LMFDB - Embedded newform 8048.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78533$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.78533 q^{3} -0.701114 q^{5} +2.02991 q^{7} +0.187388 q^{9} +O(q^{10})$	$q-1.78533 q^{3} -0.701114 q^{5} +2.02991 q^{7} +0.187388 q^{9} +0.626970 q^{11} -2.93743 q^{13} +1.25172 q^{15} -2.71003 q^{17} +1.11114 q^{19} -3.62406 q^{21} +0.412395 q^{23} -4.50844 q^{25} +5.02143 q^{27} +6.46349 q^{29} +4.14074 q^{31} -1.11935 q^{33} -1.42320 q^{35} +2.98634 q^{37} +5.24427 q^{39} -0.135430 q^{41} +0.861393 q^{43} -0.131380 q^{45} -1.67307 q^{47} -2.87945 q^{49} +4.83829 q^{51} -8.51721 q^{53} -0.439578 q^{55} -1.98374 q^{57} +0.406685 q^{59} -8.53132 q^{61} +0.380381 q^{63} +2.05948 q^{65} +13.0293 q^{67} -0.736259 q^{69} -2.78095 q^{71} -2.11734 q^{73} +8.04903 q^{75} +1.27270 q^{77} +0.317368 q^{79} -9.52705 q^{81} +5.05120 q^{83} +1.90004 q^{85} -11.5394 q^{87} +16.4699 q^{89} -5.96273 q^{91} -7.39256 q^{93} -0.779033 q^{95} +7.68873 q^{97} +0.117486 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.78533	−1.03076	−0.515379	−	0.856962i	$-0.672349\pi$
$3$	−1.78533	−1.03076	−0.515379	+	0.856962i	$0.672349\pi$
$4$	0	0
$5$	−0.701114	−0.313548	−0.156774	−	0.987635i	$-0.550109\pi$
$5$	−0.701114	−0.313548	−0.156774	+	0.987635i	$0.550109\pi$
$6$	0	0
$7$	2.02991	0.767235	0.383618	−	0.923492i	$-0.374678\pi$
$7$	2.02991	0.767235	0.383618	+	0.923492i	$0.374678\pi$
$8$	0	0
$9$	0.187388	0.0624625
$10$	0	0
$11$	0.626970	0.189039	0.0945194	−	0.995523i	$-0.469869\pi$
$11$	0.626970	0.189039	0.0945194	+	0.995523i	$0.469869\pi$
$12$	0	0
$13$	−2.93743	−0.814697	−0.407349	−	0.913273i	$-0.633547\pi$
$13$	−2.93743	−0.814697	−0.407349	+	0.913273i	$0.633547\pi$
$14$	0	0
$15$	1.25172	0.323192
$16$	0	0
$17$	−2.71003	−0.657280	−0.328640	−	0.944455i	$-0.606590\pi$
$17$	−2.71003	−0.657280	−0.328640	+	0.944455i	$0.606590\pi$
$18$	0	0
$19$	1.11114	0.254912	0.127456	−	0.991844i	$-0.459319\pi$
$19$	1.11114	0.254912	0.127456	+	0.991844i	$0.459319\pi$
$20$	0	0
$21$	−3.62406	−0.790834
$22$	0	0
$23$	0.412395	0.0859902	0.0429951	−	0.999075i	$-0.486310\pi$
$23$	0.412395	0.0859902	0.0429951	+	0.999075i	$0.486310\pi$
$24$	0	0
$25$	−4.50844	−0.901688
$26$	0	0
$27$	5.02143	0.966374
$28$	0	0
$29$	6.46349	1.20024	0.600120	−	0.799910i	$-0.295120\pi$
$29$	6.46349	1.20024	0.600120	+	0.799910i	$0.295120\pi$
$30$	0	0
$31$	4.14074	0.743698	0.371849	−	0.928293i	$-0.378724\pi$
$31$	4.14074	0.743698	0.371849	+	0.928293i	$0.378724\pi$
$32$	0	0
$33$	−1.11935	−0.194853
$34$	0	0
$35$	−1.42320	−0.240565
$36$	0	0
$37$	2.98634	0.490951	0.245475	−	0.969403i	$-0.421056\pi$
$37$	2.98634	0.490951	0.245475	+	0.969403i	$0.421056\pi$
$38$	0	0
$39$	5.24427	0.839756
$40$	0	0
$41$	−0.135430	−0.0211505	−0.0105753	−	0.999944i	$-0.503366\pi$
$41$	−0.135430	−0.0211505	−0.0105753	+	0.999944i	$0.503366\pi$
$42$	0	0
$43$	0.861393	0.131361	0.0656806	−	0.997841i	$-0.479078\pi$
$43$	0.861393	0.131361	0.0656806	+	0.997841i	$0.479078\pi$
$44$	0	0
$45$	−0.131380	−0.0195850
$46$	0	0
$47$	−1.67307	−0.244043	−0.122021	−	0.992527i	$-0.538938\pi$
$47$	−1.67307	−0.244043	−0.122021	+	0.992527i	$0.538938\pi$
$48$	0	0
$49$	−2.87945	−0.411350
$50$	0	0
$51$	4.83829	0.677497
$52$	0	0
$53$	−8.51721	−1.16993	−0.584964	−	0.811059i	$-0.698892\pi$
$53$	−8.51721	−1.16993	−0.584964	+	0.811059i	$0.698892\pi$
$54$	0	0
$55$	−0.439578	−0.0592727
$56$	0	0
$57$	−1.98374	−0.262753
$58$	0	0
$59$	0.406685	0.0529459	0.0264729	−	0.999650i	$-0.491572\pi$
$59$	0.406685	0.0529459	0.0264729	+	0.999650i	$0.491572\pi$
$60$	0	0
$61$	−8.53132	−1.09232	−0.546162	−	0.837680i	$-0.683912\pi$
$61$	−8.53132	−1.09232	−0.546162	+	0.837680i	$0.683912\pi$
$62$	0	0
$63$	0.380381	0.0479234
$64$	0	0
$65$	2.05948	0.255446
$66$	0	0
$67$	13.0293	1.59178	0.795889	−	0.605443i	$-0.207004\pi$
$67$	13.0293	1.59178	0.795889	+	0.605443i	$0.207004\pi$
$68$	0	0
$69$	−0.736259	−0.0886352
$70$	0	0
$71$	−2.78095	−0.330039	−0.165019	−	0.986290i	$-0.552769\pi$
$71$	−2.78095	−0.330039	−0.165019	+	0.986290i	$0.552769\pi$
$72$	0	0
$73$	−2.11734	−0.247816	−0.123908	−	0.992294i	$-0.539543\pi$
$73$	−2.11734	−0.247816	−0.123908	+	0.992294i	$0.539543\pi$
$74$	0	0
$75$	8.04903	0.929422
$76$	0	0
$77$	1.27270	0.145037
$78$	0	0
$79$	0.317368	0.0357067	0.0178534	−	0.999841i	$-0.494317\pi$
$79$	0.317368	0.0357067	0.0178534	+	0.999841i	$0.494317\pi$
$80$	0	0
$81$	−9.52705	−1.05856
$82$	0	0
$83$	5.05120	0.554442	0.277221	−	0.960806i	$-0.410587\pi$
