LMFDB - Embedded newform 1176.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,4,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, 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("1176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1176.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:	$69.3862461668$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.58461.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 65x - 126 $ x^3 - x^2 - 65*x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.37106$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -21.3959 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -21.3959 q^{5} +9.00000 q^{9} +10.7035 q^{11} +8.31878 q^{13} -64.1878 q^{15} +1.48423 q^{17} +161.519 q^{19} -123.945 q^{23} +332.786 q^{25} +27.0000 q^{27} -222.817 q^{29} +203.851 q^{31} +32.1106 q^{33} -341.224 q^{37} +24.9563 q^{39} +147.127 q^{41} +51.3401 q^{43} -192.563 q^{45} +269.510 q^{47} +4.45269 q^{51} -432.500 q^{53} -229.012 q^{55} +484.556 q^{57} +30.7341 q^{59} -592.517 q^{61} -177.988 q^{65} +414.052 q^{67} -371.836 q^{69} +294.854 q^{71} -594.063 q^{73} +998.357 q^{75} -831.386 q^{79} +81.0000 q^{81} +326.218 q^{83} -31.7564 q^{85} -668.451 q^{87} -1261.10 q^{89} +611.552 q^{93} -3455.84 q^{95} -1059.98 q^{97} +96.3319 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) $ 3 * q + 9 * q^3 - 11 * q^5 + 27 * q^9	$ 3 q + 9 q^{3} - 11 q^{5} + 27 q^{9} - 19 q^{11} - 22 q^{13} - 33 q^{15} - 104 q^{17} + 202 q^{19} - 280 q^{23} + 186 q^{25} + 81 q^{27} - 73 q^{29} - 131 q^{31} - 57 q^{33} - 326 q^{37} - 66 q^{39} - 516 q^{41} + 36 q^{43} - 99 q^{45} - 126 q^{47} - 312 q^{51} - 385 q^{53} - 611 q^{55} + 606 q^{57} + 285 q^{59} - 34 q^{61} - 920 q^{65} - 100 q^{67} - 840 q^{69} + 34 q^{71} - 108 q^{73} + 558 q^{75} - 2463 q^{79} + 243 q^{81} - 115 q^{83} - 500 q^{85} - 219 q^{87} - 110 q^{89} - 393 q^{93} - 2064 q^{95} - 2941 q^{97} - 171 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 11 * q^5 + 27 * q^9 - 19 * q^11 - 22 * q^13 - 33 * q^15 - 104 * q^17 + 202 * q^19 - 280 * q^23 + 186 * q^25 + 81 * q^27 - 73 * q^29 - 131 * q^31 - 57 * q^33 - 326 * q^37 - 66 * q^39 - 516 * q^41 + 36 * q^43 - 99 * q^45 - 126 * q^47 - 312 * q^51 - 385 * q^53 - 611 * q^55 + 606 * q^57 + 285 * q^59 - 34 * q^61 - 920 * q^65 - 100 * q^67 - 840 * q^69 + 34 * q^71 - 108 * q^73 + 558 * q^75 - 2463 * q^79 + 243 * q^81 - 115 * q^83 - 500 * q^85 - 219 * q^87 - 110 * q^89 - 393 * q^93 - 2064 * q^95 - 2941 * q^97 - 171 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−21.3959	−1.91371	−0.956855	−	0.290567i	$-0.906156\pi$
$5$	−21.3959	−1.91371	−0.956855	+	0.290567i	$0.906156\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	10.7035	0.293385	0.146693	−	0.989182i	$-0.453137\pi$
$11$	10.7035	0.293385	0.146693	+	0.989182i	$0.453137\pi$
$12$	0	0
$13$	8.31878	0.177478	0.0887390	−	0.996055i	$-0.471716\pi$
$13$	8.31878	0.177478	0.0887390	+	0.996055i	$0.471716\pi$
$14$	0	0
$15$	−64.1878	−1.10488
$16$	0	0
$17$	1.48423	0.0211752	0.0105876	−	0.999944i	$-0.496630\pi$
$17$	1.48423	0.0211752	0.0105876	+	0.999944i	$0.496630\pi$
$18$	0	0
$19$	161.519	1.95026	0.975130	−	0.221635i	$-0.0711395\pi$
$19$	161.519	1.95026	0.975130	+	0.221635i	$0.0711395\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−123.945	−1.12367	−0.561834	−	0.827250i	$-0.689904\pi$
$23$	−123.945	−1.12367	−0.561834	+	0.827250i	$0.689904\pi$
$24$	0	0
$25$	332.786	2.66228
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−222.817	−1.42676	−0.713381	−	0.700777i	$-0.752837\pi$
$29$	−222.817	−1.42676	−0.713381	+	0.700777i	$0.752837\pi$
$30$	0	0
$31$	203.851	1.18105	0.590526	−	0.807018i	$-0.298920\pi$
$31$	203.851	1.18105	0.590526	+	0.807018i	$0.298920\pi$
$32$	0	0
$33$	32.1106	0.169386
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−341.224	−1.51613	−0.758066	−	0.652178i	$-0.773856\pi$
$37$	−341.224	−1.51613	−0.758066	+	0.652178i	$0.773856\pi$
$38$	0	0
$39$	24.9563	0.102467
$40$	0	0
$41$	147.127	0.560422	0.280211	−	0.959938i	$-0.409596\pi$
$41$	147.127	0.560422	0.280211	+	0.959938i	$0.409596\pi$
$42$	0	0
$43$	51.3401	0.182077	0.0910383	−	0.995847i	$-0.470981\pi$
$43$	51.3401	0.182077	0.0910383	+	0.995847i	$0.470981\pi$
$44$	0	0
$45$	−192.563	−0.637903
$46$	0	0
$47$	269.510	0.836428	0.418214	−	0.908348i	$-0.362656\pi$
$47$	269.510	0.836428	0.418214	+	0.908348i	$0.362656\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.45269	0.0122255
$52$	0	0
$53$	−432.500	−1.12091	−0.560457	−	0.828183i	$-0.689375\pi$
$53$	−432.500	−1.12091	−0.560457	+	0.828183i	$0.689375\pi$
$54$	0	0
$55$	−229.012	−0.561454
$56$	0	0
$57$	484.556	1.12598
$58$	0	0
$59$	30.7341	0.0678176	0.0339088	−	0.999425i	$-0.489204\pi$
$59$	30.7341	0.0678176	0.0339088	+	0.999425i	$0.489204\pi$
