LMFDB - Embedded newform 6039.2.a.l.1.12

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:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q+0.404310 q^{2} -1.83653 q^{4} -0.0975766 q^{5} +1.38056 q^{7} -1.55115 q^{8} +O(q^{10})$	\(q+0.404310 q^{2} -1.83653 q^{4} -0.0975766 q^{5} +1.38056 q^{7} -1.55115 q^{8} -0.0394512 q^{10} +1.00000 q^{11} +1.91382 q^{13} +0.558177 q^{14} +3.04592 q^{16} -6.39070 q^{17} +2.68107 q^{19} +0.179203 q^{20} +0.404310 q^{22} -3.27987 q^{23} -4.99048 q^{25} +0.773778 q^{26} -2.53545 q^{28} -3.68854 q^{29} +8.77643 q^{31} +4.33380 q^{32} -2.58383 q^{34} -0.134711 q^{35} +8.11523 q^{37} +1.08399 q^{38} +0.151356 q^{40} +7.68669 q^{41} +0.587590 q^{43} -1.83653 q^{44} -1.32609 q^{46} +3.82618 q^{47} -5.09404 q^{49} -2.01770 q^{50} -3.51480 q^{52} -12.1754 q^{53} -0.0975766 q^{55} -2.14146 q^{56} -1.49132 q^{58} -4.38430 q^{59} +1.00000 q^{61} +3.54840 q^{62} -4.33964 q^{64} -0.186744 q^{65} +2.09197 q^{67} +11.7367 q^{68} -0.0544650 q^{70} +7.30897 q^{71} +7.39795 q^{73} +3.28107 q^{74} -4.92388 q^{76} +1.38056 q^{77} -15.9826 q^{79} -0.297210 q^{80} +3.10781 q^{82} +0.649661 q^{83} +0.623582 q^{85} +0.237569 q^{86} -1.55115 q^{88} +8.11324 q^{89} +2.64215 q^{91} +6.02360 q^{92} +1.54696 q^{94} -0.261610 q^{95} +6.04986 q^{97} -2.05957 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * 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 $

$2$	0.404310	0.285891	0.142945	−	0.989731i	$-0.454343\pi$
$2$	0.404310	0.285891	0.142945	+	0.989731i	$0.454343\pi$
$3$	0	0
$3$	0	0
$4$	−1.83653	−0.918267
$5$	−0.0975766	−0.0436376	−0.0218188	−	0.999762i	$-0.506946\pi$
$5$	−0.0975766	−0.0436376	−0.0218188	+	0.999762i	$0.506946\pi$
$6$	0	0
$7$	1.38056	0.521804	0.260902	−	0.965365i	$-0.415980\pi$
$7$	1.38056	0.521804	0.260902	+	0.965365i	$0.415980\pi$
$8$	−1.55115	−0.548414
$9$	0	0
$10$	−0.0394512	−0.0124756
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.91382	0.530799	0.265399	−	0.964139i	$-0.414496\pi$
$13$	1.91382	0.530799	0.265399	+	0.964139i	$0.414496\pi$
$14$	0.558177	0.149179
$15$	0	0
$16$	3.04592	0.761480
$17$	−6.39070	−1.54997	−0.774986	−	0.631979i	$-0.782243\pi$
$17$	−6.39070	−1.54997	−0.774986	+	0.631979i	$0.782243\pi$
$18$	0	0
$19$	2.68107	0.615080	0.307540	−	0.951535i	$-0.400494\pi$
$19$	2.68107	0.615080	0.307540	+	0.951535i	$0.400494\pi$
$20$	0.179203	0.0400709
$21$	0	0
$22$	0.404310	0.0861993
$23$	−3.27987	−0.683901	−0.341951	−	0.939718i	$-0.611088\pi$
$23$	−3.27987	−0.683901	−0.341951	+	0.939718i	$0.611088\pi$
$24$	0	0
$25$	−4.99048	−0.998096
$26$	0.773778	0.151750
$27$	0	0
$28$	−2.53545	−0.479156
$29$	−3.68854	−0.684945	−0.342472	−	0.939528i	$-0.611264\pi$
$29$	−3.68854	−0.684945	−0.342472	+	0.939528i	$0.611264\pi$
$30$	0	0
$31$	8.77643	1.57629	0.788147	−	0.615487i	$-0.211041\pi$
$31$	8.77643	1.57629	0.788147	+	0.615487i	$0.211041\pi$
$32$	4.33380	0.766114
$33$	0	0
$34$	−2.58383	−0.443122
$35$	−0.134711	−0.0227703
$36$	0	0
$37$	8.11523	1.33414	0.667068	−	0.744997i	$-0.267549\pi$
$37$	8.11523	1.33414	0.667068	+	0.744997i	$0.267549\pi$
$38$	1.08399	0.175846
$39$	0	0
$40$	0.151356	0.0239315
$41$	7.68669	1.20046	0.600230	−	0.799827i	$-0.295076\pi$
$41$	7.68669	1.20046	0.600230	+	0.799827i	$0.295076\pi$
$42$	0	0
$43$	0.587590	0.0896067	0.0448033	−	0.998996i	$-0.485734\pi$
$43$	0.587590	0.0896067	0.0448033	+	0.998996i	$0.485734\pi$
$44$	−1.83653	−0.276868
$45$	0	0
$46$	−1.32609	−0.195521
$47$	3.82618	0.558106	0.279053	−	0.960276i	$-0.409980\pi$
$47$	3.82618	0.558106	0.279053	+	0.960276i	$0.409980\pi$
$48$	0	0
$49$	−5.09404	−0.727720
$50$	−2.01770	−0.285346
$51$	0	0
$52$	−3.51480	−0.487415
$53$	−12.1754	−1.67242	−0.836210	−	0.548410i	$-0.815233\pi$
$53$	−12.1754	−1.67242	−0.836210	+	0.548410i	$0.815233\pi$
$54$	0	0
$55$	−0.0975766	−0.0131572
$56$	−2.14146	−0.286165
$57$	0	0
$58$	−1.49132	−0.195819
$59$	−4.38430	−0.570787	−0.285394	−	0.958410i	$-0.592124\pi$
$59$	−4.38430	−0.570787	−0.285394	+	0.958410i	$0.592124\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	3.54840	0.450648
$63$	0	0
$64$	−4.33964	−0.542455
$65$	−0.186744	−0.0231628
$66$	0	0
$67$	2.09197	0.255575	0.127787	−	0.991802i	$-0.459212\pi$
$67$	2.09197	0.255575	0.127787	+	0.991802i	$0.459212\pi$
$68$	11.7367	1.42329
$69$	0	0
$70$	−0.0544650	−0.00650981
$71$	7.30897	0.867416	0.433708	−	0.901054i	$-0.357205\pi$
$71$	7.30897	0.867416	0.433708	+	0.901054i	$0.357205\pi$
$72$	0	0
$73$	7.39795	0.865865	0.432933	−	0.901426i	$-0.357479\pi$
$73$	7.39795	0.865865	0.432933	+	0.901426i	$0.357479\pi$
$74$	3.28107	0.381417
$75$	0	0
$76$	−4.92388	−0.564807
$77$	1.38056	0.157330
$78$	0	0
$79$	−15.9826	−1.79819	−0.899093	−	0.437757i	$-0.855773\pi$
$79$	−15.9826	−1.79819	−0.899093	+	0.437757i	$0.855773\pi$
$80$	−0.297210	−0.0332291
$81$	0	0
$82$	3.10781	0.343200
$83$	0.649661	0.0713096	0.0356548	−	0.999364i	$-0.488648\pi$
