LMFDB - Embedded newform 3200.2.c.bb.2049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(2049,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3200.2049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.c (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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2049.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3200.2049
Dual form			3200.2.c.bb.2049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{3} -0.828427i q^{7} -2.82843 q^{9} +O(q^{10})$	$q+2.41421i q^{3} -0.828427i q^{7} -2.82843 q^{9} +5.24264 q^{11} +5.65685i q^{13} +0.171573i q^{17} -1.58579 q^{19} +2.00000 q^{21} -4.82843i q^{23} +0.414214i q^{27} +8.00000 q^{29} -0.828427 q^{31} +12.6569i q^{33} -7.65685i q^{37} -13.6569 q^{39} +10.6569 q^{41} +10.0000i q^{43} +9.65685i q^{47} +6.31371 q^{49} -0.414214 q^{51} -7.65685i q^{53} -3.82843i q^{57} +3.65685 q^{59} -6.00000 q^{61} +2.34315i q^{63} +2.75736i q^{67} +11.6569 q^{69} -9.65685 q^{71} +5.82843i q^{73} -4.34315i q^{77} -12.8284 q^{79} -9.48528 q^{81} +1.24264i q^{83} +19.3137i q^{87} -6.17157 q^{89} +4.68629 q^{91} -2.00000i q^{93} +17.3137i q^{97} -14.8284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 4 q^{11} - 12 q^{19} + 8 q^{21} + 32 q^{29} + 8 q^{31} - 32 q^{39} + 20 q^{41} - 20 q^{49} + 4 q^{51} - 8 q^{59} - 24 q^{61} + 24 q^{69} - 16 q^{71} - 40 q^{79} - 4 q^{81} - 36 q^{89} + 64 q^{91} - 48 q^{99}+O(q^{100}) $ 4 * q + 4 * q^11 - 12 * q^19 + 8 * q^21 + 32 * q^29 + 8 * q^31 - 32 * q^39 + 20 * q^41 - 20 * q^49 + 4 * q^51 - 8 * q^59 - 24 * q^61 + 24 * q^69 - 16 * q^71 - 40 * q^79 - 4 * q^81 - 36 * q^89 + 64 * q^91 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.41421i	1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$	2.41421i	1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.828427i	− 0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$	− 0.828427i	− 0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	5.24264	1.58072	0.790358	−	0.612646i	$-0.209895\pi$
$11$	5.24264	1.58072	0.790358	+	0.612646i	$0.209895\pi$
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.171573i	0.0416125i	0.999784	+	0.0208063i	$0.00662332\pi$
$17$	0.171573i	0.0416125i	−0.999784	+	0.0208063i	$0.993377\pi$
$18$	0	0
$19$	−1.58579	−0.363804	−0.181902	−	0.983317i	$-0.558225\pi$
$19$	−1.58579	−0.363804	−0.181902	+	0.983317i	$0.558225\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	− 4.82843i	− 1.00680i	−0.864054	−	0.503398i	$-0.832083\pi$
$23$	− 4.82843i	− 1.00680i	0.864054	−	0.503398i	$-0.167917\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.414214i	0.0797154i
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−0.828427	−0.148790	−0.0743950	−	0.997229i	$-0.523703\pi$
$31$	−0.828427	−0.148790	−0.0743950	+	0.997229i	$0.523703\pi$
$32$	0	0
$33$	12.6569i	2.20328i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.65685i	− 1.25878i	−0.777090	−	0.629390i	$-0.783305\pi$
$37$	− 7.65685i	− 1.25878i	0.777090	−	0.629390i	$-0.216695\pi$
$38$	0	0
$39$	−13.6569	−2.18685
$40$	0	0
$41$	10.6569	1.66432	0.832161	−	0.554535i	$-0.187104\pi$
$41$	10.6569	1.66432	0.832161	+	0.554535i	$0.187104\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.65685i	1.40860i	0.709904	+	0.704298i	$0.248738\pi$
$47$	9.65685i	1.40860i	−0.709904	+	0.704298i	$0.751262\pi$
$48$	0	0
$49$	6.31371	0.901958
$50$	0	0
$51$	−0.414214	−0.0580015
$52$	0	0
$53$	− 7.65685i	− 1.05175i	−0.850562	−	0.525875i	$-0.823738\pi$
$53$	− 7.65685i	− 1.05175i	0.850562	−	0.525875i	$-0.176262\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.82843i	− 0.507088i
$58$	0	0
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	2.34315i	0.295209i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.75736i	0.336865i	0.985713	+	0.168433i	$0.0538705\pi$
$67$	2.75736i	0.336865i	−0.985713	+	0.168433i	$0.946129\pi$
$68$	0	0
$69$	11.6569	1.40332
$70$	0	0
$71$	−9.65685	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$71$	−9.65685	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$72$	0	0
$73$	5.82843i	0.682166i	0.940033	+	0.341083i	$0.110794\pi$
$73$	5.82843i	0.682166i	−0.940033	+	0.341083i	$0.889206\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.34315i	− 0.494947i
$78$	0	0
$79$	−12.8284	−1.44331	−0.721655	−	0.692252i	$-0.756618\pi$
$79$	−12.8284	−1.44331	−0.721655	+	0.692252i	$0.756618\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	1.24264i	0.136398i	0.997672	+	0.0681988i	$0.0217252\pi$
