LMFDB - Embedded newform 232.4.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [232,4,Mod(1,232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(232, 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("232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 232 = 2^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	232.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:	$13.6884431213$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 $ x^6 - 40x^4 - 88x^3 - 8x^2 + 48x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.90366$ of defining polynomial
Character	$\chi$	$=$	232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.88453 q^{3} -3.33508 q^{5} +0.0162797 q^{7} -18.6795 q^{9} +O(q^{10})$	$q+2.88453 q^{3} -3.33508 q^{5} +0.0162797 q^{7} -18.6795 q^{9} -38.3105 q^{11} -47.9093 q^{13} -9.62012 q^{15} +23.3405 q^{17} +80.2019 q^{19} +0.0469593 q^{21} -161.058 q^{23} -113.877 q^{25} -131.764 q^{27} -29.0000 q^{29} +43.4141 q^{31} -110.508 q^{33} -0.0542942 q^{35} +154.830 q^{37} -138.196 q^{39} +330.941 q^{41} -271.653 q^{43} +62.2976 q^{45} -314.037 q^{47} -343.000 q^{49} +67.3263 q^{51} -105.453 q^{53} +127.769 q^{55} +231.344 q^{57} -419.410 q^{59} +240.890 q^{61} -0.304097 q^{63} +159.781 q^{65} -252.123 q^{67} -464.577 q^{69} -842.623 q^{71} +1191.14 q^{73} -328.482 q^{75} -0.623685 q^{77} +125.254 q^{79} +124.271 q^{81} +1048.17 q^{83} -77.8424 q^{85} -83.6512 q^{87} +730.280 q^{89} -0.779951 q^{91} +125.229 q^{93} -267.480 q^{95} +1741.46 q^{97} +715.622 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9	$ 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9 - 19 * q^11 + 13 * q^13 - 191 * q^15 - 218 * q^17 - 290 * q^19 - 266 * q^21 - 196 * q^23 - 13 * q^25 - 437 * q^27 - 174 * q^29 - 675 * q^31 + 291 * q^33 - 466 * q^35 - 238 * q^37 - 1297 * q^39 - 464 * q^41 - 579 * q^43 - 148 * q^45 - 975 * q^47 + 914 * q^49 - 576 * q^51 + 515 * q^53 - 1605 * q^55 - 340 * q^57 - 108 * q^59 + 1158 * q^61 - 1136 * q^63 + 1239 * q^65 - 80 * q^67 + 2568 * q^69 + 438 * q^71 + 262 * q^73 + 1766 * q^75 + 194 * q^77 - 237 * q^79 + 2554 * q^81 - 1288 * q^83 + 3112 * q^85 + 145 * q^87 - 252 * q^89 + 2450 * q^91 + 2131 * q^93 + 180 * q^95 + 380 * q^97 + 2264 * 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
$3$	2.88453	0.555127	0.277564	−	0.960707i	$-0.410473\pi$
$3$	2.88453	0.555127	0.277564	+	0.960707i	$0.410473\pi$
$4$	0	0
$5$	−3.33508	−0.298298	−0.149149	−	0.988815i	$-0.547653\pi$
$5$	−3.33508	−0.298298	−0.149149	+	0.988815i	$0.547653\pi$
$6$	0	0
$7$	0.0162797	0.000879023 0	0.000439511	−	1.00000i	$-0.499860\pi$
$7$	0.0162797	0.000879023 0	0.000439511		1.00000i	$0.499860\pi$
$8$	0	0
$9$	−18.6795	−0.691834
$10$	0	0
$11$	−38.3105	−1.05010	−0.525048	−	0.851072i	$-0.675953\pi$
$11$	−38.3105	−1.05010	−0.525048	+	0.851072i	$0.675953\pi$
$12$	0	0
$13$	−47.9093	−1.02213	−0.511064	−	0.859543i	$-0.670748\pi$
$13$	−47.9093	−1.02213	−0.511064	+	0.859543i	$0.670748\pi$
$14$	0	0
$15$	−9.62012	−0.165594
$16$	0	0
$17$	23.3405	0.332994	0.166497	−	0.986042i	$-0.446754\pi$
$17$	23.3405	0.332994	0.166497	+	0.986042i	$0.446754\pi$
$18$	0	0
$19$	80.2019	0.968399	0.484200	−	0.874958i	$-0.339111\pi$
$19$	80.2019	0.968399	0.484200	+	0.874958i	$0.339111\pi$
$20$	0	0
$21$	0.0469593	0.000487969 0
$22$	0	0
$23$	−161.058	−1.46013	−0.730065	−	0.683377i	$-0.760510\pi$
$23$	−161.058	−1.46013	−0.730065	+	0.683377i	$0.760510\pi$
$24$	0	0
$25$	−113.877	−0.911018
$26$	0	0
$27$	−131.764	−0.939183
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	43.4141	0.251529	0.125764	−	0.992060i	$-0.459862\pi$
$31$	43.4141	0.251529	0.125764	+	0.992060i	$0.459862\pi$
$32$	0	0
$33$	−110.508	−0.582937
$34$	0	0
$35$	−0.0542942	−0.000262211 0
$36$	0	0
$37$	154.830	0.687945	0.343972	−	0.938980i	$-0.388227\pi$
$37$	154.830	0.687945	0.343972	+	0.938980i	$0.388227\pi$
$38$	0	0
$39$	−138.196	−0.567411
$40$	0	0
$41$	330.941	1.26059	0.630296	−	0.776355i	$-0.282933\pi$
$41$	330.941	1.26059	0.630296	+	0.776355i	$0.282933\pi$
$42$	0	0
$43$	−271.653	−0.963412	−0.481706	−	0.876333i	$-0.659983\pi$
$43$	−271.653	−0.963412	−0.481706	+	0.876333i	$0.659983\pi$
$44$	0	0
$45$	62.2976	0.206373
$46$	0	0
$47$	−314.037	−0.974616	−0.487308	−	0.873230i	$-0.662021\pi$
$47$	−314.037	−0.974616	−0.487308	+	0.873230i	$0.662021\pi$
$48$	0	0
$49$	−343.000	−0.999999
$50$	0	0
$51$	67.3263	0.184854
$52$	0	0
$53$	−105.453	−0.273303	−0.136651	−	0.990619i	$-0.543634\pi$
$53$	−105.453	−0.273303	−0.136651	+	0.990619i	$0.543634\pi$
$54$	0	0
$55$	127.769	0.313242
$56$	0	0
$57$	231.344	0.537585
$58$	0	0
$59$	−419.410	−0.925466	−0.462733	−	0.886498i	$-0.653131\pi$
$59$	−419.410	−0.925466	−0.462733	+	0.886498i	$0.653131\pi$
$60$	0	0
$61$	240.890	0.505620	0.252810	−	0.967516i	$-0.418645\pi$
