LMFDB - Embedded newform 1872.4.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.1231 q^{5} -15.6155 q^{7} -55.7235 q^{11} +13.0000 q^{13} -23.4773 q^{17} +25.6458 q^{19} -189.477 q^{23} +47.2159 q^{25} +236.739 q^{29} -47.0625 q^{31} -204.924 q^{35} -154.648 q^{37} +34.6610 q^{41} +398.833 q^{43} +582.773 q^{47} -99.1553 q^{49} +361.015 q^{53} -731.265 q^{55} +396.985 q^{59} -211.322 q^{61} +170.600 q^{65} -85.8277 q^{67} -651.602 q^{71} +927.329 q^{73} +870.152 q^{77} +1117.98 q^{79} +391.629 q^{83} -308.095 q^{85} +745.960 q^{89} -203.002 q^{91} +336.553 q^{95} -173.231 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{5} + 10 q^{7} + 4 q^{11} + 26 q^{13} + 52 q^{17} + 142 q^{19} - 280 q^{23} - 54 q^{25} + 424 q^{29} + 178 q^{31} - 80 q^{35} + 136 q^{37} + 226 q^{41} + 72 q^{43} + 176 q^{47} + 214 q^{49} + 788 q^{53}+ \cdots - 264 q^{97}+O(q^{100}) $ 2 * q + 18 * q^5 + 10 * q^7 + 4 * q^11 + 26 * q^13 + 52 * q^17 + 142 * q^19 - 280 * q^23 - 54 * q^25 + 424 * q^29 + 178 * q^31 - 80 * q^35 + 136 * q^37 + 226 * q^41 + 72 * q^43 + 176 * q^47 + 214 * q^49 + 788 * q^53 - 440 * q^55 + 728 * q^59 - 736 * q^61 + 234 * q^65 - 1054 * q^67 - 660 * q^71 + 24 * q^73 + 2400 * q^77 - 40 * q^79 + 1344 * q^83 + 60 * q^85 - 314 * q^89 + 130 * q^91 + 904 * q^95 - 264 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	13.1231	1.17377	0.586883	−	0.809672i	$-0.300355\pi$
$5$	13.1231	1.17377	0.586883	+	0.809672i	$0.300355\pi$
$6$	0	0
$7$	−15.6155	−0.843159	−0.421580	−	0.906791i	$-0.638524\pi$
$7$	−15.6155	−0.843159	−0.421580	+	0.906791i	$0.638524\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−55.7235	−1.52739	−0.763694	−	0.645579i	$-0.776616\pi$
$11$	−55.7235	−1.52739	−0.763694	+	0.645579i	$0.776616\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−23.4773	−0.334946	−0.167473	−	0.985877i	$-0.553561\pi$
$17$	−23.4773	−0.334946	−0.167473	+	0.985877i	$0.553561\pi$
$18$	0	0
$19$	25.6458	0.309661	0.154830	−	0.987941i	$-0.450517\pi$
$19$	25.6458	0.309661	0.154830	+	0.987941i	$0.450517\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−189.477	−1.71777	−0.858886	−	0.512167i	$-0.828843\pi$
$23$	−189.477	−1.71777	−0.858886	+	0.512167i	$0.828843\pi$
$24$	0	0
$25$	47.2159	0.377727
$26$	0	0
$27$	0	0
$28$	0	0
$29$	236.739	1.51591	0.757953	−	0.652309i	$-0.226200\pi$
$29$	236.739	1.51591	0.757953	+	0.652309i	$0.226200\pi$
$30$	0	0
$31$	−47.0625	−0.272667	−0.136333	−	0.990663i	$-0.543532\pi$
$31$	−47.0625	−0.272667	−0.136333	+	0.990663i	$0.543532\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−204.924	−0.989672
$36$	0	0
$37$	−154.648	−0.687133	−0.343567	−	0.939128i	$-0.611635\pi$
$37$	−154.648	−0.687133	−0.343567	+	0.939128i	$0.611635\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	34.6610	0.132028	0.0660139	−	0.997819i	$-0.478972\pi$
$41$	34.6610	0.132028	0.0660139	+	0.997819i	$0.478972\pi$
$42$	0	0
$43$	398.833	1.41445	0.707227	−	0.706987i	$-0.249946\pi$
$43$	398.833	1.41445	0.707227	+	0.706987i	$0.249946\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	582.773	1.80864	0.904321	−	0.426854i	$-0.140378\pi$
$47$	582.773	1.80864	0.904321	+	0.426854i	$0.140378\pi$
$48$	0	0
$49$	−99.1553	−0.289082
$50$	0	0
$51$	0	0
$52$	0	0
$53$	361.015	0.935646	0.467823	−	0.883822i	$-0.345038\pi$
$53$	361.015	0.935646	0.467823	+	0.883822i	$0.345038\pi$
$54$	0	0
$55$	−731.265	−1.79280
$56$	0	0
$57$	0	0
$58$	0	0
$59$	396.985	0.875983	0.437992	−	0.898979i	$-0.355690\pi$
$59$	396.985	0.875983	0.437992	+	0.898979i	$0.355690\pi$
$60$	0	0
$61$	−211.322	−0.443558	−0.221779	−	0.975097i	$-0.571186\pi$
$61$	−211.322	−0.443558	−0.221779	+	0.975097i	$0.571186\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	170.600	0.325544
$66$	0	0
$67$	−85.8277	−0.156500	−0.0782502	−	0.996934i	$-0.524933\pi$
$67$	−85.8277	−0.156500	−0.0782502	+	0.996934i	$0.524933\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−651.602	−1.08917	−0.544584	−	0.838706i	$-0.683313\pi$
$71$	−651.602	−1.08917	−0.544584	+	0.838706i	$0.683313\pi$
$72$	0	0
$73$	927.329	1.48679	0.743395	−	0.668852i	$-0.233214\pi$
$73$	927.329	1.48679	0.743395	+	0.668852i	$0.233214\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	870.152	1.28783
$78$	0	0
$79$	1117.98	1.59218	0.796090	−	0.605178i	$-0.206898\pi$
$79$	1117.98	1.59218	0.796090	+	0.605178i	$0.206898\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	391.629	0.517914	0.258957	−	0.965889i	$-0.416621\pi$
$83$	391.629	0.517914	0.258957	+	0.965889i	$0.416621\pi$
