LMFDB - Embedded newform 1440.4.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(1,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, 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("1440.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.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:	$84.9627504083$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +16.1245 q^{7} +16.1245 q^{11} -8.00000 q^{13} -26.0000 q^{17} -96.7471 q^{19} -128.996 q^{23} +25.0000 q^{25} -54.0000 q^{29} -128.996 q^{31} +80.6226 q^{35} -124.000 q^{37} -152.000 q^{41} -193.494 q^{43} -64.4981 q^{47} -83.0000 q^{49} -78.0000 q^{53} +80.6226 q^{55} +596.607 q^{59} -470.000 q^{61} -40.0000 q^{65} +741.728 q^{67} +225.743 q^{71} -162.000 q^{73} +260.000 q^{77} +128.996 q^{79} -1193.21 q^{83} -130.000 q^{85} -124.000 q^{89} -128.996 q^{91} -483.735 q^{95} -646.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} - 16 q^{13} - 52 q^{17} + 50 q^{25} - 108 q^{29} - 248 q^{37} - 304 q^{41} - 166 q^{49} - 156 q^{53} - 940 q^{61} - 80 q^{65} - 324 q^{73} + 520 q^{77} - 260 q^{85} - 248 q^{89} - 1292 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 - 16 * q^13 - 52 * q^17 + 50 * q^25 - 108 * q^29 - 248 * q^37 - 304 * q^41 - 166 * q^49 - 156 * q^53 - 940 * q^61 - 80 * q^65 - 324 * q^73 + 520 * q^77 - 260 * q^85 - 248 * q^89 - 1292 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	16.1245	0.870642	0.435321	−	0.900275i	$-0.356635\pi$
$7$	16.1245	0.870642	0.435321	+	0.900275i	$0.356635\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.1245	0.441975	0.220987	−	0.975277i	$-0.429072\pi$
$11$	16.1245	0.441975	0.220987	+	0.975277i	$0.429072\pi$
$12$	0	0
$13$	−8.00000	−0.170677	−0.0853385	−	0.996352i	$-0.527197\pi$
$13$	−8.00000	−0.170677	−0.0853385	+	0.996352i	$0.527197\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.0000	−0.370937	−0.185468	−	0.982650i	$-0.559380\pi$
$17$	−26.0000	−0.370937	−0.185468	+	0.982650i	$0.559380\pi$
$18$	0	0
$19$	−96.7471	−1.16817	−0.584087	−	0.811691i	$-0.698547\pi$
$19$	−96.7471	−1.16817	−0.584087	+	0.811691i	$0.698547\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−128.996	−1.16946	−0.584729	−	0.811228i	$-0.698799\pi$
$23$	−128.996	−1.16946	−0.584729	+	0.811228i	$0.698799\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	−128.996	−0.747367	−0.373684	−	0.927556i	$-0.621905\pi$
$31$	−128.996	−0.747367	−0.373684	+	0.927556i	$0.621905\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	80.6226	0.389363
$36$	0	0
$37$	−124.000	−0.550959	−0.275479	−	0.961307i	$-0.588837\pi$
$37$	−124.000	−0.550959	−0.275479	+	0.961307i	$0.588837\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−152.000	−0.578986	−0.289493	−	0.957180i	$-0.593487\pi$
$41$	−152.000	−0.578986	−0.289493	+	0.957180i	$0.593487\pi$
$42$	0	0
$43$	−193.494	−0.686223	−0.343111	−	0.939295i	$-0.611481\pi$
$43$	−193.494	−0.686223	−0.343111	+	0.939295i	$0.611481\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−64.4981	−0.200170	−0.100085	−	0.994979i	$-0.531912\pi$
$47$	−64.4981	−0.200170	−0.100085	+	0.994979i	$0.531912\pi$
$48$	0	0
$49$	−83.0000	−0.241983
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−78.0000	−0.202153	−0.101077	−	0.994879i	$-0.532229\pi$
$53$	−78.0000	−0.202153	−0.101077	+	0.994879i	$0.532229\pi$
$54$	0	0
$55$	80.6226	0.197657
$56$	0	0
$57$	0	0
$58$	0	0
$59$	596.607	1.31647	0.658234	−	0.752813i	$-0.271304\pi$
$59$	596.607	1.31647	0.658234	+	0.752813i	$0.271304\pi$
$60$	0	0
$61$	−470.000	−0.986514	−0.493257	−	0.869884i	$-0.664194\pi$
$61$	−470.000	−0.986514	−0.493257	+	0.869884i	$0.664194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−40.0000	−0.0763291
$66$	0	0
$67$	741.728	1.35248	0.676242	−	0.736679i	$-0.263607\pi$
$67$	741.728	1.35248	0.676242	+	0.736679i	$0.263607\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	225.743	0.377335	0.188668	−	0.982041i	$-0.439583\pi$
$71$	225.743	0.377335	0.188668	+	0.982041i	$0.439583\pi$
$72$	0	0
$73$	−162.000	−0.259735	−0.129868	−	0.991531i	$-0.541455\pi$
$73$	−162.000	−0.259735	−0.129868	+	0.991531i	$0.541455\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	260.000	0.384802
$78$	0	0
$79$	128.996	0.183711	0.0918557	−	0.995772i	$-0.470720\pi$
$79$	128.996	0.183711	0.0918557	+	0.995772i	$0.470720\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1193.21	−1.57798	−0.788990	−	0.614407i	$-0.789395\pi$
$83$	−1193.21	−1.57798	−0.788990	+	0.614407i	$0.789395\pi$