$61$	240.890	0.505620	0.252810	+	0.967516i	$0.418645\pi$
$62$	0	0
$63$	−0.304097	−0.000608138 0
$64$	0	0
$65$	159.781	0.304899
$66$	0	0
$67$	−252.123	−0.459727	−0.229863	−	0.973223i	$-0.573828\pi$
$67$	−252.123	−0.459727	−0.229863	+	0.973223i	$0.573828\pi$
$68$	0	0
$69$	−464.577	−0.810558
$70$	0	0
$71$	−842.623	−1.40846	−0.704232	−	0.709969i	$-0.748709\pi$
$71$	−842.623	−1.40846	−0.704232	+	0.709969i	$0.748709\pi$
$72$	0	0
$73$	1191.14	1.90976	0.954879	−	0.296996i	$-0.0959849\pi$
$73$	1191.14	1.90976	0.954879	+	0.296996i	$0.0959849\pi$
$74$	0	0
$75$	−328.482	−0.505731
$76$	0	0
$77$	−0.623685	−0.000923059 0
$78$	0	0
$79$	125.254	0.178382	0.0891909	−	0.996015i	$-0.471572\pi$
$79$	125.254	0.178382	0.0891909	+	0.996015i	$0.471572\pi$
$80$	0	0
$81$	124.271	0.170468
$82$	0	0
$83$	1048.17	1.38617	0.693085	−	0.720856i	$-0.256251\pi$
$83$	1048.17	1.38617	0.693085	+	0.720856i	$0.256251\pi$
$84$	0	0
$85$	−77.8424	−0.0993317
$86$	0	0
$87$	−83.6512	−0.103085
$88$	0	0
$89$	730.280	0.869769	0.434885	−	0.900486i	$-0.356789\pi$
$89$	730.280	0.869769	0.434885	+	0.900486i	$0.356789\pi$
$90$	0	0
$91$	−0.779951	−0.000898473 0
$92$	0	0
$93$	125.229	0.139631
$94$	0	0
$95$	−267.480	−0.288872
$96$	0	0
$97$	1741.46	1.82288	0.911438	−	0.411438i	$-0.134973\pi$
$97$	1741.46	1.82288	0.911438	+	0.411438i	$0.134973\pi$
$98$	0	0
$99$	715.622	0.726492
$100$	0	0
$101$	7.67099	0.00755734	0.00377867	−	0.999993i	$-0.498797\pi$
$101$	7.67099	0.00755734	0.00377867	+	0.999993i	$0.498797\pi$
$102$	0	0
$103$	−937.601	−0.896937	−0.448469	−	0.893799i	$-0.648030\pi$
$103$	−937.601	−0.896937	−0.448469	+	0.893799i	$0.648030\pi$
$104$	0	0
$105$	−0.156613	−0.000145561 0
$106$	0	0
$107$	1489.09	1.34538	0.672691	−	0.739924i	$-0.265138\pi$
$107$	1489.09	1.34538	0.672691	+	0.739924i	$0.265138\pi$
$108$	0	0
$109$	586.261	0.515171	0.257586	−	0.966256i	$-0.417073\pi$
$109$	586.261	0.515171	0.257586	+	0.966256i	$0.417073\pi$
$110$	0	0
$111$	446.612	0.381897
$112$	0	0
$113$	−887.719	−0.739023	−0.369511	−	0.929226i	$-0.620475\pi$
$113$	−887.719	−0.739023	−0.369511	+	0.929226i	$0.620475\pi$
$114$	0	0
$115$	537.142	0.435555
$116$	0	0
$117$	894.923	0.707142
$118$	0	0
$119$	0.379977	0.000292710 0
$120$	0	0
$121$	136.697	0.102703
$122$	0	0
$123$	954.608	0.699789
$124$	0	0
$125$	796.674	0.570054
$126$	0	0
$127$	−327.440	−0.228785	−0.114392	−	0.993436i	$-0.536492\pi$
$127$	−327.440	−0.228785	−0.114392	+	0.993436i	$0.536492\pi$
$128$	0	0
$129$	−783.591	−0.534816
$130$	0	0
$131$	−954.765	−0.636780	−0.318390	−	0.947960i	$-0.603142\pi$
$131$	−954.765	−0.636780	−0.318390	+	0.947960i	$0.603142\pi$
$132$	0	0
$133$	1.30567	0.000851245 0
$134$	0	0
$135$	439.442	0.280157
$136$	0	0
$137$	−1516.83	−0.945926	−0.472963	−	0.881082i	$-0.656816\pi$
$137$	−1516.83	−0.945926	−0.472963	+	0.881082i	$0.656816\pi$
$138$	0	0
$139$	1490.43	0.909469	0.454735	−	0.890627i	$-0.349734\pi$
$139$	1490.43	0.909469	0.454735	+	0.890627i	$0.349734\pi$
$140$	0	0
$141$	−905.846	−0.541036
$142$	0	0
$143$	1835.43	1.07333
$144$	0	0
$145$	96.7172	0.0553926
$146$	0	0
$147$	−989.391	−0.555127
$148$	0	0
$149$	−363.775	−0.200011	−0.100005	−	0.994987i	$-0.531886\pi$
$149$	−363.775	−0.200011	−0.100005	+	0.994987i	$0.531886\pi$
$150$	0	0
$151$	−2480.58	−1.33687	−0.668433	−	0.743773i	$-0.733035\pi$
$151$	−2480.58	−1.33687	−0.668433	+	0.743773i	$0.733035\pi$
$152$	0	0
$153$	−435.989	−0.230377
$154$	0	0
$155$	−144.789	−0.0750307
$156$	0	0
$157$	−75.1885	−0.0382210	−0.0191105	−	0.999817i	$-0.506083\pi$
$157$	−75.1885	−0.0382210	−0.0191105	+	0.999817i	$0.506083\pi$
$158$	0	0
$159$	−304.181	−0.151718
$160$	0	0
$161$	−2.62199	−0.00128349
$162$	0	0
$163$	2475.52	1.18955	0.594777	−	0.803890i	$-0.297240\pi$
$163$	2475.52	1.18955	0.594777	+	0.803890i	$0.297240\pi$
$164$	0	0
$165$	368.552	0.173889
$166$	0	0
$167$	−985.043	−0.456437	−0.228218	−	0.973610i	$-0.573290\pi$
$167$	−985.043	−0.456437	−0.228218	+	0.973610i	$0.573290\pi$
$168$	0	0
$169$	98.3029	0.0447441
$170$	0	0
$171$	−1498.13	−0.669971
$172$	0	0
$173$	−1631.37	−0.716943	−0.358471	−	0.933541i	$-0.616702\pi$
$173$	−1631.37	−0.716943	−0.358471	+	0.933541i	$0.616702\pi$
$174$	0	0
$175$	−1.85389	−0.000800806 0
$176$	0	0
$177$	−1209.80	−0.513751
$178$	0	0
$179$	−516.369	−0.215616	−0.107808	−	0.994172i	$-0.534383\pi$
$179$	−516.369	−0.215616	−0.107808	+	0.994172i	$0.534383\pi$
$180$	0	0
$181$	−943.564	−0.387484	−0.193742	−	0.981053i	$-0.562062\pi$
$181$	−943.564	−0.387484	−0.193742	+	0.981053i	$0.562062\pi$
$182$	0	0
$183$	694.854	0.280684
$184$	0	0
$185$	−516.371	−0.205213
$186$	0	0
$187$	−894.188	−0.349676
$188$	0	0
$189$	−2.14508	−0.000825563 0
$190$	0	0
$191$	−2522.43	−0.955585	−0.477793	−	0.878473i	$-0.658563\pi$
$191$	−2522.43	−0.955585	−0.477793	+	0.878473i	$0.658563\pi$
