LMFDB - Embedded newform 668.3.d.a.333.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,3,Mod(333,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("668.333");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	668.d (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:	$18.2016816593$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			333.2
Character	$\chi$	$=$	668.333
Dual form			668.3.d.a.333.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.13795 q^{3} -2.98582i q^{5} -2.74320 q^{7} +17.3985 q^{9} +O(q^{10})$	$q-5.13795 q^{3} -2.98582i q^{5} -2.74320 q^{7} +17.3985 q^{9} -7.75727 q^{11} -8.84051i q^{13} +15.3410i q^{15} -28.0839i q^{17} +1.39470 q^{19} +14.0944 q^{21} -19.5614i q^{23} +16.0849 q^{25} -43.1511 q^{27} -0.918142 q^{29} -1.57609 q^{31} +39.8564 q^{33} +8.19071i q^{35} +45.6201i q^{37} +45.4221i q^{39} +2.61350i q^{41} +0.219270i q^{43} -51.9488i q^{45} -68.0374 q^{47} -41.4748 q^{49} +144.294i q^{51} +5.57248i q^{53} +23.1618i q^{55} -7.16591 q^{57} +99.4606i q^{59} -80.1157 q^{61} -47.7276 q^{63} -26.3962 q^{65} +99.2534i q^{67} +100.506i q^{69} +22.8280i q^{71} -87.9528i q^{73} -82.6432 q^{75} +21.2797 q^{77} -50.0441i q^{79} +65.1214 q^{81} -60.1360i q^{83} -83.8536 q^{85} +4.71737 q^{87} -31.5060 q^{89} +24.2513i q^{91} +8.09785 q^{93} -4.16433i q^{95} +137.186 q^{97} -134.965 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 64 q^{9}+O(q^{10}) $ 28 * q + 64 * q^9	$ 28 q + 64 q^{9} - 2 q^{11} + 10 q^{19} + 64 q^{21} - 100 q^{25} + 48 q^{27} + 38 q^{29} + 38 q^{31} - 92 q^{33} + 46 q^{47} + 184 q^{49} + 12 q^{57} + 38 q^{61} + 190 q^{63} - 10 q^{65} + 102 q^{75} + 8 q^{77} - 108 q^{81} + 10 q^{85} - 118 q^{87} + 62 q^{89} - 44 q^{93} - 204 q^{97} + 254 q^{99}+O(q^{100}) $ 28 * q + 64 * q^9 - 2 * q^11 + 10 * q^19 + 64 * q^21 - 100 * q^25 + 48 * q^27 + 38 * q^29 + 38 * q^31 - 92 * q^33 + 46 * q^47 + 184 * q^49 + 12 * q^57 + 38 * q^61 + 190 * q^63 - 10 * q^65 + 102 * q^75 + 8 * q^77 - 108 * q^81 + 10 * q^85 - 118 * q^87 + 62 * q^89 - 44 * q^93 - 204 * q^97 + 254 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/668\mathbb{Z}\right)^\times$.

$n$	$5$	$335$
$\chi(n)$	$-1$	$1$

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
$3$	−5.13795			−1.71265			−0.856325	−	0.516438i	$-0.827258\pi$
$3$	−5.13795			−1.71265			−0.856325	+	0.516438i	$0.827258\pi$
$4$		0			0
$5$		−	2.98582i		−	0.597164i	−0.954384	−	0.298582i	$-0.903486\pi$
$5$		−	2.98582i		−	0.597164i	0.954384	−	0.298582i	$-0.0965137\pi$
$6$		0			0
$7$	−2.74320			−0.391886			−0.195943	−	0.980615i	$-0.562777\pi$
$7$	−2.74320			−0.391886			−0.195943	+	0.980615i	$0.562777\pi$
$8$		0			0
$9$	17.3985			1.93317
$10$		0			0
$11$	−7.75727			−0.705206			−0.352603	−	0.935773i	$-0.614703\pi$
$11$	−7.75727			−0.705206			−0.352603	+	0.935773i	$0.614703\pi$
$12$		0			0
$13$		−	8.84051i		−	0.680039i	−0.940418	−	0.340019i	$-0.889566\pi$
$13$		−	8.84051i		−	0.680039i	0.940418	−	0.340019i	$-0.110434\pi$
$14$		0			0
$15$			15.3410i			1.02273i
$16$		0			0
$17$		−	28.0839i		−	1.65200i	−0.563672	−	0.825998i	$-0.690612\pi$
$17$		−	28.0839i		−	1.65200i	0.563672	−	0.825998i	$-0.309388\pi$
$18$		0			0
$19$	1.39470			0.0734054			0.0367027	−	0.999326i	$-0.488315\pi$
$19$	1.39470			0.0734054			0.0367027	+	0.999326i	$0.488315\pi$
$20$		0			0
$21$	14.0944			0.671163
$22$		0			0
$23$		−	19.5614i		−	0.850496i	−0.905077	−	0.425248i	$-0.860187\pi$
$23$		−	19.5614i		−	0.850496i	0.905077	−	0.425248i	$-0.139813\pi$
$24$		0			0
$25$	16.0849			0.643395
$26$		0			0
$27$	−43.1511			−1.59819
$28$		0			0
$29$	−0.918142			−0.0316601			−0.0158300	−	0.999875i	$-0.505039\pi$
$29$	−0.918142			−0.0316601			−0.0158300	+	0.999875i	$0.505039\pi$
$30$		0			0
$31$	−1.57609			−0.0508415			−0.0254207	−	0.999677i	$-0.508093\pi$
$31$	−1.57609			−0.0508415			−0.0254207	+	0.999677i	$0.508093\pi$
$32$		0			0
$33$	39.8564			1.20777
$34$		0			0
$35$			8.19071i			0.234020i
$36$		0			0
$37$			45.6201i			1.23298i	0.787364	+	0.616488i	$0.211445\pi$
$37$			45.6201i			1.23298i	−0.787364	+	0.616488i	$0.788555\pi$
$38$		0			0
$39$			45.4221i			1.16467i
$40$		0			0
$41$			2.61350i			0.0637439i	0.999492	+	0.0318719i	$0.0101469\pi$
$41$			2.61350i			0.0637439i	−0.999492	+	0.0318719i	$0.989853\pi$
$42$		0			0
$43$			0.219270i			0.00509930i	0.999997	+	0.00254965i	$0.000811579\pi$
$43$			0.219270i			0.00509930i	−0.999997	+	0.00254965i	$0.999188\pi$
$44$		0			0
$45$		−	51.9488i		−	1.15442i
$46$		0			0
$47$	−68.0374			−1.44760			−0.723802	−	0.690007i	$-0.757607\pi$
$47$	−68.0374			−1.44760			−0.723802	+	0.690007i	$0.757607\pi$
$48$		0			0
$49$	−41.4748			−0.846425
$50$		0			0
$51$			144.294i			2.82929i
$52$		0			0
$53$			5.57248i			0.105141i	0.998617	+	0.0525706i	$0.0167415\pi$
$53$			5.57248i			0.105141i	−0.998617	+	0.0525706i	$0.983259\pi$
$54$		0			0
$55$			23.1618i			0.421124i
$56$		0			0
$57$	−7.16591			−0.125718
$58$		0			0
$59$			99.4606i			1.68577i	0.538092	+	0.842886i	$0.319145\pi$
$59$			99.4606i			1.68577i	−0.538092	+	0.842886i	$0.680855\pi$
$60$		0			0
$61$	−80.1157			−1.31337			−0.656686	−	0.754164i	$-0.728042\pi$
$61$	−80.1157			−1.31337			−0.656686	+	0.754164i	$0.728042\pi$
$62$		0			0
$63$	−47.7276			−0.757581
$64$		0			0
$65$	−26.3962			−0.406095
$66$		0			0
