LMFDB - Embedded newform 6008.2.a.d.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04212 q^{3} -3.77557 q^{5} +1.06595 q^{7} -1.91399 q^{9} +O(q^{10})$	$q-1.04212 q^{3} -3.77557 q^{5} +1.06595 q^{7} -1.91399 q^{9} +3.78244 q^{11} -4.91846 q^{13} +3.93459 q^{15} +3.62707 q^{17} -6.79098 q^{19} -1.11085 q^{21} -6.61469 q^{23} +9.25495 q^{25} +5.12096 q^{27} -0.497600 q^{29} -6.78005 q^{31} -3.94175 q^{33} -4.02458 q^{35} -4.78420 q^{37} +5.12561 q^{39} +10.4743 q^{41} +1.82057 q^{43} +7.22642 q^{45} -6.43651 q^{47} -5.86375 q^{49} -3.77983 q^{51} -11.9977 q^{53} -14.2809 q^{55} +7.07700 q^{57} +1.88440 q^{59} +7.47287 q^{61} -2.04022 q^{63} +18.5700 q^{65} -0.911857 q^{67} +6.89328 q^{69} -10.0023 q^{71} -14.4666 q^{73} -9.64474 q^{75} +4.03190 q^{77} -3.44309 q^{79} +0.405340 q^{81} +3.28548 q^{83} -13.6943 q^{85} +0.518558 q^{87} +2.12781 q^{89} -5.24283 q^{91} +7.06561 q^{93} +25.6398 q^{95} +17.1341 q^{97} -7.23956 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * 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
$3$	−1.04212	−0.601667	−0.300833	−	0.953677i	$-0.597265\pi$
$3$	−1.04212	−0.601667	−0.300833	+	0.953677i	$0.597265\pi$
$4$	0	0
$5$	−3.77557	−1.68849	−0.844244	−	0.535959i	$-0.819950\pi$
$5$	−3.77557	−1.68849	−0.844244	+	0.535959i	$0.819950\pi$
$6$	0	0
$7$	1.06595	0.402892	0.201446	−	0.979500i	$-0.435436\pi$
$7$	1.06595	0.402892	0.201446	+	0.979500i	$0.435436\pi$
$8$	0	0
$9$	−1.91399	−0.637997
$10$	0	0
$11$	3.78244	1.14045	0.570224	−	0.821489i	$-0.306856\pi$
$11$	3.78244	1.14045	0.570224	+	0.821489i	$0.306856\pi$
$12$	0	0
$13$	−4.91846	−1.36413	−0.682067	−	0.731289i	$-0.738919\pi$
$13$	−4.91846	−1.36413	−0.682067	+	0.731289i	$0.738919\pi$
$14$	0	0
$15$	3.93459	1.01591
$16$	0	0
$17$	3.62707	0.879693	0.439847	−	0.898073i	$-0.355033\pi$
$17$	3.62707	0.879693	0.439847	+	0.898073i	$0.355033\pi$
$18$	0	0
$19$	−6.79098	−1.55796	−0.778979	−	0.627050i	$-0.784262\pi$
$19$	−6.79098	−1.55796	−0.778979	+	0.627050i	$0.784262\pi$
$20$	0	0
$21$	−1.11085	−0.242406
$22$	0	0
$23$	−6.61469	−1.37926	−0.689629	−	0.724163i	$-0.742226\pi$
$23$	−6.61469	−1.37926	−0.689629	+	0.724163i	$0.742226\pi$
$24$	0	0
$25$	9.25495	1.85099
$26$	0	0
$27$	5.12096	0.985528
$28$	0	0
$29$	−0.497600	−0.0924020	−0.0462010	−	0.998932i	$-0.514711\pi$
$29$	−0.497600	−0.0924020	−0.0462010	+	0.998932i	$0.514711\pi$
$30$	0	0
$31$	−6.78005	−1.21773	−0.608867	−	0.793273i	$-0.708376\pi$
$31$	−6.78005	−1.21773	−0.608867	+	0.793273i	$0.708376\pi$
$32$	0	0
$33$	−3.94175	−0.686170
$34$	0	0
$35$	−4.02458	−0.680277
$36$	0	0
$37$	−4.78420	−0.786518	−0.393259	−	0.919428i	$-0.628652\pi$
$37$	−4.78420	−0.786518	−0.393259	+	0.919428i	$0.628652\pi$
$38$	0	0
$39$	5.12561	0.820754
$40$	0	0
$41$	10.4743	1.63581	0.817905	−	0.575354i	$-0.195136\pi$
$41$	10.4743	1.63581	0.817905	+	0.575354i	$0.195136\pi$
$42$	0	0
$43$	1.82057	0.277635	0.138818	−	0.990318i	$-0.455670\pi$
$43$	1.82057	0.277635	0.138818	+	0.990318i	$0.455670\pi$
$44$	0	0
$45$	7.22642	1.07725
$46$	0	0
$47$	−6.43651	−0.938862	−0.469431	−	0.882969i	$-0.655541\pi$
$47$	−6.43651	−0.938862	−0.469431	+	0.882969i	$0.655541\pi$
$48$	0	0
$49$	−5.86375	−0.837678
$50$	0	0
$51$	−3.77983	−0.529282
$52$	0	0
$53$	−11.9977	−1.64800	−0.824002	−	0.566586i	$-0.808264\pi$
$53$	−11.9977	−1.64800	−0.824002	+	0.566586i	$0.808264\pi$
$54$	0	0
$55$	−14.2809	−1.92563
$56$	0	0
$57$	7.07700	0.937371
$58$	0	0
$59$	1.88440	0.245328	0.122664	−	0.992448i	$-0.460856\pi$
$59$	1.88440	0.245328	0.122664	+	0.992448i	$0.460856\pi$
$60$	0	0
$61$	7.47287	0.956803	0.478401	−	0.878141i	$-0.341216\pi$
$61$	7.47287	0.956803	0.478401	+	0.878141i	$0.341216\pi$
$62$	0	0
$63$	−2.04022	−0.257044
$64$	0	0
$65$	18.5700	2.30332
$66$	0	0
$67$	−0.911857	−0.111401	−0.0557005	−	0.998448i	$-0.517739\pi$
$67$	−0.911857	−0.111401	−0.0557005	+	0.998448i	$0.517739\pi$
$68$	0	0
$69$	6.89328	0.829854
$70$	0	0
$71$	−10.0023	−1.18705	−0.593527	−	0.804814i	$-0.702265\pi$
$71$	−10.0023	−1.18705	−0.593527	+	0.804814i	$0.702265\pi$
$72$	0	0
$73$	−14.4666	−1.69319	−0.846596	−	0.532235i	$-0.821352\pi$
$73$	−14.4666	−1.69319	−0.846596	+	0.532235i	$0.821352\pi$
$74$	0	0
$75$	−9.64474	−1.11368
$76$	0	0
$77$	4.03190	0.459477
$78$	0	0
$79$	−3.44309	−0.387378	−0.193689	−	0.981063i	$-0.562045\pi$
$79$	−3.44309	−0.387378	−0.193689	+	0.981063i	$0.562045\pi$
$80$	0	0
$81$	0.405340	0.0450377
$82$	0	0
$83$	3.28548	0.360628	0.180314	−	0.983609i	$-0.442289\pi$
