LMFDB - Embedded newform 6039.2.a.e.1.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 $ x^12 - x^11 - 16x^10 + 13x^9 + 93x^8 - 59x^7 - 238x^6 + 108x^5 + 257x^4 - 71x^3 - 93x^2 + 13x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.22938$ of defining polynomial
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)

$f(q)$	$=$	$q+2.22938 q^{2} +2.97016 q^{4} -3.25707 q^{5} -0.940570 q^{7} +2.16285 q^{8} +O(q^{10})$	\(q+2.22938 q^{2} +2.97016 q^{4} -3.25707 q^{5} -0.940570 q^{7} +2.16285 q^{8} -7.26127 q^{10} +1.00000 q^{11} +1.71970 q^{13} -2.09689 q^{14} -1.11849 q^{16} +0.0942226 q^{17} +3.20842 q^{19} -9.67401 q^{20} +2.22938 q^{22} +6.07120 q^{23} +5.60852 q^{25} +3.83387 q^{26} -2.79364 q^{28} +3.73900 q^{29} -9.85260 q^{31} -6.81924 q^{32} +0.210058 q^{34} +3.06350 q^{35} -8.79309 q^{37} +7.15280 q^{38} -7.04456 q^{40} -3.45448 q^{41} -9.13045 q^{43} +2.97016 q^{44} +13.5350 q^{46} +4.55689 q^{47} -6.11533 q^{49} +12.5035 q^{50} +5.10777 q^{52} -2.28023 q^{53} -3.25707 q^{55} -2.03431 q^{56} +8.33567 q^{58} -0.677326 q^{59} -1.00000 q^{61} -21.9652 q^{62} -12.9657 q^{64} -5.60118 q^{65} -8.66449 q^{67} +0.279856 q^{68} +6.82973 q^{70} -7.54892 q^{71} +10.5995 q^{73} -19.6032 q^{74} +9.52950 q^{76} -0.940570 q^{77} -4.70041 q^{79} +3.64299 q^{80} -7.70136 q^{82} -0.268842 q^{83} -0.306890 q^{85} -20.3553 q^{86} +2.16285 q^{88} -16.6456 q^{89} -1.61750 q^{91} +18.0324 q^{92} +10.1591 q^{94} -10.4500 q^{95} -19.6340 q^{97} -13.6334 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * 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 $

$2$	2.22938	1.57641	0.788206	−	0.615411i	$-0.211010\pi$
$2$	2.22938	1.57641	0.788206	+	0.615411i	$0.211010\pi$
$3$	0	0
$3$	0	0
$4$	2.97016	1.48508
$5$	−3.25707	−1.45661	−0.728303	−	0.685255i	$-0.759691\pi$
$5$	−3.25707	−1.45661	−0.728303	+	0.685255i	$0.759691\pi$
$6$	0	0
$7$	−0.940570	−0.355502	−0.177751	−	0.984075i	$-0.556882\pi$
$7$	−0.940570	−0.355502	−0.177751	+	0.984075i	$0.556882\pi$
$8$	2.16285	0.764683
$9$	0	0
$10$	−7.26127	−2.29621
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.71970	0.476958	0.238479	−	0.971148i	$-0.423351\pi$
$13$	1.71970	0.476958	0.238479	+	0.971148i	$0.423351\pi$
$14$	−2.09689	−0.560418
$15$	0	0
$16$	−1.11849	−0.279622
$17$	0.0942226	0.0228523	0.0114262	−	0.999935i	$-0.496363\pi$
$17$	0.0942226	0.0228523	0.0114262	+	0.999935i	$0.496363\pi$
$18$	0	0
$19$	3.20842	0.736062	0.368031	−	0.929814i	$-0.380032\pi$
$19$	3.20842	0.736062	0.368031	+	0.929814i	$0.380032\pi$
$20$	−9.67401	−2.16317
$21$	0	0
$22$	2.22938	0.475306
$23$	6.07120	1.26593	0.632966	−	0.774179i	$-0.281837\pi$
$23$	6.07120	1.26593	0.632966	+	0.774179i	$0.281837\pi$
$24$	0	0
$25$	5.60852	1.12170
$26$	3.83387	0.751883
$27$	0	0
$28$	−2.79364	−0.527948
$29$	3.73900	0.694315	0.347157	−	0.937807i	$-0.387147\pi$
$29$	3.73900	0.694315	0.347157	+	0.937807i	$0.387147\pi$
$30$	0	0
$31$	−9.85260	−1.76958	−0.884789	−	0.465991i	$-0.845698\pi$
$31$	−9.85260	−1.76958	−0.884789	+	0.465991i	$0.845698\pi$
$32$	−6.81924	−1.20548
$33$	0	0
$34$	0.210058	0.0360247
$35$	3.06350	0.517827
$36$	0	0
$37$	−8.79309	−1.44558	−0.722788	−	0.691070i	$-0.757140\pi$
$37$	−8.79309	−1.44558	−0.722788	+	0.691070i	$0.757140\pi$
$38$	7.15280	1.16034
$39$	0	0
$40$	−7.04456	−1.11384
$41$	−3.45448	−0.539499	−0.269749	−	0.962931i	$-0.586941\pi$
$41$	−3.45448	−0.539499	−0.269749	+	0.962931i	$0.586941\pi$
$42$	0	0
$43$	−9.13045	−1.39238	−0.696190	−	0.717857i	$-0.745123\pi$
$43$	−9.13045	−1.39238	−0.696190	+	0.717857i	$0.745123\pi$
$44$	2.97016	0.447768
$45$	0	0
$46$	13.5350	1.99563
$47$	4.55689	0.664691	0.332345	−	0.943158i	$-0.392160\pi$
$47$	4.55689	0.664691	0.332345	+	0.943158i	$0.392160\pi$
$48$	0	0
$49$	−6.11533	−0.873618
$50$	12.5035	1.76827
$51$	0	0
$52$	5.10777	0.708320
$53$	−2.28023	−0.313214	−0.156607	−	0.987661i	$-0.550056\pi$
$53$	−2.28023	−0.313214	−0.156607	+	0.987661i	$0.550056\pi$
$54$	0	0
$55$	−3.25707	−0.439184
$56$	−2.03431	−0.271846
$57$	0	0
$58$	8.33567	1.09453
$59$	−0.677326	−0.0881803	−0.0440902	−	0.999028i	$-0.514039\pi$
$59$	−0.677326	−0.0881803	−0.0440902	+	0.999028i	$0.514039\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−21.9652	−2.78959
$63$	0	0
$64$	−12.9657	−1.62072
$65$	−5.60118	−0.694741
$66$	0	0
$67$	−8.66449	−1.05854	−0.529268	−	0.848455i	$-0.677533\pi$
$67$	−8.66449	−1.05854	−0.529268	+	0.848455i	$0.677533\pi$
$68$	0.279856	0.0339375
$69$	0	0
$70$	6.82973	0.816309
$71$	−7.54892	−0.895892	−0.447946	−	0.894061i	$-0.647844\pi$
$71$	−7.54892	−0.895892	−0.447946	+	0.894061i	$0.647844\pi$
$72$	0	0
$73$	10.5995	1.24057	0.620287	−	0.784375i	$-0.287016\pi$
$73$	10.5995	1.24057	0.620287	+	0.784375i	$0.287016\pi$
$74$	−19.6032	−2.27882
$75$	0	0
$76$	9.52950	1.09311
$77$	−0.940570	−0.107188
$78$	0	0
$79$	−4.70041	−0.528837	−0.264419	−	0.964408i	$-0.585180\pi$
