LMFDB - Embedded newform 4024.2.a.g.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.71136 q^{3} -1.51865 q^{5} -2.20669 q^{7} +4.35148 q^{9} +O(q^{10})$	$q-2.71136 q^{3} -1.51865 q^{5} -2.20669 q^{7} +4.35148 q^{9} +6.02812 q^{11} +4.45544 q^{13} +4.11762 q^{15} -3.55509 q^{17} +3.54849 q^{19} +5.98314 q^{21} -4.23827 q^{23} -2.69369 q^{25} -3.66435 q^{27} +1.06909 q^{29} +6.50114 q^{31} -16.3444 q^{33} +3.35120 q^{35} -10.6956 q^{37} -12.0803 q^{39} +6.73973 q^{41} +10.4465 q^{43} -6.60839 q^{45} +5.48724 q^{47} -2.13051 q^{49} +9.63914 q^{51} -5.47411 q^{53} -9.15463 q^{55} -9.62123 q^{57} -9.11526 q^{59} -5.04792 q^{61} -9.60238 q^{63} -6.76626 q^{65} -0.260331 q^{67} +11.4915 q^{69} +14.2134 q^{71} -15.3949 q^{73} +7.30358 q^{75} -13.3022 q^{77} -12.2884 q^{79} -3.11906 q^{81} -16.8494 q^{83} +5.39895 q^{85} -2.89870 q^{87} +11.3252 q^{89} -9.83178 q^{91} -17.6269 q^{93} -5.38892 q^{95} +8.93984 q^{97} +26.2313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−2.71136	−1.56541	−0.782703	−	0.622396i	$-0.786159\pi$
$3$	−2.71136	−1.56541	−0.782703	+	0.622396i	$0.786159\pi$
$4$	0	0
$5$	−1.51865	−0.679162	−0.339581	−	0.940577i	$-0.610285\pi$
$5$	−1.51865	−0.679162	−0.339581	+	0.940577i	$0.610285\pi$
$6$	0	0
$7$	−2.20669	−0.834052	−0.417026	−	0.908895i	$-0.636928\pi$
$7$	−2.20669	−0.834052	−0.417026	+	0.908895i	$0.636928\pi$
$8$	0	0
$9$	4.35148	1.45049
$10$	0	0
$11$	6.02812	1.81755	0.908774	−	0.417289i	$-0.137020\pi$
$11$	6.02812	1.81755	0.908774	+	0.417289i	$0.137020\pi$
$12$	0	0
$13$	4.45544	1.23572	0.617858	−	0.786290i	$-0.288001\pi$
$13$	4.45544	1.23572	0.617858	+	0.786290i	$0.288001\pi$
$14$	0	0
$15$	4.11762	1.06316
$16$	0	0
$17$	−3.55509	−0.862237	−0.431118	−	0.902295i	$-0.641881\pi$
$17$	−3.55509	−0.862237	−0.431118	+	0.902295i	$0.641881\pi$
$18$	0	0
$19$	3.54849	0.814079	0.407039	−	0.913411i	$-0.366561\pi$
$19$	3.54849	0.814079	0.407039	+	0.913411i	$0.366561\pi$
$20$	0	0
$21$	5.98314	1.30563
$22$	0	0
$23$	−4.23827	−0.883740	−0.441870	−	0.897079i	$-0.645685\pi$
$23$	−4.23827	−0.883740	−0.441870	+	0.897079i	$0.645685\pi$
$24$	0	0
$25$	−2.69369	−0.538739
$26$	0	0
$27$	−3.66435	−0.705204
$28$	0	0
$29$	1.06909	0.198526	0.0992629	−	0.995061i	$-0.468352\pi$
$29$	1.06909	0.198526	0.0992629	+	0.995061i	$0.468352\pi$
$30$	0	0
$31$	6.50114	1.16764	0.583820	−	0.811883i	$-0.301557\pi$
$31$	6.50114	1.16764	0.583820	+	0.811883i	$0.301557\pi$
$32$	0	0
$33$	−16.3444	−2.84520
$34$	0	0
$35$	3.35120	0.566456
$36$	0	0
$37$	−10.6956	−1.75835	−0.879174	−	0.476501i	$-0.841905\pi$
$37$	−10.6956	−1.75835	−0.879174	+	0.476501i	$0.841905\pi$
$38$	0	0
$39$	−12.0803	−1.93440
$40$	0	0
$41$	6.73973	1.05257	0.526285	−	0.850308i	$-0.323585\pi$
$41$	6.73973	1.05257	0.526285	+	0.850308i	$0.323585\pi$
$42$	0	0
$43$	10.4465	1.59308	0.796541	−	0.604585i	$-0.206661\pi$
$43$	10.4465	1.59308	0.796541	+	0.604585i	$0.206661\pi$
$44$	0	0
$45$	−6.60839	−0.985120
$46$	0	0
$47$	5.48724	0.800396	0.400198	−	0.916429i	$-0.368941\pi$
$47$	5.48724	0.800396	0.400198	+	0.916429i	$0.368941\pi$
$48$	0	0
$49$	−2.13051	−0.304358
$50$	0	0
$51$	9.63914	1.34975
$52$	0	0
$53$	−5.47411	−0.751926	−0.375963	−	0.926635i	$-0.622688\pi$
$53$	−5.47411	−0.751926	−0.375963	+	0.926635i	$0.622688\pi$
$54$	0	0
$55$	−9.15463	−1.23441
$56$	0	0
$57$	−9.62123	−1.27436
$58$	0	0
$59$	−9.11526	−1.18671	−0.593353	−	0.804942i	$-0.702196\pi$
$59$	−9.11526	−1.18671	−0.593353	+	0.804942i	$0.702196\pi$
$60$	0	0
$61$	−5.04792	−0.646319	−0.323160	−	0.946344i	$-0.604745\pi$
$61$	−5.04792	−0.646319	−0.323160	+	0.946344i	$0.604745\pi$
$62$	0	0
$63$	−9.60238	−1.20979
$64$	0	0
$65$	−6.76626	−0.839251
$66$	0	0
$67$	−0.260331	−0.0318045	−0.0159022	−	0.999874i	$-0.505062\pi$
$67$	−0.260331	−0.0318045	−0.0159022	+	0.999874i	$0.505062\pi$
$68$	0	0
$69$	11.4915	1.38341
$70$	0	0
$71$	14.2134	1.68682	0.843412	−	0.537268i	$-0.180544\pi$
$71$	14.2134	1.68682	0.843412	+	0.537268i	$0.180544\pi$
$72$	0	0
$73$	−15.3949	−1.80184	−0.900919	−	0.433988i	$-0.857106\pi$
$73$	−15.3949	−1.80184	−0.900919	+	0.433988i	$0.857106\pi$
$74$	0	0
$75$	7.30358	0.843344
$76$	0	0
$77$	−13.3022	−1.51593
$78$	0	0
$79$	−12.2884	−1.38255	−0.691277	−	0.722590i	$-0.742952\pi$
$79$	−12.2884	−1.38255	−0.691277	+	0.722590i	$0.742952\pi$
$80$	0	0
$81$	−3.11906	−0.346563
$82$	0	0
$83$	−16.8494	−1.84947	−0.924733	−	0.380617i	$-0.875712\pi$
$83$	−16.8494	−1.84947	−0.924733	+	0.380617i	$0.875712\pi$
