LMFDB - Embedded newform 4014.2.a.z.1.1

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			1.1
Root			$-2.00550$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.95370 q^{5} -4.39233 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.95370 q^{5} -4.39233 q^{7} +1.00000 q^{8} -3.95370 q^{10} -2.05400 q^{11} -3.34096 q^{13} -4.39233 q^{14} +1.00000 q^{16} -4.04232 q^{17} -1.21363 q^{19} -3.95370 q^{20} -2.05400 q^{22} +7.31540 q^{23} +10.6318 q^{25} -3.34096 q^{26} -4.39233 q^{28} +4.01099 q^{29} -3.43169 q^{31} +1.00000 q^{32} -4.04232 q^{34} +17.3660 q^{35} -6.60976 q^{37} -1.21363 q^{38} -3.95370 q^{40} -9.29255 q^{41} -7.19553 q^{43} -2.05400 q^{44} +7.31540 q^{46} -7.37615 q^{47} +12.2925 q^{49} +10.6318 q^{50} -3.34096 q^{52} +4.69378 q^{53} +8.12092 q^{55} -4.39233 q^{56} +4.01099 q^{58} +6.57533 q^{59} +11.9614 q^{61} -3.43169 q^{62} +1.00000 q^{64} +13.2092 q^{65} -0.0992052 q^{67} -4.04232 q^{68} +17.3660 q^{70} -1.48015 q^{71} -8.99532 q^{73} -6.60976 q^{74} -1.21363 q^{76} +9.02186 q^{77} +15.9463 q^{79} -3.95370 q^{80} -9.29255 q^{82} +1.21977 q^{83} +15.9821 q^{85} -7.19553 q^{86} -2.05400 q^{88} +16.9753 q^{89} +14.6746 q^{91} +7.31540 q^{92} -7.37615 q^{94} +4.79833 q^{95} +7.93027 q^{97} +12.2925 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.95370	−1.76815	−0.884075	−	0.467345i	$-0.845211\pi$
$5$	−3.95370	−1.76815	−0.884075	+	0.467345i	$0.845211\pi$
$6$	0	0
$7$	−4.39233	−1.66014	−0.830072	−	0.557656i	$-0.811701\pi$
$7$	−4.39233	−1.66014	−0.830072	+	0.557656i	$0.811701\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.95370	−1.25027
$11$	−2.05400	−0.619306	−0.309653	−	0.950850i	$-0.600213\pi$
$11$	−2.05400	−0.619306	−0.309653	+	0.950850i	$0.600213\pi$
$12$	0	0
$13$	−3.34096	−0.926615	−0.463307	−	0.886198i	$-0.653337\pi$
$13$	−3.34096	−0.926615	−0.463307	+	0.886198i	$0.653337\pi$
$14$	−4.39233	−1.17390
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.04232	−0.980407	−0.490204	−	0.871608i	$-0.663078\pi$
$17$	−4.04232	−0.980407	−0.490204	+	0.871608i	$0.663078\pi$
$18$	0	0
$19$	−1.21363	−0.278426	−0.139213	−	0.990262i	$-0.544457\pi$
$19$	−1.21363	−0.278426	−0.139213	+	0.990262i	$0.544457\pi$
$20$	−3.95370	−0.884075
$21$	0	0
$22$	−2.05400	−0.437915
$23$	7.31540	1.52537	0.762684	−	0.646772i	$-0.223881\pi$
$23$	7.31540	1.52537	0.762684	+	0.646772i	$0.223881\pi$
$24$	0	0
$25$	10.6318	2.12636
$26$	−3.34096	−0.655216
$27$	0	0
$28$	−4.39233	−0.830072
$29$	4.01099	0.744823	0.372411	−	0.928068i	$-0.378531\pi$
$29$	4.01099	0.744823	0.372411	+	0.928068i	$0.378531\pi$
$30$	0	0
$31$	−3.43169	−0.616350	−0.308175	−	0.951330i	$-0.599718\pi$
$31$	−3.43169	−0.616350	−0.308175	+	0.951330i	$0.599718\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.04232	−0.693253
$35$	17.3660	2.93538
$36$	0	0
$37$	−6.60976	−1.08664	−0.543319	−	0.839526i	$-0.682833\pi$
$37$	−6.60976	−1.08664	−0.543319	+	0.839526i	$0.682833\pi$
$38$	−1.21363	−0.196877
$39$	0	0
$40$	−3.95370	−0.625136
$41$	−9.29255	−1.45125	−0.725626	−	0.688089i	$-0.758450\pi$
$41$	−9.29255	−1.45125	−0.725626	+	0.688089i	$0.758450\pi$
$42$	0	0
$43$	−7.19553	−1.09731	−0.548654	−	0.836049i	$-0.684860\pi$
$43$	−7.19553	−1.09731	−0.548654	+	0.836049i	$0.684860\pi$
$44$	−2.05400	−0.309653
$45$	0	0
$46$	7.31540	1.07860
$47$	−7.37615	−1.07592	−0.537961	−	0.842970i	$-0.680805\pi$
$47$	−7.37615	−1.07592	−0.537961	+	0.842970i	$0.680805\pi$
$48$	0	0
$49$	12.2925	1.75608
$50$	10.6318	1.50356
$51$	0	0
$52$	−3.34096	−0.463307
$53$	4.69378	0.644740	0.322370	−	0.946614i	$-0.395520\pi$
$53$	4.69378	0.644740	0.322370	+	0.946614i	$0.395520\pi$
$54$	0	0
$55$	8.12092	1.09503
$56$	−4.39233	−0.586950
$57$	0	0
$58$	4.01099	0.526669
$59$	6.57533	0.856035	0.428017	−	0.903770i	$-0.359212\pi$
$59$	6.57533	0.856035	0.428017	+	0.903770i	$0.359212\pi$
$60$	0	0
$61$	11.9614	1.53150	0.765751	−	0.643137i	$-0.222368\pi$
$61$	11.9614	1.53150	0.765751	+	0.643137i	$0.222368\pi$
$62$	−3.43169	−0.435825
$63$	0	0
$64$	1.00000	0.125000
$65$	13.2092	1.63839
$66$	0	0
$67$	−0.0992052	−0.0121198	−0.00605992	−	0.999982i	$-0.501929\pi$
$67$	−0.0992052	−0.0121198	−0.00605992	+	0.999982i	$0.501929\pi$
$68$	−4.04232	−0.490204
$69$	0	0
$70$	17.3660	2.07563
$71$	−1.48015	−0.175662	−0.0878308	−	0.996135i	$-0.527993\pi$
$71$	−1.48015	−0.175662	−0.0878308	+	0.996135i	$0.527993\pi$
$72$	0	0
$73$	−8.99532	−1.05282	−0.526412	−	0.850230i	$-0.676463\pi$
$73$	−8.99532	−1.05282	−0.526412	+	0.850230i	$0.676463\pi$
$74$	−6.60976	−0.768369
$75$	0	0
$76$	−1.21363	−0.139213
$77$	9.02186	1.02814
$78$	0	0
$79$	15.9463	1.79410	0.897052	−	0.441926i	$-0.145705\pi$
$79$	15.9463	1.79410	0.897052	+	0.441926i	$0.145705\pi$
$80$	−3.95370	−0.442038
$81$	0	0
$82$	−9.29255	−1.02619
$83$	1.21977	0.133887	0.0669434	−	0.997757i	$-0.478675\pi$