$79$	−4.70041	−0.528837	−0.264419	+	0.964408i	$0.585180\pi$
$80$	3.64299	0.407299
$81$	0	0
$82$	−7.70136	−0.850473
$83$	−0.268842	−0.0295092	−0.0147546	−	0.999891i	$-0.504697\pi$
$83$	−0.268842	−0.0295092	−0.0147546	+	0.999891i	$0.504697\pi$
$84$	0	0
$85$	−0.306890	−0.0332869
$86$	−20.3553	−2.19497
$87$	0	0
$88$	2.16285	0.230561
$89$	−16.6456	−1.76443	−0.882216	−	0.470845i	$-0.843949\pi$
$89$	−16.6456	−1.76443	−0.882216	+	0.470845i	$0.843949\pi$
$90$	0	0
$91$	−1.61750	−0.169560
$92$	18.0324	1.88001
$93$	0	0
$94$	10.1591	1.04783
$95$	−10.4500	−1.07215
$96$	0	0
$97$	−19.6340	−1.99353	−0.996765	−	0.0803703i	$-0.974390\pi$
$97$	−19.6340	−1.99353	−0.996765	+	0.0803703i	$0.974390\pi$
$98$	−13.6334	−1.37718
$99$	0	0
$100$	16.6582	1.66582
$101$	0.515342	0.0512785	0.0256392	−	0.999671i	$-0.491838\pi$
$101$	0.515342	0.0512785	0.0256392	+	0.999671i	$0.491838\pi$
$102$	0	0
$103$	5.31984	0.524179	0.262090	−	0.965044i	$-0.415588\pi$
$103$	5.31984	0.524179	0.262090	+	0.965044i	$0.415588\pi$
$104$	3.71945	0.364722
$105$	0	0
$106$	−5.08352	−0.493755
$107$	−7.25532	−0.701398	−0.350699	−	0.936488i	$-0.614056\pi$
$107$	−7.25532	−0.701398	−0.350699	+	0.936488i	$0.614056\pi$
$108$	0	0
$109$	8.69057	0.832405	0.416203	−	0.909272i	$-0.363361\pi$
$109$	8.69057	0.832405	0.416203	+	0.909272i	$0.363361\pi$
$110$	−7.26127	−0.692335
$111$	0	0
$112$	1.05202	0.0994061
$113$	9.97832	0.938682	0.469341	−	0.883017i	$-0.344492\pi$
$113$	9.97832	0.938682	0.469341	+	0.883017i	$0.344492\pi$
$114$	0	0
$115$	−19.7743	−1.84397
$116$	11.1054	1.03111
$117$	0	0
$118$	−1.51002	−0.139009
$119$	−0.0886229	−0.00812405
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.22938	−0.201839
$123$	0	0
$124$	−29.2637	−2.62796
$125$	−1.98199	−0.177274
$126$	0	0
$127$	−3.47812	−0.308633	−0.154317	−	0.988021i	$-0.549318\pi$
$127$	−3.47812	−0.308633	−0.154317	+	0.988021i	$0.549318\pi$
$128$	−15.2671	−1.34944
$129$	0	0
$130$	−12.4872	−1.09520
$131$	10.8571	0.948593	0.474296	−	0.880365i	$-0.342702\pi$
$131$	10.8571	0.948593	0.474296	+	0.880365i	$0.342702\pi$
$132$	0	0
$133$	−3.01774	−0.261671
$134$	−19.3165	−1.66869
$135$	0	0
$136$	0.203789	0.0174748
$137$	−3.79172	−0.323949	−0.161974	−	0.986795i	$-0.551786\pi$
$137$	−3.79172	−0.323949	−0.161974	+	0.986795i	$0.551786\pi$
$138$	0	0
$139$	18.3918	1.55997	0.779986	−	0.625797i	$-0.215226\pi$
$139$	18.3918	1.55997	0.779986	+	0.625797i	$0.215226\pi$
$140$	9.09908	0.769013
$141$	0	0
$142$	−16.8294	−1.41230
$143$	1.71970	0.143808
$144$	0	0
$145$	−12.1782	−1.01134
$146$	23.6303	1.95566
$147$	0	0
$148$	−26.1169	−2.14679
$149$	−9.96514	−0.816376	−0.408188	−	0.912898i	$-0.633839\pi$
$149$	−9.96514	−0.816376	−0.408188	+	0.912898i	$0.633839\pi$
$150$	0	0
$151$	−13.3148	−1.08354	−0.541771	−	0.840526i	$-0.682246\pi$
$151$	−13.3148	−1.08354	−0.541771	+	0.840526i	$0.682246\pi$
$152$	6.93933	0.562854
$153$	0	0
$154$	−2.09689	−0.168972
$155$	32.0906	2.57758
$156$	0	0
$157$	−4.79158	−0.382410	−0.191205	−	0.981550i	$-0.561239\pi$
$157$	−4.79158	−0.382410	−0.191205	+	0.981550i	$0.561239\pi$
$158$	−10.4790	−0.833666
$159$	0	0
$160$	22.2107	1.75591
$161$	−5.71039	−0.450041
$162$	0	0
$163$	−11.7986	−0.924141	−0.462071	−	0.886843i	$-0.652893\pi$
$163$	−11.7986	−0.924141	−0.462071	+	0.886843i	$0.652893\pi$
$164$	−10.2603	−0.801198
$165$	0	0
$166$	−0.599352	−0.0465188
$167$	−0.136780	−0.0105844	−0.00529218	−	0.999986i	$-0.501685\pi$
$167$	−0.136780	−0.0105844	−0.00529218	+	0.999986i	$0.501685\pi$
$168$	0	0
$169$	−10.0426	−0.772511
$170$	−0.684175	−0.0524739
$171$	0	0
$172$	−27.1189	−2.06779
$173$	0.642288	0.0488323	0.0244161	−	0.999702i	$-0.492227\pi$
$173$	0.642288	0.0488323	0.0244161	+	0.999702i	$0.492227\pi$
$174$	0	0
$175$	−5.27520	−0.398768
$176$	−1.11849	−0.0843092
$177$	0	0
$178$	−37.1095	−2.78147
$179$	−10.1771	−0.760674	−0.380337	−	0.924848i	$-0.624192\pi$
$179$	−10.1771	−0.760674	−0.380337	+	0.924848i	$0.624192\pi$
$180$	0	0
$181$	−9.82204	−0.730066	−0.365033	−	0.930994i	$-0.618942\pi$
$181$	−9.82204	−0.730066	−0.365033	+	0.930994i	$0.618942\pi$
$182$	−3.60602	−0.267296
$183$	0	0
$184$	13.1311	0.968037
$185$	28.6397	2.10564
$186$	0	0
$187$	0.0942226	0.00689024
$188$	13.5347	0.987117
$189$	0	0
$190$	−23.2972	−1.69015
$191$	−1.28961	−0.0933132	−0.0466566	−	0.998911i	$-0.514857\pi$
$191$	−1.28961	−0.0933132	−0.0466566	+	0.998911i	$0.514857\pi$
$192$	0	0
$193$	−21.3274	−1.53518	−0.767589	−	0.640942i	$-0.778544\pi$
$193$	−21.3274	−1.53518	−0.767589	+	0.640942i	$0.778544\pi$
$194$	−43.7717	−3.14263
$195$	0	0
$196$	−18.1635	−1.29739
$197$	−0.106914	−0.00761731	−0.00380866	−	0.999993i	$-0.501212\pi$
$197$	−0.106914	−0.00761731	−0.00380866	+	0.999993i	$0.501212\pi$
$198$	0	0
$199$	−3.91939	−0.277838	−0.138919	−	0.990304i	$-0.544363\pi$
$199$	−3.91939	−0.277838	−0.138919	+	0.990304i	$0.544363\pi$
