LMFDB - Embedded newform 4050.2.c.z.649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(649,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4050.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 810)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.z.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{7} -1.00000i q^{8} +3.46410 q^{11} +3.73205i q^{13} +3.73205 q^{14} +1.00000 q^{16} +3.46410i q^{17} +7.92820 q^{19} +3.46410i q^{22} -0.464102i q^{23} -3.73205 q^{26} +3.73205i q^{28} -6.92820 q^{29} +5.46410 q^{31} +1.00000i q^{32} -3.46410 q^{34} -8.00000i q^{37} +7.92820i q^{38} -5.19615 q^{41} -1.46410i q^{43} -3.46410 q^{44} +0.464102 q^{46} -6.46410i q^{47} -6.92820 q^{49} -3.73205i q^{52} +12.4641i q^{53} -3.73205 q^{56} -6.92820i q^{58} +8.66025 q^{59} -8.39230 q^{61} +5.46410i q^{62} -1.00000 q^{64} -8.00000i q^{67} -3.46410i q^{68} +6.00000 q^{71} +6.39230i q^{73} +8.00000 q^{74} -7.92820 q^{76} -12.9282i q^{77} -6.39230 q^{79} -5.19615i q^{82} +8.53590i q^{83} +1.46410 q^{86} -3.46410i q^{88} +12.0000 q^{89} +13.9282 q^{91} +0.464102i q^{92} +6.46410 q^{94} -1.07180i q^{97} -6.92820i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{14} + 4 q^{16} + 4 q^{19} - 8 q^{26} + 8 q^{31} - 12 q^{46} - 8 q^{56} + 8 q^{61} - 4 q^{64} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 16 q^{79} - 8 q^{86} + 48 q^{89} + 28 q^{91} + 12 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^14 + 4 * q^16 + 4 * q^19 - 8 * q^26 + 8 * q^31 - 12 * q^46 - 8 * q^56 + 8 * q^61 - 4 * q^64 + 24 * q^71 + 32 * q^74 - 4 * q^76 + 16 * q^79 - 8 * q^86 + 48 * q^89 + 28 * q^91 + 12 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$.

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

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	0
$5$	0	0
$6$	0	0
$7$	− 3.73205i	− 1.41058i	−0.708918	−	0.705291i	$-0.750816\pi$
$7$	− 3.73205i	− 1.41058i	0.708918	−	0.705291i	$-0.249184\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	3.73205i	1.03508i	0.855658	+	0.517542i	$0.173153\pi$
$13$	3.73205i	1.03508i	−0.855658	+	0.517542i	$0.826847\pi$
$14$	3.73205	0.997433
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$	3.46410i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$	0	0
$19$	7.92820	1.81885	0.909427	−	0.415863i	$-0.136520\pi$
$19$	7.92820	1.81885	0.909427	+	0.415863i	$0.136520\pi$
$20$	0	0
$21$	0	0
$22$	3.46410i	0.738549i
$23$	− 0.464102i	− 0.0967719i	−0.998829	−	0.0483859i	$-0.984592\pi$
$23$	− 0.464102i	− 0.0967719i	0.998829	−	0.0483859i	$-0.0154077\pi$
$24$	0	0
$25$	0	0
$26$	−3.73205	−0.731915
$27$	0	0
$28$	3.73205i	0.705291i
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−3.46410	−0.594089
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	7.92820i	1.28612i
$39$	0	0
$40$	0	0
$41$	−5.19615	−0.811503	−0.405751	−	0.913984i	$-0.632990\pi$
$41$	−5.19615	−0.811503	−0.405751	+	0.913984i	$0.632990\pi$
$42$	0	0
$43$	− 1.46410i	− 0.223273i	−0.993749	−	0.111637i	$-0.964391\pi$
$43$	− 1.46410i	− 0.223273i	0.993749	−	0.111637i	$-0.0356093\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	0.464102	0.0684280
$47$	− 6.46410i	− 0.942886i	−0.881897	−	0.471443i	$-0.843733\pi$
$47$	− 6.46410i	− 0.942886i	0.881897	−	0.471443i	$-0.156267\pi$
$48$	0	0
$49$	−6.92820	−0.989743
$50$	0	0
$51$	0	0
$52$	− 3.73205i	− 0.517542i
$53$	12.4641i	1.71208i	0.516913	+	0.856038i	$0.327081\pi$
$53$	12.4641i	1.71208i	−0.516913	+	0.856038i	$0.672919\pi$
$54$	0	0
$55$	0	0
$56$	−3.73205	−0.498716
$57$	0	0
$58$	− 6.92820i	− 0.909718i
$59$	8.66025	1.12747	0.563735	−	0.825956i	$-0.309364\pi$
$59$	8.66025	1.12747	0.563735	+	0.825956i	$0.309364\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	5.46410i	0.693942i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 3.46410i	− 0.420084i
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	6.39230i	0.748163i	0.927396	+	0.374081i	$0.122042\pi$
$73$	6.39230i	0.748163i	−0.927396	+	0.374081i	$0.877958\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−7.92820	−0.909427
$77$	− 12.9282i	− 1.47331i
$78$	0	0
$79$	−6.39230	−0.719190	−0.359595	−	0.933108i	$-0.617085\pi$
$79$	−6.39230	−0.719190	−0.359595	+	0.933108i	$0.617085\pi$
$80$	0	0
$81$	0	0
$82$	− 5.19615i	− 0.573819i
$83$	8.53590i	0.936937i	0.883480	+	0.468468i	$0.155194\pi$
$83$	8.53590i	0.936937i	−0.883480	+	0.468468i	$0.844806\pi$
$84$	0	0
$85$	0	0
$86$	1.46410	0.157878
$87$	0	0
$88$	− 3.46410i	− 0.369274i
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	13.9282	1.46007
$92$	0.464102i	0.0483859i
$93$	0	0
$94$	6.46410	0.666721
$95$	0	0
$96$	0	0
