LMFDB - Embedded newform 1305.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,4,Mod(1,1305)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1305.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,1,0,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.9974925575$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 $ x^7 - x^6 - 35x^5 + 18x^4 + 329x^3 - 167x^2 - 767*x + 638
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.52131$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.52131 q^{2} +4.39964 q^{4} +5.00000 q^{5} -8.86571 q^{7} +12.6780 q^{8} -17.6066 q^{10} +61.9492 q^{11} +31.6208 q^{13} +31.2189 q^{14} -79.8403 q^{16} +48.6694 q^{17} -127.727 q^{19} +21.9982 q^{20} -218.143 q^{22} -184.026 q^{23} +25.0000 q^{25} -111.347 q^{26} -39.0059 q^{28} +29.0000 q^{29} -251.985 q^{31} +179.719 q^{32} -171.380 q^{34} -44.3285 q^{35} +197.740 q^{37} +449.765 q^{38} +63.3899 q^{40} +163.446 q^{41} -33.2231 q^{43} +272.555 q^{44} +648.012 q^{46} -528.945 q^{47} -264.399 q^{49} -88.0328 q^{50} +139.120 q^{52} +240.256 q^{53} +309.746 q^{55} -112.399 q^{56} -102.118 q^{58} +202.976 q^{59} +492.192 q^{61} +887.318 q^{62} +5.87624 q^{64} +158.104 q^{65} -229.556 q^{67} +214.128 q^{68} +156.095 q^{70} -21.2064 q^{71} -1145.48 q^{73} -696.303 q^{74} -561.952 q^{76} -549.224 q^{77} +95.6307 q^{79} -399.201 q^{80} -575.545 q^{82} -1239.97 q^{83} +243.347 q^{85} +116.989 q^{86} +785.391 q^{88} -116.863 q^{89} -280.341 q^{91} -809.648 q^{92} +1862.58 q^{94} -638.633 q^{95} -275.722 q^{97} +931.032 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

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 $

$2$	−3.52131	−1.24497	−0.622486	−	0.782631i	$-0.713877\pi$
$2$	−3.52131	−1.24497	−0.622486	+	0.782631i	$0.713877\pi$
$3$	0	0
$3$	0	0
$4$	4.39964	0.549955
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−8.86571	−0.478703	−0.239352	−	0.970933i	$-0.576935\pi$
$7$	−8.86571	−0.478703	−0.239352	+	0.970933i	$0.576935\pi$
$8$	12.6780	0.560293
$9$	0	0
$10$	−17.6066	−0.556768
$11$	61.9492	1.69804	0.849018	−	0.528364i	$-0.177194\pi$
$11$	61.9492	1.69804	0.849018	+	0.528364i	$0.177194\pi$
$12$	0	0
$13$	31.6208	0.674618	0.337309	−	0.941394i	$-0.390483\pi$
$13$	31.6208	0.674618	0.337309	+	0.941394i	$0.390483\pi$
$14$	31.2189	0.595972
$15$	0	0
$16$	−79.8403	−1.24750
$17$	48.6694	0.694357	0.347178	−	0.937799i	$-0.387140\pi$
$17$	48.6694	0.694357	0.347178	+	0.937799i	$0.387140\pi$
$18$	0	0
$19$	−127.727	−1.54224	−0.771118	−	0.636692i	$-0.780302\pi$
$19$	−127.727	−1.54224	−0.771118	+	0.636692i	$0.780302\pi$
$20$	21.9982	0.245948
$21$	0	0
$22$	−218.143	−2.11401
$23$	−184.026	−1.66835	−0.834175	−	0.551501i	$-0.814055\pi$
$23$	−184.026	−1.66835	−0.834175	+	0.551501i	$0.814055\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−111.347	−0.839881
$27$	0	0
$28$	−39.0059	−0.263265
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−251.985	−1.45993	−0.729965	−	0.683484i	$-0.760464\pi$
$31$	−251.985	−1.45993	−0.729965	+	0.683484i	$0.760464\pi$
$32$	179.719	0.992815
$33$	0	0
$34$	−171.380	−0.864455
$35$	−44.3285	−0.214083
$36$	0	0
$37$	197.740	0.878600	0.439300	−	0.898340i	$-0.355226\pi$
$37$	197.740	0.878600	0.439300	+	0.898340i	$0.355226\pi$
$38$	449.765	1.92004
$39$	0	0
$40$	63.3899	0.250571
$41$	163.446	0.622585	0.311293	−	0.950314i	$-0.399238\pi$
$41$	163.446	0.622585	0.311293	+	0.950314i	$0.399238\pi$
$42$	0	0
$43$	−33.2231	−0.117825	−0.0589124	−	0.998263i	$-0.518763\pi$
$43$	−33.2231	−0.117825	−0.0589124	+	0.998263i	$0.518763\pi$
$44$	272.555	0.933844
$45$	0	0
$46$	648.012	2.07705
$47$	−528.945	−1.64159	−0.820793	−	0.571226i	$-0.806468\pi$
$47$	−528.945	−1.64159	−0.820793	+	0.571226i	$0.806468\pi$
$48$	0	0
$49$	−264.399	−0.770843
$50$	−88.0328	−0.248994
$51$	0	0
$52$	139.120	0.371010
$53$	240.256	0.622673	0.311336	−	0.950300i	$-0.399223\pi$
$53$	240.256	0.622673	0.311336	+	0.950300i	$0.399223\pi$
$54$	0	0
$55$	309.746	0.759385
$56$	−112.399	−0.268214
$57$	0	0
$58$	−102.118	−0.231186
$59$	202.976	0.447885	0.223942	−	0.974602i	$-0.428107\pi$
$59$	202.976	0.447885	0.223942	+	0.974602i	$0.428107\pi$
$60$	0	0
$61$	492.192	1.03309	0.516547	−	0.856259i	$-0.327217\pi$
$61$	492.192	1.03309	0.516547	+	0.856259i	$0.327217\pi$
$62$	887.318	1.81757
$63$	0	0
$64$	5.87624	0.0114770
$65$	158.104	0.301698
$66$	0	0
$67$	−229.556	−0.418577	−0.209289	−	0.977854i	$-0.567115\pi$
$67$	−229.556	−0.418577	−0.209289	+	0.977854i	$0.567115\pi$
$68$	214.128	0.381865
$69$	0	0
$70$	156.095	0.266527
$71$	−21.2064	−0.0354471	−0.0177235	−	0.999843i	$-0.505642\pi$
$71$	−21.2064	−0.0354471	−0.0177235	+	0.999843i	$0.505642\pi$
$72$	0	0
$73$	−1145.48	−1.83656	−0.918279	−	0.395934i	$-0.870421\pi$
$73$	−1145.48	−1.83656	−0.918279	+	0.395934i	$0.870421\pi$
$74$	−696.303	−1.09383
$75$	0	0
$76$	−561.952	−0.848161