$200$	12.1304	0.857747
$201$	0	0
$202$	1.14890	0.0808360
$203$	−3.51679	−0.246830
$204$	0	0
$205$	11.2515	0.785838
$206$	11.8600	0.826323
$207$	0	0
$208$	−1.92346	−0.133368
$209$	3.20842	0.221931
$210$	0	0
$211$	−14.2112	−0.978339	−0.489170	−	0.872189i	$-0.662700\pi$
$211$	−14.2112	−0.978339	−0.489170	+	0.872189i	$0.662700\pi$
$212$	−6.77265	−0.465148
$213$	0	0
$214$	−16.1749	−1.10569
$215$	29.7385	2.02815
$216$	0	0
$217$	9.26705	0.629089
$218$	19.3746	1.31221
$219$	0	0
$220$	−9.67401	−0.652222
$221$	0.162034	0.0108996
$222$	0	0
$223$	6.06987	0.406469	0.203234	−	0.979130i	$-0.434855\pi$
$223$	6.06987	0.406469	0.203234	+	0.979130i	$0.434855\pi$
$224$	6.41397	0.428551
$225$	0	0
$226$	22.2455	1.47975
$227$	1.93408	0.128370	0.0641848	−	0.997938i	$-0.479555\pi$
$227$	1.93408	0.128370	0.0641848	+	0.997938i	$0.479555\pi$
$228$	0	0
$229$	17.5914	1.16247	0.581235	−	0.813736i	$-0.302570\pi$
$229$	17.5914	1.16247	0.581235	+	0.813736i	$0.302570\pi$
$230$	−44.0846	−2.90685
$231$	0	0
$232$	8.08689	0.530931
$233$	17.0416	1.11643	0.558216	−	0.829696i	$-0.311486\pi$
$233$	17.0416	1.11643	0.558216	+	0.829696i	$0.311486\pi$
$234$	0	0
$235$	−14.8421	−0.968193
$236$	−2.01176	−0.130955
$237$	0	0
$238$	−0.197575	−0.0128069
$239$	16.1060	1.04181	0.520904	−	0.853615i	$-0.325595\pi$
$239$	16.1060	1.04181	0.520904	+	0.853615i	$0.325595\pi$
$240$	0	0
$241$	29.8082	1.92011	0.960056	−	0.279807i	$-0.0902703\pi$
$241$	29.8082	1.92011	0.960056	+	0.279807i	$0.0902703\pi$
$242$	2.22938	0.143310
$243$	0	0
$244$	−2.97016	−0.190145
$245$	19.9181	1.27252
$246$	0	0
$247$	5.51751	0.351071
$248$	−21.3097	−1.35317
$249$	0	0
$250$	−4.41861	−0.279458
$251$	−7.33297	−0.462853	−0.231426	−	0.972852i	$-0.574339\pi$
$251$	−7.33297	−0.462853	−0.231426	+	0.972852i	$0.574339\pi$
$252$	0	0
$253$	6.07120	0.381693
$254$	−7.75407	−0.486533
$255$	0	0
$256$	−8.10482	−0.506551
$257$	13.1315	0.819123	0.409562	−	0.912282i	$-0.365682\pi$
$257$	13.1315	0.819123	0.409562	+	0.912282i	$0.365682\pi$
$258$	0	0
$259$	8.27052	0.513905
$260$	−16.6364	−1.03174
$261$	0	0
$262$	24.2047	1.49537
$263$	18.5825	1.14585	0.572924	−	0.819609i	$-0.305809\pi$
$263$	18.5825	1.14585	0.572924	+	0.819609i	$0.305809\pi$
$264$	0	0
$265$	7.42689	0.456230
$266$	−6.72771	−0.412502
$267$	0	0
$268$	−25.7349	−1.57201
$269$	10.2610	0.625624	0.312812	−	0.949815i	$-0.398729\pi$
$269$	10.2610	0.625624	0.312812	+	0.949815i	$0.398729\pi$
$270$	0	0
$271$	18.9803	1.15297	0.576484	−	0.817108i	$-0.304424\pi$
$271$	18.9803	1.15297	0.576484	+	0.817108i	$0.304424\pi$
$272$	−0.105387	−0.00639001
$273$	0	0
$274$	−8.45320	−0.510677
$275$	5.60852	0.338206
$276$	0	0
$277$	−6.21274	−0.373287	−0.186644	−	0.982428i	$-0.559761\pi$
$277$	−6.21274	−0.373287	−0.186644	+	0.982428i	$0.559761\pi$
$278$	41.0024	2.45916
$279$	0	0
$280$	6.62590	0.395973
$281$	32.9356	1.96477	0.982385	−	0.186866i	$-0.0598330\pi$
$281$	32.9356	1.96477	0.982385	+	0.186866i	$0.0598330\pi$
$282$	0	0
$283$	−14.7796	−0.878558	−0.439279	−	0.898351i	$-0.644766\pi$
$283$	−14.7796	−0.878558	−0.439279	+	0.898351i	$0.644766\pi$
$284$	−22.4215	−1.33047
$285$	0	0
$286$	3.83387	0.226701
$287$	3.24918	0.191793
$288$	0	0
$289$	−16.9911	−0.999478
$290$	−27.1499	−1.59430
$291$	0	0
$292$	31.4821	1.84235
$293$	−30.6080	−1.78814	−0.894070	−	0.447928i	$-0.852162\pi$
$293$	−30.6080	−1.78814	−0.894070	+	0.447928i	$0.852162\pi$
$294$	0	0
$295$	2.20610	0.128444
$296$	−19.0181	−1.10541
$297$	0	0
$298$	−22.2161	−1.28695
$299$	10.4406	0.603797
$300$	0	0
$301$	8.58783	0.494994
$302$	−29.6838	−1.70811
$303$	0	0
$304$	−3.58858	−0.205819
$305$	3.25707	0.186499
$306$	0	0
$307$	−25.9344	−1.48016	−0.740078	−	0.672521i	$-0.765211\pi$
$307$	−25.9344	−1.48016	−0.740078	+	0.672521i	$0.765211\pi$
$308$	−2.79364	−0.159182
$309$	0	0
$310$	71.5423	4.06333
$311$	−8.92790	−0.506255	−0.253127	−	0.967433i	$-0.581459\pi$
$311$	−8.92790	−0.506255	−0.253127	+	0.967433i	$0.581459\pi$
$312$	0	0
$313$	8.19920	0.463446	0.231723	−	0.972782i	$-0.425564\pi$
$313$	8.19920	0.463446	0.231723	+	0.972782i	$0.425564\pi$
$314$	−10.6823	−0.602835
$315$	0	0
$316$	−13.9609	−0.785365
$317$	−25.6246	−1.43922	−0.719611	−	0.694377i	$-0.755680\pi$
$317$	−25.6246	−1.43922	−0.719611	+	0.694377i	$0.755680\pi$
$318$	0	0
$319$	3.73900	0.209344
$320$	42.2303	2.36075
$321$	0	0
$322$	−12.7306	−0.709451
$323$	0.302305	0.0168207
$324$	0	0
$325$	9.64496	0.535006
$326$	−26.3037	−1.45683
$327$	0	0
$328$	−7.47152	−0.412545
$329$	−4.28607	−0.236299
$330$	0	0
$331$	14.5405	0.799216	0.399608	−	0.916686i	$-0.369146\pi$
$331$	14.5405	0.799216	0.399608	+	0.916686i	$0.369146\pi$
$332$	−0.798503	−0.0438235
$333$	0	0
$334$	−0.304935	−0.0166853
$335$	28.2209	1.54187
$336$	0	0
$337$	2.60533	0.141922	0.0709608	−	0.997479i	$-0.477393\pi$
$337$	2.60533	0.141922	0.0709608	+	0.997479i	$0.477393\pi$
$338$	−22.3889	−1.21780
