LMFDB - Embedded newform 8048.2.a.q.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.16835$ 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)

$f(q)$	$=$	$q-3.16835 q^{3} +0.310718 q^{5} -0.631809 q^{7} +7.03843 q^{9} +O(q^{10})$	$q-3.16835 q^{3} +0.310718 q^{5} -0.631809 q^{7} +7.03843 q^{9} +4.43341 q^{11} +0.596176 q^{13} -0.984464 q^{15} -0.105355 q^{17} +2.07478 q^{19} +2.00179 q^{21} +3.00283 q^{23} -4.90345 q^{25} -12.7951 q^{27} +0.274898 q^{29} -3.01698 q^{31} -14.0466 q^{33} -0.196315 q^{35} -7.47331 q^{37} -1.88889 q^{39} +2.70111 q^{41} +2.91701 q^{43} +2.18697 q^{45} -7.70935 q^{47} -6.60082 q^{49} +0.333800 q^{51} -3.12377 q^{53} +1.37754 q^{55} -6.57361 q^{57} -13.3963 q^{59} +1.63953 q^{61} -4.44694 q^{63} +0.185243 q^{65} +6.02998 q^{67} -9.51402 q^{69} +1.40259 q^{71} +5.27962 q^{73} +15.5358 q^{75} -2.80106 q^{77} +1.02675 q^{79} +19.4242 q^{81} +3.69784 q^{83} -0.0327356 q^{85} -0.870972 q^{87} +5.76596 q^{89} -0.376669 q^{91} +9.55883 q^{93} +0.644671 q^{95} -2.89347 q^{97} +31.2042 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * 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
$3$	−3.16835	−1.82925	−0.914623	−	0.404307i	$-0.867513\pi$
$3$	−3.16835	−1.82925	−0.914623	+	0.404307i	$0.867513\pi$
$4$	0	0
$5$	0.310718	0.138958	0.0694788	−	0.997583i	$-0.477866\pi$
$5$	0.310718	0.138958	0.0694788	+	0.997583i	$0.477866\pi$
$6$	0	0
$7$	−0.631809	−0.238801	−0.119401	−	0.992846i	$-0.538097\pi$
$7$	−0.631809	−0.238801	−0.119401	+	0.992846i	$0.538097\pi$
$8$	0	0
$9$	7.03843	2.34614
$10$	0	0
$11$	4.43341	1.33672	0.668361	−	0.743837i	$-0.266996\pi$
$11$	4.43341	1.33672	0.668361	+	0.743837i	$0.266996\pi$
$12$	0	0
$13$	0.596176	0.165349	0.0826747	−	0.996577i	$-0.473654\pi$
$13$	0.596176	0.165349	0.0826747	+	0.996577i	$0.473654\pi$
$14$	0	0
$15$	−0.984464	−0.254188
$16$	0	0
$17$	−0.105355	−0.0255523	−0.0127761	−	0.999918i	$-0.504067\pi$
$17$	−0.105355	−0.0255523	−0.0127761	+	0.999918i	$0.504067\pi$
$18$	0	0
$19$	2.07478	0.475986	0.237993	−	0.971267i	$-0.423510\pi$
$19$	2.07478	0.475986	0.237993	+	0.971267i	$0.423510\pi$
$20$	0	0
$21$	2.00179	0.436826
$22$	0	0
$23$	3.00283	0.626134	0.313067	−	0.949731i	$-0.398644\pi$
$23$	3.00283	0.626134	0.313067	+	0.949731i	$0.398644\pi$
$24$	0	0
$25$	−4.90345	−0.980691
$26$	0	0
$27$	−12.7951	−2.46243
$28$	0	0
$29$	0.274898	0.0510472	0.0255236	−	0.999674i	$-0.491875\pi$
$29$	0.274898	0.0510472	0.0255236	+	0.999674i	$0.491875\pi$
$30$	0	0
$31$	−3.01698	−0.541865	−0.270932	−	0.962598i	$-0.587332\pi$
$31$	−3.01698	−0.541865	−0.270932	+	0.962598i	$0.587332\pi$
$32$	0	0
$33$	−14.0466	−2.44519
$34$	0	0
$35$	−0.196315	−0.0331832
$36$	0	0
$37$	−7.47331	−1.22861	−0.614303	−	0.789070i	$-0.710563\pi$
$37$	−7.47331	−1.22861	−0.614303	+	0.789070i	$0.710563\pi$
$38$	0	0
$39$	−1.88889	−0.302465
$40$	0	0
$41$	2.70111	0.421842	0.210921	−	0.977503i	$-0.432354\pi$
$41$	2.70111	0.421842	0.210921	+	0.977503i	$0.432354\pi$
$42$	0	0
$43$	2.91701	0.444841	0.222420	−	0.974951i	$-0.428604\pi$
$43$	2.91701	0.444841	0.222420	+	0.974951i	$0.428604\pi$
$44$	0	0
$45$	2.18697	0.326014
$46$	0	0
$47$	−7.70935	−1.12452	−0.562262	−	0.826959i	$-0.690069\pi$
$47$	−7.70935	−1.12452	−0.562262	+	0.826959i	$0.690069\pi$
$48$	0	0
$49$	−6.60082	−0.942974
$50$	0	0
$51$	0.333800	0.0467414
$52$	0	0
$53$	−3.12377	−0.429083	−0.214542	−	0.976715i	$-0.568826\pi$
$53$	−3.12377	−0.429083	−0.214542	+	0.976715i	$0.568826\pi$
$54$	0	0
$55$	1.37754	0.185748
$56$	0	0
$57$	−6.57361	−0.870696
$58$	0	0
$59$	−13.3963	−1.74405	−0.872025	−	0.489462i	$-0.837193\pi$
$59$	−13.3963	−1.74405	−0.872025	+	0.489462i	$0.837193\pi$
$60$	0	0
$61$	1.63953	0.209920	0.104960	−	0.994476i	$-0.466528\pi$
$61$	1.63953	0.209920	0.104960	+	0.994476i	$0.466528\pi$
$62$	0	0
$63$	−4.44694	−0.560262
$64$	0	0
$65$	0.185243	0.0229765
$66$	0	0
$67$	6.02998	0.736679	0.368340	−	0.929691i	$-0.379926\pi$
$67$	6.02998	0.736679	0.368340	+	0.929691i	$0.379926\pi$
$68$	0	0
$69$	−9.51402	−1.14535
$70$	0	0
$71$	1.40259	0.166457	0.0832284	−	0.996530i	$-0.473477\pi$
$71$	1.40259	0.166457	0.0832284	+	0.996530i	$0.473477\pi$
$72$	0	0
$73$	5.27962	0.617933	0.308967	−	0.951073i	$-0.400017\pi$
$73$	5.27962	0.617933	0.308967	+	0.951073i	$0.400017\pi$
$74$	0	0
$75$	15.5358	1.79393
$76$	0	0
$77$	−2.80106	−0.319211
$78$	0	0
$79$	1.02675	0.115518	0.0577590	−	0.998331i	$-0.481605\pi$
$79$	1.02675	0.115518	0.0577590	+	0.998331i	$0.481605\pi$
$80$	0	0
$81$	19.4242	2.15824
$82$	0	0