$84$	0	0
$85$	5.39895	0.585598
$86$	0	0
$87$	−2.89870	−0.310773
$88$	0	0
$89$	11.3252	1.20047	0.600237	−	0.799823i	$-0.295073\pi$
$89$	11.3252	1.20047	0.600237	+	0.799823i	$0.295073\pi$
$90$	0	0
$91$	−9.83178	−1.03065
$92$	0	0
$93$	−17.6269	−1.82783
$94$	0	0
$95$	−5.38892	−0.552891
$96$	0	0
$97$	8.93984	0.907703	0.453852	−	0.891077i	$-0.350050\pi$
$97$	8.93984	0.907703	0.453852	+	0.891077i	$0.350050\pi$
$98$	0	0
$99$	26.2313	2.63634
$100$	0	0
$101$	−5.49714	−0.546986	−0.273493	−	0.961874i	$-0.588179\pi$
$101$	−5.49714	−0.546986	−0.273493	+	0.961874i	$0.588179\pi$
$102$	0	0
$103$	−1.00554	−0.0990788	−0.0495394	−	0.998772i	$-0.515775\pi$
$103$	−1.00554	−0.0990788	−0.0495394	+	0.998772i	$0.515775\pi$
$104$	0	0
$105$	−9.08631	−0.886734
$106$	0	0
$107$	9.36161	0.905021	0.452510	−	0.891759i	$-0.350529\pi$
$107$	9.36161	0.905021	0.452510	+	0.891759i	$0.350529\pi$
$108$	0	0
$109$	6.64810	0.636773	0.318386	−	0.947961i	$-0.396859\pi$
$109$	6.64810	0.636773	0.318386	+	0.947961i	$0.396859\pi$
$110$	0	0
$111$	28.9997	2.75253
$112$	0	0
$113$	7.66649	0.721202	0.360601	−	0.932720i	$-0.382572\pi$
$113$	7.66649	0.721202	0.360601	+	0.932720i	$0.382572\pi$
$114$	0	0
$115$	6.43646	0.600203
$116$	0	0
$117$	19.3877	1.79240
$118$	0	0
$119$	7.84500	0.719150
$120$	0	0
$121$	25.3383	2.30348
$122$	0	0
$123$	−18.2739	−1.64770
$124$	0	0
$125$	11.6840	1.04505
$126$	0	0
$127$	10.4852	0.930407	0.465203	−	0.885204i	$-0.345981\pi$
$127$	10.4852	0.930407	0.465203	+	0.885204i	$0.345981\pi$
$128$	0	0
$129$	−28.3243	−2.49382
$130$	0	0
$131$	−17.4179	−1.52181	−0.760906	−	0.648863i	$-0.775245\pi$
$131$	−17.4179	−1.52181	−0.760906	+	0.648863i	$0.775245\pi$
$132$	0	0
$133$	−7.83042	−0.678983
$134$	0	0
$135$	5.56487	0.478948
$136$	0	0
$137$	14.2686	1.21905	0.609526	−	0.792766i	$-0.291360\pi$
$137$	14.2686	1.21905	0.609526	+	0.792766i	$0.291360\pi$
$138$	0	0
$139$	15.4465	1.31016	0.655078	−	0.755562i	$-0.272636\pi$
$139$	15.4465	1.31016	0.655078	+	0.755562i	$0.272636\pi$
$140$	0	0
$141$	−14.8779	−1.25294
$142$	0	0
$143$	26.8579	2.24597
$144$	0	0
$145$	−1.62358	−0.134831
$146$	0	0
$147$	5.77657	0.476444
$148$	0	0
$149$	11.5364	0.945096	0.472548	−	0.881305i	$-0.343334\pi$
$149$	11.5364	0.945096	0.472548	+	0.881305i	$0.343334\pi$
$150$	0	0
$151$	−1.13285	−0.0921903	−0.0460951	−	0.998937i	$-0.514678\pi$
$151$	−1.13285	−0.0921903	−0.0460951	+	0.998937i	$0.514678\pi$
$152$	0	0
$153$	−15.4699	−1.25067
$154$	0	0
$155$	−9.87298	−0.793017
$156$	0	0
$157$	−17.2388	−1.37581	−0.687904	−	0.725802i	$-0.741469\pi$
$157$	−17.2388	−1.37581	−0.687904	+	0.725802i	$0.741469\pi$
$158$	0	0
$159$	14.8423	1.17707
$160$	0	0
$161$	9.35256	0.737085
$162$	0	0
$163$	16.7800	1.31431	0.657156	−	0.753754i	$-0.271759\pi$
$163$	16.7800	1.31431	0.657156	+	0.753754i	$0.271759\pi$
$164$	0	0
$165$	24.8215	1.93235
$166$	0	0
$167$	13.7269	1.06222	0.531108	−	0.847304i	$-0.321776\pi$
$167$	13.7269	1.06222	0.531108	+	0.847304i	$0.321776\pi$
$168$	0	0
$169$	6.85092	0.526994
$170$	0	0
$171$	15.4412	1.18082
$172$	0	0
$173$	−5.19705	−0.395124	−0.197562	−	0.980290i	$-0.563302\pi$
$173$	−5.19705	−0.395124	−0.197562	+	0.980290i	$0.563302\pi$
$174$	0	0
$175$	5.94416	0.449336
$176$	0	0
$177$	24.7148	1.85768
$178$	0	0
$179$	1.70886	0.127726	0.0638632	−	0.997959i	$-0.479658\pi$
$179$	1.70886	0.127726	0.0638632	+	0.997959i	$0.479658\pi$
$180$	0	0
$181$	5.16396	0.383834	0.191917	−	0.981411i	$-0.438530\pi$
$181$	5.16396	0.383834	0.191917	+	0.981411i	$0.438530\pi$
$182$	0	0
$183$	13.6867	1.01175
$184$	0	0
$185$	16.2429	1.19420
$186$	0	0
$187$	−21.4305	−1.56716
$188$	0	0
$189$	8.08609	0.588177
$190$	0	0
$191$	20.5089	1.48397	0.741986	−	0.670416i	$-0.233884\pi$
$191$	20.5089	1.48397	0.741986	+	0.670416i	$0.233884\pi$
$192$	0	0
$193$	−10.1690	−0.731981	−0.365990	−	0.930619i	$-0.619270\pi$
$193$	−10.1690	−0.731981	−0.365990	+	0.930619i	$0.619270\pi$
$194$	0	0
$195$	18.3458	1.31377
$196$	0	0
$197$	−13.8898	−0.989606	−0.494803	−	0.869005i	$-0.664760\pi$
$197$	−13.8898	−0.989606	−0.494803	+	0.869005i	$0.664760\pi$
$198$	0	0
$199$	−18.6873	−1.32471	−0.662353	−	0.749192i	$-0.730442\pi$
$199$	−18.6873	−1.32471	−0.662353	+	0.749192i	$0.730442\pi$
$200$	0	0
$201$	0.705851	0.0497869
$202$	0	0
$203$	−2.35916	−0.165581
$204$	0	0
$205$	−10.2353	−0.714866
$206$	0	0
$207$	−18.4427	−1.28186
$208$	0	0
$209$	21.3907	1.47963
$210$	0	0
$211$	16.5941	1.14238	0.571191	−	0.820817i	$-0.306482\pi$
