LMFDB - Embedded newform 208.4.f.a.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-433}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 433 $ x^2 + 433
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-20.8087i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.4.f.a.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.00000 q^{3} -20.8087i q^{5} -20.8087i q^{7} +22.0000 q^{9} +O(q^{10})$	$q-7.00000 q^{3} -20.8087i q^{5} -20.8087i q^{7} +22.0000 q^{9} +(42.0000 + 20.8087i) q^{13} +145.661i q^{15} -77.0000 q^{17} -124.852i q^{19} +145.661i q^{21} -34.0000 q^{23} -308.000 q^{25} +35.0000 q^{27} -64.0000 q^{29} +166.469i q^{31} -433.000 q^{35} -145.661i q^{37} +(-294.000 - 145.661i) q^{39} +332.938i q^{41} -55.0000 q^{43} -457.790i q^{45} +312.130i q^{47} -90.0000 q^{49} +539.000 q^{51} +594.000 q^{53} +873.963i q^{57} +541.025i q^{59} -280.000 q^{61} -457.790i q^{63} +(433.000 - 873.963i) q^{65} -582.642i q^{67} +238.000 q^{69} -436.982i q^{71} -457.790i q^{73} +2156.00 q^{75} -594.000 q^{79} -839.000 q^{81} +457.790i q^{83} +1602.27i q^{85} +448.000 q^{87} +124.852i q^{89} +(433.000 - 873.963i) q^{91} -1165.28i q^{93} -2598.00 q^{95} -749.111i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} + 44 q^{9}+O(q^{10}) $ 2 * q - 14 * q^3 + 44 * q^9	$ 2 q - 14 q^{3} + 44 q^{9} + 84 q^{13} - 154 q^{17} - 68 q^{23} - 616 q^{25} + 70 q^{27} - 128 q^{29} - 866 q^{35} - 588 q^{39} - 110 q^{43} - 180 q^{49} + 1078 q^{51} + 1188 q^{53} - 560 q^{61} + 866 q^{65} + 476 q^{69} + 4312 q^{75} - 1188 q^{79} - 1678 q^{81} + 896 q^{87} + 866 q^{91} - 5196 q^{95}+O(q^{100}) $ 2 * q - 14 * q^3 + 44 * q^9 + 84 * q^13 - 154 * q^17 - 68 * q^23 - 616 * q^25 + 70 * q^27 - 128 * q^29 - 866 * q^35 - 588 * q^39 - 110 * q^43 - 180 * q^49 + 1078 * q^51 + 1188 * q^53 - 560 * q^61 + 866 * q^65 + 476 * q^69 + 4312 * q^75 - 1188 * q^79 - 1678 * q^81 + 896 * q^87 + 866 * q^91 - 5196 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−7.00000			−1.34715			−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000			−1.34715			−0.673575	+	0.739119i	$0.735242\pi$
$4$		0			0
$5$		−	20.8087i		−	1.86118i	−0.366060	−	0.930591i	$-0.619294\pi$
$5$		−	20.8087i		−	1.86118i	0.366060	−	0.930591i	$-0.380706\pi$
$6$		0			0
$7$		−	20.8087i		−	1.12356i	−0.827286	−	0.561781i	$-0.810116\pi$
$7$		−	20.8087i		−	1.12356i	0.827286	−	0.561781i	$-0.189884\pi$
$8$		0			0
$9$	22.0000			0.814815
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	42.0000	+	20.8087i	0.896054	+	0.443945i
$13$	42.0000	+	20.8087i	0.896054	+	0.443945i
$14$		0			0
$15$			145.661i			2.50729i
$16$		0			0
$17$	−77.0000			−1.09854			−0.549272	−	0.835644i	$-0.685095\pi$
$17$	−77.0000			−1.09854			−0.549272	+	0.835644i	$0.685095\pi$
$18$		0			0
$19$		−	124.852i		−	1.50753i	−0.657146	−	0.753763i	$-0.728237\pi$
$19$		−	124.852i		−	1.50753i	0.657146	−	0.753763i	$-0.271763\pi$
$20$		0			0
$21$			145.661i			1.51361i
$22$		0			0
$23$	−34.0000			−0.308239			−0.154119	−	0.988052i	$-0.549254\pi$
$23$	−34.0000			−0.308239			−0.154119	+	0.988052i	$0.549254\pi$
$24$		0			0
$25$	−308.000			−2.46400
$26$		0			0
$27$	35.0000			0.249472
$28$		0			0
$29$	−64.0000			−0.409810			−0.204905	−	0.978782i	$-0.565689\pi$
$29$	−64.0000			−0.409810			−0.204905	+	0.978782i	$0.565689\pi$
$30$		0			0
$31$			166.469i			0.964476i	0.876040	+	0.482238i	$0.160176\pi$
$31$			166.469i			0.964476i	−0.876040	+	0.482238i	$0.839824\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−433.000			−2.09115
$36$		0			0
$37$		−	145.661i		−	0.647201i	−0.946194	−	0.323601i	$-0.895107\pi$
$37$		−	145.661i		−	0.647201i	0.946194	−	0.323601i	$-0.104893\pi$
$38$		0			0
$39$	−294.000	−	145.661i	−1.20712	−	0.598060i
$40$		0			0
$41$			332.938i			1.26820i	0.773251	+	0.634101i	$0.218629\pi$
$41$			332.938i			1.26820i	−0.773251	+	0.634101i	$0.781371\pi$
$42$		0			0
$43$	−55.0000			−0.195056			−0.0975282	−	0.995233i	$-0.531094\pi$
$43$	−55.0000			−0.195056			−0.0975282	+	0.995233i	$0.531094\pi$
$44$		0			0
$45$		−	457.790i		−	1.51652i
$46$		0			0
$47$			312.130i			0.968698i	0.874875	+	0.484349i	$0.160944\pi$
$47$			312.130i			0.968698i	−0.874875	+	0.484349i	$0.839056\pi$
$48$		0			0
$49$	−90.0000			−0.262391
$50$		0			0
$51$	539.000			1.47990
$52$		0			0
$53$	594.000			1.53947			0.769737	−	0.638361i	$-0.220387\pi$
$53$	594.000			1.53947			0.769737	+	0.638361i	$0.220387\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			873.963i			2.03086i
$58$		0			0
$59$			541.025i			1.19382i	0.802308	+	0.596911i	$0.203605\pi$
$59$			541.025i			1.19382i	−0.802308	+	0.596911i	$0.796395\pi$
$60$		0			0
$61$	−280.000			−0.587710			−0.293855	−	0.955850i	$-0.594938\pi$
$61$	−280.000			−0.587710			−0.293855	+	0.955850i	$0.594938\pi$
$62$		0			0
$63$		−	457.790i		−	0.915495i
$64$		0			0
$65$	433.000	−	873.963i	0.826262	−	1.66772i
$66$		0			0
$67$		−	582.642i		−	1.06240i	−0.847245	−	0.531202i	$-0.821740\pi$
$67$		−	582.642i		−	1.06240i	0.847245	−	0.531202i	$-0.178260\pi$
$68$		0			0
$69$	238.000			0.415244
$70$		0			0
$71$		−	436.982i		−	0.730425i	−0.930924	−	0.365213i	$-0.880996\pi$
$71$		−	436.982i		−	0.730425i	0.930924	−	0.365213i	$-0.119004\pi$
$72$		0			0
$73$		−	457.790i		−	0.733977i	−0.930226	−	0.366988i	$-0.880389\pi$
$73$		−	457.790i		−	0.733977i	0.930226	−	0.366988i	$-0.119611\pi$
$74$		0			0
$75$	2156.00			3.31938
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−594.000			−0.845952			−0.422976	−	0.906141i	$-0.639015\pi$
