LMFDB - Embedded newform 4900.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(1,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.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:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 700)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.20440 q^{3} +7.26819 q^{9} +O(q^{10})$	$q+3.20440 q^{3} +7.26819 q^{9} +4.20440 q^{11} -0.204402 q^{13} +5.06379 q^{17} +1.06379 q^{19} -2.14061 q^{23} +13.6770 q^{27} -7.47259 q^{29} -8.47259 q^{31} +13.4726 q^{33} +10.6132 q^{37} -0.654985 q^{39} +10.5494 q^{41} -8.26819 q^{43} -3.26819 q^{47} +16.2264 q^{51} -5.67699 q^{53} +3.40880 q^{57} +1.20440 q^{59} +1.65498 q^{61} +12.4088 q^{67} -6.85939 q^{69} -0.591197 q^{71} +4.00000 q^{73} +6.54942 q^{79} +22.0220 q^{81} -3.88139 q^{83} -23.9452 q^{87} -9.26819 q^{89} -27.1496 q^{93} -1.33198 q^{97} +30.5584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 + 8 * q^9	$ 3 q + q^{3} + 8 q^{9} + 4 q^{11} + 8 q^{13} + 10 q^{17} - 2 q^{19} - 3 q^{23} + 10 q^{27} - 3 q^{31} + 18 q^{33} + 6 q^{37} - 14 q^{39} + 11 q^{41} - 11 q^{43} + 4 q^{47} - 3 q^{51} + 14 q^{53} - 7 q^{57} - 5 q^{59} + 17 q^{61} + 20 q^{67} - 24 q^{69} - 19 q^{71} + 12 q^{73} - q^{79} + 23 q^{81} + 28 q^{83} - 27 q^{87} - 14 q^{89} - 28 q^{93} + 15 q^{97} + 21 q^{99}+O(q^{100}) $ 3 * q + q^3 + 8 * q^9 + 4 * q^11 + 8 * q^13 + 10 * q^17 - 2 * q^19 - 3 * q^23 + 10 * q^27 - 3 * q^31 + 18 * q^33 + 6 * q^37 - 14 * q^39 + 11 * q^41 - 11 * q^43 + 4 * q^47 - 3 * q^51 + 14 * q^53 - 7 * q^57 - 5 * q^59 + 17 * q^61 + 20 * q^67 - 24 * q^69 - 19 * q^71 + 12 * q^73 - q^79 + 23 * q^81 + 28 * q^83 - 27 * q^87 - 14 * q^89 - 28 * q^93 + 15 * q^97 + 21 * 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
$3$	3.20440	1.85006	0.925031	−	0.379892i	$-0.124039\pi$
$3$	3.20440	1.85006	0.925031	+	0.379892i	$0.124039\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.26819	2.42273
$10$	0	0
$11$	4.20440	1.26767	0.633837	−	0.773466i	$-0.281479\pi$
$11$	4.20440	1.26767	0.633837	+	0.773466i	$0.281479\pi$
$12$	0	0
$13$	−0.204402	−0.0566908	−0.0283454	−	0.999598i	$-0.509024\pi$
$13$	−0.204402	−0.0566908	−0.0283454	+	0.999598i	$0.509024\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.06379	1.22815	0.614074	−	0.789248i	$-0.289529\pi$
$17$	5.06379	1.22815	0.614074	+	0.789248i	$0.289529\pi$
$18$	0	0
$19$	1.06379	0.244050	0.122025	−	0.992527i	$-0.461061\pi$
$19$	1.06379	0.244050	0.122025	+	0.992527i	$0.461061\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.14061	−0.446349	−0.223174	−	0.974779i	$-0.571642\pi$
$23$	−2.14061	−0.446349	−0.223174	+	0.974779i	$0.571642\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	13.6770	2.63214
$28$	0	0
$29$	−7.47259	−1.38763	−0.693813	−	0.720156i	$-0.744070\pi$
$29$	−7.47259	−1.38763	−0.693813	+	0.720156i	$0.744070\pi$
$30$	0	0
$31$	−8.47259	−1.52172	−0.760861	−	0.648915i	$-0.775223\pi$
$31$	−8.47259	−1.52172	−0.760861	+	0.648915i	$0.775223\pi$
$32$	0	0
$33$	13.4726	2.34528
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.6132	1.74480	0.872400	−	0.488793i	$-0.162563\pi$
$37$	10.6132	1.74480	0.872400	+	0.488793i	$0.162563\pi$
$38$	0	0
$39$	−0.654985	−0.104881
$40$	0	0
$41$	10.5494	1.64754	0.823771	−	0.566923i	$-0.191866\pi$
$41$	10.5494	1.64754	0.823771	+	0.566923i	$0.191866\pi$
$42$	0	0
$43$	−8.26819	−1.26089	−0.630444	−	0.776235i	$-0.717127\pi$
$43$	−8.26819	−1.26089	−0.630444	+	0.776235i	$0.717127\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.26819	−0.476714	−0.238357	−	0.971178i	$-0.576609\pi$
$47$	−3.26819	−0.476714	−0.238357	+	0.971178i	$0.576609\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.2264	2.27215
$52$	0	0
$53$	−5.67699	−0.779795	−0.389897	−	0.920858i	$-0.627490\pi$
$53$	−5.67699	−0.779795	−0.389897	+	0.920858i	$0.627490\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.40880	0.451507
$58$	0	0
$59$	1.20440	0.156800	0.0783999	−	0.996922i	$-0.475019\pi$
$59$	1.20440	0.156800	0.0783999	+	0.996922i	$0.475019\pi$
$60$	0	0
$61$	1.65498	0.211899	0.105950	−	0.994372i	$-0.466212\pi$
$61$	1.65498	0.211899	0.105950	+	0.994372i	$0.466212\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.4088	1.51598	0.757988	−	0.652268i	$-0.226182\pi$
$67$	12.4088	1.51598	0.757988	+	0.652268i	$0.226182\pi$
$68$	0	0
$69$	−6.85939	−0.825773
$70$	0	0
$71$	−0.591197	−0.0701622	−0.0350811	−	0.999384i	$-0.511169\pi$
$71$	−0.591197	−0.0701622	−0.0350811	+	0.999384i	$0.511169\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.54942	0.736867	0.368433	−	0.929654i	$-0.379894\pi$
