LMFDB - Embedded newform 6867.2.a.r.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6867,2,Mod(1,6867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6867, 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("6867.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6867 = 3^{2} \cdot 7 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6867.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:	$54.8332710680$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 7 x^{16} - 4 x^{15} + 127 x^{14} - 159 x^{13} - 833 x^{12} + 1766 x^{11} + 2316 x^{10} + \cdots + 8 $ x^17 - 7x^16 - 4x^15 + 127x^14 - 159x^13 - 833x^12 + 1766x^11 + 2316x^10 - 7456x^9 - 1858x^8 + 14879x^7 - 2693x^6 - 13874x^5 + 4498x^4 + 5607x^3 - 1594x^2 - 708x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 763)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.60194$ of defining polynomial
Character	$\chi$	$=$	6867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60194 q^{2} +0.566208 q^{4} +1.99604 q^{5} +1.00000 q^{7} +2.29685 q^{8} +O(q^{10})$	\(q-1.60194 q^{2} +0.566208 q^{4} +1.99604 q^{5} +1.00000 q^{7} +2.29685 q^{8} -3.19754 q^{10} -6.12170 q^{11} +1.39341 q^{13} -1.60194 q^{14} -4.81182 q^{16} +1.07390 q^{17} +0.472899 q^{19} +1.13018 q^{20} +9.80658 q^{22} -9.11283 q^{23} -1.01581 q^{25} -2.23216 q^{26} +0.566208 q^{28} +1.75159 q^{29} +6.15626 q^{31} +3.11455 q^{32} -1.72033 q^{34} +1.99604 q^{35} +9.41647 q^{37} -0.757555 q^{38} +4.58461 q^{40} -9.16143 q^{41} +6.65304 q^{43} -3.46615 q^{44} +14.5982 q^{46} +12.6169 q^{47} +1.00000 q^{49} +1.62727 q^{50} +0.788961 q^{52} -5.38247 q^{53} -12.2192 q^{55} +2.29685 q^{56} -2.80594 q^{58} +12.8654 q^{59} -4.35578 q^{61} -9.86195 q^{62} +4.63432 q^{64} +2.78131 q^{65} -4.31999 q^{67} +0.608054 q^{68} -3.19754 q^{70} -4.50243 q^{71} -12.0212 q^{73} -15.0846 q^{74} +0.267759 q^{76} -6.12170 q^{77} -17.3092 q^{79} -9.60461 q^{80} +14.6761 q^{82} -10.2123 q^{83} +2.14356 q^{85} -10.6578 q^{86} -14.0606 q^{88} -3.26884 q^{89} +1.39341 q^{91} -5.15976 q^{92} -20.2116 q^{94} +0.943927 q^{95} +17.3289 q^{97} -1.60194 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 7 q^{2} + 23 q^{4} - 12 q^{5} + 17 q^{7} - 18 q^{8}+O(q^{10}) $ 17 * q - 7 * q^2 + 23 * q^4 - 12 * q^5 + 17 * q^7 - 18 * q^8	$ 17 q - 7 q^{2} + 23 q^{4} - 12 q^{5} + 17 q^{7} - 18 q^{8} - 6 q^{10} - 20 q^{11} + 11 q^{13} - 7 q^{14} + 23 q^{16} - 19 q^{17} - 4 q^{19} - 10 q^{20} + 3 q^{22} - 22 q^{23} + 21 q^{25} + 12 q^{26} + 23 q^{28} - 37 q^{29} - 10 q^{31} - 18 q^{32} - 7 q^{34} - 12 q^{35} + 13 q^{37} - 18 q^{38} - 17 q^{40} + 6 q^{41} + 9 q^{43} - 35 q^{44} - 17 q^{46} + 4 q^{47} + 17 q^{49} - 37 q^{50} + 36 q^{52} - 45 q^{53} - 13 q^{55} - 18 q^{56} + 4 q^{58} + 12 q^{59} + 6 q^{61} + 24 q^{62} + 20 q^{64} - 15 q^{65} - 3 q^{67} - 60 q^{68} - 6 q^{70} - 36 q^{71} - 4 q^{73} - 13 q^{74} - 74 q^{76} - 20 q^{77} + 5 q^{79} + 25 q^{80} - 24 q^{82} - 7 q^{83} + 8 q^{85} - 2 q^{86} - 55 q^{88} - 13 q^{89} + 11 q^{91} - 36 q^{92} - 37 q^{94} + 34 q^{95} - 23 q^{97} - 7 q^{98}+O(q^{100}) $ 17 * q - 7 * q^2 + 23 * q^4 - 12 * q^5 + 17 * q^7 - 18 * q^8 - 6 * q^10 - 20 * q^11 + 11 * q^13 - 7 * q^14 + 23 * q^16 - 19 * q^17 - 4 * q^19 - 10 * q^20 + 3 * q^22 - 22 * q^23 + 21 * q^25 + 12 * q^26 + 23 * q^28 - 37 * q^29 - 10 * q^31 - 18 * q^32 - 7 * q^34 - 12 * q^35 + 13 * q^37 - 18 * q^38 - 17 * q^40 + 6 * q^41 + 9 * q^43 - 35 * q^44 - 17 * q^46 + 4 * q^47 + 17 * q^49 - 37 * q^50 + 36 * q^52 - 45 * q^53 - 13 * q^55 - 18 * q^56 + 4 * q^58 + 12 * q^59 + 6 * q^61 + 24 * q^62 + 20 * q^64 - 15 * q^65 - 3 * q^67 - 60 * q^68 - 6 * q^70 - 36 * q^71 - 4 * q^73 - 13 * q^74 - 74 * q^76 - 20 * q^77 + 5 * q^79 + 25 * q^80 - 24 * q^82 - 7 * q^83 + 8 * q^85 - 2 * q^86 - 55 * q^88 - 13 * q^89 + 11 * q^91 - 36 * q^92 - 37 * q^94 + 34 * q^95 - 23 * q^97 - 7 * 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 $

$2$	−1.60194	−1.13274	−0.566371	−	0.824150i	$-0.691653\pi$
$2$	−1.60194	−1.13274	−0.566371	+	0.824150i	$0.691653\pi$
$3$	0	0
$3$	0	0
$4$	0.566208	0.283104
$5$	1.99604	0.892658	0.446329	−	0.894869i	$-0.352731\pi$
$5$	1.99604	0.892658	0.446329	+	0.894869i	$0.352731\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.29685	0.812058
$9$	0	0
$10$	−3.19754	−1.01115
$11$	−6.12170	−1.84576	−0.922880	−	0.385087i	$-0.874171\pi$
$11$	−6.12170	−1.84576	−0.922880	+	0.385087i	$0.874171\pi$
$12$	0	0
$13$	1.39341	0.386463	0.193232	−	0.981153i	$-0.438103\pi$
$13$	1.39341	0.386463	0.193232	+	0.981153i	$0.438103\pi$
$14$	−1.60194	−0.428136
$15$	0	0
$16$	−4.81182	−1.20296
$17$	1.07390	0.260460	0.130230	−	0.991484i	$-0.458428\pi$
$17$	1.07390	0.260460	0.130230	+	0.991484i	$0.458428\pi$
$18$	0	0
$19$	0.472899	0.108490	0.0542452	−	0.998528i	$-0.482725\pi$
$19$	0.472899	0.108490	0.0542452	+	0.998528i	$0.482725\pi$
$20$	1.13018	0.252715
$21$	0	0
$22$	9.80658	2.09077
$23$	−9.11283	−1.90016	−0.950078	−	0.312011i	$-0.898997\pi$
$23$	−9.11283	−1.90016	−0.950078	+	0.312011i	$0.898997\pi$
$24$	0	0
$25$	−1.01581	−0.203162
$26$	−2.23216	−0.437763
$27$	0	0
$28$	0.566208	0.107003
$29$	1.75159	0.325262	0.162631	−	0.986687i	$-0.448002\pi$
$29$	1.75159	0.325262	0.162631	+	0.986687i	$0.448002\pi$
$30$	0	0
$31$	6.15626	1.10570	0.552849	−	0.833282i	$-0.313541\pi$
$31$	6.15626	1.10570	0.552849	+	0.833282i	$0.313541\pi$
$32$	3.11455	0.550581
$33$	0	0
