LMFDB - Embedded newform 4014.2.d.a.4013.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(4013,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.8
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.125404 q^{5} +3.10184 q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +0.125404 q^{5} +3.10184 q^{7} -1.00000i q^{8} +0.125404i q^{10} +4.94283 q^{11} +4.70774i q^{13} +3.10184i q^{14} +1.00000 q^{16} +2.61065i q^{17} -2.28223 q^{19} -0.125404 q^{20} +4.94283i q^{22} -5.47440 q^{23} -4.98427 q^{25} -4.70774 q^{26} -3.10184 q^{28} +2.70943i q^{29} -0.559979 q^{31} +1.00000i q^{32} -2.61065 q^{34} +0.388984 q^{35} -5.88432 q^{37} -2.28223i q^{38} -0.125404i q^{40} +11.5748i q^{41} +3.19781 q^{43} -4.94283 q^{44} -5.47440i q^{46} +2.06882i q^{47} +2.62139 q^{49} -4.98427i q^{50} -4.70774i q^{52} +7.30917i q^{53} +0.619852 q^{55} -3.10184i q^{56} -2.70943 q^{58} +14.2791 q^{59} -1.60819i q^{61} -0.559979i q^{62} -1.00000 q^{64} +0.590371i q^{65} -1.97287i q^{67} -2.61065i q^{68} +0.388984i q^{70} -9.30587 q^{71} +13.3671 q^{73} -5.88432i q^{74} +2.28223 q^{76} +15.3318 q^{77} -17.2377i q^{79} +0.125404 q^{80} -11.5748 q^{82} -3.82366i q^{83} +0.327387i q^{85} +3.19781i q^{86} -4.94283i q^{88} +10.2585i q^{89} +14.6026i q^{91} +5.47440 q^{92} -2.06882 q^{94} -0.286201 q^{95} +3.12236i q^{97} +2.62139i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) $ 72 * q - 72 * q^4 + 16 * q^7	$ 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) $ 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	0.125404			0.0560825			0.0280413	−	0.999607i	$-0.491073\pi$
$5$	0.125404			0.0560825			0.0280413	+	0.999607i	$0.491073\pi$
$6$		0			0
$7$	3.10184			1.17238			0.586192	−	0.810172i	$-0.300626\pi$
$7$	3.10184			1.17238			0.586192	+	0.810172i	$0.300626\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			0.125404i			0.0396563i
$11$	4.94283			1.49032			0.745159	−	0.666887i	$-0.232373\pi$
$11$	4.94283			1.49032			0.745159	+	0.666887i	$0.232373\pi$
$12$		0			0
$13$			4.70774i			1.30569i	0.757491	+	0.652845i	$0.226425\pi$
$13$			4.70774i			1.30569i	−0.757491	+	0.652845i	$0.773575\pi$
$14$			3.10184i			0.829001i
$15$		0			0
$16$	1.00000			0.250000
$17$			2.61065i			0.633175i	0.948563	+	0.316587i	$0.102537\pi$
$17$			2.61065i			0.633175i	−0.948563	+	0.316587i	$0.897463\pi$
$18$		0			0
$19$	−2.28223			−0.523579			−0.261789	−	0.965125i	$-0.584313\pi$
$19$	−2.28223			−0.523579			−0.261789	+	0.965125i	$0.584313\pi$
$20$	−0.125404			−0.0280413
$21$		0			0
$22$			4.94283i			1.05381i
$23$	−5.47440			−1.14149			−0.570746	−	0.821127i	$-0.693346\pi$
$23$	−5.47440			−1.14149			−0.570746	+	0.821127i	$0.693346\pi$
$24$		0			0
$25$	−4.98427			−0.996855
$26$	−4.70774			−0.923263
$27$		0			0
$28$	−3.10184			−0.586192
$29$			2.70943i			0.503129i	0.967841	+	0.251564i	$0.0809450\pi$
$29$			2.70943i			0.503129i	−0.967841	+	0.251564i	$0.919055\pi$
$30$		0			0
$31$	−0.559979			−0.100575			−0.0502876	−	0.998735i	$-0.516014\pi$
$31$	−0.559979			−0.100575			−0.0502876	+	0.998735i	$0.516014\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−2.61065			−0.447722
$35$	0.388984			0.0657503
$36$		0			0
$37$	−5.88432			−0.967377			−0.483688	−	0.875240i	$-0.660703\pi$
$37$	−5.88432			−0.967377			−0.483688	+	0.875240i	$0.660703\pi$
$38$		−	2.28223i		−	0.370226i
$39$		0			0
$40$		−	0.125404i		−	0.0198282i
$41$			11.5748i			1.80767i	0.427878	+	0.903836i	$0.359261\pi$
$41$			11.5748i			1.80767i	−0.427878	+	0.903836i	$0.640739\pi$
$42$		0			0
$43$	3.19781			0.487661			0.243830	−	0.969818i	$-0.421596\pi$
$43$	3.19781			0.487661			0.243830	+	0.969818i	$0.421596\pi$
$44$	−4.94283			−0.745159
$45$		0			0
$46$		−	5.47440i		−	0.807157i
$47$			2.06882i			0.301768i	0.988551	+	0.150884i	$0.0482120\pi$
$47$			2.06882i			0.301768i	−0.988551	+	0.150884i	$0.951788\pi$
$48$		0			0
$49$	2.62139			0.374484
$50$		−	4.98427i		−	0.704883i
$51$		0			0
$52$		−	4.70774i		−	0.652845i
$53$			7.30917i			1.00399i	0.864870	+	0.501996i	$0.167401\pi$
$53$			7.30917i			1.00399i	−0.864870	+	0.501996i	$0.832599\pi$
$54$		0			0
$55$	0.619852			0.0835808
$56$		−	3.10184i		−	0.414500i
$57$		0			0
$58$	−2.70943			−0.355766
$59$	14.2791			1.85898			0.929492	−	0.368843i	$-0.120246\pi$
$59$	14.2791			1.85898			0.929492	+	0.368843i	$0.120246\pi$
$60$		0			0
$61$		−	1.60819i		−	0.205908i	−0.994686	−	0.102954i	$-0.967171\pi$
$61$		−	1.60819i		−	0.205908i	0.994686	−	0.102954i	$-0.0328294\pi$
$62$		−	0.559979i		−	0.0711174i
$63$		0			0
$64$	−1.00000			−0.125000
$65$			0.590371i			0.0732265i
$66$		0			0
$67$		−	1.97287i		−	0.241024i	−0.992712	−	0.120512i	$-0.961546\pi$
$67$		−	1.97287i		−	0.241024i	0.992712	−	0.120512i	$-0.0384537\pi$
$68$		−	2.61065i		−	0.316587i
$69$		0			0
$70$			0.388984i			0.0464925i
$71$	−9.30587			−1.10440			−0.552202	−	0.833710i	$-0.686212\pi$
$71$	−9.30587			−1.10440			−0.552202	+	0.833710i	$0.686212\pi$
$72$		0			0
$73$	13.3671			1.56450			0.782251	−	0.622964i	$-0.214072\pi$
$73$	13.3671			1.56450			0.782251	+	0.622964i	$0.214072\pi$
$74$		−	5.88432i		−	0.684039i
$75$		0			0
$76$	2.28223			0.261789
$77$	15.3318			1.74723
$78$		0			0
$79$		−	17.2377i		−	1.93940i	−0.244307	−	0.969698i	$-0.578561\pi$
$79$		−	17.2377i		−	1.93940i	0.244307	−	0.969698i	$-0.421439\pi$
$80$	0.125404			0.0140206
$81$		0			0
$82$	−11.5748			−1.27822
$83$		−	3.82366i		−	0.419701i	−0.977733	−	0.209851i	$-0.932702\pi$
$83$		−	3.82366i		−	0.419701i	0.977733	−	0.209851i	$-0.0672978\pi$
