LMFDB - Embedded newform 8024.2.a.y.1.23

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:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
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)

$f(q)$	$=$	$q+3.05634 q^{3} -3.42736 q^{5} +2.67222 q^{7} +6.34124 q^{9} +O(q^{10})$	$q+3.05634 q^{3} -3.42736 q^{5} +2.67222 q^{7} +6.34124 q^{9} +2.12013 q^{11} -1.43177 q^{13} -10.4752 q^{15} -1.00000 q^{17} -4.85580 q^{19} +8.16724 q^{21} -9.15178 q^{23} +6.74681 q^{25} +10.2120 q^{27} -4.91364 q^{29} -5.96779 q^{31} +6.47985 q^{33} -9.15868 q^{35} -3.49138 q^{37} -4.37597 q^{39} +4.45741 q^{41} -9.79683 q^{43} -21.7337 q^{45} -12.3841 q^{47} +0.140783 q^{49} -3.05634 q^{51} -4.08742 q^{53} -7.26645 q^{55} -14.8410 q^{57} -1.00000 q^{59} +8.60453 q^{61} +16.9452 q^{63} +4.90718 q^{65} -10.4779 q^{67} -27.9710 q^{69} +13.5031 q^{71} +7.95741 q^{73} +20.6206 q^{75} +5.66546 q^{77} -8.63574 q^{79} +12.1876 q^{81} +4.37754 q^{83} +3.42736 q^{85} -15.0178 q^{87} +11.9201 q^{89} -3.82600 q^{91} -18.2396 q^{93} +16.6426 q^{95} -12.2732 q^{97} +13.4443 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	3.05634	1.76458	0.882291	−	0.470705i	$-0.156000\pi$
$3$	3.05634	1.76458	0.882291	+	0.470705i	$0.156000\pi$
$4$	0	0
$5$	−3.42736	−1.53276	−0.766381	−	0.642386i	$-0.777945\pi$
$5$	−3.42736	−1.53276	−0.766381	+	0.642386i	$0.777945\pi$
$6$	0	0
$7$	2.67222	1.01001	0.505003	−	0.863118i	$-0.331491\pi$
$7$	2.67222	1.01001	0.505003	+	0.863118i	$0.331491\pi$
$8$	0	0
$9$	6.34124	2.11375
$10$	0	0
$11$	2.12013	0.639243	0.319622	−	0.947545i	$-0.396444\pi$
$11$	2.12013	0.639243	0.319622	+	0.947545i	$0.396444\pi$
$12$	0	0
$13$	−1.43177	−0.397100	−0.198550	−	0.980091i	$-0.563623\pi$
$13$	−1.43177	−0.397100	−0.198550	+	0.980091i	$0.563623\pi$
$14$	0	0
$15$	−10.4752	−2.70468
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.85580	−1.11400	−0.556999	−	0.830513i	$-0.688047\pi$
$19$	−4.85580	−1.11400	−0.556999	+	0.830513i	$0.688047\pi$
$20$	0	0
$21$	8.16724	1.78224
$22$	0	0
$23$	−9.15178	−1.90828	−0.954139	−	0.299364i	$-0.903225\pi$
$23$	−9.15178	−1.90828	−0.954139	+	0.299364i	$0.903225\pi$
$24$	0	0
$25$	6.74681	1.34936
$26$	0	0
$27$	10.2120	1.96530
$28$	0	0
$29$	−4.91364	−0.912441	−0.456220	−	0.889867i	$-0.650797\pi$
$29$	−4.91364	−0.912441	−0.456220	+	0.889867i	$0.650797\pi$
$30$	0	0
$31$	−5.96779	−1.07185	−0.535923	−	0.844267i	$-0.680036\pi$
$31$	−5.96779	−1.07185	−0.535923	+	0.844267i	$0.680036\pi$
$32$	0	0
$33$	6.47985	1.12800
$34$	0	0
$35$	−9.15868	−1.54810
$36$	0	0
$37$	−3.49138	−0.573980	−0.286990	−	0.957934i	$-0.592655\pi$
$37$	−3.49138	−0.573980	−0.286990	+	0.957934i	$0.592655\pi$
$38$	0	0
$39$	−4.37597	−0.700716
$40$	0	0
$41$	4.45741	0.696130	0.348065	−	0.937470i	$-0.386839\pi$
$41$	4.45741	0.696130	0.348065	+	0.937470i	$0.386839\pi$
$42$	0	0
$43$	−9.79683	−1.49400	−0.747002	−	0.664822i	$-0.768507\pi$
$43$	−9.79683	−1.49400	−0.747002	+	0.664822i	$0.768507\pi$
$44$	0	0
$45$	−21.7337	−3.23987
$46$	0	0
$47$	−12.3841	−1.80640	−0.903202	−	0.429215i	$-0.858790\pi$
$47$	−12.3841	−1.80640	−0.903202	+	0.429215i	$0.858790\pi$
$48$	0	0
$49$	0.140783	0.0201118
$50$	0	0
$51$	−3.05634	−0.427974
$52$	0	0
$53$	−4.08742	−0.561450	−0.280725	−	0.959788i	$-0.590575\pi$
$53$	−4.08742	−0.561450	−0.280725	+	0.959788i	$0.590575\pi$
$54$	0	0
$55$	−7.26645	−0.979808
$56$	0	0
$57$	−14.8410	−1.96574
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	8.60453	1.10170	0.550849	−	0.834605i	$-0.314304\pi$
$61$	8.60453	1.10170	0.550849	+	0.834605i	$0.314304\pi$
$62$	0	0
$63$	16.9452	2.13490
$64$	0	0
$65$	4.90718	0.608660
$66$	0	0
$67$	−10.4779	−1.28008	−0.640041	−	0.768341i	$-0.721083\pi$
$67$	−10.4779	−1.28008	−0.640041	+	0.768341i	$0.721083\pi$
$68$	0	0
$69$	−27.9710	−3.36731
$70$	0	0
$71$	13.5031	1.60252	0.801259	−	0.598318i	$-0.204164\pi$
$71$	13.5031	1.60252	0.801259	+	0.598318i	$0.204164\pi$
$72$	0	0
$73$	7.95741	0.931344	0.465672	−	0.884957i	$-0.345813\pi$
$73$	7.95741	0.931344	0.465672	+	0.884957i	$0.345813\pi$
$74$	0	0
$75$	20.6206	2.38106
$76$	0	0
$77$	5.66546	0.645640
$78$	0	0
$79$	−8.63574	−0.971596	−0.485798	−	0.874071i	$-0.661471\pi$
$79$	−8.63574	−0.971596	−0.485798	+	0.874071i	$0.661471\pi$
$80$	0	0
$81$	12.1876	1.35418
$82$	0	0
$83$	4.37754	0.480497	0.240249	−	0.970711i	$-0.422771\pi$
$83$	4.37754	0.480497	0.240249	+	0.970711i	$0.422771\pi$
$84$	0	0
$85$	3.42736	0.371750
$86$	0	0
