LMFDB - Embedded newform 1127.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1127.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1127.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:	$8.99914030780$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 10x^{5} - x^{4} + 29x^{3} + 9x^{2} - 24x - 11 $ x^7 - 10x^5 - x^4 + 29x^3 + 9x^2 - 24x - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.509818$ of defining polynomial
Character	$\chi$	$=$	1127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.509818 q^{2} -0.958194 q^{3} -1.74009 q^{4} +3.43532 q^{5} -0.488505 q^{6} -1.90676 q^{8} -2.08186 q^{9} +1.75139 q^{10} +1.49322 q^{11} +1.66734 q^{12} -6.52938 q^{13} -3.29171 q^{15} +2.50807 q^{16} +1.20673 q^{17} -1.06137 q^{18} -3.68561 q^{19} -5.97776 q^{20} +0.761272 q^{22} +1.00000 q^{23} +1.82705 q^{24} +6.80146 q^{25} -3.32880 q^{26} +4.86941 q^{27} -9.34953 q^{29} -1.67817 q^{30} -9.45489 q^{31} +5.09218 q^{32} -1.43080 q^{33} +0.615215 q^{34} +3.62262 q^{36} +7.59075 q^{37} -1.87899 q^{38} +6.25642 q^{39} -6.55035 q^{40} -4.57126 q^{41} +2.55057 q^{43} -2.59834 q^{44} -7.15188 q^{45} +0.509818 q^{46} -6.54116 q^{47} -2.40322 q^{48} +3.46750 q^{50} -1.15629 q^{51} +11.3617 q^{52} -5.30196 q^{53} +2.48251 q^{54} +5.12971 q^{55} +3.53153 q^{57} -4.76656 q^{58} -2.48414 q^{59} +5.72786 q^{60} -5.20864 q^{61} -4.82027 q^{62} -2.42005 q^{64} -22.4306 q^{65} -0.729446 q^{66} -3.44270 q^{67} -2.09982 q^{68} -0.958194 q^{69} +2.37214 q^{71} +3.96962 q^{72} -5.31204 q^{73} +3.86990 q^{74} -6.51712 q^{75} +6.41328 q^{76} +3.18963 q^{78} -1.43707 q^{79} +8.61604 q^{80} +1.57975 q^{81} -2.33051 q^{82} -12.6478 q^{83} +4.14552 q^{85} +1.30033 q^{86} +8.95867 q^{87} -2.84722 q^{88} +3.95879 q^{89} -3.64615 q^{90} -1.74009 q^{92} +9.05962 q^{93} -3.33480 q^{94} -12.6613 q^{95} -4.87930 q^{96} -6.18384 q^{97} -3.10869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{3} + 6 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 9 q^{12} - 14 q^{13} - 3 q^{15} - 8 q^{16} - 4 q^{17} + 19 q^{18} - 9 q^{19} - 12 q^{20} - 10 q^{22} + 7 q^{23} + 4 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) $ 7 * q - 5 * q^3 + 6 * q^4 - 4 * q^5 - 6 * q^6 - 3 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 9 * q^12 - 14 * q^13 - 3 * q^15 - 8 * q^16 - 4 * q^17 + 19 * q^18 - 9 * q^19 - 12 * q^20 - 10 * q^22 + 7 * q^23 + 4 * q^24 + 3 * q^25 - 14 * q^26 - 20 * q^27 - 5 * q^29 - 17 * q^30 - 10 * q^31 + 19 * q^32 - 3 * q^33 - 20 * q^34 + 4 * q^36 - 5 * q^37 + 17 * q^38 + 18 * q^39 - 4 * q^40 - 23 * q^41 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 4 * q^47 - 11 * q^48 - 9 * q^50 + 7 * q^51 - 12 * q^52 + 2 * q^53 - 15 * q^54 - 17 * q^55 - 24 * q^57 - 7 * q^58 - 7 * q^59 + 32 * q^60 - 12 * q^61 + 4 * q^62 - 9 * q^64 + 9 * q^65 + 27 * q^66 + 2 * q^67 + 12 * q^68 - 5 * q^69 - 28 * q^71 - 20 * q^72 - 59 * q^73 + 2 * q^74 + 22 * q^75 - q^76 - 17 * q^78 - 4 * q^79 + 22 * q^80 + 11 * q^81 - 54 * q^82 + 9 * q^83 + 9 * q^85 - 15 * q^86 - 11 * q^87 - 33 * q^88 - q^89 - 2 * q^90 + 6 * q^92 + 15 * q^93 + 14 * q^94 - 16 * q^95 - 20 * q^96 - 35 * q^97 + 18 * q^99

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 $

$2$	0.509818	0.360496	0.180248	−	0.983621i	$-0.442310\pi$
$2$	0.509818	0.360496	0.180248	+	0.983621i	$0.442310\pi$
$3$	−0.958194	−0.553214	−0.276607	−	0.960983i	$-0.589210\pi$
$3$	−0.958194	−0.553214	−0.276607	+	0.960983i	$0.589210\pi$
$4$	−1.74009	−0.870043
$5$	3.43532	1.53632	0.768162	−	0.640256i	$-0.221172\pi$
$5$	3.43532	1.53632	0.768162	+	0.640256i	$0.221172\pi$
$6$	−0.488505	−0.199431
$7$	0	0
$7$	0	0
$8$	−1.90676	−0.674142
$9$	−2.08186	−0.693954
$10$	1.75139	0.553838
$11$	1.49322	0.450224	0.225112	−	0.974333i	$-0.427725\pi$
$11$	1.49322	0.450224	0.225112	+	0.974333i	$0.427725\pi$
$12$	1.66734	0.481320
$13$	−6.52938	−1.81093	−0.905463	−	0.424426i	$-0.860476\pi$
$13$	−6.52938	−1.81093	−0.905463	+	0.424426i	$0.860476\pi$
$14$	0	0
$15$	−3.29171	−0.849916
$16$	2.50807	0.627018
$17$	1.20673	0.292676	0.146338	−	0.989235i	$-0.453251\pi$
$17$	1.20673	0.292676	0.146338	+	0.989235i	$0.453251\pi$
$18$	−1.06137	−0.250168
$19$	−3.68561	−0.845538	−0.422769	−	0.906238i	$-0.638942\pi$
$19$	−3.68561	−0.845538	−0.422769	+	0.906238i	$0.638942\pi$
$20$	−5.97776	−1.33667
$21$	0	0
$22$	0.761272	0.162304
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.82705	0.372945
$25$	6.80146	1.36029
$26$	−3.32880	−0.652831
$27$	4.86941	0.937119
$28$	0	0
$29$	−9.34953	−1.73616	−0.868082	−	0.496421i	$-0.834647\pi$
$29$	−9.34953	−1.73616	−0.868082	+	0.496421i	$0.834647\pi$
$30$	−1.67817	−0.306391
$31$	−9.45489	−1.69815	−0.849074	−	0.528274i	$-0.822840\pi$
$31$	−9.45489	−1.69815	−0.849074	+	0.528274i	$0.822840\pi$
$32$	5.09218	0.900179
$33$	−1.43080	−0.249070
$34$	0.615215	0.105508
$35$	0	0
$36$	3.62262	0.603770
$37$	7.59075	1.24791	0.623956	−	0.781459i	$-0.285524\pi$
$37$	7.59075	1.24791	0.623956	+	0.781459i	$0.285524\pi$
$38$	−1.87899	−0.304813
$39$	6.25642	1.00183
$40$	−6.55035	−1.03570
$41$	−4.57126	−0.713911	−0.356955	−	0.934121i	$-0.616185\pi$
$41$	−4.57126	−0.713911	−0.356955	+	0.934121i	$0.616185\pi$
$42$	0	0
$43$	2.55057	0.388958	0.194479	−	0.980907i	$-0.437698\pi$
$43$	2.55057	0.388958	0.194479	+	0.980907i	$0.437698\pi$
$44$	−2.59834	−0.391714
$45$	−7.15188	−1.06614
$46$	0.509818	0.0751685
$47$	−6.54116	−0.954126	−0.477063	−	0.878869i	$-0.658299\pi$
$47$	−6.54116	−0.954126	−0.477063	+	0.878869i	$0.658299\pi$
$48$	−2.40322	−0.346875
$49$	0	0
$50$	3.46750	0.490379
$51$	−1.15629	−0.161912
$52$	11.3617	1.57558