$83$	5.05120	0.554442	0.277221	+	0.960806i	$0.410587\pi$
$84$	0	0
$85$	1.90004	0.206089
$86$	0	0
$87$	−11.5394	−1.23716
$88$	0	0
$89$	16.4699	1.74580	0.872901	−	0.487897i	$-0.162236\pi$
$89$	16.4699	1.74580	0.872901	+	0.487897i	$0.162236\pi$
$90$	0	0
$91$	−5.96273	−0.625064
$92$	0	0
$93$	−7.39256	−0.766573
$94$	0	0
$95$	−0.779033	−0.0799271
$96$	0	0
$97$	7.68873	0.780673	0.390336	−	0.920672i	$-0.372359\pi$
$97$	7.68873	0.780673	0.390336	+	0.920672i	$0.372359\pi$
$98$	0	0
$99$	0.117486	0.0118078
$100$	0	0
$101$	−0.549020	−0.0546296	−0.0273148	−	0.999627i	$-0.508696\pi$
$101$	−0.549020	−0.0546296	−0.0273148	+	0.999627i	$0.508696\pi$
$102$	0	0
$103$	−1.32758	−0.130810	−0.0654051	−	0.997859i	$-0.520834\pi$
$103$	−1.32758	−0.130810	−0.0654051	+	0.997859i	$0.520834\pi$
$104$	0	0
$105$	2.54088	0.247964
$106$	0	0
$107$	−2.57528	−0.248961	−0.124481	−	0.992222i	$-0.539726\pi$
$107$	−2.57528	−0.248961	−0.124481	+	0.992222i	$0.539726\pi$
$108$	0	0
$109$	8.89011	0.851518	0.425759	−	0.904837i	$-0.360007\pi$
$109$	8.89011	0.851518	0.425759	+	0.904837i	$0.360007\pi$
$110$	0	0
$111$	−5.33159	−0.506052
$112$	0	0
$113$	−1.91987	−0.180606	−0.0903032	−	0.995914i	$-0.528784\pi$
$113$	−1.91987	−0.180606	−0.0903032	+	0.995914i	$0.528784\pi$
$114$	0	0
$115$	−0.289136	−0.0269621
$116$	0	0
$117$	−0.550438	−0.0508880
$118$	0	0
$119$	−5.50114	−0.504288
$120$	0	0
$121$	−10.6069	−0.964264
$122$	0	0
$123$	0.241786	0.0218011
$124$	0	0
$125$	6.66650	0.596270
$126$	0	0
$127$	−21.8010	−1.93452	−0.967261	−	0.253782i	$-0.918325\pi$
$127$	−21.8010	−1.93452	−0.967261	+	0.253782i	$0.918325\pi$
$128$	0	0
$129$	−1.53787	−0.135402
$130$	0	0
$131$	−8.78145	−0.767239	−0.383620	−	0.923491i	$-0.625323\pi$
$131$	−8.78145	−0.767239	−0.383620	+	0.923491i	$0.625323\pi$
$132$	0	0
$133$	2.25551	0.195577
$134$	0	0
$135$	−3.52060	−0.303005
$136$	0	0
$137$	−4.26127	−0.364065	−0.182032	−	0.983293i	$-0.558268\pi$
$137$	−4.26127	−0.364065	−0.182032	+	0.983293i	$0.558268\pi$
$138$	0	0
$139$	−13.4951	−1.14464	−0.572321	−	0.820030i	$-0.693957\pi$
$139$	−13.4951	−1.14464	−0.572321	+	0.820030i	$0.693957\pi$
$140$	0	0
$141$	2.98698	0.251549
$142$	0	0
$143$	−1.84168	−0.154009
$144$	0	0
$145$	−4.53164	−0.376332
$146$	0	0
$147$	5.14076	0.424003
$148$	0	0
$149$	11.2453	0.921253	0.460626	−	0.887594i	$-0.347625\pi$
$149$	11.2453	0.921253	0.460626	+	0.887594i	$0.347625\pi$
$150$	0	0
$151$	2.36107	0.192141	0.0960705	−	0.995375i	$-0.469373\pi$
$151$	2.36107	0.192141	0.0960705	+	0.995375i	$0.469373\pi$
$152$	0	0
$153$	−0.507827	−0.0410554
$154$	0	0
$155$	−2.90313	−0.233185
$156$	0	0
$157$	−17.5242	−1.39859	−0.699293	−	0.714835i	$-0.746502\pi$
$157$	−17.5242	−1.39859	−0.699293	+	0.714835i	$0.746502\pi$
$158$	0	0
$159$	15.2060	1.20591
$160$	0	0
$161$	0.837126	0.0659747
$162$	0	0
$163$	7.12153	0.557801	0.278901	−	0.960320i	$-0.410030\pi$
$163$	7.12153	0.557801	0.278901	+	0.960320i	$0.410030\pi$
$164$	0	0
$165$	0.784790	0.0610958
$166$	0	0
$167$	−4.56080	−0.352925	−0.176463	−	0.984307i	$-0.556465\pi$
$167$	−4.56080	−0.352925	−0.176463	+	0.984307i	$0.556465\pi$
$168$	0	0
$169$	−4.37149	−0.336269
$170$	0	0
$171$	0.208213	0.0159224
$172$	0	0
$173$	2.91357	0.221514	0.110757	−	0.993847i	$-0.464672\pi$
$173$	2.91357	0.221514	0.110757	+	0.993847i	$0.464672\pi$
$174$	0	0
$175$	−9.15174	−0.691807
$176$	0	0
$177$	−0.726065	−0.0545744
$178$	0	0
$179$	9.01141	0.673544	0.336772	−	0.941586i	$-0.390665\pi$
$179$	9.01141	0.673544	0.336772	+	0.941586i	$0.390665\pi$
$180$	0	0
$181$	14.4065	1.07082	0.535412	−	0.844591i	$-0.320156\pi$
$181$	14.4065	1.07082	0.535412	+	0.844591i	$0.320156\pi$
$182$	0	0
$183$	15.2312	1.12592
$184$	0	0
$185$	−2.09376	−0.153937
$186$	0	0
$187$	−1.69911	−0.124251
$188$	0	0
$189$	10.1931	0.741437
$190$	0	0
$191$	−4.40730	−0.318901	−0.159450	−	0.987206i	$-0.550972\pi$
$191$	−4.40730	−0.318901	−0.159450	+	0.987206i	$0.550972\pi$
$192$	0	0
$193$	−2.90027	−0.208766	−0.104383	−	0.994537i	$-0.533287\pi$
$193$	−2.90027	−0.208766	−0.104383	+	0.994537i	$0.533287\pi$
$194$	0	0
$195$	−3.67683	−0.263304
$196$	0	0
$197$	5.91510	0.421433	0.210717	−	0.977547i	$-0.432420\pi$
$197$	5.91510	0.421433	0.210717	+	0.977547i	$0.432420\pi$
$198$	0	0
$199$	4.73473	0.335636	0.167818	−	0.985818i	$-0.446328\pi$
$199$	4.73473	0.335636	0.167818	+	0.985818i	$0.446328\pi$
$200$	0	0
$201$	−23.2615	−1.64074
$202$	0	0
$203$	13.1203	0.920866
$204$	0	0
$205$	0.0949516	0.00663171
$206$	0	0
$207$	0.0772776	0.00537117
$208$	0	0
$209$	0.696649	0.0481882