$83$	1.21977	0.133887	0.0669434	+	0.997757i	$0.478675\pi$
$84$	0	0
$85$	15.9821	1.73351
$86$	−7.19553	−0.775914
$87$	0	0
$88$	−2.05400	−0.218958
$89$	16.9753	1.79938	0.899689	−	0.436531i	$-0.143793\pi$
$89$	16.9753	1.79938	0.899689	+	0.436531i	$0.143793\pi$
$90$	0	0
$91$	14.6746	1.53831
$92$	7.31540	0.762684
$93$	0	0
$94$	−7.37615	−0.760792
$95$	4.79833	0.492298
$96$	0	0
$97$	7.93027	0.805196	0.402598	−	0.915377i	$-0.368107\pi$
$97$	7.93027	0.805196	0.402598	+	0.915377i	$0.368107\pi$
$98$	12.2925	1.24173
$99$	0	0
$100$	10.6318	1.06318
$101$	7.37750	0.734089	0.367044	−	0.930203i	$-0.380370\pi$
$101$	7.37750	0.734089	0.367044	+	0.930203i	$0.380370\pi$
$102$	0	0
$103$	1.56426	0.154131	0.0770657	−	0.997026i	$-0.475445\pi$
$103$	1.56426	0.154131	0.0770657	+	0.997026i	$0.475445\pi$
$104$	−3.34096	−0.327608
$105$	0	0
$106$	4.69378	0.455900
$107$	9.02835	0.872804	0.436402	−	0.899752i	$-0.356253\pi$
$107$	9.02835	0.872804	0.436402	+	0.899752i	$0.356253\pi$
$108$	0	0
$109$	9.29931	0.890713	0.445356	−	0.895353i	$-0.353077\pi$
$109$	9.29931	0.890713	0.445356	+	0.895353i	$0.353077\pi$
$110$	8.12092	0.774300
$111$	0	0
$112$	−4.39233	−0.415036
$113$	−10.8654	−1.02213	−0.511065	−	0.859542i	$-0.670749\pi$
$113$	−10.8654	−1.02213	−0.511065	+	0.859542i	$0.670749\pi$
$114$	0	0
$115$	−28.9229	−2.69708
$116$	4.01099	0.372411
$117$	0	0
$118$	6.57533	0.605308
$119$	17.7552	1.62762
$120$	0	0
$121$	−6.78107	−0.616461
$122$	11.9614	1.08294
$123$	0	0
$124$	−3.43169	−0.308175
$125$	−22.2664	−1.99157
$126$	0	0
$127$	7.98106	0.708205	0.354102	−	0.935207i	$-0.384786\pi$
$127$	7.98106	0.708205	0.354102	+	0.935207i	$0.384786\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	13.2092	1.15852
$131$	−5.86338	−0.512286	−0.256143	−	0.966639i	$-0.582452\pi$
$131$	−5.86338	−0.512286	−0.256143	+	0.966639i	$0.582452\pi$
$132$	0	0
$133$	5.33066	0.462227
$134$	−0.0992052	−0.00857002
$135$	0	0
$136$	−4.04232	−0.346626
$137$	0.529905	0.0452729	0.0226364	−	0.999744i	$-0.492794\pi$
$137$	0.529905	0.0452729	0.0226364	+	0.999744i	$0.492794\pi$
$138$	0	0
$139$	−7.82752	−0.663922	−0.331961	−	0.943293i	$-0.607710\pi$
$139$	−7.82752	−0.663922	−0.331961	+	0.943293i	$0.607710\pi$
$140$	17.3660	1.46769
$141$	0	0
$142$	−1.48015	−0.124211
$143$	6.86234	0.573858
$144$	0	0
$145$	−15.8583	−1.31696
$146$	−8.99532	−0.744458
$147$	0	0
$148$	−6.60976	−0.543319
$149$	17.2024	1.40927	0.704637	−	0.709568i	$-0.251110\pi$
$149$	17.2024	1.40927	0.704637	+	0.709568i	$0.251110\pi$
$150$	0	0
$151$	−9.37719	−0.763105	−0.381553	−	0.924347i	$-0.624610\pi$
$151$	−9.37719	−0.763105	−0.381553	+	0.924347i	$0.624610\pi$
$152$	−1.21363	−0.0984383
$153$	0	0
$154$	9.02186	0.727002
$155$	13.5679	1.08980
$156$	0	0
$157$	−4.20717	−0.335768	−0.167884	−	0.985807i	$-0.553693\pi$
$157$	−4.20717	−0.335768	−0.167884	+	0.985807i	$0.553693\pi$
$158$	15.9463	1.26862
$159$	0	0
$160$	−3.95370	−0.312568
$161$	−32.1317	−2.53233
$162$	0	0
$163$	−7.18925	−0.563105	−0.281553	−	0.959546i	$-0.590849\pi$
$163$	−7.18925	−0.563105	−0.281553	+	0.959546i	$0.590849\pi$
$164$	−9.29255	−0.725626
$165$	0	0
$166$	1.21977	0.0946723
$167$	−10.1210	−0.783190	−0.391595	−	0.920138i	$-0.628076\pi$
$167$	−10.1210	−0.783190	−0.391595	+	0.920138i	$0.628076\pi$
$168$	0	0
$169$	−1.83800	−0.141385
$170$	15.9821	1.22577
$171$	0	0
$172$	−7.19553	−0.548654
$173$	−23.1854	−1.76275	−0.881376	−	0.472415i	$-0.843383\pi$
$173$	−23.1854	−1.76275	−0.881376	+	0.472415i	$0.843383\pi$
$174$	0	0
$175$	−46.6983	−3.53006
$176$	−2.05400	−0.154826
$177$	0	0
$178$	16.9753	1.27235
$179$	−6.82749	−0.510311	−0.255155	−	0.966900i	$-0.582127\pi$
$179$	−6.82749	−0.510311	−0.255155	+	0.966900i	$0.582127\pi$
$180$	0	0
$181$	22.9681	1.70721	0.853604	−	0.520923i	$-0.174412\pi$
$181$	22.9681	1.70721	0.853604	+	0.520923i	$0.174412\pi$
$182$	14.6746	1.08775
$183$	0	0
$184$	7.31540	0.539299
$185$	26.1330	1.92134
$186$	0	0
$187$	8.30295	0.607172
$188$	−7.37615	−0.537961
$189$	0	0
$190$	4.79833	0.348107
$191$	8.49941	0.614996	0.307498	−	0.951549i	$-0.400508\pi$
$191$	8.49941	0.614996	0.307498	+	0.951549i	$0.400508\pi$
$192$	0	0
$193$	−16.8446	−1.21250	−0.606252	−	0.795273i	$-0.707328\pi$
$193$	−16.8446	−1.21250	−0.606252	+	0.795273i	$0.707328\pi$
$194$	7.93027	0.569360
$195$	0	0
$196$	12.2925	0.878039
$197$	6.64703	0.473581	0.236791	−	0.971561i	$-0.423904\pi$
$197$	6.64703	0.473581	0.236791	+	0.971561i	$0.423904\pi$
$198$	0	0
$199$	10.8114	0.766397	0.383198	−	0.923666i	$-0.374823\pi$
$199$	10.8114	0.766397	0.383198	+	0.923666i	$0.374823\pi$
$200$	10.6318	0.751780
$201$	0	0
$202$	7.37750	0.519079
$203$	−17.6176	−1.23651
$204$	0	0
$205$	36.7400	2.56603
$206$	1.56426	0.108987
$207$	0	0
$208$	−3.34096	−0.231654
$209$	2.49280	0.172430
$210$	0	0
