LMFDB - Embedded newform 1216.4.a.bf.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.39349$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.45545 q^{3} +11.5945 q^{5} +15.9032 q^{7} +62.4055 q^{9} +19.4189 q^{11} +69.4979 q^{13} +109.631 q^{15} -112.461 q^{17} +19.0000 q^{19} +150.372 q^{21} +57.3974 q^{23} +9.43209 q^{25} +334.775 q^{27} +59.8778 q^{29} -223.000 q^{31} +183.614 q^{33} +184.390 q^{35} +258.002 q^{37} +657.133 q^{39} -305.175 q^{41} -322.813 q^{43} +723.559 q^{45} -213.936 q^{47} -90.0873 q^{49} -1063.37 q^{51} -348.651 q^{53} +225.152 q^{55} +179.654 q^{57} +56.3816 q^{59} -87.2763 q^{61} +992.449 q^{63} +805.792 q^{65} -1074.10 q^{67} +542.718 q^{69} +725.349 q^{71} +889.389 q^{73} +89.1846 q^{75} +308.823 q^{77} -1258.90 q^{79} +1480.50 q^{81} -227.650 q^{83} -1303.93 q^{85} +566.171 q^{87} -927.529 q^{89} +1105.24 q^{91} -2108.56 q^{93} +220.295 q^{95} +603.489 q^{97} +1211.84 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 17 q^{5} + 42 q^{7} + 86 q^{9} + 33 q^{11} + 35 q^{13} + 120 q^{15} + 66 q^{17} + 133 q^{19} - 33 q^{21} + 389 q^{23} + 44 q^{25} + 39 q^{27} - 233 q^{29} + 158 q^{31} - 206 q^{33} - 123 q^{35}+ \cdots + 1591 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 17 * q^5 + 42 * q^7 + 86 * q^9 + 33 * q^11 + 35 * q^13 + 120 * q^15 + 66 * q^17 + 133 * q^19 - 33 * q^21 + 389 * q^23 + 44 * q^25 + 39 * q^27 - 233 * q^29 + 158 * q^31 - 206 * q^33 - 123 * q^35 + 436 * q^37 + 807 * q^39 - 94 * q^41 - 645 * q^43 - 103 * q^45 + 1451 * q^47 + 93 * q^49 - 1741 * q^51 - 3 * q^53 + 1971 * q^55 - 57 * q^57 - 297 * q^59 - 93 * q^61 + 2999 * q^63 - 788 * q^65 - 1641 * q^67 - 945 * q^69 + 2392 * q^71 + 324 * q^73 - 1909 * q^75 + 711 * q^77 + 2492 * q^79 + 143 * q^81 - 310 * q^83 - 2353 * q^85 + 4795 * q^87 - 440 * q^89 + 107 * q^91 - 900 * q^93 + 323 * q^95 - 532 * q^97 + 1591 * 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
$3$	9.45545	1.81970	0.909851	−	0.414935i	$-0.136196\pi$
$3$	9.45545	1.81970	0.909851	+	0.414935i	$0.136196\pi$
$4$	0	0
$5$	11.5945	1.03704	0.518521	−	0.855065i	$-0.326483\pi$
$5$	11.5945	1.03704	0.518521	+	0.855065i	$0.326483\pi$
$6$	0	0
$7$	15.9032	0.858694	0.429347	−	0.903140i	$-0.358744\pi$
$7$	15.9032	0.858694	0.429347	+	0.903140i	$0.358744\pi$
$8$	0	0
$9$	62.4055	2.31131
$10$	0	0
$11$	19.4189	0.532274	0.266137	−	0.963935i	$-0.414253\pi$
$11$	19.4189	0.532274	0.266137	+	0.963935i	$0.414253\pi$
$12$	0	0
$13$	69.4979	1.48271	0.741355	−	0.671113i	$-0.234183\pi$
$13$	69.4979	1.48271	0.741355	+	0.671113i	$0.234183\pi$
$14$	0	0
$15$	109.631	1.88711
$16$	0	0
$17$	−112.461	−1.60446	−0.802230	−	0.597015i	$-0.796353\pi$
$17$	−112.461	−1.60446	−0.802230	+	0.597015i	$0.796353\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	150.372	1.56257
$22$	0	0
$23$	57.3974	0.520356	0.260178	−	0.965561i	$-0.416219\pi$
$23$	57.3974	0.520356	0.260178	+	0.965561i	$0.416219\pi$
$24$	0	0
$25$	9.43209	0.0754567
$26$	0	0
$27$	334.775	2.38620
$28$	0	0
$29$	59.8778	0.383415	0.191707	−	0.981452i	$-0.438598\pi$
$29$	59.8778	0.383415	0.191707	+	0.981452i	$0.438598\pi$
$30$	0	0
$31$	−223.000	−1.29200	−0.645998	−	0.763339i	$-0.723559\pi$
$31$	−223.000	−1.29200	−0.645998	+	0.763339i	$0.723559\pi$
$32$	0	0
$33$	183.614	0.968580
$34$	0	0
$35$	184.390	0.890502
$36$	0	0
$37$	258.002	1.14636	0.573180	−	0.819429i	$-0.305709\pi$
$37$	258.002	1.14636	0.573180	+	0.819429i	$0.305709\pi$
$38$	0	0
$39$	657.133	2.69809
$40$	0	0
$41$	−305.175	−1.16245	−0.581224	−	0.813743i	$-0.697426\pi$
$41$	−305.175	−1.16245	−0.581224	+	0.813743i	$0.697426\pi$
$42$	0	0
$43$	−322.813	−1.14485	−0.572424	−	0.819958i	$-0.693997\pi$
$43$	−322.813	−1.14485	−0.572424	+	0.819958i	$0.693997\pi$
$44$	0	0
$45$	723.559	2.39693
$46$	0	0
$47$	−213.936	−0.663952	−0.331976	−	0.943288i	$-0.607715\pi$
$47$	−213.936	−0.663952	−0.331976	+	0.943288i	$0.607715\pi$
$48$	0	0
$49$	−90.0873	−0.262645
$50$	0	0
$51$	−1063.37	−2.91964
$52$	0	0
$53$	−348.651	−0.903603	−0.451802	−	0.892118i	$-0.649218\pi$
$53$	−348.651	−0.903603	−0.451802	+	0.892118i	$0.649218\pi$
$54$	0	0
$55$	225.152	0.551990
$56$	0	0
$57$	179.654	0.417468
$58$	0	0
$59$	56.3816	0.124411	0.0622056	−	0.998063i	$-0.480187\pi$
$59$	56.3816	0.124411	0.0622056	+	0.998063i	$0.480187\pi$
$60$	0	0
$61$	−87.2763	−0.183190	−0.0915950	−	0.995796i	$-0.529197\pi$
$61$	−87.2763	−0.183190	−0.0915950	+	0.995796i	$0.529197\pi$
$62$	0	0
$63$	992.449	1.98471
$64$	0	0
$65$	805.792	1.53763
$66$	0	0
$67$	−1074.10	−1.95854	−0.979272	−	0.202549i	$-0.935078\pi$
$67$	−1074.10	−1.95854	−0.979272	+	0.202549i	$0.935078\pi$
$68$	0	0
$69$	542.718	0.946892
$70$	0	0
