LMFDB - Embedded newform 8008.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.244558277.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 29x^{2} + 5x - 4 $ x^6 - x^5 - 13x^4 + 11x^3 + 29x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.25115$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.25115 q^{3} -0.215704 q^{5} +1.00000 q^{7} +7.56995 q^{9} +O(q^{10})$	$q+3.25115 q^{3} -0.215704 q^{5} +1.00000 q^{7} +7.56995 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.701286 q^{15} -1.35393 q^{17} +5.10414 q^{19} +3.25115 q^{21} +4.33416 q^{23} -4.95347 q^{25} +14.8576 q^{27} -0.702326 q^{29} +9.03648 q^{31} -3.25115 q^{33} -0.215704 q^{35} -2.18611 q^{37} -3.25115 q^{39} +3.79997 q^{41} +2.91699 q^{43} -1.63287 q^{45} +10.5856 q^{47} +1.00000 q^{49} -4.40181 q^{51} -8.94833 q^{53} +0.215704 q^{55} +16.5943 q^{57} -6.12859 q^{59} -1.26507 q^{61} +7.56995 q^{63} +0.215704 q^{65} -10.2368 q^{67} +14.0910 q^{69} +11.0722 q^{71} -10.6272 q^{73} -16.1045 q^{75} -1.00000 q^{77} +11.7524 q^{79} +25.5943 q^{81} -13.6773 q^{83} +0.292048 q^{85} -2.28336 q^{87} +6.89690 q^{89} -1.00000 q^{91} +29.3789 q^{93} -1.10098 q^{95} -9.18517 q^{97} -7.56995 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * 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
$3$	3.25115	1.87705	0.938525	−	0.345211i	$-0.112193\pi$
$3$	3.25115	1.87705	0.938525	+	0.345211i	$0.112193\pi$
$4$	0	0
$5$	−0.215704	−0.0964658	−0.0482329	−	0.998836i	$-0.515359\pi$
$5$	−0.215704	−0.0964658	−0.0482329	+	0.998836i	$0.515359\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.56995	2.52332
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.701286	−0.181071
$16$	0	0
$17$	−1.35393	−0.328376	−0.164188	−	0.986429i	$-0.552500\pi$
$17$	−1.35393	−0.328376	−0.164188	+	0.986429i	$0.552500\pi$
$18$	0	0
$19$	5.10414	1.17097	0.585485	−	0.810683i	$-0.300904\pi$
$19$	5.10414	1.17097	0.585485	+	0.810683i	$0.300904\pi$
$20$	0	0
$21$	3.25115	0.709458
$22$	0	0
$23$	4.33416	0.903734	0.451867	−	0.892085i	$-0.350758\pi$
$23$	4.33416	0.903734	0.451867	+	0.892085i	$0.350758\pi$
$24$	0	0
$25$	−4.95347	−0.990694
$26$	0	0
$27$	14.8576	2.85934
$28$	0	0
$29$	−0.702326	−0.130419	−0.0652093	−	0.997872i	$-0.520772\pi$
$29$	−0.702326	−0.130419	−0.0652093	+	0.997872i	$0.520772\pi$
$30$	0	0
$31$	9.03648	1.62300	0.811500	−	0.584352i	$-0.198651\pi$
$31$	9.03648	1.62300	0.811500	+	0.584352i	$0.198651\pi$
$32$	0	0
$33$	−3.25115	−0.565952
$34$	0	0
$35$	−0.215704	−0.0364607
$36$	0	0
$37$	−2.18611	−0.359394	−0.179697	−	0.983722i	$-0.557512\pi$
$37$	−2.18611	−0.359394	−0.179697	+	0.983722i	$0.557512\pi$
$38$	0	0
$39$	−3.25115	−0.520600
$40$	0	0
$41$	3.79997	0.593455	0.296728	−	0.954962i	$-0.404105\pi$
$41$	3.79997	0.593455	0.296728	+	0.954962i	$0.404105\pi$
$42$	0	0
$43$	2.91699	0.444837	0.222418	−	0.974951i	$-0.428605\pi$
$43$	2.91699	0.444837	0.222418	+	0.974951i	$0.428605\pi$
$44$	0	0
$45$	−1.63287	−0.243414
$46$	0	0
$47$	10.5856	1.54407	0.772036	−	0.635579i	$-0.219239\pi$
$47$	10.5856	1.54407	0.772036	+	0.635579i	$0.219239\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.40181	−0.616377
$52$	0	0
$53$	−8.94833	−1.22915	−0.614574	−	0.788859i	$-0.710672\pi$
$53$	−8.94833	−1.22915	−0.614574	+	0.788859i	$0.710672\pi$
$54$	0	0
$55$	0.215704	0.0290855
$56$	0	0
$57$	16.5943	2.19797
$58$	0	0
$59$	−6.12859	−0.797875	−0.398937	−	0.916978i	$-0.630621\pi$
$59$	−6.12859	−0.797875	−0.398937	+	0.916978i	$0.630621\pi$
$60$	0	0
$61$	−1.26507	−0.161975	−0.0809877	−	0.996715i	$-0.525807\pi$
$61$	−1.26507	−0.161975	−0.0809877	+	0.996715i	$0.525807\pi$
$62$	0	0
$63$	7.56995	0.953724
$64$	0	0
$65$	0.215704	0.0267548
$66$	0	0
$67$	−10.2368	−1.25063	−0.625314	−	0.780373i	$-0.715029\pi$
$67$	−10.2368	−1.25063	−0.625314	+	0.780373i	$0.715029\pi$
$68$	0	0
$69$	14.0910	1.69635
$70$	0	0
$71$	11.0722	1.31403	0.657017	−	0.753876i	$-0.271818\pi$
$71$	11.0722	1.31403	0.657017	+	0.753876i	$0.271818\pi$
$72$	0	0
$73$	−10.6272	−1.24382	−0.621912	−	0.783087i	$-0.713644\pi$
$73$	−10.6272	−1.24382	−0.621912	+	0.783087i	$0.713644\pi$
$74$	0	0
$75$	−16.1045	−1.85958
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.7524	1.32225	0.661124	−	0.750277i	$-0.270080\pi$
$79$	11.7524	1.32225	0.661124	+	0.750277i	$0.270080\pi$
$80$	0	0
$81$	25.5943	2.84381
$82$	0	0
$83$	−13.6773	−1.50128	−0.750640	−	0.660711i	$-0.770255\pi$
$83$	−13.6773	−1.50128	−0.750640	+	0.660711i	$0.770255\pi$