$83$	3.69784	0.405890	0.202945	−	0.979190i	$-0.434949\pi$
$83$	3.69784	0.405890	0.202945	+	0.979190i	$0.434949\pi$
$84$	0	0
$85$	−0.0327356	−0.00355068
$86$	0	0
$87$	−0.870972	−0.0933780
$88$	0	0
$89$	5.76596	0.611191	0.305595	−	0.952161i	$-0.401145\pi$
$89$	5.76596	0.611191	0.305595	+	0.952161i	$0.401145\pi$
$90$	0	0
$91$	−0.376669	−0.0394856
$92$	0	0
$93$	9.55883	0.991204
$94$	0	0
$95$	0.644671	0.0661419
$96$	0	0
$97$	−2.89347	−0.293787	−0.146894	−	0.989152i	$-0.546928\pi$
$97$	−2.89347	−0.293787	−0.146894	+	0.989152i	$0.546928\pi$
$98$	0	0
$99$	31.2042	3.13614
$100$	0	0
$101$	5.48114	0.545394	0.272697	−	0.962100i	$-0.412084\pi$
$101$	5.48114	0.545394	0.272697	+	0.962100i	$0.412084\pi$
$102$	0	0
$103$	3.63336	0.358006	0.179003	−	0.983849i	$-0.442713\pi$
$103$	3.63336	0.358006	0.179003	+	0.983849i	$0.442713\pi$
$104$	0	0
$105$	0.621993	0.0607003
$106$	0	0
$107$	−16.5418	−1.59916	−0.799578	−	0.600562i	$-0.794944\pi$
$107$	−16.5418	−1.59916	−0.799578	+	0.600562i	$0.794944\pi$
$108$	0	0
$109$	−3.78618	−0.362650	−0.181325	−	0.983423i	$-0.558039\pi$
$109$	−3.78618	−0.362650	−0.181325	+	0.983423i	$0.558039\pi$
$110$	0	0
$111$	23.6781	2.24742
$112$	0	0
$113$	−9.03891	−0.850309	−0.425155	−	0.905121i	$-0.639780\pi$
$113$	−9.03891	−0.850309	−0.425155	+	0.905121i	$0.639780\pi$
$114$	0	0
$115$	0.933036	0.0870061
$116$	0	0
$117$	4.19614	0.387933
$118$	0	0
$119$	0.0665640	0.00610191
$120$	0	0
$121$	8.65508	0.786826
$122$	0	0
$123$	−8.55805	−0.771654
$124$	0	0
$125$	−3.07719	−0.275232
$126$	0	0
$127$	11.9787	1.06294	0.531470	−	0.847077i	$-0.321640\pi$
$127$	11.9787	1.06294	0.531470	+	0.847077i	$0.321640\pi$
$128$	0	0
$129$	−9.24212	−0.813723
$130$	0	0
$131$	1.19340	0.104268	0.0521338	−	0.998640i	$-0.483398\pi$
$131$	1.19340	0.104268	0.0521338	+	0.998640i	$0.483398\pi$
$132$	0	0
$133$	−1.31086	−0.113666
$134$	0	0
$135$	−3.97569	−0.342173
$136$	0	0
$137$	−18.2450	−1.55878	−0.779389	−	0.626540i	$-0.784470\pi$
$137$	−18.2450	−1.55878	−0.779389	+	0.626540i	$0.784470\pi$
$138$	0	0
$139$	13.3154	1.12939	0.564697	−	0.825298i	$-0.308993\pi$
$139$	13.3154	1.12939	0.564697	+	0.825298i	$0.308993\pi$
$140$	0	0
$141$	24.4259	2.05703
$142$	0	0
$143$	2.64309	0.221026
$144$	0	0
$145$	0.0854158	0.00709340
$146$	0	0
$147$	20.9137	1.72493
$148$	0	0
$149$	1.87076	0.153259	0.0766295	−	0.997060i	$-0.475584\pi$
$149$	1.87076	0.153259	0.0766295	+	0.997060i	$0.475584\pi$
$150$	0	0
$151$	−11.4790	−0.934149	−0.467074	−	0.884218i	$-0.654692\pi$
$151$	−11.4790	−0.934149	−0.467074	+	0.884218i	$0.654692\pi$
$152$	0	0
$153$	−0.741531	−0.0599492
$154$	0	0
$155$	−0.937430	−0.0752962
$156$	0	0
$157$	20.4529	1.63232	0.816159	−	0.577827i	$-0.196099\pi$
$157$	20.4529	1.63232	0.816159	+	0.577827i	$0.196099\pi$
$158$	0	0
$159$	9.89720	0.784899
$160$	0	0
$161$	−1.89722	−0.149522
$162$	0	0
$163$	−17.4714	−1.36847	−0.684234	−	0.729262i	$-0.739863\pi$
$163$	−17.4714	−1.36847	−0.684234	+	0.729262i	$0.739863\pi$
$164$	0	0
$165$	−4.36453	−0.339778
$166$	0	0
$167$	−24.3731	−1.88605	−0.943023	−	0.332727i	$-0.892031\pi$
$167$	−24.3731	−1.88605	−0.943023	+	0.332727i	$0.892031\pi$
$168$	0	0
$169$	−12.6446	−0.972660
$170$	0	0
$171$	14.6032	1.11673
$172$	0	0
$173$	10.8858	0.827634	0.413817	−	0.910360i	$-0.364195\pi$
$173$	10.8858	0.827634	0.413817	+	0.910360i	$0.364195\pi$
$174$	0	0
$175$	3.09805	0.234190
$176$	0	0
$177$	42.4441	3.19030
$178$	0	0
$179$	1.55067	0.115903	0.0579513	−	0.998319i	$-0.481543\pi$
$179$	1.55067	0.115903	0.0579513	+	0.998319i	$0.481543\pi$
$180$	0	0
$181$	22.1198	1.64415	0.822074	−	0.569380i	$-0.192817\pi$
$181$	22.1198	1.64415	0.822074	+	0.569380i	$0.192817\pi$
$182$	0	0
$183$	−5.19460	−0.383996
$184$	0	0
$185$	−2.32210	−0.170724
$186$	0	0
$187$	−0.467080	−0.0341563
$188$	0	0
$189$	8.08408	0.588030
$190$	0	0
$191$	−14.0885	−1.01941	−0.509705	−	0.860349i	$-0.670246\pi$
$191$	−14.0885	−1.01941	−0.509705	+	0.860349i	$0.670246\pi$
$192$	0	0
$193$	−8.12874	−0.585119	−0.292560	−	0.956247i	$-0.594507\pi$
$193$	−8.12874	−0.585119	−0.292560	+	0.956247i	$0.594507\pi$
$194$	0	0
$195$	−0.586913	−0.0420297
$196$	0	0
$197$	14.9889	1.06791	0.533957	−	0.845512i	$-0.320704\pi$
$197$	14.9889	1.06791	0.533957	+	0.845512i	$0.320704\pi$
$198$	0	0
$199$	23.5897	1.67223	0.836115	−	0.548554i	$-0.184822\pi$
$199$	23.5897	1.67223	0.836115	+	0.548554i	$0.184822\pi$
$200$	0	0
$201$	−19.1051	−1.34757
$202$	0	0
$203$	−0.173683	−0.0121901
$204$	0	0
$205$	0.839285	0.0586182
$206$	0	0
$207$	21.1352	1.46900