$60$	0	0
$61$	−592.517	−1.24367	−0.621836	−	0.783147i	$-0.713613\pi$
$61$	−592.517	−1.24367	−0.621836	+	0.783147i	$0.713613\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−177.988	−0.339641
$66$	0	0
$67$	414.052	0.754993	0.377497	−	0.926011i	$-0.376785\pi$
$67$	414.052	0.754993	0.377497	+	0.926011i	$0.376785\pi$
$68$	0	0
$69$	−371.836	−0.648750
$70$	0	0
$71$	294.854	0.492856	0.246428	−	0.969161i	$-0.420743\pi$
$71$	294.854	0.492856	0.246428	+	0.969161i	$0.420743\pi$
$72$	0	0
$73$	−594.063	−0.952462	−0.476231	−	0.879320i	$-0.657997\pi$
$73$	−594.063	−0.952462	−0.476231	+	0.879320i	$0.657997\pi$
$74$	0	0
$75$	998.357	1.53707
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−831.386	−1.18403	−0.592014	−	0.805928i	$-0.701667\pi$
$79$	−831.386	−1.18403	−0.592014	+	0.805928i	$0.701667\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	326.218	0.431410	0.215705	−	0.976459i	$-0.430795\pi$
$83$	326.218	0.431410	0.215705	+	0.976459i	$0.430795\pi$
$84$	0	0
$85$	−31.7564	−0.0405232
$86$	0	0
$87$	−668.451	−0.823741
$88$	0	0
$89$	−1261.10	−1.50198	−0.750989	−	0.660315i	$-0.770423\pi$
$89$	−1261.10	−1.50198	−0.750989	+	0.660315i	$0.770423\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	611.552	0.681881
$94$	0	0
$95$	−3455.84	−3.73223
$96$	0	0
$97$	−1059.98	−1.10953	−0.554766	−	0.832007i	$-0.687192\pi$
$97$	−1059.98	−1.10953	−0.554766	+	0.832007i	$0.687192\pi$
$98$	0	0
$99$	96.3319	0.0977951
$100$	0	0
$101$	180.948	0.178267	0.0891337	−	0.996020i	$-0.471590\pi$
$101$	180.948	0.178267	0.0891337	+	0.996020i	$0.471590\pi$
$102$	0	0
$103$	67.0461	0.0641384	0.0320692	−	0.999486i	$-0.489790\pi$
$103$	67.0461	0.0641384	0.0320692	+	0.999486i	$0.489790\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1060.10	−0.957794	−0.478897	−	0.877871i	$-0.658963\pi$
$107$	−1060.10	−0.957794	−0.478897	+	0.877871i	$0.658963\pi$
$108$	0	0
$109$	−777.222	−0.682976	−0.341488	−	0.939886i	$-0.610931\pi$
$109$	−777.222	−0.682976	−0.341488	+	0.939886i	$0.610931\pi$
$110$	0	0
$111$	−1023.67	−0.875340
$112$	0	0
$113$	−1021.93	−0.850749	−0.425375	−	0.905017i	$-0.639858\pi$
$113$	−1021.93	−0.850749	−0.425375	+	0.905017i	$0.639858\pi$
$114$	0	0
$115$	2651.92	2.15037
$116$	0	0
$117$	74.8690	0.0591593
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1216.43	−0.913925
$122$	0	0
$123$	441.380	0.323560
$124$	0	0
$125$	−4445.76	−3.18113
$126$	0	0
$127$	1176.21	0.821824	0.410912	−	0.911675i	$-0.365210\pi$
$127$	1176.21	0.821824	0.410912	+	0.911675i	$0.365210\pi$
$128$	0	0
$129$	154.020	0.105122
$130$	0	0
$131$	−1900.95	−1.26784	−0.633920	−	0.773398i	$-0.718555\pi$
$131$	−1900.95	−1.26784	−0.633920	+	0.773398i	$0.718555\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−577.690	−0.368294
$136$	0	0
$137$	−619.637	−0.386417	−0.193209	−	0.981158i	$-0.561889\pi$
$137$	−619.637	−0.386417	−0.193209	+	0.981158i	$0.561889\pi$
$138$	0	0
$139$	−1248.11	−0.761604	−0.380802	−	0.924657i	$-0.624352\pi$
$139$	−1248.11	−0.761604	−0.380802	+	0.924657i	$0.624352\pi$
$140$	0	0
$141$	808.531	0.482912
$142$	0	0
$143$	89.0404	0.0520694
$144$	0	0
$145$	4767.38	2.73041
$146$	0	0
$147$	0	0
$148$	0	0
$149$	521.479	0.286720	0.143360	−	0.989671i	$-0.454209\pi$
$149$	521.479	0.286720	0.143360	+	0.989671i	$0.454209\pi$
$150$	0	0
$151$	1261.33	0.679773	0.339886	−	0.940467i	$-0.389611\pi$
$151$	1261.33	0.679773	0.339886	+	0.940467i	$0.389611\pi$
$152$	0	0
$153$	13.3581	0.00705840
$154$	0	0
$155$	−4361.57	−2.26019
$156$	0	0
$157$	−584.137	−0.296938	−0.148469	−	0.988917i	$-0.547434\pi$
$157$	−584.137	−0.296938	−0.148469	+	0.988917i	$0.547434\pi$
$158$	0	0
$159$	−1297.50	−0.647160
$160$	0	0
$161$	0	0
$162$	0	0
$163$	164.133	0.0788704	0.0394352	−	0.999222i	$-0.487444\pi$
$163$	164.133	0.0788704	0.0394352	+	0.999222i	$0.487444\pi$
$164$	0	0
$165$	−687.036	−0.324156
$166$	0	0
$167$	3292.41	1.52560	0.762798	−	0.646636i	$-0.223825\pi$
$167$	3292.41	1.52560	0.762798	+	0.646636i	$0.223825\pi$
$168$	0	0
$169$	−2127.80	−0.968502
$170$	0	0
$171$	1453.67	0.650086
$172$	0	0
$173$	2602.09	1.14355	0.571773	−	0.820412i	$-0.306256\pi$
$173$	2602.09	1.14355	0.571773	+	0.820412i	$0.306256\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	92.2023	0.0391545
$178$	0	0
$179$	−302.031	−0.126116	−0.0630582	−	0.998010i	$-0.520085\pi$
$179$	−302.031	−0.126116	−0.0630582	+	0.998010i	$0.520085\pi$
$180$	0	0
$181$	609.020	0.250100	0.125050	−	0.992150i	$-0.460091\pi$
$181$	609.020	0.250100	0.125050	+	0.992150i	$0.460091\pi$
$182$	0	0
$183$	−1777.55	−0.718035
$184$	0	0
$185$	7300.81	2.90144
$186$	0	0
$187$	15.8865	0.00621249
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4801.60	−1.81901	−0.909507	−	0.415688i	$-0.863541\pi$
$191$	−4801.60	−1.81901	−0.909507	+	0.415688i	$0.863541\pi$
$192$	0	0
