LMFDB - Embedded newform 4024.2.a.f.1.1

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			1.1
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)

$f(q)$	$=$	$q-3.25521 q^{3} -2.40628 q^{5} -4.10646 q^{7} +7.59642 q^{9} +O(q^{10})$	$q-3.25521 q^{3} -2.40628 q^{5} -4.10646 q^{7} +7.59642 q^{9} +5.28963 q^{11} -1.26582 q^{13} +7.83294 q^{15} -2.46967 q^{17} -4.09764 q^{19} +13.3674 q^{21} -0.252316 q^{23} +0.790160 q^{25} -14.9623 q^{27} -2.53768 q^{29} +0.649483 q^{31} -17.2189 q^{33} +9.88126 q^{35} +6.06797 q^{37} +4.12051 q^{39} -9.02841 q^{41} -8.38864 q^{43} -18.2791 q^{45} -10.6116 q^{47} +9.86298 q^{49} +8.03929 q^{51} +5.73201 q^{53} -12.7283 q^{55} +13.3387 q^{57} -11.3778 q^{59} +11.7202 q^{61} -31.1944 q^{63} +3.04590 q^{65} -9.70586 q^{67} +0.821344 q^{69} -10.4974 q^{71} -7.98428 q^{73} -2.57214 q^{75} -21.7217 q^{77} -14.1209 q^{79} +25.9163 q^{81} +2.38495 q^{83} +5.94269 q^{85} +8.26068 q^{87} -2.16486 q^{89} +5.19802 q^{91} -2.11421 q^{93} +9.86004 q^{95} +8.81629 q^{97} +40.1823 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.25521	−1.87940	−0.939699	−	0.342002i	$-0.888895\pi$
$3$	−3.25521	−1.87940	−0.939699	+	0.342002i	$0.888895\pi$
$4$	0	0
$5$	−2.40628	−1.07612	−0.538059	−	0.842907i	$-0.680842\pi$
$5$	−2.40628	−1.07612	−0.538059	+	0.842907i	$0.680842\pi$
$6$	0	0
$7$	−4.10646	−1.55209	−0.776047	−	0.630675i	$-0.782778\pi$
$7$	−4.10646	−1.55209	−0.776047	+	0.630675i	$0.782778\pi$
$8$	0	0
$9$	7.59642	2.53214
$10$	0	0
$11$	5.28963	1.59488	0.797442	−	0.603395i	$-0.206186\pi$
$11$	5.28963	1.59488	0.797442	+	0.603395i	$0.206186\pi$
$12$	0	0
$13$	−1.26582	−0.351075	−0.175537	−	0.984473i	$-0.556166\pi$
$13$	−1.26582	−0.351075	−0.175537	+	0.984473i	$0.556166\pi$
$14$	0	0
$15$	7.83294	2.02246
$16$	0	0
$17$	−2.46967	−0.598982	−0.299491	−	0.954099i	$-0.596817\pi$
$17$	−2.46967	−0.598982	−0.299491	+	0.954099i	$0.596817\pi$
$18$	0	0
$19$	−4.09764	−0.940063	−0.470031	−	0.882650i	$-0.655757\pi$
$19$	−4.09764	−0.940063	−0.470031	+	0.882650i	$0.655757\pi$
$20$	0	0
$21$	13.3674	2.91700
$22$	0	0
$23$	−0.252316	−0.0526116	−0.0263058	−	0.999654i	$-0.508374\pi$
$23$	−0.252316	−0.0526116	−0.0263058	+	0.999654i	$0.508374\pi$
$24$	0	0
$25$	0.790160	0.158032
$26$	0	0
$27$	−14.9623	−2.87950
$28$	0	0
$29$	−2.53768	−0.471235	−0.235617	−	0.971846i	$-0.575711\pi$
$29$	−2.53768	−0.471235	−0.235617	+	0.971846i	$0.575711\pi$
$30$	0	0
$31$	0.649483	0.116651	0.0583253	−	0.998298i	$-0.481424\pi$
$31$	0.649483	0.116651	0.0583253	+	0.998298i	$0.481424\pi$
$32$	0	0
$33$	−17.2189	−2.99742
$34$	0	0
$35$	9.88126	1.67024
$36$	0	0
$37$	6.06797	0.997568	0.498784	−	0.866726i	$-0.333780\pi$
$37$	6.06797	0.997568	0.498784	+	0.866726i	$0.333780\pi$
$38$	0	0
$39$	4.12051	0.659809
$40$	0	0
$41$	−9.02841	−1.41000	−0.705000	−	0.709207i	$-0.749053\pi$
$41$	−9.02841	−1.41000	−0.705000	+	0.709207i	$0.749053\pi$
$42$	0	0
$43$	−8.38864	−1.27926	−0.639628	−	0.768685i	$-0.720911\pi$
$43$	−8.38864	−1.27926	−0.639628	+	0.768685i	$0.720911\pi$
$44$	0	0
$45$	−18.2791	−2.72488
$46$	0	0
$47$	−10.6116	−1.54787	−0.773933	−	0.633267i	$-0.781713\pi$
$47$	−10.6116	−1.54787	−0.773933	+	0.633267i	$0.781713\pi$
$48$	0	0
$49$	9.86298	1.40900
$50$	0	0
$51$	8.03929	1.12573
$52$	0	0
$53$	5.73201	0.787352	0.393676	−	0.919249i	$-0.371203\pi$
$53$	5.73201	0.787352	0.393676	+	0.919249i	$0.371203\pi$
$54$	0	0
$55$	−12.7283	−1.71629
$56$	0	0
$57$	13.3387	1.76675
$58$	0	0
$59$	−11.3778	−1.48126	−0.740632	−	0.671911i	$-0.765474\pi$
$59$	−11.3778	−1.48126	−0.740632	+	0.671911i	$0.765474\pi$
$60$	0	0
$61$	11.7202	1.50062	0.750312	−	0.661084i	$-0.229903\pi$
$61$	11.7202	1.50062	0.750312	+	0.661084i	$0.229903\pi$
$62$	0	0
$63$	−31.1944	−3.93012
$64$	0	0
$65$	3.04590	0.377798
$66$	0	0
$67$	−9.70586	−1.18576	−0.592880	−	0.805291i	$-0.702009\pi$
$67$	−9.70586	−1.18576	−0.592880	+	0.805291i	$0.702009\pi$
$68$	0	0
$69$	0.821344	0.0988782
$70$	0	0
$71$	−10.4974	−1.24582	−0.622908	−	0.782295i	$-0.714049\pi$
$71$	−10.4974	−1.24582	−0.622908	+	0.782295i	$0.714049\pi$
$72$	0	0
$73$	−7.98428	−0.934489	−0.467244	−	0.884128i	$-0.654753\pi$
$73$	−7.98428	−0.934489	−0.467244	+	0.884128i	$0.654753\pi$
$74$	0	0
$75$	−2.57214	−0.297005
$76$	0	0
$77$	−21.7217	−2.47541
$78$	0	0
$79$	−14.1209	−1.58873	−0.794364	−	0.607442i	$-0.792196\pi$
$79$	−14.1209	−1.58873	−0.794364	+	0.607442i	$0.792196\pi$
$80$	0	0
$81$	25.9163	2.87959
$82$	0	0
