LMFDB - Embedded newform 4477.2.a.q.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4477,2,Mod(1,4477)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4477.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4477, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4477 = 11^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4477.a (trivial)

Newform invariants

comment:select newform

sage:traces = [28,0,-16,20,-12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$35.7490249849$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4477.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.394005 q^{2} +0.257216 q^{3} -1.84476 q^{4} -3.35100 q^{5} -0.101344 q^{6} -4.46729 q^{7} +1.51486 q^{8} -2.93384 q^{9} +1.32031 q^{10} -0.474502 q^{12} +3.38744 q^{13} +1.76014 q^{14} -0.861932 q^{15} +3.09266 q^{16} -4.89399 q^{17} +1.15595 q^{18} +2.38188 q^{19} +6.18180 q^{20} -1.14906 q^{21} +5.42031 q^{23} +0.389645 q^{24} +6.22923 q^{25} -1.33467 q^{26} -1.52628 q^{27} +8.24108 q^{28} +1.24704 q^{29} +0.339605 q^{30} +8.84266 q^{31} -4.24823 q^{32} +1.92826 q^{34} +14.9699 q^{35} +5.41223 q^{36} -1.00000 q^{37} -0.938474 q^{38} +0.871304 q^{39} -5.07629 q^{40} -1.22867 q^{41} +0.452735 q^{42} +7.36613 q^{43} +9.83131 q^{45} -2.13563 q^{46} -8.47310 q^{47} +0.795481 q^{48} +12.9567 q^{49} -2.45435 q^{50} -1.25881 q^{51} -6.24902 q^{52} -4.00750 q^{53} +0.601361 q^{54} -6.76730 q^{56} +0.612658 q^{57} -0.491341 q^{58} -4.03136 q^{59} +1.59006 q^{60} +4.64879 q^{61} -3.48405 q^{62} +13.1063 q^{63} -4.51149 q^{64} -11.3513 q^{65} +15.5176 q^{67} +9.02824 q^{68} +1.39419 q^{69} -5.89822 q^{70} -8.82283 q^{71} -4.44434 q^{72} -7.10968 q^{73} +0.394005 q^{74} +1.60226 q^{75} -4.39400 q^{76} -0.343298 q^{78} -3.08252 q^{79} -10.3635 q^{80} +8.40894 q^{81} +0.484101 q^{82} -5.87738 q^{83} +2.11974 q^{84} +16.3998 q^{85} -2.90229 q^{86} +0.320759 q^{87} -0.260185 q^{89} -3.87359 q^{90} -15.1327 q^{91} -9.99918 q^{92} +2.27447 q^{93} +3.33844 q^{94} -7.98170 q^{95} -1.09271 q^{96} -14.9487 q^{97} -5.10500 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 16 q^{3} + 20 q^{4} - 12 q^{5} + 16 q^{9} - 22 q^{12} - 6 q^{14} - 16 q^{15} + 36 q^{16} - 50 q^{20} - 28 q^{23} + 12 q^{25} - 68 q^{26} - 34 q^{27} - 8 q^{31} + 44 q^{34} - 16 q^{36} - 28 q^{37}+ \cdots - 14 q^{97}+O(q^{100}) $ 28 * q - 16 * q^3 + 20 * q^4 - 12 * q^5 + 16 * q^9 - 22 * q^12 - 6 * q^14 - 16 * q^15 + 36 * q^16 - 50 * q^20 - 28 * q^23 + 12 * q^25 - 68 * q^26 - 34 * q^27 - 8 * q^31 + 44 * q^34 - 16 * q^36 - 28 * q^37 - 14 * q^38 - 40 * q^42 + 34 * q^45 - 108 * q^47 - 128 * q^48 + 4 * q^49 - 6 * q^53 + 26 * q^56 - 6 * q^58 - 94 * q^59 + 102 * q^60 + 48 * q^64 - 28 * q^67 + 10 * q^69 - 120 * q^70 - 62 * q^71 - 14 * q^75 + 120 * q^78 - 128 * q^80 + 36 * q^81 - 20 * q^86 - 32 * q^89 - 58 * q^91 - 120 * q^92 - 32 * q^93 - 14 * q^97

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 $

$2$	−0.394005	−0.278604	−0.139302	−	0.990250i	$-0.544486\pi$
$2$	−0.394005	−0.278604	−0.139302	+	0.990250i	$0.544486\pi$
$3$	0.257216	0.148504	0.0742518	−	0.997240i	$-0.476343\pi$
$3$	0.257216	0.148504	0.0742518	+	0.997240i	$0.476343\pi$
$4$	−1.84476	−0.922380
$5$	−3.35100	−1.49861	−0.749307	−	0.662222i	$-0.769613\pi$
$5$	−3.35100	−1.49861	−0.749307	+	0.662222i	$0.769613\pi$
$6$	−0.101344	−0.0413737
$7$	−4.46729	−1.68848	−0.844239	−	0.535968i	$-0.819947\pi$
$7$	−4.46729	−1.68848	−0.844239	+	0.535968i	$0.819947\pi$
$8$	1.51486	0.535582
$9$	−2.93384	−0.977947
$10$	1.32031	0.417520
$11$	0	0
$11$	0	0
$12$	−0.474502	−0.136977
$13$	3.38744	0.939508	0.469754	−	0.882797i	$-0.344343\pi$
$13$	3.38744	0.939508	0.469754	+	0.882797i	$0.344343\pi$
$14$	1.76014	0.470416
$15$	−0.861932	−0.222550
$16$	3.09266	0.773165
$17$	−4.89399	−1.18697	−0.593483	−	0.804846i	$-0.702248\pi$
$17$	−4.89399	−1.18697	−0.593483	+	0.804846i	$0.702248\pi$
$18$	1.15595	0.272460
$19$	2.38188	0.546441	0.273221	−	0.961951i	$-0.411911\pi$
$19$	2.38188	0.546441	0.273221	+	0.961951i	$0.411911\pi$
$20$	6.18180	1.38229
$21$	−1.14906	−0.250745
$22$	0	0
$23$	5.42031	1.13021	0.565107	−	0.825018i	$-0.308835\pi$
$23$	5.42031	1.13021	0.565107	+	0.825018i	$0.308835\pi$
$24$	0.389645	0.0795359
$25$	6.22923	1.24585
$26$	−1.33467	−0.261750
$27$	−1.52628	−0.293732
$28$	8.24108	1.55742
$29$	1.24704	0.231570	0.115785	−	0.993274i	$-0.463062\pi$
$29$	1.24704	0.231570	0.115785	+	0.993274i	$0.463062\pi$
$30$	0.339605	0.0620032
$31$	8.84266	1.58819	0.794094	−	0.607795i	$-0.207946\pi$
$31$	8.84266	1.58819	0.794094	+	0.607795i	$0.207946\pi$
$32$	−4.24823	−0.750989
$33$	0	0
$34$	1.92826	0.330693
$35$	14.9699	2.53038
$36$	5.41223	0.902038
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.938474	−0.152241
$39$	0.871304	0.139520
$40$	−5.07629	−0.802631
$41$	−1.22867	−0.191886	−0.0959428	−	0.995387i	$-0.530587\pi$
$41$	−1.22867	−0.191886	−0.0959428	+	0.995387i	$0.530587\pi$
$42$	0.452735	0.0698585
$43$	7.36613	1.12332	0.561662	−	0.827367i	$-0.310162\pi$
$43$	7.36613	1.12332	0.561662	+	0.827367i	$0.310162\pi$
$44$	0	0
$45$	9.83131	1.46557
$46$	−2.13563	−0.314882
$47$	−8.47310	−1.23593	−0.617964	−	0.786206i	$-0.712042\pi$
$47$	−8.47310	−1.23593	−0.617964	+	0.786206i	$0.712042\pi$
$48$	0.795481	0.114818
$49$	12.9567	1.85095
$50$	−2.45435	−0.347097
$51$	−1.25881	−0.176269
$52$	−6.24902	−0.866583
$53$	−4.00750	−0.550472	−0.275236	−	0.961377i	$-0.588756\pi$
$53$	−4.00750	−0.550472	−0.275236	+	0.961377i	$0.588756\pi$
$54$	0.601361	0.0818349
$55$	0	0
$56$	−6.76730	−0.904318
$57$	0.612658	0.0811485
$58$	−0.491341	−0.0645162
