LMFDB - Embedded newform 2214.2.a.t.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2214,2,Mod(1,2214)] 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("2214.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2214, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2214 = 2 \cdot 3^{3} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2214.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,5,0,5,0,0,2,5,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.6788790075$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.7857909.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 16x^{3} + 14x^{2} + 63x + 33 $ x^5 - 2x^4 - 16x^3 + 14x^2 + 63x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.01671$ of defining polynomial
Character	$\chi$	$=$	2214.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.66155 q^{5} -0.624556 q^{7} +1.00000 q^{8} +1.66155 q^{10} -1.18557 q^{11} -5.30281 q^{13} -0.624556 q^{14} +1.00000 q^{16} +5.64126 q^{17} +6.03268 q^{19} +1.66155 q^{20} -1.18557 q^{22} +1.43899 q^{23} -2.23927 q^{25} -5.30281 q^{26} -0.624556 q^{28} +3.06967 q^{29} +9.86456 q^{31} +1.00000 q^{32} +5.64126 q^{34} -1.03773 q^{35} +9.47241 q^{37} +6.03268 q^{38} +1.66155 q^{40} +1.00000 q^{41} +5.81012 q^{43} -1.18557 q^{44} +1.43899 q^{46} -1.56101 q^{47} -6.60993 q^{49} -2.23927 q^{50} -5.30281 q^{52} +10.8813 q^{53} -1.96987 q^{55} -0.624556 q^{56} +3.06967 q^{58} -4.40334 q^{59} +3.43899 q^{61} +9.86456 q^{62} +1.00000 q^{64} -8.81086 q^{65} -8.14992 q^{67} +5.64126 q^{68} -1.03773 q^{70} -12.2036 q^{71} +1.94704 q^{73} +9.47241 q^{74} +6.03268 q^{76} +0.740453 q^{77} +12.6899 q^{79} +1.66155 q^{80} +1.00000 q^{82} +2.74180 q^{83} +9.37322 q^{85} +5.81012 q^{86} -1.18557 q^{88} -12.1507 q^{89} +3.31190 q^{91} +1.43899 q^{92} -1.56101 q^{94} +10.0236 q^{95} +4.47672 q^{97} -6.60993 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 2 q^{7} + 5 q^{8} - q^{11} + 5 q^{13} + 2 q^{14} + 5 q^{16} + 5 q^{17} + 12 q^{19} - q^{22} + 7 q^{23} + 15 q^{25} + 5 q^{26} + 2 q^{28} - 6 q^{29} + 5 q^{32} + 5 q^{34} + 16 q^{35}+ \cdots + 15 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 2 * q^7 + 5 * q^8 - q^11 + 5 * q^13 + 2 * q^14 + 5 * q^16 + 5 * q^17 + 12 * q^19 - q^22 + 7 * q^23 + 15 * q^25 + 5 * q^26 + 2 * q^28 - 6 * q^29 + 5 * q^32 + 5 * q^34 + 16 * q^35 + 11 * q^37 + 12 * q^38 + 5 * q^41 + 19 * q^43 - q^44 + 7 * q^46 - 8 * q^47 + 15 * q^49 + 15 * q^50 + 5 * q^52 - 13 * q^53 + 21 * q^55 + 2 * q^56 - 6 * q^58 + 18 * q^59 + 17 * q^61 + 5 * q^64 - 16 * q^65 + 4 * q^67 + 5 * q^68 + 16 * q^70 + 5 * q^71 + 13 * q^73 + 11 * q^74 + 12 * q^76 + 15 * q^77 + 22 * q^79 + 5 * q^82 - 18 * q^83 - 24 * q^85 + 19 * q^86 - q^88 + 2 * q^89 + 27 * q^91 + 7 * q^92 - 8 * q^94 - 2 * q^95 + q^97 + 15 * q^98

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.66155	0.743066	0.371533	−	0.928420i	$-0.378832\pi$
$5$	1.66155	0.743066	0.371533	+	0.928420i	$0.378832\pi$
$6$	0	0
$7$	−0.624556	−0.236060	−0.118030	−	0.993010i	$-0.537658\pi$
$7$	−0.624556	−0.236060	−0.118030	+	0.993010i	$0.537658\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.66155	0.525427
$11$	−1.18557	−0.357462	−0.178731	−	0.983898i	$-0.557199\pi$
$11$	−1.18557	−0.357462	−0.178731	+	0.983898i	$0.557199\pi$
$12$	0	0
$13$	−5.30281	−1.47073	−0.735367	−	0.677669i	$-0.762990\pi$
$13$	−5.30281	−1.47073	−0.735367	+	0.677669i	$0.762990\pi$
$14$	−0.624556	−0.166920
$15$	0	0
$16$	1.00000	0.250000
$17$	5.64126	1.36821	0.684104	−	0.729385i	$-0.260193\pi$
$17$	5.64126	1.36821	0.684104	+	0.729385i	$0.260193\pi$
$18$	0	0
$19$	6.03268	1.38399	0.691996	−	0.721901i	$-0.256732\pi$
$19$	6.03268	1.38399	0.691996	+	0.721901i	$0.256732\pi$
$20$	1.66155	0.371533
$21$	0	0
$22$	−1.18557	−0.252764
$23$	1.43899	0.300050	0.150025	−	0.988682i	$-0.452065\pi$
$23$	1.43899	0.300050	0.150025	+	0.988682i	$0.452065\pi$
$24$	0	0
$25$	−2.23927	−0.447853
$26$	−5.30281	−1.03997
$27$	0	0
$28$	−0.624556	−0.118030
$29$	3.06967	0.570023	0.285012	−	0.958524i	$-0.408003\pi$
$29$	3.06967	0.570023	0.285012	+	0.958524i	$0.408003\pi$
$30$	0	0
$31$	9.86456	1.77173	0.885864	−	0.463946i	$-0.153567\pi$
$31$	9.86456	1.77173	0.885864	+	0.463946i	$0.153567\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.64126	0.967469
$35$	−1.03773	−0.175408
$36$	0	0
$37$	9.47241	1.55725	0.778627	−	0.627487i	$-0.215916\pi$
$37$	9.47241	1.55725	0.778627	+	0.627487i	$0.215916\pi$
$38$	6.03268	0.978630
$39$	0	0
$40$	1.66155	0.262713
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	5.81012	0.886035	0.443018	−	0.896513i	$-0.353908\pi$
$43$	5.81012	0.886035	0.443018	+	0.896513i	$0.353908\pi$
$44$	−1.18557	−0.178731
$45$	0	0
$46$	1.43899	0.212167
$47$	−1.56101	−0.227697	−0.113848	−	0.993498i	$-0.536318\pi$
$47$	−1.56101	−0.227697	−0.113848	+	0.993498i	$0.536318\pi$
$48$	0	0
$49$	−6.60993	−0.944276
$50$	−2.23927	−0.316680
$51$	0	0
$52$	−5.30281	−0.735367
$53$	10.8813	1.49466	0.747329	−	0.664455i	$-0.231336\pi$
$53$	10.8813	1.49466	0.747329	+	0.664455i	$0.231336\pi$
$54$	0	0
$55$	−1.96987	−0.265618
$56$	−0.624556	−0.0834598
$57$	0	0
