LMFDB - Embedded newform 800.2.f.e.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.2.f.e.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{3} -0.732051i q^{7} +4.46410 q^{9} -2.00000i q^{11} +3.46410 q^{13} +3.46410i q^{17} -0.535898i q^{19} -2.00000i q^{21} -6.19615i q^{23} +4.00000 q^{27} +6.92820i q^{29} +5.46410 q^{31} -5.46410i q^{33} -2.00000 q^{37} +9.46410 q^{39} +1.46410 q^{41} -5.26795 q^{43} -3.26795i q^{47} +6.46410 q^{49} +9.46410i q^{51} -11.4641 q^{53} -1.46410i q^{57} +7.46410i q^{59} +8.92820i q^{61} -3.26795i q^{63} -10.7321 q^{67} -16.9282i q^{69} -5.46410 q^{71} -7.46410i q^{73} -1.46410 q^{77} -1.07180 q^{79} -2.46410 q^{81} +1.26795 q^{83} +18.9282i q^{87} -8.92820 q^{89} -2.53590i q^{91} +14.9282 q^{93} -14.3923i q^{97} -8.92820i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 8 q^{31} - 8 q^{37} + 24 q^{39} - 8 q^{41} - 28 q^{43} + 12 q^{49} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 8 q^{77} - 32 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{89} + 32 q^{93}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 16 * q^27 + 8 * q^31 - 8 * q^37 + 24 * q^39 - 8 * q^41 - 28 * q^43 + 12 * q^49 - 32 * q^53 - 36 * q^67 - 8 * q^71 + 8 * q^77 - 32 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^89 + 32 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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	0
$2$	0	0
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.732051i	− 0.276689i	−0.990384	−	0.138345i	$-0.955822\pi$
$7$	− 0.732051i	− 0.276689i	0.990384	−	0.138345i	$-0.0441781\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$	3.46410i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$	0	0
$19$	− 0.535898i	− 0.122944i	−0.998109	−	0.0614718i	$-0.980421\pi$
$19$	− 0.535898i	− 0.122944i	0.998109	−	0.0614718i	$-0.0195794\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	− 6.19615i	− 1.29199i	−0.763343	−	0.645994i	$-0.776443\pi$
$23$	− 6.19615i	− 1.29199i	0.763343	−	0.645994i	$-0.223557\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	6.92820i	1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$	6.92820i	1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	0	0
$33$	− 5.46410i	− 0.951178i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	9.46410	1.51547
$40$	0	0
$41$	1.46410	0.228654	0.114327	−	0.993443i	$-0.463529\pi$
$41$	1.46410	0.228654	0.114327	+	0.993443i	$0.463529\pi$
$42$	0	0
$43$	−5.26795	−0.803355	−0.401677	−	0.915781i	$-0.631573\pi$
$43$	−5.26795	−0.803355	−0.401677	+	0.915781i	$0.631573\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.26795i	− 0.476679i	−0.971182	−	0.238340i	$-0.923397\pi$
$47$	− 3.26795i	− 0.476679i	0.971182	−	0.238340i	$-0.0766032\pi$
$48$	0	0
$49$	6.46410	0.923443
$50$	0	0
$51$	9.46410i	1.32524i
$52$	0	0
$53$	−11.4641	−1.57472	−0.787358	−	0.616496i	$-0.788551\pi$
$53$	−11.4641	−1.57472	−0.787358	+	0.616496i	$0.788551\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.46410i	− 0.193925i
$58$	0	0
$59$	7.46410i	0.971743i	0.874030	+	0.485872i	$0.161498\pi$
$59$	7.46410i	0.971743i	−0.874030	+	0.485872i	$0.838502\pi$
$60$	0	0
$61$	8.92820i	1.14314i	0.820554	+	0.571570i	$0.193665\pi$
$61$	8.92820i	1.14314i	−0.820554	+	0.571570i	$0.806335\pi$
$62$	0	0
$63$	− 3.26795i	− 0.411723i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.7321	−1.31113	−0.655564	−	0.755139i	$-0.727569\pi$
$67$	−10.7321	−1.31113	−0.655564	+	0.755139i	$0.727569\pi$
$68$	0	0
$69$	− 16.9282i	− 2.03792i
$70$	0	0
$71$	−5.46410	−0.648470	−0.324235	−	0.945977i	$-0.605107\pi$
$71$	−5.46410	−0.648470	−0.324235	+	0.945977i	$0.605107\pi$
$72$	0	0
$73$	− 7.46410i	− 0.873607i	−0.899557	−	0.436804i	$-0.856111\pi$
$73$	− 7.46410i	− 0.873607i	0.899557	−	0.436804i	$-0.143889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.46410	−0.166850
$78$	0	0
$79$	−1.07180	−0.120587	−0.0602933	−	0.998181i	$-0.519204\pi$
$79$	−1.07180	−0.120587	−0.0602933	+	0.998181i	$0.519204\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	1.26795	0.139176	0.0695878	−	0.997576i	$-0.477832\pi$
$83$	1.26795	0.139176	0.0695878	+	0.997576i	$0.477832\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.9282i	2.02932i
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	− 2.53590i	− 0.265834i
$92$	0	0
$93$	14.9282	1.54798
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.3923i	− 1.46132i	−0.682743	−	0.730659i	$-0.739213\pi$
$97$	− 14.3923i	− 1.46132i	0.682743	−	0.730659i	$-0.260787\pi$
$98$	0	0
$99$	− 8.92820i	− 0.897318i
$100$	0	0
$101$	− 2.92820i	− 0.291367i	−0.989331	−	0.145684i	$-0.953462\pi$
$101$	− 2.92820i	− 0.291367i	0.989331	−	0.145684i	$-0.0465381\pi$
$102$	0	0
$103$	15.6603i	1.54305i	0.636199	+	0.771525i	$0.280506\pi$
