LMFDB - Embedded newform 2304.2.l.h.1727.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(575,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("2304.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.l (of order $4$, degree $2$, 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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1727.1
Root			$1.28897 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	2304.1727
Dual form			2304.2.l.h.575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.57794 - 2.57794i) q^{5} +3.74166 q^{7} +O(q^{10})$	$q+(-2.57794 - 2.57794i) q^{5} +3.74166 q^{7} +(3.64575 - 3.64575i) q^{11} +(4.64575 + 4.64575i) q^{13} +0.913230i q^{17} +(1.41421 - 1.41421i) q^{19} +6.00000i q^{23} +8.29150i q^{25} +(1.66471 - 1.66471i) q^{29} +6.57008i q^{31} +(-9.64575 - 9.64575i) q^{35} +(1.00000 - 1.00000i) q^{37} -0.913230 q^{41} +(-3.74166 - 3.74166i) q^{43} +6.00000 q^{47} +7.00000 q^{49} +(3.49117 + 3.49117i) q^{53} -18.7970 q^{55} +(-1.29150 + 1.29150i) q^{59} +(-0.291503 - 0.291503i) q^{61} -23.9529i q^{65} +(3.32941 - 3.32941i) q^{67} +13.2915i q^{71} -8.58301i q^{73} +(13.6412 - 13.6412i) q^{77} -8.39655i q^{79} +(4.93725 + 4.93725i) q^{83} +(2.35425 - 2.35425i) q^{85} +14.5544 q^{89} +(17.3828 + 17.3828i) q^{91} -7.29150 q^{95} +13.2915 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} + 16 q^{13} - 56 q^{35} + 8 q^{37} + 48 q^{47} + 56 q^{49} + 32 q^{59} + 40 q^{61} - 24 q^{83} + 40 q^{85} - 16 q^{95} + 64 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 + 16 * q^13 - 56 * q^35 + 8 * q^37 + 48 * q^47 + 56 * q^49 + 32 * q^59 + 40 * q^61 - 24 * q^83 + 40 * q^85 - 16 * q^95 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.57794	−	2.57794i	−1.15289	−	1.15289i	−0.985971	−	0.166917i	$-0.946619\pi$
$5$	−2.57794	−	2.57794i	−1.15289	−	1.15289i	−0.166917	−	0.985971i	$-0.553381\pi$
$6$		0			0
$7$	3.74166			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$7$	3.74166			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.64575	−	3.64575i	1.09924	−	1.09924i	0.104735	−	0.994500i	$-0.466600\pi$
$11$	3.64575	−	3.64575i	1.09924	−	1.09924i	0.994500	−	0.104735i	$-0.0333995\pi$
$12$		0			0
$13$	4.64575	+	4.64575i	1.28850	+	1.28850i	0.935698	+	0.352801i	$0.114771\pi$
$13$	4.64575	+	4.64575i	1.28850	+	1.28850i	0.352801	+	0.935698i	$0.385229\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.913230i			0.221491i	0.993849	+	0.110745i	$0.0353238\pi$
$17$			0.913230i			0.221491i	−0.993849	+	0.110745i	$0.964676\pi$
$18$		0			0
$19$	1.41421	−	1.41421i	0.324443	−	0.324443i	−0.526026	−	0.850469i	$-0.676318\pi$
$19$	1.41421	−	1.41421i	0.324443	−	0.324443i	0.850469	+	0.526026i	$0.176318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$			8.29150i			1.65830i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.66471	−	1.66471i	0.309128	−	0.309128i	−0.535443	−	0.844571i	$-0.679855\pi$
$29$	1.66471	−	1.66471i	0.309128	−	0.309128i	0.844571	+	0.535443i	$0.179855\pi$
$30$		0			0
$31$			6.57008i			1.18002i	0.807395	+	0.590011i	$0.200877\pi$
$31$			6.57008i			1.18002i	−0.807395	+	0.590011i	$0.799123\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.64575	−	9.64575i	−1.63043	−	1.63043i
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.913230			−0.142623			−0.0713113	−	0.997454i	$-0.522718\pi$
$41$	−0.913230			−0.142623			−0.0713113	+	0.997454i	$0.522718\pi$
$42$		0			0
$43$	−3.74166	−	3.74166i	−0.570597	−	0.570597i	0.361698	−	0.932295i	$-0.382197\pi$
$43$	−3.74166	−	3.74166i	−0.570597	−	0.570597i	−0.932295	+	0.361698i	$0.882197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.49117	+	3.49117i	0.479548	+	0.479548i	0.904987	−	0.425439i	$-0.139880\pi$
$53$	3.49117	+	3.49117i	0.479548	+	0.479548i	−0.425439	+	0.904987i	$0.639880\pi$
$54$		0			0
$55$	−18.7970			−2.53459
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.29150	+	1.29150i	−0.168139	+	0.168139i	−0.786161	−	0.618022i	$-0.787934\pi$
$59$	−1.29150	+	1.29150i	−0.168139	+	0.168139i	0.618022	+	0.786161i	$0.287934\pi$
$60$		0			0
$61$	−0.291503	−	0.291503i	−0.0373231	−	0.0373231i	0.688199	−	0.725522i	$-0.258402\pi$
$61$	−0.291503	−	0.291503i	−0.0373231	−	0.0373231i	−0.725522	+	0.688199i	$0.758402\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	23.9529i		−	2.97099i
$66$		0			0
$67$	3.32941	−	3.32941i	0.406752	−	0.406752i	−0.473852	−	0.880604i	$-0.657137\pi$
$67$	3.32941	−	3.32941i	0.406752	−	0.406752i	0.880604	+	0.473852i	$0.157137\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.2915i			1.57741i	0.614771	+	0.788706i	$0.289248\pi$
$71$			13.2915i			1.57741i	−0.614771	+	0.788706i	$0.710752\pi$
$72$		0			0
$73$		−	8.58301i		−	1.00456i	−0.864704	−	0.502282i	$-0.832494\pi$
$73$		−	8.58301i		−	1.00456i	0.864704	−	0.502282i	$-0.167506\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.6412	−	13.6412i	1.55455	−	1.55455i
$78$		0			0
$79$		−	8.39655i		−	0.944685i	−0.881415	−	0.472343i	$-0.843409\pi$
$79$		−	8.39655i		−	0.944685i	0.881415	−	0.472343i	$-0.156591\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.93725	+	4.93725i	0.541934	+	0.541934i	0.924096	−	0.382162i	$-0.124820\pi$
$83$	4.93725	+	4.93725i	0.541934	+	0.541934i	−0.382162	+	0.924096i	$0.624820\pi$
$84$		0			0
$85$	2.35425	−	2.35425i	0.255354	−	0.255354i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.5544			1.54276			0.771381	−	0.636374i	$-0.219566\pi$
$89$	14.5544			1.54276			0.771381	+	0.636374i	$0.219566\pi$
$90$		0			0
