LMFDB - Embedded newform 3432.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3432.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.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:	$27.4046579737$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.70164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 10x - 2 $ x^4 - x^3 - 10x^2 + 10x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.279954$ of defining polynomial
Character	$\chi$	$=$	3432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.279954 q^{5} -4.36642 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.279954 q^{5} -4.36642 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.279954 q^{15} +2.64638 q^{17} +4.58412 q^{19} +4.36642 q^{21} +1.80651 q^{23} -4.92163 q^{25} -1.00000 q^{27} +3.30887 q^{29} -1.50235 q^{31} -1.00000 q^{33} -1.22240 q^{35} -0.559909 q^{37} +1.00000 q^{39} +6.36642 q^{41} -6.97918 q^{43} +0.279954 q^{45} +2.73285 q^{47} +12.0656 q^{49} -2.64638 q^{51} +2.99529 q^{53} +0.279954 q^{55} -4.58412 q^{57} -8.39063 q^{59} -1.10729 q^{61} -4.36642 q^{63} -0.279954 q^{65} -4.39507 q^{67} -1.80651 q^{69} -2.99529 q^{71} +13.6498 q^{73} +4.92163 q^{75} -4.36642 q^{77} -14.9344 q^{79} +1.00000 q^{81} -13.7281 q^{83} +0.740865 q^{85} -3.30887 q^{87} +2.49765 q^{89} +4.36642 q^{91} +1.50235 q^{93} +1.28334 q^{95} -16.0162 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} - 6 q^{17} + q^{21} - 9 q^{23} + q^{25} - 4 q^{27} - q^{29} - 8 q^{31} - 4 q^{33} - 7 q^{35} - 2 q^{37} + 4 q^{39} + 9 q^{41} - 5 q^{43} + q^{45} - 22 q^{47} + 9 q^{49} + 6 q^{51} + 8 q^{53} + q^{55} + q^{59} - 11 q^{61} - q^{63} - q^{65} - 13 q^{67} + 9 q^{69} - 8 q^{71} - 3 q^{73} - q^{75} - q^{77} - 6 q^{79} + 4 q^{81} - 18 q^{83} + 26 q^{85} + q^{87} + 8 q^{89} + q^{91} + 8 q^{93} - 36 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - q^15 - 6 * q^17 + q^21 - 9 * q^23 + q^25 - 4 * q^27 - q^29 - 8 * q^31 - 4 * q^33 - 7 * q^35 - 2 * q^37 + 4 * q^39 + 9 * q^41 - 5 * q^43 + q^45 - 22 * q^47 + 9 * q^49 + 6 * q^51 + 8 * q^53 + q^55 + q^59 - 11 * q^61 - q^63 - q^65 - 13 * q^67 + 9 * q^69 - 8 * q^71 - 3 * q^73 - q^75 - q^77 - 6 * q^79 + 4 * q^81 - 18 * q^83 + 26 * q^85 + q^87 + 8 * q^89 + q^91 + 8 * q^93 - 36 * q^95 + 10 * q^97 + 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.279954	0.125199	0.0625997	−	0.998039i	$-0.480061\pi$
$5$	0.279954	0.125199	0.0625997	+	0.998039i	$0.480061\pi$
$6$	0	0
$7$	−4.36642	−1.65035	−0.825176	−	0.564875i	$-0.808924\pi$
$7$	−4.36642	−1.65035	−0.825176	+	0.564875i	$0.808924\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.279954	−0.0722839
$16$	0	0
$17$	2.64638	0.641841	0.320920	−	0.947106i	$-0.396008\pi$
$17$	2.64638	0.641841	0.320920	+	0.947106i	$0.396008\pi$
$18$	0	0
$19$	4.58412	1.05167	0.525834	−	0.850587i	$-0.323753\pi$
$19$	4.58412	1.05167	0.525834	+	0.850587i	$0.323753\pi$
$20$	0	0
$21$	4.36642	0.952832
$22$	0	0
$23$	1.80651	0.376684	0.188342	−	0.982103i	$-0.439689\pi$
$23$	1.80651	0.376684	0.188342	+	0.982103i	$0.439689\pi$
$24$	0	0
$25$	−4.92163	−0.984325
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.30887	0.614441	0.307221	−	0.951638i	$-0.400601\pi$
$29$	3.30887	0.614441	0.307221	+	0.951638i	$0.400601\pi$
$30$	0	0
$31$	−1.50235	−0.269831	−0.134915	−	0.990857i	$-0.543076\pi$
$31$	−1.50235	−0.269831	−0.134915	+	0.990857i	$0.543076\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.22240	−0.206623
$36$	0	0
$37$	−0.559909	−0.0920484	−0.0460242	−	0.998940i	$-0.514655\pi$
$37$	−0.559909	−0.0920484	−0.0460242	+	0.998940i	$0.514655\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	6.36642	0.994268	0.497134	−	0.867674i	$-0.334386\pi$
$41$	6.36642	0.994268	0.497134	+	0.867674i	$0.334386\pi$
$42$	0	0
$43$	−6.97918	−1.06432	−0.532158	−	0.846645i	$-0.678619\pi$
$43$	−6.97918	−1.06432	−0.532158	+	0.846645i	$0.678619\pi$
$44$	0	0
$45$	0.279954	0.0417331
$46$	0	0
$47$	2.73285	0.398627	0.199313	−	0.979936i	$-0.436129\pi$
$47$	2.73285	0.398627	0.199313	+	0.979936i	$0.436129\pi$
$48$	0	0
$49$	12.0656	1.72366
$50$	0	0
$51$	−2.64638	−0.370567
$52$	0	0
$53$	2.99529	0.411435	0.205718	−	0.978611i	$-0.434047\pi$
$53$	2.99529	0.411435	0.205718	+	0.978611i	$0.434047\pi$
$54$	0	0
$55$	0.279954	0.0377490
$56$	0	0
$57$	−4.58412	−0.607181
$58$	0	0
$59$	−8.39063	−1.09237	−0.546184	−	0.837666i	$-0.683920\pi$
$59$	−8.39063	−1.09237	−0.546184	+	0.837666i	$0.683920\pi$
$60$	0	0
$61$	−1.10729	−0.141774	−0.0708868	−	0.997484i	$-0.522583\pi$
$61$	−1.10729	−0.141774	−0.0708868	+	0.997484i	$0.522583\pi$
$62$	0	0
$63$	−4.36642	−0.550118
$64$	0	0
$65$	−0.279954	−0.0347241
$66$	0	0
$67$	−4.39507	−0.536943	−0.268471	−	0.963288i	$-0.586518\pi$
