LMFDB - Embedded newform 8024.2.a.x.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.484590 q^{3} +2.40120 q^{5} -0.883019 q^{7} -2.76517 q^{9} +O(q^{10})$	$q-0.484590 q^{3} +2.40120 q^{5} -0.883019 q^{7} -2.76517 q^{9} +4.64571 q^{11} +4.91748 q^{13} -1.16360 q^{15} -1.00000 q^{17} -2.42009 q^{19} +0.427902 q^{21} +0.882843 q^{23} +0.765762 q^{25} +2.79375 q^{27} -5.13890 q^{29} -8.47067 q^{31} -2.25127 q^{33} -2.12030 q^{35} -8.19936 q^{37} -2.38296 q^{39} -4.50967 q^{41} -4.63235 q^{43} -6.63973 q^{45} -6.46389 q^{47} -6.22028 q^{49} +0.484590 q^{51} -9.64029 q^{53} +11.1553 q^{55} +1.17275 q^{57} +1.00000 q^{59} +4.99185 q^{61} +2.44170 q^{63} +11.8078 q^{65} -1.77232 q^{67} -0.427817 q^{69} -2.58133 q^{71} -11.5050 q^{73} -0.371081 q^{75} -4.10225 q^{77} +12.8120 q^{79} +6.94170 q^{81} -7.00194 q^{83} -2.40120 q^{85} +2.49026 q^{87} -8.29555 q^{89} -4.34222 q^{91} +4.10480 q^{93} -5.81112 q^{95} -11.7077 q^{97} -12.8462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	−0.484590	−0.279778	−0.139889	−	0.990167i	$-0.544675\pi$
$3$	−0.484590	−0.279778	−0.139889	+	0.990167i	$0.544675\pi$
$4$	0	0
$5$	2.40120	1.07385	0.536925	−	0.843630i	$-0.319586\pi$
$5$	2.40120	1.07385	0.536925	+	0.843630i	$0.319586\pi$
$6$	0	0
$7$	−0.883019	−0.333750	−0.166875	−	0.985978i	$-0.553368\pi$
$7$	−0.883019	−0.333750	−0.166875	+	0.985978i	$0.553368\pi$
$8$	0	0
$9$	−2.76517	−0.921724
$10$	0	0
$11$	4.64571	1.40073	0.700367	−	0.713783i	$-0.253020\pi$
$11$	4.64571	1.40073	0.700367	+	0.713783i	$0.253020\pi$
$12$	0	0
$13$	4.91748	1.36386	0.681931	−	0.731416i	$-0.261140\pi$
$13$	4.91748	1.36386	0.681931	+	0.731416i	$0.261140\pi$
$14$	0	0
$15$	−1.16360	−0.300440
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.42009	−0.555207	−0.277603	−	0.960696i	$-0.589540\pi$
$19$	−2.42009	−0.555207	−0.277603	+	0.960696i	$0.589540\pi$
$20$	0	0
$21$	0.427902	0.0933759
$22$	0	0
$23$	0.882843	0.184085	0.0920427	−	0.995755i	$-0.470660\pi$
$23$	0.882843	0.184085	0.0920427	+	0.995755i	$0.470660\pi$
$24$	0	0
$25$	0.765762	0.153152
$26$	0	0
$27$	2.79375	0.537657
$28$	0	0
$29$	−5.13890	−0.954270	−0.477135	−	0.878830i	$-0.658325\pi$
$29$	−5.13890	−0.954270	−0.477135	+	0.878830i	$0.658325\pi$
$30$	0	0
$31$	−8.47067	−1.52138	−0.760689	−	0.649117i	$-0.775139\pi$
$31$	−8.47067	−1.52138	−0.760689	+	0.649117i	$0.775139\pi$
$32$	0	0
$33$	−2.25127	−0.391895
$34$	0	0
$35$	−2.12030	−0.358397
$36$	0	0
$37$	−8.19936	−1.34797	−0.673983	−	0.738747i	$-0.735418\pi$
$37$	−8.19936	−1.34797	−0.673983	+	0.738747i	$0.735418\pi$
$38$	0	0
$39$	−2.38296	−0.381579
$40$	0	0
$41$	−4.50967	−0.704292	−0.352146	−	0.935945i	$-0.614548\pi$
$41$	−4.50967	−0.704292	−0.352146	+	0.935945i	$0.614548\pi$
$42$	0	0
$43$	−4.63235	−0.706426	−0.353213	−	0.935543i	$-0.614911\pi$
$43$	−4.63235	−0.706426	−0.353213	+	0.935543i	$0.614911\pi$
$44$	0	0
$45$	−6.63973	−0.989793
$46$	0	0
$47$	−6.46389	−0.942855	−0.471427	−	0.881905i	$-0.656261\pi$
$47$	−6.46389	−0.942855	−0.471427	+	0.881905i	$0.656261\pi$
$48$	0	0
$49$	−6.22028	−0.888611
$50$	0	0
$51$	0.484590	0.0678562
$52$	0	0
$53$	−9.64029	−1.32420	−0.662098	−	0.749418i	$-0.730334\pi$
$53$	−9.64029	−1.32420	−0.662098	+	0.749418i	$0.730334\pi$
$54$	0	0
$55$	11.1553	1.50418
$56$	0	0
$57$	1.17275	0.155335
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	4.99185	0.639141	0.319571	−	0.947562i	$-0.396461\pi$
$61$	4.99185	0.639141	0.319571	+	0.947562i	$0.396461\pi$
$62$	0	0
$63$	2.44170	0.307625
$64$	0	0
$65$	11.8078	1.46458
$66$	0	0
$67$	−1.77232	−0.216523	−0.108262	−	0.994122i	$-0.534528\pi$
$67$	−1.77232	−0.216523	−0.108262	+	0.994122i	$0.534528\pi$
$68$	0	0
$69$	−0.427817	−0.0515031
$70$	0	0
$71$	−2.58133	−0.306347	−0.153174	−	0.988199i	$-0.548949\pi$
$71$	−2.58133	−0.306347	−0.153174	+	0.988199i	$0.548949\pi$
$72$	0	0
$73$	−11.5050	−1.34656	−0.673280	−	0.739388i	$-0.735115\pi$
$73$	−11.5050	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$74$	0	0
$75$	−0.371081	−0.0428487
$76$	0	0
$77$	−4.10225	−0.467495
$78$	0	0
$79$	12.8120	1.44146	0.720732	−	0.693214i	$-0.243806\pi$
$79$	12.8120	1.44146	0.720732	+	0.693214i	$0.243806\pi$
$80$	0	0
$81$	6.94170	0.771300
$82$	0	0
$83$	−7.00194	−0.768563	−0.384281	−	0.923216i	$-0.625551\pi$
$83$	−7.00194	−0.768563	−0.384281	+	0.923216i	$0.625551\pi$
$84$	0	0
$85$	−2.40120	−0.260447
$86$	0	0
$87$	2.49026	0.266984
$88$	0	0
$89$	−8.29555	−0.879327	−0.439663	−	0.898163i	$-0.644902\pi$