$210$	0	0
$211$	1.50671	0.103726	0.0518631	−	0.998654i	$-0.483484\pi$
$211$	1.50671	0.103726	0.0518631	+	0.998654i	$0.483484\pi$
$212$	0	0
$213$	4.96491	0.340190
$214$	0	0
$215$	−0.603935	−0.0411880
$216$	0	0
$217$	8.40534	0.570591
$218$	0	0
$219$	3.78014	0.255438
$220$	0	0
$221$	7.96054	0.535484
$222$	0	0
$223$	9.79220	0.655734	0.327867	−	0.944724i	$-0.393670\pi$
$223$	9.79220	0.655734	0.327867	+	0.944724i	$0.393670\pi$
$224$	0	0
$225$	−0.844825	−0.0563217
$226$	0	0
$227$	−6.88512	−0.456981	−0.228491	−	0.973546i	$-0.573379\pi$
$227$	−6.88512	−0.456981	−0.228491	+	0.973546i	$0.573379\pi$
$228$	0	0
$229$	12.1880	0.805405	0.402703	−	0.915331i	$-0.368071\pi$
$229$	12.1880	0.805405	0.402703	+	0.915331i	$0.368071\pi$
$230$	0	0
$231$	−2.27218	−0.149498
$232$	0	0
$233$	−21.0547	−1.37934	−0.689669	−	0.724125i	$-0.742244\pi$
$233$	−21.0547	−1.37934	−0.689669	+	0.724125i	$0.742244\pi$
$234$	0	0
$235$	1.17301	0.0765190
$236$	0	0
$237$	−0.566606	−0.0368050
$238$	0	0
$239$	−9.11944	−0.589888	−0.294944	−	0.955515i	$-0.595301\pi$
$239$	−9.11944	−0.589888	−0.294944	+	0.955515i	$0.595301\pi$
$240$	0	0
$241$	22.7352	1.46450	0.732252	−	0.681034i	$-0.238469\pi$
$241$	22.7352	1.46450	0.732252	+	0.681034i	$0.238469\pi$
$242$	0	0
$243$	1.94460	0.124746
$244$	0	0
$245$	2.01882	0.128978
$246$	0	0
$247$	−3.26388	−0.207676
$248$	0	0
$249$	−9.01804	−0.571495
$250$	0	0
$251$	11.6824	0.737385	0.368692	−	0.929551i	$-0.379806\pi$
$251$	11.6824	0.737385	0.368692	+	0.929551i	$0.379806\pi$
$252$	0	0
$253$	0.258559	0.0162555
$254$	0	0
$255$	−3.39220	−0.212428
$256$	0	0
$257$	−16.3010	−1.01683	−0.508415	−	0.861112i	$-0.669768\pi$
$257$	−16.3010	−1.01683	−0.508415	+	0.861112i	$0.669768\pi$
$258$	0	0
$259$	6.06201	0.376675
$260$	0	0
$261$	1.21118	0.0749700
$262$	0	0
$263$	15.7955	0.973993	0.486996	−	0.873404i	$-0.338093\pi$
$263$	15.7955	0.973993	0.486996	+	0.873404i	$0.338093\pi$
$264$	0	0
$265$	5.97154	0.366829
$266$	0	0
$267$	−29.4041	−1.79950
$268$	0	0
$269$	11.2401	0.685323	0.342662	−	0.939459i	$-0.388672\pi$
$269$	11.2401	0.685323	0.342662	+	0.939459i	$0.388672\pi$
$270$	0	0
$271$	−22.7114	−1.37962	−0.689809	−	0.723991i	$-0.742306\pi$
$271$	−22.7114	−1.37962	−0.689809	+	0.723991i	$0.742306\pi$
$272$	0	0
$273$	10.6454	0.644290
$274$	0	0
$275$	−2.82666	−0.170454
$276$	0	0
$277$	−3.42609	−0.205854	−0.102927	−	0.994689i	$-0.532821\pi$
$277$	−3.42609	−0.205854	−0.102927	+	0.994689i	$0.532821\pi$
$278$	0	0
$279$	0.775923	0.0464533
$280$	0	0
$281$	−6.41168	−0.382488	−0.191244	−	0.981542i	$-0.561252\pi$
$281$	−6.41168	−0.382488	−0.191244	+	0.981542i	$0.561252\pi$
$282$	0	0
$283$	22.3462	1.32834	0.664172	−	0.747580i	$-0.268784\pi$
$283$	22.3462	1.32834	0.664172	+	0.747580i	$0.268784\pi$
$284$	0	0
$285$	1.39083	0.0823855
$286$	0	0
$287$	−0.274910	−0.0162274
$288$	0	0
$289$	−9.65571	−0.567983
$290$	0	0
$291$	−13.7269	−0.804685
$292$	0	0
$293$	4.95366	0.289396	0.144698	−	0.989476i	$-0.453779\pi$
$293$	4.95366	0.289396	0.144698	+	0.989476i	$0.453779\pi$
$294$	0	0
$295$	−0.285133	−0.0166011
$296$	0	0
$297$	3.14829	0.182682
$298$	0	0
$299$	−1.21138	−0.0700560
$300$	0	0
$301$	1.74855	0.100785
$302$	0	0
$303$	0.980180	0.0563099
$304$	0	0
$305$	5.98143	0.342496
$306$	0	0
$307$	1.86937	0.106690	0.0533452	−	0.998576i	$-0.483012\pi$
$307$	1.86937	0.106690	0.0533452	+	0.998576i	$0.483012\pi$
$308$	0	0
$309$	2.37016	0.134834
$310$	0	0
$311$	4.29662	0.243639	0.121819	−	0.992552i	$-0.461127\pi$
$311$	4.29662	0.243639	0.121819	+	0.992552i	$0.461127\pi$
$312$	0	0
$313$	−24.2378	−1.37000	−0.684999	−	0.728544i	$-0.740198\pi$
$313$	−24.2378	−1.37000	−0.684999	+	0.728544i	$0.740198\pi$
$314$	0	0
$315$	−0.266690	−0.0150263
$316$	0	0
$317$	−28.1902	−1.58332	−0.791661	−	0.610961i	$-0.790783\pi$
$317$	−28.1902	−1.58332	−0.791661	+	0.610961i	$0.790783\pi$
$318$	0	0
$319$	4.05242	0.226892
$320$	0	0
$321$	4.59771	0.256619
$322$	0	0
$323$	−3.01121	−0.167548
$324$	0	0
$325$	13.2432	0.734602
$326$	0	0
$327$	−15.8717	−0.877709
$328$	0	0
$329$	−3.39619	−0.187238
$330$	0	0
$331$	−13.5185	−0.743046	−0.371523	−	0.928424i	$-0.621164\pi$
$331$	−13.5185	−0.743046	−0.371523	+	0.928424i	$0.621164\pi$
$332$	0	0
$333$	0.559603	0.0306660
$334$	0	0
$335$	−9.13500	−0.499098
$336$	0	0
$337$	−1.73199	−0.0943475	−0.0471738	−	0.998887i	$-0.515021\pi$
$337$	−1.73199	−0.0943475	−0.0471738	+	0.998887i	$0.515021\pi$
$338$	0	0
$339$	3.42760	0.186161
$340$	0	0
$341$	2.59612	0.140588
$342$	0	0
$343$	−20.0544	−1.08284
$344$	0	0
