LMFDB - Embedded newform 4014.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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([0, 0]))

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

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

chi := DirichletCharacter("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.a (trivial)

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:	yes
Analytic conductor:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.60555 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.60555 q^{7} +1.00000 q^{8} +6.30278 q^{11} +5.30278 q^{13} +2.60555 q^{14} +1.00000 q^{16} +4.60555 q^{17} +0.697224 q^{19} +6.30278 q^{22} -4.00000 q^{23} -5.00000 q^{25} +5.30278 q^{26} +2.60555 q^{28} +1.69722 q^{29} -0.605551 q^{31} +1.00000 q^{32} +4.60555 q^{34} -2.60555 q^{37} +0.697224 q^{38} -4.00000 q^{41} -6.90833 q^{43} +6.30278 q^{44} -4.00000 q^{46} +4.30278 q^{47} -0.211103 q^{49} -5.00000 q^{50} +5.30278 q^{52} -3.90833 q^{53} +2.60555 q^{56} +1.69722 q^{58} -3.30278 q^{59} -6.30278 q^{61} -0.605551 q^{62} +1.00000 q^{64} -2.60555 q^{67} +4.60555 q^{68} -3.21110 q^{71} +1.90833 q^{73} -2.60555 q^{74} +0.697224 q^{76} +16.4222 q^{77} -9.51388 q^{79} -4.00000 q^{82} +9.21110 q^{83} -6.90833 q^{86} +6.30278 q^{88} +2.78890 q^{89} +13.8167 q^{91} -4.00000 q^{92} +4.30278 q^{94} -15.2111 q^{97} -0.211103 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 9 q^{11} + 7 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 5 q^{19} + 9 q^{22} - 8 q^{23} - 10 q^{25} + 7 q^{26} - 2 q^{28} + 7 q^{29} + 6 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{37} + 5 q^{38} - 8 q^{41} - 3 q^{43} + 9 q^{44} - 8 q^{46} + 5 q^{47} + 14 q^{49} - 10 q^{50} + 7 q^{52} + 3 q^{53} - 2 q^{56} + 7 q^{58} - 3 q^{59} - 9 q^{61} + 6 q^{62} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} - 7 q^{73} + 2 q^{74} + 5 q^{76} + 4 q^{77} - q^{79} - 8 q^{82} + 4 q^{83} - 3 q^{86} + 9 q^{88} + 20 q^{89} + 6 q^{91} - 8 q^{92} + 5 q^{94} - 16 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 9 * q^11 + 7 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 5 * q^19 + 9 * q^22 - 8 * q^23 - 10 * q^25 + 7 * q^26 - 2 * q^28 + 7 * q^29 + 6 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^37 + 5 * q^38 - 8 * q^41 - 3 * q^43 + 9 * q^44 - 8 * q^46 + 5 * q^47 + 14 * q^49 - 10 * q^50 + 7 * q^52 + 3 * q^53 - 2 * q^56 + 7 * q^58 - 3 * q^59 - 9 * q^61 + 6 * q^62 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 - 7 * q^73 + 2 * q^74 + 5 * q^76 + 4 * q^77 - q^79 - 8 * q^82 + 4 * q^83 - 3 * q^86 + 9 * q^88 + 20 * q^89 + 6 * q^91 - 8 * q^92 + 5 * q^94 - 16 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	2.60555	0.984806	0.492403	−	0.870367i	$-0.336119\pi$
$7$	2.60555	0.984806	0.492403	+	0.870367i	$0.336119\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	6.30278	1.90036	0.950179	−	0.311704i	$-0.100900\pi$
$11$	6.30278	1.90036	0.950179	+	0.311704i	$0.100900\pi$
$12$	0	0
$13$	5.30278	1.47073	0.735363	−	0.677674i	$-0.237012\pi$
$13$	5.30278	1.47073	0.735363	+	0.677674i	$0.237012\pi$
$14$	2.60555	0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	4.60555	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$17$	4.60555	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$18$	0	0
$19$	0.697224	0.159954	0.0799771	−	0.996797i	$-0.474515\pi$
$19$	0.697224	0.159954	0.0799771	+	0.996797i	$0.474515\pi$
$20$	0	0
$21$	0	0
$22$	6.30278	1.34376
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	5.30278	1.03996
$27$	0	0
$28$	2.60555	0.492403
$29$	1.69722	0.315167	0.157583	−	0.987506i	$-0.449630\pi$
$29$	1.69722	0.315167	0.157583	+	0.987506i	$0.449630\pi$
$30$	0	0
$31$	−0.605551	−0.108760	−0.0543801	−	0.998520i	$-0.517318\pi$
$31$	−0.605551	−0.108760	−0.0543801	+	0.998520i	$0.517318\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.60555	0.789846
$35$	0	0
$36$	0	0
$37$	−2.60555	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$37$	−2.60555	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$38$	0.697224	0.113105
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−6.90833	−1.05351	−0.526755	−	0.850017i	$-0.676591\pi$
$43$	−6.90833	−1.05351	−0.526755	+	0.850017i	$0.676591\pi$
$44$	6.30278	0.950179
$45$	0	0
$46$	−4.00000	−0.589768
$47$	4.30278	0.627624	0.313812	−	0.949485i	$-0.398394\pi$
$47$	4.30278	0.627624	0.313812	+	0.949485i	$0.398394\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	−5.00000	−0.707107
$51$	0	0
$52$	5.30278	0.735363
$53$	−3.90833	−0.536850	−0.268425	−	0.963301i	$-0.586503\pi$
$53$	−3.90833	−0.536850	−0.268425	+	0.963301i	$0.586503\pi$
$54$	0	0
$55$	0	0
$56$	2.60555	0.348181
$57$	0	0
$58$	1.69722	0.222856
$59$	−3.30278	−0.429985	−0.214992	−	0.976616i	$-0.568973\pi$
