LMFDB - Embedded newform 312.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, 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("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.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:	$18.4085959218$
Analytic rank:	$1$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -4.87689 q^{5} -25.6155 q^{7} +9.00000 q^{9} +59.7235 q^{11} +13.0000 q^{13} -14.6307 q^{15} -75.4773 q^{17} -116.354 q^{19} -76.8466 q^{21} -90.5227 q^{23} -101.216 q^{25} +27.0000 q^{27} -187.261 q^{29} -225.062 q^{31} +179.170 q^{33} +124.924 q^{35} +290.648 q^{37} +39.0000 q^{39} -191.339 q^{41} +326.833 q^{43} -43.8920 q^{45} -406.773 q^{47} +313.155 q^{49} -226.432 q^{51} -426.985 q^{53} -291.265 q^{55} -349.062 q^{57} +331.015 q^{59} -524.678 q^{61} -230.540 q^{63} -63.3996 q^{65} +968.172 q^{67} -271.568 q^{69} -8.39776 q^{71} -903.329 q^{73} -303.648 q^{75} -1529.85 q^{77} +1157.98 q^{79} +81.0000 q^{81} +952.371 q^{83} +368.095 q^{85} -561.784 q^{87} +1059.96 q^{89} -333.002 q^{91} -675.187 q^{93} +567.447 q^{95} -90.7689 q^{97} +537.511 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 18 q^{5} - 10 q^{7} + 18 q^{9} + 4 q^{11} + 26 q^{13} - 54 q^{15} - 52 q^{17} - 142 q^{19} - 30 q^{21} - 280 q^{23} - 54 q^{25} + 54 q^{27} - 424 q^{29} - 178 q^{31} + 12 q^{33} - 80 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 18 * q^5 - 10 * q^7 + 18 * q^9 + 4 * q^11 + 26 * q^13 - 54 * q^15 - 52 * q^17 - 142 * q^19 - 30 * q^21 - 280 * q^23 - 54 * q^25 + 54 * q^27 - 424 * q^29 - 178 * q^31 + 12 * q^33 - 80 * q^35 + 136 * q^37 + 78 * q^39 - 226 * q^41 - 72 * q^43 - 162 * q^45 + 176 * q^47 + 214 * q^49 - 156 * q^51 - 788 * q^53 + 440 * q^55 - 426 * q^57 + 728 * q^59 - 736 * q^61 - 90 * q^63 - 234 * q^65 + 1054 * q^67 - 840 * q^69 - 660 * q^71 + 24 * q^73 - 162 * q^75 - 2400 * q^77 + 40 * q^79 + 162 * q^81 + 1344 * q^83 + 60 * q^85 - 1272 * q^87 + 314 * q^89 - 130 * q^91 - 534 * q^93 + 904 * q^95 - 264 * q^97 + 36 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−4.87689	−0.436203	−0.218101	−	0.975926i	$-0.569986\pi$
$5$	−4.87689	−0.436203	−0.218101	+	0.975926i	$0.569986\pi$
$6$	0	0
$7$	−25.6155	−1.38311	−0.691554	−	0.722325i	$-0.743074\pi$
$7$	−25.6155	−1.38311	−0.691554	+	0.722325i	$0.743074\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	59.7235	1.63703	0.818514	−	0.574487i	$-0.194798\pi$
$11$	59.7235	1.63703	0.818514	+	0.574487i	$0.194798\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−14.6307	−0.251842
$16$	0	0
$17$	−75.4773	−1.07682	−0.538410	−	0.842683i	$-0.680975\pi$
$17$	−75.4773	−1.07682	−0.538410	+	0.842683i	$0.680975\pi$
$18$	0	0
$19$	−116.354	−1.40492	−0.702460	−	0.711723i	$-0.747915\pi$
$19$	−116.354	−1.40492	−0.702460	+	0.711723i	$0.747915\pi$
$20$	0	0
$21$	−76.8466	−0.798538
$22$	0	0
$23$	−90.5227	−0.820665	−0.410332	−	0.911936i	$-0.634587\pi$
$23$	−90.5227	−0.820665	−0.410332	+	0.911936i	$0.634587\pi$
$24$	0	0
$25$	−101.216	−0.809727
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−187.261	−1.19909	−0.599544	−	0.800342i	$-0.704651\pi$
$29$	−187.261	−1.19909	−0.599544	+	0.800342i	$0.704651\pi$
$30$	0	0
$31$	−225.062	−1.30395	−0.651974	−	0.758241i	$-0.726059\pi$
$31$	−225.062	−1.30395	−0.651974	+	0.758241i	$0.726059\pi$
$32$	0	0
$33$	179.170	0.945138
$34$	0	0
$35$	124.924	0.603316
$36$	0	0
$37$	290.648	1.29141	0.645705	−	0.763587i	$-0.276563\pi$
$37$	290.648	1.29141	0.645705	+	0.763587i	$0.276563\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−191.339	−0.728833	−0.364416	−	0.931236i	$-0.618731\pi$
$41$	−191.339	−0.728833	−0.364416	+	0.931236i	$0.618731\pi$
$42$	0	0
$43$	326.833	1.15911	0.579554	−	0.814934i	$-0.303227\pi$
$43$	326.833	1.15911	0.579554	+	0.814934i	$0.303227\pi$
$44$	0	0
$45$	−43.8920	−0.145401
$46$	0	0
$47$	−406.773	−1.26242	−0.631212	−	0.775611i	$-0.717442\pi$
$47$	−406.773	−1.26242	−0.631212	+	0.775611i	$0.717442\pi$
$48$	0	0
$49$	313.155	0.912989
$50$	0	0
$51$	−226.432	−0.621702
$52$	0	0
$53$	−426.985	−1.10662	−0.553310	−	0.832975i	$-0.686636\pi$
$53$	−426.985	−1.10662	−0.553310	+	0.832975i	$0.686636\pi$
$54$	0	0
$55$	−291.265	−0.714076
$56$	0	0
$57$	−349.062	−0.811131
$58$	0	0
$59$	331.015	0.730415	0.365208	−	0.930926i	$-0.380998\pi$
$59$	331.015	0.730415	0.365208	+	0.930926i	$0.380998\pi$
$60$	0	0
$61$	−524.678	−1.10128	−0.550640	−	0.834743i	$-0.685616\pi$
$61$	−524.678	−1.10128	−0.550640	+	0.834743i	$0.685616\pi$
$62$	0	0
$63$	−230.540	−0.461036
$64$	0	0
$65$	−63.3996	−0.120981
$66$	0	0
$67$	968.172	1.76539	0.882695	−	0.469947i	$-0.155727\pi$
$67$	968.172	1.76539	0.882695	+	0.469947i	$0.155727\pi$
$68$	0	0
