LMFDB - Embedded newform 1160.4.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,4,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, 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("1160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1160.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:	$68.4422156067$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 123x^{7} + 288x^{6} + 5092x^{5} - 8936x^{4} - 85488x^{3} + 115600x^{2} + 478480x - 646400 $ x^9 - 3x^8 - 123x^7 + 288x^6 + 5092x^5 - 8936x^4 - 85488x^3 + 115600x^2 + 478480x - 646400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.20744$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.20744 q^{3} -5.00000 q^{5} +15.1907 q^{7} +0.117398 q^{9} -3.79422 q^{11} +21.8525 q^{13} +26.0372 q^{15} +5.42391 q^{17} -129.830 q^{19} -79.1048 q^{21} +58.3381 q^{23} +25.0000 q^{25} +139.989 q^{27} +29.0000 q^{29} -36.2252 q^{31} +19.7582 q^{33} -75.9536 q^{35} +182.678 q^{37} -113.796 q^{39} -189.168 q^{41} +238.654 q^{43} -0.586992 q^{45} +94.4059 q^{47} -112.242 q^{49} -28.2447 q^{51} +123.135 q^{53} +18.9711 q^{55} +676.083 q^{57} +836.442 q^{59} +667.146 q^{61} +1.78337 q^{63} -109.263 q^{65} -657.774 q^{67} -303.792 q^{69} -313.728 q^{71} -573.529 q^{73} -130.186 q^{75} -57.6370 q^{77} +408.329 q^{79} -732.156 q^{81} -273.810 q^{83} -27.1196 q^{85} -151.016 q^{87} -1441.55 q^{89} +331.955 q^{91} +188.641 q^{93} +649.152 q^{95} -73.6142 q^{97} -0.445436 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} - 45 q^{5} - 27 q^{7} + 12 q^{9} + 88 q^{11} - 65 q^{13} + 15 q^{15} + 43 q^{17} + 88 q^{19} + 212 q^{21} - 73 q^{23} + 225 q^{25} - 108 q^{27} + 261 q^{29} - 3 q^{31} + 76 q^{33} + 135 q^{35}+ \cdots - 442 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 - 45 * q^5 - 27 * q^7 + 12 * q^9 + 88 * q^11 - 65 * q^13 + 15 * q^15 + 43 * q^17 + 88 * q^19 + 212 * q^21 - 73 * q^23 + 225 * q^25 - 108 * q^27 + 261 * q^29 - 3 * q^31 + 76 * q^33 + 135 * q^35 + 50 * q^37 - 49 * q^39 - 184 * q^41 - 389 * q^43 - 60 * q^45 - 216 * q^47 + 628 * q^49 - 218 * q^51 - 1003 * q^53 - 440 * q^55 - 98 * q^57 + 477 * q^59 - 203 * q^61 - 744 * q^63 + 325 * q^65 - 1028 * q^67 - 903 * q^69 - 484 * q^71 - 1905 * q^73 - 75 * q^75 - 3202 * q^77 - 869 * q^79 - 3151 * q^81 - 850 * q^83 - 215 * q^85 - 87 * q^87 - 804 * q^89 - 8 * q^91 - 3822 * q^93 - 440 * q^95 - 1795 * q^97 - 442 * 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
$3$	−5.20744	−1.00217	−0.501086	−	0.865398i	$-0.667066\pi$
$3$	−5.20744	−1.00217	−0.501086	+	0.865398i	$0.667066\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	15.1907	0.820222	0.410111	−	0.912036i	$-0.365490\pi$
$7$	15.1907	0.820222	0.410111	+	0.912036i	$0.365490\pi$
$8$	0	0
$9$	0.117398	0.00434809
$10$	0	0
$11$	−3.79422	−0.104000	−0.0520000	−	0.998647i	$-0.516560\pi$
$11$	−3.79422	−0.104000	−0.0520000	+	0.998647i	$0.516560\pi$
$12$	0	0
$13$	21.8525	0.466215	0.233107	−	0.972451i	$-0.425111\pi$
$13$	21.8525	0.466215	0.233107	+	0.972451i	$0.425111\pi$
$14$	0	0
$15$	26.0372	0.448185
$16$	0	0
$17$	5.42391	0.0773819	0.0386909	−	0.999251i	$-0.487681\pi$
$17$	5.42391	0.0773819	0.0386909	+	0.999251i	$0.487681\pi$
$18$	0	0
$19$	−129.830	−1.56764	−0.783819	−	0.620989i	$-0.786731\pi$
$19$	−129.830	−1.56764	−0.783819	+	0.620989i	$0.786731\pi$
$20$	0	0
$21$	−79.1048	−0.822004
$22$	0	0
$23$	58.3381	0.528884	0.264442	−	0.964402i	$-0.414812\pi$
$23$	58.3381	0.528884	0.264442	+	0.964402i	$0.414812\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	139.989	0.997814
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−36.2252	−0.209879	−0.104939	−	0.994479i	$-0.533465\pi$
$31$	−36.2252	−0.209879	−0.104939	+	0.994479i	$0.533465\pi$
$32$	0	0
$33$	19.7582	0.104226
$34$	0	0
$35$	−75.9536	−0.366815
$36$	0	0
$37$	182.678	0.811679	0.405839	−	0.913944i	$-0.366979\pi$
$37$	182.678	0.811679	0.405839	+	0.913944i	$0.366979\pi$
$38$	0	0
$39$	−113.796	−0.467227
$40$	0	0
$41$	−189.168	−0.720561	−0.360281	−	0.932844i	$-0.617319\pi$
$41$	−189.168	−0.720561	−0.360281	+	0.932844i	$0.617319\pi$
$42$	0	0
$43$	238.654	0.846382	0.423191	−	0.906041i	$-0.360910\pi$
$43$	238.654	0.846382	0.423191	+	0.906041i	$0.360910\pi$
$44$	0	0
$45$	−0.586992	−0.00194453
$46$	0	0
$47$	94.4059	0.292990	0.146495	−	0.989211i	$-0.453201\pi$
$47$	94.4059	0.292990	0.146495	+	0.989211i	$0.453201\pi$
$48$	0	0
$49$	−112.242	−0.327235
$50$	0	0
$51$	−28.2447	−0.0775499
$52$	0	0
$53$	123.135	0.319129	0.159565	−	0.987188i	$-0.448991\pi$
$53$	123.135	0.319129	0.159565	+	0.987188i	$0.448991\pi$
$54$	0	0
$55$	18.9711	0.0465102
$56$	0	0
$57$	676.083	1.57104
$58$	0	0
$59$	836.442	1.84569	0.922843	−	0.385177i	$-0.125859\pi$
$59$	836.442	1.84569	0.922843	+	0.385177i	$0.125859\pi$
