LMFDB - Embedded newform 4024.2.a.f.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05780 q^{3} -1.22779 q^{5} +1.00021 q^{7} +1.23452 q^{9} +O(q^{10})$	$q-2.05780 q^{3} -1.22779 q^{5} +1.00021 q^{7} +1.23452 q^{9} -1.98481 q^{11} -6.56514 q^{13} +2.52655 q^{15} -4.79710 q^{17} +4.50804 q^{19} -2.05822 q^{21} -7.49395 q^{23} -3.49252 q^{25} +3.63299 q^{27} -1.71329 q^{29} -6.65018 q^{31} +4.08432 q^{33} -1.22805 q^{35} +8.10380 q^{37} +13.5097 q^{39} +9.42468 q^{41} -1.99402 q^{43} -1.51574 q^{45} -3.28783 q^{47} -5.99959 q^{49} +9.87145 q^{51} +10.2835 q^{53} +2.43693 q^{55} -9.27663 q^{57} -9.00727 q^{59} -4.23679 q^{61} +1.23478 q^{63} +8.06064 q^{65} -11.4352 q^{67} +15.4210 q^{69} +5.48836 q^{71} +2.80051 q^{73} +7.18689 q^{75} -1.98521 q^{77} -12.0281 q^{79} -11.1795 q^{81} -2.22574 q^{83} +5.88986 q^{85} +3.52559 q^{87} -6.27120 q^{89} -6.56649 q^{91} +13.6847 q^{93} -5.53495 q^{95} -15.5350 q^{97} -2.45028 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * 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
$3$	−2.05780	−1.18807	−0.594034	−	0.804440i	$-0.702466\pi$
$3$	−2.05780	−1.18807	−0.594034	+	0.804440i	$0.702466\pi$
$4$	0	0
$5$	−1.22779	−0.549086	−0.274543	−	0.961575i	$-0.588527\pi$
$5$	−1.22779	−0.549086	−0.274543	+	0.961575i	$0.588527\pi$
$6$	0	0
$7$	1.00021	0.378042	0.189021	−	0.981973i	$-0.439469\pi$
$7$	1.00021	0.378042	0.189021	+	0.981973i	$0.439469\pi$
$8$	0	0
$9$	1.23452	0.411507
$10$	0	0
$11$	−1.98481	−0.598441	−0.299221	−	0.954184i	$-0.596727\pi$
$11$	−1.98481	−0.598441	−0.299221	+	0.954184i	$0.596727\pi$
$12$	0	0
$13$	−6.56514	−1.82084	−0.910421	−	0.413683i	$-0.864242\pi$
$13$	−6.56514	−1.82084	−0.910421	+	0.413683i	$0.864242\pi$
$14$	0	0
$15$	2.52655	0.652352
$16$	0	0
$17$	−4.79710	−1.16347	−0.581734	−	0.813379i	$-0.697626\pi$
$17$	−4.79710	−1.16347	−0.581734	+	0.813379i	$0.697626\pi$
$18$	0	0
$19$	4.50804	1.03422	0.517108	−	0.855920i	$-0.327009\pi$
$19$	4.50804	1.03422	0.517108	+	0.855920i	$0.327009\pi$
$20$	0	0
$21$	−2.05822	−0.449140
$22$	0	0
$23$	−7.49395	−1.56260	−0.781298	−	0.624158i	$-0.785442\pi$
$23$	−7.49395	−1.56260	−0.781298	+	0.624158i	$0.785442\pi$
$24$	0	0
$25$	−3.49252	−0.698504
$26$	0	0
$27$	3.63299	0.699170
$28$	0	0
$29$	−1.71329	−0.318149	−0.159075	−	0.987267i	$-0.550851\pi$
$29$	−1.71329	−0.318149	−0.159075	+	0.987267i	$0.550851\pi$
$30$	0	0
$31$	−6.65018	−1.19441	−0.597204	−	0.802089i	$-0.703722\pi$
$31$	−6.65018	−1.19441	−0.597204	+	0.802089i	$0.703722\pi$
$32$	0	0
$33$	4.08432	0.710989
$34$	0	0
$35$	−1.22805	−0.207578
$36$	0	0
$37$	8.10380	1.33226	0.666128	−	0.745837i	$-0.267950\pi$
$37$	8.10380	1.33226	0.666128	+	0.745837i	$0.267950\pi$
$38$	0	0
$39$	13.5097	2.16329
$40$	0	0
$41$	9.42468	1.47189	0.735944	−	0.677043i	$-0.236739\pi$
$41$	9.42468	1.47189	0.735944	+	0.677043i	$0.236739\pi$
$42$	0	0
$43$	−1.99402	−0.304085	−0.152043	−	0.988374i	$-0.548585\pi$
$43$	−1.99402	−0.304085	−0.152043	+	0.988374i	$0.548585\pi$
$44$	0	0
$45$	−1.51574	−0.225953
$46$	0	0
$47$	−3.28783	−0.479579	−0.239790	−	0.970825i	$-0.577078\pi$
$47$	−3.28783	−0.479579	−0.239790	+	0.970825i	$0.577078\pi$
$48$	0	0
$49$	−5.99959	−0.857084
$50$	0	0
$51$	9.87145	1.38228
$52$	0	0
$53$	10.2835	1.41255	0.706274	−	0.707939i	$-0.250375\pi$
$53$	10.2835	1.41255	0.706274	+	0.707939i	$0.250375\pi$
$54$	0	0
$55$	2.43693	0.328596
$56$	0	0
$57$	−9.27663	−1.22872
$58$	0	0
$59$	−9.00727	−1.17265	−0.586324	−	0.810077i	$-0.699425\pi$
$59$	−9.00727	−1.17265	−0.586324	+	0.810077i	$0.699425\pi$
$60$	0	0
$61$	−4.23679	−0.542465	−0.271232	−	0.962514i	$-0.587431\pi$
$61$	−4.23679	−0.542465	−0.271232	+	0.962514i	$0.587431\pi$
$62$	0	0
$63$	1.23478	0.155567
$64$	0	0
$65$	8.06064	0.999799
$66$	0	0
$67$	−11.4352	−1.39704	−0.698518	−	0.715592i	$-0.746157\pi$
$67$	−11.4352	−1.39704	−0.698518	+	0.715592i	$0.746157\pi$
$68$	0	0
$69$	15.4210	1.85647
$70$	0	0
$71$	5.48836	0.651348	0.325674	−	0.945482i	$-0.394409\pi$
$71$	5.48836	0.651348	0.325674	+	0.945482i	$0.394409\pi$
$72$	0	0
$73$	2.80051	0.327775	0.163888	−	0.986479i	$-0.447597\pi$
$73$	2.80051	0.327775	0.163888	+	0.986479i	$0.447597\pi$
$74$	0	0
$75$	7.18689	0.829871
$76$	0	0
$77$	−1.98521	−0.226236
$78$	0	0
$79$	−12.0281	−1.35327	−0.676636	−	0.736318i	$-0.736563\pi$
$79$	−12.0281	−1.35327	−0.676636	+	0.736318i	$0.736563\pi$
$80$	0	0
$81$	−11.1795	−1.24217
$82$	0	0
$83$	−2.22574	−0.244307	−0.122153	−	0.992511i	$-0.538980\pi$