$79$	−594.000			−0.845952			−0.422976	+	0.906141i	$0.639015\pi$
$80$		0			0
$81$	−839.000			−1.15089
$82$		0			0
$83$			457.790i			0.605410i	0.953084	+	0.302705i	$0.0978897\pi$
$83$			457.790i			0.605410i	−0.953084	+	0.302705i	$0.902110\pi$
$84$		0			0
$85$			1602.27i			2.04459i
$86$		0			0
$87$	448.000			0.552076
$88$		0			0
$89$			124.852i			0.148700i	0.997232	+	0.0743499i	$0.0236882\pi$
$89$			124.852i			0.148700i	−0.997232	+	0.0743499i	$0.976312\pi$
$90$		0			0
$91$	433.000	−	873.963i	0.498799	−	1.00677i
$92$		0			0
$93$		−	1165.28i		−	1.29929i
$94$		0			0
$95$	−2598.00			−2.80578
$96$		0			0
$97$		−	749.111i		−	0.784131i	−0.919937	−	0.392066i	$-0.871761\pi$
$97$		−	749.111i		−	0.784131i	0.919937	−	0.392066i	$-0.128239\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−532.000			−0.524119			−0.262059	−	0.965052i	$-0.584402\pi$
$101$	−532.000			−0.524119			−0.262059	+	0.965052i	$0.584402\pi$
$102$		0			0
$103$	154.000			0.147321			0.0736605	−	0.997283i	$-0.476532\pi$
$103$	154.000			0.147321			0.0736605	+	0.997283i	$0.476532\pi$
$104$		0			0
$105$	3031.00			2.81710
$106$		0			0
$107$	−44.0000			−0.0397537			−0.0198768	−	0.999802i	$-0.506327\pi$
$107$	−44.0000			−0.0397537			−0.0198768	+	0.999802i	$0.506327\pi$
$108$		0			0
$109$		−	1602.27i		−	1.40797i	−0.710212	−	0.703987i	$-0.751401\pi$
$109$		−	1602.27i		−	1.40797i	0.710212	−	0.703987i	$-0.248599\pi$
$110$		0			0
$111$			1019.62i			0.871878i
$112$		0			0
$113$	−1030.00			−0.857471			−0.428736	−	0.903430i	$-0.641041\pi$
$113$	−1030.00			−0.857471			−0.428736	+	0.903430i	$0.641041\pi$
$114$		0			0
$115$			707.494i			0.573688i
$116$		0			0
$117$	924.000	+	457.790i	0.730118	+	0.361733i
$118$		0			0
$119$			1602.27i			1.23428i
$120$		0			0
$121$	1331.00			1.00000
$122$		0			0
$123$		−	2330.57i		−	1.70846i
$124$		0			0
$125$			3807.98i			2.72477i
$126$		0			0
$127$	2266.00			1.58327			0.791634	−	0.610996i	$-0.209231\pi$
$127$	2266.00			1.58327			0.791634	+	0.610996i	$0.209231\pi$
$128$		0			0
$129$	385.000			0.262770
$130$		0			0
$131$	371.000			0.247438			0.123719	−	0.992317i	$-0.460518\pi$
$131$	371.000			0.247438			0.123719	+	0.992317i	$0.460518\pi$
$132$		0			0
$133$	−2598.00			−1.69380
$134$		0			0
$135$		−	728.303i		−	0.464314i
$136$		0			0
$137$		−	2039.25i		−	1.27171i	−0.771807	−	0.635857i	$-0.780647\pi$
$137$		−	2039.25i		−	1.27171i	0.771807	−	0.635857i	$-0.219353\pi$
$138$		0			0
$139$	−805.000			−0.491217			−0.245609	−	0.969369i	$-0.578988\pi$
$139$	−805.000			−0.491217			−0.245609	+	0.969369i	$0.578988\pi$
$140$		0			0
$141$		−	2184.91i		−	1.30498i
$142$		0			0
$143$		0			0
$144$		0			0
$145$			1331.75i			0.762732i
$146$		0			0
$147$	630.000			0.353480
$148$		0			0
$149$			291.321i			0.160174i	0.996788	+	0.0800871i	$0.0255198\pi$
$149$			291.321i			0.160174i	−0.996788	+	0.0800871i	$0.974480\pi$
$150$		0			0
$151$			1602.27i			0.863514i	0.901990	+	0.431757i	$0.142106\pi$
$151$			1602.27i			0.863514i	−0.901990	+	0.431757i	$0.857894\pi$
$152$		0			0
$153$	−1694.00			−0.895110
$154$		0			0
$155$	3464.00			1.79507
$156$		0			0
$157$	994.000			0.505286			0.252643	−	0.967560i	$-0.418700\pi$
$157$	994.000			0.505286			0.252643	+	0.967560i	$0.418700\pi$
$158$		0			0
$159$	−4158.00			−2.07390
$160$		0			0
$161$			707.494i			0.346325i
$162$		0			0
$163$		−	3204.53i		−	1.53987i	−0.638124	−	0.769934i	$-0.720289\pi$
$163$		−	3204.53i		−	1.53987i	0.638124	−	0.769934i	$-0.279711\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2746.74i		−	1.27275i	−0.771380	−	0.636375i	$-0.780433\pi$
$167$		−	2746.74i		−	1.27275i	0.771380	−	0.636375i	$-0.219567\pi$
$168$		0			0
$169$	1331.00	+	1747.93i	0.605826	+	0.795597i
$170$		0			0
$171$		−	2746.74i		−	1.22835i
$172$		0			0
$173$	−4158.00			−1.82732			−0.913662	−	0.406476i	$-0.866758\pi$
$173$	−4158.00			−1.82732			−0.913662	+	0.406476i	$0.866758\pi$
$174$		0			0
$175$			6409.06i			2.76846i
$176$		0			0
$177$		−	3787.17i		−	1.60826i
$178$		0			0
$179$	−2295.00			−0.958304			−0.479152	−	0.877732i	$-0.659056\pi$
$179$	−2295.00			−0.958304			−0.479152	+	0.877732i	$0.659056\pi$
$180$		0			0
$181$	−1190.00			−0.488685			−0.244343	−	0.969689i	$-0.578572\pi$
$181$	−1190.00			−0.488685			−0.244343	+	0.969689i	$0.578572\pi$
$182$		0			0
$183$	1960.00			0.791734
$184$		0			0
$185$	−3031.00			−1.20456
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	728.303i		−	0.280298i
$190$		0			0
$191$	−4188.00			−1.58656			−0.793280	−	0.608857i	$-0.791628\pi$
$191$	−4188.00			−1.58656			−0.793280	+	0.608857i	$0.791628\pi$
$192$		0			0
$193$			3204.53i			1.19517i	0.801807	+	0.597584i	$0.203872\pi$
$193$			3204.53i			1.19517i	−0.801807	+	0.597584i	$0.796128\pi$
$194$		0			0
$195$	−3031.00	+	6117.74i	−1.11310	+	2.24667i
$196$		0			0
$197$		−	145.661i		−	0.0526796i	−0.999653	−	0.0263398i	$-0.991615\pi$
$197$		−	145.661i		−	0.0526796i	0.999653	−	0.0263398i	$-0.00838519\pi$
$198$		0			0
$199$	−3276.00			−1.16698			−0.583491	−	0.812119i	$-0.698314\pi$
$199$	−3276.00			−1.16698			−0.583491	+	0.812119i	$0.698314\pi$
$200$		0			0
$201$			4078.50i			1.43122i
$202$		0			0
$203$			1331.75i			0.460447i
$204$		0			0
$205$	6928.00			2.36035
$206$		0			0
$207$	−748.000			−0.251157
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−825.000			−0.269172			−0.134586	−	0.990902i	$-0.542970\pi$
$211$	−825.000			−0.269172			−0.134586	+	0.990902i	$0.542970\pi$