$87$	−15.0178	−1.61008
$88$	0	0
$89$	11.9201	1.26352	0.631762	−	0.775162i	$-0.282332\pi$
$89$	11.9201	1.26352	0.631762	+	0.775162i	$0.282332\pi$
$90$	0	0
$91$	−3.82600	−0.401074
$92$	0	0
$93$	−18.2396	−1.89136
$94$	0	0
$95$	16.6426	1.70749
$96$	0	0
$97$	−12.2732	−1.24616	−0.623078	−	0.782160i	$-0.714118\pi$
$97$	−12.2732	−1.24616	−0.623078	+	0.782160i	$0.714118\pi$
$98$	0	0
$99$	13.4443	1.35120
$100$	0	0
$101$	7.23074	0.719486	0.359743	−	0.933052i	$-0.382864\pi$
$101$	7.23074	0.719486	0.359743	+	0.933052i	$0.382864\pi$
$102$	0	0
$103$	12.8103	1.26224	0.631120	−	0.775685i	$-0.282595\pi$
$103$	12.8103	1.26224	0.631120	+	0.775685i	$0.282595\pi$
$104$	0	0
$105$	−27.9921	−2.73175
$106$	0	0
$107$	−5.27002	−0.509472	−0.254736	−	0.967011i	$-0.581989\pi$
$107$	−5.27002	−0.509472	−0.254736	+	0.967011i	$0.581989\pi$
$108$	0	0
$109$	−10.7525	−1.02991	−0.514953	−	0.857219i	$-0.672191\pi$
$109$	−10.7525	−1.02991	−0.514953	+	0.857219i	$0.672191\pi$
$110$	0	0
$111$	−10.6709	−1.01283
$112$	0	0
$113$	16.2093	1.52484	0.762422	−	0.647080i	$-0.224010\pi$
$113$	16.2093	1.52484	0.762422	+	0.647080i	$0.224010\pi$
$114$	0	0
$115$	31.3665	2.92494
$116$	0	0
$117$	−9.07917	−0.839370
$118$	0	0
$119$	−2.67222	−0.244962
$120$	0	0
$121$	−6.50505	−0.591368
$122$	0	0
$123$	13.6234	1.22838
$124$	0	0
$125$	−5.98693	−0.535488
$126$	0	0
$127$	−5.02187	−0.445619	−0.222810	−	0.974862i	$-0.571523\pi$
$127$	−5.02187	−0.445619	−0.222810	+	0.974862i	$0.571523\pi$
$128$	0	0
$129$	−29.9425	−2.63629
$130$	0	0
$131$	−7.31728	−0.639314	−0.319657	−	0.947533i	$-0.603568\pi$
$131$	−7.31728	−0.639314	−0.319657	+	0.947533i	$0.603568\pi$
$132$	0	0
$133$	−12.9758	−1.12514
$134$	0	0
$135$	−35.0002	−3.01234
$136$	0	0
$137$	14.0495	1.20033	0.600164	−	0.799877i	$-0.295102\pi$
$137$	14.0495	1.20033	0.600164	+	0.799877i	$0.295102\pi$
$138$	0	0
$139$	−12.5737	−1.06649	−0.533245	−	0.845961i	$-0.679028\pi$
$139$	−12.5737	−1.06649	−0.533245	+	0.845961i	$0.679028\pi$
$140$	0	0
$141$	−37.8500	−3.18755
$142$	0	0
$143$	−3.03553	−0.253844
$144$	0	0
$145$	16.8408	1.39856
$146$	0	0
$147$	0.430281	0.0354890
$148$	0	0
$149$	13.6977	1.12216	0.561081	−	0.827761i	$-0.310386\pi$
$149$	13.6977	1.12216	0.561081	+	0.827761i	$0.310386\pi$
$150$	0	0
$151$	9.10468	0.740928	0.370464	−	0.928847i	$-0.379199\pi$
$151$	9.10468	0.740928	0.370464	+	0.928847i	$0.379199\pi$
$152$	0	0
$153$	−6.34124	−0.512659
$154$	0	0
$155$	20.4538	1.64289
$156$	0	0
$157$	−4.01984	−0.320818	−0.160409	−	0.987051i	$-0.551281\pi$
$157$	−4.01984	−0.320818	−0.160409	+	0.987051i	$0.551281\pi$
$158$	0	0
$159$	−12.4926	−0.990725
$160$	0	0
$161$	−24.4556	−1.92737
$162$	0	0
$163$	3.89955	0.305436	0.152718	−	0.988270i	$-0.451197\pi$
$163$	3.89955	0.305436	0.152718	+	0.988270i	$0.451197\pi$
$164$	0	0
$165$	−22.2088	−1.72895
$166$	0	0
$167$	−7.28249	−0.563536	−0.281768	−	0.959483i	$-0.590921\pi$
$167$	−7.28249	−0.563536	−0.281768	+	0.959483i	$0.590921\pi$
$168$	0	0
$169$	−10.9500	−0.842311
$170$	0	0
$171$	−30.7918	−2.35471
$172$	0	0
$173$	1.18141	0.0898211	0.0449106	−	0.998991i	$-0.485700\pi$
$173$	1.18141	0.0898211	0.0449106	+	0.998991i	$0.485700\pi$
$174$	0	0
$175$	18.0290	1.36286
$176$	0	0
$177$	−3.05634	−0.229729
$178$	0	0
$179$	20.8765	1.56038	0.780191	−	0.625541i	$-0.215122\pi$
$179$	20.8765	1.56038	0.780191	+	0.625541i	$0.215122\pi$
$180$	0	0
$181$	−22.5721	−1.67777	−0.838884	−	0.544310i	$-0.816792\pi$
$181$	−22.5721	−1.67777	−0.838884	+	0.544310i	$0.816792\pi$
$182$	0	0
$183$	26.2984	1.94403
$184$	0	0
$185$	11.9662	0.879775
$186$	0	0
$187$	−2.12013	−0.155039
$188$	0	0
$189$	27.2887	1.98496
$190$	0	0
$191$	7.68484	0.556055	0.278028	−	0.960573i	$-0.410319\pi$
$191$	7.68484	0.556055	0.278028	+	0.960573i	$0.410319\pi$
$192$	0	0
$193$	−7.63402	−0.549509	−0.274755	−	0.961514i	$-0.588597\pi$
$193$	−7.63402	−0.549509	−0.274755	+	0.961514i	$0.588597\pi$
$194$	0	0
$195$	14.9980	1.07403
$196$	0	0
$197$	0.632876	0.0450906	0.0225453	−	0.999746i	$-0.492823\pi$
$197$	0.632876	0.0450906	0.0225453	+	0.999746i	$0.492823\pi$
$198$	0	0
$199$	5.13741	0.364181	0.182091	−	0.983282i	$-0.441714\pi$
$199$	5.13741	0.364181	0.182091	+	0.983282i	$0.441714\pi$
$200$	0	0
$201$	−32.0241	−2.25881
$202$	0	0
$203$	−13.1304	−0.921571
$204$	0	0
$205$	−15.2771	−1.06700
$206$	0	0
$207$	−58.0337	−4.03362
$208$	0	0
$209$	−10.2949	−0.712116
$210$	0	0
$211$	10.8495	0.746911	0.373455	−	0.927648i	$-0.378173\pi$