$103$	15.6603i	1.54305i	−0.636199	+	0.771525i	$0.719494\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.73205	−0.264117	−0.132059	−	0.991242i	$-0.542159\pi$
$107$	−2.73205	−0.264117	−0.132059	+	0.991242i	$0.542159\pi$
$108$	0	0
$109$	16.9282i	1.62143i	0.585443	+	0.810714i	$0.300921\pi$
$109$	16.9282i	1.62143i	−0.585443	+	0.810714i	$0.699079\pi$
$110$	0	0
$111$	−5.46410	−0.518630
$112$	0	0
$113$	12.9282i	1.21618i	0.793867	+	0.608092i	$0.208065\pi$
$113$	12.9282i	1.21618i	−0.793867	+	0.608092i	$0.791935\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	15.4641	1.42966
$118$	0	0
$119$	2.53590	0.232465
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16.7321i	− 1.48473i	−0.669996	−	0.742365i	$-0.733704\pi$
$127$	− 16.7321i	− 1.48473i	0.669996	−	0.742365i	$-0.266296\pi$
$128$	0	0
$129$	−14.3923	−1.26717
$130$	0	0
$131$	− 19.8564i	− 1.73486i	−0.497557	−	0.867431i	$-0.665770\pi$
$131$	− 19.8564i	− 1.73486i	0.497557	−	0.867431i	$-0.334230\pi$
$132$	0	0
$133$	−0.392305	−0.0340171
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.92820i	0.421045i	0.977589	+	0.210522i	$0.0675165\pi$
$137$	4.92820i	0.421045i	−0.977589	+	0.210522i	$0.932484\pi$
$138$	0	0
$139$	0.535898i	0.0454543i	0.999742	+	0.0227272i	$0.00723490\pi$
$139$	0.535898i	0.0454543i	−0.999742	+	0.0227272i	$0.992765\pi$
$140$	0	0
$141$	− 8.92820i	− 0.751890i
$142$	0	0
$143$	− 6.92820i	− 0.579365i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.6603	1.45659
$148$	0	0
$149$	− 7.85641i	− 0.643622i	−0.946804	−	0.321811i	$-0.895708\pi$
$149$	− 7.85641i	− 0.643622i	0.946804	−	0.321811i	$-0.104292\pi$
$150$	0	0
$151$	12.3923	1.00847	0.504236	−	0.863566i	$-0.331774\pi$
$151$	12.3923	1.00847	0.504236	+	0.863566i	$0.331774\pi$
$152$	0	0
$153$	15.4641i	1.25020i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.07180	−0.245156	−0.122578	−	0.992459i	$-0.539116\pi$
$157$	−3.07180	−0.245156	−0.122578	+	0.992459i	$0.539116\pi$
$158$	0	0
$159$	−31.3205	−2.48388
$160$	0	0
$161$	−4.53590	−0.357479
$162$	0	0
$163$	−0.196152	−0.0153638	−0.00768192	−	0.999970i	$-0.502445\pi$
$163$	−0.196152	−0.0153638	−0.00768192	+	0.999970i	$0.502445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.80385i	0.758645i	0.925265	+	0.379322i	$0.123843\pi$
$167$	9.80385i	0.758645i	−0.925265	+	0.379322i	$0.876157\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 2.39230i	− 0.182944i
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.3923i	1.53278i
$178$	0	0
$179$	8.53590i	0.638003i	0.947754	+	0.319002i	$0.103348\pi$
$179$	8.53590i	0.638003i	−0.947754	+	0.319002i	$0.896652\pi$
$180$	0	0
$181$	− 16.0000i	− 1.18927i	−0.803996	−	0.594635i	$-0.797296\pi$
$181$	− 16.0000i	− 1.18927i	0.803996	−	0.594635i	$-0.202704\pi$
$182$	0	0
$183$	24.3923i	1.80313i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.92820	0.506640
$188$	0	0
$189$	− 2.92820i	− 0.212995i
$190$	0	0
$191$	−15.3205	−1.10855	−0.554277	−	0.832333i	$-0.687005\pi$
$191$	−15.3205	−1.10855	−0.554277	+	0.832333i	$0.687005\pi$
$192$	0	0
$193$	− 0.535898i	− 0.0385748i	−0.999814	−	0.0192874i	$-0.993860\pi$
$193$	− 0.535898i	− 0.0385748i	0.999814	−	0.0192874i	$-0.00613975\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.4641	−1.38676	−0.693380	−	0.720572i	$-0.743879\pi$
$197$	−19.4641	−1.38676	−0.693380	+	0.720572i	$0.743879\pi$
$198$	0	0
$199$	−1.85641	−0.131597	−0.0657986	−	0.997833i	$-0.520959\pi$
$199$	−1.85641	−0.131597	−0.0657986	+	0.997833i	$0.520959\pi$
$200$	0	0
$201$	−29.3205	−2.06811
$202$	0	0
$203$	5.07180	0.355970
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 27.6603i	− 1.92252i
$208$	0	0
$209$	−1.07180	−0.0741377
$210$	0	0
$211$	26.7846i	1.84393i	0.387275	+	0.921964i	$0.373416\pi$
$211$	26.7846i	1.84393i	−0.387275	+	0.921964i	$0.626584\pi$
$212$	0	0
$213$	−14.9282	−1.02286
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	0	0
$219$	− 20.3923i	− 1.37798i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	− 5.80385i	− 0.388654i	−0.980937	−	0.194327i	$-0.937748\pi$
$223$	− 5.80385i	− 0.388654i	0.980937	−	0.194327i	$-0.0622523\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.0526	−0.667212	−0.333606	−	0.942713i	$-0.608265\pi$
$227$	−10.0526	−0.667212	−0.333606	+	0.942713i	$0.608265\pi$
$228$	0	0
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	5.32051i	0.348558i	0.984696	+	0.174279i	$0.0557595\pi$
$233$	5.32051i	0.348558i	−0.984696	+	0.174279i	$0.944241\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.92820	−0.190207
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	16.3923	1.05592	0.527961	−	0.849269i	$-0.322957\pi$
$241$	16.3923	1.05592	0.527961	+	0.849269i	$0.322957\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.85641i	− 0.118120i