$193$	1912.28	0.713208	0.356604	−	0.934256i	$-0.383935\pi$
$193$	1912.28	0.713208	0.356604	+	0.934256i	$0.383935\pi$
$194$	0	0
$195$	−533.964	−0.196092
$196$	0	0
$197$	4085.92	1.47771	0.738857	−	0.673862i	$-0.235366\pi$
$197$	4085.92	1.47771	0.738857	+	0.673862i	$0.235366\pi$
$198$	0	0
$199$	498.536	0.177589	0.0887946	−	0.996050i	$-0.471699\pi$
$199$	498.536	0.177589	0.0887946	+	0.996050i	$0.471699\pi$
$200$	0	0
$201$	1242.16	0.435896
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3147.91	−1.07249
$206$	0	0
$207$	−1115.51	−0.374556
$208$	0	0
$209$	1728.82	0.572177
$210$	0	0
$211$	−2033.67	−0.663523	−0.331762	−	0.943363i	$-0.607643\pi$
$211$	−2033.67	−0.663523	−0.331762	+	0.943363i	$0.607643\pi$
$212$	0	0
$213$	884.563	0.284551
$214$	0	0
$215$	−1098.47	−0.348442
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1782.19	−0.549904
$220$	0	0
$221$	12.3470	0.00375813
$222$	0	0
$223$	2858.45	0.858367	0.429184	−	0.903217i	$-0.358801\pi$
$223$	2858.45	0.858367	0.429184	+	0.903217i	$0.358801\pi$
$224$	0	0
$225$	2995.07	0.887428
$226$	0	0
$227$	−2994.18	−0.875467	−0.437733	−	0.899105i	$-0.644219\pi$
$227$	−2994.18	−0.875467	−0.437733	+	0.899105i	$0.644219\pi$
$228$	0	0
$229$	4174.68	1.20467	0.602337	−	0.798242i	$-0.294236\pi$
$229$	4174.68	1.20467	0.602337	+	0.798242i	$0.294236\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2466.76	−0.693576	−0.346788	−	0.937944i	$-0.612728\pi$
$233$	−2466.76	−0.693576	−0.346788	+	0.937944i	$0.612728\pi$
$234$	0	0
$235$	−5766.42	−1.60068
$236$	0	0
$237$	−2494.16	−0.683599
$238$	0	0
$239$	447.762	0.121185	0.0605927	−	0.998163i	$-0.480701\pi$
$239$	447.762	0.121185	0.0605927	+	0.998163i	$0.480701\pi$
$240$	0	0
$241$	−5404.61	−1.44457	−0.722286	−	0.691595i	$-0.756908\pi$
$241$	−5404.61	−1.44457	−0.722286	+	0.691595i	$0.756908\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1343.64	0.346128
$248$	0	0
$249$	978.654	0.249075
$250$	0	0
$251$	−6654.56	−1.67343	−0.836717	−	0.547635i	$-0.815528\pi$
$251$	−6654.56	−1.67343	−0.836717	+	0.547635i	$0.815528\pi$
$252$	0	0
$253$	−1326.65	−0.329668
$254$	0	0
$255$	−95.2693	−0.0233961
$256$	0	0
$257$	5185.32	1.25857	0.629283	−	0.777177i	$-0.283349\pi$
$257$	5185.32	1.25857	0.629283	+	0.777177i	$0.283349\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2005.35	−0.475587
$262$	0	0
$263$	−1234.78	−0.289504	−0.144752	−	0.989468i	$-0.546238\pi$
$263$	−1234.78	−0.289504	−0.144752	+	0.989468i	$0.546238\pi$
$264$	0	0
$265$	9253.74	2.14510
$266$	0	0
$267$	−3783.29	−0.867168
$268$	0	0
$269$	−2165.24	−0.490771	−0.245385	−	0.969426i	$-0.578914\pi$
$269$	−2165.24	−0.490771	−0.245385	+	0.969426i	$0.578914\pi$
$270$	0	0
$271$	−1767.98	−0.396300	−0.198150	−	0.980172i	$-0.563493\pi$
$271$	−1767.98	−0.396300	−0.198150	+	0.980172i	$0.563493\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3561.98	0.781075
$276$	0	0
$277$	−5539.01	−1.20147	−0.600735	−	0.799449i	$-0.705125\pi$
$277$	−5539.01	−1.20147	−0.600735	+	0.799449i	$0.705125\pi$
$278$	0	0
$279$	1834.65	0.393684
$280$	0	0
$281$	4869.88	1.03385	0.516926	−	0.856030i	$-0.327076\pi$
$281$	4869.88	1.03385	0.516926	+	0.856030i	$0.327076\pi$
$282$	0	0
$283$	8375.10	1.75918	0.879590	−	0.475732i	$-0.157817\pi$
$283$	8375.10	1.75918	0.879590	+	0.475732i	$0.157817\pi$
$284$	0	0
$285$	−10367.5	−2.15480
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4910.80	−0.999552
$290$	0	0
$291$	−3179.94	−0.640588
$292$	0	0
$293$	−9398.40	−1.87393	−0.936963	−	0.349428i	$-0.886376\pi$
$293$	−9398.40	−1.87393	−0.936963	+	0.349428i	$0.886376\pi$
$294$	0	0
$295$	−657.585	−0.129783
$296$	0	0
$297$	288.996	0.0564620
$298$	0	0
$299$	−1031.07	−0.199426
$300$	0	0
$301$	0	0
$302$	0	0
$303$	542.844	0.102923
$304$	0	0
$305$	12677.5	2.38003
$306$	0	0
$307$	−102.680	−0.0190888	−0.00954438	−	0.999954i	$-0.503038\pi$
$307$	−102.680	−0.0190888	−0.00954438	+	0.999954i	$0.503038\pi$
$308$	0	0
$309$	201.138	0.0370303
$310$	0	0
$311$	−5365.79	−0.978347	−0.489173	−	0.872187i	$-0.662701\pi$
$311$	−5365.79	−0.978347	−0.489173	+	0.872187i	$0.662701\pi$
$312$	0	0
$313$	4165.06	0.752150	0.376075	−	0.926589i	$-0.377273\pi$
$313$	4165.06	0.752150	0.376075	+	0.926589i	$0.377273\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2339.76	0.414556	0.207278	−	0.978282i	$-0.433540\pi$
$317$	2339.76	0.414556	0.207278	+	0.978282i	$0.433540\pi$
$318$	0	0
$319$	−2384.93	−0.418591
$320$	0	0
$321$	−3180.31	−0.552983
$322$	0	0
$323$	239.731	0.0412971
$324$	0	0
$325$	2768.37	0.472497
$326$	0	0
$327$	−2331.67	−0.394316
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3721.80	−0.618033	−0.309016	−	0.951057i	$-0.600000\pi$
$331$	−3721.80	−0.618033	−0.309016	+	0.951057i	$0.600000\pi$
$332$	0	0
$333$	−3071.02	−0.505378
$334$	0	0
$335$	−8859.03	−1.44484
$336$	0	0
$337$	632.972	0.102315	0.0511575	−	0.998691i	$-0.483709\pi$
