LMFDB - Embedded newform 2008.4.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,4,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2008.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2008.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:	$118.475835292$
Analytic rank:	$0$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.86273 q^{3} -21.2394 q^{5} +15.5711 q^{7} +7.37159 q^{9} +O(q^{10})$	$q-5.86273 q^{3} -21.2394 q^{5} +15.5711 q^{7} +7.37159 q^{9} +54.0725 q^{11} -11.0637 q^{13} +124.521 q^{15} +131.327 q^{17} -26.8871 q^{19} -91.2894 q^{21} +99.4551 q^{23} +326.113 q^{25} +115.076 q^{27} -226.855 q^{29} +328.862 q^{31} -317.013 q^{33} -330.722 q^{35} -394.694 q^{37} +64.8635 q^{39} -269.555 q^{41} -103.200 q^{43} -156.568 q^{45} +156.929 q^{47} -100.540 q^{49} -769.932 q^{51} -613.389 q^{53} -1148.47 q^{55} +157.632 q^{57} +221.572 q^{59} +917.017 q^{61} +114.784 q^{63} +234.987 q^{65} +644.395 q^{67} -583.078 q^{69} +939.697 q^{71} -227.180 q^{73} -1911.91 q^{75} +841.971 q^{77} -298.737 q^{79} -873.693 q^{81} +578.892 q^{83} -2789.30 q^{85} +1329.99 q^{87} -525.709 q^{89} -172.274 q^{91} -1928.03 q^{93} +571.067 q^{95} +339.349 q^{97} +398.601 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 5 q^{3} + 33 q^{5} + 4 q^{7} + 641 q^{9}+O(q^{10}) $ 54 * q + 5 * q^3 + 33 * q^5 + 4 * q^7 + 641 * q^9	$ 54 q + 5 q^{3} + 33 q^{5} + 4 q^{7} + 641 q^{9} + 55 q^{11} + 71 q^{13} + 101 q^{15} + 301 q^{17} - 122 q^{19} + 195 q^{21} + 317 q^{23} + 2111 q^{25} + 245 q^{27} + 253 q^{29} - 31 q^{31} + 994 q^{33} + 602 q^{35} + 482 q^{37} - 157 q^{39} + 1049 q^{41} - 218 q^{43} + 942 q^{45} + 1349 q^{47} + 5000 q^{49} - 689 q^{51} + 1701 q^{53} - 162 q^{55} + 2429 q^{57} + 710 q^{59} + 539 q^{61} + 798 q^{63} + 2357 q^{65} + 130 q^{67} + 546 q^{69} + 1347 q^{71} + 3185 q^{73} - 316 q^{75} + 2862 q^{77} - 1470 q^{79} + 10610 q^{81} + 2366 q^{83} + 1376 q^{85} - 32 q^{87} + 3566 q^{89} - 1986 q^{91} + 2984 q^{93} + 1655 q^{95} + 4344 q^{97} + 308 q^{99}+O(q^{100}) $ 54 * q + 5 * q^3 + 33 * q^5 + 4 * q^7 + 641 * q^9 + 55 * q^11 + 71 * q^13 + 101 * q^15 + 301 * q^17 - 122 * q^19 + 195 * q^21 + 317 * q^23 + 2111 * q^25 + 245 * q^27 + 253 * q^29 - 31 * q^31 + 994 * q^33 + 602 * q^35 + 482 * q^37 - 157 * q^39 + 1049 * q^41 - 218 * q^43 + 942 * q^45 + 1349 * q^47 + 5000 * q^49 - 689 * q^51 + 1701 * q^53 - 162 * q^55 + 2429 * q^57 + 710 * q^59 + 539 * q^61 + 798 * q^63 + 2357 * q^65 + 130 * q^67 + 546 * q^69 + 1347 * q^71 + 3185 * q^73 - 316 * q^75 + 2862 * q^77 - 1470 * q^79 + 10610 * q^81 + 2366 * q^83 + 1376 * q^85 - 32 * q^87 + 3566 * q^89 - 1986 * q^91 + 2984 * q^93 + 1655 * q^95 + 4344 * q^97 + 308 * 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
$3$	−5.86273	−1.12828	−0.564141	−	0.825678i	$-0.690793\pi$
$3$	−5.86273	−1.12828	−0.564141	+	0.825678i	$0.690793\pi$
$4$	0	0
$5$	−21.2394	−1.89971	−0.949856	−	0.312687i	$-0.898771\pi$
$5$	−21.2394	−1.89971	−0.949856	+	0.312687i	$0.898771\pi$
$6$	0	0
$7$	15.5711	0.840763	0.420381	−	0.907348i	$-0.361896\pi$
$7$	15.5711	0.840763	0.420381	+	0.907348i	$0.361896\pi$
$8$	0	0
$9$	7.37159	0.273022
$10$	0	0
$11$	54.0725	1.48213	0.741067	−	0.671431i	$-0.234320\pi$
$11$	54.0725	1.48213	0.741067	+	0.671431i	$0.234320\pi$
$12$	0	0
$13$	−11.0637	−0.236040	−0.118020	−	0.993011i	$-0.537655\pi$
$13$	−11.0637	−0.236040	−0.118020	+	0.993011i	$0.537655\pi$
$14$	0	0
$15$	124.521	2.14341
$16$	0	0
$17$	131.327	1.87361	0.936805	−	0.349851i	$-0.113768\pi$
$17$	131.327	1.87361	0.936805	+	0.349851i	$0.113768\pi$
$18$	0	0
$19$	−26.8871	−0.324649	−0.162324	−	0.986737i	$-0.551899\pi$
$19$	−26.8871	−0.324649	−0.162324	+	0.986737i	$0.551899\pi$
$20$	0	0
$21$	−91.2894	−0.948618
$22$	0	0
$23$	99.4551	0.901644	0.450822	−	0.892614i	$-0.351131\pi$
$23$	99.4551	0.901644	0.450822	+	0.892614i	$0.351131\pi$
$24$	0	0
$25$	326.113	2.60891
$26$	0	0
$27$	115.076	0.820237
$28$	0	0
$29$	−226.855	−1.45262	−0.726308	−	0.687370i	$-0.758765\pi$
$29$	−226.855	−1.45262	−0.726308	+	0.687370i	$0.758765\pi$
$30$	0	0
$31$	328.862	1.90533	0.952666	−	0.304019i	$-0.0983285\pi$
$31$	328.862	1.90533	0.952666	+	0.304019i	$0.0983285\pi$
$32$	0	0
$33$	−317.013	−1.67227
$34$	0	0
$35$	−330.722	−1.59721
$36$	0	0
$37$	−394.694	−1.75371	−0.876855	−	0.480754i	$-0.840363\pi$
$37$	−394.694	−1.75371	−0.876855	+	0.480754i	$0.840363\pi$
$38$	0	0
$39$	64.8635	0.266320
$40$	0	0
$41$	−269.555	−1.02677	−0.513383	−	0.858160i	$-0.671608\pi$
$41$	−269.555	−1.02677	−0.513383	+	0.858160i	$0.671608\pi$
$42$	0	0
$43$	−103.200	−0.365995	−0.182998	−	0.983113i	$-0.558580\pi$
$43$	−103.200	−0.365995	−0.182998	+	0.983113i	$0.558580\pi$
$44$	0	0
$45$	−156.568	−0.518663
$46$	0	0
$47$	156.929	0.487032	0.243516	−	0.969897i	$-0.421699\pi$
$47$	156.929	0.487032	0.243516	+	0.969897i	$0.421699\pi$