$339$	0	0
$340$	−0.911510	−0.0494336
$341$	−9.85260	−0.533548
$342$	0	0
$343$	12.3359	0.666075
$344$	−19.7478	−1.06473
$345$	0	0
$346$	1.43191	0.0769798
$347$	29.8097	1.60027	0.800134	−	0.599821i	$-0.204761\pi$
$347$	29.8097	1.60027	0.800134	+	0.599821i	$0.204761\pi$
$348$	0	0
$349$	19.3635	1.03650	0.518252	−	0.855228i	$-0.326583\pi$
$349$	19.3635	1.03650	0.518252	+	0.855228i	$0.326583\pi$
$350$	−11.7605	−0.628623
$351$	0	0
$352$	−6.81924	−0.363467
$353$	36.9595	1.96716	0.983579	−	0.180479i	$-0.0577649\pi$
$353$	36.9595	1.96716	0.983579	+	0.180479i	$0.0577649\pi$
$354$	0	0
$355$	24.5874	1.30496
$356$	−49.4401	−2.62032
$357$	0	0
$358$	−22.6887	−1.19914
$359$	−22.1341	−1.16819	−0.584095	−	0.811685i	$-0.698551\pi$
$359$	−22.1341	−1.16819	−0.584095	+	0.811685i	$0.698551\pi$
$360$	0	0
$361$	−8.70605	−0.458213
$362$	−21.8971	−1.15089
$363$	0	0
$364$	−4.80421	−0.251809
$365$	−34.5232	−1.80703
$366$	0	0
$367$	−0.689150	−0.0359734	−0.0179867	−	0.999838i	$-0.505726\pi$
$367$	−0.689150	−0.0359734	−0.0179867	+	0.999838i	$0.505726\pi$
$368$	−6.79056	−0.353982
$369$	0	0
$370$	63.8490	3.31935
$371$	2.14472	0.111348
$372$	0	0
$373$	22.2275	1.15090	0.575449	−	0.817838i	$-0.304827\pi$
$373$	22.2275	1.15090	0.575449	+	0.817838i	$0.304827\pi$
$374$	0.210058	0.0108619
$375$	0	0
$376$	9.85586	0.508277
$377$	6.42995	0.331159
$378$	0	0
$379$	8.98118	0.461332	0.230666	−	0.973033i	$-0.425909\pi$
$379$	8.98118	0.461332	0.230666	+	0.973033i	$0.425909\pi$
$380$	−31.0383	−1.59223
$381$	0	0
$382$	−2.87505	−0.147100
$383$	−17.8870	−0.913984	−0.456992	−	0.889471i	$-0.651073\pi$
$383$	−17.8870	−0.913984	−0.456992	+	0.889471i	$0.651073\pi$
$384$	0	0
$385$	3.06350	0.156131
$386$	−47.5469	−2.42007
$387$	0	0
$388$	−58.3160	−2.96055
$389$	−0.798321	−0.0404765	−0.0202382	−	0.999795i	$-0.506442\pi$
$389$	−0.798321	−0.0404765	−0.0202382	+	0.999795i	$0.506442\pi$
$390$	0	0
$391$	0.572044	0.0289295
$392$	−13.2265	−0.668041
$393$	0	0
$394$	−0.238353	−0.0120080
$395$	15.3096	0.770308
$396$	0	0
$397$	21.1170	1.05983	0.529916	−	0.848050i	$-0.322223\pi$
$397$	21.1170	1.05983	0.529916	+	0.848050i	$0.322223\pi$
$398$	−8.73782	−0.437987
$399$	0	0
$400$	−6.27306	−0.313653
$401$	2.28998	0.114356	0.0571780	−	0.998364i	$-0.481790\pi$
$401$	2.28998	0.114356	0.0571780	+	0.998364i	$0.481790\pi$
$402$	0	0
$403$	−16.9435	−0.844015
$404$	1.53065	0.0761525
$405$	0	0
$406$	−7.84028	−0.389106
$407$	−8.79309	−0.435857
$408$	0	0
$409$	10.8076	0.534401	0.267201	−	0.963641i	$-0.413901\pi$
$409$	10.8076	0.534401	0.267201	+	0.963641i	$0.413901\pi$
$410$	25.0839	1.23880
$411$	0	0
$412$	15.8007	0.778447
$413$	0.637072	0.0313483
$414$	0	0
$415$	0.875638	0.0429834
$416$	−11.7270	−0.574965
$417$	0	0
$418$	7.15280	0.349855
$419$	−21.2186	−1.03660	−0.518299	−	0.855199i	$-0.673435\pi$
$419$	−21.2186	−1.03660	−0.518299	+	0.855199i	$0.673435\pi$
$420$	0	0
$421$	−28.2916	−1.37885	−0.689424	−	0.724358i	$-0.742136\pi$
$421$	−28.2916	−1.37885	−0.689424	+	0.724358i	$0.742136\pi$
$422$	−31.6822	−1.54227
$423$	0	0
$424$	−4.93180	−0.239510
$425$	0.528449	0.0256336
$426$	0	0
$427$	0.940570	0.0455174
$428$	−21.5494	−1.04163
$429$	0	0
$430$	66.2986	3.19720
$431$	−11.8456	−0.570583	−0.285292	−	0.958441i	$-0.592090\pi$
$431$	−11.8456	−0.570583	−0.285292	+	0.958441i	$0.592090\pi$
$432$	0	0
$433$	−3.52344	−0.169326	−0.0846629	−	0.996410i	$-0.526981\pi$
$433$	−3.52344	−0.169326	−0.0846629	+	0.996410i	$0.526981\pi$
$434$	20.6598	0.991703
$435$	0	0
$436$	25.8123	1.23619
$437$	19.4789	0.931804
$438$	0	0
$439$	−21.8842	−1.04448	−0.522238	−	0.852800i	$-0.674903\pi$
$439$	−21.8842	−1.04448	−0.522238	+	0.852800i	$0.674903\pi$
$440$	−7.04456	−0.335836
$441$	0	0
$442$	0.361237	0.0171823
$443$	35.7962	1.70073	0.850365	−	0.526194i	$-0.176381\pi$
$443$	35.7962	1.70073	0.850365	+	0.526194i	$0.176381\pi$
$444$	0	0
$445$	54.2160	2.57008
$446$	13.5321	0.640762
$447$	0	0
$448$	12.1952	0.576168
$449$	7.17154	0.338446	0.169223	−	0.985578i	$-0.445874\pi$
$449$	7.17154	0.338446	0.169223	+	0.985578i	$0.445874\pi$
$450$	0	0
$451$	−3.45448	−0.162665
$452$	29.6372	1.39402
$453$	0	0
$454$	4.31181	0.202363
$455$	5.26830	0.246982
$456$	0	0
$457$	−4.11540	−0.192510	−0.0962552	−	0.995357i	$-0.530686\pi$
$457$	−4.11540	−0.192510	−0.0962552	+	0.995357i	$0.530686\pi$
$458$	39.2179	1.83253
$459$	0	0
$460$	−58.7328	−2.73843
$461$	30.0145	1.39792	0.698958	−	0.715162i	$-0.253647\pi$
$461$	30.0145	1.39792	0.698958	+	0.715162i	$0.253647\pi$
$462$	0	0
$463$	−38.9007	−1.80787	−0.903935	−	0.427669i	$-0.859335\pi$
$463$	−38.9007	−1.80787	−0.903935	+	0.427669i	$0.859335\pi$
$464$	−4.18202	−0.194146
$465$	0	0
$466$	37.9922	1.75996
$467$	−36.8088	−1.70331	−0.851654	−	0.524105i	$-0.824400\pi$
$467$	−36.8088	−1.70331	−0.851654	+	0.524105i	$0.824400\pi$
$468$	0	0
$469$	8.14956	0.376312
$470$	−33.0888	−1.52627
$471$	0	0