$60$	0	0
$61$	667.146	1.40032	0.700158	−	0.713988i	$-0.253113\pi$
$61$	667.146	1.40032	0.700158	+	0.713988i	$0.253113\pi$
$62$	0	0
$63$	1.78337	0.00356640
$64$	0	0
$65$	−109.263	−0.208498
$66$	0	0
$67$	−657.774	−1.19940	−0.599700	−	0.800225i	$-0.704714\pi$
$67$	−657.774	−1.19940	−0.599700	+	0.800225i	$0.704714\pi$
$68$	0	0
$69$	−303.792	−0.530032
$70$	0	0
$71$	−313.728	−0.524404	−0.262202	−	0.965013i	$-0.584449\pi$
$71$	−313.728	−0.524404	−0.262202	+	0.965013i	$0.584449\pi$
$72$	0	0
$73$	−573.529	−0.919542	−0.459771	−	0.888038i	$-0.652068\pi$
$73$	−573.529	−0.919542	−0.459771	+	0.888038i	$0.652068\pi$
$74$	0	0
$75$	−130.186	−0.200434
$76$	0	0
$77$	−57.6370	−0.0853031
$78$	0	0
$79$	408.329	0.581527	0.290764	−	0.956795i	$-0.406091\pi$
$79$	408.329	0.581527	0.290764	+	0.956795i	$0.406091\pi$
$80$	0	0
$81$	−732.156	−1.00433
$82$	0	0
$83$	−273.810	−0.362103	−0.181051	−	0.983474i	$-0.557950\pi$
$83$	−273.810	−0.362103	−0.181051	+	0.983474i	$0.557950\pi$
$84$	0	0
$85$	−27.1196	−0.0346062
$86$	0	0
$87$	−151.016	−0.186099
$88$	0	0
$89$	−1441.55	−1.71690	−0.858450	−	0.512898i	$-0.828572\pi$
$89$	−1441.55	−1.71690	−0.858450	+	0.512898i	$0.828572\pi$
$90$	0	0
$91$	331.955	0.382400
$92$	0	0
$93$	188.641	0.210335
$94$	0	0
$95$	649.152	0.701069
$96$	0	0
$97$	−73.6142	−0.0770555	−0.0385278	−	0.999258i	$-0.512267\pi$
$97$	−73.6142	−0.0770555	−0.0385278	+	0.999258i	$0.512267\pi$
$98$	0	0
$99$	−0.445436	−0.000452202 0
$100$	0	0
$101$	1098.16	1.08189	0.540946	−	0.841057i	$-0.318066\pi$
$101$	1098.16	1.08189	0.540946	+	0.841057i	$0.318066\pi$
$102$	0	0
$103$	−1053.69	−1.00799	−0.503994	−	0.863707i	$-0.668137\pi$
$103$	−1053.69	−1.00799	−0.503994	+	0.863707i	$0.668137\pi$
$104$	0	0
$105$	395.524	0.367611
$106$	0	0
$107$	−479.844	−0.433536	−0.216768	−	0.976223i	$-0.569551\pi$
$107$	−479.844	−0.433536	−0.216768	+	0.976223i	$0.569551\pi$
$108$	0	0
$109$	−1.18932	−0.00104510	−0.000522551	−	1.00000i	$-0.500166\pi$
$109$	−1.18932	−0.00104510	−0.000522551		1.00000i	$0.500166\pi$
$110$	0	0
$111$	−951.285	−0.813441
$112$	0	0
$113$	−1728.39	−1.43888	−0.719441	−	0.694554i	$-0.755602\pi$
$113$	−1728.39	−1.43888	−0.719441	+	0.694554i	$0.755602\pi$
$114$	0	0
$115$	−291.690	−0.236524
$116$	0	0
$117$	2.56545	0.00202715
$118$	0	0
$119$	82.3932	0.0634703
$120$	0	0
$121$	−1316.60	−0.989184
$122$	0	0
$123$	985.078	0.722126
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2195.47	1.53399	0.766994	−	0.641655i	$-0.221752\pi$
$127$	2195.47	1.53399	0.766994	+	0.641655i	$0.221752\pi$
$128$	0	0
$129$	−1242.78	−0.848220
$130$	0	0
$131$	1858.21	1.23933	0.619667	−	0.784865i	$-0.287268\pi$
$131$	1858.21	1.23933	0.619667	+	0.784865i	$0.287268\pi$
$132$	0	0
$133$	−1972.22	−1.28581
$134$	0	0
$135$	−699.947	−0.446236
$136$	0	0
$137$	−2328.20	−1.45191	−0.725956	−	0.687741i	$-0.758602\pi$
$137$	−2328.20	−1.45191	−0.725956	+	0.687741i	$0.758602\pi$
$138$	0	0
$139$	982.079	0.599272	0.299636	−	0.954054i	$-0.403135\pi$
$139$	982.079	0.599272	0.299636	+	0.954054i	$0.403135\pi$
$140$	0	0
$141$	−491.613	−0.293626
$142$	0	0
$143$	−82.9132	−0.0484864
$144$	0	0
$145$	−145.000	−0.0830455
$146$	0	0
$147$	584.492	0.327946
$148$	0	0
$149$	−2477.25	−1.36204	−0.681020	−	0.732265i	$-0.738463\pi$
$149$	−2477.25	−1.36204	−0.681020	+	0.732265i	$0.738463\pi$
$150$	0	0
$151$	−1708.79	−0.920922	−0.460461	−	0.887680i	$-0.652316\pi$
$151$	−1708.79	−0.920922	−0.460461	+	0.887680i	$0.652316\pi$
$152$	0	0
$153$	0.636759	0.000336463 0
$154$	0	0
$155$	181.126	0.0938607
$156$	0	0
$157$	−1146.64	−0.582879	−0.291440	−	0.956589i	$-0.594134\pi$
$157$	−1146.64	−0.582879	−0.291440	+	0.956589i	$0.594134\pi$
$158$	0	0
$159$	−641.216	−0.319822
$160$	0	0
$161$	886.198	0.433802
$162$	0	0
$163$	−3764.73	−1.80906	−0.904529	−	0.426413i	$-0.859777\pi$
$163$	−3764.73	−1.80906	−0.904529	+	0.426413i	$0.859777\pi$
$164$	0	0
$165$	−98.7908	−0.0466112
$166$	0	0
$167$	111.820	0.0518139	0.0259069	−	0.999664i	$-0.491753\pi$
$167$	111.820	0.0518139	0.0259069	+	0.999664i	$0.491753\pi$
$168$	0	0
$169$	−1719.47	−0.782644
$170$	0	0
$171$	−15.2419	−0.00681624
$172$	0	0
$173$	−2917.03	−1.28195	−0.640975	−	0.767562i	$-0.721470\pi$
$173$	−2917.03	−1.28195	−0.640975	+	0.767562i	$0.721470\pi$
$174$	0	0
$175$	379.768	0.164044
$176$	0	0
$177$	−4355.72	−1.84969
$178$	0	0
$179$	−4114.12	−1.71790	−0.858948	−	0.512062i	$-0.828882\pi$
$179$	−4114.12	−1.71790	−0.858948	+	0.512062i	$0.828882\pi$
$180$	0	0
$181$	−3406.11	−1.39875	−0.699376	−	0.714754i	$-0.746539\pi$
$181$	−3406.11	−1.39875	−0.699376	+	0.714754i	$0.746539\pi$
$182$	0	0
$183$	−3474.12	−1.40336
$184$	0	0
$185$	−913.391	−0.362994
$186$	0	0
$187$	−20.5795	−0.00804772
$188$	0	0
$189$	2126.54	0.818429
$190$	0	0