$48$	0	0
$49$	−100.540	−0.293118
$50$	0	0
$51$	−769.932	−2.11396
$52$	0	0
$53$	−613.389	−1.58973	−0.794863	−	0.606789i	$-0.792457\pi$
$53$	−613.389	−1.58973	−0.794863	+	0.606789i	$0.792457\pi$
$54$	0	0
$55$	−1148.47	−2.81563
$56$	0	0
$57$	157.632	0.366296
$58$	0	0
$59$	221.572	0.488919	0.244460	−	0.969659i	$-0.421389\pi$
$59$	221.572	0.488919	0.244460	+	0.969659i	$0.421389\pi$
$60$	0	0
$61$	917.017	1.92479	0.962393	−	0.271661i	$-0.0875730\pi$
$61$	917.017	1.92479	0.962393	+	0.271661i	$0.0875730\pi$
$62$	0	0
$63$	114.784	0.229547
$64$	0	0
$65$	234.987	0.448408
$66$	0	0
$67$	644.395	1.17500	0.587502	−	0.809222i	$-0.300111\pi$
$67$	644.395	1.17500	0.587502	+	0.809222i	$0.300111\pi$
$68$	0	0
$69$	−583.078	−1.01731
$70$	0	0
$71$	939.697	1.57072	0.785362	−	0.619036i	$-0.212477\pi$
$71$	939.697	1.57072	0.785362	+	0.619036i	$0.212477\pi$
$72$	0	0
$73$	−227.180	−0.364239	−0.182119	−	0.983276i	$-0.558296\pi$
$73$	−227.180	−0.364239	−0.182119	+	0.983276i	$0.558296\pi$
$74$	0	0
$75$	−1911.91	−2.94358
$76$	0	0
$77$	841.971	1.24612
$78$	0	0
$79$	−298.737	−0.425450	−0.212725	−	0.977112i	$-0.568234\pi$
$79$	−298.737	−0.425450	−0.212725	+	0.977112i	$0.568234\pi$
$80$	0	0
$81$	−873.693	−1.19848
$82$	0	0
$83$	578.892	0.765563	0.382781	−	0.923839i	$-0.374966\pi$
$83$	578.892	0.765563	0.382781	+	0.923839i	$0.374966\pi$
$84$	0	0
$85$	−2789.30	−3.55932
$86$	0	0
$87$	1329.99	1.63896
$88$	0	0
$89$	−525.709	−0.626124	−0.313062	−	0.949733i	$-0.601355\pi$
$89$	−525.709	−0.626124	−0.313062	+	0.949733i	$0.601355\pi$
$90$	0	0
$91$	−172.274	−0.198453
$92$	0	0
$93$	−1928.03	−2.14975
$94$	0	0
$95$	571.067	0.616739
$96$	0	0
$97$	339.349	0.355213	0.177606	−	0.984102i	$-0.443165\pi$
$97$	339.349	0.355213	0.177606	+	0.984102i	$0.443165\pi$
$98$	0	0
$99$	398.601	0.404655
$100$	0	0
$101$	−1383.24	−1.36275	−0.681376	−	0.731934i	$-0.738618\pi$
$101$	−1383.24	−1.36275	−0.681376	+	0.731934i	$0.738618\pi$
$102$	0	0
$103$	−1691.68	−1.61832	−0.809158	−	0.587591i	$-0.800076\pi$
$103$	−1691.68	−1.61832	−0.809158	+	0.587591i	$0.800076\pi$
$104$	0	0
$105$	1938.93	1.80210
$106$	0	0
$107$	−37.6928	−0.0340551	−0.0170276	−	0.999855i	$-0.505420\pi$
$107$	−37.6928	−0.0340551	−0.0170276	+	0.999855i	$0.505420\pi$
$108$	0	0
$109$	333.918	0.293427	0.146714	−	0.989179i	$-0.453130\pi$
$109$	333.918	0.293427	0.146714	+	0.989179i	$0.453130\pi$
$110$	0	0
$111$	2313.98	1.97868
$112$	0	0
$113$	531.943	0.442841	0.221420	−	0.975178i	$-0.428931\pi$
$113$	531.943	0.442841	0.221420	+	0.975178i	$0.428931\pi$
$114$	0	0
$115$	−2112.37	−1.71286
$116$	0	0
$117$	−81.5571	−0.0644441
$118$	0	0
$119$	2044.91	1.57526
$120$	0	0
$121$	1592.84	1.19672
$122$	0	0
$123$	1580.33	1.15848
$124$	0	0
$125$	−4271.53	−3.05646
$126$	0	0
$127$	−153.771	−0.107441	−0.0537203	−	0.998556i	$-0.517108\pi$
$127$	−153.771	−0.107441	−0.0537203	+	0.998556i	$0.517108\pi$
$128$	0	0
$129$	605.031	0.412946
$130$	0	0
$131$	614.019	0.409520	0.204760	−	0.978812i	$-0.434359\pi$
$131$	614.019	0.409520	0.204760	+	0.978812i	$0.434359\pi$
$132$	0	0
$133$	−418.663	−0.272952
$134$	0	0
$135$	−2444.15	−1.55821
$136$	0	0
$137$	2670.94	1.66565	0.832823	−	0.553539i	$-0.186723\pi$
$137$	2670.94	1.66565	0.832823	+	0.553539i	$0.186723\pi$
$138$	0	0
$139$	−1764.59	−1.07677	−0.538385	−	0.842699i	$-0.680965\pi$
$139$	−1764.59	−1.07677	−0.538385	+	0.842699i	$0.680965\pi$
$140$	0	0
$141$	−920.035	−0.549510
$142$	0	0
$143$	−598.242	−0.349843
$144$	0	0
$145$	4818.26	2.75955
$146$	0	0
$147$	589.436	0.330720
$148$	0	0
$149$	2973.43	1.63485	0.817425	−	0.576036i	$-0.195401\pi$
$149$	2973.43	1.63485	0.817425	+	0.576036i	$0.195401\pi$
$150$	0	0
$151$	−311.359	−0.167801	−0.0839007	−	0.996474i	$-0.526738\pi$
$151$	−311.359	−0.167801	−0.0839007	+	0.996474i	$0.526738\pi$
$152$	0	0
$153$	968.087	0.511537
$154$	0	0
$155$	−6984.83	−3.61958
$156$	0	0
$157$	−1624.55	−0.825815	−0.412908	−	0.910773i	$-0.635487\pi$
$157$	−1624.55	−0.825815	−0.412908	+	0.910773i	$0.635487\pi$
$158$	0	0
$159$	3596.13	1.79366
$160$	0	0
$161$	1548.63	0.758069
$162$	0	0
$163$	3596.40	1.72817	0.864085	−	0.503346i	$-0.167898\pi$
$163$	3596.40	1.72817	0.864085	+	0.503346i	$0.167898\pi$
$164$	0	0
$165$	6733.17	3.17683
$166$	0	0
$167$	−1208.35	−0.559908	−0.279954	−	0.960013i	$-0.590319\pi$
$167$	−1208.35	−0.559908	−0.279954	+	0.960013i	$0.590319\pi$
$168$	0	0
$169$	−2074.59	−0.944285
$170$	0	0
$171$	−198.201	−0.0886362
$172$	0	0
$173$	341.414	0.150042	0.0750209	−	0.997182i	$-0.476098\pi$
$173$	341.414	0.150042	0.0750209	+	0.997182i	$0.476098\pi$
$174$	0	0
$175$	5077.96	2.19347
$176$	0	0
$177$	−1299.02	−0.551639
$178$	0	0
$179$	918.587	0.383567	0.191783	−	0.981437i	$-0.438573\pi$
$179$	918.587	0.383567	0.191783	+	0.981437i	$0.438573\pi$
$180$	0	0
$181$	2144.39	0.880614	0.440307	−	0.897847i	$-0.354870\pi$
$181$	2144.39	0.880614	0.440307	+	0.897847i	$0.354870\pi$