$97$	− 1.07180i	− 0.108824i	−0.998519	−	0.0544122i	$-0.982671\pi$
$97$	− 1.07180i	− 0.108824i	0.998519	−	0.0544122i	$-0.0173285\pi$
$98$	− 6.92820i	− 0.699854i
$99$	0	0
$100$	0	0
$101$	9.46410	0.941713	0.470857	−	0.882210i	$-0.343945\pi$
$101$	9.46410	0.941713	0.470857	+	0.882210i	$0.343945\pi$
$102$	0	0
$103$	− 15.1962i	− 1.49732i	−0.662953	−	0.748661i	$-0.730697\pi$
$103$	− 15.1962i	− 1.49732i	0.662953	−	0.748661i	$-0.269303\pi$
$104$	3.73205	0.365958
$105$	0	0
$106$	−12.4641	−1.21062
$107$	− 8.53590i	− 0.825196i	−0.910913	−	0.412598i	$-0.864621\pi$
$107$	− 8.53590i	− 0.825196i	0.910913	−	0.412598i	$-0.135379\pi$
$108$	0	0
$109$	17.8564	1.71033	0.855167	−	0.518353i	$-0.173455\pi$
$109$	17.8564	1.71033	0.855167	+	0.518353i	$0.173455\pi$
$110$	0	0
$111$	0	0
$112$	− 3.73205i	− 0.352646i
$113$	12.9282i	1.21618i	0.793867	+	0.608092i	$0.208065\pi$
$113$	12.9282i	1.21618i	−0.793867	+	0.608092i	$0.791935\pi$
$114$	0	0
$115$	0	0
$116$	6.92820	0.643268
$117$	0	0
$118$	8.66025i	0.797241i
$119$	12.9282	1.18513
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 8.39230i	− 0.759804i
$123$	0	0
$124$	−5.46410	−0.490691
$125$	0	0
$126$	0	0
$127$	15.1962i	1.34844i	0.738530	+	0.674220i	$0.235520\pi$
$127$	15.1962i	1.34844i	−0.738530	+	0.674220i	$0.764480\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	13.7321	1.19977	0.599887	−	0.800084i	$-0.295212\pi$
$131$	13.7321	1.19977	0.599887	+	0.800084i	$0.295212\pi$
$132$	0	0
$133$	− 29.5885i	− 2.56564i
$134$	8.00000	0.691095
$135$	0	0
$136$	3.46410	0.297044
$137$	10.3923i	0.887875i	0.896058	+	0.443937i	$0.146419\pi$
$137$	10.3923i	0.887875i	−0.896058	+	0.443937i	$0.853581\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	12.9282i	1.08111i
$144$	0	0
$145$	0	0
$146$	−6.39230	−0.529031
$147$	0	0
$148$	8.00000i	0.657596i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	18.3923	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$152$	− 7.92820i	− 0.643062i
$153$	0	0
$154$	12.9282	1.04178
$155$	0	0
$156$	0	0
$157$	− 14.1244i	− 1.12725i	−0.826032	−	0.563623i	$-0.809407\pi$
$157$	− 14.1244i	− 1.12725i	0.826032	−	0.563623i	$-0.190593\pi$
$158$	− 6.39230i	− 0.508544i
$159$	0	0
$160$	0	0
$161$	−1.73205	−0.136505
$162$	0	0
$163$	9.85641i	0.772013i	0.922496	+	0.386007i	$0.126146\pi$
$163$	9.85641i	0.772013i	−0.922496	+	0.386007i	$0.873854\pi$
$164$	5.19615	0.405751
$165$	0	0
$166$	−8.53590	−0.662514
$167$	− 6.92820i	− 0.536120i	−0.963402	−	0.268060i	$-0.913617\pi$
$167$	− 6.92820i	− 0.536120i	0.963402	−	0.268060i	$-0.0863826\pi$
$168$	0	0
$169$	−0.928203	−0.0714002
$170$	0	0
$171$	0	0
$172$	1.46410i	0.111637i
$173$	− 20.3205i	− 1.54494i	−0.635051	−	0.772470i	$-0.719021\pi$
$173$	− 20.3205i	− 1.54494i	0.635051	−	0.772470i	$-0.280979\pi$
$174$	0	0
$175$	0	0
$176$	3.46410	0.261116
$177$	0	0
$178$	12.0000i	0.899438i
$179$	−12.1244	−0.906217	−0.453108	−	0.891455i	$-0.649685\pi$
$179$	−12.1244	−0.906217	−0.453108	+	0.891455i	$0.649685\pi$
$180$	0	0
$181$	−12.5359	−0.931786	−0.465893	−	0.884841i	$-0.654267\pi$
$181$	−12.5359	−0.931786	−0.465893	+	0.884841i	$0.654267\pi$
$182$	13.9282i	1.03243i
$183$	0	0
$184$	−0.464102	−0.0342140
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	6.46410i	0.471443i
$189$	0	0
$190$	0	0
$191$	2.53590	0.183491	0.0917456	−	0.995782i	$-0.470755\pi$
$191$	2.53590	0.183491	0.0917456	+	0.995782i	$0.470755\pi$
$192$	0	0
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	1.07180	0.0769505
$195$	0	0
$196$	6.92820	0.494872
$197$	11.5359i	0.821899i	0.911658	+	0.410949i	$0.134803\pi$
$197$	11.5359i	0.821899i	−0.911658	+	0.410949i	$0.865197\pi$
$198$	0	0
$199$	22.2487	1.57717	0.788585	−	0.614926i	$-0.210814\pi$
$199$	22.2487	1.57717	0.788585	+	0.614926i	$0.210814\pi$
$200$	0	0
$201$	0	0
$202$	9.46410i	0.665892i
$203$	25.8564i	1.81476i
$204$	0	0
$205$	0	0
$206$	15.1962	1.05877
$207$	0	0
$208$	3.73205i	0.258771i
$209$	27.4641	1.89973
$210$	0	0
$211$	−8.85641	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$211$	−8.85641	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$212$	− 12.4641i	− 0.856038i
$213$	0	0
$214$	8.53590	0.583502
$215$	0	0
$216$	0	0
$217$	− 20.3923i	− 1.38432i
$218$	17.8564i	1.20939i
$219$	0	0
$220$	0	0
$221$	−12.9282	−0.869645
$222$	0	0
$223$	− 12.5359i	− 0.839466i	−0.907648	−	0.419733i	$-0.862124\pi$
$223$	− 12.5359i	− 0.839466i	0.907648	−	0.419733i	$-0.137876\pi$
$224$	3.73205	0.249358
$225$	0	0
$226$	−12.9282	−0.859971
$227$	− 16.3923i	− 1.08800i	−0.839087	−	0.543998i	$-0.816910\pi$
$227$	− 16.3923i	− 1.08800i	0.839087	−	0.543998i	$-0.183090\pi$
$228$	0	0