$89$	−8.29555	−0.879327	−0.439663	+	0.898163i	$0.644902\pi$
$90$	0	0
$91$	−4.34222	−0.455189
$92$	0	0
$93$	4.10480	0.425648
$94$	0	0
$95$	−5.81112	−0.596208
$96$	0	0
$97$	−11.7077	−1.18873	−0.594367	−	0.804194i	$-0.702597\pi$
$97$	−11.7077	−1.18873	−0.594367	+	0.804194i	$0.702597\pi$
$98$	0	0
$99$	−12.8462	−1.29109
$100$	0	0
$101$	0.602623	0.0599633	0.0299816	−	0.999550i	$-0.490455\pi$
$101$	0.602623	0.0599633	0.0299816	+	0.999550i	$0.490455\pi$
$102$	0	0
$103$	17.9243	1.76613	0.883066	−	0.469249i	$-0.155475\pi$
$103$	17.9243	1.76613	0.883066	+	0.469249i	$0.155475\pi$
$104$	0	0
$105$	1.02748	0.100272
$106$	0	0
$107$	19.2119	1.85728	0.928641	−	0.370980i	$-0.120978\pi$
$107$	19.2119	1.85728	0.928641	+	0.370980i	$0.120978\pi$
$108$	0	0
$109$	11.9943	1.14884	0.574422	−	0.818559i	$-0.305227\pi$
$109$	11.9943	1.14884	0.574422	+	0.818559i	$0.305227\pi$
$110$	0	0
$111$	3.97333	0.377132
$112$	0	0
$113$	15.4597	1.45433	0.727165	−	0.686463i	$-0.240837\pi$
$113$	15.4597	1.45433	0.727165	+	0.686463i	$0.240837\pi$
$114$	0	0
$115$	2.11988	0.197680
$116$	0	0
$117$	−13.5977	−1.25711
$118$	0	0
$119$	0.883019	0.0809462
$120$	0	0
$121$	10.5826	0.962057
$122$	0	0
$123$	2.18534	0.197046
$124$	0	0
$125$	−10.1673	−0.909387
$126$	0	0
$127$	2.16948	0.192510	0.0962552	−	0.995357i	$-0.469314\pi$
$127$	2.16948	0.192510	0.0962552	+	0.995357i	$0.469314\pi$
$128$	0	0
$129$	2.24479	0.197643
$130$	0	0
$131$	−1.63283	−0.142661	−0.0713306	−	0.997453i	$-0.522725\pi$
$131$	−1.63283	−0.142661	−0.0713306	+	0.997453i	$0.522725\pi$
$132$	0	0
$133$	2.13698	0.185300
$134$	0	0
$135$	6.70834	0.577362
$136$	0	0
$137$	1.77555	0.151696	0.0758478	−	0.997119i	$-0.475834\pi$
$137$	1.77555	0.151696	0.0758478	+	0.997119i	$0.475834\pi$
$138$	0	0
$139$	−2.63756	−0.223714	−0.111857	−	0.993724i	$-0.535680\pi$
$139$	−2.63756	−0.223714	−0.111857	+	0.993724i	$0.535680\pi$
$140$	0	0
$141$	3.13234	0.263790
$142$	0	0
$143$	22.8452	1.91041
$144$	0	0
$145$	−12.3395	−1.02474
$146$	0	0
$147$	3.01429	0.248614
$148$	0	0
$149$	−5.19983	−0.425987	−0.212993	−	0.977054i	$-0.568321\pi$
$149$	−5.19983	−0.425987	−0.212993	+	0.977054i	$0.568321\pi$
$150$	0	0
$151$	−9.35697	−0.761460	−0.380730	−	0.924686i	$-0.624327\pi$
$151$	−9.35697	−0.761460	−0.380730	+	0.924686i	$0.624327\pi$
$152$	0	0
$153$	2.76517	0.223551
$154$	0	0
$155$	−20.3398	−1.63373
$156$	0	0
$157$	16.6992	1.33274	0.666371	−	0.745620i	$-0.267847\pi$
$157$	16.6992	1.33274	0.666371	+	0.745620i	$0.267847\pi$
$158$	0	0
$159$	4.67159	0.370481
$160$	0	0
$161$	−0.779567	−0.0614385
$162$	0	0
$163$	−19.9937	−1.56603	−0.783014	−	0.622004i	$-0.786319\pi$
$163$	−19.9937	−1.56603	−0.783014	+	0.622004i	$0.786319\pi$
$164$	0	0
$165$	−5.40574	−0.420836
$166$	0	0
$167$	−4.34827	−0.336479	−0.168240	−	0.985746i	$-0.553808\pi$
$167$	−4.34827	−0.336479	−0.168240	+	0.985746i	$0.553808\pi$
$168$	0	0
$169$	11.1816	0.860122
$170$	0	0
$171$	6.69196	0.511747
$172$	0	0
$173$	−14.6484	−1.11370	−0.556850	−	0.830613i	$-0.687990\pi$
$173$	−14.6484	−1.11370	−0.556850	+	0.830613i	$0.687990\pi$
$174$	0	0
$175$	−0.676182	−0.0511146
$176$	0	0
$177$	−0.484590	−0.0364240
$178$	0	0
$179$	15.2528	1.14005	0.570024	−	0.821628i	$-0.306934\pi$
$179$	15.2528	1.14005	0.570024	+	0.821628i	$0.306934\pi$
$180$	0	0
$181$	−24.7119	−1.83682	−0.918412	−	0.395625i	$-0.870528\pi$
$181$	−24.7119	−1.83682	−0.918412	+	0.395625i	$0.870528\pi$
$182$	0	0
$183$	−2.41900	−0.178818
$184$	0	0
$185$	−19.6883	−1.44751
$186$	0	0
$187$	−4.64571	−0.339728
$188$	0	0
$189$	−2.46693	−0.179443
$190$	0	0
$191$	−23.7143	−1.71591	−0.857954	−	0.513726i	$-0.828265\pi$
$191$	−23.7143	−1.71591	−0.857954	+	0.513726i	$0.828265\pi$
$192$	0	0
$193$	−2.85899	−0.205795	−0.102897	−	0.994692i	$-0.532811\pi$
$193$	−2.85899	−0.205795	−0.102897	+	0.994692i	$0.532811\pi$
$194$	0	0
$195$	−5.72197	−0.409758
$196$	0	0
$197$	−8.77391	−0.625115	−0.312558	−	0.949899i	$-0.601186\pi$
$197$	−8.77391	−0.625115	−0.312558	+	0.949899i	$0.601186\pi$
$198$	0	0
$199$	−4.46141	−0.316261	−0.158131	−	0.987418i	$-0.550547\pi$
$199$	−4.46141	−0.316261	−0.158131	+	0.987418i	$0.550547\pi$
$200$	0	0
$201$	0.858849	0.0605785
$202$	0	0
$203$	4.53775	0.318487
$204$	0	0
$205$	−10.8286	−0.756304
$206$	0	0
$207$	−2.44121	−0.169676
$208$	0	0
$209$	−11.2430	−0.777697
$210$	0	0
$211$	−6.18557	−0.425832	−0.212916	−	0.977071i	$-0.568296\pi$
$211$	−6.18557	−0.425832	−0.212916	+	0.977071i	$0.568296\pi$
$212$	0	0
$213$	1.25089	0.0857093