$83$	1.24264i	0.136398i	−0.997672	+	0.0681988i	$0.978275\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.3137i	2.07065i
$88$	0	0
$89$	−6.17157	−0.654185	−0.327093	−	0.944992i	$-0.606069\pi$
$89$	−6.17157	−0.654185	−0.327093	+	0.944992i	$0.606069\pi$
$90$	0	0
$91$	4.68629	0.491257
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.3137i	1.75794i	0.476876	+	0.878970i	$0.341769\pi$
$97$	17.3137i	1.75794i	−0.476876	+	0.878970i	$0.658231\pi$
$98$	0	0
$99$	−14.8284	−1.49031
$100$	0	0
$101$	18.9706	1.88764	0.943821	−	0.330458i	$-0.107203\pi$
$101$	18.9706	1.88764	0.943821	+	0.330458i	$0.107203\pi$
$102$	0	0
$103$	1.65685i	0.163255i	0.996663	+	0.0816274i	$0.0260117\pi$
$103$	1.65685i	0.163255i	−0.996663	+	0.0816274i	$0.973988\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.92893i	0.186477i	0.995644	+	0.0932385i	$0.0297219\pi$
$107$	1.92893i	0.186477i	−0.995644	+	0.0932385i	$0.970278\pi$
$108$	0	0
$109$	11.6569	1.11652	0.558262	−	0.829665i	$-0.311468\pi$
$109$	11.6569	1.11652	0.558262	+	0.829665i	$0.311468\pi$
$110$	0	0
$111$	18.4853	1.75455
$112$	0	0
$113$	3.34315i	0.314497i	0.987559	+	0.157248i	$0.0502623\pi$
$113$	3.34315i	0.314497i	−0.987559	+	0.157248i	$0.949738\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 16.0000i	− 1.47920i
$118$	0	0
$119$	0.142136	0.0130296
$120$	0	0
$121$	16.4853	1.49866
$122$	0	0
$123$	25.7279i	2.31981i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 20.1421i	− 1.78733i	−0.448739	−	0.893663i	$-0.648127\pi$
$127$	− 20.1421i	− 1.78733i	0.448739	−	0.893663i	$-0.351873\pi$
$128$	0	0
$129$	−24.1421	−2.12560
$130$	0	0
$131$	5.31371	0.464261	0.232130	−	0.972685i	$-0.425430\pi$
$131$	5.31371	0.464261	0.232130	+	0.972685i	$0.425430\pi$
$132$	0	0
$133$	1.31371i	0.113913i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000i	0.427179i	0.976924	+	0.213589i	$0.0685155\pi$
$137$	5.00000i	0.427179i	−0.976924	+	0.213589i	$0.931485\pi$
$138$	0	0
$139$	−10.8995	−0.924483	−0.462242	−	0.886754i	$-0.652955\pi$
$139$	−10.8995	−0.924483	−0.462242	+	0.886754i	$0.652955\pi$
$140$	0	0
$141$	−23.3137	−1.96337
$142$	0	0
$143$	29.6569i	2.48003i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.2426i	1.25719i
$148$	0	0
$149$	−3.31371	−0.271470	−0.135735	−	0.990745i	$-0.543340\pi$
$149$	−3.31371	−0.271470	−0.135735	+	0.990745i	$0.543340\pi$
$150$	0	0
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	0	0
$153$	− 0.485281i	− 0.0392327i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.6569i	1.56879i	0.620263	+	0.784394i	$0.287026\pi$
$157$	19.6569i	1.56879i	−0.620263	+	0.784394i	$0.712974\pi$
$158$	0	0
$159$	18.4853	1.46598
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	− 19.7279i	− 1.54521i	−0.634887	−	0.772605i	$-0.718953\pi$
$163$	− 19.7279i	− 1.54521i	0.634887	−	0.772605i	$-0.281047\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.34315i	0.181318i	0.995882	+	0.0906590i	$0.0288973\pi$
$167$	2.34315i	0.181318i	−0.995882	+	0.0906590i	$0.971103\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	4.48528	0.342998
$172$	0	0
$173$	− 9.65685i	− 0.734197i	−0.930182	−	0.367099i	$-0.880351\pi$
$173$	− 9.65685i	− 0.734197i	0.930182	−	0.367099i	$-0.119649\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.82843i	0.663585i
$178$	0	0
$179$	−21.5858	−1.61340	−0.806699	−	0.590963i	$-0.798748\pi$
$179$	−21.5858	−1.61340	−0.806699	+	0.590963i	$0.798748\pi$
$180$	0	0
$181$	−15.6569	−1.16376	−0.581882	−	0.813273i	$-0.697684\pi$
$181$	−15.6569	−1.16376	−0.581882	+	0.813273i	$0.697684\pi$
$182$	0	0
$183$	− 14.4853i	− 1.07078i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.899495i	0.0657776i
$188$	0	0
$189$	0.343146	0.0249602
$190$	0	0
$191$	1.51472	0.109601	0.0548006	−	0.998497i	$-0.482548\pi$
$191$	1.51472	0.109601	0.0548006	+	0.998497i	$0.482548\pi$
$192$	0	0
$193$	− 8.17157i	− 0.588203i	−0.955774	−	0.294101i	$-0.904980\pi$
$193$	− 8.17157i	− 0.588203i	0.955774	−	0.294101i	$-0.0950203\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.65685i	0.545528i	0.962081	+	0.272764i	$0.0879379\pi$
$197$	7.65685i	0.545528i	−0.962081	+	0.272764i	$0.912062\pi$
$198$	0	0
$199$	−25.6569	−1.81877	−0.909383	−	0.415960i	$-0.863446\pi$
$199$	−25.6569	−1.81877	−0.909383	+	0.415960i	$0.863446\pi$
$200$	0	0
$201$	−6.65685	−0.469538
$202$	0	0
$203$	− 6.62742i	− 0.465153i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.6569i	0.949217i
$208$	0	0
$209$	−8.31371	−0.575071
$210$	0	0
$211$	−10.8995	−0.750352	−0.375176	−	0.926954i	$-0.622418\pi$
$211$	−10.8995	−0.750352	−0.375176	+	0.926954i	$0.622418\pi$