$84$		0			0
$85$			0.327387i			0.0355101i
$86$			3.19781i			0.344828i
$87$		0			0
$88$		−	4.94283i		−	0.526907i
$89$			10.2585i			1.08740i	0.839279	+	0.543700i	$0.182977\pi$
$89$			10.2585i			1.08740i	−0.839279	+	0.543700i	$0.817023\pi$
$90$		0			0
$91$			14.6026i			1.53077i
$92$	5.47440			0.570746
$93$		0			0
$94$	−2.06882			−0.213382
$95$	−0.286201			−0.0293636
$96$		0			0
$97$			3.12236i			0.317028i	0.987357	+	0.158514i	$0.0506702\pi$
$97$			3.12236i			0.317028i	−0.987357	+	0.158514i	$0.949330\pi$
$98$			2.62139i			0.264800i
$99$		0			0
$100$	4.98427			0.498427
$101$			2.14378i			0.213314i	0.994296	+	0.106657i	$0.0340146\pi$
$101$			2.14378i			0.213314i	−0.994296	+	0.106657i	$0.965985\pi$
$102$		0			0
$103$		−	9.20787i		−	0.907279i	−0.891185	−	0.453639i	$-0.850125\pi$
$103$		−	9.20787i		−	0.907279i	0.891185	−	0.453639i	$-0.149875\pi$
$104$	4.70774			0.461631
$105$		0			0
$106$	−7.30917			−0.709930
$107$	7.88792			0.762554			0.381277	−	0.924461i	$-0.375484\pi$
$107$	7.88792			0.762554			0.381277	+	0.924461i	$0.375484\pi$
$108$		0			0
$109$	2.09553			0.200716			0.100358	−	0.994951i	$-0.468001\pi$
$109$	2.09553			0.200716			0.100358	+	0.994951i	$0.468001\pi$
$110$			0.619852i			0.0591006i
$111$		0			0
$112$	3.10184			0.293096
$113$	−7.37996			−0.694248			−0.347124	−	0.937819i	$-0.612842\pi$
$113$	−7.37996			−0.694248			−0.347124	+	0.937819i	$0.612842\pi$
$114$		0			0
$115$	−0.686514			−0.0640178
$116$		−	2.70943i		−	0.251564i
$117$		0			0
$118$			14.2791i			1.31450i
$119$			8.09780i			0.742324i
$120$		0			0
$121$	13.4315			1.22105
$122$	1.60819			0.145599
$123$		0			0
$124$	0.559979			0.0502876
$125$	−1.25207			−0.111989
$126$		0			0
$127$	−14.5511			−1.29120			−0.645600	−	0.763675i	$-0.723393\pi$
$127$	−14.5511			−1.29120			−0.645600	+	0.763675i	$0.723393\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−0.590371			−0.0517789
$131$			22.3830i			1.95562i	0.209505	+	0.977808i	$0.432815\pi$
$131$			22.3830i			1.95562i	−0.209505	+	0.977808i	$0.567185\pi$
$132$		0			0
$133$	−7.07909			−0.613835
$134$	1.97287			0.170430
$135$		0			0
$136$	2.61065			0.223861
$137$	8.77797			0.749953			0.374976	−	0.927034i	$-0.377651\pi$
$137$	8.77797			0.749953			0.374976	+	0.927034i	$0.377651\pi$
$138$		0			0
$139$	4.07568			0.345694			0.172847	−	0.984949i	$-0.444703\pi$
$139$	4.07568			0.345694			0.172847	+	0.984949i	$0.444703\pi$
$140$	−0.388984			−0.0328751
$141$		0			0
$142$		−	9.30587i		−	0.780931i
$143$			23.2695i			1.94590i
$144$		0			0
$145$			0.339774i			0.0282167i
$146$			13.3671i			1.10627i
$147$		0			0
$148$	5.88432			0.483688
$149$	−18.2274			−1.49325			−0.746623	−	0.665247i	$-0.768326\pi$
$149$	−18.2274			−1.49325			−0.746623	+	0.665247i	$0.768326\pi$
$150$		0			0
$151$		−	4.76442i		−	0.387723i	−0.981029	−	0.193861i	$-0.937899\pi$
$151$		−	4.76442i		−	0.387723i	0.981029	−	0.193861i	$-0.0621012\pi$
$152$			2.28223i			0.185113i
$153$		0			0
$154$			15.3318i			1.23548i
$155$	−0.0702238			−0.00564051
$156$		0			0
$157$			18.3897i			1.46766i	0.679334	+	0.733830i	$0.262269\pi$
$157$			18.3897i			1.46766i	−0.679334	+	0.733830i	$0.737731\pi$
$158$	17.2377			1.37136
$159$		0			0
$160$			0.125404i			0.00991408i
$161$	−16.9807			−1.33827
$162$		0			0
$163$		−	2.88236i		−	0.225764i	−0.993608	−	0.112882i	$-0.963992\pi$
$163$		−	2.88236i		−	0.225764i	0.993608	−	0.112882i	$-0.0360082\pi$
$164$		−	11.5748i		−	0.903836i
$165$		0			0
$166$	3.82366			0.296774
$167$	−14.8630			−1.15013			−0.575067	−	0.818106i	$-0.695024\pi$
$167$	−14.8630			−1.15013			−0.575067	+	0.818106i	$0.695024\pi$
$168$		0			0
$169$	−9.16277			−0.704829
$170$	−0.327387			−0.0251094
$171$		0			0
$172$	−3.19781			−0.243830
$173$	13.3207			1.01276			0.506379	−	0.862311i	$-0.330984\pi$
$173$	13.3207			1.01276			0.506379	+	0.862311i	$0.330984\pi$
$174$		0			0
$175$	−15.4604			−1.16870
$176$	4.94283			0.372580
$177$		0			0
$178$	−10.2585			−0.768909
$179$		−	6.75787i		−	0.505107i	−0.967583	−	0.252553i	$-0.918730\pi$
$179$		−	6.75787i		−	0.505107i	0.967583	−	0.252553i	$-0.0812703\pi$
$180$		0			0
$181$	−2.89344			−0.215068			−0.107534	−	0.994201i	$-0.534295\pi$
$181$	−2.89344			−0.215068			−0.107534	+	0.994201i	$0.534295\pi$
$182$	−14.6026			−1.08242
$183$		0			0
$184$			5.47440i			0.403578i
$185$	−0.737920			−0.0542529
$186$		0			0
$187$			12.9040i			0.943632i
$188$		−	2.06882i		−	0.150884i
$189$		0			0
$190$		−	0.286201i		−	0.0207632i
$191$	2.74524			0.198639			0.0993193	−	0.995056i	$-0.468333\pi$
$191$	2.74524			0.198639			0.0993193	+	0.995056i	$0.468333\pi$
$192$		0			0
$193$			6.48195i			0.466581i	0.972407	+	0.233291i	$0.0749493\pi$
$193$			6.48195i			0.466581i	−0.972407	+	0.233291i	$0.925051\pi$
$194$	−3.12236			−0.224172
$195$		0			0
$196$	−2.62139			−0.187242
$197$			19.6838i			1.40241i	0.712957	+	0.701207i	$0.247355\pi$
$197$			19.6838i			1.40241i	−0.712957	+	0.701207i	$0.752645\pi$
$198$		0			0
$199$	15.2324			1.07980			0.539899	−	0.841730i	$-0.318462\pi$
$199$	15.2324			1.07980			0.539899	+	0.841730i	$0.318462\pi$
$200$			4.98427i			0.352441i
$201$		0			0
$202$	−2.14378			−0.150836
$203$			8.40421i			0.589860i
$204$		0			0
$205$			1.45152i			0.101379i
$206$	9.20787			0.641543
$207$		0			0
$208$			4.70774i			0.326423i
$209$	−11.2806			−0.780299
$210$		0			0
$211$	−22.5654			−1.55347			−0.776733	−	0.629830i	$-0.783124\pi$
$211$	−22.5654			−1.55347			−0.776733	+	0.629830i	$0.783124\pi$
$212$		−	7.30917i		−	0.501996i