$83$	0.649661	0.0713096	0.0356548	+	0.999364i	$0.488648\pi$
$84$	0	0
$85$	0.623582	0.0676370
$86$	0.237569	0.0256177
$87$	0	0
$88$	−1.55115	−0.165353
$89$	8.11324	0.860001	0.430001	−	0.902829i	$-0.358513\pi$
$89$	8.11324	0.860001	0.430001	+	0.902829i	$0.358513\pi$
$90$	0	0
$91$	2.64215	0.276973
$92$	6.02360	0.628004
$93$	0	0
$94$	1.54696	0.159557
$95$	−0.261610	−0.0268406
$96$	0	0
$97$	6.04986	0.614271	0.307135	−	0.951666i	$-0.400630\pi$
$97$	6.04986	0.614271	0.307135	+	0.951666i	$0.400630\pi$
$98$	−2.05957	−0.208048
$99$	0	0
$100$	9.16518	0.916518
$101$	13.4229	1.33563	0.667815	−	0.744327i	$-0.267230\pi$
$101$	13.4229	1.33563	0.667815	+	0.744327i	$0.267230\pi$
$102$	0	0
$103$	13.3184	1.31230	0.656151	−	0.754630i	$-0.272183\pi$
$103$	13.3184	1.31230	0.656151	+	0.754630i	$0.272183\pi$
$104$	−2.96863	−0.291098
$105$	0	0
$106$	−4.92264	−0.478129
$107$	11.4259	1.10458	0.552290	−	0.833652i	$-0.313754\pi$
$107$	11.4259	1.10458	0.552290	+	0.833652i	$0.313754\pi$
$108$	0	0
$109$	−19.4090	−1.85904	−0.929521	−	0.368768i	$-0.879780\pi$
$109$	−19.4090	−1.85904	−0.929521	+	0.368768i	$0.879780\pi$
$110$	−0.0394512	−0.00376153
$111$	0	0
$112$	4.20509	0.397344
$113$	−11.7630	−1.10657	−0.553286	−	0.832992i	$-0.686626\pi$
$113$	−11.7630	−1.10657	−0.553286	+	0.832992i	$0.686626\pi$
$114$	0	0
$115$	0.320039	0.0298438
$116$	6.77413	0.628962
$117$	0	0
$118$	−1.77262	−0.163183
$119$	−8.82277	−0.808782
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.404310	0.0366045
$123$	0	0
$124$	−16.1182	−1.44746
$125$	0.974837	0.0871921
$126$	0	0
$127$	12.7715	1.13329	0.566645	−	0.823962i	$-0.308241\pi$
$127$	12.7715	1.13329	0.566645	+	0.823962i	$0.308241\pi$
$128$	−10.4222	−0.921197
$129$	0	0
$130$	−0.0755026	−0.00662202
$131$	−11.1301	−0.972440	−0.486220	−	0.873836i	$-0.661625\pi$
$131$	−11.1301	−0.972440	−0.486220	+	0.873836i	$0.661625\pi$
$132$	0	0
$133$	3.70139	0.320952
$134$	0.845806	0.0730665
$135$	0	0
$136$	9.91293	0.850027
$137$	6.41607	0.548162	0.274081	−	0.961707i	$-0.411626\pi$
$137$	6.41607	0.548162	0.274081	+	0.961707i	$0.411626\pi$
$138$	0	0
$139$	10.3340	0.876518	0.438259	−	0.898849i	$-0.355595\pi$
$139$	10.3340	0.876518	0.438259	+	0.898849i	$0.355595\pi$
$140$	0.247401	0.0209092
$141$	0	0
$142$	2.95509	0.247986
$143$	1.91382	0.160042
$144$	0	0
$145$	0.359915	0.0298893
$146$	2.99107	0.247543
$147$	0	0
$148$	−14.9039	−1.22509
$149$	10.9469	0.896806	0.448403	−	0.893832i	$-0.351993\pi$
$149$	10.9469	0.896806	0.448403	+	0.893832i	$0.351993\pi$
$150$	0	0
$151$	−14.0261	−1.14143	−0.570715	−	0.821149i	$-0.693334\pi$
$151$	−14.0261	−1.14143	−0.570715	+	0.821149i	$0.693334\pi$
$152$	−4.15875	−0.337319
$153$	0	0
$154$	0.558177	0.0449792
$155$	−0.856375	−0.0687857
$156$	0	0
$157$	−20.5792	−1.64240	−0.821201	−	0.570639i	$-0.806695\pi$
$157$	−20.5792	−1.64240	−0.821201	+	0.570639i	$0.806695\pi$
$158$	−6.46194	−0.514085
$159$	0	0
$160$	−0.422877	−0.0334314
$161$	−4.52808	−0.356863
$162$	0	0
$163$	1.51365	0.118559	0.0592793	−	0.998241i	$-0.481120\pi$
$163$	1.51365	0.118559	0.0592793	+	0.998241i	$0.481120\pi$
$164$	−14.1169	−1.10234
$165$	0	0
$166$	0.262665	0.0203867
$167$	16.5914	1.28388	0.641941	−	0.766754i	$-0.278130\pi$
$167$	16.5914	1.28388	0.641941	+	0.766754i	$0.278130\pi$
$168$	0	0
$169$	−9.33729	−0.718253
$170$	0.252121	0.0193368
$171$	0	0
$172$	−1.07913	−0.0822828
$173$	17.7599	1.35026	0.675128	−	0.737700i	$-0.264088\pi$
$173$	17.7599	1.35026	0.675128	+	0.737700i	$0.264088\pi$
$174$	0	0
$175$	−6.88968	−0.520811
$176$	3.04592	0.229595
$177$	0	0
$178$	3.28027	0.245866
$179$	19.8327	1.48236	0.741181	−	0.671305i	$-0.234266\pi$
$179$	19.8327	1.48236	0.741181	+	0.671305i	$0.234266\pi$
$180$	0	0
$181$	7.18951	0.534392	0.267196	−	0.963642i	$-0.413903\pi$
$181$	7.18951	0.534392	0.267196	+	0.963642i	$0.413903\pi$
$182$	1.06825	0.0791840
$183$	0	0
$184$	5.08758	0.375061
$185$	−0.791857	−0.0582185
$186$	0	0
$187$	−6.39070	−0.467334
$188$	−7.02691	−0.512490
$189$	0	0
$190$	−0.105772	−0.00767348
$191$	10.4136	0.753503	0.376751	−	0.926314i	$-0.377041\pi$
$191$	10.4136	0.753503	0.376751	+	0.926314i	$0.377041\pi$
$192$	0	0
$193$	−6.92074	−0.498166	−0.249083	−	0.968482i	$-0.580129\pi$
$193$	−6.92074	−0.498166	−0.249083	+	0.968482i	$0.580129\pi$
$194$	2.44602	0.175614
$195$	0	0
$196$	9.35537	0.668241
$197$	−5.12309	−0.365005	−0.182502	−	0.983205i	$-0.558420\pi$
$197$	−5.12309	−0.365005	−0.182502	+	0.983205i	$0.558420\pi$
$198$	0	0
$199$	−9.17574	−0.650451	−0.325226	−	0.945636i	$-0.605440\pi$
$199$	−9.17574	−0.650451	−0.325226	+	0.945636i	$0.605440\pi$
$200$	7.74098	0.547370
$201$	0	0
$202$	5.42702	0.381844
$203$	−5.09227	−0.357407
$204$	0	0
$205$	−0.750041	−0.0523852
$206$	5.38477	0.375175
$207$	0	0
$208$	5.82935	0.404193
$209$	2.68107	0.185454
$210$	0	0
$211$	10.0749	0.693588	0.346794	−	0.937941i	$-0.387270\pi$