$337$	632.972	0.102315	0.0511575	+	0.998691i	$0.483709\pi$
$338$	0	0
$339$	−3065.78	−0.491180
$340$	0	0
$341$	2181.92	0.346504
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7955.76	1.24152
$346$	0	0
$347$	−2699.12	−0.417569	−0.208784	−	0.977962i	$-0.566951\pi$
$347$	−2699.12	−0.417569	−0.208784	+	0.977962i	$0.566951\pi$
$348$	0	0
$349$	−9047.70	−1.38771	−0.693857	−	0.720113i	$-0.744090\pi$
$349$	−9047.70	−1.38771	−0.693857	+	0.720113i	$0.744090\pi$
$350$	0	0
$351$	224.607	0.0341556
$352$	0	0
$353$	−9024.27	−1.36066	−0.680331	−	0.732905i	$-0.738164\pi$
$353$	−9024.27	−1.36066	−0.680331	+	0.732905i	$0.738164\pi$
$354$	0	0
$355$	−6308.68	−0.943183
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5082.12	0.747142	0.373571	−	0.927602i	$-0.378133\pi$
$359$	5082.12	0.747142	0.373571	+	0.927602i	$0.378133\pi$
$360$	0	0
$361$	19229.3	2.80351
$362$	0	0
$363$	−3649.30	−0.527655
$364$	0	0
$365$	12710.5	1.82274
$366$	0	0
$367$	−7142.75	−1.01594	−0.507968	−	0.861376i	$-0.669603\pi$
$367$	−7142.75	−1.01594	−0.507968	+	0.861376i	$0.669603\pi$
$368$	0	0
$369$	1324.14	0.186807
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3819.86	0.530255	0.265127	−	0.964213i	$-0.414586\pi$
$373$	3819.86	0.530255	0.265127	+	0.964213i	$0.414586\pi$
$374$	0	0
$375$	−13337.3	−1.83663
$376$	0	0
$377$	−1853.57	−0.253219
$378$	0	0
$379$	−5857.71	−0.793906	−0.396953	−	0.917839i	$-0.629932\pi$
$379$	−5857.71	−0.793906	−0.396953	+	0.917839i	$0.629932\pi$
$380$	0	0
$381$	3528.63	0.474480
$382$	0	0
$383$	−7404.66	−0.987886	−0.493943	−	0.869494i	$-0.664445\pi$
$383$	−7404.66	−0.987886	−0.493943	+	0.869494i	$0.664445\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	462.061	0.0606922
$388$	0	0
$389$	−7002.31	−0.912677	−0.456339	−	0.889806i	$-0.650839\pi$
$389$	−7002.31	−0.912677	−0.456339	+	0.889806i	$0.650839\pi$
$390$	0	0
$391$	−183.963	−0.0237939
$392$	0	0
$393$	−5702.86	−0.731988
$394$	0	0
$395$	17788.3	2.26589
$396$	0	0
$397$	−7423.72	−0.938503	−0.469252	−	0.883065i	$-0.655476\pi$
$397$	−7423.72	−0.938503	−0.469252	+	0.883065i	$0.655476\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2484.62	−0.309417	−0.154708	−	0.987960i	$-0.549444\pi$
$401$	−2484.62	−0.309417	−0.154708	+	0.987960i	$0.549444\pi$
$402$	0	0
$403$	1695.79	0.209611
$404$	0	0
$405$	−1733.07	−0.212634
$406$	0	0
$407$	−3652.31	−0.444811
$408$	0	0
$409$	3509.66	0.424307	0.212153	−	0.977236i	$-0.431952\pi$
$409$	3509.66	0.424307	0.212153	+	0.977236i	$0.431952\pi$
$410$	0	0
$411$	−1858.91	−0.223098
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6979.73	−0.825594
$416$	0	0
$417$	−3744.32	−0.439712
$418$	0	0
$419$	6969.64	0.812624	0.406312	−	0.913734i	$-0.366815\pi$
$419$	6969.64	0.812624	0.406312	+	0.913734i	$0.366815\pi$
$420$	0	0
$421$	−12700.3	−1.47024	−0.735122	−	0.677935i	$-0.762875\pi$
$421$	−12700.3	−1.47024	−0.735122	+	0.677935i	$0.762875\pi$
$422$	0	0
$423$	2425.59	0.278809
$424$	0	0
$425$	493.930	0.0563744
$426$	0	0
$427$	0	0
$428$	0	0
$429$	267.121	0.0300623
$430$	0	0
$431$	894.527	0.0999718	0.0499859	−	0.998750i	$-0.484082\pi$
$431$	894.527	0.0999718	0.0499859	+	0.998750i	$0.484082\pi$
$432$	0	0
$433$	15813.7	1.75510	0.877552	−	0.479482i	$-0.159176\pi$
$433$	15813.7	1.75510	0.877552	+	0.479482i	$0.159176\pi$
$434$	0	0
$435$	14302.1	1.57640
$436$	0	0
$437$	−20019.5	−2.19144
$438$	0	0
$439$	14221.9	1.54618	0.773090	−	0.634297i	$-0.218710\pi$
$439$	14221.9	1.54618	0.773090	+	0.634297i	$0.218710\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9422.08	1.01051	0.505256	−	0.862970i	$-0.331398\pi$
$443$	9422.08	1.01051	0.505256	+	0.862970i	$0.331398\pi$
$444$	0	0
$445$	26982.3	2.87435
$446$	0	0
$447$	1564.44	0.165538
$448$	0	0
$449$	−3223.63	−0.338825	−0.169412	−	0.985545i	$-0.554187\pi$
$449$	−3223.63	−0.338825	−0.169412	+	0.985545i	$0.554187\pi$
$450$	0	0
$451$	1574.78	0.164420
$452$	0	0
$453$	3783.99	0.392467
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6219.19	0.636589	0.318295	−	0.947992i	$-0.396890\pi$
$457$	6219.19	0.636589	0.318295	+	0.947992i	$0.396890\pi$
$458$	0	0
$459$	40.0742	0.00407517
$460$	0	0
$461$	10858.6	1.09704	0.548521	−	0.836137i	$-0.315191\pi$
$461$	10858.6	1.09704	0.548521	+	0.836137i	$0.315191\pi$
$462$	0	0
$463$	19067.3	1.91389	0.956946	−	0.290266i	$-0.0937440\pi$
$463$	19067.3	1.91389	0.956946	+	0.290266i	$0.0937440\pi$
$464$	0	0
$465$	−13084.7	−1.30492
$466$	0	0
$467$	3919.79	0.388407	0.194204	−	0.980961i	$-0.437788\pi$
$467$	3919.79	0.388407	0.194204	+	0.980961i	$0.437788\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1752.41	−0.171437
$472$	0	0
$473$	549.521	0.0534186
$474$	0	0
$475$	53751.1	5.19214
$476$	0	0
$477$	−3892.50	−0.373638
$478$	0	0
$479$	−8477.26	−0.808634	−0.404317	−	0.914619i	$-0.632491\pi$