$83$	−2.22574	−0.244307	−0.122153	+	0.992511i	$0.538980\pi$
$84$	0	0
$85$	5.88986	0.638844
$86$	0	0
$87$	3.52559	0.377983
$88$	0	0
$89$	−6.27120	−0.664746	−0.332373	−	0.943148i	$-0.607849\pi$
$89$	−6.27120	−0.664746	−0.332373	+	0.943148i	$0.607849\pi$
$90$	0	0
$91$	−6.56649	−0.688355
$92$	0	0
$93$	13.6847	1.41904
$94$	0	0
$95$	−5.53495	−0.567874
$96$	0	0
$97$	−15.5350	−1.57734	−0.788670	−	0.614816i	$-0.789230\pi$
$97$	−15.5350	−1.57734	−0.788670	+	0.614816i	$0.789230\pi$
$98$	0	0
$99$	−2.45028	−0.246263
$100$	0	0
$101$	12.6225	1.25599	0.627995	−	0.778217i	$-0.283876\pi$
$101$	12.6225	1.25599	0.627995	+	0.778217i	$0.283876\pi$
$102$	0	0
$103$	−8.13734	−0.801796	−0.400898	−	0.916123i	$-0.631302\pi$
$103$	−8.13734	−0.801796	−0.400898	+	0.916123i	$0.631302\pi$
$104$	0	0
$105$	2.52707	0.246617
$106$	0	0
$107$	−9.13591	−0.883202	−0.441601	−	0.897211i	$-0.645589\pi$
$107$	−9.13591	−0.883202	−0.441601	+	0.897211i	$0.645589\pi$
$108$	0	0
$109$	12.0175	1.15107	0.575536	−	0.817776i	$-0.304793\pi$
$109$	12.0175	1.15107	0.575536	+	0.817776i	$0.304793\pi$
$110$	0	0
$111$	−16.6760	−1.58281
$112$	0	0
$113$	19.6383	1.84741	0.923706	−	0.383103i	$-0.125145\pi$
$113$	19.6383	1.84741	0.923706	+	0.383103i	$0.125145\pi$
$114$	0	0
$115$	9.20103	0.858001
$116$	0	0
$117$	−8.10480	−0.749289
$118$	0	0
$119$	−4.79809	−0.439840
$120$	0	0
$121$	−7.06055	−0.641868
$122$	0	0
$123$	−19.3941	−1.74870
$124$	0	0
$125$	10.4271	0.932626
$126$	0	0
$127$	2.50466	0.222253	0.111126	−	0.993806i	$-0.464554\pi$
$127$	2.50466	0.222253	0.111126	+	0.993806i	$0.464554\pi$
$128$	0	0
$129$	4.10328	0.361274
$130$	0	0
$131$	9.29553	0.812154	0.406077	−	0.913839i	$-0.366896\pi$
$131$	9.29553	0.812154	0.406077	+	0.913839i	$0.366896\pi$
$132$	0	0
$133$	4.50897	0.390977
$134$	0	0
$135$	−4.46057	−0.383905
$136$	0	0
$137$	12.0055	1.02570	0.512852	−	0.858477i	$-0.328589\pi$
$137$	12.0055	1.02570	0.512852	+	0.858477i	$0.328589\pi$
$138$	0	0
$139$	−6.95204	−0.589664	−0.294832	−	0.955549i	$-0.595264\pi$
$139$	−6.95204	−0.589664	−0.294832	+	0.955549i	$0.595264\pi$
$140$	0	0
$141$	6.76568	0.569773
$142$	0	0
$143$	13.0305	1.08967
$144$	0	0
$145$	2.10356	0.174691
$146$	0	0
$147$	12.3459	1.01827
$148$	0	0
$149$	23.6152	1.93463	0.967316	−	0.253573i	$-0.0816060\pi$
$149$	23.6152	1.93463	0.967316	+	0.253573i	$0.0816060\pi$
$150$	0	0
$151$	18.7830	1.52854	0.764268	−	0.644898i	$-0.223100\pi$
$151$	18.7830	1.52854	0.764268	+	0.644898i	$0.223100\pi$
$152$	0	0
$153$	−5.92213	−0.478775
$154$	0	0
$155$	8.16506	0.655833
$156$	0	0
$157$	−11.3959	−0.909492	−0.454746	−	0.890621i	$-0.650270\pi$
$157$	−11.3959	−0.909492	−0.454746	+	0.890621i	$0.650270\pi$
$158$	0	0
$159$	−21.1613	−1.67820
$160$	0	0
$161$	−7.49550	−0.590728
$162$	0	0
$163$	−8.13814	−0.637428	−0.318714	−	0.947851i	$-0.603251\pi$
$163$	−8.13814	−0.637428	−0.318714	+	0.947851i	$0.603251\pi$
$164$	0	0
$165$	−5.01471	−0.390395
$166$	0	0
$167$	−3.46211	−0.267906	−0.133953	−	0.990988i	$-0.542767\pi$
$167$	−3.46211	−0.267906	−0.133953	+	0.990988i	$0.542767\pi$
$168$	0	0
$169$	30.1010	2.31546
$170$	0	0
$171$	5.56528	0.425587
$172$	0	0
$173$	−4.47063	−0.339895	−0.169948	−	0.985453i	$-0.554360\pi$
$173$	−4.47063	−0.339895	−0.169948	+	0.985453i	$0.554360\pi$
$174$	0	0
$175$	−3.49324	−0.264064
$176$	0	0
$177$	18.5351	1.39319
$178$	0	0
$179$	23.1982	1.73391	0.866956	−	0.498384i	$-0.166073\pi$
$179$	23.1982	1.73391	0.866956	+	0.498384i	$0.166073\pi$
$180$	0	0
$181$	22.5075	1.67297	0.836486	−	0.547989i	$-0.184606\pi$
$181$	22.5075	1.67297	0.836486	+	0.547989i	$0.184606\pi$
$182$	0	0
$183$	8.71844	0.644485
$184$	0	0
$185$	−9.94980	−0.731524
$186$	0	0
$187$	9.52131	0.696267
$188$	0	0
$189$	3.63374	0.264316
$190$	0	0
$191$	−15.0215	−1.08691	−0.543457	−	0.839437i	$-0.682885\pi$
$191$	−15.0215	−1.08691	−0.543457	+	0.839437i	$0.682885\pi$
$192$	0	0
$193$	−7.46441	−0.537300	−0.268650	−	0.963238i	$-0.586577\pi$
$193$	−7.46441	−0.537300	−0.268650	+	0.963238i	$0.586577\pi$
$194$	0	0
$195$	−16.5871	−1.18783
$196$	0	0
$197$	−9.91985	−0.706760	−0.353380	−	0.935480i	$-0.614968\pi$
$197$	−9.91985	−0.706760	−0.353380	+	0.935480i	$0.614968\pi$
$198$	0	0
$199$	−12.4257	−0.880836	−0.440418	−	0.897793i	$-0.645170\pi$
$199$	−12.4257	−0.880836	−0.440418	+	0.897793i	$0.645170\pi$
$200$	0	0
$201$	23.5314	1.65978
$202$	0	0
$203$	−1.71364	−0.120274
$204$	0	0
$205$	−11.5716	−0.808193
$206$	0	0
$207$	−9.25144	−0.643020
$208$	0	0
$209$	−8.94759	−0.618917
$210$	0	0
