LMFDB - Embedded newform 2188.2.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2188,2,Mod(1,2188)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2188, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2188.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2188 = 2^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2188.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:	$17.4712679623$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	2188.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.518689 q^{3} -4.15918 q^{5} -3.03794 q^{7} -2.73096 q^{9} +O(q^{10})$	$q+0.518689 q^{3} -4.15918 q^{5} -3.03794 q^{7} -2.73096 q^{9} -4.16036 q^{11} -6.95992 q^{13} -2.15732 q^{15} +5.57340 q^{17} +2.60014 q^{19} -1.57575 q^{21} +2.78593 q^{23} +12.2987 q^{25} -2.97259 q^{27} -7.68512 q^{29} -7.14386 q^{31} -2.15793 q^{33} +12.6353 q^{35} +4.89247 q^{37} -3.61003 q^{39} +6.93664 q^{41} +3.39339 q^{43} +11.3586 q^{45} +6.56863 q^{47} +2.22910 q^{49} +2.89086 q^{51} -9.31047 q^{53} +17.3037 q^{55} +1.34867 q^{57} +3.22128 q^{59} +3.48742 q^{61} +8.29651 q^{63} +28.9475 q^{65} -12.3064 q^{67} +1.44503 q^{69} -10.1512 q^{71} +1.65469 q^{73} +6.37923 q^{75} +12.6389 q^{77} +7.53645 q^{79} +6.65104 q^{81} -10.3278 q^{83} -23.1807 q^{85} -3.98619 q^{87} +3.37411 q^{89} +21.1438 q^{91} -3.70544 q^{93} -10.8144 q^{95} -17.4156 q^{97} +11.3618 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 12 q^{3} - 3 q^{5} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 23 * q + 12 * q^3 - 3 * q^5 + 5 * q^7 + 35 * q^9	$ 23 q + 12 q^{3} - 3 q^{5} + 5 q^{7} + 35 q^{9} + 14 q^{11} + 9 q^{15} + 18 q^{17} + 10 q^{19} + 38 q^{23} + 18 q^{25} + 45 q^{27} + 17 q^{31} + 20 q^{33} + 19 q^{35} + 4 q^{37} + 19 q^{39} + 13 q^{41} + 22 q^{43} + 22 q^{45} + 49 q^{47} + 30 q^{49} + 31 q^{51} + 6 q^{53} + 21 q^{55} + 16 q^{57} + 52 q^{59} + 6 q^{61} + 57 q^{63} + 34 q^{65} + 17 q^{67} + 10 q^{69} + 31 q^{71} + 8 q^{73} + 51 q^{75} - q^{77} + 14 q^{79} + 51 q^{81} + 43 q^{83} - 41 q^{85} + 31 q^{87} + 38 q^{89} + 31 q^{91} - 8 q^{93} + 42 q^{95} + 32 q^{97} + 57 q^{99}+O(q^{100}) $ 23 * q + 12 * q^3 - 3 * q^5 + 5 * q^7 + 35 * q^9 + 14 * q^11 + 9 * q^15 + 18 * q^17 + 10 * q^19 + 38 * q^23 + 18 * q^25 + 45 * q^27 + 17 * q^31 + 20 * q^33 + 19 * q^35 + 4 * q^37 + 19 * q^39 + 13 * q^41 + 22 * q^43 + 22 * q^45 + 49 * q^47 + 30 * q^49 + 31 * q^51 + 6 * q^53 + 21 * q^55 + 16 * q^57 + 52 * q^59 + 6 * q^61 + 57 * q^63 + 34 * q^65 + 17 * q^67 + 10 * q^69 + 31 * q^71 + 8 * q^73 + 51 * q^75 - q^77 + 14 * q^79 + 51 * q^81 + 43 * q^83 - 41 * q^85 + 31 * q^87 + 38 * q^89 + 31 * q^91 - 8 * q^93 + 42 * q^95 + 32 * q^97 + 57 * 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
$3$	0.518689	0.299465	0.149733	−	0.988727i	$-0.452159\pi$
$3$	0.518689	0.299465	0.149733	+	0.988727i	$0.452159\pi$
$4$	0	0
$5$	−4.15918	−1.86004	−0.930020	−	0.367509i	$-0.880211\pi$
$5$	−4.15918	−1.86004	−0.930020	+	0.367509i	$0.880211\pi$
$6$	0	0
$7$	−3.03794	−1.14823	−0.574117	−	0.818773i	$-0.694655\pi$
$7$	−3.03794	−1.14823	−0.574117	+	0.818773i	$0.694655\pi$
$8$	0	0
$9$	−2.73096	−0.910321
$10$	0	0
$11$	−4.16036	−1.25440	−0.627198	−	0.778860i	$-0.715798\pi$
$11$	−4.16036	−1.25440	−0.627198	+	0.778860i	$0.715798\pi$
$12$	0	0
$13$	−6.95992	−1.93033	−0.965167	−	0.261635i	$-0.915738\pi$
$13$	−6.95992	−1.93033	−0.965167	+	0.261635i	$0.915738\pi$
$14$	0	0
$15$	−2.15732	−0.557018
$16$	0	0
$17$	5.57340	1.35175	0.675874	−	0.737018i	$-0.263767\pi$
$17$	5.57340	1.35175	0.675874	+	0.737018i	$0.263767\pi$
$18$	0	0
$19$	2.60014	0.596513	0.298257	−	0.954486i	$-0.403595\pi$
$19$	2.60014	0.596513	0.298257	+	0.954486i	$0.403595\pi$
$20$	0	0
$21$	−1.57575	−0.343857
$22$	0	0
$23$	2.78593	0.580906	0.290453	−	0.956889i	$-0.406194\pi$
$23$	2.78593	0.580906	0.290453	+	0.956889i	$0.406194\pi$
$24$	0	0
$25$	12.2987	2.45975
$26$	0	0
$27$	−2.97259	−0.572075
$28$	0	0
$29$	−7.68512	−1.42709	−0.713546	−	0.700609i	$-0.752912\pi$
$29$	−7.68512	−1.42709	−0.713546	+	0.700609i	$0.752912\pi$
$30$	0	0
$31$	−7.14386	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$31$	−7.14386	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$32$	0	0
$33$	−2.15793	−0.375648
$34$	0	0
$35$	12.6353	2.13576
$36$	0	0
$37$	4.89247	0.804317	0.402159	−	0.915570i	$-0.368260\pi$
$37$	4.89247	0.804317	0.402159	+	0.915570i	$0.368260\pi$
$38$	0	0
$39$	−3.61003	−0.578068
$40$	0	0
$41$	6.93664	1.08332	0.541660	−	0.840598i	$-0.317796\pi$
$41$	6.93664	1.08332	0.541660	+	0.840598i	$0.317796\pi$
$42$	0	0
$43$	3.39339	0.517488	0.258744	−	0.965946i	$-0.416691\pi$
$43$	3.39339	0.517488	0.258744	+	0.965946i	$0.416691\pi$
$44$	0	0
$45$	11.3586	1.69323
$46$	0	0
$47$	6.56863	0.958134	0.479067	−	0.877778i	$-0.340975\pi$
$47$	6.56863	0.958134	0.479067	+	0.877778i	$0.340975\pi$