$191$	4998.15	1.89347	0.946737	−	0.322009i	$-0.104358\pi$
$191$	4998.15	1.89347	0.946737	+	0.322009i	$0.104358\pi$
$192$	0	0
$193$	−2560.69	−0.955038	−0.477519	−	0.878621i	$-0.658464\pi$
$193$	−2560.69	−0.955038	−0.477519	+	0.878621i	$0.658464\pi$
$194$	0	0
$195$	568.978	0.208950
$196$	0	0
$197$	2770.55	1.00200	0.500998	−	0.865448i	$-0.332966\pi$
$197$	2770.55	1.00200	0.500998	+	0.865448i	$0.332966\pi$
$198$	0	0
$199$	1963.69	0.699509	0.349754	−	0.936841i	$-0.386265\pi$
$199$	1963.69	0.699509	0.349754	+	0.936841i	$0.386265\pi$
$200$	0	0
$201$	3425.31	1.20201
$202$	0	0
$203$	440.531	0.152311
$204$	0	0
$205$	945.838	0.322245
$206$	0	0
$207$	6.84880	0.00229964
$208$	0	0
$209$	492.605	0.163034
$210$	0	0
$211$	−2382.01	−0.777176	−0.388588	−	0.921412i	$-0.627037\pi$
$211$	−2382.01	−0.777176	−0.388588	+	0.921412i	$0.627037\pi$
$212$	0	0
$213$	1633.72	0.525543
$214$	0	0
$215$	−1193.27	−0.378513
$216$	0	0
$217$	−550.288	−0.172147
$218$	0	0
$219$	2986.62	0.921539
$220$	0	0
$221$	118.526	0.0360766
$222$	0	0
$223$	−2364.04	−0.709901	−0.354950	−	0.934885i	$-0.615502\pi$
$223$	−2364.04	−0.709901	−0.354950	+	0.934885i	$0.615502\pi$
$224$	0	0
$225$	2.93496	0.000869618 0
$226$	0	0
$227$	6059.37	1.77169	0.885847	−	0.463978i	$-0.153578\pi$
$227$	6059.37	1.77169	0.885847	+	0.463978i	$0.153578\pi$
$228$	0	0
$229$	−3537.38	−1.02077	−0.510385	−	0.859946i	$-0.670497\pi$
$229$	−3537.38	−1.02077	−0.510385	+	0.859946i	$0.670497\pi$
$230$	0	0
$231$	300.141	0.0854884
$232$	0	0
$233$	−3295.40	−0.926561	−0.463280	−	0.886212i	$-0.653328\pi$
$233$	−3295.40	−0.926561	−0.463280	+	0.886212i	$0.653328\pi$
$234$	0	0
$235$	−472.029	−0.131029
$236$	0	0
$237$	−2126.35	−0.582790
$238$	0	0
$239$	2581.89	0.698781	0.349391	−	0.936977i	$-0.386389\pi$
$239$	2581.89	0.698781	0.349391	+	0.936977i	$0.386389\pi$
$240$	0	0
$241$	−1556.14	−0.415934	−0.207967	−	0.978136i	$-0.566685\pi$
$241$	−1556.14	−0.415934	−0.207967	+	0.978136i	$0.566685\pi$
$242$	0	0
$243$	32.9409	0.00869612
$244$	0	0
$245$	561.209	0.146344
$246$	0	0
$247$	−2837.12	−0.730856
$248$	0	0
$249$	1425.85	0.362889
$250$	0	0
$251$	−4075.13	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$251$	−4075.13	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$252$	0	0
$253$	−221.347	−0.0550039
$254$	0	0
$255$	141.223	0.0346814
$256$	0	0
$257$	2936.60	0.712763	0.356382	−	0.934340i	$-0.384010\pi$
$257$	2936.60	0.712763	0.356382	+	0.934340i	$0.384010\pi$
$258$	0	0
$259$	2775.01	0.665757
$260$	0	0
$261$	3.40456	0.000807420 0
$262$	0	0
$263$	7894.34	1.85090	0.925448	−	0.378874i	$-0.123688\pi$
$263$	7894.34	1.85090	0.925448	+	0.378874i	$0.123688\pi$
$264$	0	0
$265$	−615.673	−0.142719
$266$	0	0
$267$	7506.78	1.72063
$268$	0	0
$269$	309.360	0.0701191	0.0350596	−	0.999385i	$-0.488838\pi$
$269$	309.360	0.0701191	0.0350596	+	0.999385i	$0.488838\pi$
$270$	0	0
$271$	151.636	0.0339898	0.0169949	−	0.999856i	$-0.494590\pi$
$271$	151.636	0.0339898	0.0169949	+	0.999856i	$0.494590\pi$
$272$	0	0
$273$	−1728.64	−0.383230
$274$	0	0
$275$	−94.8555	−0.0208000
$276$	0	0
$277$	−4067.92	−0.882373	−0.441187	−	0.897415i	$-0.645442\pi$
$277$	−4067.92	−0.882373	−0.441187	+	0.897415i	$0.645442\pi$
$278$	0	0
$279$	−4.25279	−0.000912572 0
$280$	0	0
$281$	−516.096	−0.109565	−0.0547824	−	0.998498i	$-0.517447\pi$
$281$	−516.096	−0.109565	−0.0547824	+	0.998498i	$0.517447\pi$
$282$	0	0
$283$	−8451.75	−1.77528	−0.887640	−	0.460537i	$-0.847657\pi$
$283$	−8451.75	−1.77528	−0.887640	+	0.460537i	$0.847657\pi$
$284$	0	0
$285$	−3380.42	−0.702592
$286$	0	0
$287$	−2873.59	−0.591020
$288$	0	0
$289$	−4883.58	−0.994012
$290$	0	0
$291$	383.341	0.0772229
$292$	0	0
$293$	5462.68	1.08919	0.544595	−	0.838699i	$-0.316683\pi$
$293$	5462.68	1.08919	0.544595	+	0.838699i	$0.316683\pi$
$294$	0	0
$295$	−4182.21	−0.825416
$296$	0	0
$297$	−531.151	−0.103773
$298$	0	0
$299$	1274.83	0.246574
$300$	0	0
$301$	3625.33	0.694221
$302$	0	0
$303$	−5718.61	−1.08424
$304$	0	0
$305$	−3335.73	−0.626241
$306$	0	0
$307$	2136.17	0.397127	0.198563	−	0.980088i	$-0.436372\pi$
$307$	2136.17	0.397127	0.198563	+	0.980088i	$0.436372\pi$
$308$	0	0
$309$	5487.00	1.01018
$310$	0	0
$311$	−6303.14	−1.14926	−0.574628	−	0.818415i	$-0.694853\pi$
$311$	−6303.14	−1.14926	−0.574628	+	0.818415i	$0.694853\pi$
$312$	0	0
$313$	289.779	0.0523300	0.0261650	−	0.999658i	$-0.491670\pi$
$313$	289.779	0.0523300	0.0261650	+	0.999658i	$0.491670\pi$
$314$	0	0
$315$	−8.91684	−0.00159494
$316$	0	0
$317$	−1427.87	−0.252989	−0.126494	−	0.991967i	$-0.540373\pi$
$317$	−1427.87	−0.252989	−0.126494	+	0.991967i	$0.540373\pi$
$318$	0	0
$319$	−110.032	−0.0193123
$320$	0	0
$321$	2498.76	0.434477
$322$	0	0
$323$	−704.188	−0.121307
$324$	0	0
$325$	546.313	0.0932430
$326$	0	0
$327$	6.19330	0.00104737
$328$	0	0