$83$	2.38495	0.261782	0.130891	−	0.991397i	$-0.458216\pi$
$83$	2.38495	0.261782	0.130891	+	0.991397i	$0.458216\pi$
$84$	0	0
$85$	5.94269	0.644576
$86$	0	0
$87$	8.26068	0.885638
$88$	0	0
$89$	−2.16486	−0.229474	−0.114737	−	0.993396i	$-0.536603\pi$
$89$	−2.16486	−0.229474	−0.114737	+	0.993396i	$0.536603\pi$
$90$	0	0
$91$	5.19802	0.544901
$92$	0	0
$93$	−2.11421	−0.219233
$94$	0	0
$95$	9.86004	1.01162
$96$	0	0
$97$	8.81629	0.895158	0.447579	−	0.894244i	$-0.352286\pi$
$97$	8.81629	0.895158	0.447579	+	0.894244i	$0.352286\pi$
$98$	0	0
$99$	40.1823	4.03847
$100$	0	0
$101$	1.63682	0.162870	0.0814348	−	0.996679i	$-0.474050\pi$
$101$	1.63682	0.162870	0.0814348	+	0.996679i	$0.474050\pi$
$102$	0	0
$103$	−8.44485	−0.832096	−0.416048	−	0.909343i	$-0.636585\pi$
$103$	−8.44485	−0.832096	−0.416048	+	0.909343i	$0.636585\pi$
$104$	0	0
$105$	−32.1656	−3.13904
$106$	0	0
$107$	−14.2332	−1.37598	−0.687989	−	0.725721i	$-0.741506\pi$
$107$	−14.2332	−1.37598	−0.687989	+	0.725721i	$0.741506\pi$
$108$	0	0
$109$	−5.69013	−0.545016	−0.272508	−	0.962154i	$-0.587853\pi$
$109$	−5.69013	−0.545016	−0.272508	+	0.962154i	$0.587853\pi$
$110$	0	0
$111$	−19.7525	−1.87483
$112$	0	0
$113$	−15.3843	−1.44724	−0.723618	−	0.690200i	$-0.757522\pi$
$113$	−15.3843	−1.44724	−0.723618	+	0.690200i	$0.757522\pi$
$114$	0	0
$115$	0.607143	0.0566163
$116$	0	0
$117$	−9.61568	−0.888970
$118$	0	0
$119$	10.1416	0.929676
$120$	0	0
$121$	16.9802	1.54366
$122$	0	0
$123$	29.3894	2.64995
$124$	0	0
$125$	10.1300	0.906058
$126$	0	0
$127$	7.18636	0.637687	0.318843	−	0.947807i	$-0.396706\pi$
$127$	7.18636	0.637687	0.318843	+	0.947807i	$0.396706\pi$
$128$	0	0
$129$	27.3068	2.40423
$130$	0	0
$131$	17.6507	1.54215	0.771075	−	0.636744i	$-0.219719\pi$
$131$	17.6507	1.54215	0.771075	+	0.636744i	$0.219719\pi$
$132$	0	0
$133$	16.8268	1.45907
$134$	0	0
$135$	36.0035	3.09869
$136$	0	0
$137$	−1.77593	−0.151728	−0.0758640	−	0.997118i	$-0.524171\pi$
$137$	−1.77593	−0.151728	−0.0758640	+	0.997118i	$0.524171\pi$
$138$	0	0
$139$	−5.95724	−0.505287	−0.252643	−	0.967559i	$-0.581300\pi$
$139$	−5.95724	−0.505287	−0.252643	+	0.967559i	$0.581300\pi$
$140$	0	0
$141$	34.5432	2.90906
$142$	0	0
$143$	−6.69571	−0.559924
$144$	0	0
$145$	6.10635	0.507105
$146$	0	0
$147$	−32.1061	−2.64807
$148$	0	0
$149$	2.90990	0.238388	0.119194	−	0.992871i	$-0.461969\pi$
$149$	2.90990	0.238388	0.119194	+	0.992871i	$0.461969\pi$
$150$	0	0
$151$	19.3331	1.57331	0.786653	−	0.617395i	$-0.211812\pi$
$151$	19.3331	1.57331	0.786653	+	0.617395i	$0.211812\pi$
$152$	0	0
$153$	−18.7606	−1.51671
$154$	0	0
$155$	−1.56284	−0.125530
$156$	0	0
$157$	16.6310	1.32730	0.663649	−	0.748044i	$-0.269007\pi$
$157$	16.6310	1.32730	0.663649	+	0.748044i	$0.269007\pi$
$158$	0	0
$159$	−18.6589	−1.47975
$160$	0	0
$161$	1.03613	0.0816582
$162$	0	0
$163$	9.87957	0.773828	0.386914	−	0.922116i	$-0.373541\pi$
$163$	9.87957	0.773828	0.386914	+	0.922116i	$0.373541\pi$
$164$	0	0
$165$	41.4334	3.22559
$166$	0	0
$167$	13.3025	1.02938	0.514688	−	0.857377i	$-0.327908\pi$
$167$	13.3025	1.02938	0.514688	+	0.857377i	$0.327908\pi$
$168$	0	0
$169$	−11.3977	−0.876747
$170$	0	0
$171$	−31.1274	−2.38037
$172$	0	0
$173$	0.428435	0.0325733	0.0162867	−	0.999867i	$-0.494816\pi$
$173$	0.428435	0.0325733	0.0162867	+	0.999867i	$0.494816\pi$
$174$	0	0
$175$	−3.24476	−0.245281
$176$	0	0
$177$	37.0372	2.78389
$178$	0	0
$179$	−0.266575	−0.0199248	−0.00996238	−	0.999950i	$-0.503171\pi$
$179$	−0.266575	−0.0199248	−0.00996238	+	0.999950i	$0.503171\pi$
$180$	0	0
$181$	−16.6324	−1.23628	−0.618138	−	0.786069i	$-0.712113\pi$
$181$	−16.6324	−1.23628	−0.618138	+	0.786069i	$0.712113\pi$
$182$	0	0
$183$	−38.1519	−2.82027
$184$	0	0
$185$	−14.6012	−1.07350
$186$	0	0
$187$	−13.0636	−0.955307
$188$	0	0
$189$	61.4421	4.46926
$190$	0	0
$191$	−3.36977	−0.243828	−0.121914	−	0.992541i	$-0.538903\pi$
$191$	−3.36977	−0.243828	−0.121914	+	0.992541i	$0.538903\pi$
$192$	0	0
$193$	−23.3264	−1.67907	−0.839534	−	0.543307i	$-0.817172\pi$
$193$	−23.3264	−1.67907	−0.839534	+	0.543307i	$0.817172\pi$
$194$	0	0
$195$	−9.91507	−0.710033
$196$	0	0
$197$	15.9865	1.13899	0.569496	−	0.821994i	$-0.307138\pi$
$197$	15.9865	1.13899	0.569496	+	0.821994i	$0.307138\pi$
$198$	0	0
$199$	6.39616	0.453412	0.226706	−	0.973963i	$-0.427204\pi$
$199$	6.39616	0.453412	0.226706	+	0.973963i	$0.427204\pi$
$200$	0	0
$201$	31.5947	2.22852
$202$	0	0
$203$	10.4209	0.731401
$204$	0	0
$205$	21.7248	1.51733
$206$	0	0
$207$	−1.91670	−0.133220
