LMFDB - Embedded newform 4024.2.a.e.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.20984 q^{3} -0.810761 q^{5} +4.21793 q^{7} +1.88340 q^{9} +O(q^{10})$	$q-2.20984 q^{3} -0.810761 q^{5} +4.21793 q^{7} +1.88340 q^{9} +4.98070 q^{11} +0.624206 q^{13} +1.79165 q^{15} -2.63691 q^{17} -4.79574 q^{19} -9.32095 q^{21} -3.72764 q^{23} -4.34267 q^{25} +2.46750 q^{27} -7.24824 q^{29} -6.65725 q^{31} -11.0066 q^{33} -3.41973 q^{35} +8.59972 q^{37} -1.37940 q^{39} -2.64586 q^{41} +0.862734 q^{43} -1.52699 q^{45} -0.217045 q^{47} +10.7909 q^{49} +5.82716 q^{51} -5.03238 q^{53} -4.03816 q^{55} +10.5978 q^{57} -0.0696885 q^{59} +11.6589 q^{61} +7.94405 q^{63} -0.506082 q^{65} -12.0757 q^{67} +8.23749 q^{69} -8.17370 q^{71} -0.949696 q^{73} +9.59661 q^{75} +21.0082 q^{77} +6.08968 q^{79} -11.1030 q^{81} -1.19674 q^{83} +2.13790 q^{85} +16.0175 q^{87} +13.3510 q^{89} +2.63285 q^{91} +14.7115 q^{93} +3.88820 q^{95} -2.98843 q^{97} +9.38067 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.20984	−1.27585	−0.637926	−	0.770097i	$-0.720208\pi$
$3$	−2.20984	−1.27585	−0.637926	+	0.770097i	$0.720208\pi$
$4$	0	0
$5$	−0.810761	−0.362583	−0.181292	−	0.983429i	$-0.558028\pi$
$5$	−0.810761	−0.362583	−0.181292	+	0.983429i	$0.558028\pi$
$6$	0	0
$7$	4.21793	1.59423	0.797113	−	0.603830i	$-0.206359\pi$
$7$	4.21793	1.59423	0.797113	+	0.603830i	$0.206359\pi$
$8$	0	0
$9$	1.88340	0.627801
$10$	0	0
$11$	4.98070	1.50174	0.750869	−	0.660451i	$-0.229635\pi$
$11$	4.98070	1.50174	0.750869	+	0.660451i	$0.229635\pi$
$12$	0	0
$13$	0.624206	0.173124	0.0865618	−	0.996246i	$-0.472412\pi$
$13$	0.624206	0.173124	0.0865618	+	0.996246i	$0.472412\pi$
$14$	0	0
$15$	1.79165	0.462603
$16$	0	0
$17$	−2.63691	−0.639545	−0.319773	−	0.947494i	$-0.603606\pi$
$17$	−2.63691	−0.639545	−0.319773	+	0.947494i	$0.603606\pi$
$18$	0	0
$19$	−4.79574	−1.10022	−0.550109	−	0.835093i	$-0.685414\pi$
$19$	−4.79574	−1.10022	−0.550109	+	0.835093i	$0.685414\pi$
$20$	0	0
$21$	−9.32095	−2.03400
$22$	0	0
$23$	−3.72764	−0.777266	−0.388633	−	0.921393i	$-0.627053\pi$
$23$	−3.72764	−0.777266	−0.388633	+	0.921393i	$0.627053\pi$
$24$	0	0
$25$	−4.34267	−0.868533
$26$	0	0
$27$	2.46750	0.474872
$28$	0	0
$29$	−7.24824	−1.34596	−0.672982	−	0.739659i	$-0.734987\pi$
$29$	−7.24824	−1.34596	−0.672982	+	0.739659i	$0.734987\pi$
$30$	0	0
$31$	−6.65725	−1.19568	−0.597839	−	0.801616i	$-0.703974\pi$
$31$	−6.65725	−1.19568	−0.597839	+	0.801616i	$0.703974\pi$
$32$	0	0
$33$	−11.0066	−1.91600
$34$	0	0
$35$	−3.41973	−0.578040
$36$	0	0
$37$	8.59972	1.41379	0.706893	−	0.707321i	$-0.250096\pi$
$37$	8.59972	1.41379	0.706893	+	0.707321i	$0.250096\pi$
$38$	0	0
$39$	−1.37940	−0.220880
$40$	0	0
$41$	−2.64586	−0.413213	−0.206607	−	0.978424i	$-0.566242\pi$
$41$	−2.64586	−0.413213	−0.206607	+	0.978424i	$0.566242\pi$
$42$	0	0
$43$	0.862734	0.131566	0.0657829	−	0.997834i	$-0.479046\pi$
$43$	0.862734	0.131566	0.0657829	+	0.997834i	$0.479046\pi$
$44$	0	0
$45$	−1.52699	−0.227630
$46$	0	0
$47$	−0.217045	−0.0316593	−0.0158296	−	0.999875i	$-0.505039\pi$
$47$	−0.217045	−0.0316593	−0.0158296	+	0.999875i	$0.505039\pi$
$48$	0	0
$49$	10.7909	1.54156
$50$	0	0
$51$	5.82716	0.815965
$52$	0	0
$53$	−5.03238	−0.691251	−0.345626	−	0.938372i	$-0.612333\pi$
$53$	−5.03238	−0.691251	−0.345626	+	0.938372i	$0.612333\pi$
$54$	0	0
$55$	−4.03816	−0.544505
$56$	0	0
$57$	10.5978	1.40372
$58$	0	0
$59$	−0.0696885	−0.00907267	−0.00453633	−	0.999990i	$-0.501444\pi$
$59$	−0.0696885	−0.00907267	−0.00453633	+	0.999990i	$0.501444\pi$
$60$	0	0
$61$	11.6589	1.49277	0.746387	−	0.665512i	$-0.231787\pi$
$61$	11.6589	1.49277	0.746387	+	0.665512i	$0.231787\pi$
$62$	0	0
$63$	7.94405	1.00086
$64$	0	0
$65$	−0.506082	−0.0627717
$66$	0	0
$67$	−12.0757	−1.47528	−0.737641	−	0.675193i	$-0.764060\pi$
$67$	−12.0757	−1.47528	−0.737641	+	0.675193i	$0.764060\pi$
$68$	0	0
$69$	8.23749	0.991677
$70$	0	0
$71$	−8.17370	−0.970040	−0.485020	−	0.874503i	$-0.661188\pi$
$71$	−8.17370	−0.970040	−0.485020	+	0.874503i	$0.661188\pi$
$72$	0	0
$73$	−0.949696	−0.111153	−0.0555767	−	0.998454i	$-0.517700\pi$
$73$	−0.949696	−0.111153	−0.0555767	+	0.998454i	$0.517700\pi$
$74$	0	0
$75$	9.59661	1.10812
$76$	0	0
$77$	21.0082	2.39411
$78$	0	0
$79$	6.08968	0.685143	0.342571	−	0.939492i	$-0.388702\pi$
$79$	6.08968	0.685143	0.342571	+	0.939492i	$0.388702\pi$
$80$	0	0
$81$	−11.1030	−1.23367
$82$	0	0
$83$	−1.19674	−0.131359	−0.0656795	−	0.997841i	$-0.520921\pi$
$83$	−1.19674	−0.131359	−0.0656795	+	0.997841i	$0.520921\pi$