$59$	−4.03136	−0.524838	−0.262419	−	0.964954i	$-0.584520\pi$
$59$	−4.03136	−0.524838	−0.262419	+	0.964954i	$0.584520\pi$
$60$	1.59006	0.205275
$61$	4.64879	0.595217	0.297608	−	0.954688i	$-0.403811\pi$
$61$	4.64879	0.595217	0.297608	+	0.954688i	$0.403811\pi$
$62$	−3.48405	−0.442475
$63$	13.1063	1.65124
$64$	−4.51149	−0.563937
$65$	−11.3513	−1.40796
$66$	0	0
$67$	15.5176	1.89578	0.947890	−	0.318598i	$-0.103212\pi$
$67$	15.5176	1.89578	0.947890	+	0.318598i	$0.103212\pi$
$68$	9.02824	1.09483
$69$	1.39419	0.167841
$70$	−5.89822	−0.704972
$71$	−8.82283	−1.04708	−0.523539	−	0.852002i	$-0.675389\pi$
$71$	−8.82283	−1.04708	−0.523539	+	0.852002i	$0.675389\pi$
$72$	−4.44434	−0.523771
$73$	−7.10968	−0.832125	−0.416062	−	0.909336i	$-0.636590\pi$
$73$	−7.10968	−0.832125	−0.416062	+	0.909336i	$0.636590\pi$
$74$	0.394005	0.0458022
$75$	1.60226	0.185013
$76$	−4.39400	−0.504027
$77$	0	0
$78$	−0.343298	−0.0388709
$79$	−3.08252	−0.346810	−0.173405	−	0.984851i	$-0.555477\pi$
$79$	−3.08252	−0.346810	−0.173405	+	0.984851i	$0.555477\pi$
$80$	−10.3635	−1.15868
$81$	8.40894	0.934326
$82$	0.484101	0.0534601
$83$	−5.87738	−0.645126	−0.322563	−	0.946548i	$-0.604544\pi$
$83$	−5.87738	−0.645126	−0.322563	+	0.946548i	$0.604544\pi$
$84$	2.11974	0.231282
$85$	16.3998	1.77881
$86$	−2.90229	−0.312962
$87$	0.320759	0.0343889
$88$	0	0
$89$	−0.260185	−0.0275795	−0.0137898	−	0.999905i	$-0.504390\pi$
$89$	−0.260185	−0.0275795	−0.0137898	+	0.999905i	$0.504390\pi$
$90$	−3.87359	−0.408312
$91$	−15.1327	−1.58634
$92$	−9.99918	−1.04249
$93$	2.27447	0.235852
$94$	3.33844	0.344334
$95$	−7.98170	−0.818905
$96$	−1.09271	−0.111525
$97$	−14.9487	−1.51781	−0.758907	−	0.651199i	$-0.774266\pi$
$97$	−14.9487	−1.51781	−0.758907	+	0.651199i	$0.774266\pi$
$98$	−5.10500	−0.515683
$99$	0	0
$100$	−11.4914	−1.14914
$101$	6.50599	0.647370	0.323685	−	0.946165i	$-0.395078\pi$
$101$	6.50599	0.647370	0.323685	+	0.946165i	$0.395078\pi$
$102$	0.495978	0.0491092
$103$	3.15493	0.310864	0.155432	−	0.987847i	$-0.450323\pi$
$103$	3.15493	0.310864	0.155432	+	0.987847i	$0.450323\pi$
$104$	5.13149	0.503184
$105$	3.85050	0.375770
$106$	1.57898	0.153364
$107$	18.1542	1.75503	0.877517	−	0.479546i	$-0.159199\pi$
$107$	18.1542	1.75503	0.877517	+	0.479546i	$0.159199\pi$
$108$	2.81562	0.270933
$109$	−6.56447	−0.628762	−0.314381	−	0.949297i	$-0.601797\pi$
$109$	−6.56447	−0.628762	−0.314381	+	0.949297i	$0.601797\pi$
$110$	0	0
$111$	−0.257216	−0.0244139
$112$	−13.8158	−1.30547
$113$	−19.3464	−1.81996	−0.909978	−	0.414657i	$-0.863902\pi$
$113$	−19.3464	−1.81996	−0.909978	+	0.414657i	$0.863902\pi$
$114$	−0.241390	−0.0226083
$115$	−18.1635	−1.69375
$116$	−2.30049	−0.213595
$117$	−9.93822	−0.918789
$118$	1.58838	0.146222
$119$	21.8629	2.00417
$120$	−1.30570	−0.119194
$121$	0	0
$122$	−1.83165	−0.165830
$123$	−0.316033	−0.0284957
$124$	−16.3126	−1.46491
$125$	−4.11915	−0.368428
$126$	−5.16396	−0.460042
$127$	16.9314	1.50242	0.751210	−	0.660063i	$-0.229470\pi$
$127$	16.9314	1.50242	0.751210	+	0.660063i	$0.229470\pi$
$128$	10.2740	0.908104
$129$	1.89469	0.166818
$130$	4.47249	0.392263
$131$	10.2637	0.896745	0.448372	−	0.893847i	$-0.352004\pi$
$131$	10.2637	0.896745	0.448372	+	0.893847i	$0.352004\pi$
$132$	0	0
$133$	−10.6406	−0.922654
$134$	−6.11403	−0.528171
$135$	5.11456	0.440192
$136$	−7.41368	−0.635718
$137$	12.0653	1.03081	0.515404	−	0.856947i	$-0.327642\pi$
$137$	12.0653	1.03081	0.515404	+	0.856947i	$0.327642\pi$
$138$	−0.549318	−0.0467611
$139$	12.8721	1.09180	0.545899	−	0.837851i	$-0.316188\pi$
$139$	12.8721	1.09180	0.545899	+	0.837851i	$0.316188\pi$
$140$	−27.6159	−2.33397
$141$	−2.17942	−0.183540
$142$	3.47624	0.291720
$143$	0	0
$144$	−9.07337	−0.756114
$145$	−4.17884	−0.347034
$146$	2.80125	0.231833
$147$	3.33266	0.274874
$148$	1.84476	0.151638
$149$	21.0738	1.72644	0.863218	−	0.504831i	$-0.168445\pi$
$149$	21.0738	1.72644	0.863218	+	0.504831i	$0.168445\pi$
$150$	−0.631297	−0.0515452
$151$	−9.65588	−0.785784	−0.392892	−	0.919585i	$-0.628525\pi$
$151$	−9.65588	−0.785784	−0.392892	+	0.919585i	$0.628525\pi$
$152$	3.60821	0.292664
$153$	14.3582	1.16079
$154$	0	0
$155$	−29.6318	−2.38008
$156$	−1.60735	−0.128691
$157$	−19.0386	−1.51945	−0.759723	−	0.650247i	$-0.774665\pi$
$157$	−19.0386	−1.51945	−0.759723	+	0.650247i	$0.774665\pi$
$158$	1.21453	0.0966226
$159$	−1.03079	−0.0817472
$160$	14.2358	1.12544
$161$	−24.2141	−1.90834
$162$	−3.31316	−0.260307
$163$	−6.12768	−0.479957	−0.239979	−	0.970778i	$-0.577140\pi$
$163$	−6.12768	−0.479957	−0.239979	+	0.970778i	$0.577140\pi$
$164$	2.26660	0.176992
$165$	0	0
$166$	2.31572	0.179734
$167$	−20.4323	−1.58110	−0.790548	−	0.612401i	$-0.790204\pi$
$167$	−20.4323	−1.58110	−0.790548	+	0.612401i	$0.790204\pi$
$168$	−1.74066	−0.134295
$169$	−1.52522	−0.117325
$170$	−6.46160	−0.495582
$171$	−6.98806	−0.534391
$172$	−13.5887	−1.03613
$173$	8.23198	0.625866	0.312933	−	0.949775i	$-0.398689\pi$
$173$	8.23198	0.625866	0.312933	+	0.949775i	$0.398689\pi$
$174$	−0.126381	−0.00958089
$175$	−27.8278	−2.10358
$176$	0	0
$177$	−1.03693	−0.0779404
$178$	0.102514	0.00768375
$179$	6.30791	0.471476	0.235738	−	0.971817i	$-0.424249\pi$
$179$	6.30791	0.471476	0.235738	+	0.971817i	$0.424249\pi$
$180$	−18.1364	−1.35181
$181$	−22.0248	−1.63709	−0.818545	−	0.574443i	$-0.805219\pi$
$181$	−22.0248	−1.63709	−0.818545	+	0.574443i	$0.805219\pi$
$182$	5.96236	0.441960
$183$	1.19574	0.0883918
$184$	8.21099	0.605322
$185$	3.35100	0.246371
$186$	−0.896154	−0.0657092
$187$	0	0
$188$	15.6308	1.14000
$189$	6.81833	0.495960
$190$	3.14483	0.228150
$191$	5.52363	0.399676	0.199838	−	0.979829i	$-0.435958\pi$
$191$	5.52363	0.399676	0.199838	+	0.979829i	$0.435958\pi$
