LMFDB - Embedded newform 4024.2.a.g.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.65797 q^{3} -0.641790 q^{5} -1.04800 q^{7} -0.251130 q^{9} +O(q^{10})$	$q-1.65797 q^{3} -0.641790 q^{5} -1.04800 q^{7} -0.251130 q^{9} -3.12924 q^{11} -1.00410 q^{13} +1.06407 q^{15} -4.94923 q^{17} -8.16731 q^{19} +1.73756 q^{21} +1.05538 q^{23} -4.58811 q^{25} +5.39028 q^{27} -10.2423 q^{29} +7.21776 q^{31} +5.18819 q^{33} +0.672597 q^{35} -2.17217 q^{37} +1.66477 q^{39} +9.09285 q^{41} +1.16620 q^{43} +0.161173 q^{45} +3.67876 q^{47} -5.90169 q^{49} +8.20569 q^{51} -5.62192 q^{53} +2.00831 q^{55} +13.5412 q^{57} +9.72966 q^{59} +0.384625 q^{61} +0.263185 q^{63} +0.644420 q^{65} -6.44309 q^{67} -1.74979 q^{69} -10.5596 q^{71} -9.94843 q^{73} +7.60695 q^{75} +3.27945 q^{77} +9.57187 q^{79} -8.18354 q^{81} -5.35877 q^{83} +3.17637 q^{85} +16.9815 q^{87} -15.2348 q^{89} +1.05230 q^{91} -11.9668 q^{93} +5.24170 q^{95} +4.42965 q^{97} +0.785847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.65797	−0.957230	−0.478615	−	0.878025i	$-0.658861\pi$
$3$	−1.65797	−0.957230	−0.478615	+	0.878025i	$0.658861\pi$
$4$	0	0
$5$	−0.641790	−0.287017	−0.143509	−	0.989649i	$-0.545838\pi$
$5$	−0.641790	−0.287017	−0.143509	+	0.989649i	$0.545838\pi$
$6$	0	0
$7$	−1.04800	−0.396108	−0.198054	−	0.980191i	$-0.563462\pi$
$7$	−1.04800	−0.396108	−0.198054	+	0.980191i	$0.563462\pi$
$8$	0	0
$9$	−0.251130	−0.0837101
$10$	0	0
$11$	−3.12924	−0.943501	−0.471750	−	0.881732i	$-0.656378\pi$
$11$	−3.12924	−0.943501	−0.471750	+	0.881732i	$0.656378\pi$
$12$	0	0
$13$	−1.00410	−0.278487	−0.139243	−	0.990258i	$-0.544467\pi$
$13$	−1.00410	−0.278487	−0.139243	+	0.990258i	$0.544467\pi$
$14$	0	0
$15$	1.06407	0.274741
$16$	0	0
$17$	−4.94923	−1.20037	−0.600183	−	0.799863i	$-0.704905\pi$
$17$	−4.94923	−1.20037	−0.600183	+	0.799863i	$0.704905\pi$
$18$	0	0
$19$	−8.16731	−1.87371	−0.936855	−	0.349717i	$-0.886278\pi$
$19$	−8.16731	−1.87371	−0.936855	+	0.349717i	$0.886278\pi$
$20$	0	0
$21$	1.73756	0.379166
$22$	0	0
$23$	1.05538	0.220062	0.110031	−	0.993928i	$-0.464905\pi$
$23$	1.05538	0.220062	0.110031	+	0.993928i	$0.464905\pi$
$24$	0	0
$25$	−4.58811	−0.917621
$26$	0	0
$27$	5.39028	1.03736
$28$	0	0
$29$	−10.2423	−1.90195	−0.950976	−	0.309264i	$-0.899917\pi$
$29$	−10.2423	−1.90195	−0.950976	+	0.309264i	$0.899917\pi$
$30$	0	0
$31$	7.21776	1.29635	0.648174	−	0.761492i	$-0.275533\pi$
$31$	7.21776	1.29635	0.648174	+	0.761492i	$0.275533\pi$
$32$	0	0
$33$	5.18819	0.903148
$34$	0	0
$35$	0.672597	0.113690
$36$	0	0
$37$	−2.17217	−0.357103	−0.178552	−	0.983931i	$-0.557141\pi$
$37$	−2.17217	−0.357103	−0.178552	+	0.983931i	$0.557141\pi$
$38$	0	0
$39$	1.66477	0.266576
$40$	0	0
$41$	9.09285	1.42006	0.710032	−	0.704169i	$-0.248680\pi$
$41$	9.09285	1.42006	0.710032	+	0.704169i	$0.248680\pi$
$42$	0	0
$43$	1.16620	0.177843	0.0889217	−	0.996039i	$-0.471658\pi$
$43$	1.16620	0.177843	0.0889217	+	0.996039i	$0.471658\pi$
$44$	0	0
$45$	0.161173	0.0240262
$46$	0	0
$47$	3.67876	0.536602	0.268301	−	0.963335i	$-0.413538\pi$
$47$	3.67876	0.536602	0.268301	+	0.963335i	$0.413538\pi$
$48$	0	0
$49$	−5.90169	−0.843099
$50$	0	0
$51$	8.20569	1.14903
$52$	0	0
$53$	−5.62192	−0.772230	−0.386115	−	0.922451i	$-0.626183\pi$
$53$	−5.62192	−0.772230	−0.386115	+	0.922451i	$0.626183\pi$
$54$	0	0
$55$	2.00831	0.270801
$56$	0	0
$57$	13.5412	1.79357
$58$	0	0
$59$	9.72966	1.26669	0.633347	−	0.773868i	$-0.281681\pi$
$59$	9.72966	1.26669	0.633347	+	0.773868i	$0.281681\pi$
$60$	0	0
$61$	0.384625	0.0492462	0.0246231	−	0.999697i	$-0.492161\pi$
$61$	0.384625	0.0492462	0.0246231	+	0.999697i	$0.492161\pi$
$62$	0	0
$63$	0.263185	0.0331582
$64$	0	0
$65$	0.644420	0.0799304
$66$	0	0
$67$	−6.44309	−0.787149	−0.393575	−	0.919293i	$-0.628762\pi$
$67$	−6.44309	−0.787149	−0.393575	+	0.919293i	$0.628762\pi$
$68$	0	0
$69$	−1.74979	−0.210650
$70$	0	0
$71$	−10.5596	−1.25319	−0.626595	−	0.779345i	$-0.715552\pi$
$71$	−10.5596	−1.25319	−0.626595	+	0.779345i	$0.715552\pi$
$72$	0	0
$73$	−9.94843	−1.16438	−0.582188	−	0.813054i	$-0.697803\pi$
$73$	−9.94843	−1.16438	−0.582188	+	0.813054i	$0.697803\pi$
$74$	0	0
$75$	7.60695	0.878375
$76$	0	0
$77$	3.27945	0.373728
$78$	0	0
$79$	9.57187	1.07692	0.538460	−	0.842651i	$-0.319006\pi$
$79$	9.57187	1.07692	0.538460	+	0.842651i	$0.319006\pi$
$80$	0	0
$81$	−8.18354	−0.909283
$82$	0	0
$83$	−5.35877	−0.588201	−0.294100	−	0.955775i	$-0.595020\pi$
$83$	−5.35877	−0.588201	−0.294100	+	0.955775i	$0.595020\pi$
