LMFDB - Embedded newform 2288.2.b.c.2287.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2288,2,Mod(2287,2288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2288, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2288.2287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2288 = 2^{4} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2288.b (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:	$18.2697719825$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2287.11
Character	$\chi$	$=$	2288.2287
Dual form			2288.2.b.c.2287.46

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08094i q^{3} +4.28515i q^{5} -3.10047i q^{7} -1.33031 q^{9} +O(q^{10})$	$q-2.08094i q^{3} +4.28515i q^{5} -3.10047i q^{7} -1.33031 q^{9} +(-1.55800 - 2.92791i) q^{11} +(-3.54217 - 0.673063i) q^{13} +8.91714 q^{15} -3.56760i q^{17} +5.02006i q^{19} -6.45190 q^{21} -0.541694i q^{23} -13.3625 q^{25} -3.47453i q^{27} +1.94427i q^{29} +2.14410 q^{31} +(-6.09280 + 3.24210i) q^{33} +13.2860 q^{35} +8.93120i q^{37} +(-1.40060 + 7.37105i) q^{39} +9.19930 q^{41} -10.7202 q^{43} -5.70058i q^{45} +1.98486 q^{47} -2.61294 q^{49} -7.42395 q^{51} -5.94221 q^{53} +(12.5465 - 6.67625i) q^{55} +10.4464 q^{57} -11.4260 q^{59} -12.0928i q^{61} +4.12459i q^{63} +(2.88418 - 15.1787i) q^{65} -9.01909 q^{67} -1.12723 q^{69} -9.50892 q^{71} -11.8221 q^{73} +27.8066i q^{75} +(-9.07790 + 4.83053i) q^{77} -4.82604 q^{79} -11.2212 q^{81} -10.4470i q^{83} +15.2877 q^{85} +4.04590 q^{87} -2.71165i q^{89} +(-2.08681 + 10.9824i) q^{91} -4.46174i q^{93} -21.5117 q^{95} +2.77051i q^{97} +(2.07262 + 3.89502i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 72 q^{9}+O(q^{10}) $ 56 * q - 72 * q^9	$ 56 q - 72 q^{9} - 72 q^{25} - 8 q^{49} + 72 q^{53} + 8 q^{69} - 48 q^{77} + 88 q^{81}+O(q^{100}) $ 56 * q - 72 * q^9 - 72 * q^25 - 8 * q^49 + 72 * q^53 + 8 * q^69 - 48 * q^77 + 88 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2288\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$1717$	$2081$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	2.08094i		−	1.20143i	−0.799463	−	0.600715i	$-0.794882\pi$
$3$		−	2.08094i		−	1.20143i	0.799463	−	0.600715i	$-0.205118\pi$
$4$		0			0
$5$			4.28515i			1.91638i	0.286136	+	0.958189i	$0.407629\pi$
$5$			4.28515i			1.91638i	−0.286136	+	0.958189i	$0.592371\pi$
$6$		0			0
$7$		−	3.10047i		−	1.17187i	−0.810358	−	0.585935i	$-0.800728\pi$
$7$		−	3.10047i		−	1.17187i	0.810358	−	0.585935i	$-0.199272\pi$
$8$		0			0
$9$	−1.33031			−0.443436
$10$		0			0
$11$	−1.55800	−	2.92791i	−0.469754	−	0.882797i
$11$	−1.55800	−	2.92791i	−0.469754	−	0.882797i
$12$		0			0
$13$	−3.54217	−	0.673063i	−0.982422	−	0.186674i
$13$	−3.54217	−	0.673063i	−0.982422	−	0.186674i
$14$		0			0
$15$	8.91714			2.30240
$16$		0			0
$17$		−	3.56760i		−	0.865269i	−0.901569	−	0.432634i	$-0.857584\pi$
$17$		−	3.56760i		−	0.865269i	0.901569	−	0.432634i	$-0.142416\pi$
$18$		0			0
$19$			5.02006i			1.15168i	0.817562	+	0.575840i	$0.195325\pi$
$19$			5.02006i			1.15168i	−0.817562	+	0.575840i	$0.804675\pi$
$20$		0			0
$21$	−6.45190			−1.40792
$22$		0			0
$23$		−	0.541694i		−	0.112951i	−0.998404	−	0.0564755i	$-0.982014\pi$
$23$		−	0.541694i		−	0.112951i	0.998404	−	0.0564755i	$-0.0179863\pi$
$24$		0			0
$25$	−13.3625			−2.67250
$26$		0			0
$27$		−	3.47453i		−	0.668673i
$28$		0			0
$29$			1.94427i			0.361041i	0.983571	+	0.180521i	$0.0577782\pi$
$29$			1.94427i			0.361041i	−0.983571	+	0.180521i	$0.942222\pi$
$30$		0			0
$31$	2.14410			0.385091			0.192546	−	0.981288i	$-0.438326\pi$
$31$	2.14410			0.385091			0.192546	+	0.981288i	$0.438326\pi$
$32$		0			0
$33$	−6.09280	+	3.24210i	−1.06062	+	0.564377i
$34$		0			0
$35$	13.2860			2.24574
$36$		0			0
$37$			8.93120i			1.46828i	0.678998	+	0.734140i	$0.262415\pi$
$37$			8.93120i			1.46828i	−0.678998	+	0.734140i	$0.737585\pi$
$38$		0			0
$39$	−1.40060	+	7.37105i	−0.224276	+	1.18031i
$40$		0			0
$41$	9.19930			1.43669			0.718345	−	0.695687i	$-0.244900\pi$
$41$	9.19930			1.43669			0.718345	+	0.695687i	$0.244900\pi$
$42$		0			0
$43$	−10.7202			−1.63481			−0.817407	−	0.576061i	$-0.804589\pi$
$43$	−10.7202			−1.63481			−0.817407	+	0.576061i	$0.804589\pi$
$44$		0			0
$45$		−	5.70058i		−	0.849792i
$46$		0			0
$47$	1.98486			0.289522			0.144761	−	0.989467i	$-0.453759\pi$
$47$	1.98486			0.289522			0.144761	+	0.989467i	$0.453759\pi$
$48$		0			0
$49$	−2.61294			−0.373277
$50$		0			0
$51$	−7.42395			−1.03956
$52$		0			0
$53$	−5.94221			−0.816226			−0.408113	−	0.912932i	$-0.633813\pi$
$53$	−5.94221			−0.816226			−0.408113	+	0.912932i	$0.633813\pi$
$54$		0			0
$55$	12.5465	−	6.67625i	1.69177	−	0.900226i
$56$		0			0
$57$	10.4464			1.38366
$58$		0			0
$59$	−11.4260			−1.48754			−0.743770	−	0.668436i	$-0.766964\pi$
$59$	−11.4260			−1.48754			−0.743770	+	0.668436i	$0.766964\pi$
$60$		0			0
$61$		−	12.0928i		−	1.54832i	−0.632987	−	0.774162i	$-0.718171\pi$
$61$		−	12.0928i		−	1.54832i	0.632987	−	0.774162i	$-0.281829\pi$
$62$		0			0
$63$			4.12459i			0.519649i
$64$		0			0
$65$	2.88418	−	15.1787i	0.357738	−	1.88269i
$66$		0			0
$67$	−9.01909			−1.10186			−0.550929	−	0.834552i	$-0.685726\pi$
$67$	−9.01909			−1.10186			−0.550929	+	0.834552i	$0.685726\pi$
$68$		0			0
$69$	−1.12723			−0.135703
$70$		0			0
$71$	−9.50892			−1.12850			−0.564250	−	0.825604i	$-0.690835\pi$
$71$	−9.50892			−1.12850			−0.564250	+	0.825604i	$0.690835\pi$
$72$		0			0
$73$	−11.8221			−1.38367			−0.691835	−	0.722056i	$-0.743197\pi$