$84$	0	0
$85$	2.13790	0.231888
$86$	0	0
$87$	16.0175	1.71725
$88$	0	0
$89$	13.3510	1.41520	0.707601	−	0.706612i	$-0.249777\pi$
$89$	13.3510	1.41520	0.707601	+	0.706612i	$0.249777\pi$
$90$	0	0
$91$	2.63285	0.275998
$92$	0	0
$93$	14.7115	1.52551
$94$	0	0
$95$	3.88820	0.398921
$96$	0	0
$97$	−2.98843	−0.303429	−0.151715	−	0.988424i	$-0.548480\pi$
$97$	−2.98843	−0.303429	−0.151715	+	0.988424i	$0.548480\pi$
$98$	0	0
$99$	9.38067	0.942793
$100$	0	0
$101$	−2.76405	−0.275033	−0.137517	−	0.990499i	$-0.543912\pi$
$101$	−2.76405	−0.275033	−0.137517	+	0.990499i	$0.543912\pi$
$102$	0	0
$103$	−11.2909	−1.11252	−0.556261	−	0.831007i	$-0.687765\pi$
$103$	−11.2909	−1.11252	−0.556261	+	0.831007i	$0.687765\pi$
$104$	0	0
$105$	7.55706	0.737494
$106$	0	0
$107$	−4.95815	−0.479323	−0.239661	−	0.970857i	$-0.577036\pi$
$107$	−4.95815	−0.479323	−0.239661	+	0.970857i	$0.577036\pi$
$108$	0	0
$109$	10.5775	1.01314	0.506569	−	0.862199i	$-0.330914\pi$
$109$	10.5775	1.01314	0.506569	+	0.862199i	$0.330914\pi$
$110$	0	0
$111$	−19.0040	−1.80378
$112$	0	0
$113$	−10.0314	−0.943670	−0.471835	−	0.881687i	$-0.656408\pi$
$113$	−10.0314	−0.943670	−0.471835	+	0.881687i	$0.656408\pi$
$114$	0	0
$115$	3.02222	0.281824
$116$	0	0
$117$	1.17563	0.108687
$118$	0	0
$119$	−11.1223	−1.01958
$120$	0	0
$121$	13.8074	1.25522
$122$	0	0
$123$	5.84693	0.527199
$124$	0	0
$125$	7.57467	0.677499
$126$	0	0
$127$	−21.9317	−1.94613	−0.973064	−	0.230536i	$-0.925952\pi$
$127$	−21.9317	−1.94613	−0.973064	+	0.230536i	$0.925952\pi$
$128$	0	0
$129$	−1.90651	−0.167859
$130$	0	0
$131$	−13.5170	−1.18099	−0.590493	−	0.807043i	$-0.701067\pi$
$131$	−13.5170	−1.18099	−0.590493	+	0.807043i	$0.701067\pi$
$132$	0	0
$133$	−20.2281	−1.75400
$134$	0	0
$135$	−2.00056	−0.172181
$136$	0	0
$137$	−8.79887	−0.751738	−0.375869	−	0.926673i	$-0.622656\pi$
$137$	−8.79887	−0.751738	−0.375869	+	0.926673i	$0.622656\pi$
$138$	0	0
$139$	21.3431	1.81030	0.905150	−	0.425092i	$-0.139758\pi$
$139$	21.3431	1.81030	0.905150	+	0.425092i	$0.139758\pi$
$140$	0	0
$141$	0.479635	0.0403926
$142$	0	0
$143$	3.10898	0.259986
$144$	0	0
$145$	5.87659	0.488024
$146$	0	0
$147$	−23.8462	−1.96680
$148$	0	0
$149$	−14.4667	−1.18516	−0.592580	−	0.805511i	$-0.701891\pi$
$149$	−14.4667	−1.18516	−0.592580	+	0.805511i	$0.701891\pi$
$150$	0	0
$151$	0.684236	0.0556823	0.0278412	−	0.999612i	$-0.491137\pi$
$151$	0.684236	0.0556823	0.0278412	+	0.999612i	$0.491137\pi$
$152$	0	0
$153$	−4.96637	−0.401507
$154$	0	0
$155$	5.39744	0.433533
$156$	0	0
$157$	−9.94746	−0.793893	−0.396947	−	0.917842i	$-0.629930\pi$
$157$	−9.94746	−0.793893	−0.396947	+	0.917842i	$0.629930\pi$
$158$	0	0
$159$	11.1208	0.881935
$160$	0	0
$161$	−15.7229	−1.23914
$162$	0	0
$163$	−20.2822	−1.58863	−0.794314	−	0.607507i	$-0.792170\pi$
$163$	−20.2822	−1.58863	−0.794314	+	0.607507i	$0.792170\pi$
$164$	0	0
$165$	8.92370	0.694709
$166$	0	0
$167$	12.7217	0.984436	0.492218	−	0.870472i	$-0.336186\pi$
$167$	12.7217	0.984436	0.492218	+	0.870472i	$0.336186\pi$
$168$	0	0
$169$	−12.6104	−0.970028
$170$	0	0
$171$	−9.03230	−0.690718
$172$	0	0
$173$	2.61574	0.198871	0.0994355	−	0.995044i	$-0.468296\pi$
$173$	2.61574	0.198871	0.0994355	+	0.995044i	$0.468296\pi$
$174$	0	0
$175$	−18.3171	−1.38464
$176$	0	0
$177$	0.154001	0.0115754
$178$	0	0
$179$	18.6799	1.39620	0.698102	−	0.715999i	$-0.254028\pi$
$179$	18.6799	1.39620	0.698102	+	0.715999i	$0.254028\pi$
$180$	0	0
$181$	20.4484	1.51992	0.759959	−	0.649971i	$-0.225219\pi$
$181$	20.4484	1.51992	0.759959	+	0.649971i	$0.225219\pi$
$182$	0	0
$183$	−25.7644	−1.90456
$184$	0	0
$185$	−6.97232	−0.512615
$186$	0	0
$187$	−13.1337	−0.960429
$188$	0	0
$189$	10.4078	0.757053
$190$	0	0
$191$	−20.6293	−1.49268	−0.746341	−	0.665564i	$-0.768191\pi$
$191$	−20.6293	−1.49268	−0.746341	+	0.665564i	$0.768191\pi$
$192$	0	0
$193$	16.6930	1.20159	0.600795	−	0.799403i	$-0.294851\pi$
$193$	16.6930	1.20159	0.600795	+	0.799403i	$0.294851\pi$
$194$	0	0
$195$	1.11836	0.0800874
$196$	0	0
$197$	−12.7041	−0.905129	−0.452565	−	0.891732i	$-0.649491\pi$
$197$	−12.7041	−0.905129	−0.452565	+	0.891732i	$0.649491\pi$
$198$	0	0
$199$	−7.47274	−0.529729	−0.264864	−	0.964286i	$-0.585327\pi$
$199$	−7.47274	−0.529729	−0.264864	+	0.964286i	$0.585327\pi$
$200$	0	0
$201$	26.6854	1.88224
$202$	0	0
$203$	−30.5726	−2.14577
$204$	0	0
$205$	2.14516	0.149824
$206$	0	0
$207$	−7.02064	−0.487968
$208$	0	0
$209$	−23.8862	−1.65224
$210$	0	0
$211$	−21.6653	−1.49150	−0.745751	−	0.666225i	$-0.767909\pi$