$79$	6.54942	0.736867	0.368433	+	0.929654i	$0.379894\pi$
$80$	0	0
$81$	22.0220	2.44689
$82$	0	0
$83$	−3.88139	−0.426038	−0.213019	−	0.977048i	$-0.568330\pi$
$83$	−3.88139	−0.426038	−0.213019	+	0.977048i	$0.568330\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−23.9452	−2.56719
$88$	0	0
$89$	−9.26819	−0.982426	−0.491213	−	0.871039i	$-0.663446\pi$
$89$	−9.26819	−0.982426	−0.491213	+	0.871039i	$0.663446\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−27.1496	−2.81528
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.33198	−0.135242	−0.0676209	−	0.997711i	$-0.521541\pi$
$97$	−1.33198	−0.135242	−0.0676209	+	0.997711i	$0.521541\pi$
$98$	0	0
$99$	30.5584	3.07123
$100$	0	0
$101$	6.19136	0.616064	0.308032	−	0.951376i	$-0.400330\pi$
$101$	6.19136	0.616064	0.308032	+	0.951376i	$0.400330\pi$
$102$	0	0
$103$	−9.74078	−0.959788	−0.479894	−	0.877327i	$-0.659325\pi$
$103$	−9.74078	−0.959788	−0.479894	+	0.877327i	$0.659325\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.85939	−0.953143	−0.476571	−	0.879136i	$-0.658121\pi$
$107$	−9.85939	−0.953143	−0.476571	+	0.879136i	$0.658121\pi$
$108$	0	0
$109$	−4.07683	−0.390489	−0.195245	−	0.980755i	$-0.562550\pi$
$109$	−4.07683	−0.390489	−0.195245	+	0.980755i	$0.562550\pi$
$110$	0	0
$111$	34.0090	3.22799
$112$	0	0
$113$	−2.21744	−0.208599	−0.104300	−	0.994546i	$-0.533260\pi$
$113$	−2.21744	−0.208599	−0.104300	+	0.994546i	$0.533260\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.48563	−0.137346
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.67699	0.606999
$122$	0	0
$123$	33.8046	3.04806
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.54942	−0.581167	−0.290583	−	0.956850i	$-0.593849\pi$
$127$	−6.54942	−0.581167	−0.290583	+	0.956850i	$0.593849\pi$
$128$	0	0
$129$	−26.4946	−2.33272
$130$	0	0
$131$	−17.0090	−1.48608	−0.743040	−	0.669247i	$-0.766617\pi$
$131$	−17.0090	−1.48608	−0.743040	+	0.669247i	$0.766617\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.0858	1.45974	0.729869	−	0.683587i	$-0.239581\pi$
$137$	17.0858	1.45974	0.729869	+	0.683587i	$0.239581\pi$
$138$	0	0
$139$	0.740780	0.0628322	0.0314161	−	0.999506i	$-0.489998\pi$
$139$	0.740780	0.0628322	0.0314161	+	0.999506i	$0.489998\pi$
$140$	0	0
$141$	−10.4726	−0.881951
$142$	0	0
$143$	−0.859386	−0.0718655
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.4178	−1.34500	−0.672498	−	0.740099i	$-0.734779\pi$
$149$	−16.4178	−1.34500	−0.672498	+	0.740099i	$0.734779\pi$
$150$	0	0
$151$	0.527409	0.0429199	0.0214600	−	0.999770i	$-0.493169\pi$
$151$	0.527409	0.0429199	0.0214600	+	0.999770i	$0.493169\pi$
$152$	0	0
$153$	36.8046	2.97547
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.48563	0.757036	0.378518	−	0.925594i	$-0.376434\pi$
$157$	9.48563	0.757036	0.378518	+	0.925594i	$0.376434\pi$
$158$	0	0
$159$	−18.1914	−1.44267
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.85939	−0.693921	−0.346960	−	0.937880i	$-0.612786\pi$
$163$	−8.85939	−0.693921	−0.346960	+	0.937880i	$0.612786\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.33198	−0.644748	−0.322374	−	0.946612i	$-0.604481\pi$
$167$	−8.33198	−0.644748	−0.322374	+	0.946612i	$0.604481\pi$
$168$	0	0
$169$	−12.9582	−0.996786
$170$	0	0
$171$	7.73181	0.591266
$172$	0	0
$173$	18.3450	1.39475	0.697373	−	0.716709i	$-0.254352\pi$
$173$	18.3450	1.39475	0.697373	+	0.716709i	$0.254352\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.85939	0.290089
$178$	0	0
$179$	−4.28123	−0.319994	−0.159997	−	0.987118i	$-0.551148\pi$
$179$	−4.28123	−0.319994	−0.159997	+	0.987118i	$0.551148\pi$
$180$	0	0
$181$	2.93621	0.218247	0.109123	−	0.994028i	$-0.465196\pi$
$181$	2.93621	0.218247	0.109123	+	0.994028i	$0.465196\pi$
$182$	0	0
$183$	5.30324	0.392026
$184$	0	0
$185$	0	0
$186$	0	0
$187$	21.2902	1.55689
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.9452	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$191$	−11.9452	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$192$	0	0
$193$	−7.08580	−0.510047	−0.255023	−	0.966935i	$-0.582083\pi$
$193$	−7.08580	−0.510047	−0.255023	+	0.966935i	$0.582083\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.48563	0.390835	0.195417	−	0.980720i	$-0.437394\pi$
$197$	5.48563	0.390835	0.195417	+	0.980720i	$0.437394\pi$
$198$	0	0
$199$	16.0858	1.14029	0.570146	−	0.821543i	$-0.306887\pi$
$199$	16.0858	1.14029	0.570146	+	0.821543i	$0.306887\pi$
$200$	0	0
$201$	39.7628	2.80465
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−15.5584	−1.08138