$214$	0	0
$215$	−11.1232	−0.758595
$216$	0	0
$217$	7.47976	0.507759
$218$	0	0
$219$	5.57521	0.376738
$220$	0	0
$221$	−4.91748	−0.330785
$222$	0	0
$223$	29.6012	1.98224	0.991121	−	0.132960i	$-0.0424481\pi$
$223$	29.6012	1.98224	0.991121	+	0.132960i	$0.0424481\pi$
$224$	0	0
$225$	−2.11746	−0.141164
$226$	0	0
$227$	−15.9929	−1.06149	−0.530743	−	0.847533i	$-0.678087\pi$
$227$	−15.9929	−1.06149	−0.530743	+	0.847533i	$0.678087\pi$
$228$	0	0
$229$	0.126697	0.00837235	0.00418617	−	0.999991i	$-0.498667\pi$
$229$	0.126697	0.00837235	0.00418617	+	0.999991i	$0.498667\pi$
$230$	0	0
$231$	1.98791	0.130795
$232$	0	0
$233$	15.6980	1.02841	0.514204	−	0.857668i	$-0.328087\pi$
$233$	15.6980	1.02841	0.514204	+	0.857668i	$0.328087\pi$
$234$	0	0
$235$	−15.5211	−1.01248
$236$	0	0
$237$	−6.20858	−0.403290
$238$	0	0
$239$	24.5504	1.58803	0.794015	−	0.607898i	$-0.207987\pi$
$239$	24.5504	1.58803	0.794015	+	0.607898i	$0.207987\pi$
$240$	0	0
$241$	24.5959	1.58436	0.792181	−	0.610286i	$-0.208946\pi$
$241$	24.5959	1.58436	0.792181	+	0.610286i	$0.208946\pi$
$242$	0	0
$243$	−11.7451	−0.753449
$244$	0	0
$245$	−14.9361	−0.954235
$246$	0	0
$247$	−11.9007	−0.757226
$248$	0	0
$249$	3.39307	0.215027
$250$	0	0
$251$	−2.73903	−0.172886	−0.0864432	−	0.996257i	$-0.527550\pi$
$251$	−2.73903	−0.172886	−0.0864432	+	0.996257i	$0.527550\pi$
$252$	0	0
$253$	4.10143	0.257855
$254$	0	0
$255$	1.16360	0.0728673
$256$	0	0
$257$	−30.2874	−1.88927	−0.944637	−	0.328118i	$-0.893586\pi$
$257$	−30.2874	−1.88927	−0.944637	+	0.328118i	$0.893586\pi$
$258$	0	0
$259$	7.24019	0.449883
$260$	0	0
$261$	14.2099	0.879574
$262$	0	0
$263$	4.48855	0.276776	0.138388	−	0.990378i	$-0.455808\pi$
$263$	4.48855	0.276776	0.138388	+	0.990378i	$0.455808\pi$
$264$	0	0
$265$	−23.1483	−1.42199
$266$	0	0
$267$	4.01994	0.246016
$268$	0	0
$269$	−0.398291	−0.0242842	−0.0121421	−	0.999926i	$-0.503865\pi$
$269$	−0.398291	−0.0242842	−0.0121421	+	0.999926i	$0.503865\pi$
$270$	0	0
$271$	−2.71518	−0.164936	−0.0824678	−	0.996594i	$-0.526280\pi$
$271$	−2.71518	−0.164936	−0.0824678	+	0.996594i	$0.526280\pi$
$272$	0	0
$273$	2.10420	0.127352
$274$	0	0
$275$	3.55751	0.214526
$276$	0	0
$277$	−23.1176	−1.38900	−0.694500	−	0.719492i	$-0.744375\pi$
$277$	−23.1176	−1.38900	−0.694500	+	0.719492i	$0.744375\pi$
$278$	0	0
$279$	23.4229	1.40229
$280$	0	0
$281$	15.8319	0.944454	0.472227	−	0.881477i	$-0.343450\pi$
$281$	15.8319	0.944454	0.472227	+	0.881477i	$0.343450\pi$
$282$	0	0
$283$	23.9819	1.42557	0.712787	−	0.701381i	$-0.247433\pi$
$283$	23.9819	1.42557	0.712787	+	0.701381i	$0.247433\pi$
$284$	0	0
$285$	2.81601	0.166806
$286$	0	0
$287$	3.98212	0.235057
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.67342	0.332582
$292$	0	0
$293$	−26.2141	−1.53144	−0.765721	−	0.643173i	$-0.777618\pi$
$293$	−26.2141	−1.53144	−0.765721	+	0.643173i	$0.777618\pi$
$294$	0	0
$295$	2.40120	0.139803
$296$	0	0
$297$	12.9789	0.753114
$298$	0	0
$299$	4.34136	0.251067
$300$	0	0
$301$	4.09045	0.235770
$302$	0	0
$303$	−0.292025	−0.0167764
$304$	0	0
$305$	11.9864	0.686341
$306$	0	0
$307$	11.2344	0.641180	0.320590	−	0.947218i	$-0.396119\pi$
$307$	11.2344	0.641180	0.320590	+	0.947218i	$0.396119\pi$
$308$	0	0
$309$	−8.68593	−0.494125
$310$	0	0
$311$	1.33802	0.0758722	0.0379361	−	0.999280i	$-0.487922\pi$
$311$	1.33802	0.0758722	0.0379361	+	0.999280i	$0.487922\pi$
$312$	0	0
$313$	2.93901	0.166123	0.0830614	−	0.996544i	$-0.473530\pi$
$313$	2.93901	0.166123	0.0830614	+	0.996544i	$0.473530\pi$
$314$	0	0
$315$	5.86301	0.330343
$316$	0	0
$317$	−3.40311	−0.191138	−0.0955689	−	0.995423i	$-0.530467\pi$
$317$	−3.40311	−0.191138	−0.0955689	+	0.995423i	$0.530467\pi$
$318$	0	0
$319$	−23.8739	−1.33668
$320$	0	0
$321$	−9.30988	−0.519627
$322$	0	0
$323$	2.42009	0.134657
$324$	0	0
$325$	3.76562	0.208879
$326$	0	0
$327$	−5.81232	−0.321422
$328$	0	0
$329$	5.70773	0.314678
$330$	0	0
$331$	−7.34335	−0.403627	−0.201813	−	0.979424i	$-0.564683\pi$
$331$	−7.34335	−0.403627	−0.201813	+	0.979424i	$0.564683\pi$
$332$	0	0
$333$	22.6726	1.24245
$334$	0	0
$335$	−4.25569	−0.232513
$336$	0	0
$337$	5.57540	0.303711	0.151856	−	0.988403i	$-0.451475\pi$
$337$	5.57540	0.303711	0.151856	+	0.988403i	$0.451475\pi$
$338$	0	0
$339$	−7.49164	−0.406890
$340$	0	0
$341$	−39.3523	−2.13105
$342$	0	0
$343$	11.6738	0.630323
$344$	0	0
$345$	−1.02727	−0.0553066
$346$	0	0
$347$	10.8049	0.580039	0.290019	−	0.957021i	$-0.406338\pi$
$347$	10.8049	0.580039	0.290019	+	0.957021i	$0.406338\pi$
