LMFDB - Embedded newform 6039.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.81809 q^{2} +5.94161 q^{4} -0.621459 q^{5} +3.29277 q^{7} -11.1078 q^{8} +O(q^{10})$	\(q-2.81809 q^{2} +5.94161 q^{4} -0.621459 q^{5} +3.29277 q^{7} -11.1078 q^{8} +1.75133 q^{10} +1.00000 q^{11} -1.50990 q^{13} -9.27931 q^{14} +19.4195 q^{16} -6.84217 q^{17} -4.68898 q^{19} -3.69247 q^{20} -2.81809 q^{22} +6.01878 q^{23} -4.61379 q^{25} +4.25503 q^{26} +19.5644 q^{28} -0.884432 q^{29} -0.545433 q^{31} -32.5102 q^{32} +19.2818 q^{34} -2.04632 q^{35} +6.78889 q^{37} +13.2139 q^{38} +6.90304 q^{40} +1.45630 q^{41} +1.14017 q^{43} +5.94161 q^{44} -16.9614 q^{46} +9.26757 q^{47} +3.84234 q^{49} +13.0021 q^{50} -8.97124 q^{52} +3.50889 q^{53} -0.621459 q^{55} -36.5754 q^{56} +2.49241 q^{58} -9.56011 q^{59} +1.00000 q^{61} +1.53708 q^{62} +52.7776 q^{64} +0.938342 q^{65} -9.11848 q^{67} -40.6535 q^{68} +5.76671 q^{70} +6.11087 q^{71} +2.67201 q^{73} -19.1317 q^{74} -27.8601 q^{76} +3.29277 q^{77} -5.91843 q^{79} -12.0684 q^{80} -4.10398 q^{82} +14.3638 q^{83} +4.25213 q^{85} -3.21309 q^{86} -11.1078 q^{88} -4.13117 q^{89} -4.97176 q^{91} +35.7612 q^{92} -26.1168 q^{94} +2.91401 q^{95} +17.8387 q^{97} -10.8281 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

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 $

$2$	−2.81809	−1.99269	−0.996344	−	0.0854342i	$-0.972772\pi$
$2$	−2.81809	−1.99269	−0.996344	+	0.0854342i	$0.972772\pi$
$3$	0	0
$3$	0	0
$4$	5.94161	2.97080
$5$	−0.621459	−0.277925	−0.138963	−	0.990298i	$-0.544377\pi$
$5$	−0.621459	−0.277925	−0.138963	+	0.990298i	$0.544377\pi$
$6$	0	0
$7$	3.29277	1.24455	0.622275	−	0.782798i	$-0.286208\pi$
$7$	3.29277	1.24455	0.622275	+	0.782798i	$0.286208\pi$
$8$	−11.1078	−3.92720
$9$	0	0
$10$	1.75133	0.553818
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.50990	−0.418771	−0.209386	−	0.977833i	$-0.567146\pi$
$13$	−1.50990	−0.418771	−0.209386	+	0.977833i	$0.567146\pi$
$14$	−9.27931	−2.48000
$15$	0	0
$16$	19.4195	4.85487
$17$	−6.84217	−1.65947	−0.829735	−	0.558158i	$-0.811508\pi$
$17$	−6.84217	−1.65947	−0.829735	+	0.558158i	$0.811508\pi$
$18$	0	0
$19$	−4.68898	−1.07572	−0.537862	−	0.843033i	$-0.680768\pi$
$19$	−4.68898	−1.07572	−0.537862	+	0.843033i	$0.680768\pi$
$20$	−3.69247	−0.825661
$21$	0	0
$22$	−2.81809	−0.600818
$23$	6.01878	1.25500	0.627501	−	0.778616i	$-0.284078\pi$
$23$	6.01878	1.25500	0.627501	+	0.778616i	$0.284078\pi$
$24$	0	0
$25$	−4.61379	−0.922758
$26$	4.25503	0.834480
$27$	0	0
$28$	19.5644	3.69732
$29$	−0.884432	−0.164235	−0.0821175	−	0.996623i	$-0.526168\pi$
$29$	−0.884432	−0.164235	−0.0821175	+	0.996623i	$0.526168\pi$
$30$	0	0
$31$	−0.545433	−0.0979626	−0.0489813	−	0.998800i	$-0.515597\pi$
$31$	−0.545433	−0.0979626	−0.0489813	+	0.998800i	$0.515597\pi$
$32$	−32.5102	−5.74705
$33$	0	0
$34$	19.2818	3.30681
$35$	−2.04632	−0.345892
$36$	0	0
$37$	6.78889	1.11609	0.558043	−	0.829812i	$-0.311552\pi$
$37$	6.78889	1.11609	0.558043	+	0.829812i	$0.311552\pi$
$38$	13.2139	2.14358
$39$	0	0
$40$	6.90304	1.09147
$41$	1.45630	0.227436	0.113718	−	0.993513i	$-0.463724\pi$
$41$	1.45630	0.227436	0.113718	+	0.993513i	$0.463724\pi$
$42$	0	0
$43$	1.14017	0.173874	0.0869369	−	0.996214i	$-0.472292\pi$
$43$	1.14017	0.173874	0.0869369	+	0.996214i	$0.472292\pi$
$44$	5.94161	0.895731
$45$	0	0
$46$	−16.9614	−2.50083
$47$	9.26757	1.35181	0.675907	−	0.736987i	$-0.263752\pi$
$47$	9.26757	1.35181	0.675907	+	0.736987i	$0.263752\pi$
$48$	0	0
$49$	3.84234	0.548906
$50$	13.0021	1.83877
$51$	0	0
$52$	−8.97124	−1.24409
$53$	3.50889	0.481983	0.240992	−	0.970527i	$-0.422527\pi$
$53$	3.50889	0.481983	0.240992	+	0.970527i	$0.422527\pi$
$54$	0	0
$55$	−0.621459	−0.0837975
$56$	−36.5754	−4.88760
$57$	0	0
$58$	2.49241	0.327269
$59$	−9.56011	−1.24462	−0.622310	−	0.782771i	$-0.713806\pi$
$59$	−9.56011	−1.24462	−0.622310	+	0.782771i	$0.713806\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	1.53708	0.195209
$63$	0	0
$64$	52.7776	6.59720
$65$	0.938342	0.116387
$66$	0	0
$67$	−9.11848	−1.11400	−0.557000	−	0.830513i	$-0.688048\pi$
$67$	−9.11848	−1.11400	−0.557000	+	0.830513i	$0.688048\pi$
$68$	−40.6535	−4.92996
$69$	0	0
$70$	5.76671	0.689254
$71$	6.11087	0.725227	0.362613	−	0.931940i	$-0.381885\pi$
$71$	6.11087	0.725227	0.362613	+	0.931940i	$0.381885\pi$
$72$	0	0
$73$	2.67201	0.312735	0.156367	−	0.987699i	$-0.450022\pi$
$73$	2.67201	0.312735	0.156367	+	0.987699i	$0.450022\pi$
$74$	−19.1317	−2.22401
$75$	0	0
$76$	−27.8601	−3.19577
$77$	3.29277	0.375246
$78$	0	0
$79$	−5.91843	−0.665875	−0.332938	−	0.942949i	$-0.608040\pi$
$79$	−5.91843	−0.665875	−0.332938	+	0.942949i	$0.608040\pi$
$80$	−12.0684	−1.34929
$81$	0	0
$82$	−4.10398	−0.453209
$83$	14.3638	1.57664	0.788318	−	0.615268i	$-0.210952\pi$
$83$	14.3638	1.57664	0.788318	+	0.615268i	$0.210952\pi$