$212$		0			0
$213$			3058.87i			0.983993i
$214$		0			0
$215$			1144.48i			0.363035i
$216$		0			0
$217$	3464.00			1.08365
$218$		0			0
$219$			3204.53i			0.988777i
$220$		0			0
$221$	−3234.00	−	1602.27i	−0.984355	−	0.487693i
$222$		0			0
$223$		−	270.512i		−	0.0812325i	−0.999175	−	0.0406163i	$-0.987068\pi$
$223$		−	270.512i		−	0.0812325i	0.999175	−	0.0406163i	$-0.0129321\pi$
$224$		0			0
$225$	−6776.00			−2.00770
$226$		0			0
$227$		−	915.581i		−	0.267706i	−0.991001	−	0.133853i	$-0.957265\pi$
$227$		−	915.581i		−	0.267706i	0.991001	−	0.133853i	$-0.0427350\pi$
$228$		0			0
$229$		−	3599.90i		−	1.03881i	−0.854528	−	0.519406i	$-0.826153\pi$
$229$		−	3599.90i		−	1.03881i	0.854528	−	0.519406i	$-0.173847\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3333.00			−0.937133			−0.468567	−	0.883428i	$-0.655229\pi$
$233$	−3333.00			−0.937133			−0.468567	+	0.883428i	$0.655229\pi$
$234$		0			0
$235$	6495.00			1.80292
$236$		0			0
$237$	4158.00			1.13962
$238$		0			0
$239$		−	4224.16i		−	1.14326i	−0.820513	−	0.571628i	$-0.806312\pi$
$239$		−	4224.16i		−	1.14326i	0.820513	−	0.571628i	$-0.193688\pi$
$240$		0			0
$241$			1290.14i			0.344834i	0.985024	+	0.172417i	$0.0551577\pi$
$241$			1290.14i			0.344834i	−0.985024	+	0.172417i	$0.944842\pi$
$242$		0			0
$243$	4928.00			1.30095
$244$		0			0
$245$			1872.78i			0.488357i
$246$		0			0
$247$	2598.00	−	5243.78i	0.669258	−	1.35082i
$248$		0			0
$249$		−	3204.53i		−	0.815578i
$250$		0			0
$251$	28.0000			0.00704121			0.00352061	−	0.999994i	$-0.498879\pi$
$251$	28.0000			0.00704121			0.00352061	+	0.999994i	$0.498879\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	11215.9i		−	2.75437i
$256$		0			0
$257$	1491.00			0.361891			0.180946	−	0.983493i	$-0.442084\pi$
$257$	1491.00			0.361891			0.180946	+	0.983493i	$0.442084\pi$
$258$		0			0
$259$	−3031.00			−0.727171
$260$		0			0
$261$	−1408.00			−0.333920
$262$		0			0
$263$	2808.00			0.658360			0.329180	−	0.944267i	$-0.393228\pi$
$263$	2808.00			0.658360			0.329180	+	0.944267i	$0.393228\pi$
$264$		0			0
$265$		−	12360.3i		−	2.86524i
$266$		0			0
$267$		−	873.963i		−	0.200321i
$268$		0			0
$269$	1232.00			0.279243			0.139621	−	0.990205i	$-0.455411\pi$
$269$	1232.00			0.279243			0.139621	+	0.990205i	$0.455411\pi$
$270$		0			0
$271$			3891.22i			0.872231i	0.899891	+	0.436116i	$0.143646\pi$
$271$			3891.22i			0.872231i	−0.899891	+	0.436116i	$0.856354\pi$
$272$		0			0
$273$	−3031.00	+	6117.74i	−0.671958	+	1.35627i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1016.00			−0.220381			−0.110190	−	0.993910i	$-0.535146\pi$
$277$	−1016.00			−0.220381			−0.110190	+	0.993910i	$0.535146\pi$
$278$		0			0
$279$			3662.32i			0.785869i
$280$		0			0
$281$			291.321i			0.0618461i	0.999522	+	0.0309231i	$0.00984469\pi$
$281$			291.321i			0.0618461i	−0.999522	+	0.0309231i	$0.990155\pi$
$282$		0			0
$283$	−4060.00			−0.852798			−0.426399	−	0.904535i	$-0.640218\pi$
$283$	−4060.00			−0.852798			−0.426399	+	0.904535i	$0.640218\pi$
$284$		0			0
$285$	18186.0			3.77981
$286$		0			0
$287$	6928.00			1.42490
$288$		0			0
$289$	1016.00			0.206798
$290$		0			0
$291$			5243.78i			1.05634i
$292$		0			0
$293$		−	3225.34i		−	0.643094i	−0.946894	−	0.321547i	$-0.895797\pi$
$293$		−	3225.34i		−	0.643094i	0.946894	−	0.321547i	$-0.104203\pi$
$294$		0			0
$295$	11258.0			2.22192
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1428.00	−	707.494i	−0.276199	−	0.136841i
$300$		0			0
$301$			1144.48i			0.219158i
$302$		0			0
$303$	3724.00			0.706067
$304$		0			0
$305$			5826.42i			1.09384i
$306$		0			0
$307$		−	5951.27i		−	1.10638i	−0.833057	−	0.553188i	$-0.813411\pi$
$307$		−	5951.27i		−	1.10638i	0.833057	−	0.553188i	$-0.186589\pi$
$308$		0			0
$309$	−1078.00			−0.198464
$310$		0			0
$311$	−3542.00			−0.645815			−0.322907	−	0.946431i	$-0.604660\pi$
$311$	−3542.00			−0.645815			−0.322907	+	0.946431i	$0.604660\pi$
$312$		0			0
$313$	8659.00			1.56369			0.781846	−	0.623472i	$-0.214278\pi$
$313$	8659.00			1.56369			0.781846	+	0.623472i	$0.214278\pi$
$314$		0			0
$315$	−9526.00			−1.70390
$316$		0			0
$317$			6117.74i			1.08393i	0.840400	+	0.541966i	$0.182320\pi$
$317$			6117.74i			1.08393i	−0.840400	+	0.541966i	$0.817680\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	308.000			0.0535542
$322$		0			0
$323$			9613.60i			1.65608i
$324$		0			0
$325$	−12936.0	−	6409.06i	−2.20788	−	1.09388i
$326$		0			0
$327$			11215.9i			1.89675i
$328$		0			0
$329$	6495.00			1.08839
$330$		0			0
$331$		−	9030.95i		−	1.49966i	−0.661633	−	0.749828i	$-0.730136\pi$
$331$		−	9030.95i		−	1.49966i	0.661633	−	0.749828i	$-0.269864\pi$
$332$		0			0
$333$		−	3204.53i		−	0.527349i
$334$		0			0
$335$	−12124.0			−1.97733
$336$		0			0
$337$	4213.00			0.680999			0.340500	−	0.940245i	$-0.389404\pi$
$337$	4213.00			0.680999			0.340500	+	0.940245i	$0.389404\pi$
$338$		0			0
$339$	7210.00			1.15514
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	5264.59i		−	0.828750i
$344$		0			0
$345$		−	4952.46i		−	0.772845i
$346$		0			0
$347$	3855.00			0.596390			0.298195	−	0.954505i	$-0.403615\pi$
$347$	3855.00			0.596390			0.298195	+	0.954505i	$0.403615\pi$
$348$		0			0
$349$			4349.01i			0.667040i	0.942743	+	0.333520i	$0.108237\pi$
$349$			4349.01i			0.667040i	−0.942743	+	0.333520i	$0.891763\pi$
$350$		0			0
$351$	1470.00	+	728.303i	0.223541	+	0.110752i
$352$		0			0
$353$			3537.47i			0.533373i	0.963783	+	0.266686i	$0.0859288\pi$