$208$	0	0
$209$	4.47259	0.309376
$210$	0	0
$211$	16.6002	1.14280	0.571401	−	0.820671i	$-0.306400\pi$
$211$	16.6002	1.14280	0.571401	+	0.820671i	$0.306400\pi$
$212$	0	0
$213$	−1.89443	−0.129804
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.8176	0.866134
$220$	0	0
$221$	−1.03505	−0.0696247
$222$	0	0
$223$	15.4856	1.03699	0.518497	−	0.855079i	$-0.326492\pi$
$223$	15.4856	1.03699	0.518497	+	0.855079i	$0.326492\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.78256	0.251057	0.125529	−	0.992090i	$-0.459937\pi$
$227$	3.78256	0.251057	0.125529	+	0.992090i	$0.459937\pi$
$228$	0	0
$229$	−11.2044	−0.740408	−0.370204	−	0.928951i	$-0.620712\pi$
$229$	−11.2044	−0.740408	−0.370204	+	0.928951i	$0.620712\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.9452	0.782555	0.391277	−	0.920273i	$-0.372033\pi$
$233$	11.9452	0.782555	0.391277	+	0.920273i	$0.372033\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	20.9870	1.36325
$238$	0	0
$239$	15.3450	0.992587	0.496293	−	0.868155i	$-0.334694\pi$
$239$	15.3450	0.992587	0.496293	+	0.868155i	$0.334694\pi$
$240$	0	0
$241$	−14.3958	−0.927313	−0.463656	−	0.886015i	$-0.653463\pi$
$241$	−14.3958	−0.927313	−0.463656	+	0.886015i	$0.653463\pi$
$242$	0	0
$243$	29.5364	1.89476
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.217440	−0.0138354
$248$	0	0
$249$	−12.4375	−0.788197
$250$	0	0
$251$	−19.4178	−1.22564	−0.612819	−	0.790223i	$-0.709965\pi$
$251$	−19.4178	−1.22564	−0.612819	+	0.790223i	$0.709965\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.8724	0.865338	0.432669	−	0.901553i	$-0.357572\pi$
$257$	13.8724	0.865338	0.432669	+	0.901553i	$0.357572\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−54.3122	−3.36184
$262$	0	0
$263$	−3.45955	−0.213325	−0.106663	−	0.994295i	$-0.534016\pi$
$263$	−3.45955	−0.213325	−0.106663	+	0.994295i	$0.534016\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−29.6990	−1.81755
$268$	0	0
$269$	25.6640	1.56476	0.782379	−	0.622802i	$-0.214006\pi$
$269$	25.6640	1.56476	0.782379	+	0.622802i	$0.214006\pi$
$270$	0	0
$271$	19.6770	1.19529	0.597646	−	0.801760i	$-0.296103\pi$
$271$	19.6770	1.19529	0.597646	+	0.801760i	$0.296103\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.1496	−1.09050	−0.545251	−	0.838273i	$-0.683566\pi$
$277$	−18.1496	−1.09050	−0.545251	+	0.838273i	$0.683566\pi$
$278$	0	0
$279$	−61.5804	−3.68672
$280$	0	0
$281$	−18.5364	−1.10579	−0.552894	−	0.833252i	$-0.686476\pi$
$281$	−18.5364	−1.10579	−0.552894	+	0.833252i	$0.686476\pi$
$282$	0	0
$283$	28.5584	1.69762	0.848810	−	0.528698i	$-0.177320\pi$
$283$	28.5584	1.69762	0.848810	+	0.528698i	$0.177320\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.64195	0.508350
$290$	0	0
$291$	−4.26819	−0.250206
$292$	0	0
$293$	25.5494	1.49261	0.746306	−	0.665603i	$-0.231825\pi$
$293$	25.5494	1.49261	0.746306	+	0.665603i	$0.231825\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	57.5036	3.33670
$298$	0	0
$299$	0.437545	0.0253039
$300$	0	0
$301$	0	0
$302$	0	0
$303$	19.8396	1.13976
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.1145	0.577267	0.288634	−	0.957440i	$-0.406799\pi$
$307$	10.1145	0.577267	0.288634	+	0.957440i	$0.406799\pi$
$308$	0	0
$309$	−31.2134	−1.77567
$310$	0	0
$311$	34.1716	1.93769	0.968847	−	0.247662i	$-0.0796621\pi$
$311$	34.1716	1.93769	0.968847	+	0.247662i	$0.0796621\pi$
$312$	0	0
$313$	−8.61320	−0.486847	−0.243424	−	0.969920i	$-0.578270\pi$
$313$	−8.61320	−0.486847	−0.243424	+	0.969920i	$0.578270\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.81761	−0.439081	−0.219540	−	0.975603i	$-0.570456\pi$
$317$	−7.81761	−0.439081	−0.219540	+	0.975603i	$0.570456\pi$
$318$	0	0
$319$	−31.4178	−1.75906
$320$	0	0
$321$	−31.5934	−1.76337
$322$	0	0
$323$	5.38680	0.299729
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.0638	−0.722429
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.0988	−1.21466	−0.607331	−	0.794449i	$-0.707760\pi$
$331$	−22.0988	−1.21466	−0.607331	+	0.794449i	$0.707760\pi$
$332$	0	0
$333$	77.1388	4.22718
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.29020	0.0702815	0.0351408	−	0.999382i	$-0.488812\pi$
$337$	1.29020	0.0702815	0.0351408	+	0.999382i	$0.488812\pi$
$338$	0	0
$339$	−7.10557	−0.385921
$340$	0	0
$341$	−35.6222	−1.92905
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.45955	0.507815	0.253908	−	0.967228i	$-0.418284\pi$
$347$	9.45955	0.507815	0.253908	+	0.967228i	$0.418284\pi$
$348$	0	0
$349$	15.5494	0.832341	0.416171	−	0.909287i	$-0.363372\pi$
$349$	15.5494	0.832341	0.416171	+	0.909287i	$0.363372\pi$
