LMFDB - Embedded newform 4024.2.a.e.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:	$1$
Dimension:	$29$
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)

$f(q)$	$=$	$q-2.89251 q^{3} +3.16304 q^{5} -0.0867661 q^{7} +5.36660 q^{9} +O(q^{10})$	$q-2.89251 q^{3} +3.16304 q^{5} -0.0867661 q^{7} +5.36660 q^{9} +1.65036 q^{11} -1.35137 q^{13} -9.14912 q^{15} +3.42774 q^{17} +0.324957 q^{19} +0.250972 q^{21} -8.51147 q^{23} +5.00483 q^{25} -6.84542 q^{27} -8.13675 q^{29} -6.31055 q^{31} -4.77369 q^{33} -0.274445 q^{35} +0.255613 q^{37} +3.90885 q^{39} +3.09635 q^{41} -4.19428 q^{43} +16.9748 q^{45} -0.838419 q^{47} -6.99247 q^{49} -9.91478 q^{51} +10.3655 q^{53} +5.22017 q^{55} -0.939942 q^{57} -7.20574 q^{59} -14.0694 q^{61} -0.465639 q^{63} -4.27444 q^{65} +3.88189 q^{67} +24.6195 q^{69} +0.709282 q^{71} -5.35573 q^{73} -14.4765 q^{75} -0.143196 q^{77} +6.01256 q^{79} +3.70062 q^{81} -4.45317 q^{83} +10.8421 q^{85} +23.5356 q^{87} -3.14173 q^{89} +0.117253 q^{91} +18.2533 q^{93} +1.02785 q^{95} +12.9322 q^{97} +8.85684 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−2.89251	−1.66999	−0.834995	−	0.550257i	$-0.814530\pi$
$3$	−2.89251	−1.66999	−0.834995	+	0.550257i	$0.814530\pi$
$4$	0	0
$5$	3.16304	1.41456	0.707278	−	0.706936i	$-0.249923\pi$
$5$	3.16304	1.41456	0.707278	+	0.706936i	$0.249923\pi$
$6$	0	0
$7$	−0.0867661	−0.0327945	−0.0163973	−	0.999866i	$-0.505220\pi$
$7$	−0.0867661	−0.0327945	−0.0163973	+	0.999866i	$0.505220\pi$
$8$	0	0
$9$	5.36660	1.78887
$10$	0	0
$11$	1.65036	0.497603	0.248801	−	0.968554i	$-0.419963\pi$
$11$	1.65036	0.497603	0.248801	+	0.968554i	$0.419963\pi$
$12$	0	0
$13$	−1.35137	−0.374803	−0.187401	−	0.982283i	$-0.560007\pi$
$13$	−1.35137	−0.374803	−0.187401	+	0.982283i	$0.560007\pi$
$14$	0	0
$15$	−9.14912	−2.36229
$16$	0	0
$17$	3.42774	0.831350	0.415675	−	0.909513i	$-0.363545\pi$
$17$	3.42774	0.831350	0.415675	+	0.909513i	$0.363545\pi$
$18$	0	0
$19$	0.324957	0.0745504	0.0372752	−	0.999305i	$-0.488132\pi$
$19$	0.324957	0.0745504	0.0372752	+	0.999305i	$0.488132\pi$
$20$	0	0
$21$	0.250972	0.0547665
$22$	0	0
$23$	−8.51147	−1.77476	−0.887382	−	0.461035i	$-0.847478\pi$
$23$	−8.51147	−1.77476	−0.887382	+	0.461035i	$0.847478\pi$
$24$	0	0
$25$	5.00483	1.00097
$26$	0	0
$27$	−6.84542	−1.31740
$28$	0	0
$29$	−8.13675	−1.51096	−0.755478	−	0.655174i	$-0.772595\pi$
$29$	−8.13675	−1.51096	−0.755478	+	0.655174i	$0.772595\pi$
$30$	0	0
$31$	−6.31055	−1.13341	−0.566704	−	0.823921i	$-0.691782\pi$
$31$	−6.31055	−1.13341	−0.566704	+	0.823921i	$0.691782\pi$
$32$	0	0
$33$	−4.77369	−0.830992
$34$	0	0
$35$	−0.274445	−0.0463896
$36$	0	0
$37$	0.255613	0.0420226	0.0210113	−	0.999779i	$-0.493311\pi$
$37$	0.255613	0.0420226	0.0210113	+	0.999779i	$0.493311\pi$
$38$	0	0
$39$	3.90885	0.625917
$40$	0	0
$41$	3.09635	0.483569	0.241785	−	0.970330i	$-0.422267\pi$
$41$	3.09635	0.483569	0.241785	+	0.970330i	$0.422267\pi$
$42$	0	0
$43$	−4.19428	−0.639622	−0.319811	−	0.947481i	$-0.603619\pi$
$43$	−4.19428	−0.639622	−0.319811	+	0.947481i	$0.603619\pi$
$44$	0	0
$45$	16.9748	2.53045
$46$	0	0
$47$	−0.838419	−0.122296	−0.0611480	−	0.998129i	$-0.519476\pi$
$47$	−0.838419	−0.122296	−0.0611480	+	0.998129i	$0.519476\pi$
$48$	0	0
$49$	−6.99247	−0.998925
$50$	0	0
$51$	−9.91478	−1.38835
$52$	0	0
$53$	10.3655	1.42381	0.711907	−	0.702274i	$-0.247832\pi$
$53$	10.3655	1.42381	0.711907	+	0.702274i	$0.247832\pi$
$54$	0	0
$55$	5.22017	0.703887
$56$	0	0
$57$	−0.939942	−0.124498
$58$	0	0
$59$	−7.20574	−0.938107	−0.469054	−	0.883170i	$-0.655405\pi$
$59$	−7.20574	−0.938107	−0.469054	+	0.883170i	$0.655405\pi$
$60$	0	0
$61$	−14.0694	−1.80140	−0.900698	−	0.434446i	$-0.856944\pi$
$61$	−14.0694	−1.80140	−0.900698	+	0.434446i	$0.856944\pi$
$62$	0	0
$63$	−0.465639	−0.0586650
$64$	0	0
$65$	−4.27444	−0.530179
$66$	0	0
$67$	3.88189	0.474248	0.237124	−	0.971479i	$-0.423795\pi$
$67$	3.88189	0.474248	0.237124	+	0.971479i	$0.423795\pi$
$68$	0	0
$69$	24.6195	2.96384
$70$	0	0
$71$	0.709282	0.0841763	0.0420881	−	0.999114i	$-0.486599\pi$
$71$	0.709282	0.0841763	0.0420881	+	0.999114i	$0.486599\pi$
$72$	0	0
$73$	−5.35573	−0.626841	−0.313421	−	0.949614i	$-0.601475\pi$
$73$	−5.35573	−0.626841	−0.313421	+	0.949614i	$0.601475\pi$
$74$	0	0
$75$	−14.4765	−1.67160
$76$	0	0
$77$	−0.143196	−0.0163186
$78$	0	0
$79$	6.01256	0.676466	0.338233	−	0.941062i	$-0.390171\pi$
$79$	6.01256	0.676466	0.338233	+	0.941062i	$0.390171\pi$
$80$	0	0
$81$	3.70062	0.411180
$82$	0	0