$211$	16.5941	1.14238	0.571191	+	0.820817i	$0.306482\pi$
$212$	0	0
$213$	−38.5377	−2.64056
$214$	0	0
$215$	−15.8647	−1.08196
$216$	0	0
$217$	−14.3460	−0.973872
$218$	0	0
$219$	41.7411	2.82061
$220$	0	0
$221$	−15.8395	−1.06548
$222$	0	0
$223$	−15.4325	−1.03344	−0.516718	−	0.856156i	$-0.672846\pi$
$223$	−15.4325	−1.03344	−0.516718	+	0.856156i	$0.672846\pi$
$224$	0	0
$225$	−11.7216	−0.781437
$226$	0	0
$227$	27.2487	1.80856	0.904280	−	0.426940i	$-0.140408\pi$
$227$	27.2487	1.80856	0.904280	+	0.426940i	$0.140408\pi$
$228$	0	0
$229$	12.7827	0.844706	0.422353	−	0.906431i	$-0.361204\pi$
$229$	12.7827	0.844706	0.422353	+	0.906431i	$0.361204\pi$
$230$	0	0
$231$	36.0671	2.37304
$232$	0	0
$233$	9.27252	0.607463	0.303731	−	0.952758i	$-0.401767\pi$
$233$	9.27252	0.607463	0.303731	+	0.952758i	$0.401767\pi$
$234$	0	0
$235$	−8.33321	−0.543598
$236$	0	0
$237$	33.3183	2.16426
$238$	0	0
$239$	−17.9121	−1.15864	−0.579320	−	0.815100i	$-0.696682\pi$
$239$	−17.9121	−1.15864	−0.579320	+	0.815100i	$0.696682\pi$
$240$	0	0
$241$	12.1744	0.784220	0.392110	−	0.919918i	$-0.371745\pi$
$241$	12.1744	0.784220	0.392110	+	0.919918i	$0.371745\pi$
$242$	0	0
$243$	19.4500	1.24772
$244$	0	0
$245$	3.23550	0.206708
$246$	0	0
$247$	15.8101	1.00597
$248$	0	0
$249$	45.6849	2.89516
$250$	0	0
$251$	−15.6575	−0.988291	−0.494145	−	0.869379i	$-0.664519\pi$
$251$	−15.6575	−0.988291	−0.494145	+	0.869379i	$0.664519\pi$
$252$	0	0
$253$	−25.5488	−1.60624
$254$	0	0
$255$	−14.6385	−0.916699
$256$	0	0
$257$	−7.91970	−0.494017	−0.247009	−	0.969013i	$-0.579448\pi$
$257$	−7.91970	−0.494017	−0.247009	+	0.969013i	$0.579448\pi$
$258$	0	0
$259$	23.6019	1.46655
$260$	0	0
$261$	4.65214	0.287960
$262$	0	0
$263$	1.11569	0.0687966	0.0343983	−	0.999408i	$-0.489049\pi$
$263$	1.11569	0.0687966	0.0343983	+	0.999408i	$0.489049\pi$
$264$	0	0
$265$	8.31327	0.510680
$266$	0	0
$267$	−30.7068	−1.87923
$268$	0	0
$269$	−6.93486	−0.422826	−0.211413	−	0.977397i	$-0.567806\pi$
$269$	−6.93486	−0.422826	−0.211413	+	0.977397i	$0.567806\pi$
$270$	0	0
$271$	16.9637	1.03047	0.515236	−	0.857048i	$-0.327704\pi$
$271$	16.9637	1.03047	0.515236	+	0.857048i	$0.327704\pi$
$272$	0	0
$273$	26.6575	1.61339
$274$	0	0
$275$	−16.2379	−0.979183
$276$	0	0
$277$	28.4674	1.71044	0.855220	−	0.518266i	$-0.173422\pi$
$277$	28.4674	1.71044	0.855220	+	0.518266i	$0.173422\pi$
$278$	0	0
$279$	28.2896	1.69365
$280$	0	0
$281$	26.0015	1.55112	0.775558	−	0.631276i	$-0.217468\pi$
$281$	26.0015	1.55112	0.775558	+	0.631276i	$0.217468\pi$
$282$	0	0
$283$	14.7441	0.876445	0.438223	−	0.898866i	$-0.355608\pi$
$283$	14.7441	0.876445	0.438223	+	0.898866i	$0.355608\pi$
$284$	0	0
$285$	14.6113	0.865499
$286$	0	0
$287$	−14.8725	−0.877897
$288$	0	0
$289$	−4.36132	−0.256548
$290$	0	0
$291$	−24.2391	−1.42092
$292$	0	0
$293$	−25.4914	−1.48922	−0.744611	−	0.667499i	$-0.767365\pi$
$293$	−25.4914	−1.48922	−0.744611	+	0.667499i	$0.767365\pi$
$294$	0	0
$295$	13.8429	0.805966
$296$	0	0
$297$	−22.0892	−1.28174
$298$	0	0
$299$	−18.8833	−1.09205
$300$	0	0
$301$	−23.0523	−1.32871
$302$	0	0
$303$	14.9047	0.856255
$304$	0	0
$305$	7.66603	0.438956
$306$	0	0
$307$	17.2075	0.982082	0.491041	−	0.871136i	$-0.336617\pi$
$307$	17.2075	0.982082	0.491041	+	0.871136i	$0.336617\pi$
$308$	0	0
$309$	2.72638	0.155098
$310$	0	0
$311$	−3.84209	−0.217865	−0.108933	−	0.994049i	$-0.534743\pi$
$311$	−3.84209	−0.217865	−0.108933	+	0.994049i	$0.534743\pi$
$312$	0	0
$313$	−1.93766	−0.109523	−0.0547614	−	0.998499i	$-0.517440\pi$
$313$	−1.93766	−0.109523	−0.0547614	+	0.998499i	$0.517440\pi$
$314$	0	0
$315$	14.5827	0.821641
$316$	0	0
$317$	−4.00466	−0.224924	−0.112462	−	0.993656i	$-0.535874\pi$
$317$	−4.00466	−0.224924	−0.112462	+	0.993656i	$0.535874\pi$
$318$	0	0
$319$	6.44463	0.360830
$320$	0	0
$321$	−25.3827	−1.41672
$322$	0	0
$323$	−12.6152	−0.701928
$324$	0	0
$325$	−12.0016	−0.665728
$326$	0	0
$327$	−18.0254	−0.996807
$328$	0	0
$329$	−12.1086	−0.667571
$330$	0	0
$331$	−32.1505	−1.76715	−0.883575	−	0.468290i	$-0.844870\pi$
$331$	−32.1505	−1.76715	−0.883575	+	0.468290i	$0.844870\pi$
$332$	0	0
$333$	−46.5417	−2.55047
$334$	0	0
$335$	0.395352	0.0216004
$336$	0	0
$337$	29.6361	1.61438	0.807191	−	0.590290i	$-0.200987\pi$
$337$	29.6361	1.61438	0.807191	+	0.590290i	$0.200987\pi$
$338$	0	0
$339$	−20.7866	−1.12897
$340$	0	0
$341$	39.1897	2.12224
$342$	0	0
$343$	20.1482	1.08790
$344$	0	0
$345$	−17.4516	−0.939561