$73$	−11.8221			−1.38367			−0.691835	+	0.722056i	$0.743197\pi$
$74$		0			0
$75$			27.8066i			3.21083i
$76$		0			0
$77$	−9.07790	+	4.83053i	−1.03452	+	0.550490i
$78$		0			0
$79$	−4.82604			−0.542972			−0.271486	−	0.962442i	$-0.587515\pi$
$79$	−4.82604			−0.542972			−0.271486	+	0.962442i	$0.587515\pi$
$80$		0			0
$81$	−11.2212			−1.24680
$82$		0			0
$83$		−	10.4470i		−	1.14671i	−0.819308	−	0.573354i	$-0.805642\pi$
$83$		−	10.4470i		−	1.14671i	0.819308	−	0.573354i	$-0.194358\pi$
$84$		0			0
$85$	15.2877			1.65818
$86$		0			0
$87$	4.04590			0.433766
$88$		0			0
$89$		−	2.71165i		−	0.287434i	−0.989619	−	0.143717i	$-0.954094\pi$
$89$		−	2.71165i		−	0.287434i	0.989619	−	0.143717i	$-0.0459055\pi$
$90$		0			0
$91$	−2.08681	+	10.9824i	−0.218758	+	1.15127i
$92$		0			0
$93$		−	4.46174i		−	0.462661i
$94$		0			0
$95$	−21.5117			−2.20705
$96$		0			0
$97$			2.77051i			0.281303i	0.990059	+	0.140651i	$0.0449197\pi$
$97$			2.77051i			0.281303i	−0.990059	+	0.140651i	$0.955080\pi$
$98$		0			0
$99$	2.07262	+	3.89502i	0.208306	+	0.391464i
$100$		0			0
$101$			10.9014i			1.08473i	0.840144	+	0.542363i	$0.182470\pi$
$101$			10.9014i			1.08473i	−0.840144	+	0.542363i	$0.817530\pi$
$102$		0			0
$103$		−	6.40451i		−	0.631055i	−0.948916	−	0.315527i	$-0.897819\pi$
$103$		−	6.40451i		−	0.631055i	0.948916	−	0.315527i	$-0.102181\pi$
$104$		0			0
$105$		−	27.6474i		−	2.69811i
$106$		0			0
$107$	−12.0734			−1.16718			−0.583590	−	0.812049i	$-0.698352\pi$
$107$	−12.0734			−1.16718			−0.583590	+	0.812049i	$0.698352\pi$
$108$		0			0
$109$	−6.75577			−0.647085			−0.323542	−	0.946214i	$-0.604874\pi$
$109$	−6.75577			−0.647085			−0.323542	+	0.946214i	$0.604874\pi$
$110$		0			0
$111$	18.5853			1.76404
$112$		0			0
$113$	−13.1594			−1.23794			−0.618968	−	0.785416i	$-0.712449\pi$
$113$	−13.1594			−1.23794			−0.618968	+	0.785416i	$0.712449\pi$
$114$		0			0
$115$	2.32124			0.216457
$116$		0			0
$117$	4.71218	+	0.895382i	0.435642	+	0.0827781i
$118$		0			0
$119$	−11.0612			−1.01398
$120$		0			0
$121$	−6.14529	+	9.12335i	−0.558663	+	0.829395i
$122$		0			0
$123$		−	19.1432i		−	1.72608i
$124$		0			0
$125$		−	35.8347i		−	3.20515i
$126$		0			0
$127$	17.4658			1.54984			0.774920	−	0.632060i	$-0.217790\pi$
$127$	17.4658			1.54984			0.774920	+	0.632060i	$0.217790\pi$
$128$		0			0
$129$			22.3081i			1.96412i
$130$		0			0
$131$	−2.81169			−0.245658			−0.122829	−	0.992428i	$-0.539197\pi$
$131$	−2.81169			−0.245658			−0.122829	+	0.992428i	$0.539197\pi$
$132$		0			0
$133$	15.5646			1.34962
$134$		0			0
$135$	14.8889			1.28143
$136$		0			0
$137$		−	4.85331i		−	0.414646i	−0.978273	−	0.207323i	$-0.933525\pi$
$137$		−	4.85331i		−	0.414646i	0.978273	−	0.207323i	$-0.0664751\pi$
$138$		0			0
$139$	−0.883564			−0.0749429			−0.0374715	−	0.999298i	$-0.511930\pi$
$139$	−0.883564			−0.0749429			−0.0374715	+	0.999298i	$0.511930\pi$
$140$		0			0
$141$		−	4.13038i		−	0.347841i
$142$		0			0
$143$	3.54803	+	11.4198i	0.296701	+	0.954970i
$144$		0			0
$145$	−8.33148			−0.691891
$146$		0			0
$147$			5.43737i			0.448467i
$148$		0			0
$149$	12.3264			1.00982			0.504909	−	0.863173i	$-0.331526\pi$
$149$	12.3264			1.00982			0.504909	+	0.863173i	$0.331526\pi$
$150$		0			0
$151$		−	8.86095i		−	0.721094i	−0.932741	−	0.360547i	$-0.882590\pi$
$151$		−	8.86095i		−	0.721094i	0.932741	−	0.360547i	$-0.117410\pi$
$152$		0			0
$153$			4.74600i			0.383692i
$154$		0			0
$155$			9.18778i			0.737980i
$156$		0			0
$157$	19.2494			1.53627			0.768137	−	0.640286i	$-0.221184\pi$
$157$	19.2494			1.53627			0.768137	+	0.640286i	$0.221184\pi$
$158$		0			0
$159$			12.3654i			0.980639i
$160$		0			0
$161$	−1.67951			−0.132364
$162$		0			0
$163$	20.5129			1.60669			0.803346	−	0.595513i	$-0.203051\pi$
$163$	20.5129			1.60669			0.803346	+	0.595513i	$0.203051\pi$
$164$		0			0
$165$	−13.8929	−	26.1086i	−1.08156	−	2.03255i
$166$		0			0
$167$			10.2930i			0.796499i	0.917277	+	0.398249i	$0.130382\pi$
$167$			10.2930i			0.796499i	−0.917277	+	0.398249i	$0.869618\pi$
$168$		0			0
$169$	12.0940	+	4.76821i	0.930306	+	0.366785i
$170$		0			0
$171$		−	6.67823i		−	0.510697i
$172$		0			0
$173$		−	8.50733i		−	0.646801i	−0.946262	−	0.323400i	$-0.895174\pi$
$173$		−	8.50733i		−	0.646801i	0.946262	−	0.323400i	$-0.104826\pi$
$174$		0			0
$175$			41.4302i			3.13183i
$176$		0			0
$177$			23.7768i			1.78718i
$178$		0			0
$179$			12.5310i			0.936614i	0.883566	+	0.468307i	$0.155136\pi$
$179$			12.5310i			0.936614i	−0.883566	+	0.468307i	$0.844864\pi$
$180$		0			0
$181$	2.05683			0.152883			0.0764413	−	0.997074i	$-0.475644\pi$
$181$	2.05683			0.152883			0.0764413	+	0.997074i	$0.475644\pi$
$182$		0			0
$183$	−25.1644			−1.86020
$184$		0			0
$185$	−38.2716			−2.81378
$186$		0			0
$187$	−10.4456	+	5.55830i	−0.763857	+	0.406463i
$188$		0			0
$189$	−10.7727			−0.783597
$190$		0			0
$191$		−	5.57546i		−	0.403426i	−0.979445	−	0.201713i	$-0.935349\pi$
$191$		−	5.57546i		−	0.403426i	0.979445	−	0.201713i	$-0.0646508\pi$
$192$		0			0
$193$	−5.46449			−0.393343			−0.196671	−	0.980469i	$-0.563013\pi$
$193$	−5.46449			−0.393343			−0.196671	+	0.980469i	$0.563013\pi$
$194$		0			0
$195$	−31.5861	−	6.00180i	−2.26192	−	0.429798i
$196$		0			0
$197$	−18.7783			−1.33790			−0.668949	−	0.743308i	$-0.733255\pi$
$197$	−18.7783			−1.33790			−0.668949	+	0.743308i	$0.733255\pi$
$198$		0			0
$199$		−	1.41073i		−	0.100004i	−0.998749	−	0.0500020i	$-0.984077\pi$
$199$		−	1.41073i		−	0.100004i	0.998749	−	0.0500020i	$-0.0159228\pi$
$200$		0			0