$71$	725.349	1.21244	0.606219	−	0.795298i	$-0.292685\pi$
$71$	725.349	1.21244	0.606219	+	0.795298i	$0.292685\pi$
$72$	0	0
$73$	889.389	1.42596	0.712980	−	0.701184i	$-0.247345\pi$
$73$	889.389	1.42596	0.712980	+	0.701184i	$0.247345\pi$
$74$	0	0
$75$	89.1846	0.137309
$76$	0	0
$77$	308.823	0.457060
$78$	0	0
$79$	−1258.90	−1.79288	−0.896439	−	0.443168i	$-0.853855\pi$
$79$	−1258.90	−1.79288	−0.896439	+	0.443168i	$0.853855\pi$
$80$	0	0
$81$	1480.50	2.03086
$82$	0	0
$83$	−227.650	−0.301058	−0.150529	−	0.988606i	$-0.548098\pi$
$83$	−227.650	−0.301058	−0.150529	+	0.988606i	$0.548098\pi$
$84$	0	0
$85$	−1303.93	−1.66389
$86$	0	0
$87$	566.171	0.697700
$88$	0	0
$89$	−927.529	−1.10470	−0.552348	−	0.833614i	$-0.686268\pi$
$89$	−927.529	−1.10470	−0.552348	+	0.833614i	$0.686268\pi$
$90$	0	0
$91$	1105.24	1.27319
$92$	0	0
$93$	−2108.56	−2.35105
$94$	0	0
$95$	220.295	0.237914
$96$	0	0
$97$	603.489	0.631701	0.315851	−	0.948809i	$-0.397710\pi$
$97$	603.489	0.631701	0.315851	+	0.948809i	$0.397710\pi$
$98$	0	0
$99$	1211.84	1.23025
$100$	0	0
$101$	1035.55	1.02021	0.510106	−	0.860112i	$-0.329606\pi$
$101$	1035.55	1.02021	0.510106	+	0.860112i	$0.329606\pi$
$102$	0	0
$103$	666.961	0.638035	0.319018	−	0.947749i	$-0.396647\pi$
$103$	666.961	0.638035	0.319018	+	0.947749i	$0.396647\pi$
$104$	0	0
$105$	1743.49	1.62045
$106$	0	0
$107$	1369.29	1.23714	0.618572	−	0.785728i	$-0.287711\pi$
$107$	1369.29	1.23714	0.618572	+	0.785728i	$0.287711\pi$
$108$	0	0
$109$	−1326.45	−1.16560	−0.582800	−	0.812615i	$-0.698043\pi$
$109$	−1326.45	−1.16560	−0.582800	+	0.812615i	$0.698043\pi$
$110$	0	0
$111$	2439.53	2.08603
$112$	0	0
$113$	−1204.31	−1.00258	−0.501291	−	0.865279i	$-0.667141\pi$
$113$	−1204.31	−1.00258	−0.501291	+	0.865279i	$0.667141\pi$
$114$	0	0
$115$	665.493	0.539631
$116$	0	0
$117$	4337.05	3.42701
$118$	0	0
$119$	−1788.49	−1.37774
$120$	0	0
$121$	−953.907	−0.716685
$122$	0	0
$123$	−2885.57	−2.11531
$124$	0	0
$125$	−1339.95	−0.958790
$126$	0	0
$127$	375.658	0.262475	0.131237	−	0.991351i	$-0.458105\pi$
$127$	375.658	0.262475	0.131237	+	0.991351i	$0.458105\pi$
$128$	0	0
$129$	−3052.34	−2.08328
$130$	0	0
$131$	2063.01	1.37592	0.687962	−	0.725747i	$-0.258505\pi$
$131$	2063.01	1.37592	0.687962	+	0.725747i	$0.258505\pi$
$132$	0	0
$133$	302.161	0.196998
$134$	0	0
$135$	3881.54	2.47459
$136$	0	0
$137$	1905.46	1.18828	0.594141	−	0.804361i	$-0.297492\pi$
$137$	1905.46	1.18828	0.594141	+	0.804361i	$0.297492\pi$
$138$	0	0
$139$	−1869.18	−1.14059	−0.570293	−	0.821441i	$-0.693170\pi$
$139$	−1869.18	−1.14059	−0.570293	+	0.821441i	$0.693170\pi$
$140$	0	0
$141$	−2022.86	−1.20819
$142$	0	0
$143$	1349.57	0.789208
$144$	0	0
$145$	694.252	0.397617
$146$	0	0
$147$	−851.816	−0.477936
$148$	0	0
$149$	−2041.70	−1.12257	−0.561284	−	0.827623i	$-0.689692\pi$
$149$	−2041.70	−1.12257	−0.561284	+	0.827623i	$0.689692\pi$
$150$	0	0
$151$	1970.09	1.06175	0.530874	−	0.847451i	$-0.321864\pi$
$151$	1970.09	1.06175	0.530874	+	0.847451i	$0.321864\pi$
$152$	0	0
$153$	−7018.19	−3.70841
$154$	0	0
$155$	−2585.56	−1.33986
$156$	0	0
$157$	−945.030	−0.480392	−0.240196	−	0.970724i	$-0.577212\pi$
$157$	−945.030	−0.480392	−0.240196	+	0.970724i	$0.577212\pi$
$158$	0	0
$159$	−3296.66	−1.64429
$160$	0	0
$161$	912.803	0.446826
$162$	0	0
$163$	345.511	0.166028	0.0830139	−	0.996548i	$-0.473545\pi$
$163$	345.511	0.166028	0.0830139	+	0.996548i	$0.473545\pi$
$164$	0	0
$165$	2128.91	1.00446
$166$	0	0
$167$	1554.01	0.720079	0.360040	−	0.932937i	$-0.382763\pi$
$167$	1554.01	0.720079	0.360040	+	0.932937i	$0.382763\pi$
$168$	0	0
$169$	2632.95	1.19843
$170$	0	0
$171$	1185.70	0.530252
$172$	0	0
$173$	2065.34	0.907658	0.453829	−	0.891089i	$-0.350058\pi$
$173$	2065.34	0.907658	0.453829	+	0.891089i	$0.350058\pi$
$174$	0	0
$175$	150.001	0.0647942
$176$	0	0
$177$	533.113	0.226391
$178$	0	0
$179$	3565.91	1.48899	0.744493	−	0.667631i	$-0.232691\pi$
$179$	3565.91	1.48899	0.744493	+	0.667631i	$0.232691\pi$
$180$	0	0
$181$	−1620.30	−0.665393	−0.332697	−	0.943034i	$-0.607958\pi$
$181$	−1620.30	−0.665393	−0.332697	+	0.943034i	$0.607958\pi$
$182$	0	0
$183$	−825.237	−0.333351
$184$	0	0
$185$	2991.41	1.18882
$186$	0	0
$187$	−2183.87	−0.854012
$188$	0	0
$189$	5324.00	2.04902
$190$	0	0
$191$	−2138.73	−0.810226	−0.405113	−	0.914267i	$-0.632768\pi$
$191$	−2138.73	−0.810226	−0.405113	+	0.914267i	$0.632768\pi$
$192$	0	0
$193$	−3704.25	−1.38154	−0.690772	−	0.723073i	$-0.742729\pi$
$193$	−3704.25	−1.38154	−0.690772	+	0.723073i	$0.742729\pi$
$194$	0	0
$195$	7619.12	2.79803
$196$	0	0
$197$	520.563	0.188267	0.0941334	−	0.995560i	$-0.469992\pi$
$197$	520.563	0.188267	0.0941334	+	0.995560i	$0.469992\pi$
$198$	0	0