$346$	0	0
$347$	25.5324	1.37065	0.685327	−	0.728236i	$-0.259659\pi$
$347$	25.5324	1.37065	0.685327	+	0.728236i	$0.259659\pi$
$348$	0	0
$349$	−14.6639	−0.784938	−0.392469	−	0.919765i	$-0.628379\pi$
$349$	−14.6639	−0.784938	−0.392469	+	0.919765i	$0.628379\pi$
$350$	0	0
$351$	−16.3263	−0.871432
$352$	0	0
$353$	−15.2573	−0.812064	−0.406032	−	0.913859i	$-0.633088\pi$
$353$	−15.2573	−0.812064	−0.406032	+	0.913859i	$0.633088\pi$
$354$	0	0
$355$	−21.5853	−1.14563
$356$	0	0
$357$	−21.2706	−1.12576
$358$	0	0
$359$	−27.5569	−1.45440	−0.727198	−	0.686428i	$-0.759178\pi$
$359$	−27.5569	−1.45440	−0.727198	+	0.686428i	$0.759178\pi$
$360$	0	0
$361$	−6.40825	−0.337276
$362$	0	0
$363$	−68.7012	−3.60588
$364$	0	0
$365$	23.3795	1.22374
$366$	0	0
$367$	11.6198	0.606550	0.303275	−	0.952903i	$-0.401920\pi$
$367$	11.6198	0.606550	0.303275	+	0.952903i	$0.401920\pi$
$368$	0	0
$369$	29.3278	1.52675
$370$	0	0
$371$	12.0797	0.627145
$372$	0	0
$373$	−14.7329	−0.762842	−0.381421	−	0.924402i	$-0.624565\pi$
$373$	−14.7329	−0.762842	−0.381421	+	0.924402i	$0.624565\pi$
$374$	0	0
$375$	−31.6797	−1.63593
$376$	0	0
$377$	4.76328	0.245322
$378$	0	0
$379$	−9.41628	−0.483682	−0.241841	−	0.970316i	$-0.577751\pi$
$379$	−9.41628	−0.483682	−0.241841	+	0.970316i	$0.577751\pi$
$380$	0	0
$381$	−28.4290	−1.45646
$382$	0	0
$383$	21.5746	1.10241	0.551205	−	0.834370i	$-0.314168\pi$
$383$	21.5746	1.10241	0.551205	+	0.834370i	$0.314168\pi$
$384$	0	0
$385$	20.2015	1.02956
$386$	0	0
$387$	45.4579	2.31075
$388$	0	0
$389$	25.4378	1.28975	0.644873	−	0.764290i	$-0.276910\pi$
$389$	25.4378	1.28975	0.644873	+	0.764290i	$0.276910\pi$
$390$	0	0
$391$	15.0674	0.761993
$392$	0	0
$393$	47.2263	2.38225
$394$	0	0
$395$	18.6618	0.938979
$396$	0	0
$397$	−23.7383	−1.19139	−0.595696	−	0.803210i	$-0.703124\pi$
$397$	−23.7383	−1.19139	−0.595696	+	0.803210i	$0.703124\pi$
$398$	0	0
$399$	21.2311	1.06288
$400$	0	0
$401$	23.6958	1.18331	0.591657	−	0.806190i	$-0.298474\pi$
$401$	23.6958	1.18331	0.591657	+	0.806190i	$0.298474\pi$
$402$	0	0
$403$	28.9654	1.44287
$404$	0	0
$405$	4.73678	0.235372
$406$	0	0
$407$	−64.4745	−3.19588
$408$	0	0
$409$	28.1477	1.39181	0.695907	−	0.718132i	$-0.255003\pi$
$409$	28.1477	1.39181	0.695907	+	0.718132i	$0.255003\pi$
$410$	0	0
$411$	−38.6874	−1.90831
$412$	0	0
$413$	20.1146	0.989774
$414$	0	0
$415$	25.5884	1.25609
$416$	0	0
$417$	−41.8810	−2.05092
$418$	0	0
$419$	−8.14326	−0.397824	−0.198912	−	0.980017i	$-0.563741\pi$
$419$	−8.14326	−0.397824	−0.198912	+	0.980017i	$0.563741\pi$
$420$	0	0
$421$	0.350111	0.0170634	0.00853168	−	0.999964i	$-0.497284\pi$
$421$	0.350111	0.0170634	0.00853168	+	0.999964i	$0.497284\pi$
$422$	0	0
$423$	23.8776	1.16097
$424$	0	0
$425$	9.57633	0.464520
$426$	0	0
$427$	11.1392	0.539064
$428$	0	0
$429$	−72.8215	−3.51586
$430$	0	0
$431$	7.64273	0.368137	0.184069	−	0.982913i	$-0.441073\pi$
$431$	7.64273	0.368137	0.184069	+	0.982913i	$0.441073\pi$
$432$	0	0
$433$	−20.1243	−0.967113	−0.483557	−	0.875313i	$-0.660655\pi$
$433$	−20.1243	−0.967113	−0.483557	+	0.875313i	$0.660655\pi$
$434$	0	0
$435$	4.40212	0.211065
$436$	0	0
$437$	−15.0394	−0.719434
$438$	0	0
$439$	11.6638	0.556681	0.278340	−	0.960483i	$-0.410216\pi$
$439$	11.6638	0.556681	0.278340	+	0.960483i	$0.410216\pi$
$440$	0	0
$441$	−9.27085	−0.441469
$442$	0	0
$443$	6.91501	0.328542	0.164271	−	0.986415i	$-0.447473\pi$
$443$	6.91501	0.328542	0.164271	+	0.986415i	$0.447473\pi$
$444$	0	0
$445$	−17.1991	−0.815316
$446$	0	0
$447$	−31.2793	−1.47946
$448$	0	0
$449$	25.3281	1.19531	0.597654	−	0.801754i	$-0.296100\pi$
$449$	25.3281	1.19531	0.597654	+	0.801754i	$0.296100\pi$
$450$	0	0
$451$	40.6280	1.91310
$452$	0	0
$453$	3.07157	0.144315
$454$	0	0
$455$	14.9311	0.699979
$456$	0	0
$457$	−17.6528	−0.825765	−0.412883	−	0.910784i	$-0.635478\pi$
$457$	−17.6528	−0.825765	−0.412883	+	0.910784i	$0.635478\pi$
$458$	0	0
$459$	13.0271	0.608053
$460$	0	0
$461$	7.79862	0.363218	0.181609	−	0.983371i	$-0.441869\pi$
$461$	7.79862	0.363218	0.181609	+	0.983371i	$0.441869\pi$
$462$	0	0
$463$	16.2226	0.753927	0.376963	−	0.926228i	$-0.376968\pi$
$463$	16.2226	0.753927	0.376963	+	0.926228i	$0.376968\pi$
$464$	0	0
$465$	26.7692	1.24139
$466$	0	0
$467$	26.6470	1.23307	0.616537	−	0.787326i	$-0.288535\pi$
$467$	26.6470	1.23307	0.616537	+	0.787326i	$0.288535\pi$
$468$	0	0
$469$	0.574470	0.0265266
$470$	0	0
$471$	46.7407	2.15370
$472$	0	0
$473$	62.9730	2.89550
