LMFDB - Embedded newform 4024.2.a.f.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.23369 q^{3} -1.90057 q^{5} +0.0272535 q^{7} +1.98935 q^{9} +O(q^{10})$	$q-2.23369 q^{3} -1.90057 q^{5} +0.0272535 q^{7} +1.98935 q^{9} -3.42657 q^{11} +6.41103 q^{13} +4.24528 q^{15} +0.693638 q^{17} -6.06610 q^{19} -0.0608757 q^{21} +2.26235 q^{23} -1.38783 q^{25} +2.25747 q^{27} -0.515741 q^{29} -9.76295 q^{31} +7.65388 q^{33} -0.0517972 q^{35} +2.77471 q^{37} -14.3202 q^{39} -3.95440 q^{41} +4.04875 q^{43} -3.78091 q^{45} -3.28818 q^{47} -6.99926 q^{49} -1.54937 q^{51} +0.768994 q^{53} +6.51243 q^{55} +13.5498 q^{57} +5.88230 q^{59} -2.19160 q^{61} +0.0542168 q^{63} -12.1846 q^{65} -5.70029 q^{67} -5.05339 q^{69} -5.79918 q^{71} -6.45318 q^{73} +3.09998 q^{75} -0.0933859 q^{77} -13.7220 q^{79} -11.0105 q^{81} +8.22906 q^{83} -1.31831 q^{85} +1.15200 q^{87} +7.71962 q^{89} +0.174723 q^{91} +21.8074 q^{93} +11.5290 q^{95} +9.17764 q^{97} -6.81666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.23369	−1.28962	−0.644810	−	0.764343i	$-0.723063\pi$
$3$	−2.23369	−1.28962	−0.644810	+	0.764343i	$0.723063\pi$
$4$	0	0
$5$	−1.90057	−0.849961	−0.424980	−	0.905203i	$-0.639719\pi$
$5$	−1.90057	−0.849961	−0.424980	+	0.905203i	$0.639719\pi$
$6$	0	0
$7$	0.0272535	0.0103008	0.00515042	−	0.999987i	$-0.498361\pi$
$7$	0.0272535	0.0103008	0.00515042	+	0.999987i	$0.498361\pi$
$8$	0	0
$9$	1.98935	0.663118
$10$	0	0
$11$	−3.42657	−1.03315	−0.516574	−	0.856242i	$-0.672793\pi$
$11$	−3.42657	−1.03315	−0.516574	+	0.856242i	$0.672793\pi$
$12$	0	0
$13$	6.41103	1.77810	0.889050	−	0.457810i	$-0.151366\pi$
$13$	6.41103	1.77810	0.889050	+	0.457810i	$0.151366\pi$
$14$	0	0
$15$	4.24528	1.09613
$16$	0	0
$17$	0.693638	0.168232	0.0841159	−	0.996456i	$-0.473193\pi$
$17$	0.693638	0.168232	0.0841159	+	0.996456i	$0.473193\pi$
$18$	0	0
$19$	−6.06610	−1.39166	−0.695829	−	0.718207i	$-0.744963\pi$
$19$	−6.06610	−1.39166	−0.695829	+	0.718207i	$0.744963\pi$
$20$	0	0
$21$	−0.0608757	−0.0132842
$22$	0	0
$23$	2.26235	0.471733	0.235867	−	0.971785i	$-0.424207\pi$
$23$	2.26235	0.471733	0.235867	+	0.971785i	$0.424207\pi$
$24$	0	0
$25$	−1.38783	−0.277566
$26$	0	0
$27$	2.25747	0.434449
$28$	0	0
$29$	−0.515741	−0.0957708	−0.0478854	−	0.998853i	$-0.515248\pi$
$29$	−0.515741	−0.0957708	−0.0478854	+	0.998853i	$0.515248\pi$
$30$	0	0
$31$	−9.76295	−1.75348	−0.876739	−	0.480966i	$-0.840286\pi$
$31$	−9.76295	−1.75348	−0.876739	+	0.480966i	$0.840286\pi$
$32$	0	0
$33$	7.65388	1.33237
$34$	0	0
$35$	−0.0517972	−0.00875532
$36$	0	0
$37$	2.77471	0.456160	0.228080	−	0.973642i	$-0.426755\pi$
$37$	2.77471	0.456160	0.228080	+	0.973642i	$0.426755\pi$
$38$	0	0
$39$	−14.3202	−2.29307
$40$	0	0
$41$	−3.95440	−0.617574	−0.308787	−	0.951131i	$-0.599923\pi$
$41$	−3.95440	−0.617574	−0.308787	+	0.951131i	$0.599923\pi$
$42$	0	0
$43$	4.04875	0.617428	0.308714	−	0.951155i	$-0.400101\pi$
$43$	4.04875	0.617428	0.308714	+	0.951155i	$0.400101\pi$
$44$	0	0
$45$	−3.78091	−0.563624
$46$	0	0
$47$	−3.28818	−0.479631	−0.239815	−	0.970819i	$-0.577087\pi$
$47$	−3.28818	−0.479631	−0.239815	+	0.970819i	$0.577087\pi$
$48$	0	0
$49$	−6.99926	−0.999894
$50$	0	0
$51$	−1.54937	−0.216955
$52$	0	0
$53$	0.768994	0.105629	0.0528147	−	0.998604i	$-0.483181\pi$
$53$	0.768994	0.105629	0.0528147	+	0.998604i	$0.483181\pi$
$54$	0	0
$55$	6.51243	0.878136
$56$	0	0
$57$	13.5498	1.79471
$58$	0	0
$59$	5.88230	0.765810	0.382905	−	0.923788i	$-0.374924\pi$
$59$	5.88230	0.765810	0.382905	+	0.923788i	$0.374924\pi$
$60$	0	0
$61$	−2.19160	−0.280605	−0.140302	−	0.990109i	$-0.544808\pi$
$61$	−2.19160	−0.280605	−0.140302	+	0.990109i	$0.544808\pi$
$62$	0	0
$63$	0.0542168	0.00683068
$64$	0	0
$65$	−12.1846	−1.51132
$66$	0	0
$67$	−5.70029	−0.696401	−0.348201	−	0.937420i	$-0.613207\pi$
$67$	−5.70029	−0.696401	−0.348201	+	0.937420i	$0.613207\pi$
$68$	0	0
$69$	−5.05339	−0.608356
$70$	0	0
$71$	−5.79918	−0.688236	−0.344118	−	0.938926i	$-0.611822\pi$
$71$	−5.79918	−0.688236	−0.344118	+	0.938926i	$0.611822\pi$
$72$	0	0
$73$	−6.45318	−0.755287	−0.377644	−	0.925951i	$-0.623266\pi$
$73$	−6.45318	−0.755287	−0.377644	+	0.925951i	$0.623266\pi$
$74$	0	0
$75$	3.09998	0.357955
$76$	0	0
$77$	−0.0933859	−0.0106423
$78$	0	0
$79$	−13.7220	−1.54384	−0.771922	−	0.635717i	$-0.780705\pi$
$79$	−13.7220	−1.54384	−0.771922	+	0.635717i	$0.780705\pi$
$80$	0	0
$81$	−11.0105	−1.22339
$82$	0	0
$83$	8.22906	0.903256	0.451628	−	0.892206i	$-0.350843\pi$
$83$	8.22906	0.903256	0.451628	+	0.892206i	$0.350843\pi$