$211$	10.0749	0.693588	0.346794	+	0.937941i	$0.387270\pi$
$212$	22.3605	1.53573
$213$	0	0
$214$	4.61959	0.315789
$215$	−0.0573350	−0.00391022
$216$	0	0
$217$	12.1164	0.822517
$218$	−7.84725	−0.531483
$219$	0	0
$220$	0.179203	0.0120818
$221$	−12.2307	−0.822723
$222$	0	0
$223$	25.6563	1.71807	0.859037	−	0.511913i	$-0.171063\pi$
$223$	25.6563	1.71807	0.859037	+	0.511913i	$0.171063\pi$
$224$	5.98309	0.399762
$225$	0	0
$226$	−4.75591	−0.316358
$227$	22.7936	1.51286	0.756431	−	0.654074i	$-0.226941\pi$
$227$	22.7936	1.51286	0.756431	+	0.654074i	$0.226941\pi$
$228$	0	0
$229$	22.0797	1.45907	0.729535	−	0.683943i	$-0.239736\pi$
$229$	22.0797	1.45907	0.729535	+	0.683943i	$0.239736\pi$
$230$	0.129395	0.00853206
$231$	0	0
$232$	5.72148	0.375634
$233$	−24.2302	−1.58737	−0.793686	−	0.608327i	$-0.791841\pi$
$233$	−24.2302	−1.58737	−0.793686	+	0.608327i	$0.791841\pi$
$234$	0	0
$235$	−0.373346	−0.0243544
$236$	8.05191	0.524135
$237$	0	0
$238$	−3.56714	−0.231223
$239$	22.4004	1.44896	0.724481	−	0.689295i	$-0.242080\pi$
$239$	22.4004	1.44896	0.724481	+	0.689295i	$0.242080\pi$
$240$	0	0
$241$	17.1080	1.10202	0.551011	−	0.834498i	$-0.314242\pi$
$241$	17.1080	1.10202	0.551011	+	0.834498i	$0.314242\pi$
$242$	0.404310	0.0259901
$243$	0	0
$244$	−1.83653	−0.117572
$245$	0.497059	0.0317559
$246$	0	0
$247$	5.13109	0.326484
$248$	−13.6136	−0.864463
$249$	0	0
$250$	0.394137	0.0249274
$251$	21.1591	1.33555	0.667774	−	0.744364i	$-0.267247\pi$
$251$	21.1591	1.33555	0.667774	+	0.744364i	$0.267247\pi$
$252$	0	0
$253$	−3.27987	−0.206204
$254$	5.16366	0.323997
$255$	0	0
$256$	4.46549	0.279093
$257$	−18.3294	−1.14336	−0.571680	−	0.820477i	$-0.693708\pi$
$257$	−18.3294	−1.14336	−0.571680	+	0.820477i	$0.693708\pi$
$258$	0	0
$259$	11.2036	0.696158
$260$	0.342962	0.0212696
$261$	0	0
$262$	−4.50001	−0.278012
$263$	7.41077	0.456968	0.228484	−	0.973548i	$-0.426623\pi$
$263$	7.41077	0.456968	0.228484	+	0.973548i	$0.426623\pi$
$264$	0	0
$265$	1.18803	0.0729803
$266$	1.49651	0.0917570
$267$	0	0
$268$	−3.84197	−0.234686
$269$	28.5189	1.73883	0.869415	−	0.494082i	$-0.164496\pi$
$269$	28.5189	1.73883	0.869415	+	0.494082i	$0.164496\pi$
$270$	0	0
$271$	23.7868	1.44495	0.722473	−	0.691399i	$-0.243005\pi$
$271$	23.7868	1.44495	0.722473	+	0.691399i	$0.243005\pi$
$272$	−19.4655	−1.18027
$273$	0	0
$274$	2.59408	0.156714
$275$	−4.99048	−0.300937
$276$	0	0
$277$	10.2932	0.618460	0.309230	−	0.950987i	$-0.399929\pi$
$277$	10.2932	0.618460	0.309230	+	0.950987i	$0.399929\pi$
$278$	4.17814	0.250588
$279$	0	0
$280$	0.208957	0.0124876
$281$	20.1829	1.20401	0.602004	−	0.798493i	$-0.294369\pi$
$281$	20.1829	1.20401	0.602004	+	0.798493i	$0.294369\pi$
$282$	0	0
$283$	5.16490	0.307021	0.153511	−	0.988147i	$-0.450942\pi$
$283$	5.16490	0.307021	0.153511	+	0.988147i	$0.450942\pi$
$284$	−13.4232	−0.796519
$285$	0	0
$286$	0.773778	0.0457545
$287$	10.6120	0.626405
$288$	0	0
$289$	23.8410	1.40241
$290$	0.145517	0.00854508
$291$	0	0
$292$	−13.5866	−0.795095
$293$	−1.92764	−0.112614	−0.0563071	−	0.998413i	$-0.517933\pi$
$293$	−1.92764	−0.112614	−0.0563071	+	0.998413i	$0.517933\pi$
$294$	0	0
$295$	0.427805	0.0249078
$296$	−12.5879	−0.731660
$297$	0	0
$298$	4.42595	0.256388
$299$	−6.27710	−0.363014
$300$	0	0
$301$	0.811206	0.0467572
$302$	−5.67090	−0.326324
$303$	0	0
$304$	8.16633	0.468371
$305$	−0.0975766	−0.00558722
$306$	0	0
$307$	11.9619	0.682699	0.341350	−	0.939936i	$-0.389116\pi$
$307$	11.9619	0.682699	0.341350	+	0.939936i	$0.389116\pi$
$308$	−2.53545	−0.144471
$309$	0	0
$310$	−0.346241	−0.0196652
$311$	−8.08067	−0.458213	−0.229106	−	0.973401i	$-0.573580\pi$
$311$	−8.08067	−0.458213	−0.229106	+	0.973401i	$0.573580\pi$
$312$	0	0
$313$	29.4242	1.66316	0.831578	−	0.555408i	$-0.187438\pi$
$313$	29.4242	1.66316	0.831578	+	0.555408i	$0.187438\pi$
$314$	−8.32040	−0.469547
$315$	0	0
$316$	29.3526	1.65121
$317$	−22.5448	−1.26624	−0.633120	−	0.774054i	$-0.718226\pi$
$317$	−22.5448	−1.26624	−0.633120	+	0.774054i	$0.718226\pi$
$318$	0	0
$319$	−3.68854	−0.206519
$320$	0.423447	0.0236714
$321$	0	0
$322$	−1.83075	−0.102024
$323$	−17.1339	−0.953357
$324$	0	0
$325$	−9.55089	−0.529788
$326$	0.611986	0.0338948
$327$	0	0
$328$	−11.9232	−0.658350
$329$	5.28229	0.291222
$330$	0	0
$331$	11.1529	0.613020	0.306510	−	0.951867i	$-0.400839\pi$
$331$	11.1529	0.613020	0.306510	+	0.951867i	$0.400839\pi$
$332$	−1.19312	−0.0654812
$333$	0	0
$334$	6.70808	0.367050
$335$	−0.204127	−0.0111527
$336$	0	0
$337$	−14.9848	−0.816274	−0.408137	−	0.912921i	$-0.633821\pi$
$337$	−14.9848	−0.816274	−0.408137	+	0.912921i	$0.633821\pi$
$338$	−3.77516	−0.205342
$339$	0	0
$340$	−1.14523	−0.0621088
$341$	8.77643	0.475271
$342$	0	0
$343$	−16.6966	−0.901532
$344$	−0.911441	−0.0491416
$345$	0	0
$346$	7.18049	0.386026
$347$	−14.5439	−0.780760	−0.390380	−	0.920654i	$-0.627656\pi$