$84$	0	0
$85$	4.25213	0.461208
$86$	−3.21309	−0.346476
$87$	0	0
$88$	−11.1078	−1.18409
$89$	−4.13117	−0.437904	−0.218952	−	0.975736i	$-0.570264\pi$
$89$	−4.13117	−0.437904	−0.218952	+	0.975736i	$0.570264\pi$
$90$	0	0
$91$	−4.97176	−0.521182
$92$	35.7612	3.72837
$93$	0	0
$94$	−26.1168	−2.69374
$95$	2.91401	0.298971
$96$	0	0
$97$	17.8387	1.81125	0.905623	−	0.424083i	$-0.139404\pi$
$97$	17.8387	1.81125	0.905623	+	0.424083i	$0.139404\pi$
$98$	−10.8281	−1.09380
$99$	0	0
$100$	−27.4133	−2.74133
$101$	7.40379	0.736704	0.368352	−	0.929686i	$-0.379922\pi$
$101$	7.40379	0.736704	0.368352	+	0.929686i	$0.379922\pi$
$102$	0	0
$103$	−7.73458	−0.762111	−0.381055	−	0.924552i	$-0.624439\pi$
$103$	−7.73458	−0.762111	−0.381055	+	0.924552i	$0.624439\pi$
$104$	16.7717	1.64460
$105$	0	0
$106$	−9.88835	−0.960442
$107$	0.703087	0.0679700	0.0339850	−	0.999422i	$-0.489180\pi$
$107$	0.703087	0.0679700	0.0339850	+	0.999422i	$0.489180\pi$
$108$	0	0
$109$	−15.8469	−1.51786	−0.758930	−	0.651172i	$-0.774278\pi$
$109$	−15.8469	−1.51786	−0.758930	+	0.651172i	$0.774278\pi$
$110$	1.75133	0.166982
$111$	0	0
$112$	63.9439	6.04213
$113$	−14.6639	−1.37947	−0.689733	−	0.724064i	$-0.742272\pi$
$113$	−14.6639	−1.37947	−0.689733	+	0.724064i	$0.742272\pi$
$114$	0	0
$115$	−3.74043	−0.348797
$116$	−5.25495	−0.487910
$117$	0	0
$118$	26.9412	2.48014
$119$	−22.5297	−2.06529
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.81809	−0.255138
$123$	0	0
$124$	−3.24075	−0.291028
$125$	5.97458	0.534382
$126$	0	0
$127$	−5.33650	−0.473537	−0.236769	−	0.971566i	$-0.576088\pi$
$127$	−5.33650	−0.473537	−0.236769	+	0.971566i	$0.576088\pi$
$128$	−83.7113	−7.39911
$129$	0	0
$130$	−2.64433	−0.231923
$131$	9.15557	0.799926	0.399963	−	0.916531i	$-0.369023\pi$
$131$	9.15557	0.799926	0.399963	+	0.916531i	$0.369023\pi$
$132$	0	0
$133$	−15.4397	−1.33879
$134$	25.6967	2.21985
$135$	0	0
$136$	76.0014	6.51706
$137$	−8.88931	−0.759465	−0.379733	−	0.925096i	$-0.623984\pi$
$137$	−8.88931	−0.759465	−0.379733	+	0.925096i	$0.623984\pi$
$138$	0	0
$139$	−1.87243	−0.158817	−0.0794085	−	0.996842i	$-0.525303\pi$
$139$	−1.87243	−0.158817	−0.0794085	+	0.996842i	$0.525303\pi$
$140$	−12.1585	−1.02758
$141$	0	0
$142$	−17.2210	−1.44515
$143$	−1.50990	−0.126264
$144$	0	0
$145$	0.549639	0.0456450
$146$	−7.52994	−0.623183
$147$	0	0
$148$	40.3369	3.31567
$149$	−17.6307	−1.44436	−0.722180	−	0.691705i	$-0.756860\pi$
$149$	−17.6307	−1.44436	−0.722180	+	0.691705i	$0.756860\pi$
$150$	0	0
$151$	−15.3259	−1.24721	−0.623603	−	0.781741i	$-0.714332\pi$
$151$	−15.3259	−1.24721	−0.623603	+	0.781741i	$0.714332\pi$
$152$	52.0842	4.22458
$153$	0	0
$154$	−9.27931	−0.747748
$155$	0.338964	0.0272262
$156$	0	0
$157$	1.06872	0.0852927	0.0426464	−	0.999090i	$-0.486421\pi$
$157$	1.06872	0.0852927	0.0426464	+	0.999090i	$0.486421\pi$
$158$	16.6786	1.32688
$159$	0	0
$160$	20.2038	1.59725
$161$	19.8185	1.56191
$162$	0	0
$163$	−2.32929	−0.182444	−0.0912220	−	0.995831i	$-0.529077\pi$
$163$	−2.32929	−0.182444	−0.0912220	+	0.995831i	$0.529077\pi$
$164$	8.65277	0.675668
$165$	0	0
$166$	−40.4785	−3.14174
$167$	−19.0252	−1.47221	−0.736106	−	0.676866i	$-0.763338\pi$
$167$	−19.0252	−1.47221	−0.736106	+	0.676866i	$0.763338\pi$
$168$	0	0
$169$	−10.7202	−0.824631
$170$	−11.9829	−0.919044
$171$	0	0
$172$	6.77443	0.516545
$173$	−10.6230	−0.807654	−0.403827	−	0.914835i	$-0.632320\pi$
$173$	−10.6230	−0.807654	−0.403827	+	0.914835i	$0.632320\pi$
$174$	0	0
$175$	−15.1922	−1.14842
$176$	19.4195	1.46380
$177$	0	0
$178$	11.6420	0.872605
$179$	−22.4785	−1.68012	−0.840061	−	0.542492i	$-0.817481\pi$
$179$	−22.4785	−1.68012	−0.840061	+	0.542492i	$0.817481\pi$
$180$	0	0
$181$	−15.0553	−1.11905	−0.559525	−	0.828813i	$-0.689016\pi$
$181$	−15.0553	−1.11905	−0.559525	+	0.828813i	$0.689016\pi$
$182$	14.0108	1.03855
$183$	0	0
$184$	−66.8553	−4.92864
$185$	−4.21902	−0.310188
$186$	0	0
$187$	−6.84217	−0.500349
$188$	55.0643	4.01597
$189$	0	0
$190$	−8.21192	−0.595755
$191$	19.0426	1.37787	0.688936	−	0.724822i	$-0.258078\pi$
$191$	19.0426	1.37787	0.688936	+	0.724822i	$0.258078\pi$
$192$	0	0
$193$	15.3784	1.10696	0.553480	−	0.832862i	$-0.313299\pi$
$193$	15.3784	1.10696	0.553480	+	0.832862i	$0.313299\pi$
$194$	−50.2710	−3.60925
$195$	0	0
$196$	22.8297	1.63069
$197$	10.1697	0.724558	0.362279	−	0.932070i	$-0.381999\pi$
$197$	10.1697	0.724558	0.362279	+	0.932070i	$0.381999\pi$
$198$	0	0
$199$	−20.1794	−1.43048	−0.715240	−	0.698879i	$-0.753683\pi$
$199$	−20.1794	−1.43048	−0.715240	+	0.698879i	$0.753683\pi$
$200$	51.2490	3.62385
$201$	0	0
$202$	−20.8645	−1.46802
$203$	−2.91223	−0.204399
$204$	0	0
$205$	−0.905031	−0.0632101
$206$	21.7967	1.51865
$207$	0	0
$208$	−29.3215	−2.03308
$209$	−4.68898	−0.324343
$210$	0	0
$211$	15.4971	1.06687	0.533433	−	0.845842i	$-0.320901\pi$
