LMFDB - Embedded newform 320.3.e.b.159.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(159,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("320.159");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.e (of order $2$, degree $1$, 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:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 $ x^16 + 19x^14 + 301x^12 + 1102x^10 + 3238x^8 + 1102x^6 + 301x^4 + 19*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{34} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			159.9
Root			$-1.94376 - 3.36668i$ of defining polynomial
Character	$\chi$	$=$	320.159
Dual form			320.3.e.b.159.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90845i q^{3} +(-4.37937 - 2.41269i) q^{5} +3.95502 q^{7} +5.35782 q^{9} +O(q^{10})$	$q+1.90845i q^{3} +(-4.37937 - 2.41269i) q^{5} +3.95502 q^{7} +5.35782 q^{9} +6.18667 q^{11} -4.94185 q^{13} +(4.60451 - 8.35782i) q^{15} +22.4311i q^{17} +10.3923 q^{19} +7.54796i q^{21} +39.6083 q^{23} +(13.3578 + 21.1322i) q^{25} +27.4012i q^{27} +30.4354i q^{29} -24.7156i q^{31} +11.8070i q^{33} +(-17.3205 - 9.54225i) q^{35} -24.0264 q^{37} -9.43127i q^{39} +31.0735 q^{41} +52.2110i q^{43} +(-23.4639 - 12.9268i) q^{45} +13.1640 q^{47} -33.3578 q^{49} -42.8087 q^{51} +17.9596 q^{53} +(-27.0938 - 14.9265i) q^{55} +19.8332i q^{57} +104.421 q^{59} -57.2849i q^{61} +21.1903 q^{63} +(21.6422 + 11.9232i) q^{65} -99.3796i q^{67} +75.5905i q^{69} +16.7156i q^{71} -96.3357i q^{73} +(-40.3297 + 25.4927i) q^{75} +24.4684 q^{77} +139.578i q^{79} -4.07345 q^{81} -91.7458i q^{83} +(54.1195 - 98.2344i) q^{85} -58.0844 q^{87} -94.7156 q^{89} -19.5451 q^{91} +47.1686 q^{93} +(-45.5118 - 25.0735i) q^{95} -143.796i q^{97} +33.1471 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{9}+O(q^{10}) $ 16 * q - 96 * q^9	$ 16 q - 96 q^{9} + 32 q^{25} - 48 q^{41} - 352 q^{49} + 528 q^{65} + 480 q^{81} - 1152 q^{89}+O(q^{100}) $ 16 * q - 96 * q^9 + 32 * q^25 - 48 * q^41 - 352 * q^49 + 528 * q^65 + 480 * q^81 - 1152 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			1.90845i			0.636150i	0.948066	+	0.318075i	$0.103036\pi$
$3$			1.90845i			0.636150i	−0.948066	+	0.318075i	$0.896964\pi$
$4$		0			0
$5$	−4.37937	−	2.41269i	−0.875875	−	0.482539i
$5$	−4.37937	−	2.41269i	−0.875875	−	0.482539i
$6$		0			0
$7$	3.95502			0.565003			0.282501	−	0.959267i	$-0.408836\pi$
$7$	3.95502			0.565003			0.282501	+	0.959267i	$0.408836\pi$
$8$		0			0
$9$	5.35782			0.595313
$10$		0			0
$11$	6.18667			0.562425			0.281212	−	0.959646i	$-0.409263\pi$
$11$	6.18667			0.562425			0.281212	+	0.959646i	$0.409263\pi$
$12$		0			0
$13$	−4.94185			−0.380142			−0.190071	−	0.981770i	$-0.560872\pi$
$13$	−4.94185			−0.380142			−0.190071	+	0.981770i	$0.560872\pi$
$14$		0			0
$15$	4.60451	−	8.35782i	0.306967	−	0.557188i
$16$		0			0
$17$			22.4311i			1.31948i	0.751494	+	0.659740i	$0.229333\pi$
$17$			22.4311i			1.31948i	−0.751494	+	0.659740i	$0.770667\pi$
$18$		0			0
$19$	10.3923			0.546963			0.273482	−	0.961877i	$-0.411825\pi$
$19$	10.3923			0.546963			0.273482	+	0.961877i	$0.411825\pi$
$20$		0			0
$21$			7.54796i			0.359427i
$22$		0			0
$23$	39.6083			1.72210			0.861050	−	0.508520i	$-0.169807\pi$
$23$	39.6083			1.72210			0.861050	+	0.508520i	$0.169807\pi$
$24$		0			0
$25$	13.3578	+	21.1322i	0.534313	+	0.845287i
$26$		0			0
$27$			27.4012i			1.01486i
$28$		0			0
$29$			30.4354i			1.04950i	0.851258	+	0.524748i	$0.175840\pi$
$29$			30.4354i			1.04950i	−0.851258	+	0.524748i	$0.824160\pi$
$30$		0			0
$31$		−	24.7156i		−	0.797278i	−0.917108	−	0.398639i	$-0.869483\pi$
$31$		−	24.7156i		−	0.797278i	0.917108	−	0.398639i	$-0.130517\pi$
$32$		0			0
$33$			11.8070i			0.357787i
$34$		0			0
$35$	−17.3205	−	9.54225i	−0.494872	−	0.272636i
$36$		0			0
$37$	−24.0264			−0.649361			−0.324680	−	0.945824i	$-0.605257\pi$
$37$	−24.0264			−0.649361			−0.324680	+	0.945824i	$0.605257\pi$
$38$		0			0
$39$		−	9.43127i		−	0.241827i
$40$		0			0
$41$	31.0735			0.757889			0.378945	−	0.925419i	$-0.376287\pi$
$41$	31.0735			0.757889			0.378945	+	0.925419i	$0.376287\pi$
$42$		0			0
$43$			52.2110i			1.21421i	0.794622	+	0.607105i	$0.207669\pi$
$43$			52.2110i			1.21421i	−0.794622	+	0.607105i	$0.792331\pi$
$44$		0			0
$45$	−23.4639	−	12.9268i	−0.521420	−	0.287262i
$46$		0			0
$47$	13.1640			0.280086			0.140043	−	0.990145i	$-0.455276\pi$
$47$	13.1640			0.280086			0.140043	+	0.990145i	$0.455276\pi$
$48$		0			0
$49$	−33.3578			−0.680772
$50$		0			0
$51$	−42.8087			−0.839387
$52$		0			0
$53$	17.9596			0.338860			0.169430	−	0.985542i	$-0.445807\pi$
$53$	17.9596			0.338860			0.169430	+	0.985542i	$0.445807\pi$
$54$		0			0
$55$	−27.0938	−	14.9265i	−0.492614	−	0.271392i
$56$		0			0
$57$			19.8332i			0.347951i
$58$		0			0
$59$	104.421			1.76985			0.884924	−	0.465735i	$-0.154210\pi$
$59$	104.421			1.76985			0.884924	+	0.465735i	$0.154210\pi$
$60$		0			0
$61$		−	57.2849i		−	0.939097i	−0.882907	−	0.469548i	$-0.844417\pi$
$61$		−	57.2849i		−	0.939097i	0.882907	−	0.469548i	$-0.155583\pi$
$62$		0			0
$63$	21.1903			0.336354
$64$		0			0
$65$	21.6422	+	11.9232i	0.332957	+	0.183433i
$66$		0			0
$67$		−	99.3796i		−	1.48328i	−0.670799	−	0.741639i	$-0.734049\pi$
$67$		−	99.3796i		−	1.48328i	0.670799	−	0.741639i	$-0.265951\pi$
$68$		0			0
$69$			75.5905i			1.09551i
$70$		0			0
$71$			16.7156i			0.235431i	0.993047	+	0.117716i	$0.0375572\pi$
$71$			16.7156i			0.235431i	−0.993047	+	0.117716i	$0.962443\pi$
$72$		0			0
$73$		−	96.3357i		−	1.31967i	−0.751412	−	0.659833i	$-0.770627\pi$
$73$		−	96.3357i		−	1.31967i	0.751412	−	0.659833i	$-0.229373\pi$
$74$		0			0
$75$	−40.3297	+	25.4927i	−0.537729	+	0.339903i
$76$		0			0