$348$	0	0
$349$	−30.4558	−1.63026	−0.815131	−	0.579277i	$-0.803335\pi$
$349$	−30.4558	−1.63026	−0.815131	+	0.579277i	$0.803335\pi$
$350$	0	0
$351$	13.7382	0.733290
$352$	0	0
$353$	−17.0570	−0.907855	−0.453927	−	0.891039i	$-0.649977\pi$
$353$	−17.0570	−0.907855	−0.453927	+	0.891039i	$0.649977\pi$
$354$	0	0
$355$	−6.19829	−0.328971
$356$	0	0
$357$	−0.427902	−0.0226470
$358$	0	0
$359$	−17.0774	−0.901311	−0.450655	−	0.892698i	$-0.648810\pi$
$359$	−17.0774	−0.901311	−0.450655	+	0.892698i	$0.648810\pi$
$360$	0	0
$361$	−13.1432	−0.691746
$362$	0	0
$363$	−5.12824	−0.269163
$364$	0	0
$365$	−27.6258	−1.44600
$366$	0	0
$367$	−17.2445	−0.900154	−0.450077	−	0.892990i	$-0.648604\pi$
$367$	−17.2445	−0.900154	−0.450077	+	0.892990i	$0.648604\pi$
$368$	0	0
$369$	12.4700	0.649163
$370$	0	0
$371$	8.51255	0.441950
$372$	0	0
$373$	−13.6801	−0.708329	−0.354165	−	0.935183i	$-0.615235\pi$
$373$	−13.6801	−0.708329	−0.354165	+	0.935183i	$0.615235\pi$
$374$	0	0
$375$	4.92695	0.254427
$376$	0	0
$377$	−25.2704	−1.30149
$378$	0	0
$379$	0.727777	0.0373834	0.0186917	−	0.999825i	$-0.494050\pi$
$379$	0.727777	0.0373834	0.0186917	+	0.999825i	$0.494050\pi$
$380$	0	0
$381$	−1.05131	−0.0538602
$382$	0	0
$383$	9.28383	0.474381	0.237191	−	0.971463i	$-0.423773\pi$
$383$	9.28383	0.474381	0.237191	+	0.971463i	$0.423773\pi$
$384$	0	0
$385$	−9.85032	−0.502019
$386$	0	0
$387$	12.8092	0.651130
$388$	0	0
$389$	−37.9854	−1.92594	−0.962968	−	0.269616i	$-0.913103\pi$
$389$	−37.9854	−1.92594	−0.962968	+	0.269616i	$0.913103\pi$
$390$	0	0
$391$	−0.882843	−0.0446473
$392$	0	0
$393$	0.791254	0.0399135
$394$	0	0
$395$	30.7642	1.54792
$396$	0	0
$397$	18.7545	0.941261	0.470630	−	0.882330i	$-0.344027\pi$
$397$	18.7545	0.941261	0.470630	+	0.882330i	$0.344027\pi$
$398$	0	0
$399$	−1.03556	−0.0518429
$400$	0	0
$401$	−13.4489	−0.671606	−0.335803	−	0.941932i	$-0.609008\pi$
$401$	−13.4489	−0.671606	−0.335803	+	0.941932i	$0.609008\pi$
$402$	0	0
$403$	−41.6543	−2.07495
$404$	0	0
$405$	16.6684	0.828260
$406$	0	0
$407$	−38.0919	−1.88814
$408$	0	0
$409$	17.7975	0.880029	0.440014	−	0.897991i	$-0.354973\pi$
$409$	17.7975	0.880029	0.440014	+	0.897991i	$0.354973\pi$
$410$	0	0
$411$	−0.860414	−0.0424411
$412$	0	0
$413$	−0.883019	−0.0434505
$414$	0	0
$415$	−16.8131	−0.825321
$416$	0	0
$417$	1.27813	0.0625904
$418$	0	0
$419$	−0.691716	−0.0337925	−0.0168963	−	0.999857i	$-0.505379\pi$
$419$	−0.691716	−0.0337925	−0.0168963	+	0.999857i	$0.505379\pi$
$420$	0	0
$421$	9.08827	0.442935	0.221468	−	0.975168i	$-0.428915\pi$
$421$	9.08827	0.442935	0.221468	+	0.975168i	$0.428915\pi$
$422$	0	0
$423$	17.8738	0.869052
$424$	0	0
$425$	−0.765762	−0.0371449
$426$	0	0
$427$	−4.40790	−0.213313
$428$	0	0
$429$	−11.0705	−0.534491
$430$	0	0
$431$	23.6787	1.14056	0.570281	−	0.821450i	$-0.306834\pi$
$431$	23.6787	1.14056	0.570281	+	0.821450i	$0.306834\pi$
$432$	0	0
$433$	37.7394	1.81364	0.906819	−	0.421520i	$-0.138503\pi$
$433$	37.7394	1.81364	0.906819	+	0.421520i	$0.138503\pi$
$434$	0	0
$435$	5.97961	0.286701
$436$	0	0
$437$	−2.13656	−0.102205
$438$	0	0
$439$	1.73433	0.0827750	0.0413875	−	0.999143i	$-0.486822\pi$
$439$	1.73433	0.0827750	0.0413875	+	0.999143i	$0.486822\pi$
$440$	0	0
$441$	17.2001	0.819054
$442$	0	0
$443$	−1.82356	−0.0866402	−0.0433201	−	0.999061i	$-0.513794\pi$
$443$	−1.82356	−0.0866402	−0.0433201	+	0.999061i	$0.513794\pi$
$444$	0	0
$445$	−19.9193	−0.944264
$446$	0	0
$447$	2.51979	0.119182
$448$	0	0
$449$	13.8627	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$449$	13.8627	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$450$	0	0
$451$	−20.9506	−0.986526
$452$	0	0
$453$	4.53430	0.213040
$454$	0	0
$455$	−10.4265	−0.488804
$456$	0	0
$457$	−13.9226	−0.651274	−0.325637	−	0.945495i	$-0.605579\pi$
$457$	−13.9226	−0.651274	−0.325637	+	0.945495i	$0.605579\pi$
$458$	0	0
$459$	−2.79375	−0.130401
$460$	0	0
$461$	4.21090	0.196121	0.0980605	−	0.995180i	$-0.468736\pi$
$461$	4.21090	0.196121	0.0980605	+	0.995180i	$0.468736\pi$
$462$	0	0
$463$	13.7713	0.640007	0.320004	−	0.947416i	$-0.396316\pi$
$463$	13.7713	0.640007	0.320004	+	0.947416i	$0.396316\pi$
$464$	0	0
$465$	9.85645	0.457082
$466$	0	0
$467$	30.5538	1.41386	0.706930	−	0.707283i	$-0.250079\pi$
$467$	30.5538	1.41386	0.706930	+	0.707283i	$0.250079\pi$
$468$	0	0
$469$	1.56499	0.0722646
$470$	0	0
$471$	−8.09227	−0.372872
$472$	0	0
$473$	−21.5205	−0.989516
$474$	0	0
$475$	−1.85321	−0.0850312
$476$	0	0