$53$	−5.30196	−0.728280	−0.364140	−	0.931344i	$-0.618637\pi$
$53$	−5.30196	−0.728280	−0.364140	+	0.931344i	$0.618637\pi$
$54$	2.48251	0.337827
$55$	5.12971	0.691690
$56$	0	0
$57$	3.53153	0.467763
$58$	−4.76656	−0.625879
$59$	−2.48414	−0.323407	−0.161704	−	0.986839i	$-0.551699\pi$
$59$	−2.48414	−0.323407	−0.161704	+	0.986839i	$0.551699\pi$
$60$	5.72786	0.739463
$61$	−5.20864	−0.666898	−0.333449	−	0.942768i	$-0.608213\pi$
$61$	−5.20864	−0.666898	−0.333449	+	0.942768i	$0.608213\pi$
$62$	−4.82027	−0.612175
$63$	0	0
$64$	−2.42005	−0.302507
$65$	−22.4306	−2.78217
$66$	−0.729446	−0.0897887
$67$	−3.44270	−0.420593	−0.210296	−	0.977638i	$-0.567443\pi$
$67$	−3.44270	−0.420593	−0.210296	+	0.977638i	$0.567443\pi$
$68$	−2.09982	−0.254641
$69$	−0.958194	−0.115353
$70$	0	0
$71$	2.37214	0.281521	0.140760	−	0.990044i	$-0.455045\pi$
$71$	2.37214	0.281521	0.140760	+	0.990044i	$0.455045\pi$
$72$	3.96962	0.467824
$73$	−5.31204	−0.621727	−0.310864	−	0.950455i	$-0.600618\pi$
$73$	−5.31204	−0.621727	−0.310864	+	0.950455i	$0.600618\pi$
$74$	3.86990	0.449867
$75$	−6.51712	−0.752532
$76$	6.41328	0.735654
$77$	0	0
$78$	3.18963	0.361155
$79$	−1.43707	−0.161683	−0.0808416	−	0.996727i	$-0.525761\pi$
$79$	−1.43707	−0.161683	−0.0808416	+	0.996727i	$0.525761\pi$
$80$	8.61604	0.963302
$81$	1.57975	0.175527
$82$	−2.33051	−0.257362
$83$	−12.6478	−1.38827	−0.694136	−	0.719844i	$-0.744213\pi$
$83$	−12.6478	−1.38827	−0.694136	+	0.719844i	$0.744213\pi$
$84$	0	0
$85$	4.14552	0.449645
$86$	1.30033	0.140218
$87$	8.95867	0.960470
$88$	−2.84722	−0.303515
$89$	3.95879	0.419631	0.209816	−	0.977741i	$-0.432714\pi$
$89$	3.95879	0.419631	0.209816	+	0.977741i	$0.432714\pi$
$90$	−3.64615	−0.384338
$91$	0	0
$92$	−1.74009	−0.181416
$93$	9.05962	0.939439
$94$	−3.33480	−0.343958
$95$	−12.6613	−1.29902
$96$	−4.87930	−0.497992
$97$	−6.18384	−0.627873	−0.313937	−	0.949444i	$-0.601648\pi$
$97$	−6.18384	−0.627873	−0.313937	+	0.949444i	$0.601648\pi$
$98$	0	0
$99$	−3.10869	−0.312435
$100$	−11.8351	−1.18351
$101$	−0.664901	−0.0661601	−0.0330800	−	0.999453i	$-0.510532\pi$
$101$	−0.664901	−0.0661601	−0.0330800	+	0.999453i	$0.510532\pi$
$102$	−0.589495	−0.0583687
$103$	17.0569	1.68067	0.840333	−	0.542071i	$-0.182360\pi$
$103$	17.0569	1.68067	0.840333	+	0.542071i	$0.182360\pi$
$104$	12.4500	1.22082
$105$	0	0
$106$	−2.70303	−0.262542
$107$	9.08281	0.878068	0.439034	−	0.898470i	$-0.355321\pi$
$107$	9.08281	0.878068	0.439034	+	0.898470i	$0.355321\pi$
$108$	−8.47320	−0.815334
$109$	−7.58949	−0.726942	−0.363471	−	0.931606i	$-0.618408\pi$
$109$	−7.58949	−0.726942	−0.363471	+	0.931606i	$0.618408\pi$
$110$	2.61522	0.249351
$111$	−7.27342	−0.690362
$112$	0	0
$113$	−20.0045	−1.88186	−0.940931	−	0.338599i	$-0.890047\pi$
$113$	−20.0045	−1.88186	−0.940931	+	0.338599i	$0.890047\pi$
$114$	1.80044	0.168627
$115$	3.43532	0.320346
$116$	16.2690	1.51054
$117$	13.5933	1.25670
$118$	−1.26646	−0.116587
$119$	0	0
$120$	6.27651	0.572964
$121$	−8.77028	−0.797299
$122$	−2.65546	−0.240414
$123$	4.38015	0.394945
$124$	16.4523	1.47746
$125$	6.18859	0.553524
$126$	0	0
$127$	8.78156	0.779237	0.389619	−	0.920976i	$-0.372607\pi$
$127$	8.78156	0.779237	0.389619	+	0.920976i	$0.372607\pi$
$128$	−11.4182	−1.00923
$129$	−2.44394	−0.215177
$130$	−11.4355	−1.00296
$131$	−1.57911	−0.137968	−0.0689838	−	0.997618i	$-0.521976\pi$
$131$	−1.57911	−0.137968	−0.0689838	+	0.997618i	$0.521976\pi$
$132$	2.48971	0.216702
$133$	0	0
$134$	−1.75515	−0.151622
$135$	16.7280	1.43972
$136$	−2.30096	−0.197305
$137$	−2.66167	−0.227402	−0.113701	−	0.993515i	$-0.536271\pi$
$137$	−2.66167	−0.227402	−0.113701	+	0.993515i	$0.536271\pi$
$138$	−0.488505	−0.0415843
$139$	6.26608	0.531482	0.265741	−	0.964044i	$-0.414383\pi$
$139$	6.26608	0.531482	0.265741	+	0.964044i	$0.414383\pi$
$140$	0	0
$141$	6.26770	0.527836
$142$	1.20936	0.101487
$143$	−9.74983	−0.815322
$144$	−5.22146	−0.435122
$145$	−32.1187	−2.66731
$146$	−2.70817	−0.224130
$147$	0	0
$148$	−13.2086	−1.08574
$149$	10.0849	0.826184	0.413092	−	0.910689i	$-0.364449\pi$
$149$	10.0849	0.826184	0.413092	+	0.910689i	$0.364449\pi$
$150$	−3.32254	−0.271284
$151$	10.1979	0.829897	0.414948	−	0.909845i	$-0.363800\pi$
$151$	10.1979	0.829897	0.414948	+	0.909845i	$0.363800\pi$
$152$	7.02759	0.570013
$153$	−2.51226	−0.203104
$154$	0	0
$155$	−32.4806	−2.60891
$156$	−10.8867	−0.871634
$157$	10.7030	0.854196	0.427098	−	0.904205i	$-0.359536\pi$
$157$	10.7030	0.854196	0.427098	+	0.904205i	$0.359536\pi$
$158$	−0.732645	−0.0582861
$159$	5.08031	0.402895
$160$	17.4933	1.38297
$161$	0	0
$162$	0.805382	0.0632768
$163$	12.5142	0.980190	0.490095	−	0.871669i	$-0.336962\pi$
$163$	12.5142	0.980190	0.490095	+	0.871669i	$0.336962\pi$
$164$	7.95438	0.621133
$165$	−4.91526	−0.382652
$166$	−6.44805	−0.500466
$167$	−6.86257	−0.531042	−0.265521	−	0.964105i	$-0.585544\pi$
$167$	−6.86257	−0.531042	−0.265521	+	0.964105i	$0.585544\pi$
$168$	0	0
$169$	29.6328	2.27945
$170$	2.11346	0.162095
$171$	7.67294	0.586765
$172$	−4.43821	−0.338410
$173$	10.7000	0.813510	0.406755	−	0.913537i	$-0.366660\pi$
$173$	10.7000	0.813510	0.406755	+	0.913537i	$0.366660\pi$
$174$	4.56729	0.346245
$175$	0	0
$176$	3.74511	0.282298
$177$	2.38029	0.178913
$178$	2.01826	0.151275
$179$	22.0198	1.64584	0.822919	−	0.568159i	$-0.192344\pi$
$179$	22.0198	1.64584	0.822919	+	0.568159i	$0.192344\pi$
$180$	12.4449	0.927587
$181$	5.10153	0.379194	0.189597	−	0.981862i	$-0.439282\pi$
$181$	5.10153	0.379194	0.189597	+	0.981862i	$0.439282\pi$
$182$	0	0
$183$	4.99089	0.368937
$184$	−1.90676	−0.140568
$185$	26.0767	1.91720
$186$	4.61876	0.338664
$187$	1.80192	0.131770
