LMFDB - Embedded newform 4014.2.a.z.1.7

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 12x^{6} + 46x^{5} + 54x^{4} - 148x^{3} - 98x^{2} + 126x + 69 $ x^8 - 4x^7 - 12x^6 + 46x^5 + 54x^4 - 148x^3 - 98x^2 + 126*x + 69
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.36314$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.80312 q^{5} +1.62712 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.80312 q^{5} +1.62712 q^{7} +1.00000 q^{8} +2.80312 q^{10} +4.88403 q^{11} +3.62240 q^{13} +1.62712 q^{14} +1.00000 q^{16} -7.41973 q^{17} +5.70702 q^{19} +2.80312 q^{20} +4.88403 q^{22} -1.28060 q^{23} +2.85748 q^{25} +3.62240 q^{26} +1.62712 q^{28} +4.72629 q^{29} -9.52740 q^{31} +1.00000 q^{32} -7.41973 q^{34} +4.56101 q^{35} +2.72697 q^{37} +5.70702 q^{38} +2.80312 q^{40} +7.35248 q^{41} -5.14186 q^{43} +4.88403 q^{44} -1.28060 q^{46} +3.87590 q^{47} -4.35248 q^{49} +2.85748 q^{50} +3.62240 q^{52} -1.89922 q^{53} +13.6905 q^{55} +1.62712 q^{56} +4.72629 q^{58} -2.18450 q^{59} -8.49027 q^{61} -9.52740 q^{62} +1.00000 q^{64} +10.1540 q^{65} -7.77136 q^{67} -7.41973 q^{68} +4.56101 q^{70} -1.80780 q^{71} +1.22647 q^{73} +2.72697 q^{74} +5.70702 q^{76} +7.94690 q^{77} -6.91055 q^{79} +2.80312 q^{80} +7.35248 q^{82} -14.8463 q^{83} -20.7984 q^{85} -5.14186 q^{86} +4.88403 q^{88} -7.10986 q^{89} +5.89409 q^{91} -1.28060 q^{92} +3.87590 q^{94} +15.9975 q^{95} +2.46564 q^{97} -4.35248 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 4 * q^5 + 4 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8} - 4 q^{10} - 2 q^{11} + 10 q^{13} + 4 q^{14} + 8 q^{16} - 8 q^{17} + 16 q^{19} - 4 q^{20} - 2 q^{22} + 4 q^{23} + 24 q^{25} + 10 q^{26} + 4 q^{28} - 8 q^{29} + 12 q^{31} + 8 q^{32} - 8 q^{34} + 24 q^{35} + 16 q^{37} + 16 q^{38} - 4 q^{40} - 16 q^{41} + 12 q^{43} - 2 q^{44} + 4 q^{46} + 4 q^{47} + 40 q^{49} + 24 q^{50} + 10 q^{52} + 8 q^{53} + 8 q^{55} + 4 q^{56} - 8 q^{58} + 26 q^{61} + 12 q^{62} + 8 q^{64} + 24 q^{65} - 10 q^{67} - 8 q^{68} + 24 q^{70} - 8 q^{71} + 8 q^{73} + 16 q^{74} + 16 q^{76} + 56 q^{77} + 16 q^{79} - 4 q^{80} - 16 q^{82} + 44 q^{83} - 18 q^{85} + 12 q^{86} - 2 q^{88} + 4 q^{91} + 4 q^{92} + 4 q^{94} + 48 q^{95} + 12 q^{97} + 40 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 4 * q^5 + 4 * q^7 + 8 * q^8 - 4 * q^10 - 2 * q^11 + 10 * q^13 + 4 * q^14 + 8 * q^16 - 8 * q^17 + 16 * q^19 - 4 * q^20 - 2 * q^22 + 4 * q^23 + 24 * q^25 + 10 * q^26 + 4 * q^28 - 8 * q^29 + 12 * q^31 + 8 * q^32 - 8 * q^34 + 24 * q^35 + 16 * q^37 + 16 * q^38 - 4 * q^40 - 16 * q^41 + 12 * q^43 - 2 * q^44 + 4 * q^46 + 4 * q^47 + 40 * q^49 + 24 * q^50 + 10 * q^52 + 8 * q^53 + 8 * q^55 + 4 * q^56 - 8 * q^58 + 26 * q^61 + 12 * q^62 + 8 * q^64 + 24 * q^65 - 10 * q^67 - 8 * q^68 + 24 * q^70 - 8 * q^71 + 8 * q^73 + 16 * q^74 + 16 * q^76 + 56 * q^77 + 16 * q^79 - 4 * q^80 - 16 * q^82 + 44 * q^83 - 18 * q^85 + 12 * q^86 - 2 * q^88 + 4 * q^91 + 4 * q^92 + 4 * q^94 + 48 * q^95 + 12 * q^97 + 40 * 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
$5$	2.80312	1.25359	0.626797	−	0.779183i	$-0.284366\pi$
$5$	2.80312	1.25359	0.626797	+	0.779183i	$0.284366\pi$
$6$	0	0
$7$	1.62712	0.614994	0.307497	−	0.951549i	$-0.400509\pi$
$7$	1.62712	0.614994	0.307497	+	0.951549i	$0.400509\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.80312	0.886424
$11$	4.88403	1.47259	0.736295	−	0.676661i	$-0.236574\pi$
$11$	4.88403	1.47259	0.736295	+	0.676661i	$0.236574\pi$
$12$	0	0
$13$	3.62240	1.00467	0.502337	−	0.864672i	$-0.332474\pi$
$13$	3.62240	1.00467	0.502337	+	0.864672i	$0.332474\pi$
$14$	1.62712	0.434866
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.41973	−1.79955	−0.899774	−	0.436356i	$-0.856269\pi$
$17$	−7.41973	−1.79955	−0.899774	+	0.436356i	$0.856269\pi$
$18$	0	0
$19$	5.70702	1.30928	0.654640	−	0.755941i	$-0.272820\pi$
$19$	5.70702	1.30928	0.654640	+	0.755941i	$0.272820\pi$
$20$	2.80312	0.626797
$21$	0	0
$22$	4.88403	1.04128
$23$	−1.28060	−0.267024	−0.133512	−	0.991047i	$-0.542625\pi$
$23$	−1.28060	−0.267024	−0.133512	+	0.991047i	$0.542625\pi$
$24$	0	0
$25$	2.85748	0.571496
$26$	3.62240	0.710412
$27$	0	0
$28$	1.62712	0.307497
$29$	4.72629	0.877650	0.438825	−	0.898573i	$-0.355395\pi$
$29$	4.72629	0.877650	0.438825	+	0.898573i	$0.355395\pi$
$30$	0	0
$31$	−9.52740	−1.71117	−0.855585	−	0.517662i	$-0.826803\pi$
$31$	−9.52740	−1.71117	−0.855585	+	0.517662i	$0.826803\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.41973	−1.27247
$35$	4.56101	0.770952
$36$	0	0
$37$	2.72697	0.448311	0.224156	−	0.974553i	$-0.428038\pi$
$37$	2.72697	0.448311	0.224156	+	0.974553i	$0.428038\pi$
$38$	5.70702	0.925801
$39$	0	0
$40$	2.80312	0.443212
$41$	7.35248	1.14826	0.574132	−	0.818763i	$-0.305340\pi$
$41$	7.35248	1.14826	0.574132	+	0.818763i	$0.305340\pi$
$42$	0	0
$43$	−5.14186	−0.784127	−0.392063	−	0.919938i	$-0.628239\pi$
$43$	−5.14186	−0.784127	−0.392063	+	0.919938i	$0.628239\pi$
$44$	4.88403	0.736295
$45$	0	0
$46$	−1.28060	−0.188814
$47$	3.87590	0.565358	0.282679	−	0.959215i	$-0.408777\pi$
$47$	3.87590	0.565358	0.282679	+	0.959215i	$0.408777\pi$
$48$	0	0
$49$	−4.35248	−0.621783
$50$	2.85748	0.404109
$51$	0	0
$52$	3.62240	0.502337
