LMFDB - Embedded newform 8024.2.a.y.1.7

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			1.7
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)

$f(q)$	$=$	$q-1.61430 q^{3} -1.23431 q^{5} +4.85046 q^{7} -0.394052 q^{9} +O(q^{10})$	$q-1.61430 q^{3} -1.23431 q^{5} +4.85046 q^{7} -0.394052 q^{9} -1.78231 q^{11} -5.69601 q^{13} +1.99254 q^{15} -1.00000 q^{17} +2.51169 q^{19} -7.83007 q^{21} +6.91882 q^{23} -3.47648 q^{25} +5.47900 q^{27} -7.19652 q^{29} +5.87284 q^{31} +2.87718 q^{33} -5.98697 q^{35} +3.49518 q^{37} +9.19504 q^{39} -1.25490 q^{41} +1.45306 q^{43} +0.486382 q^{45} +2.43142 q^{47} +16.5270 q^{49} +1.61430 q^{51} +2.66779 q^{53} +2.19993 q^{55} -4.05460 q^{57} -1.00000 q^{59} -8.88620 q^{61} -1.91133 q^{63} +7.03064 q^{65} -3.95895 q^{67} -11.1690 q^{69} -2.75208 q^{71} -9.97414 q^{73} +5.61206 q^{75} -8.64504 q^{77} +1.01525 q^{79} -7.66257 q^{81} -2.01954 q^{83} +1.23431 q^{85} +11.6173 q^{87} +11.9291 q^{89} -27.6283 q^{91} -9.48050 q^{93} -3.10020 q^{95} -3.33104 q^{97} +0.702323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.61430	−0.932014	−0.466007	−	0.884781i	$-0.654308\pi$
$3$	−1.61430	−0.932014	−0.466007	+	0.884781i	$0.654308\pi$
$4$	0	0
$5$	−1.23431	−0.552000	−0.276000	−	0.961158i	$-0.589009\pi$
$5$	−1.23431	−0.552000	−0.276000	+	0.961158i	$0.589009\pi$
$6$	0	0
$7$	4.85046	1.83330	0.916651	−	0.399689i	$-0.130882\pi$
$7$	4.85046	1.83330	0.916651	+	0.399689i	$0.130882\pi$
$8$	0	0
$9$	−0.394052	−0.131351
$10$	0	0
$11$	−1.78231	−0.537388	−0.268694	−	0.963226i	$-0.586592\pi$
$11$	−1.78231	−0.537388	−0.268694	+	0.963226i	$0.586592\pi$
$12$	0	0
$13$	−5.69601	−1.57979	−0.789894	−	0.613243i	$-0.789865\pi$
$13$	−5.69601	−1.57979	−0.789894	+	0.613243i	$0.789865\pi$
$14$	0	0
$15$	1.99254	0.514472
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.51169	0.576221	0.288110	−	0.957597i	$-0.406973\pi$
$19$	2.51169	0.576221	0.288110	+	0.957597i	$0.406973\pi$
$20$	0	0
$21$	−7.83007	−1.70866
$22$	0	0
$23$	6.91882	1.44267	0.721337	−	0.692584i	$-0.243528\pi$
$23$	6.91882	1.44267	0.721337	+	0.692584i	$0.243528\pi$
$24$	0	0
$25$	−3.47648	−0.695296
$26$	0	0
$27$	5.47900	1.05443
$28$	0	0
$29$	−7.19652	−1.33636	−0.668180	−	0.743999i	$-0.732926\pi$
$29$	−7.19652	−1.33636	−0.668180	+	0.743999i	$0.732926\pi$
$30$	0	0
$31$	5.87284	1.05479	0.527397	−	0.849619i	$-0.323168\pi$
$31$	5.87284	1.05479	0.527397	+	0.849619i	$0.323168\pi$
$32$	0	0
$33$	2.87718	0.500853
$34$	0	0
$35$	−5.98697	−1.01198
$36$	0	0
$37$	3.49518	0.574604	0.287302	−	0.957840i	$-0.407242\pi$
$37$	3.49518	0.574604	0.287302	+	0.957840i	$0.407242\pi$
$38$	0	0
$39$	9.19504	1.47238
$40$	0	0
$41$	−1.25490	−0.195982	−0.0979909	−	0.995187i	$-0.531242\pi$
$41$	−1.25490	−0.195982	−0.0979909	+	0.995187i	$0.531242\pi$
$42$	0	0
$43$	1.45306	0.221590	0.110795	−	0.993843i	$-0.464660\pi$
$43$	1.45306	0.221590	0.110795	+	0.993843i	$0.464660\pi$
$44$	0	0
$45$	0.486382	0.0725055
$46$	0	0
$47$	2.43142	0.354659	0.177330	−	0.984151i	$-0.443254\pi$
$47$	2.43142	0.354659	0.177330	+	0.984151i	$0.443254\pi$
$48$	0	0
$49$	16.5270	2.36100
$50$	0	0
$51$	1.61430	0.226047
$52$	0	0
$53$	2.66779	0.366450	0.183225	−	0.983071i	$-0.441346\pi$
$53$	2.66779	0.366450	0.183225	+	0.983071i	$0.441346\pi$
$54$	0	0
$55$	2.19993	0.296638
$56$	0	0
$57$	−4.05460	−0.537045
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−8.88620	−1.13776	−0.568881	−	0.822420i	$-0.692623\pi$
$61$	−8.88620	−1.13776	−0.568881	+	0.822420i	$0.692623\pi$
$62$	0	0
$63$	−1.91133	−0.240805
$64$	0	0
$65$	7.03064	0.872043
$66$	0	0
$67$	−3.95895	−0.483663	−0.241831	−	0.970318i	$-0.577748\pi$
$67$	−3.95895	−0.483663	−0.241831	+	0.970318i	$0.577748\pi$
$68$	0	0
$69$	−11.1690	−1.34459
$70$	0	0
$71$	−2.75208	−0.326611	−0.163306	−	0.986576i	$-0.552216\pi$
$71$	−2.75208	−0.326611	−0.163306	+	0.986576i	$0.552216\pi$
$72$	0	0
$73$	−9.97414	−1.16739	−0.583693	−	0.811975i	$-0.698393\pi$
$73$	−9.97414	−1.16739	−0.583693	+	0.811975i	$0.698393\pi$
$74$	0	0
$75$	5.61206	0.648025
$76$	0	0
$77$	−8.64504	−0.985194
$78$	0	0
$79$	1.01525	0.114224	0.0571120	−	0.998368i	$-0.481811\pi$
$79$	1.01525	0.114224	0.0571120	+	0.998368i	$0.481811\pi$
$80$	0	0
$81$	−7.66257	−0.851396
$82$	0	0
$83$	−2.01954	−0.221673	−0.110836	−	0.993839i	$-0.535353\pi$
$83$	−2.01954	−0.221673	−0.110836	+	0.993839i	$0.535353\pi$
$84$	0	0
$85$	1.23431	0.133880
$86$	0	0
$87$	11.6173	1.24551
$88$	0	0