$329$	1434.09	0.240317
$330$	0	0
$331$	4168.63	0.692232	0.346116	−	0.938192i	$-0.387500\pi$
$331$	4168.63	0.692232	0.346116	+	0.938192i	$0.387500\pi$
$332$	0	0
$333$	21.4461	0.00352925
$334$	0	0
$335$	3288.87	0.536388
$336$	0	0
$337$	−1108.83	−0.179233	−0.0896166	−	0.995976i	$-0.528564\pi$
$337$	−1108.83	−0.179233	−0.0896166	+	0.995976i	$0.528564\pi$
$338$	0	0
$339$	9000.50	1.44201
$340$	0	0
$341$	137.446	0.0218274
$342$	0	0
$343$	−6915.45	−1.08863
$344$	0	0
$345$	1518.96	0.237038
$346$	0	0
$347$	−3248.20	−0.502514	−0.251257	−	0.967920i	$-0.580844\pi$
$347$	−3248.20	−0.502514	−0.251257	+	0.967920i	$0.580844\pi$
$348$	0	0
$349$	11502.7	1.76426	0.882132	−	0.471003i	$-0.156108\pi$
$349$	11502.7	1.76426	0.882132	+	0.471003i	$0.156108\pi$
$350$	0	0
$351$	3059.12	0.465196
$352$	0	0
$353$	−1492.97	−0.225107	−0.112553	−	0.993646i	$-0.535903\pi$
$353$	−1492.97	−0.225107	−0.112553	+	0.993646i	$0.535903\pi$
$354$	0	0
$355$	1568.64	0.234521
$356$	0	0
$357$	−429.057	−0.0636082
$358$	0	0
$359$	703.707	0.103455	0.0517274	−	0.998661i	$-0.483527\pi$
$359$	703.707	0.103455	0.0517274	+	0.998661i	$0.483527\pi$
$360$	0	0
$361$	9996.92	1.45749
$362$	0	0
$363$	6856.13	0.991332
$364$	0	0
$365$	2867.65	0.411232
$366$	0	0
$367$	−1855.13	−0.263862	−0.131931	−	0.991259i	$-0.542118\pi$
$367$	−1855.13	−0.263862	−0.131931	+	0.991259i	$0.542118\pi$
$368$	0	0
$369$	−22.2080	−0.00313307
$370$	0	0
$371$	1870.50	0.261757
$372$	0	0
$373$	−4666.05	−0.647718	−0.323859	−	0.946105i	$-0.604980\pi$
$373$	−4666.05	−0.647718	−0.323859	+	0.946105i	$0.604980\pi$
$374$	0	0
$375$	650.930	0.0896370
$376$	0	0
$377$	633.723	0.0865739
$378$	0	0
$379$	5572.81	0.755293	0.377646	−	0.925950i	$-0.376734\pi$
$379$	5572.81	0.755293	0.377646	+	0.925950i	$0.376734\pi$
$380$	0	0
$381$	−11432.8	−1.53732
$382$	0	0
$383$	−12192.8	−1.62669	−0.813344	−	0.581783i	$-0.802355\pi$
$383$	−12192.8	−1.62669	−0.813344	+	0.581783i	$0.802355\pi$
$384$	0	0
$385$	288.185	0.0381487
$386$	0	0
$387$	28.0176	0.00368014
$388$	0	0
$389$	2289.86	0.298458	0.149229	−	0.988803i	$-0.452321\pi$
$389$	2289.86	0.298458	0.149229	+	0.988803i	$0.452321\pi$
$390$	0	0
$391$	316.420	0.0409260
$392$	0	0
$393$	−9676.52	−1.24202
$394$	0	0
$395$	−2041.65	−0.260067
$396$	0	0
$397$	5087.21	0.643123	0.321562	−	0.946889i	$-0.395792\pi$
$397$	5087.21	0.643123	0.321562	+	0.946889i	$0.395792\pi$
$398$	0	0
$399$	10270.2	1.28860
$400$	0	0
$401$	435.207	0.0541974	0.0270987	−	0.999633i	$-0.491373\pi$
$401$	435.207	0.0541974	0.0270987	+	0.999633i	$0.491373\pi$
$402$	0	0
$403$	−791.612	−0.0978486
$404$	0	0
$405$	3660.78	0.449150
$406$	0	0
$407$	−693.121	−0.0844146
$408$	0	0
$409$	−4670.56	−0.564656	−0.282328	−	0.959318i	$-0.591107\pi$
$409$	−4670.56	−0.564656	−0.282328	+	0.959318i	$0.591107\pi$
$410$	0	0
$411$	12124.0	1.45506
$412$	0	0
$413$	12706.2	1.51387
$414$	0	0
$415$	1369.05	0.161937
$416$	0	0
$417$	−5114.11	−0.600574
$418$	0	0
$419$	−6522.47	−0.760486	−0.380243	−	0.924887i	$-0.624160\pi$
$419$	−6522.47	−0.760486	−0.380243	+	0.924887i	$0.624160\pi$
$420$	0	0
$421$	5854.71	0.677769	0.338885	−	0.940828i	$-0.389950\pi$
$421$	5854.71	0.677769	0.338885	+	0.940828i	$0.389950\pi$
$422$	0	0
$423$	11.0831	0.00127395
$424$	0	0
$425$	135.598	0.0154764
$426$	0	0
$427$	10134.4	1.14857
$428$	0	0
$429$	431.765	0.0485917
$430$	0	0
$431$	−1815.37	−0.202884	−0.101442	−	0.994841i	$-0.532346\pi$
$431$	−1815.37	−0.202884	−0.101442	+	0.994841i	$0.532346\pi$
$432$	0	0
$433$	−5560.46	−0.617133	−0.308567	−	0.951203i	$-0.599849\pi$
$433$	−5560.46	−0.617133	−0.308567	+	0.951203i	$0.599849\pi$
$434$	0	0
$435$	755.078	0.0832258
$436$	0	0
$437$	−7574.05	−0.829099
$438$	0	0
$439$	7209.04	0.783755	0.391878	−	0.920017i	$-0.371826\pi$
$439$	7209.04	0.783755	0.391878	+	0.920017i	$0.371826\pi$
$440$	0	0
$441$	−13.1770	−0.00142285
$442$	0	0
$443$	−600.141	−0.0643647	−0.0321824	−	0.999482i	$-0.510246\pi$
$443$	−600.141	−0.0643647	−0.0321824	+	0.999482i	$0.510246\pi$
$444$	0	0
$445$	7207.75	0.767821
$446$	0	0
$447$	12900.1	1.36500
$448$	0	0
$449$	−2528.32	−0.265743	−0.132872	−	0.991133i	$-0.542420\pi$
$449$	−2528.32	−0.265743	−0.132872	+	0.991133i	$0.542420\pi$
$450$	0	0
$451$	717.743	0.0749384
$452$	0	0
$453$	8898.40	0.922922
$454$	0	0
$455$	−1659.78	−0.171014
$456$	0	0
$457$	−13383.6	−1.36993	−0.684967	−	0.728574i	$-0.740183\pi$
$457$	−13383.6	−1.36993	−0.684967	+	0.728574i	$0.740183\pi$
$458$	0	0
$459$	759.290	0.0772127
$460$	0	0
$461$	11602.0	1.17215	0.586074	−	0.810258i	$-0.300673\pi$
$461$	11602.0	1.17215	0.586074	+	0.810258i	$0.300673\pi$
$462$	0	0
$463$	6367.19	0.639112	0.319556	−	0.947567i	$-0.396466\pi$
$463$	6367.19	0.639112	0.319556	+	0.947567i	$0.396466\pi$
$464$	0	0
$465$	−943.203	−0.0940645
$466$	0	0