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	24.9282i	1.57345i	0.617301	+	0.786727i	$0.288226\pi$
$251$	24.9282i	1.57345i	−0.617301	+	0.786727i	$0.711774\pi$
$252$	0	0
$253$	−12.3923	−0.779098
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	1.46410i	0.0909748i
$260$	0	0
$261$	30.9282i	1.91441i
$262$	0	0
$263$	− 11.6603i	− 0.719002i	−0.933145	−	0.359501i	$-0.882947\pi$
$263$	− 11.6603i	− 0.719002i	0.933145	−	0.359501i	$-0.117053\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.3923	−1.49278
$268$	0	0
$269$	− 8.92820i	− 0.544362i	−0.962246	−	0.272181i	$-0.912255\pi$
$269$	− 8.92820i	− 0.544362i	0.962246	−	0.272181i	$-0.0877450\pi$
$270$	0	0
$271$	19.3205	1.17364	0.586819	−	0.809718i	$-0.300380\pi$
$271$	19.3205	1.17364	0.586819	+	0.809718i	$0.300380\pi$
$272$	0	0
$273$	− 6.92820i	− 0.419314i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	24.3923	1.46033
$280$	0	0
$281$	10.5359	0.628519	0.314260	−	0.949337i	$-0.398244\pi$
$281$	10.5359	0.628519	0.314260	+	0.949337i	$0.398244\pi$
$282$	0	0
$283$	−9.66025	−0.574242	−0.287121	−	0.957894i	$-0.592698\pi$
$283$	−9.66025	−0.574242	−0.287121	+	0.957894i	$0.592698\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.07180i	− 0.0632662i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	− 39.3205i	− 2.30501i
$292$	0	0
$293$	15.8564	0.926341	0.463171	−	0.886269i	$-0.346712\pi$
$293$	15.8564	0.926341	0.463171	+	0.886269i	$0.346712\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 8.00000i	− 0.464207i
$298$	0	0
$299$	− 21.4641i	− 1.24130i
$300$	0	0
$301$	3.85641i	0.222280i
$302$	0	0
$303$	− 8.00000i	− 0.459588i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.9808	1.42573	0.712864	−	0.701303i	$-0.247398\pi$
$307$	24.9808	1.42573	0.712864	+	0.701303i	$0.247398\pi$
$308$	0	0
$309$	42.7846i	2.43393i
$310$	0	0
$311$	−31.3205	−1.77602	−0.888012	−	0.459821i	$-0.847914\pi$
$311$	−31.3205	−1.77602	−0.888012	+	0.459821i	$0.847914\pi$
$312$	0	0
$313$	4.14359i	0.234210i	0.993120	+	0.117105i	$0.0373614\pi$
$313$	4.14359i	0.234210i	−0.993120	+	0.117105i	$0.962639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.53590	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$317$	8.53590	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	−7.46410	−0.416606
$322$	0	0
$323$	1.85641	0.103293
$324$	0	0
$325$	0	0
$326$	0	0
$327$	46.2487i	2.55756i
$328$	0	0
$329$	−2.39230	−0.131892
$330$	0	0
$331$	− 14.0000i	− 0.769510i	−0.923019	−	0.384755i	$-0.874286\pi$
$331$	− 14.0000i	− 0.769510i	0.923019	−	0.384755i	$-0.125714\pi$
$332$	0	0
$333$	−8.92820	−0.489263
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 19.8564i	− 1.08165i	−0.841136	−	0.540824i	$-0.818113\pi$
$337$	− 19.8564i	− 1.08165i	0.841136	−	0.540824i	$-0.181887\pi$
$338$	0	0
$339$	35.3205i	1.91835i
$340$	0	0
$341$	− 10.9282i	− 0.591795i
$342$	0	0
$343$	− 9.85641i	− 0.532196i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.66025	0.0891271	0.0445636	−	0.999007i	$-0.485810\pi$
$347$	1.66025	0.0891271	0.0445636	+	0.999007i	$0.485810\pi$
$348$	0	0
$349$	− 28.0000i	− 1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	− 28.0000i	− 1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$	0	0
$351$	13.8564	0.739600
$352$	0	0
$353$	− 12.9282i	− 0.688099i	−0.938952	−	0.344049i	$-0.888201\pi$
$353$	− 12.9282i	− 0.688099i	0.938952	−	0.344049i	$-0.111799\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.92820	0.366679
$358$	0	0
$359$	18.9282	0.998992	0.499496	−	0.866316i	$-0.333518\pi$
$359$	18.9282	0.998992	0.499496	+	0.866316i	$0.333518\pi$
$360$	0	0
$361$	18.7128	0.984885
$362$	0	0
$363$	19.1244	1.00377
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.87564i	− 0.150107i	−0.997179	−	0.0750537i	$-0.976087\pi$
$367$	− 2.87564i	− 0.150107i	0.997179	−	0.0750537i	$-0.0239128\pi$
$368$	0	0
$369$	6.53590	0.340245
$370$	0	0
$371$	8.39230i	0.435707i
$372$	0	0
$373$	25.7128	1.33136	0.665679	−	0.746238i	$-0.268142\pi$
$373$	25.7128	1.33136	0.665679	+	0.746238i	$0.268142\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	36.2487i	1.86197i	0.365056	+	0.930986i	$0.381050\pi$
$379$	36.2487i	1.86197i	−0.365056	+	0.930986i	$0.618950\pi$
$380$	0	0
$381$	− 45.7128i	− 2.34194i
$382$	0	0
$383$	21.1244i	1.07940i	0.841856	+	0.539702i	$0.181463\pi$
$383$	21.1244i	1.07940i	−0.841856	+	0.539702i	$0.818537\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−23.5167	−1.19542
$388$	0	0
$389$	6.78461i	0.343993i	0.985098	+	0.171997i	$0.0550218\pi$
$389$	6.78461i	0.343993i	−0.985098	+	0.171997i	$0.944978\pi$
$390$	0	0
$391$	21.4641	1.08549
$392$	0	0