$84$	0	0
$85$	0.292048	0.0316770
$86$	0	0
$87$	−2.28336	−0.244802
$88$	0	0
$89$	6.89690	0.731070	0.365535	−	0.930798i	$-0.380886\pi$
$89$	6.89690	0.731070	0.365535	+	0.930798i	$0.380886\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	29.3789	3.04645
$94$	0	0
$95$	−1.10098	−0.112959
$96$	0	0
$97$	−9.18517	−0.932612	−0.466306	−	0.884623i	$-0.654415\pi$
$97$	−9.18517	−0.932612	−0.466306	+	0.884623i	$0.654415\pi$
$98$	0	0
$99$	−7.56995	−0.760809
$100$	0	0
$101$	−10.2727	−1.02217	−0.511084	−	0.859531i	$-0.670756\pi$
$101$	−10.2727	−1.02217	−0.511084	+	0.859531i	$0.670756\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−0.701286	−0.0684385
$106$	0	0
$107$	12.9900	1.25579	0.627893	−	0.778300i	$-0.283918\pi$
$107$	12.9900	1.25579	0.627893	+	0.778300i	$0.283918\pi$
$108$	0	0
$109$	−1.59819	−0.153078	−0.0765392	−	0.997067i	$-0.524387\pi$
$109$	−1.59819	−0.153078	−0.0765392	+	0.997067i	$0.524387\pi$
$110$	0	0
$111$	−7.10737	−0.674601
$112$	0	0
$113$	4.77551	0.449242	0.224621	−	0.974446i	$-0.427886\pi$
$113$	4.77551	0.449242	0.224621	+	0.974446i	$0.427886\pi$
$114$	0	0
$115$	−0.934896	−0.0871795
$116$	0	0
$117$	−7.56995	−0.699842
$118$	0	0
$119$	−1.35393	−0.124114
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.3542	1.11394
$124$	0	0
$125$	2.14701	0.192034
$126$	0	0
$127$	−5.68359	−0.504337	−0.252169	−	0.967683i	$-0.581144\pi$
$127$	−5.68359	−0.504337	−0.252169	+	0.967683i	$0.581144\pi$
$128$	0	0
$129$	9.48356	0.834981
$130$	0	0
$131$	−6.86899	−0.600146	−0.300073	−	0.953916i	$-0.597011\pi$
$131$	−6.86899	−0.600146	−0.300073	+	0.953916i	$0.597011\pi$
$132$	0	0
$133$	5.10414	0.442585
$134$	0	0
$135$	−3.20484	−0.275829
$136$	0	0
$137$	15.0365	1.28465	0.642327	−	0.766431i	$-0.277969\pi$
$137$	15.0365	1.28465	0.642327	+	0.766431i	$0.277969\pi$
$138$	0	0
$139$	8.93370	0.757747	0.378873	−	0.925449i	$-0.376312\pi$
$139$	8.93370	0.757747	0.378873	+	0.925449i	$0.376312\pi$
$140$	0	0
$141$	34.4154	2.89830
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.151495	0.0125809
$146$	0	0
$147$	3.25115	0.268150
$148$	0	0
$149$	18.6028	1.52400	0.761999	−	0.647578i	$-0.224218\pi$
$149$	18.6028	1.52400	0.761999	+	0.647578i	$0.224218\pi$
$150$	0	0
$151$	−21.5665	−1.75505	−0.877527	−	0.479527i	$-0.840808\pi$
$151$	−21.5665	−1.75505	−0.877527	+	0.479527i	$0.840808\pi$
$152$	0	0
$153$	−10.2492	−0.828596
$154$	0	0
$155$	−1.94921	−0.156564
$156$	0	0
$157$	12.8997	1.02951	0.514755	−	0.857337i	$-0.327883\pi$
$157$	12.8997	1.02951	0.514755	+	0.857337i	$0.327883\pi$
$158$	0	0
$159$	−29.0923	−2.30717
$160$	0	0
$161$	4.33416	0.341579
$162$	0	0
$163$	20.7694	1.62679	0.813394	−	0.581714i	$-0.197618\pi$
$163$	20.7694	1.62679	0.813394	+	0.581714i	$0.197618\pi$
$164$	0	0
$165$	0.701286	0.0545950
$166$	0	0
$167$	24.0229	1.85895	0.929473	−	0.368890i	$-0.120262\pi$
$167$	24.0229	1.85895	0.929473	+	0.368890i	$0.120262\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	38.6381	2.95473
$172$	0	0
$173$	−9.90044	−0.752717	−0.376358	−	0.926474i	$-0.622824\pi$
$173$	−9.90044	−0.752717	−0.376358	+	0.926474i	$0.622824\pi$
$174$	0	0
$175$	−4.95347	−0.374447
$176$	0	0
$177$	−19.9250	−1.49765
$178$	0	0
$179$	−0.900882	−0.0673351	−0.0336675	−	0.999433i	$-0.510719\pi$
$179$	−0.900882	−0.0673351	−0.0336675	+	0.999433i	$0.510719\pi$
$180$	0	0
$181$	−14.5169	−1.07903	−0.539515	−	0.841976i	$-0.681392\pi$
$181$	−14.5169	−1.07903	−0.539515	+	0.841976i	$0.681392\pi$
$182$	0	0
$183$	−4.11292	−0.304036
$184$	0	0
$185$	0.471553	0.0346693
$186$	0	0
$187$	1.35393	0.0990090
$188$	0	0
$189$	14.8576	1.08073
$190$	0	0
$191$	12.7425	0.922013	0.461006	−	0.887397i	$-0.347489\pi$
$191$	12.7425	0.922013	0.461006	+	0.887397i	$0.347489\pi$
$192$	0	0
$193$	−16.5772	−1.19325	−0.596625	−	0.802520i	$-0.703492\pi$
$193$	−16.5772	−1.19325	−0.596625	+	0.802520i	$0.703492\pi$
$194$	0	0
$195$	0.701286	0.0502201
$196$	0	0
$197$	17.6276	1.25591	0.627956	−	0.778249i	$-0.283892\pi$
$197$	17.6276	1.25591	0.627956	+	0.778249i	$0.283892\pi$
$198$	0	0
$199$	8.17828	0.579743	0.289871	−	0.957066i	$-0.406387\pi$
$199$	8.17828	0.579743	0.289871	+	0.957066i	$0.406387\pi$
$200$	0	0
$201$	−33.2814	−2.34749
$202$	0	0
$203$	−0.702326	−0.0492936
$204$	0	0
$205$	−0.819669	−0.0572481
$206$	0	0
$207$	32.8093	2.28041
$208$	0	0
$209$	−5.10414	−0.353061
$210$	0	0
$211$	−17.9751	−1.23746	−0.618728	−	0.785605i	$-0.712352\pi$
$211$	−17.9751	−1.23746	−0.618728	+	0.785605i	$0.712352\pi$
