LMFDB - Embedded newform 2275.2.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2275,2,Mod(1,2275)] 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("2275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2275 = 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1659664598$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	2275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.470683 q^{2} -2.24914 q^{3} -1.77846 q^{4} +1.05863 q^{6} +1.00000 q^{7} +1.77846 q^{8} +2.05863 q^{9} -2.24914 q^{11} +4.00000 q^{12} -1.00000 q^{13} -0.470683 q^{14} +2.71982 q^{16} +1.30777 q^{17} -0.968964 q^{18} -1.47068 q^{19} -2.24914 q^{21} +1.05863 q^{22} -5.83709 q^{23} -4.00000 q^{24} +0.470683 q^{26} +2.11727 q^{27} -1.77846 q^{28} +5.22154 q^{29} -7.02760 q^{31} -4.83709 q^{32} +5.05863 q^{33} -0.615547 q^{34} -3.66119 q^{36} +2.36641 q^{37} +0.692226 q^{38} +2.24914 q^{39} +6.49828 q^{41} +1.05863 q^{42} -11.3940 q^{43} +4.00000 q^{44} +2.74742 q^{46} -8.58451 q^{47} -6.11727 q^{48} +1.00000 q^{49} -2.94137 q^{51} +1.77846 q^{52} -11.2767 q^{53} -0.996562 q^{54} +1.77846 q^{56} +3.30777 q^{57} -2.45769 q^{58} -12.1725 q^{59} -2.00000 q^{61} +3.30777 q^{62} +2.05863 q^{63} -3.16291 q^{64} -2.38101 q^{66} +15.9379 q^{67} -2.32582 q^{68} +13.1284 q^{69} +1.19051 q^{71} +3.66119 q^{72} -7.64315 q^{73} -1.11383 q^{74} +2.61555 q^{76} -2.24914 q^{77} -1.05863 q^{78} -1.33881 q^{79} -10.9379 q^{81} -3.05863 q^{82} +16.3500 q^{83} +4.00000 q^{84} +5.36297 q^{86} -11.7440 q^{87} -4.00000 q^{88} +6.91033 q^{89} -1.00000 q^{91} +10.3810 q^{92} +15.8061 q^{93} +4.04059 q^{94} +10.8793 q^{96} +3.47068 q^{97} -0.470683 q^{98} -4.63016 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 2 q^{3} + 3 q^{4} + 4 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + 2 q^{11} + 12 q^{12} - 3 q^{13} - q^{14} - q^{16} - 4 q^{17} + 15 q^{18} - 4 q^{19} + 2 q^{21} + 4 q^{22} - 10 q^{23} - 12 q^{24}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q - q^2 + 2 * q^3 + 3 * q^4 + 4 * q^6 + 3 * q^7 - 3 * q^8 + 7 * q^9 + 2 * q^11 + 12 * q^12 - 3 * q^13 - q^14 - q^16 - 4 * q^17 + 15 * q^18 - 4 * q^19 + 2 * q^21 + 4 * q^22 - 10 * q^23 - 12 * q^24 + q^26 + 8 * q^27 + 3 * q^28 + 24 * q^29 - 4 * q^31 - 7 * q^32 + 16 * q^33 + 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^41 + 4 * q^42 - 10 * q^43 + 12 * q^44 - 18 * q^46 + 8 * q^47 - 20 * q^48 + 3 * q^49 - 8 * q^51 - 3 * q^52 - 8 * q^53 + 32 * q^54 - 3 * q^56 + 2 * q^57 - 12 * q^58 - 4 * q^59 - 6 * q^61 + 2 * q^62 + 7 * q^63 - 17 * q^64 + 12 * q^66 + 12 * q^67 - 22 * q^68 - 6 * q^69 - 6 * q^71 + q^72 + 10 * q^73 + 30 * q^74 - 8 * q^76 + 2 * q^77 - 4 * q^78 - 14 * q^79 + 3 * q^81 - 10 * q^82 + 12 * q^83 + 12 * q^84 - 26 * q^86 + 26 * q^87 - 12 * q^88 + 2 * q^89 - 3 * q^91 + 12 * q^92 + 22 * q^93 - 10 * q^94 - 4 * q^96 + 10 * q^97 - q^98 + 14 * 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 $

$2$	−0.470683	−0.332823	−0.166412	−	0.986056i	$-0.553218\pi$
$2$	−0.470683	−0.332823	−0.166412	+	0.986056i	$0.553218\pi$
$3$	−2.24914	−1.29854	−0.649271	−	0.760557i	$-0.724926\pi$
$3$	−2.24914	−1.29854	−0.649271	+	0.760557i	$0.724926\pi$
$4$	−1.77846	−0.889229
$5$	0	0
$5$	0	0
$6$	1.05863	0.432185
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.77846	0.628780
$9$	2.05863	0.686211
$10$	0	0
$11$	−2.24914	−0.678141	−0.339071	−	0.940761i	$-0.610113\pi$
$11$	−2.24914	−0.678141	−0.339071	+	0.940761i	$0.610113\pi$
$12$	4.00000	1.15470
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.470683	−0.125795
$15$	0	0
$16$	2.71982	0.679956
$17$	1.30777	0.317182	0.158591	−	0.987344i	$-0.449305\pi$
$17$	1.30777	0.317182	0.158591	+	0.987344i	$0.449305\pi$
$18$	−0.968964	−0.228387
$19$	−1.47068	−0.337398	−0.168699	−	0.985668i	$-0.553957\pi$
$19$	−1.47068	−0.337398	−0.168699	+	0.985668i	$0.553957\pi$
$20$	0	0
$21$	−2.24914	−0.490803
$22$	1.05863	0.225701
$23$	−5.83709	−1.21712	−0.608559	−	0.793509i	$-0.708252\pi$
$23$	−5.83709	−1.21712	−0.608559	+	0.793509i	$0.708252\pi$
$24$	−4.00000	−0.816497
$25$	0	0
$26$	0.470683	0.0923086
$27$	2.11727	0.407468
$28$	−1.77846	−0.336097
$29$	5.22154	0.969616	0.484808	−	0.874621i	$-0.338889\pi$
$29$	5.22154	0.969616	0.484808	+	0.874621i	$0.338889\pi$
$30$	0	0
$31$	−7.02760	−1.26219	−0.631097	−	0.775704i	$-0.717395\pi$
$31$	−7.02760	−1.26219	−0.631097	+	0.775704i	$0.717395\pi$
$32$	−4.83709	−0.855085
$33$	5.05863	0.880595
$34$	−0.615547	−0.105566
$35$	0	0
$36$	−3.66119	−0.610198
$37$	2.36641	0.389035	0.194517	−	0.980899i	$-0.437686\pi$
$37$	2.36641	0.389035	0.194517	+	0.980899i	$0.437686\pi$
$38$	0.692226	0.112294
$39$	2.24914	0.360151
$40$	0	0
$41$	6.49828	1.01486	0.507431	−	0.861693i	$-0.330595\pi$
$41$	6.49828	1.01486	0.507431	+	0.861693i	$0.330595\pi$
$42$	1.05863	0.163351
$43$	−11.3940	−1.73757	−0.868785	−	0.495190i	$-0.835098\pi$
$43$	−11.3940	−1.73757	−0.868785	+	0.495190i	$0.835098\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	2.74742	0.405085
$47$	−8.58451	−1.25218	−0.626090	−	0.779751i	$-0.715346\pi$
$47$	−8.58451	−1.25218	−0.626090	+	0.779751i	$0.715346\pi$
$48$	−6.11727	−0.882951
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.94137	−0.411874
$52$	1.77846	0.246628
$53$	−11.2767	−1.54898	−0.774490	−	0.632587i	$-0.781993\pi$
$53$	−11.2767	−1.54898	−0.774490	+	0.632587i	$0.781993\pi$
$54$	−0.996562	−0.135615
$55$	0	0
$56$	1.77846	0.237656
$57$	3.30777	0.438125
$58$	−2.45769	−0.322711
$59$	−12.1725	−1.58472	−0.792360	−	0.610054i	$-0.791148\pi$