$188$	11.3822	0.830131
$189$	0	0
$190$	−6.45494	−0.468291
$191$	10.6801	0.772784	0.386392	−	0.922335i	$-0.373721\pi$
$191$	10.6801	0.772784	0.386392	+	0.922335i	$0.373721\pi$
$192$	2.31888	0.167351
$193$	2.33695	0.168217	0.0841087	−	0.996457i	$-0.473196\pi$
$193$	2.33695	0.168217	0.0841087	+	0.996457i	$0.473196\pi$
$194$	−3.15263	−0.226346
$195$	21.4928	1.53913
$196$	0	0
$197$	2.54570	0.181374	0.0906870	−	0.995879i	$-0.471094\pi$
$197$	2.54570	0.181374	0.0906870	+	0.995879i	$0.471094\pi$
$198$	−1.58486	−0.112631
$199$	−0.591206	−0.0419095	−0.0209547	−	0.999780i	$-0.506671\pi$
$199$	−0.591206	−0.0419095	−0.0209547	+	0.999780i	$0.506671\pi$
$200$	−12.9688	−0.917030
$201$	3.29878	0.232678
$202$	−0.338978	−0.0238504
$203$	0	0
$204$	2.01204	0.140871
$205$	−15.7038	−1.09680
$206$	8.69591	0.605873
$207$	−2.08186	−0.144700
$208$	−16.3762	−1.13548
$209$	−5.50344	−0.380681
$210$	0	0
$211$	19.3953	1.33523	0.667613	−	0.744508i	$-0.267316\pi$
$211$	19.3953	1.33523	0.667613	+	0.744508i	$0.267316\pi$
$212$	9.22586	0.633635
$213$	−2.27297	−0.155741
$214$	4.63058	0.316540
$215$	8.76203	0.597566
$216$	−9.28481	−0.631752
$217$	0	0
$218$	−3.86926	−0.262059
$219$	5.08997	0.343948
$220$	−8.92613	−0.601800
$221$	−7.87923	−0.530014
$222$	−3.70812	−0.248873
$223$	0.782481	0.0523988	0.0261994	−	0.999657i	$-0.491660\pi$
$223$	0.782481	0.0523988	0.0261994	+	0.999657i	$0.491660\pi$
$224$	0	0
$225$	−14.1597	−0.943980
$226$	−10.1986	−0.678403
$227$	−14.9208	−0.990327	−0.495163	−	0.868800i	$-0.664892\pi$
$227$	−14.9208	−0.990327	−0.495163	+	0.868800i	$0.664892\pi$
$228$	−6.14517	−0.406974
$229$	13.0261	0.860790	0.430395	−	0.902641i	$-0.358374\pi$
$229$	13.0261	0.860790	0.430395	+	0.902641i	$0.358374\pi$
$230$	1.75139	0.115483
$231$	0	0
$232$	17.8273	1.17042
$233$	9.54841	0.625537	0.312769	−	0.949829i	$-0.398744\pi$
$233$	9.54841	0.625537	0.312769	+	0.949829i	$0.398744\pi$
$234$	6.93010	0.453035
$235$	−22.4710	−1.46585
$236$	4.32262	0.281378
$237$	1.37699	0.0894454
$238$	0	0
$239$	20.2311	1.30864	0.654319	−	0.756218i	$-0.272955\pi$
$239$	20.2311	1.30864	0.654319	+	0.756218i	$0.272955\pi$
$240$	−8.25584	−0.532912
$241$	−9.14283	−0.588941	−0.294471	−	0.955661i	$-0.595143\pi$
$241$	−9.14283	−0.588941	−0.294471	+	0.955661i	$0.595143\pi$
$242$	−4.47125	−0.287423
$243$	−16.1219	−1.03422
$244$	9.06349	0.580230
$245$	0	0
$246$	2.23308	0.142376
$247$	24.0648	1.53121
$248$	18.0282	1.14479
$249$	12.1190	0.768011
$250$	3.15505	0.199543
$251$	−5.09300	−0.321467	−0.160734	−	0.986998i	$-0.551386\pi$
$251$	−5.09300	−0.321467	−0.160734	+	0.986998i	$0.551386\pi$
$252$	0	0
$253$	1.49322	0.0938781
$254$	4.47699	0.280912
$255$	−3.97222	−0.248750
$256$	−0.981070	−0.0613169
$257$	−2.83850	−0.177061	−0.0885303	−	0.996073i	$-0.528217\pi$
$257$	−2.83850	−0.177061	−0.0885303	+	0.996073i	$0.528217\pi$
$258$	−1.24596	−0.0775704
$259$	0	0
$260$	39.0311	2.42061
$261$	19.4644	1.20482
$262$	−0.805059	−0.0497367
$263$	−0.581050	−0.0358291	−0.0179145	−	0.999840i	$-0.505703\pi$
$263$	−0.581050	−0.0358291	−0.0179145	+	0.999840i	$0.505703\pi$
$264$	2.72819	0.167909
$265$	−18.2140	−1.11887
$266$	0	0
$267$	−3.79329	−0.232146
$268$	5.99060	0.365934
$269$	−21.4601	−1.30845	−0.654224	−	0.756301i	$-0.727005\pi$
$269$	−21.4601	−1.30845	−0.654224	+	0.756301i	$0.727005\pi$
$270$	8.52824	0.519012
$271$	15.8910	0.965312	0.482656	−	0.875810i	$-0.339672\pi$
$271$	15.8910	0.965312	0.482656	+	0.875810i	$0.339672\pi$
$272$	3.02657	0.183513
$273$	0	0
$274$	−1.35697	−0.0819773
$275$	10.1561	0.612436
$276$	1.66734	0.100362
$277$	−22.4011	−1.34595	−0.672975	−	0.739665i	$-0.734984\pi$
$277$	−22.4011	−1.34595	−0.672975	+	0.739665i	$0.734984\pi$
$278$	3.19456	0.191597
$279$	19.6838	1.17844
$280$	0	0
$281$	−2.27992	−0.136009	−0.0680043	−	0.997685i	$-0.521663\pi$
$281$	−2.27992	−0.136009	−0.0680043	+	0.997685i	$0.521663\pi$
$282$	3.19539	0.190283
$283$	−9.94463	−0.591147	−0.295573	−	0.955320i	$-0.595511\pi$
$283$	−9.94463	−0.591147	−0.295573	+	0.955320i	$0.595511\pi$
$284$	−4.12772	−0.244935
$285$	12.1320	0.718636
$286$	−4.97064	−0.293920
$287$	0	0
$288$	−10.6012	−0.624684
$289$	−15.5438	−0.914341
$290$	−16.3747	−0.961554
$291$	5.92532	0.347348
$292$	9.24341	0.540929
$293$	−3.65723	−0.213657	−0.106829	−	0.994277i	$-0.534070\pi$
$293$	−3.65723	−0.213657	−0.106829	+	0.994277i	$0.534070\pi$
$294$	0	0
$295$	−8.53383	−0.496859
$296$	−14.4738	−0.841271
$297$	7.27112	0.421913
$298$	5.14144	0.297836
$299$	−6.52938	−0.377604
$300$	11.3403	0.654735
$301$	0	0
$302$	5.19909	0.299174
$303$	0.637104	0.0366007
$304$	−9.24377	−0.530167
$305$	−17.8934	−1.02457
$306$	−1.28079	−0.0732181
$307$	4.84852	0.276719	0.138360	−	0.990382i	$-0.455817\pi$
$307$	4.84852	0.276719	0.138360	+	0.990382i	$0.455817\pi$
$308$	0	0
$309$	−16.3438	−0.929767
$310$	−16.5592	−0.940499
$311$	11.7996	0.669094	0.334547	−	0.942379i	$-0.391417\pi$
$311$	11.7996	0.669094	0.334547	+	0.942379i	$0.391417\pi$
$312$	−11.9295	−0.675375
$313$	−23.7245	−1.34099	−0.670493	−	0.741916i	$-0.733917\pi$
$313$	−23.7245	−1.34099	−0.670493	+	0.741916i	$0.733917\pi$
$314$	5.45660	0.307934
$315$	0	0
$316$	2.50063	0.140671
$317$	−22.0810	−1.24019	−0.620097	−	0.784525i	$-0.712907\pi$
$317$	−22.0810	−1.24019	−0.620097	+	0.784525i	$0.712907\pi$
$318$	2.59003	0.145242
$319$	−13.9609	−0.781662
$320$	−8.31367	−0.464748
$321$	−8.70310	−0.485759
$322$	0	0
$323$	−4.44756	−0.247469
$324$	−2.74889	−0.152716
$325$	−44.4093	−2.46339
$326$	6.37997	0.353354
$327$	7.27221	0.402154
$328$	8.71630	0.481277
$329$	0	0
$330$	−2.50589	−0.137944
$331$	6.87069	0.377647	0.188824	−	0.982011i	$-0.439533\pi$