$472$	−1.46495	−0.0674300
$473$	−9.13045	−0.419819
$474$	0	0
$475$	17.9945	0.825643
$476$	−0.263224	−0.0120648
$477$	0	0
$478$	35.9064	1.64232
$479$	35.3407	1.61476	0.807378	−	0.590034i	$-0.200886\pi$
$479$	35.3407	1.61476	0.807378	+	0.590034i	$0.200886\pi$
$480$	0	0
$481$	−15.1215	−0.689479
$482$	66.4539	3.02689
$483$	0	0
$484$	2.97016	0.135007
$485$	63.9493	2.90379
$486$	0	0
$487$	10.8666	0.492414	0.246207	−	0.969217i	$-0.420816\pi$
$487$	10.8666	0.492414	0.246207	+	0.969217i	$0.420816\pi$
$488$	−2.16285	−0.0979076
$489$	0	0
$490$	44.4050	2.00601
$491$	−23.5334	−1.06205	−0.531024	−	0.847357i	$-0.678193\pi$
$491$	−23.5334	−1.06205	−0.531024	+	0.847357i	$0.678193\pi$
$492$	0	0
$493$	0.352298	0.0158667
$494$	12.3006	0.553432
$495$	0	0
$496$	11.0200	0.494813
$497$	7.10028	0.318491
$498$	0	0
$499$	−16.8524	−0.754416	−0.377208	−	0.926129i	$-0.623116\pi$
$499$	−16.8524	−0.754416	−0.377208	+	0.926129i	$0.623116\pi$
$500$	−5.88681	−0.263266
$501$	0	0
$502$	−16.3480	−0.729647
$503$	4.27444	0.190588	0.0952940	−	0.995449i	$-0.469621\pi$
$503$	4.27444	0.190588	0.0952940	+	0.995449i	$0.469621\pi$
$504$	0	0
$505$	−1.67851	−0.0746926
$506$	13.5350	0.601706
$507$	0	0
$508$	−10.3306	−0.458344
$509$	−9.90923	−0.439219	−0.219609	−	0.975588i	$-0.570478\pi$
$509$	−9.90923	−0.439219	−0.219609	+	0.975588i	$0.570478\pi$
$510$	0	0
$511$	−9.96954	−0.441027
$512$	12.4655	0.550901
$513$	0	0
$514$	29.2753	1.29128
$515$	−17.3271	−0.763523
$516$	0	0
$517$	4.55689	0.200412
$518$	18.4382	0.810126
$519$	0	0
$520$	−12.1145	−0.531256
$521$	−41.5936	−1.82225	−0.911125	−	0.412130i	$-0.864785\pi$
$521$	−41.5936	−1.82225	−0.911125	+	0.412130i	$0.864785\pi$
$522$	0	0
$523$	−12.3627	−0.540583	−0.270291	−	0.962779i	$-0.587120\pi$
$523$	−12.3627	−0.540583	−0.270291	+	0.962779i	$0.587120\pi$
$524$	32.2474	1.40873
$525$	0	0
$526$	41.4276	1.80633
$527$	−0.928337	−0.0404390
$528$	0	0
$529$	13.8594	0.602585
$530$	16.5574	0.719207
$531$	0	0
$532$	−8.96316	−0.388602
$533$	−5.94066	−0.257318
$534$	0	0
$535$	23.6311	1.02166
$536$	−18.7400	−0.809444
$537$	0	0
$538$	22.8757	0.986242
$539$	−6.11533	−0.263406
$540$	0	0
$541$	−32.6923	−1.40555	−0.702775	−	0.711412i	$-0.748056\pi$
$541$	−32.6923	−1.40555	−0.702775	+	0.711412i	$0.748056\pi$
$542$	42.3143	1.81755
$543$	0	0
$544$	−0.642526	−0.0275481
$545$	−28.3058	−1.21249
$546$	0	0
$547$	8.17011	0.349329	0.174664	−	0.984628i	$-0.444116\pi$
$547$	8.17011	0.349329	0.174664	+	0.984628i	$0.444116\pi$
$548$	−11.2620	−0.481089
$549$	0	0
$550$	12.5035	0.533153
$551$	11.9963	0.511058
$552$	0	0
$553$	4.42106	0.188003
$554$	−13.8506	−0.588455
$555$	0	0
$556$	54.6265	2.31668
$557$	−30.4156	−1.28875	−0.644375	−	0.764709i	$-0.722883\pi$
$557$	−30.4156	−1.28875	−0.644375	+	0.764709i	$0.722883\pi$
$558$	0	0
$559$	−15.7016	−0.664107
$560$	−3.42649	−0.144796
$561$	0	0
$562$	73.4260	3.09729
$563$	−0.103567	−0.00436484	−0.00218242	−	0.999998i	$-0.500695\pi$
$563$	−0.103567	−0.00436484	−0.00218242	+	0.999998i	$0.500695\pi$
$564$	0	0
$565$	−32.5001	−1.36729
$566$	−32.9495	−1.38497
$567$	0	0
$568$	−16.3272	−0.685073
$569$	−9.77229	−0.409676	−0.204838	−	0.978796i	$-0.565667\pi$
$569$	−9.77229	−0.409676	−0.204838	+	0.978796i	$0.565667\pi$
$570$	0	0
$571$	22.9916	0.962167	0.481083	−	0.876675i	$-0.340243\pi$
$571$	22.9916	0.962167	0.481083	+	0.876675i	$0.340243\pi$
$572$	5.10777	0.213567
$573$	0	0
$574$	7.24367	0.302345
$575$	34.0504	1.42000
$576$	0	0
$577$	−22.7599	−0.947507	−0.473753	−	0.880658i	$-0.657101\pi$
$577$	−22.7599	−0.947507	−0.473753	+	0.880658i	$0.657101\pi$
$578$	−37.8797	−1.57559
$579$	0	0
$580$	−36.1711	−1.50192
$581$	0.252865	0.0104906
$582$	0	0
$583$	−2.28023	−0.0944377
$584$	22.9251	0.948646
$585$	0	0
$586$	−68.2370	−2.81885
$587$	−6.04146	−0.249358	−0.124679	−	0.992197i	$-0.539790\pi$
$587$	−6.04146	−0.249358	−0.124679	+	0.992197i	$0.539790\pi$
$588$	0	0
$589$	−31.6112	−1.30252
$590$	4.91824	0.202481
$591$	0	0
$592$	9.83497	0.404215
$593$	−16.1185	−0.661907	−0.330953	−	0.943647i	$-0.607370\pi$
$593$	−16.1185	−0.661907	−0.330953	+	0.943647i	$0.607370\pi$
$594$	0	0
$595$	0.288651	0.0118335
$596$	−29.5980	−1.21238
$597$	0	0
$598$	23.2762	0.951833
$599$	29.2142	1.19366	0.596829	−	0.802368i	$-0.296427\pi$
$599$	29.2142	1.19366	0.596829	+	0.802368i	$0.296427\pi$
$600$	0	0
$601$	15.6302	0.637571	0.318785	−	0.947827i	$-0.396725\pi$
$601$	15.6302	0.637571	0.318785	+	0.947827i	$0.396725\pi$
$602$	19.1456	0.780315
$603$	0	0
$604$	−39.5470	−1.60914
$605$	−3.25707	−0.132419
$606$	0	0
$607$	16.4060	0.665898	0.332949	−	0.942945i	$-0.391956\pi$
$607$	16.4060	0.665898	0.332949	+	0.942945i	$0.391956\pi$
$608$	−21.8790	−0.887309
$609$	0	0
$610$	7.26127	0.294000
$611$	7.83647	0.317030
$612$	0	0
$613$	20.6696	0.834839	0.417419	−	0.908714i	$-0.362935\pi$
$613$	20.6696	0.834839	0.417419	+	0.908714i	$0.362935\pi$