$91$	17.3828	+	17.3828i	1.82221	+	1.82221i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.29150			−0.748092
$96$		0			0
$97$	13.2915			1.34955			0.674774	−	0.738025i	$-0.264241\pi$
$97$	13.2915			1.34955			0.674774	+	0.738025i	$0.264241\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.82058	−	6.82058i	−0.678673	−	0.678673i	0.281027	−	0.959700i	$-0.409325\pi$
$101$	−6.82058	−	6.82058i	−0.678673	−	0.678673i	−0.959700	+	0.281027i	$0.909325\pi$
$102$		0			0
$103$	−6.57008			−0.647370			−0.323685	−	0.946165i	$-0.604922\pi$
$103$	−6.57008			−0.647370			−0.323685	+	0.946165i	$0.604922\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.29150	+	7.29150i	−0.704896	+	0.704896i	−0.965457	−	0.260561i	$-0.916092\pi$
$107$	−7.29150	+	7.29150i	−0.704896	+	0.704896i	0.260561	+	0.965457i	$0.416092\pi$
$108$		0			0
$109$	1.35425	+	1.35425i	0.129713	+	0.129713i	0.768983	−	0.639269i	$-0.220763\pi$
$109$	1.35425	+	1.35425i	0.129713	+	0.129713i	−0.639269	+	0.768983i	$0.720763\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.7279i		−	1.19734i	−0.800995	−	0.598671i	$-0.795696\pi$
$113$		−	12.7279i		−	1.19734i	0.800995	−	0.598671i	$-0.204304\pi$
$114$		0			0
$115$	15.4676	−	15.4676i	1.44236	−	1.44236i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			3.41699i			0.313235i
$120$		0			0
$121$		−	15.5830i		−	1.41664i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.48528	−	8.48528i	0.758947	−	0.758947i
$126$		0			0
$127$		−	1.91520i		−	0.169946i	−0.996383	−	0.0849731i	$-0.972920\pi$
$127$		−	1.91520i		−	0.169946i	0.996383	−	0.0849731i	$-0.0270804\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000	−	12.0000i	−1.04844	−	1.04844i	−0.998765	−	0.0496797i	$-0.984180\pi$
$131$	−12.0000	−	12.0000i	−1.04844	−	1.04844i	−0.0496797	−	0.998765i	$-0.515820\pi$
$132$		0			0
$133$	5.29150	−	5.29150i	0.458831	−	0.458831i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.39851			0.802969			0.401485	−	0.915866i	$-0.368494\pi$
$137$	9.39851			0.802969			0.401485	+	0.915866i	$0.368494\pi$
$138$		0			0
$139$	−3.32941	−	3.32941i	−0.282397	−	0.282397i	0.551667	−	0.834064i	$-0.313992\pi$
$139$	−3.32941	−	3.32941i	−0.282397	−	0.282397i	−0.834064	+	0.551667i	$0.813992\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	33.8745			2.83273
$144$		0			0
$145$	−8.58301			−0.712780
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.73381	−	7.73381i	−0.633578	−	0.633578i	0.315386	−	0.948964i	$-0.397866\pi$
$149$	−7.73381	−	7.73381i	−0.633578	−	0.633578i	−0.948964	+	0.315386i	$0.897866\pi$
$150$		0			0
$151$	−10.4005			−0.846379			−0.423189	−	0.906041i	$-0.639090\pi$
$151$	−10.4005			−0.846379			−0.423189	+	0.906041i	$0.639090\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.9373	−	16.9373i	1.36043	−	1.36043i
$156$		0			0
$157$	6.29150	+	6.29150i	0.502117	+	0.502117i	0.912095	−	0.409979i	$-0.134464\pi$
$157$	6.29150	+	6.29150i	0.502117	+	0.502117i	−0.409979	+	0.912095i	$0.634464\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			22.4499i			1.76930i
$162$		0			0
$163$	1.91520	−	1.91520i	0.150010	−	0.150010i	−0.628113	−	0.778122i	$-0.716172\pi$
$163$	1.91520	−	1.91520i	0.150010	−	0.150010i	0.778122	+	0.628113i	$0.216172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.29150i		−	0.564233i	−0.959380	−	0.282117i	$-0.908963\pi$
$167$		−	7.29150i		−	0.564233i	0.959380	−	0.282117i	$-0.0910365\pi$
$168$		0			0
$169$			30.1660i			2.32046i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.57794	−	2.57794i	0.195997	−	0.195997i	−0.602285	−	0.798281i	$-0.705743\pi$
$173$	2.57794	−	2.57794i	0.195997	−	0.195997i	0.798281	+	0.602285i	$0.205743\pi$
$174$		0			0
$175$			31.0240i			2.34519i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.8745	−	15.8745i	−1.18652	−	1.18652i	−0.978024	−	0.208492i	$-0.933144\pi$
$179$	−15.8745	−	15.8745i	−1.18652	−	1.18652i	−0.208492	−	0.978024i	$-0.566856\pi$
$180$		0			0
$181$	−1.35425	+	1.35425i	−0.100661	+	0.100661i	−0.755644	−	0.654983i	$-0.772676\pi$
$181$	−1.35425	+	1.35425i	−0.100661	+	0.100661i	0.654983	+	0.755644i	$0.272676\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.15587			−0.379067
$186$		0			0
$187$	3.32941	+	3.32941i	0.243471	+	0.243471i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.2915			−1.39588			−0.697942	−	0.716154i	$-0.745901\pi$
$191$	−19.2915			−1.39588			−0.697942	+	0.716154i	$0.745901\pi$
$192$		0			0
$193$	−4.58301			−0.329892			−0.164946	−	0.986303i	$-0.552745\pi$
$193$	−4.58301			−0.329892			−0.164946	+	0.986303i	$0.552745\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.9764	+	11.9764i	0.853287	+	0.853287i	0.990536	−	0.137250i	$-0.0438263\pi$
$197$	11.9764	+	11.9764i	0.853287	+	0.853287i	−0.137250	+	0.990536i	$0.543826\pi$
$198$		0			0
$199$	16.8818			1.19672			0.598360	−	0.801227i	$-0.295819\pi$
$199$	16.8818			1.19672			0.598360	+	0.801227i	$0.295819\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.22876	−	6.22876i	0.437173	−	0.437173i
$204$		0			0
$205$	2.35425	+	2.35425i	0.164428	+	0.164428i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	10.3117i		−	0.713278i
$210$		0			0
$211$	−15.4676	+	15.4676i	−1.06483	+	1.06483i	−0.0670873	+	0.997747i	$0.521371\pi$
$211$	−15.4676	+	15.4676i	−1.06483	+	1.06483i	−0.997747	+	0.0670873i	$0.978629\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			19.2915i			1.31567i
$216$		0			0
$217$			24.5830i			1.66880i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.24264	+	4.24264i	−0.285391	+	0.285391i
$222$		0			0
$223$		−	15.0554i		−	1.00818i	−0.863651	−	0.504091i	$-0.831828\pi$