$69$	−271.568	−0.473811
$70$	0	0
$71$	−8.39776	−0.0140371	−0.00701853	−	0.999975i	$-0.502234\pi$
$71$	−8.39776	−0.0140371	−0.00701853	+	0.999975i	$0.502234\pi$
$72$	0	0
$73$	−903.329	−1.44831	−0.724156	−	0.689637i	$-0.757770\pi$
$73$	−903.329	−1.44831	−0.724156	+	0.689637i	$0.757770\pi$
$74$	0	0
$75$	−303.648	−0.467496
$76$	0	0
$77$	−1529.85	−2.26419
$78$	0	0
$79$	1157.98	1.64915	0.824573	−	0.565755i	$-0.191415\pi$
$79$	1157.98	1.64915	0.824573	+	0.565755i	$0.191415\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	952.371	1.25947	0.629737	−	0.776809i	$-0.283163\pi$
$83$	952.371	1.25947	0.629737	+	0.776809i	$0.283163\pi$
$84$	0	0
$85$	368.095	0.469711
$86$	0	0
$87$	−561.784	−0.692294
$88$	0	0
$89$	1059.96	1.26242	0.631211	−	0.775611i	$-0.282558\pi$
$89$	1059.96	1.26242	0.631211	+	0.775611i	$0.282558\pi$
$90$	0	0
$91$	−333.002	−0.383605
$92$	0	0
$93$	−675.187	−0.752835
$94$	0	0
$95$	567.447	0.612830
$96$	0	0
$97$	−90.7689	−0.0950123	−0.0475061	−	0.998871i	$-0.515127\pi$
$97$	−90.7689	−0.0950123	−0.0475061	+	0.998871i	$0.515127\pi$
$98$	0	0
$99$	537.511	0.545676
$100$	0	0
$101$	39.4507	0.0388662	0.0194331	−	0.999811i	$-0.493814\pi$
$101$	39.4507	0.0388662	0.0194331	+	0.999811i	$0.493814\pi$
$102$	0	0
$103$	445.599	0.426273	0.213137	−	0.977022i	$-0.431632\pi$
$103$	445.599	0.426273	0.213137	+	0.977022i	$0.431632\pi$
$104$	0	0
$105$	374.773	0.348324
$106$	0	0
$107$	−1079.89	−0.975669	−0.487834	−	0.872936i	$-0.662213\pi$
$107$	−1079.89	−0.975669	−0.487834	+	0.872936i	$0.662213\pi$
$108$	0	0
$109$	−551.788	−0.484878	−0.242439	−	0.970167i	$-0.577947\pi$
$109$	−551.788	−0.484878	−0.242439	+	0.970167i	$0.577947\pi$
$110$	0	0
$111$	871.943	0.745596
$112$	0	0
$113$	−2173.98	−1.80983	−0.904916	−	0.425591i	$-0.860066\pi$
$113$	−2173.98	−1.80983	−0.904916	+	0.425591i	$0.860066\pi$
$114$	0	0
$115$	441.470	0.357976
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	1933.39	1.48936
$120$	0	0
$121$	2235.89	1.67986
$122$	0	0
$123$	−574.017	−0.420792
$124$	0	0
$125$	1103.23	0.789408
$126$	0	0
$127$	−959.360	−0.670310	−0.335155	−	0.942163i	$-0.608789\pi$
$127$	−959.360	−0.670310	−0.335155	+	0.942163i	$0.608789\pi$
$128$	0	0
$129$	980.500	0.669211
$130$	0	0
$131$	−1859.58	−1.24025	−0.620124	−	0.784504i	$-0.712918\pi$
$131$	−1859.58	−1.24025	−0.620124	+	0.784504i	$0.712918\pi$
$132$	0	0
$133$	2980.47	1.94316
$134$	0	0
$135$	−131.676	−0.0839472
$136$	0	0
$137$	−1601.07	−0.998455	−0.499227	−	0.866471i	$-0.666383\pi$
$137$	−1601.07	−0.998455	−0.499227	+	0.866471i	$0.666383\pi$
$138$	0	0
$139$	1278.56	0.780185	0.390093	−	0.920776i	$-0.372443\pi$
$139$	1278.56	0.780185	0.390093	+	0.920776i	$0.372443\pi$
$140$	0	0
$141$	−1220.32	−0.728860
$142$	0	0
$143$	776.405	0.454030
$144$	0	0
$145$	913.254	0.523046
$146$	0	0
$147$	939.466	0.527115
$148$	0	0
$149$	1928.65	1.06041	0.530205	−	0.847869i	$-0.322115\pi$
$149$	1928.65	1.06041	0.530205	+	0.847869i	$0.322115\pi$
$150$	0	0
$151$	2822.23	1.52099	0.760496	−	0.649343i	$-0.224956\pi$
$151$	2822.23	1.52099	0.760496	+	0.649343i	$0.224956\pi$
$152$	0	0
$153$	−679.295	−0.358940
$154$	0	0
$155$	1097.61	0.568786
$156$	0	0
$157$	−690.345	−0.350927	−0.175463	−	0.984486i	$-0.556142\pi$
$157$	−690.345	−0.350927	−0.175463	+	0.984486i	$0.556142\pi$
$158$	0	0
$159$	−1280.95	−0.638908
$160$	0	0
$161$	2318.79	1.13507
$162$	0	0
$163$	3606.51	1.73303	0.866515	−	0.499151i	$-0.166355\pi$
$163$	3606.51	1.73303	0.866515	+	0.499151i	$0.166355\pi$
$164$	0	0
$165$	−873.795	−0.412272
$166$	0	0
$167$	−1019.70	−0.472494	−0.236247	−	0.971693i	$-0.575917\pi$
$167$	−1019.70	−0.472494	−0.236247	+	0.971693i	$0.575917\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1047.19	−0.468307
$172$	0	0
$173$	−2395.16	−1.05260	−0.526301	−	0.850298i	$-0.676422\pi$
$173$	−2395.16	−1.05260	−0.526301	+	0.850298i	$0.676422\pi$
$174$	0	0
$175$	2592.70	1.11994
$176$	0	0
$177$	993.045	0.421705
$178$	0	0
$179$	3609.61	1.50724	0.753618	−	0.657313i	$-0.228307\pi$
$179$	3609.61	1.50724	0.753618	+	0.657313i	$0.228307\pi$
$180$	0	0
$181$	−1848.55	−0.759125	−0.379562	−	0.925166i	$-0.623925\pi$
$181$	−1848.55	−0.759125	−0.379562	+	0.925166i	$0.623925\pi$
$182$	0	0
$183$	−1574.03	−0.635825
$184$	0	0
$185$	−1417.46	−0.563317
$186$	0	0
$187$	−4507.76	−1.76278
$188$	0	0
$189$	−691.619	−0.266179
$190$	0	0
$191$	−2364.55	−0.895772	−0.447886	−	0.894091i	$-0.647823\pi$
$191$	−2364.55	−0.895772	−0.447886	+	0.894091i	$0.647823\pi$
$192$	0	0
$193$	4199.18	1.56613	0.783067	−	0.621938i	$-0.213654\pi$
$193$	4199.18	1.56613	0.783067	+	0.621938i	$0.213654\pi$
$194$	0	0
$195$	−190.199	−0.0698483
$196$	0	0
$197$	−848.869	−0.307002	−0.153501	−	0.988148i	$-0.549055\pi$