$53$	−1.89922	−0.260878	−0.130439	−	0.991456i	$-0.541639\pi$
$53$	−1.89922	−0.260878	−0.130439	+	0.991456i	$0.541639\pi$
$54$	0	0
$55$	13.6905	1.84603
$56$	1.62712	0.217433
$57$	0	0
$58$	4.72629	0.620592
$59$	−2.18450	−0.284398	−0.142199	−	0.989838i	$-0.545417\pi$
$59$	−2.18450	−0.284398	−0.142199	+	0.989838i	$0.545417\pi$
$60$	0	0
$61$	−8.49027	−1.08707	−0.543534	−	0.839387i	$-0.682914\pi$
$61$	−8.49027	−1.08707	−0.543534	+	0.839387i	$0.682914\pi$
$62$	−9.52740	−1.20998
$63$	0	0
$64$	1.00000	0.125000
$65$	10.1540	1.25945
$66$	0	0
$67$	−7.77136	−0.949423	−0.474711	−	0.880142i	$-0.657448\pi$
$67$	−7.77136	−0.949423	−0.474711	+	0.880142i	$0.657448\pi$
$68$	−7.41973	−0.899774
$69$	0	0
$70$	4.56101	0.545145
$71$	−1.80780	−0.214546	−0.107273	−	0.994230i	$-0.534212\pi$
$71$	−1.80780	−0.214546	−0.107273	+	0.994230i	$0.534212\pi$
$72$	0	0
$73$	1.22647	0.143547	0.0717737	−	0.997421i	$-0.477134\pi$
$73$	1.22647	0.143547	0.0717737	+	0.997421i	$0.477134\pi$
$74$	2.72697	0.317004
$75$	0	0
$76$	5.70702	0.654640
$77$	7.94690	0.905633
$78$	0	0
$79$	−6.91055	−0.777497	−0.388749	−	0.921344i	$-0.627093\pi$
$79$	−6.91055	−0.777497	−0.388749	+	0.921344i	$0.627093\pi$
$80$	2.80312	0.313398
$81$	0	0
$82$	7.35248	0.811946
$83$	−14.8463	−1.62960	−0.814799	−	0.579744i	$-0.803153\pi$
$83$	−14.8463	−1.62960	−0.814799	+	0.579744i	$0.803153\pi$
$84$	0	0
$85$	−20.7984	−2.25590
$86$	−5.14186	−0.554461
$87$	0	0
$88$	4.88403	0.520639
$89$	−7.10986	−0.753644	−0.376822	−	0.926286i	$-0.622983\pi$
$89$	−7.10986	−0.753644	−0.376822	+	0.926286i	$0.622983\pi$
$90$	0	0
$91$	5.89409	0.617868
$92$	−1.28060	−0.133512
$93$	0	0
$94$	3.87590	0.399768
$95$	15.9975	1.64130
$96$	0	0
$97$	2.46564	0.250348	0.125174	−	0.992135i	$-0.460051\pi$
$97$	2.46564	0.250348	0.125174	+	0.992135i	$0.460051\pi$
$98$	−4.35248	−0.439667
$99$	0	0
$100$	2.85748	0.285748
$101$	2.76208	0.274837	0.137419	−	0.990513i	$-0.456119\pi$
$101$	2.76208	0.274837	0.137419	+	0.990513i	$0.456119\pi$
$102$	0	0
$103$	1.58040	0.155722	0.0778608	−	0.996964i	$-0.475191\pi$
$103$	1.58040	0.155722	0.0778608	+	0.996964i	$0.475191\pi$
$104$	3.62240	0.355206
$105$	0	0
$106$	−1.89922	−0.184469
$107$	17.4350	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$107$	17.4350	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$108$	0	0
$109$	−12.4801	−1.19538	−0.597688	−	0.801728i	$-0.703914\pi$
$109$	−12.4801	−1.19538	−0.597688	+	0.801728i	$0.703914\pi$
$110$	13.6905	1.30534
$111$	0	0
$112$	1.62712	0.153748
$113$	−5.00641	−0.470963	−0.235482	−	0.971879i	$-0.575667\pi$
$113$	−5.00641	−0.470963	−0.235482	+	0.971879i	$0.575667\pi$
$114$	0	0
$115$	−3.58968	−0.334739
$116$	4.72629	0.438825
$117$	0	0
$118$	−2.18450	−0.201100
$119$	−12.0728	−1.10671
$120$	0	0
$121$	12.8537	1.16852
$122$	−8.49027	−0.768673
$123$	0	0
$124$	−9.52740	−0.855585
$125$	−6.00574	−0.537169
$126$	0	0
$127$	0.506922	0.0449821	0.0224910	−	0.999747i	$-0.492840\pi$
$127$	0.506922	0.0449821	0.0224910	+	0.999747i	$0.492840\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.1540	0.890568
$131$	10.7751	0.941429	0.470714	−	0.882286i	$-0.343996\pi$
$131$	10.7751	0.941429	0.470714	+	0.882286i	$0.343996\pi$
$132$	0	0
$133$	9.28601	0.805199
$134$	−7.77136	−0.671343
$135$	0	0
$136$	−7.41973	−0.636236
$137$	−8.36832	−0.714954	−0.357477	−	0.933922i	$-0.616363\pi$
$137$	−8.36832	−0.714954	−0.357477	+	0.933922i	$0.616363\pi$
$138$	0	0
$139$	−0.475069	−0.0402948	−0.0201474	−	0.999797i	$-0.506414\pi$
$139$	−0.475069	−0.0402948	−0.0201474	+	0.999797i	$0.506414\pi$
$140$	4.56101	0.385476
$141$	0	0
$142$	−1.80780	−0.151707
$143$	17.6919	1.47947
$144$	0	0
$145$	13.2484	1.10022
$146$	1.22647	0.101503
$147$	0	0
$148$	2.72697	0.224156
$149$	16.2885	1.33440	0.667201	−	0.744878i	$-0.267492\pi$
$149$	16.2885	1.33440	0.667201	+	0.744878i	$0.267492\pi$
$150$	0	0
$151$	0.513029	0.0417497	0.0208749	−	0.999782i	$-0.493355\pi$
$151$	0.513029	0.0417497	0.0208749	+	0.999782i	$0.493355\pi$
$152$	5.70702	0.462900
$153$	0	0
$154$	7.94690	0.640379
$155$	−26.7064	−2.14511
$156$	0	0
$157$	6.63432	0.529476	0.264738	−	0.964320i	$-0.414715\pi$
$157$	6.63432	0.529476	0.264738	+	0.964320i	$0.414715\pi$
$158$	−6.91055	−0.549774
$159$	0	0
$160$	2.80312	0.221606
$161$	−2.08369	−0.164218
$162$	0	0
$163$	−10.1213	−0.792759	−0.396379	−	0.918087i	$-0.629733\pi$
$163$	−10.1213	−0.792759	−0.396379	+	0.918087i	$0.629733\pi$
$164$	7.35248	0.574132
$165$	0	0
$166$	−14.8463	−1.15230
$167$	−18.5167	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$167$	−18.5167	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$168$	0	0
$169$	0.121818	0.00937058
$170$	−20.7984	−1.59516
$171$	0	0
$172$	−5.14186	−0.392063
$173$	7.89063	0.599913	0.299957	−	0.953953i	$-0.403028\pi$
$173$	7.89063	0.599913	0.299957	+	0.953953i	$0.403028\pi$
$174$	0	0
$175$	4.64947	0.351467
$176$	4.88403	0.368147
$177$	0	0
$178$	−7.10986	−0.532907
$179$	11.3018	0.844737	0.422369	−	0.906424i	$-0.361199\pi$
$179$	11.3018	0.844737	0.422369	+	0.906424i	$0.361199\pi$
$180$	0	0
$181$	6.66659	0.495524	0.247762	−	0.968821i	$-0.420305\pi$
$181$	6.66659	0.495524	0.247762	+	0.968821i	$0.420305\pi$
$182$	5.89409	0.436899
$183$	0	0
$184$	−1.28060	−0.0944072
$185$	7.64403	0.562000