$84$	0	0
$85$	−308.095	−0.393148
$86$	0	0
$87$	0	0
$88$	0	0
$89$	745.960	0.888445	0.444223	−	0.895916i	$-0.353480\pi$
$89$	745.960	0.888445	0.444223	+	0.895916i	$0.353480\pi$
$90$	0	0
$91$	−203.002	−0.233850
$92$	0	0
$93$	0	0
$94$	0	0
$95$	336.553	0.363470
$96$	0	0
$97$	−173.231	−0.181329	−0.0906647	−	0.995881i	$-0.528899\pi$
$97$	−173.231	−0.181329	−0.0906647	+	0.995881i	$0.528899\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1263.45	1.24473	0.622367	−	0.782726i	$-0.286171\pi$
$101$	1263.45	1.24473	0.622367	+	0.782726i	$0.286171\pi$
$102$	0	0
$103$	−874.401	−0.836479	−0.418240	−	0.908337i	$-0.637353\pi$
$103$	−874.401	−0.836479	−0.418240	+	0.908337i	$0.637353\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1591.89	1.43826	0.719128	−	0.694877i	$-0.244541\pi$
$107$	1591.89	1.43826	0.719128	+	0.694877i	$0.244541\pi$
$108$	0	0
$109$	371.788	0.326705	0.163352	−	0.986568i	$-0.447769\pi$
$109$	371.788	0.326705	0.163352	+	0.986568i	$0.447769\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1173.98	−0.977335	−0.488667	−	0.872470i	$-0.662517\pi$
$113$	−1173.98	−0.977335	−0.488667	+	0.872470i	$0.662517\pi$
$114$	0	0
$115$	−2486.53	−2.01626
$116$	0	0
$117$	0	0
$118$	0	0
$119$	366.610	0.282412
$120$	0	0
$121$	1774.11	1.33291
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1020.77	−0.730403
$126$	0	0
$127$	−739.360	−0.516595	−0.258298	−	0.966065i	$-0.583161\pi$
$127$	−739.360	−0.516595	−0.258298	+	0.966065i	$0.583161\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2131.58	1.42166	0.710829	−	0.703365i	$-0.248320\pi$
$131$	2131.58	1.42166	0.710829	+	0.703365i	$0.248320\pi$
$132$	0	0
$133$	−400.473	−0.261094
$134$	0	0
$135$	0	0
$136$	0	0
$137$	256.934	0.160229	0.0801144	−	0.996786i	$-0.474471\pi$
$137$	256.934	0.160229	0.0801144	+	0.996786i	$0.474471\pi$
$138$	0	0
$139$	−437.443	−0.266931	−0.133466	−	0.991053i	$-0.542611\pi$
$139$	−437.443	−0.266931	−0.133466	+	0.991053i	$0.542611\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−724.405	−0.423621
$144$	0	0
$145$	3106.75	1.77932
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3406.65	1.87304	0.936522	−	0.350609i	$-0.114025\pi$
$149$	3406.65	1.87304	0.936522	+	0.350609i	$0.114025\pi$
$150$	0	0
$151$	−3275.77	−1.76542	−0.882710	−	0.469918i	$-0.844283\pi$
$151$	−3275.77	−1.76542	−0.882710	+	0.469918i	$0.844283\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−617.606	−0.320047
$156$	0	0
$157$	1074.34	0.546128	0.273064	−	0.961996i	$-0.411963\pi$
$157$	1074.34	0.546128	0.273064	+	0.961996i	$0.411963\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2958.79	1.44835
$162$	0	0
$163$	−2031.49	−0.976186	−0.488093	−	0.872792i	$-0.662307\pi$
$163$	−2031.49	−0.976186	−0.488093	+	0.872792i	$0.662307\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	299.697	0.138870	0.0694349	−	0.997586i	$-0.477880\pi$
$167$	299.697	0.138870	0.0694349	+	0.997586i	$0.477880\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2371.16	−1.04206	−0.521028	−	0.853540i	$-0.674451\pi$
$173$	−2371.16	−1.04206	−0.521028	+	0.853540i	$0.674451\pi$
$174$	0	0
$175$	−737.301	−0.318484
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−249.613	−0.104229	−0.0521145	−	0.998641i	$-0.516596\pi$
$179$	−249.613	−0.104229	−0.0521145	+	0.998641i	$0.516596\pi$
$180$	0	0
$181$	−3151.45	−1.29417	−0.647087	−	0.762416i	$-0.724013\pi$
$181$	−3151.45	−1.29417	−0.647087	+	0.762416i	$0.724013\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2029.46	−0.806534
$186$	0	0
$187$	1308.24	0.511592
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4739.45	−1.79547	−0.897735	−	0.440535i	$-0.854789\pi$
$191$	−4739.45	−1.79547	−0.897735	+	0.440535i	$0.854789\pi$
$192$	0	0
$193$	636.818	0.237509	0.118754	−	0.992924i	$-0.462110\pi$
$193$	636.818	0.237509	0.118754	+	0.992924i	$0.462110\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3001.13	1.08539	0.542695	−	0.839930i	$-0.317404\pi$
$197$	3001.13	1.08539	0.542695	+	0.839930i	$0.317404\pi$
$198$	0	0
$199$	648.958	0.231173	0.115587	−	0.993297i	$-0.463125\pi$
$199$	648.958	0.231173	0.115587	+	0.993297i	$0.463125\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3696.80	−1.27815
$204$	0	0
$205$	454.860	0.154970
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1429.08	−0.472972
$210$	0	0
$211$	240.114	0.0783418	0.0391709	−	0.999233i	$-0.487528\pi$
$211$	240.114	0.0783418	0.0391709	+	0.999233i	$0.487528\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5233.93	1.66024
$216$	0	0