$59$	−12.1725	−1.58472	−0.792360	+	0.610054i	$0.791148\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.30777	0.420088
$63$	2.05863	0.259363
$64$	−3.16291	−0.395364
$65$	0	0
$66$	−2.38101	−0.293083
$67$	15.9379	1.94713	0.973564	−	0.228415i	$-0.0733542\pi$
$67$	15.9379	1.94713	0.973564	+	0.228415i	$0.0733542\pi$
$68$	−2.32582	−0.282047
$69$	13.1284	1.58048
$70$	0	0
$71$	1.19051	0.141287	0.0706436	−	0.997502i	$-0.477495\pi$
$71$	1.19051	0.141287	0.0706436	+	0.997502i	$0.477495\pi$
$72$	3.66119	0.431475
$73$	−7.64315	−0.894562	−0.447281	−	0.894393i	$-0.647608\pi$
$73$	−7.64315	−0.894562	−0.447281	+	0.894393i	$0.647608\pi$
$74$	−1.11383	−0.129480
$75$	0	0
$76$	2.61555	0.300024
$77$	−2.24914	−0.256313
$78$	−1.05863	−0.119867
$79$	−1.33881	−0.150628	−0.0753139	−	0.997160i	$-0.523996\pi$
$79$	−1.33881	−0.150628	−0.0753139	+	0.997160i	$0.523996\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	−3.05863	−0.337770
$83$	16.3500	1.79464	0.897322	−	0.441377i	$-0.145510\pi$
$83$	16.3500	1.79464	0.897322	+	0.441377i	$0.145510\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	5.36297	0.578304
$87$	−11.7440	−1.25909
$88$	−4.00000	−0.426401
$89$	6.91033	0.732494	0.366247	−	0.930518i	$-0.380643\pi$
$89$	6.91033	0.732494	0.366247	+	0.930518i	$0.380643\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	10.3810	1.08230
$93$	15.8061	1.63901
$94$	4.04059	0.416755
$95$	0	0
$96$	10.8793	1.11036
$97$	3.47068	0.352395	0.176197	−	0.984355i	$-0.443620\pi$
$97$	3.47068	0.352395	0.176197	+	0.984355i	$0.443620\pi$
$98$	−0.470683	−0.0475462
$99$	−4.63016	−0.465348
$100$	0	0
$101$	7.75086	0.771239	0.385620	−	0.922658i	$-0.373988\pi$
$101$	7.75086	0.771239	0.385620	+	0.922658i	$0.373988\pi$
$102$	1.38445	0.137081
$103$	16.9966	1.67472	0.837361	−	0.546651i	$-0.184098\pi$
$103$	16.9966	1.67472	0.837361	+	0.546651i	$0.184098\pi$
$104$	−1.77846	−0.174392
$105$	0	0
$106$	5.30777	0.515537
$107$	5.55691	0.537207	0.268604	−	0.963251i	$-0.413438\pi$
$107$	5.55691	0.537207	0.268604	+	0.963251i	$0.413438\pi$
$108$	−3.76547	−0.362332
$109$	7.92332	0.758917	0.379458	−	0.925209i	$-0.376110\pi$
$109$	7.92332	0.758917	0.379458	+	0.925209i	$0.376110\pi$
$110$	0	0
$111$	−5.32238	−0.505178
$112$	2.71982	0.256999
$113$	9.89229	0.930588	0.465294	−	0.885156i	$-0.345949\pi$
$113$	9.89229	0.930588	0.465294	+	0.885156i	$0.345949\pi$
$114$	−1.55691	−0.145818
$115$	0	0
$116$	−9.28629	−0.862210
$117$	−2.05863	−0.190321
$118$	5.72938	0.527432
$119$	1.30777	0.119883
$120$	0	0
$121$	−5.94137	−0.540124
$122$	0.941367	0.0852273
$123$	−14.6155	−1.31784
$124$	12.4983	1.12238
$125$	0	0
$126$	−0.968964	−0.0863222
$127$	−0.824101	−0.0731271	−0.0365635	−	0.999331i	$-0.511641\pi$
$127$	−0.824101	−0.0731271	−0.0365635	+	0.999331i	$0.511641\pi$
$128$	11.1629	0.986671
$129$	25.6267	2.25631
$130$	0	0
$131$	10.6155	0.927485	0.463742	−	0.885970i	$-0.346506\pi$
$131$	10.6155	0.927485	0.463742	+	0.885970i	$0.346506\pi$
$132$	−8.99656	−0.783050
$133$	−1.47068	−0.127524
$134$	−7.50172	−0.648050
$135$	0	0
$136$	2.32582	0.199437
$137$	−11.3630	−0.970804	−0.485402	−	0.874291i	$-0.661327\pi$
$137$	−11.3630	−0.970804	−0.485402	+	0.874291i	$0.661327\pi$
$138$	−6.17934	−0.526020
$139$	−13.9233	−1.18096	−0.590480	−	0.807052i	$-0.701062\pi$
$139$	−13.9233	−1.18096	−0.590480	+	0.807052i	$0.701062\pi$
$140$	0	0
$141$	19.3078	1.62601
$142$	−0.560352	−0.0470237
$143$	2.24914	0.188083
$144$	5.59912	0.466593
$145$	0	0
$146$	3.59750	0.297731
$147$	−2.24914	−0.185506
$148$	−4.20855	−0.345941
$149$	9.30777	0.762523	0.381261	−	0.924467i	$-0.375490\pi$
$149$	9.30777	0.762523	0.381261	+	0.924467i	$0.375490\pi$
$150$	0	0
$151$	7.07324	0.575612	0.287806	−	0.957689i	$-0.407074\pi$
$151$	7.07324	0.575612	0.287806	+	0.957689i	$0.407074\pi$
$152$	−2.61555	−0.212149
$153$	2.69223	0.217654
$154$	1.05863	0.0853071
$155$	0	0
$156$	−4.00000	−0.320256
$157$	6.04059	0.482091	0.241046	−	0.970514i	$-0.422510\pi$
$157$	6.04059	0.482091	0.241046	+	0.970514i	$0.422510\pi$
$158$	0.630155	0.0501325
$159$	25.3630	2.01141
$160$	0	0
$161$	−5.83709	−0.460027
$162$	5.14830	0.404489
$163$	−6.38101	−0.499800	−0.249900	−	0.968272i	$-0.580398\pi$
$163$	−6.38101	−0.499800	−0.249900	+	0.968272i	$0.580398\pi$
$164$	−11.5569	−0.902443
$165$	0	0
$166$	−7.69566	−0.597299
$167$	16.5845	1.28335	0.641674	−	0.766977i	$-0.278240\pi$
$167$	16.5845	1.28335	0.641674	+	0.766977i	$0.278240\pi$
$168$	−4.00000	−0.308607
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.02760	−0.231526
$172$	20.2637	1.54510
$173$	23.3009	1.77153	0.885767	−	0.464130i	$-0.153633\pi$
$173$	23.3009	1.77153	0.885767	+	0.464130i	$0.153633\pi$
$174$	5.52770	0.419054
$175$	0	0
$176$	−6.11727	−0.461106
$177$	27.3776	2.05782
$178$	−3.25258	−0.243791
$179$	21.0422	1.57277	0.786384	−	0.617738i	$-0.211951\pi$
$179$	21.0422	1.57277	0.786384	+	0.617738i	$0.211951\pi$
$180$	0	0
$181$	16.7474	1.24483	0.622413	−	0.782689i	$-0.286152\pi$
$181$	16.7474	1.24483	0.622413	+	0.782689i	$0.286152\pi$
$182$	0.470683	0.0348894
$183$	4.49828	0.332523
$184$	−10.3810	−0.765299
$185$	0	0
$186$	−7.43965	−0.545501
$187$	−2.94137	−0.215094
$188$	15.2672	1.11347
$189$	2.11727	0.154008
$190$	0	0
$191$	−7.43965	−0.538314	−0.269157	−	0.963096i	$-0.586745\pi$
$191$	−7.43965	−0.538314	−0.269157	+	0.963096i	$0.586745\pi$
$192$	7.11383	0.513396
$193$	1.50172	0.108096	0.0540480	−	0.998538i	$-0.482788\pi$
$193$	1.50172	0.108096	0.0540480	+	0.998538i	$0.482788\pi$
$194$	−1.63359	−0.117285
$195$	0	0
$196$	−1.77846	−0.127033