$211$	10.8495	0.746911	0.373455	+	0.927648i	$0.378173\pi$
$212$	0	0
$213$	41.2700	2.82777
$214$	0	0
$215$	33.5773	2.28995
$216$	0	0
$217$	−15.9473	−1.08257
$218$	0	0
$219$	24.3206	1.64343
$220$	0	0
$221$	1.43177	0.0963110
$222$	0	0
$223$	−21.5878	−1.44562	−0.722812	−	0.691045i	$-0.757151\pi$
$223$	−21.5878	−1.44562	−0.722812	+	0.691045i	$0.757151\pi$
$224$	0	0
$225$	42.7831	2.85221
$226$	0	0
$227$	−13.2084	−0.876670	−0.438335	−	0.898812i	$-0.644432\pi$
$227$	−13.2084	−0.876670	−0.438335	+	0.898812i	$0.644432\pi$
$228$	0	0
$229$	−1.51661	−0.100221	−0.0501103	−	0.998744i	$-0.515957\pi$
$229$	−1.51661	−0.100221	−0.0501103	+	0.998744i	$0.515957\pi$
$230$	0	0
$231$	17.3156	1.13928
$232$	0	0
$233$	25.2134	1.65178	0.825892	−	0.563828i	$-0.190672\pi$
$233$	25.2134	1.65178	0.825892	+	0.563828i	$0.190672\pi$
$234$	0	0
$235$	42.4447	2.76879
$236$	0	0
$237$	−26.3938	−1.71446
$238$	0	0
$239$	−13.7330	−0.888315	−0.444157	−	0.895949i	$-0.646497\pi$
$239$	−13.7330	−0.888315	−0.444157	+	0.895949i	$0.646497\pi$
$240$	0	0
$241$	−13.7040	−0.882750	−0.441375	−	0.897323i	$-0.645509\pi$
$241$	−13.7040	−0.882750	−0.441375	+	0.897323i	$0.645509\pi$
$242$	0	0
$243$	6.61365	0.424266
$244$	0	0
$245$	−0.482514	−0.0308267
$246$	0	0
$247$	6.95237	0.442369
$248$	0	0
$249$	13.3793	0.847876
$250$	0	0
$251$	14.0636	0.887689	0.443845	−	0.896104i	$-0.353614\pi$
$251$	14.0636	0.887689	0.443845	+	0.896104i	$0.353614\pi$
$252$	0	0
$253$	−19.4030	−1.21985
$254$	0	0
$255$	10.4752	0.655982
$256$	0	0
$257$	25.9337	1.61770	0.808850	−	0.588015i	$-0.200090\pi$
$257$	25.9337	1.61770	0.808850	+	0.588015i	$0.200090\pi$
$258$	0	0
$259$	−9.32976	−0.579723
$260$	0	0
$261$	−31.1586	−1.92867
$262$	0	0
$263$	−8.15225	−0.502689	−0.251345	−	0.967898i	$-0.580873\pi$
$263$	−8.15225	−0.502689	−0.251345	+	0.967898i	$0.580873\pi$
$264$	0	0
$265$	14.0091	0.860570
$266$	0	0
$267$	36.4318	2.22959
$268$	0	0
$269$	4.28312	0.261147	0.130573	−	0.991439i	$-0.458318\pi$
$269$	4.28312	0.261147	0.130573	+	0.991439i	$0.458318\pi$
$270$	0	0
$271$	−0.652064	−0.0396100	−0.0198050	−	0.999804i	$-0.506305\pi$
$271$	−0.652064	−0.0396100	−0.0198050	+	0.999804i	$0.506305\pi$
$272$	0	0
$273$	−11.6936	−0.707727
$274$	0	0
$275$	14.3041	0.862570
$276$	0	0
$277$	4.10989	0.246939	0.123470	−	0.992348i	$-0.460598\pi$
$277$	4.10989	0.246939	0.123470	+	0.992348i	$0.460598\pi$
$278$	0	0
$279$	−37.8432	−2.26561
$280$	0	0
$281$	12.3848	0.738813	0.369406	−	0.929268i	$-0.379561\pi$
$281$	12.3848	0.738813	0.369406	+	0.929268i	$0.379561\pi$
$282$	0	0
$283$	31.0253	1.84427	0.922133	−	0.386874i	$-0.126445\pi$
$283$	31.0253	1.84427	0.922133	+	0.386874i	$0.126445\pi$
$284$	0	0
$285$	50.8655	3.01301
$286$	0	0
$287$	11.9112	0.703095
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−37.5112	−2.19894
$292$	0	0
$293$	16.4101	0.958689	0.479345	−	0.877627i	$-0.340874\pi$
$293$	16.4101	0.958689	0.479345	+	0.877627i	$0.340874\pi$
$294$	0	0
$295$	3.42736	0.199549
$296$	0	0
$297$	21.6508	1.25630
$298$	0	0
$299$	13.1032	0.757778
$300$	0	0
$301$	−26.1793	−1.50895
$302$	0	0
$303$	22.0996	1.26959
$304$	0	0
$305$	−29.4908	−1.68864
$306$	0	0
$307$	9.50337	0.542386	0.271193	−	0.962525i	$-0.412582\pi$
$307$	9.50337	0.542386	0.271193	+	0.962525i	$0.412582\pi$
$308$	0	0
$309$	39.1528	2.22733
$310$	0	0
$311$	−16.5502	−0.938477	−0.469239	−	0.883071i	$-0.655472\pi$
$311$	−16.5502	−0.938477	−0.469239	+	0.883071i	$0.655472\pi$
$312$	0	0
$313$	−0.474985	−0.0268477	−0.0134239	−	0.999910i	$-0.504273\pi$
$313$	−0.474985	−0.0268477	−0.0134239	+	0.999910i	$0.504273\pi$
$314$	0	0
$315$	−58.0774	−3.27229
$316$	0	0
$317$	−11.4552	−0.643389	−0.321695	−	0.946843i	$-0.604252\pi$
$317$	−11.4552	−0.643389	−0.321695	+	0.946843i	$0.604252\pi$
$318$	0	0
$319$	−10.4176	−0.583272
$320$	0	0
$321$	−16.1070	−0.899006
$322$	0	0
$323$	4.85580	0.270184
$324$	0	0
$325$	−9.65984	−0.535832
$326$	0	0
$327$	−32.8634	−1.81735
$328$	0	0
$329$	−33.0931	−1.82448
$330$	0	0
$331$	−29.6626	−1.63041	−0.815203	−	0.579175i	$-0.803375\pi$
$331$	−29.6626	−1.63041	−0.815203	+	0.579175i	$0.803375\pi$
$332$	0	0
$333$	−22.1397	−1.21325
$334$	0	0
$335$	35.9116	1.96206
$336$	0	0
$337$	−23.7622	−1.29441	−0.647205	−	0.762316i	$-0.724062\pi$
$337$	−23.7622	−1.29441	−0.647205	+	0.762316i	$0.724062\pi$
$338$	0	0
$339$	49.5413	2.69071
$340$	0	0
$341$	−12.6525	−0.685171
$342$	0	0
$343$	−18.3294	−0.989693
$344$	0	0
$345$	95.8667	5.16129
$346$	0	0
