738$	0	0
$739$	−20535.2	−1.02219	−0.511096	−	0.859524i	$-0.670760\pi$
$739$	−20535.2	−1.02219	−0.511096	+	0.859524i	$0.670760\pi$
$740$	4680.00	0.232487
$741$	0	0
$742$	0	0
$743$	20826.2	1.02832	0.514158	−	0.857696i	$-0.328105\pi$
$743$	20826.2	1.02832	0.514158	+	0.857696i	$0.328105\pi$
$744$	0	0
$745$	−16983.0	−0.835180
$746$	0	0
$747$	−10756.0	−0.526831
$748$	−48636.0	−2.37742
$749$	−4676.54	−0.228140
$750$	0	0
$751$	4834.00	0.234880	0.117440	−	0.993080i	$-0.462531\pi$
$751$	4834.00	0.234880	0.117440	+	0.993080i	$0.462531\pi$
$752$	−4655.75	−0.225768
$753$	12312.0	0.595849
$754$	0	0
$755$	8514.00	0.410406
$756$	−2244.74	−0.107990
$757$	9046.00	0.434323	0.217161	−	0.976136i	$-0.430320\pi$
$757$	9046.00	0.434323	0.217161	+	0.976136i	$0.430320\pi$
$758$	0	0
$759$	2805.92	0.134188
$760$	0	0
$761$	12034.3	0.573249	0.286625	−	0.958043i	$-0.407467\pi$
$761$	12034.3	0.573249	0.286625	+	0.958043i	$0.407467\pi$
$762$	0	0
$763$	6192.00	0.293795
$764$	21888.0	1.03649
$765$	−5471.55	−0.258594
$766$	0	0
$767$	0	0
$768$	−12288.0	−0.577350
$769$	−37543.9	−1.76056	−0.880279	−	0.474457i	$-0.842645\pi$
$769$	−37543.9	−1.76056	−0.880279	+	0.474457i	$0.842645\pi$
$770$	0	0
$771$	5967.00	0.278724
$772$	20826.2	0.970920
$773$	−15713.2	−0.731130	−0.365565	−	0.930786i	$-0.619124\pi$
$773$	−15713.2	−0.731130	−0.365565	+	0.930786i	$0.619124\pi$
$774$	0	0
$775$	19011.0	0.881155
$776$	0	0
$777$	3510.00	0.162060
$778$	0	0
$779$	−882.000	−0.0405660
$780$	0	0
$781$	−24300.0	−1.11334
$782$	0	0
$783$	2673.00	0.121999
$784$	−15040.0	−0.685131
$785$	−6541.96	−0.297443
$786$	0	0
$787$	3755.09	0.170082	0.0850409	−	0.996377i	$-0.472898\pi$
$787$	3755.09	0.170082	0.0850409	+	0.996377i	$0.472898\pi$
$788$	29763.6	1.34554
$789$	−2214.00	−0.0998992
$790$	0	0
$791$	17677.3	0.794605
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−4068.59	−0.181507
$796$	9584.00	0.426754
$797$	7830.00	0.347996	0.173998	−	0.984746i	$-0.444331\pi$
$797$	7830.00	0.347996	0.173998	+	0.984746i	$0.444331\pi$
$798$	0	0
$799$	−8511.30	−0.376856
$800$	0	0
$801$	13655.5	0.602363
$802$	0	0
$803$	35550.0	1.56231
$804$	16877.1	0.740310
$805$	−972.000	−0.0425571
$806$	0	0
$807$	6318.00	0.275594
$808$	0	0
$809$	−6165.00	−0.267923	−0.133962	−	0.990987i	$-0.542770\pi$
$809$	−6165.00	−0.267923	−0.133962	+	0.990987i	$0.542770\pi$
$810$	0	0
$811$	29839.8	1.29201	0.646003	−	0.763335i	$-0.276440\pi$
$811$	29839.8	1.29201	0.646003	+	0.763335i	$0.276440\pi$
$812$	−8230.71	−0.355716
$813$	−2057.68	−0.0887649
$814$	0	0
$815$	15336.0	0.659137
$816$	−22464.0	−0.963722
$817$	−1988.39	−0.0851470
$818$	0	0
$819$	0	0
$820$	1512.00	0.0643919
$821$	29763.6	1.26523	0.632616	−	0.774466i	$-0.281981\pi$
$821$	29763.6	1.26523	0.632616	+	0.774466i	$0.281981\pi$
$822$	0	0
$823$	8920.00	0.377803	0.188901	−	0.981996i	$-0.439507\pi$
$823$	8920.00	0.377803	0.188901	+	0.981996i	$0.439507\pi$
$824$	0	0
$825$	15276.7	0.644686