$211$	3.87590	0.266828	0.133414	−	0.991060i	$-0.457406\pi$
$211$	3.87590	0.266828	0.133414	+	0.991060i	$0.457406\pi$
$212$	4.69378	0.322370
$213$	0	0
$214$	9.02835	0.617166
$215$	28.4490	1.94021
$216$	0	0
$217$	15.0731	1.02323
$218$	9.29931	0.629829
$219$	0	0
$220$	8.12092	0.547513
$221$	13.5052	0.908460
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−4.39233	−0.293475
$225$	0	0
$226$	−10.8654	−0.722755
$227$	−5.26769	−0.349629	−0.174814	−	0.984601i	$-0.555933\pi$
$227$	−5.26769	−0.349629	−0.174814	+	0.984601i	$0.555933\pi$
$228$	0	0
$229$	−22.1590	−1.46430	−0.732152	−	0.681141i	$-0.761484\pi$
$229$	−22.1590	−1.46430	−0.732152	+	0.681141i	$0.761484\pi$
$230$	−28.9229	−1.90712
$231$	0	0
$232$	4.01099	0.263335
$233$	−19.1483	−1.25445	−0.627223	−	0.778840i	$-0.715809\pi$
$233$	−19.1483	−1.25445	−0.627223	+	0.778840i	$0.715809\pi$
$234$	0	0
$235$	29.1631	1.90239
$236$	6.57533	0.428017
$237$	0	0
$238$	17.7552	1.15090
$239$	11.4317	0.739455	0.369727	−	0.929140i	$-0.379451\pi$
$239$	11.4317	0.739455	0.369727	+	0.929140i	$0.379451\pi$
$240$	0	0
$241$	10.0135	0.645028	0.322514	−	0.946565i	$-0.395472\pi$
$241$	10.0135	0.645028	0.322514	+	0.946565i	$0.395472\pi$
$242$	−6.78107	−0.435904
$243$	0	0
$244$	11.9614	0.765751
$245$	−48.6011	−3.10501
$246$	0	0
$247$	4.05468	0.257993
$248$	−3.43169	−0.217913
$249$	0	0
$250$	−22.2664	−1.40825
$251$	−9.94833	−0.627933	−0.313967	−	0.949434i	$-0.601658\pi$
$251$	−9.94833	−0.627933	−0.313967	+	0.949434i	$0.601658\pi$
$252$	0	0
$253$	−15.0259	−0.944668
$254$	7.98106	0.500776
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.4823	−1.58954	−0.794770	−	0.606911i	$-0.792408\pi$
$257$	−25.4823	−1.58954	−0.794770	+	0.606911i	$0.792408\pi$
$258$	0	0
$259$	29.0322	1.80398
$260$	13.2092	0.819197
$261$	0	0
$262$	−5.86338	−0.362241
$263$	23.0484	1.42122	0.710612	−	0.703584i	$-0.248418\pi$
$263$	23.0484	1.42122	0.710612	+	0.703584i	$0.248418\pi$
$264$	0	0
$265$	−18.5578	−1.14000
$266$	5.33066	0.326844
$267$	0	0
$268$	−0.0992052	−0.00605992
$269$	−23.6296	−1.44072	−0.720362	−	0.693598i	$-0.756024\pi$
$269$	−23.6296	−1.44072	−0.720362	+	0.693598i	$0.756024\pi$
$270$	0	0
$271$	−4.35752	−0.264701	−0.132350	−	0.991203i	$-0.542252\pi$
$271$	−4.35752	−0.264701	−0.132350	+	0.991203i	$0.542252\pi$
$272$	−4.04232	−0.245102
$273$	0	0
$274$	0.529905	0.0320127
$275$	−21.8377	−1.31686
$276$	0	0
$277$	16.1795	0.972130	0.486065	−	0.873923i	$-0.338432\pi$
$277$	16.1795	0.972130	0.486065	+	0.873923i	$0.338432\pi$
$278$	−7.82752	−0.469464
$279$	0	0
$280$	17.3660	1.03781
$281$	−25.4393	−1.51758	−0.758790	−	0.651335i	$-0.774209\pi$
$281$	−25.4393	−1.51758	−0.758790	+	0.651335i	$0.774209\pi$
$282$	0	0
$283$	−11.1881	−0.665065	−0.332532	−	0.943092i	$-0.607903\pi$
$283$	−11.1881	−0.665065	−0.332532	+	0.943092i	$0.607903\pi$
$284$	−1.48015	−0.0878308
$285$	0	0
$286$	6.86234	0.405779
$287$	40.8159	2.40929
$288$	0	0
$289$	−0.659630	−0.0388018
$290$	−15.8583	−0.931230
$291$	0	0
$292$	−8.99532	−0.526412
$293$	27.1236	1.58458	0.792289	−	0.610147i	$-0.208889\pi$
$293$	27.1236	1.58458	0.792289	+	0.610147i	$0.208889\pi$
$294$	0	0
$295$	−25.9969	−1.51360
$296$	−6.60976	−0.384185
$297$	0	0
$298$	17.2024	0.996507
$299$	−24.4405	−1.41343
$300$	0	0
$301$	31.6051	1.82169
$302$	−9.37719	−0.539597
$303$	0	0
$304$	−1.21363	−0.0696064
$305$	−47.2919	−2.70793
$306$	0	0
$307$	16.9347	0.966516	0.483258	−	0.875478i	$-0.339453\pi$
$307$	16.9347	0.966516	0.483258	+	0.875478i	$0.339453\pi$
$308$	9.02186	0.514068
$309$	0	0
$310$	13.5679	0.770604
$311$	−1.22400	−0.0694064	−0.0347032	−	0.999398i	$-0.511049\pi$
$311$	−1.22400	−0.0694064	−0.0347032	+	0.999398i	$0.511049\pi$
$312$	0	0
$313$	5.04550	0.285188	0.142594	−	0.989781i	$-0.454456\pi$
$313$	5.04550	0.285188	0.142594	+	0.989781i	$0.454456\pi$
$314$	−4.20717	−0.237424
$315$	0	0
$316$	15.9463	0.897052
$317$	16.1606	0.907670	0.453835	−	0.891086i	$-0.350055\pi$
$317$	16.1606	0.907670	0.453835	+	0.891086i	$0.350055\pi$
$318$	0	0
$319$	−8.23859	−0.461273
$320$	−3.95370	−0.221019
$321$	0	0
$322$	−32.1317	−1.79063
$323$	4.90588	0.272970
$324$	0	0
$325$	−35.5203	−1.97031
$326$	−7.18925	−0.398176
$327$	0	0
$328$	−9.29255	−0.513095
$329$	32.3985	1.78619
$330$	0	0
$331$	−6.62918	−0.364372	−0.182186	−	0.983264i	$-0.558317\pi$
$331$	−6.62918	−0.364372	−0.182186	+	0.983264i	$0.558317\pi$
$332$	1.21977	0.0669434
$333$	0	0
$334$	−10.1210	−0.553799
$335$	0.392228	0.0214297
$336$	0	0
$337$	0.994370	0.0541668	0.0270834	−	0.999633i	$-0.491378\pi$
$337$	0.994370	0.0541668	0.0270834	+	0.999633i	$0.491378\pi$
$338$	−1.83800	−0.0999742
$339$	0	0
$340$	15.9821	0.866754
$341$	7.04870	0.381709
$342$	0	0
$343$	−23.2466	−1.25520
$344$	−7.19553	−0.387957
$345$	0	0
$346$	−23.1854	−1.24645
$347$	−5.25951	−0.282345	−0.141173	−	0.989985i	$-0.545087\pi$
