LMFDB - Embedded newform 3675.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} -2.82843 q^{11} -1.82843 q^{12} +0.828427 q^{13} +3.00000 q^{16} +3.65685 q^{17} +0.414214 q^{18} -4.82843 q^{19} -1.17157 q^{22} -3.65685 q^{23} -1.58579 q^{24} +0.343146 q^{26} +1.00000 q^{27} +6.00000 q^{29} +10.4853 q^{31} +4.41421 q^{32} -2.82843 q^{33} +1.51472 q^{34} -1.82843 q^{36} -7.65685 q^{37} -2.00000 q^{38} +0.828427 q^{39} -0.343146 q^{41} -8.00000 q^{43} +5.17157 q^{44} -1.51472 q^{46} -5.65685 q^{47} +3.00000 q^{48} +3.65685 q^{51} -1.51472 q^{52} -8.48528 q^{53} +0.414214 q^{54} -4.82843 q^{57} +2.48528 q^{58} -13.6569 q^{59} +4.34315 q^{62} -4.17157 q^{64} -1.17157 q^{66} -6.68629 q^{68} -3.65685 q^{69} +2.82843 q^{71} -1.58579 q^{72} -16.8284 q^{73} -3.17157 q^{74} +8.82843 q^{76} +0.343146 q^{78} -8.00000 q^{79} +1.00000 q^{81} -0.142136 q^{82} +9.65685 q^{83} -3.31371 q^{86} +6.00000 q^{87} +4.48528 q^{88} -17.3137 q^{89} +6.68629 q^{92} +10.4853 q^{93} -2.34315 q^{94} +4.41421 q^{96} -10.4853 q^{97} -2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 8 q^{22} + 4 q^{23} - 6 q^{24} + 12 q^{26} + 2 q^{27} + 12 q^{29} + 4 q^{31} + 6 q^{32} + 20 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} - 12 q^{41} - 16 q^{43} + 16 q^{44} - 20 q^{46} + 6 q^{48} - 4 q^{51} - 20 q^{52} - 2 q^{54} - 4 q^{57} - 12 q^{58} - 16 q^{59} + 20 q^{62} - 14 q^{64} - 8 q^{66} - 36 q^{68} + 4 q^{69} - 6 q^{72} - 28 q^{73} - 12 q^{74} + 12 q^{76} + 12 q^{78} - 16 q^{79} + 2 q^{81} + 28 q^{82} + 8 q^{83} + 16 q^{86} + 12 q^{87} - 8 q^{88} - 12 q^{89} + 36 q^{92} + 4 q^{93} - 16 q^{94} + 6 q^{96} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 6 * q^16 - 4 * q^17 - 2 * q^18 - 4 * q^19 - 8 * q^22 + 4 * q^23 - 6 * q^24 + 12 * q^26 + 2 * q^27 + 12 * q^29 + 4 * q^31 + 6 * q^32 + 20 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 - 12 * q^41 - 16 * q^43 + 16 * q^44 - 20 * q^46 + 6 * q^48 - 4 * q^51 - 20 * q^52 - 2 * q^54 - 4 * q^57 - 12 * q^58 - 16 * q^59 + 20 * q^62 - 14 * q^64 - 8 * q^66 - 36 * q^68 + 4 * q^69 - 6 * q^72 - 28 * q^73 - 12 * q^74 + 12 * q^76 + 12 * q^78 - 16 * q^79 + 2 * q^81 + 28 * q^82 + 8 * q^83 + 16 * q^86 + 12 * q^87 - 8 * q^88 - 12 * q^89 + 36 * q^92 + 4 * q^93 - 16 * q^94 + 6 * q^96 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	0.414214	0.169102
$7$	0	0
$7$	0	0
$8$	−1.58579	−0.560660
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	−1.82843	−0.527821
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0.414214	0.0976311
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	0	0
$22$	−1.17157	−0.249780
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	−1.58579	−0.323697
$25$	0	0
$26$	0.343146	0.0672964
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	10.4853	1.88321	0.941606	−	0.336717i	$-0.109316\pi$
$31$	10.4853	1.88321	0.941606	+	0.336717i	$0.109316\pi$
$32$	4.41421	0.780330
$33$	−2.82843	−0.492366
$34$	1.51472	0.259772
$35$	0	0
$36$	−1.82843	−0.304738
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	−2.00000	−0.324443
$39$	0.828427	0.132655
$40$	0	0
$41$	−0.343146	−0.0535904	−0.0267952	−	0.999641i	$-0.508530\pi$
$41$	−0.343146	−0.0535904	−0.0267952	+	0.999641i	$0.508530\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	5.17157	0.779644
$45$	0	0
$46$	−1.51472	−0.223333
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	3.00000	0.433013
$49$	0	0
$50$	0	0
$51$	3.65685	0.512062
$52$	−1.51472	−0.210054
$53$	−8.48528	−1.16554	−0.582772	−	0.812636i	$-0.698032\pi$
$53$	−8.48528	−1.16554	−0.582772	+	0.812636i	$0.698032\pi$
$54$	0.414214	0.0563673
$55$	0	0
$56$	0	0
$57$	−4.82843	−0.639541
$58$	2.48528	0.326333
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	4.34315	0.551580
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	−1.17157	−0.144211
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−6.68629	−0.810832
$69$	−3.65685	−0.440234
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	−1.58579	−0.186887
$73$	−16.8284	−1.96962	−0.984809	−	0.173640i	$-0.944447\pi$
$73$	−16.8284	−1.96962	−0.984809	+	0.173640i	$0.944447\pi$
$74$	−3.17157	−0.368688
$75$	0	0
$76$	8.82843	1.01269
$77$	0	0
$78$	0.343146	0.0388536
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.142136	−0.0156963
$83$	9.65685	1.05998	0.529989	−	0.848005i	$-0.322196\pi$
$83$	9.65685	1.05998	0.529989	+	0.848005i	$0.322196\pi$
$84$	0	0
$85$	0	0
$86$	−3.31371	−0.357326
$87$	6.00000	0.643268
$88$	4.48528	0.478133
$89$	−17.3137	−1.83525	−0.917625	−	0.397448i	$-0.869896\pi$
$89$	−17.3137	−1.83525	−0.917625	+	0.397448i	$0.869896\pi$
$90$	0	0
$91$	0	0
$92$	6.68629	0.697094
$93$	10.4853	1.08727
$94$	−2.34315	−0.241677
$95$	0	0
$96$	4.41421	0.450524
$97$	−10.4853	−1.06462	−0.532310	−	0.846550i	$-0.678676\pi$