$199$	−67.7611	−0.0241380	−0.0120690	−	0.999927i	$-0.503842\pi$
$199$	−67.7611	−0.0241380	−0.0120690	+	0.999927i	$0.503842\pi$
$200$	0	0
$201$	−10156.1	−3.56397
$202$	0	0
$203$	952.250	0.329236
$204$	0	0
$205$	−3538.35	−1.20551
$206$	0	0
$207$	3581.91	1.20271
$208$	0	0
$209$	368.959	0.122112
$210$	0	0
$211$	860.566	0.280776	0.140388	−	0.990097i	$-0.455165\pi$
$211$	860.566	0.280776	0.140388	+	0.990097i	$0.455165\pi$
$212$	0	0
$213$	6858.50	2.20628
$214$	0	0
$215$	−3742.85	−1.18726
$216$	0	0
$217$	−3546.41	−1.10943
$218$	0	0
$219$	8409.57	2.59482
$220$	0	0
$221$	−7815.80	−2.37895
$222$	0	0
$223$	6297.10	1.89096	0.945482	−	0.325674i	$-0.105591\pi$
$223$	6297.10	1.89096	0.945482	+	0.325674i	$0.105591\pi$
$224$	0	0
$225$	588.614	0.174404
$226$	0	0
$227$	−10.8392	−0.00316928	−0.00158464	−	0.999999i	$-0.500504\pi$
$227$	−10.8392	−0.00316928	−0.00158464	+	0.999999i	$0.500504\pi$
$228$	0	0
$229$	−4424.48	−1.27676	−0.638379	−	0.769722i	$-0.720395\pi$
$229$	−4424.48	−1.27676	−0.638379	+	0.769722i	$0.720395\pi$
$230$	0	0
$231$	2920.06	0.831713
$232$	0	0
$233$	−2410.23	−0.677681	−0.338840	−	0.940844i	$-0.610035\pi$
$233$	−2410.23	−0.677681	−0.338840	+	0.940844i	$0.610035\pi$
$234$	0	0
$235$	−2480.47	−0.688546
$236$	0	0
$237$	−11903.5	−3.26250
$238$	0	0
$239$	2833.55	0.766893	0.383446	−	0.923563i	$-0.374737\pi$
$239$	2833.55	0.766893	0.383446	+	0.923563i	$0.374737\pi$
$240$	0	0
$241$	3255.23	0.870074	0.435037	−	0.900413i	$-0.356735\pi$
$241$	3255.23	0.870074	0.435037	+	0.900413i	$0.356735\pi$
$242$	0	0
$243$	4959.84	1.30936
$244$	0	0
$245$	−1044.52	−0.272374
$246$	0	0
$247$	1320.46	0.340157
$248$	0	0
$249$	−2152.53	−0.547836
$250$	0	0
$251$	4125.96	1.03756	0.518782	−	0.854907i	$-0.326386\pi$
$251$	4125.96	1.03756	0.518782	+	0.854907i	$0.326386\pi$
$252$	0	0
$253$	1114.59	0.276972
$254$	0	0
$255$	−12329.2	−3.02779
$256$	0	0
$257$	6627.29	1.60856	0.804278	−	0.594253i	$-0.202552\pi$
$257$	6627.29	1.60856	0.804278	+	0.594253i	$0.202552\pi$
$258$	0	0
$259$	4103.07	0.984372
$260$	0	0
$261$	3736.70	0.886192
$262$	0	0
$263$	−823.636	−0.193109	−0.0965543	−	0.995328i	$-0.530782\pi$
$263$	−823.636	−0.193109	−0.0965543	+	0.995328i	$0.530782\pi$
$264$	0	0
$265$	−4042.43	−0.937075
$266$	0	0
$267$	−8770.20	−2.01022
$268$	0	0
$269$	5760.24	1.30561	0.652803	−	0.757527i	$-0.273593\pi$
$269$	5760.24	1.30561	0.652803	+	0.757527i	$0.273593\pi$
$270$	0	0
$271$	8413.28	1.88587	0.942934	−	0.332980i	$-0.108054\pi$
$271$	8413.28	1.88587	0.942934	+	0.332980i	$0.108054\pi$
$272$	0	0
$273$	10450.5	2.31683
$274$	0	0
$275$	183.161	0.0401636
$276$	0	0
$277$	4635.95	1.00558	0.502792	−	0.864407i	$-0.332306\pi$
$277$	4635.95	1.00558	0.502792	+	0.864407i	$0.332306\pi$
$278$	0	0
$279$	−13916.4	−2.98621
$280$	0	0
$281$	−4108.20	−0.872153	−0.436076	−	0.899910i	$-0.643632\pi$
$281$	−4108.20	−0.872153	−0.436076	+	0.899910i	$0.643632\pi$
$282$	0	0
$283$	−5079.96	−1.06704	−0.533519	−	0.845788i	$-0.679131\pi$
$283$	−5079.96	−1.06704	−0.533519	+	0.845788i	$0.679131\pi$
$284$	0	0
$285$	2082.99	0.432932
$286$	0	0
$287$	−4853.27	−0.998187
$288$	0	0
$289$	7734.49	1.57429
$290$	0	0
$291$	5706.26	1.14951
$292$	0	0
$293$	−7988.86	−1.59288	−0.796441	−	0.604716i	$-0.793286\pi$
$293$	−7988.86	−1.59288	−0.796441	+	0.604716i	$0.793286\pi$
$294$	0	0
$295$	653.716	0.129020
$296$	0	0
$297$	6500.95	1.27011
$298$	0	0
$299$	3988.99	0.771537
$300$	0	0
$301$	−5133.77	−0.983074
$302$	0	0
$303$	9791.62	1.85648
$304$	0	0
$305$	−1011.92	−0.189976
$306$	0	0
$307$	7003.80	1.30205	0.651023	−	0.759058i	$-0.274340\pi$
$307$	7003.80	1.30205	0.651023	+	0.759058i	$0.274340\pi$
$308$	0	0
$309$	6306.42	1.16103
$310$	0	0
$311$	2835.42	0.516983	0.258492	−	0.966013i	$-0.416775\pi$
$311$	2835.42	0.516983	0.258492	+	0.966013i	$0.416775\pi$
$312$	0	0
$313$	669.394	0.120883	0.0604415	−	0.998172i	$-0.480749\pi$
$313$	669.394	0.120883	0.0604415	+	0.998172i	$0.480749\pi$
$314$	0	0
$315$	11506.9	2.05823
$316$	0	0
$317$	−5609.44	−0.993872	−0.496936	−	0.867787i	$-0.665542\pi$
$317$	−5609.44	−0.993872	−0.496936	+	0.867787i	$0.665542\pi$
$318$	0	0
$319$	1162.76	0.204082
$320$	0	0
$321$	12947.3	2.25123
$322$	0	0
$323$	−2136.76	−0.368088
$324$	0	0
$325$	655.510	0.111880
$326$	0	0
$327$	−12542.1	−2.12105
$328$	0	0
$329$	−3402.27	−0.570131
$330$	0	0
$331$	10139.4	1.68373	0.841864	−	0.539689i	$-0.181458\pi$
$331$	10139.4	1.68373	0.841864	+	0.539689i	$0.181458\pi$
$332$	0	0
$333$	16100.8	2.64960
$334$	0	0
$335$	−12453.7	−2.03109
$336$	0	0
$337$	946.788	0.153041	0.0765205	−	0.997068i	$-0.475619\pi$
$337$	946.788	0.153041	0.0765205	+	0.997068i	$0.475619\pi$
$338$	0	0
$339$	−11387.3	−1.82440