$83$	3.28548	0.360628	0.180314	+	0.983609i	$0.442289\pi$
$84$	0	0
$85$	−13.6943	−1.48535
$86$	0	0
$87$	0.518558	0.0555952
$88$	0	0
$89$	2.12781	0.225547	0.112773	−	0.993621i	$-0.464027\pi$
$89$	2.12781	0.225547	0.112773	+	0.993621i	$0.464027\pi$
$90$	0	0
$91$	−5.24283	−0.549598
$92$	0	0
$93$	7.06561	0.732670
$94$	0	0
$95$	25.6398	2.63059
$96$	0	0
$97$	17.1341	1.73971	0.869854	−	0.493309i	$-0.164213\pi$
$97$	17.1341	1.73971	0.869854	+	0.493309i	$0.164213\pi$
$98$	0	0
$99$	−7.23956	−0.727603
$100$	0	0
$101$	−14.8095	−1.47360	−0.736802	−	0.676108i	$-0.763665\pi$
$101$	−14.8095	−1.47360	−0.736802	+	0.676108i	$0.763665\pi$
$102$	0	0
$103$	8.56425	0.843861	0.421930	−	0.906628i	$-0.361353\pi$
$103$	8.56425	0.843861	0.421930	+	0.906628i	$0.361353\pi$
$104$	0	0
$105$	4.19408	0.409300
$106$	0	0
$107$	−1.30993	−0.126636	−0.0633180	−	0.997993i	$-0.520168\pi$
$107$	−1.30993	−0.126636	−0.0633180	+	0.997993i	$0.520168\pi$
$108$	0	0
$109$	−10.9542	−1.04923	−0.524613	−	0.851341i	$-0.675790\pi$
$109$	−10.9542	−1.04923	−0.524613	+	0.851341i	$0.675790\pi$
$110$	0	0
$111$	4.98570	0.473222
$112$	0	0
$113$	−0.832174	−0.0782844	−0.0391422	−	0.999234i	$-0.512463\pi$
$113$	−0.832174	−0.0782844	−0.0391422	+	0.999234i	$0.512463\pi$
$114$	0	0
$115$	24.9742	2.32886
$116$	0	0
$117$	9.41389	0.870314
$118$	0	0
$119$	3.86628	0.354421
$120$	0	0
$121$	3.30686	0.300624
$122$	0	0
$123$	−10.9154	−0.984212
$124$	0	0
$125$	−16.0649	−1.43689
$126$	0	0
$127$	−6.97743	−0.619147	−0.309573	−	0.950876i	$-0.600186\pi$
$127$	−6.97743	−0.619147	−0.309573	+	0.950876i	$0.600186\pi$
$128$	0	0
$129$	−1.89725	−0.167044
$130$	0	0
$131$	−15.6856	−1.37046	−0.685229	−	0.728328i	$-0.740298\pi$
$131$	−15.6856	−1.37046	−0.685229	+	0.728328i	$0.740298\pi$
$132$	0	0
$133$	−7.23885	−0.627688
$134$	0	0
$135$	−19.3345	−1.66405
$136$	0	0
$137$	10.9520	0.935695	0.467847	−	0.883809i	$-0.345030\pi$
$137$	10.9520	0.935695	0.467847	+	0.883809i	$0.345030\pi$
$138$	0	0
$139$	−2.19273	−0.185985	−0.0929926	−	0.995667i	$-0.529643\pi$
$139$	−2.19273	−0.185985	−0.0929926	+	0.995667i	$0.529643\pi$
$140$	0	0
$141$	6.70760	0.564882
$142$	0	0
$143$	−18.6038	−1.55573
$144$	0	0
$145$	1.87873	0.156020
$146$	0	0
$147$	6.11071	0.504003
$148$	0	0
$149$	−18.2665	−1.49645	−0.748226	−	0.663444i	$-0.769094\pi$
$149$	−18.2665	−1.49645	−0.748226	+	0.663444i	$0.769094\pi$
$150$	0	0
$151$	11.3113	0.920497	0.460249	−	0.887790i	$-0.347760\pi$
$151$	11.3113	0.920497	0.460249	+	0.887790i	$0.347760\pi$
$152$	0	0
$153$	−6.94218	−0.561242
$154$	0	0
$155$	25.5986	2.05613
$156$	0	0
$157$	14.8624	1.18615	0.593074	−	0.805148i	$-0.297914\pi$
$157$	14.8624	1.18615	0.593074	+	0.805148i	$0.297914\pi$
$158$	0	0
$159$	12.5030	0.991550
$160$	0	0
$161$	−7.05094	−0.555692
$162$	0	0
$163$	−6.12294	−0.479586	−0.239793	−	0.970824i	$-0.577080\pi$
$163$	−6.12294	−0.479586	−0.239793	+	0.970824i	$0.577080\pi$
$164$	0	0
$165$	14.8824	1.15859
$166$	0	0
$167$	5.70906	0.441780	0.220890	−	0.975299i	$-0.429104\pi$
$167$	5.70906	0.441780	0.220890	+	0.975299i	$0.429104\pi$
$168$	0	0
$169$	11.1912	0.860863
$170$	0	0
$171$	12.9979	0.993973
$172$	0	0
$173$	16.9738	1.29050	0.645249	−	0.763973i	$-0.276754\pi$
$173$	16.9738	1.29050	0.645249	+	0.763973i	$0.276754\pi$
$174$	0	0
$175$	9.86532	0.745748
$176$	0	0
$177$	−1.96377	−0.147606
$178$	0	0
$179$	−3.69696	−0.276324	−0.138162	−	0.990410i	$-0.544119\pi$
$179$	−3.69696	−0.276324	−0.138162	+	0.990410i	$0.544119\pi$
$180$	0	0
$181$	14.8845	1.10636	0.553179	−	0.833063i	$-0.313415\pi$
$181$	14.8845	1.10636	0.553179	+	0.833063i	$0.313415\pi$
$182$	0	0
$183$	−7.78761	−0.575676
$184$	0	0
$185$	18.0631	1.32803
$186$	0	0
$187$	13.7192	1.00325
$188$	0	0
$189$	5.45869	0.397061
$190$	0	0
$191$	8.67988	0.628054	0.314027	−	0.949414i	$-0.398322\pi$
$191$	8.67988	0.628054	0.314027	+	0.949414i	$0.398322\pi$
$192$	0	0
$193$	23.9511	1.72404	0.862020	−	0.506875i	$-0.169199\pi$
$193$	23.9511	1.72404	0.862020	+	0.506875i	$0.169199\pi$
$194$	0	0
$195$	−19.3521	−1.38583
$196$	0	0
$197$	−19.9710	−1.42287	−0.711437	−	0.702750i	$-0.751955\pi$
$197$	−19.9710	−1.42287	−0.711437	+	0.702750i	$0.751955\pi$
$198$	0	0
$199$	5.33268	0.378024	0.189012	−	0.981975i	$-0.439472\pi$
$199$	5.33268	0.378024	0.189012	+	0.981975i	$0.439472\pi$
$200$	0	0
$201$	0.950262	0.0670263
$202$	0	0
$203$	−0.530418	−0.0372280
$204$	0	0
$205$	−39.5464	−2.76204
$206$	0	0
$207$	12.6605	0.879963