$77$	−549.224	−0.812855
$78$	0	0
$79$	95.6307	0.136194	0.0680968	−	0.997679i	$-0.478307\pi$
$79$	95.6307	0.136194	0.0680968	+	0.997679i	$0.478307\pi$
$80$	−399.201	−0.557901
$81$	0	0
$82$	−575.545	−0.775101
$83$	−1239.97	−1.63982	−0.819908	−	0.572496i	$-0.805975\pi$
$83$	−1239.97	−1.63982	−0.819908	+	0.572496i	$0.805975\pi$
$84$	0	0
$85$	243.347	0.310526
$86$	116.989	0.146689
$87$	0	0
$88$	785.391	0.951397
$89$	−116.863	−0.139185	−0.0695927	−	0.997575i	$-0.522170\pi$
$89$	−116.863	−0.139185	−0.0695927	+	0.997575i	$0.522170\pi$
$90$	0	0
$91$	−280.341	−0.322942
$92$	−809.648	−0.917518
$93$	0	0
$94$	1862.58	2.04373
$95$	−638.633	−0.689709
$96$	0	0
$97$	−275.722	−0.288611	−0.144306	−	0.989533i	$-0.546095\pi$
$97$	−275.722	−0.288611	−0.144306	+	0.989533i	$0.546095\pi$
$98$	931.032	0.959678
$99$	0	0
$100$	109.991	0.109991
$101$	1909.39	1.88111	0.940553	−	0.339648i	$-0.110308\pi$
$101$	1909.39	1.88111	0.940553	+	0.339648i	$0.110308\pi$
$102$	0	0
$103$	1501.97	1.43683	0.718415	−	0.695614i	$-0.244868\pi$
$103$	1501.97	1.43683	0.718415	+	0.695614i	$0.244868\pi$
$104$	400.888	0.377984
$105$	0	0
$106$	−846.016	−0.775210
$107$	−1434.61	−1.29616	−0.648080	−	0.761572i	$-0.724428\pi$
$107$	−1434.61	−1.29616	−0.648080	+	0.761572i	$0.724428\pi$
$108$	0	0
$109$	−347.951	−0.305758	−0.152879	−	0.988245i	$-0.548855\pi$
$109$	−347.951	−0.305758	−0.152879	+	0.988245i	$0.548855\pi$
$110$	−1090.71	−0.945413
$111$	0	0
$112$	707.840	0.597184
$113$	1812.68	1.50905	0.754524	−	0.656272i	$-0.227868\pi$
$113$	1812.68	1.50905	0.754524	+	0.656272i	$0.227868\pi$
$114$	0	0
$115$	−920.129	−0.746108
$116$	127.590	0.102124
$117$	0	0
$118$	−714.742	−0.557604
$119$	−431.489	−0.332391
$120$	0	0
$121$	2506.71	1.88333
$122$	−1733.16	−1.28617
$123$	0	0
$124$	−1108.64	−0.802897
$125$	125.000	0.0894427
$126$	0	0
$127$	550.739	0.384804	0.192402	−	0.981316i	$-0.438372\pi$
$127$	550.739	0.384804	0.192402	+	0.981316i	$0.438372\pi$
$128$	−1458.44	−1.00710
$129$	0	0
$130$	−556.734	−0.375606
$131$	124.824	0.0832515	0.0416257	−	0.999133i	$-0.486746\pi$
$131$	124.824	0.0832515	0.0416257	+	0.999133i	$0.486746\pi$
$132$	0	0
$133$	1132.39	0.738273
$134$	808.337	0.521117
$135$	0	0
$136$	617.030	0.389043
$137$	549.879	0.342915	0.171458	−	0.985192i	$-0.445152\pi$
$137$	549.879	0.342915	0.171458	+	0.985192i	$0.445152\pi$
$138$	0	0
$139$	−2243.32	−1.36889	−0.684445	−	0.729065i	$-0.739955\pi$
$139$	−2243.32	−1.36889	−0.684445	+	0.729065i	$0.739955\pi$
$140$	−195.030	−0.117736
$141$	0	0
$142$	74.6745	0.0441306
$143$	1958.88	1.14553
$144$	0	0
$145$	145.000	0.0830455
$146$	4033.61	2.28646
$147$	0	0
$148$	869.984	0.483191
$149$	614.305	0.337757	0.168879	−	0.985637i	$-0.445985\pi$
$149$	614.305	0.337757	0.168879	+	0.985637i	$0.445985\pi$
$150$	0	0
$151$	−1121.72	−0.604531	−0.302266	−	0.953224i	$-0.597743\pi$
$151$	−1121.72	−0.604531	−0.302266	+	0.953224i	$0.597743\pi$
$152$	−1619.32	−0.864104
$153$	0	0
$154$	1933.99	1.01198
$155$	−1259.92	−0.652901
$156$	0	0
$157$	3238.34	1.64616	0.823082	−	0.567923i	$-0.192253\pi$
$157$	3238.34	1.64616	0.823082	+	0.567923i	$0.192253\pi$
$158$	−336.746	−0.169557
$159$	0	0
$160$	898.594	0.444001
$161$	1631.52	0.798644
$162$	0	0
$163$	−2973.43	−1.42881	−0.714407	−	0.699730i	$-0.753303\pi$
$163$	−2973.43	−1.42881	−0.714407	+	0.699730i	$0.753303\pi$
$164$	719.105	0.342394
$165$	0	0
$166$	4366.33	2.04152
$167$	−4133.13	−1.91515	−0.957577	−	0.288176i	$-0.906951\pi$
$167$	−4133.13	−1.91515	−0.957577	+	0.288176i	$0.906951\pi$
$168$	0	0
$169$	−1197.12	−0.544890
$170$	−856.901	−0.386596
$171$	0	0
$172$	−146.170	−0.0647984
$173$	−347.755	−0.152829	−0.0764143	−	0.997076i	$-0.524347\pi$
$173$	−347.755	−0.152829	−0.0764143	+	0.997076i	$0.524347\pi$
$174$	0	0
$175$	−221.643	−0.0957406
$176$	−4946.04	−2.11831
$177$	0	0
$178$	411.513	0.173282
$179$	−789.082	−0.329490	−0.164745	−	0.986336i	$-0.552680\pi$
$179$	−789.082	−0.329490	−0.164745	+	0.986336i	$0.552680\pi$
$180$	0	0
$181$	−4547.86	−1.86762	−0.933812	−	0.357763i	$-0.883539\pi$
$181$	−4547.86	−1.86762	−0.933812	+	0.357763i	$0.883539\pi$
$182$	987.168	0.402054
$183$	0	0
$184$	−2333.08	−0.934764
$185$	988.699	0.392922
$186$	0	0
$187$	3015.03	1.17904
$188$	−2327.17	−0.902799
$189$	0	0
$190$	2248.83	0.858669
$191$	1121.45	0.424843	0.212421	−	0.977178i	$-0.431865\pi$
$191$	1121.45	0.424843	0.212421	+	0.977178i	$0.431865\pi$
$192$	0	0
$193$	−3950.05	−1.47322	−0.736608	−	0.676320i	$-0.763574\pi$
$193$	−3950.05	−1.47322	−0.736608	+	0.676320i	$0.763574\pi$
$194$	970.902	0.359313
$195$	0	0
$196$	−1163.26	−0.423930
$197$	436.434	0.157841	0.0789205	−	0.996881i	$-0.474853\pi$
$197$	436.434	0.157841	0.0789205	+	0.996881i	$0.474853\pi$
$198$	0	0
$199$	775.787	0.276352	0.138176	−	0.990408i	$-0.455876\pi$
$199$	775.787	0.276352	0.138176	+	0.990408i	$0.455876\pi$