$350$	0	0
$351$	−2.79560	−0.149218
$352$	0	0
$353$	−15.1365	−0.805637	−0.402818	−	0.915280i	$-0.631969\pi$
$353$	−15.1365	−0.805637	−0.402818	+	0.915280i	$0.631969\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.3189	0.544613	0.272306	−	0.962211i	$-0.412214\pi$
$359$	10.3189	0.544613	0.272306	+	0.962211i	$0.412214\pi$
$360$	0	0
$361$	−17.8684	−0.940440
$362$	0	0
$363$	21.3958	1.12299
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−17.9362	−0.936263	−0.468131	−	0.883659i	$-0.655073\pi$
$367$	−17.9362	−0.936263	−0.468131	+	0.883659i	$0.655073\pi$
$368$	0	0
$369$	76.6752	3.99155
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.1914	0.838357	0.419179	−	0.907904i	$-0.362318\pi$
$373$	16.1914	0.838357	0.419179	+	0.907904i	$0.362318\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.52741	0.0786656
$378$	0	0
$379$	−12.6002	−0.647227	−0.323614	−	0.946189i	$-0.604898\pi$
$379$	−12.6002	−0.647227	−0.323614	+	0.946189i	$0.604898\pi$
$380$	0	0
$381$	−20.9870	−1.07519
$382$	0	0
$383$	−15.1757	−0.775440	−0.387720	−	0.921777i	$-0.626737\pi$
$383$	−15.1757	−0.775440	−0.387720	+	0.921777i	$0.626737\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−60.0948	−3.05479
$388$	0	0
$389$	−29.2394	−1.48250	−0.741249	−	0.671230i	$-0.765766\pi$
$389$	−29.2394	−1.48250	−0.741249	+	0.671230i	$0.765766\pi$
$390$	0	0
$391$	−10.8396	−0.548183
$392$	0	0
$393$	−54.5036	−2.74934
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.2682	−0.565534	−0.282767	−	0.959189i	$-0.591252\pi$
$397$	−11.2682	−0.565534	−0.282767	+	0.959189i	$0.591252\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0220	−0.899976	−0.449988	−	0.893035i	$-0.648572\pi$
$401$	−18.0220	−0.899976	−0.449988	+	0.893035i	$0.648572\pi$
$402$	0	0
$403$	1.73181	0.0862676
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.6222	2.21184
$408$	0	0
$409$	−15.7915	−0.780841	−0.390420	−	0.920637i	$-0.627670\pi$
$409$	−15.7915	−0.780841	−0.390420	+	0.920637i	$0.627670\pi$
$410$	0	0
$411$	54.7497	2.70061
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.37376	0.116243
$418$	0	0
$419$	−13.4726	−0.658179	−0.329090	−	0.944299i	$-0.606742\pi$
$419$	−13.4726	−0.658179	−0.329090	+	0.944299i	$0.606742\pi$
$420$	0	0
$421$	−26.8396	−1.30808	−0.654041	−	0.756459i	$-0.726928\pi$
$421$	−26.8396	−1.30808	−0.654041	+	0.756459i	$0.726928\pi$
$422$	0	0
$423$	−23.7538	−1.15495
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.75382	−0.132956
$430$	0	0
$431$	16.0507	0.773137	0.386569	−	0.922261i	$-0.373660\pi$
$431$	16.0507	0.773137	0.386569	+	0.922261i	$0.373660\pi$
$432$	0	0
$433$	−0.910136	−0.0437383	−0.0218692	−	0.999761i	$-0.506962\pi$
$433$	−0.910136	−0.0437383	−0.0218692	+	0.999761i	$0.506962\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.27716	−0.108931
$438$	0	0
$439$	−37.9034	−1.80903	−0.904515	−	0.426441i	$-0.859767\pi$
$439$	−37.9034	−1.80903	−0.904515	+	0.426441i	$0.859767\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.88139	0.326945	0.163472	−	0.986548i	$-0.447731\pi$
$443$	6.88139	0.326945	0.163472	+	0.986548i	$0.447731\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−52.6091	−2.48833
$448$	0	0
$449$	6.43754	0.303807	0.151903	−	0.988395i	$-0.451460\pi$
$449$	6.43754	0.303807	0.151903	+	0.988395i	$0.451460\pi$
$450$	0	0
$451$	44.3540	2.08855
$452$	0	0
$453$	1.69003	0.0794046
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.29427	0.434767	0.217384	−	0.976086i	$-0.430248\pi$
$457$	9.29427	0.434767	0.217384	+	0.976086i	$0.430248\pi$
$458$	0	0
$459$	69.2574	3.23266
$460$	0	0
$461$	9.29020	0.432688	0.216344	−	0.976317i	$-0.430587\pi$
$461$	9.29020	0.432688	0.216344	+	0.976317i	$0.430587\pi$
$462$	0	0
$463$	29.6091	1.37605	0.688027	−	0.725685i	$-0.258477\pi$
$463$	29.6091	1.37605	0.688027	+	0.725685i	$0.258477\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.0638	1.20609	0.603044	−	0.797708i	$-0.293954\pi$
$467$	26.0638	1.20609	0.603044	+	0.797708i	$0.293954\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.3958	1.40056
$472$	0	0
$473$	−34.7628	−1.59839
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−41.2615	−1.88923
$478$	0	0
$479$	−1.10557	−0.0505147	−0.0252573	−	0.999681i	$-0.508041\pi$
$479$	−1.10557	−0.0505147	−0.0252573	+	0.999681i	$0.508041\pi$
$480$	0	0
$481$	−2.16936	−0.0989141
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.19136	−0.235243	−0.117622	−	0.993058i	$-0.537527\pi$
$487$	−5.19136	−0.235243	−0.117622	+	0.993058i	$0.537527\pi$
$488$	0	0
$489$	−28.3890	−1.28380