$467$	7322.47	0.725575	0.362787	−	0.931872i	$-0.381825\pi$
$467$	7322.47	0.725575	0.362787	+	0.931872i	$0.381825\pi$
$468$	0	0
$469$	−9992.06	−0.983775
$470$	0	0
$471$	5971.07	0.584145
$472$	0	0
$473$	−905.506	−0.0880237
$474$	0	0
$475$	−3245.76	−0.313528
$476$	0	0
$477$	14.4558	0.00138760
$478$	0	0
$479$	7086.76	0.675997	0.337998	−	0.941147i	$-0.390250\pi$
$479$	7086.76	0.675997	0.337998	+	0.941147i	$0.390250\pi$
$480$	0	0
$481$	3991.98	0.378417
$482$	0	0
$483$	−4614.82	−0.434744
$484$	0	0
$485$	368.071	0.0344603
$486$	0	0
$487$	−16558.4	−1.54072	−0.770362	−	0.637607i	$-0.779924\pi$
$487$	−16558.4	−1.54072	−0.770362	+	0.637607i	$0.779924\pi$
$488$	0	0
$489$	19604.6	1.81299
$490$	0	0
$491$	−3437.59	−0.315960	−0.157980	−	0.987442i	$-0.550498\pi$
$491$	−3437.59	−0.315960	−0.157980	+	0.987442i	$0.550498\pi$
$492$	0	0
$493$	157.293	0.0143695
$494$	0	0
$495$	2.22718	0.000202231 0
$496$	0	0
$497$	−4765.76	−0.430128
$498$	0	0
$499$	10874.5	0.975574	0.487787	−	0.872963i	$-0.337804\pi$
$499$	10874.5	0.975574	0.487787	+	0.872963i	$0.337804\pi$
$500$	0	0
$501$	−582.297	−0.0519264
$502$	0	0
$503$	337.035	0.0298761	0.0149380	−	0.999888i	$-0.495245\pi$
$503$	337.035	0.0298761	0.0149380	+	0.999888i	$0.495245\pi$
$504$	0	0
$505$	−5490.81	−0.483837
$506$	0	0
$507$	8954.02	0.784343
$508$	0	0
$509$	9419.35	0.820246	0.410123	−	0.912030i	$-0.365486\pi$
$509$	9419.35	0.820246	0.410123	+	0.912030i	$0.365486\pi$
$510$	0	0
$511$	−8712.33	−0.754229
$512$	0	0
$513$	−18174.9	−1.56421
$514$	0	0
$515$	5268.43	0.450786
$516$	0	0
$517$	−358.197	−0.0304709
$518$	0	0
$519$	15190.2	1.28473
$520$	0	0
$521$	−5135.61	−0.431853	−0.215926	−	0.976410i	$-0.569277\pi$
$521$	−5135.61	−0.431853	−0.215926	+	0.976410i	$0.569277\pi$
$522$	0	0
$523$	9279.12	0.775808	0.387904	−	0.921700i	$-0.373199\pi$
$523$	9279.12	0.775808	0.387904	+	0.921700i	$0.373199\pi$
$524$	0	0
$525$	−1977.62	−0.164401
$526$	0	0
$527$	−196.482	−0.0162408
$528$	0	0
$529$	−8763.67	−0.720282
$530$	0	0
$531$	98.1970	0.00802521
$532$	0	0
$533$	−4133.79	−0.335936
$534$	0	0
$535$	2399.22	0.193883
$536$	0	0
$537$	21424.0	1.72163
$538$	0	0
$539$	425.870	0.0340325
$540$	0	0
$541$	−6375.38	−0.506652	−0.253326	−	0.967381i	$-0.581525\pi$
$541$	−6375.38	−0.506652	−0.253326	+	0.967381i	$0.581525\pi$
$542$	0	0
$543$	17737.1	1.40179
$544$	0	0
$545$	5.94660	0.000467384 0
$546$	0	0
$547$	−8910.61	−0.696509	−0.348254	−	0.937400i	$-0.613225\pi$
$547$	−8910.61	−0.696509	−0.348254	+	0.937400i	$0.613225\pi$
$548$	0	0
$549$	78.3219	0.00608871
$550$	0	0
$551$	−3765.08	−0.291103
$552$	0	0
$553$	6202.82	0.476982
$554$	0	0
$555$	4756.43	0.363782
$556$	0	0
$557$	9584.35	0.729088	0.364544	−	0.931186i	$-0.381225\pi$
$557$	9584.35	0.729088	0.364544	+	0.931186i	$0.381225\pi$
$558$	0	0
$559$	5215.19	0.394596
$560$	0	0
$561$	107.167	0.00806519
$562$	0	0
$563$	6005.15	0.449532	0.224766	−	0.974413i	$-0.427838\pi$
$563$	6005.15	0.449532	0.224766	+	0.974413i	$0.427838\pi$
$564$	0	0
$565$	8641.97	0.643487
$566$	0	0
$567$	−11122.0	−0.823773
$568$	0	0
$569$	−7364.86	−0.542621	−0.271310	−	0.962492i	$-0.587457\pi$
$569$	−7364.86	−0.542621	−0.271310	+	0.962492i	$0.587457\pi$
$570$	0	0
$571$	−5536.67	−0.405784	−0.202892	−	0.979201i	$-0.565034\pi$
$571$	−5536.67	−0.405784	−0.202892	+	0.979201i	$0.565034\pi$
$572$	0	0
$573$	−26027.5	−1.89759
$574$	0	0
$575$	1458.45	0.105777
$576$	0	0
$577$	8165.85	0.589166	0.294583	−	0.955626i	$-0.404819\pi$
$577$	8165.85	0.589166	0.294583	+	0.955626i	$0.404819\pi$
$578$	0	0
$579$	13334.6	0.957112
$580$	0	0
$581$	−4159.37	−0.297005
$582$	0	0
$583$	−467.200	−0.0331894
$584$	0	0
$585$	−12.8273	−0.000906567 0
$586$	0	0
$587$	−4181.78	−0.294038	−0.147019	−	0.989134i	$-0.546968\pi$
$587$	−4181.78	−0.294038	−0.147019	+	0.989134i	$0.546968\pi$
$588$	0	0
$589$	4703.13	0.329014
$590$	0	0
$591$	−14427.4	−1.00417
$592$	0	0
$593$	2943.62	0.203845	0.101922	−	0.994792i	$-0.467501\pi$
$593$	2943.62	0.203845	0.101922	+	0.994792i	$0.467501\pi$
$594$	0	0
$595$	−411.966	−0.0283848
$596$	0	0
$597$	−10225.8	−0.701028
$598$	0	0
$599$	−4721.73	−0.322078	−0.161039	−	0.986948i	$-0.551484\pi$
$599$	−4721.73	−0.322078	−0.161039	+	0.986948i	$0.551484\pi$
$600$	0	0
$601$	−13462.9	−0.913746	−0.456873	−	0.889532i	$-0.651031\pi$
$601$	−13462.9	−0.913746	−0.456873	+	0.889532i	$0.651031\pi$
$602$	0	0
$603$	−77.2216	−0.00521510
$604$	0	0
$605$	6583.02	0.442377
$606$	0	0
$607$	20636.8	1.37994	0.689970	−	0.723838i	$-0.257624\pi$
$607$	20636.8	1.37994	0.689970	+	0.723838i	$0.257624\pi$
$608$	0	0
$609$	−2294.04	−0.152642
$610$	0	0
$611$	2063.00	0.136596
$612$	0	0
$613$	−4895.44	−0.322553	−0.161276	−	0.986909i	$-0.551561\pi$
$613$	−4895.44	−0.322553	−0.161276	+	0.986909i	$0.551561\pi$
$614$	0	0
$615$	−4925.39	−0.322945