$826$	0	0
$827$	−18041.0	−0.758583	−0.379292	−	0.925277i	$-0.623832\pi$
$827$	−18041.0	−0.758583	−0.379292	+	0.925277i	$0.623832\pi$
$828$	1296.00	0.0543951
$829$	21023.0	0.880771	0.440385	−	0.897809i	$-0.354842\pi$
$829$	21023.0	0.880771	0.440385	+	0.897809i	$0.354842\pi$
$830$	0	0
$831$	10995.0	0.458980
$832$	0	0
$833$	−27495.0	−1.14363
$834$	0	0
$835$	16308.0	0.675882
$836$	10080.0	0.417014
$837$	5237.72	0.216299
$838$	0	0
$839$	22073.3	0.908288	0.454144	−	0.890928i	$-0.349945\pi$
$839$	22073.3	0.908288	0.454144	+	0.890928i	$0.349945\pi$
$840$	0	0
$841$	−14588.0	−0.598139
$842$	0	0
$843$	5159.78	0.210809
$844$	−19136.0	−0.780436
$845$	0	0
$846$	0	0
$847$	−14227.1	−0.577152
$848$	−16704.0	−0.676436
$849$	−5478.00	−0.221442
$850$	0	0
$851$	−2026.50	−0.0816304
$852$	−11223.7	−0.451311
$853$	−26609.5	−1.06810	−0.534051	−	0.845452i	$-0.679331\pi$
$853$	−26609.5	−1.06810	−0.534051	+	0.845452i	$0.679331\pi$
$854$	0	0
$855$	1134.00	0.0453590
$856$	0	0
$857$	12771.0	0.509042	0.254521	−	0.967067i	$-0.418082\pi$
$857$	12771.0	0.509042	0.254521	+	0.967067i	$0.418082\pi$
$858$	0	0
$859$	17134.0	0.680564	0.340282	−	0.940323i	$-0.389477\pi$
$859$	17134.0	0.680564	0.340282	+	0.940323i	$0.389477\pi$
$860$	3408.68	0.135157
$861$	1134.00	0.0448857
$862$	0	0
$863$	7929.33	0.312766	0.156383	−	0.987696i	$-0.450017\pi$
$863$	7929.33	0.312766	0.156383	+	0.987696i	$0.450017\pi$
$864$	0	0
$865$	22166.8	0.871322
$866$	0	0
$867$	−26328.0	−1.03131
$868$	−16128.0	−0.630668
$869$	−22863.1	−0.892493
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−10413.1	−0.403700
$874$	0	0
$875$	−12042.0	−0.465250
$876$	16419.8	0.633305
$877$	9864.03	0.379800	0.189900	−	0.981803i	$-0.439184\pi$
$877$	9864.03	0.379800	0.189900	+	0.981803i	$0.439184\pi$
$878$	0	0
$879$	−1512.08	−0.0580218
$880$	−17280.0	−0.661942
$881$	−29169.0	−1.11547	−0.557735	−	0.830019i	$-0.688329\pi$
$881$	−29169.0	−1.11547	−0.557735	+	0.830019i	$0.688329\pi$
$882$	0	0
$883$	−928.000	−0.0353677	−0.0176839	−	0.999844i	$-0.505629\pi$
$883$	−928.000	−0.0353677	−0.0176839	+	0.999844i	$0.505629\pi$
$884$	0	0
$885$	12312.0	0.467642
$886$	0	0
$887$	−14400.0	−0.545101	−0.272551	−	0.962141i	$-0.587867\pi$
$887$	−14400.0	−0.545101	−0.272551	+	0.962141i	$0.587867\pi$
$888$	0	0
$889$	−17292.8	−0.652398
$890$	0	0
$891$	4208.88	0.158252
$892$	−16295.1	−0.611661
$893$	1764.00	0.0661030
$894$	0	0
$895$	15619.6	0.583360
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19205.0	0.712483
$900$	7056.00	0.261333
$901$	−30537.0	−1.12912
$902$	0	0
$903$	2556.51	0.0942140
$904$	0	0
$905$	−9732.39	−0.357476
$906$	0	0
$907$	−19684.0	−0.720614	−0.360307	−	0.932834i	$-0.617328\pi$
$907$	−19684.0	−0.720614	−0.360307	+	0.932834i	$0.617328\pi$
$908$	−17209.7	−0.628990
$909$	−14175.0	−0.517222
$910$	0	0
$911$	24480.0	0.890295	0.445147	−	0.895457i	$-0.353151\pi$
$911$	24480.0	0.890295	0.445147	+	0.895457i	$0.353151\pi$