$211$	21.8966	1.50743	0.753714	−	0.657203i	$-0.228261\pi$
$211$	21.8966	1.50743	0.753714	+	0.657203i	$0.228261\pi$
$212$	0	0
$213$	−11.2939	−0.773847
$214$	0	0
$215$	2.44825	0.166969
$216$	0	0
$217$	−6.65156	−0.451537
$218$	0	0
$219$	−5.76288	−0.389419
$220$	0	0
$221$	31.4936	2.11849
$222$	0	0
$223$	27.3874	1.83399	0.916997	−	0.398895i	$-0.130606\pi$
$223$	27.3874	1.83399	0.916997	+	0.398895i	$0.130606\pi$
$224$	0	0
$225$	−4.31159	−0.287439
$226$	0	0
$227$	−1.10634	−0.0734306	−0.0367153	−	0.999326i	$-0.511689\pi$
$227$	−1.10634	−0.0734306	−0.0367153	+	0.999326i	$0.511689\pi$
$228$	0	0
$229$	−27.6964	−1.83023	−0.915116	−	0.403190i	$-0.867902\pi$
$229$	−27.6964	−1.83023	−0.915116	+	0.403190i	$0.867902\pi$
$230$	0	0
$231$	4.08516	0.268784
$232$	0	0
$233$	−9.47804	−0.620927	−0.310464	−	0.950585i	$-0.600484\pi$
$233$	−9.47804	−0.620927	−0.310464	+	0.950585i	$0.600484\pi$
$234$	0	0
$235$	4.03678	0.263330
$236$	0	0
$237$	24.7514	1.60778
$238$	0	0
$239$	−18.9212	−1.22391	−0.611955	−	0.790893i	$-0.709617\pi$
$239$	−18.9212	−1.22391	−0.611955	+	0.790893i	$0.709617\pi$
$240$	0	0
$241$	9.80590	0.631654	0.315827	−	0.948817i	$-0.397718\pi$
$241$	9.80590	0.631654	0.315827	+	0.948817i	$0.397718\pi$
$242$	0	0
$243$	12.1062	0.776612
$244$	0	0
$245$	7.36626	0.470613
$246$	0	0
$247$	−29.5959	−1.88314
$248$	0	0
$249$	4.58011	0.290253
$250$	0	0
$251$	16.5685	1.04580	0.522898	−	0.852395i	$-0.324851\pi$
$251$	16.5685	1.04580	0.522898	+	0.852395i	$0.324851\pi$
$252$	0	0
$253$	14.8740	0.935122
$254$	0	0
$255$	−12.1201	−0.758991
$256$	0	0
$257$	−21.6205	−1.34865	−0.674326	−	0.738434i	$-0.735566\pi$
$257$	−21.6205	−1.34865	−0.674326	+	0.738434i	$0.735566\pi$
$258$	0	0
$259$	8.10547	0.503649
$260$	0	0
$261$	−2.11509	−0.130921
$262$	0	0
$263$	27.0403	1.66738	0.833689	−	0.552235i	$-0.186225\pi$
$263$	27.0403	1.66738	0.833689	+	0.552235i	$0.186225\pi$
$264$	0	0
$265$	−12.6260	−0.775611
$266$	0	0
$267$	12.9049	0.789764
$268$	0	0
$269$	18.3234	1.11720	0.558598	−	0.829439i	$-0.311340\pi$
$269$	18.3234	1.11720	0.558598	+	0.829439i	$0.311340\pi$
$270$	0	0
$271$	17.4164	1.05797	0.528984	−	0.848632i	$-0.322573\pi$
$271$	17.4164	1.05797	0.528984	+	0.848632i	$0.322573\pi$
$272$	0	0
$273$	13.5125	0.817814
$274$	0	0
$275$	6.93197	0.418014
$276$	0	0
$277$	−28.3750	−1.70489	−0.852444	−	0.522819i	$-0.824880\pi$
$277$	−28.3750	−1.70489	−0.852444	+	0.522819i	$0.824880\pi$
$278$	0	0
$279$	−8.20980	−0.491508
$280$	0	0
$281$	−2.79248	−0.166585	−0.0832927	−	0.996525i	$-0.526544\pi$
$281$	−2.79248	−0.166585	−0.0832927	+	0.996525i	$0.526544\pi$
$282$	0	0
$283$	7.37810	0.438582	0.219291	−	0.975659i	$-0.429626\pi$
$283$	7.37810	0.438582	0.219291	+	0.975659i	$0.429626\pi$
$284$	0	0
$285$	11.3898	0.674673
$286$	0	0
$287$	9.42662	0.556436
$288$	0	0
$289$	6.01218	0.353658
$290$	0	0
$291$	31.9679	1.87399
$292$	0	0
$293$	29.1778	1.70459	0.852294	−	0.523063i	$-0.175211\pi$
$293$	29.1778	1.70459	0.852294	+	0.523063i	$0.175211\pi$
$294$	0	0
$295$	11.0591	0.643885
$296$	0	0
$297$	−7.21078	−0.418412
$298$	0	0
$299$	49.1988	2.84524
$300$	0	0
$301$	−1.99443	−0.114957
$302$	0	0
$303$	−25.9746	−1.49220
$304$	0	0
$305$	5.20190	0.297860
$306$	0	0
$307$	−2.88570	−0.164696	−0.0823479	−	0.996604i	$-0.526242\pi$
$307$	−2.88570	−0.164696	−0.0823479	+	0.996604i	$0.526242\pi$
$308$	0	0
$309$	16.7450	0.952588
$310$	0	0
$311$	3.92898	0.222792	0.111396	−	0.993776i	$-0.464468\pi$
$311$	3.92898	0.222792	0.111396	+	0.993776i	$0.464468\pi$
$312$	0	0
$313$	28.8210	1.62906	0.814530	−	0.580121i	$-0.196995\pi$
$313$	28.8210	1.62906	0.814530	+	0.580121i	$0.196995\pi$
$314$	0	0
$315$	−1.51605	−0.0854198
$316$	0	0
$317$	−3.79330	−0.213053	−0.106527	−	0.994310i	$-0.533973\pi$
$317$	−3.79330	−0.213053	−0.106527	+	0.994310i	$0.533973\pi$
$318$	0	0
$319$	3.40054	0.190394
$320$	0	0
$321$	18.7998	1.04930
$322$	0	0
$323$	−21.6255	−1.20328
$324$	0	0
$325$	22.9289	1.27187
$326$	0	0
$327$	−24.7296	−1.36755
$328$	0	0
$329$	−3.28851	−0.181301
$330$	0	0
$331$	28.4015	1.56109	0.780544	−	0.625100i	$-0.214942\pi$
$331$	28.4015	1.56109	0.780544	+	0.625100i	$0.214942\pi$
$332$	0	0
$333$	10.0043	0.548233
$334$	0	0
$335$	14.0401	0.767094
$336$	0	0
$337$	7.73676	0.421448	0.210724	−	0.977546i	$-0.432418\pi$
$337$	7.73676	0.421448	0.210724	+	0.977546i	$0.432418\pi$
$338$	0	0
$339$	−40.4115	−2.19485
$340$	0	0
$341$	13.1993	0.714783
$342$	0	0
$343$	−13.0023	−0.702057
$344$	0	0
$345$	−18.9338	−1.01936
$346$	0	0
