LMFDB - Embedded newform 8024.2.a.x.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.20210 q^{3} +1.10558 q^{5} +3.71217 q^{7} -1.55495 q^{9} +O(q^{10})$	$q-1.20210 q^{3} +1.10558 q^{5} +3.71217 q^{7} -1.55495 q^{9} -3.29413 q^{11} -0.161370 q^{13} -1.32902 q^{15} -1.00000 q^{17} -7.78255 q^{19} -4.46241 q^{21} +2.77622 q^{23} -3.77770 q^{25} +5.47552 q^{27} +4.90183 q^{29} +2.51187 q^{31} +3.95989 q^{33} +4.10408 q^{35} +7.97267 q^{37} +0.193983 q^{39} -2.25712 q^{41} +3.66705 q^{43} -1.71911 q^{45} +11.1832 q^{47} +6.78018 q^{49} +1.20210 q^{51} -3.20581 q^{53} -3.64192 q^{55} +9.35543 q^{57} +1.00000 q^{59} -2.54429 q^{61} -5.77222 q^{63} -0.178407 q^{65} +0.497288 q^{67} -3.33731 q^{69} -10.4623 q^{71} -10.6171 q^{73} +4.54119 q^{75} -12.2284 q^{77} -10.6023 q^{79} -1.91730 q^{81} -4.99436 q^{83} -1.10558 q^{85} -5.89250 q^{87} +0.726929 q^{89} -0.599031 q^{91} -3.01953 q^{93} -8.60420 q^{95} -17.2029 q^{97} +5.12220 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.20210	−0.694035	−0.347017	−	0.937859i	$-0.612806\pi$
$3$	−1.20210	−0.694035	−0.347017	+	0.937859i	$0.612806\pi$
$4$	0	0
$5$	1.10558	0.494429	0.247214	−	0.968961i	$-0.420485\pi$
$5$	1.10558	0.494429	0.247214	+	0.968961i	$0.420485\pi$
$6$	0	0
$7$	3.71217	1.40307	0.701533	−	0.712637i	$-0.252499\pi$
$7$	3.71217	1.40307	0.701533	+	0.712637i	$0.252499\pi$
$8$	0	0
$9$	−1.55495	−0.518315
$10$	0	0
$11$	−3.29413	−0.993219	−0.496609	−	0.867974i	$-0.665422\pi$
$11$	−3.29413	−0.993219	−0.496609	+	0.867974i	$0.665422\pi$
$12$	0	0
$13$	−0.161370	−0.0447559	−0.0223780	−	0.999750i	$-0.507124\pi$
$13$	−0.161370	−0.0447559	−0.0223780	+	0.999750i	$0.507124\pi$
$14$	0	0
$15$	−1.32902	−0.343151
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.78255	−1.78544	−0.892719	−	0.450613i	$-0.851205\pi$
$19$	−7.78255	−1.78544	−0.892719	+	0.450613i	$0.851205\pi$
$20$	0	0
$21$	−4.46241	−0.973778
$22$	0	0
$23$	2.77622	0.578882	0.289441	−	0.957196i	$-0.406531\pi$
$23$	2.77622	0.578882	0.289441	+	0.957196i	$0.406531\pi$
$24$	0	0
$25$	−3.77770	−0.755540
$26$	0	0
$27$	5.47552	1.05376
$28$	0	0
$29$	4.90183	0.910246	0.455123	−	0.890429i	$-0.349595\pi$
$29$	4.90183	0.910246	0.455123	+	0.890429i	$0.349595\pi$
$30$	0	0
$31$	2.51187	0.451145	0.225573	−	0.974226i	$-0.427575\pi$
$31$	2.51187	0.451145	0.225573	+	0.974226i	$0.427575\pi$
$32$	0	0
$33$	3.95989	0.689328
$34$	0	0
$35$	4.10408	0.693717
$36$	0	0
$37$	7.97267	1.31070	0.655349	−	0.755326i	$-0.272521\pi$
$37$	7.97267	1.31070	0.655349	+	0.755326i	$0.272521\pi$
$38$	0	0
$39$	0.193983	0.0310622
$40$	0	0
$41$	−2.25712	−0.352503	−0.176252	−	0.984345i	$-0.556397\pi$
$41$	−2.25712	−0.352503	−0.176252	+	0.984345i	$0.556397\pi$
$42$	0	0
$43$	3.66705	0.559219	0.279610	−	0.960114i	$-0.409795\pi$
$43$	3.66705	0.559219	0.279610	+	0.960114i	$0.409795\pi$
$44$	0	0
$45$	−1.71911	−0.256270
$46$	0	0
$47$	11.1832	1.63124	0.815622	−	0.578585i	$-0.196395\pi$
$47$	11.1832	1.63124	0.815622	+	0.578585i	$0.196395\pi$
$48$	0	0
$49$	6.78018	0.968597
$50$	0	0
$51$	1.20210	0.168328
$52$	0	0
$53$	−3.20581	−0.440351	−0.220176	−	0.975460i	$-0.570663\pi$
$53$	−3.20581	−0.440351	−0.220176	+	0.975460i	$0.570663\pi$
$54$	0	0
$55$	−3.64192	−0.491076
$56$	0	0
$57$	9.35543	1.23916
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−2.54429	−0.325763	−0.162882	−	0.986646i	$-0.552079\pi$
$61$	−2.54429	−0.325763	−0.162882	+	0.986646i	$0.552079\pi$
$62$	0	0
$63$	−5.77222	−0.727231
$64$	0	0
$65$	−0.178407	−0.0221286
$66$	0	0
$67$	0.497288	0.0607534	0.0303767	−	0.999539i	$-0.490329\pi$
$67$	0.497288	0.0607534	0.0303767	+	0.999539i	$0.490329\pi$
$68$	0	0
$69$	−3.33731	−0.401765
$70$	0	0
$71$	−10.4623	−1.24165	−0.620823	−	0.783951i	$-0.713202\pi$
$71$	−10.4623	−1.24165	−0.620823	+	0.783951i	$0.713202\pi$
$72$	0	0
$73$	−10.6171	−1.24263	−0.621317	−	0.783560i	$-0.713402\pi$
$73$	−10.6171	−1.24263	−0.621317	+	0.783560i	$0.713402\pi$
$74$	0	0
$75$	4.54119	0.524371
$76$	0	0
$77$	−12.2284	−1.39355
$78$	0	0
$79$	−10.6023	−1.19285	−0.596423	−	0.802670i	$-0.703412\pi$
$79$	−10.6023	−1.19285	−0.596423	+	0.802670i	$0.703412\pi$
$80$	0	0
$81$	−1.91730	−0.213034
$82$	0	0
$83$	−4.99436	−0.548203	−0.274101	−	0.961701i	$-0.588380\pi$
$83$	−4.99436	−0.548203	−0.274101	+	0.961701i	$0.588380\pi$
$84$	0	0
$85$	−1.10558	−0.119917
$86$	0	0
$87$	−5.89250	−0.631743
$88$	0	0
$89$	0.726929	0.0770543	0.0385272	−	0.999258i	$-0.487733\pi$