$197$	−848.869	−0.307002	−0.153501	+	0.988148i	$0.549055\pi$
$198$	0	0
$199$	620.958	0.221199	0.110599	−	0.993865i	$-0.464723\pi$
$199$	620.958	0.221199	0.110599	+	0.993865i	$0.464723\pi$
$200$	0	0
$201$	2904.52	1.01925
$202$	0	0
$203$	4796.80	1.65847
$204$	0	0
$205$	933.140	0.317919
$206$	0	0
$207$	−814.705	−0.273555
$208$	0	0
$209$	−6949.08	−2.29989
$210$	0	0
$211$	−2911.89	−0.950059	−0.475030	−	0.879970i	$-0.657563\pi$
$211$	−2911.89	−0.950059	−0.475030	+	0.879970i	$0.657563\pi$
$212$	0	0
$213$	−25.1933	−0.00810430
$214$	0	0
$215$	−1593.93	−0.505606
$216$	0	0
$217$	5765.09	1.80350
$218$	0	0
$219$	−2709.99	−0.836183
$220$	0	0
$221$	−981.204	−0.298656
$222$	0	0
$223$	−5433.63	−1.63167	−0.815836	−	0.578283i	$-0.803723\pi$
$223$	−5433.63	−1.63167	−0.815836	+	0.578283i	$0.803723\pi$
$224$	0	0
$225$	−910.943	−0.269909
$226$	0	0
$227$	−4797.03	−1.40260	−0.701300	−	0.712866i	$-0.747397\pi$
$227$	−4797.03	−1.40260	−0.701300	+	0.712866i	$0.747397\pi$
$228$	0	0
$229$	932.947	0.269218	0.134609	−	0.990899i	$-0.457022\pi$
$229$	932.947	0.269218	0.134609	+	0.990899i	$0.457022\pi$
$230$	0	0
$231$	−4589.55	−1.30723
$232$	0	0
$233$	−169.095	−0.0475441	−0.0237720	−	0.999717i	$-0.507568\pi$
$233$	−169.095	−0.0475441	−0.0237720	+	0.999717i	$0.507568\pi$
$234$	0	0
$235$	1983.79	0.550672
$236$	0	0
$237$	3473.93	0.952135
$238$	0	0
$239$	−1467.25	−0.397106	−0.198553	−	0.980090i	$-0.563624\pi$
$239$	−1467.25	−0.397106	−0.198553	+	0.980090i	$0.563624\pi$
$240$	0	0
$241$	361.064	0.0965070	0.0482535	−	0.998835i	$-0.484634\pi$
$241$	361.064	0.0965070	0.0482535	+	0.998835i	$0.484634\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1527.23	−0.398248
$246$	0	0
$247$	−1512.60	−0.389655
$248$	0	0
$249$	2857.11	0.727157
$250$	0	0
$251$	1271.80	0.319823	0.159911	−	0.987131i	$-0.448879\pi$
$251$	1271.80	0.319823	0.159911	+	0.987131i	$0.448879\pi$
$252$	0	0
$253$	−5406.33	−1.34345
$254$	0	0
$255$	1104.28	0.271188
$256$	0	0
$257$	−842.178	−0.204411	−0.102205	−	0.994763i	$-0.532590\pi$
$257$	−842.178	−0.204411	−0.102205	+	0.994763i	$0.532590\pi$
$258$	0	0
$259$	−7445.09	−1.78616
$260$	0	0
$261$	−1685.35	−0.399696
$262$	0	0
$263$	3988.33	0.935099	0.467550	−	0.883967i	$-0.345137\pi$
$263$	3988.33	0.935099	0.467550	+	0.883967i	$0.345137\pi$
$264$	0	0
$265$	2082.36	0.482711
$266$	0	0
$267$	3179.88	0.728860
$268$	0	0
$269$	4850.14	1.09932	0.549662	−	0.835387i	$-0.314756\pi$
$269$	4850.14	1.09932	0.549662	+	0.835387i	$0.314756\pi$
$270$	0	0
$271$	1997.17	0.447674	0.223837	−	0.974627i	$-0.428142\pi$
$271$	1997.17	0.447674	0.223837	+	0.974627i	$0.428142\pi$
$272$	0	0
$273$	−999.006	−0.221475
$274$	0	0
$275$	−6044.97	−1.32555
$276$	0	0
$277$	−615.909	−0.133597	−0.0667985	−	0.997766i	$-0.521278\pi$
$277$	−615.909	−0.133597	−0.0667985	+	0.997766i	$0.521278\pi$
$278$	0	0
$279$	−2025.56	−0.434650
$280$	0	0
$281$	4322.77	0.917703	0.458852	−	0.888513i	$-0.348261\pi$
$281$	4322.77	0.917703	0.458852	+	0.888513i	$0.348261\pi$
$282$	0	0
$283$	−1033.91	−0.217172	−0.108586	−	0.994087i	$-0.534632\pi$
$283$	−1033.91	−0.217172	−0.108586	+	0.994087i	$0.534632\pi$
$284$	0	0
$285$	1702.34	0.353817
$286$	0	0
$287$	4901.25	1.00805
$288$	0	0
$289$	783.818	0.159540
$290$	0	0
$291$	−272.307	−0.0548554
$292$	0	0
$293$	−8391.93	−1.67325	−0.836625	−	0.547777i	$-0.815474\pi$
$293$	−8391.93	−1.67325	−0.836625	+	0.547777i	$0.815474\pi$
$294$	0	0
$295$	−1614.33	−0.318609
$296$	0	0
$297$	1612.53	0.315046
$298$	0	0
$299$	−1176.80	−0.227612
$300$	0	0
$301$	−8372.01	−1.60317
$302$	0	0
$303$	118.352	0.0224394
$304$	0	0
$305$	2558.80	0.480382
$306$	0	0
$307$	−4374.08	−0.813166	−0.406583	−	0.913614i	$-0.633280\pi$
$307$	−4374.08	−0.813166	−0.406583	+	0.913614i	$0.633280\pi$
$308$	0	0
$309$	1336.80	0.246109
$310$	0	0
$311$	5981.42	1.09060	0.545298	−	0.838242i	$-0.316416\pi$
$311$	5981.42	1.09060	0.545298	+	0.838242i	$0.316416\pi$
$312$	0	0
$313$	1159.92	0.209466	0.104733	−	0.994500i	$-0.466601\pi$
$313$	1159.92	0.209466	0.104733	+	0.994500i	$0.466601\pi$
$314$	0	0
$315$	1124.32	0.201105
$316$	0	0
$317$	−1502.60	−0.266229	−0.133115	−	0.991101i	$-0.542498\pi$
$317$	−1502.60	−0.266229	−0.133115	+	0.991101i	$0.542498\pi$
$318$	0	0
$319$	−11183.9	−1.96294
$320$	0	0
$321$	−3239.66	−0.563303
$322$	0	0
$323$	8782.09	1.51284
$324$	0	0
$325$	−1315.81	−0.224578
$326$	0	0
$327$	−1655.36	−0.279944
$328$	0	0
$329$	10419.7	1.74607
$330$	0	0
$331$	−472.260	−0.0784221	−0.0392111	−	0.999231i	$-0.512484\pi$
$331$	−472.260	−0.0784221	−0.0392111	+	0.999231i	$0.512484\pi$
$332$	0	0
$333$	2615.83	0.430470
$334$	0	0
$335$	−4721.67	−0.770067
$336$	0	0
$337$	−8318.44	−1.34461	−0.672306	−	0.740273i	$-0.734696\pi$