$347$	−12.5331	−0.672815	−0.336407	−	0.941717i	$-0.609212\pi$
$347$	−12.5331	−0.672815	−0.336407	+	0.941717i	$0.609212\pi$
$348$	0	0
$349$	9.81542	0.525407	0.262704	−	0.964877i	$-0.415386\pi$
$349$	9.81542	0.525407	0.262704	+	0.964877i	$0.415386\pi$
$350$	0	0
$351$	−23.8511	−1.27308
$352$	0	0
$353$	−10.6031	−0.564345	−0.282173	−	0.959364i	$-0.591055\pi$
$353$	−10.6031	−0.564345	−0.282173	+	0.959364i	$0.591055\pi$
$354$	0	0
$355$	−6.73858	−0.357647
$356$	0	0
$357$	9.87349	0.522560
$358$	0	0
$359$	−23.5192	−1.24130	−0.620648	−	0.784089i	$-0.713131\pi$
$359$	−23.5192	−1.24130	−0.620648	+	0.784089i	$0.713131\pi$
$360$	0	0
$361$	1.32245	0.0696025
$362$	0	0
$363$	14.5292	0.762583
$364$	0	0
$365$	−3.43845	−0.179977
$366$	0	0
$367$	−4.76121	−0.248533	−0.124267	−	0.992249i	$-0.539658\pi$
$367$	−4.76121	−0.248533	−0.124267	+	0.992249i	$0.539658\pi$
$368$	0	0
$369$	11.6350	0.605692
$370$	0	0
$371$	10.2856	0.534003
$372$	0	0
$373$	4.04969	0.209685	0.104843	−	0.994489i	$-0.466566\pi$
$373$	4.04969	0.209685	0.104843	+	0.994489i	$0.466566\pi$
$374$	0	0
$375$	−21.4568	−1.10802
$376$	0	0
$377$	11.2480	0.579300
$378$	0	0
$379$	−16.5201	−0.848582	−0.424291	−	0.905526i	$-0.639477\pi$
$379$	−16.5201	−0.848582	−0.424291	+	0.905526i	$0.639477\pi$
$380$	0	0
$381$	−5.15408	−0.264052
$382$	0	0
$383$	13.1686	0.672883	0.336441	−	0.941704i	$-0.390777\pi$
$383$	13.1686	0.672883	0.336441	+	0.941704i	$0.390777\pi$
$384$	0	0
$385$	2.43744	0.124223
$386$	0	0
$387$	−2.46166	−0.125133
$388$	0	0
$389$	8.57861	0.434953	0.217476	−	0.976066i	$-0.430217\pi$
$389$	8.57861	0.434953	0.217476	+	0.976066i	$0.430217\pi$
$390$	0	0
$391$	35.9492	1.81803
$392$	0	0
$393$	−19.1283	−0.964895
$394$	0	0
$395$	14.7681	0.743063
$396$	0	0
$397$	3.38853	0.170066	0.0850328	−	0.996378i	$-0.472900\pi$
$397$	3.38853	0.170066	0.0850328	+	0.996378i	$0.472900\pi$
$398$	0	0
$399$	−9.27854	−0.464508
$400$	0	0
$401$	−16.6411	−0.831019	−0.415509	−	0.909589i	$-0.636397\pi$
$401$	−16.6411	−0.831019	−0.415509	+	0.909589i	$0.636397\pi$
$402$	0	0
$403$	43.6594	2.17483
$404$	0	0
$405$	13.7262	0.682058
$406$	0	0
$407$	−16.0845	−0.797277
$408$	0	0
$409$	−12.4340	−0.614824	−0.307412	−	0.951577i	$-0.599463\pi$
$409$	−12.4340	−0.614824	−0.307412	+	0.951577i	$0.599463\pi$
$410$	0	0
$411$	−24.7050	−1.21861
$412$	0	0
$413$	−9.00913	−0.443310
$414$	0	0
$415$	2.73275	0.134145
$416$	0	0
$417$	14.3059	0.700561
$418$	0	0
$419$	−19.8857	−0.971481	−0.485740	−	0.874103i	$-0.661450\pi$
$419$	−19.8857	−0.971481	−0.485740	+	0.874103i	$0.661450\pi$
$420$	0	0
$421$	30.9090	1.50641	0.753207	−	0.657784i	$-0.228506\pi$
$421$	30.9090	1.50641	0.753207	+	0.657784i	$0.228506\pi$
$422$	0	0
$423$	−4.05890	−0.197350
$424$	0	0
$425$	16.7540	0.812687
$426$	0	0
$427$	−4.23766	−0.205075
$428$	0	0
$429$	−26.8141	−1.29460
$430$	0	0
$431$	−15.4360	−0.743526	−0.371763	−	0.928328i	$-0.621247\pi$
$431$	−15.4360	−0.743526	−0.371763	+	0.928328i	$0.621247\pi$
$432$	0	0
$433$	6.65500	0.319819	0.159909	−	0.987132i	$-0.448880\pi$
$433$	6.65500	0.319819	0.159909	+	0.987132i	$0.448880\pi$
$434$	0	0
$435$	−4.32870	−0.207545
$436$	0	0
$437$	−33.7830	−1.61606
$438$	0	0
$439$	−25.6017	−1.22190	−0.610952	−	0.791668i	$-0.709213\pi$
$439$	−25.6017	−1.22190	−0.610952	+	0.791668i	$0.709213\pi$
$440$	0	0
$441$	−7.40662	−0.352696
$442$	0	0
$443$	−3.23623	−0.153758	−0.0768790	−	0.997040i	$-0.524496\pi$
$443$	−3.23623	−0.153758	−0.0768790	+	0.997040i	$0.524496\pi$
$444$	0	0
$445$	7.69975	0.365003
$446$	0	0
$447$	−48.5952	−2.29848
$448$	0	0
$449$	30.4066	1.43498	0.717488	−	0.696571i	$-0.245292\pi$
$449$	30.4066	1.43498	0.717488	+	0.696571i	$0.245292\pi$
$450$	0	0
$451$	−18.7061	−0.880838
$452$	0	0
$453$	−38.6515	−1.81601
$454$	0	0
$455$	8.06230	0.377967
$456$	0	0
$457$	−33.5537	−1.56958	−0.784789	−	0.619763i	$-0.787229\pi$
$457$	−33.5537	−1.56958	−0.784789	+	0.619763i	$0.787229\pi$
$458$	0	0
$459$	−17.4278	−0.813462
$460$	0	0
$461$	−19.1322	−0.891074	−0.445537	−	0.895264i	$-0.646987\pi$
$461$	−19.1322	−0.891074	−0.445537	+	0.895264i	$0.646987\pi$
$462$	0	0
$463$	20.4824	0.951900	0.475950	−	0.879472i	$-0.342104\pi$
$463$	20.4824	0.951900	0.475950	+	0.879472i	$0.342104\pi$
$464$	0	0
$465$	−16.8020	−0.779175
$466$	0	0
$467$	17.4191	0.806058	0.403029	−	0.915187i	$-0.367957\pi$
$467$	17.4191	0.806058	0.403029	+	0.915187i	$0.367957\pi$
$468$	0	0
$469$	−11.4376	−0.528139
$470$	0	0
$471$	23.4504	1.08054
$472$	0	0
$473$	3.95774	0.181977