$474$	0	0
$475$	−9.55853	−0.438576
$476$	0	0
$477$	−23.8205	−1.09066
$478$	0	0
$479$	−22.0003	−1.00522	−0.502609	−	0.864514i	$-0.667626\pi$
$479$	−22.0003	−1.00522	−0.502609	+	0.864514i	$0.667626\pi$
$480$	0	0
$481$	−47.6536	−2.17282
$482$	0	0
$483$	−25.3582	−1.15384
$484$	0	0
$485$	−13.5765	−0.616478
$486$	0	0
$487$	18.8406	0.853747	0.426874	−	0.904311i	$-0.359615\pi$
$487$	18.8406	0.853747	0.426874	+	0.904311i	$0.359615\pi$
$488$	0	0
$489$	−45.4967	−2.05743
$490$	0	0
$491$	25.6986	1.15976	0.579881	−	0.814701i	$-0.303099\pi$
$491$	25.6986	1.15976	0.579881	+	0.814701i	$0.303099\pi$
$492$	0	0
$493$	−3.80073	−0.171176
$494$	0	0
$495$	−39.8362	−1.79050
$496$	0	0
$497$	−31.3647	−1.40690
$498$	0	0
$499$	−13.5474	−0.606465	−0.303232	−	0.952917i	$-0.598066\pi$
$499$	−13.5474	−0.606465	−0.303232	+	0.952917i	$0.598066\pi$
$500$	0	0
$501$	−37.2185	−1.66280
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	8.34825	0.371492
$506$	0	0
$507$	−18.5753	−0.824959
$508$	0	0
$509$	4.80623	0.213032	0.106516	−	0.994311i	$-0.466030\pi$
$509$	4.80623	0.213032	0.106516	+	0.994311i	$0.466030\pi$
$510$	0	0
$511$	33.9718	1.50283
$512$	0	0
$513$	−13.0029	−0.574092
$514$	0	0
$515$	1.52707	0.0672905
$516$	0	0
$517$	33.0777	1.45476
$518$	0	0
$519$	14.0911	0.618530
$520$	0	0
$521$	−1.04814	−0.0459200	−0.0229600	−	0.999736i	$-0.507309\pi$
$521$	−1.04814	−0.0459200	−0.0229600	+	0.999736i	$0.507309\pi$
$522$	0	0
$523$	37.9458	1.65925	0.829627	−	0.558319i	$-0.188553\pi$
$523$	37.9458	1.65925	0.829627	+	0.558319i	$0.188553\pi$
$524$	0	0
$525$	−16.1168	−0.703393
$526$	0	0
$527$	−23.1122	−1.00678
$528$	0	0
$529$	−5.03708	−0.219003
$530$	0	0
$531$	−39.6649	−1.72131
$532$	0	0
$533$	30.0285	1.30068
$534$	0	0
$535$	−14.2170	−0.614656
$536$	0	0
$537$	−4.63334	−0.199943
$538$	0	0
$539$	−12.8430	−0.553185
$540$	0	0
$541$	−17.3513	−0.745991	−0.372996	−	0.927833i	$-0.621669\pi$
$541$	−17.3513	−0.745991	−0.372996	+	0.927833i	$0.621669\pi$
$542$	0	0
$543$	−14.0014	−0.600856
$544$	0	0
$545$	−10.0962	−0.432472
$546$	0	0
$547$	−43.2703	−1.85011	−0.925053	−	0.379839i	$-0.875979\pi$
$547$	−43.2703	−1.85011	−0.925053	+	0.379839i	$0.875979\pi$
$548$	0	0
$549$	−21.9659	−0.937482
$550$	0	0
$551$	3.79367	0.161616
$552$	0	0
$553$	27.1168	1.15312
$554$	0	0
$555$	−44.0404	−1.86941
$556$	0	0
$557$	−14.8000	−0.627094	−0.313547	−	0.949573i	$-0.601517\pi$
$557$	−14.8000	−0.627094	−0.313547	+	0.949573i	$0.601517\pi$
$558$	0	0
$559$	46.5439	1.96860
$560$	0	0
$561$	58.1059	2.45323
$562$	0	0
$563$	−32.2147	−1.35769	−0.678845	−	0.734282i	$-0.737519\pi$
$563$	−32.2147	−1.35769	−0.678845	+	0.734282i	$0.737519\pi$
$564$	0	0
$565$	−11.6427	−0.489813
$566$	0	0
$567$	6.88282	0.289051
$568$	0	0
$569$	−7.34751	−0.308024	−0.154012	−	0.988069i	$-0.549219\pi$
$569$	−7.34751	−0.308024	−0.154012	+	0.988069i	$0.549219\pi$
$570$	0	0
$571$	26.4824	1.10826	0.554128	−	0.832432i	$-0.313052\pi$
$571$	26.4824	1.10826	0.554128	+	0.832432i	$0.313052\pi$
$572$	0	0
$573$	−55.6070	−2.32302
$574$	0	0
$575$	11.4166	0.476105
$576$	0	0
$577$	−2.93763	−0.122295	−0.0611477	−	0.998129i	$-0.519476\pi$
$577$	−2.93763	−0.122295	−0.0611477	+	0.998129i	$0.519476\pi$
$578$	0	0
$579$	27.5718	1.14585
$580$	0	0
$581$	37.1815	1.54255
$582$	0	0
$583$	−32.9986	−1.36666
$584$	0	0
$585$	−29.4432	−1.21733
$586$	0	0
$587$	−45.1562	−1.86380	−0.931898	−	0.362721i	$-0.881848\pi$
$587$	−45.1562	−1.86380	−0.931898	+	0.362721i	$0.881848\pi$
$588$	0	0
$589$	23.0692	0.950550
$590$	0	0
$591$	37.6602	1.54913
$592$	0	0
$593$	29.7620	1.22218	0.611088	−	0.791562i	$-0.290732\pi$
$593$	29.7620	1.22218	0.611088	+	0.791562i	$0.290732\pi$
$594$	0	0
$595$	−11.9138	−0.488419
$596$	0	0
$597$	50.6679	2.07370
$598$	0	0
$599$	31.0724	1.26958	0.634791	−	0.772684i	$-0.281086\pi$
$599$	31.0724	1.26958	0.634791	+	0.772684i	$0.281086\pi$
$600$	0	0
$601$	−5.09894	−0.207990	−0.103995	−	0.994578i	$-0.533163\pi$
$601$	−5.09894	−0.207990	−0.103995	+	0.994578i	$0.533163\pi$
$602$	0	0
$603$	−1.13282	−0.0461322
$604$	0	0
$605$	−38.4800	−1.56444
$606$	0	0
$607$	27.0983	1.09988	0.549942	−	0.835203i	$-0.314650\pi$
$607$	27.0983	1.09988	0.549942	+	0.835203i	$0.314650\pi$
$608$	0	0
$609$	6.39654	0.259201
$610$	0	0
$611$	24.4480	0.989062
$612$	0	0
$613$	−14.8809	−0.601034	−0.300517	−	0.953776i	$-0.597159\pi$
$613$	−14.8809	−0.601034	−0.300517	+	0.953776i	$0.597159\pi$
$614$	0	0
$615$	27.7516	1.11905
$616$	0	0