$347$	3.59711	0.193103	0.0965515	−	0.995328i	$-0.469219\pi$
$347$	3.59711	0.193103	0.0965515	+	0.995328i	$0.469219\pi$
$348$	0	0
$349$	−2.19801	−0.117657	−0.0588284	−	0.998268i	$-0.518736\pi$
$349$	−2.19801	−0.117657	−0.0588284	+	0.998268i	$0.518736\pi$
$350$	0	0
$351$	−14.6212	−0.780421
$352$	0	0
$353$	−30.8128	−1.64000	−0.820001	−	0.572362i	$-0.806027\pi$
$353$	−30.8128	−1.64000	−0.820001	+	0.572362i	$0.806027\pi$
$354$	0	0
$355$	−46.2799	−2.45628
$356$	0	0
$357$	−8.16724	−0.432256
$358$	0	0
$359$	−3.51182	−0.185347	−0.0926735	−	0.995697i	$-0.529541\pi$
$359$	−3.51182	−0.185347	−0.0926735	+	0.995697i	$0.529541\pi$
$360$	0	0
$361$	4.57884	0.240991
$362$	0	0
$363$	−19.8817	−1.04352
$364$	0	0
$365$	−27.2729	−1.42753
$366$	0	0
$367$	33.9954	1.77455	0.887274	−	0.461243i	$-0.152596\pi$
$367$	33.9954	1.77455	0.887274	+	0.461243i	$0.152596\pi$
$368$	0	0
$369$	28.2655	1.47144
$370$	0	0
$371$	−10.9225	−0.567068
$372$	0	0
$373$	18.7726	0.972008	0.486004	−	0.873957i	$-0.338454\pi$
$373$	18.7726	0.972008	0.486004	+	0.873957i	$0.338454\pi$
$374$	0	0
$375$	−18.2981	−0.944911
$376$	0	0
$377$	7.03519	0.362330
$378$	0	0
$379$	−11.3650	−0.583781	−0.291891	−	0.956452i	$-0.594284\pi$
$379$	−11.3650	−0.583781	−0.291891	+	0.956452i	$0.594284\pi$
$380$	0	0
$381$	−15.3486	−0.786331
$382$	0	0
$383$	−13.8039	−0.705344	−0.352672	−	0.935747i	$-0.614727\pi$
$383$	−13.8039	−0.705344	−0.352672	+	0.935747i	$0.614727\pi$
$384$	0	0
$385$	−19.4176	−0.989612
$386$	0	0
$387$	−62.1241	−3.15795
$388$	0	0
$389$	−32.0553	−1.62527	−0.812635	−	0.582773i	$-0.801968\pi$
$389$	−32.0553	−1.62527	−0.812635	+	0.582773i	$0.801968\pi$
$390$	0	0
$391$	9.15178	0.462825
$392$	0	0
$393$	−22.3641	−1.12812
$394$	0	0
$395$	29.5978	1.48923
$396$	0	0
$397$	6.28497	0.315434	0.157717	−	0.987484i	$-0.449587\pi$
$397$	6.28497	0.315434	0.157717	+	0.987484i	$0.449587\pi$
$398$	0	0
$399$	−39.6585	−1.98541
$400$	0	0
$401$	−3.27578	−0.163585	−0.0817924	−	0.996649i	$-0.526064\pi$
$401$	−3.27578	−0.163585	−0.0817924	+	0.996649i	$0.526064\pi$
$402$	0	0
$403$	8.54447	0.425630
$404$	0	0
$405$	−41.7714	−2.07564
$406$	0	0
$407$	−7.40219	−0.366913
$408$	0	0
$409$	25.5821	1.26495	0.632477	−	0.774579i	$-0.282038\pi$
$409$	25.5821	1.26495	0.632477	+	0.774579i	$0.282038\pi$
$410$	0	0
$411$	42.9401	2.11808
$412$	0	0
$413$	−2.67222	−0.131492
$414$	0	0
$415$	−15.0034	−0.736488
$416$	0	0
$417$	−38.4297	−1.88191
$418$	0	0
$419$	−34.7805	−1.69914	−0.849570	−	0.527476i	$-0.823138\pi$
$419$	−34.7805	−1.69914	−0.849570	+	0.527476i	$0.823138\pi$
$420$	0	0
$421$	19.2835	0.939820	0.469910	−	0.882714i	$-0.344286\pi$
$421$	19.2835	0.939820	0.469910	+	0.882714i	$0.344286\pi$
$422$	0	0
$423$	−78.5305	−3.81828
$424$	0	0
$425$	−6.74681	−0.327268
$426$	0	0
$427$	22.9932	1.11272
$428$	0	0
$429$	−9.27762	−0.447928
$430$	0	0
$431$	30.9757	1.49205	0.746023	−	0.665921i	$-0.231961\pi$
$431$	30.9757	1.49205	0.746023	+	0.665921i	$0.231961\pi$
$432$	0	0
$433$	24.7594	1.18986	0.594930	−	0.803777i	$-0.297180\pi$
$433$	24.7594	1.18986	0.594930	+	0.803777i	$0.297180\pi$
$434$	0	0
$435$	51.4714	2.46786
$436$	0	0
$437$	44.4392	2.12582
$438$	0	0
$439$	−20.0243	−0.955708	−0.477854	−	0.878439i	$-0.658585\pi$
$439$	−20.0243	−0.955708	−0.477854	+	0.878439i	$0.658585\pi$
$440$	0	0
$441$	0.892738	0.0425113
$442$	0	0
$443$	16.3793	0.778205	0.389102	−	0.921194i	$-0.372785\pi$
$443$	16.3793	0.778205	0.389102	+	0.921194i	$0.372785\pi$
$444$	0	0
$445$	−40.8544	−1.93668
$446$	0	0
$447$	41.8650	1.98015
$448$	0	0
$449$	−30.7403	−1.45073	−0.725363	−	0.688366i	$-0.758328\pi$
$449$	−30.7403	−1.45073	−0.725363	+	0.688366i	$0.758328\pi$
$450$	0	0
$451$	9.45028	0.444996
$452$	0	0
$453$	27.8270	1.30743
$454$	0	0
$455$	13.1131	0.614751
$456$	0	0
$457$	−18.8571	−0.882098	−0.441049	−	0.897483i	$-0.645394\pi$
$457$	−18.8571	−0.882098	−0.441049	+	0.897483i	$0.645394\pi$
$458$	0	0
$459$	−10.2120	−0.476655
$460$	0	0
$461$	−28.8941	−1.34573	−0.672865	−	0.739765i	$-0.734937\pi$
$461$	−28.8941	−1.34573	−0.672865	+	0.739765i	$0.734937\pi$
$462$	0	0
$463$	−11.0215	−0.512212	−0.256106	−	0.966649i	$-0.582440\pi$
$463$	−11.0215	−0.512212	−0.256106	+	0.966649i	$0.582440\pi$
$464$	0	0
$465$	62.5138	2.89901
$466$	0	0
$467$	−39.5708	−1.83112	−0.915559	−	0.402185i	$-0.868251\pi$
$467$	−39.5708	−1.83112	−0.915559	+	0.402185i	$0.868251\pi$
$468$	0	0
$469$	−27.9994	−1.29289
$470$	0	0
$471$	−12.2860	−0.566110
$472$	0	0