$340$	0	0
$341$	−4330.40	−0.687696
$342$	0	0
$343$	−6887.49	−1.08423
$344$	0	0
$345$	6292.53	0.981967
$346$	0	0
$347$	−6022.07	−0.931648	−0.465824	−	0.884878i	$-0.654242\pi$
$347$	−6022.07	−0.931648	−0.465824	+	0.884878i	$0.654242\pi$
$348$	0	0
$349$	−2271.70	−0.348427	−0.174214	−	0.984708i	$-0.555738\pi$
$349$	−2271.70	−0.348427	−0.174214	+	0.984708i	$0.555738\pi$
$350$	0	0
$351$	23266.1	3.53805
$352$	0	0
$353$	3519.71	0.530695	0.265348	−	0.964153i	$-0.414513\pi$
$353$	3519.71	0.530695	0.265348	+	0.964153i	$0.414513\pi$
$354$	0	0
$355$	8410.05	1.25735
$356$	0	0
$357$	−16911.0	−2.50707
$358$	0	0
$359$	3485.84	0.512467	0.256234	−	0.966615i	$-0.417518\pi$
$359$	3485.84	0.512467	0.256234	+	0.966615i	$0.417518\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−9019.62	−1.30415
$364$	0	0
$365$	10312.0	1.47878
$366$	0	0
$367$	13100.4	1.86331	0.931654	−	0.363348i	$-0.118366\pi$
$367$	13100.4	1.86331	0.931654	+	0.363348i	$0.118366\pi$
$368$	0	0
$369$	−19044.6	−2.68678
$370$	0	0
$371$	−5544.68	−0.775918
$372$	0	0
$373$	13692.7	1.90075	0.950377	−	0.311101i	$-0.100698\pi$
$373$	13692.7	1.90075	0.950377	+	0.311101i	$0.100698\pi$
$374$	0	0
$375$	−12669.8	−1.74471
$376$	0	0
$377$	4161.38	0.568493
$378$	0	0
$379$	3198.56	0.433506	0.216753	−	0.976226i	$-0.430453\pi$
$379$	3198.56	0.433506	0.216753	+	0.976226i	$0.430453\pi$
$380$	0	0
$381$	3552.02	0.477625
$382$	0	0
$383$	−10724.5	−1.43080	−0.715400	−	0.698716i	$-0.753755\pi$
$383$	−10724.5	−1.43080	−0.715400	+	0.698716i	$0.753755\pi$
$384$	0	0
$385$	3580.64	0.473991
$386$	0	0
$387$	−20145.3	−2.64610
$388$	0	0
$389$	6806.29	0.887128	0.443564	−	0.896243i	$-0.353714\pi$
$389$	6806.29	0.887128	0.443564	+	0.896243i	$0.353714\pi$
$390$	0	0
$391$	−6454.97	−0.834889
$392$	0	0
$393$	19506.7	2.50377
$394$	0	0
$395$	−14596.3	−1.85929
$396$	0	0
$397$	6878.98	0.869637	0.434819	−	0.900518i	$-0.356812\pi$
$397$	6878.98	0.869637	0.434819	+	0.900518i	$0.356812\pi$
$398$	0	0
$399$	2857.07	0.358477
$400$	0	0
$401$	10972.1	1.36639	0.683194	−	0.730237i	$-0.260590\pi$
$401$	10972.1	1.36639	0.683194	+	0.730237i	$0.260590\pi$
$402$	0	0
$403$	−15498.0	−1.91566
$404$	0	0
$405$	17165.6	2.10609
$406$	0	0
$407$	5010.12	0.610178
$408$	0	0
$409$	−13952.8	−1.68685	−0.843424	−	0.537249i	$-0.819464\pi$
$409$	−13952.8	−1.68685	−0.843424	+	0.537249i	$0.819464\pi$
$410$	0	0
$411$	18017.0	2.16232
$412$	0	0
$413$	896.650	0.106831
$414$	0	0
$415$	−2639.48	−0.312210
$416$	0	0
$417$	−17673.9	−2.07553
$418$	0	0
$419$	−3901.77	−0.454925	−0.227463	−	0.973787i	$-0.573043\pi$
$419$	−3901.77	−0.454925	−0.227463	+	0.973787i	$0.573043\pi$
$420$	0	0
$421$	−5739.67	−0.664452	−0.332226	−	0.943200i	$-0.607800\pi$
$421$	−5739.67	−0.664452	−0.332226	+	0.943200i	$0.607800\pi$
$422$	0	0
$423$	−13350.8	−1.53460
$424$	0	0
$425$	−1060.74	−0.121067
$426$	0	0
$427$	−1387.98	−0.157304
$428$	0	0
$429$	12760.8	1.43612
$430$	0	0
$431$	−4016.82	−0.448917	−0.224459	−	0.974484i	$-0.572061\pi$
$431$	−4016.82	−0.448917	−0.224459	+	0.974484i	$0.572061\pi$
$432$	0	0
$433$	−10619.1	−1.17857	−0.589286	−	0.807925i	$-0.700591\pi$
$433$	−10619.1	−1.17857	−0.589286	+	0.807925i	$0.700591\pi$
$434$	0	0
$435$	6564.46	0.723545
$436$	0	0
$437$	1090.55	0.119378
$438$	0	0
$439$	1594.11	0.173309	0.0866545	−	0.996238i	$-0.472382\pi$
$439$	1594.11	0.173309	0.0866545	+	0.996238i	$0.472382\pi$
$440$	0	0
$441$	−5621.94	−0.607056
$442$	0	0
$443$	−3512.03	−0.376662	−0.188331	−	0.982106i	$-0.560308\pi$
$443$	−3512.03	−0.376662	−0.188331	+	0.982106i	$0.560308\pi$
$444$	0	0
$445$	−10754.2	−1.14562
$446$	0	0
$447$	−19305.2	−2.04274
$448$	0	0
$449$	3239.71	0.340515	0.170258	−	0.985400i	$-0.445540\pi$
$449$	3239.71	0.340515	0.170258	+	0.985400i	$0.445540\pi$
$450$	0	0
$451$	−5926.16	−0.618741
$452$	0	0
$453$	18628.1	1.93206
$454$	0	0
$455$	12814.7	1.32036
$456$	0	0
$457$	−7149.11	−0.731775	−0.365887	−	0.930659i	$-0.619234\pi$
$457$	−7149.11	−0.731775	−0.365887	+	0.930659i	$0.619234\pi$
$458$	0	0
$459$	−37649.1	−3.82856
$460$	0	0
$461$	1452.36	0.146731	0.0733657	−	0.997305i	$-0.476626\pi$
$461$	1452.36	0.146731	0.0733657	+	0.997305i	$0.476626\pi$
$462$	0	0
$463$	4887.26	0.490562	0.245281	−	0.969452i	$-0.421120\pi$
$463$	4887.26	0.490562	0.245281	+	0.969452i	$0.421120\pi$
$464$	0	0
$465$	−24447.7	−2.43814
$466$	0	0
$467$	15576.4	1.54344	0.771722	−	0.635960i	$-0.219396\pi$
$467$	15576.4	1.54344	0.771722	+	0.635960i	$0.219396\pi$
$468$	0	0
$469$	−17081.7	−1.68179
$470$	0	0
$471$	−8935.68	−0.874171
$472$	0	0
$473$	−6268.66	−0.609373
$474$	0	0
$475$	179.210	0.0173110
$476$	0	0
$477$	−21757.8	−2.08851
$478$	0	0
$479$	−9155.87	−0.873366	−0.436683	−	0.899616i	$-0.643847\pi$