$353$			3537.47i			0.533373i	−0.963783	+	0.266686i	$0.914071\pi$
$354$		0			0
$355$	−9093.00			−1.35945
$356$		0			0
$357$		−	11215.9i		−	1.66276i
$358$		0			0
$359$			11652.8i			1.71313i	0.516039	+	0.856565i	$0.327406\pi$
$359$			11652.8i			1.71313i	−0.516039	+	0.856565i	$0.672594\pi$
$360$		0			0
$361$	−8729.00			−1.27263
$362$		0			0
$363$	−9317.00			−1.34715
$364$		0			0
$365$	−9526.00			−1.36606
$366$		0			0
$367$	10024.0			1.42575			0.712873	−	0.701293i	$-0.247394\pi$
$367$	10024.0			1.42575			0.712873	+	0.701293i	$0.247394\pi$
$368$		0			0
$369$			7324.65i			1.03335i
$370$		0			0
$371$		−	12360.3i		−	1.72969i
$372$		0			0
$373$	6372.00			0.884530			0.442265	−	0.896884i	$-0.354175\pi$
$373$	6372.00			0.884530			0.442265	+	0.896884i	$0.354175\pi$
$374$		0			0
$375$		−	26655.9i		−	3.67068i
$376$		0			0
$377$	−2688.00	−	1331.75i	−0.367212	−	0.181933i
$378$		0			0
$379$		−	8448.31i		−	1.14501i	−0.819899	−	0.572507i	$-0.805971\pi$
$379$		−	8448.31i		−	1.14501i	0.819899	−	0.572507i	$-0.194029\pi$
$380$		0			0
$381$	−15862.0			−2.13290
$382$		0			0
$383$			7387.07i			0.985540i	0.870160	+	0.492770i	$0.164016\pi$
$383$			7387.07i			0.985540i	−0.870160	+	0.492770i	$0.835984\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1210.00			−0.158935
$388$		0			0
$389$	−6102.00			−0.795331			−0.397665	−	0.917531i	$-0.630180\pi$
$389$	−6102.00			−0.795331			−0.397665	+	0.917531i	$0.630180\pi$
$390$		0			0
$391$	2618.00			0.338614
$392$		0			0
$393$	−2597.00			−0.333337
$394$		0			0
$395$			12360.3i			1.57447i
$396$		0			0
$397$			4785.99i			0.605043i	0.953143	+	0.302521i	$0.0978284\pi$
$397$			4785.99i			0.605043i	−0.953143	+	0.302521i	$0.902172\pi$
$398$		0			0
$399$	18186.0			2.28180
$400$		0			0
$401$		−	9613.60i		−	1.19721i	−0.801045	−	0.598604i	$-0.795722\pi$
$401$		−	9613.60i		−	1.19721i	0.801045	−	0.598604i	$-0.204278\pi$
$402$		0			0
$403$	−3464.00	+	6991.71i	−0.428174	+	0.864223i
$404$		0			0
$405$			17458.5i			2.14202i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	4536.29i		−	0.548423i	−0.961669	−	0.274211i	$-0.911583\pi$
$409$		−	4536.29i		−	0.548423i	0.961669	−	0.274211i	$-0.0884168\pi$
$410$		0			0
$411$			14274.7i			1.71319i
$412$		0			0
$413$	11258.0			1.34133
$414$		0			0
$415$	9526.00			1.12678
$416$		0			0
$417$	5635.00			0.661744
$418$		0			0
$419$	−10087.0			−1.17609			−0.588046	−	0.808828i	$-0.700102\pi$
$419$	−10087.0			−1.17609			−0.588046	+	0.808828i	$0.700102\pi$
$420$		0			0
$421$			11215.9i			1.29840i	0.760616	+	0.649202i	$0.224897\pi$
$421$			11215.9i			1.29840i	−0.760616	+	0.649202i	$0.775103\pi$
$422$		0			0
$423$			6866.86i			0.789310i
$424$		0			0
$425$	23716.0			2.70681
$426$		0			0
$427$			5826.42i			0.660329i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			1602.27i			0.179068i	0.995984	+	0.0895342i	$0.0285378\pi$
$431$			1602.27i			0.179068i	−0.995984	+	0.0895342i	$0.971462\pi$
$432$		0			0
$433$	−4403.00			−0.488671			−0.244336	−	0.969691i	$-0.578570\pi$
$433$	−4403.00			−0.488671			−0.244336	+	0.969691i	$0.578570\pi$
$434$		0			0
$435$		−	9322.28i		−	1.02751i
$436$		0			0
$437$			4244.97i			0.464678i
$438$		0			0
$439$	−9534.00			−1.03652			−0.518261	−	0.855223i	$-0.673420\pi$
$439$	−9534.00			−1.03652			−0.518261	+	0.855223i	$0.673420\pi$
$440$		0			0
$441$	−1980.00			−0.213800
$442$		0			0
$443$	−9827.00			−1.05394			−0.526969	−	0.849884i	$-0.676672\pi$
$443$	−9827.00			−1.05394			−0.526969	+	0.849884i	$0.676672\pi$
$444$		0			0
$445$	2598.00			0.276757
$446$		0			0
$447$		−	2039.25i		−	0.215779i
$448$		0			0
$449$			3495.85i			0.367438i	0.982979	+	0.183719i	$0.0588136\pi$
$449$			3495.85i			0.367438i	−0.982979	+	0.183719i	$0.941186\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	11215.9i		−	1.16328i
$454$		0			0
$455$	−18186.0	−	9010.15i	−1.87379	−	0.928356i
$456$		0			0
$457$		−	16022.7i		−	1.64006i	−0.572319	−	0.820031i	$-0.693956\pi$
$457$		−	16022.7i		−	1.64006i	0.572319	−	0.820031i	$-0.306044\pi$
$458$		0			0
$459$	−2695.00			−0.274056
$460$		0			0
$461$		−	2434.61i		−	0.245968i	−0.992409	−	0.122984i	$-0.960754\pi$
$461$		−	2434.61i		−	0.245968i	0.992409	−	0.122984i	$-0.0392464\pi$
$462$		0			0
$463$		−	6991.71i		−	0.701797i	−0.936413	−	0.350899i	$-0.885876\pi$
$463$		−	6991.71i		−	0.701797i	0.936413	−	0.350899i	$-0.114124\pi$
$464$		0			0
$465$	−24248.0			−2.41822
$466$		0			0
$467$	12516.0			1.24020			0.620098	−	0.784524i	$-0.287093\pi$
$467$	12516.0			1.24020			0.620098	+	0.784524i	$0.287093\pi$
$468$		0			0
$469$	−12124.0			−1.19368
$470$		0			0
$471$	−6958.00			−0.680696
$472$		0			0
$473$		0			0
$474$		0			0
$475$			38454.4i			3.71454i
$476$		0			0
$477$	13068.0			1.25439
$478$		0			0
$479$			2060.06i			0.196506i	0.995161	+	0.0982530i	$0.0313254\pi$
$479$			2060.06i			0.196506i	−0.995161	+	0.0982530i	$0.968675\pi$
$480$		0			0
$481$	3031.00	−	6117.74i	0.287322	−	0.579927i
$482$		0			0
$483$		−	4952.46i		−	0.466552i
$484$		0			0
$485$	−15588.0			−1.45941
$486$		0			0
$487$		−	8739.63i		−	0.813205i	−0.913605	−	0.406602i	$-0.866713\pi$
$487$		−	8739.63i		−	0.813205i	0.913605	−	0.406602i	$-0.133287\pi$
$488$		0			0
$489$			22431.7i			2.07443i
$490$		0			0
$491$	−3971.00			−0.364987			−0.182494	−	0.983207i	$-0.558417\pi$
$491$	−3971.00			−0.364987			−0.182494	+	0.983207i	$0.558417\pi$
$492$		0			0
$493$	4928.00			0.450195
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9093.00			−0.820678
$498$		0			0