$212$	0	0
$213$	− 23.3137i	− 1.59743i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.686292i	0.0465885i
$218$	0	0
$219$	−14.0711	−0.950835
$220$	0	0
$221$	−0.970563	−0.0652871
$222$	0	0
$223$	13.6569i	0.914531i	0.889330	+	0.457265i	$0.151171\pi$
$223$	13.6569i	0.914531i	−0.889330	+	0.457265i	$0.848829\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.3137i	1.41464i	0.706893	+	0.707320i	$0.250096\pi$
$227$	21.3137i	1.41464i	−0.706893	+	0.707320i	$0.749904\pi$
$228$	0	0
$229$	0.686292	0.0453514	0.0226757	−	0.999743i	$-0.492781\pi$
$229$	0.686292	0.0453514	0.0226757	+	0.999743i	$0.492781\pi$
$230$	0	0
$231$	10.4853	0.689881
$232$	0	0
$233$	− 13.3137i	− 0.872210i	−0.899896	−	0.436105i	$-0.856358\pi$
$233$	− 13.3137i	− 0.872210i	0.899896	−	0.436105i	$-0.143642\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 30.9706i	− 2.01175i
$238$	0	0
$239$	−18.6274	−1.20491	−0.602454	−	0.798154i	$-0.705810\pi$
$239$	−18.6274	−1.20491	−0.602454	+	0.798154i	$0.705810\pi$
$240$	0	0
$241$	9.82843	0.633105	0.316552	−	0.948575i	$-0.397475\pi$
$241$	9.82843	0.633105	0.316552	+	0.948575i	$0.397475\pi$
$242$	0	0
$243$	− 21.6569i	− 1.38929i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.97056i	− 0.570783i
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	8.75736	0.552760	0.276380	−	0.961048i	$-0.410865\pi$
$251$	8.75736	0.552760	0.276380	+	0.961048i	$0.410865\pi$
$252$	0	0
$253$	− 25.3137i	− 1.59146i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.31371i	− 0.331460i	−0.986171	−	0.165730i	$-0.947002\pi$
$257$	− 5.31371i	− 0.331460i	0.986171	−	0.165730i	$-0.0529980\pi$
$258$	0	0
$259$	−6.34315	−0.394144
$260$	0	0
$261$	−22.6274	−1.40060
$262$	0	0
$263$	10.4853i	0.646550i	0.946305	+	0.323275i	$0.104784\pi$
$263$	10.4853i	0.646550i	−0.946305	+	0.323275i	$0.895216\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.8995i	− 0.911834i
$268$	0	0
$269$	−6.34315	−0.386748	−0.193374	−	0.981125i	$-0.561943\pi$
$269$	−6.34315	−0.386748	−0.193374	+	0.981125i	$0.561943\pi$
$270$	0	0
$271$	19.1716	1.16459	0.582295	−	0.812978i	$-0.302155\pi$
$271$	19.1716	1.16459	0.582295	+	0.812978i	$0.302155\pi$
$272$	0	0
$273$	11.3137i	0.684737i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.65685i	0.580224i	0.956993	+	0.290112i	$0.0936926\pi$
$277$	9.65685i	0.580224i	−0.956993	+	0.290112i	$0.906307\pi$
$278$	0	0
$279$	2.34315	0.140280
$280$	0	0
$281$	−20.6274	−1.23053	−0.615264	−	0.788321i	$-0.710951\pi$
$281$	−20.6274	−1.23053	−0.615264	+	0.788321i	$0.710951\pi$
$282$	0	0
$283$	− 9.72792i	− 0.578265i	−0.957289	−	0.289132i	$-0.906633\pi$
$283$	− 9.72792i	− 0.578265i	0.957289	−	0.289132i	$-0.0933668\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.82843i	− 0.521126i
$288$	0	0
$289$	16.9706	0.998268
$290$	0	0
$291$	−41.7990	−2.45030
$292$	0	0
$293$	− 17.3137i	− 1.01148i	−0.862687	−	0.505739i	$-0.831220\pi$
$293$	− 17.3137i	− 1.01148i	0.862687	−	0.505739i	$-0.168780\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.17157i	0.126007i
$298$	0	0
$299$	27.3137	1.57959
$300$	0	0
$301$	8.28427	0.477497
$302$	0	0
$303$	45.7990i	2.63108i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.24264i	0.185067i	0.995710	+	0.0925336i	$0.0294966\pi$
$307$	3.24264i	0.185067i	−0.995710	+	0.0925336i	$0.970503\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−22.4853	−1.27502	−0.637512	−	0.770441i	$-0.720036\pi$
$311$	−22.4853	−1.27502	−0.637512	+	0.770441i	$0.720036\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.34315i	0.131604i	0.997833	+	0.0658021i	$0.0209606\pi$
$317$	2.34315i	0.131604i	−0.997833	+	0.0658021i	$0.979039\pi$
$318$	0	0
$319$	41.9411	2.34825
$320$	0	0
$321$	−4.65685	−0.259920
$322$	0	0
$323$	− 0.272078i	− 0.0151388i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	28.1421i	1.55626i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	22.8995	1.25867	0.629335	−	0.777134i	$-0.283327\pi$
$331$	22.8995	1.25867	0.629335	+	0.777134i	$0.283327\pi$
$332$	0	0
$333$	21.6569i	1.18679i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.34315i	− 0.400007i	−0.979795	−	0.200003i	$-0.935905\pi$
$337$	− 7.34315i	− 0.400007i	0.979795	−	0.200003i	$-0.0640953\pi$
$338$	0	0
$339$	−8.07107	−0.438360
$340$	0	0
$341$	−4.34315	−0.235195
$342$	0	0
$343$	− 11.0294i	− 0.595534i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.8995i	1.01458i	0.861776	+	0.507289i	$0.169352\pi$
$347$	18.8995i	1.01458i	−0.861776	+	0.507289i	$0.830648\pi$
$348$	0	0
$349$	29.3137	1.56913	0.784563	−	0.620049i	$-0.212887\pi$