$48$	0	0
$49$	2.22910	0.318443
$50$	0	0
$51$	2.89086	0.404801
$52$	0	0
$53$	−9.31047	−1.27889	−0.639446	−	0.768836i	$-0.720836\pi$
$53$	−9.31047	−1.27889	−0.639446	+	0.768836i	$0.720836\pi$
$54$	0	0
$55$	17.3037	2.33323
$56$	0	0
$57$	1.34867	0.178635
$58$	0	0
$59$	3.22128	0.419375	0.209688	−	0.977768i	$-0.432755\pi$
$59$	3.22128	0.419375	0.209688	+	0.977768i	$0.432755\pi$
$60$	0	0
$61$	3.48742	0.446519	0.223259	−	0.974759i	$-0.428330\pi$
$61$	3.48742	0.446519	0.223259	+	0.974759i	$0.428330\pi$
$62$	0	0
$63$	8.29651	1.04526
$64$	0	0
$65$	28.9475	3.59050
$66$	0	0
$67$	−12.3064	−1.50347	−0.751734	−	0.659466i	$-0.770782\pi$
$67$	−12.3064	−1.50347	−0.751734	+	0.659466i	$0.770782\pi$
$68$	0	0
$69$	1.44503	0.173961
$70$	0	0
$71$	−10.1512	−1.20473	−0.602365	−	0.798221i	$-0.705775\pi$
$71$	−10.1512	−1.20473	−0.602365	+	0.798221i	$0.705775\pi$
$72$	0	0
$73$	1.65469	0.193666	0.0968332	−	0.995301i	$-0.469129\pi$
$73$	1.65469	0.193666	0.0968332	+	0.995301i	$0.469129\pi$
$74$	0	0
$75$	6.37923	0.736610
$76$	0	0
$77$	12.6389	1.44034
$78$	0	0
$79$	7.53645	0.847917	0.423959	−	0.905682i	$-0.360640\pi$
$79$	7.53645	0.847917	0.423959	+	0.905682i	$0.360640\pi$
$80$	0	0
$81$	6.65104	0.739004
$82$	0	0
$83$	−10.3278	−1.13363	−0.566814	−	0.823846i	$-0.691824\pi$
$83$	−10.3278	−1.13363	−0.566814	+	0.823846i	$0.691824\pi$
$84$	0	0
$85$	−23.1807	−2.51430
$86$	0	0
$87$	−3.98619	−0.427364
$88$	0	0
$89$	3.37411	0.357655	0.178827	−	0.983880i	$-0.442770\pi$
$89$	3.37411	0.357655	0.178827	+	0.983880i	$0.442770\pi$
$90$	0	0
$91$	21.1438	2.21648
$92$	0	0
$93$	−3.70544	−0.384236
$94$	0	0
$95$	−10.8144	−1.10954
$96$	0	0
$97$	−17.4156	−1.76829	−0.884144	−	0.467214i	$-0.845258\pi$
$97$	−17.4156	−1.76829	−0.884144	+	0.467214i	$0.845258\pi$
$98$	0	0
$99$	11.3618	1.14190
$100$	0	0
$101$	3.19351	0.317766	0.158883	−	0.987297i	$-0.449211\pi$
$101$	3.19351	0.317766	0.158883	+	0.987297i	$0.449211\pi$
$102$	0	0
$103$	−13.3237	−1.31283	−0.656413	−	0.754401i	$-0.727927\pi$
$103$	−13.3237	−1.31283	−0.656413	+	0.754401i	$0.727927\pi$
$104$	0	0
$105$	6.55382	0.639587
$106$	0	0
$107$	0.297070	0.0287189	0.0143594	−	0.999897i	$-0.495429\pi$
$107$	0.297070	0.0287189	0.0143594	+	0.999897i	$0.495429\pi$
$108$	0	0
$109$	2.45217	0.234875	0.117438	−	0.993080i	$-0.462532\pi$
$109$	2.45217	0.234875	0.117438	+	0.993080i	$0.462532\pi$
$110$	0	0
$111$	2.53767	0.240865
$112$	0	0
$113$	6.90825	0.649874	0.324937	−	0.945736i	$-0.394657\pi$
$113$	6.90825	0.649874	0.324937	+	0.945736i	$0.394657\pi$
$114$	0	0
$115$	−11.5872	−1.08051
$116$	0	0
$117$	19.0073	1.75722
$118$	0	0
$119$	−16.9317	−1.55212
$120$	0	0
$121$	6.30859	0.573508
$122$	0	0
$123$	3.59796	0.324417
$124$	0	0
$125$	−30.3568	−2.71519
$126$	0	0
$127$	−10.1455	−0.900264	−0.450132	−	0.892962i	$-0.648623\pi$
$127$	−10.1455	−0.900264	−0.450132	+	0.892962i	$0.648623\pi$
$128$	0	0
$129$	1.76012	0.154970
$130$	0	0
$131$	−5.59674	−0.488990	−0.244495	−	0.969651i	$-0.578622\pi$
$131$	−5.59674	−0.488990	−0.244495	+	0.969651i	$0.578622\pi$
$132$	0	0
$133$	−7.89908	−0.684937
$134$	0	0
$135$	12.3635	1.06408
$136$	0	0
$137$	0.206758	0.0176645	0.00883226	−	0.999961i	$-0.497189\pi$
$137$	0.206758	0.0176645	0.00883226	+	0.999961i	$0.497189\pi$
$138$	0	0
$139$	−3.62119	−0.307145	−0.153573	−	0.988137i	$-0.549078\pi$
$139$	−3.62119	−0.307145	−0.153573	+	0.988137i	$0.549078\pi$
$140$	0	0
$141$	3.40708	0.286928
$142$	0	0
$143$	28.9558	2.42140
$144$	0	0
$145$	31.9638	2.65445
$146$	0	0
$147$	1.15621	0.0953627
$148$	0	0
$149$	−3.02676	−0.247962	−0.123981	−	0.992285i	$-0.539566\pi$
$149$	−3.02676	−0.247962	−0.123981	+	0.992285i	$0.539566\pi$
$150$	0	0
$151$	20.3823	1.65869	0.829346	−	0.558736i	$-0.188713\pi$
$151$	20.3823	1.65869	0.829346	+	0.558736i	$0.188713\pi$
$152$	0	0
$153$	−15.2207	−1.23052
$154$	0	0
$155$	29.7126	2.38657
$156$	0	0
$157$	−14.2879	−1.14030	−0.570151	−	0.821540i	$-0.693115\pi$
$157$	−14.2879	−1.14030	−0.570151	+	0.821540i	$0.693115\pi$
$158$	0	0
$159$	−4.82924	−0.382984
$160$	0	0
$161$	−8.46349	−0.667017
$162$	0	0
$163$	0.505374	0.0395839	0.0197920	−	0.999804i	$-0.493700\pi$
$163$	0.505374	0.0395839	0.0197920	+	0.999804i	$0.493700\pi$
$164$	0	0
$165$	8.97523	0.698720
$166$	0	0
$167$	24.2651	1.87769	0.938845	−	0.344340i	$-0.111897\pi$
$167$	24.2651	1.87769	0.938845	+	0.344340i	$0.111897\pi$
$168$	0	0
$169$	35.4404	2.72619
$170$	0	0
$171$	−7.10089	−0.543018
$172$	0	0
$173$	−16.4957	−1.25414	−0.627071	−	0.778962i	$-0.715746\pi$
$173$	−16.4957	−1.25414	−0.627071	+	0.778962i	$0.715746\pi$
$174$	0	0
$175$	−37.3629	−2.82437
$176$	0	0
$177$	1.67084	0.125588
$178$	0	0
$179$	−7.86733	−0.588032	−0.294016	−	0.955801i	$-0.594992\pi$
$179$	−7.86733	−0.588032	−0.294016	+	0.955801i	$0.594992\pi$
$180$	0	0
$181$	9.83034	0.730684	0.365342	−	0.930873i	$-0.380952\pi$