$84$	0	0
$85$	−1.31831	−0.142991
$86$	0	0
$87$	1.15200	0.123508
$88$	0	0
$89$	7.71962	0.818278	0.409139	−	0.912472i	$-0.365829\pi$
$89$	7.71962	0.818278	0.409139	+	0.912472i	$0.365829\pi$
$90$	0	0
$91$	0.174723	0.0183159
$92$	0	0
$93$	21.8074	2.26132
$94$	0	0
$95$	11.5290	1.18286
$96$	0	0
$97$	9.17764	0.931848	0.465924	−	0.884825i	$-0.345722\pi$
$97$	9.17764	0.931848	0.465924	+	0.884825i	$0.345722\pi$
$98$	0	0
$99$	−6.81666	−0.685100
$100$	0	0
$101$	−2.78861	−0.277477	−0.138739	−	0.990329i	$-0.544305\pi$
$101$	−2.78861	−0.277477	−0.138739	+	0.990329i	$0.544305\pi$
$102$	0	0
$103$	−18.8420	−1.85656	−0.928280	−	0.371881i	$-0.878713\pi$
$103$	−18.8420	−1.85656	−0.928280	+	0.371881i	$0.878713\pi$
$104$	0	0
$105$	0.115699	0.0112910
$106$	0	0
$107$	12.5205	1.21040	0.605202	−	0.796072i	$-0.293092\pi$
$107$	12.5205	1.21040	0.605202	+	0.796072i	$0.293092\pi$
$108$	0	0
$109$	4.67292	0.447585	0.223792	−	0.974637i	$-0.428156\pi$
$109$	4.67292	0.447585	0.223792	+	0.974637i	$0.428156\pi$
$110$	0	0
$111$	−6.19784	−0.588273
$112$	0	0
$113$	10.9939	1.03422	0.517111	−	0.855918i	$-0.327007\pi$
$113$	10.9939	1.03422	0.517111	+	0.855918i	$0.327007\pi$
$114$	0	0
$115$	−4.29976	−0.400955
$116$	0	0
$117$	12.7538	1.17909
$118$	0	0
$119$	0.0189040	0.00173293
$120$	0	0
$121$	0.741360	0.0673964
$122$	0	0
$123$	8.83290	0.796435
$124$	0	0
$125$	12.1405	1.08588
$126$	0	0
$127$	13.7122	1.21676	0.608379	−	0.793646i	$-0.291820\pi$
$127$	13.7122	1.21676	0.608379	+	0.793646i	$0.291820\pi$
$128$	0	0
$129$	−9.04363	−0.796247
$130$	0	0
$131$	3.38834	0.296040	0.148020	−	0.988984i	$-0.452710\pi$
$131$	3.38834	0.296040	0.148020	+	0.988984i	$0.452710\pi$
$132$	0	0
$133$	−0.165322	−0.0143353
$134$	0	0
$135$	−4.29047	−0.369265
$136$	0	0
$137$	10.4367	0.891664	0.445832	−	0.895117i	$-0.352908\pi$
$137$	10.4367	0.891664	0.445832	+	0.895117i	$0.352908\pi$
$138$	0	0
$139$	4.75995	0.403734	0.201867	−	0.979413i	$-0.435299\pi$
$139$	4.75995	0.403734	0.201867	+	0.979413i	$0.435299\pi$
$140$	0	0
$141$	7.34477	0.618541
$142$	0	0
$143$	−21.9678	−1.83704
$144$	0	0
$145$	0.980203	0.0814014
$146$	0	0
$147$	15.6341	1.28948
$148$	0	0
$149$	−3.84394	−0.314908	−0.157454	−	0.987526i	$-0.550329\pi$
$149$	−3.84394	−0.314908	−0.157454	+	0.987526i	$0.550329\pi$
$150$	0	0
$151$	−14.4933	−1.17945	−0.589723	−	0.807606i	$-0.700763\pi$
$151$	−14.4933	−1.17945	−0.589723	+	0.807606i	$0.700763\pi$
$152$	0	0
$153$	1.37989	0.111558
$154$	0	0
$155$	18.5552	1.49039
$156$	0	0
$157$	−3.60514	−0.287721	−0.143861	−	0.989598i	$-0.545952\pi$
$157$	−3.60514	−0.287721	−0.143861	+	0.989598i	$0.545952\pi$
$158$	0	0
$159$	−1.71769	−0.136222
$160$	0	0
$161$	0.0616570	0.00485925
$162$	0	0
$163$	12.6718	0.992529	0.496264	−	0.868171i	$-0.334705\pi$
$163$	12.6718	0.992529	0.496264	+	0.868171i	$0.334705\pi$
$164$	0	0
$165$	−14.5467	−1.13246
$166$	0	0
$167$	8.84956	0.684799	0.342400	−	0.939554i	$-0.388760\pi$
$167$	8.84956	0.684799	0.342400	+	0.939554i	$0.388760\pi$
$168$	0	0
$169$	28.1013	2.16164
$170$	0	0
$171$	−12.0676	−0.922834
$172$	0	0
$173$	6.38515	0.485454	0.242727	−	0.970095i	$-0.421958\pi$
$173$	6.38515	0.485454	0.242727	+	0.970095i	$0.421958\pi$
$174$	0	0
$175$	−0.0378233	−0.00285917
$176$	0	0
$177$	−13.1392	−0.987604
$178$	0	0
$179$	1.57797	0.117943	0.0589713	−	0.998260i	$-0.481218\pi$
$179$	1.57797	0.117943	0.0589713	+	0.998260i	$0.481218\pi$
$180$	0	0
$181$	1.64537	0.122300	0.0611498	−	0.998129i	$-0.480523\pi$
$181$	1.64537	0.122300	0.0611498	+	0.998129i	$0.480523\pi$
$182$	0	0
$183$	4.89534	0.361874
$184$	0	0
$185$	−5.27354	−0.387718
$186$	0	0
$187$	−2.37680	−0.173809
$188$	0	0
$189$	0.0615238	0.00447520
$190$	0	0
$191$	5.68431	0.411302	0.205651	−	0.978625i	$-0.434069\pi$
$191$	5.68431	0.411302	0.205651	+	0.978625i	$0.434069\pi$
$192$	0	0
$193$	7.42071	0.534154	0.267077	−	0.963675i	$-0.413942\pi$
$193$	7.42071	0.534154	0.267077	+	0.963675i	$0.413942\pi$
$194$	0	0
$195$	27.2166	1.94902
$196$	0	0
$197$	8.33607	0.593921	0.296960	−	0.954890i	$-0.404027\pi$
$197$	8.33607	0.593921	0.296960	+	0.954890i	$0.404027\pi$
$198$	0	0
$199$	21.6695	1.53611	0.768056	−	0.640382i	$-0.221224\pi$
$199$	21.6695	1.53611	0.768056	+	0.640382i	$0.221224\pi$
$200$	0	0
$201$	12.7327	0.898092
$202$	0	0
$203$	−0.0140557	−0.000986520 0
$204$	0	0
$205$	7.51562	0.524914
$206$	0	0
$207$	4.50062	0.312815
$208$	0	0
$209$	20.7859	1.43779
$210$	0	0
$211$	−1.49984	−0.103253	−0.0516266	−	0.998666i	$-0.516441\pi$