$58$	3.06967	0.403067
$59$	−4.40334	−0.573267	−0.286633	−	0.958040i	$-0.592536\pi$
$59$	−4.40334	−0.573267	−0.286633	+	0.958040i	$0.592536\pi$
$60$	0	0
$61$	3.43899	0.440317	0.220159	−	0.975464i	$-0.429342\pi$
$61$	3.43899	0.440317	0.220159	+	0.975464i	$0.429342\pi$
$62$	9.86456	1.25280
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.81086	−1.09285
$66$	0	0
$67$	−8.14992	−0.995672	−0.497836	−	0.867271i	$-0.665872\pi$
$67$	−8.14992	−0.995672	−0.497836	+	0.867271i	$0.665872\pi$
$68$	5.64126	0.684104
$69$	0	0
$70$	−1.03773	−0.124032
$71$	−12.2036	−1.44830	−0.724152	−	0.689641i	$-0.757768\pi$
$71$	−12.2036	−1.44830	−0.724152	+	0.689641i	$0.757768\pi$
$72$	0	0
$73$	1.94704	0.227884	0.113942	−	0.993487i	$-0.463652\pi$
$73$	1.94704	0.227884	0.113942	+	0.993487i	$0.463652\pi$
$74$	9.47241	1.10114
$75$	0	0
$76$	6.03268	0.691996
$77$	0.740453	0.0843824
$78$	0	0
$79$	12.6899	1.42773	0.713864	−	0.700285i	$-0.246944\pi$
$79$	12.6899	1.42773	0.713864	+	0.700285i	$0.246944\pi$
$80$	1.66155	0.185766
$81$	0	0
$82$	1.00000	0.110432
$83$	2.74180	0.300951	0.150476	−	0.988614i	$-0.451919\pi$
$83$	2.74180	0.300951	0.150476	+	0.988614i	$0.451919\pi$
$84$	0	0
$85$	9.37322	1.01667
$86$	5.81012	0.626522
$87$	0	0
$88$	−1.18557	−0.126382
$89$	−12.1507	−1.28797	−0.643984	−	0.765039i	$-0.722720\pi$
$89$	−12.1507	−1.28797	−0.643984	+	0.765039i	$0.722720\pi$
$90$	0	0
$91$	3.31190	0.347181
$92$	1.43899	0.150025
$93$	0	0
$94$	−1.56101	−0.161006
$95$	10.0236	1.02840
$96$	0	0
$97$	4.47672	0.454542	0.227271	−	0.973832i	$-0.427020\pi$
$97$	4.47672	0.454542	0.227271	+	0.973832i	$0.427020\pi$
$98$	−6.60993	−0.667704
$99$	0	0
$100$	−2.23927	−0.223927
$101$	−12.6357	−1.25730	−0.628652	−	0.777687i	$-0.716393\pi$
$101$	−12.6357	−1.25730	−0.628652	+	0.777687i	$0.716393\pi$
$102$	0	0
$103$	−4.61471	−0.454701	−0.227350	−	0.973813i	$-0.573006\pi$
$103$	−4.61471	−0.454701	−0.227350	+	0.973813i	$0.573006\pi$
$104$	−5.30281	−0.519983
$105$	0	0
$106$	10.8813	1.05688
$107$	−15.4738	−1.49590	−0.747952	−	0.663753i	$-0.768963\pi$
$107$	−15.4738	−1.49590	−0.747952	+	0.663753i	$0.768963\pi$
$108$	0	0
$109$	−9.23018	−0.884091	−0.442045	−	0.896993i	$-0.645747\pi$
$109$	−9.23018	−0.884091	−0.442045	+	0.896993i	$0.645747\pi$
$110$	−1.96987	−0.187820
$111$	0	0
$112$	−0.624556	−0.0590150
$113$	10.3410	0.972800	0.486400	−	0.873736i	$-0.338310\pi$
$113$	10.3410	0.972800	0.486400	+	0.873736i	$0.338310\pi$
$114$	0	0
$115$	2.39094	0.222957
$116$	3.06967	0.285012
$117$	0	0
$118$	−4.40334	−0.405361
$119$	−3.52328	−0.322979
$120$	0	0
$121$	−9.59443	−0.872221
$122$	3.43899	0.311351
$123$	0	0
$124$	9.86456	0.885864
$125$	−12.0284	−1.07585
$126$	0	0
$127$	2.52894	0.224407	0.112204	−	0.993685i	$-0.464209\pi$
$127$	2.52894	0.224407	0.112204	+	0.993685i	$0.464209\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−8.81086	−0.772764
$131$	16.7069	1.45969	0.729844	−	0.683614i	$-0.239593\pi$
$131$	16.7069	1.45969	0.729844	+	0.683614i	$0.239593\pi$
$132$	0	0
$133$	−3.76774	−0.326705
$134$	−8.14992	−0.704046
$135$	0	0
$136$	5.64126	0.483734
$137$	−17.8497	−1.52500	−0.762500	−	0.646988i	$-0.776028\pi$
$137$	−17.8497	−1.52500	−0.762500	+	0.646988i	$0.776028\pi$
$138$	0	0
$139$	1.16334	0.0986732	0.0493366	−	0.998782i	$-0.484289\pi$
$139$	1.16334	0.0986732	0.0493366	+	0.998782i	$0.484289\pi$
$140$	−1.03773	−0.0877040
$141$	0	0
$142$	−12.2036	−1.02411
$143$	6.28684	0.525732
$144$	0	0
$145$	5.10040	0.423565
$146$	1.94704	0.161138
$147$	0	0
$148$	9.47241	0.778627
$149$	−16.1827	−1.32574	−0.662871	−	0.748734i	$-0.730662\pi$
$149$	−16.1827	−1.32574	−0.662871	+	0.748734i	$0.730662\pi$
$150$	0	0
$151$	10.9114	0.887957	0.443978	−	0.896038i	$-0.353567\pi$
$151$	10.9114	0.887957	0.443978	+	0.896038i	$0.353567\pi$
$152$	6.03268	0.489315
$153$	0	0
$154$	0.740453	0.0596674
$155$	16.3904	1.31651
$156$	0	0
$157$	−11.7955	−0.941383	−0.470692	−	0.882298i	$-0.655996\pi$
$157$	−11.7955	−0.941383	−0.470692	+	0.882298i	$0.655996\pi$
$158$	12.6899	1.00956
$159$	0	0
$160$	1.66155	0.131357
$161$	−0.898728	−0.0708297
$162$	0	0
$163$	9.14365	0.716186	0.358093	−	0.933686i	$-0.383427\pi$
$163$	9.14365	0.716186	0.358093	+	0.933686i	$0.383427\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	2.74180	0.212805
$167$	−22.5991	−1.74877	−0.874385	−	0.485233i	$-0.838735\pi$
$167$	−22.5991	−1.74877	−0.874385	+	0.485233i	$0.838735\pi$
$168$	0	0
$169$	15.1198	1.16306
$170$	9.37322	0.718893
$171$	0	0
$172$	5.81012	0.443018
$173$	−18.7912	−1.42867	−0.714334	−	0.699805i	$-0.753270\pi$
$173$	−18.7912	−1.42867	−0.714334	+	0.699805i	$0.753270\pi$
$174$	0	0
$175$	1.39855	0.105720
$176$	−1.18557	−0.0893655
$177$	0	0
$178$	−12.1507	−0.910731
$179$	16.4176	1.22711	0.613553	−	0.789654i	$-0.289740\pi$
$179$	16.4176	1.22711	0.613553	+	0.789654i	$0.289740\pi$
$180$	0	0
$181$	23.1771	1.72274	0.861370	−	0.507979i	$-0.169607\pi$
$181$	23.1771	1.72274	0.861370	+	0.507979i	$0.169607\pi$
$182$	3.31190	0.245494
$183$	0	0
$184$	1.43899	0.106084
$185$	15.7388	1.15714
$186$	0	0
$187$	−6.68810	−0.489082
$188$	−1.56101	−0.113848
$189$	0	0
$190$	10.0236	0.727186