$217$	734.906	0.229902
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−305.204	−0.0928972
$222$	0	0
$223$	1104.37	0.331633	0.165816	−	0.986157i	$-0.446974\pi$
$223$	1104.37	0.331633	0.165816	+	0.986157i	$0.446974\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	497.034	0.145327	0.0726636	−	0.997357i	$-0.476850\pi$
$227$	497.034	0.145327	0.0726636	+	0.997357i	$0.476850\pi$
$228$	0	0
$229$	2879.05	0.830800	0.415400	−	0.909639i	$-0.363642\pi$
$229$	2879.05	0.830800	0.415400	+	0.909639i	$0.363642\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3846.91	1.08163	0.540814	−	0.841142i	$-0.318116\pi$
$233$	3846.91	1.08163	0.540814	+	0.841142i	$0.318116\pi$
$234$	0	0
$235$	7647.79	2.12292
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2903.25	0.785755	0.392877	−	0.919591i	$-0.371480\pi$
$239$	2903.25	0.785755	0.392877	+	0.919591i	$0.371480\pi$
$240$	0	0
$241$	−4801.06	−1.28325	−0.641626	−	0.767018i	$-0.721740\pi$
$241$	−4801.06	−1.28325	−0.641626	+	0.767018i	$0.721740\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1301.23	−0.339315
$246$	0	0
$247$	333.396	0.0858845
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4768.20	1.19907	0.599534	−	0.800350i	$-0.295353\pi$
$251$	4768.20	1.19907	0.599534	+	0.800350i	$0.295353\pi$
$252$	0	0
$253$	10558.3	2.62370
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6002.18	−1.45683	−0.728415	−	0.685136i	$-0.759743\pi$
$257$	−6002.18	−1.45683	−0.728415	+	0.685136i	$0.759743\pi$
$258$	0	0
$259$	2414.91	0.579363
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3268.33	−0.766289	−0.383145	−	0.923688i	$-0.625159\pi$
$263$	−3268.33	−0.766289	−0.383145	+	0.923688i	$0.625159\pi$
$264$	0	0
$265$	4737.64	1.09823
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8690.14	1.96969	0.984846	−	0.173431i	$-0.0554854\pi$
$269$	8690.14	1.96969	0.984846	+	0.173431i	$0.0554854\pi$
$270$	0	0
$271$	3239.17	0.726073	0.363036	−	0.931775i	$-0.381740\pi$
$271$	3239.17	0.726073	0.363036	+	0.931775i	$0.381740\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2631.03	−0.576936
$276$	0	0
$277$	−220.091	−0.0477400	−0.0238700	−	0.999715i	$-0.507599\pi$
$277$	−220.091	−0.0477400	−0.0238700	+	0.999715i	$0.507599\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−587.233	−0.124667	−0.0623334	−	0.998055i	$-0.519854\pi$
$281$	−587.233	−0.124667	−0.0623334	+	0.998055i	$0.519854\pi$
$282$	0	0
$283$	8274.09	1.73796	0.868981	−	0.494845i	$-0.164775\pi$
$283$	8274.09	1.73796	0.868981	+	0.494845i	$0.164775\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−541.250	−0.111320
$288$	0	0
$289$	−4361.82	−0.887812
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5382.07	1.07312	0.536559	−	0.843863i	$-0.319724\pi$
$293$	5382.07	1.07312	0.536559	+	0.843863i	$0.319724\pi$
$294$	0	0
$295$	5209.67	1.02820
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2463.20	−0.476424
$300$	0	0
$301$	−6227.99	−1.19261
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2773.20	−0.520633
$306$	0	0
$307$	3095.92	0.575549	0.287774	−	0.957698i	$-0.407085\pi$
$307$	3095.92	0.575549	0.287774	+	0.957698i	$0.407085\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3474.58	0.633521	0.316761	−	0.948506i	$-0.397405\pi$
$311$	3474.58	0.633521	0.316761	+	0.948506i	$0.397405\pi$
$312$	0	0
$313$	−3523.92	−0.636370	−0.318185	−	0.948029i	$-0.603073\pi$
$313$	−3523.92	−0.636370	−0.318185	+	0.948029i	$0.603073\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	323.396	0.0572988	0.0286494	−	0.999590i	$-0.490879\pi$
$317$	323.396	0.0572988	0.0286494	+	0.999590i	$0.490879\pi$
$318$	0	0
$319$	−13191.9	−2.31537
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−602.094	−0.103720
$324$	0	0
$325$	613.807	0.104763
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9100.30	−1.52497
$330$	0	0
$331$	1057.74	0.175646	0.0878228	−	0.996136i	$-0.472009\pi$
$331$	1057.74	0.175646	0.0878228	+	0.996136i	$0.472009\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1126.33	−0.183695
$336$	0	0
$337$	11538.4	1.86510	0.932550	−	0.361041i	$-0.117578\pi$
$337$	11538.4	1.86510	0.932550	+	0.361041i	$0.117578\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2622.49	0.416468
$342$	0	0
$343$	6904.49	1.08690
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5211.75	0.806286	0.403143	−	0.915137i	$-0.367918\pi$
$347$	5211.75	0.806286	0.403143	+	0.915137i	$0.367918\pi$
$348$	0	0
$349$	−2369.81	−0.363476	−0.181738	−	0.983347i	$-0.558172\pi$