$211$	−21.6653	−1.49150	−0.745751	+	0.666225i	$0.767909\pi$
$212$	0	0
$213$	18.0626	1.23763
$214$	0	0
$215$	−0.699471	−0.0477036
$216$	0	0
$217$	−28.0798	−1.90618
$218$	0	0
$219$	2.09868	0.141815
$220$	0	0
$221$	−1.64598	−0.110720
$222$	0	0
$223$	−23.6947	−1.58671	−0.793356	−	0.608758i	$-0.791668\pi$
$223$	−23.6947	−1.58671	−0.793356	+	0.608758i	$0.791668\pi$
$224$	0	0
$225$	−8.17899	−0.545266
$226$	0	0
$227$	29.4930	1.95752	0.978761	−	0.205005i	$-0.0657211\pi$
$227$	29.4930	1.95752	0.978761	+	0.205005i	$0.0657211\pi$
$228$	0	0
$229$	−7.20554	−0.476156	−0.238078	−	0.971246i	$-0.576517\pi$
$229$	−7.20554	−0.476156	−0.238078	+	0.971246i	$0.576517\pi$
$230$	0	0
$231$	−46.4249	−3.05453
$232$	0	0
$233$	8.08876	0.529912	0.264956	−	0.964260i	$-0.414643\pi$
$233$	8.08876	0.529912	0.264956	+	0.964260i	$0.414643\pi$
$234$	0	0
$235$	0.175972	0.0114791
$236$	0	0
$237$	−13.4572	−0.874141
$238$	0	0
$239$	−21.1574	−1.36856	−0.684280	−	0.729219i	$-0.739883\pi$
$239$	−21.1574	−1.36856	−0.684280	+	0.729219i	$0.739883\pi$
$240$	0	0
$241$	4.46457	0.287588	0.143794	−	0.989608i	$-0.454070\pi$
$241$	4.46457	0.287588	0.143794	+	0.989608i	$0.454070\pi$
$242$	0	0
$243$	17.1334	1.09911
$244$	0	0
$245$	−8.74885	−0.558943
$246$	0	0
$247$	−2.99353	−0.190474
$248$	0	0
$249$	2.64460	0.167595
$250$	0	0
$251$	−6.42122	−0.405304	−0.202652	−	0.979251i	$-0.564956\pi$
$251$	−6.42122	−0.405304	−0.202652	+	0.979251i	$0.564956\pi$
$252$	0	0
$253$	−18.5663	−1.16725
$254$	0	0
$255$	−4.72443	−0.295855
$256$	0	0
$257$	26.8560	1.67523	0.837616	−	0.546259i	$-0.183949\pi$
$257$	26.8560	1.67523	0.837616	+	0.546259i	$0.183949\pi$
$258$	0	0
$259$	36.2730	2.25389
$260$	0	0
$261$	−13.6514	−0.844997
$262$	0	0
$263$	22.5826	1.39251	0.696253	−	0.717797i	$-0.254849\pi$
$263$	22.5826	1.39251	0.696253	+	0.717797i	$0.254849\pi$
$264$	0	0
$265$	4.08006	0.250636
$266$	0	0
$267$	−29.5036	−1.80559
$268$	0	0
$269$	−0.334275	−0.0203811	−0.0101905	−	0.999948i	$-0.503244\pi$
$269$	−0.334275	−0.0203811	−0.0101905	+	0.999948i	$0.503244\pi$
$270$	0	0
$271$	14.1431	0.859134	0.429567	−	0.903035i	$-0.358666\pi$
$271$	14.1431	0.859134	0.429567	+	0.903035i	$0.358666\pi$
$272$	0	0
$273$	−5.81819	−0.352133
$274$	0	0
$275$	−21.6295	−1.30431
$276$	0	0
$277$	4.01653	0.241330	0.120665	−	0.992693i	$-0.461497\pi$
$277$	4.01653	0.241330	0.120665	+	0.992693i	$0.461497\pi$
$278$	0	0
$279$	−12.5383	−0.750648
$280$	0	0
$281$	−22.3267	−1.33190	−0.665951	−	0.745996i	$-0.731974\pi$
$281$	−22.3267	−1.33190	−0.665951	+	0.745996i	$0.731974\pi$
$282$	0	0
$283$	12.6831	0.753933	0.376966	−	0.926227i	$-0.376967\pi$
$283$	12.6831	0.753933	0.376966	+	0.926227i	$0.376967\pi$
$284$	0	0
$285$	−8.59230	−0.508964
$286$	0	0
$287$	−11.1600	−0.658756
$288$	0	0
$289$	−10.0467	−0.590982
$290$	0	0
$291$	6.60397	0.387131
$292$	0	0
$293$	24.1047	1.40821	0.704106	−	0.710094i	$-0.251348\pi$
$293$	24.1047	1.40821	0.704106	+	0.710094i	$0.251348\pi$
$294$	0	0
$295$	0.0565007	0.00328960
$296$	0	0
$297$	12.2899	0.713133
$298$	0	0
$299$	−2.32681	−0.134563
$300$	0	0
$301$	3.63895	0.209746
$302$	0	0
$303$	6.10812	0.350902
$304$	0	0
$305$	−9.45262	−0.541255
$306$	0	0
$307$	0.154442	0.00881446	0.00440723	−	0.999990i	$-0.498597\pi$
$307$	0.154442	0.00881446	0.00440723	+	0.999990i	$0.498597\pi$
$308$	0	0
$309$	24.9510	1.41942
$310$	0	0
$311$	−32.4131	−1.83798	−0.918989	−	0.394284i	$-0.870993\pi$
$311$	−32.4131	−1.83798	−0.918989	+	0.394284i	$0.870993\pi$
$312$	0	0
$313$	8.32635	0.470633	0.235317	−	0.971919i	$-0.424387\pi$
$313$	8.32635	0.470633	0.235317	+	0.971919i	$0.424387\pi$
$314$	0	0
$315$	−6.44073	−0.362894
$316$	0	0
$317$	15.0917	0.847637	0.423818	−	0.905747i	$-0.360689\pi$
$317$	15.0917	0.847637	0.423818	+	0.905747i	$0.360689\pi$
$318$	0	0
$319$	−36.1013	−2.02129
$320$	0	0
$321$	10.9567	0.611546
$322$	0	0
$323$	12.6459	0.703639
$324$	0	0
$325$	−2.71072	−0.150364
$326$	0	0
$327$	−23.3745	−1.29262
$328$	0	0
$329$	−0.915480	−0.0504720
$330$	0	0
$331$	−1.32441	−0.0727964	−0.0363982	−	0.999337i	$-0.511588\pi$
$331$	−1.32441	−0.0727964	−0.0363982	+	0.999337i	$0.511588\pi$
$332$	0	0
$333$	16.1967	0.887575
$334$	0	0
$335$	9.79051	0.534913
$336$	0	0
$337$	−15.1190	−0.823586	−0.411793	−	0.911278i	$-0.635097\pi$
$337$	−15.1190	−0.823586	−0.411793	+	0.911278i	$0.635097\pi$
$338$	0	0
$339$	22.1677	1.20398
$340$	0	0
$341$	−33.1578	−1.79560
$342$	0	0
$343$	15.9898	0.863367
$344$	0	0
$345$	−6.67863	−0.359566
$346$	0	0