$211$	15.4971	1.06687	0.533433	+	0.845842i	$0.320901\pi$
$212$	20.8484	1.43188
$213$	0	0
$214$	−1.98136	−0.135443
$215$	−0.708567	−0.0483239
$216$	0	0
$217$	−1.79598	−0.121919
$218$	44.6580	3.02462
$219$	0	0
$220$	−3.69247	−0.248946
$221$	10.3310	0.694938
$222$	0	0
$223$	−4.79440	−0.321057	−0.160528	−	0.987031i	$-0.551320\pi$
$223$	−4.79440	−0.321057	−0.160528	+	0.987031i	$0.551320\pi$
$224$	−107.049	−7.15249
$225$	0	0
$226$	41.3242	2.74884
$227$	9.59756	0.637013	0.318506	−	0.947921i	$-0.396819\pi$
$227$	9.59756	0.637013	0.318506	+	0.947921i	$0.396819\pi$
$228$	0	0
$229$	−9.65800	−0.638219	−0.319109	−	0.947718i	$-0.603384\pi$
$229$	−9.65800	−0.638219	−0.319109	+	0.947718i	$0.603384\pi$
$230$	10.5408	0.695042
$231$	0	0
$232$	9.82409	0.644983
$233$	−8.40736	−0.550784	−0.275392	−	0.961332i	$-0.588808\pi$
$233$	−8.40736	−0.550784	−0.275392	+	0.961332i	$0.588808\pi$
$234$	0	0
$235$	−5.75942	−0.375703
$236$	−56.8024	−3.69752
$237$	0	0
$238$	63.4906	4.11549
$239$	−22.3358	−1.44479	−0.722393	−	0.691483i	$-0.756958\pi$
$239$	−22.3358	−1.44479	−0.722393	+	0.691483i	$0.756958\pi$
$240$	0	0
$241$	−1.00623	−0.0648169	−0.0324084	−	0.999475i	$-0.510318\pi$
$241$	−1.00623	−0.0648169	−0.0324084	+	0.999475i	$0.510318\pi$
$242$	−2.81809	−0.181153
$243$	0	0
$244$	5.94161	0.380372
$245$	−2.38786	−0.152555
$246$	0	0
$247$	7.07989	0.450483
$248$	6.05855	0.384718
$249$	0	0
$250$	−16.8369	−1.06486
$251$	−16.6331	−1.04987	−0.524937	−	0.851141i	$-0.675911\pi$
$251$	−16.6331	−1.04987	−0.524937	+	0.851141i	$0.675911\pi$
$252$	0	0
$253$	6.01878	0.378397
$254$	15.0387	0.943612
$255$	0	0
$256$	130.351	8.14691
$257$	11.0003	0.686178	0.343089	−	0.939303i	$-0.388527\pi$
$257$	11.0003	0.686178	0.343089	+	0.939303i	$0.388527\pi$
$258$	0	0
$259$	22.3543	1.38903
$260$	5.57526	0.345763
$261$	0	0
$262$	−25.8012	−1.59400
$263$	−24.5495	−1.51378	−0.756892	−	0.653540i	$-0.773283\pi$
$263$	−24.5495	−1.51378	−0.756892	+	0.653540i	$0.773283\pi$
$264$	0	0
$265$	−2.18063	−0.133955
$266$	43.5105	2.66780
$267$	0	0
$268$	−54.1784	−3.30947
$269$	26.9044	1.64039	0.820195	−	0.572084i	$-0.193865\pi$
$269$	26.9044	1.64039	0.820195	+	0.572084i	$0.193865\pi$
$270$	0	0
$271$	21.2616	1.29155	0.645774	−	0.763528i	$-0.276535\pi$
$271$	21.2616	1.29155	0.645774	+	0.763528i	$0.276535\pi$
$272$	−132.871	−8.05651
$273$	0	0
$274$	25.0508	1.51338
$275$	−4.61379	−0.278222
$276$	0	0
$277$	21.6602	1.30143	0.650716	−	0.759321i	$-0.274469\pi$
$277$	21.6602	1.30143	0.650716	+	0.759321i	$0.274469\pi$
$278$	5.27666	0.316473
$279$	0	0
$280$	22.7301	1.35838
$281$	−23.5451	−1.40458	−0.702292	−	0.711889i	$-0.747840\pi$
$281$	−23.5451	−1.40458	−0.702292	+	0.711889i	$0.747840\pi$
$282$	0	0
$283$	1.67043	0.0992969	0.0496484	−	0.998767i	$-0.484190\pi$
$283$	1.67043	0.0992969	0.0496484	+	0.998767i	$0.484190\pi$
$284$	36.3084	2.15451
$285$	0	0
$286$	4.25503	0.251605
$287$	4.79527	0.283056
$288$	0	0
$289$	29.8153	1.75384
$290$	−1.54893	−0.0909562
$291$	0	0
$292$	15.8760	0.929074
$293$	7.65920	0.447455	0.223728	−	0.974652i	$-0.428177\pi$
$293$	7.65920	0.447455	0.223728	+	0.974652i	$0.428177\pi$
$294$	0	0
$295$	5.94122	0.345911
$296$	−75.4096	−4.38309
$297$	0	0
$298$	49.6847	2.87816
$299$	−9.08776	−0.525559
$300$	0	0
$301$	3.75431	0.216395
$302$	43.1897	2.48529
$303$	0	0
$304$	−91.0575	−5.22251
$305$	−0.621459	−0.0355847
$306$	0	0
$307$	−30.7139	−1.75293	−0.876467	−	0.481463i	$-0.840106\pi$
$307$	−30.7139	−1.75293	−0.876467	+	0.481463i	$0.840106\pi$
$308$	19.5644	1.11478
$309$	0	0
$310$	−0.955230	−0.0542534
$311$	−2.70714	−0.153508	−0.0767538	−	0.997050i	$-0.524456\pi$
$311$	−2.70714	−0.153508	−0.0767538	+	0.997050i	$0.524456\pi$
$312$	0	0
$313$	−16.1067	−0.910406	−0.455203	−	0.890388i	$-0.650433\pi$
$313$	−16.1067	−0.910406	−0.455203	+	0.890388i	$0.650433\pi$
$314$	−3.01173	−0.169962
$315$	0	0
$316$	−35.1650	−1.97818
$317$	3.48049	0.195484	0.0977419	−	0.995212i	$-0.468838\pi$
$317$	3.48049	0.195484	0.0977419	+	0.995212i	$0.468838\pi$
$318$	0	0
$319$	−0.884432	−0.0495187
$320$	−32.7991	−1.83353
$321$	0	0
$322$	−55.8501	−3.11241
$323$	32.0828	1.78513
$324$	0	0
$325$	6.96636	0.386424
$326$	6.56414	0.363554
$327$	0	0
$328$	−16.1763	−0.893186
$329$	30.5160	1.68240
$330$	0	0
$331$	8.94451	0.491635	0.245817	−	0.969316i	$-0.420944\pi$
$331$	8.94451	0.491635	0.245817	+	0.969316i	$0.420944\pi$
$332$	85.3443	4.68388
$333$	0	0
$334$	53.6146	2.93366
$335$	5.66676	0.309608
$336$	0	0
$337$	32.8754	1.79083	0.895417	−	0.445228i	$-0.146877\pi$
$337$	32.8754	1.79083	0.895417	+	0.445228i	$0.146877\pi$
$338$	30.2104	1.64323
$339$	0	0
$340$	25.2645	1.37016
$341$	−0.545433	−0.0295368
$342$	0	0
$343$	−10.3974	−0.561409
$344$	−12.6647	−0.682837
$345$	0	0
$346$	29.9366	1.60940
$347$	−1.48157	−0.0795347	−0.0397673	−	0.999209i	$-0.512662\pi$