$84$	0	0
$85$	3.17637	0.344525
$86$	0	0
$87$	16.9815	1.82061
$88$	0	0
$89$	−15.2348	−1.61488	−0.807441	−	0.589949i	$-0.799148\pi$
$89$	−15.2348	−1.61488	−0.807441	+	0.589949i	$0.799148\pi$
$90$	0	0
$91$	1.05230	0.110311
$92$	0	0
$93$	−11.9668	−1.24090
$94$	0	0
$95$	5.24170	0.537787
$96$	0	0
$97$	4.42965	0.449763	0.224882	−	0.974386i	$-0.427800\pi$
$97$	4.42965	0.449763	0.224882	+	0.974386i	$0.427800\pi$
$98$	0	0
$99$	0.785847	0.0789806
$100$	0	0
$101$	13.2292	1.31636	0.658180	−	0.752861i	$-0.271327\pi$
$101$	13.2292	1.31636	0.658180	+	0.752861i	$0.271327\pi$
$102$	0	0
$103$	3.58032	0.352779	0.176390	−	0.984320i	$-0.443558\pi$
$103$	3.58032	0.352779	0.176390	+	0.984320i	$0.443558\pi$
$104$	0	0
$105$	−1.11515	−0.108827
$106$	0	0
$107$	−12.2825	−1.18739	−0.593696	−	0.804690i	$-0.702332\pi$
$107$	−12.2825	−1.18739	−0.593696	+	0.804690i	$0.702332\pi$
$108$	0	0
$109$	−10.5397	−1.00952	−0.504758	−	0.863261i	$-0.668418\pi$
$109$	−10.5397	−1.00952	−0.504758	+	0.863261i	$0.668418\pi$
$110$	0	0
$111$	3.60140	0.341830
$112$	0	0
$113$	13.2593	1.24733	0.623666	−	0.781691i	$-0.285643\pi$
$113$	13.2593	1.24733	0.623666	+	0.781691i	$0.285643\pi$
$114$	0	0
$115$	−0.677333	−0.0631617
$116$	0	0
$117$	0.252160	0.0233122
$118$	0	0
$119$	5.18681	0.475474
$120$	0	0
$121$	−1.20787	−0.109806
$122$	0	0
$123$	−15.0757	−1.35933
$124$	0	0
$125$	6.15355	0.550390
$126$	0	0
$127$	12.7570	1.13200	0.566000	−	0.824405i	$-0.308490\pi$
$127$	12.7570	1.13200	0.566000	+	0.824405i	$0.308490\pi$
$128$	0	0
$129$	−1.93352	−0.170237
$130$	0	0
$131$	7.72668	0.675083	0.337541	−	0.941311i	$-0.390405\pi$
$131$	7.72668	0.675083	0.337541	+	0.941311i	$0.390405\pi$
$132$	0	0
$133$	8.55937	0.742191
$134$	0	0
$135$	−3.45943	−0.297740
$136$	0	0
$137$	2.53757	0.216799	0.108400	−	0.994107i	$-0.465427\pi$
$137$	2.53757	0.216799	0.108400	+	0.994107i	$0.465427\pi$
$138$	0	0
$139$	−18.9296	−1.60559	−0.802796	−	0.596254i	$-0.796655\pi$
$139$	−18.9296	−1.60559	−0.802796	+	0.596254i	$0.796655\pi$
$140$	0	0
$141$	−6.09928	−0.513652
$142$	0	0
$143$	3.14206	0.262752
$144$	0	0
$145$	6.57342	0.545893
$146$	0	0
$147$	9.78484	0.807040
$148$	0	0
$149$	14.7166	1.20563	0.602816	−	0.797880i	$-0.294045\pi$
$149$	14.7166	1.20563	0.602816	+	0.797880i	$0.294045\pi$
$150$	0	0
$151$	−4.75027	−0.386572	−0.193286	−	0.981142i	$-0.561914\pi$
$151$	−4.75027	−0.386572	−0.193286	+	0.981142i	$0.561914\pi$
$152$	0	0
$153$	1.24290	0.100483
$154$	0	0
$155$	−4.63228	−0.372074
$156$	0	0
$157$	17.8919	1.42793	0.713965	−	0.700181i	$-0.246897\pi$
$157$	17.8919	1.42793	0.713965	+	0.700181i	$0.246897\pi$
$158$	0	0
$159$	9.32098	0.739202
$160$	0	0
$161$	−1.10604	−0.0871684
$162$	0	0
$163$	−1.90495	−0.149207	−0.0746034	−	0.997213i	$-0.523769\pi$
$163$	−1.90495	−0.149207	−0.0746034	+	0.997213i	$0.523769\pi$
$164$	0	0
$165$	−3.32972	−0.259219
$166$	0	0
$167$	15.1922	1.17561	0.587804	−	0.809004i	$-0.299993\pi$
$167$	15.1922	1.17561	0.587804	+	0.809004i	$0.299993\pi$
$168$	0	0
$169$	−11.9918	−0.922445
$170$	0	0
$171$	2.05106	0.156849
$172$	0	0
$173$	−14.8865	−1.13180	−0.565899	−	0.824475i	$-0.691471\pi$
$173$	−14.8865	−1.13180	−0.565899	+	0.824475i	$0.691471\pi$
$174$	0	0
$175$	4.80835	0.363477
$176$	0	0
$177$	−16.1315	−1.21252
$178$	0	0
$179$	3.56959	0.266803	0.133402	−	0.991062i	$-0.457410\pi$
$179$	3.56959	0.266803	0.133402	+	0.991062i	$0.457410\pi$
$180$	0	0
$181$	2.67515	0.198842	0.0994210	−	0.995045i	$-0.468301\pi$
$181$	2.67515	0.198842	0.0994210	+	0.995045i	$0.468301\pi$
$182$	0	0
$183$	−0.637697	−0.0471399
$184$	0	0
$185$	1.39408	0.102495
$186$	0	0
$187$	15.4873	1.13255
$188$	0	0
$189$	−5.64903	−0.410906
$190$	0	0
$191$	7.16004	0.518082	0.259041	−	0.965866i	$-0.416593\pi$
$191$	7.16004	0.518082	0.259041	+	0.965866i	$0.416593\pi$
$192$	0	0
$193$	−2.50532	−0.180337	−0.0901685	−	0.995927i	$-0.528741\pi$
$193$	−2.50532	−0.180337	−0.0901685	+	0.995927i	$0.528741\pi$
$194$	0	0
$195$	−1.06843	−0.0765118
$196$	0	0
$197$	12.1976	0.869043	0.434522	−	0.900661i	$-0.356917\pi$
$197$	12.1976	0.869043	0.434522	+	0.900661i	$0.356917\pi$
$198$	0	0
$199$	20.7738	1.47261	0.736306	−	0.676648i	$-0.236568\pi$
$199$	20.7738	1.47261	0.736306	+	0.676648i	$0.236568\pi$
$200$	0	0
$201$	10.6825	0.753483
$202$	0	0
$203$	10.7340	0.753378
$204$	0	0
$205$	−5.83570	−0.407583
$206$	0	0
$207$	−0.265039	−0.0184215
$208$	0	0
$209$	25.5575	1.76785
$210$	0	0
$211$	19.6956	1.35590	0.677952	−	0.735106i	$-0.262868\pi$