$345$	0.516202	0.0277914
$346$	0	0
$347$	27.7262	1.48842	0.744210	−	0.667946i	$-0.232826\pi$
$347$	27.7262	1.48842	0.744210	+	0.667946i	$0.232826\pi$
$348$	0	0
$349$	−16.0612	−0.859737	−0.429869	−	0.902891i	$-0.641440\pi$
$349$	−16.0612	−0.859737	−0.429869	+	0.902891i	$0.641440\pi$
$350$	0	0
$351$	−14.7501	−0.787302
$352$	0	0
$353$	−16.0011	−0.851652	−0.425826	−	0.904805i	$-0.640016\pi$
$353$	−16.0011	−0.851652	−0.425826	+	0.904805i	$0.640016\pi$
$354$	0	0
$355$	1.94977	0.103483
$356$	0	0
$357$	9.82132	0.519799
$358$	0	0
$359$	−29.2698	−1.54480	−0.772400	−	0.635136i	$-0.780944\pi$
$359$	−29.2698	−1.54480	−0.772400	+	0.635136i	$0.780944\pi$
$360$	0	0
$361$	−17.7654	−0.935020
$362$	0	0
$363$	18.9368	0.993923
$364$	0	0
$365$	1.48450	0.0777022
$366$	0	0
$367$	−19.5331	−1.01962	−0.509810	−	0.860287i	$-0.670284\pi$
$367$	−19.5331	−1.01962	−0.509810	+	0.860287i	$0.670284\pi$
$368$	0	0
$369$	−0.0253778	−0.00132112
$370$	0	0
$371$	−17.2892	−0.897611
$372$	0	0
$373$	−22.0592	−1.14218	−0.571091	−	0.820887i	$-0.693480\pi$
$373$	−22.0592	−1.14218	−0.571091	+	0.820887i	$0.693480\pi$
$374$	0	0
$375$	−11.9019	−0.614610
$376$	0	0
$377$	−18.9861	−0.977832
$378$	0	0
$379$	−10.7429	−0.551828	−0.275914	−	0.961182i	$-0.588980\pi$
$379$	−10.7429	−0.551828	−0.275914	+	0.961182i	$0.588980\pi$
$380$	0	0
$381$	38.9218	1.99403
$382$	0	0
$383$	−7.81876	−0.399520	−0.199760	−	0.979845i	$-0.564016\pi$
$383$	−7.81876	−0.399520	−0.199760	+	0.979845i	$0.564016\pi$
$384$	0	0
$385$	−0.892305	−0.0454761
$386$	0	0
$387$	0.161414	0.00820515
$388$	0	0
$389$	−35.0312	−1.77615	−0.888075	−	0.459698i	$-0.847957\pi$
$389$	−35.0312	−1.77615	−0.888075	+	0.459698i	$0.847957\pi$
$390$	0	0
$391$	−1.11760	−0.0565197
$392$	0	0
$393$	15.6778	0.790838
$394$	0	0
$395$	−0.222512	−0.0111958
$396$	0	0
$397$	−0.245438	−0.0123182	−0.00615909	−	0.999981i	$-0.501961\pi$
$397$	−0.245438	−0.0123182	−0.00615909	+	0.999981i	$0.501961\pi$
$398$	0	0
$399$	−4.02682	−0.201593
$400$	0	0
$401$	−16.3434	−0.816151	−0.408075	−	0.912948i	$-0.633800\pi$
$401$	−16.3434	−0.816151	−0.408075	+	0.912948i	$0.633800\pi$
$402$	0	0
$403$	−12.1631	−0.605889
$404$	0	0
$405$	6.67955	0.331909
$406$	0	0
$407$	1.87235	0.0928087
$408$	0	0
$409$	4.60893	0.227897	0.113948	−	0.993487i	$-0.463650\pi$
$409$	4.60893	0.227897	0.113948	+	0.993487i	$0.463650\pi$
$410$	0	0
$411$	7.60775	0.375263
$412$	0	0
$413$	0.825536	0.0406220
$414$	0	0
$415$	−3.54147	−0.173844
$416$	0	0
$417$	24.0932	1.17985
$418$	0	0
$419$	−11.4509	−0.559414	−0.279707	−	0.960085i	$-0.590237\pi$
$419$	−11.4509	−0.559414	−0.279707	+	0.960085i	$0.590237\pi$
$420$	0	0
$421$	−22.2626	−1.08501	−0.542505	−	0.840052i	$-0.682524\pi$
$421$	−22.2626	−1.08501	−0.542505	+	0.840052i	$0.682524\pi$
$422$	0	0
$423$	−0.313513	−0.0152435
$424$	0	0
$425$	12.2180	0.592661
$426$	0	0
$427$	−17.3178	−0.838069
$428$	0	0
$429$	3.28800	0.158746
$430$	0	0
$431$	1.13299	0.0545741	0.0272871	−	0.999628i	$-0.491313\pi$
$431$	1.13299	0.0545741	0.0272871	+	0.999628i	$0.491313\pi$
$432$	0	0
$433$	−2.65692	−0.127683	−0.0638417	−	0.997960i	$-0.520335\pi$
$433$	−2.65692	−0.127683	−0.0638417	+	0.997960i	$0.520335\pi$
$434$	0	0
$435$	8.09046	0.387908
$436$	0	0
$437$	0.458226	0.0219199
$438$	0	0
$439$	−6.57228	−0.313678	−0.156839	−	0.987624i	$-0.550130\pi$
$439$	−6.57228	−0.313678	−0.156839	+	0.987624i	$0.550130\pi$
$440$	0	0
$441$	−0.539573	−0.0256940
$442$	0	0
$443$	−1.34406	−0.0638583	−0.0319291	−	0.999490i	$-0.510165\pi$
$443$	−1.34406	−0.0638583	−0.0319291	+	0.999490i	$0.510165\pi$
$444$	0	0
$445$	−11.5473	−0.547393
$446$	0	0
$447$	−20.0766	−0.949589
$448$	0	0
$449$	26.3410	1.24311	0.621555	−	0.783371i	$-0.286501\pi$
$449$	26.3410	1.24311	0.621555	+	0.783371i	$0.286501\pi$
$450$	0	0
$451$	−0.0849103	−0.00399827
$452$	0	0
$453$	−4.21528	−0.198051
$454$	0	0
$455$	4.18056	0.195988
$456$	0	0
$457$	−30.0693	−1.40658	−0.703290	−	0.710903i	$-0.748287\pi$
$457$	−30.0693	−1.40658	−0.703290	+	0.710903i	$0.748287\pi$
$458$	0	0
$459$	−13.6082	−0.635179
$460$	0	0
$461$	−7.72477	−0.359778	−0.179889	−	0.983687i	$-0.557574\pi$
$461$	−7.72477	−0.359778	−0.179889	+	0.983687i	$0.557574\pi$
$462$	0	0
$463$	−7.82961	−0.363873	−0.181936	−	0.983310i	$-0.558236\pi$
$463$	−7.82961	−0.363873	−0.181936	+	0.983310i	$0.558236\pi$
$464$	0	0
$465$	5.18303	0.240357
$466$	0	0
$467$	7.19623	0.333002	0.166501	−	0.986041i	$-0.446753\pi$
$467$	7.19623	0.333002	0.166501	+	0.986041i	$0.446753\pi$
$468$	0	0
$469$	26.4483	1.22127
$470$	0	0