$84$	0	0
$85$	−130.000	−0.165888
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−124.000	−0.147685	−0.0738425	−	0.997270i	$-0.523526\pi$
$89$	−124.000	−0.147685	−0.0738425	+	0.997270i	$0.523526\pi$
$90$	0	0
$91$	−128.996	−0.148599
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−483.735	−0.522423
$96$	0	0
$97$	−646.000	−0.676200	−0.338100	−	0.941110i	$-0.609784\pi$
$97$	−646.000	−0.676200	−0.338100	+	0.941110i	$0.609784\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	530.000	0.522148	0.261074	−	0.965319i	$-0.415923\pi$
$101$	530.000	0.522148	0.261074	+	0.965319i	$0.415923\pi$
$102$	0	0
$103$	983.595	0.940937	0.470469	−	0.882417i	$-0.344085\pi$
$103$	983.595	0.940937	0.470469	+	0.882417i	$0.344085\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	935.222	0.844966	0.422483	−	0.906371i	$-0.361159\pi$
$107$	935.222	0.844966	0.422483	+	0.906371i	$0.361159\pi$
$108$	0	0
$109$	−1374.00	−1.20739	−0.603694	−	0.797216i	$-0.706305\pi$
$109$	−1374.00	−1.20739	−0.603694	+	0.797216i	$0.706305\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−158.000	−0.131534	−0.0657672	−	0.997835i	$-0.520949\pi$
$113$	−158.000	−0.131534	−0.0657672	+	0.997835i	$0.520949\pi$
$114$	0	0
$115$	−644.981	−0.522998
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−419.237	−0.322953
$120$	0	0
$121$	−1071.00	−0.804658
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1338.33	−0.935102	−0.467551	−	0.883966i	$-0.654864\pi$
$127$	−1338.33	−0.935102	−0.467551	+	0.883966i	$0.654864\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	951.346	0.634500	0.317250	−	0.948342i	$-0.397241\pi$
$131$	951.346	0.634500	0.317250	+	0.948342i	$0.397241\pi$
$132$	0	0
$133$	−1560.00	−1.01706
$134$	0	0
$135$	0	0
$136$	0	0
$137$	474.000	0.295595	0.147798	−	0.989018i	$-0.452782\pi$
$137$	474.000	0.295595	0.147798	+	0.989018i	$0.452782\pi$
$138$	0	0
$139$	1580.20	0.964252	0.482126	−	0.876102i	$-0.339865\pi$
$139$	1580.20	0.964252	0.482126	+	0.876102i	$0.339865\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−128.996	−0.0754349
$144$	0	0
$145$	−270.000	−0.154636
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1270.00	0.698272	0.349136	−	0.937072i	$-0.386475\pi$
$149$	1270.00	0.698272	0.349136	+	0.937072i	$0.386475\pi$
$150$	0	0
$151$	−290.241	−0.156421	−0.0782103	−	0.996937i	$-0.524921\pi$
$151$	−290.241	−0.156421	−0.0782103	+	0.996937i	$0.524921\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−644.981	−0.334233
$156$	0	0
$157$	−2496.00	−1.26881	−0.634403	−	0.773003i	$-0.718754\pi$
$157$	−2496.00	−1.26881	−0.634403	+	0.773003i	$0.718754\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2080.00	−1.01818
$162$	0	0
$163$	−999.720	−0.480393	−0.240197	−	0.970724i	$-0.577212\pi$
$163$	−999.720	−0.480393	−0.240197	+	0.970724i	$0.577212\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	644.981	0.298863	0.149431	−	0.988772i	$-0.452256\pi$
$167$	644.981	0.298863	0.149431	+	0.988772i	$0.452256\pi$
$168$	0	0
$169$	−2133.00	−0.970869
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2018.00	0.886854	0.443427	−	0.896311i	$-0.353763\pi$
$173$	2018.00	0.886854	0.443427	+	0.896311i	$0.353763\pi$
$174$	0	0
$175$	403.113	0.174128
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2886.29	−1.20520	−0.602601	−	0.798042i	$-0.705869\pi$
$179$	−2886.29	−1.20520	−0.602601	+	0.798042i	$0.705869\pi$
$180$	0	0
$181$	−1342.00	−0.551105	−0.275553	−	0.961286i	$-0.588861\pi$
$181$	−1342.00	−0.551105	−0.275553	+	0.961286i	$0.588861\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−620.000	−0.246396
$186$	0	0
$187$	−419.237	−0.163945
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4160.12	1.57600	0.788000	−	0.615675i	$-0.211117\pi$
$191$	4160.12	1.57600	0.788000	+	0.615675i	$0.211117\pi$
$192$	0	0
$193$	−4362.00	−1.62686	−0.813429	−	0.581664i	$-0.802402\pi$
$193$	−4362.00	−1.62686	−0.813429	+	0.581664i	$0.802402\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1374.00	0.496921	0.248461	−	0.968642i	$-0.420075\pi$
$197$	1374.00	0.496921	0.248461	+	0.968642i	$0.420075\pi$
$198$	0	0
$199$	−1902.69	−0.677780	−0.338890	−	0.940826i	$-0.610052\pi$
$199$	−1902.69	−0.677780	−0.338890	+	0.940826i	$0.610052\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−870.724	−0.301048
$204$	0	0
$205$	−760.000	−0.258930
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1560.00	−0.516304
$210$	0	0
$211$	−1838.19	−0.599747	−0.299873	−	0.953979i	$-0.596944\pi$