$208$	0	0
$209$	−21.6750	−1.49929
$210$	0	0
$211$	16.5924	1.14227	0.571135	−	0.820856i	$-0.306503\pi$
$211$	16.5924	1.14227	0.571135	+	0.820856i	$0.306503\pi$
$212$	0	0
$213$	34.1714	2.34138
$214$	0	0
$215$	20.1854	1.37663
$216$	0	0
$217$	−2.66707	−0.181053
$218$	0	0
$219$	25.9905	1.75628
$220$	0	0
$221$	3.12615	0.210287
$222$	0	0
$223$	−8.77535	−0.587641	−0.293821	−	0.955861i	$-0.594927\pi$
$223$	−8.77535	−0.587641	−0.293821	+	0.955861i	$0.594927\pi$
$224$	0	0
$225$	6.00238	0.400159
$226$	0	0
$227$	−26.4584	−1.75610	−0.878052	−	0.478566i	$-0.841157\pi$
$227$	−26.4584	−1.75610	−0.878052	+	0.478566i	$0.841157\pi$
$228$	0	0
$229$	14.4646	0.955847	0.477924	−	0.878401i	$-0.341390\pi$
$229$	14.4646	0.955847	0.477924	+	0.878401i	$0.341390\pi$
$230$	0	0
$231$	70.7086	4.65229
$232$	0	0
$233$	12.4922	0.818394	0.409197	−	0.912446i	$-0.365809\pi$
$233$	12.4922	0.818394	0.409197	+	0.912446i	$0.365809\pi$
$234$	0	0
$235$	25.5345	1.66569
$236$	0	0
$237$	45.9666	2.98585
$238$	0	0
$239$	12.6048	0.815339	0.407669	−	0.913130i	$-0.366342\pi$
$239$	12.6048	0.815339	0.407669	+	0.913130i	$0.366342\pi$
$240$	0	0
$241$	−3.80699	−0.245230	−0.122615	−	0.992454i	$-0.539128\pi$
$241$	−3.80699	−0.245230	−0.122615	+	0.992454i	$0.539128\pi$
$242$	0	0
$243$	−39.4762	−2.53240
$244$	0	0
$245$	−23.7330	−1.51625
$246$	0	0
$247$	5.18686	0.330032
$248$	0	0
$249$	−7.76353	−0.491994
$250$	0	0
$251$	5.38969	0.340194	0.170097	−	0.985427i	$-0.445592\pi$
$251$	5.38969	0.340194	0.170097	+	0.985427i	$0.445592\pi$
$252$	0	0
$253$	−1.33466	−0.0839095
$254$	0	0
$255$	−19.3447	−1.21141
$256$	0	0
$257$	−22.0557	−1.37580	−0.687898	−	0.725807i	$-0.741467\pi$
$257$	−22.0557	−1.37580	−0.687898	+	0.725807i	$0.741467\pi$
$258$	0	0
$259$	−24.9179	−1.54832
$260$	0	0
$261$	−19.2773	−1.19323
$262$	0	0
$263$	29.8780	1.84236	0.921179	−	0.389140i	$-0.127228\pi$
$263$	29.8780	1.84236	0.921179	+	0.389140i	$0.127228\pi$
$264$	0	0
$265$	−13.7928	−0.847284
$266$	0	0
$267$	7.04707	0.431274
$268$	0	0
$269$	7.08688	0.432095	0.216047	−	0.976383i	$-0.430683\pi$
$269$	7.08688	0.432095	0.216047	+	0.976383i	$0.430683\pi$
$270$	0	0
$271$	−14.1505	−0.859580	−0.429790	−	0.902929i	$-0.641412\pi$
$271$	−14.1505	−0.859580	−0.429790	+	0.902929i	$0.641412\pi$
$272$	0	0
$273$	−16.9207	−1.02409
$274$	0	0
$275$	4.17966	0.252043
$276$	0	0
$277$	−10.4681	−0.628965	−0.314483	−	0.949263i	$-0.601831\pi$
$277$	−10.4681	−0.628965	−0.314483	+	0.949263i	$0.601831\pi$
$278$	0	0
$279$	4.93375	0.295376
$280$	0	0
$281$	−2.74438	−0.163716	−0.0818581	−	0.996644i	$-0.526085\pi$
$281$	−2.74438	−0.163716	−0.0818581	+	0.996644i	$0.526085\pi$
$282$	0	0
$283$	23.9530	1.42386	0.711928	−	0.702253i	$-0.247822\pi$
$283$	23.9530	1.42386	0.711928	+	0.702253i	$0.247822\pi$
$284$	0	0
$285$	−32.0966	−1.90124
$286$	0	0
$287$	37.0748	2.18845
$288$	0	0
$289$	−10.9008	−0.641221
$290$	0	0
$291$	−28.6989	−1.68236
$292$	0	0
$293$	−30.7289	−1.79520	−0.897600	−	0.440810i	$-0.854691\pi$
$293$	−30.7289	−1.79520	−0.897600	+	0.440810i	$0.854691\pi$
$294$	0	0
$295$	27.3781	1.59402
$296$	0	0
$297$	−79.1453	−4.59247
$298$	0	0
$299$	0.319386	0.0184706
$300$	0	0
$301$	34.4476	1.98552
$302$	0	0
$303$	−5.32820	−0.306097
$304$	0	0
$305$	−28.2021	−1.61485
$306$	0	0
$307$	21.4645	1.22504	0.612522	−	0.790454i	$-0.290155\pi$
$307$	21.4645	1.22504	0.612522	+	0.790454i	$0.290155\pi$
$308$	0	0
$309$	27.4898	1.56384
$310$	0	0
$311$	−16.1022	−0.913072	−0.456536	−	0.889705i	$-0.650910\pi$
$311$	−16.1022	−0.913072	−0.456536	+	0.889705i	$0.650910\pi$
$312$	0	0
$313$	15.0064	0.848213	0.424107	−	0.905612i	$-0.360588\pi$
$313$	15.0064	0.848213	0.424107	+	0.905612i	$0.360588\pi$
$314$	0	0
$315$	75.0622	4.22928
$316$	0	0
$317$	21.7864	1.22364	0.611822	−	0.790996i	$-0.290437\pi$
$317$	21.7864	1.22364	0.611822	+	0.790996i	$0.290437\pi$
$318$	0	0
$319$	−13.4234	−0.751565
$320$	0	0
$321$	46.3322	2.58601
$322$	0	0
$323$	10.1198	0.563080
$324$	0	0
$325$	−1.00020	−0.0554810
$326$	0	0
$327$	18.5226	1.02430
$328$	0	0
$329$	43.5762	2.40244
$330$	0	0
$331$	−31.6668	−1.74056	−0.870282	−	0.492554i	$-0.836064\pi$
$331$	−31.6668	−1.74056	−0.870282	+	0.492554i	$0.836064\pi$
$332$	0	0
$333$	46.0949	2.52598
$334$	0	0
$335$	23.3550	1.27602
$336$	0	0
$337$	−20.1984	−1.10028	−0.550139	−	0.835073i	$-0.685425\pi$
$337$	−20.1984	−1.10028	−0.550139	+	0.835073i	$0.685425\pi$
$338$	0	0
$339$	50.0793	2.71994
$340$	0	0
$341$	3.43553	0.186044
$342$	0	0
$343$	−11.7567	−0.634802