$211$	19.6956	1.35590	0.677952	+	0.735106i	$0.262868\pi$
$212$	0	0
$213$	17.5074	1.19959
$214$	0	0
$215$	−0.748453	−0.0510441
$216$	0	0
$217$	−7.56423	−0.513493
$218$	0	0
$219$	16.4942	1.11458
$220$	0	0
$221$	4.96952	0.334286
$222$	0	0
$223$	−16.6894	−1.11760	−0.558801	−	0.829302i	$-0.688738\pi$
$223$	−16.6894	−1.11760	−0.558801	+	0.829302i	$0.688738\pi$
$224$	0	0
$225$	1.15221	0.0768142
$226$	0	0
$227$	−6.91617	−0.459042	−0.229521	−	0.973304i	$-0.573716\pi$
$227$	−6.91617	−0.459042	−0.229521	+	0.973304i	$0.573716\pi$
$228$	0	0
$229$	2.07570	0.137166	0.0685832	−	0.997645i	$-0.478152\pi$
$229$	2.07570	0.137166	0.0685832	+	0.997645i	$0.478152\pi$
$230$	0	0
$231$	−5.43723	−0.357744
$232$	0	0
$233$	−12.1354	−0.795019	−0.397509	−	0.917598i	$-0.630125\pi$
$233$	−12.1354	−0.795019	−0.397509	+	0.917598i	$0.630125\pi$
$234$	0	0
$235$	−2.36099	−0.154014
$236$	0	0
$237$	−15.8699	−1.03086
$238$	0	0
$239$	9.74955	0.630646	0.315323	−	0.948984i	$-0.397887\pi$
$239$	9.74955	0.630646	0.315323	+	0.948984i	$0.397887\pi$
$240$	0	0
$241$	−22.1689	−1.42802	−0.714011	−	0.700135i	$-0.753123\pi$
$241$	−22.1689	−1.42802	−0.714011	+	0.700135i	$0.753123\pi$
$242$	0	0
$243$	−2.60276	−0.166967
$244$	0	0
$245$	3.78764	0.241984
$246$	0	0
$247$	8.20079	0.521804
$248$	0	0
$249$	8.88468	0.563044
$250$	0	0
$251$	−3.82469	−0.241412	−0.120706	−	0.992688i	$-0.538516\pi$
$251$	−3.82469	−0.241412	−0.120706	+	0.992688i	$0.538516\pi$
$252$	0	0
$253$	−3.30254	−0.207629
$254$	0	0
$255$	−5.26632	−0.329790
$256$	0	0
$257$	18.7466	1.16938	0.584690	−	0.811257i	$-0.301216\pi$
$257$	18.7466	1.16938	0.584690	+	0.811257i	$0.301216\pi$
$258$	0	0
$259$	2.27644	0.141451
$260$	0	0
$261$	2.57216	0.159213
$262$	0	0
$263$	0.710656	0.0438209	0.0219105	−	0.999760i	$-0.493025\pi$
$263$	0.710656	0.0438209	0.0219105	+	0.999760i	$0.493025\pi$
$264$	0	0
$265$	3.60809	0.221643
$266$	0	0
$267$	25.2588	1.54581
$268$	0	0
$269$	−10.5914	−0.645770	−0.322885	−	0.946438i	$-0.604653\pi$
$269$	−10.5914	−0.645770	−0.322885	+	0.946438i	$0.604653\pi$
$270$	0	0
$271$	25.4371	1.54519	0.772596	−	0.634898i	$-0.218958\pi$
$271$	25.4371	1.54519	0.772596	+	0.634898i	$0.218958\pi$
$272$	0	0
$273$	−1.74468	−0.105593
$274$	0	0
$275$	14.3573	0.865776
$276$	0	0
$277$	−19.8585	−1.19318	−0.596590	−	0.802546i	$-0.703478\pi$
$277$	−19.8585	−1.19318	−0.596590	+	0.802546i	$0.703478\pi$
$278$	0	0
$279$	−1.81260	−0.108517
$280$	0	0
$281$	13.9347	0.831272	0.415636	−	0.909531i	$-0.363559\pi$
$281$	13.9347	0.831272	0.415636	+	0.909531i	$0.363559\pi$
$282$	0	0
$283$	7.49591	0.445586	0.222793	−	0.974866i	$-0.428483\pi$
$283$	7.49591	0.445586	0.222793	+	0.974866i	$0.428483\pi$
$284$	0	0
$285$	−8.69059	−0.514786
$286$	0	0
$287$	−9.52933	−0.562498
$288$	0	0
$289$	7.49490	0.440876
$290$	0	0
$291$	−7.34424	−0.430527
$292$	0	0
$293$	−0.725430	−0.0423801	−0.0211900	−	0.999775i	$-0.506746\pi$
$293$	−0.725430	−0.0423801	−0.0211900	+	0.999775i	$0.506746\pi$
$294$	0	0
$295$	−6.24439	−0.363563
$296$	0	0
$297$	−16.8675	−0.978750
$298$	0	0
$299$	−1.05971	−0.0612845
$300$	0	0
$301$	−1.22218	−0.0704451
$302$	0	0
$303$	−21.9337	−1.26006
$304$	0	0
$305$	−0.246848	−0.0141345
$306$	0	0
$307$	7.42344	0.423678	0.211839	−	0.977305i	$-0.432055\pi$
$307$	7.42344	0.423678	0.211839	+	0.977305i	$0.432055\pi$
$308$	0	0
$309$	−5.93606	−0.337691
$310$	0	0
$311$	−32.9749	−1.86984	−0.934918	−	0.354865i	$-0.884527\pi$
$311$	−32.9749	−1.86984	−0.934918	+	0.354865i	$0.884527\pi$
$312$	0	0
$313$	−26.4515	−1.49513	−0.747563	−	0.664191i	$-0.768776\pi$
$313$	−26.4515	−1.49513	−0.747563	+	0.664191i	$0.768776\pi$
$314$	0	0
$315$	−0.168909	−0.00951697
$316$	0	0
$317$	13.1267	0.737270	0.368635	−	0.929574i	$-0.379825\pi$
$317$	13.1267	0.737270	0.368635	+	0.929574i	$0.379825\pi$
$318$	0	0
$319$	32.0507	1.79449
$320$	0	0
$321$	20.3640	1.13661
$322$	0	0
$323$	40.4219	2.24914
$324$	0	0
$325$	4.60691	0.255545
$326$	0	0
$327$	17.4744	0.966339
$328$	0	0
$329$	−3.85535	−0.212552
$330$	0	0
$331$	−9.84341	−0.541043	−0.270521	−	0.962714i	$-0.587196\pi$
$331$	−9.84341	−0.541043	−0.270521	+	0.962714i	$0.587196\pi$
$332$	0	0
$333$	0.545499	0.0298932
$334$	0	0
$335$	4.13511	0.225925
$336$	0	0
$337$	21.2297	1.15645	0.578227	−	0.815876i	$-0.303745\pi$
$337$	21.2297	1.15645	0.578227	+	0.815876i	$0.303745\pi$
$338$	0	0
$339$	−21.9836	−1.19398
$340$	0	0
$341$	−22.5861	−1.22311
$342$	0	0
$343$	13.5210	0.730066
$344$	0	0
$345$	1.12300	0.0604603
$346$	0	0
$347$	27.8706	1.49617	0.748086	−	0.663602i	$-0.230973\pi$