$192$	−1.16043	−0.0837466
$193$	5.82976	0.419635	0.209818	−	0.977741i	$-0.432713\pi$
$193$	5.82976	0.419635	0.209818	+	0.977741i	$0.432713\pi$
$194$	5.88988	0.422869
$195$	−2.91975	−0.209087
$196$	−23.9020	−1.70728
$197$	−14.6775	−1.04573	−0.522864	−	0.852416i	$-0.675136\pi$
$197$	−14.6775	−1.04573	−0.522864	+	0.852416i	$0.675136\pi$
$198$	0	0
$199$	17.6909	1.25408	0.627038	−	0.778989i	$-0.284267\pi$
$199$	17.6909	1.25408	0.627038	+	0.778989i	$0.284267\pi$
$200$	9.43638	0.667253
$201$	3.99138	0.281530
$202$	−2.56339	−0.180360
$203$	−5.57089	−0.391000
$204$	2.32221	0.162587
$205$	4.11727	0.287563
$206$	−1.24306	−0.0866079
$207$	−15.9023	−1.10529
$208$	10.4762	0.726395
$209$	0	0
$210$	−1.51712	−0.104691
$211$	−7.31451	−0.503552	−0.251776	−	0.967786i	$-0.581015\pi$
$211$	−7.31451	−0.503552	−0.251776	+	0.967786i	$0.581015\pi$
$212$	7.39288	0.507745
$213$	−2.26937	−0.155495
$214$	−7.15285	−0.488959
$215$	−24.6839	−1.68343
$216$	−2.31209	−0.157318
$217$	−39.5027	−2.68162
$218$	2.58643	0.175175
$219$	−1.82872	−0.123574
$220$	0	0
$221$	−16.5781	−1.11516
$222$	0.101344	0.00680179
$223$	−13.3021	−0.890774	−0.445387	−	0.895338i	$-0.646934\pi$
$223$	−13.3021	−0.890774	−0.445387	+	0.895338i	$0.646934\pi$
$224$	18.9781	1.26803
$225$	−18.2756	−1.21837
$226$	7.62258	0.507046
$227$	−5.92272	−0.393105	−0.196552	−	0.980493i	$-0.562975\pi$
$227$	−5.92272	−0.393105	−0.196552	+	0.980493i	$0.562975\pi$
$228$	−1.13021	−0.0748498
$229$	17.5668	1.16085	0.580424	−	0.814314i	$-0.302887\pi$
$229$	17.5668	1.16085	0.580424	+	0.814314i	$0.302887\pi$
$230$	7.15651	0.471886
$231$	0	0
$232$	1.88909	0.124025
$233$	27.9827	1.83321	0.916603	−	0.399798i	$-0.130920\pi$
$233$	27.9827	1.83321	0.916603	+	0.399798i	$0.130920\pi$
$234$	3.91571	0.255978
$235$	28.3934	1.85218
$236$	7.43689	0.484100
$237$	−0.792872	−0.0515026
$238$	−8.61408	−0.558368
$239$	−21.1886	−1.37058	−0.685289	−	0.728271i	$-0.740324\pi$
$239$	−21.1886	−1.37058	−0.685289	+	0.728271i	$0.740324\pi$
$240$	−2.66566	−0.172068
$241$	−7.47909	−0.481771	−0.240885	−	0.970554i	$-0.577438\pi$
$241$	−7.47909	−0.481771	−0.240885	+	0.970554i	$0.577438\pi$
$242$	0	0
$243$	6.74175	0.432483
$244$	−8.57590	−0.549016
$245$	−43.4179	−2.77387
$246$	0.124519	0.00793901
$247$	8.06850	0.513386
$248$	13.3953	0.850606
$249$	−1.51175	−0.0958035
$250$	1.62297	0.102645
$251$	5.75178	0.363049	0.181525	−	0.983386i	$-0.441897\pi$
$251$	5.75178	0.363049	0.181525	+	0.983386i	$0.441897\pi$
$252$	−24.1780	−1.52307
$253$	0	0
$254$	−6.67107	−0.418580
$255$	4.21828	0.264159
$256$	4.97497	0.310936
$257$	−29.1525	−1.81848	−0.909242	−	0.416268i	$-0.863338\pi$
$257$	−29.1525	−1.81848	−0.909242	+	0.416268i	$0.863338\pi$
$258$	−0.746516	−0.0464760
$259$	4.46729	0.277584
$260$	20.9405	1.29867
$261$	−3.65862	−0.226463
$262$	−4.04396	−0.249836
$263$	17.7802	1.09638	0.548188	−	0.836355i	$-0.315318\pi$
$263$	17.7802	1.09638	0.548188	+	0.836355i	$0.315318\pi$
$264$	0	0
$265$	13.4291	0.824946
$266$	4.19244	0.257055
$267$	−0.0669236	−0.00409566
$268$	−28.6263	−1.74863
$269$	0.596339	0.0363594	0.0181797	−	0.999835i	$-0.494213\pi$
$269$	0.596339	0.0363594	0.0181797	+	0.999835i	$0.494213\pi$
$270$	−2.01516	−0.122639
$271$	20.9354	1.27174	0.635869	−	0.771797i	$-0.280642\pi$
$271$	20.9354	1.27174	0.635869	+	0.771797i	$0.280642\pi$
$272$	−15.1354	−0.917721
$273$	−3.89237	−0.235577
$274$	−4.75379	−0.287187
$275$	0	0
$276$	−2.57195	−0.154813
$277$	6.56488	0.394445	0.197223	−	0.980359i	$-0.436808\pi$
$277$	6.56488	0.394445	0.197223	+	0.980359i	$0.436808\pi$
$278$	−5.07167	−0.304179
$279$	−25.9430	−1.55316
$280$	22.6772	1.35522
$281$	11.3906	0.679507	0.339754	−	0.940514i	$-0.389656\pi$
$281$	11.3906	0.679507	0.339754	+	0.940514i	$0.389656\pi$
$282$	0.858701	0.0511349
$283$	29.8537	1.77462	0.887309	−	0.461176i	$-0.152572\pi$
$283$	29.8537	1.77462	0.887309	+	0.461176i	$0.152572\pi$
$284$	16.2760	0.965804
$285$	−2.05302	−0.121610
$286$	0	0
$287$	5.48882	0.323995
$288$	12.4636	0.734427
$289$	6.95113	0.408890
$290$	1.64648	0.0966849
$291$	−3.84505	−0.225401
$292$	13.1157	0.767535
$293$	3.61230	0.211033	0.105516	−	0.994418i	$-0.466350\pi$
$293$	3.61230	0.211033	0.105516	+	0.994418i	$0.466350\pi$
$294$	−1.31309	−0.0765808
$295$	13.5091	0.786530
$296$	−1.51486	−0.0880492
$297$	0	0
$298$	−8.30320	−0.480992
$299$	18.3610	1.06184
$300$	−2.95578	−0.170652
$301$	−32.9066	−1.89671
$302$	3.80447	0.218922
$303$	1.67344	0.0961369
$304$	7.36635	0.422489
$305$	−15.5781	−0.892000
$306$	−5.65720	−0.323400
$307$	−28.3331	−1.61705	−0.808527	−	0.588460i	$-0.799735\pi$
$307$	−28.3331	−1.61705	−0.808527	+	0.588460i	$0.799735\pi$
$308$	0	0
$309$	0.811497	0.0461645
$310$	11.6751	0.663100
$311$	−7.42361	−0.420955	−0.210477	−	0.977599i	$-0.567502\pi$
$311$	−7.42361	−0.420955	−0.210477	+	0.977599i	$0.567502\pi$
$312$	1.31990	0.0747246
$313$	4.57588	0.258644	0.129322	−	0.991603i	$-0.458720\pi$
$313$	4.57588	0.258644	0.129322	+	0.991603i	$0.458720\pi$
$314$	7.50131	0.423323
$315$	−43.9193	−2.47457
$316$	5.68650	0.319891
$317$	−8.80458	−0.494514	−0.247257	−	0.968950i	$-0.579529\pi$
$317$	−8.80458	−0.494514	−0.247257	+	0.968950i	$0.579529\pi$
$318$	0.406138	0.0227751
$319$	0	0
$320$	15.1180	0.845124
$321$	4.66955	0.260629
$322$	9.54048	0.531670
$323$	−11.6569	−0.648608
$324$	−15.5125	−0.861804
$325$	21.1012	1.17048
$326$	2.41434	0.133718
$327$	−1.68849	−0.0933735
$328$	−1.86125	−0.102771
$329$	37.8518	2.08684
$330$	0	0
$331$	13.9401	0.766215	0.383108	−	0.923704i	$-0.374854\pi$
$331$	13.9401	0.766215	0.383108	+	0.923704i	$0.374854\pi$
$332$	10.8423	0.595051