$349$	−2369.81	−0.363476	−0.181738	+	0.983347i	$0.558172\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8997.40	1.35661	0.678305	−	0.734780i	$-0.262715\pi$
$353$	8997.40	1.35661	0.678305	+	0.734780i	$0.262715\pi$
$354$	0	0
$355$	−8551.05	−1.27843
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10863.4	1.59707	0.798537	−	0.601946i	$-0.205608\pi$
$359$	10863.4	1.59707	0.798537	+	0.601946i	$0.205608\pi$
$360$	0	0
$361$	−6201.29	−0.904110
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12169.4	1.74514
$366$	0	0
$367$	13552.2	1.92757	0.963785	−	0.266682i	$-0.0859273\pi$
$367$	13552.2	1.92757	0.963785	+	0.266682i	$0.0859273\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5637.44	−0.788899
$372$	0	0
$373$	2255.64	0.313118	0.156559	−	0.987669i	$-0.449960\pi$
$373$	2255.64	0.313118	0.156559	+	0.987669i	$0.449960\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3077.60	0.420437
$378$	0	0
$379$	7698.92	1.04345	0.521724	−	0.853114i	$-0.325289\pi$
$379$	7698.92	1.04345	0.521724	+	0.853114i	$0.325289\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	211.235	0.0281818	0.0140909	−	0.999901i	$-0.495515\pi$
$383$	211.235	0.0281818	0.0140909	+	0.999901i	$0.495515\pi$
$384$	0	0
$385$	11419.1	1.51161
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3515.25	−0.458175	−0.229088	−	0.973406i	$-0.573574\pi$
$389$	−3515.25	−0.458175	−0.229088	+	0.973406i	$0.573574\pi$
$390$	0	0
$391$	4448.41	0.575360
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14671.3	1.86885
$396$	0	0
$397$	−10968.8	−1.38667	−0.693337	−	0.720613i	$-0.743860\pi$
$397$	−10968.8	−1.38667	−0.693337	+	0.720613i	$0.743860\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6508.24	−0.810488	−0.405244	−	0.914208i	$-0.632813\pi$
$401$	−6508.24	−0.810488	−0.405244	+	0.914208i	$0.632813\pi$
$402$	0	0
$403$	−611.812	−0.0756242
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8617.51	1.04952
$408$	0	0
$409$	9363.46	1.13201	0.566006	−	0.824401i	$-0.308488\pi$
$409$	9363.46	1.13201	0.566006	+	0.824401i	$0.308488\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6199.13	−0.738594
$414$	0	0
$415$	5139.39	0.607910
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12057.8	−1.40588	−0.702940	−	0.711249i	$-0.748130\pi$
$419$	−12057.8	−1.40588	−0.702940	+	0.711249i	$0.748130\pi$
$420$	0	0
$421$	−15937.1	−1.84496	−0.922479	−	0.386047i	$-0.873840\pi$
$421$	−15937.1	−1.84496	−0.922479	+	0.386047i	$0.873840\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1108.50	−0.126518
$426$	0	0
$427$	3299.90	0.373990
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6792.37	0.759111	0.379556	−	0.925169i	$-0.376077\pi$
$431$	6792.37	0.759111	0.379556	+	0.925169i	$0.376077\pi$
$432$	0	0
$433$	−3338.00	−0.370471	−0.185235	−	0.982694i	$-0.559305\pi$
$433$	−3338.00	−0.370471	−0.185235	+	0.982694i	$0.559305\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4859.30	−0.531927
$438$	0	0
$439$	−13313.1	−1.44738	−0.723688	−	0.690127i	$-0.757555\pi$
$439$	−13313.1	−1.44738	−0.723688	+	0.690127i	$0.757555\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5175.32	0.555050	0.277525	−	0.960718i	$-0.410486\pi$
$443$	5175.32	0.555050	0.277525	+	0.960718i	$0.410486\pi$
$444$	0	0
$445$	9789.31	1.04283
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4090.57	0.429947	0.214973	−	0.976620i	$-0.431034\pi$
$449$	4090.57	0.429947	0.214973	+	0.976620i	$0.431034\pi$
$450$	0	0
$451$	−1931.43	−0.201658
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2664.01	−0.274486
$456$	0	0
$457$	−4300.78	−0.440224	−0.220112	−	0.975475i	$-0.570642\pi$
$457$	−4300.78	−0.440224	−0.220112	+	0.975475i	$0.570642\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2578.07	0.260461	0.130231	−	0.991484i	$-0.458428\pi$
$461$	2578.07	0.260461	0.130231	+	0.991484i	$0.458428\pi$
$462$	0	0
$463$	4529.58	0.454660	0.227330	−	0.973818i	$-0.427000\pi$
$463$	4529.58	0.454660	0.227330	+	0.973818i	$0.427000\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4761.89	0.471850	0.235925	−	0.971771i	$-0.424188\pi$
$467$	4761.89	0.471850	0.235925	+	0.971771i	$0.424188\pi$
$468$	0	0
$469$	1340.24	0.131955
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22224.4	−2.16042
$474$	0	0
$475$	1210.89	0.116967
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11621.4	1.10855	0.554275	−	0.832333i	$-0.312995\pi$
$479$	11621.4	1.10855	0.554275	+	0.832333i	$0.312995\pi$
$480$	0	0
$481$	−2010.42	−0.190576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2273.33	−0.212838