$59$	−3.30278	−0.429985	−0.214992	+	0.976616i	$0.568973\pi$
$60$	0	0
$61$	−6.30278	−0.806988	−0.403494	−	0.914982i	$-0.632204\pi$
$61$	−6.30278	−0.806988	−0.403494	+	0.914982i	$0.632204\pi$
$62$	−0.605551	−0.0769051
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.60555	−0.318319	−0.159159	−	0.987253i	$-0.550878\pi$
$67$	−2.60555	−0.318319	−0.159159	+	0.987253i	$0.550878\pi$
$68$	4.60555	0.558505
$69$	0	0
$70$	0	0
$71$	−3.21110	−0.381088	−0.190544	−	0.981679i	$-0.561025\pi$
$71$	−3.21110	−0.381088	−0.190544	+	0.981679i	$0.561025\pi$
$72$	0	0
$73$	1.90833	0.223353	0.111676	−	0.993745i	$-0.464378\pi$
$73$	1.90833	0.223353	0.111676	+	0.993745i	$0.464378\pi$
$74$	−2.60555	−0.302889
$75$	0	0
$76$	0.697224	0.0799771
$77$	16.4222	1.87148
$78$	0	0
$79$	−9.51388	−1.07039	−0.535197	−	0.844727i	$-0.679763\pi$
$79$	−9.51388	−1.07039	−0.535197	+	0.844727i	$0.679763\pi$
$80$	0	0
$81$	0	0
$82$	−4.00000	−0.441726
$83$	9.21110	1.01105	0.505525	−	0.862812i	$-0.331299\pi$
$83$	9.21110	1.01105	0.505525	+	0.862812i	$0.331299\pi$
$84$	0	0
$85$	0	0
$86$	−6.90833	−0.744944
$87$	0	0
$88$	6.30278	0.671878
$89$	2.78890	0.295623	0.147811	−	0.989016i	$-0.452777\pi$
$89$	2.78890	0.295623	0.147811	+	0.989016i	$0.452777\pi$
$90$	0	0
$91$	13.8167	1.44838
$92$	−4.00000	−0.417029
$93$	0	0
$94$	4.30278	0.443797
$95$	0	0
$96$	0	0
$97$	−15.2111	−1.54445	−0.772227	−	0.635347i	$-0.780857\pi$
$97$	−15.2111	−1.54445	−0.772227	+	0.635347i	$0.780857\pi$
$98$	−0.211103	−0.0213246
$99$	0	0
$100$	−5.00000	−0.500000
$101$	0.0916731	0.00912181	0.00456091	−	0.999990i	$-0.498548\pi$
$101$	0.0916731	0.00912181	0.00456091	+	0.999990i	$0.498548\pi$
$102$	0	0
$103$	7.90833	0.779231	0.389615	−	0.920978i	$-0.372608\pi$
$103$	7.90833	0.779231	0.389615	+	0.920978i	$0.372608\pi$
$104$	5.30278	0.519980
$105$	0	0
$106$	−3.90833	−0.379610
$107$	−9.90833	−0.957874	−0.478937	−	0.877849i	$-0.658978\pi$
$107$	−9.90833	−0.957874	−0.478937	+	0.877849i	$0.658978\pi$
$108$	0	0
$109$	−12.6056	−1.20739	−0.603696	−	0.797214i	$-0.706306\pi$
$109$	−12.6056	−1.20739	−0.603696	+	0.797214i	$0.706306\pi$
$110$	0	0
$111$	0	0
$112$	2.60555	0.246201
$113$	19.3028	1.81585	0.907926	−	0.419130i	$-0.137665\pi$
$113$	19.3028	1.81585	0.907926	+	0.419130i	$0.137665\pi$
$114$	0	0
$115$	0	0
$116$	1.69722	0.157583
$117$	0	0
$118$	−3.30278	−0.304045
$119$	12.0000	1.10004
$120$	0	0
$121$	28.7250	2.61136
$122$	−6.30278	−0.570626
$123$	0	0
$124$	−0.605551	−0.0543801
$125$	0	0
$126$	0	0
$127$	−7.21110	−0.639882	−0.319941	−	0.947437i	$-0.603663\pi$
$127$	−7.21110	−0.639882	−0.319941	+	0.947437i	$0.603663\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−12.4222	−1.08533	−0.542667	−	0.839948i	$-0.682585\pi$
$131$	−12.4222	−1.08533	−0.542667	+	0.839948i	$0.682585\pi$
$132$	0	0
$133$	1.81665	0.157524
$134$	−2.60555	−0.225085
$135$	0	0
$136$	4.60555	0.394923
$137$	3.69722	0.315875	0.157938	−	0.987449i	$-0.449516\pi$
$137$	3.69722	0.315875	0.157938	+	0.987449i	$0.449516\pi$
$138$	0	0
$139$	−1.09167	−0.0925945	−0.0462973	−	0.998928i	$-0.514742\pi$
$139$	−1.09167	−0.0925945	−0.0462973	+	0.998928i	$0.514742\pi$
$140$	0	0
$141$	0	0
$142$	−3.21110	−0.269470
$143$	33.4222	2.79491
$144$	0	0
$145$	0	0
$146$	1.90833	0.157934
$147$	0	0
$148$	−2.60555	−0.214175
$149$	−9.21110	−0.754603	−0.377301	−	0.926090i	$-0.623148\pi$
$149$	−9.21110	−0.754603	−0.377301	+	0.926090i	$0.623148\pi$
$150$	0	0
$151$	18.5139	1.50664	0.753319	−	0.657655i	$-0.228452\pi$
$151$	18.5139	1.50664	0.753319	+	0.657655i	$0.228452\pi$
$152$	0.697224	0.0565524
$153$	0	0
$154$	16.4222	1.32334
$155$	0	0
$156$	0	0
$157$	6.30278	0.503016	0.251508	−	0.967855i	$-0.419073\pi$
$157$	6.30278	0.503016	0.251508	+	0.967855i	$0.419073\pi$
$158$	−9.51388	−0.756884
$159$	0	0
$160$	0	0
$161$	−10.4222	−0.821385
$162$	0	0
$163$	1.39445	0.109222	0.0546108	−	0.998508i	$-0.482608\pi$
$163$	1.39445	0.109222	0.0546108	+	0.998508i	$0.482608\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	9.21110	0.714920
$167$	−4.60555	−0.356388	−0.178194	−	0.983995i	$-0.557026\pi$
$167$	−4.60555	−0.356388	−0.178194	+	0.983995i	$0.557026\pi$
$168$	0	0
$169$	15.1194	1.16303
$170$	0	0
$171$	0	0
$172$	−6.90833	−0.526755
$173$	23.0278	1.75077	0.875384	−	0.483428i	$-0.160609\pi$
$173$	23.0278	1.75077	0.875384	+	0.483428i	$0.160609\pi$
$174$	0	0
$175$	−13.0278	−0.984806
$176$	6.30278	0.475090
$177$	0	0
$178$	2.78890	0.209037
$179$	−1.21110	−0.0905221	−0.0452610	−	0.998975i	$-0.514412\pi$
$179$	−1.21110	−0.0905221	−0.0452610	+	0.998975i	$0.514412\pi$
$180$	0	0
$181$	8.60555	0.639646	0.319823	−	0.947477i	$-0.396377\pi$
$181$	8.60555	0.639646	0.319823	+	0.947477i	$0.396377\pi$
$182$	13.8167	1.02416
$183$	0	0
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	29.0278	2.12272