$499$		−	6409.06i		−	0.574968i	−0.957785	−	0.287484i	$-0.907181\pi$
$499$		−	6409.06i		−	0.574968i	0.957785	−	0.287484i	$-0.0928189\pi$
$500$		0			0
$501$			19227.2i			1.71459i
$502$		0			0
$503$	−17346.0			−1.53761			−0.768807	−	0.639481i	$-0.779149\pi$
$503$	−17346.0			−1.53761			−0.768807	+	0.639481i	$0.779149\pi$
$504$		0			0
$505$			11070.2i			0.975480i
$506$		0			0
$507$	−9317.00	−	12235.5i	−0.816139	−	1.07179i
$508$		0			0
$509$		−	17354.4i		−	1.51124i	−0.655010	−	0.755620i	$-0.727336\pi$
$509$		−	17354.4i		−	1.51124i	0.655010	−	0.755620i	$-0.272664\pi$
$510$		0			0
$511$	−9526.00			−0.824668
$512$		0			0
$513$		−	4369.82i		−	0.376086i
$514$		0			0
$515$		−	3204.53i		−	0.274191i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	29106.0			2.46168
$520$		0			0
$521$	19803.0			1.66523			0.832615	−	0.553852i	$-0.186843\pi$
$521$	19803.0			1.66523			0.832615	+	0.553852i	$0.186843\pi$
$522$		0			0
$523$	10808.0			0.903634			0.451817	−	0.892111i	$-0.350776\pi$
$523$	10808.0			0.903634			0.451817	+	0.892111i	$0.350776\pi$
$524$		0			0
$525$		−	44863.5i		−	3.72953i
$526$		0			0
$527$		−	12818.1i		−	1.05952i
$528$		0			0
$529$	−11011.0			−0.904989
$530$		0			0
$531$			11902.5i			0.972743i
$532$		0			0
$533$	−6928.00	+	13983.4i	−0.563011	+	1.13638i
$534$		0			0
$535$			915.581i			0.0739888i
$536$		0			0
$537$	16065.0			1.29098
$538$		0			0
$539$		0			0
$540$		0			0
$541$			8885.29i			0.706116i	0.935601	+	0.353058i	$0.114858\pi$
$541$			8885.29i			0.706116i	−0.935601	+	0.353058i	$0.885142\pi$
$542$		0			0
$543$	8330.00			0.658332
$544$		0			0
$545$	−33341.0			−2.62050
$546$		0			0
$547$	−979.000			−0.0765247			−0.0382624	−	0.999268i	$-0.512182\pi$
$547$	−979.000			−0.0765247			−0.0382624	+	0.999268i	$0.512182\pi$
$548$		0			0
$549$	−6160.00			−0.478875
$550$		0			0
$551$			7990.52i			0.617800i
$552$		0			0
$553$			12360.3i			0.950479i
$554$		0			0
$555$	21217.0			1.62272
$556$		0			0
$557$			4806.80i			0.365656i	0.983145	+	0.182828i	$0.0585252\pi$
$557$			4806.80i			0.365656i	−0.983145	+	0.182828i	$0.941475\pi$
$558$		0			0
$559$	−2310.00	−	1144.48i	−0.174781	−	0.0865942i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12481.0			0.934301			0.467150	−	0.884178i	$-0.345281\pi$
$563$	12481.0			0.934301			0.467150	+	0.884178i	$0.345281\pi$
$564$		0			0
$565$			21432.9i			1.59591i
$566$		0			0
$567$			17458.5i			1.29310i
$568$		0			0
$569$	−10571.0			−0.778839			−0.389419	−	0.921061i	$-0.627324\pi$
$569$	−10571.0			−0.778839			−0.389419	+	0.921061i	$0.627324\pi$
$570$		0			0
$571$	−16841.0			−1.23428			−0.617140	−	0.786853i	$-0.711709\pi$
$571$	−16841.0			−1.23428			−0.617140	+	0.786853i	$0.711709\pi$
$572$		0			0
$573$	29316.0			2.13734
$574$		0			0
$575$	10472.0			0.759500
$576$		0			0
$577$			24512.6i			1.76858i	0.466935	+	0.884292i	$0.345358\pi$
$577$			24512.6i			1.76858i	−0.466935	+	0.884292i	$0.654642\pi$
$578$		0			0
$579$		−	22431.7i		−	1.61007i
$580$		0			0
$581$	9526.00			0.680215
$582$		0			0
$583$		0			0
$584$		0			0
$585$	9526.00	−	19227.2i	0.673251	−	1.35888i
$586$		0			0
$587$		−	5951.27i		−	0.418459i	−0.977867	−	0.209229i	$-0.932904\pi$
$587$		−	5951.27i		−	0.418459i	0.977867	−	0.209229i	$-0.0670956\pi$
$588$		0			0
$589$	20784.0			1.45397
$590$		0			0
$591$			1019.62i			0.0709674i
$592$		0			0
$593$		−	10071.4i		−	0.697441i	−0.937227	−	0.348720i	$-0.886616\pi$
$593$		−	10071.4i		−	0.697441i	0.937227	−	0.348720i	$-0.113384\pi$
$594$		0			0
$595$	33341.0			2.29722
$596$		0			0
$597$	22932.0			1.57210
$598$		0			0
$599$	23430.0			1.59820			0.799102	−	0.601196i	$-0.205309\pi$
$599$	23430.0			1.59820			0.799102	+	0.601196i	$0.205309\pi$
$600$		0			0
$601$	−1043.00			−0.0707901			−0.0353951	−	0.999373i	$-0.511269\pi$
$601$	−1043.00			−0.0707901			−0.0353951	+	0.999373i	$0.511269\pi$
$602$		0			0
$603$		−	12818.1i		−	0.865663i
$604$		0			0
$605$		−	27696.3i		−	1.86118i
$606$		0			0
$607$	−4690.00			−0.313610			−0.156805	−	0.987630i	$-0.550119\pi$
$607$	−4690.00			−0.313610			−0.156805	+	0.987630i	$0.550119\pi$
$608$		0			0
$609$		−	9322.28i		−	0.620292i
$610$		0			0
$611$	−6495.00	+	13109.5i	−0.430048	+	0.868006i
$612$		0			0
$613$		−	28258.1i		−	1.86189i	−0.365165	−	0.930943i	$-0.618987\pi$
$613$		−	28258.1i		−	1.86189i	0.365165	−	0.930943i	$-0.381013\pi$
$614$		0			0
$615$	−48496.0			−3.17975
$616$		0			0
$617$		−	18061.9i		−	1.17852i	−0.807944	−	0.589259i	$-0.799420\pi$
$617$		−	18061.9i		−	1.17852i	0.807944	−	0.589259i	$-0.200580\pi$
$618$		0			0
$619$			3162.92i			0.205377i	0.994714	+	0.102688i	$0.0327445\pi$
$619$			3162.92i			0.205377i	−0.994714	+	0.102688i	$0.967256\pi$
$620$		0			0
$621$	−1190.00			−0.0768970
$622$		0			0
$623$	2598.00			0.167073
$624$		0			0
$625$	40739.0			2.60730
$626$		0			0
$627$		0			0
$628$		0			0
$629$			11215.9i			0.710979i
$630$		0			0
$631$			22577.4i			1.42439i	0.701980	+	0.712196i	$0.252299\pi$
$631$			22577.4i			1.42439i	−0.701980	+	0.712196i	$0.747701\pi$
$632$		0			0
$633$	5775.00			0.362616
$634$		0			0
$635$		−	47152.4i		−	2.94675i
$636$		0			0
$637$	−3780.00	−	1872.78i	−0.235116	−	0.116487i
$638$		0			0
$639$		−	9613.60i		−	0.595161i
$640$		0			0
$641$	−8562.00			−0.527580			−0.263790	−	0.964580i	$-0.584973\pi$
$641$	−8562.00			−0.527580			−0.263790	+	0.964580i	$0.584973\pi$
$642$		0			0
$643$			10737.3i			0.658532i	0.944237	+	0.329266i	$0.106801\pi$
$643$			10737.3i			0.658532i	−0.944237	+	0.329266i	$0.893199\pi$