$474$	0	0
$475$	−15.7444	−0.722404
$476$	0	0
$477$	12.6952	0.581273
$478$	0	0
$479$	−10.6698	−0.487514	−0.243757	−	0.969836i	$-0.578380\pi$
$479$	−10.6698	−0.487514	−0.243757	+	0.969836i	$0.578380\pi$
$480$	0	0
$481$	−53.2026	−2.42583
$482$	0	0
$483$	15.4242	0.701825
$484$	0	0
$485$	19.0738	0.866096
$486$	0	0
$487$	−8.84874	−0.400975	−0.200487	−	0.979696i	$-0.564253\pi$
$487$	−8.84874	−0.400975	−0.200487	+	0.979696i	$0.564253\pi$
$488$	0	0
$489$	16.7466	0.757309
$490$	0	0
$491$	29.4926	1.33098	0.665491	−	0.746406i	$-0.268222\pi$
$491$	29.4926	1.33098	0.665491	+	0.746406i	$0.268222\pi$
$492$	0	0
$493$	8.21881	0.370157
$494$	0	0
$495$	3.00845	0.135220
$496$	0	0
$497$	5.48949	0.246237
$498$	0	0
$499$	−15.4951	−0.693658	−0.346829	−	0.937928i	$-0.612742\pi$
$499$	−15.4951	−0.693658	−0.346829	+	0.937928i	$0.612742\pi$
$500$	0	0
$501$	7.12431	0.318291
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−15.4979	−0.689647
$506$	0	0
$507$	−61.9418	−2.75093
$508$	0	0
$509$	14.2220	0.630380	0.315190	−	0.949029i	$-0.397932\pi$
$509$	14.2220	0.630380	0.315190	+	0.949029i	$0.397932\pi$
$510$	0	0
$511$	2.80109	0.123913
$512$	0	0
$513$	16.3777	0.723093
$514$	0	0
$515$	9.99098	0.440255
$516$	0	0
$517$	6.52570	0.287000
$518$	0	0
$519$	9.19964	0.403819
$520$	0	0
$521$	−12.7359	−0.557970	−0.278985	−	0.960296i	$-0.589998\pi$
$521$	−12.7359	−0.557970	−0.278985	+	0.960296i	$0.589998\pi$
$522$	0	0
$523$	7.27424	0.318080	0.159040	−	0.987272i	$-0.449160\pi$
$523$	7.27424	0.318080	0.159040	+	0.987272i	$0.449160\pi$
$524$	0	0
$525$	7.18837	0.313726
$526$	0	0
$527$	31.9016	1.38966
$528$	0	0
$529$	33.1593	1.44171
$530$	0	0
$531$	−11.1197	−0.482553
$532$	0	0
$533$	−61.8743	−2.68007
$534$	0	0
$535$	11.2170	0.484954
$536$	0	0
$537$	−47.7371	−2.06001
$538$	0	0
$539$	11.9080	0.512914
$540$	0	0
$541$	9.42110	0.405045	0.202522	−	0.979278i	$-0.435086\pi$
$541$	9.42110	0.405045	0.202522	+	0.979278i	$0.435086\pi$
$542$	0	0
$543$	−46.3159	−1.98760
$544$	0	0
$545$	−14.7551	−0.632038
$546$	0	0
$547$	−40.2952	−1.72290	−0.861449	−	0.507844i	$-0.830443\pi$
$547$	−40.2952	−1.72290	−0.861449	+	0.507844i	$0.830443\pi$
$548$	0	0
$549$	−5.23040	−0.223228
$550$	0	0
$551$	−7.72357	−0.329035
$552$	0	0
$553$	−12.0306	−0.511594
$554$	0	0
$555$	20.4747	0.869101
$556$	0	0
$557$	33.1961	1.40656	0.703282	−	0.710911i	$-0.251717\pi$
$557$	33.1961	1.40656	0.703282	+	0.710911i	$0.251717\pi$
$558$	0	0
$559$	13.0910	0.553691
$560$	0	0
$561$	−19.5929	−0.827213
$562$	0	0
$563$	−34.5965	−1.45807	−0.729035	−	0.684476i	$-0.760031\pi$
$563$	−34.5965	−1.45807	−0.729035	+	0.684476i	$0.760031\pi$
$564$	0	0
$565$	−24.1117	−1.01439
$566$	0	0
$567$	−11.1818	−0.469593
$568$	0	0
$569$	15.7194	0.658993	0.329497	−	0.944157i	$-0.393121\pi$
$569$	15.7194	0.658993	0.329497	+	0.944157i	$0.393121\pi$
$570$	0	0
$571$	−34.9809	−1.46391	−0.731953	−	0.681355i	$-0.761391\pi$
$571$	−34.9809	−1.46391	−0.731953	+	0.681355i	$0.761391\pi$
$572$	0	0
$573$	30.9111	1.29133
$574$	0	0
$575$	26.1728	1.09148
$576$	0	0
$577$	16.2601	0.676916	0.338458	−	0.940981i	$-0.390095\pi$
$577$	16.2601	0.676916	0.338458	+	0.940981i	$0.390095\pi$
$578$	0	0
$579$	15.3602	0.638349
$580$	0	0
$581$	−2.22620	−0.0923582
$582$	0	0
$583$	−20.4107	−0.845327
$584$	0	0
$585$	9.95103	0.411425
$586$	0	0
$587$	−37.1587	−1.53370	−0.766851	−	0.641825i	$-0.778178\pi$
$587$	−37.1587	−1.53370	−0.766851	+	0.641825i	$0.778178\pi$
$588$	0	0
$589$	−29.9793	−1.23528
$590$	0	0
$591$	20.4130	0.839680
$592$	0	0
$593$	11.8864	0.488114	0.244057	−	0.969761i	$-0.421522\pi$
$593$	11.8864	0.488114	0.244057	+	0.969761i	$0.421522\pi$
$594$	0	0
$595$	5.89107	0.241510
$596$	0	0
$597$	25.5696	1.04649
$598$	0	0
$599$	24.4737	0.999968	0.499984	−	0.866035i	$-0.333339\pi$
$599$	24.4737	0.999968	0.499984	+	0.866035i	$0.333339\pi$
$600$	0	0
$601$	−6.82699	−0.278479	−0.139239	−	0.990259i	$-0.544466\pi$
$601$	−6.82699	−0.278479	−0.139239	+	0.990259i	$0.544466\pi$
$602$	0	0
$603$	−14.1170	−0.574891
$604$	0	0
$605$	8.66890	0.352441
$606$	0	0
$607$	−40.2732	−1.63464	−0.817319	−	0.576185i	$-0.804541\pi$
$607$	−40.2732	−1.63464	−0.817319	+	0.576185i	$0.804541\pi$
$608$	0	0
$609$	3.52632	0.142894
$610$	0	0
$611$	21.5851	0.873238
$612$	0	0
$613$	18.1607	0.733505	0.366752	−	0.930319i	$-0.380470\pi$
$613$	18.1607	0.733505	0.366752	+	0.930319i	$0.380470\pi$
$614$	0	0
$615$	23.8119	0.960189
$616$	0	0
$617$	47.6599	1.91871	0.959357	−	0.282195i	$-0.0910626\pi$