$347$	27.8706	1.49617	0.748086	+	0.663602i	$0.230973\pi$
$348$	0	0
$349$	−1.88980	−0.101159	−0.0505794	−	0.998720i	$-0.516107\pi$
$349$	−1.88980	−0.101159	−0.0505794	+	0.998720i	$0.516107\pi$
$350$	0	0
$351$	−5.41237	−0.288891
$352$	0	0
$353$	5.38868	0.286811	0.143405	−	0.989664i	$-0.454195\pi$
$353$	5.38868	0.286811	0.143405	+	0.989664i	$0.454195\pi$
$354$	0	0
$355$	6.77702	0.359687
$356$	0	0
$357$	−8.59958	−0.455138
$358$	0	0
$359$	−0.320811	−0.0169317	−0.00846587	−	0.999964i	$-0.502695\pi$
$359$	−0.320811	−0.0169317	−0.00846587	+	0.999964i	$0.502695\pi$
$360$	0	0
$361$	47.7050	2.51079
$362$	0	0
$363$	2.00261	0.105110
$364$	0	0
$365$	6.38480	0.334196
$366$	0	0
$367$	−0.814245	−0.0425032	−0.0212516	−	0.999774i	$-0.506765\pi$
$367$	−0.814245	−0.0425032	−0.0212516	+	0.999774i	$0.506765\pi$
$368$	0	0
$369$	−2.28349	−0.118874
$370$	0	0
$371$	5.89179	0.305886
$372$	0	0
$373$	−19.7230	−1.02122	−0.510609	−	0.859813i	$-0.670580\pi$
$373$	−19.7230	−1.02122	−0.510609	+	0.859813i	$0.670580\pi$
$374$	0	0
$375$	−10.2024	−0.526850
$376$	0	0
$377$	10.2843	0.529669
$378$	0	0
$379$	−6.41914	−0.329729	−0.164864	−	0.986316i	$-0.552719\pi$
$379$	−6.41914	−0.329729	−0.164864	+	0.986316i	$0.552719\pi$
$380$	0	0
$381$	−21.1507	−1.08359
$382$	0	0
$383$	29.4287	1.50374	0.751868	−	0.659313i	$-0.229153\pi$
$383$	29.4287	1.50374	0.751868	+	0.659313i	$0.229153\pi$
$384$	0	0
$385$	−2.10472	−0.107266
$386$	0	0
$387$	−0.292867	−0.0148873
$388$	0	0
$389$	−23.4775	−1.19036	−0.595180	−	0.803593i	$-0.702919\pi$
$389$	−23.4775	−1.19036	−0.595180	+	0.803593i	$0.702919\pi$
$390$	0	0
$391$	−5.22333	−0.264155
$392$	0	0
$393$	−12.8106	−0.646210
$394$	0	0
$395$	−6.14313	−0.309094
$396$	0	0
$397$	−32.5162	−1.63194	−0.815970	−	0.578094i	$-0.803797\pi$
$397$	−32.5162	−1.63194	−0.815970	+	0.578094i	$0.803797\pi$
$398$	0	0
$399$	−14.1912	−0.710448
$400$	0	0
$401$	−4.90873	−0.245130	−0.122565	−	0.992460i	$-0.539112\pi$
$401$	−4.90873	−0.245130	−0.122565	+	0.992460i	$0.539112\pi$
$402$	0	0
$403$	−7.24734	−0.361016
$404$	0	0
$405$	5.25211	0.260980
$406$	0	0
$407$	6.79725	0.336927
$408$	0	0
$409$	9.76605	0.482900	0.241450	−	0.970413i	$-0.422377\pi$
$409$	9.76605	0.482900	0.241450	+	0.970413i	$0.422377\pi$
$410$	0	0
$411$	−4.20721	−0.207527
$412$	0	0
$413$	−10.1967	−0.501747
$414$	0	0
$415$	3.43920	0.168824
$416$	0	0
$417$	31.3848	1.53692
$418$	0	0
$419$	−14.0513	−0.686451	−0.343226	−	0.939253i	$-0.611520\pi$
$419$	−14.0513	−0.686451	−0.343226	+	0.939253i	$0.611520\pi$
$420$	0	0
$421$	19.2893	0.940104	0.470052	−	0.882639i	$-0.344235\pi$
$421$	19.2893	0.940104	0.470052	+	0.882639i	$0.344235\pi$
$422$	0	0
$423$	−0.923848	−0.0449190
$424$	0	0
$425$	22.7076	1.10148
$426$	0	0
$427$	−0.403088	−0.0195068
$428$	0	0
$429$	−5.20945	−0.251515
$430$	0	0
$431$	3.29648	0.158786	0.0793930	−	0.996843i	$-0.474702\pi$
$431$	3.29648	0.158786	0.0793930	+	0.996843i	$0.474702\pi$
$432$	0	0
$433$	−11.5424	−0.554693	−0.277347	−	0.960770i	$-0.589455\pi$
$433$	−11.5424	−0.554693	−0.277347	+	0.960770i	$0.589455\pi$
$434$	0	0
$435$	−10.8985	−0.522545
$436$	0	0
$437$	−8.61964	−0.412333
$438$	0	0
$439$	−5.76164	−0.274988	−0.137494	−	0.990503i	$-0.543905\pi$
$439$	−5.76164	−0.274988	−0.137494	+	0.990503i	$0.543905\pi$
$440$	0	0
$441$	1.48209	0.0705759
$442$	0	0
$443$	1.92306	0.0913671	0.0456836	−	0.998956i	$-0.485453\pi$
$443$	1.92306	0.0913671	0.0456836	+	0.998956i	$0.485453\pi$
$444$	0	0
$445$	9.77751	0.463498
$446$	0	0
$447$	−24.3997	−1.15407
$448$	0	0
$449$	20.9544	0.988900	0.494450	−	0.869206i	$-0.335369\pi$
$449$	20.9544	0.988900	0.494450	+	0.869206i	$0.335369\pi$
$450$	0	0
$451$	−28.4537	−1.33983
$452$	0	0
$453$	7.87582	0.370038
$454$	0	0
$455$	−0.675353	−0.0316611
$456$	0	0
$457$	32.9064	1.53930	0.769648	−	0.638468i	$-0.220432\pi$
$457$	32.9064	1.53930	0.769648	+	0.638468i	$0.220432\pi$
$458$	0	0
$459$	−26.6778	−1.24521
$460$	0	0
$461$	−27.9196	−1.30035	−0.650173	−	0.759787i	$-0.725303\pi$
$461$	−27.9196	−1.30035	−0.650173	+	0.759787i	$0.725303\pi$
$462$	0	0
$463$	23.2033	1.07835	0.539173	−	0.842195i	$-0.318737\pi$
$463$	23.2033	1.07835	0.539173	+	0.842195i	$0.318737\pi$
$464$	0	0
$465$	7.68019	0.356160
$466$	0	0
$467$	−24.1395	−1.11704	−0.558521	−	0.829490i	$-0.688631\pi$
$467$	−24.1395	−1.11704	−0.558521	+	0.829490i	$0.688631\pi$
$468$	0	0
$469$	6.75238	0.311796
$470$	0	0
$471$	−29.6643	−1.36686
$472$	0	0
$473$	−3.64931	−0.167795
$474$	0	0
$475$	37.4725	1.71936