$192$	0	0
$193$	−498.201	−0.185810	−0.0929049	−	0.995675i	$-0.529615\pi$
$193$	−498.201	−0.185810	−0.0929049	+	0.995675i	$0.529615\pi$
$194$	0	0
$195$	460.893	0.169258
$196$	0	0
$197$	−3249.20	−1.17510	−0.587552	−	0.809186i	$-0.699908\pi$
$197$	−3249.20	−1.17510	−0.587552	+	0.809186i	$0.699908\pi$
$198$	0	0
$199$	1950.85	0.694935	0.347467	−	0.937692i	$-0.387042\pi$
$199$	1950.85	0.694935	0.347467	+	0.937692i	$0.387042\pi$
$200$	0	0
$201$	−727.255	−0.255207
$202$	0	0
$203$	−0.472112	−0.000163230 0
$204$	0	0
$205$	−1103.71	−0.376033
$206$	0	0
$207$	3008.49	1.01017
$208$	0	0
$209$	−3072.58	−1.01691
$210$	0	0
$211$	2481.22	0.809546	0.404773	−	0.914417i	$-0.367350\pi$
$211$	2481.22	0.809546	0.404773	+	0.914417i	$0.367350\pi$
$212$	0	0
$213$	−2430.57	−0.781877
$214$	0	0
$215$	905.985	0.287384
$216$	0	0
$217$	0.706769	0.000221100 0
$218$	0	0
$219$	3435.87	1.06016
$220$	0	0
$221$	−1118.23	−0.340363
$222$	0	0
$223$	−4749.65	−1.42628	−0.713140	−	0.701022i	$-0.752728\pi$
$223$	−4749.65	−1.42628	−0.713140	+	0.701022i	$0.752728\pi$
$224$	0	0
$225$	2127.17	0.630273
$226$	0	0
$227$	−1330.04	−0.388891	−0.194445	−	0.980913i	$-0.562291\pi$
$227$	−1330.04	−0.388891	−0.194445	+	0.980913i	$0.562291\pi$
$228$	0	0
$229$	−410.874	−0.118565	−0.0592824	−	0.998241i	$-0.518881\pi$
$229$	−410.874	−0.118565	−0.0592824	+	0.998241i	$0.518881\pi$
$230$	0	0
$231$	−1.79904	−0.000512415 0
$232$	0	0
$233$	−5902.78	−1.65967	−0.829837	−	0.558006i	$-0.811566\pi$
$233$	−5902.78	−1.65967	−0.829837	+	0.558006i	$0.811566\pi$
$234$	0	0
$235$	1047.34	0.290726
$236$	0	0
$237$	361.298	0.0990246
$238$	0	0
$239$	−2626.81	−0.710937	−0.355468	−	0.934688i	$-0.615679\pi$
$239$	−2626.81	−0.710937	−0.355468	+	0.934688i	$0.615679\pi$
$240$	0	0
$241$	5074.60	1.35636	0.678182	−	0.734894i	$-0.262768\pi$
$241$	5074.60	1.35636	0.678182	+	0.734894i	$0.262768\pi$
$242$	0	0
$243$	3916.08	1.03381
$244$	0	0
$245$	1143.93	0.298298
$246$	0	0
$247$	−3842.42	−0.989827
$248$	0	0
$249$	3023.49	0.769501
$250$	0	0
$251$	−3776.65	−0.949720	−0.474860	−	0.880061i	$-0.657501\pi$
$251$	−3776.65	−0.949720	−0.474860	+	0.880061i	$0.657501\pi$
$252$	0	0
$253$	6170.23	1.53328
$254$	0	0
$255$	−224.538	−0.0551417
$256$	0	0
$257$	−5775.32	−1.40177	−0.700884	−	0.713275i	$-0.747211\pi$
$257$	−5775.32	−1.40177	−0.700884	+	0.713275i	$0.747211\pi$
$258$	0	0
$259$	2.52060	0.000604719 0
$260$	0	0
$261$	541.706	0.128470
$262$	0	0
$263$	−2117.41	−0.496444	−0.248222	−	0.968703i	$-0.579846\pi$
$263$	−2117.41	−0.496444	−0.248222	+	0.968703i	$0.579846\pi$
$264$	0	0
$265$	351.693	0.0815257
$266$	0	0
$267$	2106.51	0.482833
$268$	0	0
$269$	6494.97	1.47214	0.736069	−	0.676907i	$-0.236680\pi$
$269$	6494.97	1.47214	0.736069	+	0.676907i	$0.236680\pi$
$270$	0	0
$271$	−3122.71	−0.699967	−0.349983	−	0.936756i	$-0.613813\pi$
$271$	−3122.71	−0.699967	−0.349983	+	0.936756i	$0.613813\pi$
$272$	0	0
$273$	−2.24979	−0.000498767 0
$274$	0	0
$275$	4362.70	0.956657
$276$	0	0
$277$	−6431.26	−1.39501	−0.697504	−	0.716581i	$-0.745706\pi$
$277$	−6431.26	−1.39501	−0.697504	+	0.716581i	$0.745706\pi$
$278$	0	0
$279$	−810.954	−0.174016
$280$	0	0
$281$	−1186.87	−0.251968	−0.125984	−	0.992032i	$-0.540209\pi$
$281$	−1186.87	−0.251968	−0.125984	+	0.992032i	$0.540209\pi$
$282$	0	0
$283$	1829.00	0.384179	0.192089	−	0.981377i	$-0.438474\pi$
$283$	1829.00	0.384179	0.192089	+	0.981377i	$0.438474\pi$
$284$	0	0
$285$	−771.552	−0.160361
$286$	0	0
$287$	5.38763	0.00110809
$288$	0	0
$289$	−4368.22	−0.889115
$290$	0	0
$291$	5023.30	1.01193
$292$	0	0
$293$	−102.771	−0.0204914	−0.0102457	−	0.999948i	$-0.503261\pi$
$293$	−102.771	−0.0204914	−0.0102457	+	0.999948i	$0.503261\pi$
$294$	0	0
$295$	1398.76	0.276065
$296$	0	0
$297$	5047.94	0.986233
$298$	0	0
$299$	7716.20	1.49244
$300$	0	0
$301$	−4.42244	−0.000846862 0
$302$	0	0
$303$	22.1272	0.00419529
$304$	0	0
$305$	−803.388	−0.150826
$306$	0	0
$307$	3410.38	0.634009	0.317004	−	0.948424i	$-0.397323\pi$
$307$	3410.38	0.634009	0.317004	+	0.948424i	$0.397323\pi$
$308$	0	0
$309$	−2704.53	−0.497914
$310$	0	0
$311$	−6174.59	−1.12582	−0.562908	−	0.826520i	$-0.690317\pi$
$311$	−6174.59	−1.12582	−0.562908	+	0.826520i	$0.690317\pi$
$312$	0	0
$313$	−4298.77	−0.776296	−0.388148	−	0.921597i	$-0.626885\pi$
$313$	−4298.77	−0.776296	−0.388148	+	0.921597i	$0.626885\pi$
$314$	0	0
$315$	1.01419	0.000181406 0
$316$	0	0
$317$	1674.71	0.296723	0.148361	−	0.988933i	$-0.452600\pi$
$317$	1674.71	0.296723	0.148361	+	0.988933i	$0.452600\pi$
$318$	0	0
$319$	1111.01	0.194998
$320$	0	0
$321$	4295.32	0.746858
$322$	0	0
$323$	1871.95	0.322472
$324$	0	0
$325$	5455.78	0.931176
$326$	0	0
$327$	1691.09	0.285985
$328$	0	0
$329$	−5.11243	−0.000856710 0
$330$	0	0
$331$	2100.96	0.348880	0.174440	−	0.984668i	$-0.444188\pi$