$211$	−1838.19	−0.599747	−0.299873	+	0.953979i	$0.596944\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−967.471	−0.306888
$216$	0	0
$217$	−2080.00	−0.650689
$218$	0	0
$219$	0	0
$220$	0	0
$221$	208.000	0.0633104
$222$	0	0
$223$	−1402.83	−0.421258	−0.210629	−	0.977566i	$-0.567551\pi$
$223$	−1402.83	−0.421258	−0.210629	+	0.977566i	$0.567551\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−322.490	−0.0942927	−0.0471463	−	0.998888i	$-0.515013\pi$
$227$	−322.490	−0.0942927	−0.0471463	+	0.998888i	$0.515013\pi$
$228$	0	0
$229$	−1450.00	−0.418422	−0.209211	−	0.977870i	$-0.567090\pi$
$229$	−1450.00	−0.418422	−0.209211	+	0.977870i	$0.567090\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3442.00	−0.967781	−0.483890	−	0.875129i	$-0.660777\pi$
$233$	−3442.00	−0.967781	−0.483890	+	0.875129i	$0.660777\pi$
$234$	0	0
$235$	−322.490	−0.0895189
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3321.65	0.898995	0.449497	−	0.893282i	$-0.351603\pi$
$239$	3321.65	0.898995	0.449497	+	0.893282i	$0.351603\pi$
$240$	0	0
$241$	−4570.00	−1.22149	−0.610746	−	0.791827i	$-0.709130\pi$
$241$	−4570.00	−1.22149	−0.610746	+	0.791827i	$0.709130\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−415.000	−0.108218
$246$	0	0
$247$	773.977	0.199380
$248$	0	0
$249$	0	0
$250$	0	0
$251$	693.354	0.174359	0.0871795	−	0.996193i	$-0.472215\pi$
$251$	693.354	0.174359	0.0871795	+	0.996193i	$0.472215\pi$
$252$	0	0
$253$	−2080.00	−0.516871
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3214.00	−0.780093	−0.390046	−	0.920795i	$-0.627541\pi$
$257$	−3214.00	−0.780093	−0.390046	+	0.920795i	$0.627541\pi$
$258$	0	0
$259$	−1999.44	−0.479688
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2966.91	−0.695618	−0.347809	−	0.937565i	$-0.613074\pi$
$263$	−2966.91	−0.695618	−0.347809	+	0.937565i	$0.613074\pi$
$264$	0	0
$265$	−390.000	−0.0904057
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3330.00	0.754772	0.377386	−	0.926056i	$-0.376823\pi$
$269$	3330.00	0.754772	0.377386	+	0.926056i	$0.376823\pi$
$270$	0	0
$271$	8610.49	1.93007	0.965037	−	0.262113i	$-0.0844195\pi$
$271$	8610.49	1.93007	0.965037	+	0.262113i	$0.0844195\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	403.113	0.0883950
$276$	0	0
$277$	−1924.00	−0.417336	−0.208668	−	0.977987i	$-0.566913\pi$
$277$	−1924.00	−0.417336	−0.208668	+	0.977987i	$0.566913\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4908.00	−1.04195	−0.520973	−	0.853573i	$-0.674431\pi$
$281$	−4908.00	−1.04195	−0.520973	+	0.853573i	$0.674431\pi$
$282$	0	0
$283$	4740.61	0.995759	0.497880	−	0.867246i	$-0.334112\pi$
$283$	4740.61	0.995759	0.497880	+	0.867246i	$0.334112\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2450.93	−0.504089
$288$	0	0
$289$	−4237.00	−0.862406
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2542.00	0.506844	0.253422	−	0.967356i	$-0.418444\pi$
$293$	2542.00	0.506844	0.253422	+	0.967356i	$0.418444\pi$
$294$	0	0
$295$	2983.04	0.588742
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1031.97	0.199600
$300$	0	0
$301$	−3120.00	−0.597455
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2350.00	−0.441182
$306$	0	0
$307$	6449.81	1.19905	0.599527	−	0.800354i	$-0.295355\pi$
$307$	6449.81	1.19905	0.599527	+	0.800354i	$0.295355\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10222.9	−1.86395	−0.931977	−	0.362517i	$-0.881918\pi$
$311$	−10222.9	−1.86395	−0.931977	+	0.362517i	$0.881918\pi$
$312$	0	0
$313$	−1482.00	−0.267628	−0.133814	−	0.991006i	$-0.542722\pi$
$313$	−1482.00	−0.267628	−0.133814	+	0.991006i	$0.542722\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10014.0	1.77427	0.887133	−	0.461514i	$-0.152694\pi$
$317$	10014.0	1.77427	0.887133	+	0.461514i	$0.152694\pi$
$318$	0	0
$319$	−870.724	−0.152825
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2515.42	0.433319
$324$	0	0
$325$	−200.000	−0.0341354
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1040.00	−0.174277
$330$	0	0
$331$	−6224.06	−1.03355	−0.516776	−	0.856121i	$-0.672868\pi$
$331$	−6224.06	−1.03355	−0.516776	+	0.856121i	$0.672868\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3708.64	0.604849
$336$	0	0
$337$	−3546.00	−0.573184	−0.286592	−	0.958053i	$-0.592522\pi$
$337$	−3546.00	−0.573184	−0.286592	+	0.958053i	$0.592522\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2080.00	−0.330318
$342$	0	0
$343$	−6869.04	−1.08132