$89$	11.9291	1.26448	0.632242	−	0.774771i	$-0.282135\pi$
$89$	11.9291	1.26448	0.632242	+	0.774771i	$0.282135\pi$
$90$	0	0
$91$	−27.6283	−2.89623
$92$	0	0
$93$	−9.48050	−0.983082
$94$	0	0
$95$	−3.10020	−0.318074
$96$	0	0
$97$	−3.33104	−0.338216	−0.169108	−	0.985598i	$-0.554089\pi$
$97$	−3.33104	−0.338216	−0.169108	+	0.985598i	$0.554089\pi$
$98$	0	0
$99$	0.702323	0.0705861
$100$	0	0
$101$	20.0483	1.99488	0.997438	−	0.0715347i	$-0.0227897\pi$
$101$	20.0483	1.99488	0.997438	+	0.0715347i	$0.0227897\pi$
$102$	0	0
$103$	−7.51221	−0.740201	−0.370100	−	0.928992i	$-0.620677\pi$
$103$	−7.51221	−0.740201	−0.370100	+	0.928992i	$0.620677\pi$
$104$	0	0
$105$	9.66474	0.943182
$106$	0	0
$107$	9.87323	0.954481	0.477241	−	0.878773i	$-0.341637\pi$
$107$	9.87323	0.954481	0.477241	+	0.878773i	$0.341637\pi$
$108$	0	0
$109$	−17.6636	−1.69187	−0.845934	−	0.533287i	$-0.820957\pi$
$109$	−17.6636	−1.69187	−0.845934	+	0.533287i	$0.820957\pi$
$110$	0	0
$111$	−5.64225	−0.535539
$112$	0	0
$113$	−16.4550	−1.54796	−0.773979	−	0.633211i	$-0.781737\pi$
$113$	−16.4550	−1.54796	−0.773979	+	0.633211i	$0.781737\pi$
$114$	0	0
$115$	−8.53997	−0.796356
$116$	0	0
$117$	2.24452	0.207506
$118$	0	0
$119$	−4.85046	−0.444641
$120$	0	0
$121$	−7.82336	−0.711215
$122$	0	0
$123$	2.02577	0.182658
$124$	0	0
$125$	10.4626	0.935803
$126$	0	0
$127$	−14.7641	−1.31011	−0.655053	−	0.755583i	$-0.727354\pi$
$127$	−14.7641	−1.31011	−0.655053	+	0.755583i	$0.727354\pi$
$128$	0	0
$129$	−2.34567	−0.206525
$130$	0	0
$131$	14.3236	1.25146	0.625731	−	0.780039i	$-0.284801\pi$
$131$	14.3236	1.25146	0.625731	+	0.780039i	$0.284801\pi$
$132$	0	0
$133$	12.1828	1.05639
$134$	0	0
$135$	−6.76278	−0.582048
$136$	0	0
$137$	19.3455	1.65280	0.826398	−	0.563086i	$-0.190386\pi$
$137$	19.3455	1.65280	0.826398	+	0.563086i	$0.190386\pi$
$138$	0	0
$139$	−9.81724	−0.832687	−0.416344	−	0.909207i	$-0.636689\pi$
$139$	−9.81724	−0.832687	−0.416344	+	0.909207i	$0.636689\pi$
$140$	0	0
$141$	−3.92503	−0.330547
$142$	0	0
$143$	10.1521	0.848959
$144$	0	0
$145$	8.88273	0.737671
$146$	0	0
$147$	−26.6794	−2.20048
$148$	0	0
$149$	−11.7655	−0.963871	−0.481936	−	0.876207i	$-0.660066\pi$
$149$	−11.7655	−0.963871	−0.481936	+	0.876207i	$0.660066\pi$
$150$	0	0
$151$	19.1626	1.55943	0.779713	−	0.626137i	$-0.215365\pi$
$151$	19.1626	1.55943	0.779713	+	0.626137i	$0.215365\pi$
$152$	0	0
$153$	0.394052	0.0318572
$154$	0	0
$155$	−7.24891	−0.582246
$156$	0	0
$157$	−5.26747	−0.420390	−0.210195	−	0.977659i	$-0.567410\pi$
$157$	−5.26747	−0.420390	−0.210195	+	0.977659i	$0.567410\pi$
$158$	0	0
$159$	−4.30661	−0.341536
$160$	0	0
$161$	33.5595	2.64486
$162$	0	0
$163$	−21.1836	−1.65923	−0.829615	−	0.558335i	$-0.811440\pi$
$163$	−21.1836	−1.65923	−0.829615	+	0.558335i	$0.811440\pi$
$164$	0	0
$165$	−3.55133	−0.276471
$166$	0	0
$167$	22.8657	1.76940	0.884702	−	0.466157i	$-0.154362\pi$
$167$	22.8657	1.76940	0.884702	+	0.466157i	$0.154362\pi$
$168$	0	0
$169$	19.4445	1.49573
$170$	0	0
$171$	−0.989735	−0.0756869
$172$	0	0
$173$	0.613113	0.0466141	0.0233071	−	0.999728i	$-0.492580\pi$
$173$	0.613113	0.0466141	0.0233071	+	0.999728i	$0.492580\pi$
$174$	0	0
$175$	−16.8625	−1.27469
$176$	0	0
$177$	1.61430	0.121338
$178$	0	0
$179$	−7.90303	−0.590700	−0.295350	−	0.955389i	$-0.595436\pi$
$179$	−7.90303	−0.590700	−0.295350	+	0.955389i	$0.595436\pi$
$180$	0	0
$181$	−2.58128	−0.191865	−0.0959327	−	0.995388i	$-0.530583\pi$
$181$	−2.58128	−0.191865	−0.0959327	+	0.995388i	$0.530583\pi$
$182$	0	0
$183$	14.3449	1.06041
$184$	0	0
$185$	−4.31414	−0.317182
$186$	0	0
$187$	1.78231	0.130336
$188$	0	0
$189$	26.5757	1.93310
$190$	0	0
$191$	21.9845	1.59074	0.795372	−	0.606122i	$-0.207276\pi$
$191$	21.9845	1.59074	0.795372	+	0.606122i	$0.207276\pi$
$192$	0	0
$193$	6.87376	0.494784	0.247392	−	0.968915i	$-0.420426\pi$
$193$	6.87376	0.494784	0.247392	+	0.968915i	$0.420426\pi$
$194$	0	0
$195$	−11.3495	−0.812756
$196$	0	0
$197$	−4.58120	−0.326397	−0.163198	−	0.986593i	$-0.552181\pi$
$197$	−4.58120	−0.326397	−0.163198	+	0.986593i	$0.552181\pi$
$198$	0	0
$199$	−0.266704	−0.0189061	−0.00945307	−	0.999955i	$-0.503009\pi$
$199$	−0.266704	−0.0189061	−0.00945307	+	0.999955i	$0.503009\pi$
$200$	0	0
$201$	6.39091	0.450780
$202$	0	0
$203$	−34.9064	−2.44995
$204$	0	0
$205$	1.54893	0.108182
$206$	0	0
$207$	−2.72637	−0.189496
$208$	0	0
$209$	−4.47661	−0.309654
$210$	0	0
$211$	−20.0898	−1.38304	−0.691518	−	0.722359i	$-0.743058\pi$
$211$	−20.0898	−1.38304	−0.691518	+	0.722359i	$0.743058\pi$