$208$	0	0
$209$	−25.6865	−1.77677
$210$	0	0
$211$	−6.34978	−0.437137	−0.218569	−	0.975822i	$-0.570139\pi$
$211$	−6.34978	−0.437137	−0.218569	+	0.975822i	$0.570139\pi$
$212$	0	0
$213$	10.4236	0.714211
$214$	0	0
$215$	−6.87371	−0.468783
$216$	0	0
$217$	−7.22720	−0.490615
$218$	0	0
$219$	15.0759	1.01874
$220$	0	0
$221$	−17.8396	−1.20002
$222$	0	0
$223$	21.4554	1.43676	0.718380	−	0.695651i	$-0.244884\pi$
$223$	21.4554	1.43676	0.718380	+	0.695651i	$0.244884\pi$
$224$	0	0
$225$	−17.7139	−1.18093
$226$	0	0
$227$	16.0678	1.06646	0.533230	−	0.845970i	$-0.320978\pi$
$227$	16.0678	1.06646	0.533230	+	0.845970i	$0.320978\pi$
$228$	0	0
$229$	−18.4312	−1.21797	−0.608985	−	0.793181i	$-0.708423\pi$
$229$	−18.4312	−1.21797	−0.608985	+	0.793181i	$0.708423\pi$
$230$	0	0
$231$	−4.20171	−0.276452
$232$	0	0
$233$	−10.1492	−0.664897	−0.332448	−	0.943121i	$-0.607875\pi$
$233$	−10.1492	−0.664897	−0.332448	+	0.943121i	$0.607875\pi$
$234$	0	0
$235$	24.3015	1.58526
$236$	0	0
$237$	3.58811	0.233073
$238$	0	0
$239$	24.4447	1.58119	0.790597	−	0.612337i	$-0.209770\pi$
$239$	24.4447	1.58119	0.790597	+	0.612337i	$0.209770\pi$
$240$	0	0
$241$	24.4884	1.57743	0.788717	−	0.614756i	$-0.210746\pi$
$241$	24.4884	1.57743	0.788717	+	0.614756i	$0.210746\pi$
$242$	0	0
$243$	−15.7853	−1.01263
$244$	0	0
$245$	22.1390	1.41441
$246$	0	0
$247$	33.4011	2.12526
$248$	0	0
$249$	−3.42385	−0.216978
$250$	0	0
$251$	−6.67911	−0.421582	−0.210791	−	0.977531i	$-0.567604\pi$
$251$	−6.67911	−0.421582	−0.210791	+	0.977531i	$0.567604\pi$
$252$	0	0
$253$	−25.0197	−1.57297
$254$	0	0
$255$	14.2710	0.893686
$256$	0	0
$257$	14.9558	0.932914	0.466457	−	0.884544i	$-0.345530\pi$
$257$	14.9558	0.932914	0.466457	+	0.884544i	$0.345530\pi$
$258$	0	0
$259$	−5.09973	−0.316882
$260$	0	0
$261$	0.952403	0.0589523
$262$	0	0
$263$	29.2069	1.80098	0.900488	−	0.434880i	$-0.143209\pi$
$263$	29.2069	1.80098	0.900488	+	0.434880i	$0.143209\pi$
$264$	0	0
$265$	45.2980	2.78264
$266$	0	0
$267$	−2.21742	−0.135704
$268$	0	0
$269$	2.47575	0.150949	0.0754746	−	0.997148i	$-0.475953\pi$
$269$	2.47575	0.150949	0.0754746	+	0.997148i	$0.475953\pi$
$270$	0	0
$271$	24.6364	1.49656	0.748279	−	0.663384i	$-0.230881\pi$
$271$	24.6364	1.49656	0.748279	+	0.663384i	$0.230881\pi$
$272$	0	0
$273$	5.46365	0.330675
$274$	0	0
$275$	35.0063	2.11096
$276$	0	0
$277$	−21.7921	−1.30936	−0.654681	−	0.755905i	$-0.727197\pi$
$277$	−21.7921	−1.30936	−0.654681	+	0.755905i	$0.727197\pi$
$278$	0	0
$279$	12.9770	0.776910
$280$	0	0
$281$	−26.8798	−1.60352	−0.801758	−	0.597648i	$-0.796102\pi$
$281$	−26.8798	−1.60352	−0.801758	+	0.597648i	$0.796102\pi$
$282$	0	0
$283$	2.60616	0.154920	0.0774600	−	0.996995i	$-0.475319\pi$
$283$	2.60616	0.154920	0.0774600	+	0.996995i	$0.475319\pi$
$284$	0	0
$285$	−26.7197	−1.58274
$286$	0	0
$287$	11.1651	0.659054
$288$	0	0
$289$	−3.84438	−0.226140
$290$	0	0
$291$	−17.8558	−1.04672
$292$	0	0
$293$	0.497812	0.0290825	0.0145413	−	0.999894i	$-0.495371\pi$
$293$	0.497812	0.0290825	0.0145413	+	0.999894i	$0.495371\pi$
$294$	0	0
$295$	−7.11469	−0.414233
$296$	0	0
$297$	19.3697	1.12394
$298$	0	0
$299$	32.5341	1.88149
$300$	0	0
$301$	1.94064	0.111857
$302$	0	0
$303$	15.4333	0.886619
$304$	0	0
$305$	−28.2144	−1.61555
$306$	0	0
$307$	−5.47123	−0.312259	−0.156130	−	0.987737i	$-0.549902\pi$
$307$	−5.47123	−0.312259	−0.156130	+	0.987737i	$0.549902\pi$
$308$	0	0
$309$	−8.92495	−0.507723
$310$	0	0
$311$	34.2238	1.94065	0.970327	−	0.241795i	$-0.0777361\pi$
$311$	34.2238	1.94065	0.970327	+	0.241795i	$0.0777361\pi$
$312$	0	0
$313$	19.6695	1.11179	0.555893	−	0.831254i	$-0.312377\pi$
$313$	19.6695	1.11179	0.555893	+	0.831254i	$0.312377\pi$
$314$	0	0
$315$	7.70300	0.434015
$316$	0	0
$317$	15.9818	0.897629	0.448815	−	0.893625i	$-0.351846\pi$
$317$	15.9818	0.897629	0.448815	+	0.893625i	$0.351846\pi$
$318$	0	0
$319$	−1.88214	−0.105380
$320$	0	0
$321$	1.36510	0.0761927
$322$	0	0
$323$	−24.6313	−1.37052
$324$	0	0
$325$	−45.5201	−2.52500
$326$	0	0
$327$	11.4156	0.631284
$328$	0	0
$329$	−6.86101	−0.378260
$330$	0	0
$331$	9.80818	0.539107	0.269553	−	0.962985i	$-0.413124\pi$
$331$	9.80818	0.539107	0.269553	+	0.962985i	$0.413124\pi$
$332$	0	0
$333$	9.15693	0.501797
$334$	0	0
$335$	3.44278	0.188099
$336$	0	0
$337$	6.04611	0.329353	0.164676	−	0.986348i	$-0.447342\pi$
$337$	6.04611	0.329353	0.164676	+	0.986348i	$0.447342\pi$
$338$	0	0
$339$	0.867223	0.0471011
$340$	0	0
$341$	−25.6451	−1.38876
$342$	0	0