$347$	−5.25951	−0.282345	−0.141173	+	0.989985i	$0.545087\pi$
$348$	0	0
$349$	4.85681	0.259979	0.129990	−	0.991515i	$-0.458506\pi$
$349$	4.85681	0.259979	0.129990	+	0.991515i	$0.458506\pi$
$350$	−46.6983	−2.49613
$351$	0	0
$352$	−2.05400	−0.109479
$353$	16.7790	0.893055	0.446528	−	0.894770i	$-0.352660\pi$
$353$	16.7790	0.893055	0.446528	+	0.894770i	$0.352660\pi$
$354$	0	0
$355$	5.85208	0.310596
$356$	16.9753	0.899689
$357$	0	0
$358$	−6.82749	−0.360844
$359$	−16.6877	−0.880744	−0.440372	−	0.897815i	$-0.645154\pi$
$359$	−16.6877	−0.880744	−0.440372	+	0.897815i	$0.645154\pi$
$360$	0	0
$361$	−17.5271	−0.922479
$362$	22.9681	1.20718
$363$	0	0
$364$	14.6746	0.769157
$365$	35.5649	1.86155
$366$	0	0
$367$	10.5475	0.550573	0.275286	−	0.961362i	$-0.411227\pi$
$367$	10.5475	0.550573	0.275286	+	0.961362i	$0.411227\pi$
$368$	7.31540	0.381342
$369$	0	0
$370$	26.1330	1.35859
$371$	−20.6166	−1.07036
$372$	0	0
$373$	9.15124	0.473833	0.236917	−	0.971530i	$-0.423863\pi$
$373$	9.15124	0.473833	0.236917	+	0.971530i	$0.423863\pi$
$374$	8.30295	0.429335
$375$	0	0
$376$	−7.37615	−0.380396
$377$	−13.4006	−0.690164
$378$	0	0
$379$	−11.7646	−0.604305	−0.302152	−	0.953260i	$-0.597705\pi$
$379$	−11.7646	−0.604305	−0.302152	+	0.953260i	$0.597705\pi$
$380$	4.79833	0.246149
$381$	0	0
$382$	8.49941	0.434868
$383$	−23.1659	−1.18372	−0.591861	−	0.806040i	$-0.701606\pi$
$383$	−23.1659	−1.18372	−0.591861	+	0.806040i	$0.701606\pi$
$384$	0	0
$385$	−35.6698	−1.81790
$386$	−16.8446	−0.857369
$387$	0	0
$388$	7.93027	0.402598
$389$	−5.04437	−0.255759	−0.127880	−	0.991790i	$-0.540817\pi$
$389$	−5.04437	−0.255759	−0.127880	+	0.991790i	$0.540817\pi$
$390$	0	0
$391$	−29.5712	−1.49548
$392$	12.2925	0.620867
$393$	0	0
$394$	6.64703	0.334872
$395$	−63.0471	−3.17224
$396$	0	0
$397$	9.25511	0.464501	0.232250	−	0.972656i	$-0.425391\pi$
$397$	9.25511	0.464501	0.232250	+	0.972656i	$0.425391\pi$
$398$	10.8114	0.541925
$399$	0	0
$400$	10.6318	0.531589
$401$	6.83302	0.341225	0.170612	−	0.985338i	$-0.445425\pi$
$401$	6.83302	0.341225	0.170612	+	0.985338i	$0.445425\pi$
$402$	0	0
$403$	11.4651	0.571119
$404$	7.37750	0.367044
$405$	0	0
$406$	−17.6176	−0.874347
$407$	13.5765	0.672961
$408$	0	0
$409$	−12.6227	−0.624152	−0.312076	−	0.950057i	$-0.601024\pi$
$409$	−12.6227	−0.624152	−0.312076	+	0.950057i	$0.601024\pi$
$410$	36.7400	1.81446
$411$	0	0
$412$	1.56426	0.0770657
$413$	−28.8810	−1.42114
$414$	0	0
$415$	−4.82260	−0.236732
$416$	−3.34096	−0.163804
$417$	0	0
$418$	2.49280	0.121927
$419$	10.0218	0.489597	0.244798	−	0.969574i	$-0.421278\pi$
$419$	10.0218	0.489597	0.244798	+	0.969574i	$0.421278\pi$
$420$	0	0
$421$	29.1502	1.42069	0.710346	−	0.703852i	$-0.248538\pi$
$421$	29.1502	1.42069	0.710346	+	0.703852i	$0.248538\pi$
$422$	3.87590	0.188676
$423$	0	0
$424$	4.69378	0.227950
$425$	−42.9771	−2.08469
$426$	0	0
$427$	−52.5384	−2.54251
$428$	9.02835	0.436402
$429$	0	0
$430$	28.4490	1.37193
$431$	−15.9482	−0.768197	−0.384098	−	0.923292i	$-0.625488\pi$
$431$	−15.9482	−0.768197	−0.384098	+	0.923292i	$0.625488\pi$
$432$	0	0
$433$	2.78614	0.133893	0.0669466	−	0.997757i	$-0.478674\pi$
$433$	2.78614	0.133893	0.0669466	+	0.997757i	$0.478674\pi$
$434$	15.0731	0.723532
$435$	0	0
$436$	9.29931	0.445356
$437$	−8.87819	−0.424701
$438$	0	0
$439$	37.6099	1.79502	0.897511	−	0.440992i	$-0.145373\pi$
$439$	37.6099	1.79502	0.897511	+	0.440992i	$0.145373\pi$
$440$	8.12092	0.387150
$441$	0	0
$442$	13.5052	0.642378
$443$	26.3690	1.25283	0.626416	−	0.779489i	$-0.284521\pi$
$443$	26.3690	1.25283	0.626416	+	0.779489i	$0.284521\pi$
$444$	0	0
$445$	−67.1153	−3.18157
$446$	1.00000	0.0473514
$447$	0	0
$448$	−4.39233	−0.207518
$449$	−35.6329	−1.68162	−0.840811	−	0.541330i	$-0.817921\pi$
$449$	−35.6329	−1.68162	−0.840811	+	0.541330i	$0.817921\pi$
$450$	0	0
$451$	19.0869	0.898768
$452$	−10.8654	−0.511065
$453$	0	0
$454$	−5.26769	−0.247225
$455$	−58.0190	−2.71997
$456$	0	0
$457$	31.8804	1.49130	0.745652	−	0.666336i	$-0.232138\pi$
$457$	31.8804	1.49130	0.745652	+	0.666336i	$0.232138\pi$
$458$	−22.1590	−1.03542
$459$	0	0
$460$	−28.9229	−1.34854
$461$	−3.39594	−0.158165	−0.0790823	−	0.996868i	$-0.525199\pi$
$461$	−3.39594	−0.158165	−0.0790823	+	0.996868i	$0.525199\pi$
$462$	0	0
$463$	11.9556	0.555625	0.277813	−	0.960635i	$-0.410391\pi$
$463$	11.9556	0.555625	0.277813	+	0.960635i	$0.410391\pi$
$464$	4.01099	0.186206
$465$	0	0
$466$	−19.1483	−0.887027
$467$	1.23323	0.0570670	0.0285335	−	0.999593i	$-0.490916\pi$
$467$	1.23323	0.0570670	0.0285335	+	0.999593i	$0.490916\pi$
$468$	0	0
$469$	0.435742	0.0201207
$470$	29.1631	1.34519
$471$	0	0
$472$	6.57533	0.302654
$473$	14.7797	0.679569
$474$	0	0
$475$	−12.9030	−0.592032
$476$	17.7552	0.813809
$477$	0	0
$478$	11.4317	0.522873
$479$	−13.4708	−0.615498	−0.307749	−	0.951468i	$-0.599576\pi$