$331$	6.87069	0.377647	0.188824	+	0.982011i	$0.439533\pi$
$332$	22.0082	1.20786
$333$	−15.8029	−0.865994
$334$	−3.49866	−0.191438
$335$	−11.8268	−0.646167
$336$	0	0
$337$	−29.3124	−1.59675	−0.798375	−	0.602160i	$-0.794307\pi$
$337$	−29.3124	−1.59675	−0.798375	+	0.602160i	$0.794307\pi$
$338$	15.1074	0.821732
$339$	19.1682	1.04107
$340$	−7.21357	−0.391211
$341$	−14.1183	−0.764547
$342$	3.91180	0.211526
$343$	0	0
$344$	−4.86333	−0.262213
$345$	−3.29171	−0.177220
$346$	5.45508	0.293267
$347$	−1.09029	−0.0585298	−0.0292649	−	0.999572i	$-0.509317\pi$
$347$	−1.09029	−0.0585298	−0.0292649	+	0.999572i	$0.509317\pi$
$348$	−15.5888	−0.835650
$349$	−25.9232	−1.38764	−0.693818	−	0.720150i	$-0.744073\pi$
$349$	−25.9232	−1.38764	−0.693818	+	0.720150i	$0.744073\pi$
$350$	0	0
$351$	−31.7943	−1.69705
$352$	7.60377	0.405282
$353$	−24.2429	−1.29032	−0.645159	−	0.764048i	$-0.723209\pi$
$353$	−24.2429	−1.29032	−0.645159	+	0.764048i	$0.723209\pi$
$354$	1.21351	0.0644975
$355$	8.14906	0.432507
$356$	−6.88864	−0.365097
$357$	0	0
$358$	11.2261	0.593317
$359$	−27.1742	−1.43420	−0.717098	−	0.696972i	$-0.754530\pi$
$359$	−27.1742	−1.43420	−0.717098	+	0.696972i	$0.754530\pi$
$360$	13.6369	0.718729
$361$	−5.41626	−0.285066
$362$	2.60085	0.136698
$363$	8.40364	0.441077
$364$	0	0
$365$	−18.2486	−0.955174
$366$	2.54445	0.133000
$367$	3.53051	0.184291	0.0921456	−	0.995746i	$-0.470627\pi$
$367$	3.53051	0.184291	0.0921456	+	0.995746i	$0.470627\pi$
$368$	2.50807	0.130742
$369$	9.51674	0.495422
$370$	13.2944	0.691141
$371$	0	0
$372$	−15.7645	−0.817352
$373$	−9.11684	−0.472052	−0.236026	−	0.971747i	$-0.575845\pi$
$373$	−9.11684	−0.472052	−0.236026	+	0.971747i	$0.575845\pi$
$374$	0.918653	0.0475024
$375$	−5.92987	−0.306217
$376$	12.4724	0.643217
$377$	61.0467	3.14406
$378$	0	0
$379$	11.6397	0.597889	0.298944	−	0.954271i	$-0.403365\pi$
$379$	11.6397	0.597889	0.298944	+	0.954271i	$0.403365\pi$
$380$	22.0317	1.13020
$381$	−8.41444	−0.431085
$382$	5.44490	0.278585
$383$	24.5489	1.25439	0.627195	−	0.778863i	$-0.284203\pi$
$383$	24.5489	1.25439	0.627195	+	0.778863i	$0.284203\pi$
$384$	10.9408	0.558321
$385$	0	0
$386$	1.19142	0.0606416
$387$	−5.30994	−0.269919
$388$	10.7604	0.546277
$389$	−16.9701	−0.860421	−0.430210	−	0.902729i	$-0.641561\pi$
$389$	−16.9701	−0.860421	−0.430210	+	0.902729i	$0.641561\pi$
$390$	10.9574	0.554851
$391$	1.20673	0.0610272
$392$	0	0
$393$	1.51310	0.0763256
$394$	1.29785	0.0653845
$395$	−4.93681	−0.248398
$396$	5.40938	0.271832
$397$	10.9799	0.551063	0.275532	−	0.961292i	$-0.411146\pi$
$397$	10.9799	0.551063	0.275532	+	0.961292i	$0.411146\pi$
$398$	−0.301407	−0.0151082
$399$	0	0
$400$	17.0585	0.852927
$401$	37.7091	1.88310	0.941551	−	0.336870i	$-0.109368\pi$
$401$	37.7091	1.88310	0.941551	+	0.336870i	$0.109368\pi$
$402$	1.68178	0.0838794
$403$	61.7346	3.07522
$404$	1.15698	0.0575621
$405$	5.42694	0.269667
$406$	0	0
$407$	11.3347	0.561840
$408$	2.20476	0.109152
$409$	16.8207	0.831732	0.415866	−	0.909426i	$-0.363479\pi$
$409$	16.8207	0.831732	0.415866	+	0.909426i	$0.363479\pi$
$410$	−8.00606	−0.395391
$411$	2.55039	0.125802
$412$	−29.6805	−1.46225
$413$	0	0
$414$	−1.06137	−0.0521635
$415$	−43.4492	−2.13284
$416$	−33.2488	−1.63016
$417$	−6.00413	−0.294023
$418$	−2.80575	−0.137234
$419$	−2.16120	−0.105582	−0.0527908	−	0.998606i	$-0.516812\pi$
$419$	−2.16120	−0.105582	−0.0527908	+	0.998606i	$0.516812\pi$
$420$	0	0
$421$	4.50556	0.219587	0.109794	−	0.993954i	$-0.464981\pi$
$421$	4.50556	0.219587	0.109794	+	0.993954i	$0.464981\pi$
$422$	9.88806	0.481343
$423$	13.6178	0.662120
$424$	10.1096	0.490964
$425$	8.20755	0.398125
$426$	−1.15880	−0.0561440
$427$	0	0
$428$	−15.8049	−0.763957
$429$	9.34223	0.451047
$430$	4.46704	0.215420
$431$	−11.2210	−0.540496	−0.270248	−	0.962791i	$-0.587106\pi$
$431$	−11.2210	−0.540496	−0.270248	+	0.962791i	$0.587106\pi$
$432$	12.2128	0.587590
$433$	2.63305	0.126536	0.0632682	−	0.997997i	$-0.479848\pi$
$433$	2.63305	0.126536	0.0632682	+	0.997997i	$0.479848\pi$
$434$	0	0
$435$	30.7759	1.47559
$436$	13.2064	0.632470
$437$	−3.68561	−0.176307
$438$	2.59496	0.123992
$439$	−25.3713	−1.21091	−0.605454	−	0.795880i	$-0.707008\pi$
$439$	−25.3713	−1.21091	−0.605454	+	0.795880i	$0.707008\pi$
$440$	−9.78113	−0.466297
$441$	0	0
$442$	−4.01697	−0.191068
$443$	15.3032	0.727079	0.363539	−	0.931579i	$-0.381568\pi$
$443$	15.3032	0.727079	0.363539	+	0.931579i	$0.381568\pi$
$444$	12.6564	0.600645
$445$	13.5997	0.644690
$446$	0.398923	0.0188895
$447$	−9.66326	−0.457057
$448$	0	0
$449$	14.6532	0.691527	0.345764	−	0.938322i	$-0.387620\pi$
$449$	14.6532	0.691527	0.345764	+	0.938322i	$0.387620\pi$
$450$	−7.21887	−0.340301
$451$	−6.82591	−0.321420
$452$	34.8095	1.63730
$453$	−9.77161	−0.459110
$454$	−7.60688	−0.357008
$455$	0	0
$456$	−6.73380	−0.315339
$457$	−5.74711	−0.268839	−0.134419	−	0.990925i	$-0.542917\pi$
$457$	−5.74711	−0.268839	−0.134419	+	0.990925i	$0.542917\pi$
$458$	6.64094	0.310311
$459$	5.87609	0.274272
$460$	−5.97776	−0.278714
$461$	15.0938	0.702988	0.351494	−	0.936190i	$-0.385674\pi$
$461$	15.0938	0.702988	0.351494	+	0.936190i	$0.385674\pi$
$462$	0	0
$463$	9.45814	0.439557	0.219779	−	0.975550i	$-0.429466\pi$
$463$	9.45814	0.439557	0.219779	+	0.975550i	$0.429466\pi$
$464$	−23.4493	−1.08861
$465$	31.1227	1.44328
$466$	4.86795	0.225503
$467$	6.65100	0.307772	0.153886	−	0.988089i	$-0.450821\pi$
$467$	6.65100	0.307772	0.153886	+	0.988089i	$0.450821\pi$
$468$	−23.6535	−1.09338
$469$	0	0
$470$	−11.4561	−0.528431
$471$	−10.2556	−0.472553
$472$	4.73666	0.218023
$473$	3.80857	0.175118