$344$	0	0
$345$	−1.97638	−0.106405
$346$	0	0
$347$	−15.7591	−0.845993	−0.422997	−	0.906131i	$-0.639022\pi$
$347$	−15.7591	−0.845993	−0.422997	+	0.906131i	$0.639022\pi$
$348$	0	0
$349$	−29.2997	−1.56838	−0.784189	−	0.620523i	$-0.786921\pi$
$349$	−29.2997	−1.56838	−0.784189	+	0.620523i	$0.786921\pi$
$350$	0	0
$351$	18.9396	1.01092
$352$	0	0
$353$	−22.9507	−1.22154	−0.610772	−	0.791806i	$-0.709141\pi$
$353$	−22.9507	−1.22154	−0.610772	+	0.791806i	$0.709141\pi$
$354$	0	0
$355$	25.2597	1.34065
$356$	0	0
$357$	−33.0130	−1.74723
$358$	0	0
$359$	14.4211	0.761118	0.380559	−	0.924757i	$-0.375732\pi$
$359$	14.4211	0.761118	0.380559	+	0.924757i	$0.375732\pi$
$360$	0	0
$361$	−2.20936	−0.116282
$362$	0	0
$363$	−55.2743	−2.90115
$364$	0	0
$365$	19.2124	1.00562
$366$	0	0
$367$	−31.5746	−1.64818	−0.824090	−	0.566459i	$-0.808313\pi$
$367$	−31.5746	−1.64818	−0.824090	+	0.566459i	$0.808313\pi$
$368$	0	0
$369$	−68.5836	−3.57032
$370$	0	0
$371$	−23.5382	−1.22204
$372$	0	0
$373$	4.37124	0.226334	0.113167	−	0.993576i	$-0.463900\pi$
$373$	4.37124	0.226334	0.113167	+	0.993576i	$0.463900\pi$
$374$	0	0
$375$	−32.9754	−1.70284
$376$	0	0
$377$	3.21224	0.165439
$378$	0	0
$379$	22.2334	1.14205	0.571026	−	0.820932i	$-0.306546\pi$
$379$	22.2334	1.14205	0.571026	+	0.820932i	$0.306546\pi$
$380$	0	0
$381$	−23.3931	−1.19847
$382$	0	0
$383$	3.11809	0.159327	0.0796634	−	0.996822i	$-0.474615\pi$
$383$	3.11809	0.159327	0.0796634	+	0.996822i	$0.474615\pi$
$384$	0	0
$385$	52.2683	2.66384
$386$	0	0
$387$	−63.7236	−3.23925
$388$	0	0
$389$	3.36393	0.170558	0.0852789	−	0.996357i	$-0.472822\pi$
$389$	3.36393	0.170558	0.0852789	+	0.996357i	$0.472822\pi$
$390$	0	0
$391$	0.623137	0.0315134
$392$	0	0
$393$	−57.4569	−2.89832
$394$	0	0
$395$	33.9788	1.70966
$396$	0	0
$397$	−5.52178	−0.277130	−0.138565	−	0.990353i	$-0.544249\pi$
$397$	−5.52178	−0.277130	−0.138565	+	0.990353i	$0.544249\pi$
$398$	0	0
$399$	−54.7747	−2.74217
$400$	0	0
$401$	−14.7239	−0.735278	−0.367639	−	0.929969i	$-0.619834\pi$
$401$	−14.7239	−0.735278	−0.367639	+	0.929969i	$0.619834\pi$
$402$	0	0
$403$	−0.822127	−0.0409531
$404$	0	0
$405$	−62.3618	−3.09878
$406$	0	0
$407$	32.0974	1.59101
$408$	0	0
$409$	1.49688	0.0740162	0.0370081	−	0.999315i	$-0.488217\pi$
$409$	1.49688	0.0740162	0.0370081	+	0.999315i	$0.488217\pi$
$410$	0	0
$411$	5.78104	0.285158
$412$	0	0
$413$	46.7225	2.29906
$414$	0	0
$415$	−5.73885	−0.281709
$416$	0	0
$417$	19.3921	0.949636
$418$	0	0
$419$	16.8227	0.821842	0.410921	−	0.911671i	$-0.365207\pi$
$419$	16.8227	0.821842	0.410921	+	0.911671i	$0.365207\pi$
$420$	0	0
$421$	32.6728	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$421$	32.6728	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$422$	0	0
$423$	−80.6105	−3.91941
$424$	0	0
$425$	−1.95143	−0.0946583
$426$	0	0
$427$	−48.1287	−2.32911
$428$	0	0
$429$	21.7960	1.05232
$430$	0	0
$431$	3.38342	0.162974	0.0814868	−	0.996674i	$-0.474033\pi$
$431$	3.38342	0.162974	0.0814868	+	0.996674i	$0.474033\pi$
$432$	0	0
$433$	−32.1465	−1.54486	−0.772432	−	0.635098i	$-0.780960\pi$
$433$	−32.1465	−1.54486	−0.772432	+	0.635098i	$0.780960\pi$
$434$	0	0
$435$	−19.8775	−0.953052
$436$	0	0
$437$	1.03390	0.0494582
$438$	0	0
$439$	−6.94804	−0.331612	−0.165806	−	0.986158i	$-0.553023\pi$
$439$	−6.94804	−0.331612	−0.165806	+	0.986158i	$0.553023\pi$
$440$	0	0
$441$	74.9233	3.56778
$442$	0	0
$443$	−18.0116	−0.855759	−0.427879	−	0.903836i	$-0.640739\pi$
$443$	−18.0116	−0.855759	−0.427879	+	0.903836i	$0.640739\pi$
$444$	0	0
$445$	5.20924	0.246942
$446$	0	0
$447$	−9.47235	−0.448027
$448$	0	0
$449$	31.4214	1.48287	0.741435	−	0.671025i	$-0.234146\pi$
$449$	31.4214	1.48287	0.741435	+	0.671025i	$0.234146\pi$
$450$	0	0
$451$	−47.7570	−2.24879
$452$	0	0
$453$	−62.9334	−2.95687
$454$	0	0
$455$	−12.5079	−0.586378
$456$	0	0
$457$	25.4785	1.19184	0.595918	−	0.803046i	$-0.296789\pi$
$457$	25.4785	1.19184	0.595918	+	0.803046i	$0.296789\pi$
$458$	0	0
$459$	36.9519	1.72477
$460$	0	0
$461$	8.35058	0.388925	0.194463	−	0.980910i	$-0.437704\pi$
$461$	8.35058	0.388925	0.194463	+	0.980910i	$0.437704\pi$
$462$	0	0
$463$	2.11130	0.0981204	0.0490602	−	0.998796i	$-0.484377\pi$
$463$	2.11130	0.0981204	0.0490602	+	0.998796i	$0.484377\pi$
$464$	0	0
$465$	5.08736	0.235921
$466$	0	0
$467$	−3.56158	−0.164810	−0.0824052	−	0.996599i	$-0.526260\pi$
$467$	−3.56158	−0.164810	−0.0824052	+	0.996599i	$0.526260\pi$
$468$	0	0
$469$	39.8567	1.84041
$470$	0	0