$343$	−13.7121	−0.740385
$344$	0	0
$345$	−26.0261	−1.40120
$346$	0	0
$347$	2.14260	0.115021	0.0575105	−	0.998345i	$-0.481684\pi$
$347$	2.14260	0.115021	0.0575105	+	0.998345i	$0.481684\pi$
$348$	0	0
$349$	26.1589	1.40026	0.700128	−	0.714018i	$-0.253126\pi$
$349$	26.1589	1.40026	0.700128	+	0.714018i	$0.253126\pi$
$350$	0	0
$351$	−25.1872	−1.34439
$352$	0	0
$353$	−22.9251	−1.22018	−0.610089	−	0.792333i	$-0.708866\pi$
$353$	−22.9251	−1.22018	−0.610089	+	0.792333i	$0.708866\pi$
$354$	0	0
$355$	37.7644	2.00433
$356$	0	0
$357$	−4.02911	−0.213243
$358$	0	0
$359$	35.3189	1.86406	0.932029	−	0.362384i	$-0.118037\pi$
$359$	35.3189	1.86406	0.932029	+	0.362384i	$0.118037\pi$
$360$	0	0
$361$	27.1174	1.42723
$362$	0	0
$363$	−3.44614	−0.180875
$364$	0	0
$365$	54.6199	2.85893
$366$	0	0
$367$	−4.97911	−0.259907	−0.129954	−	0.991520i	$-0.541483\pi$
$367$	−4.97911	−0.259907	−0.129954	+	0.991520i	$0.541483\pi$
$368$	0	0
$369$	−20.0477	−1.04364
$370$	0	0
$371$	−12.7889	−0.663967
$372$	0	0
$373$	−10.9169	−0.565256	−0.282628	−	0.959230i	$-0.591206\pi$
$373$	−10.9169	−0.565256	−0.282628	+	0.959230i	$0.591206\pi$
$374$	0	0
$375$	16.7415	0.864526
$376$	0	0
$377$	2.44743	0.126049
$378$	0	0
$379$	−13.4802	−0.692431	−0.346216	−	0.938155i	$-0.612533\pi$
$379$	−13.4802	−0.692431	−0.346216	+	0.938155i	$0.612533\pi$
$380$	0	0
$381$	7.27130	0.372520
$382$	0	0
$383$	−35.6397	−1.82111	−0.910553	−	0.413393i	$-0.864344\pi$
$383$	−35.6397	−1.82111	−0.910553	+	0.413393i	$0.864344\pi$
$384$	0	0
$385$	−15.2227	−0.775822
$386$	0	0
$387$	−3.48457	−0.177130
$388$	0	0
$389$	13.2533	0.671967	0.335983	−	0.941868i	$-0.390931\pi$
$389$	13.2533	0.671967	0.335983	+	0.941868i	$0.390931\pi$
$390$	0	0
$391$	−23.9919	−1.21332
$392$	0	0
$393$	16.3462	0.824558
$394$	0	0
$395$	12.9996	0.654083
$396$	0	0
$397$	28.4549	1.42811	0.714055	−	0.700090i	$-0.246857\pi$
$397$	28.4549	1.42811	0.714055	+	0.700090i	$0.246857\pi$
$398$	0	0
$399$	7.54373	0.377659
$400$	0	0
$401$	−24.4178	−1.21936	−0.609682	−	0.792646i	$-0.708703\pi$
$401$	−24.4178	−1.21936	−0.609682	+	0.792646i	$0.708703\pi$
$402$	0	0
$403$	33.3474	1.66115
$404$	0	0
$405$	−1.53039	−0.0760456
$406$	0	0
$407$	−18.0960	−0.896984
$408$	0	0
$409$	−19.3792	−0.958238	−0.479119	−	0.877750i	$-0.659044\pi$
$409$	−19.3792	−0.958238	−0.479119	+	0.877750i	$0.659044\pi$
$410$	0	0
$411$	−11.4133	−0.562976
$412$	0	0
$413$	2.00868	0.0988406
$414$	0	0
$415$	−12.4046	−0.608916
$416$	0	0
$417$	2.28508	0.111901
$418$	0	0
$419$	32.5329	1.58934	0.794668	−	0.607044i	$-0.207645\pi$
$419$	32.5329	1.58934	0.794668	+	0.607044i	$0.207645\pi$
$420$	0	0
$421$	−0.0950093	−0.00463047	−0.00231524	−	0.999997i	$-0.500737\pi$
$421$	−0.0950093	−0.00463047	−0.00231524	+	0.999997i	$0.500737\pi$
$422$	0	0
$423$	12.3194	0.598991
$424$	0	0
$425$	33.5683	1.62830
$426$	0	0
$427$	7.96571	0.385488
$428$	0	0
$429$	19.3873	0.936028
$430$	0	0
$431$	−22.7145	−1.09412	−0.547059	−	0.837094i	$-0.684253\pi$
$431$	−22.7145	−1.09412	−0.547059	+	0.837094i	$0.684253\pi$
$432$	0	0
$433$	3.48799	0.167622	0.0838111	−	0.996482i	$-0.473291\pi$
$433$	3.48799	0.167622	0.0838111	+	0.996482i	$0.473291\pi$
$434$	0	0
$435$	−1.95785	−0.0938719
$436$	0	0
$437$	44.9202	2.14883
$438$	0	0
$439$	7.56345	0.360984	0.180492	−	0.983576i	$-0.442231\pi$
$439$	7.56345	0.360984	0.180492	+	0.983576i	$0.442231\pi$
$440$	0	0
$441$	11.2232	0.534436
$442$	0	0
$443$	−17.0832	−0.811648	−0.405824	−	0.913951i	$-0.633015\pi$
$443$	−17.0832	−0.811648	−0.405824	+	0.913951i	$0.633015\pi$
$444$	0	0
$445$	−8.03369	−0.380833
$446$	0	0
$447$	19.0359	0.900365
$448$	0	0
$449$	−17.0703	−0.805595	−0.402798	−	0.915289i	$-0.631962\pi$
$449$	−17.0703	−0.805595	−0.402798	+	0.915289i	$0.631962\pi$
$450$	0	0
$451$	39.6184	1.86556
$452$	0	0
$453$	−11.7877	−0.553833
$454$	0	0
$455$	19.7947	0.927990
$456$	0	0
$457$	13.4545	0.629374	0.314687	−	0.949195i	$-0.398100\pi$
$457$	13.4545	0.629374	0.314687	+	0.949195i	$0.398100\pi$
$458$	0	0
$459$	18.5741	0.866962
$460$	0	0
$461$	−32.8994	−1.53228	−0.766139	−	0.642675i	$-0.777825\pi$
$461$	−32.8994	−1.53228	−0.766139	+	0.642675i	$0.777825\pi$
$462$	0	0
$463$	−12.7003	−0.590231	−0.295116	−	0.955462i	$-0.595358\pi$
$463$	−12.7003	−0.590231	−0.295116	+	0.955462i	$0.595358\pi$
$464$	0	0
$465$	−26.6767	−1.23710
$466$	0	0
$467$	19.9145	0.921533	0.460766	−	0.887521i	$-0.347575\pi$
$467$	19.9145	0.921533	0.460766	+	0.887521i	$0.347575\pi$
$468$	0	0