$476$	0	0
$477$	1.41183	0.0646435
$478$	0	0
$479$	1.22131	0.0558031	0.0279015	−	0.999611i	$-0.491118\pi$
$479$	1.22131	0.0558031	0.0279015	+	0.999611i	$0.491118\pi$
$480$	0	0
$481$	2.18108	0.0994485
$482$	0	0
$483$	1.83379	0.0834403
$484$	0	0
$485$	−2.84291	−0.129090
$486$	0	0
$487$	−12.4944	−0.566178	−0.283089	−	0.959094i	$-0.591359\pi$
$487$	−12.4944	−0.566178	−0.283089	+	0.959094i	$0.591359\pi$
$488$	0	0
$489$	3.15834	0.142825
$490$	0	0
$491$	17.6723	0.797542	0.398771	−	0.917051i	$-0.369437\pi$
$491$	17.6723	0.797542	0.398771	+	0.917051i	$0.369437\pi$
$492$	0	0
$493$	50.6917	2.28304
$494$	0	0
$495$	−0.504348	−0.0226688
$496$	0	0
$497$	11.0664	0.496398
$498$	0	0
$499$	5.74823	0.257326	0.128663	−	0.991688i	$-0.458931\pi$
$499$	5.74823	0.257326	0.128663	+	0.991688i	$0.458931\pi$
$500$	0	0
$501$	−25.1882	−1.12533
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−8.49039	−0.377818
$506$	0	0
$507$	19.8820	0.882992
$508$	0	0
$509$	14.9636	0.663250	0.331625	−	0.943411i	$-0.392403\pi$
$509$	14.9636	0.663250	0.331625	+	0.943411i	$0.392403\pi$
$510$	0	0
$511$	10.4260	0.461218
$512$	0	0
$513$	−44.0241	−1.94371
$514$	0	0
$515$	−2.29781	−0.101254
$516$	0	0
$517$	−11.5117	−0.506285
$518$	0	0
$519$	24.6813	1.08339
$520$	0	0
$521$	29.0406	1.27229	0.636147	−	0.771568i	$-0.280527\pi$
$521$	29.0406	1.27229	0.636147	+	0.771568i	$0.280527\pi$
$522$	0	0
$523$	−26.4563	−1.15685	−0.578427	−	0.815734i	$-0.696333\pi$
$523$	−26.4563	−1.15685	−0.578427	+	0.815734i	$0.696333\pi$
$524$	0	0
$525$	−7.97210	−0.347931
$526$	0	0
$527$	−35.7224	−1.55609
$528$	0	0
$529$	−21.8862	−0.951573
$530$	0	0
$531$	−2.44341	−0.106035
$532$	0	0
$533$	−9.13012	−0.395469
$534$	0	0
$535$	7.88276	0.340801
$536$	0	0
$537$	−5.91827	−0.255392
$538$	0	0
$539$	18.4678	0.795464
$540$	0	0
$541$	−15.5711	−0.669456	−0.334728	−	0.942315i	$-0.608644\pi$
$541$	−15.5711	−0.669456	−0.334728	+	0.942315i	$0.608644\pi$
$542$	0	0
$543$	−4.43532	−0.190338
$544$	0	0
$545$	6.76424	0.289748
$546$	0	0
$547$	−17.9051	−0.765567	−0.382784	−	0.923838i	$-0.625035\pi$
$547$	−17.9051	−0.765567	−0.382784	+	0.923838i	$0.625035\pi$
$548$	0	0
$549$	−0.0965910	−0.00412240
$550$	0	0
$551$	83.6523	3.56371
$552$	0	0
$553$	−10.0313	−0.426576
$554$	0	0
$555$	−2.31134	−0.0981111
$556$	0	0
$557$	1.06170	0.0449856	0.0224928	−	0.999747i	$-0.492840\pi$
$557$	1.06170	0.0449856	0.0224928	+	0.999747i	$0.492840\pi$
$558$	0	0
$559$	−1.17098	−0.0495270
$560$	0	0
$561$	−25.6775	−1.08411
$562$	0	0
$563$	16.6582	0.702057	0.351029	−	0.936365i	$-0.385832\pi$
$563$	16.6582	0.702057	0.351029	+	0.936365i	$0.385832\pi$
$564$	0	0
$565$	−8.50969	−0.358005
$566$	0	0
$567$	8.57637	0.360174
$568$	0	0
$569$	32.9626	1.38186	0.690932	−	0.722920i	$-0.257201\pi$
$569$	32.9626	1.38186	0.690932	+	0.722920i	$0.257201\pi$
$570$	0	0
$571$	−3.84666	−0.160978	−0.0804888	−	0.996756i	$-0.525648\pi$
$571$	−3.84666	−0.160978	−0.0804888	+	0.996756i	$0.525648\pi$
$572$	0	0
$573$	−11.8711	−0.495924
$574$	0	0
$575$	−4.84221	−0.201934
$576$	0	0
$577$	−45.6072	−1.89865	−0.949326	−	0.314294i	$-0.898232\pi$
$577$	−45.6072	−1.89865	−0.949326	+	0.314294i	$0.898232\pi$
$578$	0	0
$579$	4.15375	0.172624
$580$	0	0
$581$	5.61600	0.232991
$582$	0	0
$583$	17.5923	0.728600
$584$	0	0
$585$	−0.161833	−0.00669099
$586$	0	0
$587$	−14.1104	−0.582400	−0.291200	−	0.956662i	$-0.594055\pi$
$587$	−14.1104	−0.582400	−0.291200	+	0.956662i	$0.594055\pi$
$588$	0	0
$589$	−58.9497	−2.42898
$590$	0	0
$591$	−20.2233	−0.831874
$592$	0	0
$593$	−17.0252	−0.699143	−0.349571	−	0.936910i	$-0.613673\pi$
$593$	−17.0252	−0.699143	−0.349571	+	0.936910i	$0.613673\pi$
$594$	0	0
$595$	−3.32884	−0.136469
$596$	0	0
$597$	−34.4423	−1.40963
$598$	0	0
$599$	2.20413	0.0900584	0.0450292	−	0.998986i	$-0.485662\pi$
$599$	2.20413	0.0900584	0.0450292	+	0.998986i	$0.485662\pi$
$600$	0	0
$601$	5.67832	0.231624	0.115812	−	0.993271i	$-0.463053\pi$
$601$	5.67832	0.231624	0.115812	+	0.993271i	$0.463053\pi$
$602$	0	0
$603$	1.61806	0.0658923
$604$	0	0
$605$	0.775197	0.0315162
$606$	0	0
$607$	−20.1335	−0.817193	−0.408597	−	0.912715i	$-0.633982\pi$
$607$	−20.1335	−0.817193	−0.408597	+	0.912715i	$0.633982\pi$
$608$	0	0
$609$	−17.7966	−0.721156
$610$	0	0
$611$	−3.69384	−0.149437
$612$	0	0
$613$	−36.8161	−1.48699	−0.743494	−	0.668742i	$-0.766833\pi$
$613$	−36.8161	−1.48699	−0.743494	+	0.668742i	$0.766833\pi$
$614$	0	0
$615$	9.67542	0.390151
$616$	0	0
$617$	−28.1084	−1.13160	−0.565800	−	0.824542i	$-0.691433\pi$