$34$	−1.72033	−0.295034
$35$	1.99604	0.337393
$36$	0	0
$37$	9.41647	1.54806	0.774029	−	0.633150i	$-0.218238\pi$
$37$	9.41647	1.54806	0.774029	+	0.633150i	$0.218238\pi$
$38$	−0.757555	−0.122892
$39$	0	0
$40$	4.58461	0.724890
$41$	−9.16143	−1.43078	−0.715388	−	0.698728i	$-0.753750\pi$
$41$	−9.16143	−1.43078	−0.715388	+	0.698728i	$0.753750\pi$
$42$	0	0
$43$	6.65304	1.01458	0.507290	−	0.861776i	$-0.330647\pi$
$43$	6.65304	1.01458	0.507290	+	0.861776i	$0.330647\pi$
$44$	−3.46615	−0.522542
$45$	0	0
$46$	14.5982	2.15239
$47$	12.6169	1.84037	0.920186	−	0.391482i	$-0.128038\pi$
$47$	12.6169	1.84037	0.920186	+	0.391482i	$0.128038\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.62727	0.230130
$51$	0	0
$52$	0.788961	0.109409
$53$	−5.38247	−0.739339	−0.369670	−	0.929163i	$-0.620529\pi$
$53$	−5.38247	−0.739339	−0.369670	+	0.929163i	$0.620529\pi$
$54$	0	0
$55$	−12.2192	−1.64763
$56$	2.29685	0.306929
$57$	0	0
$58$	−2.80594	−0.368437
$59$	12.8654	1.67494	0.837469	−	0.546485i	$-0.184035\pi$
$59$	12.8654	1.67494	0.837469	+	0.546485i	$0.184035\pi$
$60$	0	0
$61$	−4.35578	−0.557700	−0.278850	−	0.960335i	$-0.589953\pi$
$61$	−4.35578	−0.557700	−0.278850	+	0.960335i	$0.589953\pi$
$62$	−9.86195	−1.25247
$63$	0	0
$64$	4.63432	0.579290
$65$	2.78131	0.344979
$66$	0	0
$67$	−4.31999	−0.527771	−0.263885	−	0.964554i	$-0.585004\pi$
$67$	−4.31999	−0.527771	−0.263885	+	0.964554i	$0.585004\pi$
$68$	0.608054	0.0737373
$69$	0	0
$70$	−3.19754	−0.382179
$71$	−4.50243	−0.534340	−0.267170	−	0.963649i	$-0.586089\pi$
$71$	−4.50243	−0.534340	−0.267170	+	0.963649i	$0.586089\pi$
$72$	0	0
$73$	−12.0212	−1.40697	−0.703486	−	0.710709i	$-0.748374\pi$
$73$	−12.0212	−1.40697	−0.703486	+	0.710709i	$0.748374\pi$
$74$	−15.0846	−1.75355
$75$	0	0
$76$	0.267759	0.0307141
$77$	−6.12170	−0.697632
$78$	0	0
$79$	−17.3092	−1.94744	−0.973719	−	0.227751i	$-0.926863\pi$
$79$	−17.3092	−1.94744	−0.973719	+	0.227751i	$0.926863\pi$
$80$	−9.60461	−1.07383
$81$	0	0
$82$	14.6761	1.62070
$83$	−10.2123	−1.12094	−0.560472	−	0.828173i	$-0.689380\pi$
$83$	−10.2123	−1.12094	−0.560472	+	0.828173i	$0.689380\pi$
$84$	0	0
$85$	2.14356	0.232502
$86$	−10.6578	−1.14926
$87$	0	0
$88$	−14.0606	−1.49886
$89$	−3.26884	−0.346496	−0.173248	−	0.984878i	$-0.555426\pi$
$89$	−3.26884	−0.346496	−0.173248	+	0.984878i	$0.555426\pi$
$90$	0	0
$91$	1.39341	0.146069
$92$	−5.15976	−0.537942
$93$	0	0
$94$	−20.2116	−2.08467
$95$	0.943927	0.0968449
$96$	0	0
$97$	17.3289	1.75948	0.879739	−	0.475456i	$-0.157717\pi$
$97$	17.3289	1.75948	0.879739	+	0.475456i	$0.157717\pi$
$98$	−1.60194	−0.161820
$99$	0	0
$100$	−0.575160	−0.0575160
$101$	−10.7307	−1.06774	−0.533872	−	0.845565i	$-0.679264\pi$
$101$	−10.7307	−1.06774	−0.533872	+	0.845565i	$0.679264\pi$
$102$	0	0
$103$	9.19120	0.905636	0.452818	−	0.891603i	$-0.350419\pi$
$103$	9.19120	0.905636	0.452818	+	0.891603i	$0.350419\pi$
$104$	3.20046	0.313830
$105$	0	0
$106$	8.62239	0.837480
$107$	19.6421	1.89887	0.949436	−	0.313961i	$-0.101656\pi$
$107$	19.6421	1.89887	0.949436	+	0.313961i	$0.101656\pi$
$108$	0	0
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	19.5744	1.86634
$111$	0	0
$112$	−4.81182	−0.454675
$113$	−11.7018	−1.10082	−0.550408	−	0.834896i	$-0.685528\pi$
$113$	−11.7018	−1.10082	−0.550408	+	0.834896i	$0.685528\pi$
$114$	0	0
$115$	−18.1896	−1.69619
$116$	0.991763	0.0920829
$117$	0	0
$118$	−20.6096	−1.89727
$119$	1.07390	0.0984447
$120$	0	0
$121$	26.4752	2.40683
$122$	6.97769	0.631730
$123$	0	0
$124$	3.48572	0.313027
$125$	−12.0078	−1.07401
$126$	0	0
$127$	2.62433	0.232872	0.116436	−	0.993198i	$-0.462853\pi$
$127$	2.62433	0.232872	0.116436	+	0.993198i	$0.462853\pi$
$128$	−13.6530	−1.20677
$129$	0	0
$130$	−4.45549	−0.390773
$131$	−13.6672	−1.19411	−0.597054	−	0.802201i	$-0.703662\pi$
$131$	−13.6672	−1.19411	−0.597054	+	0.802201i	$0.703662\pi$
$132$	0	0
$133$	0.472899	0.0410055
$134$	6.92036	0.597828
$135$	0	0
$136$	2.46660	0.211509
$137$	−8.69219	−0.742624	−0.371312	−	0.928508i	$-0.621092\pi$
$137$	−8.69219	−0.742624	−0.371312	+	0.928508i	$0.621092\pi$
$138$	0	0
$139$	3.80216	0.322495	0.161247	−	0.986914i	$-0.448448\pi$
$139$	3.80216	0.322495	0.161247	+	0.986914i	$0.448448\pi$
$140$	1.13018	0.0955173
$141$	0	0
$142$	7.21262	0.605269
$143$	−8.53005	−0.713318
$144$	0	0
$145$	3.49625	0.290347
$146$	19.2572	1.59374
$147$	0	0
$148$	5.33168	0.438261
$149$	−3.10677	−0.254516	−0.127258	−	0.991870i	$-0.540618\pi$
$149$	−3.10677	−0.254516	−0.127258	+	0.991870i	$0.540618\pi$
$150$	0	0
$151$	−18.3238	−1.49117	−0.745585	−	0.666411i	$-0.767830\pi$
$151$	−18.3238	−1.49117	−0.745585	+	0.666411i	$0.767830\pi$
$152$	1.08618	0.0881005
$153$	0	0
$154$	9.80658	0.790237
$155$	12.2882	0.987009
$156$	0	0
$157$	−1.36993	−0.109332	−0.0546662	−	0.998505i	$-0.517409\pi$
$157$	−1.36993	−0.109332	−0.0546662	+	0.998505i	$0.517409\pi$
$158$	27.7283	2.20595
$159$	0	0
$160$	6.21679	0.491480
$161$	−9.11283	−0.718192
$162$	0	0
$163$	19.8865	1.55763	0.778814	−	0.627255i	$-0.215822\pi$
$163$	19.8865	1.55763	0.778814	+	0.627255i	$0.215822\pi$
$164$	−5.18728	−0.405058
$165$	0	0
$166$	16.3595	1.26974
$167$	7.40554	0.573058	0.286529	−	0.958072i	$-0.407499\pi$
$167$	7.40554	0.573058	0.286529	+	0.958072i	$0.407499\pi$
$168$	0	0
$169$	−11.0584	−0.850646
$170$	−3.43385	−0.263365
$171$	0	0
$172$	3.76701	0.287232
$173$	−22.1960	−1.68753	−0.843764	−	0.536715i	$-0.819665\pi$