$208$	0	0
$209$	9.19832	0.636261
$210$	0	0
$211$	−10.6408	−0.732546	−0.366273	−	0.930507i	$-0.619366\pi$
$211$	−10.6408	−0.732546	−0.366273	+	0.930507i	$0.619366\pi$
$212$	0	0
$213$	−4.44389	−0.304491
$214$	0	0
$215$	0.906370	0.0618139
$216$	0	0
$217$	1.90615	0.129398
$218$	0	0
$219$	−16.7277	−1.13035
$220$	0	0
$221$	−0.0628099	−0.00422505
$222$	0	0
$223$	−8.28340	−0.554697	−0.277349	−	0.960769i	$-0.589456\pi$
$223$	−8.28340	−0.554697	−0.277349	+	0.960769i	$0.589456\pi$
$224$	0	0
$225$	−34.5126	−2.30084
$226$	0	0
$227$	28.8734	1.91640	0.958198	−	0.286105i	$-0.0923605\pi$
$227$	28.8734	1.91640	0.958198	+	0.286105i	$0.0923605\pi$
$228$	0	0
$229$	−3.98795	−0.263531	−0.131765	−	0.991281i	$-0.542065\pi$
$229$	−3.98795	−0.263531	−0.131765	+	0.991281i	$0.542065\pi$
$230$	0	0
$231$	8.87475	0.583915
$232$	0	0
$233$	−21.7998	−1.42815	−0.714076	−	0.700068i	$-0.753153\pi$
$233$	−21.7998	−1.42815	−0.714076	+	0.700068i	$0.753153\pi$
$234$	0	0
$235$	−2.39544	−0.156261
$236$	0	0
$237$	−3.25309	−0.211311
$238$	0	0
$239$	−6.33395	−0.409709	−0.204854	−	0.978792i	$-0.565672\pi$
$239$	−6.33395	−0.409709	−0.204854	+	0.978792i	$0.565672\pi$
$240$	0	0
$241$	7.67568	0.494434	0.247217	−	0.968960i	$-0.420484\pi$
$241$	7.67568	0.494434	0.247217	+	0.968960i	$0.420484\pi$
$242$	0	0
$243$	−23.1571	−1.48553
$244$	0	0
$245$	−2.05100	−0.131033
$246$	0	0
$247$	1.23693	0.0787040
$248$	0	0
$249$	−11.7160	−0.742473
$250$	0	0
$251$	16.2301	1.02443	0.512216	−	0.858856i	$-0.328825\pi$
$251$	16.2301	1.02443	0.512216	+	0.858856i	$0.328825\pi$
$252$	0	0
$253$	13.3128	0.836967
$254$	0	0
$255$	0.103718	0.00649506
$256$	0	0
$257$	−28.5849	−1.78307	−0.891537	−	0.452947i	$-0.850373\pi$
$257$	−28.5849	−1.78307	−0.891537	+	0.452947i	$0.850373\pi$
$258$	0	0
$259$	4.72171	0.293393
$260$	0	0
$261$	1.93485	0.119764
$262$	0	0
$263$	−6.81772	−0.420399	−0.210199	−	0.977659i	$-0.567411\pi$
$263$	−6.81772	−0.420399	−0.210199	+	0.977659i	$0.567411\pi$
$264$	0	0
$265$	−0.970614	−0.0596244
$266$	0	0
$267$	−18.2686	−1.11802
$268$	0	0
$269$	17.6296	1.07490	0.537449	−	0.843297i	$-0.319388\pi$
$269$	17.6296	1.07490	0.537449	+	0.843297i	$0.319388\pi$
$270$	0	0
$271$	1.85049	0.112409	0.0562047	−	0.998419i	$-0.482100\pi$
$271$	1.85049	0.112409	0.0562047	+	0.998419i	$0.482100\pi$
$272$	0	0
$273$	1.19342	0.0722290
$274$	0	0
$275$	−21.7390	−1.31091
$276$	0	0
$277$	−9.00164	−0.540856	−0.270428	−	0.962740i	$-0.587165\pi$
$277$	−9.00164	−0.540856	−0.270428	+	0.962740i	$0.587165\pi$
$278$	0	0
$279$	−21.2348	−1.27129
$280$	0	0
$281$	−1.52153	−0.0907671	−0.0453835	−	0.998970i	$-0.514451\pi$
$281$	−1.52153	−0.0907671	−0.0453835	+	0.998970i	$0.514451\pi$
$282$	0	0
$283$	−14.9940	−0.891300	−0.445650	−	0.895207i	$-0.647027\pi$
$283$	−14.9940	−0.891300	−0.445650	+	0.895207i	$0.647027\pi$
$284$	0	0
$285$	−2.04254	−0.120990
$286$	0	0
$287$	−1.70658	−0.100737
$288$	0	0
$289$	−16.9889	−0.999347
$290$	0	0
$291$	9.16752	0.537410
$292$	0	0
$293$	14.4671	0.845175	0.422587	−	0.906322i	$-0.361122\pi$
$293$	14.4671	0.845175	0.422587	+	0.906322i	$0.361122\pi$
$294$	0	0
$295$	−4.16248	−0.242349
$296$	0	0
$297$	−56.7260	−3.29158
$298$	0	0
$299$	1.79022	0.103531
$300$	0	0
$301$	−1.84300	−0.106229
$302$	0	0
$303$	−17.3662	−0.997660
$304$	0	0
$305$	0.509432	0.0291700
$306$	0	0
$307$	1.50584	0.0859428	0.0429714	−	0.999076i	$-0.486318\pi$
$307$	1.50584	0.0859428	0.0429714	+	0.999076i	$0.486318\pi$
$308$	0	0
$309$	−11.5118	−0.654881
$310$	0	0
$311$	5.74838	0.325961	0.162980	−	0.986629i	$-0.447889\pi$
$311$	5.74838	0.325961	0.162980	+	0.986629i	$0.447889\pi$
$312$	0	0
$313$	−12.7695	−0.721775	−0.360887	−	0.932609i	$-0.617526\pi$
$313$	−12.7695	−0.721775	−0.360887	+	0.932609i	$0.617526\pi$
$314$	0	0
$315$	−1.38175	−0.0778526
$316$	0	0
$317$	−19.5709	−1.09921	−0.549605	−	0.835424i	$-0.685222\pi$
$317$	−19.5709	−1.09921	−0.549605	+	0.835424i	$0.685222\pi$
$318$	0	0
$319$	1.21873	0.0682360
$320$	0	0
$321$	52.4102	2.92525
$322$	0	0
$323$	−0.218587	−0.0121625
$324$	0	0
$325$	−2.92332	−0.162157
$326$	0	0
$327$	11.9959	0.663376
$328$	0	0
$329$	4.87083	0.268538
$330$	0	0
$331$	21.6328	1.18905	0.594523	−	0.804079i	$-0.297341\pi$
$331$	21.6328	1.18905	0.594523	+	0.804079i	$0.297341\pi$
$332$	0	0
$333$	−52.6004	−2.88248
$334$	0	0
$335$	1.87363	0.102367
$336$	0	0
$337$	−26.8912	−1.46486	−0.732429	−	0.680844i	$-0.761613\pi$
$337$	−26.8912	−1.46486	−0.732429	+	0.680844i	$0.761613\pi$
$338$	0	0
$339$	28.6384	1.55543
$340$	0	0