$67$	−4.39507	−0.536943	−0.268471	+	0.963288i	$0.586518\pi$
$68$	0	0
$69$	−1.80651	−0.217479
$70$	0	0
$71$	−2.99529	−0.355476	−0.177738	−	0.984078i	$-0.556878\pi$
$71$	−2.99529	−0.355476	−0.177738	+	0.984078i	$0.556878\pi$
$72$	0	0
$73$	13.6498	1.59758	0.798792	−	0.601607i	$-0.205473\pi$
$73$	13.6498	1.59758	0.798792	+	0.601607i	$0.205473\pi$
$74$	0	0
$75$	4.92163	0.568300
$76$	0	0
$77$	−4.36642	−0.497600
$78$	0	0
$79$	−14.9344	−1.68025	−0.840127	−	0.542390i	$-0.817520\pi$
$79$	−14.9344	−1.68025	−0.840127	+	0.542390i	$0.817520\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.7281	−1.50686	−0.753430	−	0.657529i	$-0.771602\pi$
$83$	−13.7281	−1.50686	−0.753430	+	0.657529i	$0.771602\pi$
$84$	0	0
$85$	0.740865	0.0803580
$86$	0	0
$87$	−3.30887	−0.354748
$88$	0	0
$89$	2.49765	0.264750	0.132375	−	0.991200i	$-0.457740\pi$
$89$	2.49765	0.264750	0.132375	+	0.991200i	$0.457740\pi$
$90$	0	0
$91$	4.36642	0.457726
$92$	0	0
$93$	1.50235	0.155787
$94$	0	0
$95$	1.28334	0.131668
$96$	0	0
$97$	−16.0162	−1.62620	−0.813099	−	0.582126i	$-0.802221\pi$
$97$	−16.0162	−1.62620	−0.813099	+	0.582126i	$0.802221\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.4896	−1.04376	−0.521879	−	0.853020i	$-0.674769\pi$
$101$	−10.4896	−1.04376	−0.521879	+	0.853020i	$0.674769\pi$
$102$	0	0
$103$	−5.10729	−0.503236	−0.251618	−	0.967827i	$-0.580963\pi$
$103$	−5.10729	−0.503236	−0.251618	+	0.967827i	$0.580963\pi$
$104$	0	0
$105$	1.22240	0.119294
$106$	0	0
$107$	−10.8354	−1.04750	−0.523750	−	0.851872i	$-0.675467\pi$
$107$	−10.8354	−1.04750	−0.523750	+	0.851872i	$0.675467\pi$
$108$	0	0
$109$	−9.58882	−0.918443	−0.459221	−	0.888322i	$-0.651871\pi$
$109$	−9.58882	−0.918443	−0.459221	+	0.888322i	$0.651871\pi$
$110$	0	0
$111$	0.559909	0.0531442
$112$	0	0
$113$	−2.77760	−0.261295	−0.130647	−	0.991429i	$-0.541706\pi$
$113$	−2.77760	−0.261295	−0.130647	+	0.991429i	$0.541706\pi$
$114$	0	0
$115$	0.505741	0.0471606
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−11.5552	−1.05926
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.36642	−0.574041
$124$	0	0
$125$	−2.77760	−0.248436
$126$	0	0
$127$	−7.08176	−0.628405	−0.314202	−	0.949356i	$-0.601737\pi$
$127$	−7.08176	−0.628405	−0.314202	+	0.949356i	$0.601737\pi$
$128$	0	0
$129$	6.97918	0.614483
$130$	0	0
$131$	−2.22710	−0.194583	−0.0972915	−	0.995256i	$-0.531018\pi$
$131$	−2.22710	−0.194583	−0.0972915	+	0.995256i	$0.531018\pi$
$132$	0	0
$133$	−20.0162	−1.73562
$134$	0	0
$135$	−0.279954	−0.0240946
$136$	0	0
$137$	18.6384	1.59238	0.796191	−	0.605045i	$-0.206845\pi$
$137$	18.6384	1.59238	0.796191	+	0.605045i	$0.206845\pi$
$138$	0	0
$139$	12.5313	1.06289	0.531444	−	0.847093i	$-0.321650\pi$
$139$	12.5313	1.06289	0.531444	+	0.847093i	$0.321650\pi$
$140$	0	0
$141$	−2.73285	−0.230147
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.926332	0.0769276
$146$	0	0
$147$	−12.0656	−0.995158
$148$	0	0
$149$	7.70393	0.631131	0.315565	−	0.948904i	$-0.397806\pi$
$149$	7.70393	0.631131	0.315565	+	0.948904i	$0.397806\pi$
$150$	0	0
$151$	4.14873	0.337619	0.168809	−	0.985649i	$-0.446008\pi$
$151$	4.14873	0.337619	0.168809	+	0.985649i	$0.446008\pi$
$152$	0	0
$153$	2.64638	0.213947
$154$	0	0
$155$	−0.420590	−0.0337826
$156$	0	0
$157$	−2.59353	−0.206986	−0.103493	−	0.994630i	$-0.533002\pi$
$157$	−2.59353	−0.206986	−0.103493	+	0.994630i	$0.533002\pi$
$158$	0	0
$159$	−2.99529	−0.237542
$160$	0	0
$161$	−7.88801	−0.621662
$162$	0	0
$163$	6.00809	0.470590	0.235295	−	0.971924i	$-0.424394\pi$
$163$	6.00809	0.470590	0.235295	+	0.971924i	$0.424394\pi$
$164$	0	0
$165$	−0.279954	−0.0217944
$166$	0	0
$167$	−16.2177	−1.25496	−0.627481	−	0.778632i	$-0.715914\pi$
$167$	−16.2177	−1.25496	−0.627481	+	0.778632i	$0.715914\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.58412	0.350556
$172$	0	0
$173$	3.42398	0.260320	0.130160	−	0.991493i	$-0.458451\pi$
$173$	3.42398	0.260320	0.130160	+	0.991493i	$0.458451\pi$
$174$	0	0
$175$	21.4899	1.62448
$176$	0	0
$177$	8.39063	0.630678
$178$	0	0
$179$	16.0498	1.19962	0.599809	−	0.800143i	$-0.295243\pi$
$179$	16.0498	1.19962	0.599809	+	0.800143i	$0.295243\pi$
$180$	0	0
$181$	1.70393	0.126652	0.0633262	−	0.997993i	$-0.479829\pi$
$181$	1.70393	0.126652	0.0633262	+	0.997993i	$0.479829\pi$
$182$	0	0
$183$	1.10729	0.0818531
$184$	0	0
$185$	−0.156749	−0.0115244
$186$	0	0
$187$	2.64638	0.193522
$188$	0	0
$189$	4.36642	0.317611
$190$	0	0
$191$	−19.5683	−1.41591	−0.707955	−	0.706257i	$-0.750382\pi$
$191$	−19.5683	−1.41591	−0.707955	+	0.706257i	$0.750382\pi$
$192$	0	0
$193$	−26.8305	−1.93130	−0.965652	−	0.259840i	$-0.916330\pi$
$193$	−26.8305	−1.93130	−0.965652	+	0.259840i	$0.916330\pi$
$194$	0	0
$195$	0.279954	0.0200479
$196$	0	0