$223$		−	15.0554i		−	1.00818i	0.863651	−	0.504091i	$-0.168172\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.35425	−	2.35425i	−0.156257	−	0.156257i	0.624649	−	0.780906i	$-0.285242\pi$
$227$	−2.35425	−	2.35425i	−0.156257	−	0.156257i	−0.780906	+	0.624649i	$0.785242\pi$
$228$		0			0
$229$	9.35425	−	9.35425i	0.618146	−	0.618146i	−0.326909	−	0.945056i	$-0.606007\pi$
$229$	9.35425	−	9.35425i	0.618146	−	0.618146i	0.945056	+	0.326909i	$0.106007\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.5544			−0.953489			−0.476745	−	0.879042i	$-0.658183\pi$
$233$	−14.5544			−0.953489			−0.476745	+	0.879042i	$0.658183\pi$
$234$		0			0
$235$	−15.4676	−	15.4676i	−1.00900	−	1.00900i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.87451			−0.638729			−0.319364	−	0.947632i	$-0.603469\pi$
$239$	−9.87451			−0.638729			−0.319364	+	0.947632i	$0.603469\pi$
$240$		0			0
$241$	13.8745			0.893736			0.446868	−	0.894600i	$-0.352539\pi$
$241$	13.8745			0.893736			0.446868	+	0.894600i	$0.352539\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−18.0455	−	18.0455i	−1.15289	−	1.15289i
$246$		0			0
$247$	13.1402			0.836089
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.35425	−	8.35425i	0.527316	−	0.527316i	−0.392455	−	0.919771i	$-0.628374\pi$
$251$	8.35425	−	8.35425i	0.527316	−	0.527316i	0.919771	+	0.392455i	$0.128374\pi$
$252$		0			0
$253$	21.8745	+	21.8745i	1.37524	+	1.37524i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.06910i		−	0.378580i	−0.981921	−	0.189290i	$-0.939381\pi$
$257$		−	6.06910i		−	0.378580i	0.981921	−	0.189290i	$-0.0606186\pi$
$258$		0			0
$259$	3.74166	−	3.74166i	0.232495	−	0.232495i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	9.41699i		−	0.580677i	−0.956924	−	0.290338i	$-0.906232\pi$
$263$		−	9.41699i		−	0.580677i	0.956924	−	0.290338i	$-0.0937679\pi$
$264$		0			0
$265$		−	18.0000i		−	1.10573i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.9588	+	18.9588i	−1.15594	+	1.15594i	−0.170596	+	0.985341i	$0.554569\pi$
$269$	−18.9588	+	18.9588i	−1.15594	+	1.15594i	−0.985341	+	0.170596i	$0.945431\pi$
$270$		0			0
$271$		−	1.91520i		−	0.116340i	−0.998307	−	0.0581700i	$-0.981473\pi$
$271$		−	1.91520i		−	0.116340i	0.998307	−	0.0581700i	$-0.0185265\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	30.2288	+	30.2288i	1.82286	+	1.82286i
$276$		0			0
$277$	−10.6458	+	10.6458i	−0.639641	+	0.639641i	−0.950467	−	0.310826i	$-0.899394\pi$
$277$	−10.6458	+	10.6458i	−0.639641	+	0.639641i	0.310826	+	0.950467i	$0.399394\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−26.6926			−1.59235			−0.796173	−	0.605069i	$-0.793146\pi$
$281$	−26.6926			−1.59235			−0.796173	+	0.605069i	$0.793146\pi$
$282$		0			0
$283$	−18.7970	−	18.7970i	−1.11737	−	1.11737i	−0.992126	−	0.125241i	$-0.960030\pi$
$283$	−18.7970	−	18.7970i	−1.11737	−	1.11737i	−0.125241	−	0.992126i	$-0.539970\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.41699			−0.201699
$288$		0			0
$289$	16.1660			0.950942
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.161755	+	0.161755i	0.00944985	+	0.00944985i	0.711816	−	0.702366i	$-0.247873\pi$
$293$	0.161755	+	0.161755i	0.00944985	+	0.00944985i	−0.702366	+	0.711816i	$0.747873\pi$
$294$		0			0
$295$	6.65882			0.387692
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−27.8745	+	27.8745i	−1.61202	+	1.61202i
$300$		0			0
$301$	−14.0000	−	14.0000i	−0.806947	−	0.806947i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.50295i			0.0860587i
$306$		0			0
$307$	−22.1264	+	22.1264i	−1.26282	+	1.26282i	−0.313103	+	0.949719i	$0.601369\pi$
$307$	−22.1264	+	22.1264i	−1.26282	+	1.26282i	−0.949719	+	0.313103i	$0.898631\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			29.1660i			1.65385i	0.562310	+	0.826926i	$0.309913\pi$
$311$			29.1660i			1.65385i	−0.562310	+	0.826926i	$0.690087\pi$
$312$		0			0
$313$		−	27.1660i		−	1.53551i	−0.640741	−	0.767757i	$-0.721373\pi$
$313$		−	27.1660i		−	1.53551i	0.640741	−	0.767757i	$-0.278627\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.49117	−	3.49117i	0.196083	−	0.196083i	−0.602235	−	0.798319i	$-0.705723\pi$
$317$	3.49117	−	3.49117i	0.196083	−	0.196083i	0.798319	+	0.602235i	$0.205723\pi$
$318$		0			0
$319$		−	12.1382i		−	0.679609i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.29150	+	1.29150i	0.0718611	+	0.0718611i
$324$		0			0
$325$	−38.5203	+	38.5203i	−2.13672	+	2.13672i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.4499			1.23771
$330$		0			0
$331$	15.4676	+	15.4676i	0.850177	+	0.850177i	0.990155	−	0.139978i	$-0.0447031\pi$
$331$	15.4676	+	15.4676i	0.850177	+	0.850177i	−0.139978	+	0.990155i	$0.544703\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.1660			−0.937879
$336$		0			0
$337$	8.58301			0.467546			0.233773	−	0.972291i	$-0.424893\pi$
$337$	8.58301			0.467546			0.233773	+	0.972291i	$0.424893\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	23.9529	+	23.9529i	1.29712	+	1.29712i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.35425	−	2.35425i	0.126383	−	0.126383i	−0.641086	−	0.767469i	$-0.721516\pi$
$347$	2.35425	−	2.35425i	0.126383	−	0.126383i	0.767469	+	0.641086i	$0.221516\pi$
$348$		0			0
$349$	−21.5830	−	21.5830i	−1.15531	−	1.15531i	−0.985472	−	0.169840i	$-0.945675\pi$
$349$	−21.5830	−	21.5830i	−1.15531	−	1.15531i	−0.169840	−	0.985472i	$-0.554325\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			23.0397i			1.22628i	0.789975	+	0.613139i	$0.210093\pi$
$353$			23.0397i			1.22628i	−0.789975	+	0.613139i	$0.789907\pi$
$354$		0			0