$89$	0.726929	0.0770543	0.0385272	+	0.999258i	$0.487733\pi$
$90$	0	0
$91$	−0.599031	−0.0627956
$92$	0	0
$93$	−3.01953	−0.313111
$94$	0	0
$95$	−8.60420	−0.882772
$96$	0	0
$97$	−17.2029	−1.74669	−0.873346	−	0.487100i	$-0.838055\pi$
$97$	−17.2029	−1.74669	−0.873346	+	0.487100i	$0.838055\pi$
$98$	0	0
$99$	5.12220	0.514801
$100$	0	0
$101$	−13.5056	−1.34386	−0.671930	−	0.740615i	$-0.734534\pi$
$101$	−13.5056	−1.34386	−0.671930	+	0.740615i	$0.734534\pi$
$102$	0	0
$103$	−15.2247	−1.50013	−0.750065	−	0.661364i	$-0.769978\pi$
$103$	−15.2247	−1.50013	−0.750065	+	0.661364i	$0.769978\pi$
$104$	0	0
$105$	−4.93353	−0.481464
$106$	0	0
$107$	−1.37569	−0.132993	−0.0664963	−	0.997787i	$-0.521182\pi$
$107$	−1.37569	−0.132993	−0.0664963	+	0.997787i	$0.521182\pi$
$108$	0	0
$109$	17.5360	1.67964	0.839822	−	0.542862i	$-0.182659\pi$
$109$	17.5360	1.67964	0.839822	+	0.542862i	$0.182659\pi$
$110$	0	0
$111$	−9.58398	−0.909671
$112$	0	0
$113$	−11.2822	−1.06134	−0.530671	−	0.847578i	$-0.678060\pi$
$113$	−11.2822	−1.06134	−0.530671	+	0.847578i	$0.678060\pi$
$114$	0	0
$115$	3.06933	0.286216
$116$	0	0
$117$	0.250921	0.0231977
$118$	0	0
$119$	−3.71217	−0.340294
$120$	0	0
$121$	−0.148686	−0.0135169
$122$	0	0
$123$	2.71330	0.244650
$124$	0	0
$125$	−9.70442	−0.867990
$126$	0	0
$127$	10.7160	0.950890	0.475445	−	0.879745i	$-0.342287\pi$
$127$	10.7160	0.950890	0.475445	+	0.879745i	$0.342287\pi$
$128$	0	0
$129$	−4.40817	−0.388118
$130$	0	0
$131$	17.8084	1.55593	0.777964	−	0.628309i	$-0.216253\pi$
$131$	17.8084	1.55593	0.777964	+	0.628309i	$0.216253\pi$
$132$	0	0
$133$	−28.8901	−2.50509
$134$	0	0
$135$	6.05360	0.521011
$136$	0	0
$137$	−5.32393	−0.454854	−0.227427	−	0.973795i	$-0.573031\pi$
$137$	−5.32393	−0.454854	−0.227427	+	0.973795i	$0.573031\pi$
$138$	0	0
$139$	12.4391	1.05507	0.527535	−	0.849533i	$-0.323116\pi$
$139$	12.4391	1.05507	0.527535	+	0.849533i	$0.323116\pi$
$140$	0	0
$141$	−13.4434	−1.13214
$142$	0	0
$143$	0.531574	0.0444524
$144$	0	0
$145$	5.41934	0.450052
$146$	0	0
$147$	−8.15048	−0.672240
$148$	0	0
$149$	17.3433	1.42082	0.710408	−	0.703790i	$-0.248511\pi$
$149$	17.3433	1.42082	0.710408	+	0.703790i	$0.248511\pi$
$150$	0	0
$151$	13.2206	1.07588	0.537938	−	0.842985i	$-0.319204\pi$
$151$	13.2206	1.07588	0.537938	+	0.842985i	$0.319204\pi$
$152$	0	0
$153$	1.55495	0.125710
$154$	0	0
$155$	2.77706	0.223059
$156$	0	0
$157$	21.5243	1.71783	0.858913	−	0.512121i	$-0.171140\pi$
$157$	21.5243	1.71783	0.858913	+	0.512121i	$0.171140\pi$
$158$	0	0
$159$	3.85371	0.305619
$160$	0	0
$161$	10.3058	0.812211
$162$	0	0
$163$	−11.8901	−0.931305	−0.465653	−	0.884968i	$-0.654180\pi$
$163$	−11.8901	−0.931305	−0.465653	+	0.884968i	$0.654180\pi$
$164$	0	0
$165$	4.37796	0.340824
$166$	0	0
$167$	−24.8496	−1.92292	−0.961461	−	0.274942i	$-0.911341\pi$
$167$	−24.8496	−1.92292	−0.961461	+	0.274942i	$0.911341\pi$
$168$	0	0
$169$	−12.9740	−0.997997
$170$	0	0
$171$	12.1014	0.925420
$172$	0	0
$173$	−21.8347	−1.66006	−0.830031	−	0.557718i	$-0.811677\pi$
$173$	−21.8347	−1.66006	−0.830031	+	0.557718i	$0.811677\pi$
$174$	0	0
$175$	−14.0235	−1.06007
$176$	0	0
$177$	−1.20210	−0.0903557
$178$	0	0
$179$	6.51531	0.486977	0.243488	−	0.969904i	$-0.421708\pi$
$179$	6.51531	0.486977	0.243488	+	0.969904i	$0.421708\pi$
$180$	0	0
$181$	3.53401	0.262681	0.131341	−	0.991337i	$-0.458072\pi$
$181$	3.53401	0.262681	0.131341	+	0.991337i	$0.458072\pi$
$182$	0	0
$183$	3.05850	0.226091
$184$	0	0
$185$	8.81439	0.648047
$186$	0	0
$187$	3.29413	0.240891
$188$	0	0
$189$	20.3260	1.47850
$190$	0	0
$191$	−15.6706	−1.13388	−0.566942	−	0.823758i	$-0.691874\pi$
$191$	−15.6706	−1.13388	−0.566942	+	0.823758i	$0.691874\pi$
$192$	0	0
$193$	16.6881	1.20124	0.600618	−	0.799536i	$-0.294921\pi$
$193$	16.6881	1.20124	0.600618	+	0.799536i	$0.294921\pi$
$194$	0	0
$195$	0.214463	0.0153580
$196$	0	0
$197$	−15.2490	−1.08644	−0.543222	−	0.839589i	$-0.682796\pi$
$197$	−15.2490	−1.08644	−0.543222	+	0.839589i	$0.682796\pi$
$198$	0	0
$199$	−17.1575	−1.21627	−0.608133	−	0.793835i	$-0.708081\pi$
$199$	−17.1575	−1.21627	−0.608133	+	0.793835i	$0.708081\pi$
$200$	0	0
$201$	−0.597792	−0.0421650
$202$	0	0
$203$	18.1964	1.27714
$204$	0	0
$205$	−2.49542	−0.174288
$206$	0	0
$207$	−4.31688	−0.300044
$208$	0	0
$209$	25.6367	1.77333
$210$	0	0
$211$	−2.27945	−0.156924	−0.0784620	−	0.996917i	$-0.525001\pi$
$211$	−2.27945	−0.156924	−0.0784620	+	0.996917i	$0.525001\pi$
$212$	0	0
$213$	12.5768	0.861746
$214$	0	0
$215$	4.05420	0.276494