$188$	4.30278	0.313812
$189$	0	0
$190$	0	0
$191$	13.8167	0.999738	0.499869	−	0.866101i	$-0.333381\pi$
$191$	13.8167	0.999738	0.499869	+	0.866101i	$0.333381\pi$
$192$	0	0
$193$	−20.4222	−1.47002	−0.735011	−	0.678055i	$-0.762823\pi$
$193$	−20.4222	−1.47002	−0.735011	+	0.678055i	$0.762823\pi$
$194$	−15.2111	−1.09209
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	−1.69722	−0.120922	−0.0604611	−	0.998171i	$-0.519257\pi$
$197$	−1.69722	−0.120922	−0.0604611	+	0.998171i	$0.519257\pi$
$198$	0	0
$199$	−18.4222	−1.30592	−0.652958	−	0.757394i	$-0.726472\pi$
$199$	−18.4222	−1.30592	−0.652958	+	0.757394i	$0.726472\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	0.0916731	0.00645010
$203$	4.42221	0.310378
$204$	0	0
$205$	0	0
$206$	7.90833	0.550999
$207$	0	0
$208$	5.30278	0.367681
$209$	4.39445	0.303970
$210$	0	0
$211$	1.69722	0.116842	0.0584209	−	0.998292i	$-0.481393\pi$
$211$	1.69722	0.116842	0.0584209	+	0.998292i	$0.481393\pi$
$212$	−3.90833	−0.268425
$213$	0	0
$214$	−9.90833	−0.677319
$215$	0	0
$216$	0	0
$217$	−1.57779	−0.107108
$218$	−12.6056	−0.853756
$219$	0	0
$220$	0	0
$221$	24.4222	1.64282
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	2.60555	0.174091
$225$	0	0
$226$	19.3028	1.28400
$227$	−17.0278	−1.13017	−0.565086	−	0.825032i	$-0.691157\pi$
$227$	−17.0278	−1.13017	−0.565086	+	0.825032i	$0.691157\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	1.69722	0.111428
$233$	−8.30278	−0.543933	−0.271966	−	0.962307i	$-0.587674\pi$
$233$	−8.30278	−0.543933	−0.271966	+	0.962307i	$0.587674\pi$
$234$	0	0
$235$	0	0
$236$	−3.30278	−0.214992
$237$	0	0
$238$	12.0000	0.777844
$239$	23.7250	1.53464	0.767321	−	0.641264i	$-0.221589\pi$
$239$	23.7250	1.53464	0.767321	+	0.641264i	$0.221589\pi$
$240$	0	0
$241$	13.5139	0.870505	0.435253	−	0.900308i	$-0.356659\pi$
$241$	13.5139	0.870505	0.435253	+	0.900308i	$0.356659\pi$
$242$	28.7250	1.84651
$243$	0	0
$244$	−6.30278	−0.403494
$245$	0	0
$246$	0	0
$247$	3.69722	0.235249
$248$	−0.605551	−0.0384525
$249$	0	0
$250$	0	0
$251$	6.42221	0.405366	0.202683	−	0.979244i	$-0.435034\pi$
$251$	6.42221	0.405366	0.202683	+	0.979244i	$0.435034\pi$
$252$	0	0
$253$	−25.2111	−1.58501
$254$	−7.21110	−0.452465
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−6.78890	−0.421842
$260$	0	0
$261$	0	0
$262$	−12.4222	−0.767446
$263$	−14.6056	−0.900617	−0.450308	−	0.892873i	$-0.648686\pi$
$263$	−14.6056	−0.900617	−0.450308	+	0.892873i	$0.648686\pi$
$264$	0	0
$265$	0	0
$266$	1.81665	0.111386
$267$	0	0
$268$	−2.60555	−0.159159
$269$	−21.0278	−1.28208	−0.641042	−	0.767505i	$-0.721498\pi$
$269$	−21.0278	−1.28208	−0.641042	+	0.767505i	$0.721498\pi$
$270$	0	0
$271$	−10.1194	−0.614712	−0.307356	−	0.951595i	$-0.599444\pi$
$271$	−10.1194	−0.614712	−0.307356	+	0.951595i	$0.599444\pi$
$272$	4.60555	0.279253
$273$	0	0
$274$	3.69722	0.223357
$275$	−31.5139	−1.90036
$276$	0	0
$277$	−3.57779	−0.214969	−0.107484	−	0.994207i	$-0.534280\pi$
$277$	−3.57779	−0.214969	−0.107484	+	0.994207i	$0.534280\pi$
$278$	−1.09167	−0.0654742
$279$	0	0
$280$	0	0
$281$	7.39445	0.441116	0.220558	−	0.975374i	$-0.429212\pi$
$281$	7.39445	0.441116	0.220558	+	0.975374i	$0.429212\pi$
$282$	0	0
$283$	−19.3028	−1.14743	−0.573715	−	0.819055i	$-0.694498\pi$
$283$	−19.3028	−1.14743	−0.573715	+	0.819055i	$0.694498\pi$
$284$	−3.21110	−0.190544
$285$	0	0
$286$	33.4222	1.97630
$287$	−10.4222	−0.615203
$288$	0	0
$289$	4.21110	0.247712
$290$	0	0
$291$	0	0
$292$	1.90833	0.111676
$293$	30.2389	1.76657	0.883287	−	0.468834i	$-0.155326\pi$
$293$	30.2389	1.76657	0.883287	+	0.468834i	$0.155326\pi$
$294$	0	0
$295$	0	0
$296$	−2.60555	−0.151445
$297$	0	0
$298$	−9.21110	−0.533585
$299$	−21.2111	−1.22667
$300$	0	0
$301$	−18.0000	−1.03750
$302$	18.5139	1.06535
$303$	0	0
$304$	0.697224	0.0399886
$305$	0	0
$306$	0	0
$307$	−3.02776	−0.172803	−0.0864016	−	0.996260i	$-0.527537\pi$
$307$	−3.02776	−0.172803	−0.0864016	+	0.996260i	$0.527537\pi$
$308$	16.4222	0.935742
$309$	0	0
$310$	0	0
$311$	3.39445	0.192482	0.0962408	−	0.995358i	$-0.469318\pi$
$311$	3.39445	0.192482	0.0962408	+	0.995358i	$0.469318\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	6.30278	0.355686
$315$	0	0
$316$	−9.51388	−0.535197
$317$	−13.3028	−0.747158	−0.373579	−	0.927598i	$-0.621870\pi$
$317$	−13.3028	−0.747158	−0.373579	+	0.927598i	$0.621870\pi$
$318$	0	0
$319$	10.6972	0.598930
$320$	0	0
$321$	0	0
$322$	−10.4222	−0.580807
$323$	3.21110	0.178671
$324$	0	0
$325$	−26.5139	−1.47073
$326$	1.39445	0.0772314
$327$	0	0
$328$	−4.00000	−0.220863
$329$	11.2111	0.618088
$330$	0	0
$331$	−4.78890	−0.263222	−0.131611	−	0.991301i	$-0.542015\pi$
$331$	−4.78890	−0.263222	−0.131611	+	0.991301i	$0.542015\pi$
$332$	9.21110	0.505525
$333$	0	0
$334$	−4.60555	−0.252005
$335$	0	0
$336$	0	0