$473$	−20.7706	−0.955032
$474$	0	0
$475$	−32.7612	−1.50319
$476$	0	0
$477$	−25.9193	−1.18676
$478$	0	0
$479$	−6.00598	−0.274420	−0.137210	−	0.990542i	$-0.543814\pi$
$479$	−6.00598	−0.274420	−0.137210	+	0.990542i	$0.543814\pi$
$480$	0	0
$481$	4.99884	0.227928
$482$	0	0
$483$	−74.7448	−3.40100
$484$	0	0
$485$	42.0647	1.91006
$486$	0	0
$487$	25.2517	1.14427	0.572133	−	0.820161i	$-0.306116\pi$
$487$	25.2517	1.14427	0.572133	+	0.820161i	$0.306116\pi$
$488$	0	0
$489$	11.9184	0.538967
$490$	0	0
$491$	−14.4847	−0.653685	−0.326843	−	0.945079i	$-0.605985\pi$
$491$	−14.4847	−0.653685	−0.326843	+	0.945079i	$0.605985\pi$
$492$	0	0
$493$	4.91364	0.221299
$494$	0	0
$495$	−46.0783	−2.07107
$496$	0	0
$497$	36.0832	1.61855
$498$	0	0
$499$	9.07074	0.406062	0.203031	−	0.979172i	$-0.434921\pi$
$499$	9.07074	0.406062	0.203031	+	0.979172i	$0.434921\pi$
$500$	0	0
$501$	−22.2578	−0.994405
$502$	0	0
$503$	36.8401	1.64262	0.821310	−	0.570483i	$-0.193244\pi$
$503$	36.8401	1.64262	0.821310	+	0.570483i	$0.193244\pi$
$504$	0	0
$505$	−24.7824	−1.10280
$506$	0	0
$507$	−33.4671	−1.48633
$508$	0	0
$509$	−42.5341	−1.88529	−0.942645	−	0.333796i	$-0.891670\pi$
$509$	−42.5341	−1.88529	−0.942645	+	0.333796i	$0.891670\pi$
$510$	0	0
$511$	21.2640	0.940663
$512$	0	0
$513$	−49.5874	−2.18934
$514$	0	0
$515$	−43.9057	−1.93472
$516$	0	0
$517$	−26.2559	−1.15473
$518$	0	0
$519$	3.61080	0.158497
$520$	0	0
$521$	−26.1588	−1.14604	−0.573020	−	0.819541i	$-0.694228\pi$
$521$	−26.1588	−1.14604	−0.573020	+	0.819541i	$0.694228\pi$
$522$	0	0
$523$	−32.6492	−1.42765	−0.713826	−	0.700323i	$-0.753039\pi$
$523$	−32.6492	−1.42765	−0.713826	+	0.700323i	$0.753039\pi$
$524$	0	0
$525$	55.1028	2.40488
$526$	0	0
$527$	5.96779	0.259961
$528$	0	0
$529$	60.7550	2.64152
$530$	0	0
$531$	−6.34124	−0.275187
$532$	0	0
$533$	−6.38196	−0.276433
$534$	0	0
$535$	18.0623	0.780900
$536$	0	0
$537$	63.8058	2.75342
$538$	0	0
$539$	0.298478	0.0128564
$540$	0	0
$541$	10.4643	0.449896	0.224948	−	0.974371i	$-0.427779\pi$
$541$	10.4643	0.449896	0.224948	+	0.974371i	$0.427779\pi$
$542$	0	0
$543$	−68.9880	−2.96056
$544$	0	0
$545$	36.8528	1.57860
$546$	0	0
$547$	−17.6047	−0.752722	−0.376361	−	0.926473i	$-0.622825\pi$
$547$	−17.6047	−0.752722	−0.376361	+	0.926473i	$0.622825\pi$
$548$	0	0
$549$	54.5634	2.32871
$550$	0	0
$551$	23.8597	1.01646
$552$	0	0
$553$	−23.0766	−0.981318
$554$	0	0
$555$	36.5729	1.55243
$556$	0	0
$557$	19.1994	0.813505	0.406752	−	0.913538i	$-0.366661\pi$
$557$	19.1994	0.813505	0.406752	+	0.913538i	$0.366661\pi$
$558$	0	0
$559$	14.0268	0.593269
$560$	0	0
$561$	−6.47985	−0.273579
$562$	0	0
$563$	−44.2513	−1.86497	−0.932485	−	0.361208i	$-0.882364\pi$
$563$	−44.2513	−1.86497	−0.932485	+	0.361208i	$0.882364\pi$
$564$	0	0
$565$	−55.5552	−2.33722
$566$	0	0
$567$	32.5681	1.36773
$568$	0	0
$569$	−1.70108	−0.0713131	−0.0356565	−	0.999364i	$-0.511352\pi$
$569$	−1.70108	−0.0713131	−0.0356565	+	0.999364i	$0.511352\pi$
$570$	0	0
$571$	−13.2533	−0.554634	−0.277317	−	0.960779i	$-0.589445\pi$
$571$	−13.2533	−0.554634	−0.277317	+	0.960779i	$0.589445\pi$
$572$	0	0
$573$	23.4875	0.981205
$574$	0	0
$575$	−61.7453	−2.57496
$576$	0	0
$577$	42.7690	1.78050	0.890249	−	0.455475i	$-0.150531\pi$
$577$	42.7690	1.78050	0.890249	+	0.455475i	$0.150531\pi$
$578$	0	0
$579$	−23.3322	−0.969653
$580$	0	0
$581$	11.6978	0.485305
$582$	0	0
$583$	−8.66586	−0.358903
$584$	0	0
$585$	31.1176	1.28655
$586$	0	0
$587$	−38.2062	−1.57694	−0.788468	−	0.615075i	$-0.789126\pi$
$587$	−38.2062	−1.57694	−0.788468	+	0.615075i	$0.789126\pi$
$588$	0	0
$589$	28.9784	1.19403
$590$	0	0
$591$	1.93429	0.0795660
$592$	0	0
$593$	32.6150	1.33934	0.669669	−	0.742660i	$-0.266436\pi$
$593$	32.6150	1.33934	0.669669	+	0.742660i	$0.266436\pi$
$594$	0	0
$595$	9.15868	0.375469
$596$	0	0
$597$	15.7017	0.642627
$598$	0	0
$599$	−0.0456629	−0.00186574	−0.000932869	−	1.00000i	$-0.500297\pi$
$599$	−0.0456629	−0.00186574	−0.000932869		1.00000i	$0.500297\pi$
$600$	0	0
$601$	6.91055	0.281887	0.140944	−	0.990018i	$-0.454986\pi$
$601$	6.91055	0.281887	0.140944	+	0.990018i	$0.454986\pi$
$602$	0	0
$603$	−66.4431	−2.70577
$604$	0	0
$605$	22.2951	0.906427
$606$	0	0
$607$	−7.37752	−0.299444	−0.149722	−	0.988728i	$-0.547838\pi$
$607$	−7.37752	−0.299444	−0.149722	+	0.988728i	$0.547838\pi$
$608$	0	0
$609$	−40.1309	−1.62619
$610$	0	0
$611$	17.7311	0.717324
$612$	0	0
$613$	35.9975	1.45392	0.726962	−	0.686678i	$-0.240932\pi$