$181$	9.83034	0.730684	0.365342	+	0.930873i	$0.380952\pi$
$182$	0	0
$183$	1.80889	0.133717
$184$	0	0
$185$	−20.3487	−1.49606
$186$	0	0
$187$	−23.1873	−1.69563
$188$	0	0
$189$	9.03056	0.656876
$190$	0	0
$191$	11.8430	0.856932	0.428466	−	0.903558i	$-0.359054\pi$
$191$	11.8430	0.856932	0.428466	+	0.903558i	$0.359054\pi$
$192$	0	0
$193$	−14.2277	−1.02413	−0.512067	−	0.858946i	$-0.671120\pi$
$193$	−14.2277	−1.02413	−0.512067	+	0.858946i	$0.671120\pi$
$194$	0	0
$195$	15.0148	1.07523
$196$	0	0
$197$	16.4666	1.17319	0.586597	−	0.809879i	$-0.300467\pi$
$197$	16.4666	1.17319	0.586597	+	0.809879i	$0.300467\pi$
$198$	0	0
$199$	−4.62434	−0.327811	−0.163905	−	0.986476i	$-0.552409\pi$
$199$	−4.62434	−0.327811	−0.163905	+	0.986476i	$0.552409\pi$
$200$	0	0
$201$	−6.38321	−0.450237
$202$	0	0
$203$	23.3470	1.63864
$204$	0	0
$205$	−28.8507	−2.01502
$206$	0	0
$207$	−7.60826	−0.528811
$208$	0	0
$209$	−10.8175	−0.748264
$210$	0	0
$211$	1.96455	0.135245	0.0676225	−	0.997711i	$-0.478459\pi$
$211$	1.96455	0.135245	0.0676225	+	0.997711i	$0.478459\pi$
$212$	0	0
$213$	−5.26534	−0.360775
$214$	0	0
$215$	−14.1137	−0.962548
$216$	0	0
$217$	21.7026	1.47327
$218$	0	0
$219$	0.858268	0.0579964
$220$	0	0
$221$	−38.7904	−2.60932
$222$	0	0
$223$	−14.1716	−0.949002	−0.474501	−	0.880255i	$-0.657372\pi$
$223$	−14.1716	−0.949002	−0.474501	+	0.880255i	$0.657372\pi$
$224$	0	0
$225$	−33.5874	−2.23916
$226$	0	0
$227$	26.7159	1.77320	0.886599	−	0.462539i	$-0.153062\pi$
$227$	26.7159	1.77320	0.886599	+	0.462539i	$0.153062\pi$
$228$	0	0
$229$	−14.2561	−0.942071	−0.471036	−	0.882114i	$-0.656120\pi$
$229$	−14.2561	−0.942071	−0.471036	+	0.882114i	$0.656120\pi$
$230$	0	0
$231$	6.55568	0.431332
$232$	0	0
$233$	12.5521	0.822316	0.411158	−	0.911564i	$-0.365125\pi$
$233$	12.5521	0.822316	0.411158	+	0.911564i	$0.365125\pi$
$234$	0	0
$235$	−27.3201	−1.78217
$236$	0	0
$237$	3.90908	0.253922
$238$	0	0
$239$	17.2488	1.11573	0.557866	−	0.829931i	$-0.311620\pi$
$239$	17.2488	1.11573	0.557866	+	0.829931i	$0.311620\pi$
$240$	0	0
$241$	−21.1364	−1.36152	−0.680758	−	0.732508i	$-0.738349\pi$
$241$	−21.1364	−1.36152	−0.680758	+	0.732508i	$0.738349\pi$
$242$	0	0
$243$	12.3676	0.793381
$244$	0	0
$245$	−9.27123	−0.592317
$246$	0	0
$247$	−18.0968	−1.15147
$248$	0	0
$249$	−5.35694	−0.339482
$250$	0	0
$251$	15.4831	0.977285	0.488642	−	0.872484i	$-0.337492\pi$
$251$	15.4831	0.977285	0.488642	+	0.872484i	$0.337492\pi$
$252$	0	0
$253$	−11.5905	−0.728686
$254$	0	0
$255$	−12.0236	−0.752947
$256$	0	0
$257$	−14.8947	−0.929104	−0.464552	−	0.885546i	$-0.653785\pi$
$257$	−14.8947	−0.929104	−0.464552	+	0.885546i	$0.653785\pi$
$258$	0	0
$259$	−14.8631	−0.923545
$260$	0	0
$261$	20.9878	1.29911
$262$	0	0
$263$	24.1126	1.48685	0.743424	−	0.668820i	$-0.233200\pi$
$263$	24.1126	1.48685	0.743424	+	0.668820i	$0.233200\pi$
$264$	0	0
$265$	38.7239	2.37879
$266$	0	0
$267$	1.75011	0.107105
$268$	0	0
$269$	20.0100	1.22003	0.610017	−	0.792389i	$-0.291163\pi$
$269$	20.0100	1.22003	0.610017	+	0.792389i	$0.291163\pi$
$270$	0	0
$271$	−10.0409	−0.609940	−0.304970	−	0.952362i	$-0.598646\pi$
$271$	−10.0409	−0.609940	−0.304970	+	0.952362i	$0.598646\pi$
$272$	0	0
$273$	10.9671	0.663758
$274$	0	0
$275$	−51.1672	−3.08550
$276$	0	0
$277$	16.8659	1.01337	0.506685	−	0.862131i	$-0.330871\pi$
$277$	16.8659	1.01337	0.506685	+	0.862131i	$0.330871\pi$
$278$	0	0
$279$	19.5096	1.16801
$280$	0	0
$281$	−12.2059	−0.728141	−0.364070	−	0.931371i	$-0.618613\pi$
$281$	−12.2059	−0.728141	−0.364070	+	0.931371i	$0.618613\pi$
$282$	0	0
$283$	20.5520	1.22169	0.610846	−	0.791750i	$-0.290830\pi$
$283$	20.5520	1.22169	0.610846	+	0.791750i	$0.290830\pi$
$284$	0	0
$285$	−5.60934	−0.332268
$286$	0	0
$287$	−21.0731	−1.24391
$288$	0	0
$289$	14.0628	0.827221
$290$	0	0
$291$	−9.03329	−0.529541
$292$	0	0
$293$	14.2307	0.831368	0.415684	−	0.909509i	$-0.363542\pi$
$293$	14.2307	0.831368	0.415684	+	0.909509i	$0.363542\pi$
$294$	0	0
$295$	−13.3979	−0.780055
$296$	0	0
$297$	12.3670	0.717608
$298$	0	0
$299$	−19.3898	−1.12134
$300$	0	0
$301$	−10.3089	−0.594197
$302$	0	0
$303$	1.65644	0.0951599
$304$	0	0
$305$	−14.5048	−0.830543
$306$	0	0
$307$	2.31465	0.132104	0.0660522	−	0.997816i	$-0.478960\pi$
$307$	2.31465	0.132104	0.0660522	+	0.997816i	$0.478960\pi$
$308$	0	0
$309$	−6.91088	−0.393146
$310$	0	0
$311$	20.3033	1.15130	0.575648	−	0.817698i	$-0.304750\pi$
$311$	20.3033	1.15130	0.575648	+	0.817698i	$0.304750\pi$
$312$	0	0
$313$	12.1612	0.687390	0.343695	−	0.939081i	$-0.388321\pi$
$313$	12.1612	0.687390	0.343695	+	0.939081i	$0.388321\pi$
$314$	0	0
$315$	−34.5066	−1.94423
$316$	0	0
$317$	27.2545	1.53077	0.765383	−	0.643575i	$-0.222550\pi$
$317$	27.2545	1.53077	0.765383	+	0.643575i	$0.222550\pi$
$318$	0	0
$319$	31.9729	1.79014
$320$	0	0
$321$	0.154087	0.00860030
$322$	0	0
$323$	14.4916	0.806335
$324$	0	0