$331$	2100.96	0.348880	0.174440	+	0.984668i	$0.444188\pi$
$332$	0	0
$333$	−2892.16	−0.475944
$334$	0	0
$335$	840.849	0.137136
$336$	0	0
$337$	8074.93	1.30525	0.652625	−	0.757681i	$-0.273668\pi$
$337$	8074.93	1.30525	0.652625	+	0.757681i	$0.273668\pi$
$338$	0	0
$339$	−2560.65	−0.410252
$340$	0	0
$341$	−1663.22	−0.264130
$342$	0	0
$343$	−11.1679	−0.00175804
$344$	0	0
$345$	1549.40	0.241788
$346$	0	0
$347$	−3094.31	−0.478707	−0.239354	−	0.970932i	$-0.576936\pi$
$347$	−3094.31	−0.478707	−0.239354	+	0.970932i	$0.576936\pi$
$348$	0	0
$349$	10451.8	1.60307	0.801537	−	0.597945i	$-0.204016\pi$
$349$	10451.8	1.60307	0.801537	+	0.597945i	$0.204016\pi$
$350$	0	0
$351$	6312.71	0.959964
$352$	0	0
$353$	2559.29	0.385884	0.192942	−	0.981210i	$-0.438197\pi$
$353$	2559.29	0.385884	0.192942	+	0.981210i	$0.438197\pi$
$354$	0	0
$355$	2810.21	0.420143
$356$	0	0
$357$	1.09605	0.000162491 0
$358$	0	0
$359$	−9083.08	−1.33534	−0.667669	−	0.744458i	$-0.732708\pi$
$359$	−9083.08	−1.33534	−0.667669	+	0.744458i	$0.732708\pi$
$360$	0	0
$361$	−426.651	−0.0622031
$362$	0	0
$363$	394.307	0.0570131
$364$	0	0
$365$	−3972.54	−0.569677
$366$	0	0
$367$	−2624.08	−0.373232	−0.186616	−	0.982433i	$-0.559752\pi$
$367$	−2624.08	−0.373232	−0.186616	+	0.982433i	$0.559752\pi$
$368$	0	0
$369$	−6181.82	−0.872121
$370$	0	0
$371$	−1.71674	−0.000240239 0
$372$	0	0
$373$	12185.5	1.69154	0.845769	−	0.533549i	$-0.179142\pi$
$373$	12185.5	1.69154	0.845769	+	0.533549i	$0.179142\pi$
$374$	0	0
$375$	2298.03	0.316452
$376$	0	0
$377$	1389.37	0.189804
$378$	0	0
$379$	3648.75	0.494521	0.247261	−	0.968949i	$-0.420470\pi$
$379$	3648.75	0.494521	0.247261	+	0.968949i	$0.420470\pi$
$380$	0	0
$381$	−944.510	−0.127005
$382$	0	0
$383$	−10017.1	−1.33642	−0.668211	−	0.743972i	$-0.732940\pi$
$383$	−10017.1	−1.33642	−0.668211	+	0.743972i	$0.732940\pi$
$384$	0	0
$385$	2.08004	0.000275347 0
$386$	0	0
$387$	5074.35	0.666521
$388$	0	0
$389$	14326.4	1.86729	0.933647	−	0.358194i	$-0.116607\pi$
$389$	14326.4	1.86729	0.933647	+	0.358194i	$0.116607\pi$
$390$	0	0
$391$	−3759.19	−0.486215
$392$	0	0
$393$	−2754.05	−0.353494
$394$	0	0
$395$	−417.732	−0.0532110
$396$	0	0
$397$	2239.26	0.283086	0.141543	−	0.989932i	$-0.454794\pi$
$397$	2239.26	0.283086	0.141543	+	0.989932i	$0.454794\pi$
$398$	0	0
$399$	3.76623	0.000472549 0
$400$	0	0
$401$	−365.390	−0.0455030	−0.0227515	−	0.999741i	$-0.507243\pi$
$401$	−365.390	−0.0455030	−0.0227515	+	0.999741i	$0.507243\pi$
$402$	0	0
$403$	−2079.94	−0.257095
$404$	0	0
$405$	−414.454	−0.0508503
$406$	0	0
$407$	−5931.64	−0.722409
$408$	0	0
$409$	5210.80	0.629969	0.314985	−	0.949097i	$-0.398001\pi$
$409$	5210.80	0.629969	0.314985	+	0.949097i	$0.398001\pi$
$410$	0	0
$411$	−4375.35	−0.525109
$412$	0	0
$413$	−6.82788	−0.000813506 0
$414$	0	0
$415$	−3495.74	−0.413492
$416$	0	0
$417$	4299.17	0.504871
$418$	0	0
$419$	−1608.20	−0.187507	−0.0937536	−	0.995595i	$-0.529887\pi$
$419$	−1608.20	−0.187507	−0.0937536	+	0.995595i	$0.529887\pi$
$420$	0	0
$421$	−1146.11	−0.132680	−0.0663398	−	0.997797i	$-0.521132\pi$
$421$	−1146.11	−0.132680	−0.0663398	+	0.997797i	$0.521132\pi$
$422$	0	0
$423$	5866.05	0.674272
$424$	0	0
$425$	−2657.95	−0.303364
$426$	0	0
$427$	3.92163	0.000444452 0
$428$	0	0
$429$	5294.35	0.595836
$430$	0	0
$431$	1620.58	0.181115	0.0905573	−	0.995891i	$-0.471135\pi$
$431$	1620.58	0.181115	0.0905573	+	0.995891i	$0.471135\pi$
$432$	0	0
$433$	10905.0	1.21030	0.605152	−	0.796110i	$-0.293112\pi$
$433$	10905.0	1.21030	0.605152	+	0.796110i	$0.293112\pi$
$434$	0	0
$435$	278.983	0.0307499
$436$	0	0
$437$	−12917.2	−1.41399
$438$	0	0
$439$	6134.48	0.666931	0.333466	−	0.942762i	$-0.391782\pi$
$439$	6134.48	0.666931	0.333466	+	0.942762i	$0.391782\pi$
$440$	0	0
$441$	6407.07	0.691833
$442$	0	0
$443$	−10556.2	−1.13215	−0.566074	−	0.824355i	$-0.691538\pi$
$443$	−10556.2	−1.13215	−0.566074	+	0.824355i	$0.691538\pi$
$444$	0	0
$445$	−2435.54	−0.259451
$446$	0	0
$447$	−1049.32	−0.111031
$448$	0	0
$449$	−18169.0	−1.90968	−0.954841	−	0.297118i	$-0.903975\pi$
$449$	−18169.0	−1.90968	−0.954841	+	0.297118i	$0.903975\pi$
$450$	0	0
$451$	−12678.5	−1.32374
$452$	0	0
$453$	−7155.29	−0.742130
$454$	0	0
$455$	2.60120	0.000268013 0
$456$	0	0
$457$	1741.47	0.178255	0.0891277	−	0.996020i	$-0.471592\pi$
$457$	1741.47	0.178255	0.0891277	+	0.996020i	$0.471592\pi$
$458$	0	0
$459$	−3075.43	−0.312743
$460$	0	0
$461$	−14702.5	−1.48539	−0.742695	−	0.669630i	$-0.766453\pi$
$461$	−14702.5	−1.48539	−0.742695	+	0.669630i	$0.766453\pi$
$462$	0	0
$463$	−8370.27	−0.840171	−0.420086	−	0.907484i	$-0.638000\pi$
$463$	−8370.27	−0.840171	−0.420086	+	0.907484i	$0.638000\pi$
$464$	0	0
$465$	−417.648	−0.0416516
$466$	0	0
$467$	−8788.34	−0.870827	−0.435413	−	0.900231i	$-0.643398\pi$