$479$	−13.4708	−0.615498	−0.307749	+	0.951468i	$0.599576\pi$
$480$	0	0
$481$	22.0829	1.00689
$482$	10.0135	0.456104
$483$	0	0
$484$	−6.78107	−0.308230
$485$	−31.3539	−1.42371
$486$	0	0
$487$	25.2900	1.14600	0.573000	−	0.819556i	$-0.305780\pi$
$487$	25.2900	1.14600	0.573000	+	0.819556i	$0.305780\pi$
$488$	11.9614	0.541468
$489$	0	0
$490$	−48.6011	−2.19557
$491$	29.9179	1.35018	0.675089	−	0.737737i	$-0.264105\pi$
$491$	29.9179	1.35018	0.675089	+	0.737737i	$0.264105\pi$
$492$	0	0
$493$	−16.2137	−0.730229
$494$	4.05468	0.182429
$495$	0	0
$496$	−3.43169	−0.154087
$497$	6.50131	0.291623
$498$	0	0
$499$	−8.86369	−0.396793	−0.198397	−	0.980122i	$-0.563573\pi$
$499$	−8.86369	−0.396793	−0.198397	+	0.980122i	$0.563573\pi$
$500$	−22.2664	−0.995783
$501$	0	0
$502$	−9.94833	−0.444016
$503$	−20.9174	−0.932662	−0.466331	−	0.884610i	$-0.654424\pi$
$503$	−20.9174	−0.932662	−0.466331	+	0.884610i	$0.654424\pi$
$504$	0	0
$505$	−29.1685	−1.29798
$506$	−15.0259	−0.667981
$507$	0	0
$508$	7.98106	0.354102
$509$	4.90252	0.217301	0.108650	−	0.994080i	$-0.465347\pi$
$509$	4.90252	0.217301	0.108650	+	0.994080i	$0.465347\pi$
$510$	0	0
$511$	39.5104	1.74784
$512$	1.00000	0.0441942
$513$	0	0
$514$	−25.4823	−1.12397
$515$	−6.18463	−0.272528
$516$	0	0
$517$	15.1506	0.666325
$518$	29.0322	1.27560
$519$	0	0
$520$	13.2092	0.579260
$521$	−24.0718	−1.05460	−0.527302	−	0.849678i	$-0.676796\pi$
$521$	−24.0718	−1.05460	−0.527302	+	0.849678i	$0.676796\pi$
$522$	0	0
$523$	32.8771	1.43762	0.718808	−	0.695209i	$-0.244688\pi$
$523$	32.8771	1.43762	0.718808	+	0.695209i	$0.244688\pi$
$524$	−5.86338	−0.256143
$525$	0	0
$526$	23.0484	1.00496
$527$	13.8720	0.604274
$528$	0	0
$529$	30.5151	1.32675
$530$	−18.5578	−0.806100
$531$	0	0
$532$	5.33066	0.231113
$533$	31.0460	1.34475
$534$	0	0
$535$	−35.6954	−1.54325
$536$	−0.0992052	−0.00428501
$537$	0	0
$538$	−23.6296	−1.01875
$539$	−25.2489	−1.08755
$540$	0	0
$541$	−3.35735	−0.144344	−0.0721719	−	0.997392i	$-0.522993\pi$
$541$	−3.35735	−0.144344	−0.0721719	+	0.997392i	$0.522993\pi$
$542$	−4.35752	−0.187172
$543$	0	0
$544$	−4.04232	−0.173313
$545$	−36.7667	−1.57491
$546$	0	0
$547$	−1.31101	−0.0560548	−0.0280274	−	0.999607i	$-0.508923\pi$
$547$	−1.31101	−0.0560548	−0.0280274	+	0.999607i	$0.508923\pi$
$548$	0.529905	0.0226364
$549$	0	0
$550$	−21.8377	−0.931163
$551$	−4.86786	−0.207378
$552$	0	0
$553$	−70.0416	−2.97847
$554$	16.1795	0.687400
$555$	0	0
$556$	−7.82752	−0.331961
$557$	9.41147	0.398777	0.199388	−	0.979921i	$-0.436104\pi$
$557$	9.41147	0.398777	0.199388	+	0.979921i	$0.436104\pi$
$558$	0	0
$559$	24.0400	1.01678
$560$	17.3660	0.733846
$561$	0	0
$562$	−25.4393	−1.07309
$563$	23.1774	0.976812	0.488406	−	0.872617i	$-0.337579\pi$
$563$	23.1774	0.976812	0.488406	+	0.872617i	$0.337579\pi$
$564$	0	0
$565$	42.9585	1.80728
$566$	−11.1881	−0.470272
$567$	0	0
$568$	−1.48015	−0.0621057
$569$	21.7513	0.911864	0.455932	−	0.890015i	$-0.349306\pi$
$569$	21.7513	0.911864	0.455932	+	0.890015i	$0.349306\pi$
$570$	0	0
$571$	33.7362	1.41182	0.705908	−	0.708303i	$-0.250539\pi$
$571$	33.7362	1.41182	0.705908	+	0.708303i	$0.250539\pi$
$572$	6.86234	0.286929
$573$	0	0
$574$	40.8159	1.70362
$575$	77.7757	3.24347
$576$	0	0
$577$	34.0424	1.41720	0.708601	−	0.705609i	$-0.249327\pi$
$577$	34.0424	1.41720	0.708601	+	0.705609i	$0.249327\pi$
$578$	−0.659630	−0.0274370
$579$	0	0
$580$	−15.8583	−0.658479
$581$	−5.35762	−0.222271
$582$	0	0
$583$	−9.64104	−0.399291
$584$	−8.99532	−0.372229
$585$	0	0
$586$	27.1236	1.12047
$587$	28.8020	1.18878	0.594392	−	0.804175i	$-0.297393\pi$
$587$	28.8020	1.18878	0.594392	+	0.804175i	$0.297393\pi$
$588$	0	0
$589$	4.16480	0.171608
$590$	−25.9969	−1.07028
$591$	0	0
$592$	−6.60976	−0.271659
$593$	15.0860	0.619507	0.309753	−	0.950817i	$-0.399754\pi$
$593$	15.0860	0.619507	0.309753	+	0.950817i	$0.399754\pi$
$594$	0	0
$595$	−70.1988	−2.87787
$596$	17.2024	0.704637
$597$	0	0
$598$	−24.4405	−0.999445
$599$	−25.4197	−1.03862	−0.519310	−	0.854586i	$-0.673811\pi$
$599$	−25.4197	−1.03862	−0.519310	+	0.854586i	$0.673811\pi$
$600$	0	0
$601$	8.52205	0.347622	0.173811	−	0.984779i	$-0.444392\pi$
$601$	8.52205	0.347622	0.173811	+	0.984779i	$0.444392\pi$
$602$	31.6051	1.28813
$603$	0	0
$604$	−9.37719	−0.381553
$605$	26.8103	1.09000
$606$	0	0
$607$	−34.9035	−1.41669	−0.708345	−	0.705867i	$-0.750558\pi$
$607$	−34.9035	−1.41669	−0.708345	+	0.705867i	$0.750558\pi$
$608$	−1.21363	−0.0492192
$609$	0	0
$610$	−47.2919	−1.91479
$611$	24.6434	0.996966
$612$	0	0
$613$	−7.41766	−0.299596	−0.149798	−	0.988717i	$-0.547862\pi$
$613$	−7.41766	−0.299596	−0.149798	+	0.988717i	$0.547862\pi$
$614$	16.9347	0.683430
$615$	0	0
$616$	9.02186	0.363501
$617$	−5.52168	−0.222294	−0.111147	−	0.993804i	$-0.535453\pi$
$617$	−5.52168	−0.222294	−0.111147	+	0.993804i	$0.535453\pi$