$325$	−85.5983	−4.74814
$326$	0	0
$327$	1.27191	0.0703369
$328$	0	0
$329$	−19.9551	−1.10016
$330$	0	0
$331$	−25.7351	−1.41453	−0.707265	−	0.706949i	$-0.750071\pi$
$331$	−25.7351	−1.41453	−0.707265	+	0.706949i	$0.750071\pi$
$332$	0	0
$333$	−13.3612	−0.732187
$334$	0	0
$335$	51.1846	2.79651
$336$	0	0
$337$	−20.6023	−1.12228	−0.561138	−	0.827722i	$-0.689636\pi$
$337$	−20.6023	−1.12228	−0.561138	+	0.827722i	$0.689636\pi$
$338$	0	0
$339$	3.58324	0.194615
$340$	0	0
$341$	29.7210	1.60948
$342$	0	0
$343$	14.4937	0.782587
$344$	0	0
$345$	−6.01014	−0.323575
$346$	0	0
$347$	8.29302	0.445192	0.222596	−	0.974911i	$-0.428547\pi$
$347$	8.29302	0.445192	0.222596	+	0.974911i	$0.428547\pi$
$348$	0	0
$349$	2.20271	0.117908	0.0589541	−	0.998261i	$-0.481223\pi$
$349$	2.20271	0.117908	0.0589541	+	0.998261i	$0.481223\pi$
$350$	0	0
$351$	20.6890	1.10430
$352$	0	0
$353$	−7.10464	−0.378142	−0.189071	−	0.981963i	$-0.560548\pi$
$353$	−7.10464	−0.378142	−0.189071	+	0.981963i	$0.560548\pi$
$354$	0	0
$355$	42.2208	2.24085
$356$	0	0
$357$	−8.78227	−0.464807
$358$	0	0
$359$	−24.8267	−1.31030	−0.655152	−	0.755497i	$-0.727395\pi$
$359$	−24.8267	−1.31030	−0.655152	+	0.755497i	$0.727395\pi$
$360$	0	0
$361$	−12.2393	−0.644172
$362$	0	0
$363$	3.27220	0.171746
$364$	0	0
$365$	−6.88213	−0.360227
$366$	0	0
$367$	17.1588	0.895682	0.447841	−	0.894113i	$-0.352193\pi$
$367$	17.1588	0.895682	0.447841	+	0.894113i	$0.352193\pi$
$368$	0	0
$369$	−18.9437	−0.986169
$370$	0	0
$371$	28.2847	1.46847
$372$	0	0
$373$	−11.4307	−0.591860	−0.295930	−	0.955210i	$-0.595630\pi$
$373$	−11.4307	−0.591860	−0.295930	+	0.955210i	$0.595630\pi$
$374$	0	0
$375$	−15.7457	−0.813106
$376$	0	0
$377$	53.4878	2.75476
$378$	0	0
$379$	−2.60527	−0.133824	−0.0669119	−	0.997759i	$-0.521315\pi$
$379$	−2.60527	−0.133824	−0.0669119	+	0.997759i	$0.521315\pi$
$380$	0	0
$381$	−5.26234	−0.269598
$382$	0	0
$383$	17.3989	0.889043	0.444522	−	0.895768i	$-0.353374\pi$
$383$	17.3989	0.889043	0.444522	+	0.895768i	$0.353374\pi$
$384$	0	0
$385$	−52.5676	−2.67909
$386$	0	0
$387$	−9.26723	−0.471080
$388$	0	0
$389$	7.51392	0.380971	0.190486	−	0.981690i	$-0.438994\pi$
$389$	7.51392	0.380971	0.190486	+	0.981690i	$0.438994\pi$
$390$	0	0
$391$	15.5271	0.785238
$392$	0	0
$393$	−2.90297	−0.146435
$394$	0	0
$395$	−31.3454	−1.57716
$396$	0	0
$397$	23.5872	1.18381	0.591905	−	0.806008i	$-0.298376\pi$
$397$	23.5872	1.18381	0.591905	+	0.806008i	$0.298376\pi$
$398$	0	0
$399$	−4.09717	−0.205115
$400$	0	0
$401$	−12.9848	−0.648431	−0.324216	−	0.945983i	$-0.605100\pi$
$401$	−12.9848	−0.648431	−0.324216	+	0.945983i	$0.605100\pi$
$402$	0	0
$403$	49.7206	2.47676
$404$	0	0
$405$	−27.6628	−1.37458
$406$	0	0
$407$	−20.3544	−1.00893
$408$	0	0
$409$	−8.11123	−0.401074	−0.200537	−	0.979686i	$-0.564269\pi$
$409$	−8.11123	−0.401074	−0.200537	+	0.979686i	$0.564269\pi$
$410$	0	0
$411$	0.107243	0.00528991
$412$	0	0
$413$	−9.78608	−0.481541
$414$	0	0
$415$	42.9553	2.10859
$416$	0	0
$417$	−1.87827	−0.0919794
$418$	0	0
$419$	23.5527	1.15062	0.575311	−	0.817935i	$-0.304881\pi$
$419$	23.5527	1.15062	0.575311	+	0.817935i	$0.304881\pi$
$420$	0	0
$421$	−2.64261	−0.128793	−0.0643965	−	0.997924i	$-0.520512\pi$
$421$	−2.64261	−0.128793	−0.0643965	+	0.997924i	$0.520512\pi$
$422$	0	0
$423$	−17.9387	−0.872209
$424$	0	0
$425$	68.5458	3.32496
$426$	0	0
$427$	−10.5946	−0.512709
$428$	0	0
$429$	15.0190	0.725126
$430$	0	0
$431$	−11.1699	−0.538034	−0.269017	−	0.963135i	$-0.586699\pi$
$431$	−11.1699	−0.538034	−0.269017	+	0.963135i	$0.586699\pi$
$432$	0	0
$433$	30.1088	1.44693	0.723467	−	0.690359i	$-0.242547\pi$
$433$	30.1088	1.44693	0.723467	+	0.690359i	$0.242547\pi$
$434$	0	0
$435$	16.5793	0.794915
$436$	0	0
$437$	7.24381	0.346518
$438$	0	0
$439$	2.24094	0.106954	0.0534772	−	0.998569i	$-0.482970\pi$
$439$	2.24094	0.106954	0.0534772	+	0.998569i	$0.482970\pi$
$440$	0	0
$441$	−6.08760	−0.289885
$442$	0	0
$443$	27.2267	1.29358	0.646790	−	0.762668i	$-0.276111\pi$
$443$	27.2267	1.29358	0.646790	+	0.762668i	$0.276111\pi$
$444$	0	0
$445$	−14.0335	−0.665253
$446$	0	0
$447$	−1.56995	−0.0742560
$448$	0	0
$449$	−20.1054	−0.948834	−0.474417	−	0.880300i	$-0.657341\pi$
$449$	−20.1054	−0.948834	−0.474417	+	0.880300i	$0.657341\pi$
$450$	0	0
$451$	−28.8589	−1.35891
$452$	0	0
$453$	10.5721	0.496721
$454$	0	0
$455$	−87.9409	−4.12274
$456$	0	0
$457$	−35.7525	−1.67243	−0.836215	−	0.548402i	$-0.815236\pi$
$457$	−35.7525	−1.67243	−0.836215	+	0.548402i	$0.815236\pi$
$458$	0	0
$459$	−16.5674	−0.773301
$460$	0	0
$461$	−20.8225	−0.969799	−0.484900	−	0.874570i	$-0.661144\pi$
$461$	−20.8225	−0.969799	−0.484900	+	0.874570i	$0.661144\pi$
$462$	0	0
$463$	−33.4019	−1.55232	−0.776158	−	0.630538i	$-0.782834\pi$
$463$	−33.4019	−1.55232	−0.776158	+	0.630538i	$0.782834\pi$
$464$	0	0
$465$	15.4116	0.714695
$466$	0	0