$490$	0	0
$491$	−28.6640	−1.29359	−0.646793	−	0.762666i	$-0.723890\pi$
$491$	−28.6640	−1.29359	−0.646793	+	0.762666i	$0.723890\pi$
$492$	0	0
$493$	−37.8396	−1.70421
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−26.0728	−1.16718	−0.583588	−	0.812050i	$-0.698352\pi$
$499$	−26.0728	−1.16718	−0.583588	+	0.812050i	$0.698352\pi$
$500$	0	0
$501$	−26.6990	−1.19282
$502$	0	0
$503$	−8.80864	−0.392758	−0.196379	−	0.980528i	$-0.562918\pi$
$503$	−8.80864	−0.392758	−0.196379	+	0.980528i	$0.562918\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−41.5233	−1.84412
$508$	0	0
$509$	−22.0179	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$509$	−22.0179	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.5494	0.642372
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.7408	−0.604319
$518$	0	0
$519$	58.7848	2.58037
$520$	0	0
$521$	−15.0988	−0.661492	−0.330746	−	0.943720i	$-0.607300\pi$
$521$	−15.0988	−0.661492	−0.330746	+	0.943720i	$0.607300\pi$
$522$	0	0
$523$	4.59120	0.200759	0.100380	−	0.994949i	$-0.467994\pi$
$523$	4.59120	0.200759	0.100380	+	0.994949i	$0.467994\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.9034	−1.86890
$528$	0	0
$529$	−18.4178	−0.800773
$530$	0	0
$531$	8.75382	0.379883
$532$	0	0
$533$	−2.15632	−0.0934005
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−13.7188	−0.592009
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13.6770	0.588020	0.294010	−	0.955802i	$-0.405010\pi$
$541$	13.6770	0.588020	0.294010	+	0.955802i	$0.405010\pi$
$542$	0	0
$543$	9.40880	0.403770
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.3581	0.485635	0.242818	−	0.970072i	$-0.421928\pi$
$547$	11.3581	0.485635	0.242818	+	0.970072i	$0.421928\pi$
$548$	0	0
$549$	12.0287	0.513374
$550$	0	0
$551$	−7.94925	−0.338649
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.3958	−0.694711	−0.347355	−	0.937734i	$-0.612920\pi$
$557$	−16.3958	−0.694711	−0.347355	+	0.937734i	$0.612920\pi$
$558$	0	0
$559$	1.69003	0.0714807
$560$	0	0
$561$	68.2223	2.88035
$562$	0	0
$563$	4.87242	0.205348	0.102674	−	0.994715i	$-0.467260\pi$
$563$	4.87242	0.205348	0.102674	+	0.994715i	$0.467260\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−37.8553	−1.58698	−0.793489	−	0.608585i	$-0.791737\pi$
$569$	−37.8553	−1.58698	−0.793489	+	0.608585i	$0.791737\pi$
$570$	0	0
$571$	−41.7408	−1.74680	−0.873399	−	0.487006i	$-0.838089\pi$
$571$	−41.7408	−1.74680	−0.873399	+	0.487006i	$0.838089\pi$
$572$	0	0
$573$	−38.2772	−1.59905
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−43.0530	−1.79232	−0.896160	−	0.443732i	$-0.853654\pi$
$577$	−43.0530	−1.79232	−0.896160	+	0.443732i	$0.853654\pi$
$578$	0	0
$579$	−22.7057	−0.943618
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−23.8684	−0.988526
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.67699	0.234315	0.117157	−	0.993113i	$-0.462622\pi$
$587$	5.67699	0.234315	0.117157	+	0.993113i	$0.462622\pi$
$588$	0	0
$589$	−9.01304	−0.371376
$590$	0	0
$591$	17.5782	0.723069
$592$	0	0
$593$	−8.94518	−0.367335	−0.183667	−	0.982988i	$-0.558797\pi$
$593$	−8.94518	−0.367335	−0.183667	+	0.982988i	$0.558797\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	51.5453	2.10961
$598$	0	0
$599$	−25.2264	−1.03072	−0.515362	−	0.856973i	$-0.672342\pi$
$599$	−25.2264	−1.03072	−0.515362	+	0.856973i	$0.672342\pi$
$600$	0	0
$601$	15.3409	0.625770	0.312885	−	0.949791i	$-0.398705\pi$
$601$	15.3409	0.625770	0.312885	+	0.949791i	$0.398705\pi$
$602$	0	0
$603$	90.1895	3.67280
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.9362	−0.849775	−0.424887	−	0.905246i	$-0.639686\pi$
$607$	−20.9362	−0.849775	−0.424887	+	0.905246i	$0.639686\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.668023	0.0270253
$612$	0	0
$613$	5.31894	0.214830	0.107415	−	0.994214i	$-0.465743\pi$
$613$	5.31894	0.214830	0.107415	+	0.994214i	$0.465743\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.6262	1.15245	0.576225	−	0.817291i	$-0.304525\pi$
$617$	28.6262	1.15245	0.576225	+	0.817291i	$0.304525\pi$
$618$	0	0
$619$	−21.5625	−0.866668	−0.433334	−	0.901233i	$-0.642663\pi$
$619$	−21.5625	−0.866668	−0.433334	+	0.901233i	$0.642663\pi$
$620$	0	0
$621$	−29.2772	−1.17485
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.3320	0.572364
$628$	0	0
$629$	53.7430	2.14287
$630$	0	0
$631$	−25.1365	−1.00067	−0.500335	−	0.865832i	$-0.666790\pi$
$631$	−25.1365	−1.00067	−0.500335	+	0.865832i	$0.666790\pi$
$632$	0	0
$633$	53.1936	2.11426
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.29693	−0.169984