$191$	11.5439	0.835287	0.417643	−	0.908611i	$-0.362856\pi$
$191$	11.5439	0.835287	0.417643	+	0.908611i	$0.362856\pi$
$192$	0	0
$193$	14.8407	1.06826	0.534129	−	0.845403i	$-0.320640\pi$
$193$	14.8407	1.06826	0.534129	+	0.845403i	$0.320640\pi$
$194$	4.47672	0.321409
$195$	0	0
$196$	−6.60993	−0.472138
$197$	25.1902	1.79473	0.897365	−	0.441290i	$-0.145479\pi$
$197$	25.1902	1.79473	0.897365	+	0.441290i	$0.145479\pi$
$198$	0	0
$199$	24.5421	1.73974	0.869871	−	0.493280i	$-0.164202\pi$
$199$	24.5421	1.73974	0.869871	+	0.493280i	$0.164202\pi$
$200$	−2.23927	−0.158340
$201$	0	0
$202$	−12.6357	−0.889048
$203$	−1.91718	−0.134560
$204$	0	0
$205$	1.66155	0.116047
$206$	−4.61471	−0.321522
$207$	0	0
$208$	−5.30281	−0.367684
$209$	−7.15215	−0.494725
$210$	0	0
$211$	5.67395	0.390610	0.195305	−	0.980743i	$-0.437430\pi$
$211$	5.67395	0.390610	0.195305	+	0.980743i	$0.437430\pi$
$212$	10.8813	0.747329
$213$	0	0
$214$	−15.4738	−1.05776
$215$	9.65378	0.658383
$216$	0	0
$217$	−6.16097	−0.418234
$218$	−9.23018	−0.625146
$219$	0	0
$220$	−1.96987	−0.132809
$221$	−29.9146	−2.01227
$222$	0	0
$223$	−10.2036	−0.683285	−0.341643	−	0.939830i	$-0.610983\pi$
$223$	−10.2036	−0.683285	−0.341643	+	0.939830i	$0.610983\pi$
$224$	−0.624556	−0.0417299
$225$	0	0
$226$	10.3410	0.687874
$227$	−14.2233	−0.944033	−0.472017	−	0.881590i	$-0.656474\pi$
$227$	−14.2233	−0.944033	−0.472017	+	0.881590i	$0.656474\pi$
$228$	0	0
$229$	−20.0188	−1.32288	−0.661440	−	0.749998i	$-0.730054\pi$
$229$	−20.0188	−1.32288	−0.661440	+	0.749998i	$0.730054\pi$
$230$	2.39094	0.157654
$231$	0	0
$232$	3.06967	0.201534
$233$	−15.2622	−0.999863	−0.499932	−	0.866065i	$-0.666642\pi$
$233$	−15.2622	−0.999863	−0.499932	+	0.866065i	$0.666642\pi$
$234$	0	0
$235$	−2.59369	−0.169194
$236$	−4.40334	−0.286633
$237$	0	0
$238$	−3.52328	−0.228381
$239$	−0.200801	−0.0129887	−0.00649436	−	0.999979i	$-0.502067\pi$
$239$	−0.200801	−0.0129887	−0.00649436	+	0.999979i	$0.502067\pi$
$240$	0	0
$241$	−17.8239	−1.14814	−0.574068	−	0.818808i	$-0.694636\pi$
$241$	−17.8239	−1.14814	−0.574068	+	0.818808i	$0.694636\pi$
$242$	−9.59443	−0.616753
$243$	0	0
$244$	3.43899	0.220159
$245$	−10.9827	−0.701659
$246$	0	0
$247$	−31.9902	−2.03549
$248$	9.86456	0.626400
$249$	0	0
$250$	−12.0284	−0.760741
$251$	9.14871	0.577462	0.288731	−	0.957410i	$-0.406767\pi$
$251$	9.14871	0.577462	0.288731	+	0.957410i	$0.406767\pi$
$252$	0	0
$253$	−1.70602	−0.107256
$254$	2.52894	0.158680
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.3522	0.895266	0.447633	−	0.894217i	$-0.352267\pi$
$257$	14.3522	0.895266	0.447633	+	0.894217i	$0.352267\pi$
$258$	0	0
$259$	−5.91605	−0.367605
$260$	−8.81086	−0.546426
$261$	0	0
$262$	16.7069	1.03216
$263$	−11.8219	−0.728970	−0.364485	−	0.931209i	$-0.618755\pi$
$263$	−11.8219	−0.728970	−0.364485	+	0.931209i	$0.618755\pi$
$264$	0	0
$265$	18.0797	1.11063
$266$	−3.76774	−0.231015
$267$	0	0
$268$	−8.14992	−0.497836
$269$	−21.8129	−1.32996	−0.664979	−	0.746862i	$-0.731560\pi$
$269$	−21.8129	−1.32996	−0.664979	+	0.746862i	$0.731560\pi$
$270$	0	0
$271$	7.44330	0.452148	0.226074	−	0.974110i	$-0.427411\pi$
$271$	7.44330	0.452148	0.226074	+	0.974110i	$0.427411\pi$
$272$	5.64126	0.342052
$273$	0	0
$274$	−17.8497	−1.07834
$275$	2.65480	0.160091
$276$	0	0
$277$	−15.2205	−0.914509	−0.457254	−	0.889336i	$-0.651167\pi$
$277$	−15.2205	−0.914509	−0.457254	+	0.889336i	$0.651167\pi$
$278$	1.16334	0.0697725
$279$	0	0
$280$	−1.03773	−0.0620161
$281$	2.28758	0.136465	0.0682327	−	0.997669i	$-0.478264\pi$
$281$	2.28758	0.136465	0.0682327	+	0.997669i	$0.478264\pi$
$282$	0	0
$283$	15.3153	0.910403	0.455201	−	0.890388i	$-0.349567\pi$
$283$	15.3153	0.910403	0.455201	+	0.890388i	$0.349567\pi$
$284$	−12.2036	−0.724152
$285$	0	0
$286$	6.28684	0.371749
$287$	−0.624556	−0.0368664
$288$	0	0
$289$	14.8239	0.871992
$290$	5.10040	0.299506
$291$	0	0
$292$	1.94704	0.113942
$293$	−31.0619	−1.81466	−0.907329	−	0.420422i	$-0.861882\pi$
$293$	−31.0619	−1.81466	−0.907329	+	0.420422i	$0.861882\pi$
$294$	0	0
$295$	−7.31636	−0.425975
$296$	9.47241	0.550572
$297$	0	0
$298$	−16.1827	−0.937441
$299$	−7.63068	−0.441294
$300$	0	0
$301$	−3.62875	−0.209157
$302$	10.9114	0.627880
$303$	0	0
$304$	6.03268	0.345998
$305$	5.71404	0.327185
$306$	0	0
$307$	−3.18738	−0.181914	−0.0909568	−	0.995855i	$-0.528993\pi$
$307$	−3.18738	−0.181914	−0.0909568	+	0.995855i	$0.528993\pi$
$308$	0.740453	0.0421912
$309$	0	0
$310$	16.3904	0.930913
$311$	18.9872	1.07667	0.538333	−	0.842732i	$-0.319054\pi$
$311$	18.9872	1.07667	0.538333	+	0.842732i	$0.319054\pi$
$312$	0	0
$313$	3.51432	0.198641	0.0993207	−	0.995055i	$-0.468333\pi$
$313$	3.51432	0.198641	0.0993207	+	0.995055i	$0.468333\pi$
$314$	−11.7955	−0.665658
$315$	0	0
$316$	12.6899	0.713864
$317$	−26.8067	−1.50561	−0.752807	−	0.658242i	$-0.771301\pi$
$317$	−26.8067	−1.50561	−0.752807	+	0.658242i	$0.771301\pi$
$318$	0	0
$319$	−3.63930	−0.203762
$320$	1.66155	0.0928832
$321$	0	0
$322$	−0.898728	−0.0500842
$323$	34.0319	1.89359
$324$	0	0
$325$	11.8744	0.658673
$326$	9.14365	0.506420
$327$	0	0
$328$	1.00000	0.0552158
$329$	0.974939	0.0537501
$330$	0	0