$616$	0	0
$617$	3102.15	0.202412	0.101206	−	0.994866i	$-0.467730\pi$
$617$	3102.15	0.202412	0.101206	+	0.994866i	$0.467730\pi$
$618$	0	0
$619$	−20003.5	−1.29888	−0.649441	−	0.760412i	$-0.724997\pi$
$619$	−20003.5	−1.29888	−0.649441	+	0.760412i	$0.724997\pi$
$620$	0	0
$621$	8166.71	0.527728
$622$	0	0
$623$	−21898.2	−1.40824
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−2565.21	−0.163388
$628$	0	0
$629$	990.830	0.0628092
$630$	0	0
$631$	10604.1	0.669005	0.334502	−	0.942395i	$-0.391432\pi$
$631$	10604.1	0.669005	0.334502	+	0.942395i	$0.391432\pi$
$632$	0	0
$633$	12404.1	0.778863
$634$	0	0
$635$	−10977.3	−0.686020
$636$	0	0
$637$	−2452.76	−0.152562
$638$	0	0
$639$	−36.8312	−0.00228016
$640$	0	0
$641$	14286.4	0.880308	0.440154	−	0.897922i	$-0.354924\pi$
$641$	14286.4	0.880308	0.440154	+	0.897922i	$0.354924\pi$
$642$	0	0
$643$	−11184.8	−0.685979	−0.342990	−	0.939339i	$-0.611440\pi$
$643$	−11184.8	−0.685979	−0.342990	+	0.939339i	$0.611440\pi$
$644$	0	0
$645$	6213.88	0.379335
$646$	0	0
$647$	10630.4	0.645944	0.322972	−	0.946409i	$-0.395318\pi$
$647$	10630.4	0.645944	0.322972	+	0.946409i	$0.395318\pi$
$648$	0	0
$649$	−3173.64	−0.191951
$650$	0	0
$651$	2865.59	0.172521
$652$	0	0
$653$	−12617.4	−0.756138	−0.378069	−	0.925777i	$-0.623412\pi$
$653$	−12617.4	−0.756138	−0.378069	+	0.925777i	$0.623412\pi$
$654$	0	0
$655$	−9291.06	−0.554247
$656$	0	0
$657$	−67.3315	−0.00399825
$658$	0	0
$659$	8311.90	0.491329	0.245664	−	0.969355i	$-0.420994\pi$
$659$	8311.90	0.491329	0.245664	+	0.969355i	$0.420994\pi$
$660$	0	0
$661$	−26450.4	−1.55643	−0.778215	−	0.627998i	$-0.783875\pi$
$661$	−26450.4	−1.55643	−0.778215	+	0.627998i	$0.783875\pi$
$662$	0	0
$663$	−617.217	−0.0361549
$664$	0	0
$665$	9861.09	0.575033
$666$	0	0
$667$	1691.80	0.0982113
$668$	0	0
$669$	12310.6	0.711442
$670$	0	0
$671$	−2531.30	−0.145633
$672$	0	0
$673$	2007.05	0.114957	0.0574785	−	0.998347i	$-0.481694\pi$
$673$	2007.05	0.114957	0.0574785	+	0.998347i	$0.481694\pi$
$674$	0	0
$675$	3499.74	0.199563
$676$	0	0
$677$	−26811.0	−1.52206	−0.761028	−	0.648720i	$-0.775305\pi$
$677$	−26811.0	−1.52206	−0.761028	+	0.648720i	$0.775305\pi$
$678$	0	0
$679$	−1118.25	−0.0632027
$680$	0	0
$681$	−31553.8	−1.77554
$682$	0	0
$683$	9323.90	0.522356	0.261178	−	0.965291i	$-0.415889\pi$
$683$	9323.90	0.522356	0.261178	+	0.965291i	$0.415889\pi$
$684$	0	0
$685$	11641.0	0.649315
$686$	0	0
$687$	18420.7	1.02299
$688$	0	0
$689$	2690.80	0.148783
$690$	0	0
$691$	9312.74	0.512697	0.256348	−	0.966584i	$-0.417481\pi$
$691$	9312.74	0.512697	0.256348	+	0.966584i	$0.417481\pi$
$692$	0	0
$693$	−6.76649	−0.000370906 0
$694$	0	0
$695$	−4910.39	−0.268003
$696$	0	0
$697$	−1026.03	−0.0557584
$698$	0	0
$699$	17160.6	0.928573
$700$	0	0
$701$	14028.6	0.755853	0.377926	−	0.925836i	$-0.376637\pi$
$701$	14028.6	0.755853	0.377926	+	0.925836i	$0.376637\pi$
$702$	0	0
$703$	−23717.2	−1.27242
$704$	0	0
$705$	2458.06	0.131314
$706$	0	0
$707$	16681.9	0.887392
$708$	0	0
$709$	30882.6	1.63585	0.817927	−	0.575322i	$-0.195123\pi$
$709$	30882.6	1.63585	0.817927	+	0.575322i	$0.195123\pi$
$710$	0	0
$711$	47.9373	0.00252853
$712$	0	0
$713$	−2113.31	−0.111002
$714$	0	0
$715$	414.566	0.0216838
$716$	0	0
$717$	−13445.0	−0.700299
$718$	0	0
$719$	10273.8	0.532890	0.266445	−	0.963850i	$-0.414151\pi$
$719$	10273.8	0.532890	0.266445	+	0.963850i	$0.414151\pi$
$720$	0	0
$721$	−16006.3	−0.826774
$722$	0	0
$723$	8103.52	0.416837
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	20429.4	1.04221	0.521104	−	0.853493i	$-0.325521\pi$
$727$	20429.4	1.04221	0.521104	+	0.853493i	$0.325521\pi$
$728$	0	0
$729$	19596.7	0.995614
$730$	0	0
$731$	1294.44	0.0654946
$732$	0	0
$733$	−32179.7	−1.62154	−0.810768	−	0.585368i	$-0.800950\pi$
$733$	−32179.7	−1.62154	−0.810768	+	0.585368i	$0.800950\pi$
$734$	0	0
$735$	−2922.46	−0.146662
$736$	0	0
$737$	2495.74	0.124738
$738$	0	0
$739$	−12103.9	−0.602503	−0.301251	−	0.953545i	$-0.597404\pi$
$739$	−12103.9	−0.602503	−0.301251	+	0.953545i	$0.597404\pi$
$740$	0	0
$741$	14774.1	0.732444
$742$	0	0
$743$	4069.71	0.200947	0.100473	−	0.994940i	$-0.467964\pi$
$743$	4069.71	0.200947	0.100473	+	0.994940i	$0.467964\pi$
$744$	0	0
$745$	12386.2	0.609122
$746$	0	0
$747$	−32.1449	−0.00157446
$748$	0	0
$749$	−7289.19	−0.355596
$750$	0	0
$751$	−22410.9	−1.08893	−0.544465	−	0.838783i	$-0.683267\pi$
$751$	−22410.9	−1.08893	−0.544465	+	0.838783i	$0.683267\pi$
$752$	0	0
$753$	21221.0	1.02701
$754$	0	0
$755$	8543.94	0.411849
$756$	0	0
$757$	11846.2	0.568769	0.284385	−	0.958710i	$-0.408211\pi$
$757$	11846.2	0.568769	0.284385	+	0.958710i	$0.408211\pi$
$758$	0	0
$759$	1152.65	0.0551234
$760$	0	0
$761$	−27216.2	−1.29643	−0.648216	−	0.761456i	$-0.724485\pi$
$761$	−27216.2	−1.29643	−0.648216	+	0.761456i	$0.724485\pi$