$393$	− 54.2487i	− 2.73649i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.2487	1.61852	0.809258	−	0.587453i	$-0.199869\pi$
$397$	32.2487	1.61852	0.809258	+	0.587453i	$0.199869\pi$
$398$	0	0
$399$	−1.07180	−0.0536570
$400$	0	0
$401$	−7.85641	−0.392330	−0.196165	−	0.980571i	$-0.562849\pi$
$401$	−7.85641	−0.392330	−0.196165	+	0.980571i	$0.562849\pi$
$402$	0	0
$403$	18.9282	0.942881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	11.3205	0.559763	0.279882	−	0.960035i	$-0.409705\pi$
$409$	11.3205	0.559763	0.279882	+	0.960035i	$0.409705\pi$
$410$	0	0
$411$	13.4641i	0.664135i
$412$	0	0
$413$	5.46410	0.268871
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.46410i	0.0716974i
$418$	0	0
$419$	− 18.3923i	− 0.898523i	−0.893400	−	0.449261i	$-0.851687\pi$
$419$	− 18.3923i	− 0.898523i	0.893400	−	0.449261i	$-0.148313\pi$
$420$	0	0
$421$	− 0.143594i	− 0.00699832i	−0.999994	−	0.00349916i	$-0.998886\pi$
$421$	− 0.143594i	− 0.00699832i	0.999994	−	0.00349916i	$-0.00111382\pi$
$422$	0	0
$423$	− 14.5885i	− 0.709315i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.53590	0.316294
$428$	0	0
$429$	− 18.9282i	− 0.913862i
$430$	0	0
$431$	21.4641	1.03389	0.516945	−	0.856019i	$-0.327069\pi$
$431$	21.4641	1.03389	0.516945	+	0.856019i	$0.327069\pi$
$432$	0	0
$433$	19.4641i	0.935385i	0.883891	+	0.467693i	$0.154915\pi$
$433$	19.4641i	0.935385i	−0.883891	+	0.467693i	$0.845085\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.32051	−0.158841
$438$	0	0
$439$	40.7846	1.94654	0.973272	−	0.229657i	$-0.0737605\pi$
$439$	40.7846	1.94654	0.973272	+	0.229657i	$0.0737605\pi$
$440$	0	0
$441$	28.8564	1.37411
$442$	0	0
$443$	20.9808	0.996826	0.498413	−	0.866940i	$-0.333916\pi$
$443$	20.9808	0.996826	0.498413	+	0.866940i	$0.333916\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 21.4641i	− 1.01522i
$448$	0	0
$449$	23.3205	1.10056	0.550281	−	0.834979i	$-0.314520\pi$
$449$	23.3205	1.10056	0.550281	+	0.834979i	$0.314520\pi$
$450$	0	0
$451$	− 2.92820i	− 0.137884i
$452$	0	0
$453$	33.8564	1.59071
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.7846i	− 1.25293i	−0.779449	−	0.626466i	$-0.784501\pi$
$457$	− 26.7846i	− 1.25293i	0.779449	−	0.626466i	$-0.215499\pi$
$458$	0	0
$459$	13.8564i	0.646762i
$460$	0	0
$461$	10.9282i	0.508977i	0.967076	+	0.254489i	$0.0819071\pi$
$461$	10.9282i	0.508977i	−0.967076	+	0.254489i	$0.918093\pi$
$462$	0	0
$463$	− 11.2679i	− 0.523666i	−0.965113	−	0.261833i	$-0.915673\pi$
$463$	− 11.2679i	− 0.523666i	0.965113	−	0.261833i	$-0.0843270\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.6603	1.18741	0.593707	−	0.804681i	$-0.297664\pi$
$467$	25.6603	1.18741	0.593707	+	0.804681i	$0.297664\pi$
$468$	0	0
$469$	7.85641i	0.362775i
$470$	0	0
$471$	−8.39230	−0.386697
$472$	0	0
$473$	10.5359i	0.484441i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−51.1769	−2.34323
$478$	0	0
$479$	5.85641	0.267586	0.133793	−	0.991009i	$-0.457284\pi$
$479$	5.85641	0.267586	0.133793	+	0.991009i	$0.457284\pi$
$480$	0	0
$481$	−6.92820	−0.315899
$482$	0	0
$483$	−12.3923	−0.563869
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.58846i	− 0.298551i	−0.988796	−	0.149276i	$-0.952306\pi$
$487$	− 6.58846i	− 0.298551i	0.988796	−	0.149276i	$-0.0476942\pi$
$488$	0	0
$489$	−0.535898	−0.0242342
$490$	0	0
$491$	− 16.9282i	− 0.763959i	−0.924171	−	0.381980i	$-0.875242\pi$
$491$	− 16.9282i	− 0.763959i	0.924171	−	0.381980i	$-0.124758\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.00000i	0.179425i
$498$	0	0
$499$	− 31.4641i	− 1.40853i	−0.709939	−	0.704263i	$-0.751277\pi$
$499$	− 31.4641i	− 1.40853i	0.709939	−	0.704263i	$-0.248723\pi$
$500$	0	0
$501$	26.7846i	1.19665i
$502$	0	0
$503$	− 0.339746i	− 0.0151485i	−0.999971	−	0.00757426i	$-0.997589\pi$
$503$	− 0.339746i	− 0.0151485i	0.999971	−	0.00757426i	$-0.00241099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.73205	−0.121335
$508$	0	0
$509$	− 1.85641i	− 0.0822838i	−0.999153	−	0.0411419i	$-0.986900\pi$
$509$	− 1.85641i	− 0.0822838i	0.999153	−	0.0411419i	$-0.0130996\pi$
$510$	0	0
$511$	−5.46410	−0.241718
$512$	0	0
$513$	− 2.14359i	− 0.0946420i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.53590	−0.287448
$518$	0	0
$519$	−5.46410	−0.239847
$520$	0	0
$521$	−43.8564	−1.92138	−0.960692	−	0.277616i	$-0.910456\pi$
$521$	−43.8564	−1.92138	−0.960692	+	0.277616i	$0.910456\pi$
$522$	0	0
$523$	−11.8038	−0.516146	−0.258073	−	0.966125i	$-0.583088\pi$
$523$	−11.8038	−0.516146	−0.258073	+	0.966125i	$0.583088\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.9282i	0.824525i
$528$	0	0
$529$	−15.3923	−0.669231
$530$	0	0
$531$	33.3205i	1.44599i
$532$	0	0
$533$	5.07180	0.219684
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3205i	1.00635i
$538$	0	0
$539$	− 12.9282i	− 0.556857i