$341$	−13.3755	−0.724323
$342$	0	0
$343$	8.59312	0.463985
$344$	0	0
$345$	−2.95618	−0.159156
$346$	0	0
$347$	−10.9343	−0.586984	−0.293492	−	0.955961i	$-0.594817\pi$
$347$	−10.9343	−0.586984	−0.293492	+	0.955961i	$0.594817\pi$
$348$	0	0
$349$	−2.66130	−0.142456	−0.0712280	−	0.997460i	$-0.522692\pi$
$349$	−2.66130	−0.142456	−0.0712280	+	0.997460i	$0.522692\pi$
$350$	0	0
$351$	−7.62815	−0.407161
$352$	0	0
$353$	18.0343	0.959871	0.479936	−	0.877304i	$-0.340660\pi$
$353$	18.0343	0.959871	0.479936	+	0.877304i	$0.340660\pi$
$354$	0	0
$355$	0.435811	0.0231304
$356$	0	0
$357$	−0.210898	−0.0111619
$358$	0	0
$359$	−16.0855	−0.848958	−0.424479	−	0.905438i	$-0.639543\pi$
$359$	−16.0855	−0.848958	−0.424479	+	0.905438i	$0.639543\pi$
$360$	0	0
$361$	−14.6953	−0.773437
$362$	0	0
$363$	−27.4223	−1.43930
$364$	0	0
$365$	1.64048	0.0858665
$366$	0	0
$367$	7.55292	0.394259	0.197130	−	0.980377i	$-0.436838\pi$
$367$	7.55292	0.394259	0.197130	+	0.980377i	$0.436838\pi$
$368$	0	0
$369$	19.0116	0.989702
$370$	0	0
$371$	1.97363	0.102466
$372$	0	0
$373$	15.5440	0.804837	0.402418	−	0.915456i	$-0.368170\pi$
$373$	15.5440	0.804837	0.402418	+	0.915456i	$0.368170\pi$
$374$	0	0
$375$	9.74959	0.503467
$376$	0	0
$377$	0.163887	0.00844063
$378$	0	0
$379$	−12.3858	−0.636216	−0.318108	−	0.948054i	$-0.603047\pi$
$379$	−12.3858	−0.636216	−0.318108	+	0.948054i	$0.603047\pi$
$380$	0	0
$381$	−37.9528	−1.94438
$382$	0	0
$383$	−19.1507	−0.978554	−0.489277	−	0.872128i	$-0.662739\pi$
$383$	−19.1507	−0.978554	−0.489277	+	0.872128i	$0.662739\pi$
$384$	0	0
$385$	−0.870342	−0.0443568
$386$	0	0
$387$	20.5312	1.04366
$388$	0	0
$389$	34.4024	1.74427	0.872135	−	0.489266i	$-0.162735\pi$
$389$	34.4024	1.74427	0.872135	+	0.489266i	$0.162735\pi$
$390$	0	0
$391$	−0.316363	−0.0159991
$392$	0	0
$393$	−3.78110	−0.190731
$394$	0	0
$395$	0.319029	0.0160521
$396$	0	0
$397$	−6.02930	−0.302602	−0.151301	−	0.988488i	$-0.548346\pi$
$397$	−6.02930	−0.302602	−0.151301	+	0.988488i	$0.548346\pi$
$398$	0	0
$399$	4.15327	0.207923
$400$	0	0
$401$	31.7568	1.58586	0.792929	−	0.609314i	$-0.208555\pi$
$401$	31.7568	1.58586	0.792929	+	0.609314i	$0.208555\pi$
$402$	0	0
$403$	−1.79865	−0.0895970
$404$	0	0
$405$	6.03545	0.299904
$406$	0	0
$407$	−33.1322	−1.64230
$408$	0	0
$409$	28.6778	1.41803	0.709014	−	0.705194i	$-0.249140\pi$
$409$	28.6778	1.41803	0.709014	+	0.705194i	$0.249140\pi$
$410$	0	0
$411$	57.8066	2.85139
$412$	0	0
$413$	8.46390	0.416481
$414$	0	0
$415$	1.14899	0.0564015
$416$	0	0
$417$	−42.1877	−2.06594
$418$	0	0
$419$	−3.24938	−0.158743	−0.0793714	−	0.996845i	$-0.525291\pi$
$419$	−3.24938	−0.158743	−0.0793714	+	0.996845i	$0.525291\pi$
$420$	0	0
$421$	−26.8760	−1.30986	−0.654928	−	0.755692i	$-0.727301\pi$
$421$	−26.8760	−1.30986	−0.654928	+	0.755692i	$0.727301\pi$
$422$	0	0
$423$	−54.2617	−2.63829
$424$	0	0
$425$	0.516602	0.0250589
$426$	0	0
$427$	−1.03587	−0.0501292
$428$	0	0
$429$	−8.37422	−0.404311
$430$	0	0
$431$	−12.8327	−0.618129	−0.309064	−	0.951041i	$-0.600016\pi$
$431$	−12.8327	−0.618129	−0.309064	+	0.951041i	$0.600016\pi$
$432$	0	0
$433$	0.323893	0.0155653	0.00778265	−	0.999970i	$-0.497523\pi$
$433$	0.323893	0.0155653	0.00778265	+	0.999970i	$0.497523\pi$
$434$	0	0
$435$	−0.270627	−0.0129756
$436$	0	0
$437$	6.23021	0.298031
$438$	0	0
$439$	−32.1675	−1.53527	−0.767635	−	0.640887i	$-0.778567\pi$
$439$	−32.1675	−1.53527	−0.767635	+	0.640887i	$0.778567\pi$
$440$	0	0
$441$	−46.4594	−2.21235
$442$	0	0
$443$	−6.96232	−0.330790	−0.165395	−	0.986227i	$-0.552890\pi$
$443$	−6.96232	−0.330790	−0.165395	+	0.986227i	$0.552890\pi$
$444$	0	0
$445$	1.79159	0.0849296
$446$	0	0
$447$	−5.92723	−0.280348
$448$	0	0
$449$	−34.5274	−1.62945	−0.814725	−	0.579848i	$-0.803112\pi$
$449$	−34.5274	−1.62945	−0.814725	+	0.579848i	$0.803112\pi$
$450$	0	0
$451$	11.9751	0.563886
$452$	0	0
$453$	36.3695	1.70879
$454$	0	0
$455$	−0.117038	−0.00548683
$456$	0	0
$457$	23.7930	1.11299	0.556494	−	0.830851i	$-0.312146\pi$
$457$	23.7930	1.11299	0.556494	+	0.830851i	$0.312146\pi$
$458$	0	0
$459$	1.34803	0.0629205
$460$	0	0
$461$	−40.2114	−1.87283	−0.936416	−	0.350892i	$-0.885879\pi$
$461$	−40.2114	−1.87283	−0.936416	+	0.350892i	$0.885879\pi$
$462$	0	0
$463$	−17.8639	−0.830208	−0.415104	−	0.909774i	$-0.636255\pi$
$463$	−17.8639	−0.830208	−0.415104	+	0.909774i	$0.636255\pi$
$464$	0	0
$465$	2.97010	0.137735
$466$	0	0
$467$	−5.28826	−0.244712	−0.122356	−	0.992486i	$-0.539045\pi$
$467$	−5.28826	−0.244712	−0.122356	+	0.992486i	$0.539045\pi$
$468$	0	0