$67$			99.2534i			1.48139i	0.671839	+	0.740697i	$0.265505\pi$
$67$			99.2534i			1.48139i	−0.671839	+	0.740697i	$0.734495\pi$
$68$		0			0
$69$			100.506i			1.45660i
$70$		0			0
$71$			22.8280i			0.321522i	0.986993	+	0.160761i	$0.0513948\pi$
$71$			22.8280i			0.321522i	−0.986993	+	0.160761i	$0.948605\pi$
$72$		0			0
$73$		−	87.9528i		−	1.20483i	−0.798182	−	0.602417i	$-0.794205\pi$
$73$		−	87.9528i		−	1.20483i	0.798182	−	0.602417i	$-0.205795\pi$
$74$		0			0
$75$	−82.6432			−1.10191
$76$		0			0
$77$	21.2797			0.276360
$78$		0			0
$79$		−	50.0441i		−	0.633470i	−0.948514	−	0.316735i	$-0.897413\pi$
$79$		−	50.0441i		−	0.633470i	0.948514	−	0.316735i	$-0.102587\pi$
$80$		0			0
$81$	65.1214			0.803968
$82$		0			0
$83$		−	60.1360i		−	0.724531i	−0.932075	−	0.362265i	$-0.882003\pi$
$83$		−	60.1360i		−	0.724531i	0.932075	−	0.362265i	$-0.117997\pi$
$84$		0			0
$85$	−83.8536			−0.986513
$86$		0			0
$87$	4.71737			0.0542226
$88$		0			0
$89$	−31.5060			−0.354000			−0.177000	−	0.984211i	$-0.556639\pi$
$89$	−31.5060			−0.354000			−0.177000	+	0.984211i	$0.556639\pi$
$90$		0			0
$91$			24.2513i			0.266498i
$92$		0			0
$93$	8.09785			0.0870736
$94$		0			0
$95$		−	4.16433i		−	0.0438351i
$96$		0			0
$97$	137.186			1.41428			0.707142	−	0.707071i	$-0.249984\pi$
$97$	137.186			1.41428			0.707142	+	0.707071i	$0.249984\pi$
$98$		0			0
$99$	−134.965			−1.36328
$100$		0			0
$101$			110.569i			1.09474i	0.836890	+	0.547371i	$0.184371\pi$
$101$			110.569i			1.09474i	−0.836890	+	0.547371i	$0.815629\pi$
$102$		0			0
$103$			130.255i			1.26461i	0.774718	+	0.632307i	$0.217892\pi$
$103$			130.255i			1.26461i	−0.774718	+	0.632307i	$0.782108\pi$
$104$		0			0
$105$		−	42.0834i		−	0.400795i
$106$		0			0
$107$	−0.747391			−0.00698496			−0.00349248	−	0.999994i	$-0.501112\pi$
$107$	−0.747391			−0.00698496			−0.00349248	+	0.999994i	$0.501112\pi$
$108$		0			0
$109$			132.824i			1.21857i	0.792953	+	0.609283i	$0.208543\pi$
$109$			132.824i			1.21857i	−0.792953	+	0.609283i	$0.791457\pi$
$110$		0			0
$111$		−	234.394i		−	2.11166i
$112$		0			0
$113$			35.8343i			0.317118i	0.987350	+	0.158559i	$0.0506847\pi$
$113$			35.8343i			0.317118i	−0.987350	+	0.158559i	$0.949315\pi$
$114$		0			0
$115$	−58.4069			−0.507886
$116$		0			0
$117$		−	153.812i		−	1.31463i
$118$		0			0
$119$			77.0399i			0.647394i
$120$		0			0
$121$	−60.8248			−0.502685
$122$		0			0
$123$		−	13.4280i		−	0.109171i
$124$		0			0
$125$		−	122.672i		−	0.981377i
$126$		0			0
$127$	5.56074			0.0437853			0.0218927	−	0.999760i	$-0.493031\pi$
$127$	5.56074			0.0437853			0.0218927	+	0.999760i	$0.493031\pi$
$128$		0			0
$129$		−	1.12660i		−	0.00873330i
$130$		0			0
$131$		−	109.243i		−	0.833915i	−0.908926	−	0.416958i	$-0.863096\pi$
$131$		−	109.243i		−	0.833915i	0.908926	−	0.416958i	$-0.136904\pi$
$132$		0			0
$133$	−3.82595			−0.0287665
$134$		0			0
$135$			128.841i			0.954380i
$136$		0			0
$137$	32.0152			0.233688			0.116844	−	0.993150i	$-0.462722\pi$
$137$	32.0152			0.233688			0.116844	+	0.993150i	$0.462722\pi$
$138$		0			0
$139$		−	1.32504i		−	0.00953266i	−0.999989	−	0.00476633i	$-0.998483\pi$
$139$		−	1.32504i		−	0.00953266i	0.999989	−	0.00476633i	$-0.00151718\pi$
$140$		0			0
$141$	349.573			2.47924
$142$		0			0
$143$			68.5782i			0.479568i
$144$		0			0
$145$			2.74141i			0.0189063i
$146$		0			0
$147$	213.096			1.44963
$148$		0			0
$149$			182.530i			1.22503i	0.790458	+	0.612516i	$0.209842\pi$
$149$			182.530i			1.22503i	−0.790458	+	0.612516i	$0.790158\pi$
$150$		0			0
$151$			0.697280i			0.00461775i	0.999997	+	0.00230887i	$0.000734938\pi$
$151$			0.697280i			0.00461775i	−0.999997	+	0.00230887i	$0.999265\pi$
$152$		0			0
$153$		−	488.619i		−	3.19359i
$154$		0			0
$155$			4.70591i			0.0303607i
$156$		0			0
$157$	70.7517			0.450648			0.225324	−	0.974284i	$-0.427656\pi$
$157$	70.7517			0.450648			0.225324	+	0.974284i	$0.427656\pi$
$158$		0			0
$159$		−	28.6311i		−	0.180070i
$160$		0			0
$161$			53.6609i			0.333297i
$162$		0			0
$163$			32.8328i			0.201428i	0.994915	+	0.100714i	$0.0321127\pi$
$163$			32.8328i			0.201428i	−0.994915	+	0.100714i	$0.967887\pi$
$164$		0			0
$165$		−	119.004i		−	0.721237i
$166$		0			0
$167$	−99.3487	−	134.234i	−0.594902	−	0.803798i
$167$	−99.3487	−	134.234i	−0.594902	−	0.803798i
$168$		0			0
$169$	90.8455			0.537547
$170$		0			0
$171$	24.2657			0.141905
$172$		0			0
$173$	−2.33359			−0.0134889			−0.00674447	−	0.999977i	$-0.502147\pi$
$173$	−2.33359			−0.0134889			−0.00674447	+	0.999977i	$0.502147\pi$
$174$		0			0
$175$	−44.1240			−0.252137
$176$		0			0
$177$		−	511.023i		−	2.88714i
$178$		0			0
$179$	−209.305			−1.16930			−0.584651	−	0.811285i	$-0.698769\pi$
$179$	−209.305			−1.16930			−0.584651	+	0.811285i	$0.698769\pi$
$180$		0			0
$181$	146.147			0.807440			0.403720	−	0.914883i	$-0.367717\pi$
$181$	146.147			0.807440			0.403720	+	0.914883i	$0.367717\pi$
$182$		0			0
$183$	411.630			2.24935
$184$		0			0
$185$	136.214			0.736289
$186$		0			0
$187$			217.855i			1.16500i
$188$		0			0
$189$	118.372			0.626307
$190$		0			0
$191$	264.140			1.38293			0.691465	−	0.722410i	$-0.256966\pi$
$191$	264.140			1.38293			0.691465	+	0.722410i	$0.256966\pi$
$192$		0			0
$193$			38.6217i			0.200113i	0.994982	+	0.100056i	$0.0319023\pi$
$193$			38.6217i			0.200113i	−0.994982	+	0.100056i	$0.968098\pi$
$194$		0			0