$200$	316.949	0.112059
$201$	0	0
$202$	−6723.57	−2.34192
$203$	−257.105	−0.0888929
$204$	0	0
$205$	817.231	0.278429
$206$	−5288.91	−1.78881
$207$	0	0
$208$	−2524.61	−0.841589
$209$	−7912.57	−2.61877
$210$	0	0
$211$	2474.44	0.807335	0.403667	−	0.914906i	$-0.367735\pi$
$211$	2474.44	0.807335	0.403667	+	0.914906i	$0.367735\pi$
$212$	1057.04	0.342442
$213$	0	0
$214$	5051.71	1.61368
$215$	−166.115	−0.0526929
$216$	0	0
$217$	2234.02	0.698873
$218$	1225.24	0.380660
$219$	0	0
$220$	1362.77	0.417628
$221$	1538.97	0.468426
$222$	0	0
$223$	−1035.62	−0.310987	−0.155494	−	0.987837i	$-0.549697\pi$
$223$	−1035.62	−0.310987	−0.155494	+	0.987837i	$0.549697\pi$
$224$	−1593.33	−0.475264
$225$	0	0
$226$	−6383.01	−1.87872
$227$	1068.43	0.312398	0.156199	−	0.987726i	$-0.450076\pi$
$227$	1068.43	0.312398	0.156199	+	0.987726i	$0.450076\pi$
$228$	0	0
$229$	−638.809	−0.184339	−0.0921696	−	0.995743i	$-0.529380\pi$
$229$	−638.809	−0.184339	−0.0921696	+	0.995743i	$0.529380\pi$
$230$	3240.06	0.928884
$231$	0	0
$232$	367.661	0.104044
$233$	−4199.81	−1.18085	−0.590427	−	0.807091i	$-0.701041\pi$
$233$	−4199.81	−1.18085	−0.590427	+	0.807091i	$0.701041\pi$
$234$	0	0
$235$	−2644.72	−0.734139
$236$	893.022	0.246317
$237$	0	0
$238$	1519.41	0.413817
$239$	−1481.20	−0.400882	−0.200441	−	0.979706i	$-0.564237\pi$
$239$	−1481.20	−0.400882	−0.200441	+	0.979706i	$0.564237\pi$
$240$	0	0
$241$	7144.57	1.90964	0.954818	−	0.297191i	$-0.0960498\pi$
$241$	7144.57	1.90964	0.954818	+	0.297191i	$0.0960498\pi$
$242$	−8826.90	−2.34469
$243$	0	0
$244$	2165.47	0.568156
$245$	−1322.00	−0.344732
$246$	0	0
$247$	−4038.82	−1.04042
$248$	−3194.66	−0.817988
$249$	0	0
$250$	−440.164	−0.111354
$251$	5209.72	1.31010	0.655049	−	0.755587i	$-0.272648\pi$
$251$	5209.72	1.31010	0.655049	+	0.755587i	$0.272648\pi$
$252$	0	0
$253$	−11400.3	−2.83292
$254$	−1939.32	−0.479071
$255$	0	0
$256$	5088.62	1.24234
$257$	−3431.65	−0.832919	−0.416460	−	0.909154i	$-0.636729\pi$
$257$	−3431.65	−0.832919	−0.416460	+	0.909154i	$0.636729\pi$
$258$	0	0
$259$	−1753.10	−0.420589
$260$	695.602	0.165921
$261$	0	0
$262$	−439.545	−0.103646
$263$	1738.29	0.407558	0.203779	−	0.979017i	$-0.434678\pi$
$263$	1738.29	0.407558	0.203779	+	0.979017i	$0.434678\pi$
$264$	0	0
$265$	1201.28	0.278468
$266$	−3987.49	−0.919130
$267$	0	0
$268$	−1009.96	−0.230199
$269$	−2829.31	−0.641287	−0.320643	−	0.947200i	$-0.603899\pi$
$269$	−2829.31	−0.641287	−0.320643	+	0.947200i	$0.603899\pi$
$270$	0	0
$271$	−1399.29	−0.313657	−0.156828	−	0.987626i	$-0.550127\pi$
$271$	−1399.29	−0.313657	−0.156828	+	0.987626i	$0.550127\pi$
$272$	−3885.78	−0.866213
$273$	0	0
$274$	−1936.30	−0.426920
$275$	1548.73	0.339607
$276$	0	0
$277$	−3526.40	−0.764912	−0.382456	−	0.923974i	$-0.624922\pi$
$277$	−3526.40	−0.764912	−0.382456	+	0.923974i	$0.624922\pi$
$278$	7899.42	1.70423
$279$	0	0
$280$	−561.996	−0.119949
$281$	857.124	0.181963	0.0909817	−	0.995853i	$-0.471000\pi$
$281$	857.124	0.181963	0.0909817	+	0.995853i	$0.471000\pi$
$282$	0	0
$283$	−7197.47	−1.51182	−0.755910	−	0.654676i	$-0.772805\pi$
$283$	−7197.47	−1.51182	−0.755910	+	0.654676i	$0.772805\pi$
$284$	−93.3008	−0.0194943
$285$	0	0
$286$	−6897.85	−1.42615
$287$	−1449.07	−0.298034
$288$	0	0
$289$	−2544.29	−0.517869
$290$	−510.590	−0.103389
$291$	0	0
$292$	−5039.72	−1.01003
$293$	−2936.05	−0.585412	−0.292706	−	0.956202i	$-0.594556\pi$
$293$	−2936.05	−0.585412	−0.292706	+	0.956202i	$0.594556\pi$
$294$	0	0
$295$	1014.88	0.200300
$296$	2506.94	0.492274
$297$	0	0
$298$	−2163.16	−0.420498
$299$	−5819.04	−1.12550
$300$	0	0
$301$	294.546	0.0564031
$302$	3949.93	0.752625
$303$	0	0
$304$	10197.7	1.92395
$305$	2460.96	0.462014
$306$	0	0
$307$	−3828.27	−0.711698	−0.355849	−	0.934544i	$-0.615808\pi$
$307$	−3828.27	−0.711698	−0.355849	+	0.934544i	$0.615808\pi$
$308$	−2416.39	−0.447034
$309$	0	0
$310$	4436.59	0.812843
$311$	8035.81	1.46517	0.732587	−	0.680673i	$-0.238313\pi$
$311$	8035.81	1.46517	0.732587	+	0.680673i	$0.238313\pi$
$312$	0	0
$313$	−1288.78	−0.232736	−0.116368	−	0.993206i	$-0.537125\pi$
$313$	−1288.78	−0.232736	−0.116368	+	0.993206i	$0.537125\pi$
$314$	−11403.2	−2.04943
$315$	0	0
$316$	420.741	0.0749004
$317$	−8166.90	−1.44700	−0.723499	−	0.690325i	$-0.757468\pi$
$317$	−8166.90	−1.44700	−0.723499	+	0.690325i	$0.757468\pi$
$318$	0	0
$319$	1796.53	0.315317
$320$	29.3812	0.00513269
$321$	0	0
$322$	−5745.09	−0.994289
$323$	−6216.38	−1.07086
$324$	0	0
$325$	790.520	0.134924
$326$	10470.4	1.77883
$327$	0	0
$328$	2072.17	0.348830
$329$	4689.47	0.785832
$330$	0	0
$331$	−5018.85	−0.833417	−0.416708	−	0.909040i	$-0.636816\pi$
$331$	−5018.85	−0.833417	−0.416708	+	0.909040i	$0.636816\pi$
$332$	−5455.44	−0.901825
$333$	0	0
$334$	14554.0	2.38431
$335$	−1147.78	−0.187194
$336$	0	0
$337$	−3702.30	−0.598448	−0.299224	−	0.954183i	$-0.596728\pi$