$77$	24.4684			0.317772
$78$		0			0
$79$			139.578i			1.76681i	0.468608	+	0.883406i	$0.344756\pi$
$79$			139.578i			1.76681i	−0.468608	+	0.883406i	$0.655244\pi$
$80$		0			0
$81$	−4.07345			−0.0502895
$82$		0			0
$83$		−	91.7458i		−	1.10537i	−0.833390	−	0.552686i	$-0.813603\pi$
$83$		−	91.7458i		−	1.10537i	0.833390	−	0.552686i	$-0.186397\pi$
$84$		0			0
$85$	54.1195	−	98.2344i	0.636700	−	1.15570i
$86$		0			0
$87$	−58.0844			−0.667637
$88$		0			0
$89$	−94.7156			−1.06422			−0.532110	−	0.846675i	$-0.678601\pi$
$89$	−94.7156			−1.06422			−0.532110	+	0.846675i	$0.678601\pi$
$90$		0			0
$91$	−19.5451			−0.214781
$92$		0			0
$93$	47.1686			0.507189
$94$		0			0
$95$	−45.5118	−	25.0735i	−0.479071	−	0.263931i
$96$		0			0
$97$		−	143.796i		−	1.48243i	−0.671267	−	0.741216i	$-0.734249\pi$
$97$		−	143.796i		−	1.48243i	0.671267	−	0.741216i	$-0.265751\pi$
$98$		0			0
$99$	33.1471			0.334819
$100$		0			0
$101$		−	66.5595i		−	0.659005i	−0.944155	−	0.329502i	$-0.893119\pi$
$101$		−	66.5595i		−	0.659005i	0.944155	−	0.329502i	$-0.106881\pi$
$102$		0			0
$103$	−153.063			−1.48605			−0.743024	−	0.669264i	$-0.766609\pi$
$103$	−153.063			−1.48605			−0.743024	+	0.669264i	$0.766609\pi$
$104$		0			0
$105$	18.2109	−	33.0553i	0.173437	−	0.314813i
$106$		0			0
$107$			75.7953i			0.708367i	0.935176	+	0.354184i	$0.115241\pi$
$107$			75.7953i			0.708367i	−0.935176	+	0.354184i	$0.884759\pi$
$108$		0			0
$109$			124.088i			1.13842i	0.822192	+	0.569211i	$0.192751\pi$
$109$			124.088i			1.13842i	−0.822192	+	0.569211i	$0.807249\pi$
$110$		0			0
$111$		−	45.8531i		−	0.413091i
$112$		0			0
$113$		−	5.19589i		−	0.0459814i	−0.999736	−	0.0229907i	$-0.992681\pi$
$113$		−	5.19589i		−	0.0459814i	0.999736	−	0.0229907i	$-0.00731881\pi$
$114$		0			0
$115$	−173.460	−	95.5627i	−1.50834	−	0.830980i
$116$		0			0
$117$	−26.4775			−0.226303
$118$		0			0
$119$			88.7156i			0.745510i
$120$		0			0
$121$	−82.7251			−0.683678
$122$		0			0
$123$			59.3021i			0.482131i
$124$		0			0
$125$	−7.51340	−	124.774i	−0.0601072	−	0.998192i
$126$		0			0
$127$	−109.383			−0.861287			−0.430644	−	0.902522i	$-0.641713\pi$
$127$	−109.383			−0.861287			−0.430644	+	0.902522i	$0.641713\pi$
$128$		0			0
$129$	−99.6422			−0.772420
$130$		0			0
$131$	−139.062			−1.06154			−0.530771	−	0.847515i	$-0.678098\pi$
$131$	−139.062			−1.06154			−0.530771	+	0.847515i	$0.678098\pi$
$132$		0			0
$133$	41.1018			0.309036
$134$		0			0
$135$	66.1107	−	120.000i	0.489709	−	0.888889i
$136$		0			0
$137$		−	5.42829i		−	0.0396226i	−0.999804	−	0.0198113i	$-0.993693\pi$
$137$		−	5.42829i		−	0.0396226i	0.999804	−	0.0198113i	$-0.00630654\pi$
$138$		0			0
$139$	191.765			1.37961			0.689803	−	0.723998i	$-0.257697\pi$
$139$	191.765			1.37961			0.689803	+	0.723998i	$0.257697\pi$
$140$		0			0
$141$			25.1229i			0.178177i
$142$		0			0
$143$	−30.5736			−0.213801
$144$		0			0
$145$	73.4313	−	133.288i	0.506423	−	0.919227i
$146$		0			0
$147$		−	63.6617i		−	0.433073i
$148$		0			0
$149$			184.959i			1.24133i	0.784075	+	0.620667i	$0.213138\pi$
$149$			184.959i			1.24133i	−0.784075	+	0.620667i	$0.786862\pi$
$150$		0			0
$151$			121.431i			0.804181i	0.915600	+	0.402090i	$0.131716\pi$
$151$			121.431i			0.804181i	−0.915600	+	0.402090i	$0.868284\pi$
$152$		0			0
$153$			120.182i			0.785503i
$154$		0			0
$155$	−59.6313	+	108.239i	−0.384718	+	0.698316i
$156$		0			0
$157$	261.156			1.66341			0.831706	−	0.555216i	$-0.187364\pi$
$157$	261.156			1.66341			0.831706	+	0.555216i	$0.187364\pi$
$158$		0			0
$159$			34.2749i			0.215566i
$160$		0			0
$161$	156.652			0.972991
$162$		0			0
$163$		−	199.302i		−	1.22271i	−0.791356	−	0.611356i	$-0.790625\pi$
$163$		−	199.302i		−	1.22271i	0.791356	−	0.611356i	$-0.209375\pi$
$164$		0			0
$165$	28.4866	−	51.7071i	0.172646	−	0.313376i
$166$		0			0
$167$	168.999			1.01197			0.505986	−	0.862542i	$-0.331129\pi$
$167$	168.999			1.01197			0.505986	+	0.862542i	$0.331129\pi$
$168$		0			0
$169$	−144.578			−0.855492
$170$		0			0
$171$	55.6801			0.325614
$172$		0			0
$173$	−295.307			−1.70697			−0.853487	−	0.521113i	$-0.825517\pi$
$173$	−295.307			−1.70697			−0.853487	+	0.521113i	$0.825517\pi$
$174$		0			0
$175$	52.8304	+	83.5782i	0.301888	+	0.477590i
$176$		0			0
$177$			199.282i			1.12589i
$178$		0			0
$179$	241.258			1.34781			0.673906	−	0.738817i	$-0.264615\pi$
$179$	241.258			1.34781			0.673906	+	0.738817i	$0.264615\pi$
$180$		0			0
$181$		−	137.833i		−	0.761511i	−0.924676	−	0.380755i	$-0.875664\pi$
$181$		−	137.833i		−	0.761511i	0.924676	−	0.380755i	$-0.124336\pi$
$182$		0			0
$183$	109.325			0.597406
$184$		0			0
$185$	105.220	+	57.9682i	0.568759	+	0.313342i
$186$		0			0
$187$			138.774i			0.742108i
$188$		0			0
$189$			108.372i			0.573398i
$190$		0			0
$191$		−	340.294i		−	1.78164i	−0.454354	−	0.890821i	$-0.650130\pi$
$191$		−	340.294i		−	1.78164i	0.454354	−	0.890821i	$-0.349870\pi$
$192$		0			0
$193$			96.5680i			0.500353i	0.968200	+	0.250176i	$0.0804886\pi$
$193$			96.5680i			0.500353i	−0.968200	+	0.250176i	$0.919511\pi$
$194$		0			0
$195$	−22.7548	+	41.3030i	−0.116691	+	0.211810i
$196$		0			0
$197$	−229.054			−1.16271			−0.581354	−	0.813651i	$-0.697477\pi$
$197$	−229.054			−1.16271			−0.581354	+	0.813651i	$0.697477\pi$
$198$		0			0
$199$			51.5782i			0.259187i	0.991567	+	0.129593i	$0.0413672\pi$
$199$			51.5782i			0.259187i	−0.991567	+	0.129593i	$0.958633\pi$
$200$		0			0
$201$	189.661			0.943587
$202$		0			0
$203$			120.373i			0.592968i
$204$		0			0
$205$	−136.082	−	74.9707i	−0.663816	−	0.365711i
$206$		0			0
$207$	212.214			1.02519
$208$		0			0
$209$	64.2938			0.307626
$210$		0			0
$211$	−77.7042			−0.368266			−0.184133	−	0.982901i	$-0.558948\pi$