$213$		0			0
$214$			7.88792i			0.539207i
$215$	0.401019			0.0273493
$216$		0			0
$217$	−1.73696			−0.117913
$218$			2.09553i			0.141927i
$219$		0			0
$220$	−0.619852			−0.0417904
$221$	−12.2902			−0.826731
$222$		0			0
$223$	14.6458	+	2.91580i	0.980752	+	0.195256i
$223$	14.6458	+	2.91580i	0.980752	+	0.195256i
$224$			3.10184i			0.207250i
$225$		0			0
$226$		−	7.37996i		−	0.490907i
$227$		−	20.7381i		−	1.37644i	−0.725504	−	0.688218i	$-0.758393\pi$
$227$		−	20.7381i		−	1.37644i	0.725504	−	0.688218i	$-0.241607\pi$
$228$		0			0
$229$		−	13.5139i		−	0.893027i	−0.894777	−	0.446513i	$-0.852666\pi$
$229$		−	13.5139i		−	0.893027i	0.894777	−	0.446513i	$-0.147334\pi$
$230$		−	0.686514i		−	0.0452674i
$231$		0			0
$232$	2.70943			0.177883
$233$	22.8439			1.49655			0.748275	−	0.663388i	$-0.230882\pi$
$233$	22.8439			1.49655			0.748275	+	0.663388i	$0.230882\pi$
$234$		0			0
$235$			0.259439i			0.0169239i
$236$	−14.2791			−0.929492
$237$		0			0
$238$	−8.09780			−0.524902
$239$		−	0.990281i		−	0.0640559i	−0.999487	−	0.0320280i	$-0.989803\pi$
$239$		−	0.990281i		−	0.0640559i	0.999487	−	0.0320280i	$-0.0101966\pi$
$240$		0			0
$241$	−28.4993			−1.83580			−0.917902	−	0.396808i	$-0.870118\pi$
$241$	−28.4993			−1.83580			−0.917902	+	0.396808i	$0.870118\pi$
$242$			13.4315i			0.863412i
$243$		0			0
$244$			1.60819i			0.102954i
$245$	0.328734			0.0210020
$246$		0			0
$247$		−	10.7441i		−	0.683632i
$248$			0.559979i			0.0355587i
$249$		0			0
$250$		−	1.25207i		−	0.0791879i
$251$			16.9668i			1.07093i	0.844557	+	0.535466i	$0.179864\pi$
$251$			16.9668i			1.07093i	−0.844557	+	0.535466i	$0.820136\pi$
$252$		0			0
$253$	−27.0590			−1.70119
$254$		−	14.5511i		−	0.913017i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	21.4477i		−	1.33787i	−0.743319	−	0.668937i	$-0.766750\pi$
$257$		−	21.4477i		−	1.33787i	0.743319	−	0.668937i	$-0.233250\pi$
$258$		0			0
$259$	−18.2522			−1.13414
$260$		−	0.590371i		−	0.0366132i
$261$		0			0
$262$	−22.3830			−1.38283
$263$	31.3943			1.93585			0.967926	−	0.251234i	$-0.0808365\pi$
$263$	31.3943			1.93585			0.967926	+	0.251234i	$0.0808365\pi$
$264$		0			0
$265$			0.916602i			0.0563064i
$266$		−	7.07909i		−	0.434047i
$267$		0			0
$268$			1.97287i			0.120512i
$269$	16.3702			0.998111			0.499056	−	0.866570i	$-0.333680\pi$
$269$	16.3702			0.998111			0.499056	+	0.866570i	$0.333680\pi$
$270$		0			0
$271$		−	28.7931i		−	1.74906i	−0.484973	−	0.874529i	$-0.661171\pi$
$271$		−	28.7931i		−	1.74906i	0.484973	−	0.874529i	$-0.338829\pi$
$272$			2.61065i			0.158294i
$273$		0			0
$274$			8.77797i			0.530297i
$275$	−24.6364			−1.48563
$276$		0			0
$277$			7.78515i			0.467764i	0.972265	+	0.233882i	$0.0751430\pi$
$277$			7.78515i			0.467764i	−0.972265	+	0.233882i	$0.924857\pi$
$278$			4.07568i			0.244443i
$279$		0			0
$280$		−	0.388984i		−	0.0232462i
$281$			11.4289i			0.681792i	0.940101	+	0.340896i	$0.110730\pi$
$281$			11.4289i			0.681792i	−0.940101	+	0.340896i	$0.889270\pi$
$282$		0			0
$283$	−11.0055			−0.654211			−0.327105	−	0.944988i	$-0.606073\pi$
$283$	−11.0055			−0.654211			−0.327105	+	0.944988i	$0.606073\pi$
$284$	9.30587			0.552202
$285$		0			0
$286$	−23.2695			−1.37596
$287$			35.9030i			2.11929i
$288$		0			0
$289$	10.1845			0.599090
$290$	−0.339774			−0.0199522
$291$		0			0
$292$	−13.3671			−0.782251
$293$	17.8128			1.04063			0.520317	−	0.853973i	$-0.325814\pi$
$293$	17.8128			1.04063			0.520317	+	0.853973i	$0.325814\pi$
$294$		0			0
$295$	1.79066			0.104257
$296$			5.88432i			0.342019i
$297$		0			0
$298$		−	18.2274i		−	1.05588i
$299$		−	25.7720i		−	1.49044i
$300$		0			0
$301$	9.91907			0.571726
$302$	4.76442			0.274161
$303$		0			0
$304$	−2.28223			−0.130895
$305$		−	0.201674i		−	0.0115478i
$306$		0			0
$307$		−	1.99746i		−	0.114001i	−0.998374	−	0.0570005i	$-0.981846\pi$
$307$		−	1.99746i		−	0.114001i	0.998374	−	0.0570005i	$-0.0181537\pi$
$308$	−15.3318			−0.873613
$309$		0			0
$310$		−	0.0702238i		−	0.00398844i
$311$	7.90499			0.448251			0.224126	−	0.974560i	$-0.428047\pi$
$311$	7.90499			0.448251			0.224126	+	0.974560i	$0.428047\pi$
$312$		0			0
$313$			11.1569i			0.630625i	0.948988	+	0.315313i	$0.102109\pi$
$313$			11.1569i			0.630625i	−0.948988	+	0.315313i	$0.897891\pi$
$314$	−18.3897			−1.03779
$315$		0			0
$316$			17.2377i			0.969698i
$317$		−	6.68185i		−	0.375290i	−0.982237	−	0.187645i	$-0.939915\pi$
$317$		−	6.68185i		−	0.375290i	0.982237	−	0.187645i	$-0.0600854\pi$
$318$		0			0
$319$			13.3922i			0.749822i
$320$	−0.125404			−0.00701032
$321$		0			0
$322$		−	16.9807i		−	0.946298i
$323$		−	5.95809i		−	0.331517i
$324$		0			0
$325$		−	23.4646i		−	1.30158i
$326$	2.88236			0.159639
$327$		0			0
$328$	11.5748			0.639109
$329$			6.41713i			0.353788i
$330$		0			0
$331$			14.9052i			0.819265i	0.912251	+	0.409632i	$0.134343\pi$
$331$			14.9052i			0.819265i	−0.912251	+	0.409632i	$0.865657\pi$
$332$			3.82366i			0.209851i
$333$		0			0
$334$		−	14.8630i		−	0.813267i
$335$		−	0.247406i		−	0.0135172i
$336$		0			0
$337$			32.7719i			1.78520i	0.450848	+	0.892601i	$0.351122\pi$
$337$			32.7719i			1.78520i	−0.450848	+	0.892601i	$0.648878\pi$
$338$		−	9.16277i		−	0.498389i
$339$		0			0
$340$		−	0.327387i		−	0.0177550i
$341$	−2.76788			−0.149889
$342$		0			0
$343$	−13.5817			−0.733345
$344$		−	3.19781i		−	0.172414i
$345$		0			0
$346$			13.3207i			0.716128i
$347$			3.79762i			0.203867i	0.994791	+	0.101933i	$0.0325028\pi$
$347$			3.79762i			0.203867i	−0.994791	+	0.101933i	$0.967497\pi$
$348$		0			0