$337$	−9.21110	−0.501761	−0.250880	−	0.968018i	$-0.580720\pi$
$337$	−9.21110	−0.501761	−0.250880	+	0.968018i	$0.580720\pi$
$338$	15.1194	0.822389
$339$	0	0
$340$	0	0
$341$	−3.81665	−0.206683
$342$	0	0
$343$	−18.7889	−1.01451
$344$	−6.90833	−0.372472
$345$	0	0
$346$	23.0278	1.23798
$347$	18.4222	0.988956	0.494478	−	0.869190i	$-0.335359\pi$
$347$	18.4222	0.988956	0.494478	+	0.869190i	$0.335359\pi$
$348$	0	0
$349$	−32.6056	−1.74534	−0.872668	−	0.488315i	$-0.837612\pi$
$349$	−32.6056	−1.74534	−0.872668	+	0.488315i	$0.837612\pi$
$350$	−13.0278	−0.696363
$351$	0	0
$352$	6.30278	0.335939
$353$	22.6056	1.20317	0.601586	−	0.798808i	$-0.294536\pi$
$353$	22.6056	1.20317	0.601586	+	0.798808i	$0.294536\pi$
$354$	0	0
$355$	0	0
$356$	2.78890	0.147811
$357$	0	0
$358$	−1.21110	−0.0640088
$359$	5.48612	0.289546	0.144773	−	0.989465i	$-0.453755\pi$
$359$	5.48612	0.289546	0.144773	+	0.989465i	$0.453755\pi$
$360$	0	0
$361$	−18.5139	−0.974415
$362$	8.60555	0.452298
$363$	0	0
$364$	13.8167	0.724189
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−10.1833	−0.528693
$372$	0	0
$373$	30.8444	1.59706	0.798532	−	0.601953i	$-0.205611\pi$
$373$	30.8444	1.59706	0.798532	+	0.601953i	$0.205611\pi$
$374$	29.0278	1.50099
$375$	0	0
$376$	4.30278	0.221899
$377$	9.00000	0.463524
$378$	0	0
$379$	36.3305	1.86617	0.933087	−	0.359651i	$-0.117104\pi$
$379$	36.3305	1.86617	0.933087	+	0.359651i	$0.117104\pi$
$380$	0	0
$381$	0	0
$382$	13.8167	0.706922
$383$	−11.8167	−0.603803	−0.301901	−	0.953339i	$-0.597621\pi$
$383$	−11.8167	−0.603803	−0.301901	+	0.953339i	$0.597621\pi$
$384$	0	0
$385$	0	0
$386$	−20.4222	−1.03946
$387$	0	0
$388$	−15.2111	−0.772227
$389$	21.9361	1.11220	0.556102	−	0.831114i	$-0.312296\pi$
$389$	21.9361	1.11220	0.556102	+	0.831114i	$0.312296\pi$
$390$	0	0
$391$	−18.4222	−0.931651
$392$	−0.211103	−0.0106623
$393$	0	0
$394$	−1.69722	−0.0855049
$395$	0	0
$396$	0	0
$397$	−19.5139	−0.979373	−0.489687	−	0.871898i	$-0.662889\pi$
$397$	−19.5139	−0.979373	−0.489687	+	0.871898i	$0.662889\pi$
$398$	−18.4222	−0.923422
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−2.97224	−0.148427	−0.0742134	−	0.997242i	$-0.523645\pi$
$401$	−2.97224	−0.148427	−0.0742134	+	0.997242i	$0.523645\pi$
$402$	0	0
$403$	−3.21110	−0.159956
$404$	0.0916731	0.00456091
$405$	0	0
$406$	4.42221	0.219470
$407$	−16.4222	−0.814018
$408$	0	0
$409$	24.2389	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$409$	24.2389	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$410$	0	0
$411$	0	0
$412$	7.90833	0.389615
$413$	−8.60555	−0.423451
$414$	0	0
$415$	0	0
$416$	5.30278	0.259990
$417$	0	0
$418$	4.39445	0.214940
$419$	3.39445	0.165830	0.0829148	−	0.996557i	$-0.473577\pi$
$419$	3.39445	0.165830	0.0829148	+	0.996557i	$0.473577\pi$
$420$	0	0
$421$	−19.7250	−0.961337	−0.480668	−	0.876902i	$-0.659606\pi$
$421$	−19.7250	−0.961337	−0.480668	+	0.876902i	$0.659606\pi$
$422$	1.69722	0.0826196
$423$	0	0
$424$	−3.90833	−0.189805
$425$	−23.0278	−1.11701
$426$	0	0
$427$	−16.4222	−0.794726
$428$	−9.90833	−0.478937
$429$	0	0
$430$	0	0
$431$	25.8167	1.24354	0.621772	−	0.783198i	$-0.286413\pi$
$431$	25.8167	1.24354	0.621772	+	0.783198i	$0.286413\pi$
$432$	0	0
$433$	21.5416	1.03522	0.517612	−	0.855615i	$-0.326821\pi$
$433$	21.5416	1.03522	0.517612	+	0.855615i	$0.326821\pi$
$434$	−1.57779	−0.0757366
$435$	0	0
$436$	−12.6056	−0.603696
$437$	−2.78890	−0.133411
$438$	0	0
$439$	−15.6333	−0.746137	−0.373069	−	0.927804i	$-0.621694\pi$
$439$	−15.6333	−0.746137	−0.373069	+	0.927804i	$0.621694\pi$
$440$	0	0
$441$	0	0
$442$	24.4222	1.16165
$443$	−8.60555	−0.408862	−0.204431	−	0.978881i	$-0.565534\pi$
$443$	−8.60555	−0.408862	−0.204431	+	0.978881i	$0.565534\pi$
$444$	0	0
$445$	0	0
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	2.60555	0.123101
$449$	−0.788897	−0.0372304	−0.0186152	−	0.999827i	$-0.505926\pi$
$449$	−0.788897	−0.0372304	−0.0186152	+	0.999827i	$0.505926\pi$
$450$	0	0
$451$	−25.2111	−1.18714
$452$	19.3028	0.907926
$453$	0	0
$454$	−17.0278	−0.799152
$455$	0	0
$456$	0	0
$457$	10.1833	0.476357	0.238178	−	0.971221i	$-0.423450\pi$
$457$	10.1833	0.476357	0.238178	+	0.971221i	$0.423450\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	27.7250	1.29128	0.645641	−	0.763641i	$-0.276590\pi$
$461$	27.7250	1.29128	0.645641	+	0.763641i	$0.276590\pi$
$462$	0	0
$463$	3.21110	0.149233	0.0746163	−	0.997212i	$-0.476227\pi$
$463$	3.21110	0.149233	0.0746163	+	0.997212i	$0.476227\pi$
$464$	1.69722	0.0787917
$465$	0	0
$466$	−8.30278	−0.384619
$467$	−2.09167	−0.0967911	−0.0483955	−	0.998828i	$-0.515411\pi$
$467$	−2.09167	−0.0967911	−0.0483955	+	0.998828i	$0.515411\pi$
$468$	0	0
$469$	−6.78890	−0.313482
$470$	0	0
$471$	0	0
$472$	−3.30278	−0.152023
$473$	−43.5416	−2.00205
$474$	0	0
$475$	−3.48612	−0.159954