$97$	−10.4853	−1.06462	−0.532310	+	0.846550i	$0.678676\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	1.51472	0.149979
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−1.31371	−0.128820
$105$	0	0
$106$	−3.51472	−0.341380
$107$	17.3137	1.67378	0.836890	−	0.547372i	$-0.184372\pi$
$107$	17.3137	1.67378	0.836890	+	0.547372i	$0.184372\pi$
$108$	−1.82843	−0.175940
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−7.65685	−0.726756
$112$	0	0
$113$	2.82843	0.266076	0.133038	−	0.991111i	$-0.457527\pi$
$113$	2.82843	0.266076	0.133038	+	0.991111i	$0.457527\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−10.9706	−1.01859
$117$	0.828427	0.0765881
$118$	−5.65685	−0.520756
$119$	0	0
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−0.343146	−0.0309404
$124$	−19.1716	−1.72166
$125$	0	0
$126$	0	0
$127$	3.31371	0.294044	0.147022	−	0.989133i	$-0.453031\pi$
$127$	3.31371	0.294044	0.147022	+	0.989133i	$0.453031\pi$
$128$	−10.5563	−0.933058
$129$	−8.00000	−0.704361
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	5.17157	0.450128
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−5.79899	−0.497259
$137$	−18.8284	−1.60862	−0.804311	−	0.594209i	$-0.797465\pi$
$137$	−18.8284	−1.60862	−0.804311	+	0.594209i	$0.797465\pi$
$138$	−1.51472	−0.128941
$139$	18.4853	1.56790	0.783951	−	0.620823i	$-0.213202\pi$
$139$	18.4853	1.56790	0.783951	+	0.620823i	$0.213202\pi$
$140$	0	0
$141$	−5.65685	−0.476393
$142$	1.17157	0.0983162
$143$	−2.34315	−0.195944
$144$	3.00000	0.250000
$145$	0	0
$146$	−6.97056	−0.576888
$147$	0	0
$148$	14.0000	1.15079
$149$	3.65685	0.299581	0.149791	−	0.988718i	$-0.452140\pi$
$149$	3.65685	0.299581	0.149791	+	0.988718i	$0.452140\pi$
$150$	0	0
$151$	16.9706	1.38104	0.690522	−	0.723311i	$-0.257381\pi$
$151$	16.9706	1.38104	0.690522	+	0.723311i	$0.257381\pi$
$152$	7.65685	0.621053
$153$	3.65685	0.295639
$154$	0	0
$155$	0	0
$156$	−1.51472	−0.121275
$157$	−16.1421	−1.28828	−0.644141	−	0.764906i	$-0.722785\pi$
$157$	−16.1421	−1.28828	−0.644141	+	0.764906i	$0.722785\pi$
$158$	−3.31371	−0.263624
$159$	−8.48528	−0.672927
$160$	0	0
$161$	0	0
$162$	0.414214	0.0325437
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0.627417	0.0489930
$165$	0	0
$166$	4.00000	0.310460
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	−4.82843	−0.369239
$172$	14.6274	1.11533
$173$	−20.6274	−1.56827	−0.784137	−	0.620588i	$-0.786894\pi$
$173$	−20.6274	−1.56827	−0.784137	+	0.620588i	$0.786894\pi$
$174$	2.48528	0.188409
$175$	0	0
$176$	−8.48528	−0.639602
$177$	−13.6569	−1.02651
$178$	−7.17157	−0.537532
$179$	−6.82843	−0.510381	−0.255190	−	0.966891i	$-0.582138\pi$
$179$	−6.82843	−0.510381	−0.255190	+	0.966891i	$0.582138\pi$
$180$	0	0
$181$	−13.6569	−1.01511	−0.507553	−	0.861621i	$-0.669450\pi$
$181$	−13.6569	−1.01511	−0.507553	+	0.861621i	$0.669450\pi$
$182$	0	0
$183$	0	0
$184$	5.79899	0.427507
$185$	0	0
$186$	4.34315	0.318455
$187$	−10.3431	−0.756366
$188$	10.3431	0.754351
$189$	0	0
$190$	0	0
$191$	−14.8284	−1.07295	−0.536474	−	0.843917i	$-0.680244\pi$
$191$	−14.8284	−1.07295	−0.536474	+	0.843917i	$0.680244\pi$
$192$	−4.17157	−0.301057
$193$	−4.34315	−0.312626	−0.156313	−	0.987708i	$-0.549961\pi$
$193$	−4.34315	−0.312626	−0.156313	+	0.987708i	$0.549961\pi$
$194$	−4.34315	−0.311820
$195$	0	0
$196$	0	0
$197$	20.4853	1.45952	0.729758	−	0.683706i	$-0.239633\pi$
$197$	20.4853	1.45952	0.729758	+	0.683706i	$0.239633\pi$
$198$	−1.17157	−0.0832601
$199$	−4.82843	−0.342278	−0.171139	−	0.985247i	$-0.554745\pi$
$199$	−4.82843	−0.342278	−0.171139	+	0.985247i	$0.554745\pi$
$200$	0	0
$201$	0	0
$202$	1.51472	0.106575
$203$	0	0
$204$	−6.68629	−0.468134
$205$	0	0
$206$	−4.97056	−0.346316
$207$	−3.65685	−0.254169
$208$	2.48528	0.172323
$209$	13.6569	0.944664
$210$	0	0
$211$	1.65685	0.114063	0.0570313	−	0.998372i	$-0.481837\pi$
$211$	1.65685	0.114063	0.0570313	+	0.998372i	$0.481837\pi$
$212$	15.5147	1.06556
$213$	2.82843	0.193801
$214$	7.17157	0.490239
$215$	0	0
$216$	−1.58579	−0.107899
$217$	0	0
$218$	2.48528	0.168324
$219$	−16.8284	−1.13716
$220$	0	0
$221$	3.02944	0.203782
$222$	−3.17157	−0.212862
$223$	12.9706	0.868573	0.434287	−	0.900775i	$-0.357001\pi$
$223$	12.9706	0.868573	0.434287	+	0.900775i	$0.357001\pi$
$224$	0	0
$225$	0	0
$226$	1.17157	0.0779319
$227$	−4.97056	−0.329908	−0.164954	−	0.986301i	$-0.552748\pi$
$227$	−4.97056	−0.329908	−0.164954	+	0.986301i	$0.552748\pi$
$228$	8.82843	0.584677
$229$	−7.31371	−0.483303	−0.241652	−	0.970363i	$-0.577689\pi$
$229$	−7.31371	−0.483303	−0.241652	+	0.970363i	$0.577689\pi$
$230$	0	0
$231$	0	0
$232$	−9.51472	−0.624672
$233$	−9.17157	−0.600850	−0.300425	−	0.953805i	$-0.597128\pi$
$233$	−9.17157	−0.600850	−0.300425	+	0.953805i	$0.597128\pi$
$234$	0.343146	0.0224321
$235$	0	0