$479$	−9155.87	−0.873366	−0.436683	+	0.899616i	$0.643847\pi$
$480$	0	0
$481$	17930.6	1.69972
$482$	0	0
$483$	8630.97	0.813090
$484$	0	0
$485$	6997.14	0.655101
$486$	0	0
$487$	−4855.86	−0.451827	−0.225914	−	0.974147i	$-0.572537\pi$
$487$	−4855.86	−0.451827	−0.225914	+	0.974147i	$0.572537\pi$
$488$	0	0
$489$	3266.96	0.302121
$490$	0	0
$491$	−13319.0	−1.22419	−0.612094	−	0.790785i	$-0.709673\pi$
$491$	−13319.0	−1.22419	−0.612094	+	0.790785i	$0.709673\pi$
$492$	0	0
$493$	−6733.92	−0.615173
$494$	0	0
$495$	14050.7	1.27582
$496$	0	0
$497$	11535.4	1.04111
$498$	0	0
$499$	9859.86	0.884545	0.442273	−	0.896881i	$-0.354172\pi$
$499$	9859.86	0.884545	0.442273	+	0.896881i	$0.354172\pi$
$500$	0	0
$501$	14693.9	1.31033
$502$	0	0
$503$	9994.26	0.885929	0.442964	−	0.896539i	$-0.353927\pi$
$503$	9994.26	0.885929	0.442964	+	0.896539i	$0.353927\pi$
$504$	0	0
$505$	12006.7	1.05800
$506$	0	0
$507$	24895.8	2.18079
$508$	0	0
$509$	−9489.87	−0.826387	−0.413194	−	0.910643i	$-0.635587\pi$
$509$	−9489.87	−0.826387	−0.413194	+	0.910643i	$0.635587\pi$
$510$	0	0
$511$	14144.2	1.22446
$512$	0	0
$513$	6360.72	0.547432
$514$	0	0
$515$	7733.07	0.661670
$516$	0	0
$517$	−4154.39	−0.353404
$518$	0	0
$519$	19528.7	1.65167
$520$	0	0
$521$	−14621.5	−1.22952	−0.614759	−	0.788715i	$-0.710747\pi$
$521$	−14621.5	−1.22952	−0.614759	+	0.788715i	$0.710747\pi$
$522$	0	0
$523$	−7500.48	−0.627099	−0.313550	−	0.949572i	$-0.601518\pi$
$523$	−7500.48	−0.627099	−0.313550	+	0.949572i	$0.601518\pi$
$524$	0	0
$525$	1418.32	0.117906
$526$	0	0
$527$	25078.8	2.07296
$528$	0	0
$529$	−8872.54	−0.729230
$530$	0	0
$531$	3518.52	0.287553
$532$	0	0
$533$	−21209.0	−1.72358
$534$	0	0
$535$	15876.2	1.28297
$536$	0	0
$537$	33717.2	2.70951
$538$	0	0
$539$	−1749.39	−0.139799
$540$	0	0
$541$	2413.87	0.191831	0.0959153	−	0.995390i	$-0.469422\pi$
$541$	2413.87	0.191831	0.0959153	+	0.995390i	$0.469422\pi$
$542$	0	0
$543$	−15320.7	−1.21082
$544$	0	0
$545$	−15379.5	−1.20878
$546$	0	0
$547$	−21120.6	−1.65092	−0.825458	−	0.564464i	$-0.809083\pi$
$547$	−21120.6	−1.65092	−0.825458	+	0.564464i	$0.809083\pi$
$548$	0	0
$549$	−5446.52	−0.423410
$550$	0	0
$551$	1137.68	0.0879614
$552$	0	0
$553$	−20020.6	−1.53953
$554$	0	0
$555$	28285.1	2.16331
$556$	0	0
$557$	−16059.4	−1.22165	−0.610823	−	0.791767i	$-0.709161\pi$
$557$	−16059.4	−1.22165	−0.610823	+	0.791767i	$0.709161\pi$
$558$	0	0
$559$	−22434.8	−1.69748
$560$	0	0
$561$	−20649.4	−1.55405
$562$	0	0
$563$	19306.8	1.44527	0.722634	−	0.691231i	$-0.242931\pi$
$563$	19306.8	1.44527	0.722634	+	0.691231i	$0.242931\pi$
$564$	0	0
$565$	−13963.3	−1.03972
$566$	0	0
$567$	23544.7	1.74389
$568$	0	0
$569$	−11379.0	−0.838373	−0.419187	−	0.907900i	$-0.637685\pi$
$569$	−11379.0	−0.838373	−0.419187	+	0.907900i	$0.637685\pi$
$570$	0	0
$571$	−16788.6	−1.23044	−0.615219	−	0.788356i	$-0.710932\pi$
$571$	−16788.6	−1.23044	−0.615219	+	0.788356i	$0.710932\pi$
$572$	0	0
$573$	−20222.7	−1.47437
$574$	0	0
$575$	541.377	0.0392643
$576$	0	0
$577$	4084.31	0.294683	0.147342	−	0.989086i	$-0.452928\pi$
$577$	4084.31	0.294683	0.147342	+	0.989086i	$0.452928\pi$
$578$	0	0
$579$	−35025.4	−2.51400
$580$	0	0
$581$	−3620.37	−0.258516
$582$	0	0
$583$	−6770.42	−0.480964
$584$	0	0
$585$	50285.8	3.55396
$586$	0	0
$587$	10687.2	0.751458	0.375729	−	0.926730i	$-0.377392\pi$
$587$	10687.2	0.751458	0.375729	+	0.926730i	$0.377392\pi$
$588$	0	0
$589$	−4236.99	−0.296404
$590$	0	0
$591$	4922.15	0.342590
$592$	0	0
$593$	−18697.2	−1.29477	−0.647387	−	0.762162i	$-0.724138\pi$
$593$	−18697.2	−1.29477	−0.647387	+	0.762162i	$0.724138\pi$
$594$	0	0
$595$	−20736.7	−1.42877
$596$	0	0
$597$	−640.712	−0.0439239
$598$	0	0
$599$	−3455.17	−0.235683	−0.117842	−	0.993032i	$-0.537598\pi$
$599$	−3455.17	−0.235683	−0.117842	+	0.993032i	$0.537598\pi$
$600$	0	0
$601$	−15030.6	−1.02015	−0.510075	−	0.860130i	$-0.670382\pi$
$601$	−15030.6	−1.02015	−0.510075	+	0.860130i	$0.670382\pi$
$602$	0	0
$603$	−67029.9	−4.52681
$604$	0	0
$605$	−11060.1	−0.743232
$606$	0	0
$607$	2229.47	0.149080	0.0745398	−	0.997218i	$-0.476251\pi$
$607$	2229.47	0.149080	0.0745398	+	0.997218i	$0.476251\pi$
$608$	0	0
$609$	9003.95	0.599111
$610$	0	0
$611$	−14868.1	−0.984448
$612$	0	0
$613$	−14821.0	−0.976534	−0.488267	−	0.872694i	$-0.662371\pi$
$613$	−14821.0	−0.976534	−0.488267	+	0.872694i	$0.662371\pi$
$614$	0	0
$615$	−33456.7	−2.19367
$616$	0	0
$617$	−19301.7	−1.25941	−0.629706	−	0.776833i	$-0.716825\pi$
$617$	−19301.7	−1.25941	−0.629706	+	0.776833i	$0.716825\pi$
$618$	0	0
$619$	−20550.4	−1.33440	−0.667198	−	0.744880i	$-0.732507\pi$
$619$	−20550.4	−1.33440	−0.667198	+	0.744880i	$0.732507\pi$