$212$	0	0
$213$	4.44266	0.304406
$214$	0	0
$215$	−1.79353	−0.122318
$216$	0	0
$217$	28.4860	1.93376
$218$	0	0
$219$	16.1012	1.08802
$220$	0	0
$221$	5.69601	0.383155
$222$	0	0
$223$	−28.2922	−1.89459	−0.947293	−	0.320369i	$-0.896193\pi$
$223$	−28.2922	−1.89459	−0.947293	+	0.320369i	$0.896193\pi$
$224$	0	0
$225$	1.36991	0.0913275
$226$	0	0
$227$	7.16434	0.475514	0.237757	−	0.971325i	$-0.423588\pi$
$227$	7.16434	0.475514	0.237757	+	0.971325i	$0.423588\pi$
$228$	0	0
$229$	−6.92016	−0.457297	−0.228649	−	0.973509i	$-0.573431\pi$
$229$	−6.92016	−0.457297	−0.228649	+	0.973509i	$0.573431\pi$
$230$	0	0
$231$	13.9556	0.918214
$232$	0	0
$233$	−14.6107	−0.957177	−0.478589	−	0.878039i	$-0.658851\pi$
$233$	−14.6107	−0.957177	−0.478589	+	0.878039i	$0.658851\pi$
$234$	0	0
$235$	−3.00113	−0.195772
$236$	0	0
$237$	−1.63891	−0.106458
$238$	0	0
$239$	12.7951	0.827643	0.413822	−	0.910358i	$-0.364194\pi$
$239$	12.7951	0.827643	0.413822	+	0.910358i	$0.364194\pi$
$240$	0	0
$241$	−19.1401	−1.23292	−0.616460	−	0.787387i	$-0.711434\pi$
$241$	−19.1401	−1.23292	−0.616460	+	0.787387i	$0.711434\pi$
$242$	0	0
$243$	−4.06736	−0.260921
$244$	0	0
$245$	−20.3994	−1.30327
$246$	0	0
$247$	−14.3066	−0.910307
$248$	0	0
$249$	3.26013	0.206602
$250$	0	0
$251$	−3.57492	−0.225647	−0.112824	−	0.993615i	$-0.535989\pi$
$251$	−3.57492	−0.225647	−0.112824	+	0.993615i	$0.535989\pi$
$252$	0	0
$253$	−12.3315	−0.775275
$254$	0	0
$255$	−1.99254	−0.124778
$256$	0	0
$257$	0.666916	0.0416011	0.0208005	−	0.999784i	$-0.493379\pi$
$257$	0.666916	0.0416011	0.0208005	+	0.999784i	$0.493379\pi$
$258$	0	0
$259$	16.9532	1.05342
$260$	0	0
$261$	2.83580	0.175532
$262$	0	0
$263$	−22.7227	−1.40114	−0.700570	−	0.713584i	$-0.747071\pi$
$263$	−22.7227	−1.40114	−0.700570	+	0.713584i	$0.747071\pi$
$264$	0	0
$265$	−3.29288	−0.202280
$266$	0	0
$267$	−19.2571	−1.17852
$268$	0	0
$269$	1.26415	0.0770764	0.0385382	−	0.999257i	$-0.487730\pi$
$269$	1.26415	0.0770764	0.0385382	+	0.999257i	$0.487730\pi$
$270$	0	0
$271$	11.8640	0.720687	0.360344	−	0.932820i	$-0.382659\pi$
$271$	11.8640	0.720687	0.360344	+	0.932820i	$0.382659\pi$
$272$	0	0
$273$	44.6002	2.69933
$274$	0	0
$275$	6.19617	0.373643
$276$	0	0
$277$	30.7805	1.84942	0.924709	−	0.380674i	$-0.124308\pi$
$277$	30.7805	1.84942	0.924709	+	0.380674i	$0.124308\pi$
$278$	0	0
$279$	−2.31420	−0.138548
$280$	0	0
$281$	−27.4165	−1.63553	−0.817766	−	0.575552i	$-0.804787\pi$
$281$	−27.4165	−1.63553	−0.817766	+	0.575552i	$0.804787\pi$
$282$	0	0
$283$	11.1308	0.661658	0.330829	−	0.943691i	$-0.392672\pi$
$283$	11.1308	0.661658	0.330829	+	0.943691i	$0.392672\pi$
$284$	0	0
$285$	5.00464	0.296449
$286$	0	0
$287$	−6.08682	−0.359294
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.37728	0.315222
$292$	0	0
$293$	−7.52573	−0.439658	−0.219829	−	0.975538i	$-0.570550\pi$
$293$	−7.52573	−0.439658	−0.219829	+	0.975538i	$0.570550\pi$
$294$	0	0
$295$	1.23431	0.0718643
$296$	0	0
$297$	−9.76529	−0.566640
$298$	0	0
$299$	−39.4097	−2.27912
$300$	0	0
$301$	7.04803	0.406242
$302$	0	0
$303$	−32.3638	−1.85925
$304$	0	0
$305$	10.9683	0.628044
$306$	0	0
$307$	3.24046	0.184943	0.0924713	−	0.995715i	$-0.470523\pi$
$307$	3.24046	0.184943	0.0924713	+	0.995715i	$0.470523\pi$
$308$	0	0
$309$	12.1269	0.689877
$310$	0	0
$311$	2.79100	0.158263	0.0791316	−	0.996864i	$-0.474785\pi$
$311$	2.79100	0.158263	0.0791316	+	0.996864i	$0.474785\pi$
$312$	0	0
$313$	−19.8043	−1.11941	−0.559703	−	0.828693i	$-0.689085\pi$
$313$	−19.8043	−1.11941	−0.559703	+	0.828693i	$0.689085\pi$
$314$	0	0
$315$	2.35918	0.132924
$316$	0	0
$317$	−27.4278	−1.54050	−0.770250	−	0.637742i	$-0.779869\pi$
$317$	−27.4278	−1.54050	−0.770250	+	0.637742i	$0.779869\pi$
$318$	0	0
$319$	12.8265	0.718143
$320$	0	0
$321$	−15.9383	−0.889590
$322$	0	0
$323$	−2.51169	−0.139754
$324$	0	0
$325$	19.8021	1.09842
$326$	0	0
$327$	28.5143	1.57684
$328$	0	0
$329$	11.7935	0.650198
$330$	0	0
$331$	0.729561	0.0401003	0.0200501	−	0.999799i	$-0.493617\pi$
$331$	0.729561	0.0401003	0.0200501	+	0.999799i	$0.493617\pi$
$332$	0	0
$333$	−1.37728	−0.0754746
$334$	0	0
$335$	4.88657	0.266982
$336$	0	0
$337$	−23.1846	−1.26294	−0.631472	−	0.775399i	$-0.717549\pi$
$337$	−23.1846	−1.26294	−0.631472	+	0.775399i	$0.717549\pi$
$338$	0	0
$339$	26.5633	1.44272
$340$	0	0
$341$	−10.4672	−0.566833
$342$	0	0
$343$	46.2102	2.49512
$344$	0	0
$345$	13.7860	0.742215
$346$	0	0
$347$	26.6414	1.43018	0.715092	−	0.699031i	$-0.246385\pi$