$201$			18.7682i			1.32381i
$202$		0			0
$203$	6.02815			0.423093
$204$		0			0
$205$			39.4204i			2.75324i
$206$		0			0
$207$			0.720621i			0.0500866i
$208$		0			0
$209$	14.6983	−	7.82124i	1.01670	−	0.541006i
$210$		0			0
$211$	5.09239			0.350575			0.175287	−	0.984517i	$-0.443915\pi$
$211$	5.09239			0.350575			0.175287	+	0.984517i	$0.443915\pi$
$212$		0			0
$213$			19.7875i			1.35582i
$214$		0			0
$215$		−	45.9376i		−	3.13292i
$216$		0			0
$217$		−	6.64772i		−	0.451277i
$218$		0			0
$219$			24.6010i			1.66238i
$220$		0			0
$221$	−2.40122	+	12.6370i	−0.161523	+	0.850059i
$222$		0			0
$223$	−14.6850			−0.983378			−0.491689	−	0.870771i	$-0.663620\pi$
$223$	−14.6850			−0.983378			−0.491689	+	0.870771i	$0.663620\pi$
$224$		0			0
$225$	17.7763			1.18509
$226$		0			0
$227$			3.68412i			0.244524i	0.992498	+	0.122262i	$0.0390147\pi$
$227$			3.68412i			0.244524i	−0.992498	+	0.122262i	$0.960985\pi$
$228$		0			0
$229$			23.9534i			1.58289i	0.611243	+	0.791443i	$0.290670\pi$
$229$			23.9534i			1.58289i	−0.611243	+	0.791443i	$0.709330\pi$
$230$		0			0
$231$	10.0520	+	18.8906i	0.661376	+	1.24291i
$232$		0			0
$233$		−	19.7216i		−	1.29201i	−0.763334	−	0.646004i	$-0.776439\pi$
$233$		−	19.7216i		−	1.29201i	0.763334	−	0.646004i	$-0.223561\pi$
$234$		0			0
$235$			8.50544i			0.554834i
$236$		0			0
$237$			10.0427i			0.652344i
$238$		0			0
$239$		−	6.00099i		−	0.388172i	−0.980985	−	0.194086i	$-0.937826\pi$
$239$		−	6.00099i		−	0.388172i	0.980985	−	0.194086i	$-0.0621740\pi$
$240$		0			0
$241$	15.6409			1.00752			0.503759	−	0.863844i	$-0.331950\pi$
$241$	15.6409			1.00752			0.503759	+	0.863844i	$0.331950\pi$
$242$		0			0
$243$			12.9271i			0.829272i
$244$		0			0
$245$		−	11.1969i		−	0.715341i
$246$		0			0
$247$	3.37882	−	17.7819i	0.214989	−	1.13144i
$248$		0			0
$249$	−21.7396			−1.37769
$250$		0			0
$251$		−	8.05122i		−	0.508189i	−0.967179	−	0.254094i	$-0.918223\pi$
$251$		−	8.05122i		−	0.508189i	0.967179	−	0.254094i	$-0.0817774\pi$
$252$		0			0
$253$	−1.58603	+	0.843958i	−0.0997129	+	0.0530592i
$254$		0			0
$255$		−	31.8127i		−	1.99219i
$256$		0			0
$257$	−18.2403			−1.13780			−0.568899	−	0.822407i	$-0.692631\pi$
$257$	−18.2403			−1.13780			−0.568899	+	0.822407i	$0.692631\pi$
$258$		0			0
$259$	27.6910			1.72063
$260$		0			0
$261$		−	2.58648i		−	0.160099i
$262$		0			0
$263$	21.2305			1.30913			0.654563	−	0.756007i	$-0.272853\pi$
$263$	21.2305			1.30913			0.654563	+	0.756007i	$0.272853\pi$
$264$		0			0
$265$		−	25.4633i		−	1.56420i
$266$		0			0
$267$	−5.64277			−0.345332
$268$		0			0
$269$	22.4461			1.36856			0.684280	−	0.729219i	$-0.260117\pi$
$269$	22.4461			1.36856			0.684280	+	0.729219i	$0.260117\pi$
$270$		0			0
$271$			19.0466i			1.15700i	0.815684	+	0.578498i	$0.196361\pi$
$271$			19.0466i			1.15700i	−0.815684	+	0.578498i	$0.803639\pi$
$272$		0			0
$273$	22.8537	+	4.34254i	1.38317	+	0.262822i
$274$		0			0
$275$	20.8188	+	39.1242i	1.25542	+	2.35928i
$276$		0			0
$277$		−	29.1220i		−	1.74977i	−0.484328	−	0.874887i	$-0.660936\pi$
$277$		−	29.1220i		−	1.74977i	0.484328	−	0.874887i	$-0.339064\pi$
$278$		0			0
$279$	−2.85231			−0.170763
$280$		0			0
$281$	−15.7005			−0.936616			−0.468308	−	0.883565i	$-0.655136\pi$
$281$	−15.7005			−0.936616			−0.468308	+	0.883565i	$0.655136\pi$
$282$		0			0
$283$	−21.1796			−1.25900			−0.629498	−	0.777002i	$-0.716740\pi$
$283$	−21.1796			−1.25900			−0.629498	+	0.777002i	$0.716740\pi$
$284$		0			0
$285$			44.7646i			2.65162i
$286$		0			0
$287$		−	28.5222i		−	1.68361i
$288$		0			0
$289$	4.27226			0.251310
$290$		0			0
$291$	5.76527			0.337966
$292$		0			0
$293$	−15.1673			−0.886082			−0.443041	−	0.896501i	$-0.646100\pi$
$293$	−15.1673			−0.886082			−0.443041	+	0.896501i	$0.646100\pi$
$294$		0			0
$295$		−	48.9622i		−	2.85069i
$296$		0			0
$297$	−10.1731	+	5.41330i	−0.590303	+	0.314112i
$298$		0			0
$299$	−0.364594	+	1.91877i	−0.0210850	+	0.110966i
$300$		0			0
$301$			33.2377i			1.91579i
$302$		0			0
$303$	22.6851			1.30322
$304$		0			0
$305$	51.8195			2.96717
$306$		0			0
$307$		−	2.22871i		−	0.127199i	−0.997975	−	0.0635997i	$-0.979742\pi$
$307$		−	2.22871i		−	0.127199i	0.997975	−	0.0635997i	$-0.0202581\pi$
$308$		0			0
$309$	−13.3274			−0.758169
$310$		0			0
$311$			24.4208i			1.38478i	0.721525	+	0.692388i	$0.243441\pi$
$311$			24.4208i			1.38478i	−0.721525	+	0.692388i	$0.756559\pi$
$312$		0			0
$313$	−4.34489			−0.245588			−0.122794	−	0.992432i	$-0.539185\pi$
$313$	−4.34489			−0.245588			−0.122794	+	0.992432i	$0.539185\pi$
$314$		0			0
$315$	−17.6745			−0.995845
$316$		0			0
$317$		−	9.56482i		−	0.537214i	−0.963250	−	0.268607i	$-0.913437\pi$
$317$		−	9.56482i		−	0.537214i	0.963250	−	0.268607i	$-0.0865633\pi$
$318$		0			0
$319$	5.69263	−	3.02916i	0.318726	−	0.169601i
$320$		0			0
$321$			25.1240i			1.40229i
$322$		0			0
$323$	17.9095			0.996513
$324$		0			0
$325$	47.3324	+	8.99382i	2.62553	+	0.498887i
$326$		0			0
$327$			14.0583i			0.777428i
$328$		0			0
$329$		−	6.15402i		−	0.339282i
$330$		0			0
$331$	19.6897			1.08224			0.541121	−	0.840945i	$-0.318000\pi$
$331$	19.6897			1.08224			0.541121	+	0.840945i	$0.318000\pi$
$332$		0			0
$333$		−	11.8813i		−	0.651089i
$334$		0			0
$335$		−	38.6482i		−	2.11158i
$336$		0			0
$337$			7.60479i			0.414259i	0.978313	+	0.207130i	$0.0664122\pi$
$337$			7.60479i			0.414259i	−0.978313	+	0.207130i	$0.933588\pi$
$338$		0			0
$339$			27.3840i			1.48730i
$340$		0			0
$341$	−3.34050	−	6.27772i	−0.180898	−	0.339958i
$342$		0			0