$337$	−8318.44	−1.34461	−0.672306	+	0.740273i	$0.734696\pi$
$338$	0	0
$339$	−6521.94	−1.04491
$340$	0	0
$341$	−13441.5	−2.13460
$342$	0	0
$343$	764.488	0.120345
$344$	0	0
$345$	1324.41	0.206678
$346$	0	0
$347$	−11115.7	−1.71967	−0.859834	−	0.510574i	$-0.829433\pi$
$347$	−11115.7	−1.71967	−0.859834	+	0.510574i	$0.829433\pi$
$348$	0	0
$349$	6057.81	0.929133	0.464566	−	0.885538i	$-0.346210\pi$
$349$	6057.81	0.929133	0.464566	+	0.885538i	$0.346210\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	675.403	0.101836	0.0509179	−	0.998703i	$-0.483785\pi$
$353$	675.403	0.101836	0.0509179	+	0.998703i	$0.483785\pi$
$354$	0	0
$355$	40.9550	0.00612300
$356$	0	0
$357$	5800.17	0.859881
$358$	0	0
$359$	−351.424	−0.0516642	−0.0258321	−	0.999666i	$-0.508224\pi$
$359$	−351.424	−0.0516642	−0.0258321	+	0.999666i	$0.508224\pi$
$360$	0	0
$361$	6679.29	0.973800
$362$	0	0
$363$	6707.68	0.969868
$364$	0	0
$365$	4405.44	0.631757
$366$	0	0
$367$	−6707.82	−0.954075	−0.477037	−	0.878883i	$-0.658289\pi$
$367$	−6707.82	−0.954075	−0.477037	+	0.878883i	$0.658289\pi$
$368$	0	0
$369$	−1722.05	−0.242944
$370$	0	0
$371$	10937.4	1.53058
$372$	0	0
$373$	7236.36	1.00452	0.502258	−	0.864718i	$-0.332503\pi$
$373$	7236.36	1.00452	0.502258	+	0.864718i	$0.332503\pi$
$374$	0	0
$375$	3309.69	0.455765
$376$	0	0
$377$	−2434.40	−0.332567
$378$	0	0
$379$	4084.92	0.553636	0.276818	−	0.960922i	$-0.410720\pi$
$379$	4084.92	0.553636	0.276818	+	0.960922i	$0.410720\pi$
$380$	0	0
$381$	−2878.08	−0.387004
$382$	0	0
$383$	7764.76	1.03593	0.517965	−	0.855402i	$-0.326690\pi$
$383$	7764.76	1.03593	0.517965	+	0.855402i	$0.326690\pi$
$384$	0	0
$385$	7460.91	0.987645
$386$	0	0
$387$	2941.50	0.386369
$388$	0	0
$389$	−1927.25	−0.251197	−0.125598	−	0.992081i	$-0.540085\pi$
$389$	−1927.25	−0.251197	−0.125598	+	0.992081i	$0.540085\pi$
$390$	0	0
$391$	6832.41	0.883708
$392$	0	0
$393$	−5578.75	−0.716058
$394$	0	0
$395$	−5647.33	−0.719362
$396$	0	0
$397$	−2607.17	−0.329597	−0.164799	−	0.986327i	$-0.552697\pi$
$397$	−2607.17	−0.329597	−0.164799	+	0.986327i	$0.552697\pi$
$398$	0	0
$399$	8941.42	1.12188
$400$	0	0
$401$	−3890.24	−0.484462	−0.242231	−	0.970219i	$-0.577879\pi$
$401$	−3890.24	−0.484462	−0.242231	+	0.970219i	$0.577879\pi$
$402$	0	0
$403$	−2925.81	−0.361650
$404$	0	0
$405$	−395.028	−0.0484670
$406$	0	0
$407$	17358.5	2.11407
$408$	0	0
$409$	5916.54	0.715291	0.357646	−	0.933857i	$-0.383580\pi$
$409$	5916.54	0.715291	0.357646	+	0.933857i	$0.383580\pi$
$410$	0	0
$411$	−4803.20	−0.576458
$412$	0	0
$413$	−8479.13	−1.01024
$414$	0	0
$415$	−4644.61	−0.549386
$416$	0	0
$417$	3835.67	0.450440
$418$	0	0
$419$	1729.83	0.201689	0.100845	−	0.994902i	$-0.467845\pi$
$419$	1729.83	0.201689	0.100845	+	0.994902i	$0.467845\pi$
$420$	0	0
$421$	−5546.89	−0.642135	−0.321067	−	0.947056i	$-0.604042\pi$
$421$	−5546.89	−0.642135	−0.321067	+	0.947056i	$0.604042\pi$
$422$	0	0
$423$	−3660.95	−0.420808
$424$	0	0
$425$	7639.50	0.871930
$426$	0	0
$427$	13439.9	1.52319
$428$	0	0
$429$	2329.22	0.262134
$430$	0	0
$431$	−11184.4	−1.24996	−0.624979	−	0.780641i	$-0.714893\pi$
$431$	−11184.4	−1.24996	−0.624979	+	0.780641i	$0.714893\pi$
$432$	0	0
$433$	4298.00	0.477017	0.238509	−	0.971140i	$-0.423341\pi$
$433$	4298.00	0.477017	0.238509	+	0.971140i	$0.423341\pi$
$434$	0	0
$435$	2739.76	0.301981
$436$	0	0
$437$	10532.7	1.15297
$438$	0	0
$439$	−12481.1	−1.35692	−0.678461	−	0.734636i	$-0.737353\pi$
$439$	−12481.1	−1.35692	−0.678461	+	0.734636i	$0.737353\pi$
$440$	0	0
$441$	2818.40	0.304330
$442$	0	0
$443$	−12999.3	−1.39417	−0.697084	−	0.716989i	$-0.745520\pi$
$443$	−12999.3	−1.39417	−0.697084	+	0.716989i	$0.745520\pi$
$444$	0	0
$445$	−5169.31	−0.550672
$446$	0	0
$447$	5785.95	0.612228
$448$	0	0
$449$	14636.6	1.53840	0.769201	−	0.639007i	$-0.220654\pi$
$449$	14636.6	1.53840	0.769201	+	0.639007i	$0.220654\pi$
$450$	0	0
$451$	−11427.4	−1.19312
$452$	0	0
$453$	8466.69	0.878145
$454$	0	0
$455$	1624.01	0.167330
$456$	0	0
$457$	−8803.22	−0.901088	−0.450544	−	0.892754i	$-0.648770\pi$
$457$	−8803.22	−0.901088	−0.450544	+	0.892754i	$0.648770\pi$
$458$	0	0
$459$	−2037.89	−0.207234
$460$	0	0
$461$	−13223.9	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$461$	−13223.9	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$462$	0	0
$463$	−14136.4	−1.41895	−0.709476	−	0.704729i	$-0.751068\pi$
$463$	−14136.4	−1.41895	−0.709476	+	0.704729i	$0.751068\pi$
$464$	0	0
$465$	3292.82	0.328389
$466$	0	0
$467$	8654.11	0.857525	0.428763	−	0.903417i	$-0.358950\pi$
$467$	8654.11	0.857525	0.428763	+	0.903417i	$0.358950\pi$
$468$	0	0
$469$	−24800.2	−2.44172
$470$	0	0
$471$	−2071.03	−0.202608
$472$	0	0
$473$	19519.6	1.89749
$474$	0	0
$475$	11776.9	1.13760