$347$	−1.48157	−0.0795347	−0.0397673	+	0.999209i	$0.512662\pi$
$348$	0	0
$349$	−22.3581	−1.19680	−0.598400	−	0.801197i	$-0.704197\pi$
$349$	−22.3581	−1.19680	−0.598400	+	0.801197i	$0.704197\pi$
$350$	42.8128	2.28844
$351$	0	0
$352$	−32.5102	−1.73280
$353$	16.9429	0.901779	0.450889	−	0.892580i	$-0.351107\pi$
$353$	16.9429	0.901779	0.450889	+	0.892580i	$0.351107\pi$
$354$	0	0
$355$	−3.79766	−0.201559
$356$	−24.5458	−1.30093
$357$	0	0
$358$	63.3463	3.34796
$359$	−30.7699	−1.62398	−0.811988	−	0.583675i	$-0.801614\pi$
$359$	−30.7699	−1.62398	−0.811988	+	0.583675i	$0.801614\pi$
$360$	0	0
$361$	2.98649	0.157184
$362$	42.4271	2.22992
$363$	0	0
$364$	−29.5402	−1.54833
$365$	−1.66054	−0.0869168
$366$	0	0
$367$	−17.3727	−0.906848	−0.453424	−	0.891295i	$-0.649798\pi$
$367$	−17.3727	−0.906848	−0.453424	+	0.891295i	$0.649798\pi$
$368$	116.882	6.09288
$369$	0	0
$370$	11.8896	0.618109
$371$	11.5540	0.599852
$372$	0	0
$373$	27.8592	1.44249	0.721246	−	0.692679i	$-0.243570\pi$
$373$	27.8592	1.44249	0.721246	+	0.692679i	$0.243570\pi$
$374$	19.2818	0.997039
$375$	0	0
$376$	−102.942	−5.30884
$377$	1.33541	0.0687769
$378$	0	0
$379$	−4.12859	−0.212072	−0.106036	−	0.994362i	$-0.533816\pi$
$379$	−4.12859	−0.212072	−0.106036	+	0.994362i	$0.533816\pi$
$380$	17.3139	0.888184
$381$	0	0
$382$	−53.6636	−2.74567
$383$	8.56824	0.437817	0.218908	−	0.975745i	$-0.429750\pi$
$383$	8.56824	0.437817	0.218908	+	0.975745i	$0.429750\pi$
$384$	0	0
$385$	−2.04632	−0.104290
$386$	−43.3376	−2.20583
$387$	0	0
$388$	105.991	5.38086
$389$	0.656063	0.0332637	0.0166319	−	0.999862i	$-0.494706\pi$
$389$	0.656063	0.0332637	0.0166319	+	0.999862i	$0.494706\pi$
$390$	0	0
$391$	−41.1815	−2.08264
$392$	−42.6800	−2.15566
$393$	0	0
$394$	−28.6590	−1.44382
$395$	3.67806	0.185063
$396$	0	0
$397$	28.2540	1.41803	0.709013	−	0.705195i	$-0.249141\pi$
$397$	28.2540	1.41803	0.709013	+	0.705195i	$0.249141\pi$
$398$	56.8673	2.85050
$399$	0	0
$400$	−89.5974	−4.47987
$401$	−29.3394	−1.46514	−0.732570	−	0.680692i	$-0.761679\pi$
$401$	−29.3394	−1.46514	−0.732570	+	0.680692i	$0.761679\pi$
$402$	0	0
$403$	0.823549	0.0410239
$404$	43.9904	2.18860
$405$	0	0
$406$	8.20693	0.407303
$407$	6.78889	0.336513
$408$	0	0
$409$	−5.54748	−0.274305	−0.137153	−	0.990550i	$-0.543795\pi$
$409$	−5.54748	−0.274305	−0.137153	+	0.990550i	$0.543795\pi$
$410$	2.55046	0.125958
$411$	0	0
$412$	−45.9558	−2.26408
$413$	−31.4793	−1.54899
$414$	0	0
$415$	−8.92654	−0.438187
$416$	49.0872	2.40670
$417$	0	0
$418$	13.2139	0.646315
$419$	3.36199	0.164244	0.0821219	−	0.996622i	$-0.473830\pi$
$419$	3.36199	0.164244	0.0821219	+	0.996622i	$0.473830\pi$
$420$	0	0
$421$	−19.3562	−0.943364	−0.471682	−	0.881769i	$-0.656353\pi$
$421$	−19.3562	−0.943364	−0.471682	+	0.881769i	$0.656353\pi$
$422$	−43.6723	−2.12593
$423$	0	0
$424$	−38.9760	−1.89284
$425$	31.5683	1.53129
$426$	0	0
$427$	3.29277	0.159348
$428$	4.17747	0.201926
$429$	0	0
$430$	1.99680	0.0962944
$431$	−2.43398	−0.117241	−0.0586204	−	0.998280i	$-0.518670\pi$
$431$	−2.43398	−0.117241	−0.0586204	+	0.998280i	$0.518670\pi$
$432$	0	0
$433$	−32.2740	−1.55099	−0.775496	−	0.631353i	$-0.782500\pi$
$433$	−32.2740	−1.55099	−0.775496	+	0.631353i	$0.782500\pi$
$434$	5.06124	0.242947
$435$	0	0
$436$	−94.1563	−4.50927
$437$	−28.2219	−1.35004
$438$	0	0
$439$	−5.56309	−0.265512	−0.132756	−	0.991149i	$-0.542383\pi$
$439$	−5.56309	−0.265512	−0.132756	+	0.991149i	$0.542383\pi$
$440$	6.90304	0.329089
$441$	0	0
$442$	−29.1136	−1.38479
$443$	−19.7866	−0.940089	−0.470045	−	0.882643i	$-0.655762\pi$
$443$	−19.7866	−0.940089	−0.470045	+	0.882643i	$0.655762\pi$
$444$	0	0
$445$	2.56736	0.121704
$446$	13.5110	0.639766
$447$	0	0
$448$	173.785	8.21055
$449$	25.6056	1.20840	0.604201	−	0.796832i	$-0.293492\pi$
$449$	25.6056	1.20840	0.604201	+	0.796832i	$0.293492\pi$
$450$	0	0
$451$	1.45630	0.0685745
$452$	−87.1273	−4.09812
$453$	0	0
$454$	−27.0468	−1.26937
$455$	3.08975	0.144849
$456$	0	0
$457$	6.14815	0.287599	0.143799	−	0.989607i	$-0.454068\pi$
$457$	6.14815	0.287599	0.143799	+	0.989607i	$0.454068\pi$
$458$	27.2171	1.27177
$459$	0	0
$460$	−22.2241	−1.03621
$461$	−9.17495	−0.427320	−0.213660	−	0.976908i	$-0.568538\pi$
$461$	−9.17495	−0.427320	−0.213660	+	0.976908i	$0.568538\pi$
$462$	0	0
$463$	9.41154	0.437391	0.218696	−	0.975793i	$-0.429820\pi$
$463$	9.41154	0.437391	0.218696	+	0.975793i	$0.429820\pi$
$464$	−17.1752	−0.797340
$465$	0	0
$466$	23.6927	1.09754
$467$	35.2266	1.63009	0.815046	−	0.579397i	$-0.196712\pi$
$467$	35.2266	1.63009	0.815046	+	0.579397i	$0.196712\pi$
$468$	0	0
$469$	−30.0251	−1.38643
$470$	16.2305	0.748659
$471$	0	0
$472$	106.192	4.88787
$473$	1.14017	0.0524249
$474$	0	0
$475$	21.6339	0.992633
$476$	−133.863	−6.13558
$477$	0	0
$478$	62.9443	2.87901