$471$	−54.1375	−2.49452
$472$	0	0
$473$	−44.3728	−2.04026
$474$	0	0
$475$	−3.23779	−0.148560
$476$	0	0
$477$	43.5427	1.99368
$478$	0	0
$479$	39.0318	1.78341	0.891704	−	0.452620i	$-0.149510\pi$
$479$	39.0318	1.78341	0.891704	+	0.452620i	$0.149510\pi$
$480$	0	0
$481$	−7.68094	−0.350221
$482$	0	0
$483$	−3.37281	−0.153468
$484$	0	0
$485$	−21.2144	−0.963297
$486$	0	0
$487$	21.0484	0.953795	0.476898	−	0.878959i	$-0.341761\pi$
$487$	21.0484	0.953795	0.476898	+	0.878959i	$0.341761\pi$
$488$	0	0
$489$	−32.1601	−1.45433
$490$	0	0
$491$	−21.2629	−0.959580	−0.479790	−	0.877383i	$-0.659287\pi$
$491$	−21.2629	−0.959580	−0.479790	+	0.877383i	$0.659287\pi$
$492$	0	0
$493$	6.26721	0.282261
$494$	0	0
$495$	−96.6896	−4.34588
$496$	0	0
$497$	43.1072	1.93362
$498$	0	0
$499$	−20.9707	−0.938776	−0.469388	−	0.882992i	$-0.655525\pi$
$499$	−20.9707	−0.938776	−0.469388	+	0.882992i	$0.655525\pi$
$500$	0	0
$501$	−43.3024	−1.93461
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−3.93864	−0.175267
$506$	0	0
$507$	37.1020	1.64776
$508$	0	0
$509$	−2.29606	−0.101771	−0.0508855	−	0.998704i	$-0.516204\pi$
$509$	−2.29606	−0.101771	−0.0508855	+	0.998704i	$0.516204\pi$
$510$	0	0
$511$	32.7871	1.45042
$512$	0	0
$513$	61.3102	2.70691
$514$	0	0
$515$	20.3206	0.895434
$516$	0	0
$517$	−56.1317	−2.46867
$518$	0	0
$519$	−1.39465	−0.0612182
$520$	0	0
$521$	25.1927	1.10371	0.551856	−	0.833939i	$-0.313920\pi$
$521$	25.1927	1.10371	0.551856	+	0.833939i	$0.313920\pi$
$522$	0	0
$523$	−24.6038	−1.07585	−0.537924	−	0.842993i	$-0.680791\pi$
$523$	−24.6038	−1.07585	−0.537924	+	0.842993i	$0.680791\pi$
$524$	0	0
$525$	10.5624	0.460980
$526$	0	0
$527$	−1.60401	−0.0698716
$528$	0	0
$529$	−22.9363	−0.997232
$530$	0	0
$531$	−86.4306	−3.75077
$532$	0	0
$533$	11.4283	0.495015
$534$	0	0
$535$	34.2491	1.48072
$536$	0	0
$537$	0.867759	0.0374466
$538$	0	0
$539$	52.1716	2.24719
$540$	0	0
$541$	6.79885	0.292305	0.146153	−	0.989262i	$-0.453311\pi$
$541$	6.79885	0.292305	0.146153	+	0.989262i	$0.453311\pi$
$542$	0	0
$543$	54.1420	2.32346
$544$	0	0
$545$	13.6920	0.586502
$546$	0	0
$547$	11.4669	0.490288	0.245144	−	0.969487i	$-0.421165\pi$
$547$	11.4669	0.490288	0.245144	+	0.969487i	$0.421165\pi$
$548$	0	0
$549$	89.0319	3.79979
$550$	0	0
$551$	10.3985	0.442990
$552$	0	0
$553$	57.9870	2.46586
$554$	0	0
$555$	47.5301	2.01754
$556$	0	0
$557$	12.2963	0.521010	0.260505	−	0.965473i	$-0.416111\pi$
$557$	12.2963	0.521010	0.260505	+	0.965473i	$0.416111\pi$
$558$	0	0
$559$	10.6185	0.449114
$560$	0	0
$561$	42.5249	1.79540
$562$	0	0
$563$	26.0148	1.09639	0.548197	−	0.836349i	$-0.315314\pi$
$563$	26.0148	1.09639	0.548197	+	0.836349i	$0.315314\pi$
$564$	0	0
$565$	37.0190	1.55740
$566$	0	0
$567$	−106.424	−4.46940
$568$	0	0
$569$	−1.80442	−0.0756452	−0.0378226	−	0.999284i	$-0.512042\pi$
$569$	−1.80442	−0.0756452	−0.0378226	+	0.999284i	$0.512042\pi$
$570$	0	0
$571$	26.1365	1.09378	0.546889	−	0.837205i	$-0.315812\pi$
$571$	26.1365	1.09378	0.546889	+	0.837205i	$0.315812\pi$
$572$	0	0
$573$	10.9693	0.458251
$574$	0	0
$575$	−0.199370	−0.00831431
$576$	0	0
$577$	2.24477	0.0934509	0.0467255	−	0.998908i	$-0.485121\pi$
$577$	2.24477	0.0934509	0.0467255	+	0.998908i	$0.485121\pi$
$578$	0	0
$579$	75.9323	3.15564
$580$	0	0
$581$	−9.79370	−0.406311
$582$	0	0
$583$	30.3202	1.25574
$584$	0	0
$585$	23.1380	0.956637
$586$	0	0
$587$	25.2957	1.04406	0.522032	−	0.852926i	$-0.325174\pi$
$587$	25.2957	1.04406	0.522032	+	0.852926i	$0.325174\pi$
$588$	0	0
$589$	−2.66135	−0.109659
$590$	0	0
$591$	−52.0395	−2.14062
$592$	0	0
$593$	29.9152	1.22847	0.614236	−	0.789122i	$-0.289464\pi$
$593$	29.9152	1.22847	0.614236	+	0.789122i	$0.289464\pi$
$594$	0	0
$595$	−24.4034	−1.00044
$596$	0	0
$597$	−20.8209	−0.852142
$598$	0	0
$599$	−10.7572	−0.439528	−0.219764	−	0.975553i	$-0.570529\pi$
$599$	−10.7572	−0.439528	−0.219764	+	0.975553i	$0.570529\pi$
$600$	0	0
$601$	−0.266854	−0.0108852	−0.00544260	−	0.999985i	$-0.501732\pi$
$601$	−0.266854	−0.0108852	−0.00544260	+	0.999985i	$0.501732\pi$
$602$	0	0
$603$	−73.7298	−3.00251
$604$	0	0
$605$	−40.8591	−1.66116
$606$	0	0
$607$	−31.9146	−1.29537	−0.647687	−	0.761906i	$-0.724264\pi$
$607$	−31.9146	−1.29537	−0.647687	+	0.761906i	$0.724264\pi$
$608$	0	0
$609$	−33.9221	−1.37459
$610$	0	0
$611$	13.4324	0.543417
$612$	0	0
$613$	−27.4937	−1.11046	−0.555231	−	0.831696i	$-0.687370\pi$
$613$	−27.4937	−1.11046	−0.555231	+	0.831696i	$0.687370\pi$
$614$	0	0
$615$	−70.7190	−2.85166