$347$	−14.5439	−0.780760	−0.390380	+	0.920654i	$0.627656\pi$
$348$	0	0
$349$	15.7833	0.844863	0.422431	−	0.906395i	$-0.361177\pi$
$349$	15.7833	0.844863	0.422431	+	0.906395i	$0.361177\pi$
$350$	−2.78557	−0.148895
$351$	0	0
$352$	4.33380	0.230992
$353$	−11.2517	−0.598869	−0.299434	−	0.954117i	$-0.596798\pi$
$353$	−11.2517	−0.598869	−0.299434	+	0.954117i	$0.596798\pi$
$354$	0	0
$355$	−0.713185	−0.0378519
$356$	−14.9002	−0.789710
$357$	0	0
$358$	8.01855	0.423793
$359$	10.7378	0.566721	0.283360	−	0.959013i	$-0.408551\pi$
$359$	10.7378	0.566721	0.283360	+	0.959013i	$0.408551\pi$
$360$	0	0
$361$	−11.8119	−0.621677
$362$	2.90679	0.152778
$363$	0	0
$364$	−4.85240	−0.254335
$365$	−0.721867	−0.0377843
$366$	0	0
$367$	−31.2795	−1.63278	−0.816388	−	0.577503i	$-0.804027\pi$
$367$	−31.2795	−1.63278	−0.816388	+	0.577503i	$0.804027\pi$
$368$	−9.99024	−0.520777
$369$	0	0
$370$	−0.320156	−0.0166441
$371$	−16.8089	−0.872676
$372$	0	0
$373$	−6.69105	−0.346449	−0.173225	−	0.984882i	$-0.555419\pi$
$373$	−6.69105	−0.346449	−0.173225	+	0.984882i	$0.555419\pi$
$374$	−2.58383	−0.133606
$375$	0	0
$376$	−5.93498	−0.306073
$377$	−7.05921	−0.363568
$378$	0	0
$379$	17.4406	0.895866	0.447933	−	0.894067i	$-0.352160\pi$
$379$	17.4406	0.895866	0.447933	+	0.894067i	$0.352160\pi$
$380$	0.480455	0.0246468
$381$	0	0
$382$	4.21033	0.215419
$383$	11.5754	0.591473	0.295736	−	0.955270i	$-0.404435\pi$
$383$	11.5754	0.591473	0.295736	+	0.955270i	$0.404435\pi$
$384$	0	0
$385$	−0.134711	−0.00686550
$386$	−2.79813	−0.142421
$387$	0	0
$388$	−11.1108	−0.564064
$389$	−14.2848	−0.724266	−0.362133	−	0.932126i	$-0.617951\pi$
$389$	−14.2848	−0.724266	−0.362133	+	0.932126i	$0.617951\pi$
$390$	0	0
$391$	20.9607	1.06003
$392$	7.90162	0.399092
$393$	0	0
$394$	−2.07132	−0.104352
$395$	1.55953	0.0784685
$396$	0	0
$397$	27.6351	1.38696	0.693482	−	0.720474i	$-0.256075\pi$
$397$	27.6351	1.38696	0.693482	+	0.720474i	$0.256075\pi$
$398$	−3.70985	−0.185958
$399$	0	0
$400$	−15.2006	−0.760030
$401$	−24.1900	−1.20799	−0.603996	−	0.796987i	$-0.706426\pi$
$401$	−24.1900	−1.20799	−0.603996	+	0.796987i	$0.706426\pi$
$402$	0	0
$403$	16.7965	0.836695
$404$	−24.6516	−1.22646
$405$	0	0
$406$	−2.05886	−0.102179
$407$	8.11523	0.402257
$408$	0	0
$409$	7.93089	0.392157	0.196079	−	0.980588i	$-0.437179\pi$
$409$	7.93089	0.392157	0.196079	+	0.980588i	$0.437179\pi$
$410$	−0.303250	−0.0149764
$411$	0	0
$412$	−24.4597	−1.20504
$413$	−6.05281	−0.297839
$414$	0	0
$415$	−0.0633917	−0.00311178
$416$	8.29412	0.406653
$417$	0	0
$418$	1.08399	0.0530195
$419$	−10.8385	−0.529496	−0.264748	−	0.964318i	$-0.585289\pi$
$419$	−10.8385	−0.529496	−0.264748	+	0.964318i	$0.585289\pi$
$420$	0	0
$421$	−10.7357	−0.523228	−0.261614	−	0.965173i	$-0.584255\pi$
$421$	−10.7357	−0.523228	−0.261614	+	0.965173i	$0.584255\pi$
$422$	4.07341	0.198290
$423$	0	0
$424$	18.8859	0.917179
$425$	31.8926	1.54702
$426$	0	0
$427$	1.38056	0.0668102
$428$	−20.9840	−1.01430
$429$	0	0
$430$	−0.0231812	−0.00111789
$431$	30.2153	1.45542	0.727710	−	0.685885i	$-0.240585\pi$
$431$	30.2153	1.45542	0.727710	+	0.685885i	$0.240585\pi$
$432$	0	0
$433$	17.6507	0.848237	0.424119	−	0.905607i	$-0.360584\pi$
$433$	17.6507	0.848237	0.424119	+	0.905607i	$0.360584\pi$
$434$	4.89880	0.235150
$435$	0	0
$436$	35.6452	1.70710
$437$	−8.79358	−0.420654
$438$	0	0
$439$	−19.7097	−0.940693	−0.470347	−	0.882482i	$-0.655871\pi$
$439$	−19.7097	−0.940693	−0.470347	+	0.882482i	$0.655871\pi$
$440$	0.151356	0.00721561
$441$	0	0
$442$	−4.94498	−0.235209
$443$	36.4139	1.73008	0.865038	−	0.501706i	$-0.167294\pi$
$443$	36.4139	1.73008	0.865038	+	0.501706i	$0.167294\pi$
$444$	0	0
$445$	−0.791662	−0.0375284
$446$	10.3731	0.491181
$447$	0	0
$448$	−5.99115	−0.283055
$449$	7.81784	0.368947	0.184473	−	0.982838i	$-0.440942\pi$
$449$	7.81784	0.368947	0.184473	+	0.982838i	$0.440942\pi$
$450$	0	0
$451$	7.68669	0.361952
$452$	21.6032	1.01613
$453$	0	0
$454$	9.21567	0.432513
$455$	−0.257812	−0.0120864
$456$	0	0
$457$	1.28959	0.0603245	0.0301622	−	0.999545i	$-0.490398\pi$
$457$	1.28959	0.0603245	0.0301622	+	0.999545i	$0.490398\pi$
$458$	8.92707	0.417135
$459$	0	0
$460$	−0.587762	−0.0274046
$461$	12.1738	0.566991	0.283495	−	0.958974i	$-0.408506\pi$
$461$	12.1738	0.566991	0.283495	+	0.958974i	$0.408506\pi$
$462$	0	0
$463$	33.2077	1.54329	0.771647	−	0.636050i	$-0.219433\pi$
$463$	33.2077	1.54329	0.771647	+	0.636050i	$0.219433\pi$
$464$	−11.2350	−0.521572
$465$	0	0
$466$	−9.79652	−0.453815
$467$	25.1563	1.16410	0.582048	−	0.813154i	$-0.302252\pi$
$467$	25.1563	1.16410	0.582048	+	0.813154i	$0.302252\pi$
$468$	0	0
$469$	2.88810	0.133360
$470$	−0.150948	−0.00696269
$471$	0	0
$472$	6.80071	0.313028
$473$	0.587590	0.0270174
$474$	0	0
$475$	−13.3798	−0.613909
$476$	16.2033	0.742677
$477$	0	0
$478$	9.05672	0.414245
$479$	14.0023	0.639779	0.319890	−	0.947455i	$-0.396354\pi$
$479$	14.0023	0.639779	0.319890	+	0.947455i	$0.396354\pi$