$467$	−8788.34	−0.870827	−0.435413	+	0.900231i	$0.643398\pi$
$468$	0	0
$469$	−4.10449	−0.000404110 0
$470$	0	0
$471$	−216.883	−0.0212175
$472$	0	0
$473$	10407.2	1.01168
$474$	0	0
$475$	−9133.18	−0.882229
$476$	0	0
$477$	1969.80	0.189080
$478$	0	0
$479$	15512.5	1.47972	0.739860	−	0.672761i	$-0.234892\pi$
$479$	15512.5	1.47972	0.739860	+	0.672761i	$0.234892\pi$
$480$	0	0
$481$	−7417.82	−0.703167
$482$	0	0
$483$	−7.56319	−0.000712499 0
$484$	0	0
$485$	−5807.92	−0.543761
$486$	0	0
$487$	−17608.2	−1.63841	−0.819204	−	0.573502i	$-0.805585\pi$
$487$	−17608.2	−1.63841	−0.819204	+	0.573502i	$0.805585\pi$
$488$	0	0
$489$	7140.69	0.660354
$490$	0	0
$491$	3399.75	0.312482	0.156241	−	0.987719i	$-0.450062\pi$
$491$	3399.75	0.312482	0.156241	+	0.987719i	$0.450062\pi$
$492$	0	0
$493$	−676.875	−0.0618355
$494$	0	0
$495$	−2386.66	−0.216711
$496$	0	0
$497$	−13.7177	−0.00123807
$498$	0	0
$499$	−17526.5	−1.57233	−0.786165	−	0.618017i	$-0.787936\pi$
$499$	−17526.5	−1.57233	−0.786165	+	0.618017i	$0.787936\pi$
$500$	0	0
$501$	−2841.38	−0.253380
$502$	0	0
$503$	−12128.1	−1.07508	−0.537538	−	0.843239i	$-0.680646\pi$
$503$	−12128.1	−1.07508	−0.537538	+	0.843239i	$0.680646\pi$
$504$	0	0
$505$	−25.5833	−0.00225434
$506$	0	0
$507$	283.557	0.0248387
$508$	0	0
$509$	5393.04	0.469631	0.234816	−	0.972040i	$-0.424551\pi$
$509$	5393.04	0.469631	0.234816	+	0.972040i	$0.424551\pi$
$510$	0	0
$511$	19.3914	0.00167872
$512$	0	0
$513$	−10567.7	−0.909504
$514$	0	0
$515$	3126.97	0.267555
$516$	0	0
$517$	12030.9	1.02344
$518$	0	0
$519$	−4705.74	−0.397994
$520$	0	0
$521$	14684.3	1.23480	0.617400	−	0.786650i	$-0.288186\pi$
$521$	14684.3	1.23480	0.617400	+	0.786650i	$0.288186\pi$
$522$	0	0
$523$	6821.07	0.570295	0.285148	−	0.958484i	$-0.407957\pi$
$523$	6821.07	0.570295	0.285148	+	0.958484i	$0.407957\pi$
$524$	0	0
$525$	−5.34760	−0.000444549 0
$526$	0	0
$527$	1013.31	0.0837577
$528$	0	0
$529$	13772.8	1.13198
$530$	0	0
$531$	7834.37	0.640269
$532$	0	0
$533$	−15855.2	−1.28849
$534$	0	0
$535$	−4966.23	−0.401325
$536$	0	0
$537$	−1489.48	−0.119694
$538$	0	0
$539$	13140.5	1.05010
$540$	0	0
$541$	−6340.33	−0.503867	−0.251934	−	0.967745i	$-0.581066\pi$
$541$	−6340.33	−0.503867	−0.251934	+	0.967745i	$0.581066\pi$
$542$	0	0
$543$	−2721.73	−0.215103
$544$	0	0
$545$	−1955.23	−0.153675
$546$	0	0
$547$	24269.2	1.89703	0.948517	−	0.316725i	$-0.102583\pi$
$547$	24269.2	1.89703	0.948517	+	0.316725i	$0.102583\pi$
$548$	0	0
$549$	−4499.71	−0.349805
$550$	0	0
$551$	−2325.86	−0.179827
$552$	0	0
$553$	2.03910	0.000156802 0
$554$	0	0
$555$	−1489.49	−0.113919
$556$	0	0
$557$	−9195.89	−0.699537	−0.349769	−	0.936836i	$-0.613740\pi$
$557$	−9195.89	−0.699537	−0.349769	+	0.936836i	$0.613740\pi$
$558$	0	0
$559$	13014.7	0.984730
$560$	0	0
$561$	−2579.31	−0.194115
$562$	0	0
$563$	6222.55	0.465807	0.232903	−	0.972500i	$-0.425177\pi$
$563$	6222.55	0.465807	0.232903	+	0.972500i	$0.425177\pi$
$564$	0	0
$565$	2960.61	0.220449
$566$	0	0
$567$	2.02310	0.000149845 0
$568$	0	0
$569$	−8002.70	−0.589614	−0.294807	−	0.955557i	$-0.595255\pi$
$569$	−8002.70	−0.589614	−0.294807	+	0.955557i	$0.595255\pi$
$570$	0	0
$571$	−20478.1	−1.50085	−0.750423	−	0.660958i	$-0.770150\pi$
$571$	−20478.1	−1.50085	−0.750423	+	0.660958i	$0.770150\pi$
$572$	0	0
$573$	−7276.02	−0.530471
$574$	0	0
$575$	18340.9	1.33021
$576$	0	0
$577$	−14014.9	−1.01117	−0.505586	−	0.862776i	$-0.668724\pi$
$577$	−14014.9	−1.01117	−0.505586	+	0.862776i	$0.668724\pi$
$578$	0	0
$579$	−1437.07	−0.103148
$580$	0	0
$581$	17.0640	0.00121847
$582$	0	0
$583$	4039.95	0.286994
$584$	0	0
$585$	−2984.64	−0.210939
$586$	0	0
$587$	8706.36	0.612181	0.306090	−	0.952002i	$-0.400979\pi$
$587$	8706.36	0.612181	0.306090	+	0.952002i	$0.400979\pi$
$588$	0	0
$589$	3481.89	0.243580
$590$	0	0
$591$	−9372.39	−0.652333
$592$	0	0
$593$	−14408.2	−0.997765	−0.498883	−	0.866670i	$-0.666256\pi$
$593$	−14408.2	−0.997765	−0.498883	+	0.866670i	$0.666256\pi$
$594$	0	0
$595$	−1.26725	−8.73148e−5 0
$596$	0	0
$597$	5627.27	0.385777
$598$	0	0
$599$	27210.1	1.85605	0.928027	−	0.372513i	$-0.121504\pi$
$599$	27210.1	1.85605	0.928027	+	0.372513i	$0.121504\pi$
$600$	0	0
$601$	−7981.32	−0.541705	−0.270853	−	0.962621i	$-0.587306\pi$
$601$	−7981.32	−0.541705	−0.270853	+	0.962621i	$0.587306\pi$
$602$	0	0
$603$	4709.53	0.318055
$604$	0	0
$605$	−455.896	−0.0306361
$606$	0	0
$607$	4107.29	0.274645	0.137323	−	0.990526i	$-0.456150\pi$
$607$	4107.29	0.274645	0.137323	+	0.990526i	$0.456150\pi$
$608$	0	0
$609$	−1.36182	−9.06137e−5 0
$610$	0	0
$611$	15045.3	0.996181
$612$	0	0
$613$	−24401.2	−1.60776	−0.803879	−	0.594793i	$-0.797234\pi$
$613$	−24401.2	−1.60776	−0.803879	+	0.594793i	$0.797234\pi$
$614$	0	0
$615$	−3183.69	−0.208746
$616$	0	0
$617$	16316.3	1.06462	0.532309	−	0.846550i	$-0.321324\pi$