$173$	−22.1960	−1.68753	−0.843764	+	0.536715i	$0.819665\pi$
$174$	0	0
$175$	−1.01581	−0.0767880
$176$	29.4565	2.22037
$177$	0	0
$178$	5.23648	0.392491
$179$	−3.67198	−0.274456	−0.137228	−	0.990539i	$-0.543819\pi$
$179$	−3.67198	−0.274456	−0.137228	+	0.990539i	$0.543819\pi$
$180$	0	0
$181$	−21.2088	−1.57644	−0.788221	−	0.615393i	$-0.788997\pi$
$181$	−21.2088	−1.57644	−0.788221	+	0.615393i	$0.788997\pi$
$182$	−2.23216	−0.165459
$183$	0	0
$184$	−20.9308	−1.54304
$185$	18.7957	1.38189
$186$	0	0
$187$	−6.57412	−0.480747
$188$	7.14382	0.521017
$189$	0	0
$190$	−1.51211	−0.109700
$191$	9.05300	0.655052	0.327526	−	0.944842i	$-0.393785\pi$
$191$	9.05300	0.655052	0.327526	+	0.944842i	$0.393785\pi$
$192$	0	0
$193$	−17.3437	−1.24842	−0.624212	−	0.781255i	$-0.714580\pi$
$193$	−17.3437	−1.24842	−0.624212	+	0.781255i	$0.714580\pi$
$194$	−27.7598	−1.99304
$195$	0	0
$196$	0.566208	0.0404434
$197$	−8.27588	−0.589632	−0.294816	−	0.955554i	$-0.595258\pi$
$197$	−8.27588	−0.589632	−0.294816	+	0.955554i	$0.595258\pi$
$198$	0	0
$199$	−5.00511	−0.354803	−0.177401	−	0.984139i	$-0.556769\pi$
$199$	−5.00511	−0.354803	−0.177401	+	0.984139i	$0.556769\pi$
$200$	−2.33316	−0.164979
$201$	0	0
$202$	17.1899	1.20948
$203$	1.75159	0.122937
$204$	0	0
$205$	−18.2866	−1.27719
$206$	−14.7237	−1.02585
$207$	0	0
$208$	−6.70486	−0.464898
$209$	−2.89494	−0.200247
$210$	0	0
$211$	8.91402	0.613666	0.306833	−	0.951763i	$-0.400731\pi$
$211$	8.91402	0.613666	0.306833	+	0.951763i	$0.400731\pi$
$212$	−3.04760	−0.209310
$213$	0	0
$214$	−31.4654	−2.15093
$215$	13.2798	0.905672
$216$	0	0
$217$	6.15626	0.417914
$218$	1.60194	0.108497
$219$	0	0
$220$	−6.91859	−0.466451
$221$	1.49639	0.100658
$222$	0	0
$223$	6.93475	0.464385	0.232193	−	0.972670i	$-0.425410\pi$
$223$	6.93475	0.464385	0.232193	+	0.972670i	$0.425410\pi$
$224$	3.11455	0.208100
$225$	0	0
$226$	18.7456	1.24694
$227$	−15.8708	−1.05338	−0.526692	−	0.850056i	$-0.676568\pi$
$227$	−15.8708	−1.05338	−0.526692	+	0.850056i	$0.676568\pi$
$228$	0	0
$229$	−4.59482	−0.303634	−0.151817	−	0.988409i	$-0.548512\pi$
$229$	−4.59482	−0.303634	−0.151817	+	0.988409i	$0.548512\pi$
$230$	29.1386	1.92135
$231$	0	0
$232$	4.02313	0.264131
$233$	8.80233	0.576660	0.288330	−	0.957531i	$-0.406900\pi$
$233$	8.80233	0.576660	0.288330	+	0.957531i	$0.406900\pi$
$234$	0	0
$235$	25.1840	1.64282
$236$	7.28452	0.474182
$237$	0	0
$238$	−1.72033	−0.111512
$239$	5.70632	0.369111	0.184556	−	0.982822i	$-0.440915\pi$
$239$	5.70632	0.369111	0.184556	+	0.982822i	$0.440915\pi$
$240$	0	0
$241$	−6.40929	−0.412859	−0.206429	−	0.978461i	$-0.566184\pi$
$241$	−6.40929	−0.412859	−0.206429	+	0.978461i	$0.566184\pi$
$242$	−42.4116	−2.72632
$243$	0	0
$244$	−2.46628	−0.157887
$245$	1.99604	0.127523
$246$	0	0
$247$	0.658943	0.0419276
$248$	14.1400	0.897890
$249$	0	0
$250$	19.2358	1.21658
$251$	−4.32925	−0.273260	−0.136630	−	0.990622i	$-0.543627\pi$
$251$	−4.32925	−0.273260	−0.136630	+	0.990622i	$0.543627\pi$
$252$	0	0
$253$	55.7860	3.50723
$254$	−4.20402	−0.263784
$255$	0	0
$256$	12.6026	0.787665
$257$	2.72511	0.169988	0.0849940	−	0.996381i	$-0.472913\pi$
$257$	2.72511	0.169988	0.0849940	+	0.996381i	$0.472913\pi$
$258$	0	0
$259$	9.41647	0.585111
$260$	1.57480	0.0976650
$261$	0	0
$262$	21.8940	1.35262
$263$	1.21325	0.0748123	0.0374062	−	0.999300i	$-0.488090\pi$
$263$	1.21325	0.0748123	0.0374062	+	0.999300i	$0.488090\pi$
$264$	0	0
$265$	−10.7436	−0.659977
$266$	−0.757555	−0.0464487
$267$	0	0
$268$	−2.44601	−0.149414
$269$	−5.24164	−0.319589	−0.159794	−	0.987150i	$-0.551083\pi$
$269$	−5.24164	−0.319589	−0.159794	+	0.987150i	$0.551083\pi$
$270$	0	0
$271$	−4.50412	−0.273606	−0.136803	−	0.990598i	$-0.543683\pi$
$271$	−4.50412	−0.273606	−0.136803	+	0.990598i	$0.543683\pi$
$272$	−5.16744	−0.313322
$273$	0	0
$274$	13.9244	0.841201
$275$	6.21848	0.374988
$276$	0	0
$277$	15.5178	0.932375	0.466187	−	0.884686i	$-0.345627\pi$
$277$	15.5178	0.932375	0.466187	+	0.884686i	$0.345627\pi$
$278$	−6.09083	−0.365303
$279$	0	0
$280$	4.58461	0.273983
$281$	−3.12989	−0.186713	−0.0933567	−	0.995633i	$-0.529760\pi$
$281$	−3.12989	−0.186713	−0.0933567	+	0.995633i	$0.529760\pi$
$282$	0	0
$283$	−9.76593	−0.580524	−0.290262	−	0.956947i	$-0.593742\pi$
$283$	−9.76593	−0.580524	−0.290262	+	0.956947i	$0.593742\pi$
$284$	−2.54931	−0.151274
$285$	0	0
$286$	13.6646	0.808006
$287$	−9.16143	−0.540782
$288$	0	0
$289$	−15.8467	−0.932160
$290$	−5.60077	−0.328889
$291$	0	0
$292$	−6.80648	−0.398319
$293$	9.68619	0.565873	0.282937	−	0.959139i	$-0.408691\pi$
$293$	9.68619	0.565873	0.282937	+	0.959139i	$0.408691\pi$
$294$	0	0
$295$	25.6800	1.49515
$296$	21.6282	1.25711
$297$	0	0
$298$	4.97685	0.288301
$299$	−12.6979	−0.734341
$300$	0	0
$301$	6.65304	0.383475
$302$	29.3536	1.68911
$303$	0	0
$304$	−2.27551	−0.130509
$305$	−8.69432	−0.497836
$306$	0	0
$307$	4.43681	0.253222	0.126611	−	0.991952i	$-0.459590\pi$
$307$	4.43681	0.253222	0.126611	+	0.991952i	$0.459590\pi$
$308$	−3.46615	−0.197502
$309$	0	0
$310$	−19.6849	−1.11803
$311$	−16.4087	−0.930451	−0.465225	−	0.885192i	$-0.654027\pi$
$311$	−16.4087	−0.930451	−0.465225	+	0.885192i	$0.654027\pi$
$312$	0	0
$313$	−32.4191	−1.83244	−0.916218	−	0.400679i	$-0.868774\pi$
$313$	−32.4191	−1.83244	−0.916218	+	0.400679i	$0.868774\pi$
$314$	2.19455	0.123845
$315$	0	0
$316$	−9.80062	−0.551328
$317$	7.58257	0.425879	0.212940	−	0.977065i	$-0.431696\pi$