$467$	13.8790	0.642245	0.321123	−	0.947038i	$-0.395940\pi$
$467$	13.8790	0.642245	0.321123	+	0.947038i	$0.395940\pi$
$468$	0	0
$469$	37.3862	1.72634
$470$	0	0
$471$	−7.41100	−0.341481
$472$	0	0
$473$	−14.1177	−0.649134
$474$	0	0
$475$	31.9785	1.46727
$476$	0	0
$477$	25.4265	1.16420
$478$	0	0
$479$	33.2926	1.52118	0.760589	−	0.649234i	$-0.224910\pi$
$479$	33.2926	1.52118	0.760589	+	0.649234i	$0.224910\pi$
$480$	0	0
$481$	−34.0512	−1.55260
$482$	0	0
$483$	−4.38992	−0.199748
$484$	0	0
$485$	72.4346	3.28909
$486$	0	0
$487$	−18.4807	−0.837440	−0.418720	−	0.908115i	$-0.637521\pi$
$487$	−18.4807	−0.837440	−0.418720	+	0.908115i	$0.637521\pi$
$488$	0	0
$489$	0.262132	0.0118540
$490$	0	0
$491$	8.32394	0.375654	0.187827	−	0.982202i	$-0.439856\pi$
$491$	8.32394	0.375654	0.187827	+	0.982202i	$0.439856\pi$
$492$	0	0
$493$	−42.8322	−1.92907
$494$	0	0
$495$	−47.2557	−2.12398
$496$	0	0
$497$	30.8389	1.38331
$498$	0	0
$499$	−27.1170	−1.21393	−0.606963	−	0.794730i	$-0.707612\pi$
$499$	−27.1170	−1.21393	−0.606963	+	0.794730i	$0.707612\pi$
$500$	0	0
$501$	12.5860	0.562303
$502$	0	0
$503$	7.82378	0.348845	0.174423	−	0.984671i	$-0.444194\pi$
$503$	7.82378	0.348845	0.174423	+	0.984671i	$0.444194\pi$
$504$	0	0
$505$	−13.2824	−0.591058
$506$	0	0
$507$	18.3826	0.816399
$508$	0	0
$509$	−17.4541	−0.773641	−0.386820	−	0.922155i	$-0.626427\pi$
$509$	−17.4541	−0.773641	−0.386820	+	0.922155i	$0.626427\pi$
$510$	0	0
$511$	−5.02685	−0.222375
$512$	0	0
$513$	−7.72915	−0.341250
$514$	0	0
$515$	55.4158	2.44191
$516$	0	0
$517$	−27.3279	−1.20188
$518$	0	0
$519$	−8.55612	−0.375572
$520$	0	0
$521$	−31.8840	−1.39686	−0.698432	−	0.715677i	$-0.746118\pi$
$521$	−31.8840	−1.39686	−0.698432	+	0.715677i	$0.746118\pi$
$522$	0	0
$523$	45.2384	1.97814	0.989069	−	0.147453i	$-0.0471074\pi$
$523$	45.2384	1.97814	0.989069	+	0.147453i	$0.0471074\pi$
$524$	0	0
$525$	−19.3797	−0.845801
$526$	0	0
$527$	−39.8155	−1.73439
$528$	0	0
$529$	−15.2386	−0.662548
$530$	0	0
$531$	−8.79720	−0.381766
$532$	0	0
$533$	−48.2784	−2.09117
$534$	0	0
$535$	−1.23557	−0.0534182
$536$	0	0
$537$	−4.08070	−0.176095
$538$	0	0
$539$	−9.27387	−0.399454
$540$	0	0
$541$	−21.6324	−0.930050	−0.465025	−	0.885298i	$-0.653955\pi$
$541$	−21.6324	−0.930050	−0.465025	+	0.885298i	$0.653955\pi$
$542$	0	0
$543$	5.09889	0.218814
$544$	0	0
$545$	−10.1990	−0.436877
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0	0
$549$	−9.52402	−0.406475
$550$	0	0
$551$	−19.9824	−0.851279
$552$	0	0
$553$	−22.8953	−0.973608
$554$	0	0
$555$	−10.5546	−0.448019
$556$	0	0
$557$	−32.2842	−1.36792	−0.683962	−	0.729517i	$-0.739745\pi$
$557$	−32.2842	−1.36792	−0.683962	+	0.729517i	$0.739745\pi$
$558$	0	0
$559$	−23.6177	−0.998924
$560$	0	0
$561$	−12.0270	−0.507781
$562$	0	0
$563$	−43.9674	−1.85300	−0.926502	−	0.376290i	$-0.877200\pi$
$563$	−43.9674	−1.85300	−0.926502	+	0.376290i	$0.877200\pi$
$564$	0	0
$565$	−28.7326	−1.20879
$566$	0	0
$567$	−20.2055	−0.848550
$568$	0	0
$569$	21.9135	0.918660	0.459330	−	0.888266i	$-0.348090\pi$
$569$	21.9135	0.918660	0.459330	+	0.888266i	$0.348090\pi$
$570$	0	0
$571$	−24.9611	−1.04459	−0.522294	−	0.852765i	$-0.674924\pi$
$571$	−24.9611	−1.04459	−0.522294	+	0.852765i	$0.674924\pi$
$572$	0	0
$573$	6.14286	0.256622
$574$	0	0
$575$	34.2634	1.42888
$576$	0	0
$577$	−34.3534	−1.43015	−0.715075	−	0.699048i	$-0.753607\pi$
$577$	−34.3534	−1.43015	−0.715075	+	0.699048i	$0.753607\pi$
$578$	0	0
$579$	−7.37976	−0.306692
$580$	0	0
$581$	31.3754	1.30167
$582$	0	0
$583$	38.7349	1.60424
$584$	0	0
$585$	−79.0546	−3.26850
$586$	0	0
$587$	30.2215	1.24737	0.623687	−	0.781674i	$-0.285634\pi$
$587$	30.2215	1.24737	0.623687	+	0.781674i	$0.285634\pi$
$588$	0	0
$589$	−18.5750	−0.765371
$590$	0	0
$591$	8.54102	0.351331
$592$	0	0
$593$	−3.15649	−0.129622	−0.0648108	−	0.997898i	$-0.520644\pi$
$593$	−3.15649	−0.129622	−0.0648108	+	0.997898i	$0.520644\pi$
$594$	0	0
$595$	70.4218	2.88701
$596$	0	0
$597$	−2.39860	−0.0981680
$598$	0	0
$599$	−2.76426	−0.112945	−0.0564723	−	0.998404i	$-0.517985\pi$
$599$	−2.76426	−0.112945	−0.0564723	+	0.998404i	$0.517985\pi$
$600$	0	0
$601$	36.1475	1.47449	0.737244	−	0.675627i	$-0.236127\pi$
$601$	36.1475	1.47449	0.737244	+	0.675627i	$0.236127\pi$
$602$	0	0
$603$	33.6084	1.36864
$604$	0	0
$605$	−26.2385	−1.06675
$606$	0	0
$607$	−16.3617	−0.664100	−0.332050	−	0.943262i	$-0.607740\pi$
$607$	−16.3617	−0.664100	−0.332050	+	0.943262i	$0.607740\pi$
$608$	0	0
$609$	12.1098	0.490715
$610$	0	0
$611$	−45.7171	−1.84952
$612$	0	0
$613$	13.6668	0.551996	0.275998	−	0.961158i	$-0.410992\pi$
$613$	13.6668	0.551996	0.275998	+	0.961158i	$0.410992\pi$
$614$	0	0
$615$	−14.9645	−0.603429
$616$	0	0
$617$	−2.11802	−0.0852684	−0.0426342	−	0.999091i	$-0.513575\pi$
$617$	−2.11802	−0.0852684	−0.0426342	+	0.999091i	$0.513575\pi$
$618$	0	0
$619$	42.2179	1.69688	0.848440	−	0.529291i	$-0.177542\pi$