$479$	−33.0843	−1.51166	−0.755831	−	0.654767i	$-0.772767\pi$
$479$	−33.0843	−1.51166	−0.755831	+	0.654767i	$0.772767\pi$
$480$	0	0
$481$	−10.2506	−0.467385
$482$	2.83564	0.129160
$483$	0	0
$484$	5.94161	0.270073
$485$	−11.0860	−0.503391
$486$	0	0
$487$	−43.4925	−1.97083	−0.985417	−	0.170156i	$-0.945573\pi$
$487$	−43.4925	−1.97083	−0.985417	+	0.170156i	$0.945573\pi$
$488$	−11.1078	−0.502826
$489$	0	0
$490$	6.72920	0.303994
$491$	6.79838	0.306807	0.153403	−	0.988164i	$-0.450977\pi$
$491$	6.79838	0.306807	0.153403	+	0.988164i	$0.450977\pi$
$492$	0	0
$493$	6.05144	0.272543
$494$	−19.9517	−0.897671
$495$	0	0
$496$	−10.5920	−0.475596
$497$	20.1217	0.902582
$498$	0	0
$499$	34.0199	1.52294	0.761470	−	0.648200i	$-0.224478\pi$
$499$	34.0199	1.52294	0.761470	+	0.648200i	$0.224478\pi$
$500$	35.4986	1.58755
$501$	0	0
$502$	46.8736	2.09207
$503$	19.9586	0.889908	0.444954	−	0.895553i	$-0.353220\pi$
$503$	19.9586	0.889908	0.444954	+	0.895553i	$0.353220\pi$
$504$	0	0
$505$	−4.60115	−0.204749
$506$	−16.9614	−0.754028
$507$	0	0
$508$	−31.7074	−1.40679
$509$	15.1464	0.671351	0.335675	−	0.941978i	$-0.391036\pi$
$509$	15.1464	0.671351	0.335675	+	0.941978i	$0.391036\pi$
$510$	0	0
$511$	8.79831	0.389214
$512$	−199.916	−8.83514
$513$	0	0
$514$	−30.9997	−1.36734
$515$	4.80673	0.211810
$516$	0	0
$517$	9.26757	0.407587
$518$	−62.9962	−2.76790
$519$	0	0
$520$	−10.4229	−0.457075
$521$	4.54954	0.199319	0.0996595	−	0.995022i	$-0.468225\pi$
$521$	4.54954	0.199319	0.0996595	+	0.995022i	$0.468225\pi$
$522$	0	0
$523$	1.19315	0.0521728	0.0260864	−	0.999660i	$-0.491695\pi$
$523$	1.19315	0.0521728	0.0260864	+	0.999660i	$0.491695\pi$
$524$	54.3988	2.37642
$525$	0	0
$526$	69.1825	3.01650
$527$	3.73194	0.162566
$528$	0	0
$529$	13.2257	0.575031
$530$	6.14521	0.266931
$531$	0	0
$532$	−91.7368	−3.97729
$533$	−2.19887	−0.0952436
$534$	0	0
$535$	−0.436940	−0.0188906
$536$	101.286	4.37489
$537$	0	0
$538$	−75.8189	−3.26878
$539$	3.84234	0.165502
$540$	0	0
$541$	−10.9139	−0.469227	−0.234613	−	0.972089i	$-0.575382\pi$
$541$	−10.9139	−0.469227	−0.234613	+	0.972089i	$0.575382\pi$
$542$	−59.9170	−2.57365
$543$	0	0
$544$	222.440	9.53705
$545$	9.84822	0.421851
$546$	0	0
$547$	−24.0217	−1.02709	−0.513546	−	0.858062i	$-0.671668\pi$
$547$	−24.0217	−1.02709	−0.513546	+	0.858062i	$0.671668\pi$
$548$	−52.8168	−2.25622
$549$	0	0
$550$	13.0021	0.554409
$551$	4.14708	0.176672
$552$	0	0
$553$	−19.4880	−0.828715
$554$	−61.0402	−2.59335
$555$	0	0
$556$	−11.1252	−0.471814
$557$	18.5390	0.785522	0.392761	−	0.919641i	$-0.371520\pi$
$557$	18.5390	0.785522	0.392761	+	0.919641i	$0.371520\pi$
$558$	0	0
$559$	−1.72154	−0.0728134
$560$	−39.7386	−1.67926
$561$	0	0
$562$	66.3521	2.79890
$563$	2.78197	0.117246	0.0586231	−	0.998280i	$-0.481329\pi$
$563$	2.78197	0.117246	0.0586231	+	0.998280i	$0.481329\pi$
$564$	0	0
$565$	9.11303	0.383388
$566$	−4.70742	−0.197868
$567$	0	0
$568$	−67.8783	−2.84811
$569$	3.20908	0.134532	0.0672658	−	0.997735i	$-0.478572\pi$
$569$	3.20908	0.134532	0.0672658	+	0.997735i	$0.478572\pi$
$570$	0	0
$571$	−28.5800	−1.19604	−0.598018	−	0.801483i	$-0.704045\pi$
$571$	−28.5800	−1.19604	−0.598018	+	0.801483i	$0.704045\pi$
$572$	−8.97124	−0.375106
$573$	0	0
$574$	−13.5135	−0.564041
$575$	−27.7694	−1.15806
$576$	0	0
$577$	0.572876	0.0238491	0.0119246	−	0.999929i	$-0.496204\pi$
$577$	0.572876	0.0238491	0.0119246	+	0.999929i	$0.496204\pi$
$578$	−84.0220	−3.49486
$579$	0	0
$580$	3.26574	0.135602
$581$	47.2968	1.96220
$582$	0	0
$583$	3.50889	0.145323
$584$	−29.6801	−1.22817
$585$	0	0
$586$	−21.5843	−0.891638
$587$	−25.2257	−1.04117	−0.520587	−	0.853808i	$-0.674287\pi$
$587$	−25.2257	−1.04117	−0.520587	+	0.853808i	$0.674287\pi$
$588$	0	0
$589$	2.55752	0.105381
$590$	−16.7429	−0.689293
$591$	0	0
$592$	131.837	5.41846
$593$	5.22615	0.214612	0.107306	−	0.994226i	$-0.465778\pi$
$593$	5.22615	0.214612	0.107306	+	0.994226i	$0.465778\pi$
$594$	0	0
$595$	14.0013	0.573997
$596$	−104.754	−4.29091
$597$	0	0
$598$	25.6101	1.04727
$599$	32.1174	1.31228	0.656140	−	0.754639i	$-0.272188\pi$
$599$	32.1174	1.31228	0.656140	+	0.754639i	$0.272188\pi$
$600$	0	0
$601$	2.18814	0.0892559	0.0446280	−	0.999004i	$-0.485790\pi$
$601$	2.18814	0.0892559	0.0446280	+	0.999004i	$0.485790\pi$
$602$	−10.5800	−0.431207
$603$	0	0
$604$	−91.0606	−3.70520
$605$	−0.621459	−0.0252659
$606$	0	0
$607$	−34.3514	−1.39428	−0.697139	−	0.716936i	$-0.745544\pi$
$607$	−34.3514	−1.39428	−0.697139	+	0.716936i	$0.745544\pi$
$608$	152.440	6.18224
$609$	0	0
$610$	1.75133	0.0709091
$611$	−13.9931	−0.566101
$612$	0	0
$613$	−32.7170	−1.32143	−0.660714	−	0.750637i	$-0.729747\pi$
$613$	−32.7170	−1.32143	−0.660714	+	0.750637i	$0.729747\pi$
$614$	86.5543	3.49305
$615$	0	0
$616$	−36.5754	−1.47367
$617$	−2.09866	−0.0844889	−0.0422445	−	0.999107i	$-0.513451\pi$