$479$	−8477.26	−0.808634	−0.404317	+	0.914619i	$0.632491\pi$
$480$	0	0
$481$	−2838.57	−0.269080
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22679.2	2.12332
$486$	0	0
$487$	15250.3	1.41901	0.709505	−	0.704701i	$-0.248919\pi$
$487$	15250.3	1.41901	0.709505	+	0.704701i	$0.248919\pi$
$488$	0	0
$489$	492.399	0.0455359
$490$	0	0
$491$	6524.81	0.599716	0.299858	−	0.953984i	$-0.403061\pi$
$491$	6524.81	0.599716	0.299858	+	0.953984i	$0.403061\pi$
$492$	0	0
$493$	−330.711	−0.0302120
$494$	0	0
$495$	−2061.11	−0.187151
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7470.88	−0.670226	−0.335113	−	0.942178i	$-0.608774\pi$
$499$	−7470.88	−0.670226	−0.335113	+	0.942178i	$0.608774\pi$
$500$	0	0
$501$	9877.24	0.880804
$502$	0	0
$503$	928.803	0.0823325	0.0411663	−	0.999152i	$-0.486893\pi$
$503$	928.803	0.0823325	0.0411663	+	0.999152i	$0.486893\pi$
$504$	0	0
$505$	−3871.55	−0.341152
$506$	0	0
$507$	−6383.39	−0.559165
$508$	0	0
$509$	8947.97	0.779198	0.389599	−	0.920985i	$-0.372614\pi$
$509$	8947.97	0.779198	0.389599	+	0.920985i	$0.372614\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4361.00	0.375328
$514$	0	0
$515$	−1434.51	−0.122742
$516$	0	0
$517$	2884.72	0.245396
$518$	0	0
$519$	7806.28	0.660227
$520$	0	0
$521$	13141.0	1.10502	0.552510	−	0.833506i	$-0.313670\pi$
$521$	13141.0	1.10502	0.552510	+	0.833506i	$0.313670\pi$
$522$	0	0
$523$	821.254	0.0686633	0.0343317	−	0.999410i	$-0.489070\pi$
$523$	821.254	0.0686633	0.0343317	+	0.999410i	$0.489070\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	302.561	0.0250090
$528$	0	0
$529$	3195.40	0.262629
$530$	0	0
$531$	276.607	0.0226059
$532$	0	0
$533$	1223.91	0.0994626
$534$	0	0
$535$	22681.9	1.83294
$536$	0	0
$537$	−906.092	−0.0728133
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9933.49	−0.789416	−0.394708	−	0.918807i	$-0.629154\pi$
$541$	−9933.49	−0.789416	−0.394708	+	0.918807i	$0.629154\pi$
$542$	0	0
$543$	1827.06	0.144395
$544$	0	0
$545$	16629.4	1.30702
$546$	0	0
$547$	−22556.2	−1.76313	−0.881565	−	0.472063i	$-0.843510\pi$
$547$	−22556.2	−1.76313	−0.881565	+	0.472063i	$0.843510\pi$
$548$	0	0
$549$	−5332.65	−0.414558
$550$	0	0
$551$	−35989.1	−2.78256
$552$	0	0
$553$	0	0
$554$	0	0
$555$	21902.4	1.67515
$556$	0	0
$557$	−13146.5	−1.00006	−0.500032	−	0.866007i	$-0.666678\pi$
$557$	−13146.5	−1.00006	−0.500032	+	0.866007i	$0.666678\pi$
$558$	0	0
$559$	427.087	0.0323146
$560$	0	0
$561$	47.6595	0.00358678
$562$	0	0
$563$	−24704.9	−1.84935	−0.924676	−	0.380754i	$-0.875665\pi$
$563$	−24704.9	−1.84935	−0.924676	+	0.380754i	$0.875665\pi$
$564$	0	0
$565$	21865.0	1.62809
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4375.65	0.322385	0.161192	−	0.986923i	$-0.448466\pi$
$569$	4375.65	0.322385	0.161192	+	0.986923i	$0.448466\pi$
$570$	0	0
$571$	−11284.7	−0.827059	−0.413529	−	0.910491i	$-0.635704\pi$
$571$	−11284.7	−0.827059	−0.413529	+	0.910491i	$0.635704\pi$
$572$	0	0
$573$	−14404.8	−1.05021
$574$	0	0
$575$	−41247.2	−2.99152
$576$	0	0
$577$	−6831.44	−0.492888	−0.246444	−	0.969157i	$-0.579262\pi$
$577$	−6831.44	−0.492888	−0.246444	+	0.969157i	$0.579262\pi$
$578$	0	0
$579$	5736.85	0.411771
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4629.28	−0.328860
$584$	0	0
$585$	−1601.89	−0.113214
$586$	0	0
$587$	1116.70	0.0785196	0.0392598	−	0.999229i	$-0.487500\pi$
$587$	1116.70	0.0785196	0.0392598	+	0.999229i	$0.487500\pi$
$588$	0	0
$589$	32925.7	2.30336
$590$	0	0
$591$	12257.8	0.853159
$592$	0	0
$593$	−9724.17	−0.673396	−0.336698	−	0.941613i	$-0.609310\pi$
$593$	−9724.17	−0.673396	−0.336698	+	0.941613i	$0.609310\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1495.61	0.102531
$598$	0	0
$599$	−798.427	−0.0544622	−0.0272311	−	0.999629i	$-0.508669\pi$
$599$	−798.427	−0.0544622	−0.0272311	+	0.999629i	$0.508669\pi$
$600$	0	0
$601$	25.5785	0.00173606	0.000868028	−	1.00000i	$-0.499724\pi$
$601$	25.5785	0.00173606	0.000868028		1.00000i	$0.499724\pi$
$602$	0	0
$603$	3726.47	0.251664
$604$	0	0
$605$	26026.7	1.74899
$606$	0	0
$607$	13746.8	0.919220	0.459610	−	0.888121i	$-0.347989\pi$
$607$	13746.8	0.919220	0.459610	+	0.888121i	$0.347989\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2242.00	0.148448
$612$	0	0
$613$	14443.4	0.951651	0.475825	−	0.879540i	$-0.342149\pi$
$613$	14443.4	0.951651	0.475825	+	0.879540i	$0.342149\pi$
$614$	0	0
$615$	−9443.73	−0.619200
$616$	0	0
$617$	27585.4	1.79991	0.899956	−	0.435980i	$-0.143598\pi$
$617$	27585.4	1.79991	0.899956	+	0.435980i	$0.143598\pi$
$618$	0	0
$619$	−17551.0	−1.13963	−0.569816	−	0.821772i	$-0.692986\pi$
$619$	−17551.0	−1.13963	−0.569816	+	0.821772i	$0.692986\pi$
$620$	0	0
$621$	−3346.52	−0.216250
$622$	0	0
$623$	0	0
$624$	0	0
$625$	53523.0	3.42547
$626$	0	0
$627$	5186.47	0.330347
$628$	0	0
$629$	−506.455	−0.0321044
$630$	0	0
$631$	−19726.5	−1.24453	−0.622265	−	0.782806i	$-0.713788\pi$