$355$	34.2646	−	34.2646i	1.81858	−	1.81858i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	37.2915i		−	1.96817i	−0.177698	−	0.984085i	$-0.556865\pi$
$359$		−	37.2915i		−	1.96817i	0.177698	−	0.984085i	$-0.443135\pi$
$360$		0			0
$361$			15.0000i			0.789474i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−22.1264	+	22.1264i	−1.15815	+	1.15815i
$366$		0			0
$367$			3.74166i			0.195313i	0.995220	+	0.0976565i	$0.0311346\pi$
$367$			3.74166i			0.195313i	−0.995220	+	0.0976565i	$0.968865\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.0627	+	13.0627i	0.678184	+	0.678184i
$372$		0			0
$373$	24.2915	−	24.2915i	1.25777	−	1.25777i	0.305609	−	0.952157i	$-0.401140\pi$
$373$	24.2915	−	24.2915i	1.25777	−	1.25777i	0.952157	−	0.305609i	$-0.0988602\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.4676			0.796623
$378$		0			0
$379$	5.24461	+	5.24461i	0.269397	+	0.269397i	0.828857	−	0.559460i	$-0.188991\pi$
$379$	5.24461	+	5.24461i	0.269397	+	0.269397i	−0.559460	+	0.828857i	$0.688991\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.4575			1.24972			0.624860	−	0.780737i	$-0.285156\pi$
$383$	24.4575			1.24972			0.624860	+	0.780737i	$0.285156\pi$
$384$		0			0
$385$	−70.3320			−3.58445
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.5485	+	19.5485i	0.991148	+	0.991148i	0.999961	−	0.00881296i	$-0.00280529\pi$
$389$	19.5485	+	19.5485i	0.991148	+	0.991148i	−0.00881296	+	0.999961i	$0.502805\pi$
$390$		0			0
$391$	−5.47938			−0.277104
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−21.6458	+	21.6458i	−1.08912	+	1.08912i
$396$		0			0
$397$	−7.58301	−	7.58301i	−0.380580	−	0.380580i	0.490731	−	0.871311i	$-0.336730\pi$
$397$	−7.58301	−	7.58301i	−0.380580	−	0.380580i	−0.871311	+	0.490731i	$0.836730\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.0573i			0.801865i	0.916108	+	0.400932i	$0.131314\pi$
$401$			16.0573i			0.801865i	−0.916108	+	0.400932i	$0.868686\pi$
$402$		0			0
$403$	−30.5230	+	30.5230i	−1.52046	+	1.52046i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.29150i		−	0.361426i
$408$		0			0
$409$		−	22.7085i		−	1.12286i	−0.827523	−	0.561431i	$-0.810251\pi$
$409$		−	22.7085i		−	1.12286i	0.827523	−	0.561431i	$-0.189749\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.83236	+	4.83236i	−0.237785	+	0.237785i
$414$		0			0
$415$		−	25.4558i		−	1.24958i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.93725	−	4.93725i	−0.241201	−	0.241201i	0.576146	−	0.817347i	$-0.304556\pi$
$419$	−4.93725	−	4.93725i	−0.241201	−	0.241201i	−0.817347	+	0.576146i	$0.804556\pi$
$420$		0			0
$421$	5.93725	−	5.93725i	0.289364	−	0.289364i	−0.547465	−	0.836829i	$-0.684407\pi$
$421$	5.93725	−	5.93725i	0.289364	−	0.289364i	0.836829	+	0.547465i	$0.184407\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.57205			−0.367298
$426$		0			0
$427$	−1.09070	−	1.09070i	−0.0527828	−	0.0527828i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.58301			0.413429			0.206714	−	0.978401i	$-0.433723\pi$
$431$	8.58301			0.413429			0.206714	+	0.978401i	$0.433723\pi$
$432$		0			0
$433$	−26.5830			−1.27750			−0.638749	−	0.769415i	$-0.720548\pi$
$433$	−26.5830			−1.27750			−0.638749	+	0.769415i	$0.720548\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.48528	+	8.48528i	0.405906	+	0.405906i
$438$		0			0
$439$	−15.0554			−0.718553			−0.359277	−	0.933231i	$-0.616977\pi$
$439$	−15.0554			−0.718553			−0.359277	+	0.933231i	$0.616977\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.35425	+	2.35425i	−0.111854	+	0.111854i	−0.760818	−	0.648965i	$-0.775202\pi$
$443$	−2.35425	+	2.35425i	−0.111854	+	0.111854i	0.648965	+	0.760818i	$0.275202\pi$
$444$		0			0
$445$	−37.5203	−	37.5203i	−1.77863	−	1.77863i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.57205i			0.357347i	0.983908	+	0.178674i	$0.0571806\pi$
$449$			7.57205i			0.357347i	−0.983908	+	0.178674i	$0.942819\pi$
$450$		0			0
$451$	−3.32941	+	3.32941i	−0.156776	+	0.156776i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	89.6235i		−	4.20162i
$456$		0			0
$457$		−	30.4575i		−	1.42474i	−0.701803	−	0.712371i	$-0.747621\pi$
$457$		−	30.4575i		−	1.42474i	0.701803	−	0.712371i	$-0.252379\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	23.2014	−	23.2014i	1.08060	−	1.08060i	0.0841442	−	0.996454i	$-0.473184\pi$
$461$	23.2014	−	23.2014i	1.08060	−	1.08060i	0.996454	−	0.0841442i	$-0.0268157\pi$
$462$		0			0
$463$			37.6828i			1.75127i	0.482976	+	0.875634i	$0.339556\pi$
$463$			37.6828i			1.75127i	−0.482976	+	0.875634i	$0.660444\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.3542	+	14.3542i	0.664235	+	0.664235i	0.956376	−	0.292140i	$-0.0943674\pi$
$467$	14.3542	+	14.3542i	0.664235	+	0.664235i	−0.292140	+	0.956376i	$0.594367\pi$
$468$		0			0
$469$	12.4575	−	12.4575i	0.575235	−	0.575235i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−27.2823			−1.25444
$474$		0			0
$475$	11.7260	+	11.7260i	0.538024	+	0.538024i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.2915			−1.15560			−0.577799	−	0.816179i	$-0.696088\pi$
$479$	−25.2915			−1.15560			−0.577799	+	0.816179i	$0.696088\pi$
$480$		0			0
$481$	9.29150			0.423656
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−34.2646	−	34.2646i	−1.55588	−	1.55588i
$486$		0			0
$487$	4.74362			0.214954			0.107477	−	0.994208i	$-0.465723\pi$
$487$	4.74362			0.214954			0.107477	+	0.994208i	$0.465723\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.29150	−	1.29150i	0.0582847	−	0.0582847i	−0.677364	−	0.735648i	$-0.736878\pi$
$491$	1.29150	−	1.29150i	0.0582847	−	0.0582847i	0.735648	+	0.677364i	$0.236878\pi$