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5611.33	0.868104	0.434052	−	0.900888i	$-0.357083\pi$
$347$	5611.33	0.868104	0.434052	+	0.900888i	$0.357083\pi$
$348$	0	0
$349$	−1710.00	−0.262276	−0.131138	−	0.991364i	$-0.541863\pi$
$349$	−1710.00	−0.262276	−0.131138	+	0.991364i	$0.541863\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2058.00	−0.310301	−0.155151	−	0.987891i	$-0.549586\pi$
$353$	−2058.00	−0.310301	−0.155151	+	0.987891i	$0.549586\pi$
$354$	0	0
$355$	1128.72	0.168749
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3805.39	0.559444	0.279722	−	0.960081i	$-0.409758\pi$
$359$	3805.39	0.559444	0.279722	+	0.960081i	$0.409758\pi$
$360$	0	0
$361$	2501.00	0.364630
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−810.000	−0.116157
$366$	0	0
$367$	−8949.11	−1.27286	−0.636430	−	0.771335i	$-0.719590\pi$
$367$	−8949.11	−1.27286	−0.636430	+	0.771335i	$0.719590\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1257.71	−0.176003
$372$	0	0
$373$	2392.00	0.332046	0.166023	−	0.986122i	$-0.446907\pi$
$373$	2392.00	0.332046	0.166023	+	0.986122i	$0.446907\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	432.000	0.0590163
$378$	0	0
$379$	3708.64	0.502638	0.251319	−	0.967904i	$-0.419136\pi$
$379$	3708.64	0.502638	0.251319	+	0.967904i	$0.419136\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	257.992	0.0344198	0.0172099	−	0.999852i	$-0.494522\pi$
$383$	257.992	0.0344198	0.0172099	+	0.999852i	$0.494522\pi$
$384$	0	0
$385$	1300.00	0.172089
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9750.00	1.27081	0.635404	−	0.772180i	$-0.280833\pi$
$389$	9750.00	1.27081	0.635404	+	0.772180i	$0.280833\pi$
$390$	0	0
$391$	3353.90	0.433795
$392$	0	0
$393$	0	0
$394$	0	0
$395$	644.981	0.0821582
$396$	0	0
$397$	−2164.00	−0.273572	−0.136786	−	0.990601i	$-0.543677\pi$
$397$	−2164.00	−0.273572	−0.136786	+	0.990601i	$0.543677\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12500.0	−1.55666	−0.778329	−	0.627856i	$-0.783933\pi$
$401$	−12500.0	−1.55666	−0.778329	+	0.627856i	$0.783933\pi$
$402$	0	0
$403$	1031.97	0.127558
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1999.44	−0.243510
$408$	0	0
$409$	−646.000	−0.0780994	−0.0390497	−	0.999237i	$-0.512433\pi$
$409$	−646.000	−0.0780994	−0.0390497	+	0.999237i	$0.512433\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9620.00	1.14617
$414$	0	0
$415$	−5966.07	−0.705694
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3079.78	0.359086	0.179543	−	0.983750i	$-0.442538\pi$
$419$	3079.78	0.359086	0.179543	+	0.983750i	$0.442538\pi$
$420$	0	0
$421$	230.000	0.0266259	0.0133130	−	0.999911i	$-0.495762\pi$
$421$	230.000	0.0266259	0.0133130	+	0.999911i	$0.495762\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−650.000	−0.0741874
$426$	0	0
$427$	−7578.52	−0.858900
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2192.93	0.245081	0.122541	−	0.992464i	$-0.460896\pi$
$431$	2192.93	0.245081	0.122541	+	0.992464i	$0.460896\pi$
$432$	0	0
$433$	5902.00	0.655039	0.327520	−	0.944844i	$-0.393787\pi$
$433$	5902.00	0.655039	0.327520	+	0.944844i	$0.393787\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12480.0	1.36613
$438$	0	0
$439$	1902.69	0.206858	0.103429	−	0.994637i	$-0.467019\pi$
$439$	1902.69	0.206858	0.103429	+	0.994637i	$0.467019\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6062.82	0.650233	0.325116	−	0.945674i	$-0.394597\pi$
$443$	6062.82	0.650233	0.325116	+	0.945674i	$0.394597\pi$
$444$	0	0
$445$	−620.000	−0.0660468
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7820.00	−0.821935	−0.410967	−	0.911650i	$-0.634809\pi$
$449$	−7820.00	−0.821935	−0.410967	+	0.911650i	$0.634809\pi$
$450$	0	0
$451$	−2450.93	−0.255897
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−644.981	−0.0664553
$456$	0	0
$457$	12146.0	1.24325	0.621626	−	0.783314i	$-0.286472\pi$
$457$	12146.0	1.24325	0.621626	+	0.783314i	$0.286472\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	750.000	0.0757722	0.0378861	−	0.999282i	$-0.487938\pi$
$461$	750.000	0.0757722	0.0378861	+	0.999282i	$0.487938\pi$
$462$	0	0
$463$	8755.61	0.878851	0.439425	−	0.898279i	$-0.355182\pi$
$463$	8755.61	0.878851	0.439425	+	0.898279i	$0.355182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16479.3	−1.63291	−0.816455	−	0.577409i	$-0.804064\pi$
$467$	−16479.3	−1.63291	−0.816455	+	0.577409i	$0.804064\pi$
$468$	0	0
$469$	11960.0	1.17753
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3120.00	−0.303293
$474$	0	0
$475$	−2418.68	−0.233635