$617$	47.6599	1.91871	0.959357	+	0.282195i	$0.0910626\pi$
$618$	0	0
$619$	23.3046	0.936690	0.468345	−	0.883546i	$-0.344850\pi$
$619$	23.3046	0.936690	0.468345	+	0.883546i	$0.344850\pi$
$620$	0	0
$621$	−27.2255	−1.09252
$622$	0	0
$623$	−6.27250	−0.251302
$624$	0	0
$625$	4.66030	0.186412
$626$	0	0
$627$	18.4123	0.735316
$628$	0	0
$629$	−38.8748	−1.55004
$630$	0	0
$631$	−21.2125	−0.844455	−0.422227	−	0.906490i	$-0.638752\pi$
$631$	−21.2125	−0.844455	−0.422227	+	0.906490i	$0.638752\pi$
$632$	0	0
$633$	−45.0588	−1.79093
$634$	0	0
$635$	−3.07521	−0.122036
$636$	0	0
$637$	39.3881	1.56061
$638$	0	0
$639$	6.77550	0.268035
$640$	0	0
$641$	3.49392	0.138002	0.0690008	−	0.997617i	$-0.478019\pi$
$641$	3.49392	0.138002	0.0690008	+	0.997617i	$0.478019\pi$
$642$	0	0
$643$	−20.5778	−0.811510	−0.405755	−	0.913982i	$-0.632991\pi$
$643$	−20.5778	−0.811510	−0.405755	+	0.913982i	$0.632991\pi$
$644$	0	0
$645$	−5.03799	−0.198371
$646$	0	0
$647$	25.8736	1.01719	0.508597	−	0.861005i	$-0.330164\pi$
$647$	25.8736	1.01719	0.508597	+	0.861005i	$0.330164\pi$
$648$	0	0
$649$	17.8777	0.701760
$650$	0	0
$651$	13.6875	0.536457
$652$	0	0
$653$	−29.7077	−1.16255	−0.581276	−	0.813706i	$-0.697446\pi$
$653$	−29.7077	−1.16255	−0.581276	+	0.813706i	$0.697446\pi$
$654$	0	0
$655$	−11.4130	−0.445943
$656$	0	0
$657$	3.45729	0.134882
$658$	0	0
$659$	−1.71376	−0.0667587	−0.0333794	−	0.999443i	$-0.510627\pi$
$659$	−1.71376	−0.0667587	−0.0333794	+	0.999443i	$0.510627\pi$
$660$	0	0
$661$	−18.2808	−0.711040	−0.355520	−	0.934669i	$-0.615696\pi$
$661$	−18.2808	−0.711040	−0.355520	+	0.934669i	$0.615696\pi$
$662$	0	0
$663$	−64.8075	−2.51691
$664$	0	0
$665$	−5.53609	−0.214680
$666$	0	0
$667$	12.8393	0.497139
$668$	0	0
$669$	−56.3576	−2.17891
$670$	0	0
$671$	8.40919	0.324633
$672$	0	0
$673$	−18.4852	−0.712553	−0.356277	−	0.934381i	$-0.615954\pi$
$673$	−18.4852	−0.712553	−0.356277	+	0.934381i	$0.615954\pi$
$674$	0	0
$675$	−12.6883	−0.488373
$676$	0	0
$677$	−25.9455	−0.997165	−0.498583	−	0.866842i	$-0.666146\pi$
$677$	−25.9455	−0.997165	−0.498583	+	0.866842i	$0.666146\pi$
$678$	0	0
$679$	−15.5382	−0.596302
$680$	0	0
$681$	2.27663	0.0872406
$682$	0	0
$683$	15.2589	0.583866	0.291933	−	0.956439i	$-0.405702\pi$
$683$	15.2589	0.583866	0.291933	+	0.956439i	$0.405702\pi$
$684$	0	0
$685$	−14.7403	−0.563200
$686$	0	0
$687$	56.9936	2.17444
$688$	0	0
$689$	−67.5126	−2.57203
$690$	0	0
$691$	−50.5257	−1.92209	−0.961045	−	0.276393i	$-0.910861\pi$
$691$	−50.5257	−1.92209	−0.961045	+	0.276393i	$0.910861\pi$
$692$	0	0
$693$	−2.45079	−0.0930978
$694$	0	0
$695$	8.53567	0.323777
$696$	0	0
$697$	−45.2111	−1.71249
$698$	0	0
$699$	19.5039	0.737704
$700$	0	0
$701$	−17.0982	−0.645791	−0.322895	−	0.946435i	$-0.604656\pi$
$701$	−17.0982	−0.645791	−0.322895	+	0.946435i	$0.604656\pi$
$702$	0	0
$703$	36.5323	1.37784
$704$	0	0
$705$	−8.30687	−0.312855
$706$	0	0
$707$	12.6252	0.474818
$708$	0	0
$709$	4.79202	0.179968	0.0899840	−	0.995943i	$-0.471318\pi$
$709$	4.79202	0.179968	0.0899840	+	0.995943i	$0.471318\pi$
$710$	0	0
$711$	−14.8490	−0.556881
$712$	0	0
$713$	49.8362	1.86638
$714$	0	0
$715$	−15.9988	−0.598321
$716$	0	0
$717$	38.9359	1.45409
$718$	0	0
$719$	−7.17697	−0.267656	−0.133828	−	0.991005i	$-0.542727\pi$
$719$	−7.17697	−0.267656	−0.133828	+	0.991005i	$0.542727\pi$
$720$	0	0
$721$	−8.13902	−0.303113
$722$	0	0
$723$	−20.1785	−0.750448
$724$	0	0
$725$	5.98369	0.222229
$726$	0	0
$727$	1.65389	0.0613393	0.0306696	−	0.999530i	$-0.490236\pi$
$727$	1.65389	0.0613393	0.0306696	+	0.999530i	$0.490236\pi$
$728$	0	0
$729$	8.62651	0.319500
$730$	0	0
$731$	9.56551	0.353793
$732$	0	0
$733$	42.0673	1.55379	0.776895	−	0.629630i	$-0.216794\pi$
$733$	42.0673	1.55379	0.776895	+	0.629630i	$0.216794\pi$
$734$	0	0
$735$	−15.1583	−0.559121
$736$	0	0
$737$	22.6967	0.836044
$738$	0	0
$739$	42.6383	1.56847	0.784237	−	0.620462i	$-0.213055\pi$
$739$	42.6383	1.56847	0.784237	+	0.620462i	$0.213055\pi$
$740$	0	0
$741$	60.9023	2.23730
$742$	0	0
$743$	−0.692178	−0.0253935	−0.0126968	−	0.999919i	$-0.504042\pi$
$743$	−0.692178	−0.0253935	−0.0126968	+	0.999919i	$0.504042\pi$
$744$	0	0
$745$	−28.9946	−1.06228
$746$	0	0
$747$	−2.74772	−0.100534
$748$	0	0
$749$	−9.13780	−0.333888
$750$	0	0
$751$	50.0260	1.82547	0.912737	−	0.408548i	$-0.133965\pi$
$751$	50.0260	1.82547	0.912737	+	0.408548i	$0.133965\pi$
$752$	0	0
$753$	−34.0946	−1.24248
$754$	0	0
$755$	−23.0616	−0.839299
$756$	0	0