$617$	−2.09866	−0.0844889	−0.0422445	+	0.999107i	$0.513451\pi$
$618$	0	0
$619$	29.0585	1.16796	0.583980	−	0.811768i	$-0.301495\pi$
$619$	29.0585	1.16796	0.583980	+	0.811768i	$0.301495\pi$
$620$	2.01399	0.0808838
$621$	0	0
$622$	7.62894	0.305893
$623$	−13.6030	−0.544993
$624$	0	0
$625$	19.3560	0.774239
$626$	45.3901	1.81415
$627$	0	0
$628$	6.34989	0.253388
$629$	−46.4507	−1.85211
$630$	0	0
$631$	−43.1066	−1.71605	−0.858024	−	0.513610i	$-0.828308\pi$
$631$	−43.1066	−1.71605	−0.858024	+	0.513610i	$0.828308\pi$
$632$	65.7406	2.61502
$633$	0	0
$634$	−9.80832	−0.389538
$635$	3.31641	0.131608
$636$	0	0
$637$	−5.80156	−0.229866
$638$	2.49241	0.0986753
$639$	0	0
$640$	52.0232	2.05640
$641$	−27.1312	−1.07162	−0.535808	−	0.844340i	$-0.679993\pi$
$641$	−27.1312	−1.07162	−0.535808	+	0.844340i	$0.679993\pi$
$642$	0	0
$643$	2.47140	0.0974624	0.0487312	−	0.998812i	$-0.484482\pi$
$643$	2.47140	0.0974624	0.0487312	+	0.998812i	$0.484482\pi$
$644$	117.754	4.64014
$645$	0	0
$646$	−90.4120	−3.55721
$647$	−46.7142	−1.83652	−0.918262	−	0.395974i	$-0.870407\pi$
$647$	−46.7142	−1.83652	−0.918262	+	0.395974i	$0.870407\pi$
$648$	0	0
$649$	−9.56011	−0.375267
$650$	−19.6318	−0.770023
$651$	0	0
$652$	−13.8397	−0.542005
$653$	−24.0167	−0.939846	−0.469923	−	0.882707i	$-0.655718\pi$
$653$	−24.0167	−0.939846	−0.469923	+	0.882707i	$0.655718\pi$
$654$	0	0
$655$	−5.68981	−0.222319
$656$	28.2806	1.10417
$657$	0	0
$658$	−85.9967	−3.35250
$659$	−0.967297	−0.0376805	−0.0188403	−	0.999823i	$-0.505997\pi$
$659$	−0.967297	−0.0376805	−0.0188403	+	0.999823i	$0.505997\pi$
$660$	0	0
$661$	−23.8179	−0.926409	−0.463205	−	0.886251i	$-0.653300\pi$
$661$	−23.8179	−0.926409	−0.463205	+	0.886251i	$0.653300\pi$
$662$	−25.2064	−0.979675
$663$	0	0
$664$	−159.551	−6.19176
$665$	9.59516	0.372084
$666$	0	0
$667$	−5.32320	−0.206115
$668$	−113.040	−4.37365
$669$	0	0
$670$	−15.9694	−0.616953
$671$	1.00000	0.0386046
$672$	0	0
$673$	28.6685	1.10509	0.552544	−	0.833483i	$-0.313657\pi$
$673$	28.6685	1.10509	0.552544	+	0.833483i	$0.313657\pi$
$674$	−92.6456	−3.56857
$675$	0	0
$676$	−63.6952	−2.44982
$677$	−38.7449	−1.48909	−0.744544	−	0.667573i	$-0.767333\pi$
$677$	−38.7449	−1.48909	−0.744544	+	0.667573i	$0.767333\pi$
$678$	0	0
$679$	58.7388	2.25419
$680$	−47.2318	−1.81126
$681$	0	0
$682$	1.53708	0.0588577
$683$	36.0064	1.37775	0.688873	−	0.724882i	$-0.258106\pi$
$683$	36.0064	1.37775	0.688873	+	0.724882i	$0.258106\pi$
$684$	0	0
$685$	5.52435	0.211074
$686$	29.3009	1.11871
$687$	0	0
$688$	22.1415	0.844135
$689$	−5.29808	−0.201841
$690$	0	0
$691$	−15.4396	−0.587350	−0.293675	−	0.955905i	$-0.594878\pi$
$691$	−15.4396	−0.587350	−0.293675	+	0.955905i	$0.594878\pi$
$692$	−63.1179	−2.39938
$693$	0	0
$694$	4.17518	0.158488
$695$	1.16364	0.0441392
$696$	0	0
$697$	−9.96426	−0.377423
$698$	63.0070	2.38485
$699$	0	0
$700$	−90.2658	−3.41173
$701$	−25.0948	−0.947818	−0.473909	−	0.880574i	$-0.657157\pi$
$701$	−25.0948	−0.947818	−0.473909	+	0.880574i	$0.657157\pi$
$702$	0	0
$703$	−31.8329	−1.20060
$704$	52.7776	1.98913
$705$	0	0
$706$	−47.7465	−1.79696
$707$	24.3790	0.916866
$708$	0	0
$709$	−7.57817	−0.284604	−0.142302	−	0.989823i	$-0.545450\pi$
$709$	−7.57817	−0.284604	−0.142302	+	0.989823i	$0.545450\pi$
$710$	10.7021	0.401644
$711$	0	0
$712$	45.8882	1.71973
$713$	−3.28284	−0.122943
$714$	0	0
$715$	0.938342	0.0350920
$716$	−133.558	−4.99131
$717$	0	0
$718$	86.7123	3.23608
$719$	−35.6628	−1.33000	−0.665000	−	0.746844i	$-0.731568\pi$
$719$	−35.6628	−1.33000	−0.665000	+	0.746844i	$0.731568\pi$
$720$	0	0
$721$	−25.4682	−0.948485
$722$	−8.41619	−0.313218
$723$	0	0
$724$	−89.4526	−3.32448
$725$	4.08058	0.151549
$726$	0	0
$727$	32.9307	1.22133	0.610666	−	0.791888i	$-0.290902\pi$
$727$	32.9307	1.22133	0.610666	+	0.791888i	$0.290902\pi$
$728$	55.2253	2.04678
$729$	0	0
$730$	4.67955	0.173198
$731$	−7.80122	−0.288538
$732$	0	0
$733$	−24.7182	−0.912987	−0.456494	−	0.889727i	$-0.650895\pi$
$733$	−24.7182	−0.912987	−0.456494	+	0.889727i	$0.650895\pi$
$734$	48.9578	1.80707
$735$	0	0
$736$	−195.672	−7.21256
$737$	−9.11848	−0.335883
$738$	0	0
$739$	7.82247	0.287754	0.143877	−	0.989596i	$-0.454043\pi$
$739$	7.82247	0.287754	0.143877	+	0.989596i	$0.454043\pi$
$740$	−25.0678	−0.921509
$741$	0	0
$742$	−32.5601	−1.19532
$743$	−49.8087	−1.82730	−0.913652	−	0.406497i	$-0.866750\pi$
$743$	−49.8087	−1.82730	−0.913652	+	0.406497i	$0.866750\pi$
$744$	0	0
$745$	10.9567	0.401424
$746$	−78.5095	−2.87444
$747$	0	0
$748$	−40.6535	−1.48644
$749$	2.31511	0.0845921
$750$	0	0
$751$	−9.92975	−0.362342	−0.181171	−	0.983452i	$-0.557989\pi$
$751$	−9.92975	−0.362342	−0.181171	+	0.983452i	$0.557989\pi$
$752$	179.971	6.56288
$753$	0	0
$754$	−3.76329	−0.137051
$755$	9.52443	0.346630
$756$	0	0
$757$	−28.3055	−1.02878	−0.514391	−	0.857556i	$-0.671982\pi$