$349$	29.3137	1.56913	0.784563	+	0.620049i	$0.212887\pi$
$350$	0	0
$351$	−2.34315	−0.125068
$352$	0	0
$353$	− 12.6274i	− 0.672090i	−0.941846	−	0.336045i	$-0.890911\pi$
$353$	− 12.6274i	− 0.672090i	0.941846	−	0.336045i	$-0.109089\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.343146i	0.0181612i
$358$	0	0
$359$	−8.82843	−0.465947	−0.232973	−	0.972483i	$-0.574845\pi$
$359$	−8.82843	−0.465947	−0.232973	+	0.972483i	$0.574845\pi$
$360$	0	0
$361$	−16.4853	−0.867646
$362$	0	0
$363$	39.7990i	2.08891i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.970563i	0.0506630i	0.999679	+	0.0253315i	$0.00806412\pi$
$367$	0.970563i	0.0506630i	−0.999679	+	0.0253315i	$0.991936\pi$
$368$	0	0
$369$	−30.1421	−1.56914
$370$	0	0
$371$	−6.34315	−0.329320
$372$	0	0
$373$	33.3137i	1.72492i	0.506127	+	0.862459i	$0.331077\pi$
$373$	33.3137i	1.72492i	−0.506127	+	0.862459i	$0.668923\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.2548i	2.33074i
$378$	0	0
$379$	16.8995	0.868069	0.434034	−	0.900896i	$-0.357090\pi$
$379$	16.8995	0.868069	0.434034	+	0.900896i	$0.357090\pi$
$380$	0	0
$381$	48.6274	2.49126
$382$	0	0
$383$	− 27.4558i	− 1.40293i	−0.712705	−	0.701464i	$-0.752530\pi$
$383$	− 27.4558i	− 1.40293i	0.712705	−	0.701464i	$-0.247470\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 28.2843i	− 1.43777i
$388$	0	0
$389$	5.31371	0.269416	0.134708	−	0.990885i	$-0.456990\pi$
$389$	5.31371	0.269416	0.134708	+	0.990885i	$0.456990\pi$
$390$	0	0
$391$	0.828427	0.0418954
$392$	0	0
$393$	12.8284i	0.647109i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8.68629i	− 0.435952i	−0.975954	−	0.217976i	$-0.930054\pi$
$397$	− 8.68629i	− 0.435952i	0.975954	−	0.217976i	$-0.0699455\pi$
$398$	0	0
$399$	−3.17157	−0.158777
$400$	0	0
$401$	20.4558	1.02152	0.510758	−	0.859724i	$-0.329365\pi$
$401$	20.4558	1.02152	0.510758	+	0.859724i	$0.329365\pi$
$402$	0	0
$403$	− 4.68629i	− 0.233441i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 40.1421i	− 1.98977i
$408$	0	0
$409$	−5.97056	−0.295225	−0.147613	−	0.989045i	$-0.547159\pi$
$409$	−5.97056	−0.295225	−0.147613	+	0.989045i	$0.547159\pi$
$410$	0	0
$411$	−12.0711	−0.595422
$412$	0	0
$413$	− 3.02944i	− 0.149069i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 26.3137i	− 1.28859i
$418$	0	0
$419$	−6.55635	−0.320299	−0.160149	−	0.987093i	$-0.551198\pi$
$419$	−6.55635	−0.320299	−0.160149	+	0.987093i	$0.551198\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	− 27.3137i	− 1.32804i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.97056i	0.240542i
$428$	0	0
$429$	−71.5980	−3.45678
$430$	0	0
$431$	12.1421	0.584866	0.292433	−	0.956286i	$-0.405535\pi$
$431$	12.1421	0.584866	0.292433	+	0.956286i	$0.405535\pi$
$432$	0	0
$433$	− 21.8284i	− 1.04901i	−0.851408	−	0.524504i	$-0.824251\pi$
$433$	− 21.8284i	− 1.04901i	0.851408	−	0.524504i	$-0.175749\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.65685i	0.366277i
$438$	0	0
$439$	−0.686292	−0.0327549	−0.0163775	−	0.999866i	$-0.505213\pi$
$439$	−0.686292	−0.0327549	−0.0163775	+	0.999866i	$0.505213\pi$
$440$	0	0
$441$	−17.8579	−0.850374
$442$	0	0
$443$	22.5563i	1.07168i	0.844318	+	0.535842i	$0.180006\pi$
$443$	22.5563i	1.07168i	−0.844318	+	0.535842i	$0.819994\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.00000i	− 0.378387i
$448$	0	0
$449$	34.4558	1.62607	0.813036	−	0.582214i	$-0.197813\pi$
$449$	34.4558	1.62607	0.813036	+	0.582214i	$0.197813\pi$
$450$	0	0
$451$	55.8701	2.63082
$452$	0	0
$453$	15.6569i	0.735623i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.3431i	− 0.811278i	−0.914033	−	0.405639i	$-0.867049\pi$
$457$	− 17.3431i	− 0.811278i	0.914033	−	0.405639i	$-0.132951\pi$
$458$	0	0
$459$	−0.0710678	−0.00331716
$460$	0	0
$461$	−5.65685	−0.263466	−0.131733	−	0.991285i	$-0.542054\pi$
$461$	−5.65685	−0.263466	−0.131733	+	0.991285i	$0.542054\pi$
$462$	0	0
$463$	− 12.9706i	− 0.602793i	−0.953499	−	0.301397i	$-0.902547\pi$
$463$	− 12.9706i	− 0.602793i	0.953499	−	0.301397i	$-0.0974528\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.31371i	− 0.430987i	−0.976505	−	0.215494i	$-0.930864\pi$
$467$	− 9.31371i	− 0.430987i	0.976505	−	0.215494i	$-0.0691360\pi$
$468$	0	0
$469$	2.28427	0.105478
$470$	0	0
$471$	−47.4558	−2.18665
$472$	0	0
$473$	52.4264i	2.41057i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21.6569i	0.991599i
$478$	0	0
$479$	35.1716	1.60703	0.803515	−	0.595284i	$-0.202961\pi$
$479$	35.1716	1.60703	0.803515	+	0.595284i	$0.202961\pi$
$480$	0	0
$481$	43.3137	1.97494
$482$	0	0