$195$	135.622			0.695498
$196$		0			0
$197$		−	324.492i		−	1.64717i	−0.567194	−	0.823584i	$-0.691971\pi$
$197$		−	324.492i		−	1.64717i	0.567194	−	0.823584i	$-0.308029\pi$
$198$		0			0
$199$	−95.3443			−0.479117			−0.239559	−	0.970882i	$-0.577003\pi$
$199$	−95.3443			−0.479117			−0.239559	+	0.970882i	$0.577003\pi$
$200$		0			0
$201$		−	509.959i		−	2.53711i
$202$		0			0
$203$	2.51865			0.0124071
$204$		0			0
$205$	7.80344			0.0380656
$206$		0			0
$207$		−	340.339i		−	1.64415i
$208$		0			0
$209$	−10.8191			−0.0517659
$210$		0			0
$211$	−113.231			−0.536640			−0.268320	−	0.963330i	$-0.586468\pi$
$211$	−113.231			−0.536640			−0.268320	+	0.963330i	$0.586468\pi$
$212$		0			0
$213$		−	117.289i		−	0.550654i
$214$		0			0
$215$	0.654700			0.00304512
$216$		0			0
$217$	4.32352			0.0199241
$218$		0			0
$219$			451.897i			2.06346i
$220$		0			0
$221$	−248.276			−1.12342
$222$		0			0
$223$	−147.797			−0.662765			−0.331383	−	0.943496i	$-0.607515\pi$
$223$	−147.797			−0.662765			−0.331383	+	0.943496i	$0.607515\pi$
$224$		0			0
$225$	279.853			1.24379
$226$		0			0
$227$			19.6221i			0.0864409i	0.999066	+	0.0432204i	$0.0137618\pi$
$227$			19.6221i			0.0864409i	−0.999066	+	0.0432204i	$0.986238\pi$
$228$		0			0
$229$	−410.342			−1.79188			−0.895942	−	0.444170i	$-0.853499\pi$
$229$	−410.342			−1.79188			−0.895942	+	0.444170i	$0.853499\pi$
$230$		0			0
$231$	−109.334			−0.473308
$232$		0			0
$233$	−258.142			−1.10791			−0.553953	−	0.832548i	$-0.686881\pi$
$233$	−258.142			−1.10791			−0.553953	+	0.832548i	$0.686881\pi$
$234$		0			0
$235$			203.148i			0.864458i
$236$		0			0
$237$			257.124i			1.08491i
$238$		0			0
$239$	−103.560			−0.433306			−0.216653	−	0.976249i	$-0.569514\pi$
$239$	−103.560			−0.433306			−0.216653	+	0.976249i	$0.569514\pi$
$240$		0			0
$241$			219.466i			0.910647i	0.890326	+	0.455323i	$0.150476\pi$
$241$			219.466i			0.910647i	−0.890326	+	0.455323i	$0.849524\pi$
$242$		0			0
$243$	53.7694			0.221273
$244$		0			0
$245$			123.836i			0.505455i
$246$		0			0
$247$		−	12.3299i		−	0.0499185i
$248$		0			0
$249$			308.976i			1.24087i
$250$		0			0
$251$	−66.5117			−0.264987			−0.132494	−	0.991184i	$-0.542298\pi$
$251$	−66.5117			−0.264987			−0.132494	+	0.991184i	$0.542298\pi$
$252$		0			0
$253$			151.743i			0.599775i
$254$		0			0
$255$	430.836			1.68955
$256$		0			0
$257$		−	0.0874739i		−	0.000340365i	−1.00000		0.000170183i	$-0.999946\pi$
$257$		−	0.0874739i		−	0.000340365i	1.00000		0.000170183i	$-5.41708e-5\pi$
$258$		0			0
$259$		−	125.145i		−	0.483186i
$260$		0			0
$261$	−15.9743			−0.0612042
$262$		0			0
$263$	−90.4728			−0.344003			−0.172001	−	0.985097i	$-0.555023\pi$
$263$	−90.4728			−0.344003			−0.172001	+	0.985097i	$0.555023\pi$
$264$		0			0
$265$	16.6384			0.0627866
$266$		0			0
$267$	161.876			0.606278
$268$		0			0
$269$		−	76.9103i		−	0.285912i	−0.989729	−	0.142956i	$-0.954339\pi$
$269$		−	76.9103i		−	0.285912i	0.989729	−	0.142956i	$-0.0456607\pi$
$270$		0			0
$271$			45.8720i			0.169269i	0.996412	+	0.0846347i	$0.0269723\pi$
$271$			45.8720i			0.169269i	−0.996412	+	0.0846347i	$0.973028\pi$
$272$		0			0
$273$		−	124.602i		−	0.456417i
$274$		0			0
$275$	−124.775			−0.453726
$276$		0			0
$277$			440.990i			1.59202i	0.605281	+	0.796012i	$0.293061\pi$
$277$			440.990i			1.59202i	−0.605281	+	0.796012i	$0.706939\pi$
$278$		0			0
$279$	−27.4215			−0.0982851
$280$		0			0
$281$	40.6203			0.144556			0.0722781	−	0.997385i	$-0.476973\pi$
$281$	40.6203			0.144556			0.0722781	+	0.997385i	$0.476973\pi$
$282$		0			0
$283$	−327.466			−1.15712			−0.578562	−	0.815638i	$-0.696386\pi$
$283$	−327.466			−1.15712			−0.578562	+	0.815638i	$0.696386\pi$
$284$		0			0
$285$			21.3961i			0.0750741i
$286$		0			0
$287$		−	7.16935i		−	0.0249803i
$288$		0			0
$289$	−499.708			−1.72909
$290$		0			0
$291$	−704.852			−2.42217
$292$		0			0
$293$	359.652			1.22748			0.613741	−	0.789508i	$-0.289664\pi$
$293$	359.652			1.22748			0.613741	+	0.789508i	$0.289664\pi$
$294$		0			0
$295$	296.971			1.00668
$296$		0			0
$297$	334.734			1.12705
$298$		0			0
$299$	−172.933			−0.578371
$300$		0			0
$301$		−	0.601501i		−	0.00199834i
$302$		0			0
$303$		−	568.097i		−	1.87491i
$304$		0			0
$305$			239.211i			0.784299i
$306$		0			0
$307$		−	64.5880i		−	0.210384i	−0.994452	−	0.105192i	$-0.966454\pi$
$307$		−	64.5880i		−	0.210384i	0.994452	−	0.105192i	$-0.0335458\pi$
$308$		0			0
$309$		−	669.244i		−	2.16584i
$310$		0			0
$311$	201.939			0.649320			0.324660	−	0.945831i	$-0.394750\pi$
$311$	201.939			0.649320			0.324660	+	0.945831i	$0.394750\pi$
$312$		0			0
$313$			298.275i			0.952956i	0.879186	+	0.476478i	$0.158087\pi$
$313$			298.275i			0.952956i	−0.879186	+	0.476478i	$0.841913\pi$
$314$		0			0
$315$			142.506i			0.452400i
$316$		0			0
$317$	202.802			0.639753			0.319877	−	0.947459i	$-0.396359\pi$
$317$	202.802			0.639753			0.319877	+	0.947459i	$0.396359\pi$
$318$		0			0
$319$	7.12227			0.0223269
$320$		0			0
$321$	3.84006			0.0119628
$322$		0			0
$323$		−	39.1688i		−	0.121265i
$324$		0			0
$325$		−	142.198i		−	0.437534i
$326$		0			0
$327$		−	682.442i		−	2.08698i
$328$		0			0
$329$	186.640			0.567296
$330$		0			0
$331$		−	193.289i		−	0.583955i	−0.956425	−	0.291977i	$-0.905687\pi$
$331$		−	193.289i		−	0.583955i	0.956425	−	0.291977i	$-0.0943132\pi$
$332$		0			0