$197$	−19.6466	−1.39977	−0.699883	−	0.714258i	$-0.746764\pi$
$197$	−19.6466	−1.39977	−0.699883	+	0.714258i	$0.746764\pi$
$198$	0	0
$199$	−7.55208	−0.535353	−0.267677	−	0.963509i	$-0.586256\pi$
$199$	−7.55208	−0.535353	−0.267677	+	0.963509i	$0.586256\pi$
$200$	0	0
$201$	4.39507	0.310004
$202$	0	0
$203$	−14.4479	−1.01404
$204$	0	0
$205$	1.78231	0.124482
$206$	0	0
$207$	1.80651	0.125561
$208$	0	0
$209$	4.58412	0.317090
$210$	0	0
$211$	17.4849	1.20371	0.601856	−	0.798605i	$-0.294428\pi$
$211$	17.4849	1.20371	0.601856	+	0.798605i	$0.294428\pi$
$212$	0	0
$213$	2.99529	0.205234
$214$	0	0
$215$	−1.95385	−0.133252
$216$	0	0
$217$	6.55991	0.445316
$218$	0	0
$219$	−13.6498	−0.922366
$220$	0	0
$221$	−2.64638	−0.178015
$222$	0	0
$223$	−7.38724	−0.494686	−0.247343	−	0.968928i	$-0.579557\pi$
$223$	−7.38724	−0.494686	−0.247343	+	0.968928i	$0.579557\pi$
$224$	0	0
$225$	−4.92163	−0.328108
$226$	0	0
$227$	7.02891	0.466525	0.233263	−	0.972414i	$-0.425060\pi$
$227$	7.02891	0.466525	0.233263	+	0.972414i	$0.425060\pi$
$228$	0	0
$229$	−13.6834	−0.904224	−0.452112	−	0.891961i	$-0.649329\pi$
$229$	−13.6834	−0.904224	−0.452112	+	0.891961i	$0.649329\pi$
$230$	0	0
$231$	4.36642	0.287290
$232$	0	0
$233$	−16.6464	−1.09054	−0.545270	−	0.838260i	$-0.683573\pi$
$233$	−16.6464	−1.09054	−0.545270	+	0.838260i	$0.683573\pi$
$234$	0	0
$235$	0.765072	0.0499078
$236$	0	0
$237$	14.9344	0.970095
$238$	0	0
$239$	−11.9795	−0.774886	−0.387443	−	0.921894i	$-0.626642\pi$
$239$	−11.9795	−0.774886	−0.387443	+	0.921894i	$0.626642\pi$
$240$	0	0
$241$	17.5808	1.13248	0.566240	−	0.824241i	$-0.308398\pi$
$241$	17.5808	1.13248	0.566240	+	0.824241i	$0.308398\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.37783	0.215802
$246$	0	0
$247$	−4.58412	−0.291680
$248$	0	0
$249$	13.7281	0.869986
$250$	0	0
$251$	−18.8480	−1.18967	−0.594836	−	0.803847i	$-0.702783\pi$
$251$	−18.8480	−1.18967	−0.594836	+	0.803847i	$0.702783\pi$
$252$	0	0
$253$	1.80651	0.113575
$254$	0	0
$255$	−0.740865	−0.0463947
$256$	0	0
$257$	−17.3953	−1.08509	−0.542546	−	0.840026i	$-0.682539\pi$
$257$	−17.3953	−1.08509	−0.542546	+	0.840026i	$0.682539\pi$
$258$	0	0
$259$	2.44480	0.151912
$260$	0	0
$261$	3.30887	0.204814
$262$	0	0
$263$	−10.1151	−0.623724	−0.311862	−	0.950127i	$-0.600953\pi$
$263$	−10.1151	−0.623724	−0.311862	+	0.950127i	$0.600953\pi$
$264$	0	0
$265$	0.838545	0.0515114
$266$	0	0
$267$	−2.49765	−0.152854
$268$	0	0
$269$	−10.8385	−0.660838	−0.330419	−	0.943834i	$-0.607190\pi$
$269$	−10.8385	−0.660838	−0.330419	+	0.943834i	$0.607190\pi$
$270$	0	0
$271$	2.03362	0.123534	0.0617668	−	0.998091i	$-0.480326\pi$
$271$	2.03362	0.123534	0.0617668	+	0.998091i	$0.480326\pi$
$272$	0	0
$273$	−4.36642	−0.264268
$274$	0	0
$275$	−4.92163	−0.296785
$276$	0	0
$277$	−1.55208	−0.0932557	−0.0466279	−	0.998912i	$-0.514847\pi$
$277$	−1.55208	−0.0932557	−0.0466279	+	0.998912i	$0.514847\pi$
$278$	0	0
$279$	−1.50235	−0.0899435
$280$	0	0
$281$	10.9747	0.654698	0.327349	−	0.944903i	$-0.393845\pi$
$281$	10.9747	0.654698	0.327349	+	0.944903i	$0.393845\pi$
$282$	0	0
$283$	23.1615	1.37681	0.688405	−	0.725326i	$-0.258311\pi$
$283$	23.1615	1.37681	0.688405	+	0.725326i	$0.258311\pi$
$284$	0	0
$285$	−1.28334	−0.0760187
$286$	0	0
$287$	−27.7985	−1.64089
$288$	0	0
$289$	−9.99669	−0.588040
$290$	0	0
$291$	16.0162	0.938886
$292$	0	0
$293$	20.6097	1.20403	0.602016	−	0.798484i	$-0.294364\pi$
$293$	20.6097	1.20403	0.602016	+	0.798484i	$0.294364\pi$
$294$	0	0
$295$	−2.34899	−0.136764
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−1.80651	−0.104473
$300$	0	0
$301$	30.4741	1.75650
$302$	0	0
$303$	10.4896	0.602613
$304$	0	0
$305$	−0.309990	−0.0177500
$306$	0	0
$307$	17.1118	0.976622	0.488311	−	0.872670i	$-0.337613\pi$
$307$	17.1118	0.976622	0.488311	+	0.872670i	$0.337613\pi$
$308$	0	0
$309$	5.10729	0.290543
$310$	0	0
$311$	−18.9873	−1.07667	−0.538335	−	0.842731i	$-0.680946\pi$
$311$	−18.9873	−1.07667	−0.538335	+	0.842731i	$0.680946\pi$
$312$	0	0
$313$	−21.2050	−1.19858	−0.599288	−	0.800534i	$-0.704549\pi$
$313$	−21.2050	−1.19858	−0.599288	+	0.800534i	$0.704549\pi$
$314$	0	0
$315$	−1.22240	−0.0688744
$316$	0	0
$317$	−25.4738	−1.43075	−0.715375	−	0.698741i	$-0.753744\pi$
$317$	−25.4738	−1.43075	−0.715375	+	0.698741i	$0.753744\pi$
$318$	0	0
$319$	3.30887	0.185261
$320$	0	0
$321$	10.8354	0.604775
$322$	0	0
$323$	12.1313	0.675004
$324$	0	0
$325$	4.92163	0.273003
$326$	0	0
$327$	9.58882	0.530263
$328$	0	0
$329$	−11.9328	−0.657874
$330$	0	0
$331$	3.03561	0.166852	0.0834262	−	0.996514i	$-0.473414\pi$
$331$	3.03561	0.166852	0.0834262	+	0.996514i	$0.473414\pi$
$332$	0	0
$333$	−0.559909	−0.0306828
$334$	0	0
$335$	−1.23042	−0.0672249
$336$	0	0
$337$	−5.58080	−0.304006	−0.152003	−	0.988380i	$-0.548572\pi$