$644$		0			0
$645$		−	8011.33i		−	0.489063i
$646$		0			0
$647$	15218.0			0.924701			0.462350	−	0.886697i	$-0.347006\pi$
$647$	15218.0			0.924701			0.462350	+	0.886697i	$0.347006\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−24248.0			−1.45984
$652$		0			0
$653$	14044.0			0.841630			0.420815	−	0.907147i	$-0.361744\pi$
$653$	14044.0			0.841630			0.420815	+	0.907147i	$0.361744\pi$
$654$		0			0
$655$		−	7720.01i		−	0.460528i
$656$		0			0
$657$		−	10071.4i		−	0.598055i
$658$		0			0
$659$	25432.0			1.50332			0.751662	−	0.659549i	$-0.229253\pi$
$659$	25432.0			1.50332			0.751662	+	0.659549i	$0.229253\pi$
$660$		0			0
$661$		−	7449.50i		−	0.438354i	−0.975685	−	0.219177i	$-0.929663\pi$
$661$		−	7449.50i		−	0.438354i	0.975685	−	0.219177i	$-0.0703372\pi$
$662$		0			0
$663$	22638.0	+	11215.9i	1.32607	+	0.656996i
$664$		0			0
$665$			54060.9i			3.15247i
$666$		0			0
$667$	2176.00			0.126319
$668$		0			0
$669$			1893.59i			0.109432i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−22737.0			−1.30230			−0.651149	−	0.758950i	$-0.725713\pi$
$673$	−22737.0			−1.30230			−0.651149	+	0.758950i	$0.725713\pi$
$674$		0			0
$675$	−10780.0			−0.614700
$676$		0			0
$677$	−9366.00			−0.531705			−0.265853	−	0.964014i	$-0.585653\pi$
$677$	−9366.00			−0.531705			−0.265853	+	0.964014i	$0.585653\pi$
$678$		0			0
$679$	−15588.0			−0.881020
$680$		0			0
$681$			6409.06i			0.360640i
$682$		0			0
$683$		−	28840.8i		−	1.61576i	−0.589349	−	0.807879i	$-0.700616\pi$
$683$		−	28840.8i		−	1.61576i	0.589349	−	0.807879i	$-0.299384\pi$
$684$		0			0
$685$	−42434.0			−2.36689
$686$		0			0
$687$			25199.3i			1.39944i
$688$		0			0
$689$	24948.0	+	12360.3i	1.37945	+	0.683442i
$690$		0			0
$691$		−	5659.95i		−	0.311599i	−0.987789	−	0.155799i	$-0.950205\pi$
$691$		−	5659.95i		−	0.311599i	0.987789	−	0.155799i	$-0.0497953\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16751.0i			0.914245i
$696$		0			0
$697$		−	25636.3i		−	1.39317i
$698$		0			0
$699$	23331.0			1.26246
$700$		0			0
$701$	−8702.00			−0.468859			−0.234429	−	0.972133i	$-0.575322\pi$
$701$	−8702.00			−0.468859			−0.234429	+	0.972133i	$0.575322\pi$
$702$		0			0
$703$	−18186.0			−0.975673
$704$		0			0
$705$	−45465.0			−2.42881
$706$		0			0
$707$			11070.2i			0.588880i
$708$		0			0
$709$		−	4369.82i		−	0.231470i	−0.993280	−	0.115735i	$-0.963078\pi$
$709$		−	4369.82i		−	0.231470i	0.993280	−	0.115735i	$-0.0369223\pi$
$710$		0			0
$711$	−13068.0			−0.689294
$712$		0			0
$713$		−	5659.95i		−	0.297289i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			29569.1i			1.54014i
$718$		0			0
$719$	−6608.00			−0.342749			−0.171375	−	0.985206i	$-0.554821\pi$
$719$	−6608.00			−0.342749			−0.171375	+	0.985206i	$0.554821\pi$
$720$		0			0
$721$		−	3204.53i		−	0.165524i
$722$		0			0
$723$		−	9030.95i		−	0.464543i
$724$		0			0
$725$	19712.0			1.00977
$726$		0			0
$727$	−13916.0			−0.709926			−0.354963	−	0.934880i	$-0.615506\pi$
$727$	−13916.0			−0.709926			−0.354963	+	0.934880i	$0.615506\pi$
$728$		0			0
$729$	−11843.0			−0.601687
$730$		0			0
$731$	4235.00			0.214278
$732$		0			0
$733$		−	3225.34i		−	0.162525i	−0.996693	−	0.0812624i	$-0.974105\pi$
$733$		−	3225.34i		−	0.162525i	0.996693	−	0.0812624i	$-0.0258952\pi$
$734$		0			0
$735$		−	13109.5i		−	0.657890i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			16896.6i			0.841072i	0.907276	+	0.420536i	$0.138158\pi$
$739$			16896.6i			0.841072i	−0.907276	+	0.420536i	$0.861842\pi$
$740$		0			0
$741$	−18186.0	+	36706.5i	−0.901592	+	1.81976i
$742$		0			0
$743$			5098.12i			0.251725i	0.992048	+	0.125863i	$0.0401699\pi$
$743$			5098.12i			0.251725i	−0.992048	+	0.125863i	$0.959830\pi$
$744$		0			0
$745$	6062.00			0.298113
$746$		0			0
$747$			10071.4i			0.493297i
$748$		0			0
$749$			915.581i			0.0446657i
$750$		0			0
$751$	−8672.00			−0.421366			−0.210683	−	0.977554i	$-0.567569\pi$
$751$	−8672.00			−0.421366			−0.210683	+	0.977554i	$0.567569\pi$
$752$		0			0
$753$	−196.000			−0.00948557
$754$		0			0
$755$	33341.0			1.60716
$756$		0			0
$757$	−6490.00			−0.311602			−0.155801	−	0.987788i	$-0.549796\pi$
$757$	−6490.00			−0.311602			−0.155801	+	0.987788i	$0.549796\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	17354.4i		−	0.826672i	−0.910579	−	0.413336i	$-0.864364\pi$
$761$		−	17354.4i		−	0.826672i	0.910579	−	0.413336i	$-0.135636\pi$
$762$		0			0
$763$	−33341.0			−1.58195
$764$		0			0
$765$			35249.9i			1.66596i
$766$		0			0
$767$	−11258.0	+	22723.0i	−0.529991	+	1.06973i
$768$		0			0
$769$		−	16480.5i		−	0.772822i	−0.922327	−	0.386411i	$-0.873715\pi$
$769$		−	16480.5i		−	0.772822i	0.922327	−	0.386411i	$-0.126285\pi$
$770$		0			0
$771$	−10437.0			−0.487522
$772$		0			0
$773$			37767.7i			1.75732i	0.477446	+	0.878661i	$0.341563\pi$
$773$			37767.7i			1.75732i	−0.477446	+	0.878661i	$0.658437\pi$
$774$		0			0
$775$		−	51272.5i		−	2.37647i
$776$		0			0
$777$	21217.0			0.979608
$778$		0			0
$779$	41568.0			1.91185
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−2240.00			−0.102236
$784$		0			0
$785$		−	20683.8i		−	0.940429i
$786$		0			0
$787$			10071.4i			0.456171i	0.973641	+	0.228085i	$0.0732465\pi$
$787$			10071.4i			0.456171i	−0.973641	+	0.228085i	$0.926754\pi$
$788$		0			0
$789$	−19656.0			−0.886910
$790$		0			0
$791$			21432.9i			0.963422i
$792$		0			0
$793$	−11760.0	−	5826.42i	−0.526620	−	0.260911i
$794$		0			0
$795$			86522.4i			3.85991i
$796$		0			0