$337$	−3702.30	−0.598448	−0.299224	+	0.954183i	$0.596728\pi$
$338$	4215.45	0.678373
$339$	0	0
$340$	1070.64	0.170775
$341$	−15610.3	−2.47901
$342$	0	0
$343$	5385.02	0.847708
$344$	−421.201	−0.0660164
$345$	0	0
$346$	1224.56	0.190267
$347$	−1084.12	−0.167719	−0.0838594	−	0.996478i	$-0.526725\pi$
$347$	−1084.12	−0.167719	−0.0838594	+	0.996478i	$0.526725\pi$
$348$	0	0
$349$	−3145.52	−0.482453	−0.241226	−	0.970469i	$-0.577550\pi$
$349$	−3145.52	−0.482453	−0.241226	+	0.970469i	$0.577550\pi$
$350$	780.473	0.119194
$351$	0	0
$352$	11133.4	1.68584
$353$	7477.34	1.12742	0.563709	−	0.825974i	$-0.309374\pi$
$353$	7477.34	1.12742	0.563709	+	0.825974i	$0.309374\pi$
$354$	0	0
$355$	−106.032	−0.0158524
$356$	−514.157	−0.0765458
$357$	0	0
$358$	2778.60	0.410206
$359$	−3987.75	−0.586255	−0.293128	−	0.956073i	$-0.594696\pi$
$359$	−3987.75	−0.586255	−0.293128	+	0.956073i	$0.594696\pi$
$360$	0	0
$361$	9455.09	1.37849
$362$	16014.5	2.32514
$363$	0	0
$364$	−1233.40	−0.177604
$365$	−5727.42	−0.821334
$366$	0	0
$367$	6261.36	0.890573	0.445286	−	0.895388i	$-0.353102\pi$
$367$	6261.36	0.890573	0.445286	+	0.895388i	$0.353102\pi$
$368$	14692.7	2.08127
$369$	0	0
$370$	−3481.52	−0.489177
$371$	−2130.04	−0.298075
$372$	0	0
$373$	−9332.46	−1.29549	−0.647743	−	0.761859i	$-0.724287\pi$
$373$	−9332.46	−1.29549	−0.647743	+	0.761859i	$0.724287\pi$
$374$	−10616.9	−1.46788
$375$	0	0
$376$	−6705.95	−0.919769
$377$	917.004	0.125273
$378$	0	0
$379$	−4570.38	−0.619432	−0.309716	−	0.950829i	$-0.600234\pi$
$379$	−4570.38	−0.619432	−0.309716	+	0.950829i	$0.600234\pi$
$380$	−2809.76	−0.379309
$381$	0	0
$382$	−3948.96	−0.528918
$383$	−2266.02	−0.302320	−0.151160	−	0.988509i	$-0.548301\pi$
$383$	−2266.02	−0.302320	−0.151160	+	0.988509i	$0.548301\pi$
$384$	0	0
$385$	−2746.12	−0.363520
$386$	13909.4	1.83411
$387$	0	0
$388$	−1213.08	−0.158723
$389$	−3874.40	−0.504987	−0.252494	−	0.967599i	$-0.581251\pi$
$389$	−3874.40	−0.504987	−0.252494	+	0.967599i	$0.581251\pi$
$390$	0	0
$391$	−8956.43	−1.15843
$392$	−3352.05	−0.431898
$393$	0	0
$394$	−1536.82	−0.196508
$395$	478.154	0.0609076
$396$	0	0
$397$	−10370.5	−1.31104	−0.655519	−	0.755179i	$-0.727550\pi$
$397$	−10370.5	−1.31104	−0.655519	+	0.755179i	$0.727550\pi$
$398$	−2731.79	−0.344051
$399$	0	0
$400$	−1996.01	−0.249501
$401$	10537.9	1.31231	0.656155	−	0.754626i	$-0.272182\pi$
$401$	10537.9	1.31231	0.656155	+	0.754626i	$0.272182\pi$
$402$	0	0
$403$	−7967.97	−0.984895
$404$	8400.65	1.03452
$405$	0	0
$406$	905.349	0.110669
$407$	12249.8	1.49189
$408$	0	0
$409$	−4052.17	−0.489894	−0.244947	−	0.969536i	$-0.578771\pi$
$409$	−4052.17	−0.489894	−0.244947	+	0.969536i	$0.578771\pi$
$410$	−2877.73	−0.346636
$411$	0	0
$412$	6608.14	0.790193
$413$	−1799.52	−0.214404
$414$	0	0
$415$	−6199.86	−0.733348
$416$	5682.85	0.669771
$417$	0	0
$418$	27862.6	3.26030
$419$	12386.0	1.44414	0.722071	−	0.691819i	$-0.243190\pi$
$419$	12386.0	1.44414	0.722071	+	0.691819i	$0.243190\pi$
$420$	0	0
$421$	−16516.0	−1.91198	−0.955989	−	0.293402i	$-0.905213\pi$
$421$	−16516.0	−1.91198	−0.955989	+	0.293402i	$0.905213\pi$
$422$	−8713.29	−1.00511
$423$	0	0
$424$	3045.96	0.348879
$425$	1216.74	0.138871
$426$	0	0
$427$	−4363.63	−0.494546
$428$	−6311.78	−0.712830
$429$	0	0
$430$	584.944	0.0656012
$431$	−11987.2	−1.33969	−0.669843	−	0.742503i	$-0.733639\pi$
$431$	−11987.2	−1.33969	−0.669843	+	0.742503i	$0.733639\pi$
$432$	0	0
$433$	−14217.4	−1.57793	−0.788965	−	0.614439i	$-0.789383\pi$
$433$	−14217.4	−1.57793	−0.788965	+	0.614439i	$0.789383\pi$
$434$	−7866.70	−0.870078
$435$	0	0
$436$	−1530.86	−0.168153
$437$	23505.0	2.57299
$438$	0	0
$439$	685.738	0.0745524	0.0372762	−	0.999305i	$-0.488132\pi$
$439$	685.738	0.0745524	0.0372762	+	0.999305i	$0.488132\pi$
$440$	3926.96	0.425478
$441$	0	0
$442$	−5419.18	−0.583177
$443$	13828.1	1.48305	0.741527	−	0.670923i	$-0.234102\pi$
$443$	13828.1	1.48305	0.741527	+	0.670923i	$0.234102\pi$
$444$	0	0
$445$	−584.317	−0.0622456
$446$	3646.74	0.387170
$447$	0	0
$448$	−52.0970	−0.00549409
$449$	4346.88	0.456887	0.228443	−	0.973557i	$-0.426636\pi$
$449$	4346.88	0.456887	0.228443	+	0.973557i	$0.426636\pi$
$450$	0	0
$451$	10125.4	1.05717
$452$	7975.14	0.829909
$453$	0	0
$454$	−3762.29	−0.388927
$455$	−1401.70	−0.144424
$456$	0	0
$457$	2375.45	0.243149	0.121574	−	0.992582i	$-0.461206\pi$
$457$	2375.45	0.243149	0.121574	+	0.992582i	$0.461206\pi$
$458$	2249.45	0.229497
$459$	0	0
$460$	−4048.24	−0.410326
$461$	5656.94	0.571519	0.285759	−	0.958301i	$-0.407754\pi$
$461$	5656.94	0.571519	0.285759	+	0.958301i	$0.407754\pi$
$462$	0	0
$463$	−11017.3	−1.10587	−0.552935	−	0.833225i	$-0.686492\pi$
$463$	−11017.3	−1.10587	−0.552935	+	0.833225i	$0.686492\pi$
$464$	−2315.37	−0.231656
$465$	0	0
$466$	14788.9	1.47013
$467$	4441.89	0.440142	0.220071	−	0.975484i	$-0.429371\pi$
$467$	4441.89	0.440142	0.220071	+	0.975484i	$0.429371\pi$