$476$	12.0000	0.550019
$477$	0	0
$478$	23.7250	1.08516
$479$	36.9083	1.68638	0.843192	−	0.537612i	$-0.180674\pi$
$479$	36.9083	1.68638	0.843192	+	0.537612i	$0.180674\pi$
$480$	0	0
$481$	−13.8167	−0.629985
$482$	13.5139	0.615540
$483$	0	0
$484$	28.7250	1.30568
$485$	0	0
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−6.30278	−0.285313
$489$	0	0
$490$	0	0
$491$	−13.9361	−0.628927	−0.314463	−	0.949270i	$-0.601825\pi$
$491$	−13.9361	−0.628927	−0.314463	+	0.949270i	$0.601825\pi$
$492$	0	0
$493$	7.81665	0.352044
$494$	3.69722	0.166346
$495$	0	0
$496$	−0.605551	−0.0271901
$497$	−8.36669	−0.375297
$498$	0	0
$499$	−6.30278	−0.282151	−0.141075	−	0.989999i	$-0.545056\pi$
$499$	−6.30278	−0.282151	−0.141075	+	0.989999i	$0.545056\pi$
$500$	0	0
$501$	0	0
$502$	6.42221	0.286637
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	0	0
$506$	−25.2111	−1.12077
$507$	0	0
$508$	−7.21110	−0.319941
$509$	−27.7527	−1.23012	−0.615059	−	0.788481i	$-0.710868\pi$
$509$	−27.7527	−1.23012	−0.615059	+	0.788481i	$0.710868\pi$
$510$	0	0
$511$	4.97224	0.219959
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	27.1194	1.19271
$518$	−6.78890	−0.298287
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−5.57779	−0.243900	−0.121950	−	0.992536i	$-0.538915\pi$
$523$	−5.57779	−0.243900	−0.121950	+	0.992536i	$0.538915\pi$
$524$	−12.4222	−0.542667
$525$	0	0
$526$	−14.6056	−0.636832
$527$	−2.78890	−0.121486
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	1.81665	0.0787619
$533$	−21.2111	−0.918755
$534$	0	0
$535$	0	0
$536$	−2.60555	−0.112543
$537$	0	0
$538$	−21.0278	−0.906571
$539$	−1.33053	−0.0573101
$540$	0	0
$541$	8.27502	0.355771	0.177885	−	0.984051i	$-0.443074\pi$
$541$	8.27502	0.355771	0.177885	+	0.984051i	$0.443074\pi$
$542$	−10.1194	−0.434667
$543$	0	0
$544$	4.60555	0.197461
$545$	0	0
$546$	0	0
$547$	9.72498	0.415810	0.207905	−	0.978149i	$-0.433335\pi$
$547$	9.72498	0.415810	0.207905	+	0.978149i	$0.433335\pi$
$548$	3.69722	0.157938
$549$	0	0
$550$	−31.5139	−1.34376
$551$	1.18335	0.0504122
$552$	0	0
$553$	−24.7889	−1.05413
$554$	−3.57779	−0.152006
$555$	0	0
$556$	−1.09167	−0.0462973
$557$	38.4222	1.62800	0.814001	−	0.580864i	$-0.197285\pi$
$557$	38.4222	1.62800	0.814001	+	0.580864i	$0.197285\pi$
$558$	0	0
$559$	−36.6333	−1.54942
$560$	0	0
$561$	0	0
$562$	7.39445	0.311916
$563$	42.5416	1.79292	0.896458	−	0.443129i	$-0.146131\pi$
$563$	42.5416	1.79292	0.896458	+	0.443129i	$0.146131\pi$
$564$	0	0
$565$	0	0
$566$	−19.3028	−0.811356
$567$	0	0
$568$	−3.21110	−0.134735
$569$	−33.5416	−1.40614	−0.703069	−	0.711121i	$-0.748188\pi$
$569$	−33.5416	−1.40614	−0.703069	+	0.711121i	$0.748188\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	33.4222	1.39745
$573$	0	0
$574$	−10.4222	−0.435014
$575$	20.0000	0.834058
$576$	0	0
$577$	−42.9638	−1.78861	−0.894304	−	0.447460i	$-0.852329\pi$
$577$	−42.9638	−1.78861	−0.894304	+	0.447460i	$0.852329\pi$
$578$	4.21110	0.175159
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	−24.6333	−1.02021
$584$	1.90833	0.0789671
$585$	0	0
$586$	30.2389	1.24916
$587$	−14.9361	−0.616478	−0.308239	−	0.951309i	$-0.599740\pi$
$587$	−14.9361	−0.616478	−0.308239	+	0.951309i	$0.599740\pi$
$588$	0	0
$589$	−0.422205	−0.0173967
$590$	0	0
$591$	0	0
$592$	−2.60555	−0.107087
$593$	−32.4222	−1.33142	−0.665710	−	0.746210i	$-0.731871\pi$
$593$	−32.4222	−1.33142	−0.665710	+	0.746210i	$0.731871\pi$
$594$	0	0
$595$	0	0
$596$	−9.21110	−0.377301
$597$	0	0
$598$	−21.2111	−0.867386
$599$	9.88057	0.403709	0.201855	−	0.979416i	$-0.435303\pi$
$599$	9.88057	0.403709	0.201855	+	0.979416i	$0.435303\pi$
$600$	0	0
$601$	21.8167	0.889920	0.444960	−	0.895550i	$-0.353218\pi$
$601$	21.8167	0.889920	0.444960	+	0.895550i	$0.353218\pi$
$602$	−18.0000	−0.733625
$603$	0	0
$604$	18.5139	0.753319
$605$	0	0
$606$	0	0
$607$	32.8444	1.33311	0.666557	−	0.745454i	$-0.267767\pi$
$607$	32.8444	1.33311	0.666557	+	0.745454i	$0.267767\pi$
$608$	0.697224	0.0282762
$609$	0	0
$610$	0	0
$611$	22.8167	0.923063
$612$	0	0
$613$	0.486122	0.0196343	0.00981714	−	0.999952i	$-0.496875\pi$
$613$	0.486122	0.0196343	0.00981714	+	0.999952i	$0.496875\pi$
$614$	−3.02776	−0.122190
$615$	0	0
$616$	16.4222	0.661669
$617$	18.4222	0.741650	0.370825	−	0.928703i	$-0.379075\pi$
$617$	18.4222	0.741650	0.370825	+	0.928703i	$0.379075\pi$
$618$	0	0
$619$	10.4222	0.418904	0.209452	−	0.977819i	$-0.432832\pi$
$619$	10.4222	0.418904	0.209452	+	0.977819i	$0.432832\pi$
$620$	0	0
$621$	0	0
$622$	3.39445	0.136105
$623$	7.26662	0.291131
$624$	0	0
$625$	25.0000	1.00000
$626$	−20.0000	−0.799361
$627$	0	0
$628$	6.30278	0.251508
$629$	−12.0000	−0.478471
$630$	0	0
$631$	34.1194	1.35827	0.679137	−	0.734012i	$-0.262354\pi$
$631$	34.1194	1.35827	0.679137	+	0.734012i	$0.262354\pi$
$632$	−9.51388	−0.378442