$617$	16316.3	1.06462	0.532309	+	0.846550i	$0.321324\pi$
$618$	0	0
$619$	−12536.5	−0.814030	−0.407015	−	0.913421i	$-0.633430\pi$
$619$	−12536.5	−0.814030	−0.407015	+	0.913421i	$0.633430\pi$
$620$	0	0
$621$	21221.7	1.37133
$622$	0	0
$623$	11.8888	0.000764547 0
$624$	0	0
$625$	11577.7	0.740972
$626$	0	0
$627$	−8862.93	−0.564516
$628$	0	0
$629$	3613.82	0.229082
$630$	0	0
$631$	13113.1	0.827295	0.413648	−	0.910437i	$-0.364255\pi$
$631$	13113.1	0.827295	0.413648	+	0.910437i	$0.364255\pi$
$632$	0	0
$633$	7157.14	0.449401
$634$	0	0
$635$	1092.04	0.0682461
$636$	0	0
$637$	16432.9	1.02213
$638$	0	0
$639$	15739.8	0.974424
$640$	0	0
$641$	−5993.67	−0.369323	−0.184661	−	0.982802i	$-0.559119\pi$
$641$	−5993.67	−0.369323	−0.184661	+	0.982802i	$0.559119\pi$
$642$	0	0
$643$	9432.16	0.578489	0.289244	−	0.957255i	$-0.406596\pi$
$643$	9432.16	0.578489	0.289244	+	0.957255i	$0.406596\pi$
$644$	0	0
$645$	2613.34	0.159535
$646$	0	0
$647$	23224.1	1.41118	0.705591	−	0.708620i	$-0.250682\pi$
$647$	23224.1	1.41118	0.705591	+	0.708620i	$0.250682\pi$
$648$	0	0
$649$	16067.8	0.971829
$650$	0	0
$651$	2.03869	0.000122738 0
$652$	0	0
$653$	12321.0	0.738375	0.369188	−	0.929355i	$-0.379636\pi$
$653$	12321.0	0.738375	0.369188	+	0.929355i	$0.379636\pi$
$654$	0	0
$655$	3184.22	0.189951
$656$	0	0
$657$	−22249.9	−1.32123
$658$	0	0
$659$	−2872.80	−0.169815	−0.0849076	−	0.996389i	$-0.527060\pi$
$659$	−2872.80	−0.169815	−0.0849076	+	0.996389i	$0.527060\pi$
$660$	0	0
$661$	−6026.52	−0.354621	−0.177310	−	0.984155i	$-0.556740\pi$
$661$	−6026.52	−0.354621	−0.177310	+	0.984155i	$0.556740\pi$
$662$	0	0
$663$	−3225.56	−0.188945
$664$	0	0
$665$	−4.35450	−0.000253925 0
$666$	0	0
$667$	4670.69	0.271139
$668$	0	0
$669$	−13700.5	−0.791767
$670$	0	0
$671$	−9228.64	−0.530950
$672$	0	0
$673$	−19959.1	−1.14319	−0.571594	−	0.820537i	$-0.693675\pi$
$673$	−19959.1	−1.14319	−0.571594	+	0.820537i	$0.693675\pi$
$674$	0	0
$675$	15004.9	0.855613
$676$	0	0
$677$	−29561.8	−1.67822	−0.839109	−	0.543963i	$-0.816923\pi$
$677$	−29561.8	−1.67822	−0.839109	+	0.543963i	$0.816923\pi$
$678$	0	0
$679$	28.3506	0.00160235
$680$	0	0
$681$	−3836.55	−0.215884
$682$	0	0
$683$	419.439	0.0234984	0.0117492	−	0.999931i	$-0.496260\pi$
$683$	419.439	0.0234984	0.0117492	+	0.999931i	$0.496260\pi$
$684$	0	0
$685$	5058.76	0.282168
$686$	0	0
$687$	−1185.18	−0.0658186
$688$	0	0
$689$	5052.17	0.279350
$690$	0	0
$691$	−34418.9	−1.89487	−0.947436	−	0.319944i	$-0.896336\pi$
$691$	−34418.9	−1.89487	−0.947436	+	0.319944i	$0.896336\pi$
$692$	0	0
$693$	11.6501	0.000638603 0
$694$	0	0
$695$	−4970.68	−0.271293
$696$	0	0
$697$	7724.33	0.419770
$698$	0	0
$699$	−17026.7	−0.921330
$700$	0	0
$701$	25856.2	1.39312	0.696559	−	0.717500i	$-0.254714\pi$
$701$	25856.2	1.39312	0.696559	+	0.717500i	$0.254714\pi$
$702$	0	0
$703$	12417.7	0.666205
$704$	0	0
$705$	3021.07	0.161390
$706$	0	0
$707$	0.124882	6.64308e−6 0
$708$	0	0
$709$	22719.4	1.20345	0.601725	−	0.798703i	$-0.294480\pi$
$709$	22719.4	1.20345	0.601725	+	0.798703i	$0.294480\pi$
$710$	0	0
$711$	−2339.68	−0.123411
$712$	0	0
$713$	−6992.20	−0.367265
$714$	0	0
$715$	−6121.31	−0.320173
$716$	0	0
$717$	−7577.09	−0.394660
$718$	0	0
$719$	17737.1	0.920006	0.460003	−	0.887917i	$-0.347848\pi$
$719$	17737.1	0.920006	0.460003	+	0.887917i	$0.347848\pi$
$720$	0	0
$721$	−15.2639	−0.000788429 0
$722$	0	0
$723$	14637.8	0.752954
$724$	0	0
$725$	3302.44	0.169172
$726$	0	0
$727$	28913.6	1.47503	0.737514	−	0.675331i	$-0.235999\pi$
$727$	28913.6	1.47503	0.737514	+	0.675331i	$0.235999\pi$
$728$	0	0
$729$	7940.72	0.403431
$730$	0	0
$731$	−6340.53	−0.320811
$732$	0	0
$733$	4413.13	0.222377	0.111189	−	0.993799i	$-0.464534\pi$
$733$	4413.13	0.222377	0.111189	+	0.993799i	$0.464534\pi$
$734$	0	0
$735$	3299.70	0.165593
$736$	0	0
$737$	9658.96	0.482758
$738$	0	0
$739$	31940.0	1.58989	0.794947	−	0.606678i	$-0.207498\pi$
$739$	31940.0	1.58989	0.794947	+	0.606678i	$0.207498\pi$
$740$	0	0
$741$	−11083.6	−0.549480
$742$	0	0
$743$	−1563.17	−0.0771832	−0.0385916	−	0.999255i	$-0.512287\pi$
$743$	−1563.17	−0.0771832	−0.0385916	+	0.999255i	$0.512287\pi$
$744$	0	0
$745$	1213.22	0.0596629
$746$	0	0
$747$	−19579.4	−0.958999
$748$	0	0
$749$	24.2420	0.00118262
$750$	0	0
$751$	−4455.80	−0.216504	−0.108252	−	0.994123i	$-0.534525\pi$
$751$	−4455.80	−0.216504	−0.108252	+	0.994123i	$0.534525\pi$
$752$	0	0
$753$	−10893.8	−0.527216
$754$	0	0
$755$	8272.92	0.398785
$756$	0	0
$757$	−32820.2	−1.57579	−0.787893	−	0.615812i	$-0.788828\pi$
$757$	−32820.2	−1.57579	−0.787893	+	0.615812i	$0.788828\pi$
$758$	0	0
$759$	17798.2	0.851164
$760$	0	0
$761$	28649.3	1.36470	0.682349	−	0.731027i	$-0.260959\pi$
$761$	28649.3	1.36470	0.682349	+	0.731027i	$0.260959\pi$
$762$	0	0
$763$	9.54417	0.000452847 0
$764$	0	0
$765$	1454.06	0.0687210