$619$	42.2179	1.69688	0.848440	+	0.529291i	$0.177542\pi$
$620$	0	0
$621$	−8.28142	−0.332322
$622$	0	0
$623$	−10.2504	−0.410672
$624$	0	0
$625$	64.7655	2.59062
$626$	0	0
$627$	−5.61093	−0.224079
$628$	0	0
$629$	27.2677	1.08723
$630$	0	0
$631$	18.3463	0.730353	0.365176	−	0.930938i	$-0.381009\pi$
$631$	18.3463	0.730353	0.365176	+	0.930938i	$0.381009\pi$
$632$	0	0
$633$	1.01899	0.0405012
$634$	0	0
$635$	42.1968	1.67453
$636$	0	0
$637$	−15.5144	−0.614702
$638$	0	0
$639$	27.7226	1.09669
$640$	0	0
$641$	−33.3825	−1.31853	−0.659265	−	0.751911i	$-0.729132\pi$
$641$	−33.3825	−1.31853	−0.659265	+	0.751911i	$0.729132\pi$
$642$	0	0
$643$	14.9128	0.588103	0.294052	−	0.955790i	$-0.404996\pi$
$643$	14.9128	0.588103	0.294052	+	0.955790i	$0.404996\pi$
$644$	0	0
$645$	−7.32064	−0.288250
$646$	0	0
$647$	25.6774	1.00948	0.504740	−	0.863271i	$-0.331588\pi$
$647$	25.6774	1.00948	0.504740	+	0.863271i	$0.331588\pi$
$648$	0	0
$649$	−13.4017	−0.526063
$650$	0	0
$651$	11.2569	0.441194
$652$	0	0
$653$	18.8774	0.738729	0.369364	−	0.929285i	$-0.379575\pi$
$653$	18.8774	0.738729	0.369364	+	0.929285i	$0.379575\pi$
$654$	0	0
$655$	23.2778	0.909540
$656$	0	0
$657$	−4.51889	−0.176299
$658$	0	0
$659$	39.9445	1.55602	0.778009	−	0.628253i	$-0.216230\pi$
$659$	39.9445	1.55602	0.778009	+	0.628253i	$0.216230\pi$
$660$	0	0
$661$	6.94469	0.270117	0.135059	−	0.990838i	$-0.456878\pi$
$661$	6.94469	0.270117	0.135059	+	0.990838i	$0.456878\pi$
$662$	0	0
$663$	−20.1201	−0.781402
$664$	0	0
$665$	32.8537	1.27401
$666$	0	0
$667$	−21.4102	−0.829006
$668$	0	0
$669$	−7.35067	−0.284193
$670$	0	0
$671$	−14.5089	−0.560111
$672$	0	0
$673$	7.47688	0.288212	0.144106	−	0.989562i	$-0.453969\pi$
$673$	7.47688	0.288212	0.144106	+	0.989562i	$0.453969\pi$
$674$	0	0
$675$	−36.5591	−1.40716
$676$	0	0
$677$	−42.7368	−1.64251	−0.821254	−	0.570563i	$-0.806725\pi$
$677$	−42.7368	−1.64251	−0.821254	+	0.570563i	$0.806725\pi$
$678$	0	0
$679$	52.9077	2.03041
$680$	0	0
$681$	13.8573	0.531011
$682$	0	0
$683$	26.5499	1.01590	0.507952	−	0.861385i	$-0.330403\pi$
$683$	26.5499	1.01590	0.507952	+	0.861385i	$0.330403\pi$
$684$	0	0
$685$	−0.859943	−0.0328567
$686$	0	0
$687$	−7.39450	−0.282118
$688$	0	0
$689$	64.8001	2.46869
$690$	0	0
$691$	−46.8515	−1.78231	−0.891157	−	0.453696i	$-0.850105\pi$
$691$	−46.8515	−1.78231	−0.891157	+	0.453696i	$0.850105\pi$
$692$	0	0
$693$	−34.5165	−1.31117
$694$	0	0
$695$	15.0612	0.571303
$696$	0	0
$697$	38.6606	1.46438
$698$	0	0
$699$	6.51064	0.246255
$700$	0	0
$701$	−32.0569	−1.21077	−0.605387	−	0.795931i	$-0.706982\pi$
$701$	−32.0569	−1.21077	−0.605387	+	0.795931i	$0.706982\pi$
$702$	0	0
$703$	12.7211	0.479786
$704$	0	0
$705$	−14.1706	−0.533697
$706$	0	0
$707$	−9.70170	−0.364870
$708$	0	0
$709$	−27.6654	−1.03900	−0.519498	−	0.854472i	$-0.673881\pi$
$709$	−27.6654	−1.03900	−0.519498	+	0.854472i	$0.673881\pi$
$710$	0	0
$711$	−20.5818	−0.771876
$712$	0	0
$713$	−19.9023	−0.745346
$714$	0	0
$715$	−120.432	−4.50390
$716$	0	0
$717$	8.94676	0.334123
$718$	0	0
$719$	11.0828	0.413318	0.206659	−	0.978413i	$-0.433741\pi$
$719$	11.0828	0.413318	0.206659	+	0.978413i	$0.433741\pi$
$720$	0	0
$721$	40.4768	1.50743
$722$	0	0
$723$	−10.9632	−0.407727
$724$	0	0
$725$	−94.5174	−3.51029
$726$	0	0
$727$	10.2406	0.379802	0.189901	−	0.981803i	$-0.439183\pi$
$727$	10.2406	0.379802	0.189901	+	0.981803i	$0.439183\pi$
$728$	0	0
$729$	−13.5382	−0.501414
$730$	0	0
$731$	18.9127	0.699513
$732$	0	0
$733$	−23.9921	−0.886168	−0.443084	−	0.896480i	$-0.646116\pi$
$733$	−23.9921	−0.886168	−0.443084	+	0.896480i	$0.646116\pi$
$734$	0	0
$735$	−4.80889	−0.177379
$736$	0	0
$737$	51.1991	1.88594
$738$	0	0
$739$	−16.7892	−0.617602	−0.308801	−	0.951127i	$-0.599928\pi$
$739$	−16.7892	−0.617602	−0.308801	+	0.951127i	$0.599928\pi$
$740$	0	0
$741$	−9.38660	−0.344825
$742$	0	0
$743$	−34.9093	−1.28070	−0.640348	−	0.768085i	$-0.721210\pi$
$743$	−34.9093	−1.28070	−0.640348	+	0.768085i	$0.721210\pi$
$744$	0	0
$745$	12.5888	0.461219
$746$	0	0
$747$	28.2049	1.03196
$748$	0	0
$749$	−0.902483	−0.0329760
$750$	0	0
$751$	−25.0384	−0.913665	−0.456832	−	0.889553i	$-0.651016\pi$
$751$	−25.0384	−0.913665	−0.456832	+	0.889553i	$0.651016\pi$
$752$	0	0
$753$	8.03092	0.292663
$754$	0	0
$755$	−84.7737	−3.08523
$756$	0	0
$757$	43.5249	1.58194	0.790969	−	0.611856i	$-0.209577\pi$
$757$	43.5249	1.58194	0.790969	+	0.611856i	$0.209577\pi$
$758$	0	0
$759$	−6.01185	−0.218216
$760$	0	0
$761$	−35.9959	−1.30485	−0.652426	−	0.757853i	$-0.726249\pi$
$761$	−35.9959	−1.30485	−0.652426	+	0.757853i	$0.726249\pi$
$762$	0	0
$763$	−7.44955	−0.269692
$764$	0	0
$765$	63.3057	2.28882
$766$	0	0
$767$	−22.4199	−0.809534
$768$	0	0
$769$	24.1757	0.871799	0.435900	−	0.899995i	$-0.356430\pi$
$769$	24.1757	0.871799	0.435900	+	0.899995i	$0.356430\pi$
$770$	0	0
$771$	−7.72571	−0.278235
$772$	0	0