$229$	−21.8564	−1.44431	−0.722156	−	0.691730i	$-0.756849\pi$
$229$	−21.8564	−1.44431	−0.722156	+	0.691730i	$0.756849\pi$
$230$	0	0
$231$	0	0
$232$	6.92820i	0.454859i
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	0	0
$236$	−8.66025	−0.563735
$237$	0	0
$238$	12.9282i	0.838011i
$239$	8.53590	0.552141	0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590	0.552141	0.276071	+	0.961137i	$0.410968\pi$
$240$	0	0
$241$	−16.9282	−1.09044	−0.545221	−	0.838293i	$-0.683554\pi$
$241$	−16.9282	−1.09044	−0.545221	+	0.838293i	$0.683554\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	8.39230	0.537262
$245$	0	0
$246$	0	0
$247$	29.5885i	1.88267i
$248$	− 5.46410i	− 0.346971i
$249$	0	0
$250$	0	0
$251$	−12.1244	−0.765283	−0.382641	−	0.923897i	$-0.624985\pi$
$251$	−12.1244	−0.765283	−0.382641	+	0.923897i	$0.624985\pi$
$252$	0	0
$253$	− 1.60770i	− 0.101075i
$254$	−15.1962	−0.953491
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.8564i	1.98715i	0.113184	+	0.993574i	$0.463895\pi$
$257$	31.8564i	1.98715i	−0.113184	+	0.993574i	$0.536105\pi$
$258$	0	0
$259$	−29.8564	−1.85519
$260$	0	0
$261$	0	0
$262$	13.7321i	0.848369i
$263$	− 8.32051i	− 0.513065i	−0.966536	−	0.256532i	$-0.917420\pi$
$263$	− 8.32051i	− 0.513065i	0.966536	−	0.256532i	$-0.0825800\pi$
$264$	0	0
$265$	0	0
$266$	29.5885	1.81418
$267$	0	0
$268$	8.00000i	0.488678i
$269$	12.9282	0.788246	0.394123	−	0.919058i	$-0.371048\pi$
$269$	12.9282	0.788246	0.394123	+	0.919058i	$0.371048\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	3.46410i	0.210042i
$273$	0	0
$274$	−10.3923	−0.627822
$275$	0	0
$276$	0	0
$277$	− 12.2679i	− 0.737110i	−0.929606	−	0.368555i	$-0.879853\pi$
$277$	− 12.2679i	− 0.737110i	0.929606	−	0.368555i	$-0.120147\pi$
$278$	13.0000i	0.779688i
$279$	0	0
$280$	0	0
$281$	7.05256	0.420720	0.210360	−	0.977624i	$-0.432536\pi$
$281$	7.05256	0.420720	0.210360	+	0.977624i	$0.432536\pi$
$282$	0	0
$283$	− 3.32051i	− 0.197384i	−0.995118	−	0.0986919i	$-0.968534\pi$
$283$	− 3.32051i	− 0.197384i	0.995118	−	0.0986919i	$-0.0314658\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−12.9282	−0.764461
$287$	19.3923i	1.14469i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	− 6.39230i	− 0.374081i
$293$	7.39230i	0.431863i	0.976409	+	0.215932i	$0.0692788\pi$
$293$	7.39230i	0.431863i	−0.976409	+	0.215932i	$0.930721\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	18.0000i	1.04271i
$299$	1.73205	0.100167
$300$	0	0
$301$	−5.46410	−0.314946
$302$	18.3923i	1.05836i
$303$	0	0
$304$	7.92820	0.454714
$305$	0	0
$306$	0	0
$307$	− 12.3923i	− 0.707266i	−0.935384	−	0.353633i	$-0.884946\pi$
$307$	− 12.3923i	− 0.707266i	0.935384	−	0.353633i	$-0.115054\pi$
$308$	12.9282i	0.736653i
$309$	0	0
$310$	0	0
$311$	8.53590	0.484026	0.242013	−	0.970273i	$-0.422192\pi$
$311$	8.53590	0.484026	0.242013	+	0.970273i	$0.422192\pi$
$312$	0	0
$313$	− 3.32051i	− 0.187686i	−0.995587	−	0.0938431i	$-0.970085\pi$
$313$	− 3.32051i	− 0.187686i	0.995587	−	0.0938431i	$-0.0299152\pi$
$314$	14.1244	0.797084
$315$	0	0
$316$	6.39230	0.359595
$317$	17.5359i	0.984914i	0.870337	+	0.492457i	$0.163901\pi$
$317$	17.5359i	0.984914i	−0.870337	+	0.492457i	$0.836099\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	− 1.73205i	− 0.0965234i
$323$	27.4641i	1.52814i
$324$	0	0
$325$	0	0
$326$	−9.85641	−0.545896
$327$	0	0
$328$	5.19615i	0.286910i
$329$	−24.1244	−1.33002
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	− 8.53590i	− 0.468468i
$333$	0	0
$334$	6.92820	0.379094
$335$	0	0
$336$	0	0
$337$	− 7.07180i	− 0.385225i	−0.981275	−	0.192613i	$-0.938304\pi$
$337$	− 7.07180i	− 0.385225i	0.981275	−	0.192613i	$-0.0616961\pi$
$338$	− 0.928203i	− 0.0504876i
$339$	0	0
$340$	0	0
$341$	18.9282	1.02502
$342$	0	0
$343$	− 0.267949i	− 0.0144679i
$344$	−1.46410	−0.0789391
$345$	0	0
$346$	20.3205	1.09244
$347$	14.7846i	0.793679i	0.917888	+	0.396840i	$0.129893\pi$
$347$	14.7846i	0.793679i	−0.917888	+	0.396840i	$0.870107\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	3.46410i	0.184637i
$353$	22.3923i	1.19182i	0.803050	+	0.595911i	$0.203209\pi$
$353$	22.3923i	1.19182i	−0.803050	+	0.595911i	$0.796791\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	− 12.1244i	− 0.640792i
$359$	8.53590	0.450507	0.225254	−	0.974300i	$-0.427679\pi$
$359$	8.53590	0.450507	0.225254	+	0.974300i	$0.427679\pi$
$360$	0	0
$361$	43.8564	2.30823
$362$	− 12.5359i	− 0.658872i
$363$	0	0
$364$	−13.9282	−0.730036
$365$	0	0
$366$	0	0
$367$	− 6.39230i	− 0.333676i	−0.985984	−	0.166838i	$-0.946644\pi$
$367$	− 6.39230i	− 0.333676i	0.985984	−	0.166838i	$-0.0533556\pi$
$368$	− 0.464102i	− 0.0241930i