$617$	43.9674	1.77006	0.885030	−	0.465534i	$-0.154138\pi$
$617$	43.9674	1.77006	0.885030	+	0.465534i	$0.154138\pi$
$618$	0	0
$619$	0.655769	0.0263576	0.0131788	−	0.999913i	$-0.495805\pi$
$619$	0.655769	0.0263576	0.0131788	+	0.999913i	$0.495805\pi$
$620$	0	0
$621$	15.5305	0.623217
$622$	0	0
$623$	−24.9913	−1.00126
$624$	0	0
$625$	−4.27555	−0.171022
$626$	0	0
$627$	−57.9980	−2.31622
$628$	0	0
$629$	38.0239	1.51611
$630$	0	0
$631$	−29.2557	−1.16465	−0.582325	−	0.812956i	$-0.697857\pi$
$631$	−29.2557	−1.16465	−0.582325	+	0.812956i	$0.697857\pi$
$632$	0	0
$633$	−44.9925	−1.78829
$634$	0	0
$635$	−15.9233	−0.631897
$636$	0	0
$637$	−9.49234	−0.376100
$638$	0	0
$639$	61.8494	2.44673
$640$	0	0
$641$	29.6981	1.17301	0.586503	−	0.809947i	$-0.300504\pi$
$641$	29.6981	1.17301	0.586503	+	0.809947i	$0.300504\pi$
$642$	0	0
$643$	−2.06146	−0.0812962	−0.0406481	−	0.999174i	$-0.512942\pi$
$643$	−2.06146	−0.0812962	−0.0406481	+	0.999174i	$0.512942\pi$
$644$	0	0
$645$	43.0148	1.69371
$646$	0	0
$647$	31.7651	1.24882	0.624408	−	0.781098i	$-0.285340\pi$
$647$	31.7651	1.24882	0.624408	+	0.781098i	$0.285340\pi$
$648$	0	0
$649$	−54.9479	−2.15689
$650$	0	0
$651$	38.8973	1.52450
$652$	0	0
$653$	29.2647	1.14521	0.572607	−	0.819830i	$-0.305932\pi$
$653$	29.2647	1.14521	0.572607	+	0.819830i	$0.305932\pi$
$654$	0	0
$655$	26.4518	1.03356
$656$	0	0
$657$	−66.9906	−2.61355
$658$	0	0
$659$	−8.42889	−0.328343	−0.164171	−	0.986432i	$-0.552495\pi$
$659$	−8.42889	−0.328343	−0.164171	+	0.986432i	$0.552495\pi$
$660$	0	0
$661$	−7.74840	−0.301378	−0.150689	−	0.988581i	$-0.548149\pi$
$661$	−7.74840	−0.301378	−0.150689	+	0.988581i	$0.548149\pi$
$662$	0	0
$663$	42.9466	1.66791
$664$	0	0
$665$	11.8917	0.461140
$666$	0	0
$667$	−4.53111	−0.175445
$668$	0	0
$669$	41.8430	1.61774
$670$	0	0
$671$	−30.4295	−1.17472
$672$	0	0
$673$	16.1757	0.623527	0.311764	−	0.950160i	$-0.399080\pi$
$673$	16.1757	0.623527	0.311764	+	0.950160i	$0.399080\pi$
$674$	0	0
$675$	9.87063	0.379921
$676$	0	0
$677$	−7.61874	−0.292812	−0.146406	−	0.989225i	$-0.546771\pi$
$677$	−7.61874	−0.292812	−0.146406	+	0.989225i	$0.546771\pi$
$678$	0	0
$679$	−19.7275	−0.757071
$680$	0	0
$681$	−73.8811	−2.83113
$682$	0	0
$683$	22.5088	0.861275	0.430638	−	0.902525i	$-0.358289\pi$
$683$	22.5088	0.861275	0.430638	+	0.902525i	$0.358289\pi$
$684$	0	0
$685$	−21.6691	−0.827934
$686$	0	0
$687$	−34.6586	−1.32231
$688$	0	0
$689$	−24.3895	−0.929168
$690$	0	0
$691$	−28.6068	−1.08825	−0.544126	−	0.839003i	$-0.683139\pi$
$691$	−28.6068	−1.08825	−0.544126	+	0.839003i	$0.683139\pi$
$692$	0	0
$693$	−57.8843	−2.19884
$694$	0	0
$695$	−23.4579	−0.889808
$696$	0	0
$697$	−23.9604	−0.907564
$698$	0	0
$699$	−25.1411	−0.950925
$700$	0	0
$701$	−7.69609	−0.290677	−0.145339	−	0.989382i	$-0.546427\pi$
$701$	−7.69609	−0.290677	−0.145339	+	0.989382i	$0.546427\pi$
$702$	0	0
$703$	−37.9532	−1.43143
$704$	0	0
$705$	22.5943	0.850952
$706$	0	0
$707$	12.1305	0.456215
$708$	0	0
$709$	22.7761	0.855376	0.427688	−	0.903927i	$-0.359328\pi$
$709$	22.7761	0.855376	0.427688	+	0.903927i	$0.359328\pi$
$710$	0	0
$711$	−53.4728	−2.00539
$712$	0	0
$713$	−27.5536	−1.03189
$714$	0	0
$715$	−40.7879	−1.52538
$716$	0	0
$717$	48.5663	1.81374
$718$	0	0
$719$	20.7004	0.771995	0.385997	−	0.922500i	$-0.373857\pi$
$719$	20.7004	0.771995	0.385997	+	0.922500i	$0.373857\pi$
$720$	0	0
$721$	2.21892	0.0826368
$722$	0	0
$723$	−33.0091	−1.22762
$724$	0	0
$725$	−2.87981	−0.106954
$726$	0	0
$727$	−28.8331	−1.06936	−0.534680	−	0.845055i	$-0.679568\pi$
$727$	−28.8331	−1.06936	−0.534680	+	0.845055i	$0.679568\pi$
$728$	0	0
$729$	−43.3787	−1.60662
$730$	0	0
$731$	−37.1384	−1.37361
$732$	0	0
$733$	24.6702	0.911214	0.455607	−	0.890181i	$-0.349422\pi$
$733$	24.6702	0.911214	0.455607	+	0.890181i	$0.349422\pi$
$734$	0	0
$735$	−8.77261	−0.323582
$736$	0	0
$737$	−1.56931	−0.0578062
$738$	0	0
$739$	−30.6495	−1.12746	−0.563730	−	0.825959i	$-0.690634\pi$
$739$	−30.6495	−1.12746	−0.563730	+	0.825959i	$0.690634\pi$
$740$	0	0
$741$	−42.8668	−1.57475
$742$	0	0
$743$	26.5112	0.972601	0.486301	−	0.873792i	$-0.338346\pi$
$743$	26.5112	0.972601	0.486301	+	0.873792i	$0.338346\pi$
$744$	0	0
$745$	−17.5197	−0.641873
$746$	0	0
$747$	−73.3200	−2.68264
$748$	0	0
$749$	−20.6582	−0.754834
$750$	0	0
$751$	33.5446	1.22406	0.612030	−	0.790835i	$-0.290353\pi$
$751$	33.5446	1.22406	0.612030	+	0.790835i	$0.290353\pi$
$752$	0	0
$753$	42.4531	1.54708
$754$	0	0