$349$	13.7085			0.733801			0.366900	−	0.930260i	$-0.380419\pi$
$349$	13.7085			0.733801			0.366900	+	0.930260i	$0.380419\pi$
$350$		−	15.4604i		−	0.826393i
$351$		0			0
$352$			4.94283i			0.263454i
$353$		−	3.71749i		−	0.197862i	−0.995094	−	0.0989310i	$-0.968458\pi$
$353$		−	3.71749i		−	0.197862i	0.995094	−	0.0989310i	$-0.0315423\pi$
$354$		0			0
$355$	−1.16700			−0.0619377
$356$		−	10.2585i		−	0.543700i
$357$		0			0
$358$	6.75787			0.357164
$359$			19.6642i			1.03784i	0.854823	+	0.518920i	$0.173666\pi$
$359$			19.6642i			1.03784i	−0.854823	+	0.518920i	$0.826334\pi$
$360$		0			0
$361$	−13.7914			−0.725865
$362$		−	2.89344i		−	0.152076i
$363$		0			0
$364$		−	14.6026i		−	0.765386i
$365$	1.67629			0.0877412
$366$		0			0
$367$	−22.3878			−1.16864			−0.584318	−	0.811525i	$-0.698638\pi$
$367$	−22.3878			−1.16864			−0.584318	+	0.811525i	$0.698638\pi$
$368$	−5.47440			−0.285373
$369$		0			0
$370$		−	0.737920i		−	0.0383626i
$371$			22.6719i			1.17706i
$372$		0			0
$373$		−	19.9166i		−	1.03124i	−0.856817	−	0.515620i	$-0.827562\pi$
$373$		−	19.9166i		−	1.03124i	0.856817	−	0.515620i	$-0.172438\pi$
$374$	−12.9040			−0.667249
$375$		0			0
$376$	2.06882			0.106691
$377$	−12.7553			−0.656930
$378$		0			0
$379$	14.5118			0.745421			0.372710	−	0.927948i	$-0.378429\pi$
$379$	14.5118			0.745421			0.372710	+	0.927948i	$0.378429\pi$
$380$	0.286201			0.0146818
$381$		0			0
$382$			2.74524i			0.140459i
$383$	7.19390			0.367591			0.183795	−	0.982965i	$-0.441162\pi$
$383$	7.19390			0.367591			0.183795	+	0.982965i	$0.441162\pi$
$384$		0			0
$385$	1.92268			0.0979888
$386$	−6.48195			−0.329923
$387$		0			0
$388$		−	3.12236i		−	0.158514i
$389$		−	3.86026i		−	0.195723i	−0.995200	−	0.0978614i	$-0.968800\pi$
$389$		−	3.86026i		−	0.195723i	0.995200	−	0.0978614i	$-0.0312002\pi$
$390$		0			0
$391$		−	14.2917i		−	0.722764i
$392$		−	2.62139i		−	0.132400i
$393$		0			0
$394$	−19.6838			−0.991657
$395$		−	2.16169i		−	0.108766i
$396$		0			0
$397$			14.1557i			0.710454i	0.934780	+	0.355227i	$0.115596\pi$
$397$			14.1557i			0.710454i	−0.934780	+	0.355227i	$0.884404\pi$
$398$			15.2324i			0.763533i
$399$		0			0
$400$	−4.98427			−0.249214
$401$			9.13138i			0.456000i	0.973661	+	0.228000i	$0.0732185\pi$
$401$			9.13138i			0.456000i	−0.973661	+	0.228000i	$0.926781\pi$
$402$		0			0
$403$		−	2.63623i		−	0.131320i
$404$		−	2.14378i		−	0.106657i
$405$		0			0
$406$	−8.40421			−0.417094
$407$	−29.0852			−1.44170
$408$		0			0
$409$		−	3.35299i		−	0.165795i	−0.996558	−	0.0828974i	$-0.973583\pi$
$409$		−	3.35299i		−	0.165795i	0.996558	−	0.0828974i	$-0.0264174\pi$
$410$	−1.45152			−0.0716857
$411$		0			0
$412$			9.20787i			0.453639i
$413$	44.2915			2.17944
$414$		0			0
$415$		−	0.479504i		−	0.0235379i
$416$	−4.70774			−0.230816
$417$		0			0
$418$		−	11.2806i		−	0.551755i
$419$		−	15.3296i		−	0.748900i	−0.927247	−	0.374450i	$-0.877832\pi$
$419$		−	15.3296i		−	0.748900i	0.927247	−	0.374450i	$-0.122168\pi$
$420$		0			0
$421$		−	21.3157i		−	1.03886i	−0.854512	−	0.519431i	$-0.826144\pi$
$421$		−	21.3157i		−	1.03886i	0.854512	−	0.519431i	$-0.173856\pi$
$422$		−	22.5654i		−	1.09847i
$423$		0			0
$424$	7.30917			0.354965
$425$		−	13.0122i		−	0.631183i
$426$		0			0
$427$		−	4.98835i		−	0.241403i
$428$	−7.88792			−0.381277
$429$		0			0
$430$			0.401019i			0.0193388i
$431$	2.11310			0.101784			0.0508922	−	0.998704i	$-0.483794\pi$
$431$	2.11310			0.101784			0.0508922	+	0.998704i	$0.483794\pi$
$432$		0			0
$433$	20.1543			0.968554			0.484277	−	0.874915i	$-0.339083\pi$
$433$	20.1543			0.968554			0.484277	+	0.874915i	$0.339083\pi$
$434$		−	1.73696i		−	0.0833769i
$435$		0			0
$436$	−2.09553			−0.100358
$437$	12.4938			0.597661
$438$		0			0
$439$		−	6.85559i		−	0.327199i	−0.986527	−	0.163600i	$-0.947689\pi$
$439$		−	6.85559i		−	0.327199i	0.986527	−	0.163600i	$-0.0523106\pi$
$440$		−	0.619852i		−	0.0295503i
$441$		0			0
$442$		−	12.2902i		−	0.584587i
$443$		−	7.04707i		−	0.334816i	−0.985888	−	0.167408i	$-0.946460\pi$
$443$		−	7.04707i		−	0.334816i	0.985888	−	0.167408i	$-0.0535397\pi$
$444$		0			0
$445$			1.28646i			0.0609842i
$446$	−2.91580	+	14.6458i	−0.138067	+	0.693497i
$447$		0			0
$448$	−3.10184			−0.146548
$449$	29.1368			1.37505			0.687526	−	0.726160i	$-0.258697\pi$
$449$	29.1368			1.37505			0.687526	+	0.726160i	$0.258697\pi$
$450$		0			0
$451$			57.2120i			2.69401i
$452$	7.37996			0.347124
$453$		0			0
$454$	20.7381			0.973288
$455$			1.83123i			0.0858495i
$456$		0			0
$457$			29.6622i			1.38754i	0.720196	+	0.693770i	$0.244052\pi$
$457$			29.6622i			1.38754i	−0.720196	+	0.693770i	$0.755948\pi$
$458$	13.5139			0.631465
$459$		0			0
$460$	0.686514			0.0320089
$461$			26.0390i			1.21276i	0.795176	+	0.606379i	$0.207379\pi$
$461$			26.0390i			1.21276i	−0.795176	+	0.606379i	$0.792621\pi$
$462$		0			0
$463$	35.7308			1.66055			0.830275	−	0.557354i	$-0.188184\pi$
$463$	35.7308			1.66055			0.830275	+	0.557354i	$0.188184\pi$
$464$			2.70943i			0.125782i
$465$		0			0
$466$			22.8439i			1.05822i
$467$	34.0680			1.57648			0.788239	−	0.615369i	$-0.210993\pi$
$467$	34.0680			1.57648			0.788239	+	0.615369i	$0.210993\pi$
$468$		0			0
$469$		−	6.11951i		−	0.282573i
$470$	−0.259439			−0.0119670
$471$		0			0
$472$		−	14.2791i		−	0.657250i
$473$	15.8062			0.726770
$474$		0			0
$475$	11.3752			0.521932
$476$		−	8.09780i		−	0.371162i
$477$		0			0
$478$	0.990281			0.0452944
$479$		−	16.2890i		−	0.744266i	−0.928179	−	0.372133i	$-0.878627\pi$
$479$		−	16.2890i		−	0.744266i	0.928179	−	0.372133i	$-0.121373\pi$