$369$	0	0
$370$	0	0
$371$	46.5167	2.41502
$372$	0	0
$373$	− 10.9282i	− 0.565841i	−0.959143	−	0.282920i	$-0.908697\pi$
$373$	− 10.9282i	− 0.565841i	0.959143	−	0.282920i	$-0.0913032\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	−6.46410	−0.333361
$377$	− 25.8564i	− 1.33167i
$378$	0	0
$379$	−36.8564	−1.89319	−0.946593	−	0.322430i	$-0.895500\pi$
$379$	−36.8564	−1.89319	−0.946593	+	0.322430i	$0.895500\pi$
$380$	0	0
$381$	0	0
$382$	2.53590i	0.129748i
$383$	− 6.46410i	− 0.330300i	−0.986268	−	0.165150i	$-0.947189\pi$
$383$	− 6.46410i	− 0.330300i	0.986268	−	0.165150i	$-0.0528109\pi$
$384$	0	0
$385$	0	0
$386$	16.0000	0.814379
$387$	0	0
$388$	1.07180i	0.0544122i
$389$	−23.3205	−1.18240	−0.591198	−	0.806526i	$-0.701345\pi$
$389$	−23.3205	−1.18240	−0.591198	+	0.806526i	$0.701345\pi$
$390$	0	0
$391$	1.60770	0.0813046
$392$	6.92820i	0.349927i
$393$	0	0
$394$	−11.5359	−0.581170
$395$	0	0
$396$	0	0
$397$	− 9.85641i	− 0.494679i	−0.968929	−	0.247339i	$-0.920444\pi$
$397$	− 9.85641i	− 0.494679i	0.968929	−	0.247339i	$-0.0795563\pi$
$398$	22.2487i	1.11523i
$399$	0	0
$400$	0	0
$401$	−39.5885	−1.97695	−0.988477	−	0.151374i	$-0.951630\pi$
$401$	−39.5885	−1.97695	−0.988477	+	0.151374i	$0.951630\pi$
$402$	0	0
$403$	20.3923i	1.01581i
$404$	−9.46410	−0.470857
$405$	0	0
$406$	−25.8564	−1.28323
$407$	− 27.7128i	− 1.37367i
$408$	0	0
$409$	13.9282	0.688705	0.344353	−	0.938840i	$-0.388098\pi$
$409$	13.9282	0.688705	0.344353	+	0.938840i	$0.388098\pi$
$410$	0	0
$411$	0	0
$412$	15.1962i	0.748661i
$413$	− 32.3205i	− 1.59039i
$414$	0	0
$415$	0	0
$416$	−3.73205	−0.182979
$417$	0	0
$418$	27.4641i	1.34331i
$419$	3.46410	0.169232	0.0846162	−	0.996414i	$-0.473034\pi$
$419$	3.46410	0.169232	0.0846162	+	0.996414i	$0.473034\pi$
$420$	0	0
$421$	9.60770	0.468250	0.234125	−	0.972206i	$-0.424777\pi$
$421$	9.60770	0.468250	0.234125	+	0.972206i	$0.424777\pi$
$422$	− 8.85641i	− 0.431123i
$423$	0	0
$424$	12.4641	0.605310
$425$	0	0
$426$	0	0
$427$	31.3205i	1.51571i
$428$	8.53590i	0.412598i
$429$	0	0
$430$	0	0
$431$	−28.3923	−1.36761	−0.683805	−	0.729665i	$-0.739676\pi$
$431$	−28.3923	−1.36761	−0.683805	+	0.729665i	$0.739676\pi$
$432$	0	0
$433$	2.92820i	0.140720i	0.997522	+	0.0703602i	$0.0224149\pi$
$433$	2.92820i	0.140720i	−0.997522	+	0.0703602i	$0.977585\pi$
$434$	20.3923	0.978862
$435$	0	0
$436$	−17.8564	−0.855167
$437$	− 3.67949i	− 0.176014i
$438$	0	0
$439$	−18.3923	−0.877817	−0.438908	−	0.898532i	$-0.644635\pi$
$439$	−18.3923	−0.877817	−0.438908	+	0.898532i	$0.644635\pi$
$440$	0	0
$441$	0	0
$442$	− 12.9282i	− 0.614932i
$443$	− 31.8564i	− 1.51354i	−0.653679	−	0.756772i	$-0.726775\pi$
$443$	− 31.8564i	− 1.51354i	0.653679	−	0.756772i	$-0.273225\pi$
$444$	0	0
$445$	0	0
$446$	12.5359	0.593592
$447$	0	0
$448$	3.73205i	0.176323i
$449$	32.9090	1.55307	0.776535	−	0.630074i	$-0.216975\pi$
$449$	32.9090	1.55307	0.776535	+	0.630074i	$0.216975\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	− 12.9282i	− 0.608092i
$453$	0	0
$454$	16.3923	0.769329
$455$	0	0
$456$	0	0
$457$	20.3923i	0.953912i	0.878927	+	0.476956i	$0.158260\pi$
$457$	20.3923i	0.953912i	−0.878927	+	0.476956i	$0.841740\pi$
$458$	− 21.8564i	− 1.02128i
$459$	0	0
$460$	0	0
$461$	8.78461	0.409140	0.204570	−	0.978852i	$-0.434420\pi$
$461$	8.78461	0.409140	0.204570	+	0.978852i	$0.434420\pi$
$462$	0	0
$463$	− 4.80385i	− 0.223254i	−0.993750	−	0.111627i	$-0.964394\pi$
$463$	− 4.80385i	− 0.223254i	0.993750	−	0.111627i	$-0.0356061\pi$
$464$	−6.92820	−0.321634
$465$	0	0
$466$	12.0000	0.555889
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	−29.8564	−1.37864
$470$	0	0
$471$	0	0
$472$	− 8.66025i	− 0.398621i
$473$	− 5.07180i	− 0.233201i
$474$	0	0
$475$	0	0
$476$	−12.9282	−0.592563
$477$	0	0
$478$	8.53590i	0.390423i
$479$	−12.9282	−0.590705	−0.295352	−	0.955388i	$-0.595437\pi$
$479$	−12.9282	−0.590705	−0.295352	+	0.955388i	$0.595437\pi$
$480$	0	0
$481$	29.8564	1.36133
$482$	− 16.9282i	− 0.771059i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	27.4449i	1.24365i	0.783158	+	0.621823i	$0.213608\pi$
$487$	27.4449i	1.24365i	−0.783158	+	0.621823i	$0.786392\pi$
$488$	8.39230i	0.379902i
$489$	0	0
$490$	0	0
$491$	−12.1244	−0.547165	−0.273582	−	0.961849i	$-0.588209\pi$
$491$	−12.1244	−0.547165	−0.273582	+	0.961849i	$0.588209\pi$
$492$	0	0
$493$	− 24.0000i	− 1.08091i
$494$	−29.5885	−1.33125
$495$	0	0
$496$	5.46410	0.245345
$497$	− 22.3923i	− 1.00443i
$498$	0	0
$499$	3.78461	0.169422	0.0847112	−	0.996406i	$-0.473003\pi$
$499$	3.78461	0.169422	0.0847112	+	0.996406i	$0.473003\pi$
$500$	0	0
$501$	0	0
$502$	− 12.1244i	− 0.541136i