$620$	0	0
$621$	19215.2	1.24167
$622$	0	0
$623$	−14750.7	−0.948595
$624$	0	0
$625$	−16715.0	−1.06976
$626$	0	0
$627$	3488.67	0.222207
$628$	0	0
$629$	−29015.2	−1.83929
$630$	0	0
$631$	−6019.78	−0.379784	−0.189892	−	0.981805i	$-0.560814\pi$
$631$	−6019.78	−0.379784	−0.189892	+	0.981805i	$0.560814\pi$
$632$	0	0
$633$	8137.04	0.510929
$634$	0	0
$635$	4355.56	0.272197
$636$	0	0
$637$	−6260.88	−0.389427
$638$	0	0
$639$	45265.8	2.80233
$640$	0	0
$641$	554.836	0.0341883	0.0170941	−	0.999854i	$-0.494559\pi$
$641$	554.836	0.0341883	0.0170941	+	0.999854i	$0.494559\pi$
$642$	0	0
$643$	−25135.5	−1.54160	−0.770800	−	0.637077i	$-0.780143\pi$
$643$	−25135.5	−1.54160	−0.770800	+	0.637077i	$0.780143\pi$
$644$	0	0
$645$	−35390.3	−2.16045
$646$	0	0
$647$	17081.4	1.03792	0.518962	−	0.854797i	$-0.326318\pi$
$647$	17081.4	1.03792	0.518962	+	0.854797i	$0.326318\pi$
$648$	0	0
$649$	1094.87	0.0662208
$650$	0	0
$651$	−33532.9	−2.01883
$652$	0	0
$653$	17375.8	1.04130	0.520649	−	0.853771i	$-0.325690\pi$
$653$	17375.8	1.04130	0.520649	+	0.853771i	$0.325690\pi$
$654$	0	0
$655$	23919.5	1.42689
$656$	0	0
$657$	55502.8	3.29584
$658$	0	0
$659$	14944.2	0.883377	0.441688	−	0.897169i	$-0.354380\pi$
$659$	14944.2	0.883377	0.441688	+	0.897169i	$0.354380\pi$
$660$	0	0
$661$	2829.77	0.166513	0.0832565	−	0.996528i	$-0.473468\pi$
$661$	2829.77	0.166513	0.0832565	+	0.996528i	$0.473468\pi$
$662$	0	0
$663$	−73901.9	−4.32898
$664$	0	0
$665$	3503.41	0.204295
$666$	0	0
$667$	3436.83	0.199512
$668$	0	0
$669$	59541.9	3.44099
$670$	0	0
$671$	−1694.81	−0.0975072
$672$	0	0
$673$	−18325.9	−1.04965	−0.524823	−	0.851211i	$-0.675869\pi$
$673$	−18325.9	−1.04965	−0.524823	+	0.851211i	$0.675869\pi$
$674$	0	0
$675$	3157.62	0.180055
$676$	0	0
$677$	−12887.9	−0.731645	−0.365823	−	0.930685i	$-0.619212\pi$
$677$	−12887.9	−0.731645	−0.365823	+	0.930685i	$0.619212\pi$
$678$	0	0
$679$	9597.42	0.542438
$680$	0	0
$681$	−102.490	−0.00576714
$682$	0	0
$683$	21490.8	1.20399	0.601993	−	0.798501i	$-0.294374\pi$
$683$	21490.8	1.20399	0.601993	+	0.798501i	$0.294374\pi$
$684$	0	0
$685$	22092.8	1.23230
$686$	0	0
$687$	−41835.4	−2.32332
$688$	0	0
$689$	−24230.5	−1.33978
$690$	0	0
$691$	7795.43	0.429164	0.214582	−	0.976706i	$-0.431161\pi$
$691$	7795.43	0.429164	0.214582	+	0.976706i	$0.431161\pi$
$692$	0	0
$693$	19272.2	1.05641
$694$	0	0
$695$	−21672.1	−1.18284
$696$	0	0
$697$	34320.3	1.86510
$698$	0	0
$699$	−22789.8	−1.23318
$700$	0	0
$701$	4462.67	0.240446	0.120223	−	0.992747i	$-0.461639\pi$
$701$	4462.67	0.240446	0.120223	+	0.992747i	$0.461639\pi$
$702$	0	0
$703$	4902.05	0.262993
$704$	0	0
$705$	−23454.0	−1.25295
$706$	0	0
$707$	16468.6	0.876050
$708$	0	0
$709$	27093.8	1.43516	0.717581	−	0.696475i	$-0.245249\pi$
$709$	27093.8	1.43516	0.717581	+	0.696475i	$0.245249\pi$
$710$	0	0
$711$	−78562.3	−4.14390
$712$	0	0
$713$	−12799.6	−0.672298
$714$	0	0
$715$	15647.6	0.818442
$716$	0	0
$717$	26792.5	1.39552
$718$	0	0
$719$	−5743.87	−0.297928	−0.148964	−	0.988843i	$-0.547594\pi$
$719$	−5743.87	−0.297928	−0.148964	+	0.988843i	$0.547594\pi$
$720$	0	0
$721$	10606.8	0.547877
$722$	0	0
$723$	30779.6	1.58327
$724$	0	0
$725$	564.772	0.0289312
$726$	0	0
$727$	−33173.9	−1.69237	−0.846185	−	0.532889i	$-0.821106\pi$
$727$	−33173.9	−1.69237	−0.846185	+	0.532889i	$0.821106\pi$
$728$	0	0
$729$	6924.11	0.351781
$730$	0	0
$731$	36303.9	1.83686
$732$	0	0
$733$	−9406.08	−0.473972	−0.236986	−	0.971513i	$-0.576160\pi$
$733$	−9406.08	−0.473972	−0.236986	+	0.971513i	$0.576160\pi$
$734$	0	0
$735$	−9876.37	−0.495640
$736$	0	0
$737$	−20857.9	−1.04248
$738$	0	0
$739$	−11407.9	−0.567858	−0.283929	−	0.958845i	$-0.591638\pi$
$739$	−11407.9	−0.567858	−0.283929	+	0.958845i	$0.591638\pi$
$740$	0	0
$741$	12485.5	0.618985
$742$	0	0
$743$	25469.4	1.25758	0.628790	−	0.777575i	$-0.283551\pi$
$743$	25469.4	1.25758	0.628790	+	0.777575i	$0.283551\pi$
$744$	0	0
$745$	−23672.5	−1.16415
$746$	0	0
$747$	−14206.6	−0.695839
$748$	0	0
$749$	21776.2	1.06233
$750$	0	0
$751$	23819.5	1.15737	0.578686	−	0.815550i	$-0.303566\pi$
$751$	23819.5	1.15737	0.578686	+	0.815550i	$0.303566\pi$
$752$	0	0
$753$	39012.8	1.88806
$754$	0	0
$755$	22842.2	1.10108
$756$	0	0
$757$	−6611.66	−0.317444	−0.158722	−	0.987323i	$-0.550737\pi$
$757$	−6611.66	−0.317444	−0.158722	+	0.987323i	$0.550737\pi$
$758$	0	0
$759$	10539.0	0.504006
$760$	0	0
$761$	29643.4	1.41205	0.706026	−	0.708185i	$-0.250486\pi$
$761$	29643.4	1.41205	0.706026	+	0.708185i	$0.250486\pi$
$762$	0	0
$763$	−21094.8	−1.00089
$764$	0	0
$765$	−81372.3	−3.84578
$766$	0	0
$767$	3918.40	0.184466
$768$	0	0
$769$	−6329.72	−0.296821	−0.148411	−	0.988926i	$-0.547416\pi$