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17317.7	−1.65191	−0.825957	−	0.563732i	$-0.809365\pi$
$479$	−17317.7	−1.65191	−0.825957	+	0.563732i	$0.809365\pi$
$480$	0	0
$481$	992.000	0.0940360
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3230.00	−0.302406
$486$	0	0
$487$	112.872	0.0105025	0.00525123	−	0.999986i	$-0.498328\pi$
$487$	112.872	0.0105025	0.00525123	+	0.999986i	$0.498328\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10271.3	−0.944069	−0.472035	−	0.881580i	$-0.656480\pi$
$491$	−10271.3	−0.944069	−0.472035	+	0.881580i	$0.656480\pi$
$492$	0	0
$493$	1404.00	0.128262
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3640.00	0.328524
$498$	0	0
$499$	−2612.17	−0.234342	−0.117171	−	0.993112i	$-0.537383\pi$
$499$	−2612.17	−0.234342	−0.117171	+	0.993112i	$0.537383\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21090.9	1.86957	0.934787	−	0.355210i	$-0.115591\pi$
$503$	21090.9	1.86957	0.934787	+	0.355210i	$0.115591\pi$
$504$	0	0
$505$	2650.00	0.233512
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11834.0	1.03052	0.515258	−	0.857035i	$-0.327696\pi$
$509$	11834.0	1.03052	0.515258	+	0.857035i	$0.327696\pi$
$510$	0	0
$511$	−2612.17	−0.226136
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4917.98	0.420800
$516$	0	0
$517$	−1040.00	−0.0884703
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17860.0	−1.50184	−0.750922	−	0.660391i	$-0.770391\pi$
$521$	−17860.0	−1.50184	−0.750922	+	0.660391i	$0.770391\pi$
$522$	0	0
$523$	−5321.09	−0.444885	−0.222443	−	0.974946i	$-0.571403\pi$
$523$	−5321.09	−0.444885	−0.222443	+	0.974946i	$0.571403\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3353.90	0.277226
$528$	0	0
$529$	4473.00	0.367634
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1216.00	0.0988195
$534$	0	0
$535$	4676.11	0.377880
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1338.33	−0.106950
$540$	0	0
$541$	11642.0	0.925192	0.462596	−	0.886569i	$-0.346918\pi$
$541$	11642.0	0.925192	0.462596	+	0.886569i	$0.346918\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6870.00	−0.539960
$546$	0	0
$547$	11416.2	0.892358	0.446179	−	0.894944i	$-0.352785\pi$
$547$	11416.2	0.892358	0.446179	+	0.894944i	$0.352785\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5224.34	0.403928
$552$	0	0
$553$	2080.00	0.159947
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20026.0	1.52339	0.761696	−	0.647935i	$-0.224367\pi$
$557$	20026.0	1.52339	0.761696	+	0.647935i	$0.224367\pi$
$558$	0	0
$559$	1547.95	0.117122
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7868.76	0.589039	0.294519	−	0.955646i	$-0.404840\pi$
$563$	7868.76	0.589039	0.294519	+	0.955646i	$0.404840\pi$
$564$	0	0
$565$	−790.000	−0.0588240
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5060.00	0.372805	0.186403	−	0.982473i	$-0.440317\pi$
$569$	5060.00	0.372805	0.186403	+	0.982473i	$0.440317\pi$
$570$	0	0
$571$	−1709.20	−0.125267	−0.0626337	−	0.998037i	$-0.519950\pi$
$571$	−1709.20	−0.125267	−0.0626337	+	0.998037i	$0.519950\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3224.90	−0.233892
$576$	0	0
$577$	23794.0	1.71674	0.858368	−	0.513034i	$-0.171478\pi$
$577$	23794.0	1.71674	0.858368	+	0.513034i	$0.171478\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−19240.0	−1.37385
$582$	0	0
$583$	−1257.71	−0.0893467
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1064.22	0.0748296	0.0374148	−	0.999300i	$-0.488088\pi$
$587$	1064.22	0.0748296	0.0374148	+	0.999300i	$0.488088\pi$
$588$	0	0
$589$	12480.0	0.873055
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14502.0	−1.00426	−0.502130	−	0.864792i	$-0.667450\pi$
$593$	−14502.0	−1.00426	−0.502130	+	0.864792i	$0.667450\pi$
$594$	0	0
$595$	−2096.19	−0.144429
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6127.32	−0.417955	−0.208978	−	0.977920i	$-0.567014\pi$
$599$	−6127.32	−0.417955	−0.208978	+	0.977920i	$0.567014\pi$
$600$	0	0
$601$	13910.0	0.944094	0.472047	−	0.881573i	$-0.343515\pi$
$601$	13910.0	0.944094	0.472047	+	0.881573i	$0.343515\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5355.00	−0.359854
$606$	0	0
$607$	11529.0	0.770921	0.385460	−	0.922724i	$-0.374043\pi$
$607$	11529.0	0.770921	0.385460	+	0.922724i	$0.374043\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	515.984	0.0341645
$612$	0	0
$613$	13108.0	0.863666	0.431833	−	0.901954i	$-0.357867\pi$