$480$	0	0
$481$	15.5311	0.708158
$482$	6.91694	0.315058
$483$	0	0
$484$	−1.83653	−0.0834788
$485$	−0.590325	−0.0268053
$486$	0	0
$487$	−28.1486	−1.27553	−0.637767	−	0.770229i	$-0.720142\pi$
$487$	−28.1486	−1.27553	−0.637767	+	0.770229i	$0.720142\pi$
$488$	−1.55115	−0.0702173
$489$	0	0
$490$	0.200966	0.00907873
$491$	−27.6795	−1.24916	−0.624580	−	0.780961i	$-0.714730\pi$
$491$	−27.6795	−1.24916	−0.624580	+	0.780961i	$0.714730\pi$
$492$	0	0
$493$	23.5723	1.06164
$494$	2.07455	0.0933386
$495$	0	0
$496$	26.7323	1.20032
$497$	10.0905	0.452621
$498$	0	0
$499$	−14.1755	−0.634582	−0.317291	−	0.948328i	$-0.602773\pi$
$499$	−14.1755	−0.634582	−0.317291	+	0.948328i	$0.602773\pi$
$500$	−1.79032	−0.0800655
$501$	0	0
$502$	8.55483	0.381821
$503$	−26.0229	−1.16030	−0.580152	−	0.814508i	$-0.697007\pi$
$503$	−26.0229	−1.16030	−0.580152	+	0.814508i	$0.697007\pi$
$504$	0	0
$505$	−1.30976	−0.0582836
$506$	−1.32609	−0.0589518
$507$	0	0
$508$	−23.4553	−1.04066
$509$	−7.54307	−0.334341	−0.167170	−	0.985928i	$-0.553463\pi$
$509$	−7.54307	−0.334341	−0.167170	+	0.985928i	$0.553463\pi$
$510$	0	0
$511$	10.2134	0.451812
$512$	22.6498	1.00099
$513$	0	0
$514$	−7.41079	−0.326876
$515$	−1.29957	−0.0572657
$516$	0	0
$517$	3.82618	0.168275
$518$	4.52973	0.199025
$519$	0	0
$520$	0.289668	0.0127028
$521$	−33.3174	−1.45966	−0.729831	−	0.683628i	$-0.760401\pi$
$521$	−33.3174	−1.45966	−0.729831	+	0.683628i	$0.760401\pi$
$522$	0	0
$523$	−39.4616	−1.72554	−0.862768	−	0.505601i	$-0.831271\pi$
$523$	−39.4616	−1.72554	−0.862768	+	0.505601i	$0.831271\pi$
$524$	20.4408	0.892959
$525$	0	0
$526$	2.99625	0.130643
$527$	−56.0875	−2.44321
$528$	0	0
$529$	−12.2424	−0.532279
$530$	0.480334	0.0208644
$531$	0	0
$532$	−6.79773	−0.294719
$533$	14.7110	0.637202
$534$	0	0
$535$	−1.11490	−0.0482012
$536$	−3.24496	−0.140161
$537$	0	0
$538$	11.5305	0.497115
$539$	−5.09404	−0.219416
$540$	0	0
$541$	−13.8024	−0.593409	−0.296705	−	0.954969i	$-0.595888\pi$
$541$	−13.8024	−0.593409	−0.296705	+	0.954969i	$0.595888\pi$
$542$	9.61726	0.413097
$543$	0	0
$544$	−27.6960	−1.18746
$545$	1.89386	0.0811241
$546$	0	0
$547$	−24.3353	−1.04050	−0.520250	−	0.854014i	$-0.674161\pi$
$547$	−24.3353	−1.04050	−0.520250	+	0.854014i	$0.674161\pi$
$548$	−11.7833	−0.503358
$549$	0	0
$550$	−2.01770	−0.0860351
$551$	−9.88924	−0.421296
$552$	0	0
$553$	−22.0651	−0.938302
$554$	4.16166	0.176812
$555$	0	0
$556$	−18.9787	−0.804877
$557$	−0.570434	−0.0241701	−0.0120850	−	0.999927i	$-0.503847\pi$
$557$	−0.570434	−0.0241701	−0.0120850	+	0.999927i	$0.503847\pi$
$558$	0	0
$559$	1.12454	0.0475631
$560$	−0.410318	−0.0173391
$561$	0	0
$562$	8.16014	0.344215
$563$	−26.8221	−1.13042	−0.565209	−	0.824948i	$-0.691204\pi$
$563$	−26.8221	−1.13042	−0.565209	+	0.824948i	$0.691204\pi$
$564$	0	0
$565$	1.14779	0.0482881
$566$	2.08822	0.0877745
$567$	0	0
$568$	−11.3373	−0.475703
$569$	1.76168	0.0738534	0.0369267	−	0.999318i	$-0.488243\pi$
$569$	1.76168	0.0738534	0.0369267	+	0.999318i	$0.488243\pi$
$570$	0	0
$571$	−26.0955	−1.09206	−0.546031	−	0.837765i	$-0.683862\pi$
$571$	−26.0955	−1.09206	−0.546031	+	0.837765i	$0.683862\pi$
$572$	−3.51480	−0.146961
$573$	0	0
$574$	4.29053	0.179083
$575$	16.3681	0.682599
$576$	0	0
$577$	17.7705	0.739794	0.369897	−	0.929073i	$-0.379393\pi$
$577$	17.7705	0.739794	0.369897	+	0.929073i	$0.379393\pi$
$578$	9.63916	0.400936
$579$	0	0
$580$	−0.660996	−0.0274464
$581$	0.896899	0.0372096
$582$	0	0
$583$	−12.1754	−0.504253
$584$	−11.4753	−0.474853
$585$	0	0
$586$	−0.779367	−0.0321953
$587$	13.5033	0.557339	0.278669	−	0.960387i	$-0.410107\pi$
$587$	13.5033	0.557339	0.278669	+	0.960387i	$0.410107\pi$
$588$	0	0
$589$	23.5303	0.969547
$590$	0.172966	0.00712090
$591$	0	0
$592$	24.7184	1.01592
$593$	−9.48660	−0.389568	−0.194784	−	0.980846i	$-0.562401\pi$
$593$	−9.48660	−0.389568	−0.194784	+	0.980846i	$0.562401\pi$
$594$	0	0
$595$	0.860896	0.0352933
$596$	−20.1044	−0.823507
$597$	0	0
$598$	−2.53790	−0.103782
$599$	5.01686	0.204983	0.102492	−	0.994734i	$-0.467319\pi$
$599$	5.01686	0.204983	0.102492	+	0.994734i	$0.467319\pi$
$600$	0	0
$601$	9.73474	0.397088	0.198544	−	0.980092i	$-0.436379\pi$
$601$	9.73474	0.397088	0.198544	+	0.980092i	$0.436379\pi$
$602$	0.327979	0.0133674
$603$	0	0
$604$	25.7594	1.04814
$605$	−0.0975766	−0.00396705
$606$	0	0
$607$	20.1566	0.818130	0.409065	−	0.912505i	$-0.365855\pi$
$607$	20.1566	0.818130	0.409065	+	0.912505i	$0.365855\pi$
$608$	11.6192	0.471222
$609$	0	0
$610$	−0.0394512	−0.00159733
$611$	7.32263	0.296242
$612$	0	0
$613$	1.56278	0.0631202	0.0315601	−	0.999502i	$-0.489952\pi$
$613$	1.56278	0.0631202	0.0315601	+	0.999502i	$0.489952\pi$
$614$	4.83630	0.195177
$615$	0	0
$616$	−2.14146	−0.0862820
$617$	28.1019	1.13134	0.565670	−	0.824632i	$-0.308618\pi$
$617$	28.1019	1.13134	0.565670	+	0.824632i	$0.308618\pi$
$618$	0	0