$182$	0	0
$183$	−5376.22	−2.17170
$184$	0	0
$185$	8383.08	3.33155
$186$	0	0
$187$	7101.16	2.77694
$188$	0	0
$189$	1791.86	0.689624
$190$	0	0
$191$	1653.47	0.626391	0.313196	−	0.949689i	$-0.398600\pi$
$191$	1653.47	0.626391	0.313196	+	0.949689i	$0.398600\pi$
$192$	0	0
$193$	−2381.04	−0.888037	−0.444018	−	0.896018i	$-0.646447\pi$
$193$	−2381.04	−0.888037	−0.444018	+	0.896018i	$0.646447\pi$
$194$	0	0
$195$	−1377.66	−0.505931
$196$	0	0
$197$	−1971.62	−0.713057	−0.356529	−	0.934284i	$-0.616040\pi$
$197$	−1971.62	−0.713057	−0.356529	+	0.934284i	$0.616040\pi$
$198$	0	0
$199$	−2677.15	−0.953659	−0.476829	−	0.878996i	$-0.658214\pi$
$199$	−2677.15	−0.953659	−0.476829	+	0.878996i	$0.658214\pi$
$200$	0	0
$201$	−3777.91	−1.32574
$202$	0	0
$203$	−3532.39	−1.22130
$204$	0	0
$205$	5725.19	1.95056
$206$	0	0
$207$	733.143	0.246169
$208$	0	0
$209$	−1453.85	−0.481173
$210$	0	0
$211$	4511.54	1.47198	0.735989	−	0.676993i	$-0.236717\pi$
$211$	4511.54	1.47198	0.735989	+	0.676993i	$0.236717\pi$
$212$	0	0
$213$	−5509.19	−1.77222
$214$	0	0
$215$	2191.90	0.695285
$216$	0	0
$217$	5120.75	1.60193
$218$	0	0
$219$	1331.90	0.410964
$220$	0	0
$221$	−1452.96	−0.442247
$222$	0	0
$223$	−4170.32	−1.25231	−0.626155	−	0.779698i	$-0.715372\pi$
$223$	−4170.32	−1.25231	−0.626155	+	0.779698i	$0.715372\pi$
$224$	0	0
$225$	2403.98	0.712289
$226$	0	0
$227$	−4451.38	−1.30154	−0.650768	−	0.759277i	$-0.725553\pi$
$227$	−4451.38	−1.30154	−0.650768	+	0.759277i	$0.725553\pi$
$228$	0	0
$229$	2168.88	0.625867	0.312934	−	0.949775i	$-0.398688\pi$
$229$	2168.88	0.625867	0.312934	+	0.949775i	$0.398688\pi$
$230$	0	0
$231$	−4936.25	−1.40598
$232$	0	0
$233$	3181.94	0.894660	0.447330	−	0.894369i	$-0.352375\pi$
$233$	3181.94	0.894660	0.447330	+	0.894369i	$0.352375\pi$
$234$	0	0
$235$	−3333.09	−0.925221
$236$	0	0
$237$	1751.41	0.480028
$238$	0	0
$239$	−2662.91	−0.720709	−0.360355	−	0.932815i	$-0.617344\pi$
$239$	−2662.91	−0.720709	−0.360355	+	0.932815i	$0.617344\pi$
$240$	0	0
$241$	−2801.00	−0.748665	−0.374332	−	0.927295i	$-0.622128\pi$
$241$	−2801.00	−0.748665	−0.374332	+	0.927295i	$0.622128\pi$
$242$	0	0
$243$	2015.17	0.531989
$244$	0	0
$245$	2135.40	0.556841
$246$	0	0
$247$	297.471	0.0766300
$248$	0	0
$249$	−3393.89	−0.863771
$250$	0	0
$251$	251.000	0.0631194
$251$	251.000	0.0631194
$252$	0	0
$253$	5377.79	1.33636
$254$	0	0
$255$	16352.9	4.01592
$256$	0	0
$257$	−2732.94	−0.663331	−0.331666	−	0.943397i	$-0.607611\pi$
$257$	−2732.94	−0.663331	−0.331666	+	0.943397i	$0.607611\pi$
$258$	0	0
$259$	−6145.84	−1.47445
$260$	0	0
$261$	−1672.28	−0.396596
$262$	0	0
$263$	3893.58	0.912883	0.456441	−	0.889753i	$-0.349124\pi$
$263$	3893.58	0.912883	0.456441	+	0.889753i	$0.349124\pi$
$264$	0	0
$265$	13028.0	3.02002
$266$	0	0
$267$	3082.09	0.706445
$268$	0	0
$269$	3017.42	0.683922	0.341961	−	0.939714i	$-0.388909\pi$
$269$	3017.42	0.683922	0.341961	+	0.939714i	$0.388909\pi$
$270$	0	0
$271$	−675.976	−0.151523	−0.0757613	−	0.997126i	$-0.524139\pi$
$271$	−675.976	−0.151523	−0.0757613	+	0.997126i	$0.524139\pi$
$272$	0	0
$273$	1010.00	0.223912
$274$	0	0
$275$	17633.8	3.86675
$276$	0	0
$277$	2295.84	0.497991	0.248995	−	0.968505i	$-0.419900\pi$
$277$	2295.84	0.497991	0.248995	+	0.968505i	$0.419900\pi$
$278$	0	0
$279$	2424.23	0.520198
$280$	0	0
$281$	7039.06	1.49436	0.747180	−	0.664622i	$-0.231407\pi$
$281$	7039.06	1.49436	0.747180	+	0.664622i	$0.231407\pi$
$282$	0	0
$283$	7346.04	1.54303	0.771514	−	0.636213i	$-0.219500\pi$
$283$	7346.04	1.54303	0.771514	+	0.636213i	$0.219500\pi$
$284$	0	0
$285$	−3348.01	−0.695856
$286$	0	0
$287$	−4197.27	−0.863266
$288$	0	0
$289$	12333.7	2.51042
$290$	0	0
$291$	−1989.51	−0.400780
$292$	0	0
$293$	229.865	0.0458323	0.0229162	−	0.999737i	$-0.492705\pi$
$293$	229.865	0.0458323	0.0229162	+	0.999737i	$0.492705\pi$
$294$	0	0
$295$	−4706.06	−0.928806
$296$	0	0
$297$	6222.45	1.21570
$298$	0	0
$299$	−1100.34	−0.212824
$300$	0	0
$301$	−1606.94	−0.307715
$302$	0	0
$303$	8109.58	1.53757
$304$	0	0
$305$	−19476.9	−3.65654
$306$	0	0
$307$	−79.7448	−0.0148250	−0.00741251	−	0.999973i	$-0.502359\pi$
$307$	−79.7448	−0.0148250	−0.00741251	+	0.999973i	$0.502359\pi$
$308$	0	0
$309$	9917.88	1.82592
$310$	0	0
$311$	−994.499	−0.181327	−0.0906637	−	0.995882i	$-0.528899\pi$
$311$	−994.499	−0.181327	−0.0906637	+	0.995882i	$0.528899\pi$
$312$	0	0
$313$	839.400	0.151584	0.0757918	−	0.997124i	$-0.475852\pi$
$313$	839.400	0.151584	0.0757918	+	0.997124i	$0.475852\pi$
$314$	0	0
$315$	−2437.95	−0.436073
$316$	0	0
$317$	2637.16	0.467248	0.233624	−	0.972327i	$-0.424942\pi$
$317$	2637.16	0.467248	0.233624	+	0.972327i	$0.424942\pi$
$318$	0	0
$319$	−12266.6	−2.15297
$320$	0	0
$321$	220.982	0.0384238
$322$	0	0
$323$	−3530.99	−0.608265
$324$	0	0
$325$	−3608.02	−0.615806
$326$	0	0
$327$	−1957.67	−0.331069
$328$	0	0