$337$	−5.58080	−0.304006	−0.152003	+	0.988380i	$0.548572\pi$
$338$	0	0
$339$	2.77760	0.150859
$340$	0	0
$341$	−1.50235	−0.0813570
$342$	0	0
$343$	−22.1188	−1.19430
$344$	0	0
$345$	−0.505741	−0.0272282
$346$	0	0
$347$	−9.78597	−0.525338	−0.262669	−	0.964886i	$-0.584603\pi$
$347$	−9.78597	−0.525338	−0.262669	+	0.964886i	$0.584603\pi$
$348$	0	0
$349$	8.55050	0.457698	0.228849	−	0.973462i	$-0.426504\pi$
$349$	8.55050	0.457698	0.228849	+	0.973462i	$0.426504\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−30.2608	−1.61062	−0.805310	−	0.592854i	$-0.798001\pi$
$353$	−30.2608	−1.61062	−0.805310	+	0.592854i	$0.798001\pi$
$354$	0	0
$355$	−0.838545	−0.0445054
$356$	0	0
$357$	11.5552	0.611566
$358$	0	0
$359$	24.5555	1.29599	0.647996	−	0.761644i	$-0.275607\pi$
$359$	24.5555	1.29599	0.647996	+	0.761644i	$0.275607\pi$
$360$	0	0
$361$	2.01412	0.106006
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	3.82131	0.200017
$366$	0	0
$367$	−28.7006	−1.49816	−0.749080	−	0.662479i	$-0.769504\pi$
$367$	−28.7006	−1.49816	−0.749080	+	0.662479i	$0.769504\pi$
$368$	0	0
$369$	6.36642	0.331423
$370$	0	0
$371$	−13.0787	−0.679013
$372$	0	0
$373$	4.63026	0.239746	0.119873	−	0.992789i	$-0.461751\pi$
$373$	4.63026	0.239746	0.119873	+	0.992789i	$0.461751\pi$
$374$	0	0
$375$	2.77760	0.143435
$376$	0	0
$377$	−3.30887	−0.170415
$378$	0	0
$379$	25.5091	1.31032	0.655158	−	0.755492i	$-0.272602\pi$
$379$	25.5091	1.31032	0.655158	+	0.755492i	$0.272602\pi$
$380$	0	0
$381$	7.08176	0.362810
$382$	0	0
$383$	9.95836	0.508849	0.254424	−	0.967093i	$-0.418114\pi$
$383$	9.95836	0.508849	0.254424	+	0.967093i	$0.418114\pi$
$384$	0	0
$385$	−1.22240	−0.0622992
$386$	0	0
$387$	−6.97918	−0.354772
$388$	0	0
$389$	−7.34117	−0.372212	−0.186106	−	0.982530i	$-0.559587\pi$
$389$	−7.34117	−0.372212	−0.186106	+	0.982530i	$0.559587\pi$
$390$	0	0
$391$	4.78072	0.241771
$392$	0	0
$393$	2.22710	0.112343
$394$	0	0
$395$	−4.18096	−0.210367
$396$	0	0
$397$	14.8354	0.744569	0.372284	−	0.928119i	$-0.378575\pi$
$397$	14.8354	0.744569	0.372284	+	0.928119i	$0.378575\pi$
$398$	0	0
$399$	20.0162	1.00206
$400$	0	0
$401$	23.3940	1.16824	0.584121	−	0.811667i	$-0.301439\pi$
$401$	23.3940	1.16824	0.584121	+	0.811667i	$0.301439\pi$
$402$	0	0
$403$	1.50235	0.0748375
$404$	0	0
$405$	0.279954	0.0139110
$406$	0	0
$407$	−0.559909	−0.0277536
$408$	0	0
$409$	25.8549	1.27844	0.639222	−	0.769022i	$-0.279257\pi$
$409$	25.8549	1.27844	0.639222	+	0.769022i	$0.279257\pi$
$410$	0	0
$411$	−18.6384	−0.919362
$412$	0	0
$413$	36.6370	1.80279
$414$	0	0
$415$	−3.84325	−0.188658
$416$	0	0
$417$	−12.5313	−0.613659
$418$	0	0
$419$	−21.1024	−1.03092	−0.515460	−	0.856914i	$-0.672379\pi$
$419$	−21.1024	−1.03092	−0.515460	+	0.856914i	$0.672379\pi$
$420$	0	0
$421$	−12.3678	−0.602770	−0.301385	−	0.953502i	$-0.597449\pi$
$421$	−12.3678	−0.602770	−0.301385	+	0.953502i	$0.597449\pi$
$422$	0	0
$423$	2.73285	0.132876
$424$	0	0
$425$	−13.0245	−0.631780
$426$	0	0
$427$	4.83489	0.233977
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−34.2880	−1.65160	−0.825799	−	0.563965i	$-0.809275\pi$
$431$	−34.2880	−1.65160	−0.825799	+	0.563965i	$0.809275\pi$
$432$	0	0
$433$	9.08184	0.436445	0.218223	−	0.975899i	$-0.429974\pi$
$433$	9.08184	0.436445	0.218223	+	0.975899i	$0.429974\pi$
$434$	0	0
$435$	−0.926332	−0.0444142
$436$	0	0
$437$	8.28127	0.396147
$438$	0	0
$439$	33.1330	1.58135	0.790675	−	0.612236i	$-0.209730\pi$
$439$	33.1330	1.58135	0.790675	+	0.612236i	$0.209730\pi$
$440$	0	0
$441$	12.0656	0.574555
$442$	0	0
$443$	−5.11982	−0.243250	−0.121625	−	0.992576i	$-0.538811\pi$
$443$	−5.11982	−0.243250	−0.121625	+	0.992576i	$0.538811\pi$
$444$	0	0
$445$	0.699227	0.0331465
$446$	0	0
$447$	−7.70393	−0.364384
$448$	0	0
$449$	15.8320	0.747160	0.373580	−	0.927598i	$-0.378130\pi$
$449$	15.8320	0.747160	0.373580	+	0.927598i	$0.378130\pi$
$450$	0	0
$451$	6.36642	0.299783
$452$	0	0
$453$	−4.14873	−0.194924
$454$	0	0
$455$	1.22240	0.0573069
$456$	0	0
$457$	6.01253	0.281254	0.140627	−	0.990063i	$-0.455088\pi$
$457$	6.01253	0.281254	0.140627	+	0.990063i	$0.455088\pi$
$458$	0	0
$459$	−2.64638	−0.123522
$460$	0	0
$461$	−21.3998	−0.996690	−0.498345	−	0.866979i	$-0.666059\pi$
$461$	−21.3998	−0.996690	−0.498345	+	0.866979i	$0.666059\pi$
$462$	0	0
$463$	−38.2420	−1.77726	−0.888628	−	0.458629i	$-0.848341\pi$
$463$	−38.2420	−1.77726	−0.888628	+	0.458629i	$0.848341\pi$
$464$	0	0
$465$	0.420590	0.0195044
$466$	0	0
$467$	−13.8190	−0.639469	−0.319735	−	0.947507i	$-0.603594\pi$
$467$	−13.8190	−0.639469	−0.319735	+	0.947507i	$0.603594\pi$
$468$	0	0
$469$	19.1907	0.886145
$470$	0	0
$471$	2.59353	0.119503
$472$	0	0
$473$	−6.97918	−0.320903
$474$	0	0
$475$	−22.5613	−1.03518