$236$	24.9706	1.62545
$237$	−8.00000	−0.519656
$238$	0	0
$239$	26.1421	1.69099	0.845497	−	0.533980i	$-0.179304\pi$
$239$	26.1421	1.69099	0.845497	+	0.533980i	$0.179304\pi$
$240$	0	0
$241$	10.3431	0.666261	0.333130	−	0.942881i	$-0.391895\pi$
$241$	10.3431	0.666261	0.333130	+	0.942881i	$0.391895\pi$
$242$	−1.24264	−0.0798800
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	−0.142136	−0.00906224
$247$	−4.00000	−0.254514
$248$	−16.6274	−1.05584
$249$	9.65685	0.611978
$250$	0	0
$251$	−18.3431	−1.15781	−0.578905	−	0.815395i	$-0.696520\pi$
$251$	−18.3431	−1.15781	−0.578905	+	0.815395i	$0.696520\pi$
$252$	0	0
$253$	10.3431	0.650268
$254$	1.37258	0.0861235
$255$	0	0
$256$	3.97056	0.248160
$257$	−11.6569	−0.727135	−0.363567	−	0.931568i	$-0.618441\pi$
$257$	−11.6569	−0.727135	−0.363567	+	0.931568i	$0.618441\pi$
$258$	−3.31371	−0.206302
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	2.34315	0.144760
$263$	25.3137	1.56091	0.780455	−	0.625212i	$-0.214987\pi$
$263$	25.3137	1.56091	0.780455	+	0.625212i	$0.214987\pi$
$264$	4.48528	0.276050
$265$	0	0
$266$	0	0
$267$	−17.3137	−1.05958
$268$	0	0
$269$	−11.6569	−0.710731	−0.355365	−	0.934727i	$-0.615644\pi$
$269$	−11.6569	−0.710731	−0.355365	+	0.934727i	$0.615644\pi$
$270$	0	0
$271$	5.51472	0.334995	0.167498	−	0.985872i	$-0.446431\pi$
$271$	5.51472	0.334995	0.167498	+	0.985872i	$0.446431\pi$
$272$	10.9706	0.665188
$273$	0	0
$274$	−7.79899	−0.471154
$275$	0	0
$276$	6.68629	0.402467
$277$	1.31371	0.0789331	0.0394665	−	0.999221i	$-0.487434\pi$
$277$	1.31371	0.0789331	0.0394665	+	0.999221i	$0.487434\pi$
$278$	7.65685	0.459228
$279$	10.4853	0.627737
$280$	0	0
$281$	23.6569	1.41125	0.705625	−	0.708586i	$-0.250666\pi$
$281$	23.6569	1.41125	0.705625	+	0.708586i	$0.250666\pi$
$282$	−2.34315	−0.139532
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−5.17157	−0.306876
$285$	0	0
$286$	−0.970563	−0.0573906
$287$	0	0
$288$	4.41421	0.260110
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−10.4853	−0.614658
$292$	30.7696	1.80065
$293$	−23.6569	−1.38205	−0.691024	−	0.722832i	$-0.742840\pi$
$293$	−23.6569	−1.38205	−0.691024	+	0.722832i	$0.742840\pi$
$294$	0	0
$295$	0	0
$296$	12.1421	0.705747
$297$	−2.82843	−0.164122
$298$	1.51472	0.0877453
$299$	−3.02944	−0.175197
$300$	0	0
$301$	0	0
$302$	7.02944	0.404499
$303$	3.65685	0.210081
$304$	−14.4853	−0.830788
$305$	0	0
$306$	1.51472	0.0865907
$307$	4.97056	0.283685	0.141843	−	0.989889i	$-0.454697\pi$
$307$	4.97056	0.283685	0.141843	+	0.989889i	$0.454697\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	1.65685	0.0939516	0.0469758	−	0.998896i	$-0.485042\pi$
$311$	1.65685	0.0939516	0.0469758	+	0.998896i	$0.485042\pi$
$312$	−1.31371	−0.0743741
$313$	−14.4853	−0.818757	−0.409378	−	0.912365i	$-0.634254\pi$
$313$	−14.4853	−0.818757	−0.409378	+	0.912365i	$0.634254\pi$
$314$	−6.68629	−0.377329
$315$	0	0
$316$	14.6274	0.822856
$317$	6.14214	0.344977	0.172488	−	0.985012i	$-0.444819\pi$
$317$	6.14214	0.344977	0.172488	+	0.985012i	$0.444819\pi$
$318$	−3.51472	−0.197096
$319$	−16.9706	−0.950169
$320$	0	0
$321$	17.3137	0.966357
$322$	0	0
$323$	−17.6569	−0.982454
$324$	−1.82843	−0.101579
$325$	0	0
$326$	4.97056	0.275294
$327$	6.00000	0.331801
$328$	0.544156	0.0300460
$329$	0	0
$330$	0	0
$331$	−9.65685	−0.530789	−0.265394	−	0.964140i	$-0.585502\pi$
$331$	−9.65685	−0.530789	−0.265394	+	0.964140i	$0.585502\pi$
$332$	−17.6569	−0.969046
$333$	−7.65685	−0.419593
$334$	−4.68629	−0.256422
$335$	0	0
$336$	0	0
$337$	−26.9706	−1.46918	−0.734590	−	0.678511i	$-0.762626\pi$
$337$	−26.9706	−1.46918	−0.734590	+	0.678511i	$0.762626\pi$
$338$	−5.10051	−0.277431
$339$	2.82843	0.153619
$340$	0	0
$341$	−29.6569	−1.60601
$342$	−2.00000	−0.108148
$343$	0	0
$344$	12.6863	0.683999
$345$	0	0
$346$	−8.54416	−0.459337
$347$	−10.9706	−0.588931	−0.294465	−	0.955662i	$-0.595142\pi$
$347$	−10.9706	−0.588931	−0.294465	+	0.955662i	$0.595142\pi$
$348$	−10.9706	−0.588084
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	0.828427	0.0442182
$352$	−12.4853	−0.665468
$353$	−2.68629	−0.142977	−0.0714884	−	0.997441i	$-0.522775\pi$
$353$	−2.68629	−0.142977	−0.0714884	+	0.997441i	$0.522775\pi$
$354$	−5.65685	−0.300658
$355$	0	0
$356$	31.6569	1.67781
$357$	0	0
$358$	−2.82843	−0.149487
$359$	23.7990	1.25606	0.628031	−	0.778188i	$-0.283861\pi$
$359$	23.7990	1.25606	0.628031	+	0.778188i	$0.283861\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	−5.65685	−0.297318
$363$	−3.00000	−0.157459
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.9706	1.09465	0.547327	−	0.836919i	$-0.315645\pi$
$367$	20.9706	1.09465	0.547327	+	0.836919i	$0.315645\pi$
$368$	−10.9706	−0.571880
$369$	−0.343146	−0.0178635
$370$	0	0
$371$	0	0
$372$	−19.1716	−0.994000
$373$	−20.6274	−1.06805	−0.534024	−	0.845470i	$-0.679321\pi$