$614$	−57.8178	−2.33334
$615$	0	0
$616$	−2.03431	−0.0819647
$617$	−3.79507	−0.152784	−0.0763919	−	0.997078i	$-0.524340\pi$
$617$	−3.79507	−0.152784	−0.0763919	+	0.997078i	$0.524340\pi$
$618$	0	0
$619$	−20.8607	−0.838463	−0.419231	−	0.907879i	$-0.637700\pi$
$619$	−20.8607	−0.838463	−0.419231	+	0.907879i	$0.637700\pi$
$620$	95.3141	3.82791
$621$	0	0
$622$	−19.9037	−0.798066
$623$	15.6564	0.627259
$624$	0	0
$625$	−21.5871	−0.863485
$626$	18.2792	0.730583
$627$	0	0
$628$	−14.2317	−0.567908
$629$	−0.828508	−0.0330348
$630$	0	0
$631$	12.1694	0.484457	0.242229	−	0.970219i	$-0.422122\pi$
$631$	12.1694	0.484457	0.242229	+	0.970219i	$0.422122\pi$
$632$	−10.1663	−0.404393
$633$	0	0
$634$	−57.1271	−2.26881
$635$	11.3285	0.449557
$636$	0	0
$637$	−10.5165	−0.416679
$638$	8.33567	0.330012
$639$	0	0
$640$	49.7261	1.96560
$641$	−46.3210	−1.82957	−0.914784	−	0.403944i	$-0.867639\pi$
$641$	−46.3210	−1.82957	−0.914784	+	0.403944i	$0.867639\pi$
$642$	0	0
$643$	−9.76662	−0.385158	−0.192579	−	0.981281i	$-0.561685\pi$
$643$	−9.76662	−0.385158	−0.192579	+	0.981281i	$0.561685\pi$
$644$	−16.9607	−0.668346
$645$	0	0
$646$	0.673955	0.0265164
$647$	11.7424	0.461641	0.230820	−	0.972996i	$-0.425859\pi$
$647$	11.7424	0.461641	0.230820	+	0.972996i	$0.425859\pi$
$648$	0	0
$649$	−0.677326	−0.0265874
$650$	21.5023	0.843390
$651$	0	0
$652$	−35.0438	−1.37242
$653$	31.2686	1.22363	0.611817	−	0.790999i	$-0.290439\pi$
$653$	31.2686	1.22363	0.611817	+	0.790999i	$0.290439\pi$
$654$	0	0
$655$	−35.3625	−1.38173
$656$	3.86379	0.150856
$657$	0	0
$658$	−9.55530	−0.372504
$659$	34.9988	1.36336	0.681679	−	0.731651i	$-0.261250\pi$
$659$	34.9988	1.36336	0.681679	+	0.731651i	$0.261250\pi$
$660$	0	0
$661$	23.7226	0.922701	0.461350	−	0.887218i	$-0.347365\pi$
$661$	23.7226	0.922701	0.461350	+	0.887218i	$0.347365\pi$
$662$	32.4163	1.25989
$663$	0	0
$664$	−0.581465	−0.0225652
$665$	9.82900	0.381152
$666$	0	0
$667$	22.7002	0.878955
$668$	−0.406258	−0.0157186
$669$	0	0
$670$	62.9152	2.43063
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−25.3130	−0.975745	−0.487873	−	0.872915i	$-0.662227\pi$
$673$	−25.3130	−0.975745	−0.487873	+	0.872915i	$0.662227\pi$
$674$	5.80829	0.223727
$675$	0	0
$676$	−29.8282	−1.14724
$677$	−15.0519	−0.578490	−0.289245	−	0.957255i	$-0.593404\pi$
$677$	−15.0519	−0.578490	−0.289245	+	0.957255i	$0.593404\pi$
$678$	0	0
$679$	18.4671	0.708704
$680$	−0.663756	−0.0254539
$681$	0	0
$682$	−21.9652	−0.841092
$683$	17.3227	0.662834	0.331417	−	0.943484i	$-0.392473\pi$
$683$	17.3227	0.662834	0.331417	+	0.943484i	$0.392473\pi$
$684$	0	0
$685$	12.3499	0.471866
$686$	27.5014	1.05001
$687$	0	0
$688$	10.2123	0.389340
$689$	−3.92131	−0.149390
$690$	0	0
$691$	22.5667	0.858478	0.429239	−	0.903191i	$-0.358782\pi$
$691$	22.5667	0.858478	0.429239	+	0.903191i	$0.358782\pi$
$692$	1.90770	0.0725197
$693$	0	0
$694$	66.4573	2.52268
$695$	−59.9034	−2.27227
$696$	0	0
$697$	−0.325490	−0.0123288
$698$	43.1687	1.63396
$699$	0	0
$700$	−15.6682	−0.592201
$701$	−17.2289	−0.650728	−0.325364	−	0.945589i	$-0.605487\pi$
$701$	−17.2289	−0.650728	−0.325364	+	0.945589i	$0.605487\pi$
$702$	0	0
$703$	−28.2119	−1.06403
$704$	−12.9657	−0.488664
$705$	0	0
$706$	82.3970	3.10105
$707$	−0.484715	−0.0182296
$708$	0	0
$709$	22.9553	0.862106	0.431053	−	0.902327i	$-0.358142\pi$
$709$	22.9553	0.862106	0.431053	+	0.902327i	$0.358142\pi$
$710$	54.8147	2.05716
$711$	0	0
$712$	−36.0020	−1.34923
$713$	−59.8171	−2.24017
$714$	0	0
$715$	−5.60118	−0.209472
$716$	−30.2277	−1.12966
$717$	0	0
$718$	−49.3453	−1.84155
$719$	21.2501	0.792495	0.396247	−	0.918144i	$-0.370312\pi$
$719$	21.2501	0.792495	0.396247	+	0.918144i	$0.370312\pi$
$720$	0	0
$721$	−5.00368	−0.186347
$722$	−19.4091	−0.722333
$723$	0	0
$724$	−29.1730	−1.08421
$725$	20.9702	0.778815
$726$	0	0
$727$	21.8267	0.809509	0.404754	−	0.914425i	$-0.367357\pi$
$727$	21.8267	0.809509	0.404754	+	0.914425i	$0.367357\pi$
$728$	−3.49840	−0.129659
$729$	0	0
$730$	−76.9656	−2.84862
$731$	−0.860295	−0.0318192
$732$	0	0
$733$	35.6165	1.31553	0.657763	−	0.753225i	$-0.271503\pi$
$733$	35.6165	1.31553	0.657763	+	0.753225i	$0.271503\pi$
$734$	−1.53638	−0.0567089
$735$	0	0
$736$	−41.4009	−1.52606
$737$	−8.66449	−0.319161
$738$	0	0
$739$	−43.1364	−1.58680	−0.793400	−	0.608701i	$-0.791691\pi$
$739$	−43.1364	−1.58680	−0.793400	+	0.608701i	$0.791691\pi$
$740$	85.0645	3.12703
$741$	0	0
$742$	4.78140	0.175531
$743$	0.287472	0.0105463	0.00527316	−	0.999986i	$-0.498321\pi$
$743$	0.287472	0.0105463	0.00527316	+	0.999986i	$0.498321\pi$
$744$	0	0
$745$	32.4572	1.18914
$746$	49.5537	1.81429
$747$	0	0
$748$	0.279856	0.0102325
$749$	6.82413	0.249348
$750$	0	0
$751$	21.0685	0.768799	0.384399	−	0.923167i	$-0.374409\pi$
$751$	21.0685	0.768799	0.384399	+	0.923167i	$0.374409\pi$
$752$	−5.09682	−0.185862
$753$	0	0
$754$	14.3348	0.522044
$755$	43.3672	1.57830