$476$	0	0
$477$	2.99529	0.137145
$478$	0	0
$479$	35.4546	1.61996	0.809980	−	0.586457i	$-0.199478\pi$
$479$	35.4546	1.61996	0.809980	+	0.586457i	$0.199478\pi$
$480$	0	0
$481$	0.559909	0.0255296
$482$	0	0
$483$	7.88801	0.358917
$484$	0	0
$485$	−4.48380	−0.203599
$486$	0	0
$487$	29.6175	1.34210	0.671048	−	0.741414i	$-0.265845\pi$
$487$	29.6175	1.34210	0.671048	+	0.741414i	$0.265845\pi$
$488$	0	0
$489$	−6.00809	−0.271695
$490$	0	0
$491$	−20.5892	−0.929176	−0.464588	−	0.885527i	$-0.653798\pi$
$491$	−20.5892	−0.929176	−0.464588	+	0.885527i	$0.653798\pi$
$492$	0	0
$493$	8.75651	0.394373
$494$	0	0
$495$	0.279954	0.0125830
$496$	0	0
$497$	13.0787	0.586661
$498$	0	0
$499$	1.45760	0.0652510	0.0326255	−	0.999468i	$-0.489613\pi$
$499$	1.45760	0.0652510	0.0326255	+	0.999468i	$0.489613\pi$
$500$	0	0
$501$	16.2177	0.724553
$502$	0	0
$503$	8.10570	0.361415	0.180708	−	0.983537i	$-0.442161\pi$
$503$	8.10570	0.361415	0.180708	+	0.983537i	$0.442161\pi$
$504$	0	0
$505$	−2.93662	−0.130678
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−31.2145	−1.38356	−0.691778	−	0.722110i	$-0.743172\pi$
$509$	−31.2145	−1.38356	−0.691778	+	0.722110i	$0.743172\pi$
$510$	0	0
$511$	−59.6007	−2.63658
$512$	0	0
$513$	−4.58412	−0.202394
$514$	0	0
$515$	−1.42981	−0.0630048
$516$	0	0
$517$	2.73285	0.120190
$518$	0	0
$519$	−3.42398	−0.150296
$520$	0	0
$521$	−35.9620	−1.57552	−0.787762	−	0.615979i	$-0.788761\pi$
$521$	−35.9620	−1.57552	−0.787762	+	0.615979i	$0.788761\pi$
$522$	0	0
$523$	−17.6579	−0.772124	−0.386062	−	0.922473i	$-0.626165\pi$
$523$	−17.6579	−0.772124	−0.386062	+	0.922473i	$0.626165\pi$
$524$	0	0
$525$	−21.4899	−0.937896
$526$	0	0
$527$	−3.97579	−0.173188
$528$	0	0
$529$	−19.7365	−0.858109
$530$	0	0
$531$	−8.39063	−0.364122
$532$	0	0
$533$	−6.36642	−0.275760
$534$	0	0
$535$	−3.03342	−0.131146
$536$	0	0
$537$	−16.0498	−0.692600
$538$	0	0
$539$	12.0656	0.519704
$540$	0	0
$541$	−23.6771	−1.01796	−0.508979	−	0.860779i	$-0.669977\pi$
$541$	−23.6771	−1.01796	−0.508979	+	0.860779i	$0.669977\pi$
$542$	0	0
$543$	−1.70393	−0.0731228
$544$	0	0
$545$	−2.68443	−0.114988
$546$	0	0
$547$	−3.24217	−0.138625	−0.0693126	−	0.997595i	$-0.522081\pi$
$547$	−3.24217	−0.138625	−0.0693126	+	0.997595i	$0.522081\pi$
$548$	0	0
$549$	−1.10729	−0.0472579
$550$	0	0
$551$	15.1682	0.646188
$552$	0	0
$553$	65.2100	2.77301
$554$	0	0
$555$	0.156749	0.00665361
$556$	0	0
$557$	38.9550	1.65058	0.825289	−	0.564710i	$-0.191012\pi$
$557$	38.9550	1.65058	0.825289	+	0.564710i	$0.191012\pi$
$558$	0	0
$559$	6.97918	0.295188
$560$	0	0
$561$	−2.64638	−0.111730
$562$	0	0
$563$	−0.246259	−0.0103786	−0.00518930	−	0.999987i	$-0.501652\pi$
$563$	−0.246259	−0.0103786	−0.00518930	+	0.999987i	$0.501652\pi$
$564$	0	0
$565$	−0.777601	−0.0327139
$566$	0	0
$567$	−4.36642	−0.183373
$568$	0	0
$569$	3.13905	0.131596	0.0657979	−	0.997833i	$-0.479041\pi$
$569$	3.13905	0.131596	0.0657979	+	0.997833i	$0.479041\pi$
$570$	0	0
$571$	44.8318	1.87615	0.938077	−	0.346426i	$-0.112605\pi$
$571$	44.8318	1.87615	0.938077	+	0.346426i	$0.112605\pi$
$572$	0	0
$573$	19.5683	0.817476
$574$	0	0
$575$	−8.89099	−0.370780
$576$	0	0
$577$	−23.1933	−0.965549	−0.482775	−	0.875745i	$-0.660371\pi$
$577$	−23.1933	−0.965549	−0.482775	+	0.875745i	$0.660371\pi$
$578$	0	0
$579$	26.8305	1.11504
$580$	0	0
$581$	59.9429	2.48685
$582$	0	0
$583$	2.99529	0.124052
$584$	0	0
$585$	−0.279954	−0.0115747
$586$	0	0
$587$	3.27081	0.135001	0.0675005	−	0.997719i	$-0.478498\pi$
$587$	3.27081	0.135001	0.0675005	+	0.997719i	$0.478498\pi$
$588$	0	0
$589$	−6.88696	−0.283772
$590$	0	0
$591$	19.6466	0.808155
$592$	0	0
$593$	−4.85598	−0.199411	−0.0997055	−	0.995017i	$-0.531790\pi$
$593$	−4.85598	−0.199411	−0.0997055	+	0.995017i	$0.531790\pi$
$594$	0	0
$595$	−3.23493	−0.132619
$596$	0	0
$597$	7.55208	0.309086
$598$	0	0
$599$	10.9089	0.445726	0.222863	−	0.974850i	$-0.428460\pi$
$599$	10.9089	0.445726	0.222863	+	0.974850i	$0.428460\pi$
$600$	0	0
$601$	23.9011	0.974945	0.487473	−	0.873138i	$-0.337919\pi$
$601$	23.9011	0.974945	0.487473	+	0.873138i	$0.337919\pi$
$602$	0	0
$603$	−4.39507	−0.178981
$604$	0	0
$605$	0.279954	0.0113818
$606$	0	0
$607$	7.13905	0.289765	0.144883	−	0.989449i	$-0.453720\pi$
$607$	7.13905	0.289765	0.144883	+	0.989449i	$0.453720\pi$
$608$	0	0
$609$	14.4479	0.585459
$610$	0	0
$611$	−2.73285	−0.110559
$612$	0	0
$613$	35.6306	1.43911	0.719553	−	0.694437i	$-0.244347\pi$
$613$	35.6306	1.43911	0.719553	+	0.694437i	$0.244347\pi$
$614$	0	0
$615$	−1.78231	−0.0718696
$616$	0	0
$617$	24.6133	0.990894	0.495447	−	0.868638i	$-0.335004\pi$
$617$	24.6133	0.990894	0.495447	+	0.868638i	$0.335004\pi$
$618$	0	0
$619$	17.6300	0.708609	0.354305	−	0.935130i	$-0.384718\pi$