$640$	0	0
$641$	44.1078	1.74215	0.871077	−	0.491147i	$-0.163422\pi$
$641$	44.1078	1.74215	0.871077	+	0.491147i	$0.163422\pi$
$642$	0	0
$643$	16.4816	0.649969	0.324985	−	0.945719i	$-0.394641\pi$
$643$	16.4816	0.649969	0.324985	+	0.945719i	$0.394641\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.4398	1.23603	0.618013	−	0.786168i	$-0.287938\pi$
$647$	31.4398	1.23603	0.618013	+	0.786168i	$0.287938\pi$
$648$	0	0
$649$	5.06379	0.198771
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−40.8773	−1.59965	−0.799827	−	0.600231i	$-0.795075\pi$
$653$	−40.8773	−1.59965	−0.799827	+	0.600231i	$0.795075\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	29.0728	1.13424
$658$	0	0
$659$	−46.6860	−1.81863	−0.909313	−	0.416112i	$-0.863392\pi$
$659$	−46.6860	−1.81863	−0.909313	+	0.416112i	$0.863392\pi$
$660$	0	0
$661$	0.127575	0.00496211	0.00248106	−	0.999997i	$-0.499210\pi$
$661$	0.127575	0.00496211	0.00248106	+	0.999997i	$0.499210\pi$
$662$	0	0
$663$	−3.31670	−0.128810
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.9959	0.619365
$668$	0	0
$669$	49.6222	1.91850
$670$	0	0
$671$	6.95822	0.268619
$672$	0	0
$673$	−48.1936	−1.85773	−0.928863	−	0.370423i	$-0.879213\pi$
$673$	−48.1936	−1.85773	−0.928863	+	0.370423i	$0.879213\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.4178	1.78398	0.891990	−	0.452055i	$-0.149309\pi$
$677$	46.4178	1.78398	0.891990	+	0.452055i	$0.149309\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.1208	0.464472
$682$	0	0
$683$	−43.5494	−1.66637	−0.833186	−	0.552993i	$-0.813486\pi$
$683$	−43.5494	−1.66637	−0.833186	+	0.552993i	$0.813486\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−35.9034	−1.36980
$688$	0	0
$689$	1.16039	0.0442072
$690$	0	0
$691$	−26.0948	−0.992692	−0.496346	−	0.868125i	$-0.665325\pi$
$691$	−26.0948	−0.992692	−0.496346	+	0.868125i	$0.665325\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	53.4200	2.02343
$698$	0	0
$699$	38.2772	1.44778
$700$	0	0
$701$	21.7277	0.820645	0.410323	−	0.911940i	$-0.365416\pi$
$701$	21.7277	0.820645	0.410323	+	0.911940i	$0.365416\pi$
$702$	0	0
$703$	11.2902	0.425818
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.0768	0.641334	0.320667	−	0.947192i	$-0.396093\pi$
$709$	17.0768	0.641334	0.320667	+	0.947192i	$0.396093\pi$
$710$	0	0
$711$	47.6024	1.78523
$712$	0	0
$713$	18.1365	0.679219
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.1716	1.83635
$718$	0	0
$719$	27.2462	1.01611	0.508056	−	0.861324i	$-0.330364\pi$
$719$	27.2462	1.01611	0.508056	+	0.861324i	$0.330364\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−46.1298	−1.71559
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.9232	−0.738910	−0.369455	−	0.929249i	$-0.620456\pi$
$727$	−19.9232	−0.738910	−0.369455	+	0.929249i	$0.620456\pi$
$728$	0	0
$729$	28.5804	1.05853
$730$	0	0
$731$	−41.8684	−1.54856
$732$	0	0
$733$	−31.1626	−1.15102	−0.575509	−	0.817796i	$-0.695196\pi$
$733$	−31.1626	−1.15102	−0.575509	+	0.817796i	$0.695196\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.1716	1.92177
$738$	0	0
$739$	1.90117	0.0699355	0.0349678	−	0.999388i	$-0.488867\pi$
$739$	1.90117	0.0699355	0.0349678	+	0.999388i	$0.488867\pi$
$740$	0	0
$741$	−0.696765	−0.0255963
$742$	0	0
$743$	−36.0660	−1.32313	−0.661567	−	0.749886i	$-0.730108\pi$
$743$	−36.0660	−1.32313	−0.661567	+	0.749886i	$0.730108\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−28.2107	−1.03218
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.769522	−0.0280803	−0.0140401	−	0.999901i	$-0.504469\pi$
$751$	−0.769522	−0.0280803	−0.0140401	+	0.999901i	$0.504469\pi$
$752$	0	0
$753$	−62.2223	−2.26751
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14.9542	0.543518	0.271759	−	0.962365i	$-0.412395\pi$
$757$	14.9542	0.543518	0.271759	+	0.962365i	$0.412395\pi$
$758$	0	0
$759$	−28.8396	−1.04681
$760$	0	0
$761$	19.1716	0.694970	0.347485	−	0.937686i	$-0.387036\pi$
$761$	19.1716	0.694970	0.347485	+	0.937686i	$0.387036\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.246182	−0.00888910
$768$	0	0
$769$	36.3189	1.30969	0.654847	−	0.755761i	$-0.272733\pi$
$769$	36.3189	1.30969	0.654847	+	0.755761i	$0.272733\pi$
$770$	0	0
$771$	44.4528	1.60093
$772$	0	0
$773$	47.4685	1.70732	0.853662	−	0.520827i	$-0.174376\pi$
$773$	47.4685	1.70732	0.853662	+	0.520827i	$0.174376\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.2223	0.402082
$780$	0	0
$781$	−2.48563	−0.0889428
$782$	0	0
$783$	−102.203	−3.65242
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.88546	−0.281086	−0.140543	−	0.990075i	$-0.544885\pi$