$613$	13108.0	0.863666	0.431833	+	0.901954i	$0.357867\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13334.0	0.870027	0.435013	−	0.900424i	$-0.356744\pi$
$617$	13334.0	0.870027	0.435013	+	0.900424i	$0.356744\pi$
$618$	0	0
$619$	16027.8	1.04073	0.520364	−	0.853945i	$-0.325796\pi$
$619$	16027.8	1.04073	0.520364	+	0.853945i	$0.325796\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1999.44	−0.128581
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3224.00	0.204371
$630$	0	0
$631$	−10255.2	−0.646993	−0.323497	−	0.946229i	$-0.604858\pi$
$631$	−10255.2	−0.646993	−0.323497	+	0.946229i	$0.604858\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6691.67	−0.418191
$636$	0	0
$637$	664.000	0.0413008
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28288.0	−1.74307	−0.871536	−	0.490332i	$-0.836876\pi$
$641$	−28288.0	−1.74307	−0.871536	+	0.490332i	$0.836876\pi$
$642$	0	0
$643$	225.743	0.0138452	0.00692258	−	0.999976i	$-0.497796\pi$
$643$	225.743	0.0138452	0.00692258	+	0.999976i	$0.497796\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12448.1	0.756393	0.378197	−	0.925725i	$-0.376544\pi$
$647$	12448.1	0.756393	0.378197	+	0.925725i	$0.376544\pi$
$648$	0	0
$649$	9620.00	0.581846
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11542.0	0.691690	0.345845	−	0.938292i	$-0.387592\pi$
$653$	11542.0	0.691690	0.345845	+	0.938292i	$0.387592\pi$
$654$	0	0
$655$	4756.73	0.283757
$656$	0	0
$657$	0	0
$658$	0	0
$659$	628.856	0.0371726	0.0185863	−	0.999827i	$-0.494083\pi$
$659$	628.856	0.0371726	0.0185863	+	0.999827i	$0.494083\pi$
$660$	0	0
$661$	13930.0	0.819689	0.409844	−	0.912155i	$-0.365583\pi$
$661$	13930.0	0.819689	0.409844	+	0.912155i	$0.365583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7800.00	−0.454844
$666$	0	0
$667$	6965.79	0.404373
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7578.52	−0.436014
$672$	0	0
$673$	7862.00	0.450309	0.225154	−	0.974323i	$-0.427711\pi$
$673$	7862.00	0.450309	0.225154	+	0.974323i	$0.427711\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6526.00	0.370479	0.185240	−	0.982693i	$-0.440694\pi$
$677$	6526.00	0.370479	0.185240	+	0.982693i	$0.440694\pi$
$678$	0	0
$679$	−10416.4	−0.588728
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20284.6	−1.13641	−0.568207	−	0.822886i	$-0.692363\pi$
$683$	−20284.6	−1.13641	−0.568207	+	0.822886i	$0.692363\pi$
$684$	0	0
$685$	2370.00	0.132194
$686$	0	0
$687$	0	0
$688$	0	0
$689$	624.000	0.0345029
$690$	0	0
$691$	18736.7	1.03152	0.515758	−	0.856734i	$-0.327510\pi$
$691$	18736.7	1.03152	0.515758	+	0.856734i	$0.327510\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7901.01	0.431227
$696$	0	0
$697$	3952.00	0.214767
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8398.00	−0.452479	−0.226240	−	0.974072i	$-0.572643\pi$
$701$	−8398.00	−0.452479	−0.226240	+	0.974072i	$0.572643\pi$
$702$	0	0
$703$	11996.6	0.643616
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8545.99	0.454604
$708$	0	0
$709$	10550.0	0.558834	0.279417	−	0.960170i	$-0.409859\pi$
$709$	10550.0	0.558834	0.279417	+	0.960170i	$0.409859\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16640.0	0.874015
$714$	0	0
$715$	−644.981	−0.0337355
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19671.9	1.02036	0.510180	−	0.860068i	$-0.329579\pi$
$719$	19671.9	1.02036	0.510180	+	0.860068i	$0.329579\pi$
$720$	0	0
$721$	15860.0	0.819220
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1350.00	−0.0691555
$726$	0	0
$727$	11851.5	0.604606	0.302303	−	0.953212i	$-0.402245\pi$
$727$	11851.5	0.604606	0.302303	+	0.953212i	$0.402245\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5030.85	0.254545
$732$	0	0
$733$	16948.0	0.854009	0.427005	−	0.904249i	$-0.359569\pi$
$733$	16948.0	0.854009	0.427005	+	0.904249i	$0.359569\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11960.0	0.597764
$738$	0	0
$739$	−36989.6	−1.84125	−0.920626	−	0.390445i	$-0.872321\pi$
$739$	−36989.6	−1.84125	−0.920626	+	0.390445i	$0.872321\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11093.7	−0.547762	−0.273881	−	0.961764i	$-0.588307\pi$
$743$	−11093.7	−0.547762	−0.273881	+	0.961764i	$0.588307\pi$
$744$	0	0
$745$	6350.00	0.312277
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15080.0	0.735662
$750$	0	0
$751$	−24186.8	−1.17522	−0.587608	−	0.809145i	$-0.699930\pi$
$751$	−24186.8	−1.17522	−0.587608	+	0.809145i	$0.699930\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1451.21	−0.0699534
$756$	0	0
$757$	12716.0	0.610530	0.305265	−	0.952268i	$-0.401255\pi$