$331$	−8.97925	−0.493544	−0.246772	−	0.969074i	$-0.579370\pi$
$331$	−8.97925	−0.493544	−0.246772	+	0.969074i	$0.579370\pi$
$332$	2.74180	0.150476
$333$	0	0
$334$	−22.5991	−1.23657
$335$	−13.5415	−0.739849
$336$	0	0
$337$	0.964486	0.0525389	0.0262694	−	0.999655i	$-0.491637\pi$
$337$	0.964486	0.0525389	0.0262694	+	0.999655i	$0.491637\pi$
$338$	15.1198	0.822409
$339$	0	0
$340$	9.37322	0.508334
$341$	−11.6951	−0.633325
$342$	0	0
$343$	8.50016	0.458965
$344$	5.81012	0.313261
$345$	0	0
$346$	−18.7912	−1.01022
$347$	−29.4553	−1.58124	−0.790621	−	0.612305i	$-0.790242\pi$
$347$	−29.4553	−1.58124	−0.790621	+	0.612305i	$0.790242\pi$
$348$	0	0
$349$	23.3606	1.25046	0.625231	−	0.780440i	$-0.285005\pi$
$349$	23.3606	1.25046	0.625231	+	0.780440i	$0.285005\pi$
$350$	1.39855	0.0747555
$351$	0	0
$352$	−1.18557	−0.0631910
$353$	15.8665	0.844490	0.422245	−	0.906482i	$-0.361242\pi$
$353$	15.8665	0.844490	0.422245	+	0.906482i	$0.361242\pi$
$354$	0	0
$355$	−20.2769	−1.07618
$356$	−12.1507	−0.643984
$357$	0	0
$358$	16.4176	0.867695
$359$	−18.7360	−0.988848	−0.494424	−	0.869221i	$-0.664621\pi$
$359$	−18.7360	−0.988848	−0.494424	+	0.869221i	$0.664621\pi$
$360$	0	0
$361$	17.3932	0.915433
$362$	23.1771	1.21816
$363$	0	0
$364$	3.31190	0.173591
$365$	3.23509	0.169333
$366$	0	0
$367$	11.0567	0.577157	0.288579	−	0.957456i	$-0.406817\pi$
$367$	11.0567	0.577157	0.288579	+	0.957456i	$0.406817\pi$
$368$	1.43899	0.0750124
$369$	0	0
$370$	15.7388	0.818223
$371$	−6.79596	−0.352829
$372$	0	0
$373$	−20.3705	−1.05475	−0.527373	−	0.849634i	$-0.676823\pi$
$373$	−20.3705	−1.05475	−0.527373	+	0.849634i	$0.676823\pi$
$374$	−6.68810	−0.345833
$375$	0	0
$376$	−1.56101	−0.0805030
$377$	−16.2779	−0.838353
$378$	0	0
$379$	3.10067	0.159271	0.0796354	−	0.996824i	$-0.474624\pi$
$379$	3.10067	0.159271	0.0796354	+	0.996824i	$0.474624\pi$
$380$	10.0236	0.514198
$381$	0	0
$382$	11.5439	0.590637
$383$	13.2399	0.676526	0.338263	−	0.941052i	$-0.390161\pi$
$383$	13.2399	0.676526	0.338263	+	0.941052i	$0.390161\pi$
$384$	0	0
$385$	1.23030	0.0627017
$386$	14.8407	0.755372
$387$	0	0
$388$	4.47672	0.227271
$389$	−15.2282	−0.772101	−0.386051	−	0.922478i	$-0.626161\pi$
$389$	−15.2282	−0.772101	−0.386051	+	0.922478i	$0.626161\pi$
$390$	0	0
$391$	8.11771	0.410530
$392$	−6.60993	−0.333852
$393$	0	0
$394$	25.1902	1.26907
$395$	21.0849	1.06090
$396$	0	0
$397$	−14.7646	−0.741016	−0.370508	−	0.928829i	$-0.620816\pi$
$397$	−14.7646	−0.741016	−0.370508	+	0.928829i	$0.620816\pi$
$398$	24.5421	1.23018
$399$	0	0
$400$	−2.23927	−0.111963
$401$	15.1156	0.754839	0.377419	−	0.926042i	$-0.376812\pi$
$401$	15.1156	0.754839	0.377419	+	0.926042i	$0.376812\pi$
$402$	0	0
$403$	−52.3099	−2.60574
$404$	−12.6357	−0.628652
$405$	0	0
$406$	−1.91718	−0.0951480
$407$	−11.2302	−0.556659
$408$	0	0
$409$	−16.7673	−0.829091	−0.414545	−	0.910029i	$-0.636059\pi$
$409$	−16.7673	−0.829091	−0.414545	+	0.910029i	$0.636059\pi$
$410$	1.66155	0.0820579
$411$	0	0
$412$	−4.61471	−0.227350
$413$	2.75013	0.135325
$414$	0	0
$415$	4.55562	0.223627
$416$	−5.30281	−0.259992
$417$	0	0
$418$	−7.15215	−0.349823
$419$	30.9245	1.51076	0.755381	−	0.655286i	$-0.227452\pi$
$419$	30.9245	1.51076	0.755381	+	0.655286i	$0.227452\pi$
$420$	0	0
$421$	−18.4812	−0.900719	−0.450360	−	0.892847i	$-0.648704\pi$
$421$	−18.4812	−0.900719	−0.450360	+	0.892847i	$0.648704\pi$
$422$	5.67395	0.276203
$423$	0	0
$424$	10.8813	0.528441
$425$	−12.6323	−0.612756
$426$	0	0
$427$	−2.14784	−0.103941
$428$	−15.4738	−0.747952
$429$	0	0
$430$	9.65378	0.465547
$431$	−26.6535	−1.28385	−0.641927	−	0.766765i	$-0.721865\pi$
$431$	−26.6535	−1.28385	−0.641927	+	0.766765i	$0.721865\pi$
$432$	0	0
$433$	−31.6022	−1.51870	−0.759352	−	0.650679i	$-0.774484\pi$
$433$	−31.6022	−1.51870	−0.759352	+	0.650679i	$0.774484\pi$
$434$	−6.16097	−0.295736
$435$	0	0
$436$	−9.23018	−0.442045
$437$	8.68096	0.415266
$438$	0	0
$439$	−27.8868	−1.33096	−0.665482	−	0.746414i	$-0.731774\pi$
$439$	−27.8868	−1.33096	−0.665482	+	0.746414i	$0.731774\pi$
$440$	−1.96987	−0.0939101
$441$	0	0
$442$	−29.9146	−1.42289
$443$	9.20646	0.437412	0.218706	−	0.975791i	$-0.429816\pi$
$443$	9.20646	0.437412	0.218706	+	0.975791i	$0.429816\pi$
$444$	0	0
$445$	−20.1889	−0.957045
$446$	−10.2036	−0.483156
$447$	0	0
$448$	−0.624556	−0.0295075
$449$	25.8346	1.21921	0.609605	−	0.792706i	$-0.291328\pi$
$449$	25.8346	1.21921	0.609605	+	0.792706i	$0.291328\pi$
$450$	0	0
$451$	−1.18557	−0.0558262
$452$	10.3410	0.486400
$453$	0	0
$454$	−14.2233	−0.667532
$455$	5.50287	0.257979
$456$	0	0
$457$	−24.8590	−1.16286	−0.581428	−	0.813598i	$-0.697506\pi$
$457$	−24.8590	−1.16286	−0.581428	+	0.813598i	$0.697506\pi$
$458$	−20.0188	−0.935417
$459$	0	0
$460$	2.39094	0.111478
$461$	−3.22390	−0.150152	−0.0750759	−	0.997178i	$-0.523920\pi$
$461$	−3.22390	−0.150152	−0.0750759	+	0.997178i	$0.523920\pi$
$462$	0	0
$463$	−16.6985	−0.776047	−0.388023	−	0.921650i	$-0.626842\pi$
$463$	−16.6985	−0.776047	−0.388023	+	0.921650i	$0.626842\pi$
$464$	3.06967	0.142506
$465$	0	0
$466$	−15.2622	−0.707010
$467$	−3.18752	−0.147501	−0.0737504	−	0.997277i	$-0.523497\pi$