$468$	0	0
$469$	2035.17	0.200374
$470$	9312.90	0.913983
$471$	0	0
$472$	2573.32	0.250947
$473$	−2058.14	−0.200071
$474$	0	0
$475$	−3193.17	−0.308447
$476$	−1898.40	−0.182800
$477$	0	0
$478$	5215.76	0.499087
$479$	6919.43	0.660035	0.330017	−	0.943975i	$-0.392945\pi$
$479$	6919.43	0.660035	0.330017	+	0.943975i	$0.392945\pi$
$480$	0	0
$481$	6252.69	0.592720
$482$	−25158.3	−2.37744
$483$	0	0
$484$	11028.6	1.03575
$485$	−1378.61	−0.129071
$486$	0	0
$487$	4439.83	0.413116	0.206558	−	0.978434i	$-0.433774\pi$
$487$	4439.83	0.413116	0.206558	+	0.978434i	$0.433774\pi$
$488$	6240.01	0.578836
$489$	0	0
$490$	4655.16	0.429181
$491$	−12089.1	−1.11115	−0.555575	−	0.831466i	$-0.687502\pi$
$491$	−12089.1	−1.11115	−0.555575	+	0.831466i	$0.687502\pi$
$492$	0	0
$493$	1411.41	0.128939
$494$	14221.9	1.29529
$495$	0	0
$496$	20118.6	1.82127
$497$	188.010	0.0169686
$498$	0	0
$499$	7312.25	0.655994	0.327997	−	0.944679i	$-0.393626\pi$
$499$	7312.25	0.655994	0.327997	+	0.944679i	$0.393626\pi$
$500$	549.955	0.0491895
$501$	0	0
$502$	−18345.0	−1.63103
$503$	−7487.34	−0.663706	−0.331853	−	0.943331i	$-0.607674\pi$
$503$	−7487.34	−0.663706	−0.331853	+	0.943331i	$0.607674\pi$
$504$	0	0
$505$	9546.96	0.841256
$506$	40143.9	3.52690
$507$	0	0
$508$	2423.05	0.211625
$509$	16901.9	1.47184	0.735919	−	0.677070i	$-0.236750\pi$
$509$	16901.9	1.47184	0.735919	+	0.677070i	$0.236750\pi$
$510$	0	0
$511$	10155.5	0.879166
$512$	−6251.09	−0.539574
$513$	0	0
$514$	12083.9	1.03696
$515$	7509.85	0.642570
$516$	0	0
$517$	−32767.7	−2.78747
$518$	6173.22	0.523621
$519$	0	0
$520$	2004.44	0.169039
$521$	8351.65	0.702289	0.351144	−	0.936321i	$-0.385793\pi$
$521$	8351.65	0.702289	0.351144	+	0.936321i	$0.385793\pi$
$522$	0	0
$523$	5420.89	0.453230	0.226615	−	0.973984i	$-0.427234\pi$
$523$	5420.89	0.453230	0.226615	+	0.973984i	$0.427234\pi$
$524$	549.182	0.0457846
$525$	0	0
$526$	−6121.07	−0.507398
$527$	−12264.0	−1.01371
$528$	0	0
$529$	21698.5	1.78339
$530$	−4230.08	−0.346685
$531$	0	0
$532$	4982.10	0.406018
$533$	5168.30	0.420007
$534$	0	0
$535$	−7173.05	−0.579660
$536$	−2910.30	−0.234526
$537$	0	0
$538$	9962.89	0.798384
$539$	−16379.3	−1.30892
$540$	0	0
$541$	14428.9	1.14667	0.573333	−	0.819323i	$-0.305650\pi$
$541$	14428.9	1.14667	0.573333	+	0.819323i	$0.305650\pi$
$542$	4927.34	0.390494
$543$	0	0
$544$	8746.81	0.689368
$545$	−1739.75	−0.136739
$546$	0	0
$547$	4990.35	0.390077	0.195038	−	0.980796i	$-0.437517\pi$
$547$	4990.35	0.390077	0.195038	+	0.980796i	$0.437517\pi$
$548$	2419.27	0.188588
$549$	0	0
$550$	−5453.57	−0.422801
$551$	−3704.07	−0.286386
$552$	0	0
$553$	−847.834	−0.0651963
$554$	12417.5	0.952294
$555$	0	0
$556$	−9869.80	−0.752828
$557$	−23481.4	−1.78625	−0.893123	−	0.449812i	$-0.851491\pi$
$557$	−23481.4	−1.78625	−0.893123	+	0.449812i	$0.851491\pi$
$558$	0	0
$559$	−1050.54	−0.0794868
$560$	3539.20	0.267069
$561$	0	0
$562$	−3018.20	−0.226539
$563$	−8869.38	−0.663943	−0.331972	−	0.943289i	$-0.607714\pi$
$563$	−8869.38	−0.663943	−0.331972	+	0.943289i	$0.607714\pi$
$564$	0	0
$565$	9063.39	0.674867
$566$	25344.5	1.88217
$567$	0	0
$568$	−268.855	−0.0198607
$569$	−25722.7	−1.89517	−0.947585	−	0.319503i	$-0.896484\pi$
$569$	−25722.7	−1.89517	−0.947585	+	0.319503i	$0.896484\pi$
$570$	0	0
$571$	26056.8	1.90971	0.954853	−	0.297078i	$-0.0960122\pi$
$571$	26056.8	1.90971	0.954853	+	0.297078i	$0.0960122\pi$
$572$	8618.40	0.629988
$573$	0	0
$574$	5102.61	0.371043
$575$	−4600.64	−0.333670
$576$	0	0
$577$	−25737.2	−1.85694	−0.928471	−	0.371406i	$-0.878876\pi$
$577$	−25737.2	−1.85694	−0.928471	+	0.371406i	$0.878876\pi$
$578$	8959.23	0.644732
$579$	0	0
$580$	637.948	0.0456713
$581$	10993.2	0.784985
$582$	0	0
$583$	14883.7	1.05732
$584$	−14522.4	−1.02901
$585$	0	0
$586$	10338.7	0.728822
$587$	6862.44	0.482527	0.241263	−	0.970460i	$-0.422438\pi$
$587$	6862.44	0.482527	0.241263	+	0.970460i	$0.422438\pi$
$588$	0	0
$589$	32185.2	2.25156
$590$	−3573.71	−0.249368
$591$	0	0
$592$	−15787.6	−1.09606
$593$	−4016.32	−0.278129	−0.139065	−	0.990283i	$-0.544410\pi$
$593$	−4016.32	−0.278129	−0.139065	+	0.990283i	$0.544410\pi$
$594$	0	0
$595$	−2157.44	−0.148650
$596$	2702.72	0.185751
$597$	0	0
$598$	20490.7	1.40121
$599$	1729.64	0.117982	0.0589911	−	0.998259i	$-0.481212\pi$
$599$	1729.64	0.117982	0.0589911	+	0.998259i	$0.481212\pi$
$600$	0	0
$601$	2597.41	0.176291	0.0881453	−	0.996108i	$-0.471906\pi$
$601$	2597.41	0.176291	0.0881453	+	0.996108i	$0.471906\pi$
$602$	−1037.19	−0.0702203
$603$	0	0
$604$	−4935.17	−0.332465
$605$	12533.5	0.842249
$606$	0	0
$607$	−13844.6	−0.925755	−0.462877	−	0.886422i	$-0.653183\pi$
$607$	−13844.6	−0.925755	−0.462877	+	0.886422i	$0.653183\pi$
$608$	−22954.9	−1.53116
$609$	0	0
$610$	−8665.82	−0.575195
$611$	−16725.7	−1.10744
$612$	0	0
$613$	1127.16	0.0742667	0.0371334	−	0.999310i	$-0.488177\pi$