$211$	−77.7042			−0.368266			−0.184133	+	0.982901i	$0.558948\pi$
$212$		0			0
$213$	−31.9010			−0.149770
$214$		0			0
$215$	125.969	−	228.652i	0.585904	−	1.06350i
$216$		0			0
$217$		−	97.7508i		−	0.450465i
$218$		0			0
$219$	183.852			0.839506
$220$		0			0
$221$		−	110.851i		−	0.501589i
$222$		0			0
$223$	−98.9917			−0.443909			−0.221954	−	0.975057i	$-0.571244\pi$
$223$	−98.9917			−0.443909			−0.221954	+	0.975057i	$0.571244\pi$
$224$		0			0
$225$	71.5687	+	113.222i	0.318083	+	0.503210i
$226$		0			0
$227$			172.584i			0.760280i	0.924929	+	0.380140i	$0.124124\pi$
$227$			172.584i			0.760280i	−0.924929	+	0.380140i	$0.875876\pi$
$228$		0			0
$229$			151.690i			0.662401i	0.943560	+	0.331201i	$0.107454\pi$
$229$			151.690i			0.662401i	−0.943560	+	0.331201i	$0.892546\pi$
$230$		0			0
$231$			46.6968i			0.202151i
$232$		0			0
$233$			289.007i			1.24037i	0.784454	+	0.620187i	$0.212943\pi$
$233$			289.007i			1.24037i	−0.784454	+	0.620187i	$0.787057\pi$
$234$		0			0
$235$	−57.6502	−	31.7608i	−0.245320	−	0.135152i
$236$		0			0
$237$	−266.378			−1.12396
$238$		0			0
$239$			17.5593i			0.0734699i	0.999325	+	0.0367349i	$0.0116957\pi$
$239$			17.5593i			0.0734699i	−0.999325	+	0.0367349i	$0.988304\pi$
$240$		0			0
$241$	−101.936			−0.422971			−0.211485	−	0.977381i	$-0.567830\pi$
$241$	−101.936			−0.422971			−0.211485	+	0.977381i	$0.567830\pi$
$242$		0			0
$243$			238.837i			0.982867i
$244$		0			0
$245$	146.086	+	80.4822i	0.596271	+	0.328499i
$246$		0			0
$247$	−51.3572			−0.207924
$248$		0			0
$249$	175.092			0.703182
$250$		0			0
$251$	−123.966			−0.493889			−0.246944	−	0.969030i	$-0.579427\pi$
$251$	−123.966			−0.493889			−0.246944	+	0.969030i	$0.579427\pi$
$252$		0			0
$253$	245.044			0.968552
$254$		0			0
$255$	187.475	+	103.284i	0.735198	+	0.405037i
$256$		0			0
$257$			118.534i			0.461223i	0.973046	+	0.230612i	$0.0740727\pi$
$257$			118.534i			0.461223i	−0.973046	+	0.230612i	$0.925927\pi$
$258$		0			0
$259$	−95.0247			−0.366891
$260$		0			0
$261$			163.067i			0.624779i
$262$		0			0
$263$	223.071			0.848177			0.424089	−	0.905621i	$-0.360595\pi$
$263$	223.071			0.848177			0.424089	+	0.905621i	$0.360595\pi$
$264$		0			0
$265$	−78.6516	−	43.3309i	−0.296799	−	0.163513i
$266$		0			0
$267$		−	180.760i		−	0.677004i
$268$		0			0
$269$			249.061i			0.925877i	0.886390	+	0.462938i	$0.153205\pi$
$269$			249.061i			0.925877i	−0.886390	+	0.462938i	$0.846795\pi$
$270$		0			0
$271$		−	130.991i		−	0.483360i	−0.970356	−	0.241680i	$-0.922302\pi$
$271$		−	130.991i		−	0.483360i	0.970356	−	0.241680i	$-0.0776984\pi$
$272$		0			0
$273$		−	37.3008i		−	0.136633i
$274$		0			0
$275$	82.6405	+	130.738i	0.300511	+	0.475410i
$276$		0			0
$277$	39.7756			0.143594			0.0717971	−	0.997419i	$-0.477127\pi$
$277$	39.7756			0.143594			0.0717971	+	0.997419i	$0.477127\pi$
$278$		0			0
$279$		−	132.422i		−	0.474630i
$280$		0			0
$281$	58.0640			0.206634			0.103317	−	0.994649i	$-0.467054\pi$
$281$	58.0640			0.206634			0.103317	+	0.994649i	$0.467054\pi$
$282$		0			0
$283$		−	241.288i		−	0.852607i	−0.904580	−	0.426304i	$-0.859816\pi$
$283$		−	241.288i		−	0.852607i	0.904580	−	0.426304i	$-0.140184\pi$
$284$		0			0
$285$	47.8514	−	86.8570i	0.167900	−	0.304761i
$286$		0			0
$287$	122.896			0.428209
$288$		0			0
$289$	−214.156			−0.741025
$290$		0			0
$291$	274.427			0.943049
$292$		0			0
$293$	−87.8283			−0.299755			−0.149878	−	0.988705i	$-0.547888\pi$
$293$	−87.8283			−0.299755			−0.149878	+	0.988705i	$0.547888\pi$
$294$		0			0
$295$	−457.299	−	251.936i	−1.55016	−	0.854020i
$296$		0			0
$297$			169.522i			0.570782i
$298$		0			0
$299$	−195.738			−0.654642
$300$		0			0
$301$			206.496i			0.686032i
$302$		0			0
$303$	127.025			0.419226
$304$		0			0
$305$	−138.211	+	250.872i	−0.453151	+	0.822531i
$306$		0			0
$307$		−	138.914i		−	0.452490i	−0.974070	−	0.226245i	$-0.927355\pi$
$307$		−	138.914i		−	0.452490i	0.974070	−	0.226245i	$-0.0726450\pi$
$308$		0			0
$309$		−	292.113i		−	0.945350i
$310$		0			0
$311$			330.863i			1.06387i	0.846786	+	0.531933i	$0.178534\pi$
$311$			330.863i			1.06387i	−0.846786	+	0.531933i	$0.821466\pi$
$312$		0			0
$313$		−	385.575i		−	1.23187i	−0.787797	−	0.615935i	$-0.788779\pi$
$313$		−	385.575i		−	1.23187i	0.787797	−	0.615935i	$-0.211221\pi$
$314$		0			0
$315$	−92.8001	−	51.1256i	−0.294604	−	0.162304i
$316$		0			0
$317$	414.112			1.30635			0.653174	−	0.757208i	$-0.273437\pi$
$317$	414.112			1.30635			0.653174	+	0.757208i	$0.273437\pi$
$318$		0			0
$319$			188.294i			0.590263i
$320$		0			0
$321$	−144.652			−0.450628
$322$		0			0
$323$			233.111i			0.721707i
$324$		0			0
$325$	−66.0123	−	104.432i	−0.203115	−	0.321329i
$326$		0			0
$327$	−236.816			−0.724207
$328$		0			0
$329$	52.0640			0.158249
$330$		0			0
$331$	−52.4704			−0.158521			−0.0792604	−	0.996854i	$-0.525256\pi$
$331$	−52.4704			−0.158521			−0.0792604	+	0.996854i	$0.525256\pi$
$332$		0			0
$333$	−128.729			−0.386573
$334$		0			0
$335$	−239.773	+	435.220i	−0.715739	+	1.29917i
$336$		0			0
$337$		−	469.871i		−	1.39428i	−0.716937	−	0.697138i	$-0.754456\pi$
$337$		−	469.871i		−	1.39428i	0.716937	−	0.697138i	$-0.245544\pi$
$338$		0			0
$339$	9.91611			0.0292511
$340$		0			0
$341$		−	152.908i		−	0.448409i
$342$		0			0
$343$	−325.727			−0.949641
$344$		0			0
$345$	182.377	−	331.039i	0.528628	−	0.959533i
$346$		0			0
$347$		−	267.604i		−	0.771192i	−0.922668	−	0.385596i	$-0.873996\pi$
$347$		−	267.604i		−	0.771192i	0.922668	−	0.385596i	$-0.126004\pi$
$348$		0			0
$349$		−	662.916i		−	1.89947i	−0.313053	−	0.949736i	$-0.601352\pi$
$349$		−	662.916i		−	1.89947i	0.313053	−	0.949736i	$-0.398648\pi$
$350$		0			0
$351$		−	135.412i		−	0.385790i