$618$	0	0
$619$	38.6714	1.55433	0.777167	−	0.629295i	$-0.216656\pi$
$619$	38.6714	1.55433	0.777167	+	0.629295i	$0.216656\pi$
$620$	13.5679	0.544899
$621$	0	0
$622$	−1.22400	−0.0490777
$623$	−74.5611	−2.98723
$624$	0	0
$625$	34.8758	1.39503
$626$	5.04550	0.201659
$627$	0	0
$628$	−4.20717	−0.167884
$629$	26.7188	1.06535
$630$	0	0
$631$	−33.3637	−1.32819	−0.664094	−	0.747649i	$-0.731183\pi$
$631$	−33.3637	−1.32819	−0.664094	+	0.747649i	$0.731183\pi$
$632$	15.9463	0.634311
$633$	0	0
$634$	16.1606	0.641820
$635$	−31.5548	−1.25221
$636$	0	0
$637$	−41.0689	−1.62721
$638$	−8.23859	−0.326169
$639$	0	0
$640$	−3.95370	−0.156284
$641$	33.9514	1.34100	0.670500	−	0.741910i	$-0.266080\pi$
$641$	33.9514	1.34100	0.670500	+	0.741910i	$0.266080\pi$
$642$	0	0
$643$	12.3323	0.486339	0.243170	−	0.969984i	$-0.421813\pi$
$643$	12.3323	0.486339	0.243170	+	0.969984i	$0.421813\pi$
$644$	−32.1317	−1.26616
$645$	0	0
$646$	4.90588	0.193019
$647$	16.3751	0.643774	0.321887	−	0.946778i	$-0.395683\pi$
$647$	16.3751	0.643774	0.321887	+	0.946778i	$0.395683\pi$
$648$	0	0
$649$	−13.5058	−0.530147
$650$	−35.5203	−1.39322
$651$	0	0
$652$	−7.18925	−0.281553
$653$	−25.3752	−0.993008	−0.496504	−	0.868034i	$-0.665383\pi$
$653$	−25.3752	−0.993008	−0.496504	+	0.868034i	$0.665383\pi$
$654$	0	0
$655$	23.1821	0.905798
$656$	−9.29255	−0.362813
$657$	0	0
$658$	32.3985	1.26302
$659$	4.87099	0.189747	0.0948734	−	0.995489i	$-0.469755\pi$
$659$	4.87099	0.189747	0.0948734	+	0.995489i	$0.469755\pi$
$660$	0	0
$661$	35.2605	1.37148	0.685738	−	0.727849i	$-0.259480\pi$
$661$	35.2605	1.37148	0.685738	+	0.727849i	$0.259480\pi$
$662$	−6.62918	−0.257650
$663$	0	0
$664$	1.21977	0.0473361
$665$	−21.0758	−0.817286
$666$	0	0
$667$	29.3420	1.13613
$668$	−10.1210	−0.391595
$669$	0	0
$670$	0.392228	0.0151531
$671$	−24.5688	−0.948468
$672$	0	0
$673$	−11.5863	−0.446618	−0.223309	−	0.974748i	$-0.571686\pi$
$673$	−11.5863	−0.446618	−0.223309	+	0.974748i	$0.571686\pi$
$674$	0.994370	0.0383017
$675$	0	0
$676$	−1.83800	−0.0706924
$677$	−16.3608	−0.628797	−0.314398	−	0.949291i	$-0.601803\pi$
$677$	−16.3608	−0.628797	−0.314398	+	0.949291i	$0.601803\pi$
$678$	0	0
$679$	−34.8323	−1.33674
$680$	15.9821	0.612887
$681$	0	0
$682$	7.04870	0.269909
$683$	10.9444	0.418774	0.209387	−	0.977833i	$-0.432853\pi$
$683$	10.9444	0.418774	0.209387	+	0.977833i	$0.432853\pi$
$684$	0	0
$685$	−2.09509	−0.0800492
$686$	−23.2466	−0.887560
$687$	0	0
$688$	−7.19553	−0.274327
$689$	−15.6817	−0.597426
$690$	0	0
$691$	−23.3106	−0.886778	−0.443389	−	0.896329i	$-0.646224\pi$
$691$	−23.3106	−0.886778	−0.443389	+	0.896329i	$0.646224\pi$
$692$	−23.1854	−0.881376
$693$	0	0
$694$	−5.25951	−0.199648
$695$	30.9477	1.17391
$696$	0	0
$697$	37.5635	1.42282
$698$	4.85681	0.183833
$699$	0	0
$700$	−46.6983	−1.76503
$701$	39.0425	1.47461	0.737307	−	0.675558i	$-0.236097\pi$
$701$	39.0425	1.47461	0.737307	+	0.675558i	$0.236097\pi$
$702$	0	0
$703$	8.02180	0.302548
$704$	−2.05400	−0.0774132
$705$	0	0
$706$	16.7790	0.631485
$707$	−32.4044	−1.21869
$708$	0	0
$709$	−45.4660	−1.70751	−0.853755	−	0.520674i	$-0.825681\pi$
$709$	−45.4660	−1.70751	−0.853755	+	0.520674i	$0.825681\pi$
$710$	5.85208	0.219625
$711$	0	0
$712$	16.9753	0.636176
$713$	−25.1042	−0.940160
$714$	0	0
$715$	−27.1317	−1.01467
$716$	−6.82749	−0.255155
$717$	0	0
$718$	−16.6877	−0.622780
$719$	37.8579	1.41186	0.705931	−	0.708280i	$-0.250529\pi$
$719$	37.8579	1.41186	0.705931	+	0.708280i	$0.250529\pi$
$720$	0	0
$721$	−6.87076	−0.255880
$722$	−17.5271	−0.652291
$723$	0	0
$724$	22.9681	0.853604
$725$	42.6440	1.58376
$726$	0	0
$727$	−10.0135	−0.371380	−0.185690	−	0.982608i	$-0.559452\pi$
$727$	−10.0135	−0.371380	−0.185690	+	0.982608i	$0.559452\pi$
$728$	14.6746	0.543876
$729$	0	0
$730$	35.5649	1.31631
$731$	29.0867	1.07581
$732$	0	0
$733$	−7.13889	−0.263681	−0.131840	−	0.991271i	$-0.542089\pi$
$733$	−7.13889	−0.263681	−0.131840	+	0.991271i	$0.542089\pi$
$734$	10.5475	0.389314
$735$	0	0
$736$	7.31540	0.269649
$737$	0.203768	0.00750588
$738$	0	0
$739$	32.2687	1.18702	0.593512	−	0.804825i	$-0.297741\pi$
$739$	32.2687	1.18702	0.593512	+	0.804825i	$0.297741\pi$
$740$	26.1330	0.960670
$741$	0	0
$742$	−20.6166	−0.756860
$743$	39.5138	1.44962	0.724810	−	0.688949i	$-0.241927\pi$
$743$	39.5138	1.44962	0.724810	+	0.688949i	$0.241927\pi$
$744$	0	0
$745$	−68.0131	−2.49181
$746$	9.15124	0.335051
$747$	0	0
$748$	8.30295	0.303586
$749$	−39.6555	−1.44898
$750$	0	0
$751$	11.2956	0.412184	0.206092	−	0.978533i	$-0.433925\pi$
$751$	11.2956	0.412184	0.206092	+	0.978533i	$0.433925\pi$
$752$	−7.37615	−0.268981
$753$	0	0
$754$	−13.4006	−0.488019
$755$	37.0746	1.34928
$756$	0	0
$757$	−21.1277	−0.767898	−0.383949	−	0.923354i	$-0.625436\pi$
$757$	−21.1277	−0.767898	−0.383949	+	0.923354i	$0.625436\pi$