$619$	7.18499	0.288789	0.144394	−	0.989520i	$-0.453877\pi$
$619$	7.18499	0.288789	0.144394	+	0.989520i	$0.453877\pi$
$620$	1.57276	0.0631636
$621$	0	0
$622$	−3.26710	−0.130999
$623$	11.2008	0.448753
$624$	0	0
$625$	24.8573	0.994291
$626$	11.8965	0.475481
$627$	0	0
$628$	37.7944	1.50816
$629$	−51.8620	−2.06787
$630$	0	0
$631$	−41.4917	−1.65176	−0.825879	−	0.563848i	$-0.809321\pi$
$631$	−41.4917	−1.65176	−0.825879	+	0.563848i	$0.809321\pi$
$632$	24.7915	0.986152
$633$	0	0
$634$	−9.11508	−0.362006
$635$	−1.24620	−0.0494540
$636$	0	0
$637$	−9.74909	−0.386273
$638$	−1.49132	−0.0590417
$639$	0	0
$640$	1.01696	0.0401988
$641$	−5.36615	−0.211950	−0.105975	−	0.994369i	$-0.533796\pi$
$641$	−5.36615	−0.211950	−0.105975	+	0.994369i	$0.533796\pi$
$642$	0	0
$643$	−16.8713	−0.665341	−0.332670	−	0.943043i	$-0.607950\pi$
$643$	−16.8713	−0.665341	−0.332670	+	0.943043i	$0.607950\pi$
$644$	8.31597	0.327695
$645$	0	0
$646$	−6.92742	−0.272556
$647$	−15.1912	−0.597229	−0.298614	−	0.954374i	$-0.596524\pi$
$647$	−15.1912	−0.597229	−0.298614	+	0.954374i	$0.596524\pi$
$648$	0	0
$649$	−4.38430	−0.172099
$650$	−3.86152	−0.151461
$651$	0	0
$652$	−2.77988	−0.108868
$653$	−10.8434	−0.424334	−0.212167	−	0.977233i	$-0.568052\pi$
$653$	−10.8434	−0.424334	−0.212167	+	0.977233i	$0.568052\pi$
$654$	0	0
$655$	1.08604	0.0424349
$656$	23.4131	0.914126
$657$	0	0
$658$	2.13569	0.0832577
$659$	39.8700	1.55312	0.776558	−	0.630046i	$-0.216964\pi$
$659$	39.8700	1.55312	0.776558	+	0.630046i	$0.216964\pi$
$660$	0	0
$661$	2.58629	0.100595	0.0502975	−	0.998734i	$-0.483983\pi$
$661$	2.58629	0.100595	0.0502975	+	0.998734i	$0.483983\pi$
$662$	4.50924	0.175257
$663$	0	0
$664$	−1.00772	−0.0391072
$665$	−0.361169	−0.0140055
$666$	0	0
$667$	12.0979	0.468434
$668$	−30.4707	−1.17895
$669$	0	0
$670$	−0.0825308	−0.00318844
$671$	1.00000	0.0386046
$672$	0	0
$673$	−40.7325	−1.57012	−0.785060	−	0.619419i	$-0.787368\pi$
$673$	−40.7325	−1.57012	−0.785060	+	0.619419i	$0.787368\pi$
$674$	−6.05851	−0.233365
$675$	0	0
$676$	17.1482	0.659548
$677$	−42.3469	−1.62752	−0.813761	−	0.581199i	$-0.802584\pi$
$677$	−42.3469	−1.62752	−0.813761	+	0.581199i	$0.802584\pi$
$678$	0	0
$679$	8.35223	0.320529
$680$	−0.967270	−0.0370931
$681$	0	0
$682$	3.54840	0.135875
$683$	11.4726	0.438985	0.219492	−	0.975614i	$-0.429560\pi$
$683$	11.4726	0.438985	0.219492	+	0.975614i	$0.429560\pi$
$684$	0	0
$685$	−0.626058	−0.0239204
$686$	−6.75061	−0.257740
$687$	0	0
$688$	1.78975	0.0682337
$689$	−23.3015	−0.887718
$690$	0	0
$691$	−4.48114	−0.170471	−0.0852354	−	0.996361i	$-0.527164\pi$
$691$	−4.48114	−0.170471	−0.0852354	+	0.996361i	$0.527164\pi$
$692$	−32.6166	−1.23990
$693$	0	0
$694$	−5.88027	−0.223212
$695$	−1.00836	−0.0382491
$696$	0	0
$697$	−49.1233	−1.86068
$698$	6.38137	0.241538
$699$	0	0
$700$	12.6531	0.478243
$701$	23.8141	0.899445	0.449722	−	0.893168i	$-0.351523\pi$
$701$	23.8141	0.899445	0.449722	+	0.893168i	$0.351523\pi$
$702$	0	0
$703$	21.7575	0.820601
$704$	−4.33964	−0.163556
$705$	0	0
$706$	−4.54919	−0.171211
$707$	18.5312	0.696937
$708$	0	0
$709$	40.1995	1.50972	0.754862	−	0.655884i	$-0.227704\pi$
$709$	40.1995	1.50972	0.754862	+	0.655884i	$0.227704\pi$
$710$	−0.288348	−0.0108215
$711$	0	0
$712$	−12.5849	−0.471637
$713$	−28.7856	−1.07803
$714$	0	0
$715$	−0.186744	−0.00698384
$716$	−36.4233	−1.36120
$717$	0	0
$718$	4.34142	0.162020
$719$	20.7734	0.774719	0.387359	−	0.921929i	$-0.373387\pi$
$719$	20.7734	0.774719	0.387359	+	0.921929i	$0.373387\pi$
$720$	0	0
$721$	18.3869	0.684765
$722$	−4.77566	−0.177732
$723$	0	0
$724$	−13.2038	−0.490714
$725$	18.4076	0.683640
$726$	0	0
$727$	−22.1783	−0.822549	−0.411275	−	0.911512i	$-0.634916\pi$
$727$	−22.1783	−0.822549	−0.411275	+	0.911512i	$0.634916\pi$
$728$	−4.09838	−0.151896
$729$	0	0
$730$	−0.291858	−0.0108022
$731$	−3.75511	−0.138888
$732$	0	0
$733$	−22.3172	−0.824304	−0.412152	−	0.911115i	$-0.635223\pi$
$733$	−22.3172	−0.824304	−0.412152	+	0.911115i	$0.635223\pi$
$734$	−12.6466	−0.466796
$735$	0	0
$736$	−14.2143	−0.523947
$737$	2.09197	0.0770587
$738$	0	0
$739$	−48.0693	−1.76826	−0.884129	−	0.467243i	$-0.845247\pi$
$739$	−48.0693	−1.76826	−0.884129	+	0.467243i	$0.845247\pi$
$740$	1.45427	0.0534601
$741$	0	0
$742$	−6.79602	−0.249490
$743$	25.5828	0.938542	0.469271	−	0.883054i	$-0.344517\pi$
$743$	25.5828	0.938542	0.469271	+	0.883054i	$0.344517\pi$
$744$	0	0
$745$	−1.06816	−0.0391344
$746$	−2.70526	−0.0990467
$747$	0	0
$748$	11.7367	0.429137
$749$	15.7741	0.576374
$750$	0	0
$751$	45.0859	1.64521	0.822604	−	0.568615i	$-0.192521\pi$
$751$	45.0859	1.64521	0.822604	+	0.568615i	$0.192521\pi$
$752$	11.6542	0.424986
$753$	0	0
$754$	−2.85411	−0.103941
$755$	1.36862	0.0498092
$756$	0	0
$757$	−8.81976	−0.320560	−0.160280	−	0.987072i	$-0.551240\pi$
$757$	−8.81976	−0.320560	−0.160280	+	0.987072i	$0.551240\pi$
$758$	7.05143	0.256120
$759$	0	0