$83$	−4.45317	−0.488799	−0.244399	−	0.969675i	$-0.578591\pi$
$83$	−4.45317	−0.488799	−0.244399	+	0.969675i	$0.578591\pi$
$84$	0	0
$85$	10.8421	1.17599
$86$	0	0
$87$	23.5356	2.52328
$88$	0	0
$89$	−3.14173	−0.333023	−0.166512	−	0.986040i	$-0.553250\pi$
$89$	−3.14173	−0.333023	−0.166512	+	0.986040i	$0.553250\pi$
$90$	0	0
$91$	0.117253	0.0122915
$92$	0	0
$93$	18.2533	1.89278
$94$	0	0
$95$	1.02785	0.105456
$96$	0	0
$97$	12.9322	1.31306	0.656532	−	0.754298i	$-0.272023\pi$
$97$	12.9322	1.31306	0.656532	+	0.754298i	$0.272023\pi$
$98$	0	0
$99$	8.85684	0.890146
$100$	0	0
$101$	5.51614	0.548876	0.274438	−	0.961605i	$-0.411508\pi$
$101$	5.51614	0.548876	0.274438	+	0.961605i	$0.411508\pi$
$102$	0	0
$103$	−1.82461	−0.179784	−0.0898921	−	0.995952i	$-0.528652\pi$
$103$	−1.82461	−0.179784	−0.0898921	+	0.995952i	$0.528652\pi$
$104$	0	0
$105$	0.793834	0.0774702
$106$	0	0
$107$	7.01628	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$107$	7.01628	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$108$	0	0
$109$	0.700666	0.0671116	0.0335558	−	0.999437i	$-0.489317\pi$
$109$	0.700666	0.0671116	0.0335558	+	0.999437i	$0.489317\pi$
$110$	0	0
$111$	−0.739364	−0.0701773
$112$	0	0
$113$	−9.92272	−0.933451	−0.466726	−	0.884402i	$-0.654566\pi$
$113$	−9.92272	−0.933451	−0.466726	+	0.884402i	$0.654566\pi$
$114$	0	0
$115$	−26.9221	−2.51050
$116$	0	0
$117$	−7.25227	−0.670473
$118$	0	0
$119$	−0.297412	−0.0272637
$120$	0	0
$121$	−8.27630	−0.752391
$122$	0	0
$123$	−8.95623	−0.807556
$124$	0	0
$125$	0.0152862	0.00136724
$126$	0	0
$127$	2.53856	0.225260	0.112630	−	0.993637i	$-0.464072\pi$
$127$	2.53856	0.225260	0.112630	+	0.993637i	$0.464072\pi$
$128$	0	0
$129$	12.1320	1.06816
$130$	0	0
$131$	5.98614	0.523011	0.261506	−	0.965202i	$-0.415781\pi$
$131$	5.98614	0.523011	0.261506	+	0.965202i	$0.415781\pi$
$132$	0	0
$133$	−0.0281953	−0.00244484
$134$	0	0
$135$	−21.6523	−1.86354
$136$	0	0
$137$	−5.45258	−0.465845	−0.232923	−	0.972495i	$-0.574829\pi$
$137$	−5.45258	−0.465845	−0.232923	+	0.972495i	$0.574829\pi$
$138$	0	0
$139$	−3.85805	−0.327236	−0.163618	−	0.986524i	$-0.552316\pi$
$139$	−3.85805	−0.327236	−0.163618	+	0.986524i	$0.552316\pi$
$140$	0	0
$141$	2.42513	0.204233
$142$	0	0
$143$	−2.23025	−0.186503
$144$	0	0
$145$	−25.7369	−2.13733
$146$	0	0
$147$	20.2258	1.66819
$148$	0	0
$149$	4.90206	0.401593	0.200796	−	0.979633i	$-0.435647\pi$
$149$	4.90206	0.401593	0.200796	+	0.979633i	$0.435647\pi$
$150$	0	0
$151$	4.64883	0.378317	0.189158	−	0.981947i	$-0.439424\pi$
$151$	4.64883	0.378317	0.189158	+	0.981947i	$0.439424\pi$
$152$	0	0
$153$	18.3953	1.48717
$154$	0	0
$155$	−19.9605	−1.60327
$156$	0	0
$157$	9.85356	0.786399	0.393200	−	0.919453i	$-0.371368\pi$
$157$	9.85356	0.786399	0.393200	+	0.919453i	$0.371368\pi$
$158$	0	0
$159$	−29.9824	−2.37776
$160$	0	0
$161$	0.738507	0.0582025
$162$	0	0
$163$	3.58380	0.280705	0.140353	−	0.990102i	$-0.455176\pi$
$163$	3.58380	0.280705	0.140353	+	0.990102i	$0.455176\pi$
$164$	0	0
$165$	−15.0994	−1.17548
$166$	0	0
$167$	20.0614	1.55239	0.776197	−	0.630491i	$-0.217146\pi$
$167$	20.0614	1.55239	0.776197	+	0.630491i	$0.217146\pi$
$168$	0	0
$169$	−11.1738	−0.859523
$170$	0	0
$171$	1.74392	0.133361
$172$	0	0
$173$	7.17174	0.545257	0.272628	−	0.962119i	$-0.412107\pi$
$173$	7.17174	0.545257	0.272628	+	0.962119i	$0.412107\pi$
$174$	0	0
$175$	−0.434250	−0.0328262
$176$	0	0
$177$	20.8427	1.56663
$178$	0	0
$179$	−26.1953	−1.95793	−0.978963	−	0.204040i	$-0.934593\pi$
$179$	−26.1953	−1.95793	−0.978963	+	0.204040i	$0.934593\pi$
$180$	0	0
$181$	12.6015	0.936659	0.468330	−	0.883554i	$-0.344856\pi$
$181$	12.6015	0.936659	0.468330	+	0.883554i	$0.344856\pi$
$182$	0	0
$183$	40.6957	3.00831
$184$	0	0
$185$	0.808516	0.0594433
$186$	0	0
$187$	5.65702	0.413682
$188$	0	0
$189$	0.593950	0.0432035
$190$	0	0
$191$	−15.7086	−1.13663	−0.568317	−	0.822810i	$-0.692405\pi$
$191$	−15.7086	−1.13663	−0.568317	+	0.822810i	$0.692405\pi$
$192$	0	0
$193$	−17.3679	−1.25017	−0.625085	−	0.780557i	$-0.714936\pi$
$193$	−17.3679	−1.25017	−0.625085	+	0.780557i	$0.714936\pi$
$194$	0	0
$195$	12.3639	0.885394
$196$	0	0
$197$	11.2161	0.799114	0.399557	−	0.916708i	$-0.369164\pi$
$197$	11.2161	0.799114	0.399557	+	0.916708i	$0.369164\pi$
$198$	0	0
$199$	−19.6819	−1.39521	−0.697606	−	0.716482i	$-0.745751\pi$
$199$	−19.6819	−1.39521	−0.697606	+	0.716482i	$0.745751\pi$
$200$	0	0
$201$	−11.2284	−0.791990
$202$	0	0
$203$	0.705994	0.0495510
$204$	0	0
$205$	9.79390	0.684036
$206$	0	0
$207$	−45.6777	−3.17482