$503$	− 5.07180i	− 0.226140i	−0.993587	−	0.113070i	$-0.963932\pi$
$503$	− 5.07180i	− 0.226140i	0.993587	−	0.113070i	$-0.0360685\pi$
$504$	0	0
$505$	0	0
$506$	1.60770	0.0714708
$507$	0	0
$508$	− 15.1962i	− 0.674220i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	23.8564	1.05535
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−31.8564	−1.40513
$515$	0	0
$516$	0	0
$517$	− 22.3923i	− 0.984812i
$518$	− 29.8564i	− 1.31182i
$519$	0	0
$520$	0	0
$521$	−17.1962	−0.753377	−0.376689	−	0.926340i	$-0.622937\pi$
$521$	−17.1962	−0.753377	−0.376689	+	0.926340i	$0.622937\pi$
$522$	0	0
$523$	2.00000i	0.0874539i	0.999044	+	0.0437269i	$0.0139232\pi$
$523$	2.00000i	0.0874539i	−0.999044	+	0.0437269i	$0.986077\pi$
$524$	−13.7321	−0.599887
$525$	0	0
$526$	8.32051	0.362791
$527$	18.9282i	0.824525i
$528$	0	0
$529$	22.7846	0.990635
$530$	0	0
$531$	0	0
$532$	29.5885i	1.28282i
$533$	− 19.3923i	− 0.839974i
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	12.9282i	0.557374i
$539$	−24.0000	−1.03375
$540$	0	0
$541$	26.9282	1.15773	0.578867	−	0.815422i	$-0.303495\pi$
$541$	26.9282	1.15773	0.578867	+	0.815422i	$0.303495\pi$
$542$	14.0000i	0.601351i
$543$	0	0
$544$	−3.46410	−0.148522
$545$	0	0
$546$	0	0
$547$	− 12.3923i	− 0.529857i	−0.964268	−	0.264928i	$-0.914652\pi$
$547$	− 12.3923i	− 0.529857i	0.964268	−	0.264928i	$-0.0853483\pi$
$548$	− 10.3923i	− 0.443937i
$549$	0	0
$550$	0	0
$551$	−54.9282	−2.34002
$552$	0	0
$553$	23.8564i	1.01448i
$554$	12.2679	0.521215
$555$	0	0
$556$	−13.0000	−0.551323
$557$	− 22.1769i	− 0.939666i	−0.882755	−	0.469833i	$-0.844314\pi$
$557$	− 22.1769i	− 0.939666i	0.882755	−	0.469833i	$-0.155686\pi$
$558$	0	0
$559$	5.46410	0.231107
$560$	0	0
$561$	0	0
$562$	7.05256i	0.297494i
$563$	13.8564i	0.583978i	0.956422	+	0.291989i	$0.0943171\pi$
$563$	13.8564i	0.583978i	−0.956422	+	0.291989i	$0.905683\pi$
$564$	0	0
$565$	0	0
$566$	3.32051	0.139571
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	12.1244	0.508279	0.254140	−	0.967168i	$-0.418208\pi$
$569$	12.1244	0.508279	0.254140	+	0.967168i	$0.418208\pi$
$570$	0	0
$571$	14.9282	0.624726	0.312363	−	0.949963i	$-0.398880\pi$
$571$	14.9282	0.624726	0.312363	+	0.949963i	$0.398880\pi$
$572$	− 12.9282i	− 0.540555i
$573$	0	0
$574$	−19.3923	−0.809419
$575$	0	0
$576$	0	0
$577$	24.7846i	1.03180i	0.856650	+	0.515898i	$0.172542\pi$
$577$	24.7846i	1.03180i	−0.856650	+	0.515898i	$0.827458\pi$
$578$	5.00000i	0.207973i
$579$	0	0
$580$	0	0
$581$	31.8564	1.32163
$582$	0	0
$583$	43.1769i	1.78821i
$584$	6.39230	0.264515
$585$	0	0
$586$	−7.39230	−0.305373
$587$	− 22.3923i	− 0.924229i	−0.886820	−	0.462115i	$-0.847091\pi$
$587$	− 22.3923i	− 0.924229i	0.886820	−	0.462115i	$-0.152909\pi$
$588$	0	0
$589$	43.3205	1.78499
$590$	0	0
$591$	0	0
$592$	− 8.00000i	− 0.328798i
$593$	3.46410i	0.142254i	0.997467	+	0.0711268i	$0.0226595\pi$
$593$	3.46410i	0.142254i	−0.997467	+	0.0711268i	$0.977341\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	1.73205i	0.0708288i
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	5.92820	0.241816	0.120908	−	0.992664i	$-0.461419\pi$
$601$	5.92820	0.241816	0.120908	+	0.992664i	$0.461419\pi$
$602$	− 5.46410i	− 0.222700i
$603$	0	0
$604$	−18.3923	−0.748372
$605$	0	0
$606$	0	0
$607$	− 27.1769i	− 1.10308i	−0.834149	−	0.551538i	$-0.814041\pi$
$607$	− 27.1769i	− 1.10308i	0.834149	−	0.551538i	$-0.185959\pi$
$608$	7.92820i	0.321531i
$609$	0	0
$610$	0	0
$611$	24.1244	0.975967
$612$	0	0
$613$	− 11.9808i	− 0.483898i	−0.970289	−	0.241949i	$-0.922213\pi$
$613$	− 11.9808i	− 0.483898i	0.970289	−	0.241949i	$-0.0777867\pi$
$614$	12.3923	0.500113
$615$	0	0
$616$	−12.9282	−0.520892
$617$	22.3923i	0.901480i	0.892655	+	0.450740i	$0.148840\pi$
$617$	22.3923i	0.901480i	−0.892655	+	0.450740i	$0.851160\pi$
$618$	0	0
$619$	15.7846	0.634437	0.317219	−	0.948352i	$-0.397251\pi$
$619$	15.7846	0.634437	0.317219	+	0.948352i	$0.397251\pi$
$620$	0	0
$621$	0	0
$622$	8.53590i	0.342258i
$623$	− 44.7846i	− 1.79426i
$624$	0	0
$625$	0	0
$626$	3.32051	0.132714
$627$	0	0
$628$	14.1244i	0.563623i
$629$	27.7128	1.10498
$630$	0	0
$631$	−19.7128	−0.784755	−0.392377	−	0.919804i	$-0.628347\pi$
$631$	−19.7128	−0.784755	−0.392377	+	0.919804i	$0.628347\pi$
$632$	6.39230i	0.254272i
$633$	0	0
$634$	−17.5359	−0.696439
$635$	0	0
$636$	0	0
$637$	− 25.8564i	− 1.02447i
$638$	− 24.0000i	− 0.950169i
$639$	0	0
$640$	0	0
$641$	3.21539	0.127000	0.0635001	−	0.997982i	$-0.479774\pi$
$641$	3.21539	0.127000	0.0635001	+	0.997982i	$0.479774\pi$
$642$	0	0
$643$	0.392305i	0.0154710i	0.999970	+	0.00773550i	$0.00246231\pi$
$643$	0.392305i	0.0154710i	−0.999970	+	0.00773550i	$0.997538\pi$