$483$	− 9.65685i	− 0.439402i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.1127i	1.13796i	0.822350	+	0.568982i	$0.192663\pi$
$487$	25.1127i	1.13796i	−0.822350	+	0.568982i	$0.807337\pi$
$488$	0	0
$489$	47.6274	2.15379
$490$	0	0
$491$	−16.6274	−0.750385	−0.375192	−	0.926947i	$-0.622423\pi$
$491$	−16.6274	−0.750385	−0.375192	+	0.926947i	$0.622423\pi$
$492$	0	0
$493$	1.37258i	0.0618180i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	−5.65685	−0.252730
$502$	0	0
$503$	− 16.9706i	− 0.756680i	−0.925667	−	0.378340i	$-0.876495\pi$
$503$	− 16.9706i	− 0.756680i	0.925667	−	0.378340i	$-0.123505\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 45.8701i	− 2.03716i
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	4.82843	0.213597
$512$	0	0
$513$	− 0.656854i	− 0.0290008i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	50.6274i	2.22659i
$518$	0	0
$519$	23.3137	1.02336
$520$	0	0
$521$	16.3137	0.714717	0.357358	−	0.933967i	$-0.383678\pi$
$521$	16.3137	0.714717	0.357358	+	0.933967i	$0.383678\pi$
$522$	0	0
$523$	− 19.5858i	− 0.856427i	−0.903678	−	0.428213i	$-0.859143\pi$
$523$	− 19.5858i	− 0.856427i	0.903678	−	0.428213i	$-0.140857\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.142136i	− 0.00619153i
$528$	0	0
$529$	−0.313708	−0.0136395
$530$	0	0
$531$	−10.3431	−0.448854
$532$	0	0
$533$	60.2843i	2.61120i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 52.1127i	− 2.24883i
$538$	0	0
$539$	33.1005	1.42574
$540$	0	0
$541$	17.6569	0.759127	0.379564	−	0.925166i	$-0.376074\pi$
$541$	17.6569	0.759127	0.379564	+	0.925166i	$0.376074\pi$
$542$	0	0
$543$	− 37.7990i	− 1.62211i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.5563i	0.622385i	0.950347	+	0.311192i	$0.100728\pi$
$547$	14.5563i	0.622385i	−0.950347	+	0.311192i	$0.899272\pi$
$548$	0	0
$549$	16.9706	0.724286
$550$	0	0
$551$	−12.6863	−0.540454
$552$	0	0
$553$	10.6274i	0.451924i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 26.3431i	− 1.11619i	−0.829775	−	0.558097i	$-0.811532\pi$
$557$	− 26.3431i	− 1.11619i	0.829775	−	0.558097i	$-0.188468\pi$
$558$	0	0
$559$	−56.5685	−2.39259
$560$	0	0
$561$	−2.17157	−0.0916839
$562$	0	0
$563$	− 6.97056i	− 0.293774i	−0.989153	−	0.146887i	$-0.953075\pi$
$563$	− 6.97056i	− 0.293774i	0.989153	−	0.146887i	$-0.0469254\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.85786i	0.329999i
$568$	0	0
$569$	26.3137	1.10313	0.551564	−	0.834133i	$-0.314031\pi$
$569$	26.3137	1.10313	0.551564	+	0.834133i	$0.314031\pi$
$570$	0	0
$571$	31.9411	1.33669	0.668347	−	0.743849i	$-0.267002\pi$
$571$	31.9411	1.33669	0.668347	+	0.743849i	$0.267002\pi$
$572$	0	0
$573$	3.65685i	0.152767i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 41.2843i	− 1.71869i	−0.511399	−	0.859343i	$-0.670873\pi$
$577$	− 41.2843i	− 1.71869i	0.511399	−	0.859343i	$-0.329127\pi$
$578$	0	0
$579$	19.7279	0.819864
$580$	0	0
$581$	1.02944	0.0427083
$582$	0	0
$583$	− 40.1421i	− 1.66252i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.0710678i	0.00293328i	0.999999	+	0.00146664i	$0.000466847\pi$
$587$	0.0710678i	0.00293328i	−0.999999	+	0.00146664i	$0.999533\pi$
$588$	0	0
$589$	1.31371	0.0541304
$590$	0	0
$591$	−18.4853	−0.760383
$592$	0	0
$593$	− 7.00000i	− 0.287456i	−0.989617	−	0.143728i	$-0.954091\pi$
$593$	− 7.00000i	− 0.287456i	0.989617	−	0.143728i	$-0.0459090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 61.9411i	− 2.53508i
$598$	0	0
$599$	−26.7696	−1.09377	−0.546887	−	0.837206i	$-0.684187\pi$
$599$	−26.7696	−1.09377	−0.546887	+	0.837206i	$0.684187\pi$
$600$	0	0
$601$	35.1421	1.43348	0.716739	−	0.697342i	$-0.245634\pi$
$601$	35.1421	1.43348	0.716739	+	0.697342i	$0.245634\pi$
$602$	0	0
$603$	− 7.79899i	− 0.317599i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 36.0000i	− 1.46119i	−0.682808	−	0.730597i	$-0.739242\pi$
$607$	− 36.0000i	− 1.46119i	0.682808	−	0.730597i	$-0.260758\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	0	0
$611$	−54.6274	−2.20999
$612$	0	0
$613$	6.68629i	0.270057i	0.990842	+	0.135028i	$0.0431126\pi$
$613$	6.68629i	0.270057i	−0.990842	+	0.135028i	$0.956887\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	13.3137	0.535123	0.267562	−	0.963541i	$-0.413782\pi$
$619$	13.3137	0.535123	0.267562	+	0.963541i	$0.413782\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	5.11270i	0.204836i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 20.0711i	− 0.801561i
$628$	0	0
$629$	1.31371	0.0523810
$630$	0	0
$631$	15.8579	0.631292	0.315646	−	0.948877i	$-0.397779\pi$
$631$	15.8579	0.631292	0.315646	+	0.948877i	$0.397779\pi$