$208$	0	0
$209$	0.536298	0.0370965
$210$	0	0
$211$	0.950289	0.0654206	0.0327103	−	0.999465i	$-0.489586\pi$
$211$	0.950289	0.0654206	0.0327103	+	0.999465i	$0.489586\pi$
$212$	0	0
$213$	−2.05160	−0.140574
$214$	0	0
$215$	−13.2667	−0.904781
$216$	0	0
$217$	0.547542	0.0371696
$218$	0	0
$219$	15.4915	1.04682
$220$	0	0
$221$	−4.63215	−0.311592
$222$	0	0
$223$	−25.0527	−1.67765	−0.838825	−	0.544401i	$-0.816757\pi$
$223$	−25.0527	−1.67765	−0.838825	+	0.544401i	$0.816757\pi$
$224$	0	0
$225$	26.8589	1.79060
$226$	0	0
$227$	−21.8298	−1.44890	−0.724449	−	0.689329i	$-0.757905\pi$
$227$	−21.8298	−1.44890	−0.724449	+	0.689329i	$0.757905\pi$
$228$	0	0
$229$	−7.30329	−0.482615	−0.241308	−	0.970449i	$-0.577576\pi$
$229$	−7.30329	−0.482615	−0.241308	+	0.970449i	$0.577576\pi$
$230$	0	0
$231$	0.414194	0.0272520
$232$	0	0
$233$	−2.98100	−0.195292	−0.0976459	−	0.995221i	$-0.531131\pi$
$233$	−2.98100	−0.195292	−0.0976459	+	0.995221i	$0.531131\pi$
$234$	0	0
$235$	−2.65196	−0.172994
$236$	0	0
$237$	−17.3914	−1.12969
$238$	0	0
$239$	21.1398	1.36742	0.683711	−	0.729753i	$-0.260365\pi$
$239$	21.1398	1.36742	0.683711	+	0.729753i	$0.260365\pi$
$240$	0	0
$241$	−19.6132	−1.26340	−0.631699	−	0.775214i	$-0.717642\pi$
$241$	−19.6132	−1.26340	−0.631699	+	0.775214i	$0.717642\pi$
$242$	0	0
$243$	9.83219	0.630735
$244$	0	0
$245$	−22.1175	−1.41303
$246$	0	0
$247$	−0.439138	−0.0279417
$248$	0	0
$249$	12.8808	0.816289
$250$	0	0
$251$	−8.54437	−0.539316	−0.269658	−	0.962956i	$-0.586911\pi$
$251$	−8.54437	−0.539316	−0.269658	+	0.962956i	$0.586911\pi$
$252$	0	0
$253$	−14.0470	−0.883128
$254$	0	0
$255$	−31.3608	−1.96389
$256$	0	0
$257$	4.91812	0.306784	0.153392	−	0.988165i	$-0.450980\pi$
$257$	4.91812	0.306784	0.153392	+	0.988165i	$0.450980\pi$
$258$	0	0
$259$	−0.0221786	−0.00137811
$260$	0	0
$261$	−43.6667	−2.70290
$262$	0	0
$263$	−0.968529	−0.0597220	−0.0298610	−	0.999554i	$-0.509506\pi$
$263$	−0.968529	−0.0597220	−0.0298610	+	0.999554i	$0.509506\pi$
$264$	0	0
$265$	32.7866	2.01406
$266$	0	0
$267$	9.08749	0.556145
$268$	0	0
$269$	−2.33243	−0.142211	−0.0711055	−	0.997469i	$-0.522653\pi$
$269$	−2.33243	−0.142211	−0.0711055	+	0.997469i	$0.522653\pi$
$270$	0	0
$271$	−25.8369	−1.56948	−0.784740	−	0.619825i	$-0.787203\pi$
$271$	−25.8369	−1.56948	−0.784740	+	0.619825i	$0.787203\pi$
$272$	0	0
$273$	−0.339156	−0.0205266
$274$	0	0
$275$	8.25979	0.498084
$276$	0	0
$277$	−8.37790	−0.503379	−0.251690	−	0.967808i	$-0.580986\pi$
$277$	−8.37790	−0.503379	−0.251690	+	0.967808i	$0.580986\pi$
$278$	0	0
$279$	−33.8662	−2.02752
$280$	0	0
$281$	19.5962	1.16901	0.584505	−	0.811390i	$-0.301289\pi$
$281$	19.5962	1.16901	0.584505	+	0.811390i	$0.301289\pi$
$282$	0	0
$283$	2.45452	0.145906	0.0729531	−	0.997335i	$-0.476758\pi$
$283$	2.45452	0.145906	0.0729531	+	0.997335i	$0.476758\pi$
$284$	0	0
$285$	−2.97308	−0.176110
$286$	0	0
$287$	−0.268659	−0.0158584
$288$	0	0
$289$	−5.25058	−0.308857
$290$	0	0
$291$	−37.4064	−2.19280
$292$	0	0
$293$	9.07269	0.530032	0.265016	−	0.964244i	$-0.414623\pi$
$293$	9.07269	0.530032	0.265016	+	0.964244i	$0.414623\pi$
$294$	0	0
$295$	−22.7920	−1.32700
$296$	0	0
$297$	−11.2974	−0.655543
$298$	0	0
$299$	11.5022	0.665187
$300$	0	0
$301$	0.363922	0.0209761
$302$	0	0
$303$	−15.9555	−0.916618
$304$	0	0
$305$	−44.5019	−2.54817
$306$	0	0
$307$	1.54459	0.0881546	0.0440773	−	0.999028i	$-0.485965\pi$
$307$	1.54459	0.0881546	0.0440773	+	0.999028i	$0.485965\pi$
$308$	0	0
$309$	5.27770	0.300238
$310$	0	0
$311$	−13.9975	−0.793725	−0.396862	−	0.917878i	$-0.629901\pi$
$311$	−13.9975	−0.793725	−0.396862	+	0.917878i	$0.629901\pi$
$312$	0	0
$313$	−20.4919	−1.15827	−0.579136	−	0.815231i	$-0.696610\pi$
$313$	−20.4919	−1.15827	−0.579136	+	0.815231i	$0.696610\pi$
$314$	0	0
$315$	−1.47284	−0.0829849
$316$	0	0
$317$	−33.6266	−1.88866	−0.944330	−	0.328999i	$-0.893289\pi$
$317$	−33.6266	−1.88866	−0.944330	+	0.328999i	$0.893289\pi$
$318$	0	0
$319$	−13.4286	−0.751856
$320$	0	0
$321$	−20.2946	−1.13274
$322$	0	0
$323$	1.11387	0.0619774
$324$	0	0
$325$	−6.76339	−0.375165
$326$	0	0
$327$	−2.02668	−0.112076
$328$	0	0
$329$	0.0727464	0.00401064
$330$	0	0
$331$	19.3004	1.06085	0.530423	−	0.847733i	$-0.322033\pi$
$331$	19.3004	1.06085	0.530423	+	0.847733i	$0.322033\pi$
$332$	0	0
$333$	1.37178	0.0751728
$334$	0	0
$335$	12.2786	0.670851
$336$	0	0
$337$	−21.3479	−1.16289	−0.581446	−	0.813585i	$-0.697513\pi$
$337$	−21.3479	−1.16289	−0.581446	+	0.813585i	$0.697513\pi$
$338$	0	0