$616$	0	0
$617$	34.9882	1.40857	0.704285	−	0.709917i	$-0.251268\pi$
$617$	34.9882	1.40857	0.704285	+	0.709917i	$0.251268\pi$
$618$	0	0
$619$	25.3070	1.01717	0.508587	−	0.861011i	$-0.330168\pi$
$619$	25.3070	1.01717	0.508587	+	0.861011i	$0.330168\pi$
$620$	0	0
$621$	3.77524	0.151495
$622$	0	0
$623$	8.88989	0.356166
$624$	0	0
$625$	−28.3264	−1.13306
$626$	0	0
$627$	70.5568	2.81777
$628$	0	0
$629$	−14.9859	−0.597525
$630$	0	0
$631$	32.6669	1.30045	0.650225	−	0.759742i	$-0.274675\pi$
$631$	32.6669	1.30045	0.650225	+	0.759742i	$0.274675\pi$
$632$	0	0
$633$	−54.0119	−2.14678
$634$	0	0
$635$	−17.2924	−0.686227
$636$	0	0
$637$	−12.4847	−0.494663
$638$	0	0
$639$	−79.7429	−3.15458
$640$	0	0
$641$	28.6682	1.13233	0.566163	−	0.824294i	$-0.308427\pi$
$641$	28.6682	1.13233	0.566163	+	0.824294i	$0.308427\pi$
$642$	0	0
$643$	45.4939	1.79410	0.897052	−	0.441924i	$-0.145704\pi$
$643$	45.4939	1.79410	0.897052	+	0.441924i	$0.145704\pi$
$644$	0	0
$645$	−65.7077	−2.58724
$646$	0	0
$647$	−20.9384	−0.823172	−0.411586	−	0.911371i	$-0.635025\pi$
$647$	−20.9384	−0.823172	−0.411586	+	0.911371i	$0.635025\pi$
$648$	0	0
$649$	−60.1844	−2.36245
$650$	0	0
$651$	8.68190	0.340270
$652$	0	0
$653$	35.8939	1.40464	0.702319	−	0.711862i	$-0.252148\pi$
$653$	35.8939	1.40464	0.702319	+	0.711862i	$0.252148\pi$
$654$	0	0
$655$	−42.4725	−1.65954
$656$	0	0
$657$	−60.6519	−2.36626
$658$	0	0
$659$	−4.77565	−0.186033	−0.0930165	−	0.995665i	$-0.529651\pi$
$659$	−4.77565	−0.186033	−0.0930165	+	0.995665i	$0.529651\pi$
$660$	0	0
$661$	45.8545	1.78353	0.891766	−	0.452496i	$-0.149466\pi$
$661$	45.8545	1.78353	0.891766	+	0.452496i	$0.149466\pi$
$662$	0	0
$663$	−10.1763	−0.395214
$664$	0	0
$665$	−40.4898	−1.57013
$666$	0	0
$667$	0.640298	0.0247924
$668$	0	0
$669$	28.5657	1.10441
$670$	0	0
$671$	61.9958	2.39332
$672$	0	0
$673$	7.53461	0.290438	0.145219	−	0.989400i	$-0.453611\pi$
$673$	7.53461	0.290438	0.145219	+	0.989400i	$0.453611\pi$
$674$	0	0
$675$	−11.8226	−0.455053
$676$	0	0
$677$	−0.974883	−0.0374678	−0.0187339	−	0.999825i	$-0.505964\pi$
$677$	−0.974883	−0.0374678	−0.0187339	+	0.999825i	$0.505964\pi$
$678$	0	0
$679$	−36.2037	−1.38937
$680$	0	0
$681$	86.1276	3.30042
$682$	0	0
$683$	21.2762	0.814111	0.407056	−	0.913403i	$-0.366555\pi$
$683$	21.2762	0.814111	0.407056	+	0.913403i	$0.366555\pi$
$684$	0	0
$685$	4.27338	0.163277
$686$	0	0
$687$	−47.0854	−1.79642
$688$	0	0
$689$	−7.25567	−0.276419
$690$	0	0
$691$	37.1922	1.41486	0.707429	−	0.706784i	$-0.249855\pi$
$691$	37.1922	1.41486	0.707429	+	0.706784i	$0.249855\pi$
$692$	0	0
$693$	−165.007	−6.26809
$694$	0	0
$695$	14.3348	0.543749
$696$	0	0
$697$	22.2971	0.844565
$698$	0	0
$699$	−40.6649	−1.53809
$700$	0	0
$701$	5.74162	0.216858	0.108429	−	0.994104i	$-0.465418\pi$
$701$	5.74162	0.216858	0.108429	+	0.994104i	$0.465418\pi$
$702$	0	0
$703$	−24.8643	−0.937777
$704$	0	0
$705$	−83.1203	−3.13049
$706$	0	0
$707$	−6.72153	−0.252789
$708$	0	0
$709$	−21.7887	−0.818293	−0.409147	−	0.912469i	$-0.634174\pi$
$709$	−21.7887	−0.818293	−0.409147	+	0.912469i	$0.634174\pi$
$710$	0	0
$711$	−107.268	−4.02288
$712$	0	0
$713$	−0.163875	−0.00613718
$714$	0	0
$715$	16.1117	0.602544
$716$	0	0
$717$	−41.0314	−1.53235
$718$	0	0
$719$	7.81695	0.291523	0.145761	−	0.989320i	$-0.453437\pi$
$719$	7.81695	0.291523	0.145761	+	0.989320i	$0.453437\pi$
$720$	0	0
$721$	34.6784	1.29149
$722$	0	0
$723$	12.3926	0.460884
$724$	0	0
$725$	−2.00517	−0.0744702
$726$	0	0
$727$	−14.5520	−0.539703	−0.269852	−	0.962902i	$-0.586975\pi$
$727$	−14.5520	−0.539703	−0.269852	+	0.962902i	$0.586975\pi$
$728$	0	0
$729$	50.7545	1.87980
$730$	0	0
$731$	20.7171	0.766251
$732$	0	0
$733$	−24.2639	−0.896209	−0.448105	−	0.893981i	$-0.647901\pi$
$733$	−24.2639	−0.896209	−0.448105	+	0.893981i	$0.647901\pi$
$734$	0	0
$735$	77.2561	2.84964
$736$	0	0
$737$	−51.3405	−1.89115
$738$	0	0
$739$	−32.3852	−1.19131	−0.595654	−	0.803241i	$-0.703107\pi$
$739$	−32.3852	−1.19131	−0.595654	+	0.803241i	$0.703107\pi$
$740$	0	0
$741$	−16.8843	−0.620262
$742$	0	0
$743$	31.2046	1.14479	0.572394	−	0.819979i	$-0.306015\pi$
$743$	31.2046	1.14479	0.572394	+	0.819979i	$0.306015\pi$
$744$	0	0
$745$	−7.00202	−0.256534
$746$	0	0
$747$	18.1171	0.662870
$748$	0	0
$749$	58.4481	2.13565
$750$	0	0
$751$	5.88757	0.214841	0.107420	−	0.994214i	$-0.465741\pi$
$751$	5.88757	0.214841	0.107420	+	0.994214i	$0.465741\pi$
$752$	0	0
$753$	−17.5446	−0.639361
$754$	0	0
$755$	−46.5208	−1.69306