$632$	0	0
$633$	− 26.3137i	− 1.04588i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35.7157i	1.41511i
$638$	0	0
$639$	27.3137	1.08051
$640$	0	0
$641$	−45.3137	−1.78978	−0.894892	−	0.446283i	$-0.852748\pi$
$641$	−45.3137	−1.78978	−0.894892	+	0.446283i	$0.852748\pi$
$642$	0	0
$643$	− 14.6863i	− 0.579171i	−0.957152	−	0.289585i	$-0.906483\pi$
$643$	− 14.6863i	− 0.579171i	0.957152	−	0.289585i	$-0.0935174\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.65685i	0.222394i	0.993798	+	0.111197i	$0.0354684\pi$
$647$	5.65685i	0.222394i	−0.993798	+	0.111197i	$0.964532\pi$
$648$	0	0
$649$	19.1716	0.752550
$650$	0	0
$651$	−1.65685	−0.0649372
$652$	0	0
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 16.4853i	− 0.643152i
$658$	0	0
$659$	51.1838	1.99384	0.996918	−	0.0784478i	$-0.0249964\pi$
$659$	51.1838	1.99384	0.996918	+	0.0784478i	$0.0249964\pi$
$660$	0	0
$661$	−6.34315	−0.246720	−0.123360	−	0.992362i	$-0.539367\pi$
$661$	−6.34315	−0.246720	−0.123360	+	0.992362i	$0.539367\pi$
$662$	0	0
$663$	− 2.34315i	− 0.0910002i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 38.6274i	− 1.49566i
$668$	0	0
$669$	−32.9706	−1.27472
$670$	0	0
$671$	−31.4558	−1.21434
$672$	0	0
$673$	− 25.3137i	− 0.975772i	−0.872907	−	0.487886i	$-0.837768\pi$
$673$	− 25.3137i	− 0.975772i	0.872907	−	0.487886i	$-0.162232\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6863i	1.10250i	0.834339	+	0.551252i	$0.185850\pi$
$677$	28.6863i	1.10250i	−0.834339	+	0.551252i	$0.814150\pi$
$678$	0	0
$679$	14.3431	0.550439
$680$	0	0
$681$	−51.4558	−1.97179
$682$	0	0
$683$	16.6985i	0.638950i	0.947595	+	0.319475i	$0.103507\pi$
$683$	16.6985i	0.638950i	−0.947595	+	0.319475i	$0.896493\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.65685i	0.0632129i
$688$	0	0
$689$	43.3137	1.65012
$690$	0	0
$691$	36.2132	1.37762	0.688808	−	0.724944i	$-0.258134\pi$
$691$	36.2132	1.37762	0.688808	+	0.724944i	$0.258134\pi$
$692$	0	0
$693$	12.2843i	0.466641i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.82843i	0.0692566i
$698$	0	0
$699$	32.1421	1.21573
$700$	0	0
$701$	47.3137	1.78701	0.893507	−	0.449049i	$-0.148237\pi$
$701$	47.3137	1.78701	0.893507	+	0.449049i	$0.148237\pi$
$702$	0	0
$703$	12.1421i	0.457949i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.7157i	− 0.591051i
$708$	0	0
$709$	−21.9411	−0.824016	−0.412008	−	0.911180i	$-0.635172\pi$
$709$	−21.9411	−0.824016	−0.412008	+	0.911180i	$0.635172\pi$
$710$	0	0
$711$	36.2843	1.36077
$712$	0	0
$713$	4.00000i	0.149801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 44.9706i	− 1.67946i
$718$	0	0
$719$	23.4558	0.874755	0.437378	−	0.899278i	$-0.355907\pi$
$719$	23.4558	0.874755	0.437378	+	0.899278i	$0.355907\pi$
$720$	0	0
$721$	1.37258	0.0511177
$722$	0	0
$723$	23.7279i	0.882451i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	23.8284	0.882534
$730$	0	0
$731$	−1.71573	−0.0634585
$732$	0	0
$733$	26.3431i	0.973006i	0.873679	+	0.486503i	$0.161728\pi$
$733$	26.3431i	0.973006i	−0.873679	+	0.486503i	$0.838272\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.4558i	0.532488i
$738$	0	0
$739$	28.6274	1.05308	0.526538	−	0.850151i	$-0.323490\pi$
$739$	28.6274	1.05308	0.526538	+	0.850151i	$0.323490\pi$
$740$	0	0
$741$	21.6569	0.795584
$742$	0	0
$743$	25.7990i	0.946473i	0.880935	+	0.473237i	$0.156914\pi$
$743$	25.7990i	0.946473i	−0.880935	+	0.473237i	$0.843086\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.51472i	− 0.128597i
$748$	0	0
$749$	1.59798	0.0583889
$750$	0	0
$751$	−34.6274	−1.26357	−0.631786	−	0.775143i	$-0.717678\pi$
$751$	−34.6274	−1.26357	−0.631786	+	0.775143i	$0.717678\pi$
$752$	0	0
$753$	21.1421i	0.770462i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.6569i	0.932514i	0.884649	+	0.466257i	$0.154398\pi$
$757$	25.6569i	0.932514i	−0.884649	+	0.466257i	$0.845602\pi$
$758$	0	0
$759$	61.1127	2.21825
$760$	0	0
$761$	−39.2843	−1.42405	−0.712027	−	0.702152i	$-0.752223\pi$
$761$	−39.2843	−1.42405	−0.712027	+	0.702152i	$0.752223\pi$
$762$	0	0
$763$	− 9.65685i	− 0.349602i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.6863i	0.746939i
$768$	0	0
$769$	−34.1127	−1.23014	−0.615068	−	0.788474i	$-0.710871\pi$
$769$	−34.1127	−1.23014	−0.615068	+	0.788474i	$0.710871\pi$
$770$	0	0
$771$	12.8284	0.462005
$772$	0	0
$773$	− 19.3137i	− 0.694666i	−0.937742	−	0.347333i	$-0.887087\pi$
$773$	− 19.3137i	− 0.694666i	0.937742	−	0.347333i	$-0.112913\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 15.3137i	− 0.549376i
$778$	0	0
$779$	−16.8995	−0.605487