$762$	0	0
$763$	−18.0666	−0.000857216 0
$764$	0	0
$765$	−3.18379	−0.000150471 0
$766$	0	0
$767$	18278.4	0.860486
$768$	0	0
$769$	39079.9	1.83259	0.916293	−	0.400508i	$-0.131166\pi$
$769$	39079.9	1.83259	0.916293	+	0.400508i	$0.131166\pi$
$770$	0	0
$771$	−15292.2	−0.714311
$772$	0	0
$773$	20855.1	0.970384	0.485192	−	0.874408i	$-0.338750\pi$
$773$	20855.1	0.970384	0.485192	+	0.874408i	$0.338750\pi$
$774$	0	0
$775$	−905.631	−0.0419758
$776$	0	0
$777$	−14450.7	−0.667203
$778$	0	0
$779$	24559.7	1.12958
$780$	0	0
$781$	1190.35	0.0545380
$782$	0	0
$783$	4059.69	0.185289
$784$	0	0
$785$	5733.21	0.260672
$786$	0	0
$787$	15857.5	0.718244	0.359122	−	0.933291i	$-0.383076\pi$
$787$	15857.5	0.718244	0.359122	+	0.933291i	$0.383076\pi$
$788$	0	0
$789$	−41109.3	−1.85492
$790$	0	0
$791$	−26255.6	−1.18020
$792$	0	0
$793$	14578.8	0.652849
$794$	0	0
$795$	3206.08	0.143029
$796$	0	0
$797$	−34420.6	−1.52978	−0.764892	−	0.644158i	$-0.777208\pi$
$797$	−34420.6	−1.52978	−0.764892	+	0.644158i	$0.777208\pi$
$798$	0	0
$799$	512.049	0.0226721
$800$	0	0
$801$	−169.236	−0.00746524
$802$	0	0
$803$	2176.10	0.0956324
$804$	0	0
$805$	−4430.99	−0.194002
$806$	0	0
$807$	−1610.98	−0.0702714
$808$	0	0
$809$	−19841.6	−0.862289	−0.431144	−	0.902283i	$-0.641890\pi$
$809$	−19841.6	−0.862289	−0.431144	+	0.902283i	$0.641890\pi$
$810$	0	0
$811$	−12635.4	−0.547087	−0.273543	−	0.961860i	$-0.588196\pi$
$811$	−12635.4	−0.547087	−0.273543	+	0.961860i	$0.588196\pi$
$812$	0	0
$813$	−789.635	−0.0340636
$814$	0	0
$815$	18823.6	0.809035
$816$	0	0
$817$	−30984.5	−1.32682
$818$	0	0
$819$	38.9711	0.00166271
$820$	0	0
$821$	41953.8	1.78343	0.891716	−	0.452595i	$-0.149502\pi$
$821$	41953.8	1.78343	0.891716	+	0.452595i	$0.149502\pi$
$822$	0	0
$823$	−32285.7	−1.36745	−0.683724	−	0.729741i	$-0.739641\pi$
$823$	−32285.7	−1.36745	−0.683724	+	0.729741i	$0.739641\pi$
$824$	0	0
$825$	493.954	0.0208452
$826$	0	0
$827$	−13277.6	−0.558290	−0.279145	−	0.960249i	$-0.590051\pi$
$827$	−13277.6	−0.558290	−0.279145	+	0.960249i	$0.590051\pi$
$828$	0	0
$829$	7888.29	0.330485	0.165242	−	0.986253i	$-0.447159\pi$
$829$	7888.29	0.330485	0.165242	+	0.986253i	$0.447159\pi$
$830$	0	0
$831$	21183.4	0.884290
$832$	0	0
$833$	−608.789	−0.0253221
$834$	0	0
$835$	−559.102	−0.0231719
$836$	0	0
$837$	−5071.15	−0.209420
$838$	0	0
$839$	11675.4	0.480430	0.240215	−	0.970720i	$-0.422782\pi$
$839$	11675.4	0.480430	0.240215	+	0.970720i	$0.422782\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	2687.54	0.109803
$844$	0	0
$845$	8597.34	0.350009
$846$	0	0
$847$	−20000.2	−0.811351
$848$	0	0
$849$	44012.0	1.77914
$850$	0	0
$851$	10657.1	0.429284
$852$	0	0
$853$	18187.7	0.730053	0.365026	−	0.930997i	$-0.381060\pi$
$853$	18187.7	0.730053	0.365026	+	0.930997i	$0.381060\pi$
$854$	0	0
$855$	76.2094	0.00304831
$856$	0	0
$857$	16155.0	0.643926	0.321963	−	0.946752i	$-0.395657\pi$
$857$	16155.0	0.643926	0.321963	+	0.946752i	$0.395657\pi$
$858$	0	0
$859$	−6942.41	−0.275753	−0.137877	−	0.990449i	$-0.544028\pi$
$859$	−6942.41	−0.275753	−0.137877	+	0.990449i	$0.544028\pi$
$860$	0	0
$861$	14964.1	0.592304
$862$	0	0
$863$	−24632.9	−0.971628	−0.485814	−	0.874062i	$-0.661477\pi$
$863$	−24632.9	−0.971628	−0.485814	+	0.874062i	$0.661477\pi$
$864$	0	0
$865$	14585.1	0.573305
$866$	0	0
$867$	25430.9	0.996171
$868$	0	0
$869$	−1549.29	−0.0604788
$870$	0	0
$871$	−14374.0	−0.559179
$872$	0	0
$873$	−8.64219	−0.000335044 0
$874$	0	0
$875$	−1898.84	−0.0733629
$876$	0	0
$877$	−10567.7	−0.406893	−0.203447	−	0.979086i	$-0.565214\pi$
$877$	−10567.7	−0.406893	−0.203447	+	0.979086i	$0.565214\pi$
$878$	0	0
$879$	−28446.5	−1.09156
$880$	0	0
$881$	−27130.1	−1.03750	−0.518748	−	0.854927i	$-0.673602\pi$
$881$	−27130.1	−1.03750	−0.518748	+	0.854927i	$0.673602\pi$
$882$	0	0
$883$	−41743.8	−1.59093	−0.795465	−	0.605999i	$-0.792774\pi$
$883$	−41743.8	−1.59093	−0.795465	+	0.605999i	$0.792774\pi$
$884$	0	0
$885$	21778.6	0.827208
$886$	0	0
$887$	−3233.84	−0.122414	−0.0612072	−	0.998125i	$-0.519495\pi$
$887$	−3233.84	−0.122414	−0.0612072	+	0.998125i	$0.519495\pi$
$888$	0	0
$889$	33350.8	1.25821
$890$	0	0
$891$	2777.96	0.104450
$892$	0	0
$893$	−12256.8	−0.459302
$894$	0	0
$895$	20570.6	0.768267
$896$	0	0
$897$	−6638.61	−0.247109
$898$	0	0
$899$	−1050.53	−0.0389735
$900$	0	0
$901$	667.871	0.0246948
$902$	0	0
$903$	−18878.7	−0.695729
$904$	0	0
$905$	17030.6	0.625541
$906$	0	0
$907$	−6543.88	−0.239566	−0.119783	−	0.992800i	$-0.538220\pi$
$907$	−6543.88	−0.239566	−0.119783	+	0.992800i	$0.538220\pi$
$908$	0	0
$909$	128.922	0.00470417
$910$	0	0
$911$	−9902.90	−0.360151	−0.180076	−	0.983653i	$-0.557634\pi$
$911$	−9902.90	−0.360151	−0.180076	+	0.983653i	$0.557634\pi$
$912$	0	0
$913$	1038.90	0.0376587
$914$	0	0