$492$		0			0
$493$	1.52026	+	1.52026i	0.0684690	+	0.0684690i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			49.7322i			2.23080i
$498$		0			0
$499$	−6.65882	+	6.65882i	−0.298090	+	0.298090i	−0.840265	−	0.542176i	$-0.817601\pi$
$499$	−6.65882	+	6.65882i	−0.298090	+	0.298090i	0.542176	+	0.840265i	$0.317601\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	27.8745i		−	1.24286i	−0.783469	−	0.621431i	$-0.786551\pi$
$503$		−	27.8745i		−	1.24286i	0.783469	−	0.621431i	$-0.213449\pi$
$504$		0			0
$505$			35.1660i			1.56487i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.49117	+	3.49117i	−0.154743	+	0.154743i	−0.780233	−	0.625489i	$-0.784899\pi$
$509$	−3.49117	+	3.49117i	−0.154743	+	0.154743i	0.625489	+	0.780233i	$0.284899\pi$
$510$		0			0
$511$		−	32.1147i		−	1.42067i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.9373	+	16.9373i	0.746345	+	0.746345i
$516$		0			0
$517$	21.8745	−	21.8745i	0.962040	−	0.962040i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.39851			0.411756			0.205878	−	0.978578i	$-0.433995\pi$
$521$	9.39851			0.411756			0.205878	+	0.978578i	$0.433995\pi$
$522$		0			0
$523$	13.5524	+	13.5524i	0.592606	+	0.592606i	0.938335	−	0.345729i	$-0.112368\pi$
$523$	13.5524	+	13.5524i	0.592606	+	0.592606i	−0.345729	+	0.938335i	$0.612368\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000			−0.261364
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.24264	−	4.24264i	−0.183769	−	0.183769i
$534$		0			0
$535$	37.5940			1.62533
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.5203	−	25.5203i	1.09924	−	1.09924i
$540$		0			0
$541$	−3.35425	−	3.35425i	−0.144210	−	0.144210i	0.631316	−	0.775526i	$-0.282515\pi$
$541$	−3.35425	−	3.35425i	−0.144210	−	0.144210i	−0.775526	+	0.631316i	$0.782515\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.98233i		−	0.299090i
$546$		0			0
$547$	−22.5387	+	22.5387i	−0.963684	+	0.963684i	−0.999363	−	0.0356789i	$-0.988641\pi$
$547$	−22.5387	+	22.5387i	−0.963684	+	0.963684i	0.0356789	+	0.999363i	$0.488641\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	4.70850i		−	0.200589i
$552$		0			0
$553$		−	31.4170i		−	1.33599i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.8779	+	22.8779i	−0.969368	+	0.969368i	−0.999545	−	0.0301765i	$-0.990393\pi$
$557$	−22.8779	+	22.8779i	−0.969368	+	0.969368i	0.0301765	+	0.999545i	$0.490393\pi$
$558$		0			0
$559$		−	34.7656i		−	1.47043i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.6458	+	15.6458i	0.659390	+	0.659390i	0.955236	−	0.295846i	$-0.0956015\pi$
$563$	15.6458	+	15.6458i	0.659390	+	0.659390i	−0.295846	+	0.955236i	$0.595601\pi$
$564$		0			0
$565$	−32.8118	+	32.8118i	−1.38040	+	1.38040i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.3632			0.979435			0.489718	−	0.871881i	$-0.337100\pi$
$569$	23.3632			0.979435			0.489718	+	0.871881i	$0.337100\pi$
$570$		0			0
$571$	−18.7970	−	18.7970i	−0.786631	−	0.786631i	0.194309	−	0.980940i	$-0.437753\pi$
$571$	−18.7970	−	18.7970i	−0.786631	−	0.786631i	−0.980940	+	0.194309i	$0.937753\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−49.7490			−2.07468
$576$		0			0
$577$	−9.41699			−0.392035			−0.196017	−	0.980600i	$-0.562801\pi$
$577$	−9.41699			−0.392035			−0.196017	+	0.980600i	$0.562801\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.4735	+	18.4735i	0.766410	+	0.766410i
$582$		0			0
$583$	25.4558			1.05427
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.2915	+	13.2915i	−0.548599	+	0.548599i	−0.926035	−	0.377436i	$-0.876806\pi$
$587$	−13.2915	+	13.2915i	−0.548599	+	0.548599i	0.377436	+	0.926035i	$0.376806\pi$
$588$		0			0
$589$	9.29150	+	9.29150i	0.382850	+	0.382850i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	4.24264i		−	0.174224i	−0.996199	−	0.0871122i	$-0.972236\pi$
$593$		−	4.24264i		−	0.174224i	0.996199	−	0.0871122i	$-0.0277639\pi$
$594$		0			0
$595$	8.80879	−	8.80879i	0.361125	−	0.361125i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			23.1660i			0.946538i	0.880918	+	0.473269i	$0.156926\pi$
$599$			23.1660i			0.946538i	−0.880918	+	0.473269i	$0.843074\pi$
$600$		0			0
$601$			9.41699i			0.384127i	0.981382	+	0.192064i	$0.0615180\pi$
$601$			9.41699i			0.384127i	−0.981382	+	0.192064i	$0.938482\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−40.1720	+	40.1720i	−1.63322	+	1.63322i
$606$		0			0
$607$			1.09070i			0.0442703i	0.999755	+	0.0221351i	$0.00704641\pi$
$607$			1.09070i			0.0442703i	−0.999755	+	0.0221351i	$0.992954\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	27.8745	+	27.8745i	1.12768	+	1.12768i
$612$		0			0
$613$	−7.58301	+	7.58301i	−0.306275	+	0.306275i	−0.843463	−	0.537188i	$-0.819487\pi$
$613$	−7.58301	+	7.58301i	−0.306275	+	0.306275i	0.537188	+	0.843463i	$0.319487\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.5190			−1.14813			−0.574067	−	0.818808i	$-0.694635\pi$
$617$	−28.5190			−1.14813			−0.574067	+	0.818808i	$0.694635\pi$
$618$		0			0
$619$	−12.1382	−	12.1382i	−0.487876	−	0.487876i	0.419760	−	0.907635i	$-0.362114\pi$
$619$	−12.1382	−	12.1382i	−0.487876	−	0.487876i	−0.907635	+	0.419760i	$0.862114\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	54.4575			2.18179
$624$		0			0
$625$	−2.29150			−0.0916601
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.913230	+	0.913230i	0.0364129	+	0.0364129i
$630$		0			0
$631$	26.1916			1.04267			0.521336	−	0.853352i	$-0.325434\pi$
$631$	26.1916			1.04267			0.521336	+	0.853352i	$0.325434\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.93725	+	4.93725i	−0.195929	+	0.195929i