$343$		−	13.6020i		−	0.734437i
$344$		0			0
$345$		−	4.83036i		−	0.260058i
$346$		0			0
$347$	4.69917			0.252265			0.126132	−	0.992013i	$-0.459744\pi$
$347$	4.69917			0.252265			0.126132	+	0.992013i	$0.459744\pi$
$348$		0			0
$349$	16.7299			0.895528			0.447764	−	0.894152i	$-0.352220\pi$
$349$	16.7299			0.895528			0.447764	+	0.894152i	$0.352220\pi$
$350$		0			0
$351$	−2.33858	+	12.3074i	−0.124824	+	0.656919i
$352$		0			0
$353$		−	33.6048i		−	1.78861i	−0.447463	−	0.894303i	$-0.647672\pi$
$353$		−	33.6048i		−	1.78861i	0.447463	−	0.894303i	$-0.352328\pi$
$354$		0			0
$355$		−	40.7471i		−	2.16263i
$356$		0			0
$357$			23.0178i			1.21823i
$358$		0			0
$359$		−	13.9272i		−	0.735051i	−0.930013	−	0.367526i	$-0.880205\pi$
$359$		−	13.9272i		−	0.735051i	0.930013	−	0.367526i	$-0.119795\pi$
$360$		0			0
$361$	−6.20098			−0.326367
$362$		0			0
$363$	18.9851	+	12.7880i	0.996461	+	0.671194i
$364$		0			0
$365$		−	50.6594i		−	2.65163i
$366$		0			0
$367$			2.49880i			0.130437i	0.997871	+	0.0652183i	$0.0207744\pi$
$367$			2.49880i			0.130437i	−0.997871	+	0.0652183i	$0.979226\pi$
$368$		0			0
$369$	−12.2379			−0.637081
$370$		0			0
$371$			18.4237i			0.956510i
$372$		0			0
$373$			9.67276i			0.500836i	0.968138	+	0.250418i	$0.0805681\pi$
$373$			9.67276i			0.500836i	−0.968138	+	0.250418i	$0.919432\pi$
$374$		0			0
$375$	−74.5698			−3.85077
$376$		0			0
$377$	1.30861	−	6.88693i	0.0673970	−	0.354695i
$378$		0			0
$379$	0.728681			0.0374298			0.0187149	−	0.999825i	$-0.494043\pi$
$379$	0.728681			0.0374298			0.0187149	+	0.999825i	$0.494043\pi$
$380$		0			0
$381$		−	36.3453i		−	1.86203i
$382$		0			0
$383$	−14.3889			−0.735238			−0.367619	−	0.929977i	$-0.619827\pi$
$383$	−14.3889			−0.735238			−0.367619	+	0.929977i	$0.619827\pi$
$384$		0			0
$385$	−20.6996	−	38.9002i	−1.05495	−	1.98254i
$386$		0			0
$387$	14.2612			0.724936
$388$		0			0
$389$	13.5337			0.686187			0.343093	−	0.939301i	$-0.388525\pi$
$389$	13.5337			0.686187			0.343093	+	0.939301i	$0.388525\pi$
$390$		0			0
$391$	−1.93255			−0.0977330
$392$		0			0
$393$			5.85095i			0.295142i
$394$		0			0
$395$		−	20.6803i		−	1.04054i
$396$		0			0
$397$			12.8184i			0.643337i	0.946852	+	0.321669i	$0.104244\pi$
$397$			12.8184i			0.643337i	−0.946852	+	0.321669i	$0.895756\pi$
$398$		0			0
$399$		−	32.3889i		−	1.62147i
$400$		0			0
$401$		−	21.5061i		−	1.07396i	−0.843594	−	0.536981i	$-0.819565\pi$
$401$		−	21.5061i		−	1.07396i	0.843594	−	0.536981i	$-0.180435\pi$
$402$		0			0
$403$	−7.59476	−	1.44311i	−0.378322	−	0.0718866i
$404$		0			0
$405$		−	48.0846i		−	2.38934i
$406$		0			0
$407$	26.1497	−	13.9148i	1.29619	−	0.689730i
$408$		0			0
$409$	−14.9792			−0.740674			−0.370337	−	0.928897i	$-0.620758\pi$
$409$	−14.9792			−0.740674			−0.370337	+	0.928897i	$0.620758\pi$
$410$		0			0
$411$	−10.0994			−0.498169
$412$		0			0
$413$			35.4261i			1.74320i
$414$		0			0
$415$	44.7670			2.19753
$416$		0			0
$417$			1.83864i			0.0900387i
$418$		0			0
$419$			10.9824i			0.536526i	0.963346	+	0.268263i	$0.0864496\pi$
$419$			10.9824i			0.536526i	−0.963346	+	0.268263i	$0.913550\pi$
$420$		0			0
$421$		−	5.77502i		−	0.281457i	−0.990048	−	0.140729i	$-0.955056\pi$
$421$		−	5.77502i		−	0.281457i	0.990048	−	0.140729i	$-0.0449445\pi$
$422$		0			0
$423$	−2.64048			−0.128385
$424$		0			0
$425$			47.6721i			2.31244i
$426$		0			0
$427$	−37.4934			−1.81443
$428$		0			0
$429$	23.7639	−	7.38323i	1.14733	−	0.356466i
$430$		0			0
$431$			0.836564i			0.0402959i	0.999797	+	0.0201479i	$0.00641372\pi$
$431$			0.836564i			0.0402959i	−0.999797	+	0.0201479i	$0.993586\pi$
$432$		0			0
$433$	13.6220			0.654631			0.327316	−	0.944915i	$-0.393856\pi$
$433$	13.6220			0.654631			0.327316	+	0.944915i	$0.393856\pi$
$434$		0			0
$435$			17.3373i			0.831260i
$436$		0			0
$437$	2.71934			0.130083
$438$		0			0
$439$	−5.41288			−0.258343			−0.129171	−	0.991622i	$-0.541232\pi$
$439$	−5.41288			−0.258343			−0.129171	+	0.991622i	$0.541232\pi$
$440$		0			0
$441$	3.47602			0.165525
$442$		0			0
$443$			0.992538i			0.0471569i	0.999722	+	0.0235785i	$0.00750595\pi$
$443$			0.992538i			0.0471569i	−0.999722	+	0.0235785i	$0.992494\pi$
$444$		0			0
$445$	11.6198			0.550832
$446$		0			0
$447$		−	25.6505i		−	1.21323i
$448$		0			0
$449$			10.2799i			0.485139i	0.970134	+	0.242570i	$0.0779903\pi$
$449$			10.2799i			0.485139i	−0.970134	+	0.242570i	$0.922010\pi$
$450$		0			0
$451$	−14.3325	−	26.9347i	−0.674891	−	1.26831i
$452$		0			0
$453$	−18.4391			−0.866345
$454$		0			0
$455$	−47.0613	−	8.94232i	−2.20627	−	0.419222i
$456$		0			0
$457$	32.0244			1.49804			0.749019	−	0.662548i	$-0.230525\pi$
$457$	32.0244			1.49804			0.749019	+	0.662548i	$0.230525\pi$
$458$		0			0
$459$	−12.3957			−0.578582
$460$		0			0
$461$	−28.2365			−1.31510			−0.657552	−	0.753409i	$-0.728408\pi$
$461$	−28.2365			−1.31510			−0.657552	+	0.753409i	$0.728408\pi$
$462$		0			0
$463$	23.2418			1.08014			0.540068	−	0.841621i	$-0.318398\pi$
$463$	23.2418			1.08014			0.540068	+	0.841621i	$0.318398\pi$
$464$		0			0
$465$	19.1192			0.886632
$466$		0			0
$467$		−	23.1424i		−	1.07090i	−0.844566	−	0.535452i	$-0.820141\pi$
$467$		−	23.1424i		−	1.07090i	0.844566	−	0.535452i	$-0.179859\pi$
$468$		0			0
$469$			27.9635i			1.29123i
$470$		0			0
$471$		−	40.0569i		−	1.84573i
$472$		0			0
$473$	16.7020	+	31.3877i	0.767960	+	1.44321i
$474$		0			0
$475$		−	67.0806i		−	3.07787i
$476$		0			0
$477$	7.90498			0.361944
$478$		0			0
$479$		−	22.0062i		−	1.00549i	−0.864435	−	0.502745i	$-0.832324\pi$