$352$		0			0
$353$		−	366.925i		−	1.03945i	−0.854335	−	0.519723i	$-0.826035\pi$
$353$		−	366.925i		−	1.03945i	0.854335	−	0.519723i	$-0.173965\pi$
$354$		0			0
$355$	40.3297	−	73.2040i	0.113605	−	0.206208i
$356$		0			0
$357$	−169.309			−0.474256
$358$		0			0
$359$		−	234.863i		−	0.654213i	−0.944987	−	0.327107i	$-0.893926\pi$
$359$		−	234.863i		−	0.654213i	0.944987	−	0.327107i	$-0.106074\pi$
$360$		0			0
$361$	−253.000			−0.700831
$362$		0			0
$363$		−	157.877i		−	0.434922i
$364$		0			0
$365$	−232.428	+	421.890i	−0.636790	+	1.15586i
$366$		0			0
$367$	−455.060			−1.23994			−0.619972	−	0.784624i	$-0.712856\pi$
$367$	−455.060			−1.23994			−0.619972	+	0.784624i	$0.712856\pi$
$368$		0			0
$369$	166.486			0.451181
$370$		0			0
$371$	71.0304			0.191457
$372$		0			0
$373$	518.534			1.39017			0.695086	−	0.718926i	$-0.255366\pi$
$373$	518.534			1.39017			0.695086	+	0.718926i	$0.255366\pi$
$374$		0			0
$375$	238.125	−	14.3389i	0.635000	−	0.0382372i
$376$		0			0
$377$		−	150.407i		−	0.398957i
$378$		0			0
$379$	−563.918			−1.48791			−0.743955	−	0.668230i	$-0.767052\pi$
$379$	−563.918			−1.48791			−0.743955	+	0.668230i	$0.767052\pi$
$380$		0			0
$381$		−	208.753i		−	0.547908i
$382$		0			0
$383$	56.8436			0.148417			0.0742083	−	0.997243i	$-0.476357\pi$
$383$	56.8436			0.148417			0.0742083	+	0.997243i	$0.476357\pi$
$384$		0			0
$385$	−107.156	−	59.0348i	−0.278328	−	0.153337i
$386$		0			0
$387$			279.737i			0.722835i
$388$		0			0
$389$		−	188.677i		−	0.485031i	−0.970147	−	0.242516i	$-0.922027\pi$
$389$		−	188.677i		−	0.485031i	0.970147	−	0.242516i	$-0.0779726\pi$
$390$		0			0
$391$			888.460i			2.27228i
$392$		0			0
$393$		−	265.393i		−	0.675300i
$394$		0			0
$395$	336.759	−	611.265i	0.852555	−	1.54751i
$396$		0			0
$397$	−143.716			−0.362005			−0.181003	−	0.983483i	$-0.557934\pi$
$397$	−143.716			−0.362005			−0.181003	+	0.983483i	$0.557934\pi$
$398$		0			0
$399$			78.4407i			0.196593i
$400$		0			0
$401$	1.28437			0.00320291			0.00160145	−	0.999999i	$-0.499490\pi$
$401$	1.28437			0.00320291			0.00160145	+	0.999999i	$0.499490\pi$
$402$		0			0
$403$			122.141i			0.303079i
$404$		0			0
$405$	17.8392	+	9.82799i	0.0440473	+	0.0242666i
$406$		0			0
$407$	−148.643			−0.365217
$408$		0			0
$409$	−76.3767			−0.186740			−0.0933700	−	0.995631i	$-0.529764\pi$
$409$	−76.3767			−0.186740			−0.0933700	+	0.995631i	$0.529764\pi$
$410$		0			0
$411$	10.3596			0.0252059
$412$		0			0
$413$	412.987			0.999969
$414$		0			0
$415$	−221.355	+	401.789i	−0.533384	+	0.968166i
$416$		0			0
$417$			365.974i			0.877636i
$418$		0			0
$419$	−33.3906			−0.0796912			−0.0398456	−	0.999206i	$-0.512687\pi$
$419$	−33.3906			−0.0796912			−0.0398456	+	0.999206i	$0.512687\pi$
$420$		0			0
$421$			392.318i			0.931871i	0.884819	+	0.465935i	$0.154282\pi$
$421$			392.318i			0.931871i	−0.884819	+	0.465935i	$0.845718\pi$
$422$		0			0
$423$	70.5305			0.166739
$424$		0			0
$425$	−474.019	+	299.631i	−1.11534	+	0.705014i
$426$		0			0
$427$		−	226.563i		−	0.530592i
$428$		0			0
$429$		−	58.3482i		−	0.136010i
$430$		0			0
$431$		−	840.460i		−	1.95002i	−0.222156	−	0.975011i	$-0.571309\pi$
$431$		−	840.460i		−	1.95002i	0.222156	−	0.975011i	$-0.428691\pi$
$432$		0			0
$433$		−	275.552i		−	0.636380i	−0.948027	−	0.318190i	$-0.896925\pi$
$433$		−	275.552i		−	0.636380i	0.948027	−	0.318190i	$-0.103075\pi$
$434$		0			0
$435$	254.373	+	140.140i	0.584766	+	0.322161i
$436$		0			0
$437$	411.622			0.941926
$438$		0			0
$439$		−	477.725i		−	1.08821i	−0.839016	−	0.544106i	$-0.816869\pi$
$439$		−	477.725i		−	1.08821i	0.839016	−	0.544106i	$-0.183131\pi$
$440$		0			0
$441$	−178.725			−0.405272
$442$		0			0
$443$		−	625.709i		−	1.41244i	−0.707994	−	0.706218i	$-0.750400\pi$
$443$		−	625.709i		−	1.41244i	0.707994	−	0.706218i	$-0.249600\pi$
$444$		0			0
$445$	414.795	+	228.520i	0.932124	+	0.513528i
$446$		0			0
$447$	−352.984			−0.789674
$448$		0			0
$449$	104.377			0.232465			0.116232	−	0.993222i	$-0.462918\pi$
$449$	104.377			0.232465			0.116232	+	0.993222i	$0.462918\pi$
$450$		0			0
$451$	192.241			0.426256
$452$		0			0
$453$	−231.746			−0.511580
$454$		0			0
$455$	85.5953	+	47.1563i	0.188121	+	0.103640i
$456$		0			0
$457$		−	198.564i		−	0.434495i	−0.976117	−	0.217248i	$-0.930292\pi$
$457$		−	198.564i		−	0.434495i	0.976117	−	0.217248i	$-0.0697079\pi$
$458$		0			0
$459$	−614.640			−1.33908
$460$		0			0
$461$			11.1338i			0.0241515i	0.999927	+	0.0120757i	$0.00384392\pi$
$461$			11.1338i			0.0241515i	−0.999927	+	0.0120757i	$0.996156\pi$
$462$		0			0
$463$	15.9944			0.0345451			0.0172725	−	0.999851i	$-0.494502\pi$
$463$	15.9944			0.0345451			0.0172725	+	0.999851i	$0.494502\pi$
$464$		0			0
$465$	−206.569	−	113.803i	−0.444234	−	0.244738i
$466$		0			0
$467$			539.567i			1.15539i	0.816253	+	0.577695i	$0.196048\pi$
$467$			539.567i			1.15539i	−0.816253	+	0.577695i	$0.803952\pi$
$468$		0			0
$469$		−	393.048i		−	0.838056i
$470$		0			0
$471$			498.403i			1.05818i
$472$		0			0
$473$			323.013i			0.682902i
$474$		0			0
$475$	138.819	+	219.612i	0.292249	+	0.462341i
$476$		0			0
$477$	96.2240			0.201728
$478$		0			0
$479$			127.744i			0.266689i	0.991070	+	0.133344i	$0.0425716\pi$
$479$			127.744i			0.266689i	−0.991070	+	0.133344i	$0.957428\pi$
$480$		0			0
$481$	118.735			0.246849
$482$		0			0
$483$			298.962i			0.618969i
$484$		0			0
$485$	−346.935	+	629.736i	−0.715331	+	1.29842i
$486$		0			0
$487$	624.466			1.28227			0.641135	−	0.767428i	$-0.278464\pi$
$487$	624.466			1.28227			0.641135	+	0.767428i	$0.278464\pi$
$488$		0			0
$489$	380.358			0.777828
$490$		0			0
$491$	−424.601			−0.864769			−0.432384	−	0.901689i	$-0.642328\pi$
$491$	−424.601			−0.864769			−0.432384	+	0.901689i	$0.642328\pi$