$633$	0	0
$634$	−13.3028	−0.528321
$635$	0	0
$636$	0	0
$637$	−1.11943	−0.0443534
$638$	10.6972	0.423507
$639$	0	0
$640$	0	0
$641$	−40.5139	−1.60020	−0.800101	−	0.599865i	$-0.795221\pi$
$641$	−40.5139	−1.60020	−0.800101	+	0.599865i	$0.795221\pi$
$642$	0	0
$643$	30.3028	1.19502	0.597512	−	0.801860i	$-0.296156\pi$
$643$	30.3028	1.19502	0.597512	+	0.801860i	$0.296156\pi$
$644$	−10.4222	−0.410692
$645$	0	0
$646$	3.21110	0.126339
$647$	24.0917	0.947141	0.473571	−	0.880756i	$-0.342965\pi$
$647$	24.0917	0.947141	0.473571	+	0.880756i	$0.342965\pi$
$648$	0	0
$649$	−20.8167	−0.817125
$650$	−26.5139	−1.03996
$651$	0	0
$652$	1.39445	0.0546108
$653$	−7.63331	−0.298714	−0.149357	−	0.988783i	$-0.547720\pi$
$653$	−7.63331	−0.298714	−0.149357	+	0.988783i	$0.547720\pi$
$654$	0	0
$655$	0	0
$656$	−4.00000	−0.156174
$657$	0	0
$658$	11.2111	0.437054
$659$	−21.6333	−0.842714	−0.421357	−	0.906895i	$-0.638446\pi$
$659$	−21.6333	−0.842714	−0.421357	+	0.906895i	$0.638446\pi$
$660$	0	0
$661$	−37.9361	−1.47554	−0.737771	−	0.675051i	$-0.764122\pi$
$661$	−37.9361	−1.47554	−0.737771	+	0.675051i	$0.764122\pi$
$662$	−4.78890	−0.186126
$663$	0	0
$664$	9.21110	0.357460
$665$	0	0
$666$	0	0
$667$	−6.78890	−0.262867
$668$	−4.60555	−0.178194
$669$	0	0
$670$	0	0
$671$	−39.7250	−1.53357
$672$	0	0
$673$	11.1194	0.428623	0.214311	−	0.976765i	$-0.431249\pi$
$673$	11.1194	0.428623	0.214311	+	0.976765i	$0.431249\pi$
$674$	−9.21110	−0.354798
$675$	0	0
$676$	15.1194	0.581517
$677$	−20.7889	−0.798982	−0.399491	−	0.916737i	$-0.630813\pi$
$677$	−20.7889	−0.798982	−0.399491	+	0.916737i	$0.630813\pi$
$678$	0	0
$679$	−39.6333	−1.52099
$680$	0	0
$681$	0	0
$682$	−3.81665	−0.146147
$683$	−45.8167	−1.75313	−0.876563	−	0.481288i	$-0.840169\pi$
$683$	−45.8167	−1.75313	−0.876563	+	0.481288i	$0.840169\pi$
$684$	0	0
$685$	0	0
$686$	−18.7889	−0.717363
$687$	0	0
$688$	−6.90833	−0.263377
$689$	−20.7250	−0.789559
$690$	0	0
$691$	−0.183346	−0.00697482	−0.00348741	−	0.999994i	$-0.501110\pi$
$691$	−0.183346	−0.00697482	−0.00348741	+	0.999994i	$0.501110\pi$
$692$	23.0278	0.875384
$693$	0	0
$694$	18.4222	0.699297
$695$	0	0
$696$	0	0
$697$	−18.4222	−0.697791
$698$	−32.6056	−1.23414
$699$	0	0
$700$	−13.0278	−0.492403
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−1.81665	−0.0685164
$704$	6.30278	0.237545
$705$	0	0
$706$	22.6056	0.850771
$707$	0.238859	0.00898321
$708$	0	0
$709$	−3.57779	−0.134367	−0.0671835	−	0.997741i	$-0.521401\pi$
$709$	−3.57779	−0.134367	−0.0671835	+	0.997741i	$0.521401\pi$
$710$	0	0
$711$	0	0
$712$	2.78890	0.104518
$713$	2.42221	0.0907123
$714$	0	0
$715$	0	0
$716$	−1.21110	−0.0452610
$717$	0	0
$718$	5.48612	0.204740
$719$	21.2111	0.791041	0.395520	−	0.918457i	$-0.370564\pi$
$719$	21.2111	0.791041	0.395520	+	0.918457i	$0.370564\pi$
$720$	0	0
$721$	20.6056	0.767391
$722$	−18.5139	−0.689015
$723$	0	0
$724$	8.60555	0.319823
$725$	−8.48612	−0.315167
$726$	0	0
$727$	−14.1833	−0.526031	−0.263016	−	0.964792i	$-0.584717\pi$
$727$	−14.1833	−0.526031	−0.263016	+	0.964792i	$0.584717\pi$
$728$	13.8167	0.512079
$729$	0	0
$730$	0	0
$731$	−31.8167	−1.17678
$732$	0	0
$733$	20.4222	0.754311	0.377156	−	0.926150i	$-0.376902\pi$
$733$	20.4222	0.754311	0.377156	+	0.926150i	$0.376902\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−16.4222	−0.604920
$738$	0	0
$739$	43.4500	1.59833	0.799166	−	0.601110i	$-0.205275\pi$
$739$	43.4500	1.59833	0.799166	+	0.601110i	$0.205275\pi$
$740$	0	0
$741$	0	0
$742$	−10.1833	−0.373842
$743$	16.9361	0.621325	0.310662	−	0.950520i	$-0.399449\pi$
$743$	16.9361	0.621325	0.310662	+	0.950520i	$0.399449\pi$
$744$	0	0
$745$	0	0
$746$	30.8444	1.12929
$747$	0	0
$748$	29.0278	1.06136
$749$	−25.8167	−0.943320
$750$	0	0
$751$	43.4500	1.58551	0.792756	−	0.609539i	$-0.208646\pi$
$751$	43.4500	1.58551	0.792756	+	0.609539i	$0.208646\pi$
$752$	4.30278	0.156906
$753$	0	0
$754$	9.00000	0.327761
$755$	0	0
$756$	0	0
$757$	−6.33053	−0.230087	−0.115044	−	0.993360i	$-0.536701\pi$
$757$	−6.33053	−0.230087	−0.115044	+	0.993360i	$0.536701\pi$
$758$	36.3305	1.31958
$759$	0	0
$760$	0	0
$761$	−26.0917	−0.945822	−0.472911	−	0.881110i	$-0.656797\pi$
$761$	−26.0917	−0.945822	−0.472911	+	0.881110i	$0.656797\pi$
$762$	0	0
$763$	−32.8444	−1.18905
$764$	13.8167	0.499869
$765$	0	0
$766$	−11.8167	−0.426953
$767$	−17.5139	−0.632389
$768$	0	0
$769$	−34.1194	−1.23038	−0.615189	−	0.788380i	$-0.710920\pi$
$769$	−34.1194	−1.23038	−0.615189	+	0.788380i	$0.710920\pi$
$770$	0	0
$771$	0	0
$772$	−20.4222	−0.735011
$773$	−28.2389	−1.01568	−0.507841	−	0.861451i	$-0.669556\pi$
$773$	−28.2389	−1.01568	−0.507841	+	0.861451i	$0.669556\pi$
$774$	0	0
$775$	3.02776	0.108760
$776$	−15.2111	−0.546047
$777$	0	0
$778$	21.9361	0.786447
$779$	−2.78890	−0.0999226
$780$	0	0
$781$	−20.2389	−0.724203