$769$	−6329.72	−0.296821	−0.148411	+	0.988926i	$0.547416\pi$
$770$	0	0
$771$	62664.0	2.92709
$772$	0	0
$773$	24504.8	1.14020	0.570101	−	0.821575i	$-0.306904\pi$
$773$	24504.8	1.14020	0.570101	+	0.821575i	$0.306904\pi$
$774$	0	0
$775$	−2103.35	−0.0974898
$776$	0	0
$777$	38796.4	1.79126
$778$	0	0
$779$	−5798.33	−0.266684
$780$	0	0
$781$	14085.5	0.645349
$782$	0	0
$783$	20045.6	0.914904
$784$	0	0
$785$	−10957.1	−0.498187
$786$	0	0
$787$	−33851.3	−1.53325	−0.766627	−	0.642093i	$-0.778066\pi$
$787$	−33851.3	−1.53325	−0.766627	+	0.642093i	$0.778066\pi$
$788$	0	0
$789$	−7787.84	−0.351400
$790$	0	0
$791$	−19152.4	−0.860910
$792$	0	0
$793$	−6065.52	−0.271618
$794$	0	0
$795$	−38223.0	−1.70520
$796$	0	0
$797$	18282.7	0.812556	0.406278	−	0.913750i	$-0.366826\pi$
$797$	18282.7	0.812556	0.406278	+	0.913750i	$0.366826\pi$
$798$	0	0
$799$	24059.4	1.06528
$800$	0	0
$801$	−57882.9	−2.55330
$802$	0	0
$803$	17270.9	0.759002
$804$	0	0
$805$	10583.5	0.463377
$806$	0	0
$807$	54465.7	2.37582
$808$	0	0
$809$	−12733.6	−0.553386	−0.276693	−	0.960958i	$-0.589238\pi$
$809$	−12733.6	−0.553386	−0.276693	+	0.960958i	$0.589238\pi$
$810$	0	0
$811$	−10126.7	−0.438469	−0.219234	−	0.975672i	$-0.570356\pi$
$811$	−10126.7	−0.438469	−0.219234	+	0.975672i	$0.570356\pi$
$812$	0	0
$813$	79551.3	3.43172
$814$	0	0
$815$	4006.03	0.172178
$816$	0	0
$817$	−6133.44	−0.262646
$818$	0	0
$819$	68973.1	2.94275
$820$	0	0
$821$	1944.98	0.0826802	0.0413401	−	0.999145i	$-0.486837\pi$
$821$	1944.98	0.0826802	0.0413401	+	0.999145i	$0.486837\pi$
$822$	0	0
$823$	31712.7	1.34318	0.671589	−	0.740924i	$-0.265612\pi$
$823$	31712.7	1.34318	0.671589	+	0.740924i	$0.265612\pi$
$824$	0	0
$825$	1731.86	0.0730858
$826$	0	0
$827$	3728.06	0.156756	0.0783781	−	0.996924i	$-0.475026\pi$
$827$	3728.06	0.156756	0.0783781	+	0.996924i	$0.475026\pi$
$828$	0	0
$829$	−13200.4	−0.553037	−0.276518	−	0.961009i	$-0.589181\pi$
$829$	−13200.4	−0.553037	−0.276518	+	0.961009i	$0.589181\pi$
$830$	0	0
$831$	43834.9	1.82986
$832$	0	0
$833$	10131.3	0.421404
$834$	0	0
$835$	18018.0	0.746753
$836$	0	0
$837$	−74654.6	−3.08296
$838$	0	0
$839$	−11445.0	−0.470949	−0.235475	−	0.971881i	$-0.575664\pi$
$839$	−11445.0	−0.470949	−0.235475	+	0.971881i	$0.575664\pi$
$840$	0	0
$841$	−20803.7	−0.852993
$842$	0	0
$843$	−38844.9	−1.58706
$844$	0	0
$845$	30527.7	1.24282
$846$	0	0
$847$	−15170.2	−0.615413
$848$	0	0
$849$	−48033.2	−1.94169
$850$	0	0
$851$	14808.7	0.596515
$852$	0	0
$853$	8701.48	0.349276	0.174638	−	0.984633i	$-0.444124\pi$
$853$	8701.48	0.349276	0.174638	+	0.984633i	$0.444124\pi$
$854$	0	0
$855$	13747.6	0.549894
$856$	0	0
$857$	−720.765	−0.0287291	−0.0143646	−	0.999897i	$-0.504573\pi$
$857$	−720.765	−0.0287291	−0.0143646	+	0.999897i	$0.504573\pi$
$858$	0	0
$859$	−4678.63	−0.185836	−0.0929179	−	0.995674i	$-0.529619\pi$
$859$	−4678.63	−0.185836	−0.0929179	+	0.995674i	$0.529619\pi$
$860$	0	0
$861$	−45889.9	−1.81640
$862$	0	0
$863$	7009.92	0.276501	0.138250	−	0.990397i	$-0.455852\pi$
$863$	7009.92	0.276501	0.138250	+	0.990397i	$0.455852\pi$
$864$	0	0
$865$	23946.5	0.941280
$866$	0	0
$867$	73133.0	2.86474
$868$	0	0
$869$	−24446.4	−0.954302
$870$	0	0
$871$	−74647.8	−2.90396
$872$	0	0
$873$	37661.0	1.46006
$874$	0	0
$875$	−21309.5	−0.823307
$876$	0	0
$877$	22737.0	0.875453	0.437727	−	0.899108i	$-0.355784\pi$
$877$	22737.0	0.875453	0.437727	+	0.899108i	$0.355784\pi$
$878$	0	0
$879$	−75538.3	−2.89857
$880$	0	0
$881$	7697.02	0.294346	0.147173	−	0.989111i	$-0.452983\pi$
$881$	7697.02	0.294346	0.147173	+	0.989111i	$0.452983\pi$
$882$	0	0
$883$	7842.91	0.298907	0.149454	−	0.988769i	$-0.452249\pi$
$883$	7842.91	0.298907	0.149454	+	0.988769i	$0.452249\pi$
$884$	0	0
$885$	6181.18	0.234777
$886$	0	0
$887$	41079.7	1.55504	0.777521	−	0.628857i	$-0.216477\pi$
$887$	41079.7	1.55504	0.777521	+	0.628857i	$0.216477\pi$
$888$	0	0
$889$	5974.18	0.225385
$890$	0	0
$891$	28749.6	1.08097
$892$	0	0
$893$	−4064.78	−0.152321
$894$	0	0
$895$	41344.9	1.54414
$896$	0	0
$897$	37717.7	1.40397
$898$	0	0
$899$	−13352.7	−0.495370
$900$	0	0
$901$	39209.7	1.44979
$902$	0	0
$903$	−48542.0	−1.78890
$904$	0	0
$905$	−18786.6	−0.690041
$906$	0	0
$907$	38639.4	1.41455	0.707277	−	0.706937i	$-0.249924\pi$
$907$	38639.4	1.41455	0.707277	+	0.706937i	$0.249924\pi$
$908$	0	0
$909$	64624.2	2.35803
$910$	0	0
$911$	18398.0	0.669103	0.334551	−	0.942377i	$-0.391415\pi$
$911$	18398.0	0.669103	0.334551	+	0.942377i	$0.391415\pi$
$912$	0	0
$913$	−4420.70	−0.160245
$914$	0	0
$915$	−9568.19	−0.345699
$916$	0	0
$917$	32808.5	1.18150
$918$	0	0
$919$	24502.3	0.879495	0.439748	−	0.898121i	$-0.355068\pi$
$919$	24502.3	0.879495	0.439748	+	0.898121i	$0.355068\pi$