$756$	0	0
$757$	−21.5609	−0.783646	−0.391823	−	0.920041i	$-0.628155\pi$
$757$	−21.5609	−0.783646	−0.391823	+	0.920041i	$0.628155\pi$
$758$	20.0225	0.727250
$759$	0	0
$760$	−22.6019	−0.819856
$761$	12.7499	0.462185	0.231093	−	0.972932i	$-0.425770\pi$
$761$	12.7499	0.462185	0.231093	+	0.972932i	$0.425770\pi$
$762$	0	0
$763$	−8.17408	−0.295922
$764$	−3.83035	−0.138577
$765$	0	0
$766$	−39.8771	−1.44082
$767$	−1.16480	−0.0420583
$768$	0	0
$769$	37.8909	1.36638	0.683190	−	0.730241i	$-0.260592\pi$
$769$	37.8909	1.36638	0.683190	+	0.730241i	$0.260592\pi$
$770$	6.82973	0.246126
$771$	0	0
$772$	−63.3456	−2.27986
$773$	17.1918	0.618345	0.309172	−	0.951006i	$-0.399948\pi$
$773$	17.1918	0.618345	0.309172	+	0.951006i	$0.399948\pi$
$774$	0	0
$775$	−55.2585	−1.98494
$776$	−42.4654	−1.52442
$777$	0	0
$778$	−1.77976	−0.0638077
$779$	−11.0834	−0.397104
$780$	0	0
$781$	−7.54892	−0.270122
$782$	1.27531	0.0456049
$783$	0	0
$784$	6.83992	0.244283
$785$	15.6065	0.557020
$786$	0	0
$787$	34.9880	1.24719	0.623593	−	0.781749i	$-0.285672\pi$
$787$	34.9880	1.24719	0.623593	+	0.781749i	$0.285672\pi$
$788$	−0.317551	−0.0113123
$789$	0	0
$790$	34.1309	1.21432
$791$	−9.38531	−0.333703
$792$	0	0
$793$	−1.71970	−0.0610682
$794$	47.0779	1.67073
$795$	0	0
$796$	−11.6412	−0.412611
$797$	49.3845	1.74929	0.874644	−	0.484766i	$-0.161095\pi$
$797$	49.3845	1.74929	0.874644	+	0.484766i	$0.161095\pi$
$798$	0	0
$799$	0.429362	0.0151897
$800$	−38.2458	−1.35219
$801$	0	0
$802$	5.10524	0.180272
$803$	10.5995	0.374047
$804$	0	0
$805$	18.5991	0.655533
$806$	−37.7735	−1.33052
$807$	0	0
$808$	1.11461	0.0392118
$809$	28.9188	1.01673	0.508366	−	0.861141i	$-0.330250\pi$
$809$	28.9188	1.01673	0.508366	+	0.861141i	$0.330250\pi$
$810$	0	0
$811$	30.1416	1.05841	0.529207	−	0.848493i	$-0.322490\pi$
$811$	30.1416	1.05841	0.529207	+	0.848493i	$0.322490\pi$
$812$	−10.4454	−0.366562
$813$	0	0
$814$	−19.6032	−0.687091
$815$	38.4290	1.34611
$816$	0	0
$817$	−29.2943	−1.02488
$818$	24.0943	0.842437
$819$	0	0
$820$	33.4187	1.16703
$821$	25.6829	0.896340	0.448170	−	0.893948i	$-0.352076\pi$
$821$	25.6829	0.896340	0.448170	+	0.893948i	$0.352076\pi$
$822$	0	0
$823$	−17.9881	−0.627026	−0.313513	−	0.949584i	$-0.601506\pi$
$823$	−17.9881	−0.627026	−0.313513	+	0.949584i	$0.601506\pi$
$824$	11.5060	0.400831
$825$	0	0
$826$	1.42028	0.0494178
$827$	−3.49812	−0.121642	−0.0608208	−	0.998149i	$-0.519372\pi$
$827$	−3.49812	−0.121642	−0.0608208	+	0.998149i	$0.519372\pi$
$828$	0	0
$829$	37.1722	1.29105	0.645523	−	0.763741i	$-0.276640\pi$
$829$	37.1722	1.29105	0.645523	+	0.763741i	$0.276640\pi$
$830$	1.95213	0.0677595
$831$	0	0
$832$	−22.2971	−0.773014
$833$	−0.576202	−0.0199642
$834$	0	0
$835$	0.445502	0.0154172
$836$	9.52950	0.329585
$837$	0	0
$838$	−47.3045	−1.63411
$839$	−51.7898	−1.78798	−0.893991	−	0.448084i	$-0.852107\pi$
$839$	−51.7898	−1.78798	−0.893991	+	0.448084i	$0.852107\pi$
$840$	0	0
$841$	−15.0199	−0.517927
$842$	−63.0728	−2.17363
$843$	0	0
$844$	−42.2095	−1.45291
$845$	32.7096	1.12524
$846$	0	0
$847$	−0.940570	−0.0323184
$848$	2.55041	0.0875816
$849$	0	0
$850$	1.17812	0.0404091
$851$	−53.3846	−1.83000
$852$	0	0
$853$	−10.9091	−0.373521	−0.186761	−	0.982405i	$-0.559799\pi$
$853$	−10.9091	−0.373521	−0.186761	+	0.982405i	$0.559799\pi$
$854$	2.09689	0.0717542
$855$	0	0
$856$	−15.6922	−0.536347
$857$	−21.6232	−0.738634	−0.369317	−	0.929303i	$-0.620408\pi$
$857$	−21.6232	−0.738634	−0.369317	+	0.929303i	$0.620408\pi$
$858$	0	0
$859$	−3.86862	−0.131996	−0.0659978	−	0.997820i	$-0.521023\pi$
$859$	−3.86862	−0.131996	−0.0659978	+	0.997820i	$0.521023\pi$
$860$	88.3281	3.01196
$861$	0	0
$862$	−26.4084	−0.899475
$863$	−42.3462	−1.44148	−0.720740	−	0.693205i	$-0.756198\pi$
$863$	−42.3462	−1.44148	−0.720740	+	0.693205i	$0.756198\pi$
$864$	0	0
$865$	−2.09198	−0.0711294
$866$	−7.85511	−0.266927
$867$	0	0
$868$	27.5246	0.934245
$869$	−4.70041	−0.159450
$870$	0	0
$871$	−14.9003	−0.504878
$872$	18.7964	0.636526
$873$	0	0
$874$	43.4261	1.46891
$875$	1.86420	0.0630214
$876$	0	0
$877$	−3.28466	−0.110915	−0.0554575	−	0.998461i	$-0.517662\pi$
$877$	−3.28466	−0.110915	−0.0554575	+	0.998461i	$0.517662\pi$
$878$	−48.7883	−1.64652
$879$	0	0
$880$	3.64299	0.122805
$881$	30.9433	1.04251	0.521253	−	0.853402i	$-0.325465\pi$
$881$	30.9433	1.04251	0.521253	+	0.853402i	$0.325465\pi$
$882$	0	0
$883$	−25.3509	−0.853124	−0.426562	−	0.904458i	$-0.640275\pi$
$883$	−25.3509	−0.853124	−0.426562	+	0.904458i	$0.640275\pi$
$884$	0.481267	0.0161868
$885$	0	0
$886$	79.8035	2.68105
$887$	−24.6058	−0.826180	−0.413090	−	0.910690i	$-0.635551\pi$
$887$	−24.6058	−0.826180	−0.413090	+	0.910690i	$0.635551\pi$
$888$	0	0
$889$	3.27141	0.109720
$890$	120.868	4.05151
$891$	0	0
$892$	18.0285	0.603637
$893$	14.6204	0.489253
$894$	0	0
$895$	33.1476	1.10800
$896$	14.3598	0.479727
$897$	0	0
$898$	15.9881	0.533531