$333$			793.722i			2.38355i
$334$		0			0
$335$	296.353			0.884635
$336$		0			0
$337$	−183.245			−0.543753			−0.271877	−	0.962332i	$-0.587644\pi$
$337$	−183.245			−0.543753			−0.271877	+	0.962332i	$0.587644\pi$
$338$		0			0
$339$		−	184.115i		−	0.543111i
$340$		0			0
$341$	12.2261			0.0358537
$342$		0			0
$343$	248.191			0.723588
$344$		0			0
$345$	300.092			0.869830
$346$		0			0
$347$		−	466.940i		−	1.34565i	−0.739802	−	0.672825i	$-0.765081\pi$
$347$		−	466.940i		−	1.34565i	0.739802	−	0.672825i	$-0.234919\pi$
$348$		0			0
$349$			9.89216i			0.0283443i	0.999900	+	0.0141721i	$0.00451129\pi$
$349$			9.89216i			0.0283443i	−0.999900	+	0.0141721i	$0.995489\pi$
$350$		0			0
$351$			381.477i			1.08683i
$352$		0			0
$353$	302.034			0.855620			0.427810	−	0.903869i	$-0.359285\pi$
$353$	302.034			0.855620			0.427810	+	0.903869i	$0.359285\pi$
$354$		0			0
$355$	68.1605			0.192001
$356$		0			0
$357$		−	395.827i		−	1.10876i
$358$		0			0
$359$	430.611			1.19947			0.599736	−	0.800198i	$-0.295272\pi$
$359$	430.611			1.19947			0.599736	+	0.800198i	$0.295272\pi$
$360$		0			0
$361$	−359.055			−0.994612
$362$		0			0
$363$	312.515			0.860922
$364$		0			0
$365$	−262.611			−0.719483
$366$		0			0
$367$	−56.3517			−0.153547			−0.0767735	−	0.997049i	$-0.524462\pi$
$367$	−56.3517			−0.153547			−0.0767735	+	0.997049i	$0.524462\pi$
$368$		0			0
$369$			45.4710i			0.123228i
$370$		0			0
$371$		−	15.2864i		−	0.0412034i
$372$		0			0
$373$		−	149.186i		−	0.399962i	−0.979800	−	0.199981i	$-0.935912\pi$
$373$		−	149.186i		−	0.399962i	0.979800	−	0.199981i	$-0.0640880\pi$
$374$		0			0
$375$			630.283i			1.68075i
$376$		0			0
$377$			8.11684i			0.0215301i
$378$		0			0
$379$			494.285i			1.30418i	0.758140	+	0.652091i	$0.226108\pi$
$379$			494.285i			1.30418i	−0.758140	+	0.652091i	$0.773892\pi$
$380$		0			0
$381$	−28.5708			−0.0749889
$382$		0			0
$383$	493.548			1.28864			0.644318	−	0.764758i	$-0.277141\pi$
$383$	493.548			1.28864			0.644318	+	0.764758i	$0.277141\pi$
$384$		0			0
$385$		−	63.5375i		−	0.165032i
$386$		0			0
$387$			3.81496i			0.00985779i
$388$		0			0
$389$			440.263i			1.13178i	0.824480	+	0.565891i	$0.191468\pi$
$389$			440.263i			1.13178i	−0.824480	+	0.565891i	$0.808532\pi$
$390$		0			0
$391$	−549.362			−1.40502
$392$		0			0
$393$			561.284i			1.42820i
$394$		0			0
$395$	−149.423			−0.378285
$396$		0			0
$397$	418.770			1.05484			0.527418	−	0.849606i	$-0.323160\pi$
$397$	418.770			1.05484			0.527418	+	0.849606i	$0.323160\pi$
$398$		0			0
$399$	19.6575			0.0492670
$400$		0			0
$401$			295.324i			0.736468i	0.929733	+	0.368234i	$0.120037\pi$
$401$			295.324i			0.736468i	−0.929733	+	0.368234i	$0.879963\pi$
$402$		0			0
$403$			13.9334i			0.0345742i
$404$		0			0
$405$		−	194.441i		−	0.480101i
$406$		0			0
$407$		−	353.887i		−	0.869502i
$408$		0			0
$409$	−748.982			−1.83125			−0.915625	−	0.402033i	$-0.868304\pi$
$409$	−748.982			−1.83125			−0.915625	+	0.402033i	$0.868304\pi$
$410$		0			0
$411$	−164.493			−0.400225
$412$		0			0
$413$		−	272.840i		−	0.660630i
$414$		0			0
$415$	−179.555			−0.432664
$416$		0			0
$417$			6.80798i			0.0163261i
$418$		0			0
$419$	−156.993			−0.374685			−0.187343	−	0.982295i	$-0.559987\pi$
$419$	−156.993			−0.374685			−0.187343	+	0.982295i	$0.559987\pi$
$420$		0			0
$421$	−472.419			−1.12214			−0.561068	−	0.827770i	$-0.689609\pi$
$421$	−472.419			−1.12214			−0.561068	+	0.827770i	$0.689609\pi$
$422$		0			0
$423$	−1183.75			−2.79846
$424$		0			0
$425$		−	451.727i		−	1.06289i
$426$		0			0
$427$	219.774			0.514692
$428$		0			0
$429$		−	352.351i		−	0.821331i
$430$		0			0
$431$	145.784			0.338245			0.169122	−	0.985595i	$-0.445907\pi$
$431$	145.784			0.338245			0.169122	+	0.985595i	$0.445907\pi$
$432$		0			0
$433$	−84.4449			−0.195023			−0.0975114	−	0.995234i	$-0.531088\pi$
$433$	−84.4449			−0.195023			−0.0975114	+	0.995234i	$0.531088\pi$
$434$		0			0
$435$		−	14.0852i		−	0.0323798i
$436$		0			0
$437$		−	27.2824i		−	0.0624310i
$438$		0			0
$439$			362.166i			0.824980i	0.910962	+	0.412490i	$0.135341\pi$
$439$			362.166i			0.824980i	−0.910962	+	0.412490i	$0.864659\pi$
$440$		0			0
$441$	−721.600			−1.63628
$442$		0			0
$443$			617.266i			1.39338i	0.717374	+	0.696688i	$0.245344\pi$
$443$			617.266i			1.39338i	−0.717374	+	0.696688i	$0.754656\pi$
$444$		0			0
$445$			94.0713i			0.211396i
$446$		0			0
$447$		−	937.828i		−	2.09805i
$448$		0			0
$449$	638.548			1.42216			0.711078	−	0.703113i	$-0.248207\pi$
$449$	638.548			1.42216			0.711078	+	0.703113i	$0.248207\pi$
$450$		0			0
$451$		−	20.2736i		−	0.0449526i
$452$		0			0
$453$		−	3.58259i		−	0.00790858i
$454$		0			0
$455$	72.4100			0.159143
$456$		0			0
$457$			117.697i			0.257542i	0.991674	+	0.128771i	$0.0411032\pi$
$457$			117.697i			0.257542i	−0.991674	+	0.128771i	$0.958897\pi$
$458$		0			0
$459$			1211.85i			2.64020i
$460$		0			0
$461$	−242.781			−0.526641			−0.263320	−	0.964708i	$-0.584818\pi$
$461$	−242.781			−0.526641			−0.263320	+	0.964708i	$0.584818\pi$
$462$		0			0
$463$		−	49.0769i		−	0.105998i	−0.998595	−	0.0529989i	$-0.983122\pi$
$463$		−	49.0769i		−	0.105998i	0.998595	−	0.0529989i	$-0.0168780\pi$
$464$		0			0
$465$		−	24.1787i		−	0.0519973i
$466$		0			0
$467$	793.848			1.69989			0.849945	−	0.526872i	$-0.176635\pi$
$467$	793.848			1.69989			0.849945	+	0.526872i	$0.176635\pi$