$216$	0	0
$217$	9.32448	0.632987
$218$	0	0
$219$	12.7628	0.862431
$220$	0	0
$221$	0.161370	0.0108549
$222$	0	0
$223$	7.65256	0.512454	0.256227	−	0.966617i	$-0.417521\pi$
$223$	7.65256	0.512454	0.256227	+	0.966617i	$0.417521\pi$
$224$	0	0
$225$	5.87412	0.391608
$226$	0	0
$227$	19.5032	1.29447	0.647235	−	0.762291i	$-0.275925\pi$
$227$	19.5032	1.29447	0.647235	+	0.762291i	$0.275925\pi$
$228$	0	0
$229$	−27.4954	−1.81695	−0.908473	−	0.417944i	$-0.862751\pi$
$229$	−27.4954	−1.81695	−0.908473	+	0.417944i	$0.862751\pi$
$230$	0	0
$231$	14.6998	0.967174
$232$	0	0
$233$	−28.6941	−1.87981	−0.939906	−	0.341432i	$-0.889088\pi$
$233$	−28.6941	−1.87981	−0.939906	+	0.341432i	$0.889088\pi$
$234$	0	0
$235$	12.3639	0.806534
$236$	0	0
$237$	12.7450	0.827877
$238$	0	0
$239$	3.38865	0.219193	0.109597	−	0.993976i	$-0.465044\pi$
$239$	3.38865	0.219193	0.109597	+	0.993976i	$0.465044\pi$
$240$	0	0
$241$	−10.3759	−0.668369	−0.334184	−	0.942508i	$-0.608461\pi$
$241$	−10.3759	−0.668369	−0.334184	+	0.942508i	$0.608461\pi$
$242$	0	0
$243$	−14.1218	−0.905911
$244$	0	0
$245$	7.49600	0.478902
$246$	0	0
$247$	1.25587	0.0799090
$248$	0	0
$249$	6.00374	0.380472
$250$	0	0
$251$	−6.78765	−0.428433	−0.214216	−	0.976786i	$-0.568720\pi$
$251$	−6.78765	−0.428433	−0.214216	+	0.976786i	$0.568720\pi$
$252$	0	0
$253$	−9.14524	−0.574957
$254$	0	0
$255$	1.32902	0.0832263
$256$	0	0
$257$	5.31628	0.331620	0.165810	−	0.986158i	$-0.446976\pi$
$257$	5.31628	0.331620	0.165810	+	0.986158i	$0.446976\pi$
$258$	0	0
$259$	29.5959	1.83900
$260$	0	0
$261$	−7.62208	−0.471795
$262$	0	0
$263$	11.0560	0.681741	0.340870	−	0.940110i	$-0.389278\pi$
$263$	11.0560	0.681741	0.340870	+	0.940110i	$0.389278\pi$
$264$	0	0
$265$	−3.54426	−0.217722
$266$	0	0
$267$	−0.873844	−0.0534784
$268$	0	0
$269$	−20.6132	−1.25681	−0.628405	−	0.777886i	$-0.716292\pi$
$269$	−20.6132	−1.25681	−0.628405	+	0.777886i	$0.716292\pi$
$270$	0	0
$271$	−14.0255	−0.851987	−0.425993	−	0.904726i	$-0.640075\pi$
$271$	−14.0255	−0.851987	−0.425993	+	0.904726i	$0.640075\pi$
$272$	0	0
$273$	0.720098	0.0435823
$274$	0	0
$275$	12.4443	0.750417
$276$	0	0
$277$	−20.8605	−1.25338	−0.626692	−	0.779267i	$-0.715592\pi$
$277$	−20.8605	−1.25338	−0.626692	+	0.779267i	$0.715592\pi$
$278$	0	0
$279$	−3.90582	−0.233836
$280$	0	0
$281$	−4.94786	−0.295165	−0.147582	−	0.989050i	$-0.547149\pi$
$281$	−4.94786	−0.295165	−0.147582	+	0.989050i	$0.547149\pi$
$282$	0	0
$283$	−9.95548	−0.591792	−0.295896	−	0.955220i	$-0.595618\pi$
$283$	−9.95548	−0.591792	−0.295896	+	0.955220i	$0.595618\pi$
$284$	0	0
$285$	10.3431	0.612675
$286$	0	0
$287$	−8.37881	−0.494586
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	20.6797	1.21227
$292$	0	0
$293$	11.3326	0.662057	0.331029	−	0.943621i	$-0.392604\pi$
$293$	11.3326	0.662057	0.331029	+	0.943621i	$0.392604\pi$
$294$	0	0
$295$	1.10558	0.0643691
$296$	0	0
$297$	−18.0371	−1.04662
$298$	0	0
$299$	−0.447998	−0.0259084
$300$	0	0
$301$	13.6127	0.784622
$302$	0	0
$303$	16.2352	0.932686
$304$	0	0
$305$	−2.81291	−0.161067
$306$	0	0
$307$	−14.1173	−0.805715	−0.402858	−	0.915263i	$-0.631983\pi$
$307$	−14.1173	−0.805715	−0.402858	+	0.915263i	$0.631983\pi$
$308$	0	0
$309$	18.3016	1.04114
$310$	0	0
$311$	2.87332	0.162931	0.0814656	−	0.996676i	$-0.474040\pi$
$311$	2.87332	0.162931	0.0814656	+	0.996676i	$0.474040\pi$
$312$	0	0
$313$	−17.7077	−1.00090	−0.500450	−	0.865766i	$-0.666832\pi$
$313$	−17.7077	−1.00090	−0.500450	+	0.865766i	$0.666832\pi$
$314$	0	0
$315$	−6.38163	−0.359564
$316$	0	0
$317$	2.74045	0.153919	0.0769595	−	0.997034i	$-0.475479\pi$
$317$	2.74045	0.153919	0.0769595	+	0.997034i	$0.475479\pi$
$318$	0	0
$319$	−16.1473	−0.904074
$320$	0	0
$321$	1.65372	0.0923015
$322$	0	0
$323$	7.78255	0.433032
$324$	0	0
$325$	0.609607	0.0338149
$326$	0	0
$327$	−21.0801	−1.16573
$328$	0	0
$329$	41.5141	2.28874
$330$	0	0
$331$	12.5624	0.690493	0.345246	−	0.938512i	$-0.387795\pi$
$331$	12.5624	0.690493	0.345246	+	0.938512i	$0.387795\pi$
$332$	0	0
$333$	−12.3971	−0.679355
$334$	0	0
$335$	0.549790	0.0300382
$336$	0	0
$337$	−4.83332	−0.263288	−0.131644	−	0.991297i	$-0.542026\pi$
$337$	−4.83332	−0.263288	−0.131644	+	0.991297i	$0.542026\pi$
$338$	0	0
$339$	13.5624	0.736609
$340$	0	0
$341$	−8.27443	−0.448086
$342$	0	0
$343$	−0.816019	−0.0440609
$344$	0	0
$345$	−3.68965	−0.198644
$346$	0	0
$347$	3.80579	0.204306	0.102153	−	0.994769i	$-0.467427\pi$
$347$	3.80579	0.204306	0.102153	+	0.994769i	$0.467427\pi$