$766$	0	0
$767$	20093.6	0.945944
$768$	0	0
$769$	1729.41	0.0810975	0.0405487	−	0.999178i	$-0.487089\pi$
$769$	1729.41	0.0810975	0.0405487	+	0.999178i	$0.487089\pi$
$770$	0	0
$771$	−16659.1	−0.778160
$772$	0	0
$773$	−4920.92	−0.228969	−0.114485	−	0.993425i	$-0.536522\pi$
$773$	−4920.92	−0.228969	−0.114485	+	0.993425i	$0.536522\pi$
$774$	0	0
$775$	−4943.87	−0.229147
$776$	0	0
$777$	7.27073	0.000335696 0
$778$	0	0
$779$	26542.1	1.22076
$780$	0	0
$781$	32281.4	1.47902
$782$	0	0
$783$	3821.15	0.174402
$784$	0	0
$785$	250.759	0.0114013
$786$	0	0
$787$	−16135.9	−0.730857	−0.365428	−	0.930839i	$-0.619078\pi$
$787$	−16135.9	−0.730857	−0.365428	+	0.930839i	$0.619078\pi$
$788$	0	0
$789$	−6107.71	−0.275590
$790$	0	0
$791$	−14.4518	−0.000649618 0
$792$	0	0
$793$	−11540.9	−0.516808
$794$	0	0
$795$	1014.47	0.0452571
$796$	0	0
$797$	−27007.2	−1.20031	−0.600153	−	0.799885i	$-0.704894\pi$
$797$	−27007.2	−1.20031	−0.600153	+	0.799885i	$0.704894\pi$
$798$	0	0
$799$	−7329.77	−0.324542
$800$	0	0
$801$	−13641.3	−0.601736
$802$	0	0
$803$	−45633.2	−2.00543
$804$	0	0
$805$	8.74453	0.000382862 0
$806$	0	0
$807$	18734.9	0.817224
$808$	0	0
$809$	10936.4	0.475283	0.237642	−	0.971353i	$-0.423626\pi$
$809$	10936.4	0.475283	0.237642	+	0.971353i	$0.423626\pi$
$810$	0	0
$811$	−30566.1	−1.32345	−0.661727	−	0.749745i	$-0.730176\pi$
$811$	−30566.1	−1.32345	−0.661727	+	0.749745i	$0.730176\pi$
$812$	0	0
$813$	−9007.53	−0.388570
$814$	0	0
$815$	−8256.04	−0.354842
$816$	0	0
$817$	−21787.1	−0.932968
$818$	0	0
$819$	14.5691	0.000621594 0
$820$	0	0
$821$	27903.1	1.18615	0.593073	−	0.805149i	$-0.297915\pi$
$821$	27903.1	1.18615	0.593073	+	0.805149i	$0.297915\pi$
$822$	0	0
$823$	−41229.3	−1.74625	−0.873125	−	0.487497i	$-0.837910\pi$
$823$	−41229.3	−1.74625	−0.873125	+	0.487497i	$0.837910\pi$
$824$	0	0
$825$	12584.3	0.531066
$826$	0	0
$827$	8031.88	0.337722	0.168861	−	0.985640i	$-0.445991\pi$
$827$	8031.88	0.337722	0.168861	+	0.985640i	$0.445991\pi$
$828$	0	0
$829$	35924.7	1.50509	0.752543	−	0.658543i	$-0.228827\pi$
$829$	35924.7	1.50509	0.752543	+	0.658543i	$0.228827\pi$
$830$	0	0
$831$	−18551.1	−0.774407
$832$	0	0
$833$	−8005.79	−0.332994
$834$	0	0
$835$	3285.19	0.136154
$836$	0	0
$837$	−5720.40	−0.236232
$838$	0	0
$839$	36856.8	1.51661	0.758306	−	0.651899i	$-0.226027\pi$
$839$	36856.8	1.51661	0.758306	+	0.651899i	$0.226027\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−3423.57	−0.139874
$844$	0	0
$845$	−327.848	−0.0133471
$846$	0	0
$847$	2.22540	9.02781e−5 0
$848$	0	0
$849$	5275.79	0.213268
$850$	0	0
$851$	−24936.7	−1.00449
$852$	0	0
$853$	−23018.6	−0.923965	−0.461982	−	0.886889i	$-0.652862\pi$
$853$	−23018.6	−0.923965	−0.461982	+	0.886889i	$0.652862\pi$
$854$	0	0
$855$	4996.39	0.199851
$856$	0	0
$857$	−20706.3	−0.825336	−0.412668	−	0.910882i	$-0.635403\pi$
$857$	−20706.3	−0.825336	−0.412668	+	0.910882i	$0.635403\pi$
$858$	0	0
$859$	−9778.26	−0.388394	−0.194197	−	0.980963i	$-0.562210\pi$
$859$	−9778.26	−0.388394	−0.194197	+	0.980963i	$0.562210\pi$
$860$	0	0
$861$	15.5408	0.000615131 0
$862$	0	0
$863$	−16757.7	−0.660994	−0.330497	−	0.943807i	$-0.607216\pi$
$863$	−16757.7	−0.660994	−0.330497	+	0.943807i	$0.607216\pi$
$864$	0	0
$865$	5440.76	0.213863
$866$	0	0
$867$	−12600.2	−0.493572
$868$	0	0
$869$	−4798.55	−0.187318
$870$	0	0
$871$	12079.0	0.469899
$872$	0	0
$873$	−32529.7	−1.26113
$874$	0	0
$875$	12.9696	0.000501090 0
$876$	0	0
$877$	25977.3	1.00022	0.500108	−	0.865963i	$-0.333293\pi$
$877$	25977.3	1.00022	0.500108	+	0.865963i	$0.333293\pi$
$878$	0	0
$879$	−296.447	−0.0113753
$880$	0	0
$881$	−18677.5	−0.714258	−0.357129	−	0.934055i	$-0.616244\pi$
$881$	−18677.5	−0.714258	−0.357129	+	0.934055i	$0.616244\pi$
$882$	0	0
$883$	−682.100	−0.0259960	−0.0129980	−	0.999916i	$-0.504138\pi$
$883$	−682.100	−0.0259960	−0.0129980	+	0.999916i	$0.504138\pi$
$884$	0	0
$885$	4034.77	0.153251
$886$	0	0
$887$	−11241.3	−0.425530	−0.212765	−	0.977103i	$-0.568247\pi$
$887$	−11241.3	−0.425530	−0.212765	+	0.977103i	$0.568247\pi$
$888$	0	0
$889$	−5.33064	−0.000201107 0
$890$	0	0
$891$	−4760.89	−0.179008
$892$	0	0
$893$	−25186.3	−0.943817
$894$	0	0
$895$	1722.13	0.0643178
$896$	0	0
$897$	22257.6	0.828494
$898$	0	0
$899$	−1259.01	−0.0467077
$900$	0	0
$901$	−2461.32	−0.0910083
$902$	0	0
$903$	−12.7566	−0.000470116 0
$904$	0	0
$905$	3146.86	0.115586
$906$	0	0
$907$	−2531.90	−0.0926907	−0.0463454	−	0.998925i	$-0.514757\pi$
$907$	−2531.90	−0.0926907	−0.0463454	+	0.998925i	$0.514757\pi$
$908$	0	0
$909$	−143.290	−0.00522842
$910$	0	0
$911$	19862.3	0.722358	0.361179	−	0.932497i	$-0.382374\pi$
$911$	19862.3	0.722358	0.361179	+	0.932497i	$0.382374\pi$
$912$	0	0
$913$	−40156.1	−1.45561
$914$	0	0
$915$	−2317.39	−0.0837275
$916$	0	0
$917$	−15.5433	−0.000559745 0
$918$	0	0