$758$	−11.7646	−0.427308
$759$	0	0
$760$	4.79833	0.174054
$761$	12.7553	0.462377	0.231189	−	0.972909i	$-0.425739\pi$
$761$	12.7553	0.462377	0.231189	+	0.972909i	$0.425739\pi$
$762$	0	0
$763$	−40.8456	−1.47871
$764$	8.49941	0.307498
$765$	0	0
$766$	−23.1659	−0.837017
$767$	−21.9679	−0.793215
$768$	0	0
$769$	−32.0517	−1.15581	−0.577907	−	0.816102i	$-0.696130\pi$
$769$	−32.0517	−1.15581	−0.577907	+	0.816102i	$0.696130\pi$
$770$	−35.6698	−1.28545
$771$	0	0
$772$	−16.8446	−0.606252
$773$	21.6833	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$773$	21.6833	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$774$	0	0
$775$	−36.4850	−1.31058
$776$	7.93027	0.284680
$777$	0	0
$778$	−5.04437	−0.180849
$779$	11.2777	0.404066
$780$	0	0
$781$	3.04023	0.108788
$782$	−29.5712	−1.05746
$783$	0	0
$784$	12.2925	0.439020
$785$	16.6339	0.593689
$786$	0	0
$787$	−39.7428	−1.41668	−0.708339	−	0.705873i	$-0.750555\pi$
$787$	−39.7428	−1.41668	−0.708339	+	0.705873i	$0.750555\pi$
$788$	6.64703	0.236791
$789$	0	0
$790$	−63.0471	−2.24312
$791$	47.7243	1.69688
$792$	0	0
$793$	−39.9626	−1.41911
$794$	9.25511	0.328452
$795$	0	0
$796$	10.8114	0.383198
$797$	−9.95706	−0.352697	−0.176349	−	0.984328i	$-0.556429\pi$
$797$	−9.95706	−0.352697	−0.176349	+	0.984328i	$0.556429\pi$
$798$	0	0
$799$	29.8168	1.05484
$800$	10.6318	0.375890
$801$	0	0
$802$	6.83302	0.241282
$803$	18.4764	0.652019
$804$	0	0
$805$	127.039	4.47754
$806$	11.4651	0.403842
$807$	0	0
$808$	7.37750	0.259540
$809$	19.5727	0.688139	0.344070	−	0.938944i	$-0.388194\pi$
$809$	19.5727	0.688139	0.344070	+	0.938944i	$0.388194\pi$
$810$	0	0
$811$	29.1233	1.02266	0.511329	−	0.859385i	$-0.329153\pi$
$811$	29.1233	1.02266	0.511329	+	0.859385i	$0.329153\pi$
$812$	−17.6176	−0.618256
$813$	0	0
$814$	13.5765	0.475855
$815$	28.4242	0.995655
$816$	0	0
$817$	8.73271	0.305519
$818$	−12.6227	−0.441342
$819$	0	0
$820$	36.7400	1.28302
$821$	8.63350	0.301311	0.150656	−	0.988586i	$-0.451862\pi$
$821$	8.63350	0.301311	0.150656	+	0.988586i	$0.451862\pi$
$822$	0	0
$823$	−43.7675	−1.52564	−0.762820	−	0.646611i	$-0.776186\pi$
$823$	−43.7675	−1.52564	−0.762820	+	0.646611i	$0.776186\pi$
$824$	1.56426	0.0544937
$825$	0	0
$826$	−28.8810	−1.00490
$827$	43.7434	1.52111	0.760553	−	0.649275i	$-0.224928\pi$
$827$	43.7434	1.52111	0.760553	+	0.649275i	$0.224928\pi$
$828$	0	0
$829$	22.7878	0.791453	0.395727	−	0.918368i	$-0.370493\pi$
$829$	22.7878	0.791453	0.395727	+	0.918368i	$0.370493\pi$
$830$	−4.82260	−0.167395
$831$	0	0
$832$	−3.34096	−0.115827
$833$	−49.6904	−1.72167
$834$	0	0
$835$	40.0156	1.38480
$836$	2.49280	0.0862152
$837$	0	0
$838$	10.0218	0.346197
$839$	14.3238	0.494514	0.247257	−	0.968950i	$-0.420471\pi$
$839$	14.3238	0.494514	0.247257	+	0.968950i	$0.420471\pi$
$840$	0	0
$841$	−12.9119	−0.445239
$842$	29.1502	1.00458
$843$	0	0
$844$	3.87590	0.133414
$845$	7.26692	0.249990
$846$	0	0
$847$	29.7847	1.02341
$848$	4.69378	0.161185
$849$	0	0
$850$	−42.9771	−1.47410
$851$	−48.3531	−1.65752
$852$	0	0
$853$	17.1400	0.586864	0.293432	−	0.955980i	$-0.405203\pi$
$853$	17.1400	0.586864	0.293432	+	0.955980i	$0.405203\pi$
$854$	−52.5384	−1.79783
$855$	0	0
$856$	9.02835	0.308583
$857$	−2.76996	−0.0946201	−0.0473101	−	0.998880i	$-0.515065\pi$
$857$	−2.76996	−0.0946201	−0.0473101	+	0.998880i	$0.515065\pi$
$858$	0	0
$859$	32.0643	1.09402	0.547010	−	0.837126i	$-0.315766\pi$
$859$	32.0643	1.09402	0.547010	+	0.837126i	$0.315766\pi$
$860$	28.4490	0.970103
$861$	0	0
$862$	−15.9482	−0.543197
$863$	50.1088	1.70572	0.852861	−	0.522138i	$-0.174865\pi$
$863$	50.1088	1.70572	0.852861	+	0.522138i	$0.174865\pi$
$864$	0	0
$865$	91.6682	3.11681
$866$	2.78614	0.0946767
$867$	0	0
$868$	15.0731	0.511615
$869$	−32.7538	−1.11110
$870$	0	0
$871$	0.331440	0.0112304
$872$	9.29931	0.314914
$873$	0	0
$874$	−8.87819	−0.300309
$875$	97.8012	3.30629
$876$	0	0
$877$	−49.2107	−1.66173	−0.830863	−	0.556477i	$-0.812153\pi$
$877$	−49.2107	−1.66173	−0.830863	+	0.556477i	$0.812153\pi$
$878$	37.6099	1.26927
$879$	0	0
$880$	8.12092	0.273756
$881$	17.7923	0.599438	0.299719	−	0.954027i	$-0.403107\pi$
$881$	17.7923	0.599438	0.299719	+	0.954027i	$0.403107\pi$
$882$	0	0
$883$	20.7711	0.699005	0.349502	−	0.936936i	$-0.386351\pi$
$883$	20.7711	0.699005	0.349502	+	0.936936i	$0.386351\pi$
$884$	13.5052	0.454230
$885$	0	0
$886$	26.3690	0.885885
$887$	−15.8301	−0.531522	−0.265761	−	0.964039i	$-0.585623\pi$
$887$	−15.8301	−0.531522	−0.265761	+	0.964039i	$0.585623\pi$
$888$	0	0
$889$	−35.0554	−1.17572
$890$	−67.1153	−2.24971
$891$	0	0
$892$	1.00000	0.0334825
$893$	8.95191	0.299564
$894$	0	0
$895$	26.9939	0.902306
$896$	−4.39233	−0.146737
$897$	0	0
$898$	−35.6329	−1.18909
$899$	−13.7645	−0.459071
$900$	0	0
$901$	−18.9738	−0.632108
$902$	19.0869	0.635525
$903$	0	0
$904$	−10.8654	−0.361377