$540$	0	0
$541$	26.9282i	1.15773i	0.815422	+	0.578867i	$0.196505\pi$
$541$	26.9282i	1.15773i	−0.815422	+	0.578867i	$0.803495\pi$
$542$	0	0
$543$	− 43.7128i	− 1.87590i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−33.2679	−1.42243	−0.711217	−	0.702972i	$-0.751856\pi$
$547$	−33.2679	−1.42243	−0.711217	+	0.702972i	$0.751856\pi$
$548$	0	0
$549$	39.8564i	1.70103i
$550$	0	0
$551$	3.71281	0.158171
$552$	0	0
$553$	0.784610i	0.0333650i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7846	0.626444	0.313222	−	0.949680i	$-0.398592\pi$
$557$	14.7846	0.626444	0.313222	+	0.949680i	$0.398592\pi$
$558$	0	0
$559$	−18.2487	−0.771838
$560$	0	0
$561$	18.9282	0.799149
$562$	0	0
$563$	−22.0526	−0.929405	−0.464702	−	0.885467i	$-0.653839\pi$
$563$	−22.0526	−0.929405	−0.464702	+	0.885467i	$0.653839\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.80385i	0.0757545i
$568$	0	0
$569$	13.4641	0.564445	0.282222	−	0.959349i	$-0.408928\pi$
$569$	13.4641	0.564445	0.282222	+	0.959349i	$0.408928\pi$
$570$	0	0
$571$	6.78461i	0.283927i	0.989872	+	0.141964i	$0.0453416\pi$
$571$	6.78461i	0.283927i	−0.989872	+	0.141964i	$0.954658\pi$
$572$	0	0
$573$	−41.8564	−1.74858
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.5692i	1.64729i	0.567107	+	0.823644i	$0.308063\pi$
$577$	39.5692i	1.64729i	−0.567107	+	0.823644i	$0.691937\pi$
$578$	0	0
$579$	− 1.46410i	− 0.0608460i
$580$	0	0
$581$	− 0.928203i	− 0.0385084i
$582$	0	0
$583$	22.9282i	0.949589i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.80385	0.157002	0.0785008	−	0.996914i	$-0.474987\pi$
$587$	3.80385	0.157002	0.0785008	+	0.996914i	$0.474987\pi$
$588$	0	0
$589$	− 2.92820i	− 0.120655i
$590$	0	0
$591$	−53.1769	−2.18741
$592$	0	0
$593$	− 32.6410i	− 1.34041i	−0.742178	−	0.670203i	$-0.766207\pi$
$593$	− 32.6410i	− 1.34041i	0.742178	−	0.670203i	$-0.233793\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.07180	−0.207575
$598$	0	0
$599$	−34.6410	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$599$	−34.6410	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$600$	0	0
$601$	18.5359	0.756095	0.378048	−	0.925786i	$-0.376596\pi$
$601$	18.5359	0.756095	0.378048	+	0.925786i	$0.376596\pi$
$602$	0	0
$603$	−47.9090	−1.95100
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 30.9808i	− 1.25747i	−0.777619	−	0.628735i	$-0.783573\pi$
$607$	− 30.9808i	− 1.25747i	0.777619	−	0.628735i	$-0.216427\pi$
$608$	0	0
$609$	13.8564	0.561490
$610$	0	0
$611$	− 11.3205i	− 0.457979i
$612$	0	0
$613$	−26.3923	−1.06598	−0.532988	−	0.846123i	$-0.678931\pi$
$613$	−26.3923	−1.06598	−0.532988	+	0.846123i	$0.678931\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.5359i	− 0.826744i	−0.910562	−	0.413372i	$-0.864351\pi$
$617$	− 20.5359i	− 0.826744i	0.910562	−	0.413372i	$-0.135649\pi$
$618$	0	0
$619$	− 1.32051i	− 0.0530757i	−0.999648	−	0.0265379i	$-0.991552\pi$
$619$	− 1.32051i	− 0.0530757i	0.999648	−	0.0265379i	$-0.00844825\pi$
$620$	0	0
$621$	− 24.7846i	− 0.994572i
$622$	0	0
$623$	6.53590i	0.261855i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.92820	−0.116941
$628$	0	0
$629$	− 6.92820i	− 0.276246i
$630$	0	0
$631$	23.3205	0.928375	0.464187	−	0.885737i	$-0.346346\pi$
$631$	23.3205	0.928375	0.464187	+	0.885737i	$0.346346\pi$
$632$	0	0
$633$	73.1769i	2.90852i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.3923	0.887215
$638$	0	0
$639$	−24.3923	−0.964945
$640$	0	0
$641$	0.392305	0.0154951	0.00774755	−	0.999970i	$-0.497534\pi$
$641$	0.392305	0.0154951	0.00774755	+	0.999970i	$0.497534\pi$
$642$	0	0
$643$	39.1244	1.54291	0.771457	−	0.636281i	$-0.219528\pi$
$643$	39.1244	1.54291	0.771457	+	0.636281i	$0.219528\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.7321i	− 0.657805i	−0.944364	−	0.328902i	$-0.893321\pi$
$647$	− 16.7321i	− 0.657805i	0.944364	−	0.328902i	$-0.106679\pi$
$648$	0	0
$649$	14.9282	0.585983
$650$	0	0
$651$	− 10.9282i	− 0.428310i
$652$	0	0
$653$	−12.2487	−0.479329	−0.239665	−	0.970856i	$-0.577037\pi$
$653$	−12.2487	−0.479329	−0.239665	+	0.970856i	$0.577037\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 33.3205i	− 1.29996i
$658$	0	0
$659$	17.3205i	0.674711i	0.941377	+	0.337356i	$0.109532\pi$
$659$	17.3205i	0.674711i	−0.941377	+	0.337356i	$0.890468\pi$
$660$	0	0
$661$	8.14359i	0.316749i	0.987379	+	0.158375i	$0.0506253\pi$
$661$	8.14359i	0.316749i	−0.987379	+	0.158375i	$0.949375\pi$
$662$	0	0
$663$	32.7846i	1.27325i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42.9282	1.66219
$668$	0	0
$669$	− 15.8564i	− 0.613044i
$670$	0	0
$671$	17.8564	0.689339
$672$	0	0
$673$	− 12.5359i	− 0.483223i	−0.970373	−	0.241612i	$-0.922324\pi$
$673$	− 12.5359i	− 0.483223i	0.970373	−	0.241612i	$-0.0776760\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.6077	0.676719	0.338359	−	0.941017i	$-0.390128\pi$