$486$	0	0
$487$	−7836.62	−0.729181	−0.364590	−	0.931168i	$-0.618791\pi$
$487$	−7836.62	−0.729181	−0.364590	+	0.931168i	$0.618791\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14163.9	−1.30185	−0.650925	−	0.759142i	$-0.725619\pi$
$491$	−14163.9	−1.30185	−0.650925	+	0.759142i	$0.725619\pi$
$492$	0	0
$493$	−5557.98	−0.507746
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10175.1	0.918342
$498$	0	0
$499$	811.953	0.0728417	0.0364208	−	0.999337i	$-0.488404\pi$
$499$	811.953	0.0728417	0.0364208	+	0.999337i	$0.488404\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9718.41	0.861476	0.430738	−	0.902477i	$-0.358253\pi$
$503$	9718.41	0.861476	0.430738	+	0.902477i	$0.358253\pi$
$504$	0	0
$505$	16580.4	1.46103
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10546.5	0.918403	0.459201	−	0.888332i	$-0.348136\pi$
$509$	10546.5	0.918403	0.459201	+	0.888332i	$0.348136\pi$
$510$	0	0
$511$	−14480.7	−1.25360
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11474.9	−0.981831
$516$	0	0
$517$	−32474.1	−2.76250
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1157.10	−0.0973005	−0.0486503	−	0.998816i	$-0.515492\pi$
$521$	−1157.10	−0.0973005	−0.0486503	+	0.998816i	$0.515492\pi$
$522$	0	0
$523$	−19193.0	−1.60468	−0.802342	−	0.596865i	$-0.796413\pi$
$523$	−19193.0	−1.60468	−0.802342	+	0.596865i	$0.796413\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1104.90	0.0913285
$528$	0	0
$529$	23734.6	1.95074
$530$	0	0
$531$	0	0
$532$	0	0
$533$	450.593	0.0366179
$534$	0	0
$535$	20890.5	1.68818
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5525.28	0.441541
$540$	0	0
$541$	−7809.48	−0.620620	−0.310310	−	0.950635i	$-0.600433\pi$
$541$	−7809.48	−0.620620	−0.310310	+	0.950635i	$0.600433\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4879.01	0.383475
$546$	0	0
$547$	−4979.12	−0.389199	−0.194599	−	0.980883i	$-0.562341\pi$
$547$	−4979.12	−0.389199	−0.194599	+	0.980883i	$0.562341\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6071.36	0.469417
$552$	0	0
$553$	−17457.8	−1.34246
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24508.2	−1.86435	−0.932176	−	0.362005i	$-0.882092\pi$
$557$	−24508.2	−1.86435	−0.932176	+	0.362005i	$0.882092\pi$
$558$	0	0
$559$	5184.83	0.392299
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14957.3	1.11967	0.559836	−	0.828604i	$-0.310864\pi$
$563$	14957.3	1.11967	0.559836	+	0.828604i	$0.310864\pi$
$564$	0	0
$565$	−15406.3	−1.14716
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5756.83	−0.424146	−0.212073	−	0.977254i	$-0.568021\pi$
$569$	−5756.83	−0.424146	−0.212073	+	0.977254i	$0.568021\pi$
$570$	0	0
$571$	7292.96	0.534502	0.267251	−	0.963627i	$-0.413885\pi$
$571$	7292.96	0.534502	0.267251	+	0.963627i	$0.413885\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8946.34	−0.648849
$576$	0	0
$577$	20749.7	1.49709	0.748544	−	0.663085i	$-0.230753\pi$
$577$	20749.7	1.49709	0.748544	+	0.663085i	$0.230753\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6115.49	−0.436684
$582$	0	0
$583$	−20117.0	−1.42909
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2046.37	0.143889	0.0719444	−	0.997409i	$-0.477080\pi$
$587$	2046.37	0.143889	0.0719444	+	0.997409i	$0.477080\pi$
$588$	0	0
$589$	−1206.96	−0.0844343
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12664.1	−0.876988	−0.438494	−	0.898734i	$-0.644488\pi$
$593$	−12664.1	−0.876988	−0.438494	+	0.898734i	$0.644488\pi$
$594$	0	0
$595$	4811.06	0.331486
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5788.48	0.394843	0.197421	−	0.980319i	$-0.436743\pi$
$599$	5788.48	0.394843	0.197421	+	0.980319i	$0.436743\pi$
$600$	0	0
$601$	−19622.2	−1.33179	−0.665897	−	0.746044i	$-0.731951\pi$
$601$	−19622.2	−1.33179	−0.665897	+	0.746044i	$0.731951\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23281.8	1.56453
$606$	0	0
$607$	11386.5	0.761389	0.380695	−	0.924701i	$-0.375685\pi$
$607$	11386.5	0.761389	0.380695	+	0.924701i	$0.375685\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7576.04	0.501627
$612$	0	0
$613$	−16046.0	−1.05725	−0.528623	−	0.848857i	$-0.677291\pi$
$613$	−16046.0	−1.05725	−0.528623	+	0.848857i	$0.677291\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25943.8	1.69280	0.846402	−	0.532545i	$-0.178764\pi$
$617$	25943.8	1.69280	0.846402	+	0.532545i	$0.178764\pi$
$618$	0	0
$619$	−22044.1	−1.43139	−0.715693	−	0.698415i	$-0.753889\pi$
$619$	−22044.1	−1.43139	−0.715693	+	0.698415i	$0.753889\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11648.6	−0.749101