$757$	−16.2033	−0.588919	−0.294460	−	0.955664i	$-0.595140\pi$
$757$	−16.2033	−0.588919	−0.294460	+	0.955664i	$0.595140\pi$
$758$	0	0
$759$	−30.6077	−1.11099
$760$	0	0
$761$	−5.11057	−0.185258	−0.0926290	−	0.995701i	$-0.529527\pi$
$761$	−5.11057	−0.185258	−0.0926290	+	0.995701i	$0.529527\pi$
$762$	0	0
$763$	12.0200	0.435154
$764$	0	0
$765$	7.27115	0.262889
$766$	0	0
$767$	59.1340	2.13520
$768$	0	0
$769$	−17.8082	−0.642182	−0.321091	−	0.947048i	$-0.604049\pi$
$769$	−17.8082	−0.642182	−0.321091	+	0.947048i	$0.604049\pi$
$770$	0	0
$771$	44.4907	1.60229
$772$	0	0
$773$	−39.2651	−1.41227	−0.706133	−	0.708079i	$-0.749562\pi$
$773$	−39.2651	−1.41227	−0.706133	+	0.708079i	$0.749562\pi$
$774$	0	0
$775$	23.2259	0.834299
$776$	0	0
$777$	−16.6794	−0.598370
$778$	0	0
$779$	42.4868	1.52225
$780$	0	0
$781$	−10.8933	−0.389794
$782$	0	0
$783$	−6.22436	−0.222440
$784$	0	0
$785$	13.9918	0.499390
$786$	0	0
$787$	−8.39892	−0.299389	−0.149695	−	0.988732i	$-0.547829\pi$
$787$	−8.39892	−0.299389	−0.149695	+	0.988732i	$0.547829\pi$
$788$	0	0
$789$	−55.6434	−1.98096
$790$	0	0
$791$	19.6423	0.698400
$792$	0	0
$793$	27.8151	0.987743
$794$	0	0
$795$	25.9818	0.921479
$796$	0	0
$797$	35.4432	1.25546	0.627731	−	0.778431i	$-0.283984\pi$
$797$	35.4432	1.25546	0.627731	+	0.778431i	$0.283984\pi$
$798$	0	0
$799$	15.7721	0.557975
$800$	0	0
$801$	−7.74194	−0.273548
$802$	0	0
$803$	−5.55847	−0.196154
$804$	0	0
$805$	9.20293	0.324361
$806$	0	0
$807$	−37.7057	−1.32730
$808$	0	0
$809$	−14.0783	−0.494965	−0.247482	−	0.968892i	$-0.579603\pi$
$809$	−14.0783	−0.494965	−0.247482	+	0.968892i	$0.579603\pi$
$810$	0	0
$811$	−3.14673	−0.110497	−0.0552483	−	0.998473i	$-0.517595\pi$
$811$	−3.14673	−0.110497	−0.0552483	+	0.998473i	$0.517595\pi$
$812$	0	0
$813$	−35.8393	−1.25694
$814$	0	0
$815$	9.99196	0.350003
$816$	0	0
$817$	−8.98912	−0.314490
$818$	0	0
$819$	−8.10648	−0.283263
$820$	0	0
$821$	43.6000	1.52165	0.760824	−	0.648958i	$-0.224795\pi$
$821$	43.6000	1.52165	0.760824	+	0.648958i	$0.224795\pi$
$822$	0	0
$823$	−31.0002	−1.08060	−0.540300	−	0.841473i	$-0.681689\pi$
$823$	−31.0002	−1.08060	−0.540300	+	0.841473i	$0.681689\pi$
$824$	0	0
$825$	−14.2646	−0.496629
$826$	0	0
$827$	−19.3533	−0.672981	−0.336490	−	0.941687i	$-0.609240\pi$
$827$	−19.3533	−0.672981	−0.336490	+	0.941687i	$0.609240\pi$
$828$	0	0
$829$	55.6139	1.93155	0.965775	−	0.259379i	$-0.0835180\pi$
$829$	55.6139	1.93155	0.965775	+	0.259379i	$0.0835180\pi$
$830$	0	0
$831$	58.3899	2.02552
$832$	0	0
$833$	28.7806	0.997190
$834$	0	0
$835$	4.25076	0.147103
$836$	0	0
$837$	−24.1601	−0.835094
$838$	0	0
$839$	−1.05873	−0.0365515	−0.0182757	−	0.999833i	$-0.505818\pi$
$839$	−1.05873	−0.0365515	−0.0182757	+	0.999833i	$0.505818\pi$
$840$	0	0
$841$	−26.0646	−0.898781
$842$	0	0
$843$	5.74635	0.197915
$844$	0	0
$845$	−36.9579	−1.27139
$846$	0	0
$847$	−7.06201	−0.242653
$848$	0	0
$849$	−15.1826	−0.521066
$850$	0	0
$851$	−60.7295	−2.08178
$852$	0	0
$853$	0.206776	0.00707987	0.00353994	−	0.999994i	$-0.498873\pi$
$853$	0.206776	0.00707987	0.00353994	+	0.999994i	$0.498873\pi$
$854$	0	0
$855$	−6.83302	−0.233684
$856$	0	0
$857$	9.18665	0.313810	0.156905	−	0.987614i	$-0.449848\pi$
$857$	9.18665	0.313810	0.156905	+	0.987614i	$0.449848\pi$
$858$	0	0
$859$	−55.9282	−1.90825	−0.954123	−	0.299414i	$-0.903209\pi$
$859$	−55.9282	−1.90825	−0.954123	+	0.299414i	$0.903209\pi$
$860$	0	0
$861$	−19.3981	−0.661084
$862$	0	0
$863$	−35.9794	−1.22475	−0.612377	−	0.790566i	$-0.709786\pi$
$863$	−35.9794	−1.22475	−0.612377	+	0.790566i	$0.709786\pi$
$864$	0	0
$865$	5.48901	0.186632
$866$	0	0
$867$	−12.3718	−0.420170
$868$	0	0
$869$	23.8735	0.809853
$870$	0	0
$871$	75.0739	2.54378
$872$	0	0
$873$	−19.1783	−0.649087
$874$	0	0
$875$	10.4292	0.352572
$876$	0	0
$877$	−12.8761	−0.434794	−0.217397	−	0.976083i	$-0.569757\pi$
$877$	−12.8761	−0.434794	−0.217397	+	0.976083i	$0.569757\pi$
$878$	0	0
$879$	−60.0420	−2.02517
$880$	0	0
$881$	31.7352	1.06919	0.534593	−	0.845110i	$-0.320465\pi$
$881$	31.7352	1.06919	0.534593	+	0.845110i	$0.320465\pi$
$882$	0	0
$883$	−36.5875	−1.23127	−0.615634	−	0.788032i	$-0.711100\pi$
$883$	−36.5875	−1.23127	−0.615634	+	0.788032i	$0.711100\pi$
$884$	0	0
$885$	−22.7573	−0.764979
$886$	0	0
$887$	12.9644	0.435302	0.217651	−	0.976027i	$-0.430161\pi$
$887$	12.9644	0.435302	0.217651	+	0.976027i	$0.430161\pi$
$888$	0	0
$889$	2.50518	0.0840210
$890$	0	0
$891$	22.1892	0.743365
$892$	0	0
$893$	−14.8217	−0.495989
$894$	0	0
$895$	−28.4826	−0.952068