$467$	−3.18752	−0.147501	−0.0737504	+	0.997277i	$0.523497\pi$
$468$	0	0
$469$	5.09008	0.235038
$470$	−2.59369	−0.119638
$471$	0	0
$472$	−4.40334	−0.202680
$473$	−6.88829	−0.316724
$474$	0	0
$475$	−13.5088	−0.619825
$476$	−3.52328	−0.161489
$477$	0	0
$478$	−0.200801	−0.00918441
$479$	21.8671	0.999135	0.499567	−	0.866275i	$-0.333492\pi$
$479$	21.8671	0.999135	0.499567	+	0.866275i	$0.333492\pi$
$480$	0	0
$481$	−50.2304	−2.29031
$482$	−17.8239	−0.811855
$483$	0	0
$484$	−9.59443	−0.436110
$485$	7.43827	0.337754
$486$	0	0
$487$	20.0372	0.907972	0.453986	−	0.891009i	$-0.350002\pi$
$487$	20.0372	0.907972	0.453986	+	0.891009i	$0.350002\pi$
$488$	3.43899	0.155676
$489$	0	0
$490$	−10.9827	−0.496148
$491$	−1.00641	−0.0454187	−0.0227093	−	0.999742i	$-0.507229\pi$
$491$	−1.00641	−0.0454187	−0.0227093	+	0.999742i	$0.507229\pi$
$492$	0	0
$493$	17.3168	0.779910
$494$	−31.9902	−1.43931
$495$	0	0
$496$	9.86456	0.442932
$497$	7.62184	0.341886
$498$	0	0
$499$	17.5170	0.784169	0.392084	−	0.919929i	$-0.371754\pi$
$499$	17.5170	0.784169	0.392084	+	0.919929i	$0.371754\pi$
$500$	−12.0284	−0.537925
$501$	0	0
$502$	9.14871	0.408327
$503$	5.36170	0.239066	0.119533	−	0.992830i	$-0.461860\pi$
$503$	5.36170	0.239066	0.119533	+	0.992830i	$0.461860\pi$
$504$	0	0
$505$	−20.9949	−0.934259
$506$	−1.70602	−0.0758417
$507$	0	0
$508$	2.52894	0.112204
$509$	−31.3580	−1.38992	−0.694959	−	0.719049i	$-0.744578\pi$
$509$	−31.3580	−1.38992	−0.694959	+	0.719049i	$0.744578\pi$
$510$	0	0
$511$	−1.21603	−0.0537942
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.3522	0.633048
$515$	−7.66755	−0.337873
$516$	0	0
$517$	1.85068	0.0813930
$518$	−5.91605	−0.259936
$519$	0	0
$520$	−8.81086	−0.386382
$521$	31.8750	1.39647	0.698234	−	0.715869i	$-0.253969\pi$
$521$	31.8750	1.39647	0.698234	+	0.715869i	$0.253969\pi$
$522$	0	0
$523$	14.6589	0.640990	0.320495	−	0.947250i	$-0.396151\pi$
$523$	14.6589	0.640990	0.320495	+	0.947250i	$0.396151\pi$
$524$	16.7069	0.729844
$525$	0	0
$526$	−11.8219	−0.515460
$527$	55.6486	2.42409
$528$	0	0
$529$	−20.9293	−0.909970
$530$	18.0797	0.785333
$531$	0	0
$532$	−3.76774	−0.163352
$533$	−5.30281	−0.229690
$534$	0	0
$535$	−25.7103	−1.11156
$536$	−8.14992	−0.352023
$537$	0	0
$538$	−21.8129	−0.940423
$539$	7.83652	0.337543
$540$	0	0
$541$	5.47059	0.235199	0.117600	−	0.993061i	$-0.462480\pi$
$541$	5.47059	0.235199	0.117600	+	0.993061i	$0.462480\pi$
$542$	7.44330	0.319717
$543$	0	0
$544$	5.64126	0.241867
$545$	−15.3364	−0.656937
$546$	0	0
$547$	30.0931	1.28669	0.643344	−	0.765577i	$-0.277546\pi$
$547$	30.0931	1.28669	0.643344	+	0.765577i	$0.277546\pi$
$548$	−17.8497	−0.762500
$549$	0	0
$550$	2.65480	0.113201
$551$	18.5183	0.788908
$552$	0	0
$553$	−7.92556	−0.337029
$554$	−15.2205	−0.646655
$555$	0	0
$556$	1.16334	0.0493366
$557$	31.1517	1.31994	0.659971	−	0.751291i	$-0.270569\pi$
$557$	31.1517	1.31994	0.659971	+	0.751291i	$0.270569\pi$
$558$	0	0
$559$	−30.8100	−1.30312
$560$	−1.03773	−0.0438520
$561$	0	0
$562$	2.28758	0.0964956
$563$	−31.0326	−1.30787	−0.653933	−	0.756552i	$-0.726882\pi$
$563$	−31.0326	−1.30787	−0.653933	+	0.756552i	$0.726882\pi$
$564$	0	0
$565$	17.1821	0.722855
$566$	15.3153	0.643752
$567$	0	0
$568$	−12.2036	−0.512053
$569$	14.8594	0.622938	0.311469	−	0.950256i	$-0.399179\pi$
$569$	14.8594	0.622938	0.311469	+	0.950256i	$0.399179\pi$
$570$	0	0
$571$	−27.2837	−1.14179	−0.570895	−	0.821023i	$-0.693404\pi$
$571$	−27.2837	−1.14179	−0.570895	+	0.821023i	$0.693404\pi$
$572$	6.28684	0.262866
$573$	0	0
$574$	−0.624556	−0.0260685
$575$	−3.22228	−0.134378
$576$	0	0
$577$	12.7967	0.532734	0.266367	−	0.963872i	$-0.414177\pi$
$577$	12.7967	0.532734	0.266367	+	0.963872i	$0.414177\pi$
$578$	14.8239	0.616592
$579$	0	0
$580$	5.10040	0.211782
$581$	−1.71241	−0.0710426
$582$	0	0
$583$	−12.9005	−0.534283
$584$	1.94704	0.0805690
$585$	0	0
$586$	−31.0619	−1.28316
$587$	−13.5909	−0.560955	−0.280478	−	0.959861i	$-0.590493\pi$
$587$	−13.5909	−0.560955	−0.280478	+	0.959861i	$0.590493\pi$
$588$	0	0
$589$	59.5097	2.45206
$590$	−7.31636	−0.301210
$591$	0	0
$592$	9.47241	0.389314
$593$	7.88349	0.323736	0.161868	−	0.986812i	$-0.448248\pi$
$593$	7.88349	0.323736	0.161868	+	0.986812i	$0.448248\pi$
$594$	0	0
$595$	−5.85410	−0.239995
$596$	−16.1827	−0.662871
$597$	0	0
$598$	−7.63068	−0.312042
$599$	13.6972	0.559651	0.279826	−	0.960051i	$-0.409723\pi$
$599$	13.6972	0.559651	0.279826	+	0.960051i	$0.409723\pi$
$600$	0	0
$601$	11.3146	0.461532	0.230766	−	0.973009i	$-0.425877\pi$
$601$	11.3146	0.461532	0.230766	+	0.973009i	$0.425877\pi$
$602$	−3.62875	−0.147897
$603$	0	0
$604$	10.9114	0.443978
$605$	−15.9416	−0.648117
$606$	0	0
$607$	−0.578589	−0.0234842	−0.0117421	−	0.999931i	$-0.503738\pi$
$607$	−0.578589	−0.0234842	−0.0117421	+	0.999931i	$0.503738\pi$
$608$	6.03268	0.244657
$609$	0	0
$610$	5.71404	0.231355
$611$	8.27775	0.334882
$612$	0	0
$613$	−29.4201	−1.18826	−0.594132	−	0.804367i	$-0.702504\pi$
$613$	−29.4201	−1.18826	−0.594132	+	0.804367i	$0.702504\pi$
$614$	−3.18738	−0.128632
$615$	0	0
$616$	0.740453	0.0298337
$617$	27.8164	1.11985	0.559924	−	0.828544i	$-0.310830\pi$