$347$	26.6414	1.43018	0.715092	+	0.699031i	$0.246385\pi$
$348$	0	0
$349$	−21.5689	−1.15456	−0.577278	−	0.816548i	$-0.695885\pi$
$349$	−21.5689	−1.15456	−0.577278	+	0.816548i	$0.695885\pi$
$350$	0	0
$351$	−31.2084	−1.66578
$352$	0	0
$353$	−18.7378	−0.997314	−0.498657	−	0.866799i	$-0.666173\pi$
$353$	−18.7378	−0.997314	−0.498657	+	0.866799i	$0.666173\pi$
$354$	0	0
$355$	3.39691	0.180289
$356$	0	0
$357$	7.83007	0.414412
$358$	0	0
$359$	9.27611	0.489574	0.244787	−	0.969577i	$-0.421282\pi$
$359$	9.27611	0.489574	0.244787	+	0.969577i	$0.421282\pi$
$360$	0	0
$361$	−12.6914	−0.667970
$362$	0	0
$363$	12.6292	0.662862
$364$	0	0
$365$	12.3112	0.644397
$366$	0	0
$367$	−17.8028	−0.929298	−0.464649	−	0.885495i	$-0.653819\pi$
$367$	−17.8028	−0.929298	−0.464649	+	0.885495i	$0.653819\pi$
$368$	0	0
$369$	0.494494	0.0257423
$370$	0	0
$371$	12.9400	0.671813
$372$	0	0
$373$	−2.76975	−0.143412	−0.0717060	−	0.997426i	$-0.522844\pi$
$373$	−2.76975	−0.143412	−0.0717060	+	0.997426i	$0.522844\pi$
$374$	0	0
$375$	−16.8897	−0.872182
$376$	0	0
$377$	40.9914	2.11117
$378$	0	0
$379$	19.8785	1.02109	0.510546	−	0.859851i	$-0.329443\pi$
$379$	19.8785	1.02109	0.510546	+	0.859851i	$0.329443\pi$
$380$	0	0
$381$	23.8337	1.22104
$382$	0	0
$383$	−23.5542	−1.20357	−0.601783	−	0.798660i	$-0.705543\pi$
$383$	−23.5542	−1.20357	−0.601783	+	0.798660i	$0.705543\pi$
$384$	0	0
$385$	10.6707	0.543827
$386$	0	0
$387$	−0.572582	−0.0291060
$388$	0	0
$389$	−3.68939	−0.187060	−0.0935298	−	0.995616i	$-0.529815\pi$
$389$	−3.68939	−0.187060	−0.0935298	+	0.995616i	$0.529815\pi$
$390$	0	0
$391$	−6.91882	−0.349900
$392$	0	0
$393$	−23.1226	−1.16638
$394$	0	0
$395$	−1.25313	−0.0630517
$396$	0	0
$397$	−27.3140	−1.37085	−0.685425	−	0.728143i	$-0.740384\pi$
$397$	−27.3140	−1.37085	−0.685425	+	0.728143i	$0.740384\pi$
$398$	0	0
$399$	−19.6667	−0.984566
$400$	0	0
$401$	9.50420	0.474617	0.237309	−	0.971434i	$-0.423735\pi$
$401$	9.50420	0.474617	0.237309	+	0.971434i	$0.423735\pi$
$402$	0	0
$403$	−33.4518	−1.66635
$404$	0	0
$405$	9.45798	0.469971
$406$	0	0
$407$	−6.22951	−0.308785
$408$	0	0
$409$	−24.4497	−1.20896	−0.604479	−	0.796621i	$-0.706619\pi$
$409$	−24.4497	−1.20896	−0.604479	+	0.796621i	$0.706619\pi$
$410$	0	0
$411$	−31.2293	−1.54043
$412$	0	0
$413$	−4.85046	−0.238676
$414$	0	0
$415$	2.49273	0.122363
$416$	0	0
$417$	15.8479	0.776076
$418$	0	0
$419$	−20.1650	−0.985126	−0.492563	−	0.870277i	$-0.663940\pi$
$419$	−20.1650	−0.985126	−0.492563	+	0.870277i	$0.663940\pi$
$420$	0	0
$421$	28.7819	1.40274	0.701372	−	0.712796i	$-0.252571\pi$
$421$	28.7819	1.40274	0.701372	+	0.712796i	$0.252571\pi$
$422$	0	0
$423$	−0.958106	−0.0465847
$424$	0	0
$425$	3.47648	0.168634
$426$	0	0
$427$	−43.1022	−2.08586
$428$	0	0
$429$	−16.3884	−0.791241
$430$	0	0
$431$	19.0386	0.917059	0.458529	−	0.888679i	$-0.348376\pi$
$431$	19.0386	0.917059	0.458529	+	0.888679i	$0.348376\pi$
$432$	0	0
$433$	19.1501	0.920297	0.460148	−	0.887842i	$-0.347796\pi$
$433$	19.1501	0.920297	0.460148	+	0.887842i	$0.347796\pi$
$434$	0	0
$435$	−14.3394	−0.687519
$436$	0	0
$437$	17.3779	0.831299
$438$	0	0
$439$	−23.0343	−1.09937	−0.549684	−	0.835372i	$-0.685252\pi$
$439$	−23.0343	−1.09937	−0.549684	+	0.835372i	$0.685252\pi$
$440$	0	0
$441$	−6.51248	−0.310118
$442$	0	0
$443$	−28.1446	−1.33719	−0.668595	−	0.743626i	$-0.733104\pi$
$443$	−28.1446	−1.33719	−0.668595	+	0.743626i	$0.733104\pi$
$444$	0	0
$445$	−14.7242	−0.697995
$446$	0	0
$447$	18.9931	0.898341
$448$	0	0
$449$	−23.2186	−1.09576	−0.547878	−	0.836558i	$-0.684564\pi$
$449$	−23.2186	−1.09576	−0.547878	+	0.836558i	$0.684564\pi$
$450$	0	0
$451$	2.23662	0.105318
$452$	0	0
$453$	−30.9340	−1.45341
$454$	0	0
$455$	34.1018	1.59872
$456$	0	0
$457$	39.7866	1.86114	0.930569	−	0.366115i	$-0.119312\pi$
$457$	39.7866	1.86114	0.930569	+	0.366115i	$0.119312\pi$
$458$	0	0
$459$	−5.47900	−0.255738
$460$	0	0
$461$	−23.5524	−1.09694	−0.548471	−	0.836169i	$-0.684790\pi$
$461$	−23.5524	−1.09694	−0.548471	+	0.836169i	$0.684790\pi$
$462$	0	0
$463$	17.2771	0.802937	0.401469	−	0.915873i	$-0.368500\pi$
$463$	17.2771	0.802937	0.401469	+	0.915873i	$0.368500\pi$
$464$	0	0
$465$	11.7019	0.542662
$466$	0	0
$467$	−11.6604	−0.539577	−0.269789	−	0.962920i	$-0.586954\pi$
$467$	−11.6604	−0.539577	−0.269789	+	0.962920i	$0.586954\pi$
$468$	0	0
$469$	−19.2027	−0.886700
$470$	0	0
$471$	8.50326	0.391809
$472$	0	0
$473$	−2.58981	−0.119080
$474$	0	0