$476$	0	0
$477$	−3842.86	−0.368873
$478$	0	0
$479$	3622.59	0.345554	0.172777	−	0.984961i	$-0.444726\pi$
$479$	3622.59	0.345554	0.172777	+	0.984961i	$0.444726\pi$
$480$	0	0
$481$	3778.42	0.358173
$482$	0	0
$483$	6956.36	0.655332
$484$	0	0
$485$	442.671	0.0414446
$486$	0	0
$487$	20857.4	1.94074	0.970368	−	0.241632i	$-0.0776828\pi$
$487$	20857.4	1.94074	0.970368	+	0.241632i	$0.0776828\pi$
$488$	0	0
$489$	10819.5	1.00057
$490$	0	0
$491$	−5060.09	−0.465089	−0.232545	−	0.972586i	$-0.574705\pi$
$491$	−5060.09	−0.465089	−0.232545	+	0.972586i	$0.574705\pi$
$492$	0	0
$493$	14134.0	1.29120
$494$	0	0
$495$	−2621.39	−0.238025
$496$	0	0
$497$	215.113	0.0194148
$498$	0	0
$499$	−1150.05	−0.103173	−0.0515864	−	0.998669i	$-0.516428\pi$
$499$	−1150.05	−0.103173	−0.0515864	+	0.998669i	$0.516428\pi$
$500$	0	0
$501$	−3059.09	−0.272795
$502$	0	0
$503$	7145.59	0.633412	0.316706	−	0.948524i	$-0.397423\pi$
$503$	7145.59	0.633412	0.316706	+	0.948524i	$0.397423\pi$
$504$	0	0
$505$	−192.397	−0.0169536
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	7240.54	0.630513	0.315257	−	0.949006i	$-0.397909\pi$
$509$	7240.54	0.630513	0.315257	+	0.949006i	$0.397909\pi$
$510$	0	0
$511$	23139.3	2.00317
$512$	0	0
$513$	−3141.56	−0.270377
$514$	0	0
$515$	−2173.14	−0.185941
$516$	0	0
$517$	−24293.9	−2.06662
$518$	0	0
$519$	−7185.47	−0.607720
$520$	0	0
$521$	2690.90	0.226277	0.113139	−	0.993579i	$-0.463910\pi$
$521$	2690.90	0.226277	0.113139	+	0.993579i	$0.463910\pi$
$522$	0	0
$523$	507.043	0.0423928	0.0211964	−	0.999775i	$-0.493252\pi$
$523$	507.043	0.0423928	0.0211964	+	0.999775i	$0.493252\pi$
$524$	0	0
$525$	7778.10	0.646598
$526$	0	0
$527$	16987.1	1.40412
$528$	0	0
$529$	−3972.63	−0.326509
$530$	0	0
$531$	2979.14	0.243472
$532$	0	0
$533$	−2487.41	−0.202142
$534$	0	0
$535$	5266.49	0.425589
$536$	0	0
$537$	10828.8	0.870203
$538$	0	0
$539$	18702.7	1.49459
$540$	0	0
$541$	997.477	0.0792696	0.0396348	−	0.999214i	$-0.487381\pi$
$541$	997.477	0.0792696	0.0396348	+	0.999214i	$0.487381\pi$
$542$	0	0
$543$	−5545.65	−0.438281
$544$	0	0
$545$	2691.01	0.211505
$546$	0	0
$547$	−15191.1	−1.18743	−0.593716	−	0.804675i	$-0.702340\pi$
$547$	−15191.1	−1.18743	−0.593716	+	0.804675i	$0.702340\pi$
$548$	0	0
$549$	−4722.10	−0.367094
$550$	0	0
$551$	21788.6	1.68462
$552$	0	0
$553$	−29662.2	−2.28095
$554$	0	0
$555$	−4252.37	−0.325231
$556$	0	0
$557$	−1426.17	−0.108489	−0.0542447	−	0.998528i	$-0.517275\pi$
$557$	−1426.17	−0.108489	−0.0542447	+	0.998528i	$0.517275\pi$
$558$	0	0
$559$	4248.83	0.321478
$560$	0	0
$561$	−13523.3	−1.01774
$562$	0	0
$563$	14066.7	1.05300	0.526502	−	0.850174i	$-0.323503\pi$
$563$	14066.7	1.05300	0.526502	+	0.850174i	$0.323503\pi$
$564$	0	0
$565$	10602.3	0.789453
$566$	0	0
$567$	−2074.86	−0.153679
$568$	0	0
$569$	−20020.8	−1.47507	−0.737537	−	0.675307i	$-0.764011\pi$
$569$	−20020.8	−1.47507	−0.737537	+	0.675307i	$0.764011\pi$
$570$	0	0
$571$	−6023.04	−0.441430	−0.220715	−	0.975338i	$-0.570839\pi$
$571$	−6023.04	−0.441430	−0.220715	+	0.975338i	$0.570839\pi$
$572$	0	0
$573$	−7093.64	−0.517174
$574$	0	0
$575$	9162.34	0.664515
$576$	0	0
$577$	−6825.67	−0.492472	−0.246236	−	0.969210i	$-0.579194\pi$
$577$	−6825.67	−0.492472	−0.246236	+	0.969210i	$0.579194\pi$
$578$	0	0
$579$	12597.5	0.904207
$580$	0	0
$581$	−24395.5	−1.74199
$582$	0	0
$583$	−25501.0	−1.81157
$584$	0	0
$585$	−570.597	−0.0403270
$586$	0	0
$587$	−24638.4	−1.73243	−0.866213	−	0.499675i	$-0.833453\pi$
$587$	−24638.4	−1.73243	−0.866213	+	0.499675i	$0.833453\pi$
$588$	0	0
$589$	26187.0	1.83194
$590$	0	0
$591$	−2546.61	−0.177248
$592$	0	0
$593$	−4694.14	−0.325068	−0.162534	−	0.986703i	$-0.551967\pi$
$593$	−4694.14	−0.325068	−0.162534	+	0.986703i	$0.551967\pi$
$594$	0	0
$595$	−9428.94	−0.649662
$596$	0	0
$597$	1862.87	0.127709
$598$	0	0
$599$	10043.5	0.685087	0.342544	−	0.939502i	$-0.388712\pi$
$599$	10043.5	0.685087	0.342544	+	0.939502i	$0.388712\pi$
$600$	0	0
$601$	−20677.8	−1.40343	−0.701717	−	0.712456i	$-0.747583\pi$
$601$	−20677.8	−1.40343	−0.701717	+	0.712456i	$0.747583\pi$
$602$	0	0
$603$	8713.55	0.588463
$604$	0	0
$605$	−10904.2	−0.732760
$606$	0	0
$607$	4990.49	0.333703	0.166851	−	0.985982i	$-0.446640\pi$
$607$	4990.49	0.333703	0.166851	+	0.985982i	$0.446640\pi$
$608$	0	0
$609$	14390.4	0.957518
$610$	0	0
$611$	−5288.04	−0.350133
$612$	0	0
$613$	−27970.0	−1.84290	−0.921450	−	0.388496i	$-0.872995\pi$
$613$	−27970.0	−1.84290	−0.921450	+	0.388496i	$0.872995\pi$
$614$	0	0
$615$	2799.42	0.183550
$616$	0	0
$617$	−2978.15	−0.194321	−0.0971603	−	0.995269i	$-0.530976\pi$
$617$	−2978.15	−0.194321	−0.0971603	+	0.995269i	$0.530976\pi$
$618$	0	0
$619$	29605.9	1.92239	0.961196	−	0.275865i	$-0.0889643\pi$