$474$	0.702017	0.0322447
$475$	−25.0675	−1.15018
$476$	0	0
$477$	11.0380	0.505393
$478$	10.3142	0.471759
$479$	34.4642	1.57471	0.787356	−	0.616499i	$-0.211450\pi$
$479$	34.4642	1.57471	0.787356	+	0.616499i	$0.211450\pi$
$480$	−16.7620	−0.765077
$481$	−49.5629	−2.25988
$482$	−4.66118	−0.212311
$483$	0	0
$484$	15.2610	0.693684
$485$	−21.2435	−0.964617
$486$	−8.21925	−0.372833
$487$	−18.0310	−0.817063	−0.408532	−	0.912744i	$-0.633959\pi$
$487$	−18.0310	−0.817063	−0.408532	+	0.912744i	$0.633959\pi$
$488$	9.93165	0.449584
$489$	−11.9911	−0.542255
$490$	0	0
$491$	−17.8672	−0.806337	−0.403169	−	0.915126i	$-0.632091\pi$
$491$	−17.8672	−0.806337	−0.403169	+	0.915126i	$0.632091\pi$
$492$	−7.62185	−0.343619
$493$	−11.2824	−0.508134
$494$	12.2687	0.551993
$495$	−10.6793	−0.480001
$496$	−23.7135	−1.06477
$497$	0	0
$498$	6.17849	0.276865
$499$	−6.87725	−0.307868	−0.153934	−	0.988081i	$-0.549194\pi$
$499$	−6.87725	−0.307868	−0.153934	+	0.988081i	$0.549194\pi$
$500$	−10.7687	−0.481590
$501$	6.57568	0.293780
$502$	−2.59650	−0.115888
$503$	7.33863	0.327213	0.163607	−	0.986526i	$-0.447687\pi$
$503$	7.33863	0.327213	0.163607	+	0.986526i	$0.447687\pi$
$504$	0	0
$505$	−2.28415	−0.101643
$506$	0.761272	0.0338427
$507$	−28.3940	−1.26102
$508$	−15.2807	−0.677970
$509$	−27.9520	−1.23895	−0.619475	−	0.785017i	$-0.712654\pi$
$509$	−27.9520	−1.23895	−0.619475	+	0.785017i	$0.712654\pi$
$510$	−2.02511	−0.0896733
$511$	0	0
$512$	22.3361	0.987127
$513$	−17.9468	−0.792369
$514$	−1.44712	−0.0638296
$515$	58.5960	2.58205
$516$	4.25267	0.187213
$517$	−9.76741	−0.429570
$518$	0	0
$519$	−10.2527	−0.450045
$520$	42.7697	1.87558
$521$	−18.8305	−0.824981	−0.412491	−	0.910962i	$-0.635341\pi$
$521$	−18.8305	−0.824981	−0.412491	+	0.910962i	$0.635341\pi$
$522$	9.92332	0.434332
$523$	17.5606	0.767872	0.383936	−	0.923360i	$-0.374568\pi$
$523$	17.5606	0.767872	0.383936	+	0.923360i	$0.374568\pi$
$524$	2.74779	0.120038
$525$	0	0
$526$	−0.296230	−0.0129162
$527$	−11.4095	−0.497007
$528$	−3.58854	−0.156171
$529$	1.00000	0.0434783
$530$	−9.28580	−0.403349
$531$	5.17164	0.224430
$532$	0	0
$533$	29.8475	1.29284
$534$	−1.93389	−0.0836876
$535$	31.2024	1.34900
$536$	6.56441	0.283540
$537$	−21.0993	−0.910500
$538$	−10.9408	−0.471690
$539$	0	0
$540$	−29.1082	−1.25262
$541$	18.2254	0.783570	0.391785	−	0.920057i	$-0.371858\pi$
$541$	18.2254	0.783570	0.391785	+	0.920057i	$0.371858\pi$
$542$	8.10154	0.347991
$543$	−4.88826	−0.209775
$544$	6.14491	0.263461
$545$	−26.0724	−1.11682
$546$	0	0
$547$	−44.1231	−1.88657	−0.943284	−	0.331987i	$-0.892281\pi$
$547$	−44.1231	−1.88657	−0.943284	+	0.331987i	$0.892281\pi$
$548$	4.63153	0.197849
$549$	10.8437	0.462797
$550$	5.17776	0.220780
$551$	34.4587	1.46799
$552$	1.82705	0.0777644
$553$	0	0
$554$	−11.4205	−0.485209
$555$	−24.9866	−1.06062
$556$	−10.9035	−0.462412
$557$	1.24160	0.0526082	0.0263041	−	0.999654i	$-0.491626\pi$
$557$	1.24160	0.0526082	0.0263041	+	0.999654i	$0.491626\pi$
$558$	10.0351	0.424822
$559$	−16.6536	−0.704374
$560$	0	0
$561$	−1.72659	−0.0728968
$562$	−1.16234	−0.0490305
$563$	23.6164	0.995314	0.497657	−	0.867374i	$-0.334194\pi$
$563$	23.6164	0.995314	0.497657	+	0.867374i	$0.334194\pi$
$564$	−10.9063	−0.459240
$565$	−68.7218	−2.89115
$566$	−5.06995	−0.213106
$567$	0	0
$568$	−4.52310	−0.189785
$569$	−22.2390	−0.932307	−0.466153	−	0.884704i	$-0.654361\pi$
$569$	−22.2390	−0.932307	−0.466153	+	0.884704i	$0.654361\pi$
$570$	6.18509	0.259065
$571$	−36.8065	−1.54030	−0.770152	−	0.637861i	$-0.779820\pi$
$571$	−36.8065	−1.54030	−0.770152	+	0.637861i	$0.779820\pi$
$572$	16.9655	0.709365
$573$	−10.2336	−0.427515
$574$	0	0
$575$	6.80146	0.283640
$576$	5.03822	0.209926
$577$	−3.26446	−0.135901	−0.0679505	−	0.997689i	$-0.521646\pi$
$577$	−3.26446	−0.135901	−0.0679505	+	0.997689i	$0.521646\pi$
$578$	−7.92450	−0.329616
$579$	−2.23925	−0.0930602
$580$	55.8892	2.32067
$581$	0	0
$582$	3.02083	0.125218
$583$	−7.91701	−0.327889
$584$	10.1288	0.419133
$585$	46.6973	1.93070
$586$	−1.86452	−0.0770226
$587$	14.8229	0.611809	0.305904	−	0.952062i	$-0.401041\pi$
$587$	14.8229	0.611809	0.305904	+	0.952062i	$0.401041\pi$
$588$	0	0
$589$	34.8471	1.43585
$590$	−4.35070	−0.179115
$591$	−2.43928	−0.100339
$592$	19.0381	0.782463
$593$	−29.7501	−1.22169	−0.610845	−	0.791750i	$-0.709170\pi$
$593$	−29.7501	−1.22169	−0.610845	+	0.791750i	$0.709170\pi$
$594$	3.70695	0.152098
$595$	0	0
$596$	−17.5485	−0.718816
$597$	0.566490	0.0231849
$598$	−3.32880	−0.136125
$599$	13.6928	0.559473	0.279736	−	0.960077i	$-0.409753\pi$
$599$	13.6928	0.559473	0.279736	+	0.960077i	$0.409753\pi$
$600$	12.4266	0.507314
$601$	−27.5265	−1.12283	−0.561414	−	0.827535i	$-0.689742\pi$
$601$	−27.5265	−1.12283	−0.561414	+	0.827535i	$0.689742\pi$
$602$	0	0
$603$	7.16724	0.291872
$604$	−17.7453	−0.722046
$605$	−30.1288	−1.22491
$606$	0.324807	0.0131944
$607$	−27.3800	−1.11132	−0.555660	−	0.831410i	$-0.687534\pi$
$607$	−27.3800	−1.11132	−0.555660	+	0.831410i	$0.687534\pi$
$608$	−18.7678	−0.761135
$609$	0	0
$610$	−9.12237	−0.369354
$611$	42.7097	1.72785
$612$	4.37154	0.176709
$613$	−20.9862	−0.847624	−0.423812	−	0.905750i	$-0.639308\pi$
$613$	−20.9862	−0.847624	−0.423812	+	0.905750i	$0.639308\pi$
$614$	2.47186	0.0997561
$615$	15.0473	0.606764
$616$	0	0
$617$	43.5925	1.75497	0.877483	−	0.479607i	$-0.159221\pi$
$617$	43.5925	1.75497	0.877483	+	0.479607i	$0.159221\pi$
$618$	−8.33237	−0.335177
$619$	−35.8937	−1.44269	−0.721345	−	0.692575i	$-0.756476\pi$
$619$	−35.8937	−1.44269	−0.721345	+	0.692575i	$0.756476\pi$
$620$	56.5191	2.26986
$621$	4.86941	0.195403
$622$	6.01565	0.241206
$623$	0	0