$212$	0	0
$213$	35.9975	2.46651
$214$	0	0
$215$	−0.629207	−0.0429116
$216$	0	0
$217$	9.03648	0.613436
$218$	0	0
$219$	−34.5507	−2.33472
$220$	0	0
$221$	1.35393	0.0910750
$222$	0	0
$223$	21.3676	1.43088	0.715441	−	0.698673i	$-0.246226\pi$
$223$	21.3676	1.43088	0.715441	+	0.698673i	$0.246226\pi$
$224$	0	0
$225$	−37.4975	−2.49984
$226$	0	0
$227$	−9.91374	−0.657998	−0.328999	−	0.944330i	$-0.606711\pi$
$227$	−9.91374	−0.657998	−0.328999	+	0.944330i	$0.606711\pi$
$228$	0	0
$229$	−17.0346	−1.12568	−0.562840	−	0.826566i	$-0.690291\pi$
$229$	−17.0346	−1.12568	−0.562840	+	0.826566i	$0.690291\pi$
$230$	0	0
$231$	−3.25115	−0.213910
$232$	0	0
$233$	−13.9903	−0.916533	−0.458267	−	0.888815i	$-0.651530\pi$
$233$	−13.9903	−0.916533	−0.458267	+	0.888815i	$0.651530\pi$
$234$	0	0
$235$	−2.28336	−0.148950
$236$	0	0
$237$	38.2088	2.48193
$238$	0	0
$239$	3.31371	0.214346	0.107173	−	0.994240i	$-0.465820\pi$
$239$	3.31371	0.214346	0.107173	+	0.994240i	$0.465820\pi$
$240$	0	0
$241$	10.5209	0.677711	0.338856	−	0.940838i	$-0.389960\pi$
$241$	10.5209	0.677711	0.338856	+	0.940838i	$0.389960\pi$
$242$	0	0
$243$	38.6381	2.47863
$244$	0	0
$245$	−0.215704	−0.0137808
$246$	0	0
$247$	−5.10414	−0.324769
$248$	0	0
$249$	−44.4670	−2.81798
$250$	0	0
$251$	−10.8840	−0.686993	−0.343497	−	0.939154i	$-0.611611\pi$
$251$	−10.8840	−0.686993	−0.343497	+	0.939154i	$0.611611\pi$
$252$	0	0
$253$	−4.33416	−0.272486
$254$	0	0
$255$	0.949490	0.0594594
$256$	0	0
$257$	−22.0396	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$257$	−22.0396	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$258$	0	0
$259$	−2.18611	−0.135838
$260$	0	0
$261$	−5.31657	−0.329087
$262$	0	0
$263$	−3.88772	−0.239727	−0.119863	−	0.992790i	$-0.538246\pi$
$263$	−3.88772	−0.239727	−0.119863	+	0.992790i	$0.538246\pi$
$264$	0	0
$265$	1.93019	0.118571
$266$	0	0
$267$	22.4228	1.37225
$268$	0	0
$269$	−20.9399	−1.27673	−0.638363	−	0.769735i	$-0.720388\pi$
$269$	−20.9399	−1.27673	−0.638363	+	0.769735i	$0.720388\pi$
$270$	0	0
$271$	16.5926	1.00793	0.503963	−	0.863725i	$-0.331875\pi$
$271$	16.5926	1.00793	0.503963	+	0.863725i	$0.331875\pi$
$272$	0	0
$273$	−3.25115	−0.196768
$274$	0	0
$275$	4.95347	0.298706
$276$	0	0
$277$	14.1362	0.849364	0.424682	−	0.905343i	$-0.360386\pi$
$277$	14.1362	0.849364	0.424682	+	0.905343i	$0.360386\pi$
$278$	0	0
$279$	68.4057	4.09534
$280$	0	0
$281$	8.49250	0.506620	0.253310	−	0.967385i	$-0.418481\pi$
$281$	8.49250	0.506620	0.253310	+	0.967385i	$0.418481\pi$
$282$	0	0
$283$	11.4242	0.679097	0.339549	−	0.940588i	$-0.389726\pi$
$283$	11.4242	0.679097	0.339549	+	0.940588i	$0.389726\pi$
$284$	0	0
$285$	−3.57946	−0.212029
$286$	0	0
$287$	3.79997	0.224305
$288$	0	0
$289$	−15.1669	−0.892169
$290$	0	0
$291$	−29.8623	−1.75056
$292$	0	0
$293$	−28.5641	−1.66873	−0.834366	−	0.551211i	$-0.814166\pi$
$293$	−28.5641	−1.66873	−0.834366	+	0.551211i	$0.814166\pi$
$294$	0	0
$295$	1.32196	0.0769677
$296$	0	0
$297$	−14.8576	−0.862124
$298$	0	0
$299$	−4.33416	−0.250651
$300$	0	0
$301$	2.91699	0.168133
$302$	0	0
$303$	−33.3979	−1.91866
$304$	0	0
$305$	0.272881	0.0156251
$306$	0	0
$307$	−15.3257	−0.874682	−0.437341	−	0.899296i	$-0.644080\pi$
$307$	−15.3257	−0.874682	−0.437341	+	0.899296i	$0.644080\pi$
$308$	0	0
$309$	26.0092	1.47961
$310$	0	0
$311$	16.7201	0.948111	0.474055	−	0.880495i	$-0.342790\pi$
$311$	16.7201	0.948111	0.474055	+	0.880495i	$0.342790\pi$
$312$	0	0
$313$	−22.1688	−1.25305	−0.626527	−	0.779400i	$-0.715524\pi$
$313$	−22.1688	−1.25305	−0.626527	+	0.779400i	$0.715524\pi$
$314$	0	0
$315$	−1.63287	−0.0920018
$316$	0	0
$317$	4.11034	0.230860	0.115430	−	0.993316i	$-0.463175\pi$
$317$	4.11034	0.230860	0.115430	+	0.993316i	$0.463175\pi$
$318$	0	0
$319$	0.702326	0.0393227
$320$	0	0
$321$	42.2322	2.35717
$322$	0	0
$323$	−6.91063	−0.384518
$324$	0	0
$325$	4.95347	0.274769
$326$	0	0
$327$	−5.19593	−0.287336
$328$	0	0
$329$	10.5856	0.583604
$330$	0	0
$331$	−9.19240	−0.505260	−0.252630	−	0.967563i	$-0.581296\pi$
$331$	−9.19240	−0.505260	−0.252630	+	0.967563i	$0.581296\pi$
$332$	0	0
$333$	−16.5488	−0.906866
$334$	0	0
$335$	2.20813	0.120643
$336$	0	0
$337$	−16.8468	−0.917704	−0.458852	−	0.888513i	$-0.651739\pi$
$337$	−16.8468	−0.917704	−0.458852	+	0.888513i	$0.651739\pi$
$338$	0	0
$339$	15.5259	0.843251
$340$	0	0
$341$	−9.03648	−0.489353
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.03948	−0.163640
$346$	0	0
$347$	−0.693747	−0.0372423	−0.0186211	−	0.999827i	$-0.505928\pi$