$619$	29605.9	1.92239	0.961196	+	0.275865i	$0.0889643\pi$
$620$	0	0
$621$	−2444.11	−0.157937
$622$	0	0
$623$	−27151.4	−1.74607
$624$	0	0
$625$	7271.65	0.465385
$626$	0	0
$627$	−20847.2	−1.32784
$628$	0	0
$629$	−21937.3	−1.39062
$630$	0	0
$631$	−937.256	−0.0591309	−0.0295654	−	0.999563i	$-0.509412\pi$
$631$	−937.256	−0.0591309	−0.0295654	+	0.999563i	$0.509412\pi$
$632$	0	0
$633$	−8735.66	−0.548517
$634$	0	0
$635$	4678.70	0.292391
$636$	0	0
$637$	4071.02	0.253218
$638$	0	0
$639$	−75.5799	−0.00467902
$640$	0	0
$641$	−8879.08	−0.547118	−0.273559	−	0.961855i	$-0.588201\pi$
$641$	−8879.08	−0.547118	−0.273559	+	0.961855i	$0.588201\pi$
$642$	0	0
$643$	−8098.91	−0.496718	−0.248359	−	0.968668i	$-0.579891\pi$
$643$	−8098.91	−0.496718	−0.248359	+	0.968668i	$0.579891\pi$
$644$	0	0
$645$	−4781.79	−0.291912
$646$	0	0
$647$	−16069.3	−0.976429	−0.488214	−	0.872724i	$-0.662352\pi$
$647$	−16069.3	−0.976429	−0.488214	+	0.872724i	$0.662352\pi$
$648$	0	0
$649$	19769.4	1.19571
$650$	0	0
$651$	17295.3	1.04125
$652$	0	0
$653$	4610.46	0.276296	0.138148	−	0.990412i	$-0.455885\pi$
$653$	4610.46	0.276296	0.138148	+	0.990412i	$0.455885\pi$
$654$	0	0
$655$	9068.99	0.541000
$656$	0	0
$657$	−8129.97	−0.482770
$658$	0	0
$659$	−24960.1	−1.47543	−0.737714	−	0.675113i	$-0.764095\pi$
$659$	−24960.1	−1.47543	−0.737714	+	0.675113i	$0.764095\pi$
$660$	0	0
$661$	24437.3	1.43798	0.718988	−	0.695023i	$-0.244606\pi$
$661$	24437.3	1.43798	0.718988	+	0.695023i	$0.244606\pi$
$662$	0	0
$663$	−2943.61	−0.172429
$664$	0	0
$665$	−14535.5	−0.847610
$666$	0	0
$667$	16951.4	0.984050
$668$	0	0
$669$	−16300.9	−0.942046
$670$	0	0
$671$	−31335.6	−1.80283
$672$	0	0
$673$	2702.96	0.154816	0.0774081	−	0.996999i	$-0.475336\pi$
$673$	2702.96	0.154816	0.0774081	+	0.996999i	$0.475336\pi$
$674$	0	0
$675$	−2732.83	−0.155832
$676$	0	0
$677$	11071.8	0.628544	0.314272	−	0.949333i	$-0.398240\pi$
$677$	11071.8	0.628544	0.314272	+	0.949333i	$0.398240\pi$
$678$	0	0
$679$	2325.09	0.131412
$680$	0	0
$681$	−14391.1	−0.809792
$682$	0	0
$683$	−11778.4	−0.659864	−0.329932	−	0.944005i	$-0.607026\pi$
$683$	−11778.4	−0.659864	−0.329932	+	0.944005i	$0.607026\pi$
$684$	0	0
$685$	7808.23	0.435529
$686$	0	0
$687$	2798.84	0.155433
$688$	0	0
$689$	−5550.80	−0.306921
$690$	0	0
$691$	30715.1	1.69097	0.845483	−	0.534002i	$-0.179313\pi$
$691$	30715.1	1.69097	0.845483	+	0.534002i	$0.179313\pi$
$692$	0	0
$693$	−13768.6	−0.754729
$694$	0	0
$695$	−6235.39	−0.340319
$696$	0	0
$697$	14441.7	0.784821
$698$	0	0
$699$	−507.285	−0.0274496
$700$	0	0
$701$	741.115	0.0399308	0.0199654	−	0.999801i	$-0.493644\pi$
$701$	741.115	0.0399308	0.0199654	+	0.999801i	$0.493644\pi$
$702$	0	0
$703$	−33818.1	−1.81433
$704$	0	0
$705$	5951.36	0.317931
$706$	0	0
$707$	−1010.55	−0.0537562
$708$	0	0
$709$	−24269.6	−1.28556	−0.642781	−	0.766050i	$-0.722219\pi$
$709$	−24269.6	−1.28556	−0.642781	+	0.766050i	$0.722219\pi$
$710$	0	0
$711$	10421.8	0.549716
$712$	0	0
$713$	20373.3	1.07011
$714$	0	0
$715$	−3786.45	−0.198049
$716$	0	0
$717$	−4401.74	−0.229269
$718$	0	0
$719$	15598.6	0.809083	0.404541	−	0.914520i	$-0.367431\pi$
$719$	15598.6	0.809083	0.404541	+	0.914520i	$0.367431\pi$
$720$	0	0
$721$	−11414.2	−0.589582
$722$	0	0
$723$	1083.19	0.0557183
$724$	0	0
$725$	18953.8	0.970934
$726$	0	0
$727$	−32399.2	−1.65285	−0.826423	−	0.563050i	$-0.809628\pi$
$727$	−32399.2	−1.65285	−0.826423	+	0.563050i	$0.809628\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−24668.5	−1.24815
$732$	0	0
$733$	7851.92	0.395658	0.197829	−	0.980237i	$-0.436611\pi$
$733$	7851.92	0.395658	0.197829	+	0.980237i	$0.436611\pi$
$734$	0	0
$735$	−4581.68	−0.229929
$736$	0	0
$737$	57822.6	2.88999
$738$	0	0
$739$	13348.4	0.664451	0.332226	−	0.943200i	$-0.392200\pi$
$739$	13348.4	0.664451	0.332226	+	0.943200i	$0.392200\pi$
$740$	0	0
$741$	−4537.81	−0.224967
$742$	0	0
$743$	−9303.61	−0.459376	−0.229688	−	0.973264i	$-0.573771\pi$
$743$	−9303.61	−0.459376	−0.229688	+	0.973264i	$0.573771\pi$
$744$	0	0
$745$	−9405.82	−0.462554
$746$	0	0
$747$	8571.34	0.419825
$748$	0	0
$749$	27661.9	1.34946
$750$	0	0
$751$	−9774.38	−0.474930	−0.237465	−	0.971396i	$-0.576316\pi$
$751$	−9774.38	−0.474930	−0.237465	+	0.971396i	$0.576316\pi$
$752$	0	0
$753$	3815.41	0.184650
$754$	0	0
$755$	−13763.7	−0.663461
$756$	0	0
$757$	24541.6	1.17831	0.589154	−	0.808021i	$-0.299461\pi$
$757$	24541.6	1.17831	0.589154	+	0.808021i	$0.299461\pi$
$758$	0	0
$759$	−16219.0	−0.775642
$760$	0	0
$761$	−14409.4	−0.686389	−0.343194	−	0.939264i	$-0.611509\pi$
$761$	−14409.4	−0.686389	−0.343194	+	0.939264i	$0.611509\pi$
$762$	0	0
$763$	14134.3	0.670639
$764$	0	0
$765$	3312.85	0.156570
$766$	0	0
$767$	4303.20	0.202581