$477$	26.6571	1.22054
$478$	0	0
$479$	−2.96584	−0.135513	−0.0677563	−	0.997702i	$-0.521584\pi$
$479$	−2.96584	−0.135513	−0.0677563	+	0.997702i	$0.521584\pi$
$480$	0	0
$481$	−40.3202	−1.83844
$482$	0	0
$483$	0.377770	0.0171891
$484$	0	0
$485$	−28.1125	−1.27652
$486$	0	0
$487$	−14.4499	−0.654788	−0.327394	−	0.944888i	$-0.606170\pi$
$487$	−14.4499	−0.654788	−0.327394	+	0.944888i	$0.606170\pi$
$488$	0	0
$489$	9.68875	0.438141
$490$	0	0
$491$	14.2554	0.643339	0.321669	−	0.946852i	$-0.395756\pi$
$491$	14.2554	0.643339	0.321669	+	0.946852i	$0.395756\pi$
$492$	0	0
$493$	5.13890	0.231444
$494$	0	0
$495$	−30.8463	−1.38644
$496$	0	0
$497$	2.27936	0.102243
$498$	0	0
$499$	21.0075	0.940424	0.470212	−	0.882554i	$-0.344178\pi$
$499$	21.0075	0.940424	0.470212	+	0.882554i	$0.344178\pi$
$500$	0	0
$501$	2.10713	0.0941396
$502$	0	0
$503$	13.5793	0.605471	0.302736	−	0.953075i	$-0.402100\pi$
$503$	13.5793	0.605471	0.302736	+	0.953075i	$0.402100\pi$
$504$	0	0
$505$	1.44702	0.0643915
$506$	0	0
$507$	−5.41849	−0.240643
$508$	0	0
$509$	−14.4382	−0.639963	−0.319982	−	0.947424i	$-0.603677\pi$
$509$	−14.4382	−0.639963	−0.319982	+	0.947424i	$0.603677\pi$
$510$	0	0
$511$	10.1591	0.449414
$512$	0	0
$513$	−6.76111	−0.298510
$514$	0	0
$515$	43.0398	1.89656
$516$	0	0
$517$	−30.0294	−1.32069
$518$	0	0
$519$	7.09849	0.311589
$520$	0	0
$521$	−4.25855	−0.186571	−0.0932853	−	0.995639i	$-0.529737\pi$
$521$	−4.25855	−0.186571	−0.0932853	+	0.995639i	$0.529737\pi$
$522$	0	0
$523$	−18.8840	−0.825738	−0.412869	−	0.910790i	$-0.635473\pi$
$523$	−18.8840	−0.825738	−0.412869	+	0.910790i	$0.635473\pi$
$524$	0	0
$525$	0.327671	0.0143007
$526$	0	0
$527$	8.47067	0.368988
$528$	0	0
$529$	−22.2206	−0.966113
$530$	0	0
$531$	−2.76517	−0.119998
$532$	0	0
$533$	−22.1762	−0.960558
$534$	0	0
$535$	46.1315	1.99444
$536$	0	0
$537$	−7.39136	−0.318961
$538$	0	0
$539$	−28.8976	−1.24471
$540$	0	0
$541$	38.4285	1.65217	0.826085	−	0.563545i	$-0.190563\pi$
$541$	38.4285	1.65217	0.826085	+	0.563545i	$0.190563\pi$
$542$	0	0
$543$	11.9752	0.513903
$544$	0	0
$545$	28.8007	1.23369
$546$	0	0
$547$	−21.8825	−0.935629	−0.467814	−	0.883827i	$-0.654958\pi$
$547$	−21.8825	−0.935629	−0.467814	+	0.883827i	$0.654958\pi$
$548$	0	0
$549$	−13.8033	−0.589112
$550$	0	0
$551$	12.4366	0.529817
$552$	0	0
$553$	−11.3132	−0.481088
$554$	0	0
$555$	9.54076	0.404983
$556$	0	0
$557$	−34.8284	−1.47573	−0.737864	−	0.674950i	$-0.764165\pi$
$557$	−34.8284	−1.47573	−0.737864	+	0.674950i	$0.764165\pi$
$558$	0	0
$559$	−22.7795	−0.963469
$560$	0	0
$561$	2.25127	0.0950485
$562$	0	0
$563$	35.1129	1.47983	0.739917	−	0.672698i	$-0.234865\pi$
$563$	35.1129	1.47983	0.739917	+	0.672698i	$0.234865\pi$
$564$	0	0
$565$	37.1219	1.56173
$566$	0	0
$567$	−6.12965	−0.257421
$568$	0	0
$569$	28.8247	1.20839	0.604197	−	0.796835i	$-0.293494\pi$
$569$	28.8247	1.20839	0.604197	+	0.796835i	$0.293494\pi$
$570$	0	0
$571$	−8.61227	−0.360412	−0.180206	−	0.983629i	$-0.557676\pi$
$571$	−8.61227	−0.360412	−0.180206	+	0.983629i	$0.557676\pi$
$572$	0	0
$573$	11.4917	0.480074
$574$	0	0
$575$	0.676048	0.0281931
$576$	0	0
$577$	42.9089	1.78632	0.893161	−	0.449736i	$-0.148482\pi$
$577$	42.9089	1.78632	0.893161	+	0.449736i	$0.148482\pi$
$578$	0	0
$579$	1.38544	0.0575769
$580$	0	0
$581$	6.18284	0.256508
$582$	0	0
$583$	−44.7860	−1.85485
$584$	0	0
$585$	−32.6507	−1.34994
$586$	0	0
$587$	−19.8085	−0.817585	−0.408793	−	0.912627i	$-0.634050\pi$
$587$	−19.8085	−0.817585	−0.408793	+	0.912627i	$0.634050\pi$
$588$	0	0
$589$	20.4998	0.844679
$590$	0	0
$591$	4.25175	0.174894
$592$	0	0
$593$	−0.797081	−0.0327322	−0.0163661	−	0.999866i	$-0.505210\pi$
$593$	−0.797081	−0.0327322	−0.0163661	+	0.999866i	$0.505210\pi$
$594$	0	0
$595$	2.12030	0.0869240
$596$	0	0
$597$	2.16196	0.0884830
$598$	0	0
$599$	5.52359	0.225688	0.112844	−	0.993613i	$-0.464004\pi$
$599$	5.52359	0.225688	0.112844	+	0.993613i	$0.464004\pi$
$600$	0	0
$601$	−9.46980	−0.386281	−0.193141	−	0.981171i	$-0.561867\pi$
$601$	−9.46980	−0.386281	−0.193141	+	0.981171i	$0.561867\pi$
$602$	0	0
$603$	4.90077	0.199575
$604$	0	0
$605$	25.4110	1.03310
$606$	0	0
$607$	−18.5020	−0.750971	−0.375486	−	0.926828i	$-0.622524\pi$
$607$	−18.5020	−0.750971	−0.375486	+	0.926828i	$0.622524\pi$
$608$	0	0
$609$	−2.19895	−0.0891058
$610$	0	0
$611$	−31.7860	−1.28592
$612$	0	0
$613$	26.2267	1.05929	0.529643	−	0.848221i	$-0.322326\pi$
$613$	26.2267	1.05929	0.529643	+	0.848221i	$0.322326\pi$
$614$	0	0