$475$	−8.73183	−0.400644
$476$	0	0
$477$	−1.05125	−0.0481334
$478$	0	0
$479$	−13.7278	−0.627240	−0.313620	−	0.949549i	$-0.601542\pi$
$479$	−13.7278	−0.627240	−0.313620	+	0.949549i	$0.601542\pi$
$480$	0	0
$481$	−19.9086	−0.907753
$482$	0	0
$483$	−54.1749	−2.46504
$484$	0	0
$485$	4.11153	0.186695
$486$	0	0
$487$	−8.37770	−0.379630	−0.189815	−	0.981820i	$-0.560789\pi$
$487$	−8.37770	−0.379630	−0.189815	+	0.981820i	$0.560789\pi$
$488$	0	0
$489$	34.1966	1.54643
$490$	0	0
$491$	−10.1742	−0.459154	−0.229577	−	0.973290i	$-0.573734\pi$
$491$	−10.1742	−0.459154	−0.229577	+	0.973290i	$0.573734\pi$
$492$	0	0
$493$	7.19652	0.324115
$494$	0	0
$495$	−0.866884	−0.0389636
$496$	0	0
$497$	−13.3488	−0.598777
$498$	0	0
$499$	6.06056	0.271308	0.135654	−	0.990756i	$-0.456686\pi$
$499$	6.06056	0.271308	0.135654	+	0.990756i	$0.456686\pi$
$500$	0	0
$501$	−36.9121	−1.64911
$502$	0	0
$503$	−3.10211	−0.138316	−0.0691581	−	0.997606i	$-0.522031\pi$
$503$	−3.10211	−0.138316	−0.0691581	+	0.997606i	$0.522031\pi$
$504$	0	0
$505$	−24.7458	−1.10117
$506$	0	0
$507$	−31.3892	−1.39404
$508$	0	0
$509$	−33.8803	−1.50172	−0.750859	−	0.660462i	$-0.770360\pi$
$509$	−33.8803	−1.50172	−0.750859	+	0.660462i	$0.770360\pi$
$510$	0	0
$511$	−48.3792	−2.14017
$512$	0	0
$513$	13.7615	0.607587
$514$	0	0
$515$	9.27240	0.408591
$516$	0	0
$517$	−4.33356	−0.190590
$518$	0	0
$519$	−0.989745	−0.0434450
$520$	0	0
$521$	−10.0899	−0.442046	−0.221023	−	0.975269i	$-0.570940\pi$
$521$	−10.0899	−0.442046	−0.221023	+	0.975269i	$0.570940\pi$
$522$	0	0
$523$	3.78130	0.165345	0.0826723	−	0.996577i	$-0.473655\pi$
$523$	3.78130	0.165345	0.0826723	+	0.996577i	$0.473655\pi$
$524$	0	0
$525$	27.2211	1.18803
$526$	0	0
$527$	−5.87284	−0.255825
$528$	0	0
$529$	24.8701	1.08131
$530$	0	0
$531$	0.394052	0.0171004
$532$	0	0
$533$	7.14790	0.309610
$534$	0	0
$535$	−12.1866	−0.526874
$536$	0	0
$537$	12.7578	0.550541
$538$	0	0
$539$	−29.4562	−1.26877
$540$	0	0
$541$	36.8884	1.58596	0.792979	−	0.609249i	$-0.208529\pi$
$541$	36.8884	1.58596	0.792979	+	0.609249i	$0.208529\pi$
$542$	0	0
$543$	4.16695	0.178821
$544$	0	0
$545$	21.8024	0.933911
$546$	0	0
$547$	13.3980	0.572855	0.286428	−	0.958102i	$-0.407532\pi$
$547$	13.3980	0.572855	0.286428	+	0.958102i	$0.407532\pi$
$548$	0	0
$549$	3.50162	0.149446
$550$	0	0
$551$	−18.0754	−0.770038
$552$	0	0
$553$	4.92441	0.209407
$554$	0	0
$555$	6.96429	0.295618
$556$	0	0
$557$	36.2789	1.53719	0.768594	−	0.639737i	$-0.220957\pi$
$557$	36.2789	1.53719	0.768594	+	0.639737i	$0.220957\pi$
$558$	0	0
$559$	−8.27666	−0.350065
$560$	0	0
$561$	−2.87718	−0.121475
$562$	0	0
$563$	−20.1913	−0.850964	−0.425482	−	0.904967i	$-0.639895\pi$
$563$	−20.1913	−0.850964	−0.425482	+	0.904967i	$0.639895\pi$
$564$	0	0
$565$	20.3106	0.854473
$566$	0	0
$567$	−37.1670	−1.56087
$568$	0	0
$569$	2.37441	0.0995405	0.0497702	−	0.998761i	$-0.484151\pi$
$569$	2.37441	0.0995405	0.0497702	+	0.998761i	$0.484151\pi$
$570$	0	0
$571$	−33.7880	−1.41398	−0.706991	−	0.707222i	$-0.749948\pi$
$571$	−33.7880	−1.41398	−0.706991	+	0.707222i	$0.749948\pi$
$572$	0	0
$573$	−35.4895	−1.48260
$574$	0	0
$575$	−24.0531	−1.00309
$576$	0	0
$577$	−33.9418	−1.41302	−0.706509	−	0.707704i	$-0.749731\pi$
$577$	−33.9418	−1.41302	−0.706509	+	0.707704i	$0.749731\pi$
$578$	0	0
$579$	−11.0963	−0.461146
$580$	0	0
$581$	−9.79568	−0.406393
$582$	0	0
$583$	−4.75484	−0.196926
$584$	0	0
$585$	−2.77043	−0.114543
$586$	0	0
$587$	−37.8512	−1.56229	−0.781143	−	0.624353i	$-0.785363\pi$
$587$	−37.8512	−1.56229	−0.781143	+	0.624353i	$0.785363\pi$
$588$	0	0
$589$	14.7507	0.607794
$590$	0	0
$591$	7.39540	0.304206
$592$	0	0
$593$	29.3239	1.20419	0.602094	−	0.798425i	$-0.294333\pi$
$593$	29.3239	1.20419	0.602094	+	0.798425i	$0.294333\pi$
$594$	0	0
$595$	5.98697	0.245442
$596$	0	0
$597$	0.430539	0.0176208
$598$	0	0
$599$	14.0944	0.575882	0.287941	−	0.957648i	$-0.407029\pi$
$599$	14.0944	0.575882	0.287941	+	0.957648i	$0.407029\pi$
$600$	0	0
$601$	−36.5380	−1.49042	−0.745208	−	0.666832i	$-0.767650\pi$
$601$	−36.5380	−1.49042	−0.745208	+	0.666832i	$0.767650\pi$
$602$	0	0
$603$	1.56003	0.0635294
$604$	0	0
$605$	9.65645	0.392590
$606$	0	0
$607$	−14.9790	−0.607977	−0.303988	−	0.952676i	$-0.598318\pi$
$607$	−14.9790	−0.607977	−0.303988	+	0.952676i	$0.598318\pi$
$608$	0	0
$609$	56.3493	2.28339
$610$	0	0
$611$	−13.8494	−0.560287
$612$	0	0
$613$	−40.9844	−1.65534	−0.827672	−	0.561212i	$-0.810335\pi$