$912$	4655.75	0.169043
$913$	−62100.0	−2.25105
$914$	0	0
$915$	−11208.1	−0.404949
$916$	−27768.2	−1.00162
$917$	15339.0	0.552388
$918$	0	0
$919$	38608.0	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$919$	38608.0	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$920$	0	0
$921$	−5850.87	−0.209330
$922$	0	0
$923$	0	0
$924$	−12960.0	−0.461421
$925$	−11033.2	−0.392182
$926$	0	0
$927$	−7146.00	−0.253188
$928$	0	0
$929$	10210.4	0.360596	0.180298	−	0.983612i	$-0.442294\pi$
$929$	10210.4	0.360596	0.180298	+	0.983612i	$0.442294\pi$
$930$	0	0
$931$	5698.45	0.200600
$932$	−14832.0	−0.521286
$933$	−11394.0	−0.399810
$934$	0	0
$935$	−31590.0	−1.10492
$936$	0	0
$937$	28495.0	0.993480	0.496740	−	0.867899i	$-0.334530\pi$
$937$	28495.0	0.993480	0.496740	+	0.867899i	$0.334530\pi$
$938$	0	0
$939$	−4134.00	−0.143672
$940$	−3024.00	−0.104928
$941$	−10724.9	−0.371541	−0.185771	−	0.982593i	$-0.559478\pi$
$941$	−10724.9	−0.371541	−0.185771	+	0.982593i	$0.559478\pi$
$942$	0	0
$943$	−654.715	−0.0226092
$944$	50548.2	1.74280
$945$	−1458.00	−0.0501891
$946$	0	0
$947$	33962.1	1.16538	0.582692	−	0.812693i	$-0.301999\pi$
$947$	33962.1	1.16538	0.582692	+	0.812693i	$0.301999\pi$
$948$	−10560.0	−0.361786
$949$	0	0
$950$	0	0
$951$	21309.4	0.726609
$952$	0	0
$953$	−23814.0	−0.809456	−0.404728	−	0.914437i	$-0.632634\pi$
$953$	−23814.0	−0.809456	−0.404728	+	0.914437i	$0.632634\pi$
$954$	0	0
$955$	14216.7	0.481718
$956$	−35666.4	−1.20662
$957$	15432.6	0.521279
$958$	0	0
$959$	10530.0	0.354569
$960$	−7981.29	−0.268328
$961$	7841.00	0.263200
$962$	0	0
$963$	4050.00	0.135524
$964$	−3339.39	−0.111571
$965$	13527.0	0.451243
$966$	0	0
$967$	−51549.3	−1.71429	−0.857143	−	0.515079i	$-0.827762\pi$
$967$	−51549.3	−1.71429	−0.857143	+	0.515079i	$0.827762\pi$
$968$	0	0
$969$	8511.30	0.282170
$970$	0	0
$971$	−12312.0	−0.406911	−0.203456	−	0.979084i	$-0.565217\pi$
$971$	−12312.0	−0.406911	−0.203456	+	0.979084i	$0.565217\pi$
$972$	1944.00	0.0641500
$973$	−11681.0	−0.384865
$974$	0	0
$975$	0	0
$976$	−46016.0	−1.50916
$977$	−21538.1	−0.705285	−0.352642	−	0.935758i	$-0.614717\pi$
$977$	−21538.1	−0.705285	−0.352642	+	0.935758i	$0.614717\pi$
$978$	0	0
$979$	78840.0	2.57379
$980$	−9768.77	−0.318420
$981$	−5362.43	−0.174525
$982$	0	0
$983$	−32611.1	−1.05812	−0.529060	−	0.848585i	$-0.677455\pi$
$983$	−32611.1	−1.05812	−0.529060	+	0.848585i	$0.677455\pi$
$984$	0	0
$985$	19332.0	0.625349
$986$	0	0
$987$	−2268.00	−0.0731421
$988$	0	0
$989$	−1476.00	−0.0474561
$990$	0	0
$991$	−22330.0	−0.715778	−0.357889	−	0.933764i	$-0.616503\pi$
$991$	−22330.0	−0.715778	−0.357889	+	0.933764i	$0.616503\pi$
$992$	0	0
$993$	−30220.8	−0.965789
$994$	0	0
$995$	6224.99	0.198337
$996$	−28682.8	−0.912498
$997$	−24931.0	−0.791949	−0.395974	−	0.918262i	$-0.629593\pi$
$997$	−24931.0	−0.791949	−0.395974	+	0.918262i	$0.629593\pi$
$998$	0	0
$999$	−3039.75	−0.0962697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.g.1.1		2