$615$	5.24744	0.211597
$616$	0	0
$617$	−7.25061	−0.291899	−0.145949	−	0.989292i	$-0.546624\pi$
$617$	−7.25061	−0.291899	−0.145949	+	0.989292i	$0.546624\pi$
$618$	0	0
$619$	−3.72428	−0.149691	−0.0748457	−	0.997195i	$-0.523846\pi$
$619$	−3.72428	−0.149691	−0.0748457	+	0.997195i	$0.523846\pi$
$620$	0	0
$621$	2.46644	0.0989748
$622$	0	0
$623$	7.32513	0.293475
$624$	0	0
$625$	−28.2424	−1.12970
$626$	0	0
$627$	5.44826	0.217583
$628$	0	0
$629$	8.19936	0.326930
$630$	0	0
$631$	42.3282	1.68506	0.842529	−	0.538651i	$-0.181066\pi$
$631$	42.3282	1.68506	0.842529	+	0.538651i	$0.181066\pi$
$632$	0	0
$633$	2.99746	0.119139
$634$	0	0
$635$	5.20936	0.206727
$636$	0	0
$637$	−30.5881	−1.21194
$638$	0	0
$639$	7.13782	0.282368
$640$	0	0
$641$	−3.52901	−0.139387	−0.0696937	−	0.997568i	$-0.522202\pi$
$641$	−3.52901	−0.139387	−0.0696937	+	0.997568i	$0.522202\pi$
$642$	0	0
$643$	−12.2705	−0.483899	−0.241950	−	0.970289i	$-0.577787\pi$
$643$	−12.2705	−0.483899	−0.241950	+	0.970289i	$0.577787\pi$
$644$	0	0
$645$	5.39019	0.212238
$646$	0	0
$647$	−22.2242	−0.873725	−0.436863	−	0.899528i	$-0.643910\pi$
$647$	−22.2242	−0.873725	−0.436863	+	0.899528i	$0.643910\pi$
$648$	0	0
$649$	4.64571	0.182360
$650$	0	0
$651$	−3.62462	−0.142060
$652$	0	0
$653$	−32.1463	−1.25798	−0.628991	−	0.777412i	$-0.716532\pi$
$653$	−32.1463	−1.25798	−0.628991	+	0.777412i	$0.716532\pi$
$654$	0	0
$655$	−3.92075	−0.153197
$656$	0	0
$657$	31.8133	1.24116
$658$	0	0
$659$	45.9407	1.78959	0.894797	−	0.446472i	$-0.147320\pi$
$659$	45.9407	1.78959	0.894797	+	0.446472i	$0.147320\pi$
$660$	0	0
$661$	0.872959	0.0339542	0.0169771	−	0.999856i	$-0.494596\pi$
$661$	0.872959	0.0339542	0.0169771	+	0.999856i	$0.494596\pi$
$662$	0	0
$663$	2.38296	0.0925465
$664$	0	0
$665$	5.13133	0.198984
$666$	0	0
$667$	−4.53684	−0.175667
$668$	0	0
$669$	−14.3444	−0.554588
$670$	0	0
$671$	23.1907	0.895267
$672$	0	0
$673$	−17.5371	−0.676005	−0.338002	−	0.941145i	$-0.609751\pi$
$673$	−17.5371	−0.676005	−0.338002	+	0.941145i	$0.609751\pi$
$674$	0	0
$675$	2.13934	0.0823434
$676$	0	0
$677$	23.0540	0.886037	0.443018	−	0.896513i	$-0.353908\pi$
$677$	23.0540	0.886037	0.443018	+	0.896513i	$0.353908\pi$
$678$	0	0
$679$	10.3381	0.396740
$680$	0	0
$681$	7.75000	0.296981
$682$	0	0
$683$	−31.9963	−1.22430	−0.612152	−	0.790740i	$-0.709696\pi$
$683$	−31.9963	−1.22430	−0.612152	+	0.790740i	$0.709696\pi$
$684$	0	0
$685$	4.26345	0.162898
$686$	0	0
$687$	−0.0613959	−0.00234240
$688$	0	0
$689$	−47.4059	−1.80602
$690$	0	0
$691$	6.60499	0.251266	0.125633	−	0.992077i	$-0.459904\pi$
$691$	6.60499	0.251266	0.125633	+	0.992077i	$0.459904\pi$
$692$	0	0
$693$	11.3434	0.430901
$694$	0	0
$695$	−6.33330	−0.240236
$696$	0	0
$697$	4.50967	0.170816
$698$	0	0
$699$	−7.60708	−0.287726
$700$	0	0
$701$	22.4089	0.846373	0.423187	−	0.906043i	$-0.360911\pi$
$701$	22.4089	0.846373	0.423187	+	0.906043i	$0.360911\pi$
$702$	0	0
$703$	19.8432	0.748400
$704$	0	0
$705$	7.52137	0.283271
$706$	0	0
$707$	−0.532128	−0.0200127
$708$	0	0
$709$	21.3709	0.802601	0.401300	−	0.915947i	$-0.368558\pi$
$709$	21.3709	0.802601	0.401300	+	0.915947i	$0.368558\pi$
$710$	0	0
$711$	−35.4274	−1.32863
$712$	0	0
$713$	−7.47827	−0.280064
$714$	0	0
$715$	54.8558	2.05149
$716$	0	0
$717$	−11.8969	−0.444296
$718$	0	0
$719$	−13.0936	−0.488311	−0.244155	−	0.969736i	$-0.578511\pi$
$719$	−13.0936	−0.488311	−0.244155	+	0.969736i	$0.578511\pi$
$720$	0	0
$721$	−15.8275	−0.589446
$722$	0	0
$723$	−11.9189	−0.443270
$724$	0	0
$725$	−3.93518	−0.146149
$726$	0	0
$727$	−27.4250	−1.01714	−0.508568	−	0.861022i	$-0.669825\pi$
$727$	−27.4250	−1.01714	−0.508568	+	0.861022i	$0.669825\pi$
$728$	0	0
$729$	−15.1335	−0.560501
$730$	0	0
$731$	4.63235	0.171334
$732$	0	0
$733$	14.4158	0.532459	0.266229	−	0.963910i	$-0.414222\pi$
$733$	14.4158	0.532459	0.266229	+	0.963910i	$0.414222\pi$
$734$	0	0
$735$	7.23790	0.266974
$736$	0	0
$737$	−8.23369	−0.303292
$738$	0	0
$739$	48.6432	1.78937	0.894685	−	0.446697i	$-0.147400\pi$
$739$	48.6432	1.78937	0.894685	+	0.446697i	$0.147400\pi$
$740$	0	0
$741$	5.76698	0.211855
$742$	0	0
$743$	24.3890	0.894746	0.447373	−	0.894347i	$-0.352360\pi$
$743$	24.3890	0.894746	0.447373	+	0.894347i	$0.352360\pi$
$744$	0	0
$745$	−12.4858	−0.457446
$746$	0	0
$747$	19.3616	0.708403
$748$	0	0
$749$	−16.9644	−0.619867
$750$	0	0
$751$	−23.3398	−0.851682	−0.425841	−	0.904798i	$-0.640022\pi$
$751$	−23.3398	−0.851682	−0.425841	+	0.904798i	$0.640022\pi$
$752$	0	0