$919$	−26719.5	−0.959080	−0.479540	−	0.877520i	$-0.659196\pi$
$919$	−26719.5	−0.959080	−0.479540	+	0.877520i	$0.659196\pi$
$920$	0	0
$921$	9837.33	0.351955
$922$	0	0
$923$	40369.5	1.43963
$924$	0	0
$925$	−17631.7	−0.626730
$926$	0	0
$927$	17513.9	0.620532
$928$	0	0
$929$	21382.4	0.755151	0.377575	−	0.925979i	$-0.376758\pi$
$929$	21382.4	0.755151	0.377575	+	0.925979i	$0.376758\pi$
$930$	0	0
$931$	−27509.2	−0.968398
$932$	0	0
$933$	−17810.7	−0.624971
$934$	0	0
$935$	2982.18	0.104308
$936$	0	0
$937$	56156.7	1.95791	0.978954	−	0.204083i	$-0.0654213\pi$
$937$	56156.7	1.95791	0.978954	+	0.204083i	$0.0654213\pi$
$938$	0	0
$939$	−12399.9	−0.430943
$940$	0	0
$941$	54215.3	1.87818	0.939091	−	0.343669i	$-0.111670\pi$
$941$	54215.3	1.87818	0.939091	+	0.343669i	$0.111670\pi$
$942$	0	0
$943$	−53300.8	−1.84063
$944$	0	0
$945$	7.15400	0.000246264 0
$946$	0	0
$947$	−45924.8	−1.57588	−0.787939	−	0.615753i	$-0.788852\pi$
$947$	−45924.8	−1.57588	−0.787939	+	0.615753i	$0.788852\pi$
$948$	0	0
$949$	−57066.7	−1.95201
$950$	0	0
$951$	4830.75	0.164719
$952$	0	0
$953$	−19078.1	−0.648478	−0.324239	−	0.945975i	$-0.605108\pi$
$953$	−19078.1	−0.648478	−0.324239	+	0.945975i	$0.605108\pi$
$954$	0	0
$955$	8412.51	0.285050
$956$	0	0
$957$	3204.72	0.108249
$958$	0	0
$959$	−24.6937	−0.000831491 0
$960$	0	0
$961$	−27906.2	−0.936733
$962$	0	0
$963$	−27815.5	−0.930781
$964$	0	0
$965$	1661.54	0.0554268
$966$	0	0
$967$	−56312.9	−1.87270	−0.936351	−	0.351067i	$-0.885819\pi$
$967$	−56312.9	−1.87270	−0.936351	+	0.351067i	$0.885819\pi$
$968$	0	0
$969$	5399.70	0.179013
$970$	0	0
$971$	−25944.8	−0.857474	−0.428737	−	0.903429i	$-0.641041\pi$
$971$	−25944.8	−0.857474	−0.428737	+	0.903429i	$0.641041\pi$
$972$	0	0
$973$	24.2637	0.000799444 0
$974$	0	0
$975$	15737.3	0.516921
$976$	0	0
$977$	16786.3	0.549684	0.274842	−	0.961489i	$-0.411374\pi$
$977$	16786.3	0.549684	0.274842	+	0.961489i	$0.411374\pi$
$978$	0	0
$979$	−27977.4	−0.913342
$980$	0	0
$981$	−10951.1	−0.356413
$982$	0	0
$983$	54504.3	1.76848	0.884241	−	0.467030i	$-0.154676\pi$
$983$	54504.3	1.76848	0.884241	+	0.467030i	$0.154676\pi$
$984$	0	0
$985$	10836.3	0.350532
$986$	0	0
$987$	−14.7469	−0.000475583 0
$988$	0	0
$989$	43752.0	1.40671
$990$	0	0
$991$	3346.09	0.107258	0.0536288	−	0.998561i	$-0.482921\pi$
$991$	3346.09	0.107258	0.0536288	+	0.998561i	$0.482921\pi$
$992$	0	0
$993$	6060.28	0.193673
$994$	0	0
$995$	−6506.23	−0.207298
$996$	0	0
$997$	38893.6	1.23548	0.617740	−	0.786382i	$-0.288048\pi$
$997$	38893.6	1.23548	0.617740	+	0.786382i	$0.288048\pi$
$998$	0	0
$999$	−20401.0	−0.646106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.4.a.e.1.5	✓	6
3.2	odd	2		2088.4.a.l.1.4		6
4.3	odd	2		464.4.a.n.1.2		6
8.3	odd	2		1856.4.a.bc.1.5		6
8.5	even	2		1856.4.a.bd.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.e.1.5	✓	6	1.1	even	1	trivial
464.4.a.n.1.2		6	4.3	odd	2
1856.4.a.bc.1.5		6	8.3	odd	2
1856.4.a.bd.1.2		6	8.5	even	2
2088.4.a.l.1.4		6	3.2	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:	\( N \)	\(=\)	\( 232 = 2^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	232.a (trivial)

\(f(q)\)	\(=\)	\(q+2.88453 q^{3} -3.33508 q^{5} +0.0162797 q^{7} -18.6795 q^{9} +O(q^{10})\)	\(q+2.88453 q^{3} -3.33508 q^{5} +0.0162797 q^{7} -18.6795 q^{9} -38.3105 q^{11} -47.9093 q^{13} -9.62012 q^{15} +23.3405 q^{17} +80.2019 q^{19} +0.0469593 q^{21} -161.058 q^{23} -113.877 q^{25} -131.764 q^{27} -29.0000 q^{29} +43.4141 q^{31} -110.508 q^{33} -0.0542942 q^{35} +154.830 q^{37} -138.196 q^{39} +330.941 q^{41} -271.653 q^{43} +62.2976 q^{45} -314.037 q^{47} -343.000 q^{49} +67.3263 q^{51} -105.453 q^{53} +127.769 q^{55} +231.344 q^{57} -419.410 q^{59} +240.890 q^{61} -0.304097 q^{63} +159.781 q^{65} -252.123 q^{67} -464.577 q^{69} -842.623 q^{71} +1191.14 q^{73} -328.482 q^{75} -0.623685 q^{77} +125.254 q^{79} +124.271 q^{81} +1048.17 q^{83} -77.8424 q^{85} -83.6512 q^{87} +730.280 q^{89} -0.779951 q^{91} +125.229 q^{93} -267.480 q^{95} +1741.46 q^{97} +715.622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9 - 19 * q^11 + 13 * q^13 - 191 * q^15 - 218 * q^17 - 290 * q^19 - 266 * q^21 - 196 * q^23 - 13 * q^25 - 437 * q^27 - 174 * q^29 - 675 * q^31 + 291 * q^33 - 466 * q^35 - 238 * q^37 - 1297 * q^39 - 464 * q^41 - 579 * q^43 - 148 * q^45 - 975 * q^47 + 914 * q^49 - 576 * q^51 + 515 * q^53 - 1605 * q^55 - 340 * q^57 - 108 * q^59 + 1158 * q^61 - 1136 * q^63 + 1239 * q^65 - 80 * q^67 + 2568 * q^69 + 438 * q^71 + 262 * q^73 + 1766 * q^75 + 194 * q^77 - 237 * q^79 + 2554 * q^81 - 1288 * q^83 + 3112 * q^85 + 145 * q^87 - 252 * q^89 + 2450 * q^91 + 2131 * q^93 + 180 * q^95 + 380 * q^97 + 2264 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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