$624$	15.6915	0.628164
$625$	−12.7475	−0.509899
$626$	−12.0952	−0.483420
$627$	5.27337	0.210598
$628$	−18.6242	−0.743187
$629$	9.16002	0.365234
$630$	0	0
$631$	−5.68711	−0.226400	−0.113200	−	0.993572i	$-0.536110\pi$
$631$	−5.68711	−0.226400	−0.113200	+	0.993572i	$0.536110\pi$
$632$	2.74016	0.108998
$633$	−18.5845	−0.738666
$634$	−11.2573	−0.447085
$635$	30.1675	1.19716
$636$	−8.84017	−0.350536
$637$	0	0
$638$	−7.11753	−0.281786
$639$	−4.93846	−0.195363
$640$	−39.2251	−1.55051
$641$	36.2495	1.43177	0.715885	−	0.698218i	$-0.246024\pi$
$641$	36.2495	1.43177	0.715885	+	0.698218i	$0.246024\pi$
$642$	−4.43699	−0.175114
$643$	−19.0700	−0.752047	−0.376024	−	0.926610i	$-0.622709\pi$
$643$	−19.0700	−0.752047	−0.376024	+	0.926610i	$0.622709\pi$
$644$	0	0
$645$	−8.39573	−0.330582
$646$	−2.26744	−0.0892114
$647$	−0.338385	−0.0133033	−0.00665164	−	0.999978i	$-0.502117\pi$
$647$	−0.338385	−0.0133033	−0.00665164	+	0.999978i	$0.502117\pi$
$648$	−3.01220	−0.118330
$649$	−3.70938	−0.145606
$650$	−22.6407	−0.888040
$651$	0	0
$652$	−21.7758	−0.852807
$653$	3.32508	0.130120	0.0650602	−	0.997881i	$-0.479276\pi$
$653$	3.32508	0.130120	0.0650602	+	0.997881i	$0.479276\pi$
$654$	3.70750	0.144975
$655$	−5.42476	−0.211963
$656$	−11.4650	−0.447635
$657$	11.0589	0.431450
$658$	0	0
$659$	−19.7596	−0.769725	−0.384862	−	0.922974i	$-0.625751\pi$
$659$	−19.7596	−0.769725	−0.384862	+	0.922974i	$0.625751\pi$
$660$	8.55297	0.332924
$661$	12.6166	0.490731	0.245365	−	0.969431i	$-0.421092\pi$
$661$	12.6166	0.490731	0.245365	+	0.969431i	$0.421092\pi$
$662$	3.50280	0.136140
$663$	7.54984	0.293211
$664$	24.1163	0.935893
$665$	0	0
$666$	−8.05661	−0.312187
$667$	−9.34953	−0.362015
$668$	11.9415	0.462029
$669$	−0.749769	−0.0289877
$670$	−6.02951	−0.232940
$671$	−7.77767	−0.300254
$672$	0	0
$673$	7.12771	0.274753	0.137376	−	0.990519i	$-0.456133\pi$
$673$	7.12771	0.274753	0.137376	+	0.990519i	$0.456133\pi$
$674$	−14.9440	−0.575621
$675$	33.1191	1.27475
$676$	−51.5637	−1.98322
$677$	43.8383	1.68484	0.842421	−	0.538820i	$-0.181130\pi$
$677$	43.8383	1.68484	0.842421	+	0.538820i	$0.181130\pi$
$678$	9.77227	0.375302
$679$	0	0
$680$	−7.90453	−0.303125
$681$	14.2970	0.547862
$682$	−7.19774	−0.275616
$683$	−32.7059	−1.25146	−0.625728	−	0.780041i	$-0.715198\pi$
$683$	−32.7059	−1.25146	−0.625728	+	0.780041i	$0.715198\pi$
$684$	−13.3516	−0.510510
$685$	−9.14369	−0.349363
$686$	0	0
$687$	−12.4815	−0.476201
$688$	6.39701	0.243884
$689$	34.6185	1.31886
$690$	−1.67817	−0.0638869
$691$	38.2634	1.45561	0.727805	−	0.685784i	$-0.240541\pi$
$691$	38.2634	1.45561	0.727805	+	0.685784i	$0.240541\pi$
$692$	−18.6190	−0.707788
$693$	0	0
$694$	−0.555849	−0.0210997
$695$	21.5260	0.816529
$696$	−17.0820	−0.647493
$697$	−5.51630	−0.208945
$698$	−13.2161	−0.500237
$699$	−9.14923	−0.346056
$700$	0	0
$701$	−26.5559	−1.00300	−0.501501	−	0.865157i	$-0.667219\pi$
$701$	−26.5559	−1.00300	−0.501501	+	0.865157i	$0.667219\pi$
$702$	−16.2093	−0.611780
$703$	−27.9766	−1.05516
$704$	−3.61368	−0.136196
$705$	21.5316	0.810927
$706$	−12.3595	−0.465154
$707$	0	0
$708$	−4.14191	−0.155662
$709$	44.5580	1.67341	0.836706	−	0.547652i	$-0.184478\pi$
$709$	44.5580	1.67341	0.836706	+	0.547652i	$0.184478\pi$
$710$	4.15453	0.155917
$711$	2.99179	0.112201
$712$	−7.54848	−0.282891
$713$	−9.45489	−0.354088
$714$	0	0
$715$	−33.4938	−1.25260
$716$	−38.3164	−1.43195
$717$	−19.3853	−0.723957
$718$	−13.8539	−0.517022
$719$	23.0725	0.860460	0.430230	−	0.902719i	$-0.358433\pi$
$719$	23.0725	0.860460	0.430230	+	0.902719i	$0.358433\pi$
$720$	−17.9374	−0.668488
$721$	0	0
$722$	−2.76131	−0.102765
$723$	8.76061	0.325811
$724$	−8.87710	−0.329915
$725$	−63.5904	−2.36169
$726$	4.28432	0.159006
$727$	−19.2060	−0.712311	−0.356155	−	0.934427i	$-0.615913\pi$
$727$	−19.2060	−0.712311	−0.356155	+	0.934427i	$0.615913\pi$
$728$	0	0
$729$	10.7087	0.396619
$730$	−9.30345	−0.344336
$731$	3.07786	0.113839
$732$	−8.68458	−0.320991
$733$	−20.2070	−0.746364	−0.373182	−	0.927758i	$-0.621733\pi$
$733$	−20.2070	−0.746364	−0.373182	+	0.927758i	$0.621733\pi$
$734$	1.79992	0.0664362
$735$	0	0
$736$	5.09218	0.187700
$737$	−5.14072	−0.189361
$738$	4.85180	0.178597
$739$	0.592441	0.0217933	0.0108966	−	0.999941i	$-0.496531\pi$
$739$	0.592441	0.0217933	0.0108966	+	0.999941i	$0.496531\pi$
$740$	−45.3757	−1.66804
$741$	−23.0587	−0.847084
$742$	0	0
$743$	6.43191	0.235964	0.117982	−	0.993016i	$-0.462358\pi$
$743$	6.43191	0.235964	0.117982	+	0.993016i	$0.462358\pi$
$744$	−17.2745	−0.633316
$745$	34.6448	1.26929
$746$	−4.64793	−0.170173
$747$	26.3309	0.963398
$748$	−3.13550	−0.114645
$749$	0	0
$750$	−3.02315	−0.110390
$751$	−51.4557	−1.87765	−0.938823	−	0.344400i	$-0.888082\pi$
$751$	−51.4557	−1.87765	−0.938823	+	0.344400i	$0.888082\pi$
$752$	−16.4057	−0.598254
$753$	4.88008	0.177840
$754$	31.1227	1.13342
$755$	35.0333	1.27499
$756$	0	0
$757$	19.3207	0.702224	0.351112	−	0.936333i	$-0.385804\pi$
$757$	19.3207	0.702224	0.351112	+	0.936333i	$0.385804\pi$
$758$	5.93410	0.215536
$759$	−1.43080	−0.0519347
$760$	24.1420	0.875724
$761$	−10.2602	−0.371931	−0.185965	−	0.982556i	$-0.559541\pi$
$761$	−10.2602	−0.371931	−0.185965	+	0.982556i	$0.559541\pi$
$762$	−4.28983	−0.155404
$763$	0	0
$764$	−18.5843	−0.672355
$765$	−8.63042	−0.312033
$766$	12.5155	0.452202
$767$	16.2199	0.585667
$768$	0.940056	0.0339213
$769$	51.8423	1.86948	0.934741	−	0.355329i	$-0.115631\pi$
$769$	51.8423	1.86948	0.934741	+	0.355329i	$0.115631\pi$
$770$	0	0
$771$	2.71983	0.0979523
$772$	−4.06649	−0.146356
$773$	45.9169	1.65152	0.825759	−	0.564023i	$-0.190747\pi$
$773$	45.9169	1.65152	0.825759	+	0.564023i	$0.190747\pi$