$347$	−0.693747	−0.0372423	−0.0186211	+	0.999827i	$0.505928\pi$
$348$	0	0
$349$	−21.6150	−1.15703	−0.578513	−	0.815674i	$-0.696367\pi$
$349$	−21.6150	−1.15703	−0.578513	+	0.815674i	$0.696367\pi$
$350$	0	0
$351$	−14.8576	−0.793039
$352$	0	0
$353$	33.7393	1.79576	0.897882	−	0.440236i	$-0.145105\pi$
$353$	33.7393	1.79576	0.897882	+	0.440236i	$0.145105\pi$
$354$	0	0
$355$	−2.38833	−0.126759
$356$	0	0
$357$	−4.40181	−0.232969
$358$	0	0
$359$	−15.5388	−0.820107	−0.410054	−	0.912061i	$-0.634490\pi$
$359$	−15.5388	−0.820107	−0.410054	+	0.912061i	$0.634490\pi$
$360$	0	0
$361$	7.05225	0.371171
$362$	0	0
$363$	3.25115	0.170641
$364$	0	0
$365$	2.29234	0.119987
$366$	0	0
$367$	−10.9705	−0.572656	−0.286328	−	0.958132i	$-0.592435\pi$
$367$	−10.9705	−0.572656	−0.286328	+	0.958132i	$0.592435\pi$
$368$	0	0
$369$	28.7656	1.49748
$370$	0	0
$371$	−8.94833	−0.464574
$372$	0	0
$373$	−7.36436	−0.381312	−0.190656	−	0.981657i	$-0.561062\pi$
$373$	−7.36436	−0.381312	−0.190656	+	0.981657i	$0.561062\pi$
$374$	0	0
$375$	6.98023	0.360457
$376$	0	0
$377$	0.702326	0.0361716
$378$	0	0
$379$	−10.7926	−0.554381	−0.277190	−	0.960815i	$-0.589403\pi$
$379$	−10.7926	−0.554381	−0.277190	+	0.960815i	$0.589403\pi$
$380$	0	0
$381$	−18.4782	−0.946667
$382$	0	0
$383$	21.0587	1.07605	0.538024	−	0.842929i	$-0.319171\pi$
$383$	21.0587	1.07605	0.538024	+	0.842929i	$0.319171\pi$
$384$	0	0
$385$	0.215704	0.0109933
$386$	0	0
$387$	22.0815	1.12246
$388$	0	0
$389$	−8.12674	−0.412042	−0.206021	−	0.978548i	$-0.566052\pi$
$389$	−8.12674	−0.412042	−0.206021	+	0.978548i	$0.566052\pi$
$390$	0	0
$391$	−5.86813	−0.296764
$392$	0	0
$393$	−22.3321	−1.12650
$394$	0	0
$395$	−2.53504	−0.127552
$396$	0	0
$397$	−32.7843	−1.64540	−0.822698	−	0.568479i	$-0.807532\pi$
$397$	−32.7843	−1.64540	−0.822698	+	0.568479i	$0.807532\pi$
$398$	0	0
$399$	16.5943	0.830754
$400$	0	0
$401$	3.80205	0.189865	0.0949326	−	0.995484i	$-0.469736\pi$
$401$	3.80205	0.189865	0.0949326	+	0.995484i	$0.469736\pi$
$402$	0	0
$403$	−9.03648	−0.450139
$404$	0	0
$405$	−5.52080	−0.274331
$406$	0	0
$407$	2.18611	0.108361
$408$	0	0
$409$	−12.4744	−0.616822	−0.308411	−	0.951253i	$-0.599797\pi$
$409$	−12.4744	−0.616822	−0.308411	+	0.951253i	$0.599797\pi$
$410$	0	0
$411$	48.8858	2.41136
$412$	0	0
$413$	−6.12859	−0.301568
$414$	0	0
$415$	2.95025	0.144822
$416$	0	0
$417$	29.0448	1.42233
$418$	0	0
$419$	8.41603	0.411150	0.205575	−	0.978641i	$-0.434094\pi$
$419$	8.41603	0.411150	0.205575	+	0.978641i	$0.434094\pi$
$420$	0	0
$421$	30.1244	1.46818	0.734088	−	0.679055i	$-0.237610\pi$
$421$	30.1244	1.46818	0.734088	+	0.679055i	$0.237610\pi$
$422$	0	0
$423$	80.1326	3.89618
$424$	0	0
$425$	6.70664	0.325320
$426$	0	0
$427$	−1.26507	−0.0612210
$428$	0	0
$429$	3.25115	0.156967
$430$	0	0
$431$	26.1605	1.26011	0.630054	−	0.776551i	$-0.283033\pi$
$431$	26.1605	1.26011	0.630054	+	0.776551i	$0.283033\pi$
$432$	0	0
$433$	18.3284	0.880808	0.440404	−	0.897800i	$-0.354835\pi$
$433$	18.3284	0.880808	0.440404	+	0.897800i	$0.354835\pi$
$434$	0	0
$435$	0.492531	0.0236151
$436$	0	0
$437$	22.1221	1.05825
$438$	0	0
$439$	−18.5239	−0.884098	−0.442049	−	0.896991i	$-0.645748\pi$
$439$	−18.5239	−0.884098	−0.442049	+	0.896991i	$0.645748\pi$
$440$	0	0
$441$	7.56995	0.360474
$442$	0	0
$443$	−7.89994	−0.375337	−0.187669	−	0.982232i	$-0.560093\pi$
$443$	−7.89994	−0.375337	−0.187669	+	0.982232i	$0.560093\pi$
$444$	0	0
$445$	−1.48769	−0.0705233
$446$	0	0
$447$	60.4803	2.86062
$448$	0	0
$449$	2.05604	0.0970305	0.0485153	−	0.998822i	$-0.484551\pi$
$449$	2.05604	0.0970305	0.0485153	+	0.998822i	$0.484551\pi$
$450$	0	0
$451$	−3.79997	−0.178933
$452$	0	0
$453$	−70.1158	−3.29433
$454$	0	0
$455$	0.215704	0.0101124
$456$	0	0
$457$	−8.68043	−0.406053	−0.203027	−	0.979173i	$-0.565078\pi$
$457$	−8.68043	−0.406053	−0.203027	+	0.979173i	$0.565078\pi$
$458$	0	0
$459$	−20.1161	−0.938938
$460$	0	0
$461$	0.873465	0.0406813	0.0203407	−	0.999793i	$-0.493525\pi$
$461$	0.873465	0.0406813	0.0203407	+	0.999793i	$0.493525\pi$
$462$	0	0
$463$	−17.6693	−0.821160	−0.410580	−	0.911825i	$-0.634674\pi$
$463$	−17.6693	−0.821160	−0.410580	+	0.911825i	$0.634674\pi$
$464$	0	0
$465$	−6.33716	−0.293879
$466$	0	0
$467$	13.7534	0.636433	0.318217	−	0.948018i	$-0.396916\pi$
$467$	13.7534	0.636433	0.318217	+	0.948018i	$0.396916\pi$
$468$	0	0
$469$	−10.2368	−0.472693
$470$	0	0
$471$	41.9389	1.93244
$472$	0	0
$473$	−2.91699	−0.134123
$474$	0	0