$644$	1.73205	0.0682524
$645$	0	0
$646$	−27.4641	−1.08056
$647$	22.6410i	0.890110i	0.895503	+	0.445055i	$0.146816\pi$
$647$	22.6410i	0.890110i	−0.895503	+	0.445055i	$0.853184\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	0	0
$652$	− 9.85641i	− 0.386007i
$653$	12.9282i	0.505920i	0.967477	+	0.252960i	$0.0814041\pi$
$653$	12.9282i	0.505920i	−0.967477	+	0.252960i	$0.918596\pi$
$654$	0	0
$655$	0	0
$656$	−5.19615	−0.202876
$657$	0	0
$658$	− 24.1244i	− 0.940465i
$659$	−4.94744	−0.192725	−0.0963625	−	0.995346i	$-0.530721\pi$
$659$	−4.94744	−0.192725	−0.0963625	+	0.995346i	$0.530721\pi$
$660$	0	0
$661$	−11.6077	−0.451487	−0.225744	−	0.974187i	$-0.572481\pi$
$661$	−11.6077	−0.451487	−0.225744	+	0.974187i	$0.572481\pi$
$662$	8.00000i	0.310929i
$663$	0	0
$664$	8.53590	0.331257
$665$	0	0
$666$	0	0
$667$	3.21539i	0.124500i
$668$	6.92820i	0.268060i
$669$	0	0
$670$	0	0
$671$	−29.0718	−1.12230
$672$	0	0
$673$	39.1769i	1.51016i	0.655633	+	0.755080i	$0.272402\pi$
$673$	39.1769i	1.51016i	−0.655633	+	0.755080i	$0.727598\pi$
$674$	7.07180	0.272395
$675$	0	0
$676$	0.928203	0.0357001
$677$	15.2487i	0.586056i	0.956104	+	0.293028i	$0.0946628\pi$
$677$	15.2487i	0.586056i	−0.956104	+	0.293028i	$0.905337\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	18.9282i	0.724798i
$683$	21.7128i	0.830818i	0.909635	+	0.415409i	$0.136361\pi$
$683$	21.7128i	0.830818i	−0.909635	+	0.415409i	$0.863639\pi$
$684$	0	0
$685$	0	0
$686$	0.267949	0.0102303
$687$	0	0
$688$	− 1.46410i	− 0.0558184i
$689$	−46.5167	−1.77214
$690$	0	0
$691$	−39.7846	−1.51348	−0.756739	−	0.653717i	$-0.773209\pi$
$691$	−39.7846	−1.51348	−0.756739	+	0.653717i	$0.773209\pi$
$692$	20.3205i	0.772470i
$693$	0	0
$694$	−14.7846	−0.561216
$695$	0	0
$696$	0	0
$697$	− 18.0000i	− 0.681799i
$698$	− 2.00000i	− 0.0757011i
$699$	0	0
$700$	0	0
$701$	−41.3205	−1.56065	−0.780327	−	0.625372i	$-0.784947\pi$
$701$	−41.3205	−1.56065	−0.780327	+	0.625372i	$0.784947\pi$
$702$	0	0
$703$	− 63.4256i	− 2.39214i
$704$	−3.46410	−0.130558
$705$	0	0
$706$	−22.3923	−0.842746
$707$	− 35.3205i	− 1.32836i
$708$	0	0
$709$	21.3205	0.800708	0.400354	−	0.916360i	$-0.368887\pi$
$709$	21.3205	0.800708	0.400354	+	0.916360i	$0.368887\pi$
$710$	0	0
$711$	0	0
$712$	− 12.0000i	− 0.449719i
$713$	− 2.53590i	− 0.0949701i
$714$	0	0
$715$	0	0
$716$	12.1244	0.453108
$717$	0	0
$718$	8.53590i	0.318557i
$719$	−35.5692	−1.32651	−0.663254	−	0.748394i	$-0.730825\pi$
$719$	−35.5692	−1.32651	−0.663254	+	0.748394i	$0.730825\pi$
$720$	0	0
$721$	−56.7128	−2.11210
$722$	43.8564i	1.63217i
$723$	0	0
$724$	12.5359	0.465893
$725$	0	0
$726$	0	0
$727$	28.8038i	1.06828i	0.845397	+	0.534138i	$0.179364\pi$
$727$	28.8038i	1.06828i	−0.845397	+	0.534138i	$0.820636\pi$
$728$	− 13.9282i	− 0.516214i
$729$	0	0
$730$	0	0
$731$	5.07180	0.187587
$732$	0	0
$733$	− 16.0000i	− 0.590973i	−0.955347	−	0.295487i	$-0.904518\pi$
$733$	− 16.0000i	− 0.590973i	0.955347	−	0.295487i	$-0.0954818\pi$
$734$	6.39230	0.235944
$735$	0	0
$736$	0.464102	0.0171070
$737$	− 27.7128i	− 1.02081i
$738$	0	0
$739$	−37.5692	−1.38201	−0.691003	−	0.722852i	$-0.742831\pi$
$739$	−37.5692	−1.38201	−0.691003	+	0.722852i	$0.742831\pi$
$740$	0	0
$741$	0	0
$742$	46.5167i	1.70768i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	10.9282	0.400110
$747$	0	0
$748$	− 12.0000i	− 0.438763i
$749$	−31.8564	−1.16401
$750$	0	0
$751$	22.7846	0.831422	0.415711	−	0.909497i	$-0.363533\pi$
$751$	22.7846	0.831422	0.415711	+	0.909497i	$0.363533\pi$
$752$	− 6.46410i	− 0.235722i
$753$	0	0
$754$	25.8564	0.941635
$755$	0	0
$756$	0	0
$757$	− 8.80385i	− 0.319981i	−0.987119	−	0.159991i	$-0.948854\pi$
$757$	− 8.80385i	− 0.319981i	0.987119	−	0.159991i	$-0.0511464\pi$
$758$	− 36.8564i	− 1.33868i
$759$	0	0
$760$	0	0
$761$	13.7321	0.497786	0.248893	−	0.968531i	$-0.419933\pi$
$761$	13.7321	0.497786	0.248893	+	0.968531i	$0.419933\pi$
$762$	0	0
$763$	− 66.6410i	− 2.41257i
$764$	−2.53590	−0.0917456
$765$	0	0
$766$	6.46410	0.233557
$767$	32.3205i	1.16703i
$768$	0	0
$769$	27.0718	0.976234	0.488117	−	0.872778i	$-0.337684\pi$
$769$	27.0718	0.976234	0.488117	+	0.872778i	$0.337684\pi$
$770$	0	0
$771$	0	0
$772$	16.0000i	0.575853i
$773$	24.9282i	0.896605i	0.893882	+	0.448303i	$0.147971\pi$
$773$	24.9282i	0.896605i	−0.893882	+	0.448303i	$0.852029\pi$
$774$	0	0
$775$	0	0
$776$	−1.07180	−0.0384753
$777$	0	0
$778$	− 23.3205i	− 0.836081i
$779$	−41.1962	−1.47601
$780$	0	0
$781$	20.7846	0.743732
$782$	1.60770i	0.0574911i
$783$	0	0
$784$	−6.92820	−0.247436
$785$	0	0
$786$	0	0
$787$	34.0000i	1.21197i	0.795476	+	0.605985i	$0.207221\pi$