$677$	17.6077	0.676719	0.338359	+	0.941017i	$0.390128\pi$
$678$	0	0
$679$	−10.5359	−0.404331
$680$	0	0
$681$	−27.4641	−1.05243
$682$	0	0
$683$	−16.9808	−0.649751	−0.324875	−	0.945757i	$-0.605322\pi$
$683$	−16.9808	−0.649751	−0.324875	+	0.945757i	$0.605322\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10.9282i	− 0.416937i
$688$	0	0
$689$	−39.7128	−1.51294
$690$	0	0
$691$	18.0000i	0.684752i	0.939563	+	0.342376i	$0.111232\pi$
$691$	18.0000i	0.684752i	−0.939563	+	0.342376i	$0.888768\pi$
$692$	0	0
$693$	−6.53590	−0.248278
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.07180i	0.192108i
$698$	0	0
$699$	14.5359i	0.549798i
$700$	0	0
$701$	− 19.0718i	− 0.720332i	−0.932888	−	0.360166i	$-0.882720\pi$
$701$	− 19.0718i	− 0.720332i	0.932888	−	0.360166i	$-0.117280\pi$
$702$	0	0
$703$	1.07180i	0.0404236i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.14359	−0.0806181
$708$	0	0
$709$	− 12.7846i	− 0.480136i	−0.970756	−	0.240068i	$-0.922830\pi$
$709$	− 12.7846i	− 0.480136i	0.970756	−	0.240068i	$-0.0771698\pi$
$710$	0	0
$711$	−4.78461	−0.179437
$712$	0	0
$713$	− 33.8564i	− 1.26793i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−54.6410	−2.04061
$718$	0	0
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	0	0
$721$	11.4641	0.426945
$722$	0	0
$723$	44.7846	1.66556
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.0526i	0.892060i	0.895018	+	0.446030i	$0.147163\pi$
$727$	24.0526i	0.892060i	−0.895018	+	0.446030i	$0.852837\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	− 18.2487i	− 0.674953i
$732$	0	0
$733$	35.0718	1.29541	0.647703	−	0.761893i	$-0.275730\pi$
$733$	35.0718	1.29541	0.647703	+	0.761893i	$0.275730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.4641i	0.790640i
$738$	0	0
$739$	− 29.3205i	− 1.07857i	−0.842123	−	0.539286i	$-0.818694\pi$
$739$	− 29.3205i	− 1.07857i	0.842123	−	0.539286i	$-0.181306\pi$
$740$	0	0
$741$	− 5.07180i	− 0.186317i
$742$	0	0
$743$	10.9808i	0.402845i	0.979504	+	0.201423i	$0.0645564\pi$
$743$	10.9808i	0.402845i	−0.979504	+	0.201423i	$0.935444\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.66025	0.207098
$748$	0	0
$749$	2.00000i	0.0730784i
$750$	0	0
$751$	−26.2487	−0.957829	−0.478915	−	0.877862i	$-0.658970\pi$
$751$	−26.2487	−0.957829	−0.478915	+	0.877862i	$0.658970\pi$
$752$	0	0
$753$	68.1051i	2.48189i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19.0718	−0.693176	−0.346588	−	0.938017i	$-0.612660\pi$
$757$	−19.0718	−0.693176	−0.346588	+	0.938017i	$0.612660\pi$
$758$	0	0
$759$	−33.8564	−1.22891
$760$	0	0
$761$	−5.71281	−0.207089	−0.103545	−	0.994625i	$-0.533018\pi$
$761$	−5.71281	−0.207089	−0.103545	+	0.994625i	$0.533018\pi$
$762$	0	0
$763$	12.3923	0.448632
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.8564i	0.933621i
$768$	0	0
$769$	−12.9282	−0.466203	−0.233101	−	0.972452i	$-0.574887\pi$
$769$	−12.9282	−0.466203	−0.233101	+	0.972452i	$0.574887\pi$
$770$	0	0
$771$	− 5.46410i	− 0.196785i
$772$	0	0
$773$	22.3923	0.805395	0.402698	−	0.915333i	$-0.368073\pi$
$773$	22.3923	0.805395	0.402698	+	0.915333i	$0.368073\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000i	0.143499i
$778$	0	0
$779$	− 0.784610i	− 0.0281116i
$780$	0	0
$781$	10.9282i	0.391042i
$782$	0	0
$783$	27.7128i	0.990375i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.5885	−0.591315	−0.295657	−	0.955294i	$-0.595539\pi$
$787$	−16.5885	−0.591315	−0.295657	+	0.955294i	$0.595539\pi$
$788$	0	0
$789$	− 31.8564i	− 1.13412i
$790$	0	0
$791$	9.46410	0.336505
$792$	0	0
$793$	30.9282i	1.09829i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.1051	1.77481	0.887407	−	0.460986i	$-0.152504\pi$
$797$	50.1051	1.77481	0.887407	+	0.460986i	$0.152504\pi$
$798$	0	0
$799$	11.3205	0.400491
$800$	0	0
$801$	−39.8564	−1.40826
$802$	0	0
$803$	−14.9282	−0.526805
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 24.3923i	− 0.858650i
$808$	0	0
$809$	−23.8564	−0.838747	−0.419373	−	0.907814i	$-0.637750\pi$
$809$	−23.8564	−0.838747	−0.419373	+	0.907814i	$0.637750\pi$
$810$	0	0
$811$	− 28.9282i	− 1.01581i	−0.861414	−	0.507903i	$-0.830421\pi$
$811$	− 28.9282i	− 1.01581i	0.861414	−	0.507903i	$-0.169579\pi$
$812$	0	0
$813$	52.7846	1.85124
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.82309i	0.0987673i
$818$	0	0
$819$	− 11.3205i	− 0.395571i
$820$	0	0
$821$	− 34.7846i	− 1.21399i	−0.794705	−	0.606996i	$-0.792375\pi$
$821$	− 34.7846i	− 1.21399i	0.794705	−	0.606996i	$-0.207625\pi$
$822$	0	0
$823$	− 9.12436i	− 0.318055i	−0.987274	−	0.159028i	$-0.949164\pi$
$823$	− 9.12436i	− 0.318055i	0.987274	−	0.159028i	$-0.0508359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.1244	−0.804113	−0.402056	−	0.915615i	$-0.631704\pi$