$613$	35.9975	1.45392	0.726962	+	0.686678i	$0.240932\pi$
$614$	0	0
$615$	−46.6922	−1.88281
$616$	0	0
$617$	12.5574	0.505541	0.252771	−	0.967526i	$-0.418658\pi$
$617$	12.5574	0.505541	0.252771	+	0.967526i	$0.418658\pi$
$618$	0	0
$619$	15.9207	0.639906	0.319953	−	0.947433i	$-0.396333\pi$
$619$	15.9207	0.639906	0.319953	+	0.947433i	$0.396333\pi$
$620$	0	0
$621$	−93.4579	−3.75034
$622$	0	0
$623$	31.8531	1.27617
$624$	0	0
$625$	−13.2146	−0.528586
$626$	0	0
$627$	−31.4649	−1.25659
$628$	0	0
$629$	3.49138	0.139211
$630$	0	0
$631$	3.29453	0.131153	0.0655766	−	0.997848i	$-0.479111\pi$
$631$	3.29453	0.131153	0.0655766	+	0.997848i	$0.479111\pi$
$632$	0	0
$633$	33.1598	1.31798
$634$	0	0
$635$	17.2118	0.683028
$636$	0	0
$637$	−0.201568	−0.00798641
$638$	0	0
$639$	85.6262	3.38732
$640$	0	0
$641$	5.50052	0.217258	0.108629	−	0.994082i	$-0.465354\pi$
$641$	5.50052	0.217258	0.108629	+	0.994082i	$0.465354\pi$
$642$	0	0
$643$	−0.680366	−0.0268310	−0.0134155	−	0.999910i	$-0.504270\pi$
$643$	−0.680366	−0.0268310	−0.0134155	+	0.999910i	$0.504270\pi$
$644$	0	0
$645$	102.624	4.04081
$646$	0	0
$647$	22.5324	0.885842	0.442921	−	0.896561i	$-0.353942\pi$
$647$	22.5324	0.885842	0.442921	+	0.896561i	$0.353942\pi$
$648$	0	0
$649$	−2.12013	−0.0832224
$650$	0	0
$651$	−48.7403	−1.91028
$652$	0	0
$653$	−16.1295	−0.631197	−0.315598	−	0.948893i	$-0.602205\pi$
$653$	−16.1295	−0.631197	−0.315598	+	0.948893i	$0.602205\pi$
$654$	0	0
$655$	25.0790	0.979916
$656$	0	0
$657$	50.4599	1.96863
$658$	0	0
$659$	34.6414	1.34944	0.674719	−	0.738075i	$-0.264265\pi$
$659$	34.6414	1.34944	0.674719	+	0.738075i	$0.264265\pi$
$660$	0	0
$661$	−19.4781	−0.757610	−0.378805	−	0.925477i	$-0.623665\pi$
$661$	−19.4781	−0.757610	−0.378805	+	0.925477i	$0.623665\pi$
$662$	0	0
$663$	4.37597	0.169949
$664$	0	0
$665$	44.4727	1.72458
$666$	0	0
$667$	44.9686	1.74119
$668$	0	0
$669$	−65.9797	−2.55092
$670$	0	0
$671$	18.2427	0.704253
$672$	0	0
$673$	−21.8434	−0.842000	−0.421000	−	0.907061i	$-0.638321\pi$
$673$	−21.8434	−0.842000	−0.421000	+	0.907061i	$0.638321\pi$
$674$	0	0
$675$	68.8983	2.65190
$676$	0	0
$677$	−8.59870	−0.330475	−0.165237	−	0.986254i	$-0.552839\pi$
$677$	−8.59870	−0.330475	−0.165237	+	0.986254i	$0.552839\pi$
$678$	0	0
$679$	−32.7968	−1.25862
$680$	0	0
$681$	−40.3693	−1.54696
$682$	0	0
$683$	31.1518	1.19199	0.595995	−	0.802988i	$-0.296758\pi$
$683$	31.1518	1.19199	0.595995	+	0.802988i	$0.296758\pi$
$684$	0	0
$685$	−48.1526	−1.83982
$686$	0	0
$687$	−4.63529	−0.176847
$688$	0	0
$689$	5.85223	0.222952
$690$	0	0
$691$	−29.8660	−1.13615	−0.568077	−	0.822975i	$-0.692313\pi$
$691$	−29.8660	−1.13615	−0.568077	+	0.822975i	$0.692313\pi$
$692$	0	0
$693$	35.9261	1.36472
$694$	0	0
$695$	43.0947	1.63468
$696$	0	0
$697$	−4.45741	−0.168836
$698$	0	0
$699$	77.0608	2.91471
$700$	0	0
$701$	−5.72015	−0.216047	−0.108024	−	0.994148i	$-0.534452\pi$
$701$	−5.72015	−0.216047	−0.108024	+	0.994148i	$0.534452\pi$
$702$	0	0
$703$	16.9535	0.639412
$704$	0	0
$705$	129.726	4.88575
$706$	0	0
$707$	19.3222	0.726685
$708$	0	0
$709$	26.5961	0.998837	0.499418	−	0.866361i	$-0.333547\pi$
$709$	26.5961	0.998837	0.499418	+	0.866361i	$0.333547\pi$
$710$	0	0
$711$	−54.7613	−2.05371
$712$	0	0
$713$	54.6159	2.04538
$714$	0	0
$715$	10.4039	0.389082
$716$	0	0
$717$	−41.9728	−1.56750
$718$	0	0
$719$	−7.46983	−0.278578	−0.139289	−	0.990252i	$-0.544482\pi$
$719$	−7.46983	−0.278578	−0.139289	+	0.990252i	$0.544482\pi$
$720$	0	0
$721$	34.2321	1.27487
$722$	0	0
$723$	−41.8840	−1.55768
$724$	0	0
$725$	−33.1514	−1.23121
$726$	0	0
$727$	−0.679575	−0.0252040	−0.0126020	−	0.999921i	$-0.504011\pi$
$727$	−0.679575	−0.0252040	−0.0126020	+	0.999921i	$0.504011\pi$
$728$	0	0
$729$	−16.3493	−0.605530
$730$	0	0
$731$	9.79683	0.362349
$732$	0	0
$733$	−8.70988	−0.321707	−0.160853	−	0.986978i	$-0.551425\pi$
$733$	−8.70988	−0.321707	−0.160853	+	0.986978i	$0.551425\pi$
$734$	0	0
$735$	−1.47473	−0.0543962
$736$	0	0
$737$	−22.2146	−0.818284
$738$	0	0
$739$	−8.64151	−0.317883	−0.158942	−	0.987288i	$-0.550808\pi$
$739$	−8.64151	−0.317883	−0.158942	+	0.987288i	$0.550808\pi$
$740$	0	0
$741$	21.2488	0.780596
$742$	0	0
$743$	8.53311	0.313050	0.156525	−	0.987674i	$-0.449971\pi$
$743$	8.53311	0.313050	0.156525	+	0.987674i	$0.449971\pi$
$744$	0	0
$745$	−46.9471	−1.72001
$746$	0	0
$747$	27.7590	1.01565
$748$	0	0
$749$	−14.0827	−0.514570
$750$	0	0
$751$	4.24950	0.155066	0.0775332	−	0.996990i	$-0.475296\pi$