$624$	0	0
$625$	−19297.6	−1.23505
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3630.71	0.230152
$630$	0	0
$631$	6948.74	0.438392	0.219196	−	0.975681i	$-0.429657\pi$
$631$	6948.74	0.438392	0.219196	+	0.975681i	$0.429657\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9702.70	−0.606362
$636$	0	0
$637$	−1289.02	−0.0801770
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17916.9	1.10402	0.552009	−	0.833838i	$-0.313861\pi$
$641$	17916.9	1.10402	0.552009	+	0.833838i	$0.313861\pi$
$642$	0	0
$643$	−10248.9	−0.628581	−0.314290	−	0.949327i	$-0.601767\pi$
$643$	−10248.9	−0.628581	−0.314290	+	0.949327i	$0.601767\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13034.7	−0.792035	−0.396018	−	0.918243i	$-0.629608\pi$
$647$	−13034.7	−0.792035	−0.396018	+	0.918243i	$0.629608\pi$
$648$	0	0
$649$	−22121.4	−1.33797
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5517.54	−0.330656	−0.165328	−	0.986239i	$-0.552868\pi$
$653$	−5517.54	−0.330656	−0.165328	+	0.986239i	$0.552868\pi$
$654$	0	0
$655$	27973.0	1.66869
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31919.9	−1.88683	−0.943416	−	0.331610i	$-0.892408\pi$
$659$	−31919.9	−1.88683	−0.943416	+	0.331610i	$0.892408\pi$
$660$	0	0
$661$	−3517.33	−0.206972	−0.103486	−	0.994631i	$-0.533000\pi$
$661$	−3517.33	−0.206972	−0.103486	+	0.994631i	$0.533000\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5255.45	−0.306463
$666$	0	0
$667$	−44856.6	−2.60398
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11775.6	0.677484
$672$	0	0
$673$	−33399.0	−1.91298	−0.956490	−	0.291766i	$-0.905757\pi$
$673$	−33399.0	−1.91298	−0.956490	+	0.291766i	$0.905757\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15843.8	0.899450	0.449725	−	0.893167i	$-0.351522\pi$
$677$	15843.8	0.899450	0.449725	+	0.893167i	$0.351522\pi$
$678$	0	0
$679$	2705.09	0.152890
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10145.6	−0.568392	−0.284196	−	0.958766i	$-0.591727\pi$
$683$	−10145.6	−0.568392	−0.284196	+	0.958766i	$0.591727\pi$
$684$	0	0
$685$	3371.77	0.188071
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4693.20	0.259502
$690$	0	0
$691$	−18650.9	−1.02679	−0.513396	−	0.858152i	$-0.671613\pi$
$691$	−18650.9	−1.02679	−0.513396	+	0.858152i	$0.671613\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5740.61	−0.313315
$696$	0	0
$697$	−813.745	−0.0442221
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25182.9	−1.35684	−0.678420	−	0.734675i	$-0.737335\pi$
$701$	−25182.9	−1.35684	−0.678420	+	0.734675i	$0.737335\pi$
$702$	0	0
$703$	−3966.07	−0.212778
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19729.4	−1.04951
$708$	0	0
$709$	16697.6	0.884473	0.442236	−	0.896899i	$-0.354185\pi$
$709$	16697.6	0.884473	0.442236	+	0.896899i	$0.354185\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8917.27	0.468379
$714$	0	0
$715$	−9506.45	−0.497232
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22657.4	1.17521	0.587606	−	0.809147i	$-0.300070\pi$
$719$	22657.4	1.17521	0.587606	+	0.809147i	$0.300070\pi$
$720$	0	0
$721$	13654.2	0.705285
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11177.8	0.572599
$726$	0	0
$727$	32052.8	1.63518	0.817588	−	0.575803i	$-0.195310\pi$
$727$	32052.8	1.63518	0.817588	+	0.575803i	$0.195310\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9363.52	−0.473765
$732$	0	0
$733$	22728.1	1.14527	0.572634	−	0.819811i	$-0.305922\pi$
$733$	22728.1	1.14527	0.572634	+	0.819811i	$0.305922\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4782.62	0.239037
$738$	0	0
$739$	−3741.58	−0.186247	−0.0931234	−	0.995655i	$-0.529685\pi$
$739$	−3741.58	−0.186247	−0.0931234	+	0.995655i	$0.529685\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23932.4	−1.18169	−0.590844	−	0.806786i	$-0.701205\pi$
$743$	−23932.4	−1.18169	−0.590844	+	0.806786i	$0.701205\pi$
$744$	0	0
$745$	44705.8	2.19852
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24858.1	−1.21268
$750$	0	0
$751$	15777.6	0.766623	0.383311	−	0.923619i	$-0.374784\pi$
$751$	15777.6	0.766623	0.383311	+	0.923619i	$0.374784\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−42988.3	−2.07219
$756$	0	0
$757$	32474.4	1.55919	0.779593	−	0.626287i	$-0.215426\pi$
$757$	32474.4	1.55919	0.779593	+	0.626287i	$0.215426\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24471.4	−1.16569	−0.582844	−	0.812584i	$-0.698060\pi$
$761$	−24471.4	−1.16569	−0.582844	+	0.812584i	$0.698060\pi$
$762$	0	0
$763$	−5805.66	−0.275464