$339$	28.7016	1.55885
$340$	0	0
$341$	−10.4147	−0.563988
$342$	0	0
$343$	1.21407	0.0655537
$344$	0	0
$345$	77.8725	4.19251
$346$	0	0
$347$	20.9888	1.12674	0.563370	−	0.826205i	$-0.309505\pi$
$347$	20.9888	1.12674	0.563370	+	0.826205i	$0.309505\pi$
$348$	0	0
$349$	−15.8450	−0.848163	−0.424082	−	0.905624i	$-0.639403\pi$
$349$	−15.8450	−0.848163	−0.424082	+	0.905624i	$0.639403\pi$
$350$	0	0
$351$	9.25070	0.493766
$352$	0	0
$353$	−4.84418	−0.257830	−0.128915	−	0.991656i	$-0.541149\pi$
$353$	−4.84418	−0.257830	−0.128915	+	0.991656i	$0.541149\pi$
$354$	0	0
$355$	2.24349	0.119072
$356$	0	0
$357$	0.860266	0.0455301
$358$	0	0
$359$	10.2357	0.540218	0.270109	−	0.962830i	$-0.412940\pi$
$359$	10.2357	0.540218	0.270109	+	0.962830i	$0.412940\pi$
$360$	0	0
$361$	−18.8944	−0.994442
$362$	0	0
$363$	23.9393	1.25649
$364$	0	0
$365$	−16.9404	−0.886702
$366$	0	0
$367$	26.6133	1.38921	0.694603	−	0.719394i	$-0.255580\pi$
$367$	26.6133	1.38921	0.694603	+	0.719394i	$0.255580\pi$
$368$	0	0
$369$	16.6169	0.865042
$370$	0	0
$371$	−0.899376	−0.0466933
$372$	0	0
$373$	36.1254	1.87050	0.935252	−	0.353982i	$-0.115173\pi$
$373$	36.1254	1.87050	0.935252	+	0.353982i	$0.115173\pi$
$374$	0	0
$375$	−0.0442155	−0.00228328
$376$	0	0
$377$	10.9958	0.566311
$378$	0	0
$379$	−29.1879	−1.49928	−0.749640	−	0.661845i	$-0.769773\pi$
$379$	−29.1879	−1.49928	−0.749640	+	0.661845i	$0.769773\pi$
$380$	0	0
$381$	−7.34279	−0.376183
$382$	0	0
$383$	27.0607	1.38274	0.691368	−	0.722503i	$-0.257008\pi$
$383$	27.0607	1.38274	0.691368	+	0.722503i	$0.257008\pi$
$384$	0	0
$385$	−0.452933	−0.0230836
$386$	0	0
$387$	−22.5091	−1.14420
$388$	0	0
$389$	9.47982	0.480646	0.240323	−	0.970693i	$-0.422747\pi$
$389$	9.47982	0.480646	0.240323	+	0.970693i	$0.422747\pi$
$390$	0	0
$391$	−29.1751	−1.47545
$392$	0	0
$393$	−17.3150	−0.873424
$394$	0	0
$395$	19.0180	0.956899
$396$	0	0
$397$	−8.24883	−0.413997	−0.206998	−	0.978341i	$-0.566369\pi$
$397$	−8.24883	−0.413997	−0.206998	+	0.978341i	$0.566369\pi$
$398$	0	0
$399$	0.0815551	0.00408286
$400$	0	0
$401$	7.22273	0.360686	0.180343	−	0.983604i	$-0.442279\pi$
$401$	7.22273	0.360686	0.180343	+	0.983604i	$0.442279\pi$
$402$	0	0
$403$	8.52790	0.424805
$404$	0	0
$405$	11.7052	0.581636
$406$	0	0
$407$	0.421855	0.0209106
$408$	0	0
$409$	−6.11636	−0.302434	−0.151217	−	0.988501i	$-0.548319\pi$
$409$	−6.11636	−0.302434	−0.151217	+	0.988501i	$0.548319\pi$
$410$	0	0
$411$	15.7716	0.777957
$412$	0	0
$413$	0.625214	0.0307648
$414$	0	0
$415$	−14.0856	−0.691433
$416$	0	0
$417$	11.1594	0.546480
$418$	0	0
$419$	14.2107	0.694238	0.347119	−	0.937821i	$-0.387160\pi$
$419$	14.2107	0.694238	0.347119	+	0.937821i	$0.387160\pi$
$420$	0	0
$421$	−3.57680	−0.174322	−0.0871612	−	0.996194i	$-0.527780\pi$
$421$	−3.57680	−0.174322	−0.0871612	+	0.996194i	$0.527780\pi$
$422$	0	0
$423$	−4.49946	−0.218771
$424$	0	0
$425$	17.1553	0.832153
$426$	0	0
$427$	1.22074	0.0590759
$428$	0	0
$429$	6.45102	0.311458
$430$	0	0
$431$	−12.7372	−0.613528	−0.306764	−	0.951786i	$-0.599246\pi$
$431$	−12.7372	−0.613528	−0.306764	+	0.951786i	$0.599246\pi$
$432$	0	0
$433$	−27.2636	−1.31020	−0.655102	−	0.755540i	$-0.727374\pi$
$433$	−27.2636	−1.31020	−0.655102	+	0.755540i	$0.727374\pi$
$434$	0	0
$435$	74.4441	3.56932
$436$	0	0
$437$	−2.76587	−0.132309
$438$	0	0
$439$	2.80779	0.134008	0.0670042	−	0.997753i	$-0.478656\pi$
$439$	2.80779	0.134008	0.0670042	+	0.997753i	$0.478656\pi$
$440$	0	0
$441$	−37.5258	−1.78694
$442$	0	0
$443$	19.4362	0.923441	0.461721	−	0.887025i	$-0.347232\pi$
$443$	19.4362	0.923441	0.461721	+	0.887025i	$0.347232\pi$
$444$	0	0
$445$	−9.93743	−0.471079
$446$	0	0
$447$	−14.1793	−0.670656
$448$	0	0
$449$	−20.4801	−0.966516	−0.483258	−	0.875478i	$-0.660547\pi$
$449$	−20.4801	−0.966516	−0.483258	+	0.875478i	$0.660547\pi$
$450$	0	0
$451$	5.11011	0.240626
$452$	0	0
$453$	−13.4468	−0.631785
$454$	0	0
$455$	0.370877	0.0173870
$456$	0	0
$457$	−4.14429	−0.193862	−0.0969308	−	0.995291i	$-0.530903\pi$
$457$	−4.14429	−0.193862	−0.0969308	+	0.995291i	$0.530903\pi$
$458$	0	0
$459$	−23.4643	−1.09522
$460$	0	0
$461$	5.62269	0.261875	0.130937	−	0.991391i	$-0.458201\pi$
$461$	5.62269	0.261875	0.130937	+	0.991391i	$0.458201\pi$
$462$	0	0
$463$	10.8435	0.503942	0.251971	−	0.967735i	$-0.418921\pi$
$463$	10.8435	0.503942	0.251971	+	0.967735i	$0.418921\pi$
$464$	0	0
$465$	57.7360	2.67744
$466$	0	0
$467$	−5.41417	−0.250538	−0.125269	−	0.992123i	$-0.539979\pi$
$467$	−5.41417	−0.250538	−0.125269	+	0.992123i	$0.539979\pi$