$905$	−90.8092	−3.01860
$906$	0	0
$907$	−10.0036	−0.332163	−0.166082	−	0.986112i	$-0.553112\pi$
$907$	−10.0036	−0.332163	−0.166082	+	0.986112i	$0.553112\pi$
$908$	−5.26769	−0.174814
$909$	0	0
$910$	−58.0190	−1.92331
$911$	−5.37392	−0.178046	−0.0890230	−	0.996030i	$-0.528374\pi$
$911$	−5.37392	−0.178046	−0.0890230	+	0.996030i	$0.528374\pi$
$912$	0	0
$913$	−2.50541	−0.0829169
$914$	31.8804	1.05451
$915$	0	0
$916$	−22.1590	−0.732152
$917$	25.7539	0.850468
$918$	0	0
$919$	−16.2545	−0.536188	−0.268094	−	0.963393i	$-0.586394\pi$
$919$	−16.2545	−0.536188	−0.268094	+	0.963393i	$0.586394\pi$
$920$	−28.9229	−0.953561
$921$	0	0
$922$	−3.39594	−0.111839
$923$	4.94512	0.162771
$924$	0	0
$925$	−70.2735	−2.31058
$926$	11.9556	0.392886
$927$	0	0
$928$	4.01099	0.131667
$929$	34.5401	1.13322	0.566611	−	0.823985i	$-0.308254\pi$
$929$	34.5401	1.13322	0.566611	+	0.823985i	$0.308254\pi$
$930$	0	0
$931$	−14.9186	−0.488937
$932$	−19.1483	−0.627223
$933$	0	0
$934$	1.23323	0.0403525
$935$	−32.8274	−1.07357
$936$	0	0
$937$	−55.2476	−1.80486	−0.902431	−	0.430835i	$-0.858219\pi$
$937$	−55.2476	−1.80486	−0.902431	+	0.430835i	$0.858219\pi$
$938$	0.435742	0.0142275
$939$	0	0
$940$	29.1631	0.951196
$941$	−12.4623	−0.406261	−0.203130	−	0.979152i	$-0.565112\pi$
$941$	−12.4623	−0.406261	−0.203130	+	0.979152i	$0.565112\pi$
$942$	0	0
$943$	−67.9787	−2.21369
$944$	6.57533	0.214009
$945$	0	0
$946$	14.7797	0.480528
$947$	28.8830	0.938570	0.469285	−	0.883047i	$-0.344512\pi$
$947$	28.8830	0.938570	0.469285	+	0.883047i	$0.344512\pi$
$948$	0	0
$949$	30.0530	0.975562
$950$	−12.9030	−0.418630
$951$	0	0
$952$	17.7552	0.575450
$953$	−7.16880	−0.232220	−0.116110	−	0.993236i	$-0.537043\pi$
$953$	−7.16880	−0.232220	−0.116110	+	0.993236i	$0.537043\pi$
$954$	0	0
$955$	−33.6042	−1.08741
$956$	11.4317	0.369727
$957$	0	0
$958$	−13.4708	−0.435223
$959$	−2.32752	−0.0751595
$960$	0	0
$961$	−19.2235	−0.620113
$962$	22.0829	0.711982
$963$	0	0
$964$	10.0135	0.322514
$965$	66.5987	2.14389
$966$	0	0
$967$	−28.0789	−0.902957	−0.451479	−	0.892282i	$-0.649103\pi$
$967$	−28.0789	−0.902957	−0.451479	+	0.892282i	$0.649103\pi$
$968$	−6.78107	−0.217952
$969$	0	0
$970$	−31.3539	−1.00671
$971$	−57.1450	−1.83387	−0.916935	−	0.399036i	$-0.869345\pi$
$971$	−57.1450	−1.83387	−0.916935	+	0.399036i	$0.869345\pi$
$972$	0	0
$973$	34.3811	1.10221
$974$	25.2900	0.810344
$975$	0	0
$976$	11.9614	0.382875
$977$	−8.31117	−0.265898	−0.132949	−	0.991123i	$-0.542445\pi$
$977$	−8.31117	−0.265898	−0.132949	+	0.991123i	$0.542445\pi$
$978$	0	0
$979$	−34.8673	−1.11437
$980$	−48.6011	−1.55251
$981$	0	0
$982$	29.9179	0.954719
$983$	13.2663	0.423130	0.211565	−	0.977364i	$-0.432144\pi$
$983$	13.2663	0.423130	0.211565	+	0.977364i	$0.432144\pi$
$984$	0	0
$985$	−26.2804	−0.837362
$986$	−16.2137	−0.516350
$987$	0	0
$988$	4.05468	0.128997
$989$	−52.6382	−1.67380
$990$	0	0
$991$	1.89534	0.0602076	0.0301038	−	0.999547i	$-0.490416\pi$
$991$	1.89534	0.0602076	0.0301038	+	0.999547i	$0.490416\pi$
$992$	−3.43169	−0.108956
$993$	0	0
$994$	6.50131	0.206209
$995$	−42.7449	−1.35511
$996$	0	0
$997$	3.93163	0.124516	0.0622579	−	0.998060i	$-0.480170\pi$
$997$	3.93163	0.124516	0.0622579	+	0.998060i	$0.480170\pi$
$998$	−8.86369	−0.280575
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.z.1.1		8
3.2	odd	2		446.2.a.f.1.7	✓	8
12.11	even	2		3568.2.a.n.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.f.1.7	✓	8	3.2	odd	2
3568.2.a.n.1.2		8	12.11	even	2
4014.2.a.z.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.95370 q^{5} -4.39233 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.95370 q^{5} -4.39233 q^{7} +1.00000 q^{8} -3.95370 q^{10} -2.05400 q^{11} -3.34096 q^{13} -4.39233 q^{14} +1.00000 q^{16} -4.04232 q^{17} -1.21363 q^{19} -3.95370 q^{20} -2.05400 q^{22} +7.31540 q^{23} +10.6318 q^{25} -3.34096 q^{26} -4.39233 q^{28} +4.01099 q^{29} -3.43169 q^{31} +1.00000 q^{32} -4.04232 q^{34} +17.3660 q^{35} -6.60976 q^{37} -1.21363 q^{38} -3.95370 q^{40} -9.29255 q^{41} -7.19553 q^{43} -2.05400 q^{44} +7.31540 q^{46} -7.37615 q^{47} +12.2925 q^{49} +10.6318 q^{50} -3.34096 q^{52} +4.69378 q^{53} +8.12092 q^{55} -4.39233 q^{56} +4.01099 q^{58} +6.57533 q^{59} +11.9614 q^{61} -3.43169 q^{62} +1.00000 q^{64} +13.2092 q^{65} -0.0992052 q^{67} -4.04232 q^{68} +17.3660 q^{70} -1.48015 q^{71} -8.99532 q^{73} -6.60976 q^{74} -1.21363 q^{76} +9.02186 q^{77} +15.9463 q^{79} -3.95370 q^{80} -9.29255 q^{82} +1.21977 q^{83} +15.9821 q^{85} -7.19553 q^{86} -2.05400 q^{88} +16.9753 q^{89} +14.6746 q^{91} +7.31540 q^{92} -7.37615 q^{94} +4.79833 q^{95} +7.93027 q^{97} +12.2925 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) 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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