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5160.80	0.242954
$768$	0	0
$769$	5185.33	0.243157	0.121578	−	0.992582i	$-0.461204\pi$
$769$	5185.33	0.243157	0.121578	+	0.992582i	$0.461204\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6218.64	−0.289352	−0.144676	−	0.989479i	$-0.546214\pi$
$773$	−6218.64	−0.289352	−0.144676	+	0.989479i	$0.546214\pi$
$774$	0	0
$775$	−2222.10	−0.102994
$776$	0	0
$777$	0	0
$778$	0	0
$779$	888.910	0.0408838
$780$	0	0
$781$	36309.5	1.66358
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14098.7	0.641026
$786$	0	0
$787$	15933.8	0.721702	0.360851	−	0.932623i	$-0.382486\pi$
$787$	15933.8	0.721702	0.360851	+	0.932623i	$0.382486\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18332.3	0.824049
$792$	0	0
$793$	−2747.19	−0.123021
$794$	0	0
$795$	0	0
$796$	0	0
$797$	825.379	0.0366831	0.0183415	−	0.999832i	$-0.494161\pi$
$797$	825.379	0.0366831	0.0183415	+	0.999832i	$0.494161\pi$
$798$	0	0
$799$	−13681.9	−0.605796
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−51674.0	−2.27090
$804$	0	0
$805$	38828.5	1.70003
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23981.8	−1.04222	−0.521110	−	0.853490i	$-0.674482\pi$
$809$	−23981.8	−1.04222	−0.521110	+	0.853490i	$0.674482\pi$
$810$	0	0
$811$	1359.77	0.0588753	0.0294376	−	0.999567i	$-0.490628\pi$
$811$	1359.77	0.0588753	0.0294376	+	0.999567i	$0.490628\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−26659.4	−1.14581
$816$	0	0
$817$	10228.4	0.438001
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29276.9	1.24455	0.622273	−	0.782801i	$-0.286209\pi$
$821$	29276.9	1.24455	0.622273	+	0.782801i	$0.286209\pi$
$822$	0	0
$823$	3971.15	0.168196	0.0840982	−	0.996457i	$-0.473199\pi$
$823$	3971.15	0.168196	0.0840982	+	0.996457i	$0.473199\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18920.5	−0.795564	−0.397782	−	0.917480i	$-0.630220\pi$
$827$	−18920.5	−0.795564	−0.397782	+	0.917480i	$0.630220\pi$
$828$	0	0
$829$	29902.3	1.25277	0.626387	−	0.779512i	$-0.284533\pi$
$829$	29902.3	1.25277	0.626387	+	0.779512i	$0.284533\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2327.90	0.0968269
$834$	0	0
$835$	3932.95	0.163001
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41125.0	1.69224	0.846121	−	0.532991i	$-0.178932\pi$
$839$	41125.0	1.69224	0.846121	+	0.532991i	$0.178932\pi$
$840$	0	0
$841$	31656.2	1.29797
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2217.80	0.0902897
$846$	0	0
$847$	−27703.6	−1.12386
$848$	0	0
$849$	0	0
$850$	0	0
$851$	29302.2	1.18034
$852$	0	0
$853$	2252.79	0.0904266	0.0452133	−	0.998977i	$-0.485603\pi$
$853$	2252.79	0.0904266	0.0452133	+	0.998977i	$0.485603\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16608.6	−0.662005	−0.331002	−	0.943630i	$-0.607387\pi$
$857$	−16608.6	−0.662005	−0.331002	+	0.943630i	$0.607387\pi$
$858$	0	0
$859$	14112.0	0.560529	0.280265	−	0.959923i	$-0.409578\pi$
$859$	14112.0	0.560529	0.280265	+	0.959923i	$0.409578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40154.6	1.58387	0.791934	−	0.610607i	$-0.209075\pi$
$863$	40154.6	1.58387	0.791934	+	0.610607i	$0.209075\pi$
$864$	0	0
$865$	−31116.9	−1.22313
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−62297.6	−2.43188
$870$	0	0
$871$	−1115.76	−0.0434054
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15939.8	0.615846
$876$	0	0
$877$	−22495.6	−0.866161	−0.433080	−	0.901355i	$-0.642573\pi$
$877$	−22495.6	−0.866161	−0.433080	+	0.901355i	$0.642573\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42296.3	1.61748	0.808740	−	0.588166i	$-0.200150\pi$
$881$	42296.3	1.61748	0.808740	+	0.588166i	$0.200150\pi$
$882$	0	0
$883$	37440.9	1.42694	0.713469	−	0.700687i	$-0.247123\pi$
$883$	37440.9	1.42694	0.713469	+	0.700687i	$0.247123\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3372.82	0.127675	0.0638377	−	0.997960i	$-0.479666\pi$
$887$	3372.82	0.127675	0.0638377	+	0.997960i	$0.479666\pi$
$888$	0	0
$889$	11545.5	0.435572
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14945.7	0.560066
$894$	0	0
$895$	−3275.70	−0.122340
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11141.5	−0.413337
$900$	0	0
$901$	−8475.65	−0.313390
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−41356.8	−1.51906
$906$	0	0
$907$	−30130.3	−1.10304	−0.551522	−	0.834160i	$-0.685953\pi$
$907$	−30130.3	−1.10304	−0.551522	+	0.834160i	$0.685953\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26089.9	−0.948844	−0.474422	−	0.880297i	$-0.657343\pi$