$756$	0	0
$757$	−43.3540	−1.57573	−0.787863	−	0.615850i	$-0.788813\pi$
$757$	−43.3540	−1.57573	−0.787863	+	0.615850i	$0.788813\pi$
$758$	0	0
$759$	4.34461	0.157699
$760$	0	0
$761$	−27.2883	−0.989199	−0.494599	−	0.869121i	$-0.664685\pi$
$761$	−27.2883	−0.989199	−0.494599	+	0.869121i	$0.664685\pi$
$762$	0	0
$763$	23.3663	0.845916
$764$	0	0
$765$	45.1432	1.63216
$766$	0	0
$767$	14.4022	0.520034
$768$	0	0
$769$	−33.4898	−1.20767	−0.603837	−	0.797108i	$-0.706362\pi$
$769$	−33.4898	−1.20767	−0.603837	+	0.797108i	$0.706362\pi$
$770$	0	0
$771$	71.7960	2.58567
$772$	0	0
$773$	−25.4526	−0.915468	−0.457734	−	0.889089i	$-0.651339\pi$
$773$	−25.4526	−0.915468	−0.457734	+	0.889089i	$0.651339\pi$
$774$	0	0
$775$	0.513195	0.0184345
$776$	0	0
$777$	81.1130	2.90991
$778$	0	0
$779$	36.9951	1.32549
$780$	0	0
$781$	−55.5276	−1.98693
$782$	0	0
$783$	37.9696	1.35692
$784$	0	0
$785$	−40.0188	−1.42833
$786$	0	0
$787$	−44.1253	−1.57290	−0.786448	−	0.617656i	$-0.788082\pi$
$787$	−44.1253	−1.57290	−0.786448	+	0.617656i	$0.788082\pi$
$788$	0	0
$789$	−97.2593	−3.46252
$790$	0	0
$791$	63.1751	2.24625
$792$	0	0
$793$	−14.8357	−0.526831
$794$	0	0
$795$	44.8985	1.59238
$796$	0	0
$797$	−2.71322	−0.0961071	−0.0480535	−	0.998845i	$-0.515302\pi$
$797$	−2.71322	−0.0961071	−0.0480535	+	0.998845i	$0.515302\pi$
$798$	0	0
$799$	26.2072	0.927144
$800$	0	0
$801$	−16.4452	−0.581061
$802$	0	0
$803$	−42.2339	−1.49040
$804$	0	0
$805$	−2.49320	−0.0878739
$806$	0	0
$807$	−23.0693	−0.812078
$808$	0	0
$809$	−19.0664	−0.670339	−0.335170	−	0.942158i	$-0.608794\pi$
$809$	−19.0664	−0.670339	−0.335170	+	0.942158i	$0.608794\pi$
$810$	0	0
$811$	−3.04639	−0.106973	−0.0534865	−	0.998569i	$-0.517033\pi$
$811$	−3.04639	−0.106973	−0.0534865	+	0.998569i	$0.517033\pi$
$812$	0	0
$813$	46.0628	1.61549
$814$	0	0
$815$	−23.7730	−0.832730
$816$	0	0
$817$	34.3736	1.20258
$818$	0	0
$819$	39.4864	1.37977
$820$	0	0
$821$	23.5826	0.823040	0.411520	−	0.911401i	$-0.364998\pi$
$821$	23.5826	0.823040	0.411520	+	0.911401i	$0.364998\pi$
$822$	0	0
$823$	−50.0761	−1.74554	−0.872772	−	0.488128i	$-0.837680\pi$
$823$	−50.0761	−1.74554	−0.872772	+	0.488128i	$0.837680\pi$
$824$	0	0
$825$	−13.6057	−0.473689
$826$	0	0
$827$	6.20587	0.215799	0.107900	−	0.994162i	$-0.465587\pi$
$827$	6.20587	0.215799	0.107900	+	0.994162i	$0.465587\pi$
$828$	0	0
$829$	26.3173	0.914039	0.457019	−	0.889457i	$-0.348917\pi$
$829$	26.3173	0.914039	0.457019	+	0.889457i	$0.348917\pi$
$830$	0	0
$831$	34.0758	1.18208
$832$	0	0
$833$	−24.3583	−0.843964
$834$	0	0
$835$	−32.0094	−1.10773
$836$	0	0
$837$	−9.71778	−0.335896
$838$	0	0
$839$	26.8788	0.927960	0.463980	−	0.885846i	$-0.346421\pi$
$839$	26.8788	0.927960	0.463980	+	0.885846i	$0.346421\pi$
$840$	0	0
$841$	−22.5602	−0.777938
$842$	0	0
$843$	8.93356	0.307688
$844$	0	0
$845$	27.4260	0.943484
$846$	0	0
$847$	−69.7286	−2.39590
$848$	0	0
$849$	−77.9720	−2.67599
$850$	0	0
$851$	−1.53105	−0.0524837
$852$	0	0
$853$	17.6574	0.604576	0.302288	−	0.953217i	$-0.402250\pi$
$853$	17.6574	0.604576	0.302288	+	0.953217i	$0.402250\pi$
$854$	0	0
$855$	74.9010	2.56156
$856$	0	0
$857$	−12.3209	−0.420875	−0.210437	−	0.977607i	$-0.567489\pi$
$857$	−12.3209	−0.420875	−0.210437	+	0.977607i	$0.567489\pi$
$858$	0	0
$859$	−27.9895	−0.954990	−0.477495	−	0.878635i	$-0.658455\pi$
$859$	−27.9895	−0.954990	−0.477495	+	0.878635i	$0.658455\pi$
$860$	0	0
$861$	−120.686	−4.11298
$862$	0	0
$863$	41.7895	1.42253	0.711265	−	0.702924i	$-0.248123\pi$
$863$	41.7895	1.42253	0.711265	+	0.702924i	$0.248123\pi$
$864$	0	0
$865$	−1.03093	−0.0350528
$866$	0	0
$867$	35.4843	1.20511
$868$	0	0
$869$	−74.6946	−2.53384
$870$	0	0
$871$	12.2859	0.416290
$872$	0	0
$873$	66.9722	2.26667
$874$	0	0
$875$	−41.5985	−1.40629
$876$	0	0
$877$	10.7739	0.363809	0.181905	−	0.983316i	$-0.441774\pi$
$877$	10.7739	0.363809	0.181905	+	0.983316i	$0.441774\pi$
$878$	0	0
$879$	100.029	3.37390
$880$	0	0
$881$	13.6210	0.458904	0.229452	−	0.973320i	$-0.426307\pi$
$881$	13.6210	0.458904	0.229452	+	0.973320i	$0.426307\pi$
$882$	0	0
$883$	−2.70995	−0.0911969	−0.0455985	−	0.998960i	$-0.514519\pi$
$883$	−2.70995	−0.0911969	−0.0455985	+	0.998960i	$0.514519\pi$
$884$	0	0
$885$	−89.1217	−2.99579
$886$	0	0
$887$	14.4039	0.483636	0.241818	−	0.970322i	$-0.422256\pi$
$887$	14.4039	0.483636	0.241818	+	0.970322i	$0.422256\pi$
$888$	0	0
$889$	−29.5105	−0.989750
$890$	0	0
$891$	137.088	4.59262
$892$	0	0
$893$	43.4827	1.45509
$894$	0	0