$211$	−1.49984	−0.103253	−0.0516266	+	0.998666i	$0.516441\pi$
$212$	0	0
$213$	12.9535	0.887563
$214$	0	0
$215$	−7.69493	−0.524790
$216$	0	0
$217$	−0.266074	−0.0180623
$218$	0	0
$219$	14.4144	0.974033
$220$	0	0
$221$	4.44693	0.299133
$222$	0	0
$223$	−17.6250	−1.18026	−0.590130	−	0.807309i	$-0.700923\pi$
$223$	−17.6250	−1.18026	−0.590130	+	0.807309i	$0.700923\pi$
$224$	0	0
$225$	−2.76089	−0.184059
$226$	0	0
$227$	16.1259	1.07032	0.535158	−	0.844752i	$-0.320252\pi$
$227$	16.1259	1.07032	0.535158	+	0.844752i	$0.320252\pi$
$228$	0	0
$229$	−0.536348	−0.0354429	−0.0177214	−	0.999843i	$-0.505641\pi$
$229$	−0.536348	−0.0354429	−0.0177214	+	0.999843i	$0.505641\pi$
$230$	0	0
$231$	0.208595	0.0137245
$232$	0	0
$233$	−15.8731	−1.03988	−0.519941	−	0.854202i	$-0.674046\pi$
$233$	−15.8731	−1.03988	−0.519941	+	0.854202i	$0.674046\pi$
$234$	0	0
$235$	6.24942	0.407667
$236$	0	0
$237$	30.6506	1.99097
$238$	0	0
$239$	13.3322	0.862388	0.431194	−	0.902259i	$-0.358092\pi$
$239$	13.3322	0.862388	0.431194	+	0.902259i	$0.358092\pi$
$240$	0	0
$241$	27.6260	1.77955	0.889774	−	0.456400i	$-0.150862\pi$
$241$	27.6260	1.77955	0.889774	+	0.456400i	$0.150862\pi$
$242$	0	0
$243$	17.8217	1.14326
$244$	0	0
$245$	13.3026	0.849871
$246$	0	0
$247$	−38.8899	−2.47451
$248$	0	0
$249$	−18.3811	−1.16486
$250$	0	0
$251$	−17.0856	−1.07843	−0.539217	−	0.842167i	$-0.681280\pi$
$251$	−17.0856	−1.07843	−0.539217	+	0.842167i	$0.681280\pi$
$252$	0	0
$253$	−7.75210	−0.487371
$254$	0	0
$255$	2.94469	0.184403
$256$	0	0
$257$	−20.7259	−1.29285	−0.646423	−	0.762979i	$-0.723736\pi$
$257$	−20.7259	−1.29285	−0.646423	+	0.762979i	$0.723736\pi$
$258$	0	0
$259$	0.0756206	0.00469883
$260$	0	0
$261$	−1.02599	−0.0635073
$262$	0	0
$263$	2.17910	0.134369	0.0671844	−	0.997741i	$-0.478598\pi$
$263$	2.17910	0.134369	0.0671844	+	0.997741i	$0.478598\pi$
$264$	0	0
$265$	−1.46153	−0.0897809
$266$	0	0
$267$	−17.2432	−1.05527
$268$	0	0
$269$	14.6207	0.891443	0.445721	−	0.895172i	$-0.352947\pi$
$269$	14.6207	0.891443	0.445721	+	0.895172i	$0.352947\pi$
$270$	0	0
$271$	−20.5421	−1.24785	−0.623923	−	0.781486i	$-0.714462\pi$
$271$	−20.5421	−1.24785	−0.623923	+	0.781486i	$0.714462\pi$
$272$	0	0
$273$	−0.390276	−0.0236206
$274$	0	0
$275$	4.75550	0.286767
$276$	0	0
$277$	−1.13936	−0.0684575	−0.0342287	−	0.999414i	$-0.510897\pi$
$277$	−1.13936	−0.0684575	−0.0342287	+	0.999414i	$0.510897\pi$
$278$	0	0
$279$	−19.4220	−1.16276
$280$	0	0
$281$	20.7389	1.23718	0.618588	−	0.785715i	$-0.287705\pi$
$281$	20.7389	1.23718	0.618588	+	0.785715i	$0.287705\pi$
$282$	0	0
$283$	−7.73489	−0.459792	−0.229896	−	0.973215i	$-0.573839\pi$
$283$	−7.73489	−0.459792	−0.229896	+	0.973215i	$0.573839\pi$
$284$	0	0
$285$	−25.7523	−1.52543
$286$	0	0
$287$	−0.107771	−0.00636154
$288$	0	0
$289$	−16.5189	−0.971698
$290$	0	0
$291$	−20.5000	−1.20173
$292$	0	0
$293$	20.9643	1.22475	0.612373	−	0.790569i	$-0.290215\pi$
$293$	20.9643	1.22475	0.612373	+	0.790569i	$0.290215\pi$
$294$	0	0
$295$	−11.1797	−0.650909
$296$	0	0
$297$	−7.73536	−0.448851
$298$	0	0
$299$	14.5040	0.838789
$300$	0	0
$301$	0.110342	0.00636003
$302$	0	0
$303$	6.22889	0.357840
$304$	0	0
$305$	4.16528	0.238503
$306$	0	0
$307$	−19.4810	−1.11184	−0.555921	−	0.831235i	$-0.687634\pi$
$307$	−19.4810	−1.11184	−0.555921	+	0.831235i	$0.687634\pi$
$308$	0	0
$309$	42.0872	2.39426
$310$	0	0
$311$	24.3549	1.38104	0.690519	−	0.723315i	$-0.257382\pi$
$311$	24.3549	1.38104	0.690519	+	0.723315i	$0.257382\pi$
$312$	0	0
$313$	−5.17014	−0.292234	−0.146117	−	0.989267i	$-0.546678\pi$
$313$	−5.17014	−0.292234	−0.146117	+	0.989267i	$0.546678\pi$
$314$	0	0
$315$	−0.103043	−0.00580581
$316$	0	0
$317$	23.6584	1.32879	0.664395	−	0.747382i	$-0.268689\pi$
$317$	23.6584	1.32879	0.664395	+	0.747382i	$0.268689\pi$
$318$	0	0
$319$	1.76722	0.0989455
$320$	0	0
$321$	−27.9669	−1.56096
$322$	0	0
$323$	−4.20767	−0.234121
$324$	0	0
$325$	−8.89743	−0.493541
$326$	0	0
$327$	−10.4378	−0.577214
$328$	0	0
$329$	−0.0896144	−0.00494060
$330$	0	0
$331$	16.2578	0.893610	0.446805	−	0.894631i	$-0.352562\pi$
$331$	16.2578	0.893610	0.446805	+	0.894631i	$0.352562\pi$
$332$	0	0
$333$	5.51989	0.302488
$334$	0	0
$335$	10.8338	0.591914
$336$	0	0
$337$	30.3639	1.65403	0.827013	−	0.562183i	$-0.190038\pi$
$337$	30.3639	1.65403	0.827013	+	0.562183i	$0.190038\pi$
$338$	0	0
$339$	−24.5570	−1.33375
$340$	0	0
$341$	33.4534	1.81160
$342$	0	0
$343$	−0.381528	−0.0206006
$344$	0	0
$345$	9.60432	0.517079
$346$	0	0