$782$	−18.4222	−0.658777
$783$	0	0
$784$	−0.211103	−0.00753938
$785$	0	0
$786$	0	0
$787$	29.2111	1.04126	0.520632	−	0.853781i	$-0.325696\pi$
$787$	29.2111	1.04126	0.520632	+	0.853781i	$0.325696\pi$
$788$	−1.69722	−0.0604611
$789$	0	0
$790$	0	0
$791$	50.2944	1.78826
$792$	0	0
$793$	−33.4222	−1.18686
$794$	−19.5139	−0.692522
$795$	0	0
$796$	−18.4222	−0.652958
$797$	−12.4222	−0.440017	−0.220009	−	0.975498i	$-0.570609\pi$
$797$	−12.4222	−0.440017	−0.220009	+	0.975498i	$0.570609\pi$
$798$	0	0
$799$	19.8167	0.701063
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−2.97224	−0.104954
$803$	12.0278	0.424450
$804$	0	0
$805$	0	0
$806$	−3.21110	−0.113106
$807$	0	0
$808$	0.0916731	0.00322505
$809$	14.8444	0.521902	0.260951	−	0.965352i	$-0.415964\pi$
$809$	14.8444	0.521902	0.260951	+	0.965352i	$0.415964\pi$
$810$	0	0
$811$	4.23886	0.148846	0.0744232	−	0.997227i	$-0.476288\pi$
$811$	4.23886	0.148846	0.0744232	+	0.997227i	$0.476288\pi$
$812$	4.42221	0.155189
$813$	0	0
$814$	−16.4222	−0.575598
$815$	0	0
$816$	0	0
$817$	−4.81665	−0.168513
$818$	24.2389	0.847492
$819$	0	0
$820$	0	0
$821$	−1.45837	−0.0508973	−0.0254487	−	0.999676i	$-0.508101\pi$
$821$	−1.45837	−0.0508973	−0.0254487	+	0.999676i	$0.508101\pi$
$822$	0	0
$823$	−51.2666	−1.78704	−0.893521	−	0.449022i	$-0.851773\pi$
$823$	−51.2666	−1.78704	−0.893521	+	0.449022i	$0.851773\pi$
$824$	7.90833	0.275500
$825$	0	0
$826$	−8.60555	−0.299425
$827$	46.9361	1.63213	0.816064	−	0.577962i	$-0.196152\pi$
$827$	46.9361	1.63213	0.816064	+	0.577962i	$0.196152\pi$
$828$	0	0
$829$	−51.5139	−1.78915	−0.894575	−	0.446917i	$-0.852522\pi$
$829$	−51.5139	−1.78915	−0.894575	+	0.446917i	$0.852522\pi$
$830$	0	0
$831$	0	0
$832$	5.30278	0.183841
$833$	−0.972244	−0.0336862
$834$	0	0
$835$	0	0
$836$	4.39445	0.151985
$837$	0	0
$838$	3.39445	0.117259
$839$	−16.6056	−0.573287	−0.286644	−	0.958037i	$-0.592540\pi$
$839$	−16.6056	−0.573287	−0.286644	+	0.958037i	$0.592540\pi$
$840$	0	0
$841$	−26.1194	−0.900670
$842$	−19.7250	−0.679768
$843$	0	0
$844$	1.69722	0.0584209
$845$	0	0
$846$	0	0
$847$	74.8444	2.57168
$848$	−3.90833	−0.134212
$849$	0	0
$850$	−23.0278	−0.789846
$851$	10.4222	0.357269
$852$	0	0
$853$	28.1194	0.962791	0.481395	−	0.876504i	$-0.340130\pi$
$853$	28.1194	0.962791	0.481395	+	0.876504i	$0.340130\pi$
$854$	−16.4222	−0.561956
$855$	0	0
$856$	−9.90833	−0.338660
$857$	−7.39445	−0.252590	−0.126295	−	0.991993i	$-0.540309\pi$
$857$	−7.39445	−0.252590	−0.126295	+	0.991993i	$0.540309\pi$
$858$	0	0
$859$	−55.0278	−1.87752	−0.938761	−	0.344568i	$-0.888025\pi$
$859$	−55.0278	−1.87752	−0.938761	+	0.344568i	$0.888025\pi$
$860$	0	0
$861$	0	0
$862$	25.8167	0.879319
$863$	35.2111	1.19860	0.599300	−	0.800525i	$-0.295446\pi$
$863$	35.2111	1.19860	0.599300	+	0.800525i	$0.295446\pi$
$864$	0	0
$865$	0	0
$866$	21.5416	0.732015
$867$	0	0
$868$	−1.57779	−0.0535538
$869$	−59.9638	−2.03413
$870$	0	0
$871$	−13.8167	−0.468159
$872$	−12.6056	−0.426878
$873$	0	0
$874$	−2.78890	−0.0943359
$875$	0	0
$876$	0	0
$877$	40.1194	1.35474	0.677368	−	0.735644i	$-0.263120\pi$
$877$	40.1194	1.35474	0.677368	+	0.735644i	$0.263120\pi$
$878$	−15.6333	−0.527599
$879$	0	0
$880$	0	0
$881$	18.7889	0.633014	0.316507	−	0.948590i	$-0.397490\pi$
$881$	18.7889	0.633014	0.316507	+	0.948590i	$0.397490\pi$
$882$	0	0
$883$	32.0555	1.07875	0.539377	−	0.842064i	$-0.318660\pi$
$883$	32.0555	1.07875	0.539377	+	0.842064i	$0.318660\pi$
$884$	24.4222	0.821408
$885$	0	0
$886$	−8.60555	−0.289109
$887$	−0.275019	−0.00923424	−0.00461712	−	0.999989i	$-0.501470\pi$
$887$	−0.275019	−0.00923424	−0.00461712	+	0.999989i	$0.501470\pi$
$888$	0	0
$889$	−18.7889	−0.630159
$890$	0	0
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	2.60555	0.0870454
$897$	0	0
$898$	−0.788897	−0.0263258
$899$	−1.02776	−0.0342776
$900$	0	0
$901$	−18.0000	−0.599667
$902$	−25.2111	−0.839438
$903$	0	0
$904$	19.3028	0.642001
$905$	0	0
$906$	0	0
$907$	28.3305	0.940700	0.470350	−	0.882480i	$-0.344128\pi$
$907$	28.3305	0.940700	0.470350	+	0.882480i	$0.344128\pi$
$908$	−17.0278	−0.565086
$909$	0	0
$910$	0	0
$911$	9.57779	0.317327	0.158663	−	0.987333i	$-0.449282\pi$
$911$	9.57779	0.317327	0.158663	+	0.987333i	$0.449282\pi$
$912$	0	0
$913$	58.0555	1.92136
$914$	10.1833	0.336835
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−32.3667	−1.06884
$918$	0	0
$919$	−7.54163	−0.248776	−0.124388	−	0.992234i	$-0.539697\pi$
$919$	−7.54163	−0.248776	−0.124388	+	0.992234i	$0.539697\pi$
$920$	0	0
$921$	0	0
$922$	27.7250	0.913074
$923$	−17.0278	−0.560475
$924$	0	0
$925$	13.0278	0.428350
$926$	3.21110	0.105523
$927$	0	0
$928$	1.69722	0.0557141
$929$	−22.9722	−0.753695	−0.376847	−	0.926275i	$-0.622992\pi$
$929$	−22.9722	−0.753695	−0.376847	+	0.926275i	$0.622992\pi$
$930$	0	0
$931$	−0.147186	−0.00482382