$920$	0	0
$921$	66224.1	2.36934
$922$	0	0
$923$	50410.2	1.79770
$924$	0	0
$925$	2433.50	0.0865006
$926$	0	0
$927$	41622.0	1.47470
$928$	0	0
$929$	−47616.6	−1.68165	−0.840823	−	0.541310i	$-0.817928\pi$
$929$	−47616.6	−1.68165	−0.840823	+	0.541310i	$0.817928\pi$
$930$	0	0
$931$	−1711.66	−0.0602549
$932$	0	0
$933$	26810.1	0.940755
$934$	0	0
$935$	−25320.8	−0.885646
$936$	0	0
$937$	21360.6	0.744739	0.372369	−	0.928085i	$-0.378545\pi$
$937$	21360.6	0.744739	0.372369	+	0.928085i	$0.378545\pi$
$938$	0	0
$939$	6329.42	0.219971
$940$	0	0
$941$	−23923.3	−0.828776	−0.414388	−	0.910100i	$-0.636004\pi$
$941$	−23923.3	−0.828776	−0.414388	+	0.910100i	$0.636004\pi$
$942$	0	0
$943$	−17516.3	−0.604887
$944$	0	0
$945$	61729.0	2.12492
$946$	0	0
$947$	23330.5	0.800569	0.400285	−	0.916391i	$-0.368911\pi$
$947$	23330.5	0.800569	0.400285	+	0.916391i	$0.368911\pi$
$948$	0	0
$949$	61810.7	2.11429
$950$	0	0
$951$	−53039.8	−1.80855
$952$	0	0
$953$	−25756.2	−0.875472	−0.437736	−	0.899104i	$-0.644220\pi$
$953$	−25756.2	−0.875472	−0.437736	+	0.899104i	$0.644220\pi$
$954$	0	0
$955$	−24797.5	−0.840238
$956$	0	0
$957$	10994.4	0.371368
$958$	0	0
$959$	30303.0	1.02037
$960$	0	0
$961$	19937.8	0.669255
$962$	0	0
$963$	85451.3	2.85943
$964$	0	0
$965$	−42948.9	−1.43272
$966$	0	0
$967$	2488.74	0.0827635	0.0413818	−	0.999143i	$-0.486824\pi$
$967$	2488.74	0.0827635	0.0413818	+	0.999143i	$0.486824\pi$
$968$	0	0
$969$	−20204.0	−0.669811
$970$	0	0
$971$	4909.90	0.162272	0.0811360	−	0.996703i	$-0.474145\pi$
$971$	4909.90	0.162272	0.0811360	+	0.996703i	$0.474145\pi$
$972$	0	0
$973$	−29726.0	−0.979415
$974$	0	0
$975$	6198.14	0.203589
$976$	0	0
$977$	47452.4	1.55388	0.776938	−	0.629578i	$-0.216772\pi$
$977$	47452.4	1.55388	0.776938	+	0.629578i	$0.216772\pi$
$978$	0	0
$979$	−18011.6	−0.588000
$980$	0	0
$981$	−82777.5	−2.69407
$982$	0	0
$983$	−27802.2	−0.902088	−0.451044	−	0.892502i	$-0.648948\pi$
$983$	−27802.2	−0.902088	−0.451044	+	0.892502i	$0.648948\pi$
$984$	0	0
$985$	6035.66	0.195241
$986$	0	0
$987$	−32170.0	−1.03747
$988$	0	0
$989$	−18528.6	−0.595728
$990$	0	0
$991$	−26200.5	−0.839845	−0.419922	−	0.907560i	$-0.637943\pi$
$991$	−26200.5	−0.839845	−0.419922	+	0.907560i	$0.637943\pi$
$992$	0	0
$993$	95873.0	3.06388
$994$	0	0
$995$	−785.655	−0.0250321
$996$	0	0
$997$	−25254.6	−0.802227	−0.401113	−	0.916028i	$-0.631377\pi$
$997$	−25254.6	−0.802227	−0.401113	+	0.916028i	$0.631377\pi$
$998$	0	0
$999$	86372.7	2.73545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bf.1.7		7
4.3	odd	2		1216.4.a.bg.1.1		7
8.3	odd	2		608.4.a.j.1.7	✓	7
8.5	even	2		608.4.a.k.1.1	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.7	✓	7	8.3	odd	2
608.4.a.k.1.1	yes	7	8.5	even	2
1216.4.a.bf.1.7		7	1.1	even	1	trivial
1216.4.a.bg.1.1		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+9.45545 q^{3} +11.5945 q^{5} +15.9032 q^{7} +62.4055 q^{9} +19.4189 q^{11} +69.4979 q^{13} +109.631 q^{15} -112.461 q^{17} +19.0000 q^{19} +150.372 q^{21} +57.3974 q^{23} +9.43209 q^{25} +334.775 q^{27} +59.8778 q^{29} -223.000 q^{31} +183.614 q^{33} +184.390 q^{35} +258.002 q^{37} +657.133 q^{39} -305.175 q^{41} -322.813 q^{43} +723.559 q^{45} -213.936 q^{47} -90.0873 q^{49} -1063.37 q^{51} -348.651 q^{53} +225.152 q^{55} +179.654 q^{57} +56.3816 q^{59} -87.2763 q^{61} +992.449 q^{63} +805.792 q^{65} -1074.10 q^{67} +542.718 q^{69} +725.349 q^{71} +889.389 q^{73} +89.1846 q^{75} +308.823 q^{77} -1258.90 q^{79} +1480.50 q^{81} -227.650 q^{83} -1303.93 q^{85} +566.171 q^{87} -927.529 q^{89} +1105.24 q^{91} -2108.56 q^{93} +220.295 q^{95} +603.489 q^{97} +1211.84 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 17 q^{5} + 42 q^{7} + 86 q^{9} + 33 q^{11} + 35 q^{13} + 120 q^{15} + 66 q^{17} + 133 q^{19} - 33 q^{21} + 389 q^{23} + 44 q^{25} + 39 q^{27} - 233 q^{29} + 158 q^{31} - 206 q^{33} - 123 q^{35}+ \cdots + 1591 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 17 * q^5 + 42 * q^7 + 86 * q^9 + 33 * q^11 + 35 * q^13 + 120 * q^15 + 66 * q^17 + 133 * q^19 - 33 * q^21 + 389 * q^23 + 44 * q^25 + 39 * q^27 - 233 * q^29 + 158 * q^31 - 206 * q^33 - 123 * q^35 + 436 * q^37 + 807 * q^39 - 94 * q^41 - 645 * q^43 - 103 * q^45 + 1451 * q^47 + 93 * q^49 - 1741 * q^51 - 3 * q^53 + 1971 * q^55 - 57 * q^57 - 297 * q^59 - 93 * q^61 + 2999 * q^63 - 788 * q^65 - 1641 * q^67 - 945 * q^69 + 2392 * q^71 + 324 * q^73 - 1909 * q^75 + 711 * q^77 + 2492 * q^79 + 143 * q^81 - 310 * q^83 - 2353 * q^85 + 4795 * q^87 - 440 * q^89 + 107 * q^91 - 900 * q^93 + 323 * q^95 - 532 * q^97 + 1591 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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