$631$	−19726.5	−1.24453	−0.622265	+	0.782806i	$0.713788\pi$
$632$	0	0
$633$	−6101.00	−0.383085
$634$	0	0
$635$	−25166.1	−1.57273
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2653.69	0.164285
$640$	0	0
$641$	−1808.43	−0.111433	−0.0557166	−	0.998447i	$-0.517744\pi$
$641$	−1808.43	−0.111433	−0.0557166	+	0.998447i	$0.517744\pi$
$642$	0	0
$643$	−6017.67	−0.369073	−0.184536	−	0.982826i	$-0.559078\pi$
$643$	−6017.67	−0.369073	−0.184536	+	0.982826i	$0.559078\pi$
$644$	0	0
$645$	−3295.41	−0.201173
$646$	0	0
$647$	21123.6	1.28355	0.641774	−	0.766894i	$-0.278199\pi$
$647$	21123.6	1.28355	0.641774	+	0.766894i	$0.278199\pi$
$648$	0	0
$649$	328.964	0.0198967
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24226.0	−1.45182	−0.725908	−	0.687792i	$-0.758580\pi$
$653$	−24226.0	−1.45182	−0.725908	+	0.687792i	$0.758580\pi$
$654$	0	0
$655$	40672.7	2.42628
$656$	0	0
$657$	−5346.56	−0.317487
$658$	0	0
$659$	11836.0	0.699643	0.349822	−	0.936816i	$-0.386242\pi$
$659$	11836.0	0.699643	0.349822	+	0.936816i	$0.386242\pi$
$660$	0	0
$661$	18763.4	1.10410	0.552051	−	0.833810i	$-0.313845\pi$
$661$	18763.4	1.10410	0.552051	+	0.833810i	$0.313845\pi$
$662$	0	0
$663$	37.0409	0.00216976
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27617.1	1.60321
$668$	0	0
$669$	8575.35	0.495579
$670$	0	0
$671$	−6342.03	−0.364875
$672$	0	0
$673$	−2130.24	−0.122013	−0.0610065	−	0.998137i	$-0.519431\pi$
$673$	−2130.24	−0.122013	−0.0610065	+	0.998137i	$0.519431\pi$
$674$	0	0
$675$	8985.21	0.512357
$676$	0	0
$677$	5386.83	0.305809	0.152904	−	0.988241i	$-0.451137\pi$
$677$	5386.83	0.305809	0.152904	+	0.988241i	$0.451137\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8982.55	−0.505451
$682$	0	0
$683$	−3421.99	−0.191711	−0.0958556	−	0.995395i	$-0.530559\pi$
$683$	−3421.99	−0.191711	−0.0958556	+	0.995395i	$0.530559\pi$
$684$	0	0
$685$	13257.7	0.739491
$686$	0	0
$687$	12524.0	0.695519
$688$	0	0
$689$	−3597.87	−0.198938
$690$	0	0
$691$	−3373.63	−0.185730	−0.0928648	−	0.995679i	$-0.529602\pi$
$691$	−3373.63	−0.185730	−0.0928648	+	0.995679i	$0.529602\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26704.4	1.45749
$696$	0	0
$697$	218.370	0.0118671
$698$	0	0
$699$	−7400.29	−0.400436
$700$	0	0
$701$	−28103.0	−1.51417	−0.757087	−	0.653314i	$-0.773378\pi$
$701$	−28103.0	−1.51417	−0.757087	+	0.653314i	$0.773378\pi$
$702$	0	0
$703$	−55114.1	−2.95685
$704$	0	0
$705$	−17299.3	−0.924154
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30036.9	−1.59106	−0.795528	−	0.605917i	$-0.792806\pi$
$709$	−30036.9	−1.59106	−0.795528	+	0.605917i	$0.792806\pi$
$710$	0	0
$711$	−7482.47	−0.394676
$712$	0	0
$713$	−25266.3	−1.32711
$714$	0	0
$715$	−1905.10	−0.0996458
$716$	0	0
$717$	1343.29	0.0699665
$718$	0	0
$719$	−14102.6	−0.731486	−0.365743	−	0.930716i	$-0.619185\pi$
$719$	−14102.6	−0.731486	−0.365743	+	0.930716i	$0.619185\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−16213.8	−0.834024
$724$	0	0
$725$	−74150.3	−3.79845
$726$	0	0
$727$	−3339.23	−0.170351	−0.0851756	−	0.996366i	$-0.527145\pi$
$727$	−3339.23	−0.170351	−0.0851756	+	0.996366i	$0.527145\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	76.2005	0.00385551
$732$	0	0
$733$	−25436.3	−1.28174	−0.640868	−	0.767651i	$-0.721425\pi$
$733$	−25436.3	−1.28174	−0.640868	+	0.767651i	$0.721425\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4431.83	0.221504
$738$	0	0
$739$	−21842.5	−1.08727	−0.543633	−	0.839323i	$-0.682952\pi$
$739$	−21842.5	−1.08727	−0.543633	+	0.839323i	$0.682952\pi$
$740$	0	0
$741$	4030.91	0.199837
$742$	0	0
$743$	1480.51	0.0731019	0.0365510	−	0.999332i	$-0.488363\pi$
$743$	1480.51	0.0731019	0.0365510	+	0.999332i	$0.488363\pi$
$744$	0	0
$745$	−11157.5	−0.548698
$746$	0	0
$747$	2935.96	0.143803
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7767.42	0.377413	0.188706	−	0.982034i	$-0.439571\pi$
$751$	7767.42	0.377413	0.188706	+	0.982034i	$0.439571\pi$
$752$	0	0
$753$	−19963.7	−0.966158
$754$	0	0
$755$	−26987.3	−1.30089
$756$	0	0
$757$	17950.6	0.861856	0.430928	−	0.902386i	$-0.358186\pi$
$757$	17950.6	0.861856	0.430928	+	0.902386i	$0.358186\pi$
$758$	0	0
$759$	−3979.96	−0.190334
$760$	0	0
$761$	−19870.2	−0.946509	−0.473255	−	0.880926i	$-0.656921\pi$
$761$	−19870.2	−0.946509	−0.473255	+	0.880926i	$0.656921\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−285.808	−0.0135077
$766$	0	0
$767$	255.670	0.0120361
$768$	0	0
$769$	−13321.1	−0.624668	−0.312334	−	0.949972i	$-0.601111\pi$
$769$	−13321.1	−0.624668	−0.312334	+	0.949972i	$0.601111\pi$
$770$	0	0
$771$	15556.0	0.726633
$772$	0	0
$773$	−3413.53	−0.158831	−0.0794153	−	0.996842i	$-0.525305\pi$
$773$	−3413.53	−0.158831	−0.0794153	+	0.996842i	$0.525305\pi$
$774$	0	0
$775$	67838.5	3.14430
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23763.7	1.09297