$773$	−39.3633	−1.41580	−0.707899	−	0.706313i	$-0.750357\pi$
$773$	−39.3633	−1.41580	−0.707899	+	0.706313i	$0.750357\pi$
$774$	0	0
$775$	−87.8605	−3.15604
$776$	0	0
$777$	−7.70931	−0.276570
$778$	0	0
$779$	18.0362	0.646215
$780$	0	0
$781$	42.2328	1.51121
$782$	0	0
$783$	22.8447	0.816403
$784$	0	0
$785$	59.4261	2.12101
$786$	0	0
$787$	51.9472	1.85172	0.925860	−	0.377868i	$-0.123343\pi$
$787$	51.9472	1.85172	0.925860	+	0.377868i	$0.123343\pi$
$788$	0	0
$789$	12.5070	0.445260
$790$	0	0
$791$	−20.9869	−0.746208
$792$	0	0
$793$	−24.2722	−0.861930
$794$	0	0
$795$	20.0857	0.712365
$796$	0	0
$797$	−31.4011	−1.11228	−0.556142	−	0.831087i	$-0.687719\pi$
$797$	−31.4011	−1.11228	−0.556142	+	0.831087i	$0.687719\pi$
$798$	0	0
$799$	36.6096	1.29515
$800$	0	0
$801$	−9.21456	−0.325581
$802$	0	0
$803$	−6.88409	−0.242934
$804$	0	0
$805$	35.2012	1.24068
$806$	0	0
$807$	10.3790	0.365358
$808$	0	0
$809$	13.5035	0.474758	0.237379	−	0.971417i	$-0.423712\pi$
$809$	13.5035	0.474758	0.237379	+	0.971417i	$0.423712\pi$
$810$	0	0
$811$	−21.9510	−0.770804	−0.385402	−	0.922749i	$-0.625937\pi$
$811$	−21.9510	−0.770804	−0.385402	+	0.922749i	$0.625937\pi$
$812$	0	0
$813$	−5.20809	−0.182656
$814$	0	0
$815$	−2.10194	−0.0736277
$816$	0	0
$817$	8.82330	0.308688
$818$	0	0
$819$	−57.7430	−2.01770
$820$	0	0
$821$	−24.7368	−0.863321	−0.431661	−	0.902036i	$-0.642072\pi$
$821$	−24.7368	−0.863321	−0.431661	+	0.902036i	$0.642072\pi$
$822$	0	0
$823$	19.5382	0.681059	0.340529	−	0.940234i	$-0.389394\pi$
$823$	19.5382	0.681059	0.340529	+	0.940234i	$0.389394\pi$
$824$	0	0
$825$	−26.5399	−0.924000
$826$	0	0
$827$	−37.8037	−1.31456	−0.657282	−	0.753645i	$-0.728294\pi$
$827$	−37.8037	−1.31456	−0.657282	+	0.753645i	$0.728294\pi$
$828$	0	0
$829$	14.2353	0.494412	0.247206	−	0.968963i	$-0.420488\pi$
$829$	14.2353	0.494412	0.247206	+	0.968963i	$0.420488\pi$
$830$	0	0
$831$	8.74814	0.303469
$832$	0	0
$833$	12.4237	0.430455
$834$	0	0
$835$	−100.923	−3.49258
$836$	0	0
$837$	21.2357	0.734015
$838$	0	0
$839$	21.8912	0.755767	0.377883	−	0.925853i	$-0.376652\pi$
$839$	21.8912	0.755767	0.377883	+	0.925853i	$0.376652\pi$
$840$	0	0
$841$	30.0611	1.03659
$842$	0	0
$843$	−6.33105	−0.218053
$844$	0	0
$845$	−147.403	−5.07082
$846$	0	0
$847$	−19.1651	−0.658522
$848$	0	0
$849$	10.6601	0.365854
$850$	0	0
$851$	13.6301	0.467233
$852$	0	0
$853$	50.6115	1.73290	0.866452	−	0.499261i	$-0.166395\pi$
$853$	50.6115	1.73290	0.866452	+	0.499261i	$0.166395\pi$
$854$	0	0
$855$	29.5338	1.01004
$856$	0	0
$857$	25.5316	0.872143	0.436071	−	0.899912i	$-0.356369\pi$
$857$	25.5316	0.872143	0.436071	+	0.899912i	$0.356369\pi$
$858$	0	0
$859$	−20.3462	−0.694204	−0.347102	−	0.937827i	$-0.612834\pi$
$859$	−20.3462	−0.694204	−0.347102	+	0.937827i	$0.612834\pi$
$860$	0	0
$861$	−10.9304	−0.372507
$862$	0	0
$863$	35.4571	1.20697	0.603486	−	0.797374i	$-0.293778\pi$
$863$	35.4571	1.20697	0.603486	+	0.797374i	$0.293778\pi$
$864$	0	0
$865$	68.6083	2.33275
$866$	0	0
$867$	7.29420	0.247724
$868$	0	0
$869$	−31.3544	−1.06362
$870$	0	0
$871$	85.6517	2.90220
$872$	0	0
$873$	47.5614	1.60971
$874$	0	0
$875$	92.2222	3.11768
$876$	0	0
$877$	38.9912	1.31664	0.658320	−	0.752738i	$-0.271267\pi$
$877$	38.9912	1.31664	0.658320	+	0.752738i	$0.271267\pi$
$878$	0	0
$879$	7.38133	0.248966
$880$	0	0
$881$	21.0514	0.709240	0.354620	−	0.935011i	$-0.384610\pi$
$881$	21.0514	0.709240	0.354620	+	0.935011i	$0.384610\pi$
$882$	0	0
$883$	7.34069	0.247034	0.123517	−	0.992342i	$-0.460583\pi$
$883$	7.34069	0.247034	0.123517	+	0.992342i	$0.460583\pi$
$884$	0	0
$885$	−6.94934	−0.233599
$886$	0	0
$887$	−13.7002	−0.460007	−0.230004	−	0.973190i	$-0.573874\pi$
$887$	−13.7002	−0.460007	−0.230004	+	0.973190i	$0.573874\pi$
$888$	0	0
$889$	30.8213	1.03371
$890$	0	0
$891$	−27.6707	−0.927003
$892$	0	0
$893$	17.0794	0.571539
$894$	0	0
$895$	32.7216	1.09376
$896$	0	0
$897$	−10.0573	−0.335803
$898$	0	0
$899$	54.9014	1.83106
$900$	0	0
$901$	−51.8910	−1.72874
$902$	0	0
$903$	−5.34714	−0.177942
$904$	0	0
$905$	−40.8861	−1.35910
$906$	0	0
$907$	12.2027	0.405183	0.202591	−	0.979263i	$-0.435064\pi$
$907$	12.2027	0.405183	0.202591	+	0.979263i	$0.435064\pi$
$908$	0	0
$909$	−8.72135	−0.289269
$910$	0	0
$911$	−25.3238	−0.839015	−0.419507	−	0.907752i	$-0.637797\pi$
$911$	−25.3238	−0.839015	−0.419507	+	0.907752i	$0.637797\pi$
$912$	0	0
$913$	42.9675	1.42202
$914$	0	0
$915$	−7.52349	−0.248719
$916$	0	0
$917$	17.0026	0.561475
$918$	0	0
$919$	−27.2280	−0.898169	−0.449084	−	0.893489i	$-0.648250\pi$
$919$	−27.2280	−0.898169	−0.449084	+	0.893489i	$0.648250\pi$
$920$	0	0
$921$	1.20059	0.0395607
$922$	0	0
$923$	70.6518	2.32553
$924$	0	0
$925$	60.1713	1.97842
$926$	0	0
$927$	36.3866	1.19509
$928$	0	0
$929$	15.2891	0.501618	0.250809	−	0.968037i	$-0.419303\pi$
$929$	15.2891	0.501618	0.250809	+	0.968037i	$0.419303\pi$