$468$		0			0
$469$		−	272.272i		−	0.580537i
$470$		0			0
$471$	−363.519			−0.771802
$472$		0			0
$473$		−	1.70093i		−	0.00359605i
$474$		0			0
$475$	22.4336			0.0472287
$476$		0			0
$477$			96.9529i			0.203256i
$478$		0			0
$479$		−	833.244i		−	1.73955i	−0.493450	−	0.869774i	$-0.664264\pi$
$479$		−	833.244i		−	1.73955i	0.493450	−	0.869774i	$-0.335736\pi$
$480$		0			0
$481$	403.305			0.838472
$482$		0			0
$483$		−	275.707i		−	0.570822i
$484$		0			0
$485$		−	409.612i		−	0.844560i
$486$		0			0
$487$			708.169i			1.45415i	0.686560	+	0.727073i	$0.259120\pi$
$487$			708.169i			1.45415i	−0.686560	+	0.727073i	$0.740880\pi$
$488$		0			0
$489$		−	168.693i		−	0.344976i
$490$		0			0
$491$	−406.903			−0.828723			−0.414362	−	0.910112i	$-0.635995\pi$
$491$	−406.903			−0.828723			−0.414362	+	0.910112i	$0.635995\pi$
$492$		0			0
$493$			25.7851i			0.0523023i
$494$		0			0
$495$			402.981i			0.814103i
$496$		0			0
$497$		−	62.6219i		−	0.126000i
$498$		0			0
$499$			370.778i			0.743042i	0.928424	+	0.371521i	$0.121164\pi$
$499$			370.778i			0.743042i	−0.928424	+	0.371521i	$0.878836\pi$
$500$		0			0
$501$	510.448	+	689.689i	1.01886	+	1.37662i
$502$		0			0
$503$	−10.0132			−0.0199069			−0.00995343	−	0.999950i	$-0.503168\pi$
$503$	−10.0132			−0.0199069			−0.00995343	+	0.999950i	$0.503168\pi$
$504$		0			0
$505$	330.139			0.653740
$506$		0			0
$507$	−466.759			−0.920630
$508$		0			0
$509$	−301.479			−0.592297			−0.296149	−	0.955142i	$-0.595702\pi$
$509$	−301.479			−0.592297			−0.296149	+	0.955142i	$0.595702\pi$
$510$		0			0
$511$			241.272i			0.472157i
$512$		0			0
$513$	−60.1829			−0.117316
$514$		0			0
$515$	388.919			0.755182
$516$		0			0
$517$	527.784			1.02086
$518$		0			0
$519$	11.9898			0.0231018
$520$		0			0
$521$			144.175i			0.276727i	0.990382	+	0.138363i	$0.0441842\pi$
$521$			144.175i			0.276727i	−0.990382	+	0.138363i	$0.955816\pi$
$522$		0			0
$523$	−704.323			−1.34670			−0.673349	−	0.739325i	$-0.735145\pi$
$523$	−704.323			−1.34670			−0.673349	+	0.739325i	$0.735145\pi$
$524$		0			0
$525$	226.707			0.431823
$526$		0			0
$527$			44.2627i			0.0839900i
$528$		0			0
$529$	146.351			0.276656
$530$		0			0
$531$			1730.46i			3.25888i
$532$		0			0
$533$	23.1046			0.0433483
$534$		0			0
$535$			2.23158i			0.00417117i
$536$		0			0
$537$	1075.40			2.00260
$538$		0			0
$539$	321.731			0.596904
$540$		0			0
$541$			378.892i			0.700355i	0.936683	+	0.350178i	$0.113879\pi$
$541$			378.892i			0.700355i	−0.936683	+	0.350178i	$0.886121\pi$
$542$		0			0
$543$	−750.894			−1.38286
$544$		0			0
$545$	396.588			0.727684
$546$		0			0
$547$		−	681.276i		−	1.24548i	−0.782430	−	0.622739i	$-0.786020\pi$
$547$		−	681.276i		−	1.24548i	0.782430	−	0.622739i	$-0.213980\pi$
$548$		0			0
$549$	−1393.89			−2.53897
$550$		0			0
$551$	−1.28054			−0.00232402
$552$		0			0
$553$			137.281i			0.248248i
$554$		0			0
$555$	−699.858			−1.26101
$556$		0			0
$557$	−1020.15			−1.83150			−0.915750	−	0.401749i	$-0.868402\pi$
$557$	−1020.15			−1.83150			−0.915750	+	0.401749i	$0.868402\pi$
$558$		0			0
$559$	1.93845			0.00346772
$560$		0			0
$561$		−	1119.33i		−	1.99523i
$562$		0			0
$563$	−588.188			−1.04474			−0.522370	−	0.852719i	$-0.674952\pi$
$563$	−588.188			−1.04474			−0.522370	+	0.852719i	$0.674952\pi$
$564$		0			0
$565$	106.995			0.189371
$566$		0			0
$567$	−178.641			−0.315064
$568$		0			0
$569$			379.440i			0.666855i	0.942776	+	0.333427i	$0.108205\pi$
$569$			379.440i			0.666855i	−0.942776	+	0.333427i	$0.891795\pi$
$570$		0			0
$571$			611.578i			1.07107i	0.844515	+	0.535533i	$0.179889\pi$
$571$			611.578i			1.07107i	−0.844515	+	0.535533i	$0.820111\pi$
$572$		0			0
$573$	−1357.13			−2.36847
$574$		0			0
$575$		−	314.643i		−	0.547205i
$576$		0			0
$577$	462.941			0.802324			0.401162	−	0.916007i	$-0.368607\pi$
$577$	462.941			0.802324			0.401162	+	0.916007i	$0.368607\pi$
$578$		0			0
$579$		−	198.436i		−	0.342723i
$580$		0			0
$581$			164.965i			0.283933i
$582$		0			0
$583$		−	43.2272i		−	0.0741462i
$584$		0			0
$585$	−459.254			−0.785049
$586$		0			0
$587$		−	846.233i		−	1.44162i	−0.693131	−	0.720812i	$-0.743769\pi$
$587$		−	846.233i		−	1.44162i	0.693131	−	0.720812i	$-0.256231\pi$
$588$		0			0
$589$	−2.19817			−0.00373204
$590$		0			0
$591$			1667.22i			2.82102i
$592$		0			0
$593$			948.313i			1.59918i	0.600547	+	0.799590i	$0.294950\pi$
$593$			948.313i			1.59918i	−0.600547	+	0.799590i	$0.705050\pi$
$594$		0			0
$595$	230.027			0.386601
$596$		0			0
$597$	489.874			0.820560
$598$		0			0
$599$	−470.840			−0.786044			−0.393022	−	0.919529i	$-0.628570\pi$
$599$	−470.840			−0.786044			−0.393022	+	0.919529i	$0.628570\pi$
$600$		0			0
$601$	79.8951			0.132937			0.0664685	−	0.997789i	$-0.478827\pi$
$601$	79.8951			0.132937			0.0664685	+	0.997789i	$0.478827\pi$
$602$		0			0
$603$			1726.86i			2.86378i
$604$		0			0
$605$			181.612i			0.300185i
$606$		0			0
$607$		−	965.185i		−	1.59009i	−0.606550	−	0.795045i	$-0.707447\pi$
$607$		−	965.185i		−	1.59009i	0.606550	−	0.795045i	$-0.292553\pi$
$608$		0			0
$609$	−12.9407			−0.0212491
$610$		0			0
$611$			601.485i			0.984427i
$612$		0			0
$613$	−580.752			−0.947393			−0.473696	−	0.880688i	$-0.657081\pi$
$613$	−580.752			−0.947393			−0.473696	+	0.880688i	$0.657081\pi$
$614$		0			0
$615$	−40.0937			−0.0651929
$616$		0			0