$613$	−40.9844	−1.65534	−0.827672	+	0.561212i	$0.810335\pi$
$614$	0	0
$615$	−2.50043	−0.100827
$616$	0	0
$617$	−28.9541	−1.16565	−0.582824	−	0.812599i	$-0.698052\pi$
$617$	−28.9541	−1.16565	−0.582824	+	0.812599i	$0.698052\pi$
$618$	0	0
$619$	−26.1513	−1.05111	−0.525554	−	0.850760i	$-0.676142\pi$
$619$	−26.1513	−1.05111	−0.525554	+	0.850760i	$0.676142\pi$
$620$	0	0
$621$	37.9082	1.52121
$622$	0	0
$623$	57.8617	2.31818
$624$	0	0
$625$	4.46831	0.178733
$626$	0	0
$627$	7.22657	0.288602
$628$	0	0
$629$	−3.49518	−0.139362
$630$	0	0
$631$	4.46223	0.177638	0.0888192	−	0.996048i	$-0.471691\pi$
$631$	4.46223	0.177638	0.0888192	+	0.996048i	$0.471691\pi$
$632$	0	0
$633$	32.4308	1.28901
$634$	0	0
$635$	18.2235	0.723179
$636$	0	0
$637$	−94.1378	−3.72987
$638$	0	0
$639$	1.08446	0.0429006
$640$	0	0
$641$	−37.7471	−1.49092	−0.745460	−	0.666551i	$-0.767770\pi$
$641$	−37.7471	−1.49092	−0.745460	+	0.666551i	$0.767770\pi$
$642$	0	0
$643$	10.0647	0.396914	0.198457	−	0.980110i	$-0.436407\pi$
$643$	10.0647	0.396914	0.198457	+	0.980110i	$0.436407\pi$
$644$	0	0
$645$	2.89529	0.114002
$646$	0	0
$647$	1.36824	0.0537909	0.0268954	−	0.999638i	$-0.491438\pi$
$647$	1.36824	0.0537909	0.0268954	+	0.999638i	$0.491438\pi$
$648$	0	0
$649$	1.78231	0.0699619
$650$	0	0
$651$	−45.9848	−1.80229
$652$	0	0
$653$	−38.3956	−1.50254	−0.751269	−	0.659996i	$-0.770558\pi$
$653$	−38.3956	−1.50254	−0.751269	+	0.659996i	$0.770558\pi$
$654$	0	0
$655$	−17.6798	−0.690807
$656$	0	0
$657$	3.93033	0.153337
$658$	0	0
$659$	25.3253	0.986533	0.493266	−	0.869878i	$-0.335803\pi$
$659$	25.3253	0.986533	0.493266	+	0.869878i	$0.335803\pi$
$660$	0	0
$661$	−5.39400	−0.209802	−0.104901	−	0.994483i	$-0.533453\pi$
$661$	−5.39400	−0.209802	−0.104901	+	0.994483i	$0.533453\pi$
$662$	0	0
$663$	−9.19504	−0.357106
$664$	0	0
$665$	−15.0374	−0.583125
$666$	0	0
$667$	−49.7915	−1.92793
$668$	0	0
$669$	45.6720	1.76578
$670$	0	0
$671$	15.8380	0.611419
$672$	0	0
$673$	18.9020	0.728620	0.364310	−	0.931278i	$-0.381305\pi$
$673$	18.9020	0.728620	0.364310	+	0.931278i	$0.381305\pi$
$674$	0	0
$675$	−19.0476	−0.733144
$676$	0	0
$677$	−25.4561	−0.978356	−0.489178	−	0.872184i	$-0.662703\pi$
$677$	−25.4561	−0.978356	−0.489178	+	0.872184i	$0.662703\pi$
$678$	0	0
$679$	−16.1571	−0.620051
$680$	0	0
$681$	−11.5654	−0.443186
$682$	0	0
$683$	−0.245640	−0.00939916	−0.00469958	−	0.999989i	$-0.501496\pi$
$683$	−0.245640	−0.00939916	−0.00469958	+	0.999989i	$0.501496\pi$
$684$	0	0
$685$	−23.8783	−0.912344
$686$	0	0
$687$	11.1712	0.426207
$688$	0	0
$689$	−15.1958	−0.578913
$690$	0	0
$691$	−42.1558	−1.60368	−0.801840	−	0.597538i	$-0.796146\pi$
$691$	−42.1558	−1.60368	−0.801840	+	0.597538i	$0.796146\pi$
$692$	0	0
$693$	3.40659	0.129406
$694$	0	0
$695$	12.1175	0.459643
$696$	0	0
$697$	1.25490	0.0475326
$698$	0	0
$699$	23.5859	0.892102
$700$	0	0
$701$	−24.5288	−0.926442	−0.463221	−	0.886243i	$-0.653306\pi$
$701$	−24.5288	−0.926442	−0.463221	+	0.886243i	$0.653306\pi$
$702$	0	0
$703$	8.77880	0.331099
$704$	0	0
$705$	4.84471	0.182462
$706$	0	0
$707$	97.2433	3.65721
$708$	0	0
$709$	46.2949	1.73864	0.869321	−	0.494249i	$-0.164557\pi$
$709$	46.2949	1.73864	0.869321	+	0.494249i	$0.164557\pi$
$710$	0	0
$711$	−0.400059	−0.0150034
$712$	0	0
$713$	40.6332	1.52172
$714$	0	0
$715$	−12.5308	−0.468625
$716$	0	0
$717$	−20.6550	−0.771375
$718$	0	0
$719$	39.5347	1.47440	0.737198	−	0.675676i	$-0.236148\pi$
$719$	39.5347	1.47440	0.737198	+	0.675676i	$0.236148\pi$
$720$	0	0
$721$	−36.4377	−1.35701
$722$	0	0
$723$	30.8977	1.14910
$724$	0	0
$725$	25.0186	0.929166
$726$	0	0
$727$	−21.5557	−0.799456	−0.399728	−	0.916634i	$-0.630895\pi$
$727$	−21.5557	−0.799456	−0.399728	+	0.916634i	$0.630895\pi$
$728$	0	0
$729$	29.5536	1.09458
$730$	0	0
$731$	−1.45306	−0.0537435
$732$	0	0
$733$	−2.88456	−0.106544	−0.0532719	−	0.998580i	$-0.516965\pi$
$733$	−2.88456	−0.106544	−0.0532719	+	0.998580i	$0.516965\pi$
$734$	0	0
$735$	32.9306	1.21467
$736$	0	0
$737$	7.05609	0.259914
$738$	0	0
$739$	−24.7043	−0.908761	−0.454380	−	0.890808i	$-0.650139\pi$
$739$	−24.7043	−0.908761	−0.454380	+	0.890808i	$0.650139\pi$
$740$	0	0
$741$	23.0951	0.848418
$742$	0	0
$743$	−34.5812	−1.26866	−0.634331	−	0.773061i	$-0.718725\pi$
$743$	−34.5812	−1.26866	−0.634331	+	0.773061i	$0.718725\pi$
$744$	0	0
$745$	14.5223	0.532057
$746$	0	0
$747$	0.795801	0.0291169
$748$	0	0
$749$	47.8897	1.74985
$750$	0	0
$751$	−5.19955	−0.189734	−0.0948672	−	0.995490i	$-0.530243\pi$