$333$	2.93384	0.160773
$334$	8.05041	0.440499
$335$	−51.9996	−2.84104
$336$	−3.55364	−0.193867
$337$	12.0934	0.658772	0.329386	−	0.944195i	$-0.393158\pi$
$337$	12.0934	0.658772	0.329386	+	0.944195i	$0.393158\pi$
$338$	0.600945	0.0326871
$339$	−4.97620	−0.270270
$340$	−30.2537	−1.64073
$341$	0	0
$342$	2.75333	0.148883
$343$	−26.6102	−1.43682
$344$	11.1586	0.601632
$345$	−4.67194	−0.251529
$346$	−3.24344	−0.174368
$347$	−20.0609	−1.07693	−0.538464	−	0.842649i	$-0.680995\pi$
$347$	−20.0609	−1.07693	−0.538464	+	0.842649i	$0.680995\pi$
$348$	−0.591723	−0.0317197
$349$	19.3800	1.03739	0.518694	−	0.854960i	$-0.326419\pi$
$349$	19.3800	1.03739	0.518694	+	0.854960i	$0.326419\pi$
$350$	10.9643	0.586066
$351$	−5.17018	−0.275964
$352$	0	0
$353$	−20.4031	−1.08595	−0.542973	−	0.839750i	$-0.682702\pi$
$353$	−20.4031	−1.08595	−0.542973	+	0.839750i	$0.682702\pi$
$354$	0.408555	0.0217145
$355$	29.5654	1.56917
$356$	0.479978	0.0254388
$357$	5.62348	0.297626
$358$	−2.48535	−0.131355
$359$	−17.5124	−0.924272	−0.462136	−	0.886809i	$-0.652917\pi$
$359$	−17.5124	−0.924272	−0.462136	+	0.886809i	$0.652917\pi$
$360$	14.8930	0.784931
$361$	−13.3266	−0.701402
$362$	8.67788	0.456099
$363$	0	0
$364$	27.9162	1.46321
$365$	23.8246	1.24703
$366$	−0.471129	−0.0246263
$367$	−12.4763	−0.651259	−0.325629	−	0.945497i	$-0.605576\pi$
$367$	−12.4763	−0.651259	−0.325629	+	0.945497i	$0.605576\pi$
$368$	16.7632	0.873841
$369$	3.60471	0.187654
$370$	−1.32031	−0.0686398
$371$	17.9027	0.929460
$372$	−4.19586	−0.217545
$373$	5.64775	0.292429	0.146215	−	0.989253i	$-0.453291\pi$
$373$	5.64775	0.292429	0.146215	+	0.989253i	$0.453291\pi$
$374$	0	0
$375$	−1.05951	−0.0547129
$376$	−12.8355	−0.661941
$377$	4.22428	0.217562
$378$	−2.68646	−0.138176
$379$	18.2232	0.936065	0.468033	−	0.883711i	$-0.344963\pi$
$379$	18.2232	0.936065	0.468033	+	0.883711i	$0.344963\pi$
$380$	14.7243	0.755342
$381$	4.35503	0.223115
$382$	−2.17634	−0.111351
$383$	25.2864	1.29207	0.646037	−	0.763306i	$-0.276425\pi$
$383$	25.2864	1.29207	0.646037	+	0.763306i	$0.276425\pi$
$384$	2.64264	0.134857
$385$	0	0
$386$	−2.29695	−0.116912
$387$	−21.6110	−1.09855
$388$	27.5768	1.40000
$389$	−13.2886	−0.673760	−0.336880	−	0.941548i	$-0.609372\pi$
$389$	−13.2886	−0.673760	−0.336880	+	0.941548i	$0.609372\pi$
$390$	1.15039	0.0582525
$391$	−26.5270	−1.34153
$392$	19.6275	0.991338
$393$	2.63999	0.133170
$394$	5.78300	0.291343
$395$	10.3295	0.519735
$396$	0	0
$397$	30.7397	1.54278	0.771392	−	0.636360i	$-0.219561\pi$
$397$	30.7397	1.54278	0.771392	+	0.636360i	$0.219561\pi$
$398$	−6.97031	−0.349390
$399$	−2.73692	−0.137017
$400$	19.2649	0.963244
$401$	−18.3325	−0.915481	−0.457740	−	0.889086i	$-0.651341\pi$
$401$	−18.3325	−0.915481	−0.457740	+	0.889086i	$0.651341\pi$
$402$	−1.57262	−0.0784354
$403$	29.9540	1.49212
$404$	−12.0020	−0.597121
$405$	−28.1784	−1.40020
$406$	2.19496	0.108934
$407$	0	0
$408$	−1.90692	−0.0944065
$409$	−23.4962	−1.16181	−0.580907	−	0.813970i	$-0.697302\pi$
$409$	−23.4962	−1.16181	−0.580907	+	0.813970i	$0.697302\pi$
$410$	−1.62223	−0.0801160
$411$	3.10339	0.153079
$412$	−5.82008	−0.286735
$413$	18.0092	0.886177
$414$	6.26560	0.307937
$415$	19.6951	0.966795
$416$	−14.3907	−0.705560
$417$	3.31091	0.162136
$418$	0	0
$419$	−28.2071	−1.37801	−0.689003	−	0.724759i	$-0.741951\pi$
$419$	−28.2071	−1.37801	−0.689003	+	0.724759i	$0.741951\pi$
$420$	−7.10325	−0.346603
$421$	7.39330	0.360327	0.180164	−	0.983637i	$-0.442337\pi$
$421$	7.39330	0.360327	0.180164	+	0.983637i	$0.442337\pi$
$422$	2.88196	0.140291
$423$	24.8587	1.20867
$424$	−6.07078	−0.294823
$425$	−30.4858	−1.47878
$426$	0.894145	0.0433214
$427$	−20.7675	−1.00501
$428$	−33.4902	−1.61881
$429$	0	0
$430$	9.72560	0.469010
$431$	0.975758	0.0470006	0.0235003	−	0.999724i	$-0.492519\pi$
$431$	0.975758	0.0470006	0.0235003	+	0.999724i	$0.492519\pi$
$432$	−4.72026	−0.227103
$433$	−6.47164	−0.311007	−0.155503	−	0.987835i	$-0.549700\pi$
$433$	−6.47164	−0.311007	−0.155503	+	0.987835i	$0.549700\pi$
$434$	15.5643	0.747109
$435$	−1.07486	−0.0515358
$436$	12.1099	0.579958
$437$	12.9105	0.617595
$438$	0.720526	0.0344281
$439$	20.2248	0.965276	0.482638	−	0.875820i	$-0.339679\pi$
$439$	20.2248	0.965276	0.482638	+	0.875820i	$0.339679\pi$
$440$	0	0
$441$	−38.0128	−1.81014
$442$	6.53186	0.310689
$443$	3.00110	0.142586	0.0712932	−	0.997455i	$-0.477287\pi$
$443$	3.00110	0.142586	0.0712932	+	0.997455i	$0.477287\pi$
$444$	0.474502	0.0225188
$445$	0.871880	0.0413311
$446$	5.24110	0.248173
$447$	5.42053	0.256382
$448$	20.1541	0.952194
$449$	−20.7881	−0.981053	−0.490527	−	0.871426i	$-0.663196\pi$
$449$	−20.7881	−0.981053	−0.490527	+	0.871426i	$0.663196\pi$
$450$	7.20067	0.339443
$451$	0	0
$452$	35.6895	1.67869
$453$	−2.48365	−0.116692
$454$	2.33358	0.109520
$455$	50.7097	2.37731
$456$	0.928088	0.0434617
$457$	−19.2699	−0.901410	−0.450705	−	0.892673i	$-0.648827\pi$
$457$	−19.2699	−0.901410	−0.450705	+	0.892673i	$0.648827\pi$
$458$	−6.92142	−0.323417
$459$	7.46959	0.348650
$460$	33.5073	1.56228
$461$	32.0285	1.49172	0.745858	−	0.666105i	$-0.232040\pi$
$461$	32.0285	1.49172	0.745858	+	0.666105i	$0.232040\pi$
$462$	0	0
$463$	−28.4412	−1.32178	−0.660888	−	0.750484i	$-0.729820\pi$
$463$	−28.4412	−1.32178	−0.660888	+	0.750484i	$0.729820\pi$
$464$	3.85667	0.179042
$465$	−7.62177	−0.353451
$466$	−11.0253	−0.510738
$467$	−10.8605	−0.502562	−0.251281	−	0.967914i	$-0.580852\pi$
$467$	−10.8605	−0.502562	−0.251281	+	0.967914i	$0.580852\pi$
$468$	18.3336	0.847472
$469$	−69.3217	−3.20098
$470$	−11.1871	−0.516024
$471$	−4.89703	−0.225643
$472$	−6.10692	−0.281094
$473$	0	0
$474$	0.312396	0.0143488