$317$	7.58257	0.425879	0.212940	+	0.977065i	$0.431696\pi$
$318$	0	0
$319$	−10.7227	−0.600355
$320$	9.25031	0.517108
$321$	0	0
$322$	14.5982	0.813526
$323$	0.507848	0.0282574
$324$	0	0
$325$	−1.41544	−0.0785146
$326$	−31.8569	−1.76439
$327$	0	0
$328$	−21.0424	−1.16187
$329$	12.6169	0.695595
$330$	0	0
$331$	−22.3483	−1.22837	−0.614186	−	0.789161i	$-0.710516\pi$
$331$	−22.3483	−1.22837	−0.614186	+	0.789161i	$0.710516\pi$
$332$	−5.78228	−0.317344
$333$	0	0
$334$	−11.8632	−0.649126
$335$	−8.62288	−0.471118
$336$	0	0
$337$	14.9903	0.816576	0.408288	−	0.912853i	$-0.366126\pi$
$337$	14.9903	0.816576	0.408288	+	0.912853i	$0.366126\pi$
$338$	17.7149	0.963563
$339$	0	0
$340$	1.21370	0.0658222
$341$	−37.6868	−2.04085
$342$	0	0
$343$	1.00000	0.0539949
$344$	15.2810	0.823897
$345$	0	0
$346$	35.5566	1.91153
$347$	22.0354	1.18292	0.591462	−	0.806333i	$-0.298551\pi$
$347$	22.0354	1.18292	0.591462	+	0.806333i	$0.298551\pi$
$348$	0	0
$349$	0.610848	0.0326979	0.0163490	−	0.999866i	$-0.494796\pi$
$349$	0.610848	0.0326979	0.0163490	+	0.999866i	$0.494796\pi$
$350$	1.62727	0.0869810
$351$	0	0
$352$	−19.0664	−1.01624
$353$	−7.61170	−0.405130	−0.202565	−	0.979269i	$-0.564928\pi$
$353$	−7.61170	−0.405130	−0.202565	+	0.979269i	$0.564928\pi$
$354$	0	0
$355$	−8.98705	−0.476983
$356$	−1.85084	−0.0980945
$357$	0	0
$358$	5.88228	0.310888
$359$	13.3788	0.706105	0.353052	−	0.935604i	$-0.385144\pi$
$359$	13.3788	0.706105	0.353052	+	0.935604i	$0.385144\pi$
$360$	0	0
$361$	−18.7764	−0.988230
$362$	33.9753	1.78570
$363$	0	0
$364$	0.788961	0.0413528
$365$	−23.9948	−1.25594
$366$	0	0
$367$	4.50858	0.235346	0.117673	−	0.993052i	$-0.462457\pi$
$367$	4.50858	0.235346	0.117673	+	0.993052i	$0.462457\pi$
$368$	43.8494	2.28581
$369$	0	0
$370$	−30.1095	−1.56532
$371$	−5.38247	−0.279444
$372$	0	0
$373$	15.0141	0.777403	0.388702	−	0.921364i	$-0.372924\pi$
$373$	15.0141	0.777403	0.388702	+	0.921364i	$0.372924\pi$
$374$	10.5313	0.544562
$375$	0	0
$376$	28.9792	1.49449
$377$	2.44068	0.125702
$378$	0	0
$379$	2.28578	0.117413	0.0587063	−	0.998275i	$-0.481302\pi$
$379$	2.28578	0.117413	0.0587063	+	0.998275i	$0.481302\pi$
$380$	0.534459	0.0274172
$381$	0	0
$382$	−14.5024	−0.742005
$383$	27.3077	1.39536	0.697679	−	0.716411i	$-0.254216\pi$
$383$	27.3077	1.39536	0.697679	+	0.716411i	$0.254216\pi$
$384$	0	0
$385$	−12.2192	−0.622747
$386$	27.7835	1.41414
$387$	0	0
$388$	9.81174	0.498116
$389$	−4.87520	−0.247182	−0.123591	−	0.992333i	$-0.539441\pi$
$389$	−4.87520	−0.247182	−0.123591	+	0.992333i	$0.539441\pi$
$390$	0	0
$391$	−9.78632	−0.494915
$392$	2.29685	0.116008
$393$	0	0
$394$	13.2575	0.667901
$395$	−34.5499	−1.73840
$396$	0	0
$397$	1.79380	0.0900284	0.0450142	−	0.998986i	$-0.485667\pi$
$397$	1.79380	0.0900284	0.0450142	+	0.998986i	$0.485667\pi$
$398$	8.01788	0.401900
$399$	0	0
$400$	4.88790	0.244395
$401$	13.7982	0.689050	0.344525	−	0.938777i	$-0.388040\pi$
$401$	13.7982	0.689050	0.344525	+	0.938777i	$0.388040\pi$
$402$	0	0
$403$	8.57821	0.427311
$404$	−6.07581	−0.302283
$405$	0	0
$406$	−2.80594	−0.139256
$407$	−57.6447	−2.85734
$408$	0	0
$409$	−30.6819	−1.51712	−0.758562	−	0.651601i	$-0.774098\pi$
$409$	−30.6819	−1.51712	−0.758562	+	0.651601i	$0.774098\pi$
$410$	29.2941	1.44673
$411$	0	0
$412$	5.20413	0.256389
$413$	12.8654	0.633067
$414$	0	0
$415$	−20.3842	−1.00062
$416$	4.33986	0.212779
$417$	0	0
$418$	4.63752	0.226829
$419$	22.6245	1.10528	0.552640	−	0.833420i	$-0.313620\pi$
$419$	22.6245	1.10528	0.552640	+	0.833420i	$0.313620\pi$
$420$	0	0
$421$	−20.8482	−1.01608	−0.508040	−	0.861333i	$-0.669630\pi$
$421$	−20.8482	−1.01608	−0.508040	+	0.861333i	$0.669630\pi$
$422$	−14.2797	−0.695125
$423$	0	0
$424$	−12.3627	−0.600386
$425$	−1.09088	−0.0529156
$426$	0	0
$427$	−4.35578	−0.210791
$428$	11.1215	0.537578
$429$	0	0
$430$	−21.2734	−1.02589
$431$	13.9803	0.673408	0.336704	−	0.941610i	$-0.390688\pi$
$431$	13.9803	0.673408	0.336704	+	0.941610i	$0.390688\pi$
$432$	0	0
$433$	−20.4512	−0.982821	−0.491411	−	0.870928i	$-0.663519\pi$
$433$	−20.4512	−0.982821	−0.491411	+	0.870928i	$0.663519\pi$
$434$	−9.86195	−0.473389
$435$	0	0
$436$	−0.566208	−0.0271164
$437$	−4.30945	−0.206149
$438$	0	0
$439$	23.0301	1.09917	0.549583	−	0.835439i	$-0.314787\pi$
$439$	23.0301	1.09917	0.549583	+	0.835439i	$0.314787\pi$
$440$	−28.0656	−1.33797
$441$	0	0
$442$	−2.39713	−0.114020
$443$	−1.97532	−0.0938504	−0.0469252	−	0.998898i	$-0.514942\pi$
$443$	−1.97532	−0.0938504	−0.0469252	+	0.998898i	$0.514942\pi$
$444$	0	0
$445$	−6.52475	−0.309303
$446$	−11.1090	−0.526028
$447$	0	0
$448$	4.63432	0.218951
$449$	−10.5385	−0.497343	−0.248672	−	0.968588i	$-0.579994\pi$
$449$	−10.5385	−0.497343	−0.248672	+	0.968588i	$0.579994\pi$
$450$	0	0
$451$	56.0835	2.64087
$452$	−6.62567	−0.311645
$453$	0	0
$454$	25.4241	1.19321
$455$	2.78131	0.130390
$456$	0	0
$457$	−34.6026	−1.61864	−0.809320	−	0.587368i	$-0.800164\pi$
$457$	−34.6026	−1.61864	−0.809320	+	0.587368i	$0.800164\pi$
$458$	7.36061	0.343939
$459$	0	0
$460$	−10.2991	−0.480198
$461$	−38.3892	−1.78796	−0.893982	−	0.448102i	$-0.852100\pi$
$461$	−38.3892	−1.78796	−0.893982	+	0.448102i	$0.852100\pi$
$462$	0	0
$463$	−3.68302	−0.171164	−0.0855822	−	0.996331i	$-0.527275\pi$
$463$	−3.68302	−0.171164	−0.0855822	+	0.996331i	$0.527275\pi$
$464$	−8.42833	−0.391275
$465$	0	0
$466$	−14.1008	−0.653207
$467$	23.0281	1.06561	0.532807	−	0.846237i	$-0.321137\pi$