$617$	−418.484			−0.678257			−0.339128	−	0.940740i	$-0.610132\pi$
$617$	−418.484			−0.678257			−0.339128	+	0.940740i	$0.610132\pi$
$618$		0			0
$619$		−	293.782i		−	0.474608i	−0.971435	−	0.237304i	$-0.923736\pi$
$619$		−	293.782i		−	0.474608i	0.971435	−	0.237304i	$-0.0762637\pi$
$620$		0			0
$621$			844.096i			1.35925i
$622$		0			0
$623$	86.4273			0.138728
$624$		0			0
$625$	35.8449			0.0573518
$626$		0			0
$627$	55.5879			0.0886569
$628$		0			0
$629$	1281.19			2.03687
$630$		0			0
$631$	332.289			0.526607			0.263304	−	0.964713i	$-0.415188\pi$
$631$	332.289			0.526607			0.263304	+	0.964713i	$0.415188\pi$
$632$		0			0
$633$	581.775			0.919075
$634$		0			0
$635$		−	16.6034i		−	0.0261470i
$636$		0			0
$637$			366.659i			0.575602i
$638$		0			0
$639$			397.174i			0.621555i
$640$		0			0
$641$		−	757.168i		−	1.18123i	−0.806953	−	0.590615i	$-0.798885\pi$
$641$		−	757.168i		−	1.18123i	0.806953	−	0.590615i	$-0.201115\pi$
$642$		0			0
$643$			1209.88i			1.88162i	0.338931	+	0.940811i	$0.389935\pi$
$643$			1209.88i			1.88162i	−0.338931	+	0.940811i	$0.610065\pi$
$644$		0			0
$645$	−3.36381			−0.00521522
$646$		0			0
$647$		−	547.530i		−	0.846260i	−0.906069	−	0.423130i	$-0.860931\pi$
$647$		−	547.530i		−	0.846260i	0.906069	−	0.423130i	$-0.139069\pi$
$648$		0			0
$649$		−	771.542i		−	1.18882i
$650$		0			0
$651$	−22.2140			−0.0341229
$652$		0			0
$653$	−927.059			−1.41969			−0.709846	−	0.704357i	$-0.751235\pi$
$653$	−927.059			−1.41969			−0.709846	+	0.704357i	$0.751235\pi$
$654$		0			0
$655$	−326.180			−0.497984
$656$		0			0
$657$		−	1530.25i		−	2.32914i
$658$		0			0
$659$		−	842.809i		−	1.27892i	−0.768824	−	0.639461i	$-0.779158\pi$
$659$		−	842.809i		−	1.27892i	0.768824	−	0.639461i	$-0.220842\pi$
$660$		0			0
$661$			948.207i			1.43450i	0.696814	+	0.717252i	$0.254600\pi$
$661$			948.207i			1.43450i	−0.696814	+	0.717252i	$0.745400\pi$
$662$		0			0
$663$	1275.63			1.92403
$664$		0			0
$665$			11.4236i			0.0171783i
$666$		0			0
$667$			17.9602i			0.0269268i
$668$		0			0
$669$	759.372			1.13508
$670$		0			0
$671$	621.479			0.926198
$672$		0			0
$673$		−	969.460i		−	1.44051i	−0.693712	−	0.720253i	$-0.744026\pi$
$673$		−	969.460i		−	1.44051i	0.693712	−	0.720253i	$-0.255974\pi$
$674$		0			0
$675$	−694.079			−1.02827
$676$		0			0
$677$	734.917			1.08555			0.542774	−	0.839878i	$-0.317374\pi$
$677$	734.917			1.08555			0.542774	+	0.839878i	$0.317374\pi$
$678$		0			0
$679$	−376.328			−0.554238
$680$		0			0
$681$		−	100.817i		−	0.148043i
$682$		0			0
$683$		−	1002.58i		−	1.46791i	−0.679196	−	0.733957i	$-0.737671\pi$
$683$		−	1002.58i		−	1.46791i	0.679196	−	0.733957i	$-0.262329\pi$
$684$		0			0
$685$		−	95.5918i		−	0.139550i
$686$		0			0
$687$	2108.31			3.06887
$688$		0			0
$689$	49.2636			0.0715001
$690$		0			0
$691$		−	525.650i		−	0.760709i	−0.924841	−	0.380355i	$-0.875802\pi$
$691$		−	525.650i		−	0.760709i	0.924841	−	0.380355i	$-0.124198\pi$
$692$		0			0
$693$	370.236			0.534251
$694$		0			0
$695$	−3.95633			−0.00569256
$696$		0			0
$697$	73.3973			0.105305
$698$		0			0
$699$	1326.32			1.89745
$700$		0			0
$701$	−785.725			−1.12086			−0.560432	−	0.828201i	$-0.689365\pi$
$701$	−785.725			−1.12086			−0.560432	+	0.828201i	$0.689365\pi$
$702$		0			0
$703$			63.6265i			0.0905071i
$704$		0			0
$705$		−	1043.76i		−	1.48051i
$706$		0			0
$707$		−	303.313i		−	0.429014i
$708$		0			0
$709$			426.085i			0.600966i	0.953787	+	0.300483i	$0.0971479\pi$
$709$			426.085i			0.600966i	−0.953787	+	0.300483i	$0.902852\pi$
$710$		0			0
$711$		−	870.693i		−	1.22460i
$712$		0			0
$713$			30.8305i			0.0432405i
$714$		0			0
$715$	204.762			0.286381
$716$		0			0
$717$	532.087			0.742102
$718$		0			0
$719$		−	985.518i		−	1.37068i	−0.728224	−	0.685339i	$-0.759654\pi$
$719$		−	985.518i		−	1.37068i	0.728224	−	0.685339i	$-0.240346\pi$
$720$		0			0
$721$		−	357.316i		−	0.495584i
$722$		0			0
$723$		−	1127.60i		−	1.55962i
$724$		0			0
$725$	−14.7682			−0.0203699
$726$		0			0
$727$		−	639.367i		−	0.879459i	−0.898130	−	0.439730i	$-0.855074\pi$
$727$		−	639.367i		−	0.879459i	0.898130	−	0.439730i	$-0.144926\pi$
$728$		0			0
$729$	−862.357			−1.18293
$730$		0			0
$731$	6.15796			0.00842402
$732$		0			0
$733$	−514.175			−0.701467			−0.350733	−	0.936475i	$-0.614068\pi$
$733$	−514.175			−0.701467			−0.350733	+	0.936475i	$0.614068\pi$
$734$		0			0
$735$		−	636.265i		−	0.865667i
$736$		0			0
$737$		−	769.935i		−	1.04469i
$738$		0			0
$739$		−	642.519i		−	0.869444i	−0.900565	−	0.434722i	$-0.856847\pi$
$739$		−	642.519i		−	0.869444i	0.900565	−	0.434722i	$-0.143153\pi$
$740$		0			0
$741$			63.3503i			0.0854929i
$742$		0			0
$743$	−16.2507			−0.0218718			−0.0109359	−	0.999940i	$-0.503481\pi$
$743$	−16.2507			−0.0218718			−0.0109359	+	0.999940i	$0.503481\pi$
$744$		0			0
$745$	545.001			0.731545
$746$		0			0
$747$		−	1046.28i		−	1.40064i
$748$		0			0
$749$	2.05024			0.00273731
$750$		0			0
$751$			1269.89i			1.69094i	0.534027	+	0.845468i	$0.320678\pi$
$751$			1269.89i			1.69094i	−0.534027	+	0.845468i	$0.679322\pi$
$752$		0			0
$753$	341.734			0.453830
$754$		0			0
$755$	2.08195			0.00275755
$756$		0			0
$757$	285.373			0.376979			0.188490	−	0.982075i	$-0.439641\pi$
$757$	285.373			0.376979			0.188490	+	0.982075i	$0.439641\pi$
$758$		0			0
$759$		−	779.648i		−	1.02720i
$760$		0			0