$475$	14.8373	0.680782
$476$	−40.3317	−1.84860
$477$	11.7574	0.538333
$478$	8.34843	0.381848
$479$	−1.00411	−0.0458790	−0.0229395	−	0.999737i	$-0.507303\pi$
$479$	−1.00411	−0.0458790	−0.0229395	+	0.999737i	$0.507303\pi$
$480$	3.66169	0.167132
$481$	−3.38744	−0.154454
$482$	2.94680	0.134223
$483$	−6.22825	−0.283395
$484$	0	0
$485$	50.0933	2.27462
$486$	−2.65628	−0.120491
$487$	21.6089	0.979192	0.489596	−	0.871949i	$-0.337144\pi$
$487$	21.6089	0.979192	0.489596	+	0.871949i	$0.337144\pi$
$488$	7.04224	0.318787
$489$	−1.57614	−0.0712754
$490$	17.1069	0.772810
$491$	13.5829	0.612986	0.306493	−	0.951873i	$-0.400844\pi$
$491$	13.5829	0.612986	0.306493	+	0.951873i	$0.400844\pi$
$492$	0.583005	0.0262839
$493$	−6.10301	−0.274866
$494$	−3.17903	−0.143031
$495$	0	0
$496$	27.3473	1.22793
$497$	39.4142	1.76797
$498$	0.595639	0.0266912
$499$	0.496755	0.0222378	0.0111189	−	0.999938i	$-0.496461\pi$
$499$	0.496755	0.0222378	0.0111189	+	0.999938i	$0.496461\pi$
$500$	7.59885	0.339831
$501$	−5.25550	−0.234798
$502$	−2.26623	−0.101147
$503$	−15.8286	−0.705761	−0.352881	−	0.935668i	$-0.614798\pi$
$503$	−15.8286	−0.705761	−0.352881	+	0.935668i	$0.614798\pi$
$504$	19.8542	0.884375
$505$	−21.8016	−0.970159
$506$	0	0
$507$	−0.392311	−0.0174231
$508$	−31.2344	−1.38580
$509$	2.64322	0.117159	0.0585793	−	0.998283i	$-0.481343\pi$
$509$	2.64322	0.117159	0.0585793	+	0.998283i	$0.481343\pi$
$510$	−1.66203	−0.0735957
$511$	31.7610	1.40502
$512$	−22.5082	−0.994731
$513$	−3.63542	−0.160507
$514$	11.4862	0.506636
$515$	−10.5722	−0.465866
$516$	−3.49524	−0.153869
$517$	0	0
$518$	−1.76014	−0.0773359
$519$	2.11739	0.0929433
$520$	−17.1956	−0.754079
$521$	−31.8365	−1.39478	−0.697392	−	0.716690i	$-0.745656\pi$
$521$	−31.8365	−1.39478	−0.697392	+	0.716690i	$0.745656\pi$
$522$	1.44151	0.0630934
$523$	−33.8053	−1.47820	−0.739102	−	0.673593i	$-0.764750\pi$
$523$	−33.8053	−1.47820	−0.739102	+	0.673593i	$0.764750\pi$
$524$	−18.9341	−0.827139
$525$	−7.15775	−0.312390
$526$	−7.00550	−0.305454
$527$	−43.2759	−1.88513
$528$	0	0
$529$	6.37979	0.277382
$530$	−5.29115	−0.229833
$531$	11.8274	0.513264
$532$	19.6293	0.851037
$533$	−4.16204	−0.180278
$534$	0.0263682	0.00114107
$535$	−60.8348	−2.63012
$536$	23.5070	1.01535
$537$	1.62250	0.0700159
$538$	−0.234961	−0.0101299
$539$	0	0
$540$	−9.43514	−0.406024
$541$	13.3324	0.573206	0.286603	−	0.958049i	$-0.407474\pi$
$541$	13.3324	0.573206	0.286603	+	0.958049i	$0.407474\pi$
$542$	−8.24867	−0.354311
$543$	−5.66512	−0.243114
$544$	20.7908	0.891399
$545$	21.9976	0.942272
$546$	1.53361	0.0656326
$547$	−25.9923	−1.11135	−0.555675	−	0.831400i	$-0.687540\pi$
$547$	−25.9923	−1.11135	−0.555675	+	0.831400i	$0.687540\pi$
$548$	−22.2576	−0.950796
$549$	−13.6388	−0.582090
$550$	0	0
$551$	2.97031	0.126539
$552$	2.11200	0.0898925
$553$	13.7705	0.585581
$554$	−2.58660	−0.109894
$555$	0.861932	0.0365870
$556$	−23.7459	−1.00705
$557$	31.8916	1.35129	0.675645	−	0.737227i	$-0.263865\pi$
$557$	31.8916	1.35129	0.675645	+	0.737227i	$0.263865\pi$
$558$	10.2217	0.432717
$559$	24.9524	1.05537
$560$	46.2968	1.95640
$561$	0	0
$562$	−4.48796	−0.189313
$563$	−0.364149	−0.0153471	−0.00767353	−	0.999971i	$-0.502443\pi$
$563$	−0.364149	−0.0153471	−0.00767353	+	0.999971i	$0.502443\pi$
$564$	4.02050	0.169294
$565$	64.8299	2.72741
$566$	−11.7625	−0.494415
$567$	−37.5652	−1.57759
$568$	−13.3653	−0.560796
$569$	36.4088	1.52634	0.763168	−	0.646200i	$-0.223643\pi$
$569$	36.4088	1.52634	0.763168	+	0.646200i	$0.223643\pi$
$570$	0.808900	0.0338811
$571$	−6.98227	−0.292199	−0.146100	−	0.989270i	$-0.546672\pi$
$571$	−6.98227	−0.292199	−0.146100	+	0.989270i	$0.546672\pi$
$572$	0	0
$573$	1.42076	0.0593533
$574$	−2.16262	−0.0902661
$575$	33.7644	1.40807
$576$	13.2360	0.551500
$577$	10.8690	0.452482	0.226241	−	0.974071i	$-0.427356\pi$
$577$	10.8690	0.452482	0.226241	+	0.974071i	$0.427356\pi$
$578$	−2.73878	−0.113918
$579$	1.49951	0.0623174
$580$	7.70896	0.320097
$581$	26.2559	1.08928
$582$	1.51497	0.0627975
$583$	0	0
$584$	−10.7701	−0.445671
$585$	33.3030	1.37691
$586$	−1.42326	−0.0587944
$587$	18.2776	0.754398	0.377199	−	0.926132i	$-0.376887\pi$
$587$	18.2776	0.754398	0.377199	+	0.926132i	$0.376887\pi$
$588$	−6.14797	−0.253538
$589$	21.0622	0.867852
$590$	−5.32265	−0.219130
$591$	−3.77528	−0.155294
$592$	−3.09266	−0.127108
$593$	−23.7291	−0.974436	−0.487218	−	0.873280i	$-0.661988\pi$
$593$	−23.7291	−0.974436	−0.487218	+	0.873280i	$0.661988\pi$
$594$	0	0
$595$	−73.2626	−3.00347
$596$	−38.8762	−1.59243
$597$	4.55039	0.186235
$598$	−7.23433	−0.295834
$599$	−0.779253	−0.0318394	−0.0159197	−	0.999873i	$-0.505068\pi$
$599$	−0.779253	−0.0318394	−0.0159197	+	0.999873i	$0.505068\pi$
$600$	2.42719	0.0990895
$601$	5.63712	0.229943	0.114971	−	0.993369i	$-0.463322\pi$
$601$	5.63712	0.229943	0.114971	+	0.993369i	$0.463322\pi$
$602$	12.9654	0.528430
$603$	−45.5262	−1.85397
$604$	17.8128	0.724792
$605$	0	0
$606$	−0.659346	−0.0267841
$607$	−17.2043	−0.698302	−0.349151	−	0.937066i	$-0.613530\pi$
$607$	−17.2043	−0.698302	−0.349151	+	0.937066i	$0.613530\pi$
$608$	−10.1188	−0.410371
$609$	−1.43292	−0.0580650
$610$	6.13786	0.248515
$611$	−28.7022	−1.16116
$612$	−26.4874	−1.07069
$613$	−42.7716	−1.72753	−0.863765	−	0.503894i	$-0.831900\pi$
$613$	−42.7716	−1.72753	−0.863765	+	0.503894i	$0.831900\pi$
$614$	11.1634	0.450517
$615$	1.05903	0.0427041
$616$	0	0
$617$	9.18489	0.369770	0.184885	−	0.982760i	$-0.440809\pi$
$617$	9.18489	0.369770	0.184885	+	0.982760i	$0.440809\pi$
$618$	−0.319734	−0.0128616
$619$	2.09319	0.0841323	0.0420662	−	0.999115i	$-0.486606\pi$
$619$	2.09319	0.0841323	0.0420662	+	0.999115i	$0.486606\pi$