$774$	−2.70710	−0.0973047
$775$	−64.3070	−2.30998
$776$	11.7911	0.423276
$777$	0	0
$778$	−8.65168	−0.310178
$779$	16.8479	0.603638
$780$	−37.3994	−1.33911
$781$	3.54213	0.126747
$782$	0.615215	0.0220000
$783$	−45.5267	−1.62699
$784$	0	0
$785$	36.7684	1.31232
$786$	0.771403	0.0275150
$787$	−29.2964	−1.04430	−0.522152	−	0.852852i	$-0.674871\pi$
$787$	−29.2964	−1.04430	−0.522152	+	0.852852i	$0.674871\pi$
$788$	−4.42974	−0.157803
$789$	0.556759	0.0198211
$790$	−2.51687	−0.0895464
$791$	0	0
$792$	5.92753	0.210626
$793$	34.0092	1.20770
$794$	5.59773	0.198656
$795$	17.4525	0.618977
$796$	1.02875	0.0364630
$797$	−7.17039	−0.253988	−0.126994	−	0.991903i	$-0.540533\pi$
$797$	−7.17039	−0.253988	−0.126994	+	0.991903i	$0.540533\pi$
$798$	0	0
$799$	−7.89344	−0.279250
$800$	34.6343	1.22451
$801$	−8.24167	−0.291205
$802$	19.2248	0.678850
$803$	−7.93206	−0.279916
$804$	−5.74016	−0.202440
$805$	0	0
$806$	31.4734	1.10860
$807$	20.5630	0.723851
$808$	1.26781	0.0446013
$809$	19.0325	0.669146	0.334573	−	0.942370i	$-0.391408\pi$
$809$	19.0325	0.669146	0.334573	+	0.942370i	$0.391408\pi$
$810$	2.76675	0.0972137
$811$	22.8493	0.802347	0.401173	−	0.916002i	$-0.368603\pi$
$811$	22.8493	0.802347	0.401173	+	0.916002i	$0.368603\pi$
$812$	0	0
$813$	−15.2267	−0.534024
$814$	5.77863	0.202541
$815$	42.9904	1.50589
$816$	−2.90005	−0.101522
$817$	−9.40041	−0.328879
$818$	8.57551	0.299836
$819$	0	0
$820$	27.3259	0.954261
$821$	−32.2479	−1.12546	−0.562730	−	0.826640i	$-0.690249\pi$
$821$	−32.2479	−1.12546	−0.562730	+	0.826640i	$0.690249\pi$
$822$	1.30024	0.0453510
$823$	26.5759	0.926378	0.463189	−	0.886260i	$-0.346705\pi$
$823$	26.5759	0.926378	0.463189	+	0.886260i	$0.346705\pi$
$824$	−32.5234	−1.13301
$825$	−9.73151	−0.338808
$826$	0	0
$827$	−40.0017	−1.39099	−0.695497	−	0.718529i	$-0.744816\pi$
$827$	−40.0017	−1.39099	−0.695497	+	0.718529i	$0.744816\pi$
$828$	3.62262	0.125895
$829$	−34.0508	−1.18263	−0.591317	−	0.806439i	$-0.701392\pi$
$829$	−34.0508	−1.18263	−0.591317	+	0.806439i	$0.701392\pi$
$830$	−22.1512	−0.768878
$831$	21.4646	0.744598
$832$	15.8015	0.547817
$833$	0	0
$834$	−3.06101	−0.105994
$835$	−23.5752	−0.815852
$836$	9.57646	0.331209
$837$	−46.0398	−1.59137
$838$	−1.10182	−0.0380617
$839$	−15.8357	−0.546707	−0.273354	−	0.961914i	$-0.588133\pi$
$839$	−15.8357	−0.546707	−0.273354	+	0.961914i	$0.588133\pi$
$840$	0	0
$841$	58.4137	2.01426
$842$	2.29701	0.0791603
$843$	2.18461	0.0752419
$844$	−33.7495	−1.16170
$845$	101.798	3.50197
$846$	6.94260	0.238691
$847$	0	0
$848$	−13.2977	−0.456644
$849$	9.52889	0.327031
$850$	4.18436	0.143522
$851$	7.59075	0.260208
$852$	3.95516	0.135501
$853$	5.63858	0.193061	0.0965306	−	0.995330i	$-0.469225\pi$
$853$	5.63858	0.193061	0.0965306	+	0.995330i	$0.469225\pi$
$854$	0	0
$855$	26.3590	0.901460
$856$	−17.3188	−0.591943
$857$	−46.0120	−1.57174	−0.785870	−	0.618391i	$-0.787785\pi$
$857$	−46.0120	−1.57174	−0.785870	+	0.618391i	$0.787785\pi$
$858$	4.76284	0.162601
$859$	−29.9515	−1.02193	−0.510965	−	0.859601i	$-0.670712\pi$
$859$	−29.9515	−1.02193	−0.510965	+	0.859601i	$0.670712\pi$
$860$	−15.2467	−0.519908
$861$	0	0
$862$	−5.72065	−0.194846
$863$	−50.0745	−1.70456	−0.852278	−	0.523090i	$-0.824779\pi$
$863$	−50.0745	−1.70456	−0.852278	+	0.523090i	$0.824779\pi$
$864$	24.7959	0.843575
$865$	36.7581	1.24981
$866$	1.34238	0.0456158
$867$	14.8940	0.505826
$868$	0	0
$869$	−2.14587	−0.0727937
$870$	15.6901	0.531945
$871$	22.4787	0.761662
$872$	14.4714	0.490062
$873$	12.8739	0.435716
$874$	−1.87899	−0.0635578
$875$	0	0
$876$	−8.85698	−0.299250
$877$	11.0626	0.373558	0.186779	−	0.982402i	$-0.440195\pi$
$877$	11.0626	0.373558	0.186779	+	0.982402i	$0.440195\pi$
$878$	−12.9348	−0.436527
$879$	3.50433	0.118198
$880$	12.8657	0.433701
$881$	45.8121	1.54345	0.771725	−	0.635956i	$-0.219394\pi$
$881$	45.8121	1.54345	0.771725	+	0.635956i	$0.219394\pi$
$882$	0	0
$883$	−52.9427	−1.78167	−0.890833	−	0.454332i	$-0.849878\pi$
$883$	−52.9427	−1.78167	−0.890833	+	0.454332i	$0.849878\pi$
$884$	13.7105	0.461135
$885$	8.17706	0.274869
$886$	7.80187	0.262109
$887$	−25.7034	−0.863036	−0.431518	−	0.902104i	$-0.642022\pi$
$887$	−25.7034	−0.863036	−0.431518	+	0.902104i	$0.642022\pi$
$888$	13.8687	0.465402
$889$	0	0
$890$	6.93339	0.232408
$891$	2.35891	0.0790266
$892$	−1.36158	−0.0455892
$893$	24.1082	0.806750
$894$	−4.92650	−0.164767
$895$	75.6452	2.52854
$896$	0	0
$897$	6.25642	0.208896
$898$	7.47046	0.249293
$899$	88.3988	2.94826
$900$	24.6391	0.821303
$901$	−6.39806	−0.213150
$902$	−3.47997	−0.115870
$903$	0	0
$904$	38.1438	1.26864
$905$	17.5254	0.582564
$906$	−4.98174	−0.165507
$907$	11.0112	0.365620	0.182810	−	0.983148i	$-0.441481\pi$
$907$	11.0112	0.365620	0.182810	+	0.983148i	$0.441481\pi$
$908$	25.9634	0.861627
$909$	1.38423	0.0459121
$910$	0	0
$911$	−47.1465	−1.56203	−0.781017	−	0.624510i	$-0.785299\pi$
$911$	−47.1465	−1.56203	−0.781017	+	0.624510i	$0.785299\pi$
$912$	8.85733	0.293296
$913$	−18.8859	−0.625033
$914$	−2.92998	−0.0969151
$915$	17.1453	0.566807
$916$	−22.6666	−0.748924
$917$	0	0
$918$	2.99573	0.0988740
$919$	−38.5881	−1.27291	−0.636453	−	0.771316i	$-0.719599\pi$
$919$	−38.5881	−1.27291	−0.636453	+	0.771316i	$0.719599\pi$
$920$	−6.55035	−0.215959
$921$	−4.64582	−0.153085
$922$	7.69508	0.253424
$923$	−15.4886	−0.509813
$924$	0	0
$925$	51.6282	1.69752
$926$	4.82193	0.158458
$927$	−35.5101	−1.16631
$928$	−47.6095	−1.56286
$929$	−6.32784	−0.207610	−0.103805	−	0.994598i	$-0.533102\pi$
$929$	−6.32784	−0.207610	−0.103805	+	0.994598i	$0.533102\pi$
$930$	15.8669	0.520297
$931$	0	0