$787$	−7.88546	−0.281086	−0.140543	+	0.990075i	$0.544885\pi$
$788$	0	0
$789$	−11.0858	−0.394665
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.338281	−0.0120127
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.6132	1.19064	0.595320	−	0.803488i	$-0.297025\pi$
$797$	33.6132	1.19064	0.595320	+	0.803488i	$0.297025\pi$
$798$	0	0
$799$	−16.5494	−0.585476
$800$	0	0
$801$	−67.3630	−2.38015
$802$	0	0
$803$	16.8176	0.593480
$804$	0	0
$805$	0	0
$806$	0	0
$807$	82.2376	2.89490
$808$	0	0
$809$	−9.19136	−0.323151	−0.161576	−	0.986860i	$-0.551658\pi$
$809$	−9.19136	−0.323151	−0.161576	+	0.986860i	$0.551658\pi$
$810$	0	0
$811$	21.6483	0.760173	0.380086	−	0.924951i	$-0.375894\pi$
$811$	21.6483	0.760173	0.380086	+	0.924951i	$0.375894\pi$
$812$	0	0
$813$	63.0530	2.21136
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.79560	−0.307719
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.2615	1.12593	0.562966	−	0.826480i	$-0.309660\pi$
$821$	32.2615	1.12593	0.562966	+	0.826480i	$0.309660\pi$
$822$	0	0
$823$	−11.4438	−0.398908	−0.199454	−	0.979907i	$-0.563917\pi$
$823$	−11.4438	−0.398908	−0.199454	+	0.979907i	$0.563917\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.82658	−0.341704	−0.170852	−	0.985297i	$-0.554652\pi$
$827$	−9.82658	−0.341704	−0.170852	+	0.985297i	$0.554652\pi$
$828$	0	0
$829$	38.2574	1.32873	0.664367	−	0.747407i	$-0.268701\pi$
$829$	38.2574	1.32873	0.664367	+	0.747407i	$0.268701\pi$
$830$	0	0
$831$	−58.1586	−2.01750
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−115.880	−4.00538
$838$	0	0
$839$	30.6900	1.05954	0.529769	−	0.848142i	$-0.322279\pi$
$839$	30.6900	1.05954	0.529769	+	0.848142i	$0.322279\pi$
$840$	0	0
$841$	26.8396	0.925504
$842$	0	0
$843$	−59.3980	−2.04578
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	91.5125	3.14070
$850$	0	0
$851$	−22.7188	−0.778789
$852$	0	0
$853$	−39.4569	−1.35098	−0.675489	−	0.737370i	$-0.736067\pi$
$853$	−39.4569	−1.35098	−0.675489	+	0.737370i	$0.736067\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.8594	1.36157	0.680785	−	0.732483i	$-0.261639\pi$
$857$	39.8594	1.36157	0.680785	+	0.732483i	$0.261639\pi$
$858$	0	0
$859$	−19.5144	−0.665822	−0.332911	−	0.942958i	$-0.608031\pi$
$859$	−19.5144	−0.665822	−0.332911	+	0.942958i	$0.608031\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.9321	1.29123	0.645613	−	0.763665i	$-0.276602\pi$
$863$	37.9321	1.29123	0.645613	+	0.763665i	$0.276602\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	27.6923	0.940479
$868$	0	0
$869$	27.5364	0.934108
$870$	0	0
$871$	−2.53638	−0.0859419
$872$	0	0
$873$	−9.68106	−0.327654
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.9452	−0.369593	−0.184796	−	0.982777i	$-0.559163\pi$
$877$	−10.9452	−0.369593	−0.184796	+	0.982777i	$0.559163\pi$
$878$	0	0
$879$	81.8706	2.76143
$880$	0	0
$881$	−16.2592	−0.547787	−0.273894	−	0.961760i	$-0.588312\pi$
$881$	−16.2592	−0.547787	−0.273894	+	0.961760i	$0.588312\pi$
$882$	0	0
$883$	−23.2511	−0.782461	−0.391231	−	0.920293i	$-0.627951\pi$
$883$	−23.2511	−0.782461	−0.391231	+	0.920293i	$0.627951\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.2394	0.780304	0.390152	−	0.920750i	$-0.372422\pi$
$887$	23.2394	0.780304	0.390152	+	0.920750i	$0.372422\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	92.5894	3.10186
$892$	0	0
$893$	−3.47666	−0.116342
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.40207	0.0468137
$898$	0	0
$899$	63.3122	2.11158
$900$	0	0
$901$	−28.7471	−0.957704
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.2264	1.46851	0.734257	−	0.678872i	$-0.237531\pi$
$907$	44.2264	1.46851	0.734257	+	0.678872i	$0.237531\pi$
$908$	0	0
$909$	45.0000	1.49256
$910$	0	0
$911$	−27.4218	−0.908526	−0.454263	−	0.890868i	$-0.650097\pi$
$911$	−27.4218	−0.908526	−0.454263	+	0.890868i	$0.650097\pi$
$912$	0	0
$913$	−16.3189	−0.540078
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−33.3163	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$919$	−33.3163	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$920$	0	0
$921$	32.4110	1.06798
$922$	0	0
$923$	0.120842	0.00397755
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−70.7978	−2.32531
$928$	0	0
$929$	−13.2484	−0.434666	−0.217333	−	0.976097i	$-0.569736\pi$
$929$	−13.2484	−0.434666	−0.217333	+	0.976097i	$0.569736\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	109.499	3.58485
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.1365	0.331146	0.165573	−	0.986197i	$-0.447053\pi$