$467$	23.0281	1.06561	0.532807	+	0.846237i	$0.321137\pi$
$468$	0	0
$469$	−4.31999	−0.199479
$470$	−40.3432	−1.86089
$471$	0	0
$472$	29.5499	1.36015
$473$	−40.7279	−1.87267
$474$	0	0
$475$	−0.480375	−0.0220411
$476$	0.608054	0.0278701
$477$	0	0
$478$	−9.14117	−0.418108
$479$	−29.8640	−1.36452	−0.682261	−	0.731108i	$-0.739003\pi$
$479$	−29.8640	−1.36452	−0.682261	+	0.731108i	$0.739003\pi$
$480$	0	0
$481$	13.1210	0.598267
$482$	10.2673	0.467663
$483$	0	0
$484$	14.9904	0.681384
$485$	34.5892	1.57061
$486$	0	0
$487$	−2.67472	−0.121203	−0.0606015	−	0.998162i	$-0.519302\pi$
$487$	−2.67472	−0.121203	−0.0606015	+	0.998162i	$0.519302\pi$
$488$	−10.0046	−0.452885
$489$	0	0
$490$	−3.19754	−0.144450
$491$	−1.85728	−0.0838177	−0.0419088	−	0.999121i	$-0.513344\pi$
$491$	−1.85728	−0.0838177	−0.0419088	+	0.999121i	$0.513344\pi$
$492$	0	0
$493$	1.88104	0.0847177
$494$	−1.05559	−0.0474931
$495$	0	0
$496$	−29.6229	−1.33011
$497$	−4.50243	−0.201962
$498$	0	0
$499$	−27.1990	−1.21759	−0.608797	−	0.793326i	$-0.708347\pi$
$499$	−27.1990	−1.21759	−0.608797	+	0.793326i	$0.708347\pi$
$500$	−6.79892	−0.304057
$501$	0	0
$502$	6.93519	0.309533
$503$	5.67751	0.253148	0.126574	−	0.991957i	$-0.459602\pi$
$503$	5.67751	0.253148	0.126574	+	0.991957i	$0.459602\pi$
$504$	0	0
$505$	−21.4189	−0.953130
$506$	−89.3657	−3.97279
$507$	0	0
$508$	1.48592	0.0659269
$509$	−33.6191	−1.49014	−0.745071	−	0.666985i	$-0.767584\pi$
$509$	−33.6191	−1.49014	−0.745071	+	0.666985i	$0.767584\pi$
$510$	0	0
$511$	−12.0212	−0.531785
$512$	7.11736	0.314546
$513$	0	0
$514$	−4.36547	−0.192552
$515$	18.3460	0.808423
$516$	0	0
$517$	−77.2371	−3.39688
$518$	−15.0846	−0.662780
$519$	0	0
$520$	6.38825	0.280143
$521$	−0.398531	−0.0174599	−0.00872997	−	0.999962i	$-0.502779\pi$
$521$	−0.398531	−0.0174599	−0.00872997	+	0.999962i	$0.502779\pi$
$522$	0	0
$523$	29.0930	1.27215	0.636073	−	0.771629i	$-0.280558\pi$
$523$	29.0930	1.27215	0.636073	+	0.771629i	$0.280558\pi$
$524$	−7.73847	−0.338057
$525$	0	0
$526$	−1.94356	−0.0847431
$527$	6.61124	0.287990
$528$	0	0
$529$	60.0437	2.61060
$530$	17.2107	0.747583
$531$	0	0
$532$	0.267759	0.0116088
$533$	−12.7657	−0.552942
$534$	0	0
$535$	39.2065	1.69504
$536$	−9.92235	−0.428580
$537$	0	0
$538$	8.39679	0.362011
$539$	−6.12170	−0.263680
$540$	0	0
$541$	−2.01105	−0.0864617	−0.0432309	−	0.999065i	$-0.513765\pi$
$541$	−2.01105	−0.0864617	−0.0432309	+	0.999065i	$0.513765\pi$
$542$	7.21532	0.309925
$543$	0	0
$544$	3.34474	0.143404
$545$	−1.99604	−0.0855011
$546$	0	0
$547$	−2.65049	−0.113327	−0.0566633	−	0.998393i	$-0.518046\pi$
$547$	−2.65049	−0.113327	−0.0566633	+	0.998393i	$0.518046\pi$
$548$	−4.92159	−0.210240
$549$	0	0
$550$	−9.96162	−0.424765
$551$	0.828324	0.0352878
$552$	0	0
$553$	−17.3092	−0.736063
$554$	−24.8586	−1.05614
$555$	0	0
$556$	2.15281	0.0912996
$557$	−29.0053	−1.22900	−0.614498	−	0.788919i	$-0.710641\pi$
$557$	−29.0053	−1.22900	−0.614498	+	0.788919i	$0.710641\pi$
$558$	0	0
$559$	9.27043	0.392098
$560$	−9.60461	−0.405869
$561$	0	0
$562$	5.01389	0.211498
$563$	1.80605	0.0761158	0.0380579	−	0.999276i	$-0.487883\pi$
$563$	1.80605	0.0761158	0.0380579	+	0.999276i	$0.487883\pi$
$564$	0	0
$565$	−23.3574	−0.982652
$566$	15.6444	0.657584
$567$	0	0
$568$	−10.3414	−0.433915
$569$	−28.7429	−1.20496	−0.602481	−	0.798133i	$-0.705821\pi$
$569$	−28.7429	−1.20496	−0.602481	+	0.798133i	$0.705821\pi$
$570$	0	0
$571$	−8.32124	−0.348233	−0.174117	−	0.984725i	$-0.555707\pi$
$571$	−8.32124	−0.348233	−0.174117	+	0.984725i	$0.555707\pi$
$572$	−4.82978	−0.201943
$573$	0	0
$574$	14.6761	0.612567
$575$	9.25690	0.386040
$576$	0	0
$577$	16.2484	0.676432	0.338216	−	0.941069i	$-0.390177\pi$
$577$	16.2484	0.676432	0.338216	+	0.941069i	$0.390177\pi$
$578$	25.3855	1.05590
$579$	0	0
$580$	1.97960	0.0821985
$581$	−10.2123	−0.423677
$582$	0	0
$583$	32.9498	1.36464
$584$	−27.6108	−1.14254
$585$	0	0
$586$	−15.5167	−0.640989
$587$	−2.71317	−0.111985	−0.0559923	−	0.998431i	$-0.517832\pi$
$587$	−2.71317	−0.111985	−0.0559923	+	0.998431i	$0.517832\pi$
$588$	0	0
$589$	2.91129	0.119958
$590$	−41.1378	−1.69361
$591$	0	0
$592$	−45.3104	−1.86225
$593$	−16.1388	−0.662740	−0.331370	−	0.943501i	$-0.607511\pi$
$593$	−16.1388	−0.662740	−0.331370	+	0.943501i	$0.607511\pi$
$594$	0	0
$595$	2.14356	0.0878774
$596$	−1.75908	−0.0720546
$597$	0	0
$598$	20.3413	0.831818
$599$	3.28946	0.134404	0.0672019	−	0.997739i	$-0.478593\pi$
$599$	3.28946	0.134404	0.0672019	+	0.997739i	$0.478593\pi$
$600$	0	0
$601$	−6.05430	−0.246960	−0.123480	−	0.992347i	$-0.539405\pi$
$601$	−6.05430	−0.246960	−0.123480	+	0.992347i	$0.539405\pi$
$602$	−10.6578	−0.434378
$603$	0	0
$604$	−10.3751	−0.422156
$605$	52.8456	2.14848
$606$	0	0
$607$	−31.3177	−1.27115	−0.635573	−	0.772041i	$-0.719236\pi$
$607$	−31.3177	−1.27115	−0.635573	+	0.772041i	$0.719236\pi$
$608$	1.47287	0.0597327
$609$	0	0
$610$	13.9278	0.563919
$611$	17.5806	0.711236
$612$	0	0
$613$	5.87627	0.237340	0.118670	−	0.992934i	$-0.462137\pi$
$613$	5.87627	0.237340	0.118670	+	0.992934i	$0.462137\pi$
$614$	−7.10749	−0.286835
$615$	0	0
$616$	−14.0606	−0.566518
$617$	−10.9834	−0.442173	−0.221087	−	0.975254i	$-0.570960\pi$
$617$	−10.9834	−0.442173	−0.221087	+	0.975254i	$0.570960\pi$
$618$	0	0
$619$	−7.17635	−0.288442	−0.144221	−	0.989546i	$-0.546068\pi$
$619$	−7.17635	−0.288442	−0.144221	+	0.989546i	$0.546068\pi$
$620$	6.95766	0.279426
$621$	0	0
$622$	26.2857	1.05396