$617$	−28.1084	−1.13160	−0.565800	+	0.824542i	$0.691433\pi$
$618$	0	0
$619$	7.64584	0.307312	0.153656	−	0.988124i	$-0.450895\pi$
$619$	7.64584	0.307312	0.153656	+	0.988124i	$0.450895\pi$
$620$	0	0
$621$	5.68881	0.228284
$622$	0	0
$623$	15.9661	0.639667
$624$	0	0
$625$	18.9912	0.759650
$626$	0	0
$627$	−42.3736	−1.69224
$628$	0	0
$629$	10.7506	0.428654
$630$	0	0
$631$	−16.7645	−0.667383	−0.333692	−	0.942682i	$-0.608294\pi$
$631$	−16.7645	−0.667383	−0.333692	+	0.942682i	$0.608294\pi$
$632$	0	0
$633$	−32.6548	−1.29791
$634$	0	0
$635$	−8.18731	−0.324903
$636$	0	0
$637$	5.92588	0.234792
$638$	0	0
$639$	2.65183	0.104905
$640$	0	0
$641$	−9.93466	−0.392395	−0.196198	−	0.980564i	$-0.562859\pi$
$641$	−9.93466	−0.392395	−0.196198	+	0.980564i	$0.562859\pi$
$642$	0	0
$643$	3.03775	0.119797	0.0598986	−	0.998204i	$-0.480922\pi$
$643$	3.03775	0.119797	0.0598986	+	0.998204i	$0.480922\pi$
$644$	0	0
$645$	1.24091	0.0488609
$646$	0	0
$647$	−37.4903	−1.47390	−0.736949	−	0.675949i	$-0.763734\pi$
$647$	−37.4903	−1.47390	−0.736949	+	0.675949i	$0.763734\pi$
$648$	0	0
$649$	−30.4464	−1.19513
$650$	0	0
$651$	12.5413	0.491531
$652$	0	0
$653$	4.16723	0.163076	0.0815381	−	0.996670i	$-0.474017\pi$
$653$	4.16723	0.163076	0.0815381	+	0.996670i	$0.474017\pi$
$654$	0	0
$655$	−4.95890	−0.193760
$656$	0	0
$657$	2.49835	0.0974700
$658$	0	0
$659$	−10.1072	−0.393722	−0.196861	−	0.980431i	$-0.563075\pi$
$659$	−10.1072	−0.393722	−0.196861	+	0.980431i	$0.563075\pi$
$660$	0	0
$661$	2.83634	0.110321	0.0551603	−	0.998478i	$-0.482433\pi$
$661$	2.83634	0.110321	0.0551603	+	0.998478i	$0.482433\pi$
$662$	0	0
$663$	−8.23932	−0.319989
$664$	0	0
$665$	−5.49331	−0.213021
$666$	0	0
$667$	−10.8096	−0.418548
$668$	0	0
$669$	27.6705	1.06980
$670$	0	0
$671$	−1.20358	−0.0464638
$672$	0	0
$673$	34.5726	1.33267	0.666337	−	0.745651i	$-0.267861\pi$
$673$	34.5726	1.33267	0.666337	+	0.745651i	$0.267861\pi$
$674$	0	0
$675$	−24.7312	−0.951904
$676$	0	0
$677$	−0.408256	−0.0156905	−0.00784527	−	0.999969i	$-0.502497\pi$
$677$	−0.408256	−0.0156905	−0.00784527	+	0.999969i	$0.502497\pi$
$678$	0	0
$679$	−4.64229	−0.178155
$680$	0	0
$681$	11.4668	0.439409
$682$	0	0
$683$	11.6726	0.446639	0.223320	−	0.974745i	$-0.428311\pi$
$683$	11.6726	0.446639	0.223320	+	0.974745i	$0.428311\pi$
$684$	0	0
$685$	−1.62858	−0.0622250
$686$	0	0
$687$	−3.44146	−0.131300
$688$	0	0
$689$	5.64496	0.215056
$690$	0	0
$691$	−15.5939	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$691$	−15.5939	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$692$	0	0
$693$	−0.823569	−0.0312848
$694$	0	0
$695$	12.1488	0.460832
$696$	0	0
$697$	−45.0026	−1.70460
$698$	0	0
$699$	20.1202	0.761016
$700$	0	0
$701$	−8.22169	−0.310529	−0.155264	−	0.987873i	$-0.549623\pi$
$701$	−8.22169	−0.310529	−0.155264	+	0.987873i	$0.549623\pi$
$702$	0	0
$703$	17.7408	0.669108
$704$	0	0
$705$	3.91445	0.147427
$706$	0	0
$707$	−13.8643	−0.521420
$708$	0	0
$709$	10.1527	0.381294	0.190647	−	0.981659i	$-0.438941\pi$
$709$	10.1527	0.381294	0.190647	+	0.981659i	$0.438941\pi$
$710$	0	0
$711$	−2.40379	−0.0901491
$712$	0	0
$713$	7.61750	0.285277
$714$	0	0
$715$	−2.01654	−0.0754144
$716$	0	0
$717$	−16.1645	−0.603674
$718$	0	0
$719$	−44.2502	−1.65026	−0.825128	−	0.564947i	$-0.808897\pi$
$719$	−44.2502	−1.65026	−0.825128	+	0.564947i	$0.808897\pi$
$720$	0	0
$721$	−3.75218	−0.139738
$722$	0	0
$723$	36.7553	1.36695
$724$	0	0
$725$	46.9929	1.74527
$726$	0	0
$727$	−19.4098	−0.719871	−0.359935	−	0.932977i	$-0.617201\pi$
$727$	−19.4098	−0.719871	−0.359935	+	0.932977i	$0.617201\pi$
$728$	0	0
$729$	28.8659	1.06911
$730$	0	0
$731$	−5.77178	−0.213477
$732$	0	0
$733$	−5.43153	−0.200618	−0.100309	−	0.994956i	$-0.531983\pi$
$733$	−5.43153	−0.200618	−0.100309	+	0.994956i	$0.531983\pi$
$734$	0	0
$735$	−6.27981	−0.231634
$736$	0	0
$737$	20.1620	0.742676
$738$	0	0
$739$	8.54513	0.314338	0.157169	−	0.987572i	$-0.449763\pi$
$739$	8.54513	0.314338	0.157169	+	0.987572i	$0.449763\pi$
$740$	0	0
$741$	−13.5967	−0.499486
$742$	0	0
$743$	36.1454	1.32605	0.663024	−	0.748598i	$-0.269273\pi$
$743$	36.1454	1.32605	0.663024	+	0.748598i	$0.269273\pi$
$744$	0	0
$745$	−9.44497	−0.346037
$746$	0	0
$747$	1.34575	0.0492384
$748$	0	0
$749$	12.8721	0.470335
$750$	0	0
$751$	2.75984	0.100708	0.0503539	−	0.998731i	$-0.483965\pi$
$751$	2.75984	0.100708	0.0503539	+	0.998731i	$0.483965\pi$
$752$	0	0
$753$	6.34123	0.231087
$754$	0	0
$755$	3.04868	0.110953
$756$	0	0