$911$	−26089.9	−0.948844	−0.474422	+	0.880297i	$0.657343\pi$
$912$	0	0
$913$	−21822.9	−0.791055
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33285.8	−1.19868
$918$	0	0
$919$	5142.55	0.184589	0.0922944	−	0.995732i	$-0.470580\pi$
$919$	5142.55	0.184589	0.0922944	+	0.995732i	$0.470580\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8470.83	−0.302081
$924$	0	0
$925$	−7301.83	−0.259549
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23311.6	−0.823282	−0.411641	−	0.911346i	$-0.635044\pi$
$929$	−23311.6	−0.823282	−0.411641	+	0.911346i	$0.635044\pi$
$930$	0	0
$931$	−2542.92	−0.0895176
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17168.1	0.600489
$936$	0	0
$937$	−17214.9	−0.600199	−0.300099	−	0.953908i	$-0.597020\pi$
$937$	−17214.9	−0.600199	−0.300099	+	0.953908i	$0.597020\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27871.2	0.965541	0.482770	−	0.875747i	$-0.339631\pi$
$941$	27871.2	0.965541	0.482770	+	0.875747i	$0.339631\pi$
$942$	0	0
$943$	−6567.47	−0.226793
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14956.7	−0.513227	−0.256614	−	0.966514i	$-0.582607\pi$
$947$	−14956.7	−0.513227	−0.256614	+	0.966514i	$0.582607\pi$
$948$	0	0
$949$	12055.3	0.412361
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2056.14	0.0698898	0.0349449	−	0.999389i	$-0.488874\pi$
$953$	2056.14	0.0698898	0.0349449	+	0.999389i	$0.488874\pi$
$954$	0	0
$955$	−62196.4	−2.10746
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4012.16	−0.135098
$960$	0	0
$961$	−27576.1	−0.925653
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8357.03	0.278780
$966$	0	0
$967$	−25282.1	−0.840764	−0.420382	−	0.907347i	$-0.638104\pi$
$967$	−25282.1	−0.840764	−0.420382	+	0.907347i	$0.638104\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22919.6	0.757493	0.378747	−	0.925500i	$-0.376355\pi$
$971$	22919.6	0.757493	0.378747	+	0.925500i	$0.376355\pi$
$972$	0	0
$973$	6830.91	0.225066
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42964.8	−1.40693	−0.703463	−	0.710732i	$-0.748364\pi$
$977$	−42964.8	−1.40693	−0.703463	+	0.710732i	$0.748364\pi$
$978$	0	0
$979$	−41567.5	−1.35700
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53757.9	−1.74426	−0.872132	−	0.489271i	$-0.837263\pi$
$983$	−53757.9	−1.74426	−0.872132	+	0.489271i	$0.837263\pi$
$984$	0	0
$985$	39384.2	1.27399
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−75569.8	−2.42971
$990$	0	0
$991$	28021.7	0.898224	0.449112	−	0.893475i	$-0.351740\pi$
$991$	28021.7	0.898224	0.449112	+	0.893475i	$0.351740\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8516.35	0.271343
$996$	0	0
$997$	7057.01	0.224170	0.112085	−	0.993699i	$-0.464247\pi$
$997$	7057.01	0.224170	0.112085	+	0.993699i	$0.464247\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bi.1.2		2
3.2	odd	2		624.4.a.j.1.1		2
4.3	odd	2		936.4.a.j.1.2		2
12.11	even	2		312.4.a.d.1.1	✓	2
24.5	odd	2		2496.4.a.bj.1.2		2
24.11	even	2		2496.4.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.d.1.1	✓	2	12.11	even	2
624.4.a.j.1.1		2	3.2	odd	2
936.4.a.j.1.2		2	4.3	odd	2
1872.4.a.bi.1.2		2	1.1	even	1	trivial
2496.4.a.ba.1.2		2	24.11	even	2
2496.4.a.bj.1.2		2	24.5	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+13.1231 q^{5} -15.6155 q^{7} -55.7235 q^{11} +13.0000 q^{13} -23.4773 q^{17} +25.6458 q^{19} -189.477 q^{23} +47.2159 q^{25} +236.739 q^{29} -47.0625 q^{31} -204.924 q^{35} -154.648 q^{37} +34.6610 q^{41} +398.833 q^{43} +582.773 q^{47} -99.1553 q^{49} +361.015 q^{53} -731.265 q^{55} +396.985 q^{59} -211.322 q^{61} +170.600 q^{65} -85.8277 q^{67} -651.602 q^{71} +927.329 q^{73} +870.152 q^{77} +1117.98 q^{79} +391.629 q^{83} -308.095 q^{85} +745.960 q^{89} -203.002 q^{91} +336.553 q^{95} -173.231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{5} + 10 q^{7} + 4 q^{11} + 26 q^{13} + 52 q^{17} + 142 q^{19} - 280 q^{23} - 54 q^{25} + 424 q^{29} + 178 q^{31} - 80 q^{35} + 136 q^{37} + 226 q^{41} + 72 q^{43} + 176 q^{47} + 214 q^{49} + 788 q^{53}+ \cdots - 264 q^{97}+O(q^{100}) \) 2 * q + 18 * q^5 + 10 * q^7 + 4 * q^11 + 26 * q^13 + 52 * q^17 + 142 * q^19 - 280 * q^23 - 54 * q^25 + 424 * q^29 + 178 * q^31 - 80 * q^35 + 136 * q^37 + 226 * q^41 + 72 * q^43 + 176 * q^47 + 214 * q^49 + 788 * q^53 - 440 * q^55 + 728 * q^59 - 736 * q^61 + 234 * q^65 - 1054 * q^67 - 660 * q^71 + 24 * q^73 + 2400 * q^77 - 40 * q^79 + 1344 * q^83 + 60 * q^85 - 314 * q^89 + 130 * q^91 + 904 * q^95 - 264 * q^97

Introduction

L-functions

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