$613$	1127.16	0.0742667	0.0371334	+	0.999310i	$0.488177\pi$
$614$	13480.6	0.886044
$615$	0	0
$616$	−6963.05	−0.455437
$617$	−29575.8	−1.92978	−0.964892	−	0.262645i	$-0.915405\pi$
$617$	−29575.8	−1.92978	−0.964892	+	0.262645i	$0.915405\pi$
$618$	0	0
$619$	12535.0	0.813935	0.406967	−	0.913443i	$-0.366586\pi$
$619$	12535.0	0.813935	0.406967	+	0.913443i	$0.366586\pi$
$620$	−5543.22	−0.359066
$621$	0	0
$622$	−28296.6	−1.82410
$623$	1036.08	0.0666285
$624$	0	0
$625$	625.000	0.0400000
$626$	4538.20	0.289749
$627$	0	0
$628$	14247.5	0.905316
$629$	9623.88	0.610062
$630$	0	0
$631$	15162.1	0.956569	0.478284	−	0.878205i	$-0.341259\pi$
$631$	15162.1	0.956569	0.478284	+	0.878205i	$0.341259\pi$
$632$	1212.40	0.0763083
$633$	0	0
$634$	28758.2	1.80147
$635$	2753.69	0.172090
$636$	0	0
$637$	−8360.52	−0.520025
$638$	−6326.14	−0.392561
$639$	0	0
$640$	−7292.21	−0.450391
$641$	10138.5	0.624722	0.312361	−	0.949964i	$-0.398880\pi$
$641$	10138.5	0.624722	0.312361	+	0.949964i	$0.398880\pi$
$642$	0	0
$643$	29580.6	1.81423	0.907113	−	0.420888i	$-0.138282\pi$
$643$	29580.6	1.81423	0.907113	+	0.420888i	$0.138282\pi$
$644$	7178.10	0.439219
$645$	0	0
$646$	21889.8	1.33319
$647$	−30140.7	−1.83146	−0.915728	−	0.401798i	$-0.868385\pi$
$647$	−30140.7	−1.83146	−0.915728	+	0.401798i	$0.868385\pi$
$648$	0	0
$649$	12574.2	0.760525
$650$	−2783.67	−0.167976
$651$	0	0
$652$	−13082.0	−0.785784
$653$	10094.0	0.604911	0.302456	−	0.953163i	$-0.402194\pi$
$653$	10094.0	0.604911	0.302456	+	0.953163i	$0.402194\pi$
$654$	0	0
$655$	624.121	0.0372312
$656$	−13049.6	−0.776678
$657$	0	0
$658$	−16513.1	−0.978339
$659$	−16019.7	−0.946947	−0.473473	−	0.880808i	$-0.657000\pi$
$659$	−16019.7	−0.946947	−0.473473	+	0.880808i	$0.657000\pi$
$660$	0	0
$661$	−26006.5	−1.53031	−0.765157	−	0.643844i	$-0.777338\pi$
$661$	−26006.5	−1.53031	−0.765157	+	0.643844i	$0.777338\pi$
$662$	17672.9	1.03758
$663$	0	0
$664$	−15720.3	−0.918777
$665$	5661.93	0.330166
$666$	0	0
$667$	−5336.75	−0.309805
$668$	−18184.3	−1.05325
$669$	0	0
$670$	4041.69	0.233051
$671$	30490.9	1.75423
$672$	0	0
$673$	30662.4	1.75624	0.878119	−	0.478443i	$-0.158799\pi$
$673$	30662.4	1.75624	0.878119	+	0.478443i	$0.158799\pi$
$674$	13036.9	0.745051
$675$	0	0
$676$	−5266.92	−0.299665
$677$	4128.37	0.234366	0.117183	−	0.993110i	$-0.462614\pi$
$677$	4128.37	0.234366	0.117183	+	0.993110i	$0.462614\pi$
$678$	0	0
$679$	2444.47	0.138159
$680$	3085.15	0.173985
$681$	0	0
$682$	54968.7	3.08630
$683$	19130.6	1.07176	0.535881	−	0.844294i	$-0.319980\pi$
$683$	19130.6	1.07176	0.535881	+	0.844294i	$0.319980\pi$
$684$	0	0
$685$	2749.40	0.153356
$686$	−18962.4	−1.05537
$687$	0	0
$688$	2652.54	0.146987
$689$	7597.08	0.420066
$690$	0	0
$691$	27493.6	1.51361	0.756807	−	0.653638i	$-0.226758\pi$
$691$	27493.6	1.51361	0.756807	+	0.653638i	$0.226758\pi$
$692$	−1530.00	−0.0840489
$693$	0	0
$694$	3817.51	0.208805
$695$	−11216.6	−0.612186
$696$	0	0
$697$	7954.83	0.432296
$698$	11076.4	0.600640
$699$	0	0
$700$	−975.149	−0.0526531
$701$	−13029.3	−0.702010	−0.351005	−	0.936374i	$-0.614160\pi$
$701$	−13029.3	−0.702010	−0.351005	+	0.936374i	$0.614160\pi$
$702$	0	0
$703$	−25256.6	−1.35501
$704$	364.029	0.0194884
$705$	0	0
$706$	−26330.0	−1.40360
$707$	−16928.1	−0.900491
$708$	0	0
$709$	−33140.8	−1.75547	−0.877735	−	0.479146i	$-0.840946\pi$
$709$	−33140.8	−1.75547	−0.877735	+	0.479146i	$0.840946\pi$
$710$	373.373	0.0197358
$711$	0	0
$712$	−1481.59	−0.0779846
$713$	46371.7	2.43567
$714$	0	0
$715$	9794.42	0.512295
$716$	−3471.68	−0.181205
$717$	0	0
$718$	14042.1	0.729871
$719$	−23672.8	−1.22788	−0.613940	−	0.789353i	$-0.710416\pi$
$719$	−23672.8	−1.22788	−0.613940	+	0.789353i	$0.710416\pi$
$720$	0	0
$721$	−13316.0	−0.687815
$722$	−33294.3	−1.71619
$723$	0	0
$724$	−20009.0	−1.02711
$725$	725.000	0.0371391
$726$	0	0
$727$	16424.5	0.837895	0.418947	−	0.908010i	$-0.362399\pi$
$727$	16424.5	0.837895	0.418947	+	0.908010i	$0.362399\pi$
$728$	−3554.16	−0.180942
$729$	0	0
$730$	20168.0	1.02254
$731$	−1616.95	−0.0818125
$732$	0	0
$733$	19399.9	0.977559	0.488780	−	0.872407i	$-0.337442\pi$
$733$	19399.9	0.977559	0.488780	+	0.872407i	$0.337442\pi$
$734$	−22048.2	−1.10874
$735$	0	0
$736$	−33072.9	−1.65636
$737$	−14220.8	−0.710759
$738$	0	0
$739$	−5065.65	−0.252156	−0.126078	−	0.992020i	$-0.540239\pi$
$739$	−5065.65	−0.252156	−0.126078	+	0.992020i	$0.540239\pi$
$740$	4349.92	0.216090
$741$	0	0
$742$	7500.53	0.371096
$743$	15186.2	0.749833	0.374917	−	0.927059i	$-0.377671\pi$
$743$	15186.2	0.749833	0.374917	+	0.927059i	$0.377671\pi$
$744$	0	0
$745$	3071.53	0.151050
$746$	32862.5	1.61284
$747$	0	0
$748$	13265.1	0.648421
$749$	12718.8	0.620476
$750$	0	0
$751$	−25919.9	−1.25943	−0.629713	−	0.776828i	$-0.716828\pi$
$751$	−25919.9	−1.25943	−0.629713	+	0.776828i	$0.716828\pi$
$752$	42231.1	2.04789
$753$	0	0
$754$	−3229.06	−0.155962