$348$	0	0
$349$	−15.6451	−0.837464	−0.418732	−	0.908110i	$-0.637525\pi$
$349$	−15.6451	−0.837464	−0.418732	+	0.908110i	$0.637525\pi$
$350$	0	0
$351$	−0.883583	−0.0471622
$352$	0	0
$353$	−4.43276	−0.235932	−0.117966	−	0.993018i	$-0.537637\pi$
$353$	−4.43276	−0.235932	−0.117966	+	0.993018i	$0.537637\pi$
$354$	0	0
$355$	−11.5669	−0.613906
$356$	0	0
$357$	4.46241	0.236176
$358$	0	0
$359$	−32.2287	−1.70097	−0.850483	−	0.526003i	$-0.823690\pi$
$359$	−32.2287	−1.70097	−0.850483	+	0.526003i	$0.823690\pi$
$360$	0	0
$361$	41.5680	2.18779
$362$	0	0
$363$	0.178736	0.00938119
$364$	0	0
$365$	−11.7380	−0.614394
$366$	0	0
$367$	12.7311	0.664557	0.332278	−	0.943181i	$-0.392183\pi$
$367$	12.7311	0.664557	0.332278	+	0.943181i	$0.392183\pi$
$368$	0	0
$369$	3.50970	0.182708
$370$	0	0
$371$	−11.9005	−0.617843
$372$	0	0
$373$	−30.2739	−1.56753	−0.783763	−	0.621060i	$-0.786702\pi$
$373$	−30.2739	−1.56753	−0.783763	+	0.621060i	$0.786702\pi$
$374$	0	0
$375$	11.6657	0.602415
$376$	0	0
$377$	−0.791007	−0.0407389
$378$	0	0
$379$	−14.8819	−0.764433	−0.382216	−	0.924073i	$-0.624839\pi$
$379$	−14.8819	−0.764433	−0.382216	+	0.924073i	$0.624839\pi$
$380$	0	0
$381$	−12.8817	−0.659951
$382$	0	0
$383$	14.6536	0.748763	0.374382	−	0.927275i	$-0.377855\pi$
$383$	14.6536	0.748763	0.374382	+	0.927275i	$0.377855\pi$
$384$	0	0
$385$	−13.5194	−0.689012
$386$	0	0
$387$	−5.70206	−0.289852
$388$	0	0
$389$	2.23425	0.113281	0.0566404	−	0.998395i	$-0.481961\pi$
$389$	2.23425	0.113281	0.0566404	+	0.998395i	$0.481961\pi$
$390$	0	0
$391$	−2.77622	−0.140400
$392$	0	0
$393$	−21.4076	−1.07987
$394$	0	0
$395$	−11.7216	−0.589778
$396$	0	0
$397$	−27.0859	−1.35940	−0.679701	−	0.733490i	$-0.737890\pi$
$397$	−27.0859	−1.35940	−0.679701	+	0.733490i	$0.737890\pi$
$398$	0	0
$399$	34.7289	1.73862
$400$	0	0
$401$	−36.2744	−1.81146	−0.905728	−	0.423859i	$-0.860675\pi$
$401$	−36.2744	−1.81146	−0.905728	+	0.423859i	$0.860675\pi$
$402$	0	0
$403$	−0.405340	−0.0201914
$404$	0	0
$405$	−2.11972	−0.105330
$406$	0	0
$407$	−26.2630	−1.30181
$408$	0	0
$409$	−14.6416	−0.723979	−0.361989	−	0.932182i	$-0.617902\pi$
$409$	−14.6416	−0.723979	−0.361989	+	0.932182i	$0.617902\pi$
$410$	0	0
$411$	6.39992	0.315685
$412$	0	0
$413$	3.71217	0.182664
$414$	0	0
$415$	−5.52165	−0.271047
$416$	0	0
$417$	−14.9531	−0.732256
$418$	0	0
$419$	22.1617	1.08267	0.541335	−	0.840807i	$-0.317919\pi$
$419$	22.1617	1.08267	0.541335	+	0.840807i	$0.317919\pi$
$420$	0	0
$421$	34.7889	1.69551	0.847754	−	0.530390i	$-0.177955\pi$
$421$	34.7889	1.69551	0.847754	+	0.530390i	$0.177955\pi$
$422$	0	0
$423$	−17.3893	−0.845499
$424$	0	0
$425$	3.77770	0.183245
$426$	0	0
$427$	−9.44484	−0.457068
$428$	0	0
$429$	−0.639007	−0.0308515
$430$	0	0
$431$	22.4063	1.07927	0.539636	−	0.841898i	$-0.318562\pi$
$431$	22.4063	1.07927	0.539636	+	0.841898i	$0.318562\pi$
$432$	0	0
$433$	−34.2586	−1.64636	−0.823181	−	0.567779i	$-0.807803\pi$
$433$	−34.2586	−1.64636	−0.823181	+	0.567779i	$0.807803\pi$
$434$	0	0
$435$	−6.51461	−0.312352
$436$	0	0
$437$	−21.6061	−1.03356
$438$	0	0
$439$	26.2281	1.25180	0.625900	−	0.779904i	$-0.284732\pi$
$439$	26.2281	1.25180	0.625900	+	0.779904i	$0.284732\pi$
$440$	0	0
$441$	−10.5428	−0.502039
$442$	0	0
$443$	−18.7824	−0.892379	−0.446190	−	0.894938i	$-0.647219\pi$
$443$	−18.7824	−0.892379	−0.446190	+	0.894938i	$0.647219\pi$
$444$	0	0
$445$	0.803676	0.0380979
$446$	0	0
$447$	−20.8484	−0.986096
$448$	0	0
$449$	36.3160	1.71386	0.856930	−	0.515433i	$-0.172369\pi$
$449$	36.3160	1.71386	0.856930	+	0.515433i	$0.172369\pi$
$450$	0	0
$451$	7.43526	0.350113
$452$	0	0
$453$	−15.8925	−0.746695
$454$	0	0
$455$	−0.662275	−0.0310479
$456$	0	0
$457$	−37.6471	−1.76106	−0.880528	−	0.473995i	$-0.842812\pi$
$457$	−37.6471	−1.76106	−0.880528	+	0.473995i	$0.842812\pi$
$458$	0	0
$459$	−5.47552	−0.255575
$460$	0	0
$461$	6.20782	0.289127	0.144563	−	0.989496i	$-0.453822\pi$
$461$	6.20782	0.289127	0.144563	+	0.989496i	$0.453822\pi$
$462$	0	0
$463$	−4.45092	−0.206852	−0.103426	−	0.994637i	$-0.532980\pi$
$463$	−4.45092	−0.206852	−0.103426	+	0.994637i	$0.532980\pi$
$464$	0	0
$465$	−3.33832	−0.154811
$466$	0	0
$467$	12.8305	0.593723	0.296862	−	0.954921i	$-0.404060\pi$
$467$	12.8305	0.593723	0.296862	+	0.954921i	$0.404060\pi$
$468$	0	0
$469$	1.84602	0.0852411
$470$	0	0
$471$	−25.8744	−1.19223
$472$	0	0
$473$	−12.0797	−0.555427
$474$	0	0
$475$	29.4001	1.34897
$476$	0	0
$477$	4.98486	0.228241
$478$	0	0