$751$	4.24950	0.155066	0.0775332	+	0.996990i	$0.475296\pi$
$752$	0	0
$753$	42.9833	1.56640
$754$	0	0
$755$	−31.2050	−1.13567
$756$	0	0
$757$	−34.0334	−1.23696	−0.618482	−	0.785799i	$-0.712252\pi$
$757$	−34.0334	−1.23696	−0.618482	+	0.785799i	$0.712252\pi$
$758$	0	0
$759$	−59.3021	−2.15253
$760$	0	0
$761$	−41.5669	−1.50680	−0.753399	−	0.657563i	$-0.771587\pi$
$761$	−41.5669	−1.50680	−0.753399	+	0.657563i	$0.771587\pi$
$762$	0	0
$763$	−28.7332	−1.04021
$764$	0	0
$765$	21.7337	0.785785
$766$	0	0
$767$	1.43177	0.0516981
$768$	0	0
$769$	30.9341	1.11551	0.557756	−	0.830005i	$-0.311663\pi$
$769$	30.9341	1.11551	0.557756	+	0.830005i	$0.311663\pi$
$770$	0	0
$771$	79.2624	2.85456
$772$	0	0
$773$	27.3225	0.982720	0.491360	−	0.870956i	$-0.336500\pi$
$773$	27.3225	0.982720	0.491360	+	0.870956i	$0.336500\pi$
$774$	0	0
$775$	−40.2635	−1.44631
$776$	0	0
$777$	−28.5150	−1.02297
$778$	0	0
$779$	−21.6443	−0.775487
$780$	0	0
$781$	28.6282	1.02440
$782$	0	0
$783$	−50.1781	−1.79322
$784$	0	0
$785$	13.7774	0.491738
$786$	0	0
$787$	38.2499	1.36346	0.681731	−	0.731603i	$-0.261227\pi$
$787$	38.2499	1.36346	0.681731	+	0.731603i	$0.261227\pi$
$788$	0	0
$789$	−24.9161	−0.887036
$790$	0	0
$791$	43.3149	1.54010
$792$	0	0
$793$	−12.3197	−0.437484
$794$	0	0
$795$	42.8165	1.51855
$796$	0	0
$797$	3.71142	0.131465	0.0657326	−	0.997837i	$-0.479062\pi$
$797$	3.71142	0.131465	0.0657326	+	0.997837i	$0.479062\pi$
$798$	0	0
$799$	12.3841	0.438117
$800$	0	0
$801$	75.5880	2.67077
$802$	0	0
$803$	16.8707	0.595356
$804$	0	0
$805$	83.8182	2.95420
$806$	0	0
$807$	13.0907	0.460815
$808$	0	0
$809$	36.5218	1.28404	0.642019	−	0.766688i	$-0.278097\pi$
$809$	36.5218	1.28404	0.642019	+	0.766688i	$0.278097\pi$
$810$	0	0
$811$	30.8814	1.08439	0.542197	−	0.840251i	$-0.317593\pi$
$811$	30.8814	1.08439	0.542197	+	0.840251i	$0.317593\pi$
$812$	0	0
$813$	−1.99293	−0.0698952
$814$	0	0
$815$	−13.3652	−0.468161
$816$	0	0
$817$	47.5715	1.66432
$818$	0	0
$819$	−24.2616	−0.847769
$820$	0	0
$821$	5.97266	0.208447	0.104224	−	0.994554i	$-0.466764\pi$
$821$	5.97266	0.208447	0.104224	+	0.994554i	$0.466764\pi$
$822$	0	0
$823$	−23.6299	−0.823688	−0.411844	−	0.911254i	$-0.635115\pi$
$823$	−23.6299	−0.823688	−0.411844	+	0.911254i	$0.635115\pi$
$824$	0	0
$825$	43.7183	1.52208
$826$	0	0
$827$	−10.8204	−0.376261	−0.188131	−	0.982144i	$-0.560243\pi$
$827$	−10.8204	−0.376261	−0.188131	+	0.982144i	$0.560243\pi$
$828$	0	0
$829$	−41.1832	−1.43035	−0.715176	−	0.698944i	$-0.753654\pi$
$829$	−41.1832	−1.43035	−0.715176	+	0.698944i	$0.753654\pi$
$830$	0	0
$831$	12.5612	0.435745
$832$	0	0
$833$	−0.140783	−0.00487784
$834$	0	0
$835$	24.9597	0.863767
$836$	0	0
$837$	−60.9430	−2.10650
$838$	0	0
$839$	11.4113	0.393961	0.196980	−	0.980407i	$-0.436886\pi$
$839$	11.4113	0.393961	0.196980	+	0.980407i	$0.436886\pi$
$840$	0	0
$841$	−4.85610	−0.167452
$842$	0	0
$843$	37.8521	1.30370
$844$	0	0
$845$	37.5298	1.29106
$846$	0	0
$847$	−17.3829	−0.597285
$848$	0	0
$849$	94.8242	3.25436
$850$	0	0
$851$	31.9524	1.09531
$852$	0	0
$853$	44.3122	1.51722	0.758611	−	0.651544i	$-0.225878\pi$
$853$	44.3122	1.51722	0.758611	+	0.651544i	$0.225878\pi$
$854$	0	0
$855$	105.535	3.60921
$856$	0	0
$857$	0.546695	0.0186748	0.00933738	−	0.999956i	$-0.497028\pi$
$857$	0.546695	0.0186748	0.00933738	+	0.999956i	$0.497028\pi$
$858$	0	0
$859$	31.4032	1.07146	0.535731	−	0.844388i	$-0.320036\pi$
$859$	31.4032	1.07146	0.535731	+	0.844388i	$0.320036\pi$
$860$	0	0
$861$	36.4047	1.24067
$862$	0	0
$863$	12.8162	0.436270	0.218135	−	0.975919i	$-0.430003\pi$
$863$	12.8162	0.436270	0.218135	+	0.975919i	$0.430003\pi$
$864$	0	0
$865$	−4.04913	−0.137674
$866$	0	0
$867$	3.05634	0.103799
$868$	0	0
$869$	−18.3089	−0.621086
$870$	0	0
$871$	15.0019	0.508321
$872$	0	0
$873$	−77.8274	−2.63406
$874$	0	0
$875$	−15.9984	−0.540846
$876$	0	0
$877$	3.36259	0.113547	0.0567734	−	0.998387i	$-0.481919\pi$
$877$	3.36259	0.113547	0.0567734	+	0.998387i	$0.481919\pi$
$878$	0	0
$879$	50.1550	1.69169
$880$	0	0
$881$	30.3283	1.02179	0.510893	−	0.859644i	$-0.329315\pi$
$881$	30.3283	1.02179	0.510893	+	0.859644i	$0.329315\pi$
$882$	0	0
$883$	−25.7965	−0.868122	−0.434061	−	0.900884i	$-0.642920\pi$
$883$	−25.7965	−0.868122	−0.434061	+	0.900884i	$0.642920\pi$
$884$	0	0
$885$	10.4752	0.352120
$886$	0	0
$887$	8.75940	0.294112	0.147056	−	0.989128i	$-0.453020\pi$
$887$	8.75940	0.294112	0.147056	+	0.989128i	$0.453020\pi$
$888$	0	0
$889$	−13.4196	−0.450078