$619$	17.6300	0.708609	0.354305	+	0.935130i	$0.384718\pi$
$620$	0	0
$621$	−1.80651	−0.0724929
$622$	0	0
$623$	−10.9058	−0.436931
$624$	0	0
$625$	23.8305	0.953221
$626$	0	0
$627$	−4.58412	−0.183072
$628$	0	0
$629$	−1.48173	−0.0590804
$630$	0	0
$631$	−20.8919	−0.831695	−0.415847	−	0.909434i	$-0.636515\pi$
$631$	−20.8919	−0.831695	−0.415847	+	0.909434i	$0.636515\pi$
$632$	0	0
$633$	−17.4849	−0.694963
$634$	0	0
$635$	−1.98257	−0.0786759
$636$	0	0
$637$	−12.0656	−0.478058
$638$	0	0
$639$	−2.99529	−0.118492
$640$	0	0
$641$	11.4594	0.452619	0.226310	−	0.974055i	$-0.427334\pi$
$641$	11.4594	0.452619	0.226310	+	0.974055i	$0.427334\pi$
$642$	0	0
$643$	−18.8275	−0.742484	−0.371242	−	0.928536i	$-0.621068\pi$
$643$	−18.8275	−0.742484	−0.371242	+	0.928536i	$0.621068\pi$
$644$	0	0
$645$	1.95385	0.0769328
$646$	0	0
$647$	−17.2739	−0.679108	−0.339554	−	0.940587i	$-0.610276\pi$
$647$	−17.2739	−0.679108	−0.339554	+	0.940587i	$0.610276\pi$
$648$	0	0
$649$	−8.39063	−0.329361
$650$	0	0
$651$	−6.55991	−0.257103
$652$	0	0
$653$	14.0512	0.549866	0.274933	−	0.961463i	$-0.411344\pi$
$653$	14.0512	0.549866	0.274933	+	0.961463i	$0.411344\pi$
$654$	0	0
$655$	−0.623487	−0.0243617
$656$	0	0
$657$	13.6498	0.532528
$658$	0	0
$659$	20.5120	0.799035	0.399518	−	0.916725i	$-0.369178\pi$
$659$	20.5120	0.799035	0.399518	+	0.916725i	$0.369178\pi$
$660$	0	0
$661$	19.5230	0.759356	0.379678	−	0.925119i	$-0.376035\pi$
$661$	19.5230	0.759356	0.379678	+	0.925119i	$0.376035\pi$
$662$	0	0
$663$	2.64638	0.102777
$664$	0	0
$665$	−5.60362	−0.217299
$666$	0	0
$667$	5.97752	0.231450
$668$	0	0
$669$	7.38724	0.285607
$670$	0	0
$671$	−1.10729	−0.0427464
$672$	0	0
$673$	−1.65467	−0.0637827	−0.0318914	−	0.999491i	$-0.510153\pi$
$673$	−1.65467	−0.0637827	−0.0318914	+	0.999491i	$0.510153\pi$
$674$	0	0
$675$	4.92163	0.189433
$676$	0	0
$677$	−26.6209	−1.02313	−0.511563	−	0.859246i	$-0.670933\pi$
$677$	−26.6209	−1.02313	−0.511563	+	0.859246i	$0.670933\pi$
$678$	0	0
$679$	69.9335	2.68380
$680$	0	0
$681$	−7.02891	−0.269349
$682$	0	0
$683$	23.8563	0.912837	0.456418	−	0.889765i	$-0.349132\pi$
$683$	23.8563	0.912837	0.456418	+	0.889765i	$0.349132\pi$
$684$	0	0
$685$	5.21789	0.199365
$686$	0	0
$687$	13.6834	0.522054
$688$	0	0
$689$	−2.99529	−0.114112
$690$	0	0
$691$	16.5811	0.630774	0.315387	−	0.948963i	$-0.397866\pi$
$691$	16.5811	0.630774	0.315387	+	0.948963i	$0.397866\pi$
$692$	0	0
$693$	−4.36642	−0.165867
$694$	0	0
$695$	3.50818	0.133073
$696$	0	0
$697$	16.8480	0.638162
$698$	0	0
$699$	16.6464	0.629624
$700$	0	0
$701$	43.8622	1.65665	0.828325	−	0.560247i	$-0.189294\pi$
$701$	43.8622	1.65665	0.828325	+	0.560247i	$0.189294\pi$
$702$	0	0
$703$	−2.56669	−0.0968044
$704$	0	0
$705$	−0.765072	−0.0288143
$706$	0	0
$707$	45.8022	1.72257
$708$	0	0
$709$	−17.1557	−0.644296	−0.322148	−	0.946689i	$-0.604405\pi$
$709$	−17.1557	−0.644296	−0.322148	+	0.946689i	$0.604405\pi$
$710$	0	0
$711$	−14.9344	−0.560085
$712$	0	0
$713$	−2.71402	−0.101641
$714$	0	0
$715$	−0.279954	−0.0104697
$716$	0	0
$717$	11.9795	0.447381
$718$	0	0
$719$	16.4256	0.612573	0.306287	−	0.951939i	$-0.400913\pi$
$719$	16.4256	0.612573	0.306287	+	0.951939i	$0.400913\pi$
$720$	0	0
$721$	22.3006	0.830517
$722$	0	0
$723$	−17.5808	−0.653837
$724$	0	0
$725$	−16.2850	−0.604810
$726$	0	0
$727$	−8.65221	−0.320893	−0.160446	−	0.987045i	$-0.551293\pi$
$727$	−8.65221	−0.320893	−0.160446	+	0.987045i	$0.551293\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.4695	−0.683121
$732$	0	0
$733$	−21.9758	−0.811694	−0.405847	−	0.913941i	$-0.633023\pi$
$733$	−21.9758	−0.811694	−0.405847	+	0.913941i	$0.633023\pi$
$734$	0	0
$735$	−3.37783	−0.124593
$736$	0	0
$737$	−4.39507	−0.161894
$738$	0	0
$739$	19.9857	0.735188	0.367594	−	0.929986i	$-0.380182\pi$
$739$	19.9857	0.735188	0.367594	+	0.929986i	$0.380182\pi$
$740$	0	0
$741$	4.58412	0.168402
$742$	0	0
$743$	25.4021	0.931913	0.465957	−	0.884808i	$-0.345710\pi$
$743$	25.4021	0.931913	0.465957	+	0.884808i	$0.345710\pi$
$744$	0	0
$745$	2.15675	0.0790172
$746$	0	0
$747$	−13.7281	−0.502286
$748$	0	0
$749$	47.3121	1.72875
$750$	0	0
$751$	17.2158	0.628212	0.314106	−	0.949388i	$-0.398295\pi$
$751$	17.2158	0.628212	0.314106	+	0.949388i	$0.398295\pi$
$752$	0	0
$753$	18.8480	0.686858
$754$	0	0
$755$	1.16145	0.0422697
$756$	0	0
$757$	32.1076	1.16697	0.583486	−	0.812123i	$-0.301688\pi$
$757$	32.1076	1.16697	0.583486	+	0.812123i	$0.301688\pi$
$758$	0	0
$759$	−1.80651	−0.0655723
$760$	0	0
$761$	−12.2353	−0.443528	−0.221764	−	0.975100i	$-0.571182\pi$
$761$	−12.2353	−0.443528	−0.221764	+	0.975100i	$0.571182\pi$
$762$	0	0
$763$	41.8689	1.51575
$764$	0	0
$765$	0.740865	0.0267860
$766$	0	0
$767$	8.39063	0.302968
$768$	0	0