$479$	25.4369	1.16224	0.581122	−	0.813817i	$-0.302614\pi$
$479$	25.4369	1.16224	0.581122	+	0.813817i	$0.302614\pi$
$480$	0	0
$481$	−1.28655	−0.0586615
$482$	0	0
$483$	−12.3886	−0.563703
$484$	0	0
$485$	−19.0192	−0.863615
$486$	0	0
$487$	−15.7377	−0.713142	−0.356571	−	0.934268i	$-0.616054\pi$
$487$	−15.7377	−0.713142	−0.356571	+	0.934268i	$0.616054\pi$
$488$	0	0
$489$	14.2931	0.646358
$490$	0	0
$491$	33.6274	1.51758	0.758791	−	0.651334i	$-0.225790\pi$
$491$	33.6274	1.51758	0.758791	+	0.651334i	$0.225790\pi$
$492$	0	0
$493$	−4.90183	−0.220767
$494$	0	0
$495$	5.66298	0.254532
$496$	0	0
$497$	−38.8378	−1.74211
$498$	0	0
$499$	−1.71391	−0.0767251	−0.0383626	−	0.999264i	$-0.512214\pi$
$499$	−1.71391	−0.0767251	−0.0383626	+	0.999264i	$0.512214\pi$
$500$	0	0
$501$	29.8718	1.33457
$502$	0	0
$503$	40.7094	1.81514	0.907571	−	0.419898i	$-0.137934\pi$
$503$	40.7094	1.81514	0.907571	+	0.419898i	$0.137934\pi$
$504$	0	0
$505$	−14.9315	−0.664443
$506$	0	0
$507$	15.5960	0.692645
$508$	0	0
$509$	−16.4105	−0.727381	−0.363690	−	0.931520i	$-0.618483\pi$
$509$	−16.4105	−0.727381	−0.363690	+	0.931520i	$0.618483\pi$
$510$	0	0
$511$	−39.4123	−1.74350
$512$	0	0
$513$	−42.6135	−1.88143
$514$	0	0
$515$	−16.8320	−0.741708
$516$	0	0
$517$	−36.8391	−1.62018
$518$	0	0
$519$	26.2476	1.15214
$520$	0	0
$521$	4.76149	0.208605	0.104302	−	0.994546i	$-0.466739\pi$
$521$	4.76149	0.208605	0.104302	+	0.994546i	$0.466739\pi$
$522$	0	0
$523$	28.3186	1.23828	0.619142	−	0.785279i	$-0.287481\pi$
$523$	28.3186	1.23828	0.619142	+	0.785279i	$0.287481\pi$
$524$	0	0
$525$	16.8576	0.735728
$526$	0	0
$527$	−2.51187	−0.109419
$528$	0	0
$529$	−15.2926	−0.664895
$530$	0	0
$531$	−1.55495	−0.0674789
$532$	0	0
$533$	0.364231	0.0157766
$534$	0	0
$535$	−1.52093	−0.0657553
$536$	0	0
$537$	−7.83208	−0.337979
$538$	0	0
$539$	−22.3348	−0.962028
$540$	0	0
$541$	31.1823	1.34063	0.670317	−	0.742075i	$-0.266158\pi$
$541$	31.1823	1.34063	0.670317	+	0.742075i	$0.266158\pi$
$542$	0	0
$543$	−4.24825	−0.182310
$544$	0	0
$545$	19.3874	0.830464
$546$	0	0
$547$	−12.8842	−0.550887	−0.275444	−	0.961317i	$-0.588825\pi$
$547$	−12.8842	−0.550887	−0.275444	+	0.961317i	$0.588825\pi$
$548$	0	0
$549$	3.95624	0.168848
$550$	0	0
$551$	−38.1487	−1.62519
$552$	0	0
$553$	−39.3573	−1.67364
$554$	0	0
$555$	−10.5958	−0.449767
$556$	0	0
$557$	31.4051	1.33068	0.665338	−	0.746542i	$-0.268287\pi$
$557$	31.4051	1.33068	0.665338	+	0.746542i	$0.268287\pi$
$558$	0	0
$559$	−0.591750	−0.0250284
$560$	0	0
$561$	−3.95989	−0.167187
$562$	0	0
$563$	−6.32363	−0.266509	−0.133255	−	0.991082i	$-0.542543\pi$
$563$	−6.32363	−0.266509	−0.133255	+	0.991082i	$0.542543\pi$
$564$	0	0
$565$	−12.4734	−0.524758
$566$	0	0
$567$	−7.11735	−0.298900
$568$	0	0
$569$	30.8472	1.29318	0.646592	−	0.762836i	$-0.276194\pi$
$569$	30.8472	1.29318	0.646592	+	0.762836i	$0.276194\pi$
$570$	0	0
$571$	40.8921	1.71128	0.855640	−	0.517571i	$-0.173164\pi$
$571$	40.8921	1.71128	0.855640	+	0.517571i	$0.173164\pi$
$572$	0	0
$573$	18.8377	0.786955
$574$	0	0
$575$	−10.4877	−0.437369
$576$	0	0
$577$	22.8559	0.951503	0.475752	−	0.879580i	$-0.342176\pi$
$577$	22.8559	0.951503	0.475752	+	0.879580i	$0.342176\pi$
$578$	0	0
$579$	−20.0608	−0.833700
$580$	0	0
$581$	−18.5399	−0.769165
$582$	0	0
$583$	10.5604	0.437365
$584$	0	0
$585$	0.277413	0.0114696
$586$	0	0
$587$	27.0223	1.11533	0.557665	−	0.830066i	$-0.311698\pi$
$587$	27.0223	1.11533	0.557665	+	0.830066i	$0.311698\pi$
$588$	0	0
$589$	−19.5487	−0.805492
$590$	0	0
$591$	18.3308	0.754030
$592$	0	0
$593$	7.30196	0.299856	0.149928	−	0.988697i	$-0.452096\pi$
$593$	7.30196	0.299856	0.149928	+	0.988697i	$0.452096\pi$
$594$	0	0
$595$	−4.10408	−0.168251
$596$	0	0
$597$	20.6251	0.844131
$598$	0	0
$599$	−1.24495	−0.0508673	−0.0254337	−	0.999677i	$-0.508097\pi$
$599$	−1.24495	−0.0508673	−0.0254337	+	0.999677i	$0.508097\pi$
$600$	0	0
$601$	−35.3353	−1.44136	−0.720679	−	0.693268i	$-0.756170\pi$
$601$	−35.3353	−1.44136	−0.720679	+	0.693268i	$0.756170\pi$
$602$	0	0
$603$	−0.773256	−0.0314894
$604$	0	0
$605$	−0.164383	−0.00668314
$606$	0	0
$607$	16.6946	0.677614	0.338807	−	0.940856i	$-0.389977\pi$
$607$	16.6946	0.677614	0.338807	+	0.940856i	$0.389977\pi$
$608$	0	0
$609$	−21.8740	−0.886377
$610$	0	0
$611$	−1.80464	−0.0730078
$612$	0	0
$613$	−19.6488	−0.793609	−0.396804	−	0.917903i	$-0.629881\pi$
$613$	−19.6488	−0.793609	−0.396804	+	0.917903i	$0.629881\pi$
$614$	0	0
$615$	2.99976	0.120962
$616$	0	0