$768$	0	0
$769$	4426.67	0.207581	0.103791	−	0.994599i	$-0.466903\pi$
$769$	4426.67	0.207581	0.103791	+	0.994599i	$0.466903\pi$
$770$	0	0
$771$	−2526.53	−0.118017
$772$	0	0
$773$	−4608.64	−0.214439	−0.107219	−	0.994235i	$-0.534195\pi$
$773$	−4608.64	−0.214439	−0.107219	+	0.994235i	$0.534195\pi$
$774$	0	0
$775$	22779.9	1.05584
$776$	0	0
$777$	−22335.3	−1.03124
$778$	0	0
$779$	22263.1	1.02395
$780$	0	0
$781$	−501.544	−0.0229790
$782$	0	0
$783$	−5056.06	−0.230765
$784$	0	0
$785$	3366.74	0.153075
$786$	0	0
$787$	2743.83	0.124278	0.0621392	−	0.998067i	$-0.480208\pi$
$787$	2743.83	0.124278	0.0621392	+	0.998067i	$0.480208\pi$
$788$	0	0
$789$	11965.0	0.539880
$790$	0	0
$791$	55687.7	2.50319
$792$	0	0
$793$	−6820.81	−0.305440
$794$	0	0
$795$	6247.08	0.278693
$796$	0	0
$797$	−2474.62	−0.109982	−0.0549910	−	0.998487i	$-0.517513\pi$
$797$	−2474.62	−0.109982	−0.0549910	+	0.998487i	$0.517513\pi$
$798$	0	0
$799$	30702.1	1.35940
$800$	0	0
$801$	9539.64	0.420807
$802$	0	0
$803$	−53950.0	−2.37093
$804$	0	0
$805$	−11308.5	−0.495120
$806$	0	0
$807$	14550.4	0.634695
$808$	0	0
$809$	13410.2	0.582790	0.291395	−	0.956603i	$-0.405881\pi$
$809$	13410.2	0.582790	0.291395	+	0.956603i	$0.405881\pi$
$810$	0	0
$811$	24145.8	1.04547	0.522733	−	0.852497i	$-0.324913\pi$
$811$	24145.8	1.04547	0.522733	+	0.852497i	$0.324913\pi$
$812$	0	0
$813$	5991.52	0.258465
$814$	0	0
$815$	−17588.6	−0.755952
$816$	0	0
$817$	−38028.4	−1.62845
$818$	0	0
$819$	−2997.02	−0.127868
$820$	0	0
$821$	−5569.07	−0.236738	−0.118369	−	0.992970i	$-0.537767\pi$
$821$	−5569.07	−0.236738	−0.118369	+	0.992970i	$0.537767\pi$
$822$	0	0
$823$	17139.2	0.725921	0.362961	−	0.931804i	$-0.381766\pi$
$823$	17139.2	0.725921	0.362961	+	0.931804i	$0.381766\pi$
$824$	0	0
$825$	−18134.9	−0.765304
$826$	0	0
$827$	3756.54	0.157954	0.0789769	−	0.996876i	$-0.474835\pi$
$827$	3756.54	0.157954	0.0789769	+	0.996876i	$0.474835\pi$
$828$	0	0
$829$	−15534.3	−0.650819	−0.325410	−	0.945573i	$-0.605502\pi$
$829$	−15534.3	−0.650819	−0.325410	+	0.945573i	$0.605502\pi$
$830$	0	0
$831$	−1847.73	−0.0771323
$832$	0	0
$833$	−23636.1	−0.983124
$834$	0	0
$835$	4972.95	0.206103
$836$	0	0
$837$	−6076.69	−0.250945
$838$	0	0
$839$	−31953.0	−1.31483	−0.657413	−	0.753531i	$-0.728349\pi$
$839$	−31953.0	−1.31483	−0.657413	+	0.753531i	$0.728349\pi$
$840$	0	0
$841$	10677.8	0.437813
$842$	0	0
$843$	12968.3	0.529836
$844$	0	0
$845$	−824.195	−0.0335541
$846$	0	0
$847$	−57273.6	−2.32343
$848$	0	0
$849$	−3101.74	−0.125384
$850$	0	0
$851$	−26310.2	−1.05982
$852$	0	0
$853$	41587.2	1.66931	0.834654	−	0.550775i	$-0.185668\pi$
$853$	41587.2	1.66931	0.834654	+	0.550775i	$0.185668\pi$
$854$	0	0
$855$	5107.02	0.204277
$856$	0	0
$857$	−8080.58	−0.322086	−0.161043	−	0.986947i	$-0.551486\pi$
$857$	−8080.58	−0.322086	−0.161043	+	0.986947i	$0.551486\pi$
$858$	0	0
$859$	33848.0	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$859$	33848.0	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$860$	0	0
$861$	14703.7	0.582000
$862$	0	0
$863$	−27002.6	−1.06510	−0.532548	−	0.846400i	$-0.678766\pi$
$863$	−27002.6	−1.06510	−0.532548	+	0.846400i	$0.678766\pi$
$864$	0	0
$865$	11680.9	0.459148
$866$	0	0
$867$	2351.45	0.0921102
$868$	0	0
$869$	69158.4	2.69970
$870$	0	0
$871$	12586.2	0.489631
$872$	0	0
$873$	−816.920	−0.0316708
$874$	0	0
$875$	−28259.8	−1.09184
$876$	0	0
$877$	45535.6	1.75328	0.876641	−	0.481145i	$-0.159779\pi$
$877$	45535.6	1.75328	0.876641	+	0.481145i	$0.159779\pi$
$878$	0	0
$879$	−25175.8	−0.966051
$880$	0	0
$881$	864.335	0.0330536	0.0165268	−	0.999863i	$-0.494739\pi$
$881$	864.335	0.0330536	0.0165268	+	0.999863i	$0.494739\pi$
$882$	0	0
$883$	−1075.10	−0.0409741	−0.0204871	−	0.999790i	$-0.506522\pi$
$883$	−1075.10	−0.0409741	−0.0204871	+	0.999790i	$0.506522\pi$
$884$	0	0
$885$	−4842.98	−0.183949
$886$	0	0
$887$	−36604.8	−1.38565	−0.692824	−	0.721107i	$-0.743634\pi$
$887$	−36604.8	−1.38565	−0.692824	+	0.721107i	$0.743634\pi$
$888$	0	0
$889$	24574.5	0.927112
$890$	0	0
$891$	4837.60	0.181892
$892$	0	0
$893$	47329.7	1.77360
$894$	0	0
$895$	−17603.7	−0.657460
$896$	0	0
$897$	−3530.39	−0.131412
$898$	0	0
$899$	42145.5	1.56355
$900$	0	0
$901$	32227.6	1.19163
$902$	0	0
$903$	−25116.0	−0.925591
$904$	0	0
$905$	9015.18	0.331132
$906$	0	0
$907$	19657.7	0.719649	0.359825	−	0.933020i	$-0.382837\pi$
$907$	19657.7	0.719649	0.359825	+	0.933020i	$0.382837\pi$
$908$	0	0
$909$	355.056	0.0129554
$910$	0	0
$911$	−8278.09	−0.301060	−0.150530	−	0.988605i	$-0.548098\pi$
$911$	−8278.09	−0.301060	−0.150530	+	0.988605i	$0.548098\pi$
$912$	0	0
$913$	56878.9	2.06179
$914$	0	0
$915$	7676.40	0.277349
$916$	0	0
$917$	47634.2	1.71540
$918$	0	0