$636$		0			0
$637$	32.5203	+	32.5203i	1.28850	+	1.28850i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			24.5426i			0.969375i	0.874687	+	0.484687i	$0.161067\pi$
$641$			24.5426i			0.969375i	−0.874687	+	0.484687i	$0.838933\pi$
$642$		0			0
$643$	13.2289	−	13.2289i	0.521697	−	0.521697i	−0.396387	−	0.918084i	$-0.629736\pi$
$643$	13.2289	−	13.2289i	0.521697	−	0.521697i	0.918084	+	0.396387i	$0.129736\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			40.3320i			1.58562i	0.609472	+	0.792808i	$0.291382\pi$
$647$			40.3320i			1.58562i	−0.609472	+	0.792808i	$0.708618\pi$
$648$		0			0
$649$			9.41699i			0.369649i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.57794	+	2.57794i	−0.100882	+	0.100882i	−0.755747	−	0.654864i	$-0.772726\pi$
$653$	−2.57794	+	2.57794i	−0.100882	+	0.100882i	0.654864	+	0.755747i	$0.272726\pi$
$654$		0			0
$655$			61.8705i			2.41748i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.1660	−	23.1660i	−0.902420	−	0.902420i	0.0932254	−	0.995645i	$-0.470282\pi$
$659$	−23.1660	−	23.1660i	−0.902420	−	0.902420i	−0.995645	+	0.0932254i	$0.970282\pi$
$660$		0			0
$661$	−23.4575	+	23.4575i	−0.912392	+	0.912392i	−0.996460	−	0.0840685i	$-0.973209\pi$
$661$	−23.4575	+	23.4575i	−0.912392	+	0.912392i	0.0840685	+	0.996460i	$0.473209\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−27.2823			−1.05796
$666$		0			0
$667$	9.98823	+	9.98823i	0.386746	+	0.386746i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.12549			−0.0820537
$672$		0			0
$673$	43.7490			1.68640			0.843200	−	0.537600i	$-0.180669\pi$
$673$	43.7490			1.68640			0.843200	+	0.537600i	$0.180669\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.23086	+	6.23086i	0.239471	+	0.239471i	0.816631	−	0.577160i	$-0.195839\pi$
$677$	6.23086	+	6.23086i	0.239471	+	0.239471i	−0.577160	+	0.816631i	$0.695839\pi$
$678$		0			0
$679$	49.7322			1.90855
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.6458	+	21.6458i	−0.828252	+	0.828252i	−0.987275	−	0.159023i	$-0.949166\pi$
$683$	−21.6458	+	21.6458i	−0.828252	+	0.828252i	0.159023	+	0.987275i	$0.449166\pi$
$684$		0			0
$685$	−24.2288	−	24.2288i	−0.925733	−	0.925733i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			32.4382i			1.23580i
$690$		0			0
$691$	−15.8799	+	15.8799i	−0.604098	+	0.604098i	−0.941398	−	0.337299i	$-0.890487\pi$
$691$	−15.8799	+	15.8799i	−0.604098	+	0.604098i	0.337299	+	0.941398i	$0.390487\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			17.1660i			0.651144i
$696$		0			0
$697$		−	0.833990i		−	0.0315896i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.98822	+	1.98822i	−0.0750939	+	0.0750939i	−0.743656	−	0.668562i	$-0.766910\pi$
$701$	−1.98822	+	1.98822i	−0.0750939	+	0.0750939i	0.668562	+	0.743656i	$0.266910\pi$
$702$		0			0
$703$		−	2.82843i		−	0.106676i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.5203	−	25.5203i	−0.959788	−	0.959788i
$708$		0			0
$709$	−28.6458	+	28.6458i	−1.07581	+	1.07581i	−0.0789339	+	0.996880i	$0.525152\pi$
$709$	−28.6458	+	28.6458i	−1.07581	+	1.07581i	−0.996880	+	0.0789339i	$0.974848\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−39.4205			−1.47631
$714$		0			0
$715$	−87.3263	−	87.3263i	−3.26582	−	3.26582i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.4575			−0.688349			−0.344175	−	0.938906i	$-0.611841\pi$
$719$	−18.4575			−0.688349			−0.344175	+	0.938906i	$0.611841\pi$
$720$		0			0
$721$	−24.5830			−0.915519
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.8029	+	13.8029i	0.512627	+	0.512627i
$726$		0			0
$727$	−45.9906			−1.70570			−0.852848	−	0.522159i	$-0.825127\pi$
$727$	−45.9906			−1.70570			−0.852848	+	0.522159i	$0.825127\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.41699	−	3.41699i	0.126382	−	0.126382i
$732$		0			0
$733$	−21.9373	−	21.9373i	−0.810271	−	0.810271i	0.174403	−	0.984674i	$-0.444200\pi$
$733$	−21.9373	−	21.9373i	−0.810271	−	0.810271i	−0.984674	+	0.174403i	$0.944200\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	24.2764i		−	0.894233i
$738$		0			0
$739$	−18.7970	+	18.7970i	−0.691460	+	0.691460i	−0.962553	−	0.271093i	$-0.912615\pi$
$739$	−18.7970	+	18.7970i	−0.691460	+	0.691460i	0.271093	+	0.962553i	$0.412615\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			39.0405i			1.43226i	0.697968	+	0.716129i	$0.254088\pi$
$743$			39.0405i			1.43226i	−0.697968	+	0.716129i	$0.745912\pi$
$744$		0			0
$745$			39.8745i			1.46089i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27.2823	+	27.2823i	−0.996874	+	0.996874i
$750$		0			0
$751$		−	17.7063i		−	0.646113i	−0.946380	−	0.323056i	$-0.895290\pi$
$751$		−	17.7063i		−	0.646113i	0.946380	−	0.323056i	$-0.104710\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.8118	+	26.8118i	0.975780	+	0.975780i
$756$		0			0
$757$	−14.6458	+	14.6458i	−0.532309	+	0.532309i	−0.921259	−	0.388950i	$-0.872838\pi$
$757$	−14.6458	+	14.6458i	−0.532309	+	0.532309i	0.388950	+	0.921259i	$0.372838\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	45.1661			1.63727			0.818635	−	0.574314i	$-0.194731\pi$
$761$	45.1661			1.63727			0.818635	+	0.574314i	$0.194731\pi$
$762$		0			0
$763$	5.06713	+	5.06713i	0.183443	+	0.183443i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.0000			−0.433295
$768$		0			0
$769$	31.1660			1.12388			0.561938	−	0.827180i	$-0.310056\pi$
$769$	31.1660			1.12388			0.561938	+	0.827180i	$0.310056\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.49117	−	3.49117i	−0.125569	−	0.125569i	0.641530	−	0.767098i	$-0.278300\pi$