$617$	24.5846	0.989737	0.494869	−	0.868968i	$-0.335216\pi$
$617$	24.5846	0.989737	0.494869	+	0.868968i	$0.335216\pi$
$618$	0	0
$619$	−1.22017	−0.0490426	−0.0245213	−	0.999699i	$-0.507806\pi$
$619$	−1.22017	−0.0490426	−0.0245213	+	0.999699i	$0.507806\pi$
$620$	0	0
$621$	15.2013	0.610005
$622$	0	0
$623$	2.69848	0.108112
$624$	0	0
$625$	8.15953	0.326381
$626$	0	0
$627$	−30.8180	−1.23075
$628$	0	0
$629$	−7.97267	−0.317891
$630$	0	0
$631$	−38.9183	−1.54931	−0.774657	−	0.632382i	$-0.782077\pi$
$631$	−38.9183	−1.54931	−0.774657	+	0.632382i	$0.782077\pi$
$632$	0	0
$633$	2.74014	0.108911
$634$	0	0
$635$	11.8473	0.470148
$636$	0	0
$637$	−1.09412	−0.0433504
$638$	0	0
$639$	16.2683	0.643565
$640$	0	0
$641$	−44.0892	−1.74142	−0.870710	−	0.491797i	$-0.836340\pi$
$641$	−44.0892	−1.74142	−0.870710	+	0.491797i	$0.836340\pi$
$642$	0	0
$643$	33.2599	1.31164	0.655821	−	0.754916i	$-0.272322\pi$
$643$	33.2599	1.31164	0.655821	+	0.754916i	$0.272322\pi$
$644$	0	0
$645$	−4.87357	−0.191897
$646$	0	0
$647$	18.5755	0.730278	0.365139	−	0.930953i	$-0.381021\pi$
$647$	18.5755	0.730278	0.365139	+	0.930953i	$0.381021\pi$
$648$	0	0
$649$	−3.29413	−0.129306
$650$	0	0
$651$	−11.2090	−0.439315
$652$	0	0
$653$	37.9770	1.48616	0.743078	−	0.669205i	$-0.233365\pi$
$653$	37.9770	1.48616	0.743078	+	0.669205i	$0.233365\pi$
$654$	0	0
$655$	19.6886	0.769295
$656$	0	0
$657$	16.5090	0.644076
$658$	0	0
$659$	6.67505	0.260023	0.130012	−	0.991512i	$-0.458499\pi$
$659$	6.67505	0.260023	0.130012	+	0.991512i	$0.458499\pi$
$660$	0	0
$661$	−46.9344	−1.82554	−0.912769	−	0.408477i	$-0.866060\pi$
$661$	−46.9344	−1.82554	−0.912769	+	0.408477i	$0.866060\pi$
$662$	0	0
$663$	−0.193983	−0.00753368
$664$	0	0
$665$	−31.9402	−1.23859
$666$	0	0
$667$	13.6086	0.526925
$668$	0	0
$669$	−9.19918	−0.355661
$670$	0	0
$671$	8.38124	0.323554
$672$	0	0
$673$	27.8357	1.07299	0.536493	−	0.843905i	$-0.319749\pi$
$673$	27.8357	1.07299	0.536493	+	0.843905i	$0.319749\pi$
$674$	0	0
$675$	−20.6849	−0.796161
$676$	0	0
$677$	−40.4900	−1.55616	−0.778078	−	0.628167i	$-0.783805\pi$
$677$	−40.4900	−1.55616	−0.778078	+	0.628167i	$0.783805\pi$
$678$	0	0
$679$	−63.8601	−2.45073
$680$	0	0
$681$	−23.4448	−0.898407
$682$	0	0
$683$	13.2211	0.505891	0.252945	−	0.967481i	$-0.418601\pi$
$683$	13.2211	0.505891	0.252945	+	0.967481i	$0.418601\pi$
$684$	0	0
$685$	−5.88601	−0.224893
$686$	0	0
$687$	33.0523	1.26102
$688$	0	0
$689$	0.517320	0.0197083
$690$	0	0
$691$	−13.2911	−0.505617	−0.252809	−	0.967516i	$-0.581354\pi$
$691$	−13.2911	−0.505617	−0.252809	+	0.967516i	$0.581354\pi$
$692$	0	0
$693$	19.0145	0.722300
$694$	0	0
$695$	13.7524	0.521657
$696$	0	0
$697$	2.25712	0.0854946
$698$	0	0
$699$	34.4933	1.30466
$700$	0	0
$701$	36.4792	1.37780	0.688899	−	0.724857i	$-0.258094\pi$
$701$	36.4792	1.37780	0.688899	+	0.724857i	$0.258094\pi$
$702$	0	0
$703$	−62.0477	−2.34017
$704$	0	0
$705$	−14.8627	−0.559763
$706$	0	0
$707$	−50.1351	−1.88553
$708$	0	0
$709$	−47.7392	−1.79288	−0.896442	−	0.443161i	$-0.853857\pi$
$709$	−47.7392	−1.79288	−0.896442	+	0.443161i	$0.853857\pi$
$710$	0	0
$711$	16.4859	0.618271
$712$	0	0
$713$	6.97351	0.261160
$714$	0	0
$715$	0.587695	0.0219786
$716$	0	0
$717$	−4.07351	−0.152128
$718$	0	0
$719$	−30.6628	−1.14353	−0.571765	−	0.820418i	$-0.693741\pi$
$719$	−30.6628	−1.14353	−0.571765	+	0.820418i	$0.693741\pi$
$720$	0	0
$721$	−56.5165	−2.10478
$722$	0	0
$723$	12.4729	0.463871
$724$	0	0
$725$	−18.5176	−0.687728
$726$	0	0
$727$	17.8632	0.662510	0.331255	−	0.943541i	$-0.392528\pi$
$727$	17.8632	0.662510	0.331255	+	0.943541i	$0.392528\pi$
$728$	0	0
$729$	22.7277	0.841768
$730$	0	0
$731$	−3.66705	−0.135631
$732$	0	0
$733$	−38.5143	−1.42256	−0.711279	−	0.702910i	$-0.751884\pi$
$733$	−38.5143	−1.42256	−0.711279	+	0.702910i	$0.751884\pi$
$734$	0	0
$735$	−9.01098	−0.332375
$736$	0	0
$737$	−1.63813	−0.0603414
$738$	0	0
$739$	−12.5311	−0.460962	−0.230481	−	0.973077i	$-0.574030\pi$
$739$	−12.5311	−0.460962	−0.230481	+	0.973077i	$0.574030\pi$
$740$	0	0
$741$	−1.50968	−0.0554596
$742$	0	0
$743$	−50.1724	−1.84065	−0.920323	−	0.391159i	$-0.872074\pi$
$743$	−50.1724	−1.84065	−0.920323	+	0.391159i	$0.872074\pi$
$744$	0	0
$745$	19.1743	0.702492
$746$	0	0
$747$	7.76597	0.284142
$748$	0	0
$749$	−5.10677	−0.186597
$750$	0	0
$751$	−15.2985	−0.558249	−0.279125	−	0.960255i	$-0.590044\pi$
$751$	−15.2985	−0.558249	−0.279125	+	0.960255i	$0.590044\pi$
$752$	0	0
$753$	8.15946	0.297347
$754$	0	0