$753$	1.32731	0.0483698
$754$	0	0
$755$	−22.4680	−0.817693
$756$	0	0
$757$	−53.2474	−1.93531	−0.967655	−	0.252277i	$-0.918821\pi$
$757$	−53.2474	−1.93531	−0.967655	+	0.252277i	$0.918821\pi$
$758$	0	0
$759$	−1.98751	−0.0721422
$760$	0	0
$761$	−33.5224	−1.21519	−0.607594	−	0.794248i	$-0.707865\pi$
$761$	−33.5224	−1.21519	−0.607594	+	0.794248i	$0.707865\pi$
$762$	0	0
$763$	−10.5912	−0.383427
$764$	0	0
$765$	6.63973	0.240060
$766$	0	0
$767$	4.91748	0.177560
$768$	0	0
$769$	−11.0670	−0.399086	−0.199543	−	0.979889i	$-0.563946\pi$
$769$	−11.0670	−0.399086	−0.199543	+	0.979889i	$0.563946\pi$
$770$	0	0
$771$	14.6770	0.528578
$772$	0	0
$773$	15.0837	0.542524	0.271262	−	0.962506i	$-0.412559\pi$
$773$	15.0837	0.542524	0.271262	+	0.962506i	$0.412559\pi$
$774$	0	0
$775$	−6.48652	−0.233003
$776$	0	0
$777$	−3.50852	−0.125868
$778$	0	0
$779$	10.9138	0.391028
$780$	0	0
$781$	−11.9921	−0.429111
$782$	0	0
$783$	−14.3568	−0.513070
$784$	0	0
$785$	40.0981	1.43116
$786$	0	0
$787$	−53.7710	−1.91673	−0.958365	−	0.285548i	$-0.907825\pi$
$787$	−53.7710	−1.91673	−0.958365	+	0.285548i	$0.907825\pi$
$788$	0	0
$789$	−2.17511	−0.0774358
$790$	0	0
$791$	−13.6512	−0.485382
$792$	0	0
$793$	24.5473	0.871701
$794$	0	0
$795$	11.2174	0.397841
$796$	0	0
$797$	8.29212	0.293722	0.146861	−	0.989157i	$-0.453083\pi$
$797$	8.29212	0.293722	0.146861	+	0.989157i	$0.453083\pi$
$798$	0	0
$799$	6.46389	0.228676
$800$	0	0
$801$	22.9386	0.810497
$802$	0	0
$803$	−53.4490	−1.88617
$804$	0	0
$805$	−1.87190	−0.0659757
$806$	0	0
$807$	0.193008	0.00679419
$808$	0	0
$809$	−16.7391	−0.588514	−0.294257	−	0.955726i	$-0.595072\pi$
$809$	−16.7391	−0.588514	−0.294257	+	0.955726i	$0.595072\pi$
$810$	0	0
$811$	1.27457	0.0447563	0.0223781	−	0.999750i	$-0.492876\pi$
$811$	1.27457	0.0447563	0.0223781	+	0.999750i	$0.492876\pi$
$812$	0	0
$813$	1.31575	0.0461454
$814$	0	0
$815$	−48.0089	−1.68168
$816$	0	0
$817$	11.2107	0.392213
$818$	0	0
$819$	12.0070	0.419559
$820$	0	0
$821$	21.0137	0.733385	0.366692	−	0.930342i	$-0.380490\pi$
$821$	21.0137	0.733385	0.366692	+	0.930342i	$0.380490\pi$
$822$	0	0
$823$	26.9612	0.939807	0.469903	−	0.882718i	$-0.344289\pi$
$823$	26.9612	0.939807	0.469903	+	0.882718i	$0.344289\pi$
$824$	0	0
$825$	−1.72393	−0.0600197
$826$	0	0
$827$	−13.1569	−0.457510	−0.228755	−	0.973484i	$-0.573465\pi$
$827$	−13.1569	−0.457510	−0.228755	+	0.973484i	$0.573465\pi$
$828$	0	0
$829$	8.31107	0.288655	0.144328	−	0.989530i	$-0.453898\pi$
$829$	8.31107	0.288655	0.144328	+	0.989530i	$0.453898\pi$
$830$	0	0
$831$	11.2026	0.388612
$832$	0	0
$833$	6.22028	0.215520
$834$	0	0
$835$	−10.4411	−0.361328
$836$	0	0
$837$	−23.6649	−0.817979
$838$	0	0
$839$	−24.1535	−0.833872	−0.416936	−	0.908936i	$-0.636896\pi$
$839$	−24.1535	−0.833872	−0.416936	+	0.908936i	$0.636896\pi$
$840$	0	0
$841$	−2.59169	−0.0893686
$842$	0	0
$843$	−7.67199	−0.264238
$844$	0	0
$845$	26.8492	0.923642
$846$	0	0
$847$	−9.34466	−0.321086
$848$	0	0
$849$	−11.6214	−0.398844
$850$	0	0
$851$	−7.23875	−0.248141
$852$	0	0
$853$	45.4733	1.55698	0.778488	−	0.627660i	$-0.215987\pi$
$853$	45.4733	1.55698	0.778488	+	0.627660i	$0.215987\pi$
$854$	0	0
$855$	16.0687	0.549539
$856$	0	0
$857$	−21.7641	−0.743448	−0.371724	−	0.928343i	$-0.621233\pi$
$857$	−21.7641	−0.743448	−0.371724	+	0.928343i	$0.621233\pi$
$858$	0	0
$859$	7.71598	0.263266	0.131633	−	0.991299i	$-0.457978\pi$
$859$	7.71598	0.263266	0.131633	+	0.991299i	$0.457978\pi$
$860$	0	0
$861$	−1.92970	−0.0657639
$862$	0	0
$863$	−19.1556	−0.652066	−0.326033	−	0.945358i	$-0.605712\pi$
$863$	−19.1556	−0.652066	−0.326033	+	0.945358i	$0.605712\pi$
$864$	0	0
$865$	−35.1738	−1.19595
$866$	0	0
$867$	−0.484590	−0.0164575
$868$	0	0
$869$	59.5209	2.01911
$870$	0	0
$871$	−8.71534	−0.295308
$872$	0	0
$873$	32.3737	1.09569
$874$	0	0
$875$	8.97787	0.303508
$876$	0	0
$877$	−0.177040	−0.00597823	−0.00298912	−	0.999996i	$-0.500951\pi$
$877$	−0.177040	−0.00597823	−0.00298912	+	0.999996i	$0.500951\pi$
$878$	0	0
$879$	12.7031	0.428464
$880$	0	0
$881$	−29.5674	−0.996151	−0.498075	−	0.867134i	$-0.665960\pi$
$881$	−29.5674	−0.996151	−0.498075	+	0.867134i	$0.665960\pi$
$882$	0	0
$883$	14.1742	0.476998	0.238499	−	0.971143i	$-0.423345\pi$
$883$	14.1742	0.476998	0.238499	+	0.971143i	$0.423345\pi$
$884$	0	0
$885$	−1.16360	−0.0391139
$886$	0	0
$887$	−5.65842	−0.189991	−0.0949956	−	0.995478i	$-0.530284\pi$
$887$	−5.65842	−0.189991	−0.0949956	+	0.995478i	$0.530284\pi$
$888$	0	0
$889$	−1.91569	−0.0642503