$475$	−25.2832	−1.16007
$476$	0	0
$477$	−67.7384	−3.10153
$478$	0	0
$479$	−36.2058	−1.65429	−0.827143	−	0.561991i	$-0.810036\pi$
$479$	−36.2058	−1.65429	−0.827143	+	0.561991i	$0.810036\pi$
$480$	0	0
$481$	2.18611	0.0996781
$482$	0	0
$483$	14.0910	0.641162
$484$	0	0
$485$	1.98128	0.0899652
$486$	0	0
$487$	2.21948	0.100574	0.0502872	−	0.998735i	$-0.483986\pi$
$487$	2.21948	0.100574	0.0502872	+	0.998735i	$0.483986\pi$
$488$	0	0
$489$	67.5244	3.05356
$490$	0	0
$491$	−38.9913	−1.75965	−0.879825	−	0.475297i	$-0.842341\pi$
$491$	−38.9913	−1.75965	−0.879825	+	0.475297i	$0.842341\pi$
$492$	0	0
$493$	0.950898	0.0428263
$494$	0	0
$495$	1.63287	0.0733921
$496$	0	0
$497$	11.0722	0.496658
$498$	0	0
$499$	41.4187	1.85416	0.927079	−	0.374867i	$-0.122312\pi$
$499$	41.4187	1.85416	0.927079	+	0.374867i	$0.122312\pi$
$500$	0	0
$501$	78.1019	3.48934
$502$	0	0
$503$	−9.03035	−0.402643	−0.201322	−	0.979525i	$-0.564524\pi$
$503$	−9.03035	−0.402643	−0.201322	+	0.979525i	$0.564524\pi$
$504$	0	0
$505$	2.21586	0.0986043
$506$	0	0
$507$	3.25115	0.144388
$508$	0	0
$509$	32.4570	1.43863	0.719316	−	0.694683i	$-0.244455\pi$
$509$	32.4570	1.43863	0.719316	+	0.694683i	$0.244455\pi$
$510$	0	0
$511$	−10.6272	−0.470122
$512$	0	0
$513$	75.8352	3.34820
$514$	0	0
$515$	−1.72563	−0.0760405
$516$	0	0
$517$	−10.5856	−0.465555
$518$	0	0
$519$	−32.1878	−1.41289
$520$	0	0
$521$	15.6556	0.685883	0.342941	−	0.939357i	$-0.388577\pi$
$521$	15.6556	0.685883	0.342941	+	0.939357i	$0.388577\pi$
$522$	0	0
$523$	20.8745	0.912779	0.456389	−	0.889780i	$-0.349142\pi$
$523$	20.8745	0.912779	0.456389	+	0.889780i	$0.349142\pi$
$524$	0	0
$525$	−16.1045	−0.702856
$526$	0	0
$527$	−12.2347	−0.532954
$528$	0	0
$529$	−4.21509	−0.183265
$530$	0	0
$531$	−46.3932	−2.01329
$532$	0	0
$533$	−3.79997	−0.164595
$534$	0	0
$535$	−2.80199	−0.121140
$536$	0	0
$537$	−2.92890	−0.126391
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	31.5755	1.35754	0.678768	−	0.734353i	$-0.262514\pi$
$541$	31.5755	1.35754	0.678768	+	0.734353i	$0.262514\pi$
$542$	0	0
$543$	−47.1964	−2.02539
$544$	0	0
$545$	0.344735	0.0147668
$546$	0	0
$547$	−16.2961	−0.696771	−0.348385	−	0.937351i	$-0.613270\pi$
$547$	−16.2961	−0.696771	−0.348385	+	0.937351i	$0.613270\pi$
$548$	0	0
$549$	−9.57651	−0.408715
$550$	0	0
$551$	−3.58477	−0.152716
$552$	0	0
$553$	11.7524	0.499763
$554$	0	0
$555$	1.53309	0.0650760
$556$	0	0
$557$	17.0716	0.723349	0.361674	−	0.932305i	$-0.382205\pi$
$557$	17.0716	0.723349	0.361674	+	0.932305i	$0.382205\pi$
$558$	0	0
$559$	−2.91699	−0.123376
$560$	0	0
$561$	4.40181	0.185845
$562$	0	0
$563$	−32.5537	−1.37197	−0.685987	−	0.727614i	$-0.740629\pi$
$563$	−32.5537	−1.37197	−0.685987	+	0.727614i	$0.740629\pi$
$564$	0	0
$565$	−1.03010	−0.0433366
$566$	0	0
$567$	25.5943	1.07486
$568$	0	0
$569$	−36.9412	−1.54866	−0.774328	−	0.632785i	$-0.781912\pi$
$569$	−36.9412	−1.54866	−0.774328	+	0.632785i	$0.781912\pi$
$570$	0	0
$571$	−27.3288	−1.14368	−0.571838	−	0.820367i	$-0.693769\pi$
$571$	−27.3288	−1.14368	−0.571838	+	0.820367i	$0.693769\pi$
$572$	0	0
$573$	41.4276	1.73066
$574$	0	0
$575$	−21.4691	−0.895324
$576$	0	0
$577$	4.97239	0.207003	0.103502	−	0.994629i	$-0.466995\pi$
$577$	4.97239	0.207003	0.103502	+	0.994629i	$0.466995\pi$
$578$	0	0
$579$	−53.8948	−2.23979
$580$	0	0
$581$	−13.6773	−0.567431
$582$	0	0
$583$	8.94833	0.370602
$584$	0	0
$585$	1.63287	0.0675109
$586$	0	0
$587$	6.25355	0.258112	0.129056	−	0.991637i	$-0.458805\pi$
$587$	6.25355	0.258112	0.129056	+	0.991637i	$0.458805\pi$
$588$	0	0
$589$	46.1235	1.90048
$590$	0	0
$591$	57.3098	2.35741
$592$	0	0
$593$	−5.49143	−0.225506	−0.112753	−	0.993623i	$-0.535967\pi$
$593$	−5.49143	−0.225506	−0.112753	+	0.993623i	$0.535967\pi$
$594$	0	0
$595$	0.292048	0.0119728
$596$	0	0
$597$	26.5888	1.08821
$598$	0	0
$599$	12.4086	0.507001	0.253501	−	0.967335i	$-0.418418\pi$
$599$	12.4086	0.507001	0.253501	+	0.967335i	$0.418418\pi$
$600$	0	0
$601$	−46.9461	−1.91497	−0.957486	−	0.288481i	$-0.906850\pi$
$601$	−46.9461	−1.91497	−0.957486	+	0.288481i	$0.906850\pi$
$602$	0	0
$603$	−77.4923	−3.15573
$604$	0	0
$605$	−0.215704	−0.00876962
$606$	0	0
$607$	−6.31974	−0.256510	−0.128255	−	0.991741i	$-0.540938\pi$
$607$	−6.31974	−0.256510	−0.128255	+	0.991741i	$0.540938\pi$
$608$	0	0
$609$	−2.28336	−0.0925265
$610$	0	0
$611$	−10.5856	−0.428248
$612$	0	0
$613$	3.89919	0.157487	0.0787434	−	0.996895i	$-0.474909\pi$
$613$	3.89919	0.157487	0.0787434	+	0.996895i	$0.474909\pi$