$492$		0			0
$493$	−682.701			−1.38479
$494$		0			0
$495$	−145.163	−	79.9737i	−0.293259	−	0.161563i
$496$		0			0
$497$			66.1107i			0.133019i
$498$		0			0
$499$	−591.631			−1.18563			−0.592816	−	0.805338i	$-0.701984\pi$
$499$	−591.631			−1.18563			−0.592816	+	0.805338i	$0.701984\pi$
$500$		0			0
$501$			322.527i			0.643766i
$502$		0			0
$503$	−5.71879			−0.0113694			−0.00568468	−	0.999984i	$-0.501809\pi$
$503$	−5.71879			−0.0113694			−0.00568468	+	0.999984i	$0.501809\pi$
$504$		0			0
$505$	−160.588	+	291.489i	−0.317995	+	0.577205i
$506$		0			0
$507$		−	275.920i		−	0.544221i
$508$		0			0
$509$		−	439.708i		−	0.863867i	−0.901905	−	0.431933i	$-0.857832\pi$
$509$		−	439.708i		−	0.863867i	0.901905	−	0.431933i	$-0.142168\pi$
$510$		0			0
$511$		−	381.009i		−	0.745615i
$512$		0			0
$513$			284.761i			0.555091i
$514$		0			0
$515$	670.320	+	369.294i	1.30159	+	0.717076i
$516$		0			0
$517$	81.4416			0.157527
$518$		0			0
$519$		−	563.578i		−	1.08589i
$520$		0			0
$521$	80.5687			0.154642			0.0773212	−	0.997006i	$-0.475363\pi$
$521$	80.5687			0.154642			0.0773212	+	0.997006i	$0.475363\pi$
$522$		0			0
$523$			585.089i			1.11872i	0.828926	+	0.559359i	$0.188953\pi$
$523$			585.089i			1.11872i	−0.828926	+	0.559359i	$0.811047\pi$
$524$		0			0
$525$	−159.505	+	100.824i	−0.303819	+	0.192046i
$526$		0			0
$527$	554.400			1.05199
$528$		0			0
$529$	1039.82			1.96563
$530$		0			0
$531$	559.469			1.05361
$532$		0			0
$533$	−153.560			−0.288105
$534$		0			0
$535$	182.871	−	331.936i	0.341815	−	0.620441i
$536$		0			0
$537$			460.430i			0.857411i
$538$		0			0
$539$	−206.374			−0.382883
$540$		0			0
$541$			248.685i			0.459676i	0.973229	+	0.229838i	$0.0738196\pi$
$541$			248.685i			0.459676i	−0.973229	+	0.229838i	$0.926180\pi$
$542$		0			0
$543$	263.048			0.484435
$544$		0			0
$545$	299.386	−	543.427i	0.549332	−	0.997114i
$546$		0			0
$547$		−	47.9916i		−	0.0877360i	−0.999037	−	0.0438680i	$-0.986032\pi$
$547$		−	47.9916i		−	0.0877360i	0.999037	−	0.0438680i	$-0.0139681\pi$
$548$		0			0
$549$		−	306.922i		−	0.559056i
$550$		0			0
$551$			316.294i			0.574036i
$552$		0			0
$553$			552.034i			0.998254i
$554$		0			0
$555$	−110.629	+	200.808i	−0.199332	+	0.361816i
$556$		0			0
$557$	−1091.71			−1.95998			−0.979990	−	0.199045i	$-0.936216\pi$
$557$	−1091.71			−1.95998			−0.979990	+	0.199045i	$0.936216\pi$
$558$		0			0
$559$		−	258.019i		−	0.461572i
$560$		0			0
$561$	−264.844			−0.472092
$562$		0			0
$563$			427.231i			0.758846i	0.925223	+	0.379423i	$0.123878\pi$
$563$			427.231i			0.758846i	−0.925223	+	0.379423i	$0.876122\pi$
$564$		0			0
$565$	−12.5361	+	22.7548i	−0.0221878	+	0.0402739i
$566$		0			0
$567$	−16.1106			−0.0284137
$568$		0			0
$569$	54.2298			0.0953072			0.0476536	−	0.998864i	$-0.484826\pi$
$569$	54.2298			0.0953072			0.0476536	+	0.998864i	$0.484826\pi$
$570$		0			0
$571$	1050.42			1.83961			0.919806	−	0.392373i	$-0.128346\pi$
$571$	1050.42			1.83961			0.919806	+	0.392373i	$0.128346\pi$
$572$		0			0
$573$	649.434			1.13339
$574$		0			0
$575$	529.080	+	837.009i	0.920140	+	1.45567i
$576$		0			0
$577$			161.031i			0.279083i	0.990216	+	0.139542i	$0.0445629\pi$
$577$			161.031i			0.279083i	−0.990216	+	0.139542i	$0.955437\pi$
$578$		0			0
$579$	−184.295			−0.318299
$580$		0			0
$581$		−	362.856i		−	0.624538i
$582$		0			0
$583$	111.110			0.190583
$584$		0			0
$585$	115.955	+	63.8821i	0.198213	+	0.109200i
$586$		0			0
$587$		−	186.363i		−	0.317484i	−0.987320	−	0.158742i	$-0.949256\pi$
$587$		−	186.363i		−	0.317484i	0.987320	−	0.158742i	$-0.0507439\pi$
$588$		0			0
$589$		−	256.852i		−	0.436082i
$590$		0			0
$591$		−	437.137i		−	0.739657i
$592$		0			0
$593$			1011.51i			1.70576i	0.522109	+	0.852879i	$0.325145\pi$
$593$			1011.51i			1.70576i	−0.522109	+	0.852879i	$0.674855\pi$
$594$		0			0
$595$	214.044	−	388.519i	0.359737	−	0.652973i
$596$		0			0
$597$	−98.4344			−0.164882
$598$		0			0
$599$		−	647.616i		−	1.08116i	−0.841292	−	0.540581i	$-0.818204\pi$
$599$		−	647.616i		−	1.08116i	0.841292	−	0.540581i	$-0.181796\pi$
$600$		0			0
$601$	40.3767			0.0671825			0.0335913	−	0.999436i	$-0.489306\pi$
$601$	40.3767			0.0671825			0.0335913	+	0.999436i	$0.489306\pi$
$602$		0			0
$603$		−	532.458i		−	0.883014i
$604$		0			0
$605$	362.284	+	199.590i	0.598816	+	0.329901i
$606$		0			0
$607$	799.902			1.31780			0.658898	−	0.752233i	$-0.271023\pi$
$607$	799.902			1.31780			0.658898	+	0.752233i	$0.271023\pi$
$608$		0			0
$609$	−229.725			−0.377217
$610$		0			0
$611$	−65.0546			−0.106472
$612$		0			0
$613$	−900.584			−1.46914			−0.734571	−	0.678532i	$-0.762616\pi$
$613$	−900.584			−1.46914			−0.734571	+	0.678532i	$0.762616\pi$
$614$		0			0
$615$	143.078	−	259.706i	0.232647	−	0.422287i
$616$		0			0
$617$		−	255.001i		−	0.413292i	−0.978416	−	0.206646i	$-0.933745\pi$
$617$		−	255.001i		−	0.413292i	0.978416	−	0.206646i	$-0.0662549\pi$
$618$		0			0
$619$	168.789			0.272679			0.136340	−	0.990662i	$-0.456466\pi$
$619$	168.789			0.272679			0.136340	+	0.990662i	$0.456466\pi$
$620$		0			0
$621$			1085.31i			1.74769i
$622$		0			0
$623$	−374.602			−0.601288
$624$		0			0
$625$	−268.137	+	564.559i	−0.429020	+	0.903295i
$626$		0			0
$627$			122.702i			0.195696i
$628$		0			0
$629$		−	538.939i		−	0.856818i
$630$		0			0
$631$			665.431i			1.05457i	0.849690	+	0.527283i	$0.176789\pi$
$631$			665.431i			1.05457i	−0.849690	+	0.527283i	$0.823211\pi$
$632$		0			0
$633$		−	148.295i		−	0.234273i
$634$		0			0
$635$	479.031	+	263.909i	0.754380	+	0.415605i
$636$		0			0
$637$	164.849			0.258790
$638$		0			0
$639$			89.5593i			0.140155i
$640$		0			0
$641$	552.633			0.862142			0.431071	−	0.902318i	$-0.358136\pi$