$480$		0			0
$481$		−	27.7018i		−	1.26310i
$482$		−	28.4993i		−	1.29811i
$483$		0			0
$484$	−13.4315			−0.610524
$485$			0.391557i			0.0177797i
$486$		0			0
$487$	3.93370			0.178253			0.0891266	−	0.996020i	$-0.471592\pi$
$487$	3.93370			0.178253			0.0891266	+	0.996020i	$0.471592\pi$
$488$	−1.60819			−0.0727994
$489$		0			0
$490$			0.328734i			0.0148507i
$491$	−28.2547			−1.27512			−0.637558	−	0.770402i	$-0.720055\pi$
$491$	−28.2547			−1.27512			−0.637558	+	0.770402i	$0.720055\pi$
$492$		0			0
$493$	−7.07337			−0.318568
$494$	10.7441			0.483401
$495$		0			0
$496$	−0.559979			−0.0251438
$497$	−28.8653			−1.29479
$498$		0			0
$499$	29.1303			1.30405			0.652025	−	0.758197i	$-0.273920\pi$
$499$	29.1303			1.30405			0.652025	+	0.758197i	$0.273920\pi$
$500$	1.25207			0.0559943
$501$		0			0
$502$	−16.9668			−0.757263
$503$	−32.8878			−1.46639			−0.733197	−	0.680016i	$-0.761973\pi$
$503$	−32.8878			−1.46639			−0.733197	+	0.680016i	$0.761973\pi$
$504$		0			0
$505$			0.268839i			0.0119632i
$506$		−	27.0590i		−	1.20292i
$507$		0			0
$508$	14.5511			0.645600
$509$		−	26.6157i		−	1.17972i	−0.807506	−	0.589860i	$-0.799183\pi$
$509$		−	26.6157i		−	1.17972i	0.807506	−	0.589860i	$-0.200817\pi$
$510$		0			0
$511$	41.4626			1.83420
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	21.4477			0.946019
$515$		−	1.15471i		−	0.0508825i
$516$		0			0
$517$			10.2258i			0.449730i
$518$		−	18.2522i		−	0.801956i
$519$		0			0
$520$	0.590371			0.0258895
$521$	−27.1834			−1.19093			−0.595463	−	0.803383i	$-0.703031\pi$
$521$	−27.1834			−1.19093			−0.595463	+	0.803383i	$0.703031\pi$
$522$		0			0
$523$			1.85328i			0.0810382i	0.999179	+	0.0405191i	$0.0129012\pi$
$523$			1.85328i			0.0810382i	−0.999179	+	0.0405191i	$0.987099\pi$
$524$		−	22.3830i		−	0.977808i
$525$		0			0
$526$			31.3943i			1.36885i
$527$		−	1.46191i		−	0.0636817i
$528$		0			0
$529$	6.96909			0.303004
$530$	−0.916602			−0.0398146
$531$		0			0
$532$	7.07909			0.306918
$533$	−54.4909			−2.36026
$534$		0			0
$535$	0.989179			0.0427659
$536$	−1.97287			−0.0852149
$537$		0			0
$538$			16.3702i			0.705771i
$539$	12.9571			0.558101
$540$		0			0
$541$			0.238445i			0.0102515i	0.999987	+	0.00512577i	$0.00163159\pi$
$541$			0.238445i			0.0102515i	−0.999987	+	0.00512577i	$0.998368\pi$
$542$	28.7931			1.23677
$543$		0			0
$544$	−2.61065			−0.111931
$545$	0.262789			0.0112566
$546$		0			0
$547$	31.8393			1.36135			0.680676	−	0.732585i	$-0.261686\pi$
$547$	31.8393			1.36135			0.680676	+	0.732585i	$0.261686\pi$
$548$	−8.77797			−0.374976
$549$		0			0
$550$		−	24.6364i		−	1.05050i
$551$		−	6.18353i		−	0.263427i
$552$		0			0
$553$		−	53.4686i		−	2.27372i
$554$	−7.78515			−0.330759
$555$		0			0
$556$	−4.07568			−0.172847
$557$	−3.66450			−0.155270			−0.0776349	−	0.996982i	$-0.524737\pi$
$557$	−3.66450			−0.155270			−0.0776349	+	0.996982i	$0.524737\pi$
$558$		0			0
$559$			15.0544i			0.636735i
$560$	0.388984			0.0164376
$561$		0			0
$562$	−11.4289			−0.482099
$563$	−15.5605			−0.655798			−0.327899	−	0.944713i	$-0.606341\pi$
$563$	−15.5605			−0.655798			−0.327899	+	0.944713i	$0.606341\pi$
$564$		0			0
$565$	−0.925479			−0.0389352
$566$		−	11.0055i		−	0.462597i
$567$		0			0
$568$			9.30587i			0.390466i
$569$	−34.9138			−1.46366			−0.731832	−	0.681485i	$-0.761335\pi$
$569$	−34.9138			−1.46366			−0.731832	+	0.681485i	$0.761335\pi$
$570$		0			0
$571$			24.2204i			1.01359i	0.862066	+	0.506795i	$0.169170\pi$
$571$			24.2204i			1.01359i	−0.862066	+	0.506795i	$0.830830\pi$
$572$		−	23.2695i		−	0.972948i
$573$		0			0
$574$	−35.9030			−1.49856
$575$	27.2859			1.13790
$576$		0			0
$577$	−18.6272			−0.775459			−0.387730	−	0.921773i	$-0.626741\pi$
$577$	−18.6272			−0.775459			−0.387730	+	0.921773i	$0.626741\pi$
$578$			10.1845i			0.423620i
$579$		0			0
$580$		−	0.339774i		−	0.0141084i
$581$		−	11.8604i		−	0.492051i
$582$		0			0
$583$			36.1280i			1.49627i
$584$		−	13.3671i		−	0.553135i
$585$		0			0
$586$			17.8128i			0.735840i
$587$	27.2066			1.12294			0.561468	−	0.827498i	$-0.310237\pi$
$587$	27.2066			1.12294			0.561468	+	0.827498i	$0.310237\pi$
$588$		0			0
$589$	1.27800			0.0526590
$590$			1.79066i			0.0737205i
$591$		0			0
$592$	−5.88432			−0.241844
$593$	12.2042			0.501165			0.250582	−	0.968095i	$-0.419378\pi$
$593$	12.2042			0.501165			0.250582	+	0.968095i	$0.419378\pi$
$594$		0			0
$595$			1.01550i			0.0416314i
$596$	18.2274			0.746623
$597$		0			0
$598$	25.7720			1.05390
$599$		−	36.5851i		−	1.49483i	−0.664360	−	0.747413i	$-0.731296\pi$
$599$		−	36.5851i		−	1.49483i	0.664360	−	0.747413i	$-0.268704\pi$
$600$		0			0
$601$		−	0.550615i		−	0.0224601i	−0.999937	−	0.0112300i	$-0.996425\pi$
$601$		−	0.550615i		−	0.0224601i	0.999937	−	0.0112300i	$-0.00357470\pi$
$602$			9.91907i			0.404271i
$603$		0			0
$604$			4.76442i			0.193861i
$605$	1.68437			0.0684795
$606$		0			0
$607$			22.1300i			0.898231i	0.893474	+	0.449115i	$0.148261\pi$
$607$			22.1300i			0.898231i	−0.893474	+	0.449115i	$0.851739\pi$
$608$		−	2.28223i		−	0.0925565i
$609$		0			0
$610$	0.201674			0.00816555
$611$	−9.73944			−0.394016
$612$		0			0
$613$		−	31.7818i		−	1.28365i	−0.766849	−	0.641827i	$-0.778177\pi$
$613$		−	31.7818i		−	1.28365i	0.766849	−	0.641827i	$-0.221823\pi$
$614$	1.99746			0.0806109
$615$		0			0
$616$		−	15.3318i		−	0.617738i
$617$		−	32.2082i		−	1.29665i	−0.761363	−	0.648326i	$-0.775469\pi$
$617$		−	32.2082i		−	1.29665i	0.761363	−	0.648326i	$-0.224531\pi$
$618$		0			0