$896$	0	0
$897$	−101.241	−3.38034
$898$	0	0
$899$	11.3937	0.380000
$900$	0	0
$901$	−49.3310	−1.64345
$902$	0	0
$903$	4.10413	0.136577
$904$	0	0
$905$	−27.6346	−0.918606
$906$	0	0
$907$	4.97952	0.165342	0.0826712	−	0.996577i	$-0.473655\pi$
$907$	4.97952	0.165342	0.0826712	+	0.996577i	$0.473655\pi$
$908$	0	0
$909$	15.5828	0.516849
$910$	0	0
$911$	10.6107	0.351547	0.175774	−	0.984431i	$-0.443757\pi$
$911$	10.6107	0.351547	0.175774	+	0.984431i	$0.443757\pi$
$912$	0	0
$913$	4.41766	0.146203
$914$	0	0
$915$	−10.7044	−0.353878
$916$	0	0
$917$	9.29745	0.307029
$918$	0	0
$919$	6.52892	0.215369	0.107685	−	0.994185i	$-0.465656\pi$
$919$	6.52892	0.215369	0.107685	+	0.994185i	$0.465656\pi$
$920$	0	0
$921$	5.93819	0.195670
$922$	0	0
$923$	−36.0318	−1.18600
$924$	0	0
$925$	−28.3027	−0.930587
$926$	0	0
$927$	−10.0457	−0.329945
$928$	0	0
$929$	−1.98762	−0.0652118	−0.0326059	−	0.999468i	$-0.510381\pi$
$929$	−1.98762	−0.0652118	−0.0326059	+	0.999468i	$0.510381\pi$
$930$	0	0
$931$	−27.0464	−0.886410
$932$	0	0
$933$	−8.08504	−0.264692
$934$	0	0
$935$	−11.6902	−0.382311
$936$	0	0
$937$	−60.2767	−1.96916	−0.984578	−	0.174948i	$-0.944024\pi$
$937$	−60.2767	−1.96916	−0.984578	+	0.174948i	$0.944024\pi$
$938$	0	0
$939$	−59.3078	−1.93544
$940$	0	0
$941$	−40.4218	−1.31771	−0.658857	−	0.752268i	$-0.728960\pi$
$941$	−40.4218	−1.31771	−0.658857	+	0.752268i	$0.728960\pi$
$942$	0	0
$943$	−70.6281	−2.29997
$944$	0	0
$945$	−4.46149	−0.145132
$946$	0	0
$947$	25.6111	0.832250	0.416125	−	0.909307i	$-0.363388\pi$
$947$	25.6111	0.832250	0.416125	+	0.909307i	$0.363388\pi$
$948$	0	0
$949$	−18.3857	−0.596827
$950$	0	0
$951$	7.80584	0.253122
$952$	0	0
$953$	40.8546	1.32341	0.661705	−	0.749765i	$-0.269833\pi$
$953$	40.8546	1.32341	0.661705	+	0.749765i	$0.269833\pi$
$954$	0	0
$955$	18.4433	0.596810
$956$	0	0
$957$	−6.99761	−0.226201
$958$	0	0
$959$	12.0080	0.387759
$960$	0	0
$961$	13.2250	0.426611
$962$	0	0
$963$	−11.2785	−0.363444
$964$	0	0
$965$	9.16476	0.295024
$966$	0	0
$967$	−30.1154	−0.968445	−0.484222	−	0.874945i	$-0.660897\pi$
$967$	−30.1154	−0.968445	−0.484222	+	0.874945i	$0.660897\pi$
$968$	0	0
$969$	44.5009	1.42958
$970$	0	0
$971$	−47.1319	−1.51253	−0.756267	−	0.654264i	$-0.772979\pi$
$971$	−47.1319	−1.51253	−0.756267	+	0.654264i	$0.772979\pi$
$972$	0	0
$973$	−6.95347	−0.222918
$974$	0	0
$975$	−47.1829	−1.51106
$976$	0	0
$977$	−7.18220	−0.229779	−0.114889	−	0.993378i	$-0.536651\pi$
$977$	−7.18220	−0.229779	−0.114889	+	0.993378i	$0.536651\pi$
$978$	0	0
$979$	12.4471	0.397812
$980$	0	0
$981$	14.8359	0.473674
$982$	0	0
$983$	49.7331	1.58624	0.793120	−	0.609065i	$-0.208455\pi$
$983$	49.7331	1.58624	0.793120	+	0.609065i	$0.208455\pi$
$984$	0	0
$985$	12.1795	0.388072
$986$	0	0
$987$	6.76708	0.215398
$988$	0	0
$989$	14.9431	0.475162
$990$	0	0
$991$	34.6263	1.09994	0.549971	−	0.835184i	$-0.314639\pi$
$991$	34.6263	1.09994	0.549971	+	0.835184i	$0.314639\pi$
$992$	0	0
$993$	−58.4445	−1.85468
$994$	0	0
$995$	15.2562	0.483655
$996$	0	0
$997$	−61.1638	−1.93708	−0.968538	−	0.248866i	$-0.919942\pi$
$997$	−61.1638	−1.93708	−0.968538	+	0.248866i	$0.919942\pi$
$998$	0	0
$999$	29.4410	0.931474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.10	✓	33
4.3	odd	2		8048.2.a.y.1.24		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.10	✓	33	1.1	even	1	trivial
8048.2.a.y.1.24		33	4.3	odd	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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05780 q^{3} -1.22779 q^{5} +1.00021 q^{7} +1.23452 q^{9} +O(q^{10})\)	\(q-2.05780 q^{3} -1.22779 q^{5} +1.00021 q^{7} +1.23452 q^{9} -1.98481 q^{11} -6.56514 q^{13} +2.52655 q^{15} -4.79710 q^{17} +4.50804 q^{19} -2.05822 q^{21} -7.49395 q^{23} -3.49252 q^{25} +3.63299 q^{27} -1.71329 q^{29} -6.65018 q^{31} +4.08432 q^{33} -1.22805 q^{35} +8.10380 q^{37} +13.5097 q^{39} +9.42468 q^{41} -1.99402 q^{43} -1.51574 q^{45} -3.28783 q^{47} -5.99959 q^{49} +9.87145 q^{51} +10.2835 q^{53} +2.43693 q^{55} -9.27663 q^{57} -9.00727 q^{59} -4.23679 q^{61} +1.23478 q^{63} +8.06064 q^{65} -11.4352 q^{67} +15.4210 q^{69} +5.48836 q^{71} +2.80051 q^{73} +7.18689 q^{75} -1.98521 q^{77} -12.0281 q^{79} -11.1795 q^{81} -2.22574 q^{83} +5.88986 q^{85} +3.52559 q^{87} -6.27120 q^{89} -6.56649 q^{91} +13.6847 q^{93} -5.53495 q^{95} -15.5350 q^{97} -2.45028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Introduction

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