$620$	54.6635	2.19534
$621$	−8.27290	−0.331980
$622$	2.92494	0.117279
$623$	1.16232	0.0465674
$624$	2.69465	0.107872
$625$	−17.3428	−0.693714
$626$	−1.80292	−0.0720591
$627$	0	0
$628$	35.1216	1.40151
$629$	4.89399	0.195136
$630$	17.3044	0.689425
$631$	−20.7494	−0.826019	−0.413009	−	0.910727i	$-0.635522\pi$
$631$	−20.7494	−0.826019	−0.413009	+	0.910727i	$0.635522\pi$
$632$	−4.66957	−0.185745
$633$	−1.88141	−0.0747793
$634$	3.46905	0.137774
$635$	−56.7373	−2.25155
$636$	1.90157	0.0754020
$637$	43.8900	1.73899
$638$	0	0
$639$	25.8848	1.02399
$640$	−34.4283	−1.36090
$641$	23.0266	0.909497	0.454749	−	0.890620i	$-0.349729\pi$
$641$	23.0266	0.909497	0.454749	+	0.890620i	$0.349729\pi$
$642$	−1.83983	−0.0726122
$643$	3.70078	0.145944	0.0729722	−	0.997334i	$-0.476752\pi$
$643$	3.70078	0.145944	0.0729722	+	0.997334i	$0.476752\pi$
$644$	44.6692	1.76021
$645$	−6.34910	−0.249996
$646$	4.59288	0.180705
$647$	−9.81647	−0.385925	−0.192963	−	0.981206i	$-0.561810\pi$
$647$	−9.81647	−0.385925	−0.192963	+	0.981206i	$0.561810\pi$
$648$	12.7383	0.500409
$649$	0	0
$650$	−8.31397	−0.326101
$651$	−10.1607	−0.398230
$652$	11.3041	0.442703
$653$	29.9965	1.17385	0.586927	−	0.809640i	$-0.300338\pi$
$653$	29.9965	1.17385	0.586927	+	0.809640i	$0.300338\pi$
$654$	0.665272	0.0260142
$655$	−34.3937	−1.34387
$656$	−3.79985	−0.148359
$657$	20.8587	0.813774
$658$	−14.9138	−0.581401
$659$	−20.1702	−0.785719	−0.392860	−	0.919598i	$-0.628514\pi$
$659$	−20.1702	−0.785719	−0.392860	+	0.919598i	$0.628514\pi$
$660$	0	0
$661$	−25.4552	−0.990093	−0.495046	−	0.868867i	$-0.664849\pi$
$661$	−25.4552	−0.990093	−0.495046	+	0.868867i	$0.664849\pi$
$662$	−5.49246	−0.213470
$663$	−4.26415	−0.165606
$664$	−8.90337	−0.345518
$665$	35.6566	1.38270
$666$	−1.15595	−0.0447921
$667$	6.75935	0.261723
$668$	37.6926	1.45837
$669$	−3.42151	−0.132283
$670$	20.4881	0.791525
$671$	0	0
$672$	4.88147	0.188307
$673$	−40.4306	−1.55849	−0.779243	−	0.626722i	$-0.784396\pi$
$673$	−40.4306	−1.55849	−0.779243	+	0.626722i	$0.784396\pi$
$674$	−4.76488	−0.183536
$675$	−9.50753	−0.365945
$676$	2.81367	0.108218
$677$	16.0904	0.618405	0.309202	−	0.950996i	$-0.399938\pi$
$677$	16.0904	0.618405	0.309202	+	0.950996i	$0.399938\pi$
$678$	1.96065	0.0752983
$679$	66.7803	2.56279
$680$	24.8433	0.952697
$681$	−1.52342	−0.0583775
$682$	0	0
$683$	−12.0307	−0.460342	−0.230171	−	0.973150i	$-0.573929\pi$
$683$	−12.0307	−0.460342	−0.230171	+	0.973150i	$0.573929\pi$
$684$	12.8913	0.492911
$685$	−40.4309	−1.54478
$686$	10.4846	0.400303
$687$	4.51847	0.172390
$688$	22.7809	0.868515
$689$	−13.5752	−0.517173
$690$	1.84077	0.0700768
$691$	20.5205	0.780636	0.390318	−	0.920680i	$-0.372365\pi$
$691$	20.5205	0.780636	0.390318	+	0.920680i	$0.372365\pi$
$692$	−15.1860	−0.577286
$693$	0	0
$694$	7.90412	0.300036
$695$	−43.1345	−1.63618
$696$	0.485903	0.0184181
$697$	6.01309	0.227762
$698$	−7.63582	−0.289020
$699$	7.19759	0.272238
$700$	51.3356	1.94030
$701$	−37.9071	−1.43173	−0.715866	−	0.698238i	$-0.753968\pi$
$701$	−37.9071	−1.43173	−0.715866	+	0.698238i	$0.753968\pi$
$702$	2.03708	0.0768846
$703$	−2.38188	−0.0898344
$704$	0	0
$705$	7.30323	0.275056
$706$	8.03892	0.302549
$707$	−29.0642	−1.09307
$708$	1.91289	0.0718906
$709$	−47.3525	−1.77836	−0.889181	−	0.457556i	$-0.848725\pi$
$709$	−47.3525	−1.77836	−0.889181	+	0.457556i	$0.848725\pi$
$710$	−11.6489	−0.437175
$711$	9.04361	0.339162
$712$	−0.394142	−0.0147711
$713$	47.9300	1.79499
$714$	−2.21568	−0.0829197
$715$	0	0
$716$	−11.6366	−0.434880
$717$	−5.45005	−0.203536
$718$	6.89999	0.257506
$719$	−45.8517	−1.70998	−0.854990	−	0.518644i	$-0.826437\pi$
$719$	−45.8517	−1.70998	−0.854990	+	0.518644i	$0.826437\pi$
$720$	30.4049	1.13312
$721$	−14.0940	−0.524887
$722$	5.25076	0.195413
$723$	−1.92374	−0.0715447
$724$	40.6304	1.51002
$725$	7.76810	0.288500
$726$	0	0
$727$	32.3655	1.20037	0.600184	−	0.799862i	$-0.295094\pi$
$727$	32.3655	1.20037	0.600184	+	0.799862i	$0.295094\pi$
$728$	−22.9238	−0.849614
$729$	−23.4927	−0.870101
$730$	−9.38700	−0.347428
$731$	−36.0498	−1.33335
$732$	−2.20586	−0.0815309
$733$	43.7794	1.61703	0.808514	−	0.588477i	$-0.200272\pi$
$733$	43.7794	1.61703	0.808514	+	0.588477i	$0.200272\pi$
$734$	4.91574	0.181443
$735$	−11.1678	−0.411930
$736$	−23.0268	−0.848777
$737$	0	0
$738$	−1.42028	−0.0522811
$739$	26.9288	0.990591	0.495296	−	0.868725i	$-0.335060\pi$
$739$	26.9288	0.990591	0.495296	+	0.868725i	$0.335060\pi$
$740$	−6.18180	−0.227247
$741$	2.07535	0.0762397
$742$	−7.05374	−0.258951
$743$	19.9939	0.733506	0.366753	−	0.930318i	$-0.380469\pi$
$743$	19.9939	0.733506	0.366753	+	0.930318i	$0.380469\pi$
$744$	3.44550	0.126318
$745$	−70.6185	−2.58726
$746$	−2.22524	−0.0814719
$747$	17.2433	0.630899
$748$	0	0
$749$	−81.1001	−2.96333
$750$	0.417453	0.0152432
$751$	−2.08451	−0.0760647	−0.0380324	−	0.999277i	$-0.512109\pi$
$751$	−2.08451	−0.0760647	−0.0380324	+	0.999277i	$0.512109\pi$
$752$	−26.2044	−0.955576
$753$	1.47945	0.0539142
$754$	−1.66439	−0.0606135
$755$	32.3569	1.17759
$756$	−12.5782	−0.457464
$757$	54.7719	1.99072	0.995359	−	0.0962355i	$-0.0306802\pi$
$757$	54.7719	1.99072	0.995359	+	0.0962355i	$0.0306802\pi$
$758$	−7.18005	−0.260791
$759$	0	0
$760$	−12.0911	−0.438591
$761$	−28.7718	−1.04298	−0.521489	−	0.853258i	$-0.674623\pi$
$761$	−28.7718	−1.04298	−0.521489	+	0.853258i	$0.674623\pi$
$762$	−1.71590	−0.0621607
$763$	29.3254	1.06165
$764$	−10.1898	−0.368653
$765$	−48.1143	−1.73958
$766$	−9.96297	−0.359977
$767$	−13.6560	−0.493090
$768$	1.27964	0.0461751
$769$	−42.4659	−1.53136	−0.765679	−	0.643222i	$-0.777597\pi$
$769$	−42.4659	−1.53136	−0.765679	+	0.643222i	$0.777597\pi$