$197$	−23.9931	−1.70944	−0.854720	−	0.519090i	$-0.826271\pi$
$197$	−23.9931	−1.70944	−0.854720	+	0.519090i	$0.826271\pi$
$198$	2.17934	0.154879
$199$	−2.01461	−0.142812	−0.0714059	−	0.997447i	$-0.522749\pi$
$199$	−2.01461	−0.142812	−0.0714059	+	0.997447i	$0.522749\pi$
$200$	0	0
$201$	−35.8466	−2.52843
$202$	−3.64820	−0.256687
$203$	5.22154	0.366480
$204$	5.23109	0.366250
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−12.0164	−0.835199
$208$	−2.71982	−0.188586
$209$	3.30777	0.228803
$210$	0	0
$211$	10.1008	0.695370	0.347685	−	0.937611i	$-0.386968\pi$
$211$	10.1008	0.695370	0.347685	+	0.937611i	$0.386968\pi$
$212$	20.0552	1.37740
$213$	−2.67762	−0.183467
$214$	−2.61555	−0.178795
$215$	0	0
$216$	3.76547	0.256208
$217$	−7.02760	−0.477064
$218$	−3.72938	−0.252585
$219$	17.1905	1.16163
$220$	0	0
$221$	−1.30777	−0.0879704
$222$	2.50516	0.168135
$223$	−10.1414	−0.679120	−0.339560	−	0.940584i	$-0.610278\pi$
$223$	−10.1414	−0.679120	−0.339560	+	0.940584i	$0.610278\pi$
$224$	−4.83709	−0.323192
$225$	0	0
$226$	−4.65613	−0.309721
$227$	5.38445	0.357379	0.178689	−	0.983906i	$-0.442814\pi$
$227$	5.38445	0.357379	0.178689	+	0.983906i	$0.442814\pi$
$228$	−5.88273	−0.389594
$229$	3.32238	0.219549	0.109775	−	0.993957i	$-0.464987\pi$
$229$	3.32238	0.219549	0.109775	+	0.993957i	$0.464987\pi$
$230$	0	0
$231$	5.05863	0.332834
$232$	9.28629	0.609675
$233$	−13.7198	−0.898816	−0.449408	−	0.893327i	$-0.648365\pi$
$233$	−13.7198	−0.898816	−0.449408	+	0.893327i	$0.648365\pi$
$234$	0.968964	0.0633432
$235$	0	0
$236$	21.6482	1.40918
$237$	3.01117	0.195597
$238$	−0.615547	−0.0399000
$239$	−3.50172	−0.226507	−0.113254	−	0.993566i	$-0.536127\pi$
$239$	−3.50172	−0.226507	−0.113254	+	0.993566i	$0.536127\pi$
$240$	0	0
$241$	1.58795	0.102289	0.0511444	−	0.998691i	$-0.483713\pi$
$241$	1.58795	0.102289	0.0511444	+	0.998691i	$0.483713\pi$
$242$	2.79650	0.179766
$243$	18.2491	1.17068
$244$	3.55691	0.227708
$245$	0	0
$246$	6.87930	0.438608
$247$	1.47068	0.0935773
$248$	−12.4983	−0.793642
$249$	−36.7734	−2.33042
$250$	0	0
$251$	−4.92676	−0.310974	−0.155487	−	0.987838i	$-0.549695\pi$
$251$	−4.92676	−0.310974	−0.155487	+	0.987838i	$0.549695\pi$
$252$	−3.66119	−0.230633
$253$	13.1284	0.825378
$254$	0.387890	0.0243384
$255$	0	0
$256$	1.07162	0.0669764
$257$	8.01461	0.499938	0.249969	−	0.968254i	$-0.419580\pi$
$257$	8.01461	0.499938	0.249969	+	0.968254i	$0.419580\pi$
$258$	−12.0621	−0.750952
$259$	2.36641	0.147041
$260$	0	0
$261$	10.7492	0.665361
$262$	−4.99656	−0.308689
$263$	−1.60256	−0.0988179	−0.0494090	−	0.998779i	$-0.515734\pi$
$263$	−1.60256	−0.0988179	−0.0494090	+	0.998779i	$0.515734\pi$
$264$	8.99656	0.553700
$265$	0	0
$266$	0.692226	0.0424431
$267$	−15.5423	−0.951174
$268$	−28.3449	−1.73144
$269$	−11.8207	−0.720719	−0.360359	−	0.932814i	$-0.617346\pi$
$269$	−11.8207	−0.720719	−0.360359	+	0.932814i	$0.617346\pi$
$270$	0	0
$271$	21.8827	1.32928	0.664641	−	0.747163i	$-0.268585\pi$
$271$	21.8827	1.32928	0.664641	+	0.747163i	$0.268585\pi$
$272$	3.55691	0.215670
$273$	2.24914	0.136124
$274$	5.34836	0.323106
$275$	0	0
$276$	−23.3484	−1.40541
$277$	10.2181	0.613946	0.306973	−	0.951718i	$-0.400684\pi$
$277$	10.2181	0.613946	0.306973	+	0.951718i	$0.400684\pi$
$278$	6.55348	0.393051
$279$	−14.4672	−0.866131
$280$	0	0
$281$	1.54231	0.0920063	0.0460031	−	0.998941i	$-0.485352\pi$
$281$	1.54231	0.0920063	0.0460031	+	0.998941i	$0.485352\pi$
$282$	−9.08785	−0.541174
$283$	−15.8466	−0.941985	−0.470993	−	0.882137i	$-0.656104\pi$
$283$	−15.8466	−0.941985	−0.470993	+	0.882137i	$0.656104\pi$
$284$	−2.11727	−0.125637
$285$	0	0
$286$	−1.05863	−0.0625983
$287$	6.49828	0.383581
$288$	−9.95779	−0.586769
$289$	−15.2897	−0.899396
$290$	0	0
$291$	−7.80605	−0.457599
$292$	13.5930	0.795470
$293$	−11.0828	−0.647464	−0.323732	−	0.946149i	$-0.604938\pi$
$293$	−11.0828	−0.647464	−0.323732	+	0.946149i	$0.604938\pi$
$294$	1.05863	0.0617407
$295$	0	0
$296$	4.20855	0.244617
$297$	−4.76203	−0.276321
$298$	−4.38101	−0.253785
$299$	5.83709	0.337568
$300$	0	0
$301$	−11.3940	−0.656740
$302$	−3.32926	−0.191577
$303$	−17.4328	−1.00149
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−1.26719	−0.0724402
$307$	−20.4121	−1.16498	−0.582489	−	0.812839i	$-0.697921\pi$
$307$	−20.4121	−1.16498	−0.582489	+	0.812839i	$0.697921\pi$
$308$	4.00000	0.227921
$309$	−38.2277	−2.17470
$310$	0	0
$311$	−1.92332	−0.109062	−0.0545308	−	0.998512i	$-0.517366\pi$
$311$	−1.92332	−0.109062	−0.0545308	+	0.998512i	$0.517366\pi$
$312$	4.00000	0.226455
$313$	14.3664	0.812037	0.406019	−	0.913865i	$-0.366917\pi$
$313$	14.3664	0.812037	0.406019	+	0.913865i	$0.366917\pi$
$314$	−2.84320	−0.160451
$315$	0	0
$316$	2.38101	0.133943
$317$	−15.5569	−0.873763	−0.436882	−	0.899519i	$-0.643917\pi$
$317$	−15.5569	−0.873763	−0.436882	+	0.899519i	$0.643917\pi$
$318$	−11.9379	−0.669446
$319$	−11.7440	−0.657537
$320$	0	0
$321$	−12.4983	−0.697586
$322$	2.74742	0.153108
$323$	−1.92332	−0.107016
$324$	19.4526	1.08070
$325$	0	0
$326$	3.00344	0.166345
$327$	−17.8207	−0.985485
$328$	11.5569	0.638124
$329$	−8.58451	−0.473279
$330$	0	0
$331$	31.5500	1.73415	0.867073	−	0.498180i	$-0.165998\pi$
$331$	31.5500	1.73415	0.867073	+	0.498180i	$0.165998\pi$
$332$	−29.0777	−1.59585
$333$	4.87156	0.266960
$334$	−7.80605	−0.427128
$335$	0	0
$336$	−6.11727	−0.333724
$337$	8.42666	0.459029	0.229515	−	0.973305i	$-0.426286\pi$
$337$	8.42666	0.459029	0.229515	+	0.973305i	$0.426286\pi$
$338$	−0.470683	−0.0256018
$339$	−22.2491	−1.20841
$340$	0	0
$341$	15.8061	0.855946
$342$	1.42504	0.0770573