$769$	−28.1027	−1.01341	−0.506705	−	0.862119i	$-0.669137\pi$
$769$	−28.1027	−1.01341	−0.506705	+	0.862119i	$0.669137\pi$
$770$	0	0
$771$	17.3953	0.626478
$772$	0	0
$773$	46.8686	1.68575	0.842873	−	0.538113i	$-0.180863\pi$
$773$	46.8686	1.68575	0.842873	+	0.538113i	$0.180863\pi$
$774$	0	0
$775$	7.39402	0.265601
$776$	0	0
$777$	−2.44480	−0.0877066
$778$	0	0
$779$	29.1844	1.04564
$780$	0	0
$781$	−2.99529	−0.107180
$782$	0	0
$783$	−3.30887	−0.118249
$784$	0	0
$785$	−0.726069	−0.0259145
$786$	0	0
$787$	−41.9415	−1.49505	−0.747526	−	0.664233i	$-0.768758\pi$
$787$	−41.9415	−1.49505	−0.747526	+	0.664233i	$0.768758\pi$
$788$	0	0
$789$	10.1151	0.360107
$790$	0	0
$791$	12.1282	0.431229
$792$	0	0
$793$	1.10729	0.0393209
$794$	0	0
$795$	−0.838545	−0.0297401
$796$	0	0
$797$	9.31836	0.330073	0.165037	−	0.986287i	$-0.447226\pi$
$797$	9.31836	0.330073	0.165037	+	0.986287i	$0.447226\pi$
$798$	0	0
$799$	7.23214	0.255855
$800$	0	0
$801$	2.49765	0.0882500
$802$	0	0
$803$	13.6498	0.481690
$804$	0	0
$805$	−2.20828	−0.0778317
$806$	0	0
$807$	10.8385	0.381535
$808$	0	0
$809$	−26.8839	−0.945188	−0.472594	−	0.881280i	$-0.656682\pi$
$809$	−26.8839	−0.945188	−0.472594	+	0.881280i	$0.656682\pi$
$810$	0	0
$811$	21.9763	0.771693	0.385847	−	0.922563i	$-0.373909\pi$
$811$	21.9763	0.771693	0.385847	+	0.922563i	$0.373909\pi$
$812$	0	0
$813$	−2.03362	−0.0713221
$814$	0	0
$815$	1.68199	0.0589176
$816$	0	0
$817$	−31.9934	−1.11931
$818$	0	0
$819$	4.36642	0.152575
$820$	0	0
$821$	−15.7362	−0.549196	−0.274598	−	0.961559i	$-0.588545\pi$
$821$	−15.7362	−0.549196	−0.274598	+	0.961559i	$0.588545\pi$
$822$	0	0
$823$	24.0932	0.839835	0.419918	−	0.907562i	$-0.362059\pi$
$823$	24.0932	0.839835	0.419918	+	0.907562i	$0.362059\pi$
$824$	0	0
$825$	4.92163	0.171349
$826$	0	0
$827$	−0.491819	−0.0171022	−0.00855111	−	0.999963i	$-0.502722\pi$
$827$	−0.491819	−0.0171022	−0.00855111	+	0.999963i	$0.502722\pi$
$828$	0	0
$829$	−5.35945	−0.186141	−0.0930707	−	0.995660i	$-0.529668\pi$
$829$	−5.35945	−0.186141	−0.0930707	+	0.995660i	$0.529668\pi$
$830$	0	0
$831$	1.55208	0.0538412
$832$	0	0
$833$	31.9303	1.10632
$834$	0	0
$835$	−4.54021	−0.157121
$836$	0	0
$837$	1.50235	0.0519289
$838$	0	0
$839$	−20.9442	−0.723076	−0.361538	−	0.932357i	$-0.617748\pi$
$839$	−20.9442	−0.723076	−0.361538	+	0.932357i	$0.617748\pi$
$840$	0	0
$841$	−18.0514	−0.622462
$842$	0	0
$843$	−10.9747	−0.377990
$844$	0	0
$845$	0.279954	0.00963072
$846$	0	0
$847$	−4.36642	−0.150032
$848$	0	0
$849$	−23.1615	−0.794902
$850$	0	0
$851$	−1.01148	−0.0346732
$852$	0	0
$853$	−52.4288	−1.79513	−0.897563	−	0.440886i	$-0.854664\pi$
$853$	−52.4288	−1.79513	−0.897563	+	0.440886i	$0.854664\pi$
$854$	0	0
$855$	1.28334	0.0438894
$856$	0	0
$857$	−27.5616	−0.941485	−0.470743	−	0.882271i	$-0.656014\pi$
$857$	−27.5616	−0.941485	−0.470743	+	0.882271i	$0.656014\pi$
$858$	0	0
$859$	14.6428	0.499606	0.249803	−	0.968297i	$-0.419634\pi$
$859$	14.6428	0.499606	0.249803	+	0.968297i	$0.419634\pi$
$860$	0	0
$861$	27.7985	0.947370
$862$	0	0
$863$	−54.7396	−1.86336	−0.931679	−	0.363282i	$-0.881656\pi$
$863$	−54.7396	−1.86336	−0.931679	+	0.363282i	$0.881656\pi$
$864$	0	0
$865$	0.958557	0.0325919
$866$	0	0
$867$	9.99669	0.339505
$868$	0	0
$869$	−14.9344	−0.506616
$870$	0	0
$871$	4.39507	0.148921
$872$	0	0
$873$	−16.0162	−0.542066
$874$	0	0
$875$	12.1282	0.410007
$876$	0	0
$877$	−9.14542	−0.308819	−0.154409	−	0.988007i	$-0.549347\pi$
$877$	−9.14542	−0.308819	−0.154409	+	0.988007i	$0.549347\pi$
$878$	0	0
$879$	−20.6097	−0.695148
$880$	0	0
$881$	−45.8631	−1.54517	−0.772584	−	0.634913i	$-0.781036\pi$
$881$	−45.8631	−1.54517	−0.772584	+	0.634913i	$0.781036\pi$
$882$	0	0
$883$	26.3131	0.885507	0.442753	−	0.896643i	$-0.354002\pi$
$883$	26.3131	0.885507	0.442753	+	0.896643i	$0.354002\pi$
$884$	0	0
$885$	2.34899	0.0789605
$886$	0	0
$887$	−2.09230	−0.0702525	−0.0351262	−	0.999383i	$-0.511183\pi$
$887$	−2.09230	−0.0702525	−0.0351262	+	0.999383i	$0.511183\pi$
$888$	0	0
$889$	30.9220	1.03709
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	12.5277	0.419223
$894$	0	0
$895$	4.49321	0.150192
$896$	0	0
$897$	1.80651	0.0603178
$898$	0	0
$899$	−4.97109	−0.165795
$900$	0	0
$901$	7.92668	0.264076
$902$	0	0
$903$	−30.4741	−1.01411
$904$	0	0
$905$	0.477023	0.0158568
$906$	0	0
$907$	−11.6641	−0.387299	−0.193650	−	0.981071i	$-0.562033\pi$
$907$	−11.6641	−0.387299	−0.193650	+	0.981071i	$0.562033\pi$
$908$	0	0
$909$	−10.4896	−0.347919
$910$	0	0
$911$	−10.1635	−0.336733	−0.168366	−	0.985724i	$-0.553849\pi$
$911$	−10.1635	−0.336733	−0.168366	+	0.985724i	$0.553849\pi$
$912$	0	0
$913$	−13.7281	−0.454335
$914$	0	0
$915$	0.309990	0.0102479
$916$	0	0
$917$	9.72448	0.321131
$918$	0	0
$919$	50.6215	1.66985	0.834924	−	0.550365i	$-0.185511\pi$