$780$	0	0
$781$	3155.99	0.144597
$782$	0	0
$783$	−6016.06	−0.274580
$784$	0	0
$785$	12498.1	0.568252
$786$	0	0
$787$	4392.25	0.198941	0.0994706	−	0.995040i	$-0.468285\pi$
$787$	4392.25	0.198941	0.0994706	+	0.995040i	$0.468285\pi$
$788$	0	0
$789$	−3704.33	−0.167145
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4929.02	−0.220725
$794$	0	0
$795$	27761.2	1.23848
$796$	0	0
$797$	10628.2	0.472357	0.236179	−	0.971710i	$-0.424105\pi$
$797$	10628.2	0.472357	0.236179	+	0.971710i	$0.424105\pi$
$798$	0	0
$799$	400.015	0.0177115
$800$	0	0
$801$	−11349.9	−0.500659
$802$	0	0
$803$	−6358.57	−0.279439
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6495.73	−0.283347
$808$	0	0
$809$	31455.7	1.36702	0.683512	−	0.729939i	$-0.260452\pi$
$809$	31455.7	1.36702	0.683512	+	0.729939i	$0.260452\pi$
$810$	0	0
$811$	15106.2	0.654070	0.327035	−	0.945012i	$-0.393951\pi$
$811$	15106.2	0.654070	0.327035	+	0.945012i	$0.393951\pi$
$812$	0	0
$813$	−5303.95	−0.228804
$814$	0	0
$815$	−3511.77	−0.150935
$816$	0	0
$817$	8292.39	0.355097
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44912.6	−1.90921	−0.954605	−	0.297875i	$-0.903722\pi$
$821$	−44912.6	−1.90921	−0.954605	+	0.297875i	$0.903722\pi$
$822$	0	0
$823$	17546.9	0.743191	0.371595	−	0.928395i	$-0.378811\pi$
$823$	17546.9	0.743191	0.371595	+	0.928395i	$0.378811\pi$
$824$	0	0
$825$	10686.0	0.450954
$826$	0	0
$827$	6540.96	0.275032	0.137516	−	0.990500i	$-0.456088\pi$
$827$	6540.96	0.275032	0.137516	+	0.990500i	$0.456088\pi$
$828$	0	0
$829$	7490.22	0.313807	0.156903	−	0.987614i	$-0.449849\pi$
$829$	7490.22	0.313807	0.156903	+	0.987614i	$0.449849\pi$
$830$	0	0
$831$	−16617.0	−0.693668
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−70444.2	−2.91955
$836$	0	0
$837$	5503.96	0.227294
$838$	0	0
$839$	35766.1	1.47173	0.735867	−	0.677127i	$-0.236775\pi$
$839$	35766.1	1.47173	0.735867	+	0.677127i	$0.236775\pi$
$840$	0	0
$841$	25258.4	1.03565
$842$	0	0
$843$	14609.6	0.596895
$844$	0	0
$845$	45526.2	1.85343
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25125.3	1.01566
$850$	0	0
$851$	42293.1	1.70363
$852$	0	0
$853$	25697.7	1.03150	0.515751	−	0.856738i	$-0.327513\pi$
$853$	25697.7	1.03150	0.515751	+	0.856738i	$0.327513\pi$
$854$	0	0
$855$	−31102.6	−1.24408
$856$	0	0
$857$	−4707.36	−0.187632	−0.0938159	−	0.995590i	$-0.529906\pi$
$857$	−4707.36	−0.187632	−0.0938159	+	0.995590i	$0.529906\pi$
$858$	0	0
$859$	15093.5	0.599514	0.299757	−	0.954016i	$-0.403094\pi$
$859$	15093.5	0.599514	0.299757	+	0.954016i	$0.403094\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42516.6	−1.67704	−0.838519	−	0.544873i	$-0.816578\pi$
$863$	−42516.6	−1.67704	−0.838519	+	0.544873i	$0.816578\pi$
$864$	0	0
$865$	−55674.2	−2.18842
$866$	0	0
$867$	−14732.4	−0.577091
$868$	0	0
$869$	−8898.77	−0.347376
$870$	0	0
$871$	3444.41	0.133995
$872$	0	0
$873$	−9539.81	−0.369844
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4181.79	−0.161014	−0.0805069	−	0.996754i	$-0.525654\pi$
$877$	−4181.79	−0.161014	−0.0805069	+	0.996754i	$0.525654\pi$
$878$	0	0
$879$	−28195.2	−1.08191
$880$	0	0
$881$	−3070.34	−0.117415	−0.0587073	−	0.998275i	$-0.518698\pi$
$881$	−3070.34	−0.117415	−0.0587073	+	0.998275i	$0.518698\pi$
$882$	0	0
$883$	146.842	0.00559640	0.00279820	−	0.999996i	$-0.499109\pi$
$883$	146.842	0.00559640	0.00279820	+	0.999996i	$0.499109\pi$
$884$	0	0
$885$	−1972.75	−0.0749304
$886$	0	0
$887$	10739.3	0.406527	0.203263	−	0.979124i	$-0.434845\pi$
$887$	10739.3	0.406527	0.203263	+	0.979124i	$0.434845\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	866.987	0.0325984
$892$	0	0
$893$	43531.0	1.63125
$894$	0	0
$895$	6462.23	0.241350
$896$	0	0
$897$	−3093.22	−0.115139
$898$	0	0
$899$	−45421.4	−1.68508
$900$	0	0
$901$	−641.929	−0.0237356
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13030.6	−0.478619
$906$	0	0
$907$	−30615.1	−1.12079	−0.560396	−	0.828225i	$-0.689351\pi$
$907$	−30615.1	−1.12079	−0.560396	+	0.828225i	$0.689351\pi$
$908$	0	0
$909$	1628.53	0.0594225
$910$	0	0
$911$	45026.6	1.63754	0.818770	−	0.574121i	$-0.194656\pi$
$911$	45026.6	1.63754	0.818770	+	0.574121i	$0.194656\pi$
$912$	0	0
$913$	3491.69	0.126569
$914$	0	0
$915$	38032.4	1.37411
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22816.1	−0.818970	−0.409485	−	0.912317i	$-0.634292\pi$
$919$	−22816.1	−0.818970	−0.409485	+	0.912317i	$0.634292\pi$
$920$	0	0
$921$	−308.040	−0.0110209
$922$	0	0
$923$	2452.83	0.0874711
$924$	0	0
$925$	−113554.	−4.03638
$926$	0	0
$927$	603.415	0.0213795
$928$	0	0
$929$	10944.1	0.386507	0.193254	−	0.981149i	$-0.438096\pi$
$929$	10944.1	0.386507	0.193254	+	0.981149i	$0.438096\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16097.4	−0.564849
$934$	0	0
$935$	−339.906	−0.0118889
$936$	0	0