$186$	0	0
$187$	−36.2381	−2.65000
$188$	3.87590	0.282679
$189$	0	0
$190$	15.9975	1.16058
$191$	−9.93188	−0.718645	−0.359323	−	0.933213i	$-0.616992\pi$
$191$	−9.93188	−0.718645	−0.359323	+	0.933213i	$0.616992\pi$
$192$	0	0
$193$	12.3928	0.892052	0.446026	−	0.895020i	$-0.352839\pi$
$193$	12.3928	0.892052	0.446026	+	0.895020i	$0.352839\pi$
$194$	2.46564	0.177023
$195$	0	0
$196$	−4.35248	−0.310891
$197$	5.56956	0.396814	0.198407	−	0.980120i	$-0.436423\pi$
$197$	5.56956	0.396814	0.198407	+	0.980120i	$0.436423\pi$
$198$	0	0
$199$	21.3068	1.51040	0.755199	−	0.655496i	$-0.227540\pi$
$199$	21.3068	1.51040	0.755199	+	0.655496i	$0.227540\pi$
$200$	2.85748	0.202054
$201$	0	0
$202$	2.76208	0.194339
$203$	7.69024	0.539749
$204$	0	0
$205$	20.6099	1.43946
$206$	1.58040	0.110112
$207$	0	0
$208$	3.62240	0.251169
$209$	27.8732	1.92803
$210$	0	0
$211$	4.56855	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$211$	4.56855	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$212$	−1.89922	−0.130439
$213$	0	0
$214$	17.4350	1.19183
$215$	−14.4133	−0.982976
$216$	0	0
$217$	−15.5022	−1.05236
$218$	−12.4801	−0.845259
$219$	0	0
$220$	13.6905	0.923014
$221$	−26.8773	−1.80796
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	1.62712	0.108717
$225$	0	0
$226$	−5.00641	−0.333021
$227$	1.37598	0.0913268	0.0456634	−	0.998957i	$-0.485460\pi$
$227$	1.37598	0.0913268	0.0456634	+	0.998957i	$0.485460\pi$
$228$	0	0
$229$	−10.4332	−0.689442	−0.344721	−	0.938705i	$-0.612026\pi$
$229$	−10.4332	−0.689442	−0.344721	+	0.938705i	$0.612026\pi$
$230$	−3.58968	−0.236697
$231$	0	0
$232$	4.72629	0.310296
$233$	3.28891	0.215464	0.107732	−	0.994180i	$-0.465641\pi$
$233$	3.28891	0.215464	0.107732	+	0.994180i	$0.465641\pi$
$234$	0	0
$235$	10.8646	0.708729
$236$	−2.18450	−0.142199
$237$	0	0
$238$	−12.0728	−0.782562
$239$	17.5274	1.13375	0.566877	−	0.823803i	$-0.308152\pi$
$239$	17.5274	1.13375	0.566877	+	0.823803i	$0.308152\pi$
$240$	0	0
$241$	−0.255243	−0.0164417	−0.00822083	−	0.999966i	$-0.502617\pi$
$241$	−0.255243	−0.0164417	−0.00822083	+	0.999966i	$0.502617\pi$
$242$	12.8537	0.826268
$243$	0	0
$244$	−8.49027	−0.543534
$245$	−12.2005	−0.779463
$246$	0	0
$247$	20.6731	1.31540
$248$	−9.52740	−0.604990
$249$	0	0
$250$	−6.00574	−0.379836
$251$	−5.33942	−0.337021	−0.168510	−	0.985700i	$-0.553896\pi$
$251$	−5.33942	−0.337021	−0.168510	+	0.985700i	$0.553896\pi$
$252$	0	0
$253$	−6.25449	−0.393217
$254$	0.506922	0.0318071
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.3493	−0.957462	−0.478731	−	0.877962i	$-0.658903\pi$
$257$	−15.3493	−0.957462	−0.478731	+	0.877962i	$0.658903\pi$
$258$	0	0
$259$	4.43711	0.275709
$260$	10.1540	0.629727
$261$	0	0
$262$	10.7751	0.665691
$263$	26.6345	1.64235	0.821177	−	0.570673i	$-0.193318\pi$
$263$	26.6345	1.64235	0.821177	+	0.570673i	$0.193318\pi$
$264$	0	0
$265$	−5.32374	−0.327035
$266$	9.28601	0.569362
$267$	0	0
$268$	−7.77136	−0.474711
$269$	−8.87441	−0.541083	−0.270541	−	0.962708i	$-0.587203\pi$
$269$	−8.87441	−0.541083	−0.270541	+	0.962708i	$0.587203\pi$
$270$	0	0
$271$	14.9484	0.908051	0.454025	−	0.890989i	$-0.349988\pi$
$271$	14.9484	0.908051	0.454025	+	0.890989i	$0.349988\pi$
$272$	−7.41973	−0.449887
$273$	0	0
$274$	−8.36832	−0.505549
$275$	13.9560	0.841580
$276$	0	0
$277$	−10.4679	−0.628957	−0.314479	−	0.949265i	$-0.601830\pi$
$277$	−10.4679	−0.628957	−0.314479	+	0.949265i	$0.601830\pi$
$278$	−0.475069	−0.0284927
$279$	0	0
$280$	4.56101	0.272573
$281$	3.55084	0.211826	0.105913	−	0.994375i	$-0.466224\pi$
$281$	3.55084	0.211826	0.105913	+	0.994375i	$0.466224\pi$
$282$	0	0
$283$	32.0270	1.90381	0.951903	−	0.306400i	$-0.0991245\pi$
$283$	32.0270	1.90381	0.951903	+	0.306400i	$0.0991245\pi$
$284$	−1.80780	−0.107273
$285$	0	0
$286$	17.6919	1.04615
$287$	11.9634	0.706175
$288$	0	0
$289$	38.0523	2.23837
$290$	13.2484	0.777970
$291$	0	0
$292$	1.22647	0.0717737
$293$	−33.2339	−1.94154	−0.970772	−	0.240003i	$-0.922852\pi$
$293$	−33.2339	−1.94154	−0.970772	+	0.240003i	$0.922852\pi$
$294$	0	0
$295$	−6.12342	−0.356519
$296$	2.72697	0.158502
$297$	0	0
$298$	16.2885	0.943565
$299$	−4.63886	−0.268272
$300$	0	0
$301$	−8.36643	−0.482233
$302$	0.513029	0.0295215
$303$	0	0
$304$	5.70702	0.327320
$305$	−23.7992	−1.36274
$306$	0	0
$307$	−20.4035	−1.16449	−0.582244	−	0.813014i	$-0.697825\pi$
$307$	−20.4035	−1.16449	−0.582244	+	0.813014i	$0.697825\pi$
$308$	7.94690	0.452817
$309$	0	0
$310$	−26.7064	−1.51682
$311$	8.99243	0.509914	0.254957	−	0.966952i	$-0.417939\pi$
$311$	8.99243	0.509914	0.254957	+	0.966952i	$0.417939\pi$
$312$	0	0
$313$	−4.30737	−0.243467	−0.121734	−	0.992563i	$-0.538845\pi$
$313$	−4.30737	−0.243467	−0.121734	+	0.992563i	$0.538845\pi$
$314$	6.63432	0.374396
$315$	0	0
$316$	−6.91055	−0.388749
$317$	3.84895	0.216179	0.108089	−	0.994141i	$-0.465527\pi$
$317$	3.84895	0.216179	0.108089	+	0.994141i	$0.465527\pi$
$318$	0	0
$319$	23.0833	1.29242
$320$	2.80312	0.156699
$321$	0	0
$322$	−2.08369	−0.116120
$323$	−42.3445	−2.35611
$324$	0	0
$325$	10.3510	0.574168
$326$	−10.1213	−0.560565
$327$	0	0
$328$	7.35248	0.405973
$329$	6.30655	0.347692
$330$	0	0
$331$	−8.07055	−0.443598	−0.221799	−	0.975092i	$-0.571193\pi$