$827$	−23.1244	−0.804113	−0.402056	+	0.915615i	$0.631704\pi$
$828$	0	0
$829$	28.9282i	1.00472i	0.864659	+	0.502359i	$0.167534\pi$
$829$	28.9282i	1.00472i	−0.864659	+	0.502359i	$0.832466\pi$
$830$	0	0
$831$	5.46410	0.189548
$832$	0	0
$833$	22.3923i	0.775847i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21.8564	0.755468
$838$	0	0
$839$	24.7846	0.855660	0.427830	−	0.903859i	$-0.359278\pi$
$839$	24.7846	0.855660	0.427830	+	0.903859i	$0.359278\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	28.7846	0.991395
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.12436i	− 0.176075i
$848$	0	0
$849$	−26.3923	−0.905782
$850$	0	0
$851$	12.3923i	0.424803i
$852$	0	0
$853$	21.6077	0.739833	0.369917	−	0.929065i	$-0.379386\pi$
$853$	21.6077	0.739833	0.369917	+	0.929065i	$0.379386\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19.8564i	− 0.678282i	−0.940736	−	0.339141i	$-0.889864\pi$
$857$	− 19.8564i	− 0.678282i	0.940736	−	0.339141i	$-0.110136\pi$
$858$	0	0
$859$	− 28.2487i	− 0.963834i	−0.876217	−	0.481917i	$-0.839941\pi$
$859$	− 28.2487i	− 0.963834i	0.876217	−	0.481917i	$-0.160059\pi$
$860$	0	0
$861$	− 2.92820i	− 0.0997929i
$862$	0	0
$863$	47.6603i	1.62237i	0.584787	+	0.811187i	$0.301178\pi$
$863$	47.6603i	1.62237i	−0.584787	+	0.811187i	$0.698822\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.6603	0.463927
$868$	0	0
$869$	2.14359i	0.0727164i
$870$	0	0
$871$	−37.1769	−1.25969
$872$	0	0
$873$	− 64.2487i	− 2.17449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.71281	−0.0578376	−0.0289188	−	0.999582i	$-0.509206\pi$
$877$	−1.71281	−0.0578376	−0.0289188	+	0.999582i	$0.509206\pi$
$878$	0	0
$879$	43.3205	1.46116
$880$	0	0
$881$	9.46410	0.318854	0.159427	−	0.987210i	$-0.449035\pi$
$881$	9.46410	0.318854	0.159427	+	0.987210i	$0.449035\pi$
$882$	0	0
$883$	−27.9090	−0.939211	−0.469606	−	0.882876i	$-0.655604\pi$
$883$	−27.9090	−0.939211	−0.469606	+	0.882876i	$0.655604\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.9090i	0.467017i	0.972355	+	0.233509i	$0.0750207\pi$
$887$	13.9090i	0.467017i	−0.972355	+	0.233509i	$0.924979\pi$
$888$	0	0
$889$	−12.2487	−0.410809
$890$	0	0
$891$	4.92820i	0.165101i
$892$	0	0
$893$	−1.75129	−0.0586046
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 58.6410i	− 1.95797i
$898$	0	0
$899$	37.8564i	1.26258i
$900$	0	0
$901$	− 39.7128i	− 1.32303i
$902$	0	0
$903$	10.5359i	0.350613i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.87564	−0.161893	−0.0809466	−	0.996718i	$-0.525794\pi$
$907$	−4.87564	−0.161893	−0.0809466	+	0.996718i	$0.525794\pi$
$908$	0	0
$909$	− 13.0718i	− 0.433564i
$910$	0	0
$911$	49.1769	1.62930	0.814652	−	0.579950i	$-0.196928\pi$
$911$	49.1769	1.62930	0.814652	+	0.579950i	$0.196928\pi$
$912$	0	0
$913$	− 2.53590i	− 0.0839260i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.5359	−0.480018
$918$	0	0
$919$	−38.9282	−1.28412	−0.642061	−	0.766653i	$-0.721921\pi$
$919$	−38.9282	−1.28412	−0.642061	+	0.766653i	$0.721921\pi$
$920$	0	0
$921$	68.2487	2.24887
$922$	0	0
$923$	−18.9282	−0.623029
$924$	0	0
$925$	0	0
$926$	0	0
$927$	69.9090i	2.29611i
$928$	0	0
$929$	17.4641	0.572979	0.286489	−	0.958083i	$-0.407512\pi$
$929$	17.4641	0.572979	0.286489	+	0.958083i	$0.407512\pi$
$930$	0	0
$931$	− 3.46410i	− 0.113531i
$932$	0	0
$933$	−85.5692	−2.80141
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.24871i	0.138799i	0.997589	+	0.0693997i	$0.0221084\pi$
$937$	4.24871i	0.138799i	−0.997589	+	0.0693997i	$0.977892\pi$
$938$	0	0
$939$	11.3205i	0.369431i
$940$	0	0
$941$	32.0000i	1.04317i	0.853199	+	0.521585i	$0.174659\pi$
$941$	32.0000i	1.04317i	−0.853199	+	0.521585i	$0.825341\pi$
$942$	0	0
$943$	− 9.07180i	− 0.295418i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.12436	0.101528	0.0507640	−	0.998711i	$-0.483834\pi$
$947$	3.12436	0.101528	0.0507640	+	0.998711i	$0.483834\pi$
$948$	0	0
$949$	− 25.8564i	− 0.839334i
$950$	0	0
$951$	23.3205	0.756219
$952$	0	0
$953$	17.2154i	0.557661i	0.960340	+	0.278831i	$0.0899468\pi$
$953$	17.2154i	0.557661i	−0.960340	+	0.278831i	$0.910053\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	37.8564	1.22372
$958$	0	0
$959$	3.60770	0.116499
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	−12.1962	−0.393016
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 16.3397i	− 0.525451i	−0.964871	−	0.262725i	$-0.915379\pi$
$967$	− 16.3397i	− 0.525451i	0.964871	−	0.262725i	$-0.0846213\pi$
$968$	0	0
$969$	5.07180	0.162930
$970$	0	0
$971$	36.9282i	1.18508i	0.805540	+	0.592541i	$0.201875\pi$
$971$	36.9282i	1.18508i	−0.805540	+	0.592541i	$0.798125\pi$
$972$	0	0
$973$	0.392305	0.0125767