$347$	−10.0566	−0.539866	−0.269933	−	0.962879i	$-0.587002\pi$
$347$	−10.0566	−0.539866	−0.269933	+	0.962879i	$0.587002\pi$
$348$	0	0
$349$	23.0726	1.23505	0.617524	−	0.786552i	$-0.288136\pi$
$349$	23.0726	1.23505	0.617524	+	0.786552i	$0.288136\pi$
$350$	0	0
$351$	1.54023	0.0822114
$352$	0	0
$353$	−27.5918	−1.46857	−0.734283	−	0.678844i	$-0.762481\pi$
$353$	−27.5918	−1.46857	−0.734283	+	0.678844i	$0.762481\pi$
$354$	0	0
$355$	6.62692	0.351720
$356$	0	0
$357$	24.5785	1.30083
$358$	0	0
$359$	−11.0620	−0.583832	−0.291916	−	0.956444i	$-0.594293\pi$
$359$	−11.0620	−0.583832	−0.291916	+	0.956444i	$0.594293\pi$
$360$	0	0
$361$	3.99911	0.210479
$362$	0	0
$363$	−30.5122	−1.60147
$364$	0	0
$365$	0.769976	0.0403024
$366$	0	0
$367$	16.2072	0.846009	0.423005	−	0.906128i	$-0.360975\pi$
$367$	16.2072	0.846009	0.423005	+	0.906128i	$0.360975\pi$
$368$	0	0
$369$	−4.98321	−0.259416
$370$	0	0
$371$	−21.2262	−1.10201
$372$	0	0
$373$	16.7339	0.866450	0.433225	−	0.901286i	$-0.357375\pi$
$373$	16.7339	0.866450	0.433225	+	0.901286i	$0.357375\pi$
$374$	0	0
$375$	−16.7388	−0.864389
$376$	0	0
$377$	−4.52439	−0.233018
$378$	0	0
$379$	26.9249	1.38304	0.691521	−	0.722357i	$-0.256941\pi$
$379$	26.9249	1.38304	0.691521	+	0.722357i	$0.256941\pi$
$380$	0	0
$381$	48.4657	2.48297
$382$	0	0
$383$	−27.3625	−1.39816	−0.699080	−	0.715044i	$-0.746407\pi$
$383$	−27.3625	−1.39816	−0.699080	+	0.715044i	$0.746407\pi$
$384$	0	0
$385$	−17.0327	−0.868065
$386$	0	0
$387$	1.62488	0.0825971
$388$	0	0
$389$	−1.17797	−0.0597254	−0.0298627	−	0.999554i	$-0.509507\pi$
$389$	−1.17797	−0.0597254	−0.0298627	+	0.999554i	$0.509507\pi$
$390$	0	0
$391$	9.82945	0.497097
$392$	0	0
$393$	29.8705	1.50676
$394$	0	0
$395$	−4.93728	−0.248421
$396$	0	0
$397$	−22.1124	−1.10979	−0.554895	−	0.831921i	$-0.687241\pi$
$397$	−22.1124	−1.10979	−0.554895	+	0.831921i	$0.687241\pi$
$398$	0	0
$399$	44.7009	2.23784
$400$	0	0
$401$	−18.8155	−0.939602	−0.469801	−	0.882772i	$-0.655674\pi$
$401$	−18.8155	−0.939602	−0.469801	+	0.882772i	$0.655674\pi$
$402$	0	0
$403$	−4.15550	−0.207000
$404$	0	0
$405$	9.00188	0.447307
$406$	0	0
$407$	42.8327	2.12314
$408$	0	0
$409$	−11.7383	−0.580424	−0.290212	−	0.956962i	$-0.593726\pi$
$409$	−11.7383	−0.580424	−0.290212	+	0.956962i	$0.593726\pi$
$410$	0	0
$411$	19.4441	0.959107
$412$	0	0
$413$	−0.293941	−0.0144639
$414$	0	0
$415$	0.970267	0.0476286
$416$	0	0
$417$	−47.1649	−2.30968
$418$	0	0
$419$	−30.3054	−1.48052	−0.740258	−	0.672323i	$-0.765297\pi$
$419$	−30.3054	−1.48052	−0.740258	+	0.672323i	$0.765297\pi$
$420$	0	0
$421$	−24.2112	−1.17998	−0.589990	−	0.807410i	$-0.700868\pi$
$421$	−24.2112	−1.17998	−0.589990	+	0.807410i	$0.700868\pi$
$422$	0	0
$423$	−0.408783	−0.0198757
$424$	0	0
$425$	11.4512	0.555466
$426$	0	0
$427$	49.1766	2.37982
$428$	0	0
$429$	−6.87036	−0.331704
$430$	0	0
$431$	−13.0789	−0.629988	−0.314994	−	0.949094i	$-0.602003\pi$
$431$	−13.0789	−0.629988	−0.314994	+	0.949094i	$0.602003\pi$
$432$	0	0
$433$	12.0019	0.576776	0.288388	−	0.957514i	$-0.406881\pi$
$433$	12.0019	0.576776	0.288388	+	0.957514i	$0.406881\pi$
$434$	0	0
$435$	−12.9863	−0.622647
$436$	0	0
$437$	17.8768	0.855162
$438$	0	0
$439$	−5.24867	−0.250505	−0.125253	−	0.992125i	$-0.539974\pi$
$439$	−5.24867	−0.250505	−0.125253	+	0.992125i	$0.539974\pi$
$440$	0	0
$441$	20.3236	0.967792
$442$	0	0
$443$	−27.4894	−1.30606	−0.653032	−	0.757331i	$-0.726503\pi$
$443$	−27.4894	−1.30606	−0.653032	+	0.757331i	$0.726503\pi$
$444$	0	0
$445$	−10.8245	−0.513129
$446$	0	0
$447$	31.9692	1.51209
$448$	0	0
$449$	−18.7815	−0.886353	−0.443176	−	0.896434i	$-0.646148\pi$
$449$	−18.7815	−0.886353	−0.443176	+	0.896434i	$0.646148\pi$
$450$	0	0
$451$	−13.1782	−0.620538
$452$	0	0
$453$	−1.51205	−0.0710425
$454$	0	0
$455$	−2.13462	−0.100072
$456$	0	0
$457$	41.2414	1.92919	0.964596	−	0.263731i	$-0.0849531\pi$
$457$	41.2414	1.92919	0.964596	+	0.263731i	$0.0849531\pi$
$458$	0	0
$459$	−6.50659	−0.303702
$460$	0	0
$461$	−11.5323	−0.537112	−0.268556	−	0.963264i	$-0.586546\pi$
$461$	−11.5323	−0.537112	−0.268556	+	0.963264i	$0.586546\pi$
$462$	0	0
$463$	15.2893	0.710555	0.355278	−	0.934761i	$-0.384386\pi$
$463$	15.2893	0.710555	0.355278	+	0.934761i	$0.384386\pi$
$464$	0	0
$465$	−11.9275	−0.553124
$466$	0	0
$467$	−32.9364	−1.52411	−0.762057	−	0.647510i	$-0.775810\pi$
$467$	−32.9364	−1.52411	−0.762057	+	0.647510i	$0.775810\pi$
$468$	0	0
$469$	−50.9344	−2.35193
$470$	0	0
$471$	21.9823	1.01289
$472$	0	0
$473$	4.29702	0.197577