$757$	−28.3055	−1.02878	−0.514391	+	0.857556i	$0.671982\pi$
$758$	11.6347	0.422592
$759$	0	0
$760$	−32.3682	−1.17412
$761$	24.3630	0.883157	0.441578	−	0.897223i	$-0.354419\pi$
$761$	24.3630	0.883157	0.441578	+	0.897223i	$0.354419\pi$
$762$	0	0
$763$	−52.1803	−1.88905
$764$	113.143	4.09339
$765$	0	0
$766$	−24.1460	−0.872432
$767$	14.4348	0.521211
$768$	0	0
$769$	15.3379	0.553097	0.276549	−	0.961000i	$-0.410809\pi$
$769$	15.3379	0.553097	0.276549	+	0.961000i	$0.410809\pi$
$770$	5.76671	0.207818
$771$	0	0
$772$	91.3723	3.28856
$773$	−42.4817	−1.52796	−0.763980	−	0.645240i	$-0.776757\pi$
$773$	−42.4817	−1.52796	−0.763980	+	0.645240i	$0.776757\pi$
$774$	0	0
$775$	2.51651	0.0903957
$776$	−198.149	−7.11312
$777$	0	0
$778$	−1.84884	−0.0662842
$779$	−6.82856	−0.244658
$780$	0	0
$781$	6.11087	0.218664
$782$	116.053	4.15005
$783$	0	0
$784$	74.6164	2.66487
$785$	−0.664163	−0.0237050
$786$	0	0
$787$	−37.4983	−1.33667	−0.668334	−	0.743861i	$-0.732992\pi$
$787$	−37.4983	−1.33667	−0.668334	+	0.743861i	$0.732992\pi$
$788$	60.4241	2.15252
$789$	0	0
$790$	−10.3651	−0.368773
$791$	−48.2849	−1.71681
$792$	0	0
$793$	−1.50990	−0.0536182
$794$	−79.6221	−2.82568
$795$	0	0
$796$	−119.898	−4.24968
$797$	−26.1393	−0.925901	−0.462951	−	0.886384i	$-0.653209\pi$
$797$	−26.1393	−0.925901	−0.462951	+	0.886384i	$0.653209\pi$
$798$	0	0
$799$	−63.4103	−2.24329
$800$	149.995	5.30313
$801$	0	0
$802$	82.6809	2.91956
$803$	2.67201	0.0942931
$804$	0	0
$805$	−12.3164	−0.434095
$806$	−2.32083	−0.0817478
$807$	0	0
$808$	−82.2397	−2.89318
$809$	−24.4357	−0.859115	−0.429558	−	0.903039i	$-0.641330\pi$
$809$	−24.4357	−0.859115	−0.429558	+	0.903039i	$0.641330\pi$
$810$	0	0
$811$	55.7391	1.95726	0.978632	−	0.205620i	$-0.0659209\pi$
$811$	55.7391	1.95726	0.978632	+	0.205620i	$0.0659209\pi$
$812$	−17.3034	−0.607229
$813$	0	0
$814$	−19.1317	−0.670565
$815$	1.44756	0.0507058
$816$	0	0
$817$	−5.34622	−0.187040
$818$	15.6333	0.546605
$819$	0	0
$820$	−5.37734	−0.187785
$821$	21.8507	0.762595	0.381298	−	0.924452i	$-0.375477\pi$
$821$	21.8507	0.762595	0.381298	+	0.924452i	$0.375477\pi$
$822$	0	0
$823$	11.9653	0.417083	0.208541	−	0.978014i	$-0.433128\pi$
$823$	11.9653	0.417083	0.208541	+	0.978014i	$0.433128\pi$
$824$	85.9141	2.99296
$825$	0	0
$826$	88.7112	3.08666
$827$	4.09861	0.142522	0.0712612	−	0.997458i	$-0.477298\pi$
$827$	4.09861	0.142522	0.0712612	+	0.997458i	$0.477298\pi$
$828$	0	0
$829$	15.0804	0.523763	0.261881	−	0.965100i	$-0.415657\pi$
$829$	15.0804	0.523763	0.261881	+	0.965100i	$0.415657\pi$
$830$	25.1558	0.873169
$831$	0	0
$832$	−79.6889	−2.76272
$833$	−26.2900	−0.910894
$834$	0	0
$835$	11.8234	0.409165
$836$	−27.8601	−0.963560
$837$	0	0
$838$	−9.47437	−0.327287
$839$	48.3907	1.67063	0.835317	−	0.549769i	$-0.185284\pi$
$839$	48.3907	1.67063	0.835317	+	0.549769i	$0.185284\pi$
$840$	0	0
$841$	−28.2178	−0.973027
$842$	54.5475	1.87983
$843$	0	0
$844$	92.0779	3.16945
$845$	6.66217	0.229185
$846$	0	0
$847$	3.29277	0.113141
$848$	68.1408	2.33997
$849$	0	0
$850$	−88.9622	−3.05138
$851$	40.8608	1.40069
$852$	0	0
$853$	0.606579	0.0207689	0.0103844	−	0.999946i	$-0.496694\pi$
$853$	0.606579	0.0207689	0.0103844	+	0.999946i	$0.496694\pi$
$854$	−9.27931	−0.317532
$855$	0	0
$856$	−7.80975	−0.266932
$857$	−6.43140	−0.219693	−0.109846	−	0.993949i	$-0.535036\pi$
$857$	−6.43140	−0.219693	−0.109846	+	0.993949i	$0.535036\pi$
$858$	0	0
$859$	−9.38353	−0.320162	−0.160081	−	0.987104i	$-0.551176\pi$
$859$	−9.38353	−0.320162	−0.160081	+	0.987104i	$0.551176\pi$
$860$	−4.21003	−0.143561
$861$	0	0
$862$	6.85917	0.233624
$863$	−18.7547	−0.638418	−0.319209	−	0.947684i	$-0.603417\pi$
$863$	−18.7547	−0.638418	−0.319209	+	0.947684i	$0.603417\pi$
$864$	0	0
$865$	6.60178	0.224467
$866$	90.9510	3.09064
$867$	0	0
$868$	−10.6710	−0.362199
$869$	−5.91843	−0.200769
$870$	0	0
$871$	13.7680	0.466511
$872$	176.024	5.96094
$873$	0	0
$874$	79.5318	2.69020
$875$	19.6729	0.665066
$876$	0	0
$877$	−7.91424	−0.267245	−0.133622	−	0.991032i	$-0.542661\pi$
$877$	−7.91424	−0.267245	−0.133622	+	0.991032i	$0.542661\pi$
$878$	15.6773	0.529082
$879$	0	0
$880$	−12.0684	−0.406826
$881$	−8.43022	−0.284021	−0.142011	−	0.989865i	$-0.545357\pi$
$881$	−8.43022	−0.284021	−0.142011	+	0.989865i	$0.545357\pi$
$882$	0	0
$883$	54.9609	1.84958	0.924791	−	0.380476i	$-0.124240\pi$
$883$	54.9609	1.84958	0.924791	+	0.380476i	$0.124240\pi$
$884$	61.3827	2.06453
$885$	0	0
$886$	55.7603	1.87330
$887$	−17.0744	−0.573302	−0.286651	−	0.958035i	$-0.592542\pi$
$887$	−17.0744	−0.573302	−0.286651	+	0.958035i	$0.592542\pi$
$888$	0	0
$889$	−17.5719	−0.589341
$890$	−7.23503	−0.242519
$891$	0	0
$892$	−28.4865	−0.953797
$893$	−43.4554	−1.45418
$894$	0	0
$895$	13.9695	0.466948
$896$	−275.642	−9.20856
$897$	0	0
$898$	−72.1587	−2.40797
$899$	0.482398	0.0160889
$900$	0	0