$347$	−16.8347	−0.903736	−0.451868	−	0.892085i	$-0.649242\pi$
$347$	−16.8347	−0.903736	−0.451868	+	0.892085i	$0.649242\pi$
$348$	0	0
$349$	6.23432	0.333715	0.166858	−	0.985981i	$-0.446638\pi$
$349$	6.23432	0.333715	0.166858	+	0.985981i	$0.446638\pi$
$350$	0	0
$351$	14.4727	0.772495
$352$	0	0
$353$	33.3874	1.77703	0.888516	−	0.458845i	$-0.151737\pi$
$353$	33.3874	1.77703	0.888516	+	0.458845i	$0.151737\pi$
$354$	0	0
$355$	11.0218	0.584974
$356$	0	0
$357$	−0.0422257	−0.00223482
$358$	0	0
$359$	−3.04882	−0.160911	−0.0804553	−	0.996758i	$-0.525637\pi$
$359$	−3.04882	−0.160911	−0.0804553	+	0.996758i	$0.525637\pi$
$360$	0	0
$361$	17.7975	0.936713
$362$	0	0
$363$	−1.65597	−0.0869157
$364$	0	0
$365$	12.2647	0.641965
$366$	0	0
$367$	27.6034	1.44089	0.720443	−	0.693514i	$-0.243938\pi$
$367$	27.6034	1.44089	0.720443	+	0.693514i	$0.243938\pi$
$368$	0	0
$369$	−7.86671	−0.409525
$370$	0	0
$371$	0.0209578	0.00108807
$372$	0	0
$373$	18.3863	0.952004	0.476002	−	0.879444i	$-0.342085\pi$
$373$	18.3863	0.952004	0.476002	+	0.879444i	$0.342085\pi$
$374$	0	0
$375$	−27.1181	−1.40037
$376$	0	0
$377$	−3.30643	−0.170290
$378$	0	0
$379$	−35.5217	−1.82463	−0.912314	−	0.409492i	$-0.865706\pi$
$379$	−35.5217	−1.82463	−0.912314	+	0.409492i	$0.865706\pi$
$380$	0	0
$381$	−30.6287	−1.56916
$382$	0	0
$383$	7.57845	0.387241	0.193620	−	0.981077i	$-0.437977\pi$
$383$	7.57845	0.387241	0.193620	+	0.981077i	$0.437977\pi$
$384$	0	0
$385$	0.177486	0.00904555
$386$	0	0
$387$	8.05439	0.409428
$388$	0	0
$389$	−18.0286	−0.914088	−0.457044	−	0.889444i	$-0.651092\pi$
$389$	−18.0286	−0.914088	−0.457044	+	0.889444i	$0.651092\pi$
$390$	0	0
$391$	1.56925	0.0793606
$392$	0	0
$393$	−7.56848	−0.381779
$394$	0	0
$395$	26.0796	1.31221
$396$	0	0
$397$	16.3607	0.821122	0.410561	−	0.911833i	$-0.365333\pi$
$397$	16.3607	0.821122	0.410561	+	0.911833i	$0.365333\pi$
$398$	0	0
$399$	0.369278	0.0184870
$400$	0	0
$401$	−4.52710	−0.226073	−0.113036	−	0.993591i	$-0.536058\pi$
$401$	−4.52710	−0.226073	−0.113036	+	0.993591i	$0.536058\pi$
$402$	0	0
$403$	−62.5906	−3.11786
$404$	0	0
$405$	20.9263	1.03984
$406$	0	0
$407$	−9.50774	−0.471281
$408$	0	0
$409$	−1.45865	−0.0721255	−0.0360627	−	0.999350i	$-0.511482\pi$
$409$	−1.45865	−0.0721255	−0.0360627	+	0.999350i	$0.511482\pi$
$410$	0	0
$411$	−23.3122	−1.14991
$412$	0	0
$413$	0.160313	0.00788849
$414$	0	0
$415$	−15.6399	−0.767733
$416$	0	0
$417$	−10.6322	−0.520663
$418$	0	0
$419$	−11.7922	−0.576089	−0.288045	−	0.957617i	$-0.593005\pi$
$419$	−11.7922	−0.576089	−0.288045	+	0.957617i	$0.593005\pi$
$420$	0	0
$421$	−10.1641	−0.495370	−0.247685	−	0.968841i	$-0.579670\pi$
$421$	−10.1641	−0.495370	−0.247685	+	0.968841i	$0.579670\pi$
$422$	0	0
$423$	−6.54136	−0.318052
$424$	0	0
$425$	−0.962653	−0.0466955
$426$	0	0
$427$	−0.0597286	−0.00289047
$428$	0	0
$429$	49.0692	2.36909
$430$	0	0
$431$	20.1107	0.968697	0.484348	−	0.874875i	$-0.339057\pi$
$431$	20.1107	0.968697	0.484348	+	0.874875i	$0.339057\pi$
$432$	0	0
$433$	6.64257	0.319222	0.159611	−	0.987180i	$-0.448976\pi$
$433$	6.64257	0.319222	0.159611	+	0.987180i	$0.448976\pi$
$434$	0	0
$435$	−2.18947	−0.104977
$436$	0	0
$437$	−13.7237	−0.656492
$438$	0	0
$439$	10.3223	0.492658	0.246329	−	0.969186i	$-0.420776\pi$
$439$	10.3223	0.492658	0.246329	+	0.969186i	$0.420776\pi$
$440$	0	0
$441$	−13.9240	−0.663048
$442$	0	0
$443$	36.9438	1.75525	0.877627	−	0.479344i	$-0.159125\pi$
$443$	36.9438	1.75525	0.877627	+	0.479344i	$0.159125\pi$
$444$	0	0
$445$	−14.6717	−0.695504
$446$	0	0
$447$	8.58616	0.406112
$448$	0	0
$449$	3.71503	0.175323	0.0876616	−	0.996150i	$-0.472061\pi$
$449$	3.71503	0.175323	0.0876616	+	0.996150i	$0.472061\pi$
$450$	0	0
$451$	13.5500	0.638046
$452$	0	0
$453$	32.3734	1.52104
$454$	0	0
$455$	−0.332073	−0.0155678
$456$	0	0
$457$	−12.7400	−0.595950	−0.297975	−	0.954574i	$-0.596311\pi$
$457$	−12.7400	−0.595950	−0.297975	+	0.954574i	$0.596311\pi$
$458$	0	0
$459$	1.56586	0.0730882
$460$	0	0
$461$	1.96330	0.0914401	0.0457200	−	0.998954i	$-0.485442\pi$
$461$	1.96330	0.0914401	0.0457200	+	0.998954i	$0.485442\pi$
$462$	0	0
$463$	14.0480	0.652864	0.326432	−	0.945221i	$-0.394154\pi$
$463$	14.0480	0.652864	0.326432	+	0.945221i	$0.394154\pi$
$464$	0	0
$465$	−41.4465	−1.92203
$466$	0	0
$467$	−11.2043	−0.518472	−0.259236	−	0.965814i	$-0.583471\pi$
$467$	−11.2043	−0.518472	−0.259236	+	0.965814i	$0.583471\pi$
$468$	0	0
$469$	−0.155353	−0.00717352
$470$	0	0
$471$	8.05275	0.371051
$472$	0	0
$473$	−13.8733	−0.637895
$474$	0	0