$797$	−42854.0			−1.90460			−0.952300	−	0.305163i	$-0.901289\pi$
$797$	−42854.0			−1.90460			−0.952300	+	0.305163i	$0.901289\pi$
$798$		0			0
$799$		−	24034.0i		−	1.06416i
$800$		0			0
$801$			2746.74i			0.121163i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	14722.0			0.644574
$806$		0			0
$807$	−8624.00			−0.376182
$808$		0			0
$809$	38863.0			1.68894			0.844469	−	0.535605i	$-0.179916\pi$
$809$	38863.0			1.68894			0.844469	+	0.535605i	$0.179916\pi$
$810$		0			0
$811$		−	7324.65i		−	0.317143i	−0.987347	−	0.158572i	$-0.949311\pi$
$811$		−	7324.65i		−	0.317143i	0.987347	−	0.158572i	$-0.0506889\pi$
$812$		0			0
$813$		−	27238.5i		−	1.17503i
$814$		0			0
$815$	−66682.0			−2.86597
$816$		0			0
$817$			6866.86i			0.294052i
$818$		0			0
$819$	9526.00	−	19227.2i	0.406429	−	0.820333i
$820$		0			0
$821$		−	33647.6i		−	1.43034i	−0.698951	−	0.715170i	$-0.746349\pi$
$821$		−	33647.6i		−	1.43034i	0.698951	−	0.715170i	$-0.253651\pi$
$822$		0			0
$823$	10554.0			0.447010			0.223505	−	0.974703i	$-0.428250\pi$
$823$	10554.0			0.447010			0.223505	+	0.974703i	$0.428250\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37580.4i		−	1.58017i	−0.612998	−	0.790084i	$-0.710037\pi$
$827$		−	37580.4i		−	1.58017i	0.612998	−	0.790084i	$-0.289963\pi$
$828$		0			0
$829$	32508.0			1.36194			0.680971	−	0.732311i	$-0.261558\pi$
$829$	32508.0			1.36194			0.680971	+	0.732311i	$0.261558\pi$
$830$		0			0
$831$	7112.00			0.296886
$832$		0			0
$833$	6930.00			0.288248
$834$		0			0
$835$	−57156.0			−2.36882
$836$		0			0
$837$			5826.42i			0.240610i
$838$		0			0
$839$		−	9655.21i		−	0.397300i	−0.980070	−	0.198650i	$-0.936344\pi$
$839$		−	9655.21i		−	0.397300i	0.980070	−	0.198650i	$-0.0636558\pi$
$840$		0			0
$841$	−20293.0			−0.832055
$842$		0			0
$843$		−	2039.25i		−	0.0833160i
$844$		0			0
$845$	36372.0	−	27696.3i	1.48075	−	1.12755i
$846$		0			0
$847$		−	27696.3i		−	1.12356i
$848$		0			0
$849$	28420.0			1.14885
$850$		0			0
$851$			4952.46i			0.199492i
$852$		0			0
$853$		−	47214.8i		−	1.89520i	−0.319460	−	0.947600i	$-0.603502\pi$
$853$		−	47214.8i		−	1.89520i	0.319460	−	0.947600i	$-0.396498\pi$
$854$		0			0
$855$	−57156.0			−2.28619
$856$		0			0
$857$	770.000			0.0306916			0.0153458	−	0.999882i	$-0.495115\pi$
$857$	770.000			0.0306916			0.0153458	+	0.999882i	$0.495115\pi$
$858$		0			0
$859$	33824.0			1.34349			0.671746	−	0.740781i	$-0.265545\pi$
$859$	33824.0			1.34349			0.671746	+	0.740781i	$0.265545\pi$
$860$		0			0
$861$	−48496.0			−1.91956
$862$		0			0
$863$			40056.7i			1.58001i	0.613104	+	0.790003i	$0.289921\pi$
$863$			40056.7i			1.58001i	−0.613104	+	0.790003i	$0.710079\pi$
$864$		0			0
$865$			86522.4i			3.40098i
$866$		0			0
$867$	−7112.00			−0.278588
$868$		0			0
$869$		0			0
$870$		0			0
$871$	12124.0	−	24471.0i	0.471649	−	0.951972i
$872$		0			0
$873$		−	16480.5i		−	0.638922i
$874$		0			0
$875$	79239.0			3.06145
$876$		0			0
$877$			13837.8i			0.532802i	0.963862	+	0.266401i	$0.0858346\pi$
$877$			13837.8i			0.532802i	−0.963862	+	0.266401i	$0.914165\pi$
$878$		0			0
$879$			22577.4i			0.866344i
$880$		0			0
$881$	28623.0			1.09459			0.547295	−	0.836940i	$-0.315658\pi$
$881$	28623.0			1.09459			0.547295	+	0.836940i	$0.315658\pi$
$882$		0			0
$883$	−24453.0			−0.931947			−0.465973	−	0.884799i	$-0.654296\pi$
$883$	−24453.0			−0.931947			−0.465973	+	0.884799i	$0.654296\pi$
$884$		0			0
$885$	−78806.0			−2.99326
$886$		0			0
$887$	26796.0			1.01434			0.507171	−	0.861845i	$-0.330691\pi$
$887$	26796.0			1.01434			0.507171	+	0.861845i	$0.330691\pi$
$888$		0			0
$889$		−	47152.4i		−	1.77890i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	38970.0			1.46034
$894$		0			0
$895$			47755.9i			1.78358i
$896$		0			0
$897$	9996.00	+	4952.46i	0.372081	+	0.184345i
$898$		0			0
$899$		−	10654.0i		−	0.395252i
$900$		0			0
$901$	−45738.0			−1.69118
$902$		0			0
$903$		−	8011.33i		−	0.295239i
$904$		0			0
$905$			24762.3i			0.909532i
$906$		0			0
$907$	40601.0			1.48637			0.743183	−	0.669088i	$-0.233315\pi$
$907$	40601.0			1.48637			0.743183	+	0.669088i	$0.233315\pi$
$908$		0			0
$909$	−11704.0			−0.427060
$910$		0			0
$911$	−9372.00			−0.340843			−0.170422	−	0.985371i	$-0.554513\pi$
$911$	−9372.00			−0.340843			−0.170422	+	0.985371i	$0.554513\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	40785.0i		−	1.47356i
$916$		0			0
$917$		−	7720.01i		−	0.278012i
$918$		0			0
$919$	52932.0			1.89996			0.949981	−	0.312307i	$-0.101102\pi$
$919$	52932.0			1.89996			0.949981	+	0.312307i	$0.101102\pi$
$920$		0			0
$921$			41658.9i			1.49045i
$922$		0			0
$923$	9093.00	−	18353.2i	0.324268	−	0.654500i
$924$		0			0
$925$			44863.5i			1.59470i
$926$		0			0
$927$	3388.00			0.120039
$928$		0			0
$929$			2746.74i			0.0970050i	0.998823	+	0.0485025i	$0.0154449\pi$
$929$			2746.74i			0.0970050i	−0.998823	+	0.0485025i	$0.984555\pi$
$930$		0			0
$931$			11236.7i			0.395561i
$932$		0			0
$933$	24794.0			0.870010
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−26642.0			−0.928875			−0.464438	−	0.885606i	$-0.653744\pi$
$937$	−26642.0			−0.928875			−0.464438	+	0.885606i	$0.653744\pi$
$938$		0			0
$939$	−60613.0			−2.10653
$940$		0			0
$941$			42803.4i			1.48284i	0.671042	+	0.741419i	$0.265847\pi$
$941$			42803.4i			1.48284i	−0.671042	+	0.741419i	$0.734153\pi$
$942$		0			0
$943$		−	11319.9i		−	0.390909i
$944$		0			0
$945$	−15155.0			−0.521685
$946$		0			0
$947$		−	32628.0i		−	1.11961i	−0.828626	−	0.559803i	$-0.810877\pi$