$469$	−0.971995	−0.0448826
$470$	0	0
$471$	−15.4883	−0.713665
$472$	0	0
$473$	6.88622	0.316629
$474$	0	0
$475$	−62.8502	−2.88376
$476$	0	0
$477$	22.9634	1.05142
$478$	0	0
$479$	28.4099	1.29808	0.649040	−	0.760754i	$-0.275171\pi$
$479$	28.4099	1.29808	0.649040	+	0.760754i	$0.275171\pi$
$480$	0	0
$481$	23.5309	1.07292
$482$	0	0
$483$	7.34790	0.334341
$484$	0	0
$485$	−64.6912	−2.93748
$486$	0	0
$487$	15.8015	0.716037	0.358018	−	0.933715i	$-0.383453\pi$
$487$	15.8015	0.716037	0.358018	+	0.933715i	$0.383453\pi$
$488$	0	0
$489$	6.38083	0.288551
$490$	0	0
$491$	24.2930	1.09633	0.548164	−	0.836371i	$-0.315327\pi$
$491$	24.2930	1.09633	0.548164	+	0.836371i	$0.315327\pi$
$492$	0	0
$493$	−1.80483	−0.0812854
$494$	0	0
$495$	27.3335	1.22855
$496$	0	0
$497$	−10.6620	−0.478254
$498$	0	0
$499$	−22.3723	−1.00152	−0.500761	−	0.865585i	$-0.666947\pi$
$499$	−22.3723	−1.00152	−0.500761	+	0.865585i	$0.666947\pi$
$500$	0	0
$501$	−5.94951	−0.265804
$502$	0	0
$503$	−9.15386	−0.408150	−0.204075	−	0.978955i	$-0.565419\pi$
$503$	−9.15386	−0.408150	−0.204075	+	0.978955i	$0.565419\pi$
$504$	0	0
$505$	55.9145	2.48816
$506$	0	0
$507$	−11.6626	−0.517952
$508$	0	0
$509$	−1.49487	−0.0662587	−0.0331294	−	0.999451i	$-0.510547\pi$
$509$	−1.49487	−0.0662587	−0.0331294	+	0.999451i	$0.510547\pi$
$510$	0	0
$511$	−15.4207	−0.682173
$512$	0	0
$513$	−34.7763	−1.53541
$514$	0	0
$515$	−32.3349	−1.42485
$516$	0	0
$517$	−24.3457	−1.07072
$518$	0	0
$519$	−17.6887	−0.776449
$520$	0	0
$521$	−0.984211	−0.0431191	−0.0215595	−	0.999768i	$-0.506863\pi$
$521$	−0.984211	−0.0431191	−0.0215595	+	0.999768i	$0.506863\pi$
$522$	0	0
$523$	27.0888	1.18451	0.592256	−	0.805750i	$-0.298237\pi$
$523$	27.0888	1.18451	0.592256	+	0.805750i	$0.298237\pi$
$524$	0	0
$525$	−10.2808	−0.448692
$526$	0	0
$527$	−24.5917	−1.07123
$528$	0	0
$529$	20.7541	0.902353
$530$	0	0
$531$	−3.60673	−0.156519
$532$	0	0
$533$	−51.5173	−2.23146
$534$	0	0
$535$	4.94575	0.213823
$536$	0	0
$537$	3.85266	0.166255
$538$	0	0
$539$	−22.1793	−0.955329
$540$	0	0
$541$	21.1927	0.911144	0.455572	−	0.890199i	$-0.349435\pi$
$541$	21.1927	0.911144	0.455572	+	0.890199i	$0.349435\pi$
$542$	0	0
$543$	−15.5114	−0.665659
$544$	0	0
$545$	41.3585	1.77160
$546$	0	0
$547$	21.9529	0.938638	0.469319	−	0.883029i	$-0.344499\pi$
$547$	21.9529	0.938638	0.469319	+	0.883029i	$0.344499\pi$
$548$	0	0
$549$	−14.3030	−0.610438
$550$	0	0
$551$	3.37919	0.143959
$552$	0	0
$553$	−3.67017	−0.156071
$554$	0	0
$555$	−18.8239	−0.799029
$556$	0	0
$557$	26.7465	1.13329	0.566643	−	0.823963i	$-0.308242\pi$
$557$	26.7465	1.13329	0.566643	+	0.823963i	$0.308242\pi$
$558$	0	0
$559$	−8.95442	−0.378732
$560$	0	0
$561$	−14.2970	−0.603619
$562$	0	0
$563$	−2.48438	−0.104704	−0.0523521	−	0.998629i	$-0.516672\pi$
$563$	−2.48438	−0.104704	−0.0523521	+	0.998629i	$0.516672\pi$
$564$	0	0
$565$	3.14193	0.132182
$566$	0	0
$567$	0.432072	0.0181453
$568$	0	0
$569$	−10.5255	−0.441253	−0.220626	−	0.975358i	$-0.570810\pi$
$569$	−10.5255	−0.441253	−0.220626	+	0.975358i	$0.570810\pi$
$570$	0	0
$571$	−12.2697	−0.513473	−0.256736	−	0.966481i	$-0.582647\pi$
$571$	−12.2697	−0.513473	−0.256736	+	0.966481i	$0.582647\pi$
$572$	0	0
$573$	−9.04546	−0.377879
$574$	0	0
$575$	−61.2186	−2.55299
$576$	0	0
$577$	−7.43466	−0.309509	−0.154754	−	0.987953i	$-0.549459\pi$
$577$	−7.43466	−0.309509	−0.154754	+	0.987953i	$0.549459\pi$
$578$	0	0
$579$	−24.9599	−1.03730
$580$	0	0
$581$	3.50216	0.145294
$582$	0	0
$583$	−45.3804	−1.87947
$584$	0	0
$585$	−35.5428	−1.46951
$586$	0	0
$587$	−17.4175	−0.718896	−0.359448	−	0.933165i	$-0.617035\pi$
$587$	−17.4175	−0.718896	−0.359448	+	0.933165i	$0.617035\pi$
$588$	0	0
$589$	46.0432	1.89718
$590$	0	0
$591$	20.8121	0.856096
$592$	0	0
$593$	−2.13753	−0.0877780	−0.0438890	−	0.999036i	$-0.513975\pi$
$593$	−2.13753	−0.0877780	−0.0438890	+	0.999036i	$0.513975\pi$
$594$	0	0
$595$	−14.5974	−0.598435
$596$	0	0
$597$	−5.55728	−0.227444
$598$	0	0
$599$	−24.0165	−0.981289	−0.490645	−	0.871360i	$-0.663239\pi$
$599$	−24.0165	−0.981289	−0.490645	+	0.871360i	$0.663239\pi$
$600$	0	0
$601$	−25.3962	−1.03593	−0.517967	−	0.855401i	$-0.673311\pi$
$601$	−25.3962	−1.03593	−0.517967	+	0.855401i	$0.673311\pi$
$602$	0	0
$603$	1.74529	0.0710736
$604$	0	0
$605$	−12.4853	−0.507599
$606$	0	0
$607$	−48.2406	−1.95803	−0.979014	−	0.203795i	$-0.934672\pi$
$607$	−48.2406	−1.95803	−0.979014	+	0.203795i	$0.934672\pi$
$608$	0	0
$609$	0.552757	0.0223989
$610$	0	0