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.5359i	− 0.784973i	−0.919758	−	0.392486i	$-0.871615\pi$
$977$	− 24.5359i	− 0.784973i	0.919758	−	0.392486i	$-0.128385\pi$
$978$	0	0
$979$	17.8564i	0.570693i
$980$	0	0
$981$	75.5692i	2.41274i
$982$	0	0
$983$	− 48.7321i	− 1.55431i	−0.629309	−	0.777156i	$-0.716662\pi$
$983$	− 48.7321i	− 1.55431i	0.629309	−	0.777156i	$-0.283338\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.53590	−0.208040
$988$	0	0
$989$	32.6410i	1.03792i
$990$	0	0
$991$	−41.4641	−1.31715	−0.658575	−	0.752515i	$-0.728841\pi$
$991$	−41.4641	−1.31715	−0.658575	+	0.752515i	$0.728841\pi$
$992$	0	0
$993$	− 38.2487i	− 1.21379i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.1769	−0.353976	−0.176988	−	0.984213i	$-0.556635\pi$
$997$	−11.1769	−0.353976	−0.176988	+	0.984213i	$0.556635\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.f.e.49.3		4
3.2	odd	2		7200.2.d.n.2449.2		4
4.3	odd	2		200.2.f.e.149.1		4
5.2	odd	4		800.2.d.e.401.1		4
5.3	odd	4		160.2.d.a.81.4		4
5.4	even	2		800.2.f.c.49.2		4
8.3	odd	2		200.2.f.c.149.3		4
8.5	even	2		800.2.f.c.49.1		4
12.11	even	2		1800.2.d.l.1549.4		4
15.2	even	4		7200.2.k.j.3601.4		4
15.8	even	4		1440.2.k.e.721.1		4
15.14	odd	2		7200.2.d.o.2449.3		4
20.3	even	4		40.2.d.a.21.2	yes	4
20.7	even	4		200.2.d.f.101.3		4
20.19	odd	2		200.2.f.c.149.4		4
24.5	odd	2		7200.2.d.o.2449.2		4
24.11	even	2		1800.2.d.p.1549.2		4
40.3	even	4		40.2.d.a.21.1	✓	4
40.13	odd	4		160.2.d.a.81.1		4
40.19	odd	2		200.2.f.e.149.2		4
40.27	even	4		200.2.d.f.101.4		4
40.29	even	2	inner	800.2.f.e.49.4		4
40.37	odd	4		800.2.d.e.401.4		4
60.23	odd	4		360.2.k.e.181.3		4
60.47	odd	4		1800.2.k.j.901.2		4
60.59	even	2		1800.2.d.p.1549.1		4
80.3	even	4		1280.2.a.o.1.2		2
80.13	odd	4		1280.2.a.d.1.1		2
80.27	even	4		6400.2.a.ce.1.2		2
80.37	odd	4		6400.2.a.be.1.1		2
80.43	even	4		1280.2.a.a.1.1		2
80.53	odd	4		1280.2.a.n.1.2		2
80.67	even	4		6400.2.a.z.1.1		2
80.77	odd	4		6400.2.a.cj.1.2		2
120.29	odd	2		7200.2.d.n.2449.3		4
120.53	even	4		1440.2.k.e.721.3		4
120.59	even	2		1800.2.d.l.1549.3		4
120.77	even	4		7200.2.k.j.3601.3		4
120.83	odd	4		360.2.k.e.181.4		4
120.107	odd	4		1800.2.k.j.901.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.d.a.21.1	✓	4	40.3	even	4
40.2.d.a.21.2	yes	4	20.3	even	4
160.2.d.a.81.1		4	40.13	odd	4
160.2.d.a.81.4		4	5.3	odd	4
200.2.d.f.101.3		4	20.7	even	4
200.2.d.f.101.4		4	40.27	even	4
200.2.f.c.149.3		4	8.3	odd	2
200.2.f.c.149.4		4	20.19	odd	2
200.2.f.e.149.1		4	4.3	odd	2
200.2.f.e.149.2		4	40.19	odd	2
360.2.k.e.181.3		4	60.23	odd	4
360.2.k.e.181.4		4	120.83	odd	4
800.2.d.e.401.1		4	5.2	odd	4
800.2.d.e.401.4		4	40.37	odd	4
800.2.f.c.49.1		4	8.5	even	2
800.2.f.c.49.2		4	5.4	even	2
800.2.f.e.49.3		4	1.1	even	1	trivial
800.2.f.e.49.4		4	40.29	even	2	inner
1280.2.a.a.1.1		2	80.43	even	4
1280.2.a.d.1.1		2	80.13	odd	4
1280.2.a.n.1.2		2	80.53	odd	4
1280.2.a.o.1.2		2	80.3	even	4
1440.2.k.e.721.1		4	15.8	even	4
1440.2.k.e.721.3		4	120.53	even	4
1800.2.d.l.1549.3		4	120.59	even	2
1800.2.d.l.1549.4		4	12.11	even	2
1800.2.d.p.1549.1		4	60.59	even	2
1800.2.d.p.1549.2		4	24.11	even	2
1800.2.k.j.901.1		4	120.107	odd	4
1800.2.k.j.901.2		4	60.47	odd	4
6400.2.a.z.1.1		2	80.67	even	4
6400.2.a.be.1.1		2	80.37	odd	4
6400.2.a.ce.1.2		2	80.27	even	4
6400.2.a.cj.1.2		2	80.77	odd	4
7200.2.d.n.2449.2		4	3.2	odd	2
7200.2.d.n.2449.3		4	120.29	odd	2
7200.2.d.o.2449.2		4	24.5	odd	2
7200.2.d.o.2449.3		4	15.14	odd	2
7200.2.k.j.3601.3		4	120.77	even	4
7200.2.k.j.3601.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} -0.732051i q^{7} +4.46410 q^{9} -2.00000i q^{11} +3.46410 q^{13} +3.46410i q^{17} -0.535898i q^{19} -2.00000i q^{21} -6.19615i q^{23} +4.00000 q^{27} +6.92820i q^{29} +5.46410 q^{31} -5.46410i q^{33} -2.00000 q^{37} +9.46410 q^{39} +1.46410 q^{41} -5.26795 q^{43} -3.26795i q^{47} +6.46410 q^{49} +9.46410i q^{51} -11.4641 q^{53} -1.46410i q^{57} +7.46410i q^{59} +8.92820i q^{61} -3.26795i q^{63} -10.7321 q^{67} -16.9282i q^{69} -5.46410 q^{71} -7.46410i q^{73} -1.46410 q^{77} -1.07180 q^{79} -2.46410 q^{81} +1.26795 q^{83} +18.9282i q^{87} -8.92820 q^{89} -2.53590i q^{91} +14.9282 q^{93} -14.3923i q^{97} -8.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 8 q^{31} - 8 q^{37} + 24 q^{39} - 8 q^{41} - 28 q^{43} + 12 q^{49} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 8 q^{77} - 32 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{89} + 32 q^{93}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 16 * q^27 + 8 * q^31 - 8 * q^37 + 24 * q^39 - 8 * q^41 - 28 * q^43 + 12 * q^49 - 32 * q^53 - 36 * q^67 - 8 * q^71 + 8 * q^77 - 32 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^89 + 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more