$468$	0	0
$469$	−0.336816	−0.0155527
$470$	0	0
$471$	−28.5015	−1.31328
$472$	0	0
$473$	−6.92209	−0.318278
$474$	0	0
$475$	1.62636	0.0746224
$476$	0	0
$477$	55.6276	2.54701
$478$	0	0
$479$	−30.2707	−1.38310	−0.691552	−	0.722327i	$-0.743073\pi$
$479$	−30.2707	−1.38310	−0.691552	+	0.722327i	$0.743073\pi$
$480$	0	0
$481$	−0.345428	−0.0157502
$482$	0	0
$483$	−2.13614	−0.0971976
$484$	0	0
$485$	40.9050	1.85740
$486$	0	0
$487$	8.01454	0.363173	0.181587	−	0.983375i	$-0.441877\pi$
$487$	8.01454	0.363173	0.181587	+	0.983375i	$0.441877\pi$
$488$	0	0
$489$	−10.3662	−0.468775
$490$	0	0
$491$	−24.9119	−1.12426	−0.562130	−	0.827049i	$-0.690018\pi$
$491$	−24.9119	−1.12426	−0.562130	+	0.827049i	$0.690018\pi$
$492$	0	0
$493$	−27.8907	−1.25613
$494$	0	0
$495$	28.0146	1.25916
$496$	0	0
$497$	−0.0615416	−0.00276052
$498$	0	0
$499$	6.62946	0.296776	0.148388	−	0.988929i	$-0.452592\pi$
$499$	6.62946	0.296776	0.148388	+	0.988929i	$0.452592\pi$
$500$	0	0
$501$	−58.0276	−2.59248
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	17.4478	0.776415
$506$	0	0
$507$	32.3203	1.43539
$508$	0	0
$509$	17.7505	0.786776	0.393388	−	0.919373i	$-0.371303\pi$
$509$	17.7505	0.786776	0.393388	+	0.919373i	$0.371303\pi$
$510$	0	0
$511$	0.464696	0.0205569
$512$	0	0
$513$	−2.22447	−0.0982127
$514$	0	0
$515$	−5.77132	−0.254315
$516$	0	0
$517$	−1.38370	−0.0608549
$518$	0	0
$519$	−20.7443	−0.910574
$520$	0	0
$521$	24.8175	1.08727	0.543637	−	0.839320i	$-0.317047\pi$
$521$	24.8175	1.08727	0.543637	+	0.839320i	$0.317047\pi$
$522$	0	0
$523$	−0.998861	−0.0436771	−0.0218386	−	0.999762i	$-0.506952\pi$
$523$	−0.998861	−0.0436771	−0.0218386	+	0.999762i	$0.506952\pi$
$524$	0	0
$525$	1.25607	0.0548194
$526$	0	0
$527$	−21.6310	−0.942259
$528$	0	0
$529$	49.4451	2.14979
$530$	0	0
$531$	−38.6703	−1.67815
$532$	0	0
$533$	−4.18432	−0.181243
$534$	0	0
$535$	22.1928	0.959478
$536$	0	0
$537$	75.7700	3.26972
$538$	0	0
$539$	−11.5401	−0.497068
$540$	0	0
$541$	30.4572	1.30946	0.654728	−	0.755864i	$-0.272783\pi$
$541$	30.4572	1.30946	0.654728	+	0.755864i	$0.272783\pi$
$542$	0	0
$543$	−36.4498	−1.56421
$544$	0	0
$545$	2.21624	0.0949331
$546$	0	0
$547$	41.2020	1.76167	0.880836	−	0.473422i	$-0.156981\pi$
$547$	41.2020	1.76167	0.880836	+	0.473422i	$0.156981\pi$
$548$	0	0
$549$	−75.5046	−3.22246
$550$	0	0
$551$	−2.64410	−0.112642
$552$	0	0
$553$	−0.521687	−0.0221844
$554$	0	0
$555$	−2.33864	−0.0992697
$556$	0	0
$557$	16.7702	0.710577	0.355288	−	0.934757i	$-0.384383\pi$
$557$	16.7702	0.710577	0.355288	+	0.934757i	$0.384383\pi$
$558$	0	0
$559$	5.66803	0.239732
$560$	0	0
$561$	−16.3630	−0.690845
$562$	0	0
$563$	7.39659	0.311729	0.155865	−	0.987778i	$-0.450184\pi$
$563$	7.39659	0.311729	0.155865	+	0.987778i	$0.450184\pi$
$564$	0	0
$565$	−31.3860	−1.32042
$566$	0	0
$567$	−0.321088	−0.0134844
$568$	0	0
$569$	−36.9403	−1.54862	−0.774308	−	0.632808i	$-0.781902\pi$
$569$	−36.9403	−1.54862	−0.774308	+	0.632808i	$0.781902\pi$
$570$	0	0
$571$	−4.25238	−0.177956	−0.0889782	−	0.996034i	$-0.528360\pi$
$571$	−4.25238	−0.177956	−0.0889782	+	0.996034i	$0.528360\pi$
$572$	0	0
$573$	45.4372	1.89817
$574$	0	0
$575$	−42.5985	−1.77648
$576$	0	0
$577$	11.5494	0.480807	0.240403	−	0.970673i	$-0.422720\pi$
$577$	11.5494	0.480807	0.240403	+	0.970673i	$0.422720\pi$
$578$	0	0
$579$	50.2368	2.08777
$580$	0	0
$581$	0.386384	0.0160299
$582$	0	0
$583$	17.1069	0.708494
$584$	0	0
$585$	−22.9392	−0.948421
$586$	0	0
$587$	−19.3350	−0.798041	−0.399020	−	0.916942i	$-0.630650\pi$
$587$	−19.3350	−0.798041	−0.399020	+	0.916942i	$0.630650\pi$
$588$	0	0
$589$	−2.05066	−0.0844960
$590$	0	0
$591$	−32.4426	−1.33451
$592$	0	0
$593$	−41.7203	−1.71325	−0.856623	−	0.515942i	$-0.827442\pi$
$593$	−41.7203	−1.71325	−0.856623	+	0.515942i	$0.827442\pi$
$594$	0	0
$595$	−0.940726	−0.0385660
$596$	0	0
$597$	56.9300	2.32999
$598$	0	0
$599$	24.4441	0.998757	0.499378	−	0.866384i	$-0.333562\pi$
$599$	24.4441	0.998757	0.499378	+	0.866384i	$0.333562\pi$
$600$	0	0
$601$	−35.5699	−1.45093	−0.725464	−	0.688260i	$-0.758375\pi$
$601$	−35.5699	−1.45093	−0.725464	+	0.688260i	$0.758375\pi$
$602$	0	0
$603$	20.8326	0.848368
$604$	0	0
$605$	−26.1783	−1.06430
$606$	0	0
$607$	26.2333	1.06478	0.532389	−	0.846500i	$-0.321294\pi$
$607$	26.2333	1.06478	0.532389	+	0.846500i	$0.321294\pi$
$608$	0	0
$609$	−2.04209	−0.0827498
$610$	0	0
$611$	1.13302	0.0458369
$612$	0	0
$613$	−12.0076	−0.484982	−0.242491	−	0.970154i	$-0.577965\pi$
$613$	−12.0076	−0.484982	−0.242491	+	0.970154i	$0.577965\pi$