$343$	1.00000	0.0539949
$344$	−20.2637	−1.09255
$345$	0	0
$346$	−10.9673	−0.589608
$347$	−19.3484	−1.03867	−0.519337	−	0.854569i	$-0.673821\pi$
$347$	−19.3484	−1.03867	−0.519337	+	0.854569i	$0.673821\pi$
$348$	20.8862	1.11962
$349$	27.2553	1.45894	0.729470	−	0.684013i	$-0.239767\pi$
$349$	27.2553	1.45894	0.729470	+	0.684013i	$0.239767\pi$
$350$	0	0
$351$	−2.11727	−0.113011
$352$	10.8793	0.579868
$353$	25.6742	1.36650	0.683249	−	0.730185i	$-0.260566\pi$
$353$	25.6742	1.36650	0.683249	+	0.730185i	$0.260566\pi$
$354$	−12.8862	−0.684892
$355$	0	0
$356$	−12.2897	−0.651354
$357$	−2.94137	−0.155674
$358$	−9.90422	−0.523454
$359$	−23.4182	−1.23596	−0.617982	−	0.786192i	$-0.712049\pi$
$359$	−23.4182	−1.23596	−0.617982	+	0.786192i	$0.712049\pi$
$360$	0	0
$361$	−16.8371	−0.886163
$362$	−7.88273	−0.414307
$363$	13.3630	0.701374
$364$	1.77846	0.0932165
$365$	0	0
$366$	−2.11727	−0.110671
$367$	14.6854	0.766569	0.383285	−	0.923630i	$-0.374793\pi$
$367$	14.6854	0.766569	0.383285	+	0.923630i	$0.374793\pi$
$368$	−15.8759	−0.827586
$369$	13.3776	0.696409
$370$	0	0
$371$	−11.2767	−0.585459
$372$	−28.1104	−1.45746
$373$	23.6673	1.22545	0.612723	−	0.790298i	$-0.290074\pi$
$373$	23.6673	1.22545	0.612723	+	0.790298i	$0.290074\pi$
$374$	1.38445	0.0715883
$375$	0	0
$376$	−15.2672	−0.787345
$377$	−5.22154	−0.268923
$378$	−0.996562	−0.0512576
$379$	−32.7405	−1.68177	−0.840884	−	0.541215i	$-0.817965\pi$
$379$	−32.7405	−1.68177	−0.840884	+	0.541215i	$0.817965\pi$
$380$	0	0
$381$	1.85352	0.0949586
$382$	3.50172	0.179164
$383$	22.6155	1.15560	0.577800	−	0.816178i	$-0.303911\pi$
$383$	22.6155	1.15560	0.577800	+	0.816178i	$0.303911\pi$
$384$	−25.1070	−1.28123
$385$	0	0
$386$	−0.706834	−0.0359769
$387$	−23.4561	−1.19234
$388$	−6.17246	−0.313359
$389$	38.0483	1.92913	0.964563	−	0.263852i	$-0.0849930\pi$
$389$	38.0483	1.92913	0.964563	+	0.263852i	$0.0849930\pi$
$390$	0	0
$391$	−7.63359	−0.386047
$392$	1.77846	0.0898256
$393$	−23.8759	−1.20438
$394$	11.2932	0.568941
$395$	0	0
$396$	8.23453	0.413801
$397$	11.7052	0.587468	0.293734	−	0.955887i	$-0.405102\pi$
$397$	11.7052	0.587468	0.293734	+	0.955887i	$0.405102\pi$
$398$	0.948243	0.0475311
$399$	3.30777	0.165596
$400$	0	0
$401$	3.55691	0.177624	0.0888119	−	0.996048i	$-0.471693\pi$
$401$	3.55691	0.177624	0.0888119	+	0.996048i	$0.471693\pi$
$402$	16.8724	0.841520
$403$	7.02760	0.350070
$404$	−13.7846	−0.685808
$405$	0	0
$406$	−2.45769	−0.121973
$407$	−5.32238	−0.263821
$408$	−5.23109	−0.258978
$409$	5.26213	0.260196	0.130098	−	0.991501i	$-0.458471\pi$
$409$	5.26213	0.260196	0.130098	+	0.991501i	$0.458471\pi$
$410$	0	0
$411$	25.5569	1.26063
$412$	−30.2277	−1.48921
$413$	−12.1725	−0.598968
$414$	5.65593	0.277974
$415$	0	0
$416$	4.83709	0.237158
$417$	31.3155	1.53353
$418$	−1.55691	−0.0761512
$419$	26.0337	1.27183	0.635915	−	0.771759i	$-0.280623\pi$
$419$	26.0337	1.27183	0.635915	+	0.771759i	$0.280623\pi$
$420$	0	0
$421$	22.2423	1.08402	0.542011	−	0.840372i	$-0.317663\pi$
$421$	22.2423	1.08402	0.542011	+	0.840372i	$0.317663\pi$
$422$	−4.75430	−0.231436
$423$	−17.6724	−0.859260
$424$	−20.0552	−0.973966
$425$	0	0
$426$	1.26031	0.0610622
$427$	−2.00000	−0.0967868
$428$	−9.88273	−0.477700
$429$	−5.05863	−0.244233
$430$	0	0
$431$	27.6742	1.33302	0.666509	−	0.745497i	$-0.267788\pi$
$431$	27.6742	1.33302	0.666509	+	0.745497i	$0.267788\pi$
$432$	5.75859	0.277060
$433$	12.7880	0.614552	0.307276	−	0.951620i	$-0.400582\pi$
$433$	12.7880	0.614552	0.307276	+	0.951620i	$0.400582\pi$
$434$	3.30777	0.158778
$435$	0	0
$436$	−14.0913	−0.674850
$437$	8.58451	0.410653
$438$	−8.09129	−0.386617
$439$	−18.1656	−0.866996	−0.433498	−	0.901155i	$-0.642721\pi$
$439$	−18.1656	−0.866996	−0.433498	+	0.901155i	$0.642721\pi$
$440$	0	0
$441$	2.05863	0.0980302
$442$	0.615547	0.0292786
$443$	−0.107714	−0.00511767	−0.00255883	−	0.999997i	$-0.500815\pi$
$443$	−0.107714	−0.00511767	−0.00255883	+	0.999997i	$0.500815\pi$
$444$	9.46563	0.449219
$445$	0	0
$446$	4.77340	0.226027
$447$	−20.9345	−0.990167
$448$	−3.16291	−0.149433
$449$	−22.1725	−1.04638	−0.523192	−	0.852215i	$-0.675259\pi$
$449$	−22.1725	−1.04638	−0.523192	+	0.852215i	$0.675259\pi$
$450$	0	0
$451$	−14.6155	−0.688219
$452$	−17.5930	−0.827505
$453$	−15.9087	−0.747457
$454$	−2.53437	−0.118944
$455$	0	0
$456$	5.88273	0.275484
$457$	−4.35953	−0.203930	−0.101965	−	0.994788i	$-0.532513\pi$
$457$	−4.35953	−0.203930	−0.101965	+	0.994788i	$0.532513\pi$
$458$	−1.56379	−0.0730711
$459$	2.76891	0.129241
$460$	0	0
$461$	32.3810	1.50813	0.754067	−	0.656797i	$-0.228089\pi$
$461$	32.3810	1.50813	0.754067	+	0.656797i	$0.228089\pi$
$462$	−2.38101	−0.110775
$463$	−8.36641	−0.388820	−0.194410	−	0.980920i	$-0.562279\pi$
$463$	−8.36641	−0.388820	−0.194410	+	0.980920i	$0.562279\pi$
$464$	14.2017	0.659296
$465$	0	0
$466$	6.45769	0.299147
$467$	19.5423	0.904310	0.452155	−	0.891939i	$-0.350655\pi$
$467$	19.5423	0.904310	0.452155	+	0.891939i	$0.350655\pi$
$468$	3.66119	0.169239
$469$	15.9379	0.735945
$470$	0	0
$471$	−13.5861	−0.626016
$472$	−21.6482	−0.996439
$473$	25.6267	1.17832
$474$	−1.41731	−0.0650991
$475$	0	0
$476$	−2.32582	−0.106604
$477$	−23.2147	−1.06293
$478$	1.64820	0.0753870
$479$	−28.5224	−1.30322	−0.651612	−	0.758553i	$-0.725907\pi$
$479$	−28.5224	−1.30322	−0.651612	+	0.758553i	$0.725907\pi$
$480$	0	0
$481$	−2.36641	−0.107899
$482$	−0.747422	−0.0340441
$483$	13.1284	0.597365
$484$	10.5665	0.480294
$485$	0	0
$486$	−8.58957	−0.389631
$487$	−24.8241	−1.12489	−0.562444	−	0.826836i	$-0.690139\pi$