$930$	0	0
$931$	5.79598	0.189956
$932$	0	0
$933$	10.5311	0.344773
$934$	0	0
$935$	96.4402	3.15393
$936$	0	0
$937$	33.8189	1.10481	0.552407	−	0.833574i	$-0.313709\pi$
$937$	33.8189	1.10481	0.552407	+	0.833574i	$0.313709\pi$
$938$	0	0
$939$	6.30787	0.205849
$940$	0	0
$941$	8.41013	0.274163	0.137081	−	0.990560i	$-0.456228\pi$
$941$	8.41013	0.274163	0.137081	+	0.990560i	$0.456228\pi$
$942$	0	0
$943$	19.3250	0.629308
$944$	0	0
$945$	−37.5597	−1.22182
$946$	0	0
$947$	−38.6320	−1.25537	−0.627686	−	0.778466i	$-0.715998\pi$
$947$	−38.6320	−1.25537	−0.627686	+	0.778466i	$0.715998\pi$
$948$	0	0
$949$	−11.5165	−0.373841
$950$	0	0
$951$	14.1366	0.458411
$952$	0	0
$953$	12.4958	0.404780	0.202390	−	0.979305i	$-0.435129\pi$
$953$	12.4958	0.404780	0.202390	+	0.979305i	$0.435129\pi$
$954$	0	0
$955$	−49.2573	−1.59393
$956$	0	0
$957$	16.5840	0.536084
$958$	0	0
$959$	−0.628119	−0.0202830
$960$	0	0
$961$	20.0347	0.646280
$962$	0	0
$963$	−0.811287	−0.0261434
$964$	0	0
$965$	59.1756	1.90493
$966$	0	0
$967$	30.0156	0.965236	0.482618	−	0.875831i	$-0.339686\pi$
$967$	30.0156	0.965236	0.482618	+	0.875831i	$0.339686\pi$
$968$	0	0
$969$	7.51665	0.241469
$970$	0	0
$971$	56.4260	1.81080	0.905398	−	0.424563i	$-0.139572\pi$
$971$	56.4260	1.81080	0.905398	+	0.424563i	$0.139572\pi$
$972$	0	0
$973$	11.0010	0.352675
$974$	0	0
$975$	−44.3989	−1.42190
$976$	0	0
$977$	−53.8222	−1.72192	−0.860962	−	0.508669i	$-0.830138\pi$
$977$	−53.8222	−1.72192	−0.860962	+	0.508669i	$0.830138\pi$
$978$	0	0
$979$	−14.0375	−0.448641
$980$	0	0
$981$	−6.69677	−0.213812
$982$	0	0
$983$	1.40310	0.0447519	0.0223759	−	0.999750i	$-0.492877\pi$
$983$	1.40310	0.0447519	0.0223759	+	0.999750i	$0.492877\pi$
$984$	0	0
$985$	−68.4873	−2.18219
$986$	0	0
$987$	−10.3505	−0.329461
$988$	0	0
$989$	9.45375	0.300612
$990$	0	0
$991$	−30.9529	−0.983251	−0.491625	−	0.870807i	$-0.663597\pi$
$991$	−30.9529	−0.983251	−0.491625	+	0.870807i	$0.663597\pi$
$992$	0	0
$993$	−13.3485	−0.423603
$994$	0	0
$995$	19.2334	0.609741
$996$	0	0
$997$	22.3377	0.707443	0.353721	−	0.935351i	$-0.384916\pi$
$997$	22.3377	0.707443	0.353721	+	0.935351i	$0.384916\pi$
$998$	0	0
$999$	−14.5433	−0.460130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2188.2.a.e.1.11	✓	23
4.3	odd	2		8752.2.a.u.1.13		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2188.2.a.e.1.11	✓	23	1.1	even	1	trivial
8752.2.a.u.1.13		23	4.3	odd	2

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:	\( N \)	\(=\)	\( 2188 = 2^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2188.a (trivial)

\(f(q)\)	\(=\)	\(q+0.518689 q^{3} -4.15918 q^{5} -3.03794 q^{7} -2.73096 q^{9} +O(q^{10})\)	\(q+0.518689 q^{3} -4.15918 q^{5} -3.03794 q^{7} -2.73096 q^{9} -4.16036 q^{11} -6.95992 q^{13} -2.15732 q^{15} +5.57340 q^{17} +2.60014 q^{19} -1.57575 q^{21} +2.78593 q^{23} +12.2987 q^{25} -2.97259 q^{27} -7.68512 q^{29} -7.14386 q^{31} -2.15793 q^{33} +12.6353 q^{35} +4.89247 q^{37} -3.61003 q^{39} +6.93664 q^{41} +3.39339 q^{43} +11.3586 q^{45} +6.56863 q^{47} +2.22910 q^{49} +2.89086 q^{51} -9.31047 q^{53} +17.3037 q^{55} +1.34867 q^{57} +3.22128 q^{59} +3.48742 q^{61} +8.29651 q^{63} +28.9475 q^{65} -12.3064 q^{67} +1.44503 q^{69} -10.1512 q^{71} +1.65469 q^{73} +6.37923 q^{75} +12.6389 q^{77} +7.53645 q^{79} +6.65104 q^{81} -10.3278 q^{83} -23.1807 q^{85} -3.98619 q^{87} +3.37411 q^{89} +21.1438 q^{91} -3.70544 q^{93} -10.8144 q^{95} -17.4156 q^{97} +11.3618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 12 q^{3} - 3 q^{5} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 23 * q + 12 * q^3 - 3 * q^5 + 5 * q^7 + 35 * q^9	\( 23 q + 12 q^{3} - 3 q^{5} + 5 q^{7} + 35 q^{9} + 14 q^{11} + 9 q^{15} + 18 q^{17} + 10 q^{19} + 38 q^{23} + 18 q^{25} + 45 q^{27} + 17 q^{31} + 20 q^{33} + 19 q^{35} + 4 q^{37} + 19 q^{39} + 13 q^{41} + 22 q^{43} + 22 q^{45} + 49 q^{47} + 30 q^{49} + 31 q^{51} + 6 q^{53} + 21 q^{55} + 16 q^{57} + 52 q^{59} + 6 q^{61} + 57 q^{63} + 34 q^{65} + 17 q^{67} + 10 q^{69} + 31 q^{71} + 8 q^{73} + 51 q^{75} - q^{77} + 14 q^{79} + 51 q^{81} + 43 q^{83} - 41 q^{85} + 31 q^{87} + 38 q^{89} + 31 q^{91} - 8 q^{93} + 42 q^{95} + 32 q^{97} + 57 q^{99}+O(q^{100}) \) 23 * q + 12 * q^3 - 3 * q^5 + 5 * q^7 + 35 * q^9 + 14 * q^11 + 9 * q^15 + 18 * q^17 + 10 * q^19 + 38 * q^23 + 18 * q^25 + 45 * q^27 + 17 * q^31 + 20 * q^33 + 19 * q^35 + 4 * q^37 + 19 * q^39 + 13 * q^41 + 22 * q^43 + 22 * q^45 + 49 * q^47 + 30 * q^49 + 31 * q^51 + 6 * q^53 + 21 * q^55 + 16 * q^57 + 52 * q^59 + 6 * q^61 + 57 * q^63 + 34 * q^65 + 17 * q^67 + 10 * q^69 + 31 * q^71 + 8 * q^73 + 51 * q^75 - q^77 + 14 * q^79 + 51 * q^81 + 43 * q^83 - 41 * q^85 + 31 * q^87 + 38 * q^89 + 31 * q^91 - 8 * q^93 + 42 * q^95 + 32 * q^97 + 57 * q^99

Introduction

L-functions

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