$614$	0	0
$615$	−28.3289	−1.14233
$616$	0	0
$617$	−1.23915	−0.0498865	−0.0249432	−	0.999689i	$-0.507941\pi$
$617$	−1.23915	−0.0498865	−0.0249432	+	0.999689i	$0.507941\pi$
$618$	0	0
$619$	20.9312	0.841297	0.420648	−	0.907224i	$-0.361803\pi$
$619$	20.9312	0.841297	0.420648	+	0.907224i	$0.361803\pi$
$620$	0	0
$621$	58.2646	2.33808
$622$	0	0
$623$	0.272596	0.0109213
$624$	0	0
$625$	−24.9758	−0.999033
$626$	0	0
$627$	−1.55125	−0.0619508
$628$	0	0
$629$	0.876177	0.0349355
$630$	0	0
$631$	29.6292	1.17952	0.589761	−	0.807578i	$-0.299222\pi$
$631$	29.6292	1.17952	0.589761	+	0.807578i	$0.299222\pi$
$632$	0	0
$633$	−2.74872	−0.109252
$634$	0	0
$635$	8.02956	0.318643
$636$	0	0
$637$	9.44942	0.374400
$638$	0	0
$639$	3.80643	0.150580
$640$	0	0
$641$	−8.62403	−0.340629	−0.170314	−	0.985390i	$-0.554478\pi$
$641$	−8.62403	−0.340629	−0.170314	+	0.985390i	$0.554478\pi$
$642$	0	0
$643$	−6.52546	−0.257339	−0.128670	−	0.991688i	$-0.541071\pi$
$643$	−6.52546	−0.257339	−0.128670	+	0.991688i	$0.541071\pi$
$644$	0	0
$645$	38.3740	1.51098
$646$	0	0
$647$	−7.93948	−0.312133	−0.156067	−	0.987747i	$-0.549881\pi$
$647$	−7.93948	−0.312133	−0.156067	+	0.987747i	$0.549881\pi$
$648$	0	0
$649$	−11.8921	−0.466805
$650$	0	0
$651$	−1.58377	−0.0620728
$652$	0	0
$653$	−24.0164	−0.939834	−0.469917	−	0.882710i	$-0.655716\pi$
$653$	−24.0164	−0.939834	−0.469917	+	0.882710i	$0.655716\pi$
$654$	0	0
$655$	18.9344	0.739829
$656$	0	0
$657$	−28.7421	−1.12134
$658$	0	0
$659$	−0.746995	−0.0290988	−0.0145494	−	0.999894i	$-0.504631\pi$
$659$	−0.746995	−0.0290988	−0.0145494	+	0.999894i	$0.504631\pi$
$660$	0	0
$661$	50.7158	1.97262	0.986308	−	0.164915i	$-0.0527349\pi$
$661$	50.7158	1.97262	0.986308	+	0.164915i	$0.0527349\pi$
$662$	0	0
$663$	13.3985	0.520356
$664$	0	0
$665$	−0.0891829	−0.00345836
$666$	0	0
$667$	69.2557	2.68159
$668$	0	0
$669$	72.4650	2.80166
$670$	0	0
$671$	−23.2195	−0.896380
$672$	0	0
$673$	−28.8044	−1.11033	−0.555165	−	0.831740i	$-0.687345\pi$
$673$	−28.8044	−1.11033	−0.555165	+	0.831740i	$0.687345\pi$
$674$	0	0
$675$	−34.2602	−1.31867
$676$	0	0
$677$	12.5708	0.483135	0.241567	−	0.970384i	$-0.422339\pi$
$677$	12.5708	0.483135	0.241567	+	0.970384i	$0.422339\pi$
$678$	0	0
$679$	−1.12207	−0.0430613
$680$	0	0
$681$	63.1430	2.41964
$682$	0	0
$683$	8.12933	0.311060	0.155530	−	0.987831i	$-0.450291\pi$
$683$	8.12933	0.311060	0.155530	+	0.987831i	$0.450291\pi$
$684$	0	0
$685$	−17.2467	−0.658964
$686$	0	0
$687$	21.1248	0.805963
$688$	0	0
$689$	−14.0077	−0.533650
$690$	0	0
$691$	19.2922	0.733911	0.366955	−	0.930239i	$-0.380400\pi$
$691$	19.2922	0.733911	0.366955	+	0.930239i	$0.380400\pi$
$692$	0	0
$693$	−0.768473	−0.0291919
$694$	0	0
$695$	−12.2032	−0.462893
$696$	0	0
$697$	10.6135	0.402015
$698$	0	0
$699$	8.62257	0.326135
$700$	0	0
$701$	−21.5766	−0.814938	−0.407469	−	0.913219i	$-0.633589\pi$
$701$	−21.5766	−0.814938	−0.407469	+	0.913219i	$0.633589\pi$
$702$	0	0
$703$	0.0830635	0.00313280
$704$	0	0
$705$	7.67080	0.288899
$706$	0	0
$707$	−0.478614	−0.0180001
$708$	0	0
$709$	−37.1740	−1.39610	−0.698049	−	0.716050i	$-0.745948\pi$
$709$	−37.1740	−1.39610	−0.698049	+	0.716050i	$0.745948\pi$
$710$	0	0
$711$	32.2670	1.21011
$712$	0	0
$713$	53.7121	2.01153
$714$	0	0
$715$	−7.05438	−0.263819
$716$	0	0
$717$	−61.1472	−2.28358
$718$	0	0
$719$	−23.8741	−0.890353	−0.445177	−	0.895443i	$-0.646859\pi$
$719$	−23.8741	−0.890353	−0.445177	+	0.895443i	$0.646859\pi$
$720$	0	0
$721$	0.158314	0.00589594
$722$	0	0
$723$	56.7314	2.10986
$724$	0	0
$725$	−40.7231	−1.51242
$726$	0	0
$727$	50.8913	1.88745	0.943726	−	0.330727i	$-0.107294\pi$
$727$	50.8913	1.88745	0.943726	+	0.330727i	$0.107294\pi$
$728$	0	0
$729$	−39.5415	−1.46450
$730$	0	0
$731$	−14.3769	−0.531750
$732$	0	0
$733$	42.4031	1.56620	0.783098	−	0.621899i	$-0.213639\pi$
$733$	42.4031	1.56620	0.783098	+	0.621899i	$0.213639\pi$
$734$	0	0
$735$	63.9750	2.35975
$736$	0	0
$737$	6.40653	0.235987
$738$	0	0
$739$	−8.78547	−0.323179	−0.161589	−	0.986858i	$-0.551662\pi$
$739$	−8.78547	−0.323179	−0.161589	+	0.986858i	$0.551662\pi$
$740$	0	0
$741$	1.27021	0.0466623
$742$	0	0
$743$	−45.5350	−1.67052	−0.835258	−	0.549858i	$-0.814682\pi$
$743$	−45.5350	−1.67052	−0.835258	+	0.549858i	$0.814682\pi$
$744$	0	0
$745$	15.5054	0.568075
$746$	0	0
$747$	−23.8984	−0.874396
$748$	0	0
$749$	−0.608775	−0.0222442
$750$	0	0
$751$	41.0516	1.49799	0.748996	−	0.662574i	$-0.230536\pi$
$751$	41.0516	1.49799	0.748996	+	0.662574i	$0.230536\pi$