$331$	−8.07055	−0.443598	−0.221799	+	0.975092i	$0.571193\pi$
$332$	−14.8463	−0.814799
$333$	0	0
$334$	−18.5167	−1.01319
$335$	−21.7841	−1.19019
$336$	0	0
$337$	10.2899	0.560525	0.280263	−	0.959923i	$-0.409578\pi$
$337$	10.2899	0.560525	0.280263	+	0.959923i	$0.409578\pi$
$338$	0.121818	0.00662600
$339$	0	0
$340$	−20.7984	−1.12795
$341$	−46.5321	−2.51985
$342$	0	0
$343$	−18.4718	−0.997386
$344$	−5.14186	−0.277231
$345$	0	0
$346$	7.89063	0.424203
$347$	13.7951	0.740560	0.370280	−	0.928920i	$-0.379262\pi$
$347$	13.7951	0.740560	0.370280	+	0.928920i	$0.379262\pi$
$348$	0	0
$349$	−11.7721	−0.630145	−0.315072	−	0.949068i	$-0.602029\pi$
$349$	−11.7721	−0.630145	−0.315072	+	0.949068i	$0.602029\pi$
$350$	4.64947	0.248524
$351$	0	0
$352$	4.88403	0.260319
$353$	−19.8217	−1.05501	−0.527503	−	0.849553i	$-0.676871\pi$
$353$	−19.8217	−1.05501	−0.527503	+	0.849553i	$0.676871\pi$
$354$	0	0
$355$	−5.06748	−0.268954
$356$	−7.10986	−0.376822
$357$	0	0
$358$	11.3018	0.597320
$359$	6.89554	0.363933	0.181966	−	0.983305i	$-0.441754\pi$
$359$	6.89554	0.363933	0.181966	+	0.983305i	$0.441754\pi$
$360$	0	0
$361$	13.5701	0.714214
$362$	6.66659	0.350388
$363$	0	0
$364$	5.89409	0.308934
$365$	3.43794	0.179950
$366$	0	0
$367$	11.9728	0.624973	0.312486	−	0.949922i	$-0.398838\pi$
$367$	11.9728	0.624973	0.312486	+	0.949922i	$0.398838\pi$
$368$	−1.28060	−0.0667560
$369$	0	0
$370$	7.64403	0.397394
$371$	−3.09026	−0.160438
$372$	0	0
$373$	−17.7096	−0.916970	−0.458485	−	0.888702i	$-0.651608\pi$
$373$	−17.7096	−0.916970	−0.458485	+	0.888702i	$0.651608\pi$
$374$	−36.2381	−1.87383
$375$	0	0
$376$	3.87590	0.199884
$377$	17.1205	0.881752
$378$	0	0
$379$	6.18691	0.317800	0.158900	−	0.987295i	$-0.449205\pi$
$379$	6.18691	0.317800	0.158900	+	0.987295i	$0.449205\pi$
$380$	15.9975	0.820652
$381$	0	0
$382$	−9.93188	−0.508159
$383$	−12.2112	−0.623963	−0.311982	−	0.950088i	$-0.600993\pi$
$383$	−12.2112	−0.623963	−0.311982	+	0.950088i	$0.600993\pi$
$384$	0	0
$385$	22.2761	1.13530
$386$	12.3928	0.630776
$387$	0	0
$388$	2.46564	0.125174
$389$	−20.3603	−1.03231	−0.516154	−	0.856496i	$-0.672637\pi$
$389$	−20.3603	−1.03231	−0.516154	+	0.856496i	$0.672637\pi$
$390$	0	0
$391$	9.50171	0.480522
$392$	−4.35248	−0.219833
$393$	0	0
$394$	5.56956	0.280590
$395$	−19.3711	−0.974665
$396$	0	0
$397$	−26.7165	−1.34087	−0.670433	−	0.741970i	$-0.733891\pi$
$397$	−26.7165	−1.34087	−0.670433	+	0.741970i	$0.733891\pi$
$398$	21.3068	1.06801
$399$	0	0
$400$	2.85748	0.142874
$401$	−20.2413	−1.01080	−0.505400	−	0.862885i	$-0.668655\pi$
$401$	−20.2413	−1.01080	−0.505400	+	0.862885i	$0.668655\pi$
$402$	0	0
$403$	−34.5121	−1.71917
$404$	2.76208	0.137419
$405$	0	0
$406$	7.69024	0.381660
$407$	13.3186	0.660178
$408$	0	0
$409$	29.0470	1.43628	0.718140	−	0.695899i	$-0.244994\pi$
$409$	29.0470	1.43628	0.718140	+	0.695899i	$0.244994\pi$
$410$	20.6099	1.01785
$411$	0	0
$412$	1.58040	0.0778608
$413$	−3.55445	−0.174903
$414$	0	0
$415$	−41.6161	−2.04285
$416$	3.62240	0.177603
$417$	0	0
$418$	27.8732	1.36332
$419$	9.10873	0.444990	0.222495	−	0.974934i	$-0.428580\pi$
$419$	9.10873	0.444990	0.222495	+	0.974934i	$0.428580\pi$
$420$	0	0
$421$	29.6256	1.44386	0.721931	−	0.691966i	$-0.243255\pi$
$421$	29.6256	1.44386	0.721931	+	0.691966i	$0.243255\pi$
$422$	4.56855	0.222394
$423$	0	0
$424$	−1.89922	−0.0922343
$425$	−21.2017	−1.02844
$426$	0	0
$427$	−13.8147	−0.668539
$428$	17.4350	0.842754
$429$	0	0
$430$	−14.4133	−0.695069
$431$	9.05749	0.436284	0.218142	−	0.975917i	$-0.430000\pi$
$431$	9.05749	0.436284	0.218142	+	0.975917i	$0.430000\pi$
$432$	0	0
$433$	−15.1574	−0.728418	−0.364209	−	0.931317i	$-0.618661\pi$
$433$	−15.1574	−0.728418	−0.364209	+	0.931317i	$0.618661\pi$
$434$	−15.5022	−0.744130
$435$	0	0
$436$	−12.4801	−0.597688
$437$	−7.30842	−0.349609
$438$	0	0
$439$	22.8847	1.09223	0.546114	−	0.837711i	$-0.316106\pi$
$439$	22.8847	1.09223	0.546114	+	0.837711i	$0.316106\pi$
$440$	13.6905	0.652670
$441$	0	0
$442$	−26.8773	−1.27842
$443$	14.8447	0.705295	0.352647	−	0.935756i	$-0.385282\pi$
$443$	14.8447	0.705295	0.352647	+	0.935756i	$0.385282\pi$
$444$	0	0
$445$	−19.9298	−0.944763
$446$	1.00000	0.0473514
$447$	0	0
$448$	1.62712	0.0768742
$449$	−13.8348	−0.652905	−0.326453	−	0.945214i	$-0.605853\pi$
$449$	−13.8348	−0.652905	−0.326453	+	0.945214i	$0.605853\pi$
$450$	0	0
$451$	35.9097	1.69092
$452$	−5.00641	−0.235482
$453$	0	0
$454$	1.37598	0.0645778
$455$	16.5218	0.774556
$456$	0	0
$457$	35.0332	1.63879	0.819393	−	0.573232i	$-0.194311\pi$
$457$	35.0332	1.63879	0.819393	+	0.573232i	$0.194311\pi$
$458$	−10.4332	−0.487509
$459$	0	0
$460$	−3.58968	−0.167370
$461$	−34.9345	−1.62706	−0.813532	−	0.581520i	$-0.802458\pi$
$461$	−34.9345	−1.62706	−0.813532	+	0.581520i	$0.802458\pi$
$462$	0	0
$463$	−9.21484	−0.428250	−0.214125	−	0.976806i	$-0.568690\pi$
$463$	−9.21484	−0.428250	−0.214125	+	0.976806i	$0.568690\pi$
$464$	4.72629	0.219412
$465$	0	0
$466$	3.28891	0.152356
$467$	7.43435	0.344021	0.172010	−	0.985095i	$-0.444974\pi$
$467$	7.43435	0.344021	0.172010	+	0.985095i	$0.444974\pi$
$468$	0	0
$469$	−12.6449	−0.583889
$470$	10.8646	0.501147
$471$	0	0
$472$	−2.18450	−0.100550
$473$	−25.1130	−1.15470
$474$	0	0
$475$	16.3077	0.748249
$476$	−12.0728	−0.553355