$901$	−24.0084	−0.799836
$902$	−4.10398	−0.136648
$903$	0	0
$904$	162.884	5.41743
$905$	9.35624	0.311012
$906$	0	0
$907$	18.6932	0.620697	0.310349	−	0.950623i	$-0.399554\pi$
$907$	18.6932	0.620697	0.310349	+	0.950623i	$0.399554\pi$
$908$	57.0250	1.89244
$909$	0	0
$910$	−8.70717	−0.288640
$911$	37.4430	1.24054	0.620271	−	0.784388i	$-0.287023\pi$
$911$	37.4430	1.24054	0.620271	+	0.784388i	$0.287023\pi$
$912$	0	0
$913$	14.3638	0.475374
$914$	−17.3260	−0.573094
$915$	0	0
$916$	−57.3841	−1.89602
$917$	30.1472	0.995548
$918$	0	0
$919$	−26.0957	−0.860816	−0.430408	−	0.902634i	$-0.641630\pi$
$919$	−26.0957	−0.860816	−0.430408	+	0.902634i	$0.641630\pi$
$920$	41.5479	1.36979
$921$	0	0
$922$	25.8558	0.851515
$923$	−9.22681	−0.303704
$924$	0	0
$925$	−31.3225	−1.02988
$926$	−26.5225	−0.871584
$927$	0	0
$928$	28.7531	0.943866
$929$	−54.1120	−1.77536	−0.887679	−	0.460462i	$-0.847684\pi$
$929$	−54.1120	−1.77536	−0.887679	+	0.460462i	$0.847684\pi$
$930$	0	0
$931$	−18.0167	−0.590472
$932$	−49.9532	−1.63627
$933$	0	0
$934$	−99.2715	−3.24826
$935$	4.25213	0.139059
$936$	0	0
$937$	−55.0180	−1.79736	−0.898680	−	0.438604i	$-0.855473\pi$
$937$	−55.0180	−1.79736	−0.898680	+	0.438604i	$0.855473\pi$
$938$	84.6132	2.76272
$939$	0	0
$940$	−34.2202	−1.11614
$941$	20.5266	0.669149	0.334574	−	0.942369i	$-0.391407\pi$
$941$	20.5266	0.669149	0.334574	+	0.942369i	$0.391407\pi$
$942$	0	0
$943$	8.76515	0.285433
$944$	−185.652	−6.04247
$945$	0	0
$946$	−3.21309	−0.104467
$947$	−21.2760	−0.691378	−0.345689	−	0.938349i	$-0.612355\pi$
$947$	−21.2760	−0.691378	−0.345689	+	0.938349i	$0.612355\pi$
$948$	0	0
$949$	−4.03447	−0.130964
$950$	−60.9663	−1.97801
$951$	0	0
$952$	250.255	8.11082
$953$	5.15917	0.167122	0.0835610	−	0.996503i	$-0.473371\pi$
$953$	5.15917	0.167122	0.0835610	+	0.996503i	$0.473371\pi$
$954$	0	0
$955$	−11.8342	−0.382945
$956$	−132.711	−4.29217
$957$	0	0
$958$	93.2345	3.01227
$959$	−29.2705	−0.945193
$960$	0	0
$961$	−30.7025	−0.990403
$962$	28.8869	0.931352
$963$	0	0
$964$	−5.97861	−0.192558
$965$	−9.55704	−0.307652
$966$	0	0
$967$	6.63634	0.213410	0.106705	−	0.994291i	$-0.465970\pi$
$967$	6.63634	0.213410	0.106705	+	0.994291i	$0.465970\pi$
$968$	−11.1078	−0.357018
$969$	0	0
$970$	31.2414	1.00310
$971$	−0.732434	−0.0235049	−0.0117525	−	0.999931i	$-0.503741\pi$
$971$	−0.732434	−0.0235049	−0.0117525	+	0.999931i	$0.503741\pi$
$972$	0	0
$973$	−6.16547	−0.197656
$974$	122.566	3.92726
$975$	0	0
$976$	19.4195	0.621603
$977$	33.8174	1.08192	0.540958	−	0.841050i	$-0.318062\pi$
$977$	33.8174	1.08192	0.540958	+	0.841050i	$0.318062\pi$
$978$	0	0
$979$	−4.13117	−0.132033
$980$	−14.1877	−0.453210
$981$	0	0
$982$	−19.1584	−0.611370
$983$	−6.58921	−0.210163	−0.105082	−	0.994464i	$-0.533510\pi$
$983$	−6.58921	−0.210163	−0.105082	+	0.994464i	$0.533510\pi$
$984$	0	0
$985$	−6.32003	−0.201373
$986$	−17.0535	−0.543093
$987$	0	0
$988$	42.0659	1.33830
$989$	6.86241	0.218212
$990$	0	0
$991$	−35.6319	−1.13189	−0.565943	−	0.824445i	$-0.691488\pi$
$991$	−35.6319	−1.13189	−0.565943	+	0.824445i	$0.691488\pi$
$992$	17.7321	0.562996
$993$	0	0
$994$	−56.7047	−1.79856
$995$	12.5407	0.397566
$996$	0	0
$997$	61.7366	1.95522	0.977609	−	0.210429i	$-0.0674861\pi$
$997$	61.7366	1.95522	0.977609	+	0.210429i	$0.0674861\pi$
$998$	−95.8710	−3.03474
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.1	✓	25
3.2	odd	2		6039.2.a.p.1.25	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.1	✓	25	1.1	even	1	trivial
6039.2.a.p.1.25	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.81809 q^{2} +5.94161 q^{4} -0.621459 q^{5} +3.29277 q^{7} -11.1078 q^{8} +O(q^{10})\)	\(q-2.81809 q^{2} +5.94161 q^{4} -0.621459 q^{5} +3.29277 q^{7} -11.1078 q^{8} +1.75133 q^{10} +1.00000 q^{11} -1.50990 q^{13} -9.27931 q^{14} +19.4195 q^{16} -6.84217 q^{17} -4.68898 q^{19} -3.69247 q^{20} -2.81809 q^{22} +6.01878 q^{23} -4.61379 q^{25} +4.25503 q^{26} +19.5644 q^{28} -0.884432 q^{29} -0.545433 q^{31} -32.5102 q^{32} +19.2818 q^{34} -2.04632 q^{35} +6.78889 q^{37} +13.2139 q^{38} +6.90304 q^{40} +1.45630 q^{41} +1.14017 q^{43} +5.94161 q^{44} -16.9614 q^{46} +9.26757 q^{47} +3.84234 q^{49} +13.0021 q^{50} -8.97124 q^{52} +3.50889 q^{53} -0.621459 q^{55} -36.5754 q^{56} +2.49241 q^{58} -9.56011 q^{59} +1.00000 q^{61} +1.53708 q^{62} +52.7776 q^{64} +0.938342 q^{65} -9.11848 q^{67} -40.6535 q^{68} +5.76671 q^{70} +6.11087 q^{71} +2.67201 q^{73} -19.1317 q^{74} -27.8601 q^{76} +3.29277 q^{77} -5.91843 q^{79} -12.0684 q^{80} -4.10398 q^{82} +14.3638 q^{83} +4.25213 q^{85} -3.21309 q^{86} -11.1078 q^{88} -4.13117 q^{89} -4.97176 q^{91} +35.7612 q^{92} -26.1168 q^{94} +2.91401 q^{95} +17.8387 q^{97} -10.8281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

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