$479$		−	22.0062i		−	1.00549i	0.864435	−	0.502745i	$-0.167676\pi$
$480$		0			0
$481$	6.01126	−	31.6359i	0.274090	−	1.44247i
$482$		0			0
$483$			3.49496i			0.159026i
$484$		0			0
$485$	−11.8721			−0.539083
$486$		0			0
$487$	−32.1947			−1.45888			−0.729441	−	0.684044i	$-0.760220\pi$
$487$	−32.1947			−1.45888			−0.729441	+	0.684044i	$0.760220\pi$
$488$		0			0
$489$		−	42.6860i		−	1.93033i
$490$		0			0
$491$	20.4908			0.924736			0.462368	−	0.886688i	$-0.347000\pi$
$491$	20.4908			0.924736			0.462368	+	0.886688i	$0.347000\pi$
$492$		0			0
$493$	6.93636			0.312398
$494$		0			0
$495$	−16.6908	+	8.88148i	−0.750194	+	0.399193i
$496$		0			0
$497$			29.4821i			1.32246i
$498$		0			0
$499$	16.2514			0.727511			0.363756	−	0.931494i	$-0.381494\pi$
$499$	16.2514			0.727511			0.363756	+	0.931494i	$0.381494\pi$
$500$		0			0
$501$	21.4192			0.956938
$502$		0			0
$503$	−33.8633			−1.50989			−0.754945	−	0.655788i	$-0.772336\pi$
$503$	−33.8633			−1.50989			−0.754945	+	0.655788i	$0.772336\pi$
$504$		0			0
$505$	−46.7140			−2.07874
$506$		0			0
$507$	9.92236	−	25.1668i	0.440667	−	1.11770i
$508$		0			0
$509$			17.1485i			0.760092i	0.924968	+	0.380046i	$0.124092\pi$
$509$			17.1485i			0.760092i	−0.924968	+	0.380046i	$0.875908\pi$
$510$		0			0
$511$			36.6541i			1.62148i
$512$		0			0
$513$	17.4423			0.770097
$514$		0			0
$515$	27.4443			1.20934
$516$		0			0
$517$	−3.09241	−	5.81150i	−0.136004	−	0.255589i
$518$		0			0
$519$	−17.7032			−0.777086
$520$		0			0
$521$	9.02281			0.395297			0.197648	−	0.980273i	$-0.436670\pi$
$521$	9.02281			0.395297			0.197648	+	0.980273i	$0.436670\pi$
$522$		0			0
$523$	29.1135			1.27304			0.636522	−	0.771259i	$-0.280372\pi$
$523$	29.1135			1.27304			0.636522	+	0.771259i	$0.280372\pi$
$524$		0			0
$525$	86.2137			3.76267
$526$		0			0
$527$		−	7.64927i		−	0.333207i
$528$		0			0
$529$	22.7066			0.987242
$530$		0			0
$531$	15.2001			0.659629
$532$		0			0
$533$	−32.5855	−	6.19171i	−1.41144	−	0.268193i
$534$		0			0
$535$		−	51.7363i		−	2.23676i
$536$		0			0
$537$	26.0763			1.12528
$538$		0			0
$539$	4.07096	+	7.65045i	0.175349	+	0.329528i
$540$		0			0
$541$	−28.4104			−1.22146			−0.610730	−	0.791839i	$-0.709124\pi$
$541$	−28.4104			−1.22146			−0.610730	+	0.791839i	$0.709124\pi$
$542$		0			0
$543$		−	4.28013i		−	0.183678i
$544$		0			0
$545$		−	28.9495i		−	1.24006i
$546$		0			0
$547$	37.2991			1.59480			0.797398	−	0.603454i	$-0.206209\pi$
$547$	37.2991			1.59480			0.797398	+	0.603454i	$0.206209\pi$
$548$		0			0
$549$			16.0872i			0.686583i
$550$		0			0
$551$	−9.76033			−0.415804
$552$		0			0
$553$			14.9630i			0.636293i
$554$		0			0
$555$			79.6408i			3.38056i
$556$		0			0
$557$	3.61902			0.153343			0.0766714	−	0.997056i	$-0.475571\pi$
$557$	3.61902			0.153343			0.0766714	+	0.997056i	$0.475571\pi$
$558$		0			0
$559$	37.9728	+	7.21536i	1.60608	+	0.305177i
$560$		0			0
$561$	11.5665	+	21.7366i	0.488338	+	0.917722i
$562$		0			0
$563$	−18.2342			−0.768481			−0.384241	−	0.923233i	$-0.625537\pi$
$563$	−18.2342			−0.768481			−0.384241	+	0.923233i	$0.625537\pi$
$564$		0			0
$565$		−	56.3902i		−	2.37235i
$566$		0			0
$567$			34.7911i			1.46109i
$568$		0			0
$569$			3.94924i			0.165561i	0.996568	+	0.0827803i	$0.0263800\pi$
$569$			3.94924i			0.165561i	−0.996568	+	0.0827803i	$0.973620\pi$
$570$		0			0
$571$	0.105264			0.00440515			0.00220258	−	0.999998i	$-0.499299\pi$
$571$	0.105264			0.00440515			0.00220258	+	0.999998i	$0.499299\pi$
$572$		0			0
$573$	−11.6022			−0.484689
$574$		0			0
$575$			7.23840i			0.301862i
$576$		0			0
$577$			21.7958i			0.907372i	0.891162	+	0.453686i	$0.149891\pi$
$577$			21.7958i			0.907372i	−0.891162	+	0.453686i	$0.850109\pi$
$578$		0			0
$579$			11.3713i			0.472574i
$580$		0			0
$581$	−32.3907			−1.34379
$582$		0			0
$583$	9.25795	+	17.3982i	0.383425	+	0.720562i
$584$		0			0
$585$	−3.83685	+	20.1924i	−0.158634	+	0.834854i
$586$		0			0
$587$	21.9528			0.906087			0.453044	−	0.891488i	$-0.350338\pi$
$587$	21.9528			0.906087			0.453044	+	0.891488i	$0.350338\pi$
$588$		0			0
$589$			10.7635i			0.443502i
$590$		0			0
$591$			39.0765i			1.60739i
$592$		0			0
$593$	−15.7973			−0.648719			−0.324360	−	0.945934i	$-0.605149\pi$
$593$	−15.7973			−0.648719			−0.324360	+	0.945934i	$0.605149\pi$
$594$		0			0
$595$		−	47.3991i		−	1.94317i
$596$		0			0
$597$	−2.93564			−0.120148
$598$		0			0
$599$		−	19.7920i		−	0.808679i	−0.914609	−	0.404339i	$-0.867501\pi$
$599$		−	19.7920i		−	0.808679i	0.914609	−	0.404339i	$-0.132499\pi$
$600$		0			0
$601$		−	20.8510i		−	0.850529i	−0.905069	−	0.425265i	$-0.860181\pi$
$601$		−	20.8510i		−	0.850529i	0.905069	−	0.425265i	$-0.139819\pi$
$602$		0			0
$603$	11.9982			0.488604
$604$		0			0
$605$	−39.0949	−	26.3335i	−1.58943	−	1.07061i
$606$		0			0
$607$	−17.3977			−0.706149			−0.353075	−	0.935595i	$-0.614864\pi$
$607$	−17.3977			−0.706149			−0.353075	+	0.935595i	$0.614864\pi$
$608$		0			0
$609$		−	12.5442i		−	0.508317i
$610$		0			0
$611$	−7.03073	−	1.33594i	−0.284433	−	0.0540463i
$612$		0			0
$613$	−3.35288			−0.135422			−0.0677109	−	0.997705i	$-0.521570\pi$
$613$	−3.35288			−0.135422			−0.0677109	+	0.997705i	$0.521570\pi$
$614$		0			0
$615$	82.0315			3.30783
$616$		0			0
$617$			33.8911i			1.36440i	0.731164	+	0.682201i	$0.238977\pi$
$617$			33.8911i			1.36440i	−0.731164	+	0.682201i	$0.761023\pi$
$618$		0			0
$619$	11.8464			0.476148			0.238074	−	0.971247i	$-0.423484\pi$
$619$	11.8464			0.476148			0.238074	+	0.971247i	$0.423484\pi$
$620$		0			0