$611$	31.6577	1.28073
$612$	0	0
$613$	30.9473	1.24995	0.624974	−	0.780645i	$-0.285109\pi$
$613$	30.9473	1.24995	0.624974	+	0.780645i	$0.285109\pi$
$614$	0	0
$615$	41.2120	1.66183
$616$	0	0
$617$	−40.3984	−1.62638	−0.813189	−	0.582000i	$-0.802270\pi$
$617$	−40.3984	−1.62638	−0.813189	+	0.582000i	$0.802270\pi$
$618$	0	0
$619$	−0.609242	−0.0244875	−0.0122437	−	0.999925i	$-0.503897\pi$
$619$	−0.609242	−0.0244875	−0.0122437	+	0.999925i	$0.503897\pi$
$620$	0	0
$621$	−33.8735	−1.35930
$622$	0	0
$623$	2.26814	0.0908710
$624$	0	0
$625$	14.3793	0.575174
$626$	0	0
$627$	26.7683	1.06902
$628$	0	0
$629$	−17.3526	−0.691895
$630$	0	0
$631$	29.0046	1.15465	0.577327	−	0.816513i	$-0.304096\pi$
$631$	29.0046	1.15465	0.577327	+	0.816513i	$0.304096\pi$
$632$	0	0
$633$	6.61722	0.263011
$634$	0	0
$635$	26.3438	1.04542
$636$	0	0
$637$	28.8406	1.14271
$638$	0	0
$639$	19.1443	0.757337
$640$	0	0
$641$	11.7251	0.463114	0.231557	−	0.972821i	$-0.425618\pi$
$641$	11.7251	0.463114	0.231557	+	0.972821i	$0.425618\pi$
$642$	0	0
$643$	−33.2429	−1.31097	−0.655485	−	0.755208i	$-0.727536\pi$
$643$	−33.2429	−1.31097	−0.655485	+	0.755208i	$0.727536\pi$
$644$	0	0
$645$	7.16321	0.282051
$646$	0	0
$647$	−6.69921	−0.263373	−0.131687	−	0.991291i	$-0.542039\pi$
$647$	−6.69921	−0.263373	−0.131687	+	0.991291i	$0.542039\pi$
$648$	0	0
$649$	7.12763	0.279784
$650$	0	0
$651$	7.53159	0.295186
$652$	0	0
$653$	−35.5238	−1.39016	−0.695078	−	0.718935i	$-0.744630\pi$
$653$	−35.5238	−1.39016	−0.695078	+	0.718935i	$0.744630\pi$
$654$	0	0
$655$	59.2221	2.31400
$656$	0	0
$657$	27.6890	1.08025
$658$	0	0
$659$	−36.5629	−1.42429	−0.712143	−	0.702034i	$-0.752275\pi$
$659$	−36.5629	−1.42429	−0.712143	+	0.702034i	$0.752275\pi$
$660$	0	0
$661$	16.6865	0.649030	0.324515	−	0.945881i	$-0.394799\pi$
$661$	16.6865	0.649030	0.324515	+	0.945881i	$0.394799\pi$
$662$	0	0
$663$	18.5909	0.722012
$664$	0	0
$665$	27.3308	1.05984
$666$	0	0
$667$	3.29147	0.127446
$668$	0	0
$669$	−22.3590	−0.864450
$670$	0	0
$671$	28.2657	1.09118
$672$	0	0
$673$	−32.6137	−1.25717	−0.628584	−	0.777742i	$-0.716365\pi$
$673$	−32.6137	−1.25717	−0.628584	+	0.777742i	$0.716365\pi$
$674$	0	0
$675$	47.3942	1.82420
$676$	0	0
$677$	−20.3775	−0.783170	−0.391585	−	0.920142i	$-0.628073\pi$
$677$	−20.3775	−0.783170	−0.391585	+	0.920142i	$0.628073\pi$
$678$	0	0
$679$	18.2642	0.700914
$680$	0	0
$681$	−16.7446	−0.641654
$682$	0	0
$683$	9.74343	0.372822	0.186411	−	0.982472i	$-0.440314\pi$
$683$	9.74343	0.372822	0.186411	+	0.982472i	$0.440314\pi$
$684$	0	0
$685$	−41.3502	−1.57991
$686$	0	0
$687$	19.2075	0.732812
$688$	0	0
$689$	59.0100	2.24810
$690$	0	0
$691$	−24.2373	−0.922030	−0.461015	−	0.887392i	$-0.652515\pi$
$691$	−24.2373	−0.922030	−0.461015	+	0.887392i	$0.652515\pi$
$692$	0	0
$693$	−7.71702	−0.293145
$694$	0	0
$695$	8.27882	0.314034
$696$	0	0
$697$	37.9910	1.43901
$698$	0	0
$699$	10.5767	0.400046
$700$	0	0
$701$	−16.6917	−0.630436	−0.315218	−	0.949019i	$-0.602078\pi$
$701$	−16.6917	−0.630436	−0.315218	+	0.949019i	$0.602078\pi$
$702$	0	0
$703$	32.4894	1.22536
$704$	0	0
$705$	−25.3250	−0.953796
$706$	0	0
$707$	−15.7862	−0.593703
$708$	0	0
$709$	40.7761	1.53138	0.765689	−	0.643211i	$-0.222398\pi$
$709$	40.7761	1.53138	0.765689	+	0.643211i	$0.222398\pi$
$710$	0	0
$711$	6.59005	0.247146
$712$	0	0
$713$	44.8479	1.67957
$714$	0	0
$715$	70.2399	2.62682
$716$	0	0
$717$	−25.4742	−0.951352
$718$	0	0
$719$	−20.2203	−0.754088	−0.377044	−	0.926195i	$-0.623059\pi$
$719$	−20.2203	−0.754088	−0.377044	+	0.926195i	$0.623059\pi$
$720$	0	0
$721$	9.12907	0.339984
$722$	0	0
$723$	−25.5197	−0.949090
$724$	0	0
$725$	−4.60527	−0.171035
$726$	0	0
$727$	50.3397	1.86700	0.933498	−	0.358582i	$-0.116740\pi$
$727$	50.3397	1.86700	0.933498	+	0.358582i	$0.116740\pi$
$728$	0	0
$729$	15.2341	0.564226
$730$	0	0
$731$	6.60335	0.244234
$732$	0	0
$733$	31.3725	1.15877	0.579386	−	0.815053i	$-0.303293\pi$
$733$	31.3725	1.15877	0.579386	+	0.815053i	$0.303293\pi$
$734$	0	0
$735$	−23.0714	−0.851003
$736$	0	0
$737$	−3.44905	−0.127047
$738$	0	0
$739$	40.5507	1.49168	0.745840	−	0.666125i	$-0.232048\pi$
$739$	40.5507	1.49168	0.745840	+	0.666125i	$0.232048\pi$
$740$	0	0
$741$	−34.8079	−1.27870
$742$	0	0
$743$	−8.77347	−0.321867	−0.160934	−	0.986965i	$-0.551451\pi$
$743$	−8.77347	−0.321867	−0.160934	+	0.986965i	$0.551451\pi$
$744$	0	0
$745$	68.9666	2.52674
$746$	0	0
$747$	−6.28838	−0.230080
$748$	0	0
$749$	−1.39632	−0.0510206