$471$	31.2865	1.44160
$472$	0	0
$473$	0.540068	0.0248323
$474$	0	0
$475$	−5.00948	−0.229851
$476$	0	0
$477$	−1.59602	−0.0730767
$478$	0	0
$479$	12.1955	0.557229	0.278614	−	0.960403i	$-0.410125\pi$
$479$	12.1955	0.557229	0.278614	+	0.960403i	$0.410125\pi$
$480$	0	0
$481$	−8.77217	−0.399976
$482$	0	0
$483$	−1.49454	−0.0680040
$484$	0	0
$485$	−5.39068	−0.244778
$486$	0	0
$487$	−35.9684	−1.62988	−0.814942	−	0.579543i	$-0.803231\pi$
$487$	−35.9684	−1.62988	−0.814942	+	0.579543i	$0.803231\pi$
$488$	0	0
$489$	−12.7142	−0.574958
$490$	0	0
$491$	17.3618	0.783527	0.391763	−	0.920066i	$-0.371865\pi$
$491$	17.3618	0.783527	0.391763	+	0.920066i	$0.371865\pi$
$492$	0	0
$493$	−17.5163	−0.788893
$494$	0	0
$495$	−0.0823714	−0.00370232
$496$	0	0
$497$	−5.64510	−0.253217
$498$	0	0
$499$	30.0817	1.34664	0.673320	−	0.739351i	$-0.264867\pi$
$499$	30.0817	1.34664	0.673320	+	0.739351i	$0.264867\pi$
$500$	0	0
$501$	8.14251	0.363781
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	0.384926	0.0171290
$506$	0	0
$507$	7.80454	0.346612
$508$	0	0
$509$	−32.5495	−1.44273	−0.721366	−	0.692554i	$-0.756485\pi$
$509$	−32.5495	−1.44273	−0.721366	+	0.692554i	$0.756485\pi$
$510$	0	0
$511$	−4.29802	−0.190133
$512$	0	0
$513$	5.57949	0.246340
$514$	0	0
$515$	0.930784	0.0410152
$516$	0	0
$517$	−1.04897	−0.0461335
$518$	0	0
$519$	−5.20166	−0.228328
$520$	0	0
$521$	−12.6474	−0.554094	−0.277047	−	0.960856i	$-0.589356\pi$
$521$	−12.6474	−0.554094	−0.277047	+	0.960856i	$0.589356\pi$
$522$	0	0
$523$	−14.7294	−0.644073	−0.322036	−	0.946727i	$-0.604367\pi$
$523$	−14.7294	−0.644073	−0.322036	+	0.946727i	$0.604367\pi$
$524$	0	0
$525$	16.3388	0.713085
$526$	0	0
$527$	−11.2215	−0.488818
$528$	0	0
$529$	−22.8299	−0.992606
$530$	0	0
$531$	0.0762077	0.00330713
$532$	0	0
$533$	0.397815	0.0172313
$534$	0	0
$535$	1.80556	0.0780613
$536$	0	0
$537$	−16.0883	−0.694261
$538$	0	0
$539$	−1.80533	−0.0777611
$540$	0	0
$541$	−27.4172	−1.17876	−0.589379	−	0.807857i	$-0.700627\pi$
$541$	−27.4172	−1.17876	−0.589379	+	0.807857i	$0.700627\pi$
$542$	0	0
$543$	−25.7202	−1.10376
$544$	0	0
$545$	−6.23298	−0.266992
$546$	0	0
$547$	16.1884	0.692164	0.346082	−	0.938204i	$-0.387512\pi$
$547$	16.1884	0.692164	0.346082	+	0.938204i	$0.387512\pi$
$548$	0	0
$549$	−1.59866	−0.0682293
$550$	0	0
$551$	7.18181	0.305955
$552$	0	0
$553$	0.644231	0.0273955
$554$	0	0
$555$	3.73805	0.158671
$556$	0	0
$557$	−13.3858	−0.567176	−0.283588	−	0.958946i	$-0.591525\pi$
$557$	−13.3858	−0.567176	−0.283588	+	0.958946i	$0.591525\pi$
$558$	0	0
$559$	−2.53028	−0.107020
$560$	0	0
$561$	3.03347	0.128073
$562$	0	0
$563$	−23.3845	−0.985538	−0.492769	−	0.870160i	$-0.664015\pi$
$563$	−23.3845	−0.985538	−0.492769	+	0.870160i	$0.664015\pi$
$564$	0	0
$565$	1.34605	0.0566287
$566$	0	0
$567$	−19.3391	−0.812165
$568$	0	0
$569$	43.0171	1.80337	0.901685	−	0.432393i	$-0.142331\pi$
$569$	43.0171	1.80337	0.901685	+	0.432393i	$0.142331\pi$
$570$	0	0
$571$	1.24920	0.0522772	0.0261386	−	0.999658i	$-0.491679\pi$
$571$	1.24920	0.0522772	0.0261386	+	0.999658i	$0.491679\pi$
$572$	0	0
$573$	7.86846	0.328710
$574$	0	0
$575$	−1.85926	−0.0775364
$576$	0	0
$577$	−38.6392	−1.60857	−0.804284	−	0.594244i	$-0.797451\pi$
$577$	−38.6392	−1.60857	−0.804284	+	0.594244i	$0.797451\pi$
$578$	0	0
$579$	5.17793	0.215187
$580$	0	0
$581$	10.2535	0.425387
$582$	0	0
$583$	−5.34004	−0.221162
$584$	0	0
$585$	0.385920	0.0159558
$586$	0	0
$587$	13.5179	0.557943	0.278972	−	0.960299i	$-0.410006\pi$
$587$	13.5179	0.557943	0.278972	+	0.960299i	$0.410006\pi$
$588$	0	0
$589$	4.60092	0.189578
$590$	0	0
$591$	−10.5604	−0.434396
$592$	0	0
$593$	−6.72274	−0.276070	−0.138035	−	0.990427i	$-0.544079\pi$
$593$	−6.72274	−0.276070	−0.138035	+	0.990427i	$0.544079\pi$
$594$	0	0
$595$	3.85692	0.158118
$596$	0	0
$597$	−8.45303	−0.345960
$598$	0	0
$599$	−4.15139	−0.169621	−0.0848106	−	0.996397i	$-0.527029\pi$
$599$	−4.15139	−0.169621	−0.0848106	+	0.996397i	$0.527029\pi$
$600$	0	0
$601$	−39.8754	−1.62655	−0.813276	−	0.581878i	$-0.802318\pi$
$601$	−39.8754	−1.62655	−0.813276	+	0.581878i	$0.802318\pi$
$602$	0	0
$603$	2.44152	0.0994264
$604$	0	0
$605$	7.43665	0.302343
$606$	0	0
$607$	27.9713	1.13532	0.567660	−	0.823263i	$-0.307849\pi$
$607$	27.9713	1.13532	0.567660	+	0.823263i	$0.307849\pi$
$608$	0	0
$609$	−23.4240	−0.949190
$610$	0	0
$611$	4.91454	0.198821
$612$	0	0
$613$	−30.7521	−1.24207	−0.621033	−	0.783785i	$-0.713287\pi$
$613$	−30.7521	−1.24207	−0.621033	+	0.783785i	$0.713287\pi$
$614$	0	0