$621$	−1.88213			−0.0755273
$622$		0			0
$623$	−8.40739			−0.336835
$624$		0			0
$625$	86.7444			3.46978
$626$		0			0
$627$	−16.2755	−	30.5862i	−0.649982	−	1.22150i
$628$		0			0
$629$	31.8629			1.27046
$630$		0			0
$631$	34.4272			1.37053			0.685263	−	0.728296i	$-0.259687\pi$
$631$	34.4272			1.37053			0.685263	+	0.728296i	$0.259687\pi$
$632$		0			0
$633$		−	10.5970i		−	0.421191i
$634$		0			0
$635$			74.8436i			2.97008i
$636$		0			0
$637$	9.25549	+	1.75867i	0.366716	+	0.0696812i
$638$		0			0
$639$	12.6498			0.500418
$640$		0			0
$641$	8.98522			0.354895			0.177447	−	0.984130i	$-0.443216\pi$
$641$	8.98522			0.354895			0.177447	+	0.984130i	$0.443216\pi$
$642$		0			0
$643$	−44.5317			−1.75616			−0.878080	−	0.478514i	$-0.841176\pi$
$643$	−44.5317			−1.75616			−0.878080	+	0.478514i	$0.841176\pi$
$644$		0			0
$645$	−95.5934			−3.76399
$646$		0			0
$647$			19.3528i			0.760837i	0.924814	+	0.380419i	$0.124220\pi$
$647$			19.3528i			0.760837i	−0.924814	+	0.380419i	$0.875780\pi$
$648$		0			0
$649$	17.8017	+	33.4543i	0.698778	+	1.31320i
$650$		0			0
$651$	−13.8335			−0.542178
$652$		0			0
$653$	48.6327			1.90314			0.951572	−	0.307425i	$-0.0994672\pi$
$653$	48.6327			1.90314			0.951572	+	0.307425i	$0.0994672\pi$
$654$		0			0
$655$		−	12.0485i		−	0.470774i
$656$		0			0
$657$	15.7270			0.613569
$658$		0			0
$659$	−18.3530			−0.714932			−0.357466	−	0.933926i	$-0.616359\pi$
$659$	−18.3530			−0.714932			−0.357466	+	0.933926i	$0.616359\pi$
$660$		0			0
$661$		−	3.88425i		−	0.151080i	−0.997143	−	0.0755399i	$-0.975932\pi$
$661$		−	3.88425i		−	0.151080i	0.997143	−	0.0755399i	$-0.0240680\pi$
$662$		0			0
$663$	26.2969	+	4.99679i	1.02129	+	0.194059i
$664$		0			0
$665$			66.6965i			2.58638i
$666$		0			0
$667$	1.05320			0.0407800
$668$		0			0
$669$			30.5585i			1.18146i
$670$		0			0
$671$	−35.4066	+	18.8406i	−1.36686	+	0.727331i
$672$		0			0
$673$			42.2955i			1.63037i	0.579201	+	0.815185i	$0.303365\pi$
$673$			42.2955i			1.63037i	−0.579201	+	0.815185i	$0.696635\pi$
$674$		0			0
$675$			46.4284i			1.78703i
$676$		0			0
$677$			16.1036i			0.618913i	0.950914	+	0.309457i	$0.100147\pi$
$677$			16.1036i			0.618913i	−0.950914	+	0.309457i	$0.899853\pi$
$678$		0			0
$679$	8.58990			0.329650
$680$		0			0
$681$	7.66643			0.293778
$682$		0			0
$683$	14.9120			0.570591			0.285295	−	0.958440i	$-0.407908\pi$
$683$	14.9120			0.570591			0.285295	+	0.958440i	$0.407908\pi$
$684$		0			0
$685$	20.7972			0.794619
$686$		0			0
$687$	49.8456			1.90173
$688$		0			0
$689$	21.0483	+	3.99948i	0.801878	+	0.152368i
$690$		0			0
$691$	−39.5014			−1.50270			−0.751352	−	0.659902i	$-0.770598\pi$
$691$	−39.5014			−1.50270			−0.751352	+	0.659902i	$0.770598\pi$
$692$		0			0
$693$	12.0764	−	6.42610i	0.458745	−	0.244107i
$694$		0			0
$695$		−	3.78621i		−	0.143619i
$696$		0			0
$697$		−	32.8194i		−	1.24312i
$698$		0			0
$699$	−41.0395			−1.55226
$700$		0			0
$701$		−	3.87079i		−	0.146198i	−0.997325	−	0.0730989i	$-0.976711\pi$
$701$		−	3.87079i		−	0.146198i	0.997325	−	0.0730989i	$-0.0232889\pi$
$702$		0			0
$703$	−44.8351			−1.69099
$704$		0			0
$705$	17.6993			0.666594
$706$		0			0
$707$	33.7994			1.27116
$708$		0			0
$709$		−	2.99859i		−	0.112614i	−0.998413	−	0.0563072i	$-0.982067\pi$
$709$		−	2.99859i		−	0.112614i	0.998413	−	0.0563072i	$-0.0179326\pi$
$710$		0			0
$711$	6.42013			0.240774
$712$		0			0
$713$		−	1.16145i		−	0.0434965i
$714$		0			0
$715$	−48.9355	+	15.2038i	−1.83008	+	0.568591i
$716$		0			0
$717$	−12.4877			−0.466362
$718$		0			0
$719$		−	12.8814i		−	0.480397i	−0.970724	−	0.240198i	$-0.922787\pi$
$719$		−	12.8814i		−	0.480397i	0.970724	−	0.240198i	$-0.0772125\pi$
$720$		0			0
$721$	−19.8570			−0.739514
$722$		0			0
$723$		−	32.5477i		−	1.21046i
$724$		0			0
$725$		−	25.9803i		−	0.964884i
$726$		0			0
$727$		−	51.4961i		−	1.90989i	−0.296790	−	0.954943i	$-0.595916\pi$
$727$		−	51.4961i		−	1.90989i	0.296790	−	0.954943i	$-0.404084\pi$
$728$		0			0
$729$	−6.76316			−0.250488
$730$		0			0
$731$			38.2453i			1.41455i
$732$		0			0
$733$	−25.9838			−0.959734			−0.479867	−	0.877341i	$-0.659315\pi$
$733$	−25.9838			−0.959734			−0.479867	+	0.877341i	$0.659315\pi$
$734$		0			0
$735$	−23.3000			−0.859432
$736$		0			0
$737$	14.0517	+	26.4071i	0.517602	+	0.972717i
$738$		0			0
$739$		−	20.9635i		−	0.771155i	−0.922675	−	0.385578i	$-0.874002\pi$
$739$		−	20.9635i		−	0.771155i	0.922675	−	0.385578i	$-0.125998\pi$
$740$		0			0
$741$	−37.0031	−	7.03111i	−1.35934	−	0.258294i
$742$		0			0
$743$		−	3.51692i		−	0.129023i	−0.997917	−	0.0645116i	$-0.979451\pi$
$743$		−	3.51692i		−	0.129023i	0.997917	−	0.0645116i	$-0.0205490\pi$
$744$		0			0
$745$			52.8204i			1.93519i
$746$		0			0
$747$			13.8977i			0.508492i
$748$		0			0
$749$			37.4333i			1.36778i
$750$		0			0
$751$			5.03675i			0.183794i	0.995769	+	0.0918968i	$0.0292930\pi$
$751$			5.03675i			0.183794i	−0.995769	+	0.0918968i	$0.970707\pi$
$752$		0			0
$753$	−16.7541			−0.610554
$754$		0			0
$755$	37.9705			1.38189
$756$		0			0
$757$	29.1106			1.05804			0.529022	−	0.848608i	$-0.322559\pi$
$757$	29.1106			1.05804			0.529022	+	0.848608i	$0.322559\pi$
$758$		0			0
$759$	1.75623	+	3.30043i	0.0637470	+	0.119798i
$760$		0			0
$761$	−15.6962			−0.568988			−0.284494	−	0.958678i	$-0.591826\pi$
$761$	−15.6962			−0.568988			−0.284494	+	0.958678i	$0.591826\pi$
$762$		0			0
$763$			20.9461i			0.758299i
$764$		0			0
$765$	−20.3373			−0.735298
$766$		0			0
$767$	40.4729	+	7.69043i	1.46139	+	0.277685i