$919$	50.6215	1.66985	0.834924	+	0.550365i	$0.185511\pi$
$920$	0	0
$921$	−17.1118	−0.563853
$922$	0	0
$923$	2.99529	0.0985913
$924$	0	0
$925$	2.75566	0.0906055
$926$	0	0
$927$	−5.10729	−0.167745
$928$	0	0
$929$	23.3524	0.766167	0.383083	−	0.923714i	$-0.374862\pi$
$929$	23.3524	0.766167	0.383083	+	0.923714i	$0.374862\pi$
$930$	0	0
$931$	55.3103	1.81272
$932$	0	0
$933$	18.9873	0.621615
$934$	0	0
$935$	0.740865	0.0242289
$936$	0	0
$937$	30.8553	1.00800	0.503999	−	0.863704i	$-0.331862\pi$
$937$	30.8553	1.00800	0.503999	+	0.863704i	$0.331862\pi$
$938$	0	0
$939$	21.2050	0.691998
$940$	0	0
$941$	−16.8237	−0.548439	−0.274219	−	0.961667i	$-0.588419\pi$
$941$	−16.8237	−0.548439	−0.274219	+	0.961667i	$0.588419\pi$
$942$	0	0
$943$	11.5010	0.374525
$944$	0	0
$945$	1.22240	0.0397646
$946$	0	0
$947$	45.3991	1.47527	0.737637	−	0.675197i	$-0.235942\pi$
$947$	45.3991	1.47527	0.737637	+	0.675197i	$0.235942\pi$
$948$	0	0
$949$	−13.6498	−0.443090
$950$	0	0
$951$	25.4738	0.826044
$952$	0	0
$953$	−49.2871	−1.59657	−0.798283	−	0.602283i	$-0.794258\pi$
$953$	−49.2871	−1.59657	−0.798283	+	0.602283i	$0.794258\pi$
$954$	0	0
$955$	−5.47822	−0.177271
$956$	0	0
$957$	−3.30887	−0.106960
$958$	0	0
$959$	−81.3830	−2.62799
$960$	0	0
$961$	−28.7429	−0.927191
$962$	0	0
$963$	−10.8354	−0.349167
$964$	0	0
$965$	−7.51132	−0.241798
$966$	0	0
$967$	−36.9587	−1.18851	−0.594256	−	0.804276i	$-0.702553\pi$
$967$	−36.9587	−1.18851	−0.594256	+	0.804276i	$0.702553\pi$
$968$	0	0
$969$	−12.1313	−0.389713
$970$	0	0
$971$	40.2638	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$971$	40.2638	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$972$	0	0
$973$	−54.7168	−1.75414
$974$	0	0
$975$	−4.92163	−0.157618
$976$	0	0
$977$	−21.7237	−0.695003	−0.347501	−	0.937679i	$-0.612970\pi$
$977$	−21.7237	−0.695003	−0.347501	+	0.937679i	$0.612970\pi$
$978$	0	0
$979$	2.49765	0.0798251
$980$	0	0
$981$	−9.58882	−0.306148
$982$	0	0
$983$	7.72868	0.246507	0.123253	−	0.992375i	$-0.460667\pi$
$983$	7.72868	0.246507	0.123253	+	0.992375i	$0.460667\pi$
$984$	0	0
$985$	−5.50016	−0.175250
$986$	0	0
$987$	11.9328	0.379824
$988$	0	0
$989$	−12.6080	−0.400911
$990$	0	0
$991$	−29.8563	−0.948417	−0.474209	−	0.880412i	$-0.657266\pi$
$991$	−29.8563	−0.948417	−0.474209	+	0.880412i	$0.657266\pi$
$992$	0	0
$993$	−3.03561	−0.0963323
$994$	0	0
$995$	−2.11424	−0.0670259
$996$	0	0
$997$	−26.9856	−0.854641	−0.427320	−	0.904100i	$-0.640542\pi$
$997$	−26.9856	−0.854641	−0.427320	+	0.904100i	$0.640542\pi$
$998$	0	0
$999$	0.559909	0.0177147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.a.r.1.2	✓	4
4.3	odd	2		6864.2.a.ca.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.r.1.2	✓	4	1.1	even	1	trivial
6864.2.a.ca.1.2		4	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:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.279954 q^{5} -4.36642 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.279954 q^{5} -4.36642 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.279954 q^{15} +2.64638 q^{17} +4.58412 q^{19} +4.36642 q^{21} +1.80651 q^{23} -4.92163 q^{25} -1.00000 q^{27} +3.30887 q^{29} -1.50235 q^{31} -1.00000 q^{33} -1.22240 q^{35} -0.559909 q^{37} +1.00000 q^{39} +6.36642 q^{41} -6.97918 q^{43} +0.279954 q^{45} +2.73285 q^{47} +12.0656 q^{49} -2.64638 q^{51} +2.99529 q^{53} +0.279954 q^{55} -4.58412 q^{57} -8.39063 q^{59} -1.10729 q^{61} -4.36642 q^{63} -0.279954 q^{65} -4.39507 q^{67} -1.80651 q^{69} -2.99529 q^{71} +13.6498 q^{73} +4.92163 q^{75} -4.36642 q^{77} -14.9344 q^{79} +1.00000 q^{81} -13.7281 q^{83} +0.740865 q^{85} -3.30887 q^{87} +2.49765 q^{89} +4.36642 q^{91} +1.50235 q^{93} +1.28334 q^{95} -16.0162 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} - 6 q^{17} + q^{21} - 9 q^{23} + q^{25} - 4 q^{27} - q^{29} - 8 q^{31} - 4 q^{33} - 7 q^{35} - 2 q^{37} + 4 q^{39} + 9 q^{41} - 5 q^{43} + q^{45} - 22 q^{47} + 9 q^{49} + 6 q^{51} + 8 q^{53} + q^{55} + q^{59} - 11 q^{61} - q^{63} - q^{65} - 13 q^{67} + 9 q^{69} - 8 q^{71} - 3 q^{73} - q^{75} - q^{77} - 6 q^{79} + 4 q^{81} - 18 q^{83} + 26 q^{85} + q^{87} + 8 q^{89} + q^{91} + 8 q^{93} - 36 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - q^15 - 6 * q^17 + q^21 - 9 * q^23 + q^25 - 4 * q^27 - q^29 - 8 * q^31 - 4 * q^33 - 7 * q^35 - 2 * q^37 + 4 * q^39 + 9 * q^41 - 5 * q^43 + q^45 - 22 * q^47 + 9 * q^49 + 6 * q^51 + 8 * q^53 + q^55 + q^59 - 11 * q^61 - q^63 - q^65 - 13 * q^67 + 9 * q^69 - 8 * q^71 - 3 * q^73 - q^75 - q^77 - 6 * q^79 + 4 * q^81 - 18 * q^83 + 26 * q^85 + q^87 + 8 * q^89 + q^91 + 8 * q^93 - 36 * q^95 + 10 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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