$619$		−	30.5234i		−	1.22684i	−0.789757	−	0.613420i	$-0.789793\pi$
$619$		−	30.5234i		−	1.22684i	0.789757	−	0.613420i	$-0.210207\pi$
$620$	0.0702238			0.00282026
$621$		0			0
$622$			7.90499i			0.316961i
$623$			31.8202i			1.27485i
$624$		0			0
$625$	24.7644			0.990574
$626$	−11.1569			−0.445920
$627$		0			0
$628$		−	18.3897i		−	0.733830i
$629$		−	15.3619i		−	0.612519i
$630$		0			0
$631$		−	27.9902i		−	1.11427i	−0.830421	−	0.557137i	$-0.811900\pi$
$631$		−	27.9902i		−	1.11427i	0.830421	−	0.557137i	$-0.188100\pi$
$632$	−17.2377			−0.685680
$633$		0			0
$634$	6.68185			0.265370
$635$	−1.82477			−0.0724138
$636$		0			0
$637$			12.3408i			0.488961i
$638$	−13.3922			−0.530204
$639$		0			0
$640$		−	0.125404i		−	0.00495704i
$641$	−16.7730			−0.662495			−0.331248	−	0.943544i	$-0.607470\pi$
$641$	−16.7730			−0.662495			−0.331248	+	0.943544i	$0.607470\pi$
$642$		0			0
$643$	−16.1692			−0.637650			−0.318825	−	0.947814i	$-0.603288\pi$
$643$	−16.1692			−0.637650			−0.318825	+	0.947814i	$0.603288\pi$
$644$	16.9807			0.669134
$645$		0			0
$646$	5.95809			0.234418
$647$			18.8203i			0.739901i	0.929051	+	0.369951i	$0.120625\pi$
$647$			18.8203i			0.739901i	−0.929051	+	0.369951i	$0.879375\pi$
$648$		0			0
$649$	70.5792			2.77048
$650$	23.4646			0.920359
$651$		0			0
$652$			2.88236i			0.112882i
$653$	−37.6022			−1.47149			−0.735745	−	0.677259i	$-0.763168\pi$
$653$	−37.6022			−1.47149			−0.735745	+	0.677259i	$0.763168\pi$
$654$		0			0
$655$			2.80693i			0.109676i
$656$			11.5748i			0.451918i
$657$		0			0
$658$	−6.41713			−0.250166
$659$			17.9324i			0.698548i	0.937021	+	0.349274i	$0.113572\pi$
$659$			17.9324i			0.698548i	−0.937021	+	0.349274i	$0.886428\pi$
$660$		0			0
$661$		−	36.0185i		−	1.40096i	−0.713673	−	0.700479i	$-0.752970\pi$
$661$		−	36.0185i		−	1.40096i	0.713673	−	0.700479i	$-0.247030\pi$
$662$	−14.9052			−0.579308
$663$		0			0
$664$	−3.82366			−0.148387
$665$	−0.887749			−0.0344254
$666$		0			0
$667$		−	14.8325i		−	0.574317i
$668$	14.8630			0.575067
$669$		0			0
$670$	0.247406			0.00955814
$671$		−	7.94901i		−	0.306868i
$672$		0			0
$673$	−18.9293			−0.729669			−0.364835	−	0.931072i	$-0.618874\pi$
$673$	−18.9293			−0.729669			−0.364835	+	0.931072i	$0.618874\pi$
$674$	−32.7719			−1.26233
$675$		0			0
$676$	9.16277			0.352414
$677$		−	5.10294i		−	0.196122i	−0.995180	−	0.0980609i	$-0.968736\pi$
$677$		−	5.10294i		−	0.196122i	0.995180	−	0.0980609i	$-0.0312640\pi$
$678$		0			0
$679$			9.68505i			0.371678i
$680$	0.327387			0.0125547
$681$		0			0
$682$		−	2.76788i		−	0.105988i
$683$		−	18.0739i		−	0.691580i	−0.938312	−	0.345790i	$-0.887611\pi$
$683$		−	18.0739i		−	0.691580i	0.938312	−	0.345790i	$-0.112389\pi$
$684$		0			0
$685$	1.10080			0.0420593
$686$		−	13.5817i		−	0.518553i
$687$		0			0
$688$	3.19781			0.121915
$689$	−34.4096			−1.31090
$690$		0			0
$691$		−	14.3107i		−	0.544405i	−0.962240	−	0.272203i	$-0.912248\pi$
$691$		−	14.3107i		−	0.544405i	0.962240	−	0.272203i	$-0.0877521\pi$
$692$	−13.3207			−0.506379
$693$		0			0
$694$	−3.79762			−0.144156
$695$	0.511108			0.0193874
$696$		0			0
$697$	−30.2176			−1.14457
$698$			13.7085i			0.518875i
$699$		0			0
$700$	15.4604			0.584348
$701$		−	26.6108i		−	1.00508i	−0.864555	−	0.502539i	$-0.832399\pi$
$701$		−	26.6108i		−	1.00508i	0.864555	−	0.502539i	$-0.167601\pi$
$702$		0			0
$703$	13.4294			0.506498
$704$	−4.94283			−0.186290
$705$		0			0
$706$	3.71749			0.139910
$707$			6.64964i			0.250086i
$708$		0			0
$709$			4.37219i			0.164201i	0.996624	+	0.0821005i	$0.0261628\pi$
$709$			4.37219i			0.164201i	−0.996624	+	0.0821005i	$0.973837\pi$
$710$		−	1.16700i		−	0.0437966i
$711$		0			0
$712$	10.2585			0.384454
$713$	3.06555			0.114806
$714$		0			0
$715$			2.91810i			0.109131i
$716$			6.75787i			0.252553i
$717$		0			0
$718$	−19.6642			−0.733863
$719$			51.2998i			1.91316i	0.291470	+	0.956580i	$0.405856\pi$
$719$			51.2998i			1.91316i	−0.291470	+	0.956580i	$0.594144\pi$
$720$		0			0
$721$		−	28.5613i		−	1.06368i
$722$		−	13.7914i		−	0.513264i
$723$		0			0
$724$	2.89344			0.107534
$725$		−	13.5045i		−	0.501546i
$726$		0			0
$727$	41.9387			1.55542			0.777710	−	0.628624i	$-0.216381\pi$
$727$	41.9387			1.55542			0.777710	+	0.628624i	$0.216381\pi$
$728$	14.6026			0.541209
$729$		0			0
$730$			1.67629i			0.0620424i
$731$			8.34835i			0.308775i
$732$		0			0
$733$	12.6547			0.467410			0.233705	−	0.972308i	$-0.424915\pi$
$733$	12.6547			0.467410			0.233705	+	0.972308i	$0.424915\pi$
$734$		−	22.3878i		−	0.826350i
$735$		0			0
$736$		−	5.47440i		−	0.201789i
$737$		−	9.75154i		−	0.359203i
$738$		0			0
$739$			17.7683i			0.653619i	0.945090	+	0.326809i	$0.105973\pi$
$739$			17.7683i			0.653619i	−0.945090	+	0.326809i	$0.894027\pi$
$740$	0.737920			0.0271265
$741$		0			0
$742$	−22.6719			−0.832310
$743$			27.9044i			1.02371i	0.859071	+	0.511857i	$0.171042\pi$
$743$			27.9044i			1.02371i	−0.859071	+	0.511857i	$0.828958\pi$
$744$		0			0
$745$	−2.28580			−0.0837451
$746$	19.9166			0.729197
$747$		0			0
$748$		−	12.9040i		−	0.471816i
$749$	24.4670			0.894006
$750$		0			0
$751$	37.0445			1.35177			0.675886	−	0.737006i	$-0.263761\pi$
$751$	37.0445			1.35177			0.675886	+	0.737006i	$0.263761\pi$
$752$			2.06882i			0.0754420i
$753$		0			0
$754$		−	12.7553i		−	0.464520i
$755$		−	0.597479i		−	0.0217445i
$756$		0			0
$757$		−	30.3191i		−	1.10196i	−0.834517	−	0.550982i	$-0.814253\pi$
$757$		−	30.3191i		−	1.10196i	0.834517	−	0.550982i	$-0.185747\pi$
$758$			14.5118i			0.527092i
$759$		0			0