$373$	−20.6274	−1.06805	−0.534024	+	0.845470i	$0.679321\pi$
$374$	−4.28427	−0.221534
$375$	0	0
$376$	8.97056	0.462621
$377$	4.97056	0.255997
$378$	0	0
$379$	12.9706	0.666253	0.333127	−	0.942882i	$-0.391896\pi$
$379$	12.9706	0.666253	0.333127	+	0.942882i	$0.391896\pi$
$380$	0	0
$381$	3.31371	0.169766
$382$	−6.14214	−0.314259
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−10.5563	−0.538701
$385$	0	0
$386$	−1.79899	−0.0915662
$387$	−8.00000	−0.406663
$388$	19.1716	0.973289
$389$	−0.343146	−0.0173982	−0.00869909	−	0.999962i	$-0.502769\pi$
$389$	−0.343146	−0.0173982	−0.00869909	+	0.999962i	$0.502769\pi$
$390$	0	0
$391$	−13.3726	−0.676281
$392$	0	0
$393$	5.65685	0.285351
$394$	8.48528	0.427482
$395$	0	0
$396$	5.17157	0.259881
$397$	−19.1716	−0.962194	−0.481097	−	0.876667i	$-0.659761\pi$
$397$	−19.1716	−0.962194	−0.481097	+	0.876667i	$0.659761\pi$
$398$	−2.00000	−0.100251
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	8.68629	0.432695
$404$	−6.68629	−0.332655
$405$	0	0
$406$	0	0
$407$	21.6569	1.07349
$408$	−5.79899	−0.287093
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	0	0
$411$	−18.8284	−0.928738
$412$	21.9411	1.08096
$413$	0	0
$414$	−1.51472	−0.0744444
$415$	0	0
$416$	3.65685	0.179292
$417$	18.4853	0.905228
$418$	5.65685	0.276686
$419$	23.3137	1.13895	0.569475	−	0.822009i	$-0.307147\pi$
$419$	23.3137	1.13895	0.569475	+	0.822009i	$0.307147\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0.686292	0.0334081
$423$	−5.65685	−0.275046
$424$	13.4558	0.653474
$425$	0	0
$426$	1.17157	0.0567629
$427$	0	0
$428$	−31.6569	−1.53019
$429$	−2.34315	−0.113128
$430$	0	0
$431$	−5.17157	−0.249106	−0.124553	−	0.992213i	$-0.539750\pi$
$431$	−5.17157	−0.249106	−0.124553	+	0.992213i	$0.539750\pi$
$432$	3.00000	0.144338
$433$	12.8284	0.616495	0.308247	−	0.951306i	$-0.400258\pi$
$433$	12.8284	0.616495	0.308247	+	0.951306i	$0.400258\pi$
$434$	0	0
$435$	0	0
$436$	−10.9706	−0.525395
$437$	17.6569	0.844642
$438$	−6.97056	−0.333066
$439$	30.4853	1.45498	0.727492	−	0.686117i	$-0.240686\pi$
$439$	30.4853	1.45498	0.727492	+	0.686117i	$0.240686\pi$
$440$	0	0
$441$	0	0
$442$	1.25483	0.0596864
$443$	−12.3431	−0.586441	−0.293220	−	0.956045i	$-0.594727\pi$
$443$	−12.3431	−0.586441	−0.293220	+	0.956045i	$0.594727\pi$
$444$	14.0000	0.664411
$445$	0	0
$446$	5.37258	0.254399
$447$	3.65685	0.172963
$448$	0	0
$449$	32.6274	1.53978	0.769892	−	0.638175i	$-0.220310\pi$
$449$	32.6274	1.53978	0.769892	+	0.638175i	$0.220310\pi$
$450$	0	0
$451$	0.970563	0.0457020
$452$	−5.17157	−0.243250
$453$	16.9706	0.797347
$454$	−2.05887	−0.0966278
$455$	0	0
$456$	7.65685	0.358565
$457$	−27.6569	−1.29373	−0.646867	−	0.762603i	$-0.723921\pi$
$457$	−27.6569	−1.29373	−0.646867	+	0.762603i	$0.723921\pi$
$458$	−3.02944	−0.141556
$459$	3.65685	0.170687
$460$	0	0
$461$	9.31371	0.433783	0.216891	−	0.976196i	$-0.430408\pi$
$461$	9.31371	0.433783	0.216891	+	0.976196i	$0.430408\pi$
$462$	0	0
$463$	6.62742	0.308002	0.154001	−	0.988071i	$-0.450784\pi$
$463$	6.62742	0.308002	0.154001	+	0.988071i	$0.450784\pi$
$464$	18.0000	0.835629
$465$	0	0
$466$	−3.79899	−0.175985
$467$	32.2843	1.49394	0.746969	−	0.664859i	$-0.231508\pi$
$467$	32.2843	1.49394	0.746969	+	0.664859i	$0.231508\pi$
$468$	−1.51472	−0.0700179
$469$	0	0
$470$	0	0
$471$	−16.1421	−0.743790
$472$	21.6569	0.996838
$473$	22.6274	1.04041
$474$	−3.31371	−0.152204
$475$	0	0
$476$	0	0
$477$	−8.48528	−0.388514
$478$	10.8284	0.495281
$479$	25.6569	1.17229	0.586146	−	0.810206i	$-0.300645\pi$
$479$	25.6569	1.17229	0.586146	+	0.810206i	$0.300645\pi$
$480$	0	0
$481$	−6.34315	−0.289223
$482$	4.28427	0.195143
$483$	0	0
$484$	5.48528	0.249331
$485$	0	0
$486$	0.414214	0.0187891
$487$	37.9411	1.71928	0.859638	−	0.510903i	$-0.170689\pi$
$487$	37.9411	1.71928	0.859638	+	0.510903i	$0.170689\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−10.1421	−0.457708	−0.228854	−	0.973461i	$-0.573498\pi$
$491$	−10.1421	−0.457708	−0.228854	+	0.973461i	$0.573498\pi$
$492$	0.627417	0.0282861
$493$	21.9411	0.988179
$494$	−1.65685	−0.0745454
$495$	0	0
$496$	31.4558	1.41241
$497$	0	0
$498$	4.00000	0.179244
$499$	−28.9706	−1.29690	−0.648450	−	0.761257i	$-0.724583\pi$
$499$	−28.9706	−1.29690	−0.648450	+	0.761257i	$0.724583\pi$
$500$	0	0
$501$	−11.3137	−0.505459
$502$	−7.59798	−0.339114
$503$	27.3137	1.21786	0.608929	−	0.793225i	$-0.291599\pi$
$503$	27.3137	1.21786	0.608929	+	0.793225i	$0.291599\pi$
$504$	0	0
$505$	0	0
$506$	4.28427	0.190459
$507$	−12.3137	−0.546871
$508$	−6.05887	−0.268819
$509$	39.6569	1.75776	0.878880	−	0.477044i	$-0.158292\pi$
$509$	39.6569	1.75776	0.878880	+	0.477044i	$0.158292\pi$
$510$	0	0
$511$	0	0
$512$	22.7574	1.00574
$513$	−4.82843	−0.213180
$514$	−4.82843	−0.212973
$515$	0	0