$475$	8.41872	0.386278
$476$	0	0
$477$	1.52980	0.0700448
$478$	0	0
$479$	1.01568	0.0464075	0.0232037	−	0.999731i	$-0.492613\pi$
$479$	1.01568	0.0464075	0.0232037	+	0.999731i	$0.492613\pi$
$480$	0	0
$481$	17.7888	0.811098
$482$	0	0
$483$	−0.137722	−0.00626659
$484$	0	0
$485$	−17.4428	−0.792035
$486$	0	0
$487$	10.0928	0.457346	0.228673	−	0.973503i	$-0.426561\pi$
$487$	10.0928	0.457346	0.228673	+	0.973503i	$0.426561\pi$
$488$	0	0
$489$	−28.3047	−1.27998
$490$	0	0
$491$	19.4058	0.875771	0.437886	−	0.899031i	$-0.355728\pi$
$491$	19.4058	0.875771	0.437886	+	0.899031i	$0.355728\pi$
$492$	0	0
$493$	−0.357738	−0.0161117
$494$	0	0
$495$	12.9555	0.582308
$496$	0	0
$497$	−0.158048	−0.00708941
$498$	0	0
$499$	−9.34761	−0.418456	−0.209228	−	0.977867i	$-0.567095\pi$
$499$	−9.34761	−0.418456	−0.209228	+	0.977867i	$0.567095\pi$
$500$	0	0
$501$	−19.7671	−0.883131
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	5.29996	0.235845
$506$	0	0
$507$	−62.7696	−2.78769
$508$	0	0
$509$	−3.58757	−0.159016	−0.0795082	−	0.996834i	$-0.525335\pi$
$509$	−3.58757	−0.159016	−0.0795082	+	0.996834i	$0.525335\pi$
$510$	0	0
$511$	−0.175872	−0.00778010
$512$	0	0
$513$	−13.6940	−0.604605
$514$	0	0
$515$	35.8106	1.57800
$516$	0	0
$517$	11.2672	0.495530
$518$	0	0
$519$	−14.2624	−0.626051
$520$	0	0
$521$	7.88058	0.345254	0.172627	−	0.984987i	$-0.444774\pi$
$521$	7.88058	0.345254	0.172627	+	0.984987i	$0.444774\pi$
$522$	0	0
$523$	−12.1925	−0.533141	−0.266570	−	0.963815i	$-0.585891\pi$
$523$	−12.1925	−0.533141	−0.266570	+	0.963815i	$0.585891\pi$
$524$	0	0
$525$	0.0844853	0.00368724
$526$	0	0
$527$	−6.77195	−0.294991
$528$	0	0
$529$	−17.8818	−0.777468
$530$	0	0
$531$	11.7020	0.507823
$532$	0	0
$533$	−25.3518	−1.09811
$534$	0	0
$535$	−23.7961	−1.02880
$536$	0	0
$537$	−3.52468	−0.152101
$538$	0	0
$539$	23.9834	1.03304
$540$	0	0
$541$	−1.32187	−0.0568317	−0.0284158	−	0.999596i	$-0.509046\pi$
$541$	−1.32187	−0.0568317	−0.0284158	+	0.999596i	$0.509046\pi$
$542$	0	0
$543$	−3.67525	−0.157720
$544$	0	0
$545$	−8.88122	−0.380429
$546$	0	0
$547$	10.2763	0.439381	0.219690	−	0.975570i	$-0.429495\pi$
$547$	10.2763	0.439381	0.219690	+	0.975570i	$0.429495\pi$
$548$	0	0
$549$	−4.35986	−0.186074
$550$	0	0
$551$	3.12854	0.133280
$552$	0	0
$553$	−0.373972	−0.0159029
$554$	0	0
$555$	11.7794	0.500009
$556$	0	0
$557$	17.3361	0.734556	0.367278	−	0.930111i	$-0.380290\pi$
$557$	17.3361	0.734556	0.367278	+	0.930111i	$0.380290\pi$
$558$	0	0
$559$	25.9566	1.09785
$560$	0	0
$561$	5.30902	0.224147
$562$	0	0
$563$	−32.5202	−1.37056	−0.685282	−	0.728278i	$-0.740321\pi$
$563$	−32.5202	−1.37056	−0.685282	+	0.728278i	$0.740321\pi$
$564$	0	0
$565$	−20.8947	−0.879048
$566$	0	0
$567$	−0.300075	−0.0126020
$568$	0	0
$569$	−15.9303	−0.667832	−0.333916	−	0.942603i	$-0.608370\pi$
$569$	−15.9303	−0.667832	−0.333916	+	0.942603i	$0.608370\pi$
$570$	0	0
$571$	32.6026	1.36438	0.682189	−	0.731176i	$-0.261028\pi$
$571$	32.6026	1.36438	0.682189	+	0.731176i	$0.261028\pi$
$572$	0	0
$573$	−12.6970	−0.530423
$574$	0	0
$575$	−3.13977	−0.130937
$576$	0	0
$577$	−14.4125	−0.600001	−0.300001	−	0.953939i	$-0.596987\pi$
$577$	−14.4125	−0.600001	−0.300001	+	0.953939i	$0.596987\pi$
$578$	0	0
$579$	−16.5755	−0.688856
$580$	0	0
$581$	0.224270	0.00930431
$582$	0	0
$583$	−2.63501	−0.109131
$584$	0	0
$585$	−24.2395	−1.00218
$586$	0	0
$587$	−19.5819	−0.808231	−0.404116	−	0.914708i	$-0.632421\pi$
$587$	−19.5819	−0.808231	−0.404116	+	0.914708i	$0.632421\pi$
$588$	0	0
$589$	59.2230	2.44024
$590$	0	0
$591$	−18.6202	−0.765932
$592$	0	0
$593$	−44.2596	−1.81752	−0.908761	−	0.417316i	$-0.862971\pi$
$593$	−44.2596	−1.81752	−0.908761	+	0.417316i	$0.862971\pi$
$594$	0	0
$595$	−0.0359285	−0.00147292
$596$	0	0
$597$	−48.4029	−1.98100
$598$	0	0
$599$	−18.6044	−0.760156	−0.380078	−	0.924954i	$-0.624103\pi$
$599$	−18.6044	−0.760156	−0.380078	+	0.924954i	$0.624103\pi$
$600$	0	0
$601$	−22.6978	−0.925861	−0.462930	−	0.886395i	$-0.653202\pi$
$601$	−22.6978	−0.925861	−0.462930	+	0.886395i	$0.653202\pi$
$602$	0	0
$603$	−11.3399	−0.461796
$604$	0	0
$605$	−1.40901	−0.0572843
$606$	0	0
$607$	42.1731	1.71175	0.855876	−	0.517182i	$-0.173019\pi$
$607$	42.1731	1.71175	0.855876	+	0.517182i	$0.173019\pi$
$608$	0	0
$609$	0.0313961	0.00127224
$610$	0	0
$611$	−21.0806	−0.852831
$612$	0	0
$613$	3.75440	0.151639	0.0758194	−	0.997122i	$-0.475843\pi$
$613$	3.75440	0.151639	0.0758194	+	0.997122i	$0.475843\pi$
$614$	0	0
$615$	−16.7875	−0.676939
$616$	0	0