$780$	0	0
$781$	−50.6274	−1.81159
$782$	0	0
$783$	3.31371i	0.118422i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.6274i	0.592703i	0.955079	+	0.296352i	$0.0957701\pi$
$787$	16.6274i	0.592703i	−0.955079	+	0.296352i	$0.904230\pi$
$788$	0	0
$789$	−25.3137	−0.901192
$790$	0	0
$791$	2.76955	0.0984740
$792$	0	0
$793$	− 33.9411i	− 1.20528i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	−1.65685	−0.0586153
$800$	0	0
$801$	17.4558	0.616772
$802$	0	0
$803$	30.5563i	1.07831i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 15.3137i	− 0.539068i
$808$	0	0
$809$	−40.6274	−1.42838	−0.714192	−	0.699950i	$-0.753206\pi$
$809$	−40.6274	−1.42838	−0.714192	+	0.699950i	$0.753206\pi$
$810$	0	0
$811$	−35.9411	−1.26206	−0.631032	−	0.775757i	$-0.717368\pi$
$811$	−35.9411	−1.26206	−0.631032	+	0.775757i	$0.717368\pi$
$812$	0	0
$813$	46.2843i	1.62326i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 15.8579i	− 0.554796i
$818$	0	0
$819$	−13.2548	−0.463161
$820$	0	0
$821$	40.6274	1.41791	0.708953	−	0.705255i	$-0.249168\pi$
$821$	40.6274	1.41791	0.708953	+	0.705255i	$0.249168\pi$
$822$	0	0
$823$	− 12.9706i	− 0.452125i	−0.974113	−	0.226063i	$-0.927415\pi$
$823$	− 12.9706i	− 0.452125i	0.974113	−	0.226063i	$-0.0725854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.0711i	1.67159i	0.549038	+	0.835797i	$0.314994\pi$
$827$	48.0711i	1.67159i	−0.549038	+	0.835797i	$0.685006\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	−23.3137	−0.808744
$832$	0	0
$833$	1.08326i	0.0375328i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 0.343146i	− 0.0118609i
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	− 49.7990i	− 1.71517i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 13.6569i	− 0.469255i
$848$	0	0
$849$	23.4853	0.806013
$850$	0	0
$851$	−36.9706	−1.26733
$852$	0	0
$853$	− 16.6274i	− 0.569312i	−0.958630	−	0.284656i	$-0.908121\pi$
$853$	− 16.6274i	− 0.569312i	0.958630	−	0.284656i	$-0.0918794\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.0000i	− 0.375753i	−0.982193	−	0.187876i	$-0.939840\pi$
$857$	− 11.0000i	− 0.375753i	0.982193	−	0.187876i	$-0.0601604\pi$
$858$	0	0
$859$	−54.2132	−1.84973	−0.924865	−	0.380295i	$-0.875823\pi$
$859$	−54.2132	−1.84973	−0.924865	+	0.380295i	$0.875823\pi$
$860$	0	0
$861$	21.3137	0.726369
$862$	0	0
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	40.9706i	1.39143i
$868$	0	0
$869$	−67.2548	−2.28146
$870$	0	0
$871$	−15.5980	−0.528517
$872$	0	0
$873$	− 48.9706i	− 1.65740i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 26.3431i	− 0.889545i	−0.895644	−	0.444772i	$-0.853285\pi$
$877$	− 26.3431i	− 0.889545i	0.895644	−	0.444772i	$-0.146715\pi$
$878$	0	0
$879$	41.7990	1.40984
$880$	0	0
$881$	−14.6863	−0.494794	−0.247397	−	0.968914i	$-0.579575\pi$
$881$	−14.6863	−0.494794	−0.247397	+	0.968914i	$0.579575\pi$
$882$	0	0
$883$	− 43.2426i	− 1.45523i	−0.685985	−	0.727615i	$-0.740629\pi$
$883$	− 43.2426i	− 1.45523i	0.685985	−	0.727615i	$-0.259371\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.3431i	− 0.750209i	−0.926982	−	0.375105i	$-0.877607\pi$
$887$	− 22.3431i	− 0.750209i	0.926982	−	0.375105i	$-0.122393\pi$
$888$	0	0
$889$	−16.6863	−0.559640
$890$	0	0
$891$	−49.7279	−1.66595
$892$	0	0
$893$	− 15.3137i	− 0.512454i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	65.9411i	2.20171i
$898$	0	0
$899$	−6.62742	−0.221037
$900$	0	0
$901$	1.31371	0.0437660
$902$	0	0
$903$	20.0000i	0.665558i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 45.3137i	− 1.50462i	−0.658811	−	0.752308i	$-0.728940\pi$
$907$	− 45.3137i	− 1.50462i	0.658811	−	0.752308i	$-0.271060\pi$
$908$	0	0
$909$	−53.6569	−1.77969
$910$	0	0
$911$	36.2843	1.20215	0.601076	−	0.799192i	$-0.294739\pi$
$911$	36.2843	1.20215	0.601076	+	0.799192i	$0.294739\pi$
$912$	0	0
$913$	6.51472i	0.215606i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.40202i	− 0.145368i
$918$	0	0
$919$	39.1716	1.29215	0.646075	−	0.763274i	$-0.276409\pi$
$919$	39.1716	1.29215	0.646075	+	0.763274i	$0.276409\pi$
$920$	0	0
$921$	−7.82843	−0.257955
$922$	0	0
$923$	− 54.6274i	− 1.79808i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.68629i	− 0.153918i
$928$	0	0
$929$	−0.627417	−0.0205849	−0.0102924	−	0.999947i	$-0.503276\pi$
$929$	−0.627417	−0.0205849	−0.0102924	+	0.999947i	$0.503276\pi$
$930$	0	0
$931$	−10.0122	−0.328136
$932$	0	0
$933$	− 54.2843i	− 1.77719i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.1127i	0.657053i	0.944495	+	0.328527i	$0.106552\pi$
$937$	20.1127i	0.657053i	−0.944495	+	0.328527i	$0.893448\pi$