$516$	14.6274	0.643936
$517$	16.0000	0.703679
$518$	0	0
$519$	−20.6274	−0.905443
$520$	0	0
$521$	34.2843	1.50202	0.751011	−	0.660290i	$-0.229567\pi$
$521$	34.2843	1.50202	0.751011	+	0.660290i	$0.229567\pi$
$522$	2.48528	0.108778
$523$	−20.2843	−0.886969	−0.443485	−	0.896282i	$-0.646258\pi$
$523$	−20.2843	−0.886969	−0.443485	+	0.896282i	$0.646258\pi$
$524$	−10.3431	−0.451842
$525$	0	0
$526$	10.4853	0.457180
$527$	38.3431	1.67025
$528$	−8.48528	−0.369274
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−13.6569	−0.592657
$532$	0	0
$533$	−0.284271	−0.0123132
$534$	−7.17157	−0.310344
$535$	0	0
$536$	0	0
$537$	−6.82843	−0.294668
$538$	−4.82843	−0.208168
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	2.28427	0.0981179
$543$	−13.6569	−0.586072
$544$	16.1421	0.692088
$545$	0	0
$546$	0	0
$547$	4.68629	0.200371	0.100186	−	0.994969i	$-0.468056\pi$
$547$	4.68629	0.200371	0.100186	+	0.994969i	$0.468056\pi$
$548$	34.4264	1.47062
$549$	0	0
$550$	0	0
$551$	−28.9706	−1.23419
$552$	5.79899	0.246821
$553$	0	0
$554$	0.544156	0.0231190
$555$	0	0
$556$	−33.7990	−1.43340
$557$	−21.1716	−0.897068	−0.448534	−	0.893766i	$-0.648054\pi$
$557$	−21.1716	−0.897068	−0.448534	+	0.893766i	$0.648054\pi$
$558$	4.34315	0.183860
$559$	−6.62742	−0.280310
$560$	0	0
$561$	−10.3431	−0.436688
$562$	9.79899	0.413345
$563$	31.3137	1.31972	0.659858	−	0.751391i	$-0.270617\pi$
$563$	31.3137	1.31972	0.659858	+	0.751391i	$0.270617\pi$
$564$	10.3431	0.435525
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−4.48528	−0.188198
$569$	33.3137	1.39658	0.698292	−	0.715813i	$-0.253944\pi$
$569$	33.3137	1.39658	0.698292	+	0.715813i	$0.253944\pi$
$570$	0	0
$571$	−39.3137	−1.64523	−0.822614	−	0.568601i	$-0.807485\pi$
$571$	−39.3137	−1.64523	−0.822614	+	0.568601i	$0.807485\pi$
$572$	4.28427	0.179134
$573$	−14.8284	−0.619466
$574$	0	0
$575$	0	0
$576$	−4.17157	−0.173816
$577$	1.51472	0.0630586	0.0315293	−	0.999503i	$-0.489962\pi$
$577$	1.51472	0.0630586	0.0315293	+	0.999503i	$0.489962\pi$
$578$	−1.50253	−0.0624968
$579$	−4.34315	−0.180495
$580$	0	0
$581$	0	0
$582$	−4.34315	−0.180029
$583$	24.0000	0.993978
$584$	26.6863	1.10429
$585$	0	0
$586$	−9.79899	−0.404793
$587$	−31.3137	−1.29246	−0.646228	−	0.763145i	$-0.723654\pi$
$587$	−31.3137	−1.29246	−0.646228	+	0.763145i	$0.723654\pi$
$588$	0	0
$589$	−50.6274	−2.08607
$590$	0	0
$591$	20.4853	0.842652
$592$	−22.9706	−0.944084
$593$	40.6274	1.66837	0.834184	−	0.551486i	$-0.185939\pi$
$593$	40.6274	1.66837	0.834184	+	0.551486i	$0.185939\pi$
$594$	−1.17157	−0.0480702
$595$	0	0
$596$	−6.68629	−0.273881
$597$	−4.82843	−0.197614
$598$	−1.25483	−0.0513140
$599$	−27.5147	−1.12422	−0.562110	−	0.827062i	$-0.690010\pi$
$599$	−27.5147	−1.12422	−0.562110	+	0.827062i	$0.690010\pi$
$600$	0	0
$601$	−22.3431	−0.911396	−0.455698	−	0.890134i	$-0.650610\pi$
$601$	−22.3431	−0.911396	−0.455698	+	0.890134i	$0.650610\pi$
$602$	0	0
$603$	0	0
$604$	−31.0294	−1.26257
$605$	0	0
$606$	1.51472	0.0615312
$607$	−36.9706	−1.50059	−0.750294	−	0.661104i	$-0.770088\pi$
$607$	−36.9706	−1.50059	−0.750294	+	0.661104i	$0.770088\pi$
$608$	−21.3137	−0.864385
$609$	0	0
$610$	0	0
$611$	−4.68629	−0.189587
$612$	−6.68629	−0.270277
$613$	−43.9411	−1.77477	−0.887383	−	0.461034i	$-0.847479\pi$
$613$	−43.9411	−1.77477	−0.887383	+	0.461034i	$0.847479\pi$
$614$	2.05887	0.0830894
$615$	0	0
$616$	0	0
$617$	37.4558	1.50792	0.753958	−	0.656923i	$-0.228142\pi$
$617$	37.4558	1.50792	0.753958	+	0.656923i	$0.228142\pi$
$618$	−4.97056	−0.199945
$619$	19.8579	0.798155	0.399077	−	0.916917i	$-0.369330\pi$
$619$	19.8579	0.798155	0.399077	+	0.916917i	$0.369330\pi$
$620$	0	0
$621$	−3.65685	−0.146745
$622$	0.686292	0.0275178
$623$	0	0
$624$	2.48528	0.0994909
$625$	0	0
$626$	−6.00000	−0.239808
$627$	13.6569	0.545402
$628$	29.5147	1.17777
$629$	−28.0000	−1.11643
$630$	0	0
$631$	−19.3137	−0.768867	−0.384433	−	0.923153i	$-0.625603\pi$
$631$	−19.3137	−0.768867	−0.384433	+	0.923153i	$0.625603\pi$
$632$	12.6863	0.504634
$633$	1.65685	0.0658540
$634$	2.54416	0.101041
$635$	0	0
$636$	15.5147	0.615199
$637$	0	0
$638$	−7.02944	−0.278298
$639$	2.82843	0.111891
$640$	0	0
$641$	22.2843	0.880176	0.440088	−	0.897955i	$-0.354947\pi$
$641$	22.2843	0.880176	0.440088	+	0.897955i	$0.354947\pi$
$642$	7.17157	0.283039
$643$	−28.9706	−1.14249	−0.571244	−	0.820780i	$-0.693539\pi$
$643$	−28.9706	−1.14249	−0.571244	+	0.820780i	$0.693539\pi$
$644$	0	0
$645$	0	0
$646$	−7.31371	−0.287754
$647$	3.31371	0.130275	0.0651377	−	0.997876i	$-0.479251\pi$
$647$	3.31371	0.130275	0.0651377	+	0.997876i	$0.479251\pi$
$648$	−1.58579	−0.0622956
$649$	38.6274	1.51626
$650$	0	0
$651$	0	0
$652$	−21.9411	−0.859281
$653$	19.7990	0.774794	0.387397	−	0.921913i	$-0.373374\pi$
$653$	19.7990	0.774794	0.387397	+	0.921913i	$0.373374\pi$