$617$	29.9117	1.20420	0.602099	−	0.798421i	$-0.294331\pi$
$617$	29.9117	1.20420	0.602099	+	0.798421i	$0.294331\pi$
$618$	0	0
$619$	−9.29474	−0.373587	−0.186793	−	0.982399i	$-0.559810\pi$
$619$	−9.29474	−0.373587	−0.186793	+	0.982399i	$0.559810\pi$
$620$	0	0
$621$	5.10718	0.204944
$622$	0	0
$623$	0.210386	0.00842896
$624$	0	0
$625$	−16.1348	−0.645391
$626$	0	0
$627$	−46.4292	−1.85420
$628$	0	0
$629$	1.92465	0.0767407
$630$	0	0
$631$	41.8438	1.66577	0.832887	−	0.553443i	$-0.186686\pi$
$631$	41.8438	1.66577	0.832887	+	0.553443i	$0.186686\pi$
$632$	0	0
$633$	3.35017	0.133157
$634$	0	0
$635$	−26.0609	−1.03420
$636$	0	0
$637$	−44.8725	−1.77791
$638$	0	0
$639$	−11.5366	−0.456382
$640$	0	0
$641$	−43.5052	−1.71835	−0.859175	−	0.511681i	$-0.829023\pi$
$641$	−43.5052	−1.71835	−0.859175	+	0.511681i	$0.829023\pi$
$642$	0	0
$643$	−40.0775	−1.58050	−0.790251	−	0.612783i	$-0.790050\pi$
$643$	−40.0775	−1.58050	−0.790251	+	0.612783i	$0.790050\pi$
$644$	0	0
$645$	17.1881	0.676779
$646$	0	0
$647$	−25.9820	−1.02146	−0.510729	−	0.859742i	$-0.670624\pi$
$647$	−25.9820	−1.02146	−0.510729	+	0.859742i	$0.670624\pi$
$648$	0	0
$649$	−20.1561	−0.791196
$650$	0	0
$651$	0.594327	0.0232935
$652$	0	0
$653$	46.6700	1.82634	0.913169	−	0.407582i	$-0.133628\pi$
$653$	46.6700	1.82634	0.913169	+	0.407582i	$0.133628\pi$
$654$	0	0
$655$	−6.43977	−0.251623
$656$	0	0
$657$	−12.8377	−0.500845
$658$	0	0
$659$	7.54117	0.293762	0.146881	−	0.989154i	$-0.453076\pi$
$659$	7.54117	0.293762	0.146881	+	0.989154i	$0.453076\pi$
$660$	0	0
$661$	−15.5700	−0.605602	−0.302801	−	0.953054i	$-0.597922\pi$
$661$	−15.5700	−0.605602	−0.302801	+	0.953054i	$0.597922\pi$
$662$	0	0
$663$	−9.93306	−0.385768
$664$	0	0
$665$	0.314207	0.0121844
$666$	0	0
$667$	−1.16679	−0.0451783
$668$	0	0
$669$	39.3688	1.52209
$670$	0	0
$671$	7.50965	0.289907
$672$	0	0
$673$	44.4791	1.71454	0.857271	−	0.514865i	$-0.172158\pi$
$673$	44.4791	1.71454	0.857271	+	0.514865i	$0.172158\pi$
$674$	0	0
$675$	−3.13298	−0.120589
$676$	0	0
$677$	−45.8838	−1.76346	−0.881729	−	0.471756i	$-0.843620\pi$
$677$	−45.8838	−1.76346	−0.881729	+	0.471756i	$0.843620\pi$
$678$	0	0
$679$	0.250123	0.00959883
$680$	0	0
$681$	−36.0203	−1.38030
$682$	0	0
$683$	10.1759	0.389370	0.194685	−	0.980866i	$-0.437632\pi$
$683$	10.1759	0.389370	0.194685	+	0.980866i	$0.437632\pi$
$684$	0	0
$685$	−19.8356	−0.757880
$686$	0	0
$687$	1.19803	0.0457078
$688$	0	0
$689$	4.93004	0.187820
$690$	0	0
$691$	30.9082	1.17580	0.587902	−	0.808932i	$-0.299954\pi$
$691$	30.9082	1.17580	0.587902	+	0.808932i	$0.299954\pi$
$692$	0	0
$693$	−0.185778	−0.00705711
$694$	0	0
$695$	−9.04662	−0.343158
$696$	0	0
$697$	−2.74292	−0.103896
$698$	0	0
$699$	35.4555	1.34105
$700$	0	0
$701$	−28.9806	−1.09458	−0.547290	−	0.836943i	$-0.684341\pi$
$701$	−28.9806	−1.09458	−0.547290	+	0.836943i	$0.684341\pi$
$702$	0	0
$703$	−16.8317	−0.634819
$704$	0	0
$705$	−13.9592	−0.525736
$706$	0	0
$707$	−0.0759994	−0.00285825
$708$	0	0
$709$	−12.8021	−0.480794	−0.240397	−	0.970675i	$-0.577278\pi$
$709$	−12.8021	−0.480794	−0.240397	+	0.970675i	$0.577278\pi$
$710$	0	0
$711$	−27.2979	−1.02375
$712$	0	0
$713$	−22.0872	−0.827174
$714$	0	0
$715$	41.7514	1.56141
$716$	0	0
$717$	−29.7800	−1.11215
$718$	0	0
$719$	−13.8124	−0.515114	−0.257557	−	0.966263i	$-0.582918\pi$
$719$	−13.8124	−0.515114	−0.257557	+	0.966263i	$0.582918\pi$
$720$	0	0
$721$	−0.513511	−0.0191241
$722$	0	0
$723$	−61.7079	−2.29494
$724$	0	0
$725$	0.715762	0.0265828
$726$	0	0
$727$	4.95693	0.183842	0.0919212	−	0.995766i	$-0.470699\pi$
$727$	4.95693	0.183842	0.0919212	+	0.995766i	$0.470699\pi$
$728$	0	0
$729$	−6.77644	−0.250979
$730$	0	0
$731$	2.80836	0.103871
$732$	0	0
$733$	−16.1193	−0.595381	−0.297691	−	0.954662i	$-0.596216\pi$
$733$	−16.1193	−0.595381	−0.297691	+	0.954662i	$0.596216\pi$
$734$	0	0
$735$	−29.7138	−1.09601
$736$	0	0
$737$	19.5324	0.719486
$738$	0	0
$739$	−5.76676	−0.212134	−0.106067	−	0.994359i	$-0.533826\pi$
$739$	−5.76676	−0.212134	−0.106067	+	0.994359i	$0.533826\pi$
$740$	0	0
$741$	86.8679	3.19117
$742$	0	0
$743$	−35.2149	−1.29191	−0.645954	−	0.763376i	$-0.723540\pi$
$743$	−35.2149	−1.29191	−0.645954	+	0.763376i	$0.723540\pi$
$744$	0	0
$745$	7.30569	0.267660
$746$	0	0
$747$	16.3705	0.598966
$748$	0	0
$749$	0.341228	0.0124682
$750$	0	0
$751$	24.3852	0.889829	0.444914	−	0.895573i	$-0.353234\pi$
$751$	24.3852	0.889829	0.444914	+	0.895573i	$0.353234\pi$
$752$	0	0
$753$	38.1639	1.39077
$754$	0	0
$755$	27.5455	1.00248
$756$	0	0