$760$			0.286201i			0.0103816i
$761$	34.1243			1.23700			0.618502	−	0.785783i	$-0.287740\pi$
$761$	34.1243			1.23700			0.618502	+	0.785783i	$0.287740\pi$
$762$		0			0
$763$	6.50000			0.235316
$764$	−2.74524			−0.0993193
$765$		0			0
$766$			7.19390i			0.259926i
$767$			67.2223i			2.42726i
$768$		0			0
$769$	−8.33070			−0.300413			−0.150206	−	0.988655i	$-0.547994\pi$
$769$	−8.33070			−0.300413			−0.150206	+	0.988655i	$0.547994\pi$
$770$			1.92268i			0.0692886i
$771$		0			0
$772$		−	6.48195i		−	0.233291i
$773$	32.3077			1.16203			0.581013	−	0.813894i	$-0.302656\pi$
$773$	32.3077			1.16203			0.581013	+	0.813894i	$0.302656\pi$
$774$		0			0
$775$	2.79109			0.100259
$776$	3.12236			0.112086
$777$		0			0
$778$	3.86026			0.138397
$779$		−	26.4162i		−	0.946459i
$780$		0			0
$781$	−45.9973			−1.64591
$782$	14.2917			0.511071
$783$		0			0
$784$	2.62139			0.0936211
$785$			2.30615i			0.0823100i
$786$		0			0
$787$			34.0079i			1.21225i	0.795369	+	0.606126i	$0.207277\pi$
$787$			34.0079i			1.21225i	−0.795369	+	0.606126i	$0.792723\pi$
$788$		−	19.6838i		−	0.701207i
$789$		0			0
$790$	2.16169			0.0769093
$791$	−22.8914			−0.813925
$792$		0			0
$793$	7.57094			0.268852
$794$	−14.1557			−0.502367
$795$		0			0
$796$	−15.2324			−0.539899
$797$		−	21.2027i		−	0.751040i	−0.926814	−	0.375520i	$-0.877464\pi$
$797$		−	21.2027i		−	0.751040i	0.926814	−	0.375520i	$-0.122536\pi$
$798$		0			0
$799$	−5.40095			−0.191072
$800$		−	4.98427i		−	0.176221i
$801$		0			0
$802$	−9.13138			−0.322440
$803$	66.0713			2.33161
$804$		0			0
$805$	−2.12945			−0.0750534
$806$	2.63623			0.0928574
$807$		0			0
$808$	2.14378			0.0754178
$809$	−4.45681			−0.156693			−0.0783466	−	0.996926i	$-0.524964\pi$
$809$	−4.45681			−0.156693			−0.0783466	+	0.996926i	$0.524964\pi$
$810$		0			0
$811$		−	15.2843i		−	0.536703i	−0.963321	−	0.268351i	$-0.913521\pi$
$811$		−	15.2843i		−	0.536703i	0.963321	−	0.268351i	$-0.0864788\pi$
$812$		−	8.40421i		−	0.294930i
$813$		0			0
$814$		−	29.0852i		−	1.01944i
$815$		−	0.361461i		−	0.0126614i
$816$		0			0
$817$	−7.29812			−0.255329
$818$	3.35299			0.117235
$819$		0			0
$820$		−	1.45152i		−	0.0506894i
$821$		−	23.7776i		−	0.829844i	−0.909857	−	0.414922i	$-0.863809\pi$
$821$		−	23.7776i		−	0.829844i	0.909857	−	0.414922i	$-0.136191\pi$
$822$		0			0
$823$		−	38.4474i		−	1.34019i	−0.742275	−	0.670096i	$-0.766253\pi$
$823$		−	38.4474i		−	1.34019i	0.742275	−	0.670096i	$-0.233747\pi$
$824$	−9.20787			−0.320771
$825$		0			0
$826$			44.2915i			1.54110i
$827$	15.4051			0.535687			0.267844	−	0.963462i	$-0.413689\pi$
$827$	15.4051			0.535687			0.267844	+	0.963462i	$0.413689\pi$
$828$		0			0
$829$			13.3737i			0.464487i	0.972658	+	0.232244i	$0.0746066\pi$
$829$			13.3737i			0.464487i	−0.972658	+	0.232244i	$0.925393\pi$
$830$	0.479504			0.0166438
$831$		0			0
$832$		−	4.70774i		−	0.163211i
$833$			6.84352i			0.237114i
$834$		0			0
$835$	−1.86388			−0.0645024
$836$	11.2806			0.390149
$837$		0			0
$838$	15.3296			0.529552
$839$	18.0560			0.623362			0.311681	−	0.950187i	$-0.399108\pi$
$839$	18.0560			0.623362			0.311681	+	0.950187i	$0.399108\pi$
$840$		0			0
$841$	21.6590			0.746862
$842$	21.3157			0.734587
$843$		0			0
$844$	22.5654			0.776733
$845$	−1.14905			−0.0395286
$846$		0			0
$847$	41.6624			1.43154
$848$			7.30917i			0.250998i
$849$		0			0
$850$	13.0122			0.446314
$851$	32.2132			1.10425
$852$		0			0
$853$			30.1073i			1.03086i	0.856933	+	0.515428i	$0.172367\pi$
$853$			30.1073i			1.03086i	−0.856933	+	0.515428i	$0.827633\pi$
$854$	4.98835			0.170698
$855$		0			0
$856$		−	7.88792i		−	0.269603i
$857$		−	6.32662i		−	0.216113i	−0.994145	−	0.108057i	$-0.965537\pi$
$857$		−	6.32662i		−	0.216113i	0.994145	−	0.108057i	$-0.0344628\pi$
$858$		0			0
$859$		−	43.3903i		−	1.48046i	−0.672354	−	0.740230i	$-0.734717\pi$
$859$		−	43.3903i		−	1.48046i	0.672354	−	0.740230i	$-0.265283\pi$
$860$	−0.401019			−0.0136746
$861$		0			0
$862$			2.11310i			0.0719724i
$863$	−31.6544			−1.07753			−0.538765	−	0.842456i	$-0.681109\pi$
$863$	−31.6544			−1.07753			−0.538765	+	0.842456i	$0.681109\pi$
$864$		0			0
$865$	1.67048			0.0567980
$866$			20.1543i			0.684871i
$867$		0			0
$868$	1.73696			0.0589564
$869$		−	85.2031i		−	2.89032i
$870$		0			0
$871$	9.28774			0.314703
$872$		−	2.09553i		−	0.0709637i
$873$		0			0
$874$			12.4938i			0.422610i
$875$	−3.88372			−0.131294
$876$		0			0
$877$			0.359854i			0.0121514i	0.999982	+	0.00607571i	$0.00193397\pi$
$877$			0.359854i			0.0121514i	−0.999982	+	0.00607571i	$0.998066\pi$
$878$	6.85559			0.231365
$879$		0			0
$880$	0.619852			0.0208952
$881$		−	16.3787i		−	0.551811i	−0.961185	−	0.275906i	$-0.911022\pi$
$881$		−	16.3787i		−	0.551811i	0.961185	−	0.275906i	$-0.0889777\pi$
$882$		0			0
$883$		−	21.3770i		−	0.719392i	−0.933070	−	0.359696i	$-0.882880\pi$
$883$		−	21.3770i		−	0.719392i	0.933070	−	0.359696i	$-0.117120\pi$
$884$	12.2902			0.413365
$885$		0			0
$886$	7.04707			0.236751
$887$			35.4632i			1.19074i	0.803453	+	0.595369i	$0.202994\pi$
$887$			35.4632i			1.19074i	−0.803453	+	0.595369i	$0.797006\pi$
$888$		0			0
$889$	−45.1351			−1.51378
$890$	−1.28646			−0.0431223
$891$		0			0
$892$	−14.6458	−	2.91580i	−0.490376	−	0.0976281i
$893$		−	4.72151i		−	0.157999i
$894$		0			0
$895$		−	0.847466i		−	0.0283277i
$896$		−	3.10184i		−	0.103625i
$897$		0			0
$898$			29.1368i			0.972309i
$899$		−	1.51722i		−	0.0506023i
$900$		0			0
$901$	−19.0817			−0.635703
$902$	−57.2120			−1.90495
$903$		0			0
$904$			7.37996i			0.245454i