$617$	27.8164	1.11985	0.559924	+	0.828544i	$0.310830\pi$
$618$	0	0
$619$	−33.8146	−1.35912	−0.679562	−	0.733618i	$-0.737830\pi$
$619$	−33.8146	−1.35912	−0.679562	+	0.733618i	$0.737830\pi$
$620$	16.3904	0.658255
$621$	0	0
$622$	18.9872	0.761318
$623$	7.58876	0.304037
$624$	0	0
$625$	−8.78935	−0.351574
$626$	3.51432	0.140461
$627$	0	0
$628$	−11.7955	−0.470692
$629$	53.4364	2.13065
$630$	0	0
$631$	−27.6911	−1.10237	−0.551183	−	0.834384i	$-0.685823\pi$
$631$	−27.6911	−1.10237	−0.551183	+	0.834384i	$0.685823\pi$
$632$	12.6899	0.504778
$633$	0	0
$634$	−26.8067	−1.06463
$635$	4.20195	0.166749
$636$	0	0
$637$	35.0512	1.38878
$638$	−3.63930	−0.144081
$639$	0	0
$640$	1.66155	0.0656784
$641$	−6.05058	−0.238984	−0.119492	−	0.992835i	$-0.538127\pi$
$641$	−6.05058	−0.238984	−0.119492	+	0.992835i	$0.538127\pi$
$642$	0	0
$643$	30.0701	1.18585	0.592925	−	0.805258i	$-0.297973\pi$
$643$	30.0701	1.18585	0.592925	+	0.805258i	$0.297973\pi$
$644$	−0.898728	−0.0354149
$645$	0	0
$646$	34.0319	1.33897
$647$	−23.3893	−0.919529	−0.459765	−	0.888041i	$-0.652066\pi$
$647$	−23.3893	−0.919529	−0.459765	+	0.888041i	$0.652066\pi$
$648$	0	0
$649$	5.22046	0.204921
$650$	11.8744	0.465752
$651$	0	0
$652$	9.14365	0.358093
$653$	−28.0066	−1.09598	−0.547991	−	0.836484i	$-0.684607\pi$
$653$	−28.0066	−1.09598	−0.547991	+	0.836484i	$0.684607\pi$
$654$	0	0
$655$	27.7593	1.08464
$656$	1.00000	0.0390434
$657$	0	0
$658$	0.974939	0.0380071
$659$	39.1796	1.52622	0.763111	−	0.646267i	$-0.223671\pi$
$659$	39.1796	1.52622	0.763111	+	0.646267i	$0.223671\pi$
$660$	0	0
$661$	32.0681	1.24730	0.623651	−	0.781703i	$-0.285649\pi$
$661$	32.0681	1.24730	0.623651	+	0.781703i	$0.285649\pi$
$662$	−8.97925	−0.348988
$663$	0	0
$664$	2.74180	0.106402
$665$	−6.26028	−0.242763
$666$	0	0
$667$	4.41722	0.171035
$668$	−22.5991	−0.874385
$669$	0	0
$670$	−13.5415	−0.523153
$671$	−4.07715	−0.157397
$672$	0	0
$673$	−4.32982	−0.166902	−0.0834511	−	0.996512i	$-0.526594\pi$
$673$	−4.32982	−0.166902	−0.0834511	+	0.996512i	$0.526594\pi$
$674$	0.964486	0.0371506
$675$	0	0
$676$	15.1198	0.581531
$677$	25.3887	0.975768	0.487884	−	0.872908i	$-0.337769\pi$
$677$	25.3887	0.975768	0.487884	+	0.872908i	$0.337769\pi$
$678$	0	0
$679$	−2.79596	−0.107299
$680$	9.37322	0.359447
$681$	0	0
$682$	−11.6951	−0.447829
$683$	−1.89771	−0.0726139	−0.0363070	−	0.999341i	$-0.511559\pi$
$683$	−1.89771	−0.0726139	−0.0363070	+	0.999341i	$0.511559\pi$
$684$	0	0
$685$	−29.6580	−1.13318
$686$	8.50016	0.324538
$687$	0	0
$688$	5.81012	0.221509
$689$	−57.7013	−2.19824
$690$	0	0
$691$	23.3194	0.887111	0.443556	−	0.896247i	$-0.353717\pi$
$691$	23.3194	0.887111	0.443556	+	0.896247i	$0.353717\pi$
$692$	−18.7912	−0.714334
$693$	0	0
$694$	−29.4553	−1.11811
$695$	1.93294	0.0733207
$696$	0	0
$697$	5.64126	0.213678
$698$	23.3606	0.884210
$699$	0	0
$700$	1.39855	0.0528601
$701$	−39.3874	−1.48764	−0.743821	−	0.668378i	$-0.766989\pi$
$701$	−39.3874	−1.48764	−0.743821	+	0.668378i	$0.766989\pi$
$702$	0	0
$703$	57.1440	2.15523
$704$	−1.18557	−0.0446828
$705$	0	0
$706$	15.8665	0.597144
$707$	7.89173	0.296799
$708$	0	0
$709$	−38.9598	−1.46316	−0.731582	−	0.681753i	$-0.761218\pi$
$709$	−38.9598	−1.46316	−0.731582	+	0.681753i	$0.761218\pi$
$710$	−20.2769	−0.760978
$711$	0	0
$712$	−12.1507	−0.455365
$713$	14.1950	0.531606
$714$	0	0
$715$	10.4459	0.390653
$716$	16.4176	0.613553
$717$	0	0
$718$	−18.7360	−0.699221
$719$	−8.64692	−0.322476	−0.161238	−	0.986916i	$-0.551549\pi$
$719$	−8.64692	−0.322476	−0.161238	+	0.986916i	$0.551549\pi$
$720$	0	0
$721$	2.88214	0.107337
$722$	17.3932	0.647309
$723$	0	0
$724$	23.1771	0.861370
$725$	−6.87381	−0.255287
$726$	0	0
$727$	−7.30861	−0.271061	−0.135531	−	0.990773i	$-0.543274\pi$
$727$	−7.30861	−0.271061	−0.135531	+	0.990773i	$0.543274\pi$
$728$	3.31190	0.122747
$729$	0	0
$730$	3.23509	0.119736
$731$	32.7764	1.21228
$732$	0	0
$733$	25.0208	0.924163	0.462081	−	0.886837i	$-0.347103\pi$
$733$	25.0208	0.924163	0.462081	+	0.886837i	$0.347103\pi$
$734$	11.0567	0.408112
$735$	0	0
$736$	1.43899	0.0530418
$737$	9.66228	0.355915
$738$	0	0
$739$	8.31621	0.305917	0.152958	−	0.988233i	$-0.451120\pi$
$739$	8.31621	0.305917	0.152958	+	0.988233i	$0.451120\pi$
$740$	15.7388	0.578571
$741$	0	0
$742$	−6.79596	−0.249487
$743$	22.7109	0.833184	0.416592	−	0.909094i	$-0.363224\pi$
$743$	22.7109	0.833184	0.416592	+	0.909094i	$0.363224\pi$
$744$	0	0
$745$	−26.8883	−0.985113
$746$	−20.3705	−0.745818
$747$	0	0
$748$	−6.68810	−0.244541
$749$	9.66422	0.353123
$750$	0	0
$751$	−49.0495	−1.78984	−0.894920	−	0.446227i	$-0.852768\pi$
$751$	−49.0495	−1.78984	−0.894920	+	0.446227i	$0.852768\pi$
$752$	−1.56101	−0.0569242
$753$	0	0
$754$	−16.2779	−0.592805
$755$	18.1298	0.659810
$756$	0	0
$757$	1.54673	0.0562168	0.0281084	−	0.999605i	$-0.491052\pi$
$757$	1.54673	0.0562168	0.0281084	+	0.999605i	$0.491052\pi$
$758$	3.10067	0.112621
$759$	0	0
$760$	10.0236	0.363593
$761$	24.8410	0.900487	0.450244	−	0.892906i	$-0.351337\pi$
$761$	24.8410	0.900487	0.450244	+	0.892906i	$0.351337\pi$
$762$	0	0
$763$	5.76476	0.208698
$764$	11.5439	0.417643
$765$	0	0
$766$	13.2399	0.478376
$767$	23.3501	0.843123