$755$	−5608.60	−0.270355
$756$	0	0
$757$	25691.7	1.23353	0.616765	−	0.787148i	$-0.288443\pi$
$757$	25691.7	1.23353	0.616765	+	0.787148i	$0.288443\pi$
$758$	16093.7	0.771175
$759$	0	0
$760$	−8096.58	−0.386439
$761$	−3453.84	−0.164523	−0.0822613	−	0.996611i	$-0.526214\pi$
$761$	−3453.84	−0.164523	−0.0822613	+	0.996611i	$0.526214\pi$
$762$	0	0
$763$	3084.83	0.146367
$764$	4933.96	0.233645
$765$	0	0
$766$	7979.38	0.376379
$767$	6418.26	0.302151
$768$	0	0
$769$	24995.5	1.17212	0.586060	−	0.810268i	$-0.300678\pi$
$769$	24995.5	1.17212	0.586060	+	0.810268i	$0.300678\pi$
$770$	9669.94	0.452572
$771$	0	0
$772$	−17378.8	−0.810203
$773$	−15444.9	−0.718648	−0.359324	−	0.933213i	$-0.616993\pi$
$773$	−15444.9	−0.718648	−0.359324	+	0.933213i	$0.616993\pi$
$774$	0	0
$775$	−6299.62	−0.291986
$776$	−3495.59	−0.161707
$777$	0	0
$778$	13643.0	0.628695
$779$	−20876.4	−0.960174
$780$	0	0
$781$	−1313.72	−0.0601904
$782$	31538.4	1.44221
$783$	0	0
$784$	21109.7	0.961630
$785$	16191.7	0.736187
$786$	0	0
$787$	−30204.3	−1.36807	−0.684033	−	0.729451i	$-0.739776\pi$
$787$	−30204.3	−1.36807	−0.684033	+	0.729451i	$0.739776\pi$
$788$	1920.16	0.0868055
$789$	0	0
$790$	−1683.73	−0.0758283
$791$	−16070.7	−0.722386
$792$	0	0
$793$	15563.5	0.696944
$794$	36517.9	1.63220
$795$	0	0
$796$	3413.19	0.151981
$797$	−44520.9	−1.97869	−0.989343	−	0.145606i	$-0.953487\pi$
$797$	−44520.9	−1.97869	−0.989343	+	0.145606i	$0.953487\pi$
$798$	0	0
$799$	−25743.4	−1.13985
$800$	4492.97	0.198563
$801$	0	0
$802$	−37107.1	−1.63379
$803$	−70961.8	−3.11854
$804$	0	0
$805$	8157.59	0.357164
$806$	28057.7	1.22617
$807$	0	0
$808$	24207.2	1.05397
$809$	10369.6	0.450651	0.225326	−	0.974284i	$-0.427655\pi$
$809$	10369.6	0.450651	0.225326	+	0.974284i	$0.427655\pi$
$810$	0	0
$811$	24341.6	1.05394	0.526972	−	0.849883i	$-0.323327\pi$
$811$	24341.6	1.05394	0.526972	+	0.849883i	$0.323327\pi$
$812$	−1131.17	−0.0488872
$813$	0	0
$814$	−43135.5	−1.85737
$815$	−14867.1	−0.638985
$816$	0	0
$817$	4243.47	0.181714
$818$	14268.9	0.609905
$819$	0	0
$820$	3595.52	0.153123
$821$	17287.1	0.734864	0.367432	−	0.930050i	$-0.380237\pi$
$821$	17287.1	0.734864	0.367432	+	0.930050i	$0.380237\pi$
$822$	0	0
$823$	5724.85	0.242474	0.121237	−	0.992624i	$-0.461314\pi$
$823$	5724.85	0.242474	0.121237	+	0.992624i	$0.461314\pi$
$824$	19042.0	0.805046
$825$	0	0
$826$	6336.69	0.266927
$827$	−25964.5	−1.09175	−0.545874	−	0.837867i	$-0.683802\pi$
$827$	−25964.5	−1.09175	−0.545874	+	0.837867i	$0.683802\pi$
$828$	0	0
$829$	−13876.2	−0.581349	−0.290675	−	0.956822i	$-0.593880\pi$
$829$	−13876.2	−0.581349	−0.290675	+	0.956822i	$0.593880\pi$
$830$	21831.7	0.912997
$831$	0	0
$832$	185.812	0.00774262
$833$	−12868.2	−0.535240
$834$	0	0
$835$	−20665.6	−0.856483
$836$	−34812.5	−1.44021
$837$	0	0
$838$	−43615.0	−1.79792
$839$	20190.6	0.830818	0.415409	−	0.909635i	$-0.363638\pi$
$839$	20190.6	0.830818	0.415409	+	0.909635i	$0.363638\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	58158.2	2.38036
$843$	0	0
$844$	10886.7	0.443998
$845$	−5985.62	−0.243682
$846$	0	0
$847$	−22223.7	−0.901554
$848$	−19182.1	−0.776787
$849$	0	0
$850$	−4284.51	−0.172891
$851$	−36389.2	−1.46581
$852$	0	0
$853$	−14388.2	−0.577542	−0.288771	−	0.957398i	$-0.593247\pi$
$853$	−14388.2	−0.577542	−0.288771	+	0.957398i	$0.593247\pi$
$854$	15365.7	0.615696
$855$	0	0
$856$	−18188.0	−0.726229
$857$	12095.7	0.482125	0.241063	−	0.970510i	$-0.422504\pi$
$857$	12095.7	0.482125	0.241063	+	0.970510i	$0.422504\pi$
$858$	0	0
$859$	−16863.4	−0.669815	−0.334907	−	0.942251i	$-0.608705\pi$
$859$	−16863.4	−0.669815	−0.334907	+	0.942251i	$0.608705\pi$
$860$	−730.848	−0.0289787
$861$	0	0
$862$	42210.8	1.66787
$863$	−6575.91	−0.259382	−0.129691	−	0.991554i	$-0.541399\pi$
$863$	−6575.91	−0.259382	−0.129691	+	0.991554i	$0.541399\pi$
$864$	0	0
$865$	−1738.78	−0.0683470
$866$	50063.8	1.96448
$867$	0	0
$868$	9828.91	0.384349
$869$	5924.25	0.231262
$870$	0	0
$871$	−7258.74	−0.282380
$872$	−4411.31	−0.171314
$873$	0	0
$874$	−82768.4	−3.20330
$875$	−1108.21	−0.0428165
$876$	0	0
$877$	−2474.27	−0.0952683	−0.0476341	−	0.998865i	$-0.515168\pi$
$877$	−2474.27	−0.0952683	−0.0476341	+	0.998865i	$0.515168\pi$
$878$	−2414.70	−0.0928157
$879$	0	0
$880$	−24730.2	−0.947336
$881$	−39922.4	−1.52670	−0.763349	−	0.645987i	$-0.776446\pi$
$881$	−39922.4	−1.52670	−0.763349	+	0.645987i	$0.776446\pi$
$882$	0	0
$883$	15872.9	0.604944	0.302472	−	0.953158i	$-0.402188\pi$
$883$	15872.9	0.604944	0.302472	+	0.953158i	$0.402188\pi$
$884$	6770.90	0.257613
$885$	0	0
$886$	−48693.0	−1.84636
$887$	48655.2	1.84181	0.920903	−	0.389792i	$-0.127453\pi$
$887$	48655.2	1.84181	0.920903	+	0.389792i	$0.127453\pi$
$888$	0	0
$889$	−4882.69	−0.184207
$890$	2057.56	0.0774940
$891$	0	0
$892$	−4556.35	−0.171029
$893$	67560.3	2.53171
$894$	0	0
$895$	−3945.41	−0.147352
$896$	12930.1	0.482104