$641$	552.633			0.862142			0.431071	+	0.902318i	$0.358136\pi$
$642$		0			0
$643$		−	901.490i		−	1.40201i	−0.713159	−	0.701003i	$-0.752736\pi$
$643$		−	901.490i		−	1.40201i	0.713159	−	0.701003i	$-0.247264\pi$
$644$		0			0
$645$	436.370	+	240.406i	0.676543	+	0.372723i
$646$		0			0
$647$	−67.8163			−0.104817			−0.0524083	−	0.998626i	$-0.516690\pi$
$647$	−67.8163			−0.104817			−0.0524083	+	0.998626i	$0.516690\pi$
$648$		0			0
$649$	646.019			0.995407
$650$		0			0
$651$	186.553			0.286563
$652$		0			0
$653$	−124.351			−0.190431			−0.0952153	−	0.995457i	$-0.530354\pi$
$653$	−124.351			−0.190431			−0.0952153	+	0.995457i	$0.530354\pi$
$654$		0			0
$655$	609.005	+	335.514i	0.929778	+	0.512235i
$656$		0			0
$657$		−	516.149i		−	0.785615i
$658$		0			0
$659$	−260.582			−0.395420			−0.197710	−	0.980261i	$-0.563350\pi$
$659$	−260.582			−0.395420			−0.197710	+	0.980261i	$0.563350\pi$
$660$		0			0
$661$			447.965i			0.677708i	0.940839	+	0.338854i	$0.110039\pi$
$661$			447.965i			0.677708i	−0.940839	+	0.338854i	$0.889961\pi$
$662$		0			0
$663$	211.554			0.319086
$664$		0			0
$665$	−180.000	−	99.1660i	−0.270677	−	0.149122i
$666$		0			0
$667$			1205.49i			1.80734i
$668$		0			0
$669$		−	188.921i		−	0.282393i
$670$		0			0
$671$		−	354.403i		−	0.528171i
$672$		0			0
$673$			288.775i			0.429086i	0.976715	+	0.214543i	$0.0688261\pi$
$673$			288.775i			0.429086i	−0.976715	+	0.214543i	$0.931174\pi$
$674$		0			0
$675$	−579.047	+	366.020i	−0.857847	+	0.542252i
$676$		0			0
$677$	248.339			0.366823			0.183412	−	0.983036i	$-0.441286\pi$
$677$	248.339			0.366823			0.183412	+	0.983036i	$0.441286\pi$
$678$		0			0
$679$		−	568.716i		−	0.837578i
$680$		0			0
$681$	−329.367			−0.483652
$682$		0			0
$683$			451.900i			0.661640i	0.943694	+	0.330820i	$0.107325\pi$
$683$			451.900i			0.661640i	−0.943694	+	0.330820i	$0.892675\pi$
$684$		0			0
$685$	−13.0968	+	23.7725i	−0.0191194	+	0.0347044i
$686$		0			0
$687$	−289.493			−0.421387
$688$		0			0
$689$	−88.7534			−0.128815
$690$		0			0
$691$	−193.513			−0.280048			−0.140024	−	0.990148i	$-0.544718\pi$
$691$	−193.513			−0.280048			−0.140024	+	0.990148i	$0.544718\pi$
$692$		0			0
$693$	131.097			0.189174
$694$		0			0
$695$	−839.811	−	462.670i	−1.20836	−	0.665713i
$696$		0			0
$697$			697.013i			1.00002i
$698$		0			0
$699$	−551.555			−0.789064
$700$		0			0
$701$		−	928.025i		−	1.32386i	−0.749566	−	0.661929i	$-0.769738\pi$
$701$		−	928.025i		−	1.32386i	0.749566	−	0.661929i	$-0.230262\pi$
$702$		0			0
$703$	−249.689			−0.355177
$704$		0			0
$705$	60.6139	−	110.023i	0.0859771	−	0.156060i
$706$		0			0
$707$		−	263.244i		−	0.372339i
$708$		0			0
$709$			75.9885i			0.107177i	0.998563	+	0.0535885i	$0.0170659\pi$
$709$			75.9885i			0.107177i	−0.998563	+	0.0535885i	$0.982934\pi$
$710$		0			0
$711$			747.834i			1.05181i
$712$		0			0
$713$		−	978.944i		−	1.37299i
$714$		0			0
$715$	133.893	+	73.7647i	0.187263	+	0.103167i
$716$		0			0
$717$	−33.5111			−0.0467379
$718$		0			0
$719$			395.578i			0.550178i	0.961419	+	0.275089i	$0.0887074\pi$
$719$			395.578i			0.550178i	−0.961419	+	0.275089i	$0.911293\pi$
$720$		0			0
$721$	−605.367			−0.839622
$722$		0			0
$723$		−	194.540i		−	0.269073i
$724$		0			0
$725$	−643.166	+	406.550i	−0.887125	+	0.560759i
$726$		0			0
$727$	−352.113			−0.484337			−0.242168	−	0.970234i	$-0.577859\pi$
$727$	−352.113			−0.484337			−0.242168	+	0.970234i	$0.577859\pi$
$728$		0			0
$729$	−492.469			−0.675540
$730$		0			0
$731$	−1171.15			−1.60213
$732$		0			0
$733$	−156.532			−0.213550			−0.106775	−	0.994283i	$-0.534053\pi$
$733$	−156.532			−0.213550			−0.106775	+	0.994283i	$0.534053\pi$
$734$		0			0
$735$	−153.596	+	278.799i	−0.208975	+	0.379318i
$736$		0			0
$737$		−	614.829i		−	0.834232i
$738$		0			0
$739$	−183.376			−0.248140			−0.124070	−	0.992273i	$-0.539595\pi$
$739$	−183.376			−0.248140			−0.124070	+	0.992273i	$0.539595\pi$
$740$		0			0
$741$		−	98.0126i		−	0.132271i
$742$		0			0
$743$	−1028.34			−1.38404			−0.692021	−	0.721878i	$-0.743279\pi$
$743$	−1028.34			−1.38404			−0.692021	+	0.721878i	$0.743279\pi$
$744$		0			0
$745$	446.249	−	810.003i	0.598991	−	1.08725i
$746$		0			0
$747$		−	491.557i		−	0.658042i
$748$		0			0
$749$			299.772i			0.400230i
$750$		0			0
$751$		−	740.038i		−	0.985403i	−0.870198	−	0.492702i	$-0.836009\pi$
$751$		−	740.038i		−	0.985403i	0.870198	−	0.492702i	$-0.163991\pi$
$752$		0			0
$753$		−	236.583i		−	0.314188i
$754$		0			0
$755$	292.976	−	531.793i	0.388048	−	0.704361i
$756$		0			0
$757$	934.857			1.23495			0.617475	−	0.786591i	$-0.288156\pi$
$757$	934.857			1.23495			0.617475	+	0.786591i	$0.288156\pi$
$758$		0			0
$759$			467.654i			0.616144i
$760$		0			0
$761$	436.735			0.573896			0.286948	−	0.957946i	$-0.407359\pi$
$761$	436.735			0.573896			0.286948	+	0.957946i	$0.407359\pi$
$762$		0			0
$763$			490.770i			0.643211i
$764$		0			0
$765$	289.962	−	526.322i	0.379036	−	0.688002i
$766$		0			0
$767$	−516.033			−0.672793
$768$		0			0
$769$	−768.275			−0.999057			−0.499529	−	0.866297i	$-0.666493\pi$
$769$	−768.275			−0.999057			−0.499529	+	0.866297i	$0.666493\pi$
$770$		0			0
$771$	−226.217			−0.293407
$772$		0			0
$773$	792.266			1.02492			0.512462	−	0.858710i	$-0.328734\pi$
$773$	792.266			1.02492			0.512462	+	0.858710i	$0.328734\pi$
$774$		0			0
$775$	522.295	−	330.147i	0.673929	−	0.425996i
$776$		0			0
$777$		−	181.350i		−	0.233398i
$778$		0			0
$779$	322.925			0.414538
$780$		0			0
$781$			103.414i			0.132413i
$782$		0			0
$783$	−833.966			−1.06509
$784$		0			0
$785$	−1143.70	−	630.089i	−1.45694	−	0.802661i
$786$		0			0
$787$		−	965.852i		−	1.22726i	−0.789594	−	0.613629i	$-0.789709\pi$