$768$		0			0
$769$	27.0006			0.973665			0.486833	−	0.873495i	$-0.338152\pi$
$769$	27.0006			0.973665			0.486833	+	0.873495i	$0.338152\pi$
$770$		0			0
$771$			37.9570i			1.36699i
$772$		0			0
$773$			10.1763i			0.366016i	0.983111	+	0.183008i	$0.0585835\pi$
$773$			10.1763i			0.366016i	−0.983111	+	0.183008i	$0.941417\pi$
$774$		0			0
$775$	−28.6505			−1.02916
$776$		0			0
$777$		−	57.6232i		−	2.06722i
$778$		0			0
$779$			46.1810i			1.65461i
$780$		0			0
$781$	14.8149	+	27.8412i	0.530118	+	0.996237i
$782$		0			0
$783$	6.75540			0.241418
$784$		0			0
$785$			82.4868i			2.94408i
$786$		0			0
$787$		−	15.9052i		−	0.566960i	−0.958978	−	0.283480i	$-0.908511\pi$
$787$		−	15.9052i		−	0.566960i	0.958978	−	0.283480i	$-0.0914890\pi$
$788$		0			0
$789$		−	44.1793i		−	1.57282i
$790$		0			0
$791$			40.8005i			1.45070i
$792$		0			0
$793$	−8.13922	+	42.8348i	−0.289032	+	1.52111i
$794$		0			0
$795$	−52.9875			−1.87927
$796$		0			0
$797$	−10.8184			−0.383207			−0.191603	−	0.981472i	$-0.561369\pi$
$797$	−10.8184			−0.383207			−0.191603	+	0.981472i	$0.561369\pi$
$798$		0			0
$799$		−	7.08119i		−	0.250514i
$800$		0			0
$801$			3.60733i			0.127459i
$802$		0			0
$803$	18.4188	+	34.6140i	0.649984	+	1.22150i
$804$		0			0
$805$		−	7.19695i		−	0.253659i
$806$		0			0
$807$		−	46.7089i		−	1.64423i
$808$		0			0
$809$		−	17.8199i		−	0.626515i	−0.949668	−	0.313258i	$-0.898580\pi$
$809$		−	17.8199i		−	0.626515i	0.949668	−	0.313258i	$-0.101420\pi$
$810$		0			0
$811$		−	21.4177i		−	0.752078i	−0.926604	−	0.376039i	$-0.877286\pi$
$811$		−	21.4177i		−	0.752078i	0.926604	−	0.376039i	$-0.122714\pi$
$812$		0			0
$813$	39.6347			1.39005
$814$		0			0
$815$			87.9008i			3.07903i
$816$		0			0
$817$		−	53.8160i		−	1.88278i
$818$		0			0
$819$	2.77611	−	14.6100i	0.0970051	−	0.510515i
$820$		0			0
$821$	−23.3726			−0.815710			−0.407855	−	0.913047i	$-0.633723\pi$
$821$	−23.3726			−0.815710			−0.407855	+	0.913047i	$0.633723\pi$
$822$		0			0
$823$			5.90745i			0.205921i	0.994685	+	0.102960i	$0.0328315\pi$
$823$			5.90745i			0.205921i	−0.994685	+	0.102960i	$0.967169\pi$
$824$		0			0
$825$	81.4152	−	43.3226i	2.83451	−	1.50830i
$826$		0			0
$827$		−	51.0312i		−	1.77453i	−0.461260	−	0.887265i	$-0.652603\pi$
$827$		−	51.0312i		−	1.77453i	0.461260	−	0.887265i	$-0.347397\pi$
$828$		0			0
$829$	25.6839			0.892039			0.446020	−	0.895023i	$-0.352841\pi$
$829$	25.6839			0.892039			0.446020	+	0.895023i	$0.352841\pi$
$830$		0			0
$831$	−60.6012			−2.10223
$832$		0			0
$833$			9.32192i			0.322985i
$834$		0			0
$835$	−44.1072			−1.52639
$836$		0			0
$837$		−	7.44972i		−	0.257500i
$838$		0			0
$839$	8.40722			0.290249			0.145125	−	0.989413i	$-0.453642\pi$
$839$	8.40722			0.290249			0.145125	+	0.989413i	$0.453642\pi$
$840$		0			0
$841$	25.2198			0.869649
$842$		0			0
$843$			32.6719i			1.12528i
$844$		0			0
$845$	−20.4325	+	51.8245i	−0.702900	+	1.78282i
$846$		0			0
$847$	28.2867	+	19.0533i	0.971943	+	0.654679i
$848$		0			0
$849$			44.0735i			1.51260i
$850$		0			0
$851$	4.83798			0.165844
$852$		0			0
$853$	3.47719			0.119057			0.0595284	−	0.998227i	$-0.481040\pi$
$853$	3.47719			0.119057			0.0595284	+	0.998227i	$0.481040\pi$
$854$		0			0
$855$	28.6172			0.978688
$856$		0			0
$857$		−	56.3032i		−	1.92328i	−0.274311	−	0.961641i	$-0.588450\pi$
$857$		−	56.3032i		−	1.92328i	0.274311	−	0.961641i	$-0.411550\pi$
$858$		0			0
$859$			36.1680i			1.23404i	0.786949	+	0.617019i	$0.211660\pi$
$859$			36.1680i			1.23404i	−0.786949	+	0.617019i	$0.788340\pi$
$860$		0			0
$861$	−59.3530			−2.02274
$862$		0			0
$863$	26.9972			0.918995			0.459498	−	0.888179i	$-0.348029\pi$
$863$	26.9972			0.918995			0.459498	+	0.888179i	$0.348029\pi$
$864$		0			0
$865$	36.4552			1.23951
$866$		0			0
$867$		−	8.89032i		−	0.301931i
$868$		0			0
$869$	7.51896	+	14.1302i	0.255063	+	0.479335i
$870$		0			0
$871$	31.9472	+	6.07042i	1.08249	+	0.205688i
$872$		0			0
$873$		−	3.68564i		−	0.124740i
$874$		0			0
$875$	−111.105			−3.75602
$876$		0			0
$877$	−25.3482			−0.855949			−0.427974	−	0.903791i	$-0.640773\pi$
$877$	−25.3482			−0.855949			−0.427974	+	0.903791i	$0.640773\pi$
$878$		0			0
$879$			31.5622i			1.06457i
$880$		0			0
$881$	−14.2695			−0.480752			−0.240376	−	0.970680i	$-0.577271\pi$
$881$	−14.2695			−0.480752			−0.240376	+	0.970680i	$0.577271\pi$
$882$		0			0
$883$		−	55.3201i		−	1.86167i	−0.365442	−	0.930834i	$-0.619082\pi$
$883$		−	55.3201i		−	1.86167i	0.365442	−	0.930834i	$-0.380918\pi$
$884$		0			0
$885$	−101.887			−3.42491
$886$		0			0
$887$	54.8398			1.84134			0.920670	−	0.390341i	$-0.127643\pi$
$887$	54.8398			1.84134			0.920670	+	0.390341i	$0.127643\pi$
$888$		0			0
$889$		−	54.1523i		−	1.81621i
$890$		0			0
$891$	17.4826	+	32.8547i	0.585689	+	1.10067i
$892$		0			0
$893$			9.96413i			0.333437i
$894$		0			0
$895$	−53.6974			−1.79491
$896$		0			0
$897$	3.99285	+	0.758699i	0.133317	+	0.0253322i
$898$		0			0
$899$			4.16870i			0.139034i
$900$		0			0
$901$			21.1994i			0.706255i
$902$		0			0
$903$	69.1656			2.30169
$904$		0			0
$905$			8.81381i			0.292981i
$906$		0			0
$907$		−	37.6971i		−	1.25171i	−0.779939	−	0.625855i	$-0.784750\pi$
$907$		−	37.6971i		−	1.25171i	0.779939	−	0.625855i	$-0.215250\pi$
$908$		0			0
$909$		−	14.5022i		−	0.481007i
$910$		0			0
$911$		−	36.3316i		−	1.20372i	−0.798602	−	0.601859i	$-0.794427\pi$
$911$		−	36.3316i		−	1.20372i	0.798602	−	0.601859i	$-0.205573\pi$
$912$		0			0
$913$	−30.5879	+	16.2764i	−1.01231	+	0.538670i
$914$		0			0