$905$	−0.362850			−0.0120615
$906$		0			0
$907$	4.83863			0.160664			0.0803321	−	0.996768i	$-0.474402\pi$
$907$	4.83863			0.160664			0.0803321	+	0.996768i	$0.474402\pi$
$908$			20.7381i			0.688218i
$909$		0			0
$910$	−1.83123			−0.0607048
$911$			26.9069i			0.891465i	0.895166	+	0.445732i	$0.147057\pi$
$911$			26.9069i			0.891465i	−0.895166	+	0.445732i	$0.852943\pi$
$912$		0			0
$913$		−	18.8997i		−	0.625489i
$914$	−29.6622			−0.981139
$915$		0			0
$916$			13.5139i			0.446513i
$917$			69.4285i			2.29273i
$918$		0			0
$919$			29.5449i			0.974595i	0.873236	+	0.487297i	$0.162017\pi$
$919$			29.5449i			0.974595i	−0.873236	+	0.487297i	$0.837983\pi$
$920$			0.686514i			0.0226337i
$921$		0			0
$922$	−26.0390			−0.857550
$923$		−	43.8096i		−	1.44201i
$924$		0			0
$925$	29.3291			0.964334
$926$			35.7308i			1.17419i
$927$		0			0
$928$	−2.70943			−0.0889414
$929$			16.1767i			0.530741i	0.964146	+	0.265371i	$0.0854943\pi$
$929$			16.1767i			0.530741i	−0.964146	+	0.265371i	$0.914506\pi$
$930$		0			0
$931$	−5.98261			−0.196072
$932$	−22.8439			−0.748275
$933$		0			0
$934$			34.0680i			1.11474i
$935$			1.61821i			0.0529213i
$936$		0			0
$937$			3.09356i			0.101062i	0.998722	+	0.0505311i	$0.0160914\pi$
$937$			3.09356i			0.101062i	−0.998722	+	0.0505311i	$0.983909\pi$
$938$	6.11951			0.199809
$939$		0			0
$940$		−	0.259439i		−	0.00846196i
$941$		−	25.1151i		−	0.818727i	−0.912371	−	0.409364i	$-0.865751\pi$
$941$		−	25.1151i		−	0.818727i	0.912371	−	0.409364i	$-0.134249\pi$
$942$		0			0
$943$		−	63.3649i		−	2.06344i
$944$	14.2791			0.464746
$945$		0			0
$946$			15.8062i			0.513904i
$947$			26.8238i			0.871657i	0.900030	+	0.435828i	$0.143545\pi$
$947$			26.8238i			0.871657i	−0.900030	+	0.435828i	$0.856455\pi$
$948$		0			0
$949$			62.9288i			2.04276i
$950$			11.3752i			0.369062i
$951$		0			0
$952$	8.09780			0.262451
$953$	26.3835			0.854645			0.427323	−	0.904099i	$-0.359457\pi$
$953$	26.3835			0.854645			0.427323	+	0.904099i	$0.359457\pi$
$954$		0			0
$955$	0.344265			0.0111402
$956$			0.990281i			0.0320280i
$957$		0			0
$958$	16.2890			0.526275
$959$	27.2278			0.879233
$960$		0			0
$961$	−30.6864			−0.989885
$962$	27.7018			0.893143
$963$		0			0
$964$	28.4993			0.917902
$965$			0.812865i			0.0261671i
$966$		0			0
$967$			40.0050i			1.28648i	0.765667	+	0.643238i	$0.222409\pi$
$967$			40.0050i			1.28648i	−0.765667	+	0.643238i	$0.777591\pi$
$968$		−	13.4315i		−	0.431706i
$969$		0			0
$970$	−0.391557			−0.0125722
$971$	−12.3094			−0.395027			−0.197514	−	0.980300i	$-0.563287\pi$
$971$	−12.3094			−0.395027			−0.197514	+	0.980300i	$0.563287\pi$
$972$		0			0
$973$	12.6421			0.405287
$974$			3.93370i			0.126044i
$975$		0			0
$976$		−	1.60819i		−	0.0514769i
$977$	28.4520			0.910259			0.455130	−	0.890425i	$-0.349593\pi$
$977$	28.4520			0.910259			0.455130	+	0.890425i	$0.349593\pi$
$978$		0			0
$979$			50.7061i			1.62057i
$980$	−0.328734			−0.0105010
$981$		0			0
$982$		−	28.2547i		−	0.901643i
$983$	−10.8522			−0.346132			−0.173066	−	0.984910i	$-0.555367\pi$
$983$	−10.8522			−0.346132			−0.173066	+	0.984910i	$0.555367\pi$
$984$		0			0
$985$			2.46844i			0.0786510i
$986$		−	7.07337i		−	0.225262i
$987$		0			0
$988$			10.7441i			0.341816i
$989$	−17.5061			−0.556661
$990$		0			0
$991$			19.2863i			0.612651i	0.951927	+	0.306325i	$0.0990996\pi$
$991$			19.2863i			0.612651i	−0.951927	+	0.306325i	$0.900900\pi$
$992$		−	0.559979i		−	0.0177794i
$993$		0			0
$994$		−	28.8653i		−	0.915551i
$995$	1.91021			0.0605579
$996$		0			0
$997$	−22.0400			−0.698014			−0.349007	−	0.937120i	$-0.613481\pi$
$997$	−22.0400			−0.698014			−0.349007	+	0.937120i	$0.613481\pi$
$998$			29.1303i			0.922103i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.8	yes	72
3.2	odd	2	inner	4014.2.d.a.4013.27	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.28	yes	72
669.668	even	2	inner	4014.2.d.a.4013.7	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.7	✓	72	669.668	even	2	inner
4014.2.d.a.4013.8	yes	72	1.1	even	1	trivial
4014.2.d.a.4013.27	yes	72	3.2	odd	2	inner
4014.2.d.a.4013.28	yes	72	223.222	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.125404 q^{5} +3.10184 q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.125404 q^{5} +3.10184 q^{7} -1.00000i q^{8} +0.125404i q^{10} +4.94283 q^{11} +4.70774i q^{13} +3.10184i q^{14} +1.00000 q^{16} +2.61065i q^{17} -2.28223 q^{19} -0.125404 q^{20} +4.94283i q^{22} -5.47440 q^{23} -4.98427 q^{25} -4.70774 q^{26} -3.10184 q^{28} +2.70943i q^{29} -0.559979 q^{31} +1.00000i q^{32} -2.61065 q^{34} +0.388984 q^{35} -5.88432 q^{37} -2.28223i q^{38} -0.125404i q^{40} +11.5748i q^{41} +3.19781 q^{43} -4.94283 q^{44} -5.47440i q^{46} +2.06882i q^{47} +2.62139 q^{49} -4.98427i q^{50} -4.70774i q^{52} +7.30917i q^{53} +0.619852 q^{55} -3.10184i q^{56} -2.70943 q^{58} +14.2791 q^{59} -1.60819i q^{61} -0.559979i q^{62} -1.00000 q^{64} +0.590371i q^{65} -1.97287i q^{67} -2.61065i q^{68} +0.388984i q^{70} -9.30587 q^{71} +13.3671 q^{73} -5.88432i q^{74} +2.28223 q^{76} +15.3318 q^{77} -17.2377i q^{79} +0.125404 q^{80} -11.5748 q^{82} -3.82366i q^{83} +0.327387i q^{85} +3.19781i q^{86} -4.94283i q^{88} +10.2585i q^{89} +14.6026i q^{91} +5.47440 q^{92} -2.06882 q^{94} -0.286201 q^{95} +3.12236i q^{97} +2.62139i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 72 * q - 72 * q^4 + 16 * q^7	\( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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