$623$	−3.26884	−0.130963
$624$	0	0
$625$	−18.8891	−0.755563
$626$	51.9334	2.07568
$627$	0	0
$628$	−0.775666	−0.0309525
$629$	10.1124	0.403207
$630$	0	0
$631$	−27.1529	−1.08094	−0.540469	−	0.841364i	$-0.681753\pi$
$631$	−27.1529	−1.08094	−0.540469	+	0.841364i	$0.681753\pi$
$632$	−39.7566	−1.58143
$633$	0	0
$634$	−12.1468	−0.482411
$635$	5.23828	0.207875
$636$	0	0
$637$	1.39341	0.0552090
$638$	17.1771	0.680047
$639$	0	0
$640$	−27.2520	−1.07723
$641$	−2.45267	−0.0968748	−0.0484374	−	0.998826i	$-0.515424\pi$
$641$	−2.45267	−0.0968748	−0.0484374	+	0.998826i	$0.515424\pi$
$642$	0	0
$643$	−32.7507	−1.29156	−0.645780	−	0.763523i	$-0.723468\pi$
$643$	−32.7507	−1.29156	−0.645780	+	0.763523i	$0.723468\pi$
$644$	−5.15976	−0.203323
$645$	0	0
$646$	−0.813542	−0.0320084
$647$	−4.95628	−0.194852	−0.0974258	−	0.995243i	$-0.531061\pi$
$647$	−4.95628	−0.194852	−0.0974258	+	0.995243i	$0.531061\pi$
$648$	0	0
$649$	−78.7583	−3.09153
$650$	2.26745	0.0889368
$651$	0	0
$652$	11.2599	0.440971
$653$	31.5005	1.23271	0.616354	−	0.787469i	$-0.288609\pi$
$653$	31.5005	1.23271	0.616354	+	0.787469i	$0.288609\pi$
$654$	0	0
$655$	−27.2803	−1.06593
$656$	44.0832	1.72116
$657$	0	0
$658$	−20.2116	−0.787930
$659$	4.91001	0.191267	0.0956334	−	0.995417i	$-0.469512\pi$
$659$	4.91001	0.191267	0.0956334	+	0.995417i	$0.469512\pi$
$660$	0	0
$661$	−14.9711	−0.582309	−0.291155	−	0.956676i	$-0.594039\pi$
$661$	−14.9711	−0.582309	−0.291155	+	0.956676i	$0.594039\pi$
$662$	35.8006	1.39143
$663$	0	0
$664$	−23.4561	−0.910272
$665$	0.943927	0.0366039
$666$	0	0
$667$	−15.9619	−0.618048
$668$	4.19308	0.162235
$669$	0	0
$670$	13.8133	0.533656
$671$	26.6647	1.02938
$672$	0	0
$673$	−37.5332	−1.44680	−0.723400	−	0.690429i	$-0.757422\pi$
$673$	−37.5332	−1.44680	−0.723400	+	0.690429i	$0.757422\pi$
$674$	−24.0136	−0.924970
$675$	0	0
$676$	−6.26136	−0.240821
$677$	−34.1871	−1.31392	−0.656958	−	0.753927i	$-0.728157\pi$
$677$	−34.1871	−1.31392	−0.656958	+	0.753927i	$0.728157\pi$
$678$	0	0
$679$	17.3289	0.665020
$680$	4.92343	0.188805
$681$	0	0
$682$	60.3719	2.31176
$683$	3.65035	0.139677	0.0698384	−	0.997558i	$-0.477752\pi$
$683$	3.65035	0.139677	0.0698384	+	0.997558i	$0.477752\pi$
$684$	0	0
$685$	−17.3500	−0.662909
$686$	−1.60194	−0.0611623
$687$	0	0
$688$	−32.0133	−1.22049
$689$	−7.50000	−0.285727
$690$	0	0
$691$	24.4230	0.929095	0.464548	−	0.885548i	$-0.346217\pi$
$691$	24.4230	0.929095	0.464548	+	0.885548i	$0.346217\pi$
$692$	−12.5675	−0.477746
$693$	0	0
$694$	−35.2994	−1.33995
$695$	7.58927	0.287878
$696$	0	0
$697$	−9.83851	−0.372660
$698$	−0.978541	−0.0370383
$699$	0	0
$700$	−0.575160	−0.0217390
$701$	41.2310	1.55727	0.778636	−	0.627476i	$-0.215912\pi$
$701$	41.2310	1.55727	0.778636	+	0.627476i	$0.215912\pi$
$702$	0	0
$703$	4.45304	0.167949
$704$	−28.3699	−1.06923
$705$	0	0
$706$	12.1935	0.458908
$707$	−10.7307	−0.403569
$708$	0	0
$709$	14.3586	0.539249	0.269625	−	0.962966i	$-0.413100\pi$
$709$	14.3586	0.539249	0.269625	+	0.962966i	$0.413100\pi$
$710$	14.3967	0.540298
$711$	0	0
$712$	−7.50803	−0.281375
$713$	−56.1010	−2.10100
$714$	0	0
$715$	−17.0263	−0.636749
$716$	−2.07910	−0.0776997
$717$	0	0
$718$	−21.4320	−0.799834
$719$	37.0201	1.38062	0.690308	−	0.723515i	$-0.257475\pi$
$719$	37.0201	1.38062	0.690308	+	0.723515i	$0.257475\pi$
$720$	0	0
$721$	9.19120	0.342298
$722$	30.0786	1.11941
$723$	0	0
$724$	−12.0086	−0.446297
$725$	−1.77928	−0.0660808
$726$	0	0
$727$	−5.83724	−0.216491	−0.108246	−	0.994124i	$-0.534523\pi$
$727$	−5.83724	−0.216491	−0.108246	+	0.994124i	$0.534523\pi$
$728$	3.20046	0.118617
$729$	0	0
$730$	38.4382	1.42266
$731$	7.14474	0.264258
$732$	0	0
$733$	17.6337	0.651316	0.325658	−	0.945488i	$-0.394414\pi$
$733$	17.6337	0.651316	0.325658	+	0.945488i	$0.394414\pi$
$734$	−7.22247	−0.266586
$735$	0	0
$736$	−28.3824	−1.04619
$737$	26.4456	0.974138
$738$	0	0
$739$	−14.3976	−0.529625	−0.264813	−	0.964300i	$-0.585310\pi$
$739$	−14.3976	−0.529625	−0.264813	+	0.964300i	$0.585310\pi$
$740$	10.6423	0.391218
$741$	0	0
$742$	8.62239	0.316538
$743$	−22.6064	−0.829350	−0.414675	−	0.909970i	$-0.636105\pi$
$743$	−22.6064	−0.829350	−0.414675	+	0.909970i	$0.636105\pi$
$744$	0	0
$745$	−6.20124	−0.227196
$746$	−24.0517	−0.880597
$747$	0	0
$748$	−3.72232	−0.136101
$749$	19.6421	0.717706
$750$	0	0
$751$	−5.29574	−0.193244	−0.0966222	−	0.995321i	$-0.530804\pi$
$751$	−5.29574	−0.193244	−0.0966222	+	0.995321i	$0.530804\pi$
$752$	−60.7105	−2.21389
$753$	0	0
$754$	−3.90983	−0.142387
$755$	−36.5751	−1.33110
$756$	0	0
$757$	50.5713	1.83805	0.919023	−	0.394203i	$-0.128979\pi$
$757$	50.5713	1.83805	0.919023	+	0.394203i	$0.128979\pi$
$758$	−3.66168	−0.132998
$759$	0	0
$760$	2.16806	0.0786436
$761$	−16.0567	−0.582056	−0.291028	−	0.956715i	$-0.593997\pi$
$761$	−16.0567	−0.582056	−0.291028	+	0.956715i	$0.593997\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	5.12588	0.185448
$765$	0	0
$766$	−43.7452	−1.58058
$767$	17.9269	0.647302
$768$	0	0
$769$	−27.1569	−0.979303	−0.489651	−	0.871918i	$-0.662876\pi$
$769$	−27.1569	−0.979303	−0.489651	+	0.871918i	$0.662876\pi$
$770$	19.5744	0.705411
$771$	0	0
$772$	−9.82013	−0.353434
$773$	8.22488	0.295828	0.147914	−	0.989000i	$-0.452744\pi$
$773$	8.22488	0.295828	0.147914	+	0.989000i	$0.452744\pi$
$774$	0	0
$775$	−6.25359	−0.224636
$776$	39.8017	1.42880
$777$	0	0
$778$	7.80977	0.279994
$779$	−4.33243	−0.155225
$780$	0	0
$781$	27.5625	0.986264