$761$	484.791			0.637045			0.318522	−	0.947915i	$-0.396813\pi$
$761$	484.791			0.637045			0.318522	+	0.947915i	$0.396813\pi$
$762$		0			0
$763$		−	364.362i		−	0.477539i
$764$		0			0
$765$	−1458.93			−1.90710
$766$		0			0
$767$	879.282			1.14639
$768$		0			0
$769$		−	1302.86i		−	1.69422i	−0.531416	−	0.847111i	$-0.678340\pi$
$769$		−	1302.86i		−	1.69422i	0.531416	−	0.847111i	$-0.321660\pi$
$770$		0			0
$771$			0.449436i			0.000582926i
$772$		0			0
$773$		−	767.717i		−	0.993165i	−0.867989	−	0.496583i	$-0.834588\pi$
$773$		−	767.717i		−	0.993165i	0.867989	−	0.496583i	$-0.165412\pi$
$774$		0			0
$775$	−25.3511			−0.0327111
$776$		0			0
$777$			642.989i			0.827528i
$778$		0			0
$779$			3.64505i			0.00467914i
$780$		0			0
$781$		−	177.083i		−	0.226739i
$782$		0			0
$783$	39.6188			0.0505987
$784$		0			0
$785$		−	211.252i		−	0.269111i
$786$		0			0
$787$		−	1121.01i		−	1.42441i	−0.701971	−	0.712206i	$-0.747696\pi$
$787$		−	1121.01i		−	1.42441i	0.701971	−	0.712206i	$-0.252304\pi$
$788$		0			0
$789$	464.844			0.589156
$790$		0			0
$791$		−	98.3006i		−	0.124274i
$792$		0			0
$793$			708.264i			0.893145i
$794$		0			0
$795$	−85.4874			−0.107531
$796$		0			0
$797$			1299.17i			1.63007i	0.579409	+	0.815037i	$0.303283\pi$
$797$			1299.17i			1.63007i	−0.579409	+	0.815037i	$0.696717\pi$
$798$		0			0
$799$			1910.76i			2.39144i
$800$		0			0
$801$	−548.157			−0.684341
$802$		0			0
$803$			682.273i			0.849655i
$804$		0			0
$805$	160.222			0.199033
$806$		0			0
$807$			395.161i			0.489667i
$808$		0			0
$809$	770.396			0.952282			0.476141	−	0.879369i	$-0.342035\pi$
$809$	770.396			0.952282			0.476141	+	0.879369i	$0.342035\pi$
$810$		0			0
$811$		−	1162.55i		−	1.43348i	−0.697343	−	0.716738i	$-0.745634\pi$
$811$		−	1162.55i		−	1.43348i	0.697343	−	0.716738i	$-0.254366\pi$
$812$		0			0
$813$		−	235.688i		−	0.289899i
$814$		0			0
$815$	98.0328			0.120286
$816$		0			0
$817$			0.305816i			0.000374316i
$818$		0			0
$819$			421.936i			0.515184i
$820$		0			0
$821$		−	897.560i		−	1.09325i	−0.837377	−	0.546626i	$-0.815912\pi$
$821$		−	897.560i		−	1.09325i	0.837377	−	0.546626i	$-0.184088\pi$
$822$		0			0
$823$			150.769i			0.183195i	0.995796	+	0.0915975i	$0.0291973\pi$
$823$			150.769i			0.183195i	−0.995796	+	0.0915975i	$0.970803\pi$
$824$		0			0
$825$	641.085			0.777073
$826$		0			0
$827$		−	345.162i		−	0.417366i	−0.977983	−	0.208683i	$-0.933082\pi$
$827$		−	345.162i		−	0.417366i	0.977983	−	0.208683i	$-0.0669177\pi$
$828$		0			0
$829$			1258.91i			1.51858i	0.650750	+	0.759292i	$0.274455\pi$
$829$			1258.91i			1.51858i	−0.650750	+	0.759292i	$0.725545\pi$
$830$		0			0
$831$		−	2265.79i		−	2.72658i
$832$		0			0
$833$			1164.78i			1.39829i
$834$		0			0
$835$	−400.799	+	296.637i	−0.479999	+	0.355254i
$836$		0			0
$837$	68.0098			0.0812542
$838$		0			0
$839$	1095.87			1.30616			0.653081	−	0.757288i	$-0.273476\pi$
$839$	1095.87			1.30616			0.653081	+	0.757288i	$0.273476\pi$
$840$		0			0
$841$	−840.157			−0.998998
$842$		0			0
$843$	−208.705			−0.247574
$844$		0			0
$845$		−	271.248i		−	0.321004i
$846$		0			0
$847$	166.855			0.196995
$848$		0			0
$849$	1682.50			1.98175
$850$		0			0
$851$	892.394			1.04864
$852$		0			0
$853$	392.118			0.459693			0.229847	−	0.973227i	$-0.426178\pi$
$853$	392.118			0.459693			0.229847	+	0.973227i	$0.426178\pi$
$854$		0			0
$855$		−	72.4531i		−	0.0847405i
$856$		0			0
$857$	−1701.64			−1.98558			−0.992788	−	0.119882i	$-0.961748\pi$
$857$	−1701.64			−1.98558			−0.992788	+	0.119882i	$0.961748\pi$
$858$		0			0
$859$	1539.47			1.79216			0.896081	−	0.443890i	$-0.146402\pi$
$859$	1539.47			1.79216			0.896081	+	0.443890i	$0.146402\pi$
$860$		0			0
$861$			36.8357i			0.0427825i
$862$		0			0
$863$	−1072.16			−1.24236			−0.621180	−	0.783668i	$-0.713346\pi$
$863$	−1072.16			−1.24236			−0.621180	+	0.783668i	$0.713346\pi$
$864$		0			0
$865$			6.96767i			0.00805511i
$866$		0			0
$867$	2567.47			2.96133
$868$		0			0
$869$			388.205i			0.446727i
$870$		0			0
$871$	877.450			1.00741
$872$		0			0
$873$	2386.82			2.73405
$874$		0			0
$875$			336.514i			0.384588i
$876$		0			0
$877$	−1488.11			−1.69681			−0.848407	−	0.529345i	$-0.822438\pi$
$877$	−1488.11			−1.69681			−0.848407	+	0.529345i	$0.822438\pi$
$878$		0			0
$879$	−1847.87			−2.10225
$880$		0			0
$881$			263.738i			0.299362i	0.988734	+	0.149681i	$0.0478247\pi$
$881$			263.738i			0.299362i	−0.988734	+	0.149681i	$0.952175\pi$
$882$		0			0
$883$	−1490.50			−1.68799			−0.843996	−	0.536349i	$-0.819803\pi$
$883$	−1490.50			−1.68799			−0.843996	+	0.536349i	$0.819803\pi$
$884$		0			0
$885$	−1525.82			−1.72409
$886$		0			0
$887$		−	1045.93i		−	1.17918i	−0.807704	−	0.589588i	$-0.799290\pi$
$887$		−	1045.93i		−	1.17918i	0.807704	−	0.589588i	$-0.200710\pi$
$888$		0			0
$889$	−15.2542			−0.0171588
$890$		0			0
$891$	−505.164			−0.566963
$892$		0			0
$893$	−94.8920			−0.106262
$894$		0			0
$895$			624.947i			0.698265i
$896$		0			0
$897$	888.520			0.990546
$898$		0			0
$899$	1.44707			0.00160964
$900$		0			0
$901$	156.497			0.173693
$902$		0			0
$903$			3.09048i			0.00342246i
$904$		0			0
$905$		−	436.368i		−	0.482174i
$906$		0			0
$907$	−224.212			−0.247202			−0.123601	−	0.992332i	$-0.539444\pi$
$907$	−224.212			−0.247202			−0.123601	+	0.992332i	$0.539444\pi$
$908$		0			0
$909$			1923.73i			2.11632i
$910$		0			0