$757$	−43.7211	−1.58907	−0.794534	−	0.607219i	$-0.792285\pi$
$757$	−43.7211	−1.58907	−0.794534	+	0.607219i	$0.792285\pi$
$758$	0	0
$759$	17.3158	0.628523
$760$	0	0
$761$	30.1317	1.09227	0.546136	−	0.837697i	$-0.316098\pi$
$761$	30.1317	1.09227	0.546136	+	0.837697i	$0.316098\pi$
$762$	0	0
$763$	0.127353	0.00461050
$764$	0	0
$765$	−2.62258	−0.0948196
$766$	0	0
$767$	37.7116	1.36169
$768$	0	0
$769$	0.306214	0.0110424	0.00552118	−	0.999985i	$-0.498243\pi$
$769$	0.306214	0.0110424	0.00552118	+	0.999985i	$0.498243\pi$
$770$	0	0
$771$	46.2951	1.66728
$772$	0	0
$773$	25.4423	0.915095	0.457548	−	0.889185i	$-0.348728\pi$
$773$	25.4423	0.915095	0.457548	+	0.889185i	$0.348728\pi$
$774$	0	0
$775$	13.5493	0.486707
$776$	0	0
$777$	−0.168913	−0.00605971
$778$	0	0
$779$	23.9878	0.859452
$780$	0	0
$781$	19.8713	0.711050
$782$	0	0
$783$	−1.16427	−0.0416076
$784$	0	0
$785$	6.85182	0.244552
$786$	0	0
$787$	26.0334	0.927990	0.463995	−	0.885838i	$-0.346416\pi$
$787$	26.0334	0.927990	0.463995	+	0.885838i	$0.346416\pi$
$788$	0	0
$789$	−4.86741	−0.173285
$790$	0	0
$791$	0.299623	0.0106534
$792$	0	0
$793$	−14.0504	−0.498944
$794$	0	0
$795$	3.26459	0.115783
$796$	0	0
$797$	−35.9759	−1.27433	−0.637167	−	0.770726i	$-0.719894\pi$
$797$	−35.9759	−1.27433	−0.637167	+	0.770726i	$0.719894\pi$
$798$	0	0
$799$	−2.28081	−0.0806892
$800$	0	0
$801$	15.3571	0.542615
$802$	0	0
$803$	22.1122	0.780324
$804$	0	0
$805$	−0.117183	−0.00413017
$806$	0	0
$807$	−32.6582	−1.14962
$808$	0	0
$809$	−13.0235	−0.457880	−0.228940	−	0.973441i	$-0.573526\pi$
$809$	−13.0235	−0.457880	−0.228940	+	0.973441i	$0.573526\pi$
$810$	0	0
$811$	11.3948	0.400126	0.200063	−	0.979783i	$-0.435885\pi$
$811$	11.3948	0.400126	0.200063	+	0.979783i	$0.435885\pi$
$812$	0	0
$813$	45.8847	1.60925
$814$	0	0
$815$	−24.0836	−0.843611
$816$	0	0
$817$	−24.5601	−0.859249
$818$	0	0
$819$	0.347586	0.0121456
$820$	0	0
$821$	−24.4995	−0.855037	−0.427518	−	0.904007i	$-0.640612\pi$
$821$	−24.4995	−0.855037	−0.427518	+	0.904007i	$0.640612\pi$
$822$	0	0
$823$	−24.5839	−0.856940	−0.428470	−	0.903556i	$-0.640947\pi$
$823$	−24.5839	−0.856940	−0.428470	+	0.903556i	$0.640947\pi$
$824$	0	0
$825$	−10.6223	−0.369821
$826$	0	0
$827$	37.4433	1.30203	0.651016	−	0.759064i	$-0.274343\pi$
$827$	37.4433	1.30203	0.651016	+	0.759064i	$0.274343\pi$
$828$	0	0
$829$	11.8484	0.411511	0.205755	−	0.978603i	$-0.434035\pi$
$829$	11.8484	0.411511	0.205755	+	0.978603i	$0.434035\pi$
$830$	0	0
$831$	2.54497	0.0882841
$832$	0	0
$833$	−4.85495	−0.168214
$834$	0	0
$835$	−16.8192	−0.582053
$836$	0	0
$837$	−22.0395	−0.761798
$838$	0	0
$839$	−46.9904	−1.62229	−0.811144	−	0.584846i	$-0.801155\pi$
$839$	−46.9904	−1.62229	−0.811144	+	0.584846i	$0.801155\pi$
$840$	0	0
$841$	−28.7340	−0.990828
$842$	0	0
$843$	−46.3241	−1.59549
$844$	0	0
$845$	−53.4086	−1.83731
$846$	0	0
$847$	0.0202046	0.000694240 0
$848$	0	0
$849$	17.2773	0.592956
$850$	0	0
$851$	6.27738	0.215186
$852$	0	0
$853$	−0.145158	−0.00497013	−0.00248506	−	0.999997i	$-0.500791\pi$
$853$	−0.145158	−0.00497013	−0.00248506	+	0.999997i	$0.500791\pi$
$854$	0	0
$855$	22.9354	0.784373
$856$	0	0
$857$	−39.0523	−1.33400	−0.667001	−	0.745056i	$-0.732423\pi$
$857$	−39.0523	−1.33400	−0.667001	+	0.745056i	$0.732423\pi$
$858$	0	0
$859$	53.5578	1.82737	0.913684	−	0.406426i	$-0.133225\pi$
$859$	53.5578	1.82737	0.913684	+	0.406426i	$0.133225\pi$
$860$	0	0
$861$	0.240727	0.00820396
$862$	0	0
$863$	41.7330	1.42061	0.710305	−	0.703894i	$-0.248557\pi$
$863$	41.7330	1.42061	0.710305	+	0.703894i	$0.248557\pi$
$864$	0	0
$865$	−12.1354	−0.412617
$866$	0	0
$867$	36.8980	1.25312
$868$	0	0
$869$	47.0193	1.59502
$870$	0	0
$871$	−36.5447	−1.23827
$872$	0	0
$873$	18.2576	0.617925
$874$	0	0
$875$	0.330872	0.0111855
$876$	0	0
$877$	−7.63641	−0.257863	−0.128932	−	0.991653i	$-0.541155\pi$
$877$	−7.63641	−0.257863	−0.128932	+	0.991653i	$0.541155\pi$
$878$	0	0
$879$	−46.8276	−1.57946
$880$	0	0
$881$	−43.5104	−1.46590	−0.732950	−	0.680282i	$-0.761857\pi$
$881$	−43.5104	−1.46590	−0.732950	+	0.680282i	$0.761857\pi$
$882$	0	0
$883$	29.5763	0.995323	0.497662	−	0.867371i	$-0.334192\pi$
$883$	29.5763	0.995323	0.497662	+	0.867371i	$0.334192\pi$
$884$	0	0
$885$	24.9720	0.839425
$886$	0	0
$887$	−12.6535	−0.424862	−0.212431	−	0.977176i	$-0.568138\pi$
$887$	−12.6535	−0.424862	−0.212431	+	0.977176i	$0.568138\pi$
$888$	0	0
$889$	0.373704	0.0125336
$890$	0	0
$891$	37.7283	1.26395
$892$	0	0
$893$	19.9464	0.667482
$894$	0	0