$755$	1.72041	0.0626122
$756$	0	0
$757$	5.87110	0.213389	0.106694	−	0.994292i	$-0.465973\pi$
$757$	5.87110	0.213389	0.106694	+	0.994292i	$0.465973\pi$
$758$	0	0
$759$	69.2721	2.51442
$760$	0	0
$761$	13.6982	0.496559	0.248280	−	0.968688i	$-0.420135\pi$
$761$	13.6982	0.496559	0.248280	+	0.968688i	$0.420135\pi$
$762$	0	0
$763$	−14.6703	−0.531101
$764$	0	0
$765$	23.4934	0.849407
$766$	0	0
$767$	−40.6125	−1.46643
$768$	0	0
$769$	43.8547	1.58144	0.790720	−	0.612178i	$-0.209706\pi$
$769$	43.8547	1.58144	0.790720	+	0.612178i	$0.209706\pi$
$770$	0	0
$771$	21.4732	0.773337
$772$	0	0
$773$	23.0462	0.828915	0.414458	−	0.910069i	$-0.363971\pi$
$773$	23.0462	0.828915	0.414458	+	0.910069i	$0.363971\pi$
$774$	0	0
$775$	−17.5121	−0.629053
$776$	0	0
$777$	−63.9934	−2.29575
$778$	0	0
$779$	23.9159	0.856874
$780$	0	0
$781$	85.6803	3.06588
$782$	0	0
$783$	−3.91753	−0.140001
$784$	0	0
$785$	26.1798	0.934396
$786$	0	0
$787$	−33.6277	−1.19870	−0.599349	−	0.800488i	$-0.704574\pi$
$787$	−33.6277	−1.19870	−0.599349	+	0.800488i	$0.704574\pi$
$788$	0	0
$789$	−3.02505	−0.107695
$790$	0	0
$791$	−16.9176	−0.601520
$792$	0	0
$793$	−22.4907	−0.798667
$794$	0	0
$795$	−22.5403	−0.799421
$796$	0	0
$797$	38.8708	1.37688	0.688438	−	0.725296i	$-0.258297\pi$
$797$	38.8708	1.37688	0.688438	+	0.725296i	$0.258297\pi$
$798$	0	0
$799$	−19.5076	−0.690130
$800$	0	0
$801$	49.2816	1.74128
$802$	0	0
$803$	−92.8024	−3.27493
$804$	0	0
$805$	−14.2033	−0.500600
$806$	0	0
$807$	18.8029	0.661894
$808$	0	0
$809$	−35.1603	−1.23617	−0.618085	−	0.786112i	$-0.712091\pi$
$809$	−35.1603	−1.23617	−0.618085	+	0.786112i	$0.712091\pi$
$810$	0	0
$811$	−29.6700	−1.04185	−0.520927	−	0.853601i	$-0.674414\pi$
$811$	−29.6700	−1.04185	−0.520927	+	0.853601i	$0.674414\pi$
$812$	0	0
$813$	−45.9947	−1.61311
$814$	0	0
$815$	−25.4830	−0.892631
$816$	0	0
$817$	37.0694	1.29689
$818$	0	0
$819$	−42.7828	−1.49495
$820$	0	0
$821$	39.8763	1.39169	0.695845	−	0.718192i	$-0.255030\pi$
$821$	39.8763	1.39169	0.695845	+	0.718192i	$0.255030\pi$
$822$	0	0
$823$	−30.2160	−1.05326	−0.526632	−	0.850094i	$-0.676545\pi$
$823$	−30.2160	−1.05326	−0.526632	+	0.850094i	$0.676545\pi$
$824$	0	0
$825$	44.0269	1.53282
$826$	0	0
$827$	53.8618	1.87296	0.936480	−	0.350721i	$-0.114063\pi$
$827$	53.8618	1.87296	0.936480	+	0.350721i	$0.114063\pi$
$828$	0	0
$829$	25.6487	0.890815	0.445408	−	0.895328i	$-0.353059\pi$
$829$	25.6487	0.890815	0.445408	+	0.895328i	$0.353059\pi$
$830$	0	0
$831$	−77.1854	−2.67753
$832$	0	0
$833$	7.57415	0.262429
$834$	0	0
$835$	−20.8463	−0.721417
$836$	0	0
$837$	−23.8225	−0.823424
$838$	0	0
$839$	−19.9417	−0.688464	−0.344232	−	0.938885i	$-0.611861\pi$
$839$	−19.9417	−0.688464	−0.344232	+	0.938885i	$0.611861\pi$
$840$	0	0
$841$	−27.8570	−0.960587
$842$	0	0
$843$	−70.4993	−2.42813
$844$	0	0
$845$	−10.4042	−0.357914
$846$	0	0
$847$	−55.9138	−1.92122
$848$	0	0
$849$	−39.9766	−1.37199
$850$	0	0
$851$	45.3309	1.55392
$852$	0	0
$853$	−9.68944	−0.331760	−0.165880	−	0.986146i	$-0.553046\pi$
$853$	−9.68944	−0.331760	−0.165880	+	0.986146i	$0.553046\pi$
$854$	0	0
$855$	−23.4498	−0.801965
$856$	0	0
$857$	−10.5844	−0.361556	−0.180778	−	0.983524i	$-0.557862\pi$
$857$	−10.5844	−0.361556	−0.180778	+	0.983524i	$0.557862\pi$
$858$	0	0
$859$	0.0309792	0.00105700	0.000528499	−	1.00000i	$-0.499832\pi$
$859$	0.0309792	0.00105700	0.000528499		1.00000i	$0.499832\pi$
$860$	0	0
$861$	40.3248	1.37427
$862$	0	0
$863$	−10.4945	−0.357236	−0.178618	−	0.983918i	$-0.557163\pi$
$863$	−10.4945	−0.357236	−0.178618	+	0.983918i	$0.557163\pi$
$864$	0	0
$865$	7.89251	0.268354
$866$	0	0
$867$	11.8251	0.401602
$868$	0	0
$869$	−74.0761	−2.51286
$870$	0	0
$871$	−1.15989	−0.0393013
$872$	0	0
$873$	38.9015	1.31662
$874$	0	0
$875$	−25.7831	−0.871628
$876$	0	0
$877$	−46.8999	−1.58370	−0.791848	−	0.610718i	$-0.790881\pi$
$877$	−46.8999	−1.58370	−0.791848	+	0.610718i	$0.790881\pi$
$878$	0	0
$879$	69.1163	2.33124
$880$	0	0
$881$	−11.1824	−0.376746	−0.188373	−	0.982098i	$-0.560321\pi$
$881$	−11.1824	−0.376746	−0.188373	+	0.982098i	$0.560321\pi$
$882$	0	0
$883$	−8.16819	−0.274881	−0.137441	−	0.990510i	$-0.543888\pi$
$883$	−8.16819	−0.274881	−0.137441	+	0.990510i	$0.543888\pi$
$884$	0	0
$885$	−37.5331	−1.26166
$886$	0	0
$887$	52.5194	1.76343	0.881715	−	0.471782i	$-0.156389\pi$
$887$	52.5194	1.76343	0.881715	+	0.471782i	$0.156389\pi$
$888$	0	0
$889$	−23.1375	−0.776007
$890$	0	0
$891$	−18.8021	−0.629894
$892$	0	0