$787$		−	965.852i		−	1.22726i	0.789594	−	0.613629i	$-0.210291\pi$
$788$		0			0
$789$			425.719i			0.539568i
$790$		0			0
$791$		−	20.5499i		−	0.0259796i
$792$		0			0
$793$			283.093i			0.356990i
$794$		0			0
$795$	82.6949	−	150.103i	0.104019	−	0.188808i
$796$		0			0
$797$	358.505			0.449818			0.224909	−	0.974380i	$-0.427792\pi$
$797$	358.505			0.449818			0.224909	+	0.974380i	$0.427792\pi$
$798$		0			0
$799$			295.284i			0.369567i
$800$		0			0
$801$	−507.469			−0.633544
$802$		0			0
$803$		−	595.997i		−	0.742213i
$804$		0			0
$805$	−686.036	−	377.952i	−0.852218	−	0.469506i
$806$		0			0
$807$	−475.320			−0.588997
$808$		0			0
$809$	222.881			0.275502			0.137751	−	0.990467i	$-0.456013\pi$
$809$	222.881			0.275502			0.137751	+	0.990467i	$0.456013\pi$
$810$		0			0
$811$	1292.40			1.59358			0.796792	−	0.604253i	$-0.206529\pi$
$811$	1292.40			1.59358			0.796792	+	0.604253i	$0.206529\pi$
$812$		0			0
$813$	249.989			0.307490
$814$		0			0
$815$	−480.854	+	872.817i	−0.590005	+	1.07094i
$816$		0			0
$817$			542.593i			0.664129i
$818$		0			0
$819$	−104.719			−0.127862
$820$		0			0
$821$		−	1276.69i		−	1.55505i	−0.628855	−	0.777523i	$-0.716476\pi$
$821$		−	1276.69i		−	1.55505i	0.628855	−	0.777523i	$-0.283524\pi$
$822$		0			0
$823$	1177.68			1.43096			0.715482	−	0.698631i	$-0.246207\pi$
$823$	1177.68			1.43096			0.715482	+	0.698631i	$0.246207\pi$
$824$		0			0
$825$	−249.507	+	157.715i	−0.302432	+	0.191170i
$826$		0			0
$827$			154.182i			0.186435i	0.995646	+	0.0932176i	$0.0297152\pi$
$827$			154.182i			0.186435i	−0.995646	+	0.0932176i	$0.970285\pi$
$828$		0			0
$829$		−	243.838i		−	0.294135i	−0.989126	−	0.147067i	$-0.953017\pi$
$829$		−	243.838i		−	0.294135i	0.989126	−	0.147067i	$-0.0469834\pi$
$830$		0			0
$831$			75.9097i			0.0913474i
$832$		0			0
$833$		−	748.254i		−	0.898264i
$834$		0			0
$835$	−740.111	−	407.744i	−0.886360	−	0.488316i
$836$		0			0
$837$	677.238			0.809125
$838$		0			0
$839$			1521.51i			1.81348i	0.421694	+	0.906738i	$0.361436\pi$
$839$			1521.51i			1.81348i	−0.421694	+	0.906738i	$0.638564\pi$
$840$		0			0
$841$	−85.3127			−0.101442
$842$		0			0
$843$			110.812i			0.131450i
$844$		0			0
$845$	633.162	+	348.823i	0.749304	+	0.412808i
$846$		0			0
$847$	−327.179			−0.386280
$848$		0			0
$849$	460.486			0.542386
$850$		0			0
$851$	−951.643			−1.11826
$852$		0			0
$853$	59.6651			0.0699474			0.0349737	−	0.999388i	$-0.488865\pi$
$853$	59.6651			0.0699474			0.0349737	+	0.999388i	$0.488865\pi$
$854$		0			0
$855$	−243.844	−	134.339i	−0.285197	−	0.157122i
$856$		0			0
$857$		−	1263.47i		−	1.47430i	−0.675730	−	0.737149i	$-0.736172\pi$
$857$		−	1263.47i		−	1.47430i	0.675730	−	0.737149i	$-0.263828\pi$
$858$		0			0
$859$	1067.99			1.24330			0.621649	−	0.783296i	$-0.286463\pi$
$859$	1067.99			1.24330			0.621649	+	0.783296i	$0.286463\pi$
$860$		0			0
$861$			234.541i			0.272406i
$862$		0			0
$863$	439.103			0.508810			0.254405	−	0.967098i	$-0.418120\pi$
$863$	439.103			0.508810			0.254405	+	0.967098i	$0.418120\pi$
$864$		0			0
$865$	1293.26	+	712.485i	1.49510	+	0.823682i
$866$		0			0
$867$		−	408.707i		−	0.471403i
$868$		0			0
$869$			863.525i			0.993699i
$870$		0			0
$871$			491.119i			0.563856i
$872$		0			0
$873$		−	770.432i		−	0.882511i
$874$		0			0
$875$	−29.7156	−	493.484i	−0.0339607	−	0.563981i
$876$		0			0
$877$	−932.204			−1.06295			−0.531473	−	0.847075i	$-0.678361\pi$
$877$	−932.204			−1.06295			−0.531473	+	0.847075i	$0.678361\pi$
$878$		0			0
$879$		−	167.616i		−	0.190689i
$880$		0			0
$881$	223.073			0.253205			0.126602	−	0.991954i	$-0.459593\pi$
$881$	223.073			0.253205			0.126602	+	0.991954i	$0.459593\pi$
$882$		0			0
$883$		−	790.958i		−	0.895762i	−0.894093	−	0.447881i	$-0.852179\pi$
$883$		−	790.958i		−	0.895762i	0.894093	−	0.447881i	$-0.147821\pi$
$884$		0			0
$885$	480.807	−	872.732i	0.543285	−	0.986138i
$886$		0			0
$887$	233.462			0.263204			0.131602	−	0.991303i	$-0.457988\pi$
$887$	233.462			0.263204			0.131602	+	0.991303i	$0.457988\pi$
$888$		0			0
$889$	−432.614			−0.486630
$890$		0			0
$891$	−25.2011			−0.0282841
$892$		0			0
$893$	136.805			0.153197
$894$		0			0
$895$	−1056.56	−	582.083i	−1.18052	−	0.650372i
$896$		0			0
$897$		−	373.556i		−	0.416451i
$898$		0			0
$899$	752.230			0.836741
$900$		0			0
$901$			402.854i			0.447118i
$902$		0			0
$903$	−394.087			−0.436420
$904$		0			0
$905$	−332.550	+	603.624i	−0.367458	+	0.666988i
$906$		0			0
$907$			1679.77i			1.85201i	0.377514	+	0.926004i	$0.376779\pi$
$907$			1679.77i			1.85201i	−0.377514	+	0.926004i	$0.623221\pi$
$908$		0			0
$909$		−	356.613i		−	0.392314i
$910$		0			0
$911$		−	44.0903i		−	0.0483977i	−0.999707	−	0.0241988i	$-0.992297\pi$
$911$		−	44.0903i		−	0.0483977i	0.999707	−	0.0241988i	$-0.00770348\pi$
$912$		0			0
$913$		−	567.601i		−	0.621688i
$914$		0			0
$915$	−478.777	−	263.769i	−0.523253	−	0.288272i
$916$		0			0
$917$	−549.993			−0.599774
$918$		0			0
$919$		−	153.303i		−	0.166815i	−0.996516	−	0.0834076i	$-0.973420\pi$
$919$		−	153.303i		−	0.166815i	0.996516	−	0.0834076i	$-0.0265804\pi$
$920$		0			0
$921$	265.111			0.287851
$922$		0			0
$923$		−	82.6061i		−	0.0894974i
$924$		0			0
$925$	−320.940	−	507.729i	−0.346962	−	0.548896i
$926$		0			0
$927$	−820.084			−0.884664
$928$		0			0
$929$	−1413.68			−1.52172			−0.760861	−	0.648915i	$-0.775223\pi$
$929$	−1413.68			−1.52172			−0.760861	+	0.648915i	$0.775223\pi$
$930$		0			0
$931$	−346.665			−0.372357
$932$		0			0
$933$	−631.435			−0.676779
$934$		0			0
$935$	334.820	−	607.744i	0.358096	−	0.649994i
$936$		0			0
$937$			90.2102i			0.0962755i	0.998841	+	0.0481378i	$0.0153287\pi$
$937$			90.2102i			0.0962755i	−0.998841	+	0.0481378i	$0.984671\pi$
$938$		0			0