$329$	2443.57	0.409478
$330$	0	0
$331$	−3587.84	−0.595787	−0.297893	−	0.954599i	$-0.596284\pi$
$331$	−3587.84	−0.595787	−0.297893	+	0.954599i	$0.596284\pi$
$332$	0	0
$333$	−2909.52	−0.478802
$334$	0	0
$335$	−13686.6	−2.23217
$336$	0	0
$337$	4810.16	0.777526	0.388763	−	0.921338i	$-0.372903\pi$
$337$	4810.16	0.777526	0.388763	+	0.921338i	$0.372903\pi$
$338$	0	0
$339$	−3118.64	−0.499650
$340$	0	0
$341$	17782.4	2.82396
$342$	0	0
$343$	−6906.42	−1.08721
$344$	0	0
$345$	12384.2	1.93260
$346$	0	0
$347$	600.929	0.0929670	0.0464835	−	0.998919i	$-0.485199\pi$
$347$	600.929	0.0929670	0.0464835	+	0.998919i	$0.485199\pi$
$348$	0	0
$349$	−4880.98	−0.748632	−0.374316	−	0.927301i	$-0.622123\pi$
$349$	−4880.98	−0.748632	−0.374316	+	0.927301i	$0.622123\pi$
$350$	0	0
$351$	−1273.17	−0.193609
$352$	0	0
$353$	−2853.28	−0.430212	−0.215106	−	0.976591i	$-0.569010\pi$
$353$	−2853.28	−0.430212	−0.215106	+	0.976591i	$0.569010\pi$
$354$	0	0
$355$	−19958.6	−2.98393
$356$	0	0
$357$	−11988.7	−1.77734
$358$	0	0
$359$	6109.01	0.898109	0.449055	−	0.893504i	$-0.351761\pi$
$359$	6109.01	0.898109	0.449055	+	0.893504i	$0.351761\pi$
$360$	0	0
$361$	−6136.08	−0.894603
$362$	0	0
$363$	−9338.39	−1.35024
$364$	0	0
$365$	4825.18	0.691949
$366$	0	0
$367$	2726.85	0.387848	0.193924	−	0.981017i	$-0.437878\pi$
$367$	2726.85	0.387848	0.193924	+	0.981017i	$0.437878\pi$
$368$	0	0
$369$	−1987.05	−0.280330
$370$	0	0
$371$	−9551.17	−1.33658
$372$	0	0
$373$	579.349	0.0804224	0.0402112	−	0.999191i	$-0.487197\pi$
$373$	579.349	0.0804224	0.0402112	+	0.999191i	$0.487197\pi$
$374$	0	0
$375$	25042.8	3.44855
$376$	0	0
$377$	2509.85	0.342875
$378$	0	0
$379$	10689.3	1.44875	0.724373	−	0.689408i	$-0.242129\pi$
$379$	10689.3	1.44875	0.724373	+	0.689408i	$0.242129\pi$
$380$	0	0
$381$	901.517	0.121223
$382$	0	0
$383$	2080.11	0.277516	0.138758	−	0.990326i	$-0.455689\pi$
$383$	2080.11	0.277516	0.138758	+	0.990326i	$0.455689\pi$
$384$	0	0
$385$	−17883.0	−2.36728
$386$	0	0
$387$	−760.745	−0.0999247
$388$	0	0
$389$	−5444.38	−0.709618	−0.354809	−	0.934939i	$-0.615454\pi$
$389$	−5444.38	−0.709618	−0.354809	+	0.934939i	$0.615454\pi$
$390$	0	0
$391$	13061.1	1.68933
$392$	0	0
$393$	−3599.83	−0.462054
$394$	0	0
$395$	6345.00	0.808232
$396$	0	0
$397$	−1277.08	−0.161448	−0.0807239	−	0.996737i	$-0.525723\pi$
$397$	−1277.08	−0.161448	−0.0807239	+	0.996737i	$0.525723\pi$
$398$	0	0
$399$	2454.51	0.307968
$400$	0	0
$401$	−2482.23	−0.309118	−0.154559	−	0.987984i	$-0.549396\pi$
$401$	−2482.23	−0.309118	−0.154559	+	0.987984i	$0.549396\pi$
$402$	0	0
$403$	−3638.43	−0.449734
$404$	0	0
$405$	18556.7	2.27677
$406$	0	0
$407$	−21342.1	−2.59924
$408$	0	0
$409$	5238.17	0.633278	0.316639	−	0.948546i	$-0.397446\pi$
$409$	5238.17	0.633278	0.316639	+	0.948546i	$0.397446\pi$
$410$	0	0
$411$	−15659.0	−1.87932
$412$	0	0
$413$	3450.13	0.411065
$414$	0	0
$415$	−12295.3	−1.45435
$416$	0	0
$417$	10345.3	1.21490
$418$	0	0
$419$	−10107.4	−1.17847	−0.589237	−	0.807960i	$-0.700572\pi$
$419$	−10107.4	−1.17847	−0.589237	+	0.807960i	$0.700572\pi$
$420$	0	0
$421$	−16502.4	−1.91039	−0.955197	−	0.295971i	$-0.904357\pi$
$421$	−16502.4	−1.91039	−0.955197	+	0.295971i	$0.904357\pi$
$422$	0	0
$423$	1156.82	0.132971
$424$	0	0
$425$	42827.4	4.88808
$426$	0	0
$427$	14279.0	1.61829
$428$	0	0
$429$	3507.33	0.394722
$430$	0	0
$431$	16703.8	1.86680	0.933402	−	0.358832i	$-0.116825\pi$
$431$	16703.8	1.86680	0.933402	+	0.358832i	$0.116825\pi$
$432$	0	0
$433$	−3116.79	−0.345920	−0.172960	−	0.984929i	$-0.555333\pi$
$433$	−3116.79	−0.345920	−0.172960	+	0.984929i	$0.555333\pi$
$434$	0	0
$435$	−28248.2	−3.11355
$436$	0	0
$437$	−2674.06	−0.292718
$438$	0	0
$439$	3148.27	0.342275	0.171138	−	0.985247i	$-0.445256\pi$
$439$	3148.27	0.342275	0.171138	+	0.985247i	$0.445256\pi$
$440$	0	0
$441$	−741.137	−0.0800278
$442$	0	0
$443$	483.337	0.0518376	0.0259188	−	0.999664i	$-0.491749\pi$
$443$	483.337	0.0518376	0.0259188	+	0.999664i	$0.491749\pi$
$444$	0	0
$445$	11165.8	1.18946
$446$	0	0
$447$	−17432.4	−1.84457
$448$	0	0
$449$	−14657.1	−1.54056	−0.770281	−	0.637704i	$-0.779884\pi$
$449$	−14657.1	−1.54056	−0.770281	+	0.637704i	$0.779884\pi$
$450$	0	0
$451$	−14575.5	−1.52180
$452$	0	0
$453$	1825.41	0.189327
$454$	0	0
$455$	3659.01	0.377004
$456$	0	0
$457$	−12225.6	−1.25140	−0.625699	−	0.780065i	$-0.715186\pi$
$457$	−12225.6	−1.25140	−0.625699	+	0.780065i	$0.715186\pi$
$458$	0	0
$459$	15112.5	1.53680
$460$	0	0
$461$	12727.2	1.28582	0.642912	−	0.765940i	$-0.277726\pi$
$461$	12727.2	1.28582	0.642912	+	0.765940i	$0.277726\pi$
$462$	0	0
$463$	−12511.9	−1.25589	−0.627945	−	0.778257i	$-0.716104\pi$
$463$	−12511.9	−1.25589	−0.627945	+	0.778257i	$0.716104\pi$
$464$	0	0
$465$	40950.2	4.08391
$466$	0	0
$467$	11778.0	1.16707	0.583535	−	0.812088i	$-0.301669\pi$
$467$	11778.0	1.16707	0.583535	+	0.812088i	$0.301669\pi$
$468$	0	0