$615$	−0.169520	−0.00683569
$616$	0	0
$617$	−24.4669	−0.984999	−0.492500	−	0.870313i	$-0.663917\pi$
$617$	−24.4669	−0.984999	−0.492500	+	0.870313i	$0.663917\pi$
$618$	0	0
$619$	−37.1492	−1.49315	−0.746575	−	0.665301i	$-0.768303\pi$
$619$	−37.1492	−1.49315	−0.746575	+	0.665301i	$0.768303\pi$
$620$	0	0
$621$	2.07081	0.0830988
$622$	0	0
$623$	33.4324	1.33944
$624$	0	0
$625$	17.8682	0.714729
$626$	0	0
$627$	−1.24375	−0.0496704
$628$	0	0
$629$	−8.09308	−0.322692
$630$	0	0
$631$	15.7642	0.627563	0.313781	−	0.949495i	$-0.398404\pi$
$631$	15.7642	0.627563	0.313781	+	0.949495i	$0.398404\pi$
$632$	0	0
$633$	−2.68997	−0.106917
$634$	0	0
$635$	15.2850	0.606565
$636$	0	0
$637$	8.45819	0.335126
$638$	0	0
$639$	−0.521116	−0.0206150
$640$	0	0
$641$	48.3528	1.90982	0.954911	−	0.296892i	$-0.0959503\pi$
$641$	48.3528	1.90982	0.954911	+	0.296892i	$0.0959503\pi$
$642$	0	0
$643$	−3.30378	−0.130288	−0.0651441	−	0.997876i	$-0.520751\pi$
$643$	−3.30378	−0.130288	−0.0651441	+	0.997876i	$0.520751\pi$
$644$	0	0
$645$	1.07822	0.0424549
$646$	0	0
$647$	−34.3017	−1.34854	−0.674269	−	0.738486i	$-0.735541\pi$
$647$	−34.3017	−1.34854	−0.674269	+	0.738486i	$0.735541\pi$
$648$	0	0
$649$	0.254980	0.0100088
$650$	0	0
$651$	−15.0063	−0.588142
$652$	0	0
$653$	−13.5211	−0.529120	−0.264560	−	0.964369i	$-0.585227\pi$
$653$	−13.5211	−0.529120	−0.264560	+	0.964369i	$0.585227\pi$
$654$	0	0
$655$	6.15680	0.240566
$656$	0	0
$657$	−0.396763	−0.0154792
$658$	0	0
$659$	−29.2605	−1.13983	−0.569914	−	0.821704i	$-0.693023\pi$
$659$	−29.2605	−1.13983	−0.569914	+	0.821704i	$0.693023\pi$
$660$	0	0
$661$	36.0006	1.40026	0.700131	−	0.714015i	$-0.253125\pi$
$661$	36.0006	1.40026	0.700131	+	0.714015i	$0.253125\pi$
$662$	0	0
$663$	−14.2122	−0.551955
$664$	0	0
$665$	−1.58137	−0.0613229
$666$	0	0
$667$	2.66551	0.103209
$668$	0	0
$669$	−17.4823	−0.675903
$670$	0	0
$671$	−5.34889	−0.206491
$672$	0	0
$673$	14.3874	0.554595	0.277298	−	0.960784i	$-0.410561\pi$
$673$	14.3874	0.554595	0.277298	+	0.960784i	$0.410561\pi$
$674$	0	0
$675$	−22.6388	−0.871368
$676$	0	0
$677$	25.3712	0.975094	0.487547	−	0.873097i	$-0.337892\pi$
$677$	25.3712	0.975094	0.487547	+	0.873097i	$0.337892\pi$
$678$	0	0
$679$	15.6075	0.598960
$680$	0	0
$681$	12.2922	0.471037
$682$	0	0
$683$	−31.2549	−1.19594	−0.597968	−	0.801520i	$-0.704025\pi$
$683$	−31.2549	−1.19594	−0.597968	+	0.801520i	$0.704025\pi$
$684$	0	0
$685$	2.98764	0.114152
$686$	0	0
$687$	−21.7595	−0.830178
$688$	0	0
$689$	25.0187	0.953138
$690$	0	0
$691$	1.92143	0.0730946	0.0365473	−	0.999332i	$-0.488364\pi$
$691$	1.92143	0.0730946	0.0365473	+	0.999332i	$0.488364\pi$
$692$	0	0
$693$	0.238487	0.00905939
$694$	0	0
$695$	9.46162	0.358900
$696$	0	0
$697$	0.367019	0.0139018
$698$	0	0
$699$	37.5894	1.42176
$700$	0	0
$701$	5.22129	0.197205	0.0986027	−	0.995127i	$-0.468563\pi$
$701$	5.22129	0.197205	0.0986027	+	0.995127i	$0.468563\pi$
$702$	0	0
$703$	3.31823	0.125149
$704$	0	0
$705$	−2.09421	−0.0788726
$706$	0	0
$707$	−1.11446	−0.0419137
$708$	0	0
$709$	1.68414	0.0632493	0.0316247	−	0.999500i	$-0.489932\pi$
$709$	1.68414	0.0632493	0.0316247	+	0.999500i	$0.489932\pi$
$710$	0	0
$711$	0.0594709	0.00223033
$712$	0	0
$713$	1.70762	0.0639508
$714$	0	0
$715$	1.29123	0.0482893
$716$	0	0
$717$	16.2812	0.608032
$718$	0	0
$719$	38.7868	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$719$	38.7868	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$720$	0	0
$721$	−2.69487	−0.100362
$722$	0	0
$723$	−40.5898	−1.50955
$724$	0	0
$725$	−29.1402	−1.08224
$726$	0	0
$727$	18.0472	0.669333	0.334666	−	0.942337i	$-0.391376\pi$
$727$	18.0472	0.669333	0.334666	+	0.942337i	$0.391376\pi$
$728$	0	0
$729$	25.1094	0.929978
$730$	0	0
$731$	−2.33440	−0.0863410
$732$	0	0
$733$	2.36945	0.0875175	0.0437587	−	0.999042i	$-0.486067\pi$
$733$	2.36945	0.0875175	0.0437587	+	0.999042i	$0.486067\pi$
$734$	0	0
$735$	−3.60426	−0.132945
$736$	0	0
$737$	8.16896	0.300907
$738$	0	0
$739$	−23.0868	−0.849260	−0.424630	−	0.905367i	$-0.639596\pi$
$739$	−23.0868	−0.849260	−0.424630	+	0.905367i	$0.639596\pi$
$740$	0	0
$741$	5.82710	0.214064
$742$	0	0
$743$	−2.24456	−0.0823449	−0.0411725	−	0.999152i	$-0.513109\pi$
$743$	−2.24456	−0.0823449	−0.0411725	+	0.999152i	$0.513109\pi$
$744$	0	0
$745$	−7.88426	−0.288857
$746$	0	0
$747$	0.946533	0.0346318
$748$	0	0
$749$	−5.22759	−0.191012
$750$	0	0
$751$	−54.7798	−1.99894	−0.999472	−	0.0324826i	$-0.989659\pi$
$751$	−54.7798	−1.99894	−0.999472	+	0.0324826i	$0.989659\pi$
$752$	0	0
$753$	−20.8568	−0.760065