$760$	0.405796	0.0147198
$761$	−43.9572	−1.59345	−0.796723	−	0.604344i	$-0.793435\pi$
$761$	−43.9572	−1.59345	−0.796723	+	0.604344i	$0.793435\pi$
$762$	0	0
$763$	−26.7953	−0.970057
$764$	−19.1249	−0.691916
$765$	0	0
$766$	4.68004	0.169097
$767$	−8.39077	−0.302973
$768$	0	0
$769$	3.56562	0.128580	0.0642898	−	0.997931i	$-0.479522\pi$
$769$	3.56562	0.128580	0.0642898	+	0.997931i	$0.479522\pi$
$770$	−0.0544650	−0.00196278
$771$	0	0
$772$	12.7102	0.457449
$773$	47.7439	1.71723	0.858614	−	0.512622i	$-0.171326\pi$
$773$	47.7439	1.71723	0.858614	+	0.512622i	$0.171326\pi$
$774$	0	0
$775$	−43.7986	−1.57329
$776$	−9.38425	−0.336875
$777$	0	0
$778$	−5.77548	−0.207061
$779$	20.6086	0.738379
$780$	0	0
$781$	7.30897	0.261536
$782$	8.47462	0.303052
$783$	0	0
$784$	−15.5160	−0.554144
$785$	2.00805	0.0716704
$786$	0	0
$787$	−14.5262	−0.517803	−0.258902	−	0.965904i	$-0.583361\pi$
$787$	−14.5262	−0.517803	−0.258902	+	0.965904i	$0.583361\pi$
$788$	9.40872	0.335172
$789$	0	0
$790$	0.630535	0.0224334
$791$	−16.2396	−0.577414
$792$	0	0
$793$	1.91382	0.0679618
$794$	11.1732	0.396520
$795$	0	0
$796$	16.8516	0.597288
$797$	25.7378	0.911679	0.455839	−	0.890062i	$-0.349339\pi$
$797$	25.7378	0.911679	0.455839	+	0.890062i	$0.349339\pi$
$798$	0	0
$799$	−24.4520	−0.865048
$800$	−21.6277	−0.764656
$801$	0	0
$802$	−9.78028	−0.345354
$803$	7.39795	0.261068
$804$	0	0
$805$	0.441835	0.0155726
$806$	6.79101	0.239203
$807$	0	0
$808$	−20.8210	−0.732479
$809$	−16.1314	−0.567150	−0.283575	−	0.958950i	$-0.591521\pi$
$809$	−16.1314	−0.567150	−0.283575	+	0.958950i	$0.591521\pi$
$810$	0	0
$811$	−37.2088	−1.30658	−0.653289	−	0.757108i	$-0.726611\pi$
$811$	−37.2088	−1.30658	−0.653289	+	0.757108i	$0.726611\pi$
$812$	9.35212	0.328195
$813$	0	0
$814$	3.28107	0.115002
$815$	−0.147697	−0.00517361
$816$	0	0
$817$	1.57537	0.0551153
$818$	3.20654	0.112114
$819$	0	0
$820$	1.37748	0.0481035
$821$	24.8527	0.867366	0.433683	−	0.901065i	$-0.357214\pi$
$821$	24.8527	0.867366	0.433683	+	0.901065i	$0.357214\pi$
$822$	0	0
$823$	−27.3472	−0.953262	−0.476631	−	0.879104i	$-0.658142\pi$
$823$	−27.3472	−0.953262	−0.476631	+	0.879104i	$0.658142\pi$
$824$	−20.6589	−0.719685
$825$	0	0
$826$	−2.44721	−0.0851495
$827$	−3.62847	−0.126174	−0.0630871	−	0.998008i	$-0.520095\pi$
$827$	−3.62847	−0.126174	−0.0630871	+	0.998008i	$0.520095\pi$
$828$	0	0
$829$	26.8941	0.934070	0.467035	−	0.884239i	$-0.345322\pi$
$829$	26.8941	0.934070	0.467035	+	0.884239i	$0.345322\pi$
$830$	−0.0256299	−0.000889628 0
$831$	0	0
$832$	−8.30530	−0.287934
$833$	32.5545	1.12795
$834$	0	0
$835$	−1.61893	−0.0560255
$836$	−4.92388	−0.170296
$837$	0	0
$838$	−4.38212	−0.151378
$839$	−8.01173	−0.276596	−0.138298	−	0.990391i	$-0.544163\pi$
$839$	−8.01173	−0.276596	−0.138298	+	0.990391i	$0.544163\pi$
$840$	0	0
$841$	−15.3947	−0.530851
$842$	−4.34057	−0.149586
$843$	0	0
$844$	−18.5030	−0.636899
$845$	0.911101	0.0313428
$846$	0	0
$847$	1.38056	0.0474368
$848$	−37.0853	−1.27351
$849$	0	0
$850$	12.8945	0.442279
$851$	−26.6170	−0.912417
$852$	0	0
$853$	27.7986	0.951807	0.475904	−	0.879497i	$-0.342121\pi$
$853$	27.7986	0.951807	0.475904	+	0.879497i	$0.342121\pi$
$854$	0.558177	0.0191004
$855$	0	0
$856$	−17.7232	−0.605767
$857$	36.1854	1.23607	0.618035	−	0.786150i	$-0.287929\pi$
$857$	36.1854	1.23607	0.618035	+	0.786150i	$0.287929\pi$
$858$	0	0
$859$	−36.5369	−1.24662	−0.623312	−	0.781974i	$-0.714213\pi$
$859$	−36.5369	−1.24662	−0.623312	+	0.781974i	$0.714213\pi$
$860$	0.105298	0.00359062
$861$	0	0
$862$	12.2164	0.416091
$863$	−17.8996	−0.609309	−0.304655	−	0.952463i	$-0.598541\pi$
$863$	−17.8996	−0.609309	−0.304655	+	0.952463i	$0.598541\pi$
$864$	0	0
$865$	−1.73295	−0.0589219
$866$	7.13635	0.242503
$867$	0	0
$868$	−22.2522	−0.755290
$869$	−15.9826	−0.542174
$870$	0	0
$871$	4.00366	0.135659
$872$	30.1062	1.01953
$873$	0	0
$874$	−3.55534	−0.120261
$875$	1.34583	0.0454972
$876$	0	0
$877$	−28.7263	−0.970019	−0.485009	−	0.874509i	$-0.661184\pi$
$877$	−28.7263	−0.970019	−0.485009	+	0.874509i	$0.661184\pi$
$878$	−7.96884	−0.268935
$879$	0	0
$880$	−0.297210	−0.0100190
$881$	−18.4462	−0.621467	−0.310734	−	0.950497i	$-0.600575\pi$
$881$	−18.4462	−0.621467	−0.310734	+	0.950497i	$0.600575\pi$
$882$	0	0
$883$	−8.21016	−0.276294	−0.138147	−	0.990412i	$-0.544115\pi$
$883$	−8.21016	−0.276294	−0.138147	+	0.990412i	$0.544115\pi$
$884$	22.4620	0.755479
$885$	0	0
$886$	14.7225	0.494613
$887$	1.69374	0.0568703	0.0284352	−	0.999596i	$-0.490948\pi$
$887$	1.69374	0.0568703	0.0284352	+	0.999596i	$0.490948\pi$
$888$	0	0
$889$	17.6319	0.591356
$890$	−0.320077	−0.0107290
$891$	0	0
$892$	−47.1187	−1.57765
$893$	10.2583	0.343280
$894$	0	0
$895$	−1.93520	−0.0646867
$896$	−14.3885	−0.480685
$897$	0	0
$898$	3.16083	0.105478
$899$	−32.3722	−1.07967
$900$	0	0
$901$	77.8093	2.59220
$902$	3.10781	0.103479
$903$	0	0
$904$	18.2462	0.606860