$469$	10034.0	0.987900
$470$	0	0
$471$	9524.28	0.931753
$472$	0	0
$473$	−5580.26	−0.542454
$474$	0	0
$475$	−8768.25	−0.846978
$476$	0	0
$477$	−4521.65	−0.434030
$478$	0	0
$479$	1571.21	0.149876	0.0749380	−	0.997188i	$-0.476124\pi$
$479$	1571.21	0.149876	0.0749380	+	0.997188i	$0.476124\pi$
$480$	0	0
$481$	4366.78	0.413946
$482$	0	0
$483$	−9079.19	−0.855316
$484$	0	0
$485$	−7207.57	−0.674802
$486$	0	0
$487$	9585.31	0.891893	0.445946	−	0.895060i	$-0.352867\pi$
$487$	9585.31	0.891893	0.445946	+	0.895060i	$0.352867\pi$
$488$	0	0
$489$	−21084.7	−1.94986
$490$	0	0
$491$	10773.8	0.990258	0.495129	−	0.868819i	$-0.335121\pi$
$491$	10773.8	0.990258	0.495129	+	0.868819i	$0.335121\pi$
$492$	0	0
$493$	−29792.1	−2.72164
$494$	0	0
$495$	−8466.05	−0.768729
$496$	0	0
$497$	14632.1	1.32061
$498$	0	0
$499$	11067.5	0.992882	0.496441	−	0.868070i	$-0.334640\pi$
$499$	11067.5	0.992882	0.496441	+	0.868070i	$0.334640\pi$
$500$	0	0
$501$	7084.21	0.631735
$502$	0	0
$503$	7187.47	0.637124	0.318562	−	0.947902i	$-0.396800\pi$
$503$	7187.47	0.637124	0.318562	+	0.947902i	$0.396800\pi$
$504$	0	0
$505$	29379.3	2.58884
$506$	0	0
$507$	12162.8	1.06542
$508$	0	0
$509$	3665.39	0.319186	0.159593	−	0.987183i	$-0.448982\pi$
$509$	3665.39	0.319186	0.159593	+	0.987183i	$0.448982\pi$
$510$	0	0
$511$	−3537.46	−0.306238
$512$	0	0
$513$	−3094.06	−0.266289
$514$	0	0
$515$	35930.4	3.07433
$516$	0	0
$517$	8485.57	0.721847
$518$	0	0
$519$	−2001.62	−0.169290
$520$	0	0
$521$	19091.2	1.60538	0.802688	−	0.596399i	$-0.203402\pi$
$521$	19091.2	1.60538	0.802688	+	0.596399i	$0.203402\pi$
$522$	0	0
$523$	−6602.04	−0.551983	−0.275991	−	0.961160i	$-0.589006\pi$
$523$	−6602.04	−0.551983	−0.275991	+	0.961160i	$0.589006\pi$
$524$	0	0
$525$	−29770.7	−2.47486
$526$	0	0
$527$	43188.3	3.56985
$528$	0	0
$529$	−2275.68	−0.187037
$530$	0	0
$531$	1633.34	0.133486
$532$	0	0
$533$	2982.27	0.242358
$534$	0	0
$535$	800.573	0.0646949
$536$	0	0
$537$	−5385.43	−0.432772
$538$	0	0
$539$	−5436.43	−0.434441
$540$	0	0
$541$	−20117.4	−1.59873	−0.799367	−	0.600843i	$-0.794832\pi$
$541$	−20117.4	−1.59873	−0.799367	+	0.600843i	$0.794832\pi$
$542$	0	0
$543$	−12572.0	−0.993582
$544$	0	0
$545$	−7092.24	−0.557428
$546$	0	0
$547$	−2899.29	−0.226626	−0.113313	−	0.993559i	$-0.536146\pi$
$547$	−2899.29	−0.226626	−0.113313	+	0.993559i	$0.536146\pi$
$548$	0	0
$549$	6759.87	0.525509
$550$	0	0
$551$	6099.47	0.471590
$552$	0	0
$553$	−4651.67	−0.357702
$554$	0	0
$555$	−49147.7	−3.75893
$556$	0	0
$557$	17784.2	1.35286	0.676429	−	0.736508i	$-0.263526\pi$
$557$	17784.2	1.35286	0.676429	+	0.736508i	$0.263526\pi$
$558$	0	0
$559$	1141.77	0.0863894
$560$	0	0
$561$	−41632.2	−3.13318
$562$	0	0
$563$	6478.16	0.484941	0.242471	−	0.970159i	$-0.422042\pi$
$563$	6478.16	0.484941	0.242471	+	0.970159i	$0.422042\pi$
$564$	0	0
$565$	−11298.2	−0.841270
$566$	0	0
$567$	−13604.4	−1.00764
$568$	0	0
$569$	1671.94	0.123183	0.0615917	−	0.998101i	$-0.480382\pi$
$569$	1671.94	0.123183	0.0615917	+	0.998101i	$0.480382\pi$
$570$	0	0
$571$	21407.0	1.56892	0.784462	−	0.620177i	$-0.212939\pi$
$571$	21407.0	1.56892	0.784462	+	0.620177i	$0.212939\pi$
$572$	0	0
$573$	−9693.83	−0.706746
$574$	0	0
$575$	32433.6	2.35231
$576$	0	0
$577$	−7553.30	−0.544970	−0.272485	−	0.962160i	$-0.587846\pi$
$577$	−7553.30	−0.544970	−0.272485	+	0.962160i	$0.587846\pi$
$578$	0	0
$579$	13959.4	1.00196
$580$	0	0
$581$	9014.01	0.643656
$582$	0	0
$583$	−33167.5	−2.35619
$584$	0	0
$585$	1732.23	0.122425
$586$	0	0
$587$	−942.676	−0.0662835	−0.0331417	−	0.999451i	$-0.510551\pi$
$587$	−942.676	−0.0662835	−0.0331417	+	0.999451i	$0.510551\pi$
$588$	0	0
$589$	−8842.14	−0.618564
$590$	0	0
$591$	11559.1	0.804530
$592$	0	0
$593$	4667.33	0.323211	0.161606	−	0.986855i	$-0.448333\pi$
$593$	4667.33	0.323211	0.161606	+	0.986855i	$0.448333\pi$
$594$	0	0
$595$	−43432.6	−2.99254
$596$	0	0
$597$	15695.4	1.07600
$598$	0	0
$599$	2257.29	0.153974	0.0769870	−	0.997032i	$-0.475470\pi$
$599$	2257.29	0.153974	0.0769870	+	0.997032i	$0.475470\pi$
$600$	0	0
$601$	−2224.02	−0.150948	−0.0754738	−	0.997148i	$-0.524047\pi$
$601$	−2224.02	−0.150948	−0.0754738	+	0.997148i	$0.524047\pi$
$602$	0	0
$603$	4750.22	0.320802
$604$	0	0
$605$	−33831.0	−2.27343
$606$	0	0
$607$	−6590.37	−0.440683	−0.220342	−	0.975423i	$-0.570717\pi$
$607$	−6590.37	−0.440683	−0.220342	+	0.975423i	$0.570717\pi$
$608$	0	0
$609$	20709.4	1.37798
$610$	0	0
$611$	−1736.22	−0.114959
$612$	0	0
$613$	−16222.3	−1.06886	−0.534430	−	0.845213i	$-0.679474\pi$
$613$	−16222.3	−1.06886	−0.534430	+	0.845213i	$0.679474\pi$
$614$	0	0
$615$	−33565.2	−2.20078
$616$	0	0
$617$	25831.8	1.68549	0.842747	−	0.538310i	$-0.180937\pi$
$617$	25831.8	1.68549	0.842747	+	0.538310i	$0.180937\pi$
$618$	0	0
$619$	−23968.6	−1.55635	−0.778175	−	0.628048i	$-0.783854\pi$
$619$	−23968.6	−1.55635	−0.778175	+	0.628048i	$0.783854\pi$