$477$	0	0
$478$	17.5274	0.801684
$479$	0.658732	0.0300982	0.0150491	−	0.999887i	$-0.495210\pi$
$479$	0.658732	0.0300982	0.0150491	+	0.999887i	$0.495210\pi$
$480$	0	0
$481$	9.87819	0.450407
$482$	−0.255243	−0.0116260
$483$	0	0
$484$	12.8537	0.584260
$485$	6.91148	0.313834
$486$	0	0
$487$	41.8859	1.89803	0.949015	−	0.315231i	$-0.102082\pi$
$487$	41.8859	1.89803	0.949015	+	0.315231i	$0.102082\pi$
$488$	−8.49027	−0.384336
$489$	0	0
$490$	−12.2005	−0.551164
$491$	27.2206	1.22845	0.614225	−	0.789131i	$-0.289469\pi$
$491$	27.2206	1.22845	0.614225	+	0.789131i	$0.289469\pi$
$492$	0	0
$493$	−35.0678	−1.57937
$494$	20.6731	0.930128
$495$	0	0
$496$	−9.52740	−0.427793
$497$	−2.94151	−0.131945
$498$	0	0
$499$	−38.6603	−1.73067	−0.865336	−	0.501192i	$-0.832895\pi$
$499$	−38.6603	−1.73067	−0.865336	+	0.501192i	$0.832895\pi$
$500$	−6.00574	−0.268585
$501$	0	0
$502$	−5.33942	−0.238310
$503$	24.2740	1.08233	0.541163	−	0.840918i	$-0.317984\pi$
$503$	24.2740	1.08233	0.541163	+	0.840918i	$0.317984\pi$
$504$	0	0
$505$	7.74245	0.344534
$506$	−6.25449	−0.278046
$507$	0	0
$508$	0.506922	0.0224910
$509$	4.37168	0.193771	0.0968856	−	0.995296i	$-0.469112\pi$
$509$	4.37168	0.193771	0.0968856	+	0.995296i	$0.469112\pi$
$510$	0	0
$511$	1.99561	0.0882807
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.3493	−0.677028
$515$	4.43005	0.195212
$516$	0	0
$517$	18.9300	0.832540
$518$	4.43711	0.194955
$519$	0	0
$520$	10.1540	0.445284
$521$	39.8560	1.74612	0.873061	−	0.487611i	$-0.162131\pi$
$521$	39.8560	1.74612	0.873061	+	0.487611i	$0.162131\pi$
$522$	0	0
$523$	−33.6725	−1.47240	−0.736198	−	0.676766i	$-0.763381\pi$
$523$	−33.6725	−1.47240	−0.736198	+	0.676766i	$0.763381\pi$
$524$	10.7751	0.470714
$525$	0	0
$526$	26.6345	1.16132
$527$	70.6907	3.07933
$528$	0	0
$529$	−21.3601	−0.928698
$530$	−5.32374	−0.231249
$531$	0	0
$532$	9.28601	0.402599
$533$	26.6337	1.15363
$534$	0	0
$535$	48.8725	2.11294
$536$	−7.77136	−0.335672
$537$	0	0
$538$	−8.87441	−0.382603
$539$	−21.2576	−0.915631
$540$	0	0
$541$	−10.9916	−0.472566	−0.236283	−	0.971684i	$-0.575929\pi$
$541$	−10.9916	−0.472566	−0.236283	+	0.971684i	$0.575929\pi$
$542$	14.9484	0.642089
$543$	0	0
$544$	−7.41973	−0.318118
$545$	−34.9832	−1.49852
$546$	0	0
$547$	−2.76693	−0.118305	−0.0591527	−	0.998249i	$-0.518840\pi$
$547$	−2.76693	−0.118305	−0.0591527	+	0.998249i	$0.518840\pi$
$548$	−8.36832	−0.357477
$549$	0	0
$550$	13.9560	0.595087
$551$	26.9730	1.14909
$552$	0	0
$553$	−11.2443	−0.478156
$554$	−10.4679	−0.444740
$555$	0	0
$556$	−0.475069	−0.0201474
$557$	−34.5169	−1.46253	−0.731265	−	0.682094i	$-0.761070\pi$
$557$	−34.5169	−1.46253	−0.731265	+	0.682094i	$0.761070\pi$
$558$	0	0
$559$	−18.6259	−0.787792
$560$	4.56101	0.192738
$561$	0	0
$562$	3.55084	0.149783
$563$	21.6499	0.912435	0.456217	−	0.889868i	$-0.349204\pi$
$563$	21.6499	0.912435	0.456217	+	0.889868i	$0.349204\pi$
$564$	0	0
$565$	−14.0336	−0.590397
$566$	32.0270	1.34619
$567$	0	0
$568$	−1.80780	−0.0758536
$569$	24.5140	1.02768	0.513840	−	0.857886i	$-0.328223\pi$
$569$	24.5140	1.02768	0.513840	+	0.857886i	$0.328223\pi$
$570$	0	0
$571$	34.8592	1.45881	0.729406	−	0.684081i	$-0.239796\pi$
$571$	34.8592	1.45881	0.729406	+	0.684081i	$0.239796\pi$
$572$	17.6919	0.739736
$573$	0	0
$574$	11.9634	0.499341
$575$	−3.65930	−0.152603
$576$	0	0
$577$	−2.20534	−0.0918096	−0.0459048	−	0.998946i	$-0.514617\pi$
$577$	−2.20534	−0.0918096	−0.0459048	+	0.998946i	$0.514617\pi$
$578$	38.0523	1.58277
$579$	0	0
$580$	13.2484	0.550108
$581$	−24.1568	−1.00219
$582$	0	0
$583$	−9.27584	−0.384166
$584$	1.22647	0.0507517
$585$	0	0
$586$	−33.2339	−1.37288
$587$	24.4096	1.00749	0.503746	−	0.863852i	$-0.331955\pi$
$587$	24.4096	1.00749	0.503746	+	0.863852i	$0.331955\pi$
$588$	0	0
$589$	−54.3730	−2.24040
$590$	−6.12342	−0.252097
$591$	0	0
$592$	2.72697	0.112078
$593$	−20.2832	−0.832930	−0.416465	−	0.909152i	$-0.636731\pi$
$593$	−20.2832	−0.832930	−0.416465	+	0.909152i	$0.636731\pi$
$594$	0	0
$595$	−33.8415	−1.38736
$596$	16.2885	0.667201
$597$	0	0
$598$	−4.63886	−0.189697
$599$	−21.2505	−0.868274	−0.434137	−	0.900847i	$-0.642947\pi$
$599$	−21.2505	−0.868274	−0.434137	+	0.900847i	$0.642947\pi$
$600$	0	0
$601$	1.51881	0.0619535	0.0309768	−	0.999520i	$-0.490138\pi$
$601$	1.51881	0.0619535	0.0309768	+	0.999520i	$0.490138\pi$
$602$	−8.36643	−0.340990
$603$	0	0
$604$	0.513029	0.0208749
$605$	36.0305	1.46485
$606$	0	0
$607$	−46.9670	−1.90633	−0.953165	−	0.302450i	$-0.902195\pi$
$607$	−46.9670	−1.90633	−0.953165	+	0.302450i	$0.902195\pi$
$608$	5.70702	0.231450
$609$	0	0
$610$	−23.7992	−0.963603
$611$	14.0401	0.568001
$612$	0	0
$613$	−31.0349	−1.25349	−0.626744	−	0.779225i	$-0.715613\pi$
$613$	−31.0349	−1.25349	−0.626744	+	0.779225i	$0.715613\pi$
$614$	−20.4035	−0.823417
$615$	0	0
$616$	7.94690	0.320190
$617$	−32.8342	−1.32185	−0.660927	−	0.750450i	$-0.729837\pi$
$617$	−32.8342	−1.32185	−0.660927	+	0.750450i	$0.729837\pi$
$618$	0	0
$619$	24.7996	0.996780	0.498390	−	0.866953i	$-0.333925\pi$
$619$	24.7996	0.996780	0.498390	+	0.866953i	$0.333925\pi$
$620$	−26.7064	−1.07256
$621$	0	0
$622$	8.99243	0.360564
$623$	−11.5686	−0.463486
$624$	0	0
$625$	−31.1222	−1.24489
$626$	−4.30737	−0.172157