$751$	−5.19955	−0.189734	−0.0948672	+	0.995490i	$0.530243\pi$
$752$	0	0
$753$	5.77098	0.210306
$754$	0	0
$755$	−23.6525	−0.860804
$756$	0	0
$757$	−26.1870	−0.951783	−0.475892	−	0.879504i	$-0.657875\pi$
$757$	−26.1870	−0.951783	−0.475892	+	0.879504i	$0.657875\pi$
$758$	0	0
$759$	19.9067	0.722567
$760$	0	0
$761$	−14.4960	−0.525480	−0.262740	−	0.964867i	$-0.584626\pi$
$761$	−14.4960	−0.525480	−0.262740	+	0.964867i	$0.584626\pi$
$762$	0	0
$763$	−85.6767	−3.10171
$764$	0	0
$765$	−0.486382	−0.0175852
$766$	0	0
$767$	5.69601	0.205671
$768$	0	0
$769$	−39.3720	−1.41979	−0.709895	−	0.704308i	$-0.751257\pi$
$769$	−39.3720	−1.41979	−0.709895	+	0.704308i	$0.751257\pi$
$770$	0	0
$771$	−1.07660	−0.0387728
$772$	0	0
$773$	5.98155	0.215141	0.107571	−	0.994197i	$-0.465693\pi$
$773$	5.98155	0.215141	0.107571	+	0.994197i	$0.465693\pi$
$774$	0	0
$775$	−20.4168	−0.733394
$776$	0	0
$777$	−27.3675	−0.981805
$778$	0	0
$779$	−3.15191	−0.112929
$780$	0	0
$781$	4.90506	0.175517
$782$	0	0
$783$	−39.4297	−1.40910
$784$	0	0
$785$	6.50169	0.232055
$786$	0	0
$787$	−7.77371	−0.277103	−0.138551	−	0.990355i	$-0.544245\pi$
$787$	−7.77371	−0.277103	−0.138551	+	0.990355i	$0.544245\pi$
$788$	0	0
$789$	36.6811	1.30588
$790$	0	0
$791$	−79.8145	−2.83788
$792$	0	0
$793$	50.6159	1.79742
$794$	0	0
$795$	5.31569	0.188528
$796$	0	0
$797$	−26.2899	−0.931237	−0.465619	−	0.884985i	$-0.654168\pi$
$797$	−26.2899	−0.931237	−0.465619	+	0.884985i	$0.654168\pi$
$798$	0	0
$799$	−2.43142	−0.0860175
$800$	0	0
$801$	−4.70069	−0.166091
$802$	0	0
$803$	17.7770	0.627338
$804$	0	0
$805$	−41.4228	−1.45996
$806$	0	0
$807$	−2.04071	−0.0718362
$808$	0	0
$809$	−4.73126	−0.166342	−0.0831711	−	0.996535i	$-0.526505\pi$
$809$	−4.73126	−0.166342	−0.0831711	+	0.996535i	$0.526505\pi$
$810$	0	0
$811$	12.4726	0.437973	0.218987	−	0.975728i	$-0.429725\pi$
$811$	12.4726	0.437973	0.218987	+	0.975728i	$0.429725\pi$
$812$	0	0
$813$	−19.1520	−0.671690
$814$	0	0
$815$	26.1472	0.915895
$816$	0	0
$817$	3.64964	0.127685
$818$	0	0
$819$	10.8870	0.380421
$820$	0	0
$821$	48.1030	1.67881	0.839404	−	0.543509i	$-0.182904\pi$
$821$	48.1030	1.67881	0.839404	+	0.543509i	$0.182904\pi$
$822$	0	0
$823$	19.3001	0.672761	0.336380	−	0.941726i	$-0.390797\pi$
$823$	19.3001	0.672761	0.336380	+	0.941726i	$0.390797\pi$
$824$	0	0
$825$	−10.0025	−0.348241
$826$	0	0
$827$	13.4333	0.467120	0.233560	−	0.972342i	$-0.424962\pi$
$827$	13.4333	0.467120	0.233560	+	0.972342i	$0.424962\pi$
$828$	0	0
$829$	7.72366	0.268254	0.134127	−	0.990964i	$-0.457177\pi$
$829$	7.72366	0.268254	0.134127	+	0.990964i	$0.457177\pi$
$830$	0	0
$831$	−49.6887	−1.72368
$832$	0	0
$833$	−16.5270	−0.572626
$834$	0	0
$835$	−28.2234	−0.976711
$836$	0	0
$837$	32.1773	1.11221
$838$	0	0
$839$	−42.3323	−1.46147	−0.730737	−	0.682659i	$-0.760823\pi$
$839$	−42.3323	−1.46147	−0.730737	+	0.682659i	$0.760823\pi$
$840$	0	0
$841$	22.7899	0.785859
$842$	0	0
$843$	44.2583	1.52434
$844$	0	0
$845$	−24.0006	−0.825644
$846$	0	0
$847$	−37.9469	−1.30387
$848$	0	0
$849$	−17.9684	−0.616674
$850$	0	0
$851$	24.1825	0.828967
$852$	0	0
$853$	50.1307	1.71644	0.858221	−	0.513281i	$-0.171570\pi$
$853$	50.1307	1.71644	0.858221	+	0.513281i	$0.171570\pi$
$854$	0	0
$855$	1.22164	0.0417792
$856$	0	0
$857$	−32.9976	−1.12718	−0.563588	−	0.826056i	$-0.690580\pi$
$857$	−32.9976	−1.12718	−0.563588	+	0.826056i	$0.690580\pi$
$858$	0	0
$859$	54.0597	1.84449	0.922246	−	0.386604i	$-0.126352\pi$
$859$	54.0597	1.84449	0.922246	+	0.386604i	$0.126352\pi$
$860$	0	0
$861$	9.82593	0.334867
$862$	0	0
$863$	−14.7938	−0.503587	−0.251794	−	0.967781i	$-0.581020\pi$
$863$	−14.7938	−0.503587	−0.251794	+	0.967781i	$0.581020\pi$
$864$	0	0
$865$	−0.756771	−0.0257310
$866$	0	0
$867$	−1.61430	−0.0548243
$868$	0	0
$869$	−1.80949	−0.0613826
$870$	0	0
$871$	22.5502	0.764085
$872$	0	0
$873$	1.31260	0.0444248
$874$	0	0
$875$	50.7484	1.71561
$876$	0	0
$877$	23.0894	0.779673	0.389837	−	0.920884i	$-0.372531\pi$
$877$	23.0894	0.779673	0.389837	+	0.920884i	$0.372531\pi$
$878$	0	0
$879$	12.1487	0.409767
$880$	0	0
$881$	34.0516	1.14723	0.573613	−	0.819126i	$-0.305541\pi$
$881$	34.0516	1.14723	0.573613	+	0.819126i	$0.305541\pi$
$882$	0	0
$883$	−27.0301	−0.909634	−0.454817	−	0.890585i	$-0.650295\pi$
$883$	−27.0301	−0.909634	−0.454817	+	0.890585i	$0.650295\pi$
$884$	0	0
$885$	−1.99254	−0.0669785
$886$	0	0
$887$	56.0157	1.88082	0.940412	−	0.340037i	$-0.110440\pi$
$887$	56.0157	1.88082	0.940412	+	0.340037i	$0.110440\pi$