$937$	10110.2	0.352494	0.176247	−	0.984346i	$-0.443604\pi$
$937$	10110.2	0.352494	0.176247	+	0.984346i	$0.443604\pi$
$938$	0	0
$939$	12495.2	0.434254
$940$	0	0
$941$	24417.0	0.845879	0.422939	−	0.906158i	$-0.360998\pi$
$941$	24417.0	0.845879	0.422939	+	0.906158i	$0.360998\pi$
$942$	0	0
$943$	−18235.6	−0.629728
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3131.71	0.107462	0.0537311	−	0.998555i	$-0.482889\pi$
$947$	3131.71	0.107462	0.0537311	+	0.998555i	$0.482889\pi$
$948$	0	0
$949$	−4941.87	−0.169041
$950$	0	0
$951$	7019.29	0.239344
$952$	0	0
$953$	−6544.00	−0.222436	−0.111218	−	0.993796i	$-0.535475\pi$
$953$	−6544.00	−0.222436	−0.111218	+	0.993796i	$0.535475\pi$
$954$	0	0
$955$	102735.	3.48107
$956$	0	0
$957$	−7154.79	−0.241674
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11764.0	0.394886
$962$	0	0
$963$	−9540.92	−0.319265
$964$	0	0
$965$	−40915.0	−1.36487
$966$	0	0
$967$	45805.7	1.52328	0.761640	−	0.648000i	$-0.224395\pi$
$967$	45805.7	1.52328	0.761640	+	0.648000i	$0.224395\pi$
$968$	0	0
$969$	719.192	0.0238429
$970$	0	0
$971$	34822.0	1.15087	0.575434	−	0.817848i	$-0.304833\pi$
$971$	34822.0	1.15087	0.575434	+	0.817848i	$0.304833\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8305.10	0.272796
$976$	0	0
$977$	−5213.81	−0.170731	−0.0853657	−	0.996350i	$-0.527206\pi$
$977$	−5213.81	−0.170731	−0.0853657	+	0.996350i	$0.527206\pi$
$978$	0	0
$979$	−13498.2	−0.440658
$980$	0	0
$981$	−6995.00	−0.227659
$982$	0	0
$983$	−21152.9	−0.686341	−0.343171	−	0.939273i	$-0.611501\pi$
$983$	−21152.9	−0.686341	−0.343171	+	0.939273i	$0.611501\pi$
$984$	0	0
$985$	−87422.0	−2.82792
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6363.36	−0.204594
$990$	0	0
$991$	42635.1	1.36665	0.683324	−	0.730116i	$-0.260534\pi$
$991$	42635.1	1.36665	0.683324	+	0.730116i	$0.260534\pi$
$992$	0	0
$993$	−11165.4	−0.356821
$994$	0	0
$995$	−10666.6	−0.339854
$996$	0	0
$997$	−14197.8	−0.451003	−0.225502	−	0.974243i	$-0.572402\pi$
$997$	−14197.8	−0.451003	−0.225502	+	0.974243i	$0.572402\pi$
$998$	0	0
$999$	−9213.05	−0.291780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.y.1.1		3
4.3	odd	2		2352.4.a.ch.1.1		3
7.2	even	3		168.4.q.e.25.3	✓	6
7.4	even	3		168.4.q.e.121.3	yes	6
7.6	odd	2		1176.4.a.x.1.3		3
21.2	odd	6		504.4.s.g.361.1		6
21.11	odd	6		504.4.s.g.289.1		6
28.11	odd	6		336.4.q.l.289.3		6
28.23	odd	6		336.4.q.l.193.3		6
28.27	even	2		2352.4.a.cj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.e.25.3	✓	6	7.2	even	3
168.4.q.e.121.3	yes	6	7.4	even	3
336.4.q.l.193.3		6	28.23	odd	6
336.4.q.l.289.3		6	28.11	odd	6
504.4.s.g.289.1		6	21.11	odd	6
504.4.s.g.361.1		6	21.2	odd	6
1176.4.a.x.1.3		3	7.6	odd	2
1176.4.a.y.1.1		3	1.1	even	1	trivial
2352.4.a.ch.1.1		3	4.3	odd	2
2352.4.a.cj.1.3		3	28.27	even	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -21.3959 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -21.3959 q^{5} +9.00000 q^{9} +10.7035 q^{11} +8.31878 q^{13} -64.1878 q^{15} +1.48423 q^{17} +161.519 q^{19} -123.945 q^{23} +332.786 q^{25} +27.0000 q^{27} -222.817 q^{29} +203.851 q^{31} +32.1106 q^{33} -341.224 q^{37} +24.9563 q^{39} +147.127 q^{41} +51.3401 q^{43} -192.563 q^{45} +269.510 q^{47} +4.45269 q^{51} -432.500 q^{53} -229.012 q^{55} +484.556 q^{57} +30.7341 q^{59} -592.517 q^{61} -177.988 q^{65} +414.052 q^{67} -371.836 q^{69} +294.854 q^{71} -594.063 q^{73} +998.357 q^{75} -831.386 q^{79} +81.0000 q^{81} +326.218 q^{83} -31.7564 q^{85} -668.451 q^{87} -1261.10 q^{89} +611.552 q^{93} -3455.84 q^{95} -1059.98 q^{97} +96.3319 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) \) 3 * q + 9 * q^3 - 11 * q^5 + 27 * q^9	\( 3 q + 9 q^{3} - 11 q^{5} + 27 q^{9} - 19 q^{11} - 22 q^{13} - 33 q^{15} - 104 q^{17} + 202 q^{19} - 280 q^{23} + 186 q^{25} + 81 q^{27} - 73 q^{29} - 131 q^{31} - 57 q^{33} - 326 q^{37} - 66 q^{39} - 516 q^{41} + 36 q^{43} - 99 q^{45} - 126 q^{47} - 312 q^{51} - 385 q^{53} - 611 q^{55} + 606 q^{57} + 285 q^{59} - 34 q^{61} - 920 q^{65} - 100 q^{67} - 840 q^{69} + 34 q^{71} - 108 q^{73} + 558 q^{75} - 2463 q^{79} + 243 q^{81} - 115 q^{83} - 500 q^{85} - 219 q^{87} - 110 q^{89} - 393 q^{93} - 2064 q^{95} - 2941 q^{97} - 171 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 11 * q^5 + 27 * q^9 - 19 * q^11 - 22 * q^13 - 33 * q^15 - 104 * q^17 + 202 * q^19 - 280 * q^23 + 186 * q^25 + 81 * q^27 - 73 * q^29 - 131 * q^31 - 57 * q^33 - 326 * q^37 - 66 * q^39 - 516 * q^41 + 36 * q^43 - 99 * q^45 - 126 * q^47 - 312 * q^51 - 385 * q^53 - 611 * q^55 + 606 * q^57 + 285 * q^59 - 34 * q^61 - 920 * q^65 - 100 * q^67 - 840 * q^69 + 34 * q^71 - 108 * q^73 + 558 * q^75 - 2463 * q^79 + 243 * q^81 - 115 * q^83 - 500 * q^85 - 219 * q^87 - 110 * q^89 - 393 * q^93 - 2064 * q^95 - 2941 * q^97 - 171 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more