$905$	−0.701527	−0.0233196
$906$	0	0
$907$	41.0996	1.36469	0.682345	−	0.731031i	$-0.260960\pi$
$907$	41.0996	1.36469	0.682345	+	0.731031i	$0.260960\pi$
$908$	−41.8611	−1.38921
$909$	0	0
$910$	−0.104236	−0.00345540
$911$	10.5470	0.349437	0.174719	−	0.984618i	$-0.444098\pi$
$911$	10.5470	0.349437	0.174719	+	0.984618i	$0.444098\pi$
$912$	0	0
$913$	0.649661	0.0215006
$914$	0.521395	0.0172462
$915$	0	0
$916$	−40.5502	−1.33982
$917$	−15.3658	−0.507424
$918$	0	0
$919$	−2.90139	−0.0957080	−0.0478540	−	0.998854i	$-0.515238\pi$
$919$	−2.90139	−0.0957080	−0.0478540	+	0.998854i	$0.515238\pi$
$920$	−0.496429	−0.0163668
$921$	0	0
$922$	4.92200	0.162097
$923$	13.9881	0.460423
$924$	0	0
$925$	−40.4989	−1.33160
$926$	13.4262	0.441214
$927$	0	0
$928$	−15.9854	−0.524746
$929$	37.3322	1.22483	0.612416	−	0.790536i	$-0.290198\pi$
$929$	37.3322	1.22483	0.612416	+	0.790536i	$0.290198\pi$
$930$	0	0
$931$	−13.6575	−0.447606
$932$	44.4995	1.45763
$933$	0	0
$934$	10.1710	0.332804
$935$	0.623582	0.0203933
$936$	0	0
$937$	1.64643	0.0537867	0.0268933	−	0.999638i	$-0.491439\pi$
$937$	1.64643	0.0537867	0.0268933	+	0.999638i	$0.491439\pi$
$938$	1.16769	0.0381264
$939$	0	0
$940$	0.685662	0.0223638
$941$	55.6461	1.81401	0.907005	−	0.421120i	$-0.138363\pi$
$941$	55.6461	1.81401	0.907005	+	0.421120i	$0.138363\pi$
$942$	0	0
$943$	−25.2114	−0.820996
$944$	−13.3542	−0.434643
$945$	0	0
$946$	0.237569	0.00772403
$947$	27.5185	0.894230	0.447115	−	0.894476i	$-0.352451\pi$
$947$	27.5185	0.894230	0.447115	+	0.894476i	$0.352451\pi$
$948$	0	0
$949$	14.1584	0.459600
$950$	−5.40961	−0.175511
$951$	0	0
$952$	13.6854	0.443548
$953$	−26.8576	−0.870003	−0.435002	−	0.900430i	$-0.643252\pi$
$953$	−26.8576	−0.870003	−0.435002	+	0.900430i	$0.643252\pi$
$954$	0	0
$955$	−1.01612	−0.0328810
$956$	−41.1391	−1.33053
$957$	0	0
$958$	5.66126	0.182907
$959$	8.85780	0.286033
$960$	0	0
$961$	46.0258	1.48470
$962$	6.27939	0.202456
$963$	0	0
$964$	−31.4194	−1.01195
$965$	0.675302	0.0217387
$966$	0	0
$967$	−7.13631	−0.229488	−0.114744	−	0.993395i	$-0.536605\pi$
$967$	−7.13631	−0.229488	−0.114744	+	0.993395i	$0.536605\pi$
$968$	−1.55115	−0.0498559
$969$	0	0
$970$	−0.238675	−0.00766338
$971$	−48.6208	−1.56032	−0.780158	−	0.625582i	$-0.784862\pi$
$971$	−48.6208	−1.56032	−0.780158	+	0.625582i	$0.784862\pi$
$972$	0	0
$973$	14.2668	0.457371
$974$	−11.3808	−0.364663
$975$	0	0
$976$	3.04592	0.0974975
$977$	−37.7471	−1.20764	−0.603819	−	0.797122i	$-0.706355\pi$
$977$	−37.7471	−1.20764	−0.603819	+	0.797122i	$0.706355\pi$
$978$	0	0
$979$	8.11324	0.259300
$980$	−0.912866	−0.0291604
$981$	0	0
$982$	−11.1911	−0.357123
$983$	31.2734	0.997467	0.498734	−	0.866755i	$-0.333799\pi$
$983$	31.2734	0.997467	0.498734	+	0.866755i	$0.333799\pi$
$984$	0	0
$985$	0.499894	0.0159279
$986$	9.53054	0.303514
$987$	0	0
$988$	−9.42342	−0.299799
$989$	−1.92722	−0.0612821
$990$	0	0
$991$	−34.5374	−1.09712	−0.548558	−	0.836112i	$-0.684823\pi$
$991$	−34.5374	−1.09712	−0.548558	+	0.836112i	$0.684823\pi$
$992$	38.0353	1.20762
$993$	0	0
$994$	4.07970	0.129400
$995$	0.895338	0.0283841
$996$	0	0
$997$	1.18905	0.0376577	0.0188288	−	0.999823i	$-0.494006\pi$
$997$	1.18905	0.0376577	0.0188288	+	0.999823i	$0.494006\pi$
$998$	−5.73130	−0.181421
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.12		21
3.2	odd	2		671.2.a.d.1.10	✓	21
33.32	even	2		7381.2.a.j.1.12		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.10	✓	21	3.2	odd	2
6039.2.a.l.1.12		21	1.1	even	1	trivial
7381.2.a.j.1.12		21	33.32	even	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)

\(f(q)\)	\(=\)	\(q+0.404310 q^{2} -1.83653 q^{4} -0.0975766 q^{5} +1.38056 q^{7} -1.55115 q^{8} +O(q^{10})\)	\(q+0.404310 q^{2} -1.83653 q^{4} -0.0975766 q^{5} +1.38056 q^{7} -1.55115 q^{8} -0.0394512 q^{10} +1.00000 q^{11} +1.91382 q^{13} +0.558177 q^{14} +3.04592 q^{16} -6.39070 q^{17} +2.68107 q^{19} +0.179203 q^{20} +0.404310 q^{22} -3.27987 q^{23} -4.99048 q^{25} +0.773778 q^{26} -2.53545 q^{28} -3.68854 q^{29} +8.77643 q^{31} +4.33380 q^{32} -2.58383 q^{34} -0.134711 q^{35} +8.11523 q^{37} +1.08399 q^{38} +0.151356 q^{40} +7.68669 q^{41} +0.587590 q^{43} -1.83653 q^{44} -1.32609 q^{46} +3.82618 q^{47} -5.09404 q^{49} -2.01770 q^{50} -3.51480 q^{52} -12.1754 q^{53} -0.0975766 q^{55} -2.14146 q^{56} -1.49132 q^{58} -4.38430 q^{59} +1.00000 q^{61} +3.54840 q^{62} -4.33964 q^{64} -0.186744 q^{65} +2.09197 q^{67} +11.7367 q^{68} -0.0544650 q^{70} +7.30897 q^{71} +7.39795 q^{73} +3.28107 q^{74} -4.92388 q^{76} +1.38056 q^{77} -15.9826 q^{79} -0.297210 q^{80} +3.10781 q^{82} +0.649661 q^{83} +0.623582 q^{85} +0.237569 q^{86} -1.55115 q^{88} +8.11324 q^{89} +2.64215 q^{91} +6.02360 q^{92} +1.54696 q^{94} -0.261610 q^{95} +6.04986 q^{97} -2.05957 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Introduction

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