$893$	19.4714	0.651585
$894$	0	0
$895$	−2.59517	−0.0867469
$896$	0	0
$897$	51.1996	1.70950
$898$	0	0
$899$	6.95033	0.231807
$900$	0	0
$901$	19.4610	0.648338
$902$	0	0
$903$	62.5031	2.07997
$904$	0	0
$905$	−7.84226	−0.260685
$906$	0	0
$907$	27.7070	0.919995	0.459998	−	0.887920i	$-0.347850\pi$
$907$	27.7070	0.919995	0.459998	+	0.887920i	$0.347850\pi$
$908$	0	0
$909$	−23.9207	−0.793400
$910$	0	0
$911$	51.0121	1.69011	0.845053	−	0.534682i	$-0.179569\pi$
$911$	51.0121	1.69011	0.845053	+	0.534682i	$0.179569\pi$
$912$	0	0
$913$	−101.570	−3.36149
$914$	0	0
$915$	−20.7854	−0.687144
$916$	0	0
$917$	38.4360	1.26927
$918$	0	0
$919$	−29.3396	−0.967825	−0.483912	−	0.875116i	$-0.660785\pi$
$919$	−29.3396	−0.967825	−0.483912	+	0.875116i	$0.660785\pi$
$920$	0	0
$921$	−46.6557	−1.53736
$922$	0	0
$923$	63.3270	2.08443
$924$	0	0
$925$	28.8107	0.947290
$926$	0	0
$927$	−4.37559	−0.143713
$928$	0	0
$929$	−24.1908	−0.793673	−0.396837	−	0.917889i	$-0.629892\pi$
$929$	−24.1908	−0.793673	−0.396837	+	0.917889i	$0.629892\pi$
$930$	0	0
$931$	−7.56007	−0.247771
$932$	0	0
$933$	10.4173	0.341047
$934$	0	0
$935$	32.5455	1.06435
$936$	0	0
$937$	−14.0561	−0.459194	−0.229597	−	0.973286i	$-0.573741\pi$
$937$	−14.0561	−0.459194	−0.229597	+	0.973286i	$0.573741\pi$
$938$	0	0
$939$	5.25368	0.171448
$940$	0	0
$941$	−40.7092	−1.32708	−0.663541	−	0.748140i	$-0.730947\pi$
$941$	−40.7092	−1.32708	−0.663541	+	0.748140i	$0.730947\pi$
$942$	0	0
$943$	−28.5648	−0.930198
$944$	0	0
$945$	−12.2800	−0.399467
$946$	0	0
$947$	11.1048	0.360858	0.180429	−	0.983588i	$-0.442251\pi$
$947$	11.1048	0.360858	0.180429	+	0.983588i	$0.442251\pi$
$948$	0	0
$949$	−68.5910	−2.22656
$950$	0	0
$951$	10.8581	0.352097
$952$	0	0
$953$	31.4626	1.01917	0.509586	−	0.860420i	$-0.329798\pi$
$953$	31.4626	1.01917	0.509586	+	0.860420i	$0.329798\pi$
$954$	0	0
$955$	−31.1459	−1.00786
$956$	0	0
$957$	−17.4737	−0.564845
$958$	0	0
$959$	−31.4865	−1.01675
$960$	0	0
$961$	11.2648	0.363382
$962$	0	0
$963$	40.7368	1.31273
$964$	0	0
$965$	15.4432	0.497134
$966$	0	0
$967$	1.75252	0.0563573	0.0281787	−	0.999603i	$-0.491029\pi$
$967$	1.75252	0.0563573	0.0281787	+	0.999603i	$0.491029\pi$
$968$	0	0
$969$	34.2044	1.09880
$970$	0	0
$971$	44.0033	1.41213	0.706067	−	0.708145i	$-0.250468\pi$
$971$	44.0033	1.41213	0.706067	+	0.708145i	$0.250468\pi$
$972$	0	0
$973$	−34.0857	−1.09274
$974$	0	0
$975$	32.5406	1.04213
$976$	0	0
$977$	48.9393	1.56571	0.782854	−	0.622206i	$-0.213763\pi$
$977$	48.9393	1.56571	0.782854	+	0.622206i	$0.213763\pi$
$978$	0	0
$979$	68.2700	2.18192
$980$	0	0
$981$	28.9291	0.923635
$982$	0	0
$983$	15.4655	0.493272	0.246636	−	0.969108i	$-0.420675\pi$
$983$	15.4655	0.493272	0.246636	+	0.969108i	$0.420675\pi$
$984$	0	0
$985$	21.0938	0.672103
$986$	0	0
$987$	32.8309	1.04502
$988$	0	0
$989$	−44.2752	−1.40787
$990$	0	0
$991$	41.9949	1.33401	0.667006	−	0.745053i	$-0.267576\pi$
$991$	41.9949	1.33401	0.667006	+	0.745053i	$0.267576\pi$
$992$	0	0
$993$	87.1716	2.76631
$994$	0	0
$995$	28.3795	0.899690
$996$	0	0
$997$	−41.1817	−1.30424	−0.652119	−	0.758116i	$-0.726120\pi$
$997$	−41.1817	−1.30424	−0.652119	+	0.758116i	$0.726120\pi$
$998$	0	0
$999$	39.1925	1.23999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.3	✓	33
4.3	odd	2		8048.2.a.x.1.31		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.3	✓	33	1.1	even	1	trivial
8048.2.a.x.1.31		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)

\(f(q)\)	\(=\)	\(q-2.71136 q^{3} -1.51865 q^{5} -2.20669 q^{7} +4.35148 q^{9} +O(q^{10})\)	\(q-2.71136 q^{3} -1.51865 q^{5} -2.20669 q^{7} +4.35148 q^{9} +6.02812 q^{11} +4.45544 q^{13} +4.11762 q^{15} -3.55509 q^{17} +3.54849 q^{19} +5.98314 q^{21} -4.23827 q^{23} -2.69369 q^{25} -3.66435 q^{27} +1.06909 q^{29} +6.50114 q^{31} -16.3444 q^{33} +3.35120 q^{35} -10.6956 q^{37} -12.0803 q^{39} +6.73973 q^{41} +10.4465 q^{43} -6.60839 q^{45} +5.48724 q^{47} -2.13051 q^{49} +9.63914 q^{51} -5.47411 q^{53} -9.15463 q^{55} -9.62123 q^{57} -9.11526 q^{59} -5.04792 q^{61} -9.60238 q^{63} -6.76626 q^{65} -0.260331 q^{67} +11.4915 q^{69} +14.2134 q^{71} -15.3949 q^{73} +7.30358 q^{75} -13.3022 q^{77} -12.2884 q^{79} -3.11906 q^{81} -16.8494 q^{83} +5.39895 q^{85} -2.89870 q^{87} +11.3252 q^{89} -9.83178 q^{91} -17.6269 q^{93} -5.38892 q^{95} +8.93984 q^{97} +26.2313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

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