$469$	−3.80980	−0.175920
$470$	0	0
$471$	−64.8018	−2.98591
$472$	0	0
$473$	12.9323	0.594628
$474$	0	0
$475$	−10.1736	−0.466795
$476$	0	0
$477$	−21.9865	−1.00669
$478$	0	0
$479$	−3.58434	−0.163772	−0.0818862	−	0.996642i	$-0.526094\pi$
$479$	−3.58434	−0.163772	−0.0818862	+	0.996642i	$0.526094\pi$
$480$	0	0
$481$	−4.45541	−0.203149
$482$	0	0
$483$	6.01104	0.273512
$484$	0	0
$485$	−0.899055	−0.0408240
$486$	0	0
$487$	−12.0478	−0.545936	−0.272968	−	0.962023i	$-0.588005\pi$
$487$	−12.0478	−0.545936	−0.272968	+	0.962023i	$0.588005\pi$
$488$	0	0
$489$	55.3556	2.50327
$490$	0	0
$491$	−15.4239	−0.696070	−0.348035	−	0.937482i	$-0.613151\pi$
$491$	−15.4239	−0.696070	−0.348035	+	0.937482i	$0.613151\pi$
$492$	0	0
$493$	−0.0289618	−0.00130437
$494$	0	0
$495$	9.69572	0.435790
$496$	0	0
$497$	−0.886169	−0.0397501
$498$	0	0
$499$	−17.9322	−0.802755	−0.401378	−	0.915913i	$-0.631468\pi$
$499$	−17.9322	−0.802755	−0.401378	+	0.915913i	$0.631468\pi$
$500$	0	0
$501$	77.2224	3.45004
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	1.70309	0.0757866
$506$	0	0
$507$	40.0624	1.77923
$508$	0	0
$509$	31.2374	1.38457	0.692287	−	0.721623i	$-0.256603\pi$
$509$	31.2374	1.38457	0.692287	+	0.721623i	$0.256603\pi$
$510$	0	0
$511$	−3.33571	−0.147563
$512$	0	0
$513$	−26.5470	−1.17208
$514$	0	0
$515$	1.12895	0.0497476
$516$	0	0
$517$	−34.1787	−1.50318
$518$	0	0
$519$	−34.4901	−1.51395
$520$	0	0
$521$	−9.62993	−0.421895	−0.210948	−	0.977497i	$-0.567655\pi$
$521$	−9.62993	−0.421895	−0.210948	+	0.977497i	$0.567655\pi$
$522$	0	0
$523$	−25.8465	−1.13019	−0.565093	−	0.825027i	$-0.691160\pi$
$523$	−25.8465	−1.13019	−0.565093	+	0.825027i	$0.691160\pi$
$524$	0	0
$525$	−9.81569	−0.428392
$526$	0	0
$527$	0.317852	0.0138459
$528$	0	0
$529$	−13.9830	−0.607956
$530$	0	0
$531$	−94.2889	−4.09179
$532$	0	0
$533$	1.61034	0.0697514
$534$	0	0
$535$	−5.13984	−0.222215
$536$	0	0
$537$	−4.91306	−0.212014
$538$	0	0
$539$	−29.2641	−1.26049
$540$	0	0
$541$	−45.7706	−1.96783	−0.983916	−	0.178630i	$-0.942834\pi$
$541$	−45.7706	−1.96783	−0.983916	+	0.178630i	$0.942834\pi$
$542$	0	0
$543$	−70.0831	−3.00755
$544$	0	0
$545$	−1.17643	−0.0503929
$546$	0	0
$547$	−7.98219	−0.341294	−0.170647	−	0.985332i	$-0.554586\pi$
$547$	−7.98219	−0.341294	−0.170647	+	0.985332i	$0.554586\pi$
$548$	0	0
$549$	11.5397	0.492503
$550$	0	0
$551$	0.570351	0.0242978
$552$	0	0
$553$	−0.648707	−0.0275858
$554$	0	0
$555$	7.35721	0.312296
$556$	0	0
$557$	10.3743	0.439572	0.219786	−	0.975548i	$-0.429464\pi$
$557$	10.3743	0.439572	0.219786	+	0.975548i	$0.429464\pi$
$558$	0	0
$559$	1.73905	0.0735541
$560$	0	0
$561$	1.47987	0.0624802
$562$	0	0
$563$	−24.9276	−1.05057	−0.525287	−	0.850925i	$-0.676042\pi$
$563$	−24.9276	−1.05057	−0.525287	+	0.850925i	$0.676042\pi$
$564$	0	0
$565$	−2.80856	−0.118157
$566$	0	0
$567$	−12.2724	−0.515391
$568$	0	0
$569$	−3.30340	−0.138486	−0.0692429	−	0.997600i	$-0.522058\pi$
$569$	−3.30340	−0.138486	−0.0692429	+	0.997600i	$0.522058\pi$
$570$	0	0
$571$	−4.43137	−0.185447	−0.0927235	−	0.995692i	$-0.529557\pi$
$571$	−4.43137	−0.185447	−0.0927235	+	0.995692i	$0.529557\pi$
$572$	0	0
$573$	44.6374	1.86475
$574$	0	0
$575$	−14.7243	−0.614044
$576$	0	0
$577$	−7.23087	−0.301025	−0.150513	−	0.988608i	$-0.548092\pi$
$577$	−7.23087	−0.301025	−0.150513	+	0.988608i	$0.548092\pi$
$578$	0	0
$579$	25.7547	1.07033
$580$	0	0
$581$	−2.33633	−0.0969271
$582$	0	0
$583$	−13.8490	−0.573565
$584$	0	0
$585$	1.30382	0.0539062
$586$	0	0
$587$	−32.1758	−1.32804	−0.664018	−	0.747717i	$-0.731150\pi$
$587$	−32.1758	−1.32804	−0.664018	+	0.747717i	$0.731150\pi$
$588$	0	0
$589$	−6.25955	−0.257920
$590$	0	0
$591$	−47.4900	−1.95348
$592$	0	0
$593$	13.9250	0.571830	0.285915	−	0.958255i	$-0.407703\pi$
$593$	13.9250	0.571830	0.285915	+	0.958255i	$0.407703\pi$
$594$	0	0
$595$	0.0206827	0.000847906 0
$596$	0	0
$597$	−74.7404	−3.05892
$598$	0	0
$599$	40.6318	1.66017	0.830085	−	0.557637i	$-0.188292\pi$
$599$	40.6318	1.66017	0.830085	+	0.557637i	$0.188292\pi$
$600$	0	0
$601$	8.75658	0.357188	0.178594	−	0.983923i	$-0.442845\pi$
$601$	8.75658	0.357188	0.178594	+	0.983923i	$0.442845\pi$
$602$	0	0
$603$	42.4416	1.72835
$604$	0	0
$605$	2.68929	0.109335
$606$	0	0
$607$	−11.3026	−0.458758	−0.229379	−	0.973337i	$-0.573670\pi$
$607$	−11.3026	−0.458758	−0.229379	+	0.973337i	$0.573670\pi$
$608$	0	0
$609$	0.550288	0.0222988
$610$	0	0
$611$	−4.59613	−0.185939
$612$	0	0
$613$	18.5865	0.750703	0.375351	−	0.926883i	$-0.377522\pi$