$754$	0	0
$755$	−1.65538	−0.0602454
$756$	0	0
$757$	−5.96003	−0.216621	−0.108311	−	0.994117i	$-0.534544\pi$
$757$	−5.96003	−0.216621	−0.108311	+	0.994117i	$0.534544\pi$
$758$	0	0
$759$	−0.461613	−0.0167555
$760$	0	0
$761$	−33.5210	−1.21513	−0.607567	−	0.794268i	$-0.707854\pi$
$761$	−33.5210	−1.21513	−0.607567	+	0.794268i	$0.707854\pi$
$762$	0	0
$763$	18.0461	0.653314
$764$	0	0
$765$	0.356045	0.0128728
$766$	0	0
$767$	−1.19461	−0.0431349
$768$	0	0
$769$	−41.9889	−1.51416	−0.757080	−	0.653322i	$-0.773375\pi$
$769$	−41.9889	−1.51416	−0.757080	+	0.653322i	$0.773375\pi$
$770$	0	0
$771$	29.1026	1.04810
$772$	0	0
$773$	11.9695	0.430512	0.215256	−	0.976558i	$-0.430941\pi$
$773$	11.9695	0.430512	0.215256	+	0.976558i	$0.430941\pi$
$774$	0	0
$775$	−18.6683	−0.670584
$776$	0	0
$777$	−10.8227	−0.388261
$778$	0	0
$779$	−0.150481	−0.00539153
$780$	0	0
$781$	−1.74358	−0.0623901
$782$	0	0
$783$	32.4559	1.15988
$784$	0	0
$785$	12.2865	0.438523
$786$	0	0
$787$	14.5883	0.520017	0.260008	−	0.965606i	$-0.416275\pi$
$787$	14.5883	0.520017	0.260008	+	0.965606i	$0.416275\pi$
$788$	0	0
$789$	−28.2001	−1.00395
$790$	0	0
$791$	−3.89717	−0.138568
$792$	0	0
$793$	25.0602	0.889913
$794$	0	0
$795$	−10.6611	−0.378112
$796$	0	0
$797$	13.0225	0.461281	0.230641	−	0.973039i	$-0.425918\pi$
$797$	13.0225	0.461281	0.230641	+	0.973039i	$0.425918\pi$
$798$	0	0
$799$	4.53408	0.160404
$800$	0	0
$801$	3.08625	0.109047
$802$	0	0
$803$	−1.32751	−0.0468468
$804$	0	0
$805$	−0.586921	−0.0206862
$806$	0	0
$807$	−20.0673	−0.706403
$808$	0	0
$809$	23.5857	0.829230	0.414615	−	0.909997i	$-0.363916\pi$
$809$	23.5857	0.829230	0.414615	+	0.909997i	$0.363916\pi$
$810$	0	0
$811$	−44.3230	−1.55639	−0.778195	−	0.628023i	$-0.783864\pi$
$811$	−44.3230	−1.55639	−0.778195	+	0.628023i	$0.783864\pi$
$812$	0	0
$813$	40.5472	1.42205
$814$	0	0
$815$	−4.99300	−0.174897
$816$	0	0
$817$	0.957124	0.0334855
$818$	0	0
$819$	−1.11734	−0.0390431
$820$	0	0
$821$	−11.0481	−0.385581	−0.192791	−	0.981240i	$-0.561754\pi$
$821$	−11.0481	−0.385581	−0.192791	+	0.981240i	$0.561754\pi$
$822$	0	0
$823$	−8.97374	−0.312805	−0.156403	−	0.987693i	$-0.549990\pi$
$823$	−8.97374	−0.312805	−0.156403	+	0.987693i	$0.549990\pi$
$824$	0	0
$825$	5.04650	0.175697
$826$	0	0
$827$	−36.0996	−1.25531	−0.627654	−	0.778493i	$-0.715985\pi$
$827$	−36.0996	−1.25531	−0.627654	+	0.778493i	$0.715985\pi$
$828$	0	0
$829$	−5.63971	−0.195875	−0.0979375	−	0.995193i	$-0.531225\pi$
$829$	−5.63971	−0.195875	−0.0979375	+	0.995193i	$0.531225\pi$
$830$	0	0
$831$	6.11669	0.212186
$832$	0	0
$833$	7.80341	0.270372
$834$	0	0
$835$	3.19764	0.110659
$836$	0	0
$837$	20.7924	0.718691
$838$	0	0
$839$	19.2580	0.664859	0.332429	−	0.943128i	$-0.392132\pi$
$839$	19.2580	0.664859	0.332429	+	0.943128i	$0.392132\pi$
$840$	0	0
$841$	12.7767	0.440575
$842$	0	0
$843$	11.4469	0.394253
$844$	0	0
$845$	3.06492	0.105436
$846$	0	0
$847$	−21.5311	−0.739818
$848$	0	0
$849$	−39.8953	−1.36920
$850$	0	0
$851$	1.23155	0.0422170
$852$	0	0
$853$	29.1126	0.996796	0.498398	−	0.866948i	$-0.333922\pi$
$853$	29.1126	0.996796	0.498398	+	0.866948i	$0.333922\pi$
$854$	0	0
$855$	−0.145981	−0.00499245
$856$	0	0
$857$	−18.6286	−0.636342	−0.318171	−	0.948033i	$-0.603069\pi$
$857$	−18.6286	−0.636342	−0.318171	+	0.948033i	$0.603069\pi$
$858$	0	0
$859$	−18.5140	−0.631690	−0.315845	−	0.948811i	$-0.602288\pi$
$859$	−18.5140	−0.631690	−0.315845	+	0.948811i	$0.602288\pi$
$860$	0	0
$861$	0.490804	0.0167266
$862$	0	0
$863$	43.5150	1.48127	0.740635	−	0.671908i	$-0.234525\pi$
$863$	43.5150	1.48127	0.740635	+	0.671908i	$0.234525\pi$
$864$	0	0
$865$	−2.04274	−0.0694553
$866$	0	0
$867$	17.2386	0.585453
$868$	0	0
$869$	0.198981	0.00674996
$870$	0	0
$871$	−38.2726	−1.29682
$872$	0	0
$873$	1.44077	0.0487628
$874$	0	0
$875$	13.5324	0.457479
$876$	0	0
$877$	11.9837	0.404662	0.202331	−	0.979317i	$-0.435148\pi$
$877$	11.9837	0.404662	0.202331	+	0.979317i	$0.435148\pi$
$878$	0	0
$879$	−8.84390	−0.298297
$880$	0	0
$881$	−8.33770	−0.280904	−0.140452	−	0.990087i	$-0.544856\pi$
$881$	−8.33770	−0.280904	−0.140452	+	0.990087i	$0.544856\pi$
$882$	0	0
$883$	−14.7773	−0.497296	−0.248648	−	0.968594i	$-0.579986\pi$
$883$	−14.7773	−0.497296	−0.248648	+	0.968594i	$0.579986\pi$
$884$	0	0
$885$	0.509055	0.0171117
$886$	0	0
$887$	26.6636	0.895278	0.447639	−	0.894214i	$-0.352265\pi$
$887$	26.6636	0.895278	0.447639	+	0.894214i	$0.352265\pi$
$888$	0	0
$889$	−44.2541	−1.48423
$890$	0	0
$891$	−5.97318	−0.200109
$892$	0	0