$937$	10.1365	0.331146	0.165573	+	0.986197i	$0.447053\pi$
$938$	0	0
$939$	−27.6002	−0.900697
$940$	0	0
$941$	−32.8684	−1.07148	−0.535739	−	0.844384i	$-0.679967\pi$
$941$	−32.8684	−1.07148	−0.535739	+	0.844384i	$0.679967\pi$
$942$	0	0
$943$	−22.5822	−0.735379
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.3252	0.725473	0.362736	−	0.931892i	$-0.381843\pi$
$947$	22.3252	0.725473	0.362736	+	0.931892i	$0.381843\pi$
$948$	0	0
$949$	−0.817606	−0.0265406
$950$	0	0
$951$	−25.0507	−0.812326
$952$	0	0
$953$	47.4685	1.53766	0.768828	−	0.639455i	$-0.220840\pi$
$953$	47.4685	1.53766	0.768828	+	0.639455i	$0.220840\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−100.675	−3.25437
$958$	0	0
$959$	0	0
$960$	0	0
$961$	40.7848	1.31564
$962$	0	0
$963$	−71.6599	−2.30921
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.3189	−0.685571	−0.342785	−	0.939414i	$-0.611370\pi$
$967$	−21.3189	−0.685571	−0.342785	+	0.939414i	$0.611370\pi$
$968$	0	0
$969$	17.2615	0.554518
$970$	0	0
$971$	−29.5625	−0.948704	−0.474352	−	0.880335i	$-0.657318\pi$
$971$	−29.5625	−0.948704	−0.474352	+	0.880335i	$0.657318\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.83961	0.0588545	0.0294272	−	0.999567i	$-0.490632\pi$
$977$	1.83961	0.0588545	0.0294272	+	0.999567i	$0.490632\pi$
$978$	0	0
$979$	−38.9672	−1.24540
$980$	0	0
$981$	−29.6311	−0.946050
$982$	0	0
$983$	11.3760	0.362838	0.181419	−	0.983406i	$-0.441931\pi$
$983$	11.3760	0.362838	0.181419	+	0.983406i	$0.441931\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.6990	0.562795
$990$	0	0
$991$	7.79153	0.247506	0.123753	−	0.992313i	$-0.460507\pi$
$991$	7.79153	0.247506	0.123753	+	0.992313i	$0.460507\pi$
$992$	0	0
$993$	−70.8135	−2.24720
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.4218	−0.361733	−0.180867	−	0.983508i	$-0.557890\pi$
$997$	−11.4218	−0.361733	−0.180867	+	0.983508i	$0.557890\pi$
$998$	0	0
$999$	145.157	4.59256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bc.1.3		3
5.2	odd	4		4900.2.e.t.2549.1		6
5.3	odd	4		4900.2.e.t.2549.6		6
5.4	even	2		4900.2.a.ba.1.1		3
7.2	even	3		700.2.i.d.501.1	yes	6
7.4	even	3		700.2.i.d.401.1	✓	6
7.6	odd	2		4900.2.a.bb.1.1		3
35.2	odd	12		700.2.r.d.249.6		12
35.4	even	6		700.2.i.e.401.3	yes	6
35.9	even	6		700.2.i.e.501.3	yes	6
35.13	even	4		4900.2.e.s.2549.1		6
35.18	odd	12		700.2.r.d.149.6		12
35.23	odd	12		700.2.r.d.249.1		12
35.27	even	4		4900.2.e.s.2549.6		6
35.32	odd	12		700.2.r.d.149.1		12
35.34	odd	2		4900.2.a.bd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.1	✓	6	7.4	even	3
700.2.i.d.501.1	yes	6	7.2	even	3
700.2.i.e.401.3	yes	6	35.4	even	6
700.2.i.e.501.3	yes	6	35.9	even	6
700.2.r.d.149.1		12	35.32	odd	12
700.2.r.d.149.6		12	35.18	odd	12
700.2.r.d.249.1		12	35.23	odd	12
700.2.r.d.249.6		12	35.2	odd	12
4900.2.a.ba.1.1		3	5.4	even	2
4900.2.a.bb.1.1		3	7.6	odd	2
4900.2.a.bc.1.3		3	1.1	even	1	trivial
4900.2.a.bd.1.3		3	35.34	odd	2
4900.2.e.s.2549.1		6	35.13	even	4
4900.2.e.s.2549.6		6	35.27	even	4
4900.2.e.t.2549.1		6	5.2	odd	4
4900.2.e.t.2549.6		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q+3.20440 q^{3} +7.26819 q^{9} +O(q^{10})\)	\(q+3.20440 q^{3} +7.26819 q^{9} +4.20440 q^{11} -0.204402 q^{13} +5.06379 q^{17} +1.06379 q^{19} -2.14061 q^{23} +13.6770 q^{27} -7.47259 q^{29} -8.47259 q^{31} +13.4726 q^{33} +10.6132 q^{37} -0.654985 q^{39} +10.5494 q^{41} -8.26819 q^{43} -3.26819 q^{47} +16.2264 q^{51} -5.67699 q^{53} +3.40880 q^{57} +1.20440 q^{59} +1.65498 q^{61} +12.4088 q^{67} -6.85939 q^{69} -0.591197 q^{71} +4.00000 q^{73} +6.54942 q^{79} +22.0220 q^{81} -3.88139 q^{83} -23.9452 q^{87} -9.26819 q^{89} -27.1496 q^{93} -1.33198 q^{97} +30.5584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 + 8 * q^9	\( 3 q + q^{3} + 8 q^{9} + 4 q^{11} + 8 q^{13} + 10 q^{17} - 2 q^{19} - 3 q^{23} + 10 q^{27} - 3 q^{31} + 18 q^{33} + 6 q^{37} - 14 q^{39} + 11 q^{41} - 11 q^{43} + 4 q^{47} - 3 q^{51} + 14 q^{53} - 7 q^{57} - 5 q^{59} + 17 q^{61} + 20 q^{67} - 24 q^{69} - 19 q^{71} + 12 q^{73} - q^{79} + 23 q^{81} + 28 q^{83} - 27 q^{87} - 14 q^{89} - 28 q^{93} + 15 q^{97} + 21 q^{99}+O(q^{100}) \) 3 * q + q^3 + 8 * q^9 + 4 * q^11 + 8 * q^13 + 10 * q^17 - 2 * q^19 - 3 * q^23 + 10 * q^27 - 3 * q^31 + 18 * q^33 + 6 * q^37 - 14 * q^39 + 11 * q^41 - 11 * q^43 + 4 * q^47 - 3 * q^51 + 14 * q^53 - 7 * q^57 - 5 * q^59 + 17 * q^61 + 20 * q^67 - 24 * q^69 - 19 * q^71 + 12 * q^73 - q^79 + 23 * q^81 + 28 * q^83 - 27 * q^87 - 14 * q^89 - 28 * q^93 + 15 * q^97 + 21 * q^99

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