$613$	18.5865	0.750703	0.375351	+	0.926883i	$0.377522\pi$
$614$	0	0
$615$	−2.65915	−0.107227
$616$	0	0
$617$	−1.38124	−0.0556066	−0.0278033	−	0.999613i	$-0.508851\pi$
$617$	−1.38124	−0.0556066	−0.0278033	+	0.999613i	$0.508851\pi$
$618$	0	0
$619$	−7.98058	−0.320766	−0.160383	−	0.987055i	$-0.551273\pi$
$619$	−7.98058	−0.320766	−0.160383	+	0.987055i	$0.551273\pi$
$620$	0	0
$621$	−38.4217	−1.54181
$622$	0	0
$623$	−3.64299	−0.145953
$624$	0	0
$625$	23.5611	0.942445
$626$	0	0
$627$	−29.1435	−1.16388
$628$	0	0
$629$	0.787348	0.0313936
$630$	0	0
$631$	−3.54280	−0.141036	−0.0705182	−	0.997510i	$-0.522465\pi$
$631$	−3.54280	−0.141036	−0.0705182	+	0.997510i	$0.522465\pi$
$632$	0	0
$633$	33.7139	1.34001
$634$	0	0
$635$	3.72201	0.147704
$636$	0	0
$637$	−3.93525	−0.155920
$638$	0	0
$639$	9.87203	0.390531
$640$	0	0
$641$	8.43696	0.333240	0.166620	−	0.986021i	$-0.446715\pi$
$641$	8.43696	0.333240	0.166620	+	0.986021i	$0.446715\pi$
$642$	0	0
$643$	11.6623	0.459917	0.229959	−	0.973200i	$-0.426141\pi$
$643$	11.6623	0.459917	0.229959	+	0.973200i	$0.426141\pi$
$644$	0	0
$645$	−2.87170	−0.113073
$646$	0	0
$647$	−9.85308	−0.387364	−0.193682	−	0.981064i	$-0.562043\pi$
$647$	−9.85308	−0.387364	−0.193682	+	0.981064i	$0.562043\pi$
$648$	0	0
$649$	−59.3912	−2.33131
$650$	0	0
$651$	−6.03935	−0.236701
$652$	0	0
$653$	−12.7572	−0.499227	−0.249613	−	0.968346i	$-0.580304\pi$
$653$	−12.7572	−0.499227	−0.249613	+	0.968346i	$0.580304\pi$
$654$	0	0
$655$	0.370811	0.0144888
$656$	0	0
$657$	37.1602	1.44976
$658$	0	0
$659$	30.9267	1.20473	0.602367	−	0.798219i	$-0.294224\pi$
$659$	30.9267	1.20473	0.602367	+	0.798219i	$0.294224\pi$
$660$	0	0
$661$	12.3382	0.479901	0.239951	−	0.970785i	$-0.422869\pi$
$661$	12.3382	0.479901	0.239951	+	0.970785i	$0.422869\pi$
$662$	0	0
$663$	0.199003	0.00772865
$664$	0	0
$665$	−0.407309	−0.0157948
$666$	0	0
$667$	0.825472	0.0319624
$668$	0	0
$669$	26.2447	1.01468
$670$	0	0
$671$	7.26870	0.280605
$672$	0	0
$673$	3.95963	0.152632	0.0763162	−	0.997084i	$-0.475684\pi$
$673$	3.95963	0.152632	0.0763162	+	0.997084i	$0.475684\pi$
$674$	0	0
$675$	62.7404	2.41488
$676$	0	0
$677$	12.2376	0.470329	0.235164	−	0.971956i	$-0.424437\pi$
$677$	12.2376	0.470329	0.235164	+	0.971956i	$0.424437\pi$
$678$	0	0
$679$	1.82812	0.0701568
$680$	0	0
$681$	−91.4811	−3.50556
$682$	0	0
$683$	18.3679	0.702830	0.351415	−	0.936220i	$-0.385701\pi$
$683$	18.3679	0.702830	0.351415	+	0.936220i	$0.385701\pi$
$684$	0	0
$685$	−5.66907	−0.216604
$686$	0	0
$687$	12.6352	0.482063
$688$	0	0
$689$	−1.86232	−0.0709487
$690$	0	0
$691$	43.0212	1.63660	0.818301	−	0.574789i	$-0.194916\pi$
$691$	43.0212	1.63660	0.818301	+	0.574789i	$0.194916\pi$
$692$	0	0
$693$	−19.7151	−0.748914
$694$	0	0
$695$	4.13733	0.156938
$696$	0	0
$697$	−0.284574	−0.0107790
$698$	0	0
$699$	69.0693	2.61244
$700$	0	0
$701$	−30.6940	−1.15930	−0.579648	−	0.814867i	$-0.696810\pi$
$701$	−30.6940	−1.15930	−0.579648	+	0.814867i	$0.696810\pi$
$702$	0	0
$703$	−15.5055	−0.584799
$704$	0	0
$705$	7.58958	0.285840
$706$	0	0
$707$	−3.46303	−0.130241
$708$	0	0
$709$	26.9461	1.01198	0.505992	−	0.862538i	$-0.331127\pi$
$709$	26.9461	1.01198	0.505992	+	0.862538i	$0.331127\pi$
$710$	0	0
$711$	7.22668	0.271022
$712$	0	0
$713$	−9.05948	−0.339280
$714$	0	0
$715$	0.821256	0.0307132
$716$	0	0
$717$	20.0681	0.749459
$718$	0	0
$719$	33.2312	1.23931	0.619657	−	0.784873i	$-0.287272\pi$
$719$	33.2312	1.23931	0.619657	+	0.784873i	$0.287272\pi$
$720$	0	0
$721$	−2.29559	−0.0854923
$722$	0	0
$723$	−24.3192	−0.904442
$724$	0	0
$725$	−1.34795	−0.0500616
$726$	0	0
$727$	−30.4877	−1.13072	−0.565362	−	0.824843i	$-0.691264\pi$
$727$	−30.4877	−1.13072	−0.565362	+	0.824843i	$0.691264\pi$
$728$	0	0
$729$	15.0972	0.559156
$730$	0	0
$731$	−0.307321	−0.0113667
$732$	0	0
$733$	−10.7027	−0.395312	−0.197656	−	0.980271i	$-0.563333\pi$
$733$	−10.7027	−0.395312	−0.197656	+	0.980271i	$0.563333\pi$
$734$	0	0
$735$	6.49827	0.239692
$736$	0	0
$737$	26.7333	0.984736
$738$	0	0
$739$	36.2270	1.33263	0.666316	−	0.745670i	$-0.267870\pi$
$739$	36.2270	1.33263	0.666316	+	0.745670i	$0.267870\pi$
$740$	0	0
$741$	−3.91903	−0.143969
$742$	0	0
$743$	−17.2977	−0.634590	−0.317295	−	0.948327i	$-0.602774\pi$
$743$	−17.2977	−0.634590	−0.317295	+	0.948327i	$0.602774\pi$
$744$	0	0
$745$	0.581281	0.0212965
$746$	0	0
$747$	26.0269	0.952276
$748$	0	0
$749$	10.4513	0.381881
$750$	0	0
$751$	33.4549	1.22079	0.610394	−	0.792098i	$-0.291011\pi$
$751$	33.4549	1.22079	0.610394	+	0.792098i	$0.291011\pi$
$752$	0	0
$753$	−51.4225	−1.87394