$487$	−24.8241	−1.12489	−0.562444	+	0.826836i	$0.690139\pi$
$488$	−3.55691	−0.161014
$489$	14.3518	0.649011
$490$	0	0
$491$	29.1690	1.31638	0.658190	−	0.752852i	$-0.271322\pi$
$491$	29.1690	1.31638	0.658190	+	0.752852i	$0.271322\pi$
$492$	25.9931	1.17186
$493$	6.82860	0.307545
$494$	−0.692226	−0.0311447
$495$	0	0
$496$	−19.1138	−0.858236
$497$	1.19051	0.0534016
$498$	17.3086	0.775618
$499$	−33.3009	−1.49075	−0.745376	−	0.666644i	$-0.767730\pi$
$499$	−33.3009	−1.49075	−0.745376	+	0.666644i	$0.767730\pi$
$500$	0	0
$501$	−37.3009	−1.66648
$502$	2.31894	0.103500
$503$	12.3258	0.549581	0.274791	−	0.961504i	$-0.411391\pi$
$503$	12.3258	0.549581	0.274791	+	0.961504i	$0.411391\pi$
$504$	3.66119	0.163082
$505$	0	0
$506$	−6.17934	−0.274705
$507$	−2.24914	−0.0998878
$508$	1.46563	0.0650267
$509$	23.7052	1.05072	0.525358	−	0.850882i	$-0.323932\pi$
$509$	23.7052	1.05072	0.525358	+	0.850882i	$0.323932\pi$
$510$	0	0
$511$	−7.64315	−0.338113
$512$	−22.8302	−1.00896
$513$	−3.11383	−0.137479
$514$	−3.77234	−0.166391
$515$	0	0
$516$	−45.5760	−2.00637
$517$	19.3078	0.849155
$518$	−1.11383	−0.0489388
$519$	−52.4070	−2.30041
$520$	0	0
$521$	43.9018	1.92337	0.961687	−	0.274149i	$-0.0883962\pi$
$521$	43.9018	1.92337	0.961687	+	0.274149i	$0.0883962\pi$
$522$	−5.05949	−0.221448
$523$	−37.4328	−1.63682	−0.818410	−	0.574634i	$-0.805144\pi$
$523$	−37.4328	−1.63682	−0.818410	+	0.574634i	$0.805144\pi$
$524$	−18.8793	−0.824746
$525$	0	0
$526$	0.754297	0.0328889
$527$	−9.19051	−0.400345
$528$	13.7586	0.598766
$529$	11.0716	0.481375
$530$	0	0
$531$	−25.0586	−1.08745
$532$	2.61555	0.113398
$533$	−6.49828	−0.281472
$534$	7.31551	0.316573
$535$	0	0
$536$	28.3449	1.22431
$537$	−47.3269	−2.04231
$538$	5.56379	0.239872
$539$	−2.24914	−0.0968773
$540$	0	0
$541$	−34.9751	−1.50370	−0.751848	−	0.659336i	$-0.770837\pi$
$541$	−34.9751	−1.50370	−0.751848	+	0.659336i	$0.770837\pi$
$542$	−10.2998	−0.442416
$543$	−37.6673	−1.61646
$544$	−6.32582	−0.271217
$545$	0	0
$546$	−1.05863	−0.0453053
$547$	−6.50783	−0.278255	−0.139127	−	0.990274i	$-0.544430\pi$
$547$	−6.50783	−0.278255	−0.139127	+	0.990274i	$0.544430\pi$
$548$	20.2086	0.863267
$549$	−4.11727	−0.175721
$550$	0	0
$551$	−7.67924	−0.327146
$552$	23.3484	0.993772
$553$	−1.33881	−0.0569320
$554$	−4.80949	−0.204336
$555$	0	0
$556$	24.7620	1.05014
$557$	43.4328	1.84031	0.920153	−	0.391559i	$-0.128064\pi$
$557$	43.4328	1.84031	0.920153	+	0.391559i	$0.128064\pi$
$558$	6.80949	0.288269
$559$	11.3940	0.481915
$560$	0	0
$561$	6.61555	0.279309
$562$	−0.725938	−0.0306218
$563$	33.8827	1.42799	0.713993	−	0.700152i	$-0.246885\pi$
$563$	33.8827	1.42799	0.713993	+	0.700152i	$0.246885\pi$
$564$	−34.3380	−1.44589
$565$	0	0
$566$	7.45875	0.313515
$567$	−10.9379	−0.459350
$568$	2.11727	0.0888385
$569$	19.2147	0.805521	0.402760	−	0.915305i	$-0.368051\pi$
$569$	19.2147	0.805521	0.402760	+	0.915305i	$0.368051\pi$
$570$	0	0
$571$	20.8268	0.871573	0.435787	−	0.900050i	$-0.356470\pi$
$571$	20.8268	0.871573	0.435787	+	0.900050i	$0.356470\pi$
$572$	−4.00000	−0.167248
$573$	16.7328	0.699023
$574$	−3.05863	−0.127665
$575$	0	0
$576$	−6.51127	−0.271303
$577$	−28.6448	−1.19250	−0.596249	−	0.802800i	$-0.703343\pi$
$577$	−28.6448	−1.19250	−0.596249	+	0.802800i	$0.703343\pi$
$578$	7.19662	0.299340
$579$	−3.37758	−0.140367
$580$	0	0
$581$	16.3500	0.678311
$582$	3.67418	0.152300
$583$	25.3630	1.05043
$584$	−13.5930	−0.562483
$585$	0	0
$586$	5.21649	0.215491
$587$	4.32076	0.178337	0.0891685	−	0.996017i	$-0.471579\pi$
$587$	4.32076	0.178337	0.0891685	+	0.996017i	$0.471579\pi$
$588$	4.00000	0.164957
$589$	10.3354	0.425862
$590$	0	0
$591$	53.9639	2.21978
$592$	6.43621	0.264527
$593$	15.9690	0.655767	0.327883	−	0.944718i	$-0.393665\pi$
$593$	15.9690	0.655767	0.327883	+	0.944718i	$0.393665\pi$
$594$	2.24141	0.0919661
$595$	0	0
$596$	−16.5535	−0.678057
$597$	4.53114	0.185447
$598$	−2.74742	−0.112350
$599$	−16.8697	−0.689279	−0.344640	−	0.938735i	$-0.611999\pi$
$599$	−16.8697	−0.689279	−0.344640	+	0.938735i	$0.611999\pi$
$600$	0	0
$601$	−15.3415	−0.625792	−0.312896	−	0.949787i	$-0.601299\pi$
$601$	−15.3415	−0.625792	−0.312896	+	0.949787i	$0.601299\pi$
$602$	5.36297	0.218578
$603$	32.8103	1.33614
$604$	−12.5795	−0.511851
$605$	0	0
$606$	8.20532	0.333318
$607$	−35.8353	−1.45451	−0.727254	−	0.686368i	$-0.759204\pi$
$607$	−35.8353	−1.45451	−0.727254	+	0.686368i	$0.759204\pi$
$608$	7.11383	0.288504
$609$	−11.7440	−0.475890
$610$	0	0
$611$	8.58451	0.347292
$612$	−4.78801	−0.193544
$613$	−19.6673	−0.794355	−0.397177	−	0.917742i	$-0.630010\pi$
$613$	−19.6673	−0.794355	−0.397177	+	0.917742i	$0.630010\pi$
$614$	9.60761	0.387732
$615$	0	0
$616$	−4.00000	−0.161165
$617$	41.4588	1.66907	0.834533	−	0.550958i	$-0.185737\pi$
$617$	41.4588	1.66907	0.834533	+	0.550958i	$0.185737\pi$
$618$	17.9931	0.723790
$619$	10.8793	0.437276	0.218638	−	0.975806i	$-0.429839\pi$
$619$	10.8793	0.437276	0.218638	+	0.975806i	$0.429839\pi$
$620$	0	0
$621$	−12.3587	−0.495937
$622$	0.905275	0.0362982
$623$	6.91033	0.276857
$624$	6.11727	0.244887
$625$	0	0
$626$	−6.76203	−0.270265
$627$	−7.43965	−0.297111
$628$	−10.7429	−0.428689
$629$	3.09472	0.123395
$630$	0	0
$631$	−31.4396	−1.25159	−0.625796	−	0.779987i	$-0.715226\pi$
$631$	−31.4396	−1.25159	−0.625796	+	0.779987i	$0.715226\pi$
$632$	−2.38101	−0.0947117
$633$	−22.7182	−0.902968
$634$	7.32238	0.290809
$635$	0	0
$636$	−45.1070	−1.78861
$637$	−1.00000	−0.0396214
$638$	5.52770	0.218844
$639$	2.45082	0.0969529
$640$	0	0
$641$	3.04221	0.120160	0.0600799	−	0.998194i	$-0.480864\pi$