$620$	0	0
$621$	11444.9	0.739562
$622$	0	0
$623$	−8185.89	−0.526422
$624$	0	0
$625$	49960.8	3.19749
$626$	0	0
$627$	8523.55	0.542899
$628$	0	0
$629$	−51833.8	−3.28577
$630$	0	0
$631$	−21907.2	−1.38211	−0.691054	−	0.722803i	$-0.742853\pi$
$631$	−21907.2	−1.38211	−0.691054	+	0.722803i	$0.742853\pi$
$632$	0	0
$633$	−26450.0	−1.66081
$634$	0	0
$635$	3266.01	0.204106
$636$	0	0
$637$	1112.34	0.0691876
$638$	0	0
$639$	6927.06	0.428842
$640$	0	0
$641$	−7862.26	−0.484463	−0.242231	−	0.970219i	$-0.577879\pi$
$641$	−7862.26	−0.484463	−0.242231	+	0.970219i	$0.577879\pi$
$642$	0	0
$643$	24360.5	1.49407	0.747033	−	0.664787i	$-0.231478\pi$
$643$	24360.5	1.49407	0.747033	+	0.664787i	$0.231478\pi$
$644$	0	0
$645$	−12850.5	−0.784479
$646$	0	0
$647$	1185.92	0.0720606	0.0360303	−	0.999351i	$-0.488529\pi$
$647$	1185.92	0.0720606	0.0360303	+	0.999351i	$0.488529\pi$
$648$	0	0
$649$	11981.0	0.724644
$650$	0	0
$651$	−30021.6	−1.80743
$652$	0	0
$653$	−7723.88	−0.462877	−0.231439	−	0.972850i	$-0.574343\pi$
$653$	−7723.88	−0.462877	−0.231439	+	0.972850i	$0.574343\pi$
$654$	0	0
$655$	−13041.4	−0.777970
$656$	0	0
$657$	−1674.68	−0.0994452
$658$	0	0
$659$	27069.9	1.60014	0.800070	−	0.599906i	$-0.204795\pi$
$659$	27069.9	1.60014	0.800070	+	0.599906i	$0.204795\pi$
$660$	0	0
$661$	23919.0	1.40747	0.703737	−	0.710460i	$-0.251513\pi$
$661$	23919.0	1.40747	0.703737	+	0.710460i	$0.251513\pi$
$662$	0	0
$663$	8518.30	0.498979
$664$	0	0
$665$	8892.16	0.518531
$666$	0	0
$667$	−22561.9	−1.30974
$668$	0	0
$669$	24449.5	1.41296
$670$	0	0
$671$	49585.4	2.85279
$672$	0	0
$673$	1418.14	0.0812262	0.0406131	−	0.999175i	$-0.487069\pi$
$673$	1418.14	0.0812262	0.0406131	+	0.999175i	$0.487069\pi$
$674$	0	0
$675$	37527.8	2.13992
$676$	0	0
$677$	−20064.8	−1.13907	−0.569537	−	0.821966i	$-0.692877\pi$
$677$	−20064.8	−1.13907	−0.569537	+	0.821966i	$0.692877\pi$
$678$	0	0
$679$	5284.05	0.298650
$680$	0	0
$681$	26097.2	1.46850
$682$	0	0
$683$	34744.3	1.94649	0.973246	−	0.229766i	$-0.0737959\pi$
$683$	34744.3	1.94649	0.973246	+	0.229766i	$0.0737959\pi$
$684$	0	0
$685$	−56729.2	−3.16425
$686$	0	0
$687$	−12715.6	−0.706155
$688$	0	0
$689$	6786.35	0.375239
$690$	0	0
$691$	−21068.2	−1.15987	−0.579936	−	0.814662i	$-0.696922\pi$
$691$	−21068.2	−1.15987	−0.579936	+	0.814662i	$0.696922\pi$
$692$	0	0
$693$	6206.67	0.340219
$694$	0	0
$695$	37479.0	2.04555
$696$	0	0
$697$	−35399.7	−1.92376
$698$	0	0
$699$	−18654.9	−1.00943
$700$	0	0
$701$	−19544.6	−1.05305	−0.526525	−	0.850159i	$-0.676505\pi$
$701$	−19544.6	−1.05305	−0.526525	+	0.850159i	$0.676505\pi$
$702$	0	0
$703$	10612.2	0.569340
$704$	0	0
$705$	19541.0	1.04391
$706$	0	0
$707$	−21538.7	−1.14575
$708$	0	0
$709$	10728.5	0.568289	0.284144	−	0.958781i	$-0.408290\pi$
$709$	10728.5	0.568289	0.284144	+	0.958781i	$0.408290\pi$
$710$	0	0
$711$	−2202.17	−0.116157
$712$	0	0
$713$	32707.0	1.71793
$714$	0	0
$715$	12706.3	0.664601
$716$	0	0
$717$	15611.9	0.813164
$718$	0	0
$719$	23045.2	1.19533	0.597664	−	0.801747i	$-0.296096\pi$
$719$	23045.2	1.19533	0.597664	+	0.801747i	$0.296096\pi$
$720$	0	0
$721$	−26341.4	−1.36062
$722$	0	0
$723$	16421.5	0.844705
$724$	0	0
$725$	−73980.3	−3.78974
$726$	0	0
$727$	−13008.3	−0.663620	−0.331810	−	0.943346i	$-0.607659\pi$
$727$	−13008.3	−0.663620	−0.331810	+	0.943346i	$0.607659\pi$
$728$	0	0
$729$	11775.3	0.598247
$730$	0	0
$731$	−13552.9	−0.685732
$732$	0	0
$733$	−28380.9	−1.43011	−0.715056	−	0.699067i	$-0.753599\pi$
$733$	−28380.9	−1.43011	−0.715056	+	0.699067i	$0.753599\pi$
$734$	0	0
$735$	−12519.3	−0.628274
$736$	0	0
$737$	34844.0	1.74152
$738$	0	0
$739$	2986.40	0.148655	0.0743277	−	0.997234i	$-0.476319\pi$
$739$	2986.40	0.148655	0.0743277	+	0.997234i	$0.476319\pi$
$740$	0	0
$741$	−1743.99	−0.0864603
$742$	0	0
$743$	−2872.95	−0.141855	−0.0709276	−	0.997481i	$-0.522596\pi$
$743$	−2872.95	−0.141855	−0.0709276	+	0.997481i	$0.522596\pi$
$744$	0	0
$745$	−63153.9	−3.10574
$746$	0	0
$747$	4267.36	0.209015
$748$	0	0
$749$	−586.919	−0.0286323
$750$	0	0
$751$	39142.6	1.90191	0.950955	−	0.309330i	$-0.100105\pi$
$751$	39142.6	1.90191	0.950955	+	0.309330i	$0.100105\pi$
$752$	0	0
$753$	−1471.55	−0.0712166
$754$	0	0
$755$	6613.08	0.318774
$756$	0	0
$757$	19533.9	0.937878	0.468939	−	0.883231i	$-0.344636\pi$
$757$	19533.9	0.937878	0.468939	+	0.883231i	$0.344636\pi$
$758$	0	0
$759$	−31528.5	−1.50779
$760$	0	0
$761$	−22561.7	−1.07472	−0.537360	−	0.843353i	$-0.680578\pi$
$761$	−22561.7	−1.07472	−0.537360	+	0.843353i	$0.680578\pi$
$762$	0	0
$763$	5199.49	0.246703
$764$	0	0
$765$	−20561.6	−0.971773
$766$	0	0
$767$	−2451.41	−0.115404
$768$	0	0
$769$	−103.150	−0.00483703	−0.00241851	−	0.999997i	$-0.500770\pi$
$769$	−103.150	−0.00483703	−0.00241851	+	0.999997i	$0.500770\pi$
$770$	0	0
$771$	16022.5	0.748425
$772$	0	0
$773$	22202.9	1.03310	0.516549	−	0.856258i	$-0.327217\pi$