$614$	0	0
$615$	−2.66486	−0.107458
$616$	0	0
$617$	18.3877	0.740260	0.370130	−	0.928980i	$-0.379313\pi$
$617$	18.3877	0.740260	0.370130	+	0.928980i	$0.379313\pi$
$618$	0	0
$619$	−30.8782	−1.24110	−0.620549	−	0.784168i	$-0.713090\pi$
$619$	−30.8782	−1.24110	−0.620549	+	0.784168i	$0.713090\pi$
$620$	0	0
$621$	64.3951	2.58408
$622$	0	0
$623$	6.89690	0.276318
$624$	0	0
$625$	24.3042	0.972170
$626$	0	0
$627$	−16.5943	−0.662713
$628$	0	0
$629$	2.95983	0.118016
$630$	0	0
$631$	−16.0835	−0.640276	−0.320138	−	0.947371i	$-0.603729\pi$
$631$	−16.0835	−0.640276	−0.320138	+	0.947371i	$0.603729\pi$
$632$	0	0
$633$	−58.4396	−2.32277
$634$	0	0
$635$	1.22598	0.0486513
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	83.8163	3.31572
$640$	0	0
$641$	−3.38611	−0.133743	−0.0668716	−	0.997762i	$-0.521302\pi$
$641$	−3.38611	−0.133743	−0.0668716	+	0.997762i	$0.521302\pi$
$642$	0	0
$643$	45.0545	1.77678	0.888388	−	0.459094i	$-0.151826\pi$
$643$	45.0545	1.77678	0.888388	+	0.459094i	$0.151826\pi$
$644$	0	0
$645$	−2.04564	−0.0805471
$646$	0	0
$647$	2.86125	0.112487	0.0562437	−	0.998417i	$-0.482088\pi$
$647$	2.86125	0.112487	0.0562437	+	0.998417i	$0.482088\pi$
$648$	0	0
$649$	6.12859	0.240568
$650$	0	0
$651$	29.3789	1.15145
$652$	0	0
$653$	−23.6377	−0.925013	−0.462506	−	0.886616i	$-0.653050\pi$
$653$	−23.6377	−0.925013	−0.462506	+	0.886616i	$0.653050\pi$
$654$	0	0
$655$	1.48167	0.0578936
$656$	0	0
$657$	−80.4477	−3.13856
$658$	0	0
$659$	7.87286	0.306683	0.153342	−	0.988173i	$-0.450997\pi$
$659$	7.87286	0.306683	0.153342	+	0.988173i	$0.450997\pi$
$660$	0	0
$661$	−33.0419	−1.28518	−0.642591	−	0.766210i	$-0.722140\pi$
$661$	−33.0419	−1.28518	−0.642591	+	0.766210i	$0.722140\pi$
$662$	0	0
$663$	4.40181	0.170952
$664$	0	0
$665$	−1.10098	−0.0426943
$666$	0	0
$667$	−3.04399	−0.117864
$668$	0	0
$669$	69.4693	2.68584
$670$	0	0
$671$	1.26507	0.0488374
$672$	0	0
$673$	13.1226	0.505838	0.252919	−	0.967487i	$-0.418609\pi$
$673$	13.1226	0.505838	0.252919	+	0.967487i	$0.418609\pi$
$674$	0	0
$675$	−73.5966	−2.83273
$676$	0	0
$677$	−8.59089	−0.330175	−0.165087	−	0.986279i	$-0.552791\pi$
$677$	−8.59089	−0.330175	−0.165087	+	0.986279i	$0.552791\pi$
$678$	0	0
$679$	−9.18517	−0.352494
$680$	0	0
$681$	−32.2310	−1.23509
$682$	0	0
$683$	−17.9611	−0.687263	−0.343632	−	0.939105i	$-0.611657\pi$
$683$	−17.9611	−0.687263	−0.343632	+	0.939105i	$0.611657\pi$
$684$	0	0
$685$	−3.24343	−0.123925
$686$	0	0
$687$	−55.3821	−2.11296
$688$	0	0
$689$	8.94833	0.340904
$690$	0	0
$691$	−3.51104	−0.133566	−0.0667831	−	0.997768i	$-0.521274\pi$
$691$	−3.51104	−0.133566	−0.0667831	+	0.997768i	$0.521274\pi$
$692$	0	0
$693$	−7.56995	−0.287559
$694$	0	0
$695$	−1.92704	−0.0730967
$696$	0	0
$697$	−5.14488	−0.194876
$698$	0	0
$699$	−45.4844	−1.72038
$700$	0	0
$701$	−27.1928	−1.02706	−0.513529	−	0.858072i	$-0.671662\pi$
$701$	−27.1928	−1.02706	−0.513529	+	0.858072i	$0.671662\pi$
$702$	0	0
$703$	−11.1582	−0.420840
$704$	0	0
$705$	−7.42355	−0.279587
$706$	0	0
$707$	−10.2727	−0.386343
$708$	0	0
$709$	−29.4670	−1.10666	−0.553328	−	0.832963i	$-0.686642\pi$
$709$	−29.4670	−1.10666	−0.553328	+	0.832963i	$0.686642\pi$
$710$	0	0
$711$	88.9651	3.33645
$712$	0	0
$713$	39.1655	1.46676
$714$	0	0
$715$	−0.215704	−0.00806688
$716$	0	0
$717$	10.7734	0.402339
$718$	0	0
$719$	−52.3195	−1.95119	−0.975593	−	0.219585i	$-0.929530\pi$
$719$	−52.3195	−1.95119	−0.975593	+	0.219585i	$0.929530\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	34.2050	1.27210
$724$	0	0
$725$	3.47895	0.129205
$726$	0	0
$727$	−9.69341	−0.359509	−0.179754	−	0.983712i	$-0.557530\pi$
$727$	−9.69341	−0.359509	−0.179754	+	0.983712i	$0.557530\pi$
$728$	0	0
$729$	48.8352	1.80871
$730$	0	0
$731$	−3.94939	−0.146074
$732$	0	0
$733$	25.6850	0.948696	0.474348	−	0.880337i	$-0.342684\pi$
$733$	25.6850	0.948696	0.474348	+	0.880337i	$0.342684\pi$
$734$	0	0
$735$	−0.701286	−0.0258673
$736$	0	0
$737$	10.2368	0.377079
$738$	0	0
$739$	32.7900	1.20620	0.603100	−	0.797665i	$-0.293932\pi$
$739$	32.7900	1.20620	0.603100	+	0.797665i	$0.293932\pi$
$740$	0	0
$741$	−16.5943	−0.609607
$742$	0	0
$743$	25.6814	0.942158	0.471079	−	0.882091i	$-0.343865\pi$
$743$	25.6814	0.942158	0.471079	+	0.882091i	$0.343865\pi$
$744$	0	0
$745$	−4.01270	−0.147014
$746$	0	0
$747$	−103.537	−3.78821
$748$	0	0
$749$	12.9900	0.474643
$750$	0	0
$751$	−34.6188	−1.26326	−0.631629	−	0.775271i	$-0.717613\pi$
$751$	−34.6188	−1.26326	−0.631629	+	0.775271i	$0.717613\pi$