$641$	3.04221	0.120160	0.0600799	+	0.998194i	$0.480864\pi$
$642$	5.88273	0.232173
$643$	−8.02922	−0.316641	−0.158321	−	0.987388i	$-0.550608\pi$
$643$	−8.02922	−0.316641	−0.158321	+	0.987388i	$0.550608\pi$
$644$	10.3810	0.409069
$645$	0	0
$646$	0.905275	0.0356176
$647$	−7.07324	−0.278078	−0.139039	−	0.990287i	$-0.544401\pi$
$647$	−7.07324	−0.278078	−0.139039	+	0.990287i	$0.544401\pi$
$648$	−19.4526	−0.764172
$649$	27.3776	1.07466
$650$	0	0
$651$	15.8061	0.619488
$652$	11.3484	0.444436
$653$	15.7586	0.616681	0.308341	−	0.951276i	$-0.400226\pi$
$653$	15.7586	0.616681	0.308341	+	0.951276i	$0.400226\pi$
$654$	8.38789	0.327992
$655$	0	0
$656$	17.6742	0.690061
$657$	−15.7344	−0.613859
$658$	4.04059	0.157518
$659$	−12.2181	−0.475950	−0.237975	−	0.971271i	$-0.576484\pi$
$659$	−12.2181	−0.475950	−0.237975	+	0.971271i	$0.576484\pi$
$660$	0	0
$661$	3.73443	0.145253	0.0726263	−	0.997359i	$-0.476862\pi$
$661$	3.73443	0.145253	0.0726263	+	0.997359i	$0.476862\pi$
$662$	−14.8501	−0.577165
$663$	2.94137	0.114233
$664$	29.0777	1.12844
$665$	0	0
$666$	−2.29296	−0.0888506
$667$	−30.4786	−1.18014
$668$	−29.4948	−1.14119
$669$	22.8095	0.881866
$670$	0	0
$671$	4.49828	0.173654
$672$	10.8793	0.419678
$673$	5.65775	0.218090	0.109045	−	0.994037i	$-0.465221\pi$
$673$	5.65775	0.218090	0.109045	+	0.994037i	$0.465221\pi$
$674$	−3.96629	−0.152776
$675$	0	0
$676$	−1.77846	−0.0684022
$677$	−9.39906	−0.361235	−0.180618	−	0.983553i	$-0.557810\pi$
$677$	−9.39906	−0.361235	−0.180618	+	0.983553i	$0.557810\pi$
$678$	10.4723	0.402186
$679$	3.47068	0.133193
$680$	0	0
$681$	−12.1104	−0.464071
$682$	−7.43965	−0.284879
$683$	20.7328	0.793319	0.396660	−	0.917966i	$-0.370169\pi$
$683$	20.7328	0.793319	0.396660	+	0.917966i	$0.370169\pi$
$684$	5.38445	0.205880
$685$	0	0
$686$	−0.470683	−0.0179708
$687$	−7.47250	−0.285094
$688$	−30.9897	−1.18147
$689$	11.2767	0.429610
$690$	0	0
$691$	16.0862	0.611949	0.305975	−	0.952040i	$-0.401018\pi$
$691$	16.0862	0.611949	0.305975	+	0.952040i	$0.401018\pi$
$692$	−41.4396	−1.57530
$693$	−4.63016	−0.175885
$694$	9.10695	0.345695
$695$	0	0
$696$	−20.8862	−0.791688
$697$	8.49828	0.321895
$698$	−12.8286	−0.485570
$699$	30.8578	1.16715
$700$	0	0
$701$	−6.98013	−0.263636	−0.131818	−	0.991274i	$-0.542081\pi$
$701$	−6.98013	−0.263636	−0.131818	+	0.991274i	$0.542081\pi$
$702$	0.996562	0.0376128
$703$	−3.48024	−0.131260
$704$	7.11383	0.268112
$705$	0	0
$706$	−12.0844	−0.454803
$707$	7.75086	0.291501
$708$	−48.6898	−1.82988
$709$	8.39239	0.315183	0.157591	−	0.987504i	$-0.449627\pi$
$709$	8.39239	0.315183	0.157591	+	0.987504i	$0.449627\pi$
$710$	0	0
$711$	−2.75612	−0.103362
$712$	12.2897	0.460577
$713$	41.0207	1.53624
$714$	1.38445	0.0518118
$715$	0	0
$716$	−37.4227	−1.39855
$717$	7.87586	0.294129
$718$	11.0225	0.411358
$719$	−5.16129	−0.192484	−0.0962418	−	0.995358i	$-0.530682\pi$
$719$	−5.16129	−0.192484	−0.0962418	+	0.995358i	$0.530682\pi$
$720$	0	0
$721$	16.9966	0.632985
$722$	7.92494	0.294936
$723$	−3.57152	−0.132826
$724$	−29.7846	−1.10693
$725$	0	0
$726$	−6.28973	−0.233434
$727$	40.4362	1.49970	0.749848	−	0.661610i	$-0.230127\pi$
$727$	40.4362	1.49970	0.749848	+	0.661610i	$0.230127\pi$
$728$	−1.77846	−0.0659140
$729$	−8.23109	−0.304855
$730$	0	0
$731$	−14.9008	−0.551125
$732$	−8.00000	−0.295689
$733$	39.1311	1.44534	0.722670	−	0.691193i	$-0.242915\pi$
$733$	39.1311	1.44534	0.722670	+	0.691193i	$0.242915\pi$
$734$	−6.91215	−0.255132
$735$	0	0
$736$	28.2345	1.04074
$737$	−35.8466	−1.32043
$738$	−6.29660	−0.231781
$739$	−7.13531	−0.262477	−0.131238	−	0.991351i	$-0.541895\pi$
$739$	−7.13531	−0.262477	−0.131238	+	0.991351i	$0.541895\pi$
$740$	0	0
$741$	−3.30777	−0.121514
$742$	5.30777	0.194855
$743$	−13.8827	−0.509308	−0.254654	−	0.967032i	$-0.581962\pi$
$743$	−13.8827	−0.509308	−0.254654	+	0.967032i	$0.581962\pi$
$744$	28.1104	1.03058
$745$	0	0
$746$	−11.1398	−0.407857
$747$	33.6586	1.23150
$748$	5.23109	0.191268
$749$	5.55691	0.203045
$750$	0	0
$751$	−37.3251	−1.36201	−0.681005	−	0.732278i	$-0.738457\pi$
$751$	−37.3251	−1.36201	−0.681005	+	0.732278i	$0.738457\pi$
$752$	−23.3484	−0.851427
$753$	11.0810	0.403813
$754$	2.45769	0.0895039
$755$	0	0
$756$	−3.76547	−0.136949
$757$	7.10428	0.258209	0.129105	−	0.991631i	$-0.458790\pi$
$757$	7.10428	0.258209	0.129105	+	0.991631i	$0.458790\pi$
$758$	15.4104	0.559732
$759$	−29.5277	−1.07179
$760$	0	0
$761$	−25.9621	−0.941125	−0.470562	−	0.882367i	$-0.655949\pi$
$761$	−25.9621	−0.941125	−0.470562	+	0.882367i	$0.655949\pi$
$762$	−0.872420	−0.0316044
$763$	7.92332	0.286843
$764$	13.2311	0.478684
$765$	0	0
$766$	−10.6448	−0.384611
$767$	12.1725	0.439522
$768$	−2.41023	−0.0869717
$769$	−21.4638	−0.774005	−0.387002	−	0.922079i	$-0.626489\pi$
$769$	−21.4638	−0.774005	−0.387002	+	0.922079i	$0.626489\pi$
$770$	0	0
$771$	−18.0260	−0.649190
$772$	−2.67074	−0.0961221
$773$	40.4914	1.45637	0.728187	−	0.685378i	$-0.240363\pi$
$773$	40.4914	1.45637	0.728187	+	0.685378i	$0.240363\pi$
$774$	11.0404	0.396838
$775$	0	0
$776$	6.17246	0.221578
$777$	−5.32238	−0.190939
$778$	−17.9087	−0.642058
$779$	−9.55691	−0.342412
$780$	0	0
$781$	−2.67762	−0.0958127
$782$	3.59301	0.128486
$783$	11.0554	0.395088
$784$	2.71982	0.0971366
$785$	0	0
$786$	11.2380	0.400845
$787$	5.02072	0.178969	0.0894847	−	0.995988i	$-0.471478\pi$
$787$	5.02072	0.178969	0.0894847	+	0.995988i	$0.471478\pi$
$788$	42.6707	1.52008
$789$	3.60438	0.128319
$790$	0	0
$791$	9.89229	0.351729
$792$	−8.23453	−0.292601
$793$	2.00000	0.0710221
$794$	−5.50945	−0.195523
$795$	0	0
$796$	3.58289	0.126992