$932$	−16.6151	−0.544244
$933$	−11.3063	−0.370152
$934$	3.39080	0.110950
$935$	6.19019	0.202441
$936$	−25.9192	−0.847194
$937$	−13.5040	−0.441155	−0.220578	−	0.975369i	$-0.570794\pi$
$937$	−13.5040	−0.441155	−0.220578	+	0.975369i	$0.570794\pi$
$938$	0	0
$939$	22.7326	0.741852
$940$	39.1015	1.27535
$941$	−19.8342	−0.646576	−0.323288	−	0.946301i	$-0.604788\pi$
$941$	−19.8342	−0.646576	−0.323288	+	0.946301i	$0.604788\pi$
$942$	−5.22849	−0.170353
$943$	−4.57126	−0.148861
$944$	−6.23040	−0.202782
$945$	0	0
$946$	1.94168	0.0631294
$947$	−11.5510	−0.375356	−0.187678	−	0.982231i	$-0.560096\pi$
$947$	−11.5510	−0.375356	−0.187678	+	0.982231i	$0.560096\pi$
$948$	−2.39609	−0.0778214
$949$	34.6843	1.12590
$950$	−12.7799	−0.414634
$951$	21.1579	0.686093
$952$	0	0
$953$	−16.8320	−0.545243	−0.272622	−	0.962121i	$-0.587891\pi$
$953$	−16.8320	−0.545243	−0.272622	+	0.962121i	$0.587891\pi$
$954$	5.62735	0.182192
$955$	36.6896	1.18725
$956$	−35.2038	−1.13857
$957$	13.3773	0.432426
$958$	17.5705	0.567677
$959$	0	0
$960$	7.96611	0.257105
$961$	58.3950	1.88371
$962$	−25.2681	−0.814675
$963$	−18.9092	−0.609339
$964$	15.9093	0.512404
$965$	8.02818	0.258436
$966$	0	0
$967$	−14.2153	−0.457133	−0.228567	−	0.973528i	$-0.573404\pi$
$967$	−14.2153	−0.457133	−0.228567	+	0.973528i	$0.573404\pi$
$968$	16.7228	0.537493
$969$	4.26162	0.136903
$970$	−10.8303	−0.347740
$971$	−34.1487	−1.09588	−0.547941	−	0.836517i	$-0.684588\pi$
$971$	−34.1487	−1.09588	−0.547941	+	0.836517i	$0.684588\pi$
$972$	28.0536	0.899818
$973$	0	0
$974$	−9.19253	−0.294548
$975$	42.5528	1.36278
$976$	−13.0636	−0.418157
$977$	9.63902	0.308380	0.154190	−	0.988041i	$-0.450723\pi$
$977$	9.63902	0.308380	0.154190	+	0.988041i	$0.450723\pi$
$978$	−6.11326	−0.195480
$979$	5.91136	0.188928
$980$	0	0
$981$	15.8003	0.504464
$982$	−9.10904	−0.290681
$983$	−51.6806	−1.64836	−0.824178	−	0.566332i	$-0.808362\pi$
$983$	−51.6806	−1.64836	−0.824178	+	0.566332i	$0.808362\pi$
$984$	−8.35191	−0.266249
$985$	8.74532	0.278649
$986$	−5.75197	−0.183180
$987$	0	0
$988$	−41.8748	−1.33221
$989$	2.55057	0.0811034
$990$	−5.44452	−0.173038
$991$	37.1602	1.18043	0.590216	−	0.807245i	$-0.299043\pi$
$991$	37.1602	1.18043	0.590216	+	0.807245i	$0.299043\pi$
$992$	−48.1460	−1.52864
$993$	−6.58346	−0.208920
$994$	0	0
$995$	−2.03098	−0.0643865
$996$	−21.0881	−0.668203
$997$	41.9411	1.32829	0.664144	−	0.747605i	$-0.268796\pi$
$997$	41.9411	1.32829	0.664144	+	0.747605i	$0.268796\pi$
$998$	−3.50615	−0.110985
$999$	36.9625	1.16944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.a.k.1.4		7
7.3	odd	6		161.2.e.a.93.4	✓	14
7.5	odd	6		161.2.e.a.116.4	yes	14
7.6	odd	2		1127.2.a.n.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.e.a.93.4	✓	14	7.3	odd	6
161.2.e.a.116.4	yes	14	7.5	odd	6
1127.2.a.k.1.4		7	1.1	even	1	trivial
1127.2.a.n.1.4		7	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q+0.509818 q^{2} -0.958194 q^{3} -1.74009 q^{4} +3.43532 q^{5} -0.488505 q^{6} -1.90676 q^{8} -2.08186 q^{9} +1.75139 q^{10} +1.49322 q^{11} +1.66734 q^{12} -6.52938 q^{13} -3.29171 q^{15} +2.50807 q^{16} +1.20673 q^{17} -1.06137 q^{18} -3.68561 q^{19} -5.97776 q^{20} +0.761272 q^{22} +1.00000 q^{23} +1.82705 q^{24} +6.80146 q^{25} -3.32880 q^{26} +4.86941 q^{27} -9.34953 q^{29} -1.67817 q^{30} -9.45489 q^{31} +5.09218 q^{32} -1.43080 q^{33} +0.615215 q^{34} +3.62262 q^{36} +7.59075 q^{37} -1.87899 q^{38} +6.25642 q^{39} -6.55035 q^{40} -4.57126 q^{41} +2.55057 q^{43} -2.59834 q^{44} -7.15188 q^{45} +0.509818 q^{46} -6.54116 q^{47} -2.40322 q^{48} +3.46750 q^{50} -1.15629 q^{51} +11.3617 q^{52} -5.30196 q^{53} +2.48251 q^{54} +5.12971 q^{55} +3.53153 q^{57} -4.76656 q^{58} -2.48414 q^{59} +5.72786 q^{60} -5.20864 q^{61} -4.82027 q^{62} -2.42005 q^{64} -22.4306 q^{65} -0.729446 q^{66} -3.44270 q^{67} -2.09982 q^{68} -0.958194 q^{69} +2.37214 q^{71} +3.96962 q^{72} -5.31204 q^{73} +3.86990 q^{74} -6.51712 q^{75} +6.41328 q^{76} +3.18963 q^{78} -1.43707 q^{79} +8.61604 q^{80} +1.57975 q^{81} -2.33051 q^{82} -12.6478 q^{83} +4.14552 q^{85} +1.30033 q^{86} +8.95867 q^{87} -2.84722 q^{88} +3.95879 q^{89} -3.64615 q^{90} -1.74009 q^{92} +9.05962 q^{93} -3.33480 q^{94} -12.6613 q^{95} -4.87930 q^{96} -6.18384 q^{97} -3.10869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{3} + 6 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 9 q^{12} - 14 q^{13} - 3 q^{15} - 8 q^{16} - 4 q^{17} + 19 q^{18} - 9 q^{19} - 12 q^{20} - 10 q^{22} + 7 q^{23} + 4 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) 7 * q - 5 * q^3 + 6 * q^4 - 4 * q^5 - 6 * q^6 - 3 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 9 * q^12 - 14 * q^13 - 3 * q^15 - 8 * q^16 - 4 * q^17 + 19 * q^18 - 9 * q^19 - 12 * q^20 - 10 * q^22 + 7 * q^23 + 4 * q^24 + 3 * q^25 - 14 * q^26 - 20 * q^27 - 5 * q^29 - 17 * q^30 - 10 * q^31 + 19 * q^32 - 3 * q^33 - 20 * q^34 + 4 * q^36 - 5 * q^37 + 17 * q^38 + 18 * q^39 - 4 * q^40 - 23 * q^41 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 4 * q^47 - 11 * q^48 - 9 * q^50 + 7 * q^51 - 12 * q^52 + 2 * q^53 - 15 * q^54 - 17 * q^55 - 24 * q^57 - 7 * q^58 - 7 * q^59 + 32 * q^60 - 12 * q^61 + 4 * q^62 - 9 * q^64 + 9 * q^65 + 27 * q^66 + 2 * q^67 + 12 * q^68 - 5 * q^69 - 28 * q^71 - 20 * q^72 - 59 * q^73 + 2 * q^74 + 22 * q^75 - q^76 - 17 * q^78 - 4 * q^79 + 22 * q^80 + 11 * q^81 - 54 * q^82 + 9 * q^83 + 9 * q^85 - 15 * q^86 - 11 * q^87 - 33 * q^88 - q^89 - 2 * q^90 + 6 * q^92 + 15 * q^93 + 14 * q^94 - 16 * q^95 - 20 * q^96 - 35 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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