$754$	0	0
$755$	−3.56674	−0.129807
$756$	0	0
$757$	−43.7026	−1.58840	−0.794200	−	0.607657i	$-0.792110\pi$
$757$	−43.7026	−1.58840	−0.794200	+	0.607657i	$0.792110\pi$
$758$	0	0
$759$	−42.1795	−1.53102
$760$	0	0
$761$	37.3977	1.35567	0.677833	−	0.735216i	$-0.262919\pi$
$761$	37.3977	1.35567	0.677833	+	0.735216i	$0.262919\pi$
$762$	0	0
$763$	2.39214	0.0866013
$764$	0	0
$765$	−0.230407	−0.00833039
$766$	0	0
$767$	−7.98655	−0.288377
$768$	0	0
$769$	−15.5188	−0.559623	−0.279812	−	0.960055i	$-0.590272\pi$
$769$	−15.5188	−0.559623	−0.279812	+	0.960055i	$0.590272\pi$
$770$	0	0
$771$	90.5668	3.26168
$772$	0	0
$773$	43.2988	1.55735	0.778674	−	0.627429i	$-0.215893\pi$
$773$	43.2988	1.55735	0.778674	+	0.627429i	$0.215893\pi$
$774$	0	0
$775$	14.7936	0.531402
$776$	0	0
$777$	−14.9600	−0.536687
$778$	0	0
$779$	5.60420	0.200791
$780$	0	0
$781$	6.21825	0.222507
$782$	0	0
$783$	−3.51735	−0.125700
$784$	0	0
$785$	6.35509	0.226823
$786$	0	0
$787$	−35.5133	−1.26591	−0.632956	−	0.774187i	$-0.718159\pi$
$787$	−35.5133	−1.26591	−0.632956	+	0.774187i	$0.718159\pi$
$788$	0	0
$789$	21.6009	0.769013
$790$	0	0
$791$	5.71086	0.203055
$792$	0	0
$793$	0.977448	0.0347102
$794$	0	0
$795$	3.07524	0.109068
$796$	0	0
$797$	−53.6545	−1.90054	−0.950270	−	0.311429i	$-0.899193\pi$
$797$	−53.6545	−1.90054	−0.950270	+	0.311429i	$0.899193\pi$
$798$	0	0
$799$	0.812215	0.0287341
$800$	0	0
$801$	40.5833	1.43394
$802$	0	0
$803$	23.4067	0.826005
$804$	0	0
$805$	−0.589500	−0.0207772
$806$	0	0
$807$	−55.8568	−1.96625
$808$	0	0
$809$	56.5487	1.98815	0.994073	−	0.108715i	$-0.0346735\pi$
$809$	56.5487	1.98815	0.994073	+	0.108715i	$0.0346735\pi$
$810$	0	0
$811$	−34.5774	−1.21418	−0.607089	−	0.794634i	$-0.707663\pi$
$811$	−34.5774	−1.21418	−0.607089	+	0.794634i	$0.707663\pi$
$812$	0	0
$813$	−5.86300	−0.205624
$814$	0	0
$815$	−5.42870	−0.190159
$816$	0	0
$817$	6.05215	0.211738
$818$	0	0
$819$	−2.65116	−0.0926389
$820$	0	0
$821$	46.5117	1.62327	0.811636	−	0.584164i	$-0.198578\pi$
$821$	46.5117	1.62327	0.811636	+	0.584164i	$0.198578\pi$
$822$	0	0
$823$	−45.4567	−1.58452	−0.792261	−	0.610182i	$-0.791096\pi$
$823$	−45.4567	−1.58452	−0.792261	+	0.610182i	$0.791096\pi$
$824$	0	0
$825$	68.8767	2.39798
$826$	0	0
$827$	−40.6228	−1.41259	−0.706296	−	0.707917i	$-0.749635\pi$
$827$	−40.6228	−1.41259	−0.706296	+	0.707917i	$0.749635\pi$
$828$	0	0
$829$	−43.6345	−1.51549	−0.757744	−	0.652551i	$-0.773699\pi$
$829$	−43.6345	−1.51549	−0.757744	+	0.652551i	$0.773699\pi$
$830$	0	0
$831$	28.5203	0.989359
$832$	0	0
$833$	0.695427	0.0240951
$834$	0	0
$835$	−7.57317	−0.262080
$836$	0	0
$837$	38.6026	1.33430
$838$	0	0
$839$	−10.1056	−0.348885	−0.174442	−	0.984667i	$-0.555812\pi$
$839$	−10.1056	−0.348885	−0.174442	+	0.984667i	$0.555812\pi$
$840$	0	0
$841$	−28.9244	−0.997394
$842$	0	0
$843$	4.82075	0.166035
$844$	0	0
$845$	−3.92890	−0.135158
$846$	0	0
$847$	−5.46836	−0.187895
$848$	0	0
$849$	47.5062	1.63041
$850$	0	0
$851$	−22.4411	−0.769272
$852$	0	0
$853$	−9.53832	−0.326586	−0.163293	−	0.986578i	$-0.552212\pi$
$853$	−9.53832	−0.326586	−0.163293	+	0.986578i	$0.552212\pi$
$854$	0	0
$855$	4.53747	0.155178
$856$	0	0
$857$	−26.7656	−0.914297	−0.457148	−	0.889390i	$-0.651129\pi$
$857$	−26.7656	−0.914297	−0.457148	+	0.889390i	$0.651129\pi$
$858$	0	0
$859$	35.4365	1.20908	0.604538	−	0.796576i	$-0.293358\pi$
$859$	35.4365	1.20908	0.604538	+	0.796576i	$0.293358\pi$
$860$	0	0
$861$	5.40705	0.184272
$862$	0	0
$863$	−13.5796	−0.462257	−0.231128	−	0.972923i	$-0.574242\pi$
$863$	−13.5796	−0.462257	−0.231128	+	0.972923i	$0.574242\pi$
$864$	0	0
$865$	3.38243	0.115006
$866$	0	0
$867$	53.8267	1.82805
$868$	0	0
$869$	4.55198	0.154415
$870$	0	0
$871$	3.59493	0.121809
$872$	0	0
$873$	−20.3655	−0.689267
$874$	0	0
$875$	1.94419	0.0657257
$876$	0	0
$877$	−9.51364	−0.321253	−0.160626	−	0.987015i	$-0.551351\pi$
$877$	−9.51364	−0.321253	−0.160626	+	0.987015i	$0.551351\pi$
$878$	0	0
$879$	−45.8367	−1.54603
$880$	0	0
$881$	43.7393	1.47361	0.736806	−	0.676104i	$-0.236333\pi$
$881$	43.7393	1.47361	0.736806	+	0.676104i	$0.236333\pi$
$882$	0	0
$883$	18.4288	0.620180	0.310090	−	0.950707i	$-0.399641\pi$
$883$	18.4288	0.620180	0.310090	+	0.950707i	$0.399641\pi$
$884$	0	0
$885$	13.1882	0.443316
$886$	0	0
$887$	40.5584	1.36182	0.680909	−	0.732368i	$-0.261585\pi$
$887$	40.5584	1.36182	0.680909	+	0.732368i	$0.261585\pi$
$888$	0	0
$889$	−7.56827	−0.253832
$890$	0	0
$891$	86.1152	2.88497
$892$	0	0
$893$	−15.9952	−0.535258
$894$	0	0