$768$	0	0
$769$	10.7601	0.388020	0.194010	−	0.981000i	$-0.437851\pi$
$769$	10.7601	0.388020	0.194010	+	0.981000i	$0.437851\pi$
$770$	1.23030	0.0443368
$771$	0	0
$772$	14.8407	0.534129
$773$	−30.0712	−1.08159	−0.540793	−	0.841155i	$-0.681876\pi$
$773$	−30.0712	−1.08159	−0.540793	+	0.841155i	$0.681876\pi$
$774$	0	0
$775$	−22.0894	−0.793474
$776$	4.47672	0.160705
$777$	0	0
$778$	−15.2282	−0.545958
$779$	6.03268	0.216143
$780$	0	0
$781$	14.4682	0.517714
$782$	8.11771	0.290289
$783$	0	0
$784$	−6.60993	−0.236069
$785$	−19.5988	−0.699510
$786$	0	0
$787$	48.8635	1.74180	0.870899	−	0.491463i	$-0.163538\pi$
$787$	48.8635	1.74180	0.870899	+	0.491463i	$0.163538\pi$
$788$	25.1902	0.897365
$789$	0	0
$790$	21.0849	0.750166
$791$	−6.45854	−0.229639
$792$	0	0
$793$	−18.2363	−0.647590
$794$	−14.7646	−0.523977
$795$	0	0
$796$	24.5421	0.869871
$797$	−14.1065	−0.499680	−0.249840	−	0.968287i	$-0.580378\pi$
$797$	−14.1065	−0.499680	−0.249840	+	0.968287i	$0.580378\pi$
$798$	0	0
$799$	−8.80608	−0.311537
$800$	−2.23927	−0.0791700
$801$	0	0
$802$	15.1156	0.533752
$803$	−2.30835	−0.0814598
$804$	0	0
$805$	−1.49328	−0.0526311
$806$	−52.3099	−1.84254
$807$	0	0
$808$	−12.6357	−0.444524
$809$	14.7037	0.516953	0.258477	−	0.966018i	$-0.416780\pi$
$809$	14.7037	0.516953	0.258477	+	0.966018i	$0.416780\pi$
$810$	0	0
$811$	−8.12649	−0.285360	−0.142680	−	0.989769i	$-0.545572\pi$
$811$	−8.12649	−0.285360	−0.142680	+	0.989769i	$0.545572\pi$
$812$	−1.91718	−0.0672798
$813$	0	0
$814$	−11.2302	−0.393617
$815$	15.1926	0.532173
$816$	0	0
$817$	35.0506	1.22627
$818$	−16.7673	−0.586256
$819$	0	0
$820$	1.66155	0.0580237
$821$	20.8934	0.729185	0.364592	−	0.931167i	$-0.381208\pi$
$821$	20.8934	0.729185	0.364592	+	0.931167i	$0.381208\pi$
$822$	0	0
$823$	−29.2963	−1.02120	−0.510602	−	0.859817i	$-0.670577\pi$
$823$	−29.2963	−1.02120	−0.510602	+	0.859817i	$0.670577\pi$
$824$	−4.61471	−0.160761
$825$	0	0
$826$	2.75013	0.0956894
$827$	−42.6398	−1.48273	−0.741365	−	0.671102i	$-0.765821\pi$
$827$	−42.6398	−1.48273	−0.741365	+	0.671102i	$0.765821\pi$
$828$	0	0
$829$	−11.8692	−0.412235	−0.206117	−	0.978527i	$-0.566083\pi$
$829$	−11.8692	−0.412235	−0.206117	+	0.978527i	$0.566083\pi$
$830$	4.55562	0.158128
$831$	0	0
$832$	−5.30281	−0.183842
$833$	−37.2884	−1.29197
$834$	0	0
$835$	−37.5494	−1.29945
$836$	−7.15215	−0.247362
$837$	0	0
$838$	30.9245	1.06827
$839$	−12.3808	−0.427431	−0.213716	−	0.976896i	$-0.568557\pi$
$839$	−12.3808	−0.427431	−0.213716	+	0.976896i	$0.568557\pi$
$840$	0	0
$841$	−19.5771	−0.675073
$842$	−18.4812	−0.636905
$843$	0	0
$844$	5.67395	0.195305
$845$	25.1222	0.864231
$846$	0	0
$847$	5.99226	0.205896
$848$	10.8813	0.373664
$849$	0	0
$850$	−12.6323	−0.433284
$851$	13.6307	0.467254
$852$	0	0
$853$	23.6736	0.810568	0.405284	−	0.914191i	$-0.367173\pi$
$853$	23.6736	0.810568	0.405284	+	0.914191i	$0.367173\pi$
$854$	−2.14784	−0.0734976
$855$	0	0
$856$	−15.4738	−0.528882
$857$	−14.3623	−0.490607	−0.245303	−	0.969446i	$-0.578888\pi$
$857$	−14.3623	−0.490607	−0.245303	+	0.969446i	$0.578888\pi$
$858$	0	0
$859$	38.5591	1.31562	0.657810	−	0.753184i	$-0.271483\pi$
$859$	38.5591	1.31562	0.657810	+	0.753184i	$0.271483\pi$
$860$	9.65378	0.329191
$861$	0	0
$862$	−26.6535	−0.907823
$863$	12.8607	0.437783	0.218892	−	0.975749i	$-0.429756\pi$
$863$	12.8607	0.437783	0.218892	+	0.975749i	$0.429756\pi$
$864$	0	0
$865$	−31.2224	−1.06159
$866$	−31.6022	−1.07389
$867$	0	0
$868$	−6.16097	−0.209117
$869$	−15.0448	−0.510358
$870$	0	0
$871$	43.2175	1.46437
$872$	−9.23018	−0.312573
$873$	0	0
$874$	8.68096	0.293638
$875$	7.51239	0.253965
$876$	0	0
$877$	33.3161	1.12500	0.562502	−	0.826796i	$-0.309839\pi$
$877$	33.3161	1.12500	0.562502	+	0.826796i	$0.309839\pi$
$878$	−27.8868	−0.941134
$879$	0	0
$880$	−1.96987	−0.0664044
$881$	2.18138	0.0734925	0.0367462	−	0.999325i	$-0.488301\pi$
$881$	2.18138	0.0734925	0.0367462	+	0.999325i	$0.488301\pi$
$882$	0	0
$883$	−41.9639	−1.41220	−0.706099	−	0.708113i	$-0.749547\pi$
$883$	−41.9639	−1.41220	−0.706099	+	0.708113i	$0.749547\pi$
$884$	−29.9146	−1.00614
$885$	0	0
$886$	9.20646	0.309297
$887$	−31.4222	−1.05505	−0.527526	−	0.849539i	$-0.676880\pi$
$887$	−31.4222	−1.05505	−0.527526	+	0.849539i	$0.676880\pi$
$888$	0	0
$889$	−1.57946	−0.0529735
$890$	−20.1889	−0.676733
$891$	0	0
$892$	−10.2036	−0.341643
$893$	−9.41708	−0.315131
$894$	0	0
$895$	27.2785	0.911821
$896$	−0.624556	−0.0208649
$897$	0	0
$898$	25.8346	0.862111
$899$	30.2809	1.00993
$900$	0	0
$901$	61.3841	2.04500
$902$	−1.18557	−0.0394751
$903$	0	0
$904$	10.3410	0.343937
$905$	38.5098	1.28011
$906$	0	0
$907$	−29.5341	−0.980663	−0.490331	−	0.871536i	$-0.663124\pi$
$907$	−29.5341	−0.980663	−0.490331	+	0.871536i	$0.663124\pi$
$908$	−14.2233	−0.472017
$909$	0	0
$910$	5.50287	0.182418
$911$	45.9390	1.52203	0.761014	−	0.648736i	$-0.224702\pi$
$911$	45.9390	1.52203	0.761014	+	0.648736i	$0.224702\pi$
$912$	0	0
$913$	−3.25059	−0.107579
$914$	−24.8590	−0.822264
$915$	0	0
$916$	−20.0188	−0.661440
$917$	−10.4344	−0.344574
$918$	0	0
$919$	34.4360	1.13594	0.567970	−	0.823049i	$-0.307729\pi$