$757$	−13.3009	−0.483428	−0.241714	−	0.970348i	$-0.577710\pi$
$757$	−13.3009	−0.483428	−0.241714	+	0.970348i	$0.577710\pi$
$758$	0	0
$759$	5.47552	0.198749
$760$	0	0
$761$	−21.1916	−0.768197	−0.384098	−	0.923292i	$-0.625488\pi$
$761$	−21.1916	−0.768197	−0.384098	+	0.923292i	$0.625488\pi$
$762$	0	0
$763$	11.0456	0.399877
$764$	0	0
$765$	−0.797682	−0.0288402
$766$	0	0
$767$	−9.76954	−0.352758
$768$	0	0
$769$	22.7330	0.819772	0.409886	−	0.912137i	$-0.365568\pi$
$769$	22.7330	0.819772	0.409886	+	0.912137i	$0.365568\pi$
$770$	0	0
$771$	−31.0813	−1.11937
$772$	0	0
$773$	−10.7932	−0.388206	−0.194103	−	0.980981i	$-0.562180\pi$
$773$	−10.7932	−0.388206	−0.194103	+	0.980981i	$0.562180\pi$
$774$	0	0
$775$	−33.1158	−1.18956
$776$	0	0
$777$	−3.77428	−0.135402
$778$	0	0
$779$	−74.2642	−2.66079
$780$	0	0
$781$	33.0434	1.18239
$782$	0	0
$783$	−55.2090	−1.97301
$784$	0	0
$785$	−11.4828	−0.409840
$786$	0	0
$787$	−32.1698	−1.14673	−0.573364	−	0.819301i	$-0.694362\pi$
$787$	−32.1698	−1.14673	−0.573364	+	0.819301i	$0.694362\pi$
$788$	0	0
$789$	−1.17825	−0.0419467
$790$	0	0
$791$	−13.8958	−0.494077
$792$	0	0
$793$	−0.386201	−0.0137144
$794$	0	0
$795$	−5.98211	−0.212164
$796$	0	0
$797$	−33.4049	−1.18326	−0.591631	−	0.806209i	$-0.701516\pi$
$797$	−33.4049	−1.18326	−0.591631	+	0.806209i	$0.701516\pi$
$798$	0	0
$799$	−18.2070	−0.644119
$800$	0	0
$801$	3.82591	0.135182
$802$	0	0
$803$	31.1310	1.09859
$804$	0	0
$805$	0.709847	0.0250188
$806$	0	0
$807$	17.5603	0.618151
$808$	0	0
$809$	46.5252	1.63574	0.817869	−	0.575405i	$-0.195155\pi$
$809$	46.5252	1.63574	0.817869	+	0.575405i	$0.195155\pi$
$810$	0	0
$811$	−33.6202	−1.18056	−0.590282	−	0.807197i	$-0.700983\pi$
$811$	−33.6202	−1.18056	−0.590282	+	0.807197i	$0.700983\pi$
$812$	0	0
$813$	−42.1739	−1.47910
$814$	0	0
$815$	1.22257	0.0428249
$816$	0	0
$817$	−9.52470	−0.333227
$818$	0	0
$819$	−0.264264	−0.00923412
$820$	0	0
$821$	28.4031	0.991276	0.495638	−	0.868529i	$-0.334934\pi$
$821$	28.4031	0.991276	0.495638	+	0.868529i	$0.334934\pi$
$822$	0	0
$823$	−48.2284	−1.68114	−0.840568	−	0.541706i	$-0.817779\pi$
$823$	−48.2284	−1.68114	−0.840568	+	0.541706i	$0.817779\pi$
$824$	0	0
$825$	−23.8040	−0.828747
$826$	0	0
$827$	1.85957	0.0646638	0.0323319	−	0.999477i	$-0.489707\pi$
$827$	1.85957	0.0646638	0.0323319	+	0.999477i	$0.489707\pi$
$828$	0	0
$829$	10.9628	0.380754	0.190377	−	0.981711i	$-0.439029\pi$
$829$	10.9628	0.380754	0.190377	+	0.981711i	$0.439029\pi$
$830$	0	0
$831$	32.9248	1.14215
$832$	0	0
$833$	29.2088	1.01203
$834$	0	0
$835$	−9.75019	−0.337419
$836$	0	0
$837$	38.9058	1.34478
$838$	0	0
$839$	−6.41321	−0.221408	−0.110704	−	0.993853i	$-0.535311\pi$
$839$	−6.41321	−0.221408	−0.110704	+	0.993853i	$0.535311\pi$
$840$	0	0
$841$	75.9053	2.61742
$842$	0	0
$843$	−23.1033	−0.795719
$844$	0	0
$845$	7.69620	0.264757
$846$	0	0
$847$	1.26585	0.0434951
$848$	0	0
$849$	−12.4280	−0.426528
$850$	0	0
$851$	−2.29248	−0.0785850
$852$	0	0
$853$	−32.6378	−1.11750	−0.558749	−	0.829337i	$-0.688718\pi$
$853$	−32.6378	−1.11750	−0.558749	+	0.829337i	$0.688718\pi$
$854$	0	0
$855$	−1.31635	−0.0450182
$856$	0	0
$857$	−47.1493	−1.61059	−0.805295	−	0.592874i	$-0.797993\pi$
$857$	−47.1493	−1.61059	−0.805295	+	0.592874i	$0.797993\pi$
$858$	0	0
$859$	37.4406	1.27746	0.638729	−	0.769432i	$-0.279461\pi$
$859$	37.4406	1.27746	0.638729	+	0.769432i	$0.279461\pi$
$860$	0	0
$861$	15.7994	0.538441
$862$	0	0
$863$	−7.47429	−0.254428	−0.127214	−	0.991875i	$-0.540603\pi$
$863$	−7.47429	−0.254428	−0.127214	+	0.991875i	$0.540603\pi$
$864$	0	0
$865$	9.55398	0.324845
$866$	0	0
$867$	−12.4263	−0.422020
$868$	0	0
$869$	−29.9527	−1.01607
$870$	0	0
$871$	6.46950	0.219211
$872$	0	0
$873$	−1.11242	−0.0376497
$874$	0	0
$875$	−6.44893	−0.218014
$876$	0	0
$877$	7.18580	0.242647	0.121324	−	0.992613i	$-0.461286\pi$
$877$	7.18580	0.242647	0.121324	+	0.992613i	$0.461286\pi$
$878$	0	0
$879$	1.20274	0.0405675
$880$	0	0
$881$	52.2278	1.75960	0.879799	−	0.475346i	$-0.157677\pi$
$881$	52.2278	1.75960	0.879799	+	0.475346i	$0.157677\pi$
$882$	0	0
$883$	22.9672	0.772908	0.386454	−	0.922309i	$-0.373700\pi$
$883$	22.9672	0.772908	0.386454	+	0.922309i	$0.373700\pi$
$884$	0	0
$885$	10.3530	0.348013
$886$	0	0
$887$	30.0966	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$887$	30.0966	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$888$	0	0
$889$	−13.3694	−0.448394
$890$	0	0
$891$	25.6083	0.857909
$892$	0	0
$893$	−30.0456	−1.00544
$894$	0	0
$895$	−2.29092	−0.0765771