$654$	2.48528	0.0971822
$655$	0	0
$656$	−1.02944	−0.0401928
$657$	−16.8284	−0.656539
$658$	0	0
$659$	−19.1127	−0.744525	−0.372263	−	0.928127i	$-0.621418\pi$
$659$	−19.1127	−0.744525	−0.372263	+	0.928127i	$0.621418\pi$
$660$	0	0
$661$	−3.31371	−0.128888	−0.0644442	−	0.997921i	$-0.520527\pi$
$661$	−3.31371	−0.128888	−0.0644442	+	0.997921i	$0.520527\pi$
$662$	−4.00000	−0.155464
$663$	3.02944	0.117654
$664$	−15.3137	−0.594287
$665$	0	0
$666$	−3.17157	−0.122896
$667$	−21.9411	−0.849564
$668$	20.6863	0.800377
$669$	12.9706	0.501471
$670$	0	0
$671$	0	0
$672$	0	0
$673$	10.9706	0.422884	0.211442	−	0.977391i	$-0.432184\pi$
$673$	10.9706	0.422884	0.211442	+	0.977391i	$0.432184\pi$
$674$	−11.1716	−0.430313
$675$	0	0
$676$	22.5147	0.865951
$677$	−4.34315	−0.166921	−0.0834603	−	0.996511i	$-0.526597\pi$
$677$	−4.34315	−0.166921	−0.0834603	+	0.996511i	$0.526597\pi$
$678$	1.17157	0.0449940
$679$	0	0
$680$	0	0
$681$	−4.97056	−0.190472
$682$	−12.2843	−0.470389
$683$	27.6569	1.05826	0.529130	−	0.848541i	$-0.322518\pi$
$683$	27.6569	1.05826	0.529130	+	0.848541i	$0.322518\pi$
$684$	8.82843	0.337563
$685$	0	0
$686$	0	0
$687$	−7.31371	−0.279035
$688$	−24.0000	−0.914991
$689$	−7.02944	−0.267800
$690$	0	0
$691$	−10.4853	−0.398879	−0.199439	−	0.979910i	$-0.563912\pi$
$691$	−10.4853	−0.398879	−0.199439	+	0.979910i	$0.563912\pi$
$692$	37.7157	1.43374
$693$	0	0
$694$	−4.54416	−0.172494
$695$	0	0
$696$	−9.51472	−0.360654
$697$	−1.25483	−0.0475302
$698$	−6.62742	−0.250851
$699$	−9.17157	−0.346901
$700$	0	0
$701$	−32.6274	−1.23232	−0.616160	−	0.787621i	$-0.711313\pi$
$701$	−32.6274	−1.23232	−0.616160	+	0.787621i	$0.711313\pi$
$702$	0.343146	0.0129512
$703$	36.9706	1.39437
$704$	11.7990	0.444691
$705$	0	0
$706$	−1.11270	−0.0418770
$707$	0	0
$708$	24.9706	0.938451
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	27.4558	1.02895
$713$	−38.3431	−1.43596
$714$	0	0
$715$	0	0
$716$	12.4853	0.466597
$717$	26.1421	0.976296
$718$	9.85786	0.367892
$719$	−1.65685	−0.0617902	−0.0308951	−	0.999523i	$-0.509836\pi$
$719$	−1.65685	−0.0617902	−0.0308951	+	0.999523i	$0.509836\pi$
$720$	0	0
$721$	0	0
$722$	1.78680	0.0664977
$723$	10.3431	0.384666
$724$	24.9706	0.928024
$725$	0	0
$726$	−1.24264	−0.0461187
$727$	−21.6569	−0.803208	−0.401604	−	0.915813i	$-0.631547\pi$
$727$	−21.6569	−0.803208	−0.401604	+	0.915813i	$0.631547\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−29.2548	−1.08203
$732$	0	0
$733$	47.4558	1.75282	0.876411	−	0.481564i	$-0.159931\pi$
$733$	47.4558	1.75282	0.876411	+	0.481564i	$0.159931\pi$
$734$	8.68629	0.320617
$735$	0	0
$736$	−16.1421	−0.595007
$737$	0	0
$738$	−0.142136	−0.00523208
$739$	37.9411	1.39569	0.697843	−	0.716250i	$-0.254143\pi$
$739$	37.9411	1.39569	0.697843	+	0.716250i	$0.254143\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	6.68629	0.245296	0.122648	−	0.992450i	$-0.460861\pi$
$743$	6.68629	0.245296	0.122648	+	0.992450i	$0.460861\pi$
$744$	−16.6274	−0.609591
$745$	0	0
$746$	−8.54416	−0.312824
$747$	9.65685	0.353326
$748$	18.9117	0.691480
$749$	0	0
$750$	0	0
$751$	−26.3431	−0.961275	−0.480638	−	0.876919i	$-0.659595\pi$
$751$	−26.3431	−0.961275	−0.480638	+	0.876919i	$0.659595\pi$
$752$	−16.9706	−0.618853
$753$	−18.3431	−0.668461
$754$	2.05887	0.0749798
$755$	0	0
$756$	0	0
$757$	15.6569	0.569058	0.284529	−	0.958667i	$-0.408163\pi$
$757$	15.6569	0.569058	0.284529	+	0.958667i	$0.408163\pi$
$758$	5.37258	0.195141
$759$	10.3431	0.375432
$760$	0	0
$761$	13.3137	0.482622	0.241311	−	0.970448i	$-0.422423\pi$
$761$	13.3137	0.482622	0.241311	+	0.970448i	$0.422423\pi$
$762$	1.37258	0.0497234
$763$	0	0
$764$	27.1127	0.980903
$765$	0	0
$766$	0	0
$767$	−11.3137	−0.408514
$768$	3.97056	0.143275
$769$	−36.9706	−1.33319	−0.666596	−	0.745419i	$-0.732249\pi$
$769$	−36.9706	−1.33319	−0.666596	+	0.745419i	$0.732249\pi$
$770$	0	0
$771$	−11.6569	−0.419811
$772$	7.94113	0.285807
$773$	−12.6274	−0.454177	−0.227088	−	0.973874i	$-0.572921\pi$
$773$	−12.6274	−0.454177	−0.227088	+	0.973874i	$0.572921\pi$
$774$	−3.31371	−0.119109
$775$	0	0
$776$	16.6274	0.596889
$777$	0	0
$778$	−0.142136	−0.00509581
$779$	1.65685	0.0593630
$780$	0	0
$781$	−8.00000	−0.286263
$782$	−5.53911	−0.198078
$783$	6.00000	0.214423
$784$	0	0
$785$	0	0
$786$	2.34315	0.0835772
$787$	2.34315	0.0835241	0.0417621	−	0.999128i	$-0.486703\pi$
$787$	2.34315	0.0835241	0.0417621	+	0.999128i	$0.486703\pi$
$788$	−37.4558	−1.33431
$789$	25.3137	0.901192
$790$	0	0
$791$	0	0
$792$	4.48528	0.159378
$793$	0	0
$794$	−7.94113	−0.281820
$795$	0	0
$796$	8.82843	0.312915
$797$	−52.6274	−1.86416	−0.932079	−	0.362254i	$-0.882007\pi$
$797$	−52.6274	−1.86416	−0.932079	+	0.362254i	$0.882007\pi$
$798$	0	0
$799$	−20.6863	−0.731828
$800$	0	0
$801$	−17.3137	−0.611750
$802$	12.4264	0.438792