$770$	0	0
$771$	−7.49849	−0.270052
$772$	−10.7545	−0.387063
$773$	−14.6392	−0.526537	−0.263268	−	0.964723i	$-0.584800\pi$
$773$	−14.6392	−0.526537	−0.263268	+	0.964723i	$0.584800\pi$
$774$	8.51486	0.306060
$775$	55.0830	1.97864
$776$	−22.6452	−0.812914
$777$	1.14906	0.0412222
$778$	5.23579	0.187712
$779$	−2.92654	−0.104854
$780$	5.38623	0.192858
$781$	0	0
$782$	10.4518	0.373754
$783$	−1.90333	−0.0680195
$784$	40.0706	1.43109
$785$	63.7984	2.27706
$786$	−1.04017	−0.0371016
$787$	−54.0710	−1.92742	−0.963711	−	0.266946i	$-0.913985\pi$
$787$	−54.0710	−1.92742	−0.963711	+	0.266946i	$0.913985\pi$
$788$	27.0764	0.964558
$789$	4.57336	0.162816
$790$	−4.06989	−0.144800
$791$	86.4260	3.07295
$792$	0	0
$793$	15.7475	0.559211
$794$	−12.1116	−0.429825
$795$	3.45419	0.122508
$796$	−32.6355	−1.15673
$797$	−9.79568	−0.346981	−0.173490	−	0.984836i	$-0.555505\pi$
$797$	−9.79568	−0.346981	−0.173490	+	0.984836i	$0.555505\pi$
$798$	1.07836	0.0381736
$799$	41.4673	1.46701
$800$	−26.4632	−0.935616
$801$	0.763340	0.0269713
$802$	7.22310	0.255056
$803$	0	0
$804$	−7.36314	−0.259678
$805$	81.1416	2.85987
$806$	−11.8020	−0.415709
$807$	0.153388	0.00539951
$808$	9.85564	0.346720
$809$	−24.2524	−0.852668	−0.426334	−	0.904566i	$-0.640195\pi$
$809$	−24.2524	−0.852668	−0.426334	+	0.904566i	$0.640195\pi$
$810$	11.1024	0.390100
$811$	−10.6049	−0.372389	−0.186195	−	0.982513i	$-0.559615\pi$
$811$	−10.6049	−0.372389	−0.186195	+	0.982513i	$0.559615\pi$
$812$	10.2770	0.360651
$813$	5.38493	0.188858
$814$	0	0
$815$	20.5339	0.719271
$816$	−3.89308	−0.136285
$817$	17.5453	0.613831
$818$	9.25764	0.323686
$819$	44.3969	1.55135
$820$	−7.59538	−0.265242
$821$	3.97237	0.138637	0.0693184	−	0.997595i	$-0.477918\pi$
$821$	3.97237	0.138637	0.0693184	+	0.997595i	$0.477918\pi$
$822$	−1.22275	−0.0426483
$823$	−18.3419	−0.639359	−0.319680	−	0.947526i	$-0.603575\pi$
$823$	−18.3419	−0.639359	−0.319680	+	0.947526i	$0.603575\pi$
$824$	4.77926	0.166493
$825$	0	0
$826$	−7.09574	−0.246892
$827$	9.20838	0.320207	0.160103	−	0.987100i	$-0.448817\pi$
$827$	9.20838	0.320207	0.160103	+	0.987100i	$0.448817\pi$
$828$	29.3360	1.01950
$829$	17.9251	0.622563	0.311282	−	0.950318i	$-0.399242\pi$
$829$	17.9251	0.622563	0.311282	+	0.950318i	$0.399242\pi$
$830$	−7.75997	−0.269353
$831$	1.68859	0.0585766
$832$	−15.2824	−0.529823
$833$	−63.4099	−2.19702
$834$	−1.30452	−0.0451717
$835$	68.4686	2.36945
$836$	0	0
$837$	−13.4964	−0.466502
$838$	11.1137	0.383917
$839$	14.3368	0.494961	0.247480	−	0.968893i	$-0.420397\pi$
$839$	14.3368	0.494961	0.247480	+	0.968893i	$0.420397\pi$
$840$	5.83295	0.201256
$841$	−27.4449	−0.946375
$842$	−2.91300	−0.100389
$843$	2.92985	0.100909
$844$	13.4935	0.464466
$845$	5.11102	0.175824
$846$	−9.79446	−0.336741
$847$	0	0
$848$	−12.3938	−0.425606
$849$	7.67884	0.263537
$850$	12.0116	0.411993
$851$	−5.42031	−0.185806
$852$	4.18645	0.143425
$853$	36.8544	1.26187	0.630935	−	0.775836i	$-0.282672\pi$
$853$	36.8544	1.26187	0.630935	+	0.775836i	$0.282672\pi$
$854$	8.18250	0.279999
$855$	23.4170	0.800845
$856$	27.5010	0.939965
$857$	0.886786	0.0302920	0.0151460	−	0.999885i	$-0.495179\pi$
$857$	0.886786	0.0302920	0.0151460	+	0.999885i	$0.495179\pi$
$858$	0	0
$859$	−0.166226	−0.00567155	−0.00283577	−	0.999996i	$-0.500903\pi$
$859$	−0.166226	−0.00567155	−0.00283577	+	0.999996i	$0.500903\pi$
$860$	45.5359	1.55276
$861$	1.41181	0.0481144
$862$	−0.384454	−0.0130945
$863$	−46.2413	−1.57407	−0.787036	−	0.616907i	$-0.788385\pi$
$863$	−46.2413	−1.57407	−0.787036	+	0.616907i	$0.788385\pi$
$864$	6.48399	0.220590
$865$	−27.5854	−0.937931
$866$	2.54986	0.0866477
$867$	1.78794	0.0607217
$868$	72.8731	2.47347
$869$	0	0
$870$	0.423502	0.0143581
$871$	52.5651	1.78110
$872$	−9.94422	−0.336754
$873$	43.8572	1.48434
$874$	−5.08682	−0.172064
$875$	18.4015	0.622083
$876$	3.37355	0.113982
$877$	−11.7141	−0.395557	−0.197779	−	0.980247i	$-0.563373\pi$
$877$	−11.7141	−0.395557	−0.197779	+	0.980247i	$0.563373\pi$
$878$	−7.96866	−0.268929
$879$	0.929140	0.0313391
$880$	0	0
$881$	−32.3155	−1.08874	−0.544368	−	0.838846i	$-0.683231\pi$
$881$	−32.3155	−1.08874	−0.544368	+	0.838846i	$0.683231\pi$
$882$	14.9773	0.504310
$883$	9.15504	0.308092	0.154046	−	0.988064i	$-0.450770\pi$
$883$	9.15504	0.308092	0.154046	+	0.988064i	$0.450770\pi$
$884$	30.5826	1.02861
$885$	3.47475	0.116803
$886$	−1.18245	−0.0397251
$887$	16.1634	0.542715	0.271357	−	0.962479i	$-0.412527\pi$
$887$	16.1634	0.542715	0.271357	+	0.962479i	$0.412527\pi$
$888$	−0.389645	−0.0130756
$889$	−75.6376	−2.53680
$890$	−0.343525	−0.0115150
$891$	0	0
$892$	24.5392	0.821633
$893$	−20.1819	−0.675363
$894$	−2.13572	−0.0714290
$895$	−21.1378	−0.706560
$896$	−45.8970	−1.53331
$897$	4.72274	0.157688
$898$	8.19064	0.273325
$899$	11.0272	0.367776
$900$	33.7140	1.12380
$901$	19.6127	0.653393
$902$	0	0
$903$	−8.46411	−0.281668
$904$	−29.3070	−0.974736
$905$	73.8051	2.45337
$906$	0.978569	0.0325108
$907$	19.0290	0.631847	0.315924	−	0.948785i	$-0.397686\pi$
$907$	19.0290	0.631847	0.315924	+	0.948785i	$0.397686\pi$
$908$	10.9260	0.362592
$909$	−19.0875	−0.633094
$910$	−19.9799	−0.662327
$911$	37.6266	1.24663	0.623313	−	0.781973i	$-0.285786\pi$
$911$	37.6266	1.24663	0.623313	+	0.781973i	$0.285786\pi$
$912$	1.89474	0.0627412
$913$	0	0
$914$	7.59246	0.251136
$915$	−4.00694	−0.132465
$916$	−32.4066	−1.07074
$917$	−45.8510	−1.51413
$918$	−2.94306	−0.0971353
$919$	−44.2948	−1.46115	−0.730575	−	0.682833i	$-0.760748\pi$
$919$	−44.2948	−1.46115	−0.730575	+	0.682833i	$0.760748\pi$
$920$	−27.5151	−0.907145
$921$	−7.28771	−0.240138
$922$	−12.6194	−0.415598