$474$	0	0
$475$	20.8263	0.955576
$476$	0	0
$477$	−9.47800	−0.433968
$478$	0	0
$479$	−27.4502	−1.25423	−0.627116	−	0.778926i	$-0.715765\pi$
$479$	−27.4502	−1.25423	−0.627116	+	0.778926i	$0.715765\pi$
$480$	0	0
$481$	5.36799	0.244759
$482$	0	0
$483$	34.7451	1.58096
$484$	0	0
$485$	2.42291	0.110018
$486$	0	0
$487$	13.3285	0.603970	0.301985	−	0.953313i	$-0.402351\pi$
$487$	13.3285	0.603970	0.301985	+	0.953313i	$0.402351\pi$
$488$	0	0
$489$	44.8206	2.02686
$490$	0	0
$491$	−23.4231	−1.05707	−0.528534	−	0.848912i	$-0.677258\pi$
$491$	−23.4231	−1.05707	−0.528534	+	0.848912i	$0.677258\pi$
$492$	0	0
$493$	19.1130	0.860805
$494$	0	0
$495$	−7.60548	−0.341841
$496$	0	0
$497$	−34.4761	−1.54646
$498$	0	0
$499$	2.48648	0.111310	0.0556550	−	0.998450i	$-0.482275\pi$
$499$	2.48648	0.111310	0.0556550	+	0.998450i	$0.482275\pi$
$500$	0	0
$501$	−28.1130	−1.25600
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	2.24099	0.0997226
$506$	0	0
$507$	27.8669	1.23761
$508$	0	0
$509$	−17.6918	−0.784175	−0.392088	−	0.919928i	$-0.628247\pi$
$509$	−17.6918	−0.784175	−0.392088	+	0.919928i	$0.628247\pi$
$510$	0	0
$511$	−4.00575	−0.177204
$512$	0	0
$513$	−11.8335	−0.522462
$514$	0	0
$515$	9.15420	0.403382
$516$	0	0
$517$	−1.08104	−0.0475439
$518$	0	0
$519$	−5.78037	−0.253730
$520$	0	0
$521$	24.9268	1.09206	0.546031	−	0.837765i	$-0.316138\pi$
$521$	24.9268	1.09206	0.546031	+	0.837765i	$0.316138\pi$
$522$	0	0
$523$	5.58878	0.244380	0.122190	−	0.992507i	$-0.461008\pi$
$523$	5.58878	0.244380	0.122190	+	0.992507i	$0.461008\pi$
$524$	0	0
$525$	40.4778	1.76660
$526$	0	0
$527$	17.5546	0.764690
$528$	0	0
$529$	−9.10473	−0.395858
$530$	0	0
$531$	−0.131251	−0.00569583
$532$	0	0
$533$	−1.65156	−0.0715369
$534$	0	0
$535$	4.01988	0.173794
$536$	0	0
$537$	−41.2797	−1.78135
$538$	0	0
$539$	53.7463	2.31502
$540$	0	0
$541$	−16.7052	−0.718214	−0.359107	−	0.933296i	$-0.616919\pi$
$541$	−16.7052	−0.718214	−0.359107	+	0.933296i	$0.616919\pi$
$542$	0	0
$543$	−45.1877	−1.93919
$544$	0	0
$545$	−8.57580	−0.367347
$546$	0	0
$547$	18.6770	0.798571	0.399285	−	0.916827i	$-0.369258\pi$
$547$	18.6770	0.798571	0.399285	+	0.916827i	$0.369258\pi$
$548$	0	0
$549$	21.9585	0.937165
$550$	0	0
$551$	34.7607	1.48085
$552$	0	0
$553$	25.6858	1.09227
$554$	0	0
$555$	15.4077	0.654021
$556$	0	0
$557$	−17.6111	−0.746207	−0.373104	−	0.927790i	$-0.621706\pi$
$557$	−17.6111	−0.746207	−0.373104	+	0.927790i	$0.621706\pi$
$558$	0	0
$559$	0.538524	0.0227771
$560$	0	0
$561$	29.0233	1.22537
$562$	0	0
$563$	26.7608	1.12783	0.563916	−	0.825832i	$-0.309294\pi$
$563$	26.7608	1.12783	0.563916	+	0.825832i	$0.309294\pi$
$564$	0	0
$565$	8.13303	0.342159
$566$	0	0
$567$	−46.8317	−1.96674
$568$	0	0
$569$	34.2838	1.43725	0.718625	−	0.695398i	$-0.244772\pi$
$569$	34.2838	1.43725	0.718625	+	0.695398i	$0.244772\pi$
$570$	0	0
$571$	35.5328	1.48700	0.743501	−	0.668735i	$-0.233164\pi$
$571$	35.5328	1.48700	0.743501	+	0.668735i	$0.233164\pi$
$572$	0	0
$573$	45.5874	1.90444
$574$	0	0
$575$	16.1879	0.675081
$576$	0	0
$577$	23.7507	0.988755	0.494377	−	0.869247i	$-0.335396\pi$
$577$	23.7507	0.988755	0.494377	+	0.869247i	$0.335396\pi$
$578$	0	0
$579$	−36.8890	−1.53305
$580$	0	0
$581$	−5.04775	−0.209416
$582$	0	0
$583$	−25.0648	−1.03808
$584$	0	0
$585$	−0.953155	−0.0394081
$586$	0	0
$587$	7.60667	0.313961	0.156980	−	0.987602i	$-0.449824\pi$
$587$	7.60667	0.313961	0.156980	+	0.987602i	$0.449824\pi$
$588$	0	0
$589$	31.9265	1.31551
$590$	0	0
$591$	28.0740	1.15481
$592$	0	0
$593$	−29.5405	−1.21308	−0.606542	−	0.795051i	$-0.707444\pi$
$593$	−29.5405	−1.21308	−0.606542	+	0.795051i	$0.707444\pi$
$594$	0	0
$595$	9.01753	0.369683
$596$	0	0
$597$	16.5136	0.675856
$598$	0	0
$599$	−38.2936	−1.56463	−0.782316	−	0.622881i	$-0.785962\pi$
$599$	−38.2936	−1.56463	−0.782316	+	0.622881i	$0.785962\pi$
$600$	0	0
$601$	3.94764	0.161028	0.0805138	−	0.996753i	$-0.474344\pi$
$601$	3.94764	0.161028	0.0805138	+	0.996753i	$0.474344\pi$
$602$	0	0
$603$	−22.7434	−0.926183
$604$	0	0
$605$	−11.1945	−0.455121
$606$	0	0
$607$	−9.78533	−0.397174	−0.198587	−	0.980083i	$-0.563635\pi$
$607$	−9.78533	−0.397174	−0.198587	+	0.980083i	$0.563635\pi$
$608$	0	0
$609$	67.5605	2.73769
$610$	0	0
$611$	−0.135481	−0.00548096
$612$	0	0
$613$	−18.4607	−0.745619	−0.372810	−	0.927908i	$-0.621606\pi$
$613$	−18.4607	−0.745619	−0.372810	+	0.927908i	$0.621606\pi$
$614$	0	0
$615$	−4.74046	−0.191154
$616$	0	0
$617$	−48.0950	−1.93623	−0.968116	−	0.250504i	$-0.919404\pi$