$803$	47.5980	1.67970
$804$	0	0
$805$	0	0
$806$	3.59798	0.126733
$807$	−11.6569	−0.410341
$808$	−5.79899	−0.204008
$809$	−7.65685	−0.269201	−0.134600	−	0.990900i	$-0.542975\pi$
$809$	−7.65685	−0.269201	−0.134600	+	0.990900i	$0.542975\pi$
$810$	0	0
$811$	−7.45584	−0.261810	−0.130905	−	0.991395i	$-0.541788\pi$
$811$	−7.45584	−0.261810	−0.130905	+	0.991395i	$0.541788\pi$
$812$	0	0
$813$	5.51472	0.193410
$814$	8.97056	0.314418
$815$	0	0
$816$	10.9706	0.384047
$817$	38.6274	1.35140
$818$	−4.97056	−0.173792
$819$	0	0
$820$	0	0
$821$	0.627417	0.0218970	0.0109485	−	0.999940i	$-0.496515\pi$
$821$	0.627417	0.0218970	0.0109485	+	0.999940i	$0.496515\pi$
$822$	−7.79899	−0.272021
$823$	−25.9411	−0.904251	−0.452125	−	0.891954i	$-0.649334\pi$
$823$	−25.9411	−0.904251	−0.452125	+	0.891954i	$0.649334\pi$
$824$	19.0294	0.662922
$825$	0	0
$826$	0	0
$827$	17.3137	0.602057	0.301028	−	0.953615i	$-0.402670\pi$
$827$	17.3137	0.602057	0.301028	+	0.953615i	$0.402670\pi$
$828$	6.68629	0.232365
$829$	31.5980	1.09744	0.548722	−	0.836005i	$-0.315115\pi$
$829$	31.5980	1.09744	0.548722	+	0.836005i	$0.315115\pi$
$830$	0	0
$831$	1.31371	0.0455720
$832$	−3.45584	−0.119810
$833$	0	0
$834$	7.65685	0.265135
$835$	0	0
$836$	−24.9706	−0.863625
$837$	10.4853	0.362424
$838$	9.65685	0.333590
$839$	1.37258	0.0473868	0.0236934	−	0.999719i	$-0.492457\pi$
$839$	1.37258	0.0473868	0.0236934	+	0.999719i	$0.492457\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−2.48528	−0.0856485
$843$	23.6569	0.814785
$844$	−3.02944	−0.104278
$845$	0	0
$846$	−2.34315	−0.0805590
$847$	0	0
$848$	−25.4558	−0.874157
$849$	0	0
$850$	0	0
$851$	28.0000	0.959828
$852$	−5.17157	−0.177175
$853$	−6.48528	−0.222052	−0.111026	−	0.993818i	$-0.535414\pi$
$853$	−6.48528	−0.222052	−0.111026	+	0.993818i	$0.535414\pi$
$854$	0	0
$855$	0	0
$856$	−27.4558	−0.938421
$857$	−0.627417	−0.0214322	−0.0107161	−	0.999943i	$-0.503411\pi$
$857$	−0.627417	−0.0214322	−0.0107161	+	0.999943i	$0.503411\pi$
$858$	−0.970563	−0.0331345
$859$	6.20101	0.211576	0.105788	−	0.994389i	$-0.466264\pi$
$859$	6.20101	0.211576	0.105788	+	0.994389i	$0.466264\pi$
$860$	0	0
$861$	0	0
$862$	−2.14214	−0.0729614
$863$	−13.0294	−0.443527	−0.221764	−	0.975100i	$-0.571181\pi$
$863$	−13.0294	−0.443527	−0.221764	+	0.975100i	$0.571181\pi$
$864$	4.41421	0.150175
$865$	0	0
$866$	5.31371	0.180567
$867$	−3.62742	−0.123194
$868$	0	0
$869$	22.6274	0.767583
$870$	0	0
$871$	0	0
$872$	−9.51472	−0.322209
$873$	−10.4853	−0.354873
$874$	7.31371	0.247390
$875$	0	0
$876$	30.7696	1.03961
$877$	39.9411	1.34872	0.674358	−	0.738405i	$-0.264420\pi$
$877$	39.9411	1.34872	0.674358	+	0.738405i	$0.264420\pi$
$878$	12.6274	0.426155
$879$	−23.6569	−0.797926
$880$	0	0
$881$	−47.2548	−1.59206	−0.796028	−	0.605260i	$-0.793069\pi$
$881$	−47.2548	−1.59206	−0.796028	+	0.605260i	$0.793069\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−5.53911	−0.186300
$885$	0	0
$886$	−5.11270	−0.171764
$887$	−40.9706	−1.37566	−0.687828	−	0.725873i	$-0.741436\pi$
$887$	−40.9706	−1.37566	−0.687828	+	0.725873i	$0.741436\pi$
$888$	12.1421	0.407463
$889$	0	0
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	−23.7157	−0.794061
$893$	27.3137	0.914018
$894$	1.51472	0.0506598
$895$	0	0
$896$	0	0
$897$	−3.02944	−0.101150
$898$	13.5147	0.450992
$899$	62.9117	2.09822
$900$	0	0
$901$	−31.0294	−1.03374
$902$	0.402020	0.0133858
$903$	0	0
$904$	−4.48528	−0.149178
$905$	0	0
$906$	7.02944	0.233537
$907$	16.6863	0.554059	0.277030	−	0.960861i	$-0.410650\pi$
$907$	16.6863	0.554059	0.277030	+	0.960861i	$0.410650\pi$
$908$	9.08831	0.301606
$909$	3.65685	0.121290
$910$	0	0
$911$	31.7990	1.05355	0.526774	−	0.850006i	$-0.323402\pi$
$911$	31.7990	1.05355	0.526774	+	0.850006i	$0.323402\pi$
$912$	−14.4853	−0.479656
$913$	−27.3137	−0.903952
$914$	−11.4558	−0.378926
$915$	0	0
$916$	13.3726	0.441843
$917$	0	0
$918$	1.51472	0.0499932
$919$	−12.2843	−0.405221	−0.202610	−	0.979259i	$-0.564942\pi$
$919$	−12.2843	−0.405221	−0.202610	+	0.979259i	$0.564942\pi$
$920$	0	0
$921$	4.97056	0.163786
$922$	3.85786	0.127052
$923$	2.34315	0.0771256
$924$	0	0
$925$	0	0
$926$	2.74517	0.0902118
$927$	−12.0000	−0.394132
$928$	26.4853	0.869422
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	0	0
$932$	16.7696	0.549305
$933$	1.65685	0.0542430
$934$	13.3726	0.437564
$935$	0	0
$936$	−1.31371	−0.0429399
$937$	−35.4558	−1.15829	−0.579146	−	0.815224i	$-0.696614\pi$
$937$	−35.4558	−1.15829	−0.579146	+	0.815224i	$0.696614\pi$
$938$	0	0
$939$	−14.4853	−0.472709
$940$	0	0
$941$	−44.6274	−1.45481	−0.727406	−	0.686207i	$-0.759274\pi$
$941$	−44.6274	−1.45481	−0.727406	+	0.686207i	$0.759274\pi$
$942$	−6.68629	−0.217851
$943$	1.25483	0.0408630
$944$	−40.9706	−1.33348
$945$	0	0
$946$	9.37258	0.304729