$752$	0	0
$753$	24.7147	0.900653
$754$	0	0
$755$	14.7045	0.535150
$756$	0	0
$757$	−18.1117	−0.658283	−0.329141	−	0.944281i	$-0.606759\pi$
$757$	−18.1117	−0.658283	−0.329141	+	0.944281i	$0.606759\pi$
$758$	0	0
$759$	40.6311	1.47482
$760$	0	0
$761$	−14.8401	−0.537955	−0.268977	−	0.963147i	$-0.586686\pi$
$761$	−14.8401	−0.537955	−0.268977	+	0.963147i	$0.586686\pi$
$762$	0	0
$763$	−0.0607940	−0.00220089
$764$	0	0
$765$	58.1852	2.10369
$766$	0	0
$767$	9.73763	0.351605
$768$	0	0
$769$	47.2779	1.70489	0.852443	−	0.522820i	$-0.175120\pi$
$769$	47.2779	1.70489	0.852443	+	0.522820i	$0.175120\pi$
$770$	0	0
$771$	−14.2257	−0.512326
$772$	0	0
$773$	22.9669	0.826063	0.413031	−	0.910717i	$-0.364470\pi$
$773$	22.9669	0.826063	0.413031	+	0.910717i	$0.364470\pi$
$774$	0	0
$775$	−31.5833	−1.13450
$776$	0	0
$777$	0.0641517	0.00230143
$778$	0	0
$779$	1.00618	0.0360503
$780$	0	0
$781$	1.17057	0.0418864
$782$	0	0
$783$	55.6994	1.99054
$784$	0	0
$785$	31.1672	1.11241
$786$	0	0
$787$	34.0346	1.21320	0.606601	−	0.795006i	$-0.292533\pi$
$787$	34.0346	1.21320	0.606601	+	0.795006i	$0.292533\pi$
$788$	0	0
$789$	2.80148	0.0997352
$790$	0	0
$791$	0.860956	0.0306121
$792$	0	0
$793$	19.0129	0.675168
$794$	0	0
$795$	−94.8354	−3.36347
$796$	0	0
$797$	28.3698	1.00491	0.502455	−	0.864603i	$-0.332430\pi$
$797$	28.3698	1.00491	0.502455	+	0.864603i	$0.332430\pi$
$798$	0	0
$799$	−2.87389	−0.101671
$800$	0	0
$801$	−16.8604	−0.595734
$802$	0	0
$803$	−8.83890	−0.311918
$804$	0	0
$805$	2.33593	0.0823307
$806$	0	0
$807$	6.74659	0.237491
$808$	0	0
$809$	17.7726	0.624853	0.312426	−	0.949942i	$-0.398858\pi$
$809$	17.7726	0.624853	0.312426	+	0.949942i	$0.398858\pi$
$810$	0	0
$811$	2.18540	0.0767398	0.0383699	−	0.999264i	$-0.487783\pi$
$811$	2.18540	0.0767398	0.0383699	+	0.999264i	$0.487783\pi$
$812$	0	0
$813$	74.7334	2.62102
$814$	0	0
$815$	11.3357	0.397073
$816$	0	0
$817$	−1.36296	−0.0476841
$818$	0	0
$819$	0.629251	0.0219878
$820$	0	0
$821$	−37.6191	−1.31292	−0.656458	−	0.754363i	$-0.727946\pi$
$821$	−37.6191	−1.31292	−0.656458	+	0.754363i	$0.727946\pi$
$822$	0	0
$823$	−43.0117	−1.49929	−0.749647	−	0.661838i	$-0.769777\pi$
$823$	−43.0117	−1.49929	−0.749647	+	0.661838i	$0.769777\pi$
$824$	0	0
$825$	−23.8915	−0.831795
$826$	0	0
$827$	29.1066	1.01213	0.506067	−	0.862494i	$-0.331099\pi$
$827$	29.1066	1.01213	0.506067	+	0.862494i	$0.331099\pi$
$828$	0	0
$829$	2.85870	0.0992866	0.0496433	−	0.998767i	$-0.484192\pi$
$829$	2.85870	0.0992866	0.0496433	+	0.998767i	$0.484192\pi$
$830$	0	0
$831$	24.2332	0.840639
$832$	0	0
$833$	−23.9684	−0.830456
$834$	0	0
$835$	63.4549	2.19595
$836$	0	0
$837$	43.1984	1.49315
$838$	0	0
$839$	21.1794	0.731195	0.365598	−	0.930773i	$-0.380865\pi$
$839$	21.1794	0.731195	0.365598	+	0.930773i	$0.380865\pi$
$840$	0	0
$841$	37.2066	1.28299
$842$	0	0
$843$	−56.6821	−1.95224
$844$	0	0
$845$	−35.3432	−1.21584
$846$	0	0
$847$	0.718103	0.0246743
$848$	0	0
$849$	−7.09973	−0.243662
$850$	0	0
$851$	−2.17565	−0.0745802
$852$	0	0
$853$	29.4353	1.00784	0.503922	−	0.863749i	$-0.331890\pi$
$853$	29.4353	1.00784	0.503922	+	0.863749i	$0.331890\pi$
$854$	0	0
$855$	5.51608	0.188646
$856$	0	0
$857$	−33.1237	−1.13148	−0.565742	−	0.824582i	$-0.691410\pi$
$857$	−33.1237	−1.13148	−0.565742	+	0.824582i	$0.691410\pi$
$858$	0	0
$859$	22.1817	0.756831	0.378416	−	0.925636i	$-0.376469\pi$
$859$	22.1817	0.756831	0.378416	+	0.925636i	$0.376469\pi$
$860$	0	0
$861$	0.777097	0.0264834
$862$	0	0
$863$	7.65441	0.260559	0.130280	−	0.991477i	$-0.458412\pi$
$863$	7.65441	0.260559	0.130280	+	0.991477i	$0.458412\pi$
$864$	0	0
$865$	22.6845	0.771296
$866$	0	0
$867$	15.1873	0.515789
$868$	0	0
$869$	9.92291	0.336612
$870$	0	0
$871$	−5.24587	−0.177750
$872$	0	0
$873$	69.4018	2.34890
$874$	0	0
$875$	−0.00132632	−4.48380e−5 0
$876$	0	0
$877$	0.405191	0.0136823	0.00684117	−	0.999977i	$-0.497822\pi$
$877$	0.405191	0.0136823	0.00684117	+	0.999977i	$0.497822\pi$
$878$	0	0
$879$	−26.2428	−0.885149
$880$	0	0
$881$	13.1508	0.443062	0.221531	−	0.975153i	$-0.428895\pi$
$881$	13.1508	0.443062	0.221531	+	0.975153i	$0.428895\pi$
$882$	0	0
$883$	−37.9590	−1.27742	−0.638711	−	0.769447i	$-0.720532\pi$
$883$	−37.9590	−1.27742	−0.638711	+	0.769447i	$0.720532\pi$
$884$	0	0
$885$	65.9262	2.21608
$886$	0	0
$887$	−37.6438	−1.26395	−0.631977	−	0.774987i	$-0.717756\pi$
$887$	−37.6438	−1.26395	−0.631977	+	0.774987i	$0.717756\pi$
$888$	0	0
$889$	−0.220261	−0.00738730
$890$	0	0
$891$	6.10736	0.204604
$892$	0	0