$919$	34.4360	1.13594	0.567970	+	0.823049i	$0.307729\pi$
$920$	2.39094	0.0788271
$921$	0	0
$922$	−3.22390	−0.106173
$923$	64.7135	2.13007
$924$	0	0
$925$	−21.2112	−0.697421
$926$	−16.6985	−0.548748
$927$	0	0
$928$	3.06967	0.100767
$929$	−23.2946	−0.764273	−0.382136	−	0.924106i	$-0.624812\pi$
$929$	−23.2946	−0.764273	−0.382136	+	0.924106i	$0.624812\pi$
$930$	0	0
$931$	−39.8756	−1.30687
$932$	−15.2622	−0.499932
$933$	0	0
$934$	−3.18752	−0.104299
$935$	−11.1126	−0.363420
$936$	0	0
$937$	9.30052	0.303835	0.151917	−	0.988393i	$-0.451455\pi$
$937$	9.30052	0.303835	0.151917	+	0.988393i	$0.451455\pi$
$938$	5.09008	0.166197
$939$	0	0
$940$	−2.59369	−0.0845969
$941$	10.9894	0.358245	0.179122	−	0.983827i	$-0.442674\pi$
$941$	10.9894	0.358245	0.179122	+	0.983827i	$0.442674\pi$
$942$	0	0
$943$	1.43899	0.0468599
$944$	−4.40334	−0.143317
$945$	0	0
$946$	−6.88829	−0.223958
$947$	−44.9162	−1.45958	−0.729791	−	0.683671i	$-0.760383\pi$
$947$	−44.9162	−1.45958	−0.729791	+	0.683671i	$0.760383\pi$
$948$	0	0
$949$	−10.3248	−0.335156
$950$	−13.5088	−0.438283
$951$	0	0
$952$	−3.52328	−0.114190
$953$	0.799194	0.0258884	0.0129442	−	0.999916i	$-0.495880\pi$
$953$	0.799194	0.0258884	0.0129442	+	0.999916i	$0.495880\pi$
$954$	0	0
$955$	19.1807	0.620673
$956$	−0.200801	−0.00649436
$957$	0	0
$958$	21.8671	0.706495
$959$	11.1481	0.359991
$960$	0	0
$961$	66.3095	2.13902
$962$	−50.2304	−1.61949
$963$	0	0
$964$	−17.8239	−0.574068
$965$	24.6585	0.793786
$966$	0	0
$967$	−22.8706	−0.735468	−0.367734	−	0.929931i	$-0.619866\pi$
$967$	−22.8706	−0.735468	−0.367734	+	0.929931i	$0.619866\pi$
$968$	−9.59443	−0.308377
$969$	0	0
$970$	7.43827	0.238828
$971$	0.825350	0.0264867	0.0132434	−	0.999912i	$-0.495784\pi$
$971$	0.825350	0.0264867	0.0132434	+	0.999912i	$0.495784\pi$
$972$	0	0
$973$	−0.726571	−0.0232928
$974$	20.0372	0.642033
$975$	0	0
$976$	3.43899	0.110079
$977$	−47.9034	−1.53256	−0.766282	−	0.642504i	$-0.777896\pi$
$977$	−47.9034	−1.53256	−0.766282	+	0.642504i	$0.777896\pi$
$978$	0	0
$979$	14.4054	0.460399
$980$	−10.9827	−0.350829
$981$	0	0
$982$	−1.00641	−0.0321158
$983$	−0.310883	−0.00991564	−0.00495782	−	0.999988i	$-0.501578\pi$
$983$	−0.310883	−0.00991564	−0.00495782	+	0.999988i	$0.501578\pi$
$984$	0	0
$985$	41.8547	1.33360
$986$	17.3168	0.551480
$987$	0	0
$988$	−31.9902	−1.01774
$989$	8.36070	0.265855
$990$	0	0
$991$	55.1840	1.75298	0.876488	−	0.481423i	$-0.159880\pi$
$991$	55.1840	1.75298	0.876488	+	0.481423i	$0.159880\pi$
$992$	9.86456	0.313200
$993$	0	0
$994$	7.62184	0.241750
$995$	40.7778	1.29274
$996$	0	0
$997$	54.5181	1.72660	0.863302	−	0.504687i	$-0.168392\pi$
$997$	54.5181	1.72660	0.863302	+	0.504687i	$0.168392\pi$
$998$	17.5170	0.554491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2214.2.a.t.1.4	yes	5
3.2	odd	2		2214.2.a.q.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2214.2.a.q.1.2	✓	5	3.2	odd	2
2214.2.a.t.1.4	yes	5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2214 = 2 \cdot 3^{3} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2214.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.66155 q^{5} -0.624556 q^{7} +1.00000 q^{8} +1.66155 q^{10} -1.18557 q^{11} -5.30281 q^{13} -0.624556 q^{14} +1.00000 q^{16} +5.64126 q^{17} +6.03268 q^{19} +1.66155 q^{20} -1.18557 q^{22} +1.43899 q^{23} -2.23927 q^{25} -5.30281 q^{26} -0.624556 q^{28} +3.06967 q^{29} +9.86456 q^{31} +1.00000 q^{32} +5.64126 q^{34} -1.03773 q^{35} +9.47241 q^{37} +6.03268 q^{38} +1.66155 q^{40} +1.00000 q^{41} +5.81012 q^{43} -1.18557 q^{44} +1.43899 q^{46} -1.56101 q^{47} -6.60993 q^{49} -2.23927 q^{50} -5.30281 q^{52} +10.8813 q^{53} -1.96987 q^{55} -0.624556 q^{56} +3.06967 q^{58} -4.40334 q^{59} +3.43899 q^{61} +9.86456 q^{62} +1.00000 q^{64} -8.81086 q^{65} -8.14992 q^{67} +5.64126 q^{68} -1.03773 q^{70} -12.2036 q^{71} +1.94704 q^{73} +9.47241 q^{74} +6.03268 q^{76} +0.740453 q^{77} +12.6899 q^{79} +1.66155 q^{80} +1.00000 q^{82} +2.74180 q^{83} +9.37322 q^{85} +5.81012 q^{86} -1.18557 q^{88} -12.1507 q^{89} +3.31190 q^{91} +1.43899 q^{92} -1.56101 q^{94} +10.0236 q^{95} +4.47672 q^{97} -6.60993 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 2 q^{7} + 5 q^{8} - q^{11} + 5 q^{13} + 2 q^{14} + 5 q^{16} + 5 q^{17} + 12 q^{19} - q^{22} + 7 q^{23} + 15 q^{25} + 5 q^{26} + 2 q^{28} - 6 q^{29} + 5 q^{32} + 5 q^{34} + 16 q^{35}+ \cdots + 15 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 2 * q^7 + 5 * q^8 - q^11 + 5 * q^13 + 2 * q^14 + 5 * q^16 + 5 * q^17 + 12 * q^19 - q^22 + 7 * q^23 + 15 * q^25 + 5 * q^26 + 2 * q^28 - 6 * q^29 + 5 * q^32 + 5 * q^34 + 16 * q^35 + 11 * q^37 + 12 * q^38 + 5 * q^41 + 19 * q^43 - q^44 + 7 * q^46 - 8 * q^47 + 15 * q^49 + 15 * q^50 + 5 * q^52 - 13 * q^53 + 21 * q^55 + 2 * q^56 - 6 * q^58 + 18 * q^59 + 17 * q^61 + 5 * q^64 - 16 * q^65 + 4 * q^67 + 5 * q^68 + 16 * q^70 + 5 * q^71 + 13 * q^73 + 11 * q^74 + 12 * q^76 + 15 * q^77 + 22 * q^79 + 5 * q^82 - 18 * q^83 - 24 * q^85 + 19 * q^86 - q^88 + 2 * q^89 + 27 * q^91 + 7 * q^92 - 8 * q^94 - 2 * q^95 + q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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