$895$	−2.99904	−0.100247
$896$	0	0
$897$	−32.3974	−1.08172
$898$	0	0
$899$	5.03516	0.167932
$900$	0	0
$901$	0.533403	0.0177702
$902$	0	0
$903$	−0.246470	−0.00820202
$904$	0	0
$905$	−3.12715	−0.103950
$906$	0	0
$907$	30.7875	1.02228	0.511141	−	0.859497i	$-0.329223\pi$
$907$	30.7875	1.02228	0.511141	+	0.859497i	$0.329223\pi$
$908$	0	0
$909$	−5.54754	−0.184000
$910$	0	0
$911$	14.5709	0.482754	0.241377	−	0.970431i	$-0.422401\pi$
$911$	14.5709	0.482754	0.241377	+	0.970431i	$0.422401\pi$
$912$	0	0
$913$	−28.1974	−0.933198
$914$	0	0
$915$	−9.30393	−0.307578
$916$	0	0
$917$	0.0923440	0.00304947
$918$	0	0
$919$	22.9178	0.755988	0.377994	−	0.925808i	$-0.376614\pi$
$919$	22.9178	0.755988	0.377994	+	0.925808i	$0.376614\pi$
$920$	0	0
$921$	43.5145	1.43385
$922$	0	0
$923$	−37.1787	−1.22375
$924$	0	0
$925$	−3.85084	−0.126615
$926$	0	0
$927$	−37.4835	−1.23112
$928$	0	0
$929$	−29.0366	−0.952660	−0.476330	−	0.879267i	$-0.658033\pi$
$929$	−29.0366	−0.952660	−0.476330	+	0.879267i	$0.658033\pi$
$930$	0	0
$931$	42.4582	1.39151
$932$	0	0
$933$	−54.4011	−1.78101
$934$	0	0
$935$	4.51727	0.147730
$936$	0	0
$937$	6.57117	0.214671	0.107335	−	0.994223i	$-0.465768\pi$
$937$	6.57117	0.214671	0.107335	+	0.994223i	$0.465768\pi$
$938$	0	0
$939$	11.5485	0.376870
$940$	0	0
$941$	31.8108	1.03700	0.518502	−	0.855076i	$-0.326490\pi$
$941$	31.8108	1.03700	0.518502	+	0.855076i	$0.326490\pi$
$942$	0	0
$943$	−8.94626	−0.291330
$944$	0	0
$945$	−0.116930	−0.00380374
$946$	0	0
$947$	42.9192	1.39469	0.697344	−	0.716737i	$-0.254365\pi$
$947$	42.9192	1.39469	0.697344	+	0.716737i	$0.254365\pi$
$948$	0	0
$949$	−41.3715	−1.34298
$950$	0	0
$951$	−52.8455	−1.71363
$952$	0	0
$953$	−3.87077	−0.125386	−0.0626932	−	0.998033i	$-0.519969\pi$
$953$	−3.87077	−0.125386	−0.0626932	+	0.998033i	$0.519969\pi$
$954$	0	0
$955$	−10.8034	−0.349591
$956$	0	0
$957$	−3.94742	−0.127602
$958$	0	0
$959$	0.284435	0.00918490
$960$	0	0
$961$	64.3153	2.07469
$962$	0	0
$963$	24.9078	0.802641
$964$	0	0
$965$	−14.1036	−0.454010
$966$	0	0
$967$	18.3174	0.589049	0.294525	−	0.955644i	$-0.404839\pi$
$967$	18.3174	0.589049	0.294525	+	0.955644i	$0.404839\pi$
$968$	0	0
$969$	9.39863	0.301927
$970$	0	0
$971$	39.2301	1.25895	0.629477	−	0.777019i	$-0.283269\pi$
$971$	39.2301	1.25895	0.629477	+	0.777019i	$0.283269\pi$
$972$	0	0
$973$	0.129725	0.00415880
$974$	0	0
$975$	19.8741	0.636480
$976$	0	0
$977$	−49.8287	−1.59416	−0.797081	−	0.603873i	$-0.793623\pi$
$977$	−49.8287	−1.59416	−0.797081	+	0.603873i	$0.793623\pi$
$978$	0	0
$979$	−26.4518	−0.845403
$980$	0	0
$981$	9.29610	0.296801
$982$	0	0
$983$	36.4191	1.16159	0.580794	−	0.814051i	$-0.302742\pi$
$983$	36.4191	1.16159	0.580794	+	0.814051i	$0.302742\pi$
$984$	0	0
$985$	−15.8433	−0.504809
$986$	0	0
$987$	0.200170	0.00637150
$988$	0	0
$989$	9.15969	0.291261
$990$	0	0
$991$	25.7519	0.818036	0.409018	−	0.912526i	$-0.365871\pi$
$991$	25.7519	0.818036	0.409018	+	0.912526i	$0.365871\pi$
$992$	0	0
$993$	−36.3148	−1.15242
$994$	0	0
$995$	−41.1845	−1.30564
$996$	0	0
$997$	53.2433	1.68623	0.843116	−	0.537732i	$-0.180719\pi$
$997$	53.2433	1.68623	0.843116	+	0.537732i	$0.180719\pi$
$998$	0	0
$999$	6.26382	0.198178

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.8	✓	33
4.3	odd	2		8048.2.a.y.1.26		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.8	✓	33	1.1	even	1	trivial
8048.2.a.y.1.26		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)

\(f(q)\)	\(=\)	\(q-2.23369 q^{3} -1.90057 q^{5} +0.0272535 q^{7} +1.98935 q^{9} +O(q^{10})\)	\(q-2.23369 q^{3} -1.90057 q^{5} +0.0272535 q^{7} +1.98935 q^{9} -3.42657 q^{11} +6.41103 q^{13} +4.24528 q^{15} +0.693638 q^{17} -6.06610 q^{19} -0.0608757 q^{21} +2.26235 q^{23} -1.38783 q^{25} +2.25747 q^{27} -0.515741 q^{29} -9.76295 q^{31} +7.65388 q^{33} -0.0517972 q^{35} +2.77471 q^{37} -14.3202 q^{39} -3.95440 q^{41} +4.04875 q^{43} -3.78091 q^{45} -3.28818 q^{47} -6.99926 q^{49} -1.54937 q^{51} +0.768994 q^{53} +6.51243 q^{55} +13.5498 q^{57} +5.88230 q^{59} -2.19160 q^{61} +0.0542168 q^{63} -12.1846 q^{65} -5.70029 q^{67} -5.05339 q^{69} -5.79918 q^{71} -6.45318 q^{73} +3.09998 q^{75} -0.0933859 q^{77} -13.7220 q^{79} -11.0105 q^{81} +8.22906 q^{83} -1.31831 q^{85} +1.15200 q^{87} +7.71962 q^{89} +0.174723 q^{91} +21.8074 q^{93} +11.5290 q^{95} +9.17764 q^{97} -6.81666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 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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