$787$	34.0000i	1.21197i	−0.795476	+	0.605985i	$0.792779\pi$
$788$	− 11.5359i	− 0.410949i
$789$	0	0
$790$	0	0
$791$	48.2487	1.71553
$792$	0	0
$793$	− 31.3205i	− 1.11222i
$794$	9.85641	0.349791
$795$	0	0
$796$	−22.2487	−0.788585
$797$	16.1436i	0.571835i	0.958254	+	0.285918i	$0.0922984\pi$
$797$	16.1436i	0.571835i	−0.958254	+	0.285918i	$0.907702\pi$
$798$	0	0
$799$	22.3923	0.792183
$800$	0	0
$801$	0	0
$802$	− 39.5885i	− 1.39792i
$803$	22.1436i	0.781430i
$804$	0	0
$805$	0	0
$806$	−20.3923	−0.718288
$807$	0	0
$808$	− 9.46410i	− 0.332946i
$809$	−51.5885	−1.81375	−0.906877	−	0.421396i	$-0.861540\pi$
$809$	−51.5885	−1.81375	−0.906877	+	0.421396i	$0.861540\pi$
$810$	0	0
$811$	−2.14359	−0.0752717	−0.0376359	−	0.999292i	$-0.511983\pi$
$811$	−2.14359	−0.0752717	−0.0376359	+	0.999292i	$0.511983\pi$
$812$	− 25.8564i	− 0.907382i
$813$	0	0
$814$	27.7128	0.971334
$815$	0	0
$816$	0	0
$817$	− 11.6077i	− 0.406102i
$818$	13.9282i	0.486988i
$819$	0	0
$820$	0	0
$821$	−41.3205	−1.44210	−0.721048	−	0.692885i	$-0.756339\pi$
$821$	−41.3205	−1.44210	−0.721048	+	0.692885i	$0.756339\pi$
$822$	0	0
$823$	25.3205i	0.882617i	0.897355	+	0.441309i	$0.145486\pi$
$823$	25.3205i	0.882617i	−0.897355	+	0.441309i	$0.854514\pi$
$824$	−15.1962	−0.529383
$825$	0	0
$826$	32.3205	1.12457
$827$	56.1051i	1.95097i	0.220075	+	0.975483i	$0.429370\pi$
$827$	56.1051i	1.95097i	−0.220075	+	0.975483i	$0.570630\pi$
$828$	0	0
$829$	−37.5692	−1.30483	−0.652416	−	0.757861i	$-0.726245\pi$
$829$	−37.5692	−1.30483	−0.652416	+	0.757861i	$0.726245\pi$
$830$	0	0
$831$	0	0
$832$	− 3.73205i	− 0.129386i
$833$	− 24.0000i	− 0.831551i
$834$	0	0
$835$	0	0
$836$	−27.4641	−0.949866
$837$	0	0
$838$	3.46410i	0.119665i
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	9.60770i	0.331103i
$843$	0	0
$844$	8.85641	0.304850
$845$	0	0
$846$	0	0
$847$	− 3.73205i	− 0.128235i
$848$	12.4641i	0.428019i
$849$	0	0
$850$	0	0
$851$	−3.71281	−0.127274
$852$	0	0
$853$	− 7.21539i	− 0.247050i	−0.992341	−	0.123525i	$-0.960580\pi$
$853$	− 7.21539i	− 0.247050i	0.992341	−	0.123525i	$-0.0394200\pi$
$854$	−31.3205	−1.07177
$855$	0	0
$856$	−8.53590	−0.291751
$857$	45.7128i	1.56152i	0.624831	+	0.780760i	$0.285168\pi$
$857$	45.7128i	1.56152i	−0.624831	+	0.780760i	$0.714832\pi$
$858$	0	0
$859$	−52.7846	−1.80099	−0.900494	−	0.434869i	$-0.856795\pi$
$859$	−52.7846	−1.80099	−0.900494	+	0.434869i	$0.856795\pi$
$860$	0	0
$861$	0	0
$862$	− 28.3923i	− 0.967046i
$863$	− 9.67949i	− 0.329494i	−0.986336	−	0.164747i	$-0.947319\pi$
$863$	− 9.67949i	− 0.329494i	0.986336	−	0.164747i	$-0.0526807\pi$
$864$	0	0
$865$	0	0
$866$	−2.92820	−0.0995044
$867$	0	0
$868$	20.3923i	0.692160i
$869$	−22.1436	−0.751170
$870$	0	0
$871$	29.8564	1.01165
$872$	− 17.8564i	− 0.604694i
$873$	0	0
$874$	3.67949	0.124461
$875$	0	0
$876$	0	0
$877$	27.4449i	0.926747i	0.886163	+	0.463374i	$0.153361\pi$
$877$	27.4449i	0.926747i	−0.886163	+	0.463374i	$0.846639\pi$
$878$	− 18.3923i	− 0.620710i
$879$	0	0
$880$	0	0
$881$	−29.5692	−0.996212	−0.498106	−	0.867116i	$-0.665971\pi$
$881$	−29.5692	−0.996212	−0.498106	+	0.867116i	$0.665971\pi$
$882$	0	0
$883$	23.4641i	0.789630i	0.918761	+	0.394815i	$0.129191\pi$
$883$	23.4641i	0.789630i	−0.918761	+	0.394815i	$0.870809\pi$
$884$	12.9282	0.434823
$885$	0	0
$886$	31.8564	1.07024
$887$	− 53.1051i	− 1.78310i	−0.452927	−	0.891548i	$-0.649620\pi$
$887$	− 53.1051i	− 1.78310i	0.452927	−	0.891548i	$-0.350380\pi$
$888$	0	0
$889$	56.7128	1.90209
$890$	0	0
$891$	0	0
$892$	12.5359i	0.419733i
$893$	− 51.2487i	− 1.71497i
$894$	0	0
$895$	0	0
$896$	−3.73205	−0.124679
$897$	0	0
$898$	32.9090i	1.09819i
$899$	−37.8564	−1.26258
$900$	0	0
$901$	−43.1769	−1.43843
$902$	− 18.0000i	− 0.599334i
$903$	0	0
$904$	12.9282	0.429986
$905$	0	0
$906$	0	0
$907$	27.5692i	0.915421i	0.889101	+	0.457710i	$0.151330\pi$
$907$	27.5692i	0.915421i	−0.889101	+	0.457710i	$0.848670\pi$
$908$	16.3923i	0.543998i
$909$	0	0
$910$	0	0
$911$	46.3923	1.53705	0.768523	−	0.639822i	$-0.220992\pi$
$911$	46.3923	1.53705	0.768523	+	0.639822i	$0.220992\pi$
$912$	0	0
$913$	29.5692i	0.978598i
$914$	−20.3923	−0.674517
$915$	0	0
$916$	21.8564	0.722156
$917$	− 51.2487i	− 1.69238i
$918$	0	0
$919$	−16.7846	−0.553673	−0.276837	−	0.960917i	$-0.589286\pi$
$919$	−16.7846	−0.553673	−0.276837	+	0.960917i	$0.589286\pi$
$920$	0	0
$921$	0	0
$922$	8.78461i	0.289306i
$923$	22.3923i	0.737052i
$924$	0	0
$925$	0	0
$926$	4.80385	0.157864
$927$	0	0
$928$	− 6.92820i	− 0.227429i
$929$	−30.9282	−1.01472	−0.507361	−	0.861734i	$-0.669379\pi$
$929$	−30.9282	−1.01472	−0.507361	+	0.861734i	$0.669379\pi$
$930$	0	0
$931$	−54.9282	−1.80020