$797$	−19.7002	−0.697815	−0.348908	−	0.937157i	$-0.613447\pi$
$797$	−19.7002	−0.697815	−0.348908	+	0.937157i	$0.613447\pi$
$798$	−1.55691	−0.0551142
$799$	−11.2266	−0.397169
$800$	0	0
$801$	14.2258	0.502645
$802$	−1.67418	−0.0591174
$803$	17.1905	0.606640
$804$	63.7517	2.24835
$805$	0	0
$806$	−3.30777	−0.116511
$807$	26.5863	0.935883
$808$	13.7846	0.484940
$809$	−2.57678	−0.0905947	−0.0452974	−	0.998974i	$-0.514424\pi$
$809$	−2.57678	−0.0905947	−0.0452974	+	0.998974i	$0.514424\pi$
$810$	0	0
$811$	41.2311	1.44782	0.723910	−	0.689895i	$-0.242343\pi$
$811$	41.2311	1.44782	0.723910	+	0.689895i	$0.242343\pi$
$812$	−9.28629	−0.325885
$813$	−49.2173	−1.72613
$814$	2.50516	0.0878057
$815$	0	0
$816$	−8.00000	−0.280056
$817$	16.7570	0.586252
$818$	−2.47680	−0.0865992
$819$	−2.05863	−0.0719345
$820$	0	0
$821$	5.26719	0.183826	0.0919130	−	0.995767i	$-0.470702\pi$
$821$	5.26719	0.183826	0.0919130	+	0.995767i	$0.470702\pi$
$822$	−12.0292	−0.419567
$823$	−36.0191	−1.25555	−0.627774	−	0.778396i	$-0.716034\pi$
$823$	−36.0191	−1.25555	−0.627774	+	0.778396i	$0.716034\pi$
$824$	30.2277	1.05303
$825$	0	0
$826$	5.72938	0.199350
$827$	16.3157	0.567353	0.283676	−	0.958920i	$-0.408446\pi$
$827$	16.3157	0.567353	0.283676	+	0.958920i	$0.408446\pi$
$828$	21.3707	0.742683
$829$	−30.3956	−1.05568	−0.527842	−	0.849343i	$-0.676999\pi$
$829$	−30.3956	−1.05568	−0.527842	+	0.849343i	$0.676999\pi$
$830$	0	0
$831$	−22.9820	−0.797235
$832$	3.16291	0.109654
$833$	1.30777	0.0453117
$834$	−14.7397	−0.510394
$835$	0	0
$836$	−5.88273	−0.203459
$837$	−14.8793	−0.514304
$838$	−12.2536	−0.423295
$839$	29.8398	1.03018	0.515092	−	0.857135i	$-0.327758\pi$
$839$	29.8398	1.03018	0.515092	+	0.857135i	$0.327758\pi$
$840$	0	0
$841$	−1.73549	−0.0598445
$842$	−10.4691	−0.360788
$843$	−3.46886	−0.119474
$844$	−17.9639	−0.618343
$845$	0	0
$846$	8.31809	0.285982
$847$	−5.94137	−0.204148
$848$	−30.6707	−1.05324
$849$	35.6413	1.22321
$850$	0	0
$851$	−13.8129	−0.473501
$852$	4.76203	0.163144
$853$	0.203497	0.00696761	0.00348380	−	0.999994i	$-0.498891\pi$
$853$	0.203497	0.00696761	0.00348380	+	0.999994i	$0.498891\pi$
$854$	0.941367	0.0322129
$855$	0	0
$856$	9.88273	0.337785
$857$	12.6155	0.430939	0.215469	−	0.976511i	$-0.430872\pi$
$857$	12.6155	0.430939	0.215469	+	0.976511i	$0.430872\pi$
$858$	2.38101	0.0812865
$859$	−27.9671	−0.954227	−0.477113	−	0.878842i	$-0.658317\pi$
$859$	−27.9671	−0.954227	−0.477113	+	0.878842i	$0.658317\pi$
$860$	0	0
$861$	−14.6155	−0.498097
$862$	−13.0258	−0.443660
$863$	2.76891	0.0942546	0.0471273	−	0.998889i	$-0.484993\pi$
$863$	2.76891	0.0942546	0.0471273	+	0.998889i	$0.484993\pi$
$864$	−10.2414	−0.348420
$865$	0	0
$866$	−6.01910	−0.204537
$867$	34.3887	1.16790
$868$	12.4983	0.424219
$869$	3.01117	0.102147
$870$	0	0
$871$	−15.9379	−0.540036
$872$	14.0913	0.477191
$873$	7.14486	0.241817
$874$	−4.04059	−0.136675
$875$	0	0
$876$	−30.5726	−1.03295
$877$	−6.71133	−0.226626	−0.113313	−	0.993559i	$-0.536146\pi$
$877$	−6.71133	−0.226626	−0.113313	+	0.993559i	$0.536146\pi$
$878$	8.55024	0.288557
$879$	24.9268	0.840759
$880$	0	0
$881$	−18.3741	−0.619040	−0.309520	−	0.950893i	$-0.600168\pi$
$881$	−18.3741	−0.619040	−0.309520	+	0.950893i	$0.600168\pi$
$882$	−0.968964	−0.0326267
$883$	−9.93105	−0.334207	−0.167103	−	0.985939i	$-0.553441\pi$
$883$	−9.93105	−0.334207	−0.167103	+	0.985939i	$0.553441\pi$
$884$	2.32582	0.0782258
$885$	0	0
$886$	0.0506994	0.00170328
$887$	51.3776	1.72509	0.862545	−	0.505980i	$-0.168869\pi$
$887$	51.3776	1.72509	0.862545	+	0.505980i	$0.168869\pi$
$888$	−9.46563	−0.317646
$889$	−0.824101	−0.0276394
$890$	0	0
$891$	24.6009	0.824162
$892$	18.0361	0.603893
$893$	12.6251	0.422483
$894$	9.85352	0.329551
$895$	0	0
$896$	11.1629	0.372927
$897$	−13.1284	−0.438346
$898$	10.4362	0.348261
$899$	−36.6949	−1.22384
$900$	0	0
$901$	−14.7474	−0.491308
$902$	6.87930	0.229055
$903$	25.6267	0.852804
$904$	17.5930	0.585135
$905$	0	0
$906$	7.48797	0.248771
$907$	−4.34225	−0.144182	−0.0720910	−	0.997398i	$-0.522967\pi$
$907$	−4.34225	−0.144182	−0.0720910	+	0.997398i	$0.522967\pi$
$908$	−9.57602	−0.317791
$909$	15.9562	0.529233
$910$	0	0
$911$	31.4853	1.04315	0.521577	−	0.853204i	$-0.325344\pi$
$911$	31.4853	1.04315	0.521577	+	0.853204i	$0.325344\pi$
$912$	8.99656	0.297906
$913$	−36.7734	−1.21702
$914$	2.05196	0.0678728
$915$	0	0
$916$	−5.90871	−0.195229
$917$	10.6155	0.350556
$918$	−1.30328	−0.0430146
$919$	43.4588	1.43357	0.716786	−	0.697293i	$-0.245612\pi$
$919$	43.4588	1.43357	0.716786	+	0.697293i	$0.245612\pi$
$920$	0	0
$921$	45.9096	1.51277
$922$	−15.2412	−0.501942
$923$	−1.19051	−0.0391860
$924$	−8.99656	−0.295965
$925$	0	0
$926$	3.93793	0.129408
$927$	34.9897	1.14921
$928$	−25.2571	−0.829104
$929$	4.40517	0.144529	0.0722645	−	0.997385i	$-0.476977\pi$
$929$	4.40517	0.144529	0.0722645	+	0.997385i	$0.476977\pi$
$930$	0	0
$931$	−1.47068	−0.0481997
$932$	24.4001	0.799252
$933$	4.32582	0.141621
$934$	−9.19824	−0.300976
$935$	0	0
$936$	−3.66119	−0.119670
$937$	34.0990	1.11397	0.556983	−	0.830524i	$-0.311959\pi$
$937$	34.0990	1.11397	0.556983	+	0.830524i	$0.311959\pi$
$938$	−7.50172	−0.244940
$939$	−32.3121	−1.05446
$940$	0	0
$941$	44.4672	1.44959	0.724795	−	0.688964i	$-0.241934\pi$
$941$	44.4672	1.44959	0.724795	+	0.688964i	$0.241934\pi$
$942$	6.39477	0.208353
$943$	−37.9311	−1.23521
$944$	−33.1070	−1.07754
$945$	0	0
$946$	−12.0621	−0.392172
$947$	−57.9311	−1.88251	−0.941253	−	0.337702i	$-0.890350\pi$
$947$	−57.9311	−1.88251	−0.941253	+	0.337702i	$0.890350\pi$