$895$	0.481822	0.0161055
$896$	0	0
$897$	−5.67203	−0.189384
$898$	0	0
$899$	−0.829360	−0.0276607
$900$	0	0
$901$	0.329104	0.0109640
$902$	0	0
$903$	5.83925	0.194318
$904$	0	0
$905$	6.87302	0.228467
$906$	0	0
$907$	24.9433	0.828229	0.414115	−	0.910225i	$-0.364091\pi$
$907$	24.9433	0.828229	0.414115	+	0.910225i	$0.364091\pi$
$908$	0	0
$909$	38.5786	1.27957
$910$	0	0
$911$	−16.9350	−0.561081	−0.280540	−	0.959842i	$-0.590514\pi$
$911$	−16.9350	−0.561081	−0.280540	+	0.959842i	$0.590514\pi$
$912$	0	0
$913$	16.3940	0.542562
$914$	0	0
$915$	−1.61406	−0.0533591
$916$	0	0
$917$	−0.754000	−0.0248993
$918$	0	0
$919$	−16.6168	−0.548138	−0.274069	−	0.961710i	$-0.588370\pi$
$919$	−16.6168	−0.548138	−0.274069	+	0.961710i	$0.588370\pi$
$920$	0	0
$921$	−4.77102	−0.157211
$922$	0	0
$923$	0.836190	0.0275235
$924$	0	0
$925$	36.6451	1.20488
$926$	0	0
$927$	25.5732	0.839933
$928$	0	0
$929$	2.31946	0.0760990	0.0380495	−	0.999276i	$-0.487886\pi$
$929$	2.31946	0.0760990	0.0380495	+	0.999276i	$0.487886\pi$
$930$	0	0
$931$	−13.6952	−0.448843
$932$	0	0
$933$	−18.2129	−0.596262
$934$	0	0
$935$	−0.145130	−0.00474627
$936$	0	0
$937$	−57.6197	−1.88235	−0.941176	−	0.337916i	$-0.890278\pi$
$937$	−57.6197	−1.88235	−0.941176	+	0.337916i	$0.890278\pi$
$938$	0	0
$939$	40.4582	1.32030
$940$	0	0
$941$	−56.3587	−1.83724	−0.918621	−	0.395140i	$-0.870696\pi$
$941$	−56.3587	−1.83724	−0.918621	+	0.395140i	$0.870696\pi$
$942$	0	0
$943$	8.11098	0.264130
$944$	0	0
$945$	2.51187	0.0817112
$946$	0	0
$947$	−18.5184	−0.601766	−0.300883	−	0.953661i	$-0.597281\pi$
$947$	−18.5184	−0.601766	−0.300883	+	0.953661i	$0.597281\pi$
$948$	0	0
$949$	3.14758	0.102175
$950$	0	0
$951$	62.0074	2.01073
$952$	0	0
$953$	−49.4083	−1.60049	−0.800246	−	0.599672i	$-0.795298\pi$
$953$	−49.4083	−1.60049	−0.800246	+	0.599672i	$0.795298\pi$
$954$	0	0
$955$	−4.37757	−0.141655
$956$	0	0
$957$	−3.86137	−0.124820
$958$	0	0
$959$	11.5274	0.372238
$960$	0	0
$961$	−21.8979	−0.706382
$962$	0	0
$963$	−116.428	−3.75185
$964$	0	0
$965$	−2.52575	−0.0813067
$966$	0	0
$967$	58.3674	1.87697	0.938484	−	0.345323i	$-0.112231\pi$
$967$	58.3674	1.87697	0.938484	+	0.345323i	$0.112231\pi$
$968$	0	0
$969$	0.692560	0.0222482
$970$	0	0
$971$	5.60086	0.179740	0.0898701	−	0.995953i	$-0.471355\pi$
$971$	5.60086	0.179740	0.0898701	+	0.995953i	$0.471355\pi$
$972$	0	0
$973$	−8.41277	−0.269701
$974$	0	0
$975$	9.26209	0.296624
$976$	0	0
$977$	14.4621	0.462685	0.231343	−	0.972872i	$-0.425688\pi$
$977$	14.4621	0.462685	0.231343	+	0.972872i	$0.425688\pi$
$978$	0	0
$979$	25.5628	0.816992
$980$	0	0
$981$	−26.6487	−0.850828
$982$	0	0
$983$	−17.7477	−0.566063	−0.283032	−	0.959111i	$-0.591340\pi$
$983$	−17.7477	−0.566063	−0.283032	+	0.959111i	$0.591340\pi$
$984$	0	0
$985$	4.65732	0.148395
$986$	0	0
$987$	−15.4325	−0.491222
$988$	0	0
$989$	8.75931	0.278530
$990$	0	0
$991$	15.8734	0.504235	0.252117	−	0.967697i	$-0.418873\pi$
$991$	15.8734	0.504235	0.252117	+	0.967697i	$0.418873\pi$
$992$	0	0
$993$	−68.5402	−2.17506
$994$	0	0
$995$	7.32976	0.232369
$996$	0	0
$997$	16.8232	0.532795	0.266397	−	0.963863i	$-0.414167\pi$
$997$	16.8232	0.532795	0.266397	+	0.963863i	$0.414167\pi$
$998$	0	0
$999$	95.6221	3.02535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.1		12
4.3	odd	2		1006.2.a.j.1.12	✓	12
12.11	even	2		9054.2.a.bi.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.12	✓	12	4.3	odd	2
8048.2.a.q.1.1		12	1.1	even	1	trivial
9054.2.a.bi.1.8		12	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-3.16835 q^{3} +0.310718 q^{5} -0.631809 q^{7} +7.03843 q^{9} +O(q^{10})\)	\(q-3.16835 q^{3} +0.310718 q^{5} -0.631809 q^{7} +7.03843 q^{9} +4.43341 q^{11} +0.596176 q^{13} -0.984464 q^{15} -0.105355 q^{17} +2.07478 q^{19} +2.00179 q^{21} +3.00283 q^{23} -4.90345 q^{25} -12.7951 q^{27} +0.274898 q^{29} -3.01698 q^{31} -14.0466 q^{33} -0.196315 q^{35} -7.47331 q^{37} -1.88889 q^{39} +2.70111 q^{41} +2.91701 q^{43} +2.18697 q^{45} -7.70935 q^{47} -6.60082 q^{49} +0.333800 q^{51} -3.12377 q^{53} +1.37754 q^{55} -6.57361 q^{57} -13.3963 q^{59} +1.63953 q^{61} -4.44694 q^{63} +0.185243 q^{65} +6.02998 q^{67} -9.51402 q^{69} +1.40259 q^{71} +5.27962 q^{73} +15.5358 q^{75} -2.80106 q^{77} +1.02675 q^{79} +19.4242 q^{81} +3.69784 q^{83} -0.0327356 q^{85} -0.870972 q^{87} +5.76596 q^{89} -0.376669 q^{91} +9.55883 q^{93} +0.644671 q^{95} -2.89347 q^{97} +31.2042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

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