$915$		−	107.833i		−	3.56486i
$916$		0			0
$917$			8.71757i			0.287879i
$918$		0			0
$919$	−12.5756			−0.414830			−0.207415	−	0.978253i	$-0.566505\pi$
$919$	−12.5756			−0.414830			−0.207415	+	0.978253i	$0.566505\pi$
$920$		0			0
$921$	−4.63782			−0.152821
$922$		0			0
$923$	33.6822	+	6.40010i	1.10866	+	0.210662i
$924$		0			0
$925$		−	119.343i		−	3.92399i
$926$		0			0
$927$			8.51997i			0.279833i
$928$		0			0
$929$		−	23.2242i		−	0.761962i	−0.924583	−	0.380981i	$-0.875586\pi$
$929$		−	23.2242i		−	0.761962i	0.924583	−	0.380981i	$-0.124414\pi$
$930$		0			0
$931$		−	13.1171i		−	0.429896i
$932$		0			0
$933$	50.8182			1.66371
$934$		0			0
$935$	−23.8182	−	44.7609i	−0.778938	−	1.46384i
$936$		0			0
$937$			51.5572i			1.68430i	0.539242	+	0.842151i	$0.318711\pi$
$937$			51.5572i			1.68430i	−0.539242	+	0.842151i	$0.681289\pi$
$938$		0			0
$939$			9.04146i			0.295057i
$940$		0			0
$941$	11.7867			0.384236			0.192118	−	0.981372i	$-0.438464\pi$
$941$	11.7867			0.384236			0.192118	+	0.981372i	$0.438464\pi$
$942$		0			0
$943$		−	4.98321i		−	0.162276i
$944$		0			0
$945$		−	46.1626i		−	1.50167i
$946$		0			0
$947$	−35.4878			−1.15320			−0.576599	−	0.817027i	$-0.695621\pi$
$947$	−35.4878			−1.15320			−0.576599	+	0.817027i	$0.695621\pi$
$948$		0			0
$949$	41.8758	+	7.95701i	1.35935	+	0.258295i
$950$		0			0
$951$	−19.9038			−0.645425
$952$		0			0
$953$		−	27.8984i		−	0.903718i	−0.892089	−	0.451859i	$-0.850761\pi$
$953$		−	27.8984i		−	0.903718i	0.892089	−	0.451859i	$-0.149239\pi$
$954$		0			0
$955$	23.8917			0.773117
$956$		0			0
$957$	−6.30350	−	11.8460i	−0.203763	−	0.382928i
$958$		0			0
$959$	−15.0476			−0.485911
$960$		0			0
$961$	−26.4028			−0.851705
$962$		0			0
$963$	16.0614			0.517570
$964$		0			0
$965$		−	23.4162i		−	0.753793i
$966$		0			0
$967$			58.2565i			1.87340i	0.350130	+	0.936701i	$0.386137\pi$
$967$			58.2565i			1.87340i	−0.350130	+	0.936701i	$0.613863\pi$
$968$		0			0
$969$		−	37.2687i		−	1.19724i
$970$		0			0
$971$			25.5721i			0.820648i	0.911940	+	0.410324i	$0.134584\pi$
$971$			25.5721i			0.820648i	−0.911940	+	0.410324i	$0.865416\pi$
$972$		0			0
$973$			2.73947i			0.0878233i
$974$		0			0
$975$	18.7156	−	98.4958i	0.599379	−	3.15439i
$976$		0			0
$977$		−	19.5165i		−	0.624387i	−0.950019	−	0.312193i	$-0.898936\pi$
$977$		−	19.5165i		−	0.624387i	0.950019	−	0.312193i	$-0.101064\pi$
$978$		0			0
$979$	−7.93945	+	4.22474i	−0.253746	+	0.135023i
$980$		0			0
$981$	8.98726			0.286941
$982$		0			0
$983$	−2.72839			−0.0870220			−0.0435110	−	0.999053i	$-0.513854\pi$
$983$	−2.72839			−0.0870220			−0.0435110	+	0.999053i	$0.513854\pi$
$984$		0			0
$985$		−	80.4678i		−	2.56392i
$986$		0			0
$987$	−12.8061			−0.407624
$988$		0			0
$989$			5.80706i			0.184654i
$990$		0			0
$991$			46.4565i			1.47574i	0.674943	+	0.737870i	$0.264168\pi$
$991$			46.4565i			1.47574i	−0.674943	+	0.737870i	$0.735832\pi$
$992$		0			0
$993$		−	40.9730i		−	1.30024i
$994$		0			0
$995$	6.04519			0.191646
$996$		0			0
$997$		−	23.9559i		−	0.758692i	−0.925255	−	0.379346i	$-0.876149\pi$
$997$		−	23.9559i		−	0.758692i	0.925255	−	0.379346i	$-0.123851\pi$
$998$		0			0
$999$	31.0317			0.981799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2288.2.b.c.2287.11	yes	56
4.3	odd	2	inner	2288.2.b.c.2287.48	yes	56
11.10	odd	2	inner	2288.2.b.c.2287.12	yes	56
13.12	even	2	inner	2288.2.b.c.2287.10	yes	56
44.43	even	2	inner	2288.2.b.c.2287.47	yes	56
52.51	odd	2	inner	2288.2.b.c.2287.45	yes	56
143.142	odd	2	inner	2288.2.b.c.2287.9	✓	56
572.571	even	2	inner	2288.2.b.c.2287.46	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2288.2.b.c.2287.9	✓	56	143.142	odd	2	inner
2288.2.b.c.2287.10	yes	56	13.12	even	2	inner
2288.2.b.c.2287.11	yes	56	1.1	even	1	trivial
2288.2.b.c.2287.12	yes	56	11.10	odd	2	inner
2288.2.b.c.2287.45	yes	56	52.51	odd	2	inner
2288.2.b.c.2287.46	yes	56	572.571	even	2	inner
2288.2.b.c.2287.47	yes	56	44.43	even	2	inner
2288.2.b.c.2287.48	yes	56	4.3	odd	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:	\( N \)	\(=\)	\( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2288.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.08094i q^{3} +4.28515i q^{5} -3.10047i q^{7} -1.33031 q^{9} +O(q^{10})\)	\(q-2.08094i q^{3} +4.28515i q^{5} -3.10047i q^{7} -1.33031 q^{9} +(-1.55800 - 2.92791i) q^{11} +(-3.54217 - 0.673063i) q^{13} +8.91714 q^{15} -3.56760i q^{17} +5.02006i q^{19} -6.45190 q^{21} -0.541694i q^{23} -13.3625 q^{25} -3.47453i q^{27} +1.94427i q^{29} +2.14410 q^{31} +(-6.09280 + 3.24210i) q^{33} +13.2860 q^{35} +8.93120i q^{37} +(-1.40060 + 7.37105i) q^{39} +9.19930 q^{41} -10.7202 q^{43} -5.70058i q^{45} +1.98486 q^{47} -2.61294 q^{49} -7.42395 q^{51} -5.94221 q^{53} +(12.5465 - 6.67625i) q^{55} +10.4464 q^{57} -11.4260 q^{59} -12.0928i q^{61} +4.12459i q^{63} +(2.88418 - 15.1787i) q^{65} -9.01909 q^{67} -1.12723 q^{69} -9.50892 q^{71} -11.8221 q^{73} +27.8066i q^{75} +(-9.07790 + 4.83053i) q^{77} -4.82604 q^{79} -11.2212 q^{81} -10.4470i q^{83} +15.2877 q^{85} +4.04590 q^{87} -2.71165i q^{89} +(-2.08681 + 10.9824i) q^{91} -4.46174i q^{93} -21.5117 q^{95} +2.77051i q^{97} +(2.07262 + 3.89502i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 72 q^{9}+O(q^{10}) \) 56 * q - 72 * q^9	\( 56 q - 72 q^{9} - 72 q^{25} - 8 q^{49} + 72 q^{53} + 8 q^{69} - 48 q^{77} + 88 q^{81}+O(q^{100}) \) 56 * q - 72 * q^9 - 72 * q^25 - 8 * q^49 + 72 * q^53 + 8 * q^69 - 48 * q^77 + 88 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more