$890$	0	0
$891$	32.2491	1.08039
$892$	0	0
$893$	15.6432	0.523479
$894$	0	0
$895$	36.6251	1.22424
$896$	0	0
$897$	−2.10378	−0.0702432
$898$	0	0
$899$	43.5299	1.45180
$900$	0	0
$901$	9.64029	0.321164
$902$	0	0
$903$	−1.98219	−0.0659632
$904$	0	0
$905$	−59.3383	−1.97247
$906$	0	0
$907$	−51.8834	−1.72276	−0.861380	−	0.507961i	$-0.830399\pi$
$907$	−51.8834	−1.72276	−0.861380	+	0.507961i	$0.830399\pi$
$908$	0	0
$909$	−1.66636	−0.0552696
$910$	0	0
$911$	8.11788	0.268957	0.134479	−	0.990916i	$-0.457064\pi$
$911$	8.11788	0.268957	0.134479	+	0.990916i	$0.457064\pi$
$912$	0	0
$913$	−32.5290	−1.07655
$914$	0	0
$915$	−5.80851	−0.192023
$916$	0	0
$917$	1.44182	0.0476131
$918$	0	0
$919$	1.00875	0.0332757	0.0166378	−	0.999862i	$-0.494704\pi$
$919$	1.00875	0.0332757	0.0166378	+	0.999862i	$0.494704\pi$
$920$	0	0
$921$	−5.44407	−0.179388
$922$	0	0
$923$	−12.6936	−0.417816
$924$	0	0
$925$	−6.27876	−0.206444
$926$	0	0
$927$	−49.5637	−1.62789
$928$	0	0
$929$	38.7659	1.27187	0.635933	−	0.771744i	$-0.280615\pi$
$929$	38.7659	1.27187	0.635933	+	0.771744i	$0.280615\pi$
$930$	0	0
$931$	15.0536	0.493363
$932$	0	0
$933$	−0.648392	−0.0212274
$934$	0	0
$935$	−11.1553	−0.364817
$936$	0	0
$937$	−22.7815	−0.744240	−0.372120	−	0.928185i	$-0.621369\pi$
$937$	−22.7815	−0.744240	−0.372120	+	0.928185i	$0.621369\pi$
$938$	0	0
$939$	−1.42422	−0.0464776
$940$	0	0
$941$	42.1791	1.37500	0.687500	−	0.726184i	$-0.258708\pi$
$941$	42.1791	1.37500	0.687500	+	0.726184i	$0.258708\pi$
$942$	0	0
$943$	−3.98133	−0.129650
$944$	0	0
$945$	−5.92359	−0.192694
$946$	0	0
$947$	−55.1713	−1.79283	−0.896413	−	0.443219i	$-0.853836\pi$
$947$	−55.1713	−1.79283	−0.896413	+	0.443219i	$0.853836\pi$
$948$	0	0
$949$	−56.5756	−1.83652
$950$	0	0
$951$	1.64911	0.0534762
$952$	0	0
$953$	18.0607	0.585042	0.292521	−	0.956259i	$-0.405506\pi$
$953$	18.0607	0.585042	0.292521	+	0.956259i	$0.405506\pi$
$954$	0	0
$955$	−56.9429	−1.84263
$956$	0	0
$957$	11.5690	0.373974
$958$	0	0
$959$	−1.56784	−0.0506283
$960$	0	0
$961$	40.7523	1.31459
$962$	0	0
$963$	−53.1241	−1.71190
$964$	0	0
$965$	−6.86501	−0.220992
$966$	0	0
$967$	−13.4819	−0.433549	−0.216775	−	0.976222i	$-0.569554\pi$
$967$	−13.4819	−0.433549	−0.216775	+	0.976222i	$0.569554\pi$
$968$	0	0
$969$	−1.17275	−0.0376742
$970$	0	0
$971$	−31.8746	−1.02290	−0.511452	−	0.859312i	$-0.670892\pi$
$971$	−31.8746	−1.02290	−0.511452	+	0.859312i	$0.670892\pi$
$972$	0	0
$973$	2.32901	0.0746646
$974$	0	0
$975$	−1.82478	−0.0584398
$976$	0	0
$977$	16.1871	0.517872	0.258936	−	0.965894i	$-0.416628\pi$
$977$	16.1871	0.517872	0.258936	+	0.965894i	$0.416628\pi$
$978$	0	0
$979$	−38.5387	−1.23170
$980$	0	0
$981$	−33.1663	−1.05892
$982$	0	0
$983$	4.25189	0.135614	0.0678071	−	0.997698i	$-0.478400\pi$
$983$	4.25189	0.135614	0.0678071	+	0.997698i	$0.478400\pi$
$984$	0	0
$985$	−21.0679	−0.671280
$986$	0	0
$987$	−2.76591	−0.0880399
$988$	0	0
$989$	−4.08964	−0.130043
$990$	0	0
$991$	−15.3615	−0.487974	−0.243987	−	0.969778i	$-0.578455\pi$
$991$	−15.3615	−0.487974	−0.243987	+	0.969778i	$0.578455\pi$
$992$	0	0
$993$	3.55851	0.112926
$994$	0	0
$995$	−10.7128	−0.339617
$996$	0	0
$997$	16.3083	0.516488	0.258244	−	0.966080i	$-0.416856\pi$
$997$	16.3083	0.516488	0.258244	+	0.966080i	$0.416856\pi$
$998$	0	0
$999$	−22.9069	−0.724743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.10	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.10	✓	22	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.484590 q^{3} +2.40120 q^{5} -0.883019 q^{7} -2.76517 q^{9} +O(q^{10})\)	\(q-0.484590 q^{3} +2.40120 q^{5} -0.883019 q^{7} -2.76517 q^{9} +4.64571 q^{11} +4.91748 q^{13} -1.16360 q^{15} -1.00000 q^{17} -2.42009 q^{19} +0.427902 q^{21} +0.882843 q^{23} +0.765762 q^{25} +2.79375 q^{27} -5.13890 q^{29} -8.47067 q^{31} -2.25127 q^{33} -2.12030 q^{35} -8.19936 q^{37} -2.38296 q^{39} -4.50967 q^{41} -4.63235 q^{43} -6.63973 q^{45} -6.46389 q^{47} -6.22028 q^{49} +0.484590 q^{51} -9.64029 q^{53} +11.1553 q^{55} +1.17275 q^{57} +1.00000 q^{59} +4.99185 q^{61} +2.44170 q^{63} +11.8078 q^{65} -1.77232 q^{67} -0.427817 q^{69} -2.58133 q^{71} -11.5050 q^{73} -0.371081 q^{75} -4.10225 q^{77} +12.8120 q^{79} +6.94170 q^{81} -7.00194 q^{83} -2.40120 q^{85} +2.49026 q^{87} -8.29555 q^{89} -4.34222 q^{91} +4.10480 q^{93} -5.81112 q^{95} -11.7077 q^{97} -12.8462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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