$947$		−	32628.0i		−	1.11961i	0.828626	−	0.559803i	$-0.189123\pi$
$948$		0			0
$949$	9526.00	−	19227.2i	0.325845	−	0.657683i
$950$		0			0
$951$		−	42824.2i		−	1.46022i
$952$		0			0
$953$	−39327.0			−1.33675			−0.668377	−	0.743823i	$-0.733011\pi$
$953$	−39327.0			−1.33675			−0.668377	+	0.743823i	$0.733011\pi$
$954$		0			0
$955$			87146.6i			2.95288i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−42434.0			−1.42885
$960$		0			0
$961$	2079.00			0.0697862
$962$		0			0
$963$	−968.000			−0.0323919
$964$		0			0
$965$	66682.0			2.22442
$966$		0			0
$967$			17624.9i			0.586121i	0.956094	+	0.293061i	$0.0946738\pi$
$967$			17624.9i			0.586121i	−0.956094	+	0.293061i	$0.905326\pi$
$968$		0			0
$969$		−	67295.2i		−	2.23099i
$970$		0			0
$971$	−29715.0			−0.982080			−0.491040	−	0.871137i	$-0.663383\pi$
$971$	−29715.0			−0.982080			−0.491040	+	0.871137i	$0.663383\pi$
$972$		0			0
$973$			16751.0i			0.551913i
$974$		0			0
$975$	90552.0	+	44863.5i	2.97434	+	1.47362i
$976$		0			0
$977$		−	22723.0i		−	0.744089i	−0.928215	−	0.372044i	$-0.878657\pi$
$977$		−	22723.0i		−	0.744089i	0.928215	−	0.372044i	$-0.121343\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	35249.9i		−	1.14724i
$982$		0			0
$983$			23035.2i			0.747414i	0.927547	+	0.373707i	$0.121913\pi$
$983$			23035.2i			0.747414i	−0.927547	+	0.373707i	$0.878087\pi$
$984$		0			0
$985$	−3031.00			−0.0980464
$986$		0			0
$987$	−45465.0			−1.46623
$988$		0			0
$989$	1870.00			0.0601239
$990$		0			0
$991$	61094.0			1.95834			0.979170	−	0.203042i	$-0.0650829\pi$
$991$	61094.0			1.95834			0.979170	+	0.203042i	$0.0650829\pi$
$992$		0			0
$993$			63216.7i			2.02026i
$994$		0			0
$995$			68169.1i			2.17197i
$996$		0			0
$997$	−3192.00			−0.101396			−0.0506979	−	0.998714i	$-0.516145\pi$
$997$	−3192.00			−0.101396			−0.0506979	+	0.998714i	$0.516145\pi$
$998$		0			0
$999$		−	5098.12i		−	0.161459i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.f.a.129.1		2
4.3	odd	2		52.4.d.b.25.1	✓	2
8.3	odd	2		832.4.f.a.129.2		2
8.5	even	2		832.4.f.g.129.2		2
12.11	even	2		468.4.b.c.181.2		2
13.12	even	2	inner	208.4.f.a.129.2		2
20.3	even	4		1300.4.d.a.649.4		4
20.7	even	4		1300.4.d.a.649.1		4
20.19	odd	2		1300.4.f.a.701.1		2
52.3	odd	6		676.4.h.c.485.1		4
52.7	even	12		676.4.e.c.653.2		4
52.11	even	12		676.4.e.c.529.2		4
52.15	even	12		676.4.e.c.529.1		4
52.19	even	12		676.4.e.c.653.1		4
52.23	odd	6		676.4.h.c.485.2		4
52.31	even	4		676.4.a.d.1.1		2
52.35	odd	6		676.4.h.c.361.1		4
52.43	odd	6		676.4.h.c.361.2		4
52.47	even	4		676.4.a.d.1.2		2
52.51	odd	2		52.4.d.b.25.2	yes	2
104.51	odd	2		832.4.f.a.129.1		2
104.77	even	2		832.4.f.g.129.1		2
156.155	even	2		468.4.b.c.181.1		2
260.103	even	4		1300.4.d.a.649.3		4
260.207	even	4		1300.4.d.a.649.2		4
260.259	odd	2		1300.4.f.a.701.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.d.b.25.1	✓	2	4.3	odd	2
52.4.d.b.25.2	yes	2	52.51	odd	2
208.4.f.a.129.1		2	1.1	even	1	trivial
208.4.f.a.129.2		2	13.12	even	2	inner
468.4.b.c.181.1		2	156.155	even	2
468.4.b.c.181.2		2	12.11	even	2
676.4.a.d.1.1		2	52.31	even	4
676.4.a.d.1.2		2	52.47	even	4
676.4.e.c.529.1		4	52.15	even	12
676.4.e.c.529.2		4	52.11	even	12
676.4.e.c.653.1		4	52.19	even	12
676.4.e.c.653.2		4	52.7	even	12
676.4.h.c.361.1		4	52.35	odd	6
676.4.h.c.361.2		4	52.43	odd	6
676.4.h.c.485.1		4	52.3	odd	6
676.4.h.c.485.2		4	52.23	odd	6
832.4.f.a.129.1		2	104.51	odd	2
832.4.f.a.129.2		2	8.3	odd	2
832.4.f.g.129.1		2	104.77	even	2
832.4.f.g.129.2		2	8.5	even	2
1300.4.d.a.649.1		4	20.7	even	4
1300.4.d.a.649.2		4	260.207	even	4
1300.4.d.a.649.3		4	260.103	even	4
1300.4.d.a.649.4		4	20.3	even	4
1300.4.f.a.701.1		2	20.19	odd	2
1300.4.f.a.701.2		2	260.259	odd	2

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.00000 q^{3} -20.8087i q^{5} -20.8087i q^{7} +22.0000 q^{9} +O(q^{10})\)	\(q-7.00000 q^{3} -20.8087i q^{5} -20.8087i q^{7} +22.0000 q^{9} +(42.0000 + 20.8087i) q^{13} +145.661i q^{15} -77.0000 q^{17} -124.852i q^{19} +145.661i q^{21} -34.0000 q^{23} -308.000 q^{25} +35.0000 q^{27} -64.0000 q^{29} +166.469i q^{31} -433.000 q^{35} -145.661i q^{37} +(-294.000 - 145.661i) q^{39} +332.938i q^{41} -55.0000 q^{43} -457.790i q^{45} +312.130i q^{47} -90.0000 q^{49} +539.000 q^{51} +594.000 q^{53} +873.963i q^{57} +541.025i q^{59} -280.000 q^{61} -457.790i q^{63} +(433.000 - 873.963i) q^{65} -582.642i q^{67} +238.000 q^{69} -436.982i q^{71} -457.790i q^{73} +2156.00 q^{75} -594.000 q^{79} -839.000 q^{81} +457.790i q^{83} +1602.27i q^{85} +448.000 q^{87} +124.852i q^{89} +(433.000 - 873.963i) q^{91} -1165.28i q^{93} -2598.00 q^{95} -749.111i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} + 44 q^{9}+O(q^{10}) \) 2 * q - 14 * q^3 + 44 * q^9	\( 2 q - 14 q^{3} + 44 q^{9} + 84 q^{13} - 154 q^{17} - 68 q^{23} - 616 q^{25} + 70 q^{27} - 128 q^{29} - 866 q^{35} - 588 q^{39} - 110 q^{43} - 180 q^{49} + 1078 q^{51} + 1188 q^{53} - 560 q^{61} + 866 q^{65} + 476 q^{69} + 4312 q^{75} - 1188 q^{79} - 1678 q^{81} + 896 q^{87} + 866 q^{91} - 5196 q^{95}+O(q^{100}) \) 2 * q - 14 * q^3 + 44 * q^9 + 84 * q^13 - 154 * q^17 - 68 * q^23 - 616 * q^25 + 70 * q^27 - 128 * q^29 - 866 * q^35 - 588 * q^39 - 110 * q^43 - 180 * q^49 + 1078 * q^51 + 1188 * q^53 - 560 * q^61 + 866 * q^65 + 476 * q^69 + 4312 * q^75 - 1188 * q^79 - 1678 * q^81 + 896 * q^87 + 866 * q^91 - 5196 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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