$888$	0	0
$889$	−71.6129	−2.40182
$890$	0	0
$891$	13.6571	0.457530
$892$	0	0
$893$	6.10697	0.204362
$894$	0	0
$895$	9.75479	0.326067
$896$	0	0
$897$	63.6188	2.12417
$898$	0	0
$899$	−42.2641	−1.40959
$900$	0	0
$901$	−2.66779	−0.0888771
$902$	0	0
$903$	−11.3776	−0.378623
$904$	0	0
$905$	3.18610	0.105910
$906$	0	0
$907$	39.2412	1.30298	0.651491	−	0.758656i	$-0.274144\pi$
$907$	39.2412	1.30298	0.651491	+	0.758656i	$0.274144\pi$
$908$	0	0
$909$	−7.90005	−0.262028
$910$	0	0
$911$	34.2943	1.13622	0.568110	−	0.822952i	$-0.307675\pi$
$911$	34.2943	1.13622	0.568110	+	0.822952i	$0.307675\pi$
$912$	0	0
$913$	3.59944	0.119124
$914$	0	0
$915$	−17.7061	−0.585346
$916$	0	0
$917$	69.4762	2.29431
$918$	0	0
$919$	−18.1283	−0.597998	−0.298999	−	0.954253i	$-0.596653\pi$
$919$	−18.1283	−0.597998	−0.298999	+	0.954253i	$0.596653\pi$
$920$	0	0
$921$	−5.23105	−0.172369
$922$	0	0
$923$	15.6758	0.515977
$924$	0	0
$925$	−12.1509	−0.399520
$926$	0	0
$927$	2.96020	0.0972257
$928$	0	0
$929$	5.79273	0.190053	0.0950266	−	0.995475i	$-0.469706\pi$
$929$	5.79273	0.190053	0.0950266	+	0.995475i	$0.469706\pi$
$930$	0	0
$931$	41.5106	1.36045
$932$	0	0
$933$	−4.50550	−0.147503
$934$	0	0
$935$	−2.19993	−0.0719453
$936$	0	0
$937$	−11.4079	−0.372679	−0.186339	−	0.982485i	$-0.559662\pi$
$937$	−11.4079	−0.372679	−0.186339	+	0.982485i	$0.559662\pi$
$938$	0	0
$939$	31.9700	1.04330
$940$	0	0
$941$	15.6788	0.511114	0.255557	−	0.966794i	$-0.417741\pi$
$941$	15.6788	0.511114	0.255557	+	0.966794i	$0.417741\pi$
$942$	0	0
$943$	−8.68240	−0.282738
$944$	0	0
$945$	−32.8026	−1.06707
$946$	0	0
$947$	−46.9839	−1.52677	−0.763386	−	0.645943i	$-0.776464\pi$
$947$	−46.9839	−1.52677	−0.763386	+	0.645943i	$0.776464\pi$
$948$	0	0
$949$	56.8128	1.84422
$950$	0	0
$951$	44.2766	1.43577
$952$	0	0
$953$	−6.19866	−0.200794	−0.100397	−	0.994947i	$-0.532011\pi$
$953$	−6.19866	−0.200794	−0.100397	+	0.994947i	$0.532011\pi$
$954$	0	0
$955$	−27.1357	−0.878091
$956$	0	0
$957$	−20.7057	−0.669319
$958$	0	0
$959$	93.8345	3.03007
$960$	0	0
$961$	3.49031	0.112590
$962$	0	0
$963$	−3.89056	−0.125372
$964$	0	0
$965$	−8.48435	−0.273121
$966$	0	0
$967$	37.6246	1.20992	0.604962	−	0.796254i	$-0.293188\pi$
$967$	37.6246	1.20992	0.604962	+	0.796254i	$0.293188\pi$
$968$	0	0
$969$	4.05460	0.130253
$970$	0	0
$971$	−33.8081	−1.08495	−0.542477	−	0.840070i	$-0.682514\pi$
$971$	−33.8081	−1.08495	−0.542477	+	0.840070i	$0.682514\pi$
$972$	0	0
$973$	−47.6181	−1.52657
$974$	0	0
$975$	−31.9664	−1.02374
$976$	0	0
$977$	11.4683	0.366903	0.183451	−	0.983029i	$-0.441273\pi$
$977$	11.4683	0.366903	0.183451	+	0.983029i	$0.441273\pi$
$978$	0	0
$979$	−21.2614	−0.679518
$980$	0	0
$981$	6.96038	0.222228
$982$	0	0
$983$	15.5235	0.495124	0.247562	−	0.968872i	$-0.420371\pi$
$983$	15.5235	0.495124	0.247562	+	0.968872i	$0.420371\pi$
$984$	0	0
$985$	5.65461	0.180171
$986$	0	0
$987$	−19.0382	−0.605993
$988$	0	0
$989$	10.0535	0.319682
$990$	0	0
$991$	57.8531	1.83776	0.918881	−	0.394534i	$-0.129094\pi$
$991$	57.8531	1.83776	0.918881	+	0.394534i	$0.129094\pi$
$992$	0	0
$993$	−1.17773	−0.0373740
$994$	0	0
$995$	0.329195	0.0104362
$996$	0	0
$997$	0.537065	0.0170090	0.00850452	−	0.999964i	$-0.497293\pi$
$997$	0.537065	0.0170090	0.00850452	+	0.999964i	$0.497293\pi$
$998$	0	0
$999$	19.1501	0.605882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.7	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-1.61430 q^{3} -1.23431 q^{5} +4.85046 q^{7} -0.394052 q^{9} +O(q^{10})\)	\(q-1.61430 q^{3} -1.23431 q^{5} +4.85046 q^{7} -0.394052 q^{9} -1.78231 q^{11} -5.69601 q^{13} +1.99254 q^{15} -1.00000 q^{17} +2.51169 q^{19} -7.83007 q^{21} +6.91882 q^{23} -3.47648 q^{25} +5.47900 q^{27} -7.19652 q^{29} +5.87284 q^{31} +2.87718 q^{33} -5.98697 q^{35} +3.49518 q^{37} +9.19504 q^{39} -1.25490 q^{41} +1.45306 q^{43} +0.486382 q^{45} +2.43142 q^{47} +16.5270 q^{49} +1.61430 q^{51} +2.66779 q^{53} +2.19993 q^{55} -4.05460 q^{57} -1.00000 q^{59} -8.88620 q^{61} -1.91133 q^{63} +7.03064 q^{65} -3.95895 q^{67} -11.1690 q^{69} -2.75208 q^{71} -9.97414 q^{73} +5.61206 q^{75} -8.64504 q^{77} +1.01525 q^{79} -7.66257 q^{81} -2.01954 q^{83} +1.23431 q^{85} +11.6173 q^{87} +11.9291 q^{89} -27.6283 q^{91} -9.48050 q^{93} -3.10020 q^{95} -3.33104 q^{97} +0.702323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 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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