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	6.96041	0.253652
$754$	0	0
$755$	−42.7065	−1.55425
$756$	0	0
$757$	35.5445	1.29189	0.645944	−	0.763385i	$-0.276464\pi$
$757$	35.5445	1.29189	0.645944	+	0.763385i	$0.276464\pi$
$758$	0	0
$759$	26.0734	0.946406
$760$	0	0
$761$	−4.68976	−0.170004	−0.0850019	−	0.996381i	$-0.527090\pi$
$761$	−4.68976	−0.170004	−0.0850019	+	0.996381i	$0.527090\pi$
$762$	0	0
$763$	−11.6767	−0.422724
$764$	0	0
$765$	26.2107	0.947650
$766$	0	0
$767$	−9.26834	−0.334660
$768$	0	0
$769$	−2.45247	−0.0884384	−0.0442192	−	0.999022i	$-0.514080\pi$
$769$	−2.45247	−0.0884384	−0.0442192	+	0.999022i	$0.514080\pi$
$770$	0	0
$771$	−15.5857	−0.561303
$772$	0	0
$773$	−37.8275	−1.36056	−0.680280	−	0.732952i	$-0.738142\pi$
$773$	−37.8275	−1.36056	−0.680280	+	0.732952i	$0.738142\pi$
$774$	0	0
$775$	−62.7490	−2.25401
$776$	0	0
$777$	5.31451	0.190657
$778$	0	0
$779$	−71.1307	−2.54852
$780$	0	0
$781$	−37.8331	−1.35377
$782$	0	0
$783$	−2.54819	−0.0910648
$784$	0	0
$785$	−56.1140	−2.00279
$786$	0	0
$787$	36.0650	1.28558	0.642790	−	0.766042i	$-0.277777\pi$
$787$	36.0650	1.28558	0.642790	+	0.766042i	$0.277777\pi$
$788$	0	0
$789$	−30.4370	−1.08359
$790$	0	0
$791$	−0.887057	−0.0315401
$792$	0	0
$793$	−36.7550	−1.30521
$794$	0	0
$795$	−47.2058	−1.67422
$796$	0	0
$797$	−3.80405	−0.134746	−0.0673732	−	0.997728i	$-0.521462\pi$
$797$	−3.80405	−0.134746	−0.0673732	+	0.997728i	$0.521462\pi$
$798$	0	0
$799$	−23.3457	−0.825910
$800$	0	0
$801$	−4.07260	−0.143898
$802$	0	0
$803$	−54.7192	−1.93100
$804$	0	0
$805$	26.6213	0.938278
$806$	0	0
$807$	−2.58002	−0.0908211
$808$	0	0
$809$	−43.7889	−1.53953	−0.769767	−	0.638325i	$-0.779628\pi$
$809$	−43.7889	−1.53953	−0.769767	+	0.638325i	$0.779628\pi$
$810$	0	0
$811$	−32.0687	−1.12608	−0.563042	−	0.826429i	$-0.690369\pi$
$811$	−32.0687	−1.12608	−0.563042	+	0.826429i	$0.690369\pi$
$812$	0	0
$813$	−25.6741	−0.900429
$814$	0	0
$815$	23.1176	0.809775
$816$	0	0
$817$	−12.3635	−0.432544
$818$	0	0
$819$	10.0347	0.350642
$820$	0	0
$821$	3.31410	0.115663	0.0578315	−	0.998326i	$-0.481581\pi$
$821$	3.31410	0.115663	0.0578315	+	0.998326i	$0.481581\pi$
$822$	0	0
$823$	−30.1928	−1.05246	−0.526228	−	0.850344i	$-0.676394\pi$
$823$	−30.1928	−1.05246	−0.526228	+	0.850344i	$0.676394\pi$
$824$	0	0
$825$	−36.4807	−1.27009
$826$	0	0
$827$	−37.5201	−1.30470	−0.652351	−	0.757917i	$-0.726217\pi$
$827$	−37.5201	−1.30470	−0.652351	+	0.757917i	$0.726217\pi$
$828$	0	0
$829$	6.59885	0.229188	0.114594	−	0.993412i	$-0.463443\pi$
$829$	6.59885	0.229188	0.114594	+	0.993412i	$0.463443\pi$
$830$	0	0
$831$	22.7100	0.787799
$832$	0	0
$833$	−21.2682	−0.736900
$834$	0	0
$835$	−21.5550	−0.745940
$836$	0	0
$837$	−34.7203	−1.20011
$838$	0	0
$839$	13.7908	0.476111	0.238056	−	0.971252i	$-0.423490\pi$
$839$	13.7908	0.476111	0.238056	+	0.971252i	$0.423490\pi$
$840$	0	0
$841$	−28.7524	−0.991462
$842$	0	0
$843$	28.0119	0.964782
$844$	0	0
$845$	−42.2533	−1.45356
$846$	0	0
$847$	3.52495	0.121119
$848$	0	0
$849$	−2.71592	−0.0932102
$850$	0	0
$851$	31.6460	1.08481
$852$	0	0
$853$	13.5533	0.464056	0.232028	−	0.972709i	$-0.425464\pi$
$853$	13.5533	0.464056	0.232028	+	0.972709i	$0.425464\pi$
$854$	0	0
$855$	−49.0745	−1.67831
$856$	0	0
$857$	41.9878	1.43428	0.717138	−	0.696931i	$-0.245452\pi$
$857$	41.9878	1.43428	0.717138	+	0.696931i	$0.245452\pi$
$858$	0	0
$859$	−9.61731	−0.328138	−0.164069	−	0.986449i	$-0.552462\pi$
$859$	−9.61731	−0.328138	−0.164069	+	0.986449i	$0.552462\pi$
$860$	0	0
$861$	−11.6353	−0.396531
$862$	0	0
$863$	37.6936	1.28311	0.641553	−	0.767079i	$-0.278291\pi$
$863$	37.6936	1.28311	0.641553	+	0.767079i	$0.278291\pi$
$864$	0	0
$865$	−64.0860	−2.17899
$866$	0	0
$867$	4.00629	0.136061
$868$	0	0
$869$	−13.0233	−0.441785
$870$	0	0
$871$	4.48493	0.151966
$872$	0	0
$873$	−32.7946	−1.10993
$874$	0	0
$875$	−17.1244	−0.578909
$876$	0	0
$877$	−42.4673	−1.43402	−0.717009	−	0.697063i	$-0.754490\pi$
$877$	−42.4673	−1.43402	−0.717009	+	0.697063i	$0.754490\pi$
$878$	0	0
$879$	−0.518779	−0.0174980
$880$	0	0
$881$	12.8679	0.433529	0.216765	−	0.976224i	$-0.430450\pi$
$881$	12.8679	0.433529	0.216765	+	0.976224i	$0.430450\pi$
$882$	0	0
$883$	17.6095	0.592606	0.296303	−	0.955094i	$-0.404246\pi$
$883$	17.6095	0.592606	0.296303	+	0.955094i	$0.404246\pi$
$884$	0	0
$885$	7.41434	0.249230
$886$	0	0
$887$	−57.8473	−1.94232	−0.971161	−	0.238425i	$-0.923369\pi$
$887$	−57.8473	−1.94232	−0.971161	+	0.238425i	$0.923369\pi$