$773$	−3.49117	−	3.49117i	−0.125569	−	0.125569i	−0.767098	+	0.641530i	$0.778300\pi$
$774$		0			0
$775$	−54.4759			−1.95683
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.29150	+	1.29150i	−0.0462729	+	0.0462729i
$780$		0			0
$781$	48.4575	+	48.4575i	1.73395	+	1.73395i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	32.4382i		−	1.15777i
$786$		0			0
$787$	8.89753	−	8.89753i	0.317163	−	0.317163i	−0.530514	−	0.847676i	$-0.678001\pi$
$787$	8.89753	−	8.89753i	0.317163	−	0.317163i	0.847676	+	0.530514i	$0.178001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	47.6235i		−	1.69330i
$792$		0			0
$793$		−	2.70850i		−	0.0961816i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.8897	+	12.8897i	−0.456576	+	0.456576i	−0.897530	−	0.440954i	$-0.854640\pi$
$797$	−12.8897	+	12.8897i	−0.456576	+	0.456576i	0.440954	+	0.897530i	$0.354640\pi$
$798$		0			0
$799$			5.47938i			0.193847i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−31.2915	−	31.2915i	−1.10425	−	1.10425i
$804$		0			0
$805$	57.8745	−	57.8745i	2.03981	−	2.03981i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−40.3337			−1.41806			−0.709029	−	0.705179i	$-0.750867\pi$
$809$	−40.3337			−1.41806			−0.709029	+	0.705179i	$0.750867\pi$
$810$		0			0
$811$	24.0416	+	24.0416i	0.844216	+	0.844216i	0.989404	−	0.145188i	$-0.0463788\pi$
$811$	24.0416	+	24.0416i	0.844216	+	0.844216i	−0.145188	+	0.989404i	$0.546379\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.87451			−0.345889
$816$		0			0
$817$	−10.5830			−0.370252
$818$		0			0
$819$		0			0
$820$		0			0
$821$	32.5999	+	32.5999i	1.13775	+	1.13775i	0.988853	+	0.148892i	$0.0475707\pi$
$821$	32.5999	+	32.5999i	1.13775	+	1.13775i	0.148892	+	0.988853i	$0.452429\pi$
$822$		0			0
$823$	6.74756			0.235205			0.117603	−	0.993061i	$-0.462479\pi$
$823$	6.74756			0.235205			0.117603	+	0.993061i	$0.462479\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.8745	−	33.8745i	1.17793	−	1.17793i	0.197662	−	0.980270i	$-0.436665\pi$
$827$	33.8745	−	33.8745i	1.17793	−	1.17793i	0.980270	−	0.197662i	$-0.0633348\pi$
$828$		0			0
$829$	8.06275	+	8.06275i	0.280031	+	0.280031i	0.833121	−	0.553090i	$-0.186552\pi$
$829$	8.06275	+	8.06275i	0.280031	+	0.280031i	−0.553090	+	0.833121i	$0.686552\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			6.39261i			0.221491i
$834$		0			0
$835$	−18.7970	+	18.7970i	−0.650498	+	0.650498i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	3.87451i		−	0.133763i	−0.997761	−	0.0668814i	$-0.978695\pi$
$839$		−	3.87451i		−	0.133763i	0.997761	−	0.0668814i	$-0.0213049\pi$
$840$		0			0
$841$			23.4575i			0.808880i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	77.7660	−	77.7660i	2.67523	−	2.67523i
$846$		0			0
$847$		−	58.3063i		−	2.00343i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	+	6.00000i	0.205677	+	0.205677i
$852$		0			0
$853$	18.2915	−	18.2915i	0.626289	−	0.626289i	−0.320843	−	0.947132i	$-0.603966\pi$
$853$	18.2915	−	18.2915i	0.626289	−	0.626289i	0.947132	+	0.320843i	$0.103966\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	28.1955			0.963141			0.481571	−	0.876407i	$-0.340067\pi$
$857$	28.1955			0.963141			0.481571	+	0.876407i	$0.340067\pi$
$858$		0			0
$859$	18.7083	+	18.7083i	0.638319	+	0.638319i	0.950141	−	0.311822i	$-0.100939\pi$
$859$	18.7083	+	18.7083i	0.638319	+	0.638319i	−0.311822	+	0.950141i	$0.600939\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27.0405			−0.920470			−0.460235	−	0.887797i	$-0.652235\pi$
$863$	−27.0405			−0.920470			−0.460235	+	0.887797i	$0.652235\pi$
$864$		0			0
$865$	−13.2915			−0.451925
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.6117	−	30.6117i	−1.03843	−	1.03843i
$870$		0			0
$871$	30.9352			1.04820
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.7490	−	31.7490i	1.07331	−	1.07331i
$876$		0			0
$877$	−33.5830	−	33.5830i	−1.13402	−	1.13402i	−0.989503	−	0.144515i	$-0.953838\pi$
$877$	−33.5830	−	33.5830i	−1.13402	−	1.13402i	−0.144515	−	0.989503i	$-0.546162\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	52.1484i		−	1.75693i	−0.477811	−	0.878463i	$-0.658570\pi$
$881$		−	52.1484i		−	1.75693i	0.477811	−	0.878463i	$-0.341430\pi$
$882$		0			0
$883$	4.74362	−	4.74362i	0.159636	−	0.159636i	−0.622770	−	0.782405i	$-0.713993\pi$
$883$	4.74362	−	4.74362i	0.159636	−	0.159636i	0.782405	+	0.622770i	$0.213993\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			6.83399i			0.229463i	0.993397	+	0.114731i	$0.0366008\pi$
$887$			6.83399i			0.229463i	−0.993397	+	0.114731i	$0.963399\pi$
$888$		0			0
$889$		−	7.16601i		−	0.240340i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.48528	−	8.48528i	0.283949	−	0.283949i
$894$		0			0
$895$			81.8469i			2.73584i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.9373	+	10.9373i	0.364778	+	0.364778i
$900$		0			0
$901$	−3.18824	+	3.18824i	−0.106216	+	0.106216i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.98233			0.232101
$906$		0			0
$907$	29.1975	+	29.1975i	0.969487	+	0.969487i	0.999548	−	0.0300609i	$-0.00957014\pi$
$907$	29.1975	+	29.1975i	0.969487	+	0.969487i	−0.0300609	+	0.999548i	$0.509570\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	41.6235			1.37905			0.689524	−	0.724262i	$-0.257820\pi$
$911$	41.6235			1.37905			0.689524	+	0.724262i	$0.257820\pi$
$912$		0			0
$913$	36.0000			1.19143
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−44.8999	−	44.8999i	−1.48272	−	1.48272i
$918$		0			0
$919$	−54.4759			−1.79699			−0.898497	−	0.438980i	$-0.855340\pi$