$773$	22202.9	1.03310	0.516549	+	0.856258i	$0.327217\pi$
$774$	0	0
$775$	107246.	4.97083
$776$	0	0
$777$	36031.4	1.66360
$778$	0	0
$779$	7247.55	0.333338
$780$	0	0
$781$	50811.8	2.32803
$782$	0	0
$783$	−26105.5	−1.19149
$784$	0	0
$785$	34504.5	1.56881
$786$	0	0
$787$	−41023.7	−1.85812	−0.929058	−	0.369935i	$-0.879380\pi$
$787$	−41023.7	−1.85812	−0.929058	+	0.369935i	$0.879380\pi$
$788$	0	0
$789$	−22827.0	−1.02999
$790$	0	0
$791$	8282.96	0.372324
$792$	0	0
$793$	−10145.6	−0.454326
$794$	0	0
$795$	−76379.8	−3.40744
$796$	0	0
$797$	−14368.7	−0.638601	−0.319300	−	0.947654i	$-0.603448\pi$
$797$	−14368.7	−0.638601	−0.319300	+	0.947654i	$0.603448\pi$
$798$	0	0
$799$	20609.0	0.912509
$800$	0	0
$801$	−3875.31	−0.170946
$802$	0	0
$803$	−12284.2	−0.539851
$804$	0	0
$805$	−32892.0	−1.44011
$806$	0	0
$807$	−17690.3	−0.771658
$808$	0	0
$809$	−24669.4	−1.07210	−0.536051	−	0.844186i	$-0.680084\pi$
$809$	−24669.4	−1.07210	−0.536051	+	0.844186i	$0.680084\pi$
$810$	0	0
$811$	433.738	0.0187800	0.00939002	−	0.999956i	$-0.497011\pi$
$811$	433.738	0.0187800	0.00939002	+	0.999956i	$0.497011\pi$
$812$	0	0
$813$	3963.06	0.170960
$814$	0	0
$815$	−76385.4	−3.28302
$816$	0	0
$817$	2774.74	0.118820
$818$	0	0
$819$	−1269.94	−0.0541822
$820$	0	0
$821$	−16421.0	−0.698049	−0.349025	−	0.937114i	$-0.613487\pi$
$821$	−16421.0	−0.698049	−0.349025	+	0.937114i	$0.613487\pi$
$822$	0	0
$823$	29900.0	1.26640	0.633200	−	0.773988i	$-0.281741\pi$
$823$	29900.0	1.26640	0.633200	+	0.773988i	$0.281741\pi$
$824$	0	0
$825$	−103382.	−4.36279
$826$	0	0
$827$	−13919.6	−0.585285	−0.292643	−	0.956222i	$-0.594535\pi$
$827$	−13919.6	−0.585285	−0.292643	+	0.956222i	$0.594535\pi$
$828$	0	0
$829$	28329.2	1.18687	0.593434	−	0.804883i	$-0.297772\pi$
$829$	28329.2	1.18687	0.593434	+	0.804883i	$0.297772\pi$
$830$	0	0
$831$	−13459.9	−0.561874
$832$	0	0
$833$	−13203.5	−0.549190
$834$	0	0
$835$	25664.6	1.06366
$836$	0	0
$837$	37844.1	1.56282
$838$	0	0
$839$	−11503.1	−0.473340	−0.236670	−	0.971590i	$-0.576056\pi$
$839$	−11503.1	−0.473340	−0.236670	+	0.971590i	$0.576056\pi$
$840$	0	0
$841$	27074.0	1.11009
$842$	0	0
$843$	−41268.1	−1.68606
$844$	0	0
$845$	44063.2	1.79387
$846$	0	0
$847$	24802.3	1.00616
$848$	0	0
$849$	−43067.9	−1.74097
$850$	0	0
$851$	−39254.3	−1.58122
$852$	0	0
$853$	−1522.18	−0.0611002	−0.0305501	−	0.999533i	$-0.509726\pi$
$853$	−1522.18	−0.0611002	−0.0305501	+	0.999533i	$0.509726\pi$
$854$	0	0
$855$	4209.67	0.168383
$856$	0	0
$857$	42861.4	1.70842	0.854211	−	0.519926i	$-0.174040\pi$
$857$	42861.4	1.70842	0.854211	+	0.519926i	$0.174040\pi$
$858$	0	0
$859$	−28703.5	−1.14010	−0.570052	−	0.821609i	$-0.693077\pi$
$859$	−28703.5	−1.14010	−0.570052	+	0.821609i	$0.693077\pi$
$860$	0	0
$861$	24607.5	0.974008
$862$	0	0
$863$	−4304.33	−0.169781	−0.0848906	−	0.996390i	$-0.527054\pi$
$863$	−4304.33	−0.169781	−0.0848906	+	0.996390i	$0.527054\pi$
$864$	0	0
$865$	−7251.44	−0.285036
$866$	0	0
$867$	−72309.0	−2.83246
$868$	0	0
$869$	−16153.5	−0.630574
$870$	0	0
$871$	−7129.39	−0.277348
$872$	0	0
$873$	2501.54	0.0969809
$874$	0	0
$875$	−66512.6	−2.56976
$876$	0	0
$877$	30422.7	1.17138	0.585691	−	0.810534i	$-0.300823\pi$
$877$	30422.7	1.17138	0.585691	+	0.810534i	$0.300823\pi$
$878$	0	0
$879$	−1347.64	−0.0517118
$880$	0	0
$881$	−24604.0	−0.940896	−0.470448	−	0.882428i	$-0.655908\pi$
$881$	−24604.0	−0.940896	−0.470448	+	0.882428i	$0.655908\pi$
$882$	0	0
$883$	20998.5	0.800288	0.400144	−	0.916452i	$-0.368960\pi$
$883$	20998.5	0.800288	0.400144	+	0.916452i	$0.368960\pi$
$884$	0	0
$885$	27590.4	1.04796
$886$	0	0
$887$	50901.4	1.92683	0.963417	−	0.268006i	$-0.0863647\pi$
$887$	50901.4	1.92683	0.963417	+	0.268006i	$0.0863647\pi$
$888$	0	0
$889$	−2394.39	−0.0903321
$890$	0	0
$891$	−47242.8	−1.77631
$892$	0	0
$893$	−4219.38	−0.158114
$894$	0	0
$895$	−19510.3	−0.728666
$896$	0	0
$897$	6451.00	0.240126
$898$	0	0
$899$	−74603.8	−2.76772
$900$	0	0
$901$	−80554.3	−2.97853
$902$	0	0
$903$	9421.03	0.347189
$904$	0	0
$905$	−45545.6	−1.67291
$906$	0	0
$907$	−31295.8	−1.14571	−0.572855	−	0.819656i	$-0.694164\pi$
$907$	−31295.8	−1.14571	−0.572855	+	0.819656i	$0.694164\pi$
$908$	0	0
$909$	−10196.7	−0.372061
$910$	0	0
$911$	19891.1	0.723404	0.361702	−	0.932294i	$-0.382196\pi$
$911$	19891.1	0.723404	0.361702	+	0.932294i	$0.382196\pi$
$912$	0	0
$913$	31302.2	1.13467
$914$	0	0
$915$	114188.	4.12561
$916$	0	0
$917$	9560.98	0.344309
$918$	0	0
$919$	10147.2	0.364229	0.182114	−	0.983277i	$-0.441706\pi$
$919$	10147.2	0.364229	0.182114	+	0.983277i	$0.441706\pi$
$920$	0	0
$921$	467.522	0.0167268
$922$	0	0
$923$	−10396.5	−0.370754
$924$	0	0
$925$	−128715.	−4.57527
$926$	0	0
$927$	−12470.4	−0.441836
$928$	0	0
$929$	−30850.7	−1.08953	−0.544767	−	0.838587i	$-0.683382\pi$
$929$	−30850.7	−1.08953	−0.544767	+	0.838587i	$0.683382\pi$
$930$	0	0
$931$	2703.22	0.0951605