$893$	−1.85901	−0.0622094
$894$	0	0
$895$	−6.31803	−0.211188
$896$	0	0
$897$	2.16271	0.0722108
$898$	0	0
$899$	26.7636	0.892616
$900$	0	0
$901$	23.0819	0.768971
$902$	0	0
$903$	−3.12174	−0.103885
$904$	0	0
$905$	−10.1006	−0.335754
$906$	0	0
$907$	41.3358	1.37253	0.686266	−	0.727351i	$-0.259249\pi$
$907$	41.3358	1.37253	0.686266	+	0.727351i	$0.259249\pi$
$908$	0	0
$909$	−0.102880	−0.00341230
$910$	0	0
$911$	32.2613	1.06887	0.534433	−	0.845211i	$-0.320525\pi$
$911$	32.2613	1.06887	0.534433	+	0.845211i	$0.320525\pi$
$912$	0	0
$913$	3.16696	0.104811
$914$	0	0
$915$	−10.6788	−0.353030
$916$	0	0
$917$	−17.8256	−0.588653
$918$	0	0
$919$	48.5538	1.60164	0.800821	−	0.598903i	$-0.204397\pi$
$919$	48.5538	1.60164	0.800821	+	0.598903i	$0.204397\pi$
$920$	0	0
$921$	−3.33743	−0.109972
$922$	0	0
$923$	8.16886	0.268881
$924$	0	0
$925$	−13.4637	−0.442684
$926$	0	0
$927$	−0.248772	−0.00817073
$928$	0	0
$929$	28.1849	0.924717	0.462358	−	0.886693i	$-0.347003\pi$
$929$	28.1849	0.924717	0.462358	+	0.886693i	$0.347003\pi$
$930$	0	0
$931$	−3.19946	−0.104858
$932$	0	0
$933$	−7.67087	−0.251133
$934$	0	0
$935$	1.19127	0.0389587
$936$	0	0
$937$	−59.4487	−1.94211	−0.971053	−	0.238865i	$-0.923225\pi$
$937$	−59.4487	−1.94211	−0.971053	+	0.238865i	$0.923225\pi$
$938$	0	0
$939$	43.2723	1.41214
$940$	0	0
$941$	24.0890	0.785277	0.392639	−	0.919693i	$-0.371562\pi$
$941$	24.0890	0.785277	0.392639	+	0.919693i	$0.371562\pi$
$942$	0	0
$943$	−0.0558504	−0.00181874
$944$	0	0
$945$	−7.14650	−0.232476
$946$	0	0
$947$	−40.0415	−1.30117	−0.650586	−	0.759432i	$-0.725477\pi$
$947$	−40.0415	−1.30117	−0.650586	+	0.759432i	$0.725477\pi$
$948$	0	0
$949$	6.21955	0.201895
$950$	0	0
$951$	50.3288	1.63202
$952$	0	0
$953$	−24.4039	−0.790521	−0.395261	−	0.918569i	$-0.629346\pi$
$953$	−24.4039	−0.790521	−0.395261	+	0.918569i	$0.629346\pi$
$954$	0	0
$955$	3.09002	0.0999907
$956$	0	0
$957$	−7.23488	−0.233870
$958$	0	0
$959$	−8.65001	−0.279323
$960$	0	0
$961$	−13.8543	−0.446913
$962$	0	0
$963$	−0.482575	−0.0155508
$964$	0	0
$965$	2.03342	0.0654581
$966$	0	0
$967$	16.2701	0.523212	0.261606	−	0.965175i	$-0.415748\pi$
$967$	16.2701	0.523212	0.261606	+	0.965175i	$0.415748\pi$
$968$	0	0
$969$	5.37600	0.172702
$970$	0	0
$971$	53.0256	1.70167	0.850837	−	0.525430i	$-0.176096\pi$
$971$	53.0256	1.70167	0.850837	+	0.525430i	$0.176096\pi$
$972$	0	0
$973$	−27.3939	−0.878209
$974$	0	0
$975$	−23.6435	−0.757197
$976$	0	0
$977$	−47.0435	−1.50505	−0.752527	−	0.658561i	$-0.771165\pi$
$977$	−47.0435	−1.50505	−0.752527	+	0.658561i	$0.771165\pi$
$978$	0	0
$979$	10.3261	0.330024
$980$	0	0
$981$	1.66590	0.0531879
$982$	0	0
$983$	10.9007	0.347678	0.173839	−	0.984774i	$-0.444383\pi$
$983$	10.9007	0.347678	0.173839	+	0.984774i	$0.444383\pi$
$984$	0	0
$985$	−4.14716	−0.132139
$986$	0	0
$987$	6.06331	0.192997
$988$	0	0
$989$	0.355234	0.0112958
$990$	0	0
$991$	48.5839	1.54332	0.771659	−	0.636036i	$-0.219427\pi$
$991$	48.5839	1.54332	0.771659	+	0.636036i	$0.219427\pi$
$992$	0	0
$993$	24.1350	0.765901
$994$	0	0
$995$	−3.31959	−0.105238
$996$	0	0
$997$	20.3630	0.644902	0.322451	−	0.946586i	$-0.395493\pi$
$997$	20.3630	0.644902	0.322451	+	0.946586i	$0.395493\pi$
$998$	0	0
$999$	14.9957	0.474442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.p.1.1		10
4.3	odd	2		503.2.a.e.1.2	✓	10
12.11	even	2		4527.2.a.k.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.2	✓	10	4.3	odd	2
4527.2.a.k.1.9		10	12.11	even	2
8048.2.a.p.1.1		10	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.78533 q^{3} -0.701114 q^{5} +2.02991 q^{7} +0.187388 q^{9} +O(q^{10})\)	\(q-1.78533 q^{3} -0.701114 q^{5} +2.02991 q^{7} +0.187388 q^{9} +0.626970 q^{11} -2.93743 q^{13} +1.25172 q^{15} -2.71003 q^{17} +1.11114 q^{19} -3.62406 q^{21} +0.412395 q^{23} -4.50844 q^{25} +5.02143 q^{27} +6.46349 q^{29} +4.14074 q^{31} -1.11935 q^{33} -1.42320 q^{35} +2.98634 q^{37} +5.24427 q^{39} -0.135430 q^{41} +0.861393 q^{43} -0.131380 q^{45} -1.67307 q^{47} -2.87945 q^{49} +4.83829 q^{51} -8.51721 q^{53} -0.439578 q^{55} -1.98374 q^{57} +0.406685 q^{59} -8.53132 q^{61} +0.380381 q^{63} +2.05948 q^{65} +13.0293 q^{67} -0.736259 q^{69} -2.78095 q^{71} -2.11734 q^{73} +8.04903 q^{75} +1.27270 q^{77} +0.317368 q^{79} -9.52705 q^{81} +5.05120 q^{83} +1.90004 q^{85} -11.5394 q^{87} +16.4699 q^{89} -5.96273 q^{91} -7.39256 q^{93} -0.779033 q^{95} +7.68873 q^{97} +0.117486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

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