$893$	−0.272451	−0.00911721
$894$	0	0
$895$	−82.8567	−2.76959
$896$	0	0
$897$	−33.2701	−1.11086
$898$	0	0
$899$	51.3474	1.71253
$900$	0	0
$901$	35.5303	1.18369
$902$	0	0
$903$	−1.05265	−0.0350299
$904$	0	0
$905$	39.8590	1.32496
$906$	0	0
$907$	−12.6513	−0.420080	−0.210040	−	0.977693i	$-0.567359\pi$
$907$	−12.6513	−0.420080	−0.210040	+	0.977693i	$0.567359\pi$
$908$	0	0
$909$	29.6029	0.981867
$910$	0	0
$911$	20.2591	0.671215	0.335607	−	0.942002i	$-0.391058\pi$
$911$	20.2591	0.671215	0.335607	+	0.942002i	$0.391058\pi$
$912$	0	0
$913$	−7.34934	−0.243228
$914$	0	0
$915$	128.722	4.25543
$916$	0	0
$917$	−0.519394	−0.0171519
$918$	0	0
$919$	31.7653	1.04784	0.523920	−	0.851768i	$-0.324469\pi$
$919$	31.7653	1.04784	0.523920	+	0.851768i	$0.324469\pi$
$920$	0	0
$921$	−4.46775	−0.147217
$922$	0	0
$923$	−0.958503	−0.0315495
$924$	0	0
$925$	1.27930	0.0420632
$926$	0	0
$927$	−9.79196	−0.321610
$928$	0	0
$929$	−5.10606	−0.167524	−0.0837621	−	0.996486i	$-0.526694\pi$
$929$	−5.10606	−0.167524	−0.0837621	+	0.996486i	$0.526694\pi$
$930$	0	0
$931$	−2.27226	−0.0744702
$932$	0	0
$933$	40.4879	1.32551
$934$	0	0
$935$	17.8934	0.585176
$936$	0	0
$937$	37.1398	1.21330	0.606652	−	0.794968i	$-0.292512\pi$
$937$	37.1398	1.21330	0.606652	+	0.794968i	$0.292512\pi$
$938$	0	0
$939$	59.2731	1.93430
$940$	0	0
$941$	−3.11400	−0.101514	−0.0507568	−	0.998711i	$-0.516163\pi$
$941$	−3.11400	−0.101514	−0.0507568	+	0.998711i	$0.516163\pi$
$942$	0	0
$943$	−26.3545	−0.858222
$944$	0	0
$945$	1.87869	0.0611138
$946$	0	0
$947$	−19.8244	−0.644207	−0.322104	−	0.946704i	$-0.604390\pi$
$947$	−19.8244	−0.644207	−0.322104	+	0.946704i	$0.604390\pi$
$948$	0	0
$949$	7.23758	0.234942
$950$	0	0
$951$	97.2654	3.15405
$952$	0	0
$953$	−22.6195	−0.732718	−0.366359	−	0.930473i	$-0.619396\pi$
$953$	−22.6195	−0.732718	−0.366359	+	0.930473i	$0.619396\pi$
$954$	0	0
$955$	−49.6869	−1.60783
$956$	0	0
$957$	38.8423	1.25559
$958$	0	0
$959$	0.473099	0.0152772
$960$	0	0
$961$	8.82307	0.284615
$962$	0	0
$963$	37.6536	1.21337
$964$	0	0
$965$	−54.9354	−1.76843
$966$	0	0
$967$	29.4065	0.945649	0.472825	−	0.881157i	$-0.343234\pi$
$967$	29.4065	0.945649	0.472825	+	0.881157i	$0.343234\pi$
$968$	0	0
$969$	−3.22188	−0.103502
$970$	0	0
$971$	41.1462	1.32044	0.660222	−	0.751070i	$-0.270462\pi$
$971$	41.1462	1.32044	0.660222	+	0.751070i	$0.270462\pi$
$972$	0	0
$973$	0.334748	0.0107315
$974$	0	0
$975$	19.5631	0.626522
$976$	0	0
$977$	−57.0787	−1.82611	−0.913055	−	0.407836i	$-0.866283\pi$
$977$	−57.0787	−1.82611	−0.913055	+	0.407836i	$0.866283\pi$
$978$	0	0
$979$	−5.18500	−0.165713
$980$	0	0
$981$	3.76020	0.120054
$982$	0	0
$983$	−8.46913	−0.270123	−0.135062	−	0.990837i	$-0.543123\pi$
$983$	−8.46913	−0.270123	−0.135062	+	0.990837i	$0.543123\pi$
$984$	0	0
$985$	35.4770	1.13039
$986$	0	0
$987$	−0.210419	−0.00669773
$988$	0	0
$989$	35.6995	1.13518
$990$	0	0
$991$	−51.9254	−1.64947	−0.824733	−	0.565522i	$-0.808675\pi$
$991$	−51.9254	−1.64947	−0.824733	+	0.565522i	$0.808675\pi$
$992$	0	0
$993$	−55.8265	−1.77160
$994$	0	0
$995$	−62.2546	−1.97360
$996$	0	0
$997$	−42.4845	−1.34550	−0.672749	−	0.739871i	$-0.734887\pi$
$997$	−42.4845	−1.34550	−0.672749	+	0.739871i	$0.734887\pi$
$998$	0	0
$999$	−1.74978	−0.0553606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.3	✓	29
4.3	odd	2		8048.2.a.w.1.27		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.3	✓	29	1.1	even	1	trivial
8048.2.a.w.1.27		29	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)

\(f(q)\)	\(=\)	\(q-2.89251 q^{3} +3.16304 q^{5} -0.0867661 q^{7} +5.36660 q^{9} +O(q^{10})\)	\(q-2.89251 q^{3} +3.16304 q^{5} -0.0867661 q^{7} +5.36660 q^{9} +1.65036 q^{11} -1.35137 q^{13} -9.14912 q^{15} +3.42774 q^{17} +0.324957 q^{19} +0.250972 q^{21} -8.51147 q^{23} +5.00483 q^{25} -6.84542 q^{27} -8.13675 q^{29} -6.31055 q^{31} -4.77369 q^{33} -0.274445 q^{35} +0.255613 q^{37} +3.90885 q^{39} +3.09635 q^{41} -4.19428 q^{43} +16.9748 q^{45} -0.838419 q^{47} -6.99247 q^{49} -9.91478 q^{51} +10.3655 q^{53} +5.22017 q^{55} -0.939942 q^{57} -7.20574 q^{59} -14.0694 q^{61} -0.465639 q^{63} -4.27444 q^{65} +3.88189 q^{67} +24.6195 q^{69} +0.709282 q^{71} -5.35573 q^{73} -14.4765 q^{75} -0.143196 q^{77} +6.01256 q^{79} +3.70062 q^{81} -4.45317 q^{83} +10.8421 q^{85} +23.5356 q^{87} -3.14173 q^{89} +0.117253 q^{91} +18.2533 q^{93} +1.02785 q^{95} +12.9322 q^{97} +8.85684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

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