$627$	0	0
$628$	6.63432	0.264738
$629$	−20.2334	−0.806757
$630$	0	0
$631$	5.87503	0.233881	0.116941	−	0.993139i	$-0.462691\pi$
$631$	5.87503	0.233881	0.116941	+	0.993139i	$0.462691\pi$
$632$	−6.91055	−0.274887
$633$	0	0
$634$	3.84895	0.152861
$635$	1.42096	0.0563892
$636$	0	0
$637$	−15.7664	−0.624689
$638$	23.0833	0.913877
$639$	0	0
$640$	2.80312	0.110803
$641$	−12.9994	−0.513446	−0.256723	−	0.966485i	$-0.582643\pi$
$641$	−12.9994	−0.513446	−0.256723	+	0.966485i	$0.582643\pi$
$642$	0	0
$643$	−30.1448	−1.18879	−0.594397	−	0.804172i	$-0.702609\pi$
$643$	−30.1448	−1.18879	−0.594397	+	0.804172i	$0.702609\pi$
$644$	−2.08369	−0.0821090
$645$	0	0
$646$	−42.3445	−1.66602
$647$	−5.45361	−0.214404	−0.107202	−	0.994237i	$-0.534189\pi$
$647$	−5.45361	−0.214404	−0.107202	+	0.994237i	$0.534189\pi$
$648$	0	0
$649$	−10.6692	−0.418801
$650$	10.3510	0.405998
$651$	0	0
$652$	−10.1213	−0.396379
$653$	19.4342	0.760520	0.380260	−	0.924880i	$-0.375835\pi$
$653$	19.4342	0.760520	0.380260	+	0.924880i	$0.375835\pi$
$654$	0	0
$655$	30.2040	1.18017
$656$	7.35248	0.287066
$657$	0	0
$658$	6.30655	0.245855
$659$	−17.1998	−0.670008	−0.335004	−	0.942217i	$-0.608738\pi$
$659$	−17.1998	−0.670008	−0.335004	+	0.942217i	$0.608738\pi$
$660$	0	0
$661$	24.5129	0.953442	0.476721	−	0.879055i	$-0.341825\pi$
$661$	24.5129	0.953442	0.476721	+	0.879055i	$0.341825\pi$
$662$	−8.07055	−0.313671
$663$	0	0
$664$	−14.8463	−0.576150
$665$	26.0298	1.00939
$666$	0	0
$667$	−6.05249	−0.234354
$668$	−18.5167	−0.716432
$669$	0	0
$670$	−21.7841	−0.841591
$671$	−41.4667	−1.60080
$672$	0	0
$673$	−15.1588	−0.584329	−0.292164	−	0.956368i	$-0.594375\pi$
$673$	−15.1588	−0.584329	−0.292164	+	0.956368i	$0.594375\pi$
$674$	10.2899	0.396351
$675$	0	0
$676$	0.121818	0.00468529
$677$	−29.3114	−1.12653	−0.563264	−	0.826277i	$-0.690455\pi$
$677$	−29.3114	−1.12653	−0.563264	+	0.826277i	$0.690455\pi$
$678$	0	0
$679$	4.01189	0.153962
$680$	−20.7984	−0.797582
$681$	0	0
$682$	−46.5321	−1.78180
$683$	−28.4148	−1.08726	−0.543630	−	0.839325i	$-0.682951\pi$
$683$	−28.4148	−1.08726	−0.543630	+	0.839325i	$0.682951\pi$
$684$	0	0
$685$	−23.4574	−0.896262
$686$	−18.4718	−0.705258
$687$	0	0
$688$	−5.14186	−0.196032
$689$	−6.87975	−0.262097
$690$	0	0
$691$	−16.2985	−0.620026	−0.310013	−	0.950732i	$-0.600333\pi$
$691$	−16.2985	−0.620026	−0.310013	+	0.950732i	$0.600333\pi$
$692$	7.89063	0.299957
$693$	0	0
$694$	13.7951	0.523655
$695$	−1.33167	−0.0505133
$696$	0	0
$697$	−54.5534	−2.06636
$698$	−11.7721	−0.445580
$699$	0	0
$700$	4.64947	0.175733
$701$	−46.1845	−1.74437	−0.872183	−	0.489180i	$-0.837296\pi$
$701$	−46.1845	−1.74437	−0.872183	+	0.489180i	$0.837296\pi$
$702$	0	0
$703$	15.5629	0.586965
$704$	4.88403	0.184074
$705$	0	0
$706$	−19.8217	−0.746001
$707$	4.49424	0.169023
$708$	0	0
$709$	27.1290	1.01885	0.509426	−	0.860515i	$-0.329858\pi$
$709$	27.1290	1.01885	0.509426	+	0.860515i	$0.329858\pi$
$710$	−5.06748	−0.190179
$711$	0	0
$712$	−7.10986	−0.266453
$713$	12.2008	0.456924
$714$	0	0
$715$	49.5926	1.85466
$716$	11.3018	0.422369
$717$	0	0
$718$	6.89554	0.257339
$719$	−6.14447	−0.229150	−0.114575	−	0.993415i	$-0.536551\pi$
$719$	−6.14447	−0.229150	−0.114575	+	0.993415i	$0.536551\pi$
$720$	0	0
$721$	2.57150	0.0957677
$722$	13.5701	0.505026
$723$	0	0
$724$	6.66659	0.247762
$725$	13.5053	0.501574
$726$	0	0
$727$	35.1476	1.30355	0.651776	−	0.758411i	$-0.274024\pi$
$727$	35.1476	1.30355	0.651776	+	0.758411i	$0.274024\pi$
$728$	5.89409	0.218449
$729$	0	0
$730$	3.43794	0.127244
$731$	38.1512	1.41107
$732$	0	0
$733$	12.3156	0.454887	0.227443	−	0.973791i	$-0.426963\pi$
$733$	12.3156	0.454887	0.227443	+	0.973791i	$0.426963\pi$
$734$	11.9728	0.441923
$735$	0	0
$736$	−1.28060	−0.0472036
$737$	−37.9555	−1.39811
$738$	0	0
$739$	−1.26153	−0.0464063	−0.0232031	−	0.999731i	$-0.507386\pi$
$739$	−1.26153	−0.0464063	−0.0232031	+	0.999731i	$0.507386\pi$
$740$	7.64403	0.281000
$741$	0	0
$742$	−3.09026	−0.113447
$743$	−43.6588	−1.60169	−0.800843	−	0.598874i	$-0.795615\pi$
$743$	−43.6588	−1.60169	−0.800843	+	0.598874i	$0.795615\pi$
$744$	0	0
$745$	45.6585	1.67280
$746$	−17.7096	−0.648396
$747$	0	0
$748$	−36.2381	−1.32500
$749$	28.3689	1.03658
$750$	0	0
$751$	44.3812	1.61949	0.809746	−	0.586780i	$-0.199605\pi$
$751$	44.3812	1.61949	0.809746	+	0.586780i	$0.199605\pi$
$752$	3.87590	0.141340
$753$	0	0
$754$	17.1205	0.623493
$755$	1.43808	0.0523372
$756$	0	0
$757$	24.7579	0.899840	0.449920	−	0.893069i	$-0.351452\pi$
$757$	24.7579	0.899840	0.449920	+	0.893069i	$0.351452\pi$
$758$	6.18691	0.224719
$759$	0	0
$760$	15.9975	0.580289
$761$	−17.9077	−0.649154	−0.324577	−	0.945859i	$-0.605222\pi$
$761$	−17.9077	−0.649154	−0.324577	+	0.945859i	$0.605222\pi$
$762$	0	0
$763$	−20.3066	−0.735149
$764$	−9.93188	−0.359323
$765$	0	0
$766$	−12.2112	−0.441209
$767$	−7.91315	−0.285727
$768$	0	0
$769$	−16.2476	−0.585903	−0.292952	−	0.956127i	$-0.594638\pi$
$769$	−16.2476	−0.585903	−0.292952	+	0.956127i	$0.594638\pi$
$770$	22.2761	0.802775
$771$	0	0
$772$	12.3928	0.446026
$773$	45.7237	1.64457	0.822284	−	0.569077i	$-0.192700\pi$
$773$	45.7237	1.64457	0.822284	+	0.569077i	$0.192700\pi$
$774$	0	0
$775$	−27.2244	−0.977928
$776$	2.46564	0.0885113
$777$	0	0