$938$	0	0
$939$	24.1421	0.787849
$940$	0	0
$941$	−40.9706	−1.33560	−0.667801	−	0.744340i	$-0.732764\pi$
$941$	−40.9706	−1.33560	−0.667801	+	0.744340i	$0.732764\pi$
$942$	0	0
$943$	− 51.4558i	− 1.67563i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 33.5980i	− 1.09179i	−0.837854	−	0.545894i	$-0.816190\pi$
$947$	− 33.5980i	− 1.09179i	0.837854	−	0.545894i	$-0.183810\pi$
$948$	0	0
$949$	−32.9706	−1.07027
$950$	0	0
$951$	−5.65685	−0.183436
$952$	0	0
$953$	− 12.3137i	− 0.398880i	−0.979910	−	0.199440i	$-0.936088\pi$
$953$	− 12.3137i	− 0.398880i	0.979910	−	0.199440i	$-0.0639123\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	101.255i	3.27310i
$958$	0	0
$959$	4.14214	0.133757
$960$	0	0
$961$	−30.3137	−0.977862
$962$	0	0
$963$	− 5.45584i	− 0.175812i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 42.6274i	− 1.37081i	−0.728164	−	0.685403i	$-0.759626\pi$
$967$	− 42.6274i	− 1.37081i	0.728164	−	0.685403i	$-0.240374\pi$
$968$	0	0
$969$	0.656854	0.0211012
$970$	0	0
$971$	42.8995	1.37671	0.688355	−	0.725374i	$-0.258333\pi$
$971$	42.8995	1.37671	0.688355	+	0.725374i	$0.258333\pi$
$972$	0	0
$973$	9.02944i	0.289470i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.1127i	− 0.771434i	−0.922617	−	0.385717i	$-0.873954\pi$
$977$	− 24.1127i	− 0.771434i	0.922617	−	0.385717i	$-0.126046\pi$
$978$	0	0
$979$	−32.3553	−1.03408
$980$	0	0
$981$	−32.9706	−1.05267
$982$	0	0
$983$	33.7990i	1.07802i	0.842299	+	0.539010i	$0.181202\pi$
$983$	33.7990i	1.07802i	−0.842299	+	0.539010i	$0.818798\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	19.3137i	0.614762i
$988$	0	0
$989$	48.2843	1.53535
$990$	0	0
$991$	62.0833	1.97214	0.986070	−	0.166331i	$-0.0531922\pi$
$991$	62.0833	1.97214	0.986070	+	0.166331i	$0.0531922\pi$
$992$	0	0
$993$	55.2843i	1.75439i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.9706i	0.537463i	0.963215	+	0.268732i	$0.0866045\pi$
$997$	16.9706i	0.537463i	−0.963215	+	0.268732i	$0.913396\pi$
$998$	0	0
$999$	3.17157	0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.c.bb.2049.4		4
4.3	odd	2		3200.2.c.z.2049.1		4
5.2	odd	4		3200.2.a.bj.1.2	yes	2
5.3	odd	4		3200.2.a.bh.1.1	yes	2
5.4	even	2	inner	3200.2.c.bb.2049.1		4
8.3	odd	2		3200.2.c.ba.2049.4		4
8.5	even	2		3200.2.c.y.2049.1		4
20.3	even	4		3200.2.a.bi.1.2	yes	2
20.7	even	4		3200.2.a.bg.1.1	yes	2
20.19	odd	2		3200.2.c.z.2049.4		4
40.3	even	4		3200.2.a.bd.1.1	yes	2
40.13	odd	4		3200.2.a.bm.1.2	yes	2
40.19	odd	2		3200.2.c.ba.2049.1		4
40.27	even	4		3200.2.a.bn.1.2	yes	2
40.29	even	2		3200.2.c.y.2049.4		4
40.37	odd	4		3200.2.a.bc.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.a.bc.1.1	✓	2	40.37	odd	4
3200.2.a.bd.1.1	yes	2	40.3	even	4
3200.2.a.bg.1.1	yes	2	20.7	even	4
3200.2.a.bh.1.1	yes	2	5.3	odd	4
3200.2.a.bi.1.2	yes	2	20.3	even	4
3200.2.a.bj.1.2	yes	2	5.2	odd	4
3200.2.a.bm.1.2	yes	2	40.13	odd	4
3200.2.a.bn.1.2	yes	2	40.27	even	4
3200.2.c.y.2049.1		4	8.5	even	2
3200.2.c.y.2049.4		4	40.29	even	2
3200.2.c.z.2049.1		4	4.3	odd	2
3200.2.c.z.2049.4		4	20.19	odd	2
3200.2.c.ba.2049.1		4	40.19	odd	2
3200.2.c.ba.2049.4		4	8.3	odd	2
3200.2.c.bb.2049.1		4	5.4	even	2	inner
3200.2.c.bb.2049.4		4	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{3} -0.828427i q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q+2.41421i q^{3} -0.828427i q^{7} -2.82843 q^{9} +5.24264 q^{11} +5.65685i q^{13} +0.171573i q^{17} -1.58579 q^{19} +2.00000 q^{21} -4.82843i q^{23} +0.414214i q^{27} +8.00000 q^{29} -0.828427 q^{31} +12.6569i q^{33} -7.65685i q^{37} -13.6569 q^{39} +10.6569 q^{41} +10.0000i q^{43} +9.65685i q^{47} +6.31371 q^{49} -0.414214 q^{51} -7.65685i q^{53} -3.82843i q^{57} +3.65685 q^{59} -6.00000 q^{61} +2.34315i q^{63} +2.75736i q^{67} +11.6569 q^{69} -9.65685 q^{71} +5.82843i q^{73} -4.34315i q^{77} -12.8284 q^{79} -9.48528 q^{81} +1.24264i q^{83} +19.3137i q^{87} -6.17157 q^{89} +4.68629 q^{91} -2.00000i q^{93} +17.3137i q^{97} -14.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 4 q^{11} - 12 q^{19} + 8 q^{21} + 32 q^{29} + 8 q^{31} - 32 q^{39} + 20 q^{41} - 20 q^{49} + 4 q^{51} - 8 q^{59} - 24 q^{61} + 24 q^{69} - 16 q^{71} - 40 q^{79} - 4 q^{81} - 36 q^{89} + 64 q^{91} - 48 q^{99}+O(q^{100}) \) 4 * q + 4 * q^11 - 12 * q^19 + 8 * q^21 + 32 * q^29 + 8 * q^31 - 32 * q^39 + 20 * q^41 - 20 * q^49 + 4 * q^51 - 8 * q^59 - 24 * q^61 + 24 * q^69 - 16 * q^71 - 40 * q^79 - 4 * q^81 - 36 * q^89 + 64 * q^91 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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