$939$	735.851			0.783654
$940$		0			0
$941$		−	1347.77i		−	1.43227i	−0.697961	−	0.716136i	$-0.745909\pi$
$941$		−	1347.77i		−	1.43227i	0.697961	−	0.716136i	$-0.254091\pi$
$942$		0			0
$943$	1230.77			1.30516
$944$		0			0
$945$	261.469	−	474.602i	0.276687	−	0.502225i
$946$		0			0
$947$		−	910.489i		−	0.961446i	−0.876873	−	0.480723i	$-0.840374\pi$
$947$		−	910.489i		−	0.961446i	0.876873	−	0.480723i	$-0.159626\pi$
$948$		0			0
$949$			476.076i			0.501661i
$950$		0			0
$951$			790.313i			0.831033i
$952$		0			0
$953$			86.1971i			0.0904481i	0.998977	+	0.0452241i	$0.0144002\pi$
$953$			86.1971i			0.0904481i	−0.998977	+	0.0452241i	$0.985600\pi$
$954$		0			0
$955$	−821.025	+	1490.27i	−0.859712	+	1.56050i
$956$		0			0
$957$	−359.349			−0.375496
$958$		0			0
$959$		−	21.4690i		−	0.0223869i
$960$		0			0
$961$	350.137			0.364347
$962$		0			0
$963$			406.097i			0.421700i
$964$		0			0
$965$	232.989	−	422.908i	0.241440	−	0.438246i
$966$		0			0
$967$	419.290			0.433599			0.216799	−	0.976216i	$-0.430438\pi$
$967$	419.290			0.433599			0.216799	+	0.976216i	$0.430438\pi$
$968$		0			0
$969$	−444.881			−0.459114
$970$		0			0
$971$	−829.403			−0.854174			−0.427087	−	0.904210i	$-0.640460\pi$
$971$	−829.403			−0.854174			−0.427087	+	0.904210i	$0.640460\pi$
$972$		0			0
$973$	758.435			0.779481
$974$		0			0
$975$	199.303	−	125.981i	0.204413	−	0.129211i
$976$		0			0
$977$			1667.21i			1.70646i	0.521533	+	0.853231i	$0.325360\pi$
$977$			1667.21i			1.70646i	−0.521533	+	0.853231i	$0.674640\pi$
$978$		0			0
$979$	−585.975			−0.598544
$980$		0			0
$981$			664.840i			0.677717i
$982$		0			0
$983$	−776.192			−0.789616			−0.394808	−	0.918764i	$-0.629189\pi$
$983$	−776.192			−0.789616			−0.394808	+	0.918764i	$0.629189\pi$
$984$		0			0
$985$	1003.11	+	552.636i	1.01839	+	0.561052i
$986$		0			0
$987$			99.3616i			0.100670i
$988$		0			0
$989$			2067.99i			2.09099i
$990$		0			0
$991$		−	1392.66i		−	1.40531i	−0.711530	−	0.702655i	$-0.751998\pi$
$991$		−	1392.66i		−	1.40531i	0.711530	−	0.702655i	$-0.248002\pi$
$992$		0			0
$993$		−	100.137i		−	0.100843i
$994$		0			0
$995$	124.442	−	225.880i	0.125068	−	0.227015i
$996$		0			0
$997$	−1400.32			−1.40453			−0.702266	−	0.711915i	$-0.747828\pi$
$997$	−1400.32			−1.40453			−0.702266	+	0.711915i	$0.747828\pi$
$998$		0			0
$999$		−	658.350i		−	0.659009i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.3.e.b.159.9	yes	16
4.3	odd	2	inner	320.3.e.b.159.5	✓	16
5.2	odd	4		1600.3.g.k.351.12		16
5.3	odd	4		1600.3.g.k.351.6		16
5.4	even	2	inner	320.3.e.b.159.7	yes	16
8.3	odd	2	inner	320.3.e.b.159.12	yes	16
8.5	even	2	inner	320.3.e.b.159.8	yes	16
16.3	odd	4		1280.3.h.n.1279.10		16
16.5	even	4		1280.3.h.n.1279.11		16
16.11	odd	4		1280.3.h.n.1279.7		16
16.13	even	4		1280.3.h.n.1279.6		16
20.3	even	4		1600.3.g.k.351.11		16
20.7	even	4		1600.3.g.k.351.5		16
20.19	odd	2	inner	320.3.e.b.159.11	yes	16
40.3	even	4		1600.3.g.k.351.8		16
40.13	odd	4		1600.3.g.k.351.9		16
40.19	odd	2	inner	320.3.e.b.159.6	yes	16
40.27	even	4		1600.3.g.k.351.10		16
40.29	even	2	inner	320.3.e.b.159.10	yes	16
40.37	odd	4		1600.3.g.k.351.7		16
80.19	odd	4		1280.3.h.n.1279.5		16
80.29	even	4		1280.3.h.n.1279.9		16
80.59	odd	4		1280.3.h.n.1279.12		16
80.69	even	4		1280.3.h.n.1279.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.b.159.5	✓	16	4.3	odd	2	inner
320.3.e.b.159.6	yes	16	40.19	odd	2	inner
320.3.e.b.159.7	yes	16	5.4	even	2	inner
320.3.e.b.159.8	yes	16	8.5	even	2	inner
320.3.e.b.159.9	yes	16	1.1	even	1	trivial
320.3.e.b.159.10	yes	16	40.29	even	2	inner
320.3.e.b.159.11	yes	16	20.19	odd	2	inner
320.3.e.b.159.12	yes	16	8.3	odd	2	inner
1280.3.h.n.1279.5		16	80.19	odd	4
1280.3.h.n.1279.6		16	16.13	even	4
1280.3.h.n.1279.7		16	16.11	odd	4
1280.3.h.n.1279.8		16	80.69	even	4
1280.3.h.n.1279.9		16	80.29	even	4
1280.3.h.n.1279.10		16	16.3	odd	4
1280.3.h.n.1279.11		16	16.5	even	4
1280.3.h.n.1279.12		16	80.59	odd	4
1600.3.g.k.351.5		16	20.7	even	4
1600.3.g.k.351.6		16	5.3	odd	4
1600.3.g.k.351.7		16	40.37	odd	4
1600.3.g.k.351.8		16	40.3	even	4
1600.3.g.k.351.9		16	40.13	odd	4
1600.3.g.k.351.10		16	40.27	even	4
1600.3.g.k.351.11		16	20.3	even	4
1600.3.g.k.351.12		16	5.2	odd	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.90845i q^{3} +(-4.37937 - 2.41269i) q^{5} +3.95502 q^{7} +5.35782 q^{9} +O(q^{10})\)	\(q+1.90845i q^{3} +(-4.37937 - 2.41269i) q^{5} +3.95502 q^{7} +5.35782 q^{9} +6.18667 q^{11} -4.94185 q^{13} +(4.60451 - 8.35782i) q^{15} +22.4311i q^{17} +10.3923 q^{19} +7.54796i q^{21} +39.6083 q^{23} +(13.3578 + 21.1322i) q^{25} +27.4012i q^{27} +30.4354i q^{29} -24.7156i q^{31} +11.8070i q^{33} +(-17.3205 - 9.54225i) q^{35} -24.0264 q^{37} -9.43127i q^{39} +31.0735 q^{41} +52.2110i q^{43} +(-23.4639 - 12.9268i) q^{45} +13.1640 q^{47} -33.3578 q^{49} -42.8087 q^{51} +17.9596 q^{53} +(-27.0938 - 14.9265i) q^{55} +19.8332i q^{57} +104.421 q^{59} -57.2849i q^{61} +21.1903 q^{63} +(21.6422 + 11.9232i) q^{65} -99.3796i q^{67} +75.5905i q^{69} +16.7156i q^{71} -96.3357i q^{73} +(-40.3297 + 25.4927i) q^{75} +24.4684 q^{77} +139.578i q^{79} -4.07345 q^{81} -91.7458i q^{83} +(54.1195 - 98.2344i) q^{85} -58.0844 q^{87} -94.7156 q^{89} -19.5451 q^{91} +47.1686 q^{93} +(-45.5118 - 25.0735i) q^{95} -143.796i q^{97} +33.1471 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{9}+O(q^{10}) \) 16 * q - 96 * q^9	\( 16 q - 96 q^{9} + 32 q^{25} - 48 q^{41} - 352 q^{49} + 528 q^{65} + 480 q^{81} - 1152 q^{89}+O(q^{100}) \) 16 * q - 96 * q^9 + 32 * q^25 - 48 * q^41 - 352 * q^49 + 528 * q^65 + 480 * q^81 - 1152 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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