$899$	−36.8388	−1.22864
$900$	0	0
$901$	−0.214850	−0.00715768
$902$	−7.70136	−0.256427
$903$	0	0
$904$	21.5816	0.717794
$905$	31.9911	1.06342
$906$	0	0
$907$	24.0589	0.798863	0.399432	−	0.916763i	$-0.369208\pi$
$907$	24.0589	0.798863	0.399432	+	0.916763i	$0.369208\pi$
$908$	5.74453	0.190639
$909$	0	0
$910$	11.7451	0.389345
$911$	1.51563	0.0502151	0.0251076	−	0.999685i	$-0.492007\pi$
$911$	1.51563	0.0502151	0.0251076	+	0.999685i	$0.492007\pi$
$912$	0	0
$913$	−0.268842	−0.00889737
$914$	−9.17481	−0.303476
$915$	0	0
$916$	52.2491	1.72636
$917$	−10.2119	−0.337227
$918$	0	0
$919$	4.20636	0.138755	0.0693775	−	0.997590i	$-0.477899\pi$
$919$	4.20636	0.138755	0.0693775	+	0.997590i	$0.477899\pi$
$920$	−42.7689	−1.41005
$921$	0	0
$922$	66.9140	2.20369
$923$	−12.9819	−0.427303
$924$	0	0
$925$	−49.3162	−1.62151
$926$	−86.7247	−2.84995
$927$	0	0
$928$	−25.4971	−0.836984
$929$	−44.3375	−1.45466	−0.727332	−	0.686285i	$-0.759240\pi$
$929$	−44.3375	−1.45466	−0.727332	+	0.686285i	$0.759240\pi$
$930$	0	0
$931$	−19.6205	−0.643037
$932$	50.6161	1.65799
$933$	0	0
$934$	−82.0609	−2.68512
$935$	−0.306890	−0.0100364
$936$	0	0
$937$	−27.5401	−0.899696	−0.449848	−	0.893105i	$-0.648522\pi$
$937$	−27.5401	−0.899696	−0.449848	+	0.893105i	$0.648522\pi$
$938$	18.1685	0.593223
$939$	0	0
$940$	−44.0834	−1.43784
$941$	−3.94839	−0.128714	−0.0643569	−	0.997927i	$-0.520500\pi$
$941$	−3.94839	−0.128714	−0.0643569	+	0.997927i	$0.520500\pi$
$942$	0	0
$943$	−20.9728	−0.682969
$944$	0.757581	0.0246572
$945$	0	0
$946$	−20.3553	−0.661807
$947$	−7.81142	−0.253837	−0.126919	−	0.991913i	$-0.540509\pi$
$947$	−7.81142	−0.253837	−0.126919	+	0.991913i	$0.540509\pi$
$948$	0	0
$949$	18.2279	0.591702
$950$	40.1166	1.30155
$951$	0	0
$952$	−0.191678	−0.00621232
$953$	−32.0753	−1.03902	−0.519510	−	0.854464i	$-0.673885\pi$
$953$	−32.0753	−1.03902	−0.519510	+	0.854464i	$0.673885\pi$
$954$	0	0
$955$	4.20037	0.135921
$956$	47.8372	1.54717
$957$	0	0
$958$	78.7880	2.54552
$959$	3.56638	0.115164
$960$	0	0
$961$	66.0736	2.13141
$962$	−33.7115	−1.08690
$963$	0	0
$964$	88.5349	2.85152
$965$	69.4648	2.23615
$966$	0	0
$967$	20.8629	0.670904	0.335452	−	0.942057i	$-0.391111\pi$
$967$	20.8629	0.670904	0.335452	+	0.942057i	$0.391111\pi$
$968$	2.16285	0.0695166
$969$	0	0
$970$	142.568	4.57757
$971$	−10.7941	−0.346399	−0.173199	−	0.984887i	$-0.555410\pi$
$971$	−10.7941	−0.346399	−0.173199	+	0.984887i	$0.555410\pi$
$972$	0	0
$973$	−17.2988	−0.554573
$974$	24.2259	0.776248
$975$	0	0
$976$	1.11849	0.0358019
$977$	−5.64466	−0.180589	−0.0902944	−	0.995915i	$-0.528781\pi$
$977$	−5.64466	−0.180589	−0.0902944	+	0.995915i	$0.528781\pi$
$978$	0	0
$979$	−16.6456	−0.531996
$980$	59.1597	1.88979
$981$	0	0
$982$	−52.4651	−1.67423
$983$	−28.2828	−0.902083	−0.451041	−	0.892503i	$-0.648947\pi$
$983$	−28.2828	−0.902083	−0.451041	+	0.892503i	$0.648947\pi$
$984$	0	0
$985$	0.348227	0.0110954
$986$	0.785408	0.0250125
$987$	0	0
$988$	16.3879	0.521367
$989$	−55.4328	−1.76266
$990$	0	0
$991$	−25.3384	−0.804901	−0.402450	−	0.915442i	$-0.631841\pi$
$991$	−25.3384	−0.804901	−0.402450	+	0.915442i	$0.631841\pi$
$992$	67.1872	2.13320
$993$	0	0
$994$	15.8293	0.502074
$995$	12.7657	0.404701
$996$	0	0
$997$	−12.0555	−0.381803	−0.190901	−	0.981609i	$-0.561141\pi$
$997$	−12.0555	−0.381803	−0.190901	+	0.981609i	$0.561141\pi$
$998$	−37.5704	−1.18927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.e.1.11		12
3.2	odd	2		2013.2.a.d.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.2	✓	12	3.2	odd	2
6039.2.a.e.1.11		12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.22938 q^{2} +2.97016 q^{4} -3.25707 q^{5} -0.940570 q^{7} +2.16285 q^{8} +O(q^{10})\)	\(q+2.22938 q^{2} +2.97016 q^{4} -3.25707 q^{5} -0.940570 q^{7} +2.16285 q^{8} -7.26127 q^{10} +1.00000 q^{11} +1.71970 q^{13} -2.09689 q^{14} -1.11849 q^{16} +0.0942226 q^{17} +3.20842 q^{19} -9.67401 q^{20} +2.22938 q^{22} +6.07120 q^{23} +5.60852 q^{25} +3.83387 q^{26} -2.79364 q^{28} +3.73900 q^{29} -9.85260 q^{31} -6.81924 q^{32} +0.210058 q^{34} +3.06350 q^{35} -8.79309 q^{37} +7.15280 q^{38} -7.04456 q^{40} -3.45448 q^{41} -9.13045 q^{43} +2.97016 q^{44} +13.5350 q^{46} +4.55689 q^{47} -6.11533 q^{49} +12.5035 q^{50} +5.10777 q^{52} -2.28023 q^{53} -3.25707 q^{55} -2.03431 q^{56} +8.33567 q^{58} -0.677326 q^{59} -1.00000 q^{61} -21.9652 q^{62} -12.9657 q^{64} -5.60118 q^{65} -8.66449 q^{67} +0.279856 q^{68} +6.82973 q^{70} -7.54892 q^{71} +10.5995 q^{73} -19.6032 q^{74} +9.52950 q^{76} -0.940570 q^{77} -4.70041 q^{79} +3.64299 q^{80} -7.70136 q^{82} -0.268842 q^{83} -0.306890 q^{85} -20.3553 q^{86} +2.16285 q^{88} -16.6456 q^{89} -1.61750 q^{91} +18.0324 q^{92} +10.1591 q^{94} -10.4500 q^{95} -19.6340 q^{97} -13.6334 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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