$895$	0.641453	0.0214414
$896$	0	0
$897$	−1.03967	−0.0347136
$898$	0	0
$899$	−1.64818	−0.0549698
$900$	0	0
$901$	−14.1561	−0.471609
$902$	0	0
$903$	−112.134	−3.73159
$904$	0	0
$905$	40.0221	1.33038
$906$	0	0
$907$	21.6036	0.717335	0.358667	−	0.933465i	$-0.383231\pi$
$907$	21.6036	0.717335	0.358667	+	0.933465i	$0.383231\pi$
$908$	0	0
$909$	12.4340	0.412409
$910$	0	0
$911$	−48.7033	−1.61361	−0.806806	−	0.590816i	$-0.798806\pi$
$911$	−48.7033	−1.61361	−0.806806	+	0.590816i	$0.798806\pi$
$912$	0	0
$913$	12.6155	0.417513
$914$	0	0
$915$	91.8040	3.03495
$916$	0	0
$917$	−72.4819	−2.39356
$918$	0	0
$919$	−52.9195	−1.74565	−0.872826	−	0.488031i	$-0.837715\pi$
$919$	−52.9195	−1.74565	−0.872826	+	0.488031i	$0.837715\pi$
$920$	0	0
$921$	−69.8715	−2.30234
$922$	0	0
$923$	13.2878	0.437374
$924$	0	0
$925$	4.79467	0.157648
$926$	0	0
$927$	−64.1506	−2.10698
$928$	0	0
$929$	28.6654	0.940482	0.470241	−	0.882538i	$-0.344167\pi$
$929$	28.6654	0.940482	0.470241	+	0.882538i	$0.344167\pi$
$930$	0	0
$931$	−40.4149	−1.32455
$932$	0	0
$933$	52.4161	1.71603
$934$	0	0
$935$	31.4347	1.02802
$936$	0	0
$937$	49.3214	1.61126	0.805630	−	0.592419i	$-0.201827\pi$
$937$	49.3214	1.61126	0.805630	+	0.592419i	$0.201827\pi$
$938$	0	0
$939$	−48.8491	−1.59413
$940$	0	0
$941$	−21.8399	−0.711961	−0.355981	−	0.934493i	$-0.615853\pi$
$941$	−21.8399	−0.711961	−0.355981	+	0.934493i	$0.615853\pi$
$942$	0	0
$943$	2.27801	0.0741824
$944$	0	0
$945$	−147.847	−4.80945
$946$	0	0
$947$	−53.2937	−1.73181	−0.865906	−	0.500208i	$-0.833257\pi$
$947$	−53.2937	−1.73181	−0.865906	+	0.500208i	$0.833257\pi$
$948$	0	0
$949$	10.1066	0.328075
$950$	0	0
$951$	−70.9192	−2.29971
$952$	0	0
$953$	8.36995	0.271129	0.135565	−	0.990769i	$-0.456715\pi$
$953$	8.36995	0.271129	0.135565	+	0.990769i	$0.456715\pi$
$954$	0	0
$955$	8.10860	0.262388
$956$	0	0
$957$	43.6960	1.41249
$958$	0	0
$959$	7.29278	0.235496
$960$	0	0
$961$	−30.5782	−0.986393
$962$	0	0
$963$	−108.122	−3.48417
$964$	0	0
$965$	56.1296	1.80688
$966$	0	0
$967$	27.0798	0.870827	0.435414	−	0.900230i	$-0.356602\pi$
$967$	27.0798	0.870827	0.435414	+	0.900230i	$0.356602\pi$
$968$	0	0
$969$	−32.9421	−1.05825
$970$	0	0
$971$	41.6669	1.33715	0.668577	−	0.743643i	$-0.266904\pi$
$971$	41.6669	1.33715	0.668577	+	0.743643i	$0.266904\pi$
$972$	0	0
$973$	24.4632	0.784253
$974$	0	0
$975$	3.25586	0.104271
$976$	0	0
$977$	−9.48703	−0.303517	−0.151758	−	0.988418i	$-0.548494\pi$
$977$	−9.48703	−0.303517	−0.151758	+	0.988418i	$0.548494\pi$
$978$	0	0
$979$	−11.4513	−0.365985
$980$	0	0
$981$	−43.2246	−1.38006
$982$	0	0
$983$	33.8921	1.08099	0.540495	−	0.841347i	$-0.318237\pi$
$983$	33.8921	1.08099	0.540495	+	0.841347i	$0.318237\pi$
$984$	0	0
$985$	−38.4680	−1.22569
$986$	0	0
$987$	−141.850	−4.51513
$988$	0	0
$989$	2.11659	0.0673037
$990$	0	0
$991$	−15.6630	−0.497553	−0.248776	−	0.968561i	$-0.580028\pi$
$991$	−15.6630	−0.497553	−0.248776	+	0.968561i	$0.580028\pi$
$992$	0	0
$993$	103.082	3.27121
$994$	0	0
$995$	−15.3909	−0.487925
$996$	0	0
$997$	45.9055	1.45384	0.726920	−	0.686722i	$-0.240951\pi$
$997$	45.9055	1.45384	0.726920	+	0.686722i	$0.240951\pi$
$998$	0	0
$999$	−90.7910	−2.87250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.1	✓	33
4.3	odd	2		8048.2.a.y.1.33		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.1	✓	33	1.1	even	1	trivial
8048.2.a.y.1.33		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)

\(f(q)\)	\(=\)	\(q-3.25521 q^{3} -2.40628 q^{5} -4.10646 q^{7} +7.59642 q^{9} +O(q^{10})\)	\(q-3.25521 q^{3} -2.40628 q^{5} -4.10646 q^{7} +7.59642 q^{9} +5.28963 q^{11} -1.26582 q^{13} +7.83294 q^{15} -2.46967 q^{17} -4.09764 q^{19} +13.3674 q^{21} -0.252316 q^{23} +0.790160 q^{25} -14.9623 q^{27} -2.53768 q^{29} +0.649483 q^{31} -17.2189 q^{33} +9.88126 q^{35} +6.06797 q^{37} +4.12051 q^{39} -9.02841 q^{41} -8.38864 q^{43} -18.2791 q^{45} -10.6116 q^{47} +9.86298 q^{49} +8.03929 q^{51} +5.73201 q^{53} -12.7283 q^{55} +13.3387 q^{57} -11.3778 q^{59} +11.7202 q^{61} -31.1944 q^{63} +3.04590 q^{65} -9.70586 q^{67} +0.821344 q^{69} -10.4974 q^{71} -7.98428 q^{73} -2.57214 q^{75} -21.7217 q^{77} -14.1209 q^{79} +25.9163 q^{81} +2.38495 q^{83} +5.94269 q^{85} +8.26068 q^{87} -2.16486 q^{89} +5.19802 q^{91} -2.11421 q^{93} +9.86004 q^{95} +8.81629 q^{97} +40.1823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 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

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