$932$	0	0
$933$	5830.48	0.204589
$934$	0	0
$935$	−150825.	−5.27539
$936$	0	0
$937$	42907.2	1.49596	0.747982	−	0.663719i	$-0.231023\pi$
$937$	42907.2	1.49596	0.747982	+	0.663719i	$0.231023\pi$
$938$	0	0
$939$	−4921.17	−0.171029
$940$	0	0
$941$	−12519.4	−0.433711	−0.216855	−	0.976204i	$-0.569580\pi$
$941$	−12519.4	−0.433711	−0.216855	+	0.976204i	$0.569580\pi$
$942$	0	0
$943$	−26808.6	−0.925777
$944$	0	0
$945$	−38058.2	−1.31009
$946$	0	0
$947$	−5773.97	−0.198130	−0.0990649	−	0.995081i	$-0.531585\pi$
$947$	−5773.97	−0.198130	−0.0990649	+	0.995081i	$0.531585\pi$
$948$	0	0
$949$	2513.45	0.0859749
$950$	0	0
$951$	−15460.9	−0.527188
$952$	0	0
$953$	11909.1	0.404800	0.202400	−	0.979303i	$-0.435126\pi$
$953$	11909.1	0.404800	0.202400	+	0.979303i	$0.435126\pi$
$954$	0	0
$955$	−35118.7	−1.18996
$956$	0	0
$957$	71915.8	2.42916
$958$	0	0
$959$	41589.5	1.40041
$960$	0	0
$961$	78359.0	2.63029
$962$	0	0
$963$	−277.856	−0.00929779
$964$	0	0
$965$	50572.0	1.68701
$966$	0	0
$967$	16843.8	0.560145	0.280072	−	0.959979i	$-0.409642\pi$
$967$	16843.8	0.560145	0.280072	+	0.959979i	$0.409642\pi$
$968$	0	0
$969$	20701.3	0.686295
$970$	0	0
$971$	37468.0	1.23832	0.619158	−	0.785266i	$-0.287474\pi$
$971$	37468.0	1.23832	0.619158	+	0.785266i	$0.287474\pi$
$972$	0	0
$973$	−27476.7	−0.905307
$974$	0	0
$975$	21152.8	0.694803
$976$	0	0
$977$	34323.8	1.12397	0.561984	−	0.827148i	$-0.310038\pi$
$977$	34323.8	1.12397	0.561984	+	0.827148i	$0.310038\pi$
$978$	0	0
$979$	−28426.4	−0.928001
$980$	0	0
$981$	2461.51	0.0801121
$982$	0	0
$983$	−3251.00	−0.105484	−0.0527420	−	0.998608i	$-0.516796\pi$
$983$	−3251.00	−0.105484	−0.0527420	+	0.998608i	$0.516796\pi$
$984$	0	0
$985$	41876.1	1.35460
$986$	0	0
$987$	−14326.0	−0.462007
$988$	0	0
$989$	−10263.7	−0.329997
$990$	0	0
$991$	−5718.24	−0.183296	−0.0916478	−	0.995791i	$-0.529213\pi$
$991$	−5718.24	−0.183296	−0.0916478	+	0.995791i	$0.529213\pi$
$992$	0	0
$993$	21034.5	0.672216
$994$	0	0
$995$	56861.1	1.81168
$996$	0	0
$997$	31666.5	1.00591	0.502954	−	0.864313i	$-0.332247\pi$
$997$	31666.5	1.00591	0.502954	+	0.864313i	$0.332247\pi$
$998$	0	0
$999$	−45419.8	−1.43846

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.4.a.d.1.14	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.4.a.d.1.14	✓	54	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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q-5.86273 q^{3} -21.2394 q^{5} +15.5711 q^{7} +7.37159 q^{9} +O(q^{10})\)	\(q-5.86273 q^{3} -21.2394 q^{5} +15.5711 q^{7} +7.37159 q^{9} +54.0725 q^{11} -11.0637 q^{13} +124.521 q^{15} +131.327 q^{17} -26.8871 q^{19} -91.2894 q^{21} +99.4551 q^{23} +326.113 q^{25} +115.076 q^{27} -226.855 q^{29} +328.862 q^{31} -317.013 q^{33} -330.722 q^{35} -394.694 q^{37} +64.8635 q^{39} -269.555 q^{41} -103.200 q^{43} -156.568 q^{45} +156.929 q^{47} -100.540 q^{49} -769.932 q^{51} -613.389 q^{53} -1148.47 q^{55} +157.632 q^{57} +221.572 q^{59} +917.017 q^{61} +114.784 q^{63} +234.987 q^{65} +644.395 q^{67} -583.078 q^{69} +939.697 q^{71} -227.180 q^{73} -1911.91 q^{75} +841.971 q^{77} -298.737 q^{79} -873.693 q^{81} +578.892 q^{83} -2789.30 q^{85} +1329.99 q^{87} -525.709 q^{89} -172.274 q^{91} -1928.03 q^{93} +571.067 q^{95} +339.349 q^{97} +398.601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 5 q^{3} + 33 q^{5} + 4 q^{7} + 641 q^{9}+O(q^{10}) \) 54 * q + 5 * q^3 + 33 * q^5 + 4 * q^7 + 641 * q^9	\( 54 q + 5 q^{3} + 33 q^{5} + 4 q^{7} + 641 q^{9} + 55 q^{11} + 71 q^{13} + 101 q^{15} + 301 q^{17} - 122 q^{19} + 195 q^{21} + 317 q^{23} + 2111 q^{25} + 245 q^{27} + 253 q^{29} - 31 q^{31} + 994 q^{33} + 602 q^{35} + 482 q^{37} - 157 q^{39} + 1049 q^{41} - 218 q^{43} + 942 q^{45} + 1349 q^{47} + 5000 q^{49} - 689 q^{51} + 1701 q^{53} - 162 q^{55} + 2429 q^{57} + 710 q^{59} + 539 q^{61} + 798 q^{63} + 2357 q^{65} + 130 q^{67} + 546 q^{69} + 1347 q^{71} + 3185 q^{73} - 316 q^{75} + 2862 q^{77} - 1470 q^{79} + 10610 q^{81} + 2366 q^{83} + 1376 q^{85} - 32 q^{87} + 3566 q^{89} - 1986 q^{91} + 2984 q^{93} + 1655 q^{95} + 4344 q^{97} + 308 q^{99}+O(q^{100}) \) 54 * q + 5 * q^3 + 33 * q^5 + 4 * q^7 + 641 * q^9 + 55 * q^11 + 71 * q^13 + 101 * q^15 + 301 * q^17 - 122 * q^19 + 195 * q^21 + 317 * q^23 + 2111 * q^25 + 245 * q^27 + 253 * q^29 - 31 * q^31 + 994 * q^33 + 602 * q^35 + 482 * q^37 - 157 * q^39 + 1049 * q^41 - 218 * q^43 + 942 * q^45 + 1349 * q^47 + 5000 * q^49 - 689 * q^51 + 1701 * q^53 - 162 * q^55 + 2429 * q^57 + 710 * q^59 + 539 * q^61 + 798 * q^63 + 2357 * q^65 + 130 * q^67 + 546 * q^69 + 1347 * q^71 + 3185 * q^73 - 316 * q^75 + 2862 * q^77 - 1470 * q^79 + 10610 * q^81 + 2366 * q^83 + 1376 * q^85 - 32 * q^87 + 3566 * q^89 - 1986 * q^91 + 2984 * q^93 + 1655 * q^95 + 4344 * q^97 + 308 * q^99

Introduction

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