$617$	−48.0950	−1.93623	−0.968116	+	0.250504i	$0.919404\pi$
$618$	0	0
$619$	16.9849	0.682681	0.341341	−	0.939940i	$-0.389119\pi$
$619$	16.9849	0.682681	0.341341	+	0.939940i	$0.389119\pi$
$620$	0	0
$621$	−9.19796	−0.369101
$622$	0	0
$623$	56.3135	2.25615
$624$	0	0
$625$	15.5721	0.622884
$626$	0	0
$627$	52.7846	2.10801
$628$	0	0
$629$	−22.6767	−0.904179
$630$	0	0
$631$	−20.0292	−0.797351	−0.398675	−	0.917092i	$-0.630530\pi$
$631$	−20.0292	−0.797351	−0.398675	+	0.917092i	$0.630530\pi$
$632$	0	0
$633$	47.8769	1.90294
$634$	0	0
$635$	17.7814	0.705633
$636$	0	0
$637$	6.73575	0.266880
$638$	0	0
$639$	−15.3944	−0.608992
$640$	0	0
$641$	6.84875	0.270510	0.135255	−	0.990811i	$-0.456815\pi$
$641$	6.84875	0.270510	0.135255	+	0.990811i	$0.456815\pi$
$642$	0	0
$643$	−31.1109	−1.22690	−0.613448	−	0.789735i	$-0.710218\pi$
$643$	−31.1109	−1.22690	−0.613448	+	0.789735i	$0.710218\pi$
$644$	0	0
$645$	1.54572	0.0608627
$646$	0	0
$647$	16.4056	0.644969	0.322485	−	0.946575i	$-0.395482\pi$
$647$	16.4056	0.644969	0.322485	+	0.946575i	$0.395482\pi$
$648$	0	0
$649$	−0.347098	−0.0136248
$650$	0	0
$651$	62.0520	2.43201
$652$	0	0
$653$	0.718479	0.0281163	0.0140581	−	0.999901i	$-0.495525\pi$
$653$	0.718479	0.0281163	0.0140581	+	0.999901i	$0.495525\pi$
$654$	0	0
$655$	10.9591	0.428206
$656$	0	0
$657$	−1.78866	−0.0697822
$658$	0	0
$659$	28.5474	1.11205	0.556025	−	0.831165i	$-0.312326\pi$
$659$	28.5474	1.11205	0.556025	+	0.831165i	$0.312326\pi$
$660$	0	0
$661$	−36.1897	−1.40762	−0.703809	−	0.710389i	$-0.748519\pi$
$661$	−36.1897	−1.40762	−0.703809	+	0.710389i	$0.748519\pi$
$662$	0	0
$663$	3.63735	0.141263
$664$	0	0
$665$	16.4001	0.635970
$666$	0	0
$667$	27.0188	1.04617
$668$	0	0
$669$	52.3615	2.02441
$670$	0	0
$671$	58.0697	2.24176
$672$	0	0
$673$	−14.5951	−0.562600	−0.281300	−	0.959620i	$-0.590766\pi$
$673$	−14.5951	−0.562600	−0.281300	+	0.959620i	$0.590766\pi$
$674$	0	0
$675$	−10.7156	−0.412442
$676$	0	0
$677$	28.8256	1.10786	0.553930	−	0.832564i	$-0.313128\pi$
$677$	28.8256	1.10786	0.553930	+	0.832564i	$0.313128\pi$
$678$	0	0
$679$	−12.6050	−0.483735
$680$	0	0
$681$	−65.1750	−2.49751
$682$	0	0
$683$	−40.3937	−1.54562	−0.772811	−	0.634636i	$-0.781150\pi$
$683$	−40.3937	−1.54562	−0.772811	+	0.634636i	$0.781150\pi$
$684$	0	0
$685$	7.13378	0.272568
$686$	0	0
$687$	15.9231	0.607505
$688$	0	0
$689$	−3.14124	−0.119672
$690$	0	0
$691$	−6.17442	−0.234886	−0.117443	−	0.993080i	$-0.537470\pi$
$691$	−6.17442	−0.234886	−0.117443	+	0.993080i	$0.537470\pi$
$692$	0	0
$693$	39.5670	1.50303
$694$	0	0
$695$	−17.3042	−0.656385
$696$	0	0
$697$	6.97689	0.264269
$698$	0	0
$699$	−17.8749	−0.676090
$700$	0	0
$701$	−39.2016	−1.48062	−0.740312	−	0.672263i	$-0.765322\pi$
$701$	−39.2016	−1.48062	−0.740312	+	0.672263i	$0.765322\pi$
$702$	0	0
$703$	−41.2420	−1.55547
$704$	0	0
$705$	−0.388869	−0.0146457
$706$	0	0
$707$	−11.6586	−0.438466
$708$	0	0
$709$	45.3388	1.70273	0.851367	−	0.524571i	$-0.175774\pi$
$709$	45.3388	1.70273	0.851367	+	0.524571i	$0.175774\pi$
$710$	0	0
$711$	11.4693	0.430133
$712$	0	0
$713$	24.8158	0.929360
$714$	0	0
$715$	−2.52064	−0.0942667
$716$	0	0
$717$	46.7546	1.74608
$718$	0	0
$719$	−10.1027	−0.376769	−0.188384	−	0.982095i	$-0.560325\pi$
$719$	−10.1027	−0.376769	−0.188384	+	0.982095i	$0.560325\pi$
$720$	0	0
$721$	−47.6241	−1.77361
$722$	0	0
$723$	−9.86599	−0.366920
$724$	0	0
$725$	31.4767	1.16902
$726$	0	0
$727$	3.72738	0.138241	0.0691205	−	0.997608i	$-0.477981\pi$
$727$	3.72738	0.138241	0.0691205	+	0.997608i	$0.477981\pi$
$728$	0	0
$729$	−4.55303	−0.168631
$730$	0	0
$731$	−2.27495	−0.0841422
$732$	0	0
$733$	−38.3406	−1.41614	−0.708071	−	0.706141i	$-0.750434\pi$
$733$	−38.3406	−1.41614	−0.708071	+	0.706141i	$0.750434\pi$
$734$	0	0
$735$	19.3336	0.713130
$736$	0	0
$737$	−60.1455	−2.21549
$738$	0	0
$739$	−9.24661	−0.340142	−0.170071	−	0.985432i	$-0.554400\pi$
$739$	−9.24661	−0.340142	−0.170071	+	0.985432i	$0.554400\pi$
$740$	0	0
$741$	6.61522	0.243016
$742$	0	0
$743$	24.6878	0.905706	0.452853	−	0.891585i	$-0.350406\pi$
$743$	24.6878	0.905706	0.452853	+	0.891585i	$0.350406\pi$
$744$	0	0
$745$	11.7291	0.429720
$746$	0	0
$747$	−2.25394	−0.0824672
$748$	0	0
$749$	−20.9131	−0.764149
$750$	0	0
$751$	−14.4083	−0.525767	−0.262884	−	0.964828i	$-0.584674\pi$
$751$	−14.4083	−0.525767	−0.262884	+	0.964828i	$0.584674\pi$
$752$	0	0
$753$	14.1899	0.517108
$754$	0	0
$755$	−0.554752	−0.0201895
$756$	0	0
$757$	−4.61991	−0.167913	−0.0839567	−	0.996469i	$-0.526756\pi$