$755$	14.6164	0.531944
$756$	0	0
$757$	−40.0134	−1.45431	−0.727156	−	0.686472i	$-0.759158\pi$
$757$	−40.0134	−1.45431	−0.727156	+	0.686472i	$0.759158\pi$
$758$	0	0
$759$	10.9935	0.399040
$760$	0	0
$761$	−44.0675	−1.59744	−0.798722	−	0.601700i	$-0.794490\pi$
$761$	−44.0675	−1.59744	−0.798722	+	0.601700i	$0.794490\pi$
$762$	0	0
$763$	65.0965	2.35665
$764$	0	0
$765$	1.71911	0.0621546
$766$	0	0
$767$	−0.161370	−0.00582672
$768$	0	0
$769$	−13.7571	−0.496094	−0.248047	−	0.968748i	$-0.579789\pi$
$769$	−13.7571	−0.496094	−0.248047	+	0.968748i	$0.579789\pi$
$770$	0	0
$771$	−6.39072	−0.230156
$772$	0	0
$773$	−12.3550	−0.444380	−0.222190	−	0.975003i	$-0.571320\pi$
$773$	−12.3550	−0.444380	−0.222190	+	0.975003i	$0.571320\pi$
$774$	0	0
$775$	−9.48909	−0.340858
$776$	0	0
$777$	−35.5773	−1.27633
$778$	0	0
$779$	17.5662	0.629373
$780$	0	0
$781$	34.4642	1.23323
$782$	0	0
$783$	26.8400	0.959185
$784$	0	0
$785$	23.7968	0.849343
$786$	0	0
$787$	30.7595	1.09646	0.548229	−	0.836328i	$-0.315302\pi$
$787$	30.7595	1.09646	0.548229	+	0.836328i	$0.315302\pi$
$788$	0	0
$789$	−13.2904	−0.473152
$790$	0	0
$791$	−41.8815	−1.48913
$792$	0	0
$793$	0.410572	0.0145798
$794$	0	0
$795$	4.26057	0.151107
$796$	0	0
$797$	4.14428	0.146798	0.0733990	−	0.997303i	$-0.476615\pi$
$797$	4.14428	0.146798	0.0733990	+	0.997303i	$0.476615\pi$
$798$	0	0
$799$	−11.1832	−0.395635
$800$	0	0
$801$	−1.13034	−0.0399385
$802$	0	0
$803$	34.9740	1.23421
$804$	0	0
$805$	11.3938	0.401580
$806$	0	0
$807$	24.7792	0.872270
$808$	0	0
$809$	−11.4323	−0.401939	−0.200969	−	0.979598i	$-0.564409\pi$
$809$	−11.4323	−0.401939	−0.200969	+	0.979598i	$0.564409\pi$
$810$	0	0
$811$	−24.2682	−0.852173	−0.426087	−	0.904682i	$-0.640108\pi$
$811$	−24.2682	−0.852173	−0.426087	+	0.904682i	$0.640108\pi$
$812$	0	0
$813$	16.8601	0.591309
$814$	0	0
$815$	−13.1454	−0.460464
$816$	0	0
$817$	−28.5390	−0.998452
$818$	0	0
$819$	0.931462	0.0325479
$820$	0	0
$821$	37.6916	1.31545	0.657723	−	0.753260i	$-0.271520\pi$
$821$	37.6916	1.31545	0.657723	+	0.753260i	$0.271520\pi$
$822$	0	0
$823$	12.8490	0.447887	0.223944	−	0.974602i	$-0.428107\pi$
$823$	12.8490	0.447887	0.223944	+	0.974602i	$0.428107\pi$
$824$	0	0
$825$	−14.9593	−0.520815
$826$	0	0
$827$	−38.4744	−1.33789	−0.668944	−	0.743313i	$-0.733253\pi$
$827$	−38.4744	−1.33789	−0.668944	+	0.743313i	$0.733253\pi$
$828$	0	0
$829$	−2.53338	−0.0879879	−0.0439939	−	0.999032i	$-0.514008\pi$
$829$	−2.53338	−0.0879879	−0.0439939	+	0.999032i	$0.514008\pi$
$830$	0	0
$831$	25.0765	0.869893
$832$	0	0
$833$	−6.78018	−0.234919
$834$	0	0
$835$	−27.4732	−0.950748
$836$	0	0
$837$	13.7538	0.475401
$838$	0	0
$839$	21.4809	0.741604	0.370802	−	0.928712i	$-0.379083\pi$
$839$	21.4809	0.741604	0.370802	+	0.928712i	$0.379083\pi$
$840$	0	0
$841$	−4.97210	−0.171452
$842$	0	0
$843$	5.94784	0.204855
$844$	0	0
$845$	−14.3437	−0.493438
$846$	0	0
$847$	−0.551946	−0.0189651
$848$	0	0
$849$	11.9675	0.410724
$850$	0	0
$851$	22.1339	0.758740
$852$	0	0
$853$	−32.5523	−1.11457	−0.557285	−	0.830321i	$-0.688157\pi$
$853$	−32.5523	−1.11457	−0.557285	+	0.830321i	$0.688157\pi$
$854$	0	0
$855$	13.3791	0.457555
$856$	0	0
$857$	42.8909	1.46513	0.732563	−	0.680699i	$-0.238324\pi$
$857$	42.8909	1.46513	0.732563	+	0.680699i	$0.238324\pi$
$858$	0	0
$859$	−45.4264	−1.54993	−0.774965	−	0.632005i	$-0.782232\pi$
$859$	−45.4264	−1.54993	−0.774965	+	0.632005i	$0.782232\pi$
$860$	0	0
$861$	10.0722	0.343260
$862$	0	0
$863$	−21.5006	−0.731890	−0.365945	−	0.930636i	$-0.619254\pi$
$863$	−21.5006	−0.731890	−0.365945	+	0.930636i	$0.619254\pi$
$864$	0	0
$865$	−24.1399	−0.820782
$866$	0	0
$867$	−1.20210	−0.0408256
$868$	0	0
$869$	34.9252	1.18476
$870$	0	0
$871$	−0.0802472	−0.00271907
$872$	0	0
$873$	26.7496	0.905338
$874$	0	0
$875$	−36.0244	−1.21785
$876$	0	0
$877$	22.6479	0.764764	0.382382	−	0.924004i	$-0.375104\pi$
$877$	22.6479	0.764764	0.382382	+	0.924004i	$0.375104\pi$
$878$	0	0
$879$	−13.6230	−0.459491
$880$	0	0
$881$	36.1189	1.21688	0.608438	−	0.793601i	$-0.291796\pi$
$881$	36.1189	1.21688	0.608438	+	0.793601i	$0.291796\pi$
$882$	0	0
$883$	−15.5079	−0.521884	−0.260942	−	0.965354i	$-0.584033\pi$
$883$	−15.5079	−0.521884	−0.260942	+	0.965354i	$0.584033\pi$
$884$	0	0
$885$	−1.32902	−0.0446744
$886$	0	0
$887$	−28.5383	−0.958224	−0.479112	−	0.877754i	$-0.659041\pi$
$887$	−28.5383	−0.958224	−0.479112	+	0.877754i	$0.659041\pi$
$888$	0	0
$889$	39.7795	1.33416
$890$	0	0
$891$	6.31585	0.211589
$892$	0	0