$757$	12716.0	0.610530	0.305265	+	0.952268i	$0.401255\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20300.0	−0.966983	−0.483492	−	0.875349i	$-0.660632\pi$
$761$	−20300.0	−0.966983	−0.483492	+	0.875349i	$0.660632\pi$
$762$	0	0
$763$	−22155.1	−1.05120
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4772.86	−0.224691
$768$	0	0
$769$	−15570.0	−0.730128	−0.365064	−	0.930982i	$-0.618953\pi$
$769$	−15570.0	−0.730128	−0.365064	+	0.930982i	$0.618953\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	438.000	0.0203800	0.0101900	−	0.999948i	$-0.496756\pi$
$773$	438.000	0.0203800	0.0101900	+	0.999948i	$0.496756\pi$
$774$	0	0
$775$	−3224.90	−0.149473
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14705.6	0.676356
$780$	0	0
$781$	3640.00	0.166773
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12480.0	−0.567427
$786$	0	0
$787$	−27572.9	−1.24888	−0.624440	−	0.781073i	$-0.714673\pi$
$787$	−27572.9	−1.24888	−0.624440	+	0.781073i	$0.714673\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2547.67	−0.114519
$792$	0	0
$793$	3760.00	0.168375
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31746.0	1.41092	0.705459	−	0.708751i	$-0.250741\pi$
$797$	31746.0	1.41092	0.705459	+	0.708751i	$0.250741\pi$
$798$	0	0
$799$	1676.95	0.0742506
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2612.17	−0.114796
$804$	0	0
$805$	−10400.0	−0.455344
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35904.0	1.56034	0.780171	−	0.625566i	$-0.215132\pi$
$809$	35904.0	1.56034	0.780171	+	0.625566i	$0.215132\pi$
$810$	0	0
$811$	18994.7	0.822433	0.411217	−	0.911538i	$-0.365104\pi$
$811$	18994.7	0.822433	0.411217	+	0.911538i	$0.365104\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4998.60	−0.214838
$816$	0	0
$817$	18720.0	0.801628
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42878.0	−1.82272	−0.911360	−	0.411611i	$-0.864966\pi$
$821$	−42878.0	−1.82272	−0.911360	+	0.411611i	$0.864966\pi$
$822$	0	0
$823$	−15431.2	−0.653580	−0.326790	−	0.945097i	$-0.605967\pi$
$823$	−15431.2	−0.653580	−0.326790	+	0.945097i	$0.605967\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20574.9	−0.865125	−0.432563	−	0.901604i	$-0.642391\pi$
$827$	−20574.9	−0.865125	−0.432563	+	0.901604i	$0.642391\pi$
$828$	0	0
$829$	36634.0	1.53480	0.767401	−	0.641167i	$-0.221549\pi$
$829$	36634.0	1.53480	0.767401	+	0.641167i	$0.221549\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2158.00	0.0897602
$834$	0	0
$835$	3224.90	0.133656
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2192.93	0.0902366	0.0451183	−	0.998982i	$-0.485634\pi$
$839$	2192.93	0.0902366	0.0451183	+	0.998982i	$0.485634\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10665.0	−0.434186
$846$	0	0
$847$	−17269.4	−0.700569
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15995.5	0.644324
$852$	0	0
$853$	9028.00	0.362383	0.181192	−	0.983448i	$-0.442005\pi$
$853$	9028.00	0.362383	0.181192	+	0.983448i	$0.442005\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44886.0	1.78912	0.894560	−	0.446947i	$-0.147489\pi$
$857$	44886.0	1.78912	0.894560	+	0.446947i	$0.147489\pi$
$858$	0	0
$859$	24864.0	0.987601	0.493800	−	0.869575i	$-0.335607\pi$
$859$	24864.0	0.987601	0.493800	+	0.869575i	$0.335607\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47599.6	−1.87753	−0.938765	−	0.344558i	$-0.888029\pi$
$863$	−47599.6	−1.87753	−0.938765	+	0.344558i	$0.888029\pi$
$864$	0	0
$865$	10090.0	0.396613
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2080.00	0.0811958
$870$	0	0
$871$	−5933.82	−0.230838
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2015.56	0.0778726
$876$	0	0
$877$	−25016.0	−0.963204	−0.481602	−	0.876390i	$-0.659945\pi$
$877$	−25016.0	−0.963204	−0.481602	+	0.876390i	$0.659945\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35532.0	−1.35880	−0.679400	−	0.733768i	$-0.737760\pi$
$881$	−35532.0	−1.35880	−0.679400	+	0.733768i	$0.737760\pi$
$882$	0	0
$883$	29249.9	1.11476	0.557382	−	0.830256i	$-0.311806\pi$
$883$	29249.9	1.11476	0.557382	+	0.830256i	$0.311806\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13609.1	−0.515162	−0.257581	−	0.966257i	$-0.582925\pi$
$887$	−13609.1	−0.515162	−0.257581	+	0.966257i	$0.582925\pi$
$888$	0	0
$889$	−21580.0	−0.814139
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6240.00	0.233834
$894$	0	0
$895$	−14431.4	−0.538983
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6965.79	0.258423