$888$	0	0
$889$	−7.43760	−0.249449
$890$	0	0
$891$	1.53317	0.0513632
$892$	0	0
$893$	43.7102	1.46271
$894$	0	0
$895$	13.9581	0.466569
$896$	0	0
$897$	−33.9043	−1.13203
$898$	0	0
$899$	3.37376	0.112521
$900$	0	0
$901$	−43.5163	−1.44974
$902$	0	0
$903$	−2.02238	−0.0673005
$904$	0	0
$905$	−56.1976	−1.86807
$906$	0	0
$907$	28.7317	0.954021	0.477010	−	0.878898i	$-0.341720\pi$
$907$	28.7317	0.954021	0.477010	+	0.878898i	$0.341720\pi$
$908$	0	0
$909$	28.3453	0.940156
$910$	0	0
$911$	27.7914	0.920771	0.460385	−	0.887719i	$-0.347711\pi$
$911$	27.7914	0.920771	0.460385	+	0.887719i	$0.347711\pi$
$912$	0	0
$913$	12.4271	0.411278
$914$	0	0
$915$	29.4027	0.972022
$916$	0	0
$917$	−16.7201	−0.552146
$918$	0	0
$919$	49.7048	1.63961	0.819805	−	0.572642i	$-0.194082\pi$
$919$	49.7048	1.63961	0.819805	+	0.572642i	$0.194082\pi$
$920$	0	0
$921$	5.70166	0.187876
$922$	0	0
$923$	49.1959	1.61930
$924$	0	0
$925$	−44.2776	−1.45584
$926$	0	0
$927$	−16.3919	−0.538381
$928$	0	0
$929$	23.2502	0.762815	0.381408	−	0.924407i	$-0.375439\pi$
$929$	23.2502	0.762815	0.381408	+	0.924407i	$0.375439\pi$
$930$	0	0
$931$	39.8206	1.30507
$932$	0	0
$933$	−35.6652	−1.16763
$934$	0	0
$935$	−51.7977	−1.69397
$936$	0	0
$937$	−37.0599	−1.21069	−0.605347	−	0.795962i	$-0.706966\pi$
$937$	−37.0599	−1.21069	−0.605347	+	0.795962i	$0.706966\pi$
$938$	0	0
$939$	−20.4979	−0.668924
$940$	0	0
$941$	−4.84951	−0.158089	−0.0790447	−	0.996871i	$-0.525187\pi$
$941$	−4.84951	−0.158089	−0.0790447	+	0.996871i	$0.525187\pi$
$942$	0	0
$943$	−69.2842	−2.25620
$944$	0	0
$945$	−20.6097	−0.670433
$946$	0	0
$947$	37.7643	1.22718	0.613588	−	0.789626i	$-0.289726\pi$
$947$	37.7643	1.22718	0.613588	+	0.789626i	$0.289726\pi$
$948$	0	0
$949$	71.1536	2.30974
$950$	0	0
$951$	−16.6549	−0.540074
$952$	0	0
$953$	2.88556	0.0934725	0.0467362	−	0.998907i	$-0.485118\pi$
$953$	2.88556	0.0934725	0.0467362	+	0.998907i	$0.485118\pi$
$954$	0	0
$955$	−32.7715	−1.06046
$956$	0	0
$957$	1.96141	0.0634035
$958$	0	0
$959$	11.6743	0.376984
$960$	0	0
$961$	14.9691	0.482874
$962$	0	0
$963$	2.50720	0.0807934
$964$	0	0
$965$	−90.4292	−2.91102
$966$	0	0
$967$	−50.5627	−1.62599	−0.812994	−	0.582273i	$-0.802164\pi$
$967$	−50.5627	−1.62599	−0.812994	+	0.582273i	$0.802164\pi$
$968$	0	0
$969$	25.6688	0.824599
$970$	0	0
$971$	−17.2136	−0.552410	−0.276205	−	0.961099i	$-0.589077\pi$
$971$	−17.2136	−0.552410	−0.276205	+	0.961099i	$0.589077\pi$
$972$	0	0
$973$	−2.33735	−0.0749319
$974$	0	0
$975$	47.4372	1.51921
$976$	0	0
$977$	19.3415	0.618788	0.309394	−	0.950934i	$-0.399874\pi$
$977$	19.3415	0.618788	0.309394	+	0.950934i	$0.399874\pi$
$978$	0	0
$979$	8.04830	0.257225
$980$	0	0
$981$	20.9663	0.669403
$982$	0	0
$983$	15.2080	0.485060	0.242530	−	0.970144i	$-0.422023\pi$
$983$	15.2080	0.485060	0.242530	+	0.970144i	$0.422023\pi$
$984$	0	0
$985$	75.4019	2.40251
$986$	0	0
$987$	7.14997	0.227586
$988$	0	0
$989$	−12.0425	−0.382930
$990$	0	0
$991$	32.1262	1.02052	0.510262	−	0.860019i	$-0.329549\pi$
$991$	32.1262	1.02052	0.510262	+	0.860019i	$0.329549\pi$
$992$	0	0
$993$	−10.2213	−0.324362
$994$	0	0
$995$	−20.1339	−0.638289
$996$	0	0
$997$	30.7183	0.972858	0.486429	−	0.873720i	$-0.338299\pi$
$997$	30.7183	0.972858	0.486429	+	0.873720i	$0.338299\pi$
$998$	0	0
$999$	−24.4997	−0.775136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.18	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.18	✓	49	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.04212 q^{3} -3.77557 q^{5} +1.06595 q^{7} -1.91399 q^{9} +O(q^{10})\)	\(q-1.04212 q^{3} -3.77557 q^{5} +1.06595 q^{7} -1.91399 q^{9} +3.78244 q^{11} -4.91846 q^{13} +3.93459 q^{15} +3.62707 q^{17} -6.79098 q^{19} -1.11085 q^{21} -6.61469 q^{23} +9.25495 q^{25} +5.12096 q^{27} -0.497600 q^{29} -6.78005 q^{31} -3.94175 q^{33} -4.02458 q^{35} -4.78420 q^{37} +5.12561 q^{39} +10.4743 q^{41} +1.82057 q^{43} +7.22642 q^{45} -6.43651 q^{47} -5.86375 q^{49} -3.77983 q^{51} -11.9977 q^{53} -14.2809 q^{55} +7.07700 q^{57} +1.88440 q^{59} +7.47287 q^{61} -2.04022 q^{63} +18.5700 q^{65} -0.911857 q^{67} +6.89328 q^{69} -10.0023 q^{71} -14.4666 q^{73} -9.64474 q^{75} +4.03190 q^{77} -3.44309 q^{79} +0.405340 q^{81} +3.28548 q^{83} -13.6943 q^{85} +0.518558 q^{87} +2.12781 q^{89} -5.24283 q^{91} +7.06561 q^{93} +25.6398 q^{95} +17.1341 q^{97} -7.23956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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