$919$	20750.6	0.744829	0.372414	−	0.928067i	$-0.378530\pi$
$919$	20750.6	0.744829	0.372414	+	0.928067i	$0.378530\pi$
$920$	0	0
$921$	−13122.2	−0.469482
$922$	0	0
$923$	−109.171	−0.00389318
$924$	0	0
$925$	−29418.2	−1.04569
$926$	0	0
$927$	4010.39	0.142091
$928$	0	0
$929$	−27905.6	−0.985526	−0.492763	−	0.870164i	$-0.664013\pi$
$929$	−27905.6	−0.985526	−0.492763	+	0.870164i	$0.664013\pi$
$930$	0	0
$931$	−36436.9	−1.28268
$932$	0	0
$933$	17944.3	0.629656
$934$	0	0
$935$	21983.9	0.768931
$936$	0	0
$937$	1734.90	0.0604873	0.0302437	−	0.999543i	$-0.490372\pi$
$937$	1734.90	0.0604873	0.0302437	+	0.999543i	$0.490372\pi$
$938$	0	0
$939$	3479.77	0.120935
$940$	0	0
$941$	18761.2	0.649943	0.324972	−	0.945724i	$-0.394645\pi$
$941$	18761.2	0.649943	0.324972	+	0.945724i	$0.394645\pi$
$942$	0	0
$943$	17320.5	0.598127
$944$	0	0
$945$	3372.95	0.116108
$946$	0	0
$947$	−20003.3	−0.686400	−0.343200	−	0.939262i	$-0.611511\pi$
$947$	−20003.3	−0.686400	−0.343200	+	0.939262i	$0.611511\pi$
$948$	0	0
$949$	−11743.3	−0.401689
$950$	0	0
$951$	−4507.81	−0.153707
$952$	0	0
$953$	−14639.9	−0.497620	−0.248810	−	0.968552i	$-0.580039\pi$
$953$	−14639.9	−0.497620	−0.248810	+	0.968552i	$0.580039\pi$
$954$	0	0
$955$	11531.6	0.390738
$956$	0	0
$957$	−33551.7	−1.13330
$958$	0	0
$959$	41012.2	1.38097
$960$	0	0
$961$	20862.1	0.700283
$962$	0	0
$963$	−9718.98	−0.325223
$964$	0	0
$965$	−20479.0	−0.683152
$966$	0	0
$967$	−29052.1	−0.966136	−0.483068	−	0.875583i	$-0.660478\pi$
$967$	−29052.1	−0.966136	−0.483068	+	0.875583i	$0.660478\pi$
$968$	0	0
$969$	26346.3	0.873441
$970$	0	0
$971$	−30911.6	−1.02163	−0.510814	−	0.859691i	$-0.670656\pi$
$971$	−30911.6	−1.02163	−0.510814	+	0.859691i	$0.670656\pi$
$972$	0	0
$973$	−32750.9	−1.07908
$974$	0	0
$975$	−3947.42	−0.129660
$976$	0	0
$977$	−34458.8	−1.12839	−0.564194	−	0.825642i	$-0.690813\pi$
$977$	−34458.8	−1.12839	−0.564194	+	0.825642i	$0.690813\pi$
$978$	0	0
$979$	63304.5	2.06662
$980$	0	0
$981$	−4966.09	−0.161626
$982$	0	0
$983$	41353.9	1.34180	0.670898	−	0.741550i	$-0.265909\pi$
$983$	41353.9	1.34180	0.670898	+	0.741550i	$0.265909\pi$
$984$	0	0
$985$	4139.85	0.133915
$986$	0	0
$987$	31259.1	1.00809
$988$	0	0
$989$	−29585.8	−0.951239
$990$	0	0
$991$	4649.74	0.149045	0.0745227	−	0.997219i	$-0.476257\pi$
$991$	4649.74	0.149045	0.0745227	+	0.997219i	$0.476257\pi$
$992$	0	0
$993$	−1416.78	−0.0452770
$994$	0	0
$995$	−3028.35	−0.0964875
$996$	0	0
$997$	−45125.0	−1.43342	−0.716712	−	0.697369i	$-0.754354\pi$
$997$	−45125.0	−1.43342	−0.716712	+	0.697369i	$0.754354\pi$
$998$	0	0
$999$	7847.49	0.248532

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.d.1.2	✓	2	1.1	even	1	trivial
624.4.a.j.1.2		2	4.3	odd	2
936.4.a.j.1.1		2	3.2	odd	2
1872.4.a.bi.1.1		2	12.11	even	2
2496.4.a.ba.1.1		2	8.5	even	2
2496.4.a.bj.1.1		2	8.3	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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -4.87689 q^{5} -25.6155 q^{7} +9.00000 q^{9} +59.7235 q^{11} +13.0000 q^{13} -14.6307 q^{15} -75.4773 q^{17} -116.354 q^{19} -76.8466 q^{21} -90.5227 q^{23} -101.216 q^{25} +27.0000 q^{27} -187.261 q^{29} -225.062 q^{31} +179.170 q^{33} +124.924 q^{35} +290.648 q^{37} +39.0000 q^{39} -191.339 q^{41} +326.833 q^{43} -43.8920 q^{45} -406.773 q^{47} +313.155 q^{49} -226.432 q^{51} -426.985 q^{53} -291.265 q^{55} -349.062 q^{57} +331.015 q^{59} -524.678 q^{61} -230.540 q^{63} -63.3996 q^{65} +968.172 q^{67} -271.568 q^{69} -8.39776 q^{71} -903.329 q^{73} -303.648 q^{75} -1529.85 q^{77} +1157.98 q^{79} +81.0000 q^{81} +952.371 q^{83} +368.095 q^{85} -561.784 q^{87} +1059.96 q^{89} -333.002 q^{91} -675.187 q^{93} +567.447 q^{95} -90.7689 q^{97} +537.511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 18 q^{5} - 10 q^{7} + 18 q^{9} + 4 q^{11} + 26 q^{13} - 54 q^{15} - 52 q^{17} - 142 q^{19} - 30 q^{21} - 280 q^{23} - 54 q^{25} + 54 q^{27} - 424 q^{29} - 178 q^{31} + 12 q^{33} - 80 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 18 * q^5 - 10 * q^7 + 18 * q^9 + 4 * q^11 + 26 * q^13 - 54 * q^15 - 52 * q^17 - 142 * q^19 - 30 * q^21 - 280 * q^23 - 54 * q^25 + 54 * q^27 - 424 * q^29 - 178 * q^31 + 12 * q^33 - 80 * q^35 + 136 * q^37 + 78 * q^39 - 226 * q^41 - 72 * q^43 - 162 * q^45 + 176 * q^47 + 214 * q^49 - 156 * q^51 - 788 * q^53 + 440 * q^55 - 426 * q^57 + 728 * q^59 - 736 * q^61 - 90 * q^63 - 234 * q^65 + 1054 * q^67 - 840 * q^69 - 660 * q^71 + 24 * q^73 - 162 * q^75 - 2400 * q^77 + 40 * q^79 + 162 * q^81 + 1344 * q^83 + 60 * q^85 - 1272 * q^87 + 314 * q^89 - 130 * q^91 - 534 * q^93 + 904 * q^95 - 264 * q^97 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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