$948$	−5.35524	−0.173930
$949$	7.64315	0.248107
$950$	0	0
$951$	34.9897	1.13462
$952$	2.32582	0.0753802
$953$	−35.3060	−1.14367	−0.571836	−	0.820368i	$-0.693769\pi$
$953$	−35.3060	−1.14367	−0.571836	+	0.820368i	$0.693769\pi$
$954$	10.9268	0.353767
$955$	0	0
$956$	6.22766	0.201417
$957$	26.4139	0.853839
$958$	13.4250	0.433743
$959$	−11.3630	−0.366929
$960$	0	0
$961$	18.3871	0.593133
$962$	1.11383	0.0359113
$963$	11.4396	0.368638
$964$	−2.82410	−0.0909582
$965$	0	0
$966$	−6.17934	−0.198817
$967$	−23.7148	−0.762616	−0.381308	−	0.924448i	$-0.624526\pi$
$967$	−23.7148	−0.762616	−0.381308	+	0.924448i	$0.624526\pi$
$968$	−10.5665	−0.339619
$969$	4.32582	0.138965
$970$	0	0
$971$	−23.9379	−0.768205	−0.384102	−	0.923291i	$-0.625489\pi$
$971$	−23.9379	−0.768205	−0.384102	+	0.923291i	$0.625489\pi$
$972$	−32.4553	−1.04100
$973$	−13.9233	−0.446361
$974$	11.6843	0.374389
$975$	0	0
$976$	−5.43965	−0.174119
$977$	16.1871	0.517870	0.258935	−	0.965895i	$-0.416628\pi$
$977$	16.1871	0.517870	0.258935	+	0.965895i	$0.416628\pi$
$978$	−6.75515	−0.216006
$979$	−15.5423	−0.496734
$980$	0	0
$981$	16.3112	0.520777
$982$	−13.7294	−0.438122
$983$	−45.7243	−1.45838	−0.729190	−	0.684312i	$-0.760103\pi$
$983$	−45.7243	−1.45838	−0.729190	+	0.684312i	$0.760103\pi$
$984$	−25.9931	−0.828631
$985$	0	0
$986$	−3.21411	−0.102358
$987$	19.3078	0.614573
$988$	−2.61555	−0.0832116
$989$	66.5078	2.11483
$990$	0	0
$991$	−5.40356	−0.171650	−0.0858248	−	0.996310i	$-0.527353\pi$
$991$	−5.40356	−0.171650	−0.0858248	+	0.996310i	$0.527353\pi$
$992$	33.9931	1.07928
$993$	−70.9605	−2.25186
$994$	−0.560352	−0.0177733
$995$	0	0
$996$	65.3999	2.07228
$997$	−6.04832	−0.191552	−0.0957761	−	0.995403i	$-0.530533\pi$
$997$	−6.04832	−0.191552	−0.0957761	+	0.995403i	$0.530533\pi$
$998$	15.6742	0.496158
$999$	5.01031	0.158519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.m.1.2		3
5.4	even	2		91.2.a.d.1.2	✓	3
15.14	odd	2		819.2.a.i.1.2		3
20.19	odd	2		1456.2.a.t.1.1		3
35.4	even	6		637.2.e.j.79.2		6
35.9	even	6		637.2.e.j.508.2		6
35.19	odd	6		637.2.e.i.508.2		6
35.24	odd	6		637.2.e.i.79.2		6
35.34	odd	2		637.2.a.j.1.2		3
40.19	odd	2		5824.2.a.bs.1.3		3
40.29	even	2		5824.2.a.by.1.1		3
65.34	odd	4		1183.2.c.f.337.4		6
65.44	odd	4		1183.2.c.f.337.3		6
65.64	even	2		1183.2.a.i.1.2		3
105.104	even	2		5733.2.a.x.1.2		3
455.454	odd	2		8281.2.a.bg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.2	✓	3	5.4	even	2
637.2.a.j.1.2		3	35.34	odd	2
637.2.e.i.79.2		6	35.24	odd	6
637.2.e.i.508.2		6	35.19	odd	6
637.2.e.j.79.2		6	35.4	even	6
637.2.e.j.508.2		6	35.9	even	6
819.2.a.i.1.2		3	15.14	odd	2
1183.2.a.i.1.2		3	65.64	even	2
1183.2.c.f.337.3		6	65.44	odd	4
1183.2.c.f.337.4		6	65.34	odd	4
1456.2.a.t.1.1		3	20.19	odd	2
2275.2.a.m.1.2		3	1.1	even	1	trivial
5733.2.a.x.1.2		3	105.104	even	2
5824.2.a.bs.1.3		3	40.19	odd	2
5824.2.a.by.1.1		3	40.29	even	2
8281.2.a.bg.1.2		3	455.454	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

\(f(q)\)	\(=\)	\(q-0.470683 q^{2} -2.24914 q^{3} -1.77846 q^{4} +1.05863 q^{6} +1.00000 q^{7} +1.77846 q^{8} +2.05863 q^{9} -2.24914 q^{11} +4.00000 q^{12} -1.00000 q^{13} -0.470683 q^{14} +2.71982 q^{16} +1.30777 q^{17} -0.968964 q^{18} -1.47068 q^{19} -2.24914 q^{21} +1.05863 q^{22} -5.83709 q^{23} -4.00000 q^{24} +0.470683 q^{26} +2.11727 q^{27} -1.77846 q^{28} +5.22154 q^{29} -7.02760 q^{31} -4.83709 q^{32} +5.05863 q^{33} -0.615547 q^{34} -3.66119 q^{36} +2.36641 q^{37} +0.692226 q^{38} +2.24914 q^{39} +6.49828 q^{41} +1.05863 q^{42} -11.3940 q^{43} +4.00000 q^{44} +2.74742 q^{46} -8.58451 q^{47} -6.11727 q^{48} +1.00000 q^{49} -2.94137 q^{51} +1.77846 q^{52} -11.2767 q^{53} -0.996562 q^{54} +1.77846 q^{56} +3.30777 q^{57} -2.45769 q^{58} -12.1725 q^{59} -2.00000 q^{61} +3.30777 q^{62} +2.05863 q^{63} -3.16291 q^{64} -2.38101 q^{66} +15.9379 q^{67} -2.32582 q^{68} +13.1284 q^{69} +1.19051 q^{71} +3.66119 q^{72} -7.64315 q^{73} -1.11383 q^{74} +2.61555 q^{76} -2.24914 q^{77} -1.05863 q^{78} -1.33881 q^{79} -10.9379 q^{81} -3.05863 q^{82} +16.3500 q^{83} +4.00000 q^{84} +5.36297 q^{86} -11.7440 q^{87} -4.00000 q^{88} +6.91033 q^{89} -1.00000 q^{91} +10.3810 q^{92} +15.8061 q^{93} +4.04059 q^{94} +10.8793 q^{96} +3.47068 q^{97} -0.470683 q^{98} -4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 2 q^{3} + 3 q^{4} + 4 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + 2 q^{11} + 12 q^{12} - 3 q^{13} - q^{14} - q^{16} - 4 q^{17} + 15 q^{18} - 4 q^{19} + 2 q^{21} + 4 q^{22} - 10 q^{23} - 12 q^{24}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q - q^2 + 2 * q^3 + 3 * q^4 + 4 * q^6 + 3 * q^7 - 3 * q^8 + 7 * q^9 + 2 * q^11 + 12 * q^12 - 3 * q^13 - q^14 - q^16 - 4 * q^17 + 15 * q^18 - 4 * q^19 + 2 * q^21 + 4 * q^22 - 10 * q^23 - 12 * q^24 + q^26 + 8 * q^27 + 3 * q^28 + 24 * q^29 - 4 * q^31 - 7 * q^32 + 16 * q^33 + 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^41 + 4 * q^42 - 10 * q^43 + 12 * q^44 - 18 * q^46 + 8 * q^47 - 20 * q^48 + 3 * q^49 - 8 * q^51 - 3 * q^52 - 8 * q^53 + 32 * q^54 - 3 * q^56 + 2 * q^57 - 12 * q^58 - 4 * q^59 - 6 * q^61 + 2 * q^62 + 7 * q^63 - 17 * q^64 + 12 * q^66 + 12 * q^67 - 22 * q^68 - 6 * q^69 - 6 * q^71 + q^72 + 10 * q^73 + 30 * q^74 - 8 * q^76 + 2 * q^77 - 4 * q^78 - 14 * q^79 + 3 * q^81 - 10 * q^82 + 12 * q^83 + 12 * q^84 - 26 * q^86 + 26 * q^87 - 12 * q^88 + 2 * q^89 - 3 * q^91 + 12 * q^92 + 22 * q^93 - 10 * q^94 - 4 * q^96 + 10 * q^97 - q^98 + 14 * q^99

Introduction

L-functions

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