$932$	−8.30278	−0.271966
$933$	0	0
$934$	−2.09167	−0.0684416
$935$	0	0
$936$	0	0
$937$	−24.7889	−0.809818	−0.404909	−	0.914357i	$-0.632697\pi$
$937$	−24.7889	−0.809818	−0.404909	+	0.914357i	$0.632697\pi$
$938$	−6.78890	−0.221665
$939$	0	0
$940$	0	0
$941$	49.1194	1.60125	0.800624	−	0.599167i	$-0.204502\pi$
$941$	49.1194	1.60125	0.800624	+	0.599167i	$0.204502\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	−3.30278	−0.107496
$945$	0	0
$946$	−43.5416	−1.41566
$947$	−25.8167	−0.838929	−0.419464	−	0.907772i	$-0.637782\pi$
$947$	−25.8167	−0.838929	−0.419464	+	0.907772i	$0.637782\pi$
$948$	0	0
$949$	10.1194	0.328491
$950$	−3.48612	−0.113105
$951$	0	0
$952$	12.0000	0.388922
$953$	44.0555	1.42710	0.713549	−	0.700605i	$-0.247087\pi$
$953$	44.0555	1.42710	0.713549	+	0.700605i	$0.247087\pi$
$954$	0	0
$955$	0	0
$956$	23.7250	0.767321
$957$	0	0
$958$	36.9083	1.19245
$959$	9.63331	0.311076
$960$	0	0
$961$	−30.6333	−0.988171
$962$	−13.8167	−0.445467
$963$	0	0
$964$	13.5139	0.435253
$965$	0	0
$966$	0	0
$967$	12.0917	0.388842	0.194421	−	0.980918i	$-0.437717\pi$
$967$	12.0917	0.388842	0.194421	+	0.980918i	$0.437717\pi$
$968$	28.7250	0.923256
$969$	0	0
$970$	0	0
$971$	−36.7250	−1.17856	−0.589280	−	0.807929i	$-0.700589\pi$
$971$	−36.7250	−1.17856	−0.589280	+	0.807929i	$0.700589\pi$
$972$	0	0
$973$	−2.84441	−0.0911876
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−6.30278	−0.201747
$977$	−30.7250	−0.982979	−0.491490	−	0.870883i	$-0.663547\pi$
$977$	−30.7250	−0.982979	−0.491490	+	0.870883i	$0.663547\pi$
$978$	0	0
$979$	17.5778	0.561789
$980$	0	0
$981$	0	0
$982$	−13.9361	−0.444718
$983$	−36.7889	−1.17338	−0.586692	−	0.809810i	$-0.699570\pi$
$983$	−36.7889	−1.17338	−0.586692	+	0.809810i	$0.699570\pi$
$984$	0	0
$985$	0	0
$986$	7.81665	0.248933
$987$	0	0
$988$	3.69722	0.117624
$989$	27.6333	0.878688
$990$	0	0
$991$	30.8806	0.980954	0.490477	−	0.871454i	$-0.336823\pi$
$991$	30.8806	0.980954	0.490477	+	0.871454i	$0.336823\pi$
$992$	−0.605551	−0.0192263
$993$	0	0
$994$	−8.36669	−0.265375
$995$	0	0
$996$	0	0
$997$	11.8167	0.374237	0.187119	−	0.982337i	$-0.440085\pi$
$997$	11.8167	0.374237	0.187119	+	0.982337i	$0.440085\pi$
$998$	−6.30278	−0.199511
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.l.1.2	yes	2
3.2	odd	2		4014.2.a.j.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.j.1.2	✓	2	3.2	odd	2
4014.2.a.l.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.60555 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.60555 q^{7} +1.00000 q^{8} +6.30278 q^{11} +5.30278 q^{13} +2.60555 q^{14} +1.00000 q^{16} +4.60555 q^{17} +0.697224 q^{19} +6.30278 q^{22} -4.00000 q^{23} -5.00000 q^{25} +5.30278 q^{26} +2.60555 q^{28} +1.69722 q^{29} -0.605551 q^{31} +1.00000 q^{32} +4.60555 q^{34} -2.60555 q^{37} +0.697224 q^{38} -4.00000 q^{41} -6.90833 q^{43} +6.30278 q^{44} -4.00000 q^{46} +4.30278 q^{47} -0.211103 q^{49} -5.00000 q^{50} +5.30278 q^{52} -3.90833 q^{53} +2.60555 q^{56} +1.69722 q^{58} -3.30278 q^{59} -6.30278 q^{61} -0.605551 q^{62} +1.00000 q^{64} -2.60555 q^{67} +4.60555 q^{68} -3.21110 q^{71} +1.90833 q^{73} -2.60555 q^{74} +0.697224 q^{76} +16.4222 q^{77} -9.51388 q^{79} -4.00000 q^{82} +9.21110 q^{83} -6.90833 q^{86} +6.30278 q^{88} +2.78890 q^{89} +13.8167 q^{91} -4.00000 q^{92} +4.30278 q^{94} -15.2111 q^{97} -0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 9 q^{11} + 7 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 5 q^{19} + 9 q^{22} - 8 q^{23} - 10 q^{25} + 7 q^{26} - 2 q^{28} + 7 q^{29} + 6 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{37} + 5 q^{38} - 8 q^{41} - 3 q^{43} + 9 q^{44} - 8 q^{46} + 5 q^{47} + 14 q^{49} - 10 q^{50} + 7 q^{52} + 3 q^{53} - 2 q^{56} + 7 q^{58} - 3 q^{59} - 9 q^{61} + 6 q^{62} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} - 7 q^{73} + 2 q^{74} + 5 q^{76} + 4 q^{77} - q^{79} - 8 q^{82} + 4 q^{83} - 3 q^{86} + 9 q^{88} + 20 q^{89} + 6 q^{91} - 8 q^{92} + 5 q^{94} - 16 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 9 * q^11 + 7 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 5 * q^19 + 9 * q^22 - 8 * q^23 - 10 * q^25 + 7 * q^26 - 2 * q^28 + 7 * q^29 + 6 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^37 + 5 * q^38 - 8 * q^41 - 3 * q^43 + 9 * q^44 - 8 * q^46 + 5 * q^47 + 14 * q^49 - 10 * q^50 + 7 * q^52 + 3 * q^53 - 2 * q^56 + 7 * q^58 - 3 * q^59 - 9 * q^61 + 6 * q^62 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 - 7 * q^73 + 2 * q^74 + 5 * q^76 + 4 * q^77 - q^79 - 8 * q^82 + 4 * q^83 - 3 * q^86 + 9 * q^88 + 20 * q^89 + 6 * q^91 - 8 * q^92 + 5 * q^94 - 16 * q^97 + 14 * q^98

Introduction

L-functions

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Database

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