$752$	0	0
$753$	−35.3855	−1.28952
$754$	0	0
$755$	4.65198	0.169303
$756$	0	0
$757$	−33.9348	−1.23338	−0.616691	−	0.787205i	$-0.711527\pi$
$757$	−33.9348	−1.23338	−0.616691	+	0.787205i	$0.711527\pi$
$758$	0	0
$759$	−14.0910	−0.511470
$760$	0	0
$761$	−42.4495	−1.53879	−0.769396	−	0.638772i	$-0.779443\pi$
$761$	−42.4495	−1.53879	−0.769396	+	0.638772i	$0.779443\pi$
$762$	0	0
$763$	−1.59819	−0.0578582
$764$	0	0
$765$	2.21079	0.0799312
$766$	0	0
$767$	6.12859	0.221291
$768$	0	0
$769$	−23.4062	−0.844050	−0.422025	−	0.906584i	$-0.638680\pi$
$769$	−23.4062	−0.844050	−0.422025	+	0.906584i	$0.638680\pi$
$770$	0	0
$771$	−71.6539	−2.58055
$772$	0	0
$773$	52.7928	1.89883	0.949413	−	0.314031i	$-0.101680\pi$
$773$	52.7928	1.89883	0.949413	+	0.314031i	$0.101680\pi$
$774$	0	0
$775$	−44.7620	−1.60790
$776$	0	0
$777$	−7.10737	−0.254975
$778$	0	0
$779$	19.3956	0.694918
$780$	0	0
$781$	−11.0722	−0.396196
$782$	0	0
$783$	−10.4349	−0.372911
$784$	0	0
$785$	−2.78252	−0.0993126
$786$	0	0
$787$	45.2805	1.61408	0.807039	−	0.590499i	$-0.201069\pi$
$787$	45.2805	1.61408	0.807039	+	0.590499i	$0.201069\pi$
$788$	0	0
$789$	−12.6395	−0.449979
$790$	0	0
$791$	4.77551	0.169798
$792$	0	0
$793$	1.26507	0.0449239
$794$	0	0
$795$	6.27534	0.222563
$796$	0	0
$797$	−17.2169	−0.609855	−0.304927	−	0.952376i	$-0.598632\pi$
$797$	−17.2169	−0.609855	−0.304927	+	0.952376i	$0.598632\pi$
$798$	0	0
$799$	−14.3322	−0.507035
$800$	0	0
$801$	52.2092	1.84472
$802$	0	0
$803$	10.6272	0.375027
$804$	0	0
$805$	−0.934896	−0.0329507
$806$	0	0
$807$	−68.0786	−2.39648
$808$	0	0
$809$	−37.0834	−1.30378	−0.651892	−	0.758312i	$-0.726024\pi$
$809$	−37.0834	−1.30378	−0.651892	+	0.758312i	$0.726024\pi$
$810$	0	0
$811$	−39.9009	−1.40111	−0.700555	−	0.713599i	$-0.747064\pi$
$811$	−39.9009	−1.40111	−0.700555	+	0.713599i	$0.747064\pi$
$812$	0	0
$813$	53.9448	1.89193
$814$	0	0
$815$	−4.48005	−0.156929
$816$	0	0
$817$	14.8887	0.520891
$818$	0	0
$819$	−7.56995	−0.264515
$820$	0	0
$821$	−23.1435	−0.807713	−0.403856	−	0.914822i	$-0.632330\pi$
$821$	−23.1435	−0.807713	−0.403856	+	0.914822i	$0.632330\pi$
$822$	0	0
$823$	11.0615	0.385581	0.192791	−	0.981240i	$-0.438246\pi$
$823$	11.0615	0.385581	0.192791	+	0.981240i	$0.438246\pi$
$824$	0	0
$825$	16.1045	0.560685
$826$	0	0
$827$	41.4752	1.44223	0.721117	−	0.692813i	$-0.243629\pi$
$827$	41.4752	1.44223	0.721117	+	0.692813i	$0.243629\pi$
$828$	0	0
$829$	−32.8401	−1.14058	−0.570292	−	0.821442i	$-0.693170\pi$
$829$	−32.8401	−1.14058	−0.570292	+	0.821442i	$0.693170\pi$
$830$	0	0
$831$	45.9590	1.59430
$832$	0	0
$833$	−1.35393	−0.0469108
$834$	0	0
$835$	−5.18184	−0.179325
$836$	0	0
$837$	134.260	4.64071
$838$	0	0
$839$	−53.0612	−1.83188	−0.915938	−	0.401319i	$-0.868552\pi$
$839$	−53.0612	−1.83188	−0.915938	+	0.401319i	$0.868552\pi$
$840$	0	0
$841$	−28.5067	−0.982991
$842$	0	0
$843$	27.6104	0.950951
$844$	0	0
$845$	−0.215704	−0.00742045
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	37.1417	1.27470
$850$	0	0
$851$	−9.47495	−0.324797
$852$	0	0
$853$	50.0001	1.71197	0.855985	−	0.517000i	$-0.172951\pi$
$853$	50.0001	1.71197	0.855985	+	0.517000i	$0.172951\pi$
$854$	0	0
$855$	−8.33440	−0.285030
$856$	0	0
$857$	−18.5101	−0.632294	−0.316147	−	0.948710i	$-0.602389\pi$
$857$	−18.5101	−0.632294	−0.316147	+	0.948710i	$0.602389\pi$
$858$	0	0
$859$	−16.3202	−0.556838	−0.278419	−	0.960460i	$-0.589810\pi$
$859$	−16.3202	−0.556838	−0.278419	+	0.960460i	$0.589810\pi$
$860$	0	0
$861$	12.3542	0.421032
$862$	0	0
$863$	29.9243	1.01863	0.509317	−	0.860579i	$-0.329898\pi$
$863$	29.9243	1.01863	0.509317	+	0.860579i	$0.329898\pi$
$864$	0	0
$865$	2.13557	0.0726115
$866$	0	0
$867$	−49.3097	−1.67465
$868$	0	0
$869$	−11.7524	−0.398673
$870$	0	0
$871$	10.2368	0.346862
$872$	0	0
$873$	−69.5313	−2.35328
$874$	0	0
$875$	2.14701	0.0725820
$876$	0	0
$877$	−50.8390	−1.71671	−0.858355	−	0.513056i	$-0.828513\pi$
$877$	−50.8390	−1.71671	−0.858355	+	0.513056i	$0.828513\pi$
$878$	0	0
$879$	−92.8660	−3.13229
$880$	0	0
$881$	45.9576	1.54835	0.774176	−	0.632971i	$-0.218165\pi$
$881$	45.9576	1.54835	0.774176	+	0.632971i	$0.218165\pi$
$882$	0	0
$883$	−24.6131	−0.828298	−0.414149	−	0.910209i	$-0.635921\pi$
$883$	−24.6131	−0.828298	−0.414149	+	0.910209i	$0.635921\pi$
$884$	0	0
$885$	4.29790	0.144472
$886$	0	0
$887$	−13.3357	−0.447770	−0.223885	−	0.974616i	$-0.571874\pi$
$887$	−13.3357	−0.447770	−0.223885	+	0.974616i	$0.571874\pi$
$888$	0	0