$778$	−20.3603	−0.729952
$779$	41.9607	1.50340
$780$	0	0
$781$	−8.82934	−0.315939
$782$	9.50171	0.339781
$783$	0	0
$784$	−4.35248	−0.155446
$785$	18.5968	0.663748
$786$	0	0
$787$	−27.8093	−0.991293	−0.495647	−	0.868524i	$-0.665069\pi$
$787$	−27.8093	−0.991293	−0.495647	+	0.868524i	$0.665069\pi$
$788$	5.56956	0.198407
$789$	0	0
$790$	−19.3711	−0.689192
$791$	−8.14603	−0.289640
$792$	0	0
$793$	−30.7552	−1.09215
$794$	−26.7165	−0.948135
$795$	0	0
$796$	21.3068	0.755199
$797$	−54.7954	−1.94095	−0.970476	−	0.241197i	$-0.922460\pi$
$797$	−54.7954	−1.94095	−0.970476	+	0.241197i	$0.922460\pi$
$798$	0	0
$799$	−28.7581	−1.01739
$800$	2.85748	0.101027
$801$	0	0
$802$	−20.2413	−0.714744
$803$	5.99011	0.211386
$804$	0	0
$805$	−5.84084	−0.205863
$806$	−34.5121	−1.21564
$807$	0	0
$808$	2.76208	0.0971697
$809$	−10.9965	−0.386617	−0.193309	−	0.981138i	$-0.561922\pi$
$809$	−10.9965	−0.386617	−0.193309	+	0.981138i	$0.561922\pi$
$810$	0	0
$811$	21.3521	0.749773	0.374887	−	0.927071i	$-0.377682\pi$
$811$	21.3521	0.749773	0.374887	+	0.927071i	$0.377682\pi$
$812$	7.69024	0.269875
$813$	0	0
$814$	13.3186	0.466817
$815$	−28.3711	−0.993797
$816$	0	0
$817$	−29.3447	−1.02664
$818$	29.0470	1.01560
$819$	0	0
$820$	20.6099	0.719728
$821$	−34.6645	−1.20980	−0.604900	−	0.796301i	$-0.706787\pi$
$821$	−34.6645	−1.20980	−0.604900	+	0.796301i	$0.706787\pi$
$822$	0	0
$823$	−43.5483	−1.51800	−0.758999	−	0.651092i	$-0.774311\pi$
$823$	−43.5483	−1.51800	−0.758999	+	0.651092i	$0.774311\pi$
$824$	1.58040	0.0550559
$825$	0	0
$826$	−3.55445	−0.123675
$827$	−18.3929	−0.639585	−0.319792	−	0.947488i	$-0.603613\pi$
$827$	−18.3929	−0.639585	−0.319792	+	0.947488i	$0.603613\pi$
$828$	0	0
$829$	24.2452	0.842071	0.421036	−	0.907044i	$-0.361667\pi$
$829$	24.2452	0.842071	0.421036	+	0.907044i	$0.361667\pi$
$830$	−41.6161	−1.44452
$831$	0	0
$832$	3.62240	0.125584
$833$	32.2942	1.11893
$834$	0	0
$835$	−51.9045	−1.79623
$836$	27.8732	0.964016
$837$	0	0
$838$	9.10873	0.314656
$839$	−29.6480	−1.02356	−0.511781	−	0.859116i	$-0.671014\pi$
$839$	−29.6480	−1.02356	−0.511781	+	0.859116i	$0.671014\pi$
$840$	0	0
$841$	−6.66219	−0.229731
$842$	29.6256	1.02096
$843$	0	0
$844$	4.56855	0.157256
$845$	0.341469	0.0117469
$846$	0	0
$847$	20.9145	0.718632
$848$	−1.89922	−0.0652195
$849$	0	0
$850$	−21.2017	−0.727213
$851$	−3.49216	−0.119710
$852$	0	0
$853$	−7.06609	−0.241938	−0.120969	−	0.992656i	$-0.538600\pi$
$853$	−7.06609	−0.241938	−0.120969	+	0.992656i	$0.538600\pi$
$854$	−13.8147	−0.472729
$855$	0	0
$856$	17.4350	0.595917
$857$	3.29313	0.112491	0.0562455	−	0.998417i	$-0.482087\pi$
$857$	3.29313	0.112491	0.0562455	+	0.998417i	$0.482087\pi$
$858$	0	0
$859$	−14.7315	−0.502633	−0.251317	−	0.967905i	$-0.580864\pi$
$859$	−14.7315	−0.502633	−0.251317	+	0.967905i	$0.580864\pi$
$860$	−14.4133	−0.491488
$861$	0	0
$862$	9.05749	0.308499
$863$	6.02541	0.205107	0.102554	−	0.994727i	$-0.467299\pi$
$863$	6.02541	0.205107	0.102554	+	0.994727i	$0.467299\pi$
$864$	0	0
$865$	22.1184	0.752047
$866$	−15.1574	−0.515069
$867$	0	0
$868$	−15.5022	−0.526180
$869$	−33.7513	−1.14493
$870$	0	0
$871$	−28.1510	−0.953861
$872$	−12.4801	−0.422630
$873$	0	0
$874$	−7.30842	−0.247211
$875$	−9.77205	−0.330356
$876$	0	0
$877$	−22.0686	−0.745203	−0.372601	−	0.927991i	$-0.621534\pi$
$877$	−22.0686	−0.745203	−0.372601	+	0.927991i	$0.621534\pi$
$878$	22.8847	0.772322
$879$	0	0
$880$	13.6905	0.461507
$881$	−1.19776	−0.0403534	−0.0201767	−	0.999796i	$-0.506423\pi$
$881$	−1.19776	−0.0403534	−0.0201767	+	0.999796i	$0.506423\pi$
$882$	0	0
$883$	50.4064	1.69631	0.848155	−	0.529748i	$-0.177713\pi$
$883$	50.4064	1.69631	0.848155	+	0.529748i	$0.177713\pi$
$884$	−26.8773	−0.903980
$885$	0	0
$886$	14.8447	0.498719
$887$	8.22912	0.276307	0.138153	−	0.990411i	$-0.455883\pi$
$887$	8.22912	0.276307	0.138153	+	0.990411i	$0.455883\pi$
$888$	0	0
$889$	0.824823	0.0276637
$890$	−19.9298	−0.668049
$891$	0	0
$892$	1.00000	0.0334825
$893$	22.1198	0.740212
$894$	0	0
$895$	31.6804	1.05896
$896$	1.62712	0.0543583
$897$	0	0
$898$	−13.8348	−0.461674
$899$	−45.0292	−1.50181
$900$	0	0
$901$	14.0917	0.469462
$902$	35.9097	1.19566
$903$	0	0
$904$	−5.00641	−0.166511
$905$	18.6872	0.621185
$906$	0	0
$907$	−11.0264	−0.366124	−0.183062	−	0.983101i	$-0.558601\pi$
$907$	−11.0264	−0.366124	−0.183062	+	0.983101i	$0.558601\pi$
$908$	1.37598	0.0456634
$909$	0	0
$910$	16.5218	0.547694
$911$	−41.2046	−1.36517	−0.682585	−	0.730806i	$-0.739144\pi$
$911$	−41.2046	−1.36517	−0.682585	+	0.730806i	$0.739144\pi$
$912$	0	0
$913$	−72.5099	−2.39973
$914$	35.0332	1.15880
$915$	0	0
$916$	−10.4332	−0.344721
$917$	17.5325	0.578973
$918$	0	0
$919$	35.3908	1.16743	0.583717	−	0.811957i	$-0.301598\pi$
$919$	35.3908	1.16743	0.583717	+	0.811957i	$0.301598\pi$
$920$	−3.58968	−0.118348
$921$	0	0
$922$	−34.9345	−1.15051
$923$	−6.54858	−0.215549
$924$	0	0
$925$	7.79227	0.256208
$926$	−9.21484	−0.302819
$927$	0	0
$928$	4.72629	0.155148
$929$	35.5904	1.16768	0.583842	−	0.811867i	$-0.301549\pi$
$929$	35.5904	1.16768	0.583842	+	0.811867i	$0.301549\pi$
$930$	0	0
$931$	−24.8397	−0.814088
$932$	3.28891	0.107732
$933$	0	0
$934$	7.43435	0.243259
$935$	−101.580	−3.32202
$936$	0	0