$932$	12.0000i	0.393073i
$933$	0	0
$934$	6.00000	0.196326
$935$	0	0
$936$	0	0
$937$	− 6.14359i	− 0.200702i	−0.994952	−	0.100351i	$-0.968003\pi$
$937$	− 6.14359i	− 0.200702i	0.994952	−	0.100351i	$-0.0319966\pi$
$938$	− 29.8564i	− 0.974846i
$939$	0	0
$940$	0	0
$941$	5.32051	0.173444	0.0867218	−	0.996233i	$-0.472361\pi$
$941$	5.32051	0.173444	0.0867218	+	0.996233i	$0.472361\pi$
$942$	0	0
$943$	2.41154i	0.0785306i
$944$	8.66025	0.281867
$945$	0	0
$946$	5.07180	0.164898
$947$	14.5359i	0.472353i	0.971710	+	0.236177i	$0.0758944\pi$
$947$	14.5359i	0.472353i	−0.971710	+	0.236177i	$0.924106\pi$
$948$	0	0
$949$	−23.8564	−0.774412
$950$	0	0
$951$	0	0
$952$	− 12.9282i	− 0.419005i
$953$	− 18.2487i	− 0.591134i	−0.955322	−	0.295567i	$-0.904491\pi$
$953$	− 18.2487i	− 0.591134i	0.955322	−	0.295567i	$-0.0955085\pi$
$954$	0	0
$955$	0	0
$956$	−8.53590	−0.276071
$957$	0	0
$958$	− 12.9282i	− 0.417691i
$959$	38.7846	1.25242
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	29.8564i	0.962609i
$963$	0	0
$964$	16.9282	0.545221
$965$	0	0
$966$	0	0
$967$	− 6.39230i	− 0.205563i	−0.994704	−	0.102781i	$-0.967226\pi$
$967$	− 6.39230i	− 0.205563i	0.994704	−	0.102781i	$-0.0327742\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	43.3013	1.38960	0.694802	−	0.719201i	$-0.255492\pi$
$971$	43.3013	1.38960	0.694802	+	0.719201i	$0.255492\pi$
$972$	0	0
$973$	− 48.5167i	− 1.55537i
$974$	−27.4449	−0.879390
$975$	0	0
$976$	−8.39230	−0.268631
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	41.5692	1.32856
$980$	0	0
$981$	0	0
$982$	− 12.1244i	− 0.386904i
$983$	36.4974i	1.16409i	0.813158	+	0.582043i	$0.197747\pi$
$983$	36.4974i	1.16409i	−0.813158	+	0.582043i	$0.802253\pi$
$984$	0	0
$985$	0	0
$986$	24.0000	0.764316
$987$	0	0
$988$	− 29.5885i	− 0.941334i
$989$	−0.679492	−0.0216066
$990$	0	0
$991$	11.2154	0.356269	0.178134	−	0.984006i	$-0.442994\pi$
$991$	11.2154	0.356269	0.178134	+	0.984006i	$0.442994\pi$
$992$	5.46410i	0.173485i
$993$	0	0
$994$	22.3923	0.710241
$995$	0	0
$996$	0	0
$997$	3.19615i	0.101223i	0.998718	+	0.0506116i	$0.0161171\pi$
$997$	3.19615i	0.101223i	−0.998718	+	0.0506116i	$0.983883\pi$
$998$	3.78461i	0.119800i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.z.649.3		4
3.2	odd	2		4050.2.c.x.649.1		4
5.2	odd	4		810.2.a.j.1.2	✓	2
5.3	odd	4		4050.2.a.bt.1.1		2
5.4	even	2	inner	4050.2.c.z.649.2		4
15.2	even	4		810.2.a.l.1.2	yes	2
15.8	even	4		4050.2.a.bk.1.1		2
15.14	odd	2		4050.2.c.x.649.4		4
20.7	even	4		6480.2.a.bb.1.1		2
45.2	even	12		810.2.e.m.271.1		4
45.7	odd	12		810.2.e.n.271.1		4
45.22	odd	12		810.2.e.n.541.1		4
45.32	even	12		810.2.e.m.541.1		4
60.47	odd	4		6480.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.2.a.j.1.2	✓	2	5.2	odd	4
810.2.a.l.1.2	yes	2	15.2	even	4
810.2.e.m.271.1		4	45.2	even	12
810.2.e.m.541.1		4	45.32	even	12
810.2.e.n.271.1		4	45.7	odd	12
810.2.e.n.541.1		4	45.22	odd	12
4050.2.a.bk.1.1		2	15.8	even	4
4050.2.a.bt.1.1		2	5.3	odd	4
4050.2.c.x.649.1		4	3.2	odd	2
4050.2.c.x.649.4		4	15.14	odd	2
4050.2.c.z.649.2		4	5.4	even	2	inner
4050.2.c.z.649.3		4	1.1	even	1	trivial
6480.2.a.bb.1.1		2	20.7	even	4
6480.2.a.bj.1.1		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{7} -1.00000i q^{8} +3.46410 q^{11} +3.73205i q^{13} +3.73205 q^{14} +1.00000 q^{16} +3.46410i q^{17} +7.92820 q^{19} +3.46410i q^{22} -0.464102i q^{23} -3.73205 q^{26} +3.73205i q^{28} -6.92820 q^{29} +5.46410 q^{31} +1.00000i q^{32} -3.46410 q^{34} -8.00000i q^{37} +7.92820i q^{38} -5.19615 q^{41} -1.46410i q^{43} -3.46410 q^{44} +0.464102 q^{46} -6.46410i q^{47} -6.92820 q^{49} -3.73205i q^{52} +12.4641i q^{53} -3.73205 q^{56} -6.92820i q^{58} +8.66025 q^{59} -8.39230 q^{61} +5.46410i q^{62} -1.00000 q^{64} -8.00000i q^{67} -3.46410i q^{68} +6.00000 q^{71} +6.39230i q^{73} +8.00000 q^{74} -7.92820 q^{76} -12.9282i q^{77} -6.39230 q^{79} -5.19615i q^{82} +8.53590i q^{83} +1.46410 q^{86} -3.46410i q^{88} +12.0000 q^{89} +13.9282 q^{91} +0.464102i q^{92} +6.46410 q^{94} -1.07180i q^{97} -6.92820i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{14} + 4 q^{16} + 4 q^{19} - 8 q^{26} + 8 q^{31} - 12 q^{46} - 8 q^{56} + 8 q^{61} - 4 q^{64} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 16 q^{79} - 8 q^{86} + 48 q^{89} + 28 q^{91} + 12 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^14 + 4 * q^16 + 4 * q^19 - 8 * q^26 + 8 * q^31 - 12 * q^46 - 8 * q^56 + 8 * q^61 - 4 * q^64 + 24 * q^71 + 32 * q^74 - 4 * q^76 + 16 * q^79 - 8 * q^86 + 48 * q^89 + 28 * q^91 + 12 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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