$900$	0	0
$901$	2028.00	0.0749861
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6710.00	−0.246462
$906$	0	0
$907$	−24186.8	−0.885456	−0.442728	−	0.896656i	$-0.645989\pi$
$907$	−24186.8	−0.885456	−0.442728	+	0.896656i	$0.645989\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22574.3	0.820989	0.410494	−	0.911863i	$-0.365356\pi$
$911$	22574.3	0.820989	0.410494	+	0.911863i	$0.365356\pi$
$912$	0	0
$913$	−19240.0	−0.697427
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15340.0	0.552422
$918$	0	0
$919$	48760.5	1.75023	0.875115	−	0.483915i	$-0.160786\pi$
$919$	48760.5	1.75023	0.875115	+	0.483915i	$0.160786\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1805.95	−0.0644024
$924$	0	0
$925$	−3100.00	−0.110192
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2500.00	0.0882910	0.0441455	−	0.999025i	$-0.485943\pi$
$929$	2500.00	0.0882910	0.0441455	+	0.999025i	$0.485943\pi$
$930$	0	0
$931$	8030.01	0.282678
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2096.19	−0.0733183
$936$	0	0
$937$	25114.0	0.875602	0.437801	−	0.899072i	$-0.355758\pi$
$937$	25114.0	0.875602	0.437801	+	0.899072i	$0.355758\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29090.0	−1.00777	−0.503883	−	0.863772i	$-0.668096\pi$
$941$	−29090.0	−1.00777	−0.503883	+	0.863772i	$0.668096\pi$
$942$	0	0
$943$	19607.4	0.677100
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56693.8	1.94541	0.972704	−	0.232051i	$-0.0745435\pi$
$947$	56693.8	1.94541	0.972704	+	0.232051i	$0.0745435\pi$
$948$	0	0
$949$	1296.00	0.0443308
$950$	0	0
$951$	0	0
$952$	0	0
$953$	17202.0	0.584709	0.292354	−	0.956310i	$-0.405561\pi$
$953$	17202.0	0.584709	0.292354	+	0.956310i	$0.405561\pi$
$954$	0	0
$955$	20800.6	0.704809
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7643.02	0.257358
$960$	0	0
$961$	−13151.0	−0.441442
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−21810.0	−0.727553
$966$	0	0
$967$	3821.51	0.127085	0.0635426	−	0.997979i	$-0.479760\pi$
$967$	3821.51	0.127085	0.0635426	+	0.997979i	$0.479760\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38811.7	1.28273	0.641363	−	0.767238i	$-0.278369\pi$
$971$	38811.7	1.28273	0.641363	+	0.767238i	$0.278369\pi$
$972$	0	0
$973$	25480.0	0.839518
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29046.0	0.951140	0.475570	−	0.879678i	$-0.342242\pi$
$977$	29046.0	0.951140	0.475570	+	0.879678i	$0.342242\pi$
$978$	0	0
$979$	−1999.44	−0.0652731
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23735.3	−0.770130	−0.385065	−	0.922889i	$-0.625821\pi$
$983$	−23735.3	−0.770130	−0.385065	+	0.922889i	$0.625821\pi$
$984$	0	0
$985$	6870.00	0.222230
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24960.0	0.802509
$990$	0	0
$991$	33377.7	1.06991	0.534954	−	0.844881i	$-0.320329\pi$
$991$	33377.7	1.06991	0.534954	+	0.844881i	$0.320329\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9513.46	−0.303113
$996$	0	0
$997$	−52336.0	−1.66249	−0.831243	−	0.555910i	$-0.812370\pi$
$997$	−52336.0	−1.66249	−0.831243	+	0.555910i	$0.812370\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.bc.1.2	yes	2
3.2	odd	2		1440.4.a.u.1.2	yes	2
4.3	odd	2	inner	1440.4.a.bc.1.1	yes	2
12.11	even	2		1440.4.a.u.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.4.a.u.1.1	✓	2	12.11	even	2
1440.4.a.u.1.2	yes	2	3.2	odd	2
1440.4.a.bc.1.1	yes	2	4.3	odd	2	inner
1440.4.a.bc.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +16.1245 q^{7} +16.1245 q^{11} -8.00000 q^{13} -26.0000 q^{17} -96.7471 q^{19} -128.996 q^{23} +25.0000 q^{25} -54.0000 q^{29} -128.996 q^{31} +80.6226 q^{35} -124.000 q^{37} -152.000 q^{41} -193.494 q^{43} -64.4981 q^{47} -83.0000 q^{49} -78.0000 q^{53} +80.6226 q^{55} +596.607 q^{59} -470.000 q^{61} -40.0000 q^{65} +741.728 q^{67} +225.743 q^{71} -162.000 q^{73} +260.000 q^{77} +128.996 q^{79} -1193.21 q^{83} -130.000 q^{85} -124.000 q^{89} -128.996 q^{91} -483.735 q^{95} -646.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} - 16 q^{13} - 52 q^{17} + 50 q^{25} - 108 q^{29} - 248 q^{37} - 304 q^{41} - 166 q^{49} - 156 q^{53} - 940 q^{61} - 80 q^{65} - 324 q^{73} + 520 q^{77} - 260 q^{85} - 248 q^{89} - 1292 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 - 16 * q^13 - 52 * q^17 + 50 * q^25 - 108 * q^29 - 248 * q^37 - 304 * q^41 - 166 * q^49 - 156 * q^53 - 940 * q^61 - 80 * q^65 - 324 * q^73 + 520 * q^77 - 260 * q^85 - 248 * q^89 - 1292 * q^97

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