$782$	15.6771	0.560611
$783$	0	0
$784$	−4.81182	−0.171851
$785$	−2.73444	−0.0975965
$786$	0	0
$787$	−10.7476	−0.383110	−0.191555	−	0.981482i	$-0.561353\pi$
$787$	−10.7476	−0.383110	−0.191555	+	0.981482i	$0.561353\pi$
$788$	−4.68587	−0.166927
$789$	0	0
$790$	55.3469	1.96915
$791$	−11.7018	−0.416069
$792$	0	0
$793$	−6.06940	−0.215531
$794$	−2.87356	−0.101979
$795$	0	0
$796$	−2.83393	−0.100446
$797$	−15.3250	−0.542841	−0.271420	−	0.962461i	$-0.587493\pi$
$797$	−15.3250	−0.542841	−0.271420	+	0.962461i	$0.587493\pi$
$798$	0	0
$799$	13.5494	0.479343
$800$	−3.16380	−0.111857
$801$	0	0
$802$	−22.1039	−0.780516
$803$	73.5899	2.59693
$804$	0	0
$805$	−18.1896	−0.641100
$806$	−13.7418	−0.484033
$807$	0	0
$808$	−24.6468	−0.867070
$809$	−44.5786	−1.56730	−0.783651	−	0.621202i	$-0.786645\pi$
$809$	−44.5786	−1.56730	−0.783651	+	0.621202i	$0.786645\pi$
$810$	0	0
$811$	−2.71546	−0.0953528	−0.0476764	−	0.998863i	$-0.515182\pi$
$811$	−2.71546	−0.0953528	−0.0476764	+	0.998863i	$0.515182\pi$
$812$	0.991763	0.0348041
$813$	0	0
$814$	92.3434	3.23663
$815$	39.6942	1.39043
$816$	0	0
$817$	3.14622	0.110072
$818$	49.1506	1.71851
$819$	0	0
$820$	−10.3540	−0.361579
$821$	33.2946	1.16199	0.580994	−	0.813908i	$-0.302664\pi$
$821$	33.2946	1.16199	0.580994	+	0.813908i	$0.302664\pi$
$822$	0	0
$823$	−19.2141	−0.669762	−0.334881	−	0.942260i	$-0.608696\pi$
$823$	−19.2141	−0.669762	−0.334881	+	0.942260i	$0.608696\pi$
$824$	21.1108	0.735429
$825$	0	0
$826$	−20.6096	−0.717101
$827$	−26.6969	−0.928342	−0.464171	−	0.885746i	$-0.653648\pi$
$827$	−26.6969	−0.928342	−0.464171	+	0.885746i	$0.653648\pi$
$828$	0	0
$829$	12.4422	0.432134	0.216067	−	0.976378i	$-0.430677\pi$
$829$	12.4422	0.432134	0.216067	+	0.976378i	$0.430677\pi$
$830$	32.6542	1.13344
$831$	0	0
$832$	6.45752	0.223874
$833$	1.07390	0.0372086
$834$	0	0
$835$	14.7818	0.511544
$836$	−1.63914	−0.0566908
$837$	0	0
$838$	−36.2431	−1.25200
$839$	−35.6063	−1.22927	−0.614633	−	0.788813i	$-0.710696\pi$
$839$	−35.6063	−1.22927	−0.614633	+	0.788813i	$0.710696\pi$
$840$	0	0
$841$	−25.9319	−0.894205
$842$	33.3976	1.15096
$843$	0	0
$844$	5.04719	0.173731
$845$	−22.0731	−0.759336
$846$	0	0
$847$	26.4752	0.909697
$848$	25.8995	0.889393
$849$	0	0
$850$	1.74753	0.0599397
$851$	−85.8107	−2.94155
$852$	0	0
$853$	−0.984186	−0.0336979	−0.0168489	−	0.999858i	$-0.505363\pi$
$853$	−0.984186	−0.0336979	−0.0168489	+	0.999858i	$0.505363\pi$
$854$	6.97769	0.238772
$855$	0	0
$856$	45.1149	1.54199
$857$	15.2667	0.521500	0.260750	−	0.965406i	$-0.416030\pi$
$857$	15.2667	0.521500	0.260750	+	0.965406i	$0.416030\pi$
$858$	0	0
$859$	−42.8825	−1.46313	−0.731566	−	0.681771i	$-0.761210\pi$
$859$	−42.8825	−1.46313	−0.731566	+	0.681771i	$0.761210\pi$
$860$	7.51911	0.256400
$861$	0	0
$862$	−22.3956	−0.762798
$863$	−31.4420	−1.07030	−0.535149	−	0.844758i	$-0.679745\pi$
$863$	−31.4420	−1.07030	−0.535149	+	0.844758i	$0.679745\pi$
$864$	0	0
$865$	−44.3041	−1.50638
$866$	32.7616	1.11328
$867$	0	0
$868$	3.48572	0.118313
$869$	105.962	3.59451
$870$	0	0
$871$	−6.01953	−0.203964
$872$	−2.29685	−0.0777811
$873$	0	0
$874$	6.90347	0.233513
$875$	−12.0078	−0.405938
$876$	0	0
$877$	24.2732	0.819647	0.409823	−	0.912165i	$-0.365590\pi$
$877$	24.2732	0.819647	0.409823	+	0.912165i	$0.365590\pi$
$878$	−36.8928	−1.24507
$879$	0	0
$880$	58.7965	1.98203
$881$	−0.807379	−0.0272013	−0.0136006	−	0.999908i	$-0.504329\pi$
$881$	−0.807379	−0.0272013	−0.0136006	+	0.999908i	$0.504329\pi$
$882$	0	0
$883$	2.76822	0.0931581	0.0465790	−	0.998915i	$-0.485168\pi$
$883$	2.76822	0.0931581	0.0465790	+	0.998915i	$0.485168\pi$
$884$	0.847270	0.0284968
$885$	0	0
$886$	3.16435	0.106308
$887$	28.5910	0.959991	0.479996	−	0.877271i	$-0.340638\pi$
$887$	28.5910	0.959991	0.479996	+	0.877271i	$0.340638\pi$
$888$	0	0
$889$	2.62433	0.0880173
$890$	10.4522	0.350360
$891$	0	0
$892$	3.92651	0.131469
$893$	5.96654	0.199663
$894$	0	0
$895$	−7.32942	−0.244996
$896$	−13.6530	−0.456115
$897$	0	0
$898$	16.8821	0.563362
$899$	10.7832	0.359641
$900$	0	0
$901$	−5.78026	−0.192568
$902$	−89.8423	−2.99142
$903$	0	0
$904$	−26.8773	−0.893926
$905$	−42.3338	−1.40722
$906$	0	0
$907$	11.5318	0.382909	0.191454	−	0.981502i	$-0.438680\pi$
$907$	11.5318	0.382909	0.191454	+	0.981502i	$0.438680\pi$
$908$	−8.98620	−0.298218
$909$	0	0
$910$	−4.45549	−0.147698
$911$	50.8597	1.68506	0.842528	−	0.538652i	$-0.181066\pi$
$911$	50.8597	1.68506	0.842528	+	0.538652i	$0.181066\pi$
$912$	0	0
$913$	62.5165	2.06900
$914$	55.4312	1.83350
$915$	0	0
$916$	−2.60162	−0.0859600
$917$	−13.6672	−0.451330
$918$	0	0
$919$	−12.9498	−0.427174	−0.213587	−	0.976924i	$-0.568515\pi$
$919$	−12.9498	−0.427174	−0.213587	+	0.976924i	$0.568515\pi$
$920$	−41.7788	−1.37740
$921$	0	0
$922$	61.4972	2.02530
$923$	−6.27374	−0.206503
$924$	0	0
$925$	−9.56534	−0.314506
$926$	5.89997	0.193885
$927$	0	0
$928$	5.45542	0.179083
$929$	−2.37168	−0.0778122	−0.0389061	−	0.999243i	$-0.512387\pi$
$929$	−2.37168	−0.0778122	−0.0389061	+	0.999243i	$0.512387\pi$
$930$	0	0
$931$	0.472899	0.0154986
$932$	4.98395	0.163255
$933$	0	0
$934$	−36.8896	−1.20706
$935$	−13.1222	−0.429143
$936$	0	0
$937$	27.1556	0.887135	0.443567	−	0.896241i	$-0.353713\pi$
$937$	27.1556	0.887135	0.443567	+	0.896241i	$0.353713\pi$
$938$	6.92036	0.225958
$939$	0	0
$940$	14.2594	0.465090
$941$	44.0537	1.43611	0.718055	−	0.695987i	$-0.245033\pi$