$757$	−4.61991	−0.167913	−0.0839567	+	0.996469i	$0.526756\pi$
$758$	0	0
$759$	41.0285	1.48924
$760$	0	0
$761$	−10.1960	−0.369604	−0.184802	−	0.982776i	$-0.559164\pi$
$761$	−10.1960	−0.369604	−0.184802	+	0.982776i	$0.559164\pi$
$762$	0	0
$763$	44.6150	1.61517
$764$	0	0
$765$	4.02653	0.145580
$766$	0	0
$767$	−0.0435000	−0.00157069
$768$	0	0
$769$	−16.9921	−0.612751	−0.306375	−	0.951911i	$-0.599116\pi$
$769$	−16.9921	−0.612751	−0.306375	+	0.951911i	$0.599116\pi$
$770$	0	0
$771$	−59.3476	−2.13735
$772$	0	0
$773$	39.3133	1.41400	0.707001	−	0.707212i	$-0.250047\pi$
$773$	39.3133	1.41400	0.707001	+	0.707212i	$0.250047\pi$
$774$	0	0
$775$	28.9102	1.03849
$776$	0	0
$777$	−80.1576	−2.87564
$778$	0	0
$779$	12.6888	0.454625
$780$	0	0
$781$	−40.7108	−1.45675
$782$	0	0
$783$	−17.8851	−0.639160
$784$	0	0
$785$	8.06501	0.287852
$786$	0	0
$787$	33.8400	1.20627	0.603133	−	0.797640i	$-0.293919\pi$
$787$	33.8400	1.20627	0.603133	+	0.797640i	$0.293919\pi$
$788$	0	0
$789$	−49.9041	−1.77663
$790$	0	0
$791$	−42.3115	−1.50442
$792$	0	0
$793$	7.27758	0.258434
$794$	0	0
$795$	−9.01629	−0.319775
$796$	0	0
$797$	20.2357	0.716785	0.358392	−	0.933571i	$-0.383325\pi$
$797$	20.2357	0.716785	0.358392	+	0.933571i	$0.383325\pi$
$798$	0	0
$799$	0.572329	0.0202475
$800$	0	0
$801$	25.1453	0.888465
$802$	0	0
$803$	−4.73015	−0.166923
$804$	0	0
$805$	12.7475	0.449291
$806$	0	0
$807$	0.738694	0.0260033
$808$	0	0
$809$	36.0145	1.26620	0.633102	−	0.774069i	$-0.281782\pi$
$809$	36.0145	1.26620	0.633102	+	0.774069i	$0.281782\pi$
$810$	0	0
$811$	−19.2852	−0.677194	−0.338597	−	0.940932i	$-0.609952\pi$
$811$	−19.2852	−0.677194	−0.338597	+	0.940932i	$0.609952\pi$
$812$	0	0
$813$	−31.2541	−1.09613
$814$	0	0
$815$	16.4441	0.576010
$816$	0	0
$817$	−4.13745	−0.144751
$818$	0	0
$819$	4.95872	0.173272
$820$	0	0
$821$	−19.5650	−0.682822	−0.341411	−	0.939914i	$-0.610905\pi$
$821$	−19.5650	−0.682822	−0.341411	+	0.939914i	$0.610905\pi$
$822$	0	0
$823$	22.8920	0.797964	0.398982	−	0.916959i	$-0.369364\pi$
$823$	22.8920	0.797964	0.398982	+	0.916959i	$0.369364\pi$
$824$	0	0
$825$	47.7979	1.66411
$826$	0	0
$827$	−2.99841	−0.104265	−0.0521324	−	0.998640i	$-0.516602\pi$
$827$	−2.99841	−0.104265	−0.0521324	+	0.998640i	$0.516602\pi$
$828$	0	0
$829$	15.9709	0.554692	0.277346	−	0.960770i	$-0.410545\pi$
$829$	15.9709	0.554692	0.277346	+	0.960770i	$0.410545\pi$
$830$	0	0
$831$	−8.87590	−0.307901
$832$	0	0
$833$	−28.4547	−0.985896
$834$	0	0
$835$	−10.3143	−0.356940
$836$	0	0
$837$	−16.4268	−0.567794
$838$	0	0
$839$	0.217848	0.00752096	0.00376048	−	0.999993i	$-0.498803\pi$
$839$	0.217848	0.00752096	0.00376048	+	0.999993i	$0.498803\pi$
$840$	0	0
$841$	23.5370	0.811620
$842$	0	0
$843$	49.3386	1.69931
$844$	0	0
$845$	10.2240	0.351716
$846$	0	0
$847$	58.2386	2.00110
$848$	0	0
$849$	−28.0277	−0.961907
$850$	0	0
$851$	−32.0566	−1.09889
$852$	0	0
$853$	53.7355	1.83987	0.919934	−	0.392074i	$-0.128242\pi$
$853$	53.7355	1.83987	0.919934	+	0.392074i	$0.128242\pi$
$854$	0	0
$855$	7.32304	0.250443
$856$	0	0
$857$	−8.48184	−0.289734	−0.144867	−	0.989451i	$-0.546275\pi$
$857$	−8.48184	−0.289734	−0.144867	+	0.989451i	$0.546275\pi$
$858$	0	0
$859$	−1.01884	−0.0347622	−0.0173811	−	0.999849i	$-0.505533\pi$
$859$	−1.01884	−0.0347622	−0.0173811	+	0.999849i	$0.505533\pi$
$860$	0	0
$861$	24.6619	0.840475
$862$	0	0
$863$	−1.74467	−0.0593892	−0.0296946	−	0.999559i	$-0.509453\pi$
$863$	−1.74467	−0.0593892	−0.0296946	+	0.999559i	$0.509453\pi$
$864$	0	0
$865$	−2.12074	−0.0721073
$866$	0	0
$867$	22.2016	0.754006
$868$	0	0
$869$	30.3309	1.02891
$870$	0	0
$871$	−7.53772	−0.255406
$872$	0	0
$873$	−5.62842	−0.190493
$874$	0	0
$875$	31.9494	1.08009
$876$	0	0
$877$	45.5653	1.53863	0.769316	−	0.638869i	$-0.220597\pi$
$877$	45.5653	1.53863	0.769316	+	0.638869i	$0.220597\pi$
$878$	0	0
$879$	−53.2676	−1.79667
$880$	0	0
$881$	−14.2948	−0.481606	−0.240803	−	0.970574i	$-0.577411\pi$
$881$	−14.2948	−0.481606	−0.240803	+	0.970574i	$0.577411\pi$
$882$	0	0
$883$	6.03950	0.203245	0.101623	−	0.994823i	$-0.467597\pi$
$883$	6.03950	0.203245	0.101623	+	0.994823i	$0.467597\pi$
$884$	0	0
$885$	−0.124858	−0.00419704
$886$	0	0
$887$	39.7866	1.33590	0.667951	−	0.744205i	$-0.267171\pi$
$887$	39.7866	1.33590	0.667951	+	0.744205i	$0.267171\pi$
$888$	0	0
$889$	−92.5065	−3.10257
$890$	0	0
$891$	−55.3008	−1.85265
$892$	0	0
$893$	1.04089	0.0348321
$894$	0	0
$895$	−15.1450	−0.506240
$896$	0	0
$897$	5.14189	0.171683
$898$	0	0
$899$	48.2534	1.60934