3.2	odd	2		1521.4.a.m.1.2		2
13.5	odd	4		507.4.b.a.337.2		2
13.6	odd	12		39.4.j.a.10.1	yes	2
13.8	odd	4		507.4.b.a.337.1		2
13.11	odd	12		39.4.j.a.4.1	✓	2
13.12	even	2	inner	507.4.a.g.1.2		2
39.11	even	12		117.4.q.b.82.1		2
39.32	even	12		117.4.q.b.10.1		2
39.38	odd	2		1521.4.a.m.1.1		2
52.11	even	12		624.4.bv.a.433.1		2
52.19	even	12		624.4.bv.a.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.a.4.1	✓	2	13.11	odd	12
39.4.j.a.10.1	yes	2	13.6	odd	12
117.4.q.b.10.1		2	39.32	even	12
117.4.q.b.82.1		2	39.11	even	12
507.4.a.g.1.1		2	1.1	even	1	trivial
507.4.a.g.1.2		2	13.12	even	2	inner
507.4.b.a.337.1		2	13.8	odd	4
507.4.b.a.337.2		2	13.5	odd	4
624.4.bv.a.49.1		2	52.19	even	12
624.4.bv.a.433.1		2	52.11	even	12
1521.4.a.m.1.1		2	39.38	odd	2
1521.4.a.m.1.2		2	3.2	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -8.00000 q^{4} -5.19615 q^{5} -10.3923 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -8.00000 q^{4} -5.19615 q^{5} -10.3923 q^{7} +9.00000 q^{9} +51.9615 q^{11} +24.0000 q^{12} +15.5885 q^{15} +64.0000 q^{16} +117.000 q^{17} -24.2487 q^{19} +41.5692 q^{20} +31.1769 q^{21} -18.0000 q^{23} -98.0000 q^{25} -27.0000 q^{27} +83.1384 q^{28} -99.0000 q^{29} -193.990 q^{31} -155.885 q^{33} +54.0000 q^{35} -72.0000 q^{36} +112.583 q^{37} +36.3731 q^{41} +82.0000 q^{43} -415.692 q^{44} -46.7654 q^{45} -72.7461 q^{47} -192.000 q^{48} -235.000 q^{49} -351.000 q^{51} -261.000 q^{53} -270.000 q^{55} +72.7461 q^{57} +789.815 q^{59} -124.708 q^{60} -719.000 q^{61} -93.5307 q^{63} -512.000 q^{64} +703.213 q^{67} -936.000 q^{68} +54.0000 q^{69} -467.654 q^{71} +684.160 q^{73} +294.000 q^{75} +193.990 q^{76} -540.000 q^{77} -440.000 q^{79} -332.554 q^{80} +81.0000 q^{81} -1195.12 q^{83} -249.415 q^{84} -607.950 q^{85} +297.000 q^{87} +1517.28 q^{89} +144.000 q^{92} +581.969 q^{93} +126.000 q^{95} -1157.01 q^{97} +467.654 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 16 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 16 * q^4 + 18 * q^9	\( 2 q - 6 q^{3} - 16 q^{4} + 18 q^{9} + 48 q^{12} + 128 q^{16} + 234 q^{17} - 36 q^{23} - 196 q^{25} - 54 q^{27} - 198 q^{29} + 108 q^{35} - 144 q^{36} + 164 q^{43} - 384 q^{48} - 470 q^{49} - 702 q^{51} - 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} - 1872 q^{68} + 108 q^{69} + 588 q^{75} - 1080 q^{77} - 880 q^{79} + 162 q^{81} + 594 q^{87} + 288 q^{92} + 252 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 - 16 * q^4 + 18 * q^9 + 48 * q^12 + 128 * q^16 + 234 * q^17 - 36 * q^23 - 196 * q^25 - 54 * q^27 - 198 * q^29 + 108 * q^35 - 144 * q^36 + 164 * q^43 - 384 * q^48 - 470 * q^49 - 702 * q^51 - 522 * q^53 - 540 * q^55 - 1438 * q^61 - 1024 * q^64 - 1872 * q^68 + 108 * q^69 + 588 * q^75 - 1080 * q^77 - 880 * q^79 + 162 * q^81 + 594 * q^87 + 288 * q^92 + 252 * q^95

Introduction

L-functions

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