$941$	44.0537	1.43611	0.718055	+	0.695987i	$0.245033\pi$
$942$	0	0
$943$	83.4866	2.71870
$944$	−61.9062	−2.01488
$945$	0	0
$946$	65.2436	2.12125
$947$	−24.3557	−0.791455	−0.395728	−	0.918368i	$-0.629508\pi$
$947$	−24.3557	−0.791455	−0.395728	+	0.918368i	$0.629508\pi$
$948$	0	0
$949$	−16.7504	−0.543743
$950$	0.769532	0.0249669
$951$	0	0
$952$	2.46660	0.0799428
$953$	−11.3397	−0.367330	−0.183665	−	0.982989i	$-0.558796\pi$
$953$	−11.3397	−0.367330	−0.183665	+	0.982989i	$0.558796\pi$
$954$	0	0
$955$	18.0702	0.584737
$956$	3.23096	0.104497
$957$	0	0
$958$	47.8404	1.54565
$959$	−8.69219	−0.280685
$960$	0	0
$961$	6.89956	0.222566
$962$	−21.0191	−0.677682
$963$	0	0
$964$	−3.62899	−0.116882
$965$	−34.6187	−1.11442
$966$	0	0
$967$	1.98292	0.0637663	0.0318832	−	0.999492i	$-0.489850\pi$
$967$	1.98292	0.0637663	0.0318832	+	0.999492i	$0.489850\pi$
$968$	60.8094	1.95449
$969$	0	0
$970$	−55.4097	−1.77910
$971$	−8.88322	−0.285076	−0.142538	−	0.989789i	$-0.545526\pi$
$971$	−8.88322	−0.285076	−0.142538	+	0.989789i	$0.545526\pi$
$972$	0	0
$973$	3.80216	0.121892
$974$	4.28474	0.137292
$975$	0	0
$976$	20.9592	0.670889
$977$	37.4644	1.19859	0.599297	−	0.800527i	$-0.295447\pi$
$977$	37.4644	1.19859	0.599297	+	0.800527i	$0.295447\pi$
$978$	0	0
$979$	20.0108	0.639549
$980$	1.13018	0.0361021
$981$	0	0
$982$	2.97524	0.0949438
$983$	−40.6491	−1.29650	−0.648252	−	0.761426i	$-0.724500\pi$
$983$	−40.6491	−1.29650	−0.648252	+	0.761426i	$0.724500\pi$
$984$	0	0
$985$	−16.5190	−0.526340
$986$	−3.01331	−0.0959633
$987$	0	0
$988$	0.373099	0.0118699
$989$	−60.6281	−1.92786
$990$	0	0
$991$	−42.4526	−1.34855	−0.674276	−	0.738480i	$-0.735544\pi$
$991$	−42.4526	−1.34855	−0.674276	+	0.738480i	$0.735544\pi$
$992$	19.1740	0.608776
$993$	0	0
$994$	7.21262	0.228770
$995$	−9.99042	−0.316718
$996$	0	0
$997$	−36.4256	−1.15361	−0.576805	−	0.816882i	$-0.695701\pi$
$997$	−36.4256	−1.15361	−0.576805	+	0.816882i	$0.695701\pi$
$998$	43.5711	1.37922
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6867.2.a.r.1.7		17
3.2	odd	2		763.2.a.e.1.11	✓	17
21.20	even	2		5341.2.a.j.1.11		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
763.2.a.e.1.11	✓	17	3.2	odd	2
5341.2.a.j.1.11		17	21.20	even	2
6867.2.a.r.1.7		17	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:	\( N \)	\(=\)	\( 6867 = 3^{2} \cdot 7 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6867.a (trivial)

\(f(q)\)	\(=\)	\(q-1.60194 q^{2} +0.566208 q^{4} +1.99604 q^{5} +1.00000 q^{7} +2.29685 q^{8} +O(q^{10})\)	\(q-1.60194 q^{2} +0.566208 q^{4} +1.99604 q^{5} +1.00000 q^{7} +2.29685 q^{8} -3.19754 q^{10} -6.12170 q^{11} +1.39341 q^{13} -1.60194 q^{14} -4.81182 q^{16} +1.07390 q^{17} +0.472899 q^{19} +1.13018 q^{20} +9.80658 q^{22} -9.11283 q^{23} -1.01581 q^{25} -2.23216 q^{26} +0.566208 q^{28} +1.75159 q^{29} +6.15626 q^{31} +3.11455 q^{32} -1.72033 q^{34} +1.99604 q^{35} +9.41647 q^{37} -0.757555 q^{38} +4.58461 q^{40} -9.16143 q^{41} +6.65304 q^{43} -3.46615 q^{44} +14.5982 q^{46} +12.6169 q^{47} +1.00000 q^{49} +1.62727 q^{50} +0.788961 q^{52} -5.38247 q^{53} -12.2192 q^{55} +2.29685 q^{56} -2.80594 q^{58} +12.8654 q^{59} -4.35578 q^{61} -9.86195 q^{62} +4.63432 q^{64} +2.78131 q^{65} -4.31999 q^{67} +0.608054 q^{68} -3.19754 q^{70} -4.50243 q^{71} -12.0212 q^{73} -15.0846 q^{74} +0.267759 q^{76} -6.12170 q^{77} -17.3092 q^{79} -9.60461 q^{80} +14.6761 q^{82} -10.2123 q^{83} +2.14356 q^{85} -10.6578 q^{86} -14.0606 q^{88} -3.26884 q^{89} +1.39341 q^{91} -5.15976 q^{92} -20.2116 q^{94} +0.943927 q^{95} +17.3289 q^{97} -1.60194 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 7 q^{2} + 23 q^{4} - 12 q^{5} + 17 q^{7} - 18 q^{8}+O(q^{10}) \) 17 * q - 7 * q^2 + 23 * q^4 - 12 * q^5 + 17 * q^7 - 18 * q^8	\( 17 q - 7 q^{2} + 23 q^{4} - 12 q^{5} + 17 q^{7} - 18 q^{8} - 6 q^{10} - 20 q^{11} + 11 q^{13} - 7 q^{14} + 23 q^{16} - 19 q^{17} - 4 q^{19} - 10 q^{20} + 3 q^{22} - 22 q^{23} + 21 q^{25} + 12 q^{26} + 23 q^{28} - 37 q^{29} - 10 q^{31} - 18 q^{32} - 7 q^{34} - 12 q^{35} + 13 q^{37} - 18 q^{38} - 17 q^{40} + 6 q^{41} + 9 q^{43} - 35 q^{44} - 17 q^{46} + 4 q^{47} + 17 q^{49} - 37 q^{50} + 36 q^{52} - 45 q^{53} - 13 q^{55} - 18 q^{56} + 4 q^{58} + 12 q^{59} + 6 q^{61} + 24 q^{62} + 20 q^{64} - 15 q^{65} - 3 q^{67} - 60 q^{68} - 6 q^{70} - 36 q^{71} - 4 q^{73} - 13 q^{74} - 74 q^{76} - 20 q^{77} + 5 q^{79} + 25 q^{80} - 24 q^{82} - 7 q^{83} + 8 q^{85} - 2 q^{86} - 55 q^{88} - 13 q^{89} + 11 q^{91} - 36 q^{92} - 37 q^{94} + 34 q^{95} - 23 q^{97} - 7 q^{98}+O(q^{100}) \) 17 * q - 7 * q^2 + 23 * q^4 - 12 * q^5 + 17 * q^7 - 18 * q^8 - 6 * q^10 - 20 * q^11 + 11 * q^13 - 7 * q^14 + 23 * q^16 - 19 * q^17 - 4 * q^19 - 10 * q^20 + 3 * q^22 - 22 * q^23 + 21 * q^25 + 12 * q^26 + 23 * q^28 - 37 * q^29 - 10 * q^31 - 18 * q^32 - 7 * q^34 - 12 * q^35 + 13 * q^37 - 18 * q^38 - 17 * q^40 + 6 * q^41 + 9 * q^43 - 35 * q^44 - 17 * q^46 + 4 * q^47 + 17 * q^49 - 37 * q^50 + 36 * q^52 - 45 * q^53 - 13 * q^55 - 18 * q^56 + 4 * q^58 + 12 * q^59 + 6 * q^61 + 24 * q^62 + 20 * q^64 - 15 * q^65 - 3 * q^67 - 60 * q^68 - 6 * q^70 - 36 * q^71 - 4 * q^73 - 13 * q^74 - 74 * q^76 - 20 * q^77 + 5 * q^79 + 25 * q^80 - 24 * q^82 - 7 * q^83 + 8 * q^85 - 2 * q^86 - 55 * q^88 - 13 * q^89 + 11 * q^91 - 36 * q^92 - 37 * q^94 + 34 * q^95 - 23 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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