$900$	0	0
$901$	13.2700	0.442086
$902$	0	0
$903$	−8.04151	−0.267605
$904$	0	0
$905$	−16.5788	−0.551097
$906$	0	0
$907$	43.1806	1.43379	0.716894	−	0.697183i	$-0.245563\pi$
$907$	43.1806	1.43379	0.716894	+	0.697183i	$0.245563\pi$
$908$	0	0
$909$	−5.20582	−0.172666
$910$	0	0
$911$	51.7635	1.71500	0.857501	−	0.514483i	$-0.172016\pi$
$911$	51.7635	1.71500	0.857501	+	0.514483i	$0.172016\pi$
$912$	0	0
$913$	−5.96059	−0.197267
$914$	0	0
$915$	20.8888	0.690562
$916$	0	0
$917$	−57.0138	−1.88276
$918$	0	0
$919$	4.46309	0.147224	0.0736119	−	0.997287i	$-0.476547\pi$
$919$	4.46309	0.147224	0.0736119	+	0.997287i	$0.476547\pi$
$920$	0	0
$921$	−0.341292	−0.0112460
$922$	0	0
$923$	−5.10207	−0.167937
$924$	0	0
$925$	−37.3457	−1.22792
$926$	0	0
$927$	−21.2652	−0.698442
$928$	0	0
$929$	−11.7999	−0.387142	−0.193571	−	0.981086i	$-0.562007\pi$
$929$	−11.7999	−0.387142	−0.193571	+	0.981086i	$0.562007\pi$
$930$	0	0
$931$	−51.7504	−1.69605
$932$	0	0
$933$	71.6278	2.34499
$934$	0	0
$935$	10.6483	0.348236
$936$	0	0
$937$	18.3004	0.597848	0.298924	−	0.954277i	$-0.403372\pi$
$937$	18.3004	0.597848	0.298924	+	0.954277i	$0.403372\pi$
$938$	0	0
$939$	−18.3999	−0.600459
$940$	0	0
$941$	26.0706	0.849879	0.424939	−	0.905222i	$-0.360295\pi$
$941$	26.0706	0.849879	0.424939	+	0.905222i	$0.360295\pi$
$942$	0	0
$943$	9.86279	0.321177
$944$	0	0
$945$	−8.43820	−0.274495
$946$	0	0
$947$	−20.4739	−0.665311	−0.332656	−	0.943048i	$-0.607945\pi$
$947$	−20.4739	−0.665311	−0.332656	+	0.943048i	$0.607945\pi$
$948$	0	0
$949$	−0.592805	−0.0192433
$950$	0	0
$951$	−33.3504	−1.08146
$952$	0	0
$953$	−11.3252	−0.366860	−0.183430	−	0.983033i	$-0.558720\pi$
$953$	−11.3252	−0.366860	−0.183430	+	0.983033i	$0.558720\pi$
$954$	0	0
$955$	16.7254	0.541221
$956$	0	0
$957$	79.7783	2.57886
$958$	0	0
$959$	−37.1130	−1.19844
$960$	0	0
$961$	13.3190	0.429646
$962$	0	0
$963$	−9.33820	−0.300919
$964$	0	0
$965$	−13.5341	−0.435677
$966$	0	0
$967$	−25.2102	−0.810704	−0.405352	−	0.914161i	$-0.632851\pi$
$967$	−25.2102	−0.810704	−0.405352	+	0.914161i	$0.632851\pi$
$968$	0	0
$969$	−27.9455	−0.897740
$970$	0	0
$971$	−33.2940	−1.06846	−0.534228	−	0.845341i	$-0.679397\pi$
$971$	−33.2940	−1.06846	−0.534228	+	0.845341i	$0.679397\pi$
$972$	0	0
$973$	90.0237	2.88603
$974$	0	0
$975$	5.99026	0.191842
$976$	0	0
$977$	7.75354	0.248058	0.124029	−	0.992279i	$-0.460418\pi$
$977$	7.75354	0.248058	0.124029	+	0.992279i	$0.460418\pi$
$978$	0	0
$979$	66.4973	2.12526
$980$	0	0
$981$	19.9216	0.636049
$982$	0	0
$983$	7.87203	0.251079	0.125539	−	0.992089i	$-0.459934\pi$
$983$	7.87203	0.251079	0.125539	+	0.992089i	$0.459934\pi$
$984$	0	0
$985$	10.3000	0.328185
$986$	0	0
$987$	2.02307	0.0643949
$988$	0	0
$989$	−3.21596	−0.102262
$990$	0	0
$991$	8.27118	0.262743	0.131371	−	0.991333i	$-0.458062\pi$
$991$	8.27118	0.262743	0.131371	+	0.991333i	$0.458062\pi$
$992$	0	0
$993$	2.92675	0.0928774
$994$	0	0
$995$	6.05861	0.192071
$996$	0	0
$997$	−25.6883	−0.813555	−0.406778	−	0.913527i	$-0.633348\pi$
$997$	−25.6883	−0.813555	−0.406778	+	0.913527i	$0.633348\pi$
$998$	0	0
$999$	21.2198	0.671366

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.7	✓	29
4.3	odd	2		8048.2.a.w.1.23		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.7	✓	29	1.1	even	1	trivial
8048.2.a.w.1.23		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)

\(f(q)\)	\(=\)	\(q-2.20984 q^{3} -0.810761 q^{5} +4.21793 q^{7} +1.88340 q^{9} +O(q^{10})\)	\(q-2.20984 q^{3} -0.810761 q^{5} +4.21793 q^{7} +1.88340 q^{9} +4.98070 q^{11} +0.624206 q^{13} +1.79165 q^{15} -2.63691 q^{17} -4.79574 q^{19} -9.32095 q^{21} -3.72764 q^{23} -4.34267 q^{25} +2.46750 q^{27} -7.24824 q^{29} -6.65725 q^{31} -11.0066 q^{33} -3.41973 q^{35} +8.59972 q^{37} -1.37940 q^{39} -2.64586 q^{41} +0.862734 q^{43} -1.52699 q^{45} -0.217045 q^{47} +10.7909 q^{49} +5.82716 q^{51} -5.03238 q^{53} -4.03816 q^{55} +10.5978 q^{57} -0.0696885 q^{59} +11.6589 q^{61} +7.94405 q^{63} -0.506082 q^{65} -12.0757 q^{67} +8.23749 q^{69} -8.17370 q^{71} -0.949696 q^{73} +9.59661 q^{75} +21.0082 q^{77} +6.08968 q^{79} -11.1030 q^{81} -1.19674 q^{83} +2.13790 q^{85} +16.0175 q^{87} +13.3510 q^{89} +2.63285 q^{91} +14.7115 q^{93} +3.88820 q^{95} -2.98843 q^{97} +9.38067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 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

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Modular forms

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