$915$	17370.6	0.627601
$916$	0	0
$917$	28227.6	1.01653
$918$	0	0
$919$	−76.8989	−0.00276024	−0.00138012	−	0.999999i	$-0.500439\pi$
$919$	−76.8989	−0.00276024	−0.00138012	+	0.999999i	$0.500439\pi$
$920$	0	0
$921$	−11124.0	−0.397989
$922$	0	0
$923$	−6855.74	−0.244485
$924$	0	0
$925$	4566.95	0.162336
$926$	0	0
$927$	−123.701	−0.00438282
$928$	0	0
$929$	−41387.6	−1.46166	−0.730831	−	0.682558i	$-0.760867\pi$
$929$	−41387.6	−1.46166	−0.730831	+	0.682558i	$0.760867\pi$
$930$	0	0
$931$	14572.4	0.512987
$932$	0	0
$933$	32823.2	1.15175
$934$	0	0
$935$	102.898	0.00359905
$936$	0	0
$937$	1361.85	0.0474809	0.0237404	−	0.999718i	$-0.492442\pi$
$937$	1361.85	0.0474809	0.0237404	+	0.999718i	$0.492442\pi$
$938$	0	0
$939$	−1509.01	−0.0524436
$940$	0	0
$941$	−16032.2	−0.555405	−0.277702	−	0.960667i	$-0.589573\pi$
$941$	−16032.2	−0.555405	−0.277702	+	0.960667i	$0.589573\pi$
$942$	0	0
$943$	−11035.7	−0.381093
$944$	0	0
$945$	−10632.7	−0.366013
$946$	0	0
$947$	−17141.3	−0.588190	−0.294095	−	0.955776i	$-0.595018\pi$
$947$	−17141.3	−0.588190	−0.294095	+	0.955776i	$0.595018\pi$
$948$	0	0
$949$	−12533.1	−0.428704
$950$	0	0
$951$	7435.56	0.253538
$952$	0	0
$953$	47724.2	1.62218	0.811091	−	0.584920i	$-0.198874\pi$
$953$	47724.2	1.62218	0.811091	+	0.584920i	$0.198874\pi$
$954$	0	0
$955$	−24990.7	−0.846787
$956$	0	0
$957$	572.987	0.0193543
$958$	0	0
$959$	−35367.1	−1.19089
$960$	0	0
$961$	−28478.7	−0.955951
$962$	0	0
$963$	−56.3330	−0.00188505
$964$	0	0
$965$	12803.4	0.427106
$966$	0	0
$967$	−48259.7	−1.60489	−0.802444	−	0.596727i	$-0.796468\pi$
$967$	−48259.7	−1.60489	−0.802444	+	0.596727i	$0.796468\pi$
$968$	0	0
$969$	3667.02	0.121570
$970$	0	0
$971$	−48301.7	−1.59637	−0.798185	−	0.602413i	$-0.794206\pi$
$971$	−48301.7	−1.59637	−0.798185	+	0.602413i	$0.794206\pi$
$972$	0	0
$973$	14918.5	0.491536
$974$	0	0
$975$	−2844.89	−0.0934455
$976$	0	0
$977$	9396.18	0.307687	0.153844	−	0.988095i	$-0.450835\pi$
$977$	9396.18	0.307687	0.153844	+	0.988095i	$0.450835\pi$
$978$	0	0
$979$	5469.56	0.178558
$980$	0	0
$981$	−0.139624	−4.54420e−6 0
$982$	0	0
$983$	21460.9	0.696334	0.348167	−	0.937433i	$-0.386804\pi$
$983$	21460.9	0.696334	0.348167	+	0.937433i	$0.386804\pi$
$984$	0	0
$985$	−13852.7	−0.448106
$986$	0	0
$987$	−7467.95	−0.240839
$988$	0	0
$989$	13922.6	0.447637
$990$	0	0
$991$	−46424.2	−1.48811	−0.744054	−	0.668120i	$-0.767099\pi$
$991$	−46424.2	−1.48811	−0.744054	+	0.668120i	$0.767099\pi$
$992$	0	0
$993$	−21707.9	−0.693735
$994$	0	0
$995$	−9818.45	−0.312830
$996$	0	0
$997$	−32624.6	−1.03634	−0.518170	−	0.855278i	$-0.673386\pi$
$997$	−32624.6	−1.03634	−0.518170	+	0.855278i	$0.673386\pi$
$998$	0	0
$999$	25573.0	0.809904

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.e.1.3	✓	9
4.3	odd	2		2320.4.a.w.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.e.1.3	✓	9	1.1	even	1	trivial
2320.4.a.w.1.7		9	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.20744 q^{3} -5.00000 q^{5} +15.1907 q^{7} +0.117398 q^{9} -3.79422 q^{11} +21.8525 q^{13} +26.0372 q^{15} +5.42391 q^{17} -129.830 q^{19} -79.1048 q^{21} +58.3381 q^{23} +25.0000 q^{25} +139.989 q^{27} +29.0000 q^{29} -36.2252 q^{31} +19.7582 q^{33} -75.9536 q^{35} +182.678 q^{37} -113.796 q^{39} -189.168 q^{41} +238.654 q^{43} -0.586992 q^{45} +94.4059 q^{47} -112.242 q^{49} -28.2447 q^{51} +123.135 q^{53} +18.9711 q^{55} +676.083 q^{57} +836.442 q^{59} +667.146 q^{61} +1.78337 q^{63} -109.263 q^{65} -657.774 q^{67} -303.792 q^{69} -313.728 q^{71} -573.529 q^{73} -130.186 q^{75} -57.6370 q^{77} +408.329 q^{79} -732.156 q^{81} -273.810 q^{83} -27.1196 q^{85} -151.016 q^{87} -1441.55 q^{89} +331.955 q^{91} +188.641 q^{93} +649.152 q^{95} -73.6142 q^{97} -0.445436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} - 45 q^{5} - 27 q^{7} + 12 q^{9} + 88 q^{11} - 65 q^{13} + 15 q^{15} + 43 q^{17} + 88 q^{19} + 212 q^{21} - 73 q^{23} + 225 q^{25} - 108 q^{27} + 261 q^{29} - 3 q^{31} + 76 q^{33} + 135 q^{35}+ \cdots - 442 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 - 45 * q^5 - 27 * q^7 + 12 * q^9 + 88 * q^11 - 65 * q^13 + 15 * q^15 + 43 * q^17 + 88 * q^19 + 212 * q^21 - 73 * q^23 + 225 * q^25 - 108 * q^27 + 261 * q^29 - 3 * q^31 + 76 * q^33 + 135 * q^35 + 50 * q^37 - 49 * q^39 - 184 * q^41 - 389 * q^43 - 60 * q^45 - 216 * q^47 + 628 * q^49 - 218 * q^51 - 1003 * q^53 - 440 * q^55 - 98 * q^57 + 477 * q^59 - 203 * q^61 - 744 * q^63 + 325 * q^65 - 1028 * q^67 - 903 * q^69 - 484 * q^71 - 1905 * q^73 - 75 * q^75 - 3202 * q^77 - 869 * q^79 - 3151 * q^81 - 850 * q^83 - 215 * q^85 - 87 * q^87 - 804 * q^89 - 8 * q^91 - 3822 * q^93 - 440 * q^95 - 1795 * q^97 - 442 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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