$889$	−5.68359	−0.190622
$890$	0	0
$891$	−25.5943	−0.857442
$892$	0	0
$893$	54.0305	1.80806
$894$	0	0
$895$	0.194324	0.00649554
$896$	0	0
$897$	−14.0910	−0.470484
$898$	0	0
$899$	−6.34655	−0.211669
$900$	0	0
$901$	12.1154	0.403622
$902$	0	0
$903$	9.48356	0.315593
$904$	0	0
$905$	3.13135	0.104089
$906$	0	0
$907$	−33.8554	−1.12415	−0.562074	−	0.827087i	$-0.689997\pi$
$907$	−33.8554	−1.12415	−0.562074	+	0.827087i	$0.689997\pi$
$908$	0	0
$909$	−77.7636	−2.57925
$910$	0	0
$911$	−6.15681	−0.203984	−0.101992	−	0.994785i	$-0.532522\pi$
$911$	−6.15681	−0.203984	−0.101992	+	0.994785i	$0.532522\pi$
$912$	0	0
$913$	13.6773	0.452653
$914$	0	0
$915$	0.887175	0.0293291
$916$	0	0
$917$	−6.86899	−0.226834
$918$	0	0
$919$	16.1255	0.531930	0.265965	−	0.963983i	$-0.414309\pi$
$919$	16.1255	0.531930	0.265965	+	0.963983i	$0.414309\pi$
$920$	0	0
$921$	−49.8260	−1.64182
$922$	0	0
$923$	−11.0722	−0.364447
$924$	0	0
$925$	10.8288	0.356050
$926$	0	0
$927$	60.5596	1.98904
$928$	0	0
$929$	−10.5610	−0.346495	−0.173247	−	0.984878i	$-0.555426\pi$
$929$	−10.5610	−0.346495	−0.173247	+	0.984878i	$0.555426\pi$
$930$	0	0
$931$	5.10414	0.167281
$932$	0	0
$933$	54.3595	1.77965
$934$	0	0
$935$	−0.292048	−0.00955098
$936$	0	0
$937$	21.5682	0.704603	0.352302	−	0.935887i	$-0.385399\pi$
$937$	21.5682	0.704603	0.352302	+	0.935887i	$0.385399\pi$
$938$	0	0
$939$	−72.0739	−2.35204
$940$	0	0
$941$	12.6688	0.412991	0.206496	−	0.978448i	$-0.433794\pi$
$941$	12.6688	0.412991	0.206496	+	0.978448i	$0.433794\pi$
$942$	0	0
$943$	16.4696	0.536326
$944$	0	0
$945$	−3.20484	−0.104254
$946$	0	0
$947$	11.0177	0.358026	0.179013	−	0.983847i	$-0.442710\pi$
$947$	11.0177	0.358026	0.179013	+	0.983847i	$0.442710\pi$
$948$	0	0
$949$	10.6272	0.344975
$950$	0	0
$951$	13.3633	0.433335
$952$	0	0
$953$	50.3089	1.62967	0.814833	−	0.579695i	$-0.196828\pi$
$953$	50.3089	1.62967	0.814833	+	0.579695i	$0.196828\pi$
$954$	0	0
$955$	−2.74860	−0.0889427
$956$	0	0
$957$	2.28336	0.0738106
$958$	0	0
$959$	15.0365	0.485553
$960$	0	0
$961$	50.6580	1.63413
$962$	0	0
$963$	98.3333	3.16875
$964$	0	0
$965$	3.57576	0.115108
$966$	0	0
$967$	−50.5440	−1.62539	−0.812694	−	0.582691i	$-0.802000\pi$
$967$	−50.5440	−1.62539	−0.812694	+	0.582691i	$0.802000\pi$
$968$	0	0
$969$	−22.4675	−0.721760
$970$	0	0
$971$	−11.0514	−0.354657	−0.177328	−	0.984152i	$-0.556745\pi$
$971$	−11.0514	−0.354657	−0.177328	+	0.984152i	$0.556745\pi$
$972$	0	0
$973$	8.93370	0.286401
$974$	0	0
$975$	16.1045	0.515755
$976$	0	0
$977$	−46.4909	−1.48737	−0.743687	−	0.668528i	$-0.766925\pi$
$977$	−46.4909	−1.48737	−0.743687	+	0.668528i	$0.766925\pi$
$978$	0	0
$979$	−6.89690	−0.220426
$980$	0	0
$981$	−12.0982	−0.386265
$982$	0	0
$983$	42.7163	1.36244	0.681219	−	0.732080i	$-0.261450\pi$
$983$	42.7163	1.36244	0.681219	+	0.732080i	$0.261450\pi$
$984$	0	0
$985$	−3.80234	−0.121153
$986$	0	0
$987$	34.4154	1.09545
$988$	0	0
$989$	12.6427	0.402014
$990$	0	0
$991$	−33.2073	−1.05487	−0.527433	−	0.849597i	$-0.676845\pi$
$991$	−33.2073	−1.05487	−0.527433	+	0.849597i	$0.676845\pi$
$992$	0	0
$993$	−29.8858	−0.948398
$994$	0	0
$995$	−1.76409	−0.0559254
$996$	0	0
$997$	−27.7407	−0.878556	−0.439278	−	0.898351i	$-0.644766\pi$
$997$	−27.7407	−0.878556	−0.439278	+	0.898351i	$0.644766\pi$
$998$	0	0
$999$	−32.4803	−1.02763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.k.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.k.1.6	✓	6	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+3.25115 q^{3} -0.215704 q^{5} +1.00000 q^{7} +7.56995 q^{9} +O(q^{10})\)	\(q+3.25115 q^{3} -0.215704 q^{5} +1.00000 q^{7} +7.56995 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.701286 q^{15} -1.35393 q^{17} +5.10414 q^{19} +3.25115 q^{21} +4.33416 q^{23} -4.95347 q^{25} +14.8576 q^{27} -0.702326 q^{29} +9.03648 q^{31} -3.25115 q^{33} -0.215704 q^{35} -2.18611 q^{37} -3.25115 q^{39} +3.79997 q^{41} +2.91699 q^{43} -1.63287 q^{45} +10.5856 q^{47} +1.00000 q^{49} -4.40181 q^{51} -8.94833 q^{53} +0.215704 q^{55} +16.5943 q^{57} -6.12859 q^{59} -1.26507 q^{61} +7.56995 q^{63} +0.215704 q^{65} -10.2368 q^{67} +14.0910 q^{69} +11.0722 q^{71} -10.6272 q^{73} -16.1045 q^{75} -1.00000 q^{77} +11.7524 q^{79} +25.5943 q^{81} -13.6773 q^{83} +0.292048 q^{85} -2.28336 q^{87} +6.89690 q^{89} -1.00000 q^{91} +29.3789 q^{93} -1.10098 q^{95} -9.18517 q^{97} -7.56995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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