$919$	−54.4759			−1.79699			−0.898497	+	0.438980i	$0.855340\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−61.7490	+	61.7490i	−2.03249	+	2.03249i
$924$		0			0
$925$	8.29150	+	8.29150i	0.272623	+	0.272623i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	22.7162i		−	0.745293i	−0.927973	−	0.372646i	$-0.878450\pi$
$929$		−	22.7162i		−	0.745293i	0.927973	−	0.372646i	$-0.121550\pi$
$930$		0			0
$931$	9.89949	−	9.89949i	0.324443	−	0.324443i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	17.1660i		−	0.561389i
$936$		0			0
$937$		−	32.3320i		−	1.05624i	−0.849169	−	0.528121i	$-0.822897\pi$
$937$		−	32.3320i		−	1.05624i	0.849169	−	0.528121i	$-0.177103\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	14.7161	−	14.7161i	0.479732	−	0.479732i	−0.425314	−	0.905046i	$-0.639836\pi$
$941$	14.7161	−	14.7161i	0.479732	−	0.479732i	0.905046	+	0.425314i	$0.139836\pi$
$942$		0			0
$943$		−	5.47938i		−	0.178433i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.58301	+	8.58301i	0.278910	+	0.278910i	0.832674	−	0.553764i	$-0.186809\pi$
$947$	8.58301	+	8.58301i	0.278910	+	0.278910i	−0.553764	+	0.832674i	$0.686809\pi$
$948$		0			0
$949$	39.8745	−	39.8745i	1.29438	−	1.29438i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.3691			−0.854178			−0.427089	−	0.904210i	$-0.640461\pi$
$953$	−26.3691			−0.854178			−0.427089	+	0.904210i	$0.640461\pi$
$954$		0			0
$955$	49.7322	+	49.7322i	1.60930	+	1.60930i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	35.1660			1.13557
$960$		0			0
$961$	−12.1660			−0.392452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.8147	+	11.8147i	0.380328	+	0.380328i
$966$		0			0
$967$	11.0475			0.355264			0.177632	−	0.984097i	$-0.443156\pi$
$967$	11.0475			0.355264			0.177632	+	0.984097i	$0.443156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.0627	−	19.0627i	0.611753	−	0.611753i	−0.331650	−	0.943403i	$-0.607605\pi$
$971$	19.0627	−	19.0627i	0.611753	−	0.611753i	0.943403	+	0.331650i	$0.107605\pi$
$972$		0			0
$973$	−12.4575	−	12.4575i	−0.399370	−	0.399370i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			31.8485i			1.01892i	0.860494	+	0.509461i	$0.170155\pi$
$977$			31.8485i			1.01892i	−0.860494	+	0.509461i	$0.829845\pi$
$978$		0			0
$979$	53.0617	−	53.0617i	1.69586	−	1.69586i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	14.5830i		−	0.465126i	−0.972581	−	0.232563i	$-0.925289\pi$
$983$		−	14.5830i		−	0.465126i	0.972581	−	0.232563i	$-0.0747111\pi$
$984$		0			0
$985$		−	61.7490i		−	1.96749i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	22.4499	−	22.4499i	0.713867	−	0.713867i
$990$		0			0
$991$		−	23.5406i		−	0.747793i	−0.927470	−	0.373897i	$-0.878021\pi$
$991$		−	23.5406i		−	0.747793i	0.927470	−	0.373897i	$-0.121979\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−43.5203	−	43.5203i	−1.37968	−	1.37968i
$996$		0			0
$997$	−0.291503	+	0.291503i	−0.00923198	+	0.00923198i	−0.711708	−	0.702476i	$-0.752078\pi$
$997$	−0.291503	+	0.291503i	−0.00923198	+	0.00923198i	0.702476	+	0.711708i	$0.252078\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.l.h.1727.1	yes	8
3.2	odd	2		2304.2.l.d.1727.4	yes	8
4.3	odd	2		2304.2.l.d.1727.1	yes	8
8.3	odd	2		2304.2.l.e.1727.4	yes	8
8.5	even	2		2304.2.l.a.1727.4	yes	8
12.11	even	2	inner	2304.2.l.h.1727.4	yes	8
16.3	odd	4		2304.2.l.e.575.1	yes	8
16.5	even	4	inner	2304.2.l.h.575.4	yes	8
16.11	odd	4		2304.2.l.d.575.4	yes	8
16.13	even	4		2304.2.l.a.575.1	✓	8
24.5	odd	2		2304.2.l.e.1727.1	yes	8
24.11	even	2		2304.2.l.a.1727.1	yes	8
48.5	odd	4		2304.2.l.d.575.1	yes	8
48.11	even	4	inner	2304.2.l.h.575.1	yes	8
48.29	odd	4		2304.2.l.e.575.4	yes	8
48.35	even	4		2304.2.l.a.575.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.l.a.575.1	✓	8	16.13	even	4
2304.2.l.a.575.4	yes	8	48.35	even	4
2304.2.l.a.1727.1	yes	8	24.11	even	2
2304.2.l.a.1727.4	yes	8	8.5	even	2
2304.2.l.d.575.1	yes	8	48.5	odd	4
2304.2.l.d.575.4	yes	8	16.11	odd	4
2304.2.l.d.1727.1	yes	8	4.3	odd	2
2304.2.l.d.1727.4	yes	8	3.2	odd	2
2304.2.l.e.575.1	yes	8	16.3	odd	4
2304.2.l.e.575.4	yes	8	48.29	odd	4
2304.2.l.e.1727.1	yes	8	24.5	odd	2
2304.2.l.e.1727.4	yes	8	8.3	odd	2
2304.2.l.h.575.1	yes	8	48.11	even	4	inner
2304.2.l.h.575.4	yes	8	16.5	even	4	inner
2304.2.l.h.1727.1	yes	8	1.1	even	1	trivial
2304.2.l.h.1727.4	yes	8	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.57794 - 2.57794i) q^{5} +3.74166 q^{7} +O(q^{10})\)	\(q+(-2.57794 - 2.57794i) q^{5} +3.74166 q^{7} +(3.64575 - 3.64575i) q^{11} +(4.64575 + 4.64575i) q^{13} +0.913230i q^{17} +(1.41421 - 1.41421i) q^{19} +6.00000i q^{23} +8.29150i q^{25} +(1.66471 - 1.66471i) q^{29} +6.57008i q^{31} +(-9.64575 - 9.64575i) q^{35} +(1.00000 - 1.00000i) q^{37} -0.913230 q^{41} +(-3.74166 - 3.74166i) q^{43} +6.00000 q^{47} +7.00000 q^{49} +(3.49117 + 3.49117i) q^{53} -18.7970 q^{55} +(-1.29150 + 1.29150i) q^{59} +(-0.291503 - 0.291503i) q^{61} -23.9529i q^{65} +(3.32941 - 3.32941i) q^{67} +13.2915i q^{71} -8.58301i q^{73} +(13.6412 - 13.6412i) q^{77} -8.39655i q^{79} +(4.93725 + 4.93725i) q^{83} +(2.35425 - 2.35425i) q^{85} +14.5544 q^{89} +(17.3828 + 17.3828i) q^{91} -7.29150 q^{95} +13.2915 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} + 16 q^{13} - 56 q^{35} + 8 q^{37} + 48 q^{47} + 56 q^{49} + 32 q^{59} + 40 q^{61} - 24 q^{83} + 40 q^{85} - 16 q^{95} + 64 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 + 16 * q^13 - 56 * q^35 + 8 * q^37 + 48 * q^47 + 56 * q^49 + 32 * q^59 + 40 * q^61 - 24 * q^83 + 40 * q^85 - 16 * q^95 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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