$937$	35.0053	1.14357	0.571786	−	0.820403i	$-0.306251\pi$
$937$	35.0053	1.14357	0.571786	+	0.820403i	$0.306251\pi$
$938$	−12.6449	−0.412872
$939$	0	0
$940$	10.8646	0.354365
$941$	48.1829	1.57072	0.785359	−	0.619041i	$-0.212478\pi$
$941$	48.1829	1.57072	0.785359	+	0.619041i	$0.212478\pi$
$942$	0	0
$943$	−9.41560	−0.306614
$944$	−2.18450	−0.0710995
$945$	0	0
$946$	−25.1130	−0.816494
$947$	3.93084	0.127735	0.0638675	−	0.997958i	$-0.479656\pi$
$947$	3.93084	0.127735	0.0638675	+	0.997958i	$0.479656\pi$
$948$	0	0
$949$	4.44277	0.144218
$950$	16.3077	0.529092
$951$	0	0
$952$	−12.0728	−0.391281
$953$	−16.8623	−0.546223	−0.273112	−	0.961982i	$-0.588053\pi$
$953$	−16.8623	−0.546223	−0.273112	+	0.961982i	$0.588053\pi$
$954$	0	0
$955$	−27.8402	−0.900889
$956$	17.5274	0.566877
$957$	0	0
$958$	0.658732	0.0212827
$959$	−13.6163	−0.439692
$960$	0	0
$961$	59.7713	1.92811
$962$	9.87819	0.318486
$963$	0	0
$964$	−0.255243	−0.00822083
$965$	34.7384	1.11827
$966$	0	0
$967$	−18.3545	−0.590242	−0.295121	−	0.955460i	$-0.595360\pi$
$967$	−18.3545	−0.590242	−0.295121	+	0.955460i	$0.595360\pi$
$968$	12.8537	0.413134
$969$	0	0
$970$	6.91148	0.221914
$971$	57.6781	1.85098	0.925489	−	0.378774i	$-0.123654\pi$
$971$	57.6781	1.85098	0.925489	+	0.378774i	$0.123654\pi$
$972$	0	0
$973$	−0.772994	−0.0247810
$974$	41.8859	1.34211
$975$	0	0
$976$	−8.49027	−0.271767
$977$	34.9645	1.11861	0.559306	−	0.828961i	$-0.311068\pi$
$977$	34.9645	1.11861	0.559306	+	0.828961i	$0.311068\pi$
$978$	0	0
$979$	−34.7248	−1.10981
$980$	−12.2005	−0.389731
$981$	0	0
$982$	27.2206	0.868645
$983$	−10.6590	−0.339968	−0.169984	−	0.985447i	$-0.554372\pi$
$983$	−10.6590	−0.339968	−0.169984	+	0.985447i	$0.554372\pi$
$984$	0	0
$985$	15.6121	0.497444
$986$	−35.0678	−1.11679
$987$	0	0
$988$	20.6731	0.657700
$989$	6.58468	0.209381
$990$	0	0
$991$	−30.3268	−0.963363	−0.481682	−	0.876346i	$-0.659974\pi$
$991$	−30.3268	−0.963363	−0.481682	+	0.876346i	$0.659974\pi$
$992$	−9.52740	−0.302495
$993$	0	0
$994$	−2.94151	−0.0932989
$995$	59.7255	1.89343
$996$	0	0
$997$	−30.9818	−0.981204	−0.490602	−	0.871384i	$-0.663223\pi$
$997$	−30.9818	−0.981204	−0.490602	+	0.871384i	$0.663223\pi$
$998$	−38.6603	−1.22377
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.z.1.7		8
3.2	odd	2		446.2.a.f.1.8	✓	8
12.11	even	2		3568.2.a.n.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.f.1.8	✓	8	3.2	odd	2
3568.2.a.n.1.1		8	12.11	even	2
4014.2.a.z.1.7		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.80312 q^{5} +1.62712 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.80312 q^{5} +1.62712 q^{7} +1.00000 q^{8} +2.80312 q^{10} +4.88403 q^{11} +3.62240 q^{13} +1.62712 q^{14} +1.00000 q^{16} -7.41973 q^{17} +5.70702 q^{19} +2.80312 q^{20} +4.88403 q^{22} -1.28060 q^{23} +2.85748 q^{25} +3.62240 q^{26} +1.62712 q^{28} +4.72629 q^{29} -9.52740 q^{31} +1.00000 q^{32} -7.41973 q^{34} +4.56101 q^{35} +2.72697 q^{37} +5.70702 q^{38} +2.80312 q^{40} +7.35248 q^{41} -5.14186 q^{43} +4.88403 q^{44} -1.28060 q^{46} +3.87590 q^{47} -4.35248 q^{49} +2.85748 q^{50} +3.62240 q^{52} -1.89922 q^{53} +13.6905 q^{55} +1.62712 q^{56} +4.72629 q^{58} -2.18450 q^{59} -8.49027 q^{61} -9.52740 q^{62} +1.00000 q^{64} +10.1540 q^{65} -7.77136 q^{67} -7.41973 q^{68} +4.56101 q^{70} -1.80780 q^{71} +1.22647 q^{73} +2.72697 q^{74} +5.70702 q^{76} +7.94690 q^{77} -6.91055 q^{79} +2.80312 q^{80} +7.35248 q^{82} -14.8463 q^{83} -20.7984 q^{85} -5.14186 q^{86} +4.88403 q^{88} -7.10986 q^{89} +5.89409 q^{91} -1.28060 q^{92} +3.87590 q^{94} +15.9975 q^{95} +2.46564 q^{97} -4.35248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 4 * q^5 + 4 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8} - 4 q^{10} - 2 q^{11} + 10 q^{13} + 4 q^{14} + 8 q^{16} - 8 q^{17} + 16 q^{19} - 4 q^{20} - 2 q^{22} + 4 q^{23} + 24 q^{25} + 10 q^{26} + 4 q^{28} - 8 q^{29} + 12 q^{31} + 8 q^{32} - 8 q^{34} + 24 q^{35} + 16 q^{37} + 16 q^{38} - 4 q^{40} - 16 q^{41} + 12 q^{43} - 2 q^{44} + 4 q^{46} + 4 q^{47} + 40 q^{49} + 24 q^{50} + 10 q^{52} + 8 q^{53} + 8 q^{55} + 4 q^{56} - 8 q^{58} + 26 q^{61} + 12 q^{62} + 8 q^{64} + 24 q^{65} - 10 q^{67} - 8 q^{68} + 24 q^{70} - 8 q^{71} + 8 q^{73} + 16 q^{74} + 16 q^{76} + 56 q^{77} + 16 q^{79} - 4 q^{80} - 16 q^{82} + 44 q^{83} - 18 q^{85} + 12 q^{86} - 2 q^{88} + 4 q^{91} + 4 q^{92} + 4 q^{94} + 48 q^{95} + 12 q^{97} + 40 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 4 * q^5 + 4 * q^7 + 8 * q^8 - 4 * q^10 - 2 * q^11 + 10 * q^13 + 4 * q^14 + 8 * q^16 - 8 * q^17 + 16 * q^19 - 4 * q^20 - 2 * q^22 + 4 * q^23 + 24 * q^25 + 10 * q^26 + 4 * q^28 - 8 * q^29 + 12 * q^31 + 8 * q^32 - 8 * q^34 + 24 * q^35 + 16 * q^37 + 16 * q^38 - 4 * q^40 - 16 * q^41 + 12 * q^43 - 2 * q^44 + 4 * q^46 + 4 * q^47 + 40 * q^49 + 24 * q^50 + 10 * q^52 + 8 * q^53 + 8 * q^55 + 4 * q^56 - 8 * q^58 + 26 * q^61 + 12 * q^62 + 8 * q^64 + 24 * q^65 - 10 * q^67 - 8 * q^68 + 24 * q^70 - 8 * q^71 + 8 * q^73 + 16 * q^74 + 16 * q^76 + 56 * q^77 + 16 * q^79 - 4 * q^80 - 16 * q^82 + 44 * q^83 - 18 * q^85 + 12 * q^86 - 2 * q^88 + 4 * q^91 + 4 * q^92 + 4 * q^94 + 48 * q^95 + 12 * q^97 + 40 * q^98

Introduction

L-functions

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