$896$	0	0
$897$	1.75697	0.0586634
$898$	0	0
$899$	−73.9266	−2.46559
$900$	0	0
$901$	27.8242	0.926958
$902$	0	0
$903$	2.02633	0.0674322
$904$	0	0
$905$	−1.71688	−0.0570710
$906$	0	0
$907$	−52.0429	−1.72806	−0.864028	−	0.503444i	$-0.832066\pi$
$907$	−52.0429	−1.72806	−0.864028	+	0.503444i	$0.832066\pi$
$908$	0	0
$909$	−3.32227	−0.110193
$910$	0	0
$911$	22.6563	0.750638	0.375319	−	0.926896i	$-0.377533\pi$
$911$	22.6563	0.750638	0.375319	+	0.926896i	$0.377533\pi$
$912$	0	0
$913$	16.7689	0.554968
$914$	0	0
$915$	0.409267	0.0135300
$916$	0	0
$917$	−8.09758	−0.267405
$918$	0	0
$919$	7.89700	0.260498	0.130249	−	0.991481i	$-0.458422\pi$
$919$	7.89700	0.260498	0.130249	+	0.991481i	$0.458422\pi$
$920$	0	0
$921$	−12.3078	−0.405557
$922$	0	0
$923$	10.6028	0.348997
$924$	0	0
$925$	9.96617	0.327686
$926$	0	0
$927$	−0.899126	−0.0295312
$928$	0	0
$929$	26.1299	0.857293	0.428647	−	0.903472i	$-0.358991\pi$
$929$	26.1299	0.857293	0.428647	+	0.903472i	$0.358991\pi$
$930$	0	0
$931$	48.2010	1.57972
$932$	0	0
$933$	54.6715	1.78986
$934$	0	0
$935$	−9.93960	−0.325060
$936$	0	0
$937$	−5.66690	−0.185130	−0.0925649	−	0.995707i	$-0.529507\pi$
$937$	−5.66690	−0.185130	−0.0925649	+	0.995707i	$0.529507\pi$
$938$	0	0
$939$	43.8558	1.43118
$940$	0	0
$941$	−33.4174	−1.08938	−0.544689	−	0.838638i	$-0.683352\pi$
$941$	−33.4174	−1.08938	−0.544689	+	0.838638i	$0.683352\pi$
$942$	0	0
$943$	9.59644	0.312503
$944$	0	0
$945$	3.62549	0.117937
$946$	0	0
$947$	−50.2675	−1.63348	−0.816738	−	0.577009i	$-0.804220\pi$
$947$	−50.2675	−1.63348	−0.816738	+	0.577009i	$0.804220\pi$
$948$	0	0
$949$	9.98920	0.324263
$950$	0	0
$951$	−21.7637	−0.705737
$952$	0	0
$953$	53.9037	1.74611	0.873056	−	0.487620i	$-0.162135\pi$
$953$	53.9037	1.74611	0.873056	+	0.487620i	$0.162135\pi$
$954$	0	0
$955$	−4.59524	−0.148698
$956$	0	0
$957$	−53.1391	−1.71774
$958$	0	0
$959$	−2.65938	−0.0858757
$960$	0	0
$961$	21.0960	0.680517
$962$	0	0
$963$	3.08450	0.0993966
$964$	0	0
$965$	1.60789	0.0517598
$966$	0	0
$967$	14.7876	0.475537	0.237769	−	0.971322i	$-0.423584\pi$
$967$	14.7876	0.475537	0.237769	+	0.971322i	$0.423584\pi$
$968$	0	0
$969$	−67.0184	−2.15294
$970$	0	0
$971$	−37.3831	−1.19968	−0.599840	−	0.800120i	$-0.704769\pi$
$971$	−37.3831	−1.19968	−0.599840	+	0.800120i	$0.704769\pi$
$972$	0	0
$973$	19.8383	0.635987
$974$	0	0
$975$	−7.63813	−0.244616
$976$	0	0
$977$	−12.1974	−0.390228	−0.195114	−	0.980781i	$-0.562508\pi$
$977$	−12.1974	−0.390228	−0.195114	+	0.980781i	$0.562508\pi$
$978$	0	0
$979$	47.6732	1.52364
$980$	0	0
$981$	2.64683	0.0845066
$982$	0	0
$983$	37.3342	1.19078	0.595388	−	0.803438i	$-0.296998\pi$
$983$	37.3342	1.19078	0.595388	+	0.803438i	$0.296998\pi$
$984$	0	0
$985$	−7.82829	−0.249430
$986$	0	0
$987$	6.39206	0.203461
$988$	0	0
$989$	1.23078	0.0391366
$990$	0	0
$991$	−36.6721	−1.16493	−0.582464	−	0.812856i	$-0.697911\pi$
$991$	−36.6721	−1.16493	−0.582464	+	0.812856i	$0.697911\pi$
$992$	0	0
$993$	16.3201	0.517903
$994$	0	0
$995$	−13.3324	−0.422665
$996$	0	0
$997$	4.94918	0.156742	0.0783710	−	0.996924i	$-0.475028\pi$
$997$	4.94918	0.156742	0.0783710	+	0.996924i	$0.475028\pi$
$998$	0	0
$999$	−11.7086	−0.370445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.9	✓	33
4.3	odd	2		8048.2.a.x.1.25		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.9	✓	33	1.1	even	1	trivial
8048.2.a.x.1.25		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)

\(f(q)\)	\(=\)	\(q-1.65797 q^{3} -0.641790 q^{5} -1.04800 q^{7} -0.251130 q^{9} +O(q^{10})\)	\(q-1.65797 q^{3} -0.641790 q^{5} -1.04800 q^{7} -0.251130 q^{9} -3.12924 q^{11} -1.00410 q^{13} +1.06407 q^{15} -4.94923 q^{17} -8.16731 q^{19} +1.73756 q^{21} +1.05538 q^{23} -4.58811 q^{25} +5.39028 q^{27} -10.2423 q^{29} +7.21776 q^{31} +5.18819 q^{33} +0.672597 q^{35} -2.17217 q^{37} +1.66477 q^{39} +9.09285 q^{41} +1.16620 q^{43} +0.161173 q^{45} +3.67876 q^{47} -5.90169 q^{49} +8.20569 q^{51} -5.62192 q^{53} +2.00831 q^{55} +13.5412 q^{57} +9.72966 q^{59} +0.384625 q^{61} +0.263185 q^{63} +0.644420 q^{65} -6.44309 q^{67} -1.74979 q^{69} -10.5596 q^{71} -9.94843 q^{73} +7.60695 q^{75} +3.27945 q^{77} +9.57187 q^{79} -8.18354 q^{81} -5.35877 q^{83} +3.17637 q^{85} +16.9815 q^{87} -15.2348 q^{89} +1.05230 q^{91} -11.9668 q^{93} +5.24170 q^{95} +4.42965 q^{97} +0.785847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 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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