$911$	−223.742			−0.245600			−0.122800	−	0.992431i	$-0.539187\pi$
$911$	−223.742			−0.245600			−0.122800	+	0.992431i	$0.539187\pi$
$912$		0			0
$913$			466.491i			0.510943i
$914$		0			0
$915$		−	1229.05i		−	1.34323i
$916$		0			0
$917$			299.675i			0.326800i
$918$		0			0
$919$	−1010.13			−1.09916			−0.549581	−	0.835440i	$-0.685213\pi$
$919$	−1010.13			−1.09916			−0.549581	+	0.835440i	$0.685213\pi$
$920$		0			0
$921$			331.850i			0.360315i
$922$		0			0
$923$	201.811			0.218647
$924$		0			0
$925$			733.794i			0.793291i
$926$		0			0
$927$			2266.24i			2.44471i
$928$		0			0
$929$	−192.826			−0.207563			−0.103781	−	0.994600i	$-0.533094\pi$
$929$	−192.826			−0.207563			−0.103781	+	0.994600i	$0.533094\pi$
$930$		0			0
$931$	−57.8451			−0.0621322
$932$		0			0
$933$	−1037.55			−1.11206
$934$		0			0
$935$	650.475			0.695695
$936$		0			0
$937$		−	913.010i		−	0.974397i	−0.873291	−	0.487199i	$-0.838019\pi$
$937$		−	913.010i		−	0.974397i	0.873291	−	0.487199i	$-0.161981\pi$
$938$		0			0
$939$		−	1532.52i		−	1.63208i
$940$		0			0
$941$			32.8555i			0.0349155i	0.999848	+	0.0174577i	$0.00555726\pi$
$941$			32.8555i			0.0349155i	−0.999848	+	0.0174577i	$0.994443\pi$
$942$		0			0
$943$	51.1237			0.0542139
$944$		0			0
$945$		−	353.438i		−	0.374008i
$946$		0			0
$947$	1075.63			1.13583			0.567913	−	0.823088i	$-0.307751\pi$
$947$	1075.63			1.13583			0.567913	+	0.823088i	$0.307751\pi$
$948$		0			0
$949$	−777.547			−0.819333
$950$		0			0
$951$	−1041.98			−1.09567
$952$		0			0
$953$			40.1775i			0.0421590i	0.999778	+	0.0210795i	$0.00671031\pi$
$953$			40.1775i			0.0421590i	−0.999778	+	0.0210795i	$0.993290\pi$
$954$		0			0
$955$		−	788.673i		−	0.825836i
$956$		0			0
$957$	−36.5939			−0.0382381
$958$		0			0
$959$	−87.8242			−0.0915790
$960$		0			0
$961$	−958.516			−0.997415
$962$		0			0
$963$	−13.0035			−0.0135031
$964$		0			0
$965$	115.318			0.119500
$966$		0			0
$967$	118.849			0.122905			0.0614524	−	0.998110i	$-0.480427\pi$
$967$	118.849			0.122905			0.0614524	+	0.998110i	$0.480427\pi$
$968$		0			0
$969$			201.247i			0.207685i
$970$		0			0
$971$		−	1100.38i		−	1.13324i	−0.823980	−	0.566620i	$-0.808251\pi$
$971$		−	1100.38i		−	1.13324i	0.823980	−	0.566620i	$-0.191749\pi$
$972$		0			0
$973$			3.63485i			0.00373571i
$974$		0			0
$975$			730.608i			0.749341i
$976$		0			0
$977$		−	1340.94i		−	1.37251i	−0.727361	−	0.686255i	$-0.759253\pi$
$977$		−	1340.94i		−	1.37251i	0.727361	−	0.686255i	$-0.240747\pi$
$978$		0			0
$979$	244.400			0.249643
$980$		0			0
$981$			2310.93i			2.35569i
$982$		0			0
$983$			456.023i			0.463910i	0.972727	+	0.231955i	$0.0745122\pi$
$983$			456.023i			0.463910i	−0.972727	+	0.231955i	$0.925488\pi$
$984$		0			0
$985$	−968.876			−0.983630
$986$		0			0
$987$	−958.948			−0.971579
$988$		0			0
$989$	4.28923			0.00433693
$990$		0			0
$991$			572.416i			0.577614i	0.957387	+	0.288807i	$0.0932587\pi$
$991$			572.416i			0.577614i	−0.957387	+	0.288807i	$0.906741\pi$
$992$		0			0
$993$			993.109i			1.00011i
$994$		0			0
$995$			284.681i			0.286112i
$996$		0			0
$997$	496.858			0.498353			0.249177	−	0.968458i	$-0.419840\pi$
$997$	496.858			0.498353			0.249177	+	0.968458i	$0.419840\pi$
$998$		0			0
$999$		−	1968.56i		−	1.97053i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.3.d.a.333.2	yes	28
167.166	odd	2	inner	668.3.d.a.333.1	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.3.d.a.333.1	✓	28	167.166	odd	2	inner
668.3.d.a.333.2	yes	28	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:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	668.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.13795 q^{3} -2.98582i q^{5} -2.74320 q^{7} +17.3985 q^{9} +O(q^{10})\)	\(q-5.13795 q^{3} -2.98582i q^{5} -2.74320 q^{7} +17.3985 q^{9} -7.75727 q^{11} -8.84051i q^{13} +15.3410i q^{15} -28.0839i q^{17} +1.39470 q^{19} +14.0944 q^{21} -19.5614i q^{23} +16.0849 q^{25} -43.1511 q^{27} -0.918142 q^{29} -1.57609 q^{31} +39.8564 q^{33} +8.19071i q^{35} +45.6201i q^{37} +45.4221i q^{39} +2.61350i q^{41} +0.219270i q^{43} -51.9488i q^{45} -68.0374 q^{47} -41.4748 q^{49} +144.294i q^{51} +5.57248i q^{53} +23.1618i q^{55} -7.16591 q^{57} +99.4606i q^{59} -80.1157 q^{61} -47.7276 q^{63} -26.3962 q^{65} +99.2534i q^{67} +100.506i q^{69} +22.8280i q^{71} -87.9528i q^{73} -82.6432 q^{75} +21.2797 q^{77} -50.0441i q^{79} +65.1214 q^{81} -60.1360i q^{83} -83.8536 q^{85} +4.71737 q^{87} -31.5060 q^{89} +24.2513i q^{91} +8.09785 q^{93} -4.16433i q^{95} +137.186 q^{97} -134.965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 28 * q + 64 * q^9	\( 28 q + 64 q^{9} - 2 q^{11} + 10 q^{19} + 64 q^{21} - 100 q^{25} + 48 q^{27} + 38 q^{29} + 38 q^{31} - 92 q^{33} + 46 q^{47} + 184 q^{49} + 12 q^{57} + 38 q^{61} + 190 q^{63} - 10 q^{65} + 102 q^{75} + 8 q^{77} - 108 q^{81} + 10 q^{85} - 118 q^{87} + 62 q^{89} - 44 q^{93} - 204 q^{97} + 254 q^{99}+O(q^{100}) \) 28 * q + 64 * q^9 - 2 * q^11 + 10 * q^19 + 64 * q^21 - 100 * q^25 + 48 * q^27 + 38 * q^29 + 38 * q^31 - 92 * q^33 + 46 * q^47 + 184 * q^49 + 12 * q^57 + 38 * q^61 + 190 * q^63 - 10 * q^65 + 102 * q^75 + 8 * q^77 - 108 * q^81 + 10 * q^85 - 118 * q^87 + 62 * q^89 - 44 * q^93 - 204 * q^97 + 254 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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