$947$	23.6569	0.768744	0.384372	−	0.923178i	$-0.374418\pi$
$947$	23.6569	0.768744	0.384372	+	0.923178i	$0.374418\pi$
$948$	14.6274	0.475076
$949$	−13.9411	−0.452548
$950$	0	0
$951$	6.14214	0.199172
$952$	0	0
$953$	22.1421	0.717254	0.358627	−	0.933481i	$-0.383245\pi$
$953$	22.1421	0.717254	0.358627	+	0.933481i	$0.383245\pi$
$954$	−3.51472	−0.113793
$955$	0	0
$956$	−47.7990	−1.54593
$957$	−16.9706	−0.548580
$958$	10.6274	0.343356
$959$	0	0
$960$	0	0
$961$	78.9411	2.54649
$962$	−2.62742	−0.0847113
$963$	17.3137	0.557926
$964$	−18.9117	−0.609104
$965$	0	0
$966$	0	0
$967$	−50.6274	−1.62807	−0.814034	−	0.580817i	$-0.802733\pi$
$967$	−50.6274	−1.62807	−0.814034	+	0.580817i	$0.802733\pi$
$968$	4.75736	0.152907
$969$	−17.6569	−0.567220
$970$	0	0
$971$	−16.6863	−0.535489	−0.267744	−	0.963490i	$-0.586278\pi$
$971$	−16.6863	−0.535489	−0.267744	+	0.963490i	$0.586278\pi$
$972$	−1.82843	−0.0586468
$973$	0	0
$974$	15.7157	0.503564
$975$	0	0
$976$	0	0
$977$	3.51472	0.112446	0.0562229	−	0.998418i	$-0.482094\pi$
$977$	3.51472	0.112446	0.0562229	+	0.998418i	$0.482094\pi$
$978$	4.97056	0.158941
$979$	48.9706	1.56511
$980$	0	0
$981$	6.00000	0.191565
$982$	−4.20101	−0.134060
$983$	10.3431	0.329895	0.164948	−	0.986302i	$-0.447255\pi$
$983$	10.3431	0.329895	0.164948	+	0.986302i	$0.447255\pi$
$984$	0.544156	0.0173471
$985$	0	0
$986$	9.08831	0.289431
$987$	0	0
$988$	7.31371	0.232680
$989$	29.2548	0.930250
$990$	0	0
$991$	−16.9706	−0.539088	−0.269544	−	0.962988i	$-0.586873\pi$
$991$	−16.9706	−0.539088	−0.269544	+	0.962988i	$0.586873\pi$
$992$	46.2843	1.46953
$993$	−9.65685	−0.306451
$994$	0	0
$995$	0	0
$996$	−17.6569	−0.559479
$997$	−4.54416	−0.143915	−0.0719574	−	0.997408i	$-0.522925\pi$
$997$	−4.54416	−0.143915	−0.0719574	+	0.997408i	$0.522925\pi$
$998$	−12.0000	−0.379853
$999$	−7.65685	−0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.t.1.2		2
5.4	even	2		735.2.a.l.1.1	✓	2
7.6	odd	2		3675.2.a.s.1.2		2
15.14	odd	2		2205.2.a.r.1.2		2
35.4	even	6		735.2.i.h.226.2		4
35.9	even	6		735.2.i.h.361.2		4
35.19	odd	6		735.2.i.g.361.2		4
35.24	odd	6		735.2.i.g.226.2		4
35.34	odd	2		735.2.a.m.1.1	yes	2
105.104	even	2		2205.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.a.l.1.1	✓	2	5.4	even	2
735.2.a.m.1.1	yes	2	35.34	odd	2
735.2.i.g.226.2		4	35.24	odd	6
735.2.i.g.361.2		4	35.19	odd	6
735.2.i.h.226.2		4	35.4	even	6
735.2.i.h.361.2		4	35.9	even	6
2205.2.a.o.1.2		2	105.104	even	2
2205.2.a.r.1.2		2	15.14	odd	2
3675.2.a.s.1.2		2	7.6	odd	2
3675.2.a.t.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} -2.82843 q^{11} -1.82843 q^{12} +0.828427 q^{13} +3.00000 q^{16} +3.65685 q^{17} +0.414214 q^{18} -4.82843 q^{19} -1.17157 q^{22} -3.65685 q^{23} -1.58579 q^{24} +0.343146 q^{26} +1.00000 q^{27} +6.00000 q^{29} +10.4853 q^{31} +4.41421 q^{32} -2.82843 q^{33} +1.51472 q^{34} -1.82843 q^{36} -7.65685 q^{37} -2.00000 q^{38} +0.828427 q^{39} -0.343146 q^{41} -8.00000 q^{43} +5.17157 q^{44} -1.51472 q^{46} -5.65685 q^{47} +3.00000 q^{48} +3.65685 q^{51} -1.51472 q^{52} -8.48528 q^{53} +0.414214 q^{54} -4.82843 q^{57} +2.48528 q^{58} -13.6569 q^{59} +4.34315 q^{62} -4.17157 q^{64} -1.17157 q^{66} -6.68629 q^{68} -3.65685 q^{69} +2.82843 q^{71} -1.58579 q^{72} -16.8284 q^{73} -3.17157 q^{74} +8.82843 q^{76} +0.343146 q^{78} -8.00000 q^{79} +1.00000 q^{81} -0.142136 q^{82} +9.65685 q^{83} -3.31371 q^{86} +6.00000 q^{87} +4.48528 q^{88} -17.3137 q^{89} +6.68629 q^{92} +10.4853 q^{93} -2.34315 q^{94} +4.41421 q^{96} -10.4853 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 8 q^{22} + 4 q^{23} - 6 q^{24} + 12 q^{26} + 2 q^{27} + 12 q^{29} + 4 q^{31} + 6 q^{32} + 20 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} - 12 q^{41} - 16 q^{43} + 16 q^{44} - 20 q^{46} + 6 q^{48} - 4 q^{51} - 20 q^{52} - 2 q^{54} - 4 q^{57} - 12 q^{58} - 16 q^{59} + 20 q^{62} - 14 q^{64} - 8 q^{66} - 36 q^{68} + 4 q^{69} - 6 q^{72} - 28 q^{73} - 12 q^{74} + 12 q^{76} + 12 q^{78} - 16 q^{79} + 2 q^{81} + 28 q^{82} + 8 q^{83} + 16 q^{86} + 12 q^{87} - 8 q^{88} - 12 q^{89} + 36 q^{92} + 4 q^{93} - 16 q^{94} + 6 q^{96} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 6 * q^16 - 4 * q^17 - 2 * q^18 - 4 * q^19 - 8 * q^22 + 4 * q^23 - 6 * q^24 + 12 * q^26 + 2 * q^27 + 12 * q^29 + 4 * q^31 + 6 * q^32 + 20 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 - 12 * q^41 - 16 * q^43 + 16 * q^44 - 20 * q^46 + 6 * q^48 - 4 * q^51 - 20 * q^52 - 2 * q^54 - 4 * q^57 - 12 * q^58 - 16 * q^59 + 20 * q^62 - 14 * q^64 - 8 * q^66 - 36 * q^68 + 4 * q^69 - 6 * q^72 - 28 * q^73 - 12 * q^74 + 12 * q^76 + 12 * q^78 - 16 * q^79 + 2 * q^81 + 28 * q^82 + 8 * q^83 + 16 * q^86 + 12 * q^87 - 8 * q^88 - 12 * q^89 + 36 * q^92 + 4 * q^93 - 16 * q^94 + 6 * q^96 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more