$897$	0	0
$898$	−15306.7	−0.568811
$899$	−7307.56	−0.271102
$900$	0	0
$901$	11693.1	0.432357
$902$	−35654.6	−1.31615
$903$	0	0
$904$	22981.1	0.845509
$905$	−22739.3	−0.835227
$906$	0	0
$907$	−26851.3	−0.983001	−0.491501	−	0.870877i	$-0.663551\pi$
$907$	−26851.3	−0.983001	−0.491501	+	0.870877i	$0.663551\pi$
$908$	4700.73	0.171805
$909$	0	0
$910$	4935.84	0.179804
$911$	−29057.6	−1.05677	−0.528387	−	0.849003i	$-0.677203\pi$
$911$	−29057.6	−1.05677	−0.528387	+	0.849003i	$0.677203\pi$
$912$	0	0
$913$	−76815.3	−2.78447
$914$	−8364.72	−0.302714
$915$	0	0
$916$	−2810.53	−0.101378
$917$	−1106.65	−0.0398527
$918$	0	0
$919$	24093.6	0.864825	0.432413	−	0.901676i	$-0.357662\pi$
$919$	24093.6	0.864825	0.432413	+	0.901676i	$0.357662\pi$
$920$	−11665.4	−0.418039
$921$	0	0
$922$	−19919.9	−0.711525
$923$	−670.565	−0.0239132
$924$	0	0
$925$	4943.49	0.175720
$926$	38795.4	1.37678
$927$	0	0
$928$	5211.84	0.184361
$929$	53631.6	1.89408	0.947038	−	0.321123i	$-0.104060\pi$
$929$	53631.6	1.89408	0.947038	+	0.321123i	$0.104060\pi$
$930$	0	0
$931$	33770.8	1.18882
$932$	−18477.7	−0.649417
$933$	0	0
$934$	−15641.3	−0.547964
$935$	15075.2	0.527284
$936$	0	0
$937$	26075.5	0.909125	0.454562	−	0.890715i	$-0.349796\pi$
$937$	26075.5	0.909125	0.454562	+	0.890715i	$0.349796\pi$
$938$	−7166.48	−0.249460
$939$	0	0
$940$	−11635.8	−0.403744
$941$	18872.4	0.653796	0.326898	−	0.945060i	$-0.393997\pi$
$941$	18872.4	0.653796	0.326898	+	0.945060i	$0.393997\pi$
$942$	0	0
$943$	−30078.3	−1.03869
$944$	−16205.7	−0.558739
$945$	0	0
$946$	7247.37	0.249083
$947$	9904.80	0.339876	0.169938	−	0.985455i	$-0.445643\pi$
$947$	9904.80	0.339876	0.169938	+	0.985455i	$0.445643\pi$
$948$	0	0
$949$	−36221.1	−1.23898
$950$	11244.1	0.384008
$951$	0	0
$952$	−5470.41	−0.186236
$953$	−11915.9	−0.405029	−0.202514	−	0.979279i	$-0.564911\pi$
$953$	−11915.9	−0.405029	−0.202514	+	0.979279i	$0.564911\pi$
$954$	0	0
$955$	5607.23	0.189996
$956$	−6516.74	−0.220467
$957$	0	0
$958$	−24365.5	−0.821725
$959$	−4875.07	−0.164155
$960$	0	0
$961$	33705.4	1.13140
$962$	−22017.7	−0.737919
$963$	0	0
$964$	31433.6	1.05021
$965$	−19750.2	−0.658842
$966$	0	0
$967$	38119.9	1.26769	0.633843	−	0.773462i	$-0.281477\pi$
$967$	38119.9	1.26769	0.633843	+	0.773462i	$0.281477\pi$
$968$	31780.0	1.05521
$969$	0	0
$970$	4854.51	0.160690
$971$	12393.2	0.409593	0.204797	−	0.978805i	$-0.434347\pi$
$971$	12393.2	0.409593	0.204797	+	0.978805i	$0.434347\pi$
$972$	0	0
$973$	19888.6	0.655292
$974$	−15634.0	−0.514318
$975$	0	0
$976$	−39296.8	−1.28879
$977$	−36796.8	−1.20495	−0.602474	−	0.798138i	$-0.705818\pi$
$977$	−36796.8	−1.20495	−0.602474	+	0.798138i	$0.705818\pi$
$978$	0	0
$979$	−7239.60	−0.236342
$980$	−5816.31	−0.189587
$981$	0	0
$982$	42569.6	1.38335
$983$	−17819.2	−0.578174	−0.289087	−	0.957303i	$-0.593352\pi$
$983$	−17819.2	−0.578174	−0.289087	+	0.957303i	$0.593352\pi$
$984$	0	0
$985$	2182.17	0.0705886
$986$	−4970.03	−0.160525
$987$	0	0
$988$	−17769.4	−0.572185
$989$	6113.90	0.196573
$990$	0	0
$991$	−832.856	−0.0266968	−0.0133484	−	0.999911i	$-0.504249\pi$
$991$	−832.856	−0.0266968	−0.0133484	+	0.999911i	$0.504249\pi$
$992$	−45286.4	−1.44944
$993$	0	0
$994$	−662.042	−0.0211255
$995$	3878.94	0.123589
$996$	0	0
$997$	−5156.04	−0.163785	−0.0818923	−	0.996641i	$-0.526096\pi$
$997$	−5156.04	−0.163785	−0.0818923	+	0.996641i	$0.526096\pi$
$998$	−25748.7	−0.816694
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.m.1.2		7
3.2	odd	2		435.4.a.j.1.6	✓	7
15.14	odd	2		2175.4.a.m.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.j.1.6	✓	7	3.2	odd	2
1305.4.a.m.1.2		7	1.1	even	1	trivial
2175.4.a.m.1.2		7	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-3.52131 q^{2} +4.39964 q^{4} +5.00000 q^{5} -8.86571 q^{7} +12.6780 q^{8} -17.6066 q^{10} +61.9492 q^{11} +31.6208 q^{13} +31.2189 q^{14} -79.8403 q^{16} +48.6694 q^{17} -127.727 q^{19} +21.9982 q^{20} -218.143 q^{22} -184.026 q^{23} +25.0000 q^{25} -111.347 q^{26} -39.0059 q^{28} +29.0000 q^{29} -251.985 q^{31} +179.719 q^{32} -171.380 q^{34} -44.3285 q^{35} +197.740 q^{37} +449.765 q^{38} +63.3899 q^{40} +163.446 q^{41} -33.2231 q^{43} +272.555 q^{44} +648.012 q^{46} -528.945 q^{47} -264.399 q^{49} -88.0328 q^{50} +139.120 q^{52} +240.256 q^{53} +309.746 q^{55} -112.399 q^{56} -102.118 q^{58} +202.976 q^{59} +492.192 q^{61} +887.318 q^{62} +5.87624 q^{64} +158.104 q^{65} -229.556 q^{67} +214.128 q^{68} +156.095 q^{70} -21.2064 q^{71} -1145.48 q^{73} -696.303 q^{74} -561.952 q^{76} -549.224 q^{77} +95.6307 q^{79} -399.201 q^{80} -575.545 q^{82} -1239.97 q^{83} +243.347 q^{85} +116.989 q^{86} +785.391 q^{88} -116.863 q^{89} -280.341 q^{91} -809.648 q^{92} +1862.58 q^{94} -638.633 q^{95} -275.722 q^{97} +931.032 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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