$923$	−29.8869	−0.983738
$924$	0	0
$925$	−6.22923	−0.204816
$926$	11.2060	0.368252
$927$	−9.25605	−0.304009
$928$	−5.29772	−0.173906
$929$	−45.2507	−1.48463	−0.742314	−	0.670052i	$-0.766272\pi$
$929$	−45.2507	−1.48463	−0.742314	+	0.670052i	$0.766272\pi$
$930$	3.00302	0.0984728
$931$	30.8613	1.01144
$932$	−51.6213	−1.69091
$933$	−1.90947	−0.0625133
$934$	4.27908	0.140016
$935$	0	0
$936$	−15.0550	−0.492087
$937$	−3.99583	−0.130538	−0.0652690	−	0.997868i	$-0.520791\pi$
$937$	−3.99583	−0.130538	−0.0652690	+	0.997868i	$0.520791\pi$
$938$	27.3131	0.891805
$939$	1.17699	0.0384096
$940$	−52.3790	−1.70841
$941$	5.20212	0.169584	0.0847921	−	0.996399i	$-0.472977\pi$
$941$	5.20212	0.169584	0.0847921	+	0.996399i	$0.472977\pi$
$942$	1.92945	0.0628650
$943$	−6.65976	−0.216872
$944$	−12.4676	−0.405786
$945$	−22.8482	−0.743253
$946$	0	0
$947$	−8.84665	−0.287478	−0.143739	−	0.989616i	$-0.545913\pi$
$947$	−8.84665	−0.287478	−0.143739	+	0.989616i	$0.545913\pi$
$948$	1.46266	0.0475050
$949$	−24.0836	−0.781788
$950$	−5.84597	−0.189668
$951$	−2.26468	−0.0734372
$952$	33.1191	1.07340
$953$	22.7420	0.736684	0.368342	−	0.929690i	$-0.379926\pi$
$953$	22.7420	0.736684	0.368342	+	0.929690i	$0.379926\pi$
$954$	−4.63246	−0.149981
$955$	−18.5097	−0.598960
$956$	39.0879	1.26419
$957$	0	0
$958$	0.395625	0.0127821
$959$	−53.8992	−1.74050
$960$	3.88860	0.125504
$961$	47.1926	1.52234
$962$	1.33467	0.0430315
$963$	−53.2615	−1.71633
$964$	13.7971	0.444375
$965$	−19.5355	−0.628871
$966$	2.45396	0.0789550
$967$	−47.4256	−1.52510	−0.762552	−	0.646927i	$-0.776054\pi$
$967$	−47.4256	−1.52510	−0.762552	+	0.646927i	$0.776054\pi$
$968$	0	0
$969$	−2.99834	−0.0963206
$970$	−19.7370	−0.633717
$971$	25.4284	0.816038	0.408019	−	0.912974i	$-0.366220\pi$
$971$	25.4284	0.816038	0.408019	+	0.912974i	$0.366220\pi$
$972$	−12.4369	−0.398914
$973$	−57.5034	−1.84347
$974$	−8.51402	−0.272807
$975$	5.42756	0.173821
$976$	14.3771	0.460200
$977$	−46.8585	−1.49914	−0.749568	−	0.661928i	$-0.769738\pi$
$977$	−46.8585	−1.49914	−0.749568	+	0.661928i	$0.769738\pi$
$978$	0.621006	0.0198576
$979$	0	0
$980$	80.0956	2.55856
$981$	19.2591	0.614896
$982$	−5.35171	−0.170780
$983$	−38.1894	−1.21805	−0.609026	−	0.793151i	$-0.708439\pi$
$983$	−38.1894	−1.21805	−0.609026	+	0.793151i	$0.708439\pi$
$984$	−0.478744	−0.0152618
$985$	49.1843	1.56714
$986$	2.40462	0.0765786
$987$	9.73608	0.309903
$988$	−14.8844	−0.473537
$989$	39.9267	1.26960
$990$	0	0
$991$	−37.1978	−1.18163	−0.590813	−	0.806808i	$-0.701193\pi$
$991$	−37.1978	−1.18163	−0.590813	+	0.806808i	$0.701193\pi$
$992$	−37.5657	−1.19271
$993$	3.58561	0.113786
$994$	−15.5294	−0.492562
$995$	−59.2824	−1.87938
$996$	2.78882	0.0883673
$997$	10.0316	0.317705	0.158852	−	0.987302i	$-0.449221\pi$
$997$	10.0316	0.317705	0.158852	+	0.987302i	$0.449221\pi$
$998$	−0.195724	−0.00619553
$999$	1.52628	0.0482893

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4477.2.a.q.1.13	✓	28
11.10	odd	2	inner	4477.2.a.q.1.16	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4477.2.a.q.1.13	✓	28	1.1	even	1	trivial
4477.2.a.q.1.16	yes	28	11.10	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4477 = 11^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4477.a (trivial)

\(f(q)\)	\(=\)	\(q-0.394005 q^{2} +0.257216 q^{3} -1.84476 q^{4} -3.35100 q^{5} -0.101344 q^{6} -4.46729 q^{7} +1.51486 q^{8} -2.93384 q^{9} +1.32031 q^{10} -0.474502 q^{12} +3.38744 q^{13} +1.76014 q^{14} -0.861932 q^{15} +3.09266 q^{16} -4.89399 q^{17} +1.15595 q^{18} +2.38188 q^{19} +6.18180 q^{20} -1.14906 q^{21} +5.42031 q^{23} +0.389645 q^{24} +6.22923 q^{25} -1.33467 q^{26} -1.52628 q^{27} +8.24108 q^{28} +1.24704 q^{29} +0.339605 q^{30} +8.84266 q^{31} -4.24823 q^{32} +1.92826 q^{34} +14.9699 q^{35} +5.41223 q^{36} -1.00000 q^{37} -0.938474 q^{38} +0.871304 q^{39} -5.07629 q^{40} -1.22867 q^{41} +0.452735 q^{42} +7.36613 q^{43} +9.83131 q^{45} -2.13563 q^{46} -8.47310 q^{47} +0.795481 q^{48} +12.9567 q^{49} -2.45435 q^{50} -1.25881 q^{51} -6.24902 q^{52} -4.00750 q^{53} +0.601361 q^{54} -6.76730 q^{56} +0.612658 q^{57} -0.491341 q^{58} -4.03136 q^{59} +1.59006 q^{60} +4.64879 q^{61} -3.48405 q^{62} +13.1063 q^{63} -4.51149 q^{64} -11.3513 q^{65} +15.5176 q^{67} +9.02824 q^{68} +1.39419 q^{69} -5.89822 q^{70} -8.82283 q^{71} -4.44434 q^{72} -7.10968 q^{73} +0.394005 q^{74} +1.60226 q^{75} -4.39400 q^{76} -0.343298 q^{78} -3.08252 q^{79} -10.3635 q^{80} +8.40894 q^{81} +0.484101 q^{82} -5.87738 q^{83} +2.11974 q^{84} +16.3998 q^{85} -2.90229 q^{86} +0.320759 q^{87} -0.260185 q^{89} -3.87359 q^{90} -15.1327 q^{91} -9.99918 q^{92} +2.27447 q^{93} +3.33844 q^{94} -7.98170 q^{95} -1.09271 q^{96} -14.9487 q^{97} -5.10500 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 16 q^{3} + 20 q^{4} - 12 q^{5} + 16 q^{9} - 22 q^{12} - 6 q^{14} - 16 q^{15} + 36 q^{16} - 50 q^{20} - 28 q^{23} + 12 q^{25} - 68 q^{26} - 34 q^{27} - 8 q^{31} + 44 q^{34} - 16 q^{36} - 28 q^{37}+ \cdots - 14 q^{97}+O(q^{100}) \) 28 * q - 16 * q^3 + 20 * q^4 - 12 * q^5 + 16 * q^9 - 22 * q^12 - 6 * q^14 - 16 * q^15 + 36 * q^16 - 50 * q^20 - 28 * q^23 + 12 * q^25 - 68 * q^26 - 34 * q^27 - 8 * q^31 + 44 * q^34 - 16 * q^36 - 28 * q^37 - 14 * q^38 - 40 * q^42 + 34 * q^45 - 108 * q^47 - 128 * q^48 + 4 * q^49 - 6 * q^53 + 26 * q^56 - 6 * q^58 - 94 * q^59 + 102 * q^60 + 48 * q^64 - 28 * q^67 + 10 * q^69 - 120 * q^70 - 62 * q^71 - 14 * q^75 + 120 * q^78 - 128 * q^80 + 36 * q^81 - 20 * q^86 - 32 * q^89 - 58 * q^91 - 120 * q^92 - 32 * q^93 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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