$890$	0	0
$891$	25.8394	0.865652
$892$	0	0
$893$	60.1347	2.01233
$894$	0	0
$895$	−71.5513	−2.39170
$896$	0	0
$897$	40.0479	1.33716
$898$	0	0
$899$	29.3236	0.977996
$900$	0	0
$901$	4.08742	0.136172
$902$	0	0
$903$	−80.0131	−2.66267
$904$	0	0
$905$	77.3626	2.57162
$906$	0	0
$907$	−44.6561	−1.48278	−0.741391	−	0.671074i	$-0.765833\pi$
$907$	−44.6561	−1.48278	−0.741391	+	0.671074i	$0.765833\pi$
$908$	0	0
$909$	45.8519	1.52081
$910$	0	0
$911$	2.74735	0.0910237	0.0455118	−	0.998964i	$-0.485508\pi$
$911$	2.74735	0.0910237	0.0455118	+	0.998964i	$0.485508\pi$
$912$	0	0
$913$	9.28095	0.307155
$914$	0	0
$915$	−90.1342	−2.97974
$916$	0	0
$917$	−19.5534	−0.645711
$918$	0	0
$919$	−37.1788	−1.22641	−0.613207	−	0.789922i	$-0.710121\pi$
$919$	−37.1788	−1.22641	−0.613207	+	0.789922i	$0.710121\pi$
$920$	0	0
$921$	29.0456	0.957084
$922$	0	0
$923$	−19.3332	−0.636360
$924$	0	0
$925$	−23.5557	−0.774506
$926$	0	0
$927$	81.2335	2.66806
$928$	0	0
$929$	9.60687	0.315191	0.157596	−	0.987504i	$-0.449626\pi$
$929$	9.60687	0.315191	0.157596	+	0.987504i	$0.449626\pi$
$930$	0	0
$931$	−0.683614	−0.0224045
$932$	0	0
$933$	−50.5832	−1.65602
$934$	0	0
$935$	7.26645	0.237638
$936$	0	0
$937$	−24.5801	−0.802998	−0.401499	−	0.915859i	$-0.631511\pi$
$937$	−24.5801	−0.802998	−0.401499	+	0.915859i	$0.631511\pi$
$938$	0	0
$939$	−1.45172	−0.0473750
$940$	0	0
$941$	1.63078	0.0531618	0.0265809	−	0.999647i	$-0.491538\pi$
$941$	1.63078	0.0531618	0.0265809	+	0.999647i	$0.491538\pi$
$942$	0	0
$943$	−40.7932	−1.32841
$944$	0	0
$945$	−93.5284	−3.04248
$946$	0	0
$947$	35.5499	1.15522	0.577608	−	0.816314i	$-0.303986\pi$
$947$	35.5499	1.15522	0.577608	+	0.816314i	$0.303986\pi$
$948$	0	0
$949$	−11.3931	−0.369837
$950$	0	0
$951$	−35.0111	−1.13531
$952$	0	0
$953$	36.6487	1.18717	0.593584	−	0.804772i	$-0.297712\pi$
$953$	36.6487	1.18717	0.593584	+	0.804772i	$0.297712\pi$
$954$	0	0
$955$	−26.3387	−0.852301
$956$	0	0
$957$	−31.8397	−1.02923
$958$	0	0
$959$	37.5434	1.21234
$960$	0	0
$961$	4.61449	0.148855
$962$	0	0
$963$	−33.4185	−1.07690
$964$	0	0
$965$	26.1646	0.842267
$966$	0	0
$967$	43.6182	1.40267	0.701333	−	0.712834i	$-0.252589\pi$
$967$	43.6182	1.40267	0.701333	+	0.712834i	$0.252589\pi$
$968$	0	0
$969$	14.8410	0.476762
$970$	0	0
$971$	−56.0409	−1.79844	−0.899219	−	0.437498i	$-0.855865\pi$
$971$	−56.0409	−1.79844	−0.899219	+	0.437498i	$0.855865\pi$
$972$	0	0
$973$	−33.5999	−1.07716
$974$	0	0
$975$	−29.5238	−0.945519
$976$	0	0
$977$	−34.2331	−1.09522	−0.547608	−	0.836735i	$-0.684461\pi$
$977$	−34.2331	−1.09522	−0.547608	+	0.836735i	$0.684461\pi$
$978$	0	0
$979$	25.2721	0.807700
$980$	0	0
$981$	−68.1844	−2.17696
$982$	0	0
$983$	−26.3524	−0.840512	−0.420256	−	0.907406i	$-0.638060\pi$
$983$	−26.3524	−0.840512	−0.420256	+	0.907406i	$0.638060\pi$
$984$	0	0
$985$	−2.16910	−0.0691131
$986$	0	0
$987$	−101.144	−3.21944
$988$	0	0
$989$	89.6585	2.85097
$990$	0	0
$991$	−41.5302	−1.31925	−0.659626	−	0.751594i	$-0.729285\pi$
$991$	−41.5302	−1.31925	−0.659626	+	0.751594i	$0.729285\pi$
$992$	0	0
$993$	−90.6592	−2.87698
$994$	0	0
$995$	−17.6078	−0.558203
$996$	0	0
$997$	34.0563	1.07857	0.539286	−	0.842123i	$-0.318694\pi$
$997$	34.0563	1.07857	0.539286	+	0.842123i	$0.318694\pi$
$998$	0	0
$999$	−35.6540	−1.12804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.23	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.23	✓	23	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)

\(f(q)\)	\(=\)	\(q+3.05634 q^{3} -3.42736 q^{5} +2.67222 q^{7} +6.34124 q^{9} +O(q^{10})\)	\(q+3.05634 q^{3} -3.42736 q^{5} +2.67222 q^{7} +6.34124 q^{9} +2.12013 q^{11} -1.43177 q^{13} -10.4752 q^{15} -1.00000 q^{17} -4.85580 q^{19} +8.16724 q^{21} -9.15178 q^{23} +6.74681 q^{25} +10.2120 q^{27} -4.91364 q^{29} -5.96779 q^{31} +6.47985 q^{33} -9.15868 q^{35} -3.49138 q^{37} -4.37597 q^{39} +4.45741 q^{41} -9.79683 q^{43} -21.7337 q^{45} -12.3841 q^{47} +0.140783 q^{49} -3.05634 q^{51} -4.08742 q^{53} -7.26645 q^{55} -14.8410 q^{57} -1.00000 q^{59} +8.60453 q^{61} +16.9452 q^{63} +4.90718 q^{65} -10.4779 q^{67} -27.9710 q^{69} +13.5031 q^{71} +7.95741 q^{73} +20.6206 q^{75} +5.66546 q^{77} -8.63574 q^{79} +12.1876 q^{81} +4.37754 q^{83} +3.42736 q^{85} -15.0178 q^{87} +11.9201 q^{89} -3.82600 q^{91} -18.2396 q^{93} +16.6426 q^{95} -12.2732 q^{97} +13.4443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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