$893$	−87.0341	−2.91249
$894$	0	0
$895$	7.20317	0.240775
$896$	0	0
$897$	0.538540	0.0179813
$898$	0	0
$899$	12.3128	0.410653
$900$	0	0
$901$	3.20581	0.106801
$902$	0	0
$903$	−16.3639	−0.544555
$904$	0	0
$905$	3.90712	0.129877
$906$	0	0
$907$	−55.8935	−1.85591	−0.927957	−	0.372688i	$-0.878436\pi$
$907$	−55.8935	−1.85591	−0.927957	+	0.372688i	$0.878436\pi$
$908$	0	0
$909$	21.0005	0.696543
$910$	0	0
$911$	4.34936	0.144101	0.0720504	−	0.997401i	$-0.477046\pi$
$911$	4.34936	0.144101	0.0720504	+	0.997401i	$0.477046\pi$
$912$	0	0
$913$	16.4521	0.544485
$914$	0	0
$915$	3.38141	0.111786
$916$	0	0
$917$	66.1078	2.18307
$918$	0	0
$919$	14.5815	0.481000	0.240500	−	0.970649i	$-0.422689\pi$
$919$	14.5815	0.481000	0.240500	+	0.970649i	$0.422689\pi$
$920$	0	0
$921$	16.9704	0.559195
$922$	0	0
$923$	1.68830	0.0555710
$924$	0	0
$925$	−30.1184	−0.990285
$926$	0	0
$927$	23.6735	0.777541
$928$	0	0
$929$	−29.1182	−0.955337	−0.477669	−	0.878540i	$-0.658518\pi$
$929$	−29.1182	−0.955337	−0.477669	+	0.878540i	$0.658518\pi$
$930$	0	0
$931$	−52.7670	−1.72937
$932$	0	0
$933$	−3.45403	−0.113080
$934$	0	0
$935$	3.64192	0.119103
$936$	0	0
$937$	−12.6765	−0.414124	−0.207062	−	0.978328i	$-0.566390\pi$
$937$	−12.6765	−0.414124	−0.207062	+	0.978328i	$0.566390\pi$
$938$	0	0
$939$	21.2865	0.694659
$940$	0	0
$941$	18.5326	0.604146	0.302073	−	0.953285i	$-0.402321\pi$
$941$	18.5326	0.604146	0.302073	+	0.953285i	$0.402321\pi$
$942$	0	0
$943$	−6.26627	−0.204058
$944$	0	0
$945$	22.4720	0.731014
$946$	0	0
$947$	−14.6376	−0.475660	−0.237830	−	0.971307i	$-0.576436\pi$
$947$	−14.6376	−0.475660	−0.237830	+	0.971307i	$0.576436\pi$
$948$	0	0
$949$	1.71327	0.0556152
$950$	0	0
$951$	−3.29431	−0.106825
$952$	0	0
$953$	26.3667	0.854102	0.427051	−	0.904228i	$-0.359552\pi$
$953$	26.3667	0.854102	0.427051	+	0.904228i	$0.359552\pi$
$954$	0	0
$955$	−17.3250	−0.560625
$956$	0	0
$957$	19.4107	0.627459
$958$	0	0
$959$	−19.7633	−0.638191
$960$	0	0
$961$	−24.6905	−0.796468
$962$	0	0
$963$	2.13912	0.0689321
$964$	0	0
$965$	18.4500	0.593926
$966$	0	0
$967$	−38.3090	−1.23194	−0.615968	−	0.787771i	$-0.711235\pi$
$967$	−38.3090	−1.23194	−0.615968	+	0.787771i	$0.711235\pi$
$968$	0	0
$969$	−9.35543	−0.300540
$970$	0	0
$971$	−12.4704	−0.400194	−0.200097	−	0.979776i	$-0.564126\pi$
$971$	−12.4704	−0.400194	−0.200097	+	0.979776i	$0.564126\pi$
$972$	0	0
$973$	46.1760	1.48033
$974$	0	0
$975$	−0.732811	−0.0234687
$976$	0	0
$977$	18.1255	0.579886	0.289943	−	0.957044i	$-0.406364\pi$
$977$	18.1255	0.579886	0.289943	+	0.957044i	$0.406364\pi$
$978$	0	0
$979$	−2.39460	−0.0765318
$980$	0	0
$981$	−27.2675	−0.870586
$982$	0	0
$983$	−3.08644	−0.0984421	−0.0492211	−	0.998788i	$-0.515674\pi$
$983$	−3.08644	−0.0984421	−0.0492211	+	0.998788i	$0.515674\pi$
$984$	0	0
$985$	−16.8589	−0.537169
$986$	0	0
$987$	−49.9042	−1.58847
$988$	0	0
$989$	10.1805	0.323722
$990$	0	0
$991$	52.1139	1.65545	0.827726	−	0.561132i	$-0.189634\pi$
$991$	52.1139	1.65545	0.827726	+	0.561132i	$0.189634\pi$
$992$	0	0
$993$	−15.1013	−0.479226
$994$	0	0
$995$	−18.9690	−0.601356
$996$	0	0
$997$	−40.0494	−1.26838	−0.634189	−	0.773178i	$-0.718666\pi$
$997$	−40.0494	−1.26838	−0.634189	+	0.773178i	$0.718666\pi$
$998$	0	0
$999$	43.6545	1.38117

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.8	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.8	✓	22	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-1.20210 q^{3} +1.10558 q^{5} +3.71217 q^{7} -1.55495 q^{9} +O(q^{10})\)	\(q-1.20210 q^{3} +1.10558 q^{5} +3.71217 q^{7} -1.55495 q^{9} -3.29413 q^{11} -0.161370 q^{13} -1.32902 q^{15} -1.00000 q^{17} -7.78255 q^{19} -4.46241 q^{21} +2.77622 q^{23} -3.77770 q^{25} +5.47552 q^{27} +4.90183 q^{29} +2.51187 q^{31} +3.95989 q^{33} +4.10408 q^{35} +7.97267 q^{37} +0.193983 q^{39} -2.25712 q^{41} +3.66705 q^{43} -1.71911 q^{45} +11.1832 q^{47} +6.78018 q^{49} +1.20210 q^{51} -3.20581 q^{53} -3.64192 q^{55} +9.35543 q^{57} +1.00000 q^{59} -2.54429 q^{61} -5.77222 q^{63} -0.178407 q^{65} +0.497288 q^{67} -3.33731 q^{69} -10.4623 q^{71} -10.6171 q^{73} +4.54119 q^{75} -12.2284 q^{77} -10.6023 q^{79} -1.91730 q^{81} -4.99436 q^{83} -1.10558 q^{85} -5.89250 q^{87} +0.726929 q^{89} -0.599031 q^{91} -3.01953 q^{93} -8.60420 q^{95} -17.2029 q^{97} +5.12220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 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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